# Properties

 Label 4025.2.a.y 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: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 2 x^{11} - 14 x^{10} + 26 x^{9} + 71 x^{8} - 120 x^{7} - 162 x^{6} + 244 x^{5} + 170 x^{4} - 210 x^{3} - 81 x^{2} + 58 x + 17$$ 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 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} - \beta_{9} - \beta_{6} - 1) q^{6} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 1) q^{8} + ( - \beta_{8} + \beta_1 + 1) q^{9}+O(q^{10})$$ q + b1 * q^2 - b6 * q^3 + (b2 + 1) * q^4 + (-b10 - b9 - b6 - 1) * q^6 - q^7 + (b10 + b9 + b7 + b6 + b5 + b4 - b1 + 1) * q^8 + (-b8 + b1 + 1) * q^9 $$q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_{10} - \beta_{9} - \beta_{6} - 1) q^{6} - q^{7} + (\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1 + 1) q^{8} + ( - \beta_{8} + \beta_1 + 1) q^{9} + (\beta_{9} + \beta_{8} - \beta_{2} - 1) q^{11} + ( - \beta_{10} - \beta_{9} - \beta_{4} - \beta_{2} - 2) q^{12} + ( - \beta_{11} + \beta_{10} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{13} - \beta_1 q^{14} + (\beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + \beta_{2}) q^{16} + (\beta_{11} - \beta_{10} - \beta_{7} - \beta_{5} - \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} + 2) q^{18} + (\beta_{10} + \beta_{8} - \beta_{5} - \beta_{2} - 3) q^{19} + \beta_{6} q^{21} + (\beta_{11} - \beta_{10} + \beta_{8} - 2 \beta_{7} - \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{22} + q^{23} + (\beta_{10} + \beta_{9} + \beta_{8} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{24} + ( - 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 2 \beta_{6} + \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots - 2) q^{26}+ \cdots + (3 \beta_{11} - \beta_{10} + 2 \beta_{8} - 3 \beta_{7} - 2 \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots - 3) q^{99}+O(q^{100})$$ q + b1 * q^2 - b6 * q^3 + (b2 + 1) * q^4 + (-b10 - b9 - b6 - 1) * q^6 - q^7 + (b10 + b9 + b7 + b6 + b5 + b4 - b1 + 1) * q^8 + (-b8 + b1 + 1) * q^9 + (b9 + b8 - b2 - 1) * q^11 + (-b10 - b9 - b4 - b2 - 2) * q^12 + (-b11 + b10 + b6 + b4 - b3 - b1) * q^13 - b1 * q^14 + (b11 + b10 + b6 + b5 + b2) * q^16 + (b11 - b10 - b7 - b5 - b4 + b3) * q^17 + (-b9 - b8 - b5 + b4 - b3 + 2*b2 + 2) * q^18 + (b10 + b8 - b5 - b2 - 3) * q^19 + b6 * q^21 + (b11 - b10 + b8 - 2*b7 - b6 - 2*b4 + b3 + b2 + 1) * q^22 + q^23 + (b10 + b9 + b8 - b4 + b3 - 2*b2 - 2*b1 - 1) * q^24 + (-2*b11 + b10 + b9 - b8 + 2*b6 + b4 - b3 - 2*b2 + b1 - 2) * q^26 + (-b11 - b10 - b4 + b3 - b2 + 1) * q^27 + (-b2 - 1) * q^28 + (-b11 - b10 - b9 - 2*b8 - b5 - b4 - b3 + b2 + b1 - 1) * q^29 + (-b11 + b9 + b5 + b4 + b3 - b2 - 2*b1 - 3) * q^31 + (-b10 + b8 - b5 - b4 + b3 + 1) * q^32 + (2*b11 + b9 + b8 - b7 + b6 - b5 - b4 + b3 + 1) * q^33 + (b11 + b9 - b7 - 2*b5 - b4 - b2 - 2) * q^34 + (-b11 + b10 + b9 - b8 + b6 - b5 + 2*b4 - b3 + 2*b1 - 1) * q^36 + (-b11 - b8 + b7 + 3*b6 - b4 - b3 + b1) * q^37 + (-b11 - b10 - b7 - b5 - b4 + b3 - 2*b2 - 2*b1) * q^38 + (2*b10 - b9 + 2*b7 + 2*b6 + 2*b5 + 2*b4 - b3 - 3*b1 - 3) * q^39 + (2*b11 + b10 + b8 - b7 - b4 + 2*b3 + b2 - 2*b1 + 1) * q^41 + (b10 + b9 + b6 + 1) * q^42 + (2*b11 - 2*b10 + b8 - 2*b6 - b5 - 2*b4 + 3*b3 - b1 + 3) * q^43 + (b9 + b8 + 2*b7 + b6 + b5 + b3 - 3*b2 - 2) * q^44 + b1 * q^46 + (2*b10 - 2*b9 + 2*b5 + 2*b4 - b3 + 2*b2 - b1 + 1) * q^47 + (b10 + b9 + 2*b8 + b4 + b3 - b2 - 3*b1 - 3) * q^48 + q^49 + (b11 - b10 + 2*b9 + 2*b8 - 2*b7 - b6 - b5 - 2*b4 + b3 + b1 - 2) * q^51 + (-b10 - b9 - 2*b8 - 2*b7 - 2*b5 - b4 - b3 + 2*b2 + 2) * q^52 + (-b11 - 2*b10 - 2*b9 - 2*b8 + b6 - b4 - 3*b3 + b2 + b1 - 3) * q^53 + (b10 - b9 + b8 + b7 + b5 - b3 - b2 - 3) * q^54 + (-b10 - b9 - b7 - b6 - b5 - b4 + b1 - 1) * q^56 + (b10 - b9 - b8 + b7 + 4*b6 + 3*b5 + 4*b4 - 3*b3 + 2*b2 - 2*b1 - 1) * q^57 + (-b11 + b10 - b9 - 2*b8 + b7 + 4*b4 - b3 - b1 - 1) * q^58 + (b11 + b10 - b9 - 2*b7 - b6 + 2*b5 - 2*b3 + b2 - 2) * q^59 + (-b11 - b10 - 2*b9 - 2*b8 + b7 + b6 + b5 - b4 - 2*b3 + b1 - 1) * q^61 + (b10 - b9 + b6 - b4 - 3*b3 - 4*b1 - 6) * q^62 + (b8 - b1 - 1) * q^63 + (-b11 - b10 + b9 + b8 - 2*b6 - b5 - b4 + b3 - 3*b2 + 2*b1) * q^64 + (2*b11 + 3*b9 + b8 - 2*b7 + b6 - 2*b5 - 2*b4 + 2*b3 + b1 + 2) * q^66 + (-b10 + b9 + b8 + 2*b4 - b2 + b1 + 3) * q^67 + (-b11 - b8 - 2*b7 - b6 - 2*b5 + b4 - b3 - b1 - 2) * q^68 - b6 * q^69 + (2*b11 + 2*b10 + b8 + 2*b7 + 2*b4 + 2*b2 - b1 - 2) * q^71 + (-b11 + b10 + 2*b9 - 2*b8 - 3*b7 + 2*b6 - b5 - b4 - b3 + b2 + b1 + 4) * q^72 + (2*b11 + 2*b10 - 2*b9 + 3*b7 + 2*b6 + 4*b5 + 3*b4 - b3 + b2 - 5*b1 - 2) * q^73 + (3*b10 + b9 + b7 + 2*b6 + 2*b5 + 3*b4 + b3 + 2*b2 - b1 + 5) * q^74 + (-b11 - 2*b10 - b9 - 2*b8 - b7 - b4 - 2*b3 - 3*b2 - 6) * q^76 + (-b9 - b8 + b2 + 