LMFDB - Embedded newform 531.5.b.a.296.48

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,5,Mod(296,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("531.296");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	531.b (of order $2$, degree $1$, 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:	$54.8894503975$
Analytic rank:	$0$
Dimension:	$76$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.48
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05329i q^{2} +11.7840 q^{4} -8.97562i q^{5} -18.7464 q^{7} +57.0487i q^{8} +O(q^{10})$	\(q+2.05329i q^{2} +11.7840 q^{4} -8.97562i q^{5} -18.7464 q^{7} +57.0487i q^{8} +18.4296 q^{10} +217.678i q^{11} -276.677 q^{13} -38.4918i q^{14} +71.4059 q^{16} +203.088i q^{17} +27.7080 q^{19} -105.769i q^{20} -446.957 q^{22} -999.650i q^{23} +544.438 q^{25} -568.100i q^{26} -220.907 q^{28} -947.379i q^{29} -379.054 q^{31} +1059.40i q^{32} -417.000 q^{34} +168.260i q^{35} -774.539 q^{37} +56.8928i q^{38} +512.047 q^{40} -579.394i q^{41} -1925.79 q^{43} +2565.11i q^{44} +2052.58 q^{46} -3513.61i q^{47} -2049.57 q^{49} +1117.89i q^{50} -3260.36 q^{52} +1995.12i q^{53} +1953.79 q^{55} -1069.46i q^{56} +1945.25 q^{58} +453.188i q^{59} -5542.38 q^{61} -778.310i q^{62} -1032.76 q^{64} +2483.35i q^{65} -7076.57 q^{67} +2393.19i q^{68} -345.488 q^{70} +6129.62i q^{71} -1180.89 q^{73} -1590.36i q^{74} +326.511 q^{76} -4080.67i q^{77} -1548.39 q^{79} -640.912i q^{80} +1189.67 q^{82} -1142.31i q^{83} +1822.84 q^{85} -3954.22i q^{86} -12418.2 q^{88} -598.920i q^{89} +5186.70 q^{91} -11779.9i q^{92} +7214.47 q^{94} -248.697i q^{95} +6266.12 q^{97} -4208.38i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) $ 76 * q - 632 * q^4 - 48 * q^7	$ 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) $ 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/531\mathbb{Z}\right)^\times$.

$n$	$119$	$415$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.05329i	0.513324i	0.966501	+	0.256662i	$0.0826227\pi$
$2$	2.05329i	0.513324i	−0.966501	+	0.256662i	$0.917377\pi$
$3$	0	0
$3$	0	0
$4$	11.7840	0.736499
$5$	− 8.97562i	− 0.359025i	−0.983756	−	0.179512i	$-0.942548\pi$
$5$	− 8.97562i	− 0.359025i	0.983756	−	0.179512i	$-0.0574520\pi$
$6$	0	0
$7$	−18.7464	−0.382579	−0.191290	−	0.981534i	$-0.561267\pi$
$7$	−18.7464	−0.382579	−0.191290	+	0.981534i	$0.561267\pi$
$8$	57.0487i	0.891386i
$9$	0	0
$10$	18.4296	0.184296
$11$	217.678i	1.79899i	0.436931	+	0.899495i	$0.356065\pi$
$11$	217.678i	1.79899i	−0.436931	+	0.899495i	$0.643935\pi$
$12$	0	0
$13$	−276.677	−1.63714	−0.818572	−	0.574404i	$-0.805234\pi$
$13$	−276.677	−1.63714	−0.818572	+	0.574404i	$0.805234\pi$
$14$	− 38.4918i	− 0.196387i
$15$	0	0
$16$	71.4059	0.278929
$17$	203.088i	0.702727i	0.936239	+	0.351363i	$0.114282\pi$
$17$	203.088i	0.702727i	−0.936239	+	0.351363i	$0.885718\pi$
$18$	0	0
$19$	27.7080	0.0767536	0.0383768	−	0.999263i	$-0.487781\pi$
$19$	27.7080	0.0767536	0.0383768	+	0.999263i	$0.487781\pi$
$20$	− 105.769i	− 0.264421i
$21$	0	0
$22$	−446.957	−0.923464
$23$	− 999.650i	− 1.88970i	−0.327507	−	0.944849i	$-0.606208\pi$
$23$	− 999.650i	− 1.88970i	0.327507	−	0.944849i	$-0.393792\pi$
$24$	0	0
$25$	544.438	0.871101
$26$	− 568.100i	− 0.840385i
$27$	0	0
$28$	−220.907	−0.281769
$29$	− 947.379i	− 1.12649i	−0.826289	−	0.563246i	$-0.809552\pi$
$29$	− 947.379i	− 1.12649i	0.826289	−	0.563246i	$-0.190448\pi$
$30$	0	0
$31$	−379.054	−0.394437	−0.197219	−	0.980360i	$-0.563191\pi$
$31$	−379.054	−0.394437	−0.197219	+	0.980360i	$0.563191\pi$
$32$	1059.40i	1.03457i
$33$	0	0
$34$	−417.000	−0.360726
$35$	168.260i	0.137355i
$36$	0	0
$37$	−774.539	−0.565770	−0.282885	−	0.959154i	$-0.591291\pi$
$37$	−774.539	−0.565770	−0.282885	+	0.959154i	$0.591291\pi$
$38$	56.8928i	0.0393994i
$39$	0	0
$40$	512.047	0.320030
$41$	− 579.394i	− 0.344672i	−0.985038	−	0.172336i	$-0.944869\pi$
$41$	− 579.394i	− 0.344672i	0.985038	−	0.172336i	$-0.0551315\pi$
$42$	0	0
$43$	−1925.79	−1.04153	−0.520766	−	0.853700i	$-0.674353\pi$
$43$	−1925.79	−1.04153	−0.520766	+	0.853700i	$0.674353\pi$
$44$	2565.11i	1.32495i
$45$	0	0
$46$	2052.58	0.970026
$47$	− 3513.61i	− 1.59059i	−0.606225	−	0.795293i	$-0.707317\pi$
$47$	− 3513.61i	− 1.59059i	0.606225	−	0.795293i	$-0.292683\pi$
$48$	0	0
$49$	−2049.57	−0.853633
$50$	1117.89i	0.447157i
$51$	0	0
$52$	−3260.36	−1.20576
$53$	1995.12i	0.710260i	0.934817	+	0.355130i	$0.115563\pi$
$53$	1995.12i	0.710260i	−0.934817	+	0.355130i	$0.884437\pi$
$54$	0	0
$55$	1953.79	0.645882
$56$	− 1069.46i	− 0.341026i
$57$	0	0
$58$	1945.25	0.578255
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	−5542.38	−1.48949	−0.744743	−	0.667351i	$-0.767428\pi$
$61$	−5542.38	−1.48949	−0.744743	+	0.667351i	$0.767428\pi$
$62$	− 778.310i	− 0.202474i
$63$	0	0
$64$	−1032.76	−0.252138
$65$	2483.35i	0.587775i
$66$	0	0
$67$	−7076.57	−1.57643	−0.788213	−	0.615403i	$-0.788993\pi$
$67$	−7076.57	−1.57643	−0.788213	+	0.615403i	$0.788993\pi$
$68$	2393.19i	0.517558i
$69$	0	0
$70$	−345.488	−0.0705077
$71$	6129.62i	1.21595i	0.793955	+	0.607977i	$0.208019\pi$
$71$	6129.62i	1.21595i	−0.793955	+	0.607977i	$0.791981\pi$
$72$	0	0
$73$	−1180.89	−0.221598	−0.110799	−	0.993843i	$-0.535341\pi$
