LMFDB - Embedded newform 2160.4.a.bq.1.3

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:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.67370$ 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)

$f(q)$	$=$	$q+5.00000 q^{5} +22.8935 q^{7} +O(q^{10})$	$q+5.00000 q^{5} +22.8935 q^{7} -11.0828 q^{11} -11.6368 q^{13} +10.0643 q^{17} -117.865 q^{19} -172.441 q^{23} +25.0000 q^{25} +178.321 q^{29} -140.528 q^{31} +114.468 q^{35} +250.074 q^{37} +361.569 q^{41} +360.707 q^{43} +600.121 q^{47} +181.114 q^{49} +201.312 q^{53} -55.4140 q^{55} -415.772 q^{59} -54.6270 q^{61} -58.1841 q^{65} +531.079 q^{67} +933.534 q^{71} -560.199 q^{73} -253.725 q^{77} -810.781 q^{79} -538.210 q^{83} +50.3214 q^{85} +686.173 q^{89} -266.408 q^{91} -589.324 q^{95} +714.655 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 15 q^{5} + 4 q^{7}+O(q^{10}) $ 3 * q + 15 * q^5 + 4 * q^7	$ 3 q + 15 q^{5} + 4 q^{7} - 5 q^{11} + 7 q^{13} + 155 q^{17} + 50 q^{19} - 285 q^{23} + 75 q^{25} + 115 q^{29} + 115 q^{31} + 20 q^{35} - 384 q^{37} + 580 q^{41} + 797 q^{43} + 145 q^{47} + 577 q^{49} - 400 q^{53} - 25 q^{55} - 380 q^{59} - 152 q^{61} + 35 q^{65} - 2 q^{67} - 40 q^{71} - 980 q^{73} + 1950 q^{77} - 1013 q^{79} - 270 q^{83} + 775 q^{85} + 1020 q^{89} + 632 q^{91} + 250 q^{95} + 720 q^{97}+O(q^{100}) $ 3 * q + 15 * q^5 + 4 * q^7 - 5 * q^11 + 7 * q^13 + 155 * q^17 + 50 * q^19 - 285 * q^23 + 75 * q^25 + 115 * q^29 + 115 * q^31 + 20 * q^35 - 384 * q^37 + 580 * q^41 + 797 * q^43 + 145 * q^47 + 577 * q^49 - 400 * q^53 - 25 * q^55 - 380 * q^59 - 152 * q^61 + 35 * q^65 - 2 * q^67 - 40 * q^71 - 980 * q^73 + 1950 * q^77 - 1013 * q^79 - 270 * q^83 + 775 * q^85 + 1020 * q^89 + 632 * q^91 + 250 * q^95 + 720 * 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
$7$	22.8935	1.23613	0.618067	−	0.786125i	$-0.287916\pi$
$7$	22.8935	1.23613	0.618067	+	0.786125i	$0.287916\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.0828	−0.303781	−0.151891	−	0.988397i	$-0.548536\pi$
$11$	−11.0828	−0.303781	−0.151891	+	0.988397i	$0.548536\pi$
$12$	0	0
$13$	−11.6368	−0.248267	−0.124134	−	0.992266i	$-0.539615\pi$
$13$	−11.6368	−0.248267	−0.124134	+	0.992266i	$0.539615\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	10.0643	0.143585	0.0717926	−	0.997420i	$-0.477128\pi$
$17$	10.0643	0.143585	0.0717926	+	0.997420i	$0.477128\pi$
$18$	0	0
$19$	−117.865	−1.42316	−0.711579	−	0.702606i	$-0.752020\pi$
$19$	−117.865	−1.42316	−0.711579	+	0.702606i	$0.752020\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−172.441	−1.56332	−0.781661	−	0.623704i	$-0.785627\pi$
$23$	−172.441	−1.56332	−0.781661	+	0.623704i	$0.785627\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	178.321	1.14184	0.570919	−	0.821006i	$-0.306587\pi$
$29$	178.321	1.14184	0.570919	+	0.821006i	$0.306587\pi$
$30$	0	0
$31$	−140.528	−0.814182	−0.407091	−	0.913388i	$-0.633457\pi$
$31$	−140.528	−0.814182	−0.407091	+	0.913388i	$0.633457\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	114.468	0.552816
$36$	0	0
$37$	250.074	1.11113	0.555566	−	0.831473i	$-0.312502\pi$
$37$	250.074	1.11113	0.555566	+	0.831473i	$0.312502\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	361.569	1.37726	0.688629	−	0.725114i	$-0.258213\pi$
$41$	361.569	1.37726	0.688629	+	0.725114i	$0.258213\pi$
$42$	0	0
$43$	360.707	1.27924	0.639620	−	0.768691i	$-0.279092\pi$
$43$	360.707	1.27924	0.639620	+	0.768691i	$0.279092\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	600.121	1.86248	0.931241	−	0.364405i	$-0.118727\pi$
$47$	600.121	1.86248	0.931241	+	0.364405i	$0.118727\pi$
$48$	0	0
$49$	181.114	0.528028
$50$	0	0
$51$	0	0
$52$	0	0
$53$	201.312	0.521742	0.260871	−	0.965374i	$-0.415990\pi$
$53$	201.312	0.521742	0.260871	+	0.965374i	$0.415990\pi$
$54$	0	0
$55$	−55.4140	−0.135855
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−415.772	−0.917438	−0.458719	−	0.888581i	$-0.651692\pi$
$59$	−415.772	−0.917438	−0.458719	+	0.888581i	$0.651692\pi$
$60$	0	0
$61$	−54.6270	−0.114660	−0.0573301	−	0.998355i	$-0.518259\pi$
$61$	−54.6270	−0.114660	−0.0573301	+	0.998355i	$0.518259\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−58.1841	−0.111029
$66$	0	0
$67$	531.079	0.968382	0.484191	−	0.874962i	$-0.339114\pi$
$67$	531.079	0.968382	0.484191	+	0.874962i	$0.339114\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	933.534	1.56042	0.780212	−	0.625515i	$-0.215111\pi$
$71$	933.534	1.56042	0.780212	+	0.625515i	$0.215111\pi$
$72$	0	0
$73$	−560.199	−0.898169	−0.449085	−	0.893489i	$-0.648250\pi$
$73$	−560.199	−0.898169	−0.449085	+	0.893489i	$0.648250\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−253.725	−0.375514
