LMFDB - Embedded newform 2160.4.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{241}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 60 $ x^2 - x - 60
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.26209$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -5.26209 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -5.26209 q^{7} -10.2621 q^{11} -58.5725 q^{13} -33.8346 q^{17} -119.883 q^{19} -182.455 q^{23} +25.0000 q^{25} -49.4071 q^{29} +200.980 q^{31} +26.3104 q^{35} -89.6896 q^{37} -136.621 q^{41} +254.552 q^{43} +61.8346 q^{47} -315.310 q^{49} +671.104 q^{53} +51.3104 q^{55} +450.290 q^{59} -126.310 q^{61} +292.863 q^{65} -466.132 q^{67} -370.056 q^{71} +151.705 q^{73} +54.0000 q^{77} -44.8626 q^{79} +855.435 q^{83} +169.173 q^{85} -1098.60 q^{89} +308.214 q^{91} +599.415 q^{95} -1207.08 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 5 q^{7}+O(q^{10}) $ 2 * q - 10 * q^5 + 5 * q^7	$ 2 q - 10 q^{5} + 5 q^{7} - 5 q^{11} - 24 q^{13} + 41 q^{17} - 69 q^{19} - 101 q^{23} + 50 q^{25} + 103 q^{29} + 107 q^{31} - 25 q^{35} - 257 q^{37} - 118 q^{41} + 121 q^{43} + 15 q^{47} - 553 q^{49} + 566 q^{53} + 25 q^{55} + 528 q^{59} - 175 q^{61} + 120 q^{65} + 201 q^{67} - 26 q^{71} - 923 q^{73} + 108 q^{77} + 376 q^{79} + 1152 q^{83} - 205 q^{85} - 148 q^{89} + 663 q^{91} + 345 q^{95} - 1343 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 + 5 * q^7 - 5 * q^11 - 24 * q^13 + 41 * q^17 - 69 * q^19 - 101 * q^23 + 50 * q^25 + 103 * q^29 + 107 * q^31 - 25 * q^35 - 257 * q^37 - 118 * q^41 + 121 * q^43 + 15 * q^47 - 553 * q^49 + 566 * q^53 + 25 * q^55 + 528 * q^59 - 175 * q^61 + 120 * q^65 + 201 * q^67 - 26 * q^71 - 923 * q^73 + 108 * q^77 + 376 * q^79 + 1152 * q^83 - 205 * q^85 - 148 * q^89 + 663 * q^91 + 345 * q^95 - 1343 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−5.26209	−0.284126	−0.142063	−	0.989858i	$-0.545374\pi$
$7$	−5.26209	−0.284126	−0.142063	+	0.989858i	$0.545374\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−10.2621	−0.281285	−0.140643	−	0.990060i	$-0.544917\pi$
$11$	−10.2621	−0.281285	−0.140643	+	0.990060i	$0.544917\pi$
$12$	0	0
$13$	−58.5725	−1.24962	−0.624811	−	0.780776i	$-0.714824\pi$
$13$	−58.5725	−1.24962	−0.624811	+	0.780776i	$0.714824\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−33.8346	−0.482712	−0.241356	−	0.970437i	$-0.577592\pi$
$17$	−33.8346	−0.482712	−0.241356	+	0.970437i	$0.577592\pi$
$18$	0	0
$19$	−119.883	−1.44753	−0.723764	−	0.690047i	$-0.757590\pi$
$19$	−119.883	−1.44753	−0.723764	+	0.690047i	$0.757590\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−182.455	−1.65411	−0.827056	−	0.562119i	$-0.809986\pi$
$23$	−182.455	−1.65411	−0.827056	+	0.562119i	$0.809986\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−49.4071	−0.316368	−0.158184	−	0.987410i	$-0.550564\pi$
$29$	−49.4071	−0.316368	−0.158184	+	0.987410i	$0.550564\pi$
$30$	0	0
$31$	200.980	1.16442	0.582210	−	0.813039i	$-0.302188\pi$
$31$	200.980	1.16442	0.582210	+	0.813039i	$0.302188\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	26.3104	0.127065
$36$	0	0
$37$	−89.6896	−0.398510	−0.199255	−	0.979948i	$-0.563852\pi$
$37$	−89.6896	−0.398510	−0.199255	+	0.979948i	$0.563852\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−136.621	−0.520405	−0.260202	−	0.965554i	$-0.583789\pi$
$41$	−136.621	−0.520405	−0.260202	+	0.965554i	$0.583789\pi$
$42$	0	0
$43$	254.552	0.902764	0.451382	−	0.892331i	$-0.350931\pi$
$43$	254.552	0.902764	0.451382	+	0.892331i	$0.350931\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	61.8346	0.191904	0.0959522	−	0.995386i	$-0.469410\pi$
$47$	61.8346	0.191904	0.0959522	+	0.995386i	$0.469410\pi$
$48$	0	0
$49$	−315.310	−0.919272
$50$	0	0
$51$	0	0
$52$	0	0
$53$	671.104	1.73931	0.869654	−	0.493663i	$-0.164342\pi$
$53$	671.104	1.73931	0.869654	+	0.493663i	$0.164342\pi$
$54$	0	0
$55$	51.3104	0.125795
$56$	0	0
$57$	0	0
$58$	0	0
$59$	450.290	0.993606	0.496803	−	0.867863i	$-0.334507\pi$
$59$	450.290	0.993606	0.496803	+	0.867863i	$0.334507\pi$
$60$	0	0
$61$	−126.310	−0.265121	−0.132561	−	0.991175i	$-0.542320\pi$
$61$	−126.310	−0.265121	−0.132561	+	0.991175i	$0.542320\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	292.863	0.558848
$66$	0	0
$67$	−466.132	−0.849957	−0.424979	−	0.905203i	$-0.639718\pi$
$67$	−466.132	−0.849957	−0.424979	+	0.905203i	$0.639718\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−370.056	−0.618557	−0.309279	−	0.950971i	$-0.600088\pi$
$71$	−370.056	−0.618557	−0.309279	+	0.950971i	$0.600088\pi$
$72$	0	0
$73$	151.705	0.243229	0.121614	−	0.992577i	$-0.461193\pi$