$73$	−1180.89	−0.221598	−0.110799	+	0.993843i	$0.535341\pi$
$74$	− 1590.36i	− 0.290423i
$75$	0	0
$76$	326.511	0.0565289
$77$	− 4080.67i	− 0.688256i
$78$	0	0
$79$	−1548.39	−0.248100	−0.124050	−	0.992276i	$-0.539588\pi$
$79$	−1548.39	−0.248100	−0.124050	+	0.992276i	$0.539588\pi$
$80$	− 640.912i	− 0.100143i
$81$	0	0
$82$	1189.67	0.176928
$83$	− 1142.31i	− 0.165817i	−0.996557	−	0.0829084i	$-0.973579\pi$
$83$	− 1142.31i	− 0.165817i	0.996557	−	0.0829084i	$-0.0264209\pi$
$84$	0	0
$85$	1822.84	0.252296
$86$	− 3954.22i	− 0.534643i
$87$	0	0
$88$	−12418.2	−1.60359
$89$	− 598.920i	− 0.0756117i	−0.999285	−	0.0378058i	$-0.987963\pi$
$89$	− 598.920i	− 0.0756117i	0.999285	−	0.0378058i	$-0.0120368\pi$
$90$	0	0
$91$	5186.70	0.626337
$92$	− 11779.9i	− 1.39176i
$93$	0	0
$94$	7214.47	0.816486
$95$	− 248.697i	− 0.0275564i
$96$	0	0
$97$	6266.12	0.665970	0.332985	−	0.942932i	$-0.391944\pi$
$97$	6266.12	0.665970	0.332985	+	0.942932i	$0.391944\pi$
$98$	− 4208.38i	− 0.438190i
$99$	0	0
$100$	6415.65	0.641565
$101$	16502.0i	1.61769i	0.588025	+	0.808843i	$0.299906\pi$
$101$	16502.0i	1.61769i	−0.588025	+	0.808843i	$0.700094\pi$
$102$	0	0
$103$	−7315.24	−0.689531	−0.344766	−	0.938689i	$-0.612042\pi$
$103$	−7315.24	−0.689531	−0.344766	+	0.938689i	$0.612042\pi$
$104$	− 15784.1i	− 1.45933i
$105$	0	0
$106$	−4096.57	−0.364593
$107$	12558.9i	1.09694i	0.836169	+	0.548472i	$0.184790\pi$
$107$	12558.9i	1.09694i	−0.836169	+	0.548472i	$0.815210\pi$
$108$	0	0
$109$	5936.96	0.499702	0.249851	−	0.968284i	$-0.419618\pi$
$109$	5936.96	0.499702	0.249851	+	0.968284i	$0.419618\pi$
$110$	4011.71i	0.331546i
$111$	0	0
$112$	−1338.60	−0.106713
$113$	− 11445.0i	− 0.896314i	−0.893955	−	0.448157i	$-0.852080\pi$
$113$	− 11445.0i	− 0.896314i	0.893955	−	0.448157i	$-0.147920\pi$
$114$	0	0
$115$	−8972.48	−0.678448
$116$	− 11163.9i	− 0.829660i
$117$	0	0
$118$	−930.528	−0.0668290
$119$	− 3807.16i	− 0.268849i
$120$	0	0
$121$	−32742.6	−2.23636
$122$	− 11380.1i	− 0.764588i
$123$	0	0
$124$	−4466.77	−0.290503
$125$	− 10496.4i	− 0.671772i
$126$	0	0
$127$	−20010.7	−1.24067	−0.620333	−	0.784339i	$-0.713002\pi$
$127$	−20010.7	−1.24067	−0.620333	+	0.784339i	$0.713002\pi$
$128$	14829.8i	0.905138i
$129$	0	0
$130$	−5099.05	−0.301719
$131$	7404.43i	0.431469i	0.976452	+	0.215734i	$0.0692145\pi$
$131$	7404.43i	0.431469i	−0.976452	+	0.215734i	$0.930785\pi$
$132$	0	0
$133$	−519.425	−0.0293643
$134$	− 14530.3i	− 0.809216i
$135$	0	0
$136$	−11585.9	−0.626401
$137$	− 18557.5i	− 0.988734i	−0.869253	−	0.494367i	$-0.835400\pi$
$137$	− 18557.5i	− 0.988734i	0.869253	−	0.494367i	$-0.164600\pi$
$138$	0	0
$139$	6403.79	0.331442	0.165721	−	0.986173i	$-0.447005\pi$
$139$	6403.79	0.331442	0.165721	+	0.986173i	$0.447005\pi$
$140$	1982.78i	0.101162i
$141$	0	0
$142$	−12585.9	−0.624178
$143$	− 60226.5i	− 2.94521i
$144$	0	0
$145$	−8503.31	−0.404438
$146$	− 2424.72i	− 0.113751i
$147$	0	0
$148$	−9127.15	−0.416689
$149$	− 7493.19i	− 0.337516i	−0.985658	−	0.168758i	$-0.946024\pi$
$149$	− 7493.19i	− 0.337516i	0.985658	−	0.168758i	$-0.0539756\pi$
$150$	0	0
$151$	−30461.2	−1.33596	−0.667980	−	0.744179i	$-0.732841\pi$
$151$	−30461.2	−1.33596	−0.667980	+	0.744179i	$0.732841\pi$
$152$	1580.71i	0.0684170i
$153$	0	0
$154$	8378.82	0.353298
$155$	3402.24i	0.141613i
$156$	0	0
$157$	32410.6	1.31488	0.657442	−	0.753505i	$-0.271638\pi$
$157$	32410.6	1.31488	0.657442	+	0.753505i	$0.271638\pi$
$158$	− 3179.31i	− 0.127356i
$159$	0	0
$160$	9508.74	0.371435
$161$	18739.8i	0.722959i
$162$	0	0
$163$	−10188.8	−0.383486	−0.191743	−	0.981445i	$-0.561414\pi$
$163$	−10188.8	−0.383486	−0.191743	+	0.981445i	$0.561414\pi$
$164$	− 6827.57i	− 0.253851i
$165$	0	0
$166$	2345.50	0.0851177
$167$	58.9172i	0.00211256i	0.999999	+	0.00105628i	$0.000336224\pi$
$167$	58.9172i	0.00211256i	−0.999999	+	0.00105628i	$0.999664\pi$
$168$	0	0
$169$	47989.4	1.68024
$170$	3742.83i	0.129510i
$171$	0	0
$172$	−22693.5	−0.767087
$173$	− 11950.4i	− 0.399291i	−0.979868	−	0.199646i	$-0.936021\pi$
$173$	− 11950.4i	− 0.399291i	0.979868	−	0.199646i	$-0.0639791\pi$
$174$	0	0
$175$	−10206.2	−0.333265
$176$	15543.5i	0.501791i
$177$	0	0
$178$	1229.76	0.0388133
$179$	− 3009.72i	− 0.0939335i	−0.998896	−	0.0469668i	$-0.985045\pi$
$179$	− 3009.72i	− 0.0939335i	0.998896	−	0.0469668i	$-0.0149555\pi$
$180$	0	0
$181$	53801.7	1.64225	0.821124	−	0.570749i	$-0.193347\pi$
$181$	53801.7	1.64225	0.821124	+	0.570749i	$0.193347\pi$
$182$	10649.8i	0.321514i
$183$	0	0
$184$	57028.7	1.68445
$185$	6951.96i	0.203125i
$186$	0	0
$187$	−44207.8	−1.26420
$188$	− 41404.3i	− 1.17147i
$189$	0	0
$190$	510.648	0.0141454
$191$	8982.37i	0.246220i	0.992393	+	0.123110i	$0.0392869\pi$
$191$	8982.37i	0.246220i	−0.992393	+	0.123110i	$0.960713\pi$
$192$	0	0
$193$	−61640.3	−1.65482	−0.827409	−	0.561599i	$-0.810186\pi$
$193$	−61640.3	−1.65482	−0.827409	+	0.561599i	$0.810186\pi$
$194$	12866.2i	0.341858i
$195$	0	0
$196$	−24152.1	−0.628700
$197$	19491.6i	0.502244i	0.967955	+	0.251122i	$0.0807996\pi$
$197$	19491.6i	0.502244i	−0.967955	+	0.251122i	$0.919200\pi$
$198$	0	0
$199$	4955.23	0.125129	0.0625644	−	0.998041i	$-0.480072\pi$
$199$	4955.23	0.125129	0.0625644	+	0.998041i	$0.480072\pi$
$200$	31059.5i	0.776487i
$201$	0	0
$202$	−33883.5	−0.830396
