LMFDB - Embedded newform 2151.4.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.32883 q^{2} +10.7387 q^{4} -4.62088 q^{5} +32.0311 q^{7} -11.8555 q^{8} +O(q^{10})$	\(q-4.32883 q^{2} +10.7387 q^{4} -4.62088 q^{5} +32.0311 q^{7} -11.8555 q^{8} +20.0030 q^{10} +7.45205 q^{11} +58.3261 q^{13} -138.657 q^{14} -34.5895 q^{16} -66.5398 q^{17} -121.974 q^{19} -49.6224 q^{20} -32.2586 q^{22} -161.202 q^{23} -103.647 q^{25} -252.483 q^{26} +343.974 q^{28} +33.6881 q^{29} -86.2107 q^{31} +244.576 q^{32} +288.039 q^{34} -148.012 q^{35} +122.943 q^{37} +528.004 q^{38} +54.7828 q^{40} -158.318 q^{41} +429.033 q^{43} +80.0255 q^{44} +697.814 q^{46} -400.858 q^{47} +682.994 q^{49} +448.672 q^{50} +626.348 q^{52} +663.917 q^{53} -34.4350 q^{55} -379.745 q^{56} -145.830 q^{58} -609.482 q^{59} +425.713 q^{61} +373.191 q^{62} -782.010 q^{64} -269.518 q^{65} +473.045 q^{67} -714.553 q^{68} +640.719 q^{70} +120.578 q^{71} -120.106 q^{73} -532.200 q^{74} -1309.85 q^{76} +238.698 q^{77} -1341.42 q^{79} +159.834 q^{80} +685.331 q^{82} +830.189 q^{83} +307.473 q^{85} -1857.21 q^{86} -88.3477 q^{88} +188.281 q^{89} +1868.25 q^{91} -1731.10 q^{92} +1735.25 q^{94} +563.627 q^{95} +1690.20 q^{97} -2956.56 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * 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 $

$2$	−4.32883	−1.53047	−0.765235	−	0.643750i	$-0.777377\pi$
$2$	−4.32883	−1.53047	−0.765235	+	0.643750i	$0.777377\pi$
$3$	0	0
$3$	0	0
$4$	10.7387	1.34234
$5$	−4.62088	−0.413304	−0.206652	−	0.978414i	$-0.566257\pi$
$5$	−4.62088	−0.413304	−0.206652	+	0.978414i	$0.566257\pi$
$6$	0	0
$7$	32.0311	1.72952	0.864759	−	0.502186i	$-0.167471\pi$
$7$	32.0311	1.72952	0.864759	+	0.502186i	$0.167471\pi$
$8$	−11.8555	−0.523944
$9$	0	0
$10$	20.0030	0.632550
$11$	7.45205	0.204262	0.102131	−	0.994771i	$-0.467434\pi$
$11$	7.45205	0.204262	0.102131	+	0.994771i	$0.467434\pi$
$12$	0	0
$13$	58.3261	1.24436	0.622182	−	0.782872i	$-0.286246\pi$
$13$	58.3261	1.24436	0.622182	+	0.782872i	$0.286246\pi$
$14$	−138.657	−2.64698
$15$	0	0
$16$	−34.5895	−0.540461
$17$	−66.5398	−0.949310	−0.474655	−	0.880172i	$-0.657427\pi$
$17$	−66.5398	−0.949310	−0.474655	+	0.880172i	$0.657427\pi$
$18$	0	0
$19$	−121.974	−1.47278	−0.736388	−	0.676559i	$-0.763470\pi$
$19$	−121.974	−1.47278	−0.736388	+	0.676559i	$0.763470\pi$
$20$	−49.6224	−0.554795
$21$	0	0
$22$	−32.2586	−0.312616
$23$	−161.202	−1.46143	−0.730715	−	0.682683i	$-0.760813\pi$
$23$	−161.202	−1.46143	−0.730715	+	0.682683i	$0.760813\pi$
$24$	0	0
$25$	−103.647	−0.829180
$26$	−252.483	−1.90446
$27$	0	0
$28$	343.974	2.32160
$29$	33.6881	0.215715	0.107857	−	0.994166i	$-0.465601\pi$
$29$	33.6881	0.215715	0.107857	+	0.994166i	$0.465601\pi$
$30$	0	0
$31$	−86.2107	−0.499481	−0.249740	−	0.968313i	$-0.580345\pi$
$31$	−86.2107	−0.499481	−0.249740	+	0.968313i	$0.580345\pi$
$32$	244.576	1.35110
$33$	0	0
$34$	288.039	1.45289
$35$	−148.012	−0.714818
$36$	0	0
$37$	122.943	0.546264	0.273132	−	0.961977i	$-0.411940\pi$
$37$	122.943	0.546264	0.273132	+	0.961977i	$0.411940\pi$
$38$	528.004	2.25404
$39$	0	0
$40$	54.7828	0.216548
$41$	−158.318	−0.603052	−0.301526	−	0.953458i	$-0.597496\pi$
$41$	−158.318	−0.603052	−0.301526	+	0.953458i	$0.597496\pi$
$42$	0	0
$43$	429.033	1.52156	0.760779	−	0.649011i	$-0.224817\pi$
$43$	429.033	1.52156	0.760779	+	0.649011i	$0.224817\pi$
$44$	80.0255	0.274189
$45$	0	0
$46$	697.814	2.23668
$47$	−400.858	−1.24407	−0.622034	−	0.782990i	$-0.713693\pi$
$47$	−400.858	−1.24407	−0.622034	+	0.782990i	$0.713693\pi$
$48$	0	0
$49$	682.994	1.99124
$50$	448.672	1.26904
$51$	0	0
$52$	626.348	1.67036
$53$	663.917	1.72068	0.860340	−	0.509721i	$-0.170251\pi$
$53$	663.917	1.72068	0.860340	+	0.509721i	$0.170251\pi$
$54$	0	0
$55$	−34.4350	−0.0844222
$56$	−379.745	−0.906170
$57$	0	0
$58$	−145.830	−0.330145
$59$	−609.482	−1.34488	−0.672439	−	0.740152i	$-0.734753\pi$
$59$	−609.482	−1.34488	−0.672439	+	0.740152i	$0.734753\pi$
$60$	0	0
$61$	425.713	0.893558	0.446779	−	0.894644i	$-0.352571\pi$
$61$	425.713	0.893558	0.446779	+	0.894644i	$0.352571\pi$
$62$	373.191	0.764441
$63$	0	0
$64$	−782.010	−1.52736
$65$	−269.518	−0.514301
$66$	0	0
$67$	473.045	0.862562	0.431281	−	0.902218i	$-0.358062\pi$
$67$	473.045	0.862562	0.431281	+	0.902218i	$0.358062\pi$
$68$	−714.553	−1.27430
$69$	0	0
$70$	640.719	1.09401
$71$	120.578	0.201549	0.100774	−	0.994909i	$-0.467868\pi$
$71$	120.578	0.201549	0.100774	+	0.994909i	$0.467868\pi$
$72$	0	0
$73$	−120.106	−0.192567	−0.0962834	−	0.995354i	$-0.530696\pi$
$73$	−120.106	−0.192567	−0.0962834	+	0.995354i	$0.530696\pi$
$74$	−532.200	−0.836041
$75$	0	0
$76$	−1309.85	−1.97697
$77$	238.698	0.353274
$78$	0	0
$79$	−1341.42	−1.91039	−0.955197	−	0.295971i	$-0.904357\pi$
$79$	−1341.42	−1.91039	−0.955197	+	0.295971i	$0.904357\pi$
