LMFDB - Embedded newform 1444.3.c.d.721.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,3,Mod(721,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("1444.721");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1444 = 2^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1444.c (of order $2$, degree $1$, 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:	$39.3461501736$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			721.20
Character	$\chi$	$=$	1444.721
Dual form			1444.3.c.d.721.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.28147i q^{3} -1.88757 q^{5} +1.12007 q^{7} -9.33100 q^{9} +O(q^{10})$	$q+4.28147i q^{3} -1.88757 q^{5} +1.12007 q^{7} -9.33100 q^{9} -9.34284 q^{11} +23.0228i q^{13} -8.08159i q^{15} +15.6171 q^{17} +4.79553i q^{21} +10.4638 q^{23} -21.4371 q^{25} -1.41718i q^{27} +45.5963i q^{29} +43.8900i q^{31} -40.0011i q^{33} -2.11421 q^{35} -60.6912i q^{37} -98.5716 q^{39} -48.7975i q^{41} +8.19532 q^{43} +17.6129 q^{45} -71.9274 q^{47} -47.7455 q^{49} +66.8640i q^{51} -28.3575i q^{53} +17.6353 q^{55} -58.0515i q^{59} -37.3828 q^{61} -10.4513 q^{63} -43.4573i q^{65} -55.7302i q^{67} +44.8007i q^{69} +5.04846i q^{71} -8.53268 q^{73} -91.7822i q^{75} -10.4646 q^{77} +6.01967i q^{79} -77.9114 q^{81} +142.629 q^{83} -29.4784 q^{85} -195.219 q^{87} +167.710i q^{89} +25.7871i q^{91} -187.914 q^{93} +110.328i q^{97} +87.1781 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	$ 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) $ 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$.

$n$	$723$	$1085$
$\chi(n)$	$1$	$-1$

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
$3$	4.28147i	1.42716i	0.700575	+	0.713579i	$0.252927\pi$
$3$	4.28147i	1.42716i	−0.700575	+	0.713579i	$0.747073\pi$
$4$	0	0
$5$	−1.88757	−0.377515	−0.188757	−	0.982024i	$-0.560446\pi$
$5$	−1.88757	−0.377515	−0.188757	+	0.982024i	$0.560446\pi$
$6$	0	0
$7$	1.12007	0.160010	0.0800048	−	0.996794i	$-0.474506\pi$
$7$	1.12007	0.160010	0.0800048	+	0.996794i	$0.474506\pi$
$8$	0	0
$9$	−9.33100	−1.03678
$10$	0	0
$11$	−9.34284	−0.849349	−0.424675	−	0.905346i	$-0.639612\pi$
$11$	−9.34284	−0.849349	−0.424675	+	0.905346i	$0.639612\pi$
$12$	0	0
$13$	23.0228i	1.77099i	0.464652	+	0.885494i	$0.346180\pi$
$13$	23.0228i	1.77099i	−0.464652	+	0.885494i	$0.653820\pi$
$14$	0	0
$15$	− 8.08159i	− 0.538773i
$16$	0	0
$17$	15.6171	0.918651	0.459326	−	0.888268i	$-0.348091\pi$
$17$	15.6171	0.918651	0.459326	+	0.888268i	$0.348091\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.79553i	0.228359i
$22$	0	0
$23$	10.4638	0.454950	0.227475	−	0.973784i	$-0.426953\pi$
$23$	10.4638	0.454950	0.227475	+	0.973784i	$0.426953\pi$
$24$	0	0
$25$	−21.4371	−0.857483
$26$	0	0
$27$	− 1.41718i	− 0.0524881i
$28$	0	0
$29$	45.5963i	1.57229i	0.618044	+	0.786144i	$0.287926\pi$
$29$	45.5963i	1.57229i	−0.618044	+	0.786144i	$0.712074\pi$
$30$	0	0
$31$	43.8900i	1.41581i	0.706309	+	0.707904i	$0.250359\pi$
$31$	43.8900i	1.41581i	−0.706309	+	0.707904i	$0.749641\pi$
$32$	0	0
$33$	− 40.0011i	− 1.21216i
$34$	0	0
$35$	−2.11421	−0.0604059
$36$	0	0
$37$	− 60.6912i	− 1.64030i	−0.572147	−	0.820151i	$-0.693889\pi$
$37$	− 60.6912i	− 1.64030i	0.572147	−	0.820151i	$-0.306111\pi$
$38$	0	0
$39$	−98.5716	−2.52748
$40$	0	0
$41$	− 48.7975i	− 1.19018i	−0.803658	−	0.595091i	$-0.797116\pi$
$41$	− 48.7975i	− 1.19018i	0.803658	−	0.595091i	$-0.202884\pi$
$42$	0	0
$43$	8.19532	0.190589	0.0952944	−	0.995449i	$-0.469621\pi$
$43$	8.19532	0.190589	0.0952944	+	0.995449i	$0.469621\pi$
$44$	0	0
$45$	17.6129	0.391399
$46$	0	0
$47$	−71.9274	−1.53037	−0.765185	−	0.643811i	$-0.777352\pi$
$47$	−71.9274	−1.53037	−0.765185	+	0.643811i	$0.777352\pi$
$48$	0	0
$49$	−47.7455	−0.974397
$50$	0	0
$51$	66.8640i	1.31106i
$52$	0	0
$53$	− 28.3575i	− 0.535047i	−0.963551	−	0.267524i	$-0.913795\pi$
$53$	− 28.3575i	− 0.535047i	0.963551	−	0.267524i	$-0.0862054\pi$
$54$	0	0
$55$	17.6353	0.320642
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 58.0515i	− 0.983923i	−0.870617	−	0.491962i	$-0.836280\pi$
$59$	− 58.0515i	− 0.983923i	0.870617	−	0.491962i	$-0.163720\pi$
$60$	0	0
$61$	−37.3828	−0.612832	−0.306416	−	0.951898i	$-0.599130\pi$
$61$	−37.3828	−0.612832	−0.306416	+	0.951898i	$0.599130\pi$
$62$	0	0
$63$	−10.4513	−0.165894
$64$	0	0
$65$	− 43.4573i	− 0.668574i
$66$	0	0
$67$	− 55.7302i	− 0.831794i	−0.909412	−	0.415897i	$-0.863468\pi$
$67$	− 55.7302i	− 0.831794i	0.909412	−	0.415897i	$-0.136532\pi$
$68$	0	0
$69$	44.8007i	0.649285i
$70$	0	0
$71$	5.04846i	0.0711051i	0.999368	+	0.0355525i	$0.0113191\pi$
$71$	5.04846i	0.0711051i	−0.999368	+	0.0355525i	$0.988681\pi$
$72$	0	0
$73$	−8.53268	−0.116886	−0.0584430	−	0.998291i	$-0.518614\pi$
