LMFDB - Embedded newform 117.4.b.d.64.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(64,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("117.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.90322347067$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.5054412.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 29x^{2} + 48 $ x^4 + 29*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.3
Root			$1.32750i$ of defining polynomial
Character	$\chi$	$=$	117.64
Dual form			117.4.b.d.64.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32750i q^{2} +6.23774 q^{4} +15.4241i q^{5} -7.96501i q^{7} +18.9006i q^{8} +O(q^{10})$	\(q+1.32750i q^{2} +6.23774 q^{4} +15.4241i q^{5} -7.96501i q^{7} +18.9006i q^{8} -20.4755 q^{10} -12.7691i q^{11} +(7.47548 + 46.2722i) q^{13} +10.5736 q^{14} +24.8113 q^{16} -54.0000 q^{17} +84.5794i q^{19} +96.2113i q^{20} +16.9510 q^{22} +122.853 q^{23} -112.902 q^{25} +(-61.4264 + 9.92371i) q^{26} -49.6837i q^{28} -140.853 q^{29} +116.439i q^{31} +184.142i q^{32} -71.6851i q^{34} +122.853 q^{35} -433.898i q^{37} -112.279 q^{38} -291.525 q^{40} -205.823i q^{41} +418.853 q^{43} -79.6501i q^{44} +163.087i q^{46} -485.861i q^{47} +279.559 q^{49} -149.877i q^{50} +(46.6301 + 288.634i) q^{52} +674.559 q^{53} +196.951 q^{55} +150.544 q^{56} -186.982i q^{58} -186.226i q^{59} -671.902 q^{61} -154.574 q^{62} -45.9584 q^{64} +(-713.706 + 115.302i) q^{65} -14.0364i q^{67} -336.838 q^{68} +163.087i q^{70} -346.789i q^{71} -832.900i q^{73} +576.000 q^{74} +527.584i q^{76} -101.706 q^{77} -335.608 q^{79} +382.691i q^{80} +273.230 q^{82} +568.797i q^{83} -832.900i q^{85} +556.028i q^{86} +241.343 q^{88} +236.671i q^{89} +(368.559 - 59.5423i) q^{91} +766.324 q^{92} +644.981 q^{94} -1304.56 q^{95} +1278.94i q^{97} +371.115i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 26 q^{4}+O(q^{10}) $ 4 * q - 26 * q^4	$ 4 q - 26 q^{4} + 20 q^{10} - 72 q^{13} + 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} - 44 q^{25} + 60 q^{26} + 48 q^{29} - 120 q^{35} + 468 q^{38} - 1268 q^{40} + 1064 q^{43} - 716 q^{49} + 1766 q^{52} + 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} - 1050 q^{64} - 1632 q^{65} + 1404 q^{68} + 2304 q^{74} + 816 q^{77} + 288 q^{79} - 28 q^{82} + 2392 q^{88} - 360 q^{91} + 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) $ 4 * q - 26 * q^4 + 20 * q^10 - 72 * q^13 + 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 - 44 * q^25 + 60 * q^26 + 48 * q^29 - 120 * q^35 + 468 * q^38 - 1268 * q^40 + 1064 * q^43 - 716 * q^49 + 1766 * q^52 + 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 - 1050 * q^64 - 1632 * q^65 + 1404 * q^68 + 2304 * q^74 + 816 * q^77 + 288 * q^79 - 28 * q^82 + 2392 * q^88 - 360 * q^91 + 8568 * q^92 + 6656 * q^94 - 3384 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$			1.32750i			0.469343i	0.972075	+	0.234671i	$0.0754014\pi$
$2$			1.32750i			0.469343i	−0.972075	+	0.234671i	$0.924599\pi$
$3$		0			0
$3$		0			0
$4$	6.23774			0.779717
$5$			15.4241i			1.37957i	0.724014	+	0.689785i	$0.242295\pi$
$5$			15.4241i			1.37957i	−0.724014	+	0.689785i	$0.757705\pi$
$6$		0			0
$7$		−	7.96501i		−	0.430070i	−0.976606	−	0.215035i	$-0.931013\pi$
$7$		−	7.96501i		−	0.430070i	0.976606	−	0.215035i	$-0.0689866\pi$
$8$			18.9006i			0.835297i
$9$		0			0
$10$	−20.4755			−0.647491
$11$		−	12.7691i		−	0.350002i	−0.984568	−	0.175001i	$-0.944007\pi$
$11$		−	12.7691i		−	0.350002i	0.984568	−	0.175001i	$-0.0559928\pi$
$12$		0			0
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$13$	7.47548	+	46.2722i	0.159487	+	0.987200i
$14$	10.5736			0.201850
$15$		0			0
$16$	24.8113			0.387677
$17$	−54.0000			−0.770407			−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000			−0.770407			−0.385204	+	0.922832i	$0.625869\pi$
$18$		0			0
$19$			84.5794i			1.02126i	0.859802	+	0.510628i	$0.170587\pi$
$19$			84.5794i			1.02126i	−0.859802	+	0.510628i	$0.829413\pi$
$20$			96.2113i			1.07568i
$21$		0			0
$22$	16.9510			0.164271
$23$	122.853			1.11376			0.556882	−	0.830591i	$-0.311997\pi$
$23$	122.853			1.11376			0.556882	+	0.830591i	$0.311997\pi$
$24$		0			0
$25$	−112.902			−0.903215
$26$	−61.4264	+	9.92371i	−0.463335	+	0.0748538i
$27$		0			0
$28$		−	49.6837i		−	0.335333i
$29$	−140.853			−0.901921			−0.450961	−	0.892544i	$-0.648919\pi$
$29$	−140.853			−0.901921			−0.450961	+	0.892544i	$0.648919\pi$
$30$		0			0
$31$			116.439i			0.674617i	0.941394	+	0.337309i	$0.109517\pi$
$31$			116.439i			0.674617i	−0.941394	+	0.337309i	$0.890483\pi$
$32$			184.142i			1.01725i
$33$		0			0
$34$		−	71.6851i		−	0.361585i
$35$	122.853			0.593312
$36$		0			0
$37$		−	433.898i		−	1.92790i	−0.266081	−	0.963951i	$-0.585729\pi$
$37$		−	433.898i		−	1.92790i	0.266081	−	0.963951i	$-0.414271\pi$
$38$	−112.279			−0.479319
$39$		0			0
