LMFDB - Embedded newform 1690.2.d.l.1351.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1690,2,Mod(1351,1690)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1690, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1690.1351"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1690 = 2 \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1690.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-4,-12,0,0,0,0,32,-12,0,4,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4947179416$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 34x^{10} + 453x^{8} + 2990x^{6} + 10094x^{4} + 15876x^{2} + 8281 $ x^12 + 34x^10 + 453x^8 + 2990x^6 + 10094x^4 + 15876*x^2 + 8281
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1351.11
Root			$-2.23727i$ of defining polynomial
Character	$\chi$	$=$	1690.1351
Dual form			1690.2.d.l.1351.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} +2.23727 q^{3} -1.00000 q^{4} +1.00000i q^{5} +2.23727i q^{6} -1.43294i q^{7} -1.00000i q^{8} +2.00539 q^{9} -1.00000 q^{10} -0.315763i q^{11} -2.23727 q^{12} +1.43294 q^{14} +2.23727i q^{15} +1.00000 q^{16} +1.69981 q^{17} +2.00539i q^{18} +5.83768i q^{19} -1.00000i q^{20} -3.20587i q^{21} +0.315763 q^{22} +9.32839 q^{23} -2.23727i q^{24} -1.00000 q^{25} -2.22522 q^{27} +1.43294i q^{28} +7.09544 q^{29} -2.23727 q^{30} +4.69438i q^{31} +1.00000i q^{32} -0.706448i q^{33} +1.69981i q^{34} +1.43294 q^{35} -2.00539 q^{36} +11.6581i q^{37} -5.83768 q^{38} +1.00000 q^{40} +2.79223i q^{41} +3.20587 q^{42} +4.77478 q^{43} +0.315763i q^{44} +2.00539i q^{45} +9.32839i q^{46} -10.4362i q^{47} +2.23727 q^{48} +4.94669 q^{49} -1.00000i q^{50} +3.80293 q^{51} -1.48663 q^{53} -2.22522i q^{54} +0.315763 q^{55} -1.43294 q^{56} +13.0605i q^{57} +7.09544i q^{58} +3.93789i q^{59} -2.23727i q^{60} +0.971828 q^{61} -4.69438 q^{62} -2.87359i q^{63} -1.00000 q^{64} +0.706448 q^{66} -12.7810i q^{67} -1.69981 q^{68} +20.8701 q^{69} +1.43294i q^{70} +4.17629i q^{71} -2.00539i q^{72} -8.83875i q^{73} -11.6581 q^{74} -2.23727 q^{75} -5.83768i q^{76} -0.452469 q^{77} -6.80085 q^{79} +1.00000i q^{80} -10.9946 q^{81} -2.79223 q^{82} +11.1336i q^{83} +3.20587i q^{84} +1.69981i q^{85} +4.77478i q^{86} +15.8744 q^{87} -0.315763 q^{88} -1.48615i q^{89} -2.00539 q^{90} -9.32839 q^{92} +10.5026i q^{93} +10.4362 q^{94} -5.83768 q^{95} +2.23727i q^{96} -11.9669i q^{97} +4.94669i q^{98} -0.633227i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} - 12 q^{4} + 32 q^{9} - 12 q^{10} + 4 q^{12} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 30 q^{22} + 6 q^{23} - 12 q^{25} - 40 q^{27} + 14 q^{29} + 4 q^{30} + 6 q^{35} - 32 q^{36} - 2 q^{38} + 12 q^{40}+ \cdots - 2 q^{95}+O(q^{100}) $ 12 * q - 4 * q^3 - 12 * q^4 + 32 * q^9 - 12 * q^10 + 4 * q^12 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 30 * q^22 + 6 * q^23 - 12 * q^25 - 40 * q^27 + 14 * q^29 + 4 * q^30 + 6 * q^35 - 32 * q^36 - 2 * q^38 + 12 * q^40 - 4 * q^42 + 44 * q^43 - 4 * q^48 - 62 * q^49 + 44 * q^51 - 32 * q^53 + 30 * q^55 - 6 * q^56 + 66 * q^61 - 12 * q^64 + 8 * q^66 - 6 * q^68 + 12 * q^69 + 12 * q^74 + 4 * q^75 + 68 * q^77 + 40 * q^79 - 4 * q^81 - 4 * q^82 - 78 * q^87 - 30 * q^88 - 32 * q^90 - 6 * q^92 + 14 * q^94 - 2 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1690\mathbb{Z}\right)^\times$.

$n$	$171$	$677$
$\chi(n)$	$-1$	$1$

Coefficient data

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

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

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

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	2.23727	1.29169	0.645845	−	0.763469i	$-0.276505\pi$
$3$	2.23727	1.29169	0.645845	+	0.763469i	$0.276505\pi$
$4$	−1.00000	−0.500000
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	2.23727i	0.913363i
$7$	− 1.43294i	− 0.541599i	−0.962636	−	0.270800i	$-0.912712\pi$
$7$	− 1.43294i	− 0.541599i	0.962636	−	0.270800i	$-0.0872881\pi$
$8$	− 1.00000i	− 0.353553i
$9$	2.00539	0.668462
$10$	−1.00000	−0.316228
$11$	− 0.315763i	− 0.0952062i	−0.998866	−	0.0476031i	$-0.984842\pi$
$11$	− 0.315763i	− 0.0952062i	0.998866	−	0.0476031i	$-0.0151583\pi$
$12$	−2.23727	−0.645845
$13$	0	0
$13$	0	0
$14$	1.43294	0.382969
$15$	2.23727i	0.577661i
$16$	1.00000	0.250000
$17$	1.69981	0.412264	0.206132	−	0.978524i	$-0.433912\pi$
$17$	1.69981	0.412264	0.206132	+	0.978524i	$0.433912\pi$
$18$	2.00539i	0.472674i
$19$	5.83768i	1.33926i	0.742696	+	0.669628i	$0.233547\pi$
$19$	5.83768i	1.33926i	−0.742696	+	0.669628i	$0.766453\pi$
$20$	− 1.00000i	− 0.223607i
$21$	− 3.20587i	− 0.699578i
$22$	0.315763	0.0673209
$23$	9.32839	1.94510	0.972552	−	0.232686i	$-0.0747516\pi$
$23$	9.32839	1.94510	0.972552	+	0.232686i	$0.0747516\pi$
$24$	− 2.23727i	− 0.456681i
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−2.22522	−0.428244
$28$	1.43294i	0.270800i
$29$	7.09544	1.31759	0.658795	−	0.752323i	$-0.271067\pi$
$29$	7.09544	1.31759	0.658795	+	0.752323i	$0.271067\pi$
$30$	−2.23727	−0.408468
$31$	4.69438i	0.843135i	0.906797	+	0.421568i	$0.138520\pi$
$31$	4.69438i	0.843135i	−0.906797	+	0.421568i	$0.861480\pi$
$32$	1.00000i	0.176777i
$33$	− 0.706448i	− 0.122977i
$34$	1.69981i	0.291514i
$35$	1.43294	0.242211
$36$	−2.00539	−0.334231
$37$	11.6581i	1.91658i	0.285802	+	0.958289i	$0.407740\pi$
$37$	11.6581i	1.91658i	−0.285802	+	0.958289i	$0.592260\pi$
$38$	−5.83768	−0.946997
$39$	0	0
$40$	1.00000	0.158114
$41$	2.79223i	0.436073i	0.975941	+	0.218037i	$0.0699652\pi$
$41$	2.79223i	0.436073i	−0.975941	+	0.218037i	$0.930035\pi$
$42$	3.20587	0.494677
$43$	4.77478	0.728147	0.364074	−	0.931370i	$-0.381386\pi$
$43$	4.77478	0.728147	0.364074	+	0.931370i	$0.381386\pi$
$44$	0.315763i	0.0476031i
$45$	2.00539i	0.298945i
$46$	9.32839i	1.37540i
$47$	− 10.4362i	− 1.52227i	−0.648592	−	0.761136i	$-0.724642\pi$
$47$	− 10.4362i	− 1.52227i	0.648592	−	0.761136i	$-0.275358\pi$
$48$	2.23727	0.322922
$49$	4.94669	0.706670
$50$	− 1.00000i	− 0.141421i
$51$	3.80293	0.532517
$52$	0	0
$53$	−1.48663	−0.204204	−0.102102	−	0.994774i	$-0.532557\pi$
