LMFDB - Embedded newform 1617.4.a.n.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1617,4,Mod(1,1617)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1617, 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("1617.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.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:	$95.4060884793$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 30x^{3} + 11x^{2} + 185x + 162 $ x^5 - x^4 - 30x^3 + 11x^2 + 185*x + 162
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.59998$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.59998 q^{2} +3.00000 q^{3} -5.44007 q^{4} +17.2762 q^{5} +4.79994 q^{6} -21.5038 q^{8} +9.00000 q^{9} +27.6416 q^{10} +11.0000 q^{11} -16.3202 q^{12} -46.1062 q^{13} +51.8287 q^{15} +9.11483 q^{16} -19.8752 q^{17} +14.3998 q^{18} -76.5707 q^{19} -93.9838 q^{20} +17.5998 q^{22} -163.205 q^{23} -64.5115 q^{24} +173.468 q^{25} -73.7690 q^{26} +27.0000 q^{27} +158.858 q^{29} +82.9248 q^{30} -170.768 q^{31} +186.614 q^{32} +33.0000 q^{33} -31.7999 q^{34} -48.9606 q^{36} -245.971 q^{37} -122.511 q^{38} -138.319 q^{39} -371.505 q^{40} +3.33673 q^{41} -122.798 q^{43} -59.8407 q^{44} +155.486 q^{45} -261.125 q^{46} -390.972 q^{47} +27.3445 q^{48} +277.545 q^{50} -59.6255 q^{51} +250.821 q^{52} -410.957 q^{53} +43.1995 q^{54} +190.038 q^{55} -229.712 q^{57} +254.169 q^{58} -408.774 q^{59} -281.951 q^{60} +21.9747 q^{61} -273.225 q^{62} +225.660 q^{64} -796.541 q^{65} +52.7993 q^{66} +618.424 q^{67} +108.122 q^{68} -489.616 q^{69} +929.041 q^{71} -193.534 q^{72} -868.090 q^{73} -393.549 q^{74} +520.404 q^{75} +416.549 q^{76} -221.307 q^{78} +152.702 q^{79} +157.470 q^{80} +81.0000 q^{81} +5.33871 q^{82} +100.924 q^{83} -343.368 q^{85} -196.475 q^{86} +476.573 q^{87} -236.542 q^{88} +1063.23 q^{89} +248.774 q^{90} +887.848 q^{92} -512.303 q^{93} -625.547 q^{94} -1322.85 q^{95} +559.843 q^{96} -1415.50 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 21 q^{5} - 3 q^{6} - 42 q^{8} + 45 q^{9} + 23 q^{10} + 55 q^{11} + 63 q^{12} - 101 q^{13} - 63 q^{15} - 7 q^{16} + 20 q^{17} - 9 q^{18} - 237 q^{19} - 85 q^{20} - 11 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 21 * q^5 - 3 * q^6 - 42 * q^8 + 45 * q^9 + 23 * q^10 + 55 * q^11 + 63 * q^12 - 101 * q^13 - 63 * q^15 - 7 * q^16 + 20 * q^17 - 9 * q^18 - 237 * q^19 - 85 * q^20 - 11 * q^22 - 80 * q^23 - 126 * q^24 + 486 * q^25 - 165 * q^26 + 135 * q^27 - 11 * q^29 + 69 * q^30 - 316 * q^31 + 453 * q^32 + 165 * q^33 - 936 * q^34 + 189 * q^36 + 319 * q^37 - 89 * q^38 - 303 * q^39 - 624 * q^40 - 1190 * q^41 + 88 * q^43 + 231 * q^44 - 189 * q^45 + 1000 * q^46 - 377 * q^47 - 21 * q^48 - 644 * q^50 + 60 * q^51 - 1001 * q^52 - 992 * q^53 - 27 * q^54 - 231 * q^55 - 711 * q^57 + 721 * q^58 - 71 * q^59 - 255 * q^60 + 574 * q^61 - 272 * q^62 - 1380 * q^64 + 589 * q^65 - 33 * q^66 - 527 * q^67 + 2974 * q^68 - 240 * q^69 - 1156 * q^71 - 378 * q^72 - 1061 * q^73 - 1609 * q^74 + 1458 * q^75 - 2399 * q^76 - 495 * q^78 + 588 * q^79 + 1643 * q^80 + 405 * q^81 + 2602 * q^82 + 212 * q^83 + 1918 * q^85 - 4760 * q^86 - 33 * q^87 - 462 * q^88 - 1030 * q^89 + 207 * q^90 - 1174 * q^92 - 948 * q^93 + 1799 * q^94 - 3593 * q^95 + 1359 * q^96 - 2488 * q^97 + 495 * 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 $

$2$	1.59998	0.565678	0.282839	−	0.959167i	$-0.408724\pi$
$2$	1.59998	0.565678	0.282839	+	0.959167i	$0.408724\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.44007	−0.680008
$5$	17.2762	1.54523	0.772616	−	0.634873i	$-0.218948\pi$
$5$	17.2762	1.54523	0.772616	+	0.634873i	$0.218948\pi$
$6$	4.79994	0.326594
$7$	0	0
$7$	0	0
$8$	−21.5038	−0.950344
$9$	9.00000	0.333333
$10$	27.6416	0.874104
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−16.3202	−0.392603
$13$	−46.1062	−0.983658	−0.491829	−	0.870692i	$-0.663672\pi$
$13$	−46.1062	−0.983658	−0.491829	+	0.870692i	$0.663672\pi$
$14$	0	0
$15$	51.8287	0.892140
$16$	9.11483	0.142419
$17$	−19.8752	−0.283555	−0.141778	−	0.989899i	$-0.545282\pi$
$17$	−19.8752	−0.283555	−0.141778	+	0.989899i	$0.545282\pi$
$18$	14.3998	0.188559
$19$	−76.5707	−0.924553	−0.462277	−	0.886736i	$-0.652967\pi$
$19$	−76.5707	−0.924553	−0.462277	+	0.886736i	$0.652967\pi$
$20$	−93.9838	−1.05077
$21$	0	0
$22$	17.5998	0.170558
$23$	−163.205	−1.47960	−0.739798	−	0.672829i	$-0.765079\pi$
$23$	−163.205	−1.47960	−0.739798	+	0.672829i	$0.765079\pi$
$24$	−64.5115	−0.548681
$25$	173.468	1.38774
$26$	−73.7690	−0.556434
$27$	27.0000	0.192450
$28$	0	0
$29$	158.858	1.01721	0.508605	−	0.861000i	$-0.330161\pi$
$29$	158.858	1.01721	0.508605	+	0.861000i	$0.330161\pi$
$30$	82.9248	0.504664
$31$	−170.768	−0.989380	−0.494690	−	0.869069i	$-0.664718\pi$
$31$	−170.768	−0.989380	−0.494690	+	0.869069i	$0.664718\pi$
$32$	186.614	1.03091
$33$	33.0000	0.174078
$34$	−31.7999	−0.160401
$35$	0	0
$36$	−48.9606	−0.226669
$37$	−245.971	−1.09290	−0.546452	−	0.837490i	$-0.684022\pi$
$37$	−245.971	−1.09290	−0.546452	+	0.837490i	$0.684022\pi$
$38$	−122.511	−0.523000
$39$	−138.319	−0.567915
$40$	−371.505	−1.46850
$41$	3.33673	0.0127100	0.00635500	−	0.999980i	$-0.497977\pi$
$41$	3.33673	0.0127100	0.00635500	+	0.999980i	$0.497977\pi$
$42$	0	0
$43$	−122.798	−0.435502	−0.217751	−	0.976004i	$-0.569872\pi$
$43$	−122.798	−0.435502	−0.217751	+	0.976004i	$0.569872\pi$
$44$	−59.8407	−0.205030
$45$	155.486	0.515077
$46$	−261.125	−0.836975
$47$	−390.972	−1.21338	−0.606692	−	0.794937i	$-0.707504\pi$
$47$	−390.972	−1.21338	−0.606692	+	0.794937i	$0.707504\pi$
$48$	27.3445	0.0822258
$49$	0	0
$50$	277.545	0.785016
$51$	−59.6255	−0.163711