$78$	0	0
$79$	−810.781	−1.15468	−0.577342	−	0.816503i	$-0.695910\pi$
$79$	−810.781	−1.15468	−0.577342	+	0.816503i	$0.695910\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−538.210	−0.711762	−0.355881	−	0.934531i	$-0.615819\pi$
$83$	−538.210	−0.711762	−0.355881	+	0.934531i	$0.615819\pi$
$84$	0	0
$85$	50.3214	0.0642132
$86$	0	0
$87$	0	0
$88$	0	0
$89$	686.173	0.817238	0.408619	−	0.912705i	$-0.366010\pi$
$89$	686.173	0.817238	0.408619	+	0.912705i	$0.366010\pi$
$90$	0	0
$91$	−266.408	−0.306892
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−589.324	−0.636456
$96$	0	0
$97$	714.655	0.748064	0.374032	−	0.927416i	$-0.377975\pi$
$97$	714.655	0.748064	0.374032	+	0.927416i	$0.377975\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	973.907	0.959479	0.479739	−	0.877411i	$-0.340731\pi$
$101$	973.907	0.959479	0.479739	+	0.877411i	$0.340731\pi$
$102$	0	0
$103$	759.229	0.726301	0.363151	−	0.931730i	$-0.381701\pi$
$103$	759.229	0.726301	0.363151	+	0.931730i	$0.381701\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1832.06	−1.65525	−0.827625	−	0.561282i	$-0.810308\pi$
$107$	−1832.06	−1.65525	−0.827625	+	0.561282i	$0.810308\pi$
$108$	0	0
$109$	1370.91	1.20467	0.602335	−	0.798244i	$-0.294237\pi$
$109$	1370.91	1.20467	0.602335	+	0.798244i	$0.294237\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	583.567	0.485817	0.242909	−	0.970049i	$-0.421898\pi$
$113$	583.567	0.485817	0.242909	+	0.970049i	$0.421898\pi$
$114$	0	0
$115$	−862.204	−0.699139
$116$	0	0
$117$	0	0
$118$	0	0
$119$	230.407	0.177491
$120$	0	0
$121$	−1208.17	−0.907717
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	2432.76	1.69978	0.849891	−	0.526958i	$-0.176667\pi$
$127$	2432.76	1.69978	0.849891	+	0.526958i	$0.176667\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−501.073	−0.334191	−0.167095	−	0.985941i	$-0.553439\pi$
$131$	−501.073	−0.334191	−0.167095	+	0.985941i	$0.553439\pi$
$132$	0	0
$133$	−2698.34	−1.75922
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2523.07	1.57343	0.786716	−	0.617316i	$-0.211780\pi$
$137$	2523.07	1.57343	0.786716	+	0.617316i	$0.211780\pi$
$138$	0	0
$139$	638.842	0.389826	0.194913	−	0.980821i	$-0.437558\pi$
$139$	638.842	0.389826	0.194913	+	0.980821i	$0.437558\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	128.969	0.0754189
$144$	0	0
$145$	891.603	0.510645
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2674.13	−1.47029	−0.735146	−	0.677909i	$-0.762886\pi$
$149$	−2674.13	−1.47029	−0.735146	+	0.677909i	$0.762886\pi$
$150$	0	0
$151$	1036.68	0.558700	0.279350	−	0.960189i	$-0.409881\pi$
$151$	1036.68	0.558700	0.279350	+	0.960189i	$0.409881\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−702.642	−0.364113
$156$	0	0
$157$	−381.858	−0.194112	−0.0970560	−	0.995279i	$-0.530943\pi$
$157$	−381.858	−0.194112	−0.0970560	+	0.995279i	$0.530943\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3947.78	−1.93248
$162$	0	0
$163$	−2421.17	−1.16344	−0.581719	−	0.813390i	$-0.697620\pi$
$163$	−2421.17	−1.16344	−0.581719	+	0.813390i	$0.697620\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3535.76	1.63836	0.819178	−	0.573540i	$-0.194430\pi$
$167$	3535.76	1.63836	0.819178	+	0.573540i	$0.194430\pi$
$168$	0	0
$169$	−2061.58	−0.938363
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3143.88	1.38165	0.690824	−	0.723023i	$-0.257248\pi$
$173$	3143.88	1.38165	0.690824	+	0.723023i	$0.257248\pi$
$174$	0	0
$175$	572.338	0.247227
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3299.06	1.37756	0.688780	−	0.724970i	$-0.258147\pi$
$179$	3299.06	1.37756	0.688780	+	0.724970i	$0.258147\pi$
$180$	0	0
$181$	1875.10	0.770026	0.385013	−	0.922911i	$-0.374197\pi$
$181$	1875.10	0.770026	0.385013	+	0.922911i	$0.374197\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1250.37	0.496913
$186$	0	0
$187$	−111.541	−0.0436185
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1361.62	0.515829	0.257915	−	0.966168i	$-0.416965\pi$
$191$	1361.62	0.515829	0.257915	+	0.966168i	$0.416965\pi$
$192$	0	0
$193$	2234.20	0.833271	0.416636	−	0.909074i	$-0.363209\pi$
$193$	2234.20	0.833271	0.416636	+	0.909074i	$0.363209\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−346.625	−0.125360	−0.0626801	−	0.998034i	$-0.519965\pi$
$197$	−346.625	−0.125360	−0.0626801	+	0.998034i	$0.519965\pi$
$198$	0	0
$199$	−4198.77	−1.49569	−0.747846	−	0.663872i	$-0.768912\pi$
$199$	−4198.77	−1.49569	−0.747846	+	0.663872i	$0.768912\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4082.39	1.41146
$204$	0	0
$205$	1807.84	0.615929
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1306.27	0.432329
$210$	0	0
$211$	−4728.46	−1.54275	−0.771375	−	0.636380i	$-0.780431\pi$
