LMFDB - Embedded newform 575.4.b.j.24.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,4,Mod(24,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("575.24");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	575.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$33.9260982533$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 83x^{12} + 2715x^{10} + 44273x^{8} + 372280x^{6} + 1482448x^{4} + 2136384x^{2} + 746496 $ x^14 + 83x^12 + 2715x^10 + 44273x^8 + 372280x^6 + 1482448x^4 + 2136384x^2 + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.6
Root			$-1.37904i$ of defining polynomial
Character	$\chi$	$=$	575.24
Dual form			575.4.b.j.24.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.37904i q^{2} +5.53432i q^{3} +6.09824 q^{4} +7.63206 q^{6} -8.23397i q^{7} -19.4421i q^{8} -3.62871 q^{9} -23.2487 q^{11} +33.7496i q^{12} +28.3577i q^{13} -11.3550 q^{14} +21.9745 q^{16} +128.381i q^{17} +5.00414i q^{18} -28.6966 q^{19} +45.5694 q^{21} +32.0610i q^{22} -23.0000i q^{23} +107.599 q^{24} +39.1064 q^{26} +129.344i q^{27} -50.2128i q^{28} +199.085 q^{29} +24.6414 q^{31} -185.840i q^{32} -128.666i q^{33} +177.043 q^{34} -22.1287 q^{36} +351.164i q^{37} +39.5738i q^{38} -156.940 q^{39} +374.394 q^{41} -62.8422i q^{42} -369.675i q^{43} -141.777 q^{44} -31.7180 q^{46} +129.790i q^{47} +121.614i q^{48} +275.202 q^{49} -710.501 q^{51} +172.932i q^{52} +582.723i q^{53} +178.371 q^{54} -160.085 q^{56} -158.816i q^{57} -274.547i q^{58} -3.89763 q^{59} -533.845 q^{61} -33.9816i q^{62} +29.8787i q^{63} -80.4857 q^{64} -177.436 q^{66} +498.110i q^{67} +782.898i q^{68} +127.289 q^{69} -300.340 q^{71} +70.5496i q^{72} +324.002i q^{73} +484.270 q^{74} -174.999 q^{76} +191.430i q^{77} +216.428i q^{78} +367.233 q^{79} -813.808 q^{81} -516.305i q^{82} +851.294i q^{83} +277.894 q^{84} -509.797 q^{86} +1101.80i q^{87} +452.004i q^{88} +819.266 q^{89} +233.496 q^{91} -140.260i q^{92} +136.374i q^{93} +178.986 q^{94} +1028.50 q^{96} -499.221i q^{97} -379.515i q^{98} +84.3629 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) $ 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.37904i	− 0.487565i	−0.969830	−	0.243783i	$-0.921612\pi$
$2$	− 1.37904i	− 0.487565i	0.969830	−	0.243783i	$-0.0783883\pi$
$3$	5.53432i	1.06508i	0.846405	+	0.532540i	$0.178763\pi$
$3$	5.53432i	1.06508i	−0.846405	+	0.532540i	$0.821237\pi$
$4$	6.09824	0.762280
$5$	0	0
$5$	0	0
$6$	7.63206	0.519296
$7$	− 8.23397i	− 0.444593i	−0.974979	−	0.222296i	$-0.928645\pi$
$7$	− 8.23397i	− 0.444593i	0.974979	−	0.222296i	$-0.0713553\pi$
$8$	− 19.4421i	− 0.859226i
$9$	−3.62871	−0.134397
$10$	0	0
$11$	−23.2487	−0.637251	−0.318626	−	0.947881i	$-0.603221\pi$
$11$	−23.2487	−0.637251	−0.318626	+	0.947881i	$0.603221\pi$
$12$	33.7496i	0.811890i
$13$	28.3577i	0.605000i	0.953149	+	0.302500i	$0.0978213\pi$
$13$	28.3577i	0.605000i	−0.953149	+	0.302500i	$0.902179\pi$
$14$	−11.3550	−0.216768
$15$	0	0
$16$	21.9745	0.343352
$17$	128.381i	1.83159i	0.401651	+	0.915793i	$0.368437\pi$
$17$	128.381i	1.83159i	−0.401651	+	0.915793i	$0.631563\pi$
$18$	5.00414i	0.0655271i
$19$	−28.6966	−0.346497	−0.173248	−	0.984878i	$-0.555426\pi$
$19$	−28.6966	−0.346497	−0.173248	+	0.984878i	$0.555426\pi$
$20$	0	0
$21$	45.5694	0.473527
$22$	32.0610i	0.310701i
$23$	− 23.0000i	− 0.208514i
$23$	− 23.0000i	− 0.208514i
$24$	107.599	0.915145
$25$	0	0
$26$	39.1064	0.294977
$27$	129.344i	0.921937i
$28$	− 50.2128i	− 0.338904i
$29$	199.085	1.27480	0.637399	−	0.770534i	$-0.280010\pi$
$29$	199.085	1.27480	0.637399	+	0.770534i	$0.280010\pi$
$30$	0	0
$31$	24.6414	0.142766	0.0713828	−	0.997449i	$-0.477259\pi$
$31$	24.6414	0.142766	0.0713828	+	0.997449i	$0.477259\pi$
$32$	− 185.840i	− 1.02663i
$33$	− 128.666i	− 0.678724i
$34$	177.043	0.893017
$35$	0	0
$36$	−22.1287	−0.102448
$37$	351.164i	1.56030i	0.625594	+	0.780148i	$0.284856\pi$
$37$	351.164i	1.56030i	−0.625594	+	0.780148i	$0.715144\pi$
$38$	39.5738i	0.168940i
$39$	−156.940	−0.644374
$40$	0	0
$41$	374.394	1.42611	0.713054	−	0.701109i	$-0.247311\pi$
$41$	374.394	1.42611	0.713054	+	0.701109i	$0.247311\pi$
$42$	− 62.8422i	− 0.230875i
$43$	− 369.675i	− 1.31104i	−0.755176	−	0.655522i	$-0.772449\pi$
$43$	− 369.675i	− 1.31104i	0.755176	−	0.655522i	$-0.227551\pi$
$44$	−141.777	−0.485764
$45$	0	0
$46$	−31.7180	−0.101664
$47$	129.790i	0.402806i	0.979508	+	0.201403i	$0.0645500\pi$
$47$	129.790i	0.402806i	−0.979508	+	0.201403i	$0.935450\pi$
$48$	121.614i	0.365697i
$49$	275.202	0.802337
$50$	0	0
$51$	−710.501	−1.95079
$52$	172.932i	0.461180i
$53$	582.723i	1.51025i	0.655582	+	0.755124i	$0.272423\pi$
$53$	582.723i	1.51025i	−0.655582	+	0.755124i	$0.727577\pi$
$54$	178.371	0.449504
$55$	0	0
$56$	−160.085	−0.382006
$57$	− 158.816i	− 0.369047i
$58$	− 274.547i	− 0.621547i
$59$	−3.89763	−0.00860048	−0.00430024	−	0.999991i	$-0.501369\pi$
$59$	−3.89763	−0.00860048	−0.00430024	+	0.999991i	$0.501369\pi$
$60$	0	0
$61$	−533.845	−1.12052	−0.560261	−	0.828316i	$-0.689299\pi$
$61$	−533.845	−1.12052	−0.560261	+	0.828316i	$0.689299\pi$
$62$	− 33.9816i	− 0.0696075i
$63$	29.8787i	0.0597517i
$64$	−80.4857	−0.157199
$65$	0	0
$66$	−177.436	−0.330922
$67$	498.110i	0.908266i	0.890934	+	0.454133i	$0.150051\pi$
$67$	498.110i	0.908266i	−0.890934	+	0.454133i	$0.849949\pi$
$68$	782.898i	1.39618i
$69$	127.289	0.222085
$70$	0	0
$71$	−300.340	−0.502025	−0.251012	−	0.967984i	$-0.580764\pi$
$71$	−300.340	−0.502025	−0.251012	+	0.967984i	$0.580764\pi$
$72$	70.5496i	0.115477i
