LMFDB - Embedded newform 536.4.a.b.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [536,4,Mod(1,536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("536.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 536 = 2^{3} \cdot 67 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	536.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$31.6250237631$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 188 x^{9} + 164 x^{8} + 12306 x^{7} - 4717 x^{6} - 349995 x^{5} - 28469 x^{4} + \cdots + 9069327 $ x^11 - x^10 - 188x^9 + 164x^8 + 12306x^7 - 4717x^6 - 349995x^5 - 28469x^4 + 4017906x^3 + 419382x^2 - 13979412*x + 9069327
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-3.45800$ of defining polynomial
Character	$\chi$	$=$	536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.45800 q^{3} +6.85373 q^{5} +17.0604 q^{7} -20.9582 q^{9} +O(q^{10})$	$q+2.45800 q^{3} +6.85373 q^{5} +17.0604 q^{7} -20.9582 q^{9} -27.7050 q^{11} -77.5135 q^{13} +16.8465 q^{15} -47.5988 q^{17} -56.3774 q^{19} +41.9343 q^{21} -59.5943 q^{23} -78.0264 q^{25} -117.881 q^{27} -152.747 q^{29} +124.456 q^{31} -68.0989 q^{33} +116.927 q^{35} +309.998 q^{37} -190.528 q^{39} +337.948 q^{41} -270.184 q^{43} -143.642 q^{45} +280.464 q^{47} -51.9444 q^{49} -116.998 q^{51} +91.9828 q^{53} -189.883 q^{55} -138.575 q^{57} -761.781 q^{59} -112.475 q^{61} -357.555 q^{63} -531.257 q^{65} +67.0000 q^{67} -146.483 q^{69} +385.015 q^{71} -29.4582 q^{73} -191.789 q^{75} -472.657 q^{77} +94.4076 q^{79} +276.121 q^{81} +269.891 q^{83} -326.229 q^{85} -375.452 q^{87} -1096.97 q^{89} -1322.41 q^{91} +305.913 q^{93} -386.395 q^{95} +1137.42 q^{97} +580.649 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 12 q^{3} - 3 q^{5} - 37 q^{7} + 93 q^{9}+O(q^{10}) $ 11 * q - 12 * q^3 - 3 * q^5 - 37 * q^7 + 93 * q^9	$ 11 q - 12 q^{3} - 3 q^{5} - 37 q^{7} + 93 q^{9} - 23 q^{11} - 96 q^{13} - 45 q^{15} + 145 q^{17} - 121 q^{19} - 104 q^{21} - 133 q^{23} + 388 q^{25} - 570 q^{27} - 388 q^{29} - 318 q^{31} + 127 q^{33} - 641 q^{35} - 353 q^{37} - 406 q^{39} + 53 q^{41} - 954 q^{43} - 658 q^{45} - 1848 q^{47} - 408 q^{49} - 1850 q^{51} - 973 q^{53} - 2300 q^{55} - 829 q^{57} - 1062 q^{59} - 1630 q^{61} - 1616 q^{63} - 1019 q^{65} + 737 q^{67} - 1600 q^{69} - 3413 q^{71} + 355 q^{73} - 4986 q^{75} - 295 q^{77} - 2981 q^{79} + 559 q^{81} - 2867 q^{83} + 623 q^{85} - 2257 q^{87} + 117 q^{89} - 3379 q^{91} - 275 q^{93} - 3253 q^{95} - 302 q^{97} - 5775 q^{99}+O(q^{100}) $ 11 * q - 12 * q^3 - 3 * q^5 - 37 * q^7 + 93 * q^9 - 23 * q^11 - 96 * q^13 - 45 * q^15 + 145 * q^17 - 121 * q^19 - 104 * q^21 - 133 * q^23 + 388 * q^25 - 570 * q^27 - 388 * q^29 - 318 * q^31 + 127 * q^33 - 641 * q^35 - 353 * q^37 - 406 * q^39 + 53 * q^41 - 954 * q^43 - 658 * q^45 - 1848 * q^47 - 408 * q^49 - 1850 * q^51 - 973 * q^53 - 2300 * q^55 - 829 * q^57 - 1062 * q^59 - 1630 * q^61 - 1616 * q^63 - 1019 * q^65 + 737 * q^67 - 1600 * q^69 - 3413 * q^71 + 355 * q^73 - 4986 * q^75 - 295 * q^77 - 2981 * q^79 + 559 * q^81 - 2867 * q^83 + 623 * q^85 - 2257 * q^87 + 117 * q^89 - 3379 * q^91 - 275 * q^93 - 3253 * q^95 - 302 * q^97 - 5775 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.45800	0.473042	0.236521	−	0.971626i	$-0.423993\pi$
$3$	2.45800	0.473042	0.236521	+	0.971626i	$0.423993\pi$
$4$	0	0
$5$	6.85373	0.613016	0.306508	−	0.951868i	$-0.400839\pi$
$5$	6.85373	0.613016	0.306508	+	0.951868i	$0.400839\pi$
$6$	0	0
$7$	17.0604	0.921172	0.460586	−	0.887615i	$-0.347639\pi$
$7$	17.0604	0.921172	0.460586	+	0.887615i	$0.347639\pi$
$8$	0	0
$9$	−20.9582	−0.776231
$10$	0	0
$11$	−27.7050	−0.759398	−0.379699	−	0.925110i	$-0.623972\pi$
$11$	−27.7050	−0.759398	−0.379699	+	0.925110i	$0.623972\pi$
$12$	0	0
$13$	−77.5135	−1.65372	−0.826861	−	0.562406i	$-0.809876\pi$
$13$	−77.5135	−1.65372	−0.826861	+	0.562406i	$0.809876\pi$
$14$	0	0
$15$	16.8465	0.289982
$16$	0	0
$17$	−47.5988	−0.679083	−0.339541	−	0.940591i	$-0.610272\pi$
$17$	−47.5988	−0.679083	−0.339541	+	0.940591i	$0.610272\pi$
$18$	0	0
$19$	−56.3774	−0.680729	−0.340365	−	0.940294i	$-0.610551\pi$
$19$	−56.3774	−0.680729	−0.340365	+	0.940294i	$0.610551\pi$
$20$	0	0
$21$	41.9343	0.435753
$22$	0	0
$23$	−59.5943	−0.540273	−0.270136	−	0.962822i	$-0.587069\pi$
$23$	−59.5943	−0.540273	−0.270136	+	0.962822i	$0.587069\pi$
$24$	0	0
$25$	−78.0264	−0.624211
$26$	0	0
$27$	−117.881	−0.840232
$28$	0	0
$29$	−152.747	−0.978084	−0.489042	−	0.872260i	$-0.662653\pi$
$29$	−152.747	−0.978084	−0.489042	+	0.872260i	$0.662653\pi$
$30$	0	0
$31$	124.456	0.721063	0.360532	−	0.932747i	$-0.382595\pi$
$31$	124.456	0.721063	0.360532	+	0.932747i	$0.382595\pi$
$32$	0	0
$33$	−68.0989	−0.359227
$34$	0	0
$35$	116.927	0.564694
$36$	0	0
$37$	309.998	1.37739	0.688695	−	0.725052i	$-0.258184\pi$
$37$	309.998	1.37739	0.688695	+	0.725052i	$0.258184\pi$
$38$	0	0
$39$	−190.528	−0.782280
$40$	0	0
$41$	337.948	1.28728	0.643642	−	0.765327i	$-0.277423\pi$
