LMFDB - Embedded newform 693.4.a.q.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
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:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.50528$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.50528 q^{2} -5.73413 q^{4} +0.155265 q^{5} -7.00000 q^{7} +20.6737 q^{8} +O(q^{10})$	\(q-1.50528 q^{2} -5.73413 q^{4} +0.155265 q^{5} -7.00000 q^{7} +20.6737 q^{8} -0.233718 q^{10} +11.0000 q^{11} +88.7135 q^{13} +10.5370 q^{14} +14.7533 q^{16} -102.195 q^{17} -26.8508 q^{19} -0.890312 q^{20} -16.5581 q^{22} -65.1580 q^{23} -124.976 q^{25} -133.539 q^{26} +40.1389 q^{28} -136.493 q^{29} -87.8011 q^{31} -187.598 q^{32} +153.832 q^{34} -1.08686 q^{35} +391.184 q^{37} +40.4180 q^{38} +3.20991 q^{40} +69.8526 q^{41} +293.400 q^{43} -63.0754 q^{44} +98.0811 q^{46} -122.585 q^{47} +49.0000 q^{49} +188.124 q^{50} -508.694 q^{52} -140.132 q^{53} +1.70792 q^{55} -144.716 q^{56} +205.460 q^{58} +653.662 q^{59} +295.112 q^{61} +132.165 q^{62} +164.361 q^{64} +13.7741 q^{65} -82.0975 q^{67} +585.998 q^{68} +1.63603 q^{70} +579.406 q^{71} -123.715 q^{73} -588.842 q^{74} +153.966 q^{76} -77.0000 q^{77} +420.453 q^{79} +2.29067 q^{80} -105.148 q^{82} -59.7076 q^{83} -15.8673 q^{85} -441.650 q^{86} +227.411 q^{88} +280.345 q^{89} -620.994 q^{91} +373.624 q^{92} +184.525 q^{94} -4.16900 q^{95} +19.1595 q^{97} -73.7588 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8	$ 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26} - 147 q^{28} - 591 q^{29} + 366 q^{31} + 87 q^{32} + 594 q^{34} + 49 q^{35} + 259 q^{37} - 175 q^{38} + 1890 q^{40} + 104 q^{41} + 224 q^{43} + 231 q^{44} + 1416 q^{46} + 453 q^{47} + 245 q^{49} - 1350 q^{50} + 815 q^{52} - 1032 q^{53} - 77 q^{55} - 84 q^{56} + 293 q^{58} + 517 q^{59} + 958 q^{61} + 3030 q^{62} + 516 q^{64} + 197 q^{65} + 361 q^{67} + 1680 q^{68} - 791 q^{70} - 548 q^{71} - 1035 q^{73} + 2555 q^{74} - 5201 q^{76} - 385 q^{77} + 776 q^{79} + 1273 q^{80} + 376 q^{82} + 1974 q^{83} + 1838 q^{85} - 1224 q^{86} + 132 q^{88} - 2710 q^{89} - 161 q^{91} + 2834 q^{92} - 563 q^{94} + 355 q^{95} + 1988 q^{97} + 49 q^{98}+O(q^{100}) $ 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * 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 $

$2$	−1.50528	−0.532197	−0.266099	−	0.963946i	$-0.585735\pi$
$2$	−1.50528	−0.532197	−0.266099	+	0.963946i	$0.585735\pi$
$3$	0	0
$3$	0	0
$4$	−5.73413	−0.716766
$5$	0.155265	0.0138874	0.00694368	−	0.999976i	$-0.497790\pi$
$5$	0.155265	0.0138874	0.00694368	+	0.999976i	$0.497790\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	20.6737	0.913658
$9$	0	0
$10$	−0.233718	−0.00739082
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	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	0
$16$	14.7533	0.230520
$17$	−102.195	−1.45799	−0.728996	−	0.684518i	$-0.760013\pi$
$17$	−102.195	−1.45799	−0.728996	+	0.684518i	$0.760013\pi$
$18$	0	0
$19$	−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
$21$	0	0
$22$	−16.5581	−0.160464
$23$	−65.1580	−0.590712	−0.295356	−	0.955387i	$-0.595438\pi$
$23$	−65.1580	−0.590712	−0.295356	+	0.955387i	$0.595438\pi$
$24$	0	0
$25$	−124.976	−0.999807
$26$	−133.539	−1.00727
$27$	0	0
$28$	40.1389	0.270912
$29$	−136.493	−0.874001	−0.437001	−	0.899461i	$-0.643959\pi$
$29$	−136.493	−0.874001	−0.437001	+	0.899461i	$0.643959\pi$
$30$	0	0
$31$	−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
$33$	0	0
$34$	153.832	0.775939
$35$	−1.08686	−0.00524893
$36$	0	0
$37$	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
$39$	0	0
$40$	3.20991	0.0126883
$41$	69.8526	0.266077	0.133038	−	0.991111i	$-0.457527\pi$
$41$	69.8526	0.266077	0.133038	+	0.991111i	$0.457527\pi$
$42$	0	0
$43$	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
$45$	0	0
$46$	98.0811	0.314376
$47$	−122.585	−0.380445	−0.190223	−	0.981741i	$-0.560921\pi$
$47$	−122.585	−0.380445	−0.190223	+	0.981741i	$0.560921\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	188.124	0.532095
$51$	0	0
$52$	−508.694	−1.35660
$53$	−140.132	−0.363181	−0.181591	−	0.983374i	$-0.558125\pi$
$53$	−140.132	−0.363181	−0.181591	+	0.983374i	$0.558125\pi$
$54$	0	0
$55$	1.70792	0.00418720
$56$	−144.716	−0.345330
$57$	0	0
$58$	205.460	0.465141
$59$	653.662	1.44237	0.721183	−	0.692745i	$-0.243599\pi$
