LMFDB - Embedded newform 2052.3.be.a.125.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2052,3,Mod(125,2052)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2052, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 4]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2052.125");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2052 = 2^{2} \cdot 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2052.be (of order $6$, degree $2$, not minimal)

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:	no
Analytic conductor:	$55.9129502467$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 684)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.1
Character	$\chi$	$=$	2052.125
Dual form			2052.3.be.a.197.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.04234 + 4.64325i) q^{5} +(-2.28067 - 3.95023i) q^{7} +O(q^{10})$	$q+(-8.04234 + 4.64325i) q^{5} +(-2.28067 - 3.95023i) q^{7} +(1.10584 - 0.638454i) q^{11} +6.61915 q^{13} +(-11.0966 - 6.40664i) q^{17} +(-18.7585 + 3.01961i) q^{19} -18.8771i q^{23} +(30.6195 - 53.0346i) q^{25} +(6.15298 + 3.55242i) q^{29} +(7.90302 - 13.6884i) q^{31} +(36.6838 + 21.1794i) q^{35} -16.9760 q^{37} +(-15.3205 + 8.84528i) q^{41} -41.7699 q^{43} +(19.7274 + 11.3896i) q^{47} +(14.0971 - 24.4169i) q^{49} +(5.86723 - 3.38745i) q^{53} +(-5.92901 + 10.2693i) q^{55} +(71.6659 - 41.3763i) q^{59} +(-23.7419 + 41.1222i) q^{61} +(-53.2335 + 30.7344i) q^{65} -42.7202 q^{67} +(-21.4678 - 12.3945i) q^{71} +(-39.6934 + 68.7509i) q^{73} +(-5.04409 - 2.91221i) q^{77} +108.471 q^{79} +(-34.7029 + 20.0357i) q^{83} +118.990 q^{85} +(-49.2445 + 28.4313i) q^{89} +(-15.0961 - 26.1472i) q^{91} +(136.842 - 111.385i) q^{95} +173.722 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + q^{7}+O(q^{10}) $ 80 * q + q^7	$ 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) $ 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2052\mathbb{Z}\right)^\times$.

$n$	$325$	$1027$	$1217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−8.04234	+	4.64325i	−1.60847	+	0.928650i	−0.618755	+	0.785584i	$0.712363\pi$
$5$	−8.04234	+	4.64325i	−1.60847	+	0.928650i	−0.989713	+	0.143066i	$0.954304\pi$
$6$		0			0
$7$	−2.28067	−	3.95023i	−0.325810	−	0.564319i	0.655866	−	0.754877i	$-0.272304\pi$
$7$	−2.28067	−	3.95023i	−0.325810	−	0.564319i	−0.981676	+	0.190558i	$0.938970\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.10584	−	0.638454i	0.100531	−	0.0580413i	−0.448892	−	0.893586i	$-0.648181\pi$
$11$	1.10584	−	0.638454i	0.100531	−	0.0580413i	0.549422	+	0.835545i	$0.314848\pi$
$12$		0			0
$13$	6.61915			0.509165			0.254583	−	0.967051i	$-0.418062\pi$
$13$	6.61915			0.509165			0.254583	+	0.967051i	$0.418062\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−11.0966	−	6.40664i	−0.652743	−	0.376861i	0.136764	−	0.990604i	$-0.456330\pi$
$17$	−11.0966	−	6.40664i	−0.652743	−	0.376861i	−0.789506	+	0.613743i	$0.789663\pi$
$18$		0			0
$19$	−18.7585	+	3.01961i	−0.987290	+	0.158927i
$19$	−18.7585	+	3.01961i	−0.987290	+	0.158927i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	18.8771i		−	0.820741i	−0.911919	−	0.410371i	$-0.865399\pi$
$23$		−	18.8771i		−	0.820741i	0.911919	−	0.410371i	$-0.134601\pi$
$24$		0			0
$25$	30.6195	−	53.0346i	1.22478	−	2.12138i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.15298	+	3.55242i	0.212172	+	0.122497i	0.602320	−	0.798255i	$-0.294243\pi$
$29$	6.15298	+	3.55242i	0.212172	+	0.122497i	−0.390149	+	0.920752i	$0.627576\pi$
$30$		0			0
$31$	7.90302	−	13.6884i	0.254936	−	0.441562i	−0.709942	−	0.704260i	$-0.751279\pi$
$31$	7.90302	−	13.6884i	0.254936	−	0.441562i	0.964878	+	0.262698i	$0.0846123\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	36.6838	+	21.1794i	1.04811	+	0.605126i
$36$		0			0
$37$	−16.9760			−0.458810			−0.229405	−	0.973331i	$-0.573678\pi$
$37$	−16.9760			−0.458810			−0.229405	+	0.973331i	$0.573678\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−15.3205	+	8.84528i	−0.373670	+	0.215739i	−0.675061	−	0.737762i	$-0.735883\pi$
$41$	−15.3205	+	8.84528i	−0.373670	+	0.215739i	0.301390	+	0.953501i	$0.402549\pi$
$42$		0			0
$43$	−41.7699			−0.971393			−0.485696	−	0.874128i	$-0.661434\pi$
$43$	−41.7699			−0.971393			−0.485696	+	0.874128i	$0.661434\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	19.7274	+	11.3896i	0.419732	+	0.242332i	0.694963	−	0.719046i	$-0.255421\pi$
$47$	19.7274	+	11.3896i	0.419732	+	0.242332i	−0.275231	+	0.961378i	$0.588754\pi$
$48$		0			0
$49$	14.0971	−	24.4169i	0.287696	−	0.498304i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	5.86723	−	3.38745i	0.110702	−	0.0639141i	−0.443626	−	0.896212i	$-0.646308\pi$
$53$	5.86723	−	3.38745i	0.110702	−	0.0639141i	0.554329	+	0.832298i	$0.312975\pi$
$54$		0			0
$55$	−5.92901	+	10.2693i	−0.107800	+	0.186715i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	71.6659	−	41.3763i	1.21468	−	0.701293i	0.250901	−	0.968013i	$-0.419273\pi$
$59$	71.6659	−	41.3763i	1.21468	−	0.701293i	0.963774	+	0.266719i	$0.0859396\pi$
$60$		0			0
$61$	−23.7419	+	41.1222i	−0.389211	+	0.674134i	−0.992344	−	0.123507i	$-0.960586\pi$
$61$	−23.7419	+	41.1222i	−0.389211	+	0.674134i	0.603132	+	0.797641i	$0.293919\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−53.2335	+	30.7344i	−0.818977	+	0.472836i
$66$		0			0
$67$	−42.7202			−0.637615			−0.318807	−	0.947820i	$-0.603282\pi$
$67$	−42.7202			−0.637615			−0.318807	+	0.947820i	$0.603282\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−21.4678	−	12.3945i	−0.302364	−	0.174570i	0.341140	−	0.940012i	$-0.389187\pi$
$71$	−21.4678	−	12.3945i	−0.302364	−	0.174570i	−0.643504	+	0.765442i	$0.722520\pi$
$72$		0			0
$73$	−39.6934	+	68.7509i	−0.543745	+	0.941793i	0.454940	+	0.890522i	$0.349661\pi$
$73$	−39.6934	+	68.7509i	−0.543745	+	0.941793i	−0.998685	+	0.0512713i	$0.983673\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.04409	−	2.91221i	−0.0655076	−	0.0378209i
