LMFDB - Embedded newform 840.2.k.a.209.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(209,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("840.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.k (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:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.12
Character	$\chi$	$=$	840.209
Dual form			840.2.k.a.209.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.325454 + 1.70120i) q^{3} +(1.34492 - 1.78639i) q^{5} +(1.53984 - 2.15149i) q^{7} +(-2.78816 - 1.10732i) q^{9} +O(q^{10})$	\(q+(-0.325454 + 1.70120i) q^{3} +(1.34492 - 1.78639i) q^{5} +(1.53984 - 2.15149i) q^{7} +(-2.78816 - 1.10732i) q^{9} +5.00998i q^{11} +1.62957 q^{13} +(2.60130 + 2.86936i) q^{15} -4.73603i q^{17} -2.13979i q^{19} +(3.15897 + 3.31978i) q^{21} +9.46270 q^{23} +(-1.38239 - 4.80510i) q^{25} +(2.79119 - 4.38283i) q^{27} +5.96788i q^{29} -4.38673i q^{31} +(-8.52298 - 1.63052i) q^{33} +(-1.77245 - 5.64433i) q^{35} -3.75323i q^{37} +(-0.530349 + 2.77222i) q^{39} +10.6255 q^{41} +4.74762i q^{43} +(-5.72796 + 3.49149i) q^{45} +5.64934i q^{47} +(-2.25781 - 6.62588i) q^{49} +(8.05694 + 1.54136i) q^{51} -4.69333 q^{53} +(8.94979 + 6.73801i) q^{55} +(3.64020 + 0.696402i) q^{57} +8.66206 q^{59} +2.09779i q^{61} +(-6.67570 + 4.29360i) q^{63} +(2.19163 - 2.91105i) q^{65} +6.20510i q^{67} +(-3.07967 + 16.0979i) q^{69} +8.65231i q^{71} -12.5394 q^{73} +(8.62434 - 0.787891i) q^{75} +(10.7789 + 7.71455i) q^{77} +1.06070 q^{79} +(6.54767 + 6.17479i) q^{81} -5.96483i q^{83} +(-8.46041 - 6.36957i) q^{85} +(-10.1525 - 1.94227i) q^{87} +0.187117 q^{89} +(2.50927 - 3.50600i) q^{91} +(7.46270 + 1.42768i) q^{93} +(-3.82250 - 2.87784i) q^{95} +10.8671 q^{97} +(5.54767 - 13.9686i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{9}+O(q^{10}) $ 24 * q - 2 * q^9	$ 24 q - 2 q^{9} - 6 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} - 2 q^{39} + 6 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} - 16 q^{65} - 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} - 32 q^{93} + 24 q^{95} - 6 q^{99}+O(q^{100}) $ 24 * q - 2 * q^9 - 6 * q^15 - 2 * q^21 + 16 * q^23 + 8 * q^25 - 8 * q^35 - 2 * q^39 + 6 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 - 16 * q^65 - 8 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^85 + 12 * q^91 - 32 * q^93 + 24 * q^95 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$-1$	$-1$	$-1$	$1$	$1$

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
$3$	−0.325454	+	1.70120i	−0.187901	+	0.982188i
$3$	−0.325454	+	1.70120i	−0.187901	+	0.982188i
$4$		0			0
$5$	1.34492	−	1.78639i	0.601465	−	0.798899i
$5$	1.34492	−	1.78639i	0.601465	−	0.798899i
$6$		0			0
$7$	1.53984	−	2.15149i	0.582003	−	0.813186i
$7$	1.53984	−	2.15149i	0.582003	−	0.813186i
$8$		0			0
$9$	−2.78816	−	1.10732i	−0.929387	−	0.369108i
$10$		0			0
$11$			5.00998i			1.51057i	0.655399	+	0.755283i	$0.272501\pi$
$11$			5.00998i			1.51057i	−0.655399	+	0.755283i	$0.727499\pi$
$12$		0			0
$13$	1.62957			0.451961			0.225980	−	0.974132i	$-0.427441\pi$
$13$	1.62957			0.451961			0.225980	+	0.974132i	$0.427441\pi$
$14$		0			0
$15$	2.60130	+	2.86936i	0.671653	+	0.740866i
$16$		0			0
$17$		−	4.73603i		−	1.14866i	−0.818625	−	0.574328i	$-0.805263\pi$
$17$		−	4.73603i		−	1.14866i	0.818625	−	0.574328i	$-0.194737\pi$
$18$		0			0
$19$		−	2.13979i		−	0.490901i	−0.969409	−	0.245450i	$-0.921064\pi$
$19$		−	2.13979i		−	0.490901i	0.969409	−	0.245450i	$-0.0789359\pi$
$20$		0			0
$21$	3.15897	+	3.31978i	0.689343	+	0.724435i
$22$		0			0
$23$	9.46270			1.97311			0.986555	−	0.163429i	$-0.0522555\pi$
$23$	9.46270			1.97311			0.986555	+	0.163429i	$0.0522555\pi$
$24$		0			0
$25$	−1.38239	−	4.80510i	−0.276479	−	0.961020i
$26$		0			0
$27$	2.79119	−	4.38283i	0.537166	−	0.843477i
$28$		0			0
$29$			5.96788i			1.10821i	0.832448	+	0.554103i	$0.186939\pi$
$29$			5.96788i			1.10821i	−0.832448	+	0.554103i	$0.813061\pi$
$30$		0			0
$31$		−	4.38673i		−	0.787880i	−0.919136	−	0.393940i	$-0.871112\pi$
$31$		−	4.38673i		−	0.787880i	0.919136	−	0.393940i	$-0.128888\pi$
$32$		0			0
$33$	−8.52298	−	1.63052i	−1.48366	−	0.283837i
$34$		0			0
$35$	−1.77245	−	5.64433i	−0.299599	−	0.954065i
$36$		0			0
$37$		−	3.75323i		−	0.617027i	−0.951220	−	0.308514i	$-0.900168\pi$
$37$		−	3.75323i		−	0.617027i	0.951220	−	0.308514i	$-0.0998315\pi$
$38$		0			0
$39$	−0.530349	+	2.77222i	−0.0849238	+	0.443911i
$40$		0			0
$41$	10.6255			1.65943			0.829716	−	0.558186i	$-0.188503\pi$
$41$	10.6255			1.65943			0.829716	+	0.558186i	$0.188503\pi$
$42$		0			0
$43$			4.74762i			0.724005i	0.932177	+	0.362002i	$0.117907\pi$
$43$			4.74762i			0.724005i	−0.932177	+	0.362002i	$0.882093\pi$
$44$		0			0
$45$	−5.72796	+	3.49149i	−0.853874	+	0.520480i
$46$		0			0
$47$			5.64934i			0.824041i	0.911175	+	0.412021i	$0.135177\pi$
$47$			5.64934i			0.824041i	−0.911175	+	0.412021i	$0.864823\pi$
$48$		0			0
$49$	−2.25781	−	6.62588i	−0.322544	−	0.946554i
$50$		0			0
$51$	8.05694	+	1.54136i	1.12820	+	0.215834i
$52$		0			0
$53$	−4.69333			−0.644679			−0.322340	−	0.946624i	$-0.604469\pi$
$53$	−4.69333			−0.644679			−0.322340	+	0.946624i	$0.604469\pi$
$54$		0			0
$55$	8.94979	+	6.73801i	1.20679	+	0.908553i
$56$		0			0
$57$	3.64020	+	0.696402i	0.482157	+	0.0922406i
$58$		0			0
$59$	8.66206			1.12770			0.563852	−	0.825876i	$-0.309319\pi$
$59$	8.66206			1.12770			0.563852	+	0.825876i	$0.309319\pi$
$60$		0			0
$61$			2.09779i			0.268594i	0.990941	+	0.134297i	$0.0428776\pi$
$61$			2.09779i			0.268594i	−0.990941	+	0.134297i	$0.957122\pi$
$62$		0			0
$63$	−6.67570	+	4.29360i	−0.841059	+	0.540943i
$64$		0			0
$65$	2.19163	−	2.91105i	0.271839	−	0.361071i
$66$		0			0
$67$			6.20510i			0.758073i	0.925382	+	0.379037i	$0.123745\pi$
$67$			6.20510i			0.758073i	−0.925382	+	0.379037i	$0.876255\pi$
$68$		0			0
$69$	−3.07967	+	16.0979i	−0.370749	+	1.93797i
$70$		0			0
$71$			8.65231i			1.02684i	0.858138	+	0.513420i	$0.171622\pi$
$71$			8.65231i			1.02684i	−0.858138	+	0.513420i	$0.828378\pi$
$72$		0			0
$73$	−12.5394			−1.46763			−0.733815	−	0.679349i	$-0.762262\pi$