$73$	324.002i	0.519474i	0.965679	+	0.259737i	$0.0836359\pi$
$73$	324.002i	0.519474i	−0.965679	+	0.259737i	$0.916364\pi$
$74$	484.270	0.760746
$75$	0	0
$76$	−174.999	−0.264128
$77$	191.430i	0.283317i
$78$	216.428i	0.314174i
$79$	367.233	0.522999	0.261500	−	0.965204i	$-0.415783\pi$
$79$	367.233	0.522999	0.261500	+	0.965204i	$0.415783\pi$
$80$	0	0
$81$	−813.808	−1.11633
$82$	− 516.305i	− 0.695321i
$83$	851.294i	1.12580i	0.826524	+	0.562901i	$0.190315\pi$
$83$	851.294i	1.12580i	−0.826524	+	0.562901i	$0.809685\pi$
$84$	277.894	0.360960
$85$	0	0
$86$	−509.797	−0.639219
$87$	1101.80i	1.35776i
$88$	452.004i	0.547543i
$89$	819.266	0.975753	0.487877	−	0.872913i	$-0.337772\pi$
$89$	819.266	0.975753	0.487877	+	0.872913i	$0.337772\pi$
$90$	0	0
$91$	233.496	0.268979
$92$	− 140.260i	− 0.158946i
$93$	136.374i	0.152057i
$94$	178.986	0.196394
$95$	0	0
$96$	1028.50	1.09345
$97$	− 499.221i	− 0.522559i	−0.965263	−	0.261280i	$-0.915856\pi$
$97$	− 499.221i	− 0.522559i	0.965263	−	0.261280i	$-0.0841444\pi$
$98$	− 379.515i	− 0.391192i
$99$	84.3629	0.0856444
$100$	0	0
$101$	190.273	0.187454	0.0937270	−	0.995598i	$-0.470122\pi$
$101$	190.273	0.187454	0.0937270	+	0.995598i	$0.470122\pi$
$102$	979.812i	0.951135i
$103$	− 458.967i	− 0.439062i	−0.975606	−	0.219531i	$-0.929547\pi$
$103$	− 458.967i	− 0.439062i	0.975606	−	0.219531i	$-0.0704527\pi$
$104$	551.332	0.519832
$105$	0	0
$106$	803.599	0.736344
$107$	− 224.154i	− 0.202521i	−0.994860	−	0.101261i	$-0.967712\pi$
$107$	− 224.154i	− 0.202521i	0.994860	−	0.101261i	$-0.0322876\pi$
$108$	788.772i	0.702775i
$109$	−942.722	−0.828407	−0.414203	−	0.910184i	$-0.635940\pi$
$109$	−942.722	−0.828407	−0.414203	+	0.910184i	$0.635940\pi$
$110$	0	0
$111$	−1943.45	−1.66184
$112$	− 180.937i	− 0.152652i
$113$	907.111i	0.755167i	0.925976	+	0.377583i	$0.123245\pi$
$113$	907.111i	0.755167i	−0.925976	+	0.377583i	$0.876755\pi$
$114$	−219.014	−0.179934
$115$	0	0
$116$	1214.07	0.971753
$117$	− 102.902i	− 0.0813100i
$118$	5.37500i	0.00419329i
$119$	1057.09	0.814310
$120$	0	0
$121$	−790.496	−0.593911
$122$	736.195i	0.546327i
$123$	2072.01i	1.51892i
$124$	150.270	0.108827
$125$	0	0
$126$	41.2040	0.0291329
$127$	− 1360.72i	− 0.950742i	−0.879785	−	0.475371i	$-0.842314\pi$
$127$	− 1360.72i	− 0.950742i	0.879785	−	0.475371i	$-0.157686\pi$
$128$	− 1375.73i	− 0.949988i
$129$	2045.90	1.39637
$130$	0	0
$131$	2406.94	1.60531	0.802653	−	0.596447i	$-0.203421\pi$
$131$	2406.94	1.60531	0.802653	+	0.596447i	$0.203421\pi$
$132$	− 784.637i	− 0.517378i
$133$	236.287i	0.154050i
$134$	686.915	0.442839
$135$	0	0
$136$	2495.99	1.57375
$137$	− 1282.53i	− 0.799811i	−0.916556	−	0.399906i	$-0.869043\pi$
$137$	− 1282.53i	− 0.799811i	0.916556	−	0.399906i	$-0.130957\pi$
$138$	− 175.537i	− 0.108281i
$139$	−1465.77	−0.894424	−0.447212	−	0.894428i	$-0.647583\pi$
$139$	−1465.77	−0.894424	−0.447212	+	0.894428i	$0.647583\pi$
$140$	0	0
$141$	−718.301	−0.429021
$142$	414.181i	0.244770i
$143$	− 659.280i	− 0.385537i
$144$	−79.7391	−0.0461453
$145$	0	0
$146$	446.813	0.253277
$147$	1523.05i	0.854554i
$148$	2141.48i	1.18938i
$149$	−265.901	−0.146198	−0.0730988	−	0.997325i	$-0.523289\pi$
$149$	−265.901	−0.146198	−0.0730988	+	0.997325i	$0.523289\pi$
$150$	0	0
$151$	−2697.76	−1.45391	−0.726956	−	0.686685i	$-0.759065\pi$
$151$	−2697.76	−1.45391	−0.726956	+	0.686685i	$0.759065\pi$
$152$	557.920i	0.297719i
$153$	− 465.857i	− 0.246159i
$154$	263.989	0.138136
$155$	0	0
$156$	−957.061	−0.491194
$157$	2071.67i	1.05310i	0.850143	+	0.526552i	$0.176515\pi$
$157$	2071.67i	1.05310i	−0.850143	+	0.526552i	$0.823485\pi$
$158$	− 506.430i	− 0.254996i
$159$	−3224.97	−1.60854
$160$	0	0
$161$	−189.381	−0.0927040
$162$	1122.28i	0.544286i
$163$	− 1216.16i	− 0.584400i	−0.956357	−	0.292200i	$-0.905613\pi$
$163$	− 1216.16i	− 0.584400i	0.956357	−	0.292200i	$-0.0943874\pi$
$164$	2283.14	1.08709
$165$	0	0
$166$	1173.97	0.548902
$167$	1222.75i	0.566583i	0.959034	+	0.283291i	$0.0914264\pi$
$167$	1222.75i	0.566583i	−0.959034	+	0.283291i	$0.908574\pi$
$168$	− 885.964i	− 0.406867i
$169$	1392.84	0.633975
$170$	0	0
$171$	104.131	0.0465680
$172$	− 2254.37i	− 0.999383i
$173$	− 2252.41i	− 0.989870i	−0.868930	−	0.494935i	$-0.835192\pi$
$173$	− 2252.41i	− 0.989870i	0.868930	−	0.494935i	$-0.164808\pi$
$174$	1519.43	0.661998
$175$	0	0
$176$	−510.880	−0.218801
$177$	− 21.5707i	− 0.00916021i
$178$	− 1129.80i	− 0.475743i
$179$	−910.084	−0.380016	−0.190008	−	0.981783i	$-0.560851\pi$
$179$	−910.084	−0.380016	−0.190008	+	0.981783i	$0.560851\pi$
$180$	0	0
$181$	1469.75	0.603568	0.301784	−	0.953376i	$-0.402418\pi$
$181$	1469.75	0.603568	0.301784	+	0.953376i	$0.402418\pi$
$182$	− 322.001i	− 0.131145i
$183$	− 2954.47i	− 1.19345i
$184$	−447.168	−0.179161
$185$	0	0
$186$	188.065	0.0741376
$187$	− 2984.70i	− 1.16718i
$188$	791.493i	0.307051i
$189$	1065.02	0.409887
$190$	0	0
$191$	3391.02	1.28463	0.642317	−	0.766439i	$-0.277973\pi$
$191$	3391.02	1.28463	0.642317	+	0.766439i	$0.277973\pi$
$192$	− 445.433i	− 0.167429i
$193$	228.972i	0.0853979i	0.999088	+	0.0426989i	$0.0135956\pi$
$193$	228.972i	0.0853979i	−0.999088	+	0.0426989i	$0.986404\pi$
$194$	−688.447	−0.254782
$195$	0	0
$196$	1678.25	0.611606
$197$	− 2952.81i	− 1.06791i	−0.845512	−	0.533956i	$-0.820705\pi$
$197$	− 2952.81i	− 1.06791i	0.845512	−	0.533956i	$-0.179295\pi$
$198$	− 116.340i	− 0.0417572i
$199$	2601.97	0.926876	0.463438	−	0.886129i	$-0.346616\pi$
$199$	2601.97	0.926876	0.463438	+	0.886129i	$0.346616\pi$
$200$	0	0
$201$	−2756.70	−0.967377
$202$	− 262.394i	− 0.0913960i
