LMFDB - Embedded newform 135.4.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(1,135)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$7.96525785077$
Analytic rank:	$1$
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			1.3
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)

$f(q)$	$=$	$q+2.58488 q^{2} -1.31841 q^{4} -5.00000 q^{5} -22.8935 q^{7} -24.0869 q^{8} +O(q^{10})$	\(q+2.58488 q^{2} -1.31841 q^{4} -5.00000 q^{5} -22.8935 q^{7} -24.0869 q^{8} -12.9244 q^{10} -11.0828 q^{11} -11.6368 q^{13} -59.1769 q^{14} -51.7145 q^{16} -10.0643 q^{17} +117.865 q^{19} +6.59207 q^{20} -28.6477 q^{22} -172.441 q^{23} +25.0000 q^{25} -30.0798 q^{26} +30.1831 q^{28} -178.321 q^{29} +140.528 q^{31} +59.0200 q^{32} -26.0149 q^{34} +114.468 q^{35} +250.074 q^{37} +304.666 q^{38} +120.435 q^{40} -361.569 q^{41} -360.707 q^{43} +14.6117 q^{44} -445.738 q^{46} +600.121 q^{47} +181.114 q^{49} +64.6219 q^{50} +15.3421 q^{52} -201.312 q^{53} +55.4140 q^{55} +551.435 q^{56} -460.937 q^{58} -415.772 q^{59} -54.6270 q^{61} +363.248 q^{62} +566.275 q^{64} +58.1841 q^{65} -531.079 q^{67} +13.2689 q^{68} +295.885 q^{70} +933.534 q^{71} -560.199 q^{73} +646.409 q^{74} -155.394 q^{76} +253.725 q^{77} +810.781 q^{79} +258.572 q^{80} -934.611 q^{82} -538.210 q^{83} +50.3214 q^{85} -932.384 q^{86} +266.951 q^{88} -686.173 q^{89} +266.408 q^{91} +227.348 q^{92} +1551.24 q^{94} -589.324 q^{95} +714.655 q^{97} +468.156 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{2} + 17 q^{4} - 15 q^{5} - 4 q^{7} - 75 q^{8}+O(q^{10}) $ 3 * q - 5 * q^2 + 17 * q^4 - 15 * q^5 - 4 * q^7 - 75 * q^8	$ 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}+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

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 $

$2$	2.58488	0.913892	0.456946	−	0.889494i	$-0.348943\pi$
$2$	2.58488	0.913892	0.456946	+	0.889494i	$0.348943\pi$
$3$	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	$-0.548536\pi$
$11$	−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	$0.539615\pi$
$14$	−59.1769	−1.12969
$15$	0	0
$16$	−51.7145	−0.808039
$17$	−10.0643	−0.143585	−0.0717926	−	0.997420i	$-0.522872\pi$
$17$	−10.0643	−0.143585	−0.0717926	+	0.997420i	$0.522872\pi$
$18$	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	$-0.785627\pi$
$23$	−172.441	−1.56332	−0.781661	+	0.623704i	$0.785627\pi$
$24$	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	$-0.693413\pi$
$29$	−178.321	−1.14184	−0.570919	+	0.821006i	$0.693413\pi$
$30$	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	$-0.741787\pi$
$41$	−361.569	−1.37726	−0.688629	+	0.725114i	$0.741787\pi$
$42$	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	$-0.118727\pi$
$47$	600.121	1.86248	0.931241	+	0.364405i	$0.118727\pi$
$48$	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	$-0.584010\pi$
$53$	−201.312	−0.521742	−0.260871	+	0.965374i	$0.584010\pi$
$54$	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	$-0.651692\pi$
$59$	−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	$0.518259\pi$
$62$	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	$-0.215111\pi$
$71$	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	$0.648250\pi$
$74$	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	$-0.615819\pi$
$83$	−538.210	−0.711762	−0.355881	+	0.934531i	$0.615819\pi$
$84$	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	$-0.633990\pi$
$89$	−686.173	−0.817238	−0.408619	+	0.912705i	$0.633990\pi$
$90$	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	$-0.659269\pi$
$101$	−973.907	−0.959479	−0.479739	+	0.877411i	$0.659269\pi$
$102$	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	$-0.810308\pi$
$107$	−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	$0.294237\pi$
