LMFDB - Embedded newform 531.5.b.a.296.60

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			296.60
Character	$\chi$	$=$	531.296
Dual form			531.5.b.a.296.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.38034i q^{2} -12.9480 q^{4} +37.0493i q^{5} -57.6896 q^{7} +16.4207i q^{8} +O(q^{10})$	\(q+5.38034i q^{2} -12.9480 q^{4} +37.0493i q^{5} -57.6896 q^{7} +16.4207i q^{8} -199.337 q^{10} +53.4543i q^{11} -303.430 q^{13} -310.389i q^{14} -295.517 q^{16} +552.137i q^{17} +321.254 q^{19} -479.714i q^{20} -287.602 q^{22} -231.205i q^{23} -747.647 q^{25} -1632.56i q^{26} +746.966 q^{28} +833.800i q^{29} +1694.99 q^{31} -1327.25i q^{32} -2970.68 q^{34} -2137.36i q^{35} -921.546 q^{37} +1728.46i q^{38} -608.375 q^{40} -951.495i q^{41} +1101.95 q^{43} -692.127i q^{44} +1243.96 q^{46} +2120.22i q^{47} +927.090 q^{49} -4022.59i q^{50} +3928.82 q^{52} +1209.43i q^{53} -1980.44 q^{55} -947.304i q^{56} -4486.12 q^{58} +453.188i q^{59} -7318.74 q^{61} +9119.59i q^{62} +2412.78 q^{64} -11241.9i q^{65} +5452.39 q^{67} -7149.08i q^{68} +11499.7 q^{70} -1924.87i q^{71} +1551.38 q^{73} -4958.22i q^{74} -4159.60 q^{76} -3083.76i q^{77} +9306.87 q^{79} -10948.7i q^{80} +5119.36 q^{82} -2067.41i q^{83} -20456.3 q^{85} +5928.89i q^{86} -877.758 q^{88} -4676.20i q^{89} +17504.8 q^{91} +2993.65i q^{92} -11407.5 q^{94} +11902.2i q^{95} +12086.4 q^{97} +4988.05i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.38034i	1.34508i	0.740059	+	0.672542i	$0.234798\pi$
$2$	5.38034i	1.34508i	−0.740059	+	0.672542i	$0.765202\pi$
$3$	0	0
$3$	0	0
$4$	−12.9480	−0.809251
$5$	37.0493i	1.48197i	0.671522	+	0.740985i	$0.265641\pi$
$5$	37.0493i	1.48197i	−0.671522	+	0.740985i	$0.734359\pi$
$6$	0	0
$7$	−57.6896	−1.17734	−0.588669	−	0.808374i	$-0.700348\pi$
$7$	−57.6896	−1.17734	−0.588669	+	0.808374i	$0.700348\pi$
$8$	16.4207i	0.256574i
$9$	0	0
$10$	−199.337	−1.99337
$11$	53.4543i	0.441771i	0.975300	+	0.220886i	$0.0708947\pi$
$11$	53.4543i	0.441771i	−0.975300	+	0.220886i	$0.929105\pi$
$12$	0	0
$13$	−303.430	−1.79545	−0.897723	−	0.440561i	$-0.854780\pi$
$13$	−303.430	−1.79545	−0.897723	+	0.440561i	$0.854780\pi$
$14$	− 310.389i	− 1.58362i
$15$	0	0
$16$	−295.517	−1.15436
$17$	552.137i	1.91051i	0.295785	+	0.955254i	$0.404419\pi$
$17$	552.137i	1.91051i	−0.295785	+	0.955254i	$0.595581\pi$
$18$	0	0
$19$	321.254	0.889901	0.444951	−	0.895555i	$-0.353221\pi$
$19$	321.254	0.889901	0.444951	+	0.895555i	$0.353221\pi$
$20$	− 479.714i	− 1.19929i
$21$	0	0
$22$	−287.602	−0.594219
$23$	− 231.205i	− 0.437061i	−0.975830	−	0.218531i	$-0.929874\pi$
$23$	− 231.205i	− 0.437061i	0.975830	−	0.218531i	$-0.0701264\pi$
$24$	0	0
$25$	−747.647	−1.19624
$26$	− 1632.56i	− 2.41503i
$27$	0	0
$28$	746.966	0.952762
$29$	833.800i	0.991438i	0.868483	+	0.495719i	$0.165095\pi$
$29$	833.800i	0.991438i	−0.868483	+	0.495719i	$0.834905\pi$
$30$	0	0
$31$	1694.99	1.76377	0.881886	−	0.471462i	$-0.156274\pi$
$31$	1694.99	1.76377	0.881886	+	0.471462i	$0.156274\pi$
$32$	− 1327.25i	− 1.29614i
$33$	0	0
$34$	−2970.68	−2.56979
$35$	− 2137.36i	− 1.74478i
$36$	0	0
$37$	−921.546	−0.673152	−0.336576	−	0.941656i	$-0.609269\pi$
$37$	−921.546	−0.673152	−0.336576	+	0.941656i	$0.609269\pi$
$38$	1728.46i	1.19699i
$39$	0	0
$40$	−608.375	−0.380235
$41$	− 951.495i	− 0.566029i	−0.959116	−	0.283014i	$-0.908666\pi$
$41$	− 951.495i	− 0.566029i	0.959116	−	0.283014i	$-0.0913344\pi$
$42$	0	0
$43$	1101.95	0.595973	0.297987	−	0.954570i	$-0.403685\pi$
$43$	1101.95	0.595973	0.297987	+	0.954570i	$0.403685\pi$
$44$	− 692.127i	− 0.357504i
$45$	0	0
$46$	1243.96	0.587884
$47$	2120.22i	0.959812i	0.877320	+	0.479906i	$0.159329\pi$
$47$	2120.22i	0.959812i	−0.877320	+	0.479906i	$0.840671\pi$
$48$	0	0
$49$	927.090	0.386126
$50$	− 4022.59i	− 1.60904i
$51$	0	0
$52$	3928.82	1.45297
$53$	1209.43i	0.430554i	0.976553	+	0.215277i	$0.0690654\pi$
$53$	1209.43i	0.430554i	−0.976553	+	0.215277i	$0.930935\pi$
$54$	0	0
$55$	−1980.44	−0.654692
$56$	− 947.304i	− 0.302074i
$57$	0	0
$58$	−4486.12	−1.33357
$59$	453.188i	0.130189i
$59$	453.188i	0.130189i
$60$	0	0
$61$	−7318.74	−1.96687	−0.983437	−	0.181252i	$-0.941985\pi$
$61$	−7318.74	−1.96687	−0.983437	+	0.181252i	$0.941985\pi$
$62$	9119.59i	2.37242i
$63$	0	0
$64$	2412.78	0.589057
$65$	− 11241.9i	− 2.66080i
$66$	0	0
$67$	5452.39	1.21461	0.607306	−	0.794468i	$-0.292250\pi$
$67$	5452.39	1.21461	0.607306	+	0.794468i	$0.292250\pi$
$68$	− 7149.08i	− 1.54608i
$69$	0	0
$70$	11499.7	2.34688
$71$	− 1924.87i	− 0.381843i	−0.981605	−	0.190922i	$-0.938852\pi$
$71$	− 1924.87i	− 0.381843i	0.981605	−	0.190922i	$-0.0611476\pi$
$72$	0	0
$73$	1551.38	0.291120	0.145560	−	0.989349i	$-0.453502\pi$
$73$	1551.38	0.291120	0.145560	+	0.989349i	$0.453502\pi$
$74$	− 4958.22i	− 0.905446i
$75$	0	0
$76$	−4159.60	−0.720153
$77$	− 3083.76i	− 0.520114i
$78$	0	0
$79$	9306.87	1.49125	0.745623	−	0.666368i	$-0.232152\pi$
$79$	9306.87	1.49125	0.745623	+	0.666368i	$0.232152\pi$
$80$	− 10948.7i	− 1.71073i
$81$	0	0
$82$	5119.36	0.761356
