LMFDB - Embedded newform 840.2.k.a.209.1

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			209.1
Character	$\chi$	$=$	840.209
Dual form			840.2.k.a.209.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70556 - 0.301745i) q^{3} +(2.04579 - 0.902636i) q^{5} +(1.30232 - 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} +O(q^{10})$	\(q+(-1.70556 - 0.301745i) q^{3} +(2.04579 - 0.902636i) q^{5} +(1.30232 - 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} -5.71375i q^{11} -3.76758 q^{13} +(-3.76159 + 0.922198i) q^{15} +3.22968i q^{17} +0.786936i q^{19} +(-2.91611 + 3.53501i) q^{21} +1.52714 q^{23} +(3.37050 - 3.69320i) q^{25} +(-4.49553 - 2.60581i) q^{27} +6.77221i q^{29} -1.56709i q^{31} +(-1.72410 + 9.74517i) q^{33} +(0.585457 - 5.88704i) q^{35} -8.83080i q^{37} +(6.42585 + 1.13685i) q^{39} -6.65914 q^{41} -8.85072i q^{43} +(6.69390 - 0.437826i) q^{45} +1.75054i q^{47} +(-3.60795 - 5.99856i) q^{49} +(0.974541 - 5.50843i) q^{51} +0.616602 q^{53} +(-5.15744 - 11.6891i) q^{55} +(0.237454 - 1.34217i) q^{57} -6.38642 q^{59} -14.4554i q^{61} +(6.04029 - 5.14926i) q^{63} +(-7.70767 + 3.40075i) q^{65} +6.26557i q^{67} +(-2.60463 - 0.460806i) q^{69} -5.09956i q^{71} +12.5631 q^{73} +(-6.86300 + 5.28197i) q^{75} +(-13.1590 - 7.44111i) q^{77} -12.8517 q^{79} +(6.88112 + 5.80088i) q^{81} +10.5003i q^{83} +(2.91523 + 6.60724i) q^{85} +(2.04348 - 11.5504i) q^{87} +15.6873 q^{89} +(-4.90658 + 8.67687i) q^{91} +(-0.472863 + 2.67278i) q^{93} +(0.710317 + 1.60990i) q^{95} -7.69604 q^{97} +(5.88112 - 16.1008i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.70556	−	0.301745i	−0.984708	−	0.174213i
$3$	−1.70556	−	0.301745i	−0.984708	−	0.174213i
$4$		0			0
$5$	2.04579	−	0.902636i	0.914904	−	0.403671i
$5$	2.04579	−	0.902636i	0.914904	−	0.403671i
$6$		0			0
$7$	1.30232	−	2.30304i	0.492229	−	0.870466i
$7$	1.30232	−	2.30304i	0.492229	−	0.870466i
$8$		0			0
$9$	2.81790	+	1.02929i	0.939300	+	0.343097i
$10$		0			0
$11$		−	5.71375i		−	1.72276i	−0.507960	−	0.861381i	$-0.669600\pi$
$11$		−	5.71375i		−	1.72276i	0.507960	−	0.861381i	$-0.330400\pi$
$12$		0			0
$13$	−3.76758			−1.04494			−0.522469	−	0.852658i	$-0.674989\pi$
$13$	−3.76758			−1.04494			−0.522469	+	0.852658i	$0.674989\pi$
$14$		0			0
$15$	−3.76159	+	0.922198i	−0.971238	+	0.238110i
$16$		0			0
$17$			3.22968i			0.783313i	0.920112	+	0.391656i	$0.128098\pi$
$17$			3.22968i			0.783313i	−0.920112	+	0.391656i	$0.871902\pi$
$18$		0			0
$19$			0.786936i			0.180536i	0.995918	+	0.0902678i	$0.0287723\pi$
$19$			0.786936i			0.180536i	−0.995918	+	0.0902678i	$0.971228\pi$
$20$		0			0
$21$	−2.91611	+	3.53501i	−0.636348	+	0.771402i
$22$		0			0
$23$	1.52714			0.318430			0.159215	−	0.987244i	$-0.449104\pi$
$23$	1.52714			0.318430			0.159215	+	0.987244i	$0.449104\pi$
$24$		0			0
$25$	3.37050	−	3.69320i	0.674099	−	0.738641i
$26$		0			0
$27$	−4.49553	−	2.60581i	−0.865164	−	0.501489i
$28$		0			0
$29$			6.77221i			1.25757i	0.777580	+	0.628784i	$0.216447\pi$
$29$			6.77221i			1.25757i	−0.777580	+	0.628784i	$0.783553\pi$
$30$		0			0
$31$		−	1.56709i		−	0.281458i	−0.990048	−	0.140729i	$-0.955055\pi$
$31$		−	1.56709i		−	0.281458i	0.990048	−	0.140729i	$-0.0449447\pi$
$32$		0			0
$33$	−1.72410	+	9.74517i	−0.300127	+	1.69642i
$34$		0			0
$35$	0.585457	−	5.88704i	0.0989603	−	0.995091i
$36$		0			0
$37$		−	8.83080i		−	1.45177i	−0.687814	−	0.725887i	$-0.741430\pi$
$37$		−	8.83080i		−	1.45177i	0.687814	−	0.725887i	$-0.258570\pi$
$38$		0			0
$39$	6.42585	+	1.13685i	1.02896	+	0.182042i
$40$		0			0
$41$	−6.65914			−1.03998			−0.519992	−	0.854171i	$-0.674065\pi$
$41$	−6.65914			−1.03998			−0.519992	+	0.854171i	$0.674065\pi$
$42$		0			0
$43$		−	8.85072i		−	1.34972i	−0.737945	−	0.674861i	$-0.764204\pi$
$43$		−	8.85072i		−	1.34972i	0.737945	−	0.674861i	$-0.235796\pi$
$44$		0			0
$45$	6.69390	−	0.437826i	0.997868	−	0.0652672i
$46$		0			0
$47$			1.75054i			0.255342i	0.991817	+	0.127671i	$0.0407502\pi$
$47$			1.75054i			0.255342i	−0.991817	+	0.127671i	$0.959250\pi$
$48$		0			0
$49$	−3.60795	−	5.99856i	−0.515421	−	0.856937i
$50$		0			0
$51$	0.974541	−	5.50843i	0.136463	−	0.771335i
$52$		0			0
$53$	0.616602			0.0846967			0.0423484	−	0.999103i	$-0.486516\pi$
$53$	0.616602			0.0846967			0.0423484	+	0.999103i	$0.486516\pi$
$54$		0			0
$55$	−5.15744	−	11.6891i	−0.695429	−	1.57616i
$56$		0			0
$57$	0.237454	−	1.34217i	0.0314516	−	0.177775i
$58$		0			0
$59$	−6.38642			−0.831441			−0.415721	−	0.909492i	$-0.636471\pi$
$59$	−6.38642			−0.831441			−0.415721	+	0.909492i	$0.636471\pi$
$60$		0			0
$61$		−	14.4554i		−	1.85082i	−0.378964	−	0.925412i	$-0.623719\pi$
$61$		−	14.4554i		−	1.85082i	0.378964	−	0.925412i	$-0.376281\pi$
$62$		0			0
$63$	6.04029	−	5.14926i	0.761005	−	0.648746i
$64$		0			0
$65$	−7.70767	+	3.40075i	−0.956019	+	0.421812i
$66$		0			0
$67$			6.26557i			0.765462i	0.923860	+	0.382731i	$0.125016\pi$
$67$			6.26557i			0.765462i	−0.923860	+	0.382731i	$0.874984\pi$
$68$		0			0
$69$	−2.60463	−	0.460806i	−0.313561	−	0.0554746i
$70$		0			0
$71$		−	5.09956i		−	0.605206i	−0.953117	−	0.302603i	$-0.902144\pi$
$71$		−	5.09956i		−	0.605206i	0.953117	−	0.302603i	$-0.0978556\pi$
$72$		0			0
$73$	12.5631			1.47039			0.735197	−	0.677853i	$-0.237089\pi$
$73$	12.5631			1.47039			0.735197	+	0.677853i	$0.237089\pi$
$74$		0			0
$75$	−6.86300	+	5.28197i	−0.792471	+	0.609909i
$76$		0			0
$77$	−13.1590	−	7.44111i	−1.49960	−	0.847993i
$78$		0			0
$79$	−12.8517			−1.44593			−0.722965	−	0.690885i	$-0.757221\pi$
$79$	−12.8517			−1.44593			−0.722965	+	0.690885i	$0.757221\pi$
$80$		0			0
$81$	6.88112	+	5.80088i	0.764569	+	0.644542i
$82$		0			0
$83$			10.5003i			1.15256i	0.817254	+	0.576278i	$0.195495\pi$
$83$			10.5003i			1.15256i	−0.817254	+	0.576278i	$0.804505\pi$
$84$		0			0
$85$	2.91523	+	6.60724i	0.316201	+	0.716656i
$86$		0			0
$87$	2.04348	−	11.5504i	0.219084	−	1.23834i
$88$		0			0
$89$	15.6873			1.66285			0.831423	−	0.555640i	$-0.187527\pi$
