LMFDB - Embedded newform 1160.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1160,4,Mod(1,1160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("1160.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$1160 = 2^{3} \cdot 5 \cdot 29$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,0,1,0,-35] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$68.4422156067$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{7} - x^{6} - 121x^{5} + 38x^{4} + 4100x^{3} + 836x^{2} - 34400x - 6912$ x^7 - x^6 - 121x^5 + 38x^4 + 4100x^3 + 836x^2 - 34400*x - 6912
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$2^{5}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.91868$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-7.91868 q^{3} -5.00000 q^{5} +30.1700 q^{7} +35.7055 q^{9} -41.5009 q^{11} +20.4606 q^{13} +39.5934 q^{15} -44.4259 q^{17} -139.197 q^{19} -238.907 q^{21} +164.416 q^{23} +25.0000 q^{25} -68.9361 q^{27} -29.0000 q^{29} +143.019 q^{31} +328.632 q^{33} -150.850 q^{35} -178.297 q^{37} -162.021 q^{39} +283.451 q^{41} -400.334 q^{43} -178.527 q^{45} +66.1270 q^{47} +567.231 q^{49} +351.794 q^{51} +707.089 q^{53} +207.504 q^{55} +1102.26 q^{57} -385.448 q^{59} +207.613 q^{61} +1077.24 q^{63} -102.303 q^{65} +1007.51 q^{67} -1301.96 q^{69} -4.72980 q^{71} +259.868 q^{73} -197.967 q^{75} -1252.08 q^{77} -954.754 q^{79} -418.166 q^{81} -883.835 q^{83} +222.129 q^{85} +229.642 q^{87} +1256.60 q^{89} +617.297 q^{91} -1132.53 q^{93} +695.984 q^{95} -396.847 q^{97} -1481.81 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$7 q + q^{3} - 35 q^{5} + 65 q^{7} + 54 q^{9} - 14 q^{11} - 137 q^{13} - 5 q^{15} - 181 q^{17} + 14 q^{19} - 84 q^{21} + 279 q^{23} + 175 q^{25} + 196 q^{27} - 203 q^{29} - 55 q^{31} - 656 q^{33} - 325 q^{35}+ \cdots - 5216 q^{99}+O(q^{100})$ 7 * q + q^3 - 35 * q^5 + 65 * q^7 + 54 * q^9 - 14 * q^11 - 137 * q^13 - 5 * q^15 - 181 * q^17 + 14 * q^19 - 84 * q^21 + 279 * q^23 + 175 * q^25 + 196 * q^27 - 203 * q^29 - 55 * q^31 - 656 * q^33 - 325 * q^35 - 408 * q^37 - 287 * q^39 + 416 * q^41 - 685 * q^43 - 270 * q^45 - 122 * q^47 - 166 * q^49 + 90 * q^51 + q^53 + 70 * q^55 + 78 * q^57 - 511 * q^59 + 623 * q^61 + 1808 * q^63 + 685 * q^65 + 552 * q^67 - 977 * q^69 + 288 * q^71 - 841 * q^73 + 25 * q^75 - 1526 * q^77 - 797 * q^79 - 1365 * q^81 - 1534 * q^83 + 905 * q^85 - 29 * q^87 + 1252 * q^89 - 2352 * q^91 - 3526 * q^93 - 70 * q^95 - 3179 * q^97 - 5216 * q^99

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
$3$	−7.91868	−1.52395	−0.761975	−	0.647606i	$-0.775770\pi$
$3$	−7.91868	−1.52395	−0.761975	+	0.647606i	$0.775770\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	30.1700	1.62903	0.814514	−	0.580144i	$-0.197004\pi$
$7$	30.1700	1.62903	0.814514	+	0.580144i	$0.197004\pi$
$8$	0	0
$9$	35.7055	1.32243
$10$	0	0
$11$	−41.5009	−1.13754	−0.568772	−	0.822495i	$-0.692581\pi$
$11$	−41.5009	−1.13754	−0.568772	+	0.822495i	$0.692581\pi$
$12$	0	0
$13$	20.4606	0.436519	0.218260	−	0.975891i	$-0.429962\pi$
$13$	20.4606	0.436519	0.218260	+	0.975891i	$0.429962\pi$
$14$	0	0
$15$	39.5934	0.681531
$16$	0	0
$17$	−44.4259	−0.633815	−0.316907	−	0.948456i	$-0.602644\pi$
$17$	−44.4259	−0.633815	−0.316907	+	0.948456i	$0.602644\pi$
$18$	0	0
$19$	−139.197	−1.68073	−0.840367	−	0.542018i	$-0.817660\pi$
$19$	−139.197	−1.68073	−0.840367	+	0.542018i	$0.817660\pi$
$20$	0	0
$21$	−238.907	−2.48256
$22$	0	0
$23$	164.416	1.49057	0.745286	−	0.666745i	$-0.232313\pi$
$23$	164.416	1.49057	0.745286	+	0.666745i	$0.232313\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−68.9361	−0.491361
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	143.019	0.828614	0.414307	−	0.910137i	$-0.364024\pi$
$31$	143.019	0.828614	0.414307	+	0.910137i	$0.364024\pi$
$32$	0	0
$33$	328.632	1.73356
$34$	0	0
$35$	−150.850	−0.728524
$36$	0	0
$37$	−178.297	−0.792212	−0.396106	−	0.918205i	$-0.629639\pi$
$37$	−178.297	−0.792212	−0.396106	+	0.918205i	$0.629639\pi$
$38$	0	0
$39$	−162.021	−0.665234
$40$	0	0
$41$	283.451	1.07970	0.539850	−	0.841761i	$-0.318481\pi$
$41$	283.451	1.07970	0.539850	+	0.841761i	$0.318481\pi$
$42$	0	0
$43$	−400.334	−1.41978	−0.709888	−	0.704314i	$-0.751255\pi$
$43$	−400.334	−1.41978	−0.709888	+	0.704314i	$0.751255\pi$
$44$	0	0
$45$	−178.527	−0.591407
$46$	0	0
$47$	66.1270	0.205226	0.102613	−	0.994721i	$-0.467280\pi$
$47$	66.1270	0.205226	0.102613	+	0.994721i	$0.467280\pi$
$48$	0	0
$49$	567.231	1.65373
$50$	0	0
$51$	351.794	0.965903
$52$	0	0