$83$	− 2067.41i	− 0.300103i	−0.988678	−	0.150052i	$-0.952056\pi$
$83$	− 2067.41i	− 0.300103i	0.988678	−	0.150052i	$-0.0479440\pi$
$84$	0	0
$85$	−20456.3	−2.83132
$86$	5928.89i	0.801634i
$87$	0	0
$88$	−877.758	−0.113347
$89$	− 4676.20i	− 0.590355i	−0.955442	−	0.295178i	$-0.904621\pi$
$89$	− 4676.20i	− 0.590355i	0.955442	−	0.295178i	$-0.0953788\pi$
$90$	0	0
$91$	17504.8	2.11385
$92$	2993.65i	0.353692i
$93$	0	0
$94$	−11407.5	−1.29103
$95$	11902.2i	1.31881i
$96$	0	0
$97$	12086.4	1.28455	0.642277	−	0.766472i	$-0.277990\pi$
$97$	12086.4	1.28455	0.642277	+	0.766472i	$0.277990\pi$
$98$	4988.05i	0.519372i
$99$	0	0
$100$	9680.54	0.968054
$101$	− 2066.91i	− 0.202618i	−0.994855	−	0.101309i	$-0.967697\pi$
$101$	− 2066.91i	− 0.202618i	0.994855	−	0.101309i	$-0.0323031\pi$
$102$	0	0
$103$	3190.96	0.300779	0.150389	−	0.988627i	$-0.451947\pi$
$103$	3190.96	0.300779	0.150389	+	0.988627i	$0.451947\pi$
$104$	− 4982.54i	− 0.460664i
$105$	0	0
$106$	−6507.11	−0.579131
$107$	− 3516.27i	− 0.307125i	−0.988139	−	0.153562i	$-0.950925\pi$
$107$	− 3516.27i	− 0.307125i	0.988139	−	0.153562i	$-0.0490746\pi$
$108$	0	0
$109$	−5632.09	−0.474042	−0.237021	−	0.971505i	$-0.576171\pi$
$109$	−5632.09	−0.474042	−0.237021	+	0.971505i	$0.576171\pi$
$110$	− 10655.4i	− 0.880615i
$111$	0	0
$112$	17048.3	1.35908
$113$	16859.7i	1.32036i	0.751107	+	0.660181i	$0.229520\pi$
$113$	16859.7i	1.32036i	−0.751107	+	0.660181i	$0.770480\pi$
$114$	0	0
$115$	8565.99	0.647712
$116$	− 10796.0i	− 0.802322i
$117$	0	0
$118$	−2438.30	−0.175115
$119$	− 31852.6i	− 2.24932i
$120$	0	0
$121$	11783.6	0.804838
$122$	− 39377.3i	− 2.64561i
$123$	0	0
$124$	−21946.7	−1.42733
$125$	− 4543.99i	− 0.290815i
$126$	0	0
$127$	−5513.34	−0.341828	−0.170914	−	0.985286i	$-0.554672\pi$
$127$	−5513.34	−0.341828	−0.170914	+	0.985286i	$0.554672\pi$
$128$	− 8254.45i	− 0.503812i
$129$	0	0
$130$	60485.0	3.57900
$131$	2507.68i	0.146127i	0.997327	+	0.0730633i	$0.0232775\pi$
$131$	2507.68i	0.146127i	−0.997327	+	0.0730633i	$0.976722\pi$
$132$	0	0
$133$	−18533.0	−1.04771
$134$	29335.7i	1.63375i
$135$	0	0
$136$	−9066.49	−0.490186
$137$	9242.55i	0.492437i	0.969214	+	0.246218i	$0.0791881\pi$
$137$	9242.55i	0.492437i	−0.969214	+	0.246218i	$0.920812\pi$
$138$	0	0
$139$	−2922.72	−0.151272	−0.0756358	−	0.997136i	$-0.524099\pi$
$139$	−2922.72	−0.151272	−0.0756358	+	0.997136i	$0.524099\pi$
$140$	27674.5i	1.41197i
$141$	0	0
$142$	10356.5	0.513611
$143$	− 16219.7i	− 0.793176i
$144$	0	0
$145$	−30891.7	−1.46928
$146$	8346.93i	0.391581i
$147$	0	0
$148$	11932.2	0.544749
$149$	− 11273.9i	− 0.507810i	−0.967229	−	0.253905i	$-0.918285\pi$
$149$	− 11273.9i	− 0.507810i	0.967229	−	0.253905i	$-0.0817150\pi$
$150$	0	0
$151$	−42531.1	−1.86532	−0.932658	−	0.360761i	$-0.882517\pi$
$151$	−42531.1	−1.86532	−0.932658	+	0.360761i	$0.882517\pi$
$152$	5275.23i	0.228325i
$153$	0	0
$154$	16591.7	0.699597
$155$	62797.9i	2.61386i
$156$	0	0
$157$	32044.6	1.30004	0.650019	−	0.759918i	$-0.274761\pi$
$157$	32044.6	1.30004	0.650019	+	0.759918i	$0.274761\pi$
$158$	50074.1i	2.00585i
$159$	0	0
$160$	49173.6	1.92084
$161$	13338.1i	0.514569i
$162$	0	0
$163$	30109.4	1.13325	0.566627	−	0.823974i	$-0.308248\pi$
$163$	30109.4	1.13325	0.566627	+	0.823974i	$0.308248\pi$
$164$	12320.0i	0.458059i
$165$	0	0
$166$	11123.4	0.403664
$167$	20538.9i	0.736451i	0.929736	+	0.368226i	$0.120035\pi$
$167$	20538.9i	0.736451i	−0.929736	+	0.368226i	$0.879965\pi$
$168$	0	0
$169$	63509.0	2.22363
$170$	− 110062.i	− 3.80836i
$171$	0	0
$172$	−14268.1	−0.482292
$173$	8586.01i	0.286879i	0.989659	+	0.143440i	$0.0458163\pi$
$173$	8586.01i	0.286879i	−0.989659	+	0.143440i	$0.954184\pi$
$174$	0	0
$175$	43131.5	1.40837
$176$	− 15796.7i	− 0.509965i
$177$	0	0
$178$	25159.5	0.794077
$179$	53646.9i	1.67432i	0.546959	+	0.837160i	$0.315786\pi$
$179$	53646.9i	1.67432i	−0.546959	+	0.837160i	$0.684214\pi$
$180$	0	0
$181$	−16319.4	−0.498135	−0.249068	−	0.968486i	$-0.580124\pi$
$181$	−16319.4	−0.498135	−0.249068	+	0.968486i	$0.580124\pi$
$182$	94181.6i	2.84330i
$183$	0	0
$184$	3796.56	0.112138
$185$	− 34142.6i	− 0.997592i
$186$	0	0
$187$	−29514.1	−0.844008
$188$	− 27452.7i	− 0.776728i
$189$	0	0
$190$	−64038.0	−1.77391
$191$	− 14185.9i	− 0.388857i	−0.980917	−	0.194428i	$-0.937715\pi$
$191$	− 14185.9i	− 0.388857i	0.980917	−	0.194428i	$-0.0622852\pi$
$192$	0	0
$193$	51635.6	1.38623	0.693113	−	0.720829i	$-0.256239\pi$
$193$	51635.6	1.38623	0.693113	+	0.720829i	$0.256239\pi$
$194$	65028.8i	1.72783i
$195$	0	0
$196$	−12004.0	−0.312473
$197$	30764.4i	0.792713i	0.918097	+	0.396356i	$0.129726\pi$
$197$	30764.4i	0.792713i	−0.918097	+	0.396356i	$0.870274\pi$
$198$	0	0
$199$	−37603.4	−0.949557	−0.474779	−	0.880105i	$-0.657472\pi$
$199$	−37603.4	−0.949557	−0.474779	+	0.880105i	$0.657472\pi$
$200$	− 12276.9i	− 0.306923i
$201$	0	0
$202$	11120.6	0.272538
$203$	− 48101.6i	− 1.16726i
$204$	0	0
$205$	35252.2	0.838838
$206$	17168.5i	0.404573i
$207$	0	0
$208$	89668.9	2.07260
$209$	17172.4i	0.393133i
$210$	0	0
$211$	−80196.5	−1.80132	−0.900659	−	0.434527i	$-0.856916\pi$
