LMFDB - Embedded newform 135.4.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(1,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.96525785077$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.67370$ of defining polynomial
Character	$\chi$	$=$	135.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.58488 q^{2} -1.31841 q^{4} +5.00000 q^{5} -22.8935 q^{7} +24.0869 q^{8} -12.9244 q^{10} +11.0828 q^{11} -11.6368 q^{13} +59.1769 q^{14} -51.7145 q^{16} +10.0643 q^{17} +117.865 q^{19} -6.59207 q^{20} -28.6477 q^{22} +172.441 q^{23} +25.0000 q^{25} +30.0798 q^{26} +30.1831 q^{28} +178.321 q^{29} +140.528 q^{31} -59.0200 q^{32} -26.0149 q^{34} -114.468 q^{35} +250.074 q^{37} -304.666 q^{38} +120.435 q^{40} +361.569 q^{41} -360.707 q^{43} -14.6117 q^{44} -445.738 q^{46} -600.121 q^{47} +181.114 q^{49} -64.6219 q^{50} +15.3421 q^{52} +201.312 q^{53} +55.4140 q^{55} -551.435 q^{56} -460.937 q^{58} +415.772 q^{59} -54.6270 q^{61} -363.248 q^{62} +566.275 q^{64} -58.1841 q^{65} -531.079 q^{67} -13.2689 q^{68} +295.885 q^{70} -933.534 q^{71} -560.199 q^{73} -646.409 q^{74} -155.394 q^{76} -253.725 q^{77} +810.781 q^{79} -258.572 q^{80} -934.611 q^{82} +538.210 q^{83} +50.3214 q^{85} +932.384 q^{86} +266.951 q^{88} +686.173 q^{89} +266.408 q^{91} -227.348 q^{92} +1551.24 q^{94} +589.324 q^{95} +714.655 q^{97} -468.156 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{2} + 17 q^{4} + 15 q^{5} - 4 q^{7} + 75 q^{8} + 25 q^{10} + 5 q^{11} + 7 q^{13} + 60 q^{14} + 161 q^{16} + 155 q^{17} - 50 q^{19} + 85 q^{20} - 229 q^{22} + 285 q^{23} + 75 q^{25} + 185 q^{26}+ \cdots - 305 q^{98}+O(q^{100}) $ 3 * q + 5 * q^2 + 17 * q^4 + 15 * q^5 - 4 * q^7 + 75 * q^8 + 25 * q^10 + 5 * q^11 + 7 * q^13 + 60 * q^14 + 161 * q^16 + 155 * q^17 - 50 * q^19 + 85 * q^20 - 229 * q^22 + 285 * q^23 + 75 * q^25 + 185 * q^26 - 334 * q^28 + 115 * q^29 - 115 * q^31 + 775 * q^32 + 413 * q^34 - 20 * q^35 - 384 * q^37 - 1150 * q^38 + 375 * q^40 + 580 * q^41 - 797 * q^43 - 1415 * q^44 - 285 * q^46 - 145 * q^47 + 577 * q^49 + 125 * q^50 + 825 * q^52 - 400 * q^53 + 25 * q^55 - 2190 * q^56 - 59 * q^58 + 380 * q^59 - 152 * q^61 - 1005 * q^62 + 2937 * q^64 + 35 * q^65 + 2 * q^67 + 475 * q^68 + 300 * q^70 + 40 * q^71 - 980 * q^73 - 2720 * q^74 - 3276 * q^76 + 1950 * q^77 + 1013 * q^79 + 805 * q^80 + 4 * q^82 + 270 * q^83 + 775 * q^85 - 1555 * q^86 - 5193 * q^88 + 1020 * q^89 - 632 * q^91 - 1215 * q^92 + 3833 * q^94 - 250 * q^95 + 720 * q^97 - 305 * q^98

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 $

$2$	−2.58488	−0.913892	−0.456946	−	0.889494i	$-0.651057\pi$
$2$	−2.58488	−0.913892	−0.456946	+	0.889494i	$0.651057\pi$
$3$	0	0
$3$	0	0
$4$	−1.31841	−0.164802
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−22.8935	−1.23613	−0.618067	−	0.786125i	$-0.712084\pi$
$7$	−22.8935	−1.23613	−0.618067	+	0.786125i	$0.712084\pi$
$8$	24.0869	1.06450
$9$	0	0
$10$	−12.9244	−0.408705
$11$	11.0828	0.303781	0.151891	−	0.988397i	$-0.451464\pi$
$11$	11.0828	0.303781	0.151891	+	0.988397i	$0.451464\pi$
$12$	0	0
$13$	−11.6368	−0.248267	−0.124134	−	0.992266i	$-0.539615\pi$
$13$	−11.6368	−0.248267	−0.124134	+	0.992266i	$0.539615\pi$
$14$	59.1769	1.12969
$15$	0	0
$16$	−51.7145	−0.808039
$17$	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	$-0.247980\pi$
$19$	117.865	1.42316	0.711579	+	0.702606i	$0.247980\pi$
$20$	−6.59207	−0.0737015
$21$	0	0
$22$	−28.6477	−0.277623
$23$	172.441	1.56332	0.781661	−	0.623704i	$-0.214373\pi$
$23$	172.441	1.56332	0.781661	+	0.623704i	$0.214373\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	30.0798	0.226889
$27$	0	0
$28$	30.1831	0.203717
$29$	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	$-0.366543\pi$
$31$	140.528	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$32$	−59.0200	−0.326043
$33$	0	0
$34$	−26.0149	−0.131221
$35$	−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	$0.312502\pi$
$38$	−304.666	−1.30061
$39$	0	0
$40$	120.435	0.476060
$41$	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	$-0.720908\pi$
$43$	−360.707	−1.27924	−0.639620	+	0.768691i	$0.720908\pi$
$44$	−14.6117	−0.0500636
$45$	0	0
$46$	−445.738	−1.42871
$47$	−600.121	−1.86248	−0.931241	−	0.364405i	$-0.881273\pi$
$47$	−600.121	−1.86248	−0.931241	+	0.364405i	$0.881273\pi$
$48$	0	0
$49$	181.114	0.528028
$50$	−64.6219	−0.182778
$51$	0	0
$52$	15.3421	0.0409149
$53$	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
