# Properties

 Label 260.2.bf.c Level $260$ Weight $2$ Character orbit 260.bf Analytic conductor $2.076$ Analytic rank $0$ Dimension $20$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.bf (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$20$$ Relative dimension: $$5$$ over $$\Q(\zeta_{12})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{20} + \cdots)$$ Defining polynomial: $$x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9$$ x^20 + 30*x^18 + 371*x^16 + 2460*x^14 + 9517*x^12 + 21870*x^10 + 29001*x^8 + 20400*x^6 + 6399*x^4 + 666*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{19}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{18} + \beta_{17} + \beta_{12} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{12} + \beta_{10} - \beta_{7} - \beta_{6}) q^{5} + (\beta_{19} + \beta_{18} + \beta_{17} + \beta_{14} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{7} + \cdots - 1) q^{7}+ \cdots + (\beta_{14} - \beta_{9} + \beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 1) q^{9}+O(q^{10})$$ q + (b18 + b17 + b12 - b7 - b6 + b5 - b4 + b3 - b2 - b1 - 1) * q^3 + (b12 + b10 - b7 - b6) * q^5 + (b19 + b18 + b17 + b14 + b12 - b11 - 2*b10 + b9 - 2*b7 - 2*b6 + 2*b5 - 2*b4 + 2*b3 - b2 - b1 - 1) * q^7 + (b14 - b9 + b7 - b6 + b4 - b3 + 1) * q^9 $$q + (\beta_{18} + \beta_{17} + \beta_{12} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{12} + \beta_{10} - \beta_{7} - \beta_{6}) q^{5} + (\beta_{19} + \beta_{18} + \beta_{17} + \beta_{14} + \beta_{12} - \beta_{11} - 2 \beta_{10} + \beta_{9} - 2 \beta_{7} + \cdots - 1) q^{7}+ \cdots + ( - 2 \beta_{19} - 4 \beta_{18} - 3 \beta_{17} + 2 \beta_{16} + \beta_{14} - 2 \beta_{13} + \cdots + 3) q^{99}+O(q^{100})$$ q + (b18 + b17 + b12 - b7 - b6 + b5 - b4 + b3 - b2 - b1 - 1) * q^3 + (b12 + b10 - b7 - b6) * q^5 + (b19 + b18 + b17 + b14 + b12 - b11 - 2*b10 + b9 - 2*b7 - 2*b6 + 2*b5 - 2*b4 + 2*b3 - b2 - b1 - 1) * q^7 + (b14 - b9 + b7 - b6 + b4 - b3 + 1) * q^9 + (-b14 - b12 - b9 - b7 - b2 - 1) * q^11 + (b19 + b14 + b12 - b11 - b10 + b8 - b7 + b5 - b4 + 2*b3 - b2) * q^13 + (b19 - b16 + b15 + b13 + b12 - b10 + b9 + b8 - 2*b6 + 2*b5) * q^15 + (b17 - b16 + b15 + b14 + b12 - 2*b11 - b8 - b7 - 2*b6 + b5 - b4 - b3 - b2 - b1 + 1) * q^17 + (-b19 - 2*b18 + b16 - b14 - 2*b13 - 2*b12 + b10 - b9 + b7 + b6 - 2*b5 + b4 - 3*b3 + 2*b2) * q^19 + (-2*b17 + b16 - 2*b15 + b14 + b13 - b12 + b11 - b9 + 2*b7 + 3*b6 - 2*b5 + 3*b4 + b3 + b1 + 1) * q^21 + (-b19 + 2*b17 + b15 + b14 - b11 - b10 - 2*b7 - b6 + b5 - b4 - b3 - b1 + 2) * q^23 + (-2*b19 - 2*b18 - 2*b17 + b16 - b14 - 3*b13 - 4*b12 + b11 + 2*b10 - 2*b9 - 2*b8 + 4*b7 + 2*b6 - 4*b5 + 3*b4 - 6*b3 + 4*b2 + 1) * q^25 + (-b19 + b17 + b16 + b12 + b11 + 2*b10 + b9 - 2*b8 - b6 - b5 - b4 - b3 - b1) * q^27 + (b19 + 2*b18 + 3*b17 + 2*b14 + 2*b13 + 3*b12 - 2*b11 - 2*b10 + b9 + b8 - 4*b7 - 3*b6 + 3*b5 - 2*b4 + 5*b3 - 2*b2 - b1 - 1) * q^29 + (-b17 - 2*b16 - 2*b14 + b12 + 2*b11 + b9 + 2*b8 + 2*b6 + b4 + 2*b1 - 1) * q^31 + (-3*b19 - b18 + b17 + 5*b16 - 2*b15 - 2*b13 - 5*b12 + b11 + 4*b10 - 4*b9 - 4*b8 + 2*b7 - b6 - 4*b5 + b4 - 5*b3 + 2*b2 - b1) * q^33 + (3*b19 + b18 - b17 - 2*b16 + 2*b14 + b13 + 2*b12 - 2*b10 + 2*b9 + 2*b6 - 2*b4 + 3*b3 - b2 + 2*b1) * q^35 + (b18 - 4*b17 - 2*b16 + b14 + b13 - b12 + b11 - b10 + 2*b8 + 3*b7 + 3*b6 + b5 + b4 + 3*b3 + b2 + 2*b1) * q^37 + (b19 + 4*b17 + 2*b15 - 2*b14 + 2*b13 + 3*b12 + 2*b9 + 2*b8 - 7*b7 - 4*b6 + 4*b5 - 3*b4 + 2*b3 - 3*b2 - 2) * q^39 + (2*b19 - 2*b17 - 2*b16 - 3*b14 + b9 + 3*b7 + 4*b6 - 2*b5 + 2*b3 + 2*b1 - 1) * q^41 + (-b17 - b16 - b15 + b12 + 2*b9 + 2*b8 + b7 + b6 + b5 + b3 + b2 + 1) * q^43 + (2*b18 - 3*b17 + 2*b16 - 2*b15 - b14 + 2*b11 + 2*b10 - 3*b8 + 4*b7 + 4*b6 - 2*b5 + b4 + b3 + b2 - b1 + 1) * q^45 + (b19 - b17 - b16 - 2*b14 - b13 + b12 + b11 - b10 + 2*b9 + 2*b8 - 2*b7 + 2*b6 + b3 + b1 - 1) * q^47 + (-2*b19 - b17 - b14 - 2*b12 - b10 - b9 + 2*b8 - b6 + b5 + b4 + b3 - 2*b1 - 2) * q^49 + (b19 - 3*b17 + b16 + b13 - 4*b12 + b11 - 2*b10 - 2*b8 + 7*b7 + 7*b6 - 4*b5 + 5*b4 + b2 + 2*b1 + 2) * q^51 + (b19 + 2*b17 - 2*b16 + 2*b15 - 6*b14 + b13 + 3*b12 - b11 - b10 + 2*b9 + 2*b8 + b7 - 5*b6 + 4*b5 - 3*b4 + b3 - 2*b2 - b1 - 3) * q^53 + (b19 - b17 - 2*b16 - 2*b14 + 2*b13 + b12 + 2*b9 + 2*b8 - b7 + b6 + 3*b5 - b4 + 2*b3 - 2*b2 + b1 - 4) * q^55 + (-b18 + b17 + b15 - 4*b14 + b12 + 3*b10 + b8 + 2*b7 + 2*b6 - b5 - b4 - 2*b3 + b2 + b1 - 1) * q^57 + (-3*b19 - b18 - b16 - b15 + b14 + b13 - 2*b12 + b10 - b9 + 2*b7 + b6 - b3 + b1 + 2) * q^59 + (-2*b19 - b18 - b17 - 2*b16 + b15 + 4*b14 - b13 - 3*b12 + 2*b10 - 2*b9 - 2*b8 + b7 + 2*b6 - 4*b5 + 2*b4 - 4*b3 + b1 + 2) * q^61 + (-3*b19 + 3*b17 + 3*b16 + 2*b14 - b12 - b10 - b8 - 5*b7 - 3*b6 - b4 + 3*b2 - 2) * q^63 + (-b18 + 2*b17 + 2*b16 - b15 - 3*b14 - 3*b13 - b12 + b11 + b10 + b9 - 2*b8 - 3*b7 - 2*b6 - 2*b5 - 2*b4 - 4*b3 + 2*b2 - b1 - 4) * q^65 + (-3*b19 - 2*b18 - 2*b17 + 3*b16 - b15 + 2*b14 - b13 - 6*b12 + 2*b11 + 2*b10 - 3*b9 - 4*b8 + 3*b7 + 3*b6 - 5*b5 + 3*b4 - 5*b3 + 5*b2 + b1) * q^67 + (b19 + b18 - 2*b16 - b15 + b13 + 2*b12 + 3*b7 + 3*b6 - b4 + 2*b3 - 3*b2) * q^69 + (b19 - 3*b18 - b17 - 2*b16 + b15 + 3*b13 + b12 - 2*b10 + 3*b9 + 3*b8 - b7 + b6 + 2*b5 + 2*b4 + b3 - b2 + 3*b1 + 2) * q^71 + (2*b19 - 2*b18 + 2*b17 + 2*b16 + 2*b15 + 4*b14 - b13 - b12 - b11 + b8 - 4*b7 - 4*b6 - 2*b3 - b1 + 3) * q^73 + (-2*b19 + b18 - 5*b17 + b16 - 2*b15 - b12 + 2*b11 + 3*b10 - b9 - 2*b8 + 11*b7 + 9*b6 - 6*b5 + 4*b4 + 4*b2 + 2*b1 + 4) * q^75 + (-b17 + 6*b16 - 4*b15 + b14 - 3*b13 - 3*b12 + 2*b11 + 4*b10 - 3*b9 - 3*b8 + 4*b7 + 5*b6 - 6*b5 + 6*b4 - 3*b3 + 4*b2 + 4) * q^77 + (-2*b17 + 4*b14 - b13 - b11 - 3*b10 - 3*b8 + 2*b7 + 2*b6 - b4 + b3 + b2 + 2) * q^79 + (b19 - 2*b18 + b17 - b16 - 2*b13 - 3*b12 - 2*b11 - 3*b10 - b9 + b8 - b5 + 3*b4 - 4*b3 + 3*b2 + 2*b1 + 1) * q^81 + (b19 + b17 - b16 + 2*b14 + 2*b13 + 3*b12 - 2*b11 - 3*b10 + 4*b8 - 3*b7 + b6 + 2*b5 + b4 + 3*b3 - b2 - b1 - 1) * q^83 + (-b19 - 2*b18 - 8*b17 + b16 + b15 - 3*b13 - 6*b12 + 3*b11 + 3*b10 - 2*b9 - 2*b8 + 4*b7 + 7*b6 - 7*b5 + 4*b4 - 4*b3 + 4*b2 + 4*b1 + 6) * q^85 + (3*b19 - b18 + b17 - 3*b16 + 2*b15 - 3*b14 + b13 + 3*b12 - 3*b10 + 2*b9 + 3*b8 - 2*b7 - 4*b6 + 3*b5 - 2*b4 - b3 - 4*b2 - 1) * q^87 + (-5*b19 - b18 + b17 + 4*b16 - b15 - 2*b13 - 6*b12 + b11 + 3*b10 - 4*b9 - 4*b8 + 4*b7 + 3*b6 - 7*b5 + 4*b4 - 8*b3 + 6*b2 + 5) * q^89 + (-b18 - 3*b17 - 3*b16 + 3*b15 - 2*b14 + b13 - b12 + 3*b8 + b7 + b6 + 2*b4 + 2*b2 + b1 - 5) * q^91 + (-3*b19 + 2*b17 - b16 - b14 + 4*b12 + 4*b10 - b9 - 4*b8 - 3*b7 - 8*b6 + b5 - 4*b4 - 3*b3 + b2 - 2*b1 - 2) * q^93 + (2*b19 + b18 + 7*b17 + b16 + b15 - b13 + 2*b12 - 2*b11 - b10 + b9 + b8 - 6*b7 - 7*b6 + 3*b5 - 5*b4 - b3 - 5*b2 - 5*b1 - 2) * q^95 + (-3*b19 + b18 + b17 + 2*b16 + b14 - b12 + b11 + 2*b10 - 3*b9 - 2*b8 + 2*b7 - 6*b6 - 2*b3 + b2 - 3*b1 - 1) * q^97 + (-2*b19 - 4*b18 - 3*b17 + 2*b16 + b14 - 2*b13 - 7*b12 + 4*b10 - 4*b9 + 6*b7 + 4*b6 - 5*b5 + 5*b4 - 7*b3 + 3*b2 + 3*b1 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$20 q - 2 q^{3} - 6 q^{5} - 6 q^{7} + 12 q^{9}+O(q^{10})$$ 20 * q - 2 * q^3 - 6 * q^5 - 6 * q^7 + 12 * q^9 $$20 q - 2 q^{3} - 6 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{13} + 20 q^{15} + 6 q^{17} - 20 q^{19} - 12 q^{21} + 30 q^{23} - 2 q^{25} - 20 q^{27} - 24 q^{29} + 8 q^{31} - 30 q^{33} + 30 q^{37} - 4 q^{39} + 6 q^{41} + 22 q^{43} + 36 q^{45} - 14 q^{49} + 30 q^{53} - 34 q^{55} + 24 q^{59} - 32 q^{61} - 84 q^{63} - 60 q^{65} - 54 q^{67} + 16 q^{69} + 26 q^{75} + 12 q^{77} + 2 q^{81} - 48 q^{83} + 74 q^{85} + 38 q^{87} + 30 q^{89} - 72 q^{91} - 16 q^{93} - 6 q^{95} - 6 q^{97} - 20 q^{99}+O(q^{100})$$ 20 * q - 2 * q^3 - 6 * q^5 - 6 * q^7 + 12 * q^9 - 6 * q^13 + 20 * q^15 + 6 * q^17 - 20 * q^19 - 12 * q^21 + 30 * q^23 - 2 * q^25 - 20 * q^27 - 24 * q^29 + 8 * q^31 - 30 * q^33 + 30 * q^37 - 4 * q^39 + 6 * q^41 + 