$211$	−80196.5	−1.80132	−0.900659	+	0.434527i	$0.856916\pi$
$212$	− 15659.7i	− 0.348426i
$213$	0	0
$214$	18918.7	0.413109
$215$	40826.6i	0.883215i
$216$	0	0
$217$	−97783.0	−2.07656
$218$	− 30302.5i	− 0.637626i
$219$	0	0
$220$	25642.8	0.529810
$221$	− 167535.i	− 3.43022i
$222$	0	0
$223$	18216.8	0.366321	0.183160	−	0.983083i	$-0.441367\pi$
$223$	18216.8	0.366321	0.183160	+	0.983083i	$0.441367\pi$
$224$	76568.5i	1.52600i
$225$	0	0
$226$	−90710.8	−1.77600
$227$	− 82315.0i	− 1.59745i	−0.601696	−	0.798725i	$-0.705508\pi$
$227$	− 82315.0i	− 1.59745i	0.601696	−	0.798725i	$-0.294492\pi$
$228$	0	0
$229$	−58051.9	−1.10699	−0.553497	−	0.832851i	$-0.686707\pi$
$229$	−58051.9	−1.10699	−0.553497	+	0.832851i	$0.686707\pi$
$230$	46087.9i	0.871227i
$231$	0	0
$232$	−13691.6	−0.254377
$233$	− 83557.6i	− 1.53913i	−0.638571	−	0.769563i	$-0.720474\pi$
$233$	− 83557.6i	− 1.53913i	0.638571	−	0.769563i	$-0.279526\pi$
$234$	0	0
$235$	−78552.7	−1.42241
$236$	− 5867.88i	− 0.105355i
$237$	0	0
$238$	171378.	3.02552
$239$	− 97444.7i	− 1.70593i	−0.521965	−	0.852967i	$-0.674801\pi$
$239$	− 97444.7i	− 1.70593i	0.521965	−	0.852967i	$-0.325199\pi$
$240$	0	0
$241$	−60187.3	−1.03626	−0.518132	−	0.855301i	$-0.673373\pi$
$241$	−60187.3	−1.03626	−0.518132	+	0.855301i	$0.673373\pi$
$242$	63399.9i	1.08257i
$243$	0	0
$244$	94763.1	1.59169
$245$	34348.0i	0.572228i
$246$	0	0
$247$	−97478.3	−1.59777
$248$	27832.9i	0.452538i
$249$	0	0
$250$	24448.2	0.391171
$251$	97.2944i	0.00154433i	1.00000		0.000772165i	$0.000245788\pi$
$251$	97.2944i	0.00154433i	−1.00000		0.000772165i	$0.999754\pi$
$252$	0	0
$253$	12358.9	0.193081
$254$	− 29663.6i	− 0.459787i
$255$	0	0
$256$	83016.2	1.26673
$257$	111244.i	1.68427i	0.539267	+	0.842135i	$0.318701\pi$
$257$	111244.i	1.68427i	−0.539267	+	0.842135i	$0.681299\pi$
$258$	0	0
$259$	53163.6	0.792528
$260$	145560.i	2.15325i
$261$	0	0
$262$	−13492.1	−0.196552
$263$	75021.6i	1.08461i	0.840181	+	0.542306i	$0.182449\pi$
$263$	75021.6i	1.08461i	−0.840181	+	0.542306i	$0.817551\pi$
$264$	0	0
$265$	−44808.3	−0.638068
$266$	− 99713.9i	− 1.40926i
$267$	0	0
$268$	−70597.6	−0.982925
$269$	− 33129.7i	− 0.457839i	−0.973445	−	0.228919i	$-0.926481\pi$
$269$	− 33129.7i	− 0.457839i	0.973445	−	0.228919i	$-0.0735192\pi$
$270$	0	0
$271$	−69539.6	−0.946877	−0.473438	−	0.880827i	$-0.656987\pi$
$271$	−69539.6	−0.946877	−0.473438	+	0.880827i	$0.656987\pi$
$272$	− 163166.i	− 2.20542i
$273$	0	0
$274$	−49728.0	−0.662369
$275$	− 39965.0i	− 0.528462i
$276$	0	0
$277$	−52233.2	−0.680749	−0.340374	−	0.940290i	$-0.610554\pi$
$277$	−52233.2	−0.680749	−0.340374	+	0.940290i	$0.610554\pi$
$278$	− 15725.2i	− 0.203473i
$279$	0	0
$280$	35096.9	0.447665
$281$	− 64507.5i	− 0.816953i	−0.912769	−	0.408477i	$-0.866060\pi$
$281$	− 64507.5i	− 0.816953i	0.912769	−	0.408477i	$-0.133940\pi$
$282$	0	0
$283$	−100618.	−1.25633	−0.628163	−	0.778082i	$-0.716193\pi$
$283$	−100618.	−1.25633	−0.628163	+	0.778082i	$0.716193\pi$
$284$	24923.2i	0.309007i
$285$	0	0
$286$	87267.2	1.06689
$287$	54891.3i	0.666408i
$288$	0	0
$289$	−221334.	−2.65004
$290$	− 166207.i	− 1.97631i
$291$	0	0
$292$	−20087.3	−0.235589
$293$	− 70741.0i	− 0.824016i	−0.911180	−	0.412008i	$-0.864828\pi$
$293$	− 70741.0i	− 0.824016i	0.911180	−	0.412008i	$-0.135172\pi$
$294$	0	0
$295$	−16790.3	−0.192936
$296$	− 15132.4i	− 0.172713i
$297$	0	0
$298$	60657.3	0.683047
$299$	70154.7i	0.784720i
$300$	0	0
$301$	−63571.3	−0.701663
$302$	− 228831.i	− 2.50901i
$303$	0	0
$304$	−94936.2	−1.02727
$305$	− 271154.i	− 2.91485i
$306$	0	0
$307$	23151.7	0.245644	0.122822	−	0.992429i	$-0.460806\pi$
$307$	23151.7	0.245644	0.122822	+	0.992429i	$0.460806\pi$
$308$	39928.5i	0.420903i
$309$	0	0
$310$	−337874.	−3.51586
$311$	− 68715.9i	− 0.710455i	−0.934780	−	0.355227i	$-0.884403\pi$
$311$	− 68715.9i	− 0.710455i	0.934780	−	0.355227i	$-0.115597\pi$
$312$	0	0
$313$	−99137.6	−1.01193	−0.505964	−	0.862555i	$-0.668863\pi$
$313$	−99137.6	−1.01193	−0.505964	+	0.862555i	$0.668863\pi$
$314$	172411.i	1.74866i
$315$	0	0
$316$	−120505.	−1.20679
$317$	118327.i	1.17751i	0.808310	+	0.588757i	$0.200383\pi$
$317$	118327.i	1.17751i	−0.808310	+	0.588757i	$0.799617\pi$
$318$	0	0
$319$	−44570.2	−0.437989
$320$	89391.6i	0.872965i
$321$	0	0
$322$	−71763.7	−0.692139
$323$	177376.i	1.70016i
$324$	0	0
$325$	226859.	2.14778
$326$	161999.i	1.52432i
$327$	0	0
$328$	15624.2	0.145228
$329$	− 122315.i	− 1.13002i
$330$	0	0
$331$	−124259.	−1.13416	−0.567078	−	0.823664i	$-0.691926\pi$
$331$	−124259.	−1.13416	−0.567078	+	0.823664i	$0.691926\pi$
$332$	26768.8i	0.242859i
$333$	0	0
$334$	−110506.	−0.990589
$335$	202007.i	1.80002i
$336$	0	0
$337$	145065.	1.27733	0.638665	−	0.769485i	$-0.279487\pi$
$337$	145065.	1.27733	0.638665	+	0.769485i	$0.279487\pi$
$338$	341700.i	2.99096i
$339$	0	0
$340$	264868.	2.29125
$341$	90604.3i	0.779184i
$342$	0	0
$343$	85029.3	0.722737
$344$	18094.9i	0.152911i
$345$	0	0
$346$	−46195.6	−0.385877
$347$	− 119715.i	− 0.994235i	−0.867683	−	0.497117i	$-0.834392\pi$