$53$	707.089	1.83257	0.916285	−	0.400527i	$-0.131173\pi$
$53$	707.089	1.83257	0.916285	+	0.400527i	$0.131173\pi$
$54$	0	0
$55$	207.504	0.508725
$56$	0	0
$57$	1102.26	2.56136
$58$	0	0
$59$	−385.448	−0.850526	−0.425263	−	0.905070i	$-0.639818\pi$
$59$	−385.448	−0.850526	−0.425263	+	0.905070i	$0.639818\pi$
$60$	0	0
$61$	207.613	0.435772	0.217886	−	0.975974i	$-0.430084\pi$
$61$	207.613	0.435772	0.217886	+	0.975974i	$0.430084\pi$
$62$	0	0
$63$	1077.24	2.15427
$64$	0	0
$65$	−102.303	−0.195217
$66$	0	0
$67$	1007.51	1.83712	0.918560	−	0.395282i	$-0.129353\pi$
$67$	1007.51	1.83712	0.918560	+	0.395282i	$0.129353\pi$
$68$	0	0
$69$	−1301.96	−2.27156
$70$	0	0
$71$	−4.72980	−0.00790597	−0.00395299	−	0.999992i	$-0.501258\pi$
$71$	−4.72980	−0.00790597	−0.00395299	+	0.999992i	$0.501258\pi$
$72$	0	0
$73$	259.868	0.416647	0.208324	−	0.978060i	$-0.433199\pi$
$73$	259.868	0.416647	0.208324	+	0.978060i	$0.433199\pi$
$74$	0	0
$75$	−197.967	−0.304790
$76$	0	0
$77$	−1252.08	−1.85309
$78$	0	0
$79$	−954.754	−1.35972	−0.679862	−	0.733340i	$-0.737960\pi$
$79$	−954.754	−1.35972	−0.679862	+	0.733340i	$0.737960\pi$
$80$	0	0
$81$	−418.166	−0.573616
$82$	0	0
$83$	−883.835	−1.16884	−0.584419	−	0.811452i	$-0.698678\pi$
$83$	−883.835	−1.16884	−0.584419	+	0.811452i	$0.698678\pi$
$84$	0	0
$85$	222.129	0.283451
$86$	0	0
$87$	229.642	0.282991
$88$	0	0
$89$	1256.60	1.49663	0.748313	−	0.663346i	$-0.230864\pi$
$89$	1256.60	1.49663	0.748313	+	0.663346i	$0.230864\pi$
$90$	0	0
$91$	617.297	0.711103
$92$	0	0
$93$	−1132.53	−1.26277
$94$	0	0
$95$	695.984	0.751647
$96$	0	0
$97$	−396.847	−0.415399	−0.207699	−	0.978193i	$-0.566598\pi$
$97$	−396.847	−0.415399	−0.207699	+	0.978193i	$0.566598\pi$
$98$	0	0
$99$	−1481.81	−1.50432
$100$	0	0
$101$	1362.77	1.34258	0.671289	−	0.741195i	$-0.265741\pi$
$101$	1362.77	1.34258	0.671289	+	0.741195i	$0.265741\pi$
$102$	0	0
$103$	−1953.67	−1.86894	−0.934470	−	0.356042i	$-0.884126\pi$
$103$	−1953.67	−1.86894	−0.934470	+	0.356042i	$0.884126\pi$
$104$	0	0
$105$	1194.53	1.11023
$106$	0	0
$107$	1773.39	1.60225	0.801123	−	0.598499i	$-0.204236\pi$
$107$	1773.39	1.60225	0.801123	+	0.598499i	$0.204236\pi$
$108$	0	0
$109$	−1705.45	−1.49865	−0.749323	−	0.662205i	$-0.769621\pi$
$109$	−1705.45	−1.49865	−0.749323	+	0.662205i	$0.769621\pi$
$110$	0	0
$111$	1411.88	1.20729
$112$	0	0
$113$	−1540.09	−1.28212	−0.641060	−	0.767490i	$-0.721505\pi$
$113$	−1540.09	−1.28212	−0.641060	+	0.767490i	$0.721505\pi$
$114$	0	0
$115$	−822.081	−0.666604
$116$	0	0
$117$	730.556	0.577265
$118$	0	0
$119$	−1340.33	−1.03250
$120$	0	0
$121$	391.325	0.294008
$122$	0	0
$123$	−2244.56	−1.64541
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−1521.76	−1.06326	−0.531630	−	0.846977i	$-0.678420\pi$
$127$	−1521.76	−1.06326	−0.531630	+	0.846977i	$0.678420\pi$
$128$	0	0
$129$	3170.12	2.16367
$130$	0	0
$131$	−319.260	−0.212931	−0.106465	−	0.994316i	$-0.533953\pi$
$131$	−319.260	−0.212931	−0.106465	+	0.994316i	$0.533953\pi$
$132$	0	0
$133$	−4199.57	−2.73796
$134$	0	0
$135$	344.680	0.219743
$136$	0	0
$137$	−1680.02	−1.04769	−0.523847	−	0.851812i	$-0.675504\pi$
$137$	−1680.02	−1.04769	−0.523847	+	0.851812i	$0.675504\pi$
$138$	0	0
$139$	−567.983	−0.346587	−0.173294	−	0.984870i	$-0.555441\pi$
$139$	−567.983	−0.346587	−0.173294	+	0.984870i	$0.555441\pi$
$140$	0	0
$141$	−523.639	−0.312754
$142$	0	0
$143$	−849.134	−0.496560
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	−4491.72	−2.52021
$148$	0	0
$149$	2330.64	1.28143	0.640716	−	0.767778i	$-0.278638\pi$
$149$	2330.64	1.28143	0.640716	+	0.767778i	$0.278638\pi$
$150$	0	0
$151$	−2335.72	−1.25880	−0.629398	−	0.777083i	$-0.716698\pi$
$151$	−2335.72	−1.25880	−0.629398	+	0.777083i	$0.716698\pi$
$152$	0	0
$153$	−1586.25	−0.838173
$154$	0	0
$155$	−715.097	−0.370568
$156$	0	0
$157$	−598.686	−0.304333	−0.152167	−	0.988355i	$-0.548625\pi$
$157$	−598.686	−0.304333	−0.152167	+	0.988355i	$0.548625\pi$
$158$	0	0
$159$	−5599.22	−2.79275
$160$	0	0
$161$	4960.44	2.42818
$162$	0	0
$163$	−2355.82	−1.13204	−0.566018	−	0.824393i	$-0.691517\pi$
$163$	−2355.82	−1.13204	−0.566018	+	0.824393i	$0.691517\pi$
$164$	0	0
$165$	−1643.16	−0.775273
$166$	0	0
$167$	−975.190	−0.451871	−0.225936	−	0.974142i	$-0.572544\pi$
$167$	−975.190	−0.451871	−0.225936	+	0.974142i	$0.572544\pi$
$168$	0	0
$169$	−1778.36	−0.809451
$170$	0	0
$171$	−4970.09	−2.22265
$172$	0	0
$173$	1750.07	0.769106	0.384553	−	0.923103i	$-0.374356\pi$
$173$	1750.07	0.769106	0.384553	+	0.923103i	$0.374356\pi$
$174$	0	0
$175$	754.251	0.325806
$176$	0	0
$177$	3052.24	1.29616