$53$	−1.48663	−0.204204	−0.102102	+	0.994774i	$0.532557\pi$
$54$	− 2.22522i	− 0.302814i
$55$	0.315763	0.0425775
$56$	−1.43294	−0.191484
$57$	13.0605i	1.72990i
$58$	7.09544i	0.931677i
$59$	3.93789i	0.512670i	0.966588	+	0.256335i	$0.0825150\pi$
$59$	3.93789i	0.512670i	−0.966588	+	0.256335i	$0.917485\pi$
$60$	− 2.23727i	− 0.288831i
$61$	0.971828	0.124430	0.0622149	−	0.998063i	$-0.480184\pi$
$61$	0.971828	0.124430	0.0622149	+	0.998063i	$0.480184\pi$
$62$	−4.69438	−0.596186
$63$	− 2.87359i	− 0.362039i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0.706448	0.0869578
$67$	− 12.7810i	− 1.56144i	−0.624879	−	0.780722i	$-0.714852\pi$
$67$	− 12.7810i	− 1.56144i	0.624879	−	0.780722i	$-0.285148\pi$
$68$	−1.69981	−0.206132
$69$	20.8701	2.51247
$70$	1.43294i	0.171269i
$71$	4.17629i	0.495635i	0.968807	+	0.247817i	$0.0797133\pi$
$71$	4.17629i	0.495635i	−0.968807	+	0.247817i	$0.920287\pi$
$72$	− 2.00539i	− 0.236337i
$73$	− 8.83875i	− 1.03450i	−0.855835	−	0.517249i	$-0.826956\pi$
$73$	− 8.83875i	− 1.03450i	0.855835	−	0.517249i	$-0.173044\pi$
$74$	−11.6581	−1.35522
$75$	−2.23727	−0.258338
$76$	− 5.83768i	− 0.669628i
$77$	−0.452469	−0.0515636
$78$	0	0
$79$	−6.80085	−0.765156	−0.382578	−	0.923923i	$-0.624964\pi$
$79$	−6.80085	−0.765156	−0.382578	+	0.923923i	$0.624964\pi$
$80$	1.00000i	0.111803i
$81$	−10.9946	−1.22162
$82$	−2.79223	−0.308350
$83$	11.1336i	1.22207i	0.791602	+	0.611037i	$0.209247\pi$
$83$	11.1336i	1.22207i	−0.791602	+	0.611037i	$0.790753\pi$
$84$	3.20587i	0.349789i
$85$	1.69981i	0.184370i
$86$	4.77478i	0.514878i
$87$	15.8744	1.70192
$88$	−0.315763	−0.0336605
$89$	− 1.48615i	− 0.157532i	−0.996893	−	0.0787659i	$-0.974902\pi$
$89$	− 1.48615i	− 0.157532i	0.996893	−	0.0787659i	$-0.0250979\pi$
$90$	−2.00539	−0.211386
$91$	0	0
$92$	−9.32839	−0.972552
$93$	10.5026i	1.08907i
$94$	10.4362	1.07641
$95$	−5.83768	−0.598934
$96$	2.23727i	0.228341i
$97$	− 11.9669i	− 1.21506i	−0.794298	−	0.607528i	$-0.792161\pi$
$97$	− 11.9669i	− 1.21506i	0.794298	−	0.607528i	$-0.207839\pi$
$98$	4.94669i	0.499691i
$99$	− 0.633227i	− 0.0636418i
$100$	1.00000	0.100000
$101$	−16.6774	−1.65947	−0.829734	−	0.558159i	$-0.811508\pi$
$101$	−16.6774	−1.65947	−0.829734	+	0.558159i	$0.811508\pi$
$102$	3.80293i	0.376546i
$103$	7.91612	0.779999	0.389999	−	0.920815i	$-0.372475\pi$
$103$	7.91612	0.779999	0.389999	+	0.920815i	$0.372475\pi$
$104$	0	0
$105$	3.20587	0.312861
$106$	− 1.48663i	− 0.144394i
$107$	−6.46441	−0.624939	−0.312469	−	0.949928i	$-0.601156\pi$
$107$	−6.46441	−0.624939	−0.312469	+	0.949928i	$0.601156\pi$
$108$	2.22522	0.214122
$109$	− 16.6406i	− 1.59388i	−0.604059	−	0.796939i	$-0.706451\pi$
$109$	− 16.6406i	− 1.59388i	0.604059	−	0.796939i	$-0.293549\pi$
$110$	0.315763i	0.0301068i
$111$	26.0823i	2.47562i
$112$	− 1.43294i	− 0.135400i
$113$	−14.2361	−1.33922	−0.669612	−	0.742711i	$-0.733540\pi$
$113$	−14.2361	−1.33922	−0.669612	+	0.742711i	$0.733540\pi$
$114$	−13.0605	−1.22323
$115$	9.32839i	0.869877i
$116$	−7.09544	−0.658795
$117$	0	0
$118$	−3.93789	−0.362512
$119$	− 2.43572i	− 0.223282i
$120$	2.23727	0.204234
$121$	10.9003	0.990936
$122$	0.971828i	0.0879852i
$123$	6.24698i	0.563271i
$124$	− 4.69438i	− 0.421568i
$125$	− 1.00000i	− 0.0894427i
$126$	2.87359	0.256000
$127$	−16.6911	−1.48109	−0.740546	−	0.672006i	$-0.765433\pi$
$127$	−16.6911	−1.48109	−0.740546	+	0.672006i	$0.765433\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	10.6825	0.940540
$130$	0	0
$131$	5.22386	0.456411	0.228205	−	0.973613i	$-0.426714\pi$
$131$	5.22386	0.456411	0.228205	+	0.973613i	$0.426714\pi$
$132$	0.706448i	0.0614884i
$133$	8.36503	0.725340
$134$	12.7810	1.10411
$135$	− 2.22522i	− 0.191516i
$136$	− 1.69981i	− 0.145757i
$137$	− 10.8426i	− 0.926345i	−0.886268	−	0.463173i	$-0.846711\pi$
$137$	− 10.8426i	− 0.926345i	0.886268	−	0.463173i	$-0.153289\pi$
$138$	20.8701i	1.77658i
$139$	12.7506	1.08149	0.540745	−	0.841187i	$-0.318142\pi$
$139$	12.7506	1.08149	0.540745	+	0.841187i	$0.318142\pi$
$140$	−1.43294	−0.121105
$141$	− 23.3486i	− 1.96630i
$142$	−4.17629	−0.350467
$143$	0	0
$144$	2.00539	0.167116
$145$	7.09544i	0.589244i
$146$	8.83875	0.731500
$147$	11.0671	0.912799
$148$	− 11.6581i	− 0.958289i
$149$	− 2.68691i	− 0.220120i	−0.993925	−	0.110060i	$-0.964896\pi$
$149$	− 2.68691i	− 0.220120i	0.993925	−	0.110060i	$-0.0351044\pi$
$150$	− 2.23727i	− 0.182673i
$151$	− 15.3443i	− 1.24870i	−0.781144	−	0.624351i	$-0.785363\pi$
$151$	− 15.3443i	− 1.24870i	0.781144	−	0.624351i	$-0.214637\pi$
$152$	5.83768	0.473499
$153$	3.40877	0.275583
$154$	− 0.452469i	− 0.0364610i
$155$	−4.69438	−0.377061
$156$	0	0
$157$	−5.59044	−0.446166	−0.223083	−	0.974799i	$-0.571612\pi$
$157$	−5.59044	−0.446166	−0.223083	+	0.974799i	$0.571612\pi$
$158$	− 6.80085i	− 0.541047i
$159$	−3.32599	−0.263768
$160$	−1.00000	−0.0790569
$161$	− 13.3670i	− 1.05347i
$162$	− 10.9946i	− 0.863816i
$163$	− 4.33627i	− 0.339643i	−0.985475	−	0.169821i	$-0.945681\pi$
$163$	− 4.33627i	− 0.339643i	0.985475	−	0.169821i	$-0.0543191\pi$
$164$	− 2.79223i	− 0.218037i
$165$	0.706448	0.0549969
$166$	−11.1336	−0.864137
$167$	21.8290i	1.68918i	0.535413	+	0.844590i	$0.320156\pi$
$167$	21.8290i	1.68918i	−0.535413	+	0.844590i	$0.679844\pi$
$168$	−3.20587	−0.247338
$169$	0	0
$170$	−1.69981	−0.130369
$171$	11.7068i	0.895243i
$172$	−4.77478	−0.364074
$173$	4.20426	0.319644	0.159822	−	0.987146i	$-0.448908\pi$
$173$	4.20426	0.319644	0.159822	+	0.987146i	$0.448908\pi$
$174$	15.8744i	1.20344i
$175$	1.43294i	0.108320i
$176$	− 0.315763i	− 0.0238015i
$177$	8.81014i	0.662210i
$178$	1.48615	0.111392
$179$	23.5902	1.76322	0.881608	−	0.471983i	$-0.156462\pi$
$179$	23.5902	1.76322	0.881608	+	0.471983i	$0.156462\pi$
$180$	− 2.00539i	− 0.149473i
$181$	−16.4103	−1.21976	−0.609882	−	0.792492i	$-0.708783\pi$
$181$	−16.4103	−1.21976	−0.609882	+	0.792492i	$0.708783\pi$
$182$	0	0
$183$	2.17424	0.160725
$184$	− 9.32839i	− 0.687698i
$185$	−11.6581	−0.857120
$186$	−10.5026	−0.770088
$187$	− 0.536736i	− 0.0392500i
$188$	10.4362i	0.761136i
$189$	3.18860i	0.231936i
