LMFDB - Embedded newform 475.4.a.g.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [475,4,Mod(1,475)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("475.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3,11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.0259072527$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1304.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 2 $ x^3 - 11*x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.182370$ of defining polynomial
Character	$\chi$	$=$	475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.817630 q^{2} -0.0745555 q^{3} -7.33148 q^{4} -0.0609588 q^{6} -12.0216 q^{7} -12.5355 q^{8} -26.9944 q^{9} -60.0987 q^{11} +0.546602 q^{12} +80.2669 q^{13} -9.82925 q^{14} +48.4025 q^{16} -17.0926 q^{17} -22.0715 q^{18} -19.0000 q^{19} +0.896278 q^{21} -49.1385 q^{22} +149.583 q^{23} +0.934589 q^{24} +65.6287 q^{26} +4.02558 q^{27} +88.1364 q^{28} +150.128 q^{29} +22.1422 q^{31} +139.859 q^{32} +4.48068 q^{33} -13.9754 q^{34} +197.909 q^{36} -68.7802 q^{37} -15.5350 q^{38} -5.98434 q^{39} -35.3560 q^{41} +0.732825 q^{42} -135.309 q^{43} +440.612 q^{44} +122.304 q^{46} -204.430 q^{47} -3.60867 q^{48} -198.480 q^{49} +1.27435 q^{51} -588.475 q^{52} +371.497 q^{53} +3.29144 q^{54} +150.697 q^{56} +1.41655 q^{57} +122.749 q^{58} +655.426 q^{59} +389.512 q^{61} +18.1042 q^{62} +324.517 q^{63} -272.866 q^{64} +3.66354 q^{66} +450.867 q^{67} +125.314 q^{68} -11.1522 q^{69} +453.060 q^{71} +338.388 q^{72} +1215.57 q^{73} -56.2368 q^{74} +139.298 q^{76} +722.484 q^{77} -4.89298 q^{78} -1281.35 q^{79} +728.550 q^{81} -28.9082 q^{82} -792.205 q^{83} -6.57105 q^{84} -110.632 q^{86} -11.1929 q^{87} +753.366 q^{88} +494.903 q^{89} -964.940 q^{91} -1096.67 q^{92} -1.65082 q^{93} -167.148 q^{94} -10.4273 q^{96} +205.030 q^{97} -162.284 q^{98} +1622.33 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 11 q^{3} + q^{4} + 25 q^{6} + 5 q^{7} + 27 q^{8} - 12 q^{9} - 4 q^{11} + 75 q^{12} + 125 q^{13} - 93 q^{14} - 7 q^{16} + 119 q^{17} + 138 q^{18} - 57 q^{19} + 35 q^{21} + 88 q^{22} + 129 q^{23}+ \cdots + 2540 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 + 11 * q^3 + q^4 + 25 * q^6 + 5 * q^7 + 27 * q^8 - 12 * q^9 - 4 * q^11 + 75 * q^12 + 125 * q^13 - 93 * q^14 - 7 * q^16 + 119 * q^17 + 138 * q^18 - 57 * q^19 + 35 * q^21 + 88 * q^22 + 129 * q^23 + 201 * q^24 + 279 * q^26 - 127 * q^27 - 63 * q^28 + 37 * q^29 + 90 * q^31 - 221 * q^32 + 360 * q^33 - 31 * q^34 + 574 * q^36 + 250 * q^37 - 57 * q^38 + 339 * q^39 - 94 * q^41 - 415 * q^42 + 210 * q^43 + 840 * q^44 + 391 * q^46 - 524 * q^47 - 301 * q^48 - 268 * q^49 + 657 * q^51 - 43 * q^52 + 1417 * q^53 + 403 * q^54 + 635 * q^56 - 209 * q^57 + 907 * q^58 + 15 * q^59 - 912 * q^61 - 1586 * q^62 - 214 * q^63 - 559 * q^64 + 1176 * q^66 - 237 * q^67 + 331 * q^68 + 47 * q^69 - 124 * q^71 + 426 * q^72 + 2457 * q^73 + 670 * q^74 - 19 * q^76 + 848 * q^77 + 1573 * q^78 - 98 * q^79 - 81 * q^81 - 1358 * q^82 + 454 * q^83 - 853 * q^84 + 1362 * q^86 - 101 * q^87 + 1752 * q^88 - 62 * q^89 - 1347 * q^91 - 547 * q^92 - 618 * q^93 - 3056 * q^94 - 2207 * q^96 + 1730 * q^97 - 1970 * q^98 + 2540 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0.817630	0.289076	0.144538	−	0.989499i	$-0.453830\pi$
$2$	0.817630	0.289076	0.144538	+	0.989499i	$0.453830\pi$
$3$	−0.0745555	−0.0143482	−0.00717410	−	0.999974i	$-0.502284\pi$
$3$	−0.0745555	−0.0143482	−0.00717410	+	0.999974i	$0.502284\pi$
$4$	−7.33148	−0.916435
$5$	0	0
$5$	0	0
$6$	−0.0609588	−0.00414772
$7$	−12.0216	−0.649107	−0.324554	−	0.945867i	$-0.605214\pi$
$7$	−12.0216	−0.649107	−0.324554	+	0.945867i	$0.605214\pi$
$8$	−12.5355	−0.553995
$9$	−26.9944	−0.999794
$10$	0	0
$11$	−60.0987	−1.64731	−0.823656	−	0.567090i	$-0.808069\pi$
$11$	−60.0987	−1.64731	−0.823656	+	0.567090i	$0.808069\pi$
$12$	0.546602	0.0131492
$13$	80.2669	1.71246	0.856232	−	0.516591i	$-0.172799\pi$
$13$	80.2669	1.71246	0.856232	+	0.516591i	$0.172799\pi$
$14$	−9.82925	−0.187641
$15$	0	0
$16$	48.4025	0.756288
$17$	−17.0926	−0.243857	−0.121928	−	0.992539i	$-0.538908\pi$
$17$	−17.0926	−0.243857	−0.121928	+	0.992539i	$0.538908\pi$
$18$	−22.0715	−0.289016
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	0.896278	0.00931352
$22$	−49.1385	−0.476198
$23$	149.583	1.35610	0.678049	−	0.735017i	$-0.262826\pi$
$23$	149.583	1.35610	0.678049	+	0.735017i	$0.262826\pi$
$24$	0.934589	0.00794884
$25$	0	0
$26$	65.6287	0.495032
$27$	4.02558	0.0286935
$28$	88.1364	0.594865
$29$	150.128	0.961314	0.480657	−	0.876909i	$-0.340398\pi$
$29$	150.128	0.961314	0.480657	+	0.876909i	$0.340398\pi$
$30$	0	0
$31$	22.1422	0.128286	0.0641429	−	0.997941i	$-0.479569\pi$
$31$	22.1422	0.128286	0.0641429	+	0.997941i	$0.479569\pi$
$32$	139.859	0.772620
$33$	4.48068	0.0236360
$34$	−13.9754	−0.0704931
$35$	0	0
$36$	197.909	0.916246
$37$	−68.7802	−0.305605	−0.152803	−	0.988257i	$-0.548830\pi$
$37$	−68.7802	−0.305605	−0.152803	+	0.988257i	$0.548830\pi$
$38$	−15.5350	−0.0663186
$39$	−5.98434	−0.0245708
$40$	0	0
$41$	−35.3560	−0.134675	−0.0673376	−	0.997730i	$-0.521450\pi$
$41$	−35.3560	−0.134675	−0.0673376	+	0.997730i	$0.521450\pi$
$42$	0.732825	0.00269232
$43$	−135.309	−0.479869	−0.239935	−	0.970789i	$-0.577126\pi$
$43$	−135.309	−0.479869	−0.239935	+	0.970789i	$0.577126\pi$
$44$	440.612	1.50965
$45$	0	0
$46$	122.304	0.392015
$47$	−204.430	−0.634450	−0.317225	−	0.948350i	$-0.602751\pi$
$47$	−204.430	−0.634450	−0.317225	+	0.948350i	$0.602751\pi$
$48$	−3.60867	−0.0108514
$49$	−198.480	−0.578660
$50$	0	0
