# Properties

 Label 637.2.r.f Level $637$ Weight $2$ Character orbit 637.r Analytic conductor $5.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.r (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.08647060876$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9$$ x^16 - 11*x^14 + 85*x^12 - 334*x^10 + 952*x^8 - 1050*x^6 + 853*x^4 - 93*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 91) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{8} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{4} + 1) q^{3} + ( - \beta_{11} - \beta_{7} + 1) q^{4} - \beta_{14} q^{5} + (\beta_{14} - \beta_{13} - \beta_{8} + \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{13} + \beta_{2}) q^{8} + ( - \beta_{12} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + 1) q^{9}+O(q^{10})$$ q + (-b8 - b1) * q^2 + (-b7 - b4 + 1) * q^3 + (-b11 - b7 + 1) * q^4 - b14 * q^5 + (b14 - b13 - b8 + b6 + b2) * q^6 + (-b13 + b2) * q^8 + (-b12 + b9 - 2*b7 + b5 - b4 + 1) * q^9 $$q + ( - \beta_{8} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{4} + 1) q^{3} + ( - \beta_{11} - \beta_{7} + 1) q^{4} - \beta_{14} q^{5} + (\beta_{14} - \beta_{13} - \beta_{8} + \beta_{6} + \beta_{2}) q^{6} + ( - \beta_{13} + \beta_{2}) q^{8} + ( - \beta_{12} + \beta_{9} - 2 \beta_{7} + \beta_{5} - \beta_{4} + 1) q^{9} + (\beta_{12} - \beta_{11} + 2 \beta_{4}) q^{10} + (\beta_{10} + \beta_1) q^{11} + ( - \beta_{12} - \beta_{11} + \beta_{9} - 3 \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3} + 1) q^{12} + ( - \beta_{14} - \beta_{13} + \beta_{8} - \beta_{6} + \beta_{3} + \beta_{2} + 1) q^{13} + ( - \beta_{14} + 2 \beta_{8} - \beta_{6}) q^{15} + (\beta_{11} + \beta_{5} - \beta_{4} + \beta_{3} + 1) q^{16} + ( - \beta_{12} + \beta_{7} - 1) q^{17} + ( - \beta_{10} + 3 \beta_{6} + 3 \beta_1) q^{18} + ( - \beta_{13} + 2 \beta_{8} + 2 \beta_1) q^{19} + (\beta_{15} - 2 \beta_{14} - 2 \beta_{6}) q^{20} + ( - \beta_{9} - \beta_{5} - \beta_{3} - 3) q^{22} + ( - \beta_{7} - \beta_{5} + \beta_{4} - 1) q^{23} + ( - \beta_{10} + \beta_{6} - \beta_{2} + 3 \beta_1) q^{24} + (\beta_{12} - 2 \beta_{11} + \beta_{4}) q^{25} + ( - \beta_{13} + \beta_{12} - \beta_{11} - \beta_{9} - 2 \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \cdots - 1) q^{26}+ \cdots + (\beta_{15} - \beta_{14} - 3 \beta_{8} - \beta_{6}) q^{99}+O(q^{100})$$ q + (-b8 - b1) * q^2 + (-b7 - b4 + 1) * q^3 + (-b11 - b7 + 1) * q^4 - b14 * q^5 + (b14 - b13 - b8 + b6 + b2) * q^6 + (-b13 + b2) * q^8 + (-b12 + b9 - 2*b7 + b5 - b4 + 1) * q^9 + (b12 - b11 + 2*b4) * q^10 + (b10 + b1) * q^11 + (-b12 - b11 + b9 - 3*b7 + b5 - b4 - b3 + 1) * q^12 + (-b14 - b13 + b8 - b6 + b3 + b2 + 1) * q^13 + (-b14 + 2*b8 - b6) * q^15 + (b11 + b5 - b4 + b3 + 1) * q^16 + (-b12 + b7 - 1) * q^17 + (-b10 + 3*b6 + 3*b1) * q^18 + (-b13 + 2*b8 + 2*b1) * q^19 + (b15 - 2*b14 - 2*b6) * q^20 + (-b9 - b5 - b3 - 3) * q^22 + (-b7 - b5 + b4 - 1) * q^23 + (-b10 + b6 - b2 + 3*b1) * q^24 + (b12 - 2*b11 + b4) * q^25 + (-b13 + b12 - b11 - b9 - 2*b8 + b7 - b5 + b4 - b3 - 2*b1 - 1) * q^26 + (b9 + b5 - 2*b3 - 2) * q^27 + (-3*b5 - 2*b3 - 3) * q^29 + (b12 + b11 - b9 + 6*b7 - 2*b5 + 2*b4 + b3 - 2) * q^30 + (b10 + b6 + b2 + b1) * q^31 + (b6 - 2*b2 - b1) * q^32 + (-b15 - b14 - b13 + b10 + b8 + b1) * q^33 + (-b15 + 2*b14 + b13 + 2*b6 - b2) * q^34 + (2*b9 + 5*b5 - 1) * q^36 + (-b15 - b14 - b13 + b10 - b8 - b1) * q^37 + (b11 + 4*b7 - b4 - 4) * q^38 + (-b12 - b11 - b10 - 3*b7 - b6 - b4 - 2*b2 + 3) * q^39 + (b12 - b9 + b7 - b5 + b4 - 1) * q^40 + (-b15 - 2*b14 + b13 + b8 - 2*b6 - b2) * q^41 + (-b9 + 2*b5 + 2) * q^43 + (b15 + 3*b14 + b13 - b10) * q^44 + (-2*b2 - 4*b1) * q^45 + (-b6 - b2 + b1) * q^46 + (b15 + 2*b14 - 2*b13 - b10 - b8 - b1) * q^47 + (b3 - 2) * q^48 + (b15 - 3*b14 - 2*b13 - b8 - 3*b6 + 2*b2) * q^50 + (-2*b11 + b7 + b5 - b4 - 2*b3 + 1) * q^51 + (-b11 + b10 - 6*b7 - b6 - b4 - b2 + b1 + 6) * q^52 + (2*b11 + 3*b7 - 3) * q^53 + (b15 - 3*b14 + 2*b13 - b10 + 6*b8 + 6*b1) * q^54 + (-b9 - 3*b5 - 4*b3 - 4) * q^55 + (-b15 - b14 + 3*b13 - b8 - b6 - 3*b2) * q^57 + (3*b14 - b13 + 2*b8 + 2*b1) * q^58 + (b10 - 2*b6 + 2*b2 - 3*b1) * q^59 + (b10 - 2*b6 - 2*b2 - 4*b1) * q^60 + (-2*b11 + b7 - 2*b3) * q^61 + (2*b5 - b3 - 2) * q^62 + (b9 - 2*b5 + 2*b3 + 5) * q^64 + (b15 - 2*b14 + b12 - b11 - b10 - b9 - 4*b7 - b3) * q^65 - b11 * q^66 + (2*b6 + 3*b2) * q^67 + (-b12 + 3*b11 + b9 - b7 + 4*b5 - 4*b4 + 3*b3 + 4) * q^68 + (-b9 + b5 + 4) * q^69 + (b14 - 3*b13 + 4*b8 + b6 + 3*b2) * q^71 + (-3*b14 + 3*b13 + 2*b8 + 2*b1) * q^72 + (-2*b10 + b6 + 2*b1) * q^73 + (-3*b11 - 6*b7 + 6) * q^74 + (-b12 + b9 - 2*b5 + 2*b4 - 2) * q^75 + (b14 + 2*b13 + b8 + b6 - 2*b2) * q^76 + (-b15 + 3*b14 - b13 - 5*b8 + 3*b6 - 3*b5 + b3 + b2) * q^78 + (b12 - b9 + 3*b7 - b5 + b4 - 1) * q^79 + (-b10 + b6 - 2*b1) * q^80 + (4*b11 + 5*b7 + 2*b4 - 5) * q^81 + (b12 - b9 + 4*b7 - 4*b5 + 4*b4 - 4) * q^82 + (b15 - b13 + 5*b8 + b2) * q^83 + (3*b14 + 2*b13 + 2*b8 + 3*b6 - 2*b2) * q^85 + (-b15 + 3*b13 + b10 - b8 - b1) * q^86 + (-b12 + 2*b11 - 8*b7 + 8) * q^87 + (2*b11 - b7 - 2*b4 + 1) * q^88 + (b15 - b14 + 3*b13 - b10 - 5*b8 - 5*b1) * q^89 + (-2*b5 + 6*b3 + 14) * q^90 + (-b9 - b5 - b3) * q^92 + (-3*b14 - 2*b13 + 6*b8 + 6*b1) * q^93 + (-b12 - b11 - 6*b7 - 5*b4 + 6) * q^94 + (-b12 + 2*b11 + b7 - 3*b4 - 1) * q^95 + (-2*b15 + 2*b14 + b13 + 2*b10 - 5*b8 - 5*b1) * q^96 + (b15 + b13 + b8 - b2) * q^97 + (b15 - b14 - 3*b8 - b6) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 4 q^{3} + 6 q^{4} - 12 q^{9}+O(q^{10})$$ 16 * q + 4 * q^3 + 6 * q^4 - 12 * q^9 $$16 q + 4 q^{3} + 6 q^{4} - 12 q^{9} + 6 q^{10} - 18 q^{12} + 12 q^{13} + 2 q^{16} - 8 q^{17} - 36 q^{22} - 12 q^{23} + 6 q^{26} - 32 q^{27} - 16 q^{29} + 38 q^{30} - 56 q^{36} - 34 q^{38} + 18 q^{39} + 4 q^{40} + 16 q^{43} - 36 q^{48} + 16 q^{51} + 42 q^{52} - 20 q^{53} - 24 q^{55} + 12 q^{61} - 44 q^{62} + 88 q^{64} - 30 q^{65} - 2 q^{66} + 2 q^{68} + 56 q^{69} + 42 q^{74} - 8 q^{75} + 20 q^{78} + 20 q^{79} - 24 q^{81} + 16 q^{82} + 68 q^{87} + 4 q^{88} + 216 q^{90} + 12 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100})$$ 16 * q + 4 * q^3 + 6 * q^4 - 12 * q^9 + 6 * q^10 - 18 * q^12 + 12 * q^13 + 2 * q^16 - 8 * q^17 - 36 * q^22 - 12 * q^23 + 6 * q^26 - 32 * q^27 - 16 * q^29 + 38 * q^30 - 56 * q^36 - 34 * q^38 + 18 * q^39 + 4 * q^40 + 16 * q^43 - 36 * q^48 + 16 * q^51 + 42 * q^52 - 20 * q^53 - 24 * q^55 + 12 * q^61 - 44 * q^62 + 88 * q^64 - 30 * q^65 - 2 * q^66 + 2 * q^68 + 56 * q^69 + 42 * q^74 - 8 * q^75 + 20 * q^78 + 20 * q^79 - 24 * q^81 + 16 * q^82 + 68 * q^87 + 4 * q^88 + 216 * q^90 + 12 * q^92 + 26 * q^94 - 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 11x^{14} + 85x^{12} - 334x^{10} + 952x^{8} - 1050x^{6} + 853x^{4} - 93x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 24498 \nu^{15} + 246060 \nu^{13} - 1852321 \nu^{11} + 6411671 \nu^{9} - 17193085 \nu^{7} + 6321845 \nu^{5} - 690027 \nu^{3} - 55035781 \nu ) / 14163622$$ (-24498*v^15 + 246060*v^13 - 1852321*v^11 + 6411671*v^9 - 17193085*v^7 + 6321845*v^5 - 690027*v^3 - 55035781*v) / 14163622 $$\beta_{3}$$ $$=$$ $$( - 24498 \nu^{14} + 246060 \nu^{12} - 1852321 \nu^{10} + 6411671 \nu^{8} - 17193085 \nu^{6} + 6321845 \nu^{4} - 690027 \nu^{2} - 40872159 ) / 14163622$$ (-24498*v^14 + 246060*v^12 - 1852321*v^10 + 6411671*v^8 - 17193085*v^6 + 6321845*v^4 - 690027*v^2 - 40872159) / 14163622 $$\beta_{4}$$ $$=$$ $$( 172099 \nu^{14} - 2170865 \nu^{12} + 17340370 \nu^{10} - 78484018 \nu^{8} + 236538400 \nu^{6} - 377649654 \nu^{4} + 218482087 \nu^{2} + \cdots - 23829231 ) / 42490866$$ (172099*v^14 - 2170865*v^12 + 17340370*v^10 - 78484018*v^8 + 236538400*v^6 - 377649654*v^4 + 218482087*v^2 - 23829231) / 42490866 $$\beta_{5}$$ $$=$$ $$( - 99072 \nu^{14} + 1000291 \nu^{12} - 7490944 \nu^{10} + 25929344 \nu^{8} - 66564370 \nu^{6} + 25566080 \nu^{4} - 2790528 \nu^{2} - 20454191 ) / 14163622$$ (-99072*v^14 + 1000291*v^12 - 7490944*v^10 + 25929344*v^8 - 66564370*v^6 + 25566080*v^4 - 2790528*v^2 - 20454191) / 14163622 $$\beta_{6}$$ $$=$$ $$( 123570 \nu^{15} - 1246351 \nu^{13} + 9343265 \nu^{11} - 32341015 \nu^{9} + 83757455 \nu^{7} - 31887925 \nu^{5} + 3480555 \nu^{3} + 61326350 \nu ) / 14163622$$ (123570*v^15 - 1246351*v^13 + 9343265*v^11 - 32341015*v^9 + 83757455*v^7 - 31887925*v^5 + 3480555*v^3 + 61326350*v) / 14163622 $$\beta_{7}$$ $$=$$ $$( 539569 \nu^{14} - 5861765 \nu^{12} + 45125185 \nu^{10} - 174659083 \nu^{8} + 494434675 \nu^{6} - 514968195 \nu^{4} + 441286822 \nu^{2} + \cdots - 5618970 ) / 42490866$$ (539569*v^14 - 5861765*v^12 + 45125185*v^10 - 174659083*v^8 + 494434675*v^6 - 514968195*v^4 + 441286822*v^2 - 5618970) / 42490866 $$\beta_{8}$$ $$=$$ $$( - 539569 \nu^{15} + 5861765 \nu^{13} - 45125185 \nu^{11} + 174659083 \nu^{9} - 494434675 \nu^{7} + 514968195 \nu^{5} - 441286822 \nu^{3} + \cdots + 5618970 \nu ) / 42490866$$ (-539569*v^15 + 5861765*v^13 - 45125185*v^11 + 174659083*v^9 - 494434675*v^7 + 514968195*v^5 - 441286822*v^3 + 5618970*v) / 42490866 $$\beta_{9}$$ $$=$$ $$( 102312 \nu^{14} - 1048444 \nu^{12} + 7735924 \nu^{10} - 26777324 \nu^{8} + 67022271 \nu^{6} - 26402180 \nu^{4} + 2881788 \nu^{2} + 23341327 ) / 7081811$$ (102312*v^14 - 1048444*v^12 + 7735924*v^10 - 26777324*v^8 + 67022271*v^6 - 26402180*v^4 + 2881788*v^2 + 23341327) / 7081811 $$\beta_{10}$$ $$=$$ $$( 476262 \nu^{15} - 4835650 \nu^{13} + 36010699 \nu^{11} - 124648349 \nu^{9} + 318752537 \nu^{7} - 122902055 \nu^{5} + 13414713 \nu^{3} + \cdots + 238534757 \nu ) / 14163622$$ (476262*v^15 - 4835650*v^13 + 36010699*v^11 - 124648349*v^9 + 318752537*v^7 - 122902055*v^5 + 13414713*v^3 + 238534757*v) / 14163622 $$\beta_{11}$$ $$=$$ $$( - 