$178$	0	0
$179$	215.951	0.0901728	0.0450864	−	0.998983i	$-0.485644\pi$
$179$	215.951	0.0901728	0.0450864	+	0.998983i	$0.485644\pi$
$180$	0	0
$181$	369.011	0.151538	0.0757690	−	0.997125i	$-0.475859\pi$
$181$	369.011	0.151538	0.0757690	+	0.997125i	$0.475859\pi$
$182$	0	0
$183$	−1644.02	−0.664096
$184$	0	0
$185$	891.485	0.354288
$186$	0	0
$187$	1843.71	0.720993
$188$	0	0
$189$	−2079.80	−0.800441
$190$	0	0
$191$	−2981.01	−1.12931	−0.564655	−	0.825327i	$-0.690991\pi$
$191$	−2981.01	−1.12931	−0.564655	+	0.825327i	$0.690991\pi$
$192$	0	0
$193$	−1421.80	−0.530277	−0.265138	−	0.964210i	$-0.585418\pi$
$193$	−1421.80	−0.530277	−0.265138	+	0.964210i	$0.585418\pi$
$194$	0	0
$195$	810.105	0.297502
$196$	0	0
$197$	866.654	0.313434	0.156717	−	0.987644i	$-0.449909\pi$
$197$	866.654	0.313434	0.156717	+	0.987644i	$0.449909\pi$
$198$	0	0
$199$	−2936.39	−1.04601	−0.523003	−	0.852331i	$-0.675188\pi$
$199$	−2936.39	−1.04601	−0.523003	+	0.852331i	$0.675188\pi$
$200$	0	0
$201$	−7978.16	−2.79968
$202$	0	0
$203$	−874.931	−0.302503
$204$	0	0
$205$	−1417.26	−0.482856
$206$	0	0
$207$	5870.56	1.97117
$208$	0	0
$209$	5776.79	1.91191
$210$	0	0
$211$	−242.085	−0.0789851	−0.0394925	−	0.999220i	$-0.512574\pi$
$211$	−242.085	−0.0789851	−0.0394925	+	0.999220i	$0.512574\pi$
$212$	0	0
$213$	37.4538	0.0120483
$214$	0	0
$215$	2001.67	0.634943
$216$	0	0
$217$	4314.90	1.34984
$218$	0	0
$219$	−2057.81	−0.634950
$220$	0	0
$221$	−908.980	−0.276673
$222$	0	0
$223$	−725.560	−0.217879	−0.108940	−	0.994048i	$-0.534746\pi$
$223$	−725.560	−0.217879	−0.108940	+	0.994048i	$0.534746\pi$
$224$	0	0
$225$	892.637	0.264485
$226$	0	0
$227$	−6224.96	−1.82011	−0.910055	−	0.414487i	$-0.863961\pi$
$227$	−6224.96	−1.82011	−0.910055	+	0.414487i	$0.863961\pi$
$228$	0	0
$229$	4182.45	1.20692	0.603459	−	0.797394i	$-0.293789\pi$
$229$	4182.45	1.20692	0.603459	+	0.797394i	$0.293789\pi$
$230$	0	0
$231$	9914.85	2.82402
$232$	0	0
$233$	−3487.32	−0.980524	−0.490262	−	0.871575i	$-0.663099\pi$
$233$	−3487.32	−0.980524	−0.490262	+	0.871575i	$0.663099\pi$
$234$	0	0
$235$	−330.635	−0.0917798
$236$	0	0
$237$	7560.39	2.07215
$238$	0	0
$239$	−1596.56	−0.432104	−0.216052	−	0.976382i	$-0.569318\pi$
$239$	−1596.56	−0.432104	−0.216052	+	0.976382i	$0.569318\pi$
$240$	0	0
$241$	6924.23	1.85074	0.925371	−	0.379062i	$-0.123753\pi$
$241$	6924.23	1.85074	0.925371	+	0.379062i	$0.123753\pi$
$242$	0	0
$243$	5172.60	1.36552
$244$	0	0
$245$	−2836.15	−0.739572
$246$	0	0
$247$	−2848.05	−0.733673
$248$	0	0
$249$	6998.81	1.78125
$250$	0	0
$251$	5781.91	1.45399	0.726994	−	0.686644i	$-0.240917\pi$
$251$	5781.91	1.45399	0.726994	+	0.686644i	$0.240917\pi$
$252$	0	0
$253$	−6823.42	−1.69559
$254$	0	0
$255$	−1758.97	−0.431965
$256$	0	0
$257$	−145.551	−0.0353277	−0.0176639	−	0.999844i	$-0.505623\pi$
$257$	−145.551	−0.0353277	−0.0176639	+	0.999844i	$0.505623\pi$
$258$	0	0
$259$	−5379.22	−1.29054
$260$	0	0
$261$	−1035.46	−0.245568
$262$	0	0
$263$	3122.63	0.732128	0.366064	−	0.930590i	$-0.380705\pi$
$263$	3122.63	0.732128	0.366064	+	0.930590i	$0.380705\pi$
$264$	0	0
$265$	−3535.45	−0.819550
$266$	0	0
$267$	−9950.64	−2.28078
$268$	0	0
$269$	750.382	0.170080	0.0850402	−	0.996378i	$-0.472898\pi$
$269$	750.382	0.170080	0.0850402	+	0.996378i	$0.472898\pi$
$270$	0	0
$271$	−4817.42	−1.07984	−0.539921	−	0.841715i	$-0.681546\pi$
$271$	−4817.42	−1.07984	−0.539921	+	0.841715i	$0.681546\pi$
$272$	0	0
$273$	−4888.18	−1.08369
$274$	0	0
$275$	−1037.52	−0.227509
$276$	0	0
$277$	−2441.36	−0.529556	−0.264778	−	0.964309i	$-0.585299\pi$
$277$	−2441.36	−0.529556	−0.264778	+	0.964309i	$0.585299\pi$
$278$	0	0
$279$	5106.58	1.09578
$280$	0	0
$281$	5596.93	1.18820	0.594101	−	0.804391i	$-0.297508\pi$
$281$	5596.93	1.18820	0.594101	+	0.804391i	$0.297508\pi$
$282$	0	0
$283$	−2573.05	−0.540467	−0.270234	−	0.962795i	$-0.587101\pi$
$283$	−2573.05	−0.540467	−0.270234	+	0.962795i	$0.587101\pi$
$284$	0	0
$285$	−5511.28	−1.14547
$286$	0	0
$287$	8551.74	1.75886
$288$	0	0
$289$	−2939.34	−0.598279
$290$	0	0
$291$	3142.50	0.633047
$292$	0	0
$293$	−4699.84	−0.937090	−0.468545	−	0.883440i	$-0.655222\pi$
$293$	−4699.84	−0.937090	−0.468545	+	0.883440i	$0.655222\pi$
$294$	0	0
$295$	1927.24	0.380367
$296$	0	0
$297$	2860.91	0.558945
$298$	0	0
$299$	3364.06	0.650664
$300$	0	0
$301$	−12078.1	−2.31286
$302$	0	0
$303$	−10791.3	−2.04602
$304$	0	0
$305$	−1038.06	−0.194883
$306$	0	0
$307$	−2818.58	−0.523989	−0.261995	−	0.965069i	$-0.584380\pi$
$307$	−2818.58	−0.523989	−0.261995	+	0.965069i	$0.584380\pi$
$308$	0	0