$73$	−8.53268	−0.116886	−0.0584430	+	0.998291i	$0.518614\pi$
$74$	0	0
$75$	− 91.7822i	− 1.22376i
$76$	0	0
$77$	−10.4646	−0.135904
$78$	0	0
$79$	6.01967i	0.0761984i	0.999274	+	0.0380992i	$0.0121303\pi$
$79$	6.01967i	0.0761984i	−0.999274	+	0.0380992i	$0.987870\pi$
$80$	0	0
$81$	−77.9114	−0.961869
$82$	0	0
$83$	142.629	1.71842	0.859209	−	0.511624i	$-0.170956\pi$
$83$	142.629	1.71842	0.859209	+	0.511624i	$0.170956\pi$
$84$	0	0
$85$	−29.4784	−0.346804
$86$	0	0
$87$	−195.219	−2.24390
$88$	0	0
$89$	167.710i	1.88438i	0.335079	+	0.942190i	$0.391237\pi$
$89$	167.710i	1.88438i	−0.335079	+	0.942190i	$0.608763\pi$
$90$	0	0
$91$	25.7871i	0.283375i
$92$	0	0
$93$	−187.914	−2.02058
$94$	0	0
$95$	0	0
$96$	0	0
$97$	110.328i	1.13740i	0.822545	+	0.568700i	$0.192553\pi$
$97$	110.328i	1.13740i	−0.822545	+	0.568700i	$0.807447\pi$
$98$	0	0
$99$	87.1781	0.880587
$100$	0	0
$101$	0.281646	0.00278858	0.00139429	−	0.999999i	$-0.499556\pi$
$101$	0.281646	0.00278858	0.00139429	+	0.999999i	$0.499556\pi$
$102$	0	0
$103$	− 93.8690i	− 0.911350i	−0.890146	−	0.455675i	$-0.849398\pi$
$103$	− 93.8690i	− 0.911350i	0.890146	−	0.455675i	$-0.150602\pi$
$104$	0	0
$105$	− 9.05192i	− 0.0862088i
$106$	0	0
$107$	− 186.594i	− 1.74387i	−0.489624	−	0.871934i	$-0.662866\pi$
$107$	− 186.594i	− 1.74387i	0.489624	−	0.871934i	$-0.337134\pi$
$108$	0	0
$109$	− 6.85241i	− 0.0628661i	−0.999506	−	0.0314331i	$-0.989993\pi$
$109$	− 6.85241i	− 0.0628661i	0.999506	−	0.0314331i	$-0.0100071\pi$
$110$	0	0
$111$	259.848	2.34097
$112$	0	0
$113$	18.6288i	0.164857i	0.996597	+	0.0824284i	$0.0262676\pi$
$113$	18.6288i	0.164857i	−0.996597	+	0.0824284i	$0.973732\pi$
$114$	0	0
$115$	−19.7513	−0.171750
$116$	0	0
$117$	− 214.826i	− 1.83612i
$118$	0	0
$119$	17.4922	0.146993
$120$	0	0
$121$	−33.7113	−0.278606
$122$	0	0
$123$	208.925	1.69858
$124$	0	0
$125$	87.6534	0.701227
$126$	0	0
$127$	− 70.4894i	− 0.555035i	−0.960721	−	0.277517i	$-0.910488\pi$
$127$	− 70.4894i	− 0.555035i	0.960721	−	0.277517i	$-0.0895116\pi$
$128$	0	0
$129$	35.0880i	0.272000i
$130$	0	0
$131$	−143.504	−1.09545	−0.547727	−	0.836657i	$-0.684507\pi$
$131$	−143.504	−1.09545	−0.547727	+	0.836657i	$0.684507\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.67503i	0.0198150i
$136$	0	0
$137$	44.9069	0.327788	0.163894	−	0.986478i	$-0.447595\pi$
$137$	44.9069	0.327788	0.163894	+	0.986478i	$0.447595\pi$
$138$	0	0
$139$	164.709	1.18496	0.592478	−	0.805587i	$-0.298150\pi$
$139$	164.709	1.18496	0.592478	+	0.805587i	$0.298150\pi$
$140$	0	0
$141$	− 307.955i	− 2.18408i
$142$	0	0
$143$	− 215.099i	− 1.50419i
$144$	0	0
$145$	− 86.0664i	− 0.593562i
$146$	0	0
$147$	− 204.421i	− 1.39062i
$148$	0	0
$149$	50.2928	0.337536	0.168768	−	0.985656i	$-0.446021\pi$
$149$	50.2928	0.337536	0.168768	+	0.985656i	$0.446021\pi$
$150$	0	0
$151$	− 107.612i	− 0.712664i	−0.934359	−	0.356332i	$-0.884027\pi$
$151$	− 107.612i	− 0.712664i	0.934359	−	0.356332i	$-0.115973\pi$
$152$	0	0
$153$	−145.723	−0.952437
$154$	0	0
$155$	− 82.8456i	− 0.534488i
$156$	0	0
$157$	−230.070	−1.46542	−0.732708	−	0.680543i	$-0.761744\pi$
$157$	−230.070	−1.46542	−0.732708	+	0.680543i	$0.761744\pi$
$158$	0	0
$159$	121.412	0.763597
$160$	0	0
$161$	11.7202	0.0727963
$162$	0	0
$163$	−200.475	−1.22991	−0.614953	−	0.788564i	$-0.710825\pi$
$163$	−200.475	−1.22991	−0.614953	+	0.788564i	$0.710825\pi$
$164$	0	0
$165$	75.5050i	0.457606i
$166$	0	0
$167$	288.051i	1.72485i	0.506182	+	0.862427i	$0.331057\pi$
$167$	288.051i	1.72485i	−0.506182	+	0.862427i	$0.668943\pi$
$168$	0	0
$169$	−361.051	−2.13640
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.4973i	0.124262i	0.998068	+	0.0621310i	$0.0197897\pi$
$173$	21.4973i	0.124262i	−0.998068	+	0.0621310i	$0.980210\pi$
$174$	0	0
$175$	−24.0109	−0.137205
$176$	0	0
$177$	248.546	1.40421
$178$	0	0
$179$	− 8.06692i	− 0.0450666i	−0.999746	−	0.0225333i	$-0.992827\pi$
$179$	− 8.06692i	− 0.0450666i	0.999746	−	0.0225333i	$-0.00717318\pi$
$180$	0	0
$181$	− 242.831i	− 1.34161i	−0.741634	−	0.670804i	$-0.765949\pi$
$181$	− 242.831i	− 1.34161i	0.741634	−	0.670804i	$-0.234051\pi$
$182$	0	0
$183$	− 160.053i	− 0.874608i
$184$	0	0
$185$	114.559i	0.619238i
$186$	0	0
$187$	−145.908	−0.780256
$188$	0	0
$189$	− 1.58733i	− 0.00839860i
$190$	0	0
$191$	267.047	1.39815	0.699075	−	0.715048i	$-0.253595\pi$
$191$	267.047	1.39815	0.699075	+	0.715048i	$0.253595\pi$
$192$	0	0
$193$	− 142.570i	− 0.738707i	−0.929289	−	0.369353i	$-0.879579\pi$
$193$	− 142.570i	− 0.738707i	0.929289	−	0.369353i	$-0.120421\pi$
$194$	0	0
$195$	186.061	0.954160
$196$	0	0
$197$	13.7319	0.0697053	0.0348527	−	0.999392i	$-0.488904\pi$
$197$	13.7319	0.0697053	0.0348527	+	0.999392i	$0.488904\pi$
$198$	0	0
$199$	−146.846	−0.737920	−0.368960	−	0.929445i	$-0.620286\pi$