$347$	− 119715.i	− 0.994235i	0.867683	−	0.497117i	$-0.165608\pi$
$348$	0	0
$349$	−13520.5	−0.111005	−0.0555024	−	0.998459i	$-0.517676\pi$
$349$	−13520.5	−0.111005	−0.0555024	+	0.998459i	$0.517676\pi$
$350$	232062.i	1.89438i
$351$	0	0
$352$	70947.2	0.572599
$353$	208001.i	1.66923i	0.550833	+	0.834615i	$0.314310\pi$
$353$	208001.i	1.66923i	−0.550833	+	0.834615i	$0.685690\pi$
$354$	0	0
$355$	71315.0	0.565880
$356$	60547.5i	0.477745i
$357$	0	0
$358$	−288638.	−2.25210
$359$	42595.2i	0.330501i	0.986252	+	0.165250i	$0.0528432\pi$
$359$	42595.2i	0.330501i	−0.986252	+	0.165250i	$0.947157\pi$
$360$	0	0
$361$	−27116.7	−0.208076
$362$	− 87803.9i	− 0.670034i
$363$	0	0
$364$	−226652.	−1.71063
$365$	57477.4i	0.431431i
$366$	0	0
$367$	−192869.	−1.43196	−0.715980	−	0.698121i	$-0.754020\pi$
$367$	−192869.	−1.43196	−0.715980	+	0.698121i	$0.754020\pi$
$368$	68325.2i	0.504528i
$369$	0	0
$370$	183699.	1.34184
$371$	− 69771.3i	− 0.506908i
$372$	0	0
$373$	227462.	1.63490	0.817449	−	0.576001i	$-0.195387\pi$
$373$	227462.	1.63490	0.817449	+	0.576001i	$0.195387\pi$
$374$	− 158796.i	− 1.13526i
$375$	0	0
$376$	−34815.6	−0.246262
$377$	− 253000.i	− 1.78007i
$378$	0	0
$379$	207028.	1.44129	0.720644	−	0.693305i	$-0.243846\pi$
$379$	207028.	1.44129	0.720644	+	0.693305i	$0.243846\pi$
$380$	− 154110.i	− 1.06725i
$381$	0	0
$382$	76324.8	0.523045
$383$	− 54887.5i	− 0.374176i	−0.982343	−	0.187088i	$-0.940095\pi$
$383$	− 54887.5i	− 0.374176i	0.982343	−	0.187088i	$-0.0599050\pi$
$384$	0	0
$385$	114251.	0.770794
$386$	277817.i	1.86459i
$387$	0	0
$388$	−156495.	−1.03953
$389$	5416.69i	0.0357960i	0.999840	+	0.0178980i	$0.00569741\pi$
$389$	5416.69i	0.0357960i	−0.999840	+	0.0178980i	$0.994303\pi$
$390$	0	0
$391$	127657.	0.835010
$392$	15223.5i	0.0990699i
$393$	0	0
$394$	−165523.	−1.06627
$395$	344812.i	2.20998i
$396$	0	0
$397$	−274690.	−1.74286	−0.871429	−	0.490521i	$-0.836806\pi$
$397$	−274690.	−1.74286	−0.871429	+	0.490521i	$0.836806\pi$
$398$	− 202319.i	− 1.27723i
$399$	0	0
$400$	220943.	1.38089
$401$	− 100756.i	− 0.626589i	−0.949656	−	0.313295i	$-0.898567\pi$
$401$	− 100756.i	− 0.626589i	0.949656	−	0.313295i	$-0.101433\pi$
$402$	0	0
$403$	−514310.	−3.16676
$404$	26762.3i	0.163969i
$405$	0	0
$406$	258803.	1.57006
$407$	− 49260.6i	− 0.297379i
$408$	0	0
$409$	231377.	1.38316	0.691582	−	0.722298i	$-0.256914\pi$
$409$	231377.	1.38316	0.691582	+	0.722298i	$0.256914\pi$
$410$	189668.i	1.12831i
$411$	0	0
$412$	−41316.6	−0.243406
$413$	− 26144.2i	− 0.153276i
$414$	0	0
$415$	76596.0	0.444744
$416$	402728.i	2.32715i
$417$	0	0
$418$	−92393.4	−0.528796
$419$	127741.i	0.727617i	0.931474	+	0.363809i	$0.118524\pi$
$419$	127741.i	0.727617i	−0.931474	+	0.363809i	$0.881476\pi$
$420$	0	0
$421$	−13036.3	−0.0735512	−0.0367756	−	0.999324i	$-0.511709\pi$
$421$	−13036.3	−0.0735512	−0.0367756	+	0.999324i	$0.511709\pi$
$422$	− 431484.i	− 2.42292i
$423$	0	0
$424$	−19859.6	−0.110469
$425$	− 412804.i	− 2.28542i
$426$	0	0
$427$	422215.	2.31568
$428$	45528.7i	0.248541i
$429$	0	0
$430$	−219661.	−1.18800
$431$	− 196851.i	− 1.05970i	−0.848092	−	0.529850i	$-0.822248\pi$
$431$	− 196851.i	− 1.05970i	0.848092	−	0.529850i	$-0.177752\pi$
$432$	0	0
$433$	−81985.8	−0.437283	−0.218642	−	0.975805i	$-0.570163\pi$
$433$	−81985.8	−0.437283	−0.218642	+	0.975805i	$0.570163\pi$
$434$	− 526106.i	− 2.79314i
$435$	0	0
$436$	72924.4	0.383619
$437$	− 74275.7i	− 0.388941i
$438$	0	0
$439$	124627.	0.646673	0.323336	−	0.946284i	$-0.395195\pi$
$439$	124627.	0.646673	0.323336	+	0.946284i	$0.395195\pi$
$440$	− 32520.3i	− 0.167977i
$441$	0	0
$442$	901395.	4.61393
$443$	74992.4i	0.382129i	0.981577	+	0.191064i	$0.0611940\pi$
$443$	74992.4i	0.382129i	−0.981577	+	0.191064i	$0.938806\pi$
$444$	0	0
$445$	173250.	0.874889
$446$	98012.4i	0.492732i
$447$	0	0
$448$	−139192.	−0.693519
$449$	364745.i	1.80924i	0.426218	+	0.904621i	$0.359846\pi$
$449$	364745.i	1.80924i	−0.426218	+	0.904621i	$0.640154\pi$
$450$	0	0
$451$	50861.5	0.250055
$452$	− 218300.i	− 1.06850i
$453$	0	0
$454$	442882.	2.14870
$455$	648539.i	3.13266i
$456$	0	0
$457$	−90669.0	−0.434137	−0.217068	−	0.976156i	$-0.569649\pi$
$457$	−90669.0	−0.434137	−0.217068	+	0.976156i	$0.569649\pi$
$458$	− 312339.i	− 1.48900i
$459$	0	0
$460$	−110913.	−0.524161
$461$	33791.0i	0.159001i	0.996835	+	0.0795004i	$0.0253325\pi$
$461$	33791.0i	0.159001i	−0.996835	+	0.0795004i	$0.974668\pi$
$462$	0	0
$463$	−369624.	−1.72424	−0.862122	−	0.506701i	$-0.830865\pi$
$463$	−369624.	−1.72424	−0.862122	+	0.506701i	$0.830865\pi$
$464$	− 246402.i	− 1.14448i
$465$	0	0
$466$	449568.	2.07025
$467$	171471.i	0.786245i	0.919486	+	0.393122i	$0.128605\pi$
$467$	171471.i	0.786245i	−0.919486	+	0.393122i	$0.871395\pi$
$468$	0	0
$469$	−314546.	−1.43001
$470$	− 422640.i	− 1.91326i
$471$	0	0
$472$	−7441.66	−0.0334030
$473$	58904.2i	0.263284i
$474$	0	0
$475$	−240185.	−1.06453
$476$	412427.i	1.82026i
$477$	0	0
$478$	524285.	2.29462
$479$	428902.i	1.86933i	0.355524	+	0.934667i	$0.384302\pi$
$479$	428902.i	1.86933i	−0.355524	+	0.934667i	$0.615698\pi$
