# Properties

 Label 4025.2.a.x Level 4025 Weight 2 Character orbit 4025.a Self dual yes Analytic conductor 32.140 Analytic rank 1 Dimension 12 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4025 = 5^{2} \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1397868136$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 805) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + \beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{6} - \beta_{9} - \beta_{10} ) q^{6} + q^{7} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{8} + ( 1 + \beta_{1} - \beta_{8} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + \beta_{6} q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 - \beta_{6} - \beta_{9} - \beta_{10} ) q^{6} + q^{7} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{8} + ( 1 + \beta_{1} - \beta_{8} ) q^{9} + ( -1 - \beta_{2} + \beta_{8} + \beta_{9} ) q^{11} + ( 2 + \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{12} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{13} -\beta_{1} q^{14} + ( \beta_{2} + \beta_{5} + \beta_{6} + \beta_{10} + \beta_{11} ) q^{16} + ( -\beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{17} + ( -2 - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{8} + \beta_{9} ) q^{18} + ( -3 - \beta_{2} - \beta_{5} + \beta_{8} + \beta_{10} ) q^{19} + \beta_{6} q^{21} + ( -1 - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} ) q^{22} - q^{23} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{24} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{26} + ( -1 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{10} + \beta_{11} ) q^{27} + ( 1 + \beta_{2} ) q^{28} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{9} - \beta_{11} ) q^{31} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{8} + \beta_{10} ) q^{32} + ( -1 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{33} + ( -2 - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} + \beta_{11} ) q^{34} + ( -1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{36} + ( -\beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} ) q^{37} + ( 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{10} + \beta_{11} ) q^{38} + ( -3 - 3 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - \beta_{9} + 2 \beta_{10} ) q^{39} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{7} + \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{41} + ( -1 - \beta_{6} - \beta_{9} - \beta_{10} ) q^{42} + ( -3 + \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{8} + 2 \beta_{10} - 2 \beta_{11} ) q^{43} + ( -2 - 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{44} + \beta_{1} q^{46} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{9} - 2 \beta_{10} ) q^{47} + ( 3 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{48} + q^{49} + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{51} + ( -2 - 2 \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{52} + ( 