$73$	−12.5394			−1.46763			−0.733815	+	0.679349i	$0.762262\pi$
$74$		0			0
$75$	8.62434	−	0.787891i	0.995853	−	0.0909778i
$76$		0			0
$77$	10.7789	+	7.71455i	1.22837	+	0.879155i
$78$		0			0
$79$	1.06070			0.119338			0.0596689	−	0.998218i	$-0.480996\pi$
$79$	1.06070			0.119338			0.0596689	+	0.998218i	$0.480996\pi$
$80$		0			0
$81$	6.54767	+	6.17479i	0.727519	+	0.686088i
$82$		0			0
$83$		−	5.96483i		−	0.654725i	−0.944899	−	0.327362i	$-0.893840\pi$
$83$		−	5.96483i		−	0.654725i	0.944899	−	0.327362i	$-0.106160\pi$
$84$		0			0
$85$	−8.46041	−	6.36957i	−0.917661	−	0.690877i
$86$		0			0
$87$	−10.1525	−	1.94227i	−1.08847	−	0.208233i
$88$		0			0
$89$	0.187117			0.0198344			0.00991719	−	0.999951i	$-0.496843\pi$
$89$	0.187117			0.0198344			0.00991719	+	0.999951i	$0.496843\pi$
$90$		0			0
$91$	2.50927	−	3.50600i	0.263043	−	0.367529i
$92$		0			0
$93$	7.46270	+	1.42768i	0.773846	+	0.148043i
$94$		0			0
$95$	−3.82250	−	2.87784i	−0.392180	−	0.295260i
$96$		0			0
$97$	10.8671			1.10339			0.551693	−	0.834047i	$-0.313982\pi$
$97$	10.8671			1.10339			0.551693	+	0.834047i	$0.313982\pi$
$98$		0			0
$99$	5.54767	−	13.9686i	0.557562	−	1.40390i
$100$		0			0
$101$	−4.64716			−0.462409			−0.231205	−	0.972905i	$-0.574267\pi$
$101$	−4.64716			−0.462409			−0.231205	+	0.972905i	$0.574267\pi$
$102$		0			0
$103$	−0.674162			−0.0664272			−0.0332136	−	0.999448i	$-0.510574\pi$
$103$	−0.674162			−0.0664272			−0.0332136	+	0.999448i	$0.510574\pi$
$104$		0			0
$105$	10.1790	−	1.17833i	0.993366	−	0.114993i
$106$		0			0
$107$	−8.45930			−0.817792			−0.408896	−	0.912581i	$-0.634086\pi$
$107$	−8.45930			−0.817792			−0.408896	+	0.912581i	$0.634086\pi$
$108$		0			0
$109$	7.32715			0.701814			0.350907	−	0.936410i	$-0.385873\pi$
$109$	7.32715			0.701814			0.350907	+	0.936410i	$0.385873\pi$
$110$		0			0
$111$	6.38499	+	1.22150i	0.606037	+	0.115940i
$112$		0			0
$113$	−10.2229			−0.961691			−0.480845	−	0.876805i	$-0.659670\pi$
$113$	−10.2229			−0.961691			−0.480845	+	0.876805i	$0.659670\pi$
$114$		0			0
$115$	12.7266	−	16.9041i	1.18676	−	1.57632i
$116$		0			0
$117$	−4.54350	−	1.80446i	−0.420046	−	0.166822i
$118$		0			0
$119$	−10.1895	−	7.29271i	−0.934072	−	0.668522i
$120$		0			0
$121$	−14.0999			−1.28181
$122$		0			0
$123$	−3.45812	+	18.0762i	−0.311808	+	1.62987i
$124$		0			0
$125$	−10.4430	−	3.99296i	−0.934050	−	0.357142i
$126$		0			0
$127$		−	19.7565i		−	1.75310i	−0.481307	−	0.876552i	$-0.659838\pi$
$127$		−	19.7565i		−	1.75310i	0.481307	−	0.876552i	$-0.340162\pi$
$128$		0			0
$129$	−8.07664	−	1.54513i	−0.711109	−	0.136041i
$130$		0			0
$131$	−17.5174			−1.53050			−0.765251	−	0.643732i	$-0.777385\pi$
$131$	−17.5174			−1.53050			−0.765251	+	0.643732i	$0.777385\pi$
$132$		0			0
$133$	−4.60373	−	3.29492i	−0.399194	−	0.285706i
$134$		0			0
$135$	−4.07553	−	10.8807i	−0.350766	−	0.936463i
$136$		0			0
$137$	−21.3696			−1.82573			−0.912863	−	0.408266i	$-0.866134\pi$
$137$	−21.3696			−1.82573			−0.912863	+	0.408266i	$0.866134\pi$
$138$		0			0
$139$		−	3.32832i		−	0.282305i	−0.989988	−	0.141152i	$-0.954919\pi$
$139$		−	3.32832i		−	0.282305i	0.989988	−	0.141152i	$-0.0450807\pi$
$140$		0			0
$141$	−9.61066	−	1.83860i	−0.809363	−	0.154838i
$142$		0			0
$143$			8.16411i			0.682717i
$144$		0			0
$145$	10.6610	+	8.02630i	0.885345	+	0.666548i
$146$		0			0
$147$	12.0068	−	1.68457i	0.990301	−	0.138941i
$148$		0			0
$149$		−	4.29916i		−	0.352201i	−0.984372	−	0.176101i	$-0.943652\pi$
$149$		−	4.29916i		−	0.352201i	0.984372	−	0.176101i	$-0.0563484\pi$
$150$		0			0
$151$	−7.59367			−0.617964			−0.308982	−	0.951068i	$-0.599988\pi$
$151$	−7.59367			−0.617964			−0.308982	+	0.951068i	$0.599988\pi$
$152$		0			0
$153$	−5.24432	+	13.2048i	−0.423978	+	1.06755i
$154$		0			0
$155$	−7.83642	−	5.89979i	−0.629436	−	0.473883i
$156$		0			0
$157$	15.4164			1.23036			0.615181	−	0.788386i	$-0.289083\pi$
$157$	15.4164			1.23036			0.615181	+	0.788386i	$0.289083\pi$
$158$		0			0
$159$	1.52746	−	7.98430i	0.121136	−	0.633196i
$160$		0			0
$161$	14.5710	−	20.3589i	1.14836	−	1.60451i
$162$		0			0
$163$			7.60297i			0.595511i	0.954642	+	0.297755i	$0.0962380\pi$
$163$			7.60297i			0.595511i	−0.954642	+	0.297755i	$0.903762\pi$
$164$		0			0
$165$	−14.3754	+	13.0325i	−1.11913	+	1.01458i
$166$		0			0
$167$			3.12412i			0.241752i	0.992668	+	0.120876i	$0.0385703\pi$
$167$			3.12412i			0.241752i	−0.992668	+	0.120876i	$0.961430\pi$
$168$		0			0
$169$	−10.3445			−0.795731
$170$		0			0
$171$	−2.36944	+	5.96607i	−0.181195	+	0.456237i
$172$		0			0
$173$			5.28993i			0.402186i	0.979572	+	0.201093i	$0.0644493\pi$
$173$			5.28993i			0.402186i	−0.979572	+	0.201093i	$0.935551\pi$
$174$		0			0
$175$	−12.4668	−	4.42486i	−0.942400	−	0.334488i
$176$		0			0
$177$	−2.81910	+	14.7359i	−0.211897	+	1.10762i
$178$		0			0
$179$			14.5042i			1.08410i	0.840348	+	0.542048i	$0.182351\pi$
$179$			14.5042i			1.08410i	−0.840348	+	0.542048i	$0.817649\pi$
$180$		0			0
$181$		−	18.5082i		−	1.37571i	−0.725850	−	0.687853i	$-0.758554\pi$
$181$		−	18.5082i		−	1.37571i	0.725850	−	0.687853i	$-0.241446\pi$
$182$		0			0
$183$	−3.56875	−	0.682732i	−0.263810	−	0.0504690i
$184$		0			0
$185$	−6.70474	−	5.04778i	−0.492942	−	0.371120i
$186$		0			0
$187$	23.7274			1.73512
$188$		0			0
$189$	−5.13164	−	12.7541i	−0.373272	−	0.927722i
$190$		0			0
$191$		−	12.7342i		−	0.921414i	−0.887552	−	0.460707i	$-0.847596\pi$
$191$		−	12.7342i		−	0.921414i	0.887552	−	0.460707i	$-0.152404\pi$
$192$		0			0
$193$		−	4.04248i		−	0.290984i	−0.989359	−	0.145492i	$-0.953523\pi$
$193$		−	4.04248i		−	0.290984i	0.989359	−	0.145492i	$-0.0464765\pi$
$194$		0			0
$195$	4.23900	+	4.67582i	0.303561	+	0.334842i
$196$		0			0
$197$	5.44077			0.387639			0.193819	−	0.981037i	$-0.437912\pi$
$197$	5.44077			0.387639			0.193819	+	0.981037i	$0.437912\pi$
$198$		0			0
$199$			15.8119i			1.12087i	0.828197	+	0.560437i	$0.189367\pi$
$199$			15.8119i			1.12087i	−0.828197	+	0.560437i	$0.810633\pi$
$200$		0			0
$201$	−10.5561	−	2.01947i	−0.744570	−	0.142443i
$202$		0			0
$203$	12.8398	+	9.18955i	0.901179	+	0.644980i
$204$		0			0
$205$	14.2905	−	18.9814i	0.998090	−	1.32572i
$206$		0			0
