# Properties

 Label 4001.2.a.a Level $4001$ Weight $2$ Character orbit 4001.a Self dual yes Analytic conductor $31.948$ Analytic rank $1$ Dimension $149$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4001,2,Mod(1,4001)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4001, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4001$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4001.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$31.9481458487$$ Analytic rank: $$1$$ Dimension: $$149$$ Twist minimal: yes Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9}+O(q^{10})$$ 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$149 q - 6 q^{2} - 28 q^{3} + 116 q^{4} - 19 q^{5} - 31 q^{6} - 47 q^{7} - 15 q^{8} + 115 q^{9} - 48 q^{10} - 31 q^{11} - 61 q^{12} - 54 q^{13} - 44 q^{14} - 65 q^{15} + 58 q^{16} - 26 q^{17} - 23 q^{18} - 86 q^{19} - 52 q^{20} - 30 q^{21} - 56 q^{22} - 63 q^{23} - 90 q^{24} + 92 q^{25} - 38 q^{26} - 103 q^{27} - 77 q^{28} - 51 q^{29} - 22 q^{30} - 256 q^{31} - 21 q^{32} - 36 q^{33} - 124 q^{34} - 50 q^{35} + 45 q^{36} - 42 q^{37} - 14 q^{38} - 119 q^{39} - 131 q^{40} - 55 q^{41} - 5 q^{42} - 55 q^{43} - 54 q^{44} - 68 q^{45} - 59 q^{46} - 82 q^{47} - 89 q^{48} + 30 q^{49} + 13 q^{50} - 83 q^{51} - 126 q^{52} - 23 q^{53} - 83 q^{54} - 244 q^{55} - 94 q^{56} - 14 q^{57} - 60 q^{58} - 93 q^{59} - 70 q^{60} - 139 q^{61} - 10 q^{62} - 120 q^{63} - 45 q^{64} - 37 q^{65} - 28 q^{66} - 110 q^{67} - 27 q^{68} - 79 q^{69} - 78 q^{70} - 123 q^{71} - 74 q^{73} - 25 q^{74} - 146 q^{75} - 192 q^{76} - q^{77} + 26 q^{78} - 273 q^{79} - 51 q^{80} + 41 q^{81} - 120 q^{82} - 27 q^{83} - 14 q^{84} - 71 q^{85} + 3 q^{86} - 98 q^{87} - 143 q^{88} - 70 q^{89} - 24 q^{90} - 256 q^{91} - 47 q^{92} + 16 q^{93} - 151 q^{94} - 83 q^{95} - 137 q^{96} - 108 q^{97} + 17 q^{98} - 131 q^{99}+O(q^{100})$$ 149 * q - 6 * q^2 - 28 * q^3 + 116 * q^4 - 19 * q^5 - 31 * q^6 - 47 * q^7 - 15 * q^8 + 115 * q^9 - 48 * q^10 - 31 * q^11 - 61 * q^12 - 54 * q^13 - 44 * q^14 - 65 * q^15 + 58 * q^16 - 26 * q^17 - 23 * q^18 - 86 * q^19 - 52 * q^20 - 30 * q^21 - 56 * q^22 - 63 * q^23 - 90 * q^24 + 92 * q^25 - 38 * q^26 - 103 * q^27 - 77 * q^28 - 51 * q^29 - 22 * q^30 - 256 * q^31 - 21 * q^32 - 36 * q^33 - 124 * q^34 - 50 * q^35 + 45 * q^36 - 42 * q^37 - 14 * q^38 - 119 * q^39 - 131 * q^40 - 55 * q^41 - 5 * q^42 - 55 * q^43 - 54 * q^44 - 68 * q^45 - 59 * q^46 - 82 * q^47 - 89 * q^48 + 30 * q^49 + 13 * q^50 - 83 * q^51 - 126 * q^52 - 23 * q^53 - 83 * q^54 - 244 * q^55 - 94 * q^56 - 14 * q^57 - 60 * q^58 - 93 * q^59 - 70 * q^60 - 139 * q^61 - 10 * q^62 - 120 * q^63 - 45 * q^64 - 37 * q^65 - 28 * q^66 - 110 * q^67 - 27 * q^68 - 79 * q^69 - 78 * q^70 - 123 * q^71 - 74 * q^73 - 25 * q^74 - 146 * q^75 - 192 * q^76 - q^77 + 26 * q^78 - 273 * q^79 - 51 * q^80 + 41 * q^81 - 120 * q^82 - 27 * q^83 - 14 * q^84 - 71 * q^85 + 3 * q^86 - 98 * q^87 - 143 * q^88 - 70 * q^89 - 24 * q^90 - 256 * q^91 - 47 * q^92 + 16 * q^93 - 151 * q^94 - 83 * q^95 - 137 * q^96 - 108 * q^97 + 17 * q^98 - 131 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1 −2.71370 0.987518 5.36416 1.22965 −2.67983 4.88451 −9.12930 −2.02481 −3.33689
1.2 −2.68009 1.97360 5.18289 2.92338 −5.28944 −1.04102 −8.53045 0.895108 −7.83492
1.3 −2.67250 0.814974 5.14227 −1.39982 −2.17802 −2.73506 −8.39771 −2.33582 3.74102
1.4 −2.66308 −1.99633 5.09200 2.13614 5.31640 2.50209 −8.23426 0.985342 −5.68872
1.5 −2.62536 1.08746 4.89252 0.733437 −2.85498 −4.12609 −7.59390 −1.81743 −1.92554
1.6 −2.59431 −3.00898 4.73047 −0.189685 7.80624 1.12756 −7.08370 6.05397 0.492103
1.7 −2.59150 −1.41020 4.71587 −1.43230 3.65455 −1.66447 −7.03818 −1.01132 3.71180
1.8 −2.58646 −1.10071 4.68977 −0.828720 2.84694 −1.43853 −6.95698 −1.78844 2.14345
1.9 −2.47526 2.78845 4.12689 −0.919151 −6.90213 −1.88527 −5.26460 4.77546 2.27513
1.10 −2.46316 2.96188 4.06716 −3.16778 −7.29560 1.16261 −5.09175 5.77276 7.80275
1.11 −2.45256 0.529200 4.01507 2.50676 −1.29790 −0.602620 −4.94208 −2.71995 −6.14798
1.12 −2.44683 −2.82076 3.98699 4.21697 6.90192 −1.62820 −4.86182 4.95666 −10.3182
1.13 −2.41976 −1.68750 3.85525 −0.342140 4.08335 2.33028 −4.48927 −0.152339 0.827898
1.14 −2.31463 1.63767 3.35751 −2.89581 −3.79059 −1.05145 −3.14214 −0.318053 6.70274
1.15 −2.27915 2.08452 3.19451 −0.864719 −4.75093 2.39591 −2.72247 1.34523 1.97082
1.16 −2.27484 −0.745234 3.17488 3.93220 1.69529 2.00217 −2.67267 −2.44463 −8.94513
1.17 −2.25752 −0.289421 3.09639 −1.87801 0.653373 3.41261 −2.47511 −2.91624 4.23965
1.18 −2.24881 −2.62223 3.05717 −2.07631 5.89691 −2.22773 −2.37738 3.87609 4.66924
1.19 −2.21148 1.63586 2.89063 1.70947 −3.61768 0.528860 −1.96961 −0.323947 −3.78045
1.20 −2.16569 −0.380840 2.69021 −3.90103 0.824780 −2.04947 −1.49478 −2.85496 8.44841
See next 80 embeddings (of 149 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.149 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$4001$$ $$+1$$

## 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 4001.2.a.a 149

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4001.2.a.a 149 1.a even 1 1 trivial