3 - \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} - \beta_{6} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{53} + ( -3 - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{54} + ( -1 + \beta_{1} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{56} + ( 1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} - 4 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} ) q^{57} + ( 1 + \beta_{1} + \beta_{3} - 4 \beta_{4} - \beta_{7} + 2 \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{58} + ( -2 + \beta_{2} - 2 \beta_{3} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{59} + ( -1 + \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{61} + ( 6 + 4 \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{62} + ( 1 + \beta_{1} - \beta_{8} ) q^{63} + ( 2 \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} ) q^{64} + ( 2 + \beta_{1} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} + 3 \beta_{9} + 2 \beta_{11} ) q^{66} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{4} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{67} + ( 2 + \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{11} ) q^{68} -\beta_{6} q^{69} + ( -2 - \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{7} + \beta_{8} + 2 \beta_{10} + 2 \beta_{11} ) q^{71} + ( -4 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} ) q^{72} + ( 2 + 5 \beta_{1} - \beta_{2} + \beta_{3} - 3 \beta_{4} - 4 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} ) q^{73} + ( 5 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} + 3 \beta_{10} ) q^{74} + ( -6 - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{76} + ( -1 - \beta_{2} + \beta_{8} + \beta_{9} ) q^{77} + ( 2 + 4 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{10} + \beta_{11} ) q^{78} + ( -5 - \beta_{1} - 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{79} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - 2 \beta_{11} ) q^{81} + ( 5 + 2 \beta_{1} + 5 \beta_{2} - \beta_{3} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{82} + ( 1 + \beta_{1} - 2 \beta_{3} + \beta_{5} + \beta_{6} - 3 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} + ( 2 + \beta_{2} + \beta_{4} + \beta_{9} + \beta_{10} ) q^{84} + ( -4 + 2 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 4 \beta_{11} ) q^{86} + ( -2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 3 \beta_{10} + 3 \beta_{11} ) q^{87} + ( -2 + 3 \beta_{1} - 3 \beta_{2} - \beta_{5} + \beta_{6} - 2 \beta_{7} + 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{88} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} - 4 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{89} + ( \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} + \beta_{11} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( -4 + \beta_{2} - \beta_{3} + 2 \beta_{4} - 6 \beta_{6} + \beta_{8} - 3 \beta_{9} - 3 \beta_{10} ) q^{93} + ( -1 - 5 \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 5 \beta_{7} + 2 \beta_{9} + \beta_{10} - 4 \beta_{11} ) q^{94} + ( -4 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{96} + ( 4 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{97} -\beta_{1} q^{98} + ( -3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + 2 \beta_{8} - \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 2q^{2} + 8q^{4} - 6q^{6} + 12q^{7} - 6q^{8} + 8q^{9} + O(q^{10})$$ $$12q - 2q^{2} + 8q^{4} - 6q^{6} + 12q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 12q^{12} - 2q^{13} - 2q^{14} - 4q^{16} + 8q^{17} - 20q^{18} - 26q^{19} - 4q^{22} - 12q^{23} - 12q^{24} - 22q^{26} - 12q^{27} + 8q^{28} - 12q^{29} - 50q^{31} - 14q^{32} - 4q^{33} - 28q^{34} - 18q^{36} - 8q^{37} + 4q^{38} - 26q^{39} - 4q^{41} - 6q^{42} - 26q^{43} - 10q^{44} + 2q^{46} - 16q^{47} + 40q^{48} + 12q^{49} - 32q^{51} - 10q^{52} + 18q^{53} - 10q^{54} - 6q^{56} + 10q^{57} + 18q^{58} - 18q^{59} + 8q^{61} + 54q^{62} + 8q^{63} + 12q^{64} - 2q^{66} - 38q^{67} + 36q^{68} - 24q^{71} - 18q^{72} + 14q^{73} + 36q^{74} - 56q^{76} - 8q^{77} + 26q^{78} - 44q^{79} - 16q^{81} + 44q^{82} + 14q^{83} + 12q^{84} - 32q^{86} - 16q^{87} - 32q^{88} - 10q^{89} - 2q^{91} - 8q^{92} - 26q^{93} + 18q^{94} - 38q^{96} + 4q^{97} - 2q^{98} - 56q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 14 x^{10} + 26 x^{9} + 71 x^{8} - 120 x^{7} - 162 x^{6} + 244 x^{5} + 170 x^{4} - 210 x^{3} - 81 x^{2} + 58 x + 17$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$17 \nu^{11} - 24 \nu^{10} - 218 \nu^{9} + 237 \nu^{8} + 966 \nu^{7} - 670 \nu^{6} - 1780 \nu^{5} + 537 \nu^{4} + 1297 \nu^{3} + 81 \nu^{2} - 241 \nu - 94$$$$)/29$$ $$\beta_{4}$$ $$=$$ $$($$$$-12 \nu^{11} + 5 \nu^{10} + 188 \nu^{9} - 53 \nu^{8} - 1035 \nu^{7} + 171 \nu^{6} + 2367 \nu^{5} - 159 \nu^{4} - 2125 \nu^{3} - 151 \nu^{2} + 571 \nu + 138$$$$)/29$$ $$\beta_{5}$$ $$=$$ $$($$$$18 \nu^{11} - 22 \nu^{10} - 253 \nu^{9} + 239 \nu^{8} + 1277 \nu^{7} - 822 \nu^{6} - 2840 \nu^{5} + 1036 \nu^{4} + 2767 \nu^{3} - 281 \nu^{2} - 958 \nu - 120$$$$)/29$$ $$\beta_{6}$$ $$=$$ $$($$$$20 \nu^{11} - 18 \nu^{10} - 294 \nu^{9} + 185 \nu^{8} + 1551 \nu^{7} - 575 \nu^{6} - 3539 \nu^{5} + 613 \nu^{4} + 3329 \nu^{3} - 77 \nu^{2} - 913 \nu - 114$$$$)/29$$ $$\beta_{7}$$ $$=$$ $$($$$$-28 \nu^{11} + 31 \nu^{10} + 400 \nu^{9} - 317 \nu^{8} - 2067 \nu^{7} + 950 \nu^{6} + 4740 \nu^{5} - 806 \nu^{4} - 4736 \nu^{3} - 275 \nu^{2} + 1574 \nu + 322$$$$)/29$$ $$\beta_{8}$$ $$=$$ $$($$$$24 \nu^{11} - 10 \nu^{10} - 376 \nu^{9} + 77 \nu^{8} + 2128 \nu^{7} - 52 \nu^{6} - 5256 \nu^{5} - 523 \nu^{4} + 5468 \nu^{3} + 1056 \nu^{2} - 1751 \nu - 508$$$$)/29$$ $$\beta_{9}$$ $$=$$ $$($$$$-33 \nu^{11} + 21 \nu^{10} + 488 \nu^{9} - 182 \nu^{8} - 2578 \nu^{7} + 347 \nu^{6} + 5864 \nu^{5} + 179 \nu^{4} - 5561 \nu^{3} - 785 \nu^{2} + 1592 \nu + 336$$$$)/29$$ $$\beta_{10}$$ $$=$$ $$($$$$35 \nu^{11} - 17 \nu^{10} - 529 \nu^{9} + 128 \nu^{8} + 2852 \nu^{7} - 71 \nu^{6} - 6592 \nu^{5} - 863 \nu^{4} + 6355 \nu^{3} + 1569 \nu^{2} - 1953 \nu - 591$$$$)/29$$ $$\beta_{11}$$ $$=$$ $$($$$$-73 \nu^{11} + 57 \nu^{10} + 1076 \nu^{9} - 552 \nu^{8} - 5680 \nu^{7} + 1468 \nu^{6} + 12971 \nu^{5} - 757 \nu^{4} - 12451 \nu^{3} - 1414 \nu^{2} + 3824 \nu + 1028$$$$)/29$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + 7 \beta_{2} + 14$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{10} + 8 \beta_{9} + \beta_{8} + 8 \beta_{7} + 8 \beta_{6} + 7 \beta_{5} + 7 \beta_{4} + \beta_{3} + 12 \beta_{1} + 9$$ $$\nu^{6}$$ $$=$$ $$9 \beta_{11} + 9 \beta_{10} + \beta_{9} + \beta_{8} + 8 \beta_{6} + 9 \beta_{5} - \beta_{4} + \beta_{3} + 43 \beta_{2} + 2 \beta_{1} + 76$$ $$\nu^{7}$$ $$=$$ $$\beta_{11} + 43 \beta_{10} + 52 \beta_{9} + 11 \beta_{8} + 53 \beta_{7} + 52 \beta_{6} + 45 \beta_{5} + 43 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 57 \beta_{1} + 67$$ $$\nu^{8}$$ $$=$$ $$63 \beta_{11} + 63 \beta_{10} + 12 \beta_{9} + 13 \beta_{8} + 4 \beta_{7} + 53 \beta_{6} + 67 \beta_{5} - 10 \beta_{4} + 12 \beta_{3} + 257 \beta_{2} + 23 \beta_{1} + 438$$ $$\nu^{9}$$ $$=$$ $$16 \beta_{11} + 259 \beta_{10} + 319 \beta_{9} + 90 \beta_{8} + 334 \beta_{7} + 318 \beta_{6} + 288 \beta_{5} + 254 \beta_{4} + 78 \beta_{3} + 32 \beta_{2} + 299 \beta_{1} + 465$$ $$\nu^{10}$$ $$=$$ $$410 \beta_{11} + 412 \beta_{10} + 106 \beta_{9} + 124 \beta_{8} + 68 \beta_{7} + 337 \beta_{6} + 474 \beta_{5} - 69 \beta_{4} + 108 \beta_{3} + 1526 \beta_{2} + 184 \beta_{1} + 2594$$ $$\nu^{11}$$ $$=$$ $$172 \beta_{11} + 1561 \beta_{10} + 1919 \beta_{9} + 667 \beta_{8} + 2073 \beta_{7} + 1905 \beta_{6} + 1851 \beta_{5} + 1473 \beta_{4} + 563 \beta_{3} + 337 \beta_{2} + 1655 \beta_{1} + 3122$$

## 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
 2.55311 2.43455 1.53413 1.31562 1.23080 0.792462 −0.271475 −0.649823 −1.11343 −1.69164 −1.76402 −2.37028
−2.55311 0.556415 4.51835 0 −1.42059 1.00000 −6.42962 −2.69040 0
1.2 −2.43455 2.82471 3.92701 0 −6.87689 1.00000 −4.69139 4.97900 0
1.3 −1.53413 −2.66039 0.353561 0 4.08139 1.00000 2.52585 4.07768 0
1.4 −1.31562 −2.16083 −0.269134 0 2.84283 1.00000 2.98533 1.66917 0
1.5 −1.23080 0.767488 −0.485126 0 −0.944626 1.00000 3.05870 −2.41096 0
1.6 −0.792462 2.09947 −1.37200 0 −1.66375 1.00000 2.67218 1.40777 0
1.7 0.271475 2.40463 −1.92630 0 0.652799 1.00000 −1.06589 2.78227 0
1.8 0.649823 −1.89575 −1.57773 0 −1.23191 1.00000 −2.32489 0.593883 0
1.9 1.11343 −2.87976 −0.760266 0 −3.20642 1.00000 −3.07337 5.29300 0
1.10 1.69164 −0.285164 0.861643 0 −0.482394 1.00000 −1.92569 −2.91868 0
1.11 1.76402 1.09514 1.11176 0 1.93184 1.00000 −1.56687 −1.80068 0
1.12 2.37028 0.134036 3.61823 0 0.317703 1.00000 3.83566 −2.98203 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

## 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 4025.2.a.x 12
5.b even 2 1 4025.2.a.y 12
