# Properties

 Label 260.2.z.a Level $260$ Weight $2$ Character orbit 260.z Analytic conductor $2.076$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561$$ x^16 - 7*x^14 + 21*x^12 - 22*x^10 - 26*x^8 - 198*x^6 + 1701*x^4 - 5103*x^2 + 6561 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + \beta_{14} q^{5} - \beta_{8} q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \beta_{10} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b3 * q^3 + b14 * q^5 - b8 * q^7 + (b15 - b14 + b12 + b10 + b1 + 1) * q^9 $$q - \beta_{3} q^{3} + \beta_{14} q^{5} - \beta_{8} q^{7} + (\beta_{15} - \beta_{14} + \beta_{12} + \beta_{10} + \beta_1 + 1) q^{9} + (\beta_{6} - \beta_{5} - \beta_1) q^{11} + (\beta_{14} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{4}) q^{13} + (\beta_{14} + \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{15} + ( - 2 \beta_{14} - 2 \beta_{12} + \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{2}) q^{17} + (\beta_{15} - \beta_{14} - 2 \beta_{13} + \beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} + \beta_{2} - 2) q^{19} + (\beta_{11} + \beta_{10} + \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2} + 2 \beta_1 + 1) q^{21} + (\beta_{14} + \beta_{12} + \beta_{11} + \beta_{10} - \beta_{8} - \beta_{6} - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{23} + (\beta_{14} + \beta_{12} + \beta_{9} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{25} + ( - 2 \beta_{14} - 2 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + \cdots + 2 \beta_{2}) q^{27}+ \cdots + ( - 2 \beta_{15} + 4 \beta_{14} + \beta_{13} - 4 \beta_{12} - 2 \beta_{11} - 3 \beta_{10} + \cdots - 3) q^{99}+O(q^{100})$$ q - b3 * q^3 + b14 * q^5 - b8 * q^7 + (b15 - b14 + b12 + b10 + b1 + 1) * q^9 + (b6 - b5 - b1) * q^11 + (b14 + b12 - b11 - b10 + b9 - b8 - b7 + b4) * q^13 + (b14 + b11 + b9 - b8 - b7 - b5 + b4 + b3 - b2 + 1) * q^15 + (-2*b14 - 2*b12 + b11 + b10 - b9 + b8 + b7 - b2) * q^17 + (b15 - b14 - 2*b13 + b12 + b10 - b9 - b7 + b2 - 2) * q^19 + (b11 + b10 + b9 + b7 - b6 + b5 - b2 + 2*b1 + 1) * q^21 + (b14 + b12 + b11 + b10 - b8 - b6 - b5 + b4 + 2*b3) * q^23 + (b14 + b12 + b9 - b5 + b4 - b3 + b2) * q^25 + (-2*b14 - 2*b12 - 2*b11 - 2*b10 - 2*b9 + 2*b8 + 2*b7 + 2*b6 + 2*b5 - 3*b4 - 4*b3 + 2*b2) * q^27 + (-b15 + b13 - 2*b10 - 2*b1) * q^29 + (b14 - b12 - b11 - b10 - b9 - b7 + b6 - b5 + b2) * q^31 + (2*b11 + 2*b10 + 2*b8 + 2*b3 - b2) * q^33 + (-b15 + 2*b14 + 2*b12 - b11 - b10 + 2*b9 - b8 - b7 - b6 + b5 + b4 + b1) * q^35 + (-2*b14 - 2*b12 - 2*b9 + b8 + 2*b7 + b6 + b5 - 3*b4 - 2*b2) * q^37 + (-b14 + b13 + b12 - b11 + 2*b10 + b9 + b7 - b6 + b5 - b2 + b1 + 1) * q^39 + (b14 - b12 + b11 - b10 - b6 + b5 - b1 - 4) * q^41 + (b9 - 2*b8 - b7 - 2*b4) * q^43 + (-b15 + 2*b14 + 2*b13 - 2*b11 - b10 - b7 - b6 - b5 + b4 + 2*b3 - 2*b1 + 1) * q^45 + (-2*b14 - 2*b12) * q^47 + (-b15 + b14 + b13 - b12 - b11 - b10 + b1) * q^49 + (-b11 - b10 - b9 - b7 - b6 + b5 + b2 + 1) * q^51 + (b14 + b12 - 2*b11 - 2*b10 + 2*b8 + b4 - b3 + b2) * q^53 + (b15 - b14 - b13 + 2*b12 + b11 - b9 + b8 - b7 + b6 - b4 - 2*b3 + b2 - 2*b1 - 1) * q^55 + (-b14 - b12 + b11 + b10 - b9 + b7 - b6 - b5 + b4 + 4*b3 + 3*b2) * q^57 + (b11 - b10 + b1 - 1) * q^59 + (-b15 - b14 + b12 - b11 - 4*b1 - 4) * q^61 + (2*b14 + 2*b12 - 2*b11 - 2*b10 + b8 + b4 - b3 + 2*b2) * q^63 + (-b13 + b12 + b10 + b9 + b8 - b6 + b5 + 2*b3 - 2*b2 - b1 - 2) * q^65 + (-5*b14 - 5*b12 + 3*b11 + 3*b10 - 2*b9 + b8 + 2*b7 + b6 + b5 - 3*b4 - 2*b3 + 2*b2) * q^67 + (-2*b15 + 4*b14 - 4*b12 + b11 - 3*b10 - 3*b1 - 3) * q^69 + (-b15 + b14 + 2*b13 - b12 - 2*b11 + b10 + b9 + b7 - b2 + 2*b1) * q^71 + (-b11 - b10 + b9 - b7 + b6 + b5 + b4 - b3) * q^73 + (b15 - b13 + 2*b12 + 2*b11 + 2*b10 + b8 - b4 + 2*b3 + 5*b1) * q^75 + (-2*b14 - 2*b12 - 2*b11 - 2*b10 - 2*b9 + 2*b8 + 2*b7 + 2*b6 + 2*b5 - 3*b4 + 2*b3 - 4*b2) * q^77 + (-b13 + 2*b11 + b10 + b9 + b7 + b6 - b5 - b2 + 1) * q^79 + (b15 - 3*b14 - b13 + 3*b12 + 5*b11 + b10 + 2*b9 + 2*b7 - b6 + b5 - 2*b2 + 6*b1 + 2) * q^81 + (-b14 - b12 + b11 + b10 - b9 + b7 - b6 - b5 - 2*b4 - b2) * q^83 + (2*b14 + 2*b12 + b11 - 4*b8 - 2*b7 - 2*b6 - 2*b5 + 2*b4 + 2*b3 + b2 + b1 - 3) * q^85 + (-b9 + b8 + b7 - 3*b2) * q^87 + (-b14 + b12 - b11 + b10 + b1 + 2) * q^89 + (-2*b14 - b13 + 2*b12 - 3*b11 + 2*b10 - b9 - b7 + b6 - b5 + b2 + 3*b1) * q^91 + (-b14 - b12 + 3*b11 + 3*b10 + 2*b9 - 2*b7 - b6 - b5 + 4*b4 + 2*b3 - 2*b2) * q^93 + (2*b15 - b14 + b12 + b11 + 2*b10 + 2*b9 + b8 - b7 - b6 + b5 + 5*b4 + b2 + b1) * q^95 + (-2*b11 - 2*b10 - b8 - 4*b3 + 2*b2) * q^97 + (-2*b15 + 4*b14 + b13 - 4*b12 - 2*b11 - 3*b10 - b9 - b7 + b6 - b5 + b2 - 6*b1 - 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 10 q^{9}+O(q^{10})$$ 16 * q + 10 * q^9 $$16 q + 10 q^{9} - 6 q^{11} + 6 q^{15} - 18 q^{19} - 14 q^{25} + 12 q^{29} + 18 q^{39} - 48 q^{41} + 45 q^{45} - 6 q^{49} + 44 q^{51} + 2 q^{55} - 30 q^{59} - 28 q^{61} - 15 q^{65} - 34 q^{69} - 18 q^{71} - 42 q^{75} - 16 q^{79} - 44 q^{81} - 45 q^{85} + 30 q^{89} - 10 