# Properties

 Label 4021.2.a.c Level 4021 Weight 2 Character orbit 4021.a Self dual yes Analytic conductor 32.108 Analytic rank 0 Dimension 182 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4021$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4021.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1078466528$$ Analytic rank: $$0$$ Dimension: $$182$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$182q + 18q^{2} + 28q^{3} + 208q^{4} + 22q^{5} + 18q^{6} + 14q^{7} + 54q^{8} + 238q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$182q + 18q^{2} + 28q^{3} + 208q^{4} + 22q^{5} + 18q^{6} + 14q^{7} + 54q^{8} + 238q^{9} + 7q^{10} + 138q^{11} + 47q^{12} + 5q^{13} + 72q^{14} + 41q^{15} + 256q^{16} + 29q^{17} + 35q^{18} + 48q^{19} + 56q^{20} + 22q^{21} + 19q^{22} + 91q^{23} + 46q^{24} + 230q^{25} + 88q^{26} + 103q^{27} + 15q^{28} + 75q^{29} + 18q^{30} + 43q^{31} + 116q^{32} + 15q^{33} + 13q^{34} + 185q^{35} + 364q^{36} + 15q^{37} + 53q^{38} + 80q^{39} - 13q^{40} + 68q^{41} + 32q^{42} + 82q^{43} + 259q^{44} + 37q^{45} + 13q^{46} + 121q^{47} + 53q^{48} + 244q^{49} + 93q^{50} + 144q^{51} - 16q^{52} + 101q^{53} + 47q^{54} + 49q^{55} + 199q^{56} + 3q^{57} + 4q^{58} + 254q^{59} + 24q^{60} + 8q^{61} + 37q^{62} + 19q^{63} + 326q^{64} + 65q^{65} + 41q^{66} + 91q^{67} + 50q^{68} + 50q^{69} + 5q^{70} + 212q^{71} + 77q^{72} + 5q^{73} + 101q^{74} + 127q^{75} + 22q^{76} + 87q^{77} - 20q^{78} + 86q^{79} + 71q^{80} + 358q^{81} - 20q^{82} + 139q^{83} - 30q^{84} + 25q^{85} + 82q^{86} + 36q^{87} - 8q^{88} + 100q^{89} - 87q^{90} + 74q^{91} + 171q^{92} + 50q^{93} - 13q^{94} + 217q^{95} + 42q^{96} + 20q^{97} + 47q^{98} + 389q^{99} + O(q^{100})$$

## 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 −2.81003 −0.712978 5.89629 1.47211 2.00349 1.95505 −10.9487 −2.49166 −4.13669
1.2 −2.80453 3.34034 5.86536 −1.76340 −9.36806 −4.23059 −10.8405 8.15785 4.94549
1.3 −2.71768 −3.37653 5.38576 2.58064 9.17630 −1.70858 −9.20140 8.40093 −7.01334
1.4 −2.69235 −2.05539 5.24877 −1.71000 5.53383 −4.72268 −8.74684 1.22462 4.60392
1.5 −2.67927 −1.86893 5.17851 2.00029 5.00738 −3.62338 −8.51610 0.492900 −5.35934
1.6 −2.67278 1.10671 5.14375 3.24617 −2.95798 −3.43903 −8.40257 −1.77520 −8.67630
1.7 −2.60649 −2.80454 4.79381 −0.526210 7.31002 5.03978 −7.28206 4.86546 1.37156
1.8 −2.58741 0.104608 4.69469 −1.37313 −0.270664 −0.569668 −6.97226 −2.98906 3.55285
1.9 −2.58653 3.00156 4.69012 3.01819 −7.76362 1.43921 −6.95807 6.00937 −7.80662
1.10 −2.57672 −1.91417 4.63947 −2.65404 4.93228 −2.26992 −6.80118 0.664061 6.83870
1.11 −2.56968 2.97740 4.60327 3.17406 −7.65098 1.41939 −6.68959 5.86491 −8.15632
1.12 −2.53727 1.21389 4.43775 −4.40547 −3.07998 −1.31615 −6.18523 −1.52646 11.1779
1.13 −2.51665 0.669973 4.33353 4.29318 −1.68609 3.31221 −5.87268 −2.55114 −10.8044
1.14 −2.51316 2.70275 4.31595 0.484778 −6.79244 2.96790 −5.82035 4.30486 −1.21832
1.15 −2.46406 1.32162 4.07159 −2.41266 −3.25655 −2.12421 −5.10452 −1.25332 5.94493
1.16 −2.43939 2.42778 3.95064 −0.884079 −5.92232 −2.32249 −4.75837 2.89413 2.15662
1.17 −2.38633 −0.134750 3.69456 −0.899112 0.321557 −0.745765 −4.04378 −2.98184 2.14558
1.18 −2.37596 −0.498603 3.64520 −0.0295149 1.18466 0.657526 −3.90895 −2.75139 0.0701264
1.19 −2.35347 0.704714 3.53883 3.49583 −1.65853 1.05101 −3.62159 −2.50338 −8.22733
1.20 −2.33739 −2.27957 3.46338 1.15085 5.32823 1.11748 −3.42047 2.19642 −2.68998
See next 80 embeddings (of 182 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.182 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 4021.2.a.c 182

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4021.2.a.c 182 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$4021$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{182} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4021))$$.