$110$	143.238	0.124157
$111$	0	0
$112$	1183.93	0.998844
$113$	−583.567	−0.485817	−0.242909	−	0.970049i	$-0.578102\pi$
$113$	−583.567	−0.485817	−0.242909	+	0.970049i	$0.578102\pi$
$114$	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	$-0.553439\pi$
$131$	−501.073	−0.334191	−0.167095	+	0.985941i	$0.553439\pi$
$132$	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	$-0.788220\pi$
$137$	−2523.07	−1.57343	−0.786716	+	0.617316i	$0.788220\pi$
$138$	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	$-0.237114\pi$
$149$	2674.13	1.47029	0.735146	+	0.677909i	$0.237114\pi$
$150$	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	$-0.194430\pi$
$167$	3535.76	1.63836	0.819178	+	0.573540i	$0.194430\pi$
$168$	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	$-0.742752\pi$
$173$	−3143.88	−1.38165	−0.690824	+	0.723023i	$0.742752\pi$
$174$	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	$-0.258147\pi$
$179$	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	$0.374197\pi$
$182$	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	$-0.416965\pi$
$191$	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	$0.363209\pi$
$194$	1847.29	0.683650
$195$	0	0
$196$	−238.782	−0.0870199
$197$	346.625	0.125360	0.0626801	−	0.998034i	$-0.480035\pi$
$197$	346.625	0.125360	0.0626801	+	0.998034i	$0.480035\pi$
$198$	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	$-0.472758\pi$
$227$	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	$0.739202\pi$
$230$	2228.69	0.638937
$231$	0	0
$232$	4295.20	1.21549
$233$	−1228.45	−0.345401	−0.172701	−	0.984974i	$-0.555249\pi$
$233$	−1228.45	−0.345401	−0.172701	+	0.984974i	$0.555249\pi$
$234$	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	$-0.494807\pi$
$239$	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	$0.425623\pi$
$242$	−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	$-0.364342\pi$
$251$	3287.82	0.826793	0.413397	+	0.910551i	$0.364342\pi$
$252$	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	$-0.442130\pi$
$257$	1489.82	0.361605	0.180803	+	0.983519i	$0.442130\pi$
$258$	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	$-0.526530\pi$
$263$	−710.144	−0.166499	−0.0832497	+	0.996529i	$0.526530\pi$
$264$	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	$-0.636445\pi$
$269$	−3667.61	−0.831294	−0.415647	+	0.909526i	$0.636445\pi$
$270$	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	$-0.709617\pi$
$281$	−5765.13	−1.22391	−0.611955	+	0.790892i	$0.709617\pi$
$282$	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	$-0.677305\pi$
$293$	−5302.82	−1.05732	−0.528659	+	0.848834i	$0.677305\pi$
$294$	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	$-0.234354\pi$
$311$	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	$0.655823\pi$
$314$	−987.056	−0.177397
$315$	0	0
$316$	−1068.94	−0.190294
$317$	3262.34	0.578016	0.289008	−	0.957327i	$-0.406675\pi$
$317$	3262.34	0.578016	0.289008	+	0.957327i	$0.406675\pi$
$318$	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	$-0.504851\pi$
$347$	−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	$0.690141\pi$
$350$	−1479.42	−0.225939
$351$	0	0
$352$	−654.107	−0.0990456
$353$	−4616.28	−0.696034	−0.348017	−	0.937488i	$-0.613145\pi$
$353$	−4616.28	−0.696034	−0.348017	+	0.937488i	$0.613145\pi$
$354$	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	$-0.472971\pi$
$359$	1153.79	0.169623	0.0848115	+	0.996397i	$0.472971\pi$
$360$	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	$-0.376449\pi$
$383$	5673.66	0.756946	0.378473	+	0.925612i	$0.376449\pi$
$384$	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	$-0.321020\pi$
$389$	8180.45	1.06623	0.533117	+	0.846041i	$0.321020\pi$
$390$	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	$-0.634399\pi$