$89$	15.6873			1.66285			0.831423	+	0.555640i	$0.187527\pi$
$90$		0			0
$91$	−4.90658	+	8.67687i	−0.514349	+	0.909583i
$92$		0			0
$93$	−0.472863	+	2.67278i	−0.0490336	+	0.277154i
$94$		0			0
$95$	0.710317	+	1.60990i	0.0728770	+	0.165173i
$96$		0			0
$97$	−7.69604			−0.781414			−0.390707	−	0.920515i	$-0.627769\pi$
$97$	−7.69604			−0.781414			−0.390707	+	0.920515i	$0.627769\pi$
$98$		0			0
$99$	5.88112	−	16.1008i	0.591075	−	1.61819i
$100$		0			0
$101$	10.2658			1.02149			0.510745	−	0.859732i	$-0.329370\pi$
$101$	10.2658			1.02149			0.510745	+	0.859732i	$0.329370\pi$
$102$		0			0
$103$	3.62328			0.357013			0.178506	−	0.983939i	$-0.442873\pi$
$103$	3.62328			0.357013			0.178506	+	0.983939i	$0.442873\pi$
$104$		0			0
$105$	−2.77492	+	9.86407i	−0.270805	+	0.962634i
$106$		0			0
$107$	8.65500			0.836710			0.418355	−	0.908284i	$-0.362607\pi$
$107$	8.65500			0.836710			0.418355	+	0.908284i	$0.362607\pi$
$108$		0			0
$109$	8.32109			0.797016			0.398508	−	0.917165i	$-0.369528\pi$
$109$	8.32109			0.797016			0.398508	+	0.917165i	$0.369528\pi$
$110$		0			0
$111$	−2.66465	+	15.0615i	−0.252917	+	1.42957i
$112$		0			0
$113$	14.0986			1.32628			0.663142	−	0.748494i	$-0.269223\pi$
$113$	14.0986			1.32628			0.663142	+	0.748494i	$0.269223\pi$
$114$		0			0
$115$	3.12420	−	1.37845i	0.291333	−	0.128541i
$116$		0			0
$117$	−10.6167	−	3.87794i	−0.981511	−	0.358516i
$118$		0			0
$119$	7.43807	+	4.20606i	0.681847	+	0.385569i
$120$		0			0
$121$	−21.6470			−1.96791
$122$		0			0
$123$	11.3576	+	2.00936i	1.02408	+	0.181178i
$124$		0			0
$125$	3.56170	−	10.5978i	0.318568	−	0.947900i
$126$		0			0
$127$			12.7643i			1.13265i	0.824181	+	0.566326i	$0.191636\pi$
$127$			12.7643i			1.13265i	−0.824181	+	0.566326i	$0.808364\pi$
$128$		0			0
$129$	−2.67066	+	15.0955i	−0.235139	+	1.32908i
$130$		0			0
$131$	−16.9991			−1.48522			−0.742610	−	0.669724i	$-0.766412\pi$
$131$	−16.9991			−1.48522			−0.742610	+	0.669724i	$0.766412\pi$
$132$		0			0
$133$	1.81234	+	1.02484i	0.157150	+	0.0888648i
$134$		0			0
$135$	−11.5490	−	1.27311i	−0.993979	−	0.109572i
$136$		0			0
$137$	−12.3946			−1.05894			−0.529470	−	0.848329i	$-0.677609\pi$
$137$	−12.3946			−1.05894			−0.529470	+	0.848329i	$0.677609\pi$
$138$		0			0
$139$		−	11.2465i		−	0.953912i	−0.878927	−	0.476956i	$-0.841740\pi$
$139$		−	11.2465i		−	0.953912i	0.878927	−	0.476956i	$-0.158260\pi$
$140$		0			0
$141$	0.528217	−	2.98566i	0.0444839	−	0.251438i
$142$		0			0
$143$			21.5270i			1.80018i
$144$		0			0
$145$	6.11284	+	13.8545i	0.507644	+	1.15055i
$146$		0			0
$147$	4.34356	+	11.3196i	0.358250	+	0.933626i
$148$		0			0
$149$			2.85057i			0.233528i	0.993160	+	0.116764i	$0.0372521\pi$
$149$			2.85057i			0.233528i	−0.993160	+	0.116764i	$0.962748\pi$
$150$		0			0
$151$	16.1515			1.31439			0.657197	−	0.753719i	$-0.271742\pi$
$151$	16.1515			1.31439			0.657197	+	0.753719i	$0.271742\pi$
$152$		0			0
$153$	−3.32428	+	9.10092i	−0.268752	+	0.735766i
$154$		0			0
$155$	−1.41452	−	3.20594i	−0.113617	−	0.257507i
$156$		0			0
$157$	7.21577			0.575881			0.287941	−	0.957648i	$-0.407029\pi$
$157$	7.21577			0.575881			0.287941	+	0.957648i	$0.407029\pi$
$158$		0			0
$159$	−1.05165	−	0.186057i	−0.0834016	−	0.0147552i
$160$		0			0
$161$	1.98881	−	3.51705i	0.156740	−	0.277182i
$162$		0			0
$163$		−	3.50516i		−	0.274545i	−0.990533	−	0.137273i	$-0.956166\pi$
$163$		−	3.50516i		−	0.274545i	0.990533	−	0.137273i	$-0.0438337\pi$
$164$		0			0
$165$	5.26921	+	21.4928i	0.410207	+	1.67321i
$166$		0			0
$167$			1.38365i			0.107070i	0.998566	+	0.0535349i	$0.0170489\pi$
$167$			1.38365i			0.107070i	−0.998566	+	0.0535349i	$0.982951\pi$
$168$		0			0
$169$	1.19466			0.0918967
$170$		0			0
$171$	−0.809987	+	2.21751i	−0.0619413	+	0.169577i
$172$		0			0
$173$			10.1282i			0.770032i	0.922910	+	0.385016i	$0.125804\pi$
$173$			10.1282i			0.770032i	−0.922910	+	0.385016i	$0.874196\pi$
$174$		0			0
$175$	−4.11613	−	12.5721i	−0.311151	−	0.950361i
$176$		0			0
$177$	10.8925	+	1.92707i	0.818727	+	0.144848i
$178$		0			0
$179$			0.542452i			0.0405447i	0.999794	+	0.0202724i	$0.00645334\pi$
$179$			0.542452i			0.0405447i	−0.999794	+	0.0202724i	$0.993547\pi$
$180$		0			0
$181$			1.15745i			0.0860323i	0.999074	+	0.0430162i	$0.0136967\pi$
$181$			1.15745i			0.0860323i	−0.999074	+	0.0430162i	$0.986303\pi$
$182$		0			0
$183$	−4.36184	+	24.6546i	−0.322437	+	1.82252i
$184$		0			0
$185$	−7.97100	−	18.0659i	−0.586039	−	1.32823i
$186$		0			0
$187$	18.4536			1.34946
$188$		0			0
$189$	−11.8559	+	6.95977i	−0.862387	+	0.506249i
$190$		0			0
$191$			10.8558i			0.785495i	0.919646	+	0.392748i	$0.128475\pi$
$191$			10.8558i			0.785495i	−0.919646	+	0.392748i	$0.871525\pi$
$192$		0			0
$193$			24.6409i			1.77369i	0.462067	+	0.886845i	$0.347108\pi$
$193$			24.6409i			1.77369i	−0.462067	+	0.886845i	$0.652892\pi$
$194$		0			0
$195$	14.1721	−	3.47445i	1.01488	−	0.248811i
$196$		0			0
$197$	3.15816			0.225009			0.112505	−	0.993651i	$-0.464113\pi$
$197$	3.15816			0.225009			0.112505	+	0.993651i	$0.464113\pi$
$198$		0			0
$199$			3.60377i			0.255464i	0.991809	+	0.127732i	$0.0407698\pi$
$199$			3.60377i			0.255464i	−0.991809	+	0.127732i	$0.959230\pi$
$200$		0			0
$201$	1.89061	−	10.6863i	0.133353	−	0.753756i
$202$		0			0
$203$	15.5966	+	8.81956i	1.09467	+	0.619011i
$204$		0			0
$205$	−13.6232	+	6.01078i	−0.951485	+	0.419811i
$206$		0			0
$207$	4.30332	+	1.57187i	0.299101	+	0.109252i
$208$		0			0
$209$	4.49636			0.311020
$210$		0			0
$211$	0.159466			0.0109781			0.00548905	−	0.999985i	$-0.498253\pi$
$211$	0.159466			0.0109781			0.00548905	+	0.999985i	$0.498253\pi$
$212$		0			0
$213$	−1.53877	+	8.69762i	−0.105435	+	0.595951i
$214$		0			0
$215$	−7.98898	−	18.1067i	−0.544844	−	1.23487i
$216$		0			0
$217$	−3.60907	−	2.04085i	−0.245000	−	0.138542i
$218$		0			0
$219$	−21.4271	−	3.79084i	−1.44791	−	0.256161i
$220$		0			0
$221$		−	12.1681i		−	0.818514i
$222$		0			0
$223$	4.39412			0.294252			0.147126	−	0.989118i	$-0.452998\pi$