$80$	159.834	0.223375
$81$	0	0
$82$	685.331	0.922953
$83$	830.189	1.09789	0.548946	−	0.835858i	$-0.315029\pi$
$83$	830.189	1.09789	0.548946	+	0.835858i	$0.315029\pi$
$84$	0	0
$85$	307.473	0.392354
$86$	−1857.21	−2.32870
$87$	0	0
$88$	−88.3477	−0.107022
$89$	188.281	0.224245	0.112122	−	0.993694i	$-0.464235\pi$
$89$	188.281	0.224245	0.112122	+	0.993694i	$0.464235\pi$
$90$	0	0
$91$	1868.25	2.15215
$92$	−1731.10	−1.96174
$93$	0	0
$94$	1735.25	1.90401
$95$	563.627	0.608705
$96$	0	0
$97$	1690.20	1.76922	0.884608	−	0.466336i	$-0.154426\pi$
$97$	1690.20	1.76922	0.884608	+	0.466336i	$0.154426\pi$
$98$	−2956.56	−3.04753
$99$	0	0
$100$	−1113.04	−1.11304
$101$	728.720	0.717924	0.358962	−	0.933352i	$-0.383131\pi$
$101$	728.720	0.717924	0.358962	+	0.933352i	$0.383131\pi$
$102$	0	0
$103$	−1683.06	−1.61006	−0.805032	−	0.593231i	$-0.797852\pi$
$103$	−1683.06	−1.61006	−0.805032	+	0.593231i	$0.797852\pi$
$104$	−691.484	−0.651977
$105$	0	0
$106$	−2873.98	−2.63345
$107$	−1294.48	−1.16956	−0.584778	−	0.811193i	$-0.698819\pi$
$107$	−1294.48	−1.16956	−0.584778	+	0.811193i	$0.698819\pi$
$108$	0	0
$109$	1224.01	1.07559	0.537793	−	0.843077i	$-0.319258\pi$
$109$	1224.01	1.07559	0.537793	+	0.843077i	$0.319258\pi$
$110$	149.063	0.129206
$111$	0	0
$112$	−1107.94	−0.934737
$113$	1912.25	1.59194	0.795969	−	0.605337i	$-0.206962\pi$
$113$	1912.25	1.59194	0.795969	+	0.605337i	$0.206962\pi$
$114$	0	0
$115$	744.894	0.604015
$116$	361.768	0.289563
$117$	0	0
$118$	2638.34	2.05830
$119$	−2131.35	−1.64185
$120$	0	0
$121$	−1275.47	−0.958277
$122$	−1842.84	−1.36756
$123$	0	0
$124$	−925.794	−0.670474
$125$	1056.55	0.756008
$126$	0	0
$127$	272.426	0.190345	0.0951727	−	0.995461i	$-0.469660\pi$
$127$	272.426	0.190345	0.0951727	+	0.995461i	$0.469660\pi$
$128$	1428.58	0.986482
$129$	0	0
$130$	1166.70	0.787123
$131$	−96.5191	−0.0643734	−0.0321867	−	0.999482i	$-0.510247\pi$
$131$	−96.5191	−0.0643734	−0.0321867	+	0.999482i	$0.510247\pi$
$132$	0	0
$133$	−3906.97	−2.54719
$134$	−2047.73	−1.32013
$135$	0	0
$136$	788.862	0.497385
$137$	1327.04	0.827568	0.413784	−	0.910375i	$-0.364207\pi$
$137$	1327.04	0.827568	0.413784	+	0.910375i	$0.364207\pi$
$138$	0	0
$139$	−1570.32	−0.958220	−0.479110	−	0.877755i	$-0.659040\pi$
$139$	−1570.32	−0.958220	−0.479110	+	0.877755i	$0.659040\pi$
$140$	−1589.46	−0.959529
$141$	0	0
$142$	−521.961	−0.308465
$143$	434.649	0.254176
$144$	0	0
$145$	−155.669	−0.0891559
$146$	519.919	0.294718
$147$	0	0
$148$	1320.26	0.733273
$149$	1.57479	0.000865851 0	0.000432925	−	1.00000i	$-0.499862\pi$
$149$	1.57479	0.000865851 0	0.000432925		1.00000i	$0.499862\pi$
$150$	0	0
$151$	3064.46	1.65154	0.825768	−	0.564010i	$-0.190742\pi$
$151$	3064.46	1.65154	0.825768	+	0.564010i	$0.190742\pi$
$152$	1446.06	0.771652
$153$	0	0
$154$	−1033.28	−0.540676
$155$	398.370	0.206438
$156$	0	0
$157$	2539.83	1.29109	0.645543	−	0.763724i	$-0.276631\pi$
$157$	2539.83	1.29109	0.645543	+	0.763724i	$0.276631\pi$
$158$	5806.76	2.92380
$159$	0	0
$160$	−1130.16	−0.558417
$161$	−5163.47	−2.52757
$162$	0	0
$163$	2374.04	1.14079	0.570397	−	0.821369i	$-0.306790\pi$
$163$	2374.04	1.14079	0.570397	+	0.821369i	$0.306790\pi$
$164$	−1700.13	−0.809501
$165$	0	0
$166$	−3593.74	−1.68029
$167$	3398.45	1.57473	0.787365	−	0.616487i	$-0.211445\pi$
$167$	3398.45	1.57473	0.787365	+	0.616487i	$0.211445\pi$
$168$	0	0
$169$	1204.93	0.548444
$170$	−1331.00	−0.600486
$171$	0	0
$172$	4607.27	2.04245
$173$	3280.90	1.44186	0.720931	−	0.693007i	$-0.243714\pi$
$173$	3280.90	1.44186	0.720931	+	0.693007i	$0.243714\pi$
$174$	0	0
$175$	−3319.95	−1.43408
$176$	−257.763	−0.110395
$177$	0	0
$178$	−815.036	−0.343200
$179$	−2457.52	−1.02616	−0.513082	−	0.858339i	$-0.671496\pi$
$179$	−2457.52	−1.02616	−0.513082	+	0.858339i	$0.671496\pi$
$180$	0	0
$181$	2503.98	1.02828	0.514142	−	0.857705i	$-0.328110\pi$
$181$	2503.98	1.02828	0.514142	+	0.857705i	$0.328110\pi$
$182$	−8087.33	−3.29381
$183$	0	0
$184$	1911.13	0.765707
$185$	−568.107	−0.225773
$186$	0	0
$187$	−495.858	−0.193908
$188$	−4304.71	−1.66996
$189$	0	0
$190$	−2439.84	−0.931605
$191$	2323.72	0.880305	0.440152	−	0.897923i	$-0.354924\pi$
$191$	2323.72	0.880305	0.440152	+	0.897923i	$0.354924\pi$
$192$	0	0
$193$	−3690.88	−1.37656	−0.688279	−	0.725446i	$-0.741633\pi$
$193$	−3690.88	−1.37656	−0.688279	+	0.725446i	$0.741633\pi$
$194$	−7316.58	−2.70773
$195$	0	0
$196$	7334.49	2.67292
$197$	3068.32	1.10969	0.554844	−	0.831954i	$-0.312778\pi$
$197$	3068.32	1.10969	0.554844	+	0.831954i	$0.312778\pi$
$198$	0	0
$199$	−2289.80	−0.815678	−0.407839	−	0.913054i	$-0.633718\pi$
$199$	−2289.80	−0.815678	−0.407839	+	0.913054i	$0.633718\pi$
$200$	1228.79	0.434443
$201$	0	0
$202$	−3154.50	−1.09876
$203$	1079.07	0.373083
$204$	0	0
$205$	731.569	0.249244
$206$	7285.67	2.46416
$207$	0	0
$208$	−2017.47	−0.672531