$73$	151.705	0.243229	0.121614	+	0.992577i	$0.461193\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	54.0000	0.0799204
$78$	0	0
$79$	−44.8626	−0.0638916	−0.0319458	−	0.999490i	$-0.510170\pi$
$79$	−44.8626	−0.0638916	−0.0319458	+	0.999490i	$0.510170\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	855.435	1.13128	0.565640	−	0.824652i	$-0.308629\pi$
$83$	855.435	1.13128	0.565640	+	0.824652i	$0.308629\pi$
$84$	0	0
$85$	169.173	0.215875
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1098.60	−1.30844	−0.654219	−	0.756306i	$-0.727002\pi$
$89$	−1098.60	−1.30844	−0.654219	+	0.756306i	$0.727002\pi$
$90$	0	0
$91$	308.214	0.355050
$92$	0	0
$93$	0	0
$94$	0	0
$95$	599.415	0.647354
$96$	0	0
$97$	−1207.08	−1.26351	−0.631757	−	0.775167i	$-0.717666\pi$
$97$	−1207.08	−1.26351	−0.631757	+	0.775167i	$0.717666\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	523.463	0.515708	0.257854	−	0.966184i	$-0.416985\pi$
$101$	523.463	0.515708	0.257854	+	0.966184i	$0.416985\pi$
$102$	0	0
$103$	969.883	0.927819	0.463910	−	0.885882i	$-0.346446\pi$
$103$	969.883	0.927819	0.463910	+	0.885882i	$0.346446\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−374.000	−0.337906	−0.168953	−	0.985624i	$-0.554039\pi$
$107$	−374.000	−0.337906	−0.168953	+	0.985624i	$0.554039\pi$
$108$	0	0
$109$	−1962.36	−1.72440	−0.862202	−	0.506564i	$-0.830915\pi$
$109$	−1962.36	−1.72440	−0.862202	+	0.506564i	$0.830915\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−386.649	−0.321884	−0.160942	−	0.986964i	$-0.551453\pi$
$113$	−386.649	−0.321884	−0.160942	+	0.986964i	$0.551453\pi$
$114$	0	0
$115$	912.277	0.739742
$116$	0	0
$117$	0	0
$118$	0	0
$119$	178.041	0.137151
$120$	0	0
$121$	−1225.69	−0.920879
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1348.83	0.942435	0.471218	−	0.882017i	$-0.343815\pi$
$127$	1348.83	0.942435	0.471218	+	0.882017i	$0.343815\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−353.148	−0.235532	−0.117766	−	0.993041i	$-0.537573\pi$
$131$	−353.148	−0.235532	−0.117766	+	0.993041i	$0.537573\pi$
$132$	0	0
$133$	630.835	0.411280
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1196.48	0.746150	0.373075	−	0.927801i	$-0.378303\pi$
$137$	1196.48	0.746150	0.373075	+	0.927801i	$0.378303\pi$
$138$	0	0
$139$	2768.71	1.68949	0.844745	−	0.535169i	$-0.179752\pi$
$139$	2768.71	1.68949	0.844745	+	0.535169i	$0.179752\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	601.076	0.351500
$144$	0	0
$145$	247.036	0.141484
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3310.89	1.82039	0.910196	−	0.414177i	$-0.135930\pi$
$149$	3310.89	1.82039	0.910196	+	0.414177i	$0.135930\pi$
$150$	0	0
$151$	−974.715	−0.525306	−0.262653	−	0.964890i	$-0.584597\pi$
$151$	−974.715	−0.525306	−0.262653	+	0.964890i	$0.584597\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1004.90	−0.520744
$156$	0	0
$157$	−930.013	−0.472759	−0.236379	−	0.971661i	$-0.575961\pi$
$157$	−930.013	−0.472759	−0.236379	+	0.971661i	$0.575961\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	960.097	0.469977
$162$	0	0
$163$	1461.81	0.702443	0.351221	−	0.936292i	$-0.385766\pi$
$163$	1461.81	0.702443	0.351221	+	0.936292i	$0.385766\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3507.59	1.62530	0.812652	−	0.582750i	$-0.198023\pi$
$167$	3507.59	1.62530	0.812652	+	0.582750i	$0.198023\pi$
$168$	0	0
$169$	1233.74	0.561557
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2407.46	−1.05801	−0.529004	−	0.848619i	$-0.677434\pi$
$173$	−2407.46	−1.05801	−0.529004	+	0.848619i	$0.677434\pi$
$174$	0	0
$175$	−131.552	−0.0568252
$176$	0	0
$177$	0	0
$178$	0	0
$179$	709.089	0.296088	0.148044	−	0.988981i	$-0.452702\pi$
$179$	709.089	0.296088	0.148044	+	0.988981i	$0.452702\pi$
$180$	0	0
$181$	−1245.94	−0.511657	−0.255828	−	0.966722i	$-0.582348\pi$
$181$	−1245.94	−0.511657	−0.255828	+	0.966722i	$0.582348\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	448.448	0.178219
$186$	0	0
$187$	347.214	0.135780
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2809.14	1.06420	0.532099	−	0.846682i	$-0.321403\pi$
$191$	2809.14	1.06420	0.532099	+	0.846682i	$0.321403\pi$
$192$	0	0
$193$	−4285.72	−1.59841	−0.799204	−	0.601059i	$-0.794746\pi$
$193$	−4285.72	−1.59841	−0.799204	+	0.601059i	$0.794746\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2741.44	0.991468	0.495734	−	0.868474i	$-0.334899\pi$
$197$	2741.44	0.991468	0.495734	+	0.868474i	$0.334899\pi$
$198$	0	0
$199$	−2922.07	−1.04090	−0.520452	−	0.853891i	$-0.674237\pi$