$40$	−291.525			−1.15235
$41$		−	205.823i		−	0.784003i	−0.919965	−	0.392002i	$-0.871783\pi$
$41$		−	205.823i		−	0.784003i	0.919965	−	0.392002i	$-0.128217\pi$
$42$		0			0
$43$	418.853			1.48545			0.742726	−	0.669595i	$-0.233532\pi$
$43$	418.853			1.48545			0.742726	+	0.669595i	$0.233532\pi$
$44$		−	79.6501i		−	0.272902i
$45$		0			0
$46$			163.087i			0.522737i
$47$		−	485.861i		−	1.50787i	−0.656947	−	0.753937i	$-0.728152\pi$
$47$		−	485.861i		−	1.50787i	0.656947	−	0.753937i	$-0.271848\pi$
$48$		0			0
$49$	279.559			0.815040
$50$		−	149.877i		−	0.423918i
$51$		0			0
$52$	46.6301	+	288.634i	0.124354	+	0.769737i
$53$	674.559			1.74826			0.874130	−	0.485693i	$-0.161433\pi$
$53$	674.559			1.74826			0.874130	+	0.485693i	$0.161433\pi$
$54$		0			0
$55$	196.951			0.482852
$56$	150.544			0.359236
$57$		0			0
$58$		−	186.982i		−	0.423310i
$59$		−	186.226i		−	0.410925i	−0.978665	−	0.205462i	$-0.934130\pi$
$59$		−	186.226i		−	0.410925i	0.978665	−	0.205462i	$-0.0658698\pi$
$60$		0			0
$61$	−671.902			−1.41030			−0.705149	−	0.709059i	$-0.749120\pi$
$61$	−671.902			−1.41030			−0.705149	+	0.709059i	$0.749120\pi$
$62$	−154.574			−0.316627
$63$		0			0
$64$	−45.9584			−0.0897626
$65$	−713.706	+	115.302i	−1.36191	+	0.220023i
$66$		0			0
$67$		−	14.0364i		−	0.0255944i	−0.999918	−	0.0127972i	$-0.995926\pi$
$67$		−	14.0364i		−	0.0255944i	0.999918	−	0.0127972i	$-0.00407358\pi$
$68$	−336.838			−0.600700
$69$		0			0
$70$			163.087i			0.278467i
$71$		−	346.789i		−	0.579665i	−0.957077	−	0.289833i	$-0.906400\pi$
$71$		−	346.789i		−	0.579665i	0.957077	−	0.289833i	$-0.0935996\pi$
$72$		0			0
$73$		−	832.900i		−	1.33539i	−0.744435	−	0.667695i	$-0.767281\pi$
$73$		−	832.900i		−	1.33539i	0.744435	−	0.667695i	$-0.232719\pi$
$74$	576.000			0.904846
$75$		0			0
$76$			527.584i			0.796290i
$77$	−101.706			−0.150525
$78$		0			0
$79$	−335.608			−0.477960			−0.238980	−	0.971025i	$-0.576813\pi$
$79$	−335.608			−0.477960			−0.238980	+	0.971025i	$0.576813\pi$
$80$			382.691i			0.534827i
$81$		0			0
$82$	273.230			0.367966
$83$			568.797i			0.752212i	0.926577	+	0.376106i	$0.122737\pi$
$83$			568.797i			0.752212i	−0.926577	+	0.376106i	$0.877263\pi$
$84$		0			0
$85$		−	832.900i		−	1.06283i
$86$			556.028i			0.697186i
$87$		0			0
$88$	241.343			0.292355
$89$			236.671i			0.281877i	0.990018	+	0.140939i	$0.0450120\pi$
$89$			236.671i			0.281877i	−0.990018	+	0.140939i	$0.954988\pi$
$90$		0			0
$91$	368.559	−	59.5423i	0.424565	−	0.0685904i
$92$	766.324			0.868422
$93$		0			0
$94$	644.981			0.707710
$95$	−1304.56			−1.40889
$96$		0			0
$97$			1278.94i			1.33873i	0.742934	+	0.669365i	$0.233434\pi$
$97$			1278.94i			1.33873i	−0.742934	+	0.669365i	$0.766566\pi$
$98$			371.115i			0.382533i
$99$		0			0
$100$	−704.253			−0.704253
$101$	632.264			0.622898			0.311449	−	0.950263i	$-0.399186\pi$
$101$	632.264			0.622898			0.311449	+	0.950263i	$0.399186\pi$
$102$		0			0
$103$	−1506.26			−1.44094			−0.720469	−	0.693487i	$-0.756073\pi$
$103$	−1506.26			−1.44094			−0.720469	+	0.693487i	$0.756073\pi$
$104$	−874.574	+	141.291i	−0.824606	+	0.133219i
$105$		0			0
$106$			895.478i			0.820533i
$107$	1268.56			1.14613			0.573066	−	0.819509i	$-0.305754\pi$
$107$	1268.56			1.14613			0.573066	+	0.819509i	$0.305754\pi$
$108$		0			0
$109$		−	347.425i		−	0.305296i	−0.988281	−	0.152648i	$-0.951220\pi$
$109$		−	347.425i		−	0.305296i	0.988281	−	0.152648i	$-0.0487801\pi$
$110$			261.453i			0.226623i
$111$		0			0
$112$		−	197.622i		−	0.166728i
$113$	−659.706			−0.549203			−0.274601	−	0.961558i	$-0.588546\pi$
$113$	−659.706			−0.549203			−0.274601	+	0.961558i	$0.588546\pi$
$114$		0			0
$115$			1894.89i			1.53652i
$116$	−878.603			−0.703244
$117$		0			0
$118$	247.215			0.192865
$119$			430.111i			0.331329i
$120$		0			0
$121$	1167.95			0.877499
$122$		−	891.951i		−	0.661913i
$123$		0			0
$124$			726.319i			0.526011i
$125$			186.602i			0.133521i
$126$		0			0
$127$	−275.019			−0.192157			−0.0960787	−	0.995374i	$-0.530630\pi$
$127$	−275.019			−0.192157			−0.0960787	+	0.995374i	$0.530630\pi$
$128$			1412.13i			0.975121i
$129$		0			0
$130$	−153.064	−	947.446i	−0.103266	−	0.639204i
$131$	1183.97			0.789648			0.394824	−	0.918757i	$-0.370805\pi$
$131$	1183.97			0.789648			0.394824	+	0.918757i	$0.370805\pi$
$132$		0			0
$133$	673.676			0.439211
$134$	18.6334			0.0120125
$135$		0			0
$136$		−	1020.63i		−	0.643519i
$137$		−	2557.36i		−	1.59482i	−0.603438	−	0.797410i	$-0.706203\pi$
$137$		−	2557.36i		−	1.59482i	0.603438	−	0.797410i	$-0.293797\pi$
$138$		0			0
$139$	545.736			0.333012			0.166506	−	0.986040i	$-0.446751\pi$
$139$	545.736			0.333012			0.166506	+	0.986040i	$0.446751\pi$
$140$	766.324			0.462616
$141$		0			0
$142$	460.362			0.272062
$143$	590.853	−	95.4549i	0.345522	−	0.0558205i
$144$		0			0
$145$		−	2172.52i		−	1.24426i
$146$	1105.68			0.626756