$199$	−146.846	−0.737920	−0.368960	+	0.929445i	$0.620286\pi$
$200$	0	0
$201$	238.607	1.18710
$202$	0	0
$203$	51.0709i	0.251581i
$204$	0	0
$205$	92.1088i	0.449311i
$206$	0	0
$207$	−97.6382	−0.471682
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 330.505i	− 1.56637i	−0.621787	−	0.783187i	$-0.713593\pi$
$211$	− 330.505i	− 1.56637i	0.621787	−	0.783187i	$-0.286407\pi$
$212$	0	0
$213$	−21.6148	−0.101478
$214$	0	0
$215$	−15.4693	−0.0719501
$216$	0	0
$217$	49.1598i	0.226543i
$218$	0	0
$219$	− 36.5324i	− 0.166815i
$220$	0	0
$221$	359.549i	1.62692i
$222$	0	0
$223$	112.946i	0.506485i	0.967403	+	0.253242i	$0.0814970\pi$
$223$	112.946i	0.506485i	−0.967403	+	0.253242i	$0.918503\pi$
$224$	0	0
$225$	200.029	0.889019
$226$	0	0
$227$	− 11.7523i	− 0.0517722i	−0.999665	−	0.0258861i	$-0.991759\pi$
$227$	− 11.7523i	− 0.0517722i	0.999665	−	0.0258861i	$-0.00824072\pi$
$228$	0	0
$229$	176.482	0.770665	0.385333	−	0.922778i	$-0.374087\pi$
$229$	176.482	0.770665	0.385333	+	0.922778i	$0.374087\pi$
$230$	0	0
$231$	− 44.8039i	− 0.193956i
$232$	0	0
$233$	−258.210	−1.10820	−0.554098	−	0.832452i	$-0.686937\pi$
$233$	−258.210	−1.10820	−0.554098	+	0.832452i	$0.686937\pi$
$234$	0	0
$235$	135.768	0.577737
$236$	0	0
$237$	−25.7731	−0.108747
$238$	0	0
$239$	−471.048	−1.97091	−0.985456	−	0.169928i	$-0.945646\pi$
$239$	−471.048	−1.97091	−0.985456	+	0.169928i	$0.945646\pi$
$240$	0	0
$241$	325.302i	1.34980i	0.737909	+	0.674900i	$0.235813\pi$
$241$	325.302i	1.34980i	−0.737909	+	0.674900i	$0.764187\pi$
$242$	0	0
$243$	− 346.330i	− 1.42523i
$244$	0	0
$245$	90.1230	0.367849
$246$	0	0
$247$	0	0
$248$	0	0
$249$	610.661i	2.45245i
$250$	0	0
$251$	100.711	0.401237	0.200619	−	0.979669i	$-0.435705\pi$
$251$	100.711	0.401237	0.200619	+	0.979669i	$0.435705\pi$
$252$	0	0
$253$	−97.7621	−0.386411
$254$	0	0
$255$	− 126.211i	− 0.494944i
$256$	0	0
$257$	377.944i	1.47060i	0.677743	+	0.735299i	$0.262958\pi$
$257$	377.944i	1.47060i	−0.677743	+	0.735299i	$0.737042\pi$
$258$	0	0
$259$	− 67.9782i	− 0.262464i
$260$	0	0
$261$	− 425.460i	− 1.63011i
$262$	0	0
$263$	−30.2137	−0.114881	−0.0574405	−	0.998349i	$-0.518294\pi$
$263$	−30.2137	−0.114881	−0.0574405	+	0.998349i	$0.518294\pi$
$264$	0	0
$265$	53.5269i	0.201988i
$266$	0	0
$267$	−718.045	−2.68931
$268$	0	0
$269$	159.383i	0.592500i	0.955110	+	0.296250i	$0.0957362\pi$
$269$	159.383i	0.592500i	−0.955110	+	0.296250i	$0.904264\pi$
$270$	0	0
$271$	177.494	0.654960	0.327480	−	0.944858i	$-0.393801\pi$
$271$	177.494	0.654960	0.327480	+	0.944858i	$0.393801\pi$
$272$	0	0
$273$	−110.407	−0.404420
$274$	0	0
$275$	200.283	0.728302
$276$	0	0
$277$	395.518	1.42786	0.713931	−	0.700216i	$-0.246913\pi$
$277$	395.518	1.42786	0.713931	+	0.700216i	$0.246913\pi$
$278$	0	0
$279$	− 409.538i	− 1.46788i
$280$	0	0
$281$	12.1060i	0.0430818i	0.999768	+	0.0215409i	$0.00685722\pi$
$281$	12.1060i	0.0430818i	−0.999768	+	0.0215409i	$0.993143\pi$
$282$	0	0
$283$	402.299	1.42155	0.710776	−	0.703418i	$-0.248344\pi$
$283$	402.299	1.42155	0.710776	+	0.703418i	$0.248344\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 54.6564i	− 0.190441i
$288$	0	0
$289$	−45.1071	−0.156080
$290$	0	0
$291$	−472.365	−1.62325
$292$	0	0
$293$	288.411i	0.984338i	0.870500	+	0.492169i	$0.163796\pi$
$293$	288.411i	0.984338i	−0.870500	+	0.492169i	$0.836204\pi$
$294$	0	0
$295$	109.576i	0.371445i
$296$	0	0
$297$	13.2405i	0.0445807i
$298$	0	0
$299$	240.907i	0.805710i
$300$	0	0
$301$	9.17931	0.0304960
$302$	0	0
$303$	1.20586i	0.00397974i
$304$	0	0
$305$	70.5627	0.231353
$306$	0	0
$307$	417.916i	1.36129i	0.732614	+	0.680644i	$0.238300\pi$
$307$	417.916i	1.36129i	−0.732614	+	0.680644i	$0.761700\pi$
$308$	0	0
$309$	401.898	1.30064
$310$	0	0
$311$	−218.502	−0.702579	−0.351290	−	0.936267i	$-0.614257\pi$
$311$	−218.502	−0.702579	−0.351290	+	0.936267i	$0.614257\pi$
$312$	0	0
$313$	−466.572	−1.49064	−0.745322	−	0.666704i	$-0.767704\pi$
$313$	−466.572	−1.49064	−0.745322	+	0.666704i	$0.767704\pi$
$314$	0	0
$315$	19.7277	0.0626275
$316$	0	0
$317$	261.372i	0.824517i	0.911067	+	0.412258i	$0.135260\pi$
$317$	261.372i	0.824517i	−0.911067	+	0.412258i	$0.864740\pi$
$318$	0	0
$319$	− 425.999i	− 1.33542i
$320$	0	0
$321$	798.896	2.48877
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 493.542i	− 1.51859i
$326$	0	0
$327$	29.3384	0.0897198
$328$	0	0
$329$	−80.5634	−0.244874
$330$	0	0
$331$	160.364i	0.484484i	0.970216	+	0.242242i	$0.0778828\pi$
$331$	160.364i	0.484484i	−0.970216	+	0.242242i	$0.922117\pi$
$332$	0	0
$333$	566.309i	1.70063i
$334$	0	0
$335$	105.195i	0.314014i
$336$	0	0
$337$	84.0647i	0.249450i	0.992191	+	0.124725i	$0.0398049\pi$
$337$	84.0647i	0.249450i	−0.992191	+	0.124725i	$0.960195\pi$
$338$	0	0
$339$	−79.7587	−0.235277
$340$	0	0
$341$	− 410.058i	− 1.20252i
