LMFDB - Embedded newform 231.4.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [231,4,Mod(1,231)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(231, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("231.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$231 = 3 \cdot 7 \cdot 11$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	231.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-1,-15,21,7] 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:	$13.6294412113$
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:	$x^{5} - x^{4} - 30x^{3} + 21x^{2} + 103x + 18$ x^5 - x^4 - 30x^3 + 21x^2 + 103*x + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50528$ of defining polynomial
Character	$\chi$	$=$	231.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50528 q^{2} -3.00000 q^{3} -5.73413 q^{4} -0.155265 q^{5} -4.51584 q^{6} -7.00000 q^{7} -20.6737 q^{8} +9.00000 q^{9} -0.233718 q^{10} -11.0000 q^{11} +17.2024 q^{12} +88.7135 q^{13} -10.5370 q^{14} +0.465796 q^{15} +14.7533 q^{16} +102.195 q^{17} +13.5475 q^{18} -26.8508 q^{19} +0.890312 q^{20} +21.0000 q^{21} -16.5581 q^{22} +65.1580 q^{23} +62.0212 q^{24} -124.976 q^{25} +133.539 q^{26} -27.0000 q^{27} +40.1389 q^{28} +136.493 q^{29} +0.701154 q^{30} -87.8011 q^{31} +187.598 q^{32} +33.0000 q^{33} +153.832 q^{34} +1.08686 q^{35} -51.6072 q^{36} +391.184 q^{37} -40.4180 q^{38} -266.140 q^{39} +3.20991 q^{40} -69.8526 q^{41} +31.6109 q^{42} +293.400 q^{43} +63.0754 q^{44} -1.39739 q^{45} +98.0811 q^{46} +122.585 q^{47} -44.2598 q^{48} +49.0000 q^{49} -188.124 q^{50} -306.584 q^{51} -508.694 q^{52} +140.132 q^{53} -40.6426 q^{54} +1.70792 q^{55} +144.716 q^{56} +80.5524 q^{57} +205.460 q^{58} -653.662 q^{59} -2.67094 q^{60} +295.112 q^{61} -132.165 q^{62} -63.0000 q^{63} +164.361 q^{64} -13.7741 q^{65} +49.6743 q^{66} -82.0975 q^{67} -585.998 q^{68} -195.474 q^{69} +1.63603 q^{70} -579.406 q^{71} -186.064 q^{72} -123.715 q^{73} +588.842 q^{74} +374.928 q^{75} +153.966 q^{76} +77.0000 q^{77} -400.616 q^{78} +420.453 q^{79} -2.29067 q^{80} +81.0000 q^{81} -105.148 q^{82} +59.7076 q^{83} -120.417 q^{84} -15.8673 q^{85} +441.650 q^{86} -409.478 q^{87} +227.411 q^{88} -280.345 q^{89} -2.10346 q^{90} -620.994 q^{91} -373.624 q^{92} +263.403 q^{93} +184.525 q^{94} +4.16900 q^{95} -562.793 q^{96} +19.1595 q^{97} +73.7588 q^{98} -99.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - q^{2} - 15 q^{3} + 21 q^{4} + 7 q^{5} + 3 q^{6} - 35 q^{7} - 12 q^{8} + 45 q^{9} + 113 q^{10} - 55 q^{11} - 63 q^{12} + 23 q^{13} + 7 q^{14} - 21 q^{15} + 281 q^{16} - 102 q^{17} - 9 q^{18} - 155 q^{19}+ \cdots - 495 q^{99}+O(q^{100})$ 5 * q - q^2 - 15 * q^3 + 21 * q^4 + 7 * q^5 + 3 * q^6 - 35 * q^7 - 12 * q^8 + 45 * q^9 + 113 * q^10 - 55 * q^11 - 63 * q^12 + 23 * q^13 + 7 * q^14 - 21 * q^15 + 281 * q^16 - 102 * q^17 - 9 * q^18 - 155 * q^19 + 291 * q^20 + 105 * q^21 + 11 * q^22 + 192 * q^23 + 36 * q^24 + 394 * q^25 + 41 * q^26 - 135 * q^27 - 147 * q^28 + 591 * q^29 - 339 * q^30 + 366 * q^31 - 87 * q^32 + 165 * q^33 + 594 * q^34 - 49 * q^35 + 189 * q^36 + 259 * q^37 + 175 * q^38 - 69 * q^39 + 1890 * q^40 - 104 * q^41 - 21 * q^42 + 224 * q^43 - 231 * q^44 + 63 * q^45 + 1416 * q^46 - 453 * q^47 - 843 * q^48 + 245 * q^49 + 1350 * q^50 + 306 * q^51 + 815 * q^52 + 1032 * q^53 + 27 * q^54 - 77 * q^55 + 84 * q^56 + 465 * q^57 + 293 * q^58 - 517 * q^59 - 873 * q^60 + 958 * q^61 - 3030 * q^62 - 315 * q^63 + 516 * q^64 - 197 * q^65 - 33 * q^66 + 361 * q^67 - 1680 * q^68 - 576 * q^69 - 791 * q^70 + 548 * q^71 - 108 * q^72 - 1035 * q^73 - 2555 * q^74 - 1182 * q^75 - 5201 * q^76 + 385 * q^77 - 123 * q^78 + 776 * q^79 - 1273 * q^80 + 405 * q^81 + 376 * q^82 - 1974 * q^83 + 441 * q^84 + 1838 * q^85 + 1224 * q^86 - 1773 * q^87 + 132 * q^88 + 2710 * q^89 + 1017 * q^90 - 161 * q^91 - 2834 * q^92 - 1098 * q^93 - 563 * q^94 - 355 * q^95 + 261 * q^96 + 1988 * q^97 - 49 * q^98 - 495 * 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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	1.50528	0.532197	0.266099	−	0.963946i	$-0.414265\pi$
$2$	1.50528	0.532197	0.266099	+	0.963946i	$0.414265\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−5.73413	−0.716766
$5$	−0.155265	−0.0138874	−0.00694368	−	0.999976i	$-0.502210\pi$
$5$	−0.155265	−0.0138874	−0.00694368	+	0.999976i	$0.502210\pi$
$6$	−4.51584	−0.307264
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−20.6737	−0.913658
$9$	9.00000	0.333333
$10$	−0.233718	−0.00739082
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	17.2024	0.413825
$13$	88.7135	1.89267	0.946334	−	0.323190i	$-0.104755\pi$
$13$	88.7135	1.89267	0.946334	+	0.323190i	$0.104755\pi$
$14$	−10.5370	−0.201152
$15$	0.465796	0.00801787
$16$	14.7533	0.230520
$17$	102.195	1.45799	0.728996	−	0.684518i	$-0.239987\pi$
$17$	102.195	1.45799	0.728996	+	0.684518i	$0.239987\pi$
$18$	13.5475	0.177399
$19$	−26.8508	−0.324210	−0.162105	−	0.986774i	$-0.551828\pi$
$19$	−26.8508	−0.324210	−0.162105	+	0.986774i	$0.551828\pi$
$20$	0.890312	0.00995399
$21$	21.0000	0.218218
$22$	−16.5581	−0.160464
$23$	65.1580	0.590712	0.295356	−	0.955387i	$-0.404562\pi$
$23$	65.1580	0.590712	0.295356	+	0.955387i	$0.404562\pi$
$24$	62.0212	0.527501
$25$	−124.976	−0.999807
$26$	133.539	1.00727
$27$	−27.0000	−0.192450
$28$	40.1389	0.270912
$29$	136.493	0.874001	0.437001	−	0.899461i	$-0.356041\pi$
$29$	136.493	0.874001	0.437001	+	0.899461i	$0.356041\pi$
$30$	0.701154	0.00426709
$31$	−87.8011	−0.508695	−0.254348	−	0.967113i	$-0.581861\pi$
$31$	−87.8011	−0.508695	−0.254348	+	0.967113i	$0.581861\pi$
$32$	187.598	1.03634
$33$	33.0000	0.174078
$34$	153.832	0.775939
$35$	1.08686	0.00524893