$203$	17759.9i	0.430972i
$204$	0	0
$205$	−5200.42	−0.123746
$206$	− 15020.3i	− 0.353953i
$207$	0	0
$208$	−19756.4	−0.456648
$209$	6031.42i	0.138079i
$210$	0	0
$211$	23894.7	0.536707	0.268353	−	0.963321i	$-0.413521\pi$
$211$	23894.7	0.536707	0.268353	+	0.963321i	$0.413521\pi$
$212$	23510.5i	0.523106i
$213$	0	0
$214$	−25787.1	−0.563087
$215$	17285.2i	0.373935i
$216$	0	0
$217$	7105.89	0.150903
$218$	12190.3i	0.256509i
$219$	0	0
$220$	23023.5	0.475691
$221$	− 56189.9i	− 1.15047i
$222$	0	0
$223$	87946.6	1.76852	0.884259	−	0.466997i	$-0.154664\pi$
$223$	87946.6	1.76852	0.884259	+	0.466997i	$0.154664\pi$
$224$	− 19859.8i	− 0.395804i
$225$	0	0
$226$	23500.0	0.460099
$227$	79520.9i	1.54323i	0.636093	+	0.771613i	$0.280550\pi$
$227$	79520.9i	1.54323i	−0.636093	+	0.771613i	$0.719450\pi$
$228$	0	0
$229$	−20491.0	−0.390744	−0.195372	−	0.980729i	$-0.562591\pi$
$229$	−20491.0	−0.390744	−0.195372	+	0.980729i	$0.562591\pi$
$230$	− 18423.1i	− 0.348263i
$231$	0	0
$232$	54046.8	1.00414
$233$	− 54284.3i	− 0.999914i	−0.866050	−	0.499957i	$-0.833349\pi$
$233$	− 54284.3i	− 0.999914i	0.866050	−	0.499957i	$-0.166651\pi$
$234$	0	0
$235$	−31536.8	−0.571060
$236$	5340.35i	0.0958840i
$237$	0	0
$238$	7817.23	0.138006
$239$	− 30534.4i	− 0.534557i	−0.963619	−	0.267279i	$-0.913876\pi$
$239$	− 30534.4i	− 0.534557i	0.963619	−	0.267279i	$-0.0861244\pi$
$240$	0	0
$241$	30624.4	0.527270	0.263635	−	0.964622i	$-0.415079\pi$
$241$	30624.4	0.527270	0.263635	+	0.964622i	$0.415079\pi$
$242$	− 67230.2i	− 1.14798i
$243$	0	0
$244$	−65311.3	−1.09700
$245$	18396.2i	0.306475i
$246$	0	0
$247$	−7666.19	−0.125657
$248$	− 21624.5i	− 0.351596i
$249$	0	0
$250$	21552.3	0.344836
$251$	75140.6i	1.19269i	0.802728	+	0.596345i	$0.203381\pi$
$251$	75140.6i	1.19269i	−0.802728	+	0.596345i	$0.796619\pi$
$252$	0	0
$253$	217602.	3.39955
$254$	− 41087.8i	− 0.636863i
$255$	0	0
$256$	−46974.1	−0.716767
$257$	81363.0i	1.23186i	0.787802	+	0.615929i	$0.211219\pi$
$257$	81363.0i	1.23186i	−0.787802	+	0.615929i	$0.788781\pi$
$258$	0	0
$259$	14519.8	0.216452
$260$	29263.8i	0.432896i
$261$	0	0
$262$	−15203.5	−0.221483
$263$	46465.3i	0.671765i	0.941904	+	0.335882i	$0.109034\pi$
$263$	46465.3i	0.671765i	−0.941904	+	0.335882i	$0.890966\pi$
$264$	0	0
$265$	17907.4	0.255001
$266$	− 1066.53i	− 0.0150734i
$267$	0	0
$268$	−83390.2	−1.16104
$269$	16724.6i	0.231127i	0.993300	+	0.115564i	$0.0368674\pi$
$269$	16724.6i	0.231127i	−0.993300	+	0.115564i	$0.963133\pi$
$270$	0	0
$271$	109332.	1.48871	0.744355	−	0.667784i	$-0.232757\pi$
$271$	109332.	1.48871	0.744355	+	0.667784i	$0.232757\pi$
$272$	14501.7i	0.196011i
$273$	0	0
$274$	38104.1	0.507541
$275$	118512.i	1.56710i
$276$	0	0
$277$	−32493.2	−0.423480	−0.211740	−	0.977326i	$-0.567913\pi$
$277$	−32493.2	−0.423480	−0.211740	+	0.977326i	$0.567913\pi$
$278$	13148.9i	0.170137i
$279$	0	0
$280$	−9599.03	−0.122437
$281$	− 113870.i	− 1.44211i	−0.692878	−	0.721055i	$-0.743658\pi$
$281$	− 113870.i	− 1.44211i	0.692878	−	0.721055i	$-0.256342\pi$
$282$	0	0
$283$	−147109.	−1.83683	−0.918413	−	0.395624i	$-0.870528\pi$
$283$	−147109.	−1.83683	−0.918413	+	0.395624i	$0.870528\pi$
$284$	72231.4i	0.895549i
$285$	0	0
$286$	123663.	1.51184
$287$	10861.5i	0.131864i
$288$	0	0
$289$	42276.2	0.506175
$290$	− 17459.8i	− 0.207608i
$291$	0	0
$292$	−13915.6	−0.163206
$293$	− 89515.5i	− 1.04271i	−0.853340	−	0.521354i	$-0.825427\pi$
$293$	− 89515.5i	− 1.04271i	0.853340	−	0.521354i	$-0.174573\pi$
$294$	0	0
$295$	4067.64	0.0467410
$296$	− 44186.4i	− 0.504319i
$297$	0	0
$298$	15385.7	0.173255
$299$	276581.i	3.09371i
$300$	0	0
$301$	36101.6	0.398468
$302$	− 62545.9i	− 0.685780i
$303$	0	0
$304$	1978.52	0.0214088
$305$	49746.3i	0.534762i
$306$	0	0
$307$	−43352.3	−0.459977	−0.229988	−	0.973193i	$-0.573869\pi$
$307$	−43352.3	−0.459977	−0.229988	+	0.973193i	$0.573869\pi$
$308$	− 48086.5i	− 0.506900i
$309$	0	0
$310$	−6985.81	−0.0726931
$311$	65034.5i	0.672393i	0.941792	+	0.336196i	$0.109141\pi$
$311$	65034.5i	0.672393i	−0.941792	+	0.336196i	$0.890859\pi$
$312$	0	0
$313$	−90676.9	−0.925568	−0.462784	−	0.886471i	$-0.653149\pi$
$313$	−90676.9	−0.925568	−0.462784	+	0.886471i	$0.653149\pi$
$314$	66548.5i	0.674962i
$315$	0	0
$316$	−18246.2	−0.182725
$317$	156855.i	1.56092i	0.625205	+	0.780461i	$0.285016\pi$
$317$	156855.i	1.56092i	−0.625205	+	0.780461i	$0.714984\pi$
$318$	0	0
$319$	206223.	2.02655
$320$	9269.64i	0.0905239i
$321$	0	0
$322$	−38478.4	−0.371112
$323$	5627.17i	0.0539368i
$324$	0	0
$325$	−150634.	−1.42612
$326$	− 20920.7i	− 0.196852i
$327$	0	0
$328$	33053.7	0.307236
$329$	65867.4i	0.608525i
$330$	0	0
$331$	−85136.4	−0.777069	−0.388534	−	0.921434i	$-0.627019\pi$
$331$	−85136.4	−0.777069	−0.388534	+	0.921434i	$0.627019\pi$
$332$	− 13461.0i	− 0.122124i
$333$	0	0
$334$	−120.974	−0.00108443
$335$	63516.6i	0.565976i
$336$	0	0
$337$	11829.0	0.104157	0.0520785	−	0.998643i	$-0.483415\pi$
$337$	11829.0	0.104157	0.0520785	+	0.998643i	$0.483415\pi$
$338$	98536.4i	0.862508i
$339$	0	0
$340$	21480.3	0.185816
$341$	− 82511.7i	− 0.709589i
$342$	0	0
$343$	83432.1	0.709161
$344$	− 109864.i	− 0.928406i
$345$	0	0
$346$	24537.7	0.204966
$347$	− 47876.4i	− 0.397615i	−0.980039	−	0.198807i	$-0.936293\pi$
