LMFDB - Embedded newform 1380.4.a.e.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1380,4,Mod(1,1380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1380.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1380.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,-15,0,-25] 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:	$81.4226358079$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 278x^{3} + 216x^{2} + 14064x - 33408 $ x^5 - x^4 - 278x^3 + 216x^2 + 14064*x - 33408
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.62011$ of defining polynomial
Character	$\chi$	$=$	1380.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -5.00000 q^{5} -10.8260 q^{7} +9.00000 q^{9} -54.7975 q^{11} -37.0769 q^{13} +15.0000 q^{15} -84.5816 q^{17} -101.690 q^{19} +32.4780 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -92.1437 q^{29} -70.5081 q^{31} +164.393 q^{33} +54.1299 q^{35} -35.7561 q^{37} +111.231 q^{39} +300.733 q^{41} -136.814 q^{43} -45.0000 q^{45} -371.796 q^{47} -225.798 q^{49} +253.745 q^{51} +42.8401 q^{53} +273.988 q^{55} +305.069 q^{57} +154.584 q^{59} -159.820 q^{61} -97.4339 q^{63} +185.385 q^{65} +16.1624 q^{67} -69.0000 q^{69} -864.131 q^{71} -730.329 q^{73} -75.0000 q^{75} +593.237 q^{77} -246.416 q^{79} +81.0000 q^{81} +779.092 q^{83} +422.908 q^{85} +276.431 q^{87} -532.933 q^{89} +401.394 q^{91} +211.524 q^{93} +508.449 q^{95} +760.277 q^{97} -493.178 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 15 q^{3} - 25 q^{5} - 7 q^{7} + 45 q^{9} + 64 q^{11} + 80 q^{13} + 75 q^{15} - 21 q^{17} - 52 q^{19} + 21 q^{21} + 115 q^{23} + 125 q^{25} - 135 q^{27} - 277 q^{29} - 289 q^{31} - 192 q^{33} + 35 q^{35}+ \cdots + 576 q^{99}+O(q^{100}) $ 5 * q - 15 * q^3 - 25 * q^5 - 7 * q^7 + 45 * q^9 + 64 * q^11 + 80 * q^13 + 75 * q^15 - 21 * q^17 - 52 * q^19 + 21 * q^21 + 115 * q^23 + 125 * q^25 - 135 * q^27 - 277 * q^29 - 289 * q^31 - 192 * q^33 + 35 * q^35 - 303 * q^37 - 240 * q^39 - 95 * q^41 - 372 * q^43 - 225 * q^45 - 462 * q^47 - 272 * q^49 + 63 * q^51 - 73 * q^53 - 320 * q^55 + 156 * q^57 + 529 * q^59 + 174 * q^61 - 63 * q^63 - 400 * q^65 - 919 * q^67 - 345 * q^69 - 141 * q^71 + 448 * q^73 - 375 * q^75 - 136 * q^77 - 496 * q^79 + 405 * q^81 + 1219 * q^83 + 105 * q^85 + 831 * q^87 + 1492 * q^89 + 1560 * q^91 + 867 * q^93 + 260 * q^95 + 574 * q^97 + 576 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−10.8260	−0.584548	−0.292274	−	0.956335i	$-0.594412\pi$
$7$	−10.8260	−0.584548	−0.292274	+	0.956335i	$0.594412\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−54.7975	−1.50201	−0.751004	−	0.660298i	$-0.770430\pi$
$11$	−54.7975	−1.50201	−0.751004	+	0.660298i	$0.770430\pi$
$12$	0	0
$13$	−37.0769	−0.791022	−0.395511	−	0.918461i	$-0.629432\pi$
$13$	−37.0769	−0.791022	−0.395511	+	0.918461i	$0.629432\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−84.5816	−1.20671	−0.603355	−	0.797473i	$-0.706170\pi$
$17$	−84.5816	−1.20671	−0.603355	+	0.797473i	$0.706170\pi$
$18$	0	0
$19$	−101.690	−1.22785	−0.613927	−	0.789363i	$-0.710411\pi$
$19$	−101.690	−1.22785	−0.613927	+	0.789363i	$0.710411\pi$
$20$	0	0
$21$	32.4780	0.337489
$22$	0	0
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−92.1437	−0.590023	−0.295011	−	0.955494i	$-0.595323\pi$
$29$	−92.1437	−0.590023	−0.295011	+	0.955494i	$0.595323\pi$
$30$	0	0
$31$	−70.5081	−0.408504	−0.204252	−	0.978918i	$-0.565476\pi$
$31$	−70.5081	−0.408504	−0.204252	+	0.978918i	$0.565476\pi$
$32$	0	0
$33$	164.393	0.867184
$34$	0	0
$35$	54.1299	0.261418
$36$	0	0
$37$	−35.7561	−0.158872	−0.0794361	−	0.996840i	$-0.525312\pi$
$37$	−35.7561	−0.158872	−0.0794361	+	0.996840i	$0.525312\pi$
$38$	0	0
$39$	111.231	0.456697
$40$	0	0
$41$	300.733	1.14553	0.572764	−	0.819720i	$-0.305871\pi$
$41$	300.733	1.14553	0.572764	+	0.819720i	$0.305871\pi$
$42$	0	0
$43$	−136.814	−0.485208	−0.242604	−	0.970125i	$-0.578001\pi$
$43$	−136.814	−0.485208	−0.242604	+	0.970125i	$0.578001\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−371.796	−1.15387	−0.576937	−	0.816789i	$-0.695752\pi$
$47$	−371.796	−1.15387	−0.576937	+	0.816789i	$0.695752\pi$
$48$	0	0
$49$	−225.798	−0.658303
$50$	0	0
$51$	253.745	0.696694
$52$	0	0
$53$	42.8401	0.111029	0.0555146	−	0.998458i	$-0.482320\pi$
$53$	42.8401	0.111029	0.0555146	+	0.998458i	$0.482320\pi$
$54$	0	0
$55$	273.988	0.671718
$56$	0	0
$57$	305.069	0.708902
$58$	0	0
$59$	154.584	0.341104	0.170552	−	0.985349i	$-0.445445\pi$
$59$	154.584	0.341104	0.170552	+	0.985349i	$0.445445\pi$
$60$	0	0
$61$	−159.820	−0.335456	−0.167728	−	0.985833i	$-0.553643\pi$
$61$	−159.820	−0.335456	−0.167728	+	0.985833i	$0.553643\pi$
$62$	0	0
$63$	−97.4339	−0.194849
$64$	0	0