$207$	−26.3835	−	10.4783i	−1.83378	−	0.728290i
$208$		0			0
$209$	10.7203			0.741538
$210$		0			0
$211$	17.7369			1.22106			0.610531	−	0.791993i	$-0.290956\pi$
$211$	17.7369			1.22106			0.610531	+	0.791993i	$0.290956\pi$
$212$		0			0
$213$	−14.7193	−	2.81593i	−1.00855	−	0.192944i
$214$		0			0
$215$	8.48111	+	6.38515i	0.578407	+	0.435464i
$216$		0			0
$217$	−9.43800	−	6.75484i	−0.640693	−	0.458549i
$218$		0			0
$219$	4.08101	−	21.3321i	0.275769	−	1.44149i
$220$		0			0
$221$		−	7.71769i		−	0.519148i
$222$		0			0
$223$	−28.0313			−1.87711			−0.938556	−	0.345126i	$-0.887836\pi$
$223$	−28.0313			−1.87711			−0.938556	+	0.345126i	$0.887836\pi$
$224$		0			0
$225$	−1.46646	+	14.9281i	−0.0977643	+	0.995210i
$226$		0			0
$227$			17.2192i			1.14288i	0.820645	+	0.571439i	$0.193615\pi$
$227$			17.2192i			1.14288i	−0.820645	+	0.571439i	$0.806385\pi$
$228$		0			0
$229$			4.89863i			0.323710i	0.986815	+	0.161855i	$0.0517477\pi$
$229$			4.89863i			0.323710i	−0.986815	+	0.161855i	$0.948252\pi$
$230$		0			0
$231$	−16.6320	+	15.8264i	−1.09431	+	1.04130i
$232$		0			0
$233$	17.0858			1.11933			0.559663	−	0.828720i	$-0.310931\pi$
$233$	17.0858			1.11933			0.559663	+	0.828720i	$0.310931\pi$
$234$		0			0
$235$	10.0919	+	7.59790i	0.658326	+	0.495632i
$236$		0			0
$237$	−0.345208	+	1.80446i	−0.0224237	+	0.117212i
$238$		0			0
$239$			20.4559i			1.32319i	0.749863	+	0.661593i	$0.230119\pi$
$239$			20.4559i			1.32319i	−0.749863	+	0.661593i	$0.769881\pi$
$240$		0			0
$241$			10.8088i			0.696253i	0.937448	+	0.348127i	$0.113182\pi$
$241$			10.8088i			0.696253i	−0.937448	+	0.348127i	$0.886818\pi$
$242$		0			0
$243$	−12.6355	+	9.12929i	−0.810568	+	0.585644i
$244$		0			0
$245$	−14.8730	−	4.87793i	−0.950200	−	0.311639i
$246$		0			0
$247$		−	3.48693i		−	0.221868i
$248$		0			0
$249$	10.1474	+	1.94128i	0.643063	+	0.123023i
$250$		0			0
$251$	−29.7407			−1.87722			−0.938609	−	0.344982i	$-0.887885\pi$
$251$	−29.7407			−1.87722			−0.938609	+	0.344982i	$0.887885\pi$
$252$		0			0
$253$			47.4080i			2.98051i
$254$		0			0
$255$	13.5894	−	12.3198i	0.851001	−	0.771499i
$256$		0			0
$257$		−	28.1376i		−	1.75518i	−0.479415	−	0.877589i	$-0.659151\pi$
$257$		−	28.1376i		−	1.75518i	0.479415	−	0.877589i	$-0.340849\pi$
$258$		0			0
$259$	−8.07503	−	5.77936i	−0.501758	−	0.359112i
$260$		0			0
$261$	6.60837	−	16.6394i	0.409048	−	1.02995i
$262$		0			0
$263$	−21.0985			−1.30099			−0.650496	−	0.759510i	$-0.725439\pi$
$263$	−21.0985			−1.30099			−0.650496	+	0.759510i	$0.725439\pi$
$264$		0			0
$265$	−6.31215	+	8.38414i	−0.387752	+	0.515033i
$266$		0			0
$267$	−0.0608979	+	0.318324i	−0.00372689	+	0.0194811i
$268$		0			0
$269$	5.42683			0.330880			0.165440	−	0.986220i	$-0.447096\pi$
$269$	5.42683			0.330880			0.165440	+	0.986220i	$0.447096\pi$
$270$		0			0
$271$			17.0004i			1.03270i	0.856377	+	0.516351i	$0.172710\pi$
$271$			17.0004i			1.03270i	−0.856377	+	0.516351i	$0.827290\pi$
$272$		0			0
$273$	5.14775	+	5.40981i	0.311556	+	0.327416i
$274$		0			0
$275$	24.0735	−	6.92577i	1.45168	−	0.417640i
$276$		0			0
$277$			0.897874i			0.0539480i	0.999636	+	0.0269740i	$0.00858714\pi$
$277$			0.897874i			0.0539480i	−0.999636	+	0.0269740i	$0.991413\pi$
$278$		0			0
$279$	−4.85753	+	12.2309i	−0.290813	+	0.732245i
$280$		0			0
$281$		−	12.6282i		−	0.753338i	−0.926348	−	0.376669i	$-0.877069\pi$
$281$		−	12.6282i		−	0.753338i	0.926348	−	0.376669i	$-0.122931\pi$
$282$		0			0
$283$	1.82856			0.108696			0.0543482	−	0.998522i	$-0.482692\pi$
$283$	1.82856			0.108696			0.0543482	+	0.998522i	$0.482692\pi$
$284$		0			0
$285$	6.13982	−	5.56623i	0.363692	−	0.329715i
$286$		0			0
$287$	16.3616	−	22.8607i	0.965794	−	1.34943i
$288$		0			0
$289$	−5.43001			−0.319413
$290$		0			0
$291$	−3.53674	+	18.4871i	−0.207327	+	1.08373i
$292$		0			0
$293$			6.26878i			0.366226i	0.983092	+	0.183113i	$0.0586175\pi$
$293$			6.26878i			0.366226i	−0.983092	+	0.183113i	$0.941383\pi$
$294$		0			0
$295$	11.6498	−	15.4738i	0.678275	−	0.900922i
$296$		0			0
$297$	21.9579	+	13.9838i	1.27413	+	0.811424i
$298$		0			0
$299$	15.4201			0.891769
$300$		0			0
$301$	10.2144	+	7.31055i	0.588751	+	0.421373i
$302$		0			0
$303$	1.51243	−	7.90574i	0.0868871	−	0.454173i
$304$		0			0
$305$	3.74747	+	2.82135i	0.214579	+	0.161550i
$306$		0			0
$307$	−18.5040			−1.05608			−0.528039	−	0.849220i	$-0.677073\pi$
$307$	−18.5040			−1.05608			−0.528039	+	0.849220i	$0.677073\pi$
$308$		0			0
$309$	0.219409	−	1.14688i	0.0124817	−	0.0652440i
$310$		0			0
$311$	−19.3348			−1.09637			−0.548187	−	0.836356i	$-0.684682\pi$
$311$	−19.3348			−1.09637			−0.548187	+	0.836356i	$0.684682\pi$
$312$		0			0
$313$	10.6125			0.599856			0.299928	−	0.953962i	$-0.403037\pi$
$313$	10.6125			0.599856			0.299928	+	0.953962i	$0.403037\pi$
$314$		0			0
$315$	−1.30822	+	17.7000i	−0.0737097	+	0.997280i
$316$		0			0
$317$	−2.68654			−0.150891			−0.0754455	−	0.997150i	$-0.524038\pi$
$317$	−2.68654			−0.150891			−0.0754455	+	0.997150i	$0.524038\pi$
$318$		0			0
$319$	−29.8990			−1.67402
$320$		0			0
$321$	2.75311	−	14.3910i	0.153664	−	0.803225i
$322$		0			0
$323$	−10.1341			−0.563877
$324$		0			0
$325$	−2.25271	−	7.83024i	−0.124958	−	0.434344i
$326$		0			0
$327$	−2.38465	+	12.4649i	−0.131871	+	0.689313i
$328$		0			0
$329$	12.1545	+	8.69906i	0.670099	+	0.479595i
$330$		0			0
$331$	−15.6637			−0.860954			−0.430477	−	0.902602i	$-0.641655\pi$
$331$	−15.6637			−0.860954			−0.430477	+	0.902602i	$0.641655\pi$
$332$		0			0
$333$	−4.15604	+	10.4646i	−0.227749	+	0.573457i
$334$		0			0
$335$	11.0847	+	8.34534i	0.605624	+	0.455955i
$336$		0			0
$337$			8.05239i			0.438642i	0.975653	+	0.219321i	$0.0703842\pi$
$337$			8.05239i			0.438642i	−0.975653	+	0.219321i	$0.929616\pi$
$338$		0			0
$339$	3.32708	−	17.3912i	0.180702	−	0.944561i
$340$		0			0
$341$	21.9774			1.19015
$342$		0			0
$343$	−17.7322	−	5.34511i	−0.957447	−	0.288609i
$344$		0			0
$345$	24.6153	+	27.1519i	1.32525	+	1.46181i
$346$		0			0
$347$	−0.814731			−0.0437370			−0.0218685	−	0.999761i	$-0.506962\pi$
$347$	−0.814731			−0.0437370			−0.0218685	+	0.999761i	$0.506962\pi$
$348$		0			0