$56$	−551.435	−1.31587
$57$	0	0
$58$	−460.937	−1.04352
$59$	415.772	0.917438	0.458719	−	0.888581i	$-0.348308\pi$
$59$	415.772	0.917438	0.458719	+	0.888581i	$0.348308\pi$
$60$	0	0
$61$	−54.6270	−0.114660	−0.0573301	−	0.998355i	$-0.518259\pi$
$61$	−54.6270	−0.114660	−0.0573301	+	0.998355i	$0.518259\pi$
$62$	−363.248	−0.744074
$63$	0	0
$64$	566.275	1.10601
$65$	−58.1841	−0.111029
$66$	0	0
$67$	−531.079	−0.968382	−0.484191	−	0.874962i	$-0.660886\pi$
$67$	−531.079	−0.968382	−0.484191	+	0.874962i	$0.660886\pi$
$68$	−13.2689	−0.0236631
$69$	0	0
$70$	295.885	0.505214
$71$	−933.534	−1.56042	−0.780212	−	0.625515i	$-0.784889\pi$
$71$	−933.534	−1.56042	−0.780212	+	0.625515i	$0.784889\pi$
$72$	0	0
$73$	−560.199	−0.898169	−0.449085	−	0.893489i	$-0.648250\pi$
$73$	−560.199	−0.898169	−0.449085	+	0.893489i	$0.648250\pi$
$74$	−646.409	−1.01545
$75$	0	0
$76$	−155.394	−0.234539
$77$	−253.725	−0.375514
$78$	0	0
$79$	810.781	1.15468	0.577342	−	0.816503i	$-0.304090\pi$
$79$	810.781	1.15468	0.577342	+	0.816503i	$0.304090\pi$
$80$	−258.572	−0.361366
$81$	0	0
$82$	−934.611	−1.25867
$83$	538.210	0.711762	0.355881	−	0.934531i	$-0.384181\pi$
$83$	538.210	0.711762	0.355881	+	0.934531i	$0.384181\pi$
$84$	0	0
$85$	50.3214	0.0642132
$86$	932.384	1.16909
$87$	0	0
$88$	266.951	0.323376
$89$	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
$92$	−227.348	−0.257638
$93$	0	0
$94$	1551.24	1.70211
$95$	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	$0.377975\pi$
$98$	−468.156	−0.482560
$99$	0	0
$100$	−32.9603	−0.0329603
$101$	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	$-0.618299\pi$
$103$	−759.229	−0.726301	−0.363151	+	0.931730i	$0.618299\pi$
$104$	−280.296	−0.264281
$105$	0	0
$106$	−520.366	−0.476815
$107$	1832.06	1.65525	0.827625	−	0.561282i	$-0.189692\pi$
$107$	1832.06	1.65525	0.827625	+	0.561282i	$0.189692\pi$
$108$	0	0
$109$	1370.91	1.20467	0.602335	−	0.798244i	$-0.294237\pi$
$109$	1370.91	1.20467	0.602335	+	0.798244i	$0.294237\pi$
$110$	−143.238	−0.124157
$111$	0	0
$112$	1183.93	0.998844
$113$	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
$116$	−235.100	−0.188177
$117$	0	0
$118$	−1074.72	−0.838439
$119$	−230.407	−0.177491
$120$	0	0
$121$	−1208.17	−0.907717
$122$	141.204	0.104787
$123$	0	0
$124$	−185.274	−0.134178
$125$	125.000	0.0894427
$126$	0	0
$127$	−2432.76	−1.69978	−0.849891	−	0.526958i	$-0.823333\pi$
$127$	−2432.76	−1.69978	−0.849891	+	0.526958i	$0.823333\pi$
$128$	−991.592	−0.684728
$129$	0	0
$130$	150.399	0.101468
$131$	501.073	0.334191	0.167095	−	0.985941i	$-0.446561\pi$
$131$	501.073	0.334191	0.167095	+	0.985941i	$0.446561\pi$
$132$	0	0
$133$	−2698.34	−1.75922
$134$	1372.77	0.884996
$135$	0	0
$136$	242.418	0.152847
$137$	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	$-0.562442\pi$
$139$	−638.842	−0.389826	−0.194913	+	0.980821i	$0.562442\pi$
$140$	150.916	0.0911050
$141$	0	0
$142$	2413.07	1.42606
$143$	−128.969	−0.0754189
$144$	0	0
$145$	891.603	0.510645
$146$	1448.05	0.820829
$147$	0	0
$148$	−329.700	−0.183116
$149$	−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	$-0.590119\pi$
$151$	−1036.68	−0.558700	−0.279350	+	0.960189i	$0.590119\pi$
$152$	2839.00	1.51496
$153$	0	0
$154$	655.847	0.343179
$155$	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	$0.530943\pi$
$158$	−2095.77	−1.05526
$159$	0	0
$160$	−295.100	−0.145811
$161$	−3947.78	−1.93248
$162$	0	0
$163$	2421.17	1.16344	0.581719	−	0.813390i	$-0.302380\pi$
$163$	2421.17	1.16344	0.581719	+	0.813390i	$0.302380\pi$
$164$	−476.697	−0.226974
$165$	0	0
$166$	−1391.21	−0.650473
$167$	−3535.76	−1.63836	−0.819178	−	0.573540i	$-0.805570\pi$
$167$	−3535.76	−1.63836	−0.819178	+	0.573540i	$0.805570\pi$
$168$	0	0
$169$	−2061.58	−0.938363
$170$	−130.075	−0.0586840
$171$	0	0
$172$	475.561	0.210821
$173$	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
$176$	−573.142	−0.245467
$177$	0	0
$178$	−1773.67	−0.746867
$179$	−3299.06	−1.37756	−0.688780	−	0.724970i	$-0.741853\pi$
$179$	−3299.06	−1.37756	−0.688780	+	0.724970i	$0.741853\pi$
$180$	0	0
$181$	1875.10	0.770026	0.385013	−	0.922911i	$-0.374197\pi$
$181$	1875.10	0.770026	0.385013	+	0.922911i	$0.374197\pi$
$182$	−688.632	−0.280466
$183$	0	0
$184$	4153.57	1.66416
$185$	1250.37	0.496913
$186$	0	0
$187$	111.541	0.0436185
$188$	791.207	0.306940
$189$	0	0
$190$	−1523.33	−0.581652
$191$	−1361.62	−0.515829	−0.257915	−	0.966168i	$-0.583035\pi$