$147$		0			0
$148$		−	2706.54i		−	1.50322i
$149$			1376.78i			0.756981i	0.925605	+	0.378491i	$0.123557\pi$
$149$			1376.78i			0.756981i	−0.925605	+	0.378491i	$0.876443\pi$
$150$		0			0
$151$			2733.47i			1.47316i	0.676352	+	0.736579i	$0.263560\pi$
$151$			2733.47i			1.47316i	−0.676352	+	0.736579i	$0.736440\pi$
$152$	−1598.60			−0.853052
$153$		0			0
$154$		−	135.015i		−	0.0706479i
$155$	−1795.97			−0.930683
$156$		0			0
$157$	1029.97			0.523570			0.261785	−	0.965126i	$-0.415689\pi$
$157$	1029.97			0.523570			0.261785	+	0.965126i	$0.415689\pi$
$158$		−	445.520i		−	0.224327i
$159$		0			0
$160$	−2840.22			−1.40337
$161$		−	978.524i		−	0.478997i
$162$		0			0
$163$		−	2882.91i		−	1.38532i	−0.721264	−	0.692660i	$-0.756439\pi$
$163$		−	2882.91i		−	1.38532i	0.721264	−	0.692660i	$-0.243561\pi$
$164$		−	1283.87i		−	0.611301i
$165$		0			0
$166$	−755.079			−0.353045
$167$		−	1153.90i		−	0.534679i	−0.963602	−	0.267340i	$-0.913855\pi$
$167$		−	1153.90i		−	0.534679i	0.963602	−	0.267340i	$-0.0861446\pi$
$168$		0			0
$169$	−2085.23	+	691.814i	−0.949128	+	0.314890i
$170$	1105.68			0.498832
$171$		0			0
$172$	2612.69			1.15823
$173$	−1688.85			−0.742203			−0.371101	−	0.928592i	$-0.621020\pi$
$173$	−1688.85			−0.742203			−0.371101	+	0.928592i	$0.621020\pi$
$174$		0			0
$175$			899.265i			0.388446i
$176$		−	316.817i		−	0.135687i
$177$		0			0
$178$	−314.181			−0.132297
$179$	−942.793			−0.393674			−0.196837	−	0.980436i	$-0.563067\pi$
$179$	−942.793			−0.393674			−0.196837	+	0.980436i	$0.563067\pi$
$180$		0			0
$181$	−482.030			−0.197950			−0.0989751	−	0.995090i	$-0.531556\pi$
$181$	−482.030			−0.197950			−0.0989751	+	0.995090i	$0.531556\pi$
$182$	79.0425	+	489.262i	0.0321924	+	0.199267i
$183$		0			0
$184$			2322.00i			0.930325i
$185$	6692.47			2.65968
$186$		0			0
$187$			689.530i			0.269644i
$188$		−	3030.67i		−	1.17572i
$189$		0			0
$190$		−	1731.80i		−	0.661254i
$191$	−4223.32			−1.59994			−0.799971	−	0.600039i	$-0.795152\pi$
$191$	−4223.32			−1.59994			−0.799971	+	0.600039i	$0.795152\pi$
$192$		0			0
$193$		−	229.092i		−	0.0854424i	−0.999087	−	0.0427212i	$-0.986397\pi$
$193$		−	229.092i		−	0.0854424i	0.999087	−	0.0427212i	$-0.0136027\pi$
$194$	−1697.80			−0.628323
$195$		0			0
$196$	1743.81			0.635501
$197$		−	228.335i		−	0.0825798i	−0.999147	−	0.0412899i	$-0.986853\pi$
$197$		−	228.335i		−	0.0825798i	0.999147	−	0.0412899i	$-0.0131467\pi$
$198$		0			0
$199$	−2939.02			−1.04694			−0.523471	−	0.852043i	$-0.675363\pi$
$199$	−2939.02			−1.04694			−0.523471	+	0.852043i	$0.675363\pi$
$200$		−	2133.92i		−	0.754453i
$201$		0			0
$202$			839.332i			0.292352i
$203$			1121.89i			0.387889i
$204$		0			0
$205$	3174.63			1.08159
$206$		−	1999.57i		−	0.676294i
$207$		0			0
$208$	185.476	+	1148.07i	0.0618292	+	0.382714i
$209$	1080.00			0.357441
$210$		0			0
$211$	−1607.02			−0.524321			−0.262161	−	0.965024i	$-0.584435\pi$
$211$	−1607.02			−0.524321			−0.262161	+	0.965024i	$0.584435\pi$
$212$	4207.72			1.36315
$213$		0			0
$214$			1684.01i			0.537929i
$215$			6460.42i			2.04929i
$216$		0			0
$217$	927.441			0.290133
$218$	461.207			0.143288
$219$		0			0
$220$	1228.53			0.376488
$221$	−403.676	−	2498.70i	−0.122870	−	0.760546i
$222$		0			0
$223$		−	130.867i		−	0.0392981i	−0.999807	−	0.0196490i	$-0.993745\pi$
$223$		−	130.867i		−	0.0392981i	0.999807	−	0.0196490i	$-0.00625488\pi$
$224$	1466.69			0.437489
$225$		0			0
$226$		−	875.761i		−	0.257764i
$227$			4325.19i			1.26464i	0.774708	+	0.632319i	$0.217897\pi$
$227$			4325.19i			1.26464i	−0.774708	+	0.632319i	$0.782103\pi$
$228$		0			0
$229$		−	2621.57i		−	0.756499i	−0.925704	−	0.378250i	$-0.876526\pi$
$229$		−	2621.57i		−	0.756499i	0.925704	−	0.378250i	$-0.123474\pi$
$230$	−2515.47			−0.721153
$231$		0			0
$232$		−	2662.21i		−	0.753373i
$233$	−4643.12			−1.30550			−0.652748	−	0.757575i	$-0.726384\pi$
$233$	−4643.12			−1.30550			−0.652748	+	0.757575i	$0.726384\pi$
$234$		0			0
$235$	7493.95			2.08022
$236$		−	1161.63i		−	0.320405i
$237$		0			0
$238$	−570.972			−0.155507
$239$			6696.69i			1.81244i	0.422809	+	0.906219i	$0.361044\pi$
$239$			6696.69i			1.81244i	−0.422809	+	0.906219i	$0.638956\pi$
$240$		0			0
$241$			2301.47i			0.615148i	0.951524	+	0.307574i	$0.0995171\pi$
$241$			2301.47i			0.615148i	−0.951524	+	0.307574i	$0.900483\pi$
$242$			1550.46i			0.411848i
$243$		0			0
$244$	−4191.15			−1.09963
$245$			4311.93i			1.12440i
$246$		0			0
$247$	−3913.68	+	632.272i	−1.00818	+	0.162876i
$248$	−2200.78			−0.563506
$249$		0			0
$250$	−247.714			−0.0626673
$251$	828.000			0.208219			0.104109	−	0.994566i	$-0.466801\pi$
$251$	828.000			0.208219			0.104109	+	0.994566i	$0.466801\pi$
$252$		0			0
$253$		−	1568.72i		−	0.389820i
$254$		−	365.088i		−	0.0901877i
$255$		0			0
$256$	−2242.27			−0.547429
$257$	884.763			0.214747			0.107374	−	0.994219i	$-0.465756\pi$