$41$	337.948	1.28728	0.643642	+	0.765327i	$0.277423\pi$
$42$	0	0
$43$	−270.184	−0.958201	−0.479100	−	0.877760i	$-0.659037\pi$
$43$	−270.184	−0.958201	−0.479100	+	0.877760i	$0.659037\pi$
$44$	0	0
$45$	−143.642	−0.475843
$46$	0	0
$47$	280.464	0.870424	0.435212	−	0.900328i	$-0.356673\pi$
$47$	280.464	0.870424	0.435212	+	0.900328i	$0.356673\pi$
$48$	0	0
$49$	−51.9444	−0.151442
$50$	0	0
$51$	−116.998	−0.321234
$52$	0	0
$53$	91.9828	0.238393	0.119196	−	0.992871i	$-0.461968\pi$
$53$	91.9828	0.238393	0.119196	+	0.992871i	$0.461968\pi$
$54$	0	0
$55$	−189.883	−0.465523
$56$	0	0
$57$	−138.575	−0.322014
$58$	0	0
$59$	−761.781	−1.68094	−0.840470	−	0.541859i	$-0.817721\pi$
$59$	−761.781	−1.68094	−0.840470	+	0.541859i	$0.817721\pi$
$60$	0	0
$61$	−112.475	−0.236082	−0.118041	−	0.993009i	$-0.537661\pi$
$61$	−112.475	−0.236082	−0.118041	+	0.993009i	$0.537661\pi$
$62$	0	0
$63$	−357.555	−0.715043
$64$	0	0
$65$	−531.257	−1.01376
$66$	0	0
$67$	67.0000	0.122169
$67$	67.0000	0.122169
$68$	0	0
$69$	−146.483	−0.255572
$70$	0	0
$71$	385.015	0.643562	0.321781	−	0.946814i	$-0.395719\pi$
$71$	385.015	0.643562	0.321781	+	0.946814i	$0.395719\pi$
$72$	0	0
$73$	−29.4582	−0.0472304	−0.0236152	−	0.999721i	$-0.507518\pi$
$73$	−29.4582	−0.0472304	−0.0236152	+	0.999721i	$0.507518\pi$
$74$	0	0
$75$	−191.789	−0.295278
$76$	0	0
$77$	−472.657	−0.699537
$78$	0	0
$79$	94.4076	0.134452	0.0672258	−	0.997738i	$-0.478585\pi$
$79$	94.4076	0.134452	0.0672258	+	0.997738i	$0.478585\pi$
$80$	0	0
$81$	276.121	0.378767
$82$	0	0
$83$	269.891	0.356920	0.178460	−	0.983947i	$-0.442888\pi$
$83$	269.891	0.356920	0.178460	+	0.983947i	$0.442888\pi$
$84$	0	0
$85$	−326.229	−0.416289
$86$	0	0
$87$	−375.452	−0.462675
$88$	0	0
$89$	−1096.97	−1.30650	−0.653251	−	0.757141i	$-0.726596\pi$
$89$	−1096.97	−1.30650	−0.653251	+	0.757141i	$0.726596\pi$
$90$	0	0
$91$	−1322.41	−1.52336
$92$	0	0
$93$	305.913	0.341093
$94$	0	0
$95$	−386.395	−0.417298
$96$	0	0
$97$	1137.42	1.19059	0.595295	−	0.803507i	$-0.297035\pi$
$97$	1137.42	1.19059	0.595295	+	0.803507i	$0.297035\pi$
$98$	0	0
$99$	580.649	0.589469
$100$	0	0
$101$	644.909	0.635355	0.317678	−	0.948199i	$-0.397097\pi$
$101$	644.909	0.635355	0.317678	+	0.948199i	$0.397097\pi$
$102$	0	0
$103$	−1297.58	−1.24130	−0.620651	−	0.784087i	$-0.713132\pi$
$103$	−1297.58	−1.24130	−0.620651	+	0.784087i	$0.713132\pi$
$104$	0	0
$105$	287.406	0.267124
$106$	0	0
$107$	−1051.60	−0.950113	−0.475056	−	0.879955i	$-0.657572\pi$
$107$	−1051.60	−0.950113	−0.475056	+	0.879955i	$0.657572\pi$
$108$	0	0
$109$	394.520	0.346680	0.173340	−	0.984862i	$-0.444544\pi$
$109$	394.520	0.346680	0.173340	+	0.984862i	$0.444544\pi$
$110$	0	0
$111$	761.975	0.651563
$112$	0	0
$113$	442.195	0.368126	0.184063	−	0.982914i	$-0.441075\pi$
$113$	442.195	0.368126	0.184063	+	0.982914i	$0.441075\pi$
$114$	0	0
$115$	−408.444	−0.331196
$116$	0	0
$117$	1624.55	1.28367
$118$	0	0
$119$	−812.052	−0.625552
$120$	0	0
$121$	−563.432	−0.423315
$122$	0	0
$123$	830.676	0.608939
$124$	0	0
$125$	−1391.49	−0.995668
$126$	0	0
$127$	−236.713	−0.165393	−0.0826964	−	0.996575i	$-0.526353\pi$
$127$	−236.713	−0.165393	−0.0826964	+	0.996575i	$0.526353\pi$
$128$	0	0
$129$	−664.111	−0.453269
$130$	0	0
$131$	1139.41	0.759927	0.379963	−	0.925002i	$-0.375937\pi$
$131$	1139.41	0.759927	0.379963	+	0.925002i	$0.375937\pi$
$132$	0	0
$133$	−961.818	−0.627069
$134$	0	0
$135$	−807.926	−0.515076
$136$	0	0
$137$	−1124.53	−0.701280	−0.350640	−	0.936510i	$-0.614036\pi$
$137$	−1124.53	−0.701280	−0.350640	+	0.936510i	$0.614036\pi$
$138$	0	0
$139$	−900.684	−0.549604	−0.274802	−	0.961501i	$-0.588612\pi$
$139$	−900.684	−0.549604	−0.274802	+	0.961501i	$0.588612\pi$
$140$	0	0
$141$	689.381	0.411747
$142$	0	0
$143$	2147.51	1.25583
$144$	0	0
$145$	−1046.89	−0.599581
$146$	0	0
$147$	−127.679	−0.0716382
$148$	0	0
$149$	1474.83	0.810889	0.405444	−	0.914120i	$-0.367117\pi$
$149$	1474.83	0.810889	0.405444	+	0.914120i	$0.367117\pi$
$150$	0	0
$151$	−2854.60	−1.53844	−0.769220	−	0.638984i	$-0.779355\pi$
$151$	−2854.60	−1.53844	−0.769220	+	0.638984i	$0.779355\pi$
$152$	0	0
$153$	997.587	0.527125
$154$	0	0
$155$	852.988	0.442024
$156$	0	0
$157$	1128.11	0.573459	0.286730	−	0.958012i	$-0.407432\pi$
$157$	1128.11	0.573459	0.286730	+	0.958012i	$0.407432\pi$
$158$	0	0
$159$	226.094	0.112770
$160$	0	0
$161$	−1016.70	−0.497685
$162$	0	0
$163$	−4105.59	−1.97285	−0.986425	−	0.164212i	$-0.947492\pi$
$163$	−4105.59	−1.97285	−0.986425	+	0.164212i	$0.947492\pi$
$164$	0	0
$165$	−466.731	−0.220212
$166$	0	0
$167$	−1171.78	−0.542964	−0.271482	−	0.962443i	$-0.587514\pi$
$167$	−1171.78	−0.542964	−0.271482	+	0.962443i	$0.587514\pi$
$168$	0	0
$169$	3811.35	1.73480
$170$	0	0
$171$	1181.57	0.528404
$172$	0	0
$173$	4095.25	1.79974	0.899872	−	0.436154i	$-0.143660\pi$
$173$	4095.25	1.79974	0.899872	+	0.436154i	$0.143660\pi$
$174$	0	0
$175$	−1331.16	−0.575006
$176$	0	0
$177$	−1872.46	−0.795155