$209$	−908.956	−0.300832
$210$	0	0
$211$	−193.560	−0.0631525	−0.0315763	−	0.999501i	$-0.510053\pi$
$211$	−193.560	−0.0631525	−0.0315763	+	0.999501i	$0.510053\pi$
$212$	7129.63	2.30974
$213$	0	0
$214$	5603.60	1.78997
$215$	−1982.51	−0.628866
$216$	0	0
$217$	−2761.43	−0.863861
$218$	−5298.52	−1.64615
$219$	0	0
$220$	−369.789	−0.113323
$221$	−3881.01	−1.18129
$222$	0	0
$223$	−3521.33	−1.05743	−0.528713	−	0.848801i	$-0.677325\pi$
$223$	−3521.33	−1.05743	−0.528713	+	0.848801i	$0.677325\pi$
$224$	7834.04	2.33676
$225$	0	0
$226$	−8277.78	−2.43642
$227$	5071.90	1.48297	0.741485	−	0.670970i	$-0.234122\pi$
$227$	5071.90	1.48297	0.741485	+	0.670970i	$0.234122\pi$
$228$	0	0
$229$	−4004.06	−1.15544	−0.577719	−	0.816236i	$-0.696057\pi$
$229$	−4004.06	−1.15544	−0.577719	+	0.816236i	$0.696057\pi$
$230$	−3224.52	−0.924428
$231$	0	0
$232$	−399.389	−0.113022
$233$	6232.98	1.75252	0.876258	−	0.481843i	$-0.160032\pi$
$233$	6232.98	1.75252	0.876258	+	0.481843i	$0.160032\pi$
$234$	0	0
$235$	1852.32	0.514179
$236$	−6545.07	−1.80529
$237$	0	0
$238$	9226.22	2.51280
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−4671.87	−1.24872	−0.624360	−	0.781137i	$-0.714640\pi$
$241$	−4671.87	−1.24872	−0.624360	+	0.781137i	$0.714640\pi$
$242$	5521.27	1.46662
$243$	0	0
$244$	4571.62	1.19946
$245$	−3156.03	−0.822986
$246$	0	0
$247$	−7114.26	−1.83267
$248$	1022.07	0.261700
$249$	0	0
$250$	−4573.63	−1.15705
$251$	−3088.70	−0.776722	−0.388361	−	0.921507i	$-0.626959\pi$
$251$	−3088.70	−0.776722	−0.388361	+	0.921507i	$0.626959\pi$
$252$	0	0
$253$	−1201.28	−0.298514
$254$	−1179.28	−0.291318
$255$	0	0
$256$	72.0124	0.0175811
$257$	5100.24	1.23791	0.618957	−	0.785425i	$-0.287555\pi$
$257$	5100.24	1.23791	0.618957	+	0.785425i	$0.287555\pi$
$258$	0	0
$259$	3938.02	0.944774
$260$	−2894.28	−0.690368
$261$	0	0
$262$	417.814	0.0985216
$263$	−2708.83	−0.635110	−0.317555	−	0.948240i	$-0.602862\pi$
$263$	−2708.83	−0.635110	−0.317555	+	0.948240i	$0.602862\pi$
$264$	0	0
$265$	−3067.88	−0.711164
$266$	16912.6	3.89841
$267$	0	0
$268$	5079.90	1.15785
$269$	−330.026	−0.0748031	−0.0374015	−	0.999300i	$-0.511908\pi$
$269$	−330.026	−0.0748031	−0.0374015	+	0.999300i	$0.511908\pi$
$270$	0	0
$271$	786.279	0.176247	0.0881237	−	0.996110i	$-0.471913\pi$
$271$	786.279	0.176247	0.0881237	+	0.996110i	$0.471913\pi$
$272$	2301.58	0.513065
$273$	0	0
$274$	−5744.53	−1.26657
$275$	−772.386	−0.169370
$276$	0	0
$277$	3777.12	0.819297	0.409648	−	0.912244i	$-0.365651\pi$
$277$	3777.12	0.819297	0.409648	+	0.912244i	$0.365651\pi$
$278$	6797.63	1.46653
$279$	0	0
$280$	1754.76	0.374524
$281$	5756.83	1.22215	0.611074	−	0.791573i	$-0.290738\pi$
$281$	5756.83	1.22215	0.611074	+	0.791573i	$0.290738\pi$
$282$	0	0
$283$	3205.05	0.673217	0.336609	−	0.941645i	$-0.390720\pi$
$283$	3205.05	0.673217	0.336609	+	0.941645i	$0.390720\pi$
$284$	1294.85	0.270547
$285$	0	0
$286$	−1881.52	−0.389009
$287$	−5071.11	−1.04299
$288$	0	0
$289$	−485.454	−0.0988101
$290$	673.864	0.136450
$291$	0	0
$292$	−1289.79	−0.258490
$293$	−745.526	−0.148649	−0.0743243	−	0.997234i	$-0.523680\pi$
$293$	−745.526	−0.148649	−0.0743243	+	0.997234i	$0.523680\pi$
$294$	0	0
$295$	2816.35	0.555844
$296$	−1457.55	−0.286211
$297$	0	0
$298$	−6.81699	−0.00132516
$299$	−9402.26	−1.81855
$300$	0	0
$301$	13742.4	2.63156
$302$	−13265.5	−2.52763
$303$	0	0
$304$	4219.02	0.795978
$305$	−1967.17	−0.369311
$306$	0	0
$307$	−647.637	−0.120399	−0.0601997	−	0.998186i	$-0.519174\pi$
$307$	−647.637	−0.120399	−0.0601997	+	0.998186i	$0.519174\pi$
$308$	2563.31	0.474215
$309$	0	0
$310$	−1724.47	−0.315947
$311$	3054.17	0.556869	0.278434	−	0.960455i	$-0.410185\pi$
$311$	3054.17	0.556869	0.278434	+	0.960455i	$0.410185\pi$
$312$	0	0
$313$	5846.73	1.05584	0.527918	−	0.849295i	$-0.322973\pi$
$313$	5846.73	1.05584	0.527918	+	0.849295i	$0.322973\pi$
$314$	−10994.5	−1.97597
$315$	0	0
$316$	−14405.1	−2.56440
$317$	−8125.38	−1.43964	−0.719822	−	0.694159i	$-0.755776\pi$
$317$	−8125.38	−1.43964	−0.719822	+	0.694159i	$0.755776\pi$
$318$	0	0
$319$	251.046	0.0440622
$320$	3613.58	0.631266
$321$	0	0
$322$	22351.8	3.86837
$323$	8116.13	1.39812
$324$	0	0
$325$	−6045.35	−1.03180
$326$	−10276.8	−1.74595
$327$	0	0
$328$	1876.94	0.315965
$329$	−12839.9	−2.15164
$330$	0	0
$331$	−4937.67	−0.819936	−0.409968	−	0.912100i	$-0.634460\pi$
$331$	−4937.67	−0.819936	−0.409968	+	0.912100i	$0.634460\pi$
$332$	8915.17	1.47375
$333$	0	0
$334$	−14711.3	−2.41008
$335$	−2185.89	−0.356501
$336$	0	0
$337$	127.949	0.0206820	0.0103410	−	0.999947i	$-0.496708\pi$
$337$	127.949	0.0206820	0.0103410	+	0.999947i	$0.496708\pi$
$338$	−5215.94	−0.839377
$339$	0	0
$340$	3301.87	0.526673
$341$	−642.447	−0.102025
$342$	0	0
$343$	10890.4	1.71436
$344$	−5086.40	−0.797210
$345$	0	0
$346$	−14202.4	−2.20673