1) * q^77 + (-b11 + b10 + 2*b7 + b6 + 2*b5 + b3 - b2 - 4*b1 - 2) * q^78 + (-b10 - b9 + b7 + 2*b5 + 3*b4 - b3 - 2*b2 - b1 - 5) * q^79 + (-2*b11 + b9 + b6 - b4 - b3 + b2 - b1 - 1) * q^81 + (b10 + 3*b9 + 2*b8 + 3*b7 + 3*b6 + b3 - 5*b2 - 2*b1 - 5) * q^82 + (b11 - b10 + b9 + 3*b8 - b6 - b5 + 2*b3 - b1 - 1) * q^83 + (b10 + b9 + b4 + b2 + 2) * q^84 + (4*b11 - b10 - b9 + 2*b8 - 3*b6 - 3*b4 + 2*b3 + 2*b1 - 4) * q^86 + (-3*b11 - 3*b10 - 2*b9 - 2*b8 + b7 - 2*b4 - b2 + 2*b1) * q^87 + (b11 + b10 - 3*b9 + 2*b7 - b6 + b5 + 3*b2 - 3*b1 + 2) * q^88 + (2*b11 - 4*b10 - 4*b9 - 2*b7 - 3*b6 - 2*b5 - b4 + 3*b3 + 3*b2 + b1) * q^89 + (b11 - b10 - b6 - b4 + b3 + b1) * q^91 + (b2 + 1) * q^92 + (3*b10 + 3*b9 - b8 + 6*b6 - 2*b4 + b3 - b2 + 4) * q^93 + (-4*b11 + b10 + 2*b9 + 5*b7 + 3*b6 + 2*b5 + 2*b4 - b3 - 5*b2 - 1) * q^94 + (-2*b10 - b9 - b8 - b7 + b6 - b4 - 2*b3 + b2 + 2*b1 - 4) * q^96 + (-b11 + 2*b10 + 4*b9 + 2*b8 + 2*b6 - 2*b5 + b4 + b3 - 4*b2) * q^97 + b1 * q^98 + (3*b11 - b10 + 2*b8 - 3*b7 - 2*b6 - b5 - 2*b4 + 2*b3 + 2*b2 - 2*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 2 q^{2} + 8 q^{4} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 8 q^{9}+O(q^{10})$$ 12 * q + 2 * q^2 + 8 * q^4 - 6 * q^6 - 12 * q^7 + 6 * q^8 + 8 * q^9 $$12 q + 2 q^{2} + 8 q^{4} - 6 q^{6} - 12 q^{7} + 6 q^{8} + 8 q^{9} - 8 q^{11} - 12 q^{12} + 2 q^{13} - 2 q^{14} - 4 q^{16} - 8 q^{17} + 20 q^{18} - 26 q^{19} + 4 q^{22} + 12 q^{23} - 12 q^{24} - 22 q^{26} + 12 q^{27} - 8 q^{28} - 12 q^{29} - 50 q^{31} + 14 q^{32} + 4 q^{33} - 28 q^{34} - 18 q^{36} + 8 q^{37} - 4 q^{38} - 26 q^{39} - 4 q^{41} + 6 q^{42} + 26 q^{43} - 10 q^{44} + 2 q^{46} + 16 q^{47} - 40 q^{48} + 12 q^{49} - 32 q^{51} + 10 q^{52} - 18 q^{53} - 10 q^{54} - 6 q^{56} - 10 q^{57} - 18 q^{58} - 18 q^{59} + 8 q^{61} - 54 q^{62} - 8 q^{63} + 12 q^{64} - 2 q^{66} + 38 q^{67} - 36 q^{68} - 24 q^{71} + 18 q^{72} - 14 q^{73} + 36 q^{74} - 56 q^{76} + 8 q^{77} - 26 q^{78} - 44 q^{79} - 16 q^{81} - 44 q^{82} - 14 q^{83} + 12 q^{84} - 32 q^{86} + 16 q^{87} + 32 q^{88} - 10 q^{89} - 2 q^{91} + 8 q^{92} + 26 q^{93} + 18 q^{94} - 38 q^{96} - 4 q^{97} + 2 q^{98} - 56 q^{99}+O(q^{100})$$ 12 * q + 2 * q^2 + 8 * q^4 - 6 * q^6 - 12 * q^7 + 6 * q^8 + 8 * q^9 - 8 * q^11 - 12 * q^12 + 2 * q^13 - 2 * q^14 - 4 * q^16 - 8 * q^17 + 20 * q^18 - 26 * q^19 + 4 * q^22 + 12 * q^23 - 12 * q^24 - 22 * q^26 + 12 * q^27 - 8 * q^28 - 12 * q^29 - 50 * q^31 + 14 * q^32 + 4 * q^33 - 28 * q^34 - 18 * q^36 + 8 * q^37 - 4 * q^38 - 26 * q^39 - 4 * q^41 + 6 * q^42 + 26 * q^43 - 10 * q^44 + 2 * q^46 + 16 * q^47 - 40 * q^48 + 12 * q^49 - 32 * q^51 + 10 * q^52 - 18 * q^53 - 10 * q^54 - 6 * q^56 - 10 * q^57 - 18 * q^58 - 18 * q^59 + 8 * q^61 - 54 * q^62 - 8 * q^63 + 12 * q^64 - 2 * q^66 + 38 * q^67 - 36 * q^68 - 24 * q^71 + 18 * q^72 - 14 * q^73 + 36 * q^74 - 56 * q^76 + 8 * q^77 - 26 * q^78 - 44 * q^79 - 16 * q^81 - 44 * q^82 - 14 * q^83 + 12 * q^84 - 32 * q^86 + 16 * q^87 + 32 * q^88 - 10 * q^89 - 2 * q^91 + 8 * q^92 + 26 * q^93 + 18 * q^94 - 38 * q^96 - 4 * q^97 + 2 * q^98 - 56 * q^99

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

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$( 17 \nu^{11} - 24 \nu^{10} - 218 \nu^{9} + 237 \nu^{8} + 966 \nu^{7} - 670 \nu^{6} - 1780 \nu^{5} + 537 \nu^{4} + 1297 \nu^{3} + 81 \nu^{2} - 241 \nu - 94 ) / 29$$ (17*v^11 - 24*v^10 - 218*v^9 + 237*v^8 + 966*v^7 - 670*v^6 - 1780*v^5 + 537*v^4 + 1297*v^3 + 81*v^2 - 241*v - 94) / 29 $$\beta_{4}$$ $$=$$ $$( - 12 \nu^{11} + 5 \nu^{10} + 188 \nu^{9} - 53 \nu^{8} - 1035 \nu^{7} + 171 \nu^{6} + 2367 \nu^{5} - 159 \nu^{4} - 2125 \nu^{3} - 151 \nu^{2} + 571 \nu + 138 ) / 29$$ (-12*v^11 + 5*v^10 + 188*v^9 - 53*v^8 - 1035*v^7 + 171*v^6 + 2367*v^5 - 159*v^4 - 2125*v^3 - 151*v^2 + 571*v + 138) / 29 $$\beta_{5}$$ $$=$$ $$( 18 \nu^{11} - 22 \nu^{10} - 253 \nu^{9} + 239 \nu^{8} + 1277 \nu^{7} - 822 \nu^{6} - 2840 \nu^{5} + 1036 \nu^{4} + 2767 \nu^{3} - 281 \nu^{2} - 958 \nu - 120 ) / 29$$ (18*v^11 - 22*v^10 - 253*v^9 + 239*v^8 + 1277*v^7 - 822*v^6 - 2840*v^5 + 1036*v^4 + 2767*v^3 - 281*v^2 - 958*v - 120) / 29 $$\beta_{6}$$ $$=$$ $$( 20 \nu^{11} - 18 \nu^{10} - 294 \nu^{9} + 185 \nu^{8} + 1551 \nu^{7} - 575 \nu^{6} - 3539 \nu^{5} + 613 \nu^{4} + 3329 \nu^{3} - 77 \nu^{2} - 913 \nu - 114 ) / 29$$ (20*v^11 - 18*v^10 - 294*v^9 + 185*v^8 + 1551*v^7 - 575*v^6 - 3539*v^5 + 613*v^4 + 3329*v^3 - 77*v^2 - 913*v - 114) / 29 $$\beta_{7}$$ $$=$$ $$( - 28 \nu^{11} + 31 \nu^{10} + 400 \nu^{9} - 317 \nu^{8} - 2067 \nu^{7} + 950 \nu^{6} + 4740 \nu^{5} - 806 \nu^{4} - 4736 \nu^{3} - 275 \nu^{2} + 1574 \nu + 322 ) / 29$$ (-28*v^11 + 31*v^10 + 400*v^9 - 317*v^8 - 2067*v^7 + 950*v^6 + 4740*v^5 - 806*v^4 - 4736*v^3 - 275*v^2 + 1574*v + 322) / 29 $$\beta_{8}$$ $$=$$ $$( 24 \nu^{11} - 10 \nu^{10} - 376 \nu^{9} + 77 \nu^{8} + 2128 \nu^{7} - 52 \nu^{6} - 5256 \nu^{5} - 523 \nu^{4} + 5468 \nu^{3} + 1056 \nu^{2} - 1751 \nu - 508 ) / 29$$ (24*v^11 - 10*v^10 - 376*v^9 + 77*v^8 + 2128*v^7 - 52*v^6 - 5256*v^5 - 523*v^4 + 5468*v^3 + 1056*v^2 - 1751*v - 508) / 29 $$\beta_{9}$$ $$=$$ $$( - 33 \nu^{11} + 21 \nu^{10} + 488 \nu^{9} - 182 \nu^{8} - 2578 \nu^{7} + 347 \nu^{6} + 5864 \nu^{5} + 179 \nu^{4} - 5561 \nu^{3} - 785 \nu^{2} + 1592 \nu + 336 ) / 29$$ (-33*v^11 + 21*v^10 + 488*v^9 - 182*v^8 - 2578*v^7 + 347*v^6 + 5864*v^5 + 179*v^4 - 5561*v^3 - 785*v^2 + 1592*v + 336) / 29 $$\beta_{10}$$ $$=$$ $$( 35 \nu^{11} - 17 \nu^{10} - 529 \nu^{9} + 128 \nu^{8} + 2852 \nu^{7} - 71 \nu^{6} - 6592 \nu^{5} - 863 \nu^{4} + 6355 \nu^{3} + 1569 \nu^{2} - 1953 \nu - 591 ) / 29$$ (35*v^11 - 17*v^10 - 529*v^9 + 128*v^8 + 2852*v^7 - 71*v^6 - 6592*v^5 - 863*v^4 + 6355*v^3 + 1569*v^2 - 1953*v - 591) / 29 $$\beta_{11}$$ $$=$$ $$( - 73 \nu^{11} + 57 \nu^{10} + 1076 \nu^{9} - 552 \nu^{8} - 5680 \nu^{7} + 1468 \nu^{6} + 12971 \nu^{5} - 757 \nu^{4} - 12451 \nu^{3} - 1414 \nu^{2} + 3824 \nu + 1028 ) / 29$$ (-73*v^11 + 57*v^10 + 1076*v^9 - 552*v^8 - 5680*v^7 + 1468*v^6 + 12971*v^5 - 757*v^4 - 12451*v^3 - 1414*v^2 + 3824*v + 1028) / 29
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3\beta _1 + 1$$ b10 + b9 + b7 + b6 + b5 + b4 + 3*b1 + 1 $$\nu^{4}$$ $$=$$ $$\beta_{11} + \beta_{10} + \beta_{6} + \beta_{5} + 7\beta_{2} + 14$$ b11 + b10 + b6 + b5 + 7*b2 + 14 $$\nu^{5}$$ $$=$$ $$7\beta_{10} + 8\beta_{9} + \beta_{8} + 8\beta_{7} + 8\beta_{6} + 7\beta_{5} + 7\beta_{4} + \beta_{3} + 12\beta _1 + 9$$ 7*b10 + 8*b9 + b8 + 8*b7 + 8*b6 + 7*b5 + 7*b4 + b3 + 12*b1 + 9 $$\nu^{6}$$ $$=$$ $$9 \beta_{11} + 9 \beta_{10} + \beta_{9} + \beta_{8} + 8 \beta_{6} + 9 \beta_{5} - \beta_{4} + \beta_{3} + 43 \beta_{2} + 2 \beta _1 + 76$$ 9*b11 + 9*b10 + b9 + b8 + 8*b6 + 9*b5 - b4 + b3 + 43*b2 + 2*b1 + 76 $$\nu^{7}$$ $$=$$ $$\beta_{11} + 43 \beta_{10} + 52 \beta_{9} + 11 \beta_{8} + 53 \beta_{7} + 52 \beta_{6} + 45 \beta_{5} + 43 \beta_{4} + 10 \beta_{3} + 2 \beta_{2} + 57 \beta _1 + 67$$ b11 + 43*b10 + 52*b9 + 11*b8 + 53*b7 + 52*b6 + 45*b5 + 43*b4 + 10*b3 + 2*b2 + 57*b1 + 67 $$\nu^{8}$$ $$=$$ $$63 \beta_{11} + 63 \beta_{10} + 12 \beta_{9} + 13 \beta_{8} + 4 \beta_{7} + 53 \beta_{6} + 67 \beta_{5} - 10 \beta_{4} + 12 \beta_{3} + 