$191$	−1361.62	−0.515829	−0.257915	+	0.966168i	$0.583035\pi$
$192$	0	0
$193$	2234.20	0.833271	0.416636	−	0.909074i	$-0.363209\pi$
$193$	2234.20	0.833271	0.416636	+	0.909074i	$0.363209\pi$
$194$	−1847.29	−0.683650
$195$	0	0
$196$	−238.782	−0.0870199
$197$	−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	$-0.231088\pi$
$199$	4198.77	1.49569	0.747846	+	0.663872i	$0.231088\pi$
$200$	602.174	0.212901
$201$	0	0
$202$	−2517.43	−0.876860
$203$	−4082.39	−1.41146
$204$	0	0
$205$	1807.84	0.615929
$206$	1962.51	0.663761
$207$	0	0
$208$	601.792	0.200610
$209$	1306.27	0.432329
$210$	0	0
$211$	4728.46	1.54275	0.771375	−	0.636380i	$-0.219569\pi$
$211$	4728.46	1.54275	0.771375	+	0.636380i	$0.219569\pi$
$212$	−265.412	−0.0859839
$213$	0	0
$214$	−4735.64	−1.51272
$215$	−1803.54	−0.572094
$216$	0	0
$217$	−3217.19	−1.00644
$218$	−3543.62	−1.10094
$219$	0	0
$220$	−73.0586	−0.0223891
$221$	−117.116	−0.0356475
$222$	0	0
$223$	2430.16	0.729756	0.364878	−	0.931055i	$-0.381111\pi$
$223$	2430.16	0.729756	0.364878	+	0.931055i	$0.381111\pi$
$224$	1351.18	0.403032
$225$	0	0
$226$	−1508.45	−0.443985
$227$	−584.687	−0.170956	−0.0854780	−	0.996340i	$-0.527242\pi$
$227$	−584.687	−0.170956	−0.0854780	+	0.996340i	$0.527242\pi$
$228$	0	0
$229$	−4731.77	−1.36543	−0.682717	−	0.730683i	$-0.739202\pi$
$229$	−4731.77	−1.36543	−0.682717	+	0.730683i	$0.739202\pi$
$230$	−2228.69	−0.638937
$231$	0	0
$232$	4295.20	1.21549
$233$	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
$236$	−548.159	−0.151195
$237$	0	0
$238$	595.574	0.162207
$239$	−120.545	−0.0326253	−0.0163126	−	0.999867i	$-0.505193\pi$
$239$	−120.545	−0.0326253	−0.0163126	+	0.999867i	$0.505193\pi$
$240$	0	0
$241$	1732.56	0.463086	0.231543	−	0.972825i	$-0.425623\pi$
$241$	1732.56	0.463086	0.231543	+	0.972825i	$0.425623\pi$
$242$	3122.97	0.829555
$243$	0	0
$244$	72.0209	0.0188962
$245$	905.568	0.236141
$246$	0	0
$247$	−1371.57	−0.353324
$248$	3384.90	0.866699
$249$	0	0
$250$	−323.110	−0.0817410
$251$	−3287.82	−0.826793	−0.413397	−	0.910551i	$-0.635658\pi$
$251$	−3287.82	−0.826793	−0.413397	+	0.910551i	$0.635658\pi$
$252$	0	0
$253$	1911.13	0.474908
$254$	6288.38	1.55342
$255$	0	0
$256$	−1967.06	−0.480239
$257$	−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
$260$	76.7107	0.0182977
$261$	0	0
$262$	−1295.21	−0.305414
$263$	710.144	0.166499	0.0832497	−	0.996529i	$-0.473470\pi$
$263$	710.144	0.166499	0.0832497	+	0.996529i	$0.473470\pi$
$264$	0	0
$265$	1006.56	0.233330
$266$	6974.87	1.60773
$267$	0	0
$268$	700.181	0.159591
$269$	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	$-0.571612\pi$
$271$	−1990.45	−0.446166	−0.223083	+	0.974799i	$0.571612\pi$
$272$	−520.469	−0.116022
$273$	0	0
$274$	−6521.81	−1.43795
$275$	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	$0.142268\pi$
$278$	1651.33	0.356259
$279$	0	0
$280$	−2757.18	−0.588474
$281$	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	$-0.484698\pi$
$283$	457.561	0.0961102	0.0480551	+	0.998845i	$0.484698\pi$
$284$	1230.78	0.257160
$285$	0	0
$286$	333.368	0.0689247
$287$	−8277.59	−1.70248
$288$	0	0
$289$	−4811.71	−0.979383
$290$	−2304.68	−0.466675
$291$	0	0
$292$	738.574	0.148020
$293$	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
$296$	6023.51	1.18280
$297$	0	0
$298$	6912.30	1.34369
$299$	−2006.66	−0.388122
$300$	0	0
$301$	8257.86	1.58131
$302$	2679.68	0.510591
$303$	0	0
$304$	−6095.31	−1.14997
$305$	−273.135	−0.0512776
$306$	0	0
$307$	−6583.54	−1.22392	−0.611959	−	0.790890i	$-0.709618\pi$
$307$	−6583.54	−1.22392	−0.611959	+	0.790890i	$0.709618\pi$
$308$	334.514	0.0618853
$309$	0	0
$310$	−1816.24	−0.332760
$311$	−8128.05	−1.48199	−0.740996	−	0.671510i	$-0.765646\pi$
$311$	−8128.05	−1.48199	−0.740996	+	0.671510i	$0.765646\pi$
$312$	0	0
$313$	−5207.63	−0.940424	−0.470212	−	0.882554i	$-0.655823\pi$
$313$	−5207.63	−0.940424	−0.470212	+	0.882554i	$0.655823\pi$
$314$	987.056	0.177397
$315$	0	0
$316$	−1068.94	−0.190294
$317$	−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
$320$	2831.38	0.494621
$321$	0	0
$322$	10204.5	1.76607
$323$	1186.22	0.204345
$324$	0	0
$325$	−290.921	−0.0496535
$326$	−6258.41	−1.06326
$327$	0	0
$328$	8709.09	1.46610
$329$	13738.9	2.30228
$330$	0	0
$331$	10360.5	1.72043	0.860216	−	0.509930i	$-0.170329\pi$
$331$	10360.5	1.72043	0.860216	+	0.509930i	$0.170329\pi$
$332$	−709.583	−0.117299
$333$	0	0
$334$	9139.51	1.49728