$211$	−4728.46	−1.54275	−0.771375	+	0.636380i	$0.780431\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1803.54	0.572094
$216$	0	0
$217$	−3217.19	−1.00644
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−117.116	−0.0356475
$222$	0	0
$223$	−2430.16	−0.729756	−0.364878	−	0.931055i	$-0.618889\pi$
$223$	−2430.16	−0.729756	−0.364878	+	0.931055i	$0.618889\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	584.687	0.170956	0.0854780	−	0.996340i	$-0.472758\pi$
$227$	584.687	0.170956	0.0854780	+	0.996340i	$0.472758\pi$
$228$	0	0
$229$	−4731.77	−1.36543	−0.682717	−	0.730683i	$-0.739202\pi$
$229$	−4731.77	−1.36543	−0.682717	+	0.730683i	$0.739202\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1228.45	0.345401	0.172701	−	0.984974i	$-0.444751\pi$
$233$	1228.45	0.345401	0.172701	+	0.984974i	$0.444751\pi$
$234$	0	0
$235$	3000.60	0.832927
$236$	0	0
$237$	0	0
$238$	0	0
$239$	120.545	0.0326253	0.0163126	−	0.999867i	$-0.494807\pi$
$239$	120.545	0.0326253	0.0163126	+	0.999867i	$0.494807\pi$
$240$	0	0
$241$	1732.56	0.463086	0.231543	−	0.972825i	$-0.425623\pi$
$241$	1732.56	0.463086	0.231543	+	0.972825i	$0.425623\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	905.568	0.236141
$246$	0	0
$247$	1371.57	0.353324
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3287.82	0.826793	0.413397	−	0.910551i	$-0.364342\pi$
$251$	3287.82	0.826793	0.413397	+	0.910551i	$0.364342\pi$
$252$	0	0
$253$	1911.13	0.474908
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1489.82	−0.361605	−0.180803	−	0.983519i	$-0.557870\pi$
$257$	−1489.82	−0.361605	−0.180803	+	0.983519i	$0.557870\pi$
$258$	0	0
$259$	5725.07	1.37351
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−710.144	−0.166499	−0.0832497	−	0.996529i	$-0.526530\pi$
$263$	−710.144	−0.166499	−0.0832497	+	0.996529i	$0.526530\pi$
$264$	0	0
$265$	1006.56	0.233330
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3667.61	0.831294	0.415647	−	0.909526i	$-0.363555\pi$
$269$	3667.61	0.831294	0.415647	+	0.909526i	$0.363555\pi$
$270$	0	0
$271$	1990.45	0.446166	0.223083	−	0.974799i	$-0.428388\pi$
$271$	1990.45	0.446166	0.223083	+	0.974799i	$0.428388\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−277.070	−0.0607562
$276$	0	0
$277$	8314.68	1.80354	0.901770	−	0.432215i	$-0.142268\pi$
$277$	8314.68	1.80354	0.901770	+	0.432215i	$0.142268\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5765.13	1.22391	0.611955	−	0.790892i	$-0.290383\pi$
$281$	5765.13	1.22391	0.611955	+	0.790892i	$0.290383\pi$
$282$	0	0
$283$	−457.561	−0.0961102	−0.0480551	−	0.998845i	$-0.515302\pi$
$283$	−457.561	−0.0961102	−0.0480551	+	0.998845i	$0.515302\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8277.59	1.70248
$288$	0	0
$289$	−4811.71	−0.979383
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5302.82	1.05732	0.528659	−	0.848834i	$-0.322695\pi$
$293$	5302.82	1.05732	0.528659	+	0.848834i	$0.322695\pi$
$294$	0	0
$295$	−2078.86	−0.410291
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2006.66	0.388122
$300$	0	0
$301$	8257.86	1.58131
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−273.135	−0.0512776
$306$	0	0
$307$	6583.54	1.22392	0.611959	−	0.790890i	$-0.290382\pi$
$307$	6583.54	1.22392	0.611959	+	0.790890i	$0.290382\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8128.05	1.48199	0.740996	−	0.671510i	$-0.234354\pi$
$311$	8128.05	1.48199	0.740996	+	0.671510i	$0.234354\pi$
$312$	0	0
$313$	−5207.63	−0.940424	−0.470212	−	0.882554i	$-0.655823\pi$
$313$	−5207.63	−0.940424	−0.470212	+	0.882554i	$0.655823\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3262.34	−0.578016	−0.289008	−	0.957327i	$-0.593325\pi$
$317$	−3262.34	−0.578016	−0.289008	+	0.957327i	$0.593325\pi$
$318$	0	0
$319$	−1976.29	−0.346869
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1186.22	−0.204345
$324$	0	0
$325$	−290.921	−0.0496535
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13738.9	2.30228
$330$	0	0
$331$	−10360.5	−1.72043	−0.860216	−	0.509930i	$-0.829671\pi$
$331$	−10360.5	−1.72043	−0.860216	+	0.509930i	$0.829671\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2655.39	0.433074
$336$	0	0
$337$	−3735.26	−0.603777	−0.301888	−	0.953343i	$-0.597617\pi$
$337$	−3735.26	−0.603777	−0.301888	+	0.953343i	$0.597617\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1557.45	0.247333
$342$	0	0
$343$	−3706.15	−0.583421
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−197.010	−0.0304785	−0.0152393	−	0.999884i	$-0.504851\pi$
$347$	−197.010	−0.0304785	−0.0152393	+	0.999884i	$0.504851\pi$
$348$	0	0
$349$	−7334.20	−1.12490	−0.562451	−	0.826831i	$-0.690141\pi$
