Properties

 Label 6003.2.a.n Level $6003$ Weight $2$ Character orbit 6003.a Self dual yes Analytic conductor $47.934$ Analytic rank $0$ Dimension $12$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6003 = 3^{2} \cdot 23 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6003.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$47.9341963334$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 667) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 54 x^{8} - 188 x^{7} - 77 x^{6} + 342 x^{5} + 13 x^{4} - 215 x^{3} + 9 x^{2} + 37 x - 5$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$69 \nu^{11} - 2268 \nu^{10} + 4126 \nu^{9} + 30343 \nu^{8} - 62480 \nu^{7} - 131017 \nu^{6} + 277030 \nu^{5} + 194815 \nu^{4} - 433403 \nu^{3} - 31183 \nu^{2} + 158295 \nu - 20718$$$$)/6469$$ $$\beta_{3}$$ $$=$$ $$($$$$-127 \nu^{11} - 607 \nu^{10} + 4875 \nu^{9} + 7810 \nu^{8} - 51132 \nu^{7} - 30082 \nu^{6} + 209382 \nu^{5} + 24036 \nu^{4} - 328362 \nu^{3} + 49707 \nu^{2} + 117880 \nu - 29932$$$$)/6469$$ $$\beta_{4}$$ $$=$$ $$($$$$-353 \nu^{11} - 210 \nu^{10} + 8049 \nu^{9} + 773 \nu^{8} - 63527 \nu^{7} + 13625 \nu^{6} + 210856 \nu^{5} - 84627 \nu^{4} - 276488 \nu^{3} + 135717 \nu^{2} + 84019 \nu - 25638$$$$)/6469$$ $$\beta_{5}$$ $$=$$ $$($$$$549 \nu^{11} - 1451 \nu^{10} - 8798 \nu^{9} + 21760 \nu^{8} + 52179 \nu^{7} - 114560 \nu^{6} - 143208 \nu^{5} + 255406 \nu^{4} + 177916 \nu^{3} - 216607 \nu^{2} - 75949 \nu + 34852$$$$)/6469$$ $$\beta_{6}$$ $$=$$ $$($$$$-874 \nu^{11} + 2852 \nu^{10} + 10271 \nu^{9} - 37175 \nu^{8} - 32306 \nu^{7} + 154428 \nu^{6} + 1464 \nu^{5} - 218601 \nu^{4} + 86000 \nu^{3} + 34877 \nu^{2} - 19087 \nu + 23075$$$$)/6469$$ $$\beta_{7}$$ $$=$$ $$($$$$1153 \nu^{11} - 2741 \nu^{10} - 16651 \nu^{9} + 36674 \nu^{8} + 84275 \nu^{7} - 163016 \nu^{6} - 182412 \nu^{5} + 282647 \nu^{4} + 156809 \nu^{3} - 153652 \nu^{2} - 43159 \nu + 10438$$$$)/6469$$ $$\beta_{8}$$ $$=$$ $$($$$$-1227 \nu^{11} + 2642 \nu^{10} + 18320 \nu^{9} - 36402 \nu^{8} - 95833 \nu^{7} + 168053 \nu^{6} + 212320 \nu^{5} - 303228 \nu^{4} - 190488 \nu^{3} + 177063 \nu^{2} + 64932 \nu - 21970$$$$)/6469$$ $$\beta_{9}$$ $$=$$ $$($$$$1231 \nu^{11} - 3336 \nu^{10} - 16487 \nu^{9} + 44255 \nu^{8} + 73554 \nu^{7} - 190180 \nu^{6} - 130539 \nu^{5} + 291927 \nu^{4} + 89329 \nu^{3} - 89336 \nu^{2} - 43380 \nu - 10451$$$$)/6469$$ $$\beta_{10}$$ $$=$$ $$($$$$2257 \nu^{11} - 6684 \nu^{10} - 28263 \nu^{9} + 88739 \nu^{8} + 106697 \nu^{7} - 