257 \beta_{2} + 23 \beta _1 + 438$$ 63*b11 + 63*b10 + 12*b9 + 13*b8 + 4*b7 + 53*b6 + 67*b5 - 10*b4 + 12*b3 + 257*b2 + 23*b1 + 438 $$\nu^{9}$$ $$=$$ $$16 \beta_{11} + 259 \beta_{10} + 319 \beta_{9} + 90 \beta_{8} + 334 \beta_{7} + 318 \beta_{6} + 288 \beta_{5} + 254 \beta_{4} + 78 \beta_{3} + 32 \beta_{2} + 299 \beta _1 + 465$$ 16*b11 + 259*b10 + 319*b9 + 90*b8 + 334*b7 + 318*b6 + 288*b5 + 254*b4 + 78*b3 + 32*b2 + 299*b1 + 465 $$\nu^{10}$$ $$=$$ $$410 \beta_{11} + 412 \beta_{10} + 106 \beta_{9} + 124 \beta_{8} + 68 \beta_{7} + 337 \beta_{6} + 474 \beta_{5} - 69 \beta_{4} + 108 \beta_{3} + 1526 \beta_{2} + 184 \beta _1 + 2594$$ 410*b11 + 412*b10 + 106*b9 + 124*b8 + 68*b7 + 337*b6 + 474*b5 - 69*b4 + 108*b3 + 1526*b2 + 184*b1 + 2594 $$\nu^{11}$$ $$=$$ $$172 \beta_{11} + 1561 \beta_{10} + 1919 \beta_{9} + 667 \beta_{8} + 2073 \beta_{7} + 1905 \beta_{6} + 1851 \beta_{5} + 1473 \beta_{4} + 563 \beta_{3} + 337 \beta_{2} + 1655 \beta _1 + 3122$$ 172*b11 + 1561*b10 + 1919*b9 + 667*b8 + 2073*b7 + 1905*b6 + 1851*b5 + 1473*b4 + 563*b3 + 337*b2 + 1655*b1 + 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
 −2.37028 −1.76402 −1.69164 −1.11343 −0.649823 −0.271475 0.792462 1.23080 1.31562 1.53413 2.43455 2.55311
−2.37028 −0.134036 3.61823 0 0.317703 −1.00000 −3.83566 −2.98203 0
1.2 −1.76402 −1.09514 1.11176 0 1.93184 −1.00000 1.56687 −1.80068 0
1.3 −1.69164 0.285164 0.861643 0 −0.482394 −1.00000 1.92569 −2.91868 0
1.4 −1.11343 2.87976 −0.760266 0 −3.20642 −1.00000 3.07337 5.29300 0
1.5 −0.649823 1.89575 −1.57773 0 −1.23191 −1.00000 2.32489 0.593883 0
1.6 −0.271475 −2.40463 −1.92630 0 0.652799 −1.00000 1.06589 2.78227 0
1.7 0.792462 −2.09947 −1.37200 0 −1.66375 −1.00000 −2.67218 1.40777 0
1.8 1.23080 −0.767488 −0.485126 0 −0.944626 −1.00000 −3.05870 −2.41096 0
1.9 1.31562 2.16083 −0.269134 0 2.84283 −1.00000 −2.98533 1.66917 0
1.10 1.53413 2.66039 0.353561 0 4.08139 −1.00000 −2.52585 4.07768 0
1.11 2.43455 −2.82471 3.92701 0 −6.87689 −1.00000 4.69139 4.97900 0
1.12 2.55311 −0.556415 4.51835 0 −1.42059 −1.00000 6.42962 −2.69040 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
$$5$$ $$-1$$
$$7$$ $$1$$
$$23$$ $$-1$$

## 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 4025.2.a.y 12
5.b even 2 1 4025.2.a.x 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 5.b even 2 1