$349$	−7334.20	−1.12490	−0.562451	+	0.826831i	$0.690141\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	4616.28	0.696034	0.348017	−	0.937488i	$-0.386855\pi$
$353$	4616.28	0.696034	0.348017	+	0.937488i	$0.386855\pi$
$354$	0	0
$355$	4667.67	0.697843
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1153.79	0.169623	0.0848115	−	0.996397i	$-0.472971\pi$
$359$	1153.79	0.169623	0.0848115	+	0.996397i	$0.472971\pi$
$360$	0	0
$361$	7033.09	1.02538
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2801.00	−0.401673
$366$	0	0
$367$	3449.05	0.490569	0.245285	−	0.969451i	$-0.421119\pi$
$367$	3449.05	0.490569	0.245285	+	0.969451i	$0.421119\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4608.74	0.644943
$372$	0	0
$373$	362.880	0.0503732	0.0251866	−	0.999683i	$-0.491982\pi$
$373$	362.880	0.0503732	0.0251866	+	0.999683i	$0.491982\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2075.09	−0.283481
$378$	0	0
$379$	7719.79	1.04628	0.523139	−	0.852248i	$-0.324761\pi$
$379$	7719.79	1.04628	0.523139	+	0.852248i	$0.324761\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5673.66	0.756946	0.378473	−	0.925612i	$-0.376449\pi$
$383$	5673.66	0.756946	0.378473	+	0.925612i	$0.376449\pi$
$384$	0	0
$385$	−1268.62	−0.167935
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−8180.45	−1.06623	−0.533117	−	0.846041i	$-0.678980\pi$
$389$	−8180.45	−1.06623	−0.533117	+	0.846041i	$0.678980\pi$
$390$	0	0
$391$	−1735.49	−0.224470
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4053.91	−0.516390
$396$	0	0
$397$	−12940.5	−1.63594	−0.817968	−	0.575263i	$-0.804900\pi$
$397$	−12940.5	−1.63594	−0.817968	+	0.575263i	$0.804900\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6581.29	0.819586	0.409793	−	0.912179i	$-0.365601\pi$
$401$	6581.29	0.819586	0.409793	+	0.912179i	$0.365601\pi$
$402$	0	0
$403$	1635.30	0.202135
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2771.52	−0.337541
$408$	0	0
$409$	4921.85	0.595036	0.297518	−	0.954716i	$-0.403841\pi$
$409$	4921.85	0.595036	0.297518	+	0.954716i	$0.403841\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9518.48	−1.13408
$414$	0	0
$415$	−2691.05	−0.318310
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3914.29	0.456385	0.228193	−	0.973616i	$-0.426718\pi$
$419$	3914.29	0.456385	0.228193	+	0.973616i	$0.426718\pi$
$420$	0	0
$421$	−5258.76	−0.608780	−0.304390	−	0.952547i	$-0.598453\pi$
$421$	−5258.76	−0.608780	−0.304390	+	0.952547i	$0.598453\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	251.607	0.0287170
$426$	0	0
$427$	−1250.60	−0.141735
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14350.1	−1.60376	−0.801881	−	0.597484i	$-0.796167\pi$
$431$	−14350.1	−1.60376	−0.801881	+	0.597484i	$0.796167\pi$
$432$	0	0
$433$	863.149	0.0957974	0.0478987	−	0.998852i	$-0.484748\pi$
$433$	863.149	0.0957974	0.0478987	+	0.998852i	$0.484748\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20324.7	2.22486
$438$	0	0
$439$	16142.1	1.75494	0.877470	−	0.479632i	$-0.159230\pi$
$439$	16142.1	1.75494	0.877470	+	0.479632i	$0.159230\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5884.09	−0.631065	−0.315532	−	0.948915i	$-0.602183\pi$
$443$	−5884.09	−0.631065	−0.315532	+	0.948915i	$0.602183\pi$
$444$	0	0
$445$	3430.87	0.365480
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16858.4	−1.77193	−0.885965	−	0.463753i	$-0.846503\pi$
$449$	−16858.4	−1.77193	−0.885965	+	0.463753i	$0.846503\pi$
$450$	0	0
$451$	−4007.20	−0.418385
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1332.04	−0.137246
$456$	0	0
$457$	7623.04	0.780286	0.390143	−	0.920754i	$-0.372426\pi$
$457$	7623.04	0.780286	0.390143	+	0.920754i	$0.372426\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4079.39	−0.412139	−0.206070	−	0.978537i	$-0.566067\pi$
$461$	−4079.39	−0.412139	−0.206070	+	0.978537i	$0.566067\pi$
$462$	0	0
$463$	5499.22	0.551988	0.275994	−	0.961159i	$-0.410993\pi$
$463$	5499.22	0.551988	0.275994	+	0.961159i	$0.410993\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6422.51	−0.636399	−0.318199	−	0.948024i	$-0.603078\pi$
$467$	−6422.51	−0.636399	−0.318199	+	0.948024i	$0.603078\pi$
$468$	0	0
$469$	12158.3	1.19705
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3997.65	−0.388609
$474$	0	0
$475$	−2946.62	−0.284632
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−198.760	−0.0189595	−0.00947973	−	0.999955i	$-0.503018\pi$
$479$	−198.760	−0.0189595	−0.00947973	+	0.999955i	$0.503018\pi$
$480$	0	0
$481$	−2910.06	−0.275858
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3573.27	0.334544
$486$	0	0
$487$	1308.73	0.121774	0.0608871	−	0.998145i	$-0.480607\pi$
