LMFDB - Embedded newform 2070.2.d.a.829.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(829,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2070.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.d (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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 690)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2070.829
Dual form			2070.2.d.a.829.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +(2.12132 - 0.707107i) q^{10} +4.24264 q^{11} +0.828427i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.82843i q^{17} -6.24264 q^{19} +(-0.707107 - 2.12132i) q^{20} -4.24264i q^{22} -1.00000i q^{23} +(-4.00000 + 3.00000i) q^{25} +0.828427 q^{26} +2.00000i q^{28} +3.65685 q^{29} +6.00000 q^{31} -1.00000i q^{32} -6.82843 q^{34} +(4.24264 - 1.41421i) q^{35} +0.585786i q^{37} +6.24264i q^{38} +(-2.12132 + 0.707107i) q^{40} +6.82843 q^{41} -10.2426i q^{43} -4.24264 q^{44} -1.00000 q^{46} +0.828427i q^{47} +3.00000 q^{49} +(3.00000 + 4.00000i) q^{50} -0.828427i q^{52} +10.5858i q^{53} +(3.00000 + 9.00000i) q^{55} +2.00000 q^{56} -3.65685i q^{58} +8.48528 q^{59} -0.585786 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(-1.75736 + 0.585786i) q^{65} -3.41421i q^{67} +6.82843i q^{68} +(-1.41421 - 4.24264i) q^{70} +5.65685 q^{71} -7.65685i q^{73} +0.585786 q^{74} +6.24264 q^{76} -8.48528i q^{77} +3.65685 q^{79} +(0.707107 + 2.12132i) q^{80} -6.82843i q^{82} +1.41421i q^{83} +(14.4853 - 4.82843i) q^{85} -10.2426 q^{86} +4.24264i q^{88} +9.17157 q^{89} +1.65685 q^{91} +1.00000i q^{92} +0.828427 q^{94} +(-4.41421 - 13.2426i) q^{95} -6.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 8 q^{14} + 4 q^{16} - 8 q^{19} - 16 q^{25} - 8 q^{26} - 8 q^{29} + 24 q^{31} - 16 q^{34} + 16 q^{41} - 4 q^{46} + 12 q^{49} + 12 q^{50} + 12 q^{55} + 8 q^{56} - 8 q^{61} - 4 q^{64} - 24 q^{65} + 8 q^{74} + 8 q^{76} - 8 q^{79} + 24 q^{85} - 24 q^{86} + 48 q^{89} - 16 q^{91} - 8 q^{94} - 12 q^{95}+O(q^{100}) $ 4 * q - 4 * q^4 - 8 * q^14 + 4 * q^16 - 8 * q^19 - 16 * q^25 - 8 * q^26 - 8 * q^29 + 24 * q^31 - 16 * q^34 + 16 * q^41 - 4 * q^46 + 12 * q^49 + 12 * q^50 + 12 * q^55 + 8 * q^56 - 8 * q^61 - 4 * q^64 - 24 * q^65 + 8 * q^74 + 8 * q^76 - 8 * q^79 + 24 * q^85 - 24 * q^86 + 48 * q^89 - 16 * q^91 - 8 * q^94 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2070\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$1891$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$5$	0.707107	+	2.12132i	0.316228	+	0.948683i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	2.12132	−	0.707107i	0.670820	−	0.223607i
$11$	4.24264			1.27920			0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264			1.27920			0.639602	+	0.768706i	$0.279099\pi$
$12$		0			0
$13$			0.828427i			0.229764i	0.993379	+	0.114882i	$0.0366490\pi$
$13$			0.828427i			0.229764i	−0.993379	+	0.114882i	$0.963351\pi$
$14$	−2.00000			−0.534522
$15$		0			0
$16$	1.00000			0.250000
$17$		−	6.82843i		−	1.65614i	−0.560627	−	0.828068i	$-0.689440\pi$
$17$		−	6.82843i		−	1.65614i	0.560627	−	0.828068i	$-0.310560\pi$
$18$		0			0
$19$	−6.24264			−1.43216			−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264			−1.43216			−0.716080	+	0.698018i	$0.754065\pi$
$20$	−0.707107	−	2.12132i	−0.158114	−	0.474342i
$21$		0			0
$22$		−	4.24264i		−	0.904534i
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	−4.00000	+	3.00000i	−0.800000	+	0.600000i
$26$	0.828427			0.162468
$27$		0			0
$28$			2.00000i			0.377964i
$29$	3.65685			0.679061			0.339530	−	0.940595i	$-0.389732\pi$
$29$	3.65685			0.679061			0.339530	+	0.940595i	$0.389732\pi$
$30$		0			0
$31$	6.00000			1.07763			0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000			1.07763			0.538816	+	0.842424i	$0.318872\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−6.82843			−1.17107
$35$	4.24264	−	1.41421i	0.717137	−	0.239046i
$36$		0			0
$37$			0.585786i			0.0963027i	0.998840	+	0.0481513i	$0.0153330\pi$
$37$			0.585786i			0.0963027i	−0.998840	+	0.0481513i	$0.984667\pi$
$38$			6.24264i			1.01269i
$39$		0			0
$40$	−2.12132	+	0.707107i	−0.335410	+	0.111803i
$41$	6.82843			1.06642			0.533211	−	0.845983i	$-0.320985\pi$
$41$	6.82843			1.06642			0.533211	+	0.845983i	$0.320985\pi$
$42$		0			0
$43$		−	10.2426i		−	1.56199i	−0.624538	−	0.780994i	$-0.714713\pi$
$43$		−	10.2426i		−	1.56199i	0.624538	−	0.780994i	$-0.285287\pi$
$44$	−4.24264			−0.639602
$45$		0			0
$46$	−1.00000			−0.147442
$47$			0.828427i			0.120839i	0.998173	+	0.0604193i	$0.0192438\pi$
$47$			0.828427i			0.120839i	−0.998173	+	0.0604193i	$0.980756\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$	3.00000	+	4.00000i	0.424264	+	0.565685i
$51$		0			0
$52$		−	0.828427i		−	0.114882i
$53$			10.5858i			1.45407i	0.686601	+	0.727035i	$0.259102\pi$
$53$			10.5858i			1.45407i	−0.686601	+	0.727035i	$0.740898\pi$
$54$		0			0
$55$	3.00000	+	9.00000i	0.404520	+	1.21356i
$56$	2.00000			0.267261
$57$		0			0
$58$		−	3.65685i		−	0.480168i
$59$	8.48528			1.10469			0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528			1.10469			0.552345	+	0.833616i	$0.313733\pi$
$60$		0			0
$61$	−0.585786			−0.0750023			−0.0375011	−	0.999297i	$-0.511940\pi$
$61$	−0.585786			−0.0750023			−0.0375011	+	0.999297i	$0.511940\pi$
$62$		−	6.00000i		−	0.762001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	−1.75736	+	0.585786i	−0.217974	+	0.0726579i
$66$		0			0
$67$		−	3.41421i		−	0.417113i	−0.978010	−	0.208556i	$-0.933124\pi$
$67$		−	3.41421i		−	0.417113i	0.978010	−	0.208556i	$-0.0668764\pi$
$68$			6.82843i			0.828068i
$69$		0			0
$70$	−1.41421	−	4.24264i	−0.169031	−	0.507093i
$71$	5.65685			0.671345			0.335673	−	0.941979i	$-0.391036\pi$