$347$	11998.2	1.85619	0.928096	−	0.372342i	$-0.121445\pi$
$347$	11998.2	1.85619	0.928096	+	0.372342i	$0.121445\pi$
$348$	0	0
$349$	7363.95	1.12947	0.564733	−	0.825274i	$-0.308979\pi$
$349$	7363.95	1.12947	0.564733	+	0.825274i	$0.308979\pi$
$350$	14371.5	2.19482
$351$	0	0
$352$	1822.59	0.275978
$353$	−2223.93	−0.335320	−0.167660	−	0.985845i	$-0.553621\pi$
$353$	−2223.93	−0.335320	−0.167660	+	0.985845i	$0.553621\pi$
$354$	0	0
$355$	−557.176	−0.0833010
$356$	2021.90	0.301013
$357$	0	0
$358$	10638.2	1.57052
$359$	−9375.28	−1.37830	−0.689148	−	0.724620i	$-0.742015\pi$
$359$	−9375.28	−1.37830	−0.689148	+	0.724620i	$0.742015\pi$
$360$	0	0
$361$	8018.65	1.16907
$362$	−10839.3	−1.57376
$363$	0	0
$364$	20062.6	2.88892
$365$	554.997	0.0795887
$366$	0	0
$367$	−8068.15	−1.14756	−0.573779	−	0.819010i	$-0.694523\pi$
$367$	−8068.15	−1.14756	−0.573779	+	0.819010i	$0.694523\pi$
$368$	5575.89	0.789846
$369$	0	0
$370$	2459.24	0.345539
$371$	21266.0	2.97595
$372$	0	0
$373$	6797.13	0.943545	0.471773	−	0.881720i	$-0.343614\pi$
$373$	6797.13	0.943545	0.471773	+	0.881720i	$0.343614\pi$
$374$	2146.48	0.296770
$375$	0	0
$376$	4752.37	0.651821
$377$	1964.90	0.268428
$378$	0	0
$379$	−195.503	−0.0264969	−0.0132484	−	0.999912i	$-0.504217\pi$
$379$	−195.503	−0.0264969	−0.0132484	+	0.999912i	$0.504217\pi$
$380$	6052.64	0.817090
$381$	0	0
$382$	−10059.0	−1.34728
$383$	3469.50	0.462881	0.231440	−	0.972849i	$-0.425656\pi$
$383$	3469.50	0.462881	0.231440	+	0.972849i	$0.425656\pi$
$384$	0	0
$385$	−1102.99	−0.146010
$386$	15977.2	2.10678
$387$	0	0
$388$	18150.6	2.37489
$389$	−6676.65	−0.870230	−0.435115	−	0.900375i	$-0.643292\pi$
$389$	−6676.65	−0.870230	−0.435115	+	0.900375i	$0.643292\pi$
$390$	0	0
$391$	10726.3	1.38735
$392$	−8097.22	−1.04329
$393$	0	0
$394$	−13282.2	−1.69835
$395$	6198.53	0.789574
$396$	0	0
$397$	12937.2	1.63552	0.817761	−	0.575558i	$-0.195215\pi$
$397$	12937.2	1.63552	0.817761	+	0.575558i	$0.195215\pi$
$398$	9912.16	1.24837
$399$	0	0
$400$	3585.11	0.448139
$401$	−12980.1	−1.61645	−0.808223	−	0.588877i	$-0.799570\pi$
$401$	−12980.1	−1.61645	−0.808223	+	0.588877i	$0.799570\pi$
$402$	0	0
$403$	−5028.33	−0.621536
$404$	7825.52	0.963699
$405$	0	0
$406$	−4671.10	−0.570992
$407$	916.180	0.111581
$408$	0	0
$409$	−1564.27	−0.189115	−0.0945576	−	0.995519i	$-0.530144\pi$
$409$	−1564.27	−0.189115	−0.0945576	+	0.995519i	$0.530144\pi$
$410$	−3166.83	−0.381461
$411$	0	0
$412$	−18073.9	−2.16126
$413$	−19522.4	−2.32599
$414$	0	0
$415$	−3836.20	−0.453764
$416$	14265.1	1.68127
$417$	0	0
$418$	3934.71	0.460414
$419$	10262.3	1.19653	0.598264	−	0.801299i	$-0.295857\pi$
$419$	10262.3	1.19653	0.598264	+	0.801299i	$0.295857\pi$
$420$	0	0
$421$	−5433.32	−0.628988	−0.314494	−	0.949259i	$-0.601835\pi$
$421$	−5433.32	−0.628988	−0.314494	+	0.949259i	$0.601835\pi$
$422$	837.885	0.0966531
$423$	0	0
$424$	−7871.06	−0.901539
$425$	6896.68	0.787149
$426$	0	0
$427$	13636.1	1.54542
$428$	−13901.1	−1.56994
$429$	0	0
$430$	8581.95	0.962462
$431$	−7961.32	−0.889753	−0.444876	−	0.895592i	$-0.646752\pi$
$431$	−7961.32	−0.889753	−0.444876	+	0.895592i	$0.646752\pi$
$432$	0	0
$433$	−2838.56	−0.315040	−0.157520	−	0.987516i	$-0.550350\pi$
$433$	−2838.56	−0.315040	−0.157520	+	0.987516i	$0.550350\pi$
$434$	11953.7	1.32211
$435$	0	0
$436$	13144.3	1.44380
$437$	19662.4	2.15236
$438$	0	0
$439$	15562.1	1.69189	0.845943	−	0.533273i	$-0.179038\pi$
$439$	15562.1	1.69189	0.845943	+	0.533273i	$0.179038\pi$
$440$	408.244	0.0442325
$441$	0	0
$442$	16800.2	1.80793
$443$	−7667.31	−0.822314	−0.411157	−	0.911565i	$-0.634875\pi$
$443$	−7667.31	−0.822314	−0.411157	+	0.911565i	$0.634875\pi$
$444$	0	0
$445$	−870.025	−0.0926812
$446$	15243.2	1.61836
$447$	0	0
$448$	−25048.7	−2.64160
$449$	13178.9	1.38519	0.692595	−	0.721327i	$-0.256468\pi$
$449$	13178.9	1.38519	0.692595	+	0.721327i	$0.256468\pi$
$450$	0	0
$451$	−1179.79	−0.123180
$452$	20535.1	2.13693
$453$	0	0
$454$	−21955.4	−2.26964
$455$	−8632.97	−0.889494
$456$	0	0
$457$	−8191.84	−0.838508	−0.419254	−	0.907869i	$-0.637708\pi$
$457$	−8191.84	−0.838508	−0.419254	+	0.907869i	$0.637708\pi$
$458$	17332.9	1.76837
$459$	0	0
$460$	7999.22	0.810794
$461$	8439.41	0.852630	0.426315	−	0.904575i	$-0.359811\pi$
$461$	8439.41	0.852630	0.426315	+	0.904575i	$0.359811\pi$
$462$	0	0
$463$	−497.661	−0.0499530	−0.0249765	−	0.999688i	$-0.507951\pi$
$463$	−497.661	−0.0499530	−0.0249765	+	0.999688i	$0.507951\pi$
$464$	−1165.26	−0.116585
$465$	0	0
$466$	−26981.5	−2.68217
$467$	−10057.1	−0.996547	−0.498273	−	0.867020i	$-0.666032\pi$
$467$	−10057.1	−0.996547	−0.498273	+	0.867020i	$0.666032\pi$
$468$	0	0
$469$	15152.2	1.49182
$470$	−8018.37	−0.786935
$471$	0	0
$472$	7225.71	0.704641
$473$	3197.18	0.310796
$474$	0	0
$475$	12642.3	1.22120
$476$	−22887.9	−2.20392
$477$	0	0
$478$	−1034.59	−0.0989979