$52$	250.821	0.668896
$53$	−410.957	−1.06508	−0.532541	−	0.846404i	$-0.678763\pi$
$53$	−410.957	−1.06508	−0.532541	+	0.846404i	$0.678763\pi$
$54$	43.1995	0.108865
$55$	190.038	0.465905
$56$	0	0
$57$	−229.712	−0.533791
$58$	254.169	0.575414
$59$	−408.774	−0.901998	−0.450999	−	0.892524i	$-0.648932\pi$
$59$	−408.774	−0.901998	−0.450999	+	0.892524i	$0.648932\pi$
$60$	−281.951	−0.606663
$61$	21.9747	0.0461241	0.0230621	−	0.999734i	$-0.492658\pi$
$61$	21.9747	0.0461241	0.0230621	+	0.999734i	$0.492658\pi$
$62$	−273.225	−0.559671
$63$	0	0
$64$	225.660	0.440743
$65$	−796.541	−1.51998
$66$	52.7993	0.0984719
$67$	618.424	1.12765	0.563824	−	0.825895i	$-0.309329\pi$
$67$	618.424	1.12765	0.563824	+	0.825895i	$0.309329\pi$
$68$	108.122	0.192820
$69$	−489.616	−0.854245
$70$	0	0
$71$	929.041	1.55291	0.776457	−	0.630171i	$-0.217015\pi$
$71$	929.041	1.55291	0.776457	+	0.630171i	$0.217015\pi$
$72$	−193.534	−0.316781
$73$	−868.090	−1.39181	−0.695906	−	0.718133i	$-0.744997\pi$
$73$	−868.090	−1.39181	−0.695906	+	0.718133i	$0.744997\pi$
$74$	−393.549	−0.618232
$75$	520.404	0.801214
$76$	416.549	0.628704
$77$	0	0
$78$	−221.307	−0.321257
$79$	152.702	0.217472	0.108736	−	0.994071i	$-0.465320\pi$
$79$	152.702	0.217472	0.108736	+	0.994071i	$0.465320\pi$
$80$	157.470	0.220071
$81$	81.0000	0.111111
$82$	5.33871	0.00718977
$83$	100.924	0.133469	0.0667343	−	0.997771i	$-0.478742\pi$
$83$	100.924	0.133469	0.0667343	+	0.997771i	$0.478742\pi$
$84$	0	0
$85$	−343.368	−0.438159
$86$	−196.475	−0.246354
$87$	476.573	0.587287
$88$	−236.542	−0.286540
$89$	1063.23	1.26631	0.633156	−	0.774024i	$-0.281759\pi$
$89$	1063.23	1.26631	0.633156	+	0.774024i	$0.281759\pi$
$90$	248.774	0.291368
$91$	0	0
$92$	887.848	1.00614
$93$	−512.303	−0.571219
$94$	−625.547	−0.686385
$95$	−1322.85	−1.42865
$96$	559.843	0.595195
$97$	−1415.50	−1.48167	−0.740836	−	0.671686i	$-0.765570\pi$
$97$	−1415.50	−1.48167	−0.740836	+	0.671686i	$0.765570\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	−943.677	−0.943677
$101$	774.140	0.762671	0.381336	−	0.924437i	$-0.375464\pi$
$101$	774.140	0.762671	0.381336	+	0.924437i	$0.375464\pi$
$102$	−95.3997	−0.0926076
$103$	−759.159	−0.726235	−0.363118	−	0.931743i	$-0.618288\pi$
$103$	−759.159	−0.726235	−0.363118	+	0.931743i	$0.618288\pi$
$104$	991.460	0.934814
$105$	0	0
$106$	−657.523	−0.602493
$107$	−1181.22	−1.06722	−0.533612	−	0.845729i	$-0.679166\pi$
$107$	−1181.22	−1.06722	−0.533612	+	0.845729i	$0.679166\pi$
$108$	−146.882	−0.130868
$109$	738.091	0.648590	0.324295	−	0.945956i	$-0.394873\pi$
$109$	738.091	0.648590	0.324295	+	0.945956i	$0.394873\pi$
$110$	304.058	0.263552
$111$	−737.914	−0.630989
$112$	0	0
$113$	620.181	0.516298	0.258149	−	0.966105i	$-0.416887\pi$
$113$	620.181	0.516298	0.258149	+	0.966105i	$0.416887\pi$
$114$	−367.534	−0.301954
$115$	−2819.57	−2.28632
$116$	−864.195	−0.691711
$117$	−414.956	−0.327886
$118$	−654.031	−0.510241
$119$	0	0
$120$	−1114.51	−0.847840
$121$	121.000	0.0909091
$122$	35.1591	0.0260914
$123$	10.0102	0.00733813
$124$	928.988	0.672787
$125$	837.342	0.599153
$126$	0	0
$127$	−2291.79	−1.60129	−0.800645	−	0.599139i	$-0.795510\pi$
$127$	−2291.79	−1.60129	−0.800645	+	0.599139i	$0.795510\pi$
$128$	−1131.86	−0.781589
$129$	−368.395	−0.251437
$130$	−1274.45	−0.859820
$131$	1491.77	0.994938	0.497469	−	0.867482i	$-0.334263\pi$
$131$	1491.77	0.994938	0.497469	+	0.867482i	$0.334263\pi$
$132$	−179.522	−0.118374
$133$	0	0
$134$	989.465	0.637886
$135$	466.458	0.297380
$136$	427.393	0.269475
$137$	2590.79	1.61566	0.807831	−	0.589414i	$-0.200641\pi$
$137$	2590.79	1.61566	0.807831	+	0.589414i	$0.200641\pi$
$138$	−783.376	−0.483228
$139$	−2674.14	−1.63178	−0.815890	−	0.578207i	$-0.803752\pi$
$139$	−2674.14	−1.63178	−0.815890	+	0.578207i	$0.803752\pi$
$140$	0	0
$141$	−1172.92	−0.700548
$142$	1486.45	0.878449
$143$	−507.168	−0.296584
$144$	82.0335	0.0474731
$145$	2744.46	1.57183
$146$	−1388.93	−0.787318
$147$	0	0
$148$	1338.10	0.743184
$149$	822.355	0.452147	0.226074	−	0.974110i	$-0.427411\pi$
$149$	822.355	0.452147	0.226074	+	0.974110i	$0.427411\pi$
$150$	832.635	0.453229
$151$	−138.803	−0.0748053	−0.0374027	−	0.999300i	$-0.511908\pi$
$151$	−138.803	−0.0748053	−0.0374027	+	0.999300i	$0.511908\pi$
$152$	1646.56	0.878644
$153$	−178.877	−0.0945184
$154$	0	0
$155$	−2950.22	−1.52882
$156$	752.462	0.386187
$157$	634.746	0.322664	0.161332	−	0.986900i	$-0.448421\pi$
$157$	634.746	0.322664	0.161332	+	0.986900i	$0.448421\pi$
$158$	244.320	0.123019
$159$	−1232.87	−0.614925
$160$	3223.99	1.59299
$161$	0	0
$162$	129.598	0.0628531
$163$	−608.367	−0.292337	−0.146169	−	0.989260i	$-0.546694\pi$
$163$	−608.367	−0.292337	−0.146169	+	0.989260i	$0.546694\pi$
$164$	−18.1520	−0.00864291
$165$	570.115	0.268990
$166$	161.477	0.0755003
$167$	1545.50	0.716135	0.358067	−	0.933696i	$-0.383436\pi$
$167$	1545.50	0.716135	0.358067	+	0.933696i	$0.383436\pi$
$168$	0	0
$169$	−71.2185	−0.0324163
$170$	−549.382	−0.247857
$171$	−689.136	−0.308184
$172$	668.031	0.296145
$173$	−126.598	−0.0556362	−0.0278181	−	0.999613i	$-0.508856\pi$
$173$	−126.598	−0.0556362	−0.0278181	+	0.999613i	$0.508856\pi$
$174$	762.506	0.332215
$175$	0	0
$176$	100.263	0.0429410
$177$	−1226.32	−0.520769
$178$	1701.14	0.716325
$179$	−144.263	−0.0602387	−0.0301194	−	0.999546i	$-0.509589\pi$
$179$	−144.263	−0.0602387	−0.0301194	+	0.999546i	$0.509589\pi$
$180$	−845.854	−0.350257
$181$	−2925.01	−1.20118	−0.600591	−	0.799556i	$-0.705068\pi$
$181$	−2925.01	−1.20118	−0.600591	+	0.799556i	$0.705068\pi$
$182$	0	0
$183$	65.9241	0.0266298
$184$	3509.54	1.40612
$185$	−4249.46	−1.68879