$223$	4.39412			0.294252			0.147126	+	0.989118i	$0.452998\pi$
$224$		0			0
$225$	13.2991	−	6.93786i	0.886607	−	0.462524i
$226$		0			0
$227$		−	19.1642i		−	1.27197i	−0.771700	−	0.635987i	$-0.780593\pi$
$227$		−	19.1642i		−	1.27197i	0.771700	−	0.635987i	$-0.219407\pi$
$228$		0			0
$229$			1.25651i			0.0830328i	0.999138	+	0.0415164i	$0.0132189\pi$
$229$			1.25651i			0.0830328i	−0.999138	+	0.0415164i	$0.986781\pi$
$230$		0			0
$231$	20.1982	+	16.6619i	1.32894	+	1.09628i
$232$		0			0
$233$	5.73752			0.375878			0.187939	−	0.982181i	$-0.439819\pi$
$233$	5.73752			0.375878			0.187939	+	0.982181i	$0.439819\pi$
$234$		0			0
$235$	1.58010	+	3.58123i	0.103074	+	0.233614i
$236$		0			0
$237$	21.9194	+	3.87794i	1.42382	+	0.251899i
$238$		0			0
$239$			18.1836i			1.17620i	0.808789	+	0.588099i	$0.200123\pi$
$239$			18.1836i			1.17620i	−0.808789	+	0.588099i	$0.799877\pi$
$240$		0			0
$241$		−	18.1259i		−	1.16759i	−0.811901	−	0.583795i	$-0.801567\pi$
$241$		−	18.1259i		−	1.16759i	0.811901	−	0.583795i	$-0.198433\pi$
$242$		0			0
$243$	−9.98580	−	11.9701i	−0.640589	−	0.767884i
$244$		0			0
$245$	−12.7956	−	9.01511i	−0.817482	−	0.575954i
$246$		0			0
$247$		−	2.96485i		−	0.188649i
$248$		0			0
$249$	3.16841	−	17.9089i	0.200790	−	1.13493i
$250$		0			0
$251$	0.598823			0.0377974			0.0188987	−	0.999821i	$-0.493984\pi$
$251$	0.598823			0.0377974			0.0188987	+	0.999821i	$0.493984\pi$
$252$		0			0
$253$		−	8.72568i		−	0.548579i
$254$		0			0
$255$	−2.97840	−	12.1487i	−0.186515	−	0.760783i
$256$		0			0
$257$		−	15.4070i		−	0.961063i	−0.876977	−	0.480532i	$-0.840444\pi$
$257$		−	15.4070i		−	0.961063i	0.876977	−	0.480532i	$-0.159556\pi$
$258$		0			0
$259$	−20.3376	−	11.5005i	−1.26372	−	0.714605i
$260$		0			0
$261$	−6.97058	+	19.0834i	−0.431468	+	1.18123i
$262$		0			0
$263$	9.66296			0.595843			0.297922	−	0.954590i	$-0.403707\pi$
$263$	9.66296			0.595843			0.297922	+	0.954590i	$0.403707\pi$
$264$		0			0
$265$	1.26144	−	0.556567i	0.0774894	−	0.0341896i
$266$		0			0
$267$	−26.7556	−	4.73355i	−1.63742	−	0.289689i
$268$		0			0
$269$	−13.2397			−0.807241			−0.403620	−	0.914927i	$-0.632248\pi$
$269$	−13.2397			−0.807241			−0.403620	+	0.914927i	$0.632248\pi$
$270$		0			0
$271$			15.6372i			0.949890i	0.880016	+	0.474945i	$0.157532\pi$
$271$			15.6372i			0.949890i	−0.880016	+	0.474945i	$0.842468\pi$
$272$		0			0
$273$	10.9867	−	13.3184i	0.664944	−	0.806068i
$274$		0			0
$275$	−21.1021	−	19.2582i	−1.27250	−	1.16131i
$276$		0			0
$277$			3.48524i			0.209408i	0.994503	+	0.104704i	$0.0333895\pi$
$277$			3.48524i			0.209408i	−0.994503	+	0.104704i	$0.966611\pi$
$278$		0			0
$279$	1.61300	−	4.41591i	0.0965676	−	0.264374i
$280$		0			0
$281$			21.4311i			1.27847i	0.769011	+	0.639236i	$0.220749\pi$
$281$			21.4311i			1.27847i	−0.769011	+	0.639236i	$0.779251\pi$
$282$		0			0
$283$	−7.97240			−0.473910			−0.236955	−	0.971521i	$-0.576149\pi$
$283$	−7.97240			−0.473910			−0.236955	+	0.971521i	$0.576149\pi$
$284$		0			0
$285$	−0.725711	−	2.96013i	−0.0429874	−	0.175343i
$286$		0			0
$287$	−8.67230	+	15.3362i	−0.511910	+	0.905270i
$288$		0			0
$289$	6.56915			0.386421
$290$		0			0
$291$	13.1261	+	2.32224i	0.769465	+	0.136132i
$292$		0			0
$293$			12.8728i			0.752035i	0.926613	+	0.376018i	$0.122707\pi$
$293$			12.8728i			0.752035i	−0.926613	+	0.376018i	$0.877293\pi$
$294$		0			0
$295$	−13.0653	+	5.76461i	−0.760689	+	0.335629i
$296$		0			0
$297$	−14.8890	+	25.6863i	−0.863945	+	1.49047i
$298$		0			0
$299$	−5.75361			−0.332740
$300$		0			0
$301$	−20.3835	−	11.5264i	−1.17489	−	0.664372i
$302$		0			0
$303$	−17.5091	−	3.09767i	−1.00587	−	0.177956i
$304$		0			0
$305$	−13.0480	−	29.5727i	−0.747124	−	1.69333i
$306$		0			0
$307$	−7.16507			−0.408932			−0.204466	−	0.978874i	$-0.565546\pi$
$307$	−7.16507			−0.408932			−0.204466	+	0.978874i	$0.565546\pi$
$308$		0			0
$309$	−6.17974	−	1.09331i	−0.351553	−	0.0621961i
$310$		0			0
$311$	34.4684			1.95453			0.977263	−	0.212030i	$-0.0680076\pi$
$311$	34.4684			1.95453			0.977263	+	0.212030i	$0.0680076\pi$
$312$		0			0
$313$	28.5587			1.61423			0.807117	−	0.590392i	$-0.201027\pi$
$313$	28.5587			1.61423			0.807117	+	0.590392i	$0.201027\pi$
$314$		0			0
$315$	7.70924	−	15.9865i	0.434366	−	0.900736i
$316$		0			0
$317$	20.9809			1.17840			0.589202	−	0.807986i	$-0.299442\pi$
$317$	20.9809			1.17840			0.589202	+	0.807986i	$0.299442\pi$
$318$		0			0
$319$	38.6947			2.16649
$320$		0			0
$321$	−14.7617	−	2.61160i	−0.823915	−	0.145766i
$322$		0			0
$323$	−2.54155			−0.141416
$324$		0			0
$325$	−12.6986	+	13.9144i	−0.704392	+	0.771834i
$326$		0			0
$327$	−14.1922	−	2.51085i	−0.784828	−	0.138850i
$328$		0			0
$329$	4.03156	+	2.27975i	0.222267	+	0.125687i
$330$		0			0
$331$	10.9404			0.601340			0.300670	−	0.953728i	$-0.402790\pi$
$331$	10.9404			0.601340			0.300670	+	0.953728i	$0.402790\pi$
$332$		0			0
$333$	9.08946	−	24.8843i	0.498100	−	1.36365i
$334$		0			0
$335$	5.65553	+	12.8180i	0.308995	+	0.700324i
$336$		0			0
$337$			5.98022i			0.325763i	0.986646	+	0.162882i	$0.0520789\pi$
$337$			5.98022i			0.325763i	−0.986646	+	0.162882i	$0.947921\pi$
$338$		0			0
$339$	−24.0460	−	4.25418i	−1.30600	−	0.231055i
$340$		0			0
$341$	−8.95399			−0.484886
$342$		0			0
$343$	−18.5136	+	0.497225i	−0.999640	+	0.0268476i
$344$		0			0
$345$	−5.74446	+	1.40832i	−0.309271	+	0.0758215i
$346$		0			0
$347$	32.3200			1.73503			0.867514	−	0.497412i	$-0.165716\pi$
$347$	32.3200			1.73503			0.867514	+	0.497412i	$0.165716\pi$
$348$		0			0
$349$			12.9630i			0.693896i	0.937884	+	0.346948i	$0.112782\pi$
$349$			12.9630i			0.693896i	−0.937884	+	0.346948i	$0.887218\pi$
$350$		0			0
$351$	16.9373	+	9.81760i	0.904044	+	0.524025i
$352$		0			0
$353$		−	18.4705i		−	0.983083i	−0.870854	−	0.491541i	$-0.836434\pi$
$353$		−	18.4705i		−	0.983083i	0.870854	−	0.491541i	$-0.163566\pi$
$354$		0			0
$355$	−4.60304	−	10.4326i	−0.244304	−	0.553705i
$356$		0			0
$357$	−11.4170	−	9.41812i	−0.604249	−	0.498460i