$59$	653.662	1.44237	0.721183	+	0.692745i	$0.243599\pi$
$60$	0	0
$61$	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
$63$	0	0
$64$	164.361	0.321018
$65$	13.7741	0.0262842
$66$	0	0
$67$	−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
$69$	0	0
$70$	1.63603	0.00279347
$71$	579.406	0.968491	0.484245	−	0.874932i	$-0.339094\pi$
$71$	579.406	0.968491	0.484245	+	0.874932i	$0.339094\pi$
$72$	0	0
$73$	−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
$75$	0	0
$76$	153.966	0.232383
$77$	−77.0000	−0.113961
$78$	0	0
$79$	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
$81$	0	0
$82$	−105.148	−0.141605
$83$	−59.7076	−0.0789610	−0.0394805	−	0.999220i	$-0.512570\pi$
$83$	−59.7076	−0.0789610	−0.0394805	+	0.999220i	$0.512570\pi$
$84$	0	0
$85$	−15.8673	−0.0202477
$86$	−441.650	−0.553771
$87$	0	0
$88$	227.411	0.275478
$89$	280.345	0.333894	0.166947	−	0.985966i	$-0.446609\pi$
$89$	280.345	0.333894	0.166947	+	0.985966i	$0.446609\pi$
$90$	0	0
$91$	−620.994	−0.715361
$92$	373.624	0.423403
$93$	0	0
$94$	184.525	0.202472
$95$	−4.16900	−0.00450242
$96$	0	0
$97$	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
$99$	0	0
$100$	716.628	0.716628
$101$	35.9731	0.0354401	0.0177201	−	0.999843i	$-0.494359\pi$
$101$	35.9731	0.0354401	0.0177201	+	0.999843i	$0.494359\pi$
$102$	0	0
$103$	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
$105$	0	0
$106$	210.938	0.193284
$107$	−992.428	−0.896651	−0.448326	−	0.893870i	$-0.647979\pi$
$107$	−992.428	−0.896651	−0.448326	+	0.893870i	$0.647979\pi$
$108$	0	0
$109$	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
$111$	0	0
$112$	−103.273	−0.0871282
$113$	876.628	0.729790	0.364895	−	0.931049i	$-0.381105\pi$
$113$	876.628	0.729790	0.364895	+	0.931049i	$0.381105\pi$
$114$	0	0
$115$	−10.1168	−0.00820344
$116$	782.666	0.626454
$117$	0	0
$118$	−983.945	−0.767623
$119$	715.363	0.551069
$120$	0	0
$121$	121.000	0.0909091
$122$	−444.227	−0.329659
$123$	0	0
$124$	503.463	0.364615
$125$	−38.8126	−0.0277720
$126$	0	0
$127$	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
$129$	0	0
$130$	−20.7339	−0.0139884
$131$	112.396	0.0749628	0.0374814	−	0.999297i	$-0.488067\pi$
$131$	112.396	0.0749628	0.0374814	+	0.999297i	$0.488067\pi$
$132$	0	0
$133$	187.956	0.122540
$134$	123.580	0.0796692
$135$	0	0
$136$	−2112.75	−1.33211
$137$	1417.41	0.883925	0.441963	−	0.897033i	$-0.354282\pi$
$137$	1417.41	0.883925	0.441963	+	0.897033i	$0.354282\pi$
$138$	0	0
$139$	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
$141$	0	0
$142$	−872.169	−0.515428
$143$	975.848	0.570661
$144$	0	0
$145$	−21.1926	−0.0121376
$146$	186.226	0.105563
$147$	0	0
$148$	−2243.10	−1.24582
$149$	−3130.80	−1.72138	−0.860689	−	0.509132i	$-0.829967\pi$
$149$	−3130.80	−1.72138	−0.860689	+	0.509132i	$0.829967\pi$
$150$	0	0
$151$	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
$153$	0	0
$154$	115.907	0.0606495
$155$	−13.6325	−0.00706443
$156$	0	0
$157$	−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
$159$	0	0
$160$	−29.1274	−0.0143920
$161$	456.106	0.223268
$162$	0	0
$163$	−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
$165$	0	0
$166$	89.8767	0.0420228
$167$	−2606.86	−1.20793	−0.603967	−	0.797009i	$-0.706414\pi$
$167$	−2606.86	−1.20793	−0.603967	+	0.797009i	$0.706414\pi$
$168$	0	0
$169$	5673.08	2.58219
$170$	23.8848	0.0107758
$171$	0	0
$172$	−1682.39	−0.745822
$173$	4498.50	1.97696	0.988481	−	0.151343i	$-0.0483599\pi$
$173$	4498.50	1.97696	0.988481	+	0.151343i	$0.0483599\pi$
$174$	0	0
$175$	874.831	0.377892
$176$	162.286	0.0695043
$177$	0	0
$178$	−421.998	−0.177697
$179$	−589.770	−0.246265	−0.123133	−	0.992390i	$-0.539294\pi$
$179$	−589.770	−0.246265	−0.123133	+	0.992390i	$0.539294\pi$
$180$	0	0
$181$	−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
$183$	0	0
$184$	−1347.06	−0.539709
$185$	60.7374	0.0241379
$186$	0	0
$187$	−1124.14	−0.439601
$188$	702.920	0.272690
$189$	0	0
$190$	6.27552	0.00239618
$191$	3698.73	1.40121	0.700605	−	0.713550i	$-0.252914\pi$
$191$	3698.73	1.40121	0.700605	+	0.713550i	$0.252914\pi$
$192$	0	0
$193$	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
$195$	0	0
$196$	−280.972	−0.102395
$197$	−1356.16	−0.490469	−0.245234	−	0.969464i	$-0.578865\pi$
$197$	−1356.16	−0.490469	−0.245234	+	0.969464i	$0.578865\pi$
$198$	0	0