$71$	5.65685			0.671345			0.335673	+	0.941979i	$0.391036\pi$
$72$		0			0
$73$		−	7.65685i		−	0.896167i	−0.893992	−	0.448084i	$-0.852107\pi$
$73$		−	7.65685i		−	0.896167i	0.893992	−	0.448084i	$-0.147893\pi$
$74$	0.585786			0.0680963
$75$		0			0
$76$	6.24264			0.716080
$77$		−	8.48528i		−	0.966988i
$78$		0			0
$79$	3.65685			0.411428			0.205714	−	0.978612i	$-0.434048\pi$
$79$	3.65685			0.411428			0.205714	+	0.978612i	$0.434048\pi$
$80$	0.707107	+	2.12132i	0.0790569	+	0.237171i
$81$		0			0
$82$		−	6.82843i		−	0.754074i
$83$			1.41421i			0.155230i	0.996983	+	0.0776151i	$0.0247305\pi$
$83$			1.41421i			0.155230i	−0.996983	+	0.0776151i	$0.975269\pi$
$84$		0			0
$85$	14.4853	−	4.82843i	1.57115	−	0.523716i
$86$	−10.2426			−1.10449
$87$		0			0
$88$			4.24264i			0.452267i
$89$	9.17157			0.972185			0.486092	−	0.873907i	$-0.338422\pi$
$89$	9.17157			0.972185			0.486092	+	0.873907i	$0.338422\pi$
$90$		0			0
$91$	1.65685			0.173686
$92$			1.00000i			0.104257i
$93$		0			0
$94$	0.828427			0.0854457
$95$	−4.41421	−	13.2426i	−0.452889	−	1.35867i
$96$		0			0
$97$		−	6.00000i		−	0.609208i	−0.952479	−	0.304604i	$-0.901476\pi$
$97$		−	6.00000i		−	0.609208i	0.952479	−	0.304604i	$-0.0985241\pi$
$98$		−	3.00000i		−	0.303046i
$99$		0			0
$100$	4.00000	−	3.00000i	0.400000	−	0.300000i
$101$	−13.3137			−1.32476			−0.662382	−	0.749166i	$-0.730454\pi$
$101$	−13.3137			−1.32476			−0.662382	+	0.749166i	$0.730454\pi$
$102$		0			0
$103$		−	16.1421i		−	1.59053i	−0.606261	−	0.795266i	$-0.707331\pi$
$103$		−	16.1421i		−	1.59053i	0.606261	−	0.795266i	$-0.292669\pi$
$104$	−0.828427			−0.0812340
$105$		0			0
$106$	10.5858			1.02818
$107$		−	0.928932i		−	0.0898033i	−0.998991	−	0.0449016i	$-0.985703\pi$
$107$		−	0.928932i		−	0.0898033i	0.998991	−	0.0449016i	$-0.0142974\pi$
$108$		0			0
$109$	−12.5858			−1.20550			−0.602750	−	0.797930i	$-0.705928\pi$
$109$	−12.5858			−1.20550			−0.602750	+	0.797930i	$0.705928\pi$
$110$	9.00000	−	3.00000i	0.858116	−	0.286039i
$111$		0			0
$112$		−	2.00000i		−	0.188982i
$113$		−	16.9706i		−	1.59646i	−0.602355	−	0.798228i	$-0.705771\pi$
$113$		−	16.9706i		−	1.59646i	0.602355	−	0.798228i	$-0.294229\pi$
$114$		0			0
$115$	2.12132	−	0.707107i	0.197814	−	0.0659380i
$116$	−3.65685			−0.339530
$117$		0			0
$118$		−	8.48528i		−	0.781133i
$119$	−13.6569			−1.25192
$120$		0			0
$121$	7.00000			0.636364
$122$			0.585786i			0.0530346i
$123$		0			0
$124$	−6.00000			−0.538816
$125$	−9.19239	−	6.36396i	−0.822192	−	0.569210i
$126$		0			0
$127$		−	2.00000i		−	0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$		−	2.00000i		−	0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	0.585786	+	1.75736i	0.0513769	+	0.154131i
$131$	21.6569			1.89217			0.946084	−	0.323921i	$-0.105001\pi$
$131$	21.6569			1.89217			0.946084	+	0.323921i	$0.105001\pi$
$132$		0			0
$133$			12.4853i			1.08261i
$134$	−3.41421			−0.294943
$135$		0			0
$136$	6.82843			0.585533
$137$			8.48528i			0.724947i	0.931994	+	0.362473i	$0.118068\pi$
$137$			8.48528i			0.724947i	−0.931994	+	0.362473i	$0.881932\pi$
$138$		0			0
$139$	16.9706			1.43942			0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706			1.43942			0.719712	+	0.694273i	$0.244274\pi$
$140$	−4.24264	+	1.41421i	−0.358569	+	0.119523i
$141$		0			0
$142$		−	5.65685i		−	0.474713i
$143$			3.51472i			0.293916i
$144$		0			0
$145$	2.58579	+	7.75736i	0.214738	+	0.644214i
$146$	−7.65685			−0.633686
$147$		0			0
$148$		−	0.585786i		−	0.0481513i
$149$	12.7279			1.04271			0.521356	−	0.853339i	$-0.325426\pi$
$149$	12.7279			1.04271			0.521356	+	0.853339i	$0.325426\pi$
$150$		0			0
$151$	−4.34315			−0.353440			−0.176720	−	0.984261i	$-0.556549\pi$
$151$	−4.34315			−0.353440			−0.176720	+	0.984261i	$0.556549\pi$
$152$		−	6.24264i		−	0.506345i
$153$		0			0
$154$	−8.48528			−0.683763
$155$	4.24264	+	12.7279i	0.340777	+	1.02233i
$156$		0			0
$157$			21.0711i			1.68165i	0.541304	+	0.840827i	$0.317931\pi$
$157$			21.0711i			1.68165i	−0.541304	+	0.840827i	$0.682069\pi$
$158$		−	3.65685i		−	0.290924i
$159$		0			0
$160$	2.12132	−	0.707107i	0.167705	−	0.0559017i
$161$	−2.00000			−0.157622
$162$		0			0
$163$		−	6.82843i		−	0.534844i	−0.963580	−	0.267422i	$-0.913828\pi$
$163$		−	6.82843i		−	0.534844i	0.963580	−	0.267422i	$-0.0861717\pi$
$164$	−6.82843			−0.533211
$165$		0			0
$166$	1.41421			0.109764
$167$			3.17157i			0.245424i	0.992442	+	0.122712i	$0.0391591\pi$
$167$			3.17157i			0.245424i	−0.992442	+	0.122712i	$0.960841\pi$
$168$		0			0
$169$	12.3137			0.947208
$170$	−4.82843	−	14.4853i	−0.370323	−	1.11097i
$171$		0			0
$172$			10.2426i			0.780994i
$173$			8.82843i			0.671213i	0.942002	+	0.335606i	$0.108941\pi$
$173$			8.82843i			0.671213i	−0.942002	+	0.335606i	$0.891059\pi$
$174$		0			0
$175$	6.00000	+	8.00000i	0.453557	+	0.604743i
$176$	4.24264			0.319801
$177$		0			0
$178$		−	9.17157i		−	0.687438i
$179$	−21.6569			−1.61871			−0.809355	−	0.587320i	$-0.800183\pi$
$179$	−21.6569			−1.61871			−0.809355	+	0.587320i	$0.800183\pi$
$180$		0			0
$181$	−20.3848			−1.51519			−0.757594	−	0.652726i	$-0.773625\pi$
$181$	−20.3848			−1.51519			−0.757594	+	0.652726i	$0.773625\pi$
$182$		−	1.65685i		−	0.122814i
$183$		0			0
$184$	1.00000			0.0737210
$185$	−1.24264	+	0.414214i	−0.0913608	+	0.0304536i
$186$		0			0
$187$		−	28.9706i		−	2.11854i
$188$		−	0.828427i		−	0.0604193i
$189$		0			0
$190$	−13.2426	+	4.41421i	−0.960722	+	0.320241i
$191$	10.8284			0.783517			0.391759	−	0.920068i	$-0.371867\pi$
$191$	10.8284			0.783517			0.391759	+	0.920068i	$0.371867\pi$
$192$		0			0
$193$			20.9706i			1.50949i	0.656016	+	0.754747i	$0.272240\pi$
$193$			20.9706i			1.50949i	−0.656016	+	0.754747i	$0.727760\pi$
$194$	−6.00000			−0.430775
$195$		0			0
$196$	−3.00000			−0.214286