$51$	1.27435	0.00349891
$52$	−588.475	−1.56936
$53$	371.497	0.962813	0.481407	−	0.876497i	$-0.340126\pi$
$53$	371.497	0.962813	0.481407	+	0.876497i	$0.340126\pi$
$54$	3.29144	0.00829459
$55$	0	0
$56$	150.697	0.359602
$57$	1.41655	0.00329170
$58$	122.749	0.277893
$59$	655.426	1.44626	0.723128	−	0.690714i	$-0.242704\pi$
$59$	655.426	1.44626	0.723128	+	0.690714i	$0.242704\pi$
$60$	0	0
$61$	389.512	0.817573	0.408786	−	0.912630i	$-0.365952\pi$
$61$	389.512	0.817573	0.408786	+	0.912630i	$0.365952\pi$
$62$	18.1042	0.0370844
$63$	324.517	0.648974
$64$	−272.866	−0.532942
$65$	0	0
$66$	3.66354	0.00683259
$67$	450.867	0.822122	0.411061	−	0.911608i	$-0.365158\pi$
$67$	450.867	0.822122	0.411061	+	0.911608i	$0.365158\pi$
$68$	125.314	0.223479
$69$	−11.1522	−0.0194576
$70$	0	0
$71$	453.060	0.757301	0.378651	−	0.925540i	$-0.376388\pi$
$71$	453.060	0.757301	0.378651	+	0.925540i	$0.376388\pi$
$72$	338.388	0.553881
$73$	1215.57	1.94894	0.974468	−	0.224528i	$-0.0720840\pi$
$73$	1215.57	1.94894	0.974468	+	0.224528i	$0.0720840\pi$
$74$	−56.2368	−0.0883431
$75$	0	0
$76$	139.298	0.210245
$77$	722.484	1.06928
$78$	−4.89298	−0.00710283
$79$	−1281.35	−1.82485	−0.912425	−	0.409245i	$-0.865792\pi$
$79$	−1281.35	−1.82485	−0.912425	+	0.409245i	$0.865792\pi$
$80$	0	0
$81$	728.550	0.999382
$82$	−28.9082	−0.0389314
$83$	−792.205	−1.04766	−0.523830	−	0.851823i	$-0.675497\pi$
$83$	−792.205	−1.04766	−0.523830	+	0.851823i	$0.675497\pi$
$84$	−6.57105	−0.00853524
$85$	0	0
$86$	−110.632	−0.138719
$87$	−11.1929	−0.0137931
$88$	753.366	0.912603
$89$	494.903	0.589434	0.294717	−	0.955585i	$-0.404775\pi$
$89$	494.903	0.589434	0.294717	+	0.955585i	$0.404775\pi$
$90$	0	0
$91$	−964.940	−1.11157
$92$	−1096.67	−1.24278
$93$	−1.65082	−0.00184067
$94$	−167.148	−0.183404
$95$	0	0
$96$	−10.4273	−0.0110857
$97$	205.030	0.214615	0.107308	−	0.994226i	$-0.465777\pi$
$97$	205.030	0.214615	0.107308	+	0.994226i	$0.465777\pi$
$98$	−162.284	−0.167277
$99$	1622.33	1.64697
$100$	0	0
$101$	−228.137	−0.224758	−0.112379	−	0.993665i	$-0.535847\pi$
$101$	−228.137	−0.224758	−0.112379	+	0.993665i	$0.535847\pi$
$102$	1.04194	0.00101145
$103$	5.89094	0.00563545	0.00281772	−	0.999996i	$-0.499103\pi$
$103$	5.89094	0.00563545	0.00281772	+	0.999996i	$0.499103\pi$
$104$	−1006.18	−0.948698
$105$	0	0
$106$	303.748	0.278326
$107$	−1515.79	−1.36951	−0.684753	−	0.728775i	$-0.740090\pi$
$107$	−1515.79	−1.36951	−0.684753	+	0.728775i	$0.740090\pi$
$108$	−29.5135	−0.0262957
$109$	−1811.18	−1.59156	−0.795779	−	0.605587i	$-0.792938\pi$
$109$	−1811.18	−1.59156	−0.795779	+	0.605587i	$0.792938\pi$
$110$	0	0
$111$	5.12794	0.00438489
$112$	−581.877	−0.490912
$113$	2118.32	1.76349	0.881747	−	0.471722i	$-0.156367\pi$
$113$	2118.32	1.76349	0.881747	+	0.471722i	$0.156367\pi$
$114$	1.15822	0.000951553 0
$115$	0	0
$116$	−1100.66	−0.880982
$117$	−2166.76	−1.71211
$118$	535.896	0.418078
$119$	205.481	0.158289
$120$	0	0
$121$	2280.85	1.71364
$122$	318.477	0.236341
$123$	2.63599	0.00193235
$124$	−162.335	−0.117566
$125$	0	0
$126$	265.335	0.187603
$127$	1533.99	1.07181	0.535903	−	0.844280i	$-0.319971\pi$
$127$	1533.99	1.07181	0.535903	+	0.844280i	$0.319971\pi$
$128$	−1341.98	−0.926681
$129$	10.0880	0.00688526
$130$	0	0
$131$	1015.30	0.677153	0.338576	−	0.940939i	$-0.390055\pi$
$131$	1015.30	0.677153	0.338576	+	0.940939i	$0.390055\pi$
$132$	−32.8500	−0.0216608
$133$	228.411	0.148915
$134$	368.643	0.237656
$135$	0	0
$136$	214.264	0.135096
$137$	−140.178	−0.0874176	−0.0437088	−	0.999044i	$-0.513917\pi$
$137$	−140.178	−0.0874176	−0.0437088	+	0.999044i	$0.513917\pi$
$138$	−9.11841	−0.00562472
$139$	2624.32	1.60138	0.800690	−	0.599079i	$-0.204467\pi$
$139$	2624.32	1.60138	0.800690	+	0.599079i	$0.204467\pi$
$140$	0	0
$141$	15.2414	0.00910323
$142$	370.436	0.218918
$143$	−4823.94	−2.82096
$144$	−1306.60	−0.756133
$145$	0	0
$146$	993.891	0.563390
$147$	14.7978	0.00830273
$148$	504.261	0.280067
$149$	−1110.97	−0.610836	−0.305418	−	0.952218i	$-0.598796\pi$
$149$	−1110.97	−0.610836	−0.305418	+	0.952218i	$0.598796\pi$
$150$	0	0
$151$	−1483.33	−0.799414	−0.399707	−	0.916643i	$-0.630888\pi$
$151$	−1483.33	−0.799414	−0.399707	+	0.916643i	$0.630888\pi$
$152$	238.174	0.127095
$153$	461.405	0.243807
$154$	590.725	0.309104
$155$	0	0
$156$	43.8741	0.0225175
$157$	2152.02	1.09395	0.546974	−	0.837150i	$-0.315780\pi$
$157$	2152.02	1.09395	0.546974	+	0.837150i	$0.315780\pi$
$158$	−1047.67	−0.527520
$159$	−27.6972	−0.0138146
$160$	0	0
$161$	−1798.23	−0.880253
$162$	595.684	0.288897
$163$	−1056.21	−0.507540	−0.253770	−	0.967265i	$-0.581671\pi$
$163$	−1056.21	−0.507540	−0.253770	+	0.967265i	$0.581671\pi$
$164$	259.212	0.123421
$165$	0	0
$166$	−647.731	−0.302854
$167$	1147.86	0.531881	0.265941	−	0.963989i	$-0.414317\pi$
$167$	1147.86	0.531881	0.265941	+	0.963989i	$0.414317\pi$
$168$	−11.2353	−0.00515965
$169$	4245.78	1.93254
$170$	0	0
$171$	512.894	0.229369
$172$	992.013	0.439769
$173$	−1945.74	−0.855098	−0.427549	−	0.903992i	$-0.640623\pi$
$173$	−1945.74	−0.855098	−0.427549	+	0.903992i	$0.640623\pi$
$174$	−9.15164	−0.00398726
$175$	0	0
$176$	−2908.92	−1.24584
$177$	−48.8656	−0.0207512
$178$	404.648	0.170391
$179$	−2159.46	−0.901707	−0.450854	−	0.892598i	$-0.648880\pi$
$179$	−2159.46	−0.901707	−0.450854	+	0.892598i	$0.648880\pi$
$180$	0	0
$181$	−2082.08	−0.855026	−0.427513	−	0.904009i	$-0.640610\pi$
$181$	−2082.08	−0.855026	−0.427513	+	0.904009i	$0.640610\pi$
$182$	−788.964	−0.321329
$183$	−29.0403	−0.0117307
$184$	−1875.10	−0.751272
$185$	0	0
$186$	−1.34976	−0.000532094 0
$187$	1027.24	0.401708