$480$	0	0
$481$	279625.	1.20861
$482$	− 323828.i	− 1.39386i
$483$	0	0
$484$	−152575.	−0.651316
$485$	447791.i	1.90367i
$486$	0	0
$487$	194228.	0.818945	0.409473	−	0.912322i	$-0.365713\pi$
$487$	194228.	0.818945	0.409473	+	0.912322i	$0.365713\pi$
$488$	− 120179.i	− 0.504648i
$489$	0	0
$490$	−184804.	−0.769695
$491$	− 345800.i	− 1.43437i	−0.696881	−	0.717186i	$-0.745430\pi$
$491$	− 345800.i	− 1.43437i	0.696881	−	0.717186i	$-0.254570\pi$
$492$	0	0
$493$	−460372.	−1.89415
$494$	− 524466.i	− 2.14913i
$495$	0	0
$496$	−500897.	−2.03604
$497$	111045.i	0.449559i
$498$	0	0
$499$	275983.	1.10836	0.554180	−	0.832397i	$-0.313032\pi$
$499$	275983.	1.10836	0.554180	+	0.832397i	$0.313032\pi$
$500$	58835.6i	0.235342i
$501$	0	0
$502$	−523.476	−0.00207725
$503$	120055.i	0.474508i	0.971448	+	0.237254i	$0.0762473\pi$
$503$	120055.i	0.474508i	−0.971448	+	0.237254i	$0.923753\pi$
$504$	0	0
$505$	76577.3	0.300274
$506$	66495.2i	0.259710i
$507$	0	0
$508$	71386.7	0.276624
$509$	− 307543.i	− 1.18705i	−0.804814	−	0.593527i	$-0.797735\pi$
$509$	− 307543.i	− 1.18705i	0.804814	−	0.593527i	$-0.202265\pi$
$510$	0	0
$511$	−89498.3	−0.342747
$512$	314584.i	1.20004i
$513$	0	0
$514$	−598532.	−2.26548
$515$	118223.i	0.445745i
$516$	0	0
$517$	−113335.	−0.424017
$518$	286038.i	1.06602i
$519$	0	0
$520$	184600.	0.682690
$521$	51907.9i	0.191231i	0.995418	+	0.0956154i	$0.0304819\pi$
$521$	51907.9i	0.191231i	−0.995418	+	0.0956154i	$0.969518\pi$
$522$	0	0
$523$	−239850.	−0.876872	−0.438436	−	0.898763i	$-0.644467\pi$
$523$	−239850.	−0.876872	−0.438436	+	0.898763i	$0.644467\pi$
$524$	− 32469.4i	− 0.118253i
$525$	0	0
$526$	−403641.	−1.45890
$527$	935864.i	3.36970i
$528$	0	0
$529$	226385.	0.808977
$530$	− 241084.i	− 0.858255i
$531$	0	0
$532$	239966.	0.847864
$533$	288712.i	1.01627i
$534$	0	0
$535$	130275.	0.455150
$536$	89532.1i	0.311637i
$537$	0	0
$538$	178249.	0.615831
$539$	49556.9i	0.170580i
$540$	0	0
$541$	521337.	1.78124	0.890622	−	0.454744i	$-0.150269\pi$
$541$	521337.	1.78124	0.890622	+	0.454744i	$0.150269\pi$
$542$	− 374146.i	− 1.27363i
$543$	0	0
$544$	732824.	2.47629
$545$	− 208665.i	− 0.702516i
$546$	0	0
$547$	−28870.5	−0.0964895	−0.0482448	−	0.998836i	$-0.515363\pi$
$547$	−28870.5	−0.0964895	−0.0482448	+	0.998836i	$0.515363\pi$
$548$	− 119673.i	− 0.398505i
$549$	0	0
$550$	215025.	0.710826
$551$	267862.i	0.882282i
$552$	0	0
$553$	−536909.	−1.75570
$554$	− 281032.i	− 0.915664i
$555$	0	0
$556$	37843.4	0.122417
$557$	40542.8i	0.130678i	0.997863	+	0.0653392i	$0.0208129\pi$
$557$	40542.8i	0.130678i	−0.997863	+	0.0653392i	$0.979187\pi$
$558$	0	0
$559$	−334367.	−1.07004
$560$	631626.i	2.01411i
$561$	0	0
$562$	347072.	1.09887
$563$	− 197241.i	− 0.622272i	−0.950365	−	0.311136i	$-0.899290\pi$
$563$	− 197241.i	− 0.622272i	0.950365	−	0.311136i	$-0.100710\pi$
$564$	0	0
$565$	−624639.	−1.95674
$566$	− 541358.i	− 1.68986i
$567$	0	0
$568$	31607.8	0.0979709
$569$	− 152088.i	− 0.469755i	−0.972025	−	0.234878i	$-0.924531\pi$
$569$	− 152088.i	− 0.469755i	0.972025	−	0.234878i	$-0.0754689\pi$
$570$	0	0
$571$	−240204.	−0.736731	−0.368365	−	0.929681i	$-0.620082\pi$
$571$	−240204.	−0.736731	−0.368365	+	0.929681i	$0.620082\pi$
$572$	210012.i	0.641879i
$573$	0	0
$574$	−295334.	−0.896374
$575$	172860.i	0.522828i
$576$	0	0
$577$	−194615.	−0.584553	−0.292277	−	0.956334i	$-0.594413\pi$
$577$	−194615.	−0.584553	−0.292277	+	0.956334i	$0.594413\pi$
$578$	− 1.19085e6i	− 3.56453i
$579$	0	0
$580$	399986.	1.18902
$581$	119268.i	0.353323i
$582$	0	0
$583$	−64649.0	−0.190206
$584$	25474.7i	0.0746937i
$585$	0	0
$586$	380610.	1.10837
$587$	− 408852.i	− 1.18656i	−0.804996	−	0.593281i	$-0.797832\pi$
$587$	− 408852.i	− 1.18656i	0.804996	−	0.593281i	$-0.202168\pi$
$588$	0	0
$589$	544521.	1.56958
$590$	− 90337.2i	− 0.259515i
$591$	0	0
$592$	272333.	0.777063
$593$	− 16959.2i	− 0.0482275i	−0.999709	−	0.0241138i	$-0.992324\pi$
$593$	− 16959.2i	− 0.0482275i	0.999709	−	0.0241138i	$-0.00767639\pi$
$594$	0	0
$595$	1.18011e6	3.33342
$596$	145974.i	0.410945i
$597$	0	0
$598$	−377456.	−1.05551
$599$	− 96105.6i	− 0.267852i	−0.990991	−	0.133926i	$-0.957242\pi$
$599$	− 96105.6i	− 0.267852i	0.990991	−	0.133926i	$-0.0427585\pi$
$600$	0	0
$601$	118617.	0.328397	0.164198	−	0.986427i	$-0.447496\pi$
$601$	118617.	0.328397	0.164198	+	0.986427i	$0.447496\pi$
$602$	− 342035.i	− 0.943795i
$603$	0	0
$604$	550693.	1.50951
$605$	436575.i	1.19275i
$606$	0	0
$607$	531966.	1.44380	0.721899	−	0.691999i	$-0.243270\pi$
$607$	531966.	1.44380	0.721899	+	0.691999i	$0.243270\pi$
$608$	− 426385.i	− 1.15344i
$609$	0	0
$610$	1.45890e6	3.92071
$611$	− 643340.i	− 1.72329i
$612$	0	0
$613$	−621052.	−1.65275	−0.826375	−	0.563121i	$-0.809601\pi$
$613$	−621052.	−1.65275	−0.826375	+	0.563121i	$0.809601\pi$
$614$	124564.i	0.330412i
$615$	0	0
$616$	50637.5	0.133448
$617$	416360.i	1.09370i	0.837231	+	0.546850i	$0.184173\pi$
$617$	416360.i	1.09370i	−0.837231	+	0.546850i	$0.815827\pi$
$618$	0	0
$619$	−176137.	−0.459695	−0.229847	−	0.973227i	$-0.573823\pi$