$349$		−	20.6841i		−	1.10719i	−0.832785	−	0.553597i	$-0.813255\pi$
$349$		−	20.6841i		−	1.10719i	0.832785	−	0.553597i	$-0.186745\pi$
$350$		0			0
$351$	4.54844	−	7.14213i	0.242778	−	0.381219i
$352$		0			0
$353$		−	13.1361i		−	0.699163i	−0.936906	−	0.349581i	$-0.886324\pi$
$353$		−	13.1361i		−	0.699163i	0.936906	−	0.349581i	$-0.113676\pi$
$354$		0			0
$355$	15.4564	+	11.6366i	0.820341	+	0.617609i
$356$		0			0
$357$	15.7226	−	14.9610i	0.832127	−	0.791819i
$358$		0			0
$359$			15.6128i			0.824014i	0.911181	+	0.412007i	$0.135172\pi$
$359$			15.6128i			0.824014i	−0.911181	+	0.412007i	$0.864828\pi$
$360$		0			0
$361$	14.4213			0.759016
$362$		0			0
$363$	4.58887	−	23.9868i	0.240853	−	1.25898i
$364$		0			0
$365$	−16.8645	+	22.4004i	−0.882729	+	1.17249i
$366$		0			0
$367$	−9.40534			−0.490955			−0.245477	−	0.969402i	$-0.578945\pi$
$367$	−9.40534			−0.490955			−0.245477	+	0.969402i	$0.578945\pi$
$368$		0			0
$369$	−29.6257	−	11.7659i	−1.54225	−	0.612509i
$370$		0			0
$371$	−7.22697	+	10.0977i	−0.375205	+	0.524244i
$372$		0			0
$373$			8.98592i			0.465273i	0.972564	+	0.232637i	$0.0747353\pi$
$373$			8.98592i			0.465273i	−0.972564	+	0.232637i	$0.925265\pi$
$374$		0			0
$375$	10.1915	−	16.4661i	0.526289	−	0.850306i
$376$		0			0
$377$			9.72506i			0.500866i
$378$		0			0
$379$	17.6705			0.907671			0.453836	−	0.891085i	$-0.350055\pi$
$379$	17.6705			0.907671			0.453836	+	0.891085i	$0.350055\pi$
$380$		0			0
$381$	33.6097	+	6.42982i	1.72188	+	0.329410i
$382$		0			0
$383$		−	9.60898i		−	0.490996i	−0.969397	−	0.245498i	$-0.921049\pi$
$383$		−	9.60898i		−	0.490996i	0.969397	−	0.245498i	$-0.0789515\pi$
$384$		0			0
$385$	28.2780	−	8.87995i	1.44118	−	0.452564i
$386$		0			0
$387$	5.25715	−	13.2371i	0.267236	−	0.672880i
$388$		0			0
$389$			31.1675i			1.58026i	0.612942	+	0.790128i	$0.289986\pi$
$389$			31.1675i			1.58026i	−0.612942	+	0.790128i	$0.710014\pi$
$390$		0			0
$391$		−	44.8157i		−	2.26643i
$392$		0			0
$393$	5.70110	−	29.8006i	0.287583	−	1.50324i
$394$		0			0
$395$	1.42655	−	1.89482i	0.0717776	−	0.0953389i
$396$		0			0
$397$	−5.58833			−0.280470			−0.140235	−	0.990118i	$-0.544786\pi$
$397$	−5.58833			−0.280470			−0.140235	+	0.990118i	$0.544786\pi$
$398$		0			0
$399$	7.10362	−	6.75952i	0.355626	−	0.338399i
$400$		0			0
$401$		−	19.0883i		−	0.953224i	−0.879114	−	0.476612i	$-0.841865\pi$
$401$		−	19.0883i		−	0.953224i	0.879114	−	0.476612i	$-0.158135\pi$
$402$		0			0
$403$		−	7.14848i		−	0.356091i
$404$		0			0
$405$	19.8367	−	3.39213i	0.985692	−	0.168556i
$406$		0			0
$407$	18.8036			0.932060
$408$		0			0
$409$			5.53326i			0.273602i	0.990599	+	0.136801i	$0.0436821\pi$
$409$			5.53326i			0.273602i	−0.990599	+	0.136801i	$0.956318\pi$
$410$		0			0
$411$	6.95481	−	36.3539i	0.343055	−	1.79321i
$412$		0			0
$413$	13.3382	−	18.6363i	0.656328	−	0.917034i
$414$		0			0
$415$	−10.6555	−	8.02220i	−0.523059	−	0.393794i
$416$		0			0
$417$	5.66214	+	1.08321i	0.277276	+	0.0530452i
$418$		0			0
$419$	17.0661			0.833734			0.416867	−	0.908968i	$-0.363128\pi$
$419$	17.0661			0.833734			0.416867	+	0.908968i	$0.363128\pi$
$420$		0			0
$421$	−26.1421			−1.27409			−0.637045	−	0.770827i	$-0.719843\pi$
$421$	−26.1421			−1.27409			−0.637045	+	0.770827i	$0.719843\pi$
$422$		0			0
$423$	6.25565	−	15.7513i	0.304160	−	0.765853i
$424$		0			0
$425$	−22.7571	+	6.54706i	−1.10388	+	0.317579i
$426$		0			0
$427$	4.51336	+	3.23024i	0.218417	+	0.156323i
$428$		0			0
$429$	−13.8888	−	2.65704i	−0.670557	−	0.128283i
$430$		0			0
$431$			15.4016i			0.741869i	0.928659	+	0.370934i	$0.120963\pi$
$431$			15.4016i			0.741869i	−0.928659	+	0.370934i	$0.879037\pi$
$432$		0			0
$433$	−15.0966			−0.725494			−0.362747	−	0.931888i	$-0.618161\pi$
$433$	−15.0966			−0.725494			−0.362747	+	0.931888i	$0.618161\pi$
$434$		0			0
$435$	−17.1240	+	15.5242i	−0.821032	+	0.744330i
$436$		0			0
$437$		−	20.2482i		−	0.968601i
$438$		0			0
$439$			17.0500i			0.813751i	0.913484	+	0.406875i	$0.133382\pi$
$439$			17.0500i			0.813751i	−0.913484	+	0.406875i	$0.866618\pi$
$440$		0			0
$441$	−1.04185	+	20.9741i	−0.0496121	+	0.998769i
$442$		0			0
$443$	21.8460			1.03793			0.518967	−	0.854794i	$-0.326317\pi$
$443$	21.8460			1.03793			0.518967	+	0.854794i	$0.326317\pi$
$444$		0			0
$445$	0.251657	−	0.334265i	0.0119297	−	0.0158457i
$446$		0			0
$447$	7.31374	+	1.39918i	0.345928	+	0.0661789i
$448$		0			0
$449$			14.4240i			0.680710i	0.940297	+	0.340355i	$0.110547\pi$
$449$			14.4240i			0.680710i	−0.940297	+	0.340355i	$0.889453\pi$
$450$		0			0
$451$			53.2338i			2.50668i
$452$		0			0
$453$	2.47139	−	12.9184i	0.116116	−	0.606957i
$454$		0			0
$455$	−2.88833	−	9.19782i	−0.135407	−	0.431200i
$456$		0			0
$457$			9.40452i			0.439925i	0.975508	+	0.219962i	$0.0705935\pi$
$457$			9.40452i			0.439925i	−0.975508	+	0.219962i	$0.929406\pi$
$458$		0			0
$459$	−20.7572	−	13.2192i	−0.968865	−	0.617019i
$460$		0			0
$461$	−8.63026			−0.401951			−0.200976	−	0.979596i	$-0.564411\pi$
$461$	−8.63026			−0.401951			−0.200976	+	0.979596i	$0.564411\pi$
$462$		0			0
$463$		−	24.9809i		−	1.16096i	−0.814275	−	0.580479i	$-0.802865\pi$
$463$		−	24.9809i		−	1.16096i	0.814275	−	0.580479i	$-0.197135\pi$
$464$		0			0
$465$	12.5871	−	11.4112i	0.583713	−	0.529182i
$466$		0			0
$467$			5.52170i			0.255514i	0.991806	+	0.127757i	$0.0407777\pi$
$467$			5.52170i			0.255514i	−0.991806	+	0.127757i	$0.959222\pi$
$468$		0			0
$469$	13.3502	+	9.55483i	0.616455	+	0.441201i
$470$		0			0
$471$	−5.01732	+	26.2264i	−0.231186	+	1.20845i
$472$		0			0
$473$	−23.7855			−1.09366
$474$		0			0
$475$	−10.2819	+	2.95803i	−0.471766	+	0.135724i
$476$		0			0
$477$	13.0858	+	5.19704i	0.599156	+	0.237956i
$478$		0			0
$479$	3.15027			0.143940			0.0719699	−	0.997407i	$-0.477071\pi$
$479$	3.15027			0.143940			0.0719699	+	0.997407i	$0.477071\pi$
$480$		0			0
$481$		−	6.11614i		−	0.278872i
$482$		0			0
$483$	29.8924	+	31.4141i	1.36015	+	1.42939i
$484$		0			0
$485$	14.6153	−	19.4129i	0.663649	−	0.881494i
$486$		0			0
$487$			31.1480i			1.41145i	0.708485	+	0.705726i	$0.249379\pi$
$487$			31.1480i			1.41145i	−0.708485	+	0.705726i	$0.750621\pi$
$488$		0			0