$65$	185.385	0.353756
$66$	0	0
$67$	16.1624	0.0294709	0.0147354	−	0.999891i	$-0.495309\pi$
$67$	16.1624	0.0294709	0.0147354	+	0.999891i	$0.495309\pi$
$68$	0	0
$69$	−69.0000	−0.120386
$70$	0	0
$71$	−864.131	−1.44442	−0.722208	−	0.691676i	$-0.756872\pi$
$71$	−864.131	−1.44442	−0.722208	+	0.691676i	$0.756872\pi$
$72$	0	0
$73$	−730.329	−1.17094	−0.585469	−	0.810695i	$-0.699090\pi$
$73$	−730.329	−1.17094	−0.585469	+	0.810695i	$0.699090\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	593.237	0.877996
$78$	0	0
$79$	−246.416	−0.350936	−0.175468	−	0.984485i	$-0.556144\pi$
$79$	−246.416	−0.350936	−0.175468	+	0.984485i	$0.556144\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	779.092	1.03032	0.515159	−	0.857095i	$-0.327733\pi$
$83$	779.092	1.03032	0.515159	+	0.857095i	$0.327733\pi$
$84$	0	0
$85$	422.908	0.539657
$86$	0	0
$87$	276.431	0.340650
$88$	0	0
$89$	−532.933	−0.634727	−0.317364	−	0.948304i	$-0.602798\pi$
$89$	−532.933	−0.634727	−0.317364	+	0.948304i	$0.602798\pi$
$90$	0	0
$91$	401.394	0.462390
$92$	0	0
$93$	211.524	0.235850
$94$	0	0
$95$	508.449	0.549113
$96$	0	0
$97$	760.277	0.795819	0.397909	−	0.917425i	$-0.369736\pi$
$97$	760.277	0.795819	0.397909	+	0.917425i	$0.369736\pi$
$98$	0	0
$99$	−493.178	−0.500669
$100$	0	0
$101$	434.376	0.427941	0.213971	−	0.976840i	$-0.431360\pi$
$101$	434.376	0.427941	0.213971	+	0.976840i	$0.431360\pi$
$102$	0	0
$103$	−1517.38	−1.45157	−0.725784	−	0.687923i	$-0.758523\pi$
$103$	−1517.38	−1.45157	−0.725784	+	0.687923i	$0.758523\pi$
$104$	0	0
$105$	−162.390	−0.150930
$106$	0	0
$107$	18.5570	0.0167661	0.00838307	−	0.999965i	$-0.497332\pi$
$107$	18.5570	0.0167661	0.00838307	+	0.999965i	$0.497332\pi$
$108$	0	0
$109$	1638.74	1.44003	0.720013	−	0.693960i	$-0.244136\pi$
$109$	1638.74	1.44003	0.720013	+	0.693960i	$0.244136\pi$
$110$	0	0
$111$	107.268	0.0917250
$112$	0	0
$113$	−1459.93	−1.21538	−0.607691	−	0.794173i	$-0.707904\pi$
$113$	−1459.93	−1.21538	−0.607691	+	0.794173i	$0.707904\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	−333.692	−0.263674
$118$	0	0
$119$	915.679	0.705380
$120$	0	0
$121$	1671.77	1.25603
$122$	0	0
$123$	−902.200	−0.661371
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−916.657	−0.640474	−0.320237	−	0.947337i	$-0.603763\pi$
$127$	−916.657	−0.640474	−0.320237	+	0.947337i	$0.603763\pi$
$128$	0	0
$129$	410.442	0.280135
$130$	0	0
$131$	1962.33	1.30878	0.654389	−	0.756158i	$-0.272926\pi$
$131$	1962.33	1.30878	0.654389	+	0.756158i	$0.272926\pi$
$132$	0	0
$133$	1100.89	0.717740
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	653.968	0.407827	0.203913	−	0.978989i	$-0.434634\pi$
$137$	653.968	0.407827	0.203913	+	0.978989i	$0.434634\pi$
$138$	0	0
$139$	989.239	0.603641	0.301821	−	0.953365i	$-0.402406\pi$
$139$	989.239	0.603641	0.301821	+	0.953365i	$0.402406\pi$
$140$	0	0
$141$	1115.39	0.666189
$142$	0	0
$143$	2031.72	1.18812
$144$	0	0
$145$	460.718	0.263866
$146$	0	0
$147$	677.394	0.380072
$148$	0	0
$149$	3011.34	1.65570	0.827848	−	0.560952i	$-0.189565\pi$
$149$	3011.34	1.65570	0.827848	+	0.560952i	$0.189565\pi$
$150$	0	0
$151$	−1773.97	−0.956049	−0.478024	−	0.878347i	$-0.658647\pi$
$151$	−1773.97	−0.956049	−0.478024	+	0.878347i	$0.658647\pi$
$152$	0	0
$153$	−761.235	−0.402236
$154$	0	0
$155$	352.540	0.182689
$156$	0	0
$157$	1666.19	0.846982	0.423491	−	0.905900i	$-0.360804\pi$
$157$	1666.19	0.846982	0.423491	+	0.905900i	$0.360804\pi$
$158$	0	0
$159$	−128.520	−0.0641027
$160$	0	0
$161$	−248.998	−0.121887
$162$	0	0
$163$	600.652	0.288630	0.144315	−	0.989532i	$-0.453902\pi$
$163$	600.652	0.288630	0.144315	+	0.989532i	$0.453902\pi$
$164$	0	0
$165$	−821.963	−0.387817
$166$	0	0
$167$	−545.020	−0.252544	−0.126272	−	0.991996i	$-0.540301\pi$
$167$	−545.020	−0.252544	−0.126272	+	0.991996i	$0.540301\pi$
$168$	0	0
$169$	−822.303	−0.374284
$170$	0	0
$171$	−915.208	−0.409285
$172$	0	0
$173$	506.110	0.222421	0.111210	−	0.993797i	$-0.464527\pi$
$173$	506.110	0.222421	0.111210	+	0.993797i	$0.464527\pi$
$174$	0	0
$175$	−270.650	−0.116910
$176$	0	0
$177$	−463.753	−0.196937
$178$	0	0
$179$	−4583.58	−1.91393	−0.956963	−	0.290211i	$-0.906275\pi$
$179$	−4583.58	−1.91393	−0.956963	+	0.290211i	$0.906275\pi$
$180$	0	0
$181$	2077.09	0.852975	0.426488	−	0.904493i	$-0.359751\pi$
$181$	2077.09	0.852975	0.426488	+	0.904493i	$0.359751\pi$
$182$	0	0
$183$	479.459	0.193675
$184$	0	0
$185$	178.781	0.0710498
$186$	0	0
$187$	4634.87	1.81249
$188$	0	0
$189$	292.302	0.112496
$190$	0	0
$191$	4036.84	1.52930	0.764648	−	0.644448i	$-0.222913\pi$
$191$	4036.84	1.52930	0.764648	+	0.644448i	$0.222913\pi$
$192$	0	0
$193$	−1543.62	−0.575710	−0.287855	−	0.957674i	$-0.592942\pi$
$193$	−1543.62	−0.575710	−0.287855	+	0.957674i	$0.592942\pi$