$197$		−	20.6274i		−	1.46964i	−0.678261	−	0.734821i	$-0.737266\pi$
$197$		−	20.6274i		−	1.46964i	0.678261	−	0.734821i	$-0.262734\pi$
$198$		0			0
$199$	14.0000			0.992434			0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000			0.992434			0.496217	+	0.868199i	$0.334722\pi$
$200$	−3.00000	−	4.00000i	−0.212132	−	0.282843i
$201$		0			0
$202$			13.3137i			0.936749i
$203$		−	7.31371i		−	0.513322i
$204$		0			0
$205$	4.82843	+	14.4853i	0.337232	+	1.01170i
$206$	−16.1421			−1.12468
$207$		0			0
$208$			0.828427i			0.0574411i
$209$	−26.4853			−1.83203
$210$		0			0
$211$	−3.51472			−0.241963			−0.120982	−	0.992655i	$-0.538604\pi$
$211$	−3.51472			−0.241963			−0.120982	+	0.992655i	$0.538604\pi$
$212$		−	10.5858i		−	0.727035i
$213$		0			0
$214$	−0.928932			−0.0635005
$215$	21.7279	−	7.24264i	1.48183	−	0.493944i
$216$		0			0
$217$		−	12.0000i		−	0.814613i
$218$			12.5858i			0.852417i
$219$		0			0
$220$	−3.00000	−	9.00000i	−0.202260	−	0.606780i
$221$	5.65685			0.380521
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−2.00000			−0.133631
$225$		0			0
$226$	−16.9706			−1.12887
$227$			22.3848i			1.48573i	0.669441	+	0.742865i	$0.266534\pi$
$227$			22.3848i			1.48573i	−0.669441	+	0.742865i	$0.733466\pi$
$228$		0			0
$229$	−10.2426			−0.676853			−0.338426	−	0.940993i	$-0.609895\pi$
$229$	−10.2426			−0.676853			−0.338426	+	0.940993i	$0.609895\pi$
$230$	−0.707107	−	2.12132i	−0.0466252	−	0.139876i
$231$		0			0
$232$			3.65685i			0.240084i
$233$			2.14214i			0.140336i	0.997535	+	0.0701680i	$0.0223535\pi$
$233$			2.14214i			0.140336i	−0.997535	+	0.0701680i	$0.977646\pi$
$234$		0			0
$235$	−1.75736	+	0.585786i	−0.114637	+	0.0382125i
$236$	−8.48528			−0.552345
$237$		0			0
$238$			13.6569i			0.885242i
$239$	−16.8284			−1.08854			−0.544270	−	0.838910i	$-0.683193\pi$
$239$	−16.8284			−1.08854			−0.544270	+	0.838910i	$0.683193\pi$
$240$		0			0
$241$	6.48528			0.417754			0.208877	−	0.977942i	$-0.433019\pi$
$241$	6.48528			0.417754			0.208877	+	0.977942i	$0.433019\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	0.585786			0.0375011
$245$	2.12132	+	6.36396i	0.135526	+	0.406579i
$246$		0			0
$247$		−	5.17157i		−	0.329059i
$248$			6.00000i			0.381000i
$249$		0			0
$250$	−6.36396	+	9.19239i	−0.402492	+	0.581378i
$251$	−0.727922			−0.0459460			−0.0229730	−	0.999736i	$-0.507313\pi$
$251$	−0.727922			−0.0459460			−0.0229730	+	0.999736i	$0.507313\pi$
$252$		0			0
$253$		−	4.24264i		−	0.266733i
$254$	−2.00000			−0.125491
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	1.17157			0.0727980
$260$	1.75736	−	0.585786i	0.108987	−	0.0363289i
$261$		0			0
$262$		−	21.6569i		−	1.33796i
$263$		−	6.34315i		−	0.391135i	−0.980690	−	0.195568i	$-0.937345\pi$
$263$		−	6.34315i		−	0.391135i	0.980690	−	0.195568i	$-0.0626549\pi$
$264$		0			0
$265$	−22.4558	+	7.48528i	−1.37945	+	0.459817i
$266$	12.4853			0.765522
$267$		0			0
$268$			3.41421i			0.208556i
$269$	−19.1716			−1.16891			−0.584456	−	0.811426i	$-0.698692\pi$
$269$	−19.1716			−1.16891			−0.584456	+	0.811426i	$0.698692\pi$
$270$		0			0
$271$	9.65685			0.586612			0.293306	−	0.956019i	$-0.405245\pi$
$271$	9.65685			0.586612			0.293306	+	0.956019i	$0.405245\pi$
$272$		−	6.82843i		−	0.414034i
$273$		0			0
$274$	8.48528			0.512615
$275$	−16.9706	+	12.7279i	−1.02336	+	0.767523i
$276$		0			0
$277$			13.5147i			0.812021i	0.913868	+	0.406010i	$0.133080\pi$
$277$			13.5147i			0.812021i	−0.913868	+	0.406010i	$0.866920\pi$
$278$		−	16.9706i		−	1.01783i
$279$		0			0
$280$	1.41421	+	4.24264i	0.0845154	+	0.253546i
$281$	1.85786			0.110831			0.0554154	−	0.998463i	$-0.482352\pi$
$281$	1.85786			0.110831			0.0554154	+	0.998463i	$0.482352\pi$
$282$		0			0
$283$		−	15.2132i		−	0.904331i	−0.891934	−	0.452166i	$-0.850652\pi$
$283$		−	15.2132i		−	0.904331i	0.891934	−	0.452166i	$-0.149348\pi$
$284$	−5.65685			−0.335673
$285$		0			0
$286$	3.51472			0.207830
$287$		−	13.6569i		−	0.806139i
$288$		0			0
$289$	−29.6274			−1.74279
$290$	7.75736	−	2.58579i	0.455528	−	0.151843i
$291$		0			0
$292$			7.65685i			0.448084i
$293$			20.0416i			1.17084i	0.810729	+	0.585422i	$0.199071\pi$
$293$			20.0416i			1.17084i	−0.810729	+	0.585422i	$0.800929\pi$
$294$		0			0
$295$	6.00000	+	18.0000i	0.349334	+	1.04800i
$296$	−0.585786			−0.0340481
$297$		0			0
$298$		−	12.7279i		−	0.737309i
$299$	0.828427			0.0479092
$300$		0			0
$301$	−20.4853			−1.18075
$302$			4.34315i			0.249920i
$303$		0			0
$304$	−6.24264			−0.358040
$305$	−0.414214	−	1.24264i	−0.0237178	−	0.0711534i
$306$		0			0
$307$			25.4558i			1.45284i	0.687250	+	0.726421i	$0.258818\pi$
$307$			25.4558i			1.45284i	−0.687250	+	0.726421i	$0.741182\pi$
$308$			8.48528i			0.483494i
$309$		0			0
$310$	12.7279	−	4.24264i	0.722897	−	0.240966i
$311$	21.7990			1.23611			0.618054	−	0.786136i	$-0.287921\pi$
$311$	21.7990			1.23611			0.618054	+	0.786136i	$0.287921\pi$
$312$		0			0
$313$		−	18.0000i		−	1.01742i	−0.860938	−	0.508710i	$-0.830123\pi$
$313$		−	18.0000i		−	1.01742i	0.860938	−	0.508710i	$-0.169877\pi$
$314$	21.0711			1.18911
$315$		0			0
$316$	−3.65685			−0.205714
$317$			14.4853i			0.813574i	0.913523	+	0.406787i	$0.133351\pi$
$317$			14.4853i			0.813574i	−0.913523	+	0.406787i	$0.866649\pi$
$318$		0			0
$319$	15.5147			0.868657
$320$	−0.707107	−	2.12132i	−0.0395285	−	0.118585i
$321$		0			0
$322$			2.00000i			0.111456i
$323$			42.6274i			2.37185i
$324$		0			0
$325$	−2.48528	−	3.31371i	−0.137859	−	0.183811i
$326$	−6.82843			−0.378192
$327$		0			0
$328$			6.82843i			0.377037i
$329$	1.65685			0.0913453
$330$		0			0
$331$	−20.4853			−1.12597			−0.562986	−	0.826466i	$-0.690348\pi$
$331$	−20.4853			−1.12597			−0.562986	+	0.826466i	$0.690348\pi$
$332$		−	1.41421i		−	0.0776151i
$333$		0			0
$334$	3.17157			0.173541