$199$	−2922.07	−1.04090	−0.520452	+	0.853891i	$0.674237\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	259.985	0.0898884
$204$	0	0
$205$	683.104	0.232732
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1230.25	0.407168
$210$	0	0
$211$	3525.37	1.15022	0.575111	−	0.818076i	$-0.304959\pi$
$211$	3525.37	1.15022	0.575111	+	0.818076i	$0.304959\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1272.76	−0.403728
$216$	0	0
$217$	−1057.57	−0.330842
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1981.78	0.603207
$222$	0	0
$223$	3631.29	1.09044	0.545222	−	0.838292i	$-0.316445\pi$
$223$	3631.29	1.09044	0.545222	+	0.838292i	$0.316445\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3795.86	−1.10987	−0.554934	−	0.831895i	$-0.687256\pi$
$227$	−3795.86	−1.10987	−0.554934	+	0.831895i	$0.687256\pi$
$228$	0	0
$229$	−257.913	−0.0744253	−0.0372126	−	0.999307i	$-0.511848\pi$
$229$	−257.913	−0.0744253	−0.0372126	+	0.999307i	$0.511848\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1579.73	0.444170	0.222085	−	0.975027i	$-0.428714\pi$
$233$	1579.73	0.444170	0.222085	+	0.975027i	$0.428714\pi$
$234$	0	0
$235$	−309.173	−0.0858222
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2892.29	0.782790	0.391395	−	0.920223i	$-0.371993\pi$
$239$	2892.29	0.782790	0.391395	+	0.920223i	$0.371993\pi$
$240$	0	0
$241$	−2028.30	−0.542135	−0.271067	−	0.962560i	$-0.587377\pi$
$241$	−2028.30	−0.542135	−0.271067	+	0.962560i	$0.587377\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1576.55	0.411111
$246$	0	0
$247$	7021.85	1.80886
$248$	0	0
$249$	0	0
$250$	0	0
$251$	6509.83	1.63704	0.818520	−	0.574479i	$-0.194795\pi$
$251$	6509.83	1.63704	0.818520	+	0.574479i	$0.194795\pi$
$252$	0	0
$253$	1872.37	0.465277
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1966.13	0.477214	0.238607	−	0.971116i	$-0.423309\pi$
$257$	1966.13	0.477214	0.238607	+	0.971116i	$0.423309\pi$
$258$	0	0
$259$	471.954	0.113227
$260$	0	0
$261$	0	0
$262$	0	0
$263$	38.8089	0.00909909	0.00454955	−	0.999990i	$-0.498552\pi$
$263$	38.8089	0.00909909	0.00454955	+	0.999990i	$0.498552\pi$
$264$	0	0
$265$	−3355.52	−0.777842
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−2109.23	−0.478074	−0.239037	−	0.971010i	$-0.576832\pi$
$269$	−2109.23	−0.478074	−0.239037	+	0.971010i	$0.576832\pi$
$270$	0	0
$271$	−607.852	−0.136252	−0.0681262	−	0.997677i	$-0.521702\pi$
$271$	−607.852	−0.136252	−0.0681262	+	0.997677i	$0.521702\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−256.552	−0.0562570
$276$	0	0
$277$	8360.69	1.81352	0.906760	−	0.421647i	$-0.138548\pi$
$277$	8360.69	1.81352	0.906760	+	0.421647i	$0.138548\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	443.944	0.0942473	0.0471236	−	0.998889i	$-0.484995\pi$
$281$	443.944	0.0942473	0.0471236	+	0.998889i	$0.484995\pi$
$282$	0	0
$283$	−6190.56	−1.30032	−0.650160	−	0.759798i	$-0.725298\pi$
$283$	−6190.56	−1.30032	−0.650160	+	0.759798i	$0.725298\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	718.911	0.147861
$288$	0	0
$289$	−3768.22	−0.766989
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3187.87	0.635623	0.317811	−	0.948154i	$-0.397052\pi$
$293$	3187.87	0.635623	0.317811	+	0.948154i	$0.397052\pi$
$294$	0	0
$295$	−2251.45	−0.444354
$296$	0	0
$297$	0	0
$298$	0	0
$299$	10686.9	2.06702
$300$	0	0
$301$	−1339.48	−0.256499
$302$	0	0
$303$	0	0
$304$	0	0
$305$	631.552	0.118566
$306$	0	0
$307$	−28.2874	−0.00525879	−0.00262940	−	0.999997i	$-0.500837\pi$
$307$	−28.2874	−0.00525879	−0.00262940	+	0.999997i	$0.500837\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	229.923	0.0419220	0.0209610	−	0.999780i	$-0.493327\pi$
$311$	229.923	0.0419220	0.0209610	+	0.999780i	$0.493327\pi$
$312$	0	0
$313$	7115.56	1.28497	0.642484	−	0.766299i	$-0.277904\pi$
$313$	7115.56	1.28497	0.642484	+	0.766299i	$0.277904\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4346.34	0.770078	0.385039	−	0.922900i	$-0.374188\pi$
$317$	4346.34	0.770078	0.385039	+	0.922900i	$0.374188\pi$
$318$	0	0
$319$	507.020	0.0889896
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4056.19	0.698739
$324$	0	0
$325$	−1464.31	−0.249925
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−325.379	−0.0545250
$330$	0	0
$331$	−5567.01	−0.924443	−0.462222	−	0.886764i	$-0.652948\pi$
$331$	−5567.01	−0.924443	−0.462222	+	0.886764i	$0.652948\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2330.66	0.380112
$336$	0	0
$337$	−2803.11	−0.453101	−0.226551	−	0.973999i	$-0.572745\pi$
$337$	−2803.11	−0.453101	−0.226551	+	0.973999i	$0.572745\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2062.47	−0.327534
$342$	0	0
