# Properties

 Label 2001.2.a.m Level $2001$ Weight $2$ Character orbit 2001.a Self dual yes Analytic conductor $15.978$ Analytic rank $1$ Dimension $14$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2001 = 3 \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2001.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.9780654445$$ Analytic rank: $$1$$ Dimension: $$14$$ Coefficient field: $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ Defining polynomial: $$x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} - 939 x^{5} - 717 x^{4} + 604 x^{3} + 352 x^{2} - 128 x - 64$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{13}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{11} q^{5} + \beta_{1} q^{6} + ( \beta_{3} - \beta_{11} - \beta_{13} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + \beta_{11} q^{5} + \beta_{1} q^{6} + ( \beta_{3} - \beta_{11} - \beta_{13} ) q^{7} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{8} + q^{9} + ( -1 + \beta_{10} - \beta_{11} ) q^{10} + ( -1 - \beta_{6} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( 1 - \beta_{8} ) q^{13} + ( \beta_{1} - 2 \beta_{2} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{14} -\beta_{11} q^{15} + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{16} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{11} + \beta_{13} ) q^{17} -\beta_{1} q^{18} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{19} + ( 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{20} + ( -\beta_{3} + \beta_{11} + \beta_{13} ) q^{21} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} + \beta_{13} ) q^{22} - q^{23} + ( 1 + \beta_{1} + \beta_{3} + \beta_{4} ) q^{24} + ( 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{13} ) q^{25} + ( -1 - 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{7} - \beta_{9} - \beta_{13} ) q^{26} - q^{27} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{10} - \beta_{11} ) q^{28} - q^{29} + ( 1 - \beta_{10} + \beta_{11} ) q^{30} + ( -3 - \beta_{3} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{31} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{13} ) q^{32} + ( 1 + \beta_{6} ) q^{33} + ( \beta_{2} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{34} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{13} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{37} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} ) q^{38} + ( -1 + \beta_{8} ) q^{39} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{40} + ( -3 - \beta_{2} - \beta_{4} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{41} + ( -\beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{42} + ( 1 + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{43} + ( -3 + \beta_{1} - 3 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{44} + \beta_{11} q^{45} + \beta_{1} q^{46} + ( -2 - \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} + \beta_{12} ) q^{47} + ( -\beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{48} + ( 2 - \beta_{2} - \beta_{4} - 2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{12} ) q^{49} + ( -3 + 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{13} ) q^{50} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{11} - \beta_{13} ) q^{51} + ( 2 + 3 