LMFDB - Embedded newform 468.4.t.g.361.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [468,4,Mod(361,468)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("468.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 5])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	468.t (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$27.6128938827$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 51x^{6} - 224x^{5} + 2520x^{4} - 5712x^{3} + 16675x^{2} + 9072x + 6561 $ x^8 + 51x^6 - 224x^5 + 2520x^4 - 5712x^3 + 16675x^2 + 9072x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	no (minimal twist has level 52)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			361.3
Root			$-0.287051 - 0.497187i$ of defining polynomial
Character	$\chi$	$=$	468.361
Dual form			468.4.t.g.433.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.49640i q^{5} +(-15.5619 - 8.98467i) q^{7} +(-49.6032 + 28.6384i) q^{11} +(46.6395 + 4.66469i) q^{13} +(8.75292 - 15.1605i) q^{17} +(19.1586 + 11.0612i) q^{19} +(65.5963 + 113.616i) q^{23} +118.768 q^{25} +(37.1264 + 64.3048i) q^{29} -110.463i q^{31} +(-22.4294 + 38.8488i) q^{35} +(209.104 - 120.726i) q^{37} +(369.537 - 213.352i) q^{41} +(-104.148 + 180.390i) q^{43} +158.624i q^{47} +(-10.0516 - 17.4098i) q^{49} -47.9103 q^{53} +(71.4931 + 123.830i) q^{55} +(382.291 + 220.716i) q^{59} +(-434.699 + 752.921i) q^{61} +(11.6449 - 116.431i) q^{65} +(693.726 - 400.523i) q^{67} +(773.901 + 446.812i) q^{71} +413.180i q^{73} +1029.23 q^{77} -246.527 q^{79} -333.857i q^{83} +(-37.8467 - 21.8508i) q^{85} +(328.376 - 189.588i) q^{89} +(-683.888 - 491.632i) q^{91} +(27.6133 - 47.8276i) q^{95} +(815.158 + 470.632i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{7} - 72 q^{11} + 62 q^{13} - 88 q^{17} - 144 q^{19} + 20 q^{23} - 84 q^{25} + 484 q^{29} - 40 q^{35} + 996 q^{37} - 156 q^{41} + 504 q^{43} + 922 q^{49} + 1164 q^{53} - 1128 q^{55} - 600 q^{59}+ \cdots - 3042 q^{97}+O(q^{100}) $ 8 * q - 36 * q^7 - 72 * q^11 + 62 * q^13 - 88 * q^17 - 144 * q^19 + 20 * q^23 - 84 * q^25 + 484 * q^29 - 40 * q^35 + 996 * q^37 - 156 * q^41 + 504 * q^43 + 922 * q^49 + 1164 * q^53 - 1128 * q^55 - 600 * q^59 - 1224 * q^61 - 670 * q^65 + 960 * q^67 + 2964 * q^71 + 3972 * q^77 - 3968 * q^79 + 3870 * q^85 - 5430 * q^89 - 1720 * q^91 - 2400 * q^95 - 3042 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		−	2.49640i		−	0.223285i	−0.993748	−	0.111643i	$-0.964389\pi$
$5$		−	2.49640i		−	0.223285i	0.993748	−	0.111643i	$-0.0356112\pi$
$6$		0			0
$7$	−15.5619	−	8.98467i	−0.840264	−	0.485126i	0.0170903	−	0.999854i	$-0.494560\pi$
$7$	−15.5619	−	8.98467i	−0.840264	−	0.485126i	−0.857354	+	0.514728i	$0.827893\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−49.6032	+	28.6384i	−1.35963	+	0.784983i	−0.989574	−	0.144026i	$-0.953995\pi$
$11$	−49.6032	+	28.6384i	−1.35963	+	0.784983i	−0.370057	+	0.929009i	$0.620662\pi$
$12$		0			0
$13$	46.6395	+	4.66469i	0.995036	+	0.0995194i
$13$	46.6395	+	4.66469i	0.995036	+	0.0995194i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8.75292	−	15.1605i	0.124876	−	0.216292i	−0.796808	−	0.604232i	$-0.793480\pi$
$17$	8.75292	−	15.1605i	0.124876	−	0.216292i	0.921685	+	0.387940i	$0.126813\pi$
$18$		0			0
$19$	19.1586	+	11.0612i	0.231331	+	0.133559i	0.611186	−	0.791487i	$-0.290693\pi$
$19$	19.1586	+	11.0612i	0.231331	+	0.133559i	−0.379855	+	0.925046i	$0.624026\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	65.5963	+	113.616i	0.594686	+	1.03003i	0.993591	+	0.113034i	$0.0360570\pi$
$23$	65.5963	+	113.616i	0.594686	+	1.03003i	−0.398905	+	0.916992i	$0.630610\pi$
$24$		0			0
$25$	118.768			0.950144
$26$		0			0
$27$		0			0
$28$		0			0
$29$	37.1264	+	64.3048i	0.237731	+	0.411762i	0.960063	−	0.279785i	$-0.0902631\pi$
$29$	37.1264	+	64.3048i	0.237731	+	0.411762i	−0.722332	+	0.691546i	$0.756930\pi$
$30$		0			0
$31$		−	110.463i		−	0.639994i	−0.947419	−	0.319997i	$-0.896318\pi$
$31$		−	110.463i		−	0.639994i	0.947419	−	0.319997i	$-0.103682\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−22.4294	+	38.8488i	−0.108322	+	0.187618i
$36$		0			0
$37$	209.104	−	120.726i	0.929093	−	0.536412i	0.0425684	−	0.999094i	$-0.486446\pi$
$37$	209.104	−	120.726i	0.929093	−	0.536412i	0.886525	+	0.462681i	$0.153113\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	369.537	−	213.352i	1.40761	−	0.812683i	0.412451	−	0.910980i	$-0.364673\pi$
$41$	369.537	−	213.352i	1.40761	−	0.812683i	0.995157	+	0.0982971i	$0.0313395\pi$
$42$		0			0
$43$	−104.148	+	180.390i	−0.369359	+	0.639749i	−0.989465	−	0.144769i	$-0.953756\pi$
$43$	−104.148	+	180.390i	−0.369359	+	0.639749i	0.620106	+	0.784518i	$0.287089\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			158.624i			0.492292i	0.969233	+	0.246146i	$0.0791642\pi$
$47$			158.624i			0.492292i	−0.969233	+	0.246146i	$0.920836\pi$
$48$		0			0
$49$	−10.0516	−	17.4098i	−0.0293048	−	0.0507574i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−47.9103			−0.124170			−0.0620848	−	0.998071i	$-0.519775\pi$
$53$	−47.9103			−0.124170			−0.0620848	+	0.998071i	$0.519775\pi$
$54$		0			0
$55$	71.4931	+	123.830i	0.175275	+	0.303585i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	382.291	+	220.716i	0.843561	+	0.487030i	0.858473	−	0.512859i	$-0.171414\pi$
$59$	382.291	+	220.716i	0.843561	+	0.487030i	−0.0149121	+	0.999889i	$0.504747\pi$
$60$		0			0
$61$	−434.699	+	752.921i	−0.912419	+	1.58036i	−0.101781	+	0.994807i	$0.532454\pi$
$61$	−434.699	+	752.921i	−0.912419	+	1.58036i	−0.810637	+	0.585549i	$0.800879\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	11.6449	−	116.431i	0.0222212	−	0.222177i
$66$		0			0
$67$	693.726	−	400.523i	1.26496	−	0.730323i	0.290927	−	0.956745i	$-0.406036\pi$
$67$	693.726	−	400.523i	1.26496	−	0.730323i	0.974029	+	0.226422i	$0.0727029\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	773.901	+	446.812i	1.29359	+	0.746857i	0.979289	−	0.202465i	$-0.0648954\pi$
$71$	773.901	+	446.812i	1.29359	+	0.746857i	0.314305	+	0.949322i	$0.398229\pi$