$257$	884.763			0.214747			0.107374	+	0.994219i	$0.465756\pi$
$258$		0			0
$259$	−3456.00			−0.829133
$260$	−4451.91	+	719.226i	−1.06191	+	0.171556i
$261$		0			0
$262$			1571.72i			0.370616i
$263$	8343.94			1.95631			0.978155	−	0.207878i	$-0.0666557\pi$
$263$	8343.94			1.95631			0.978155	+	0.207878i	$0.0666557\pi$
$264$		0			0
$265$			10404.4i			2.41185i
$266$			894.306i			0.206141i
$267$		0			0
$268$		−	87.5556i		−	0.0199564i
$269$	−2762.56			−0.626157			−0.313078	−	0.949727i	$-0.601360\pi$
$269$	−2762.56			−0.626157			−0.313078	+	0.949727i	$0.601360\pi$
$270$		0			0
$271$		−	3116.54i		−	0.698585i	−0.937014	−	0.349293i	$-0.886422\pi$
$271$		−	3116.54i		−	0.698585i	0.937014	−	0.349293i	$-0.113578\pi$
$272$	−1339.81			−0.298669
$273$		0			0
$274$	3394.91			0.748517
$275$			1441.65i			0.316127i
$276$		0			0
$277$	502.060			0.108902			0.0544510	−	0.998516i	$-0.482659\pi$
$277$	502.060			0.108902			0.0544510	+	0.998516i	$0.482659\pi$
$278$			724.465i			0.156297i
$279$		0			0
$280$			2322.00i			0.495592i
$281$		−	6607.56i		−	1.40275i	−0.712791	−	0.701377i	$-0.752569\pi$
$281$		−	6607.56i		−	1.40275i	0.712791	−	0.701377i	$-0.247431\pi$
$282$		0			0
$283$	4368.98			0.917699			0.458850	−	0.888514i	$-0.348262\pi$
$283$	4368.98			0.917699			0.458850	+	0.888514i	$0.348262\pi$
$284$		−	2163.18i		−	0.451975i
$285$		0			0
$286$	126.717	+	784.358i	0.0261990	+	0.162168i
$287$	−1639.38			−0.337176
$288$		0			0
$289$	−1997.00			−0.406473
$290$	2884.03			0.583986
$291$		0			0
$292$		−	5195.41i		−	1.04123i
$293$		−	5348.12i		−	1.06635i	−0.846005	−	0.533175i	$-0.820999\pi$
$293$		−	5348.12i		−	1.06635i	0.846005	−	0.533175i	$-0.179001\pi$
$294$		0			0
$295$	2872.36			0.566900
$296$	8200.94			1.61037
$297$		0			0
$298$	−1827.68			−0.355284
$299$	918.384	+	5684.67i	0.177630	+	1.09951i
$300$		0			0
$301$		−	3336.17i		−	0.638849i
$302$	−3628.69			−0.691416
$303$		0			0
$304$			2098.53i			0.395917i
$305$		−	10363.5i		−	1.94561i
$306$		0			0
$307$			4502.46i			0.837032i	0.908210	+	0.418516i	$0.137450\pi$
$307$			4502.46i			0.837032i	−0.908210	+	0.418516i	$0.862550\pi$
$308$	−634.414			−0.117367
$309$		0			0
$310$		−	2384.15i		−	0.436809i
$311$	−7447.20			−1.35785			−0.678926	−	0.734207i	$-0.737554\pi$
$311$	−7447.20			−1.35785			−0.678926	+	0.734207i	$0.737554\pi$
$312$		0			0
$313$	−6508.93			−1.17542			−0.587710	−	0.809072i	$-0.699970\pi$
$313$	−6508.93			−1.17542			−0.587710	+	0.809072i	$0.699970\pi$
$314$			1367.29i			0.245734i
$315$		0			0
$316$	−2093.43			−0.372673
$317$		−	2465.57i		−	0.436846i	−0.975854	−	0.218423i	$-0.929909\pi$
$317$		−	2465.57i		−	0.436846i	0.975854	−	0.218423i	$-0.0700913\pi$
$318$		0			0
$319$			1798.56i			0.315674i
$320$		−	708.866i		−	0.123834i
$321$		0			0
$322$	1298.99			0.224814
$323$		−	4567.29i		−	0.786782i
$324$		0			0
$325$	−843.996	−	5224.22i	−0.144051	−	0.891654i
$326$	3827.07			0.650190
$327$		0			0
$328$	3890.18			0.654876
$329$	−3869.89			−0.648491
$330$		0			0
$331$		−	4114.84i		−	0.683300i	−0.939827	−	0.341650i	$-0.889014\pi$
$331$		−	4114.84i		−	0.683300i	0.939827	−	0.341650i	$-0.110986\pi$
$332$			3548.01i			0.586513i
$333$		0			0
$334$	1531.80			0.250948
$335$	216.499			0.0353092
$336$		0			0
$337$	−4798.05			−0.775568			−0.387784	−	0.921750i	$-0.626759\pi$
$337$	−4798.05			−0.775568			−0.387784	+	0.921750i	$0.626759\pi$
$338$	−918.384	−	2768.15i	−0.147791	−	0.445466i
$339$		0			0
$340$		−	5195.41i		−	0.828708i
$341$	1486.82			0.236117
$342$		0			0
$343$		−	4958.69i		−	0.780594i
$344$			7916.58i			1.24079i
$345$		0			0
$346$		−	2241.96i		−	0.348348i
$347$	3314.76			0.512812			0.256406	−	0.966569i	$-0.417462\pi$
$347$	3314.76			0.512812			0.256406	+	0.966569i	$0.417462\pi$
$348$		0			0
$349$			371.740i			0.0570166i	0.999594	+	0.0285083i	$0.00907570\pi$
$349$			371.740i			0.0570166i	−0.999594	+	0.0285083i	$0.990924\pi$
$350$	−1193.78			−0.182314
$351$		0			0
$352$	2351.32			0.356039
$353$		−	7539.10i		−	1.13673i	−0.822776	−	0.568365i	$-0.807576\pi$
$353$		−	7539.10i		−	1.13673i	0.822776	−	0.568365i	$-0.192424\pi$
$354$		0			0
$355$	5348.89			0.799689
$356$			1476.29i			0.219785i
$357$		0			0
$358$		−	1251.56i		−	0.184768i
$359$			12741.5i			1.87317i	0.350437	+	0.936586i	$0.386033\pi$
$359$			12741.5i			1.87317i	−0.350437	+	0.936586i	$0.613967\pi$
$360$		0			0
$361$	−294.676			−0.0429619
$362$		−	639.896i		−	0.0929065i
$363$		0			0
$364$	2298.97	−	371.409i	0.331041	−	0.0534811i
$365$	12846.7			1.84227
$366$		0			0
$367$	7187.26			1.02227			0.511134	−	0.859501i	$-0.329226\pi$
$367$	7187.26			1.02227			0.511134	+	0.859501i	$0.329226\pi$
$368$	3048.14			0.431781
$369$		0			0
$370$			8884.26i			1.24830i
$371$		−	5372.87i		−	0.751874i
$372$		0			0
$373$	−2087.99			−0.289845			−0.144922	−	0.989443i	$-0.546293\pi$