$186$	−819.675	−0.323126
$187$	−218.627	−0.0854951
$188$	2126.91	0.825111
$189$	0	0
$190$	−2116.54	−0.808156
$191$	−3771.70	−1.42885	−0.714426	−	0.699711i	$-0.753312\pi$
$191$	−3771.70	−1.42885	−0.714426	+	0.699711i	$0.753312\pi$
$192$	676.981	0.254463
$193$	−1282.92	−0.478480	−0.239240	−	0.970960i	$-0.576898\pi$
$193$	−1282.92	−0.478480	−0.239240	+	0.970960i	$0.576898\pi$
$194$	−2264.77	−0.838149
$195$	−2389.62	−0.877561
$196$	0	0
$197$	3052.31	1.10390	0.551949	−	0.833878i	$-0.313884\pi$
$197$	3052.31	1.10390	0.551949	+	0.833878i	$0.313884\pi$
$198$	158.398	0.0568528
$199$	−5040.28	−1.79546	−0.897729	−	0.440547i	$-0.854784\pi$
$199$	−5040.28	−1.79546	−0.897729	+	0.440547i	$0.854784\pi$
$200$	−3730.22	−1.31883
$201$	1855.27	0.651048
$202$	1238.61	0.431426
$203$	0	0
$204$	324.367	0.111325
$205$	57.6462	0.0196399
$206$	−1214.64	−0.410815
$207$	−1468.85	−0.493198
$208$	−420.250	−0.140092
$209$	−842.277	−0.278763
$210$	0	0
$211$	3556.28	1.16031	0.580153	−	0.814508i	$-0.302993\pi$
$211$	3556.28	1.16031	0.580153	+	0.814508i	$0.302993\pi$
$212$	2235.63	0.724264
$213$	2787.12	0.896575
$214$	−1889.93	−0.603706
$215$	−2121.49	−0.672952
$216$	−580.603	−0.182894
$217$	0	0
$218$	1180.93	0.366893
$219$	−2604.27	−0.803563
$220$	−1033.82	−0.316819
$221$	916.369	0.278922
$222$	−1180.65	−0.356937
$223$	−1767.85	−0.530870	−0.265435	−	0.964129i	$-0.585516\pi$
$223$	−1767.85	−0.530870	−0.265435	+	0.964129i	$0.585516\pi$
$224$	0	0
$225$	1561.21	0.462581
$226$	992.277	0.292059
$227$	−4836.74	−1.41421	−0.707105	−	0.707109i	$-0.749999\pi$
$227$	−4836.74	−1.41421	−0.707105	+	0.707109i	$0.749999\pi$
$228$	1249.65	0.362982
$229$	−4941.30	−1.42590	−0.712949	−	0.701216i	$-0.752641\pi$
$229$	−4941.30	−1.42590	−0.712949	+	0.701216i	$0.752641\pi$
$230$	−4511.26	−1.29332
$231$	0	0
$232$	−3416.04	−0.966700
$233$	4822.14	1.35583	0.677916	−	0.735140i	$-0.262883\pi$
$233$	4822.14	1.35583	0.677916	+	0.735140i	$0.262883\pi$
$234$	−663.921	−0.185478
$235$	−6754.51	−1.87496
$236$	2223.76	0.613366
$237$	458.106	0.125558
$238$	0	0
$239$	4821.80	1.30501	0.652503	−	0.757786i	$-0.273719\pi$
$239$	4821.80	1.30501	0.652503	+	0.757786i	$0.273719\pi$
$240$	472.409	0.127058
$241$	−1544.41	−0.412798	−0.206399	−	0.978468i	$-0.566175\pi$
$241$	−1544.41	−0.412798	−0.206399	+	0.978468i	$0.566175\pi$
$242$	193.598	0.0514253
$243$	243.000	0.0641500
$244$	−119.544	−0.0313648
$245$	0	0
$246$	16.0161	0.00415102
$247$	3530.38	0.909445
$248$	3672.16	0.940252
$249$	302.773	0.0770581
$250$	1339.73	0.338928
$251$	7568.20	1.90319	0.951595	−	0.307354i	$-0.0994436\pi$
$251$	7568.20	1.90319	0.951595	+	0.307354i	$0.0994436\pi$
$252$	0	0
$253$	−1795.26	−0.446115
$254$	−3666.82	−0.905815
$255$	−1030.10	−0.252971
$256$	−3616.24	−0.882871
$257$	−1415.19	−0.343490	−0.171745	−	0.985141i	$-0.554941\pi$
$257$	−1415.19	−0.343490	−0.171745	+	0.985141i	$0.554941\pi$
$258$	−589.425	−0.142233
$259$	0	0
$260$	4333.23	1.03360
$261$	1429.72	0.339070
$262$	2386.81	0.562815
$263$	−7875.57	−1.84650	−0.923248	−	0.384204i	$-0.874476\pi$
$263$	−7875.57	−1.84650	−0.923248	+	0.384204i	$0.874476\pi$
$264$	−709.626	−0.165434
$265$	−7099.79	−1.64580
$266$	0	0
$267$	3189.68	0.731105
$268$	−3364.27	−0.766810
$269$	2062.01	0.467371	0.233685	−	0.972312i	$-0.424921\pi$
$269$	2062.01	0.467371	0.233685	+	0.972312i	$0.424921\pi$
$270$	746.323	0.168221
$271$	8247.07	1.84861	0.924306	−	0.381651i	$-0.124645\pi$
$271$	8247.07	1.84861	0.924306	+	0.381651i	$0.124645\pi$
$272$	−181.159	−0.0403837
$273$	0	0
$274$	4145.21	0.913945
$275$	1908.15	0.418420
$276$	2663.55	0.580893
$277$	−6637.33	−1.43971	−0.719853	−	0.694126i	$-0.755791\pi$
$277$	−6637.33	−1.43971	−0.719853	+	0.694126i	$0.755791\pi$
$278$	−4278.57	−0.923062
$279$	−1536.91	−0.329793
$280$	0	0
$281$	5600.82	1.18903	0.594515	−	0.804085i	$-0.297344\pi$
$281$	5600.82	1.18903	0.594515	+	0.804085i	$0.297344\pi$
$282$	−1876.64	−0.396285
$283$	934.745	0.196342	0.0981711	−	0.995170i	$-0.468701\pi$
$283$	934.745	0.196342	0.0981711	+	0.995170i	$0.468701\pi$
$284$	−5054.04	−1.05599
$285$	−3968.56	−0.824831
$286$	−811.459	−0.167771
$287$	0	0
$288$	1679.53	0.343636
$289$	−4517.98	−0.919596
$290$	4391.08	0.889148
$291$	−4246.50	−0.855444
$292$	4722.47	0.946444
$293$	−9169.86	−1.82836	−0.914179	−	0.405311i	$-0.867163\pi$
$293$	−9169.86	−1.82836	−0.914179	+	0.405311i	$0.867163\pi$
$294$	0	0
$295$	−7062.08	−1.39380
$296$	5289.33	1.03864
$297$	297.000	0.0580259
$298$	1315.75	0.255770
$299$	7524.78	1.45542
$300$	−2831.03	−0.544832
$301$	0	0
$302$	−222.081	−0.0423157
$303$	2322.42	0.440328
$304$	−697.929	−0.131674
$305$	379.640	0.0712725
$306$	−286.199	−0.0534670
$307$	6193.71	1.15144	0.575722	−	0.817645i	$-0.304721\pi$
$307$	6193.71	1.15144	0.575722	+	0.817645i	$0.304721\pi$
$308$	0	0
$309$	−2277.48	−0.419292
$310$	−4720.29	−0.864822
$311$	3373.58	0.615106	0.307553	−	0.951531i	$-0.400490\pi$
$311$	3373.58	0.615106	0.307553	+	0.951531i	$0.400490\pi$
$312$	2974.38	0.539715
$313$	−7175.09	−1.29572	−0.647860	−	0.761760i	$-0.724336\pi$
$313$	−7175.09	−1.29572	−0.647860	+	0.761760i	$0.724336\pi$
$314$	1015.58	0.182524
$315$	0	0
$316$	−830.709	−0.147883
$317$	−6302.06	−1.11659	−0.558295	−	0.829643i	$-0.688544\pi$
$317$	−6302.06	−1.11659	−0.558295	+	0.829643i	$0.688544\pi$
$318$	−1972.57	−0.347850
$319$	1747.43	0.306700
$320$	3898.56	0.681050
$321$	−3543.66	−0.616162
$322$	0	0
$323$	1521.86	0.262162
$324$	−440.645	−0.0755565
$325$	−7997.94	−1.36506
$326$	−973.375	−0.165369
$327$	2214.27	0.374463
$328$	−71.7526	−0.0120789
$329$	0	0
$330$	912.173	0.152162
$331$	−7404.03	−1.22949	−0.614747	−	0.788725i	$-0.710742\pi$