515071 \nu^{14} + 5615705 \nu^{12} - 43272864 \nu^{10} + 168247412 \nu^{8} - 477241590 \nu^{6} + 508646350 \nu^{4} - 426433173 \nu^{2} + \cdots + 46491129 ) / 14163622$$ (-515071*v^14 + 5615705*v^12 - 43272864*v^10 + 168247412*v^8 - 477241590*v^6 + 508646350*v^4 - 426433173*v^2 + 46491129) / 14163622 $$\beta_{12}$$ $$=$$ $$( - 3598 \nu^{14} + 40712 \nu^{12} - 317920 \nu^{10} + 1287475 \nu^{8} - 3722089 \nu^{6} + 4460568 \nu^{4} - 3361729 \nu^{2} + 366555 ) / 95271$$ (-3598*v^14 + 40712*v^12 - 317920*v^10 + 1287475*v^8 - 3722089*v^6 + 4460568*v^4 - 3361729*v^2 + 366555) / 95271 $$\beta_{13}$$ $$=$$ $$( 2084782 \nu^{15} - 22708880 \nu^{13} + 174943777 \nu^{11} - 679401319 \nu^{9} + 1926159445 \nu^{7} - 2040907245 \nu^{5} + 1720586341 \nu^{3} + \cdots - 187583223 \nu ) / 42490866$$ (2084782*v^15 - 22708880*v^13 + 174943777*v^11 - 679401319*v^9 + 1926159445*v^7 - 2040907245*v^5 + 1720586341*v^3 - 187583223*v) / 42490866 $$\beta_{14}$$ $$=$$ $$( 205171 \nu^{15} - 2261795 \nu^{13} + 17480377 \nu^{11} - 68898667 \nu^{9} + 196608895 \nu^{7} - 219868809 \nu^{5} + 176278948 \nu^{3} + \cdots - 19219314 \nu ) / 3862806$$ (205171*v^15 - 2261795*v^13 + 17480377*v^11 - 68898667*v^9 + 196608895*v^7 - 219868809*v^5 + 176278948*v^3 - 19219314*v) / 3862806 $$\beta_{15}$$ $$=$$ $$( 10171607 \nu^{15} - 110994637 \nu^{13} + 854461673 \nu^{11} - 3317989973 \nu^{9} + 9362299115 \nu^{7} - 9751108731 \nu^{5} + 7579585292 \nu^{3} + \cdots - 106397226 \nu ) / 42490866$$ (10171607*v^15 - 110994637*v^13 + 854461673*v^11 - 3317989973*v^9 + 9362299115*v^7 - 9751108731*v^5 + 7579585292*v^3 - 106397226*v) / 42490866
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + 3\beta_{7} + \beta_{3}$$ b11 + 3*b7 + b3 $$\nu^{3}$$ $$=$$ $$-\beta_{13} - 4\beta_{8} + \beta_{2}$$ -b13 - 4*b8 + b2 $$\nu^{4}$$ $$=$$ $$5\beta_{11} + 14\beta_{7} + \beta_{4} - 14$$ 5*b11 + 14*b7 + b4 - 14 $$\nu^{5}$$ $$=$$ $$\beta_{14} - 6\beta_{13} - 19\beta_{8} - 19\beta_1$$ b14 - 6*b13 - 19*b8 - 19*b1 $$\nu^{6}$$ $$=$$ $$\beta_{9} + 8\beta_{5} - 24\beta_{3} - 61$$ b9 + 8*b5 - 24*b3 - 61 $$\nu^{7}$$ $$=$$ $$\beta_{10} - 10\beta_{6} - 31\beta_{2} - 94\beta_1$$ b10 - 10*b6 - 31*b2 - 94*b1 $$\nu^{8}$$ $$=$$ $$-11\beta_{12} - 115\beta_{11} + 11\beta_{9} - 345\beta_{7} + 52\beta_{5} - 52\beta_{4} - 115\beta_{3} + 52$$ -11*b12 - 115*b11 + 11*b9 - 345*b7 + 52*b5 - 52*b4 - 115*b3 + 52 $$\nu^{9}$$ $$=$$ $$11\beta_{15} - 74\beta_{14} + 156\beta_{13} + 471\beta_{8} - 74\beta_{6} - 156\beta_{2}$$ 11*b15 - 74*b14 + 156*b13 + 471*b8 - 74*b6 - 156*b2 $$\nu^{10}$$ $$=$$ $$-85\beta_{12} - 553\beta_{11} - 1736\beta_{7} - 315\beta_{4} + 1736$$ -85*b12 - 553*b11 - 1736*b7 - 315*b4 + 1736 $$\nu^{11}$$ $$=$$ $$85\beta_{15} - 485\beta_{14} + 783\beta_{13} - 85\beta_{10} + 2374\beta_{8} + 2374\beta_1$$ 85*b15 - 485*b14 + 783*b13 - 85*b10 + 2374*b8 + 2374*b1 $$\nu^{12}$$ $$=$$ $$-570\beta_{9} - 1838\beta_{5} + 2672\beta_{3} + 6935$$ -570*b9 - 1838*b5 + 2672*b3 + 6935 $$\nu^{13}$$ $$=$$ $$-570\beta_{10} + 2978\beta_{6} + 3940\beta_{2} + 12015\beta_1$$ -570*b10 + 2978*b6 + 3940*b2 + 12015*b1 $$\nu^{14}$$ $$=$$ $$3548 \beta_{12} + 12977 \beta_{11} - 3548 \beta_{9} + 44495 \beta_{7} - 10466 \beta_{5} + 10466 \beta_{4} + 12977 \beta_{3} - 10466$$ 3548*b12 + 12977*b11 - 3548*b9 + 44495*b7 - 10466*b5 + 10466*b4 + 12977*b3 - 10466 $$\nu^{15}$$ $$=$$ $$-3548\beta_{15} + 17562\beta_{14} - 19895\beta_{13} - 61020\beta_{8} + 17562\beta_{6} + 19895\beta_{2}$$ -3548*b15 + 17562*b14 - 19895*b13 - 61020*b8 + 17562*b6 + 19895*b2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$248$$ $$\chi(n)$$ $$-1$$ $$-\beta_{7}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
116.1
 1.97871 + 1.14241i 1.84073 + 1.06275i 0.929293 + 0.536527i 0.287846 + 0.166188i −0.287846 − 0.166188i −0.929293 − 0.536527i −1.84073 − 1.06275i −1.97871 − 1.14241i 1.97871 − 1.14241i 1.84073 − 1.06275i 0.929293 − 0.536527i 0.287846 − 0.166188i −0.287846 + 0.166188i −0.929293 + 0.536527i −1.84073 + 1.06275i −1.97871 + 1.14241i
−1.97871 + 1.14241i 1.57521 2.72835i 1.61019 2.78892i 1.84030 1.06250i 7.19813i 0 2.78832i −3.46258 5.99736i −2.42760 + 4.20473i
116.2 −1.84073 + 1.06275i −0.0894272 + 0.154892i 1.25885 2.18040i −3.12291 + 1.80301i 0.380153i 0 1.10038i 1.48401 + 2.57037i 3.83229 6.63772i
116.3 −0.929293 + 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i −0.541640 + 0.312716i 2.60903i 0 3.05665i −1.45586 2.52163i 0.335561 0.581209i
116.4 −0.287846 + 0.166188i 0.729919 1.26426i −0.944763 + 1.63638i 1.25195 0.722811i 0.485214i 0 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.5 0.287846 0.166188i 0.729919 1.26426i −0.944763 + 1.63638i −1.25195 + 0.722811i 0.485214i 0 1.29278i 0.434437 + 0.752468i −0.240245 + 0.416116i