$178$	0	0
$179$	692.949	0.289349	0.144674	−	0.989479i	$-0.453787\pi$
$179$	692.949	0.289349	0.144674	+	0.989479i	$0.453787\pi$
$180$	0	0
$181$	1984.07	0.814778	0.407389	−	0.913255i	$-0.366439\pi$
$181$	1984.07	0.814778	0.407389	+	0.913255i	$0.366439\pi$
$182$	0	0
$183$	−276.465	−0.111677
$184$	0	0
$185$	2124.64	0.844362
$186$	0	0
$187$	1318.73	0.515694
$188$	0	0
$189$	−2011.10	−0.773998
$190$	0	0
$191$	−309.021	−0.117068	−0.0585339	−	0.998285i	$-0.518643\pi$
$191$	−309.021	−0.117068	−0.0585339	+	0.998285i	$0.518643\pi$
$192$	0	0
$193$	2845.48	1.06125	0.530627	−	0.847605i	$-0.321957\pi$
$193$	2845.48	1.06125	0.530627	+	0.847605i	$0.321957\pi$
$194$	0	0
$195$	−1305.83	−0.479550
$196$	0	0
$197$	−2930.99	−1.06002	−0.530010	−	0.847991i	$-0.677812\pi$
$197$	−2930.99	−1.06002	−0.530010	+	0.847991i	$0.677812\pi$
$198$	0	0
$199$	−887.027	−0.315978	−0.157989	−	0.987441i	$-0.550501\pi$
$199$	−887.027	−0.315978	−0.157989	+	0.987441i	$0.550501\pi$
$200$	0	0
$201$	164.686	0.0577913
$202$	0	0
$203$	−2605.92	−0.900984
$204$	0	0
$205$	2316.21	0.789126
$206$	0	0
$207$	1248.99	0.419377
$208$	0	0
$209$	1561.94	0.516945
$210$	0	0
$211$	5141.93	1.67765	0.838827	−	0.544398i	$-0.183242\pi$
$211$	5141.93	1.67765	0.838827	+	0.544398i	$0.183242\pi$
$212$	0	0
$213$	946.366	0.304432
$214$	0	0
$215$	−1851.77	−0.587393
$216$	0	0
$217$	2123.26	0.664224
$218$	0	0
$219$	−72.4082	−0.0223420
$220$	0	0
$221$	3689.55	1.12301
$222$	0	0
$223$	−5781.95	−1.73627	−0.868135	−	0.496328i	$-0.834681\pi$
$223$	−5781.95	−1.73627	−0.868135	+	0.496328i	$0.834681\pi$
$224$	0	0
$225$	1635.30	0.484532
$226$	0	0
$227$	−3322.11	−0.971349	−0.485675	−	0.874140i	$-0.661426\pi$
$227$	−3322.11	−0.971349	−0.485675	+	0.874140i	$0.661426\pi$
$228$	0	0
$229$	−1652.08	−0.476736	−0.238368	−	0.971175i	$-0.576613\pi$
$229$	−1652.08	−0.476736	−0.238368	+	0.971175i	$0.576613\pi$
$230$	0	0
$231$	−1161.79	−0.330910
$232$	0	0
$233$	3800.67	1.06863	0.534313	−	0.845287i	$-0.320570\pi$
$233$	3800.67	1.06863	0.534313	+	0.845287i	$0.320570\pi$
$234$	0	0
$235$	1922.23	0.533584
$236$	0	0
$237$	232.054	0.0636013
$238$	0	0
$239$	1865.35	0.504851	0.252425	−	0.967616i	$-0.418772\pi$
$239$	1865.35	0.504851	0.252425	+	0.967616i	$0.418772\pi$
$240$	0	0
$241$	5240.73	1.40077	0.700384	−	0.713767i	$-0.253012\pi$
$241$	5240.73	1.40077	0.700384	+	0.713767i	$0.253012\pi$
$242$	0	0
$243$	3861.50	1.01940
$244$	0	0
$245$	−356.013	−0.0928361
$246$	0	0
$247$	4370.01	1.12574
$248$	0	0
$249$	663.391	0.168838
$250$	0	0
$251$	1911.11	0.480590	0.240295	−	0.970700i	$-0.422756\pi$
$251$	1911.11	0.480590	0.240295	+	0.970700i	$0.422756\pi$
$252$	0	0
$253$	1651.06	0.410282
$254$	0	0
$255$	−801.871	−0.196922
$256$	0	0
$257$	−1270.30	−0.308323	−0.154162	−	0.988046i	$-0.549268\pi$
$257$	−1270.30	−0.308323	−0.154162	+	0.988046i	$0.549268\pi$
$258$	0	0
$259$	5288.68	1.26881
$260$	0	0
$261$	3201.31	0.759219
$262$	0	0
$263$	1726.55	0.404806	0.202403	−	0.979302i	$-0.435125\pi$
$263$	1726.55	0.404806	0.202403	+	0.979302i	$0.435125\pi$
$264$	0	0
$265$	630.426	0.146139
$266$	0	0
$267$	−2696.35	−0.618031
$268$	0	0
$269$	−4513.31	−1.02298	−0.511490	−	0.859289i	$-0.670906\pi$
$269$	−4513.31	−1.02298	−0.511490	+	0.859289i	$0.670906\pi$
$270$	0	0
$271$	1987.79	0.445572	0.222786	−	0.974867i	$-0.428485\pi$
$271$	1987.79	0.445572	0.222786	+	0.974867i	$0.428485\pi$
$272$	0	0
$273$	−3250.48	−0.720615
$274$	0	0
$275$	2161.72	0.474025
$276$	0	0
$277$	−6354.24	−1.37830	−0.689150	−	0.724618i	$-0.742016\pi$
$277$	−6354.24	−1.37830	−0.689150	+	0.724618i	$0.742016\pi$
$278$	0	0
$279$	−2608.38	−0.559712
$280$	0	0
$281$	5805.82	1.23255	0.616275	−	0.787531i	$-0.288641\pi$
$281$	5805.82	1.23255	0.616275	+	0.787531i	$0.288641\pi$
$282$	0	0
$283$	4799.13	1.00805	0.504026	−	0.863688i	$-0.331852\pi$
$283$	4799.13	1.00805	0.504026	+	0.863688i	$0.331852\pi$
$284$	0	0
$285$	−949.759	−0.197400
$286$	0	0
$287$	5765.51	1.18581
$288$	0	0
$289$	−2647.35	−0.538847
$290$	0	0
$291$	2795.77	0.563199
$292$	0	0
$293$	−5157.68	−1.02838	−0.514189	−	0.857677i	$-0.671907\pi$
$293$	−5157.68	−1.02838	−0.514189	+	0.857677i	$0.671907\pi$
$294$	0	0
$295$	−5221.04	−1.03044
$296$	0	0
$297$	3265.90	0.638070
$298$	0	0
$299$	4619.37	0.893461
$300$	0	0
$301$	−4609.43	−0.882668
$302$	0	0
$303$	1585.19	0.300550
$304$	0	0
$305$	−770.877	−0.144722
$306$	0	0
$307$	−7473.08	−1.38929	−0.694644	−	0.719354i	$-0.744438\pi$
$307$	−7473.08	−1.38929	−0.694644	+	0.719354i	$0.744438\pi$
$308$	0	0
$309$	−3189.44	−0.587188
$310$	0	0
$311$	1829.49	0.333572	0.166786	−	0.985993i	$-0.446661\pi$
$311$	1829.49	0.333572	0.166786	+	0.985993i	$0.446661\pi$
$312$	0	0
$313$	−380.029	−0.0686278	−0.0343139	−	0.999411i	$-0.510925\pi$
$313$	−380.029	−0.0686278	−0.0343139	+	0.999411i	$0.510925\pi$
$314$	0	0
$315$	−2450.59	−0.438333
$316$	0	0
$317$	−4463.72	−0.790876	−0.395438	−	0.918493i	$-0.629407\pi$
$317$	−4463.72	−0.790876	−0.395438	+	0.918493i	$0.629407\pi$