$78$		0			0
$79$	108.471			1.37304			0.686522	−	0.727109i	$-0.259136\pi$
$79$	108.471			1.37304			0.686522	+	0.727109i	$0.259136\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−34.7029	+	20.0357i	−0.418107	+	0.241394i	−0.694267	−	0.719717i	$-0.744271\pi$
$83$	−34.7029	+	20.0357i	−0.418107	+	0.241394i	0.276160	+	0.961112i	$0.410938\pi$
$84$		0			0
$85$	118.990			1.39989
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−49.2445	+	28.4313i	−0.553309	+	0.319453i	−0.750455	−	0.660921i	$-0.770166\pi$
$89$	−49.2445	+	28.4313i	−0.553309	+	0.319453i	0.197147	+	0.980374i	$0.436832\pi$
$90$		0			0
$91$	−15.0961	−	26.1472i	−0.165891	−	0.287332i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	136.842	−	111.385i	1.44044	−	1.17248i
$96$		0			0
$97$	173.722			1.79095			0.895477	−	0.445109i	$-0.146835\pi$
$97$	173.722			1.79095			0.895477	+	0.445109i	$0.146835\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−76.9543	−	44.4296i	−0.761924	−	0.439897i	0.0680622	−	0.997681i	$-0.478318\pi$
$101$	−76.9543	−	44.4296i	−0.761924	−	0.439897i	−0.829986	+	0.557784i	$0.811652\pi$
$102$		0			0
$103$	43.5924	−	75.5043i	0.423227	−	0.733051i	−0.573026	−	0.819537i	$-0.694231\pi$
$103$	43.5924	−	75.5043i	0.423227	−	0.733051i	0.996253	+	0.0864862i	$0.0275639\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			85.1392i			0.795693i	0.917452	+	0.397847i	$0.130242\pi$
$107$			85.1392i			0.795693i	−0.917452	+	0.397847i	$0.869758\pi$
$108$		0			0
$109$	−103.317	+	178.950i	−0.947861	+	1.64174i	−0.197941	+	0.980214i	$0.563426\pi$
$109$	−103.317	+	178.950i	−0.947861	+	1.64174i	−0.749919	+	0.661529i	$0.769908\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	180.905	+	104.445i	1.60093	+	0.924296i	0.991302	+	0.131605i	$0.0420132\pi$
$113$	180.905	+	104.445i	1.60093	+	0.924296i	0.609625	+	0.792690i	$0.291320\pi$
$114$		0			0
$115$	87.6509	+	151.816i	0.762181	+	1.32014i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			58.4457i			0.491140i
$120$		0			0
$121$	−59.6848	+	103.377i	−0.493262	+	0.854356i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			336.534i			2.69227i
$126$		0			0
$127$	56.4818	+	97.8294i	0.444739	+	0.770310i	0.998034	−	0.0626751i	$-0.0199632\pi$
$127$	56.4818	+	97.8294i	0.444739	+	0.770310i	−0.553295	+	0.832985i	$0.686630\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	7.37198	−	4.25621i	0.0562746	−	0.0324902i	−0.471599	−	0.881813i	$-0.656323\pi$
$131$	7.37198	−	4.25621i	0.0562746	−	0.0324902i	0.527873	+	0.849323i	$0.322990\pi$
$132$		0			0
$133$	54.7101	+	67.2138i	0.411354	+	0.505367i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	185.497	+	107.097i	1.35399	+	0.781729i	0.988806	−	0.149204i	$-0.0476712\pi$
$137$	185.497	+	107.097i	1.35399	+	0.781729i	0.365189	+	0.930934i	$0.381005\pi$
$138$		0			0
$139$	−51.5431			−0.370813			−0.185407	−	0.982662i	$-0.559360\pi$
$139$	−51.5431			−0.370813			−0.185407	+	0.982662i	$0.559360\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	7.31969	−	4.22603i	0.0511867	−	0.0295526i
$144$		0			0
$145$	−65.9792			−0.455029
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−124.543	+	71.9047i	−0.835857	+	0.482582i	−0.855854	−	0.517218i	$-0.826968\pi$
$149$	−124.543	+	71.9047i	−0.835857	+	0.482582i	0.0199971	+	0.999800i	$0.493634\pi$
$150$		0			0
$151$	109.749	+	190.090i	0.726813	+	1.25888i	0.958223	+	0.286021i	$0.0923326\pi$
$151$	109.749	+	190.090i	0.726813	+	1.25888i	−0.231411	+	0.972856i	$0.574334\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			146.783i			0.946985i
$156$		0			0
$157$	15.7375	+	27.2581i	0.100239	+	0.173619i	0.911783	−	0.410672i	$-0.134706\pi$
$157$	15.7375	+	27.2581i	0.100239	+	0.173619i	−0.811544	+	0.584291i	$0.801373\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−74.5688	+	43.0523i	−0.463160	+	0.267406i
$162$		0			0
$163$	−27.3667			−0.167894			−0.0839469	−	0.996470i	$-0.526753\pi$
$163$	−27.3667			−0.167894			−0.0839469	+	0.996470i	$0.526753\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	231.660i		−	1.38718i	−0.720368	−	0.693592i	$-0.756027\pi$
$167$		−	231.660i		−	1.38718i	0.720368	−	0.693592i	$-0.243973\pi$
$168$		0			0
$169$	−125.187			−0.740751
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	119.666i		−	0.691712i	−0.938288	−	0.345856i	$-0.887588\pi$
$173$		−	119.666i		−	0.691712i	0.938288	−	0.345856i	$-0.112412\pi$
$174$		0			0
$175$	−279.332			−1.59618
$176$		0			0
$177$		0			0
$178$		0			0
$179$			182.887i			1.02171i	0.859666	+	0.510856i	$0.170672\pi$
$179$			182.887i			1.02171i	−0.859666	+	0.510856i	$0.829328\pi$
$180$		0			0
$181$	77.9550	+	135.022i	0.430691	+	0.745978i	0.996933	−	0.0782606i	$-0.0249366\pi$
$181$	77.9550	+	135.022i	0.430691	+	0.745978i	−0.566242	+	0.824239i	$0.691603\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	136.527	−	78.8237i	0.737982	−	0.426074i
$186$		0			0
$187$	−16.3614			−0.0874941
$188$		0			0
$189$		0			0
$190$		0			0
$191$	292.923	−	169.119i	1.53363	−	0.885440i	0.534436	−	0.845209i	$-0.320524\pi$
$191$	292.923	−	169.119i	1.53363	−	0.885440i	0.999190	−	0.0402304i	$-0.0128092\pi$
$192$		0			0
$193$	106.504	+	184.470i	0.551832	+	0.955801i	0.998142	+	0.0609231i	$0.0194044\pi$
$193$	106.504	+	184.470i	0.551832	+	0.955801i	−0.446310	+	0.894878i	$0.647262\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			7.74954i			0.0393378i	0.999807	+	0.0196689i	$0.00626120\pi$
$197$			7.74954i			0.0393378i	−0.999807	+	0.0196689i	$0.993739\pi$
$198$		0			0
$199$	−149.979	−	259.771i	−0.753662	−	1.30538i	−0.946037	−	0.324059i	$-0.894952\pi$
$199$	−149.979	−	259.771i	−0.753662	−	1.30538i	0.192375	−	0.981321i	$-0.438381\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	32.4076i		−	0.159643i
$204$		0			0
$205$	82.1417	−	142.274i	0.400691	−	0.694018i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−18.8159	+	15.3157i	−0.0900285	+	0.0732806i