116.6 0.929293 0.536527i −1.21570 + 2.10566i −0.424277 + 0.734868i 0.541640 0.312716i 2.60903i 0 3.05665i −1.45586 2.52163i 0.335561 0.581209i
116.7 1.84073 1.06275i −0.0894272 + 0.154892i 1.25885 2.18040i 3.12291 1.80301i 0.380153i 0 1.10038i 1.48401 + 2.57037i 3.83229 6.63772i
116.8 1.97871 1.14241i 1.57521 2.72835i 1.61019 2.78892i −1.84030 + 1.06250i 7.19813i 0 2.78832i −3.46258 5.99736i −2.42760 + 4.20473i
324.1 −1.97871 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i 1.84030 + 1.06250i 7.19813i 0 2.78832i −3.46258 + 5.99736i −2.42760 4.20473i
324.2 −1.84073 1.06275i −0.0894272 0.154892i 1.25885 + 2.18040i −3.12291 1.80301i 0.380153i 0 1.10038i 1.48401 2.57037i 3.83229 + 6.63772i
324.3 −0.929293 0.536527i −1.21570 2.10566i −0.424277 0.734868i −0.541640 0.312716i 2.60903i 0 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.4 −0.287846 0.166188i 0.729919 + 1.26426i −0.944763 1.63638i 1.25195 + 0.722811i 0.485214i 0 1.29278i 0.434437 0.752468i −0.240245 0.416116i
324.5 0.287846 + 0.166188i 0.729919 + 1.26426i −0.944763 1.63638i −1.25195 0.722811i 0.485214i 0 1.29278i 0.434437 0.752468i −0.240245 0.416116i
324.6 0.929293 + 0.536527i −1.21570 2.10566i −0.424277 0.734868i 0.541640 + 0.312716i 2.60903i 0 3.05665i −1.45586 + 2.52163i 0.335561 + 0.581209i
324.7 1.84073 + 1.06275i −0.0894272 0.154892i 1.25885 + 2.18040i 3.12291 + 1.80301i 0.380153i 0 1.10038i 1.48401 2.57037i 3.83229 + 6.63772i
324.8 1.97871 + 1.14241i 1.57521 + 2.72835i 1.61019 + 2.78892i −1.84030 1.06250i 7.19813i 0 2.78832i −3.46258 + 5.99736i −2.42760 4.20473i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
13.b even 2 1 inner
91.r even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.r.f 16
7.b odd 2 1 91.2.r.a 16
7.c even 3 1 637.2.c.e 8
7.c even 3 1 inner 637.2.r.f 16
7.d odd 6 1 91.2.r.a 16
7.d odd 6 1 637.2.c.f 8
13.b even 2 1 inner 637.2.r.f 16
21.c even 2 1 819.2.dl.e 16
21.g even 6 1 819.2.dl.e 16
91.b odd 2 1 91.2.r.a 16
91.i even 4 2 1183.2.e.i 16
91.r even 6 1 637.2.c.e 8
91.r even 6 1 inner 637.2.r.f 16
91.s odd 6 1 91.2.r.a 16
91.s odd 6 1 637.2.c.f 8
91.z odd 12 2 8281.2.a.cj 8
91.bb even 12 2 1183.2.e.i 16
91.bb even 12 2 8281.2.a.ck 8
273.g even 2 1 819.2.dl.e 16
273.ba even 6 1 819.2.dl.e 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.r.a 16 7.b odd 2 1
91.2.r.a 16 7.d odd 6 1
91.2.r.a 16 91.b odd 2 1
91.2.r.a 16 91.s odd 6 1
637.2.c.e 8 7.c even 3 1
637.2.c.e 8 91.r even 6 1
637.2.c.f 8 7.d odd 6 1