$331$	−7404.03	−1.22949	−0.614747	+	0.788725i	$0.710742\pi$
$332$	−549.035	−0.0907597
$333$	−2213.74	−0.364301
$334$	2472.77	0.405102
$335$	10684.0	1.74248
$336$	0	0
$337$	7008.68	1.13290	0.566450	−	0.824096i	$-0.308316\pi$
$337$	7008.68	1.13290	0.566450	+	0.824096i	$0.308316\pi$
$338$	−113.948	−0.0183372
$339$	1860.54	0.298085
$340$	1867.94	0.297952
$341$	−1878.45	−0.298309
$342$	−1102.60	−0.174333
$343$	0	0
$344$	2640.64	0.413877
$345$	−8458.72	−1.32001
$346$	−202.554	−0.0314722
$347$	−10025.8	−1.55104	−0.775521	−	0.631322i	$-0.782513\pi$
$347$	−10025.8	−1.55104	−0.775521	+	0.631322i	$0.782513\pi$
$348$	−2592.59	−0.399360
$349$	10746.5	1.64827	0.824136	−	0.566391i	$-0.191661\pi$
$349$	10746.5	1.64827	0.824136	+	0.566391i	$0.191661\pi$
$350$	0	0
$351$	−1244.87	−0.189305
$352$	2052.76	0.310830
$353$	606.136	0.0913919	0.0456960	−	0.998955i	$-0.485449\pi$
$353$	606.136	0.0913919	0.0456960	+	0.998955i	$0.485449\pi$
$354$	−1962.09	−0.294588
$355$	16050.3	2.39961
$356$	−5784.02	−0.861102
$357$	0	0
$358$	−230.818	−0.0340757
$359$	8527.32	1.25363	0.626817	−	0.779166i	$-0.284357\pi$
$359$	8527.32	1.25363	0.626817	+	0.779166i	$0.284357\pi$
$360$	−3343.54	−0.489501
$361$	−995.934	−0.145201
$362$	−4679.95	−0.679483
$363$	363.000	0.0524864
$364$	0	0
$365$	−14997.3	−2.15067
$366$	105.477	0.0150639
$367$	−5504.16	−0.782874	−0.391437	−	0.920205i	$-0.628022\pi$
$367$	−5504.16	−0.782874	−0.391437	+	0.920205i	$0.628022\pi$
$368$	−1487.59	−0.210723
$369$	30.0306	0.00423667
$370$	−6799.05	−0.955312
$371$	0	0
$372$	2786.96	0.388434
$373$	−7091.36	−0.984387	−0.492194	−	0.870486i	$-0.663805\pi$
$373$	−7091.36	−0.984387	−0.492194	+	0.870486i	$0.663805\pi$
$374$	−349.799	−0.0483627
$375$	2512.03	0.345921
$376$	8407.39	1.15313
$377$	−7324.32	−1.00059
$378$	0	0
$379$	−77.3370	−0.0104816	−0.00524081	−	0.999986i	$-0.501668\pi$
$379$	−77.3370	−0.0104816	−0.00524081	+	0.999986i	$0.501668\pi$
$380$	7196.40	0.971493
$381$	−6875.38	−0.924505
$382$	−6034.65	−0.808271
$383$	−4843.75	−0.646224	−0.323112	−	0.946361i	$-0.604729\pi$
$383$	−4843.75	−0.646224	−0.323112	+	0.946361i	$0.604729\pi$
$384$	−3395.59	−0.451251
$385$	0	0
$386$	−2052.65	−0.270666
$387$	−1105.19	−0.145167
$388$	7700.41	1.00755
$389$	9882.00	1.28801	0.644007	−	0.765020i	$-0.277271\pi$
$389$	9882.00	1.28801	0.644007	+	0.765020i	$0.277271\pi$
$390$	−3823.35	−0.496417
$391$	3243.74	0.419547
$392$	0	0
$393$	4475.32	0.574428
$394$	4883.63	0.624451
$395$	2638.11	0.336045
$396$	−538.566	−0.0683434
$397$	10135.5	1.28133	0.640663	−	0.767823i	$-0.278660\pi$
$397$	10135.5	1.28133	0.640663	+	0.767823i	$0.278660\pi$
$398$	−8064.35	−1.01565
$399$	0	0
$400$	1581.13	0.197641
$401$	9220.96	1.14831	0.574155	−	0.818746i	$-0.305331\pi$
$401$	9220.96	1.14831	0.574155	+	0.818746i	$0.305331\pi$
$402$	2968.40	0.368284
$403$	7873.45	0.973212
$404$	−4211.37	−0.518623
$405$	1399.37	0.171692
$406$	0	0
$407$	−2705.69	−0.329523
$408$	1282.18	0.155582
$409$	5710.89	0.690429	0.345214	−	0.938524i	$-0.387806\pi$
$409$	5710.89	0.690429	0.345214	+	0.938524i	$0.387806\pi$
$410$	92.2327	0.0111099
$411$	7772.36	0.932803
$412$	4129.88	0.493846
$413$	0	0
$414$	−2350.13	−0.278992
$415$	1743.59	0.206240
$416$	−8604.07	−1.01406
$417$	−8022.41	−0.942108
$418$	−1347.63	−0.157690
$419$	−4520.73	−0.527093	−0.263547	−	0.964647i	$-0.584892\pi$
$419$	−4520.73	−0.527093	−0.263547	+	0.964647i	$0.584892\pi$
$420$	0	0
$421$	3798.26	0.439705	0.219853	−	0.975533i	$-0.429442\pi$
$421$	3798.26	0.439705	0.219853	+	0.975533i	$0.429442\pi$
$422$	5689.98	0.656359
$423$	−3518.75	−0.404462
$424$	8837.15	1.01219
$425$	−3447.71	−0.393502
$426$	4459.34	0.507173
$427$	0	0
$428$	6425.92	0.725721
$429$	−1521.50	−0.171233
$430$	−3394.35	−0.380674
$431$	−3468.51	−0.387638	−0.193819	−	0.981037i	$-0.562088\pi$
$431$	−3468.51	−0.387638	−0.193819	+	0.981037i	$0.562088\pi$
$432$	246.100	0.0274086
$433$	9859.98	1.09432	0.547160	−	0.837028i	$-0.315709\pi$
$433$	9859.98	1.09432	0.547160	+	0.837028i	$0.315709\pi$
$434$	0	0
$435$	8233.37	0.907494
$436$	−4015.26	−0.441046
$437$	12496.8	1.36796
$438$	−4166.78	−0.454558
$439$	−8464.33	−0.920228	−0.460114	−	0.887860i	$-0.652192\pi$
$439$	−8464.33	−0.920228	−0.460114	+	0.887860i	$0.652192\pi$
$440$	−4086.55	−0.442770
$441$	0	0
$442$	1466.17	0.157780
$443$	−7825.41	−0.839270	−0.419635	−	0.907693i	$-0.637842\pi$
$443$	−7825.41	−0.839270	−0.419635	+	0.907693i	$0.637842\pi$
$444$	4014.30	0.429077
$445$	18368.5	1.95675
$446$	−2828.52	−0.300302
$447$	2467.06	0.261047
$448$	0	0
$449$	1050.94	0.110461	0.0552303	−	0.998474i	$-0.482411\pi$
$449$	1050.94	0.110461	0.0552303	+	0.998474i	$0.482411\pi$
$450$	2497.91	0.261672
$451$	36.7041	0.00383221
$452$	−3373.83	−0.351087
$453$	−416.408	−0.0431889
$454$	−7738.68	−0.799988
$455$	0	0
$456$	4939.69	0.507285
$457$	14397.2	1.47369	0.736843	−	0.676064i	$-0.236316\pi$
$457$	14397.2	1.47369	0.736843	+	0.676064i	$0.236316\pi$
$458$	−7905.98	−0.806599
$459$	−536.630	−0.0545702
$460$	15338.7	1.55472
$461$	5755.83	0.581509	0.290755	−	0.956798i	$-0.406094\pi$
$461$	5755.83	0.581509	0.290755	+	0.956798i	$0.406094\pi$
$462$	0	0
$463$	−10056.0	−1.00938	−0.504689	−	0.863302i	$-0.668393\pi$
$463$	−10056.0	−1.00938	−0.504689	+	0.863302i	$0.668393\pi$
$464$	1447.96	0.144870
$465$	−8850.67	−0.882666
$466$	7715.32	0.766964
$467$	−2607.32	−0.258356	−0.129178	−	0.991621i	$-0.541234\pi$
$467$	−2607.32	−0.258356	−0.129178	+	0.991621i	$0.541234\pi$
$468$	2257.39	0.222965
$469$	0	0
$470$	−10807.1	−1.06062
$471$	1904.24	0.186290
$472$	8790.21	0.857208
$473$	−1350.78	−0.131309