$309$	15470.5	2.84817
$310$	0	0
$311$	340.672	0.0621148	0.0310574	−	0.999518i	$-0.490113\pi$
$311$	340.672	0.0621148	0.0310574	+	0.999518i	$0.490113\pi$
$312$	0	0
$313$	−6854.04	−1.23774	−0.618871	−	0.785492i	$-0.712410\pi$
$313$	−6854.04	−1.23774	−0.618871	+	0.785492i	$0.712410\pi$
$314$	0	0
$315$	−5386.18	−0.963419
$316$	0	0
$317$	−6510.76	−1.15357	−0.576783	−	0.816897i	$-0.695692\pi$
$317$	−6510.76	−1.15357	−0.576783	+	0.816897i	$0.695692\pi$
$318$	0	0
$319$	1203.53	0.211237
$320$	0	0
$321$	−14042.9	−2.44174
$322$	0	0
$323$	6183.94	1.06527
$324$	0	0
$325$	511.515	0.0873039
$326$	0	0
$327$	13504.9	2.28386
$328$	0	0
$329$	1995.05	0.334319
$330$	0	0
$331$	−3135.36	−0.520649	−0.260324	−	0.965521i	$-0.583829\pi$
$331$	−3135.36	−0.520649	−0.260324	+	0.965521i	$0.583829\pi$
$332$	0	0
$333$	−6366.18	−1.04764
$334$	0	0
$335$	−5037.55	−0.821585
$336$	0	0
$337$	5459.28	0.882451	0.441225	−	0.897396i	$-0.354544\pi$
$337$	5459.28	0.882451	0.441225	+	0.897396i	$0.354544\pi$
$338$	0	0
$339$	12195.5	1.95389
$340$	0	0
$341$	−5935.43	−0.942586
$342$	0	0
$343$	6765.04	1.06495
$344$	0	0
$345$	6509.80	1.01587
$346$	0	0
$347$	−4476.98	−0.692613	−0.346306	−	0.938121i	$-0.612564\pi$
$347$	−4476.98	−0.692613	−0.346306	+	0.938121i	$0.612564\pi$
$348$	0	0
$349$	−3769.50	−0.578157	−0.289078	−	0.957305i	$-0.593349\pi$
$349$	−3769.50	−0.578157	−0.289078	+	0.957305i	$0.593349\pi$
$350$	0	0
$351$	−1410.47	−0.214489
$352$	0	0
$353$	10801.6	1.62865	0.814323	−	0.580412i	$-0.197109\pi$
$353$	10801.6	1.62865	0.814323	+	0.580412i	$0.197109\pi$
$354$	0	0
$355$	23.6490	0.00353566
$356$	0	0
$357$	10613.6	1.57348
$358$	0	0
$359$	−83.1926	−0.0122305	−0.00611523	−	0.999981i	$-0.501947\pi$
$359$	−83.1926	−0.0122305	−0.00611523	+	0.999981i	$0.501947\pi$
$360$	0	0
$361$	12516.8	1.82487
$362$	0	0
$363$	−3098.77	−0.448054
$364$	0	0
$365$	−1299.34	−0.186330
$366$	0	0
$367$	−5161.95	−0.734201	−0.367101	−	0.930181i	$-0.619649\pi$
$367$	−5161.95	−0.734201	−0.367101	+	0.930181i	$0.619649\pi$
$368$	0	0
$369$	10120.8	1.42782
$370$	0	0
$371$	21332.9	2.98531
$372$	0	0
$373$	−3535.69	−0.490807	−0.245403	−	0.969421i	$-0.578920\pi$
$373$	−3535.69	−0.490807	−0.245403	+	0.969421i	$0.578920\pi$
$374$	0	0
$375$	989.835	0.136306
$376$	0	0
$377$	−593.358	−0.0810596
$378$	0	0
$379$	10014.9	1.35734	0.678668	−	0.734445i	$-0.262558\pi$
$379$	10014.9	1.35734	0.678668	+	0.734445i	$0.262558\pi$
$380$	0	0
$381$	12050.3	1.62035
$382$	0	0
$383$	−2999.55	−0.400183	−0.200091	−	0.979777i	$-0.564124\pi$
$383$	−2999.55	−0.400183	−0.200091	+	0.979777i	$0.564124\pi$
$384$	0	0
$385$	6260.42	0.828728
$386$	0	0
$387$	−14294.1	−1.87755
$388$	0	0
$389$	−3981.88	−0.518996	−0.259498	−	0.965744i	$-0.583557\pi$
$389$	−3981.88	−0.518996	−0.259498	+	0.965744i	$0.583557\pi$
$390$	0	0
$391$	−7304.33	−0.944747
$392$	0	0
$393$	2528.12	0.324496
$394$	0	0
$395$	4773.77	0.608087
$396$	0	0
$397$	−4023.09	−0.508597	−0.254298	−	0.967126i	$-0.581845\pi$
$397$	−4023.09	−0.508597	−0.254298	+	0.967126i	$0.581845\pi$
$398$	0	0
$399$	33255.1	4.17252
$400$	0	0
$401$	−15809.5	−1.96881	−0.984403	−	0.175930i	$-0.943707\pi$
$401$	−15809.5	−1.96881	−0.984403	+	0.175930i	$0.943707\pi$
$402$	0	0
$403$	2926.27	0.361706
$404$	0	0
$405$	2090.83	0.256529
$406$	0	0
$407$	7399.49	0.901177
$408$	0	0
$409$	−8040.42	−0.972061	−0.486031	−	0.873942i	$-0.661556\pi$
$409$	−8040.42	−0.972061	−0.486031	+	0.873942i	$0.661556\pi$
$410$	0	0
$411$	13303.6	1.59663
$412$	0	0
$413$	−11629.0	−1.38553
$414$	0	0
$415$	4419.18	0.522720
$416$	0	0
$417$	4497.67	0.528182
$418$	0	0
$419$	−16378.7	−1.90967	−0.954833	−	0.297142i	$-0.903967\pi$
$419$	−16378.7	−1.90967	−0.954833	+	0.297142i	$0.903967\pi$
$420$	0	0
$421$	11576.6	1.34016	0.670082	−	0.742287i	$-0.266259\pi$
$421$	11576.6	1.34016	0.670082	+	0.742287i	$0.266259\pi$
$422$	0	0
$423$	2361.10	0.271396
$424$	0	0
$425$	−1110.65	−0.126763
$426$	0	0
$427$	6263.69	0.709886
$428$	0	0
$429$	6724.02	0.756734
$430$	0	0
$431$	747.175	0.0835039	0.0417519	−	0.999128i	$-0.486706\pi$
$431$	747.175	0.0835039	0.0417519	+	0.999128i	$0.486706\pi$
$432$	0	0
$433$	8711.47	0.966851	0.483425	−	0.875386i	$-0.339392\pi$
$433$	8711.47	0.966851	0.483425	+	0.875386i	$0.339392\pi$
$434$	0	0
$435$	−1148.21	−0.126557
$436$	0	0
$437$	−22886.2	−2.50525
$438$	0	0
$439$	−2056.40	−0.223568	−0.111784	−	0.993733i	$-0.535657\pi$
$439$	−2056.40	−0.223568	−0.111784	+	0.993733i	$0.535657\pi$
$440$	0	0
$441$	20253.2	2.18694
$442$	0	0
$443$	−16175.5	−1.73481	−0.867404	−	0.497605i	$-0.834213\pi$