$72$		0			0
$73$			413.180i			0.662452i	0.943551	+	0.331226i	$0.107462\pi$
$73$			413.180i			0.662452i	−0.943551	+	0.331226i	$0.892538\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1029.23			1.52326
$78$		0			0
$79$	−246.527			−0.351094			−0.175547	−	0.984471i	$-0.556169\pi$
$79$	−246.527			−0.351094			−0.175547	+	0.984471i	$0.556169\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	333.857i		−	0.441513i	−0.975329	−	0.220756i	$-0.929147\pi$
$83$		−	333.857i		−	0.441513i	0.975329	−	0.220756i	$-0.0708526\pi$
$84$		0			0
$85$	−37.8467	−	21.8508i	−0.0482947	−	0.0278830i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	328.376	−	189.588i	0.391098	−	0.225801i	−0.291538	−	0.956559i	$-0.594167\pi$
$89$	328.376	−	189.588i	0.391098	−	0.225801i	0.682636	+	0.730759i	$0.260834\pi$
$90$		0			0
$91$	−683.888	−	491.632i	−0.787813	−	0.566341i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	27.6133	−	47.8276i	0.0298217	−	0.0516527i
$96$		0			0
$97$	815.158	+	470.632i	0.853265	+	0.492633i	0.861751	−	0.507331i	$-0.169368\pi$
$97$	815.158	+	470.632i	0.853265	+	0.492633i	−0.00848590	+	0.999964i	$0.502701\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−192.975	−	334.242i	−0.190116	−	0.329290i	0.755173	−	0.655526i	$-0.227553\pi$
$101$	−192.975	−	334.242i	−0.190116	−	0.329290i	−0.945288	+	0.326236i	$0.894220\pi$
$102$		0			0
$103$	395.674			0.378514			0.189257	−	0.981928i	$-0.439392\pi$
$103$	395.674			0.378514			0.189257	+	0.981928i	$0.439392\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	614.070	+	1063.60i	0.554807	+	0.960955i	0.997918	+	0.0644880i	$0.0205414\pi$
$107$	614.070	+	1063.60i	0.554807	+	0.960955i	−0.443111	+	0.896467i	$0.646125\pi$
$108$		0			0
$109$			542.176i			0.476431i	0.971212	+	0.238216i	$0.0765625\pi$
$109$			542.176i			0.476431i	−0.971212	+	0.238216i	$0.923438\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−338.706	+	586.655i	−0.281971	+	0.488388i	−0.971870	−	0.235517i	$-0.924322\pi$
$113$	−338.706	+	586.655i	−0.281971	+	0.488388i	0.689899	+	0.723906i	$0.257655\pi$
$114$		0			0
$115$	283.632	−	163.755i	0.229990	−	0.132785i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−272.424	+	157.284i	−0.209858	+	0.121161i
$120$		0			0
$121$	974.820	−	1688.44i	0.732397	−	1.26855i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	608.543i		−	0.435438i
$126$		0			0
$127$	635.930	+	1101.46i	0.444328	+	0.769599i	0.998005	−	0.0631327i	$-0.0201091\pi$
$127$	635.930	+	1101.46i	0.444328	+	0.769599i	−0.553677	+	0.832731i	$0.686776\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1249.24			−0.833179			−0.416589	−	0.909095i	$-0.636775\pi$
$131$	−1249.24			−0.833179			−0.416589	+	0.909095i	$0.636775\pi$
$132$		0			0
$133$	−198.763	−	344.267i	−0.129586	−	0.224449i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1509.40	+	871.453i	0.941290	+	0.543454i	0.890365	−	0.455248i	$-0.150450\pi$
$137$	1509.40	+	871.453i	0.941290	+	0.543454i	0.0509257	+	0.998702i	$0.483783\pi$
$138$		0			0
$139$	−494.442	+	856.398i	−0.301712	+	0.522581i	−0.976524	−	0.215409i	$-0.930891\pi$
$139$	−494.442	+	856.398i	−0.301712	+	0.522581i	0.674812	+	0.737990i	$0.264225\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2447.06	+	1104.30i	−1.43100	+	0.645776i
$144$		0			0
$145$	160.531	−	92.6824i	0.0919403	−	0.0530818i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1469.28	−	848.291i	−0.807842	−	0.466408i	0.0383642	−	0.999264i	$-0.487785\pi$
$149$	−1469.28	−	848.291i	−0.807842	−	0.466408i	−0.846206	+	0.532856i	$0.821119\pi$
$150$		0			0
$151$		−	2603.11i		−	1.40290i	−0.712719	−	0.701449i	$-0.752537\pi$
$151$		−	2603.11i		−	1.40290i	0.712719	−	0.701449i	$-0.247463\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−275.761			−0.142901
$156$		0			0
$157$	−178.988			−0.0909859			−0.0454930	−	0.998965i	$-0.514486\pi$
$157$	−178.988			−0.0909859			−0.0454930	+	0.998965i	$0.514486\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2357.44i		−	1.15399i
$162$		0			0
$163$	−1801.51	−	1040.10i	−0.865678	−	0.499799i	0.000231624	−	1.00000i	$-0.499926\pi$
$163$	−1801.51	−	1040.10i	−0.865678	−	0.499799i	−0.865910	+	0.500201i	$0.833260\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2632.63	+	1519.95i	−1.21987	+	0.704295i	−0.964891	−	0.262650i	$-0.915403\pi$
$167$	−2632.63	+	1519.95i	−1.21987	+	0.704295i	−0.254984	+	0.966945i	$0.582070\pi$
$168$		0			0
$169$	2153.48	+	435.117i	0.980192	+	0.198051i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−25.8677	+	44.8042i	−0.0113681	+	0.0196902i	−0.871654	−	0.490123i	$-0.836952\pi$
$173$	−25.8677	+	44.8042i	−0.0113681	+	0.0196902i	0.860285	+	0.509813i	$0.170285\pi$
$174$		0			0
$175$	−1848.26	−	1067.09i	−0.798371	−	0.460940i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−314.447	−	544.638i	−0.131301	−	0.227420i	0.792877	−	0.609381i	$-0.208582\pi$
$179$	−314.447	−	544.638i	−0.131301	−	0.227420i	−0.924178	+	0.381961i	$0.875249\pi$
$180$		0			0
$181$	−1661.09			−0.682145			−0.341072	−	0.940037i	$-0.610790\pi$
$181$	−1661.09			−0.682145			−0.341072	+	0.940037i	$0.610790\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−301.381	−	522.007i	−0.119773	−	0.207453i
$186$		0			0
$187$			1002.68i			0.392103i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	958.968	−	1660.98i	0.363291	−	0.629238i	−0.625210	−	0.780457i	$-0.714987\pi$
$191$	958.968	−	1660.98i	0.363291	−	0.629238i	0.988500	+	0.151219i	$0.0483199\pi$
$192$		0			0
$193$	200.131	−	115.546i	0.0746413	−	0.0430942i	−0.462215	−	0.886768i	$-0.652945\pi$
$193$	200.131	−	115.546i	0.0746413	−	0.0430942i	0.536856	+	0.843674i	$0.319612\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1411.98	−	815.208i	0.510657	−	0.294828i	−0.222447	−	0.974945i	$-0.571404\pi$
$197$	1411.98	−	815.208i	0.510657	−	0.294828i	0.733104	+	0.680117i	$0.238071\pi$
$198$		0			0
$199$	−1532.79	+	2654.86i	−0.546011	+	0.945720i	0.452531	+	0.891749i	$0.350521\pi$