$318$	0	0
$319$	4231.86	0.742755
$320$	0	0
$321$	−2584.83	−0.449443
$322$	0	0
$323$	2683.50	0.462271
$324$	0	0
$325$	6048.10	1.03227
$326$	0	0
$327$	969.729	0.163994
$328$	0	0
$329$	4784.82	0.801811
$330$	0	0
$331$	1767.91	0.293574	0.146787	−	0.989168i	$-0.453107\pi$
$331$	1767.91	0.293574	0.146787	+	0.989168i	$0.453107\pi$
$332$	0	0
$333$	−6497.02	−1.06917
$334$	0	0
$335$	459.200	0.0748919
$336$	0	0
$337$	7315.22	1.18245	0.591225	−	0.806507i	$-0.298645\pi$
$337$	7315.22	1.18245	0.591225	+	0.806507i	$0.298645\pi$
$338$	0	0
$339$	1086.91	0.174139
$340$	0	0
$341$	−3448.06	−0.547574
$342$	0	0
$343$	−6737.89	−1.06068
$344$	0	0
$345$	−1003.95	−0.156670
$346$	0	0
$347$	−8232.01	−1.27354	−0.636769	−	0.771055i	$-0.719729\pi$
$347$	−8232.01	−1.27354	−0.636769	+	0.771055i	$0.719729\pi$
$348$	0	0
$349$	−5097.53	−0.781847	−0.390924	−	0.920423i	$-0.627844\pi$
$349$	−5097.53	−0.781847	−0.390924	+	0.920423i	$0.627844\pi$
$350$	0	0
$351$	9137.39	1.38951
$352$	0	0
$353$	13026.1	1.96405	0.982025	−	0.188749i	$-0.0604434\pi$
$353$	13026.1	1.96405	0.982025	+	0.188749i	$0.0604434\pi$
$354$	0	0
$355$	2638.79	0.394514
$356$	0	0
$357$	−1996.02	−0.295912
$358$	0	0
$359$	10618.9	1.56112	0.780561	−	0.625080i	$-0.214934\pi$
$359$	10618.9	1.56112	0.780561	+	0.625080i	$0.214934\pi$
$360$	0	0
$361$	−3680.59	−0.536607
$362$	0	0
$363$	−1384.91	−0.200245
$364$	0	0
$365$	−201.899	−0.0289530
$366$	0	0
$367$	−1056.29	−0.150239	−0.0751194	−	0.997175i	$-0.523934\pi$
$367$	−1056.29	−0.150239	−0.0751194	+	0.997175i	$0.523934\pi$
$368$	0	0
$369$	−7082.80	−0.999230
$370$	0	0
$371$	1569.26	0.219601
$372$	0	0
$373$	−2667.52	−0.370293	−0.185146	−	0.982711i	$-0.559276\pi$
$373$	−2667.52	−0.370293	−0.185146	+	0.982711i	$0.559276\pi$
$374$	0	0
$375$	−3420.27	−0.470993
$376$	0	0
$377$	11840.0	1.61748
$378$	0	0
$379$	6048.25	0.819730	0.409865	−	0.912146i	$-0.365576\pi$
$379$	6048.25	0.819730	0.409865	+	0.912146i	$0.365576\pi$
$380$	0	0
$381$	−581.840	−0.0782377
$382$	0	0
$383$	−11853.5	−1.58142	−0.790712	−	0.612188i	$-0.790289\pi$
$383$	−11853.5	−1.58142	−0.790712	+	0.612188i	$0.790289\pi$
$384$	0	0
$385$	−3239.47	−0.428827
$386$	0	0
$387$	5662.58	0.743786
$388$	0	0
$389$	−5073.58	−0.661288	−0.330644	−	0.943756i	$-0.607266\pi$
$389$	−5073.58	−0.661288	−0.330644	+	0.943756i	$0.607266\pi$
$390$	0	0
$391$	2836.62	0.366890
$392$	0	0
$393$	2800.66	0.359477
$394$	0	0
$395$	647.044	0.0824211
$396$	0	0
$397$	6337.56	0.801191	0.400596	−	0.916255i	$-0.368803\pi$
$397$	6337.56	0.801191	0.400596	+	0.916255i	$0.368803\pi$
$398$	0	0
$399$	−2364.15	−0.296630
$400$	0	0
$401$	−10761.4	−1.34015	−0.670074	−	0.742295i	$-0.733737\pi$
$401$	−10761.4	−1.34015	−0.670074	+	0.742295i	$0.733737\pi$
$402$	0	0
$403$	−9647.03	−1.19244
$404$	0	0
$405$	1892.46	0.232190
$406$	0	0
$407$	−8588.51	−1.04599
$408$	0	0
$409$	−9903.03	−1.19725	−0.598623	−	0.801031i	$-0.704285\pi$
$409$	−9903.03	−1.19725	−0.598623	+	0.801031i	$0.704285\pi$
$410$	0	0
$411$	−2764.10	−0.331735
$412$	0	0
$413$	−12996.2	−1.54843
$414$	0	0
$415$	1849.76	0.218798
$416$	0	0
$417$	−2213.88	−0.259986
$418$	0	0
$419$	−5252.95	−0.612467	−0.306233	−	0.951957i	$-0.599069\pi$
$419$	−5252.95	−0.612467	−0.306233	+	0.951957i	$0.599069\pi$
$420$	0	0
$421$	−789.016	−0.0913404	−0.0456702	−	0.998957i	$-0.514542\pi$
$421$	−789.016	−0.0913404	−0.0456702	+	0.998957i	$0.514542\pi$
$422$	0	0
$423$	−5878.04	−0.675651
$424$	0	0
$425$	3713.96	0.423891
$426$	0	0
$427$	−1918.87	−0.217472
$428$	0	0
$429$	5278.59	0.594062
$430$	0	0
$431$	−1734.85	−0.193886	−0.0969429	−	0.995290i	$-0.530906\pi$
$431$	−1734.85	−0.193886	−0.0969429	+	0.995290i	$0.530906\pi$
$432$	0	0
$433$	−13966.4	−1.55008	−0.775038	−	0.631915i	$-0.782269\pi$
$433$	−13966.4	−1.55008	−0.775038	+	0.631915i	$0.782269\pi$
$434$	0	0
$435$	−2573.25	−0.283627
$436$	0	0
$437$	3359.77	0.367780
$438$	0	0
$439$	−3829.24	−0.416309	−0.208154	−	0.978096i	$-0.566746\pi$
$439$	−3829.24	−0.416309	−0.208154	+	0.978096i	$0.566746\pi$
$440$	0	0
$441$	1088.66	0.117554
$442$	0	0
$443$	−1430.22	−0.153390	−0.0766951	−	0.997055i	$-0.524437\pi$
$443$	−1430.22	−0.153390	−0.0766951	+	0.997055i	$0.524437\pi$
$444$	0	0
$445$	−7518.35	−0.800908
$446$	0	0
$447$	3625.12	0.383584
$448$	0	0
$449$	7879.17	0.828154	0.414077	−	0.910242i	$-0.364104\pi$
$449$	7879.17	0.828154	0.414077	+	0.910242i	$0.364104\pi$
$450$	0	0
$451$	−9362.86	−0.977561
$452$	0	0
$453$	−7016.61	−0.727747
$454$	0	0
$455$	−9063.43	−0.933846
$456$	0	0
$457$	−12459.1	−1.27530	−0.637649	−	0.770327i	$-0.720093\pi$
$457$	−12459.1	−1.27530	−0.637649	+	0.770327i	$0.720093\pi$
$458$	0	0
$459$	5611.01	0.570587
$460$	0	0
$461$	−1356.76	−0.137073	−0.0685365	−	0.997649i	$-0.521833\pi$
$461$	−1356.76	−0.137073	−0.0685365	+	0.997649i	$0.521833\pi$
$462$	0	0
$463$	3686.55	0.370040	0.185020	−	0.982735i	$-0.440765\pi$
$463$	3686.55	0.370040	0.185020	+	0.982735i	$0.440765\pi$
$464$	0	0