$199$	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
$201$	0	0
$202$	−54.1496	−0.0188611
$203$	955.448	0.330341
$204$	0	0
$205$	10.8457	0.00369510
$206$	−2543.98	−0.860424
$207$	0	0
$208$	1308.81	0.436297
$209$	−295.359	−0.0977530
$210$	0	0
$211$	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
$213$	0	0
$214$	1493.88	0.477195
$215$	45.5549	0.0144503
$216$	0	0
$217$	614.608	0.192269
$218$	−914.605	−0.284151
$219$	0	0
$220$	−9.79343	−0.00300124
$221$	−9066.05	−2.75950
$222$	0	0
$223$	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
$225$	0	0
$226$	−1319.57	−0.388392
$227$	5802.86	1.69669	0.848347	−	0.529441i	$-0.177598\pi$
$227$	5802.86	1.69669	0.848347	+	0.529441i	$0.177598\pi$
$228$	0	0
$229$	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
$231$	0	0
$232$	−2821.81	−0.798538
$233$	−5358.60	−1.50667	−0.753333	−	0.657639i	$-0.771555\pi$
$233$	−5358.60	−1.50667	−0.753333	+	0.657639i	$0.771555\pi$
$234$	0	0
$235$	−19.0333	−0.00528338
$236$	−3748.18	−1.03384
$237$	0	0
$238$	−1076.82	−0.293278
$239$	5162.13	1.39712	0.698558	−	0.715554i	$-0.253826\pi$
$239$	5162.13	1.39712	0.698558	+	0.715554i	$0.253826\pi$
$240$	0	0
$241$	−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
$243$	0	0
$244$	−1692.21	−0.443987
$245$	7.60801	0.00198391
$246$	0	0
$247$	−2382.03	−0.613622
$248$	−1815.18	−0.464773
$249$	0	0
$250$	58.4239	0.0147802
$251$	−308.457	−0.0775682	−0.0387841	−	0.999248i	$-0.512348\pi$
$251$	−308.457	−0.0775682	−0.0387841	+	0.999248i	$0.512348\pi$
$252$	0	0
$253$	−716.738	−0.178107
$254$	−1423.58	−0.351667
$255$	0	0
$256$	−3201.56	−0.781632
$257$	−5546.96	−1.34634	−0.673171	−	0.739487i	$-0.735068\pi$
$257$	−5546.96	−1.34634	−0.673171	+	0.739487i	$0.735068\pi$
$258$	0	0
$259$	−2738.29	−0.656946
$260$	−78.9826	−0.0188396
$261$	0	0
$262$	−169.188	−0.0398950
$263$	−885.702	−0.207661	−0.103830	−	0.994595i	$-0.533110\pi$
$263$	−885.702	−0.207661	−0.103830	+	0.994595i	$0.533110\pi$
$264$	0	0
$265$	−21.7576	−0.00504363
$266$	−282.926	−0.0652154
$267$	0	0
$268$	470.758	0.107299
$269$	−4452.77	−1.00926	−0.504629	−	0.863336i	$-0.668371\pi$
$269$	−4452.77	−1.00926	−0.504629	+	0.863336i	$0.668371\pi$
$270$	0	0
$271$	−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
$273$	0	0
$274$	−2133.60	−0.470423
$275$	−1374.73	−0.301453
$276$	0	0
$277$	−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
$279$	0	0
$280$	−22.4694	−0.00479573
$281$	8624.47	1.83094	0.915468	−	0.402391i	$-0.131821\pi$
$281$	8624.47	1.83094	0.915468	+	0.402391i	$0.131821\pi$
$282$	0	0
$283$	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
$285$	0	0
$286$	−1468.93	−0.303704
$287$	−488.968	−0.100568
$288$	0	0
$289$	5530.77	1.12574
$290$	31.9008	0.00645958
$291$	0	0
$292$	709.397	0.142172
$293$	6342.78	1.26467	0.632337	−	0.774694i	$-0.282096\pi$
$293$	6342.78	1.26467	0.632337	+	0.774694i	$0.282096\pi$
$294$	0	0
$295$	101.491	0.0200306
$296$	8087.24	1.58804
$297$	0	0
$298$	4712.74	0.916112
$299$	−5780.39	−1.11802
$300$	0	0
$301$	−2053.80	−0.393286
$302$	−5149.61	−0.981215
$303$	0	0
$304$	−396.137	−0.0747368
$305$	45.8207	0.00860226
$306$	0	0
$307$	−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
$309$	0	0
$310$	20.5207	0.00375967
$311$	5363.85	0.977993	0.488996	−	0.872286i	$-0.337363\pi$
$311$	5363.85	0.977993	0.488996	+	0.872286i	$0.337363\pi$
$312$	0	0
$313$	−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
$315$	0	0
$316$	−2410.93	−0.429195
$317$	5798.44	1.02736	0.513680	−	0.857982i	$-0.328282\pi$
$317$	5798.44	1.02736	0.513680	+	0.857982i	$0.328282\pi$
$318$	0	0
$319$	−1501.42	−0.263521
$320$	25.5196	0.00445809
$321$	0	0
$322$	−686.568	−0.118823
$323$	2744.01	0.472696
$324$	0	0
$325$	−11087.0	−1.89230
$326$	660.157	0.112156
$327$	0	0
$328$	1444.11	0.243103
$329$	858.098	0.143795
$330$	0	0
$331$	−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
$333$	0	0
$334$	3924.06	0.642859
$335$	−12.7469	−0.00207892
$336$	0	0
$337$	−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
$339$	0	0
$340$	90.9852	0.0145128
$341$	−965.812	−0.153377
$342$	0	0
$343$	−343.000	−0.0539949
$344$	6065.67	0.950695
$345$	0	0
$346$	−6771.51	−1.05213
$347$	−7248.76	−1.12142	−0.560712	−	0.828011i	$-0.689472\pi$
$347$	−7248.76	−1.12142	−0.560712	+	0.828011i	$0.689472\pi$
$348$	0	0
$349$	−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