$210$		0			0
$211$	184.127	+	318.918i	0.872641	+	1.51146i	0.859254	+	0.511549i	$0.170928\pi$
$211$	184.127	+	318.918i	0.872641	+	1.51146i	0.0133872	+	0.999910i	$0.495739\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	335.928	−	193.948i	1.56245	−	0.902083i
$216$		0			0
$217$	−72.0966			−0.332243
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−73.4502	−	42.4065i	−0.332354	−	0.191885i
$222$		0			0
$223$	−153.220			−0.687085			−0.343543	−	0.939137i	$-0.611627\pi$
$223$	−153.220			−0.687085			−0.343543	+	0.939137i	$0.611627\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−119.451	+	68.9649i	−0.526214	+	0.303810i	−0.739474	−	0.673186i	$-0.764925\pi$
$227$	−119.451	+	68.9649i	−0.526214	+	0.303810i	0.213259	+	0.976996i	$0.431592\pi$
$228$		0			0
$229$	−84.2944	+	146.002i	−0.368098	+	0.637564i	−0.989268	−	0.146112i	$-0.953324\pi$
$229$	−84.2944	+	146.002i	−0.368098	+	0.637564i	0.621170	+	0.783676i	$0.286657\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−257.617	−	148.735i	−1.10565	−	0.638348i	−0.167952	−	0.985795i	$-0.553715\pi$
$233$	−257.617	−	148.735i	−1.10565	−	0.638348i	−0.937700	+	0.347447i	$0.887049\pi$
$234$		0			0
$235$	−211.539			−0.900167
$236$		0			0
$237$		0			0
$238$		0			0
$239$	277.929	+	160.462i	1.16288	+	0.671391i	0.951993	−	0.306119i	$-0.0990307\pi$
$239$	277.929	+	160.462i	1.16288	+	0.671391i	0.210889	+	0.977510i	$0.432364\pi$
$240$		0			0
$241$	196.937	−	341.105i	0.817167	−	1.41537i	−0.0905951	−	0.995888i	$-0.528877\pi$
$241$	196.937	−	341.105i	0.817167	−	1.41537i	0.907762	−	0.419486i	$-0.137790\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			261.825i			1.06868i
$246$		0			0
$247$	−124.165	+	19.9873i	−0.502694	+	0.0809201i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	320.034	−	184.772i	1.27504	−	0.736142i	0.299104	−	0.954220i	$-0.403312\pi$
$251$	320.034	−	184.772i	1.27504	−	0.736142i	0.975931	+	0.218078i	$0.0699788\pi$
$252$		0			0
$253$	−12.0521	−	20.8749i	−0.0476369	−	0.0825096i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	183.974i		−	0.715851i	−0.933750	−	0.357925i	$-0.883484\pi$
$257$		−	183.974i		−	0.715851i	0.933750	−	0.357925i	$-0.116516\pi$
$258$		0			0
$259$	38.7166	+	67.0591i	0.149485	+	0.258915i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	169.113i		−	0.643015i	−0.946907	−	0.321507i	$-0.895811\pi$
$263$		−	169.113i		−	0.643015i	0.946907	−	0.321507i	$-0.104189\pi$
$264$		0			0
$265$	−31.4575	+	54.4860i	−0.118708	+	0.205608i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.505277	+	0.291722i	0.00187835	+	0.00108447i	0.500939	−	0.865483i	$-0.332988\pi$
$269$	0.505277	+	0.291722i	0.00187835	+	0.00108447i	−0.499061	+	0.866567i	$0.666321\pi$
$270$		0			0
$271$	−173.903	+	301.208i	−0.641707	+	1.11147i	0.343344	+	0.939210i	$0.388440\pi$
$271$	−173.903	+	301.208i	−0.641707	+	1.11147i	−0.985052	+	0.172260i	$0.944893\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	78.1967i		−	0.284351i
$276$		0			0
$277$	−115.851	−	200.660i	−0.418235	−	0.724404i	0.577527	−	0.816371i	$-0.304018\pi$
$277$	−115.851	−	200.660i	−0.418235	−	0.724404i	−0.995762	+	0.0919678i	$0.970684\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	434.273	+	250.727i	1.54545	+	0.892268i	0.998480	+	0.0551190i	$0.0175538\pi$
$281$	434.273	+	250.727i	1.54545	+	0.892268i	0.546974	+	0.837149i	$0.315780\pi$
$282$		0			0
$283$	23.4437	+	40.6057i	0.0828399	+	0.143483i	0.904469	−	0.426540i	$-0.140268\pi$
$283$	23.4437	+	40.6057i	0.0828399	+	0.143483i	−0.821629	+	0.570023i	$0.806934\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	69.8819	+	40.3463i	0.243491	+	0.140580i
$288$		0			0
$289$	−62.4099	−	108.097i	−0.215951	−	0.374039i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	67.5883	+	39.0221i	0.230677	+	0.133181i	0.610884	−	0.791720i	$-0.290814\pi$
$293$	67.5883	+	39.0221i	0.230677	+	0.133181i	−0.380207	+	0.924901i	$0.624147\pi$
$294$		0			0
$295$	−384.241	+	665.525i	−1.30251	+	2.25602i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	124.950i		−	0.417893i
$300$		0			0
$301$	95.2632	+	165.001i	0.316489	+	0.548175i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	440.958i		−	1.44576i
$306$		0			0
$307$	−13.1550	+	22.7852i	−0.0428503	+	0.0742188i	−0.886655	−	0.462431i	$-0.846977\pi$
$307$	−13.1550	+	22.7852i	−0.0428503	+	0.0742188i	0.843805	+	0.536650i	$0.180311\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	339.125	+	195.794i	1.09043	+	0.629562i	0.933692	−	0.358078i	$-0.116568\pi$
$311$	339.125	+	195.794i	1.09043	+	0.629562i	0.156741	+	0.987640i	$0.449901\pi$
$312$		0			0
$313$	46.6936	−	80.8756i	0.149181	−	0.258389i	−0.781744	−	0.623599i	$-0.785670\pi$
$313$	46.6936	−	80.8756i	0.149181	−	0.258389i	0.930925	+	0.365211i	$0.119003\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−49.4493	−	28.5496i	−0.155992	−	0.0900618i	0.419972	−	0.907537i	$-0.362040\pi$
$317$	−49.4493	−	28.5496i	−0.155992	−	0.0900618i	−0.575964	+	0.817475i	$0.695373\pi$
$318$		0			0
$319$	9.07224			0.0284396
$320$		0			0
$321$		0			0
$322$		0			0
$323$	227.502	+	86.6715i	0.704340	+	0.268333i
$324$		0			0
$325$	202.675	−	351.044i	0.623616	−	1.08013i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	103.904i		−	0.315817i
$330$		0			0
$331$	−5.97582	−	10.3504i	−0.0180538	−	0.0312702i	0.856857	−	0.515554i	$-0.172414\pi$
$331$	−5.97582	−	10.3504i	−0.0180538	−	0.0312702i	−0.874911	+	0.484283i	$0.839080\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	343.570	−	198.360i	1.02558	−	0.592121i
$336$		0			0
$337$	50.2250	+	86.9922i	0.149035	+	0.258137i	0.930871	−	0.365347i	$-0.119050\pi$
$337$	50.2250	+	86.9922i	0.149035	+	0.258137i	−0.781836	+	0.623484i	$0.785716\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	20.1829i		−	0.0591873i
$342$		0			0
$343$	−352.109			−1.02656
$344$		0			0
$345$		0			0
$346$		0			0