$335$	−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	$0.597617\pi$
$338$	5328.94	0.857563
$339$	0	0
$340$	−66.3444	−0.0105824
$341$	1557.45	0.247333
$342$	0	0
$343$	3706.15	0.583421
$344$	−8688.34	−1.36176
$345$	0	0
$346$	−8126.55	−1.26268
$347$	197.010	0.0304785	0.0152393	−	0.999884i	$-0.495149\pi$
$347$	197.010	0.0304785	0.0152393	+	0.999884i	$0.495149\pi$
$348$	0	0
$349$	−7334.20	−1.12490	−0.562451	−	0.826831i	$-0.690141\pi$
$349$	−7334.20	−1.12490	−0.562451	+	0.826831i	$0.690141\pi$
$350$	1479.42	0.225939
$351$	0	0
$352$	−654.107	−0.0990456
$353$	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
$356$	−904.659	−0.134682
$357$	0	0
$358$	8527.66	1.25894
$359$	−1153.79	−0.169623	−0.0848115	−	0.996397i	$-0.527029\pi$
$359$	−1153.79	−0.169623	−0.0848115	+	0.996397i	$0.527029\pi$
$360$	0	0
$361$	7033.09	1.02538
$362$	−4846.89	−0.703721
$363$	0	0
$364$	−351.236	−0.0505763
$365$	−2801.00	−0.401673
$366$	0	0
$367$	−3449.05	−0.490569	−0.245285	−	0.969451i	$-0.578881\pi$
$367$	−3449.05	−0.490569	−0.245285	+	0.969451i	$0.578881\pi$
$368$	−8917.69	−1.26322
$369$	0	0
$370$	−3232.05	−0.454125
$371$	−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	$0.491982\pi$
$374$	−288.319	−0.0398626
$375$	0	0
$376$	−14455.1	−1.98262
$377$	−2075.09	−0.283481
$378$	0	0
$379$	−7719.79	−1.04628	−0.523139	−	0.852248i	$-0.675239\pi$
$379$	−7719.79	−1.04628	−0.523139	+	0.852248i	$0.675239\pi$
$380$	−776.972	−0.104889
$381$	0	0
$382$	3519.62	0.471412
$383$	−5673.66	−0.756946	−0.378473	−	0.925612i	$-0.623551\pi$
$383$	−5673.66	−0.756946	−0.378473	+	0.925612i	$0.623551\pi$
$384$	0	0
$385$	−1268.62	−0.167935
$386$	−5775.13	−0.761520
$387$	0	0
$388$	−942.210	−0.123282
$389$	−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
$392$	4362.47	0.562087
$393$	0	0
$394$	895.982	0.114566
$395$	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	$0.804900\pi$
$398$	−10853.3	−1.36690
$399$	0	0
$400$	−1292.86	−0.161608
$401$	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
$404$	−1284.01	−0.158124
$405$	0	0
$406$	10552.5	1.28993
$407$	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	$0.403841\pi$
$410$	−4673.06	−0.562892
$411$	0	0
$412$	1000.98	0.119696
$413$	−9518.48	−1.13408
$414$	0	0
$415$	2691.05	0.318310
$416$	686.806	0.0809457
$417$	0	0
$418$	−3376.55	−0.395102
$419$	−3914.29	−0.456385	−0.228193	−	0.973616i	$-0.573282\pi$
$419$	−3914.29	−0.456385	−0.228193	+	0.973616i	$0.573282\pi$
$420$	0	0
$421$	−5258.76	−0.608780	−0.304390	−	0.952547i	$-0.598453\pi$
$421$	−5258.76	−0.608780	−0.304390	+	0.952547i	$0.598453\pi$
$422$	−12222.5	−1.40991
$423$	0	0
$424$	4848.99	0.555395
$425$	251.607	0.0287170
$426$	0	0
$427$	1250.60	0.141735
$428$	−2415.41	−0.272788
$429$	0	0
$430$	4661.92	0.522832
$431$	14350.1	1.60376	0.801881	−	0.597484i	$-0.203833\pi$
$431$	14350.1	1.60376	0.801881	+	0.597484i	$0.203833\pi$
$432$	0	0
$433$	863.149	0.0957974	0.0478987	−	0.998852i	$-0.484748\pi$
$433$	863.149	0.0957974	0.0478987	+	0.998852i	$0.484748\pi$
$434$	8316.04	0.919776
$435$	0	0
$436$	−1807.42	−0.198532
$437$	20324.7	2.22486
$438$	0	0
$439$	−16142.1	−1.75494	−0.877470	−	0.479632i	$-0.840770\pi$
$439$	−16142.1	−1.75494	−0.877470	+	0.479632i	$0.840770\pi$
$440$	1334.75	0.144618
$441$	0	0
$442$	302.731	0.0325780
$443$	5884.09	0.631065	0.315532	−	0.948915i	$-0.397817\pi$
$443$	5884.09	0.631065	0.315532	+	0.948915i	$0.397817\pi$
$444$	0	0
$445$	3430.87	0.365480
$446$	−6281.67	−0.666918
$447$	0	0
$448$	−12964.0	−1.36717
$449$	−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
$452$	−769.382	−0.0800635
$453$	0	0
$454$	1511.34	0.156235
$455$	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	$0.372426\pi$
$458$	12231.1	1.24786
$459$	0	0
$460$	−1136.74	−0.115219
$461$	−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	$-0.589007\pi$
$463$	−5499.22	−0.551988	−0.275994	+	0.961159i	$0.589007\pi$
$464$	−9221.75	−0.922649
$465$	0	0
$466$	−3175.40	−0.315660
$467$	6422.51	0.636399	0.318199	−	0.948024i	$-0.396922\pi$
$467$	6422.51	0.636399	0.318199	+	0.948024i	$0.396922\pi$
$468$	0	0
$469$	12158.3	1.19705
$470$	7756.19	0.761205
$471$	0	0
$472$	10014.7	0.976615
$473$	−3997.65	−0.388609
$474$	0	0
$475$	2946.62	0.284632
$476$	303.772	0.0292507
$477$	0	0
$478$	311.595	0.0298160
$479$	198.760	0.0189595	0.00947973	−	0.999955i	$-0.496982\pi$
$479$	198.760	0.0189595	0.00947973	+	0.999955i	$0.496982\pi$
$480$	0	0
$481$	−2910.06	−0.275858
$482$	−4478.44	−0.423211
