# Properties

 Label 2025.4.a.be Level $2025$ Weight $4$ Character orbit 2025.a Self dual yes Analytic conductor $119.479$ Analytic rank $1$ Dimension $12$ 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: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + 52669 x^{4} - 144610 x^{3} - 96816 x^{2} + 143136 x + 73008$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 3^{8}$$ Twist minimal: no (minimal twist has level 225) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( 4 + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{7} + ( -2 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( 4 + \beta_{2} ) q^{4} + ( -1 + \beta_{6} ) q^{7} + ( -2 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{8} + ( -2 - \beta_{1} + \beta_{5} - \beta_{6} ) q^{11} + ( -1 - \beta_{1} - \beta_{5} - \beta_{11} ) q^{13} + ( 7 - \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{14} + ( 13 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} ) q^{16} + ( -8 + 3 \beta_{1} - 3 \beta_{2} + \beta_{6} + \beta_{9} + \beta_{11} ) q^{17} + ( -9 + 5 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{19} + ( -3 + 9 \beta_{1} - \beta_{2} + 3 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{22} + ( -24 - \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} + ( 12 + 3 \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{26} + ( -8 - 3 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} - \beta_{4} - \beta_{5} + 6 \beta_{6} - 3 \beta_{7} - \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{28} + ( -10 + 8 \beta_{1} - 6 \beta_{2} - 2 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{10} + 3 \beta_{11} ) q^{29} + ( -3 + 17 \beta_{1} + \beta_{2} + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{31} + ( -77 - 9 \beta_{1} - 16 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} + 5 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{32} + ( -18 + 28 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} ) q^{34} + ( 6 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + 4 \beta_{10} + 2 \beta_{11} ) q^{37} + ( -56 + 9 \beta_{1} - 6 \beta_{2} + 7 \beta_{3} - 4 \beta_{4} - \beta_{5} - 10 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 4 \beta_{10} - 6 \beta_{11} ) q^{38} + ( 30 + 8 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 4 \beta_{7} - \beta_{8} + 2 \beta_{9} - 5 \beta_{10} + 2 \beta_{11} ) q^{41} + ( 32 + 12 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{5} + 8 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{43} + ( -44 + 2 \beta_{1} - 20 \beta_{2} + 3 \beta_{3} - 5 \beta_{4} - 12 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} - 5 \beta_{11} ) q^{44} + ( 1 + 64 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{4} - 2 \beta_{5} + 8 \beta_{6} - 5 \beta_{7} - \beta_{8} + 2 \beta_{9} + 8 \beta_{10} + 6 \beta_{11} ) q^{46} + ( -105 + 12 \beta_{1} - 11 \beta_{2} + 6 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - \beta_{9} + 3 \beta_{10} - 4 \beta_{11} ) q^{47} + ( 35 + 55 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{49} + ( 22 - 44 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} + 3 \beta_{8} - 2 \beta_{10} - 2 \beta_{11} ) q^{52} + ( -38 + 5 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} + 9 \beta_{4} - \beta_{5} + 4 \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{10} + 10 \beta_{11} ) q^{53} + ( 54 + 9 \beta_{1} + 42 \beta_{2} + 8 \beta_{3} + 13 \beta_{4} - 5 \beta_{5} - 7 \beta_{6} + 8 \beta_{7} + 2 \beta_{8} + \beta_{9} + 7 \beta_{10} - 8 \beta_{11} ) q^{56} + ( -59 + 47 \beta_{1} - 3 \beta_{2} + 7 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} - 10 \beta_{6} - 3 \beta_{7} - \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - 6 \beta_{11} ) q^{58} + ( -36 - 13 \beta_{1} - 27 \beta_{2} - 3 \beta_{4} + 3 \beta_{5} - 10 \beta_{6} + 4 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} - 2 \beta_{11} ) q^{59} + ( 30 + 8 \beta_{1} + 16 \beta_{2} + 5 \beta_{3} + 8 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 4 \beta_{7} - 3 \beta_{8} - 4 \beta_{10} - 2 \beta_{11} ) q^{61} + ( -219 + 5 \beta_{1} - 48 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 10 \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{62} + ( 43 + 130 \beta_{1} + 15 \beta_{2} + 11 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} - 12 \beta_{6} + 10 \beta_{7} + 6 \beta_{8} + 3 \beta_{9} + 4 \beta_{10} - 5 \beta_{11} ) q^{64} + ( 40 - 74 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} - 5 \beta_{4} - \beta_{5} - 14 \beta_{6} + 6 \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 9 \beta_{10} - 8 \beta_{11} ) q^{67} + ( -295 + 38 \beta_{1} - 30 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - 9 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} + 10 \beta_{10} + 4 \beta_{11} ) q^{68} + ( 74 + 27 \beta_{1} + 29 \beta_{2} - \beta_{3} - 9 \beta_{4} - 6 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} + \beta_{8} + 2 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{71} + ( 24 + 50 \beta_{1} - 7 \beta_{2} - 9 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} - 8 \beta_{6} - 4 \beta_{7} + 7 \beta_{8} + 4 \beta_{9} - 8 \beta_{10} + 9 \beta_{11} ) q^{73} + ( 30 - 52 \beta_{1} - 22 \beta_{2} - 15 \beta_{3} + 11 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 3 \beta_{7} + 9 \beta_{8} - \beta_{9} + 10 \beta_{10} - \beta_{11} ) q^{74} + ( -81 + 79 \beta_{1} - 35 \beta_{2} + 6 \beta_{3} - 26 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} - 9 \beta_{8} + 3 \beta_{9} - 19 \beta_{10} - 4 \beta_{11} ) q^{76} + ( -222 + 15 \beta_{1} - 38 \beta_{2} + 12 \beta_{3} + \beta_{4} - 12 \beta_{5} - 11 \beta_{6} + 7 \beta_{9} - 13 \beta_{10} - 5 \beta_{11} ) q^{77} + ( -5 + 80 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} + \beta_{4} - 7 \beta_{5} + 15 \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - 5 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{79} + ( -27 - 109 \beta_{1} - 14 \beta_{2} - 8 \beta_{3} + 2 \beta_{4} + 7 \beta_{5} - 7 \beta_{6} + 3 \beta_{7} - 4 \beta_{8} + 5 \beta_{9} + 9 \beta_{10} + 2 \beta_{11} ) q^{82} + ( -256 - 19 \beta_{1} - 6 \beta_{2} - \beta_{3} - 5 \beta_{4} + 5 \beta_{5} - \beta_{6} + 8 \beta_{7} + 11 \beta_{8} - \beta_{9} - 11 \beta_{10} - \beta_{11} ) q^{83} + ( -161 - 55 \beta_{1} + 13 \beta_{2} - 5 \beta_{3} - 6 \beta_{4} + 16 \beta_{6} - 2 \beta_{7} + \beta_{8} - 9 \beta_{9} - 20 \beta_{10} - 9 \beta_{11} ) q^{86} + ( -42 + 150 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} - 14 \beta_{4} + 14 \beta_{5} - 6 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} - 6 \beta_{9} + 4 \beta_{10} - 2 \beta_{11} ) q^{88} + ( 39 + 59 \beta_{1} - 31 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} + 5 \beta_{6} + 6 \beta_{8} - \beta_{9} + 8 \beta_{10} + 8 \beta_{11} ) q^{89} + ( -19 - 35 \beta_{1} - 23 \beta_{3} + 7 \beta_{4} + 11 \beta_{5} - 15 \beta_{6} + 6 \beta_{7} + 5 \beta_{8} - 7 \beta_{9} + 22 \beta_{10} - 6 \beta_{11} ) q^{91} + ( -413 - 33 \beta_{1} - 72 \beta_{2} - 9 \beta_{3} + 12 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} - 10 \beta_{7} + 5 \beta_{8} + 5 \beta_{9} + 8 \beta_{10} + 5 \beta_{11} ) q^{92} + ( -79 + 215 \beta_{1} - 25 \beta_{2} - 4 \beta_{3} - 11 \beta_{4} + 9 \beta_{5} + \beta_{6} - 6 \beta_{7} + \beta_{8} - 8 \beta_{9} - 9 \beta_{10} + 5 \beta_{11} ) q^{94} + ( 78 - 138 \beta_{1} + 23 \beta_{2} + 4 \beta_{3} + 15 \beta_{4} - 12 \beta_{5} + 7 \beta_{6} - 10 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + 10 \beta_{11} ) q^{97} + ( -590 - 87 \beta_{1} - 43 \beta_{2} - 6 \beta_{3} + 8 \beta_{4} + 6 \beta_{5} - 13 \beta_{6} + \beta_{7} + 7 \beta_{8} - 2 \beta_{9} - 7 \beta_{10} - 11 \beta_{11} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} + O(q^{10})$$ $$12 q - 4 q^{2} + 48 q^{4} - 6 q^{7} - 45 q^{8} - 29 q^{11} - 24 q^{13} + 69 q^{14} + 192 q^{16} - 79 q^{17} - 75 q^{19} + 18 q^{22} - 318 q^{23} + 154 q^{26} - 96 q^{28} - 106 q^{29} + 60 q^{31} - 914 q^{32} - 108 q^{34} + 84 q^{37} - 640 q^{38} + 353 q^{41} + 426 q^{43} - 571 q^{44} + 270 q^{46} - 1210 q^{47} + 666 q^{49} + 75 q^{52} - 448 q^{53} + 570 q^{56} - 594 q^{58} - 482 q^{59} + 402 q^{61} - 2544 q^{62} + 975 q^{64} + 201 q^{67} - 3437 q^{68} + 944 q^{71} + 453 q^{73} - 10 q^{74} - 462 q^{76} - 2652 q^{77} + 258 q^{79} - 873 q^{82} - 3012 q^{83} - 1952 q^{86} + 216 q^{88} + 738 q^{89} - 618 q^{91} - 5232 q^{92} + 63 q^{94} + 318 q^{97} - 7511 q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} - 64 x^{10} + 241 x^{9} + 1452 x^{8} - 5130 x^{7} - 13834 x^{6} + 45247 x^{5} + 52669 x^{4} - 144610 x^{3} - 96816 x^{2} + 143136 x + 73008$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 20 \nu + 10$$ $$\beta_{4}$$ $$=$$ $$($$$$37591 \nu^{11} + 423747 \nu^{10} - 4004629 \nu^{9} - 25985670 \nu^{8} + 140077854 \nu^{7} + 533695176 \nu^{6} - 2078007262 \nu^{5} - 3965272965 \nu^{4} + 12714043102 \nu^{3} + 6201852960 \nu^{2} - 19241890608 \nu + 346418736$$$$)/ 158287872$$ $$\beta_{5}$$ $$=$$ $$($$$$-59961 \nu^{11} + 592499 \nu^{10} + 2246427 \nu^{9} - 35963750 \nu^{8} + 8257662 \nu^{7} + 752424648 \nu^{6} - 1135333182 \nu^{5} - 6182499797 \nu^{4} + 12441331134 \nu^{3} + 16298881568 \nu^{2} - 27249812016 \nu - 16398173904$$$$)/ 158287872$$ $$\beta_{6}$$ $$=$$ $$($$$$65605 \nu^{11} + 48265 \nu^{10} - 4932735 \nu^{9} - 4652306 \nu^{8} + 134284506 \nu^{7} + 140925528 \nu^{6} - 1610120986 \nu^{5} - 1658564047 \nu^{4} + 8249829402 \nu^{3} + 7241968736 \nu^{2} - 13531173264 \nu - 8996491632$$$$)/ 158287872$$ $$\beta_{7}$$ $$=$$ $$($$$$-6097 \nu^{11} + 10547 \nu^{10} + 390691 \nu^{9} - 623630 \nu^{8} - 8238762 \nu^{7} + 13670592 \nu^{6} + 58917130 \nu^{5} - 128683685 \nu^{4} - 36830482 \nu^{3} + 373846496 \nu^{2} - 183919344 \nu - 264990672$$$$)/9892992$$ $$\beta_{8}$$ $$=$$ $$($$$$-59757 \nu^{11} + 119023 \nu^{10} + 4175607 \nu^{9} - 6183454 \nu^{8} - 108583914 \nu^{7} + 115983912 \nu^{6} + 1271042346 \nu^{5} - 1043697913 \nu^{4} - 6177102186 \nu^{3} + 4660906144 \nu^{2} + 6912879120 \nu - 4259211024$$$$)/79143936$$ $$\beta_{9}$$ $$=$$ $$($$$$-196583 \nu^{11} + 1709165 \nu^{10} + 9232965 \nu^{9} - 103544538 \nu^{8} - 97918782 \nu^{7} + 2176503480 \nu^{6} - 836392450 \nu^{5} - 18229949003 \nu^{4} + 15043819266 \nu^{3} + 50098565088 \nu^{2} - 27707922384 \nu - 38296603440$$$$)/ 158287872$$ $$\beta_{10}$$ $$=$$ $$($$$$232351 \nu^{11} - 1184469 \nu^{10} - 13940461 \nu^{9} + 70745130 \nu^{8} + 281453742 \nu^{7} - 1447343352 \nu^{6} - 2024460142 \nu^{5} + 11263200675 \nu^{4} + 2095604974 \nu^{3} - 22654098144 \nu^{2} + 1967766096 \nu + 5029485744$$$$)/ 158287872$$ $$\beta_{11}$$ $$=$$ $$($$$$-86739 \nu^{11} + 184561 \nu^{10} + 5649033 \nu^{9} - 9691522 \nu^{8} - 129243510 \nu^{7} + 170737080 \nu^{6} + 1216581942 \nu^{5} - 1088033191 \nu^{4} - 4240037334 \nu^{3} + 834353632 \nu^{2} + 4635050736 \nu + 2385982224$$$$)/39571968$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 20 \beta_{1} + 2$$ $$\nu^{4}$$ $$=$$ $$\beta_{7} - \beta_{5} + \beta_{4} + 2 \beta_{3} + 28 \beta_{2} + 8 \beta_{1} + 237$$ $$\nu^{5}$$ $$=$$ $$-\beta_{11} + \beta_{10} + 2 \beta_{9} - \beta_{8} + 2 \beta_{7} - 5 \beta_{6} - 3 \beta_{5} + \beta_{4} + 35 \beta_{3} + 48 \beta_{2} + 457 \beta_{1} + 141$$ $$\nu^{6}$$ $$=$$ $$-5 \beta_{11} + 4 \beta_{10} + 3 \beta_{9} + 6 \beta_{8} + 50 \beta_{7} - 12 \beta_{6} - 43 \beta_{5} + 46 \beta_{4} + 91 \beta_{3} + 751 \beta_{2} + 450 \beta_{1} + 5427$$ $$\nu^{7}$$ $$=$$ $$-69 \beta_{11} + 49 \beta_{10} + 90 \beta_{9} - 29 \beta_{8} + 133 \beta_{7} - 291 \beta_{6} - 134 \beta_{5} + 78 \beta_{4} + 1070 \beta_{3} + 1790 \beta_{2} + 11170 \beta_{1} + 6568$$ $$\nu^{8}$$ $$=$$ $$-318 \beta_{11} + 245 \beta_{10} + 177 \beta_{9} + 345 \beta_{8} + 1839 \beta_{7} - 909 \beta_{6} - 1413 \beta_{5} + 1678 \beta_{4} + 3291 \beta_{3} + 20567 \beta_{2} + 18083 \beta_{1} + 134213$$ $$\nu^{9}$$ $$=$$ $$-3213 \beta_{11} + 1736 \beta_{10} + 3075 \beta_{9} - 354 \beta_{8} + 5988 \beta_{7} - 12174 \beta_{6} - 4713 \beta_{5} + 3838 \beta_{4} + 31825 \beta_{3} + 61250 \beta_{2} + 286279 \beta_{1} + 252055$$ $$\nu^{10}$$ $$=$$ $$-14481 \beta_{11} + 10121 \beta_{10} + 7626 \beta_{9} + 14103 \beta_{8} + 61153 \beta_{7} - 44439 \beta_{6} - 43042 \beta_{5} + 56102 \beta_{4} + 110969 \beta_{3} + 576770 \beta_{2} + 640906 \beta_{1} + 3497582$$ $$\nu^{11}$$ $$=$$ $$-126046 \beta_{11} + 56109 \beta_{10} + 96569 \beta_{9} + 9401 \beta_{8} + 230378 \beta_{7} - 445997 \beta_{6} - 155160 \beta_{5} + 157649 \beta_{4} + 942794 \beta_{3} + 2012128 \beta_{2} + 7615071 \beta_{1} + 8807654$$

## 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
 5.54419 4.78880 3.63978 2.97659 2.14580 1.38827 −0.476035 −1.33965 −1.66434 −3.81662 −4.33545 −4.85133
−5.54419 0 22.7380 0 0 −25.7127 −81.7101 0 0
1.2 −4.78880 0 14.9326 0 0 30.6698 −33.1990 0 0
1.3 −3.63978 0 5.24803 0 0 3.72054 10.0166 0 0
1.4 −2.97659 0 0.860098 0 0 −24.9651 21.2526 0 0
1.5 −2.14580 0 −3.39555 0 0 23.2564 24.4526 0 0
1.6 −1.38827 0 −6.07270 0 0 −26.4332 19.5367 0 0
1.7 0.476035 0 −7.77339 0 0 −12.6809 −7.50869 0 0
1.8 1.33965 0 −6.20534 0 0 22.3290 −19.0302 0 0
1.9 1.66434 0 −5.22997 0 0 10.5672 −22.0192 0 0
1.10 3.81662 0 6.56661 0 0 −9.98804 −5.47071 0 0
1.11 4.33545 0 10.7961 0 0 −13.0015 12.1226 0 0
1.12 4.85133 0 15.5354 0 0 16.2385 36.5569 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 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.be 12
3.b odd 2 1 2025.4.a.bi 12
5.b even 2 1 2025.4.a.bj 12
9.d odd 6 2 225.4.e.e 24
15.d odd 2 1 2025.4.a.bf 12
45.h odd 6 2 225.4.e.f yes 24
45.l even 12 4 225.4.k.e 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.e.e 24 9.d odd 6 2
225.4.e.f yes 24 45.h odd 6 2
225.4.k.e 48 45.l even 12 4
2025.4.a.be 12 1.a even 1 1 trivial
2025.4.a.bf 12 15.d odd 2 1
2025.4.a.bi 12 3.b odd 2 1
2025.4.a.bj 12 5.b even 2 1

## Hecke kernels

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

 $$T_{2}^{12} + \cdots$$ $$12\!\cdots\!98$$$$T_{7}^{2} -$$$$49\!\cdots\!24$$$$T_{7} +$$$$28\!\cdots\!04$$">$$T_{7}^{12} + \cdots$$ $$18\!\cdots\!23$$$$T_{11}^{4} +$$$$34\!\cdots\!25$$$$T_{11}^{3} -$$$$31\!\cdots\!44$$$$T_{11}^{2} -$$$$12\!\cdots\!44$$$$T_{11} +$$$$27\!\cdots\!56$$">$$T_{11}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$73008 - 143136 T - 96816 T^{2} + 144610 T^{3} + 52669 T^{4} - 45247 T^{5} - 13834 T^{6} + 5130 T^{7} + 1452 T^{8} - 241 T^{9} - 64 T^{10} + 4 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$284111126595504 - 49069079780724 T - 12328245550998 T^{2} + 857927085426 T^{3} + 159370316127 T^{4} - 5001274746 T^{5} - 849150162 T^{6} + 12946194 T^{7} + 2090268 T^{8} - 14914 T^{9} - 2373 T^{10} + 6 T^{11} + T^{12}$$
$11$ $$27256136074433856 - 12280529942857944 T - 3157793940816144 T^{2} + 34317620761625 T^{3} + 18363263926723 T^{4} - 190920855323 T^{5} - 33136722397 T^{6} + 337214331 T^{7} + 25135263 T^{8} - 189035 T^{9} - 8485 T^{10} + 29 T^{11} + T^{12}$$