$347$	131.094	−	75.6869i	0.377791	−	0.218118i	−0.299066	−	0.954233i	$-0.596675\pi$
$347$	131.094	−	75.6869i	0.377791	−	0.218118i	0.676857	+	0.736115i	$0.263342\pi$
$348$		0			0
$349$	−219.737	−	380.596i	−0.629619	−	1.09053i	−0.987628	−	0.156814i	$-0.949878\pi$
$349$	−219.737	−	380.596i	−0.629619	−	1.09053i	0.358009	−	0.933718i	$-0.383456\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−524.353	+	302.736i	−1.48542	+	0.857608i	−0.999862	−	0.0165964i	$-0.994717\pi$
$353$	−524.353	+	302.736i	−1.48542	+	0.857608i	−0.485558	+	0.874204i	$0.661384\pi$
$354$		0			0
$355$	230.202			0.648457
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−84.9553	−	49.0490i	−0.236644	−	0.136627i	0.376989	−	0.926218i	$-0.376959\pi$
$359$	−84.9553	−	49.0490i	−0.236644	−	0.136627i	−0.613633	+	0.789591i	$0.710293\pi$
$360$		0			0
$361$	342.764	−	113.287i	0.949484	−	0.313814i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	737.224i		−	2.01979i
$366$		0			0
$367$	162.129	−	280.816i	0.441769	−	0.765167i	−0.556051	−	0.831148i	$-0.687684\pi$
$367$	162.129	−	280.816i	0.441769	−	0.765167i	0.997821	+	0.0659807i	$0.0210176\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−26.7624	−	15.4513i	−0.0721359	−	0.0416477i
$372$		0			0
$373$	−79.8812	+	138.358i	−0.214159	+	0.370934i	−0.953012	−	0.302933i	$-0.902034\pi$
$373$	−79.8812	+	138.358i	−0.214159	+	0.370934i	0.738853	+	0.673866i	$0.235368\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	40.7275	+	23.5140i	0.108030	+	0.0623714i
$378$		0			0
$379$	366.244			0.966343			0.483171	−	0.875526i	$-0.339485\pi$
$379$	366.244			0.966343			0.483171	+	0.875526i	$0.339485\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	107.432	−	62.0259i	0.280501	−	0.161948i	−0.353149	−	0.935567i	$-0.614889\pi$
$383$	107.432	−	62.0259i	0.280501	−	0.161948i	0.633650	+	0.773620i	$0.281556\pi$
$384$		0			0
$385$	54.0884			0.140489
$386$		0			0
$387$		0			0
$388$		0			0
$389$	359.574	+	207.600i	0.924354	+	0.533676i	0.885021	−	0.465550i	$-0.154144\pi$
$389$	359.574	+	207.600i	0.924354	+	0.533676i	0.0393324	+	0.999226i	$0.487477\pi$
$390$		0			0
$391$	−120.938	+	209.472i	−0.309306	+	0.535733i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−872.357	+	503.656i	−2.20850	+	1.27508i
$396$		0			0
$397$	247.236	−	428.225i	0.622760	−	1.07865i	−0.366209	−	0.930532i	$-0.619345\pi$
$397$	247.236	−	428.225i	0.622760	−	1.07865i	0.988969	−	0.148120i	$-0.0473220\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−259.441	+	149.789i	−0.646986	+	0.373538i	−0.787301	−	0.616569i	$-0.788522\pi$
$401$	−259.441	+	149.789i	−0.646986	+	0.373538i	0.140314	+	0.990107i	$0.455189\pi$
$402$		0			0
$403$	52.3113	−	90.6058i	0.129805	−	0.224828i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−18.7726	+	10.8384i	−0.0461244	+	0.0266299i
$408$		0			0
$409$	−160.535			−0.392506			−0.196253	−	0.980553i	$-0.562877\pi$
$409$	−160.535			−0.392506			−0.196253	+	0.980553i	$0.562877\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−326.892	−	188.731i	−0.791506	−	0.456976i
$414$		0			0
$415$	186.062	−	322.268i	0.448342	−	0.776550i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−442.295	−	255.359i	−1.05560	−	0.609449i	−0.131386	−	0.991331i	$-0.541943\pi$
$419$	−442.295	−	255.359i	−1.05560	−	0.609449i	−0.924211	+	0.381882i	$0.875276\pi$
$420$		0			0
$421$	352.159			0.836481			0.418241	−	0.908336i	$-0.362647\pi$
$421$	352.159			0.836481			0.418241	+	0.908336i	$0.362647\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−679.547	+	392.336i	−1.59893	+	0.923144i
$426$		0			0
$427$	216.590			0.507236
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−561.436	+	324.145i	−1.30264	+	0.752077i	−0.980855	−	0.194738i	$-0.937614\pi$
$431$	−561.436	+	324.145i	−1.30264	+	0.752077i	−0.321780	+	0.946815i	$0.604281\pi$
$432$		0			0
$433$	270.752	+	468.956i	0.625293	+	1.08304i	0.988484	+	0.151325i	$0.0483539\pi$
$433$	270.752	+	468.956i	0.625293	+	1.08304i	−0.363191	+	0.931715i	$0.618313\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	57.0014	+	354.106i	0.130438	+	0.810310i
$438$		0			0
$439$	−460.861			−1.04980			−0.524898	−	0.851165i	$-0.675897\pi$
$439$	−460.861			−1.04980			−0.524898	+	0.851165i	$0.675897\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	691.734	+	399.373i	1.56148	+	0.901519i	0.997108	+	0.0759950i	$0.0242133\pi$
$443$	691.734	+	399.373i	1.56148	+	0.901519i	0.564368	+	0.825524i	$0.309120\pi$
$444$		0			0
$445$	264.027	−	457.309i	0.593320	−	1.02766i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			751.768i			1.67432i	0.546962	+	0.837158i	$0.315784\pi$
$449$			751.768i			1.67432i	−0.546962	+	0.837158i	$0.684216\pi$
$450$		0			0
$451$	−11.2946	+	19.5629i	−0.0250435	+	0.0433766i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	242.816	+	140.190i	0.533661	+	0.308109i
$456$		0			0
$457$	−341.536	−	591.557i	−0.747343	−	1.29444i	−0.949092	−	0.314999i	$-0.897996\pi$
$457$	−341.536	−	591.557i	−0.747343	−	1.29444i	0.201749	−	0.979437i	$-0.435338\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	572.291i		−	1.24141i	−0.784044	−	0.620706i	$-0.786846\pi$
$461$		−	572.291i		−	1.24141i	0.784044	−	0.620706i	$-0.213154\pi$
$462$		0			0
$463$	221.398	−	383.473i	0.478182	−	0.828236i	−0.521505	−	0.853248i	$-0.674629\pi$
$463$	221.398	−	383.473i	0.478182	−	0.828236i	0.999687	+	0.0250123i	$0.00796250\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			246.829i			0.528542i	0.964449	+	0.264271i	$0.0851313\pi$
$467$			246.829i			0.528542i	−0.964449	+	0.264271i	$0.914869\pi$
$468$		0			0
$469$	97.4306	+	168.755i	0.207741	+	0.359818i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−46.1906	+	26.6682i	−0.0976546	+	0.0563809i
$474$		0			0
$475$	−414.233	+	1087.31i	−0.872069	+	2.28907i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−180.527	−	104.228i	−0.376884	−	0.217594i	0.299578	−	0.954072i	$-0.403154\pi$