$483$	0	0
$484$	1592.87	0.149593
$485$	3573.27	0.334544
$486$	0	0
$487$	−1308.73	−0.121774	−0.0608871	−	0.998145i	$-0.519393\pi$
$487$	−1308.73	−0.121774	−0.0608871	+	0.998145i	$0.519393\pi$
$488$	−1315.80	−0.122056
$489$	0	0
$490$	−2340.78	−0.215808
$491$	−12474.8	−1.14660	−0.573300	−	0.819346i	$-0.694337\pi$
$491$	−12474.8	−1.14660	−0.573300	+	0.819346i	$0.694337\pi$
$492$	0	0
$493$	1794.67	0.163951
$494$	3545.34	0.322900
$495$	0	0
$496$	−7267.35	−0.657890
$497$	21371.9	1.92889
$498$	0	0
$499$	3146.82	0.282307	0.141153	−	0.989988i	$-0.454919\pi$
$499$	3146.82	0.282307	0.141153	+	0.989988i	$0.454919\pi$
$500$	−164.802	−0.0147403
$501$	0	0
$502$	8498.60	0.755600
$503$	5419.35	0.480391	0.240196	−	0.970725i	$-0.422788\pi$
$503$	5419.35	0.480391	0.240196	+	0.970725i	$0.422788\pi$
$504$	0	0
$505$	4869.53	0.429092
$506$	−4940.03	−0.434014
$507$	0	0
$508$	3207.38	0.280127
$509$	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
$512$	13017.3	1.12361
$513$	0	0
$514$	3851.01	0.330468
$515$	−3796.14	−0.324812
$516$	0	0
$517$	−6651.02	−0.565787
$518$	14798.6	1.25524
$519$	0	0
$520$	−1401.48	−0.118190
$521$	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	$-0.427618\pi$
$523$	5392.82	0.450882	0.225441	+	0.974257i	$0.427618\pi$
$524$	−660.621	−0.0550752
$525$	0	0
$526$	−1835.63	−0.152162
$527$	1414.32	0.116904
$528$	0	0
$529$	17568.8	1.44397
$530$	−2601.83	−0.213238
$531$	0	0
$532$	3557.53	0.289922
$533$	−4207.52	−0.341928
$534$	0	0
$535$	9160.29	0.740250
$536$	−12792.1	−1.03085
$537$	0	0
$538$	−9480.32	−0.759713
$539$	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	$0.650715\pi$
$542$	5145.06	0.407747
$543$	0	0
$544$	−593.994	−0.0468149
$545$	6854.53	0.538745
$546$	0	0
$547$	−7929.09	−0.619787	−0.309893	−	0.950771i	$-0.600293\pi$
$547$	−7929.09	−0.619787	−0.309893	+	0.950771i	$0.600293\pi$
$548$	−3326.44	−0.259304
$549$	0	0
$550$	−716.192	−0.0555246
$551$	21017.7	1.62502
$552$	0	0
$553$	−18561.6	−1.42734
$554$	−21492.4	−1.64824
$555$	0	0
$556$	842.257	0.0642440
$557$	−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
$560$	5919.63	0.446697
$561$	0	0
$562$	−14902.1	−1.11852
$563$	−11922.3	−0.892476	−0.446238	−	0.894914i	$-0.647237\pi$
$563$	−11922.3	−0.892476	−0.446238	+	0.894914i	$0.647237\pi$
$564$	0	0
$565$	2917.83	0.217264
$566$	−1182.74	−0.0878343
$567$	0	0
$568$	−22486.0	−1.66108
$569$	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	$-0.402229\pi$
$571$	8250.73	0.604698	0.302349	+	0.953197i	$0.402229\pi$
$572$	170.034	0.0124292
$573$	0	0
$574$	21396.5	1.55588
$575$	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	$0.694765\pi$
$578$	12437.7	0.895050
$579$	0	0
$580$	−1175.50	−0.0841552
$581$	−12321.5	−0.879833
$582$	0	0
$583$	2231.10	0.158495
$584$	−13493.5	−0.956103
$585$	0	0
$586$	−13707.1	−0.966274
$587$	−9589.65	−0.674288	−0.337144	−	0.941453i	$-0.609461\pi$
$587$	−9589.65	−0.674288	−0.337144	+	0.941453i	$0.609461\pi$
$588$	0	0
$589$	16563.3	1.15871
$590$	−5373.59	−0.374961
$591$	0	0
$592$	−12932.4	−0.897837
$593$	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
$596$	3525.61	0.242307
$597$	0	0
$598$	5186.98	0.354701
$599$	−6346.04	−0.432875	−0.216437	−	0.976296i	$-0.569444\pi$
$599$	−6346.04	−0.432875	−0.216437	+	0.976296i	$0.569444\pi$
$600$	0	0
$601$	−9813.33	−0.666047	−0.333023	−	0.942919i	$-0.608069\pi$
$601$	−9813.33	−0.666047	−0.333023	+	0.942919i	$0.608069\pi$
$602$	−21345.6	−1.44515
$603$	0	0
$604$	1366.77	0.0920746
$605$	−6040.86	−0.405943
$606$	0	0
$607$	−4170.95	−0.278902	−0.139451	−	0.990229i	$-0.544534\pi$
$607$	−4170.95	−0.278902	−0.139451	+	0.990229i	$0.544534\pi$
$608$	−6956.38	−0.464011
$609$	0	0
$610$	706.020	0.0468621
$611$	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	$0.533169\pi$
$614$	17017.6	1.11853
$615$	0	0
$616$	−6111.45	−0.399736
$617$	−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	$-0.451283\pi$
$619$	4695.72	0.304906	0.152453	+	0.988311i	$0.451283\pi$
$620$	−926.372	−0.0600064
$621$	0	0
$622$	21010.0	1.35438
$623$	−15708.9	−1.01022
$624$	0	0
$625$	625.000	0.0400000
$626$	13461.1	0.859446
$627$	0	0
$628$	503.447	0.0319900
$629$	2516.81	0.159542
$630$	0	0
$631$	−17397.6	−1.09760	−0.548800	−	0.835954i	$-0.684915\pi$
$631$	−17397.6	−1.09760	−0.548800	+	0.835954i	$0.684915\pi$
$632$	19529.2	1.22916
$633$	0	0
$634$	8432.74	0.528244
$635$	−12163.8	−0.760166
$636$	0	0
$637$	−2107.59	−0.131092