383278 \nu^{6} - 103569 \nu^{5} + 605079 \nu^{4} - 88693 \nu^{3} - 238564 \nu^{2} + 63686 \nu + 8869$$$$)/6469$$ $$\beta_{11}$$ $$=$$ $$($$$$-2687 \nu^{11} + 10130 \nu^{10} + 28520 \nu^{9} - 134015 \nu^{8} - 63186 \nu^{7} + 578643 \nu^{6} - 126167 \nu^{5} - 926609 \nu^{4} + 441457 \nu^{3} + 391735 \nu^{2} - 161659 \nu - 20106$$$$)/6469$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{8} - \beta_{6} - \beta_{4} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{10} - \beta_{9} + 7 \beta_{8} - \beta_{7} - 7 \beta_{6} + \beta_{5} - 6 \beta_{4} - \beta_{2} + 15$$ $$\nu^{5}$$ $$=$$ $$\beta_{11} + 2 \beta_{10} + 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 10 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} - 9 \beta_{2} + 30 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$\beta_{11} + 11 \beta_{10} - 8 \beta_{9} + 47 \beta_{8} - 10 \beta_{7} - 45 \beta_{6} + 11 \beta_{5} - 35 \beta_{4} + \beta_{3} - 10 \beta_{2} + 2 \beta_{1} + 86$$ $$\nu^{7}$$ $$=$$ $$10 \beta_{11} + 21 \beta_{10} + 3 \beta_{9} + 24 \beta_{8} + 20 \beta_{7} + 22 \beta_{6} + 78 \beta_{5} + 76 \beta_{4} + 78 \beta_{3} - 68 \beta_{2} + 192 \beta_{1} + 2$$ $$\nu^{8}$$ $$=$$ $$12 \beta_{11} + 90 \beta_{10} - 50 \beta_{9} + 314 \beta_{8} - 77 \beta_{7} - 283 \beta_{6} + 94 \beta_{5} - 209 \beta_{4} + 19 \beta_{3} - 81 \beta_{2} + 32 \beta_{1} + 527$$ $$\nu^{9}$$ $$=$$ $$74 \beta_{11} + 167 \beta_{10} + 43 \beta_{9} + 214 \beta_{8} + 148 \beta_{7} + 178 \beta_{6} + 559 \beta_{5} + 524 \beta_{4} + 565 \beta_{3} - 488 \beta_{2} + 1261 \beta_{1} + 40$$ $$\nu^{10}$$ $$=$$ $$99 \beta_{11} + 661 \beta_{10} - 284 \beta_{9} + 2098 \beta_{8} - 546 \beta_{7} - 1773 \beta_{6} + 724 \beta_{5} - 1281 \beta_{4} + 227 \beta_{3} - 622 \beta_{2} + 351 \beta_{1} + 3354$$ $$\nu^{11}$$ $$=$$ $$491 \beta_{11} + 1212 \beta_{10} + 431 \beta_{9} + 1715 \beta_{8} + 980 \beta_{7} + 1286 \beta_{6} + 3859 \beta_{5} + 3450 \beta_{4} + 3981 \beta_{3} - 3436 \beta_{2} + 8380 \beta_{1} + 515$$

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
 −2.52122 −1.89304 −1.65670 −0.724122 −0.613795 0.147789 0.431373 1.08419 1.49364 1.99588 2.58646 2.66955
−2.52122 0 4.35654 3.14716 0 −3.79117 −5.94135 0 −7.93467
1.2 −1.89304 0 1.58359 4.13664 0 2.17357 0.788273 0 −7.83081
1.3 −1.65670 0 0.744648 −1.18716 0 −3.31827 2.07974 0 1.96676
1.4 −0.724122 0 −1.47565 1.17886 0 −4.09140 2.51679 0 −0.853636
1.5 −0.613795 0 −1.62326 0.782514 0 3.32687 2.22394 0 −0.480303