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{8} - \beta_{13} ) q^{52} + ( -2 + 2 \beta_{1} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{53} + \beta_{1} q^{54} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{55} + ( -5 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{56} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{7} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{57} + \beta_{1} q^{58} + ( -3 + \beta_{1} + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{59} + ( -2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{13} ) q^{60} + ( -2 + \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{7} - \beta_{8} - 2 \beta_{12} - \beta_{13} ) q^{62} + ( \beta_{3} - \beta_{11} - \beta_{13} ) q^{63} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{6} - \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{64} + ( \beta_{3} - \beta_{6} - \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{65} + ( 1 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{8} - \beta_{13} ) q^{66} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{67} + ( -1 + \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{68} + q^{69} + ( 1 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - 3 \beta_{13} ) q^{70} + ( -4 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} ) q^{71} + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} ) q^{72} + ( 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{73} + ( 1 + 2 \beta_{2} + \beta_{3} - \beta_{6} - 2 \beta_{8} - \beta_{13} ) q^{74} + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{13} ) q^{75} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{76} + ( -4 - \beta_{1} + 3 \beta_{2} - 4 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + 2 \beta_{10} + \beta_{13} ) q^{77} + ( 1 + 2 \beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{9} + \beta_{13} ) q^{78} + ( -5 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{8} - \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{79} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{10} + \beta_{12} + \beta_{13} ) q^{80} + q^{81} + ( -1 + 5 \beta_{1} + 3 \beta_{4} + \beta_{5} + \beta_{6} - 3 \beta_{7} - \beta_{9} + 4 \beta_{11} + \beta_{13} ) q^{82} + ( 3 \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{83} + ( -2 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{84} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} + 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{85} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} - \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{86} + q^{87} + ( 7 \beta_{1} - \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} + \beta_{12} + \beta_{13} ) q^{88} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} + 3 \beta_{7} - 2 \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{89} + ( -1 + \beta_{10} - \beta_{11} ) q^{90} + ( -1 + 2 \beta_{1} + 2 \beta_{3} + 2 \beta_{7} + 3 \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{91} + ( -1 - \beta_{2} ) q^{92} + ( 3 + \beta_{3} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{93} + ( 2 + 2 \beta_{1} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 2 \beta_{13} ) q^{94} + ( -5 - \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{95} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{13} ) q^{96} + ( 