$343$	3464.09	0.545315
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−5508.55	−0.852203	−0.426102	−	0.904675i	$-0.640113\pi$
$347$	−5508.55	−0.852203	−0.426102	+	0.904675i	$0.640113\pi$
$348$	0	0
$349$	−3195.98	−0.490192	−0.245096	−	0.969499i	$-0.578819\pi$
$349$	−3195.98	−0.490192	−0.245096	+	0.969499i	$0.578819\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7565.44	1.14070	0.570351	−	0.821401i	$-0.306807\pi$
$353$	7565.44	1.14070	0.570351	+	0.821401i	$0.306807\pi$
$354$	0	0
$355$	1850.28	0.276627
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3410.73	−0.501424	−0.250712	−	0.968062i	$-0.580665\pi$
$359$	−3410.73	−0.501424	−0.250712	+	0.968062i	$0.580665\pi$
$360$	0	0
$361$	7512.92	1.09534
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−758.525	−0.108775
$366$	0	0
$367$	−7358.20	−1.04658	−0.523290	−	0.852155i	$-0.675296\pi$
$367$	−7358.20	−1.04658	−0.523290	+	0.852155i	$0.675296\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3531.41	−0.494182
$372$	0	0
$373$	−10392.0	−1.44256	−0.721282	−	0.692641i	$-0.756447\pi$
$373$	−10392.0	−1.44256	−0.721282	+	0.692641i	$0.756447\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2893.90	0.395341
$378$	0	0
$379$	−3672.63	−0.497758	−0.248879	−	0.968535i	$-0.580062\pi$
$379$	−3672.63	−0.497758	−0.248879	+	0.968535i	$0.580062\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10622.3	1.41716	0.708580	−	0.705631i	$-0.249336\pi$
$383$	10622.3	1.41716	0.708580	+	0.705631i	$0.249336\pi$
$384$	0	0
$385$	−270.000	−0.0357415
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5493.15	−0.715974	−0.357987	−	0.933727i	$-0.616537\pi$
$389$	−5493.15	−0.715974	−0.357987	+	0.933727i	$0.616537\pi$
$390$	0	0
$391$	6173.31	0.798460
$392$	0	0
$393$	0	0
$394$	0	0
$395$	224.313	0.0285732
$396$	0	0
$397$	6081.41	0.768809	0.384405	−	0.923165i	$-0.374407\pi$
$397$	6081.41	0.768809	0.384405	+	0.923165i	$0.374407\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1655.04	0.206107	0.103053	−	0.994676i	$-0.467139\pi$
$401$	1655.04	0.206107	0.103053	+	0.994676i	$0.467139\pi$
$402$	0	0
$403$	−11771.9	−1.45509
$404$	0	0
$405$	0	0
$406$	0	0
$407$	920.402	0.112095
$408$	0	0
$409$	1065.37	0.128800	0.0644001	−	0.997924i	$-0.479487\pi$
$409$	1065.37	0.128800	0.0644001	+	0.997924i	$0.479487\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2369.47	−0.282309
$414$	0	0
$415$	−4277.18	−0.505924
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10677.2	1.24491	0.622453	−	0.782657i	$-0.286136\pi$
$419$	10677.2	1.24491	0.622453	+	0.782657i	$0.286136\pi$
$420$	0	0
$421$	10200.3	1.18083	0.590417	−	0.807099i	$-0.298963\pi$
$421$	10200.3	1.18083	0.590417	+	0.807099i	$0.298963\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−845.865	−0.0965423
$426$	0	0
$427$	664.657	0.0753278
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4424.46	−0.494475	−0.247237	−	0.968955i	$-0.579523\pi$
$431$	−4424.46	−0.494475	−0.247237	+	0.968955i	$0.579523\pi$
$432$	0	0
$433$	7655.90	0.849698	0.424849	−	0.905264i	$-0.360327\pi$
$433$	7655.90	0.849698	0.424849	+	0.905264i	$0.360327\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21873.3	2.39438
$438$	0	0
$439$	−13343.5	−1.45068	−0.725340	−	0.688391i	$-0.758317\pi$
$439$	−13343.5	−1.45068	−0.725340	+	0.688391i	$0.758317\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9738.80	1.04448	0.522240	−	0.852799i	$-0.325097\pi$
$443$	9738.80	1.04448	0.522240	+	0.852799i	$0.325097\pi$
$444$	0	0
$445$	5492.98	0.585151
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6559.09	−0.689404	−0.344702	−	0.938712i	$-0.612020\pi$
$449$	−6559.09	−0.689404	−0.344702	+	0.938712i	$0.612020\pi$
$450$	0	0
$451$	1402.02	0.146382
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1541.07	−0.158783
$456$	0	0
$457$	−3490.55	−0.357288	−0.178644	−	0.983914i	$-0.557171\pi$
$457$	−3490.55	−0.357288	−0.178644	+	0.983914i	$0.557171\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19441.9	1.96420	0.982102	−	0.188350i	$-0.0603138\pi$
$461$	19441.9	1.96420	0.982102	+	0.188350i	$0.0603138\pi$
$462$	0	0
$463$	3239.12	0.325129	0.162565	−	0.986698i	$-0.448023\pi$
$463$	3239.12	0.325129	0.162565	+	0.986698i	$0.448023\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13057.4	−1.29384	−0.646920	−	0.762558i	$-0.723943\pi$
$467$	−13057.4	−1.29384	−0.646920	+	0.762558i	$0.723943\pi$
$468$	0	0
$469$	2452.83	0.241495
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2612.24	−0.253934
$474$	0	0
$475$	−2997.07	−0.289506
$476$	0	0
$477$	0	0
$478$	0	0