$188$	1498.77	0.581433
$189$	−48.3941	−0.0186251
$190$	0	0
$191$	−1206.70	−0.457138	−0.228569	−	0.973528i	$-0.573405\pi$
$191$	−1206.70	−0.457138	−0.228569	+	0.973528i	$0.573405\pi$
$192$	20.3437	0.00764677
$193$	−1217.50	−0.454080	−0.227040	−	0.973885i	$-0.572905\pi$
$193$	−1217.50	−0.454080	−0.227040	+	0.973885i	$0.572905\pi$
$194$	167.639	0.0620401
$195$	0	0
$196$	1455.15	0.530304
$197$	4748.00	1.71716	0.858582	−	0.512677i	$-0.171346\pi$
$197$	4748.00	1.71716	0.858582	+	0.512677i	$0.171346\pi$
$198$	1326.47	0.476100
$199$	1095.54	0.390254	0.195127	−	0.980778i	$-0.437488\pi$
$199$	1095.54	0.390254	0.195127	+	0.980778i	$0.437488\pi$
$200$	0	0
$201$	−33.6146	−0.0117960
$202$	−186.532	−0.0649720
$203$	−1804.79	−0.623996
$204$	−9.34285	−0.00320652
$205$	0	0
$206$	4.81661	0.00162907
$207$	−4037.92	−1.35582
$208$	3885.12	1.29512
$209$	1141.87	0.377919
$210$	0	0
$211$	−2742.32	−0.894737	−0.447368	−	0.894350i	$-0.647639\pi$
$211$	−2742.32	−0.894737	−0.447368	+	0.894350i	$0.647639\pi$
$212$	−2723.63	−0.882356
$213$	−33.7781	−0.0108659
$214$	−1239.36	−0.395891
$215$	0	0
$216$	−50.4626	−0.0158960
$217$	−266.186	−0.0832713
$218$	−1480.88	−0.460081
$219$	−90.6277	−0.0279637
$220$	0	0
$221$	−1371.97	−0.417596
$222$	4.19276	0.00126757
$223$	−1017.50	−0.305546	−0.152773	−	0.988261i	$-0.548820\pi$
$223$	−1017.50	−0.305546	−0.152773	+	0.988261i	$0.548820\pi$
$224$	−1681.34	−0.501513
$225$	0	0
$226$	1732.00	0.509784
$227$	6410.55	1.87438	0.937188	−	0.348826i	$-0.113419\pi$
$227$	6410.55	1.87438	0.937188	+	0.348826i	$0.113419\pi$
$228$	−10.3854	−0.00301663
$229$	−1429.24	−0.412432	−0.206216	−	0.978506i	$-0.566115\pi$
$229$	−1429.24	−0.412432	−0.206216	+	0.978506i	$0.566115\pi$
$230$	0	0
$231$	−53.8651	−0.0153423
$232$	−1881.93	−0.532564
$233$	882.406	0.248104	0.124052	−	0.992276i	$-0.460411\pi$
$233$	882.406	0.248104	0.124052	+	0.992276i	$0.460411\pi$
$234$	−1771.61	−0.494931
$235$	0	0
$236$	−4805.24	−1.32540
$237$	95.5316	0.0261833
$238$	168.008	0.0457576
$239$	1734.23	0.469364	0.234682	−	0.972072i	$-0.424595\pi$
$239$	1734.23	0.469364	0.234682	+	0.972072i	$0.424595\pi$
$240$	0	0
$241$	−6163.94	−1.64753	−0.823764	−	0.566933i	$-0.808130\pi$
$241$	−6163.94	−1.64753	−0.823764	+	0.566933i	$0.808130\pi$
$242$	1864.89	0.495371
$243$	−163.008	−0.0430328
$244$	−2855.70	−0.749253
$245$	0	0
$246$	2.15526	0.000558596 0
$247$	−1525.07	−0.392866
$248$	−277.564	−0.0710698
$249$	59.0632	0.0150320
$250$	0	0
$251$	6996.16	1.75934	0.879668	−	0.475588i	$-0.157765\pi$
$251$	6996.16	1.75934	0.879668	+	0.475588i	$0.157765\pi$
$252$	−2379.19	−0.594742
$253$	−8989.75	−2.23392
$254$	1254.23	0.309833
$255$	0	0
$256$	1085.69	0.265061
$257$	381.354	0.0925612	0.0462806	−	0.998928i	$-0.485263\pi$
$257$	381.354	0.0925612	0.0462806	+	0.998928i	$0.485263\pi$
$258$	8.24825	0.00199036
$259$	826.850	0.198371
$260$	0	0
$261$	−4052.63	−0.961116
$262$	830.138	0.195749
$263$	−5902.60	−1.38392	−0.691958	−	0.721938i	$-0.743252\pi$
$263$	−5902.60	−1.38392	−0.691958	+	0.721938i	$0.743252\pi$
$264$	−56.1675	−0.0130942
$265$	0	0
$266$	186.756	0.0430479
$267$	−36.8977	−0.00845732
$268$	−3305.52	−0.753422
$269$	5087.44	1.15311	0.576555	−	0.817059i	$-0.304397\pi$
$269$	5087.44	1.15311	0.576555	+	0.817059i	$0.304397\pi$
$270$	0	0
$271$	5820.63	1.30472	0.652358	−	0.757911i	$-0.273780\pi$
$271$	5820.63	1.30472	0.652358	+	0.757911i	$0.273780\pi$
$272$	−827.324	−0.184426
$273$	71.9415	0.0159491
$274$	−114.614	−0.0252703
$275$	0	0
$276$	81.7625	0.0178316
$277$	−2109.79	−0.457635	−0.228817	−	0.973469i	$-0.573486\pi$
$277$	−2109.79	−0.457635	−0.228817	+	0.973469i	$0.573486\pi$
$278$	2145.72	0.462920
$279$	−597.717	−0.128259
$280$	0	0
$281$	−4144.75	−0.879912	−0.439956	−	0.898019i	$-0.645006\pi$
$281$	−4144.75	−0.879912	−0.439956	+	0.898019i	$0.645006\pi$
$282$	12.4618	0.00263152
$283$	1817.95	0.381859	0.190929	−	0.981604i	$-0.438850\pi$
$283$	1817.95	0.381859	0.190929	+	0.981604i	$0.438850\pi$
$284$	−3321.60	−0.694017
$285$	0	0
$286$	−3944.20	−0.815473
$287$	425.037	0.0874187
$288$	−3775.42	−0.772461
$289$	−4620.84	−0.940534
$290$	0	0
$291$	−15.2861	−0.00307934
$292$	−8911.96	−1.78607
$293$	4983.10	0.993569	0.496785	−	0.867874i	$-0.334514\pi$
$293$	4983.10	0.993569	0.496785	+	0.867874i	$0.334514\pi$
$294$	12.0991	0.00240012
$295$	0	0
$296$	862.193	0.169304
$297$	−241.932	−0.0472671
$298$	−908.367	−0.176578
$299$	12006.6	2.32227
$300$	0	0
$301$	1626.63	0.311487
$302$	−1212.81	−0.231091
$303$	17.0089	0.00322487
$304$	−919.647	−0.173504
$305$	0	0
$306$	377.259	0.0704786
$307$	5233.35	0.972909	0.486455	−	0.873706i	$-0.338290\pi$
$307$	5233.35	0.972909	0.486455	+	0.873706i	$0.338290\pi$
$308$	−5296.88	−0.979928
$309$	−0.439202	−8.08586e−5 0
$310$	0	0
$311$	8448.98	1.54051	0.770254	−	0.637738i	$-0.220130\pi$
$311$	8448.98	1.54051	0.770254	+	0.637738i	$0.220130\pi$
$312$	75.0166	0.0136121
$313$	10788.0	1.94815	0.974076	−	0.226223i	$-0.0726377\pi$
$313$	10788.0	1.94815	0.974076	+	0.226223i	$0.0726377\pi$
$314$	1759.56	0.316234
$315$	0	0
$316$	9394.19	1.67236
$317$	2813.67	0.498521	0.249261	−	0.968436i	$-0.419812\pi$
$317$	2813.67	0.498521	0.249261	+	0.968436i	$0.419812\pi$
$318$	−22.6460	−0.00399348
$319$	−9022.50	−1.58358
$320$	0	0
$321$	113.011	0.0196500
$322$	−1470.29	−0.254460
$323$	324.759	0.0559446
$324$	−5341.35	−0.915869
$325$	0	0
$326$	−863.592	−0.146718
$327$	135.034	0.0228360
$328$	443.205	0.0746095
$329$	2457.58	0.411826
$330$	0	0
$331$	−2235.68	−0.371252	−0.185626	−	0.982620i	$-0.559431\pi$