$358$		0			0
$359$			24.2434i			1.27952i	0.768576	+	0.639759i	$0.220966\pi$
$359$			24.2434i			1.27952i	−0.768576	+	0.639759i	$0.779034\pi$
$360$		0			0
$361$	18.3807			0.967407
$362$		0			0
$363$	36.9203	+	6.53187i	1.93781	+	0.342834i
$364$		0			0
$365$	25.7014	−	11.3399i	1.34527	−	0.593556i
$366$		0			0
$367$	−1.55647			−0.0812468			−0.0406234	−	0.999175i	$-0.512934\pi$
$367$	−1.55647			−0.0812468			−0.0406234	+	0.999175i	$0.512934\pi$
$368$		0			0
$369$	−18.7648	−	6.85420i	−0.976856	−	0.356815i
$370$		0			0
$371$	0.803010	−	1.42006i	0.0416902	−	0.0737256i
$372$		0			0
$373$		−	22.0453i		−	1.14146i	−0.821137	−	0.570731i	$-0.806660\pi$
$373$		−	22.0453i		−	1.14146i	0.821137	−	0.570731i	$-0.193340\pi$
$374$		0			0
$375$	−9.27256	+	17.0006i	−0.478833	+	0.877906i
$376$		0			0
$377$		−	25.5149i		−	1.31408i
$378$		0			0
$379$	9.42384			0.484070			0.242035	−	0.970268i	$-0.422185\pi$
$379$	9.42384			0.484070			0.242035	+	0.970268i	$0.422185\pi$
$380$		0			0
$381$	3.85158	−	21.7704i	0.197322	−	1.11533i
$382$		0			0
$383$			27.3334i			1.39667i	0.715771	+	0.698335i	$0.246075\pi$
$383$			27.3334i			1.39667i	−0.715771	+	0.698335i	$0.753925\pi$
$384$		0			0
$385$	−33.6371	−	3.34516i	−1.71430	−	0.170485i
$386$		0			0
$387$	9.10997	−	24.9404i	0.463086	−	1.26779i
$388$		0			0
$389$			13.3761i			0.678195i	0.940751	+	0.339097i	$0.110122\pi$
$389$			13.3761i			0.678195i	−0.940751	+	0.339097i	$0.889878\pi$
$390$		0			0
$391$			4.93217i			0.249430i
$392$		0			0
$393$	28.9931	+	5.12940i	1.46251	+	0.258744i
$394$		0			0
$395$	−26.2919	+	11.6004i	−1.32289	+	0.583680i
$396$		0			0
$397$	−9.36101			−0.469815			−0.234908	−	0.972018i	$-0.575479\pi$
$397$	−9.36101			−0.469815			−0.234908	+	0.972018i	$0.575479\pi$
$398$		0			0
$399$	−2.78183	−	2.29479i	−0.139266	−	0.114883i
$400$		0			0
$401$		−	24.5115i		−	1.22405i	−0.790840	−	0.612024i	$-0.790356\pi$
$401$		−	24.5115i		−	1.22405i	0.790840	−	0.612024i	$-0.209644\pi$
$402$		0			0
$403$			5.90415i			0.294107i
$404$		0			0
$405$	19.3134	+	5.65623i	0.959690	+	0.281060i
$406$		0			0
$407$	−50.4570			−2.50106
$408$		0			0
$409$		−	0.0679660i		−	0.00336070i	−0.999999	−	0.00168035i	$-0.999465\pi$
$409$		−	0.0679660i		−	0.00336070i	0.999999	−	0.00168035i	$-0.000534872\pi$
$410$		0			0
$411$	21.1397	+	3.74000i	1.04275	+	0.184481i
$412$		0			0
$413$	−8.31713	+	14.7082i	−0.409259	+	0.723741i
$414$		0			0
$415$	9.47794	+	21.4814i	0.465254	+	1.05448i
$416$		0			0
$417$	−3.39356	+	19.1816i	−0.166184	+	0.939325i
$418$		0			0
$419$	32.7469			1.59979			0.799896	−	0.600138i	$-0.204888\pi$
$419$	32.7469			1.59979			0.799896	+	0.600138i	$0.204888\pi$
$420$		0			0
$421$	16.8869			0.823017			0.411508	−	0.911406i	$-0.365002\pi$
$421$	16.8869			0.823017			0.411508	+	0.911406i	$0.365002\pi$
$422$		0			0
$423$	−1.80182	+	4.93285i	−0.0876073	+	0.239843i
$424$		0			0
$425$	11.9279	+	10.8856i	0.578587	+	0.528031i
$426$		0			0
$427$	−33.2913	−	18.8255i	−1.61108	−	0.911029i
$428$		0			0
$429$	6.49567	−	36.7157i	0.313614	−	1.77265i
$430$		0			0
$431$			29.7369i			1.43238i	0.697907	+	0.716188i	$0.254115\pi$
$431$			29.7369i			1.43238i	−0.697907	+	0.716188i	$0.745885\pi$
$432$		0			0
$433$	−15.1503			−0.728076			−0.364038	−	0.931384i	$-0.618602\pi$
$433$	−15.1503			−0.728076			−0.364038	+	0.931384i	$0.618602\pi$
$434$		0			0
$435$	−6.24532	−	25.4743i	−0.299440	−	1.22140i
$436$		0			0
$437$			1.20176i			0.0574880i
$438$		0			0
$439$			9.26454i			0.442172i	0.975254	+	0.221086i	$0.0709602\pi$
$439$			9.26454i			0.442172i	−0.975254	+	0.221086i	$0.929040\pi$
$440$		0			0
$441$	−3.99258	−	20.6170i	−0.190123	−	0.981760i
$442$		0			0
$443$	−5.88820			−0.279757			−0.139878	−	0.990169i	$-0.544671\pi$
$443$	−5.88820			−0.279757			−0.139878	+	0.990169i	$0.544671\pi$
$444$		0			0
$445$	32.0928	−	14.1599i	1.52134	−	0.671243i
$446$		0			0
$447$	0.860147	−	4.86184i	0.0406836	−	0.229957i
$448$		0			0
$449$		−	14.4606i		−	0.682439i	−0.939984	−	0.341219i	$-0.889160\pi$
$449$		−	14.4606i		−	0.682439i	0.939984	−	0.341219i	$-0.110840\pi$
$450$		0			0
$451$			38.0487i			1.79164i
$452$		0			0
$453$	−27.5475	−	4.87365i	−1.29429	−	0.228984i
$454$		0			0
$455$	−2.20576	+	22.1799i	−0.103407	+	1.03981i
$456$		0			0
$457$		−	19.9157i		−	0.931617i	−0.884885	−	0.465809i	$-0.845764\pi$
$457$		−	19.9157i		−	0.931617i	0.884885	−	0.465809i	$-0.154236\pi$
$458$		0			0
$459$	8.41594	−	14.5191i	0.392822	−	0.677694i
$460$		0			0
$461$	−42.4755			−1.97828			−0.989142	−	0.146966i	$-0.953049\pi$
$461$	−42.4755			−1.97828			−0.989142	+	0.146966i	$0.953049\pi$
$462$		0			0
$463$		−	34.4952i		−	1.60313i	−0.597910	−	0.801564i	$-0.704002\pi$
$463$		−	34.4952i		−	1.60313i	0.597910	−	0.801564i	$-0.295998\pi$
$464$		0			0
$465$	1.44517	+	5.89476i	0.0670182	+	0.273363i
$466$		0			0
$467$			5.47336i			0.253277i	0.991949	+	0.126639i	$0.0404188\pi$
$467$			5.47336i			0.253277i	−0.991949	+	0.126639i	$0.959581\pi$
$468$		0			0
$469$	14.4298	+	8.15975i	0.666308	+	0.376782i
$470$		0			0
$471$	−12.3070	−	2.17732i	−0.567075	−	0.100326i
$472$		0			0
$473$	−50.5708			−2.32525
$474$		0			0
$475$	2.90632	+	2.65237i	0.133351	+	0.121699i
$476$		0			0
$477$	1.73752	+	0.634663i	0.0795556	+	0.0290592i
$478$		0			0
$479$	25.9132			1.18400			0.592001	−	0.805937i	$-0.298338\pi$
$479$	25.9132			1.18400			0.592001	+	0.805937i	$0.298338\pi$
$480$		0			0
$481$			33.2707i			1.51701i
$482$		0			0
$483$	−4.45330	+	5.39844i	−0.202632	+	0.245638i
$484$		0			0
$485$	−15.7445	+	6.94672i	−0.714919	+	0.315434i
$486$		0			0
$487$		−	8.92087i		−	0.404243i	−0.979360	−	0.202122i	$-0.935216\pi$
$487$		−	8.92087i		−	0.404243i	0.979360	−	0.202122i	$-0.0647836\pi$
$488$		0			0
$489$	−1.05767	+	5.97828i	−0.0478293	+	0.270347i
$490$		0			0
$491$		−	36.1241i		−	1.63026i	−0.579278	−	0.815130i	$-0.696666\pi$
$491$		−	36.1241i		−	1.63026i	0.579278	−	0.815130i	$-0.303334\pi$
$492$		0			0
$493$	−21.8721			−0.985070
$494$		0			0