$479$	17292.8	1.64954	0.824770	−	0.565468i	$-0.191304\pi$
$479$	17292.8	1.64954	0.824770	+	0.565468i	$0.191304\pi$
$480$	0	0
$481$	7170.80	0.679752
$482$	20223.7	1.91113
$483$	0	0
$484$	−13696.9	−1.28634
$485$	−7810.22	−0.731224
$486$	0	0
$487$	−7283.74	−0.677737	−0.338868	−	0.940834i	$-0.610044\pi$
$487$	−7283.74	−0.677737	−0.338868	+	0.940834i	$0.610044\pi$
$488$	−5047.04	−0.468174
$489$	0	0
$490$	13661.9	1.25956
$491$	−2859.11	−0.262790	−0.131395	−	0.991330i	$-0.541946\pi$
$491$	−2859.11	−0.262790	−0.131395	+	0.991330i	$0.541946\pi$
$492$	0	0
$493$	−2241.60	−0.204780
$494$	30796.4	2.80485
$495$	0	0
$496$	2981.99	0.269950
$497$	3862.25	0.348582
$498$	0	0
$499$	−658.061	−0.0590357	−0.0295179	−	0.999564i	$-0.509397\pi$
$499$	−658.061	−0.0590357	−0.0295179	+	0.999564i	$0.509397\pi$
$500$	11346.0	1.01482
$501$	0	0
$502$	13370.5	1.18875
$503$	9770.65	0.866107	0.433054	−	0.901368i	$-0.357436\pi$
$503$	9770.65	0.866107	0.433054	+	0.901368i	$0.357436\pi$
$504$	0	0
$505$	−3367.33	−0.296721
$506$	5200.14	0.456867
$507$	0	0
$508$	2925.51	0.255509
$509$	7338.95	0.639083	0.319542	−	0.947572i	$-0.396471\pi$
$509$	7338.95	0.639083	0.319542	+	0.947572i	$0.396471\pi$
$510$	0	0
$511$	−3847.14	−0.333048
$512$	−11740.4	−1.01339
$513$	0	0
$514$	−22078.0	−1.89459
$515$	7777.21	0.665447
$516$	0	0
$517$	−2987.22	−0.254115
$518$	−17047.0	−1.44595
$519$	0	0
$520$	3195.27	0.269465
$521$	−2299.31	−0.193349	−0.0966744	−	0.995316i	$-0.530821\pi$
$521$	−2299.31	−0.193349	−0.0966744	+	0.995316i	$0.530821\pi$
$522$	0	0
$523$	10415.2	0.870797	0.435399	−	0.900238i	$-0.356607\pi$
$523$	10415.2	0.870797	0.435399	+	0.900238i	$0.356607\pi$
$524$	−1036.49	−0.0864111
$525$	0	0
$526$	11726.1	0.972017
$527$	5736.45	0.474162
$528$	0	0
$529$	13819.0	1.13578
$530$	13280.3	1.08842
$531$	0	0
$532$	−41955.9	−3.41920
$533$	−9234.07	−0.750416
$534$	0	0
$535$	5981.66	0.483383
$536$	−5608.18	−0.451934
$537$	0	0
$538$	1428.62	0.114484
$539$	5089.70	0.406733
$540$	0	0
$541$	4827.54	0.383646	0.191823	−	0.981430i	$-0.438560\pi$
$541$	4827.54	0.383646	0.191823	+	0.981430i	$0.438560\pi$
$542$	−3403.66	−0.269741
$543$	0	0
$544$	−16274.0	−1.28262
$545$	−5656.00	−0.444544
$546$	0	0
$547$	20278.5	1.58509	0.792546	−	0.609812i	$-0.208755\pi$
$547$	20278.5	1.58509	0.792546	+	0.609812i	$0.208755\pi$
$548$	14250.7	1.11088
$549$	0	0
$550$	3343.52	0.259215
$551$	−4109.08	−0.317700
$552$	0	0
$553$	−42967.1	−3.30406
$554$	−16350.5	−1.25391
$555$	0	0
$556$	−16863.2	−1.28626
$557$	18308.9	1.39277	0.696385	−	0.717669i	$-0.254791\pi$
$557$	18308.9	1.39277	0.696385	+	0.717669i	$0.254791\pi$
$558$	0	0
$559$	25023.8	1.89337
$560$	5119.66	0.386331
$561$	0	0
$562$	−24920.3	−1.87046
$563$	15463.8	1.15759	0.578793	−	0.815474i	$-0.303524\pi$
$563$	15463.8	1.15759	0.578793	+	0.815474i	$0.303524\pi$
$564$	0	0
$565$	−8836.27	−0.657955
$566$	−13874.1	−1.03034
$567$	0	0
$568$	−1429.51	−0.105600
$569$	−3325.16	−0.244988	−0.122494	−	0.992469i	$-0.539089\pi$
$569$	−3325.16	−0.244988	−0.122494	+	0.992469i	$0.539089\pi$
$570$	0	0
$571$	−1460.17	−0.107016	−0.0535079	−	0.998567i	$-0.517040\pi$
$571$	−1460.17	−0.107016	−0.0535079	+	0.998567i	$0.517040\pi$
$572$	4667.58	0.341191
$573$	0	0
$574$	21951.9	1.59627
$575$	16708.1	1.21179
$576$	0	0
$577$	−2264.66	−0.163395	−0.0816977	−	0.996657i	$-0.526034\pi$
$577$	−2264.66	−0.163395	−0.0816977	+	0.996657i	$0.526034\pi$
$578$	2101.45	0.151226
$579$	0	0
$580$	−1671.69	−0.119678
$581$	26591.9	1.89883
$582$	0	0
$583$	4947.54	0.351469
$584$	1423.92	0.100894
$585$	0	0
$586$	3227.25	0.227503
$587$	−26552.1	−1.86699	−0.933496	−	0.358588i	$-0.883258\pi$
$587$	−26552.1	−1.86699	−0.933496	+	0.358588i	$0.883258\pi$
$588$	0	0
$589$	10515.5	0.735623
$590$	−12191.5	−0.850703
$591$	0	0
$592$	−4252.55	−0.295234
$593$	18593.8	1.28761	0.643807	−	0.765188i	$-0.277354\pi$
$593$	18593.8	1.28761	0.643807	+	0.765188i	$0.277354\pi$
$594$	0	0
$595$	9848.70	0.678584
$596$	16.9112	0.00116227
$597$	0	0
$598$	40700.8	2.78324
$599$	−6749.95	−0.460427	−0.230213	−	0.973140i	$-0.573942\pi$
$599$	−6749.95	−0.460427	−0.230213	+	0.973140i	$0.573942\pi$
$600$	0	0
$601$	−8090.78	−0.549134	−0.274567	−	0.961568i	$-0.588535\pi$
$601$	−8090.78	−0.549134	−0.274567	+	0.961568i	$0.588535\pi$
$602$	−59488.6	−4.02753
$603$	0	0
$604$	32908.4	2.21693
$605$	5893.78	0.396060
$606$	0	0
$607$	7644.52	0.511172	0.255586	−	0.966786i	$-0.417732\pi$
$607$	7644.52	0.511172	0.255586	+	0.966786i	$0.417732\pi$
$608$	−29831.9	−1.98987
$609$	0	0
$610$	8515.54	0.565220
$611$	−23380.5	−1.54807
$612$	0	0
$613$	−5707.37	−0.376050	−0.188025	−	0.982164i	$-0.560209\pi$
$613$	−5707.37	−0.376050	−0.188025	+	0.982164i	$0.560209\pi$
$614$	2803.51	0.184268
$615$	0	0
$616$	−2829.88	−0.185096
$617$	2714.58	0.177123	0.0885616	−	0.996071i	$-0.471773\pi$