$347$	− 47876.4i	− 0.397615i	0.980039	−	0.198807i	$-0.0637068\pi$
$348$	0	0
$349$	20759.5	0.170438	0.0852188	−	0.996362i	$-0.472841\pi$
$349$	20759.5	0.170438	0.0852188	+	0.996362i	$0.472841\pi$
$350$	− 20956.4i	− 0.171073i
$351$	0	0
$352$	−230607.	−1.86118
$353$	− 203785.i	− 1.63540i	−0.575646	−	0.817699i	$-0.695249\pi$
$353$	− 203785.i	− 1.63540i	0.575646	−	0.817699i	$-0.304751\pi$
$354$	0	0
$355$	55017.2	0.436557
$356$	− 7057.66i	− 0.0556879i
$357$	0	0
$358$	6179.85	0.0482183
$359$	− 27860.0i	− 0.216168i	−0.994142	−	0.108084i	$-0.965528\pi$
$359$	− 27860.0i	− 0.216168i	0.994142	−	0.108084i	$-0.0344716\pi$
$360$	0	0
$361$	−129553.	−0.994109
$362$	110471.i	0.843005i
$363$	0	0
$364$	61120.0	0.461297
$365$	10599.2i	0.0795590i
$366$	0	0
$367$	172281.	1.27910	0.639552	−	0.768748i	$-0.279120\pi$
$367$	172281.	1.27910	0.639552	+	0.768748i	$0.279120\pi$
$368$	− 71380.9i	− 0.527092i
$369$	0	0
$370$	−14274.4	−0.104269
$371$	− 37401.3i	− 0.271731i
$372$	0	0
$373$	124182.	0.892568	0.446284	−	0.894891i	$-0.352747\pi$
$373$	124182.	0.892568	0.446284	+	0.894891i	$0.352747\pi$
$374$	− 90771.6i	− 0.648943i
$375$	0	0
$376$	200447.	1.41783
$377$	262118.i	1.84423i
$378$	0	0
$379$	163070.	1.13526	0.567632	−	0.823282i	$-0.307860\pi$
$379$	163070.	1.13526	0.567632	+	0.823282i	$0.307860\pi$
$380$	− 2930.64i	− 0.0202953i
$381$	0	0
$382$	−18443.4	−0.126391
$383$	− 61660.0i	− 0.420345i	−0.977664	−	0.210172i	$-0.932597\pi$
$383$	− 61660.0i	− 0.420345i	0.977664	−	0.210172i	$-0.0674025\pi$
$384$	0	0
$385$	−36626.5	−0.247101
$386$	− 126566.i	− 0.849458i
$387$	0	0
$388$	73839.8	0.490486
$389$	296961.i	1.96246i	0.192851	+	0.981228i	$0.438227\pi$
$389$	296961.i	1.96246i	−0.192851	+	0.981228i	$0.561773\pi$
$390$	0	0
$391$	203017.	1.32794
$392$	− 116925.i	− 0.760917i
$393$	0	0
$394$	−40022.0	−0.257814
$395$	13897.8i	0.0890741i
$396$	0	0
$397$	291431.	1.84908	0.924539	−	0.381088i	$-0.124451\pi$
$397$	291431.	1.84908	0.924539	+	0.381088i	$0.124451\pi$
$398$	10174.5i	0.0642316i
$399$	0	0
$400$	38876.1	0.242976
$401$	39357.3i	0.244758i	0.992483	+	0.122379i	$0.0390523\pi$
$401$	39357.3i	0.244758i	−0.992483	+	0.122379i	$0.960948\pi$
$402$	0	0
$403$	104876.	0.645751
$404$	194459.i	1.19142i
$405$	0	0
$406$	−36466.4	−0.221228
$407$	− 168600.i	− 1.01781i
$408$	0	0
$409$	2651.07	0.0158480	0.00792400	−	0.999969i	$-0.497478\pi$
$409$	2651.07	0.0158480	0.00792400	+	0.999969i	$0.497478\pi$
$410$	− 10678.0i	− 0.0635216i
$411$	0	0
$412$	−86202.6	−0.507839
$413$	− 8495.62i	− 0.0498076i
$414$	0	0
$415$	−10253.0	−0.0595323
$416$	− 293111.i	− 1.69374i
$417$	0	0
$418$	−12384.3	−0.0708791
$419$	− 251293.i	− 1.43137i	−0.698423	−	0.715685i	$-0.746115\pi$
$419$	− 251293.i	− 1.43137i	0.698423	−	0.715685i	$-0.253885\pi$
$420$	0	0
$421$	−261535.	−1.47559	−0.737796	−	0.675024i	$-0.764133\pi$
$421$	−261535.	−1.47559	−0.737796	+	0.675024i	$0.764133\pi$
$422$	49062.9i	0.275504i
$423$	0	0
$424$	−113819.	−0.633116
$425$	110569.i	0.612146i
$426$	0	0
$427$	103899.	0.569846
$428$	147994.i	0.807898i
$429$	0	0
$430$	−35491.5	−0.191950
$431$	279430.i	1.50424i	0.659025	+	0.752121i	$0.270969\pi$
$431$	279430.i	1.50424i	−0.659025	+	0.752121i	$0.729031\pi$
$432$	0	0
$433$	−320826.	−1.71117	−0.855586	−	0.517661i	$-0.826803\pi$
$433$	−320826.	−1.71117	−0.855586	+	0.517661i	$0.826803\pi$
$434$	14590.5i	0.0774623i
$435$	0	0
$436$	69961.0	0.368030
$437$	− 27698.3i	− 0.145041i
$438$	0	0
$439$	−133175.	−0.691024	−0.345512	−	0.938414i	$-0.612295\pi$
$439$	−133175.	−0.691024	−0.345512	+	0.938414i	$0.612295\pi$
$440$	111461.i	0.575730i
$441$	0	0
$442$	115374.	0.590561
$443$	− 52667.1i	− 0.268369i	−0.990956	−	0.134184i	$-0.957159\pi$
$443$	− 52667.1i	− 0.268369i	0.990956	−	0.134184i	$-0.0428415\pi$
$444$	0	0
$445$	−5375.68	−0.0271465
$446$	180580.i	0.907822i
$447$	0	0
$448$	19360.5	0.0964628
$449$	136261.i	0.675895i	0.941165	+	0.337947i	$0.109733\pi$
$449$	136261.i	0.675895i	−0.941165	+	0.337947i	$0.890267\pi$
$450$	0	0
$451$	126121.	0.620062
$452$	− 134868.i	− 0.660135i
$453$	0	0
$454$	−163280.	−0.792174
$455$	− 46553.8i	− 0.224871i
$456$	0	0
$457$	−401121.	−1.92063	−0.960313	−	0.278923i	$-0.910022\pi$
$457$	−401121.	−1.92063	−0.960313	+	0.278923i	$0.910022\pi$
$458$	− 42074.1i	− 0.200578i
$459$	0	0
$460$	−105731.	−0.499676
$461$	84093.7i	0.395696i	0.980233	+	0.197848i	$0.0633952\pi$
$461$	84093.7i	0.395696i	−0.980233	+	0.197848i	$0.936605\pi$
$462$	0	0
$463$	169642.	0.791354	0.395677	−	0.918390i	$-0.370510\pi$
$463$	169642.	0.791354	0.395677	+	0.918390i	$0.370510\pi$
$464$	− 67648.5i	− 0.314212i
$465$	0	0
$466$	111462.	0.513279
$467$	− 51697.1i	− 0.237046i	−0.992951	−	0.118523i	$-0.962184\pi$
$467$	− 51697.1i	− 0.237046i	0.992951	−	0.118523i	$-0.0378159\pi$
$468$	0	0
$469$	132660.	0.603107
$470$	− 64754.3i	− 0.293138i
$471$	0	0
$472$	−25853.8	−0.116049
$473$	− 419202.i	− 1.87370i
$474$	0	0
$475$	15085.3	0.0668601
$476$	− 44863.6i	− 0.198007i
$477$	0	0
$478$	62696.2	0.274401
$479$	− 305560.i	− 1.33176i	−0.746060	−	0.665878i	$-0.768057\pi$
$479$	− 305560.i	− 1.33176i	0.746060	−	0.665878i	$-0.231943\pi$
$480$	0	0
$481$	214297.	0.926247
$482$	62880.9i	0.270660i
$483$	0	0
$484$	−385838.	−1.64708
$485$	− 56242.3i	− 0.239100i