637.2.c.f 8 91.s odd 6 1
637.2.r.f 16 1.a even 1 1 trivial
637.2.r.f 16 7.c even 3 1 inner
637.2.r.f 16 13.b even 2 1 inner
637.2.r.f 16 91.r even 6 1 inner
819.2.dl.e 16 21.c even 2 1
819.2.dl.e 16 21.g even 6 1
819.2.dl.e 16 273.g even 2 1
819.2.dl.e 16 273.ba even 6 1
1183.2.e.i 16 91.i even 4 2
1183.2.e.i 16 91.bb even 12 2
8281.2.a.cj 8 91.z odd 12 2
8281.2.a.ck 8 91.bb even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(637, [\chi])$$:

 $$T_{2}^{16} - 11T_{2}^{14} + 85T_{2}^{12} - 334T_{2}^{10} + 952T_{2}^{8} - 1050T_{2}^{6} + 853T_{2}^{4} - 93T_{2}^{2} + 9$$ T2^16 - 11*T2^14 + 85*T2^12 - 334*T2^10 + 952*T2^8 - 1050*T2^6 + 853*T2^4 - 93*T2^2 + 9 $$T_{3}^{8} - 2T_{3}^{7} + 11T_{3}^{6} - 6T_{3}^{5} + 67T_{3}^{4} - 62T_{3}^{3} + 114T_{3}^{2} + 20T_{3} + 4$$ T3^8 - 2*T3^7 + 11*T3^6 - 6*T3^5 + 67*T3^4 - 62*T3^3 + 114*T3^2 + 20*T3 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 11 T^{14} + 85 T^{12} - 334 T^{10} + \cdots + 9$$
$3$ $$(T^{8} - 2 T^{7} + 11 T^{6} - 6 T^{5} + 67 T^{4} + \cdots + 4)^{2}$$
$5$ $$T^{16} - 20 T^{14} + 297 T^{12} + \cdots + 2304$$
$7$ $$T^{16}$$
$11$ $$T^{16} - 52 T^{14} + 2108 T^{12} + \cdots + 729$$
$13$ $$(T^{8} - 6 T^{7} + 28 T^{6} - 130 T^{5} + \cdots + 28561)^{2}$$
$17$ $$(T^{8} + 4 T^{7} + 36 T^{6} + 24 T^{5} + \cdots + 15129)^{2}$$
$19$ $$T^{16} - 44 T^{14} + 1396 T^{12} + \cdots + 10673289$$
$23$ $$(T^{8} + 6 T^{7} + 31 T^{6} + 50 T^{5} + \cdots + 36)^{2}$$
$29$ $$(T^{4} + 4 T^{3} - 63 T^{2} - 208 T + 624)^{4}$$
$31$ $$T^{16} - 80 T^{14} + \cdots + 1136229264$$
$37$ $$T^{16} - 120 T^{14} + \cdots + 76527504$$
$41$ $$(T^{8} + 132 T^{6} + 5732 T^{4} + \cdots + 292032)^{2}$$
$43$ $$(T^{4} - 4 T^{3} - 66 T^{2} - 156 T - 104)^{4}$$
$47$ $$T^{16} - 196 T^{14} + \cdots + 57728231289$$
$53$ $$(T^{8} + 10 T^{7} + 100 T^{6} + 260 T^{5} + \cdots + 7569)^{2}$$
$59$ $$T^{16} - 188 T^{14} + \cdots + 12487392009$$
$61$ $$(T^{8} - 6 T^{7} + 60 T^{6} - 44 T^{5} + \cdots + 49729)^{2}$$
$67$ $$T^{16} - 284 T^{14} + \cdots + 66330457209$$
$71$ $$(T^{8} + 292 T^{6} + 19829 T^{4} + \cdots + 397488)^{2}$$
$73$ $$T^{16} - 260 T^{14} + \cdots + 8437677133824$$
$79$ $$(T^{8} - 10 T^{7} + 91 T^{6} - 210 T^{5} + \cdots + 64)^{2}$$
$83$ $$(T^{8} + 296 T^{6} + 28692 T^{4} + \cdots + 5483712)^{2}$$
$89$ $$T^{16} - 440 T^{14} + \cdots + 58102628210064$$
$97$ $$(T^{8} + 104 T^{6} + 2740 T^{4} + \cdots + 192)^{2}$$