$351$	0	0
$352$	−2063.57	−0.312468
$353$	−1592.84	−0.240165	−0.120082	−	0.992764i	$-0.538316\pi$
$353$	−1592.84	−0.240165	−0.120082	+	0.992764i	$0.538316\pi$
$354$	0	0
$355$	89.9617	0.0134498
$356$	−1607.54	−0.239324
$357$	0	0
$358$	887.770	0.131062
$359$	8489.06	1.24801	0.624005	−	0.781420i	$-0.285504\pi$
$359$	8489.06	1.24801	0.624005	+	0.781420i	$0.285504\pi$
$360$	0	0
$361$	−6138.04	−0.894888
$362$	3579.50	0.519708
$363$	0	0
$364$	3560.86	0.512747
$365$	−19.2086	−0.00275459
$366$	0	0
$367$	−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
$369$	0	0
$370$	−91.4269	−0.0128461
$371$	980.923	0.137270
$372$	0	0
$373$	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
$375$	0	0
$376$	−2534.30	−0.347597
$377$	−12108.7	−1.65419
$378$	0	0
$379$	−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
$381$	0	0
$382$	−5567.63	−0.745720
$383$	1917.46	0.255816	0.127908	−	0.991786i	$-0.459174\pi$
$383$	1917.46	0.255816	0.127908	+	0.991786i	$0.459174\pi$
$384$	0	0
$385$	−11.9554	−0.00158261
$386$	−7826.57	−1.03203
$387$	0	0
$388$	−109.863	−0.0143749
$389$	−5140.00	−0.669944	−0.334972	−	0.942228i	$-0.608727\pi$
$389$	−5140.00	−0.669944	−0.334972	+	0.942228i	$0.608727\pi$
$390$	0	0
$391$	6658.81	0.861254
$392$	1013.01	0.130523
$393$	0	0
$394$	2041.40	0.261026
$395$	65.2819	0.00831567
$396$	0	0
$397$	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
$399$	0	0
$400$	−1843.80	−0.230475
$401$	5292.84	0.659131	0.329566	−	0.944133i	$-0.393098\pi$
$401$	5292.84	0.659131	0.329566	+	0.944133i	$0.393098\pi$
$402$	0	0
$403$	−7789.14	−0.962791
$404$	−206.274	−0.0254023
$405$	0	0
$406$	−1438.22	−0.175807
$407$	4303.03	0.524062
$408$	0	0
$409$	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
$411$	0	0
$412$	−9690.88	−1.15882
$413$	−4575.63	−0.545163
$414$	0	0
$415$	−9.27053	−0.00109656
$416$	−16642.4	−1.96145
$417$	0	0
$418$	444.598	0.0520239
$419$	−3492.31	−0.407185	−0.203593	−	0.979056i	$-0.565262\pi$
$419$	−3492.31	−0.407185	−0.203593	+	0.979056i	$0.565262\pi$
$420$	0	0
$421$	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
$423$	0	0
$424$	−2897.05	−0.331823
$425$	12771.9	1.45771
$426$	0	0
$427$	−2065.79	−0.234123
$428$	5690.71	0.642689
$429$	0	0
$430$	−68.5729	−0.00769042
$431$	6938.29	0.775420	0.387710	−	0.921781i	$-0.373266\pi$
$431$	6938.29	0.775420	0.387710	+	0.921781i	$0.373266\pi$
$432$	0	0
$433$	−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
$435$	0	0
$436$	−3484.04	−0.382696
$437$	1749.54	0.191515
$438$	0	0
$439$	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
$441$	0	0
$442$	13647.0	1.46860
$443$	−12187.2	−1.30707	−0.653536	−	0.756896i	$-0.726715\pi$
$443$	−12187.2	−1.30707	−0.653536	+	0.756896i	$0.726715\pi$
$444$	0	0
$445$	43.5279	0.00463690
$446$	−4159.96	−0.441659
$447$	0	0
$448$	−1150.53	−0.121333
$449$	−9058.45	−0.952104	−0.476052	−	0.879417i	$-0.657933\pi$
$449$	−9058.45	−0.952104	−0.476052	+	0.879417i	$0.657933\pi$
$450$	0	0
$451$	768.379	0.0802252
$452$	−5026.70	−0.523088
$453$	0	0
$454$	−8734.94	−0.902975
$455$	−96.4189	−0.00993448
$456$	0	0
$457$	−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
$459$	0	0
$460$	58.0110	0.00587995
$461$	10368.9	1.04756	0.523781	−	0.851853i	$-0.324521\pi$
$461$	10368.9	1.04756	0.523781	+	0.851853i	$0.324521\pi$
$462$	0	0
$463$	−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
$465$	0	0
$466$	8066.19	0.801844
$467$	12922.6	1.28048	0.640241	−	0.768174i	$-0.278834\pi$
$467$	12922.6	1.28048	0.640241	+	0.768174i	$0.278834\pi$
$468$	0	0
$469$	574.683	0.0565808
$470$	28.6504	0.00281180
$471$	0	0
$472$	13513.6	1.31783
$473$	3227.40	0.313734
$474$	0	0
$475$	3355.70	0.324148
$476$	−4101.98	−0.394988
$477$	0	0
$478$	−7770.46	−0.743541
$479$	14681.6	1.40046	0.700228	−	0.713920i	$-0.253082\pi$
$479$	14681.6	1.40046	0.700228	+	0.713920i	$0.253082\pi$
$480$	0	0
$481$	34703.3	3.28968
$482$	9640.87	0.911057
$483$	0	0
$484$	−693.830	−0.0651606
$485$	2.97481	0.000278514 0
$486$	0	0
$487$	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
$489$	0	0
$490$	−11.4522	−0.00105583
$491$	5026.47	0.461999	0.230999	−	0.972954i	$-0.425801\pi$
$491$	5026.47	0.461999	0.230999	+	0.972954i	$0.425801\pi$
$492$	0	0
$493$	13948.8	1.27429
$494$	3585.62	0.326568
$495$	0	0
$496$	−1295.35	−0.117264
$497$	−4055.84	−0.366055