$489$	−12.9342	−	2.47442i	−0.584903	−	0.111897i
$490$		0			0
$491$			7.82187i			0.352996i	0.984301	+	0.176498i	$0.0564769\pi$
$491$			7.82187i			0.352996i	−0.984301	+	0.176498i	$0.943523\pi$
$492$		0			0
$493$	28.2641			1.27295
$494$		0			0
$495$	−17.4923	−	28.6970i	−0.786220	−	1.28983i
$496$		0			0
$497$	18.6153	+	13.3231i	0.835012	+	0.597624i
$498$		0			0
$499$	28.8804			1.29286			0.646432	−	0.762971i	$-0.276260\pi$
$499$	28.8804			1.29286			0.646432	+	0.762971i	$0.276260\pi$
$500$		0			0
$501$	−5.31475	−	1.01676i	−0.237445	−	0.0454253i
$502$		0			0
$503$			27.7497i			1.23730i	0.785668	+	0.618648i	$0.212319\pi$
$503$			27.7497i			1.23730i	−0.785668	+	0.618648i	$0.787681\pi$
$504$		0			0
$505$	−6.25004	+	8.30164i	−0.278123	+	0.369418i
$506$		0			0
$507$	3.36666	−	17.5981i	0.149519	−	0.781558i
$508$		0			0
$509$	9.57553			0.424428			0.212214	−	0.977223i	$-0.431933\pi$
$509$	9.57553			0.424428			0.212214	+	0.977223i	$0.431933\pi$
$510$		0			0
$511$	−19.3087	+	26.9785i	−0.854166	+	1.19346i
$512$		0			0
$513$	−9.37833	−	5.97256i	−0.414063	−	0.263695i
$514$		0			0
$515$	−0.906693	+	1.20432i	−0.0399536	+	0.0530686i
$516$		0			0
$517$	−28.3031			−1.24477
$518$		0			0
$519$	−8.99923	−	1.72163i	−0.395022	−	0.0755711i
$520$		0			0
$521$	25.0936			1.09937			0.549685	−	0.835372i	$-0.314748\pi$
$521$	25.0936			1.09937			0.549685	+	0.835372i	$0.314748\pi$
$522$		0			0
$523$	−31.7296			−1.38744			−0.693720	−	0.720245i	$-0.744029\pi$
$523$	−31.7296			−1.38744			−0.693720	+	0.720245i	$0.744029\pi$
$524$		0			0
$525$	11.5849	−	19.7684i	0.505608	−	0.862763i
$526$		0			0
$527$	−20.7757			−0.905004
$528$		0			0
$529$	66.5428			2.89316
$530$		0			0
$531$	−24.1512	−	9.59170i	−1.04807	−	0.416244i
$532$		0			0
$533$	17.3151			0.749998
$534$		0			0
$535$	−11.3771	+	15.1116i	−0.491873	+	0.653333i
$536$		0			0
$537$	−24.6745	−	4.72045i	−1.06479	−	0.203702i
$538$		0			0
$539$	33.1955	−	11.3116i	1.42983	−	0.487225i
$540$		0			0
$541$	35.5457			1.52823			0.764115	−	0.645080i	$-0.223176\pi$
$541$	35.5457			1.52823			0.764115	+	0.645080i	$0.223176\pi$
$542$		0			0
$543$	31.4862	+	6.02357i	1.35120	+	0.258496i
$544$		0			0
$545$	9.85442	−	13.0892i	0.422117	−	0.560678i
$546$		0			0
$547$		−	9.24333i		−	0.395216i	−0.980281	−	0.197608i	$-0.936683\pi$
$547$		−	9.24333i		−	0.395216i	0.980281	−	0.197608i	$-0.0633174\pi$
$548$		0			0
$549$	2.32293	−	5.84896i	0.0991401	−	0.249628i
$550$		0			0
$551$	12.7700			0.544020
$552$		0			0
$553$	1.63330	−	2.28208i	0.0694550	−	0.0970439i
$554$		0			0
$555$	10.7694	−	9.76328i	0.457134	−	0.414428i
$556$		0			0
$557$	−16.8645			−0.714572			−0.357286	−	0.933995i	$-0.616298\pi$
$557$	−16.8645			−0.714572			−0.357286	+	0.933995i	$0.616298\pi$
$558$		0			0
$559$			7.73657i			0.327222i
$560$		0			0
$561$	−7.72218	+	40.3651i	−0.326031	+	1.70422i
$562$		0			0
$563$			15.9764i			0.673325i	0.941625	+	0.336662i	$0.109298\pi$
$563$			15.9764i			0.673325i	−0.941625	+	0.336662i	$0.890702\pi$
$564$		0			0
$565$	−13.7490	+	18.2621i	−0.578424	+	0.768293i
$566$		0			0
$567$	23.3673	−	4.57908i	0.981336	−	0.192303i
$568$		0			0
$569$			17.2766i			0.724272i	0.932125	+	0.362136i	$0.117952\pi$
$569$			17.2766i			0.724272i	−0.932125	+	0.362136i	$0.882048\pi$
$570$		0			0
$571$	17.5387			0.733973			0.366987	−	0.930226i	$-0.380389\pi$
$571$	17.5387			0.733973			0.366987	+	0.930226i	$0.380389\pi$
$572$		0			0
$573$	21.6634	+	4.14439i	0.905002	+	0.173134i
$574$		0			0
$575$	−13.0812	−	45.4692i	−0.545523	−	1.89620i
$576$		0			0
$577$	−16.5820			−0.690316			−0.345158	−	0.938545i	$-0.612175\pi$
$577$	−16.5820			−0.690316			−0.345158	+	0.938545i	$0.612175\pi$
$578$		0			0
$579$	6.87707	+	1.31564i	0.285801	+	0.0546761i
$580$		0			0
$581$	−12.8333	−	9.18486i	−0.532413	−	0.381052i
$582$		0			0
$583$		−	23.5135i		−	0.973831i
$584$		0			0
$585$	−9.33410	+	5.68962i	−0.385918	+	0.235237i
$586$		0			0
$587$		−	31.5221i		−	1.30106i	−0.759481	−	0.650529i	$-0.774547\pi$
$587$		−	31.5221i		−	1.30106i	0.759481	−	0.650529i	$-0.225453\pi$
$588$		0			0
$589$	−9.38667			−0.386771
$590$		0			0
$591$	−1.77072	+	9.25583i	−0.0728376	+	0.380734i
$592$		0			0
$593$			31.5290i			1.29474i	0.762176	+	0.647370i	$0.224131\pi$
$593$			31.5290i			1.29474i	−0.762176	+	0.647370i	$0.775869\pi$
$594$		0			0
$595$	−26.7317	+	8.39439i	−1.09589	+	0.344136i
$596$		0			0
$597$	−26.8992	−	5.14603i	−1.10091	−	0.210613i
$598$		0			0
$599$			34.3090i			1.40183i	0.713246	+	0.700913i	$0.247224\pi$
$599$			34.3090i			1.40183i	−0.713246	+	0.700913i	$0.752776\pi$
$600$		0			0
$601$		−	37.5135i		−	1.53021i	−0.643907	−	0.765103i	$-0.722688\pi$
$601$		−	37.5135i		−	1.53021i	0.643907	−	0.765103i	$-0.277312\pi$
$602$		0			0
$603$	6.87105	−	17.3008i	0.279811	−	0.704543i
$604$		0			0
$605$	−18.9632	+	25.1880i	−0.770965	+	1.02404i
$606$		0			0
$607$	−2.56064			−0.103933			−0.0519666	−	0.998649i	$-0.516549\pi$
$607$	−2.56064			−0.103933			−0.0519666	+	0.998649i	$0.516549\pi$
$608$		0			0
$609$	−19.8120	+	18.8523i	−0.802824	+	0.763935i
$610$		0			0
$611$			9.20599i			0.372434i
$612$		0			0
$613$			6.36050i			0.256898i	0.991716	+	0.128449i	$0.0409999\pi$
$613$			6.36050i			0.256898i	−0.991716	+	0.128449i	$0.959000\pi$
$614$		0			0
$615$	27.6402	+	30.4885i	1.11456	+	1.22942i
$616$		0			0
$617$	12.0824			0.486418			0.243209	−	0.969974i	$-0.421800\pi$
$617$	12.0824			0.486418			0.243209	+	0.969974i	$0.421800\pi$
$618$		0			0
$619$		−	38.9476i		−	1.56543i	−0.622378	−	0.782717i	$-0.713833\pi$
$619$		−	38.9476i		−	1.56543i	0.622378	−	0.782717i	$-0.286167\pi$
$620$		0			0
$621$	26.4122	−	41.4735i	1.05989	−	1.66427i
$622$		0			0
$623$	0.288130	−	0.402580i	0.0115437	−	0.0161290i
$624$		0			0
$625$	−21.1780	+	13.2851i	−0.847119	+	0.531403i
$626$		0			0
$627$	−3.48896	+	18.2374i	−0.139336	+	0.728330i
$628$		0			0
$629$	−17.7754			−0.708752
$630$		0			0
$631$	9.62832			0.383297			0.191649	−	0.981464i	$-0.438617\pi$
$631$	9.62832			0.383297			0.191649	+	0.981464i	$0.438617\pi$
$632$		0			0
$633$	−5.77255	+	30.1741i	−0.229438	+	1.19931i
$634$		0			0
$635$	−35.2928	−	26.5708i	−1.40055	−	1.05443i