$443$	−16175.5	−1.73481	−0.867404	+	0.497605i	$0.834213\pi$
$444$	0	0
$445$	−6283.02	−0.669312
$446$	0	0
$447$	−18455.6	−1.95284
$448$	0	0
$449$	−11865.6	−1.24715	−0.623575	−	0.781764i	$-0.714320\pi$
$449$	−11865.6	−1.24715	−0.623575	+	0.781764i	$0.714320\pi$
$450$	0	0
$451$	−11763.5	−1.22821
$452$	0	0
$453$	18495.8	1.91834
$454$	0	0
$455$	−3086.49	−0.318015
$456$	0	0
$457$	−7236.71	−0.740742	−0.370371	−	0.928884i	$-0.620769\pi$
$457$	−7236.71	−0.740742	−0.370371	+	0.928884i	$0.620769\pi$
$458$	0	0
$459$	3062.54	0.311432
$460$	0	0
$461$	−8760.00	−0.885019	−0.442509	−	0.896764i	$-0.645912\pi$
$461$	−8760.00	−0.885019	−0.442509	+	0.896764i	$0.645912\pi$
$462$	0	0
$463$	−14811.5	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$463$	−14811.5	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$464$	0	0
$465$	5662.63	0.564727
$466$	0	0
$467$	568.520	0.0563340	0.0281670	−	0.999603i	$-0.491033\pi$
$467$	568.520	0.0563340	0.0281670	+	0.999603i	$0.491033\pi$
$468$	0	0
$469$	30396.6	2.99272
$470$	0	0
$471$	4740.80	0.463789
$472$	0	0
$473$	16614.2	1.61506
$474$	0	0
$475$	−3479.92	−0.336147
$476$	0	0
$477$	25247.0	2.42344
$478$	0	0
$479$	−14218.8	−1.35631	−0.678157	−	0.734917i	$-0.737221\pi$
$479$	−14218.8	−1.35631	−0.678157	+	0.734917i	$0.737221\pi$
$480$	0	0
$481$	−3648.07	−0.345816
$482$	0	0
$483$	−39280.2	−3.70043
$484$	0	0
$485$	1984.23	0.185772
$486$	0	0
$487$	−2724.74	−0.253532	−0.126766	−	0.991933i	$-0.540460\pi$
$487$	−2724.74	−0.253532	−0.126766	+	0.991933i	$0.540460\pi$
$488$	0	0
$489$	18655.0	1.72517
$490$	0	0
$491$	21133.2	1.94242	0.971209	−	0.238231i	$-0.0765674\pi$
$491$	21133.2	1.94242	0.971209	+	0.238231i	$0.0765674\pi$
$492$	0	0
$493$	1288.35	0.117696
$494$	0	0
$495$	7409.05	0.672752
$496$	0	0
$497$	−142.698	−0.0128791
$498$	0	0
$499$	11748.1	1.05394	0.526970	−	0.849884i	$-0.323328\pi$
$499$	11748.1	1.05394	0.526970	+	0.849884i	$0.323328\pi$
$500$	0	0
$501$	7722.22	0.688629
$502$	0	0
$503$	20052.9	1.77757	0.888783	−	0.458328i	$-0.151552\pi$
$503$	20052.9	1.77757	0.888783	+	0.458328i	$0.151552\pi$
$504$	0	0
$505$	−6813.84	−0.600419
$506$	0	0
$507$	14082.3	1.23356
$508$	0	0
$509$	−17437.0	−1.51843	−0.759214	−	0.650841i	$-0.774417\pi$
$509$	−17437.0	−1.51843	−0.759214	+	0.650841i	$0.774417\pi$
$510$	0	0
$511$	7840.23	0.678731
$512$	0	0
$513$	9595.68	0.825847
$514$	0	0
$515$	9768.35	0.835815
$516$	0	0
$517$	−2744.33	−0.233454
$518$	0	0
$519$	−13858.2	−1.17208
$520$	0	0
$521$	13465.5	1.13231	0.566154	−	0.824299i	$-0.308431\pi$
$521$	13465.5	1.13231	0.566154	+	0.824299i	$0.308431\pi$
$522$	0	0
$523$	−226.324	−0.0189224	−0.00946122	−	0.999955i	$-0.503012\pi$
$523$	−226.324	−0.0189224	−0.00946122	+	0.999955i	$0.503012\pi$
$524$	0	0
$525$	−5972.67	−0.496512
$526$	0	0
$527$	−6353.76	−0.525188
$528$	0	0
$529$	14865.7	1.22180
$530$	0	0
$531$	−13762.6	−1.12476
$532$	0	0
$533$	5799.59	0.471310
$534$	0	0
$535$	−8866.96	−0.716546
$536$	0	0
$537$	−1710.05	−0.137419
$538$	0	0
$539$	−23540.6	−1.88120
$540$	0	0
$541$	1099.62	0.0873866	0.0436933	−	0.999045i	$-0.486088\pi$
$541$	1099.62	0.0873866	0.0436933	+	0.999045i	$0.486088\pi$
$542$	0	0
$543$	−2922.08	−0.230936
$544$	0	0
$545$	8527.24	0.670215
$546$	0	0
$547$	−325.266	−0.0254248	−0.0127124	−	0.999919i	$-0.504047\pi$
$547$	−325.266	−0.0254248	−0.0127124	+	0.999919i	$0.504047\pi$
$548$	0	0
$549$	7412.92	0.576277
$550$	0	0
$551$	4036.71	0.312104
$552$	0	0
$553$	−28805.0	−2.21503
$554$	0	0
$555$	−7059.38	−0.539917
$556$	0	0
$557$	−11078.0	−0.842715	−0.421357	−	0.906895i	$-0.638446\pi$
$557$	−11078.0	−0.842715	−0.421357	+	0.906895i	$0.638446\pi$
$558$	0	0
$559$	−8191.08	−0.619760
$560$	0	0
$561$	−14599.8	−1.09876
$562$	0	0
$563$	−4301.09	−0.321971	−0.160985	−	0.986957i	$-0.551467\pi$
$563$	−4301.09	−0.321971	−0.160985	+	0.986957i	$0.551467\pi$
$564$	0	0
$565$	7700.46	0.573382
$566$	0	0
$567$	−12616.1	−0.934436
$568$	0	0
$569$	−13912.2	−1.02501	−0.512503	−	0.858686i	$-0.671282\pi$
$569$	−13912.2	−1.02501	−0.512503	+	0.858686i	$0.671282\pi$
$570$	0	0
$571$	5249.89	0.384765	0.192383	−	0.981320i	$-0.438379\pi$
$571$	5249.89	0.384765	0.192383	+	0.981320i	$0.438379\pi$
$572$	0	0
$573$	23605.7	1.72101
$574$	0	0
$575$	4110.41	0.298114
$576$	0	0
$577$	23793.2	1.71668	0.858338	−	0.513085i	$-0.171497\pi$
$577$	23793.2	1.71668	0.858338	+	0.513085i	$0.171497\pi$
$578$	0	0
$579$	11258.8	0.808116
$580$	0	0
$581$	−26665.3	−1.90407
$582$	0	0
$583$	−29344.8	−2.08463
$584$	0	0
$585$	−3652.78	−0.258161
$586$	0	0