$36$	−51.6072	−0.238922
$37$	391.184	1.73812	0.869058	−	0.494710i	$-0.164726\pi$
$37$	391.184	1.73812	0.869058	+	0.494710i	$0.164726\pi$
$38$	−40.4180	−0.172544
$39$	−266.140	−1.09273
$40$	3.20991	0.0126883
$41$	−69.8526	−0.266077	−0.133038	−	0.991111i	$-0.542473\pi$
$41$	−69.8526	−0.266077	−0.133038	+	0.991111i	$0.542473\pi$
$42$	31.6109	0.116135
$43$	293.400	1.04054	0.520268	−	0.854003i	$-0.325832\pi$
$43$	293.400	1.04054	0.520268	+	0.854003i	$0.325832\pi$
$44$	63.0754	0.216113
$45$	−1.39739	−0.00462912
$46$	98.0811	0.314376
$47$	122.585	0.380445	0.190223	−	0.981741i	$-0.439079\pi$
$47$	122.585	0.380445	0.190223	+	0.981741i	$0.439079\pi$
$48$	−44.2598	−0.133091
$49$	49.0000	0.142857
$50$	−188.124	−0.532095
$51$	−306.584	−0.841772
$52$	−508.694	−1.35660
$53$	140.132	0.363181	0.181591	−	0.983374i	$-0.441875\pi$
$53$	140.132	0.363181	0.181591	+	0.983374i	$0.441875\pi$
$54$	−40.6426	−0.102421
$55$	1.70792	0.00418720
$56$	144.716	0.345330
$57$	80.5524	0.187183
$58$	205.460	0.465141
$59$	−653.662	−1.44237	−0.721183	−	0.692745i	$-0.756401\pi$
$59$	−653.662	−1.44237	−0.721183	+	0.692745i	$0.756401\pi$
$60$	−2.67094	−0.00574694
$61$	295.112	0.619431	0.309715	−	0.950829i	$-0.399766\pi$
$61$	295.112	0.619431	0.309715	+	0.950829i	$0.399766\pi$
$62$	−132.165	−0.270726
$63$	−63.0000	−0.125988
$64$	164.361	0.321018
$65$	−13.7741	−0.0262842
$66$	49.6743	0.0926437
$67$	−82.0975	−0.149699	−0.0748493	−	0.997195i	$-0.523848\pi$
$67$	−82.0975	−0.149699	−0.0748493	+	0.997195i	$0.523848\pi$
$68$	−585.998	−1.04504
$69$	−195.474	−0.341048
$70$	1.63603	0.00279347
$71$	−579.406	−0.968491	−0.484245	−	0.874932i	$-0.660906\pi$
$71$	−579.406	−0.968491	−0.484245	+	0.874932i	$0.660906\pi$
$72$	−186.064	−0.304553
$73$	−123.715	−0.198352	−0.0991762	−	0.995070i	$-0.531621\pi$
$73$	−123.715	−0.198352	−0.0991762	+	0.995070i	$0.531621\pi$
$74$	588.842	0.925021
$75$	374.928	0.577239
$76$	153.966	0.232383
$77$	77.0000	0.113961
$78$	−400.616	−0.581549
$79$	420.453	0.598794	0.299397	−	0.954129i	$-0.403215\pi$
$79$	420.453	0.598794	0.299397	+	0.954129i	$0.403215\pi$
$80$	−2.29067	−0.00320131
$81$	81.0000	0.111111
$82$	−105.148	−0.141605
$83$	59.7076	0.0789610	0.0394805	−	0.999220i	$-0.487430\pi$
$83$	59.7076	0.0789610	0.0394805	+	0.999220i	$0.487430\pi$
$84$	−120.417	−0.156411
$85$	−15.8673	−0.0202477
$86$	441.650	0.553771
$87$	−409.478	−0.504605
$88$	227.411	0.275478
$89$	−280.345	−0.333894	−0.166947	−	0.985966i	$-0.553391\pi$
$89$	−280.345	−0.333894	−0.166947	+	0.985966i	$0.553391\pi$
$90$	−2.10346	−0.00246361
$91$	−620.994	−0.715361
$92$	−373.624	−0.423403
$93$	263.403	0.293695
$94$	184.525	0.202472
$95$	4.16900	0.00450242
$96$	−562.793	−0.598331
$97$	19.1595	0.0200552	0.0100276	−	0.999950i	$-0.496808\pi$
$97$	19.1595	0.0200552	0.0100276	+	0.999950i	$0.496808\pi$
$98$	73.7588	0.0760282
$99$	−99.0000	−0.100504
$100$	716.628	0.716628
$101$	−35.9731	−0.0354401	−0.0177201	−	0.999843i	$-0.505641\pi$
$101$	−35.9731	−0.0354401	−0.0177201	+	0.999843i	$0.505641\pi$
$102$	−461.495	−0.447989
$103$	1690.03	1.61674	0.808369	−	0.588676i	$-0.200350\pi$
$103$	1690.03	1.61674	0.808369	+	0.588676i	$0.200350\pi$
$104$	−1834.04	−1.72925
$105$	−3.26057	−0.00303047
$106$	210.938	0.193284
$107$	992.428	0.896651	0.448326	−	0.893870i	$-0.352021\pi$
$107$	992.428	0.896651	0.448326	+	0.893870i	$0.352021\pi$
$108$	154.821	0.137942
$109$	607.598	0.533920	0.266960	−	0.963708i	$-0.413981\pi$
$109$	607.598	0.533920	0.266960	+	0.963708i	$0.413981\pi$
$110$	2.57090	0.00222841
$111$	−1173.55	−1.00350
$112$	−103.273	−0.0871282
$113$	−876.628	−0.729790	−0.364895	−	0.931049i	$-0.618895\pi$
$113$	−876.628	−0.729790	−0.364895	+	0.931049i	$0.618895\pi$
$114$	121.254	0.0996182
$115$	−10.1168	−0.00820344
$116$	−782.666	−0.626454
$117$	798.421	0.630889
$118$	−983.945	−0.767623
$119$	−715.363	−0.551069
$120$	−9.62974	−0.00732559
$121$	121.000	0.0909091
$122$	444.227	0.329659
$123$	209.558	0.153620
$124$	503.463	0.364615
$125$	38.8126	0.0277720
$126$	−94.8327	−0.0670506
$127$	945.725	0.660784	0.330392	−	0.943844i	$-0.392819\pi$
$127$	945.725	0.660784	0.330392	+	0.943844i	$0.392819\pi$
$128$	−1253.37	−0.865495
$129$	−880.200	−0.600754
$130$	−20.7339	−0.0139884
$131$	−112.396	−0.0749628	−0.0374814	−	0.999297i	$-0.511933\pi$
$131$	−112.396	−0.0749628	−0.0374814	+	0.999297i	$0.511933\pi$
$132$	−189.226	−0.124773
$133$	187.956	0.122540
$134$	−123.580	−0.0796692
$135$	4.19217	0.00267262
$136$	−2112.75	−1.33211
$137$	−1417.41	−0.883925	−0.441963	−	0.897033i	$-0.645718\pi$
$137$	−1417.41	−0.883925	−0.441963	+	0.897033i	$0.645718\pi$
$138$	−294.243	−0.181505
$139$	336.386	0.205265	0.102633	−	0.994719i	$-0.467273\pi$
$139$	336.386	0.205265	0.102633	+	0.994719i	$0.467273\pi$
$140$	−6.23218	−0.00376225
$141$	−367.756	−0.219650
$142$	−872.169	−0.515428
$143$	−975.848	−0.570661
$144$	132.779	0.0768399
$145$	−21.1926	−0.0121376
$146$	−186.226	−0.105563
$147$	−147.000	−0.0824786
$148$	−2243.10	−1.24582
$149$	3130.80	1.72138	0.860689	−	0.509132i	$-0.170033\pi$
$149$	3130.80	1.72138	0.860689	+	0.509132i	$0.170033\pi$
$150$	564.372	0.307205
$151$	3421.03	1.84371	0.921853	−	0.387540i	$-0.126675\pi$
$151$	3421.03	1.84371	0.921853	+	0.387540i	$0.126675\pi$
$152$	555.106	0.296217
$153$	919.753	0.485997
$154$	115.907	0.0606495
$155$	13.6325	0.00706443
$156$	1526.08	0.783234
$157$	−1693.77	−0.861006	−0.430503	−	0.902589i	$-0.641664\pi$
$157$	−1693.77	−0.861006	−0.430503	+	0.902589i	$0.641664\pi$
$158$	632.901	0.318676
$159$	−420.396	−0.209683
$160$	−29.1274	−0.0143920
$161$	−456.106	−0.223268
$162$	121.928	0.0591330
$163$	−438.560	−0.210740	−0.105370	−	0.994433i	$-0.533603\pi$
$163$	−438.560	−0.210740	−0.105370	+	0.994433i	$0.533603\pi$