$331$	−2235.68	−0.371252	−0.185626	+	0.982620i	$0.559431\pi$
$332$	5808.04	0.960113
$333$	1856.68	0.305542
$334$	938.527	0.153754
$335$	0	0
$336$	43.3821	0.00704371
$337$	10497.1	1.69678	0.848388	−	0.529375i	$-0.177574\pi$
$337$	10497.1	1.69678	0.848388	+	0.529375i	$0.177574\pi$
$338$	3471.48	0.558650
$339$	−157.932	−0.0253030
$340$	0	0
$341$	−1330.72	−0.211327
$342$	419.358	0.0663049
$343$	6509.48	1.02472
$344$	1696.16	0.265845
$345$	0	0
$346$	−1590.90	−0.247188
$347$	452.769	0.0700459	0.0350229	−	0.999387i	$-0.488850\pi$
$347$	452.769	0.0700459	0.0350229	+	0.999387i	$0.488850\pi$
$348$	82.0604	0.0126405
$349$	1836.22	0.281635	0.140818	−	0.990036i	$-0.455027\pi$
$349$	1836.22	0.281635	0.140818	+	0.990036i	$0.455027\pi$
$350$	0	0
$351$	323.121	0.0491365
$352$	−8405.35	−1.27275
$353$	1216.97	0.183492	0.0917460	−	0.995782i	$-0.470755\pi$
$353$	1216.97	0.183492	0.0917460	+	0.995782i	$0.470755\pi$
$354$	−39.9540	−0.00599867
$355$	0	0
$356$	−3628.37	−0.540178
$357$	−15.3197	−0.00227117
$358$	−1765.64	−0.260662
$359$	−5614.22	−0.825369	−0.412684	−	0.910874i	$-0.635409\pi$
$359$	−5614.22	−0.825369	−0.412684	+	0.910874i	$0.635409\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−1702.37	−0.247167
$363$	−170.050	−0.0245876
$364$	7074.44	1.01868
$365$	0	0
$366$	−23.7442	−0.00339107
$367$	−10320.6	−1.46793	−0.733967	−	0.679185i	$-0.762333\pi$
$367$	−10320.6	−1.46793	−0.733967	+	0.679185i	$0.762333\pi$
$368$	7240.19	1.02560
$369$	954.417	0.134648
$370$	0	0
$371$	−4466.01	−0.624969
$372$	12.1030	0.00168686
$373$	−12060.8	−1.67422	−0.837108	−	0.547037i	$-0.815756\pi$
$373$	−12060.8	−1.67422	−0.837108	+	0.547037i	$0.815756\pi$
$374$	839.905	0.116124
$375$	0	0
$376$	2562.63	0.351483
$377$	12050.3	1.64622
$378$	−39.5685	−0.00538408
$379$	3928.23	0.532400	0.266200	−	0.963918i	$-0.414232\pi$
$379$	3928.23	0.532400	0.266200	+	0.963918i	$0.414232\pi$
$380$	0	0
$381$	−114.367	−0.0153785
$382$	−986.631	−0.132148
$383$	8355.64	1.11476	0.557380	−	0.830257i	$-0.311807\pi$
$383$	8355.64	1.11476	0.557380	+	0.830257i	$0.311807\pi$
$384$	100.052	0.0132962
$385$	0	0
$386$	−995.464	−0.131264
$387$	3652.58	0.479770
$388$	−1503.18	−0.196681
$389$	−3563.21	−0.464427	−0.232213	−	0.972665i	$-0.574597\pi$
$389$	−3563.21	−0.464427	−0.232213	+	0.972665i	$0.574597\pi$
$390$	0	0
$391$	−2556.77	−0.330694
$392$	2488.05	0.320575
$393$	−75.6960	−0.00971592
$394$	3882.11	0.496391
$395$	0	0
$396$	−11894.1	−1.50934
$397$	−8044.17	−1.01694	−0.508470	−	0.861080i	$-0.669789\pi$
$397$	−8044.17	−1.01694	−0.508470	+	0.861080i	$0.669789\pi$
$398$	895.743	0.112813
$399$	−17.0293	−0.00213667
$400$	0	0
$401$	14020.8	1.74605	0.873026	−	0.487674i	$-0.162154\pi$
$401$	14020.8	1.74605	0.873026	+	0.487674i	$0.162154\pi$
$402$	−27.4843	−0.00340993
$403$	1777.29	0.219685
$404$	1672.58	0.205976
$405$	0	0
$406$	−1475.65	−0.180382
$407$	4133.60	0.503427
$408$	−15.9746	−0.00193838
$409$	−3716.96	−0.449368	−0.224684	−	0.974432i	$-0.572135\pi$
$409$	−3716.96	−0.449368	−0.224684	+	0.974432i	$0.572135\pi$
$410$	0	0
$411$	10.4510	0.00125429
$412$	−43.1893	−0.00516452
$413$	−7879.29	−0.938776
$414$	−3301.52	−0.391935
$415$	0	0
$416$	11226.1	1.32308
$417$	−195.657	−0.0229769
$418$	933.632	0.109247
$419$	−10383.1	−1.21061	−0.605306	−	0.795992i	$-0.706949\pi$
$419$	−10383.1	−1.21061	−0.605306	+	0.795992i	$0.706949\pi$
$420$	0	0
$421$	15714.9	1.81923	0.909616	−	0.415449i	$-0.136376\pi$
$421$	15714.9	1.81923	0.909616	+	0.415449i	$0.136376\pi$
$422$	−2242.21	−0.258647
$423$	5518.47	0.634320
$424$	−4656.90	−0.533394
$425$	0	0
$426$	−27.6180	−0.00314107
$427$	−4682.58	−0.530693
$428$	11113.0	1.25506
$429$	359.651	0.0404758
$430$	0	0
$431$	−555.670	−0.0621013	−0.0310507	−	0.999518i	$-0.509885\pi$
$431$	−555.670	−0.0621013	−0.0310507	+	0.999518i	$0.509885\pi$
$432$	194.848	0.0217005
$433$	13488.9	1.49708	0.748541	−	0.663089i	$-0.230755\pi$
$433$	13488.9	1.49708	0.748541	+	0.663089i	$0.230755\pi$
$434$	−217.642	−0.0240717
$435$	0	0
$436$	13278.6	1.45856
$437$	−2842.08	−0.311110
$438$	−74.1000	−0.00808364
$439$	2122.23	0.230725	0.115363	−	0.993323i	$-0.463197\pi$
$439$	2122.23	0.230725	0.115363	+	0.993323i	$0.463197\pi$
$440$	0	0
$441$	5357.86	0.578541
$442$	−1121.77	−0.120717
$443$	−6803.83	−0.729706	−0.364853	−	0.931065i	$-0.618881\pi$
$443$	−6803.83	−0.729706	−0.364853	+	0.931065i	$0.618881\pi$
$444$	−37.5954	−0.00401846
$445$	0	0
$446$	−831.937	−0.0883259
$447$	82.8292	0.00876440
$448$	3280.30	0.345937
$449$	3337.92	0.350838	0.175419	−	0.984494i	$-0.443872\pi$
$449$	3337.92	0.350838	0.175419	+	0.984494i	$0.443872\pi$
$450$	0	0
$451$	2124.85	0.221852
$452$	−15530.4	−1.61613
$453$	110.590	0.0114702
$454$	5241.46	0.541837
$455$	0	0
$456$	−17.7572	−0.00182359
$457$	−3958.48	−0.405185	−0.202593	−	0.979263i	$-0.564937\pi$
$457$	−3958.48	−0.405185	−0.202593	+	0.979263i	$0.564937\pi$
$458$	−1168.59	−0.119224
$459$	−68.8077	−0.00699709
$460$	0	0
$461$	11702.4	1.18228	0.591142	−	0.806568i	$-0.298677\pi$
$461$	11702.4	1.18228	0.591142	+	0.806568i	$0.298677\pi$
$462$	−44.0418	−0.00443508
$463$	3071.52	0.308306	0.154153	−	0.988047i	$-0.450735\pi$
$463$	3071.52	0.308306	0.154153	+	0.988047i	$0.450735\pi$
$464$	7266.57	0.727031
$465$	0	0
$466$	721.482	0.0717210
$467$	−4483.46	−0.444261	−0.222130	−	0.975017i	$-0.571301\pi$
$467$	−4483.46	−0.444261	−0.222130	+	0.975017i	$0.571301\pi$
$468$	15885.6	1.56904
$469$	−5420.16	−0.533646
$470$	0	0
$471$	−160.445	−0.0156962
$472$	−8216.08	−0.801219
$473$	8131.87	0.790494
$474$	78.1095	0.00756897
$475$	0	0
$476$	−1506.48	−0.145062