$465$	2096.64	0.209096
$466$	0	0
$467$	9887.08	0.979699	0.489850	−	0.871807i	$-0.337052\pi$
$467$	9887.08	0.979699	0.489850	+	0.871807i	$0.337052\pi$
$468$	0	0
$469$	1143.04	0.112539
$470$	0	0
$471$	2772.90	0.271270
$472$	0	0
$473$	7485.45	0.727656
$474$	0	0
$475$	4398.92	0.424919
$476$	0	0
$477$	−1927.80	−0.185048
$478$	0	0
$479$	7659.37	0.730617	0.365308	−	0.930887i	$-0.380964\pi$
$479$	7659.37	0.730617	0.365308	+	0.930887i	$0.380964\pi$
$480$	0	0
$481$	−24029.1	−2.27782
$482$	0	0
$483$	−2499.05	−0.235426
$484$	0	0
$485$	7795.56	0.729852
$486$	0	0
$487$	−5232.25	−0.486850	−0.243425	−	0.969920i	$-0.578271\pi$
$487$	−5232.25	−0.486850	−0.243425	+	0.969920i	$0.578271\pi$
$488$	0	0
$489$	−10091.5	−0.933241
$490$	0	0
$491$	−8142.85	−0.748436	−0.374218	−	0.927341i	$-0.622089\pi$
$491$	−8142.85	−0.748436	−0.374218	+	0.927341i	$0.622089\pi$
$492$	0	0
$493$	7270.58	0.664199
$494$	0	0
$495$	3979.61	0.361354
$496$	0	0
$497$	6568.49	0.592831
$498$	0	0
$499$	−1625.03	−0.145784	−0.0728921	−	0.997340i	$-0.523223\pi$
$499$	−1625.03	−0.145784	−0.0728921	+	0.997340i	$0.523223\pi$
$500$	0	0
$501$	−2880.23	−0.256845
$502$	0	0
$503$	16828.8	1.49177	0.745883	−	0.666077i	$-0.232028\pi$
$503$	16828.8	1.49177	0.745883	+	0.666077i	$0.232028\pi$
$504$	0	0
$505$	4420.04	0.389483
$506$	0	0
$507$	9368.29	0.820632
$508$	0	0
$509$	−1022.27	−0.0890203	−0.0445101	−	0.999009i	$-0.514173\pi$
$509$	−1022.27	−0.0890203	−0.0445101	+	0.999009i	$0.514173\pi$
$510$	0	0
$511$	−502.567	−0.0435074
$512$	0	0
$513$	6645.84	0.571971
$514$	0	0
$515$	−8893.25	−0.760939
$516$	0	0
$517$	−7770.28	−0.660999
$518$	0	0
$519$	10066.1	0.851355
$520$	0	0
$521$	20115.3	1.69149	0.845745	−	0.533588i	$-0.179157\pi$
$521$	20115.3	1.69149	0.845745	+	0.533588i	$0.179157\pi$
$522$	0	0
$523$	−7632.00	−0.638096	−0.319048	−	0.947739i	$-0.603363\pi$
$523$	−7632.00	−0.638096	−0.319048	+	0.947739i	$0.603363\pi$
$524$	0	0
$525$	−3271.98	−0.272002
$526$	0	0
$527$	−5923.96	−0.489662
$528$	0	0
$529$	−8615.51	−0.708105
$530$	0	0
$531$	15965.6	1.30480
$532$	0	0
$533$	−26195.6	−2.12881
$534$	0	0
$535$	−7207.39	−0.582435
$536$	0	0
$537$	1703.27	0.136874
$538$	0	0
$539$	1439.12	0.115004
$540$	0	0
$541$	−16500.9	−1.31133	−0.655663	−	0.755053i	$-0.727611\pi$
$541$	−16500.9	−1.31133	−0.655663	+	0.755053i	$0.727611\pi$
$542$	0	0
$543$	4876.84	0.385424
$544$	0	0
$545$	2703.93	0.212521
$546$	0	0
$547$	−10249.9	−0.801194	−0.400597	−	0.916254i	$-0.631197\pi$
$547$	−10249.9	−0.801194	−0.400597	+	0.916254i	$0.631197\pi$
$548$	0	0
$549$	2357.29	0.183254
$550$	0	0
$551$	8611.48	0.665810
$552$	0	0
$553$	1610.63	0.123853
$554$	0	0
$555$	5222.37	0.399419
$556$	0	0
$557$	8551.85	0.650545	0.325272	−	0.945620i	$-0.394544\pi$
$557$	8551.85	0.650545	0.325272	+	0.945620i	$0.394544\pi$
$558$	0	0
$559$	20942.9	1.58460
$560$	0	0
$561$	3241.42	0.243945
$562$	0	0
$563$	−7535.96	−0.564126	−0.282063	−	0.959396i	$-0.591019\pi$
$563$	−7535.96	−0.564126	−0.282063	+	0.959396i	$0.591019\pi$
$564$	0	0
$565$	3030.68	0.225667
$566$	0	0
$567$	4710.72	0.348909
$568$	0	0
$569$	6150.10	0.453121	0.226560	−	0.973997i	$-0.427252\pi$
$569$	6150.10	0.453121	0.226560	+	0.973997i	$0.427252\pi$
$570$	0	0
$571$	−21727.5	−1.59241	−0.796207	−	0.605024i	$-0.793163\pi$
$571$	−21727.5	−1.59241	−0.796207	+	0.605024i	$0.793163\pi$
$572$	0	0
$573$	−759.572	−0.0553780
$574$	0	0
$575$	4649.93	0.337244
$576$	0	0
$577$	−13430.9	−0.969042	−0.484521	−	0.874780i	$-0.661006\pi$
$577$	−13430.9	−0.969042	−0.484521	+	0.874780i	$0.661006\pi$
$578$	0	0
$579$	6994.18	0.502018
$580$	0	0
$581$	4604.43	0.328785
$582$	0	0
$583$	−2548.39	−0.181035
$584$	0	0
$585$	11134.2	0.786911
$586$	0	0
$587$	18734.5	1.31730	0.658649	−	0.752450i	$-0.271128\pi$
$587$	18734.5	1.31730	0.658649	+	0.752450i	$0.271128\pi$
$588$	0	0
$589$	−7016.51	−0.490849
$590$	0	0
$591$	−7204.36	−0.501434
$592$	0	0
$593$	20623.1	1.42814	0.714072	−	0.700073i	$-0.246849\pi$
$593$	20623.1	1.42814	0.714072	+	0.700073i	$0.246849\pi$
$594$	0	0
$595$	−5565.59	−0.383474
$596$	0	0
$597$	−2180.31	−0.149471
$598$	0	0
$599$	2634.18	0.179682	0.0898410	−	0.995956i	$-0.471364\pi$
$599$	2634.18	0.179682	0.0898410	+	0.995956i	$0.471364\pi$
$600$	0	0
$601$	−8851.07	−0.600737	−0.300368	−	0.953823i	$-0.597110\pi$
$601$	−8851.07	−0.600737	−0.300368	+	0.953823i	$0.597110\pi$
$602$	0	0
$603$	−1404.20	−0.0948318
$604$	0	0
$605$	−3861.61	−0.259499
$606$	0	0
$607$	23813.5	1.59236	0.796179	−	0.605061i	$-0.206851\pi$
$607$	23813.5	1.59236	0.796179	+	0.605061i	$0.206851\pi$
$608$	0	0
$609$	−6405.34	−0.426203
$610$	0	0
$611$	−21739.8	−1.43944
$612$	0	0
$613$	−23328.8	−1.53710	−0.768550	−	0.639789i	$-0.779022\pi$
$613$	−23328.8	−1.53710	−0.768550	+	0.639789i	$0.779022\pi$
$614$	0	0
$615$	5693.23	0.373290
$616$	0	0
$617$	−1390.54	−0.0907311	−0.0453656	−	0.998970i	$-0.514445\pi$
$617$	−1390.54	−0.0907311	−0.0453656	+	0.998970i	$0.514445\pi$
$618$	0	0