1.6 0.147789 0 −1.97816 −0.429801 0 0.404424 −0.587929 0 −0.0635200
1.7 0.431373 0 −1.81392 3.65141 0 2.43060 −1.64522 0 1.57512
1.8 1.08419 0 −0.824542 −0.786014 0 −4.63453 −3.06233 0 −0.852185
1.9 1.49364 0 0.230961 1.29771 0 1.16738 −2.64231 0 1.93831
1.10 1.99588 0 1.98352 −2.38535 0 −0.101071 −0.0328940 0 −4.76086
1.11 2.58646 0 4.68978 3.69848 0 −0.298143 6.95702 0 9.56597
1.12 2.66955 0 5.12647 2.89555 0 −0.268248 8.34627 0 7.72982
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 6003.2.a.n 12
3.b odd 2 1 667.2.a.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
667.2.a.b 12 3.b odd 2 1
6003.2.a.n 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6003))$$:

 $$T_{2}^{12} - \cdots$$ $$T_{5}^{12} - \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-5 + 37 T + 9 T^{2} - 215 T^{3} + 13 T^{4} + 342 T^{5} - 77 T^{6} - 188 T^{7} + 54 T^{8} + 41 T^{9} - 13 T^{10} - 3 T^{11} + T^{12}$$
$3$ $$T^{12}$$
$5$ $$583 + 300 T - 3214 T^{2} + 660 T^{3} + 4808 T^{4} - 2991 T^{5} - 1627 T^{6} + 1943 T^{7} - 396 T^{8} - 162 T^{9} + 91 T^{10} - 16 T^{11} + T^{12}$$
$7$ $$-16 - 216 T - 413 T^{2} + 1988 T^{3} + 3679 T^{4} - 5425 T^{5} - 741 T^{6} + 1631 T^{7} + 125 T^{8} - 182 T^{9} - 19 T^{10} + 7 T^{11} + T^{12}$$
$11$ $$266075 - 1063507 T + 1146714 T^{2} - 279150 T^{3} - 213703 T^{4} + 115816 T^{5} + 4140 T^{6} - 11450 T^{7} + 1046 T^{8} + 443 T^{9} - 63 T^{10} - 6 T^{11} + T^{12}$$
$13$ $$3090655 + 6460857 T + 2072413 T^{2} - 1602192 T^{3} - 915400 T^{4} + 42364 T^{5} + 102285 T^{6} + 12082 T^{7} - 3802 T^{8} - 872 T^{9} + 10 T^{10} + 15 T^{11} + T^{12}$$
$17$ $$-9596 - 31240 T + 67893 T^{2} + 103277 T^{3} - 91550 T^{4} - 39627 T^{5} + 31076 T^{6} + 1682 T^{7} - 3484 T^{8} + 436 T^{9} + 71 T^{10} - 18 T^{11} + T^{12}$$
$19$ $$-149440 + 1324256 T + 1241085 T^{2} - 870280 T^{3} - 1308005 T^{4} - 400570 T^{5} + 43638 T^{6} + 38666 T^{7} + 3160 T^{8} - 897 T^{9} - 119 T^{10} + 6 T^{11} + T^{12}$$
$23$ $$( -1 + T )^{12}$$
$29$ $$( -1 + T )^{12}$$
$31$ $$169 - 29220 T - 13018 T^{2} + 97636 T^{3} - 8705 T^{4} - 90550 T^{5} + 52027 T^{6} - 4023 T^{7} - 3333 T^{8} + 703 T^{9} + 25 T^{10} - 16 T^{11} + T^{12}$$
$37$ $$-362875340 + 492689776 T - 132885771 T^{2} - 67960955 T^{3} + 29813443 T^{4} + 2097862 T^{5} - 1538543 T^{6} - 12921 T^{7} + 32317 T^{8} - 139 T^{9} - 300 T^{10} + T^{11} + T^{12}$$