$190$	− 5.83768i	− 0.423510i
$191$	−21.8818	−1.58331	−0.791656	−	0.610967i	$-0.790781\pi$
$191$	−21.8818	−1.58331	−0.791656	+	0.610967i	$0.790781\pi$
$192$	−2.23727	−0.161461
$193$	0.440205i	0.0316866i	0.999874	+	0.0158433i	$0.00504330\pi$
$193$	0.440205i	0.0316866i	−0.999874	+	0.0158433i	$0.994957\pi$
$194$	11.9669	0.859175
$195$	0	0
$196$	−4.94669	−0.353335
$197$	13.7762i	0.981516i	0.871296	+	0.490758i	$0.163280\pi$
$197$	13.7762i	0.981516i	−0.871296	+	0.490758i	$0.836720\pi$
$198$	0.633227	0.0450015
$199$	−2.53270	−0.179539	−0.0897693	−	0.995963i	$-0.528613\pi$
$199$	−2.53270	−0.179539	−0.0897693	+	0.995963i	$0.528613\pi$
$200$	1.00000i	0.0707107i
$201$	− 28.5945i	− 2.01690i
$202$	− 16.6774i	− 1.17342i
$203$	− 10.1673i	− 0.713606i
$204$	−3.80293	−0.266258
$205$	−2.79223	−0.195018
$206$	7.91612i	0.551542i
$207$	18.7070	1.30023
$208$	0	0
$209$	1.84333	0.127505
$210$	3.20587i	0.221226i
$211$	6.69030	0.460579	0.230290	−	0.973122i	$-0.426033\pi$
$211$	6.69030	0.460579	0.230290	+	0.973122i	$0.426033\pi$
$212$	1.48663	0.102102
$213$	9.34350i	0.640206i
$214$	− 6.46441i	− 0.441898i
$215$	4.77478i	0.325637i
$216$	2.22522i	0.151407i
$217$	6.72675	0.456641
$218$	16.6406	1.12704
$219$	− 19.7747i	− 1.33625i
$220$	−0.315763	−0.0212888
$221$	0	0
$222$	−26.0823	−1.75053
$223$	− 1.36952i	− 0.0917097i	−0.998948	−	0.0458548i	$-0.985399\pi$
$223$	− 1.36952i	− 0.0917097i	0.998948	−	0.0458548i	$-0.0146012\pi$
$224$	1.43294	0.0957421
$225$	−2.00539	−0.133692
$226$	− 14.2361i	− 0.946974i
$227$	1.46657i	0.0973398i	0.998815	+	0.0486699i	$0.0154982\pi$
$227$	1.46657i	0.0973398i	−0.998815	+	0.0486699i	$0.984502\pi$
$228$	− 13.0605i	− 0.864952i
$229$	7.36084i	0.486418i	0.969974	+	0.243209i	$0.0782001\pi$
$229$	7.36084i	0.486418i	−0.969974	+	0.243209i	$0.921800\pi$
$230$	−9.32839	−0.615096
$231$	−1.01230	−0.0666042
$232$	− 7.09544i	− 0.465838i
$233$	−15.5806	−1.02072	−0.510359	−	0.859961i	$-0.670488\pi$
$233$	−15.5806	−1.02072	−0.510359	+	0.859961i	$0.670488\pi$
$234$	0	0
$235$	10.4362	0.680781
$236$	− 3.93789i	− 0.256335i
$237$	−15.2154	−0.988344
$238$	2.43572	0.157884
$239$	− 9.78398i	− 0.632873i	−0.948614	−	0.316436i	$-0.897514\pi$
$239$	− 9.78398i	− 0.632873i	0.948614	−	0.316436i	$-0.102486\pi$
$240$	2.23727i	0.144415i
$241$	11.7487i	0.756802i	0.925642	+	0.378401i	$0.123526\pi$
$241$	11.7487i	0.756802i	−0.925642	+	0.378401i	$0.876474\pi$
$242$	10.9003i	0.700697i
$243$	−17.9222	−1.14971
$244$	−0.971828	−0.0622149
$245$	4.94669i	0.316033i
$246$	−6.24698	−0.398293
$247$	0	0
$248$	4.69438	0.298093
$249$	24.9089i	1.57854i
$250$	1.00000	0.0632456
$251$	−29.0462	−1.83338	−0.916689	−	0.399602i	$-0.869148\pi$
$251$	−29.0462	−1.83338	−0.916689	+	0.399602i	$0.869148\pi$
$252$	2.87359i	0.181019i
$253$	− 2.94556i	− 0.185186i
$254$	− 16.6911i	− 1.04729i
$255$	3.80293i	0.238149i
$256$	1.00000	0.0625000
$257$	15.6549	0.976524	0.488262	−	0.872697i	$-0.337631\pi$
$257$	15.6549	0.976524	0.488262	+	0.872697i	$0.337631\pi$
$258$	10.6825i	0.665062i
$259$	16.7053	1.03802
$260$	0	0
$261$	14.2291	0.880759
$262$	5.22386i	0.322731i
$263$	−11.7545	−0.724813	−0.362407	−	0.932020i	$-0.618045\pi$
$263$	−11.7545	−0.724813	−0.362407	+	0.932020i	$0.618045\pi$
$264$	−0.706448	−0.0434789
$265$	− 1.48663i	− 0.0913227i
$266$	8.36503i	0.512893i
$267$	− 3.32492i	− 0.203482i
$268$	12.7810i	0.780722i
$269$	−0.786410	−0.0479483	−0.0239741	−	0.999713i	$-0.507632\pi$
$269$	−0.786410	−0.0479483	−0.0239741	+	0.999713i	$0.507632\pi$
$270$	2.22522	0.135423
$271$	− 1.63501i	− 0.0993200i	−0.998766	−	0.0496600i	$-0.984186\pi$
$271$	− 1.63501i	− 0.0993200i	0.998766	−	0.0496600i	$-0.0158138\pi$
$272$	1.69981	0.103066
$273$	0	0
$274$	10.8426	0.655025
$275$	0.315763i	0.0190412i
$276$	−20.8701	−1.25624
$277$	−0.567538	−0.0341000	−0.0170500	−	0.999855i	$-0.505427\pi$
$277$	−0.567538	−0.0341000	−0.0170500	+	0.999855i	$0.505427\pi$
$278$	12.7506i	0.764729i
$279$	9.41404i	0.563604i
$280$	− 1.43294i	− 0.0856344i
$281$	− 11.9390i	− 0.712221i	−0.934444	−	0.356111i	$-0.884103\pi$
$281$	− 11.9390i	− 0.712221i	0.934444	−	0.356111i	$-0.115897\pi$
$282$	23.3486	1.39039
$283$	14.0478	0.835054	0.417527	−	0.908664i	$-0.362897\pi$
$283$	14.0478	0.835054	0.417527	+	0.908664i	$0.362897\pi$
$284$	− 4.17629i	− 0.247817i
$285$	−13.0605	−0.773636
$286$	0	0
$287$	4.00109	0.236177
$288$	2.00539i	0.118169i
$289$	−14.1107	−0.830039
$290$	−7.09544	−0.416658
$291$	− 26.7733i	− 1.56948i
$292$	8.83875i	0.517249i
$293$	− 28.7352i	− 1.67873i	−0.543570	−	0.839364i	$-0.682928\pi$
$293$	− 28.7352i	− 1.67873i	0.543570	−	0.839364i	$-0.317072\pi$
$294$	11.0671i	0.645446i
$295$	−3.93789	−0.229273
$296$	11.6581	0.677612
$297$	0.702643i	0.0407715i
$298$	2.68691	0.155649
$299$	0	0
$300$	2.23727	0.129169
$301$	− 6.84196i	− 0.394364i
$302$	15.3443	0.882966
$303$	−37.3120	−2.14352
$304$	5.83768i	0.334814i
$305$	0.971828i	0.0556467i
$306$	3.40877i	0.194866i
$307$	8.72162i	0.497769i	0.968533	+	0.248885i	$0.0800640\pi$
$307$	8.72162i	0.497769i	−0.968533	+	0.248885i	$0.919936\pi$
$308$	0.452469	0.0257818
$309$	17.7105	1.00752
$310$	− 4.69438i	− 0.266623i
$311$	−23.6123	−1.33893	−0.669466	−	0.742843i	$-0.733477\pi$
$311$	−23.6123	−1.33893	−0.669466	+	0.742843i	$0.733477\pi$
$312$	0	0
$313$	22.3185	1.26152	0.630759	−	0.775979i	$-0.282744\pi$
$313$	22.3185	1.26152	0.630759	+	0.775979i	$0.282744\pi$
$314$	− 5.59044i	− 0.315487i
$315$	2.87359	0.161909
$316$	6.80085	0.382578
$317$	22.3655i	1.25617i	0.778144	+	0.628086i	$0.216161\pi$
$317$	22.3655i	1.25617i	−0.778144	+	0.628086i	$0.783839\pi$
$318$	− 3.32599i	− 0.186512i
$319$	− 2.24048i	− 0.125443i
$320$	− 1.00000i	− 0.0559017i
$321$	−14.4627	−0.807227
$322$	13.3670	0.744913
$323$	9.92293i	0.552127i
$324$	10.9946	0.610810
$325$	0	0
$326$	4.33627	0.240164
$327$	− 37.2295i	− 2.05880i
$328$	2.79223	0.154175
$329$	−14.9544	−0.824461
$330$	0.706448i	0.0388887i
$331$	15.6598i	0.860740i	0.902653	+	0.430370i	$0.141617\pi$
$331$	15.6598i	0.860740i	−0.902653	+	0.430370i	$0.858383\pi$
$332$	− 11.1336i	− 0.611037i