q^{91}+O(q^{100})$$ 16 * q + 10 * q^9 - 6 * q^11 + 6 * q^15 - 18 * q^19 - 14 * q^25 + 12 * q^29 + 18 * q^39 - 48 * q^41 + 45 * q^45 - 6 * q^49 + 44 * q^51 + 2 * q^55 - 30 * q^59 - 28 * q^61 - 15 * q^65 - 34 * q^69 - 18 * q^71 - 42 * q^75 - 16 * q^79 - 44 * q^81 - 45 * q^85 + 30 * q^89 - 10 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7x^{14} + 21x^{12} - 22x^{10} - 26x^{8} - 198x^{6} + 1701x^{4} - 5103x^{2} + 6561$$ :

 $$\beta_{1}$$ $$=$$ $$( 5\nu^{14} - 2\nu^{12} + 63\nu^{10} - 11\nu^{8} - 289\nu^{6} + 69\nu^{4} + 1674\nu^{2} - 24057 ) / 15552$$ (5*v^14 - 2*v^12 + 63*v^10 - 11*v^8 - 289*v^6 + 69*v^4 + 1674*v^2 - 24057) / 15552 $$\beta_{2}$$ $$=$$ $$( -5\nu^{15} - 46\nu^{13} + 57\nu^{11} + 515\nu^{9} + 697\nu^{7} - 7677\nu^{5} + 23166\nu^{3} - 28431\nu ) / 69984$$ (-5*v^15 - 46*v^13 + 57*v^11 + 515*v^9 + 697*v^7 - 7677*v^5 + 23166*v^3 - 28431*v) / 69984 $$\beta_{3}$$ $$=$$ $$( \nu^{15} - 10\nu^{13} + 27\nu^{11} - 7\nu^{9} - 5\nu^{7} - 519\nu^{5} + 2874\nu^{3} - 4509\nu ) / 2592$$ (v^15 - 10*v^13 + 27*v^11 - 7*v^9 - 5*v^7 - 519*v^5 + 2874*v^3 - 4509*v) / 2592 $$\beta_{4}$$ $$=$$ $$( -\nu^{15} + 7\nu^{13} - 21\nu^{11} + 22\nu^{9} + 26\nu^{7} + 198\nu^{5} - 1701\nu^{3} + 7290\nu ) / 2187$$ (-v^15 + 7*v^13 - 21*v^11 + 22*v^9 + 26*v^7 + 198*v^5 - 1701*v^3 + 7290*v) / 2187 $$\beta_{5}$$ $$=$$ $$( 43 \nu^{15} - 15 \nu^{14} - 142 \nu^{13} + 438 \nu^{12} + 465 \nu^{11} - 1917 \nu^{10} + 827 \nu^{9} + 33 \nu^{8} + 1297 \nu^{7} + 6915 \nu^{6} - 6357 \nu^{5} + 6705 \nu^{4} + 54486 \nu^{3} + \cdots + 433755 ) / 93312$$ (43*v^15 - 15*v^14 - 142*v^13 + 438*v^12 + 465*v^11 - 1917*v^10 + 827*v^9 + 33*v^8 + 1297*v^7 + 6915*v^6 - 6357*v^5 + 6705*v^4 + 54486*v^3 - 170910*v^2 - 45927*v + 433755) / 93312 $$\beta_{6}$$ $$=$$ $$( 43 \nu^{15} + 15 \nu^{14} - 142 \nu^{13} - 438 \nu^{12} + 465 \nu^{11} + 1917 \nu^{10} + 827 \nu^{9} - 33 \nu^{8} + 1297 \nu^{7} - 6915 \nu^{6} - 6357 \nu^{5} - 6705 \nu^{4} + 54486 \nu^{3} + \cdots - 433755 ) / 93312$$ (43*v^15 + 15*v^14 - 142*v^13 - 438*v^12 + 465*v^11 + 1917*v^10 + 827*v^9 - 33*v^8 + 1297*v^7 - 6915*v^6 - 6357*v^5 - 6705*v^4 + 54486*v^3 + 170910*v^2 - 45927*v - 433755) / 93312 $$\beta_{7}$$ $$=$$ $$( - 41 \nu^{15} - 79 \nu^{14} + 218 \nu^{13} + 22 \nu^{12} - 459 \nu^{11} + 195 \nu^{10} + 263 \nu^{9} + 1441 \nu^{8} - 1547 \nu^{7} - 3517 \nu^{6} + 759 \nu^{5} + 5553 \nu^{4} - 3186 \nu^{3} + \cdots - 9477 ) / 93312$$ (-41*v^15 - 79*v^14 + 218*v^13 + 22*v^12 - 459*v^11 + 195*v^10 + 263*v^9 + 1441*v^8 - 1547*v^7 - 3517*v^6 + 759*v^5 + 5553*v^4 - 3186*v^3 + 5346*v^2 + 112509*v - 9477) / 93312 $$\beta_{8}$$ $$=$$ $$( -5\nu^{15} + 8\nu^{13} - 78\nu^{11} - 52\nu^{9} + 238\nu^{7} + 882\nu^{5} - 4779\nu^{3} + 13122\nu ) / 8748$$ (-5*v^15 + 8*v^13 - 78*v^11 - 52*v^9 + 238*v^7 + 882*v^5 - 4779*v^3 + 13122*v) / 8748 $$\beta_{9}$$ $$=$$ $$( - 179 \nu^{15} - 237 \nu^{14} - 610 \nu^{13} + 66 \nu^{12} + 1911 \nu^{11} + 585 \nu^{10} - 355 \nu^{9} + 4323 \nu^{8} - 11465 \nu^{7} - 10551 \nu^{6} + 20781 \nu^{5} + 16659 \nu^{4} + \cdots - 28431 ) / 279936$$ (-179*v^15 - 237*v^14 - 610*v^13 + 66*v^12 + 1911*v^11 + 585*v^10 - 355*v^9 + 4323*v^8 - 11465*v^7 - 10551*v^6 + 20781*v^5 + 16659*v^4 + 110970*v^3 + 16038*v^2 - 414801*v - 28431) / 279936 $$\beta_{10}$$ $$=$$ $$( 47 \nu^{15} - 229 \nu^{14} - 38 \nu^{13} + 946 \nu^{12} - 51 \nu^{11} - 1263 \nu^{10} + 271 \nu^{9} - 2117 \nu^{8} + 3149 \nu^{7} + 5585 \nu^{6} - 8961 \nu^{5} + 60075 \nu^{4} + \cdots + 373977 ) / 93312$$ (47*v^15 - 229*v^14 - 38*v^13 + 946*v^12 - 51*v^11 - 1263*v^10 + 271*v^9 - 2117*v^8 + 3149*v^7 + 5585*v^6 - 8961*v^5 + 60075*v^4 - 1458*v^3 - 232794*v^2 - 6075*v + 373977) / 93312 $$\beta_{11}$$ $$=$$ $$( 47 \nu^{15} + 229 \nu^{14} - 38 \nu^{13} - 946 \nu^{12} - 51 \nu^{11} + 1263 \nu^{10} + 271 \nu^{9} + 2117 \nu^{8} + 3149 \nu^{7} - 5585 \nu^{6} - 8961 \nu^{5} - 60075 \nu^{4} + \cdots - 373977 ) / 93312$$ (47*v^15 + 229*v^14 - 38*v^13 - 946*v^12 - 51*v^11 + 1263*v^10 + 271*v^9 + 2117*v^8 + 3149*v^7 - 5585*v^6 - 8961*v^5 - 60075*v^4 - 1458*v^3 + 232794*v^2 - 6075*v - 373977) / 93312 $$\beta_{12}$$ $$=$$ $$( - 107 \nu^{15} + 399 \nu^{14} + 686 \nu^{13} - 1902 \nu^{12} - 1401 \nu^{11} + 3357 \nu^{10} - 1075 \nu^{9} + 3615 \nu^{8} + 5383 \nu^{7} - 8835 \nu^{6} + 33597 \nu^{5} + \cdots - 789507 ) / 139968$$ (-107*v^15 + 399*v^14 + 686*v^13 - 1902*v^12 - 1401*v^11 + 3357*v^10 - 1075*v^9 + 3615*v^8 + 5383*v^7 - 8835*v^6 + 33597*v^5 - 87345*v^4 - 150174*v^3 + 437886*v^2 + 285039*v - 789507) / 139968 $$\beta_{13}$$ $$=$$ $$( 47 \nu^{15} + 357 \nu^{14} - 38 \nu^{13} - 1842 \nu^{12} - 51 \nu^{11} + 3951 \nu^{10} + 271 \nu^{9} - 699 \nu^{8} + 3149 \nu^{7} - 8913 \nu^{6} - 8961 \nu^{5} - 85419 \nu^{4} + \cdots - 840537 ) / 93312$$ (47*v^15 + 357*v^14 - 38*v^13 - 1842*v^12 - 51*v^11 + 3951*v^10 + 271*v^9 - 699*v^8 + 3149*v^7 - 8913*v^6 - 8961*v^5 - 85419*v^4 - 1458*v^3 + 357210*v^2 - 6075*v - 840537) / 93312 $$\beta_{14}$$ $$=$$ $$( - 107 \nu^{15} - 399 \nu^{14} + 