$636$		0			0
$637$	−3.67926	−	10.7973i	−0.145777	−	0.427806i
$638$		0			0
$639$	9.58090	−	24.1240i	0.379015	−	0.954331i
$640$		0			0
$641$			4.14281i			0.163631i	0.996647	+	0.0818155i	$0.0260718\pi$
$641$			4.14281i			0.163631i	−0.996647	+	0.0818155i	$0.973928\pi$
$642$		0			0
$643$	21.6519			0.853869			0.426934	−	0.904283i	$-0.359594\pi$
$643$	21.6519			0.853869			0.426934	+	0.904283i	$0.359594\pi$
$644$		0			0
$645$	−13.6226	+	12.3500i	−0.536390	+	0.486280i
$646$		0			0
$647$		−	12.8648i		−	0.505766i	−0.967497	−	0.252883i	$-0.918621\pi$
$647$		−	12.8648i		−	0.505766i	0.967497	−	0.252883i	$-0.0813788\pi$
$648$		0			0
$649$			43.3968i			1.70347i
$650$		0			0
$651$	14.5630	−	13.8575i	0.570768	−	0.543120i
$652$		0			0
$653$	8.82708			0.345430			0.172715	−	0.984972i	$-0.444746\pi$
$653$	8.82708			0.345430			0.172715	+	0.984972i	$0.444746\pi$
$654$		0			0
$655$	−23.5595	+	31.2929i	−0.920544	+	1.22272i
$656$		0			0
$657$	34.9620	+	13.8852i	1.36400	+	0.541714i
$658$		0			0
$659$		−	25.3918i		−	0.989123i	−0.869143	−	0.494561i	$-0.835329\pi$
$659$		−	25.3918i		−	0.989123i	0.869143	−	0.494561i	$-0.164671\pi$
$660$		0			0
$661$			27.2661i			1.06053i	0.847833	+	0.530264i	$0.177907\pi$
$661$			27.2661i			1.06053i	−0.847833	+	0.530264i	$0.822093\pi$
$662$		0			0
$663$	13.1293	+	2.51175i	0.509901	+	0.0975483i
$664$		0			0
$665$	−12.0777	+	3.79267i	−0.468351	+	0.147073i
$666$		0			0
$667$			56.4722i			2.18661i
$668$		0			0
$669$	9.12288	−	47.6868i	0.352711	−	1.84368i
$670$		0			0
$671$	−10.5099			−0.405729
$672$		0			0
$673$			24.5127i			0.944895i	0.881359	+	0.472447i	$0.156629\pi$
$673$			24.5127i			0.944895i	−0.881359	+	0.472447i	$0.843371\pi$
$674$		0			0
$675$	−24.9185	−	7.35317i	−0.959113	−	0.283024i
$676$		0			0
$677$			31.5932i			1.21423i	0.794615	+	0.607114i	$0.207673\pi$
$677$			31.5932i			1.21423i	−0.794615	+	0.607114i	$0.792327\pi$
$678$		0			0
$679$	16.7335	−	23.3804i	0.642174	−	0.897259i
$680$		0			0
$681$	−29.2933	−	5.60405i	−1.12252	−	0.214748i
$682$		0			0
$683$	−14.3252			−0.548139			−0.274069	−	0.961710i	$-0.588370\pi$
$683$	−14.3252			−0.548139			−0.274069	+	0.961710i	$0.588370\pi$
$684$		0			0
$685$	−28.7403	+	38.1744i	−1.09811	+	1.45857i
$686$		0			0
$687$	−8.33354	−	1.59428i	−0.317944	−	0.0608254i
$688$		0			0
$689$	−7.64811			−0.291370
$690$		0			0
$691$		−	1.69810i		−	0.0645986i	−0.999478	−	0.0322993i	$-0.989717\pi$
$691$		−	1.69810i		−	0.0645986i	0.999478	−	0.0322993i	$-0.0102830\pi$
$692$		0			0
$693$	−21.5109	−	33.4452i	−0.817130	−	1.27048i
$694$		0			0
$695$	−5.94569	−	4.47632i	−0.225533	−	0.169796i
$696$		0			0
$697$		−	50.3229i		−	1.90612i
$698$		0			0
$699$	−5.56063	+	29.0663i	−0.210322	+	1.09939i
$700$		0			0
$701$		−	22.8839i		−	0.864314i	−0.901798	−	0.432157i	$-0.857753\pi$
$701$		−	22.8839i		−	0.864314i	0.901798	−	0.432157i	$-0.142247\pi$
$702$		0			0
$703$	−8.03111			−0.302899
$704$		0			0
$705$	−16.2100	+	14.6956i	−0.610504	+	0.553470i
$706$		0			0
$707$	−7.15586	+	9.99831i	−0.269124	+	0.376025i
$708$		0			0
$709$	−43.3163			−1.62678			−0.813389	−	0.581720i	$-0.802380\pi$
$709$	−43.3163			−1.62678			−0.813389	+	0.581720i	$0.802380\pi$
$710$		0			0
$711$	−2.95740	−	1.17454i	−0.110911	−	0.0440485i
$712$		0			0
$713$		−	41.5103i		−	1.55457i
$714$		0			0
$715$	14.5843	+	10.9801i	0.545422	+	0.410631i
$716$		0			0
$717$	−34.7997	−	6.65746i	−1.29962	−	0.248628i
$718$		0			0
$719$	−29.7322			−1.10882			−0.554412	−	0.832242i	$-0.687057\pi$
$719$	−29.7322			−1.10882			−0.554412	+	0.832242i	$0.687057\pi$
$720$		0			0
$721$	−1.03810	+	1.45045i	−0.0386608	+	0.0540177i
$722$		0			0
$723$	−18.3879	−	3.51775i	−0.683852	−	0.130827i
$724$		0			0
$725$	28.6762	−	8.24996i	1.06501	−	0.306396i
$726$		0			0
$727$	15.6211			0.579355			0.289677	−	0.957124i	$-0.406452\pi$
$727$	15.6211			0.579355			0.289677	+	0.957124i	$0.406452\pi$
$728$		0			0
$729$	−11.4185	−	24.4667i	−0.422906	−	0.906174i
$730$		0			0
$731$	22.4849			0.831633
$732$		0			0
$733$	31.8068			1.17481			0.587406	−	0.809292i	$-0.300149\pi$
$733$	31.8068			1.17481			0.587406	+	0.809292i	$0.300149\pi$
$734$		0			0
$735$	13.1388	−	23.7144i	0.484632	−	0.874718i
$736$		0			0
$737$	−31.0874			−1.14512
$738$		0			0
$739$	−42.0016			−1.54505			−0.772527	−	0.634982i	$-0.781008\pi$
$739$	−42.0016			−1.54505			−0.772527	+	0.634982i	$0.781008\pi$
$740$		0			0
$741$	5.93196	+	1.13483i	0.217916	+	0.0416892i
$742$		0			0
$743$	−26.5031			−0.972305			−0.486152	−	0.873874i	$-0.661600\pi$
$743$	−26.5031			−0.972305			−0.486152	+	0.873874i	$0.661600\pi$
$744$		0			0
$745$	−7.67999	−	5.78202i	−0.281373	−	0.211837i
$746$		0			0
$747$	−6.60499	+	16.6309i	−0.241664	+	0.608492i
$748$		0			0
$749$	−13.0259	+	18.2001i	−0.475958	+	0.665017i
$750$		0			0
$751$	33.4006			1.21881			0.609403	−	0.792861i	$-0.291409\pi$
$751$	33.4006			1.21881			0.609403	+	0.792861i	$0.291409\pi$
$752$		0			0
$753$	9.67923	−	50.5949i	0.352731	−	1.84378i
$754$		0			0
$755$	−10.2129	+	13.5653i	−0.371684	+	0.493691i
$756$		0			0
$757$		−	3.34496i		−	0.121575i	−0.998151	−	0.0607873i	$-0.980639\pi$
$757$		−	3.34496i		−	0.121575i	0.998151	−	0.0607873i	$-0.0193611\pi$
$758$		0			0
$759$	−80.6504	−	15.4291i	−2.92743	−	0.560041i
$760$		0			0
$761$	14.4992			0.525595			0.262798	−	0.964851i	$-0.415355\pi$
$761$	14.4992			0.525595			0.262798	+	0.964851i	$0.415355\pi$
$762$		0			0
$763$	11.2826	−	15.7643i	0.408458	−	0.570706i
$764$		0			0
$765$	16.5358	+	27.1278i	0.597853	+	0.980808i
$766$		0			0
$767$	14.1154			0.509678
$768$		0			0
$769$			28.7135i			1.03544i	0.855551	+	0.517718i	$0.173218\pi$
$769$			28.7135i			1.03544i	−0.855551	+	0.517718i	$0.826782\pi$
$770$		0			0
$771$	47.8677	+	9.15750i	1.72391	+	0.329799i
$772$		0			0
$773$		−	13.0728i		−	0.470196i	−0.971972	−	0.235098i	$-0.924459\pi$
$773$		−	13.0728i		−	0.470196i	0.971972	−	0.235098i	$-0.0755411\pi$
$774$		0			0
$775$	−21.0787	+	6.06419i	−0.757168	+	0.217832i
$776$		0			0
$777$	12.4599	−	11.8563i	0.446996	−	0.425343i
$778$		0			0
$779$		−	22.7364i		−	0.814616i
$780$		0			0
$781$	−43.3479			−1.55111
$782$		0			0
