LMFDB - Embedded newform 840.2.k.a.209.11

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

$q$-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	$-0.727499\pi$
$11$		−	5.00998i		−	1.51057i	0.655399	−	0.755283i	$-0.272501\pi$
$12$		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	$0.194737\pi$
$17$			4.73603i			1.14866i	−0.818625	+	0.574328i	$0.805263\pi$
$18$		0			0
$19$			2.13979i			0.490901i	0.969409	+	0.245450i	$0.0789359\pi$
$19$			2.13979i			0.490901i	−0.969409	+	0.245450i	$0.921064\pi$
$20$		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	$-0.813061\pi$
$29$		−	5.96788i		−	1.10821i	0.832448	−	0.554103i	$-0.186939\pi$
$30$		0			0
$31$			4.38673i			0.787880i	0.919136	+	0.393940i	$0.128888\pi$
$31$			4.38673i			0.787880i	−0.919136	+	0.393940i	$0.871112\pi$
$32$		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	$0.0998315\pi$
$37$			3.75323i			0.617027i	−0.951220	+	0.308514i	$0.900168\pi$
$38$		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	$-0.882093\pi$
$43$		−	4.74762i		−	0.724005i	0.932177	−	0.362002i	$-0.117907\pi$
$44$		0			0
$45$	−5.72796	−	3.49149i	−0.853874	−	0.520480i
$46$		0			0
$47$		−	5.64934i		−	0.824041i	−0.911175	−	0.412021i	$-0.864823\pi$
$47$		−	5.64934i		−	0.824041i	0.911175	−	0.412021i	$-0.135177\pi$
$48$		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	$-0.957122\pi$
$61$		−	2.09779i		−	0.268594i	0.990941	−	0.134297i	$-0.0428776\pi$
$62$		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	$-0.876255\pi$
$67$		−	6.20510i		−	0.758073i	0.925382	−	0.379037i	$-0.123745\pi$
$68$		0			0
$69$	−3.07967	−	16.0979i	−0.370749	−	1.93797i
$70$		0			0
$71$		−	8.65231i		−	1.02684i	−0.858138	−	0.513420i	$-0.828378\pi$
$71$		−	8.65231i		−	1.02684i	0.858138	−	0.513420i	$-0.171622\pi$
$72$		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	$0.106160\pi$
$83$			5.96483i			0.654725i	−0.944899	+	0.327362i	$0.893840\pi$
$84$		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	$0.340162\pi$
$127$			19.7565i			1.75310i	−0.481307	+	0.876552i	$0.659838\pi$
$128$		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	$0.0450807\pi$
$139$			3.32832i			0.282305i	−0.989988	+	0.141152i	$0.954919\pi$
$140$		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	$0.0563484\pi$
$149$			4.29916i			0.352201i	−0.984372	+	0.176101i	$0.943652\pi$
$150$		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	$-0.903762\pi$
$163$		−	7.60297i		−	0.595511i	0.954642	−	0.297755i	$-0.0962380\pi$
$164$		0			0
$165$	−14.3754	−	13.0325i	−1.11913	−	1.01458i
$166$		0			0
$167$		−	3.12412i		−	0.241752i	−0.992668	−	0.120876i	$-0.961430\pi$
$167$		−	3.12412i		−	0.241752i	0.992668	−	0.120876i	$-0.0385703\pi$
$168$		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	$-0.935551\pi$
$173$		−	5.28993i		−	0.402186i	0.979572	−	0.201093i	$-0.0644493\pi$
$174$		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	$-0.817649\pi$
$179$		−	14.5042i		−	1.08410i	0.840348	−	0.542048i	$-0.182351\pi$
$180$		0			0
$181$			18.5082i			1.37571i	0.725850	+	0.687853i	$0.241446\pi$
$181$			18.5082i			1.37571i	−0.725850	+	0.687853i	$0.758554\pi$
$182$		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	$0.152404\pi$
$191$			12.7342i			0.921414i	−0.887552	+	0.460707i	$0.847596\pi$
$192$		0			0
$193$			4.04248i			0.290984i	0.989359	+	0.145492i	$0.0464765\pi$
$193$			4.04248i			0.290984i	−0.989359	+	0.145492i	$0.953523\pi$