$342$	0	0
$343$	−108.361	−0.315922
$344$	0	0
$345$	− 84.5645i	− 0.245115i
$346$	0	0
$347$	−414.379	−1.19418	−0.597088	−	0.802176i	$-0.703676\pi$
$347$	−414.379	−1.19418	−0.597088	+	0.802176i	$0.703676\pi$
$348$	0	0
$349$	−25.3421	−0.0726135	−0.0363068	−	0.999341i	$-0.511559\pi$
$349$	−25.3421	−0.0726135	−0.0363068	+	0.999341i	$0.511559\pi$
$350$	0	0
$351$	32.6275	0.0929558
$352$	0	0
$353$	−355.314	−1.00656	−0.503278	−	0.864125i	$-0.667873\pi$
$353$	−355.314	−1.00656	−0.503278	+	0.864125i	$0.667873\pi$
$354$	0	0
$355$	− 9.52934i	− 0.0268432i
$356$	0	0
$357$	74.8922i	0.209782i
$358$	0	0
$359$	75.5555	0.210461	0.105231	−	0.994448i	$-0.466442\pi$
$359$	75.5555	0.210461	0.105231	+	0.994448i	$0.466442\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	− 144.334i	− 0.397614i
$364$	0	0
$365$	16.1061	0.0441262
$366$	0	0
$367$	−353.387	−0.962909	−0.481454	−	0.876471i	$-0.659891\pi$
$367$	−353.387	−0.962909	−0.481454	+	0.876471i	$0.659891\pi$
$368$	0	0
$369$	455.329i	1.23396i
$370$	0	0
$371$	− 31.7623i	− 0.0856127i
$372$	0	0
$373$	387.471i	1.03880i	0.854533	+	0.519398i	$0.173844\pi$
$373$	387.471i	1.03880i	−0.854533	+	0.519398i	$0.826156\pi$
$374$	0	0
$375$	375.285i	1.00076i
$376$	0	0
$377$	−1049.76	−2.78450
$378$	0	0
$379$	194.502i	0.513198i	0.966518	+	0.256599i	$0.0826020\pi$
$379$	194.502i	0.513198i	−0.966518	+	0.256599i	$0.917398\pi$
$380$	0	0
$381$	301.799	0.792122
$382$	0	0
$383$	303.699i	0.792948i	0.918046	+	0.396474i	$0.129766\pi$
$383$	303.699i	0.792948i	−0.918046	+	0.396474i	$0.870234\pi$
$384$	0	0
$385$	19.7527	0.0513057
$386$	0	0
$387$	−76.4706	−0.197598
$388$	0	0
$389$	−216.046	−0.555389	−0.277694	−	0.960670i	$-0.589570\pi$
$389$	−216.046	−0.555389	−0.277694	+	0.960670i	$0.589570\pi$
$390$	0	0
$391$	163.415	0.417940
$392$	0	0
$393$	− 614.410i	− 1.56338i
$394$	0	0
$395$	− 11.3626i	− 0.0287660i
$396$	0	0
$397$	35.2625	0.0888224	0.0444112	−	0.999013i	$-0.485859\pi$
$397$	35.2625	0.0888224	0.0444112	+	0.999013i	$0.485859\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 163.938i	− 0.408822i	−0.978885	−	0.204411i	$-0.934472\pi$
$401$	− 163.938i	− 0.408822i	0.978885	−	0.204411i	$-0.0655279\pi$
$402$	0	0
$403$	−1010.47	−2.50738
$404$	0	0
$405$	147.063	0.363120
$406$	0	0
$407$	567.028i	1.39319i
$408$	0	0
$409$	15.1466i	0.0370332i	0.999829	+	0.0185166i	$0.00589435\pi$
$409$	15.1466i	0.0370332i	−0.999829	+	0.0185166i	$0.994106\pi$
$410$	0	0
$411$	192.268i	0.467804i
$412$	0	0
$413$	− 65.0215i	− 0.157437i
$414$	0	0
$415$	−269.222	−0.648728
$416$	0	0
$417$	705.196i	1.69112i
$418$	0	0
$419$	635.816	1.51746	0.758731	−	0.651405i	$-0.225820\pi$
$419$	635.816	1.51746	0.758731	+	0.651405i	$0.225820\pi$
$420$	0	0
$421$	409.475i	0.972626i	0.873785	+	0.486313i	$0.161658\pi$
$421$	409.475i	0.972626i	−0.873785	+	0.486313i	$0.838342\pi$
$422$	0	0
$423$	671.154	1.58665
$424$	0	0
$425$	−334.784	−0.787727
$426$	0	0
$427$	−41.8712	−0.0980590
$428$	0	0
$429$	920.939	2.14671
$430$	0	0
$431$	− 263.100i	− 0.610442i	−0.952282	−	0.305221i	$-0.901270\pi$
$431$	− 263.100i	− 0.610442i	0.952282	−	0.305221i	$-0.0987303\pi$
$432$	0	0
$433$	34.9675i	0.0807565i	0.999184	+	0.0403782i	$0.0128563\pi$
$433$	34.9675i	0.0807565i	−0.999184	+	0.0403782i	$0.987144\pi$
$434$	0	0
$435$	368.491	0.847106
$436$	0	0
$437$	0	0
$438$	0	0
$439$	334.112i	0.761075i	0.924766	+	0.380537i	$0.124261\pi$
$439$	334.112i	0.761075i	−0.924766	+	0.380537i	$0.875739\pi$
$440$	0	0
$441$	445.513	1.01023
$442$	0	0
$443$	−477.902	−1.07878	−0.539392	−	0.842055i	$-0.681346\pi$
$443$	−477.902	−1.07878	−0.539392	+	0.842055i	$0.681346\pi$
$444$	0	0
$445$	− 316.565i	− 0.711381i
$446$	0	0
$447$	215.327i	0.481717i
$448$	0	0
$449$	198.095i	0.441192i	0.975365	+	0.220596i	$0.0708002\pi$
$449$	198.095i	0.441192i	−0.975365	+	0.220596i	$0.929200\pi$
$450$	0	0
$451$	455.907i	1.01088i
$452$	0	0
$453$	460.739	1.01708
$454$	0	0
$455$	− 48.6750i	− 0.106978i
$456$	0	0
$457$	−74.7502	−0.163567	−0.0817836	−	0.996650i	$-0.526062\pi$
$457$	−74.7502	−0.163567	−0.0817836	+	0.996650i	$0.526062\pi$
$458$	0	0
$459$	− 22.1322i	− 0.0482183i
$460$	0	0
$461$	555.469	1.20492	0.602461	−	0.798149i	$-0.294187\pi$
$461$	555.469	1.20492	0.602461	+	0.798149i	$0.294187\pi$
$462$	0	0
$463$	246.296	0.531956	0.265978	−	0.963979i	$-0.414305\pi$
$463$	246.296	0.531956	0.265978	+	0.963979i	$0.414305\pi$
$464$	0	0
$465$	354.701	0.762799
$466$	0	0
$467$	379.607	0.812862	0.406431	−	0.913681i	$-0.366773\pi$
$467$	379.607	0.812862	0.406431	+	0.913681i	$0.366773\pi$
$468$	0	0
$469$	− 62.4215i	− 0.133095i
$470$	0	0
$471$	− 985.040i	− 2.09138i
$472$	0	0
$473$	−76.5676	−0.161877
$474$	0	0
$475$	0	0
$476$	0	0
$477$	264.604i	0.554726i
$478$	0	0
$479$	61.1533	0.127669	0.0638343	−	0.997961i	$-0.479667\pi$
