# Properties

 Label 2025.4.a.z Level $2025$ Weight $4$ Character orbit 2025.a Self dual yes Analytic conductor $119.479$ Analytic rank $1$ Dimension $6$ 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: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - 2 x^{5} - 38 x^{4} + 42 x^{3} + 393 x^{2} - 72 x - 432$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 405) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 6 - \beta_{1} + \beta_{3} ) q^{4} + ( -7 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{7} + ( 12 - 4 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{8} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( 6 - \beta_{1} + \beta_{3} ) q^{4} + ( -7 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{7} + ( 12 - 4 \beta_{1} + 2 \beta_{3} - \beta_{4} ) q^{8} + ( -14 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} ) q^{11} + ( -4 - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{13} + ( -32 + 7 \beta_{1} + \beta_{2} - 4 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{14} + ( 14 - 16 \beta_{1} - 3 \beta_{4} + 2 \beta_{5} ) q^{16} + ( 19 + 6 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{17} + ( -7 + 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{19} + ( 5 + 20 \beta_{1} - 10 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{22} + ( 34 + 6 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{23} + ( -2 - 2 \beta_{1} - 15 \beta_{2} + 5 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} ) q^{26} + ( -70 + 40 \beta_{1} + 29 \beta_{2} - 13 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{28} + ( -52 + 12 \beta_{1} - 8 \beta_{2} - \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{29} + ( -13 - 16 \beta_{1} + 6 \beta_{2} - 10 \beta_{3} - 3 \beta_{4} - 4 \beta_{5} ) q^{31} + ( 114 + 18 \beta_{1} - 24 \beta_{2} + 6 \beta_{3} - \beta_{4} + 6 \beta_{5} ) q^{32} + ( -60 - 25 \beta_{1} + 38 \beta_{2} - 7 \beta_{3} + 9 \beta_{4} - 4 \beta_{5} ) q^{34} + ( 29 + 10 \beta_{1} + 28 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - 3 \beta_{5} ) q^{37} + ( -9 + 31 \beta_{1} + 20 \beta_{2} - 14 \beta_{3} + 14 \beta_{4} - 4 \beta_{5} ) q^{38} + ( -62 + 16 \beta_{1} + 7 \beta_{2} + 26 \beta_{3} + 2 \beta_{4} - 10 \beta_{5} ) q^{41} + ( -99 + 42 \beta_{1} - 38 \beta_{2} - 25 \beta_{3} + \beta_{4} - \beta_{5} ) q^{43} + ( -120 + 5 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{44} + ( -28 - 22 \beta_{1} - 74 \beta_{2} - 14 \beta_{3} - 13 \beta_{4} - 4 \beta_{5} ) q^{46} + ( 23 + 44 \beta_{1} + 14 \beta_{2} - 29 \beta_{3} - 6 \beta_{4} - 9 \beta_{5} ) q^{47} + ( 15 - 26 \beta_{1} - 12 \beta_{2} + 16 \beta_{3} - 15 \beta_{4} + 12 \beta_{5} ) q^{49} + ( 82 - 28 \beta_{1} - 75 \beta_{2} + 11 \beta_{3} - 16 \beta_{4} + 19 \beta_{5} ) q^{52} + ( 127 + 20 \beta_{1} - 58 \beta_{2} - 27 \beta_{3} - 24 \beta_{4} + 5 \beta_{5} ) q^{53} + ( -410 + 92 \beta_{1} + 57 \beta_{2} - 33 \beta_{3} + 20 \beta_{4} - 17 \beta_{5} ) q^{56} + ( -169 + 58 \beta_{1} - 44 \beta_{2} - 21 \beta_{3} - 4 \beta_{4} ) q^{58} + ( -294 - 30 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} ) q^{59} + ( -84 + 72 \beta_{1} + 30 \beta_{2} - 38 \beta_{3} + 2 \beta_{4} + 22 \beta_{5} ) q^{61} + ( 155 + 73 \beta_{1} + 54 \beta_{2} + 12 \beta_{3} + 19 \beta_{4} ) q^{62} + ( -158 - 22 \beta_{1} - 96 \beta_{2} - 10 \beta_{3} - \beta_{4} + 10 \beta_{5} ) q^{64} + ( 36 - 164 \beta_{1} + 8 \beta_{2} + 30 \beta_{3} - 12 \beta_{4} - 10 \beta_{5} ) q^{67} + ( 28 + 54 \beta_{1} + 70 \beta_{2} - 8 \beta_{3} + 20 \beta_{4} - 32 \beta_{5} ) q^{68} + ( -268 + 82 \beta_{1} - 52 \beta_{2} + 9 \beta_{3} - 40 \beta_{4} - 9 \beta_{5} ) q^{71} + ( -101 - 138 \beta_{1} + 50 \beta_{2} - 33 \beta_{3} - 23 \beta_{4} + 7 \beta_{5} ) q^{73} + ( -200 - 59 \beta_{1} + 64 \beta_{2} + 5 \beta_{3} - \beta_{4} - 18 \beta_{5} ) q^{74} + ( -374 + 77 \beta_{1} + 100 \beta_{2} - 41 \beta_{3} + 8 \beta_{4} - 32 \beta_{5} ) q^{76} + ( 123 + 46 \beta_{1} - 40 \beta_{2} + 19 \beta_{3} - 5 \beta_{4} - 9 \beta_{5} ) q^{77} + ( -172 + 144 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} - 32 \beta_{4} - 12 \beta_{5} ) q^{79} + ( -257 - 94 \beta_{1} + 127 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 11 \beta_{5} ) q^{82} + ( 369 - 6 \beta_{1} + 34 \beta_{2} + 45 \beta_{3} + 27 \beta_{4} - 19 \beta_{5} ) q^{83} + ( -552 + 249 \beta_{1} - 26 \beta_{2} - 69 \beta_{3} + 29 \beta_{4} + 36 \beta_{5} ) q^{86} + ( -214 + 2 \beta_{1} + 6 \beta_{2} - 26 \beta_{3} + 8 \beta_{4} + 12 \beta_{5} ) q^{88} + ( -474 - 96 \beta_{1} + 76 \beta_{2} - 33 \beta_{3} - 18 \beta_{4} - 21 \beta_{5} ) q^{89} + ( -538 + 106 \beta_{1} + 44 \beta_{2} - 34 \beta_{3} + 21 \beta_{4} - 18 \beta_{5} ) q^{91} + ( 142 + 64 \beta_{1} - 10 \beta_{2} + 50 \beta_{3} - 11 \beta_{4} + 52 \beta_{5} ) q^{92} + ( -644 + 151 \beta_{1} + 122 \beta_{2} - 61 \beta_{3} + 50 \beta_{4} - 2 \beta_{5} ) q^{94} + ( 125 - 30 \beta_{1} - 72 \beta_{2} + 13 \beta_{3} + 39 \beta_{4} + 41 \beta_{5} ) q^{97} + ( 345 - 111 \beta_{1} - 156 \beta_{2} + 72 \beta_{3} - 67 \beta_{4} + 42 \beta_{5} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} + O(q^{10})$$ $$6 q + 4 q^{2} + 34 q^{4} - 40 q^{7} + 66 q^{8} - 88 q^{11} - 20 q^{13} - 180 q^{14} + 58 q^{16} + 124 q^{17} - 46 q^{19} + 74 q^{22} + 210 q^{23} - 4 q^{26} - 352 q^{28} - 296 q^{29} - 104 q^{31} + 722 q^{32} - 428 q^{34} + 204 q^{37} - 20 q^{38} - 344 q^{41} - 512 q^{43} - 716 q^{44} - 186 q^{46} + 238 q^{47} + 68 q^{49} + 468 q^{52} + 850 q^{53} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} - 364 q^{61} + 1038 q^{62} - 990 q^{64} - 88 q^{67} + 236 q^{68} - 1364 q^{71} - 836 q^{73} - 1316 q^{74} - 2106 q^{76} + 840 q^{77} - 680 q^{79} - 1742 q^{82} + 2148 q^{83} - 2872 q^{86} - 1296 q^{88} - 3000 q^{89} - 3058 q^{91} + 1002 q^{92} - 3662 q^{94} + 612 q^{97} + 1982 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 38 x^{4} + 42 x^{3} + 393 x^{2} - 72 x - 432$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - 2 \nu^{4} - 26 \nu^{3} + 30 \nu^{2} + 141 \nu - 36$$$$)/24$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - \nu - 13$$ $$\beta_{4}$$ $$=$$ $$\nu^{3} - \nu^{2} - 19 \nu + 1$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{4} - \nu^{3} - 21 \nu^{2} + 3 \nu + 30$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{1} + 13$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 20 \beta_{1} + 12$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} + \beta_{4} + 22 \beta_{3} + 38 \beta_{1} + 255$$ $$\nu^{5}$$ $$=$$ $$4 \beta_{5} + 28 \beta_{4} + 40 \beta_{3} + 24 \beta_{2} + 425 \beta_{1} + 468$$

## 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
 5.11734 4.57457 1.14915 −1.07326 −3.53444 −4.23336
−4.11734 0 8.95250 0 0 20.0229 −3.92177 0 0
1.2 −3.57457 0 4.77759 0 0 −14.1597 11.5188 0 0
1.3 −0.149150 0 −7.97775 0 0 −20.1424 2.38308 0 0
1.4 2.07326 0 −3.70159 0 0 4.66112 −24.2604 0 0