686 \nu^{13} + 1902 \nu^{12} - 1401 \nu^{11} - 3357 \nu^{10} - 1075 \nu^{9} - 3615 \nu^{8} + 5383 \nu^{7} + 8835 \nu^{6} + 33597 \nu^{5} + \cdots + 789507 ) / 139968$$ (-107*v^15 - 399*v^14 + 686*v^13 + 1902*v^12 - 1401*v^11 - 3357*v^10 - 1075*v^9 - 3615*v^8 + 5383*v^7 + 8835*v^6 + 33597*v^5 + 87345*v^4 - 150174*v^3 - 437886*v^2 + 285039*v + 789507) / 139968 $$\beta_{15}$$ $$=$$ $$( - 47 \nu^{15} - 511 \nu^{14} + 38 \nu^{13} + 2038 \nu^{12} + 51 \nu^{11} - 4413 \nu^{10} - 271 \nu^{9} - 3743 \nu^{8} - 3149 \nu^{7} - 1213 \nu^{6} + 8961 \nu^{5} + 127665 \nu^{4} + \cdots + 1040283 ) / 93312$$ (-47*v^15 - 511*v^14 + 38*v^13 + 2038*v^12 + 51*v^11 - 4413*v^10 - 271*v^9 - 3743*v^8 - 3149*v^7 - 1213*v^6 + 8961*v^5 + 127665*v^4 + 1458*v^3 - 555822*v^2 + 6075*v + 1040283) / 93312
 $$\nu$$ $$=$$ $$( \beta_{4} + \beta_{3} - \beta_{2} ) / 2$$ (b4 + b3 - b2) / 2 $$\nu^{2}$$ $$=$$ $$( -\beta_{13} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} - 2\beta _1 + 1 ) / 2$$ (-b13 - b10 - b9 - b7 + b6 - b5 + b2 - 2*b1 + 1) / 2 $$\nu^{3}$$ $$=$$ $$-\beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4}$$ -b11 - b10 - b9 + b8 + b7 + b6 + b5 - b4 $$\nu^{4}$$ $$=$$ $$( 2 \beta_{15} - 6 \beta_{14} - 3 \beta_{13} + 6 \beta_{12} - 4 \beta_{11} + 3 \beta_{10} - 3 \beta_{9} - 3 \beta_{7} + \beta_{6} - \beta_{5} + 3 \beta_{2} - 1 ) / 2$$ (2*b15 - 6*b14 - 3*b13 + 6*b12 - 4*b11 + 3*b10 - 3*b9 - 3*b7 + b6 - b5 + 3*b2 - 1) / 2 $$\nu^{5}$$ $$=$$ $$( -4\beta_{11} - 4\beta_{10} + 6\beta_{8} + 8\beta_{6} + 8\beta_{5} - 5\beta_{4} - 7\beta_{3} - 7\beta_{2} ) / 2$$ (-4*b11 - 4*b10 + 6*b8 + 8*b6 + 8*b5 - 5*b4 - 7*b3 - 7*b2) / 2 $$\nu^{6}$$ $$=$$ $$- 4 \beta_{15} - 2 \beta_{14} - \beta_{13} + 2 \beta_{12} - 8 \beta_{11} + 3 \beta_{10} - \beta_{9} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - 12 \beta _1 - 14$$ -4*b15 - 2*b14 - b13 + 2*b12 - 8*b11 + 3*b10 - b9 - b7 - b6 + b5 + b2 - 12*b1 - 14 $$\nu^{7}$$ $$=$$ $$( 12 \beta_{14} + 12 \beta_{12} + 12 \beta_{11} + 12 \beta_{10} + 8 \beta_{9} + 6 \beta_{8} - 8 \beta_{7} + 8 \beta_{6} + 8 \beta_{5} - \beta_{4} + 11 \beta_{3} + 11 \beta_{2} ) / 2$$ (12*b14 + 12*b12 + 12*b11 + 12*b10 + 8*b9 + 6*b8 - 8*b7 + 8*b6 + 8*b5 - b4 + 11*b3 + 11*b2) / 2 $$\nu^{8}$$ $$=$$ $$( - 10 \beta_{15} - 22 \beta_{14} - 17 \beta_{13} + 22 \beta_{12} + 22 \beta_{11} + 51 \beta_{10} + 33 \beta_{9} + 33 \beta_{7} - 19 \beta_{6} + 19 \beta_{5} - 33 \beta_{2} + 56 \beta _1 + 7 ) / 2$$ (-10*b15 - 22*b14 - 17*b13 + 22*b12 + 22*b11 + 51*b10 + 33*b9 + 33*b7 - 19*b6 + 19*b5 - 33*b2 + 56*b1 + 7) / 2 $$\nu^{9}$$ $$=$$ $$- 6 \beta_{14} - 6 \beta_{12} - \beta_{11} - \beta_{10} + 15 \beta_{9} - 11 \beta_{8} - 15 \beta_{7} + 17 \beta_{6} + 17 \beta_{5} + 10 \beta_{4} - 49 \beta_{3} + 51 \beta_{2}$$ -6*b14 - 6*b12 - b11 - b10 + 15*b9 - 11*b8 - 15*b7 + 17*b6 + 17*b5 + 10*b4 - 49*b3 + 51*b2 $$\nu^{10}$$ $$=$$ $$( - 56 \beta_{15} + 32 \beta_{14} + 47 \beta_{13} - 32 \beta_{12} - 30 \beta_{11} + 73 \beta_{10} + 73 \beta_{9} + 73 \beta_{7} - 25 \beta_{6} + 25 \beta_{5} - 73 \beta_{2} + 250 \beta _1 + 443 ) / 2$$ (-56*b15 + 32*b14 + 47*b13 - 32*b12 - 30*b11 + 73*b10 + 73*b9 + 73*b7 - 25*b6 + 25*b5 - 73*b2 + 250*b1 + 443) / 2 $$\nu^{11}$$ $$=$$ $$( 96 \beta_{14} + 96 \beta_{12} + 80 \beta_{11} + 80 \beta_{10} + 152 \beta_{9} - 312 \beta_{8} - 152 \beta_{7} - 80 \beta_{6} - 80 \beta_{5} + 251 \beta_{4} + 261 \beta_{3} - 85 \beta_{2} ) / 2$$ (96*b14 + 96*b12 + 80*b11 + 80*b10 + 152*b9 - 312*b8 - 152*b7 - 80*b6 - 80*b5 + 251*b4 + 261*b3 - 85*b2) / 2 $$\nu^{12}$$ $$=$$ $$- 72 \beta_{15} + 140 \beta_{14} - 60 \beta_{13} - 140 \beta_{12} + 84 \beta_{11} - 112 \beta_{10} - 8 \beta_{9} - 8 \beta_{7} + 40 \beta_{6} - 40 \beta_{5} + 8 \beta_{2} + 392 \beta _1 + 445$$ -72*b15 + 140*b14 - 60*b13 - 140*b12 + 84*b11 - 112*b10 - 8*b9 - 8*b7 + 40*b6 - 40*b5 + 8*b2 + 392*b1 + 445 $$\nu^{13}$$ $$=$$ $$( 216 \beta_{14} + 216 \beta_{12} - 120 \beta_{11} - 120 \beta_{10} - 272 \beta_{9} - 552 \beta_{8} + 272 \beta_{7} - 8 \beta_{6} - 8 \beta_{5} - 183 \beta_{4} + 57 \beta_{3} + 239 \beta_{2} ) / 2$$ (216*b14 + 216*b12 - 120*b11 - 120*b10 - 272*b9 - 552*b8 + 272*b7 - 8*b6 - 8*b5 - 183*b4 + 57*b3 + 239*b2) / 2 $$\nu^{14}$$ $$=$$ $$( 136 \beta_{15} - 488 \beta_{14} - 417 \beta_{13} + 488 \beta_{12} - 376 \beta_{11} - 257 \beta_{10} - 593 \beta_{9} - 593 \beta_{7} - 159 \beta_{6} + 159 \beta_{5} + 593 \beta_{2} + 2790 \beta _1 + 2473 ) / 2$$ (136*b15 - 488*b14 - 417*b13 + 488*b12 - 376*b11 - 257*b10 - 593*b9 - 593*b7 - 159*b6 + 159*b5 + 593*b2 + 2790*b1 + 2473) / 2 $$\nu^{15}$$ $$=$$ $$- 228 \beta_{14} - 228 \beta_{12} + 179 \beta_{11} + 179 \beta_{10} - 413 \beta_{9} + 73 \beta_{8} + 413 \beta_{7} + 381 \beta_{6} + 381 \beta_{5} - 405 \beta_{4} - 524 \beta_{3} - 1344 \beta_{2}$$ -228*b14 - 228*b12 + 179*b11 + 179*b10 - 413*b9 + 73*b8 + 413*b7 + 381*b6 + 381*b5 - 405*b4 - 524*b3 - 1344*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/260\mathbb{Z}\right)^\times$$.