$619$	−10219.6	−0.663585	−0.331793	−	0.943352i	$-0.607653\pi$
$619$	−10219.6	−0.663585	−0.331793	+	0.943352i	$0.607653\pi$
$620$	0	0
$621$	7025.06	0.453955
$622$	0	0
$623$	−18714.7	−1.20351
$624$	0	0
$625$	216.410	0.0138503
$626$	0	0
$627$	3839.24	0.244536
$628$	0	0
$629$	−14755.5	−0.935361
$630$	0	0
$631$	−22379.8	−1.41193	−0.705965	−	0.708247i	$-0.749486\pi$
$631$	−22379.8	−1.41193	−0.705965	+	0.708247i	$0.749486\pi$
$632$	0	0
$633$	12638.9	0.793601
$634$	0	0
$635$	−1622.37	−0.101389
$636$	0	0
$637$	4026.40	0.250442
$638$	0	0
$639$	−8069.24	−0.499553
$640$	0	0
$641$	30736.3	1.89393	0.946967	−	0.321330i	$-0.104130\pi$
$641$	30736.3	1.89393	0.946967	+	0.321330i	$0.104130\pi$
$642$	0	0
$643$	−12285.8	−0.753505	−0.376753	−	0.926314i	$-0.622959\pi$
$643$	−12285.8	−0.753505	−0.376753	+	0.926314i	$0.622959\pi$
$644$	0	0
$645$	−4551.64	−0.277861
$646$	0	0
$647$	3102.54	0.188522	0.0942608	−	0.995548i	$-0.469951\pi$
$647$	3102.54	0.188522	0.0942608	+	0.995548i	$0.469951\pi$
$648$	0	0
$649$	21105.2	1.27650
$650$	0	0
$651$	5218.98	0.314206
$652$	0	0
$653$	−28262.6	−1.69372	−0.846861	−	0.531815i	$-0.821510\pi$
$653$	−28262.6	−1.69372	−0.846861	+	0.531815i	$0.821510\pi$
$654$	0	0
$655$	7809.18	0.465847
$656$	0	0
$657$	617.392	0.0366617
$658$	0	0
$659$	11285.8	0.667121	0.333560	−	0.942729i	$-0.391750\pi$
$659$	11285.8	0.667121	0.333560	+	0.942729i	$0.391750\pi$
$660$	0	0
$661$	−11304.6	−0.665200	−0.332600	−	0.943068i	$-0.607926\pi$
$661$	−11304.6	−0.665200	−0.332600	+	0.943068i	$0.607926\pi$
$662$	0	0
$663$	9068.91	0.531233
$664$	0	0
$665$	−6592.04	−0.384404
$666$	0	0
$667$	9102.86	0.528432
$668$	0	0
$669$	−14212.0	−0.821328
$670$	0	0
$671$	3116.14	0.179280
$672$	0	0
$673$	25203.6	1.44358	0.721789	−	0.692114i	$-0.243320\pi$
$673$	25203.6	1.44358	0.721789	+	0.692114i	$0.243320\pi$
$674$	0	0
$675$	9197.85	0.524482
$676$	0	0
$677$	8294.62	0.470884	0.235442	−	0.971888i	$-0.424346\pi$
$677$	8294.62	0.470884	0.235442	+	0.971888i	$0.424346\pi$
$678$	0	0
$679$	19404.8	1.09674
$680$	0	0
$681$	−8165.74	−0.459489
$682$	0	0
$683$	30442.3	1.70548	0.852739	−	0.522336i	$-0.174939\pi$
$683$	30442.3	1.70548	0.852739	+	0.522336i	$0.174939\pi$
$684$	0	0
$685$	−7707.25	−0.429896
$686$	0	0
$687$	−4060.81	−0.225516
$688$	0	0
$689$	−7129.91	−0.394235
$690$	0	0
$691$	−11262.0	−0.620011	−0.310006	−	0.950735i	$-0.600331\pi$
$691$	−11262.0	−0.620011	−0.310006	+	0.950735i	$0.600331\pi$
$692$	0	0
$693$	9906.07	0.543002
$694$	0	0
$695$	−6173.04	−0.336916
$696$	0	0
$697$	−16085.9	−0.874172
$698$	0	0
$699$	9342.03	0.505505
$700$	0	0
$701$	−1587.59	−0.0855385	−0.0427693	−	0.999085i	$-0.513618\pi$
$701$	−1587.59	−0.0855385	−0.0427693	+	0.999085i	$0.513618\pi$
$702$	0	0
$703$	−17476.9	−0.937629
$704$	0	0
$705$	4724.83	0.252408
$706$	0	0
$707$	11002.4	0.585272
$708$	0	0
$709$	−22769.1	−1.20608	−0.603039	−	0.797711i	$-0.706044\pi$
$709$	−22769.1	−1.20608	−0.603039	+	0.797711i	$0.706044\pi$
$710$	0	0
$711$	−1978.62	−0.104366
$712$	0	0
$713$	−7416.88	−0.389571
$714$	0	0
$715$	14718.5	0.769846
$716$	0	0
$717$	4585.02	0.238816
$718$	0	0
$719$	−32032.9	−1.66151	−0.830754	−	0.556640i	$-0.812090\pi$
$719$	−32032.9	−1.66151	−0.830754	+	0.556640i	$0.812090\pi$
$720$	0	0
$721$	−22137.1	−1.14345
$722$	0	0
$723$	12881.7	0.662622
$724$	0	0
$725$	11918.3	0.610531
$726$	0	0
$727$	2749.43	0.140263	0.0701313	−	0.997538i	$-0.477658\pi$
$727$	2749.43	0.140263	0.0701313	+	0.997538i	$0.477658\pi$
$728$	0	0
$729$	2036.29	0.103454
$730$	0	0
$731$	12860.4	0.650698
$732$	0	0
$733$	13610.8	0.685849	0.342924	−	0.939363i	$-0.388583\pi$
$733$	13610.8	0.685849	0.342924	+	0.939363i	$0.388583\pi$
$734$	0	0
$735$	−875.080	−0.0439154
$736$	0	0
$737$	−1856.24	−0.0927752
$738$	0	0
$739$	−26702.9	−1.32920	−0.664602	−	0.747197i	$-0.731399\pi$
$739$	−26702.9	−1.32920	−0.664602	+	0.747197i	$0.731399\pi$
$740$	0	0
$741$	10741.5	0.532521
$742$	0	0
$743$	1114.04	0.0550071	0.0275036	−	0.999622i	$-0.491244\pi$
$743$	1114.04	0.0550071	0.0275036	+	0.999622i	$0.491244\pi$
$744$	0	0
$745$	10108.1	0.497088
$746$	0	0
$747$	−5656.44	−0.277053
$748$	0	0
$749$	−17940.7	−0.875218
$750$	0	0
$751$	30750.5	1.49414	0.747072	−	0.664743i	$-0.231459\pi$
$751$	30750.5	1.49414	0.747072	+	0.664743i	$0.231459\pi$
$752$	0	0
$753$	4697.50	0.227339
$754$	0	0
$755$	−19564.7	−0.943089
$756$	0	0
$757$	24239.7	1.16381	0.581907	−	0.813255i	$-0.302307\pi$
$757$	24239.7	1.16381	0.581907	+	0.813255i	$0.302307\pi$
$758$	0	0
$759$	4058.31	0.194081
$760$	0	0
$761$	6044.16	0.287911	0.143956	−	0.989584i	$-0.454018\pi$
$761$	6044.16	0.287911	0.143956	+	0.989584i	$0.454018\pi$
$762$	0	0
$763$	6730.65	0.319352
$764$	0	0
$765$	6837.19	0.323136
$766$	0	0
$767$	59048.3	2.77981
$768$	0	0
$769$	27618.3	1.29511	0.647555	−	0.762019i	$-0.275792\pi$
$769$	27618.3	1.29511	0.647555	+	0.762019i	$0.275792\pi$
$770$	0	0
$771$	−3122.39	−0.145850
$772$	0	0
$773$	32659.9	1.51966	0.759828	−	0.650124i	$-0.225283\pi$