$477$	−10028.4	−0.962615
$478$	1417.96	0.135682
$479$	14760.4	1.40798	0.703989	−	0.710211i	$-0.251401\pi$
$479$	14760.4	1.40798	0.703989	+	0.710211i	$0.251401\pi$
$480$	0	0
$481$	−5520.77	−0.523338
$482$	−5039.83	−0.476261
$483$	134.068	0.0126301
$484$	−16722.0	−1.57044
$485$	0	0
$486$	−133.280	−0.0124397
$487$	8898.45	0.827982	0.413991	−	0.910281i	$-0.364134\pi$
$487$	8898.45	0.827982	0.413991	+	0.910281i	$0.364134\pi$
$488$	−4882.73	−0.452932
$489$	78.7465	0.00728229
$490$	0	0
$491$	−6680.85	−0.614058	−0.307029	−	0.951700i	$-0.599335\pi$
$491$	−6680.85	−0.614058	−0.307029	+	0.951700i	$0.599335\pi$
$492$	−19.3257	−0.00177087
$493$	−2566.08	−0.234423
$494$	−1246.95	−0.113568
$495$	0	0
$496$	1071.74	0.0970211
$497$	−5446.53	−0.491570
$498$	48.2919	0.00434540
$499$	13666.0	1.22600	0.613000	−	0.790083i	$-0.289963\pi$
$499$	13666.0	1.22600	0.613000	+	0.790083i	$0.289963\pi$
$500$	0	0
$501$	−85.5794	−0.00763154
$502$	5720.27	0.508582
$503$	2553.01	0.226308	0.113154	−	0.993577i	$-0.463905\pi$
$503$	2553.01	0.226308	0.113154	+	0.993577i	$0.463905\pi$
$504$	−4067.98	−0.359528
$505$	0	0
$506$	−7350.29	−0.645772
$507$	−316.546	−0.0277284
$508$	−11246.4	−0.982240
$509$	10680.5	0.930067	0.465034	−	0.885293i	$-0.346042\pi$
$509$	10680.5	0.930067	0.465034	+	0.885293i	$0.346042\pi$
$510$	0	0
$511$	−14613.2	−1.26507
$512$	11623.5	1.00330
$513$	−76.4860	−0.00658273
$514$	311.807	0.0267572
$515$	0	0
$516$	−73.9600	−0.00630989
$517$	12286.0	1.04514
$518$	676.058	0.0573442
$519$	145.066	0.0122691
$520$	0	0
$521$	4995.01	0.420030	0.210015	−	0.977698i	$-0.432649\pi$
$521$	4995.01	0.420030	0.210015	+	0.977698i	$0.432649\pi$
$522$	−3313.55	−0.277836
$523$	−15406.6	−1.28811	−0.644056	−	0.764979i	$-0.722750\pi$
$523$	−15406.6	−1.28811	−0.644056	+	0.764979i	$0.722750\pi$
$524$	−7443.64	−0.620566
$525$	0	0
$526$	−4826.14	−0.400057
$527$	−378.468	−0.0312834
$528$	216.876	0.0178756
$529$	10208.1	0.839002
$530$	0	0
$531$	−17692.8	−1.44596
$532$	−1674.59	−0.136471
$533$	−2837.92	−0.230627
$534$	−30.1687	−0.00244481
$535$	0	0
$536$	−5651.84	−0.455452
$537$	161.000	0.0129379
$538$	4159.64	0.333336
$539$	11928.4	0.953233
$540$	0	0
$541$	18108.2	1.43906	0.719530	−	0.694461i	$-0.244357\pi$
$541$	18108.2	1.43906	0.719530	+	0.694461i	$0.244357\pi$
$542$	4759.12	0.377162
$543$	155.230	0.0122681
$544$	−2390.56	−0.188409
$545$	0	0
$546$	58.8216	0.00461050
$547$	−5938.17	−0.464164	−0.232082	−	0.972696i	$-0.574554\pi$
$547$	−5938.17	−0.464164	−0.232082	+	0.972696i	$0.574554\pi$
$548$	1027.71	0.0801125
$549$	−10514.7	−0.817405
$550$	0	0
$551$	−2852.44	−0.220541
$552$	139.799	0.0107794
$553$	15403.9	1.18452
$554$	−1725.03	−0.132291
$555$	0	0
$556$	−19240.1	−1.46756
$557$	15724.4	1.19616	0.598081	−	0.801436i	$-0.295930\pi$
$557$	15724.4	1.19616	0.598081	+	0.801436i	$0.295930\pi$
$558$	−488.712	−0.0370767
$559$	−10860.8	−0.821759
$560$	0	0
$561$	−76.5866	−0.00576379
$562$	−3388.88	−0.254361
$563$	−2529.45	−0.189349	−0.0946747	−	0.995508i	$-0.530181\pi$
$563$	−2529.45	−0.189349	−0.0946747	+	0.995508i	$0.530181\pi$
$564$	−111.742	−0.00834251
$565$	0	0
$566$	1486.41	0.110386
$567$	−8758.36	−0.648706
$568$	−5679.33	−0.419541
$569$	13583.9	1.00082	0.500411	−	0.865788i	$-0.333182\pi$
$569$	13583.9	1.00082	0.500411	+	0.865788i	$0.333182\pi$
$570$	0	0
$571$	7889.58	0.578229	0.289114	−	0.957295i	$-0.406639\pi$
$571$	7889.58	0.578229	0.289114	+	0.957295i	$0.406639\pi$
$572$	35366.6	2.58523
$573$	89.9657	0.00655911
$574$	347.524	0.0252707
$575$	0	0
$576$	7365.88	0.532833
$577$	15658.4	1.12975	0.564877	−	0.825175i	$-0.308924\pi$
$577$	15658.4	1.12975	0.564877	+	0.825175i	$0.308924\pi$
$578$	−3778.14	−0.271886
$579$	90.7712	0.00651523
$580$	0	0
$581$	9523.60	0.680044
$582$	−12.4984	−0.000890164 0
$583$	−22326.5	−1.58605
$584$	−15237.8	−1.07970
$585$	0	0
$586$	4074.33	0.287217
$587$	10101.7	0.710296	0.355148	−	0.934810i	$-0.384430\pi$
$587$	10101.7	0.710296	0.355148	+	0.934810i	$0.384430\pi$
$588$	−108.490	−0.00760891
$589$	−420.702	−0.0294308
$590$	0	0
$591$	−353.990	−0.0246382
$592$	−3329.13	−0.231126
$593$	15962.7	1.10541	0.552707	−	0.833376i	$-0.313595\pi$
$593$	15962.7	1.10541	0.552707	+	0.833376i	$0.313595\pi$
$594$	−197.811	−0.0136638
$595$	0	0
$596$	8145.09	0.559792
$597$	−81.6782	−0.00559944
$598$	9816.95	0.671313
$599$	1146.27	0.0781893	0.0390947	−	0.999236i	$-0.487553\pi$
$599$	1146.27	0.0781893	0.0390947	+	0.999236i	$0.487553\pi$
$600$	0	0
$601$	15356.9	1.04230	0.521149	−	0.853466i	$-0.325504\pi$
$601$	15356.9	1.04230	0.521149	+	0.853466i	$0.325504\pi$
$602$	1329.98	0.0900433
$603$	−12170.9	−0.821953
$604$	10875.0	0.732611
$605$	0	0
$606$	13.9070	0.000932232 0
$607$	−4515.38	−0.301933	−0.150967	−	0.988539i	$-0.548239\pi$
$607$	−4515.38	−0.301933	−0.150967	+	0.988539i	$0.548239\pi$
$608$	−2657.32	−0.177251
$609$	134.557	0.00895322
$610$	0	0
$611$	−16409.0	−1.08647
$612$	−3382.78	−0.223433
$613$	9993.86	0.658480	0.329240	−	0.944246i	$-0.393208\pi$
$613$	9993.86	0.658480	0.329240	+	0.944246i	$0.393208\pi$
$614$	4278.95	0.281245
$615$	0	0
$616$	−9056.69	−0.592377
$617$	−17602.9	−1.14857	−0.574285	−	0.818656i	$-0.694720\pi$
$617$	−17602.9	−1.14857	−0.574285	+	0.818656i	$0.694720\pi$
$618$	−0.359105	−2.33743e−5 0
$619$	−19323.1	−1.25470	−0.627350	−	0.778738i	$-0.715860\pi$
$619$	−19323.1	−1.25470	−0.627350	+	0.778738i	$0.715860\pi$
$620$	0	0
$621$	602.159	0.0389111
$622$	6908.14	0.445324
$623$	−5949.54	−0.382606
$624$	−289.657	−0.0185826
$625$	0	0
$626$	8820.56	0.563164
$627$	−85.1330	−0.00542246
$628$	−15777.5	−1.00253