$479$	61.1533	0.127669	0.0638343	+	0.997961i	$0.479667\pi$
$480$	0	0
$481$	1397.28	2.90495
$482$	0	0
$483$	50.1797i	0.103892i
$484$	0	0
$485$	− 208.252i	− 0.429385i
$486$	0	0
$487$	− 601.378i	− 1.23486i	−0.786625	−	0.617431i	$-0.788173\pi$
$487$	− 601.378i	− 1.23486i	0.786625	−	0.617431i	$-0.211827\pi$
$488$	0	0
$489$	− 858.326i	− 1.75527i
$490$	0	0
$491$	540.115	1.10003	0.550015	−	0.835155i	$-0.314622\pi$
$491$	540.115	1.10003	0.550015	+	0.835155i	$0.314622\pi$
$492$	0	0
$493$	712.081i	1.44438i
$494$	0	0
$495$	−164.555	−0.332434
$496$	0	0
$497$	5.65461i	0.0113775i
$498$	0	0
$499$	−590.929	−1.18423	−0.592113	−	0.805855i	$-0.701706\pi$
$499$	−590.929	−1.18423	−0.592113	+	0.805855i	$0.701706\pi$
$500$	0	0
$501$	−1233.28	−2.46164
$502$	0	0
$503$	754.518	1.50004	0.750018	−	0.661418i	$-0.230045\pi$
$503$	754.518	1.50004	0.750018	+	0.661418i	$0.230045\pi$
$504$	0	0
$505$	−0.531628	−0.00105273
$506$	0	0
$507$	− 1545.83i	− 3.04897i
$508$	0	0
$509$	991.954i	1.94883i	0.224759	+	0.974414i	$0.427841\pi$
$509$	991.954i	1.94883i	−0.224759	+	0.974414i	$0.572159\pi$
$510$	0	0
$511$	−9.55717	−0.0187029
$512$	0	0
$513$	0	0
$514$	0	0
$515$	177.185i	0.344048i
$516$	0	0
$517$	672.006	1.29982
$518$	0	0
$519$	−92.0402	−0.177342
$520$	0	0
$521$	556.632i	1.06839i	0.845361	+	0.534195i	$0.179385\pi$
$521$	556.632i	1.06839i	−0.845361	+	0.534195i	$0.820615\pi$
$522$	0	0
$523$	− 144.441i	− 0.276178i	−0.990420	−	0.138089i	$-0.955904\pi$
$523$	− 144.441i	− 0.276178i	0.990420	−	0.138089i	$-0.0440960\pi$
$524$	0	0
$525$	− 102.802i	− 0.195814i
$526$	0	0
$527$	685.434i	1.30063i
$528$	0	0
$529$	−419.508	−0.793021
$530$	0	0
$531$	541.678i	1.02011i
$532$	0	0
$533$	1123.46	2.10780
$534$	0	0
$535$	352.209i	0.658335i
$536$	0	0
$537$	34.5383	0.0643171
$538$	0	0
$539$	446.078	0.827603
$540$	0	0
$541$	264.901	0.489651	0.244826	−	0.969567i	$-0.421269\pi$
$541$	264.901	0.489651	0.244826	+	0.969567i	$0.421269\pi$
$542$	0	0
$543$	1039.67	1.91469
$544$	0	0
$545$	12.9344i	0.0237329i
$546$	0	0
$547$	754.403i	1.37917i	0.724207	+	0.689583i	$0.242206\pi$
$547$	754.403i	1.37917i	−0.724207	+	0.689583i	$0.757794\pi$
$548$	0	0
$549$	348.819	0.635371
$550$	0	0
$551$	0	0
$552$	0	0
$553$	6.74243i	0.0121925i
$554$	0	0
$555$	−490.481	−0.883750
$556$	0	0
$557$	−677.731	−1.21675	−0.608376	−	0.793649i	$-0.708179\pi$
$557$	−677.731	−1.21675	−0.608376	+	0.793649i	$0.708179\pi$
$558$	0	0
$559$	188.680i	0.337530i
$560$	0	0
$561$	− 624.700i	− 1.11355i
$562$	0	0
$563$	− 741.646i	− 1.31731i	−0.752445	−	0.658655i	$-0.771126\pi$
$563$	− 741.646i	− 1.31731i	0.752445	−	0.658655i	$-0.228874\pi$
$564$	0	0
$565$	− 35.1632i	− 0.0622358i
$566$	0	0
$567$	−87.2660	−0.153908
$568$	0	0
$569$	902.938i	1.58689i	0.608645	+	0.793443i	$0.291714\pi$
$569$	902.938i	1.58689i	−0.608645	+	0.793443i	$0.708286\pi$
$570$	0	0
$571$	885.622	1.55100	0.775501	−	0.631347i	$-0.217497\pi$
$571$	885.622	1.55100	0.775501	+	0.631347i	$0.217497\pi$
$572$	0	0
$573$	1143.35i	1.99538i
$574$	0	0
$575$	−224.314	−0.390112
$576$	0	0
$577$	−276.323	−0.478896	−0.239448	−	0.970909i	$-0.576967\pi$
$577$	−276.323	−0.478896	−0.239448	+	0.970909i	$0.576967\pi$
$578$	0	0
$579$	610.411	1.05425
$580$	0	0
$581$	159.754	0.274963
$582$	0	0
$583$	264.940i	0.454442i
$584$	0	0
$585$	405.500i	0.693162i
$586$	0	0
$587$	−523.874	−0.892460	−0.446230	−	0.894918i	$-0.647234\pi$
$587$	−523.874	−0.892460	−0.446230	+	0.894918i	$0.647234\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	58.7930i	0.0994805i
$592$	0	0
$593$	803.140	1.35437	0.677184	−	0.735814i	$-0.263200\pi$
$593$	803.140	1.35437	0.677184	+	0.735814i	$0.263200\pi$
$594$	0	0
$595$	−33.0177	−0.0554920
$596$	0	0
$597$	− 628.718i	− 1.05313i
$598$	0	0
$599$	593.790i	0.991302i	0.868522	+	0.495651i	$0.165070\pi$
$599$	593.790i	0.991302i	−0.868522	+	0.495651i	$0.834930\pi$
$600$	0	0
$601$	− 541.281i	− 0.900634i	−0.892869	−	0.450317i	$-0.851311\pi$
$601$	− 541.281i	− 0.900634i	0.892869	−	0.450317i	$-0.148689\pi$
$602$	0	0
$603$	520.018i	0.862385i
$604$	0	0
$605$	63.6325	0.105178
$606$	0	0
$607$	− 856.338i	− 1.41077i	−0.708824	−	0.705385i	$-0.750774\pi$
$607$	− 856.338i	− 1.41077i	0.708824	−	0.705385i	$-0.249226\pi$
$608$	0	0
$609$	−218.659	−0.359046
$610$	0	0
$611$	− 1655.97i	− 2.71026i
$612$	0	0
$613$	−353.860	−0.577259	−0.288629	−	0.957441i	$-0.593200\pi$
$613$	−353.860	−0.577259	−0.288629	+	0.957441i	$0.593200\pi$
$614$	0	0
$615$	−394.361	−0.641238
$616$	0	0
$617$	−696.697	−1.12917	−0.564584	−	0.825376i	$-0.690963\pi$
$617$	−696.697	−1.12917	−0.564584	+	0.825376i	$0.690963\pi$
$618$	0	0
$619$	−121.961	−0.197028	−0.0985142	−	0.995136i	$-0.531409\pi$
$619$	−121.961	−0.197028	−0.0985142	+	0.995136i	$0.531409\pi$
$620$	0	0