2 + 3 \beta_{1} + \beta_{2} + \beta_{5} + \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{13} ) q^{97} + ( 1 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{98} + ( -1 - \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$14q - 2q^{2} - 14q^{3} + 12q^{4} - 3q^{5} + 2q^{6} - 3q^{7} - 6q^{8} + 14q^{9} + O(q^{10})$$ $$14q - 2q^{2} - 14q^{3} + 12q^{4} - 3q^{5} + 2q^{6} - 3q^{7} - 6q^{8} + 14q^{9} - 5q^{10} - 12q^{11} - 12q^{12} + 13q^{13} - 9q^{14} + 3q^{15} - 14q^{17} - 2q^{18} - 9q^{19} - 2q^{20} + 3q^{21} - 9q^{22} - 14q^{23} + 6q^{24} + 13q^{25} - 16q^{26} - 14q^{27} + 3q^{28} - 14q^{29} + 5q^{30} - 28q^{31} - 4q^{32} + 12q^{33} + 14q^{34} - 9q^{35} + 12q^{36} - 12q^{37} + 2q^{38} - 13q^{39} - 20q^{40} - 25q^{41} + 9q^{42} + 5q^{43} - 37q^{44} - 3q^{45} + 2q^{46} - 17q^{47} + 17q^{49} - 44q^{50} + 14q^{51} + 25q^{52} - 17q^{53} + 2q^{54} + q^{55} - 54q^{56} + 9q^{57} + 2q^{58} - 18q^{59} + 2q^{60} - 13q^{61} - 8q^{62} - 3q^{63} + 20q^{64} - 16q^{65} + 9q^{66} + 2q^{67} - 19q^{68} + 14q^{69} + 14q^{70} - 55q^{71} - 6q^{72} + 19q^{73} + 4q^{74} - 13q^{75} - 32q^{76} - 19q^{77} + 16q^{78} - 68q^{79} - 2q^{80} + 14q^{81} - 12q^{82} - 21q^{83} - 3q^{84} + 16q^{85} - 22q^{86} + 14q^{87} - 25q^{88} - 17q^{89} - 5q^{90} - 30q^{91} - 12q^{92} + 28q^{93} + 16q^{94} - 55q^{95} + 4q^{96} + 25q^{97} - 31q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{14} - 2 x^{13} - 18 x^{12} + 34 x^{11} + 124 x^{10} - 216 x^{9} - 420 x^{8} + 647 x^{7} + 750 x^{6} - 939 x^{5} - 717 x^{4} + 604 x^{3} + 352 x^{2} - 128 x - 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{13} - 2 \nu^{12} - 18 \nu^{11} + 34 \nu^{10} + 124 \nu^{9} - 216 \nu^{8} - 420 \nu^{7} + 647 \nu^{6} + 750 \nu^{5} - 939 \nu^{4} - 701 \nu^{3} + 604 \nu^{2} + 256 \nu - 128$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{13} + 2 \nu^{12} + 18 \nu^{11} - 34 \nu^{10} - 124 \nu^{9} + 216 \nu^{8} + 420 \nu^{7} - 647 \nu^{6} - 750 \nu^{5} + 939 \nu^{4} + 717 \nu^{3} - 604 \nu^{2} - 336 \nu + 112$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-21 \nu^{13} + 142 \nu^{12} + 210 \nu^{11} - 2450 \nu^{10} + 108 \nu^{9} + 15640 \nu^{8} - 7004 \nu^{7} - 46019 \nu^{6} + 25206 \nu^{5} + 63135 \nu^{4} - 30667 \nu^{3} - 37648 \nu^{2} + 10560 \nu + 8496$$$$)/368$$ $$\beta_{6}$$ $$=$$ $$($$$$11 \nu^{13} - 47 \nu^{12} - 156 \nu^{11} + 808 \nu^{10} + 686 \nu^{9} - 5152 \nu^{8} - 664 \nu^{7} + 15225 \nu^{6} - 1989 \nu^{5} - 21091 \nu^{4} + 3544 \nu^{3} + 12241 \nu^{2} - 800 \nu - 2124$$$$)/92$$ $$\beta_{7}$$ $$=$$ $$($$$$-14 \nu^{13} - 5 \nu^{12} + 278 \nu^{11} + 130 \nu^{10} - 2090 \nu^{9} - 1196 \nu^{8} + 7444 \nu^{7} + 4986 \nu^{6} - 12843 \nu^{5} - 9660 \nu^{4} + 9877 \nu^{3} + 7661 \nu^{2} - 2620 \nu - 1604$$$$)/92$$ $$\beta_{8}$$ $$=$$ $$($$$$49 \nu^{13} - 86 \nu^{12} - 858 \nu^{11} + 1362 \nu^{10} + 5636 \nu^{9} - 7728 \nu^{8} - 17452 \nu^{7} + 19119 \nu^{6} + 25826 \nu^{5} - 19251 \nu^{4} - 16273 \nu^{3} + 5168 \nu^{2} + 3144 \nu + 48$$$$)/184$$ $$\beta_{9}$$ $$=$$ $$($$$$97 \nu^{13} - 266 \nu^{12} - 1522 \nu^{11} + 4386 \nu^{10} + 8412 \nu^{9} - 26680 \nu^{8} - 19396 \nu^{7} + 74967 \nu^{6} + 15014 \nu^{5} - 97451 \nu^{4} + 2763 \nu^{3} + 49860 \nu^{2} - 3040 \nu - 6544$$$$)/368$$ $$\beta_{10}$$ $$=$$ $$($$$$10 \nu^{13} - 26 \nu^{12} - 169 \nu^{11} + 446 \nu^{10} + 1046 \nu^{9} - 2852 \nu^{8} - 2912 \nu^{7} + 8530 \nu^{6} + 3486 \nu^{5} - 12121 \nu^{4} - 1282 \nu^{3} + 7283 \nu^{2} - 31 \nu - 1266$$$$)/46$$ $$\beta_{11}$$ $$=$$ $$($$$$12 \nu^{13} - 22 \nu^{12} - 212 \nu^{11} + 365 \nu^{10} + 1407 \nu^{9} - 2231 \nu^{8} - 4419 \nu^{7} + 6257 \nu^{6} + 6727 \nu^{5} - 8027 \nu^{4} - 4510 \nu^{3} + 4020 \nu^{2} + 961 \nu - 544$$$$)/46$$ $$\beta_{12}$$ $$=$$ $$($$$$157 \nu^{13} - 146 \nu^{12} - 2858 \nu^{11} + 2002 \nu^{10} + 19748 \nu^{9} - 8832 \nu^{8} - 65388 \nu^{7} + 12435 \nu^{6} + 106310 \nu^{5} + 4761 \nu^{4} - 77057 \nu^{3} - 17164 \nu^{2} + 18072 \nu + 5088$$$$)/368$$ $$\beta_{13}$$ $$=$$ $$($$$$-177 \nu^{13} + 198 \nu^{12} + 3242 \nu^{11} - 2986 \nu^{10} - 22484 \nu^{9} + 15640 \nu^{8} + 74340 \nu^{7} - 33175 \nu^{6} - 119906 \nu^{5} + 21091 \nu^{4} + 85233 \nu^{3} + 8992 \nu^{2} - 18608 \nu - 6144$$$$)/368$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{4} + 7 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$\beta_{13} + \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 9 \beta_{4} + 9 \beta_{3} + 30 \beta_{1} + 8$$ $$\nu^{6}$$ $$=$$ $$\beta_{13} + \beta_{12} + \beta_{11} - \beta_{9} + 11 \beta_{6} + 10 \beta_{5} + 11 \beta_{4} + 45 \beta_{2} + 12 \beta_{1} + 78$$ $$\nu^{7}$$ $$=$$ $$14 \beta_{13} + \beta_{12} + 3 \beta_{11} - \beta_{9} + 13 \beta_{8} - 12 \beta_{7} + 13 \beta_{6} + 13 \beta_{5} + 70 \beta_{4} + 65 \beta_{3} + \beta_{2} + 194 \beta_{1} + 58$$ $$\nu^{8}$$ $$=$$ $$16 \beta_{13} + 13 \beta_{12} + 17 \beta_{11} - 2 \beta_{10} - 14 \beta_{9} + 3 \beta_{8} - 2 \beta_{7} + 94 \beta_{6} + 80 \beta_{5} + 96 \beta_{4} + \beta_{3} + 289 \beta_{2} + 112 \beta_{1} + 475$$ $$\nu^{9}$$ $$=$$ $$137 \beta_{13} + 16 \beta_{12} + 50 \beta_{11} - 4 \beta_{10} - 16 \beta_{9} + 120 \beta_{8} - 105 \beta_{7} + 124 \beta_{6} + 122 \beta_{5} + 520 \beta_{4} + 445 \beta_{3} + 17 \beta_{2} + 1299 \beta_{1} + 418$$ $$\nu^{10}$$ $$=$$ $$177 \beta_{13} + 121 \beta_{12} + 195 \beta_{11} - 34 \beta_{10} - 136 \beta_{9} + 51 \beta_{8} - 38 \beta_{7} + 736 \beta_{6} + 601 \beta_{5} + 777 \beta_{4} + 22 \beta_{3} + 1877 \beta_{2} + 953 \beta_{1} + 3043$$ $$\nu^{11}$$ $$=$$ $$1168 \beta_{13} + 174 \beta_{12} + 558 \beta_{11} - 74 \beta_{10} - 172 \beta_{9} + 970 \beta_{8} - 826 \beta_{7} + 1057 \beta_{6} + 1015 \beta_{5} + 3789 \beta_{4} + 3004 \beta_{3} + 204 \beta_{2} + 8860 \beta_{1} + 3035$$ $$\nu^{12}$$ $$=$$ $$1673 \beta_{13} + 998 \beta_{12} + 1878 \beta_{11} - 384 \beta_{10} - 1144 \beta_{9} + 575 \beta_{8} - 467 \beta_{7} + 5533 \beta_{6} + 4407 \beta_{5} + 6070 \beta_{4} + 306 \beta_{3} + 12339 \beta_{2} + 7743 \beta_{1} + 20110$$ $$\nu^{13}$$ $$=$$ $$9303 \beta_{13} + 1611 \beta_{12} + 5255 \beta_{11} - 880 \beta_{10} - 1573 \beta_{9} + 7354 \beta_{8} - 6212 \beta_{7} + 8528 \beta_{6} + 7981 \beta_{5} + 27353 \beta_{4} + 20239 \beta_{3} + 2122 \beta_{2} + 61084 \beta_{1} + 22213$$

## 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
 2.68441 2.31598 2.05175 1.57408 1.35292 1.14192 0.630703 −0.429717 −0.656871 −0.894954 −1.37642 −1.86019 −1.98240 −2.55124
−2.68441 −1.00000 5.20606 1.06146 2.68441 1.62145 −8.60637 1.00000 −2.84940
1.2 −2.31598 −1.00000 3.36376 −2.86683 2.31598 3.60131 −3.15843 1.00000 6.63952