$638$	−5108.47	−0.317000
$639$	0	0
$640$	−4957.96	−0.306220
$641$	−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	$-0.621341\pi$
$643$	−12132.1	−0.744079	−0.372039	+	0.928217i	$0.621341\pi$
$644$	5204.80	0.318475
$645$	0	0
$646$	−3066.24	−0.186749
$647$	16784.3	1.01988	0.509939	−	0.860211i	$-0.329668\pi$
$647$	16784.3	1.01988	0.509939	+	0.860211i	$0.329668\pi$
$648$	0	0
$649$	4607.92	0.278700
$650$	751.994	0.0453779
$651$	0	0
$652$	−3192.10	−0.191736
$653$	−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
$656$	−18698.4	−1.11288
$657$	0	0
$658$	−35513.3	−2.10403
$659$	−20227.0	−1.19565	−0.597825	−	0.801626i	$-0.703968\pi$
$659$	−20227.0	−1.19565	−0.597825	+	0.801626i	$0.703968\pi$
$660$	0	0
$661$	14986.8	0.881875	0.440938	−	0.897538i	$-0.354646\pi$
$661$	14986.8	0.881875	0.440938	+	0.897538i	$0.354646\pi$
$662$	−26780.5	−1.57229
$663$	0	0
$664$	12963.8	0.757672
$665$	−13491.7	−0.786745
$666$	0	0
$667$	30749.7	1.78506
$668$	4661.59	0.270004
$669$	0	0
$670$	6863.86	0.395782
$671$	−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	$0.437300\pi$
$674$	9655.19	0.551787
$675$	0	0
$676$	2718.02	0.154644
$677$	−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
$680$	1212.09	0.0683552
$681$	0	0
$682$	−4025.81	−0.226036
$683$	−9675.88	−0.542075	−0.271038	−	0.962569i	$-0.587367\pi$
$683$	−9675.88	−0.542075	−0.271038	+	0.962569i	$0.587367\pi$
$684$	0	0
$685$	12615.3	0.703660
$686$	−9579.94	−0.533184
$687$	0	0
$688$	18653.8	1.03368
$689$	−2342.63	−0.129531
$690$	0	0
$691$	13410.6	0.738295	0.369147	−	0.929371i	$-0.379650\pi$
$691$	13410.6	0.738295	0.369147	+	0.929371i	$0.379650\pi$
$692$	−4144.94	−0.227698
$693$	0	0
$694$	−509.247	−0.0278541
$695$	−3194.21	−0.174336
$696$	0	0
$697$	3638.93	0.197754
$698$	18958.0	1.02804
$699$	0	0
$700$	754.578	0.0407434
$701$	−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
$704$	6275.92	0.335984
$705$	0	0
$706$	−11932.5	−0.636100
$707$	−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	$0.459005\pi$
$710$	12065.4	0.637753
$711$	0	0
$712$	16527.8	0.869952
$713$	24232.8	1.27283
$714$	0	0
$715$	−644.843	−0.0337284
$716$	4349.52	0.227024
$717$	0	0
$718$	2982.40	0.155017
$719$	33988.7	1.76296	0.881479	−	0.472224i	$-0.156549\pi$
$719$	33988.7	1.76296	0.881479	+	0.472224i	$0.156549\pi$
$720$	0	0
$721$	17381.4	0.897806
$722$	−18179.7	−0.937088
$723$	0	0
$724$	−2472.15	−0.126902
$725$	4458.01	0.228368
$726$	0	0
$727$	−16312.3	−0.832170	−0.416085	−	0.909326i	$-0.636598\pi$
$727$	−16312.3	−0.832170	−0.416085	+	0.909326i	$0.636598\pi$
$728$	6416.96	0.326687
$729$	0	0
$730$	7240.23	0.367086
$731$	−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	$0.902202\pi$
$734$	8915.36	0.448327
$735$	0	0
$736$	−10177.5	−0.509710
$737$	−5885.84	−0.294176
$738$	0	0
$739$	−16860.9	−0.839295	−0.419647	−	0.907687i	$-0.637846\pi$
$739$	−16860.9	−0.839295	−0.419647	+	0.907687i	$0.637846\pi$
$740$	−1648.50	−0.0818921
$741$	0	0
$742$	11913.0	0.589408
$743$	406.077	0.0200505	0.0100253	−	0.999950i	$-0.496809\pi$
$743$	406.077	0.0200505	0.0100253	+	0.999950i	$0.496809\pi$
$744$	0	0
$745$	−13370.7	−0.657535
$746$	−937.999	−0.0460357
$747$	0	0
$748$	−147.056	−0.00718839
$749$	−41942.3	−2.04611
$750$	0	0
$751$	−15106.3	−0.734004	−0.367002	−	0.930220i	$-0.619616\pi$
$751$	−15106.3	−0.734004	−0.367002	+	0.930220i	$0.619616\pi$
$752$	31034.9	1.50496
$753$	0	0
$754$	5363.84	0.259071
$755$	−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	$0.611260\pi$
$758$	19954.7	0.956184
$759$	0	0
$760$	14195.0	0.677509
$761$	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
$764$	1795.18	0.0850095
$765$	0	0
$766$	14665.7	0.691767
$767$	−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	$0.406921\pi$
$770$	3279.23	0.153474
$771$	0	0
$772$	−2945.60	−0.137324
$773$	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
$776$	17213.9	0.796316
$777$	0	0
$778$	21145.5	0.974423
$779$	42616.2	1.96006
$780$	0	0
$781$	−10346.2	−0.474027
$782$	−4486.04	−0.205141
$783$	0	0
$784$	−9366.19	−0.426667
$785$	−1909.29	−0.0868096
$786$	0	0
$787$	32425.0	1.46865	0.734325	−	0.678798i	$-0.237499\pi$
$787$	32425.0	1.46865	0.734325	+	0.678798i	$0.237499\pi$
$788$	456.994	0.0206596
$789$	0	0
$790$	−10478.8	−0.471925
$791$	−13359.9	−0.600535
$792$	0	0
$793$	635.685	0.0284664
$794$	33449.7	1.49507
$795$	0	0
$796$	−5535.71	−0.246492
$797$	−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