$474$	732.960	0.0710253
$475$	−13282.5	−1.28304
$476$	0	0
$477$	−3698.61	−0.355027
$478$	7714.79	0.738214
$479$	5944.97	0.567083	0.283541	−	0.958960i	$-0.408491\pi$
$479$	5944.97	0.567083	0.283541	+	0.958960i	$0.408491\pi$
$480$	9671.96	0.919714
$481$	11340.8	1.07504
$482$	−2471.03	−0.233511
$483$	0	0
$484$	−658.248	−0.0618189
$485$	−24454.5	−2.28953
$486$	388.795	0.0362883
$487$	15895.1	1.47901	0.739504	−	0.673152i	$-0.235060\pi$
$487$	15895.1	1.47901	0.739504	+	0.673152i	$0.235060\pi$
$488$	−472.540	−0.0438338
$489$	−1825.10	−0.168781
$490$	0	0
$491$	3655.59	0.335997	0.167999	−	0.985787i	$-0.446270\pi$
$491$	3655.59	0.335997	0.167999	+	0.985787i	$0.446270\pi$
$492$	−54.4561	−0.00498998
$493$	−3157.32	−0.288435
$494$	5648.54	0.514453
$495$	1710.35	0.155302
$496$	−1556.52	−0.140907
$497$	0	0
$498$	484.431	0.0435901
$499$	21171.9	1.89937	0.949684	−	0.313208i	$-0.101404\pi$
$499$	21171.9	1.89937	0.949684	+	0.313208i	$0.101404\pi$
$500$	−4555.19	−0.407429
$501$	4636.50	0.413460
$502$	12109.0	1.07659
$503$	12270.3	1.08768	0.543842	−	0.839188i	$-0.316969\pi$
$503$	12270.3	1.08768	0.543842	+	0.839188i	$0.316969\pi$
$504$	0	0
$505$	13374.2	1.17850
$506$	−2872.38	−0.252357
$507$	−213.656	−0.0187155
$508$	12467.5	1.08889
$509$	3168.41	0.275908	0.137954	−	0.990439i	$-0.455947\pi$
$509$	3168.41	0.275908	0.137954	+	0.990439i	$0.455947\pi$
$510$	−1648.15	−0.143100
$511$	0	0
$512$	3268.99	0.282168
$513$	−2067.41	−0.177930
$514$	−2264.27	−0.194305
$515$	−13115.4	−1.12220
$516$	2004.09	0.170979
$517$	−4300.69	−0.365849
$518$	0	0
$519$	−379.794	−0.0321216
$520$	17128.7	1.44450
$521$	1305.16	0.109751	0.0548755	−	0.998493i	$-0.482524\pi$
$521$	1305.16	0.109751	0.0548755	+	0.998493i	$0.482524\pi$
$522$	2287.52	0.191805
$523$	−707.998	−0.0591943	−0.0295971	−	0.999562i	$-0.509422\pi$
$523$	−707.998	−0.0591943	−0.0295971	+	0.999562i	$0.509422\pi$
$524$	−8115.35	−0.676566
$525$	0	0
$526$	−12600.8	−1.04452
$527$	3394.04	0.280544
$528$	300.789	0.0247920
$529$	14469.0	1.18920
$530$	−11359.5	−0.930992
$531$	−3678.97	−0.300666
$532$	0	0
$533$	−153.844	−0.0125023
$534$	5103.42	0.413570
$535$	−20407.0	−1.64911
$536$	−13298.5	−1.07165
$537$	−432.789	−0.0347788
$538$	3299.17	0.264382
$539$	0	0
$540$	−2537.56	−0.202221
$541$	−5424.05	−0.431050	−0.215525	−	0.976498i	$-0.569146\pi$
$541$	−5424.05	−0.431050	−0.215525	+	0.976498i	$0.569146\pi$
$542$	13195.1	1.04572
$543$	−8775.02	−0.693503
$544$	−3708.99	−0.292319
$545$	12751.4	1.00222
$546$	0	0
$547$	−15353.7	−1.20014	−0.600071	−	0.799947i	$-0.704861\pi$
$547$	−15353.7	−1.20014	−0.600071	+	0.799947i	$0.704861\pi$
$548$	−14094.0	−1.09866
$549$	197.772	0.0153747
$550$	3053.00	0.236691
$551$	−12163.8	−0.940465
$552$	10528.6	0.811826
$553$	0	0
$554$	−10619.6	−0.814410
$555$	−12748.4	−0.975024
$556$	14547.5	1.10962
$557$	13685.2	1.04104	0.520519	−	0.853850i	$-0.325738\pi$
$557$	13685.2	1.04104	0.520519	+	0.853850i	$0.325738\pi$
$558$	−2459.02	−0.186557
$559$	5661.77	0.428385
$560$	0	0
$561$	−655.881	−0.0493606
$562$	8961.21	0.672608
$563$	1565.67	0.117203	0.0586015	−	0.998281i	$-0.481336\pi$
$563$	1565.67	0.117203	0.0586015	+	0.998281i	$0.481336\pi$
$564$	6380.73	0.476378
$565$	10714.4	0.797801
$566$	1495.57	0.111066
$567$	0	0
$568$	−19977.9	−1.47580
$569$	20508.1	1.51097	0.755487	−	0.655164i	$-0.227400\pi$
$569$	20508.1	1.51097	0.755487	+	0.655164i	$0.227400\pi$
$570$	−6349.61	−0.466589
$571$	971.535	0.0712040	0.0356020	−	0.999366i	$-0.488665\pi$
$571$	971.535	0.0712040	0.0356020	+	0.999366i	$0.488665\pi$
$572$	2759.03	0.201680
$573$	−11315.1	−0.824949
$574$	0	0
$575$	−28310.9	−2.05330
$576$	2030.94	0.146914
$577$	−13961.1	−1.00729	−0.503647	−	0.863910i	$-0.668009\pi$
$577$	−13961.1	−1.00729	−0.503647	+	0.863910i	$0.668009\pi$
$578$	−7228.67	−0.520196
$579$	−3848.77	−0.276251
$580$	−14930.0	−1.06885
$581$	0	0
$582$	−6794.31	−0.483906
$583$	−4520.53	−0.321134
$584$	18667.3	1.32270
$585$	−7168.87	−0.506660
$586$	−14671.6	−1.03426
$587$	23834.3	1.67589	0.837946	−	0.545753i	$-0.183756\pi$
$587$	23834.3	1.67589	0.837946	+	0.545753i	$0.183756\pi$
$588$	0	0
$589$	13075.8	0.914735
$590$	−11299.2	−0.788440
$591$	9156.93	0.637336
$592$	−2241.99	−0.155651
$593$	−25102.9	−1.73837	−0.869184	−	0.494489i	$-0.835355\pi$
$593$	−25102.9	−1.73837	−0.869184	+	0.494489i	$0.835355\pi$
$594$	475.194	0.0328240
$595$	0	0
$596$	−4473.66	−0.307464
$597$	−15120.9	−1.03661
$598$	12039.5	0.823297
$599$	−12122.2	−0.826875	−0.413437	−	0.910533i	$-0.635672\pi$
$599$	−12122.2	−0.826875	−0.413437	+	0.910533i	$0.635672\pi$
$600$	−11190.7	−0.761429
$601$	−20672.9	−1.40310	−0.701552	−	0.712619i	$-0.747509\pi$
$601$	−20672.9	−1.40310	−0.701552	+	0.712619i	$0.747509\pi$
$602$	0	0
$603$	5565.81	0.375883
$604$	755.096	0.0508682
$605$	2090.42	0.140476
$606$	3715.82	0.249084
$607$	3571.58	0.238824	0.119412	−	0.992845i	$-0.461899\pi$
$607$	3571.58	0.238824	0.119412	+	0.992845i	$0.461899\pi$
$608$	−14289.2	−0.953129
$609$	0	0
$610$	607.416	0.0403173
$611$	18026.2	1.19356
$612$	973.101	0.0642733
$613$	−25996.1	−1.71284	−0.856420	−	0.516280i	$-0.827317\pi$
$613$	−25996.1	−1.71284	−0.856420	+	0.516280i	$0.827317\pi$
$614$	9909.81	0.651347
$615$	172.938	0.0113391
$616$	0	0
$617$	7266.40	0.474123	0.237062	−	0.971495i	$-0.423816\pi$
$617$	7266.40	0.474123	0.237062	+	0.971495i	$0.423816\pi$
$618$	−3643.92	−0.237184
$619$	−16227.6	−1.05371	−0.526853	−	0.849957i	$-0.676628\pi$
$619$	−16227.6	−1.05371	−0.526853	+	0.849957i	$0.676628\pi$
$620$	16049.4	1.03961
$621$	−4406.55	−0.284748
$622$	5397.65	0.347952
$623$	0	0
$624$	−1260.75	−0.0808821
$625$	−7217.38	−0.461912