$495$	−2.50163	−	38.2473i	−0.112440	−	1.71909i
$496$		0			0
$497$	−11.7445	−	6.64123i	−0.526811	−	0.297900i
$498$		0			0
$499$	−24.8816			−1.11385			−0.556927	−	0.830562i	$-0.688020\pi$
$499$	−24.8816			−1.11385			−0.556927	+	0.830562i	$0.688020\pi$
$500$		0			0
$501$	0.417509	−	2.35990i	0.0186529	−	0.105433i
$502$		0			0
$503$		−	35.3029i		−	1.57408i	−0.616902	−	0.787040i	$-0.711612\pi$
$503$		−	35.3029i		−	1.57408i	0.616902	−	0.787040i	$-0.288388\pi$
$504$		0			0
$505$	21.0017	−	9.26632i	0.934565	−	0.412346i
$506$		0			0
$507$	−2.03756	−	0.360482i	−0.0904914	−	0.0160096i
$508$		0			0
$509$	13.6651			0.605696			0.302848	−	0.953039i	$-0.402062\pi$
$509$	13.6651			0.605696			0.302848	+	0.953039i	$0.402062\pi$
$510$		0			0
$511$	16.3611	−	28.9332i	0.723771	−	1.27993i
$512$		0			0
$513$	2.05061	−	3.53769i	0.0905365	−	0.156193i
$514$		0			0
$515$	7.41247	−	3.27051i	0.326632	−	0.144116i
$516$		0			0
$517$	10.0022			0.439894
$518$		0			0
$519$	3.05613	−	17.2743i	0.134149	−	0.758257i
$520$		0			0
$521$	9.66702			0.423520			0.211760	−	0.977322i	$-0.432081\pi$
$521$	9.66702			0.423520			0.211760	+	0.977322i	$0.432081\pi$
$522$		0			0
$523$	3.51766			0.153817			0.0769083	−	0.997038i	$-0.475495\pi$
$523$	3.51766			0.153817			0.0769083	+	0.997038i	$0.475495\pi$
$524$		0			0
$525$	3.22677	+	22.6845i	0.140828	+	0.990034i
$526$		0			0
$527$	5.06121			0.220470
$528$		0			0
$529$	−20.6679			−0.898602
$530$		0			0
$531$	−17.9963	−	6.57349i	−0.780973	−	0.285265i
$532$		0			0
$533$	25.0888			1.08672
$534$		0			0
$535$	17.7063	−	7.81231i	0.765510	−	0.337756i
$536$		0			0
$537$	0.163682	−	0.925186i	0.00706341	−	0.0399247i
$538$		0			0
$539$	−34.2743	+	20.6149i	−1.47630	+	0.887948i
$540$		0			0
$541$	−5.55020			−0.238622			−0.119311	−	0.992857i	$-0.538069\pi$
$541$	−5.55020			−0.238622			−0.119311	+	0.992857i	$0.538069\pi$
$542$		0			0
$543$	0.349254	−	1.97410i	0.0149879	−	0.0847167i
$544$		0			0
$545$	17.0232	−	7.51092i	0.729193	−	0.321732i
$546$		0			0
$547$			16.5496i			0.707609i	0.935319	+	0.353805i	$0.115112\pi$
$547$			16.5496i			0.707609i	−0.935319	+	0.353805i	$0.884888\pi$
$548$		0			0
$549$	14.8788	−	40.7338i	0.635012	−	1.73848i
$550$		0			0
$551$	−5.32930			−0.227036
$552$		0			0
$553$	−16.7370	+	29.5979i	−0.711728	+	1.25863i
$554$		0			0
$555$	8.14374	+	33.2178i	0.345682	+	1.41002i
$556$		0			0
$557$	25.7014			1.08900			0.544501	−	0.838760i	$-0.316719\pi$
$557$	25.7014			1.08900			0.544501	+	0.838760i	$0.316719\pi$
$558$		0			0
$559$			33.3458i			1.41038i
$560$		0			0
$561$	−31.4738	−	5.56829i	−1.32883	−	0.235093i
$562$		0			0
$563$		−	1.70532i		−	0.0718707i	−0.999354	−	0.0359354i	$-0.988559\pi$
$563$		−	1.70532i		−	0.0718707i	0.999354	−	0.0359354i	$-0.0114410\pi$
$564$		0			0
$565$	28.8427	−	12.7259i	1.21342	−	0.535382i
$566$		0			0
$567$	22.3210	−	8.29289i	0.937395	−	0.348268i
$568$		0			0
$569$			36.2249i			1.51863i	0.650725	+	0.759314i	$0.274465\pi$
$569$			36.2249i			1.51863i	−0.650725	+	0.759314i	$0.725535\pi$
$570$		0			0
$571$	12.2875			0.514215			0.257107	−	0.966383i	$-0.417231\pi$
$571$	12.2875			0.514215			0.257107	+	0.966383i	$0.417231\pi$
$572$		0			0
$573$	3.27567	−	18.5152i	0.136843	−	0.773483i
$574$		0			0
$575$	5.14721	−	5.64003i	0.214653	−	0.235205i
$576$		0			0
$577$	−19.9917			−0.832265			−0.416132	−	0.909304i	$-0.636615\pi$
$577$	−19.9917			−0.832265			−0.416132	+	0.909304i	$0.636615\pi$
$578$		0			0
$579$	7.43527	−	42.0266i	0.308999	−	1.74657i
$580$		0			0
$581$	24.1825	+	13.6747i	1.00326	+	0.567321i
$582$		0			0
$583$		−	3.52311i		−	0.145912i
$584$		0			0
$585$	−25.2198	+	1.64954i	−1.04271	+	0.0682003i
$586$		0			0
$587$			12.6341i			0.521465i	0.965411	+	0.260733i	$0.0839641\pi$
$587$			12.6341i			0.521465i	−0.965411	+	0.260733i	$0.916036\pi$
$588$		0			0
$589$	1.23320			0.0508132
$590$		0			0
$591$	−5.38644	−	0.952958i	−0.221568	−	0.0391995i
$592$		0			0
$593$		−	25.1383i		−	1.03231i	−0.856496	−	0.516153i	$-0.827364\pi$
$593$		−	25.1383i		−	1.03231i	0.856496	−	0.516153i	$-0.172636\pi$
$594$		0			0
$595$	19.0133	+	1.89084i	0.779468	+	0.0775169i
$596$		0			0
$597$	1.08742	−	6.14645i	0.0445051	−	0.251558i
$598$		0			0
$599$			7.03306i			0.287363i	0.989624	+	0.143682i	$0.0458941\pi$
$599$			7.03306i			0.287363i	−0.989624	+	0.143682i	$0.954106\pi$
$600$		0			0
$601$		−	41.7407i		−	1.70264i	−0.524646	−	0.851320i	$-0.675802\pi$
$601$		−	41.7407i		−	1.70264i	0.524646	−	0.851320i	$-0.324198\pi$
$602$		0			0
$603$	−6.44910	+	17.6558i	−0.262628	+	0.718998i
$604$		0			0
$605$	−44.2851	+	19.5393i	−1.80045	+	0.794387i
$606$		0			0
$607$	43.5247			1.76661			0.883306	−	0.468797i	$-0.155312\pi$
$607$	43.5247			1.76661			0.883306	+	0.468797i	$0.155312\pi$
$608$		0			0
$609$	−23.9398	−	19.7485i	−0.970091	−	0.800251i
$610$		0			0
$611$		−	6.59530i		−	0.266817i
$612$		0			0
$613$			22.7849i			0.920272i	0.887849	+	0.460136i	$0.152199\pi$
$613$			22.7849i			0.920272i	−0.887849	+	0.460136i	$0.847801\pi$
$614$		0			0
$615$	25.0489	−	6.14104i	1.01007	−	0.247631i
$616$		0			0
$617$	−8.44461			−0.339967			−0.169984	−	0.985447i	$-0.554371\pi$
$617$	−8.44461			−0.339967			−0.169984	+	0.985447i	$0.554371\pi$
$618$		0			0
$619$		−	28.1187i		−	1.13018i	−0.825028	−	0.565092i	$-0.808841\pi$
$619$		−	28.1187i		−	1.13018i	0.825028	−	0.565092i	$-0.191159\pi$
$620$		0			0
$621$	−6.86528	−	3.97943i	−0.275494	−	0.159689i
$622$		0			0
$623$	20.4298	−	36.1283i	0.818501	−	1.44745i
$624$		0			0
$625$	−2.27952	−	24.8959i	−0.0911807	−	0.995834i
$626$		0			0
$627$	−7.66883	−	1.35675i	−0.306264	−	0.0541836i
$628$		0			0
$629$	28.5207			1.13719
$630$		0			0
$631$	0.462386			0.0184073			0.00920365	−	0.999958i	$-0.497070\pi$
$631$	0.462386			0.0184073			0.00920365	+	0.999958i	$0.497070\pi$
$632$		0			0
$633$	−0.271980	−	0.0481181i	−0.0108102	−	0.00191252i
$634$		0			0
$635$	11.5216	+	26.1131i	0.457219	+	1.03627i
$636$		0			0
$637$	13.5932	+	22.6000i	0.538584	+	0.895446i