$199$	−1532.79	+	2654.86i	−0.546011	+	0.945720i	−0.998543	+	0.0539709i	$0.982812\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	1334.27i		−	0.461318i
$204$		0			0
$205$	−532.613	−	922.513i	−0.181460	−	0.314298i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1267.11			−0.419366
$210$		0			0
$211$	1724.90	+	2987.61i	0.562780	+	0.974764i	0.997252	+	0.0740789i	$0.0236017\pi$
$211$	1724.90	+	2987.61i	0.562780	+	0.974764i	−0.434472	+	0.900685i	$0.643065\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	450.326	+	259.996i	0.142846	+	0.0824724i
$216$		0			0
$217$	−992.476	+	1719.02i	−0.310478	+	0.537763i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	478.950	−	666.248i	0.145781	−	0.202790i
$222$		0			0
$223$	−2186.26	+	1262.24i	−0.656516	+	0.379040i	−0.790948	−	0.611883i	$-0.790412\pi$
$223$	−2186.26	+	1262.24i	−0.656516	+	0.379040i	0.134432	+	0.990923i	$0.457079\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−998.580	−	576.530i	−0.291974	−	0.168571i	0.346858	−	0.937918i	$-0.387249\pi$
$227$	−998.580	−	576.530i	−0.291974	−	0.168571i	−0.638832	+	0.769347i	$0.720582\pi$
$228$		0			0
$229$		−	3857.53i		−	1.11316i	−0.830795	−	0.556578i	$-0.812114\pi$
$229$		−	3857.53i		−	1.11316i	0.830795	−	0.556578i	$-0.187886\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1018.55			−0.286385			−0.143192	−	0.989695i	$-0.545737\pi$
$233$	−1018.55			−0.286385			−0.143192	+	0.989695i	$0.545737\pi$
$234$		0			0
$235$	395.990			0.109921
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	1263.24i		−	0.341893i	−0.985280	−	0.170947i	$-0.945317\pi$
$239$		−	1263.24i		−	0.341893i	0.985280	−	0.170947i	$-0.0546825\pi$
$240$		0			0
$241$	−1790.89	−	1033.97i	−0.478677	−	0.276365i	0.241188	−	0.970478i	$-0.422463\pi$
$241$	−1790.89	−	1033.97i	−0.478677	−	0.276365i	−0.719865	+	0.694114i	$0.755796\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−43.4619	+	25.0927i	−0.0113334	+	0.00654333i
$246$		0			0
$247$	841.950	+	605.259i	0.216891	+	0.155918i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2563.19	+	4439.58i	−0.644570	+	1.11643i	0.339830	+	0.940487i	$0.389630\pi$
$251$	−2563.19	+	4439.58i	−0.644570	+	1.11643i	−0.984401	+	0.175942i	$0.943703\pi$
$252$		0			0
$253$	−6507.58	−	3757.15i	−1.61711	−	0.933637i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1980.76	−	3430.77i	−0.480764	−	0.832707i	0.518993	−	0.854779i	$-0.326307\pi$
$257$	−1980.76	−	3430.77i	−0.480764	−	0.832707i	−0.999756	+	0.0220714i	$0.992974\pi$
$258$		0			0
$259$	−4338.73			−1.04091
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3552.31	−	6152.79i	−0.832871	−	1.44257i	−0.895752	−	0.444553i	$-0.853362\pi$
$263$	−3552.31	−	6152.79i	−0.832871	−	1.44257i	0.0628817	−	0.998021i	$-0.479971\pi$
$264$		0			0
$265$			119.604i			0.0277252i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2806.94	−	4861.76i	0.636216	−	1.10196i	−0.350040	−	0.936735i	$-0.613832\pi$
$269$	2806.94	−	4861.76i	0.636216	−	1.10196i	0.986256	−	0.165223i	$-0.0528345\pi$
$270$		0			0
$271$	382.095	−	220.603i	0.0856480	−	0.0494489i	−0.456564	−	0.889690i	$-0.650920\pi$
$271$	382.095	−	220.603i	0.0856480	−	0.0494489i	0.542212	+	0.840242i	$0.317587\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5891.27	+	3401.33i	−1.29184	+	0.745847i
$276$		0			0
$277$	2338.44	−	4050.29i	0.507231	−	0.878550i	−0.492734	−	0.870180i	$-0.664002\pi$
$277$	2338.44	−	4050.29i	0.507231	−	0.878550i	0.999965	−	0.00837023i	$-0.00266436\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			1253.34i			0.266079i	0.991111	+	0.133040i	$0.0424737\pi$
$281$			1253.34i			0.266079i	−0.991111	+	0.133040i	$0.957526\pi$
$282$		0			0
$283$	2086.08	+	3613.19i	0.438178	+	0.758947i	0.997549	−	0.0699707i	$-0.0222906\pi$
$283$	2086.08	+	3613.19i	0.438178	+	0.758947i	−0.559371	+	0.828917i	$0.688957\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7667.59			−1.57702
$288$		0			0
$289$	2303.27	+	3989.39i	0.468812	+	0.812006i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	8468.92	+	4889.53i	1.68860	+	0.974913i	0.955592	+	0.294692i	$0.0952171\pi$
$293$	8468.92	+	4889.53i	1.68860	+	0.974913i	0.733007	+	0.680221i	$0.238116\pi$
$294$		0			0
$295$	550.996	−	954.354i	0.108747	−	0.188355i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2529.39	+	5604.99i	0.489226	+	1.08410i
$300$		0			0
$301$	3241.48	−	1871.47i	0.620718	−	0.358372i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1879.60	+	1085.18i	0.352870	+	0.203729i
$306$		0			0
$307$		−	1641.63i		−	0.305187i	−0.988289	−	0.152594i	$-0.951237\pi$
$307$		−	1641.63i		−	0.305187i	0.988289	−	0.152594i	$-0.0487626\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5900.48			1.07584			0.537919	−	0.842996i	$-0.319211\pi$
$311$	5900.48			1.07584			0.537919	+	0.842996i	$0.319211\pi$
$312$		0			0
$313$	5268.10			0.951344			0.475672	−	0.879623i	$-0.342205\pi$
$313$	5268.10			0.951344			0.475672	+	0.879623i	$0.342205\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	20.4536i		−	0.00362395i	−0.999998	−	0.00181197i	$-0.999423\pi$
$317$		−	20.4536i		−	0.00362395i	0.999998	−	0.00181197i	$-0.000576769\pi$
$318$		0			0
$319$	−3683.18	−	2126.48i	−0.646452	−	0.373229i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	335.387	−	193.636i	0.0577754	−	0.0333566i
$324$		0			0
$325$	5539.28	+	554.016i	0.945427	+	0.0945577i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1425.19	−	2468.49i	0.238824	−	0.413655i
$330$		0			0
$331$	5458.06	+	3151.21i	0.906351	+	0.523282i	0.879255	−	0.476351i	$-0.158041\pi$
$331$	5458.06	+	3151.21i	0.906351	+	0.523282i	0.0270955	+	0.999633i	$0.491374\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−999.866	−	1731.82i	−0.163070	−	0.282446i
$336$		0			0
$337$	3060.14			0.494648			0.247324	−	0.968933i	$-0.420449\pi$
$337$	3060.14			0.494648			0.247324	+	0.968933i	$0.420449\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3163.50	+	5479.34i	0.502384	+	0.870155i
$342$		0			0