$773$	32659.9	1.51966	0.759828	+	0.650124i	$0.225283\pi$
$774$	0	0
$775$	−9710.85	−0.450096
$776$	0	0
$777$	12999.6	0.600202
$778$	0	0
$779$	−19052.6	−0.876292
$780$	0	0
$781$	−10666.8	−0.488719
$782$	0	0
$783$	18006.0	0.821817
$784$	0	0
$785$	7731.78	0.351540
$786$	0	0
$787$	31089.8	1.40817	0.704086	−	0.710115i	$-0.251357\pi$
$787$	31089.8	1.40817	0.704086	+	0.710115i	$0.251357\pi$
$788$	0	0
$789$	4243.87	0.191490
$790$	0	0
$791$	7544.00	0.339107
$792$	0	0
$793$	8718.37	0.390414
$794$	0	0
$795$	1549.58	0.0691297
$796$	0	0
$797$	−7372.64	−0.327669	−0.163834	−	0.986488i	$-0.552386\pi$
$797$	−7372.64	−0.327669	−0.163834	+	0.986488i	$0.552386\pi$
$798$	0	0
$799$	−13349.8	−0.591090
$800$	0	0
$801$	22990.6	1.01415
$802$	0	0
$803$	816.140	0.0358667
$804$	0	0
$805$	−6968.19	−0.305089
$806$	0	0
$807$	−11093.7	−0.483912
$808$	0	0
$809$	25141.0	1.09259	0.546297	−	0.837591i	$-0.316037\pi$
$809$	25141.0	1.09259	0.546297	+	0.837591i	$0.316037\pi$
$810$	0	0
$811$	21022.2	0.910219	0.455110	−	0.890435i	$-0.349600\pi$
$811$	21022.2	0.910219	0.455110	+	0.890435i	$0.349600\pi$
$812$	0	0
$813$	4885.99	0.210774
$814$	0	0
$815$	−28138.6	−1.20939
$816$	0	0
$817$	15232.3	0.652276
$818$	0	0
$819$	27715.4	1.18248
$820$	0	0
$821$	380.058	0.0161561	0.00807803	−	0.999967i	$-0.497429\pi$
$821$	380.058	0.0161561	0.00807803	+	0.999967i	$0.497429\pi$
$822$	0	0
$823$	−21191.4	−0.897552	−0.448776	−	0.893644i	$-0.648140\pi$
$823$	−21191.4	−0.897552	−0.448776	+	0.893644i	$0.648140\pi$
$824$	0	0
$825$	5313.51	0.224234
$826$	0	0
$827$	−36861.7	−1.54995	−0.774975	−	0.631992i	$-0.782237\pi$
$827$	−36861.7	−1.54995	−0.774975	+	0.631992i	$0.782237\pi$
$828$	0	0
$829$	36636.4	1.53490	0.767451	−	0.641107i	$-0.221525\pi$
$829$	36636.4	1.53490	0.767451	+	0.641107i	$0.221525\pi$
$830$	0	0
$831$	−15618.7	−0.651994
$832$	0	0
$833$	2472.49	0.102841
$834$	0	0
$835$	−8031.06	−0.332846
$836$	0	0
$837$	−14671.0	−0.605860
$838$	0	0
$839$	18677.2	0.768544	0.384272	−	0.923220i	$-0.374452\pi$
$839$	18677.2	0.768544	0.384272	+	0.923220i	$0.374452\pi$
$840$	0	0
$841$	−1057.32	−0.0433524
$842$	0	0
$843$	14270.7	0.583047
$844$	0	0
$845$	26122.0	1.06346
$846$	0	0
$847$	−9612.34	−0.389946
$848$	0	0
$849$	11796.3	0.476851
$850$	0	0
$851$	−18474.1	−0.744166
$852$	0	0
$853$	−34718.5	−1.39360	−0.696799	−	0.717266i	$-0.745393\pi$
$853$	−34718.5	−1.39360	−0.696799	+	0.717266i	$0.745393\pi$
$854$	0	0
$855$	8098.17	0.323920
$856$	0	0
$857$	21642.3	0.862644	0.431322	−	0.902198i	$-0.358047\pi$
$857$	21642.3	0.862644	0.431322	+	0.902198i	$0.358047\pi$
$858$	0	0
$859$	−30946.5	−1.22920	−0.614600	−	0.788839i	$-0.710682\pi$
$859$	−30946.5	−1.22920	−0.614600	+	0.788839i	$0.710682\pi$
$860$	0	0
$861$	14171.6	0.560938
$862$	0	0
$863$	7293.89	0.287702	0.143851	−	0.989599i	$-0.454051\pi$
$863$	7293.89	0.287702	0.143851	+	0.989599i	$0.454051\pi$
$864$	0	0
$865$	28067.7	1.10327
$866$	0	0
$867$	−6507.19	−0.254897
$868$	0	0
$869$	−2615.56	−0.102102
$870$	0	0
$871$	−5193.41	−0.202034
$872$	0	0
$873$	−23838.3	−0.924174
$874$	0	0
$875$	−23739.3	−0.917182
$876$	0	0
$877$	29977.5	1.15424	0.577120	−	0.816659i	$-0.304176\pi$
$877$	29977.5	1.15424	0.577120	+	0.816659i	$0.304176\pi$
$878$	0	0
$879$	−12677.6	−0.486466
$880$	0	0
$881$	10453.2	0.399747	0.199873	−	0.979822i	$-0.435947\pi$
$881$	10453.2	0.399747	0.199873	+	0.979822i	$0.435947\pi$
$882$	0	0
$883$	36419.6	1.38801	0.694007	−	0.719968i	$-0.255844\pi$
$883$	36419.6	1.38801	0.694007	+	0.719968i	$0.255844\pi$
$884$	0	0
$885$	−12833.3	−0.487443
$886$	0	0
$887$	24903.3	0.942694	0.471347	−	0.881948i	$-0.343768\pi$
$887$	24903.3	0.942694	0.471347	+	0.881948i	$0.343768\pi$
$888$	0	0
$889$	−4038.41	−0.152355
$890$	0	0
$891$	−7649.93	−0.287635
$892$	0	0
$893$	−15811.9	−0.592524
$894$	0	0
$895$	4749.28	0.177376
$896$	0	0
$897$	11354.4	0.422645
$898$	0	0
$899$	−19010.3	−0.705260
$900$	0	0
$901$	−4378.27	−0.161888
$902$	0	0
$903$	−11330.0	−0.417539
$904$	0	0
$905$	13598.3	0.499472
$906$	0	0
$907$	−51937.9	−1.90140	−0.950700	−	0.310111i	$-0.899634\pi$
$907$	−51937.9	−1.90140	−0.950700	+	0.310111i	$0.899634\pi$
$908$	0	0
$909$	−13516.2	−0.493183
$910$	0	0
$911$	−35627.9	−1.29572	−0.647862	−	0.761757i	$-0.724337\pi$
$911$	−35627.9	−1.29572	−0.647862	+	0.761757i	$0.724337\pi$
$912$	0	0
$913$	−7477.33	−0.271044
$914$	0	0
$915$	−1894.81	−0.0684597
$916$	0	0
$917$	19438.7	0.700023
$918$	0	0
$919$	11983.0	0.430124	0.215062	−	0.976600i	$-0.431005\pi$
$919$	11983.0	0.430124	0.215062	+	0.976600i	$0.431005\pi$
$920$	0	0
$921$	−18368.8	−0.657191
$922$	0	0
$923$	−29843.9	−1.06427
$924$	0	0
$925$	−24188.0	−0.859781
$926$	0	0
$927$	27195.0	0.963538
$928$	0	0
$929$	26118.0	0.922395	0.461197	−	0.887298i	$-0.347420\pi$
$929$	26118.0	0.922395	0.461197	+	0.887298i	$0.347420\pi$
$930$	0	0
$931$	2928.49	0.103091
$932$	0	0
$933$	4496.88	0.157793
$934$	0	0
$935$	9038.19	0.316129
$936$	0	0