$638$		0			0
$639$	5.24893	−	14.3700i	0.207644	−	0.568470i
$640$		0			0
$641$		−	38.3510i		−	1.51477i	−0.652967	−	0.757387i	$-0.726476\pi$
$641$		−	38.3510i		−	1.51477i	0.652967	−	0.757387i	$-0.273524\pi$
$642$		0			0
$643$	−23.0759			−0.910024			−0.455012	−	0.890485i	$-0.650365\pi$
$643$	−23.0759			−0.910024			−0.455012	+	0.890485i	$0.650365\pi$
$644$		0			0
$645$	8.16211	+	33.2928i	0.321383	+	1.31090i
$646$		0			0
$647$			28.2861i			1.11204i	0.831169	+	0.556020i	$0.187672\pi$
$647$			28.2861i			1.11204i	−0.831169	+	0.556020i	$0.812328\pi$
$648$		0			0
$649$			36.4904i			1.43237i
$650$		0			0
$651$	5.53969	+	4.56982i	0.217118	+	0.179105i
$652$		0			0
$653$	−18.6348			−0.729238			−0.364619	−	0.931157i	$-0.618801\pi$
$653$	−18.6348			−0.729238			−0.364619	+	0.931157i	$0.618801\pi$
$654$		0			0
$655$	−34.7766	+	15.3440i	−1.35883	+	0.599540i
$656$		0			0
$657$	35.4014	+	12.9311i	1.38114	+	0.504488i
$658$		0			0
$659$		−	5.86589i		−	0.228503i	−0.993452	−	0.114251i	$-0.963553\pi$
$659$		−	5.86589i		−	0.228503i	0.993452	−	0.114251i	$-0.0364469\pi$
$660$		0			0
$661$			6.13161i			0.238492i	0.992865	+	0.119246i	$0.0380477\pi$
$661$			6.13161i			0.238492i	−0.992865	+	0.119246i	$0.961952\pi$
$662$		0			0
$663$	−3.67166	+	20.7535i	−0.142595	+	0.805997i
$664$		0			0
$665$	4.63273	+	0.460717i	0.179649	+	0.0178659i
$666$		0			0
$667$			10.3421i			0.400448i
$668$		0			0
$669$	−7.49445	−	1.32590i	−0.289752	−	0.0512624i
$670$		0			0
$671$	−82.5945			−3.18853
$672$		0			0
$673$			8.78377i			0.338589i	0.985565	+	0.169295i	$0.0541490\pi$
$673$			8.78377i			0.338589i	−0.985565	+	0.169295i	$0.945851\pi$
$674$		0			0
$675$	−24.7759	+	7.82002i	−0.953626	+	0.300993i
$676$		0			0
$677$			38.7594i			1.48964i	0.667264	+	0.744822i	$0.267465\pi$
$677$			38.7594i			1.48964i	−0.667264	+	0.744822i	$0.732535\pi$
$678$		0			0
$679$	−10.0227	+	17.7243i	−0.384635	+	0.680194i
$680$		0			0
$681$	−5.78271	+	32.6858i	−0.221594	+	1.25252i
$682$		0			0
$683$	−4.80345			−0.183799			−0.0918994	−	0.995768i	$-0.529294\pi$
$683$	−4.80345			−0.183799			−0.0918994	+	0.995768i	$0.529294\pi$
$684$		0			0
$685$	−25.3566	+	11.1878i	−0.968828	+	0.427463i
$686$		0			0
$687$	0.379147	−	2.14307i	0.0144654	−	0.0817631i
$688$		0			0
$689$	−2.32310			−0.0885029
$690$		0			0
$691$			49.9522i			1.90027i	0.311836	+	0.950136i	$0.399056\pi$
$691$			49.9522i			1.90027i	−0.311836	+	0.950136i	$0.600944\pi$
$692$		0			0
$693$	−29.4216	−	34.5127i	−1.11763	−	1.31103i
$694$		0			0
$695$	−10.1515	−	23.0079i	−0.385067	−	0.872738i
$696$		0			0
$697$		−	21.5069i		−	0.814632i
$698$		0			0
$699$	−9.78571	−	1.73127i	−0.370130	−	0.0654826i
$700$		0			0
$701$			2.56829i			0.0970031i	0.998823	+	0.0485016i	$0.0154446\pi$
$701$			2.56829i			0.0970031i	−0.998823	+	0.0485016i	$0.984555\pi$
$702$		0			0
$703$	6.94927			0.262097
$704$		0			0
$705$	−1.61434	−	6.58481i	−0.0607997	−	0.247998i
$706$		0			0
$707$	13.3694	−	23.6426i	0.502807	−	0.889172i
$708$		0			0
$709$	2.70387			0.101546			0.0507730	−	0.998710i	$-0.483832\pi$
$709$	2.70387			0.101546			0.0507730	+	0.998710i	$0.483832\pi$
$710$		0			0
$711$	−36.2148	−	13.2281i	−1.35816	−	0.496094i
$712$		0			0
$713$		−	2.39317i		−	0.0896248i
$714$		0			0
$715$	19.4311	+	44.0397i	0.726681	+	1.64699i
$716$		0			0
$717$	5.48681	−	31.0133i	0.204909	−	1.15821i
$718$		0			0
$719$	21.3073			0.794627			0.397313	−	0.917683i	$-0.369943\pi$
$719$	21.3073			0.794627			0.397313	+	0.917683i	$0.369943\pi$
$720$		0			0
$721$	4.71866	−	8.34455i	0.175732	−	0.310767i
$722$		0			0
$723$	−5.46939	+	30.9148i	−0.203409	+	1.14973i
$724$		0			0
$725$	25.0112	+	22.8257i	0.928891	+	0.847726i
$726$		0			0
$727$	−36.9056			−1.36875			−0.684377	−	0.729129i	$-0.739926\pi$
$727$	−36.9056			−1.36875			−0.684377	+	0.729129i	$0.739926\pi$
$728$		0			0
$729$	13.4195	+	23.4290i	0.497019	+	0.867740i
$730$		0			0
$731$	28.5850			1.05725
$732$		0			0
$733$	31.8436			1.17617			0.588086	−	0.808798i	$-0.299882\pi$
$733$	31.8436			1.17617			0.588086	+	0.808798i	$0.299882\pi$
$734$		0			0
$735$	19.1035	+	19.2369i	0.704642	+	0.709563i
$736$		0			0
$737$	35.7999			1.31871
$738$		0			0
$739$	−2.76708			−0.101789			−0.0508944	−	0.998704i	$-0.516207\pi$
$739$	−2.76708			−0.101789			−0.0508944	+	0.998704i	$0.516207\pi$
$740$		0			0
$741$	−0.894628	+	5.05673i	−0.0328650	+	0.185764i
$742$		0			0
$743$	−37.7284			−1.38412			−0.692060	−	0.721840i	$-0.743297\pi$
$743$	−37.7284			−1.38412			−0.692060	+	0.721840i	$0.743297\pi$
$744$		0			0
$745$	2.57303	+	5.83167i	0.0942686	+	0.213656i
$746$		0			0
$747$	−10.8079	+	29.5887i	−0.395439	+	1.08260i
$748$		0			0
$749$	11.2715	−	19.9328i	0.411853	−	0.728328i
$750$		0			0
$751$	−10.7810			−0.393403			−0.196701	−	0.980463i	$-0.563023\pi$
$751$	−10.7810			−0.393403			−0.196701	+	0.980463i	$0.563023\pi$
$752$		0			0
$753$	−1.02133	−	0.180692i	−0.0372194	−	0.00658478i
$754$		0			0
$755$	33.0426	−	14.5790i	1.20254	−	0.530583i
$756$		0			0
$757$			22.0222i			0.800409i	0.916426	+	0.400205i	$0.131061\pi$
$757$			22.0222i			0.800409i	−0.916426	+	0.400205i	$0.868939\pi$
$758$		0			0
$759$	−2.63293	+	14.8822i	−0.0955694	+	0.540190i
$760$		0			0
$761$	0.133605			0.00484319			0.00242160	−	0.999997i	$-0.499229\pi$
$761$	0.133605			0.00484319			0.00242160	+	0.999997i	$0.499229\pi$
$762$		0			0
$763$	10.8367	−	19.1638i	0.392314	−	0.693775i
$764$		0			0
$765$	1.41404	+	21.6192i	0.0511247	+	0.781643i
$766$		0			0
$767$	24.0613			0.868805
$768$		0			0
$769$		−	26.2686i		−	0.947269i	−0.880722	−	0.473634i	$-0.842942\pi$
$769$		−	26.2686i		−	0.947269i	0.880722	−	0.473634i	$-0.157058\pi$
$770$		0			0
$771$	−4.64899	+	26.2777i	−0.167429	+	0.946367i
$772$		0			0
$773$			4.91789i			0.176884i	0.996081	+	0.0884422i	$0.0281889\pi$
$773$			4.91789i			0.176884i	−0.996081	+	0.0884422i	$0.971811\pi$
$774$		0			0
$775$	−5.78760	−	5.28188i	−0.207897	−	0.189731i
$776$		0			0
$777$	31.2169	+	25.7516i	1.11990	+	0.923833i