$335$	7.24264	−	2.41421i	0.395708	−	0.131903i
$336$		0			0
$337$		−	26.4853i		−	1.44275i	−0.692547	−	0.721373i	$-0.743512\pi$
$337$		−	26.4853i		−	1.44275i	0.692547	−	0.721373i	$-0.256488\pi$
$338$		−	12.3137i		−	0.669777i
$339$		0			0
$340$	−14.4853	+	4.82843i	−0.785575	+	0.261858i
$341$	25.4558			1.37851
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$	10.2426			0.552246
$345$		0			0
$346$	8.82843			0.474619
$347$			31.7990i			1.70706i	0.521044	+	0.853530i	$0.325543\pi$
$347$			31.7990i			1.70706i	−0.521044	+	0.853530i	$0.674457\pi$
$348$		0			0
$349$	−2.68629			−0.143794			−0.0718969	−	0.997412i	$-0.522905\pi$
$349$	−2.68629			−0.143794			−0.0718969	+	0.997412i	$0.522905\pi$
$350$	8.00000	−	6.00000i	0.427618	−	0.320713i
$351$		0			0
$352$		−	4.24264i		−	0.226134i
$353$			2.14214i			0.114014i	0.998374	+	0.0570072i	$0.0181558\pi$
$353$			2.14214i			0.114014i	−0.998374	+	0.0570072i	$0.981844\pi$
$354$		0			0
$355$	4.00000	+	12.0000i	0.212298	+	0.636894i
$356$	−9.17157			−0.486092
$357$		0			0
$358$			21.6569i			1.14460i
$359$	19.7990			1.04495			0.522475	−	0.852654i	$-0.325009\pi$
$359$	19.7990			1.04495			0.522475	+	0.852654i	$0.325009\pi$
$360$		0			0
$361$	19.9706			1.05108
$362$			20.3848i			1.07140i
$363$		0			0
$364$	−1.65685			−0.0868428
$365$	16.2426	−	5.41421i	0.850179	−	0.283393i
$366$		0			0
$367$			1.02944i			0.0537362i	0.999639	+	0.0268681i	$0.00855341\pi$
$367$			1.02944i			0.0537362i	−0.999639	+	0.0268681i	$0.991447\pi$
$368$		−	1.00000i		−	0.0521286i
$369$		0			0
$370$	0.414214	+	1.24264i	0.0215339	+	0.0646018i
$371$	21.1716			1.09917
$372$		0			0
$373$			23.2132i			1.20193i	0.799274	+	0.600967i	$0.205218\pi$
$373$			23.2132i			1.20193i	−0.799274	+	0.600967i	$0.794782\pi$
$374$	−28.9706			−1.49803
$375$		0			0
$376$	−0.828427			−0.0427229
$377$			3.02944i			0.156024i
$378$		0			0
$379$	−23.2132			−1.19238			−0.596191	−	0.802843i	$-0.703320\pi$
$379$	−23.2132			−1.19238			−0.596191	+	0.802843i	$0.703320\pi$
$380$	4.41421	+	13.2426i	0.226444	+	0.679333i
$381$		0			0
$382$		−	10.8284i		−	0.554031i
$383$			26.6274i			1.36060i	0.732935	+	0.680299i	$0.238150\pi$
$383$			26.6274i			1.36060i	−0.732935	+	0.680299i	$0.761850\pi$
$384$		0			0
$385$	18.0000	−	6.00000i	0.917365	−	0.305788i
$386$	20.9706			1.06737
$387$		0			0
$388$			6.00000i			0.304604i
$389$	−30.3848			−1.54057			−0.770285	−	0.637700i	$-0.779886\pi$
$389$	−30.3848			−1.54057			−0.770285	+	0.637700i	$0.779886\pi$
$390$		0			0
$391$	−6.82843			−0.345328
$392$			3.00000i			0.151523i
$393$		0			0
$394$	−20.6274			−1.03919
$395$	2.58579	+	7.75736i	0.130105	+	0.390315i
$396$		0			0
$397$		−	19.6569i		−	0.986549i	−0.869874	−	0.493275i	$-0.835800\pi$
$397$		−	19.6569i		−	0.986549i	0.869874	−	0.493275i	$-0.164200\pi$
$398$		−	14.0000i		−	0.701757i
$399$		0			0
$400$	−4.00000	+	3.00000i	−0.200000	+	0.150000i
$401$	−4.97056			−0.248218			−0.124109	−	0.992269i	$-0.539607\pi$
$401$	−4.97056			−0.248218			−0.124109	+	0.992269i	$0.539607\pi$
$402$		0			0
$403$			4.97056i			0.247601i
$404$	13.3137			0.662382
$405$		0			0
$406$	−7.31371			−0.362973
$407$			2.48528i			0.123191i
$408$		0			0
$409$	−18.0000			−0.890043			−0.445021	−	0.895520i	$-0.646804\pi$
$409$	−18.0000			−0.890043			−0.445021	+	0.895520i	$0.646804\pi$
$410$	14.4853	−	4.82843i	0.715377	−	0.238459i
$411$		0			0
$412$			16.1421i			0.795266i
$413$		−	16.9706i		−	0.835067i
$414$		0			0
$415$	−3.00000	+	1.00000i	−0.147264	+	0.0490881i
$416$	0.828427			0.0406170
$417$		0			0
$418$			26.4853i			1.29544i
$419$	−39.0711			−1.90875			−0.954373	−	0.298616i	$-0.903475\pi$
$419$	−39.0711			−1.90875			−0.954373	+	0.298616i	$0.903475\pi$
$420$		0			0
$421$	1.75736			0.0856485			0.0428242	−	0.999083i	$-0.486364\pi$
$421$	1.75736			0.0856485			0.0428242	+	0.999083i	$0.486364\pi$
$422$			3.51472i			0.171094i
$423$		0			0
$424$	−10.5858			−0.514091
$425$	20.4853	+	27.3137i	0.993682	+	1.32491i
$426$		0			0
$427$			1.17157i			0.0566964i
$428$			0.928932i			0.0449016i
$429$		0			0
$430$	−7.24264	−	21.7279i	−0.349271	−	1.04781i
$431$	8.48528			0.408722			0.204361	−	0.978896i	$-0.434488\pi$
$431$	8.48528			0.408722			0.204361	+	0.978896i	$0.434488\pi$
$432$		0			0
$433$			6.97056i			0.334984i	0.985873	+	0.167492i	$0.0535668\pi$
$433$			6.97056i			0.334984i	−0.985873	+	0.167492i	$0.946433\pi$
$434$	−12.0000			−0.576018
$435$		0			0
$436$	12.5858			0.602750
$437$			6.24264i			0.298626i
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−9.00000	+	3.00000i	−0.429058	+	0.143019i
$441$		0			0
$442$		−	5.65685i		−	0.269069i
$443$		−	0.686292i		−	0.0326067i	−0.999867	−	0.0163033i	$-0.994810\pi$
$443$		−	0.686292i		−	0.0326067i	0.999867	−	0.0163033i	$-0.00518975\pi$
$444$		0			0
$445$	6.48528	+	19.4558i	0.307432	+	0.922295i
$446$	−16.0000			−0.757622
$447$		0			0
$448$			2.00000i			0.0944911i
$449$	−17.1716			−0.810377			−0.405188	−	0.914233i	$-0.632794\pi$
$449$	−17.1716			−0.810377			−0.405188	+	0.914233i	$0.632794\pi$
$450$		0			0
$451$	28.9706			1.36417
$452$			16.9706i			0.798228i
$453$		0			0
$454$	22.3848			1.05057
$455$	1.17157	+	3.51472i	0.0549242	+	0.164773i
$456$		0			0
$457$			34.9706i			1.63585i	0.575322	+	0.817927i	$0.304877\pi$
$457$			34.9706i			1.63585i	−0.575322	+	0.817927i	$0.695123\pi$
$458$			10.2426i			0.478607i
$459$		0			0
$460$	−2.12132	+	0.707107i	−0.0989071	+	0.0329690i
$461$	7.17157			0.334013			0.167007	−	0.985956i	$-0.446590\pi$
$461$	7.17157			0.334013			0.167007	+	0.985956i	$0.446590\pi$
$462$		0			0
$463$			30.9706i			1.43932i	0.694325	+	0.719662i	$0.255703\pi$
$463$			30.9706i			1.43932i	−0.694325	+	0.719662i	$0.744297\pi$
$464$	3.65685			0.169765
$465$		0			0
$466$	2.14214			0.0992325