$203$	− 1639.26i	− 0.566766i
$204$	−4332.81	−1.48705
$205$	0	0
$206$	−632.935	−0.214071
$207$	83.4603i	0.0280236i
$208$	623.146i	0.207728i
$209$	667.159	0.220805
$210$	0	0
$211$	−1900.58	−0.620101	−0.310050	−	0.950720i	$-0.600346\pi$
$211$	−1900.58	−0.620101	−0.310050	+	0.950720i	$0.600346\pi$
$212$	3553.58i	1.15123i
$213$	− 1662.18i	− 0.534697i
$214$	−309.118	−0.0987423
$215$	0	0
$216$	2514.72	0.792153
$217$	− 202.897i	− 0.0634726i
$218$	1300.05i	0.403902i
$219$	−1793.13	−0.553282
$220$	0	0
$221$	−3640.59	−1.10811
$222$	2680.10i	0.810256i
$223$	− 4616.71i	− 1.38636i	−0.720766	−	0.693178i	$-0.756210\pi$
$223$	− 4616.71i	− 1.38636i	0.720766	−	0.693178i	$-0.243790\pi$
$224$	−1530.20	−0.456433
$225$	0	0
$226$	1250.94	0.368193
$227$	− 2950.85i	− 0.862796i	−0.902162	−	0.431398i	$-0.858021\pi$
$227$	− 2950.85i	− 0.862796i	0.902162	−	0.431398i	$-0.141979\pi$
$228$	− 968.498i	− 0.281317i
$229$	−3209.27	−0.926089	−0.463044	−	0.886335i	$-0.653243\pi$
$229$	−3209.27	−0.926089	−0.463044	+	0.886335i	$0.653243\pi$
$230$	0	0
$231$	−1059.43	−0.301756
$232$	− 3870.62i	− 1.09534i
$233$	4672.40i	1.31373i	0.754008	+	0.656865i	$0.228118\pi$
$233$	4672.40i	1.31373i	−0.754008	+	0.656865i	$0.771882\pi$
$234$	−141.906	−0.0396439
$235$	0	0
$236$	−23.7687	−0.00655598
$237$	2032.38i	0.557036i
$238$	− 1457.77i	− 0.397029i
$239$	1937.22	0.524302	0.262151	−	0.965027i	$-0.415568\pi$
$239$	1937.22	0.524302	0.262151	+	0.965027i	$0.415568\pi$
$240$	0	0
$241$	−452.720	−0.121005	−0.0605025	−	0.998168i	$-0.519270\pi$
$241$	−452.720	−0.121005	−0.0605025	+	0.998168i	$0.519270\pi$
$242$	1090.13i	0.289570i
$243$	− 1011.58i	− 0.267048i
$244$	−3255.52	−0.854152
$245$	0	0
$246$	2857.39	0.740573
$247$	− 813.767i	− 0.209631i
$248$	− 479.081i	− 0.122668i
$249$	−4711.33	−1.19907
$250$	0	0
$251$	−4566.03	−1.14823	−0.574114	−	0.818775i	$-0.694654\pi$
$251$	−4566.03	−1.14823	−0.574114	+	0.818775i	$0.694654\pi$
$252$	182.207i	0.0455476i
$253$	534.721i	0.132876i
$254$	−1876.49	−0.463549
$255$	0	0
$256$	−2541.07	−0.620379
$257$	− 4502.83i	− 1.09291i	−0.837488	−	0.546456i	$-0.815976\pi$
$257$	− 4502.83i	− 1.09291i	0.837488	−	0.546456i	$-0.184024\pi$
$258$	− 2821.38i	− 0.680820i
$259$	2891.47	0.693697
$260$	0	0
$261$	−722.421	−0.171329
$262$	− 3319.27i	− 0.782691i
$263$	1019.88i	0.239119i	0.992827	+	0.119559i	$0.0381482\pi$
$263$	1019.88i	0.239119i	−0.992827	+	0.119559i	$0.961852\pi$
$264$	−2501.53	−0.583177
$265$	0	0
$266$	325.849	0.0751094
$267$	4534.08i	1.03926i
$268$	3037.60i	0.692353i
$269$	−7960.68	−1.80435	−0.902177	−	0.431365i	$-0.858032\pi$
$269$	−7960.68	−1.80435	−0.902177	+	0.431365i	$0.858032\pi$
$270$	0	0
$271$	3523.98	0.789913	0.394956	−	0.918700i	$-0.370760\pi$
$271$	3523.98	0.789913	0.394956	+	0.918700i	$0.370760\pi$
$272$	2821.11i	0.628878i
$273$	1292.24i	0.286484i
$274$	−1768.67	−0.389960
$275$	0	0
$276$	776.242	0.169291
$277$	− 5855.30i	− 1.27008i	−0.772481	−	0.635038i	$-0.780985\pi$
$277$	− 5855.30i	− 1.27008i	0.772481	−	0.635038i	$-0.219015\pi$
$278$	2021.36i	0.436090i
$279$	−89.4166	−0.0191872
$280$	0	0
$281$	−3241.62	−0.688180	−0.344090	−	0.938937i	$-0.611813\pi$
$281$	−3241.62	−0.688180	−0.344090	+	0.938937i	$0.611813\pi$
$282$	990.568i	0.209175i
$283$	− 568.639i	− 0.119442i	−0.998215	−	0.0597210i	$-0.980979\pi$
$283$	− 568.639i	− 0.119442i	0.998215	−	0.0597210i	$-0.0190211\pi$
$284$	−1831.54	−0.382684
$285$	0	0
$286$	−909.175	−0.187974
$287$	− 3082.75i	− 0.634037i
$288$	674.360i	0.137976i
$289$	−11568.7	−2.35471
$290$	0	0
$291$	2762.85	0.556567
$292$	1975.85i	0.395985i
$293$	− 3458.60i	− 0.689603i	−0.938676	−	0.344801i	$-0.887946\pi$
$293$	− 3458.60i	− 0.689603i	0.938676	−	0.344801i	$-0.112054\pi$
$294$	2100.36	0.416651
$295$	0	0
$296$	6827.35	1.34065
$297$	− 3007.09i	− 0.587506i
$298$	366.688i	0.0712808i
$299$	652.226	0.126151
$300$	0	0
$301$	−3043.89	−0.582880
$302$	3720.32i	0.708876i
$303$	1053.03i	0.199654i
$304$	−630.592	−0.118970
$305$	0	0
$306$	−642.437	−0.120018
$307$	− 8692.92i	− 1.61606i	−0.589139	−	0.808032i	$-0.700533\pi$
$307$	− 8692.92i	− 1.61606i	0.589139	−	0.808032i	$-0.299467\pi$
$308$	1167.38i	0.215967i
$309$	2540.07	0.467636
$310$	0	0
$311$	10831.4	1.97489	0.987446	−	0.157956i	$-0.0504905\pi$
$311$	10831.4	1.97489	0.987446	+	0.157956i	$0.0504905\pi$
$312$	3051.25i	0.553663i
$313$	8160.61i	1.47369i	0.676062	+	0.736845i	$0.263685\pi$
$313$	8160.61i	1.47369i	−0.676062	+	0.736845i	$0.736315\pi$
$314$	2856.92	0.513456
$315$	0	0
$316$	2239.48	0.398672
$317$	− 3461.14i	− 0.613239i	−0.951832	−	0.306620i	$-0.900802\pi$
$317$	− 3461.14i	− 0.613239i	0.951832	−	0.306620i	$-0.0991980\pi$
$318$	4447.38i	0.784266i
$319$	−4628.47	−0.812366
$320$	0	0
$321$	1240.54	0.215702
$322$	261.165i	0.0451992i
$323$	− 3684.09i	− 0.634639i
$324$	−4962.80	−0.850960
$325$	0	0
$326$	−1677.14	−0.284933
$327$	− 5217.32i	− 0.882320i
$328$	− 7278.99i	− 1.22535i
$329$	1068.69	0.179084
$330$	0	0
$331$	−6751.61	−1.12116	−0.560578	−	0.828102i	$-0.689421\pi$
$331$	−6751.61	−1.12116	−0.560578	+	0.828102i	$0.689421\pi$
$332$	5191.40i	0.858177i
$333$	− 1274.27i	− 0.209699i
$334$	1686.23	0.276246
$335$	0	0
$336$	1001.37	0.162586
$337$	5459.75i	0.882527i	0.897378	+	0.441263i	$0.145470\pi$
$337$	5459.75i	0.882527i	−0.897378	+	0.441263i	$0.854530\pi$
$338$	− 1920.79i	− 0.309104i
$339$	−5020.24	−0.804313
$340$	0	0
$341$	−572.883	−0.0909775
$342$	− 143.602i	− 0.0227049i
$343$	− 5090.26i	− 0.801306i
$344$	−7187.24	−1.12648
$345$	0	0
$346$	−3106.17	−0.482626