$194$		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	$-0.810633\pi$
$199$		−	15.8119i		−	1.12087i	0.828197	−	0.560437i	$-0.189367\pi$
$200$		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	$-0.806385\pi$
$227$		−	17.2192i		−	1.14288i	0.820645	−	0.571439i	$-0.193615\pi$
$228$		0			0
$229$		−	4.89863i		−	0.323710i	−0.986815	−	0.161855i	$-0.948252\pi$
$229$		−	4.89863i		−	0.323710i	0.986815	−	0.161855i	$-0.0517477\pi$
$230$		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	$-0.769881\pi$
$239$		−	20.4559i		−	1.32319i	0.749863	−	0.661593i	$-0.230119\pi$
$240$		0			0
$241$		−	10.8088i		−	0.696253i	−0.937448	−	0.348127i	$-0.886818\pi$
$241$		−	10.8088i		−	0.696253i	0.937448	−	0.348127i	$-0.113182\pi$
$242$		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	$0.340849\pi$
$257$			28.1376i			1.75518i	−0.479415	+	0.877589i	$0.659151\pi$
$258$		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	$-0.827290\pi$
$271$		−	17.0004i		−	1.03270i	0.856377	−	0.516351i	$-0.172710\pi$
$272$		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	$-0.991413\pi$
$277$		−	0.897874i		−	0.0539480i	0.999636	−	0.0269740i	$-0.00858714\pi$
$278$		0			0
$279$	−4.85753	−	12.2309i	−0.290813	−	0.732245i
$280$		0			0
$281$			12.6282i			0.753338i	0.926348	+	0.376669i	$0.122931\pi$
$281$			12.6282i			0.753338i	−0.926348	+	0.376669i	$0.877069\pi$
$282$		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	$-0.941383\pi$
$293$		−	6.26878i		−	0.366226i	0.983092	−	0.183113i	$-0.0586175\pi$
$294$		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	$-0.929616\pi$
$337$		−	8.05239i		−	0.438642i	0.975653	−	0.219321i	$-0.0703842\pi$
$338$		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	$0.186745\pi$
$349$			20.6841i			1.10719i	−0.832785	+	0.553597i	$0.813255\pi$
$350$		0			0
$351$	4.54844	+	7.14213i	0.242778	+	0.381219i
$352$		0			0
$353$			13.1361i			0.699163i	0.936906	+	0.349581i	$0.113676\pi$
$353$			13.1361i			0.699163i	−0.936906	+	0.349581i	$0.886324\pi$
$354$		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	$-0.864828\pi$
$359$		−	15.6128i		−	0.824014i	0.911181	−	0.412007i	$-0.135172\pi$
$360$		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	$-0.925265\pi$
$373$		−	8.98592i		−	0.465273i	0.972564	−	0.232637i	$-0.0747353\pi$
$374$		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	$0.0789515\pi$
$383$			9.60898i			0.490996i	−0.969397	+	0.245498i	$0.921049\pi$
$384$		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	$-0.710014\pi$
$389$		−	31.1675i		−	1.58026i	0.612942	−	0.790128i	$-0.289986\pi$
$390$		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	$0.158135\pi$
$401$			19.0883i			0.953224i	−0.879114	+	0.476612i	$0.841865\pi$
$402$		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	$-0.956318\pi$
$409$		−	5.53326i		−	0.273602i	0.990599	−	0.136801i	$-0.0436821\pi$
$410$		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	$-0.879037\pi$
$431$		−	15.4016i		−	0.741869i	0.928659	−	0.370934i	$-0.120963\pi$
$432$		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	$-0.866618\pi$
$439$		−	17.0500i		−	0.813751i	0.913484	−	0.406875i	$-0.133382\pi$
$440$		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	$-0.889453\pi$
$449$		−	14.4240i		−	0.680710i	0.940297	−	0.340355i	$-0.110547\pi$
$450$		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	$-0.929406\pi$
$457$		−	9.40452i		−	0.439925i	0.975508	−	0.219962i	$-0.0705935\pi$