$629$	1175.63	0.0745239
$630$	0	0
$631$	−19173.5	−1.20964	−0.604821	−	0.796362i	$-0.706755\pi$
$631$	−19173.5	−1.20964	−0.604821	+	0.796362i	$0.706755\pi$
$632$	16062.3	1.01096
$633$	204.455	0.0128379
$634$	2300.54	0.144111
$635$	0	0
$636$	203.061	0.0126602
$637$	−15931.4	−0.990934
$638$	−7377.07	−0.457776
$639$	−12230.1	−0.757145
$640$	0	0
$641$	−19069.3	−1.17502	−0.587512	−	0.809215i	$-0.699892\pi$
$641$	−19069.3	−1.17502	−0.587512	+	0.809215i	$0.699892\pi$
$642$	92.4009	0.00568033
$643$	13886.7	0.851690	0.425845	−	0.904796i	$-0.359977\pi$
$643$	13886.7	0.851690	0.425845	+	0.904796i	$0.359977\pi$
$644$	13183.7	0.806695
$645$	0	0
$646$	265.533	0.0161722
$647$	5517.49	0.335263	0.167631	−	0.985850i	$-0.446388\pi$
$647$	5517.49	0.335263	0.167631	+	0.985850i	$0.446388\pi$
$648$	−9132.72	−0.553653
$649$	−39390.2	−2.38244
$650$	0	0
$651$	19.8456	0.00119479
$652$	7743.61	0.465127
$653$	−14758.0	−0.884421	−0.442210	−	0.896911i	$-0.645806\pi$
$653$	−14758.0	−0.884421	−0.442210	+	0.896911i	$0.645806\pi$
$654$	110.408	0.00660134
$655$	0	0
$656$	−1711.32	−0.101853
$657$	−32813.8	−1.94853
$658$	2009.39	0.119049
$659$	−5525.29	−0.326608	−0.163304	−	0.986576i	$-0.552215\pi$
$659$	−5525.29	−0.326608	−0.163304	+	0.986576i	$0.552215\pi$
$660$	0	0
$661$	−12605.2	−0.741731	−0.370866	−	0.928687i	$-0.620939\pi$
$661$	−12605.2	−0.741731	−0.370866	+	0.928687i	$0.620939\pi$
$662$	−1827.96	−0.107320
$663$	102.288	0.00599176
$664$	9930.68	0.580399
$665$	0	0
$666$	1518.08	0.0883249
$667$	22456.7	1.30364
$668$	−8415.53	−0.487435
$669$	75.8600	0.00438403
$670$	0	0
$671$	−23409.2	−1.34680
$672$	125.353	0.00719582
$673$	−21852.1	−1.25161	−0.625807	−	0.779978i	$-0.715230\pi$
$673$	−21852.1	−1.25161	−0.625807	+	0.779978i	$0.715230\pi$
$674$	8582.75	0.490497
$675$	0	0
$676$	−31127.9	−1.77104
$677$	−29996.8	−1.70291	−0.851455	−	0.524428i	$-0.824279\pi$
$677$	−29996.8	−1.70291	−0.851455	+	0.524428i	$0.824279\pi$
$678$	−129.130	−0.00731448
$679$	−2464.80	−0.139308
$680$	0	0
$681$	−477.942	−0.0268939
$682$	−1088.04	−0.0610895
$683$	−29832.1	−1.67129	−0.835646	−	0.549269i	$-0.814906\pi$
$683$	−29832.1	−1.67129	−0.835646	+	0.549269i	$0.814906\pi$
$684$	−3760.28	−0.210201
$685$	0	0
$686$	5322.35	0.296222
$687$	106.558	0.00591766
$688$	−6549.27	−0.362919
$689$	29819.0	1.64878
$690$	0	0
$691$	−18582.1	−1.02300	−0.511502	−	0.859282i	$-0.670911\pi$
$691$	−18582.1	−1.02300	−0.511502	+	0.859282i	$0.670911\pi$
$692$	14265.2	0.783642
$693$	−19503.1	−1.06906
$694$	370.198	0.0202486
$695$	0	0
$696$	140.308	0.00764133
$697$	604.327	0.0328415
$698$	1501.35	0.0814140
$699$	−65.7882	−0.00355985
$700$	0	0
$701$	10189.2	0.548986	0.274493	−	0.961589i	$-0.411490\pi$
$701$	10189.2	0.548986	0.274493	+	0.961589i	$0.411490\pi$
$702$	264.194	0.0142042
$703$	1306.82	0.0701106
$704$	16398.9	0.877922
$705$	0	0
$706$	995.030	0.0530431
$707$	2742.58	0.145892
$708$	358.257	0.0190171
$709$	3029.58	0.160477	0.0802387	−	0.996776i	$-0.474432\pi$
$709$	3029.58	0.160477	0.0802387	+	0.996776i	$0.474432\pi$
$710$	0	0
$711$	34589.3	1.82447
$712$	−6203.85	−0.326544
$713$	3312.11	0.173968
$714$	−12.5259	−0.000656540 0
$715$	0	0
$716$	15832.0	0.826356
$717$	−129.296	−0.00673454
$718$	−4590.36	−0.238594
$719$	−16654.7	−0.863860	−0.431930	−	0.901907i	$-0.642167\pi$
$719$	−16654.7	−0.863860	−0.431930	+	0.901907i	$0.642167\pi$
$720$	0	0
$721$	−70.8187	−0.00365801
$722$	295.165	0.0152145
$723$	459.555	0.0236391
$724$	15264.7	0.783576
$725$	0	0
$726$	−139.038	−0.00710769
$727$	18968.4	0.967672	0.483836	−	0.875159i	$-0.339243\pi$
$727$	18968.4	0.967672	0.483836	+	0.875159i	$0.339243\pi$
$728$	12096.0	0.615807
$729$	−19658.7	−0.998765
$730$	0	0
$731$	2312.78	0.117019
$732$	212.908	0.0107504
$733$	7.14423	0.000359998 0	0.000179999	−	1.00000i	$-0.499943\pi$
$733$	7.14423	0.000359998 0	0.000179999		1.00000i	$0.499943\pi$
$734$	−8438.45	−0.424344
$735$	0	0
$736$	20920.6	1.04775
$737$	−27096.5	−1.35429
$738$	780.360	0.0389234
$739$	−24473.9	−1.21825	−0.609124	−	0.793075i	$-0.708479\pi$
$739$	−24473.9	−1.21825	−0.609124	+	0.793075i	$0.708479\pi$
$740$	0	0
$741$	113.702	0.00563693
$742$	−3651.54	−0.180664
$743$	−12392.3	−0.611881	−0.305941	−	0.952051i	$-0.598971\pi$
$743$	−12392.3	−0.611881	−0.305941	+	0.952051i	$0.598971\pi$
$744$	20.6939	0.00101972
$745$	0	0
$746$	−9861.25	−0.483976
$747$	21385.1	1.04744
$748$	−7531.21	−0.368139
$749$	18222.3	0.888956
$750$	0	0
$751$	−10466.8	−0.508572	−0.254286	−	0.967129i	$-0.581840\pi$
$751$	−10466.8	−0.508572	−0.254286	+	0.967129i	$0.581840\pi$
$752$	−9894.91	−0.479827
$753$	−521.602	−0.0252433
$754$	9852.72	0.475882
$755$	0	0
$756$	354.800	0.0170687
$757$	−34632.4	−1.66280	−0.831399	−	0.555676i	$-0.812459\pi$
$757$	−34632.4	−1.66280	−0.831399	+	0.555676i	$0.812459\pi$
$758$	3211.84	0.153904
$759$	670.235	0.0320527
$760$	0	0
$761$	21933.9	1.04481	0.522406	−	0.852697i	$-0.325034\pi$
$761$	21933.9	1.04481	0.522406	+	0.852697i	$0.325034\pi$
$762$	−93.5100	−0.00444555
$763$	21773.4	1.03309
$764$	8846.86	0.418938
$765$	0	0
$766$	6831.83	0.322251
$767$	52609.0	2.47666
$768$	−80.9441	−0.00380315
$769$	−18078.5	−0.847761	−0.423880	−	0.905718i	$-0.639332\pi$
$769$	−18078.5	−0.847761	−0.423880	+	0.905718i	$0.639332\pi$
$770$	0	0
$771$	−28.4321	−0.00132809
$772$	8926.07	0.416135
$773$	42118.4	1.95976	0.979879	−	0.199591i	$-0.0639613\pi$
$773$	42118.4	1.95976	0.979879	+	0.199591i	$0.0639613\pi$
$774$	2986.46	0.138690
$775$	0	0
$776$	−2570.15	−0.118896
$777$	−61.6462	−0.00284626
$778$	−2913.39	−0.134255
$779$	671.765	0.0308966