$194$	0	0
$195$	−556.154	−0.204241
$196$	0	0
$197$	885.497	0.320249	0.160125	−	0.987097i	$-0.448810\pi$
$197$	885.497	0.320249	0.160125	+	0.987097i	$0.448810\pi$
$198$	0	0
$199$	1438.25	0.512336	0.256168	−	0.966632i	$-0.417540\pi$
$199$	1438.25	0.512336	0.256168	+	0.966632i	$0.417540\pi$
$200$	0	0
$201$	−48.4871	−0.0170150
$202$	0	0
$203$	997.546	0.344897
$204$	0	0
$205$	−1503.67	−0.512296
$206$	0	0
$207$	207.000	0.0695048
$208$	0	0
$209$	5572.35	1.84425
$210$	0	0
$211$	−1671.94	−0.545504	−0.272752	−	0.962084i	$-0.587934\pi$
$211$	−1671.94	−0.545504	−0.272752	+	0.962084i	$0.587934\pi$
$212$	0	0
$213$	2592.39	0.833933
$214$	0	0
$215$	684.069	0.216991
$216$	0	0
$217$	763.319	0.238790
$218$	0	0
$219$	2190.99	0.676042
$220$	0	0
$221$	3136.02	0.954533
$222$	0	0
$223$	2522.92	0.757610	0.378805	−	0.925477i	$-0.376335\pi$
$223$	2522.92	0.757610	0.378805	+	0.925477i	$0.376335\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3712.52	−1.08550	−0.542750	−	0.839895i	$-0.682617\pi$
$227$	−3712.52	−1.08550	−0.542750	+	0.839895i	$0.682617\pi$
$228$	0	0
$229$	2762.74	0.797236	0.398618	−	0.917117i	$-0.369490\pi$
$229$	2762.74	0.797236	0.398618	+	0.917117i	$0.369490\pi$
$230$	0	0
$231$	−1779.71	−0.506911
$232$	0	0
$233$	−1722.35	−0.484269	−0.242134	−	0.970243i	$-0.577847\pi$
$233$	−1722.35	−0.484269	−0.242134	+	0.970243i	$0.577847\pi$
$234$	0	0
$235$	1858.98	0.516028
$236$	0	0
$237$	739.248	0.202613
$238$	0	0
$239$	165.745	0.0448584	0.0224292	−	0.999748i	$-0.492860\pi$
$239$	165.745	0.0448584	0.0224292	+	0.999748i	$0.492860\pi$
$240$	0	0
$241$	−3006.51	−0.803595	−0.401797	−	0.915729i	$-0.631614\pi$
$241$	−3006.51	−0.803595	−0.401797	+	0.915729i	$0.631614\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1128.99	0.294402
$246$	0	0
$247$	3770.34	0.971259
$248$	0	0
$249$	−2337.27	−0.594854
$250$	0	0
$251$	2030.25	0.510550	0.255275	−	0.966868i	$-0.417834\pi$
$251$	2030.25	0.510550	0.255275	+	0.966868i	$0.417834\pi$
$252$	0	0
$253$	−1260.34	−0.313190
$254$	0	0
$255$	−1268.72	−0.311571
$256$	0	0
$257$	−5844.20	−1.41849	−0.709244	−	0.704963i	$-0.750964\pi$
$257$	−5844.20	−1.41849	−0.709244	+	0.704963i	$0.750964\pi$
$258$	0	0
$259$	387.096	0.0928685
$260$	0	0
$261$	−829.293	−0.196674
$262$	0	0
$263$	6082.65	1.42613	0.713066	−	0.701097i	$-0.247306\pi$
$263$	6082.65	1.42613	0.713066	+	0.701097i	$0.247306\pi$
$264$	0	0
$265$	−214.201	−0.0496538
$266$	0	0
$267$	1598.80	0.366460
$268$	0	0
$269$	4553.74	1.03214	0.516071	−	0.856546i	$-0.327394\pi$
$269$	4553.74	1.03214	0.516071	+	0.856546i	$0.327394\pi$
$270$	0	0
$271$	−5783.03	−1.29629	−0.648144	−	0.761518i	$-0.724454\pi$
$271$	−5783.03	−1.29629	−0.648144	+	0.761518i	$0.724454\pi$
$272$	0	0
$273$	−1204.18	−0.266961
$274$	0	0
$275$	−1369.94	−0.300401
$276$	0	0
$277$	−8349.27	−1.81104	−0.905522	−	0.424299i	$-0.860520\pi$
$277$	−8349.27	−1.81104	−0.905522	+	0.424299i	$0.860520\pi$
$278$	0	0
$279$	−634.573	−0.136168
$280$	0	0
$281$	7536.07	1.59987	0.799936	−	0.600085i	$-0.204867\pi$
$281$	7536.07	1.59987	0.799936	+	0.600085i	$0.204867\pi$
$282$	0	0
$283$	−1036.12	−0.217636	−0.108818	−	0.994062i	$-0.534707\pi$
$283$	−1036.12	−0.217636	−0.108818	+	0.994062i	$0.534707\pi$
$284$	0	0
$285$	−1525.35	−0.317031
$286$	0	0
$287$	−3255.73	−0.669616
$288$	0	0
$289$	2241.05	0.456147
$290$	0	0
$291$	−2280.83	−0.459466
$292$	0	0
$293$	7166.52	1.42892	0.714458	−	0.699678i	$-0.246673\pi$
$293$	7166.52	1.42892	0.714458	+	0.699678i	$0.246673\pi$
$294$	0	0
$295$	−772.921	−0.152546
$296$	0	0
$297$	1479.53	0.289061
$298$	0	0
$299$	−852.769	−0.164939
$300$	0	0
$301$	1481.15	0.283627
$302$	0	0
$303$	−1303.13	−0.247072
$304$	0	0
$305$	799.098	0.150020
$306$	0	0
$307$	−4234.85	−0.787283	−0.393642	−	0.919264i	$-0.628785\pi$
$307$	−4234.85	−0.787283	−0.393642	+	0.919264i	$0.628785\pi$
$308$	0	0
$309$	4552.13	0.838063
$310$	0	0
$311$	2688.79	0.490249	0.245125	−	0.969492i	$-0.421171\pi$
$311$	2688.79	0.490249	0.245125	+	0.969492i	$0.421171\pi$
$312$	0	0
$313$	4677.80	0.844744	0.422372	−	0.906422i	$-0.361198\pi$
$313$	4677.80	0.844744	0.422372	+	0.906422i	$0.361198\pi$
$314$	0	0
$315$	487.169	0.0871393
$316$	0	0
$317$	−7045.39	−1.24829	−0.624146	−	0.781308i	$-0.714553\pi$
$317$	−7045.39	−1.24829	−0.624146	+	0.781308i	$0.714553\pi$
$318$	0	0
$319$	5049.25	0.886218
$320$	0	0
$321$	−55.6711	−0.00967994
$322$	0	0
$323$	8601.08	1.48166
$324$	0	0
$325$	−926.923	−0.158204
$326$	0	0
$327$	−4916.22	−0.831400
$328$	0	0
$329$	4025.06	0.674495
$330$	0	0
$331$	−6047.17	−1.00418	−0.502088	−	0.864816i	$-0.667435\pi$
$331$	−6047.17	−1.00418	−0.502088	+	0.864816i	$0.667435\pi$
$332$	0	0
$333$	−321.805	−0.0529574
$334$	0	0