$164$	400.544	0.190715
$165$	−5.12376	−0.00241748
$166$	89.8767	0.0420228
$167$	2606.86	1.20793	0.603967	−	0.797009i	$-0.293586\pi$
$167$	2606.86	1.20793	0.603967	+	0.797009i	$0.293586\pi$
$168$	−434.148	−0.199377
$169$	5673.08	2.58219
$170$	−23.8848	−0.0107758
$171$	−241.657	−0.108070
$172$	−1682.39	−0.745822
$173$	−4498.50	−1.97696	−0.988481	−	0.151343i	$-0.951640\pi$
$173$	−4498.50	−1.97696	−0.988481	+	0.151343i	$0.951640\pi$
$174$	−616.379	−0.268549
$175$	874.831	0.377892
$176$	−162.286	−0.0695043
$177$	1960.99	0.832750
$178$	−421.998	−0.177697
$179$	589.770	0.246265	0.123133	−	0.992390i	$-0.460706\pi$
$179$	589.770	0.246265	0.123133	+	0.992390i	$0.460706\pi$
$180$	8.01281	0.00331800
$181$	−2377.96	−0.976534	−0.488267	−	0.872694i	$-0.662371\pi$
$181$	−2377.96	−0.976534	−0.488267	+	0.872694i	$0.662371\pi$
$182$	−934.771	−0.380713
$183$	−885.337	−0.357628
$184$	−1347.06	−0.539709
$185$	−60.7374	−0.0241379
$186$	396.496	0.156304
$187$	−1124.14	−0.439601
$188$	−702.920	−0.272690
$189$	189.000	0.0727393
$190$	6.27552	0.00239618
$191$	−3698.73	−1.40121	−0.700605	−	0.713550i	$-0.747086\pi$
$191$	−3698.73	−1.40121	−0.700605	+	0.713550i	$0.747086\pi$
$192$	−493.083	−0.185340
$193$	5199.41	1.93918	0.969589	−	0.244737i	$-0.0787016\pi$
$193$	5199.41	1.93918	0.969589	+	0.244737i	$0.0787016\pi$
$194$	28.8404	0.0106733
$195$	41.3224	0.0151752
$196$	−280.972	−0.102395
$197$	1356.16	0.490469	0.245234	−	0.969464i	$-0.421135\pi$
$197$	1356.16	0.490469	0.245234	+	0.969464i	$0.421135\pi$
$198$	−149.023	−0.0534878
$199$	2506.86	0.892999	0.446500	−	0.894784i	$-0.352670\pi$
$199$	2506.86	0.892999	0.446500	+	0.894784i	$0.352670\pi$
$200$	2583.72	0.913482
$201$	246.293	0.0864286
$202$	−54.1496	−0.0188611
$203$	−955.448	−0.330341
$204$	1757.99	0.603354
$205$	10.8457	0.00369510
$206$	2543.98	0.860424
$207$	586.422	0.196904
$208$	1308.81	0.436297
$209$	295.359	0.0977530
$210$	−4.90808	−0.00161281
$211$	631.216	0.205947	0.102973	−	0.994684i	$-0.467164\pi$
$211$	631.216	0.205947	0.102973	+	0.994684i	$0.467164\pi$
$212$	−803.534	−0.260316
$213$	1738.22	0.559158
$214$	1493.88	0.477195
$215$	−45.5549	−0.0144503
$216$	558.191	0.175834
$217$	614.608	0.192269
$218$	914.605	0.284151
$219$	371.145	0.114519
$220$	−9.79343	−0.00300124
$221$	9066.05	2.75950
$222$	−1766.53	−0.534061
$223$	2763.58	0.829879	0.414939	−	0.909849i	$-0.363803\pi$
$223$	2763.58	0.829879	0.414939	+	0.909849i	$0.363803\pi$
$224$	−1313.18	−0.391700
$225$	−1124.78	−0.333269
$226$	−1319.57	−0.388392
$227$	−5802.86	−1.69669	−0.848347	−	0.529441i	$-0.822402\pi$
$227$	−5802.86	−1.69669	−0.848347	+	0.529441i	$0.822402\pi$
$228$	−461.898	−0.134166
$229$	6741.60	1.94540	0.972702	−	0.232057i	$-0.0745455\pi$
$229$	6741.60	1.94540	0.972702	+	0.232057i	$0.0745455\pi$
$230$	−15.2286	−0.00436585
$231$	−231.000	−0.0657952
$232$	−2821.81	−0.798538
$233$	5358.60	1.50667	0.753333	−	0.657639i	$-0.228445\pi$
$233$	5358.60	1.50667	0.753333	+	0.657639i	$0.228445\pi$
$234$	1201.85	0.335758
$235$	−19.0333	−0.00528338
$236$	3748.18	1.03384
$237$	−1261.36	−0.345714
$238$	−1076.82	−0.293278
$239$	−5162.13	−1.39712	−0.698558	−	0.715554i	$-0.746174\pi$
$239$	−5162.13	−1.39712	−0.698558	+	0.715554i	$0.746174\pi$
$240$	6.87201	0.00184828
$241$	−6404.70	−1.71188	−0.855939	−	0.517077i	$-0.827020\pi$
$241$	−6404.70	−1.71188	−0.855939	+	0.517077i	$0.827020\pi$
$242$	182.139	0.0483816
$243$	−243.000	−0.0641500
$244$	−1692.21	−0.443987
$245$	−7.60801	−0.00198391
$246$	315.444	0.0817559
$247$	−2382.03	−0.613622
$248$	1815.18	0.464773
$249$	−179.123	−0.0455881
$250$	58.4239	0.0147802
$251$	308.457	0.0775682	0.0387841	−	0.999248i	$-0.487652\pi$
$251$	308.457	0.0775682	0.0387841	+	0.999248i	$0.487652\pi$
$252$	361.250	0.0903040
$253$	−716.738	−0.178107
$254$	1423.58	0.351667
$255$	47.6019	0.0116900
$256$	−3201.56	−0.781632
$257$	5546.96	1.34634	0.673171	−	0.739487i	$-0.264932\pi$
$257$	5546.96	1.34634	0.673171	+	0.739487i	$0.264932\pi$
$258$	−1324.95	−0.319720
$259$	−2738.29	−0.656946
$260$	78.9826	0.0188396
$261$	1228.43	0.291334
$262$	−169.188	−0.0398950
$263$	885.702	0.207661	0.103830	−	0.994595i	$-0.466890\pi$
$263$	885.702	0.207661	0.103830	+	0.994595i	$0.466890\pi$
$264$	−682.233	−0.159047
$265$	−21.7576	−0.00504363
$266$	282.926	0.0652154
$267$	841.035	0.192774
$268$	470.758	0.107299
$269$	4452.77	1.00926	0.504629	−	0.863336i	$-0.331629\pi$
$269$	4452.77	1.00926	0.504629	+	0.863336i	$0.331629\pi$
$270$	6.31039	0.00142236
$271$	−2577.98	−0.577863	−0.288932	−	0.957350i	$-0.593300\pi$
$271$	−2577.98	−0.577863	−0.288932	+	0.957350i	$0.593300\pi$
$272$	1507.71	0.336096
$273$	1862.98	0.413014
$274$	−2133.60	−0.470423
$275$	1374.73	0.301453
$276$	1120.87	0.244452
$277$	−683.280	−0.148210	−0.0741052	−	0.997250i	$-0.523610\pi$
$277$	−683.280	−0.148210	−0.0741052	+	0.997250i	$0.523610\pi$
$278$	506.355	0.109242
$279$	−790.210	−0.169565
$280$	−22.4694	−0.00479573
$281$	−8624.47	−1.83094	−0.915468	−	0.402391i	$-0.868179\pi$
$281$	−8624.47	−1.83094	−0.915468	+	0.402391i	$0.868179\pi$
$282$	−553.576	−0.116897
$283$	1252.68	0.263123	0.131562	−	0.991308i	$-0.458001\pi$
$283$	1252.68	0.263123	0.131562	+	0.991308i	$0.458001\pi$
$284$	3322.39	0.694181
$285$	−12.5070	−0.00259948
$286$	−1468.93	−0.303704
$287$	488.968	0.100568
$288$	1688.38	0.345447
$289$	5530.77	1.12574
$290$	−31.9008	−0.00645958
$291$	−57.4785	−0.0115789
$292$	709.397	0.142172
$293$	−6342.78	−1.26467	−0.632337	−	0.774694i	$-0.717904\pi$
$293$	−6342.78	−1.26467	−0.632337	+	0.774694i	$0.717904\pi$
$294$	−221.276	−0.0438949
$295$	101.491	0.0200306
$296$	−8087.24	−1.58804
$297$	297.000	0.0580259
$298$	4712.74	0.916112
$299$	5780.39	1.11802