$619$	−176137.	−0.459695	−0.229847	+	0.973227i	$0.573823\pi$
$620$	− 813109.i	− 2.11527i
$621$	0	0
$622$	369715.	0.955622
$623$	269768.i	0.695048i
$624$	0	0
$625$	−298928.	−0.765256
$626$	− 533393.i	− 1.36113i
$627$	0	0
$628$	−414914.	−1.05206
$629$	− 508819.i	− 1.28606i
$630$	0	0
$631$	150821.	0.378795	0.189397	−	0.981901i	$-0.439347\pi$
$631$	150821.	0.378795	0.189397	+	0.981901i	$0.439347\pi$
$632$	152825.i	0.382615i
$633$	0	0
$634$	−636640.	−1.58386
$635$	− 204265.i	− 0.506578i
$636$	0	0
$637$	−281307.	−0.693269
$638$	− 239803.i	− 0.589132i
$639$	0	0
$640$	305821.	0.746634
$641$	− 254300.i	− 0.618915i	−0.950913	−	0.309458i	$-0.899853\pi$
$641$	− 254300.i	− 0.618915i	0.950913	−	0.309458i	$-0.100147\pi$
$642$	0	0
$643$	425982.	1.03031	0.515157	−	0.857096i	$-0.327734\pi$
$643$	425982.	1.03031	0.515157	+	0.857096i	$0.327734\pi$
$644$	− 172703.i	− 0.416416i
$645$	0	0
$646$	−954345.	−2.28686
$647$	− 146578.i	− 0.350155i	−0.984555	−	0.175077i	$-0.943982\pi$
$647$	− 146578.i	− 0.350155i	0.984555	−	0.175077i	$-0.0560175\pi$
$648$	0	0
$649$	−24224.8	−0.0575137
$650$	1.22058e6i	2.88894i
$651$	0	0
$652$	−389857.	−0.917087
$653$	201499.i	0.472549i	0.971686	+	0.236275i	$0.0759265\pi$
$653$	201499.i	0.472549i	−0.971686	+	0.236275i	$0.924073\pi$
$654$	0	0
$655$	−92907.6	−0.216555
$656$	281183.i	0.653403i
$657$	0	0
$658$	658095.	1.51998
$659$	111091.i	0.255805i	0.991787	+	0.127903i	$0.0408245\pi$
$659$	111091.i	0.255805i	−0.991787	+	0.127903i	$0.959176\pi$
$660$	0	0
$661$	−227169.	−0.519932	−0.259966	−	0.965618i	$-0.583711\pi$
$661$	−227169.	−0.519932	−0.259966	+	0.965618i	$0.583711\pi$
$662$	− 668557.i	− 1.52554i
$663$	0	0
$664$	33948.3	0.0769985
$665$	− 686635.i	− 1.55268i
$666$	0	0
$667$	192779.	0.433319
$668$	− 265938.i	− 0.595974i
$669$	0	0
$670$	−1.08687e6	−2.42117
$671$	− 391218.i	− 0.868908i
$672$	0	0
$673$	−220107.	−0.485963	−0.242981	−	0.970031i	$-0.578125\pi$
$673$	−220107.	−0.485963	−0.242981	+	0.970031i	$0.578125\pi$
$674$	780499.i	1.71812i
$675$	0	0
$676$	−822315.	−1.79947
$677$	624538.i	1.36264i	0.731985	+	0.681321i	$0.238594\pi$
$677$	624538.i	1.36264i	−0.731985	+	0.681321i	$0.761406\pi$
$678$	0	0
$679$	−697258.	−1.51236
$680$	− 335907.i	− 0.726442i
$681$	0	0
$682$	−487481.	−1.04807
$683$	− 188488.i	− 0.404057i	−0.979380	−	0.202028i	$-0.935247\pi$
$683$	− 188488.i	− 0.404057i	0.979380	−	0.202028i	$-0.0647533\pi$
$684$	0	0
$685$	−342429.	−0.729777
$686$	457486.i	0.972142i
$687$	0	0
$688$	−325647.	−0.687970
$689$	− 366976.i	− 0.773036i
$690$	0	0
$691$	−428936.	−0.898331	−0.449165	−	0.893449i	$-0.648279\pi$
$691$	−428936.	−0.898331	−0.449165	+	0.893449i	$0.648279\pi$
$692$	− 111172.i	− 0.232157i
$693$	0	0
$694$	644106.	1.33733
$695$	− 108285.i	− 0.224180i
$696$	0	0
$697$	525355.	1.08140
$698$	− 72744.9i	− 0.149311i
$699$	0	0
$700$	−558467.	−1.13973
$701$	− 920262.i	− 1.87273i	−0.351027	−	0.936365i	$-0.614168\pi$
$701$	− 920262.i	− 1.87273i	0.351027	−	0.936365i	$-0.385832\pi$
$702$	0	0
$703$	−296050.	−0.599039
$704$	128973.i	0.260228i
$705$	0	0
$706$	−1.11912e6	−2.24525
$707$	119239.i	0.238550i
$708$	0	0
$709$	66547.3	0.132385	0.0661924	−	0.997807i	$-0.478915\pi$
$709$	66547.3	0.132385	0.0661924	+	0.997807i	$0.478915\pi$
$710$	383699.i	0.761156i
$711$	0	0
$712$	76786.6	0.151470
$713$	− 391890.i	− 0.770877i
$714$	0	0
$715$	600926.	1.17546
$716$	− 694620.i	− 1.35494i
$717$	0	0
$718$	−229177.	−0.444551
$719$	324924.i	0.628528i	0.949336	+	0.314264i	$0.101758\pi$
$719$	324924.i	0.628528i	−0.949336	+	0.314264i	$0.898242\pi$
$720$	0	0
$721$	−184085.	−0.354119
$722$	− 145897.i	− 0.279880i
$723$	0	0
$724$	211304.	0.403116
$725$	− 623388.i	− 1.18599i
$726$	0	0
$727$	922500.	1.74541	0.872705	−	0.488248i	$-0.162364\pi$
$727$	922500.	1.74541	0.872705	+	0.488248i	$0.162364\pi$
$728$	287441.i	0.542358i
$729$	0	0
$730$	−309248.	−0.580311
$731$	608430.i	1.13861i
$732$	0	0
$733$	347662.	0.647067	0.323533	−	0.946217i	$-0.395129\pi$
$733$	347662.	0.647067	0.323533	+	0.946217i	$0.395129\pi$
$734$	− 1.03770e6i	− 1.92611i
$735$	0	0
$736$	−306867.	−0.566494
$737$	291454.i	0.536580i
$738$	0	0
$739$	440240.	0.806122	0.403061	−	0.915173i	$-0.367946\pi$
$739$	440240.	0.806122	0.403061	+	0.915173i	$0.367946\pi$
$740$	442078.i	0.807302i
$741$	0	0
$742$	375393.	0.681833
$743$	− 289334.i	− 0.524108i	−0.965053	−	0.262054i	$-0.915600\pi$
$743$	− 289334.i	− 0.524108i	0.965053	−	0.262054i	$-0.0843999\pi$
$744$	0	0
$745$	417689.	0.752559
$746$	1.22382e6i	2.19908i
$747$	0	0
$748$	382149.	0.683014
$749$	202852.i	0.361590i
$750$	0	0
$751$	939085.	1.66504	0.832521	−	0.553994i	$-0.186897\pi$
$751$	939085.	1.66504	0.832521	+	0.553994i	$0.186897\pi$
$752$	− 626563.i	− 1.10797i
$753$	0	0
$754$	1.36123e6	2.39435
$755$	− 1.57574e6i	− 2.76434i
$756$	0	0
$757$	−248298.	−0.433293	−0.216647	−	0.976250i	$-0.569512\pi$
$757$	−248298.	−0.433293	−0.216647	+	0.976250i	$0.569512\pi$
$758$	1.11388e6i	1.93865i
$759$	0	0
$760$	−195443.	−0.338371