$333$	23.3790i	1.28116i
$334$	−21.8290	−1.19443
$335$	12.7810	0.698299
$336$	− 3.20587i	− 0.174895i
$337$	28.8471	1.57140	0.785701	−	0.618606i	$-0.212302\pi$
$337$	28.8471	1.57140	0.785701	+	0.618606i	$0.212302\pi$
$338$	0	0
$339$	−31.8501	−1.72986
$340$	− 1.69981i	− 0.0921849i
$341$	1.48231	0.0802717
$342$	−11.7068	−0.633032
$343$	− 17.1189i	− 0.924331i
$344$	− 4.77478i	− 0.257439i
$345$	20.8701i	1.12361i
$346$	4.20426i	0.226023i
$347$	−6.44079	−0.345760	−0.172880	−	0.984943i	$-0.555307\pi$
$347$	−6.44079	−0.345760	−0.172880	+	0.984943i	$0.555307\pi$
$348$	−15.8744	−0.850958
$349$	21.5301i	1.15248i	0.817281	+	0.576239i	$0.195480\pi$
$349$	21.5301i	1.15248i	−0.817281	+	0.576239i	$0.804520\pi$
$350$	−1.43294	−0.0765937
$351$	0	0
$352$	0.315763	0.0168302
$353$	12.5441i	0.667657i	0.942634	+	0.333828i	$0.108341\pi$
$353$	12.5441i	0.667657i	−0.942634	+	0.333828i	$0.891659\pi$
$354$	−8.81014	−0.468253
$355$	−4.17629	−0.221655
$356$	1.48615i	0.0787659i
$357$	− 5.44936i	− 0.288411i
$358$	23.5902i	1.24678i
$359$	9.39258i	0.495721i	0.968796	+	0.247861i	$0.0797275\pi$
$359$	9.39258i	0.495721i	−0.968796	+	0.247861i	$0.920272\pi$
$360$	2.00539	0.105693
$361$	−15.0785	−0.793608
$362$	− 16.4103i	− 0.862504i
$363$	24.3869	1.27998
$364$	0	0
$365$	8.83875	0.462641
$366$	2.17424i	0.113650i
$367$	9.41908	0.491672	0.245836	−	0.969311i	$-0.420938\pi$
$367$	9.41908	0.491672	0.245836	+	0.969311i	$0.420938\pi$
$368$	9.32839	0.486276
$369$	5.59950i	0.291498i
$370$	− 11.6581i	− 0.606075i
$371$	2.13024i	0.110597i
$372$	− 10.5026i	− 0.544534i
$373$	11.4780	0.594310	0.297155	−	0.954829i	$-0.403962\pi$
$373$	11.4780	0.594310	0.297155	+	0.954829i	$0.403962\pi$
$374$	0.536736	0.0277540
$375$	− 2.23727i	− 0.115532i
$376$	−10.4362	−0.538204
$377$	0	0
$378$	−3.18860	−0.164004
$379$	− 17.6283i	− 0.905505i	−0.891636	−	0.452752i	$-0.850442\pi$
$379$	− 17.6283i	− 0.905505i	0.891636	−	0.452752i	$-0.149558\pi$
$380$	5.83768	0.299467
$381$	−37.3424	−1.91311
$382$	− 21.8818i	− 1.11957i
$383$	− 4.16819i	− 0.212985i	−0.994314	−	0.106492i	$-0.966038\pi$
$383$	− 4.16819i	− 0.212985i	0.994314	−	0.106492i	$-0.0339619\pi$
$384$	− 2.23727i	− 0.114170i
$385$	− 0.452469i	− 0.0230599i
$386$	−0.440205	−0.0224058
$387$	9.57528	0.486739
$388$	11.9669i	0.607528i
$389$	−35.7341	−1.81179	−0.905895	−	0.423503i	$-0.860800\pi$
$389$	−35.7341	−1.81179	−0.905895	+	0.423503i	$0.860800\pi$
$390$	0	0
$391$	15.8565	0.801895
$392$	− 4.94669i	− 0.249846i
$393$	11.6872	0.589541
$394$	−13.7762	−0.694037
$395$	− 6.80085i	− 0.342188i
$396$	0.633227i	0.0318209i
$397$	− 23.4645i	− 1.17765i	−0.808262	−	0.588824i	$-0.799591\pi$
$397$	− 23.4645i	− 1.17765i	0.808262	−	0.588824i	$-0.200409\pi$
$398$	− 2.53270i	− 0.126953i
$399$	18.7149	0.936915
$400$	−1.00000	−0.0500000
$401$	− 0.137896i	− 0.00688618i	−0.999994	−	0.00344309i	$-0.998904\pi$
$401$	− 0.137896i	− 0.00688618i	0.999994	−	0.00344309i	$-0.00109597\pi$
$402$	28.5945	1.42616
$403$	0	0
$404$	16.6774	0.829734
$405$	− 10.9946i	− 0.546325i
$406$	10.1673	0.504595
$407$	3.68119	0.182470
$408$	− 3.80293i	− 0.188273i
$409$	− 24.2501i	− 1.19909i	−0.800340	−	0.599546i	$-0.795348\pi$
$409$	− 24.2501i	− 1.19909i	0.800340	−	0.599546i	$-0.204652\pi$
$410$	− 2.79223i	− 0.137898i
$411$	− 24.2578i	− 1.19655i
$412$	−7.91612	−0.389999
$413$	5.64275	0.277662
$414$	18.7070i	0.919400i
$415$	−11.1336	−0.546528
$416$	0	0
$417$	28.5265	1.39695
$418$	1.84333i	0.0901600i
$419$	8.19191	0.400201	0.200100	−	0.979775i	$-0.435873\pi$
$419$	8.19191	0.400201	0.200100	+	0.979775i	$0.435873\pi$
$420$	−3.20587	−0.156430
$421$	− 24.7799i	− 1.20770i	−0.797099	−	0.603848i	$-0.793633\pi$
$421$	− 24.7799i	− 1.20770i	0.797099	−	0.603848i	$-0.206367\pi$
$422$	6.69030i	0.325679i
$423$	− 20.9286i	− 1.01758i
$424$	1.48663i	0.0721969i
$425$	−1.69981	−0.0824527
$426$	−9.34350	−0.452694
$427$	− 1.39257i	− 0.0673911i
$428$	6.46441	0.312469
$429$	0	0
$430$	−4.77478	−0.230260
$431$	− 1.94731i	− 0.0937987i	−0.998900	−	0.0468994i	$-0.985066\pi$
$431$	− 1.94731i	− 0.0937987i	0.998900	−	0.0468994i	$-0.0149340\pi$
$432$	−2.22522	−0.107061
$433$	23.7805	1.14282	0.571410	−	0.820665i	$-0.306397\pi$
$433$	23.7805	1.14282	0.571410	+	0.820665i	$0.306397\pi$
$434$	6.72675i	0.322894i
$435$	15.8744i	0.761120i
$436$	16.6406i	0.796939i
$437$	54.4562i	2.60499i
$438$	19.7747	0.944871
$439$	−33.4007	−1.59413	−0.797063	−	0.603896i	$-0.793614\pi$
$439$	−33.4007	−1.59413	−0.797063	+	0.603896i	$0.793614\pi$
$440$	− 0.315763i	− 0.0150534i
$441$	9.92003	0.472382
$442$	0	0
$443$	15.5768	0.740075	0.370038	−	0.929017i	$-0.379345\pi$
$443$	15.5768	0.740075	0.370038	+	0.929017i	$0.379345\pi$
$444$	− 26.0823i	− 1.23781i
$445$	1.48615	0.0704503
$446$	1.36952	0.0648485
$447$	− 6.01135i	− 0.284327i
$448$	1.43294i	0.0676999i
$449$	− 31.2239i	− 1.47355i	−0.676140	−	0.736773i	$-0.736348\pi$
$449$	− 31.2239i	− 1.47355i	0.676140	−	0.736773i	$-0.263652\pi$
$450$	− 2.00539i	− 0.0945349i
$451$	0.881684	0.0415169
$452$	14.2361	0.669612
$453$	− 34.3294i	− 1.61294i
$454$	−1.46657	−0.0688296
$455$	0	0
$456$	13.0605	0.611613
$457$	− 22.4950i	− 1.05227i	−0.850401	−	0.526135i	$-0.823641\pi$
$457$	− 22.4950i	− 1.05227i	0.850401	−	0.526135i	$-0.176359\pi$
$458$	−7.36084	−0.343949
$459$	−3.78244	−0.176549
$460$	− 9.32839i	− 0.434938i
$461$	− 18.4663i	− 0.860059i	−0.902815	−	0.430030i	$-0.858503\pi$
$461$	− 18.4663i	− 0.860059i	0.902815	−	0.430030i	$-0.141497\pi$
$462$	− 1.01230i	− 0.0470963i
$463$	− 20.3900i	− 0.947602i	−0.880632	−	0.473801i	$-0.842882\pi$
$463$	− 20.3900i	− 0.947602i	0.880632	−	0.473801i	$-0.157118\pi$
$464$	7.09544	0.329397
$465$	−10.5026	−0.487046
$466$	− 15.5806i	− 0.721757i
$467$	28.0498	1.29799	0.648996	−	0.760792i	$-0.275189\pi$
$467$	28.0498	1.29799	0.648996	+	0.760792i	$0.275189\pi$
$468$	0	0
$469$	−18.3143	−0.845676
$470$	10.4362i	0.481385i
$471$	−12.5073	−0.576308
$472$	3.93789	0.181256
$473$	− 1.50770i	− 0.0693241i
$474$	− 15.2154i	− 0.698865i
$475$	− 5.83768i	− 0.267851i
$476$	2.43572i	0.111641i