$487$	1308.73	0.121774	0.0608871	+	0.998145i	$0.480607\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12474.8	1.14660	0.573300	−	0.819346i	$-0.305663\pi$
$491$	12474.8	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$492$	0	0
$493$	1794.67	0.163951
$494$	0	0
$495$	0	0
$496$	0	0
$497$	21371.9	1.92889
$498$	0	0
$499$	−3146.82	−0.282307	−0.141153	−	0.989988i	$-0.545081\pi$
$499$	−3146.82	−0.282307	−0.141153	+	0.989988i	$0.545081\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5419.35	−0.480391	−0.240196	−	0.970725i	$-0.577212\pi$
$503$	−5419.35	−0.480391	−0.240196	+	0.970725i	$0.577212\pi$
$504$	0	0
$505$	4869.53	0.429092
$506$	0	0
$507$	0	0
$508$	0	0
$509$	17194.6	1.49732	0.748660	−	0.662954i	$-0.230698\pi$
$509$	17194.6	1.49732	0.748660	+	0.662954i	$0.230698\pi$
$510$	0	0
$511$	−12824.9	−1.11026
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3796.14	0.324812
$516$	0	0
$517$	−6651.02	−0.565787
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19829.5	1.66746	0.833730	−	0.552172i	$-0.186201\pi$
$521$	19829.5	1.66746	0.833730	+	0.552172i	$0.186201\pi$
$522$	0	0
$523$	−5392.82	−0.450882	−0.225441	−	0.974257i	$-0.572382\pi$
$523$	−5392.82	−0.450882	−0.225441	+	0.974257i	$0.572382\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1414.32	−0.116904
$528$	0	0
$529$	17568.8	1.44397
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4207.52	−0.341928
$534$	0	0
$535$	−9160.29	−0.740250
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2007.25	−0.160405
$540$	0	0
$541$	−11475.8	−0.911984	−0.455992	−	0.889984i	$-0.650715\pi$
$541$	−11475.8	−0.911984	−0.455992	+	0.889984i	$0.650715\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6854.53	0.538745
$546$	0	0
$547$	7929.09	0.619787	0.309893	−	0.950771i	$-0.399707\pi$
$547$	7929.09	0.619787	0.309893	+	0.950771i	$0.399707\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−21017.7	−1.62502
$552$	0	0
$553$	−18561.6	−1.42734
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9519.36	−0.724144	−0.362072	−	0.932150i	$-0.617931\pi$
$557$	−9519.36	−0.724144	−0.362072	+	0.932150i	$0.617931\pi$
$558$	0	0
$559$	−4197.49	−0.317594
$560$	0	0
$561$	0	0
$562$	0	0
$563$	11922.3	0.892476	0.446238	−	0.894914i	$-0.352763\pi$
$563$	11922.3	0.892476	0.446238	+	0.894914i	$0.352763\pi$
$564$	0	0
$565$	2917.83	0.217264
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17591.4	1.29608	0.648042	−	0.761605i	$-0.275588\pi$
$569$	17591.4	1.29608	0.648042	+	0.761605i	$0.275588\pi$
$570$	0	0
$571$	−8250.73	−0.604698	−0.302349	−	0.953197i	$-0.597771\pi$
$571$	−8250.73	−0.604698	−0.302349	+	0.953197i	$0.597771\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4311.02	−0.312664
$576$	0	0
$577$	−15922.4	−1.14880	−0.574401	−	0.818574i	$-0.694765\pi$
$577$	−15922.4	−1.14880	−0.574401	+	0.818574i	$0.694765\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12321.5	−0.879833
$582$	0	0
$583$	−2231.10	−0.158495
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9589.65	0.674288	0.337144	−	0.941453i	$-0.390539\pi$
$587$	9589.65	0.674288	0.337144	+	0.941453i	$0.390539\pi$
$588$	0	0
$589$	16563.3	1.15871
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5311.77	0.367838	0.183919	−	0.982941i	$-0.441122\pi$
$593$	5311.77	0.367838	0.183919	+	0.982941i	$0.441122\pi$
$594$	0	0
$595$	1152.04	0.0793762
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6346.04	0.432875	0.216437	−	0.976296i	$-0.430556\pi$
$599$	6346.04	0.432875	0.216437	+	0.976296i	$0.430556\pi$
$600$	0	0
$601$	−9813.33	−0.666047	−0.333023	−	0.942919i	$-0.608069\pi$
$601$	−9813.33	−0.666047	−0.333023	+	0.942919i	$0.608069\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6040.86	−0.405943
$606$	0	0
$607$	4170.95	0.278902	0.139451	−	0.990229i	$-0.455466\pi$
$607$	4170.95	0.278902	0.139451	+	0.990229i	$0.455466\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6983.50	−0.462393
$612$	0	0
$613$	−3157.32	−0.208031	−0.104016	−	0.994576i	$-0.533169\pi$
$613$	−3157.32	−0.208031	−0.104016	+	0.994576i	$0.533169\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2962.95	−0.193329	−0.0966643	−	0.995317i	$-0.530817\pi$
$617$	−2962.95	−0.193329	−0.0966643	+	0.995317i	$0.530817\pi$
$618$	0	0
$619$	−4695.72	−0.304906	−0.152453	−	0.988311i	$-0.548717\pi$
$619$	−4695.72	−0.304906	−0.152453	+	0.988311i	$0.548717\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	15708.9	1.01022
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2516.81	0.159542
$630$	0	0
$631$	17397.6	1.09760	0.548800	−	0.835954i	$-0.315085\pi$
$631$	17397.6	1.09760	0.548800	+	0.835954i	$0.315085\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12163.8	0.760166
$636$	0	0