$343$			6524.72i			1.02712i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−101.809	+	176.338i	−0.0157504	+	0.0272805i	−0.873793	−	0.486298i	$-0.838347\pi$
$347$	−101.809	+	176.338i	−0.0157504	+	0.0272805i	0.858043	+	0.513578i	$0.171680\pi$
$348$		0			0
$349$	1303.90	−	752.806i	0.199989	−	0.115464i	−0.396661	−	0.917965i	$-0.629831\pi$
$349$	1303.90	−	752.806i	0.199989	−	0.115464i	0.596650	+	0.802501i	$0.296498\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9641.82	+	5566.71i	−1.45377	+	0.839337i	−0.998693	−	0.0511134i	$-0.983723\pi$
$353$	−9641.82	+	5566.71i	−1.45377	+	0.839337i	−0.455081	+	0.890450i	$0.650390\pi$
$354$		0			0
$355$	1115.42	−	1931.97i	0.166762	−	0.288840i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	4176.37i		−	0.613985i	−0.951712	−	0.306992i	$-0.900677\pi$
$359$		−	4176.37i		−	0.613985i	0.951712	−	0.306992i	$-0.0993226\pi$
$360$		0			0
$361$	−3184.80	−	5516.23i	−0.464324	−	0.804233i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1031.46			0.147916
$366$		0			0
$367$	−2240.40	−	3880.49i	−0.318659	−	0.551934i	0.661549	−	0.749902i	$-0.269899\pi$
$367$	−2240.40	−	3880.49i	−0.318659	−	0.551934i	−0.980209	+	0.197968i	$0.936566\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	745.576	+	430.458i	0.104335	+	0.0602380i
$372$		0			0
$373$	181.930	−	315.111i	0.0252546	−	0.0437422i	−0.853122	−	0.521712i	$-0.825294\pi$
$373$	181.930	−	315.111i	0.0252546	−	0.0437422i	0.878376	+	0.477969i	$0.158627\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1431.59	+	3172.32i	0.195572	+	0.433377i
$378$		0			0
$379$	−8448.46	+	4877.72i	−1.14503	+	0.661086i	−0.947672	−	0.319245i	$-0.896571\pi$
$379$	−8448.46	+	4877.72i	−1.14503	+	0.661086i	−0.197362	+	0.980331i	$0.563238\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4548.66	+	2626.17i	0.606856	+	0.350368i	0.771734	−	0.635946i	$-0.219390\pi$
$383$	4548.66	+	2626.17i	0.606856	+	0.350368i	−0.164878	+	0.986314i	$0.552723\pi$
$384$		0			0
$385$		−	2569.37i		−	0.340122i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	11266.0			1.46841			0.734205	−	0.678928i	$-0.237555\pi$
$389$	11266.0			1.46841			0.734205	+	0.678928i	$0.237555\pi$
$390$		0			0
$391$	2296.64			0.297048
$392$		0			0
$393$		0			0
$394$		0			0
$395$			615.431i			0.0783941i
$396$		0			0
$397$	−2599.10	−	1500.59i	−0.328576	−	0.189704i	0.326632	−	0.945151i	$-0.394086\pi$
$397$	−2599.10	−	1500.59i	−0.328576	−	0.189704i	−0.655209	+	0.755448i	$0.727419\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6927.02	−	3999.31i	0.862640	−	0.498045i	−0.00225548	−	0.999997i	$-0.500718\pi$
$401$	6927.02	−	3999.31i	0.862640	−	0.498045i	0.864895	+	0.501952i	$0.167385\pi$
$402$		0			0
$403$	515.277	−	5151.95i	0.0636918	−	0.636817i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−6914.81	+	11976.8i	−0.842149	+	1.45864i
$408$		0			0
$409$	9207.67	+	5316.05i	1.11318	+	0.642694i	0.939651	−	0.342135i	$-0.111150\pi$
$409$	9207.67	+	5316.05i	1.11318	+	0.642694i	0.173528	+	0.984829i	$0.444483\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3966.12	−	6869.52i	−0.472542	−	0.818467i
$414$		0			0
$415$	−833.442			−0.0985833
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−164.769	−	285.389i	−0.0192112	−	0.0332748i	0.856260	−	0.516545i	$-0.172782\pi$
$419$	−164.769	−	285.389i	−0.0192112	−	0.0332748i	−0.875471	+	0.483270i	$0.839449\pi$
$420$		0			0
$421$			10594.7i			1.22649i	0.789891	+	0.613247i	$0.210137\pi$
$421$			10594.7i			1.22649i	−0.789891	+	0.613247i	$0.789863\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1039.57	−	1800.58i	0.118650	−	0.205508i
$426$		0			0
$427$	13529.5	−	7811.25i	1.53334	−	0.885277i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11044.8	−	6376.69i	1.23436	−	0.712655i	0.266421	−	0.963857i	$-0.414159\pi$
$431$	11044.8	−	6376.69i	1.23436	−	0.712655i	0.967935	+	0.251202i	$0.0808257\pi$
$432$		0			0
$433$	−7004.92	+	12132.9i	−0.777448	+	1.34658i	0.155961	+	0.987763i	$0.450153\pi$
$433$	−7004.92	+	12132.9i	−0.777448	+	1.34658i	−0.933408	+	0.358816i	$0.883181\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2902.30i			0.317703i
$438$		0			0
$439$	3648.09	+	6318.68i	0.396615	+	0.686957i	0.993306	−	0.115514i	$-0.0368515\pi$
$439$	3648.09	+	6318.68i	0.396615	+	0.686957i	−0.596691	+	0.802471i	$0.703518\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6098.72			0.654084			0.327042	−	0.945010i	$-0.393948\pi$
$443$	6098.72			0.654084			0.327042	+	0.945010i	$0.393948\pi$
$444$		0			0
$445$	−473.288	−	819.758i	−0.0504179	−	0.0873265i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	9393.12	+	5423.12i	0.987280	+	0.570006i	0.904460	−	0.426558i	$-0.140274\pi$
$449$	9393.12	+	5423.12i	0.987280	+	0.570006i	0.0828196	+	0.996565i	$0.473607\pi$
$450$		0			0
$451$	−12220.1	+	21165.9i	−1.27588	+	2.20990i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1227.31	+	1707.26i	−0.126455	+	0.175907i
$456$		0			0
$457$	4636.57	−	2676.92i	0.474594	−	0.274007i	−0.243567	−	0.969884i	$-0.578317\pi$
$457$	4636.57	−	2676.92i	0.474594	−	0.274007i	0.718161	+	0.695877i	$0.244984\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13074.6	+	7548.65i	1.32093	+	0.762637i	0.983876	−	0.178850i	$-0.0572376\pi$
$461$	13074.6	+	7548.65i	1.32093	+	0.762637i	0.337050	+	0.941487i	$0.390571\pi$
$462$		0			0
$463$		−	9543.47i		−	0.957932i	−0.877833	−	0.478966i	$-0.841012\pi$
$463$		−	9543.47i		−	0.957932i	0.877833	−	0.478966i	$-0.158988\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	4319.08			0.427972			0.213986	−	0.976837i	$-0.431355\pi$
$467$	4319.08			0.427972			0.213986	+	0.976837i	$0.431355\pi$
$468$		0			0
$469$	−14394.3			−1.41720
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11930.6i		−	1.15976i
$474$		0			0
$475$	2275.43	+	1313.72i	0.219798	+	0.126900i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	11469.8	−	6622.10i	1.09409	−	0.631673i	0.159428	−	0.987210i	$-0.449035\pi$
$479$	11469.8	−	6622.10i	1.09409	−	0.631673i	0.934662	+	0.355536i	$0.115702\pi$