$486$	0	0
$487$	272254.	1.14793	0.573966	−	0.818879i	$-0.305404\pi$
$487$	272254.	1.14793	0.573966	+	0.818879i	$0.305404\pi$
$488$	− 316185.i	− 1.32771i
$489$	0	0
$490$	−37772.8	−0.157321
$491$	378821.i	1.57134i	0.618644	+	0.785671i	$0.287682\pi$
$491$	378821.i	1.57134i	−0.618644	+	0.785671i	$0.712318\pi$
$492$	0	0
$493$	192401.	0.791616
$494$	− 15740.9i	− 0.0645025i
$495$	0	0
$496$	−27066.7	−0.110020
$497$	− 114908.i	− 0.465199i
$498$	0	0
$499$	−60686.7	−0.243721	−0.121860	−	0.992547i	$-0.538886\pi$
$499$	−60686.7	−0.243721	−0.121860	+	0.992547i	$0.538886\pi$
$500$	− 123690.i	− 0.494759i
$501$	0	0
$502$	−154286.	−0.612236
$503$	− 76840.6i	− 0.303707i	−0.988403	−	0.151854i	$-0.951476\pi$
$503$	− 76840.6i	− 0.303707i	0.988403	−	0.151854i	$-0.0485242\pi$
$504$	0	0
$505$	148116.	0.580789
$506$	446800.i	1.74507i
$507$	0	0
$508$	−235806.	−0.913749
$509$	− 434779.i	− 1.67816i	−0.544010	−	0.839079i	$-0.683095\pi$
$509$	− 434779.i	− 1.67816i	0.544010	−	0.839079i	$-0.316905\pi$
$510$	0	0
$511$	22137.5	0.0847786
$512$	140825.i	0.537205i
$513$	0	0
$514$	−167062.	−0.632342
$515$	65658.8i	0.247559i
$516$	0	0
$517$	764834.	2.86145
$518$	29813.4i	0.111110i
$519$	0	0
$520$	−141672.	−0.523935
$521$	272132.i	1.00255i	0.865289	+	0.501273i	$0.167135\pi$
$521$	272132.i	1.00255i	−0.865289	+	0.501273i	$0.832865\pi$
$522$	0	0
$523$	443741.	1.62228	0.811141	−	0.584850i	$-0.198847\pi$
$523$	443741.	1.62228	0.811141	+	0.584850i	$0.198847\pi$
$524$	87253.7i	0.317776i
$525$	0	0
$526$	−95406.9	−0.344833
$527$	− 76981.4i	− 0.277182i
$528$	0	0
$529$	−719459.	−2.57096
$530$	36769.2i	0.130898i
$531$	0	0
$532$	−6120.90	−0.0216268
$533$	160305.i	0.564278i
$534$	0	0
$535$	112724.	0.393830
$536$	− 403709.i	− 1.40520i
$537$	0	0
$538$	−34340.5	−0.118643
$539$	− 446147.i	− 1.53568i
$540$	0	0
$541$	−327849.	−1.12016	−0.560080	−	0.828439i	$-0.689229\pi$
$541$	−327849.	−1.12016	−0.560080	+	0.828439i	$0.689229\pi$
$542$	224492.i	0.764190i
$543$	0	0
$544$	−215151.	−0.727018
$545$	− 53287.9i	− 0.179405i
$546$	0	0
$547$	−247342.	−0.826653	−0.413327	−	0.910583i	$-0.635633\pi$
$547$	−247342.	−0.826653	−0.413327	+	0.910583i	$0.635633\pi$
$548$	− 218682.i	− 0.728201i
$549$	0	0
$550$	−243340.	−0.804431
$551$	− 26250.0i	− 0.0864622i
$552$	0	0
$553$	29026.8	0.0949179
$554$	− 66718.2i	− 0.217383i
$555$	0	0
$556$	75462.1	0.244106
$557$	2917.69i	0.00940435i	0.999989	+	0.00470217i	$0.00149675\pi$
$557$	2917.69i	0.00940435i	−0.999989	+	0.00470217i	$0.998503\pi$
$558$	0	0
$559$	532823.	1.70514
$560$	12014.8i	0.0383124i
$561$	0	0
$562$	233810.	0.740269
$563$	19911.3i	0.0628177i	0.999507	+	0.0314089i	$0.00999939\pi$
$563$	19911.3i	0.0628177i	−0.999507	+	0.0314089i	$0.990001\pi$
$564$	0	0
$565$	−102726.	−0.321799
$566$	− 302059.i	− 0.942886i
$567$	0	0
$568$	−349687.	−1.08388
$569$	− 156543.i	− 0.483514i	−0.970337	−	0.241757i	$-0.922276\pi$
$569$	− 156543.i	− 0.483514i	0.970337	−	0.241757i	$-0.0777237\pi$
$570$	0	0
$571$	−571856.	−1.75394	−0.876969	−	0.480547i	$-0.840438\pi$
$571$	−571856.	−1.75394	−0.876969	+	0.480547i	$0.840438\pi$
$572$	− 709708.i	− 2.16914i
$573$	0	0
$574$	−22301.9	−0.0676891
$575$	− 544248.i	− 1.64612i
$576$	0	0
$577$	−178708.	−0.536775	−0.268387	−	0.963311i	$-0.586491\pi$
$577$	−178708.	−0.536775	−0.268387	+	0.963311i	$0.586491\pi$
$578$	86805.6i	0.259832i
$579$	0	0
$580$	−100203.	−0.297868
$581$	21414.2i	0.0634380i
$582$	0	0
$583$	−434293.	−1.27775
$584$	− 67368.4i	− 0.197529i
$585$	0	0
$586$	183802.	0.535247
$587$	− 340594.i	− 0.988463i	−0.869330	−	0.494232i	$-0.835449\pi$
$587$	− 340594.i	− 0.988463i	0.869330	−	0.494232i	$-0.164551\pi$
$588$	0	0
$589$	−10502.8	−0.0302745
$590$	8352.06i	0.0239933i
$591$	0	0
$592$	−55306.6	−0.157810
$593$	461459.i	1.31227i	0.754642	+	0.656137i	$0.227810\pi$
$593$	461459.i	1.31227i	−0.754642	+	0.656137i	$0.772190\pi$
$594$	0	0
$595$	−34171.7	−0.0965233
$596$	− 88299.6i	− 0.248580i
$597$	0	0
$598$	−567901.	−1.58807
$599$	616737.i	1.71888i	0.511234	+	0.859442i	$0.329189\pi$
$599$	616737.i	1.71888i	−0.511234	+	0.859442i	$0.670811\pi$
$600$	0	0
$601$	162080.	0.448727	0.224363	−	0.974506i	$-0.427970\pi$
$601$	162080.	0.448727	0.224363	+	0.974506i	$0.427970\pi$
$602$	74127.2i	0.204543i
$603$	0	0
$604$	−358955.	−0.983934
$605$	293885.i	0.802910i
$606$	0	0
$607$	−618099.	−1.67757	−0.838785	−	0.544462i	$-0.816734\pi$
$607$	−618099.	−1.67757	−0.838785	+	0.544462i	$0.816734\pi$
$608$	29353.8i	0.0794067i
$609$	0	0
$610$	−102144.	−0.274506
$611$	972135.i	2.60402i
$612$	0	0
$613$	−361729.	−0.962636	−0.481318	−	0.876546i	$-0.659842\pi$
$613$	−361729.	−0.962636	−0.481318	+	0.876546i	$0.659842\pi$
$614$	− 89015.1i	− 0.236117i
$615$	0	0
$616$	232797.	0.613502
$617$	− 196981.i	− 0.517432i	−0.965953	−	0.258716i	$-0.916701\pi$
$617$	− 196981.i	− 0.517432i	0.965953	−	0.258716i	$-0.0832994\pi$
$618$	0	0
$619$	−185460.	−0.484027	−0.242013	−	0.970273i	$-0.577808\pi$
$619$	−185460.	−0.484027	−0.242013	+	0.970273i	$0.577808\pi$
$620$	40092.0i	0.104298i
$621$	0	0
$622$	−133535.	−0.345155
$623$	11227.6i	0.0289274i
$624$	0	0
$625$	246062.	0.629919
$626$	− 186186.i	− 0.475116i
$627$	0	0
$628$	381926.	0.968411
$629$	− 157300.i	− 0.397582i
$630$	0	0