$373$	−2087.99			−0.289845			−0.144922	+	0.989443i	$0.546293\pi$
$374$	−915.352			−0.126555
$375$		0			0
$376$	9183.07			1.25952
$377$	−1052.94	−	6517.57i	−0.143844	−	0.890377i
$378$		0			0
$379$			3982.08i			0.539699i	0.962902	+	0.269850i	$0.0869740\pi$
$379$			3982.08i			0.539699i	−0.962902	+	0.269850i	$0.913026\pi$
$380$	−8137.50			−1.09854
$381$		0			0
$382$		−	5606.47i		−	0.750921i
$383$		−	8638.43i		−	1.15249i	−0.817278	−	0.576244i	$-0.804518\pi$
$383$		−	8638.43i		−	1.15249i	0.817278	−	0.576244i	$-0.195482\pi$
$384$		0			0
$385$		−	1568.72i		−	0.207660i
$386$	304.120			0.0401018
$387$		0			0
$388$			7977.70i			1.04383i
$389$	−1275.74			−0.166279			−0.0831393	−	0.996538i	$-0.526495\pi$
$389$	−1275.74			−0.166279			−0.0831393	+	0.996538i	$0.526495\pi$
$390$		0			0
$391$	−6634.06			−0.858053
$392$			5283.83i			0.680801i
$393$		0			0
$394$	303.115			0.0387582
$395$		−	5176.44i		−	0.659379i
$396$		0			0
$397$			4622.65i			0.584394i	0.956358	+	0.292197i	$0.0943862\pi$
$397$			4622.65i			0.584394i	−0.956358	+	0.292197i	$0.905614\pi$
$398$		−	3901.55i		−	0.491375i
$399$		0			0
$400$	−2801.24			−0.350155
$401$		−	138.075i		−	0.0171949i	−0.999963	−	0.00859743i	$-0.997263\pi$
$401$		−	138.075i		−	0.0171949i	0.999963	−	0.00859743i	$-0.00273668\pi$
$402$		0			0
$403$	−5387.91	+	870.441i	−0.665982	+	0.107592i
$404$	3943.90			0.485684
$405$		0			0
$406$	−1489.32			−0.182053
$407$	−5540.47			−0.674769
$408$		0			0
$409$		−	1204.64i		−	0.145637i	−0.997345	−	0.0728186i	$-0.976801\pi$
$409$		−	1204.64i		−	0.145637i	0.997345	−	0.0728186i	$-0.0231994\pi$
$410$			4214.32i			0.507635i
$411$		0			0
$412$	−9395.68			−1.12352
$413$	−1483.29			−0.176726
$414$		0			0
$415$	−8773.16			−1.03773
$416$	−8520.66	+	1376.55i	−1.00423	+	0.162238i
$417$		0			0
$418$			1433.70i			0.167762i
$419$	−5199.85			−0.606275			−0.303138	−	0.952947i	$-0.598034\pi$
$419$	−5199.85			−0.606275			−0.303138	+	0.952947i	$0.598034\pi$
$420$		0			0
$421$		−	14136.5i		−	1.63651i	−0.574854	−	0.818256i	$-0.694941\pi$
$421$		−	14136.5i		−	1.63651i	0.574854	−	0.818256i	$-0.305059\pi$
$422$		−	2133.32i		−	0.246086i
$423$		0			0
$424$			12749.6i			1.46032i
$425$	6096.70			0.695844
$426$		0			0
$427$			5351.71i			0.606527i
$428$	7912.94			0.893660
$429$		0			0
$430$	−8576.21			−0.961818
$431$			2279.83i			0.254793i	0.991852	+	0.127396i	$0.0406620\pi$
$431$			2279.83i			0.254793i	−0.991852	+	0.127396i	$0.959338\pi$
$432$		0			0
$433$	13298.7			1.47597			0.737984	−	0.674819i	$-0.235778\pi$
$433$	13298.7			1.47597			0.737984	+	0.674819i	$0.235778\pi$
$434$			1231.18i			0.136172i
$435$		0			0
$436$		−	2167.14i		−	0.238045i
$437$			10390.8i			1.13744i
$438$		0			0
$439$	10452.3			1.13635			0.568177	−	0.822907i	$-0.307649\pi$
$439$	10452.3			1.13635			0.568177	+	0.822907i	$0.307649\pi$
$440$			3722.50i			0.403325i
$441$		0			0
$442$	3317.03	−	535.880i	0.356957	−	0.0576679i
$443$	−5363.50			−0.575232			−0.287616	−	0.957746i	$-0.592863\pi$
$443$	−5363.50			−0.575232			−0.287616	+	0.957746i	$0.592863\pi$
$444$		0			0
$445$	−3650.43			−0.388870
$446$	173.726			0.0184443
$447$		0			0
$448$			366.059i			0.0386042i
$449$			9681.73i			1.01762i	0.860880	+	0.508808i	$0.169914\pi$
$449$			9681.73i			1.01762i	−0.860880	+	0.508808i	$0.830086\pi$
$450$		0			0
$451$	−2628.17			−0.274402
$452$	−4115.07			−0.428223
$453$		0			0
$454$	−5741.69			−0.593548
$455$	918.384	+	5684.67i	0.0946253	+	0.585718i
$456$		0			0
$457$		−	3537.94i		−	0.362139i	−0.983470	−	0.181070i	$-0.942044\pi$
$457$		−	3537.94i		−	0.362139i	0.983470	−	0.181070i	$-0.0579560\pi$
$458$	3480.14			0.355057
$459$		0			0
$460$			11819.8i			1.19805i
$461$			15074.9i			1.52302i	0.648156	+	0.761508i	$0.275541\pi$
$461$			15074.9i			1.52302i	−0.648156	+	0.761508i	$0.724459\pi$
$462$		0			0
$463$			11070.0i			1.11116i	0.831463	+	0.555580i	$0.187504\pi$
$463$			11070.0i			1.11116i	−0.831463	+	0.555580i	$0.812496\pi$
$464$	−3494.74			−0.349654
$465$		0			0
$466$		−	6163.75i		−	0.612725i
$467$	13252.8			1.31320			0.656600	−	0.754239i	$-0.271994\pi$
$467$	13252.8			1.31320			0.656600	+	0.754239i	$0.271994\pi$
$468$		0			0
$469$	−111.800			−0.0110074
$470$			9948.23i			0.976335i
$471$		0			0
$472$	3519.79			0.343244
$473$		−	5348.36i		−	0.519911i
$474$		0			0
$475$		−	9549.18i		−	0.922413i
$476$			2682.92i			0.258343i
$477$		0			0
$478$	−8889.86			−0.850654
$479$			12241.4i			1.16769i	0.811866	+	0.583843i	$0.198452\pi$
$479$			12241.4i			1.16769i	−0.811866	+	0.583843i	$0.801548\pi$
$480$		0			0
$481$	20077.4	−	3243.59i	1.90322	−	0.307474i
$482$	−3055.20			−0.288715
$483$		0			0
$484$	7285.37			0.684201
$485$	−19726.5			−1.84687
$486$		0			0
$487$		−	13413.3i		−	1.24808i	−0.781392	−	0.624041i	$-0.785490\pi$
$487$		−	13413.3i		−	1.24808i	0.781392	−	0.624041i	$-0.214510\pi$
$488$		−	12699.4i		−	1.17802i