$13$ $$-385289000327808928 + 46160836388280048 T + 5446892780347758 T^{2} - 700629462886664 T^{3} - 8719969163409 T^{4} + 2887433528958 T^{5} - 74819695635 T^{6} - 1155502296 T^{7} + 54685350 T^{8} + 11558 T^{9} - 12534 T^{10} + 24 T^{11} + T^{12}$$
$17$ $$7181220247388950032 - 12164540926511431800 T + 598597908197941554 T^{2} + 58912433476913581 T^{3} - 191175556389239 T^{4} - 55139594039620 T^{5} - 301026745918 T^{6} + 18011903076 T^{7} + 158488608 T^{8} - 2248891 T^{9} - 24163 T^{10} + 79 T^{11} + T^{12}$$
$19$ $$-$$$$12\!\cdots\!00$$$$+$$$$33\!\cdots\!75$$$$T - 19290167939699981625 T^{2} + 48998564847805655 T^{3} + 17159088225239253 T^{4} - 150538438141749 T^{5} - 5626083860883 T^{6} + 36730419615 T^{7} + 784543635 T^{8} - 3002249 T^{9} - 47427 T^{10} + 75 T^{11} + T^{12}$$
$23$ $$-$$$$18\!\cdots\!64$$$$+$$$$62\!\cdots\!32$$$$T +$$$$30\!\cdots\!18$$$$T^{2} - 375377436918327834 T^{3} - 148822393825959147 T^{4} - 1062943053939702 T^{5} + 21386579821029 T^{6} + 246748261392 T^{7} - 730608039 T^{8} - 16782174 T^{9} - 25515 T^{10} + 318 T^{11} + T^{12}$$
$29$ $$-$$$$43\!\cdots\!56$$$$-$$$$14\!\cdots\!24$$$$T + 56090193437044967214 T^{2} +$$$$34\!\cdots\!52$$$$T^{3} + 1506350043586913377 T^{4} - 30390603945145804 T^{5} - 177481907912164 T^{6} + 1213369126746 T^{7} + 8110485012 T^{8} - 20800564 T^{9} - 157057 T^{10} + 106 T^{11} + T^{12}$$
$31$ $$95\!\cdots\!20$$$$-$$$$76\!\cdots\!28$$$$T -$$$$40\!\cdots\!86$$$$T^{2} + 70244390361946178754 T^{3} + 355499210542801401 T^{4} - 20169387935802576 T^{5} - 192804984768789 T^{6} + 682745847966 T^{7} + 10619419878 T^{8} - 1003688 T^{9} - 183282 T^{10} - 60 T^{11} + T^{12}$$
$37$ $$69\!\cdots\!84$$$$+$$$$23\!\cdots\!12$$$$T -$$$$36\!\cdots\!38$$$$T^{2} -$$$$27\!\cdots\!62$$$$T^{3} + 38348453157598002195 T^{4} + 118927993223581092 T^{5} - 1608279936474315 T^{6} - 2415968835570 T^{7} + 31736871423 T^{8} + 23197262 T^{9} - 291975 T^{10} - 84 T^{11} + T^{12}$$
$41$ $$-$$$$92\!\cdots\!30$$$$+$$$$73\!\cdots\!43$$$$T +$$$$21\!\cdots\!53$$$$T^{2} -$$$$20\!\cdots\!96$$$$T^{3} - 83619852721967185880 T^{4} + 1236618973037062340 T^{5} - 611872400517286 T^{6} - 23199108753810 T^{7} + 42325335780 T^{8} + 160562804 T^{9} - 390166 T^{10} - 353 T^{11} + T^{12}$$
$43$ $$-$$$$47\!\cdots\!87$$$$-$$$$51\!\cdots\!94$$$$T +$$$$85\!\cdots\!02$$$$T^{2} -$$$$74\!\cdots\!82$$$$T^{3} -$$$$23\!\cdots\!13$$$$T^{4} + 3696741934206708564 T^{5} - 2939186066493348 T^{6} - 51767364995604 T^{7} + 91670234319 T^{8} + 260110658 T^{9} - 558870 T^{10} - 426 T^{11} + T^{12}$$
$47$ $$32\!\cdots\!52$$$$+$$$$49\!\cdots\!12$$$$T -$$$$57\!\cdots\!92$$$$T^{2} -$$$$10\!\cdots\!62$$$$T^{3} - 70055474021037822185 T^{4} + 5089683318317523734 T^{5} + 26665265580047666 T^{6} + 18900058487112 T^{7} - 187399428681 T^{8} - 449595238 T^{9} + 85796 T^{10} + 1210 T^{11} + T^{12}$$