$631$	−309014.	−0.776102	−0.388051	−	0.921638i	$-0.626852\pi$
$631$	−309014.	−0.776102	−0.388051	+	0.921638i	$0.626852\pi$
$632$	− 88333.8i	− 0.221153i
$633$	0	0
$634$	−322070.	−0.801258
$635$	179608.i	0.445430i
$636$	0	0
$637$	567071.	1.39752
$638$	423437.i	1.04027i
$639$	0	0
$640$	133106.	0.324967
$641$	162406.i	0.395264i	0.980276	+	0.197632i	$0.0633251\pi$
$641$	162406.i	0.395264i	−0.980276	+	0.197632i	$0.936675\pi$
$642$	0	0
$643$	−504525.	−1.22028	−0.610142	−	0.792292i	$-0.708888\pi$
$643$	−504525.	−1.22028	−0.610142	+	0.792292i	$0.708888\pi$
$644$	220830.i	0.532458i
$645$	0	0
$646$	−11554.2	−0.0276870
$647$	− 64384.8i	− 0.153807i	−0.997039	−	0.0769033i	$-0.975497\pi$
$647$	− 64384.8i	− 0.153807i	0.997039	−	0.0769033i	$-0.0245033\pi$
$648$	0	0
$649$	−98648.9	−0.234209
$650$	− 309296.i	− 0.732060i
$651$	0	0
$652$	−120065.	−0.282437
$653$	− 79912.6i	− 0.187408i	−0.995600	−	0.0937042i	$-0.970129\pi$
$653$	− 79912.6i	− 0.187408i	0.995600	−	0.0937042i	$-0.0298708\pi$
$654$	0	0
$655$	66459.4	0.154908
$656$	− 41372.1i	− 0.0961392i
$657$	0	0
$658$	−135245.	−0.312370
$659$	769298.i	1.77143i	0.464229	+	0.885715i	$0.346331\pi$
$659$	769298.i	1.77143i	−0.464229	+	0.885715i	$0.653669\pi$
$660$	0	0
$661$	24910.8	0.0570145	0.0285073	−	0.999594i	$-0.490925\pi$
$661$	24910.8	0.0570145	0.0285073	+	0.999594i	$0.490925\pi$
$662$	− 174810.i	− 0.398888i
$663$	0	0
$664$	65167.4	0.147807
$665$	4662.16i	0.0105425i
$666$	0	0
$667$	−947048.	−2.12873
$668$	694.279i	0.00155590i
$669$	0	0
$670$	−130418.	−0.290529
$671$	− 1.20645e6i	− 2.67957i
$672$	0	0
$673$	466650.	1.03029	0.515147	−	0.857102i	$-0.327737\pi$
$673$	466650.	1.03029	0.515147	+	0.857102i	$0.327737\pi$
$674$	24288.4i	0.0534662i
$675$	0	0
$676$	565506.	1.23750
$677$	558691.i	1.21897i	0.792796	+	0.609487i	$0.208624\pi$
$677$	558691.i	1.21897i	−0.792796	+	0.609487i	$0.791376\pi$
$678$	0	0
$679$	−117467.	−0.254786
$680$	103991.i	0.224893i
$681$	0	0
$682$	169421.	0.364249
$683$	215225.i	0.461373i	0.973028	+	0.230686i	$0.0740971\pi$
$683$	215225.i	0.461373i	−0.973028	+	0.230686i	$0.925903\pi$
$684$	0	0
$685$	−166565.	−0.354980
$686$	171311.i	0.364029i
$687$	0	0
$688$	−137513.	−0.290514
$689$	− 552005.i	− 1.16280i
$690$	0	0
$691$	908351.	1.90238	0.951191	−	0.308603i	$-0.0998614\pi$
$691$	908351.	1.90238	0.951191	+	0.308603i	$0.0998614\pi$
$692$	− 140823.i	− 0.294077i
$693$	0	0
$694$	98304.4	0.204105
$695$	− 57477.9i	− 0.118996i
$696$	0	0
$697$	117668.	0.242210
$698$	42625.3i	0.0874897i
$699$	0	0
$700$	−120270.	−0.245449
$701$	562982.i	1.14567i	0.819672	+	0.572834i	$0.194156\pi$
$701$	562982.i	1.14567i	−0.819672	+	0.572834i	$0.805844\pi$
$702$	0	0
$703$	−21460.9	−0.0434248
$704$	− 224809.i	− 0.453594i
$705$	0	0
$706$	418431.	0.839489
$707$	− 309353.i	− 0.618893i
$708$	0	0
$709$	348911.	0.694100	0.347050	−	0.937847i	$-0.387183\pi$
$709$	348911.	0.694100	0.347050	+	0.937847i	$0.387183\pi$
$710$	112966.i	0.224095i
$711$	0	0
$712$	34167.6	0.0673992
$713$	378921.i	0.745367i
$714$	0	0
$715$	−540570.	−1.05740
$716$	− 35466.5i	− 0.0691819i
$717$	0	0
$718$	57204.8	0.110964
$719$	73708.9i	0.142581i	0.997456	+	0.0712906i	$0.0227118\pi$
$719$	73708.9i	0.142581i	−0.997456	+	0.0712906i	$0.977288\pi$
$720$	0	0
$721$	137134.	0.263800
$722$	− 266011.i	− 0.510300i
$723$	0	0
$724$	633998.	1.20951
$725$	− 515790.i	− 0.981288i
$726$	0	0
$727$	760773.	1.43942	0.719708	−	0.694277i	$-0.244276\pi$
$727$	760773.	1.43942	0.719708	+	0.694277i	$0.244276\pi$
$728$	295894.i	0.558308i
$729$	0	0
$730$	−21763.4	−0.0408395
$731$	− 391105.i	− 0.731912i
$732$	0	0
$733$	281179.	0.523329	0.261664	−	0.965159i	$-0.415729\pi$
$733$	281179.	0.523329	0.261664	+	0.965159i	$0.415729\pi$
$734$	353744.i	0.656594i
$735$	0	0
$736$	1.05903e6	1.95502
$737$	− 1.54041e6i	− 2.83597i
$738$	0	0
$739$	865064.	1.58401	0.792007	−	0.610511i	$-0.209036\pi$
$739$	865064.	1.58401	0.792007	+	0.610511i	$0.209036\pi$
$740$	81921.8i	0.149602i
$741$	0	0
$742$	76795.8	0.139486
$743$	730370.i	1.32302i	0.749938	+	0.661508i	$0.230083\pi$
$743$	730370.i	1.32302i	−0.749938	+	0.661508i	$0.769917\pi$
$744$	0	0
$745$	−67256.0	−0.121176
$746$	254982.i	0.458176i
$747$	0	0
$748$	−520943.	−0.931081
$749$	− 235434.i	− 0.419668i
$750$	0	0
$751$	−423241.	−0.750427	−0.375213	−	0.926938i	$-0.622431\pi$
$751$	−423241.	−0.750427	−0.375213	+	0.926938i	$0.622431\pi$
$752$	− 250892.i	− 0.443661i
$753$	0	0
$754$	−538206.	−0.946686
$755$	273408.i	0.479643i
$756$	0	0
$757$	458124.	0.799450	0.399725	−	0.916635i	$-0.369106\pi$
$757$	458124.	0.799450	0.399725	+	0.916635i	$0.369106\pi$
$758$	334832.i	0.582758i
$759$	0	0
$760$	14187.8	0.0245634
$761$	− 311232.i	− 0.537421i	−0.963221	−	0.268710i	$-0.913403\pi$
$761$	− 311232.i	− 0.537421i	0.963221	−	0.268710i	$-0.0865974\pi$
$762$	0	0
$763$	−111296.	−0.191175
$764$	105848.i	0.181341i
$765$	0	0
$766$	126606.	0.215773
$767$	− 125387.i	− 0.213138i
$768$	0	0
$769$	−315385.	−0.533320	−0.266660	−	0.963791i	$-0.585920\pi$
$769$	−315385.	−0.533320	−0.266660	+	0.963791i	$0.585920\pi$
$770$	− 75205.0i	− 0.126843i
$771$	0	0
$772$	−726369.	−1.21877
$773$	− 549892.i	− 0.920277i	−0.887847	−	0.460139i	$-0.847800\pi$
$773$	− 549892.i	− 0.920277i	0.887847	−	0.460139i	$-0.152200\pi$