$783$	26.1562	+	16.6575i	0.934747	+	0.595291i
$784$		0			0
$785$	20.7338	−	27.5397i	0.740020	−	0.982935i
$786$		0			0
$787$	25.4820			0.908337			0.454168	−	0.890916i	$-0.349936\pi$
$787$	25.4820			0.908337			0.454168	+	0.890916i	$0.349936\pi$
$788$		0			0
$789$	6.86660	−	35.8928i	0.244457	−	1.27782i
$790$		0			0
$791$	−15.7416	+	21.9945i	−0.559707	+	0.782034i
$792$		0			0
$793$			3.41848i			0.121394i
$794$		0			0
$795$	−12.2088	−	13.4669i	−0.433001	−	0.477621i
$796$		0			0
$797$			7.96431i			0.282110i	0.990002	+	0.141055i	$0.0450495\pi$
$797$			7.96431i			0.282110i	−0.990002	+	0.141055i	$0.954950\pi$
$798$		0			0
$799$	26.7555			0.946541
$800$		0			0
$801$	−0.521712	−	0.207199i	−0.0184338	−	0.00732102i
$802$		0			0
$803$		−	62.8224i		−	2.21695i
$804$		0			0
$805$	−16.7722	−	53.4106i	−0.591142	−	1.88248i
$806$		0			0
$807$	−1.76618	+	9.23212i	−0.0621726	+	0.324986i
$808$		0			0
$809$		−	40.8966i		−	1.43785i	−0.695088	−	0.718924i	$-0.744635\pi$
$809$		−	40.8966i		−	1.43785i	0.695088	−	0.718924i	$-0.255365\pi$
$810$		0			0
$811$		−	39.9526i		−	1.40293i	−0.712705	−	0.701464i	$-0.752530\pi$
$811$		−	39.9526i		−	1.40293i	0.712705	−	0.701464i	$-0.247470\pi$
$812$		0			0
$813$	−28.9211	−	5.53285i	−1.01431	−	0.194045i
$814$		0			0
$815$	13.5819	+	10.2254i	0.475753	+	0.358179i
$816$		0			0
$817$	10.1589			0.355415
$818$		0			0
$819$	−10.8785	+	6.99671i	−0.380126	+	0.244485i
$820$		0			0
$821$			28.4551i			0.993091i	0.868011	+	0.496546i	$0.165398\pi$
$821$			28.4551i			0.993091i	−0.868011	+	0.496546i	$0.834602\pi$
$822$		0			0
$823$		−	0.837012i		−	0.0291764i	−0.999894	−	0.0145882i	$-0.995356\pi$
$823$		−	0.837012i		−	0.0291764i	0.999894	−	0.0145882i	$-0.00464373\pi$
$824$		0			0
$825$	3.94732	+	43.2078i	0.137428	+	1.50430i
$826$		0			0
$827$	21.2414			0.738637			0.369319	−	0.929303i	$-0.379591\pi$
$827$	21.2414			0.738637			0.369319	+	0.929303i	$0.379591\pi$
$828$		0			0
$829$		−	26.3174i		−	0.914041i	−0.889456	−	0.457021i	$-0.848917\pi$
$829$		−	26.3174i		−	0.914041i	0.889456	−	0.457021i	$-0.151083\pi$
$830$		0			0
$831$	−1.52746	−	0.292216i	−0.0529871	−	0.0101369i
$832$		0			0
$833$	−31.3804	+	10.6931i	−1.08727	+	0.370493i
$834$		0			0
$835$	5.58090	+	4.20168i	0.193135	+	0.145405i
$836$		0			0
$837$	−19.2263	−	12.2442i	−0.664558	−	0.423222i
$838$		0			0
$839$	31.0138			1.07071			0.535357	−	0.844626i	$-0.320177\pi$
$839$	31.0138			1.07071			0.535357	+	0.844626i	$0.320177\pi$
$840$		0			0
$841$	−6.61554			−0.228122
$842$		0			0
$843$	21.4832	+	4.10991i	0.739919	+	0.141553i
$844$		0			0
$845$	−13.9125	+	18.4793i	−0.478605	+	0.635709i
$846$		0			0
$847$	−21.7116	+	30.3358i	−0.746018	+	1.04235i
$848$		0			0
$849$	−0.595110	+	3.11074i	−0.0204241	+	0.106760i
$850$		0			0
$851$		−	35.5157i		−	1.21746i
$852$		0			0
$853$	22.6808			0.776576			0.388288	−	0.921538i	$-0.373067\pi$
$853$	22.6808			0.776576			0.388288	+	0.921538i	$0.373067\pi$
$854$		0			0
$855$	7.47104	+	12.2566i	0.255504	+	0.419167i
$856$		0			0
$857$			5.68694i			0.194262i	0.995272	+	0.0971310i	$0.0309666\pi$
$857$			5.68694i			0.194262i	−0.995272	+	0.0971310i	$0.969033\pi$
$858$		0			0
$859$			33.2123i			1.13319i	0.823997	+	0.566595i	$0.191739\pi$
$859$			33.2123i			1.13319i	−0.823997	+	0.566595i	$0.808261\pi$
$860$		0			0
$861$	33.5657	+	35.2745i	1.14392	+	1.20215i
$862$		0			0
$863$	4.78646			0.162933			0.0814665	−	0.996676i	$-0.474040\pi$
$863$	4.78646			0.162933			0.0814665	+	0.996676i	$0.474040\pi$
$864$		0			0
$865$	9.44989	+	7.11452i	0.321306	+	0.241901i
$866$		0			0
$867$	1.76722	−	9.23754i	0.0600179	−	0.313723i
$868$		0			0
$869$			5.31408i			0.180268i
$870$		0			0
$871$			10.1116i			0.342619i
$872$		0			0
$873$	−30.2992	−	12.0334i	−1.02547	−	0.407268i
$874$		0			0
$875$	−24.6713	+	16.3195i	−0.834043	+	0.551699i
$876$		0			0
$877$		−	10.0778i		−	0.340303i	−0.985418	−	0.170151i	$-0.945574\pi$
$877$		−	10.0778i		−	0.340303i	0.985418	−	0.170151i	$-0.0544257\pi$
$878$		0			0
$879$	−10.6645	−	2.04020i	−0.359703	−	0.0688142i
$880$		0			0
$881$	−27.6439			−0.931346			−0.465673	−	0.884957i	$-0.654188\pi$
$881$	−27.6439			−0.931346			−0.465673	+	0.884957i	$0.654188\pi$
$882$		0			0
$883$		−	9.04074i		−	0.304245i	−0.988362	−	0.152123i	$-0.951389\pi$
$883$		−	9.04074i		−	0.304245i	0.988362	−	0.152123i	$-0.0486108\pi$
$884$		0			0
$885$	22.5326	+	24.8546i	0.757426	+	0.835478i
$886$		0			0
$887$			36.6139i			1.22937i	0.788771	+	0.614687i	$0.210718\pi$
$887$			36.6139i			1.22937i	−0.788771	+	0.614687i	$0.789282\pi$
$888$		0			0
$889$	−42.5059	−	30.4217i	−1.42560	−	1.02031i
$890$		0			0
$891$	−30.9356	+	32.8037i	−1.03638	+	1.09897i
$892$		0			0
$893$	12.0884			0.404523
$894$		0			0
$895$	25.9102	+	19.5070i	0.866082	+	0.652046i
$896$		0			0
$897$	−5.01854	+	26.2327i	−0.167564	+	0.875885i
$898$		0			0
$899$	26.1795			0.873134
$900$		0			0
$901$			22.2278i			0.740515i
$902$		0			0
$903$	−15.7610	+	14.9976i	−0.524494	+	0.499088i
$904$		0			0
$905$	−33.0629	−	24.8920i	−1.09905	−	0.827439i
$906$		0			0
$907$		−	5.95247i		−	0.197648i	−0.995105	−	0.0988242i	$-0.968492\pi$
$907$		−	5.95247i		−	0.197648i	0.995105	−	0.0988242i	$-0.0315081\pi$
$908$		0			0
$909$	12.9570	+	5.14591i	0.429757	+	0.170679i
$910$		0			0
$911$		−	11.3197i		−	0.375039i	−0.982261	−	0.187519i	$-0.939955\pi$
$911$		−	11.3197i		−	0.375039i	0.982261	−	0.187519i	$-0.0600447\pi$
$912$		0			0
$913$	29.8837			0.989005
$914$		0			0
$915$	−6.01930	+	5.45697i	−0.198992	+	0.180402i
$916$		0			0
$917$	−26.9739	+	37.6885i	−0.890757	+	1.24458i
$918$		0			0
$919$	7.97971			0.263226			0.131613	−	0.991301i	$-0.457984\pi$
$919$	7.97971			0.263226			0.131613	+	0.991301i	$0.457984\pi$
$920$		0			0
$921$	6.02219	−	31.4790i	0.198438	−	1.03727i
$922$		0			0
$923$			14.0995i			0.464092i
$924$		0			0
$925$	−18.0346	+	5.18844i	−0.592975	+	0.170595i
$926$		0			0
$927$	1.87967	+	0.746516i	0.0617365	+	0.0245188i
$928$		0			0
$929$	31.7309			1.04106			0.520529	−	0.853844i	$-0.325735\pi$
$929$	31.7309			1.04106			0.520529	+	0.853844i	$0.325735\pi$
$930$		0			0
$931$	−14.1780	+	4.83123i	−0.464664	+	0.158337i
$932$		0			0