$53$ $$-$$$$47\!\cdots\!48$$$$+$$$$10\!\cdots\!48$$$$T -$$$$48\!\cdots\!38$$$$T^{2} -$$$$40\!\cdots\!08$$$$T^{3} +$$$$33\!\cdots\!87$$$$T^{4} + 2270210539901990726 T^{5} - 59928808389738247 T^{6} + 44684897864436 T^{7} + 365539731594 T^{8} - 295092598 T^{9} - 982906 T^{10} + 448 T^{11} + T^{12}$$
$59$ $$-$$$$33\!\cdots\!79$$$$-$$$$15\!\cdots\!82$$$$T -$$$$11\!\cdots\!96$$$$T^{2} +$$$$79\!\cdots\!28$$$$T^{3} +$$$$61\!\cdots\!33$$$$T^{4} - 17706691267118148704 T^{5} - 78406788589879825 T^{6} + 138254723013942 T^{7} + 424568199681 T^{8} - 436910498 T^{9} - 1059103 T^{10} + 482 T^{11} + T^{12}$$
$61$ $$-$$$$15\!\cdots\!48$$$$+$$$$33\!\cdots\!72$$$$T -$$$$74\!\cdots\!60$$$$T^{2} -$$$$17\!\cdots\!92$$$$T^{3} +$$$$41\!\cdots\!97$$$$T^{4} + 28629173845059036534 T^{5} - 62160954009808797 T^{6} - 188731282550112 T^{7} + 405003637701 T^{8} + 511621144 T^{9} - 1133925 T^{10} - 402 T^{11} + T^{12}$$
$67$ $$-$$$$11\!\cdots\!76$$$$+$$$$33\!\cdots\!95$$$$T +$$$$15\!\cdots\!37$$$$T^{2} -$$$$11\!\cdots\!79$$$$T^{3} +$$$$74\!\cdots\!89$$$$T^{4} +$$$$12\!\cdots\!16$$$$T^{5} - 229265016422570820 T^{6} - 444168519276087 T^{7} + 1098662177187 T^{8} + 559514258 T^{9} - 1833576 T^{10} - 201 T^{11} + T^{12}$$
$71$ $$-$$$$53\!\cdots\!44$$$$+$$$$18\!\cdots\!00$$$$T -$$$$23\!\cdots\!54$$$$T^{2} +$$$$14\!\cdots\!40$$$$T^{3} -$$$$46\!\cdots\!95$$$$T^{4} +$$$$54\!\cdots\!58$$$$T^{5} + 716501129964507647 T^{6} - 2512456451919900 T^{7} + 1069785827760 T^{8} + 2752229720 T^{9} - 2313670 T^{10} - 944 T^{11} + T^{12}$$
$73$ $$-$$$$10\!\cdots\!02$$$$+$$$$18\!\cdots\!69$$$$T +$$$$47\!\cdots\!13$$$$T^{2} -$$$$21\!\cdots\!50$$$$T^{3} +$$$$68\!\cdots\!32$$$$T^{4} +$$$$23\!\cdots\!07$$$$T^{5} - 651783684946154823 T^{6} - 805571735249988 T^{7} + 1989839444346 T^{8} + 1073672183 T^{9} - 2432463 T^{10} - 453 T^{11} + T^{12}$$
$79$ $$16\!\cdots\!84$$$$+$$$$70\!\cdots\!48$$$$T -$$$$44\!\cdots\!12$$$$T^{2} -$$$$11\!\cdots\!42$$$$T^{3} +$$$$44\!\cdots\!27$$$$T^{4} +$$$$61\!\cdots\!88$$$$T^{5} - 1919253594695042277 T^{6} - 1297194552501660 T^{7} + 3808082445363 T^{8} + 1094395870 T^{9} - 3343407 T^{10} - 258 T^{11} + T^{12}$$
$83$ $$68\!\cdots\!39$$$$+$$$$13\!\cdots\!48$$$$T +$$$$31\!\cdots\!77$$$$T^{2} -$$$$31\!\cdots\!30$$$$T^{3} -$$$$11\!\cdots\!33$$$$T^{4} +$$$$17\!\cdots\!28$$$$T^{5} + 1059958015279203483 T^{6} + 284691924677556 T^{7} - 2918837996757 T^{8} - 2771939934 T^{9} + 1645182 T^{10} + 3012 T^{11} + T^{12}$$
$89$ $$13\!\cdots\!25$$$$-$$$$30\!\cdots\!50$$$$T +$$$$66\!\cdots\!25$$$$T^{2} +$$$$24\!\cdots\!00$$$$T^{3} -$$$$17\!\cdots\!20$$$$T^{4} - 3717622115374187898 T^{5} + 346517988733566468 T^{6} - 936059907646854 T^{7} + 78575532300 T^{8} + 2429989308 T^{9} - 2188323 T^{10} - 738 T^{11} + T^{12}$$
$97$ $$27\!\cdots\!97$$$$-$$$$55\!\cdots\!94$$$$T +$$$$31\!\cdots\!89$$$$T^{2} -$$$$30\!\cdots\!64$$$$T^{3} +$$$$38\!\cdots\!52$$$$T^{4} +$$$$27\!\cdots\!46$$$$T^{5} - 6103799304313055520 T^{6} - 2920738353253038 T^{7} + 10674843843204 T^{8} + 1525904492 T^{9} - 5808039 T^{10} - 318 T^{11} + T^{12}$$