$489$		0			0
$490$	−5724.10			−0.527731
$491$	−737.885			−0.0678214			−0.0339107	−	0.999425i	$-0.510796\pi$
$491$	−737.885			−0.0678214			−0.0339107	+	0.999425i	$0.510796\pi$
$492$		0			0
$493$	7606.06			0.694847
$494$	−839.342	−	5195.41i	−0.0764449	−	0.473183i
$495$		0			0
$496$			2889.01i			0.261533i
$497$	−2762.17			−0.249297
$498$		0			0
$499$			1865.65i			0.167370i	0.996492	+	0.0836852i	$0.0266690\pi$
$499$			1865.65i			0.167370i	−0.996492	+	0.0836852i	$0.973331\pi$
$500$			1163.97i			0.104109i
$501$		0			0
$502$			1099.17i			0.0977259i
$503$	16632.0			1.47432			0.737161	−	0.675717i	$-0.236166\pi$
$503$	16632.0			1.47432			0.737161	+	0.675717i	$0.236166\pi$
$504$		0			0
$505$			9752.09i			0.859331i
$506$	2082.47			0.182959
$507$		0			0
$508$	−1715.50			−0.149829
$509$		−	4128.91i		−	0.359549i	−0.983708	−	0.179775i	$-0.942463\pi$
$509$		−	4128.91i		−	0.359549i	0.983708	−	0.179775i	$-0.0575368\pi$
$510$		0			0
$511$	−6634.06			−0.574312
$512$			8320.40i			0.718190i
$513$		0			0
$514$			1174.52i			0.100790i
$515$		−	23232.7i		−	1.98788i
$516$		0			0
$517$	−6203.99			−0.527758
$518$		−	4587.85i		−	0.389147i
$519$		0			0
$520$	−2179.29	−	13489.5i	−0.183785	−	1.13760i
$521$	988.234			0.0831005			0.0415502	−	0.999136i	$-0.486770\pi$
$521$	988.234			0.0831005			0.0415502	+	0.999136i	$0.486770\pi$
$522$		0			0
$523$	−9441.62			−0.789394			−0.394697	−	0.918811i	$-0.629150\pi$
$523$	−9441.62			−0.789394			−0.394697	+	0.918811i	$0.629150\pi$
$524$	7385.30			0.615703
$525$		0			0
$526$			11076.6i			0.918180i
$527$		−	6287.73i		−	0.519730i
$528$		0			0
$529$	2925.83			0.240472
$530$	−13811.9			−1.13198
$531$		0			0
$532$	4202.21			0.342461
$533$	9523.88	−	1538.62i	0.773968	−	0.125038i
$534$		0			0
$535$			19566.3i			1.58117i
$536$	265.297			0.0213789
$537$		0			0
$538$		−	3667.30i		−	0.293882i
$539$		−	3569.70i		−	0.285265i
$540$		0			0
$541$		−	14001.5i		−	1.11270i	−0.830948	−	0.556351i	$-0.812201\pi$
$541$		−	14001.5i		−	1.11270i	0.830948	−	0.556351i	$-0.187799\pi$
$542$	4137.22			0.327876
$543$		0			0
$544$		−	9943.67i		−	0.783697i
$545$	5358.70			0.421177
$546$		0			0
$547$	−4244.85			−0.331804			−0.165902	−	0.986142i	$-0.553053\pi$
$547$	−4244.85			−0.331804			−0.165902	+	0.986142i	$0.553053\pi$
$548$		−	15952.2i		−	1.24351i
$549$		0			0
$550$	−1913.80			−0.148372
$551$		−	11913.3i		−	0.921092i
$552$		0			0
$553$			2673.12i			0.205556i
$554$			666.485i			0.0511124i
$555$		0			0
$556$	3404.16			0.259655
$557$		−	11732.2i		−	0.892473i	−0.894915	−	0.446236i	$-0.852764\pi$
$557$		−	11732.2i		−	0.892473i	0.894915	−	0.446236i	$-0.147236\pi$
$558$		0			0
$559$	3131.13	+	19381.2i	0.236910	+	1.46644i
$560$	3048.14			0.230013
$561$		0			0
$562$	8771.54			0.658372
$563$	−9941.80			−0.744222			−0.372111	−	0.928188i	$-0.621366\pi$
$563$	−9941.80			−0.744222			−0.372111	+	0.928188i	$0.621366\pi$
$564$		0			0
$565$		−	10175.3i		−	0.757664i
$566$			5799.83i			0.430716i
$567$		0			0
$568$	6554.52			0.484193
$569$	3690.77			0.271925			0.135962	−	0.990714i	$-0.456587\pi$
$569$	3690.77			0.271925			0.135962	+	0.990714i	$0.456587\pi$
$570$		0			0
$571$	5685.09			0.416661			0.208331	−	0.978058i	$-0.433197\pi$
$571$	5685.09			0.416661			0.208331	+	0.978058i	$0.433197\pi$
$572$	3685.59	−	595.423i	0.269409	−	0.0435242i
$573$		0			0
$574$		−	2176.28i		−	0.158251i
$575$	−13870.3			−1.00597
$576$		0			0
$577$			7746.50i			0.558910i	0.960159	+	0.279455i	$0.0901537\pi$
$577$			7746.50i			0.558910i	−0.960159	+	0.279455i	$0.909846\pi$
$578$		−	2651.02i		−	0.190775i
$579$		0			0
$580$		−	13551.6i		−	0.970175i
$581$	4530.47			0.323504
$582$		0			0
$583$		−	8613.48i		−	0.611894i
$584$	15742.3			1.11545
$585$		0			0
$586$	7099.64			0.500484
$587$			2766.54i			0.194527i	0.995259	+	0.0972635i	$0.0310089\pi$
$587$			2766.54i			0.194527i	−0.995259	+	0.0972635i	$0.968991\pi$
$588$		0			0
$589$	−9848.38			−0.688957
$590$			3813.07i			0.266070i
$591$		0			0
$592$		−	10765.6i		−	0.747402i
$593$		−	1440.79i		−	0.0997743i	−0.998755	−	0.0498871i	$-0.984114\pi$
$593$		−	1440.79i		−	0.0997743i	0.998755	−	0.0498871i	$-0.0158862\pi$
$594$		0			0
$595$	−6634.06			−0.457092
$596$			8587.99i			0.590231i
$597$		0			0
$598$	−7546.41	+	1219.16i	−0.516047	+	0.0833696i
$599$	−23837.5			−1.62600			−0.813001	−	0.582263i	$-0.802168\pi$
$599$	−23837.5			−1.62600			−0.813001	+	0.582263i	$0.802168\pi$
$600$		0			0
$601$	−6694.23			−0.454348			−0.227174	−	0.973854i	$-0.572949\pi$
$601$	−6694.23			−0.454348			−0.227174	+	0.973854i	$0.572949\pi$
$602$	4428.77			0.299839
$603$		0			0
$604$			17050.7i			1.14865i
$605$			18014.6i			1.21057i
$606$		0			0
$607$	3330.50			0.222703			0.111352	−	0.993781i	$-0.464482\pi$
$607$	3330.50			0.222703			0.111352	+	0.993781i	$0.464482\pi$
$608$	−15574.6			−1.03887
$609$		0			0