22 * q^43 + 36 * q^45 - 14 * q^49 + 30 * q^53 - 34 * q^55 + 24 * q^59 - 32 * q^61 - 84 * q^63 - 60 * q^65 - 54 * q^67 + 16 * q^69 + 26 * q^75 + 12 * q^77 + 2 * q^81 - 48 * q^83 + 74 * q^85 + 38 * q^87 + 30 * q^89 - 72 * q^91 - 16 * q^93 - 6 * q^95 - 6 * q^97 - 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{20} + 30 x^{18} + 371 x^{16} + 2460 x^{14} + 9517 x^{12} + 21870 x^{10} + 29001 x^{8} + 20400 x^{6} + 6399 x^{4} + 666 x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 33785 \nu^{19} + 39285 \nu^{18} + 894975 \nu^{17} + 1362663 \nu^{16} + 9328060 \nu^{15} + 19386102 \nu^{14} + 48734964 \nu^{13} + 145983780 \nu^{12} + \cdots - 30772215 ) / 13619376$$ (33785*v^19 + 39285*v^18 + 894975*v^17 + 1362663*v^16 + 9328060*v^15 + 19386102*v^14 + 48734964*v^13 + 145983780*v^12 + 134877863*v^11 + 626423073*v^10 + 198036087*v^9 + 1533635325*v^8 + 182934918*v^7 + 2018251380*v^6 + 208369644*v^5 + 1208743398*v^4 + 188401923*v^3 + 177442551*v^2 + 28886031*v - 30772215) / 13619376 $$\beta_{2}$$ $$=$$ $$( 33785 \nu^{19} - 39285 \nu^{18} + 894975 \nu^{17} - 1362663 \nu^{16} + 9328060 \nu^{15} - 19386102 \nu^{14} + 48734964 \nu^{13} - 145983780 \nu^{12} + \cdots + 30772215 ) / 13619376$$ (33785*v^19 - 39285*v^18 + 894975*v^17 - 1362663*v^16 + 9328060*v^15 - 19386102*v^14 + 48734964*v^13 - 145983780*v^12 + 134877863*v^11 - 626423073*v^10 + 198036087*v^9 - 1533635325*v^8 + 182934918*v^7 - 2018251380*v^6 + 208369644*v^5 - 1208743398*v^4 + 188401923*v^3 - 177442551*v^2 + 28886031*v + 30772215) / 13619376 $$\beta_{3}$$ $$=$$ $$( - 13302 \nu^{18} - 387493 \nu^{16} - 4601880 \nu^{14} - 28817129 \nu^{12} - 102464589 \nu^{10} - 206050063 \nu^{8} - 215234229 \nu^{6} - 87983292 \nu^{4} + \cdots + 1823586 ) / 1134948$$ (-13302*v^18 - 387493*v^16 - 4601880*v^14 - 28817129*v^12 - 102464589*v^10 - 206050063*v^8 - 215234229*v^6 - 87983292*v^4 + 1226733*v^2 + 1823586) / 1134948 $$\beta_{4}$$ $$=$$ $$( 7095 \nu^{19} + 54336 \nu^{18} + 236351 \nu^{17} + 1624980 \nu^{16} + 3193514 \nu^{15} + 20020756 \nu^{14} + 22344730 \nu^{13} + 132212424 \nu^{12} + \cdots + 14385840 ) / 4539792$$ (7095*v^19 + 54336*v^18 + 236351*v^17 + 1624980*v^16 + 3193514*v^15 + 20020756*v^14 + 22344730*v^13 + 132212424*v^12 + 84958189*v^11 + 509453696*v^10 + 162711893*v^9 + 1166030400*v^8 + 95995754*v^7 + 1534019320*v^6 - 124849086*v^5 + 1045511364*v^4 - 168422421*v^3 + 284298396*v^2 - 39273189*v + 14385840) / 4539792 $$\beta_{5}$$ $$=$$ $$( 15705 \nu^{19} - 7449 \nu^{18} + 402397 \nu^{17} - 223669 \nu^{16} + 3930438 \nu^{15} - 2731870 \nu^{14} + 17694998 \nu^{13} - 17453738 \nu^{12} + 30393363 \nu^{11} + \cdots - 661521 ) / 4539792$$ (15705*v^19 - 7449*v^18 + 402397*v^17 - 223669*v^16 + 3930438*v^15 - 2731870*v^14 + 17694998*v^13 - 17453738*v^12 + 30393363*v^11 - 62402747*v^10 - 29388941*v^9 - 124660699*v^8 - 180768738*v^7 - 135753310*v^6 - 223758558*v^5 - 83549682*v^4 - 88480923*v^3 - 29953629*v^2 - 7543635*v - 661521) / 4539792 $$\beta_{6}$$ $$=$$ $$( 66482 \nu^{19} + 47115 \nu^{18} + 2012742 \nu^{17} + 1207191 \nu^{16} + 25148026 \nu^{15} + 11791314 \nu^{14} + 168552744 \nu^{13} + 53084994 \nu^{12} + \cdots - 15821217 ) / 13619376$$ (66482*v^19 + 47115*v^18 + 2012742*v^17 + 1207191*v^16 + 25148026*v^15 + 11791314*v^14 + 168552744*v^13 + 53084994*v^12 + 658485308*v^11 + 91180089*v^10 + 1522503288*v^9 - 88166823*v^8 + 2016646506*v^7 - 542306214*v^6 + 1407545070*v^5 - 671275674*v^4 + 453293982*v^3 - 265442769*v^2 + 61004484*v - 15821217) / 13619376 $$\beta_{7}$$ $$=$$ $$( 66482 \nu^{19} - 47115 \nu^{18} + 2012742 \nu^{17} - 1207191 \nu^{16} + 25148026 \nu^{15} - 11791314 \nu^{14} + 168552744 \nu^{13} - 53084994 \nu^{12} + \cdots + 15821217 ) / 13619376$$ (66482*v^19 - 47115*v^18 + 2012742*v^17 - 1207191*v^16 + 25148026*v^15 - 11791314*v^14 + 168552744*v^13 - 53084994*v^12 + 658485308*v^11 - 91180089*v^10 + 1522503288*v^9 + 