$335$	−80.8119	−0.0131798
$336$	0	0
$337$	−8033.01	−1.29847	−0.649237	−	0.760586i	$-0.724912\pi$
$337$	−8033.01	−1.29847	−0.649237	+	0.760586i	$0.724912\pi$
$338$	0	0
$339$	4379.78	0.701701
$340$	0	0
$341$	3863.67	0.613576
$342$	0	0
$343$	6157.80	0.969358
$344$	0	0
$345$	345.000	0.0538382
$346$	0	0
$347$	7188.45	1.11209	0.556046	−	0.831151i	$-0.312318\pi$
$347$	7188.45	1.11209	0.556046	+	0.831151i	$0.312318\pi$
$348$	0	0
$349$	5053.84	0.775146	0.387573	−	0.921839i	$-0.373313\pi$
$349$	5053.84	0.775146	0.387573	+	0.921839i	$0.373313\pi$
$350$	0	0
$351$	1001.08	0.152232
$352$	0	0
$353$	−1769.27	−0.266767	−0.133384	−	0.991064i	$-0.542584\pi$
$353$	−1769.27	−0.266767	−0.133384	+	0.991064i	$0.542584\pi$
$354$	0	0
$355$	4320.65	0.645962
$356$	0	0
$357$	−2747.04	−0.407251
$358$	0	0
$359$	3031.16	0.445623	0.222812	−	0.974862i	$-0.428477\pi$
$359$	3031.16	0.445623	0.222812	+	0.974862i	$0.428477\pi$
$360$	0	0
$361$	3481.80	0.507625
$362$	0	0
$363$	−5015.31	−0.725167
$364$	0	0
$365$	3651.64	0.523660
$366$	0	0
$367$	−2139.70	−0.304336	−0.152168	−	0.988355i	$-0.548626\pi$
$367$	−2139.70	−0.304336	−0.152168	+	0.988355i	$0.548626\pi$
$368$	0	0
$369$	2706.60	0.381843
$370$	0	0
$371$	−463.787	−0.0649019
$372$	0	0
$373$	−7854.92	−1.09038	−0.545191	−	0.838312i	$-0.683543\pi$
$373$	−7854.92	−1.09038	−0.545191	+	0.838312i	$0.683543\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	3416.40	0.466721
$378$	0	0
$379$	3708.77	0.502656	0.251328	−	0.967902i	$-0.419133\pi$
$379$	3708.77	0.502656	0.251328	+	0.967902i	$0.419133\pi$
$380$	0	0
$381$	2749.97	0.369778
$382$	0	0
$383$	−9945.03	−1.32681	−0.663404	−	0.748262i	$-0.730889\pi$
$383$	−9945.03	−1.32681	−0.663404	+	0.748262i	$0.730889\pi$
$384$	0	0
$385$	−2966.19	−0.392652
$386$	0	0
$387$	−1231.33	−0.161736
$388$	0	0
$389$	1933.85	0.252057	0.126029	−	0.992027i	$-0.459777\pi$
$389$	1933.85	0.252057	0.126029	+	0.992027i	$0.459777\pi$
$390$	0	0
$391$	−1945.38	−0.251616
$392$	0	0
$393$	−5887.00	−0.755623
$394$	0	0
$395$	1232.08	0.156943
$396$	0	0
$397$	10794.1	1.36459	0.682293	−	0.731079i	$-0.260983\pi$
$397$	10794.1	1.36459	0.682293	+	0.731079i	$0.260983\pi$
$398$	0	0
$399$	−3302.67	−0.414387
$400$	0	0
$401$	1632.94	0.203355	0.101677	−	0.994817i	$-0.467579\pi$
$401$	1632.94	0.203355	0.101677	+	0.994817i	$0.467579\pi$
$402$	0	0
$403$	2614.22	0.323136
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	1959.35	0.238627
$408$	0	0
$409$	−2315.45	−0.279931	−0.139965	−	0.990156i	$-0.544699\pi$
$409$	−2315.45	−0.279931	−0.139965	+	0.990156i	$0.544699\pi$
$410$	0	0
$411$	−1961.90	−0.235459
$412$	0	0
$413$	−1673.53	−0.199392
$414$	0	0
$415$	−3895.46	−0.460772
$416$	0	0
$417$	−2967.72	−0.348512
$418$	0	0
$419$	11098.3	1.29401	0.647005	−	0.762486i	$-0.276021\pi$
$419$	11098.3	1.29401	0.647005	+	0.762486i	$0.276021\pi$
$420$	0	0
$421$	−1339.59	−0.155077	−0.0775386	−	0.996989i	$-0.524706\pi$
$421$	−1339.59	−0.155077	−0.0775386	+	0.996989i	$0.524706\pi$
$422$	0	0
$423$	−3346.17	−0.384625
$424$	0	0
$425$	−2114.54	−0.241342
$426$	0	0
$427$	1730.20	0.196090
$428$	0	0
$429$	−6095.17	−0.685962
$430$	0	0
$431$	−11640.5	−1.30094	−0.650468	−	0.759534i	$-0.725427\pi$
$431$	−11640.5	−1.30094	−0.650468	+	0.759534i	$0.725427\pi$
$432$	0	0
$433$	265.807	0.0295009	0.0147504	−	0.999891i	$-0.495305\pi$
$433$	265.807	0.0295009	0.0147504	+	0.999891i	$0.495305\pi$
$434$	0	0
$435$	−1382.16	−0.152343
$436$	0	0
$437$	−2338.86	−0.256025
$438$	0	0
$439$	−13319.8	−1.44810	−0.724052	−	0.689745i	$-0.757722\pi$
$439$	−13319.8	−1.44810	−0.724052	+	0.689745i	$0.757722\pi$
$440$	0	0
$441$	−2032.18	−0.219434
$442$	0	0
$443$	−7380.48	−0.791551	−0.395776	−	0.918347i	$-0.629524\pi$
$443$	−7380.48	−0.791551	−0.395776	+	0.918347i	$0.629524\pi$
$444$	0	0
$445$	2664.66	0.283859
$446$	0	0
$447$	−9034.03	−0.955917
$448$	0	0
$449$	−2149.35	−0.225911	−0.112956	−	0.993600i	$-0.536032\pi$
$449$	−2149.35	−0.225911	−0.112956	+	0.993600i	$0.536032\pi$
$450$	0	0
$451$	−16479.4	−1.72059
$452$	0	0
$453$	5321.90	0.551975
$454$	0	0
$455$	−2006.97	−0.206787
$456$	0	0
$457$	−2729.18	−0.279356	−0.139678	−	0.990197i	$-0.544607\pi$
$457$	−2729.18	−0.279356	−0.139678	+	0.990197i	$0.544607\pi$
$458$	0	0
$459$	2283.70	0.232231
$460$	0	0
$461$	−9157.59	−0.925187	−0.462594	−	0.886570i	$-0.653081\pi$
$461$	−9157.59	−0.925187	−0.462594	+	0.886570i	$0.653081\pi$
$462$	0	0
$463$	−6156.84	−0.617997	−0.308999	−	0.951063i	$-0.599994\pi$
$463$	−6156.84	−0.617997	−0.308999	+	0.951063i	$0.599994\pi$
$464$	0	0
$465$	−1057.62	−0.105475
$466$	0	0
$467$	−11199.5	−1.10974	−0.554871	−	0.831936i	$-0.687232\pi$
$467$	−11199.5	−1.10974	−0.554871	+	0.831936i	$0.687232\pi$
$468$	0	0