$778$		0			0
$779$		−	5.24032i		−	0.187754i
$780$		0			0
$781$	−29.1376			−1.04263
$782$		0			0
$783$	17.6471	−	30.4447i	0.630656	−	1.08800i
$784$		0			0
$785$	14.7619	−	6.51322i	0.526876	−	0.232467i
$786$		0			0
$787$	21.2704			0.758207			0.379104	−	0.925354i	$-0.376232\pi$
$787$	21.2704			0.758207			0.379104	+	0.925354i	$0.376232\pi$
$788$		0			0
$789$	−16.4808	−	2.91575i	−0.586732	−	0.103803i
$790$		0			0
$791$	18.3608	−	32.4695i	0.652835	−	1.15448i
$792$		0			0
$793$			54.4618i			1.93400i
$794$		0			0
$795$	−2.31940	+	0.568628i	−0.0822607	+	0.0201672i
$796$		0			0
$797$			27.3045i			0.967175i	0.875296	+	0.483588i	$0.160667\pi$
$797$			27.3045i			0.967175i	−0.875296	+	0.483588i	$0.839333\pi$
$798$		0			0
$799$	−5.65369			−0.200013
$800$		0			0
$801$	44.2051	+	16.1468i	1.56191	+	0.570518i
$802$		0			0
$803$		−	71.7822i		−	2.53314i
$804$		0			0
$805$	0.894073	−	8.99032i	0.0315119	−	0.316867i
$806$		0			0
$807$	22.5812	+	3.99503i	0.794896	+	0.140632i
$808$		0			0
$809$			37.9979i			1.33593i	0.744191	+	0.667967i	$0.232835\pi$
$809$			37.9979i			1.33593i	−0.744191	+	0.667967i	$0.767165\pi$
$810$		0			0
$811$			46.6063i			1.63657i	0.574813	+	0.818285i	$0.305075\pi$
$811$			46.6063i			1.63657i	−0.574813	+	0.818285i	$0.694925\pi$
$812$		0			0
$813$	4.71844	−	26.6702i	0.165483	−	0.935364i
$814$		0			0
$815$	−3.16389	−	7.17082i	−0.110826	−	0.251183i
$816$		0			0
$817$	6.96495			0.243673
$818$		0			0
$819$	−22.7573	+	19.4003i	−0.795203	+	0.677900i
$820$		0			0
$821$		−	13.3057i		−	0.464371i	−0.972672	−	0.232185i	$-0.925412\pi$
$821$		−	13.3057i		−	0.464371i	0.972672	−	0.232185i	$-0.0745876\pi$
$822$		0			0
$823$			6.85875i			0.239081i	0.992829	+	0.119541i	$0.0381421\pi$
$823$			6.85875i			0.239081i	−0.992829	+	0.119541i	$0.961858\pi$
$824$		0			0
$825$	30.1799	+	39.2135i	1.05073	+	1.36524i
$826$		0			0
$827$	−17.9117			−0.622852			−0.311426	−	0.950270i	$-0.600807\pi$
$827$	−17.9117			−0.622852			−0.311426	+	0.950270i	$0.600807\pi$
$828$		0			0
$829$			45.4320i			1.57792i	0.614445	+	0.788960i	$0.289380\pi$
$829$			45.4320i			1.57792i	−0.614445	+	0.788960i	$0.710620\pi$
$830$		0			0
$831$	1.05165	−	5.94430i	0.0364815	−	0.206205i
$832$		0			0
$833$	19.3734	−	11.6525i	0.671250	−	0.403736i
$834$		0			0
$835$	1.24893	+	2.83065i	0.0432210	+	0.0979587i
$836$		0			0
$837$	−4.08355	+	7.04491i	−0.141148	+	0.243508i
$838$		0			0
$839$	−8.18091			−0.282436			−0.141218	−	0.989979i	$-0.545102\pi$
$839$	−8.18091			−0.282436			−0.141218	+	0.989979i	$0.545102\pi$
$840$		0			0
$841$	−16.8629			−0.581478
$842$		0			0
$843$	6.46673	−	36.5521i	0.222726	−	1.25892i
$844$		0			0
$845$	2.44401	−	1.07834i	0.0840766	−	0.0370960i
$846$		0			0
$847$	−28.1912	+	49.8537i	−0.968660	+	1.71299i
$848$		0			0
$849$	13.5974	+	2.40563i	0.466663	+	0.0825611i
$850$		0			0
$851$		−	13.4858i		−	0.462288i
$852$		0			0
$853$	−1.25962			−0.0431287			−0.0215644	−	0.999767i	$-0.506865\pi$
$853$	−1.25962			−0.0431287			−0.0215644	+	0.999767i	$0.506865\pi$
$854$		0			0
$855$	0.344541	+	5.26767i	0.0117831	+	0.180151i
$856$		0			0
$857$		−	0.221305i		−	0.00755962i	−0.999993	−	0.00377981i	$-0.998797\pi$
$857$		−	0.221305i		−	0.00755962i	0.999993	−	0.00377981i	$-0.00120315\pi$
$858$		0			0
$859$		−	24.2694i		−	0.828062i	−0.910263	−	0.414031i	$-0.864120\pi$
$859$		−	24.2694i		−	0.828062i	0.910263	−	0.414031i	$-0.135880\pi$
$860$		0			0
$861$	19.4188	−	23.5401i	0.661791	−	0.802245i
$862$		0			0
$863$	0.515972			0.0175639			0.00878194	−	0.999961i	$-0.497205\pi$
$863$	0.515972			0.0175639			0.00878194	+	0.999961i	$0.497205\pi$
$864$		0			0
$865$	9.14207	+	20.7201i	0.310840	+	0.704506i
$866$		0			0
$867$	−11.2041	−	1.98221i	−0.380512	−	0.0673194i
$868$		0			0
$869$			73.4314i			2.49099i
$870$		0			0
$871$		−	23.6060i		−	0.799860i
$872$		0			0
$873$	−21.6867	−	7.92147i	−0.733982	−	0.268101i
$874$		0			0
$875$	−19.7688	−	22.0045i	−0.668306	−	0.743886i
$876$		0			0
$877$			45.4080i			1.53332i	0.642053	+	0.766660i	$0.278083\pi$
$877$			45.4080i			1.53332i	−0.642053	+	0.766660i	$0.721917\pi$
$878$		0			0
$879$	3.88430	−	21.9553i	0.131014	−	0.740535i
$880$		0			0
$881$	−30.6497			−1.03262			−0.516308	−	0.856403i	$-0.672694\pi$
$881$	−30.6497			−1.03262			−0.516308	+	0.856403i	$0.672694\pi$
$882$		0			0
$883$			20.1004i			0.676433i	0.941068	+	0.338217i	$0.109824\pi$
$883$			20.1004i			0.676433i	−0.941068	+	0.338217i	$0.890176\pi$
$884$		0			0
$885$	24.0231	−	5.88954i	0.807527	−	0.197975i
$886$		0			0
$887$		−	38.4432i		−	1.29080i	−0.763846	−	0.645398i	$-0.776691\pi$
$887$		−	38.4432i		−	1.29080i	0.763846	−	0.645398i	$-0.223309\pi$
$888$		0			0
$889$	29.3967	+	16.6232i	0.985934	+	0.557524i
$890$		0			0
$891$	33.1448	−	39.3170i	1.11039	−	1.31717i
$892$		0			0
$893$	−1.37756			−0.0460984
$894$		0			0
$895$	0.489636	+	1.10974i	0.0163667	+	0.0370945i
$896$		0			0
$897$	9.81315	+	1.73612i	0.327652	+	0.0579675i
$898$		0			0
$899$	10.6127			0.353953
$900$		0			0
$901$			1.99143i			0.0663440i
$902$		0			0
$903$	31.2874	+	25.8097i	1.04118	+	0.858893i
$904$		0			0
$905$	1.04475	+	2.36789i	0.0347288	+	0.0787113i
$906$		0			0
$907$		−	23.7121i		−	0.787346i	−0.919251	−	0.393673i	$-0.871204\pi$
$907$		−	23.7121i		−	0.787346i	0.919251	−	0.393673i	$-0.128796\pi$
$908$		0			0
$909$	28.9281	+	10.5665i	0.959485	+	0.350470i
$910$		0			0
$911$		−	35.4931i		−	1.17594i	−0.808883	−	0.587970i	$-0.799927\pi$
$911$		−	35.4931i		−	1.17594i	0.808883	−	0.587970i	$-0.200073\pi$
$912$		0			0
$913$	59.9960			1.98558
$914$		0			0
$915$	13.3307	+	54.3752i	0.440700	+	1.79759i
$916$		0			0
$917$	−22.1382	+	39.1496i	−0.731068	+	1.29283i
$918$		0			0
$919$	41.0530			1.35421			0.677106	−	0.735885i	$-0.263234\pi$
$919$	41.0530			1.35421			0.677106	+	0.735885i	$0.263234\pi$
$920$		0			0
$921$	12.2205	+	2.16203i	0.402679	+	0.0712412i
$922$		0			0
$923$			19.2130i			0.632403i
$924$		0			0
$925$	−32.6139	−	29.7642i	−1.07234	−	0.978640i