$498$	0	0
$499$	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
$501$	0	0
$502$	464.314	0.0412816
$503$	−12089.2	−1.07164	−0.535818	−	0.844334i	$-0.679997\pi$
$503$	−12089.2	−1.07164	−0.535818	+	0.844334i	$0.679997\pi$
$504$	0	0
$505$	5.58537	0.000492170 0
$506$	1078.89	0.0947878
$507$	0	0
$508$	−5422.91	−0.473627
$509$	6405.45	0.557792	0.278896	−	0.960321i	$-0.410031\pi$
$509$	6405.45	0.557792	0.278896	+	0.960321i	$0.410031\pi$
$510$	0	0
$511$	866.004	0.0749702
$512$	−5207.71	−0.449513
$513$	0	0
$514$	8349.74	0.716520
$515$	262.404	0.0224522
$516$	0	0
$517$	−1348.44	−0.114708
$518$	4121.90	0.349625
$519$	0	0
$520$	284.763	0.0240147
$521$	−9757.40	−0.820498	−0.410249	−	0.911974i	$-0.634558\pi$
$521$	−9757.40	−0.820498	−0.410249	+	0.911974i	$0.634558\pi$
$522$	0	0
$523$	−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
$525$	0	0
$526$	1333.23	0.110516
$527$	8972.81	0.741673
$528$	0	0
$529$	−7921.43	−0.651059
$530$	32.7514	0.00268420
$531$	0	0
$532$	−1077.76	−0.0878325
$533$	6196.87	0.503595
$534$	0	0
$535$	−154.090	−0.0124521
$536$	−1697.26	−0.136773
$537$	0	0
$538$	6702.67	0.537124
$539$	539.000	0.0430730
$540$	0	0
$541$	−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
$543$	0	0
$544$	19171.5	1.51098
$545$	94.3389	0.00741474
$546$	0	0
$547$	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
$549$	0	0
$550$	2069.36	0.160433
$551$	3664.93	0.283360
$552$	0	0
$553$	−2943.17	−0.226323
$554$	1028.53	0.0788772
$555$	0	0
$556$	−1928.88	−0.147127
$557$	4396.25	0.334425	0.167213	−	0.985921i	$-0.446523\pi$
$557$	4396.25	0.334425	0.167213	+	0.985921i	$0.446523\pi$
$558$	0	0
$559$	26028.5	1.96939
$560$	−16.0347	−0.00120998
$561$	0	0
$562$	−12982.3	−0.974419
$563$	15806.4	1.18323	0.591615	−	0.806221i	$-0.298491\pi$
$563$	15806.4	1.18323	0.591615	+	0.806221i	$0.298491\pi$
$564$	0	0
$565$	136.110	0.0101349
$566$	−1885.63	−0.140033
$567$	0	0
$568$	11978.5	0.884869
$569$	−7381.00	−0.543810	−0.271905	−	0.962324i	$-0.587654\pi$
$569$	−7381.00	−0.543810	−0.271905	+	0.962324i	$0.587654\pi$
$570$	0	0
$571$	−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
$573$	0	0
$574$	736.035	0.0535218
$575$	8143.18	0.590599
$576$	0	0
$577$	−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
$579$	0	0
$580$	121.521	0.00869980
$581$	417.953	0.0298444
$582$	0	0
$583$	−1541.45	−0.109503
$584$	−2557.65	−0.181226
$585$	0	0
$586$	−9547.67	−0.673056
$587$	26606.6	1.87082	0.935410	−	0.353565i	$-0.115031\pi$
$587$	26606.6	1.87082	0.935410	+	0.353565i	$0.115031\pi$
$588$	0	0
$589$	2357.53	0.164924
$590$	−152.773	−0.0106603
$591$	0	0
$592$	5771.24	0.400670
$593$	26279.4	1.81984	0.909920	−	0.414783i	$-0.136143\pi$
$593$	26279.4	1.81984	0.909920	+	0.414783i	$0.136143\pi$
$594$	0	0
$595$	111.071	0.00765290
$596$	17952.4	1.23382
$597$	0	0
$598$	8701.12	0.595009
$599$	−26971.4	−1.83977	−0.919884	−	0.392190i	$-0.871718\pi$
$599$	−26971.4	−1.83977	−0.919884	+	0.392190i	$0.871718\pi$
$600$	0	0
$601$	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
$603$	0	0
$604$	−19616.6	−1.32151
$605$	18.7871	0.00126249
$606$	0	0
$607$	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
$609$	0	0
$610$	−68.9731	−0.00457810
$611$	−10875.0	−0.720056
$612$	0	0
$613$	−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
$615$	0	0
$616$	−1591.88	−0.104121
$617$	−23635.8	−1.54221	−0.771104	−	0.636710i	$-0.780295\pi$
$617$	−23635.8	−1.54221	−0.771104	+	0.636710i	$0.780295\pi$
$618$	0	0
$619$	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
$621$	0	0
$622$	−8074.10	−0.520485
$623$	−1962.42	−0.126200
$624$	0	0
$625$	15616.0	0.999421
$626$	8768.37	0.559831
$627$	0	0
$628$	9712.32	0.617140
$629$	−39977.0	−2.53416
$630$	0	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
$633$	0	0
$634$	−8728.29	−0.546758
$635$	146.838	0.00917654
$636$	0	0
$637$	4346.96	0.270381
$638$	2260.06	0.140245
$639$	0	0
$640$	194.605	0.0120194
$641$	−1045.51	−0.0644231	−0.0322116	−	0.999481i	$-0.510255\pi$
$641$	−1045.51	−0.0644231	−0.0322116	+	0.999481i	$0.510255\pi$
$642$	0	0
$643$	−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
$645$	0	0
$646$	−4130.51	−0.251567
$647$	−6116.20	−0.371642	−0.185821	−	0.982584i	$-0.559494\pi$
$647$	−6116.20	−0.371642	−0.185821	+	0.982584i	$0.559494\pi$
$648$	0	0
$649$	7190.28	0.434889