$300$	−2149.88	−0.413745
$301$	−2053.80	−0.393286
$302$	5149.61	0.981215
$303$	107.919	0.0204614
$304$	−396.137	−0.0747368
$305$	−45.8207	−0.00860226
$306$	1384.49	0.258646
$307$	−5508.30	−1.02402	−0.512012	−	0.858979i	$-0.671100\pi$
$307$	−5508.30	−1.02402	−0.512012	+	0.858979i	$0.671100\pi$
$308$	−441.528	−0.0816831
$309$	−5070.10	−0.933424
$310$	20.5207	0.00375967
$311$	−5363.85	−0.977993	−0.488996	−	0.872286i	$-0.662637\pi$
$311$	−5363.85	−0.977993	−0.488996	+	0.872286i	$0.662637\pi$
$312$	5502.11	0.998384
$313$	−5825.07	−1.05192	−0.525962	−	0.850508i	$-0.676295\pi$
$313$	−5825.07	−1.05192	−0.525962	+	0.850508i	$0.676295\pi$
$314$	−2549.61	−0.458225
$315$	9.78172	0.00174964
$316$	−2410.93	−0.429195
$317$	−5798.44	−1.02736	−0.513680	−	0.857982i	$-0.671718\pi$
$317$	−5798.44	−1.02736	−0.513680	+	0.857982i	$0.671718\pi$
$318$	−632.814	−0.111593
$319$	−1501.42	−0.263521
$320$	−25.5196	−0.00445809
$321$	−2977.29	−0.517682
$322$	−686.568	−0.118823
$323$	−2744.01	−0.472696
$324$	−464.464	−0.0796407
$325$	−11087.0	−1.89230
$326$	−660.157	−0.112156
$327$	−1822.79	−0.308259
$328$	1444.11	0.243103
$329$	−858.098	−0.143795
$330$	−7.71270	−0.00128658
$331$	−9130.56	−1.51620	−0.758098	−	0.652140i	$-0.773871\pi$
$331$	−9130.56	−1.51620	−0.758098	+	0.652140i	$0.773871\pi$
$332$	−342.371	−0.0565965
$333$	3520.66	0.579372
$334$	3924.06	0.642859
$335$	12.7469	0.00207892
$336$	309.818	0.0503035
$337$	−2404.87	−0.388729	−0.194364	−	0.980929i	$-0.562264\pi$
$337$	−2404.87	−0.388729	−0.194364	+	0.980929i	$0.562264\pi$
$338$	8539.58	1.37424
$339$	2629.88	0.421344
$340$	90.9852	0.0145128
$341$	965.812	0.153377
$342$	−363.762	−0.0575146
$343$	−343.000	−0.0539949
$344$	−6065.67	−0.950695
$345$	30.3504	0.00473626
$346$	−6771.51	−1.05213
$347$	7248.76	1.12142	0.560712	−	0.828011i	$-0.310528\pi$
$347$	7248.76	1.12142	0.560712	+	0.828011i	$0.310528\pi$
$348$	2348.00	0.361684
$349$	−4150.91	−0.636657	−0.318329	−	0.947980i	$-0.603122\pi$
$349$	−4150.91	−0.636657	−0.318329	+	0.947980i	$0.603122\pi$
$350$	1316.87	0.201113
$351$	−2395.26	−0.364244
$352$	−2063.57	−0.312468
$353$	1592.84	0.240165	0.120082	−	0.992764i	$-0.461684\pi$
$353$	1592.84	0.240165	0.120082	+	0.992764i	$0.461684\pi$
$354$	2951.84	0.443187
$355$	89.9617	0.0134498
$356$	1607.54	0.239324
$357$	2146.09	0.318160
$358$	887.770	0.131062
$359$	−8489.06	−1.24801	−0.624005	−	0.781420i	$-0.714496\pi$
$359$	−8489.06	−1.24801	−0.624005	+	0.781420i	$0.714496\pi$
$360$	28.8892	0.00422943
$361$	−6138.04	−0.894888
$362$	−3579.50	−0.519708
$363$	−363.000	−0.0524864
$364$	3560.86	0.512747
$365$	19.2086	0.00275459
$366$	−1332.68	−0.190329
$367$	−6646.14	−0.945301	−0.472650	−	0.881250i	$-0.656703\pi$
$367$	−6646.14	−0.945301	−0.472650	+	0.881250i	$0.656703\pi$
$368$	961.293	0.136171
$369$	−628.674	−0.0886923
$370$	−91.4269	−0.0128461
$371$	−980.923	−0.137270
$372$	−1510.39	−0.210511
$373$	5110.79	0.709455	0.354728	−	0.934970i	$-0.384574\pi$
$373$	5110.79	0.709455	0.354728	+	0.934970i	$0.384574\pi$
$374$	−1692.15	−0.233955
$375$	−116.438	−0.0160342
$376$	−2534.30	−0.347597
$377$	12108.7	1.65419
$378$	284.498	0.0387117
$379$	−8198.11	−1.11111	−0.555553	−	0.831481i	$-0.687493\pi$
$379$	−8198.11	−1.11111	−0.555553	+	0.831481i	$0.687493\pi$
$380$	−23.9056	−0.00322718
$381$	−2837.18	−0.381504
$382$	−5567.63	−0.745720
$383$	−1917.46	−0.255816	−0.127908	−	0.991786i	$-0.540826\pi$
$383$	−1917.46	−0.255816	−0.127908	+	0.991786i	$0.540826\pi$
$384$	3760.11	0.499694
$385$	−11.9554	−0.00158261
$386$	7826.57	1.03203
$387$	2640.60	0.346846
$388$	−109.863	−0.0143749
$389$	5140.00	0.669944	0.334972	−	0.942228i	$-0.391273\pi$
$389$	5140.00	0.669944	0.334972	+	0.942228i	$0.391273\pi$
$390$	62.2018	0.00807618
$391$	6658.81	0.861254
$392$	−1013.01	−0.130523
$393$	337.189	0.0432798
$394$	2041.40	0.261026
$395$	−65.2819	−0.00831567
$396$	567.679	0.0720377
$397$	12960.0	1.63840	0.819202	−	0.573506i	$-0.194417\pi$
$397$	12960.0	1.63840	0.819202	+	0.573506i	$0.194417\pi$
$398$	3773.53	0.475252
$399$	−563.867	−0.0707485
$400$	−1843.80	−0.230475
$401$	−5292.84	−0.659131	−0.329566	−	0.944133i	$-0.606902\pi$
$401$	−5292.84	−0.659131	−0.329566	+	0.944133i	$0.606902\pi$
$402$	370.740	0.0459970
$403$	−7789.14	−0.962791
$404$	206.274	0.0254023
$405$	−12.5765	−0.00154304
$406$	−1438.22	−0.175807
$407$	−4303.03	−0.524062
$408$	6338.24	0.769092
$409$	5545.03	0.670377	0.335188	−	0.942151i	$-0.391200\pi$
$409$	5545.03	0.670377	0.335188	+	0.942151i	$0.391200\pi$
$410$	16.3258	0.00196652
$411$	4252.24	0.510334
$412$	−9690.88	−1.15882
$413$	4575.63	0.545163
$414$	882.730	0.104792
$415$	−9.27053	−0.00109656
$416$	16642.4	1.96145
$417$	−1009.16	−0.118510
$418$	444.598	0.0520239
$419$	3492.31	0.407185	0.203593	−	0.979056i	$-0.434738\pi$
$419$	3492.31	0.407185	0.203593	+	0.979056i	$0.434738\pi$
$420$	18.6965	0.00217214
$421$	15448.2	1.78836	0.894181	−	0.447706i	$-0.147759\pi$
$421$	15448.2	1.78836	0.894181	+	0.447706i	$0.147759\pi$
$422$	950.158	0.109604
$423$	1103.27	0.126815
$424$	−2897.05	−0.331823
$425$	−12771.9	−1.45771
$426$	2616.51	0.297583
$427$	−2065.79	−0.234123
$428$	−5690.71	−0.642689
$429$	2927.54	0.329471
$430$	−68.5729	−0.00769042
$431$	−6938.29	−0.775420	−0.387710	−	0.921781i	$-0.626734\pi$
$431$	−6938.29	−0.775420	−0.387710	+	0.921781i	$0.626734\pi$
$432$	−398.338	−0.0443635
$433$	−2820.31	−0.313015	−0.156508	−	0.987677i	$-0.550024\pi$
$433$	−2820.31	−0.313015	−0.156508	+	0.987677i	$0.550024\pi$
$434$	925.158	0.102325
$435$	63.5777	0.00700763
$436$	−3484.04	−0.382696
$437$	−1749.54	−0.191515
$438$	558.677	0.0609466
$439$	15521.6	1.68748	0.843740	−	0.536752i	$-0.180349\pi$
$439$	15521.6	1.68748	0.843740	+	0.536752i	$0.180349\pi$