$347$	8710.68i	1.34759i	0.738918	+	0.673795i	$0.235337\pi$
$347$	8710.68i	1.34759i	−0.738918	+	0.673795i	$0.764663\pi$
$348$	6719.04i	1.03500i
$349$	9806.36	1.50408	0.752038	−	0.659120i	$-0.229071\pi$
$349$	9806.36	1.50408	0.752038	+	0.659120i	$0.229071\pi$
$350$	0	0
$351$	−3667.90	−0.557772
$352$	4320.56i	0.654223i
$353$	− 4543.43i	− 0.685050i	−0.939509	−	0.342525i	$-0.888718\pi$
$353$	− 4543.43i	− 0.685050i	0.939509	−	0.342525i	$-0.111282\pi$
$354$	−29.7470	−0.00446620
$355$	0	0
$356$	4996.08	0.743797
$357$	5850.25i	0.867305i
$358$	1255.04i	0.185283i
$359$	11726.4	1.72394	0.861968	−	0.506962i	$-0.169232\pi$
$359$	11726.4	1.72394	0.861968	+	0.506962i	$0.169232\pi$
$360$	0	0
$361$	−6035.51	−0.879940
$362$	− 2026.85i	− 0.294279i
$363$	− 4374.86i	− 0.632563i
$364$	1423.92	0.205037
$365$	0	0
$366$	−4074.34	−0.581883
$367$	− 7650.27i	− 1.08812i	−0.839045	−	0.544061i	$-0.816886\pi$
$367$	− 7650.27i	− 1.08812i	0.839045	−	0.544061i	$-0.183114\pi$
$368$	− 505.414i	− 0.0715938i
$369$	−1358.56	−0.191664
$370$	0	0
$371$	4798.12	0.671445
$372$	831.640i	0.115910i
$373$	− 10935.0i	− 1.51795i	−0.651121	−	0.758974i	$-0.725701\pi$
$373$	− 10935.0i	− 1.51795i	0.651121	−	0.758974i	$-0.274299\pi$
$374$	−4116.02	−0.569076
$375$	0	0
$376$	2523.39	0.346101
$377$	5645.58i	0.771253i
$378$	− 1468.70i	− 0.199846i
$379$	11842.1	1.60498	0.802490	−	0.596666i	$-0.203508\pi$
$379$	11842.1	1.60498	0.802490	+	0.596666i	$0.203508\pi$
$380$	0	0
$381$	7530.65	1.01262
$382$	− 4676.35i	− 0.626343i
$383$	− 3957.65i	− 0.528007i	−0.964522	−	0.264003i	$-0.914957\pi$
$383$	− 3957.65i	− 0.528007i	0.964522	−	0.264003i	$-0.0850430\pi$
$384$	7613.73	1.01181
$385$	0	0
$386$	315.762	0.0416370
$387$	1341.44i	0.176200i
$388$	− 3044.37i	− 0.398336i
$389$	10043.6	1.30908	0.654538	−	0.756029i	$-0.272863\pi$
$389$	10043.6	1.30908	0.654538	+	0.756029i	$0.272863\pi$
$390$	0	0
$391$	2952.76	0.381912
$392$	− 5350.49i	− 0.689389i
$393$	13320.8i	1.70978i
$394$	−4072.05	−0.520677
$395$	0	0
$396$	514.466	0.0652850
$397$	− 8065.74i	− 1.01967i	−0.860273	−	0.509833i	$-0.829707\pi$
$397$	− 8065.74i	− 1.01967i	0.860273	−	0.509833i	$-0.170293\pi$
$398$	− 3588.22i	− 0.451913i
$399$	−1307.69	−0.164076
$400$	0	0
$401$	5121.53	0.637798	0.318899	−	0.947789i	$-0.396687\pi$
$401$	5121.53	0.637798	0.318899	+	0.947789i	$0.396687\pi$
$402$	3801.61i	0.471659i
$403$	698.774i	0.0863732i
$404$	1160.33	0.142892
$405$	0	0
$406$	−2260.61	−0.276335
$407$	− 8164.12i	− 0.994301i
$408$	13813.6i	1.67617i
$409$	5098.10	0.616344	0.308172	−	0.951331i	$-0.400283\pi$
$409$	5098.10	0.616344	0.308172	+	0.951331i	$0.400283\pi$
$410$	0	0
$411$	7097.94	0.851863
$412$	− 2798.89i	− 0.334688i
$413$	32.0930i	0.00382371i
$414$	115.095	0.0136633
$415$	0	0
$416$	5270.00	0.621113
$417$	− 8112.04i	− 0.952634i
$418$	− 920.040i	− 0.107657i
$419$	10714.1	1.24921	0.624605	−	0.780941i	$-0.285260\pi$
$419$	10714.1	1.24921	0.624605	+	0.780941i	$0.285260\pi$
$420$	0	0
$421$	−10385.0	−1.20222	−0.601111	−	0.799166i	$-0.705275\pi$
$421$	−10385.0	−1.20222	−0.601111	+	0.799166i	$0.705275\pi$
$422$	2620.98i	0.302339i
$423$	− 470.971i	− 0.0541357i
$424$	11329.3	1.29764
$425$	0	0
$426$	−2292.21	−0.260700
$427$	4395.66i	0.498176i
$428$	− 1366.95i	− 0.154378i
$429$	3648.67	0.410628
$430$	0	0
$431$	−1743.60	−0.194864	−0.0974320	−	0.995242i	$-0.531063\pi$
$431$	−1743.60	−0.194864	−0.0974320	+	0.995242i	$0.531063\pi$
$432$	2842.28i	0.316549i
$433$	12529.4i	1.39059i	0.718725	+	0.695294i	$0.244726\pi$
$433$	12529.4i	1.39059i	−0.718725	+	0.695294i	$0.755274\pi$
$434$	−279.804	−0.0309470
$435$	0	0
$436$	−5748.94	−0.631478
$437$	660.021i	0.0722496i
$438$	2472.81i	0.269761i
$439$	10586.1	1.15090	0.575450	−	0.817837i	$-0.304827\pi$
$439$	10586.1	1.15090	0.575450	+	0.817837i	$0.304827\pi$
$440$	0	0
$441$	−998.627	−0.107831
$442$	5020.52i	0.540276i
$443$	− 8797.57i	− 0.943533i	−0.881723	−	0.471767i	$-0.843616\pi$
$443$	− 8797.57i	− 0.943533i	0.881723	−	0.471767i	$-0.156384\pi$
$444$	−11851.7	−1.26679
$445$	0	0
$446$	−6366.63	−0.675939
$447$	− 1471.58i	− 0.155712i
$448$	662.717i	0.0698893i
$449$	7801.86	0.820028	0.410014	−	0.912079i	$-0.365524\pi$
$449$	7801.86	0.820028	0.410014	+	0.912079i	$0.365524\pi$
$450$	0	0
$451$	−8704.18	−0.908789
$452$	5531.78i	0.575649i
$453$	− 14930.3i	− 1.54853i
$454$	−4069.34	−0.420669
$455$	0	0
$456$	−3087.71	−0.317095
$457$	− 12218.9i	− 1.25071i	−0.780339	−	0.625356i	$-0.784954\pi$
$457$	− 12218.9i	− 1.25071i	0.780339	−	0.625356i	$-0.215046\pi$
$458$	4425.72i	0.451529i
$459$	−16605.3	−1.68861
$460$	0	0
$461$	10903.7	1.10160	0.550801	−	0.834637i	$-0.314322\pi$
$461$	10903.7	1.10160	0.550801	+	0.834637i	$0.314322\pi$
$462$	1461.00i	0.147125i
$463$	− 6604.91i	− 0.662972i	−0.943460	−	0.331486i	$-0.892450\pi$
$463$	− 6604.91i	− 0.662972i	0.943460	−	0.331486i	$-0.107550\pi$
$464$	4374.79	0.437704
$465$	0	0
$466$	6443.43	0.640529
$467$	4519.85i	0.447867i	0.974604	+	0.223934i	$0.0718899\pi$
$467$	4519.85i	0.447867i	−0.974604	+	0.223934i	$0.928110\pi$
$468$	− 627.520i	− 0.0619810i
$469$	4101.42	0.403809
$470$	0	0
$471$	−11465.3	−1.12164
$472$	75.7780i	0.00738976i
$473$	8594.47i	0.835464i
$474$	2802.74	0.271591
$475$	0	0
$476$	6446.36	0.620732
$477$	− 2114.53i	− 0.202972i
$478$	− 2671.50i	− 0.255631i
$479$	−14993.9	−1.43025	−0.715124	−	0.698998i	$-0.753630\pi$
$479$	−14993.9	−1.43025	−0.715124	+	0.698998i	$0.753630\pi$
$480$	0	0
$481$	−9958.19	−0.943980
$482$	624.319i	0.0589979i
$483$	− 1048.10i	− 0.0987372i
$484$	−4820.63	−0.452727
$485$	0	0