$480$		0			0
$481$	10315.6	−	4655.20i	0.977864	−	0.441286i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1174.89	−	2034.96i	0.109998	−	0.190521i
$486$		0			0
$487$	536.591	+	309.801i	0.0499286	+	0.0288263i	0.524757	−	0.851252i	$-0.324156\pi$
$487$	536.591	+	309.801i	0.0499286	+	0.0288263i	−0.474828	+	0.880079i	$0.657490\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2731.07	+	4730.35i	0.251021	+	0.434781i	0.963807	−	0.266600i	$-0.0859003\pi$
$491$	2731.07	+	4730.35i	0.251021	+	0.434781i	−0.712786	+	0.701382i	$0.752567\pi$
$492$		0			0
$493$	1299.86			0.118748
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8028.91	−	13906.5i	−0.724640	−	1.25511i
$498$		0			0
$499$		−	16005.2i		−	1.43585i	−0.696119	−	0.717926i	$-0.745092\pi$
$499$		−	16005.2i		−	1.43585i	0.696119	−	0.717926i	$-0.254908\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−5027.91	+	8708.60i	−0.445693	+	0.771963i	−0.998100	−	0.0616117i	$-0.980376\pi$
$503$	−5027.91	+	8708.60i	−0.445693	+	0.771963i	0.552407	+	0.833574i	$0.313709\pi$
$504$		0			0
$505$	−834.403	+	481.743i	−0.0735256	+	0.0424500i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−12620.1	+	7286.21i	−1.09897	+	0.634490i	−0.935950	−	0.352133i	$-0.885457\pi$
$509$	−12620.1	+	7286.21i	−1.09897	+	0.634490i	−0.163019	+	0.986623i	$0.552123\pi$
$510$		0			0
$511$	3712.28	−	6429.86i	0.321373	−	0.556635i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	987.763i		−	0.0845166i
$516$		0			0
$517$	−4542.75	−	7868.27i	−0.386441	−	0.669335i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−14409.9			−1.21172			−0.605862	−	0.795570i	$-0.707172\pi$
$521$	−14409.9			−1.21172			−0.605862	+	0.795570i	$0.707172\pi$
$522$		0			0
$523$	2124.16	+	3679.15i	0.177597	+	0.307606i	0.941057	−	0.338249i	$-0.109834\pi$
$523$	2124.16	+	3679.15i	0.177597	+	0.307606i	−0.763460	+	0.645855i	$0.776501\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1674.68	−	966.877i	−0.138425	−	0.0799199i
$528$		0			0
$529$	−2522.26	+	4368.68i	−0.207303	+	0.359060i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	18230.2	−	8226.86i	1.48150	−	0.668564i
$534$		0			0
$535$	2655.18	−	1532.97i	0.214567	−	0.123880i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	997.179	+	575.721i	0.0796874	+	0.0460076i
$540$		0			0
$541$			14369.4i			1.14194i	0.820970	+	0.570971i	$0.193433\pi$
$541$			14369.4i			1.14194i	−0.820970	+	0.570971i	$0.806567\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1353.49			0.106380
$546$		0			0
$547$	−18950.8			−1.48131			−0.740657	−	0.671884i	$-0.765485\pi$
$547$	−18950.8			−1.48131			−0.740657	+	0.671884i	$0.765485\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			1642.65i			0.127004i
$552$		0			0
$553$	3836.43	+	2214.96i	0.295012	+	0.170325i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−10264.9	+	5926.42i	−0.780855	+	0.450827i	−0.836733	−	0.547611i	$-0.815538\pi$
$557$	−10264.9	+	5926.42i	−0.780855	+	0.450827i	0.0558780	+	0.998438i	$0.482204\pi$
$558$		0			0
$559$	−5698.88	+	7927.47i	−0.431193	+	0.599814i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−5792.65	+	10033.2i	−0.433625	+	0.751061i	−0.997182	−	0.0750161i	$-0.976099\pi$
$563$	−5792.65	+	10033.2i	−0.433625	+	0.751061i	0.563557	+	0.826077i	$0.309433\pi$
$564$		0			0
$565$	1464.53	+	845.546i	0.109050	+	0.0629600i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4621.17	−	8004.10i	−0.340474	−	0.589718i	0.644047	−	0.764986i	$-0.277254\pi$
$569$	−4621.17	−	8004.10i	−0.340474	−	0.589718i	−0.984521	+	0.175268i	$0.943921\pi$
$570$		0			0
$571$	20877.1			1.53009			0.765043	−	0.643979i	$-0.222718\pi$
$571$	20877.1			1.53009			0.765043	+	0.643979i	$0.222718\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	7790.74	+	13494.0i	0.565037	+	0.978673i
$576$		0			0
$577$		−	19966.8i		−	1.44060i	−0.693661	−	0.720302i	$-0.744003\pi$
$577$		−	19966.8i		−	1.44060i	0.693661	−	0.720302i	$-0.255997\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−2999.59	+	5195.45i	−0.214190	+	0.370987i
$582$		0			0
$583$	2376.51	−	1372.08i	0.168825	−	0.0974711i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16811.2	+	9705.96i	−1.18207	+	0.682466i	−0.956492	−	0.291760i	$-0.905759\pi$
$587$	−16811.2	+	9705.96i	−1.18207	+	0.682466i	−0.225575	+	0.974226i	$0.572426\pi$
$588$		0			0
$589$	1221.86	−	2116.32i	0.0854769	−	0.148050i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	19259.0i		−	1.33368i	−0.745200	−	0.666841i	$-0.767646\pi$
$593$		−	19259.0i		−	1.33368i	0.745200	−	0.666841i	$-0.232354\pi$
$594$		0			0
$595$	392.645	+	680.080i	0.0270535	+	0.0468581i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−27804.6			−1.89660			−0.948301	−	0.317372i	$-0.897200\pi$
$599$	−27804.6			−1.89660			−0.948301	+	0.317372i	$0.897200\pi$
$600$		0			0
$601$	4963.55	+	8597.12i	0.336884	+	0.583501i	0.983845	−	0.179023i	$-0.0572935\pi$
$601$	4963.55	+	8597.12i	0.336884	+	0.583501i	−0.646961	+	0.762523i	$0.723960\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−4215.02	−	2433.54i	−0.283248	−	0.163533i
$606$		0			0
$607$	7201.01	−	12472.5i	0.481516	−	0.834010i	−0.518259	−	0.855224i	$-0.673420\pi$
$607$	7201.01	−	12472.5i	0.481516	−	0.834010i	0.999775	+	0.0212137i	$0.00675305\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−739.932	+	7398.15i	−0.0489926	+	0.489848i
$612$		0			0
$613$	−12798.3	+	7389.10i	−0.843260	+	0.486856i	−0.858371	−	0.513029i	$-0.828523\pi$
$613$	−12798.3	+	7389.10i	−0.843260	+	0.486856i	0.0151111	+	0.999886i	$0.495190\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14862.7	−	8580.96i	−0.969770	−	0.559897i	−0.0706037	−	0.997504i	$-0.522493\pi$
$617$	−14862.7	−	8580.96i	−0.969770	−	0.559897i	−0.899166	+	0.437608i	$0.855826\pi$
$618$		0			0
$619$		−	7592.09i		−	0.492975i	−0.969146	−	0.246488i	$-0.920724\pi$
$619$		−	7592.09i		−	0.492975i	0.969146	−	0.246488i	$-0.0792765\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6813.53			−0.438168
$624$		0			0