$626$	−11480.0	−0.732960
$627$	−2526.83	−0.160944
$628$	−3453.06	−0.219414
$629$	4888.73	0.309899
$630$	0	0
$631$	−3162.02	−0.199490	−0.0997450	−	0.995013i	$-0.531803\pi$
$631$	−3162.02	−0.199490	−0.0997450	+	0.995013i	$0.531803\pi$
$632$	−3283.68	−0.206674
$633$	10668.8	0.669903
$634$	−10083.2	−0.631630
$635$	−39593.5	−2.47436
$636$	6706.90	0.418154
$637$	0	0
$638$	2795.86	0.173494
$639$	8361.36	0.517638
$640$	−19554.3	−1.20774
$641$	−8679.48	−0.534819	−0.267409	−	0.963583i	$-0.586168\pi$
$641$	−8679.48	−0.534819	−0.267409	+	0.963583i	$0.586168\pi$
$642$	−5669.79	−0.348550
$643$	19226.2	1.17917	0.589586	−	0.807705i	$-0.299291\pi$
$643$	19226.2	1.17917	0.589586	+	0.807705i	$0.299291\pi$
$644$	0	0
$645$	−6364.48	−0.388529
$646$	2434.94	0.148299
$647$	−11211.6	−0.681258	−0.340629	−	0.940198i	$-0.610640\pi$
$647$	−11211.6	−0.681258	−0.340629	+	0.940198i	$0.610640\pi$
$648$	−1741.81	−0.105594
$649$	−4496.52	−0.271963
$650$	−12796.5	−0.772188
$651$	0	0
$652$	3309.56	0.198792
$653$	−12404.5	−0.743378	−0.371689	−	0.928357i	$-0.621221\pi$
$653$	−12404.5	−0.743378	−0.371689	+	0.928357i	$0.621221\pi$
$654$	3542.79	0.211826
$655$	25772.2	1.53741
$656$	30.4138	0.00181015
$657$	−7812.81	−0.463937
$658$	0	0
$659$	20954.9	1.23868	0.619339	−	0.785124i	$-0.287401\pi$
$659$	20954.9	1.23868	0.619339	+	0.785124i	$0.287401\pi$
$660$	−3101.46	−0.182916
$661$	20349.0	1.19741	0.598703	−	0.800971i	$-0.295683\pi$
$661$	20349.0	1.19741	0.598703	+	0.800971i	$0.295683\pi$
$662$	−11846.3	−0.695498
$663$	2749.11	0.161035
$664$	−2170.26	−0.126841
$665$	0	0
$666$	−3541.94	−0.206077
$667$	−25926.4	−1.50506
$668$	−8407.63	−0.486977
$669$	−5303.55	−0.306498
$670$	17094.2	0.985683
$671$	241.722	0.0139070
$672$	0	0
$673$	31712.5	1.81639	0.908193	−	0.418553i	$-0.137462\pi$
$673$	31712.5	1.81639	0.908193	+	0.418553i	$0.137462\pi$
$674$	11213.7	0.640857
$675$	4683.63	0.267071
$676$	387.433	0.0220433
$677$	−1195.61	−0.0678744	−0.0339372	−	0.999424i	$-0.510805\pi$
$677$	−1195.61	−0.0678744	−0.0339372	+	0.999424i	$0.510805\pi$
$678$	2976.83	0.168620
$679$	0	0
$680$	7383.73	0.416402
$681$	−14510.2	−0.816494
$682$	−3005.47	−0.168747
$683$	16709.8	0.936140	0.468070	−	0.883691i	$-0.344949\pi$
$683$	16709.8	0.936140	0.468070	+	0.883691i	$0.344949\pi$
$684$	3748.94	0.209568
$685$	44759.0	2.49657
$686$	0	0
$687$	−14823.9	−0.823242
$688$	−1119.29	−0.0620238
$689$	18947.7	1.04768
$690$	−13533.8	−0.746699
$691$	11154.4	0.614088	0.307044	−	0.951695i	$-0.400660\pi$
$691$	11154.4	0.614088	0.307044	+	0.951695i	$0.400660\pi$
$692$	688.701	0.0378331
$693$	0	0
$694$	−16041.0	−0.877391
$695$	−46199.0	−2.52148
$696$	−10248.1	−0.558124
$697$	−66.3182	−0.00360399
$698$	17194.2	0.932392
$699$	14466.4	0.782790
$700$	0	0
$701$	10388.9	0.559749	0.279874	−	0.960037i	$-0.409707\pi$
$701$	10388.9	0.559749	0.279874	+	0.960037i	$0.409707\pi$
$702$	−1991.76	−0.107086
$703$	18834.2	1.01045
$704$	2482.26	0.132889
$705$	−20263.5	−1.08251
$706$	969.805	0.0516984
$707$	0	0
$708$	6671.28	0.354127
$709$	−4288.90	−0.227183	−0.113592	−	0.993528i	$-0.536236\pi$
$709$	−4288.90	−0.227183	−0.113592	+	0.993528i	$0.536236\pi$
$710$	25680.2	1.35741
$711$	1374.32	0.0724908
$712$	−22863.4	−1.20343
$713$	27870.2	1.46388
$714$	0	0
$715$	−8761.95	−0.458291
$716$	784.800	0.0409628
$717$	14465.4	0.753446
$718$	13643.5	0.709153
$719$	−31876.4	−1.65339	−0.826697	−	0.562647i	$-0.809783\pi$
$719$	−31876.4	−1.65339	−0.826697	+	0.562647i	$0.809783\pi$
$720$	1417.23	0.0733569
$721$	0	0
$722$	−1593.47	−0.0821371
$723$	−4633.24	−0.238329
$724$	15912.2	0.816814
$725$	27556.7	1.41163
$726$	580.793	0.0296904
$727$	−14030.2	−0.715754	−0.357877	−	0.933769i	$-0.616499\pi$
$727$	−14030.2	−0.715754	−0.357877	+	0.933769i	$0.616499\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−23995.4	−1.21659
$731$	2440.64	0.123489
$732$	−358.631	−0.0181085
$733$	5466.09	0.275436	0.137718	−	0.990471i	$-0.456023\pi$
$733$	5466.09	0.275436	0.137718	+	0.990471i	$0.456023\pi$
$734$	−8806.54	−0.442855
$735$	0	0
$736$	−30456.5	−1.52533
$737$	6802.66	0.339999
$738$	48.0484	0.00239659
$739$	28957.0	1.44141	0.720704	−	0.693243i	$-0.243819\pi$
$739$	28957.0	1.44141	0.720704	+	0.693243i	$0.243819\pi$
$740$	23117.3	1.14839
$741$	10591.1	0.525068
$742$	0	0
$743$	−17742.1	−0.876037	−0.438019	−	0.898966i	$-0.644320\pi$
$743$	−17742.1	−0.876037	−0.438019	+	0.898966i	$0.644320\pi$
$744$	11016.5	0.542855
$745$	14207.2	0.698672
$746$	−11346.0	−0.556846
$747$	908.320	0.0444895
$748$	1189.35	0.0581374
$749$	0	0
$750$	4019.19	0.195680
$751$	30053.7	1.46029	0.730143	−	0.683294i	$-0.239453\pi$
$751$	30053.7	1.46029	0.730143	+	0.683294i	$0.239453\pi$
$752$	−3563.64	−0.172809
$753$	22704.6	1.09881
$754$	−11718.8	−0.566011
$755$	−2397.99	−0.115592
$756$	0	0
$757$	−8581.32	−0.412012	−0.206006	−	0.978551i	$-0.566047\pi$
$757$	−8581.32	−0.412012	−0.206006	+	0.978551i	$0.566047\pi$
$758$	−123.738	−0.00592923
$759$	−5385.78	−0.257565
$760$	28446.4	1.35771
$761$	−30413.3	−1.44873	−0.724363	−	0.689419i	$-0.757866\pi$
$761$	−30413.3	−1.44873	−0.724363	+	0.689419i	$0.757866\pi$
$762$	−11000.5	−0.522972
$763$	0	0
$764$	20518.3	0.971632
$765$	−3090.31	−0.146053
$766$	−7749.90	−0.365555
$767$	18847.0	0.887258
$768$	−10848.7	−0.509726
$769$	24735.2	1.15992	0.579958	−	0.814646i	$-0.303069\pi$
$769$	24735.2	1.15992	0.579958	+	0.814646i	$0.303069\pi$
$770$	0	0
$771$	−4245.56	−0.198314
$772$	6979.18	0.325371
$773$	−25108.1	−1.16827	−0.584137	−	0.811655i	$-0.698567\pi$
$773$	−25108.1	−1.16827	−0.584137	+	0.811655i	$0.698567\pi$
$774$	−1768.27	−0.0821180
$775$	−29622.7	−1.37301