$479$	−180.527	−	104.228i	−0.376884	−	0.217594i	−0.676462	+	0.736478i	$0.736488\pi$
$480$		0			0
$481$	−112.367			−0.233610
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1397.14	+	806.637i	−2.88069	+	1.66317i
$486$		0			0
$487$	467.236			0.959416			0.479708	−	0.877428i	$-0.340743\pi$
$487$	467.236			0.959416			0.479708	+	0.877428i	$0.340743\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−228.039	+	131.658i	−0.464438	+	0.268143i	−0.713908	−	0.700239i	$-0.753077\pi$
$491$	−228.039	+	131.658i	−0.464438	+	0.268143i	0.249471	+	0.968382i	$0.419743\pi$
$492$		0			0
$493$	−45.5182	−	78.8398i	−0.0923290	−	0.159919i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			113.071i			0.227506i
$498$		0			0
$499$	445.779	+	772.112i	0.893345	+	1.54732i	0.835840	+	0.548973i	$0.184981\pi$
$499$	445.779	+	772.112i	0.893345	+	1.54732i	0.0575050	+	0.998345i	$0.481685\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−383.876	+	221.631i	−0.763174	+	0.440619i	−0.830434	−	0.557117i	$-0.811908\pi$
$503$	−383.876	+	221.631i	−0.763174	+	0.440619i	0.0672604	+	0.997735i	$0.478574\pi$
$504$		0			0
$505$	825.191			1.63404
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	353.387i		−	0.694277i	−0.937814	−	0.347138i	$-0.887153\pi$
$509$		−	353.387i		−	0.694277i	0.937814	−	0.347138i	$-0.112847\pi$
$510$		0			0
$511$	362.110			0.708629
$512$		0			0
$513$		0			0
$514$		0			0
$515$			809.642i			1.57212i
$516$		0			0
$517$	29.0870			0.0562611
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	530.816i		−	1.01884i	−0.860518	−	0.509420i	$-0.829860\pi$
$521$		−	530.816i		−	1.01884i	0.860518	−	0.509420i	$-0.170140\pi$
$522$		0			0
$523$	518.783	+	898.559i	0.991938	+	1.71809i	0.605717	+	0.795680i	$0.292887\pi$
$523$	518.783	+	898.559i	0.991938	+	1.71809i	0.386221	+	0.922406i	$0.373780\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−175.394	+	101.264i	−0.332815	+	0.192151i
$528$		0			0
$529$	172.657			0.326383
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−101.409	+	58.5483i	−0.190260	+	0.109847i
$534$		0			0
$535$	−395.322	−	684.719i	−0.738920	−	1.27985i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	36.0014i		−	0.0667930i
$540$		0			0
$541$	−427.224	−	739.974i	−0.789693	−	1.36779i	−0.926155	−	0.377143i	$-0.876906\pi$
$541$	−427.224	−	739.974i	−0.789693	−	1.36779i	0.136462	−	0.990645i	$-0.456427\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1918.90i		−	3.52092i
$546$		0			0
$547$	206.144	−	357.051i	0.376862	−	0.652744i	−0.613742	−	0.789507i	$-0.710336\pi$
$547$	206.144	−	357.051i	0.376862	−	0.652744i	0.990604	+	0.136763i	$0.0436697\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−126.148	−	48.0586i	−0.228943	−	0.0872207i
$552$		0			0
$553$	−247.385	−	428.484i	−0.447351	−	0.774835i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	863.794	−	498.712i	1.55080	−	0.895353i	0.552720	−	0.833367i	$-0.313590\pi$
$557$	863.794	−	498.712i	1.55080	−	0.895353i	0.998077	−	0.0619859i	$-0.0197434\pi$
$558$		0			0
$559$	−276.481			−0.494600
$560$		0			0
$561$		0			0
$562$		0			0
$563$	305.679	+	176.484i	0.542947	+	0.313470i	0.746272	−	0.665641i	$-0.231842\pi$
$563$	305.679	+	176.484i	0.542947	+	0.313470i	−0.203326	+	0.979111i	$0.565175\pi$
$564$		0			0
$565$	−1939.86			−3.43339
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−276.835	+	159.831i	−0.486529	+	0.280897i	−0.723133	−	0.690709i	$-0.757299\pi$
$569$	−276.835	+	159.831i	−0.486529	+	0.280897i	0.236605	+	0.971606i	$0.423965\pi$
$570$		0			0
$571$	352.313	−	610.224i	0.617010	−	1.06869i	−0.373018	−	0.927824i	$-0.621677\pi$
$571$	352.313	−	610.224i	0.617010	−	1.06869i	0.990028	−	0.140869i	$-0.0449897\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1001.14	−	578.006i	−1.74111	−	1.00523i
$576$		0			0
$577$	322.338			0.558644			0.279322	−	0.960197i	$-0.409890\pi$
$577$	322.338			0.558644			0.279322	+	0.960197i	$0.409890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	158.292	+	91.3897i	0.272447	+	0.157297i
$582$		0			0
$583$	4.32546	−	7.49192i	0.00741932	−	0.0128506i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1124.82i			1.91621i	0.286409	+	0.958107i	$0.407538\pi$
$587$			1124.82i			1.91621i	−0.286409	+	0.958107i	$0.592462\pi$
$588$		0			0
$589$	−106.915	+	280.639i	−0.181520	+	0.476466i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	686.343	−	396.261i	1.15741	−	0.668230i	0.206727	−	0.978399i	$-0.433719\pi$
$593$	686.343	−	396.261i	1.15741	−	0.668230i	0.950682	+	0.310168i	$0.100385\pi$
$594$		0			0
$595$	−271.378	−	470.040i	−0.456097	−	0.789983i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			803.559i			1.34150i	0.741683	+	0.670750i	$0.234028\pi$
$599$			803.559i			1.34150i	−0.741683	+	0.670750i	$0.765972\pi$
$600$		0			0
$601$	3.87039	+	6.70371i	0.00643992	+	0.0111543i	0.869227	−	0.494412i	$-0.164617\pi$
$601$	3.87039	+	6.70371i	0.00643992	+	0.0111543i	−0.862787	+	0.505567i	$0.831283\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	1108.52i		−	1.83227i
$606$		0			0
$607$	111.655	−	193.392i	0.183945	−	0.318602i	−0.759275	−	0.650769i	$-0.774446\pi$
$607$	111.655	−	193.392i	0.183945	−	0.318602i	0.943221	+	0.332167i	$0.107780\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	130.579	+	75.3896i	0.213713	+	0.123387i
$612$		0			0
$613$	188.469	−	326.438i	0.307454	−	0.532525i	−0.670351	−	0.742044i	$-0.733856\pi$
$613$	188.469	−	326.438i	0.307454	−	0.532525i	0.977805	+	0.209519i	$0.0671898\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			467.434i			0.757591i	0.925480	+	0.378795i	$0.123662\pi$
$617$			467.434i			0.757591i	−0.925480	+	0.378795i	$0.876338\pi$
$618$		0			0
$619$	481.347	+	833.718i	0.777621	+	1.34688i	0.933310	+	0.359073i	$0.116907\pi$
$619$	481.347	+	833.718i	0.777621	+	1.34688i	−0.155689	+	0.987806i	$0.549760\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	224.621	+	129.685i	0.360547	+	0.208162i