$479$	7892.32	0.752837	0.376419	−	0.926450i	$-0.377155\pi$
$479$	7892.32	0.752837	0.376419	+	0.926450i	$0.377155\pi$
$480$	0	0
$481$	5253.34	0.497987
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6035.42	0.565060
$486$	0	0
$487$	−15479.0	−1.44029	−0.720143	−	0.693826i	$-0.755924\pi$
$487$	−15479.0	−1.44029	−0.720143	+	0.693826i	$0.755924\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14417.7	1.32518	0.662590	−	0.748983i	$-0.269457\pi$
$491$	14417.7	1.32518	0.662590	+	0.748983i	$0.269457\pi$
$492$	0	0
$493$	1671.67	0.152715
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1947.27	0.175748
$498$	0	0
$499$	3611.04	0.323953	0.161976	−	0.986795i	$-0.448213\pi$
$499$	3611.04	0.323953	0.161976	+	0.986795i	$0.448213\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−6050.08	−0.536302	−0.268151	−	0.963377i	$-0.586413\pi$
$503$	−6050.08	−0.536302	−0.268151	+	0.963377i	$0.586413\pi$
$504$	0	0
$505$	−2617.32	−0.230632
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−708.919	−0.0617334	−0.0308667	−	0.999524i	$-0.509827\pi$
$509$	−708.919	−0.0617334	−0.0308667	+	0.999524i	$0.509827\pi$
$510$	0	0
$511$	−798.284	−0.0691077
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4849.41	−0.414933
$516$	0	0
$517$	−634.552	−0.0539798
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19852.8	−1.66942	−0.834709	−	0.550691i	$-0.814364\pi$
$521$	−19852.8	−1.66942	−0.834709	+	0.550691i	$0.814364\pi$
$522$	0	0
$523$	−1684.83	−0.140865	−0.0704325	−	0.997517i	$-0.522438\pi$
$523$	−1684.83	−0.140865	−0.0704325	+	0.997517i	$0.522438\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6800.07	−0.562079
$528$	0	0
$529$	21123.0	1.73609
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8002.23	0.650310
$534$	0	0
$535$	1870.00	0.151116
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3235.74	0.258578
$540$	0	0
$541$	15054.7	1.19640	0.598200	−	0.801347i	$-0.295883\pi$
$541$	15054.7	1.19640	0.598200	+	0.801347i	$0.295883\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9811.81	0.771177
$546$	0	0
$547$	−2796.32	−0.218578	−0.109289	−	0.994010i	$-0.534857\pi$
$547$	−2796.32	−0.218578	−0.109289	+	0.994010i	$0.534857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5923.07	0.457952
$552$	0	0
$553$	236.071	0.0181533
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10530.4	0.801057	0.400529	−	0.916284i	$-0.368827\pi$
$557$	10530.4	0.801057	0.400529	+	0.916284i	$0.368827\pi$
$558$	0	0
$559$	−14909.8	−1.12811
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11472.7	0.858819	0.429410	−	0.903110i	$-0.358722\pi$
$563$	11472.7	0.858819	0.429410	+	0.903110i	$0.358722\pi$
$564$	0	0
$565$	1933.24	0.143951
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5727.73	−0.422002	−0.211001	−	0.977486i	$-0.567672\pi$
$569$	−5727.73	−0.422002	−0.211001	+	0.977486i	$0.567672\pi$
$570$	0	0
$571$	2466.83	0.180794	0.0903972	−	0.995906i	$-0.471186\pi$
$571$	2466.83	0.180794	0.0903972	+	0.995906i	$0.471186\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4561.39	−0.330823
$576$	0	0
$577$	−4094.13	−0.295392	−0.147696	−	0.989033i	$-0.547186\pi$
$577$	−4094.13	−0.295392	−0.147696	+	0.989033i	$0.547186\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4501.37	−0.321426
$582$	0	0
$583$	−6886.93	−0.489241
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9253.56	0.650657	0.325328	−	0.945601i	$-0.394525\pi$
$587$	9253.56	0.650657	0.325328	+	0.945601i	$0.394525\pi$
$588$	0	0
$589$	−24094.0	−1.68553
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23367.6	−1.61820	−0.809100	−	0.587671i	$-0.800045\pi$
$593$	−23367.6	−1.61820	−0.809100	+	0.587671i	$0.800045\pi$
$594$	0	0
$595$	−890.203	−0.0613358
$596$	0	0
$597$	0	0
$598$	0	0
$599$	9990.04	0.681439	0.340720	−	0.940165i	$-0.389329\pi$
$599$	9990.04	0.681439	0.340720	+	0.940165i	$0.389329\pi$
$600$	0	0
$601$	2726.36	0.185042	0.0925212	−	0.995711i	$-0.470507\pi$
$601$	2726.36	0.185042	0.0925212	+	0.995711i	$0.470507\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6128.45	0.411829
$606$	0	0
$607$	3765.23	0.251773	0.125886	−	0.992045i	$-0.459823\pi$
$607$	3765.23	0.251773	0.125886	+	0.992045i	$0.459823\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3621.81	−0.239808
$612$	0	0
$613$	13014.2	0.857484	0.428742	−	0.903427i	$-0.358957\pi$
$613$	13014.2	0.857484	0.428742	+	0.903427i	$0.358957\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11152.4	0.727678	0.363839	−	0.931462i	$-0.381466\pi$
$617$	11152.4	0.727678	0.363839	+	0.931462i	$0.381466\pi$
$618$	0	0
$619$	−8128.37	−0.527797	−0.263899	−	0.964550i	$-0.585008\pi$