$440$	−35.3091	−0.00382567
$441$	441.000	0.0476190
$442$	13647.0	1.46860
$443$	12187.2	1.30707	0.653536	−	0.756896i	$-0.273285\pi$
$443$	12187.2	1.30707	0.653536	+	0.756896i	$0.273285\pi$
$444$	6729.30	0.719276
$445$	43.5279	0.00463690
$446$	4159.96	0.441659
$447$	−9392.40	−0.993838
$448$	−1150.53	−0.121333
$449$	9058.45	0.952104	0.476052	−	0.879417i	$-0.342067\pi$
$449$	9058.45	0.952104	0.476052	+	0.879417i	$0.342067\pi$
$450$	−1693.11	−0.177365
$451$	768.379	0.0802252
$452$	5026.70	0.523088
$453$	−10263.1	−1.06446
$454$	−8734.94	−0.902975
$455$	96.4189	0.00993448
$456$	−1665.32	−0.171021
$457$	−1070.77	−0.109603	−0.0548013	−	0.998497i	$-0.517453\pi$
$457$	−1070.77	−0.109603	−0.0548013	+	0.998497i	$0.517453\pi$
$458$	10148.0	1.03534
$459$	−2759.26	−0.280591
$460$	58.0110	0.00587995
$461$	−10368.9	−1.04756	−0.523781	−	0.851853i	$-0.675479\pi$
$461$	−10368.9	−1.04756	−0.523781	+	0.851853i	$0.675479\pi$
$462$	−347.720	−0.0350160
$463$	−6335.12	−0.635892	−0.317946	−	0.948109i	$-0.602993\pi$
$463$	−6335.12	−0.635892	−0.317946	+	0.948109i	$0.602993\pi$
$464$	2013.71	0.201474
$465$	−40.8974	−0.00407865
$466$	8066.19	0.801844
$467$	−12922.6	−1.28048	−0.640241	−	0.768174i	$-0.721166\pi$
$467$	−12922.6	−1.28048	−0.640241	+	0.768174i	$0.721166\pi$
$468$	−4578.25	−0.452200
$469$	574.683	0.0565808
$470$	−28.6504	−0.00281180
$471$	5081.32	0.497102
$472$	13513.6	1.31783
$473$	−3227.40	−0.313734
$474$	−1898.70	−0.183988
$475$	3355.70	0.324148
$476$	4101.98	0.394988
$477$	1261.19	0.121060
$478$	−7770.46	−0.743541
$479$	−14681.6	−1.40046	−0.700228	−	0.713920i	$-0.746918\pi$
$479$	−14681.6	−1.40046	−0.700228	+	0.713920i	$0.746918\pi$
$480$	87.3823	0.00830924
$481$	34703.3	3.28968
$482$	−9640.87	−0.911057
$483$	1368.32	0.128904
$484$	−693.830	−0.0651606
$485$	−2.97481	−0.000278514 0
$486$	−365.783	−0.0341405
$487$	380.027	0.0353607	0.0176804	−	0.999844i	$-0.494372\pi$
$487$	380.027	0.0353607	0.0176804	+	0.999844i	$0.494372\pi$
$488$	−6101.07	−0.565948
$489$	1315.68	0.121671
$490$	−11.4522	−0.00105583
$491$	−5026.47	−0.461999	−0.230999	−	0.972954i	$-0.574199\pi$
$491$	−5026.47	−0.461999	−0.230999	+	0.972954i	$0.574199\pi$
$492$	−1201.63	−0.110109
$493$	13948.8	1.27429
$494$	−3585.62	−0.326568
$495$	15.3713	0.00139573
$496$	−1295.35	−0.117264
$497$	4055.84	0.366055
$498$	−269.630	−0.0242619
$499$	13825.8	1.24033	0.620167	−	0.784470i	$-0.287065\pi$
$499$	13825.8	1.24033	0.620167	+	0.784470i	$0.287065\pi$
$500$	−222.557	−0.0199061
$501$	−7820.58	−0.697401
$502$	464.314	0.0412816
$503$	12089.2	1.07164	0.535818	−	0.844334i	$-0.320003\pi$
$503$	12089.2	1.07164	0.535818	+	0.844334i	$0.320003\pi$
$504$	1302.44	0.115110
$505$	5.58537	0.000492170 0
$506$	−1078.89	−0.0947878
$507$	−17019.2	−1.49083
$508$	−5422.91	−0.473627
$509$	−6405.45	−0.557792	−0.278896	−	0.960321i	$-0.589969\pi$
$509$	−6405.45	−0.557792	−0.278896	+	0.960321i	$0.589969\pi$
$510$	71.6543	0.00622138
$511$	866.004	0.0749702
$512$	5207.71	0.449513
$513$	724.971	0.0623943
$514$	8349.74	0.716520
$515$	−262.404	−0.0224522
$516$	5047.18	0.430600
$517$	−1348.44	−0.114708
$518$	−4121.90	−0.349625
$519$	13495.5	1.14140
$520$	284.763	0.0240147
$521$	9757.40	0.820498	0.410249	−	0.911974i	$-0.365442\pi$
$521$	9757.40	0.820498	0.410249	+	0.911974i	$0.365442\pi$
$522$	1849.14	0.155047
$523$	−8930.92	−0.746696	−0.373348	−	0.927691i	$-0.621790\pi$
$523$	−8930.92	−0.746696	−0.373348	+	0.927691i	$0.621790\pi$
$524$	644.496	0.0537308
$525$	−2624.49	−0.218176
$526$	1333.23	0.110516
$527$	−8972.81	−0.741673
$528$	486.858	0.0401283
$529$	−7921.43	−0.651059
$530$	−32.7514	−0.00268420
$531$	−5882.96	−0.480788
$532$	−1077.76	−0.0878325
$533$	−6196.87	−0.503595
$534$	1265.99	0.102594
$535$	−154.090	−0.0124521
$536$	1697.26	0.136773
$537$	−1769.31	−0.142181
$538$	6702.67	0.537124
$539$	−539.000	−0.0430730
$540$	−24.0384	−0.00191565
$541$	−14163.8	−1.12560	−0.562800	−	0.826593i	$-0.690276\pi$
$541$	−14163.8	−1.12560	−0.562800	+	0.826593i	$0.690276\pi$
$542$	−3880.58	−0.307537
$543$	7133.89	0.563802
$544$	19171.5	1.51098
$545$	−94.3389	−0.00741474
$546$	2804.31	0.219805
$547$	13587.7	1.06210	0.531050	−	0.847341i	$-0.321798\pi$
$547$	13587.7	1.06210	0.531050	+	0.847341i	$0.321798\pi$
$548$	8127.63	0.633568
$549$	2656.01	0.206477
$550$	2069.36	0.160433
$551$	−3664.93	−0.283360
$552$	4041.18	0.311601
$553$	−2943.17	−0.226323
$554$	−1028.53	−0.0788772
$555$	182.212	0.0139360
$556$	−1928.88	−0.147127
$557$	−4396.25	−0.334425	−0.167213	−	0.985921i	$-0.553477\pi$
$557$	−4396.25	−0.334425	−0.167213	+	0.985921i	$0.553477\pi$
$558$	−1189.49	−0.0902420
$559$	26028.5	1.96939
$560$	16.0347	0.00120998
$561$	3372.43	0.253804
$562$	−12982.3	−0.974419
$563$	−15806.4	−1.18323	−0.591615	−	0.806221i	$-0.701509\pi$
$563$	−15806.4	−1.18323	−0.591615	+	0.806221i	$0.701509\pi$
$564$	2108.76	0.157438
$565$	136.110	0.0101349
$566$	1885.63	0.140033
$567$	−567.000	−0.0419961
$568$	11978.5	0.884869
$569$	7381.00	0.543810	0.271905	−	0.962324i	$-0.412346\pi$
$569$	7381.00	0.543810	0.271905	+	0.962324i	$0.412346\pi$
$570$	−18.8265	−0.00138343
$571$	−20365.2	−1.49257	−0.746285	−	0.665627i	$-0.768164\pi$
$571$	−20365.2	−1.49257	−0.746285	+	0.665627i	$0.768164\pi$
$572$	5595.64	0.409030
$573$	11096.2	0.808989
$574$	736.035	0.0535218
$575$	−8143.18	−0.590599
$576$	1479.25	0.107006
$577$	−2587.33	−0.186676	−0.0933378	−	0.995634i	$-0.529754\pi$
$577$	−2587.33	−0.186676	−0.0933378	+	0.995634i	$0.529754\pi$
$578$	8325.36	0.599116
$579$	−15598.2	−1.11959
$580$	121.521	0.00869980
$581$	−417.953	−0.0298444
$582$	−86.5213	−0.00616224
$583$	−1541.45	−0.109503
$584$	2557.65	0.181226
$585$	−123.967	−0.00876139