5.c odd 4 2 805.2.c.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
805.2.c.b 24 5.c odd 4 2
4025.2.a.x 12 1.a even 1 1 trivial
4025.2.a.y 12 5.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{12} + \cdots$$ $$T_{3}^{12} - \cdots$$ $$T_{11}^{12} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 10 T^{2} + 18 T^{3} + 55 T^{4} + 92 T^{5} + 214 T^{6} + 332 T^{7} + 658 T^{8} + 938 T^{9} + 1671 T^{10} + 2194 T^{11} + 3613 T^{12} + 4388 T^{13} + 6684 T^{14} + 7504 T^{15} + 10528 T^{16} + 10624 T^{17} + 13696 T^{18} + 11776 T^{19} + 14080 T^{20} + 9216 T^{21} + 10240 T^{22} + 4096 T^{23} + 4096 T^{24}$$
$3$ $$1 + 14 T^{2} + 4 T^{3} + 108 T^{4} + 40 T^{5} + 625 T^{6} + 240 T^{7} + 2874 T^{8} + 1128 T^{9} + 10992 T^{10} + 4056 T^{11} + 35812 T^{12} + 12168 T^{13} + 98928 T^{14} + 30456 T^{15} + 232794 T^{16} + 58320 T^{17} + 455625 T^{18} + 87480 T^{19} + 708588 T^{20} + 78732 T^{21} + 826686 T^{22} + 531441 T^{24}$$
$5$ 1
$7$ $$( 1 - T )^{12}$$
$11$ $$1 + 8 T + 107 T^{2} + 694 T^{3} + 5424 T^{4} + 29570 T^{5} + 173242 T^{6} + 810066 T^{7} + 3885411 T^{8} + 15777182 T^{9} + 64461251 T^{10} + 228757088 T^{11} + 811137096 T^{12} + 2516327968 T^{13} + 7799811371 T^{14} + 20999429242 T^{15} + 56886302451 T^{16} + 130461939366 T^{17} + 306908770762 T^{18} + 576235646470 T^{19} + 1162682570544 T^{20} + 1636415697554 T^{21} + 2775304432307 T^{22} + 2282493364888 T^{23} + 3138428376721 T^{24}$$
$13$ $$1 + 2 T + 81 T^{2} + 92 T^{3} + 3013 T^{4} + 982 T^{5} + 70653 T^{6} - 33340 T^{7} + 1225551 T^{8} - 1478546 T^{9} + 17756861 T^{10} - 29894704 T^{11} + 236277458 T^{12} - 388631152 T^{13} + 3000909509 T^{14} - 3248365562 T^{15} + 35002962111 T^{16} - 12378908620 T^{17} + 341028536277 T^{18} + 61619043694 T^{19} + 2457796662373 T^{20} + 975613942316 T^{21} + 11166537839769 T^{22} + 3584320788074 T^{23} + 23298085122481 T^{24}$$
$17$ $$1 - 8 T + 143 T^{2} - 1074 T^{3} + 10390 T^{4} - 69858 T^{5} + 494790 T^{6} - 2909608 T^{7} + 16897571 T^{8} - 86342196 T^{9} + 431097207 T^{10} - 1919135912 T^{11} + 8374637380 T^{12} - 32625310504 T^{13} + 124587092823 T^{14} - 424199208948 T^{15} + 1411302027491 T^{16} - 4131227286056 T^{17} + 11943027765510 T^{18} - 28665439018434 T^{19} + 72478119811990 T^{20} - 127363379357778 T^{21} + 288287127764207 T^{22} - 274175170461064 T^{23} + 582622237229761 T^{24}$$
$19$ $$1 + 26 T + 426 T^{2} + 5010 T^{3} + 47727 T^{4} + 380838 T^{5} + 2657599 T^{6} + 16489928 T^{7} + 93329855 T^{8} + 487385074 T^{9} + 2393508283 T^{10} + 11130888560 T^{11} + 49603973866 T^{12} + 211486882640 T^{13} + 864056490163 T^{14} + 3342974222566 T^{15} + 12162840033455 T^{16} + 40830694230872 T^{17} + 125029086299719 T^{18} + 340420325337282 T^{19} + 810574513257807 T^{20} + 1616665365872790 T^{21} + 2611834225823226 T^{22} + 3028746731353694 T^{23} + 2213314919066161 T^{24}$$
$23$ $$( 1 + T )^{12}$$