$617$	2714.58	0.177123	0.0885616	+	0.996071i	$0.471773\pi$
$618$	0	0
$619$	−203.146	−0.0131909	−0.00659543	−	0.999978i	$-0.502099\pi$
$619$	−203.146	−0.0131909	−0.00659543	+	0.999978i	$0.502099\pi$
$620$	4277.98	0.277110
$621$	0	0
$622$	−13221.0	−0.852271
$623$	6030.86	0.387835
$624$	0	0
$625$	8073.72	0.516718
$626$	−25309.5	−1.61593
$627$	0	0
$628$	27274.5	1.73308
$629$	−8180.63	−0.518574
$630$	0	0
$631$	−12096.9	−0.763184	−0.381592	−	0.924331i	$-0.624624\pi$
$631$	−12096.9	−0.763184	−0.381592	+	0.924331i	$0.624624\pi$
$632$	15903.1	1.00094
$633$	0	0
$634$	35173.4	2.20333
$635$	−1258.85	−0.0786706
$636$	0	0
$637$	39836.3	2.47782
$638$	−1086.73	−0.0674360
$639$	0	0
$640$	−6601.29	−0.407717
$641$	−13270.5	−0.817712	−0.408856	−	0.912599i	$-0.634072\pi$
$641$	−13270.5	−0.817712	−0.408856	+	0.912599i	$0.634072\pi$
$642$	0	0
$643$	22993.0	1.41020	0.705098	−	0.709110i	$-0.250903\pi$
$643$	22993.0	1.41020	0.705098	+	0.709110i	$0.250903\pi$
$644$	−55449.2	−3.39286
$645$	0	0
$646$	−35133.3	−2.13978
$647$	−31390.5	−1.90740	−0.953700	−	0.300760i	$-0.902760\pi$
$647$	−31390.5	−1.90740	−0.953700	+	0.300760i	$0.902760\pi$
$648$	0	0
$649$	−4541.89	−0.274707
$650$	26169.3	1.57914
$651$	0	0
$652$	25494.2	1.53133
$653$	7217.27	0.432517	0.216259	−	0.976336i	$-0.430615\pi$
$653$	7217.27	0.432517	0.216259	+	0.976336i	$0.430615\pi$
$654$	0	0
$655$	446.003	0.0266058
$656$	5476.14	0.325926
$657$	0	0
$658$	55581.9	3.29302
$659$	−17128.2	−1.01248	−0.506238	−	0.862394i	$-0.668964\pi$
$659$	−17128.2	−1.01248	−0.506238	+	0.862394i	$0.668964\pi$
$660$	0	0
$661$	13396.5	0.788298	0.394149	−	0.919047i	$-0.371039\pi$
$661$	13396.5	0.788298	0.394149	+	0.919047i	$0.371039\pi$
$662$	21374.3	1.25489
$663$	0	0
$664$	−9842.29	−0.575234
$665$	18053.6	1.05277
$666$	0	0
$667$	−5430.59	−0.315252
$668$	36495.1	2.11383
$669$	0	0
$670$	9462.32	0.545614
$671$	3172.44	0.182520
$672$	0	0
$673$	−34787.6	−1.99251	−0.996257	−	0.0864350i	$-0.972453\pi$
$673$	−34787.6	−1.99251	−0.996257	+	0.0864350i	$0.972453\pi$
$674$	−553.868	−0.0316531
$675$	0	0
$676$	12939.4	0.736199
$677$	33537.7	1.90393	0.951964	−	0.306210i	$-0.0990609\pi$
$677$	33537.7	1.90393	0.951964	+	0.306210i	$0.0990609\pi$
$678$	0	0
$679$	54139.0	3.05989
$680$	−3645.24	−0.205571
$681$	0	0
$682$	2781.04	0.156146
$683$	−1233.30	−0.0690935	−0.0345468	−	0.999403i	$-0.510999\pi$
$683$	−1233.30	−0.0690935	−0.0345468	+	0.999403i	$0.510999\pi$
$684$	0	0
$685$	−6132.11	−0.342038
$686$	−47142.6	−2.62378
$687$	0	0
$688$	−14840.1	−0.822343
$689$	38723.7	2.14115
$690$	0	0
$691$	−6693.11	−0.368478	−0.184239	−	0.982882i	$-0.558982\pi$
$691$	−6693.11	−0.368478	−0.184239	+	0.982882i	$0.558982\pi$
$692$	35232.7	1.93547
$693$	0	0
$694$	−51938.2	−2.84085
$695$	7256.25	0.396036
$696$	0	0
$697$	10534.5	0.572483
$698$	−31877.3	−1.72861
$699$	0	0
$700$	−35652.0	−1.92503
$701$	2464.62	0.132792	0.0663961	−	0.997793i	$-0.478850\pi$
$701$	2464.62	0.132792	0.0663961	+	0.997793i	$0.478850\pi$
$702$	0	0
$703$	−14995.9	−0.804525
$704$	−5827.58	−0.311982
$705$	0	0
$706$	9627.02	0.513198
$707$	23341.7	1.24166
$708$	0	0
$709$	12904.3	0.683541	0.341770	−	0.939783i	$-0.388973\pi$
$709$	12904.3	0.683541	0.341770	+	0.939783i	$0.388973\pi$
$710$	2411.92	0.127490
$711$	0	0
$712$	−2232.17	−0.117491
$713$	13897.3	0.729956
$714$	0	0
$715$	−2008.46	−0.105052
$716$	−26390.6	−1.37746
$717$	0	0
$718$	40584.0	2.10944
$719$	4872.43	0.252727	0.126364	−	0.991984i	$-0.459669\pi$
$719$	4872.43	0.252727	0.126364	+	0.991984i	$0.459669\pi$
$720$	0	0
$721$	−53910.3	−2.78464
$722$	−34711.4	−1.78923
$723$	0	0
$724$	26889.6	1.38031
$725$	−3491.69	−0.178866
$726$	0	0
$727$	22575.8	1.15171	0.575854	−	0.817553i	$-0.304670\pi$
$727$	22575.8	1.15171	0.575854	+	0.817553i	$0.304670\pi$
$728$	−22149.0	−1.12761
$729$	0	0
$730$	−2402.49	−0.121808
$731$	−28547.8	−1.44443
$732$	0	0
$733$	10997.4	0.554157	0.277079	−	0.960847i	$-0.410634\pi$
$733$	10997.4	0.554157	0.277079	+	0.960847i	$0.410634\pi$
$734$	34925.6	1.75630
$735$	0	0
$736$	−39426.0	−1.97454
$737$	3525.15	0.176188
$738$	0	0
$739$	−10588.0	−0.527046	−0.263523	−	0.964653i	$-0.584885\pi$
$739$	−10588.0	−0.527046	−0.263523	+	0.964653i	$0.584885\pi$
$740$	−6100.75	−0.303065
$741$	0	0
$742$	−92056.9	−4.55460
$743$	−20321.9	−1.00342	−0.501708	−	0.865037i	$-0.667295\pi$
$743$	−20321.9	−1.00342	−0.501708	+	0.865037i	$0.667295\pi$
$744$	0	0
$745$	−7.27692	−0.000357860 0
$746$	−29423.6	−1.44407
$747$	0	0
$748$	−5324.88	−0.260290
$749$	−41463.8	−2.02277
$750$	0	0
$751$	25621.7	1.24494	0.622469	−	0.782644i	$-0.286129\pi$
$751$	25621.7	1.24494	0.622469	+	0.782644i	$0.286129\pi$
$752$	13865.5	0.672370
$753$	0	0
$754$	−8505.70	−0.410821
$755$	−14160.5	−0.682587
$756$	0	0
$757$	−14646.9	−0.703237	−0.351619	−	0.936143i	$-0.614369\pi$