$774$	0	0
$775$	−206372.	−0.343595
$776$	357474.i	0.593637i
$777$	0	0
$778$	−609748.	−1.00738
$779$	− 16053.9i	− 0.0264548i
$780$	0	0
$781$	−1.33428e6	−2.18749
$782$	416854.i	0.681664i
$783$	0	0
$784$	−146352.	−0.238103
$785$	− 290905.i	− 0.472076i
$786$	0	0
$787$	789272.	1.27432	0.637158	−	0.770733i	$-0.280110\pi$
$787$	789272.	1.27432	0.637158	+	0.770733i	$0.280110\pi$
$788$	229689.i	0.369902i
$789$	0	0
$790$	−28536.2	−0.0457238
$791$	214553.i	0.342911i
$792$	0	0
$793$	1.53345e6	2.43850
$794$	598394.i	0.949175i
$795$	0	0
$796$	58392.3	0.0921573
$797$	− 26313.7i	− 0.0414252i	−0.999785	−	0.0207126i	$-0.993406\pi$
$797$	− 26313.7i	− 0.0414252i	0.999785	−	0.0207126i	$-0.00659350\pi$
$798$	0	0
$799$	713571.	1.11775
$800$	576776.i	0.901213i
$801$	0	0
$802$	−80812.2	−0.125640
$803$	− 257054.i	− 0.398652i
$804$	0	0
$805$	168201.	0.259560
$806$	215341.i	0.331479i
$807$	0	0
$808$	−941418.	−1.44198
$809$	957125.i	1.46242i	0.682153	+	0.731210i	$0.261044\pi$
$809$	957125.i	1.46242i	−0.682153	+	0.731210i	$0.738956\pi$
$810$	0	0
$811$	−247324.	−0.376033	−0.188016	−	0.982166i	$-0.560206\pi$
$811$	−247324.	−0.376033	−0.188016	+	0.982166i	$0.560206\pi$
$812$	209283.i	0.317410i
$813$	0	0
$814$	346185.	0.522468
$815$	91451.1i	0.137681i
$816$	0	0
$817$	−53359.9	−0.0799412
$818$	5443.43i	0.00813516i
$819$	0	0
$820$	−61281.6	−0.0911386
$821$	1.14348e6i	1.69645i	0.529637	+	0.848225i	$0.322328\pi$
$821$	1.14348e6i	1.69645i	−0.529637	+	0.848225i	$0.677672\pi$
$822$	0	0
$823$	−331225.	−0.489016	−0.244508	−	0.969647i	$-0.578626\pi$
$823$	−331225.	−0.489016	−0.244508	+	0.969647i	$0.578626\pi$
$824$	− 417325.i	− 0.614638i
$825$	0	0
$826$	17444.0	0.0255674
$827$	1.03269e6i	1.50994i	0.655758	+	0.754971i	$0.272349\pi$
$827$	1.03269e6i	1.50994i	−0.655758	+	0.754971i	$0.727651\pi$
$828$	0	0
$829$	−252981.	−0.368110	−0.184055	−	0.982916i	$-0.558923\pi$
$829$	−252981.	−0.368110	−0.184055	+	0.982916i	$0.558923\pi$
$830$	− 21052.3i	− 0.0305594i
$831$	0	0
$832$	285741.	0.412787
$833$	− 416244.i	− 0.599871i
$834$	0	0
$835$	528.818	0.000758461 0
$836$	71074.2i	0.101695i
$837$	0	0
$838$	515978.	0.734756
$839$	− 988910.i	− 1.40486i	−0.711753	−	0.702429i	$-0.752099\pi$
$839$	− 988910.i	− 1.40486i	0.711753	−	0.702429i	$-0.247901\pi$
$840$	0	0
$841$	−190247.	−0.268983
$842$	− 537009.i	− 0.757456i
$843$	0	0
$844$	281575.	0.395284
$845$	− 430734.i	− 0.603248i
$846$	0	0
$847$	613805.	0.855586
$848$	142463.i	0.198112i
$849$	0	0
$850$	−227031.	−0.314229
$851$	774268.i	1.06913i
$852$	0	0
$853$	−403846.	−0.555032	−0.277516	−	0.960721i	$-0.589511\pi$
$853$	−403846.	−0.555032	−0.277516	+	0.960721i	$0.589511\pi$
$854$	213336.i	0.292515i
$855$	0	0
$856$	−716470.	−0.977800
$857$	− 767121.i	− 1.04448i	−0.852797	−	0.522242i	$-0.825096\pi$
$857$	− 767121.i	− 1.04448i	0.852797	−	0.522242i	$-0.174904\pi$
$858$	0	0
$859$	442885.	0.600212	0.300106	−	0.953906i	$-0.402978\pi$
$859$	442885.	0.600212	0.300106	+	0.953906i	$0.402978\pi$
$860$	203688.i	0.275403i
$861$	0	0
$862$	−573751.	−0.772163
$863$	− 13598.5i	− 0.0182587i	−0.999958	−	0.00912937i	$-0.997094\pi$
$863$	− 13598.5i	− 0.0182587i	0.999958	−	0.00912937i	$-0.00290601\pi$
$864$	0	0
$865$	−107262.	−0.143355
$866$	− 658750.i	− 0.878385i
$867$	0	0
$868$	83735.7	0.111140
$869$	− 337051.i	− 0.446330i
$870$	0	0
$871$	1.95793e6	2.58084
$872$	338696.i	0.445427i
$873$	0	0
$874$	56872.8	0.0744530
$875$	196770.i	0.257006i
$876$	0	0
$877$	1.22388e6	1.59125	0.795625	−	0.605789i	$-0.207143\pi$
$877$	1.22388e6	1.59125	0.795625	+	0.605789i	$0.207143\pi$
$878$	− 273447.i	− 0.354719i
$879$	0	0
$880$	139512.	0.180155
$881$	1.09960e6i	1.41672i	0.705850	+	0.708361i	$0.250565\pi$
$881$	1.09960e6i	1.41672i	−0.705850	+	0.708361i	$0.749435\pi$
$882$	0	0
$883$	−215447.	−0.276324	−0.138162	−	0.990410i	$-0.544119\pi$
$883$	−215447.	−0.276324	−0.138162	+	0.990410i	$0.544119\pi$
$884$	− 662141.i	− 0.847316i
$885$	0	0
$886$	108141.	0.137760
$887$	341185.i	0.433653i	0.976210	+	0.216826i	$0.0695706\pi$
$887$	341185.i	0.433653i	−0.976210	+	0.216826i	$0.930429\pi$
$888$	0	0
$889$	375128.	0.474653
$890$	− 11037.8i	− 0.0139349i
$891$	0	0
$892$	1.03636e6	1.30251
$893$	− 97355.1i	− 0.122083i
$894$	0	0
$895$	−27014.1	−0.0337245
$896$	− 278005.i	− 0.346287i
$897$	0	0
$898$	−279784.	−0.346953
$899$	359108.i	0.444330i
$900$	0	0
$901$	−405185.	−0.499119
$902$	258964.i	0.318292i
$903$	0	0
$904$	652925.	0.798962
$905$	− 482904.i	− 0.589608i
$906$	0	0
$907$	−571144.	−0.694274	−0.347137	−	0.937814i	$-0.612846\pi$
$907$	−571144.	−0.694274	−0.347137	+	0.937814i	$0.612846\pi$
$908$	937072.i	1.13658i
$909$	0	0
$910$	95588.7	0.115431
$911$	659617.i	0.794795i	0.917646	+	0.397398i	$0.130087\pi$
$911$	659617.i	0.794795i	−0.917646	+	0.397398i	$0.869913\pi$
$912$	0	0
$913$	248656.	0.298303
$914$	− 823619.i	− 0.985903i
$915$	0	0
$916$	−241466.	−0.287783
$917$	− 138806.i	− 0.165071i
$918$	0	0
$919$	670479.	0.793879	0.396939	−	0.917845i	$-0.370072\pi$
$919$	670479.	0.793879	0.396939	+	0.917845i	$0.370072\pi$
$920$	− 511868.i	− 0.604759i
$921$	0	0
$922$	−172669.	−0.203120
$923$	− 1.69593e6i	− 1.99069i
$924$	0	0
$925$	−421688.	−0.492843
$926$	348325.i	0.406221i
$927$	0	0