$29$ $$1 + 12 T + 258 T^{2} + 2552 T^{3} + 31367 T^{4} + 262496 T^{5} + 2401642 T^{6} + 17323380 T^{7} + 130559371 T^{8} + 825033200 T^{9} + 5385163876 T^{10} + 30202298184 T^{11} + 174902665114 T^{12} + 875866647336 T^{13} + 4528922819716 T^{14} + 20121734714800 T^{15} + 92342162480251 T^{16} + 355322428363620 T^{17} + 1428552670293082 T^{18} + 4528023531607264 T^{19} + 15691229235347687 T^{20} + 37022236530417688 T^{21} + 108542466191451858 T^{22} + 146406117188469948 T^{23} + 353814783205469041 T^{24}$$
$31$ $$1 + 50 T + 1356 T^{2} + 25672 T^{3} + 377722 T^{4} + 4578452 T^{5} + 47514831 T^{6} + 433678672 T^{7} + 3549751230 T^{8} + 26406623386 T^{9} + 180056634294 T^{10} + 1130557827586 T^{11} + 6550831526410 T^{12} + 35047292655166 T^{13} + 173034425556534 T^{14} + 786679717292326 T^{15} + 3278269805680830 T^{16} + 12415852186167472 T^{17} + 42169587414592911 T^{18} + 125965183101736172 T^{19} + 322155708444289402 T^{20} + 678757980108745912 T^{21} + 1111415957145966156 T^{22} + 1270423844820241550 T^{23} + 787662783788549761 T^{24}$$
$37$ $$1 + 8 T + 271 T^{2} + 1588 T^{3} + 33050 T^{4} + 143046 T^{5} + 2536026 T^{6} + 8188974 T^{7} + 145490287 T^{8} + 366377890 T^{9} + 6874214803 T^{10} + 14648570878 T^{11} + 275823336652 T^{12} + 541997122486 T^{13} + 9410800065307 T^{14} + 18558139262170 T^{15} + 272672221774207 T^{16} + 567855860930118 T^{17} + 6506748882110634 T^{18} + 13579625296367118 T^{19} + 116087445952089050 T^{20} + 206379242794582276 T^{21} + 1303126364925237079 T^{22} + 1423340974235683304 T^{23} + 6582952005840035281 T^{24}$$
$41$ $$1 + 4 T + 251 T^{2} + 780 T^{3} + 29258 T^{4} + 68680 T^{5} + 2208081 T^{6} + 4031628 T^{7} + 129639772 T^{8} + 206766552 T^{9} + 6549282855 T^{10} + 9944814512 T^{11} + 288865734980 T^{12} + 407737394992 T^{13} + 11009344479255 T^{14} + 14250557530392 T^{15} + 366331011766492 T^{16} + 467089103925228 T^{17} + 10488614922571521 T^{18} + 13375723530147080 T^{19} + 233622942353622218 T^{20} + 255357908827289580 T^{21} + 3369087486848252651 T^{22} + 2201316126864993764 T^{23} + 22563490300366186081 T^{24}$$
$43$ $$1 + 26 T + 578 T^{2} + 8682 T^{3} + 117563 T^{4} + 1307620 T^{5} + 13584507 T^{6} + 124442736 T^{7} + 1088354331 T^{8} + 8668947386 T^{9} + 66529730527 T^{10} + 470551819326 T^{11} + 3210480833674 T^{12} + 20233728231018 T^{13} + 123013471744423 T^{14} + 689241999818702 T^{15} + 3720866875177131 T^{16} + 18294132862020048 T^{17} + 85872600588681843 T^{18} + 355435452255735340 T^{19} + 1374099889235606363 T^{20} + 4363509056835670926 T^{21} + 12491436777078295922 T^{22} + 24161637226251790382 T^{23} + 39959630797262576401 T^{24}$$