$486$	−1395.01	−0.130203
$487$	18262.7i	1.69930i	0.527346	+	0.849651i	$0.323187\pi$
$487$	18262.7i	1.69930i	−0.527346	+	0.849651i	$0.676813\pi$
$488$	10379.1i	0.962782i
$489$	6730.64	0.622433
$490$	0	0
$491$	4294.48	0.394719	0.197360	−	0.980331i	$-0.436763\pi$
$491$	4294.48	0.394719	0.197360	+	0.980331i	$0.436763\pi$
$492$	12635.6i	1.15784i
$493$	25558.7i	2.33490i
$494$	−1122.22	−0.102209
$495$	0	0
$496$	541.484	0.0490188
$497$	2472.99i	0.223197i
$498$	6497.13i	0.584625i
$499$	−2916.40	−0.261635	−0.130818	−	0.991406i	$-0.541760\pi$
$499$	−2916.40	−0.261635	−0.130818	+	0.991406i	$0.541760\pi$
$500$	0	0
$501$	−6767.10	−0.603457
$502$	6296.75i	0.559836i
$503$	− 8751.64i	− 0.775778i	−0.921706	−	0.387889i	$-0.873204\pi$
$503$	− 8751.64i	− 0.775778i	0.921706	−	0.387889i	$-0.126796\pi$
$504$	580.903	0.0513403
$505$	0	0
$506$	737.403	0.0647857
$507$	7708.44i	0.675234i
$508$	− 8297.99i	− 0.724732i
$509$	11287.5	0.982926	0.491463	−	0.870899i	$-0.336462\pi$
$509$	11287.5	0.982926	0.491463	+	0.870899i	$0.336462\pi$
$510$	0	0
$511$	2667.83	0.230954
$512$	− 7501.59i	− 0.647513i
$513$	− 3711.73i	− 0.319448i
$514$	−6209.59	−0.532866
$515$	0	0
$516$	12476.4	1.06442
$517$	− 3017.46i	− 0.256688i
$518$	− 3987.46i	− 0.338222i
$519$	12465.6	1.05429
$520$	0	0
$521$	−12773.3	−1.07410	−0.537051	−	0.843550i	$-0.680462\pi$
$521$	−12773.3	−1.07410	−0.537051	+	0.843550i	$0.680462\pi$
$522$	996.249i	0.0835338i
$523$	8856.68i	0.740489i	0.928934	+	0.370244i	$0.120726\pi$
$523$	8856.68i	0.740489i	−0.928934	+	0.370244i	$0.879274\pi$
$524$	14678.1	1.22369
$525$	0	0
$526$	1406.45	0.116586
$527$	3163.49i	0.261488i
$528$	− 2827.37i	− 0.233041i
$529$	−529.000	−0.0434783
$530$	0	0
$531$	14.1434	0.00115588
$532$	1440.93i	0.117429i
$533$	10616.9i	0.862796i
$534$	6252.69	0.506705
$535$	0	0
$536$	9684.29	0.780406
$537$	− 5036.70i	− 0.404748i
$538$	10978.1i	0.879740i
$539$	−6398.09	−0.511290
$540$	0	0
$541$	−16382.7	−1.30194	−0.650969	−	0.759104i	$-0.725637\pi$
$541$	−16382.7	−1.30194	−0.650969	+	0.759104i	$0.725637\pi$
$542$	− 4859.71i	− 0.385134i
$543$	8134.08i	0.642848i
$544$	23858.4	1.88037
$545$	0	0
$546$	1782.06	0.139680
$547$	− 12846.1i	− 1.00413i	−0.864830	−	0.502065i	$-0.832574\pi$
$547$	− 12846.1i	− 1.00413i	0.864830	−	0.502065i	$-0.167426\pi$
$548$	− 7821.19i	− 0.609680i
$549$	1937.17	0.150594
$550$	0	0
$551$	−5713.05	−0.441713
$552$	− 2474.77i	− 0.190821i
$553$	− 3023.79i	− 0.232522i
$554$	−8074.71	−0.619244
$555$	0	0
$556$	−8938.62	−0.681802
$557$	22330.5i	1.69870i	0.527834	+	0.849348i	$0.323004\pi$
$557$	22330.5i	1.69870i	−0.527834	+	0.849348i	$0.676996\pi$
$558$	123.309i	0.00935502i
$559$	10483.1	0.793182
$560$	0	0
$561$	16518.3	1.24314
$562$	4470.33i	0.335533i
$563$	12394.0i	0.927785i	0.885891	+	0.463893i	$0.153548\pi$
$563$	12394.0i	0.927785i	−0.885891	+	0.463893i	$0.846452\pi$
$564$	−4380.38	−0.327034
$565$	0	0
$566$	−784.177	−0.0582357
$567$	6700.87i	0.496314i
$568$	5839.23i	0.431353i
$569$	−10075.6	−0.742341	−0.371171	−	0.928565i	$-0.621043\pi$
$569$	−10075.6	−0.742341	−0.371171	+	0.928565i	$0.621043\pi$
$570$	0	0
$571$	−15792.9	−1.15746	−0.578731	−	0.815519i	$-0.696452\pi$
$571$	−15792.9	−1.15746	−0.578731	+	0.815519i	$0.696452\pi$
$572$	− 4020.45i	− 0.293887i
$573$	18767.0i	1.36824i
$574$	−4251.24	−0.309135
$575$	0	0
$576$	292.059	0.0211269
$577$	7317.48i	0.527956i	0.964529	+	0.263978i	$0.0850346\pi$
$577$	7317.48i	0.527956i	−0.964529	+	0.263978i	$0.914965\pi$
$578$	15953.7i	1.14807i
$579$	−1267.21	−0.0909556
$580$	0	0
$581$	7009.53	0.500524
$582$	− 3810.09i	− 0.271363i
$583$	− 13547.6i	− 0.962407i
$584$	6299.28	0.446346
$585$	0	0
$586$	−4769.56	−0.336226
$587$	21084.7i	1.48255i	0.671199	+	0.741277i	$0.265780\pi$
$587$	21084.7i	1.48255i	−0.671199	+	0.741277i	$0.734220\pi$
$588$	9287.96i	0.651410i
$589$	−707.125	−0.0494678
$590$	0	0
$591$	16341.8	1.13741
$592$	7716.65i	0.535730i
$593$	12354.3i	0.855533i	0.903889	+	0.427766i	$0.140699\pi$
$593$	12354.3i	0.855533i	−0.903889	+	0.427766i	$0.859301\pi$
$594$	−4146.91	−0.286447
$595$	0	0
$596$	−1621.53	−0.111444
$597$	14400.1i	0.987198i
$598$	− 899.448i	− 0.0615070i
$599$	2096.39	0.142999	0.0714994	−	0.997441i	$-0.477222\pi$
$599$	2096.39	0.142999	0.0714994	+	0.997441i	$0.477222\pi$
$600$	0	0
$601$	−21144.5	−1.43511	−0.717555	−	0.696502i	$-0.754739\pi$
$601$	−21144.5	−1.43511	−0.717555	+	0.696502i	$0.754739\pi$
$602$	4197.66i	0.284192i
$603$	− 1807.50i	− 0.122068i
$604$	−16451.6	−1.10829
$605$	0	0
$606$	1452.17	0.0973441
$607$	− 2224.25i	− 0.148731i	−0.997231	−	0.0743653i	$-0.976307\pi$
$607$	− 2224.25i	− 0.148731i	0.997231	−	0.0743653i	$-0.0236931\pi$
$608$	5332.98i	0.355725i
$609$	9072.19	0.603651
$610$	0	0
$611$	−3680.55	−0.243698
$612$	− 2840.91i	− 0.187642i
$613$	− 16796.1i	− 1.10667i	−0.832960	−	0.553334i	$-0.813355\pi$
$613$	− 16796.1i	− 1.10667i	0.832960	−	0.553334i	$-0.186645\pi$
$614$	−11987.9	−0.787936
$615$	0	0
$616$	3721.79	0.243434
$617$	− 26318.9i	− 1.71728i	−0.512581	−	0.858639i	$-0.671310\pi$
$617$	− 26318.9i	− 1.71728i	0.512581	−	0.858639i	$-0.328690\pi$
$618$	− 3502.87i	− 0.228003i
$619$	13974.7	0.907418	0.453709	−	0.891150i	$-0.350101\pi$
$619$	13974.7	0.907418	0.453709	+	0.891150i	$0.350101\pi$
$620$	0	0
$621$	2974.92	0.192237
$622$	− 14936.9i	− 0.962888i
$623$	− 6745.81i	− 0.433813i
$624$	−3448.69	−0.221247
$625$	0	0
$626$	11253.8	0.718519
$627$	3692.27i	0.235176i
$628$	12633.5i	0.802760i
$629$	−45082.8	−2.85782
$630$	0	0
$631$	−18386.9	−1.16002	−0.580008	−	0.814611i	$-0.696951\pi$