$650$	16689.1	1.00708
$651$	0	0
$652$	2514.76	0.151052
$653$	−6143.47	−0.368166	−0.184083	−	0.982911i	$-0.558932\pi$
$653$	−6143.47	−0.368166	−0.184083	+	0.982911i	$0.558932\pi$
$654$	0	0
$655$	17.4513	0.00104104
$656$	1030.55	0.0613359
$657$	0	0
$658$	−1291.68	−0.0765271
$659$	−18463.0	−1.09138	−0.545688	−	0.837989i	$-0.683731\pi$
$659$	−18463.0	−1.09138	−0.545688	+	0.837989i	$0.683731\pi$
$660$	0	0
$661$	−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
$663$	0	0
$664$	−1234.38	−0.0721433
$665$	29.1830	0.00170176
$666$	0	0
$667$	8893.59	0.516283
$668$	14948.1	0.865806
$669$	0	0
$670$	19.1877	0.00110640
$671$	3246.24	0.186765
$672$	0	0
$673$	−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
$675$	0	0
$676$	−32530.2	−1.85083
$677$	23649.2	1.34256	0.671280	−	0.741204i	$-0.265745\pi$
$677$	23649.2	1.34256	0.671280	+	0.741204i	$0.265745\pi$
$678$	0	0
$679$	−134.117	−0.00758015
$680$	−328.036	−0.0184994
$681$	0	0
$682$	1453.82	0.0816270
$683$	17165.7	0.961677	0.480839	−	0.876809i	$-0.340332\pi$
$683$	17165.7	0.961677	0.480839	+	0.876809i	$0.340332\pi$
$684$	0	0
$685$	220.075	0.0122754
$686$	516.311	0.0287360
$687$	0	0
$688$	4328.61	0.239864
$689$	−12431.6	−0.687381
$690$	0	0
$691$	−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
$693$	0	0
$694$	10911.4	0.596819
$695$	52.2291	0.00285059
$696$	0	0
$697$	−7138.57	−0.387938
$698$	6248.29	0.338827
$699$	0	0
$700$	−5016.39	−0.270860
$701$	12751.9	0.687065	0.343533	−	0.939141i	$-0.388376\pi$
$701$	12751.9	0.687065	0.343533	+	0.939141i	$0.388376\pi$
$702$	0	0
$703$	−10503.6	−0.563515
$704$	1807.97	0.0967905
$705$	0	0
$706$	2397.67	0.127815
$707$	−251.811	−0.0133951
$708$	0	0
$709$	−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
$711$	0	0
$712$	5795.78	0.305065
$713$	5720.95	0.300493
$714$	0	0
$715$	151.515	0.00792497
$716$	3381.82	0.176515
$717$	0	0
$718$	−12778.4	−0.664188
$719$	17425.0	0.903816	0.451908	−	0.892065i	$-0.350744\pi$
$719$	17425.0	0.903816	0.451908	+	0.892065i	$0.350744\pi$
$720$	0	0
$721$	−11830.2	−0.611070
$722$	9239.47	0.476257
$723$	0	0
$724$	13635.5	0.699946
$725$	17058.3	0.873833
$726$	0	0
$727$	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
$729$	0	0
$730$	28.9144	0.00146599
$731$	−29983.9	−1.51709
$732$	0	0
$733$	−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
$735$	0	0
$736$	12223.5	0.612179
$737$	−903.073	−0.0451358
$738$	0	0
$739$	−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
$741$	0	0
$742$	−1476.57	−0.0730545
$743$	−10338.8	−0.510489	−0.255245	−	0.966877i	$-0.582156\pi$
$743$	−10338.8	−0.510489	−0.255245	+	0.966877i	$0.582156\pi$
$744$	0	0
$745$	−486.105	−0.0239054
$746$	−7693.18	−0.377570
$747$	0	0
$748$	6445.98	0.315091
$749$	6947.00	0.338902
$750$	0	0
$751$	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
$753$	0	0
$754$	18227.0	0.880358
$755$	531.168	0.0256042
$756$	0	0
$757$	−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
$759$	0	0
$760$	−86.1887	−0.00411368
$761$	−26816.8	−1.27741	−0.638704	−	0.769453i	$-0.720529\pi$
$761$	−26816.8	−1.27741	−0.638704	+	0.769453i	$0.720529\pi$
$762$	0	0
$763$	−4253.18	−0.201803
$764$	−21209.0	−1.00434
$765$	0	0
$766$	−2886.31	−0.136144
$767$	57988.6	2.72992
$768$	0	0
$769$	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
$771$	0	0
$772$	−29814.1	−1.38994
$773$	−36628.4	−1.70431	−0.852156	−	0.523288i	$-0.824705\pi$
$773$	−36628.4	−1.70431	−0.852156	+	0.523288i	$0.824705\pi$
$774$	0	0
$775$	10973.0	0.508597
$776$	396.098	0.0183236
$777$	0	0
$778$	7737.14	0.356542
$779$	−1875.60	−0.0862648
$780$	0	0
$781$	6373.47	0.292011
$782$	−10023.4	−0.458357
$783$	0	0
$784$	722.910	0.0329314
$785$	−262.985	−0.0119571
$786$	0	0
$787$	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
$789$	0	0
$790$	−98.2676	−0.00442557
$791$	−6136.40	−0.275835
$792$	0	0
$793$	26180.4	1.17238
$794$	−19508.5	−0.871954
$795$	0	0
$796$	−14374.7	−0.640071
$797$	−22734.8	−1.01042	−0.505211	−	0.862996i	$-0.668585\pi$
$797$	−22734.8	−1.01042	−0.505211	+	0.862996i	$0.668585\pi$
$798$	0	0
$799$	12527.6	0.554686
$800$	23445.2	1.03614
$801$	0	0
$802$	−7967.21	−0.350788
$803$	−1360.86	−0.0598055
$804$	0	0
$805$	70.8175	0.00310061
$806$	11724.8	0.512395
$807$	0	0
$808$	743.697	0.0323802
$809$	−10636.0	−0.462228	−0.231114	−	0.972927i	$-0.574237\pi$