$637$	−2107.59	−0.131092
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6903.46	−0.425383	−0.212691	−	0.977119i	$-0.568223\pi$
$641$	−6903.46	−0.425383	−0.212691	+	0.977119i	$0.568223\pi$
$642$	0	0
$643$	12132.1	0.744079	0.372039	−	0.928217i	$-0.378659\pi$
$643$	12132.1	0.744079	0.372039	+	0.928217i	$0.378659\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16784.3	−1.01988	−0.509939	−	0.860211i	$-0.670332\pi$
$647$	−16784.3	−1.01988	−0.509939	+	0.860211i	$0.670332\pi$
$648$	0	0
$649$	4607.92	0.278700
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15711.5	−0.941561	−0.470781	−	0.882250i	$-0.656028\pi$
$653$	−15711.5	−0.941561	−0.470781	+	0.882250i	$0.656028\pi$
$654$	0	0
$655$	−2505.37	−0.149455
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20227.0	1.19565	0.597825	−	0.801626i	$-0.296032\pi$
$659$	20227.0	1.19565	0.597825	+	0.801626i	$0.296032\pi$
$660$	0	0
$661$	14986.8	0.881875	0.440938	−	0.897538i	$-0.354646\pi$
$661$	14986.8	0.881875	0.440938	+	0.897538i	$0.354646\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13491.7	−0.786745
$666$	0	0
$667$	−30749.7	−1.78506
$668$	0	0
$669$	0	0
$670$	0	0
$671$	605.420	0.0348316
$672$	0	0
$673$	6833.74	0.391414	0.195707	−	0.980662i	$-0.437300\pi$
$673$	6833.74	0.391414	0.195707	+	0.980662i	$0.437300\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30439.3	−1.72803	−0.864015	−	0.503467i	$-0.832058\pi$
$677$	−30439.3	−1.72803	−0.864015	+	0.503467i	$0.832058\pi$
$678$	0	0
$679$	16361.0	0.924707
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9675.88	0.542075	0.271038	−	0.962569i	$-0.412633\pi$
$683$	9675.88	0.542075	0.271038	+	0.962569i	$0.412633\pi$
$684$	0	0
$685$	12615.3	0.703660
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2342.63	−0.129531
$690$	0	0
$691$	−13410.6	−0.738295	−0.369147	−	0.929371i	$-0.620350\pi$
$691$	−13410.6	−0.738295	−0.369147	+	0.929371i	$0.620350\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3194.21	0.174336
$696$	0	0
$697$	3638.93	0.197754
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12796.5	−0.689466	−0.344733	−	0.938701i	$-0.612031\pi$
$701$	−12796.5	−0.689466	−0.344733	+	0.938701i	$0.612031\pi$
$702$	0	0
$703$	−29474.9	−1.58132
$704$	0	0
$705$	0	0
$706$	0	0
$707$	22296.2	1.18604
$708$	0	0
$709$	4849.33	0.256869	0.128435	−	0.991718i	$-0.459005\pi$
$709$	4849.33	0.256869	0.128435	+	0.991718i	$0.459005\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	24232.8	1.27283
$714$	0	0
$715$	644.843	0.0337284
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−33988.7	−1.76296	−0.881479	−	0.472224i	$-0.843451\pi$
$719$	−33988.7	−1.76296	−0.881479	+	0.472224i	$0.843451\pi$
$720$	0	0
$721$	17381.4	0.897806
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4458.01	0.228368
$726$	0	0
$727$	16312.3	0.832170	0.416085	−	0.909326i	$-0.363402\pi$
$727$	16312.3	0.832170	0.416085	+	0.909326i	$0.363402\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3630.26	0.183680
$732$	0	0
$733$	−37831.8	−1.90634	−0.953172	−	0.302430i	$-0.902202\pi$
$733$	−37831.8	−1.90634	−0.953172	+	0.302430i	$0.902202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5885.84	−0.294176
$738$	0	0
$739$	16860.9	0.839295	0.419647	−	0.907687i	$-0.362154\pi$
$739$	16860.9	0.839295	0.419647	+	0.907687i	$0.362154\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−406.077	−0.0200505	−0.0100253	−	0.999950i	$-0.503191\pi$
$743$	−406.077	−0.0200505	−0.0100253	+	0.999950i	$0.503191\pi$
$744$	0	0
$745$	−13370.7	−0.657535
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−41942.3	−2.04611
$750$	0	0
$751$	15106.3	0.734004	0.367002	−	0.930220i	$-0.380384\pi$
$751$	15106.3	0.734004	0.367002	+	0.930220i	$0.380384\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5183.39	0.249858
$756$	0	0
$757$	−14265.4	−0.684919	−0.342460	−	0.939533i	$-0.611260\pi$
$757$	−14265.4	−0.684919	−0.342460	+	0.939533i	$0.611260\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23001.3	1.09566	0.547828	−	0.836591i	$-0.315455\pi$
$761$	23001.3	1.09566	0.547828	+	0.836591i	$0.315455\pi$
$762$	0	0
$763$	31384.9	1.48913
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4838.26	0.227770
$768$	0	0
$769$	12294.6	0.576535	0.288267	−	0.957550i	$-0.406921\pi$
$769$	12294.6	0.576535	0.288267	+	0.957550i	$0.406921\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5938.46	0.276315	0.138158	−	0.990410i	$-0.455882\pi$
$773$	5938.46	0.276315	0.138158	+	0.990410i	$0.455882\pi$
$774$	0	0
$775$	−3513.21	−0.162836
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−42616.2	−1.96006
$780$	0	0
$781$	−10346.2	−0.474027
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−1909.29	−0.0868096