$467$			31.3553i			1.45095i	0.688247	+	0.725476i	$0.258380\pi$
$467$			31.3553i			1.45095i	−0.688247	+	0.725476i	$0.741620\pi$
$468$		0			0
$469$	−6.82843			−0.315307
$470$	0.585786	+	1.75736i	0.0270203	+	0.0810609i
$471$		0			0
$472$			8.48528i			0.390567i
$473$		−	43.4558i		−	1.99810i
$474$		0			0
$475$	24.9706	−	18.7279i	1.14573	−	0.859296i
$476$	13.6569			0.625961
$477$		0			0
$478$			16.8284i			0.769714i
$479$	27.3137			1.24800			0.623998	−	0.781426i	$-0.285507\pi$
$479$	27.3137			1.24800			0.623998	+	0.781426i	$0.285507\pi$
$480$		0			0
$481$	−0.485281			−0.0221269
$482$		−	6.48528i		−	0.295396i
$483$		0			0
$484$	−7.00000			−0.318182
$485$	12.7279	−	4.24264i	0.577945	−	0.192648i
$486$		0			0
$487$			39.9411i			1.80991i	0.425512	+	0.904953i	$0.360094\pi$
$487$			39.9411i			1.80991i	−0.425512	+	0.904953i	$0.639906\pi$
$488$		−	0.585786i		−	0.0265173i
$489$		0			0
$490$	6.36396	−	2.12132i	0.287494	−	0.0958315i
$491$	18.6274			0.840644			0.420322	−	0.907375i	$-0.361917\pi$
$491$	18.6274			0.840644			0.420322	+	0.907375i	$0.361917\pi$
$492$		0			0
$493$		−	24.9706i		−	1.12462i
$494$	−5.17157			−0.232680
$495$		0			0
$496$	6.00000			0.269408
$497$		−	11.3137i		−	0.507489i
$498$		0			0
$499$	37.4558			1.67675			0.838377	−	0.545091i	$-0.183505\pi$
$499$	37.4558			1.67675			0.838377	+	0.545091i	$0.183505\pi$
$500$	9.19239	+	6.36396i	0.411096	+	0.284605i
$501$		0			0
$502$			0.727922i			0.0324888i
$503$		−	6.14214i		−	0.273864i	−0.990580	−	0.136932i	$-0.956276\pi$
$503$		−	6.14214i		−	0.273864i	0.990580	−	0.136932i	$-0.0437242\pi$
$504$		0			0
$505$	−9.41421	−	28.2426i	−0.418927	−	1.25678i
$506$	−4.24264			−0.188608
$507$		0			0
$508$			2.00000i			0.0887357i
$509$	−10.9706			−0.486262			−0.243131	−	0.969994i	$-0.578174\pi$
$509$	−10.9706			−0.486262			−0.243131	+	0.969994i	$0.578174\pi$
$510$		0			0
$511$	−15.3137			−0.677439
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−18.0000			−0.793946
$515$	34.2426	−	11.4142i	1.50891	−	0.502970i
$516$		0			0
$517$			3.51472i			0.154577i
$518$		−	1.17157i		−	0.0514760i
$519$		0			0
$520$	−0.585786	−	1.75736i	−0.0256884	−	0.0770653i
$521$	−40.2843			−1.76489			−0.882443	−	0.470419i	$-0.844103\pi$
$521$	−40.2843			−1.76489			−0.882443	+	0.470419i	$0.844103\pi$
$522$		0			0
$523$			17.0711i			0.746466i	0.927738	+	0.373233i	$0.121751\pi$
$523$			17.0711i			0.746466i	−0.927738	+	0.373233i	$0.878249\pi$
$524$	−21.6569			−0.946084
$525$		0			0
$526$	−6.34315			−0.276574
$527$		−	40.9706i		−	1.78471i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$	7.48528	+	22.4558i	0.325140	+	0.975420i
$531$		0			0
$532$		−	12.4853i		−	0.541306i
$533$			5.65685i			0.245026i
$534$		0			0
$535$	1.97056	−	0.656854i	0.0851949	−	0.0283983i
$536$	3.41421			0.147472
$537$		0			0
$538$			19.1716i			0.826545i
$539$	12.7279			0.548230
$540$		0			0
$541$	−31.9411			−1.37326			−0.686628	−	0.727009i	$-0.740910\pi$
$541$	−31.9411			−1.37326			−0.686628	+	0.727009i	$0.740910\pi$
$542$		−	9.65685i		−	0.414797i
$543$		0			0
$544$	−6.82843			−0.292766
$545$	−8.89949	−	26.6985i	−0.381212	−	1.14364i
$546$		0			0
$547$			12.4853i			0.533832i	0.963720	+	0.266916i	$0.0860046\pi$
$547$			12.4853i			0.533832i	−0.963720	+	0.266916i	$0.913995\pi$
$548$		−	8.48528i		−	0.362473i
$549$		0			0
$550$	12.7279	+	16.9706i	0.542720	+	0.723627i
$551$	−22.8284			−0.972524
$552$		0			0
$553$		−	7.31371i		−	0.311011i
$554$	13.5147			0.574185
$555$		0			0
$556$	−16.9706			−0.719712
$557$		−	38.8701i		−	1.64698i	−0.567333	−	0.823489i	$-0.692025\pi$
$557$		−	38.8701i		−	1.64698i	0.567333	−	0.823489i	$-0.307975\pi$
$558$		0			0
$559$	8.48528			0.358889
$560$	4.24264	−	1.41421i	0.179284	−	0.0597614i
$561$		0			0
$562$		−	1.85786i		−	0.0783693i
$563$		−	2.10051i		−	0.0885257i	−0.999020	−	0.0442629i	$-0.985906\pi$
$563$		−	2.10051i		−	0.0885257i	0.999020	−	0.0442629i	$-0.0140939\pi$
$564$		0			0
$565$	36.0000	−	12.0000i	1.51453	−	0.504844i
$566$	−15.2132			−0.639459
$567$		0			0
$568$			5.65685i			0.237356i
$569$	14.3431			0.601296			0.300648	−	0.953735i	$-0.402797\pi$
$569$	14.3431			0.601296			0.300648	+	0.953735i	$0.402797\pi$
$570$		0			0
$571$	−17.2721			−0.722814			−0.361407	−	0.932408i	$-0.617703\pi$
$571$	−17.2721			−0.722814			−0.361407	+	0.932408i	$0.617703\pi$
$572$		−	3.51472i		−	0.146958i
$573$		0			0
$574$	−13.6569			−0.570026
$575$	3.00000	+	4.00000i	0.125109	+	0.166812i
$576$		0			0
$577$			24.6274i			1.02525i	0.858612	+	0.512626i	$0.171327\pi$
$577$			24.6274i			1.02525i	−0.858612	+	0.512626i	$0.828673\pi$
$578$			29.6274i			1.23234i
$579$		0			0
$580$	−2.58579	−	7.75736i	−0.107369	−	0.322107i
$581$	2.82843			0.117343
$582$		0			0
$583$			44.9117i			1.86005i
$584$	7.65685			0.316843
$585$		0			0
$586$	20.0416			0.827912
$587$		−	21.1716i		−	0.873844i	−0.899499	−	0.436922i	$-0.856069\pi$
$587$		−	21.1716i		−	0.873844i	0.899499	−	0.436922i	$-0.143931\pi$
$588$		0			0
$589$	−37.4558			−1.54334
$590$	18.0000	−	6.00000i	0.741048	−	0.247016i
$591$		0			0
$592$			0.585786i			0.0240757i
$593$		−	23.6569i		−	0.971471i	−0.874106	−	0.485735i	$-0.838552\pi$
$593$		−	23.6569i		−	0.971471i	0.874106	−	0.485735i	$-0.161448\pi$
$594$		0			0
$595$	−9.65685	−	28.9706i	−0.395892	−	1.18768i
$596$	−12.7279			−0.521356
$597$		0			0
$598$		−	0.828427i		−	0.0338769i
$599$	−11.8579			−0.484499			−0.242250	−	0.970214i	$-0.577885\pi$
$599$	−11.8579			−0.484499			−0.242250	+	0.970214i	$0.577885\pi$
$600$		0			0
$601$	12.9706			0.529080			0.264540	−	0.964375i	$-0.414780\pi$
$601$	12.9706			0.529080			0.264540	+	0.964375i	$0.414780\pi$
$602$			20.4853i			0.834918i
$603$		0			0
$604$	4.34315			0.176720
$605$	4.94975	+	14.8492i	0.201236	+	0.603708i