$469$	−174.974	−0.0172271
$470$	0	0
$471$	−4998.56	−0.489005
$472$	0	0
$473$	7497.07	0.728785
$474$	0	0
$475$	−2542.24	−0.245571
$476$	0	0
$477$	385.561	0.0370097
$478$	0	0
$479$	−5515.01	−0.526069	−0.263035	−	0.964786i	$-0.584723\pi$
$479$	−5515.01	−0.526069	−0.263035	+	0.964786i	$0.584723\pi$
$480$	0	0
$481$	1325.73	0.125671
$482$	0	0
$483$	746.993	0.0703713
$484$	0	0
$485$	−3801.38	−0.355901
$486$	0	0
$487$	−16725.2	−1.55624	−0.778122	−	0.628113i	$-0.783828\pi$
$487$	−16725.2	−1.55624	−0.778122	+	0.628113i	$0.783828\pi$
$488$	0	0
$489$	−1801.96	−0.166641
$490$	0	0
$491$	−3275.48	−0.301060	−0.150530	−	0.988605i	$-0.548098\pi$
$491$	−3275.48	−0.301060	−0.150530	+	0.988605i	$0.548098\pi$
$492$	0	0
$493$	7793.66	0.711986
$494$	0	0
$495$	2465.89	0.223906
$496$	0	0
$497$	9355.07	0.844330
$498$	0	0
$499$	−9605.61	−0.861736	−0.430868	−	0.902415i	$-0.641792\pi$
$499$	−9605.61	−0.861736	−0.430868	+	0.902415i	$0.641792\pi$
$500$	0	0
$501$	1635.06	0.145807
$502$	0	0
$503$	−15666.1	−1.38871	−0.694353	−	0.719635i	$-0.744309\pi$
$503$	−15666.1	−1.38871	−0.694353	+	0.719635i	$0.744309\pi$
$504$	0	0
$505$	−2171.88	−0.191381
$506$	0	0
$507$	2466.91	0.216093
$508$	0	0
$509$	8018.31	0.698243	0.349121	−	0.937078i	$-0.386480\pi$
$509$	8018.31	0.698243	0.349121	+	0.937078i	$0.386480\pi$
$510$	0	0
$511$	7906.53	0.684470
$512$	0	0
$513$	2745.62	0.236301
$514$	0	0
$515$	7586.88	0.649161
$516$	0	0
$517$	20373.5	1.73313
$518$	0	0
$519$	−1518.33	−0.128415
$520$	0	0
$521$	18656.8	1.56885	0.784425	−	0.620223i	$-0.212958\pi$
$521$	18656.8	1.56885	0.784425	+	0.620223i	$0.212958\pi$
$522$	0	0
$523$	−5707.10	−0.477158	−0.238579	−	0.971123i	$-0.576682\pi$
$523$	−5707.10	−0.477158	−0.238579	+	0.971123i	$0.576682\pi$
$524$	0	0
$525$	811.949	0.0674978
$526$	0	0
$527$	5963.69	0.492945
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	1391.26	0.113701
$532$	0	0
$533$	−11150.3	−0.906138
$534$	0	0
$535$	−92.7852	−0.00749805
$536$	0	0
$537$	13750.7	1.10501
$538$	0	0
$539$	12373.2	0.988777
$540$	0	0
$541$	−16964.5	−1.34817	−0.674085	−	0.738653i	$-0.735462\pi$
$541$	−16964.5	−1.34817	−0.674085	+	0.738653i	$0.735462\pi$
$542$	0	0
$543$	−6231.26	−0.492466
$544$	0	0
$545$	−8193.71	−0.644000
$546$	0	0
$547$	−16634.9	−1.30029	−0.650143	−	0.759812i	$-0.725291\pi$
$547$	−16634.9	−1.30029	−0.650143	+	0.759812i	$0.725291\pi$
$548$	0	0
$549$	−1438.38	−0.111819
$550$	0	0
$551$	9370.07	0.724461
$552$	0	0
$553$	2667.69	0.205139
$554$	0	0
$555$	−536.342	−0.0410206
$556$	0	0
$557$	−269.537	−0.0205039	−0.0102519	−	0.999947i	$-0.503263\pi$
$557$	−269.537	−0.0205039	−0.0102519	+	0.999947i	$0.503263\pi$
$558$	0	0
$559$	5072.64	0.383810
$560$	0	0
$561$	−13904.6	−1.04644
$562$	0	0
$563$	−1897.23	−0.142022	−0.0710111	−	0.997476i	$-0.522623\pi$
$563$	−1897.23	−0.142022	−0.0710111	+	0.997476i	$0.522623\pi$
$564$	0	0
$565$	7299.63	0.543536
$566$	0	0
$567$	−876.905	−0.0649498
$568$	0	0
$569$	23754.5	1.75016	0.875078	−	0.483982i	$-0.160810\pi$
$569$	23754.5	1.75016	0.875078	+	0.483982i	$0.160810\pi$
$570$	0	0
$571$	−23381.2	−1.71361	−0.856807	−	0.515638i	$-0.827555\pi$
$571$	−23381.2	−1.71361	−0.856807	+	0.515638i	$0.827555\pi$
$572$	0	0
$573$	−12110.5	−0.882940
$574$	0	0
$575$	575.000	0.0417029
$576$	0	0
$577$	−4514.83	−0.325745	−0.162873	−	0.986647i	$-0.552076\pi$
$577$	−4514.83	−0.325745	−0.162873	+	0.986647i	$0.552076\pi$
$578$	0	0
$579$	4630.85	0.332386
$580$	0	0
$581$	−8434.43	−0.602271
$582$	0	0
$583$	−2347.54	−0.166767
$584$	0	0
$585$	1668.46	0.117919
$586$	0	0
$587$	16906.6	1.18877	0.594387	−	0.804179i	$-0.297395\pi$
$587$	16906.6	1.18877	0.594387	+	0.804179i	$0.297395\pi$
$588$	0	0
$589$	7169.95	0.501583
$590$	0	0
$591$	−2656.49	−0.184896
$592$	0	0
$593$	−7881.32	−0.545779	−0.272890	−	0.962045i	$-0.587979\pi$
$593$	−7881.32	−0.545779	−0.272890	+	0.962045i	$0.587979\pi$
$594$	0	0
$595$	−4578.40	−0.315455
$596$	0	0
$597$	−4314.75	−0.295797
$598$	0	0
$599$	−19024.0	−1.29766	−0.648830	−	0.760933i	$-0.724741\pi$
$599$	−19024.0	−1.29766	−0.648830	+	0.760933i	$0.724741\pi$
$600$	0	0
$601$	24227.3	1.64435	0.822175	−	0.569235i	$-0.192761\pi$
$601$	24227.3	1.64435	0.822175	+	0.569235i	$0.192761\pi$
$602$	0	0
$603$	145.461	0.00982362
$604$	0	0
$605$	−8358.86	−0.561712
$606$	0	0
$607$	−18274.4	−1.22197	−0.610983	−	0.791644i	$-0.709226\pi$
$607$	−18274.4	−1.22197	−0.610983	+	0.791644i	$0.709226\pi$
$608$	0	0
$609$	−2992.64	−0.199126
$610$	0	0
$611$	13785.1	0.912739
$612$	0	0
$613$	23647.1	1.55807	0.779036	−	0.626980i	$-0.215709\pi$
$613$	23647.1	1.55807	0.779036	+	0.626980i	$0.215709\pi$
$614$	0	0
$615$	4511.00	0.295774
$616$	0	0
$617$	1060.90	0.0692223	0.0346112	−	0.999401i	$-0.488981\pi$