$786$	0	0
$787$	−32425.0	−1.46865	−0.734325	−	0.678798i	$-0.762501\pi$
$787$	−32425.0	−1.46865	−0.734325	+	0.678798i	$0.762501\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	13359.9	0.600535
$792$	0	0
$793$	635.685	0.0284664
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−20990.4	−0.932897	−0.466449	−	0.884548i	$-0.654467\pi$
$797$	−20990.4	−0.932897	−0.466449	+	0.884548i	$0.654467\pi$
$798$	0	0
$799$	6039.79	0.267425
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6208.58	0.272847
$804$	0	0
$805$	−19738.9	−0.864229
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4943.95	−0.214858	−0.107429	−	0.994213i	$-0.534262\pi$
$809$	−4943.95	−0.214858	−0.107429	+	0.994213i	$0.534262\pi$
$810$	0	0
$811$	19844.5	0.859230	0.429615	−	0.903012i	$-0.358649\pi$
$811$	19844.5	0.859230	0.429615	+	0.903012i	$0.358649\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12105.8	−0.520305
$816$	0	0
$817$	−42514.7	−1.82056
$818$	0	0
$819$	0	0
$820$	0	0
$821$	12608.2	0.535969	0.267985	−	0.963423i	$-0.413642\pi$
$821$	12608.2	0.535969	0.267985	+	0.963423i	$0.413642\pi$
$822$	0	0
$823$	−24843.4	−1.05223	−0.526116	−	0.850413i	$-0.676352\pi$
$823$	−24843.4	−1.05223	−0.526116	+	0.850413i	$0.676352\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−33361.1	−1.40276	−0.701379	−	0.712789i	$-0.747432\pi$
$827$	−33361.1	−1.40276	−0.701379	+	0.712789i	$0.747432\pi$
$828$	0	0
$829$	−5049.62	−0.211557	−0.105778	−	0.994390i	$-0.533733\pi$
$829$	−5049.62	−0.211557	−0.105778	+	0.994390i	$0.533733\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1822.78	0.0758170
$834$	0	0
$835$	17678.8	0.732695
$836$	0	0
$837$	0	0
$838$	0	0
$839$	17464.7	0.718650	0.359325	−	0.933213i	$-0.383007\pi$
$839$	17464.7	0.718650	0.359325	+	0.933213i	$0.383007\pi$
$840$	0	0
$841$	7409.22	0.303793
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10307.9	−0.419649
$846$	0	0
$847$	−27659.3	−1.12206
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43122.9	−1.73706
$852$	0	0
$853$	2746.47	0.110243	0.0551216	−	0.998480i	$-0.482445\pi$
$853$	2746.47	0.110243	0.0551216	+	0.998480i	$0.482445\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15656.8	0.624069	0.312034	−	0.950071i	$-0.398990\pi$
$857$	15656.8	0.624069	0.312034	+	0.950071i	$0.398990\pi$
$858$	0	0
$859$	34507.3	1.37063	0.685317	−	0.728245i	$-0.259664\pi$
$859$	34507.3	1.37063	0.685317	+	0.728245i	$0.259664\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26576.7	−1.04830	−0.524150	−	0.851626i	$-0.675617\pi$
$863$	−26576.7	−1.04830	−0.524150	+	0.851626i	$0.675617\pi$
$864$	0	0
$865$	15719.4	0.617892
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8985.73	0.350771
$870$	0	0
$871$	−6180.07	−0.240418
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2861.69	0.110563
$876$	0	0
$877$	−13399.5	−0.515927	−0.257963	−	0.966155i	$-0.583051\pi$
$877$	−13399.5	−0.515927	−0.257963	+	0.966155i	$0.583051\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44435.4	−1.69928	−0.849641	−	0.527362i	$-0.823181\pi$
$881$	−44435.4	−1.69928	−0.849641	+	0.527362i	$0.823181\pi$
$882$	0	0
$883$	11162.1	0.425408	0.212704	−	0.977117i	$-0.431773\pi$
$883$	11162.1	0.425408	0.212704	+	0.977117i	$0.431773\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11591.2	−0.438776	−0.219388	−	0.975638i	$-0.570406\pi$
$887$	−11591.2	−0.438776	−0.219388	+	0.975638i	$0.570406\pi$
$888$	0	0
$889$	55694.4	2.10116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−70733.1	−2.65061
$894$	0	0
$895$	16495.3	0.616063
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25059.1	−0.929663
$900$	0	0
$901$	2026.06	0.0749144
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9375.48	0.344366
$906$	0	0
$907$	−8652.39	−0.316756	−0.158378	−	0.987379i	$-0.550626\pi$
$907$	−8652.39	−0.316756	−0.158378	+	0.987379i	$0.550626\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36712.4	1.33517	0.667583	−	0.744536i	$-0.267329\pi$
$911$	36712.4	1.33517	0.667583	+	0.744536i	$0.267329\pi$
$912$	0	0
$913$	5964.88	0.216220
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−11471.3	−0.413104
$918$	0	0
$919$	42003.3	1.50768	0.753841	−	0.657057i	$-0.228199\pi$
$919$	42003.3	1.50768	0.753841	+	0.657057i	$0.228199\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10863.4	−0.387402
$924$	0	0
$925$	6251.84	0.222226
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−13112.6	−0.463090	−0.231545	−	0.972824i	$-0.574378\pi$
$929$	−13112.6	−0.463090	−0.231545	+	0.972824i	$0.574378\pi$
$930$	0	0
$931$	−21346.9	−0.751468
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−557.703	−0.0195068
$936$	0	0