4025.2.a.y 12 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4025))$$:

 $$T_{2}^{12} - 2 T_{2}^{11} - 14 T_{2}^{10} + 26 T_{2}^{9} + 71 T_{2}^{8} - 120 T_{2}^{7} - 162 T_{2}^{6} + 244 T_{2}^{5} + 170 T_{2}^{4} - 210 T_{2}^{3} - 81 T_{2}^{2} + 58 T_{2} + 17$$ T2^12 - 2*T2^11 - 14*T2^10 + 26*T2^9 + 71*T2^8 - 120*T2^7 - 162*T2^6 + 244*T2^5 + 170*T2^4 - 210*T2^3 - 81*T2^2 + 58*T2 + 17 $$T_{3}^{12} - 22 T_{3}^{10} - 4 T_{3}^{9} + 174 T_{3}^{8} + 68 T_{3}^{7} - 581 T_{3}^{6} - 372 T_{3}^{5} + 669 T_{3}^{4} + 672 T_{3}^{3} + 75 T_{3}^{2} - 60 T_{3} - 8$$ T3^12 - 22*T3^10 - 4*T3^9 + 174*T3^8 + 68*T3^7 - 581*T3^6 - 372*T3^5 + 669*T3^4 + 672*T3^3 + 75*T3^2 - 60*T3 - 8 $$T_{11}^{12} + 8 T_{11}^{11} - 25 T_{11}^{10} - 274 T_{11}^{9} + 188 T_{11}^{8} + 3456 T_{11}^{7} + 3 T_{11}^{6} - 19422 T_{11}^{5} - 6026 T_{11}^{4} + 45752 T_{11}^{3} + 21040 T_{11}^{2} - 28480 T_{11} - 7392$$ T11^12 + 8*T11^11 - 25*T11^10 - 274*T11^9 + 188*T11^8 + 3456*T11^7 + 3*T11^6 - 19422*T11^5 - 6026*T11^4 + 45752*T11^3 + 21040*T11^2 - 28480*T11 - 7392

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 2 T^{11} - 14 T^{10} + 26 T^{9} + \cdots + 17$$
$3$ $$T^{12} - 22 T^{10} - 4 T^{9} + 174 T^{8} + \cdots - 8$$
$5$ $$T^{12}$$
$7$ $$(T + 1)^{12}$$
$11$ $$T^{12} + 8 T^{11} - 25 T^{10} + \cdots - 7392$$
$13$ $$T^{12} - 2 T^{11} - 75 T^{10} + \cdots - 1234$$
$17$ $$T^{12} + 8 T^{11} - 61 T^{10} + \cdots + 84256$$
$19$ $$T^{12} + 26 T^{11} + 198 T^{10} + \cdots + 1987488$$
$23$ $$(T - 1)^{12}$$
$29$ $$T^{12} + 12 T^{11} - 90 T^{10} + \cdots - 6811136$$
$31$ $$T^{12} + 50 T^{11} + \cdots + 714551774$$
$37$ $$T^{12} - 8 T^{11} - 173 T^{10} + \cdots - 1732696$$
$41$ $$T^{12} + 4 T^{11} - 241 T^{10} + \cdots - 220612$$
$43$ $$T^{12} - 26 T^{11} + 62 T^{10} + \cdots + 81498208$$
$47$ $$T^{12} - 16 T^{11} - 196 T^{10} + \cdots - 23799312$$
$53$ $$T^{12} + 18 T^{11} + \cdots + 136913064$$
$59$ $$T^{12} + 18 T^{11} + \cdots - 632701728$$
$61$ $$T^{12} - 8 T^{11} - 197 T^{10} + \cdots + 7050656$$
$67$ $$T^{12} - 38 T^{11} + \cdots + 2026241824$$
$71$ $$T^{12} + 24 T^{11} - 68 T^{10} + \cdots + 65141008$$
$73$ $$T^{12} + 14 T^{11} - 487 T^{10} + \cdots + 58991502$$
$79$ $$T^{12} + 44 T^{11} + \cdots - 1360803976$$
$83$ $$T^{12} + 14 T^{11} - 261 T^{10} + \cdots - 11862896$$
$89$ $$T^{12} + 10 T^{11} + \cdots + 85724172576$$
$97$ $$T^{12} + 4 T^{11} + \cdots - 15480701824$$