# Properties

 Label 2005.2.a.g Level 2005 Weight 2 Character orbit 2005.a Self dual Yes Analytic conductor 16.010 Analytic rank 0 Dimension 37 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2005 = 5 \cdot 401$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2005.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$16.0100056053$$ Analytic rank: $$0$$ Dimension: $$37$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$37q + 11q^{2} + 13q^{3} + 43q^{4} + 37q^{5} - 2q^{6} + 34q^{7} + 27q^{8} + 50q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$37q + 11q^{2} + 13q^{3} + 43q^{4} + 37q^{5} - 2q^{6} + 34q^{7} + 27q^{8} + 50q^{9} + 11q^{10} + 38q^{11} + 24q^{12} + 17q^{13} + 14q^{14} + 13q^{15} + 47q^{16} + 22q^{17} + 18q^{18} + 6q^{19} + 43q^{20} + 4q^{21} + 18q^{22} + 45q^{23} - 19q^{24} + 37q^{25} - q^{26} + 43q^{27} + 46q^{28} + 17q^{29} - 2q^{30} - 13q^{31} + 50q^{32} + 12q^{33} - 30q^{34} + 34q^{35} + 43q^{36} + 9q^{37} + q^{38} - 7q^{39} + 27q^{40} + 12q^{41} - 38q^{42} + 53q^{43} + 36q^{44} + 50q^{45} - 15q^{46} + 39q^{47} - 6q^{48} + 37q^{49} + 11q^{50} + 38q^{51} + 17q^{52} + 19q^{53} - 62q^{54} + 38q^{55} + 18q^{56} + 6q^{57} - 13q^{58} + 33q^{59} + 24q^{60} - 11q^{61} + q^{62} + 98q^{63} + 15q^{64} + 17q^{65} - 70q^{66} + 49q^{67} + 32q^{68} - 36q^{69} + 14q^{70} + 19q^{71} + 5q^{72} + 39q^{73} + 21q^{74} + 13q^{75} - 33q^{76} + 34q^{77} - 22q^{78} - 9q^{79} + 47q^{80} + 49q^{81} + 14q^{82} + 114q^{83} - 50q^{84} + 22q^{85} + 9q^{86} + 56q^{87} + 14q^{88} + 2q^{89} + 18q^{90} - 29q^{91} + 47q^{92} - 7q^{93} - 52q^{94} + 6q^{95} - 89q^{96} + 4q^{97} + 14q^{98} + 57q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.65893 1.77446 5.06989 1.00000 −4.71815 −1.53323 −8.16262 0.148702 −2.65893
1.2 −2.47883 3.30099 4.14460 1.00000 −8.18260 4.80146 −5.31610 7.89655 −2.47883
1.3 −2.30345 −1.22130 3.30589 1.00000 2.81322 3.38528 −3.00806 −1.50842 −2.30345
1.4 −2.26463 −1.05618 3.12854 1.00000 2.39186 1.78961 −2.55571 −1.88448 −2.26463
1.5 −2.16173 −0.113518 2.67309 1.00000 0.245396 −2.73299 −1.45504 −2.98711 −2.16173
1.6 −1.99940 −2.40196 1.99760 1.00000 4.80247 1.00729 0.00479577 2.76940 −1.99940
1.7 −1.88267 2.47026 1.54444 1.00000 −4.65067 0.359672 0.857674 3.10216 −1.88267
1.8 −1.72190 2.28979 0.964945 1.00000 −3.94280 2.06448 1.78226 2.24315 −1.72190
1.9 −1.27896 1.59149 −0.364250 1.00000 −2.03546 −4.69998 3.02379 −0.467169 −1.27896
1.10 −1.20338 −1.29534 −0.551875 1.00000 1.55878 −0.246916 3.07088 −1.32210 −1.20338
1.11 −0.868310 2.12555 −1.24604 1.00000 −1.84564 3.66007 2.81857 1.51797 −0.868310
1.12 −0.831556 3.34634 −1.30851 1.00000 −2.78267 −2.41688 2.75121 8.19798 −0.831556
1.13 −0.791118 0.403039 −1.37413 1.00000 −0.318852 4.95364 2.66934 −2.83756 −0.791118
1.14 −0.728237 −2.86948 −1.46967 1.00000 2.08966 1.10622 2.52674 5.23391 −0.728237
1.15 −0.522758 −2.06441 −1.72672 1.00000 1.07919 1.40344 1.94817 1.26178 −0.522758
1.16 0.0686864 −2.29330 −1.99528 1.00000 −0.157518 −2.03277 −0.274422 2.25922 0.0686864
1.17 0.265819 −0.631423 −1.92934 1.00000 −0.167844 3.82241 −1.04449 −2.60130 0.265819
1.18 0.284641 −0.0778726 −1.91898 1.00000 −0.0221658 1.40477 −1.11550 −2.99394 0.284641
1.19 0.412837 2.39393 −1.82957 1.00000 0.988304 0.00790896 −1.58099 2.73092 0.412837
1.20 0.449264 3.11633 −1.79816 1.00000 1.40005 3.71365 −1.70638 6.71149 0.449264
See all 37 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.37 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Atkin-Lehner signs

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

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2005))$$:

 $$T_{2}^{37} - \cdots$$ $$T_{11}^{37} - \cdots$$