$625$	13326.8			0.852917
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	4226.82i		−	0.267940i
$630$		0			0
$631$	−5852.36	−	3378.86i	−0.369222	−	0.213170i	0.303897	−	0.952705i	$-0.401712\pi$
$631$	−5852.36	−	3378.86i	−0.369222	−	0.213170i	−0.673118	+	0.739535i	$0.735046\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	2749.70	−	1587.54i	0.171840	−	0.0992118i
$636$		0			0
$637$	−387.588	−	858.871i	−0.0241080	−	0.0534218i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7126.36	+	12343.2i	−0.439117	+	0.760574i	−0.997622	−	0.0689281i	$-0.978042\pi$
$641$	−7126.36	+	12343.2i	−0.439117	+	0.760574i	0.558504	+	0.829502i	$0.311375\pi$
$642$		0			0
$643$	−11160.5	−	6443.54i	−0.684492	−	0.395192i	0.117053	−	0.993126i	$-0.462655\pi$
$643$	−11160.5	−	6443.54i	−0.684492	−	0.395192i	−0.801545	+	0.597934i	$0.795988\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14176.2	−	24553.9i	−0.861398	−	1.49199i	−0.870580	−	0.492027i	$-0.836256\pi$
$647$	−14176.2	−	24553.9i	−0.861398	−	1.49199i	0.00918183	−	0.999958i	$-0.497077\pi$
$648$		0			0
$649$	−25283.8			−1.52924
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12008.1	+	20798.7i	0.719623	+	1.24642i	0.961149	+	0.276030i	$0.0890188\pi$
$653$	12008.1	+	20798.7i	0.719623	+	1.24642i	−0.241526	+	0.970394i	$0.577648\pi$
$654$		0			0
$655$			3118.60i			0.186036i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−183.020	+	317.000i	−0.0108186	+	0.0187383i	−0.871384	−	0.490602i	$-0.836777\pi$
$659$	−183.020	+	317.000i	−0.0108186	+	0.0187383i	0.860565	+	0.509340i	$0.170110\pi$
$660$		0			0
$661$	6329.00	−	3654.05i	0.372420	−	0.215017i	−0.302095	−	0.953278i	$-0.597686\pi$
$661$	6329.00	−	3654.05i	0.372420	−	0.215017i	0.674515	+	0.738261i	$0.264353\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−859.430	+	496.192i	−0.0501162	+	0.0289346i
$666$		0			0
$667$	−4870.71	+	8436.31i	−0.282750	+	0.489738i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	49796.4i		−	2.86493i
$672$		0			0
$673$	−6947.01	−	12032.6i	−0.397901	−	0.689186i	0.595565	−	0.803307i	$-0.296928\pi$
$673$	−6947.01	−	12032.6i	−0.397901	−	0.689186i	−0.993467	+	0.114121i	$0.963595\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−23804.6			−1.35138			−0.675691	−	0.737185i	$-0.736155\pi$
$677$	−23804.6			−1.35138			−0.675691	+	0.737185i	$0.736155\pi$
$678$		0			0
$679$	−8456.94	−	14647.8i	−0.477979	−	0.827883i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18071.0	+	10433.3i	1.01240	+	0.584508i	0.911893	−	0.410429i	$-0.134621\pi$
$683$	18071.0	+	10433.3i	1.01240	+	0.584508i	0.100505	+	0.994937i	$0.467954\pi$
$684$		0			0
$685$	2175.50	−	3768.07i	0.121345	−	0.210176i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2234.51	−	223.487i	−0.123553	−	0.0123573i
$690$		0			0
$691$	−4453.31	+	2571.12i	−0.245169	+	0.141549i	−0.617550	−	0.786531i	$-0.711875\pi$
$691$	−4453.31	+	2571.12i	−0.245169	+	0.141549i	0.372381	+	0.928080i	$0.378541\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2137.91	+	1234.33i	0.116685	+	0.0673678i
$696$		0			0
$697$		−	7469.81i		−	0.405939i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−8584.99			−0.462554			−0.231277	−	0.972888i	$-0.574290\pi$
$701$	−8584.99			−0.462554			−0.231277	+	0.972888i	$0.574290\pi$
$702$		0			0
$703$	5341.52			0.286571
$704$		0			0
$705$		0			0
$706$		0			0
$707$			6935.25i			0.368921i
$708$		0			0
$709$	18679.2	+	10784.4i	0.989437	+	0.571252i	0.905106	−	0.425186i	$-0.139791\pi$
$709$	18679.2	+	10784.4i	0.989437	+	0.571252i	0.0843311	+	0.996438i	$0.473125\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	12550.4	−	7245.99i	0.659211	−	0.380595i
$714$		0			0
$715$	2756.77	+	6108.84i	0.144192	+	0.319521i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16317.7	−	28263.0i	0.846379	−	1.46597i	−0.0380391	−	0.999276i	$-0.512111\pi$
$719$	16317.7	−	28263.0i	0.846379	−	1.46597i	0.884418	−	0.466695i	$-0.154556\pi$
$720$		0			0
$721$	−6157.44	−	3555.00i	−0.318052	−	0.183627i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	4409.42	+	7637.35i	0.225878	+	0.391233i
$726$		0			0
$727$	6636.77			0.338575			0.169288	−	0.985567i	$-0.445853\pi$
$727$	6636.77			0.338575			0.169288	+	0.985567i	$0.445853\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1823.20	+	3157.87i	0.0922482	+	0.159779i
$732$		0			0
$733$			18221.6i			0.918188i	0.888388	+	0.459094i	$0.151826\pi$
$733$			18221.6i			0.918188i	−0.888388	+	0.459094i	$0.848174\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−22940.7	+	39734.4i	−1.14658	+	1.98594i
$738$		0			0
$739$	−22048.9	+	12729.9i	−1.09754	+	0.633665i	−0.935574	−	0.353131i	$-0.885117\pi$
$739$	−22048.9	+	12729.9i	−1.09754	+	0.633665i	−0.161967	+	0.986796i	$0.551784\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	19536.8	−	11279.6i	0.964651	−	0.556942i	0.0670498	−	0.997750i	$-0.478641\pi$
$743$	19536.8	−	11279.6i	0.964651	−	0.556942i	0.897601	+	0.440808i	$0.145308\pi$
$744$		0			0
$745$	−2117.68	+	3667.92i	−0.104142	+	0.180379i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	22068.9i		−	1.07661i
$750$		0			0
$751$	−10904.3	−	18886.8i	−0.529830	−	0.917693i	−0.999394	−	0.0347948i	$-0.988922\pi$
$751$	−10904.3	−	18886.8i	−0.529830	−	0.917693i	0.469564	−	0.882898i	$-0.344411\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−6498.40			−0.313246
$756$		0			0
$757$	6401.86	+	11088.4i	0.307371	+	0.532382i	0.977786	−	0.209604i	$-0.0672175\pi$
$757$	6401.86	+	11088.4i	0.307371	+	0.532382i	−0.670416	+	0.741986i	$0.733884\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3597.82	−	2077.20i	−0.171381	−	0.0989467i	0.411856	−	0.911249i	$-0.364881\pi$
$761$	−3597.82	−	2077.20i	−0.171381	−	0.0989467i	−0.583237	+	0.812302i	$0.698214\pi$
$762$		0			0
$763$	4871.27	−	8437.28i	0.231129	−	0.400328i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	16800.3	+	12077.3i	0.790904	+	0.568563i
$768$		0			0
$769$	−3570.43	+	2061.39i	−0.167429	+	0.0966652i	−0.581373	−	0.813637i	$-0.697484\pi$