$586$	−9547.67	−0.673056
$587$	−26606.6	−1.87082	−0.935410	−	0.353565i	$-0.884969\pi$
$587$	−26606.6	−1.87082	−0.935410	+	0.353565i	$0.884969\pi$
$588$	842.917	0.0591179
$589$	2357.53	0.164924
$590$	152.773	0.0106603
$591$	−4068.48	−0.283172
$592$	5771.24	0.400670
$593$	−26279.4	−1.81984	−0.909920	−	0.414783i	$-0.863857\pi$
$593$	−26279.4	−1.81984	−0.909920	+	0.414783i	$0.863857\pi$
$594$	447.069	0.0308812
$595$	111.071	0.00765290
$596$	−17952.4	−1.23382
$597$	−7520.59	−0.515573
$598$	8701.12	0.595009
$599$	26971.4	1.83977	0.919884	−	0.392190i	$-0.128282\pi$
$599$	26971.4	1.83977	0.919884	+	0.392190i	$0.128282\pi$
$600$	−7751.15	−0.527399
$601$	11604.8	0.787639	0.393819	−	0.919188i	$-0.371154\pi$
$601$	11604.8	0.787639	0.393819	+	0.919188i	$0.371154\pi$
$602$	−3091.55	−0.209306
$603$	−738.878	−0.0498996
$604$	−19616.6	−1.32151
$605$	−18.7871	−0.00126249
$606$	162.449	0.0108895
$607$	1612.10	0.107797	0.0538987	−	0.998546i	$-0.482835\pi$
$607$	1612.10	0.107797	0.0538987	+	0.998546i	$0.482835\pi$
$608$	−5037.14	−0.335992
$609$	2866.34	0.190723
$610$	−68.9731	−0.00457810
$611$	10875.0	0.720056
$612$	−5273.98	−0.348346
$613$	−11448.9	−0.754348	−0.377174	−	0.926142i	$-0.623104\pi$
$613$	−11448.9	−0.754348	−0.377174	+	0.926142i	$0.623104\pi$
$614$	−8291.54	−0.544982
$615$	−32.5371	−0.00213337
$616$	−1591.88	−0.104121
$617$	23635.8	1.54221	0.771104	−	0.636710i	$-0.219705\pi$
$617$	23635.8	1.54221	0.771104	+	0.636710i	$0.219705\pi$
$618$	−7631.93	−0.496766
$619$	4143.73	0.269064	0.134532	−	0.990909i	$-0.457047\pi$
$619$	4143.73	0.269064	0.134532	+	0.990909i	$0.457047\pi$
$620$	−78.1704	−0.00506355
$621$	−1759.27	−0.113683
$622$	−8074.10	−0.520485
$623$	1962.42	0.126200
$624$	−3926.44	−0.251896
$625$	15616.0	0.999421
$626$	−8768.37	−0.559831
$627$	−886.076	−0.0564377
$628$	9712.32	0.617140
$629$	39977.0	2.53416
$630$	14.7242	0.000931155 0
$631$	16119.6	1.01697	0.508487	−	0.861070i	$-0.330205\pi$
$631$	16119.6	1.01697	0.508487	+	0.861070i	$0.330205\pi$
$632$	−8692.34	−0.547093
$633$	−1893.65	−0.118903
$634$	−8728.29	−0.546758
$635$	−146.838	−0.00917654
$636$	2410.60	0.150293
$637$	4346.96	0.270381
$638$	−2260.06	−0.140245
$639$	−5214.65	−0.322830
$640$	194.605	0.0120194
$641$	1045.51	0.0644231	0.0322116	−	0.999481i	$-0.489745\pi$
$641$	1045.51	0.0644231	0.0322116	+	0.999481i	$0.489745\pi$
$642$	−4481.65	−0.275509
$643$	−7915.19	−0.485451	−0.242725	−	0.970095i	$-0.578041\pi$
$643$	−7915.19	−0.485451	−0.242725	+	0.970095i	$0.578041\pi$
$644$	2615.37	0.160031
$645$	136.665	0.00834289
$646$	−4130.51	−0.251567
$647$	6116.20	0.371642	0.185821	−	0.982584i	$-0.440506\pi$
$647$	6116.20	0.371642	0.185821	+	0.982584i	$0.440506\pi$
$648$	−1674.57	−0.101518
$649$	7190.28	0.434889
$650$	−16689.1	−1.00708
$651$	−1843.82	−0.111006
$652$	2514.76	0.151052
$653$	6143.47	0.368166	0.184083	−	0.982911i	$-0.441068\pi$
$653$	6143.47	0.368166	0.184083	+	0.982911i	$0.441068\pi$
$654$	−2743.82	−0.164055
$655$	17.4513	0.00104104
$656$	−1030.55	−0.0613359
$657$	−1113.43	−0.0661175
$658$	−1291.68	−0.0765271
$659$	18463.0	1.09138	0.545688	−	0.837989i	$-0.316269\pi$
$659$	18463.0	1.09138	0.545688	+	0.837989i	$0.316269\pi$
$660$	29.3803	0.00173277
$661$	−22732.1	−1.33764	−0.668818	−	0.743426i	$-0.733200\pi$
$661$	−22732.1	−1.33764	−0.668818	+	0.743426i	$0.733200\pi$
$662$	−13744.1	−0.806916
$663$	−27198.1	−1.59320
$664$	−1234.38	−0.0721433
$665$	−29.1830	−0.00170176
$666$	5299.58	0.308340
$667$	8893.59	0.516283
$668$	−14948.1	−0.865806
$669$	−8290.74	−0.479131
$670$	19.1877	0.00110640
$671$	−3246.24	−0.186765
$672$	3939.55	0.226148
$673$	−5912.78	−0.338664	−0.169332	−	0.985559i	$-0.554161\pi$
$673$	−5912.78	−0.338664	−0.169332	+	0.985559i	$0.554161\pi$
$674$	−3620.00	−0.206880
$675$	3374.35	0.192413
$676$	−32530.2	−1.85083
$677$	−23649.2	−1.34256	−0.671280	−	0.741204i	$-0.734255\pi$
$677$	−23649.2	−1.34256	−0.671280	+	0.741204i	$0.734255\pi$
$678$	3958.72	0.224238
$679$	−134.117	−0.00758015
$680$	328.036	0.0184994
$681$	17408.6	0.979586
$682$	1453.82	0.0816270
$683$	−17165.7	−0.961677	−0.480839	−	0.876809i	$-0.659668\pi$
$683$	−17165.7	−0.961677	−0.480839	+	0.876809i	$0.659668\pi$
$684$	1385.69	0.0774610
$685$	220.075	0.0122754
$686$	−516.311	−0.0287360
$687$	−20224.8	−1.12318
$688$	4328.61	0.239864
$689$	12431.6	0.687381
$690$	45.6858	0.00252062
$691$	−6932.25	−0.381643	−0.190821	−	0.981625i	$-0.561115\pi$
$691$	−6932.25	−0.381643	−0.190821	+	0.981625i	$0.561115\pi$
$692$	25795.0	1.41702
$693$	693.000	0.0379869
$694$	10911.4	0.596819
$695$	−52.2291	−0.00285059
$696$	8465.43	0.461036
$697$	−7138.57	−0.387938
$698$	−6248.29	−0.338827
$699$	−16075.8	−0.869874
$700$	−5016.39	−0.270860
$701$	−12751.9	−0.687065	−0.343533	−	0.939141i	$-0.611624\pi$
$701$	−12751.9	−0.687065	−0.343533	+	0.939141i	$0.611624\pi$
$702$	−3605.54	−0.193850
$703$	−10503.6	−0.563515
$704$	−1807.97	−0.0967905
$705$	57.0998	0.00305036
$706$	2397.67	0.127815
$707$	251.811	0.0133951
$708$	−11244.5	−0.596887
$709$	−7860.09	−0.416350	−0.208175	−	0.978092i	$-0.566752\pi$
$709$	−7860.09	−0.416350	−0.208175	+	0.978092i	$0.566752\pi$
$710$	135.418	0.00715794
$711$	3784.08	0.199598
$712$	5795.78	0.305065
$713$	−5720.95	−0.300493
$714$	3230.47	0.169324
$715$	151.515	0.00792497
$716$	−3381.82	−0.176515
$717$	15486.4	0.806625
$718$	−12778.4	−0.664188
$719$	−17425.0	−0.903816	−0.451908	−	0.892065i	$-0.649256\pi$
$719$	−17425.0	−0.903816	−0.451908	+	0.892065i	$0.649256\pi$
$720$	−20.6160	−0.00106710
$721$	−11830.2	−0.611070
$722$	−9239.47	−0.476257
$723$	19214.1	0.988353
$724$	13635.5	0.699946
$725$	−17058.3	−0.873833
$726$	−546.417	−0.0279331
$727$	23685.4	1.20831	0.604156	−	0.796866i	$-0.293510\pi$