$587$	1345.64	0.0946174	0.0473087	−	0.998880i	$-0.484936\pi$
$587$	1345.64	0.0946174	0.0473087	+	0.998880i	$0.484936\pi$
$588$	0	0
$589$	−19907.9	−1.39268
$590$	0	0
$591$	−6862.76	−0.477659
$592$	0	0
$593$	−12716.6	−0.880618	−0.440309	−	0.897846i	$-0.645131\pi$
$593$	−12716.6	−0.880618	−0.440309	+	0.897846i	$0.645131\pi$
$594$	0	0
$595$	6701.65	0.461749
$596$	0	0
$597$	23252.3	1.59406
$598$	0	0
$599$	−20205.6	−1.37826	−0.689131	−	0.724637i	$-0.742008\pi$
$599$	−20205.6	−1.37826	−0.689131	+	0.724637i	$0.742008\pi$
$600$	0	0
$601$	−5770.48	−0.391652	−0.195826	−	0.980639i	$-0.562739\pi$
$601$	−5770.48	−0.391652	−0.195826	+	0.980639i	$0.562739\pi$
$602$	0	0
$603$	35973.7	2.42945
$604$	0	0
$605$	−1956.62	−0.131484
$606$	0	0
$607$	3069.56	0.205255	0.102627	−	0.994720i	$-0.467275\pi$
$607$	3069.56	0.205255	0.102627	+	0.994720i	$0.467275\pi$
$608$	0	0
$609$	6928.30	0.461000
$610$	0	0
$611$	1353.00	0.0895851
$612$	0	0
$613$	−15942.9	−1.05045	−0.525227	−	0.850962i	$-0.676020\pi$
$613$	−15942.9	−1.05045	−0.525227	+	0.850962i	$0.676020\pi$
$614$	0	0
$615$	11222.8	0.735849
$616$	0	0
$617$	7071.49	0.461406	0.230703	−	0.973024i	$-0.425897\pi$
$617$	7071.49	0.461406	0.230703	+	0.973024i	$0.425897\pi$
$618$	0	0
$619$	14764.2	0.958679	0.479339	−	0.877630i	$-0.340876\pi$
$619$	14764.2	0.958679	0.479339	+	0.877630i	$0.340876\pi$
$620$	0	0
$621$	−11334.2	−0.732409
$622$	0	0
$623$	37911.8	2.43805
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−45744.6	−2.91366
$628$	0	0
$629$	7921.00	0.502116
$630$	0	0
$631$	28574.8	1.80277	0.901383	−	0.433023i	$-0.142553\pi$
$631$	28574.8	1.80277	0.901383	+	0.433023i	$0.142553\pi$
$632$	0	0
$633$	1917.00	0.120369
$634$	0	0
$635$	7608.78	0.475504
$636$	0	0
$637$	11605.9	0.721887
$638$	0	0
$639$	−168.880	−0.0104551
$640$	0	0
$641$	−24087.2	−1.48422	−0.742112	−	0.670276i	$-0.766176\pi$
$641$	−24087.2	−1.48422	−0.742112	+	0.670276i	$0.766176\pi$
$642$	0	0
$643$	1460.69	0.0895865	0.0447932	−	0.998996i	$-0.485737\pi$
$643$	1460.69	0.0895865	0.0447932	+	0.998996i	$0.485737\pi$
$644$	0	0
$645$	−15850.6	−0.967623
$646$	0	0
$647$	6718.59	0.408246	0.204123	−	0.978945i	$-0.434566\pi$
$647$	6718.59	0.408246	0.204123	+	0.978945i	$0.434566\pi$
$648$	0	0
$649$	15996.4	0.967512
$650$	0	0
$651$	−34168.3	−2.05708
$652$	0	0
$653$	−98.3814	−0.00589580	−0.00294790	−	0.999996i	$-0.500938\pi$
$653$	−98.3814	−0.00589580	−0.00294790	+	0.999996i	$0.500938\pi$
$654$	0	0
$655$	1596.30	0.0952254
$656$	0	0
$657$	9278.72	0.550985
$658$	0	0
$659$	14606.4	0.863408	0.431704	−	0.902015i	$-0.357913\pi$
$659$	14606.4	0.863408	0.431704	+	0.902015i	$0.357913\pi$
$660$	0	0
$661$	89.9790	0.00529467	0.00264734	−	0.999996i	$-0.499157\pi$
$661$	89.9790	0.00529467	0.00264734	+	0.999996i	$0.499157\pi$
$662$	0	0
$663$	7197.92	0.421635
$664$	0	0
$665$	20997.9	1.22445
$666$	0	0
$667$	−4768.07	−0.276792
$668$	0	0
$669$	5745.48	0.332038
$670$	0	0
$671$	−8616.12	−0.495711
$672$	0	0
$673$	−13820.9	−0.791613	−0.395806	−	0.918334i	$-0.629535\pi$
$673$	−13820.9	−0.791613	−0.395806	+	0.918334i	$0.629535\pi$
$674$	0	0
$675$	−1723.40	−0.0982722
$676$	0	0
$677$	32389.0	1.83871	0.919357	−	0.393423i	$-0.128709\pi$
$677$	32389.0	1.83871	0.919357	+	0.393423i	$0.128709\pi$
$678$	0	0
$679$	−11972.9	−0.676697
$680$	0	0
$681$	49293.5	2.77376
$682$	0	0
$683$	12912.1	0.723379	0.361689	−	0.932299i	$-0.382200\pi$
$683$	12912.1	0.723379	0.361689	+	0.932299i	$0.382200\pi$
$684$	0	0
$685$	8400.12	0.468543
$686$	0	0
$687$	−33119.5	−1.83928
$688$	0	0
$689$	14467.5	0.799952
$690$	0	0
$691$	2598.12	0.143035	0.0715176	−	0.997439i	$-0.477216\pi$
$691$	2598.12	0.143035	0.0715176	+	0.997439i	$0.477216\pi$
$692$	0	0
$693$	−44706.3	−2.45058
$694$	0	0
$695$	2839.91	0.154999
$696$	0	0
$697$	−12592.6	−0.684329
$698$	0	0
$699$	27615.0	1.49427
$700$	0	0
$701$	−14722.0	−0.793212	−0.396606	−	0.917989i	$-0.629812\pi$
$701$	−14722.0	−0.793212	−0.396606	+	0.917989i	$0.629812\pi$
$702$	0	0
$703$	24818.4	1.33150
$704$	0	0
$705$	2618.19	0.139868
$706$	0	0
$707$	41114.7	2.18710
$708$	0	0
$709$	−20819.7	−1.10282	−0.551410	−	0.834234i	$-0.685910\pi$
$709$	−20819.7	−1.10282	−0.551410	+	0.834234i	$0.685910\pi$
$710$	0	0
$711$	−34090.0	−1.79813
$712$	0	0
$713$	23514.7	1.23511
$714$	0	0
$715$	4245.67	0.222069
$716$	0	0
$717$	12642.7	0.658506
$718$	0	0
$719$	9197.06	0.477041	0.238520	−	0.971137i	$-0.423338\pi$
$719$	9197.06	0.477041	0.238520	+	0.971137i	$0.423338\pi$
$720$	0	0
$721$	−58942.3	−3.04456
$722$	0	0
$723$	−54830.8	−2.82044