$458$		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	$0.197135\pi$
$463$			24.9809i			1.16096i	−0.814275	+	0.580479i	$0.802865\pi$
$464$		0			0
$465$	12.5871	+	11.4112i	0.583713	+	0.529182i
$466$		0			0
$467$		−	5.52170i		−	0.255514i	−0.991806	−	0.127757i	$-0.959222\pi$
$467$		−	5.52170i		−	0.255514i	0.991806	−	0.127757i	$-0.0407777\pi$
$468$		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	$-0.750621\pi$
$487$		−	31.1480i		−	1.41145i	0.708485	−	0.705726i	$-0.249379\pi$
$488$		0			0
$489$	−12.9342	+	2.47442i	−0.584903	+	0.111897i
$490$		0			0
$491$		−	7.82187i		−	0.352996i	−0.984301	−	0.176498i	$-0.943523\pi$
$491$		−	7.82187i		−	0.352996i	0.984301	−	0.176498i	$-0.0564769\pi$
$492$		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	$-0.787681\pi$
$503$		−	27.7497i		−	1.23730i	0.785668	−	0.618648i	$-0.212319\pi$
$504$		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	$0.0633174\pi$
$547$			9.24333i			0.395216i	−0.980281	+	0.197608i	$0.936683\pi$
$548$		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	$-0.890702\pi$
$563$		−	15.9764i		−	0.673325i	0.941625	−	0.336662i	$-0.109298\pi$
$564$		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	$-0.882048\pi$
$569$		−	17.2766i		−	0.724272i	0.932125	−	0.362136i	$-0.117952\pi$
$570$		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	$0.225453\pi$
$587$			31.5221i			1.30106i	−0.759481	+	0.650529i	$0.774547\pi$
$588$		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	$-0.775869\pi$
$593$		−	31.5290i		−	1.29474i	0.762176	−	0.647370i	$-0.224131\pi$
$594$		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	$-0.752776\pi$
$599$		−	34.3090i		−	1.40183i	0.713246	−	0.700913i	$-0.247224\pi$
$600$		0			0
$601$			37.5135i			1.53021i	0.643907	+	0.765103i	$0.277312\pi$
$601$			37.5135i			1.53021i	−0.643907	+	0.765103i	$0.722688\pi$
$602$		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	$-0.959000\pi$
$613$		−	6.36050i		−	0.256898i	0.991716	−	0.128449i	$-0.0409999\pi$
$614$		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	$0.286167\pi$
$619$			38.9476i			1.56543i	−0.622378	+	0.782717i	$0.713833\pi$
$620$		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	$-0.973928\pi$
$641$		−	4.14281i		−	0.163631i	0.996647	−	0.0818155i	$-0.0260718\pi$
$642$		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	$0.0813788\pi$
$647$			12.8648i			0.505766i	−0.967497	+	0.252883i	$0.918621\pi$
$648$		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	$0.164671\pi$
$659$			25.3918i			0.989123i	−0.869143	+	0.494561i	$0.835329\pi$
$660$		0			0
$661$		−	27.2661i		−	1.06053i	−0.847833	−	0.530264i	$-0.822093\pi$
$661$		−	27.2661i		−	1.06053i	0.847833	−	0.530264i	$-0.177907\pi$
$662$		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	$-0.843371\pi$
$673$		−	24.5127i		−	0.944895i	0.881359	−	0.472447i	$-0.156629\pi$
$674$		0			0
$675$	−24.9185	+	7.35317i	−0.959113	+	0.283024i
$676$		0			0
$677$		−	31.5932i		−	1.21423i	−0.794615	−	0.607114i	$-0.792327\pi$
$677$		−	31.5932i		−	1.21423i	0.794615	−	0.607114i	$-0.207673\pi$
$678$		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	$0.0102830\pi$
$691$			1.69810i			0.0645986i	−0.999478	+	0.0322993i	$0.989717\pi$
$692$		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	$0.142247\pi$
$701$			22.8839i			0.864314i	−0.901798	+	0.432157i	$0.857753\pi$
$702$		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	$0.0193611\pi$