$606$		0			0
$607$			16.9706i			0.688814i	0.938820	+	0.344407i	$0.111920\pi$
$607$			16.9706i			0.688814i	−0.938820	+	0.344407i	$0.888080\pi$
$608$			6.24264i			0.253173i
$609$		0			0
$610$	−1.24264	+	0.414214i	−0.0503131	+	0.0167710i
$611$	−0.686292			−0.0277644
$612$		0			0
$613$			5.75736i			0.232538i	0.993218	+	0.116269i	$0.0370934\pi$
$613$			5.75736i			0.232538i	−0.993218	+	0.116269i	$0.962907\pi$
$614$	25.4558			1.02731
$615$		0			0
$616$	8.48528			0.341882
$617$			8.68629i			0.349697i	0.984595	+	0.174848i	$0.0559436\pi$
$617$			8.68629i			0.349697i	−0.984595	+	0.174848i	$0.944056\pi$
$618$		0			0
$619$	−23.2132			−0.933017			−0.466509	−	0.884517i	$-0.654488\pi$
$619$	−23.2132			−0.933017			−0.466509	+	0.884517i	$0.654488\pi$
$620$	−4.24264	−	12.7279i	−0.170389	−	0.511166i
$621$		0			0
$622$		−	21.7990i		−	0.874060i
$623$		−	18.3431i		−	0.734903i
$624$		0			0
$625$	7.00000	−	24.0000i	0.280000	−	0.960000i
$626$	−18.0000			−0.719425
$627$		0			0
$628$		−	21.0711i		−	0.840827i
$629$	4.00000			0.159490
$630$		0			0
$631$	40.8284			1.62535			0.812677	−	0.582714i	$-0.198009\pi$
$631$	40.8284			1.62535			0.812677	+	0.582714i	$0.198009\pi$
$632$			3.65685i			0.145462i
$633$		0			0
$634$	14.4853			0.575284
$635$	4.24264	−	1.41421i	0.168364	−	0.0561214i
$636$		0			0
$637$			2.48528i			0.0984704i
$638$		−	15.5147i		−	0.614234i
$639$		0			0
$640$	−2.12132	+	0.707107i	−0.0838525	+	0.0279508i
$641$	−48.2843			−1.90711			−0.953557	−	0.301213i	$-0.902609\pi$
$641$	−48.2843			−1.90711			−0.953557	+	0.301213i	$0.902609\pi$
$642$		0			0
$643$		−	4.38478i		−	0.172919i	−0.996255	−	0.0864593i	$-0.972445\pi$
$643$		−	4.38478i		−	0.172919i	0.996255	−	0.0864593i	$-0.0275553\pi$
$644$	2.00000			0.0788110
$645$		0			0
$646$	42.6274			1.67715
$647$			18.3431i			0.721143i	0.932731	+	0.360572i	$0.117418\pi$
$647$			18.3431i			0.721143i	−0.932731	+	0.360572i	$0.882582\pi$
$648$		0			0
$649$	36.0000			1.41312
$650$	−3.31371	+	2.48528i	−0.129974	+	0.0974808i
$651$		0			0
$652$			6.82843i			0.267422i
$653$		−	7.45584i		−	0.291770i	−0.989302	−	0.145885i	$-0.953397\pi$
$653$		−	7.45584i		−	0.291770i	0.989302	−	0.145885i	$-0.0466029\pi$
$654$		0			0
$655$	15.3137	+	45.9411i	0.598356	+	1.79507i
$656$	6.82843			0.266605
$657$		0			0
$658$		−	1.65685i		−	0.0645909i
$659$	−45.8995			−1.78799			−0.893995	−	0.448076i	$-0.852109\pi$
$659$	−45.8995			−1.78799			−0.893995	+	0.448076i	$0.852109\pi$
$660$		0			0
$661$	34.7279			1.35076			0.675380	−	0.737470i	$-0.263980\pi$
$661$	34.7279			1.35076			0.675380	+	0.737470i	$0.263980\pi$
$662$			20.4853i			0.796183i
$663$		0			0
$664$	−1.41421			−0.0548821
$665$	−26.4853	+	8.82843i	−1.02706	+	0.342352i
$666$		0			0
$667$		−	3.65685i		−	0.141594i
$668$		−	3.17157i		−	0.122712i
$669$		0			0
$670$	−2.41421	−	7.24264i	−0.0932692	−	0.279808i
$671$	−2.48528			−0.0959432
$672$		0			0
$673$		−	27.9411i		−	1.07705i	−0.842609	−	0.538526i	$-0.818982\pi$
$673$		−	27.9411i		−	1.07705i	0.842609	−	0.538526i	$-0.181018\pi$
$674$	−26.4853			−1.02017
$675$		0			0
$676$	−12.3137			−0.473604
$677$			40.5269i			1.55758i	0.627287	+	0.778788i	$0.284165\pi$
$677$			40.5269i			1.55758i	−0.627287	+	0.778788i	$0.715835\pi$
$678$		0			0
$679$	−12.0000			−0.460518
$680$	4.82843	+	14.4853i	0.185162	+	0.555485i
$681$		0			0
$682$		−	25.4558i		−	0.974755i
$683$		−	8.48528i		−	0.324680i	−0.986735	−	0.162340i	$-0.948096\pi$
$683$		−	8.48528i		−	0.324680i	0.986735	−	0.162340i	$-0.0519042\pi$
$684$		0			0
$685$	−18.0000	+	6.00000i	−0.687745	+	0.229248i
$686$	−20.0000			−0.763604
$687$		0			0
$688$		−	10.2426i		−	0.390497i
$689$	−8.76955			−0.334093
$690$		0			0
$691$	−13.1716			−0.501070			−0.250535	−	0.968108i	$-0.580607\pi$
$691$	−13.1716			−0.501070			−0.250535	+	0.968108i	$0.580607\pi$
$692$		−	8.82843i		−	0.335606i
$693$		0			0
$694$	31.7990			1.20707
$695$	12.0000	+	36.0000i	0.455186	+	1.36556i
$696$		0			0
$697$		−	46.6274i		−	1.76614i
$698$			2.68629i			0.101678i
$699$		0			0
$700$	−6.00000	−	8.00000i	−0.226779	−	0.302372i
$701$	38.1838			1.44218			0.721090	−	0.692841i	$-0.243641\pi$
$701$	38.1838			1.44218			0.721090	+	0.692841i	$0.243641\pi$
$702$		0			0
$703$		−	3.65685i		−	0.137921i
$704$	−4.24264			−0.159901
$705$		0			0
$706$	2.14214			0.0806203
$707$			26.6274i			1.00143i
$708$		0			0
$709$	−32.5858			−1.22378			−0.611892	−	0.790941i	$-0.709591\pi$
$709$	−32.5858			−1.22378			−0.611892	+	0.790941i	$0.709591\pi$
$710$	12.0000	−	4.00000i	0.450352	−	0.150117i
$711$		0			0
$712$			9.17157i			0.343719i
$713$		−	6.00000i		−	0.224702i
$714$		0			0
$715$	−7.45584	+	2.48528i	−0.278833	+	0.0929443i
$716$	21.6569			0.809355
$717$		0			0
$718$		−	19.7990i		−	0.738892i
$719$	−0.142136			−0.00530076			−0.00265038	−	0.999996i	$-0.500844\pi$
$719$	−0.142136			−0.00530076			−0.00265038	+	0.999996i	$0.500844\pi$
$720$		0			0
$721$	−32.2843			−1.20233
$722$		−	19.9706i		−	0.743227i
$723$		0			0
$724$	20.3848			0.757594
$725$	−14.6274	+	10.9706i	−0.543249	+	0.407436i
$726$		0			0
$727$		−	20.3431i		−	0.754486i	−0.926114	−	0.377243i	$-0.876872\pi$
$727$		−	20.3431i		−	0.754486i	0.926114	−	0.377243i	$-0.123128\pi$
$728$			1.65685i			0.0614071i
$729$		0			0
$730$	−5.41421	−	16.2426i	−0.200389	−	0.601167i
$731$	−69.9411			−2.58687
$732$		0			0
$733$		−	11.4142i		−	0.421594i	−0.977530	−	0.210797i	$-0.932394\pi$
$733$		−	11.4142i		−	0.421594i	0.977530	−	0.210797i	$-0.0676058\pi$
$734$	1.02944			0.0379972
$735$		0			0
$736$	−1.00000			−0.0368605
$737$		−	14.4853i		−	0.533572i
$738$		0			0
$739$	2.62742			0.0966511			0.0483255	−	0.998832i	$-0.484612\pi$
$739$	2.62742			0.0966511			0.0483255	+	0.998832i	$0.484612\pi$
$740$	1.24264	−	0.414214i	0.0456804	−	0.0152268i