$937$	−33094.8	−1.15385	−0.576927	−	0.816796i	$-0.695748\pi$
$937$	−33094.8	−1.15385	−0.576927	+	0.816796i	$0.695748\pi$
$938$	0	0
$939$	−934.110	−0.0324638
$940$	0	0
$941$	4961.63	0.171886	0.0859429	−	0.996300i	$-0.472610\pi$
$941$	4961.63	0.171886	0.0859429	+	0.996300i	$0.472610\pi$
$942$	0	0
$943$	−20139.8	−0.695485
$944$	0	0
$945$	−13783.5	−0.474474
$946$	0	0
$947$	−10695.5	−0.367010	−0.183505	−	0.983019i	$-0.558744\pi$
$947$	−10695.5	−0.367010	−0.183505	+	0.983019i	$0.558744\pi$
$948$	0	0
$949$	2283.41	0.0781060
$950$	0	0
$951$	−10971.8	−0.374117
$952$	0	0
$953$	−25630.9	−0.871213	−0.435606	−	0.900137i	$-0.643466\pi$
$953$	−25630.9	−0.871213	−0.435606	+	0.900137i	$0.643466\pi$
$954$	0	0
$955$	−2117.94	−0.0717645
$956$	0	0
$957$	10401.9	0.351354
$958$	0	0
$959$	−19184.9	−0.646000
$960$	0	0
$961$	−14301.7	−0.480068
$962$	0	0
$963$	22039.7	0.737507
$964$	0	0
$965$	19502.1	0.650566
$966$	0	0
$967$	19075.8	0.634371	0.317185	−	0.948364i	$-0.397262\pi$
$967$	19075.8	0.634371	0.317185	+	0.948364i	$0.397262\pi$
$968$	0	0
$969$	6596.03	0.218674
$970$	0	0
$971$	25037.7	0.827496	0.413748	−	0.910391i	$-0.364219\pi$
$971$	25037.7	0.827496	0.413748	+	0.910391i	$0.364219\pi$
$972$	0	0
$973$	−15366.0	−0.506280
$974$	0	0
$975$	14866.2	0.488308
$976$	0	0
$977$	−26519.8	−0.868417	−0.434209	−	0.900812i	$-0.642972\pi$
$977$	−26519.8	−0.868417	−0.434209	+	0.900812i	$0.642972\pi$
$978$	0	0
$979$	30391.6	0.992156
$980$	0	0
$981$	−8268.44	−0.269104
$982$	0	0
$983$	18827.6	0.610892	0.305446	−	0.952209i	$-0.401195\pi$
$983$	18827.6	0.610892	0.305446	+	0.952209i	$0.401195\pi$
$984$	0	0
$985$	−20088.2	−0.649810
$986$	0	0
$987$	11761.1	0.379290
$988$	0	0
$989$	16101.4	0.517690
$990$	0	0
$991$	−9838.93	−0.315382	−0.157691	−	0.987488i	$-0.550405\pi$
$991$	−9838.93	−0.315382	−0.157691	+	0.987488i	$0.550405\pi$
$992$	0	0
$993$	4345.51	0.138873
$994$	0	0
$995$	−6079.44	−0.193700
$996$	0	0
$997$	52540.9	1.66899	0.834497	−	0.551013i	$-0.185758\pi$
$997$	52540.9	1.66899	0.834497	+	0.551013i	$0.185758\pi$
$998$	0	0
$999$	−36543.0	−1.15733

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	536.4.a.b.1.7	✓	11
4.3	odd	2		1072.4.a.k.1.5		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
536.4.a.b.1.7	✓	11	1.1	even	1	trivial
1072.4.a.k.1.5		11	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 536 = 2^{3} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	536.a (trivial)

\(f(q)\)	\(=\)	\(q+2.45800 q^{3} +6.85373 q^{5} +17.0604 q^{7} -20.9582 q^{9} +O(q^{10})\)	\(q+2.45800 q^{3} +6.85373 q^{5} +17.0604 q^{7} -20.9582 q^{9} -27.7050 q^{11} -77.5135 q^{13} +16.8465 q^{15} -47.5988 q^{17} -56.3774 q^{19} +41.9343 q^{21} -59.5943 q^{23} -78.0264 q^{25} -117.881 q^{27} -152.747 q^{29} +124.456 q^{31} -68.0989 q^{33} +116.927 q^{35} +309.998 q^{37} -190.528 q^{39} +337.948 q^{41} -270.184 q^{43} -143.642 q^{45} +280.464 q^{47} -51.9444 q^{49} -116.998 q^{51} +91.9828 q^{53} -189.883 q^{55} -138.575 q^{57} -761.781 q^{59} -112.475 q^{61} -357.555 q^{63} -531.257 q^{65} +67.0000 q^{67} -146.483 q^{69} +385.015 q^{71} -29.4582 q^{73} -191.789 q^{75} -472.657 q^{77} +94.4076 q^{79} +276.121 q^{81} +269.891 q^{83} -326.229 q^{85} -375.452 q^{87} -1096.97 q^{89} -1322.41 q^{91} +305.913 q^{93} -386.395 q^{95} +1137.42 q^{97} +580.649 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 12 q^{3} - 3 q^{5} - 37 q^{7} + 93 q^{9}+O(q^{10}) \) 11 * q - 12 * q^3 - 3 * q^5 - 37 * q^7 + 93 * q^9	\( 11 q - 12 q^{3} - 3 q^{5} - 37 q^{7} + 93 q^{9} - 23 q^{11} - 96 q^{13} - 45 q^{15} + 145 q^{17} - 121 q^{19} - 104 q^{21} - 133 q^{23} + 388 q^{25} - 570 q^{27} - 388 q^{29} - 318 q^{31} + 127 q^{33} - 641 q^{35} - 353 q^{37} - 406 q^{39} + 53 q^{41} - 954 q^{43} - 658 q^{45} - 1848 q^{47} - 408 q^{49} - 1850 q^{51} - 973 q^{53} - 2300 q^{55} - 829 q^{57} - 1062 q^{59} - 1630 q^{61} - 1616 q^{63} - 1019 q^{65} + 737 q^{67} - 1600 q^{69} - 3413 q^{71} + 355 q^{73} - 4986 q^{75} - 295 q^{77} - 2981 q^{79} + 559 q^{81} - 2867 q^{83} + 623 q^{85} - 2257 q^{87} + 117 q^{89} - 3379 q^{91} - 275 q^{93} - 3253 q^{95} - 302 q^{97} - 5775 q^{99}+O(q^{100}) \) 11 * q - 12 * q^3 - 3 * q^5 - 37 * q^7 + 93 * q^9 - 23 * q^11 - 96 * q^13 - 45 * q^15 + 145 * q^17 - 121 * q^19 - 104 * q^21 - 133 * q^23 + 388 * q^25 - 570 * q^27 - 388 * q^29 - 318 * q^31 + 127 * q^33 - 641 * q^35 - 353 * q^37 - 406 * q^39 + 53 * q^41 - 954 * q^43 - 658 * q^45 - 1848 * q^47 - 408 * q^49 - 1850 * q^51 - 973 * q^53 - 2300 * q^55 - 829 * q^57 - 1062 * q^59 - 1630 * q^61 - 1616 * q^63 - 1019 * q^65 + 737 * q^67 - 1600 * q^69 - 3413 * q^71 + 355 * q^73 - 4986 * q^75 - 295 * q^77 - 2981 * q^79 + 559 * q^81 - 2867 * q^83 + 623 * q^85 - 2257 * q^87 + 117 * q^89 - 3379 * q^91 - 275 * q^93 - 3253 * q^95 - 302 * q^97 - 5775 * q^99

Introduction

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Modular forms

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