$761$	− 158742.i	− 0.274108i	−0.990564	−	0.137054i	$-0.956237\pi$
$761$	− 158742.i	− 0.274108i	0.990564	−	0.137054i	$-0.0437634\pi$
$762$	0	0
$763$	324913.	0.558108
$764$	183679.i	0.314683i
$765$	0	0
$766$	295313.	0.503298
$767$	− 137511.i	− 0.233747i
$768$	0	0
$769$	598773.	1.01253	0.506267	−	0.862377i	$-0.331025\pi$
$769$	598773.	1.01253	0.506267	+	0.862377i	$0.331025\pi$
$770$	614708.i	1.03678i
$771$	0	0
$772$	−668578.	−1.12181
$773$	249332.i	0.417272i	0.977993	+	0.208636i	$0.0669024\pi$
$773$	249332.i	0.417272i	−0.977993	+	0.208636i	$0.933098\pi$
$774$	0	0
$775$	−1.26725e6	−2.10989
$776$	198467.i	0.329583i
$777$	0	0
$778$	−29143.6	−0.0481486
$779$	− 305672.i	− 0.503710i
$780$	0	0
$781$	102893.	0.168687
$782$	686838.i	1.12316i
$783$	0	0
$784$	−273971.	−0.445730
$785$	1.18723e6i	1.92662i
$786$	0	0
$787$	−488070.	−0.788012	−0.394006	−	0.919108i	$-0.628911\pi$
$787$	−488070.	−0.788012	−0.394006	+	0.919108i	$0.628911\pi$
$788$	− 398338.i	− 0.641503i
$789$	0	0
$790$	−1.85521e6	−2.97261
$791$	− 972629.i	− 1.55451i
$792$	0	0
$793$	2.22073e6	3.53141
$794$	− 1.47793e6i	− 2.34429i
$795$	0	0
$796$	486889.	0.768430
$797$	1.15987e6i	1.82597i	0.407991	+	0.912986i	$0.366229\pi$
$797$	1.15987e6i	1.82597i	−0.407991	+	0.912986i	$0.633771\pi$
$798$	0	0
$799$	−1.17065e6	−1.83373
$800$	992315.i	1.55049i
$801$	0	0
$802$	542102.	0.842815
$803$	82927.8i	0.128608i
$804$	0	0
$805$	−494168.	−0.762576
$806$	− 2.76716e6i	− 4.25956i
$807$	0	0
$808$	33940.1	0.0519864
$809$	271035.i	0.414122i	0.978328	+	0.207061i	$0.0663899\pi$
$809$	271035.i	0.414122i	−0.978328	+	0.207061i	$0.933610\pi$
$810$	0	0
$811$	−45264.8	−0.0688207	−0.0344103	−	0.999408i	$-0.510955\pi$
$811$	−45264.8	−0.0688207	−0.0344103	+	0.999408i	$0.510955\pi$
$812$	622820.i	0.944605i
$813$	0	0
$814$	265039.	0.400000
$815$	1.11553e6i	1.67945i
$816$	0	0
$817$	354008.	0.530357
$818$	1.24489e6i	1.86047i
$819$	0	0
$820$	−456445.	−0.678830
$821$	− 101878.i	− 0.151145i	−0.997140	−	0.0755726i	$-0.975922\pi$
$821$	− 101878.i	− 0.151145i	0.997140	−	0.0755726i	$-0.0240785\pi$
$822$	0	0
$823$	845835.	1.24878	0.624390	−	0.781112i	$-0.285347\pi$
$823$	845835.	1.24878	0.624390	+	0.781112i	$0.285347\pi$
$824$	52397.9i	0.0771720i
$825$	0	0
$826$	140665.	0.206170
$827$	− 610729.i	− 0.892971i	−0.894791	−	0.446485i	$-0.852675\pi$
$827$	− 610729.i	− 0.892971i	0.894791	−	0.446485i	$-0.147325\pi$
$828$	0	0
$829$	490520.	0.713753	0.356876	−	0.934152i	$-0.383842\pi$
$829$	490520.	0.713753	0.356876	+	0.934152i	$0.383842\pi$
$830$	412112.i	0.598218i
$831$	0	0
$832$	−732110.	−1.05762
$833$	511881.i	0.737698i
$834$	0	0
$835$	−760950.	−1.09140
$836$	− 222349.i	− 0.318143i
$837$	0	0
$838$	−687291.	−0.978706
$839$	− 1.03787e6i	− 1.47442i	−0.675664	−	0.737210i	$-0.736143\pi$
$839$	− 1.03787e6i	− 1.47442i	0.675664	−	0.737210i	$-0.263857\pi$
$840$	0	0
$841$	12059.2	0.0170501
$842$	− 70139.6i	− 0.0989325i
$843$	0	0
$844$	1.03839e6	1.45772
$845$	2.35296e6i	3.29535i
$846$	0	0
$847$	−679793.	−0.947567
$848$	− 357406.i	− 0.497016i
$849$	0	0
$850$	2.22102e6	3.07408
$851$	213066.i	0.294209i
$852$	0	0
$853$	−1.13660e6	−1.56210	−0.781050	−	0.624468i	$-0.785316\pi$
$853$	−1.13660e6	−1.56210	−0.781050	+	0.624468i	$0.785316\pi$
$854$	2.27166e6i	3.11478i
$855$	0	0
$856$	57739.7	0.0788001
$857$	1.19527e6i	1.62743i	0.581262	+	0.813717i	$0.302559\pi$
$857$	1.19527e6i	1.62743i	−0.581262	+	0.813717i	$0.697441\pi$
$858$	0	0
$859$	403984.	0.547491	0.273746	−	0.961802i	$-0.411737\pi$
$859$	403984.	0.547491	0.273746	+	0.961802i	$0.411737\pi$
$860$	− 528623.i	− 0.714742i
$861$	0	0
$862$	1.05912e6	1.42538
$863$	382525.i	0.513616i	0.966462	+	0.256808i	$0.0826707\pi$
$863$	382525.i	0.513616i	−0.966462	+	0.256808i	$0.917329\pi$
$864$	0	0
$865$	−318105.	−0.425147
$866$	− 441111.i	− 0.588183i
$867$	0	0
$868$	1.26610e6	1.68046
$869$	497492.i	0.658790i
$870$	0	0
$871$	−1.65442e6	−2.18077
$872$	− 92482.9i	− 0.121627i
$873$	0	0
$874$	399628.	0.523159
$875$	262141.i	0.342388i
$876$	0	0
$877$	−255168.	−0.331762	−0.165881	−	0.986146i	$-0.553047\pi$
$877$	−255168.	−0.331762	−0.165881	+	0.986146i	$0.553047\pi$
$878$	670537.i	0.869829i
$879$	0	0
$880$	585255.	0.755753
$881$	5645.35i	0.00727342i	0.999993	+	0.00363671i	$0.00115760\pi$
$881$	5645.35i	0.00727342i	−0.999993	+	0.00363671i	$0.998842\pi$
$882$	0	0
$883$	−736440.	−0.944530	−0.472265	−	0.881457i	$-0.656563\pi$
$883$	−736440.	−0.944530	−0.472265	+	0.881457i	$0.656563\pi$
$884$	2.16925e6i	2.77590i
$885$	0	0
$886$	−403484.	−0.513996
$887$	1.13088e6i	1.43737i	0.695338	+	0.718683i	$0.255255\pi$
$887$	1.13088e6i	1.43737i	−0.695338	+	0.718683i	$0.744745\pi$
$888$	0	0
$889$	318062.	0.402447
$890$	932142.i	1.17680i
$891$	0	0
$892$	−235871.	−0.296446
$893$	681131.i	0.854138i
$894$	0	0
$895$	−1.98758e6	−2.48129
$896$	476196.i	0.593157i
$897$	0	0
$898$	−1.96245e6	−2.43358
$899$	1.41328e6i	1.74867i
$900$	0	0
$901$	−667769.	−0.822577
$902$	273652.i	0.336345i
$903$	0	0
$904$	−276848.	−0.338770
$905$	− 604622.i	− 0.738221i
$906$	0	0