$809$	−10636.0	−0.462228	−0.231114	+	0.972927i	$0.574237\pi$
$810$	0	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
$813$	0	0
$814$	−6477.27	−0.278904
$815$	−68.0933	−0.00292663
$816$	0	0
$817$	−7878.02	−0.337353
$818$	−8346.83	−0.356773
$819$	0	0
$820$	−62.1906	−0.00264853
$821$	−24576.3	−1.04472	−0.522362	−	0.852724i	$-0.674949\pi$
$821$	−24576.3	−1.04472	−0.522362	+	0.852724i	$0.674949\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	6887.62	0.290134
$827$	28350.2	1.19206	0.596029	−	0.802963i	$-0.296744\pi$
$827$	28350.2	1.19206	0.596029	+	0.802963i	$0.296744\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	14581.0	0.607580
$833$	−5007.54	−0.208285
$834$	0	0
$835$	−404.755	−0.0167750
$836$	1693.62	0.0700661
$837$	0	0
$838$	5256.91	0.216703
$839$	−35911.1	−1.47770	−0.738849	−	0.673871i	$-0.764631\pi$
$839$	−35911.1	−1.47770	−0.738849	+	0.673871i	$0.764631\pi$
$840$	0	0
$841$	−5758.78	−0.236122
$842$	−23253.9	−0.951761
$843$	0	0
$844$	−3619.48	−0.147615
$845$	880.833	0.0358598
$846$	0	0
$847$	−847.000	−0.0343604
$848$	−2067.40	−0.0837204
$849$	0	0
$850$	−19225.3	−0.775790
$851$	−25488.8	−1.02673
$852$	0	0
$853$	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
$855$	0	0
$856$	−20517.2	−0.819233
$857$	41429.0	1.65133	0.825663	−	0.564164i	$-0.190801\pi$
$857$	41429.0	1.65133	0.825663	+	0.564164i	$0.190801\pi$
$858$	0	0
$859$	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
$861$	0	0
$862$	−10444.1	−0.412676
$863$	−30586.2	−1.20645	−0.603225	−	0.797571i	$-0.706118\pi$
$863$	−30586.2	−1.20645	−0.603225	+	0.797571i	$0.706118\pi$
$864$	0	0
$865$	698.461	0.0274548
$866$	4245.36	0.166586
$867$	0	0
$868$	−3524.24	−0.137812
$869$	4624.99	0.180543
$870$	0	0
$871$	−7283.16	−0.283330
$872$	12561.3	0.487821
$873$	0	0
$874$	−2633.56	−0.101924
$875$	271.688	0.0104968
$876$	0	0
$877$	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
$879$	0	0
$880$	25.1974	0.000965231 0
$881$	−34072.2	−1.30298	−0.651488	−	0.758659i	$-0.725855\pi$
$881$	−34072.2	−1.30298	−0.651488	+	0.758659i	$0.725855\pi$
$882$	0	0
$883$	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
$885$	0	0
$886$	18345.2	0.695620
$887$	5921.20	0.224142	0.112071	−	0.993700i	$-0.464252\pi$
$887$	5921.20	0.224142	0.112071	+	0.993700i	$0.464252\pi$
$888$	0	0
$889$	−6620.08	−0.249753
$890$	−65.5217	−0.00246775
$891$	0	0
$892$	−15846.7	−0.594829
$893$	3291.51	0.123344
$894$	0	0
$895$	−91.5709	−0.00341998
$896$	−8773.60	−0.327127
$897$	0	0
$898$	13635.5	0.506707
$899$	11984.2	0.444600
$900$	0	0
$901$	14320.7	0.529515
$902$	−1156.63	−0.0426956
$903$	0	0
$904$	18123.2	0.666778
$905$	−369.215	−0.0135615
$906$	0	0
$907$	−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
$909$	0	0
$910$	145.138	0.00528710
$911$	−35023.9	−1.27376	−0.636878	−	0.770964i	$-0.719775\pi$
$911$	−35023.9	−1.27376	−0.636878	+	0.770964i	$0.719775\pi$
$912$	0	0
$913$	−656.784	−0.0238076
$914$	1611.81	0.0583302
$915$	0	0
$916$	−38657.2	−1.39440
$917$	−786.775	−0.0283333
$918$	0	0
$919$	−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
$921$	0	0
$922$	−15608.0	−0.557509
$923$	51401.1	1.83303
$924$	0	0
$925$	−48888.6	−1.73778
$926$	9536.14	0.338420
$927$	0	0
$928$	25605.7	0.905763
$929$	−16251.5	−0.573945	−0.286972	−	0.957939i	$-0.592649\pi$
$929$	−16251.5	−0.573945	−0.286972	+	0.957939i	$0.592649\pi$
$930$	0	0
$931$	−1315.69	−0.0463157
$932$	30726.9	1.07993
$933$	0	0
$934$	−19452.1	−0.681469
$935$	−174.540	−0.00610490
$936$	0	0
$937$	−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
$939$	0	0
$940$	109.139	0.00378695
$941$	16381.6	0.567507	0.283754	−	0.958897i	$-0.408420\pi$
$941$	16381.6	0.567507	0.283754	+	0.958897i	$0.408420\pi$
$942$	0	0
$943$	−4551.46	−0.157175
$944$	9643.65	0.332494
$945$	0	0
$946$	−4858.15	−0.166968
$947$	−11004.5	−0.377613	−0.188807	−	0.982014i	$-0.560462\pi$
$947$	−11004.5	−0.377613	−0.188807	+	0.982014i	$0.560462\pi$
$948$	0	0
$949$	−10975.2	−0.375415
$950$	−5051.27	−0.172510
$951$	0	0
$952$	14789.2	0.503489
$953$	−45741.3	−1.55478	−0.777391	−	0.629018i	$-0.783457\pi$
$953$	−45741.3	−1.55478	−0.777391	+	0.629018i	$0.783457\pi$
$954$	0	0
$955$	574.285	0.0194591
$956$	−29600.3	−1.00140
$957$	0	0
$958$	−22099.9	−0.745318
$959$	−9921.89	−0.334092
$960$	0	0
$961$	−22082.0	−0.741229
$962$	−52238.3	−1.75076