$477$	−2.98126	−0.136503
$478$	9.78398	0.447509
$479$	8.09122i	0.369697i	0.982767	+	0.184849i	$0.0591795\pi$
$479$	8.09122i	0.369697i	−0.982767	+	0.184849i	$0.940820\pi$
$480$	−2.23727	−0.102117
$481$	0	0
$482$	−11.7487	−0.535140
$483$	− 29.9056i	− 1.36075i
$484$	−10.9003	−0.495468
$485$	11.9669	0.543390
$486$	− 17.9222i	− 0.812968i
$487$	− 14.3897i	− 0.652057i	−0.945360	−	0.326029i	$-0.894289\pi$
$487$	− 14.3897i	− 0.652057i	0.945360	−	0.326029i	$-0.105711\pi$
$488$	− 0.971828i	− 0.0439926i
$489$	− 9.70141i	− 0.438713i
$490$	−4.94669	−0.223469
$491$	10.2104	0.460789	0.230395	−	0.973097i	$-0.425998\pi$
$491$	10.2104	0.460789	0.230395	+	0.973097i	$0.425998\pi$
$492$	− 6.24698i	− 0.281636i
$493$	12.0609	0.543194
$494$	0	0
$495$	0.633227	0.0284615
$496$	4.69438i	0.210784i
$497$	5.98436	0.268435
$498$	−24.9089	−1.11620
$499$	3.13991i	0.140562i	0.997527	+	0.0702809i	$0.0223896\pi$
$499$	3.13991i	0.140562i	−0.997527	+	0.0702809i	$0.977610\pi$
$500$	1.00000i	0.0447214i
$501$	48.8375i	2.18190i
$502$	− 29.0462i	− 1.29639i
$503$	24.1794	1.07811	0.539054	−	0.842271i	$-0.318782\pi$
$503$	24.1794	1.07811	0.539054	+	0.842271i	$0.318782\pi$
$504$	−2.87359	−0.128000
$505$	− 16.6774i	− 0.742137i
$506$	2.94556	0.130946
$507$	0	0
$508$	16.6911	0.740546
$509$	− 15.4846i	− 0.686343i	−0.939273	−	0.343171i	$-0.888499\pi$
$509$	− 15.4846i	− 0.686343i	0.939273	−	0.343171i	$-0.111501\pi$
$510$	−3.80293	−0.168397
$511$	−12.6654	−0.560283
$512$	1.00000i	0.0441942i
$513$	− 12.9901i	− 0.573528i
$514$	15.6549i	0.690506i
$515$	7.91612i	0.348826i
$516$	−10.6825	−0.470270
$517$	−3.29536	−0.144930
$518$	16.7053i	0.733989i
$519$	9.40608	0.412881
$520$	0	0
$521$	8.76055	0.383807	0.191903	−	0.981414i	$-0.438534\pi$
$521$	8.76055	0.383807	0.191903	+	0.981414i	$0.438534\pi$
$522$	14.2291i	0.622791i
$523$	−3.75051	−0.163999	−0.0819993	−	0.996632i	$-0.526131\pi$
$523$	−3.75051	−0.163999	−0.0819993	+	0.996632i	$0.526131\pi$
$524$	−5.22386	−0.228205
$525$	3.20587i	0.139916i
$526$	− 11.7545i	− 0.512520i
$527$	7.97953i	0.347594i
$528$	− 0.706448i	− 0.0307442i
$529$	64.0188	2.78343
$530$	1.48663	0.0645749
$531$	7.89700i	0.342701i
$532$	−8.36503	−0.362670
$533$	0	0
$534$	3.32492	0.143884
$535$	− 6.46441i	− 0.279481i
$536$	−12.7810	−0.552053
$537$	52.7777	2.27753
$538$	− 0.786410i	− 0.0339046i
$539$	− 1.56198i	− 0.0672794i
$540$	2.22522i	0.0957582i
$541$	34.8297i	1.49745i	0.662883	+	0.748723i	$0.269333\pi$
$541$	34.8297i	1.49745i	−0.662883	+	0.748723i	$0.730667\pi$
$542$	1.63501	0.0702299
$543$	−36.7142	−1.57556
$544$	1.69981i	0.0728786i
$545$	16.6406	0.712804
$546$	0	0
$547$	−41.0027	−1.75315	−0.876575	−	0.481265i	$-0.840177\pi$
$547$	−41.0027	−1.75315	−0.876575	+	0.481265i	$0.840177\pi$
$548$	10.8426i	0.463173i
$549$	1.94889	0.0831766
$550$	−0.315763	−0.0134642
$551$	41.4209i	1.76459i
$552$	− 20.8701i	− 0.888292i
$553$	9.74519i	0.414408i
$554$	− 0.567538i	− 0.0241124i
$555$	−26.0823	−1.10713
$556$	−12.7506	−0.540745
$557$	− 12.9939i	− 0.550569i	−0.961363	−	0.275285i	$-0.911228\pi$
$557$	− 12.9939i	− 0.550569i	0.961363	−	0.275285i	$-0.0887721\pi$
$558$	−9.41404	−0.398528
$559$	0	0
$560$	1.43294	0.0605526
$561$	− 1.20083i	− 0.0506989i
$562$	11.9390	0.503617
$563$	−18.8597	−0.794840	−0.397420	−	0.917637i	$-0.630094\pi$
$563$	−18.8597	−0.794840	−0.397420	+	0.917637i	$0.630094\pi$
$564$	23.3486i	0.983152i
$565$	− 14.2361i	− 0.598919i
$566$	14.0478i	0.590472i
$567$	15.7545i	0.661629i
$568$	4.17629	0.175233
$569$	36.2095	1.51798	0.758991	−	0.651102i	$-0.225693\pi$
$569$	36.2095	1.51798	0.758991	+	0.651102i	$0.225693\pi$
$570$	− 13.0605i	− 0.547044i
$571$	−13.5330	−0.566340	−0.283170	−	0.959070i	$-0.591386\pi$
$571$	−13.5330	−0.566340	−0.283170	+	0.959070i	$0.591386\pi$
$572$	0	0
$573$	−48.9556	−2.04515
$574$	4.00109i	0.167002i
$575$	−9.32839	−0.389021
$576$	−2.00539	−0.0835578
$577$	− 9.49114i	− 0.395121i	−0.980291	−	0.197561i	$-0.936698\pi$
$577$	− 9.49114i	− 0.395121i	0.980291	−	0.197561i	$-0.0633020\pi$
$578$	− 14.1107i	− 0.586926i
$579$	0.984858i	0.0409293i
$580$	− 7.09544i	− 0.294622i
$581$	15.9538	0.661874
$582$	26.7733	1.10979
$583$	0.469422i	0.0194415i
$584$	−8.83875	−0.365750
$585$	0	0
$586$	28.7352	1.18704
$587$	− 10.2761i	− 0.424138i	−0.977255	−	0.212069i	$-0.931980\pi$
$587$	− 10.2761i	− 0.424138i	0.977255	−	0.212069i	$-0.0680202\pi$
$588$	−11.0671	−0.456399
$589$	−27.4043	−1.12917
$590$	− 3.93789i	− 0.162120i
$591$	30.8212i	1.26781i
$592$	11.6581i	0.479144i
$593$	− 36.5714i	− 1.50181i	−0.660411	−	0.750904i	$-0.729618\pi$
$593$	− 36.5714i	− 1.50181i	0.660411	−	0.750904i	$-0.270382\pi$
$594$	−0.702643	−0.0288298
$595$	2.43572	0.0998546
$596$	2.68691i	0.110060i
$597$	−5.66635	−0.231908
$598$	0	0
$599$	−42.2976	−1.72823	−0.864116	−	0.503292i	$-0.832122\pi$
$599$	−42.2976	−1.72823	−0.864116	+	0.503292i	$0.832122\pi$
$600$	2.23727i	0.0913363i
$601$	17.6798	0.721173	0.360587	−	0.932726i	$-0.382576\pi$
$601$	17.6798	0.721173	0.360587	+	0.932726i	$0.382576\pi$
$602$	6.84196	0.278857
$603$	− 25.6308i	− 1.04377i
$604$	15.3443i	0.624351i
$605$	10.9003i	0.443160i
$606$	− 37.3120i	− 1.51570i
$607$	−40.2686	−1.63445	−0.817227	−	0.576316i	$-0.804490\pi$
$607$	−40.2686	−1.63445	−0.817227	+	0.576316i	$0.804490\pi$
$608$	−5.83768	−0.236749
$609$	− 22.7471i	− 0.921757i
$610$	−0.971828	−0.0393482
$611$	0	0
$612$	−3.40877	−0.137791
$613$	4.11330i	0.166135i	0.996544	+	0.0830673i	$0.0264716\pi$
$613$	4.11330i	0.166135i	−0.996544	+	0.0830673i	$0.973528\pi$
$614$	−8.72162	−0.351976
$615$	−6.24698	−0.251903
$616$	0.452469i	0.0182305i
$617$	28.2379i	1.13682i	0.822747	+	0.568408i	$0.192440\pi$
$617$	28.2379i	1.13682i	−0.822747	+	0.568408i	$0.807560\pi$
$618$	17.7105i	0.712422i
$619$	7.31241i	0.293911i	0.989143	+	0.146955i	$0.0469473\pi$
$619$	7.31241i	0.293911i	−0.989143	+	0.146955i	$0.953053\pi$
$620$	4.69438	0.188531
$621$	−20.7577	−0.832978
$622$	− 23.6123i	− 0.946768i
$623$	−2.12956	−0.0853191
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.3185i	0.892027i
$627$	4.12402	0.164698