$741$		0			0
$742$		−	21.1716i		−	0.777233i
$743$			13.6569i			0.501021i	0.968114	+	0.250511i	$0.0805985\pi$
$743$			13.6569i			0.501021i	−0.968114	+	0.250511i	$0.919401\pi$
$744$		0			0
$745$	9.00000	+	27.0000i	0.329734	+	0.989203i
$746$	23.2132			0.849896
$747$		0			0
$748$			28.9706i			1.05927i
$749$	−1.85786			−0.0678849
$750$		0			0
$751$	−43.1716			−1.57535			−0.787677	−	0.616089i	$-0.788716\pi$
$751$	−43.1716			−1.57535			−0.787677	+	0.616089i	$0.788716\pi$
$752$			0.828427i			0.0302096i
$753$		0			0
$754$	3.02944			0.110326
$755$	−3.07107	−	9.21320i	−0.111768	−	0.335303i
$756$		0			0
$757$			33.7574i			1.22693i	0.789721	+	0.613466i	$0.210225\pi$
$757$			33.7574i			1.22693i	−0.789721	+	0.613466i	$0.789775\pi$
$758$			23.2132i			0.843142i
$759$		0			0
$760$	13.2426	−	4.41421i	0.480361	−	0.160120i
$761$	35.6569			1.29256			0.646280	−	0.763100i	$-0.276324\pi$
$761$	35.6569			1.29256			0.646280	+	0.763100i	$0.276324\pi$
$762$		0			0
$763$			25.1716i			0.911272i
$764$	−10.8284			−0.391759
$765$		0			0
$766$	26.6274			0.962088
$767$			7.02944i			0.253818i
$768$		0			0
$769$	−6.20101			−0.223614			−0.111807	−	0.993730i	$-0.535664\pi$
$769$	−6.20101			−0.223614			−0.111807	+	0.993730i	$0.535664\pi$
$770$	−6.00000	−	18.0000i	−0.216225	−	0.648675i
$771$		0			0
$772$		−	20.9706i		−	0.754747i
$773$		−	31.0711i		−	1.11755i	−0.829320	−	0.558774i	$-0.811272\pi$
$773$		−	31.0711i		−	1.11755i	0.829320	−	0.558774i	$-0.188728\pi$
$774$		0			0
$775$	−24.0000	+	18.0000i	−0.862105	+	0.646579i
$776$	6.00000			0.215387
$777$		0			0
$778$			30.3848i			1.08935i
$779$	−42.6274			−1.52729
$780$		0			0
$781$	24.0000			0.858788
$782$			6.82843i			0.244184i
$783$		0			0
$784$	3.00000			0.107143
$785$	−44.6985	+	14.8995i	−1.59536	+	0.531786i
$786$		0			0
$787$		−	22.7279i		−	0.810163i	−0.914281	−	0.405081i	$-0.867243\pi$
$787$		−	22.7279i		−	0.810163i	0.914281	−	0.405081i	$-0.132757\pi$
$788$			20.6274i			0.734821i
$789$		0			0
$790$	7.75736	−	2.58579i	0.275994	−	0.0919982i
$791$	−33.9411			−1.20681
$792$		0			0
$793$		−	0.485281i		−	0.0172328i
$794$	−19.6569			−0.697596
$795$		0			0
$796$	−14.0000			−0.496217
$797$			18.5858i			0.658342i	0.944270	+	0.329171i	$0.106769\pi$
$797$			18.5858i			0.658342i	−0.944270	+	0.329171i	$0.893231\pi$
$798$		0			0
$799$	5.65685			0.200125
$800$	3.00000	+	4.00000i	0.106066	+	0.141421i
$801$		0			0
$802$			4.97056i			0.175517i
$803$		−	32.4853i		−	1.14638i
$804$		0			0
$805$	−1.41421	−	4.24264i	−0.0498445	−	0.149533i
$806$	4.97056			0.175081
$807$		0			0
$808$		−	13.3137i		−	0.468375i
$809$	−9.31371			−0.327453			−0.163726	−	0.986506i	$-0.552351\pi$
$809$	−9.31371			−0.327453			−0.163726	+	0.986506i	$0.552351\pi$
$810$		0			0
$811$	1.17157			0.0411395			0.0205697	−	0.999788i	$-0.493452\pi$
$811$	1.17157			0.0411395			0.0205697	+	0.999788i	$0.493452\pi$
$812$			7.31371i			0.256661i
$813$		0			0
$814$	2.48528			0.0871091
$815$	14.4853	−	4.82843i	0.507397	−	0.169132i
$816$		0			0
$817$			63.9411i			2.23702i
$818$			18.0000i			0.629355i
$819$		0			0
$820$	−4.82843	−	14.4853i	−0.168616	−	0.505848i
$821$	29.3137			1.02306			0.511528	−	0.859267i	$-0.329080\pi$
$821$	29.3137			1.02306			0.511528	+	0.859267i	$0.329080\pi$
$822$		0			0
$823$			24.2843i			0.846496i	0.906014	+	0.423248i	$0.139110\pi$
$823$			24.2843i			0.846496i	−0.906014	+	0.423248i	$0.860890\pi$
$824$	16.1421			0.562338
$825$		0			0
$826$	−16.9706			−0.590481
$827$		−	37.8995i		−	1.31789i	−0.752189	−	0.658947i	$-0.771002\pi$
$827$		−	37.8995i		−	1.31789i	0.752189	−	0.658947i	$-0.228998\pi$
$828$		0			0
$829$	6.28427			0.218262			0.109131	−	0.994027i	$-0.465193\pi$
$829$	6.28427			0.218262			0.109131	+	0.994027i	$0.465193\pi$
$830$	1.00000	+	3.00000i	0.0347105	+	0.104132i
$831$		0			0
$832$		−	0.828427i		−	0.0287205i
$833$		−	20.4853i		−	0.709773i
$834$		0			0
$835$	−6.72792	+	2.24264i	−0.232829	+	0.0776098i
$836$	26.4853			0.916013
$837$		0			0
$838$			39.0711i			1.34969i
$839$	33.9411			1.17178			0.585889	−	0.810391i	$-0.300745\pi$
$839$	33.9411			1.17178			0.585889	+	0.810391i	$0.300745\pi$
$840$		0			0
$841$	−15.6274			−0.538876
$842$		−	1.75736i		−	0.0605626i
$843$		0			0
$844$	3.51472			0.120982
$845$	8.70711	+	26.1213i	0.299534	+	0.898601i
$846$		0			0
$847$		−	14.0000i		−	0.481046i
$848$			10.5858i			0.363517i
$849$		0			0
$850$	27.3137	−	20.4853i	0.936852	−	0.702639i
$851$	0.585786			0.0200805
$852$		0			0
$853$			21.3137i			0.729767i	0.931053	+	0.364884i	$0.118891\pi$
$853$			21.3137i			0.729767i	−0.931053	+	0.364884i	$0.881109\pi$
$854$	1.17157			0.0400904
$855$		0			0
$856$	0.928932			0.0317502
$857$			50.8284i			1.73627i	0.496332	+	0.868133i	$0.334680\pi$
$857$			50.8284i			1.73627i	−0.496332	+	0.868133i	$0.665320\pi$
$858$		0			0
$859$	−24.2843			−0.828569			−0.414284	−	0.910148i	$-0.635968\pi$
$859$	−24.2843			−0.828569			−0.414284	+	0.910148i	$0.635968\pi$
$860$	−21.7279	+	7.24264i	−0.740916	+	0.246972i
$861$		0			0
$862$		−	8.48528i		−	0.289010i
$863$		−	22.6274i		−	0.770246i	−0.922865	−	0.385123i	$-0.874159\pi$
$863$		−	22.6274i		−	0.770246i	0.922865	−	0.385123i	$-0.125841\pi$
$864$		0			0
$865$	−18.7279	+	6.24264i	−0.636768	+	0.212256i
$866$	6.97056			0.236869
$867$		0			0
$868$			12.0000i			0.407307i
$869$	15.5147			0.526301
$870$		0			0
$871$	2.82843			0.0958376
$872$		−	12.5858i		−	0.426209i
$873$		0			0
$874$	6.24264			0.211160
$875$	−12.7279	+	18.3848i	−0.430282	+	0.621519i
$876$		0			0
$877$		−	16.8284i		−	0.568256i	−0.958786	−	0.284128i	$-0.908296\pi$
$877$		−	16.8284i		−	0.568256i	0.958786	−	0.284128i	$-0.0917040\pi$
$878$		0			0
$879$		0			0
$880$	3.00000	+	9.00000i	0.101130	+	0.303390i