$933$	6.29257	−	32.8923i	0.206010	−	1.07685i
$934$		0			0
$935$	31.9115	−	42.3865i	1.04362	−	1.38619i
$936$		0			0
$937$	−39.1916			−1.28033			−0.640167	−	0.768236i	$-0.721135\pi$
$937$	−39.1916			−1.28033			−0.640167	+	0.768236i	$0.721135\pi$
$938$		0			0
$939$	−3.45389	+	18.0541i	−0.112713	+	0.589172i
$940$		0			0
$941$	−14.7755			−0.481667			−0.240834	−	0.970566i	$-0.577421\pi$
$941$	−14.7755			−0.481667			−0.240834	+	0.970566i	$0.577421\pi$
$942$		0			0
$943$	100.546			3.27424
$944$		0			0
$945$	−29.6854	−	7.98606i	−0.965666	−	0.259786i
$946$		0			0
$947$	−38.4136			−1.24827			−0.624137	−	0.781315i	$-0.714549\pi$
$947$	−38.4136			−1.24827			−0.624137	+	0.781315i	$0.714549\pi$
$948$		0			0
$949$	−20.4339			−0.663312
$950$		0			0
$951$	0.874343	−	4.57033i	0.0283525	−	0.148203i
$952$		0			0
$953$	−31.2446			−1.01211			−0.506056	−	0.862500i	$-0.668897\pi$
$953$	−31.2446			−1.01211			−0.506056	+	0.862500i	$0.668897\pi$
$954$		0			0
$955$	−22.7483	−	17.1264i	−0.736117	−	0.554199i
$956$		0			0
$957$	9.73072	−	50.8641i	0.314550	−	1.64420i
$958$		0			0
$959$	−32.9056	+	45.9764i	−1.06258	+	1.48466i
$960$		0			0
$961$	11.7566			0.379245
$962$		0			0
$963$	23.5859	+	9.36719i	0.760045	+	0.301853i
$964$		0			0
$965$	−7.22146	−	5.43680i	−0.232467	−	0.175017i
$966$		0			0
$967$			16.9272i			0.544341i	0.962249	+	0.272170i	$0.0877414\pi$
$967$			16.9272i			0.544341i	−0.962249	+	0.272170i	$0.912259\pi$
$968$		0			0
$969$	3.29818	−	17.2401i	0.105953	−	0.553833i
$970$		0			0
$971$	−42.4041			−1.36081			−0.680406	−	0.732836i	$-0.738196\pi$
$971$	−42.4041			−1.36081			−0.680406	+	0.732836i	$0.738196\pi$
$972$		0			0
$973$	−7.16085	−	5.12507i	−0.229566	−	0.164302i
$974$		0			0
$975$	14.0540	−	1.28392i	0.450087	−	0.0411184i
$976$		0			0
$977$	42.3945			1.35632			0.678160	−	0.734914i	$-0.262778\pi$
$977$	42.3945			1.35632			0.678160	+	0.734914i	$0.262778\pi$
$978$		0			0
$979$			0.937453i			0.0299611i
$980$		0			0
$981$	−20.4293	−	8.11353i	−0.652256	−	0.259045i
$982$		0			0
$983$		−	27.5443i		−	0.878525i	−0.898359	−	0.439263i	$-0.855240\pi$
$983$		−	27.5443i		−	0.878525i	0.898359	−	0.439263i	$-0.144760\pi$
$984$		0			0
$985$	7.31738	−	9.71935i	0.233151	−	0.309684i
$986$		0			0
$987$	−18.7546	+	17.8461i	−0.596964	+	0.568047i
$988$		0			0
$989$			44.9253i			1.42854i
$990$		0			0
$991$	33.3366			1.05897			0.529486	−	0.848319i	$-0.322385\pi$
$991$	33.3366			1.05897			0.529486	+	0.848319i	$0.322385\pi$
$992$		0			0
$993$	5.09780	−	26.6470i	0.161774	−	0.845618i
$994$		0			0
$995$	28.2462	+	21.2657i	0.895465	+	0.674167i
$996$		0			0
$997$	−42.4732			−1.34514			−0.672570	−	0.740033i	$-0.734810\pi$
$997$	−42.4732			−1.34514			−0.672570	+	0.740033i	$0.734810\pi$
$998$		0			0
$999$	−16.4498	−	10.4760i	−0.520448	−	0.331446i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.k.a.209.12	yes	24
3.2	odd	2		840.2.k.b.209.11	yes	24
4.3	odd	2		1680.2.k.i.209.13		24
5.4	even	2		840.2.k.b.209.13	yes	24
7.6	odd	2	inner	840.2.k.a.209.13	yes	24
12.11	even	2		1680.2.k.h.209.14		24
15.14	odd	2	inner	840.2.k.a.209.14	yes	24
20.19	odd	2		1680.2.k.h.209.12		24
21.20	even	2		840.2.k.b.209.14	yes	24
28.27	even	2		1680.2.k.i.209.12		24
35.34	odd	2		840.2.k.b.209.12	yes	24
60.59	even	2		1680.2.k.i.209.11		24
84.83	odd	2		1680.2.k.h.209.11		24
105.104	even	2	inner	840.2.k.a.209.11	✓	24
140.139	even	2		1680.2.k.h.209.13		24
420.419	odd	2		1680.2.k.i.209.14		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.11	✓	24	105.104	even	2	inner
840.2.k.a.209.12	yes	24	1.1	even	1	trivial
840.2.k.a.209.13	yes	24	7.6	odd	2	inner
840.2.k.a.209.14	yes	24	15.14	odd	2	inner
840.2.k.b.209.11	yes	24	3.2	odd	2
840.2.k.b.209.12	yes	24	35.34	odd	2
840.2.k.b.209.13	yes	24	5.4	even	2
840.2.k.b.209.14	yes	24	21.20	even	2
1680.2.k.h.209.11		24	84.83	odd	2
1680.2.k.h.209.12		24	20.19	odd	2
1680.2.k.h.209.13		24	140.139	even	2
1680.2.k.h.209.14		24	12.11	even	2
1680.2.k.i.209.11		24	60.59	even	2
1680.2.k.i.209.12		24	28.27	even	2
1680.2.k.i.209.13		24	4.3	odd	2
1680.2.k.i.209.14		24	420.419	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.325454 + 1.70120i) q^{3} +(1.34492 - 1.78639i) q^{5} +(1.53984 - 2.15149i) q^{7} +(-2.78816 - 1.10732i) q^{9} +O(q^{10})\)	\(q+(-0.325454 + 1.70120i) q^{3} +(1.34492 - 1.78639i) q^{5} +(1.53984 - 2.15149i) q^{7} +(-2.78816 - 1.10732i) q^{9} +5.00998i q^{11} +1.62957 q^{13} +(2.60130 + 2.86936i) q^{15} -4.73603i q^{17} -2.13979i q^{19} +(3.15897 + 3.31978i) q^{21} +9.46270 q^{23} +(-1.38239 - 4.80510i) q^{25} +(2.79119 - 4.38283i) q^{27} +5.96788i q^{29} -4.38673i q^{31} +(-8.52298 - 1.63052i) q^{33} +(-1.77245 - 5.64433i) q^{35} -3.75323i q^{37} +(-0.530349 + 2.77222i) q^{39} +10.6255 q^{41} +4.74762i q^{43} +(-5.72796 + 3.49149i) q^{45} +5.64934i q^{47} +(-2.25781 - 6.62588i) q^{49} +(8.05694 + 1.54136i) q^{51} -4.69333 q^{53} +(8.94979 + 6.73801i) q^{55} +(3.64020 + 0.696402i) q^{57} +8.66206 q^{59} +2.09779i q^{61} +(-6.67570 + 4.29360i) q^{63} +(2.19163 - 2.91105i) q^{65} +6.20510i q^{67} +(-3.07967 + 16.0979i) q^{69} +8.65231i q^{71} -12.5394 q^{73} +(8.62434 - 0.787891i) q^{75} +(10.7789 + 7.71455i) q^{77} +1.06070 q^{79} +(6.54767 + 6.17479i) q^{81} -5.96483i q^{83} +(-8.46041 - 6.36957i) q^{85} +(-10.1525 - 1.94227i) q^{87} +0.187117 q^{89} +(2.50927 - 3.50600i) q^{91} +(7.46270 + 1.42768i) q^{93} +(-3.82250 - 2.87784i) q^{95} +10.8671 q^{97} +(5.54767 - 13.9686i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9}+O(q^{10}) \) 24 * q - 2 * q^9	\( 24 q - 2 q^{9} - 6 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} - 2 q^{39} + 6 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} - 16 q^{65} - 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} - 32 q^{93} + 24 q^{95} - 6 q^{99}+O(q^{100}) \) 24 * q - 2 * q^9 - 6 * q^15 - 2 * q^21 + 16 * q^23 + 8 * q^25 - 8 * q^35 - 2 * q^39 + 6 * q^51 - 24 * q^53 - 8 * q^57 - 16 * q^63 - 16 * q^65 - 8 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^85 + 12 * q^91 - 32 * q^93 + 24 * q^95 - 6 * q^99

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