1.3 −2.05175 −1.00000 2.20969 3.75134 2.05175 −2.35999 −0.430233 1.00000 −7.69683
1.4 −1.57408 −1.00000 0.477741 0.564555 1.57408 −3.58089 2.39616 1.00000 −0.888657
1.5 −1.35292 −1.00000 −0.169600 −0.738177 1.35292 3.44152 2.93530 1.00000 0.998696
1.6 −1.14192 −1.00000 −0.696014 −4.45694 1.14192 −4.52974 3.07864 1.00000 5.08948
1.7 −0.630703 −1.00000 −1.60221 2.56371 0.630703 2.03354 2.27193 1.00000 −1.61694
1.8 0.429717 −1.00000 −1.81534 −3.93237 −0.429717 0.264416 −1.63952 1.00000 −1.68981
1.9 0.656871 −1.00000 −1.56852 3.35329 −0.656871 −2.47996 −2.34406 1.00000 2.20268
1.10 0.894954 −1.00000 −1.19906 0.474904 −0.894954 −1.99579 −2.86301 1.00000 0.425017
1.11 1.37642 −1.00000 −0.105481 −1.48313 −1.37642 4.92788 −2.89802 1.00000 −2.04140
1.12 1.86019 −1.00000 1.46029 0.608499 −1.86019 0.188796 −1.00396 1.00000 1.13192
1.13 1.98240 −1.00000 1.92989 −0.252916 −1.98240 −1.21499 −0.138982 1.00000 −0.501379
1.14 2.55124 −1.00000 4.50880 −1.64740 −2.55124 −2.91756 6.40055 1.00000 −4.20290
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$23$$ $$1$$
$$29$$ $$1$$

## 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 2001.2.a.m 14
3.b odd 2 1 6003.2.a.p 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.m 14 1.a even 1 1 trivial
6003.2.a.p 14 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2001))$$:

 $$T_{2}^{14} + \cdots$$ $$T_{5}^{14} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-64 + 128 T + 352 T^{2} - 604 T^{3} - 717 T^{4} + 939 T^{5} + 750 T^{6} - 647 T^{7} - 420 T^{8} + 216 T^{9} + 124 T^{10} - 34 T^{11} - 18 T^{12} + 2 T^{13} + T^{14}$$
$3$ $$( 1 + T )^{14}$$
$5$ $$-128 + 1920 T^{2} - 1904 T^{3} - 5256 T^{4} + 5532 T^{5} + 5276 T^{6} - 4192 T^{7} - 2540 T^{8} + 1025 T^{9} + 479 T^{10} - 96 T^{11} - 37 T^{12} + 3 T^{13} + T^{14}$$
$7$ $$-6752 + 53776 T - 63248 T^{2} - 178220 T^{3} + 18524 T^{4} + 121268 T^{5} + 15326 T^{6} - 31555 T^{7} - 7019 T^{8} + 3544 T^{9} + 963 T^{10} - 174 T^{11} - 53 T^{12} + 3 T^{13} + T^{14}$$
$11$ $$120704 + 181568 T - 890912 T^{2} - 401680 T^{3} + 782488 T^{4} + 343124 T^{5} - 205010 T^{6} - 93537 T^{7} + 21560 T^{8} + 11032 T^{9} - 744 T^{10} - 595 T^{11} - 15 T^{12} + 12 T^{13} + T^{14}$$
$13$ $$18424 - 347116 T + 1331158 T^{2} - 1839493 T^{3} + 675823 T^{4} + 487962 T^{5} - 382381 T^{6} + 2455 T^{7} + 52858 T^{8} - 9081 T^{9} - 2310 T^{10} + 710 T^{11} - 5 T^{12} - 13 T^{13} + T^{14}$$
$17$ $$740744 - 1274668 T - 2234852 T^{2} + 2561397 T^{3} + 3005317 T^{4} - 710136 T^{5} - 1103956 T^{6} - 56824 T^{7} + 135533 T^{8} + 26942 T^{9} - 3253 T^{10} - 1205 T^{11} - 30 T^{12} + 14 T^{13} + T^{14}$$
$19$ $$1130464 + 5290592 T - 1958192 T^{2} - 6597012 T^{3} + 742574 T^{4} + 2671997 T^{5} + 35017 T^{6} - 402156 T^{7} - 23996 T^{8} + 26954 T^{9} + 2278 T^{10} - 816 T^{11} - 82 T^{12} + 9 T^{13} + T^{14}$$
$23$ $$( 1 + T )^{14}$$
$29$ $$( 1 + T )^{14}$$
$31$ $$36610048 + 400362496 T + 592167680 T^{2} + 166081872 T^{3} - 125603560 T^{4} - 73057332 T^{5} - 1740456 T^{6} + 6015916 T^{7} + 1173584 T^{8} - 55935 T^{9} - 37106 T^{10} - 2957 T^{11} + 138 T^{12} + 28 T^{13} + T^{14}$$
$37$ $$5876204288 + 3235907776 T - 2000264224 T^{2} - 1168834916 T^{3} + 207444226 T^{4} + 149459809 T^{5} - 5535626 T^{6} - 8579857 T^{7} - 213614 T^{8} + 234899 T^{9} + 13299 T^{10} - 2887 T^{11} - 214 T^{12} + 12 T^{13} + T^{14}$$