$617$	1060.90	0.0692223	0.0346112	+	0.999401i	$0.488981\pi$
$618$	0	0
$619$	14526.8	0.943264	0.471632	−	0.881796i	$-0.343665\pi$
$619$	14526.8	0.943264	0.471632	+	0.881796i	$0.343665\pi$
$620$	0	0
$621$	−621.000	−0.0401286
$622$	0	0
$623$	5769.52	0.371029
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−16717.0	−1.06478
$628$	0	0
$629$	3024.31	0.191713
$630$	0	0
$631$	−6601.21	−0.416466	−0.208233	−	0.978079i	$-0.566771\pi$
$631$	−6601.21	−0.416466	−0.208233	+	0.978079i	$0.566771\pi$
$632$	0	0
$633$	5015.83	0.314947
$634$	0	0
$635$	4583.29	0.286429
$636$	0	0
$637$	8371.89	0.520732
$638$	0	0
$639$	−7777.18	−0.481472
$640$	0	0
$641$	8276.87	0.510010	0.255005	−	0.966940i	$-0.417923\pi$
$641$	8276.87	0.510010	0.255005	+	0.966940i	$0.417923\pi$
$642$	0	0
$643$	4673.00	0.286602	0.143301	−	0.989679i	$-0.454228\pi$
$643$	4673.00	0.286602	0.143301	+	0.989679i	$0.454228\pi$
$644$	0	0
$645$	−2052.21	−0.125280
$646$	0	0
$647$	−17958.0	−1.09120	−0.545598	−	0.838047i	$-0.683697\pi$
$647$	−17958.0	−1.09120	−0.545598	+	0.838047i	$0.683697\pi$
$648$	0	0
$649$	−8470.83	−0.512341
$650$	0	0
$651$	−2289.96	−0.137866
$652$	0	0
$653$	−15325.2	−0.918409	−0.459204	−	0.888331i	$-0.651865\pi$
$653$	−15325.2	−0.918409	−0.459204	+	0.888331i	$0.651865\pi$
$654$	0	0
$655$	−9811.67	−0.585303
$656$	0	0
$657$	−6572.96	−0.390313
$658$	0	0
$659$	22421.9	1.32539	0.662695	−	0.748889i	$-0.269412\pi$
$659$	22421.9	1.32539	0.662695	+	0.748889i	$0.269412\pi$
$660$	0	0
$661$	−10921.3	−0.642647	−0.321323	−	0.946970i	$-0.604128\pi$
$661$	−10921.3	−0.642647	−0.321323	+	0.946970i	$0.604128\pi$
$662$	0	0
$663$	−9408.07	−0.551100
$664$	0	0
$665$	−5504.46	−0.320983
$666$	0	0
$667$	−2119.30	−0.123028
$668$	0	0
$669$	−7568.75	−0.437406
$670$	0	0
$671$	8757.72	0.503857
$672$	0	0
$673$	8617.37	0.493574	0.246787	−	0.969070i	$-0.420625\pi$
$673$	8617.37	0.493574	0.246787	+	0.969070i	$0.420625\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	11919.7	0.676679	0.338340	−	0.941024i	$-0.390135\pi$
$677$	11919.7	0.676679	0.338340	+	0.941024i	$0.390135\pi$
$678$	0	0
$679$	−8230.75	−0.465194
$680$	0	0
$681$	11137.5	0.626713
$682$	0	0
$683$	−6568.28	−0.367977	−0.183988	−	0.982928i	$-0.558901\pi$
$683$	−6568.28	−0.367977	−0.183988	+	0.982928i	$0.558901\pi$
$684$	0	0
$685$	−3269.84	−0.182386
$686$	0	0
$687$	−8288.23	−0.460285
$688$	0	0
$689$	−1588.38	−0.0878265
$690$	0	0
$691$	−19000.6	−1.04605	−0.523023	−	0.852319i	$-0.675196\pi$
$691$	−19000.6	−1.04605	−0.523023	+	0.852319i	$0.675196\pi$
$692$	0	0
$693$	5339.14	0.292665
$694$	0	0
$695$	−4946.19	−0.269957
$696$	0	0
$697$	−25436.5	−1.38232
$698$	0	0
$699$	5167.04	0.279593
$700$	0	0
$701$	−9809.23	−0.528516	−0.264258	−	0.964452i	$-0.585127\pi$
$701$	−9809.23	−0.528516	−0.264258	+	0.964452i	$0.585127\pi$
$702$	0	0
$703$	3636.03	0.195072
$704$	0	0
$705$	−5576.94	−0.297929
$706$	0	0
$707$	−4702.55	−0.250152
$708$	0	0
$709$	−10403.1	−0.551051	−0.275525	−	0.961294i	$-0.588852\pi$
$709$	−10403.1	−0.551051	−0.275525	+	0.961294i	$0.588852\pi$
$710$	0	0
$711$	−2217.74	−0.116979
$712$	0	0
$713$	−1621.69	−0.0851790
$714$	0	0
$715$	−10158.6	−0.531344
$716$	0	0
$717$	−497.235	−0.0258990
$718$	0	0
$719$	4317.97	0.223968	0.111984	−	0.993710i	$-0.464279\pi$
$719$	4317.97	0.223968	0.111984	+	0.993710i	$0.464279\pi$
$720$	0	0
$721$	16427.1	0.848512
$722$	0	0
$723$	9019.53	0.463956
$724$	0	0
$725$	−2303.59	−0.118005
$726$	0	0
$727$	−18018.3	−0.919204	−0.459602	−	0.888125i	$-0.652008\pi$
$727$	−18018.3	−0.919204	−0.459602	+	0.888125i	$0.652008\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	11571.9	0.585504
$732$	0	0
$733$	−16659.4	−0.839466	−0.419733	−	0.907648i	$-0.637876\pi$
$733$	−16659.4	−0.839466	−0.419733	+	0.907648i	$0.637876\pi$
$734$	0	0
$735$	−3386.97	−0.169973
$736$	0	0
$737$	−885.658	−0.0442655
$738$	0	0
$739$	8311.01	0.413702	0.206851	−	0.978372i	$-0.433679\pi$
$739$	8311.01	0.413702	0.206851	+	0.978372i	$0.433679\pi$
$740$	0	0
$741$	−11311.0	−0.560757
$742$	0	0
$743$	24197.8	1.19479	0.597396	−	0.801947i	$-0.296202\pi$
$743$	24197.8	1.19479	0.597396	+	0.801947i	$0.296202\pi$
$744$	0	0
$745$	−15056.7	−0.740450
$746$	0	0
$747$	7011.82	0.343439
$748$	0	0
$749$	−200.898	−0.00980062
$750$	0	0
$751$	−26263.9	−1.27614	−0.638071	−	0.769978i	$-0.720267\pi$
$751$	−26263.9	−1.27614	−0.638071	+	0.769978i	$0.720267\pi$
$752$	0	0
$753$	−6090.74	−0.294766
$754$	0	0
$755$	8869.83	0.427558
$756$	0	0
$757$	−21151.2	−1.01553	−0.507763	−	0.861497i	$-0.669528\pi$
$757$	−21151.2	−1.01553	−0.507763	+	0.861497i	$0.669528\pi$
$758$	0	0
$759$	3781.03	0.180820
$760$	0	0
$761$	−10850.2	−0.516846	−0.258423	−	0.966032i	$-0.583203\pi$