$769$	−3570.43	+	2061.39i	−0.167429	+	0.0966652i	0.413944	+	0.910302i	$0.364151\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−29533.8	−	17051.3i	−1.37420	−	0.793394i	−0.382745	−	0.923854i	$-0.625021\pi$
$773$	−29533.8	−	17051.3i	−1.37420	−	0.793394i	−0.991454	+	0.130460i	$0.958355\pi$
$774$		0			0
$775$		−	13119.5i		−	0.608086i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9439.74			0.434164
$780$		0			0
$781$	−51184.0			−2.34508
$782$		0			0
$783$		0			0
$784$		0			0
$785$			446.826i			0.0203158i
$786$		0			0
$787$	36869.4	+	21286.6i	1.66995	+	0.964147i	0.967660	+	0.252258i	$0.0811732\pi$
$787$	36869.4	+	21286.6i	1.66995	+	0.964147i	0.702292	+	0.711889i	$0.252160\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	10541.8	−	6086.31i	0.473860	−	0.273583i
$792$		0			0
$793$	−23786.3	+	33088.1i	−1.06516	+	1.48171i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3027.38	−	5243.57i	0.134549	−	0.233045i	−0.790876	−	0.611976i	$-0.790375\pi$
$797$	3027.38	−	5243.57i	0.134549	−	0.233045i	0.925425	+	0.378931i	$0.123708\pi$
$798$		0			0
$799$	2404.82	+	1388.42i	0.106479	+	0.0614755i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11832.8	−	20495.0i	−0.520014	−	0.900690i
$804$		0			0
$805$	−5885.13			−0.257669
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18951.3	+	32824.7i	0.823601	+	1.42652i	0.902984	+	0.429675i	$0.141372\pi$
$809$	18951.3	+	32824.7i	0.823601	+	1.42652i	−0.0793826	+	0.996844i	$0.525295\pi$
$810$		0			0
$811$		−	2418.43i		−	0.104714i	−0.998628	−	0.0523568i	$-0.983327\pi$
$811$		−	2418.43i		−	0.104714i	0.998628	−	0.0523568i	$-0.0166733\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2596.52	+	4497.31i	−0.111598	+	0.193293i
$816$		0			0
$817$	−3990.67	+	2304.01i	−0.170888	+	0.0986624i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−35364.7	+	20417.8i	−1.50333	+	0.867950i	−0.503340	+	0.864088i	$0.667896\pi$
$821$	−35364.7	+	20417.8i	−1.50333	+	0.867950i	−0.999993	+	0.00386148i	$0.998771\pi$
$822$		0			0
$823$	849.678	−	1471.69i	0.0359877	−	0.0623326i	−0.847471	−	0.530842i	$-0.821876\pi$
$823$	849.678	−	1471.69i	0.0359877	−	0.0623326i	0.883458	+	0.468510i	$0.155209\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	28148.2i		−	1.18357i	−0.806097	−	0.591784i	$-0.798424\pi$
$827$		−	28148.2i		−	1.18357i	0.806097	−	0.591784i	$-0.201576\pi$
$828$		0			0
$829$	889.754	+	1541.10i	0.0372768	+	0.0645653i	0.884062	−	0.467370i	$-0.154798\pi$
$829$	889.754	+	1541.10i	0.0372768	+	0.0645653i	−0.846785	+	0.531935i	$0.821465\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−351.922			−0.0146379
$834$		0			0
$835$	3794.41	+	6572.11i	0.157259	+	0.272380i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23734.4	−	13703.1i	−0.976643	−	0.563865i	−0.0753879	−	0.997154i	$-0.524020\pi$
$839$	−23734.4	−	13703.1i	−0.976643	−	0.563865i	−0.901255	+	0.433289i	$0.857353\pi$
$840$		0			0
$841$	9437.77	−	16346.7i	0.386968	−	0.670248i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1086.23	−	5375.96i	0.0442218	−	0.218862i
$846$		0			0
$847$	−30340.1	+	17516.9i	−1.23081	+	0.710610i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	27432.9	+	15838.4i	1.10504	+	0.637994i
$852$		0			0
$853$		−	49349.2i		−	1.98087i	−0.137964	−	0.990437i	$-0.544056\pi$
$853$		−	49349.2i		−	1.98087i	0.137964	−	0.990437i	$-0.455944\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−657.558			−0.0262098			−0.0131049	−	0.999914i	$-0.504172\pi$
$857$	−657.558			−0.0262098			−0.0131049	+	0.999914i	$0.504172\pi$
$858$		0			0
$859$	2866.78			0.113869			0.0569343	−	0.998378i	$-0.481867\pi$
$859$	2866.78			0.113869			0.0569343	+	0.998378i	$0.481867\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	18625.8i		−	0.734683i	−0.930086	−	0.367341i	$-0.880268\pi$
$863$		−	18625.8i		−	0.734683i	0.930086	−	0.367341i	$-0.119732\pi$
$864$		0			0
$865$	111.849	+	64.5763i	0.00439652	+	0.00253833i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12228.5	−	7060.14i	0.477358	−	0.275603i
$870$		0			0
$871$	34223.3	−	15444.2i	1.33136	−	0.600810i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−5467.56	+	9470.09i	−0.211242	+	0.365883i
$876$		0			0
$877$	17084.9	+	9863.97i	0.657829	+	0.379798i	0.791449	−	0.611235i	$-0.209327\pi$
$877$	17084.9	+	9863.97i	0.657829	+	0.379798i	−0.133620	+	0.991033i	$0.542660\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	25212.2	+	43668.8i	0.964156	+	1.66997i	0.711866	+	0.702315i	$0.247850\pi$
$881$	25212.2	+	43668.8i	0.964156	+	1.66997i	0.252290	+	0.967652i	$0.418816\pi$
$882$		0			0
$883$	27255.8			1.03877			0.519384	−	0.854541i	$-0.326162\pi$
$883$	27255.8			1.03877			0.519384	+	0.854541i	$0.326162\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5869.03	+	10165.5i	0.222168	+	0.384806i	0.955466	−	0.295101i	$-0.0953534\pi$
$887$	5869.03	+	10165.5i	0.222168	+	0.384806i	−0.733298	+	0.679907i	$0.762020\pi$
$888$		0			0
$889$		−	22854.5i		−	0.862221i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1754.58	+	3039.02i	−0.0657500	+	0.113882i
$894$		0			0
$895$	−1359.64	+	784.986i	−0.0507795	+	0.0293175i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	7103.32	−	4101.10i	0.263525	−	0.152146i
$900$		0			0
$901$	−419.355	+	726.345i	−0.0155058	+	0.0268569i
$902$		0			0
$903$		0			0
$904$		0			0
$905$			4146.76i			0.152313i
$906$		0			0
$907$	7717.36	+	13366.9i	0.282526	+	0.489349i	0.972006	−	0.234956i	$-0.0754945\pi$
$907$	7717.36	+	13366.9i	0.282526	+	0.489349i	−0.689481	+	0.724304i	$0.742161\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	24957.2			0.907649			0.453825	−	0.891091i	$-0.350059\pi$
$911$	24957.2			0.907649			0.453825	+	0.891091i	$0.350059\pi$
$912$		0			0
$913$	9561.14	+	16560.4i	0.346580	+	0.600294i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	19440.5	+	11224.0i	0.700090	+	0.404197i
$918$		0			0
$919$	−8291.17	+	14360.7i	−0.297606	+	0.515469i	−0.975588	−	0.219610i	$-0.929522\pi$