$41$ $$245620864 - 272994880 T - 2843838976 T^{2} + 1988999712 T^{3} + 1172900360 T^{4} - 68600324 T^{5} - 89892996 T^{6} - 4857228 T^{7} + 2551756 T^{8} + 283445 T^{9} - 25769 T^{10} - 4718 T^{11} - 11 T^{12} + 25 T^{13} + T^{14}$$
$43$ $$11926919072 - 25875240400 T + 261659904 T^{2} + 5468214576 T^{3} - 585256476 T^{4} - 388084755 T^{5} + 55739017 T^{6} + 12380704 T^{7} - 2043301 T^{8} - 198428 T^{9} + 35944 T^{10} + 1576 T^{11} - 305 T^{12} - 5 T^{13} + T^{14}$$
$47$ $$-24729856 + 112177920 T + 393964464 T^{2} + 14983348 T^{3} - 444364160 T^{4} - 330856560 T^{5} - 87672874 T^{6} - 3282579 T^{7} + 2585394 T^{8} + 390878 T^{9} - 7729 T^{10} - 4878 T^{11} - 182 T^{12} + 17 T^{13} + T^{14}$$
$53$ $$-371147830784 + 156979174144 T + 52573556992 T^{2} - 25632509568 T^{3} - 2696664608 T^{4} + 1527503520 T^{5} + 84970400 T^{6} - 45022912 T^{7} - 2036968 T^{8} + 702733 T^{9} + 33171 T^{10} - 5514 T^{11} - 291 T^{12} + 17 T^{13} + T^{14}$$
$59$ $$-513236992 - 1862324224 T - 1457953280 T^{2} + 320884608 T^{3} + 647056512 T^{4} + 126929856 T^{5} - 41312160 T^{6} - 12566720 T^{7} + 651612 T^{8} + 379288 T^{9} + 5739 T^{10} - 4474 T^{11} - 188 T^{12} + 18 T^{13} + T^{14}$$
$61$ $$268644352 - 651613184 T - 1054913536 T^{2} - 61775936 T^{3} + 343496096 T^{4} + 86018896 T^{5} - 29924320 T^{6} - 10993900 T^{7} + 291418 T^{8} + 362255 T^{9} + 15877 T^{10} - 3875 T^{11} - 258 T^{12} + 13 T^{13} + T^{14}$$
$67$ $$1310603264 + 687428608 T - 11199615744 T^{2} + 5061332736 T^{3} + 2490687928 T^{4} - 1614401668 T^{5} + 136017912 T^{6} + 48225303 T^{7} - 7079611 T^{8} - 461900 T^{9} + 96278 T^{10} + 1657 T^{11} - 526 T^{12} - 2 T^{13} + T^{14}$$
$71$ $$17005748224 + 77439547392 T + 66548029824 T^{2} - 22147992112 T^{3} - 16636703308 T^{4} - 308105621 T^{5} + 1127635307 T^{6} + 214779001 T^{7} + 5265446 T^{8} - 2374038 T^{9} - 251262 T^{10} - 2777 T^{11} + 912 T^{12} + 55 T^{13} + T^{14}$$
$73$ $$-202777216 + 974110592 T - 883690464 T^{2} - 182685104 T^{3} + 408115808 T^{4} - 56181852 T^{5} - 37662560 T^{6} + 7489892 T^{7} + 1185898 T^{8} - 308901 T^{9} - 7896 T^{10} + 4809 T^{11} - 157 T^{12} - 19 T^{13} + T^{14}$$
$79$ $$-78594928544 + 70215136464 T + 22501329072 T^{2} - 20452459504 T^{3} - 5339845500 T^{4} + 1634781817 T^{5} + 647374780 T^{6} + 27278931 T^{7} - 14517654 T^{8} - 2164225 T^{9} - 25239 T^{10} + 18077 T^{11} + 1748 T^{12} + 68 T^{13} + T^{14}$$
$83$ $$2370202624 + 716367616 T - 6847328960 T^{2} - 5059155504 T^{3} + 470583520 T^{4} + 791022236 T^{5} - 15630516 T^{6} - 37712816 T^{7} - 280014 T^{8} + 767985 T^{9} + 20165 T^{10} - 6860 T^{11} - 273 T^{12} + 21 T^{13} + T^{14}$$
$89$ $$-2726914213064 - 4016991600412 T - 1914872033652 T^{2} - 230062693311 T^{3} + 67324655223 T^{4} + 16966580154 T^{5} - 422572319 T^{6} - 342921181 T^{7} - 8619574 T^{8} + 3023653 T^{9} + 134634 T^{10} - 11912 T^{11} - 641 T^{12} + 17 T^{13} + T^{14}$$
$97$ $$-3664836644864 + 2646312216576 T - 312674126080 T^{2} - 204867713984 T^{3} + 63316120448 T^{4} - 1295892592 T^{5} - 1617446336 T^{6} + 165053500 T^{7} + 12310856 T^{8} - 2382573 T^{9} + 3385 T^{10} + 13047 T^{11} - 360 T^{12} - 25 T^{13} + T^{14}$$
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