88166823*v^8 + 2016646506*v^7 + 542306214*v^6 + 1407545070*v^5 + 671275674*v^4 + 453293982*v^3 + 265442769*v^2 + 61004484*v + 15821217) / 13619376 $$\beta_{8}$$ $$=$$ $$( 65481 \nu^{19} + 34053 \nu^{18} + 1937459 \nu^{17} + 998655 \nu^{16} + 23623520 \nu^{15} + 11935630 \nu^{14} + 154800052 \nu^{13} + 75087996 \nu^{12} + \cdots - 715755 ) / 4539792$$ (65481*v^19 + 34053*v^18 + 1937459*v^17 + 998655*v^16 + 23623520*v^15 + 11935630*v^14 + 154800052*v^13 + 75087996*v^12 + 596374435*v^11 + 267331925*v^10 + 1389325403*v^9 + 536760825*v^8 + 1937754842*v^7 + 566221768*v^6 + 1541686044*v^5 + 259516266*v^4 + 627292227*v^3 + 27500163*v^2 + 97808331*v - 715755) / 4539792 $$\beta_{9}$$ $$=$$ $$( - 201140 \nu^{19} + 14217 \nu^{18} - 5787312 \nu^{17} + 295401 \nu^{16} - 67780378 \nu^{15} + 1818438 \nu^{14} - 418783860 \nu^{13} - 808986 \nu^{12} + \cdots - 9990927 ) / 13619376$$ (-201140*v^19 + 14217*v^18 - 5787312*v^17 + 295401*v^16 - 67780378*v^15 + 1818438*v^14 - 418783860*v^13 - 808986*v^12 - 1478898686*v^11 - 50124345*v^10 - 3024609738*v^9 - 196778049*v^8 - 3461988690*v^7 - 304635390*v^6 - 2012262126*v^5 - 194897718*v^4 - 441638496*v^3 - 56405943*v^2 + 13019238*v - 9990927) / 13619376 $$\beta_{10}$$ $$=$$ $$( 65481 \nu^{19} - 34053 \nu^{18} + 1937459 \nu^{17} - 998655 \nu^{16} + 23623520 \nu^{15} - 11935630 \nu^{14} + 154800052 \nu^{13} - 75087996 \nu^{12} + \cdots + 715755 ) / 4539792$$ (65481*v^19 - 34053*v^18 + 1937459*v^17 - 998655*v^16 + 23623520*v^15 - 11935630*v^14 + 154800052*v^13 - 75087996*v^12 + 596374435*v^11 - 267331925*v^10 + 1389325403*v^9 - 536760825*v^8 + 1937754842*v^7 - 566221768*v^6 + 1541686044*v^5 - 259516266*v^4 + 627292227*v^3 - 27500163*v^2 + 97808331*v + 715755) / 4539792 $$\beta_{11}$$ $$=$$ $$( 72585 \nu^{19} - 59235 \nu^{18} + 2051657 \nu^{17} - 1779699 \nu^{16} + 23339538 \nu^{15} - 22025558 \nu^{14} + 137072830 \nu^{13} - 145835154 \nu^{12} + \cdots + 8463585 ) / 4539792$$ (72585*v^19 - 59235*v^18 + 2051657*v^17 - 1779699*v^16 + 23339538*v^15 - 22025558*v^14 + 137072830*v^13 - 145835154*v^12 + 439508991*v^11 - 560140009*v^10 + 727014287*v^9 - 1259019585*v^8 + 431490234*v^7 - 1568501810*v^6 - 275073702*v^5 - 917472954*v^4 - 402330987*v^3 - 142755915*v^2 - 89801031*v + 8463585) / 4539792 $$\beta_{12}$$ $$=$$ $$( - 72576 \nu^{19} + 88389 \nu^{18} - 2173810 \nu^{17} + 2623635 \nu^{16} - 26817034 \nu^{15} + 31956386 \nu^{14} - 177144782 \nu^{13} + 207300420 \nu^{12} + \cdots + 13670085 ) / 4539792$$ (-72576*v^19 + 88389*v^18 - 2173810*v^17 + 2623635*v^16 - 26817034*v^15 + 31956386*v^14 - 177144782*v^13 + 207300420*v^12 - 681332624*v^11 + 776785621*v^10 - 1552037296*v^9 + 1702791225*v^8 - 2033750596*v^7 + 2100241088*v^6 - 1416836958*v^5 + 1305027630*v^4 - 458869806*v^3 + 311798559*v^2 - 58535142*v + 13670085) / 4539792 $$\beta_{13}$$ $$=$$ $$( 89663 \nu^{19} - 8196 \nu^{18} + 2650665 \nu^{17} - 299502 \nu^{16} + 32159852 \nu^{15} - 4457232 \nu^{14} + 207951588 \nu^{13} - 35038530 \nu^{12} + \cdots - 12592020 ) / 4539792$$ (89663*v^19 - 8196*v^18 + 2650665*v^17 - 299502*v^16 + 32159852*v^15 - 4457232*v^14 + 207951588*v^13 - 35038530*v^12 + 777713569*v^11 - 158121378*v^10 + 1704094497*v^9 - 418222590*v^8 + 2102441266*v^7 - 638267970*v^6 + 1305112428*v^5 - 530952876*v^4 + 308049909*v^3 - 200689998*v^2 + 5778177*v - 12592020) / 4539792 $$\beta_{14}$$ $$=$$ $$( - 353 \nu^{19} - 10585 \nu^{17} - 130732 \nu^{15} - 864458 \nu^{13} - 3326801 \nu^{11} - 7574149 \nu^{9} - 9886332 \nu^{7} - 6763380 \nu^{5} - 1984833 \nu^{3} + \cdots - 8376 ) / 16752$$ (-353*v^19 - 10585*v^17 - 130732*v^15 - 864458*v^13 - 3326801*v^11 - 7574149*v^9 - 9886332*v^7 - 6763380*v^5 - 1984833*v^3 - 145143*v - 8376) / 16752 $$\beta_{15}$$ $$=$$ $$( 210017 \nu^{19} + 54111 \nu^{18} + 6333330 \nu^{17} + 1652277 \nu^{16} + 78814861 \nu^{15} + 20796687 \nu^{14} + 526330083 \nu^{13} + 139985934 \nu^{12} + \cdots + 22008474 ) / 6809688$$ (210017*v^19 + 54111*v^18 + 6333330*v^17 + 1652277*v^16 + 78814861*v^15 + 20796687*v^14 + 526330083*v^13 + 139985934*v^12 + 2050478141*v^11 + 545967756*v^10 + 4730874033*v^9 + 1247438982*v^8 + 6231655746*v^7 + 1606404909*v^6 + 4213523391*v^5 + 1060506387*v^4 + 1133458272*v^3 + 298541511*v^2 + 54298512*v + 22008474) / 6809688 $$\beta_{16}$$ $$=$$ $$( 430079 \nu^{19} + 137379 \nu^{18} + 12923433 \nu^{17} + 4170825 \nu^{16} + 160317994 \nu^{15} + 52004406 \nu^{14} + 1068925596 \nu^{13} + 344662668 \nu^{12} + \cdots - 21883005 ) / 13619376$$ (430079*v^19 + 137379*v^18 + 12923433*v^17 + 4170825*v^16 + 160317994*v^15 + 52004406*v^14 + 1068925596*v^13 + 344662668*v^12 + 4173932447*v^11 + 1309752555*v^10 + 9732909123*v^9 + 2858537727*v^8 + 13180347216*v^7 + 3360969024*v^6 + 9505239378*v^5 + 1764518814*v^4 + 3001048929*v^3 + 193619313*v^2 + 248327631*v - 21883005) / 13619376 $$\beta_{17}$$ $$=$$ $$( 191039 \nu^{19} - 26604 \nu^{18} + 5681093 \nu^{17} - 774986 \nu^{16} + 69369358 \nu^{15} - 9203760 \nu^{14} + 451452628 \nu^{13} - 57634258 \nu^{12} + \cdots + 3647172 ) / 4539792$$ (191039*v^19 - 26604*v^18 + 5681093*v^17 - 774986*v^16 + 69369358*v^15 - 9203760*v^14 + 451452628*v^13 - 57634258*v^12 + 1698247247*v^11 - 204929178*v^10 + 3737954627*v^9 - 412100126*v^8 + 4628173584*v^7 - 430468458*v^6 + 2901105294*v^5 - 175966584*v^4 + 739856205*v^3 + 2453466*v^2 + 60131655*v + 3647172) / 4539792 $$\beta_{18}$$ $$=$$ $$( - 600719 \nu^{19} + 267846 \nu^{18} - 17871213 \nu^{17} + 7954470 \nu^{16} - 218545294 \nu^{15} + 96815934 \nu^{14} - 1427059092 \nu^{13} + \cdots + 28943604 ) / 13619376$$ (-600719*v^19 + 267846*v^18 - 17871213*v^17 + 7954470*v^16 - 218545294*v^15 + 96815934*v^14 - 1427059092*v^13 + 625777416*v^12 - 5401279331*v^11 + 2321510580*v^10 - 12002118807*v^9 + 4967478720*v^8 - 15017124132*v^7 + 5795620566*v^6 - 9351293946*v^5 + 3176812278*v^4 - 2066308425*v^3 + 582362226*v^2 - 28794195*v + 28943604) / 13619376 $$\beta_{19}$$ $$=$$ $$( 870232 \nu^{19} - 57567 \nu^{18} + 25941228 \nu^{17} - 1845867 \nu^{16} + 318130016 \nu^{15} - 24393126 \nu^{14} + 2086177992 \nu^{13} - 171759894 \nu^{12} + \cdots + 10941489 ) / 13619376$$ (870232*v^19 - 57567*v^18 + 25941228*v^17 - 1845867*v^16 + 318130016*v^15 - 24393126*v^14 + 2086177992*v^13 - 171759894*v^12 + 7951703572*v^11 - 694965021*v^10 + 17901766428*v^9 - 1622237349*v^8 + 23031574956*v^7 - 2069563650*v^6 + 15393465120*v^5 - 1236619062*v^4 + 4314029580*v^3 - 200979711*v^2 + 306713628*v + 10941489) / 13619376
 $$\nu$$ $$=$$ $$( \beta_{17} + \beta_{14} + \beta_{12} + \beta_{10} - \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} + 1 ) / 2$$ (b17 + b14 + b12 + b10 - b8 - b5 - b4 - b3 + 1) / 2 $$\nu^{2}$$ $$=$$ $$( - 2 \beta_{19} + \beta_{17} + 2 \beta_{16} + \beta_{14} - \beta_{12} + \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta _1 - 5 ) / 2$$ (-2*b19 + b17 + 2*b16 + b14 - b12 + b10 - 2*b9 - b8 - b5 + b4 - b3 + 2*b2 - 2*b1 - 5) / 2 $$\nu^{3}$$ $$=$$ $$- \beta_{19} - \beta_{18} - \beta_{17} - \beta_{16} + \beta_{15} - 3 \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} - 3 \beta_{10} + 3 \beta_{8} - \beta_{7} - \beta_{6} + 4 \beta_{5} + \beta_{4} + \beta_{3} - \beta _1 - 3$$ -b19 - b18 - b17 - b16 + b15 - 3*b14 - b13 - 2*b12 - b11 - 3*b10 + 3*b8 - b7 - b6 + 4*b5 + b4 + b3 - b1 - 3 $$\nu^{4}$$ $$=$$ $$( 12 \beta_{19} - 5 \beta_{17} - 12 \beta_{16} - 11 \beta_{14} - 2 \beta_{13} + 7 \beta_{12} + 2 \beta_{11} - 7 \beta_{10} + 18 \beta_{9} + 7 \beta_{8} - 8 \beta_{6} + 7 \beta_{5} - 9 \beta_{4} + \beta_{3} - 10 \beta_{2} + 12 \beta _1 + 23 ) / 2$$ (12*b19 - 5*b17 - 12*b16 - 11*b14 - 2*b13 + 7*b12 + 2*b11 - 7*b10 + 18*b9 + 7*b8 - 8*b6 + 7*b5 - 9*b4 + b3 - 10*b2 + 12*b1 + 23) / 2 $$\nu^{5}$$ $$=$$ $$( 18 \beta_{19} + 18 \beta_{18} + 7 \beta_{17} + 18 \beta_{16} - 18 \beta_{15} + 47 \beta_{14} + 18 \beta_{13} + 19 \beta_{12} + 18 \beta_{11} + 43 \beta_{10} - 37 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} - 61 \beta_{5} - \beta_{4} - 7 \beta_{3} - 2 \beta_{2} + 16 \beta _1 + 45 ) / 2$$ (18*b19 + 18*b18 + 7*b17 + 18*b16 - 18*b15 + 47*b14 + 18*b13 + 19*b12 + 18*b11 + 43*b10 - 37*b8 + 8*b7 + 8*b6 - 61*b5 - b4 - 7*b3 - 2*b2 + 16*b1 + 45) / 2 $$\nu^{6}$$ $$=$$ $$- 39 \beta_{19} - 2 \beta_{18} + 6 \beta_{17} + 39 \beta_{16} - 2 \beta_{15} + 49 \beta_{14} + 9 \beta_{13} - 30 \beta_{12} - 9 \beta_{11} + 31 \beta_{10} - 80 \beta_{9} - 28 \beta_{8} + 11 \beta_{7} + 57 \beta_{6} - 33 \beta_{5} + 45 \beta_{4} + \beta_{3} + \cdots - 63$$ -39*b19 - 2*b18 + 6*b17 + 39*b16 - 2*b15 + 49*b14 + 9*b13 - 30*b12 - 9*b11 + 31*b10 - 80*b9 - 28*b8 + 11*b7 + 57*b6 - 33*b5 + 45*b4 + b3 + 32*b2 - 37*b1 - 63 $$\nu^{7}$$ $$=$$ $$( - 136 \beta_{19} - 136 \beta_{18} - 37 \beta_{17} - 136 \beta_{16} + 136 \beta_{15} - 369 \beta_{14} - 142 \beta_{13} - 93 \beta_{12} - 142 \beta_{11} - 329 \beta_{10} + 245 \beta_{8} + 8 \beta_{7} + 8 \beta_{6} + 481 \beta_{5} + \cdots - 357 ) / 2$$ (-136*b19 - 136*b18 - 37*b17 - 136*b16 + 136*b15 - 369*b14 - 142*b13 - 93*b12 - 142*b11 - 329*b10 + 245*b8 + 8*b7 + 8*b6 + 481*b5 - 49*b4 + 55*b3 + 12*b2 - 130*b1 - 357) / 2 $$\nu^{8}$$ $$=$$ $$( 554 \beta_{19} + 76 \beta_{18} + 87 \beta_{17} - 554 \beta_{16} + 76 \beta_{15} - 833 \beta_{14} - 134 \beta_{13} + 541 \beta_{12} + 134 \beta_{11} - 565 \beta_{10} + 1398 \beta_{9} + 465 \beta_{8} - 362 \beta_{7} - 1210 \beta_{6} + \cdots + 745 ) / 2$$ (554*b19 + 76*b18 + 87*b17 - 554*b16 + 76*b15 - 833*b14 - 134*b13 + 541*b12 + 134*b11 - 565*b10 + 1398*b9 + 465*b8 - 362*b7 - 1210*b6 + 641*b5 - 875*b4 + 17*b3 - 492*b2 + 474*b1 + 745) / 2 $$\nu^{9}$$ $$=$$ $$499 \beta_{19} + 499 \beta_{18} + 124 \beta_{17} + 499 \beta_{16} - 499 \beta_{15} + 1460 \beta_{14} + 551 \beta_{13} + 217 \beta_{12} + 551 \beta_{11} + 1295 \beta_{10} - 857 \beta_{8} - 248 \beta_{7} - 248 \beta_{6} - 1935 \beta_{5} + \cdots + 1448$$ 499*b19 + 499*b18 + 124*b17 + 499*b16 - 499*b15 + 1460*b14 + 551*b13 + 217*b12 + 551*b11 + 1295*b10 - 857*b8 - 248*b7 - 248*b6 - 1935*b5 + 334*b4 - 281*b3 - 2*b2 + 549*b1 + 1448 $$\nu^{10}$$ $$=$$ $$( - 4154 \beta_{19} - 944 \beta_{18} - 1823 \beta_{17} + 4154 \beta_{16} - 944 \beta_{15} + 6965 \beta_{14} + 966 \beta_{13} - 4819 \beta_{12} - 966 \beta_{11} + 5013 \beta_{10} - 11998 \beta_{9} - 3855 \beta_{8} + \cdots - 4627 ) / 2$$ (-4154*b19 - 944*b18 - 1823*b17 + 4154*b16 - 944*b15 + 6965*b14 + 966*b13 - 4819*b12 - 966*b11 + 5013*b10 - 11998*b9 - 3855*b8 + 4168*b7 + 11476*b6 - 5977*b5 + 8101*b4 - 571*b3 + 4068*b2 - 3146*b1 - 4627) / 2 $$\nu^{11}$$ $$=$$ $$( - 7394 \beta_{19} - 7374 \beta_{18} - 1867 \beta_{17} - 7394 \beta_{16} + 7374 \beta_{15} - 23407 \beta_{14} - 8596 \beta_{13} - 1739 \beta_{12} - 8596 \beta_{11} - 20673 \beta_{10} + 12507 \beta_{8} + \cdots - 23737 ) / 2$$ (-7394*b19 - 7374*b18 - 1867*b17 - 7394*b16 + 7374*b15 - 23407*b14 - 8596*b13 - 1739*b12 - 8596*b11 - 20673*b10 + 12507*b8 + 6442*b7 + 6442*b6 + 31441*b5 - 6857*b4 + 5583*b3 - 770*b2 - 9366*b1 - 23737) / 2 $$\nu^{12}$$ $$=$$ $$16060 \beta_{19} + 4908 \beta_{18} + 10713 \beta_{17} - 16060 \beta_{16} + 4908 \beta_{15} - 28917 \beta_{14} - 3526 \beta_{13} + 21003 \beta_{12} + 3526 \beta_{11} - 21635 \beta_{10} + 50782 \beta_{9} + \cdots + 14982$$ 16060*b19 + 4908*b18 + 10713*b17 - 16060*b16 + 4908*b15 - 28917*b14 - 3526*b13 + 21003*b12 + 3526*b11 - 21635*b10 + 50782*b9 + 15865*b8 - 20776*b7 - 51432*b6 + 26773*b5 - 36069*b4 + 3867*b3 - 17110*b2 + 10820*b1 + 14982 $$\nu^{13}$$ $$=$$ $$( 55912 \beta_{19} + 55452 \beta_{18} + 14887 \beta_{17} + 55912 \beta_{16} - 55452 \beta_{15} + 189683 \beta_{14} + 67656 \beta_{13} + 3947 \beta_{12} + 67656 \beta_{11} + 166299 \beta_{10} - 94263 \beta_{8} + \cdots + 195423 ) / 2$$ (55912*b19 + 55452*b18 + 14887*b17 + 