$610$	13757.5			0.913156
$611$	22481.8	−	3632.04i	1.48857	−	0.240486i
$612$		0			0
$613$		−	13490.3i		−	0.888857i	−0.895814	−	0.444428i	$-0.853407\pi$
$613$		−	13490.3i		−	0.888857i	0.895814	−	0.444428i	$-0.146593\pi$
$614$	−5977.02			−0.392855
$615$		0			0
$616$		−	1922.30i		−	0.125733i
$617$		−	7470.76i		−	0.487458i	−0.969843	−	0.243729i	$-0.921629\pi$
$617$		−	7470.76i		−	0.487458i	0.969843	−	0.243729i	$-0.0783707\pi$
$618$		0			0
$619$		−	24806.9i		−	1.61078i	−0.592746	−	0.805389i	$-0.701956\pi$
$619$		−	24806.9i		−	1.61078i	0.592746	−	0.805389i	$-0.298044\pi$
$620$	−11202.8			−0.725669
$621$		0			0
$622$		−	9886.17i		−	0.637298i
$623$	1885.09			0.121227
$624$		0			0
$625$	−16990.9			−1.08742
$626$		−	8640.61i		−	0.551675i
$627$		0			0
$628$	6424.68			0.408237
$629$			23430.5i			1.48527i
$630$		0			0
$631$		−	314.333i		−	0.0198311i	−0.999951	−	0.00991554i	$-0.996844\pi$
$631$		−	314.333i		−	0.0198311i	0.999951	−	0.00991554i	$-0.00315627\pi$
$632$		−	6343.19i		−	0.399238i
$633$		0			0
$634$	3273.05			0.205031
$635$		−	4241.91i		−	0.265095i
$636$		0			0
$637$	2089.83	+	12935.8i	0.129988	+	0.804607i
$638$	−2387.59			−0.148159
$639$		0			0
$640$	−21780.7			−1.34525
$641$	5550.41			0.342009			0.171005	−	0.985270i	$-0.445299\pi$
$641$	5550.41			0.342009			0.171005	+	0.985270i	$0.445299\pi$
$642$		0			0
$643$			5479.48i			0.336065i	0.985781	+	0.168032i	$0.0537413\pi$
$643$			5479.48i			0.336065i	−0.985781	+	0.168032i	$0.946259\pi$
$644$		−	6103.78i		−	0.373482i
$645$		0			0
$646$	6063.08			0.369271
$647$	4724.83			0.287098			0.143549	−	0.989643i	$-0.454149\pi$
$647$	4724.83			0.287098			0.143549	+	0.989643i	$0.454149\pi$
$648$		0			0
$649$	−2377.93			−0.143824
$650$	6935.16	−	1120.41i	0.418491	−	0.0676091i
$651$		0			0
$652$		−	17982.9i		−	1.08016i
$653$	−3463.91			−0.207585			−0.103793	−	0.994599i	$-0.533098\pi$
$653$	−3463.91			−0.207585			−0.103793	+	0.994599i	$0.533098\pi$
$654$		0			0
$655$			18261.6i			1.08938i
$656$		−	5106.73i		−	0.303940i
$657$		0			0
$658$		−	5137.28i		−	0.304365i
$659$	2606.35			0.154065			0.0770327	−	0.997029i	$-0.475455\pi$
$659$	2606.35			0.154065			0.0770327	+	0.997029i	$0.475455\pi$
$660$		0			0
$661$			22436.7i			1.32025i	0.751154	+	0.660127i	$0.229498\pi$
$661$			22436.7i			1.32025i	−0.751154	+	0.660127i	$0.770502\pi$
$662$	5462.46			0.320702
$663$		0			0
$664$	−10750.6			−0.628321
$665$			10390.8i			0.605923i
$666$		0			0
$667$	−17304.2			−1.00453
$668$		−	7197.73i		−	0.416899i
$669$		0			0
$670$			287.403i			0.0165721i
$671$			8579.56i			0.493607i
$672$		0			0
$673$	633.970			0.0363117			0.0181558	−	0.999835i	$-0.494221\pi$
$673$	633.970			0.0363117			0.0181558	+	0.999835i	$0.494221\pi$
$674$		−	6369.42i		−	0.364007i
$675$		0			0
$676$	−13007.1	+	4315.35i	−0.740052	+	0.245525i
$677$	−24457.4			−1.38844			−0.694221	−	0.719762i	$-0.744251\pi$
$677$	−24457.4			−1.38844			−0.694221	+	0.719762i	$0.744251\pi$
$678$		0			0
$679$	10186.8			0.575747
$680$	15742.3			0.887780
$681$		0			0
$682$			1973.76i			0.110820i
$683$		−	12367.6i		−	0.692875i	−0.938073	−	0.346437i	$-0.887391\pi$
$683$		−	12367.6i		−	0.692875i	0.938073	−	0.346437i	$-0.112609\pi$
$684$		0			0
$685$	39445.0			2.20017
$686$	6582.66			0.366366
$687$		0			0
$688$	10392.3			0.575875
$689$	5042.65	+	31213.3i	0.278824	+	1.72588i
$690$		0			0
$691$			1050.99i			0.0578605i	0.999581	+	0.0289302i	$0.00921006\pi$
$691$			1050.99i			0.0578605i	−0.999581	+	0.0289302i	$0.990790\pi$
$692$	−10534.6			−0.578709
$693$		0			0
$694$			4400.35i			0.240685i
$695$			8417.46i			0.459414i
$696$		0			0
$697$			11114.4i			0.604002i
$698$	−493.486			−0.0267603
$699$		0			0
$700$			5609.38i			0.302878i
$701$	−24294.1			−1.30895			−0.654476	−	0.756083i	$-0.727111\pi$
$701$	−24294.1			−1.30895			−0.654476	+	0.756083i	$0.727111\pi$
$702$		0			0
$703$	36698.8			1.96888
$704$			586.846i			0.0314170i
$705$		0			0
$706$	10008.2			0.533516
$707$		−	5035.99i		−	0.267890i
$708$		0			0
$709$			27465.9i			1.45487i	0.686176	+	0.727436i	$0.259288\pi$
$709$			27465.9i			1.45487i	−0.686176	+	0.727436i	$0.740712\pi$
$710$			7100.66i			0.375328i
$711$		0			0
$712$	−4473.23			−0.235451
$713$			14304.9i			0.751365i
$714$		0			0
$715$	1472.30	+	9113.36i	0.0770084	+	0.476672i
$716$	−5880.90			−0.306955
$717$		0			0
$718$	−16914.3			−0.879160
$719$	36433.5			1.88976			0.944882	−	0.327411i	$-0.106176\pi$
$719$	36433.5			1.88976			0.944882	+	0.327411i	$0.106176\pi$
$720$		0			0
$721$			11997.4i			0.619704i
$722$		−	391.183i		−	0.0201639i
$723$		0			0
$724$	−3006.78			−0.154345
$725$	15902.6			0.814629
$726$		0			0
$727$	−551.608			−0.0281403			−0.0140701	−	0.999901i	$-0.504479\pi$
$727$	−551.608			−0.0281403			−0.0140701	+	0.999901i	$0.504479\pi$
$728$	1125.39	+	6965.99i	0.0572934	+	0.354638i
$729$		0			0