$757$			3.34496i			0.121575i	−0.998151	+	0.0607873i	$0.980639\pi$
$758$		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	$-0.826782\pi$
$769$		−	28.7135i		−	1.03544i	0.855551	−	0.517718i	$-0.173218\pi$
$770$		0			0
$771$	47.8677	−	9.15750i	1.72391	−	0.329799i
$772$		0			0
$773$			13.0728i			0.470196i	0.971972	+	0.235098i	$0.0755411\pi$
$773$			13.0728i			0.470196i	−0.971972	+	0.235098i	$0.924459\pi$
$774$		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	$-0.954950\pi$
$797$		−	7.96431i		−	0.282110i	0.990002	−	0.141055i	$-0.0450495\pi$
$798$		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	$0.255365\pi$
$809$			40.8966i			1.43785i	−0.695088	+	0.718924i	$0.744635\pi$
$810$		0			0
$811$			39.9526i			1.40293i	0.712705	+	0.701464i	$0.247470\pi$
$811$			39.9526i			1.40293i	−0.712705	+	0.701464i	$0.752530\pi$
$812$		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	$-0.834602\pi$
$821$		−	28.4551i		−	0.993091i	0.868011	−	0.496546i	$-0.165398\pi$
$822$		0			0
$823$			0.837012i			0.0291764i	0.999894	+	0.0145882i	$0.00464373\pi$
$823$			0.837012i			0.0291764i	−0.999894	+	0.0145882i	$0.995356\pi$
$824$		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	$0.151083\pi$
$829$			26.3174i			0.914041i	−0.889456	+	0.457021i	$0.848917\pi$
$830$		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	$-0.969033\pi$
$857$		−	5.68694i		−	0.194262i	0.995272	−	0.0971310i	$-0.0309666\pi$
$858$		0			0
$859$		−	33.2123i		−	1.13319i	−0.823997	−	0.566595i	$-0.808261\pi$
$859$		−	33.2123i		−	1.13319i	0.823997	−	0.566595i	$-0.191739\pi$
$860$		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	$0.0544257\pi$
$877$			10.0778i			0.340303i	−0.985418	+	0.170151i	$0.945574\pi$
$878$		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	$0.0486108\pi$
$883$			9.04074i			0.304245i	−0.988362	+	0.152123i	$0.951389\pi$
$884$		0			0
$885$	22.5326	−	24.8546i	0.757426	−	0.835478i
$886$		0			0
$887$		−	36.6139i		−	1.22937i	−0.788771	−	0.614687i	$-0.789282\pi$
$887$		−	36.6139i		−	1.22937i	0.788771	−	0.614687i	$-0.210718\pi$
$888$		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	$0.0315081\pi$
$907$			5.95247i			0.197648i	−0.995105	+	0.0988242i	$0.968492\pi$
$908$		0			0
$909$	12.9570	−	5.14591i	0.429757	−	0.170679i
$910$		0			0
$911$			11.3197i			0.375039i	0.982261	+	0.187519i	$0.0600447\pi$
$911$			11.3197i			0.375039i	−0.982261	+	0.187519i	$0.939955\pi$
$912$		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	$-0.912259\pi$
$967$		−	16.9272i		−	0.544341i	0.962249	−	0.272170i	$-0.0877414\pi$
$968$		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	$0.144760\pi$
$983$			27.5443i			0.878525i	−0.898359	+	0.439263i	$0.855240\pi$
$984$		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.209.11	✓	24
3.2	odd	2		840.2.k.b.209.12	yes	24
4.3	odd	2		1680.2.k.i.209.14		24
5.4	even	2		840.2.k.b.209.14	yes	24
7.6	odd	2	inner	840.2.k.a.209.14	yes	24
12.11	even	2		1680.2.k.h.209.13		24
15.14	odd	2	inner	840.2.k.a.209.13	yes	24
20.19	odd	2		1680.2.k.h.209.11		24
21.20	even	2		840.2.k.b.209.13	yes	24
28.27	even	2		1680.2.k.i.209.11		24
35.34	odd	2		840.2.k.b.209.11	yes	24
60.59	even	2		1680.2.k.i.209.12		24
84.83	odd	2		1680.2.k.h.209.12		24
105.104	even	2	inner	840.2.k.a.209.12	yes	24
140.139	even	2		1680.2.k.h.209.14		24
420.419	odd	2		1680.2.k.i.209.13		24

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more