$628$	5.59044	0.223083
$629$	19.8165i	0.790135i
$630$	2.87359i	0.114487i
$631$	− 36.0224i	− 1.43403i	−0.697059	−	0.717014i	$-0.745508\pi$
$631$	− 36.0224i	− 1.43403i	0.697059	−	0.717014i	$-0.254492\pi$
$632$	6.80085i	0.270523i
$633$	14.9680	0.594926
$634$	−22.3655	−0.888248
$635$	− 16.6911i	− 0.662364i
$636$	3.32599	0.131884
$637$	0	0
$638$	2.24048	0.0887014
$639$	8.37508i	0.331313i
$640$	1.00000	0.0395285
$641$	−11.3889	−0.449835	−0.224917	−	0.974378i	$-0.572211\pi$
$641$	−11.3889	−0.449835	−0.224917	+	0.974378i	$0.572211\pi$
$642$	− 14.4627i	− 0.570795i
$643$	− 14.2162i	− 0.560630i	−0.959908	−	0.280315i	$-0.909561\pi$
$643$	− 14.2162i	− 0.560630i	0.959908	−	0.280315i	$-0.0904390\pi$
$644$	13.3670i	0.526733i
$645$	10.6825i	0.420622i
$646$	−9.92293	−0.390412
$647$	2.24397	0.0882198	0.0441099	−	0.999027i	$-0.485955\pi$
$647$	2.24397	0.0882198	0.0441099	+	0.999027i	$0.485955\pi$
$648$	10.9946i	0.431908i
$649$	1.24344	0.0488093
$650$	0	0
$651$	15.0496	0.589839
$652$	4.33627i	0.169821i
$653$	21.9585	0.859301	0.429651	−	0.902995i	$-0.358637\pi$
$653$	21.9585	0.859301	0.429651	+	0.902995i	$0.358637\pi$
$654$	37.2295	1.45579
$655$	5.22386i	0.204113i
$656$	2.79223i	0.109018i
$657$	− 17.7251i	− 0.691523i
$658$	− 14.9544i	− 0.582982i
$659$	−29.0740	−1.13256	−0.566281	−	0.824212i	$-0.691618\pi$
$659$	−29.0740	−1.13256	−0.566281	+	0.824212i	$0.691618\pi$
$660$	−0.706448	−0.0274985
$661$	− 35.2273i	− 1.37018i	−0.728457	−	0.685091i	$-0.759762\pi$
$661$	− 35.2273i	− 1.37018i	0.728457	−	0.685091i	$-0.240238\pi$
$662$	−15.6598	−0.608635
$663$	0	0
$664$	11.1336	0.432068
$665$	8.36503i	0.324382i
$666$	−23.3790	−0.905917
$667$	66.1890	2.56285
$668$	− 21.8290i	− 0.844590i
$669$	− 3.06398i	− 0.118460i
$670$	12.7810i	0.493772i
$671$	− 0.306867i	− 0.0118465i
$672$	3.20587	0.123669
$673$	−13.7113	−0.528531	−0.264266	−	0.964450i	$-0.585130\pi$
$673$	−13.7113	−0.528531	−0.264266	+	0.964450i	$0.585130\pi$
$674$	28.8471i	1.11115i
$675$	2.22522	0.0856487
$676$	0	0
$677$	31.9818	1.22916	0.614581	−	0.788854i	$-0.289325\pi$
$677$	31.9818	1.22916	0.614581	+	0.788854i	$0.289325\pi$
$678$	− 31.8501i	− 1.22320i
$679$	−17.1478	−0.658074
$680$	1.69981	0.0651846
$681$	3.28112i	0.125733i
$682$	1.48231i	0.0567606i
$683$	3.53220i	0.135156i	0.997714	+	0.0675779i	$0.0215271\pi$
$683$	3.53220i	0.135156i	−0.997714	+	0.0675779i	$0.978473\pi$
$684$	− 11.7068i	− 0.447621i
$685$	10.8426	0.414274
$686$	17.1189	0.653601
$687$	16.4682i	0.628301i
$688$	4.77478	0.182037
$689$	0	0
$690$	−20.8701	−0.794513
$691$	39.4068i	1.49911i	0.661944	+	0.749553i	$0.269732\pi$
$691$	39.4068i	1.49911i	−0.661944	+	0.749553i	$0.730268\pi$
$692$	−4.20426	−0.159822
$693$	−0.907375	−0.0344683
$694$	− 6.44079i	− 0.244489i
$695$	12.7506i	0.483657i
$696$	− 15.8744i	− 0.601719i
$697$	4.74625i	0.179777i
$698$	−21.5301	−0.814925
$699$	−34.8580	−1.31845
$700$	− 1.43294i	− 0.0541599i
$701$	−26.5304	−1.00204	−0.501020	−	0.865436i	$-0.667042\pi$
$701$	−26.5304	−1.00204	−0.501020	+	0.865436i	$0.667042\pi$
$702$	0	0
$703$	−68.0562	−2.56679
$704$	0.315763i	0.0119008i
$705$	23.3486	0.879358
$706$	−12.5441	−0.472105
$707$	23.8977i	0.898767i
$708$	− 8.81014i	− 0.331105i
$709$	− 42.0988i	− 1.58106i	−0.612426	−	0.790528i	$-0.709806\pi$
$709$	− 42.0988i	− 1.58106i	0.612426	−	0.790528i	$-0.290194\pi$
$710$	− 4.17629i	− 0.156733i
$711$	−13.6383	−0.511478
$712$	−1.48615	−0.0556959
$713$	43.7910i	1.63998i
$714$	5.44936	0.203937
$715$	0	0
$716$	−23.5902	−0.881608
$717$	− 21.8894i	− 0.817475i
$718$	−9.39258	−0.350528
$719$	−21.0467	−0.784911	−0.392455	−	0.919771i	$-0.628374\pi$
$719$	−21.0467	−0.784911	−0.392455	+	0.919771i	$0.628374\pi$
$720$	2.00539i	0.0747364i
$721$	− 11.3433i	− 0.422447i
$722$	− 15.0785i	− 0.561165i
$723$	26.2851i	0.977553i
$724$	16.4103	0.609882
$725$	−7.09544	−0.263518
$726$	24.3869i	0.905084i
$727$	7.85688	0.291396	0.145698	−	0.989329i	$-0.453457\pi$
$727$	7.85688	0.291396	0.145698	+	0.989329i	$0.453457\pi$
$728$	0	0
$729$	−7.11313	−0.263449
$730$	8.83875i	0.327137i
$731$	8.11620	0.300189
$732$	−2.17424	−0.0803624
$733$	40.1151i	1.48168i	0.671679	+	0.740842i	$0.265573\pi$
$733$	40.1151i	1.48168i	−0.671679	+	0.740842i	$0.734427\pi$
$734$	9.41908i	0.347665i
$735$	11.0671i	0.408216i
$736$	9.32839i	0.343849i
$737$	−4.03576	−0.148659
$738$	−5.59950	−0.206121
$739$	34.1597i	1.25658i	0.777977	+	0.628292i	$0.216246\pi$
$739$	34.1597i	1.25658i	−0.777977	+	0.628292i	$0.783754\pi$
$740$	11.6581	0.428560
$741$	0	0
$742$	−2.13024	−0.0782036
$743$	33.6854i	1.23580i	0.786258	+	0.617898i	$0.212016\pi$
$743$	33.6854i	1.23580i	−0.786258	+	0.617898i	$0.787984\pi$
$744$	10.5026	0.385044
$745$	2.68691	0.0984408
$746$	11.4780i	0.420241i
$747$	22.3272i	0.816910i
$748$	0.536736i	0.0196250i
$749$	9.26310i	0.338466i
$750$	2.23727	0.0816936
$751$	−9.13590	−0.333374	−0.166687	−	0.986010i	$-0.553307\pi$
$751$	−9.13590	−0.333374	−0.166687	+	0.986010i	$0.553307\pi$
$752$	− 10.4362i	− 0.380568i
$753$	−64.9842	−2.36816
$754$	0	0
$755$	15.3443	0.558437
$756$	− 3.18860i	− 0.115968i
$757$	47.3897	1.72241	0.861203	−	0.508261i	$-0.169711\pi$
$757$	47.3897	1.72241	0.861203	+	0.508261i	$0.169711\pi$
$758$	17.6283	0.640289
$759$	− 6.59002i	− 0.239203i
$760$	5.83768i	0.211755i
$761$	2.34689i	0.0850747i	0.999095	+	0.0425374i	$0.0135442\pi$
$761$	2.34689i	0.0850747i	−0.999095	+	0.0425374i	$0.986456\pi$
$762$	− 37.3424i	− 1.35277i
$763$	−23.8449	−0.863244
$764$	21.8818	0.791656
$765$	3.40877i	0.123244i
$766$	4.16819	0.150603
$767$	0	0
$768$	2.23727	0.0807306
$769$	40.3538i	1.45519i	0.686004	+	0.727597i	$0.259363\pi$
$769$	40.3538i	1.45519i	−0.686004	+	0.727597i	$0.740637\pi$
$770$	0.452469	0.0163058
$771$	35.0242	1.26137
$772$	− 0.440205i	− 0.0158433i
$773$	36.3971i	1.30911i	0.756013	+	0.654556i	$0.227144\pi$
$773$	36.3971i	1.30911i	−0.756013	+	0.654556i	$0.772856\pi$
$774$	9.57528i	0.344176i
$775$	− 4.69438i	− 0.168627i
$776$	−11.9669	−0.429587
$777$	37.3743	1.34080
$778$	− 35.7341i	− 1.28113i
$779$	−16.3002	−0.584014
$780$	0	0