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−7105.85	−0.362505	−0.181253	−	0.983437i	$-0.558015\pi$
$727$	−7105.85	−0.362505	−0.181253	+	0.983437i	$0.558015\pi$
$728$	0	0
$729$	−29669.7	−1.50737
$730$	0	0
$731$	17785.2	0.899876
$732$	0	0
$733$	−999.375	−0.0503585	−0.0251792	−	0.999683i	$-0.508016\pi$
$733$	−999.375	−0.0503585	−0.0251792	+	0.999683i	$0.508016\pi$
$734$	0	0
$735$	22458.6	1.12707
$736$	0	0
$737$	−41812.6	−2.08981
$738$	0	0
$739$	31121.4	1.54914	0.774572	−	0.632486i	$-0.217965\pi$
$739$	31121.4	1.54914	0.774572	+	0.632486i	$0.217965\pi$
$740$	0	0
$741$	22552.8	1.11808
$742$	0	0
$743$	−3438.14	−0.169762	−0.0848811	−	0.996391i	$-0.527051\pi$
$743$	−3438.14	−0.169762	−0.0848811	+	0.996391i	$0.527051\pi$
$744$	0	0
$745$	−11653.2	−0.573073
$746$	0	0
$747$	−31557.8	−1.54570
$748$	0	0
$749$	53503.3	2.61010
$750$	0	0
$751$	−12142.4	−0.589992	−0.294996	−	0.955499i	$-0.595318\pi$
$751$	−12142.4	−0.589992	−0.294996	+	0.955499i	$0.595318\pi$
$752$	0	0
$753$	−45785.1	−2.21581
$754$	0	0
$755$	11678.6	0.562951
$756$	0	0
$757$	18697.6	0.897722	0.448861	−	0.893602i	$-0.351830\pi$
$757$	18697.6	0.897722	0.448861	+	0.893602i	$0.351830\pi$
$758$	0	0
$759$	54032.5	2.58400
$760$	0	0
$761$	−1401.86	−0.0667772	−0.0333886	−	0.999442i	$-0.510630\pi$
$761$	−1401.86	−0.0667772	−0.0333886	+	0.999442i	$0.510630\pi$
$762$	0	0
$763$	−51453.4	−2.44134
$764$	0	0
$765$	7931.24	0.374842
$766$	0	0
$767$	−7886.50	−0.371271
$768$	0	0
$769$	12563.3	0.589134	0.294567	−	0.955631i	$-0.404825\pi$
$769$	12563.3	0.589134	0.294567	+	0.955631i	$0.404825\pi$
$770$	0	0
$771$	1152.57	0.0538377
$772$	0	0
$773$	22987.3	1.06960	0.534798	−	0.844980i	$-0.320388\pi$
$773$	22987.3	1.06960	0.534798	+	0.844980i	$0.320388\pi$
$774$	0	0
$775$	3575.49	0.165723
$776$	0	0
$777$	42596.4	1.96671
$778$	0	0
$779$	−39455.5	−1.81469
$780$	0	0
$781$	196.291	0.00899340
$782$	0	0
$783$	1999.15	0.0912435
$784$	0	0
$785$	2993.43	0.136102
$786$	0	0
$787$	−13115.7	−0.594058	−0.297029	−	0.954868i	$-0.595996\pi$
$787$	−13115.7	−0.594058	−0.297029	+	0.954868i	$0.595996\pi$
$788$	0	0
$789$	−24727.1	−1.11573
$790$	0	0
$791$	−46464.6	−2.08861
$792$	0	0
$793$	4247.89	0.190223
$794$	0	0
$795$	27996.1	1.24895
$796$	0	0
$797$	6717.43	0.298549	0.149274	−	0.988796i	$-0.452306\pi$
$797$	6717.43	0.298549	0.149274	+	0.988796i	$0.452306\pi$
$798$	0	0
$799$	−2937.75	−0.130075
$800$	0	0
$801$	44867.7	1.97918
$802$	0	0
$803$	−10784.8	−0.473955
$804$	0	0
$805$	−24802.2	−1.08592
$806$	0	0
$807$	−5942.04	−0.259194
$808$	0	0
$809$	19473.0	0.846270	0.423135	−	0.906067i	$-0.360930\pi$
$809$	19473.0	0.846270	0.423135	+	0.906067i	$0.360930\pi$
$810$	0	0
$811$	217.936	0.00943623	0.00471811	−	0.999989i	$-0.498498\pi$
$811$	217.936	0.00943623	0.00471811	+	0.999989i	$0.498498\pi$
$812$	0	0
$813$	38147.6	1.64563
$814$	0	0
$815$	11779.1	0.506262
$816$	0	0
$817$	55725.3	2.38627
$818$	0	0
$819$	22040.9	0.940380
$820$	0	0
$821$	−6337.86	−0.269419	−0.134709	−	0.990885i	$-0.543010\pi$
$821$	−6337.86	−0.269419	−0.134709	+	0.990885i	$0.543010\pi$
$822$	0	0
$823$	33154.8	1.40426	0.702128	−	0.712051i	$-0.252233\pi$
$823$	33154.8	1.40426	0.702128	+	0.712051i	$0.252233\pi$
$824$	0	0
$825$	8215.81	0.346712
$826$	0	0
$827$	−20191.4	−0.849003	−0.424501	−	0.905427i	$-0.639551\pi$
$827$	−20191.4	−0.849003	−0.424501	+	0.905427i	$0.639551\pi$
$828$	0	0
$829$	−28296.8	−1.18551	−0.592756	−	0.805382i	$-0.701960\pi$
$829$	−28296.8	−1.18551	−0.592756	+	0.805382i	$0.701960\pi$
$830$	0	0
$831$	19332.3	0.807017
$832$	0	0
$833$	−25199.7	−1.04816
$834$	0	0
$835$	4875.95	0.202083
$836$	0	0
$837$	−9859.20	−0.407149
$838$	0	0
$839$	38620.6	1.58919	0.794596	−	0.607138i	$-0.207683\pi$
$839$	38620.6	1.58919	0.794596	+	0.607138i	$0.207683\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−44320.3	−1.81076
$844$	0	0
$845$	8891.82	0.361997
$846$	0	0
$847$	11806.3	0.478947
$848$	0	0
$849$	20375.2	0.823645
$850$	0	0
$851$	−29314.9	−1.18085
$852$	0	0
$853$	−25417.1	−1.02024	−0.510119	−	0.860104i	$-0.670399\pi$
$853$	−25417.1	−1.02024	−0.510119	+	0.860104i	$0.670399\pi$
$854$	0	0
$855$	24850.5	0.993997
$856$	0	0
$857$	25926.0	1.03339	0.516695	−	0.856170i	$-0.327162\pi$
$857$	25926.0	1.03339	0.516695	+	0.856170i	$0.327162\pi$
$858$	0	0
$859$	−7972.29	−0.316660	−0.158330	−	0.987386i	$-0.550611\pi$
$859$	−7972.29	−0.316660	−0.158330	+	0.987386i	$0.550611\pi$
$860$	0	0
$861$	−67718.5	−2.68042
$862$	0	0
$863$	−10304.2	−0.406440	−0.203220	−	0.979133i	$-0.565141\pi$
$863$	−10304.2	−0.406440	−0.203220	+	0.979133i	$0.565141\pi$