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$1 + \beta_{1}$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.517063 − 1.65307i 1.42836 − 0.979681i −1.56631 − 0.739379i 1.72890 − 0.104392i −1.72890 + 0.104392i 1.56631 + 0.739379i −1.42836 + 0.979681i 0.517063 + 1.65307i −0.517063 + 1.65307i 1.42836 + 0.979681i −1.56631 + 0.739379i 1.72890 + 0.104392i −1.72890 − 0.104392i 1.56631 − 0.739379i −1.42836 − 0.979681i 0.517063 − 1.65307i
0 −2.86320 + 1.65307i 0 0.877236 + 2.05681i 0 −0.517063 + 0.895580i 0 3.96529 6.86809i 0
49.2 0 −1.69686 + 0.979681i 0 −2.16188 0.571200i 0 1.42836 2.47400i 0 0.419550 0.726682i 0
49.3 0 −1.28064 + 0.739379i 0 −0.494086 2.18080i 0 −1.56631 + 2.71292i 0 −0.406637 + 0.704315i 0
49.4 0 −0.180812 + 0.104392i 0 1.60081 + 1.56122i 0 1.72890 2.99455i 0 −1.47820 + 2.56033i 0
49.5 0 0.180812 0.104392i 0 −1.60081 + 1.56122i 0 −1.72890 + 2.99455i 0 −1.47820 + 2.56033i 0
49.6 0 1.28064 0.739379i 0 0.494086 2.18080i 0 1.56631 2.71292i 0 −0.406637 + 0.704315i 0
49.7 0 1.69686 0.979681i 0 2.16188 0.571200i 0 −1.42836 + 2.47400i 0 0.419550 0.726682i 0
49.8 0 2.86320 1.65307i 0 −0.877236 + 2.05681i 0 0.517063 0.895580i 0 3.96529 6.86809i 0
69.1 0 −2.86320 1.65307i 0 0.877236 2.05681i 0 −0.517063 0.895580i 0 3.96529 + 6.86809i 0
69.2 0 −1.69686 0.979681i 0 −2.16188 + 0.571200i 0 1.42836 + 2.47400i 0 0.419550 + 0.726682i 0
69.3 0 −1.28064 0.739379i 0 −0.494086 + 2.18080i 0 −1.56631 2.71292i 0 −0.406637 0.704315i 0
69.4 0 −0.180812 0.104392i 0 1.60081 1.56122i 0 1.72890 + 2.99455i 0 −1.47820 2.56033i 0
69.5 0 0.180812 + 0.104392i 0 −1.60081 1.56122i 0 −1.72890 2.99455i 0 −1.47820 2.56033i 0
69.6 0 1.28064 + 0.739379i 0 0.494086 + 2.18080i 0 1.56631 + 2.71292i 0 −0.406637 0.704315i 0
69.7 0 1.69686 + 0.979681i 0 2.16188 + 0.571200i 0 −1.42836 2.47400i 0 0.419550 + 0.726682i 0
69.8 0 2.86320 + 1.65307i 0 −0.877236 2.05681i 0 0.517063 + 0.895580i 0 3.96529 + 6.86809i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 69.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.z.a 16
3.b odd 2 1 2340.2.cr.a 16
4.b odd 2 1 1040.2.df.d 16
5.b even 2 1 inner 260.2.z.a 16
5.c odd 4 2 1300.2.y.e 16
13.c even 3 1 3380.2.d.d 16
13.e even 6 1 inner 260.2.z.a 16
13.e even 6 1 3380.2.d.d 16
13.f odd 12 2 3380.2.c.e 16
15.d odd 2 1 2340.2.cr.a 16
20.d odd 2 1 1040.2.df.d 16
39.h odd 6 1 2340.2.cr.a 16