$781$	1.31872	0.0471875
$782$	15.8565i	0.567026i
$783$	−15.7889	−0.564249
$784$	4.94669	0.176668
$785$	− 5.59044i	− 0.199531i
$786$	11.6872i	0.416869i
$787$	18.5638i	0.661729i	0.943678	+	0.330864i	$0.107340\pi$
$787$	18.5638i	0.661729i	−0.943678	+	0.330864i	$0.892660\pi$
$788$	− 13.7762i	− 0.490758i
$789$	−26.2980	−0.936234
$790$	6.80085	0.241963
$791$	20.3995i	0.725323i
$792$	−0.633227	−0.0225008
$793$	0	0
$794$	23.4645	0.832722
$795$	− 3.32599i	− 0.117961i
$796$	2.53270	0.0897693
$797$	−15.9864	−0.566268	−0.283134	−	0.959080i	$-0.591374\pi$
$797$	−15.9864	−0.566268	−0.283134	+	0.959080i	$0.591374\pi$
$798$	18.7149i	0.662499i
$799$	− 17.7395i	− 0.627577i
$800$	− 1.00000i	− 0.0353553i
$801$	− 2.98031i	− 0.105304i
$802$	0.137896	0.00486927
$803$	−2.79095	−0.0984906
$804$	28.5945i	1.00845i
$805$	13.3670	0.471125
$806$	0	0
$807$	−1.75941	−0.0619343
$808$	16.6774i	0.586711i
$809$	6.97062	0.245074	0.122537	−	0.992464i	$-0.460897\pi$
$809$	6.97062	0.245074	0.122537	+	0.992464i	$0.460897\pi$
$810$	10.9946	0.386310
$811$	24.0216i	0.843511i	0.906710	+	0.421755i	$0.138586\pi$
$811$	24.0216i	0.843511i	−0.906710	+	0.421755i	$0.861414\pi$
$812$	10.1673i	0.356803i
$813$	− 3.65797i	− 0.128291i
$814$	3.68119i	0.129026i
$815$	4.33627	0.151893
$816$	3.80293	0.133129
$817$	27.8737i	0.975176i
$818$	24.2501	0.847886
$819$	0	0
$820$	2.79223	0.0975089
$821$	27.7112i	0.967126i	0.875309	+	0.483563i	$0.160658\pi$
$821$	27.7112i	0.967126i	−0.875309	+	0.483563i	$0.839342\pi$
$822$	24.2578	0.846089
$823$	24.4911	0.853706	0.426853	−	0.904321i	$-0.359622\pi$
$823$	24.4911	0.853706	0.426853	+	0.904321i	$0.359622\pi$
$824$	− 7.91612i	− 0.275771i
$825$	0.706448i	0.0245954i
$826$	5.64275i	0.196336i
$827$	18.0841i	0.628846i	0.949283	+	0.314423i	$0.101811\pi$
$827$	18.0841i	0.628846i	−0.949283	+	0.314423i	$0.898189\pi$
$828$	−18.7070	−0.650114
$829$	−11.4496	−0.397661	−0.198831	−	0.980034i	$-0.563714\pi$
$829$	−11.4496	−0.397661	−0.198831	+	0.980034i	$0.563714\pi$
$830$	− 11.1336i	− 0.386454i
$831$	−1.26974	−0.0440467
$832$	0	0
$833$	8.40842	0.291334
$834$	28.5265i	0.987792i
$835$	−21.8290	−0.755425
$836$	−1.84333	−0.0637527
$837$	− 10.4460i	− 0.361067i
$838$	8.19191i	0.282985i
$839$	− 28.4327i	− 0.981605i	−0.871271	−	0.490803i	$-0.836704\pi$
$839$	− 28.4327i	− 0.981605i	0.871271	−	0.490803i	$-0.163296\pi$
$840$	− 3.20587i	− 0.110613i
$841$	21.3452	0.736042
$842$	24.7799	0.853971
$843$	− 26.7108i	− 0.919969i
$844$	−6.69030	−0.230290
$845$	0	0
$846$	20.9286	0.719539
$847$	− 15.6194i	− 0.536690i
$848$	−1.48663	−0.0510510
$849$	31.4287	1.07863
$850$	− 1.69981i	− 0.0583029i
$851$	108.751i	3.72794i
$852$	− 9.34350i	− 0.320103i
$853$	7.19717i	0.246426i	0.992380	+	0.123213i	$0.0393199\pi$
$853$	7.19717i	0.246426i	−0.992380	+	0.123213i	$0.960680\pi$
$854$	1.39257	0.0476527
$855$	−11.7068	−0.400365
$856$	6.46441i	0.220949i
$857$	−30.2786	−1.03430	−0.517148	−	0.855896i	$-0.673006\pi$
$857$	−30.2786	−1.03430	−0.517148	+	0.855896i	$0.673006\pi$
$858$	0	0
$859$	−11.0133	−0.375769	−0.187884	−	0.982191i	$-0.560163\pi$
$859$	−11.0133	−0.375769	−0.187884	+	0.982191i	$0.560163\pi$
$860$	− 4.77478i	− 0.162819i
$861$	8.95153	0.305067
$862$	1.94731	0.0663257
$863$	− 22.5488i	− 0.767571i	−0.923422	−	0.383786i	$-0.874620\pi$
$863$	− 22.5488i	− 0.767571i	0.923422	−	0.383786i	$-0.125380\pi$
$864$	− 2.22522i	− 0.0757035i
$865$	4.20426i	0.142949i
$866$	23.7805i	0.808095i
$867$	−31.5694	−1.07215
$868$	−6.72675	−0.228321
$869$	2.14746i	0.0728476i
$870$	−15.8744	−0.538193
$871$	0	0
$872$	−16.6406	−0.563521
$873$	− 23.9983i	− 0.812220i
$874$	−54.4562	−1.84201
$875$	−1.43294	−0.0484421
$876$	19.7747i	0.668125i
$877$	55.2018i	1.86403i	0.362418	+	0.932016i	$0.381951\pi$
$877$	55.2018i	1.86403i	−0.362418	+	0.932016i	$0.618049\pi$
$878$	− 33.4007i	− 1.12722i
$879$	− 64.2885i	− 2.16840i
$880$	0.315763	0.0106444
$881$	1.80062	0.0606644	0.0303322	−	0.999540i	$-0.490343\pi$
$881$	1.80062	0.0606644	0.0303322	+	0.999540i	$0.490343\pi$
$882$	9.92003i	0.334025i
$883$	28.0786	0.944919	0.472459	−	0.881352i	$-0.343366\pi$
$883$	28.0786	0.944919	0.472459	+	0.881352i	$0.343366\pi$
$884$	0	0
$885$	−8.81014	−0.296150
$886$	15.5768i	0.523312i
$887$	−3.74577	−0.125771	−0.0628854	−	0.998021i	$-0.520030\pi$
$887$	−3.74577	−0.125771	−0.0628854	+	0.998021i	$0.520030\pi$
$888$	26.0823	0.875265
$889$	23.9172i	0.802158i
$890$	1.48615i	0.0498159i
$891$	3.47169i	0.116306i
$892$	1.36952i	0.0458548i
$893$	60.9231	2.03871
$894$	6.01135	0.201050
$895$	23.5902i	0.788534i
$896$	−1.43294	−0.0478711
$897$	0	0
$898$	31.2239	1.04195
$899$	33.3087i	1.11091i
$900$	2.00539	0.0668462
$901$	−2.52698	−0.0841858
$902$	0.881684i	0.0293569i
$903$	− 15.3073i	− 0.509396i
$904$	14.2361i	0.473487i
$905$	− 16.4103i	− 0.545495i
$906$	34.3294	1.14052
$907$	−26.8418	−0.891269	−0.445634	−	0.895215i	$-0.647022\pi$
$907$	−26.8418	−0.891269	−0.445634	+	0.895215i	$0.647022\pi$
$908$	− 1.46657i	− 0.0486699i
$909$	−33.4447	−1.10929
$910$	0	0
$911$	−22.6354	−0.749945	−0.374972	−	0.927036i	$-0.622348\pi$
$911$	−22.6354	−0.749945	−0.374972	+	0.927036i	$0.622348\pi$
$912$	13.0605i	0.432476i
$913$	3.51559	0.116349
$914$	22.4950	0.744067
$915$	2.17424i	0.0718783i
$916$	− 7.36084i	− 0.243209i
$917$	− 7.48546i	− 0.247192i
$918$	− 3.78244i	− 0.124839i
$919$	5.61518	0.185228	0.0926138	−	0.995702i	$-0.470478\pi$
$919$	5.61518	0.185228	0.0926138	+	0.995702i	$0.470478\pi$
$920$	9.32839	0.307548
$921$	19.5126i	0.642963i
$922$	18.4663	0.608154
$923$	0	0
$924$	1.01230	0.0333021
$925$	− 11.6581i	− 0.383315i
$926$	20.3900	0.670056
$927$	15.8749	0.521400
$928$	7.09544i	0.232919i
$929$	10.9655i	0.359766i	0.983688	+	0.179883i	$0.0575719\pi$
$929$	10.9655i	0.359766i	−0.983688	+	0.179883i	$0.942428\pi$
$930$	− 10.5026i	− 0.344394i
$931$	28.8772i	0.946413i
$932$	15.5806	0.510359
$933$	−52.8272	−1.72949
$934$	28.0498i	0.917819i
$935$	0.536736	0.0175532
$936$	0	0
$937$	53.8795	1.76017	0.880083	−	0.474820i	$-0.157487\pi$
$937$	53.8795	1.76017	0.880083	+	0.474820i	$0.157487\pi$