$621$	− 14.8291i	− 0.0238795i
$622$	0	0
$623$	187.846i	0.301519i
$624$	0	0
$625$	370.475	0.592759
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 947.818i	− 1.50687i
$630$	0	0
$631$	−856.168	−1.35684	−0.678422	−	0.734673i	$-0.737336\pi$
$631$	−856.168	−1.35684	−0.678422	+	0.734673i	$0.737336\pi$
$632$	0	0
$633$	1415.05	2.23546
$634$	0	0
$635$	133.054i	0.209534i
$636$	0	0
$637$	− 1099.24i	− 1.72564i
$638$	0	0
$639$	− 47.1072i	− 0.0737202i
$640$	0	0
$641$	822.385i	1.28297i	0.767135	+	0.641486i	$0.221682\pi$
$641$	822.385i	1.28297i	−0.767135	+	0.641486i	$0.778318\pi$
$642$	0	0
$643$	−380.022	−0.591015	−0.295507	−	0.955340i	$-0.595489\pi$
$643$	−380.022	−0.591015	−0.295507	+	0.955340i	$0.595489\pi$
$644$	0	0
$645$	− 66.2312i	− 0.102684i
$646$	0	0
$647$	668.561	1.03332	0.516662	−	0.856189i	$-0.327174\pi$
$647$	668.561	1.03332	0.516662	+	0.856189i	$0.327174\pi$
$648$	0	0
$649$	542.366i	0.835695i
$650$	0	0
$651$	−210.476	−0.323312
$652$	0	0
$653$	951.772	1.45754	0.728769	−	0.684760i	$-0.240093\pi$
$653$	951.772	1.45754	0.728769	+	0.684760i	$0.240093\pi$
$654$	0	0
$655$	270.875	0.413550
$656$	0	0
$657$	79.6185	0.121185
$658$	0	0
$659$	− 383.531i	− 0.581989i	−0.956725	−	0.290995i	$-0.906014\pi$
$659$	− 383.531i	− 0.581989i	0.956725	−	0.290995i	$-0.0939862\pi$
$660$	0	0
$661$	− 532.628i	− 0.805791i	−0.915246	−	0.402896i	$-0.868004\pi$
$661$	− 532.628i	− 0.805791i	0.915246	−	0.402896i	$-0.131996\pi$
$662$	0	0
$663$	−1539.40	−2.32187
$664$	0	0
$665$	0	0
$666$	0	0
$667$	477.113i	0.715312i
$668$	0	0
$669$	−483.576	−0.722833
$670$	0	0
$671$	349.261	0.520509
$672$	0	0
$673$	293.302i	0.435812i	0.975970	+	0.217906i	$0.0699227\pi$
$673$	293.302i	0.435812i	−0.975970	+	0.217906i	$0.930077\pi$
$674$	0	0
$675$	30.3802i	0.0450077i
$676$	0	0
$677$	599.339i	0.885286i	0.896698	+	0.442643i	$0.145959\pi$
$677$	599.339i	0.885286i	−0.896698	+	0.442643i	$0.854041\pi$
$678$	0	0
$679$	123.574i	0.181995i
$680$	0	0
$681$	50.3171	0.0738870
$682$	0	0
$683$	374.603i	0.548468i	0.961663	+	0.274234i	$0.0884242\pi$
$683$	374.603i	0.548468i	−0.961663	+	0.274234i	$0.911576\pi$
$684$	0	0
$685$	−84.7650	−0.123745
$686$	0	0
$687$	755.604i	1.09986i
$688$	0	0
$689$	652.870	0.947562
$690$	0	0
$691$	−1247.78	−1.80576	−0.902881	−	0.429890i	$-0.858552\pi$
$691$	−1247.78	−1.80576	−0.902881	+	0.429890i	$0.858552\pi$
$692$	0	0
$693$	97.6453	0.140902
$694$	0	0
$695$	−310.900	−0.447338
$696$	0	0
$697$	− 762.074i	− 1.09336i
$698$	0	0
$699$	− 1105.52i	− 1.58157i
$700$	0	0
$701$	637.723	0.909733	0.454867	−	0.890560i	$-0.349687\pi$
$701$	637.723	0.909733	0.454867	+	0.890560i	$0.349687\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	581.288i	0.824521i
$706$	0	0
$707$	0.315463	0.000446199 0
$708$	0	0
$709$	−835.468	−1.17837	−0.589187	−	0.807996i	$-0.700552\pi$
$709$	−835.468	−1.17837	−0.589187	+	0.807996i	$0.700552\pi$
$710$	0	0
$711$	− 56.1696i	− 0.0790008i
$712$	0	0
$713$	459.259i	0.644121i
$714$	0	0
$715$	406.015i	0.567853i
$716$	0	0
$717$	− 2016.78i	− 2.81280i
$718$	0	0
$719$	−206.453	−0.287139	−0.143570	−	0.989640i	$-0.545858\pi$
$719$	−206.453	−0.287139	−0.143570	+	0.989640i	$0.545858\pi$
$720$	0	0
$721$	− 105.140i	− 0.145825i
$722$	0	0
$723$	−1392.77	−1.92638
$724$	0	0
$725$	− 977.452i	− 1.34821i
$726$	0	0
$727$	835.272	1.14893	0.574465	−	0.818529i	$-0.305210\pi$
$727$	835.272	1.14893	0.574465	+	0.818529i	$0.305210\pi$
$728$	0	0
$729$	781.600	1.07215
$730$	0	0
$731$	127.987	0.175085
$732$	0	0
$733$	359.418	0.490338	0.245169	−	0.969480i	$-0.421157\pi$
$733$	359.418	0.490338	0.245169	+	0.969480i	$0.421157\pi$
$734$	0	0
$735$	385.859i	0.524979i
$736$	0	0
$737$	520.678i	0.706483i
$738$	0	0
$739$	−181.157	−0.245138	−0.122569	−	0.992460i	$-0.539113\pi$
$739$	−181.157	−0.245138	−0.122569	+	0.992460i	$0.539113\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 501.302i	− 0.674700i	−0.941379	−	0.337350i	$-0.890469\pi$
$743$	− 501.302i	− 0.674700i	0.941379	−	0.337350i	$-0.109531\pi$
$744$	0	0
$745$	−94.9314	−0.127425
$746$	0	0
$747$	−1330.87	−1.78162
$748$	0	0
$749$	− 208.997i	− 0.279035i
$750$	0	0
$751$	409.742i	0.545596i	0.962071	+	0.272798i	$0.0879490\pi$
$751$	409.742i	0.545596i	−0.962071	+	0.272798i	$0.912051\pi$
$752$	0	0
$753$	431.190i	0.572629i
$754$	0	0
$755$	203.126i	0.269041i
$756$	0	0
$757$	1095.98	1.44779	0.723897	−	0.689909i	$-0.242349\pi$
$757$	1095.98	1.44779	0.723897	+	0.689909i	$0.242349\pi$
$758$	0	0
$759$	− 418.566i	− 0.551470i
$760$	0	0
$761$	−509.579	−0.669617	−0.334809	−	0.942286i	$-0.608672\pi$
$761$	−509.579	−0.669617	−0.334809	+	0.942286i	$0.608672\pi$
$762$	0	0
$763$	− 7.67515i	− 0.0100592i
$764$	0	0
$765$	275.063	0.359559
$766$	0	0
$767$	1336.51	1.74252
$768$	0	0