$401$	−6581.29	−0.819586	−0.409793	+	0.912179i	$0.634399\pi$
$402$	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	$-0.426718\pi$
$419$	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	$0.598453\pi$
$422$	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	$-0.796167\pi$
$431$	−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	$0.484748\pi$
$434$	−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	$-0.602183\pi$
$443$	−5884.09	−0.631065	−0.315532	+	0.948915i	$0.602183\pi$
$444$	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	$-0.153497\pi$
$449$	16858.4	1.77193	0.885965	+	0.463753i	$0.153497\pi$
$450$	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	$-0.433933\pi$
$461$	4079.39	0.412139	0.206070	+	0.978537i	$0.433933\pi$
$462$	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	$-0.603078\pi$
$467$	−6422.51	−0.636399	−0.318199	+	0.948024i	$0.603078\pi$
$468$	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	$-0.503018\pi$
$479$	−198.760	−0.0189595	−0.00947973	+	0.999955i	$0.503018\pi$
$480$	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	$-0.305663\pi$
$491$	12474.8	1.14660	0.573300	+	0.819346i	$0.305663\pi$
$492$	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	$-0.577212\pi$
$503$	−5419.35	−0.480391	−0.240196	+	0.970725i	$0.577212\pi$
$504$	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	$-0.769302\pi$
$509$	−17194.6	−1.49732	−0.748660	+	0.662954i	$0.769302\pi$
$510$	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	$-0.813799\pi$
$521$	−19829.5	−1.66746	−0.833730	+	0.552172i	$0.813799\pi$
$522$	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	$-0.382069\pi$
$557$	9519.36	0.724144	0.362072	+	0.932150i	$0.382069\pi$
$558$	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	$-0.352763\pi$
$563$	11922.3	0.892476	0.446238	+	0.894914i	$0.352763\pi$
$564$	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	$-0.724412\pi$
$569$	−17591.4	−1.29608	−0.648042	+	0.761605i	$0.724412\pi$
$570$	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	$-0.390539\pi$
$587$	9589.65	0.674288	0.337144	+	0.941453i	$0.390539\pi$
$588$	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	$-0.558878\pi$
$593$	−5311.77	−0.367838	−0.183919	+	0.982941i	$0.558878\pi$
$594$	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	$-0.430556\pi$
$599$	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	$0.608069\pi$
$602$	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	$-0.469183\pi$
$617$	2962.95	0.193329	0.0966643	+	0.995317i	$0.469183\pi$
$618$	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	$-0.431777\pi$
$641$	6903.46	0.425383	0.212691	+	0.977119i	$0.431777\pi$
$642$	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	$-0.670332\pi$
$647$	−16784.3	−1.01988	−0.509939	+	0.860211i	$0.670332\pi$
$648$	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	$-0.343972\pi$
$653$	15711.5	0.941561	0.470781	+	0.882250i	$0.343972\pi$
$654$	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	$-0.296032\pi$
$659$	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	$0.354646\pi$
$662$	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	$-0.167942\pi$
$677$	30439.3	1.72803	0.864015	+	0.503467i	$0.167942\pi$
$678$	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	$-0.412633\pi$
$683$	9675.88	0.542075	0.271038	+	0.962569i	$0.412633\pi$
$684$	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	$-0.387969\pi$
$701$	12796.5	0.689466	0.344733	+	0.938701i	$0.387969\pi$
$702$	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	$-0.843451\pi$
$719$	−33988.7	−1.76296	−0.881479	+	0.472224i	$0.843451\pi$
$720$	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	$-0.503191\pi$