$631$	−18386.9	−1.16002	−0.580008	+	0.814611i	$0.696951\pi$
$632$	− 7139.77i	− 0.449375i
$633$	− 10518.4i	− 0.660457i
$634$	−4773.05	−0.298994
$635$	0	0
$636$	−19666.7	−1.22615
$637$	7804.08i	0.485414i
$638$	6382.86i	0.396081i
$639$	1089.85	0.0674704
$640$	0	0
$641$	−17118.6	−1.05483	−0.527414	−	0.849608i	$-0.676838\pi$
$641$	−17118.6	−1.05483	−0.527414	+	0.849608i	$0.676838\pi$
$642$	− 1710.76i	− 0.105169i
$643$	− 14156.0i	− 0.868207i	−0.900863	−	0.434103i	$-0.857065\pi$
$643$	− 14156.0i	− 0.868207i	0.900863	−	0.434103i	$-0.142935\pi$
$644$	−1154.89	−0.0706664
$645$	0	0
$646$	−5080.52	−0.309428
$647$	− 15292.3i	− 0.929215i	−0.885517	−	0.464607i	$-0.846195\pi$
$647$	− 15292.3i	− 0.929215i	0.885517	−	0.464607i	$-0.153805\pi$
$648$	15822.1i	0.959184i
$649$	90.6150	0.00548067
$650$	0	0
$651$	1122.90	0.0676034
$652$	− 7416.46i	− 0.445477i
$653$	4339.22i	0.260041i	0.991511	+	0.130021i	$0.0415043\pi$
$653$	4339.22i	0.260041i	−0.991511	+	0.130021i	$0.958496\pi$
$654$	−7194.91	−0.430188
$655$	0	0
$656$	8227.11	0.489657
$657$	− 1175.71i	− 0.0698156i
$658$	− 1473.77i	− 0.0873153i
$659$	16226.0	0.959145	0.479573	−	0.877502i	$-0.340792\pi$
$659$	16226.0	0.959145	0.479573	+	0.877502i	$0.340792\pi$
$660$	0	0
$661$	−3085.40	−0.181555	−0.0907776	−	0.995871i	$-0.528935\pi$
$661$	−3085.40	−0.181555	−0.0907776	+	0.995871i	$0.528935\pi$
$662$	9310.76i	0.546636i
$663$	− 20148.2i	− 1.18023i
$664$	16550.9	0.967319
$665$	0	0
$666$	−1757.27	−0.102242
$667$	− 4578.95i	− 0.265814i
$668$	7456.64i	0.431895i
$669$	25550.3	1.47658
$670$	0	0
$671$	12411.2	0.714054
$672$	− 8468.64i	− 0.486138i
$673$	− 25682.4i	− 1.47100i	−0.677524	−	0.735501i	$-0.736947\pi$
$673$	− 25682.4i	− 1.47100i	0.677524	−	0.735501i	$-0.263053\pi$
$674$	7529.23	0.430289
$675$	0	0
$676$	8493.89	0.483266
$677$	24845.6i	1.41048i	0.708970	+	0.705239i	$0.249160\pi$
$677$	24845.6i	1.41048i	−0.708970	+	0.705239i	$0.750840\pi$
$678$	6923.13i	0.392155i
$679$	−4110.57	−0.232326
$680$	0	0
$681$	16330.9	0.918947
$682$	790.030i	0.0443575i
$683$	− 21172.1i	− 1.18613i	−0.805153	−	0.593066i	$-0.797917\pi$
$683$	− 21172.1i	− 1.18613i	0.805153	−	0.593066i	$-0.202083\pi$
$684$	635.019	0.0354979
$685$	0	0
$686$	−7019.68	−0.390689
$687$	− 17761.1i	− 0.986359i
$688$	− 8123.42i	− 0.450149i
$689$	−16524.7	−0.913700
$690$	0	0
$691$	27174.6	1.49605	0.748025	−	0.663670i	$-0.231002\pi$
$691$	27174.6	1.49605	0.748025	+	0.663670i	$0.231002\pi$
$692$	− 13735.7i	− 0.754558i
$693$	− 694.642i	− 0.0380769i
$694$	12012.4	0.657038
$695$	0	0
$696$	21421.3	1.16663
$697$	48065.0i	2.61204i
$698$	− 13523.4i	− 0.733335i
$699$	−25858.6	−1.39923
$700$	0	0
$701$	−23144.2	−1.24700	−0.623498	−	0.781825i	$-0.714289\pi$
$701$	−23144.2	−1.24700	−0.623498	+	0.781825i	$0.714289\pi$
$702$	5058.19i	0.271950i
$703$	− 10077.2i	− 0.540638i
$704$	1871.19	0.100175
$705$	0	0
$706$	−6265.59	−0.334006
$707$	− 1566.70i	− 0.0833407i
$708$	− 131.544i	− 0.00698264i
$709$	16601.5	0.879385	0.439692	−	0.898148i	$-0.355087\pi$
$709$	16601.5	0.879385	0.439692	+	0.898148i	$0.355087\pi$
$710$	0	0
$711$	−1332.58	−0.0702893
$712$	− 15928.2i	− 0.838393i
$713$	− 566.753i	− 0.0297687i
$714$	8067.74	0.422868
$715$	0	0
$716$	−5549.91	−0.289679
$717$	10721.2i	0.558424i
$718$	− 16171.1i	− 0.840531i
$719$	−21962.8	−1.13919	−0.569593	−	0.821927i	$-0.692899\pi$
$719$	−21962.8	−1.13919	−0.569593	+	0.821927i	$0.692899\pi$
$720$	0	0
$721$	−3779.12	−0.195204
$722$	8323.22i	0.429028i
$723$	− 2505.50i	− 0.128880i
$724$	8962.90	0.460088
$725$	0	0
$726$	−6033.11	−0.308416
$727$	− 11705.5i	− 0.597158i	−0.954385	−	0.298579i	$-0.903487\pi$
$727$	− 11705.5i	− 0.597158i	0.954385	−	0.298579i	$-0.0965126\pi$
$728$	− 4539.65i	− 0.231114i
$729$	−16374.4	−0.831906
$730$	0	0
$731$	47459.2	2.40129
$732$	− 18017.1i	− 0.909741i
$733$	− 1357.48i	− 0.0684032i	−0.999415	−	0.0342016i	$-0.989111\pi$
$733$	− 1357.48i	− 0.0684032i	0.999415	−	0.0342016i	$-0.0108888\pi$
$734$	−10550.1	−0.530531
$735$	0	0
$736$	−4274.33	−0.214068
$737$	− 11580.4i	− 0.578794i
$738$	1873.52i	0.0934487i
$739$	−30410.7	−1.51377	−0.756885	−	0.653548i	$-0.773280\pi$
$739$	−30410.7	−1.51377	−0.756885	+	0.653548i	$0.773280\pi$
$740$	0	0
$741$	4503.65	0.223274
$742$	− 6616.81i	− 0.327373i
$743$	67.6266i	0.00333914i	0.999999	+	0.00166957i	$0.000531441\pi$
$743$	67.6266i	0.00333914i	−0.999999	+	0.00166957i	$0.999469\pi$
$744$	2651.39	0.130651
$745$	0	0
$746$	−15079.9	−0.740098
$747$	− 3089.10i	− 0.151304i
$748$	− 18201.4i	− 0.889718i
$749$	−1845.68	−0.0900395
$750$	0	0
$751$	−14309.2	−0.695271	−0.347636	−	0.937630i	$-0.613015\pi$
$751$	−14309.2	−0.695271	−0.347636	+	0.937630i	$0.613015\pi$
$752$	2852.08i	0.138304i
$753$	− 25269.9i	− 1.22296i
$754$	7785.50	0.376036
$755$	0	0
$756$	6494.73	0.312449
$757$	6568.31i	0.315362i	0.987490	+	0.157681i	$0.0504018\pi$
$757$	6568.31i	0.315362i	−0.987490	+	0.157681i	$0.949598\pi$
$758$	− 16330.7i	− 0.782532i
$759$	−2959.32	−0.141524
$760$	0	0
$761$	−3368.64	−0.160464	−0.0802320	−	0.996776i	$-0.525566\pi$
$761$	−3368.64	−0.160464	−0.0802320	+	0.996776i	$0.525566\pi$
$762$	− 10385.1i	− 0.493716i
$763$	7762.34i	0.368304i
$764$	20679.2	0.979252
$765$	0	0
$766$	−5457.77	−0.257438
$767$	− 110.528i	− 0.00520329i
$768$	− 14063.1i	− 0.660754i
$769$	42031.8	1.97101	0.985505	−	0.169649i	$-0.0542635\pi$
$769$	42031.8	1.97101	0.985505	+	0.169649i	$0.0542635\pi$
$770$	0	0
$771$	24920.1	1.16404
$772$	1396.33i	0.0650971i
$773$	8814.15i	0.410120i	0.978749	+	0.205060i	$0.0657390\pi$