$780$	0	0
$781$	−27228.3	−1.24751
$782$	−2090.49	−0.0955956
$783$	604.353	0.0275834
$784$	−9606.93	−0.437634
$785$	0	0
$786$	−61.8913	−0.00280864
$787$	−37733.4	−1.70909	−0.854543	−	0.519381i	$-0.826163\pi$
$787$	−37733.4	−1.70909	−0.854543	+	0.519381i	$0.826163\pi$
$788$	−34809.9	−1.57367
$789$	440.071	0.0198567
$790$	0	0
$791$	−25465.7	−1.14470
$792$	−20336.7	−0.912415
$793$	31265.0	1.40006
$794$	−6577.15	−0.293973
$795$	0	0
$796$	−8031.90	−0.357642
$797$	18290.5	0.812902	0.406451	−	0.913673i	$-0.366766\pi$
$797$	18290.5	0.812902	0.406451	+	0.913673i	$0.366766\pi$
$798$	−13.9237	−0.000617660 0
$799$	3494.24	0.154715
$800$	0	0
$801$	−13359.6	−0.589313
$802$	11463.9	0.504742
$803$	−73054.4	−3.21050
$804$	246.445	0.0108103
$805$	0	0
$806$	1453.17	0.0635057
$807$	−379.296	−0.0165450
$808$	2859.81	0.124515
$809$	−28518.3	−1.23937	−0.619685	−	0.784850i	$-0.712740\pi$
$809$	−28518.3	−1.23937	−0.619685	+	0.784850i	$0.712740\pi$
$810$	0	0
$811$	−1388.30	−0.0601108	−0.0300554	−	0.999548i	$-0.509568\pi$
$811$	−1388.30	−0.0601108	−0.0300554	+	0.999548i	$0.509568\pi$
$812$	13231.8	0.571852
$813$	−433.959	−0.0187203
$814$	3379.76	0.145529
$815$	0	0
$816$	61.6815	0.00264618
$817$	2570.86	0.110090
$818$	−3039.10	−0.129902
$819$	26048.0	1.11134
$820$	0	0
$821$	2138.62	0.0909117	0.0454558	−	0.998966i	$-0.485526\pi$
$821$	2138.62	0.0909117	0.0454558	+	0.998966i	$0.485526\pi$
$822$	8.54508	0.000362584 0
$823$	31409.7	1.33034	0.665172	−	0.746690i	$-0.268358\pi$
$823$	31409.7	1.33034	0.665172	+	0.746690i	$0.268358\pi$
$824$	−73.8458	−0.00312201
$825$	0	0
$826$	−6442.34	−0.271378
$827$	18961.2	0.797275	0.398637	−	0.917109i	$-0.369483\pi$
$827$	18961.2	0.797275	0.398637	+	0.917109i	$0.369483\pi$
$828$	29603.9	1.24252
$829$	−29730.2	−1.24556	−0.622781	−	0.782396i	$-0.713997\pi$
$829$	−29730.2	−1.24556	−0.622781	+	0.782396i	$0.713997\pi$
$830$	0	0
$831$	157.296	0.00656624
$832$	−21902.2	−0.912645
$833$	3392.55	0.141110
$834$	−159.975	−0.00664208
$835$	0	0
$836$	−8371.63	−0.346338
$837$	89.1353	0.00368097
$838$	−8489.53	−0.349959
$839$	−21698.6	−0.892873	−0.446436	−	0.894815i	$-0.647307\pi$
$839$	−21698.6	−0.892873	−0.446436	+	0.894815i	$0.647307\pi$
$840$	0	0
$841$	−1850.52	−0.0758754
$842$	12849.0	0.525897
$843$	309.014	0.0126252
$844$	20105.3	0.819968
$845$	0	0
$846$	4512.07	0.183367
$847$	−27419.5	−1.11233
$848$	17981.4	0.728164
$849$	−135.538	−0.00547899
$850$	0	0
$851$	−10288.4	−0.414431
$852$	247.644	0.00995790
$853$	−9607.26	−0.385635	−0.192817	−	0.981235i	$-0.561762\pi$
$853$	−9607.26	−0.385635	−0.192817	+	0.981235i	$0.561762\pi$
$854$	−3828.62	−0.153410
$855$	0	0
$856$	19001.2	0.758700
$857$	32356.9	1.28972	0.644861	−	0.764300i	$-0.276915\pi$
$857$	32356.9	1.28972	0.644861	+	0.764300i	$0.276915\pi$
$858$	294.061	0.0117006
$859$	−29623.1	−1.17663	−0.588316	−	0.808631i	$-0.700209\pi$
$859$	−29623.1	−1.17663	−0.588316	+	0.808631i	$0.700209\pi$
$860$	0	0
$861$	−31.6889	−0.00125430
$862$	−454.332	−0.0179520
$863$	−14067.8	−0.554896	−0.277448	−	0.960741i	$-0.589489\pi$
$863$	−14067.8	−0.554896	−0.277448	+	0.960741i	$0.589489\pi$
$864$	563.014	0.0221691
$865$	0	0
$866$	11029.0	0.432770
$867$	344.509	0.0134950
$868$	1951.54	0.0763127
$869$	77007.4	3.00610
$870$	0	0
$871$	36189.7	1.40786
$872$	22704.0	0.881716
$873$	−5534.68	−0.214571
$874$	−2323.77	−0.0899345
$875$	0	0
$876$	664.435	0.0256269
$877$	3018.99	0.116242	0.0581208	−	0.998310i	$-0.481489\pi$
$877$	3018.99	0.116242	0.0581208	+	0.998310i	$0.481489\pi$
$878$	1735.20	0.0666971
$879$	−371.517	−0.0142559
$880$	0	0
$881$	2873.05	0.109870	0.0549351	−	0.998490i	$-0.482505\pi$
$881$	2873.05	0.109870	0.0549351	+	0.998490i	$0.482505\pi$
$882$	4380.75	0.167242
$883$	34115.8	1.30021	0.650106	−	0.759843i	$-0.274724\pi$
$883$	34115.8	1.30021	0.650106	+	0.759843i	$0.274724\pi$
$884$	10058.6	0.382700
$885$	0	0
$886$	−5563.02	−0.210941
$887$	9679.32	0.366403	0.183202	−	0.983075i	$-0.441354\pi$
$887$	9679.32	0.366403	0.183202	+	0.983075i	$0.441354\pi$
$888$	−64.2812	−0.00242921
$889$	−18441.0	−0.695717
$890$	0	0
$891$	−43784.9	−1.64629
$892$	7459.76	0.280013
$893$	3884.17	0.145553
$894$	67.7237	0.00253358
$895$	0	0
$896$	16132.8	0.601515
$897$	−895.156	−0.0333204
$898$	2729.19	0.101419
$899$	3324.17	0.123323
$900$	0	0
$901$	−6349.86	−0.234789
$902$	1737.34	0.0641321
$903$	−121.274	−0.00446927
$904$	−26554.2	−0.976968
$905$	0	0
$906$	90.4219	0.00331575
$907$	18370.6	0.672531	0.336265	−	0.941767i	$-0.390836\pi$
$907$	18370.6	0.672531	0.336265	+	0.941767i	$0.390836\pi$
$908$	−46998.8	−1.71774
$909$	6158.44	0.224711
$910$	0	0
$911$	40208.1	1.46230	0.731149	−	0.682218i	$-0.238984\pi$
$911$	40208.1	1.46230	0.731149	+	0.682218i	$0.238984\pi$
$912$	68.5647	0.00248948
$913$	47610.5	1.72582
$914$	−3236.57	−0.117129
$915$	0	0
$916$	10478.5	0.377967
$917$	−12205.5	−0.439545
$918$	−56.2592	−0.00202269
$919$	39867.6	1.43103	0.715513	−	0.698600i	$-0.246193\pi$
$919$	39867.6	1.43103	0.715513	+	0.698600i	$0.246193\pi$
$920$	0	0
$921$	−390.175	−0.0139595
$922$	9568.20	0.341770
$923$	36365.8	1.29685
$924$	394.911	0.0140602
$925$	0	0
$926$	2511.37	0.0891237
$927$	−159.023	−0.00563429
$928$	20996.8	0.742731
$929$	−5755.23	−0.203254	−0.101627	−	0.994823i	$-0.532405\pi$
$929$	−5755.23	−0.203254	−0.101627	+	0.994823i	$0.532405\pi$
$930$	0	0
$931$	3771.13	0.132754
$932$	−6469.34	−0.227372
$933$	−629.918	−0.0221035
$934$	−3665.81	−0.128425
$935$	0	0
$936$	27161.4	0.948502
$937$	−23689.7	−0.825942	−0.412971	−	0.910744i	$-0.635509\pi$