$928$	1.00365e6	1.16543
$929$	− 668955.i	− 0.775114i	−0.921846	−	0.387557i	$-0.873319\pi$
$929$	− 668955.i	− 0.775114i	0.921846	−	0.387557i	$-0.126681\pi$
$930$	0	0
$931$	−56789.7	−0.0655194
$932$	− 639685.i	− 0.736435i
$933$	0	0
$934$	106149.	0.121681
$935$	396792.i	0.453878i
$936$	0	0
$937$	−503244.	−0.573191	−0.286595	−	0.958052i	$-0.592523\pi$
$937$	−503244.	−0.573191	−0.286595	+	0.958052i	$0.592523\pi$
$938$	272390.i	0.309589i
$939$	0	0
$940$	−371629.	−0.420585
$941$	186678.i	0.210821i	0.994429	+	0.105410i	$0.0336156\pi$
$941$	186678.i	0.210821i	−0.994429	+	0.105410i	$0.966384\pi$
$942$	0	0
$943$	−579191.	−0.651326
$944$	32360.3i	0.0363135i
$945$	0	0
$946$	860745.	0.961817
$947$	− 95574.4i	− 0.106572i	−0.998579	−	0.0532858i	$-0.983031\pi$
$947$	− 95574.4i	− 0.106572i	0.998579	−	0.0532858i	$-0.0169694\pi$
$948$	0	0
$949$	326726.	0.362787
$950$	30974.6i	0.0343209i
$951$	0	0
$952$	217194.	0.239648
$953$	441384.i	0.485994i	0.970027	+	0.242997i	$0.0781305\pi$
$953$	441384.i	0.485994i	−0.970027	+	0.242997i	$0.921870\pi$
$954$	0	0
$955$	80622.3	0.0883992
$956$	− 359817.i	− 0.393701i
$957$	0	0
$958$	627404.	0.683622
$959$	347887.i	0.378269i
$960$	0	0
$961$	−779839.	−0.844419
$962$	440016.i	0.475464i
$963$	0	0
$964$	360877.	0.388334
$965$	553260.i	0.594121i
$966$	0	0
$967$	200981.	0.214933	0.107466	−	0.994209i	$-0.465726\pi$
$967$	200981.	0.214933	0.107466	+	0.994209i	$0.465726\pi$
$968$	− 1.86792e6i	− 1.99346i
$969$	0	0
$970$	115482.	0.122736
$971$	− 188454.i	− 0.199878i	−0.994994	−	0.0999392i	$-0.968135\pi$
$971$	− 188454.i	− 0.199878i	0.994994	−	0.0999392i	$-0.0318648\pi$
$972$	0	0
$973$	−120048.	−0.126803
$974$	559017.i	0.589261i
$975$	0	0
$976$	−395759.	−0.415461
$977$	− 408818.i	− 0.428293i	−0.976802	−	0.214147i	$-0.931303\pi$
$977$	− 408818.i	− 0.428293i	0.976802	−	0.214147i	$-0.0686970\pi$
$978$	0	0
$979$	130372.	0.136025
$980$	216780.i	0.225719i
$981$	0	0
$982$	−777831.	−0.806607
$983$	624926.i	0.646728i	0.946275	+	0.323364i	$0.104814\pi$
$983$	624926.i	0.646728i	−0.946275	+	0.323364i	$0.895186\pi$
$984$	0	0
$985$	174949.	0.180318
$986$	395057.i	0.406355i
$987$	0	0
$988$	−90338.2	−0.0925460
$989$	1.92512e6i	1.96818i
$990$	0	0
$991$	−938374.	−0.955495	−0.477748	−	0.878497i	$-0.658547\pi$
$991$	−938374.	−0.955495	−0.477748	+	0.878497i	$0.658547\pi$
$992$	− 401569.i	− 0.408072i
$993$	0	0
$994$	235940.	0.238797
$995$	− 44476.2i	− 0.0449243i
$996$	0	0
$997$	1.09690e6	1.10351	0.551755	−	0.834006i	$-0.313958\pi$
$997$	1.09690e6	1.10351	0.551755	+	0.834006i	$0.313958\pi$
$998$	− 124608.i	− 0.125107i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.48	yes	76
3.2	odd	2	inner	531.5.b.a.296.29	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.29	✓	76	3.2	odd	2	inner
531.5.b.a.296.48	yes	76	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	531.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.05329i q^{2} +11.7840 q^{4} -8.97562i q^{5} -18.7464 q^{7} +57.0487i q^{8} +O(q^{10})\)	\(q+2.05329i q^{2} +11.7840 q^{4} -8.97562i q^{5} -18.7464 q^{7} +57.0487i q^{8} +18.4296 q^{10} +217.678i q^{11} -276.677 q^{13} -38.4918i q^{14} +71.4059 q^{16} +203.088i q^{17} +27.7080 q^{19} -105.769i q^{20} -446.957 q^{22} -999.650i q^{23} +544.438 q^{25} -568.100i q^{26} -220.907 q^{28} -947.379i q^{29} -379.054 q^{31} +1059.40i q^{32} -417.000 q^{34} +168.260i q^{35} -774.539 q^{37} +56.8928i q^{38} +512.047 q^{40} -579.394i q^{41} -1925.79 q^{43} +2565.11i q^{44} +2052.58 q^{46} -3513.61i q^{47} -2049.57 q^{49} +1117.89i q^{50} -3260.36 q^{52} +1995.12i q^{53} +1953.79 q^{55} -1069.46i q^{56} +1945.25 q^{58} +453.188i q^{59} -5542.38 q^{61} -778.310i q^{62} -1032.76 q^{64} +2483.35i q^{65} -7076.57 q^{67} +2393.19i q^{68} -345.488 q^{70} +6129.62i q^{71} -1180.89 q^{73} -1590.36i q^{74} +326.511 q^{76} -4080.67i q^{77} -1548.39 q^{79} -640.912i q^{80} +1189.67 q^{82} -1142.31i q^{83} +1822.84 q^{85} -3954.22i q^{86} -12418.2 q^{88} -598.920i q^{89} +5186.70 q^{91} -11779.9i q^{92} +7214.47 q^{94} -248.697i q^{95} +6266.12 q^{97} -4208.38i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 76 * q - 632 * q^4 - 48 * q^7	\( 76 q - 632 q^{4} - 48 q^{7} - 72 q^{10} + 448 q^{13} + 5696 q^{16} - 2240 q^{19} + 1008 q^{22} - 11180 q^{25} + 5376 q^{28} - 3440 q^{31} - 7112 q^{34} + 7016 q^{37} - 832 q^{40} - 608 q^{43} + 17728 q^{46} + 30628 q^{49} - 27056 q^{52} - 9312 q^{55} + 4616 q^{58} - 10056 q^{61} - 42864 q^{64} + 31160 q^{67} + 7544 q^{70} + 4488 q^{73} - 15632 q^{76} + 19208 q^{79} + 9768 q^{82} - 7568 q^{85} - 41736 q^{88} - 31240 q^{91} + 27360 q^{94} + 22216 q^{97}+O(q^{100}) \) 76 * q - 632 * q^4 - 48 * q^7 - 72 * q^10 + 448 * q^13 + 5696 * q^16 - 2240 * q^19 + 1008 * q^22 - 11180 * q^25 + 5376 * q^28 - 3440 * q^31 - 7112 * q^34 + 7016 * q^37 - 832 * q^40 - 608 * q^43 + 17728 * q^46 + 30628 * q^49 - 27056 * q^52 - 9312 * q^55 + 4616 * q^58 - 10056 * q^61 - 42864 * q^64 + 31160 * q^67 + 7544 * q^70 + 4488 * q^73 - 15632 * q^76 + 19208 * q^79 + 9768 * q^82 - 7568 * q^85 - 41736 * q^88 - 31240 * q^91 + 27360 * q^94 + 22216 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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