$776$	30438.7	1.40810
$777$	0	0
$778$	15811.0	0.728601
$779$	−255.496	−0.0117511
$780$	12999.7	0.596749
$781$	10219.4	0.468221
$782$	5189.92	0.237329
$783$	4289.15	0.195762
$784$	0	0
$785$	10966.0	0.498591
$786$	7160.42	0.324941
$787$	36364.0	1.64706	0.823531	−	0.567271i	$-0.192001\pi$
$787$	36364.0	1.64706	0.823531	+	0.567271i	$0.192001\pi$
$788$	−16604.8	−0.750660
$789$	−23626.7	−1.06608
$790$	4220.93	0.190094
$791$	0	0
$792$	−2128.88	−0.0955132
$793$	−1013.17	−0.0453704
$794$	16216.6	0.724818
$795$	−21299.4	−0.950202
$796$	27419.5	1.22093
$797$	18605.4	0.826897	0.413448	−	0.910528i	$-0.364324\pi$
$797$	18605.4	0.826897	0.413448	+	0.910528i	$0.364324\pi$
$798$	0	0
$799$	7770.63	0.344062
$800$	32371.6	1.43063
$801$	9569.04	0.422104
$802$	14753.3	0.649574
$803$	−9548.99	−0.419647
$804$	−10092.8	−0.442718
$805$	0	0
$806$	12597.4	0.550525
$807$	6186.02	0.269837
$808$	−16647.0	−0.724800
$809$	26846.8	1.16673	0.583364	−	0.812211i	$-0.301736\pi$
$809$	26846.8	1.16673	0.583364	+	0.812211i	$0.301736\pi$
$810$	2238.97	0.0971227
$811$	32344.3	1.40045	0.700224	−	0.713924i	$-0.253084\pi$
$811$	32344.3	1.40045	0.700224	+	0.713924i	$0.253084\pi$
$812$	0	0
$813$	24741.2	1.06730
$814$	−4329.04	−0.186404
$815$	−10510.3	−0.451729
$816$	−543.477	−0.0233156
$817$	9402.76	0.402645
$818$	9137.31	0.390561
$819$	0	0
$820$	−313.599	−0.0133553
$821$	16953.3	0.720677	0.360338	−	0.932822i	$-0.382661\pi$
$821$	16953.3	0.720677	0.360338	+	0.932822i	$0.382661\pi$
$822$	12435.6	0.527667
$823$	37814.0	1.60159	0.800797	−	0.598936i	$-0.204410\pi$
$823$	37814.0	1.60159	0.800797	+	0.598936i	$0.204410\pi$
$824$	16324.8	0.690173
$825$	5724.44	0.241575
$826$	0	0
$827$	−11894.6	−0.500141	−0.250071	−	0.968228i	$-0.580454\pi$
$827$	−11894.6	−0.500141	−0.250071	+	0.968228i	$0.580454\pi$
$828$	7990.64	0.335379
$829$	−30744.7	−1.28807	−0.644033	−	0.764998i	$-0.722740\pi$
$829$	−30744.7	−1.28807	−0.644033	+	0.764998i	$0.722740\pi$
$830$	2789.71	0.116665
$831$	−19912.0	−0.831215
$832$	−10404.3	−0.433540
$833$	0	0
$834$	−12835.7	−0.532930
$835$	26700.4	1.10659
$836$	4582.04	0.189561
$837$	−4610.73	−0.190406
$838$	−7233.08	−0.298165
$839$	24677.8	1.01546	0.507731	−	0.861516i	$-0.330484\pi$
$839$	24677.8	1.01546	0.507731	+	0.861516i	$0.330484\pi$
$840$	0	0
$841$	846.708	0.0347168
$842$	6077.14	0.248732
$843$	16802.5	0.686486
$844$	−19346.4	−0.789017
$845$	−1230.39	−0.0500907
$846$	−5629.92	−0.228795
$847$	0	0
$848$	−3745.80	−0.151688
$849$	2804.24	0.113358
$850$	−5516.26	−0.222595
$851$	40143.9	1.61706
$852$	−15162.1	−0.609678
$853$	1855.69	0.0744873	0.0372437	−	0.999306i	$-0.488142\pi$
$853$	1855.69	0.0744873	0.0372437	+	0.999306i	$0.488142\pi$
$854$	0	0
$855$	−11905.7	−0.476217
$856$	25400.8	1.01423
$857$	−3984.84	−0.158832	−0.0794162	−	0.996842i	$-0.525306\pi$
$857$	−3984.84	−0.158832	−0.0794162	+	0.996842i	$0.525306\pi$
$858$	−2434.38	−0.0968627
$859$	12526.1	0.497538	0.248769	−	0.968563i	$-0.419974\pi$
$859$	12526.1	0.497538	0.248769	+	0.968563i	$0.419974\pi$
$860$	11541.1	0.457613
$861$	0	0
$862$	−5549.54	−0.219279
$863$	−22040.9	−0.869387	−0.434693	−	0.900579i	$-0.643143\pi$
$863$	−22040.9	−0.869387	−0.434693	+	0.900579i	$0.643143\pi$
$864$	5038.58	0.198398
$865$	−2187.14	−0.0859709
$866$	15775.8	0.619033
$867$	−13553.9	−0.530929
$868$	0	0
$869$	1679.72	0.0655704
$870$	13173.2	0.513350
$871$	−28513.2	−1.10922
$872$	−15871.8	−0.616383
$873$	−12739.5	−0.493891
$874$	19994.5	0.773828
$875$	0	0
$876$	14167.4	0.546430
$877$	21039.5	0.810096	0.405048	−	0.914295i	$-0.367255\pi$
$877$	21039.5	0.810096	0.405048	+	0.914295i	$0.367255\pi$
$878$	−13542.7	−0.520553
$879$	−27509.6	−1.05560
$880$	1732.17	0.0663538
$881$	−33610.9	−1.28533	−0.642667	−	0.766146i	$-0.722172\pi$
$881$	−33610.9	−1.28533	−0.642667	+	0.766146i	$0.722172\pi$
$882$	0	0
$883$	−4618.52	−0.176020	−0.0880099	−	0.996120i	$-0.528051\pi$
$883$	−4618.52	−0.176020	−0.0880099	+	0.996120i	$0.528051\pi$
$884$	−4985.11	−0.189669
$885$	−21186.2	−0.804709
$886$	−12520.5	−0.474757
$887$	40095.6	1.51779	0.758895	−	0.651213i	$-0.225740\pi$
$887$	40095.6	1.51779	0.758895	+	0.651213i	$0.225740\pi$
$888$	15868.0	0.599656
$889$	0	0
$890$	29389.3	1.10689
$891$	891.000	0.0335013
$892$	9617.22	0.360996
$893$	29937.0	1.12184
$894$	3947.25	0.147669
$895$	−2492.32	−0.0930828
$896$	0	0
$897$	22574.4	0.840285
$898$	1681.48	0.0624852
$899$	−27127.7	−1.00641
$900$	−8493.09	−0.314559
$901$	8167.85	0.302009
$902$	58.7258	0.00216780
$903$	0	0
$904$	−13336.3	−0.490661
$905$	−50533.1	−1.85611
$906$	−666.244	−0.0244310
$907$	−52281.2	−1.91397	−0.956984	−	0.290141i	$-0.906298\pi$
$907$	−52281.2	−1.91397	−0.956984	+	0.290141i	$0.906298\pi$
$908$	26312.2	0.961674
$909$	6967.26	0.254224
$910$	0	0
$911$	30635.6	1.11416	0.557081	−	0.830458i	$-0.311921\pi$
$911$	30635.6	1.11416	0.557081	+	0.830458i	$0.311921\pi$
$912$	−2093.79	−0.0760221
$913$	1110.17	0.0402423
$914$	23035.3	0.833632
$915$	1138.92	0.0411492
$916$	26881.0	0.969622
$917$	0	0
$918$	−858.597	−0.0308692
$919$	−14574.9	−0.523159	−0.261579	−	0.965182i	$-0.584243\pi$
$919$	−14574.9	−0.523159	−0.261579	+	0.965182i	$0.584243\pi$
$920$	60631.6	2.17279
$921$	18581.1	0.664787
$922$	9209.21	0.328947
$923$	−42834.5	−1.52754
$924$	0	0
$925$	−42668.1	−1.51667
$926$	−16089.4	−0.570983
$927$	−6832.44	−0.242078
$928$	29645.1	1.04865
$929$	18955.8	0.669451	0.334726	−	0.942316i	$-0.391356\pi$
$929$	18955.8	0.669451	0.334726	+	0.942316i	$0.391356\pi$
$930$	−14160.9	−0.499305
$931$	0	0
$932$	−26232.7	−0.921976
$933$	10120.7	0.355132
$934$	−4171.66	−0.146147
$935$	−3777.05	−0.132110