1.5 4.53444 0 12.5612 0 0 2.63618 20.6823 0 0
1.6 5.23336 0 19.3881 0 0 −33.0180 59.5981 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.z 6
3.b odd 2 1 2025.4.a.y 6
5.b even 2 1 405.4.a.k 6
15.d odd 2 1 405.4.a.l yes 6
45.h odd 6 2 405.4.e.w 12
45.j even 6 2 405.4.e.x 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.k 6 5.b even 2 1
405.4.a.l yes 6 15.d odd 2 1
405.4.e.w 12 45.h odd 6 2
405.4.e.x 12 45.j even 6 2
2025.4.a.y 6 3.b odd 2 1
2025.4.a.z 6 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2025))$$:

 $$T_{2}^{6} - 4 T_{2}^{5} - 33 T_{2}^{4} + 110 T_{2}^{3} + 286 T_{2}^{2} - 684 T_{2} - 108$$ $$T_{7}^{6} + 40 T_{7}^{5} - 263 T_{7}^{4} - 18900 T_{7}^{3} - 49260 T_{7}^{2} + 1142856 T_{7} - 2316924$$ $$T_{11}^{6} + 88 T_{11}^{5} + 1518 T_{11}^{4} - 4520 T_{11}^{3} - 167759 T_{11}^{2} - 426912 T_{11} + 1197108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-108 - 684 T + 286 T^{2} + 110 T^{3} - 33 T^{4} - 4 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$-2316924 + 1142856 T - 49260 T^{2} - 18900 T^{3} - 263 T^{4} + 40 T^{5} + T^{6}$$
$11$ $$1197108 - 426912 T - 167759 T^{2} - 4520 T^{3} + 1518 T^{4} + 88 T^{5} + T^{6}$$
$13$ $$-185793728 + 11327360 T + 2055488 T^{2} - 93472 T^{3} - 4519 T^{4} + 20 T^{5} + T^{6}$$
$17$ $$105966288 - 34954656 T - 3790268 T^{2} + 762824 T^{3} - 5004 T^{4} - 124 T^{5} + T^{6}$$
$19$ $$-36821611175 + 1527158350 T + 46339391 T^{2} - 872156 T^{3} - 18169 T^{4} + 46 T^{5} + T^{6}$$
$23$ $$332569842768 - 29896763040 T + 135198072 T^{2} + 5308560 T^{3} - 25779 T^{4} - 210 T^{5} + T^{6}$$
$29$ $$635447099088 + 9092541024 T - 206203583 T^{2} - 3275896 T^{3} + 10626 T^{4} + 296 T^{5} + T^{6}$$
$31$ $$-69859854216 - 1251193716 T + 509538609 T^{2} - 1378812 T^{3} - 58274 T^{4} + 104 T^{5} + T^{6}$$
$37$ $$12008297128192 - 211123326144 T - 305431068 T^{2} + 20257048 T^{3} - 74400 T^{4} - 204 T^{5} + T^{6}$$
$41$ $$-86213802377403 + 1228171376184 T + 10310796571 T^{2} - 48204400 T^{3} - 185193 T^{4} + 344 T^{5} + T^{6}$$
$43$ $$-180465347194400 + 3519864391760 T + 8186569412 T^{2} - 89502136 T^{3} - 172552 T^{4} + 512 T^{5} + T^{6}$$
$47$ $$-49451750433900 - 1521993107760 T + 23432551972 T^{2} + 86562956 T^{3} - 427923 T^{4} - 238 T^{5} + T^{6}$$
$53$ $$15741889277692500 - 87258267127200 T - 7800157964 T^{2} + 601620140 T^{3} - 550155 T^{4} - 850 T^{5} + T^{6}$$
$59$ $$-168320389359483 + 2504531324400 T + 64312689211 T^{2} + 431982880 T^{3} + 1299447 T^{4} + 1840 T^{5} + T^{6}$$
$61$ $$-20318301594331136 + 69945317346304 T + 298491534848 T^{2} - 318570368 T^{3} - 1014088 T^{4} + 364 T^{5} + T^{6}$$
$67$ $$19496902162867200 + 171798967607040 T + 382729270992 T^{2} - 289302624 T^{3} - 1313672 T^{4} + 88 T^{5} + T^{6}$$
$71$ $$-11833359413893884 + 215551314974244 T - 225597721799 T^{2} - 1811192200 T^{3} - 995994 T^{4} + 1364 T^{5} + T^{6}$$
$73$ $$-10768933998500336 + 63694541190656 T + 254161845668 T^{2} - 781045192 T^{3} - 1130908 T^{4} + 836 T^{5} + T^{6}$$
$79$ $$-15947821725492672 + 246823997051520 T + 413753779248 T^{2} - 1090263744 T^{3} - 1627796 T^{4} + 680 T^{5} + T^{6}$$
$83$ $$4163598986306832 + 30732426001728 T - 464661188604 T^{2} + 848431368 T^{3} + 655092 T^{4} - 2148 T^{5} + T^{6}$$
$89$ $$167521946305912848 + 24233669446680 T - 1254009337479 T^{2} - 1004808240 T^{3} + 2101338 T^{4} + 3000 T^{5} + T^{6}$$
$97$ $$-979999359726233024 - 975674052130752 T + 3181554001188 T^{2} + 1545360136 T^{3} - 3167712 T^{4} - 612 T^{5} + T^{6}$$