$761$	−10850.2	−0.516846	−0.258423	+	0.966032i	$0.583203\pi$
$762$	0	0
$763$	−17741.0	−0.841765
$764$	0	0
$765$	3806.17	0.179886
$766$	0	0
$767$	−5731.50	−0.269821
$768$	0	0
$769$	−18880.2	−0.885355	−0.442677	−	0.896681i	$-0.645971\pi$
$769$	−18880.2	−0.885355	−0.442677	+	0.896681i	$0.645971\pi$
$770$	0	0
$771$	17532.6	0.818964
$772$	0	0
$773$	27164.8	1.26397	0.631985	−	0.774981i	$-0.282240\pi$
$773$	27164.8	1.26397	0.631985	+	0.774981i	$0.282240\pi$
$774$	0	0
$775$	−1762.70	−0.0817008
$776$	0	0
$777$	−1161.29	−0.0536177
$778$	0	0
$779$	−30581.5	−1.40654
$780$	0	0
$781$	47352.3	2.16952
$782$	0	0
$783$	2487.88	0.113550
$784$	0	0
$785$	−8330.94	−0.378782
$786$	0	0
$787$	−40172.2	−1.81955	−0.909775	−	0.415102i	$-0.863746\pi$
$787$	−40172.2	−1.81955	−0.909775	+	0.415102i	$0.863746\pi$
$788$	0	0
$789$	−18248.0	−0.823377
$790$	0	0
$791$	15805.1	0.710450
$792$	0	0
$793$	5925.62	0.265353
$794$	0	0
$795$	642.602	0.0286676
$796$	0	0
$797$	−8815.68	−0.391804	−0.195902	−	0.980624i	$-0.562763\pi$
$797$	−8815.68	−0.391804	−0.195902	+	0.980624i	$0.562763\pi$
$798$	0	0
$799$	31447.1	1.39239
$800$	0	0
$801$	−4796.39	−0.211576
$802$	0	0
$803$	40020.2	1.75876
$804$	0	0
$805$	1244.99	0.0545094
$806$	0	0
$807$	−13661.2	−0.595908
$808$	0	0
$809$	9213.85	0.400422	0.200211	−	0.979753i	$-0.435837\pi$
$809$	9213.85	0.400422	0.200211	+	0.979753i	$0.435837\pi$
$810$	0	0
$811$	4586.15	0.198572	0.0992859	−	0.995059i	$-0.468344\pi$
$811$	4586.15	0.198572	0.0992859	+	0.995059i	$0.468344\pi$
$812$	0	0
$813$	17349.1	0.748412
$814$	0	0
$815$	−3003.26	−0.129079
$816$	0	0
$817$	13912.6	0.595764
$818$	0	0
$819$	3612.55	0.154130
$820$	0	0
$821$	−32687.8	−1.38954	−0.694770	−	0.719232i	$-0.744494\pi$
$821$	−32687.8	−1.38954	−0.694770	+	0.719232i	$0.744494\pi$
$822$	0	0
$823$	20504.4	0.868453	0.434226	−	0.900804i	$-0.357022\pi$
$823$	20504.4	0.868453	0.434226	+	0.900804i	$0.357022\pi$
$824$	0	0
$825$	4109.82	0.173437
$826$	0	0
$827$	−4684.68	−0.196980	−0.0984900	−	0.995138i	$-0.531401\pi$
$827$	−4684.68	−0.196980	−0.0984900	+	0.995138i	$0.531401\pi$
$828$	0	0
$829$	−9147.26	−0.383230	−0.191615	−	0.981470i	$-0.561372\pi$
$829$	−9147.26	−0.383230	−0.191615	+	0.981470i	$0.561372\pi$
$830$	0	0
$831$	25047.8	1.04561
$832$	0	0
$833$	19098.4	0.794381
$834$	0	0
$835$	2725.10	0.112941
$836$	0	0
$837$	1903.72	0.0786166
$838$	0	0
$839$	14493.4	0.596387	0.298194	−	0.954505i	$-0.403616\pi$
$839$	14493.4	0.596387	0.298194	+	0.954505i	$0.403616\pi$
$840$	0	0
$841$	−15898.5	−0.651873
$842$	0	0
$843$	−22608.2	−0.923687
$844$	0	0
$845$	4111.51	0.167385
$846$	0	0
$847$	−18098.6	−0.734208
$848$	0	0
$849$	3108.36	0.125652
$850$	0	0
$851$	−822.391	−0.0331272
$852$	0	0
$853$	3353.81	0.134622	0.0673109	−	0.997732i	$-0.478558\pi$
$853$	3353.81	0.134622	0.0673109	+	0.997732i	$0.478558\pi$
$854$	0	0
$855$	4576.04	0.183038
$856$	0	0
$857$	−21865.0	−0.871522	−0.435761	−	0.900062i	$-0.643521\pi$
$857$	−21865.0	−0.871522	−0.435761	+	0.900062i	$0.643521\pi$
$858$	0	0
$859$	18902.3	0.750801	0.375400	−	0.926863i	$-0.377505\pi$
$859$	18902.3	0.750801	0.375400	+	0.926863i	$0.377505\pi$
$860$	0	0
$861$	9767.20	0.386603
$862$	0	0
$863$	7703.41	0.303855	0.151928	−	0.988392i	$-0.451452\pi$
$863$	7703.41	0.303855	0.151928	+	0.988392i	$0.451452\pi$
$864$	0	0
$865$	−2530.55	−0.0994697
$866$	0	0
$867$	−6723.15	−0.263357
$868$	0	0
$869$	13503.0	0.527109
$870$	0	0
$871$	−599.251	−0.0233121
$872$	0	0
$873$	6842.49	0.265273
$874$	0	0
$875$	1353.25	0.0522836
$876$	0	0
$877$	−34873.8	−1.34277	−0.671383	−	0.741111i	$-0.734299\pi$
$877$	−34873.8	−1.34277	−0.671383	+	0.741111i	$0.734299\pi$
$878$	0	0
$879$	−21499.6	−0.824986
$880$	0	0
$881$	30721.2	1.17483	0.587413	−	0.809287i	$-0.300146\pi$
$881$	30721.2	1.17483	0.587413	+	0.809287i	$0.300146\pi$
$882$	0	0
$883$	−9266.14	−0.353149	−0.176575	−	0.984287i	$-0.556502\pi$
$883$	−9266.14	−0.353149	−0.176575	+	0.984287i	$0.556502\pi$
$884$	0	0
$885$	2318.76	0.0880727
$886$	0	0
$887$	−517.512	−0.0195900	−0.00979501	−	0.999952i	$-0.503118\pi$
$887$	−517.512	−0.0195900	−0.00979501	+	0.999952i	$0.503118\pi$
$888$	0	0
$889$	9923.72	0.374388
$890$	0	0
$891$	−4438.60	−0.166890
$892$	0	0
$893$	37807.9	1.41679
$894$	0	0
$895$	22917.9	0.855934
$896$	0	0
$897$	2558.31	0.0952278
$898$	0	0
$899$	6496.87	0.241027
$900$	0	0
$901$	−3623.49	−0.133980
$902$	0	0
$903$	−4443.44	−0.163752
$904$	0	0
$905$	−10385.4	−0.381462
$906$	0	0
$907$	−47279.6	−1.73086	−0.865432	−	0.501026i	$-0.832956\pi$
$907$	−47279.6	−1.73086	−0.865432	+	0.501026i	$0.832956\pi$
$908$	0	0
$909$	3909.39	0.142647
$910$	0	0
$911$	24167.4	0.878925	0.439463	−	0.898261i	$-0.355169\pi$
$911$	24167.4	0.878925	0.439463	+	0.898261i	$0.355169\pi$
$912$	0	0