$963$	0	0
$964$	36725.3	1.22702
$965$	807.288	0.0269301
$966$	0	0
$967$	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
$969$	0	0
$970$	−4.47792	−0.000148224 0
$971$	−41167.9	−1.36060	−0.680299	−	0.732934i	$-0.738150\pi$
$971$	−41167.9	−1.36060	−0.680299	+	0.732934i	$0.738150\pi$
$972$	0	0
$973$	−2354.70	−0.0775829
$974$	−572.047	−0.0188189
$975$	0	0
$976$	4353.87	0.142791
$977$	34251.8	1.12161	0.560805	−	0.827948i	$-0.310492\pi$
$977$	34251.8	1.12161	0.560805	+	0.827948i	$0.310492\pi$
$978$	0	0
$979$	3083.80	0.100673
$980$	−43.6253	−0.00142200
$981$	0	0
$982$	−7566.25	−0.245874
$983$	49982.0	1.62175	0.810875	−	0.585220i	$-0.198992\pi$
$983$	49982.0	1.62175	0.810875	+	0.585220i	$0.198992\pi$
$984$	0	0
$985$	−210.565	−0.00681132
$986$	−20996.9	−0.678172
$987$	0	0
$988$	13658.8	0.439824
$989$	−19117.4	−0.614658
$990$	0	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
$993$	0	0
$994$	6105.18	0.194813
$995$	389.229	0.0124014
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.q.1.2		5
3.2	odd	2		231.4.a.j.1.4	✓	5
21.20	even	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	3.2	odd	2
693.4.a.q.1.2		5	1.1	even	1	trivial
1617.4.a.o.1.4		5	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.50528 q^{2} -5.73413 q^{4} +0.155265 q^{5} -7.00000 q^{7} +20.6737 q^{8} +O(q^{10})\)	\(q-1.50528 q^{2} -5.73413 q^{4} +0.155265 q^{5} -7.00000 q^{7} +20.6737 q^{8} -0.233718 q^{10} +11.0000 q^{11} +88.7135 q^{13} +10.5370 q^{14} +14.7533 q^{16} -102.195 q^{17} -26.8508 q^{19} -0.890312 q^{20} -16.5581 q^{22} -65.1580 q^{23} -124.976 q^{25} -133.539 q^{26} +40.1389 q^{28} -136.493 q^{29} -87.8011 q^{31} -187.598 q^{32} +153.832 q^{34} -1.08686 q^{35} +391.184 q^{37} +40.4180 q^{38} +3.20991 q^{40} +69.8526 q^{41} +293.400 q^{43} -63.0754 q^{44} +98.0811 q^{46} -122.585 q^{47} +49.0000 q^{49} +188.124 q^{50} -508.694 q^{52} -140.132 q^{53} +1.70792 q^{55} -144.716 q^{56} +205.460 q^{58} +653.662 q^{59} +295.112 q^{61} +132.165 q^{62} +164.361 q^{64} +13.7741 q^{65} -82.0975 q^{67} +585.998 q^{68} +1.63603 q^{70} +579.406 q^{71} -123.715 q^{73} -588.842 q^{74} +153.966 q^{76} -77.0000 q^{77} +420.453 q^{79} +2.29067 q^{80} -105.148 q^{82} -59.7076 q^{83} -15.8673 q^{85} -441.650 q^{86} +227.411 q^{88} +280.345 q^{89} -620.994 q^{91} +373.624 q^{92} +184.525 q^{94} -4.16900 q^{95} +19.1595 q^{97} -73.7588 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8	\( 5 q + q^{2} + 21 q^{4} - 7 q^{5} - 35 q^{7} + 12 q^{8} + 113 q^{10} + 55 q^{11} + 23 q^{13} - 7 q^{14} + 281 q^{16} + 102 q^{17} - 155 q^{19} - 291 q^{20} + 11 q^{22} - 192 q^{23} + 394 q^{25} - 41 q^{26} - 147 q^{28} - 591 q^{29} + 366 q^{31} + 87 q^{32} + 594 q^{34} + 49 q^{35} + 259 q^{37} - 175 q^{38} + 1890 q^{40} + 104 q^{41} + 224 q^{43} + 231 q^{44} + 1416 q^{46} + 453 q^{47} + 245 q^{49} - 1350 q^{50} + 815 q^{52} - 1032 q^{53} - 77 q^{55} - 84 q^{56} + 293 q^{58} + 517 q^{59} + 958 q^{61} + 3030 q^{62} + 516 q^{64} + 197 q^{65} + 361 q^{67} + 1680 q^{68} - 791 q^{70} - 548 q^{71} - 1035 q^{73} + 2555 q^{74} - 5201 q^{76} - 385 q^{77} + 776 q^{79} + 1273 q^{80} + 376 q^{82} + 1974 q^{83} + 1838 q^{85} - 1224 q^{86} + 132 q^{88} - 2710 q^{89} - 161 q^{91} + 2834 q^{92} - 563 q^{94} + 355 q^{95} + 1988 q^{97} + 49 q^{98}+O(q^{100}) \) 5 * q + q^2 + 21 * q^4 - 7 * q^5 - 35 * q^7 + 12 * q^8 + 113 * q^10 + 55 * q^11 + 23 * q^13 - 7 * q^14 + 281 * q^16 + 102 * q^17 - 155 * q^19 - 291 * q^20 + 11 * q^22 - 192 * q^23 + 394 * q^25 - 41 * q^26 - 147 * q^28 - 591 * q^29 + 366 * q^31 + 87 * q^32 + 594 * q^34 + 49 * q^35 + 259 * q^37 - 175 * q^38 + 1890 * q^40 + 104 * q^41 + 224 * q^43 + 231 * q^44 + 1416 * q^46 + 453 * q^47 + 245 * q^49 - 1350 * q^50 + 815 * q^52 - 1032 * q^53 - 77 * q^55 - 84 * q^56 + 293 * q^58 + 517 * q^59 + 958 * q^61 + 3030 * q^62 + 516 * q^64 + 197 * q^65 + 361 * q^67 + 1680 * q^68 - 791 * q^70 - 548 * q^71 - 1035 * q^73 + 2555 * q^74 - 5201 * q^76 - 385 * q^77 + 776 * q^79 + 1273 * q^80 + 376 * q^82 + 1974 * q^83 + 1838 * q^85 - 1224 * q^86 + 132 * q^88 - 2710 * q^89 - 161 * q^91 + 2834 * q^92 - 563 * q^94 + 355 * q^95 + 1988 * q^97 + 49 * q^98

Introduction

L-functions

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Database

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