$757$	−14646.9	−0.703237	−0.351619	+	0.936143i	$0.614369\pi$
$758$	846.298	0.0405527
$759$	0	0
$760$	−6682.08	−0.318927
$761$	2073.53	0.0987720	0.0493860	−	0.998780i	$-0.484274\pi$
$761$	2073.53	0.0987720	0.0493860	+	0.998780i	$0.484274\pi$
$762$	0	0
$763$	39206.4	1.86025
$764$	24953.8	1.18167
$765$	0	0
$766$	−15018.9	−0.708426
$767$	−35548.7	−1.67352
$768$	0	0
$769$	−3350.13	−0.157098	−0.0785492	−	0.996910i	$-0.525029\pi$
$769$	−3350.13	−0.157098	−0.0785492	+	0.996910i	$0.525029\pi$
$770$	4774.67	0.223464
$771$	0	0
$772$	−39635.4	−1.84781
$773$	6133.13	0.285373	0.142687	−	0.989768i	$-0.454426\pi$
$773$	6133.13	0.285373	0.142687	+	0.989768i	$0.454426\pi$
$774$	0	0
$775$	8935.52	0.414159
$776$	−20038.2	−0.926969
$777$	0	0
$778$	28902.0	1.33186
$779$	19310.7	0.888160
$780$	0	0
$781$	898.552	0.0411687
$782$	−46432.4	−2.12330
$783$	0	0
$784$	−23624.4	−1.07618
$785$	−11736.3	−0.533611
$786$	0	0
$787$	−8237.30	−0.373098	−0.186549	−	0.982446i	$-0.559730\pi$
$787$	−8237.30	−0.373098	−0.186549	+	0.982446i	$0.559730\pi$
$788$	32949.9	1.48958
$789$	0	0
$790$	−26832.3	−1.20842
$791$	61251.5	2.75329
$792$	0	0
$793$	24830.2	1.11191
$794$	−56003.1	−2.50312
$795$	0	0
$796$	−24589.6	−1.09492
$797$	−7688.23	−0.341695	−0.170848	−	0.985297i	$-0.554651\pi$
$797$	−7688.23	−0.341695	−0.170848	+	0.985297i	$0.554651\pi$
$798$	0	0
$799$	26673.0	1.18101
$800$	−25349.7	−1.12031
$801$	0	0
$802$	56188.5	2.47392
$803$	−895.038	−0.0393340
$804$	0	0
$805$	23859.8	1.04466
$806$	21766.8	0.951243
$807$	0	0
$808$	−8639.33	−0.376152
$809$	−3545.75	−0.154094	−0.0770470	−	0.997027i	$-0.524549\pi$
$809$	−3545.75	−0.154094	−0.0770470	+	0.997027i	$0.524549\pi$
$810$	0	0
$811$	−35806.2	−1.55034	−0.775169	−	0.631753i	$-0.782336\pi$
$811$	−35806.2	−1.55034	−0.775169	+	0.631753i	$0.782336\pi$
$812$	11587.8	0.500805
$813$	0	0
$814$	−3965.98	−0.170771
$815$	−10970.2	−0.471495
$816$	0	0
$817$	−52330.9	−2.24091
$818$	6771.45	0.289435
$819$	0	0
$820$	7856.12	0.334570
$821$	−5996.52	−0.254908	−0.127454	−	0.991844i	$-0.540681\pi$
$821$	−5996.52	−0.254908	−0.127454	+	0.991844i	$0.540681\pi$
$822$	0	0
$823$	26264.4	1.11242	0.556208	−	0.831043i	$-0.312256\pi$
$823$	26264.4	1.11242	0.556208	+	0.831043i	$0.312256\pi$
$824$	19953.5	0.843583
$825$	0	0
$826$	84509.1	3.55987
$827$	−5044.13	−0.212094	−0.106047	−	0.994361i	$-0.533819\pi$
$827$	−5044.13	−0.212094	−0.106047	+	0.994361i	$0.533819\pi$
$828$	0	0
$829$	−38038.1	−1.59363	−0.796814	−	0.604225i	$-0.793483\pi$
$829$	−38038.1	−1.59363	−0.796814	+	0.604225i	$0.793483\pi$
$830$	16606.3	0.694472
$831$	0	0
$832$	−45611.6	−1.90060
$833$	−45446.3	−1.89030
$834$	0	0
$835$	−15703.8	−0.650843
$836$	−9761.03	−0.403819
$837$	0	0
$838$	−44423.7	−1.83125
$839$	8573.63	0.352794	0.176397	−	0.984319i	$-0.443556\pi$
$839$	8573.63	0.352794	0.176397	+	0.984319i	$0.443556\pi$
$840$	0	0
$841$	−23254.1	−0.953467
$842$	23519.9	0.962648
$843$	0	0
$844$	−2078.58	−0.0847723
$845$	−5567.84	−0.226674
$846$	0	0
$847$	−40854.7	−1.65736
$848$	−22964.6	−0.929960
$849$	0	0
$850$	−29854.5	−1.20471
$851$	−19818.7	−0.798326
$852$	0	0
$853$	−10215.5	−0.410050	−0.205025	−	0.978757i	$-0.565728\pi$
$853$	−10215.5	−0.410050	−0.205025	+	0.978757i	$0.565728\pi$
$854$	−59028.2	−2.36523
$855$	0	0
$856$	15346.8	0.612782
$857$	−8184.20	−0.326216	−0.163108	−	0.986608i	$-0.552152\pi$
$857$	−8184.20	−0.326216	−0.163108	+	0.986608i	$0.552152\pi$
$858$	0	0
$859$	25247.4	1.00283	0.501415	−	0.865207i	$-0.332813\pi$
$859$	25247.4	1.00283	0.501415	+	0.865207i	$0.332813\pi$
$860$	−21289.7	−0.844153
$861$	0	0
$862$	34463.2	1.36174
$863$	−41577.3	−1.63999	−0.819993	−	0.572374i	$-0.806023\pi$
$863$	−41577.3	−1.63999	−0.819993	+	0.572374i	$0.806023\pi$
$864$	0	0
$865$	−15160.6	−0.595928
$866$	12287.6	0.482160
$867$	0	0
$868$	−29654.2	−1.15960
$869$	−9996.30	−0.390220
$870$	0	0
$871$	27590.9	1.07334
$872$	−14511.2	−0.563546
$873$	0	0
$874$	−85115.2	−3.29412
$875$	33842.6	1.30753
$876$	0	0
$877$	9903.31	0.381313	0.190656	−	0.981657i	$-0.438938\pi$
$877$	9903.31	0.381313	0.190656	+	0.981657i	$0.438938\pi$
$878$	−67365.6	−2.58938
$879$	0	0
$880$	1191.09	0.0456269
$881$	−21278.3	−0.813716	−0.406858	−	0.913491i	$-0.633376\pi$
$881$	−21278.3	−0.813716	−0.406858	+	0.913491i	$0.633376\pi$
$882$	0	0
$883$	47695.8	1.81777	0.908885	−	0.417046i	$-0.136935\pi$
$883$	47695.8	1.81777	0.908885	+	0.417046i	$0.136935\pi$
$884$	−41677.1	−1.58569
$885$	0	0
$886$	33190.5	1.25853
$887$	31002.5	1.17358	0.586789	−	0.809740i	$-0.300392\pi$
$887$	31002.5	1.17358	0.586789	+	0.809740i	$0.300392\pi$
$888$	0	0
$889$	8726.10	0.329206
$890$	3766.19	0.141846
$891$	0	0
$892$	−37814.6	−1.41943
$893$	48894.3	1.83223
$894$	0	0
$895$	11355.9	0.424118
$896$	45759.0	1.70614
$897$	0	0
$898$	−57049.1	−2.11999
$899$	−2904.28	−0.107745