$41$ $$-27754352 - 39955432 T + 95038325 T^{2} - 51050139 T^{3} + 5620365 T^{4} + 2820556 T^{5} - 688351 T^{6} - 36992 T^{7} + 19334 T^{8} - 100 T^{9} - 225 T^{10} + 3 T^{11} + T^{12}$$
$43$ $$-229352509 - 456409123 T - 332516092 T^{2} - 98139927 T^{3} + 120819 T^{4} + 7019312 T^{5} + 1432697 T^{6} - 4348 T^{7} - 34372 T^{8} - 3715 T^{9} + 18 T^{10} + 23 T^{11} + T^{12}$$
$47$ $$-40804117 + 3930358 T + 195233100 T^{2} + 139288371 T^{3} - 32908165 T^{4} - 14545371 T^{5} + 4020829 T^{6} - 22788 T^{7} - 68952 T^{8} + 5007 T^{9} + 222 T^{10} - 35 T^{11} + T^{12}$$
$53$ $$-142956229 + 192812894 T + 103125523 T^{2} - 155347564 T^{3} + 43362431 T^{4} + 4447789 T^{5} - 4586185 T^{6} + 947515 T^{7} - 74630 T^{8} - 1491 T^{9} + 683 T^{10} - 45 T^{11} + T^{12}$$
$59$ $$8891929600 + 5050998336 T - 1293585763 T^{2} - 690028123 T^{3} + 110629045 T^{4} + 27279008 T^{5} - 3722991 T^{6} - 472329 T^{7} + 56094 T^{8} + 3752 T^{9} - 386 T^{10} - 11 T^{11} + T^{12}$$
$61$ $$-18119916 + 27947196 T + 5427069 T^{2} - 22849850 T^{3} + 6151210 T^{4} + 2446190 T^{5} - 674995 T^{6} - 90746 T^{7} + 23370 T^{8} + 1218 T^{9} - 288 T^{10} - 4 T^{11} + T^{12}$$
$67$ $$-409376228 + 2093598060 T + 1996739575 T^{2} + 354520568 T^{3} - 86438366 T^{4} - 25317130 T^{5} + 841992 T^{6} + 575866 T^{7} + 10401 T^{8} - 5505 T^{9} - 222 T^{10} + 19 T^{11} + T^{12}$$
$71$ $$-226642180 - 276404288 T + 107691351 T^{2} + 184276655 T^{3} + 12049787 T^{4} - 17462558 T^{5} - 1037483 T^{6} + 560972 T^{7} + 24172 T^{8} - 6170 T^{9} - 295 T^{10} + 19 T^{11} + T^{12}$$
$73$ $$2131153580 - 2044166852 T - 522062685 T^{2} + 468676890 T^{3} + 138848386 T^{4} - 8381771 T^{5} - 5196543 T^{6} - 113916 T^{7} + 68014 T^{8} + 2423 T^{9} - 406 T^{10} - 10 T^{11} + T^{12}$$
$79$ $$-36136523 + 19569650 T + 115383457 T^{2} + 8905014 T^{3} - 42217641 T^{4} + 6564553 T^{5} + 1741621 T^{6} - 410928 T^{7} - 2683 T^{8} + 5302 T^{9} - 222 T^{10} - 17 T^{11} + T^{12}$$
$83$ $$14844146512 + 25209439576 T + 16164213229 T^{2} + 4686902763 T^{3} + 446283618 T^{4} - 68049836 T^{5} - 16696731 T^{6} - 220621 T^{7} + 153191 T^{8} + 4819 T^{9} - 638 T^{10} - 12 T^{11} + T^{12}$$
$89$ $$-12272416784 + 453908952 T + 5874747827 T^{2} - 1422097864 T^{3} - 227836659 T^{4} + 76569835 T^{5} + 1332308 T^{6} - 1362447 T^{7} + 35172 T^{8} + 9078 T^{9} - 384 T^{10} - 20 T^{11} + T^{12}$$
$97$ $$-6155578420 - 3227158204 T + 1049161983 T^{2} + 559443202 T^{3} - 42211711 T^{4} - 31011322 T^{5} - 422404 T^{6} + 648971 T^{7} + 33165 T^{8} - 5224 T^{9} - 368 T^{10} + 12 T^{11} + T^{12}$$