$907$	969098.	1.17802	0.589010	−	0.808125i	$-0.299518\pi$
$907$	969098.	1.17802	0.589010	+	0.808125i	$0.299518\pi$
$908$	1.06582e6i	1.29274i
$909$	0	0
$910$	−3.48936e6	−4.21369
$911$	− 1.15088e6i	− 1.38673i	−0.720587	−	0.693365i	$-0.756127\pi$
$911$	− 1.15088e6i	− 1.38673i	0.720587	−	0.693365i	$-0.243873\pi$
$912$	0	0
$913$	110512.	0.132577
$914$	− 487830.i	− 0.583950i
$915$	0	0
$916$	751657.	0.895836
$917$	− 144667.i	− 0.172040i
$918$	0	0
$919$	−357872.	−0.423737	−0.211869	−	0.977298i	$-0.567955\pi$
$919$	−357872.	−0.423737	−0.211869	+	0.977298i	$0.567955\pi$
$920$	140660.i	0.166186i
$921$	0	0
$922$	−181807.	−0.213869
$923$	584064.i	0.685578i
$924$	0	0
$925$	688991.	0.805249
$926$	− 1.98870e6i	− 2.31925i
$927$	0	0
$928$	1.10666e6	1.28505
$929$	1.34427e6i	1.55760i	0.627275	+	0.778798i	$0.284170\pi$
$929$	1.34427e6i	1.55760i	−0.627275	+	0.778798i	$0.715830\pi$
$930$	0	0
$931$	297832.	0.343614
$932$	1.08190e6i	1.24554i
$933$	0	0
$934$	−922573.	−1.05757
$935$	− 1.09348e6i	− 1.25079i
$936$	0	0
$937$	203046.	0.231268	0.115634	−	0.993292i	$-0.463110\pi$
$937$	203046.	0.231268	0.115634	+	0.993292i	$0.463110\pi$
$938$	− 1.69236e6i	− 1.92348i
$939$	0	0
$940$	1.01710e6	1.15109
$941$	− 1.68586e6i	− 1.90389i	−0.306261	−	0.951947i	$-0.599078\pi$
$941$	− 1.68586e6i	− 1.90389i	0.306261	−	0.951947i	$-0.400922\pi$
$942$	0	0
$943$	−219991.	−0.247389
$944$	− 133925.i	− 0.150285i
$945$	0	0
$946$	−316925.	−0.354139
$947$	− 1.15952e6i	− 1.29294i	−0.762939	−	0.646470i	$-0.776245\pi$
$947$	− 1.15952e6i	− 1.29294i	0.762939	−	0.646470i	$-0.223755\pi$
$948$	0	0
$949$	−470735.	−0.522690
$950$	− 1.29228e6i	− 1.43188i
$951$	0	0
$952$	523042.	0.577115
$953$	− 418205.i	− 0.460472i	−0.973135	−	0.230236i	$-0.926050\pi$
$953$	− 418205.i	− 0.460472i	0.973135	−	0.230236i	$-0.0739499\pi$
$954$	0	0
$955$	525576.	0.576274
$956$	1.26171e6i	1.38053i
$957$	0	0
$958$	−2.30764e6	−2.51441
$959$	− 533199.i	− 0.579765i
$960$	0	0
$961$	1.94945e6	2.11089
$962$	1.50448e6i	1.62568i
$963$	0	0
$964$	779306.	0.838598
$965$	1.91306e6i	2.05435i
$966$	0	0
$967$	−7004.40	−0.00749063	−0.00374531	−	0.999993i	$-0.501192\pi$
$967$	−7004.40	−0.00749063	−0.00374531	+	0.999993i	$0.501192\pi$
$968$	193496.i	0.206500i
$969$	0	0
$970$	−2.40927e6	−2.56060
$971$	360366.i	0.382213i	0.981569	+	0.191107i	$0.0612077\pi$
$971$	360366.i	0.382213i	−0.981569	+	0.191107i	$0.938792\pi$
$972$	0	0
$973$	168610.	0.178098
$974$	1.04501e6i	1.10155i
$975$	0	0
$976$	2.16281e6	2.27049
$977$	197794.i	0.207216i	0.994618	+	0.103608i	$0.0330388\pi$
$977$	197794.i	0.207216i	−0.994618	+	0.103608i	$0.966961\pi$
$978$	0	0
$979$	249963.	0.260802
$980$	− 444738.i	− 0.463076i
$981$	0	0
$982$	1.86052e6	1.92935
$983$	1.13695e6i	1.17661i	0.808639	+	0.588306i	$0.200205\pi$
$983$	1.13695e6i	1.17661i	−0.808639	+	0.588306i	$0.799795\pi$
$984$	0	0
$985$	−1.13980e6	−1.17478
$986$	− 2.47695e6i	− 2.54779i
$987$	0	0
$988$	1.26215e6	1.29300
$989$	− 254778.i	− 0.260477i
$990$	0	0
$991$	483814.	0.492642	0.246321	−	0.969188i	$-0.420778\pi$
$991$	483814.	0.492642	0.246321	+	0.969188i	$0.420778\pi$
$992$	− 2.24967e6i	− 2.28610i
$993$	0	0
$994$	−597459.	−0.604694
$995$	− 1.39318e6i	− 1.40722i
$996$	0	0
$997$	1.05706e6	1.06343	0.531716	−	0.846923i	$-0.321547\pi$
$997$	1.05706e6	1.06343	0.531716	+	0.846923i	$0.321547\pi$
$998$	1.48488e6i	1.49084i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.5.b.a.296.60	yes	76
3.2	odd	2	inner	531.5.b.a.296.17	✓	76

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
531.5.b.a.296.17	✓	76	3.2	odd	2	inner
531.5.b.a.296.60	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)

\(f(q)\)	\(=\)	\(q+5.38034i q^{2} -12.9480 q^{4} +37.0493i q^{5} -57.6896 q^{7} +16.4207i q^{8} +O(q^{10})\)	\(q+5.38034i q^{2} -12.9480 q^{4} +37.0493i q^{5} -57.6896 q^{7} +16.4207i q^{8} -199.337 q^{10} +53.4543i q^{11} -303.430 q^{13} -310.389i q^{14} -295.517 q^{16} +552.137i q^{17} +321.254 q^{19} -479.714i q^{20} -287.602 q^{22} -231.205i q^{23} -747.647 q^{25} -1632.56i q^{26} +746.966 q^{28} +833.800i q^{29} +1694.99 q^{31} -1327.25i q^{32} -2970.68 q^{34} -2137.36i q^{35} -921.546 q^{37} +1728.46i q^{38} -608.375 q^{40} -951.495i q^{41} +1101.95 q^{43} -692.127i q^{44} +1243.96 q^{46} +2120.22i q^{47} +927.090 q^{49} -4022.59i q^{50} +3928.82 q^{52} +1209.43i q^{53} -1980.44 q^{55} -947.304i q^{56} -4486.12 q^{58} +453.188i q^{59} -7318.74 q^{61} +9119.59i q^{62} +2412.78 q^{64} -11241.9i q^{65} +5452.39 q^{67} -7149.08i q^{68} +11499.7 q^{70} -1924.87i q^{71} +1551.38 q^{73} -4958.22i q^{74} -4159.60 q^{76} -3083.76i q^{77} +9306.87 q^{79} -10948.7i q^{80} +5119.36 q^{82} -2067.41i q^{83} -20456.3 q^{85} +5928.89i q^{86} -877.758 q^{88} -4676.20i q^{89} +17504.8 q^{91} +2993.65i q^{92} -11407.5 q^{94} +11902.2i q^{95} +12086.4 q^{97} +4988.05i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 76 q - 632 q^{4} - 48 q^{7}+O(q^{10}) \) 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

Related objects

Downloads

Learn more