# Properties

 Label 4010.2.a.j Level 4010 Weight 2 Character orbit 4010.a Self dual yes Analytic conductor 32.020 Analytic rank 1 Dimension 12 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4010 = 2 \cdot 5 \cdot 401$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4010.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.0200112105$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 2 x^{11} - 16 x^{10} + 30 x^{9} + 93 x^{8} - 162 x^{7} - 238 x^{6} + 391 x^{5} + 240 x^{4} - 408 x^{3} - 42 x^{2} + 120 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$($$$$11 \nu^{11} - 6 \nu^{10} - 190 \nu^{9} + 80 \nu^{8} + 1171 \nu^{7} - 374 \nu^{6} - 3046 \nu^{5} + 809 \nu^{4} + 2952 \nu^{3} - 964 \nu^{2} - 604 \nu + 278$$$$)/58$$ $$\beta_{4}$$ $$=$$ $$($$$$53 \nu^{11} - 108 \nu^{10} - 752 \nu^{9} + 1382 \nu^{8} + 3765 \nu^{7} - 5804 \nu^{6} - 8254 \nu^{5} + 9487 \nu^{4} + 7490 \nu^{3} - 5404 \nu^{2} - 2578 \nu + 596$$$$)/232$$ $$\beta_{5}$$ $$=$$ $$($$$$25 \nu^{11} + 76 \nu^{10} - 648 \nu^{9} - 994 \nu^{8} + 5361 \nu^{7} + 4196 \nu^{6} - 17542 \nu^{5} - 5965 \nu^{4} + 21130 \nu^{3} - 356 \nu^{2} - 5786 \nu + 732$$$$)/232$$ $$\beta_{6}$$ $$=$$ $$($$$$-105 \nu^{11} + 52 \nu^{10} + 1840 \nu^{9} - 558 \nu^{8} - 11705 \nu^{7} + 1540 \nu^{6} + 32334 \nu^{5} - 467 \nu^{4} - 35154 \nu^{3} + 196 \nu^{2} + 10010 \nu - 12$$$$)/232$$ $$\beta_{7}$$ $$=$$ $$($$$$-113 \nu^{11} + 88 \nu^{10} + 1936 \nu^{9} - 1038 \nu^{8} - 12177 \nu^{7} + 3784 \nu^{6} + 34022 \nu^{5} - 5147 \nu^{4} - 39062 \nu^{3} + 3892 \nu^{2} + 13402 \nu + 60$$$$)/232$$ $$\beta_{8}$$ $$=$$ $$($$$$33 \nu^{11} - 47 \nu^{10} - 512 \nu^{9} + 588 \nu^{8} + 2875 \nu^{7} - 2369 \nu^{6} - 7166 \nu^{5} + 3645 \nu^{4} + 7435 \nu^{3} - 2196 \nu^{2} - 2566 \nu + 312$$$$)/58$$ $$\beta_{9}$$ $$=$$ $$($$$$223 \nu^{11} - 148 \nu^{10} - 3952 \nu^{9} + 1954 \nu^{8} + 25511 \nu^{7} - 8684 \nu^{6} - 72138 \nu^{5} + 16253 \nu^{4} + 82734 \nu^{3} - 14924 \nu^{2} - 27446 \nu + 1908$$$$)/232$$ $$\beta_{10}$$ $$=$$ $$($$$$139 \nu^{11} - 176 \nu^{10} - 2248 \nu^{9} + 2250 \nu^{8} + 13247 \nu^{7} - 9424 \nu^{6} - 34578 \nu^{5} + 15601 \nu^{4} + 36886 \nu^{3} - 10684 \nu^{2} - 11666 \nu + 1388$$$$)/116$$ $$\beta_{11}$$ $$=$$ $$($$$$-165 \nu^{11} + 148 \nu^{10} + 2792 \nu^{9} - 1838 \nu^{8} - 17217 \nu^{7} + 7292 \nu^{6} + 46618 \nu^{5} - 11091 \nu^{4} - 50834 \nu^{3} + 8196 \nu^{2} + 15962 \nu - 1328$$$$)/116$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{11} - \beta_{10} + \beta_{6} + \beta_{4} + \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{8} + \beta_{7} + \beta_{5} - \beta_{3} + 6 \beta_{2} + \beta_{1} + 14$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{11} - 8 \beta_{10} - \beta_{8} + \beta_{7} + 7 \beta_{6} + \beta_{5} + 11 \beta_{4} - \beta_{3} + 9 \beta_{2} + 29 \beta_{1} + 10$$ $$\nu^{6}$$ $$=$$ $$-\beta_{10} + \beta_{9} + 10 \beta_{8} + 12 \beta_{7} - 2 \beta_{6} + 10 \beta_{5} + 3 \beta_{4} - 12 \beta_{3} + 37 \beta_{2} + 11 \beta_{1} + 76$$ $$\nu^{7}$$ $$=$$ $$-53 \beta_{11} - 56 \beta_{10} + 2 \beta_{9} - 8 \beta_{8} + 14 \beta_{7} + 41 \beta_{6} + 14 \beta_{5} + 93 \beta_{4} - 16 \beta_{3} + 65 \beta_{2} + 182 \beta_{1} + 77$$ $$\nu^{8}$$ $$=$$ $$-16 \beta_{10} + 15 \beta_{9} + 79 \beta_{8} + 107 \beta_{7} - 26 \beta_{6} + 82 \beta_{5} + 50 \beta_{4} - 106 \beta_{3} + 238 \beta_{2} + 98 \beta_{1} + 449$$ $$\nu^{9}$$ $$=$$ $$-336 \beta_{11} - 384 \beta_{10} + 31 \beta_{9} - 41 \beta_{8} + 141 \beta_{7} + 228 \beta_{6} + 138 \beta_{5} + 718 \beta_{4} - 165 \beta_{3} + 450 \beta_{2} + 1189 \beta_{1} + 550$$ $$\nu^{10}$$ $$=$$ $$-\beta_{11} - 180 \beta_{10} + 155 \beta_{9} + 584 \beta_{8} + 852 \beta_{7} - 245 \beta_{6} + 632 \beta_{5} + 560 \beta_{4} - 838 \beta_{3} + 1574 \beta_{2} + 820 \beta_{1} + 2799$$ $$\nu^{11}$$ $$=$$ $$-2109 \beta_{11} - 2634 \beta_{10} + 332 \beta_{9} - 123 \beta_{8} + 1243 \beta_{7} + 1231 \beta_{6} + 1185 \beta_{5} + 5323 \beta_{4} - 1439 \beta_{3} + 3109 \beta_{2} + 7941 \beta_{1} + 3857$$

