# Properties

 Label 4003.2.a.b Level $4003$ Weight $2$ Character orbit 4003.a Self dual yes Analytic conductor $31.964$ Analytic rank $1$ Dimension $152$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, 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("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4003$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4003.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: $$31.9641159291$$ Analytic rank: $$1$$ Dimension: $$152$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10})$$ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100})$$ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * 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 −2.81629 1.48630 5.93147 2.03150 −4.18584 −2.55919 −11.0722 −0.790918 −5.72129
1.2 −2.78616 −2.78181 5.76271 −2.35951 7.75059 2.45659 −10.4835 4.73848 6.57398
1.3 −2.74603 −0.470182 5.54067 −1.60967 1.29113 −2.65994 −9.72280 −2.77893 4.42020
1.4 −2.71591 −1.21901 5.37616 2.34595 3.31073 3.38769 −9.16936 −1.51401 −6.37139
1.5 −2.71344 −2.64650 5.36273 −1.98460 7.18110 −4.86510 −9.12456 4.00395 5.38509
1.6 −2.70039 3.13054 5.29213 −0.910228 −8.45368 0.335007 −8.89004 6.80026 2.45797
1.7 −2.68690 0.420690 5.21946 −4.00007 −1.13035 −1.97783 −8.65037 −2.82302 10.7478
1.8 −2.64207 1.02501 4.98054 2.32673 −2.70814 −0.493904 −7.87479 −1.94936 −6.14739
1.9 −2.60205 1.48054 4.77066 1.31197 −3.85243 3.44490 −7.20939 −0.808005 −3.41380
1.10 −2.59568 2.00123 4.73753 −2.55331 −5.19456 4.30357 −7.10575 1.00494 6.62756
1.11 −2.57418 1.68195 4.62641 −3.90988 −4.32965 1.64141 −6.76085 −0.171031 10.0647
1.12 −2.54789 −3.05680 4.49172 1.69745 7.78837 −1.95167 −6.34863 6.34402 −4.32491
1.13 −2.45287 −0.154053 4.01656 −1.18015 0.377872 4.09823 −4.94635 −2.97627 2.89476
1.14 −2.44543 −1.15615 3.98012 −3.94364 2.82729 3.67660 −4.84225 −1.66331 9.64388
1.15 −2.43441 1.77953 3.92633 −0.360928 −4.33209 −1.53438 −4.68946 0.166714 0.878646
1.16 −2.42958 −1.44088 3.90288 0.473210 3.50074 −4.22120 −4.62321 −0.923870 −1.14970
1.17 −2.41802 1.98760 3.84680 1.18517 −4.80604 −3.98093 −4.46560 0.950540 −2.86577
1.18 −2.37503 −2.00580 3.64079 0.380635 4.76385 3.69515 −3.89692 1.02324 −0.904022
1.19 −2.36790 −2.62074 3.60697 3.23023 6.20567 1.16162 −3.80516 3.86830 −7.64888
1.20 −2.36704 −1.77609 3.60290 −2.33381 4.20407 1.88575 −3.79413 0.154480 5.52424
See next 80 embeddings (of 152 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.152 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$4003$$ $$+1$$

## 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 4003.2.a.b 152

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4003.2.a.b 152 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{152} + 22 T_{2}^{151} + 21 T_{2}^{150} - 3036 T_{2}^{149} - 18712 T_{2}^{148} + 177403 T_{2}^{147} + \cdots - 67141$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4003))$$.