$937$	−23689.7	−0.825942	−0.412971	+	0.910744i	$0.635509\pi$
$938$	−4431.69	−0.154264
$939$	−804.301	−0.0279525
$940$	0	0
$941$	4109.76	0.142375	0.0711873	−	0.997463i	$-0.477321\pi$
$941$	4109.76	0.142375	0.0711873	+	0.997463i	$0.477321\pi$
$942$	−131.185	−0.00453739
$943$	−5288.67	−0.182633
$944$	31724.2	1.09379
$945$	0	0
$946$	6648.86	0.228513
$947$	−22949.1	−0.787482	−0.393741	−	0.919221i	$-0.628819\pi$
$947$	−22949.1	−0.787482	−0.393741	+	0.919221i	$0.628819\pi$
$948$	−700.388	−0.0239953
$949$	97570.5	3.33748
$950$	0	0
$951$	−209.774	−0.00715289
$952$	−2575.80	−0.0876915
$953$	34218.3	1.16311	0.581553	−	0.813509i	$-0.302445\pi$
$953$	34218.3	1.16311	0.581553	+	0.813509i	$0.302445\pi$
$954$	−8199.50	−0.278269
$955$	0	0
$956$	−12714.5	−0.430142
$957$	672.677	0.0227216
$958$	12068.6	0.407012
$959$	1685.17	0.0567434
$960$	0	0
$961$	−29300.7	−0.983543
$962$	−4513.95	−0.151285
$963$	40918.0	1.36922
$964$	45190.8	1.50985
$965$	0	0
$966$	109.618	0.00365104
$967$	−38605.0	−1.28382	−0.641909	−	0.766781i	$-0.721857\pi$
$967$	−38605.0	−1.28382	−0.641909	+	0.766781i	$0.721857\pi$
$968$	−28591.6	−0.949347
$969$	−24.2126	−0.000802704 0
$970$	0	0
$971$	22722.2	0.750967	0.375483	−	0.926829i	$-0.377477\pi$
$971$	22722.2	0.750967	0.375483	+	0.926829i	$0.377477\pi$
$972$	1195.09	0.0394368
$973$	−31548.6	−1.03947
$974$	7275.64	0.239350
$975$	0	0
$976$	18853.4	0.618321
$977$	−16931.1	−0.554425	−0.277213	−	0.960809i	$-0.589411\pi$
$977$	−16931.1	−0.554425	−0.277213	+	0.960809i	$0.589411\pi$
$978$	64.3855	0.00210513
$979$	−29743.0	−0.970982
$980$	0	0
$981$	48891.9	1.59123
$982$	−5462.46	−0.177509
$983$	2622.47	0.0850904	0.0425452	−	0.999095i	$-0.486453\pi$
$983$	2622.47	0.0850904	0.0425452	+	0.999095i	$0.486453\pi$
$984$	−33.0434	−0.00107051
$985$	0	0
$986$	−2098.11	−0.0677661
$987$	−183.226	−0.00590897
$988$	11181.0	0.360037
$989$	−20239.9	−0.650750
$990$	0	0
$991$	−40843.4	−1.30922	−0.654608	−	0.755969i	$-0.727166\pi$
$991$	−40843.4	−1.30922	−0.654608	+	0.755969i	$0.727166\pi$
$992$	3096.79	0.0991163
$993$	166.682	0.00532680
$994$	−4453.25	−0.142101
$995$	0	0
$996$	−433.021	−0.0137759
$997$	−35350.5	−1.12293	−0.561466	−	0.827500i	$-0.689762\pi$
$997$	−35350.5	−1.12293	−0.561466	+	0.827500i	$0.689762\pi$
$998$	11173.7	0.354407
$999$	−276.880	−0.00876887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.a.g.1.2		3
5.2	odd	4		475.4.b.g.324.4		6
5.3	odd	4		475.4.b.g.324.3		6
5.4	even	2		95.4.a.e.1.2	✓	3
15.14	odd	2		855.4.a.i.1.2		3
20.19	odd	2		1520.4.a.q.1.1		3
95.94	odd	2		1805.4.a.j.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.4.a.e.1.2	✓	3	5.4	even	2
475.4.a.g.1.2		3	1.1	even	1	trivial
475.4.b.g.324.3		6	5.3	odd	4
475.4.b.g.324.4		6	5.2	odd	4
855.4.a.i.1.2		3	15.14	odd	2
1520.4.a.q.1.1		3	20.19	odd	2
1805.4.a.j.1.2		3	95.94	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.817630 q^{2} -0.0745555 q^{3} -7.33148 q^{4} -0.0609588 q^{6} -12.0216 q^{7} -12.5355 q^{8} -26.9944 q^{9} -60.0987 q^{11} +0.546602 q^{12} +80.2669 q^{13} -9.82925 q^{14} +48.4025 q^{16} -17.0926 q^{17} -22.0715 q^{18} -19.0000 q^{19} +0.896278 q^{21} -49.1385 q^{22} +149.583 q^{23} +0.934589 q^{24} +65.6287 q^{26} +4.02558 q^{27} +88.1364 q^{28} +150.128 q^{29} +22.1422 q^{31} +139.859 q^{32} +4.48068 q^{33} -13.9754 q^{34} +197.909 q^{36} -68.7802 q^{37} -15.5350 q^{38} -5.98434 q^{39} -35.3560 q^{41} +0.732825 q^{42} -135.309 q^{43} +440.612 q^{44} +122.304 q^{46} -204.430 q^{47} -3.60867 q^{48} -198.480 q^{49} +1.27435 q^{51} -588.475 q^{52} +371.497 q^{53} +3.29144 q^{54} +150.697 q^{56} +1.41655 q^{57} +122.749 q^{58} +655.426 q^{59} +389.512 q^{61} +18.1042 q^{62} +324.517 q^{63} -272.866 q^{64} +3.66354 q^{66} +450.867 q^{67} +125.314 q^{68} -11.1522 q^{69} +453.060 q^{71} +338.388 q^{72} +1215.57 q^{73} -56.2368 q^{74} +139.298 q^{76} +722.484 q^{77} -4.89298 q^{78} -1281.35 q^{79} +728.550 q^{81} -28.9082 q^{82} -792.205 q^{83} -6.57105 q^{84} -110.632 q^{86} -11.1929 q^{87} +753.366 q^{88} +494.903 q^{89} -964.940 q^{91} -1096.67 q^{92} -1.65082 q^{93} -167.148 q^{94} -10.4273 q^{96} +205.030 q^{97} -162.284 q^{98} +1622.33 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 11 q^{3} + q^{4} + 25 q^{6} + 5 q^{7} + 27 q^{8} - 12 q^{9} - 4 q^{11} + 75 q^{12} + 125 q^{13} - 93 q^{14} - 7 q^{16} + 119 q^{17} + 138 q^{18} - 57 q^{19} + 35 q^{21} + 88 q^{22} + 129 q^{23}+ \cdots + 2540 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 11 * q^3 + q^4 + 25 * q^6 + 5 * q^7 + 27 * q^8 - 12 * q^9 - 4 * q^11 + 75 * q^12 + 125 * q^13 - 93 * q^14 - 7 * q^16 + 119 * q^17 + 138 * q^18 - 57 * q^19 + 35 * q^21 + 88 * q^22 + 129 * q^23 + 201 * q^24 + 279 * q^26 - 127 * q^27 - 63 * q^28 + 37 * q^29 + 90 * q^31 - 221 * q^32 + 360 * q^33 - 31 * q^34 + 574 * q^36 + 250 * q^37 - 57 * q^38 + 339 * q^39 - 94 * q^41 - 415 * q^42 + 210 * q^43 + 840 * q^44 + 391 * q^46 - 524 * q^47 - 301 * q^48 - 268 * q^49 + 657 * q^51 - 43 * q^52 + 1417 * q^53 + 403 * q^54 + 635 * q^56 - 209 * q^57 + 907 * q^58 + 15 * q^59 - 912 * q^61 - 1586 * q^62 - 214 * q^63 - 559 * q^64 + 1176 * q^66 - 237 * q^67 + 331 * q^68 + 47 * q^69 - 124 * q^71 + 426 * q^72 + 2457 * q^73 + 670 * q^74 - 19 * q^76 + 848 * q^77 + 1573 * q^78 - 98 * q^79 - 81 * q^81 - 1358 * q^82 + 454 * q^83 - 853 * q^84 + 1362 * q^86 - 101 * q^87 + 1752 * q^88 - 62 * q^89 - 1347 * q^91 - 547 * q^92 - 618 * q^93 - 3056 * q^94 - 2207 * q^96 + 1730 * q^97 - 1970 * q^98 + 2540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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