$727$	23685.4	1.20831	0.604156	+	0.796866i	$0.293510\pi$
$728$	12838.3	0.653596
$729$	729.000	0.0370370
$730$	28.9144	0.00146599
$731$	29983.9	1.51709
$732$	5076.64	0.256336
$733$	−21418.3	−1.07927	−0.539634	−	0.841899i	$-0.681438\pi$
$733$	−21418.3	−1.07927	−0.539634	+	0.841899i	$0.681438\pi$
$734$	−10004.3	−0.503087
$735$	22.8240	0.00114541
$736$	12223.5	0.612179
$737$	903.073	0.0451358
$738$	−946.331	−0.0472018
$739$	−8761.40	−0.436121	−0.218060	−	0.975935i	$-0.569973\pi$
$739$	−8761.40	−0.436121	−0.218060	+	0.975935i	$0.569973\pi$
$740$	348.276	0.0173012
$741$	7146.08	0.354275
$742$	−1476.57	−0.0730545
$743$	10338.8	0.510489	0.255245	−	0.966877i	$-0.417844\pi$
$743$	10338.8	0.510489	0.255245	+	0.966877i	$0.417844\pi$
$744$	−5445.53	−0.268337
$745$	−486.105	−0.0239054
$746$	7693.18	0.377570
$747$	537.368	0.0263203
$748$	6445.98	0.315091
$749$	−6947.00	−0.338902
$750$	−175.272	−0.00853336
$751$	14703.4	0.714426	0.357213	−	0.934023i	$-0.383727\pi$
$751$	14703.4	0.714426	0.357213	+	0.934023i	$0.383727\pi$
$752$	1808.53	0.0877001
$753$	−925.371	−0.0447840
$754$	18227.0	0.880358
$755$	−531.168	−0.0256042
$756$	−1083.75	−0.0521371
$757$	−26676.3	−1.28080	−0.640400	−	0.768041i	$-0.721232\pi$
$757$	−26676.3	−1.28080	−0.640400	+	0.768041i	$0.721232\pi$
$758$	−12340.5	−0.591327
$759$	2150.21	0.102830
$760$	−86.1887	−0.00411368
$761$	26816.8	1.27741	0.638704	−	0.769453i	$-0.279471\pi$
$761$	26816.8	1.27741	0.638704	+	0.769453i	$0.279471\pi$
$762$	−4270.75	−0.203035
$763$	−4253.18	−0.201803
$764$	21209.0	1.00434
$765$	−142.806	−0.00674922
$766$	−2886.31	−0.136144
$767$	−57988.6	−2.72992
$768$	9604.69	0.451275
$769$	40144.7	1.88251	0.941257	−	0.337690i	$-0.109646\pi$
$769$	40144.7	1.88251	0.941257	+	0.337690i	$0.109646\pi$
$770$	−17.9963	−0.000842262 0
$771$	−16640.9	−0.777311
$772$	−29814.1	−1.38994
$773$	36628.4	1.70431	0.852156	−	0.523288i	$-0.175295\pi$
$773$	36628.4	1.70431	0.852156	+	0.523288i	$0.175295\pi$
$774$	3974.85	0.184590
$775$	10973.0	0.508597
$776$	−396.098	−0.0183236
$777$	8214.87	0.379288
$778$	7737.14	0.356542
$779$	1875.60	0.0862648
$780$	−236.948	−0.0108770
$781$	6373.47	0.292011
$782$	10023.4	0.458357
$783$	−3685.30	−0.168202
$784$	722.910	0.0329314
$785$	262.985	0.0119571
$786$	507.565	0.0230334
$787$	19339.8	0.875973	0.437987	−	0.898982i	$-0.355692\pi$
$787$	19339.8	0.875973	0.437987	+	0.898982i	$0.355692\pi$
$788$	−7776.40	−0.351552
$789$	−2657.11	−0.119893
$790$	−98.2676	−0.00442557
$791$	6136.40	0.275835
$792$	2046.70	0.0918261
$793$	26180.4	1.17238
$794$	19508.5	0.871954
$795$	65.2729	0.00291194
$796$	−14374.7	−0.640071
$797$	22734.8	1.01042	0.505211	−	0.862996i	$-0.331415\pi$
$797$	22734.8	1.01042	0.505211	+	0.862996i	$0.331415\pi$
$798$	−848.778	−0.0376521
$799$	12527.6	0.554686
$800$	−23445.2	−1.03614
$801$	−2523.11	−0.111298
$802$	−7967.21	−0.350788
$803$	1360.86	0.0598055
$804$	−1412.27	−0.0619491
$805$	70.8175	0.00310061
$806$	−11724.8	−0.512395
$807$	−13358.3	−0.582695
$808$	743.697	0.0323802
$809$	10636.0	0.462228	0.231114	−	0.972927i	$-0.425763\pi$
$809$	10636.0	0.462228	0.231114	+	0.972927i	$0.425763\pi$
$810$	−18.9312	−0.000821202 0
$811$	21207.0	0.918223	0.459112	−	0.888379i	$-0.348168\pi$
$811$	21207.0	0.918223	0.459112	+	0.888379i	$0.348168\pi$
$812$	5478.66	0.236778
$813$	7733.93	0.333630
$814$	−6477.27	−0.278904
$815$	68.0933	0.00292663
$816$	−4523.12	−0.194045
$817$	−7878.02	−0.337353
$818$	8346.83	0.356773
$819$	−5588.95	−0.238454
$820$	−62.1906	−0.00264853
$821$	24576.3	1.04472	0.522362	−	0.852724i	$-0.325051\pi$
$821$	24576.3	1.04472	0.522362	+	0.852724i	$0.325051\pi$
$822$	6400.81	0.271599
$823$	−31692.5	−1.34232	−0.671161	−	0.741312i	$-0.734204\pi$
$823$	−31692.5	−1.34232	−0.671161	+	0.741312i	$0.734204\pi$
$824$	−34939.3	−1.47715
$825$	−4124.20	−0.174044
$826$	6887.62	0.290134
$827$	−28350.2	−1.19206	−0.596029	−	0.802963i	$-0.703256\pi$
$827$	−28350.2	−1.19206	−0.596029	+	0.802963i	$0.703256\pi$
$828$	−3362.62	−0.141134
$829$	−14688.3	−0.615375	−0.307688	−	0.951487i	$-0.599555\pi$
$829$	−14688.3	−0.615375	−0.307688	+	0.951487i	$0.599555\pi$
$830$	−13.9547	−0.000583586 0
$831$	2049.84	0.0855694
$832$	14581.0	0.607580
$833$	5007.54	0.208285
$834$	−1519.06	−0.0630706
$835$	−404.755	−0.0167750
$836$	−1693.62	−0.0700661
$837$	2370.63	0.0978984
$838$	5256.91	0.216703
$839$	35911.1	1.47770	0.738849	−	0.673871i	$-0.235369\pi$
$839$	35911.1	1.47770	0.738849	+	0.673871i	$0.235369\pi$
$840$	67.4082	0.00276881
$841$	−5758.78	−0.236122
$842$	23253.9	0.951761
$843$	25873.4	1.05709
$844$	−3619.48	−0.147615
$845$	−880.833	−0.0358598
$846$	1660.73	0.0674906
$847$	−847.000	−0.0343604
$848$	2067.40	0.0837204
$849$	−3758.03	−0.151914
$850$	−19225.3	−0.775790
$851$	25488.8	1.02673
$852$	−9967.16	−0.400786
$853$	37842.1	1.51898	0.759490	−	0.650519i	$-0.225449\pi$
$853$	37842.1	1.51898	0.759490	+	0.650519i	$0.225449\pi$
$854$	−3109.59	−0.124599
$855$	37.5210	0.00150081
$856$	−20517.2	−0.819233
$857$	−41429.0	−1.65133	−0.825663	−	0.564164i	$-0.809199\pi$
$857$	−41429.0	−1.65133	−0.825663	+	0.564164i	$0.809199\pi$
$858$	4406.78	0.175344
$859$	2803.59	0.111359	0.0556795	−	0.998449i	$-0.482268\pi$
$859$	2803.59	0.111359	0.0556795	+	0.998449i	$0.482268\pi$
$860$	261.218	0.0103575
$861$	−1466.91	−0.0580627
$862$	−10444.1	−0.412676
$863$	30586.2	1.20645	0.603225	−	0.797571i	$-0.293882\pi$
$863$	30586.2	1.20645	0.603225	+	0.797571i	$0.293882\pi$
$864$	−5065.14	−0.199444
$865$	698.461	0.0274548
$866$	−4245.36	−0.166586
$867$	−16592.3	−0.649947
$868$	−3524.24	−0.137812
$869$	−4624.99	−0.180543
$870$	95.7024	0.00372944
$871$	−7283.16	−0.283330
$872$	−12561.3	−0.487821
$873$	172.436	0.00668506