$769$	−779.006	−1.01301	−0.506506	−	0.862237i	$-0.669063\pi$
$769$	−779.006	−1.01301	−0.506506	+	0.862237i	$0.669063\pi$
$770$	0	0
$771$	−1618.16	−2.09877
$772$	0	0
$773$	727.726i	0.941430i	0.882285	+	0.470715i	$0.156004\pi$
$773$	727.726i	0.941430i	−0.882285	+	0.470715i	$0.843996\pi$
$774$	0	0
$775$	− 940.874i	− 1.21403i
$776$	0	0
$777$	291.047	0.374577
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 47.1670i	− 0.0603931i
$782$	0	0
$783$	64.6182	0.0825264
$784$	0	0
$785$	434.275	0.553216
$786$	0	0
$787$	− 1254.73i	− 1.59432i	−0.603768	−	0.797160i	$-0.706335\pi$
$787$	− 1254.73i	− 1.59432i	0.603768	−	0.797160i	$-0.293665\pi$
$788$	0	0
$789$	− 129.359i	− 0.163953i
$790$	0	0
$791$	20.8655i	0.0263787i
$792$	0	0
$793$	− 860.658i	− 1.08532i
$794$	0	0
$795$	−229.174	−0.288269
$796$	0	0
$797$	− 898.453i	− 1.12729i	−0.826016	−	0.563647i	$-0.809398\pi$
$797$	− 898.453i	− 1.12729i	0.826016	−	0.563647i	$-0.190602\pi$
$798$	0	0
$799$	−1123.29	−1.40588
$800$	0	0
$801$	− 1564.90i	− 1.95368i
$802$	0	0
$803$	79.7195	0.0992771
$804$	0	0
$805$	−22.1227	−0.0274817
$806$	0	0
$807$	−682.392	−0.845591
$808$	0	0
$809$	−1083.54	−1.33935	−0.669677	−	0.742653i	$-0.733567\pi$
$809$	−1083.54	−1.33935	−0.669677	+	0.742653i	$0.733567\pi$
$810$	0	0
$811$	620.911i	0.765612i	0.923829	+	0.382806i	$0.125042\pi$
$811$	620.911i	0.765612i	−0.923829	+	0.382806i	$0.874958\pi$
$812$	0	0
$813$	759.936i	0.934730i
$814$	0	0
$815$	378.410	0.464307
$816$	0	0
$817$	0	0
$818$	0	0
$819$	− 240.620i	− 0.293797i
$820$	0	0
$821$	1057.34	1.28786	0.643932	−	0.765083i	$-0.277302\pi$
$821$	1057.34	1.28786	0.643932	+	0.765083i	$0.277302\pi$
$822$	0	0
$823$	73.9820	0.0898931	0.0449465	−	0.998989i	$-0.485688\pi$
$823$	73.9820	0.0898931	0.0449465	+	0.998989i	$0.485688\pi$
$824$	0	0
$825$	857.507i	1.03940i
$826$	0	0
$827$	1192.22i	1.44162i	0.693133	+	0.720809i	$0.256230\pi$
$827$	1192.22i	1.44162i	−0.693133	+	0.720809i	$0.743770\pi$
$828$	0	0
$829$	397.548i	0.479552i	0.970828	+	0.239776i	$0.0770740\pi$
$829$	397.548i	0.479552i	−0.970828	+	0.239776i	$0.922926\pi$
$830$	0	0
$831$	1693.40i	2.03778i
$832$	0	0
$833$	−745.644	−0.895131
$834$	0	0
$835$	− 543.717i	− 0.651157i
$836$	0	0
$837$	62.2000	0.0743131
$838$	0	0
$839$	262.686i	0.313094i	0.987671	+	0.156547i	$0.0500363\pi$
$839$	262.686i	0.313094i	−0.987671	+	0.156547i	$0.949964\pi$
$840$	0	0
$841$	−1238.03	−1.47209
$842$	0	0
$843$	−51.8315	−0.0614845
$844$	0	0
$845$	681.510	0.806521
$846$	0	0
$847$	−37.7589	−0.0445795
$848$	0	0
$849$	1722.43i	2.02878i
$850$	0	0
$851$	− 635.063i	− 0.746255i
$852$	0	0
$853$	660.361	0.774162	0.387081	−	0.922046i	$-0.373483\pi$
$853$	660.361	0.774162	0.387081	+	0.922046i	$0.373483\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	276.670i	0.322835i	0.986886	+	0.161418i	$0.0516066\pi$
$857$	276.670i	0.322835i	−0.986886	+	0.161418i	$0.948393\pi$
$858$	0	0
$859$	485.320	0.564982	0.282491	−	0.959270i	$-0.408839\pi$
$859$	485.320	0.564982	0.282491	+	0.959270i	$0.408839\pi$
$860$	0	0
$861$	234.010	0.271789
$862$	0	0
$863$	646.810i	0.749490i	0.927128	+	0.374745i	$0.122270\pi$
$863$	646.810i	0.749490i	−0.927128	+	0.374745i	$0.877730\pi$
$864$	0	0
$865$	− 40.5778i	− 0.0469107i
$866$	0	0
$867$	− 193.125i	− 0.222751i
$868$	0	0
$869$	− 56.2408i	− 0.0647190i
$870$	0	0
$871$	1283.07	1.47310
$872$	0	0
$873$	− 1029.47i	− 1.17923i
$874$	0	0
$875$	98.1776	0.112203
$876$	0	0
$877$	1475.19i	1.68209i	0.540967	+	0.841044i	$0.318058\pi$
$877$	1475.19i	1.68209i	−0.540967	+	0.841044i	$0.681942\pi$
$878$	0	0
$879$	−1234.82	−1.40481
$880$	0	0
$881$	−589.063	−0.668630	−0.334315	−	0.942461i	$-0.608505\pi$
$881$	−589.063	−0.668630	−0.334315	+	0.942461i	$0.608505\pi$
$882$	0	0
$883$	−1182.00	−1.33862	−0.669308	−	0.742985i	$-0.733409\pi$
$883$	−1182.00	−1.33862	−0.669308	+	0.742985i	$0.733409\pi$
$884$	0	0
$885$	−469.148	−0.530111
$886$	0	0
$887$	1179.05i	1.32925i	0.747176	+	0.664627i	$0.231409\pi$
$887$	1179.05i	1.32925i	−0.747176	+	0.664627i	$0.768591\pi$
$888$	0	0
$889$	− 78.9529i	− 0.0888109i
$890$	0	0
$891$	727.914	0.816963
$892$	0	0
$893$	0	0
$894$	0	0
$895$	15.2269i	0.0170133i
$896$	0	0
$897$	−1031.44	−1.14988
$898$	0	0
$899$	−2001.23	−2.22606
$900$	0	0
$901$	− 442.861i	− 0.491522i
$902$	0	0
$903$	39.3009i	0.0435226i
$904$	0	0
$905$	458.362i	0.506477i
$906$	0	0
$907$	− 1442.58i	− 1.59049i	−0.606288	−	0.795245i	$-0.707342\pi$
$907$	− 1442.58i	− 1.59049i	0.606288	−	0.795245i	$-0.292658\pi$
$908$	0	0
$909$	−2.62804	−0.00289113
$910$	0	0
$911$	− 1116.15i	− 1.22520i	−0.790394	−	0.612598i	$-0.790124\pi$
$911$	− 1116.15i	− 1.22520i	0.790394	−	0.612598i	$-0.209876\pi$
$912$	0	0
$913$	−1332.56	−1.45954
$914$	0	0
$915$	302.112i	0.330177i
$916$	0	0
$917$	−160.734	−0.175283
$918$	0	0