$937$	49863.5	1.73849	0.869247	−	0.494378i	$-0.164604\pi$
$937$	49863.5	1.73849	0.869247	+	0.494378i	$0.164604\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9352.28	−0.323991	−0.161996	−	0.986791i	$-0.551793\pi$
$941$	−9352.28	−0.323991	−0.161996	+	0.986791i	$0.551793\pi$
$942$	0	0
$943$	−62349.3	−2.15310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4011.39	0.137648	0.0688239	−	0.997629i	$-0.478075\pi$
$947$	4011.39	0.137648	0.0688239	+	0.997629i	$0.478075\pi$
$948$	0	0
$949$	6518.94	0.222986
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39128.6	−1.33001	−0.665005	−	0.746839i	$-0.731571\pi$
$953$	−39128.6	−1.33001	−0.665005	+	0.746839i	$0.731571\pi$
$954$	0	0
$955$	6808.10	0.230686
$956$	0	0
$957$	0	0
$958$	0	0
$959$	57761.9	1.94497
$960$	0	0
$961$	−10042.8	−0.337108
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11171.0	0.372650
$966$	0	0
$967$	−17635.0	−0.586456	−0.293228	−	0.956043i	$-0.594729\pi$
$967$	−17635.0	−0.586456	−0.293228	+	0.956043i	$0.594729\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−26827.3	−0.886642	−0.443321	−	0.896363i	$-0.646200\pi$
$971$	−26827.3	−0.886642	−0.443321	+	0.896363i	$0.646200\pi$
$972$	0	0
$973$	14625.3	0.481877
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8343.63	0.273221	0.136610	−	0.990625i	$-0.456379\pi$
$977$	8343.63	0.273221	0.136610	+	0.990625i	$0.456379\pi$
$978$	0	0
$979$	−7604.72	−0.248261
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−36890.1	−1.19696	−0.598479	−	0.801138i	$-0.704228\pi$
$983$	−36890.1	−1.19696	−0.598479	+	0.801138i	$0.704228\pi$
$984$	0	0
$985$	−1733.12	−0.0560628
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−62200.7	−1.99987
$990$	0	0
$991$	24259.3	0.777619	0.388810	−	0.921318i	$-0.372886\pi$
$991$	24259.3	0.777619	0.388810	+	0.921318i	$0.372886\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20993.8	−0.668894
$996$	0	0
$997$	31700.2	1.00698	0.503488	−	0.864002i	$-0.332050\pi$
$997$	31700.2	1.00698	0.503488	+	0.864002i	$0.332050\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bq.1.3		3
3.2	odd	2		2160.4.a.bi.1.3		3
4.3	odd	2		135.4.a.h.1.1	yes	3
12.11	even	2		135.4.a.e.1.3	✓	3
20.3	even	4		675.4.b.n.649.5		6
20.7	even	4		675.4.b.n.649.2		6
20.19	odd	2		675.4.a.p.1.3		3
36.7	odd	6		405.4.e.q.271.3		6
36.11	even	6		405.4.e.v.271.1		6
36.23	even	6		405.4.e.v.136.1		6
36.31	odd	6		405.4.e.q.136.3		6
60.23	odd	4		675.4.b.m.649.2		6
60.47	odd	4		675.4.b.m.649.5		6
60.59	even	2		675.4.a.s.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.3	✓	3	12.11	even	2
135.4.a.h.1.1	yes	3	4.3	odd	2
405.4.e.q.136.3		6	36.31	odd	6
405.4.e.q.271.3		6	36.7	odd	6
405.4.e.v.136.1		6	36.23	even	6
405.4.e.v.271.1		6	36.11	even	6
675.4.a.p.1.3		3	20.19	odd	2
675.4.a.s.1.1		3	60.59	even	2
675.4.b.m.649.2		6	60.23	odd	4
675.4.b.m.649.5		6	60.47	odd	4
675.4.b.n.649.2		6	20.7	even	4
675.4.b.n.649.5		6	20.3	even	4
2160.4.a.bi.1.3		3	3.2	odd	2
2160.4.a.bq.1.3		3	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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} +22.8935 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} +22.8935 q^{7} -11.0828 q^{11} -11.6368 q^{13} +10.0643 q^{17} -117.865 q^{19} -172.441 q^{23} +25.0000 q^{25} +178.321 q^{29} -140.528 q^{31} +114.468 q^{35} +250.074 q^{37} +361.569 q^{41} +360.707 q^{43} +600.121 q^{47} +181.114 q^{49} +201.312 q^{53} -55.4140 q^{55} -415.772 q^{59} -54.6270 q^{61} -58.1841 q^{65} +531.079 q^{67} +933.534 q^{71} -560.199 q^{73} -253.725 q^{77} -810.781 q^{79} -538.210 q^{83} +50.3214 q^{85} +686.173 q^{89} -266.408 q^{91} -589.324 q^{95} +714.655 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 15 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q + 15 * q^5 + 4 * q^7	\( 3 q + 15 q^{5} + 4 q^{7} - 5 q^{11} + 7 q^{13} + 155 q^{17} + 50 q^{19} - 285 q^{23} + 75 q^{25} + 115 q^{29} + 115 q^{31} + 20 q^{35} - 384 q^{37} + 580 q^{41} + 797 q^{43} + 145 q^{47} + 577 q^{49} - 400 q^{53} - 25 q^{55} - 380 q^{59} - 152 q^{61} + 35 q^{65} - 2 q^{67} - 40 q^{71} - 980 q^{73} + 1950 q^{77} - 1013 q^{79} - 270 q^{83} + 775 q^{85} + 1020 q^{89} + 632 q^{91} + 250 q^{95} + 720 q^{97}+O(q^{100}) \) 3 * q + 15 * q^5 + 4 * q^7 - 5 * q^11 + 7 * q^13 + 155 * q^17 + 50 * q^19 - 285 * q^23 + 75 * q^25 + 115 * q^29 + 115 * q^31 + 20 * q^35 - 384 * q^37 + 580 * q^41 + 797 * q^43 + 145 * q^47 + 577 * q^49 - 400 * q^53 - 25 * q^55 - 380 * q^59 - 152 * q^61 + 35 * q^65 - 2 * q^67 - 40 * q^71 - 980 * q^73 + 1950 * q^77 - 1013 * q^79 - 270 * q^83 + 775 * q^85 + 1020 * q^89 + 632 * q^91 + 250 * q^95 + 720 * q^97

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