# Properties

 Label 1441.2.a.e Level $1441$ Weight $2$ Character orbit 1441.a Self dual yes Analytic conductor $11.506$ Analytic rank $0$ Dimension $28$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1441 = 11 \cdot 131$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1441.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.5064429313$$ Analytic rank: $$0$$ Dimension: $$28$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9}+O(q^{10})$$ 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q + 7 q^{2} - 2 q^{3} + 27 q^{4} + 3 q^{5} - q^{6} + 7 q^{7} + 21 q^{8} + 28 q^{9} + 14 q^{10} + 28 q^{11} - 6 q^{12} + 3 q^{13} - q^{14} + 19 q^{15} + 29 q^{16} + 9 q^{17} - 2 q^{18} + 20 q^{19} + 6 q^{20} + 6 q^{21} + 7 q^{22} + 24 q^{23} + 20 q^{24} + 23 q^{25} + 10 q^{26} - 20 q^{27} - 3 q^{28} + 43 q^{29} + 11 q^{30} + 3 q^{31} + 44 q^{32} - 2 q^{33} - 28 q^{34} + 32 q^{35} + 24 q^{36} - 4 q^{37} + 24 q^{38} + 37 q^{39} + 22 q^{40} + 32 q^{41} - 27 q^{42} + 25 q^{43} + 27 q^{44} - 36 q^{45} + 10 q^{46} + 19 q^{47} + 42 q^{48} + 17 q^{49} + q^{50} + 39 q^{51} - 19 q^{52} + 5 q^{53} + 6 q^{54} + 3 q^{55} + 8 q^{56} + 2 q^{57} + 21 q^{58} + 44 q^{59} + 65 q^{60} + 28 q^{61} + 60 q^{62} - 8 q^{63} + 5 q^{64} + 33 q^{65} - q^{66} + 7 q^{67} + 13 q^{68} - 22 q^{69} + 9 q^{70} + 117 q^{71} - 17 q^{72} + 7 q^{73} + 41 q^{74} - 40 q^{75} + 34 q^{76} + 7 q^{77} - 97 q^{78} + 48 q^{79} + 41 q^{80} + 40 q^{81} + 2 q^{82} + 22 q^{83} + 27 q^{84} + 30 q^{85} + 24 q^{86} + 37 q^{87} + 21 q^{88} - 6 q^{89} + 4 q^{90} - 33 q^{91} + 18 q^{92} + 5 q^{93} - 43 q^{94} + 64 q^{95} + 55 q^{96} - 50 q^{97} + 97 q^{98} + 28 q^{99}+O(q^{100})$$ 28 * q + 7 * q^2 - 2 * q^3 + 27 * q^4 + 3 * q^5 - q^6 + 7 * q^7 + 21 * q^8 + 28 * q^9 + 14 * q^10 + 28 * q^11 - 6 * q^12 + 3 * q^13 - q^14 + 19 * q^15 + 29 * q^16 + 9 * q^17 - 2 * q^18 + 20 * q^19 + 6 * q^20 + 6 * q^21 + 7 * q^22 + 24 * q^23 + 20 * q^24 + 23 * q^25 + 10 * q^26 - 20 * q^27 - 3 * q^28 + 43 * q^29 + 11 * q^30 + 3 * q^31 + 44 * q^32 - 2 * q^33 - 28 * q^34 + 32 * q^35 + 24 * q^36 - 4 * q^37 + 24 * q^38 + 37 * q^39 + 22 * q^40 + 32 * q^41 - 27 * q^42 + 25 * q^43 + 27 * q^44 - 36 * q^45 + 10 * q^46 + 19 * q^47 + 42 * q^48 + 17 * q^49 + q^50 + 39 * q^51 - 19 * q^52 + 5 * q^53 + 6 * q^54 + 3 * q^55 + 8 * q^56 + 2 * q^57 + 21 * q^58 + 44 * q^59 + 65 * q^60 + 28 * q^61 + 60 * q^62 - 8 * q^63 + 5 * q^64 + 33 * q^65 - q^66 + 7 * q^67 + 13 * q^68 - 22 * q^69 + 9 * q^70 + 117 * q^71 - 17 * q^72 + 7 * q^73 + 41 * q^74 - 40 * q^75 + 34 * q^76 + 7 * q^77 - 97 * q^78 + 48 * q^79 + 41 * q^80 + 40 * q^81 + 2 * q^82 + 22 * q^83 + 27 * q^84 + 30 * q^85 + 24 * q^86 + 37 * q^87 + 21 * q^88 - 6 * q^89 + 4 * q^90 - 33 * q^91 + 18 * q^92 + 5 * q^93 - 43 * q^94 + 64 * q^95 + 55 * q^96 - 50 * q^97 + 97 * q^98 + 28 * q^99