$919$	230.449	0.250760	0.125380	−	0.992109i	$-0.459985\pi$
$919$	230.449	0.250760	0.125380	+	0.992109i	$0.459985\pi$
$920$	0	0
$921$	−1789.29	−1.94277
$922$	0	0
$923$	−116.230	−0.125926
$924$	0	0
$925$	1301.04i	1.40653i
$926$	0	0
$927$	875.892i	0.944868i
$928$	0	0
$929$	−1108.41	−1.19312	−0.596562	−	0.802567i	$-0.703467\pi$
$929$	−1108.41	−1.19312	−0.596562	+	0.802567i	$0.703467\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	− 935.511i	− 1.00269i
$934$	0	0
$935$	275.412	0.294558
$936$	0	0
$937$	−811.000	−0.865528	−0.432764	−	0.901507i	$-0.642462\pi$
$937$	−811.000	−0.865528	−0.432764	+	0.901507i	$0.642462\pi$
$938$	0	0
$939$	− 1997.61i	− 2.12738i
$940$	0	0
$941$	357.628i	0.380051i	0.981779	+	0.190025i	$0.0608571\pi$
$941$	357.628i	0.380051i	−0.981779	+	0.190025i	$0.939143\pi$
$942$	0	0
$943$	− 510.609i	− 0.541473i
$944$	0	0
$945$	2.99621i	0.00317059i
$946$	0	0
$947$	−804.160	−0.849166	−0.424583	−	0.905389i	$-0.639579\pi$
$947$	−804.160	−0.849166	−0.424583	+	0.905389i	$0.639579\pi$
$948$	0	0
$949$	− 196.447i	− 0.207004i
$950$	0	0
$951$	−1119.06	−1.17671
$952$	0	0
$953$	− 1555.08i	− 1.63177i	−0.578215	−	0.815884i	$-0.696251\pi$
$953$	− 1555.08i	− 1.63177i	0.578215	−	0.815884i	$-0.303749\pi$
$954$	0	0
$955$	−504.070	−0.527822
$956$	0	0
$957$	1823.90	1.90586
$958$	0	0
$959$	50.2987	0.0524491
$960$	0	0
$961$	−965.335	−1.00451
$962$	0	0
$963$	1741.11i	1.80800i
$964$	0	0
$965$	269.112i	0.278873i
$966$	0	0
$967$	−927.048	−0.958684	−0.479342	−	0.877628i	$-0.659125\pi$
$967$	−927.048	−0.958684	−0.479342	+	0.877628i	$0.659125\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	543.963i	0.560209i	0.959969	+	0.280105i	$0.0903692\pi$
$971$	543.963i	0.560209i	−0.959969	+	0.280105i	$0.909631\pi$
$972$	0	0
$973$	184.485	0.189604
$974$	0	0
$975$	2113.09	2.16727
$976$	0	0
$977$	84.1562i	0.0861373i	0.999072	+	0.0430687i	$0.0137134\pi$
$977$	84.1562i	0.0861373i	−0.999072	+	0.0430687i	$0.986287\pi$
$978$	0	0
$979$	− 1566.89i	− 1.60050i
$980$	0	0
$981$	63.9398i	0.0651782i
$982$	0	0
$983$	− 1044.26i	− 1.06232i	−0.847272	−	0.531159i	$-0.821757\pi$
$983$	− 1044.26i	− 1.06232i	0.847272	−	0.531159i	$-0.178243\pi$
$984$	0	0
$985$	−25.9201	−0.0263148
$986$	0	0
$987$	− 344.930i	− 0.349473i
$988$	0	0
$989$	85.7546	0.0867084
$990$	0	0
$991$	− 1193.06i	− 1.20389i	−0.798537	−	0.601945i	$-0.794393\pi$
$991$	− 1193.06i	− 1.20389i	0.798537	−	0.601945i	$-0.205607\pi$
$992$	0	0
$993$	−686.595	−0.691435
$994$	0	0
$995$	277.183	0.278576
$996$	0	0
$997$	−474.406	−0.475833	−0.237917	−	0.971286i	$-0.576465\pi$
$997$	−474.406	−0.475833	−0.237917	+	0.971286i	$0.576465\pi$
$998$	0	0
$999$	−86.0103	−0.0860963

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.3.c.d.721.20	yes	24
19.18	odd	2	inner	1444.3.c.d.721.5	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.3.c.d.721.5	✓	24	19.18	odd	2	inner
1444.3.c.d.721.20	yes	24	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1444 = 2^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1444.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.28147i q^{3} -1.88757 q^{5} +1.12007 q^{7} -9.33100 q^{9} +O(q^{10})\)	\(q+4.28147i q^{3} -1.88757 q^{5} +1.12007 q^{7} -9.33100 q^{9} -9.34284 q^{11} +23.0228i q^{13} -8.08159i q^{15} +15.6171 q^{17} +4.79553i q^{21} +10.4638 q^{23} -21.4371 q^{25} -1.41718i q^{27} +45.5963i q^{29} +43.8900i q^{31} -40.0011i q^{33} -2.11421 q^{35} -60.6912i q^{37} -98.5716 q^{39} -48.7975i q^{41} +8.19532 q^{43} +17.6129 q^{45} -71.9274 q^{47} -47.7455 q^{49} +66.8640i q^{51} -28.3575i q^{53} +17.6353 q^{55} -58.0515i q^{59} -37.3828 q^{61} -10.4513 q^{63} -43.4573i q^{65} -55.7302i q^{67} +44.8007i q^{69} +5.04846i q^{71} -8.53268 q^{73} -91.7822i q^{75} -10.4646 q^{77} +6.01967i q^{79} -77.9114 q^{81} +142.629 q^{83} -29.4784 q^{85} -195.219 q^{87} +167.710i q^{89} +25.7871i q^{91} -187.914 q^{93} +110.328i q^{97} +87.1781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9}+O(q^{10}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9	\( 24 q - 12 q^{5} + 40 q^{7} - 104 q^{9} - 64 q^{11} - 60 q^{17} + 100 q^{23} + 24 q^{25} - 144 q^{35} + 68 q^{39} - 16 q^{43} + 248 q^{45} + 180 q^{47} - 208 q^{49} - 532 q^{55} + 104 q^{61} - 1216 q^{63} - 484 q^{73} + 136 q^{77} + 460 q^{81} + 744 q^{83} + 32 q^{85} - 1296 q^{87} - 256 q^{93} + 804 q^{99}+O(q^{100}) \) 24 * q - 12 * q^5 + 40 * q^7 - 104 * q^9 - 64 * q^11 - 60 * q^17 + 100 * q^23 + 24 * q^25 - 144 * q^35 + 68 * q^39 - 16 * q^43 + 248 * q^45 + 180 * q^47 - 208 * q^49 - 532 * q^55 + 104 * q^61 - 1216 * q^63 - 484 * q^73 + 136 * q^77 + 460 * q^81 + 744 * q^83 + 32 * q^85 - 1296 * q^87 - 256 * q^93 + 804 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more