$913$	−42692.3	−1.54755
$914$	0	0
$915$	−2397.29	−0.0866143
$916$	0	0
$917$	−21244.2	−0.765044
$918$	0	0
$919$	−30359.3	−1.08973	−0.544864	−	0.838524i	$-0.683419\pi$
$919$	−30359.3	−1.08973	−0.544864	+	0.838524i	$0.683419\pi$
$920$	0	0
$921$	12704.6	0.454538
$922$	0	0
$923$	32039.3	1.14256
$924$	0	0
$925$	−893.904	−0.0317745
$926$	0	0
$927$	−13656.4	−0.483856
$928$	0	0
$929$	41703.4	1.47281	0.736407	−	0.676538i	$-0.236521\pi$
$929$	41703.4	1.47281	0.736407	+	0.676538i	$0.236521\pi$
$930$	0	0
$931$	22961.3	0.808300
$932$	0	0
$933$	−8066.38	−0.283046
$934$	0	0
$935$	−23174.3	−0.810569
$936$	0	0
$937$	−10355.8	−0.361056	−0.180528	−	0.983570i	$-0.557781\pi$
$937$	−10355.8	−0.361056	−0.180528	+	0.983570i	$0.557781\pi$
$938$	0	0
$939$	−14033.4	−0.487713
$940$	0	0
$941$	−7478.39	−0.259074	−0.129537	−	0.991575i	$-0.541349\pi$
$941$	−7478.39	−0.259074	−0.129537	+	0.991575i	$0.541349\pi$
$942$	0	0
$943$	6916.87	0.238859
$944$	0	0
$945$	−1461.51	−0.0503099
$946$	0	0
$947$	18102.2	0.621165	0.310582	−	0.950546i	$-0.399476\pi$
$947$	18102.2	0.621165	0.310582	+	0.950546i	$0.399476\pi$
$948$	0	0
$949$	27078.3	0.926238
$950$	0	0
$951$	21136.2	0.720702
$952$	0	0
$953$	22335.3	0.759193	0.379597	−	0.925152i	$-0.376063\pi$
$953$	22335.3	0.759193	0.379597	+	0.925152i	$0.376063\pi$
$954$	0	0
$955$	−20184.2	−0.683922
$956$	0	0
$957$	−15147.7	−0.511658
$958$	0	0
$959$	−7079.85	−0.238394
$960$	0	0
$961$	−24819.6	−0.833125
$962$	0	0
$963$	167.013	0.00558871
$964$	0	0
$965$	7718.09	0.257465
$966$	0	0
$967$	−9714.97	−0.323074	−0.161537	−	0.986867i	$-0.551645\pi$
$967$	−9714.97	−0.323074	−0.161537	+	0.986867i	$0.551645\pi$
$968$	0	0
$969$	−25803.2	−0.855438
$970$	0	0
$971$	−4699.40	−0.155315	−0.0776575	−	0.996980i	$-0.524744\pi$
$971$	−4699.40	−0.155315	−0.0776575	+	0.996980i	$0.524744\pi$
$972$	0	0
$973$	−10709.5	−0.352857
$974$	0	0
$975$	2780.77	0.0913393
$976$	0	0
$977$	−29581.8	−0.968686	−0.484343	−	0.874878i	$-0.660941\pi$
$977$	−29581.8	−0.968686	−0.484343	+	0.874878i	$0.660941\pi$
$978$	0	0
$979$	29203.4	0.953365
$980$	0	0
$981$	14748.7	0.480009
$982$	0	0
$983$	2416.69	0.0784135	0.0392067	−	0.999231i	$-0.487517\pi$
$983$	2416.69	0.0784135	0.0392067	+	0.999231i	$0.487517\pi$
$984$	0	0
$985$	−4427.49	−0.143220
$986$	0	0
$987$	−12075.2	−0.389420
$988$	0	0
$989$	−3146.72	−0.101173
$990$	0	0
$991$	17124.7	0.548923	0.274461	−	0.961598i	$-0.411500\pi$
$991$	17124.7	0.548923	0.274461	+	0.961598i	$0.411500\pi$
$992$	0	0
$993$	18141.5	0.579762
$994$	0	0
$995$	−7191.25	−0.229124
$996$	0	0
$997$	41385.0	1.31462	0.657311	−	0.753620i	$-0.271694\pi$
$997$	41385.0	1.31462	0.657311	+	0.753620i	$0.271694\pi$
$998$	0	0
$999$	965.416	0.0305750

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.4.a.e.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.4.a.e.1.2	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -5.00000 q^{5} -10.8260 q^{7} +9.00000 q^{9} -54.7975 q^{11} -37.0769 q^{13} +15.0000 q^{15} -84.5816 q^{17} -101.690 q^{19} +32.4780 q^{21} +23.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -92.1437 q^{29} -70.5081 q^{31} +164.393 q^{33} +54.1299 q^{35} -35.7561 q^{37} +111.231 q^{39} +300.733 q^{41} -136.814 q^{43} -45.0000 q^{45} -371.796 q^{47} -225.798 q^{49} +253.745 q^{51} +42.8401 q^{53} +273.988 q^{55} +305.069 q^{57} +154.584 q^{59} -159.820 q^{61} -97.4339 q^{63} +185.385 q^{65} +16.1624 q^{67} -69.0000 q^{69} -864.131 q^{71} -730.329 q^{73} -75.0000 q^{75} +593.237 q^{77} -246.416 q^{79} +81.0000 q^{81} +779.092 q^{83} +422.908 q^{85} +276.431 q^{87} -532.933 q^{89} +401.394 q^{91} +211.524 q^{93} +508.449 q^{95} +760.277 q^{97} -493.178 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 15 q^{3} - 25 q^{5} - 7 q^{7} + 45 q^{9} + 64 q^{11} + 80 q^{13} + 75 q^{15} - 21 q^{17} - 52 q^{19} + 21 q^{21} + 115 q^{23} + 125 q^{25} - 135 q^{27} - 277 q^{29} - 289 q^{31} - 192 q^{33} + 35 q^{35}+ \cdots + 576 q^{99}+O(q^{100}) \) 5 * q - 15 * q^3 - 25 * q^5 - 7 * q^7 + 45 * q^9 + 64 * q^11 + 80 * q^13 + 75 * q^15 - 21 * q^17 - 52 * q^19 + 21 * q^21 + 115 * q^23 + 125 * q^25 - 135 * q^27 - 277 * q^29 - 289 * q^31 - 192 * q^33 + 35 * q^35 - 303 * q^37 - 240 * q^39 - 95 * q^41 - 372 * q^43 - 225 * q^45 - 462 * q^47 - 272 * q^49 + 63 * q^51 - 73 * q^53 - 320 * q^55 + 156 * q^57 + 529 * q^59 + 174 * q^61 - 63 * q^63 - 400 * q^65 - 919 * q^67 - 345 * q^69 - 141 * q^71 + 448 * q^73 - 375 * q^75 - 136 * q^77 - 496 * q^79 + 405 * q^81 + 1219 * q^83 + 105 * q^85 + 831 * q^87 + 1492 * q^89 + 1560 * q^91 + 867 * q^93 + 260 * q^95 + 574 * q^97 + 576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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