$919$	−8291.17	+	14360.7i	−0.297606	+	0.515469i	0.677981	+	0.735079i	$0.262855\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	34010.1	+	24449.1i	1.21285	+	0.871887i
$924$		0			0
$925$	24834.8	−	14338.4i	0.882772	−	0.509669i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	5118.69	+	2955.28i	0.180774	+	0.104370i	0.587656	−	0.809111i	$-0.300051\pi$
$929$	5118.69	+	2955.28i	0.180774	+	0.104370i	−0.406882	+	0.913481i	$0.633384\pi$
$930$		0			0
$931$		−	444.730i		−	0.0156557i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	2503.09			0.0875507
$936$		0			0
$937$	−7798.93			−0.271910			−0.135955	−	0.990715i	$-0.543410\pi$
$937$	−7798.93			−0.271910			−0.135955	+	0.990715i	$0.543410\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			9738.22i			0.337361i	0.985671	+	0.168681i	$0.0539507\pi$
$941$			9738.22i			0.337361i	−0.985671	+	0.168681i	$0.946049\pi$
$942$		0			0
$943$	48480.5	+	27990.2i	1.67417	+	0.966582i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	11928.9	−	6887.17i	0.409332	−	0.236328i	−0.281170	−	0.959658i	$-0.590723\pi$
$947$	11928.9	−	6887.17i	0.409332	−	0.236328i	0.690503	+	0.723330i	$0.257389\pi$
$948$		0			0
$949$	−1927.35	+	19270.5i	−0.0659268	+	0.659164i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−15302.3	+	26504.3i	−0.520135	+	0.900901i	0.479591	+	0.877492i	$0.340785\pi$
$953$	−15302.3	+	26504.3i	−0.520135	+	0.900901i	−0.999726	+	0.0234084i	$0.992548\pi$
$954$		0			0
$955$	−4146.48	−	2393.97i	−0.140499	−	0.0811174i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15659.4	−	27122.9i	−0.527288	−	0.913289i
$960$		0			0
$961$	17588.8			0.590408
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−288.449	−	499.608i	−0.00962228	−	0.0166663i
$966$		0			0
$967$		−	48790.3i		−	1.62253i	−0.584677	−	0.811266i	$-0.698779\pi$
$967$		−	48790.3i		−	1.62253i	0.584677	−	0.811266i	$-0.301221\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16538.8	+	28646.0i	−0.546607	+	0.946751i	0.451897	+	0.892070i	$0.350747\pi$
$971$	−16538.8	+	28646.0i	−0.546607	+	0.946751i	−0.998504	+	0.0546805i	$0.982586\pi$
$972$		0			0
$973$	15388.9	−	8884.79i	0.507035	−	0.292737i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−11795.8	+	6810.30i	−0.386265	+	0.223010i	−0.680540	−	0.732711i	$-0.738255\pi$
$977$	−11795.8	+	6810.30i	−0.386265	+	0.223010i	0.294276	+	0.955721i	$0.404922\pi$
$978$		0			0
$979$	−10859.0	+	18808.3i	−0.354499	+	0.614011i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	23886.5i		−	0.775037i	−0.921862	−	0.387519i	$-0.873332\pi$
$983$		−	23886.5i		−	0.775037i	0.921862	−	0.387519i	$-0.126668\pi$
$984$		0			0
$985$	−2035.09	−	3524.87i	−0.0658307	−	0.114022i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−27326.9			−0.878611
$990$		0			0
$991$	1590.09	+	2754.11i	0.0509695	+	0.0882818i	0.890385	−	0.455209i	$-0.150436\pi$
$991$	1590.09	+	2754.11i	0.0509695	+	0.0882818i	−0.839415	+	0.543491i	$0.817102\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6627.61	+	3826.45i	0.211165	+	0.121916i
$996$		0			0
$997$	26513.7	−	45923.0i	0.842223	−	1.45877i	−0.0457876	−	0.998951i	$-0.514580\pi$
$997$	26513.7	−	45923.0i	0.842223	−	1.45877i	0.888011	−	0.459822i	$-0.152087\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.4.t.g.361.3		8
3.2	odd	2		52.4.h.a.49.4	yes	8
12.11	even	2		208.4.w.c.49.1		8
13.4	even	6	inner	468.4.t.g.433.2		8
39.2	even	12		676.4.a.g.1.2		8
39.5	even	4		676.4.e.h.653.8		16
39.8	even	4		676.4.e.h.653.7		16
39.11	even	12		676.4.a.g.1.1		8
39.17	odd	6		52.4.h.a.17.4	✓	8
39.20	even	12		676.4.e.h.529.7		16
39.23	odd	6		676.4.d.d.337.1		8
39.29	odd	6		676.4.d.d.337.2		8
39.32	even	12		676.4.e.h.529.8		16
39.35	odd	6		676.4.h.e.485.4		8
39.38	odd	2		676.4.h.e.361.4		8
156.95	even	6		208.4.w.c.17.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
52.4.h.a.17.4	✓	8	39.17	odd	6
52.4.h.a.49.4	yes	8	3.2	odd	2
208.4.w.c.17.1		8	156.95	even	6
208.4.w.c.49.1		8	12.11	even	2
468.4.t.g.361.3		8	1.1	even	1	trivial
468.4.t.g.433.2		8	13.4	even	6	inner
676.4.a.g.1.1		8	39.11	even	12
676.4.a.g.1.2		8	39.2	even	12
676.4.d.d.337.1		8	39.23	odd	6
676.4.d.d.337.2		8	39.29	odd	6
676.4.e.h.529.7		16	39.20	even	12
676.4.e.h.529.8		16	39.32	even	12
676.4.e.h.653.7		16	39.8	even	4
676.4.e.h.653.8		16	39.5	even	4
676.4.h.e.361.4		8	39.38	odd	2
676.4.h.e.485.4		8	39.35	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	468.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-2.49640i q^{5} +(-15.5619 - 8.98467i) q^{7} +(-49.6032 + 28.6384i) q^{11} +(46.6395 + 4.66469i) q^{13} +(8.75292 - 15.1605i) q^{17} +(19.1586 + 11.0612i) q^{19} +(65.5963 + 113.616i) q^{23} +118.768 q^{25} +(37.1264 + 64.3048i) q^{29} -110.463i q^{31} +(-22.4294 + 38.8488i) q^{35} +(209.104 - 120.726i) q^{37} +(369.537 - 213.352i) q^{41} +(-104.148 + 180.390i) q^{43} +158.624i q^{47} +(-10.0516 - 17.4098i) q^{49} -47.9103 q^{53} +(71.4931 + 123.830i) q^{55} +(382.291 + 220.716i) q^{59} +(-434.699 + 752.921i) q^{61} +(11.6449 - 116.431i) q^{65} +(693.726 - 400.523i) q^{67} +(773.901 + 446.812i) q^{71} +413.180i q^{73} +1029.23 q^{77} -246.527 q^{79} -333.857i q^{83} +(-37.8467 - 21.8508i) q^{85} +(328.376 - 189.588i) q^{89} +(-683.888 - 491.632i) q^{91} +(27.6133 - 47.8276i) q^{95} +(815.158 + 470.632i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{7} - 72 q^{11} + 62 q^{13} - 88 q^{17} - 144 q^{19} + 20 q^{23} - 84 q^{25} + 484 q^{29} - 40 q^{35} + 996 q^{37} - 156 q^{41} + 504 q^{43} + 922 q^{49} + 1164 q^{53} - 1128 q^{55} - 600 q^{59}+ \cdots - 3042 q^{97}+O(q^{100}) \) 8 * q - 36 * q^7 - 72 * q^11 + 62 * q^13 - 88 * q^17 - 144 * q^19 + 20 * q^23 - 84 * q^25 + 484 * q^29 - 40 * q^35 + 996 * q^37 - 156 * q^41 + 504 * q^43 + 922 * q^49 + 1164 * q^53 - 1128 * q^55 - 600 * q^59 - 1224 * q^61 - 670 * q^65 + 960 * q^67 + 2964 * q^71 + 3972 * q^77 - 3968 * q^79 + 3870 * q^85 - 5430 * q^89 - 1720 * q^91 - 2400 * q^95 - 3042 * q^97

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