$938$	− 18.3143i	− 0.597984i
$939$	49.9326	1.62949
$940$	−10.4362	−0.340390
$941$	7.42918i	0.242184i	0.992641	+	0.121092i	$0.0386396\pi$
$941$	7.42918i	0.242184i	−0.992641	+	0.121092i	$0.961360\pi$
$942$	− 12.5073i	− 0.407511i
$943$	26.0470i	0.848207i
$944$	3.93789i	0.128167i
$945$	−3.18860	−0.103725
$946$	1.50770	0.0490196
$947$	− 15.5223i	− 0.504407i	−0.967674	−	0.252204i	$-0.918845\pi$
$947$	− 15.5223i	− 0.504407i	0.967674	−	0.252204i	$-0.0811553\pi$
$948$	15.2154	0.494172
$949$	0	0
$950$	5.83768	0.189399
$951$	50.0377i	1.62258i
$952$	−2.43572	−0.0789420
$953$	13.4032	0.434171	0.217085	−	0.976153i	$-0.430345\pi$
$953$	13.4032	0.434171	0.217085	+	0.976153i	$0.430345\pi$
$954$	− 2.98126i	− 0.0965219i
$955$	− 21.8818i	− 0.708079i
$956$	9.78398i	0.316436i
$957$	− 5.01256i	− 0.162033i
$958$	−8.09122	−0.261416
$959$	−15.5368	−0.501708
$960$	− 2.23727i	− 0.0722077i
$961$	8.96282	0.289123
$962$	0	0
$963$	−12.9637	−0.417748
$964$	− 11.7487i	− 0.378401i
$965$	−0.440205	−0.0141707
$966$	29.9056	0.962197
$967$	27.1373i	0.872677i	0.899783	+	0.436338i	$0.143725\pi$
$967$	27.1373i	0.872677i	−0.899783	+	0.436338i	$0.856275\pi$
$968$	− 10.9003i	− 0.350349i
$969$	22.2003i	0.713176i
$970$	11.9669i	0.384235i
$971$	−54.9714	−1.76411	−0.882057	−	0.471142i	$-0.843842\pi$
$971$	−54.9714	−1.76411	−0.882057	+	0.471142i	$0.843842\pi$
$972$	17.9222	0.574855
$973$	− 18.2708i	− 0.585734i
$974$	14.3897	0.461074
$975$	0	0
$976$	0.971828	0.0311075
$977$	58.0043i	1.85572i	0.372928	+	0.927860i	$0.378354\pi$
$977$	58.0043i	1.85572i	−0.372928	+	0.927860i	$0.621646\pi$
$978$	9.70141	0.310217
$979$	−0.469272	−0.0149980
$980$	− 4.94669i	− 0.158016i
$981$	− 33.3708i	− 1.06545i
$982$	10.2104i	0.325827i
$983$	26.4277i	0.842914i	0.906848	+	0.421457i	$0.138481\pi$
$983$	26.4277i	0.842914i	−0.906848	+	0.421457i	$0.861519\pi$
$984$	6.24698	0.199146
$985$	−13.7762	−0.438947
$986$	12.0609i	0.384096i
$987$	−33.4570	−1.06495
$988$	0	0
$989$	44.5410	1.41632
$990$	0.633227i	0.0201253i
$991$	−18.2519	−0.579790	−0.289895	−	0.957059i	$-0.593620\pi$
$991$	−18.2519	−0.579790	−0.289895	+	0.957059i	$0.593620\pi$
$992$	−4.69438	−0.149047
$993$	35.0352i	1.11181i
$994$	5.98436i	0.189812i
$995$	− 2.53270i	− 0.0802921i
$996$	− 24.9089i	− 0.789270i
$997$	−9.13634	−0.289351	−0.144675	−	0.989479i	$-0.546214\pi$
$997$	−9.13634	−0.289351	−0.144675	+	0.989479i	$0.546214\pi$
$998$	−3.13991	−0.0993922
$999$	− 25.9418i	− 0.820762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1690.2.d.l.1351.11		12
13.2	odd	12		1690.2.e.u.191.2		12
13.3	even	3		1690.2.l.n.1161.5		24
13.4	even	6		1690.2.l.n.361.5		24
13.5	odd	4		1690.2.a.w.1.5	yes	6
13.6	odd	12		1690.2.e.u.991.2		12
13.7	odd	12		1690.2.e.v.991.2		12
13.8	odd	4		1690.2.a.v.1.5	✓	6
13.9	even	3		1690.2.l.n.361.11		24
13.10	even	6		1690.2.l.n.1161.11		24
13.11	odd	12		1690.2.e.v.191.2		12
13.12	even	2	inner	1690.2.d.l.1351.5		12
65.34	odd	4		8450.2.a.cq.1.2		6
65.44	odd	4		8450.2.a.cp.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1690.2.a.v.1.5	✓	6	13.8	odd	4
1690.2.a.w.1.5	yes	6	13.5	odd	4
1690.2.d.l.1351.5		12	13.12	even	2	inner
1690.2.d.l.1351.11		12	1.1	even	1	trivial
1690.2.e.u.191.2		12	13.2	odd	12
1690.2.e.u.991.2		12	13.6	odd	12
1690.2.e.v.191.2		12	13.11	odd	12
1690.2.e.v.991.2		12	13.7	odd	12
1690.2.l.n.361.5		24	13.4	even	6
1690.2.l.n.361.11		24	13.9	even	3
1690.2.l.n.1161.5		24	13.3	even	3
1690.2.l.n.1161.11		24	13.10	even	6
8450.2.a.cp.1.2		6	65.44	odd	4
8450.2.a.cq.1.2		6	65.34	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1690 = 2 \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1690.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +2.23727 q^{3} -1.00000 q^{4} +1.00000i q^{5} +2.23727i q^{6} -1.43294i q^{7} -1.00000i q^{8} +2.00539 q^{9} -1.00000 q^{10} -0.315763i q^{11} -2.23727 q^{12} +1.43294 q^{14} +2.23727i q^{15} +1.00000 q^{16} +1.69981 q^{17} +2.00539i q^{18} +5.83768i q^{19} -1.00000i q^{20} -3.20587i q^{21} +0.315763 q^{22} +9.32839 q^{23} -2.23727i q^{24} -1.00000 q^{25} -2.22522 q^{27} +1.43294i q^{28} +7.09544 q^{29} -2.23727 q^{30} +4.69438i q^{31} +1.00000i q^{32} -0.706448i q^{33} +1.69981i q^{34} +1.43294 q^{35} -2.00539 q^{36} +11.6581i q^{37} -5.83768 q^{38} +1.00000 q^{40} +2.79223i q^{41} +3.20587 q^{42} +4.77478 q^{43} +0.315763i q^{44} +2.00539i q^{45} +9.32839i q^{46} -10.4362i q^{47} +2.23727 q^{48} +4.94669 q^{49} -1.00000i q^{50} +3.80293 q^{51} -1.48663 q^{53} -2.22522i q^{54} +0.315763 q^{55} -1.43294 q^{56} +13.0605i q^{57} +7.09544i q^{58} +3.93789i q^{59} -2.23727i q^{60} +0.971828 q^{61} -4.69438 q^{62} -2.87359i q^{63} -1.00000 q^{64} +0.706448 q^{66} -12.7810i q^{67} -1.69981 q^{68} +20.8701 q^{69} +1.43294i q^{70} +4.17629i q^{71} -2.00539i q^{72} -8.83875i q^{73} -11.6581 q^{74} -2.23727 q^{75} -5.83768i q^{76} -0.452469 q^{77} -6.80085 q^{79} +1.00000i q^{80} -10.9946 q^{81} -2.79223 q^{82} +11.1336i q^{83} +3.20587i q^{84} +1.69981i q^{85} +4.77478i q^{86} +15.8744 q^{87} -0.315763 q^{88} -1.48615i q^{89} -2.00539 q^{90} -9.32839 q^{92} +10.5026i q^{93} +10.4362 q^{94} -5.83768 q^{95} +2.23727i q^{96} -11.9669i q^{97} +4.94669i q^{98} -0.633227i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} - 12 q^{4} + 32 q^{9} - 12 q^{10} + 4 q^{12} + 6 q^{14} + 12 q^{16} + 6 q^{17} + 30 q^{22} + 6 q^{23} - 12 q^{25} - 40 q^{27} + 14 q^{29} + 4 q^{30} + 6 q^{35} - 32 q^{36} - 2 q^{38} + 12 q^{40}+ \cdots - 2 q^{95}+O(q^{100}) \) 12 * q - 4 * q^3 - 12 * q^4 + 32 * q^9 - 12 * q^10 + 4 * q^12 + 6 * q^14 + 12 * q^16 + 6 * q^17 + 30 * q^22 + 6 * q^23 - 12 * q^25 - 40 * q^27 + 14 * q^29 + 4 * q^30 + 6 * q^35 - 32 * q^36 - 2 * q^38 + 12 * q^40 - 4 * q^42 + 44 * q^43 - 4 * q^48 - 62 * q^49 + 44 * q^51 - 32 * q^53 + 30 * q^55 - 6 * q^56 + 66 * q^61 - 12 * q^64 + 8 * q^66 - 6 * q^68 + 12 * q^69 + 12 * q^74 + 4 * q^75 + 68 * q^77 + 40 * q^79 - 4 * q^81 - 4 * q^82 - 78 * q^87 - 30 * q^88 - 32 * q^90 - 6 * q^92 + 14 * q^94 - 2 * q^95

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