$619$	−8128.37	−0.527797	−0.263899	+	0.964550i	$0.585008\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5780.91	0.371761
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3034.61	0.192365
$630$	0	0
$631$	11271.8	0.711129	0.355565	−	0.934652i	$-0.384289\pi$
$631$	11271.8	0.711129	0.355565	+	0.934652i	$0.384289\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6744.15	−0.421470
$636$	0	0
$637$	18468.5	1.14874
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−19862.2	−1.22388	−0.611942	−	0.790903i	$-0.709611\pi$
$641$	−19862.2	−1.22388	−0.611942	+	0.790903i	$0.709611\pi$
$642$	0	0
$643$	28296.6	1.73547	0.867735	−	0.497026i	$-0.165575\pi$
$643$	28296.6	1.73547	0.867735	+	0.497026i	$0.165575\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20338.9	−1.23586	−0.617932	−	0.786232i	$-0.712029\pi$
$647$	−20338.9	−1.23586	−0.617932	+	0.786232i	$0.712029\pi$
$648$	0	0
$649$	−4620.92	−0.279487
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2905.09	−0.174096	−0.0870482	−	0.996204i	$-0.527743\pi$
$653$	−2905.09	−0.174096	−0.0870482	+	0.996204i	$0.527743\pi$
$654$	0	0
$655$	1765.74	0.105333
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−22802.3	−1.34788	−0.673940	−	0.738786i	$-0.735399\pi$
$659$	−22802.3	−1.34788	−0.673940	+	0.738786i	$0.735399\pi$
$660$	0	0
$661$	−21658.0	−1.27443	−0.637216	−	0.770685i	$-0.719914\pi$
$661$	−21658.0	−1.27443	−0.637216	+	0.770685i	$0.719914\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3154.17	−0.183930
$666$	0	0
$667$	9014.60	0.523309
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1296.21	0.0745746
$672$	0	0
$673$	26706.4	1.52965	0.764826	−	0.644237i	$-0.222825\pi$
$673$	26706.4	1.52965	0.764826	+	0.644237i	$0.222825\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7679.87	−0.435984	−0.217992	−	0.975951i	$-0.569951\pi$
$677$	−7679.87	−0.435984	−0.217992	+	0.975951i	$0.569951\pi$
$678$	0	0
$679$	6351.78	0.358997
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−27373.5	−1.53355	−0.766777	−	0.641914i	$-0.778141\pi$
$683$	−27373.5	−1.53355	−0.766777	+	0.641914i	$0.778141\pi$
$684$	0	0
$685$	−5982.42	−0.333688
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−39308.3	−2.17348
$690$	0	0
$691$	−21036.1	−1.15810	−0.579052	−	0.815290i	$-0.696577\pi$
$691$	−21036.1	−1.15810	−0.579052	+	0.815290i	$0.696577\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13843.6	−0.755563
$696$	0	0
$697$	4622.51	0.251205
$698$	0	0
$699$	0	0
$700$	0	0
$701$	8762.99	0.472145	0.236072	−	0.971735i	$-0.424140\pi$
$701$	8762.99	0.472145	0.236072	+	0.971735i	$0.424140\pi$
$702$	0	0
$703$	10752.3	0.576855
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2754.51	−0.146526
$708$	0	0
$709$	−12012.0	−0.636278	−0.318139	−	0.948044i	$-0.603058\pi$
$709$	−12012.0	−0.636278	−0.318139	+	0.948044i	$0.603058\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−36669.8	−1.92608
$714$	0	0
$715$	−3005.38	−0.157196
$716$	0	0
$717$	0	0
$718$	0	0
$719$	23304.8	1.20879	0.604395	−	0.796685i	$-0.293415\pi$
$719$	23304.8	1.20879	0.604395	+	0.796685i	$0.293415\pi$
$720$	0	0
$721$	−5103.61	−0.263618
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1235.18	−0.0632736
$726$	0	0
$727$	−4348.04	−0.221816	−0.110908	−	0.993831i	$-0.535376\pi$
$727$	−4348.04	−0.221816	−0.110908	+	0.993831i	$0.535376\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−8612.67	−0.435775
$732$	0	0
$733$	37182.4	1.87362	0.936809	−	0.349840i	$-0.113764\pi$
$733$	37182.4	1.87362	0.936809	+	0.349840i	$0.113764\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4783.49	0.239080
$738$	0	0
$739$	−5384.48	−0.268026	−0.134013	−	0.990980i	$-0.542786\pi$
$739$	−5384.48	−0.268026	−0.134013	+	0.990980i	$0.542786\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	5771.71	0.284984	0.142492	−	0.989796i	$-0.454488\pi$
$743$	5771.71	0.284984	0.142492	+	0.989796i	$0.454488\pi$
$744$	0	0
$745$	−16554.4	−0.814104
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1968.02	0.0960079
$750$	0	0
$751$	−24358.9	−1.18358	−0.591790	−	0.806092i	$-0.701579\pi$
$751$	−24358.9	−1.18358	−0.591790	+	0.806092i	$0.701579\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	4873.58	0.234924
$756$	0	0
$757$	−9401.29	−0.451381	−0.225691	−	0.974199i	$-0.572464\pi$
$757$	−9401.29	−0.451381	−0.225691	+	0.974199i	$0.572464\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38564.4	1.83700	0.918502	−	0.395417i	$-0.129400\pi$
$761$	38564.4	1.83700	0.918502	+	0.395417i	$0.129400\pi$
$762$	0	0
$763$	10326.1	0.489948