$47$ $$1 + 16 T + 368 T^{2} + 4392 T^{3} + 62248 T^{4} + 618200 T^{5} + 6857049 T^{6} + 59881572 T^{7} + 563243206 T^{8} + 4408165880 T^{9} + 36381823766 T^{10} + 256816993148 T^{11} + 1895036675956 T^{12} + 12070398677956 T^{13} + 80367448699094 T^{14} + 457669006159240 T^{15} + 2748447170697286 T^{16} + 13733539549511004 T^{17} + 73913607692504121 T^{18} + 313194413070226600 T^{19} + 1482204972121298728 T^{20} + 4915221037867352664 T^{21} + 19356480662785458032 T^{22} + 39554547441344196848 T^{23} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$1 - 18 T + 501 T^{2} - 6670 T^{3} + 111442 T^{4} - 1217844 T^{5} + 15555492 T^{6} - 146270876 T^{7} + 1555182919 T^{8} - 12868554898 T^{9} + 118426221347 T^{10} - 870464673070 T^{11} + 7064049896396 T^{12} - 46134627672710 T^{13} + 332659255763723 T^{14} - 1915831847549546 T^{15} + 12271141273894039 T^{16} - 61169821100361868 T^{17} + 344777542227270468 T^{18} - 1430614913383651428 T^{19} + 6938344418822892562 T^{20} - 22009423157320227110 T^{21} + 87618622653122037549 T^{22} -$$$$16\!\cdots\!46$$$$T^{23} +$$$$49\!\cdots\!41$$$$T^{24}$$
$59$ $$1 + 18 T + 443 T^{2} + 6112 T^{3} + 93728 T^{4} + 1071802 T^{5} + 12989744 T^{6} + 128973906 T^{7} + 1333184899 T^{8} + 11791088468 T^{9} + 107695184445 T^{10} + 857739367502 T^{11} + 7038996899912 T^{12} + 50606622682618 T^{13} + 374886937053045 T^{14} + 2421641958469372 T^{15} + 16154682700931539 T^{16} + 92206579336341894 T^{17} + 547914333779977904 T^{18} + 2667341638731973838 T^{19} + 13762123255777798688 T^{20} + 52948230443618987168 T^{21} +$$$$22\!\cdots\!43$$$$T^{22} +$$$$54\!\cdots\!62$$$$T^{23} +$$$$17\!\cdots\!81$$$$T^{24}$$
$61$ $$1 - 8 T + 535 T^{2} - 4494 T^{3} + 140094 T^{4} - 1173188 T^{5} + 23726874 T^{6} - 189410822 T^{7} + 2879329515 T^{8} - 21162096094 T^{9} + 261954953003 T^{10} - 1727032895578 T^{11} + 18222879036420 T^{12} - 105349006630258 T^{13} + 974734380124163 T^{14} - 4803393733512214 T^{15} + 39866738651297115 T^{16} - 159975679630569422 T^{17} + 1222417430896277514 T^{18} - 3687028182305804948 T^{19} + 26857044307041084414 T^{20} - 52553492541196629654 T^{21} +$$$$38\!\cdots\!35$$$$T^{22} -$$$$34\!\cdots\!88$$$$T^{23} +$$$$26\!\cdots\!21$$$$T^{24}$$
$67$ $$1 + 38 T + 1149 T^{2} + 24514 T^{3} + 454030 T^{4} + 7065866 T^{5} + 99332140 T^{6} + 1242394940 T^{7} + 14332969401 T^{8} + 150908335254 T^{9} + 1482069378827 T^{10} + 13442018585536 T^{11} + 114292032364392 T^{12} + 900615245230912 T^{13} + 6653009441554403 T^{14} + 45387643635998802 T^{15} + 288825400688848521 T^{16} + 1677388601303758580 T^{17} + 8985424681784611660 T^{18} + 42824176067857204718 T^{19} +$$$$18\!\cdots\!30$$$$T^{20} +$$$$66\!\cdots\!58$$$$T^{21} +$$$$20\!\cdots\!01$$$$T^{22} +$$$$46\!\cdots\!54$$$$T^{23} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$1 + 24 T + 784 T^{2} + 14490 T^{3} + 280248 T^{4} + 4197644 T^{5} + 61318753 T^{6} + 768898944 T^{7} + 9221912504 T^{8} + 98820077666 T^{9} + 1008817898632 T^{10} + 9347436776276 T^{11} + 82499882792140 T^{12} + 663668011115596 T^{13} + 5085451027003912 T^{14} + 35368792817515726 T^{15} + 234344298761559224 T^{16} + 1387270042717705344 T^{17} + 7854949668981670513 T^{18} + 38178076562149030804 T^{19} +$$$$18\!\cdots\!28$$$$T^{20} +$$$$66\!\cdots\!90$$$$T^{21} +$$$$25\!\cdots\!84$$$$T^{22} +$$$$55\!\cdots\!04$$$$T^{23} +$$$$16\!\cdots\!41$$$$T^{24}$$