55912*b16 - 55452*b15 + 189683*b14 + 67656*b13 + 3947*b12 + 67656*b11 + 166299*b10 - 94263*b8 - 65996*b7 - 65996*b6 - 256615*b5 + 63709*b4 - 52343*b3 + 11924*b2 + 79580*b1 + 195423) / 2 $$\nu^{14}$$ $$=$$ $$( - 252878 \beta_{19} - 93112 \beta_{18} - 213107 \beta_{17} + 252878 \beta_{16} - 93112 \beta_{15} + 478593 \beta_{14} + 52860 \beta_{13} - 359545 \beta_{12} - 52860 \beta_{11} + 366229 \beta_{10} + \cdots - 201853 ) / 2$$ (-252878*b19 - 93112*b18 - 213107*b17 + 252878*b16 - 93112*b15 + 478593*b14 + 52860*b13 - 359545*b12 - 52860*b11 + 366229*b10 - 851466*b9 - 259789*b8 + 384768*b7 + 892912*b6 - 465985*b5 + 625285*b4 - 81957*b3 + 287486*b2 - 154122*b1 - 201853) / 2 $$\nu^{15}$$ $$=$$ $$- 215855 \beta_{19} - 212533 \beta_{18} - 61098 \beta_{17} - 215855 \beta_{16} + 212533 \beta_{15} - 774158 \beta_{14} - 268575 \beta_{13} + 13355 \beta_{12} - 268575 \beta_{11} - 672284 \beta_{10} + \cdots - 805634$$ -215855*b19 - 212533*b18 - 61098*b17 - 215855*b16 + 212533*b15 - 774158*b14 - 268575*b13 + 13355*b12 - 268575*b11 - 672284*b10 + 364004*b8 + 307765*b7 + 307765*b6 + 1049643*b5 - 281930*b4 + 234910*b3 - 67042*b2 - 335617*b1 - 805634 $$\nu^{16}$$ $$=$$ $$( 2013708 \beta_{19} + 838112 \beta_{18} + 1962909 \beta_{17} - 2013708 \beta_{16} + 838112 \beta_{15} - 3952693 \beta_{14} - 407094 \beta_{13} + 3037049 \beta_{12} + 407094 \beta_{11} + \cdots + 1411421 ) / 2$$ (2013708*b19 + 838112*b18 + 1962909*b17 - 2013708*b16 + 838112*b15 - 3952693*b14 - 407094*b13 + 3037049*b12 + 407094*b11 - 3060729*b10 + 7091198*b9 + 2121161*b8 - 3415140*b7 - 7601876*b6 + 3976617*b5 - 5323279*b4 + 777835*b3 - 2402138*b2 + 1133008*b1 + 1411421) / 2 $$\nu^{17}$$ $$=$$ $$( 3392222 \beta_{19} + 3314446 \beta_{18} + 1016735 \beta_{17} + 3392222 \beta_{16} - 3314446 \beta_{15} + 12692143 \beta_{14} + 4296602 \beta_{13} - 559677 \beta_{12} + 4296602 \beta_{11} + \cdots + 13286633 ) / 2$$ (3392222*b19 + 3314446*b18 + 1016735*b17 + 3392222*b16 - 3314446*b15 + 12692143*b14 + 4296602*b13 - 559677*b12 + 4296602*b11 + 10911119*b10 - 5724773*b8 - 5474536*b7 - 5474536*b6 - 17195569*b5 + 4856279*b4 - 4095595*b3 + 1326318*b2 + 5622920*b1 + 13286633) / 2 $$\nu^{18}$$ $$=$$ $$- 8080497 \beta_{19} - 3654094 \beta_{18} - 8669055 \beta_{17} + 8080497 \beta_{16} - 3654094 \beta_{15} + 16301048 \beta_{14} + 1602795 \beta_{13} - 12710265 \beta_{12} + \cdots - 5104464$$ -8080497*b19 - 3654094*b18 - 8669055*b17 + 8080497*b16 - 3654094*b15 + 16301048*b14 + 1602795*b13 - 12710265*b12 - 1602795*b11 + 12690656*b10 - 29396506*b9 - 8651369*b8 + 14767651*b7 + 31966965*b6 - 16749552*b5 + 22391634*b4 - 3482128*b3 + 9984052*b2 - 4278659*b1 - 5104464 $$\nu^{19}$$ $$=$$ $$( - 27013592 \beta_{19} - 26203988 \beta_{18} - 8502479 \beta_{17} - 27013592 \beta_{16} + 26203988 \beta_{15} - 104274927 \beta_{14} - 34578050 \beta_{13} + 6564189 \beta_{12} + \cdots - 109516551 ) / 2$$ (-27013592*b19 - 26203988*b18 - 8502479*b17 - 27013592*b16 + 26203988*b15 - 104274927*b14 - 34578050*b13 + 6564189*b12 - 34578050*b11 - 88790875*b10 + 45607099*b8 + 47381200*b7 + 47381200*b6 + 140962163*b5 - 41142239*b4 + 35021537*b3 - 12269064*b2 - 46847114*b1 - 109516551) / 2

## Character values

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

 $$n$$ $$41$$ $$131$$ $$157$$ $$\chi(n)$$ $$\beta_{6}$$ $$1$$ $$-\beta_{6} - \beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 − 2.44766i 1.14923i 2.86589i − 1.70974i − 0.125665i 0.402430i 0.676406i 2.27790i 1.86950i − 1.49418i − 0.402430i − 0.676406i − 2.27790i − 1.86950i 1.49418i 2.44766i − 1.14923i − 2.86589i 1.70974i 0.125665i
0 −3.27331 0.877081i 0 −1.78557 1.34601i 0 −1.31216 + 2.27273i 0 7.34722 + 4.24192i 0
37.2 0 −1.44654 0.387600i 0 1.07994 + 1.95799i 0 −1.21558 + 2.10545i 0 −0.655821 0.378639i 0
37.3 0 −1.32537 0.355132i 0 −1.64953 + 1.50965i 0 1.96809 3.40883i 0 −0.967585 0.558635i 0