$730$			17054.0i			0.864654i
$731$	−22618.1			−1.14440
$732$		0			0
$733$		−	20317.2i		−	1.02379i	−0.859049	−	0.511893i	$-0.828945\pi$
$733$		−	20317.2i		−	1.02379i	0.859049	−	0.511893i	$-0.171055\pi$
$734$			9541.10i			0.479794i
$735$		0			0
$736$			22622.4i			1.13298i
$737$	−179.232			−0.00895807
$738$		0			0
$739$		−	29931.6i		−	1.48992i	−0.667107	−	0.744962i	$-0.732468\pi$
$739$		−	29931.6i		−	1.48992i	0.667107	−	0.744962i	$-0.267532\pi$
$740$	41745.9			2.07380
$741$		0			0
$742$	7132.49			0.352887
$743$			24376.4i			1.20361i	0.798643	+	0.601805i	$0.205552\pi$
$743$			24376.4i			1.20361i	−0.798643	+	0.601805i	$0.794448\pi$
$744$		0			0
$745$	−21235.5			−1.04431
$746$		−	2771.81i		−	0.136037i
$747$		0			0
$748$			4301.11i			0.210246i
$749$		−	10104.1i		−	0.492917i
$750$		0			0
$751$	−22692.2			−1.10260			−0.551298	−	0.834308i	$-0.685867\pi$
$751$	−22692.2			−1.10260			−0.551298	+	0.834308i	$0.685867\pi$
$752$		−	12054.8i		−	0.584567i
$753$		0			0
$754$	8652.09	−	1397.78i	0.417892	−	0.0675123i
$755$	−42161.3			−2.03232
$756$		0			0
$757$	−33063.5			−1.58747			−0.793734	−	0.608265i	$-0.791866\pi$
$757$	−33063.5			−1.58747			−0.793734	+	0.608265i	$0.791866\pi$
$758$	−5286.22			−0.253304
$759$		0			0
$760$		−	24657.0i		−	1.17685i
$761$			216.324i			0.0103045i	0.999987	+	0.00515226i	$0.00164002\pi$
$761$			216.324i			0.0103045i	−0.999987	+	0.00515226i	$0.998360\pi$
$762$		0			0
$763$	−2767.24			−0.131299
$764$	−26344.0			−1.24750
$765$		0			0
$766$	11467.5			0.540912
$767$	8617.09	−	1392.13i	0.405665	−	0.0655369i
$768$		0			0
$769$		−	19214.4i		−	0.901025i	−0.892770	−	0.450512i	$-0.851241\pi$
$769$		−	19214.4i		−	0.901025i	0.892770	−	0.450512i	$-0.148759\pi$
$770$	2082.47			0.0974638
$771$		0			0
$772$		−	1429.01i		−	0.0666209i
$773$			33175.6i			1.54365i	0.635834	+	0.771826i	$0.280656\pi$
$773$			33175.6i			1.54365i	−0.635834	+	0.771826i	$0.719344\pi$
$774$		0			0
$775$		−	13146.2i		−	0.609325i
$776$	−24172.8			−1.11824
$777$		0			0
$778$		−	1693.54i		−	0.0780416i
$779$	17408.4			0.800667
$780$		0			0
$781$	−4428.17			−0.202884
$782$		−	8806.72i		−	0.402721i
$783$		0			0
$784$	6936.21			0.315972
$785$			15886.3i			0.722302i
$786$		0			0
$787$		−	14887.9i		−	0.674330i	−0.941446	−	0.337165i	$-0.890532\pi$
$787$		−	14887.9i		−	0.674330i	0.941446	−	0.337165i	$-0.109468\pi$
$788$		−	1424.30i		−	0.0643889i
$789$		0

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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.32750i q^{2} +6.23774 q^{4} +15.4241i q^{5} -7.96501i q^{7} +18.9006i q^{8} +O(q^{10})\)	\(q+1.32750i q^{2} +6.23774 q^{4} +15.4241i q^{5} -7.96501i q^{7} +18.9006i q^{8} -20.4755 q^{10} -12.7691i q^{11} +(7.47548 + 46.2722i) q^{13} +10.5736 q^{14} +24.8113 q^{16} -54.0000 q^{17} +84.5794i q^{19} +96.2113i q^{20} +16.9510 q^{22} +122.853 q^{23} -112.902 q^{25} +(-61.4264 + 9.92371i) q^{26} -49.6837i q^{28} -140.853 q^{29} +116.439i q^{31} +184.142i q^{32} -71.6851i q^{34} +122.853 q^{35} -433.898i q^{37} -112.279 q^{38} -291.525 q^{40} -205.823i q^{41} +418.853 q^{43} -79.6501i q^{44} +163.087i q^{46} -485.861i q^{47} +279.559 q^{49} -149.877i q^{50} +(46.6301 + 288.634i) q^{52} +674.559 q^{53} +196.951 q^{55} +150.544 q^{56} -186.982i q^{58} -186.226i q^{59} -671.902 q^{61} -154.574 q^{62} -45.9584 q^{64} +(-713.706 + 115.302i) q^{65} -14.0364i q^{67} -336.838 q^{68} +163.087i q^{70} -346.789i q^{71} -832.900i q^{73} +576.000 q^{74} +527.584i q^{76} -101.706 q^{77} -335.608 q^{79} +382.691i q^{80} +273.230 q^{82} +568.797i q^{83} -832.900i q^{85} +556.028i q^{86} +241.343 q^{88} +236.671i q^{89} +(368.559 - 59.5423i) q^{91} +766.324 q^{92} +644.981 q^{94} -1304.56 q^{95} +1278.94i q^{97} +371.115i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 26 q^{4}+O(q^{10}) \) 4 * q - 26 * q^4	\( 4 q - 26 q^{4} + 20 q^{10} - 72 q^{13} + 348 q^{14} + 354 q^{16} - 216 q^{17} - 136 q^{22} - 120 q^{23} - 44 q^{25} + 60 q^{26} + 48 q^{29} - 120 q^{35} + 468 q^{38} - 1268 q^{40} + 1064 q^{43} - 716 q^{49} + 1766 q^{52} + 864 q^{53} + 584 q^{55} - 3372 q^{56} - 2280 q^{61} - 924 q^{62} - 1050 q^{64} - 1632 q^{65} + 1404 q^{68} + 2304 q^{74} + 816 q^{77} + 288 q^{79} - 28 q^{82} + 2392 q^{88} - 360 q^{91} + 8568 q^{92} + 6656 q^{94} - 3384 q^{95}+O(q^{100}) \) 4 * q - 26 * q^4 + 20 * q^10 - 72 * q^13 + 348 * q^14 + 354 * q^16 - 216 * q^17 - 136 * q^22 - 120 * q^23 - 44 * q^25 + 60 * q^26 + 48 * q^29 - 120 * q^35 + 468 * q^38 - 1268 * q^40 + 1064 * q^43 - 716 * q^49 + 1766 * q^52 + 864 * q^53 + 584 * q^55 - 3372 * q^56 - 2280 * q^61 - 924 * q^62 - 1050 * q^64 - 1632 * q^65 + 1404 * q^68 + 2304 * q^74 + 816 * q^77 + 288 * q^79 - 28 * q^82 + 2392 * q^88 - 360 * q^91 + 8568 * q^92 + 6656 * q^94 - 3384 * q^95

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