$773$	8814.15i	0.410120i	−0.978749	+	0.205060i	$0.934261\pi$
$774$	1849.90	0.0859089
$775$	0	0
$776$	−9705.89	−0.448996
$777$	16002.3i	0.738843i
$778$	− 13850.5i	− 0.638260i
$779$	−10743.8	−0.494142
$780$	0	0
$781$	6982.52	0.319916
$782$	− 4071.98i	− 0.186207i
$783$	25750.5i	1.17528i
$784$	6047.42	0.275484
$785$	0	0
$786$	18369.9	0.833629
$787$	− 35768.0i	− 1.62007i	−0.586384	−	0.810033i	$-0.699449\pi$
$787$	− 35768.0i	− 1.62007i	0.586384	−	0.810033i	$-0.300551\pi$
$788$	− 18006.9i	− 0.814049i
$789$	−5644.32	−0.254681
$790$	0	0
$791$	7469.13	0.335742
$792$	− 1640.19i	− 0.0735879i
$793$	− 15138.6i	− 0.677916i
$794$	−11123.0	−0.497154
$795$	0	0
$796$	15867.4	0.706540
$797$	5527.95i	0.245684i	0.992426	+	0.122842i	$0.0392008\pi$
$797$	5527.95i	0.245684i	−0.992426	+	0.122842i	$0.960799\pi$
$798$	1803.35i	0.0799975i
$799$	−16662.6	−0.737773
$800$	0	0
$801$	−2972.88	−0.131138
$802$	− 7062.81i	− 0.310968i
$803$	− 7532.65i	− 0.331035i
$804$	−16811.0	−0.737412
$805$	0	0
$806$	963.639	0.0421126
$807$	− 44057.0i	− 1.92178i
$808$	− 3699.30i	− 0.161065i
$809$	15062.2	0.654585	0.327293	−	0.944923i	$-0.393864\pi$
$809$	15062.2	0.654585	0.327293	+	0.944923i	$0.393864\pi$
$810$	0	0
$811$	−11471.7	−0.496703	−0.248352	−	0.968670i	$-0.579889\pi$
$811$	−11471.7	−0.496703	−0.248352	+	0.968670i	$0.579889\pi$
$812$	− 9996.60i	− 0.432034i
$813$	19502.8i	0.841321i
$814$	−11258.7	−0.484786
$815$	0	0
$816$	−15612.9	−0.669806
$817$	10608.4i	0.454272i
$818$	− 7030.49i	− 0.300508i
$819$	−847.290	−0.0361498
$820$	0	0
$821$	20154.6	0.856760	0.428380	−	0.903599i	$-0.359084\pi$
$821$	20154.6	0.856760	0.428380	+	0.903599i	$0.359084\pi$
$822$	− 9788.37i	− 0.415339i
$823$	19716.2i	0.835070i	0.908661	+	0.417535i	$0.137106\pi$
$823$	19716.2i	0.835070i	−0.908661	+	0.417535i	$0.862894\pi$
$824$	−8923.27	−0.377254
$825$	0	0
$826$	44.2576	0.00186431
$827$	13072.4i	0.549666i	0.961492	+	0.274833i	$0.0886225\pi$
$827$	13072.4i	0.549666i	−0.961492	+	0.274833i	$0.911377\pi$
$828$	508.961i	0.0213619i
$829$	−41059.4	−1.72021	−0.860104	−	0.510118i	$-0.829602\pi$
$829$	−41059.4	−1.72021	−0.860104	+	0.510118i	$0.829602\pi$
$830$	0	0
$831$	32405.1	1.35273
$832$	− 2282.39i	− 0.0951052i
$833$	35330.7i	1.46955i
$834$	−11186.8	−0.464471
$835$	0	0
$836$	4068.50	0.168316
$837$	3187.23i	0.131621i
$838$	− 14775.2i	− 0.609071i
$839$	8405.72	0.345885	0.172943	−	0.984932i	$-0.444672\pi$
$839$	8405.72	0.345885	0.172943	+	0.984932i	$0.444672\pi$
$840$	0	0
$841$	15245.8	0.625110
$842$	14321.4i	0.586161i
$843$	− 17940.1i	− 0.732967i
$844$	−11590.2	−0.472691
$845$	0	0
$846$	−649.489	−0.0263947
$847$	6508.92i	0.264049i
$848$	12805.0i	0.518546i
$849$	3147.03	0.127215
$850$	0	0
$851$	8076.77	0.325344
$852$	− 10136.4i	− 0.407589i
$853$	34913.6i	1.40143i	0.713441	+	0.700715i	$0.247136\pi$
$853$	34913.6i	1.40143i	−0.713441	+	0.700715i	$0.752864\pi$
$854$	6061.81	0.242893
$855$	0	0
$856$	−4358.02	−0.174012
$857$	− 5126.51i	− 0.204339i	−0.994767	−	0.102169i	$-0.967422\pi$
$857$	− 5126.51i	− 0.204339i	0.994767	−	0.102169i	$-0.0325784\pi$
$858$	− 5031.67i	− 0.200208i
$859$	−34625.2	−1.37532	−0.687659	−	0.726034i	$-0.741362\pi$
$859$	−34625.2	−1.37532	−0.687659	+	0.726034i	$0.741362\pi$
$860$	0	0
$861$	17060.9	0.675301
$862$	2404.50i	0.0950089i
$863$	24882.2i	0.981459i	0.871312	+	0.490729i	$0.163270\pi$
$863$	24882.2i	0.981459i	−0.871312	+	0.490729i	$0.836730\pi$
$864$	24037.4	0.946491
$865$	0	0
$866$	17278.6	0.678002
$867$	− 64024.7i	− 2.50795i
$868$	− 1237.32i	− 0.0483839i
$869$	−8537.71	−0.333282
$870$	0	0
$871$	−14125.2	−0.549501
$872$	18328.5i	0.711789i
$873$	1811.53i	0.0702302i
$874$	910.196	0.0352264
$875$	0	0
$876$	−10935.0	−0.421756
$877$	− 23428.6i	− 0.902085i	−0.892502	−	0.451043i	$-0.851052\pi$
$877$	− 23428.6i	− 0.902085i	0.892502	−	0.451043i	$-0.148948\pi$
$878$	− 14598.6i	− 0.561138i
$879$	19141.0	0.734483
$880$	0	0
$881$	17183.7	0.657134	0.328567	−	0.944481i	$-0.393434\pi$
$881$	17183.7	0.657134	0.328567	+	0.944481i	$0.393434\pi$
$882$	1377.15i	0.0525748i
$883$	16098.7i	0.613549i	0.951782	+	0.306774i	$0.0992497\pi$
$883$	16098.7i	0.613549i	−0.951782	+	0.306774i	$0.900750\pi$
$884$	−22201.2	−0.844690
$885$	0	0
$886$	−12132.2	−0.460034
$887$	31987.2i	1.21085i	0.795901	+	0.605426i	$0.206997\pi$
$887$	31987.2i	1.21085i	−0.795901	+	0.605426i	$0.793003\pi$
$888$	37784.8i	1.42790i
$889$	−11204.1	−0.422693
$890$	0	0
$891$	18920.0	0.711385
$892$	− 28153.8i	− 1.05679i
$893$	− 3724.54i	− 0.139571i
$894$	−2029.37	−0.0759198
$895$	0	0
$896$	−11327.7	−0.422358
$897$	3609.63i	0.134361i
$898$	− 10759.1i	− 0.399817i
$899$	4905.74	0.181997
$900$	0	0
$901$	−74810.5	−2.76615
$902$	12003.4i	0.443094i
$903$	− 16845.9i	− 0.620815i
$904$	17636.1	0.648859
$905$	0	0
$906$	−20589.5	−0.755010
$907$	30858.9i	1.12972i	0.825188	+	0.564858i	$0.191069\pi$
$907$	30858.9i	1.12972i	−0.825188	+	0.564858i	$0.808931\pi$
$908$	− 17995.0i	− 0.657692i
$909$	−690.445	−0.0251932
$910$	0	0
$911$	24371.5	0.886350	0.443175	−	0.896435i	$-0.353852\pi$
$911$	24371.5	0.886350	0.443175	+	0.896435i	$0.353852\pi$
$912$	− 3489.90i	− 0.126713i
$913$	− 19791.5i	− 0.717419i
$914$	−16850.4	−0.609804
$915$	0	0
$916$	−19570.9	−0.705939
$917$	− 19818.6i	− 0.713707i
$918$	22899.5i	0.823306i
$919$	−26409.1	−0.947939	−0.473970	−	0.880541i	$-0.657179\pi$
$919$	−26409.1	−0.947939	−0.473970	+	0.880541i	$0.657179\pi$
$920$	0	0
$921$	48109.4	1.72124
$922$	− 15036.7i	− 0.537102i
$923$	− 8516.94i	− 0.303725i
$924$	−6460.68	−0.230022