52.i odd 6 1 1040.2.df.d 16
65.l even 6 1 inner 260.2.z.a 16
65.l even 6 1 3380.2.d.d 16
65.n even 6 1 3380.2.d.d 16
65.r odd 12 2 1300.2.y.e 16
65.s odd 12 2 3380.2.c.e 16
195.y odd 6 1 2340.2.cr.a 16
260.w odd 6 1 1040.2.df.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.z.a 16 1.a even 1 1 trivial
260.2.z.a 16 5.b even 2 1 inner
260.2.z.a 16 13.e even 6 1 inner
260.2.z.a 16 65.l even 6 1 inner
1040.2.df.d 16 4.b odd 2 1
1040.2.df.d 16 20.d odd 2 1
1040.2.df.d 16 52.i odd 6 1
1040.2.df.d 16 260.w odd 6 1
1300.2.y.e 16 5.c odd 4 2
1300.2.y.e 16 65.r odd 12 2
2340.2.cr.a 16 3.b odd 2 1
2340.2.cr.a 16 15.d odd 2 1
2340.2.cr.a 16 39.h odd 6 1
2340.2.cr.a 16 195.y odd 6 1
3380.2.c.e 16 13.f odd 12 2
3380.2.c.e 16 65.s odd 12 2
3380.2.d.d 16 13.c even 3 1
3380.2.d.d 16 13.e even 6 1
3380.2.d.d 16 65.l even 6 1
3380.2.d.d 16 65.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 17 T^{14} + 214 T^{12} + \cdots + 16$$
$5$ $$T^{16} + 7 T^{14} + 22 T^{12} + \cdots + 390625$$
$7$ $$T^{16} + 31 T^{14} + 634 T^{12} + \cdots + 1048576$$
$11$ $$(T^{8} + 3 T^{7} - 28 T^{6} - 93 T^{5} + \cdots + 4096)^{2}$$
$13$ $$T^{16} - 29 T^{14} + \cdots + 815730721$$
$17$ $$T^{16} - 80 T^{14} + \cdots + 2750058481$$
$19$ $$(T^{8} + 9 T^{7} - 42 T^{6} - 621 T^{5} + \cdots + 412164)^{2}$$
$23$ $$T^{16} - 89 T^{14} + 5830 T^{12} + \cdots + 1336336$$
$29$ $$(T^{8} - 6 T^{7} + 84 T^{6} - 348 T^{5} + \cdots + 154449)^{2}$$
$31$ $$(T^{8} + 108 T^{6} + 2688 T^{4} + \cdots + 2304)^{2}$$
$37$ $$T^{16} + 160 T^{14} + \cdots + 85662167761$$
$41$ $$(T^{8} + 24 T^{7} + 206 T^{6} + \cdots + 1771561)^{2}$$
$43$ $$T^{16} - 237 T^{14} + \cdots + 287107358976$$
$47$ $$(T^{8} - 132 T^{6} + 4992 T^{4} + \cdots + 147456)^{2}$$
$53$ $$(T^{8} + 191 T^{6} + 8004 T^{4} + \cdots + 30976)^{2}$$
$59$ $$(T^{8} + 15 T^{7} + 80 T^{6} + 75 T^{5} + \cdots + 1936)^{2}$$
$61$ $$(T^{8} + 14 T^{7} + 184 T^{6} + \cdots + 1338649)^{2}$$
$67$ $$T^{16} + 451 T^{14} + \cdots + 722642807056$$
$71$ $$(T^{8} + 9 T^{7} - 130 T^{6} + \cdots + 3104644)^{2}$$
$73$ $$(T^{8} - 133 T^{6} + 6276 T^{4} + \cdots + 861184)^{2}$$
$79$ $$(T^{4} + 4 T^{3} - 156 T^{2} - 224 T + 4096)^{4}$$
$83$ $$(T^{8} - 228 T^{6} + 9168 T^{4} + \cdots + 36864)^{2}$$
$89$ $$(T^{8} - 15 T^{7} + 80 T^{6} - 75 T^{5} + \cdots + 1936)^{2}$$
$97$ $$T^{16} + 331 T^{14} + \cdots + 9124290174736$$