$743$	−406.077	−0.0200505	−0.0100253	+	0.999950i	$0.503191\pi$
$744$	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	$-0.684545\pi$
$761$	−23001.3	−1.09566	−0.547828	+	0.836591i	$0.684545\pi$
$762$	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	$-0.544118\pi$
$773$	−5938.46	−0.276315	−0.138158	+	0.990410i	$0.544118\pi$
$774$	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	$-0.345533\pi$
$797$	20990.4	0.932897	0.466449	+	0.884548i	$0.345533\pi$
$798$	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	$-0.465738\pi$
$809$	4943.95	0.214858	0.107429	+	0.994213i	$0.465738\pi$
$810$	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	$-0.586358\pi$
$821$	−12608.2	−0.535969	−0.267985	+	0.963423i	$0.586358\pi$
$822$	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	$-0.747432\pi$
$827$	−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	$0.533733\pi$
$830$	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	$-0.383007\pi$
$839$	17464.7	0.718650	0.359325	+	0.933213i	$0.383007\pi$
$840$	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	$-0.601010\pi$
$857$	−15656.8	−0.624069	−0.312034	+	0.950071i	$0.601010\pi$
$858$	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	$-0.675617\pi$
$863$	−26576.7	−1.04830	−0.524150	+	0.851626i	$0.675617\pi$
$864$	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	$-0.176819\pi$
$881$	44435.4	1.69928	0.849641	+	0.527362i	$0.176819\pi$
$882$	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	$-0.570406\pi$
$887$	−11591.2	−0.438776	−0.219388	+	0.975638i	$0.570406\pi$
$888$	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	$-0.267329\pi$
$911$	36712.4	1.33517	0.667583	+	0.744536i	$0.267329\pi$
$912$	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	$-0.425622\pi$
$929$	13112.6	0.463090	0.231545	+	0.972824i	$0.425622\pi$
$930$	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	$-0.448207\pi$
$941$	9352.28	0.323991	0.161996	+	0.986791i	$0.448207\pi$
$942$	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	$-0.478075\pi$
$947$	4011.39	0.137648	0.0688239	+	0.997629i	$0.478075\pi$
$948$	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	$-0.268429\pi$
$953$	39128.6	1.33001	0.665005	+	0.746839i	$0.268429\pi$
$954$	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	$-0.646200\pi$
$971$	−26827.3	−0.886642	−0.443321	+	0.896363i	$0.646200\pi$
$972$	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	$-0.543621\pi$
$977$	−8343.63	−0.273221	−0.136610	+	0.990625i	$0.543621\pi$
$978$	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	$-0.704228\pi$
$983$	−36890.1	−1.19696	−0.598479	+	0.801138i	$0.704228\pi$
$984$	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.e.1.3	✓	3
3.2	odd	2		135.4.a.h.1.1	yes	3
4.3	odd	2		2160.4.a.bi.1.3		3
5.2	odd	4		675.4.b.m.649.5		6
5.3	odd	4		675.4.b.m.649.2		6
5.4	even	2		675.4.a.s.1.1		3
9.2	odd	6		405.4.e.q.271.3		6
9.4	even	3		405.4.e.v.136.1		6
9.5	odd	6		405.4.e.q.136.3		6
9.7	even	3		405.4.e.v.271.1		6
12.11	even	2		2160.4.a.bq.1.3		3
15.2	even	4		675.4.b.n.649.2		6
15.8	even	4		675.4.b.n.649.5		6
15.14	odd	2		675.4.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.e.1.3	✓	3	1.1	even	1	trivial
135.4.a.h.1.1	yes	3	3.2	odd	2
405.4.e.q.136.3		6	9.5	odd	6
405.4.e.q.271.3		6	9.2	odd	6
405.4.e.v.136.1		6	9.4	even	3
405.4.e.v.271.1		6	9.7	even	3
675.4.a.p.1.3		3	15.14	odd	2
675.4.a.s.1.1		3	5.4	even	2
675.4.b.m.649.2		6	5.3	odd	4
675.4.b.m.649.5		6	5.2	odd	4
675.4.b.n.649.2		6	15.2	even	4
675.4.b.n.649.5		6	15.8	even	4
2160.4.a.bi.1.3		3	4.3	odd	2
2160.4.a.bq.1.3		3	12.11	even	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

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

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more