$925$	0	0
$926$	−9108.45	−0.323242
$927$	1665.46i	0.0590084i
$928$	− 36998.0i	− 1.30875i
$929$	9424.94	0.332855	0.166427	−	0.986054i	$-0.446777\pi$
$929$	9424.94	0.332855	0.166427	+	0.986054i	$0.446777\pi$
$930$	0	0
$931$	−7897.34	−0.278007
$932$	28493.4i	1.00143i
$933$	59944.4i	2.10342i
$934$	6233.07	0.218364
$935$	0	0
$936$	−2000.62	−0.0698637
$937$	− 24369.5i	− 0.849645i	−0.905277	−	0.424822i	$-0.860337\pi$
$937$	− 24369.5i	− 0.849645i	0.905277	−	0.424822i	$-0.139663\pi$
$938$	− 5656.04i	− 0.196883i
$939$	−45163.4	−1.56960
$940$	0	0
$941$	9686.67	0.335575	0.167788	−	0.985823i	$-0.446338\pi$
$941$	9686.67	0.335575	0.167788	+	0.985823i	$0.446338\pi$
$942$	15811.1i	0.546872i
$943$	− 8611.05i	− 0.297364i
$944$	−85.6485	−0.00295299
$945$	0	0
$946$	11852.1	0.407343
$947$	20646.9i	0.708483i	0.935154	+	0.354241i	$0.115261\pi$
$947$	20646.9i	0.708483i	−0.935154	+	0.354241i	$0.884739\pi$
$948$	12394.0i	0.424618i
$949$	−9187.95	−0.314282
$950$	0	0
$951$	19155.0	0.653149
$952$	− 20551.9i	− 0.699676i
$953$	5219.50i	0.177415i	0.996058	+	0.0887073i	$0.0282736\pi$
$953$	5219.50i	0.177415i	−0.996058	+	0.0887073i	$0.971726\pi$
$954$	−2916.03	−0.0989621
$955$	0	0
$956$	11813.6	0.399665
$957$	− 25615.5i	− 0.865236i
$958$	20677.2i	0.697339i
$959$	−10560.3	−0.355590
$960$	0	0
$961$	−29183.8	−0.979618
$962$	13732.8i	0.460252i
$963$	813.389i	0.0272182i
$964$	−2760.79	−0.0922398
$965$	0	0
$966$	−1445.37	−0.0481408
$967$	− 19431.9i	− 0.646211i	−0.946363	−	0.323106i	$-0.895273\pi$
$967$	− 19431.9i	− 0.646211i	0.946363	−	0.323106i	$-0.104727\pi$
$968$	15368.9i	0.510304i
$969$	20388.9	0.675941
$970$	0	0
$971$	−38774.9	−1.28151	−0.640755	−	0.767745i	$-0.721379\pi$
$971$	−38774.9	−1.28151	−0.640755	+	0.767745i	$0.721379\pi$
$972$	− 6168.85i	− 0.203566i
$973$	12069.1i	0.397654i
$974$	25185.0	0.828520
$975$	0	0
$976$	−11731.0	−0.384733
$977$	27274.0i	0.893115i	0.894755	+	0.446558i	$0.147350\pi$
$977$	27274.0i	0.893115i	−0.894755	+	0.446558i	$0.852650\pi$
$978$	− 9281.83i	− 0.303477i
$979$	−19046.9	−0.621800
$980$	0	0
$981$	3420.86	0.111335
$982$	− 5922.27i	− 0.192451i
$983$	835.449i	0.0271075i	0.999908	+	0.0135538i	$0.00431443\pi$
$983$	835.449i	0.0271075i	−0.999908	+	0.0135538i	$0.995686\pi$
$984$	40284.2	1.30510
$985$	0	0
$986$	35246.5	1.13842
$987$	5914.47i	0.190739i
$988$	− 4962.55i	− 0.159797i
$989$	−8502.52	−0.273371
$990$	0	0
$991$	42619.5	1.36615	0.683074	−	0.730349i	$-0.260643\pi$
$991$	42619.5	1.36615	0.683074	+	0.730349i	$0.260643\pi$
$992$	− 4579.38i	− 0.146568i
$993$	− 37365.6i	− 1.19412i
$994$	3410.36	0.108823
$995$	0	0
$996$	−28730.8	−0.914028
$997$	− 3926.78i	− 0.124737i	−0.998053	−	0.0623684i	$-0.980135\pi$
$997$	− 3926.78i	− 0.124737i	0.998053	−	0.0623684i	$-0.0198654\pi$
$998$	4021.84i	0.127564i
$999$	−45421.0	−1.43850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	575.4.b.j.24.6		14
5.2	odd	4		575.4.a.m.1.4	yes	7
5.3	odd	4		575.4.a.l.1.4	✓	7
5.4	even	2	inner	575.4.b.j.24.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
575.4.a.l.1.4	✓	7	5.3	odd	4
575.4.a.m.1.4	yes	7	5.2	odd	4
575.4.b.j.24.6		14	1.1	even	1	trivial
575.4.b.j.24.9		14	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	575.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.37904i q^{2} +5.53432i q^{3} +6.09824 q^{4} +7.63206 q^{6} -8.23397i q^{7} -19.4421i q^{8} -3.62871 q^{9} -23.2487 q^{11} +33.7496i q^{12} +28.3577i q^{13} -11.3550 q^{14} +21.9745 q^{16} +128.381i q^{17} +5.00414i q^{18} -28.6966 q^{19} +45.5694 q^{21} +32.0610i q^{22} -23.0000i q^{23} +107.599 q^{24} +39.1064 q^{26} +129.344i q^{27} -50.2128i q^{28} +199.085 q^{29} +24.6414 q^{31} -185.840i q^{32} -128.666i q^{33} +177.043 q^{34} -22.1287 q^{36} +351.164i q^{37} +39.5738i q^{38} -156.940 q^{39} +374.394 q^{41} -62.8422i q^{42} -369.675i q^{43} -141.777 q^{44} -31.7180 q^{46} +129.790i q^{47} +121.614i q^{48} +275.202 q^{49} -710.501 q^{51} +172.932i q^{52} +582.723i q^{53} +178.371 q^{54} -160.085 q^{56} -158.816i q^{57} -274.547i q^{58} -3.89763 q^{59} -533.845 q^{61} -33.9816i q^{62} +29.8787i q^{63} -80.4857 q^{64} -177.436 q^{66} +498.110i q^{67} +782.898i q^{68} +127.289 q^{69} -300.340 q^{71} +70.5496i q^{72} +324.002i q^{73} +484.270 q^{74} -174.999 q^{76} +191.430i q^{77} +216.428i q^{78} +367.233 q^{79} -813.808 q^{81} -516.305i q^{82} +851.294i q^{83} +277.894 q^{84} -509.797 q^{86} +1101.80i q^{87} +452.004i q^{88} +819.266 q^{89} +233.496 q^{91} -140.260i q^{92} +136.374i q^{93} +178.986 q^{94} +1028.50 q^{96} -499.221i q^{97} -379.515i q^{98} +84.3629 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 54 q^{4} - 82 q^{6} - 20 q^{9} - 104 q^{11} + 84 q^{14} - 170 q^{16} + 20 q^{19} - 404 q^{21} + 606 q^{24} - 52 q^{26} + 910 q^{29} - 1380 q^{31} + 1314 q^{34} + 408 q^{36} + 554 q^{39} - 460 q^{41}+ \cdots + 4286 q^{99}+O(q^{100}) \) 14 * q - 54 * q^4 - 82 * q^6 - 20 * q^9 - 104 * q^11 + 84 * q^14 - 170 * q^16 + 20 * q^19 - 404 * q^21 + 606 * q^24 - 52 * q^26 + 910 * q^29 - 1380 * q^31 + 1314 * q^34 + 408 * q^36 + 554 * q^39 - 460 * q^41 + 2006 * q^44 - 138 * q^46 + 1604 * q^49 - 1390 * q^51 + 2494 * q^54 - 772 * q^56 + 1102 * q^59 - 2484 * q^61 + 3002 * q^64 + 1974 * q^66 + 46 * q^69 + 372 * q^71 + 3060 * q^74 - 3550 * q^76 + 4006 * q^79 - 730 * q^81 + 2704 * q^84 + 3788 * q^86 + 5258 * q^89 - 2590 * q^91 + 6936 * q^94 - 270 * q^96 + 4286 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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