$900$	0	0
$901$	−44176.9	−1.63346
$902$	5107.12	0.188524
$903$	0	0
$904$	−22670.6	−0.834086
$905$	−11570.6	−0.424994
$906$	0	0
$907$	−6741.56	−0.246803	−0.123401	−	0.992357i	$-0.539380\pi$
$907$	−6741.56	−0.246803	−0.123401	+	0.992357i	$0.539380\pi$
$908$	54465.8	1.99065
$909$	0	0
$910$	37370.6	1.36134
$911$	−2110.49	−0.0767550	−0.0383775	−	0.999263i	$-0.512219\pi$
$911$	−2110.49	−0.0767550	−0.0383775	+	0.999263i	$0.512219\pi$
$912$	0	0
$913$	6186.61	0.224257
$914$	35461.1	1.28331
$915$	0	0
$916$	−42998.5	−1.55099
$917$	−3091.62	−0.111335
$918$	0	0
$919$	41824.6	1.50127	0.750634	−	0.660718i	$-0.229748\pi$
$919$	41824.6	1.50127	0.750634	+	0.660718i	$0.229748\pi$
$920$	−8831.08	−0.316470
$921$	0	0
$922$	−36532.7	−1.30493
$923$	7032.84	0.250800
$924$	0	0
$925$	−12742.8	−0.452951
$926$	2154.29	0.0764517
$927$	0	0
$928$	8239.30	0.291453
$929$	−38866.0	−1.37261	−0.686303	−	0.727316i	$-0.740767\pi$
$929$	−38866.0	−1.37261	−0.686303	+	0.727316i	$0.740767\pi$
$930$	0	0
$931$	−83307.5	−2.93264
$932$	66934.3	2.35247
$933$	0	0
$934$	43535.5	1.52519
$935$	2291.30	0.0801428
$936$	0	0
$937$	−54033.5	−1.88388	−0.941941	−	0.335780i	$-0.891000\pi$
$937$	−54033.5	−1.88388	−0.941941	+	0.335780i	$0.891000\pi$
$938$	−65591.1	−2.28318
$939$	0	0
$940$	19891.6	0.690203
$941$	−6944.29	−0.240571	−0.120286	−	0.992739i	$-0.538381\pi$
$941$	−6944.29	−0.240571	−0.120286	+	0.992739i	$0.538381\pi$
$942$	0	0
$943$	25521.1	0.881318
$944$	21081.7	0.726855
$945$	0	0
$946$	−13840.0	−0.475664
$947$	−18293.5	−0.627729	−0.313865	−	0.949468i	$-0.601624\pi$
$947$	−18293.5	−0.627729	−0.313865	+	0.949468i	$0.601624\pi$
$948$	0	0
$949$	−7005.33	−0.239623
$950$	−54726.3	−1.86901
$951$	0	0
$952$	25268.1	0.860237
$953$	−32671.8	−1.11054	−0.555269	−	0.831671i	$-0.687385\pi$
$953$	−32671.8	−1.11054	−0.555269	+	0.831671i	$0.687385\pi$
$954$	0	0
$955$	−10737.6	−0.363834
$956$	2566.56	0.0868288
$957$	0	0
$958$	−74857.7	−2.52457
$959$	42506.7	1.43129
$960$	0	0
$961$	−22358.7	−0.750519
$962$	−31041.2	−1.04034
$963$	0	0
$964$	−50169.9	−1.67621
$965$	17055.1	0.568937
$966$	0	0
$967$	52313.3	1.73969	0.869846	−	0.493323i	$-0.164218\pi$
$967$	52313.3	1.73969	0.869846	+	0.493323i	$0.164218\pi$
$968$	15121.3	0.502083
$969$	0	0
$970$	33809.1	1.11912
$971$	9561.98	0.316023	0.158012	−	0.987437i	$-0.449492\pi$
$971$	9561.98	0.316023	0.158012	+	0.987437i	$0.449492\pi$
$972$	0	0
$973$	−50299.0	−1.65726
$974$	31530.0	1.03726
$975$	0	0
$976$	−14725.2	−0.482933
$977$	−28210.4	−0.923779	−0.461890	−	0.886937i	$-0.652828\pi$
$977$	−28210.4	−0.923779	−0.461890	+	0.886937i	$0.652828\pi$
$978$	0	0
$979$	1403.08	0.0458045
$980$	−33891.8	−1.10473
$981$	0	0
$982$	12376.6	0.402192
$983$	45978.4	1.49185	0.745923	−	0.666033i	$-0.232009\pi$
$983$	45978.4	1.49185	0.745923	+	0.666033i	$0.232009\pi$
$984$	0	0
$985$	−14178.3	−0.458639
$986$	9703.51	0.313410
$987$	0	0
$988$	−76398.2	−2.46007
$989$	−69160.9	−2.22365
$990$	0	0
$991$	−25500.0	−0.817391	−0.408695	−	0.912671i	$-0.634016\pi$
$991$	−25500.0	−0.817391	−0.408695	+	0.912671i	$0.634016\pi$
$992$	−21085.1	−0.674850
$993$	0	0
$994$	−16719.0	−0.533495
$995$	10580.9	0.337123
$996$	0	0
$997$	9410.43	0.298928	0.149464	−	0.988767i	$-0.452245\pi$
$997$	9410.43	0.298928	0.149464	+	0.988767i	$0.452245\pi$
$998$	2848.63	0.0903525
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.6		32
3.2	odd	2		717.4.a.c.1.27	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.27	✓	32	3.2	odd	2
2151.4.a.e.1.6		32	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-4.32883 q^{2} +10.7387 q^{4} -4.62088 q^{5} +32.0311 q^{7} -11.8555 q^{8} +O(q^{10})\)	\(q-4.32883 q^{2} +10.7387 q^{4} -4.62088 q^{5} +32.0311 q^{7} -11.8555 q^{8} +20.0030 q^{10} +7.45205 q^{11} +58.3261 q^{13} -138.657 q^{14} -34.5895 q^{16} -66.5398 q^{17} -121.974 q^{19} -49.6224 q^{20} -32.2586 q^{22} -161.202 q^{23} -103.647 q^{25} -252.483 q^{26} +343.974 q^{28} +33.6881 q^{29} -86.2107 q^{31} +244.576 q^{32} +288.039 q^{34} -148.012 q^{35} +122.943 q^{37} +528.004 q^{38} +54.7828 q^{40} -158.318 q^{41} +429.033 q^{43} +80.0255 q^{44} +697.814 q^{46} -400.858 q^{47} +682.994 q^{49} +448.672 q^{50} +626.348 q^{52} +663.917 q^{53} -34.4350 q^{55} -379.745 q^{56} -145.830 q^{58} -609.482 q^{59} +425.713 q^{61} +373.191 q^{62} -782.010 q^{64} -269.518 q^{65} +473.045 q^{67} -714.553 q^{68} +640.719 q^{70} +120.578 q^{71} -120.106 q^{73} -532.200 q^{74} -1309.85 q^{76} +238.698 q^{77} -1341.42 q^{79} +159.834 q^{80} +685.331 q^{82} +830.189 q^{83} +307.473 q^{85} -1857.21 q^{86} -88.3477 q^{88} +188.281 q^{89} +1868.25 q^{91} -1731.10 q^{92} +1735.25 q^{94} +563.627 q^{95} +1690.20 q^{97} -2956.56 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

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