$624$		0			0
$625$	−797.122	−	1380.66i	−1.27539	−	2.20905i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	188.376	+	108.759i	0.299485	+	0.172908i
$630$		0			0
$631$	132.578	+	229.632i	0.210108	+	0.363917i	0.951748	−	0.306880i	$-0.0992852\pi$
$631$	132.578	+	229.632i	0.210108	+	0.363917i	−0.741640	+	0.670798i	$0.765952\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−908.492	−	524.518i	−1.43070	−	0.826013i
$636$		0			0
$637$	93.3109	−	161.619i	0.146485	−	0.253719i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	25.2502i		−	0.0393918i	−0.999806	−	0.0196959i	$-0.993730\pi$
$641$		−	25.2502i		−	0.0393918i	0.999806	−	0.0196959i	$-0.00626981\pi$
$642$		0			0
$643$	167.889	+	290.792i	0.261102	+	0.452242i	0.966535	−	0.256535i	$-0.0825808\pi$
$643$	167.889	+	290.792i	0.261102	+	0.452242i	−0.705433	+	0.708777i	$0.749247\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	905.889i		−	1.40014i	−0.714075	−	0.700069i	$-0.753152\pi$
$647$		−	905.889i		−	1.40014i	0.714075	−	0.700069i	$-0.246848\pi$
$648$		0			0
$649$	52.8338	−	91.5108i	0.0814080	−	0.141003i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−929.143	−	536.441i	−1.42288	−	0.821502i	−0.426339	−	0.904564i	$-0.640197\pi$
$653$	−929.143	−	536.441i	−1.42288	−	0.821502i	−0.996544	+	0.0830615i	$0.973530\pi$
$654$		0			0
$655$	−39.5253	+	68.4598i	−0.0603440	+	0.104519i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	897.240	+	518.021i	1.36152	+	0.786072i	0.989826	−	0.142284i	$-0.0454448\pi$
$659$	897.240	+	518.021i	1.36152	+	0.786072i	0.371691	+	0.928357i	$0.378778\pi$
$660$		0			0
$661$	−94.3544			−0.142745			−0.0713725	−	0.997450i	$-0.522738\pi$
$661$	−94.3544			−0.142745			−0.0713725	+	0.997450i	$0.522738\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−752.088	−	286.524i	−1.13096	−	0.430862i
$666$		0			0
$667$	67.0593	−	116.150i	0.100539	−	0.174138i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			60.6325i			0.0903614i
$672$		0			0
$673$	−156.533	−	271.123i	−0.232590	−	0.402858i	0.725979	−	0.687716i	$-0.241387\pi$
$673$	−156.533	−	271.123i	−0.232590	−	0.402858i	−0.958570	+	0.284858i	$0.908053\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	523.760	−	302.393i	0.773649	−	0.446666i	−0.0605259	−	0.998167i	$-0.519278\pi$
$677$	523.760	−	302.393i	0.773649	−	0.446666i	0.834175	+	0.551500i	$0.185944\pi$
$678$		0			0
$679$	−396.203	−	686.244i	−0.583510	−	1.01067i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	27.4860i		−	0.0402430i	−0.999798	−	0.0201215i	$-0.993595\pi$
$683$		−	27.4860i		−	0.0402430i	0.999798	−	0.0201215i	$-0.00640531\pi$
$684$		0			0
$685$	−1989.11			−2.90381
$686$		0			0
$687$		0			0
$688$		0			0
$689$	38.8361	−	22.4220i	0.0563659	−	0.0325429i
$690$		0			0
$691$	2.35922	+	4.08630i	0.00341422	+	0.00591360i	0.867727	−	0.497040i	$-0.165580\pi$
$691$	2.35922	+	4.08630i	0.00341422	+	0.00591360i	−0.864313	+	0.502954i	$0.832247\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	414.527	−	239.327i	0.596442	−	0.344356i
$696$		0			0
$697$	226.674			0.325214
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−81.6577	−	47.1451i	−0.116487	−	0.0672541i	0.440624	−	0.897692i	$-0.354757\pi$
$701$	−81.6577	−	47.1451i	−0.116487	−	0.0672541i	−0.557112	+	0.830438i	$0.688090\pi$
$702$		0			0
$703$	318.444	−	51.2609i	0.452979	−	0.0729173i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			405.317i			0.573291i
$708$		0			0
$709$	433.385	−	750.644i	0.611262	−	1.05874i	−0.379766	−	0.925083i	$-0.623995\pi$
$709$	433.385	−	750.644i	0.611262	−	1.05874i	0.991028	−	0.133654i	$-0.0426712\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−258.397	−	149.186i	−0.362408	−	0.209237i
$714$		0			0
$715$	−39.2450	+	67.9743i	−0.0548881	+	0.0950689i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−960.272	−	554.413i	−1.33557	−	0.771089i	−0.349419	−	0.936966i	$-0.613621\pi$
$719$	−960.272	−	554.413i	−1.33557	−	0.771089i	−0.986146	+	0.165877i	$0.946954\pi$
$720$		0			0
$721$	−397.679			−0.551566
$722$		0			0
$723$		0			0
$724$		0			0
$725$	376.802	−	217.547i	0.519728	−	0.300065i
$726$		0			0
$727$	−404.114			−0.555865			−0.277933	−	0.960601i	$-0.589649\pi$
$727$	−404.114			−0.555865			−0.277933	+	0.960601i	$0.589649\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	463.505	+	267.605i	0.634069	+	0.366080i
$732$		0			0
$733$	−564.680	+	978.054i	−0.770368	+	1.33432i	0.166994	+	0.985958i	$0.446594\pi$
$733$	−564.680	+	978.054i	−0.770368	+	1.33432i	−0.937362	+	0.348358i	$0.886739\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−47.2415	+	27.2749i	−0.0640997	+	0.0370080i
$738$		0			0
$739$	321.557	−	556.953i	0.435124	−	0.753658i	−0.562181	−	0.827014i	$-0.690038\pi$
$739$	321.557	−	556.953i	0.435124	−	0.753658i	0.997306	+	0.0733563i	$0.0233710\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1043.81	−	602.642i	1.40485	−	0.811093i	0.409969	−	0.912100i	$-0.365540\pi$
$743$	1043.81	−	602.642i	1.40485	−	0.811093i	0.994886	+	0.101006i	$0.0322063\pi$
$744$		0			0
$745$	667.743	−	1156.56i	0.896299	−	1.55244i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	336.320	−	194.174i	0.449025	−	0.259245i
$750$		0			0
$751$	247.958			0.330171			0.165086	−	0.986279i	$-0.447210\pi$
$751$	247.958			0.330171			0.165086	+	0.986279i	$0.447210\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1765.27	−	1019.18i	−2.33811	−	1.34991i
$756$		0			0
$757$	−313.951	+	543.779i	−0.414731	+	0.718335i	−0.995400	−	0.0958047i	$-0.969458\pi$
$757$	−313.951	+	543.779i	−0.414731	+	0.718335i	0.580669	+	0.814139i	$0.302791\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−856.714	−	494.624i	−1.12577	−	0.649966i	−0.182905	−	0.983131i	$-0.558550\pi$
$761$	−856.714	−	494.624i	−1.12577	−	0.649966i	−0.942869	+	0.333165i	$0.891883\pi$
$762$		0			0
$763$	942.526			1.23529
$764$		0			0
$765$		0			0
$766$		0			0