## 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 −2.46758 −2.74867 4.08895 0.488919 6.78255 −1.50214 −5.15465 4.55516 −1.20645
1.2 −2.36893 0.351154 3.61185 −1.64180 −0.831861 −1.35814 −3.81837 −2.87669 3.88932
1.3 −2.33619 2.29063 3.45776 −0.314509 −5.35135 0.543204 −3.40560 2.24701 0.734752
1.4 −2.09032 1.02488 2.36944 3.67569 −2.14232 2.35755 −0.772245 −1.94963 −7.68337
1.5 −1.79989 −0.0643329 1.23961 −3.25644 0.115792 2.45508 1.36862 −2.99586 5.86125
1.6 −1.77267 −3.40600 1.14234 −3.60328 6.03770 0.417068 1.52034 8.60085 6.38741
1.7 −1.13972 3.40925 −0.701031 −0.104086 −3.88560 1.00506 3.07843 8.62297 0.118629
1.8 −1.03351 −2.03030 −0.931861 −0.515805 2.09833 −3.45818 3.03010 1.12211 0.533089
1.9 −1.01237 −1.33231 −0.975116 3.44940 1.34878 1.50282 3.01190 −1.22496 −3.49205
1.10 −0.660505 2.12062 −1.56373 2.37769 −1.40068 3.00733 2.35386 1.49703 −1.57048
1.11 −0.574950 1.45629 −1.66943 −2.31674 −0.837294 −3.08132 2.10974 −0.879222 1.33201
1.12 −0.452030 −0.254542 −1.79567 −0.679800 0.115061 3.95437 1.71576 −2.93521 0.307290
1.13 0.00299522 0.0189616 −1.99999 1.18959 5.67941e−5 0 −5.08173 −0.0119809 −2.99964 0.00356309
1.14 0.235116 −2.49285 −1.94472 2.71737 −0.586108 3.95783 −0.927465 3.21431 0.638897
1.15 0.544027 2.32406 −1.70404 −4.20016 1.26435 −0.904548 −2.01509 2.40124 −2.28500
1.16 0.635564 −1.42688 −1.59606 2.67263 −0.906871 −2.11184 −2.28553 −0.964025 1.69862
1.17 0.851882 2.55735 −1.27430 2.24117 2.17856 −0.305679 −2.78931 3.54005 1.90921
1.18 1.06795 −2.13569 −0.859479 −3.41004 −2.28082 1.64444 −3.05379 1.56119 −3.64175
1.19 1.13327 0.404748 −0.715692 −1.18320 0.458690 4.77646 −3.07762 −2.83618 −1.34089
1.20 1.67300 −1.49184 0.798939 −1.12664 −2.49585 −0.438386 −2.00938 −0.774411 −1.88488
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$11$$ $$-1$$
$$131$$ $$1$$

## 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 1441.2.a.e 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1441.2.a.e 28 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} - 7 T_{2}^{27} - 17 T_{2}^{26} + 217 T_{2}^{25} - 45 T_{2}^{24} - 2862 T_{2}^{23} + 3378 T_{2}^{22} + 20881 T_{2}^{21} - 37507 T_{2}^{20} - 91459 T_{2}^{19} + 217251 T_{2}^{18} + 241789 T_{2}^{17} - 768563 T_{2}^{16} + \cdots + 6$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1441))$$.