$73$ $$1 - 14 T + 389 T^{2} - 4160 T^{3} + 66639 T^{4} - 553494 T^{5} + 7090493 T^{6} - 47822356 T^{7} + 554889109 T^{8} - 3067431698 T^{9} + 35193699921 T^{10} - 160607916420 T^{11} + 2266296169786 T^{12} - 11724377898660 T^{13} + 187547226879009 T^{14} - 1193283076860866 T^{15} + 15757874645657269 T^{16} - 99139167733933108 T^{17} + 1073034272162570477 T^{18} - 6114668795929074918 T^{19} + 53741694063729663759 T^{20} -$$$$24\!\cdots\!80$$$$T^{21} +$$$$16\!\cdots\!61$$$$T^{22} -$$$$43\!\cdots\!78$$$$T^{23} +$$$$22\!\cdots\!21$$$$T^{24}$$
$79$ $$1 + 44 T + 1394 T^{2} + 32652 T^{3} + 646501 T^{4} + 10969906 T^{5} + 165979585 T^{6} + 2258446966 T^{7} + 28109218637 T^{8} + 321722375442 T^{9} + 3414112084085 T^{10} + 33680849723590 T^{11} + 309968024368258 T^{12} + 2660787128163610 T^{13} + 21307473516774485 T^{14} + 158621678265548238 T^{15} + 1094856342757859597 T^{16} + 6949368688532435434 T^{17} + 40347554986081538785 T^{18} +$$$$21\!\cdots\!54$$$$T^{19} +$$$$98\!\cdots\!61$$$$T^{20} +$$$$39\!\cdots\!88$$$$T^{21} +$$$$13\!\cdots\!94$$$$T^{22} +$$$$32\!\cdots\!76$$$$T^{23} +$$$$59\!\cdots\!41$$$$T^{24}$$
$83$ $$1 - 14 T + 735 T^{2} - 9146 T^{3} + 260712 T^{4} - 2894852 T^{5} + 59256494 T^{6} - 588022944 T^{7} + 9619690971 T^{8} - 85260583062 T^{9} + 1174765790903 T^{10} - 9262183130126 T^{11} + 110625794527488 T^{12} - 768761199800458 T^{13} + 8092961533530767 T^{14} - 48750893007271794 T^{15} + 456534382022519691 T^{16} - 2316246275432512992 T^{17} + 19373340272897908286 T^{18} - 78554851479423700204 T^{19} +$$$$58\!\cdots\!92$$$$T^{20} -$$$$17\!\cdots\!38$$$$T^{21} +$$$$11\!\cdots\!15$$$$T^{22} -$$$$18\!\cdots\!38$$$$T^{23} +$$$$10\!\cdots\!61$$$$T^{24}$$
$89$ $$1 + 10 T + 363 T^{2} + 4124 T^{3} + 82060 T^{4} + 1013390 T^{5} + 14202706 T^{6} + 169764084 T^{7} + 2034510667 T^{8} + 21889141238 T^{9} + 243098243759 T^{10} + 2301367141458 T^{11} + 23914236452424 T^{12} + 204821675589762 T^{13} + 1925581188815039 T^{14} + 15431166009411622 T^{15} + 127649758585984747 T^{16} + 947972737361029716 T^{17} + 7058479163019540466 T^{18} + 44823592469780133310 T^{19} +$$$$32\!\cdots\!60$$$$T^{20} +$$$$14\!\cdots\!16$$$$T^{21} +$$$$11\!\cdots\!63$$$$T^{22} +$$$$27\!\cdots\!90$$$$T^{23} +$$$$24\!\cdots\!21$$$$T^{24}$$
$97$ $$1 - 4 T + 577 T^{2} - 3798 T^{3} + 179283 T^{4} - 1393682 T^{5} + 39663292 T^{6} - 316257962 T^{7} + 6717625877 T^{8} - 51651879538 T^{9} + 901518714519 T^{10} - 6415544013704 T^{11} + 97366354036182 T^{12} - 622307769329288 T^{13} + 8482389584909271 T^{14} - 47141275853585074 T^{15} + 594706588917804437 T^{16} - 2715814728679376234 T^{17} + 33038411859324366268 T^{18} -$$$$11\!\cdots\!66$$$$T^{19} +$$$$14\!\cdots\!63$$$$T^{20} -$$$$28\!\cdots\!66$$$$T^{21} +$$$$42\!\cdots\!73$$$$T^{22} -$$$$28\!\cdots\!12$$$$T^{23} +$$$$69\!\cdots\!41$$$$T^{24}$$