$936$	8923.14	0.311605
$937$	−6520.68	−0.227344	−0.113672	−	0.993518i	$-0.536261\pi$
$937$	−6520.68	−0.227344	−0.113672	+	0.993518i	$0.536261\pi$
$938$	0	0
$939$	−21525.3	−0.748084
$940$	36745.0	1.27499
$941$	−12631.4	−0.437589	−0.218795	−	0.975771i	$-0.570212\pi$
$941$	−12631.4	−0.437589	−0.218795	+	0.975771i	$0.570212\pi$
$942$	3046.74	0.105380
$943$	−544.573	−0.0188057
$944$	−3725.91	−0.128462
$945$	0	0
$946$	−2161.22	−0.0742785
$947$	−35170.9	−1.20686	−0.603432	−	0.797415i	$-0.706200\pi$
$947$	−35170.9	−1.20686	−0.603432	+	0.797415i	$0.706200\pi$
$948$	−2492.13	−0.0853803
$949$	40024.3	1.36907
$950$	−21251.8	−0.725789
$951$	−18906.2	−0.644663
$952$	0	0
$953$	39539.0	1.34396	0.671980	−	0.740570i	$-0.265444\pi$
$953$	39539.0	1.34396	0.671980	+	0.740570i	$0.265444\pi$
$954$	−5917.71	−0.200831
$955$	−65160.8	−2.20791
$956$	−26230.9	−0.887415
$957$	5242.30	0.177074
$958$	9511.83	0.320786
$959$	0	0
$960$	11695.7	0.393204
$961$	−629.373	−0.0211263
$962$	18145.1	0.608129
$963$	−10631.0	−0.355741
$964$	8401.70	0.280706
$965$	−22164.0	−0.739363
$966$	0	0
$967$	−29986.0	−0.997194	−0.498597	−	0.866834i	$-0.666151\pi$
$967$	−29986.0	−0.997194	−0.498597	+	0.866834i	$0.666151\pi$
$968$	−2601.96	−0.0863949
$969$	4565.57	0.151359
$970$	−39126.7	−1.29514
$971$	14879.8	0.491777	0.245889	−	0.969298i	$-0.420920\pi$
$971$	14879.8	0.491777	0.245889	+	0.969298i	$0.420920\pi$
$972$	−1321.94	−0.0436225
$973$	0	0
$974$	25431.9	0.836643
$975$	−23993.8	−0.788121
$976$	200.296	0.00656896
$977$	40864.1	1.33813	0.669067	−	0.743202i	$-0.266694\pi$
$977$	40864.1	1.33813	0.669067	+	0.743202i	$0.266694\pi$
$978$	−2920.13	−0.0954758
$979$	11695.5	0.381807
$980$	0	0
$981$	6642.82	0.216197
$982$	5848.88	0.190066
$983$	−48180.6	−1.56330	−0.781650	−	0.623718i	$-0.785621\pi$
$983$	−48180.6	−1.56330	−0.781650	+	0.623718i	$0.785621\pi$
$984$	−215.258	−0.00697374
$985$	52732.4	1.70578
$986$	−5051.65	−0.163162
$987$	0	0
$988$	−19205.5	−0.618430
$989$	20041.4	0.644367
$990$	2736.52	0.0878508
$991$	−22380.4	−0.717392	−0.358696	−	0.933454i	$-0.616779\pi$
$991$	−22380.4	−0.717392	−0.358696	+	0.933454i	$0.616779\pi$
$992$	−31867.7	−1.01996
$993$	−22212.1	−0.709848
$994$	0	0
$995$	−87077.1	−2.77440
$996$	−1647.11	−0.0524002
$997$	12425.1	0.394691	0.197346	−	0.980334i	$-0.436768\pi$
$997$	12425.1	0.394691	0.197346	+	0.980334i	$0.436768\pi$
$998$	33874.6	1.07443
$999$	−6641.23	−0.210330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.n.1.4		5
7.6	odd	2		231.4.a.k.1.4	✓	5
21.20	even	2		693.4.a.p.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.4.a.k.1.4	✓	5	7.6	odd	2
693.4.a.p.1.2		5	21.20	even	2
1617.4.a.n.1.4		5	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:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q+1.59998 q^{2} +3.00000 q^{3} -5.44007 q^{4} +17.2762 q^{5} +4.79994 q^{6} -21.5038 q^{8} +9.00000 q^{9} +27.6416 q^{10} +11.0000 q^{11} -16.3202 q^{12} -46.1062 q^{13} +51.8287 q^{15} +9.11483 q^{16} -19.8752 q^{17} +14.3998 q^{18} -76.5707 q^{19} -93.9838 q^{20} +17.5998 q^{22} -163.205 q^{23} -64.5115 q^{24} +173.468 q^{25} -73.7690 q^{26} +27.0000 q^{27} +158.858 q^{29} +82.9248 q^{30} -170.768 q^{31} +186.614 q^{32} +33.0000 q^{33} -31.7999 q^{34} -48.9606 q^{36} -245.971 q^{37} -122.511 q^{38} -138.319 q^{39} -371.505 q^{40} +3.33673 q^{41} -122.798 q^{43} -59.8407 q^{44} +155.486 q^{45} -261.125 q^{46} -390.972 q^{47} +27.3445 q^{48} +277.545 q^{50} -59.6255 q^{51} +250.821 q^{52} -410.957 q^{53} +43.1995 q^{54} +190.038 q^{55} -229.712 q^{57} +254.169 q^{58} -408.774 q^{59} -281.951 q^{60} +21.9747 q^{61} -273.225 q^{62} +225.660 q^{64} -796.541 q^{65} +52.7993 q^{66} +618.424 q^{67} +108.122 q^{68} -489.616 q^{69} +929.041 q^{71} -193.534 q^{72} -868.090 q^{73} -393.549 q^{74} +520.404 q^{75} +416.549 q^{76} -221.307 q^{78} +152.702 q^{79} +157.470 q^{80} +81.0000 q^{81} +5.33871 q^{82} +100.924 q^{83} -343.368 q^{85} -196.475 q^{86} +476.573 q^{87} -236.542 q^{88} +1063.23 q^{89} +248.774 q^{90} +887.848 q^{92} -512.303 q^{93} -625.547 q^{94} -1322.85 q^{95} +559.843 q^{96} -1415.50 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + 15 q^{3} + 21 q^{4} - 21 q^{5} - 3 q^{6} - 42 q^{8} + 45 q^{9} + 23 q^{10} + 55 q^{11} + 63 q^{12} - 101 q^{13} - 63 q^{15} - 7 q^{16} + 20 q^{17} - 9 q^{18} - 237 q^{19} - 85 q^{20} - 11 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q - q^2 + 15 * q^3 + 21 * q^4 - 21 * q^5 - 3 * q^6 - 42 * q^8 + 45 * q^9 + 23 * q^10 + 55 * q^11 + 63 * q^12 - 101 * q^13 - 63 * q^15 - 7 * q^16 + 20 * q^17 - 9 * q^18 - 237 * q^19 - 85 * q^20 - 11 * q^22 - 80 * q^23 - 126 * q^24 + 486 * q^25 - 165 * q^26 + 135 * q^27 - 11 * q^29 + 69 * q^30 - 316 * q^31 + 453 * q^32 + 165 * q^33 - 936 * q^34 + 189 * q^36 + 319 * q^37 - 89 * q^38 - 303 * q^39 - 624 * q^40 - 1190 * q^41 + 88 * q^43 + 231 * q^44 - 189 * q^45 + 1000 * q^46 - 377 * q^47 - 21 * q^48 - 644 * q^50 + 60 * q^51 - 1001 * q^52 - 992 * q^53 - 27 * q^54 - 231 * q^55 - 711 * q^57 + 721 * q^58 - 71 * q^59 - 255 * q^60 + 574 * q^61 - 272 * q^62 - 1380 * q^64 + 589 * q^65 - 33 * q^66 - 527 * q^67 + 2974 * q^68 - 240 * q^69 - 1156 * q^71 - 378 * q^72 - 1061 * q^73 - 1609 * q^74 + 1458 * q^75 - 2399 * q^76 - 495 * q^78 + 588 * q^79 + 1643 * q^80 + 405 * q^81 + 2602 * q^82 + 212 * q^83 + 1918 * q^85 - 4760 * q^86 - 33 * q^87 - 462 * q^88 - 1030 * q^89 + 207 * q^90 - 1174 * q^92 - 948 * q^93 + 1799 * q^94 - 3593 * q^95 + 1359 * q^96 - 2488 * q^97 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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