$926$		0			0
$927$	10.2101	+	3.72942i	0.335342	+	0.122490i
$928$		0			0
$929$	−17.1710			−0.563362			−0.281681	−	0.959508i	$-0.590892\pi$
$929$	−17.1710			−0.563362			−0.281681	+	0.959508i	$0.590892\pi$
$930$		0			0
$931$	4.72048	−	2.83923i	0.154708	−	0.0930519i
$932$		0			0
$933$	−58.7882	−	10.4007i	−1.92464	−	0.340503i
$934$		0			0
$935$	37.7522	−	16.6569i	1.23463	−	0.544739i
$936$		0			0
$937$	−21.1075			−0.689550			−0.344775	−	0.938685i	$-0.612045\pi$
$937$	−21.1075			−0.689550			−0.344775	+	0.938685i	$0.612045\pi$
$938$		0			0
$939$	−48.7087	−	8.61745i	−1.58955	−	0.281220i
$940$		0			0
$941$	−29.7526			−0.969908			−0.484954	−	0.874540i	$-0.661164\pi$
$941$	−29.7526			−0.969908			−0.484954	+	0.874540i	$0.661164\pi$
$942$		0			0
$943$	−10.1694			−0.331162
$944$		0			0
$945$	−17.9724	+	24.9397i	−0.584644	+	0.811290i
$946$		0			0
$947$	−15.4259			−0.501274			−0.250637	−	0.968081i	$-0.580640\pi$
$947$	−15.4259			−0.501274			−0.250637	+	0.968081i	$0.580640\pi$
$948$		0			0
$949$	−47.3323			−1.53647
$950$		0			0
$951$	−35.7842	−	6.33088i	−1.16038	−	0.205293i
$952$		0			0
$953$	−43.6225			−1.41307			−0.706535	−	0.707678i	$-0.749743\pi$
$953$	−43.6225			−1.41307			−0.706535	+	0.707678i	$0.749743\pi$
$954$		0			0
$955$	9.79880	+	22.2086i	0.317082	+	0.718653i
$956$		0			0
$957$	−65.9964	−	11.6760i	−2.13336	−	0.377430i
$958$		0			0
$959$	−16.1416	+	28.5451i	−0.521240	+	0.921770i
$960$		0			0
$961$	28.5442			0.920781
$962$		0			0
$963$	24.3889	+	8.90852i	0.785922	+	0.287073i
$964$		0			0
$965$	22.2418	+	50.4100i	0.715988	+	1.62276i
$966$		0			0
$967$		−	33.6963i		−	1.08360i	−0.840508	−	0.541799i	$-0.817743\pi$
$967$		−	33.6963i		−	1.08360i	0.840508	−	0.541799i	$-0.182257\pi$
$968$		0			0
$969$	4.33478	+	0.766902i	0.139253	+	0.0246364i
$970$		0			0
$971$	−8.74825			−0.280745			−0.140372	−	0.990099i	$-0.544830\pi$
$971$	−8.74825			−0.280745			−0.140372	+	0.990099i	$0.544830\pi$
$972$		0			0
$973$	−25.9010	−	14.6464i	−0.830348	−	0.469543i
$974$		0			0
$975$	25.8569	−	19.9002i	0.828084	−	0.637317i
$976$		0			0
$977$	−44.2690			−1.41629			−0.708145	−	0.706067i	$-0.750468\pi$
$977$	−44.2690			−1.41629			−0.708145	+	0.706067i	$0.750468\pi$
$978$		0			0
$979$		−	89.6331i		−	2.86469i
$980$		0			0
$981$	23.4480	+	8.56483i	0.748637	+	0.273454i
$982$		0			0
$983$		−	37.9368i		−	1.21000i	−0.796227	−	0.604998i	$-0.793174\pi$
$983$		−	37.9368i		−	1.21000i	0.796227	−	0.604998i	$-0.206826\pi$
$984$		0			0
$985$	6.46092	−	2.85067i	0.205862	−	0.0908298i
$986$		0			0
$987$	−6.18817	−	5.10477i	−0.196972	−	0.162487i
$988$		0			0
$989$		−	13.5163i		−	0.429792i
$990$		0			0
$991$	−45.7568			−1.45351			−0.726757	−	0.686895i	$-0.758973\pi$
$991$	−45.7568			−1.45351			−0.726757	+	0.686895i	$0.758973\pi$
$992$		0			0
$993$	−18.6596	−	3.30122i	−0.592145	−	0.104761i
$994$		0			0
$995$	3.25289	+	7.37254i	0.103124	+	0.233725i
$996$		0			0
$997$	35.7188			1.13122			0.565612	−	0.824671i	$-0.308640\pi$
$997$	35.7188			1.13122			0.565612	+	0.824671i	$0.308640\pi$
$998$		0			0
$999$	−23.0114	+	39.6991i	−0.728048	+	1.25602i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.k.a.209.1	✓	24
3.2	odd	2		840.2.k.b.209.2	yes	24
4.3	odd	2		1680.2.k.i.209.24		24
5.4	even	2		840.2.k.b.209.24	yes	24
7.6	odd	2	inner	840.2.k.a.209.24	yes	24
12.11	even	2		1680.2.k.h.209.23		24
15.14	odd	2	inner	840.2.k.a.209.23	yes	24
20.19	odd	2		1680.2.k.h.209.1		24
21.20	even	2		840.2.k.b.209.23	yes	24
28.27	even	2		1680.2.k.i.209.1		24
35.34	odd	2		840.2.k.b.209.1	yes	24
60.59	even	2		1680.2.k.i.209.2		24
84.83	odd	2		1680.2.k.h.209.2		24
105.104	even	2	inner	840.2.k.a.209.2	yes	24
140.139	even	2		1680.2.k.h.209.24		24
420.419	odd	2		1680.2.k.i.209.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.1	✓	24	1.1	even	1	trivial
840.2.k.a.209.2	yes	24	105.104	even	2	inner
840.2.k.a.209.23	yes	24	15.14	odd	2	inner
840.2.k.a.209.24	yes	24	7.6	odd	2	inner
840.2.k.b.209.1	yes	24	35.34	odd	2
840.2.k.b.209.2	yes	24	3.2	odd	2
840.2.k.b.209.23	yes	24	21.20	even	2
840.2.k.b.209.24	yes	24	5.4	even	2
1680.2.k.h.209.1		24	20.19	odd	2
1680.2.k.h.209.2		24	84.83	odd	2
1680.2.k.h.209.23		24	12.11	even	2
1680.2.k.h.209.24		24	140.139	even	2
1680.2.k.i.209.1		24	28.27	even	2
1680.2.k.i.209.2		24	60.59	even	2
1680.2.k.i.209.23		24	420.419	odd	2
1680.2.k.i.209.24		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)

\(f(q)\)	\(=\)	\(q+(-1.70556 - 0.301745i) q^{3} +(2.04579 - 0.902636i) q^{5} +(1.30232 - 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} +O(q^{10})\)	\(q+(-1.70556 - 0.301745i) q^{3} +(2.04579 - 0.902636i) q^{5} +(1.30232 - 2.30304i) q^{7} +(2.81790 + 1.02929i) q^{9} -5.71375i q^{11} -3.76758 q^{13} +(-3.76159 + 0.922198i) q^{15} +3.22968i q^{17} +0.786936i q^{19} +(-2.91611 + 3.53501i) q^{21} +1.52714 q^{23} +(3.37050 - 3.69320i) q^{25} +(-4.49553 - 2.60581i) q^{27} +6.77221i q^{29} -1.56709i q^{31} +(-1.72410 + 9.74517i) q^{33} +(0.585457 - 5.88704i) q^{35} -8.83080i q^{37} +(6.42585 + 1.13685i) q^{39} -6.65914 q^{41} -8.85072i q^{43} +(6.69390 - 0.437826i) q^{45} +1.75054i q^{47} +(-3.60795 - 5.99856i) q^{49} +(0.974541 - 5.50843i) q^{51} +0.616602 q^{53} +(-5.15744 - 11.6891i) q^{55} +(0.237454 - 1.34217i) q^{57} -6.38642 q^{59} -14.4554i q^{61} +(6.04029 - 5.14926i) q^{63} +(-7.70767 + 3.40075i) q^{65} +6.26557i q^{67} +(-2.60463 - 0.460806i) q^{69} -5.09956i q^{71} +12.5631 q^{73} +(-6.86300 + 5.28197i) q^{75} +(-13.1590 - 7.44111i) q^{77} -12.8517 q^{79} +(6.88112 + 5.80088i) q^{81} +10.5003i q^{83} +(2.91523 + 6.60724i) q^{85} +(2.04348 - 11.5504i) q^{87} +15.6873 q^{89} +(-4.90658 + 8.67687i) q^{91} +(-0.472863 + 2.67278i) q^{93} +(0.710317 + 1.60990i) q^{95} -7.69604 q^{97} +(5.88112 - 16.1008i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9}+O(q^{10}) \) 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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