$874$	−2633.56	−0.101924
$875$	−271.688	−0.0104968
$876$	−2128.19	−0.0820832
$877$	37717.7	1.45226	0.726132	−	0.687555i	$-0.241316\pi$
$877$	37717.7	1.45226	0.726132	+	0.687555i	$0.241316\pi$
$878$	23364.3	0.898072
$879$	19028.3	0.730159
$880$	25.1974	0.000965231 0
$881$	34072.2	1.30298	0.651488	−	0.758659i	$-0.274145\pi$
$881$	34072.2	1.30298	0.651488	+	0.758659i	$0.274145\pi$
$882$	663.829	0.0253427
$883$	42756.6	1.62953	0.814765	−	0.579791i	$-0.196866\pi$
$883$	42756.6	1.62953	0.814765	+	0.579791i	$0.196866\pi$
$884$	−51985.9	−1.97791
$885$	−304.473	−0.0115647
$886$	18345.2	0.695620
$887$	−5921.20	−0.224142	−0.112071	−	0.993700i	$-0.535748\pi$
$887$	−5921.20	−0.224142	−0.112071	+	0.993700i	$0.535748\pi$
$888$	24261.7	0.916858
$889$	−6620.08	−0.249753
$890$	65.5217	0.00246775
$891$	−891.000	−0.0335013
$892$	−15846.7	−0.594829
$893$	−3291.51	−0.123344
$894$	−14138.2	−0.528918
$895$	−91.5709	−0.00341998
$896$	8773.60	0.327127
$897$	−17341.2	−0.645491
$898$	13635.5	0.506707
$899$	−11984.2	−0.444600
$900$	6449.65	0.238876
$901$	14320.7	0.529515
$902$	1156.63	0.0426956
$903$	6161.40	0.227064
$904$	18123.2	0.666778
$905$	369.215	0.0135615
$906$	−15448.8	−0.566505
$907$	−42922.8	−1.57137	−0.785683	−	0.618629i	$-0.787688\pi$
$907$	−42922.8	−1.57137	−0.785683	+	0.618629i	$0.787688\pi$
$908$	33274.3	1.21613
$909$	−323.758	−0.0118134
$910$	145.138	0.00528710
$911$	35023.9	1.27376	0.636878	−	0.770964i	$-0.280225\pi$
$911$	35023.9	1.27376	0.636878	+	0.770964i	$0.280225\pi$
$912$	1188.41	0.0431493
$913$	−656.784	−0.0238076
$914$	−1611.81	−0.0583302
$915$	137.462	0.00496651
$916$	−38657.2	−1.39440
$917$	786.775	0.0283333
$918$	−4153.46	−0.149330
$919$	−48830.6	−1.75275	−0.876373	−	0.481633i	$-0.840044\pi$
$919$	−48830.6	−1.75275	−0.876373	+	0.481633i	$0.840044\pi$
$920$	209.152	0.00749514
$921$	16524.9	0.591220
$922$	−15608.0	−0.557509
$923$	−51401.1	−1.83303
$924$	1324.58	0.0471597
$925$	−48888.6	−1.73778
$926$	−9536.14	−0.338420
$927$	15210.3	0.538913
$928$	25605.7	0.905763
$929$	16251.5	0.573945	0.286972	−	0.957939i	$-0.407351\pi$
$929$	16251.5	0.573945	0.286972	+	0.957939i	$0.407351\pi$
$930$	−61.5621	−0.00217065
$931$	−1315.69	−0.0463157
$932$	−30726.9	−1.07993
$933$	16091.5	0.564644
$934$	−19452.1	−0.681469
$935$	174.540	0.00610490
$936$	−16506.3	−0.576417
$937$	−9350.37	−0.326001	−0.163001	−	0.986626i	$-0.552117\pi$
$937$	−9350.37	−0.326001	−0.163001	+	0.986626i	$0.552117\pi$
$938$	865.059	0.0301121
$939$	17475.2	0.607329
$940$	109.139	0.00378695
$941$	−16381.6	−0.567507	−0.283754	−	0.958897i	$-0.591580\pi$
$941$	−16381.6	−0.567507	−0.283754	+	0.958897i	$0.591580\pi$
$942$	7648.82	0.264556
$943$	−4551.46	−0.157175
$944$	−9643.65	−0.332494
$945$	−29.3452	−0.00101016
$946$	−4858.15	−0.166968
$947$	11004.5	0.377613	0.188807	−	0.982014i	$-0.439538\pi$
$947$	11004.5	0.377613	0.188807	+	0.982014i	$0.439538\pi$
$948$	7232.80	0.247796
$949$	−10975.2	−0.375415
$950$	5051.27	0.172510
$951$	17395.3	0.593146
$952$	14789.2	0.503489
$953$	45741.3	1.55478	0.777391	−	0.629018i	$-0.216543\pi$
$953$	45741.3	1.55478	0.777391	+	0.629018i	$0.216543\pi$
$954$	1898.44	0.0644280
$955$	574.285	0.0194591
$956$	29600.3	1.00140
$957$	4504.25	0.152144
$958$	−22099.9	−0.745318
$959$	9921.89	0.334092
$960$	76.5588	0.00257388
$961$	−22082.0	−0.741229
$962$	52238.3	1.75076
$963$	8931.86	0.298884
$964$	36725.3	1.22702
$965$	−807.288	−0.0269301
$966$	2059.70	0.0686024
$967$	32031.8	1.06522	0.532612	−	0.846359i	$-0.321210\pi$
$967$	32031.8	1.06522	0.532612	+	0.846359i	$0.321210\pi$
$968$	−2501.52	−0.0830598
$969$	8232.03	0.272911
$970$	−4.47792	−0.000148224 0
$971$	41167.9	1.36060	0.680299	−	0.732934i	$-0.261850\pi$
$971$	41167.9	1.36060	0.680299	+	0.732934i	$0.261850\pi$
$972$	1393.39	0.0459806
$973$	−2354.70	−0.0775829
$974$	572.047	0.0188189
$975$	33261.1	1.09252
$976$	4353.87	0.142791
$977$	−34251.8	−1.12161	−0.560805	−	0.827948i	$-0.689508\pi$
$977$	−34251.8	−1.12161	−0.560805	+	0.827948i	$0.689508\pi$
$978$	1980.47	0.0647530
$979$	3083.80	0.100673
$980$	43.6253	0.00142200
$981$	5468.38	0.177973
$982$	−7566.25	−0.245874
$983$	−49982.0	−1.62175	−0.810875	−	0.585220i	$-0.801008\pi$
$983$	−49982.0	−1.62175	−0.810875	+	0.585220i	$0.801008\pi$
$984$	−4332.34	−0.140356
$985$	−210.565	−0.00681132
$986$	20996.9	0.678172
$987$	2574.29	0.0830199
$988$	13658.8	0.439824
$989$	19117.4	0.614658
$990$	23.1381	0.000742805 0
$991$	1157.08	0.0370895	0.0185448	−	0.999828i	$-0.494097\pi$
$991$	1157.08	0.0370895	0.0185448	+	0.999828i	$0.494097\pi$
$992$	−16471.3	−0.527181
$993$	27391.7	0.875377
$994$	6105.18	0.194813
$995$	−389.229	−0.0124014
$996$	1027.11	0.0326760
$997$	−17659.7	−0.560972	−0.280486	−	0.959858i	$-0.590496\pi$
$997$	−17659.7	−0.560972	−0.280486	+	0.959858i	$0.590496\pi$
$998$	20811.7	0.660103
$999$	−10562.0	−0.334501

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	231.4.a.j.1.4	✓	5
3.2	odd	2		693.4.a.q.1.2		5
7.6	odd	2		1617.4.a.o.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.j.1.4	✓	5	1.1	even	1	trivial
693.4.a.q.1.2		5	3.2	odd	2
1617.4.a.o.1.4		5	7.6	odd	2

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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

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Fields

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Database

Properties

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Newspace parameters

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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	231.4.a.j.1.4

Level	$231$
Weight	$4$
Character	231.1
Self dual	yes
Analytic conductor	$13.629$
Analytic rank	$0$
Dimension	$5$
CM	no
Inner twists	$1$