## 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
 2.67561 2.34926 2.12920 1.16326 1.11338 0.803872 0.0694414 −0.660223 −1.58108 −1.73269 −1.82424 −2.50579
1.00000 −2.67561 1.00000 −1.00000 −2.67561 0.990876 1.00000 4.15891 −1.00000
1.2 1.00000 −2.34926 1.00000 −1.00000 −2.34926 −0.317129 1.00000 2.51901 −1.00000
1.3 1.00000 −2.12920 1.00000 −1.00000 −2.12920 −4.06902 1.00000 1.53348 −1.00000
1.4 1.00000 −1.16326 1.00000 −1.00000 −1.16326 −3.35475 1.00000 −1.64681 −1.00000
1.5 1.00000 −1.11338 1.00000 −1.00000 −1.11338 3.90936 1.00000 −1.76038 −1.00000
1.6 1.00000 −0.803872 1.00000 −1.00000 −0.803872 1.44710 1.00000 −2.35379 −1.00000
1.7 1.00000 −0.0694414 1.00000 −1.00000 −0.0694414 0.700507 1.00000 −2.99518 −1.00000
1.8 1.00000 0.660223 1.00000 −1.00000 0.660223 0.752610 1.00000 −2.56411 −1.00000
1.9 1.00000 1.58108 1.00000 −1.00000 1.58108 0.389201 1.00000 −0.500177 −1.00000
1.10 1.00000 1.73269 1.00000 −1.00000 1.73269 −1.89534 1.00000 0.00220907 −1.00000
1.11 1.00000 1.82424 1.00000 −1.00000 1.82424 −2.93972 1.00000 0.327838 −1.00000
1.12 1.00000 2.50579 1.00000 −1.00000 2.50579 −4.61369 1.00000 3.27901 −1.00000
 $$n$$: 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

## 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 4010.2.a.j 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.j 12 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$401$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4010))$$:

 $$T_{3}^{12} + \cdots$$ $$T_{7}^{12} + \cdots$$ $$T_{11}^{12} - \cdots$$