$767$	474.367	−	273.876i	0.618471	−	0.357074i
$768$		0			0
$769$	513.629			0.667918			0.333959	−	0.942588i	$-0.391615\pi$
$769$	513.629			0.667918			0.333959	+	0.942588i	$0.391615\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	464.596	−	268.235i	0.601030	−	0.347005i	−0.168417	−	0.985716i	$-0.553865\pi$
$773$	464.596	−	268.235i	0.601030	−	0.347005i	0.769447	+	0.638711i	$0.220532\pi$
$774$		0			0
$775$	−483.973	−	838.266i	−0.624481	−	1.08163i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	260.680	−	212.186i	0.334634	−	0.272383i
$780$		0			0
$781$	−31.6532			−0.0405291
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−253.132	−	146.146i	−0.322462	−	0.186173i
$786$		0			0
$787$	60.3896	−	104.598i	0.0767339	−	0.132907i	−0.825105	−	0.564979i	$-0.808884\pi$
$787$	60.3896	−	104.598i	0.0767339	−	0.132907i	0.901839	+	0.432072i	$0.142217\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	952.821i		−	1.20458i
$792$		0			0
$793$	−157.151	+	272.194i	−0.198173	+	0.343246i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−525.724	−	303.527i	−0.659629	−	0.380837i	0.132507	−	0.991182i	$-0.457697\pi$
$797$	−525.724	−	303.527i	−0.659629	−	0.380837i	−0.792136	+	0.610345i	$0.791031\pi$
$798$		0			0
$799$	−145.938	−	252.773i	−0.182651	−	0.316361i
$800$		0			0
$801$		0			0
$802$		0			0
$803$			101.370i			0.126239i
$804$		0			0
$805$	399.805	−	692.483i	0.496652	−	0.860227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	144.203i		−	0.178248i	−0.996021	−	0.0891241i	$-0.971593\pi$
$809$		−	144.203i		−	0.178248i	0.996021	−	0.0891241i	$-0.0284068\pi$
$810$		0			0
$811$	−487.189	−	843.837i	−0.600727	−	1.04049i	−0.992711	−	0.120518i	$-0.961545\pi$
$811$	−487.189	−	843.837i	−0.600727	−	1.04049i	0.391984	−	0.919972i	$-0.371789\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	220.092	−	127.070i	0.270052	−	0.155914i
$816$		0			0
$817$	783.541	−	126.129i	0.959047	−	0.154381i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1206.11	+	696.347i	1.46907	+	0.848169i	0.999399	−	0.0346718i	$-0.0110386\pi$
$821$	1206.11	+	696.347i	1.46907	+	0.848169i	0.469673	+	0.882841i	$0.344372\pi$
$822$		0			0
$823$	654.074			0.794744			0.397372	−	0.917658i	$-0.369922\pi$
$823$	654.074			0.794744			0.397372	+	0.917658i	$0.369922\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−966.498	+	558.008i	−1.16868	+	0.674737i	−0.953369	−	0.301808i	$-0.902410\pi$
$827$	−966.498	+	558.008i	−1.16868	+	0.674737i	−0.215310	+	0.976546i	$0.569076\pi$
$828$		0			0
$829$	−42.5592			−0.0513380			−0.0256690	−	0.999670i	$-0.508172\pi$
$829$	−42.5592			−0.0513380			−0.0256690	+	0.999670i	$0.508172\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−312.861	+	180.630i	−0.375583	+	0.216843i
$834$		0			0
$835$	1075.65	+	1863.09i	1.28821	+	2.23124i
$836$		0			0
$837$		0			0
$838$		0			0
$839$			477.884i			0.569588i	0.958589	+	0.284794i	$0.0919252\pi$
$839$			477.884i			0.569588i	−0.958589	+	0.284794i	$0.908075\pi$
$840$		0			0
$841$	−395.261	−	684.611i	−0.469989	−	0.814044i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1006.80	−	581.274i	1.19147	−	0.687898i
$846$		0			0
$847$	544.484			0.642839
$848$		0			0
$849$		0			0
$850$		0			0
$851$			320.456i			0.376565i
$852$		0			0
$853$	−881.731			−1.03368			−0.516841	−	0.856081i	$-0.672892\pi$
$853$	−881.731			−1.03368			−0.516841	+	0.856081i	$0.672892\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			840.347i			0.980568i	0.871563	+	0.490284i	$0.163107\pi$
$857$			840.347i			0.980568i	−0.871563	+	0.490284i	$0.836893\pi$
$858$		0			0
$859$	−1156.41			−1.34623			−0.673113	−	0.739540i	$-0.735043\pi$
$859$	−1156.41			−1.34623			−0.673113	+	0.739540i	$0.735043\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	267.381i		−	0.309828i	−0.987928	−	0.154914i	$-0.950490\pi$
$863$		−	267.381i		−	0.309828i	0.987928	−	0.154914i	$-0.0495100\pi$
$864$		0			0
$865$	555.640	+	962.397i	0.642359	+	1.11260i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	119.951	−	69.2535i	0.138033	−	0.0796933i
$870$		0			0
$871$	−282.771			−0.324651
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1329.39	−	767.522i	1.51930	−	0.877168i
$876$		0			0
$877$	−717.672	−	1243.04i	−0.818326	−	1.41738i	−0.906915	−	0.421314i	$-0.861569\pi$
$877$	−717.672	−	1243.04i	−0.818326	−	1.41738i	0.0885883	−	0.996068i	$-0.471764\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			920.034i			1.04431i	0.852852	+	0.522153i	$0.174871\pi$
$881$			920.034i			1.04431i	−0.852852	+	0.522153i	$0.825129\pi$
$882$		0			0
$883$	261.135	+	452.299i	0.295736	+	0.512230i	0.975156	−	0.221520i	$-0.0711018\pi$
$883$	261.135	+	452.299i	0.295736	+	0.512230i	−0.679420	+	0.733750i	$0.737768\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			276.596i			0.311833i	0.987770	+	0.155917i	$0.0498331\pi$
$887$			276.596i			0.311833i	−0.987770	+	0.155917i	$0.950167\pi$
$888$		0			0
$889$	257.633	−	446.233i	0.289800	−	0.501949i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−404.449	−	154.083i	−0.452910	−	0.172546i
$894$		0			0
$895$	−849.188	−	1470.84i	−0.948813	−	1.64339i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	97.2542	−	56.1497i	0.108180	−	0.0624580i
$900$		0			0
$901$	−86.8086			−0.0963470
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1253.88	−	723.929i	−1.38551	−	0.799922i
$906$		0			0
$907$	−342.930			−0.378093			−0.189046	−	0.981968i	$-0.560540\pi$
$907$	−342.930			−0.378093			−0.189046	+	0.981968i	$0.560540\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	629.426	−	363.399i	0.690918	−	0.398901i	−0.113038	−	0.993591i	$-0.536058\pi$
$911$	629.426	−	363.399i	0.690918	−	0.398901i	0.803956	+	0.594689i	$0.202725\pi$
$912$		0			0
$913$	−25.5838	+	44.3124i	−0.0280217	+	0.0485350i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−33.6261	−	19.4140i	−0.0366696	−	0.0211712i
$918$		0			0