37.4 0 2.10166 + 0.563138i 0 −1.38086 + 1.75876i 0 −0.724750 + 1.25530i 0 1.50178 + 0.867051i 0
37.5 0 2.57754 + 0.690650i 0 2.23602 0.0143596i 0 −1.08162 + 1.87342i 0 3.56864 + 2.06036i 0
93.1 0 −0.668040 + 2.49316i 0 −2.20341 + 0.380790i 0 −0.432607 + 0.749297i 0 −3.17149 1.83106i 0
93.2 0 −0.223531 + 0.834228i 0 −0.382788 2.20306i 0 1.19682 2.07295i 0 1.95211 + 1.12705i 0
93.3 0 −0.0651192 + 0.243028i 0 2.20744 0.356684i 0 −2.48736 + 4.30824i 0 2.54325 + 1.46835i 0
93.4 0 0.647720 2.41732i 0 −0.473860 + 2.18528i 0 1.92886 3.34088i 0 −2.82584 1.63150i 0
93.5 0 0.674996 2.51912i 0 −0.647383 2.14030i 0 −0.839682 + 1.45437i 0 −3.29226 1.90079i 0
137.1 0 −0.668040 2.49316i 0 −2.20341 0.380790i 0 −0.432607 0.749297i 0 −3.17149 + 1.83106i 0
137.2 0 −0.223531 0.834228i 0 −0.382788 + 2.20306i 0 1.19682 + 2.07295i 0 1.95211 1.12705i 0
137.3 0 −0.0651192 0.243028i 0 2.20744 + 0.356684i 0 −2.48736 4.30824i 0 2.54325 1.46835i 0
137.4 0 0.647720 + 2.41732i 0 −0.473860 2.18528i 0 1.92886 + 3.34088i 0 −2.82584 + 1.63150i 0
137.5 0 0.674996 + 2.51912i 0 −0.647383 + 2.14030i 0 −0.839682 1.45437i 0 −3.29226 + 1.90079i 0
253.1 0 −3.27331 + 0.877081i 0 −1.78557 + 1.34601i 0 −1.31216 2.27273i 0 7.34722 4.24192i 0
253.2 0 −1.44654 + 0.387600i 0 1.07994 1.95799i 0 −1.21558 2.10545i 0 −0.655821 + 0.378639i 0
253.3 0 −1.32537 + 0.355132i 0 −1.64953 1.50965i 0 1.96809 + 3.40883i 0 −0.967585 + 0.558635i 0
253.4 0 2.10166 0.563138i 0 −1.38086 1.75876i 0 −0.724750 1.25530i 0 1.50178 0.867051i 0
253.5 0 2.57754 0.690650i 0 2.23602 + 0.0143596i 0 −1.08162 1.87342i 0 3.56864 2.06036i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 253.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.bf.c 20
5.b even 2 1 1300.2.bn.d 20
5.c odd 4 1 260.2.bk.c yes 20
5.c odd 4 1 1300.2.bs.d 20
13.f odd 12 1 260.2.bk.c yes 20
65.o even 12 1 1300.2.bn.d 20
65.s odd 12 1 1300.2.bs.d 20
65.t even 12 1 inner 260.2.bf.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.bf.c 20 1.a even 1 1 trivial
260.2.bf.c 20 65.t even 12 1 inner
260.2.bk.c yes 20 5.c odd 4 1
260.2.bk.c yes 20 13.f odd 12 1
1300.2.bn.d 20 5.b even 2 1
1300.2.bn.d 20 65.o even 12 1
1300.2.bs.d 20 5.c odd 4 1
1300.2.bs.d 20 65.s odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{20} + 2 T_{3}^{19} - 4 T_{3}^{18} + 4 T_{3}^{17} - 50 T_{3}^{16} - 214 T_{3}^{15} + 268 T_{3}^{14} - 142 T_{3}^{13} + 3587 T_{3}^{12} + 8242 T_{3}^{11} - 5960 T_{3}^{10} - 1666 T_{3}^{9} - 44722 T_{3}^{8} - 208136 T_{3}^{7} + \cdots + 21904$$ acting on $$S_{2}^{\mathrm{new}}(260, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{20}$$
$3$ $$T^{20} + 2 T^{19} - 4 T^{18} + \cdots + 21904$$
$5$ $$T^{20} + 6 T^{19} + 19 T^{18} + \cdots + 9765625$$
$7$ $$T^{20} + 6 T^{19} + 60 T^{18} + \cdots + 27625536$$
$11$ $$T^{20} - 12 T^{18} + 128 T^{17} + \cdots + 8202496$$
$13$ $$T^{20} + 6 T^{19} + \cdots + 137858491849$$
$17$ $$T^{20} - 6 T^{19} + \cdots + 109980446689$$
$19$ $$T^{20} + 20 T^{19} + \cdots + 472279824$$
$23$ $$T^{20} - 30 T^{19} + 408 T^{18} + \cdots + 3139984$$
$29$ $$T^{20} + 24 T^{19} + \cdots + 341103961$$
$31$ $$T^{20} - 8 T^{19} + \cdots + 54838398976$$
$37$ $$T^{20} - 30 T^{19} + \cdots + 52\!\cdots\!69$$
$41$ $$T^{20} - 6 T^{19} + \cdots + 10017978284161$$
$43$ $$T^{20} - 22 T^{19} + \cdots + 319577657344$$
$47$ $$(T^{10} - 112 T^{8} - 248 T^{7} + \cdots - 76544)^{2}$$
$53$ $$T^{20} - 30 T^{19} + \cdots + 90\!\cdots\!64$$
$59$ $$T^{20} - 24 T^{19} + \cdots + 26\!\cdots\!96$$
$61$ $$T^{20} + 32 T^{19} + \cdots + 43\!\cdots\!41$$
$67$ $$T^{20} + 54 T^{19} + \cdots + 11546104977936$$
$71$ $$T^{20} - 24 T^{18} + \cdots + 27\!\cdots\!76$$
$73$ $$T^{20} + 730 T^{18} + \cdots + 14\!\cdots\!84$$
$79$ $$T^{20} + \cdots + 207459544743936$$
$83$ $$(T^{10} + 24 T^{9} - 160 T^{8} + \cdots + 33087744)^{2}$$
$89$ $$T^{20} + \cdots + 237919334033956$$
$97$ $$T^{20} + 6 T^{19} + \cdots + 963004643584$$