$800$	−1475.50	−0.0652085
$801$	0	0
$802$	−17011.8	−0.749013
$803$	−6208.58	−0.272847
$804$	0	0
$805$	−19738.9	−0.864229
$806$	4227.06	0.184729
$807$	0	0
$808$	23458.4	1.02137
$809$	−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	$-0.641351\pi$
$811$	−19844.5	−0.859230	−0.429615	+	0.903012i	$0.641351\pi$
$812$	5382.27	0.232612
$813$	0	0
$814$	−7164.03	−0.308476
$815$	12105.8	0.520305
$816$	0	0
$817$	−42514.7	−1.82056
$818$	−12722.4	−0.543798
$819$	0	0
$820$	−2383.49	−0.101506
$821$	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	$-0.323648\pi$
$823$	24843.4	1.05223	0.526116	+	0.850413i	$0.323648\pi$
$824$	−18287.5	−0.773149
$825$	0	0
$826$	24604.1	1.03642
$827$	33361.1	1.40276	0.701379	−	0.712789i	$-0.252568\pi$
$827$	33361.1	1.40276	0.701379	+	0.712789i	$0.252568\pi$
$828$	0	0
$829$	−5049.62	−0.211557	−0.105778	−	0.994390i	$-0.533733\pi$
$829$	−5049.62	−0.211557	−0.105778	+	0.994390i	$0.533733\pi$
$830$	−6956.03	−0.290901
$831$	0	0
$832$	−6589.65	−0.274585
$833$	1822.78	0.0758170
$834$	0	0
$835$	−17678.8	−0.732695
$836$	−1722.21	−0.0712485
$837$	0	0
$838$	10117.9	0.417087
$839$	−17464.7	−0.718650	−0.359325	−	0.933213i	$-0.616993\pi$
$839$	−17464.7	−0.718650	−0.359325	+	0.933213i	$0.616993\pi$
$840$	0	0
$841$	7409.22	0.303793
$842$	13593.3	0.556359
$843$	0	0
$844$	−6234.06	−0.254248
$845$	−10307.9	−0.419649
$846$	0	0
$847$	27659.3	1.12206
$848$	−10410.7	−0.421587
$849$	0	0
$850$	−650.373	−0.0262443
$851$	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	$0.482445\pi$
$854$	−3232.66	−0.129531
$855$	0	0
$856$	44128.7	1.76202
$857$	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	$-0.740336\pi$
$859$	−34507.3	−1.37063	−0.685317	+	0.728245i	$0.740336\pi$
$860$	2377.81	0.0942820
$861$	0	0
$862$	−37093.3	−1.46567
$863$	26576.7	1.04830	0.524150	−	0.851626i	$-0.324383\pi$
$863$	26576.7	1.04830	0.524150	+	0.851626i	$0.324383\pi$
$864$	0	0
$865$	15719.4	0.617892
$866$	−2231.13	−0.0875485
$867$	0	0
$868$	4241.58	0.165863
$869$	8985.73	0.350771
$870$	0	0
$871$	6180.07	0.240418
$872$	33020.9	1.28237
$873$	0	0
$874$	−52536.8	−2.03328
$875$	−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	$0.583051\pi$
$878$	41725.2	1.60383
$879$	0	0
$880$	−2865.71	−0.109776
$881$	−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	$-0.568227\pi$
$883$	−11162.1	−0.425408	−0.212704	+	0.977117i	$0.568227\pi$
$884$	154.408	0.00587477
$885$	0	0
$886$	−15209.7	−0.576725
$887$	11591.2	0.438776	0.219388	−	0.975638i	$-0.429594\pi$
$887$	11591.2	0.438776	0.219388	+	0.975638i	$0.429594\pi$
$888$	0	0
$889$	55694.4	2.10116
$890$	−8868.36	−0.334009
$891$	0	0
$892$	−3203.96	−0.120265
$893$	−70733.1	−2.65061
$894$	0	0
$895$	−16495.3	−0.616063
$896$	22701.0	0.846415
$897$	0	0
$898$	43576.8	1.61935
$899$	25059.1	0.929663
$900$	0	0
$901$	2026.06	0.0749144
$902$	−10358.1	−0.382359
$903$	0	0
$904$	14056.3	0.517154
$905$	9375.48	0.344366
$906$	0	0
$907$	8652.39	0.316756	0.158378	−	0.987379i	$-0.449374\pi$
$907$	8652.39	0.316756	0.158378	+	0.987379i	$0.449374\pi$
$908$	770.859	0.0281738
$909$	0	0
$910$	−3443.16	−0.125428
$911$	−36712.4	−1.33517	−0.667583	−	0.744536i	$-0.732671\pi$
$911$	−36712.4	−1.33517	−0.667583	+	0.744536i	$0.732671\pi$
$912$	0	0
$913$	5964.88	0.216220
$914$	−19704.6	−0.713097
$915$	0	0
$916$	6238.43	0.225026
$917$	−11471.3	−0.413104
$918$	0	0
$919$	−42003.3	−1.50768	−0.753841	−	0.657057i	$-0.771801\pi$
$919$	−42003.3	−1.50768	−0.753841	+	0.657057i	$0.771801\pi$
$920$	20767.9	0.744235
$921$	0	0
$922$	10544.7	0.376651
$923$	10863.4	0.387402
$924$	0	0
$925$	6251.84	0.222226
$926$	14214.8	0.504457
$927$	0	0
$928$	−10524.5	−0.372288
$929$	−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
$932$	−1619.61	−0.0569227
$933$	0	0
$934$	−16601.4	−0.581600
$935$	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	$0.164604\pi$
$938$	−31427.6	−1.09397
$939$	0	0
$940$	3956.04	0.137268
$941$	−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
$944$	−21501.4	−0.741326
$945$	0	0
$946$	10333.4	0.355147
$947$	−4011.39	−0.137648	−0.0688239	−	0.997629i	$-0.521925\pi$
$947$	−4011.39	−0.137648	−0.0688239	+	0.997629i	$0.521925\pi$
$948$	0	0
$949$	6518.94	0.222986
$950$	−7616.64	−0.260123
$951$	0	0
$952$	−5549.80	−0.188939
$953$	−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
$956$	158.929	0.00537670
$957$	0	0
$958$	−513.770	−0.0173269
$959$	−57761.9	−1.94497
$960$	0	0