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26374.6	−1.24163
$768$	0	0
$769$	−14048.8	−0.658792	−0.329396	−	0.944192i	$-0.606845\pi$
$769$	−14048.8	−0.658792	−0.329396	+	0.944192i	$0.606845\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6271.17	−0.291796	−0.145898	−	0.989300i	$-0.546607\pi$
$773$	−6271.17	−0.291796	−0.145898	+	0.989300i	$0.546607\pi$
$774$	0	0
$775$	5024.49	0.232884
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16378.5	0.753301
$780$	0	0
$781$	3797.55	0.173991
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4650.06	0.211424
$786$	0	0
$787$	−5228.24	−0.236807	−0.118403	−	0.992966i	$-0.537778\pi$
$787$	−5228.24	−0.236807	−0.118403	+	0.992966i	$0.537778\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2034.58	0.0914556
$792$	0	0
$793$	7398.32	0.331301
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40880.5	1.81689	0.908445	−	0.418004i	$-0.137270\pi$
$797$	40880.5	1.81689	0.908445	+	0.418004i	$0.137270\pi$
$798$	0	0
$799$	−2092.15	−0.0926345
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1556.81	−0.0684167
$804$	0	0
$805$	−4800.48	−0.210180
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−36230.2	−1.57452	−0.787260	−	0.616621i	$-0.788501\pi$
$809$	−36230.2	−1.57452	−0.787260	+	0.616621i	$0.788501\pi$
$810$	0	0
$811$	12939.3	0.560248	0.280124	−	0.959964i	$-0.409624\pi$
$811$	12939.3	0.560248	0.280124	+	0.959964i	$0.409624\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7309.07	−0.314142
$816$	0	0
$817$	−30516.5	−1.30678
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13429.0	0.570860	0.285430	−	0.958400i	$-0.407864\pi$
$821$	13429.0	0.570860	0.285430	+	0.958400i	$0.407864\pi$
$822$	0	0
$823$	37367.4	1.58268	0.791340	−	0.611376i	$-0.209384\pi$
$823$	37367.4	1.58268	0.791340	+	0.611376i	$0.209384\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−37017.4	−1.55650	−0.778248	−	0.627958i	$-0.783891\pi$
$827$	−37017.4	−1.55650	−0.778248	+	0.627958i	$0.783891\pi$
$828$	0	0
$829$	−13971.2	−0.585332	−0.292666	−	0.956215i	$-0.594542\pi$
$829$	−13971.2	−0.585332	−0.292666	+	0.956215i	$0.594542\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10668.4	0.443744
$834$	0	0
$835$	−17538.0	−0.726858
$836$	0	0
$837$	0	0
$838$	0	0
$839$	47630.9	1.95995	0.979976	−	0.199115i	$-0.0638068\pi$
$839$	47630.9	1.95995	0.979976	+	0.199115i	$0.0638068\pi$
$840$	0	0
$841$	−21947.9	−0.899911
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6168.70	−0.251136
$846$	0	0
$847$	6449.69	0.261646
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16364.4	0.659181
$852$	0	0
$853$	−37418.2	−1.50196	−0.750982	−	0.660323i	$-0.770420\pi$
$853$	−37418.2	−1.50196	−0.750982	+	0.660323i	$0.770420\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	21465.2	0.855587	0.427794	−	0.903876i	$-0.359291\pi$
$857$	21465.2	0.855587	0.427794	+	0.903876i	$0.359291\pi$
$858$	0	0
$859$	−32638.7	−1.29641	−0.648206	−	0.761465i	$-0.724480\pi$
$859$	−32638.7	−1.29641	−0.648206	+	0.761465i	$0.724480\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3305.92	0.130400	0.0651998	−	0.997872i	$-0.479232\pi$
$863$	3305.92	0.130400	0.0651998	+	0.997872i	$0.479232\pi$
$864$	0	0
$865$	12037.3	0.473156
$866$	0	0
$867$	0	0
$868$	0	0
$869$	460.384	0.0179718
$870$	0	0
$871$	27302.5	1.06213
$872$	0	0
$873$	0	0
$874$	0	0
$875$	657.761	0.0254130
$876$	0	0
$877$	−34721.4	−1.33690	−0.668449	−	0.743758i	$-0.733041\pi$
$877$	−34721.4	−1.33690	−0.668449	+	0.743758i	$0.733041\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−51524.8	−1.97039	−0.985195	−	0.171435i	$-0.945160\pi$
$881$	−51524.8	−1.97039	−0.985195	+	0.171435i	$0.945160\pi$
$882$	0	0
$883$	24558.5	0.935969	0.467985	−	0.883737i	$-0.344980\pi$
$883$	24558.5	0.935969	0.467985	+	0.883737i	$0.344980\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33009.1	−1.24953	−0.624767	−	0.780811i	$-0.714806\pi$
$887$	−33009.1	−1.24953	−0.624767	+	0.780811i	$0.714806\pi$
$888$	0	0
$889$	−7097.66	−0.267770
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7412.92	−0.277787
$894$	0	0
$895$	−3545.45	−0.132415
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9929.83	−0.368385
$900$	0	0
$901$	−22706.6	−0.839584
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6229.69	0.228820
$906$	0	0
$907$	26038.7	0.953253	0.476626	−	0.879106i	$-0.341859\pi$
$907$	26038.7	0.953253	0.476626	+	0.879106i	$0.341859\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48904.8	−1.77858	−0.889291	−	0.457341i	$-0.848802\pi$
$911$	−48904.8	−1.77858	−0.889291	+	0.457341i	$0.848802\pi$
$912$	0	0
$913$	−8778.55	−0.318212