$864$	0	0
$865$	−8750.34	−0.343954
$866$	0	0
$867$	23275.7	0.911747
$868$	0	0
$869$	39623.2	1.54675
$870$	0	0
$871$	20614.3	0.801939
$872$	0	0
$873$	−14169.6	−0.549334
$874$	0	0
$875$	−3771.25	−0.145705
$876$	0	0
$877$	−6900.23	−0.265683	−0.132842	−	0.991137i	$-0.542410\pi$
$877$	−6900.23	−0.265683	−0.132842	+	0.991137i	$0.542410\pi$
$878$	0	0
$879$	37216.5	1.42808
$880$	0	0
$881$	−28477.9	−1.08904	−0.544521	−	0.838747i	$-0.683288\pi$
$881$	−28477.9	−1.08904	−0.544521	+	0.838747i	$0.683288\pi$
$882$	0	0
$883$	−16775.0	−0.639326	−0.319663	−	0.947531i	$-0.603570\pi$
$883$	−16775.0	−0.639326	−0.319663	+	0.947531i	$0.603570\pi$
$884$	0	0
$885$	−15261.2	−0.579661
$886$	0	0
$887$	20332.1	0.769657	0.384829	−	0.922988i	$-0.374261\pi$
$887$	20332.1	0.769657	0.384829	+	0.922988i	$0.374261\pi$
$888$	0	0
$889$	−45911.4	−1.73208
$890$	0	0
$891$	17354.3	0.652513
$892$	0	0
$893$	−9204.67	−0.344930
$894$	0	0
$895$	−1079.75	−0.0403265
$896$	0	0
$897$	−26638.9	−0.991579
$898$	0	0
$899$	−4147.56	−0.153870
$900$	0	0
$901$	−31413.1	−1.16151
$902$	0	0
$903$	95642.6	3.52468
$904$	0	0
$905$	−1845.05	−0.0677698
$906$	0	0
$907$	33419.1	1.22344	0.611722	−	0.791073i	$-0.290477\pi$
$907$	33419.1	1.22344	0.611722	+	0.791073i	$0.290477\pi$
$908$	0	0
$909$	48658.3	1.77546
$910$	0	0
$911$	9108.80	0.331271	0.165636	−	0.986187i	$-0.447032\pi$
$911$	9108.80	0.331271	0.165636	+	0.986187i	$0.447032\pi$
$912$	0	0
$913$	36680.0	1.32961
$914$	0	0
$915$	8220.10	0.296993
$916$	0	0
$917$	−9632.09	−0.346870
$918$	0	0
$919$	26246.3	0.942095	0.471047	−	0.882108i	$-0.343876\pi$
$919$	26246.3	0.942095	0.471047	+	0.882108i	$0.343876\pi$
$920$	0	0
$921$	22319.4	0.798534
$922$	0	0
$923$	−96.7746	−0.00345111
$924$	0	0
$925$	−4457.42	−0.158442
$926$	0	0
$927$	−69756.8	−2.47153
$928$	0	0
$929$	20844.2	0.736143	0.368072	−	0.929797i	$-0.380018\pi$
$929$	20844.2	0.736143	0.368072	+	0.929797i	$0.380018\pi$
$930$	0	0
$931$	−78956.7	−2.77949
$932$	0	0
$933$	−2697.67	−0.0946600
$934$	0	0
$935$	−9218.57	−0.322438
$936$	0	0
$937$	−19289.8	−0.672540	−0.336270	−	0.941766i	$-0.609166\pi$
$937$	−19289.8	−0.672540	−0.336270	+	0.941766i	$0.609166\pi$
$938$	0	0
$939$	54275.0	1.88626
$940$	0	0
$941$	10216.4	0.353926	0.176963	−	0.984217i	$-0.443373\pi$
$941$	10216.4	0.353926	0.176963	+	0.984217i	$0.443373\pi$
$942$	0	0
$943$	46604.0	1.60937
$944$	0	0
$945$	10399.0	0.357968
$946$	0	0
$947$	−28349.2	−0.972783	−0.486391	−	0.873741i	$-0.661687\pi$
$947$	−28349.2	−0.972783	−0.486391	+	0.873741i	$0.661687\pi$
$948$	0	0
$949$	5317.06	0.181875
$950$	0	0
$951$	51556.6	1.75798
$952$	0	0
$953$	−7660.89	−0.260399	−0.130200	−	0.991488i	$-0.541562\pi$
$953$	−7660.89	−0.260399	−0.130200	+	0.991488i	$0.541562\pi$
$954$	0	0
$955$	14905.0	0.505043
$956$	0	0
$957$	−9530.34	−0.321914
$958$	0	0
$959$	−50686.4	−1.70672
$960$	0	0
$961$	−9336.44	−0.313398
$962$	0	0
$963$	63319.9	2.11885
$964$	0	0
$965$	7109.00	0.237147
$966$	0	0
$967$	46016.5	1.53029	0.765145	−	0.643858i	$-0.222667\pi$
$967$	46016.5	1.53029	0.765145	+	0.643858i	$0.222667\pi$
$968$	0	0
$969$	−48968.6	−1.62343
$970$	0	0
$971$	−41993.2	−1.38787	−0.693937	−	0.720036i	$-0.744125\pi$
$971$	−41993.2	−1.38787	−0.693937	+	0.720036i	$0.744125\pi$
$972$	0	0
$973$	−17136.1	−0.564601
$974$	0	0
$975$	−4050.53	−0.133047
$976$	0	0
$977$	8008.89	0.262259	0.131130	−	0.991365i	$-0.458140\pi$
$977$	8008.89	0.262259	0.131130	+	0.991365i	$0.458140\pi$
$978$	0	0
$979$	−52150.2	−1.70248
$980$	0	0
$981$	−60893.9	−1.98185
$982$	0	0
$983$	−36234.8	−1.17570	−0.587849	−	0.808971i	$-0.700025\pi$
$983$	−36234.8	−1.17570	−0.587849	+	0.808971i	$0.700025\pi$
$984$	0	0
$985$	−4333.27	−0.140172
$986$	0	0
$987$	−15798.2	−0.509485
$988$	0	0
$989$	−65821.4	−2.11628
$990$	0	0
$991$	−28132.9	−0.901786	−0.450893	−	0.892578i	$-0.648894\pi$
$991$	−28132.9	−0.901786	−0.450893	+	0.892578i	$0.648894\pi$
$992$	0	0
$993$	24827.9	0.793443
$994$	0	0
$995$	14682.0	0.467788
$996$	0	0
$997$	−33.2275	−0.00105549	−0.000527746	−	1.00000i	$-0.500168\pi$
$997$	−33.2275	−0.00105549	−0.000527746		1.00000i	$0.500168\pi$
$998$	0	0
$999$	12291.1	0.389262

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.c.1.1	✓	7
4.3	odd	2		2320.4.a.r.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.c.1.1	✓	7	1.1	even	1	trivial
2320.4.a.r.1.7		7	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1160.4.a.c.1.1

Level	$1160$
Weight	$4$
Character	1160.1
Self dual	yes
Analytic conductor	$68.442$
Analytic rank	$1$
Dimension	$7$
CM	no
Inner twists	$1$