$919$	−851.166			−0.926188			−0.463094	−	0.886309i	$-0.653261\pi$
$919$	−851.166			−0.926188			−0.463094	+	0.886309i	$0.653261\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−142.099	−	82.0408i	−0.153953	−	0.0888850i
$924$		0			0
$925$	−519.796	+	900.313i	−0.561942	+	0.973312i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			908.663i			0.978109i	0.872253	+	0.489055i	$0.162658\pi$
$929$			908.663i			0.978109i	−0.872253	+	0.489055i	$0.837342\pi$
$930$		0			0
$931$	−190.711	+	500.593i	−0.204846	+	0.537693i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	131.584	−	75.9700i	0.140731	−	0.0812513i
$936$		0			0
$937$	−38.1604	−	66.0958i	−0.0407262	−	0.0705398i	0.844944	−	0.534855i	$-0.179634\pi$
$937$	−38.1604	−	66.0958i	−0.0407262	−	0.0705398i	−0.885670	+	0.464315i	$0.846300\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			871.904i			0.926572i	0.886209	+	0.463286i	$0.153330\pi$
$941$			871.904i			0.926572i	−0.886209	+	0.463286i	$0.846670\pi$
$942$		0			0
$943$	166.973	+	289.206i	0.177066	+	0.306687i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			327.312i			0.345630i	0.984954	+	0.172815i	$0.0552863\pi$
$947$			327.312i			0.345630i	−0.984954	+	0.172815i	$0.944714\pi$
$948$		0			0
$949$	−262.736	+	455.073i	−0.276856	+	0.479529i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	87.9770	+	50.7935i	0.0923158	+	0.0532986i	0.545447	−	0.838145i	$-0.316360\pi$
$953$	87.9770	+	50.7935i	0.0923158	+	0.0532986i	−0.453131	+	0.891444i	$0.649693\pi$
$954$		0			0
$955$	−1570.52	+	2720.22i	−1.64453	+	2.84840i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	977.010i		−	1.01878i
$960$		0			0
$961$	355.585	+	615.891i	0.370015	+	0.640885i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1713.08	−	989.045i	−1.77521	−	1.02492i
$966$		0			0
$967$	−322.803	−	559.111i	−0.333819	−	0.578191i	0.649439	−	0.760414i	$-0.275004\pi$
$967$	−322.803	−	559.111i	−0.333819	−	0.578191i	−0.983257	+	0.182223i	$0.941671\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1455.94	+	840.588i	1.49943	+	0.865693i	1.00000		0.000663529i	$-0.000211208\pi$
$971$	1455.94	+	840.588i	1.49943	+	0.865693i	0.499425	+	0.866357i	$0.333545\pi$
$972$		0			0
$973$	117.553	+	203.607i	0.120815	+	0.209257i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−417.952	−	241.305i	−0.427792	−	0.246986i	0.270614	−	0.962688i	$-0.412773\pi$
$977$	−417.952	−	241.305i	−0.427792	−	0.246986i	−0.698405	+	0.715702i	$0.746107\pi$
$978$		0			0
$979$	−36.3042	+	62.8807i	−0.0370829	+	0.0642295i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			329.490i			0.335189i	0.985856	+	0.167594i	$0.0535999\pi$
$983$			329.490i			0.335189i	−0.985856	+	0.167594i	$0.946400\pi$
$984$		0			0
$985$	−35.9830	−	62.3244i	−0.0365310	−	0.0632735i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			788.492i			0.797262i
$990$		0			0
$991$	130.189	−	225.494i	0.131371	−	0.227542i	−0.792834	−	0.609438i	$-0.791395\pi$
$991$	130.189	−	225.494i	0.131371	−	0.227542i	0.924205	+	0.381896i	$0.124729\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2412.36	+	1392.78i	2.42448	+	1.39978i
$996$		0			0
$997$	−726.025	+	1257.51i	−0.728210	+	1.26130i	0.229429	+	0.973325i	$0.426314\pi$
$997$	−726.025	+	1257.51i	−0.728210	+	1.26130i	−0.957639	+	0.287971i	$0.907019\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2052.3.be.a.125.1		80
3.2	odd	2		684.3.be.a.581.38	yes	80
9.2	odd	6		2052.3.m.a.1493.40		80
9.7	even	3		684.3.m.a.353.12	✓	80
19.7	even	3		2052.3.m.a.881.1		80
57.26	odd	6		684.3.m.a.653.12	yes	80
171.7	even	3		684.3.be.a.425.38	yes	80
171.83	odd	6	inner	2052.3.be.a.197.1		80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.3.m.a.353.12	✓	80	9.7	even	3
684.3.m.a.653.12	yes	80	57.26	odd	6
684.3.be.a.425.38	yes	80	171.7	even	3
684.3.be.a.581.38	yes	80	3.2	odd	2
2052.3.m.a.881.1		80	19.7	even	3
2052.3.m.a.1493.40		80	9.2	odd	6
2052.3.be.a.125.1		80	1.1	even	1	trivial
2052.3.be.a.197.1		80	171.83	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2052 = 2^{2} \cdot 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2052.be (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-8.04234 + 4.64325i) q^{5} +(-2.28067 - 3.95023i) q^{7} +O(q^{10})\)	\(q+(-8.04234 + 4.64325i) q^{5} +(-2.28067 - 3.95023i) q^{7} +(1.10584 - 0.638454i) q^{11} +6.61915 q^{13} +(-11.0966 - 6.40664i) q^{17} +(-18.7585 + 3.01961i) q^{19} -18.8771i q^{23} +(30.6195 - 53.0346i) q^{25} +(6.15298 + 3.55242i) q^{29} +(7.90302 - 13.6884i) q^{31} +(36.6838 + 21.1794i) q^{35} -16.9760 q^{37} +(-15.3205 + 8.84528i) q^{41} -41.7699 q^{43} +(19.7274 + 11.3896i) q^{47} +(14.0971 - 24.4169i) q^{49} +(5.86723 - 3.38745i) q^{53} +(-5.92901 + 10.2693i) q^{55} +(71.6659 - 41.3763i) q^{59} +(-23.7419 + 41.1222i) q^{61} +(-53.2335 + 30.7344i) q^{65} -42.7202 q^{67} +(-21.4678 - 12.3945i) q^{71} +(-39.6934 + 68.7509i) q^{73} +(-5.04409 - 2.91221i) q^{77} +108.471 q^{79} +(-34.7029 + 20.0357i) q^{83} +118.990 q^{85} +(-49.2445 + 28.4313i) q^{89} +(-15.0961 - 26.1472i) q^{91} +(136.842 - 111.385i) q^{95} +173.722 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + q^{7}+O(q^{10}) \) 80 * q + q^7	\( 80 q + q^{7} - 18 q^{11} + 10 q^{13} - 9 q^{17} + 20 q^{19} + 200 q^{25} + 27 q^{29} - 8 q^{31} + 22 q^{37} + 54 q^{41} + 88 q^{43} - 198 q^{47} - 267 q^{49} - 36 q^{53} - 171 q^{59} + 7 q^{61} + 144 q^{65} + 154 q^{67} - 135 q^{71} + 43 q^{73} - 216 q^{77} + 34 q^{79} + 171 q^{83} + 216 q^{89} + 122 q^{91} + 216 q^{95} + 16 q^{97}+O(q^{100}) \) 80 * q + q^7 - 18 * q^11 + 10 * q^13 - 9 * q^17 + 20 * q^19 + 200 * q^25 + 27 * q^29 - 8 * q^31 + 22 * q^37 + 54 * q^41 + 88 * q^43 - 198 * q^47 - 267 * q^49 - 36 * q^53 - 171 * q^59 + 7 * q^61 + 144 * q^65 + 154 * q^67 - 135 * q^71 + 43 * q^73 - 216 * q^77 + 34 * q^79 + 171 * q^83 + 216 * q^89 + 122 * q^91 + 216 * q^95 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more