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1858.29	0.0669207
$918$	0	0
$919$	8429.18	0.302560	0.151280	−	0.988491i	$-0.451660\pi$
$919$	8429.18	0.302560	0.151280	+	0.988491i	$0.451660\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21675.1	0.772963
$924$	0	0
$925$	−2242.24	−0.0797020
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−30915.9	−1.09184	−0.545919	−	0.837838i	$-0.683819\pi$
$929$	−30915.9	−1.09184	−0.545919	+	0.837838i	$0.683819\pi$
$930$	0	0
$931$	37800.3	1.33067
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1736.07	−0.0607225
$936$	0	0
$937$	36984.2	1.28946	0.644729	−	0.764411i	$-0.276970\pi$
$937$	36984.2	1.28946	0.644729	+	0.764411i	$0.276970\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−24141.7	−0.836341	−0.418170	−	0.908369i	$-0.637329\pi$
$941$	−24141.7	−0.836341	−0.418170	+	0.908369i	$0.637329\pi$
$942$	0	0
$943$	24927.2	0.860808
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19388.6	−0.665306	−0.332653	−	0.943049i	$-0.607944\pi$
$947$	−19388.6	−0.665306	−0.332653	+	0.943049i	$0.607944\pi$
$948$	0	0
$949$	−8885.74	−0.303944
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−37220.6	−1.26515	−0.632577	−	0.774497i	$-0.718003\pi$
$953$	−37220.6	−1.26515	−0.632577	+	0.774497i	$0.718003\pi$
$954$	0	0
$955$	−14045.7	−0.475924
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6296.00	−0.212000
$960$	0	0
$961$	10601.8	0.355873
$962$	0	0
$963$	0	0
$964$	0	0
$965$	21428.6	0.714830
$966$	0	0
$967$	38860.3	1.29231	0.646155	−	0.763206i	$-0.276376\pi$
$967$	38860.3	1.29231	0.646155	+	0.763206i	$0.276376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−45501.5	−1.50382	−0.751912	−	0.659263i	$-0.770868\pi$
$971$	−45501.5	−1.50382	−0.751912	+	0.659263i	$0.770868\pi$
$972$	0	0
$973$	−14569.2	−0.480028
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−55908.4	−1.83078	−0.915388	−	0.402573i	$-0.868116\pi$
$977$	−55908.4	−1.83078	−0.915388	+	0.402573i	$0.868116\pi$
$978$	0	0
$979$	11273.9	0.368044
$980$	0	0
$981$	0	0
$982$	0	0
$983$	7730.62	0.250833	0.125416	−	0.992104i	$-0.459973\pi$
$983$	7730.62	0.250833	0.125416	+	0.992104i	$0.459973\pi$
$984$	0	0
$985$	−13707.2	−0.443398
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−46444.4	−1.49327
$990$	0	0
$991$	−28247.8	−0.905471	−0.452735	−	0.891645i	$-0.649552\pi$
$991$	−28247.8	−0.905471	−0.452735	+	0.891645i	$0.649552\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	14610.3	0.465507
$996$	0	0
$997$	27881.1	0.885661	0.442831	−	0.896605i	$-0.353974\pi$
$997$	27881.1	0.885661	0.442831	+	0.896605i	$0.353974\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.y.1.1		2
3.2	odd	2		2160.4.a.bd.1.1		2
4.3	odd	2		1080.4.a.a.1.2	✓	2
12.11	even	2		1080.4.a.b.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.a.1.2	✓	2	4.3	odd	2
1080.4.a.b.1.2	yes	2	12.11	even	2
2160.4.a.y.1.1		2	1.1	even	1	trivial
2160.4.a.bd.1.1		2	3.2	odd	2

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -5.26209 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -5.26209 q^{7} -10.2621 q^{11} -58.5725 q^{13} -33.8346 q^{17} -119.883 q^{19} -182.455 q^{23} +25.0000 q^{25} -49.4071 q^{29} +200.980 q^{31} +26.3104 q^{35} -89.6896 q^{37} -136.621 q^{41} +254.552 q^{43} +61.8346 q^{47} -315.310 q^{49} +671.104 q^{53} +51.3104 q^{55} +450.290 q^{59} -126.310 q^{61} +292.863 q^{65} -466.132 q^{67} -370.056 q^{71} +151.705 q^{73} +54.0000 q^{77} -44.8626 q^{79} +855.435 q^{83} +169.173 q^{85} -1098.60 q^{89} +308.214 q^{91} +599.415 q^{95} -1207.08 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 5 q^{7}+O(q^{10}) \) 2 * q - 10 * q^5 + 5 * q^7	\( 2 q - 10 q^{5} + 5 q^{7} - 5 q^{11} - 24 q^{13} + 41 q^{17} - 69 q^{19} - 101 q^{23} + 50 q^{25} + 103 q^{29} + 107 q^{31} - 25 q^{35} - 257 q^{37} - 118 q^{41} + 121 q^{43} + 15 q^{47} - 553 q^{49} + 566 q^{53} + 25 q^{55} + 528 q^{59} - 175 q^{61} + 120 q^{65} + 201 q^{67} - 26 q^{71} - 923 q^{73} + 108 q^{77} + 376 q^{79} + 1152 q^{83} - 205 q^{85} - 148 q^{89} + 663 q^{91} + 345 q^{95} - 1343 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 + 5 * q^7 - 5 * q^11 - 24 * q^13 + 41 * q^17 - 69 * q^19 - 101 * q^23 + 50 * q^25 + 103 * q^29 + 107 * q^31 - 25 * q^35 - 257 * q^37 - 118 * q^41 + 121 * q^43 + 15 * q^47 - 553 * q^49 + 566 * q^53 + 25 * q^55 + 528 * q^59 - 175 * q^61 + 120 * q^65 + 201 * q^67 - 26 * q^71 - 923 * q^73 + 108 * q^77 + 376 * q^79 + 1152 * q^83 - 205 * q^85 - 148 * q^89 + 663 * q^91 + 345 * q^95 - 1343 * q^97

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