$961$	−10042.8	−0.337108
$962$	7522.15	0.252104
$963$	0	0
$964$	−2284.22	−0.0763174
$965$	11171.0	0.372650
$966$	0	0
$967$	17635.0	0.586456	0.293228	−	0.956043i	$-0.405271\pi$
$967$	17635.0	0.586456	0.293228	+	0.956043i	$0.405271\pi$
$968$	−29101.2	−0.966267
$969$	0	0
$970$	−9236.47	−0.305737
$971$	26827.3	0.886642	0.443321	−	0.896363i	$-0.353800\pi$
$971$	26827.3	0.886642	0.443321	+	0.896363i	$0.353800\pi$
$972$	0	0
$973$	14625.3	0.481877
$974$	3382.90	0.111288
$975$	0	0
$976$	2825.01	0.0926498
$977$	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
$980$	−1193.91	−0.0389165
$981$	0	0
$982$	32245.8	1.04787
$983$	36890.1	1.19696	0.598479	−	0.801138i	$-0.295772\pi$
$983$	36890.1	1.19696	0.598479	+	0.801138i	$0.295772\pi$
$984$	0	0
$985$	−1733.12	−0.0560628
$986$	−4639.00	−0.149833
$987$	0	0
$988$	1808.30	0.0582283
$989$	−62200.7	−1.99987
$990$	0	0
$991$	−24259.3	−0.777619	−0.388810	−	0.921318i	$-0.627114\pi$
$991$	−24259.3	−0.777619	−0.388810	+	0.921318i	$0.627114\pi$
$992$	−8293.99	−0.265458
$993$	0	0
$994$	−55243.7	−1.76280
$995$	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	$0.332050\pi$
$998$	−8134.15	−0.257998
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.a.h.1.1	yes	3
3.2	odd	2		135.4.a.e.1.3	✓	3
4.3	odd	2		2160.4.a.bq.1.3		3
5.2	odd	4		675.4.b.n.649.2		6
5.3	odd	4		675.4.b.n.649.5		6
5.4	even	2		675.4.a.p.1.3		3
9.2	odd	6		405.4.e.v.271.1		6
9.4	even	3		405.4.e.q.136.3		6
9.5	odd	6		405.4.e.v.136.1		6
9.7	even	3		405.4.e.q.271.3		6
12.11	even	2		2160.4.a.bi.1.3		3
15.2	even	4		675.4.b.m.649.5		6
15.8	even	4		675.4.b.m.649.2		6
15.14	odd	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	3.2	odd	2
135.4.a.h.1.1	yes	3	1.1	even	1	trivial
405.4.e.q.136.3		6	9.4	even	3
405.4.e.q.271.3		6	9.7	even	3
405.4.e.v.136.1		6	9.5	odd	6
405.4.e.v.271.1		6	9.2	odd	6
675.4.a.p.1.3		3	5.4	even	2
675.4.a.s.1.1		3	15.14	odd	2
675.4.b.m.649.2		6	15.8	even	4
675.4.b.m.649.5		6	15.2	even	4
675.4.b.n.649.2		6	5.2	odd	4
675.4.b.n.649.5		6	5.3	odd	4
2160.4.a.bi.1.3		3	12.11	even	2
2160.4.a.bq.1.3		3	4.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

\(f(q)\)	\(=\)	\(q-2.58488 q^{2} -1.31841 q^{4} +5.00000 q^{5} -22.8935 q^{7} +24.0869 q^{8} -12.9244 q^{10} +11.0828 q^{11} -11.6368 q^{13} +59.1769 q^{14} -51.7145 q^{16} +10.0643 q^{17} +117.865 q^{19} -6.59207 q^{20} -28.6477 q^{22} +172.441 q^{23} +25.0000 q^{25} +30.0798 q^{26} +30.1831 q^{28} +178.321 q^{29} +140.528 q^{31} -59.0200 q^{32} -26.0149 q^{34} -114.468 q^{35} +250.074 q^{37} -304.666 q^{38} +120.435 q^{40} +361.569 q^{41} -360.707 q^{43} -14.6117 q^{44} -445.738 q^{46} -600.121 q^{47} +181.114 q^{49} -64.6219 q^{50} +15.3421 q^{52} +201.312 q^{53} +55.4140 q^{55} -551.435 q^{56} -460.937 q^{58} +415.772 q^{59} -54.6270 q^{61} -363.248 q^{62} +566.275 q^{64} -58.1841 q^{65} -531.079 q^{67} -13.2689 q^{68} +295.885 q^{70} -933.534 q^{71} -560.199 q^{73} -646.409 q^{74} -155.394 q^{76} -253.725 q^{77} +810.781 q^{79} -258.572 q^{80} -934.611 q^{82} +538.210 q^{83} +50.3214 q^{85} +932.384 q^{86} +266.951 q^{88} +686.173 q^{89} +266.408 q^{91} -227.348 q^{92} +1551.24 q^{94} +589.324 q^{95} +714.655 q^{97} -468.156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{2} + 17 q^{4} + 15 q^{5} - 4 q^{7} + 75 q^{8} + 25 q^{10} + 5 q^{11} + 7 q^{13} + 60 q^{14} + 161 q^{16} + 155 q^{17} - 50 q^{19} + 85 q^{20} - 229 q^{22} + 285 q^{23} + 75 q^{25} + 185 q^{26}+ \cdots - 305 q^{98}+O(q^{100}) \) 3 * q + 5 * q^2 + 17 * q^4 + 15 * q^5 - 4 * q^7 + 75 * q^8 + 25 * q^10 + 5 * q^11 + 7 * q^13 + 60 * q^14 + 161 * q^16 + 155 * q^17 - 50 * q^19 + 85 * q^20 - 229 * q^22 + 285 * q^23 + 75 * q^25 + 185 * q^26 - 334 * q^28 + 115 * q^29 - 115 * q^31 + 775 * q^32 + 413 * q^34 - 20 * q^35 - 384 * q^37 - 1150 * q^38 + 375 * q^40 + 580 * q^41 - 797 * q^43 - 1415 * q^44 - 285 * q^46 - 145 * q^47 + 577 * q^49 + 125 * q^50 + 825 * q^52 - 400 * q^53 + 25 * q^55 - 2190 * q^56 - 59 * q^58 + 380 * q^59 - 152 * q^61 - 1005 * q^62 + 2937 * q^64 + 35 * q^65 + 2 * q^67 + 475 * q^68 + 300 * q^70 + 40 * q^71 - 980 * q^73 - 2720 * q^74 - 3276 * q^76 + 1950 * q^77 + 1013 * q^79 + 805 * q^80 + 4 * q^82 + 270 * q^83 + 775 * q^85 - 1555 * q^86 - 5193 * q^88 + 1020 * q^89 - 632 * q^91 - 1215 * q^92 + 3833 * q^94 - 250 * q^95 + 720 * q^97 - 305 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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