$881$	−26.1421			−0.880751			−0.440375	−	0.897814i	$-0.645155\pi$
$881$	−26.1421			−0.880751			−0.440375	+	0.897814i	$0.645155\pi$
$882$		0			0
$883$		−	18.8284i		−	0.633627i	−0.948488	−	0.316814i	$-0.897387\pi$
$883$		−	18.8284i		−	0.633627i	0.948488	−	0.316814i	$-0.102613\pi$
$884$	−5.65685			−0.190261
$885$		0			0
$886$	−0.686292			−0.0230564
$887$		−	36.4264i		−	1.22308i	−0.791214	−	0.611540i	$-0.790551\pi$
$887$		−	36.4264i		−	1.22308i	0.791214	−	0.611540i	$-0.209449\pi$
$888$		0			0
$889$	−4.00000			−0.134156
$890$	19.4558	−	6.48528i	0.652161	−	0.217387i
$891$		0			0
$892$			16.0000i			0.535720i
$893$		−	5.17157i		−	0.173060i
$894$		0			0
$895$	−15.3137	−	45.9411i	−0.511881	−	1.53564i
$896$	2.00000			0.0668153
$897$		0			0
$898$			17.1716i			0.573023i
$899$	21.9411			0.731778
$900$		0			0
$901$	72.2843			2.40814
$902$		−	28.9706i		−	0.964614i
$903$		0			0
$904$	16.9706			0.564433
$905$	−14.4142	−	43.2426i	−0.479145	−	1.43743i
$906$		0			0
$907$		−	15.2132i		−	0.505146i	−0.967578	−	0.252573i	$-0.918723\pi$
$907$		−	15.2132i		−	0.505146i	0.967578	−	0.252573i	$-0.0812768\pi$
$908$		−	22.3848i		−	0.742865i
$909$		0			0
$910$	3.51472	−	1.17157i	0.116512	−	0.0388373i
$911$	39.7990			1.31860			0.659300	−	0.751880i	$-0.270853\pi$
$911$	39.7990			1.31860			0.659300	+	0.751880i	$0.270853\pi$
$912$		0			0
$913$			6.00000i			0.198571i
$914$	34.9706			1.15672
$915$		0			0
$916$	10.2426			0.338426
$917$		−	43.3137i		−	1.43034i
$918$		0			0
$919$	−38.9706			−1.28552			−0.642760	−	0.766068i	$-0.722211\pi$
$919$	−38.9706			−1.28552			−0.642760	+	0.766068i	$0.722211\pi$
$920$	0.707107	+	2.12132i	0.0233126	+	0.0699379i
$921$		0			0
$922$		−	7.17157i		−	0.236183i
$923$			4.68629i			0.154251i
$924$		0			0
$925$	−1.75736	−	2.34315i	−0.0577816	−	0.0770422i
$926$	30.9706			1.01776
$927$		0			0
$928$		−	3.65685i		−	0.120042i
$929$	13.8579			0.454662			0.227331	−	0.973818i	$-0.427000\pi$
$929$	13.8579			0.454662			0.227331	+	0.973818i	$0.427000\pi$
$930$		0			0
$931$	−18.7279			−0.613783
$932$		−	2.14214i		−	0.0701680i
$933$		0			0
$934$	31.3553			1.02598
$935$	61.4558	−	20.4853i	2.00982	−	0.669940i
$936$		0			0
$937$		−	40.4264i		−	1.32067i	−0.750970	−	0.660337i	$-0.770414\pi$
$937$		−	40.4264i		−	1.32067i	0.750970	−	0.660337i	$-0.229586\pi$
$938$			6.82843i			0.222956i
$939$		0			0
$940$	1.75736	−	0.585786i	0.0573187	−	0.0191062i
$941$	0.443651			0.0144626			0.00723130	−	0.999974i	$-0.497698\pi$
$941$	0.443651			0.0144626			0.00723130	+	0.999974i	$0.497698\pi$
$942$		0			0
$943$		−	6.82843i		−	0.222364i
$944$	8.48528			0.276172
$945$		0			0
$946$	−43.4558			−1.41287
$947$			49.4558i			1.60710i	0.595238	+	0.803549i	$0.297058\pi$
$947$			49.4558i			1.60710i	−0.595238	+	0.803549i	$0.702942\pi$
$948$		0			0
$949$	6.34315			0.205907
$950$	−18.7279	−	24.9706i	−0.607614	−	0.810152i
$951$		0			0
$952$		−	13.6569i		−	0.442621i
$953$		−	56.0833i		−	1.81672i	−0.418194	−	0.908358i	$-0.637337\pi$
$953$		−	56.0833i		−	1.81672i	0.418194	−	0.908358i	$-0.362663\pi$
$954$		0			0
$955$	7.65685	+	22.9706i	0.247770	+	0.743310i
$956$	16.8284			0.544270
$957$		0			0
$958$		−	27.3137i		−	0.882466i
$959$	16.9706			0.548008
$960$		0			0
$961$	5.00000			0.161290
$962$			0.485281i			0.0156461i
$963$		0			0
$964$	−6.48528			−0.208877
$965$	−44.4853	+	14.8284i	−1.43203	+	0.477344i
$966$		0			0
$967$		−	28.9706i		−	0.931630i	−0.884882	−	0.465815i	$-0.845761\pi$
$967$		−	28.9706i		−	0.931630i	0.884882	−	0.465815i	$-0.154239\pi$
$968$			7.00000i			0.224989i
$969$		0			0
$970$	−4.24264	−	12.7279i	−0.136223	−	0.408669i
$971$	54.3848			1.74529			0.872645	−	0.488355i	$-0.162403\pi$
$971$	54.3848			1.74529			0.872645	+	0.488355i	$0.162403\pi$
$972$		0			0
$973$		−	33.9411i		−	1.08810i
$974$	39.9411			1.27980
$975$		0			0
$976$	−0.585786			−0.0187506
$977$			19.7990i			0.633426i	0.948521	+	0.316713i	$0.102579\pi$
$977$			19.7990i			0.633426i	−0.948521	+	0.316713i	$0.897421\pi$
$978$		0			0
$979$	38.9117			1.24362
$980$	−2.12132	−	6.36396i	−0.0677631	−	0.203289i
$981$		0			0
$982$		−	18.6274i		−	0.594425i
$983$		−	3.51472i		−	0.112102i	−0.998428	−	0.0560511i	$-0.982149\pi$
$983$		−	3.51472i		−	0.112102i	0.998428	−	0.0560511i	$-0.0178510\pi$
$984$		0			0
$985$	43.7574	−	14.5858i	1.39423	−	0.464742i
$986$	−24.9706			−0.795225
$987$		0			0
$988$			5.17157i			0.164530i
$989$	−10.2426			−0.325697
$990$		0			0
$991$	−15.9411			−0.506387			−0.253193	−	0.967416i	$-0.581481\pi$
$991$	−15.9411			−0.506387			−0.253193	+	0.967416i	$0.581481\pi$
$992$		−	6.00000i		−	0.190500i
$993$		0			0
$994$	−11.3137			−0.358849
$995$	9.89949	+	29.6985i	0.313835	+	0.941505i
$996$		0			0
$997$		−	13.0294i		−	0.412646i	−0.978484	−	0.206323i	$-0.933850\pi$
$997$		−	13.0294i		−	0.412646i	0.978484	−	0.206323i	$-0.0661498\pi$
$998$		−	37.4558i		−	1.18564i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.d.a.829.2		4
3.2	odd	2		690.2.d.b.139.3	yes	4
5.4	even	2	inner	2070.2.d.a.829.4		4
15.2	even	4		3450.2.a.bg.1.1		2
15.8	even	4		3450.2.a.bk.1.1		2
15.14	odd	2		690.2.d.b.139.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.d.b.139.1	✓	4	15.14	odd	2
690.2.d.b.139.3	yes	4	3.2	odd	2
2070.2.d.a.829.2		4	1.1	even	1	trivial
2070.2.d.a.829.4		4	5.4	even	2	inner
3450.2.a.bg.1.1		2	15.2	even	4
3450.2.a.bk.1.1		2	15.8	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +(0.707107 + 2.12132i) q^{5} -2.00000i q^{7} +1.00000i q^{8} +(2.12132 - 0.707107i) q^{10} +4.24264 q^{11} +0.828427i q^{13} -2.00000 q^{14} +1.00000 q^{16} -6.82843i q^{17} -6.24264 q^{19} +(-0.707107 - 2.12132i) q^{20} -4.24264i q^{22} -1.00000i q^{23} +(-4.00000 + 3.00000i) q^{25} +0.828427 q^{26} +2.00000i q^{28} +3.65685 q^{29} +6.00000 q^{31} -1.00000i q^{32} -6.82843 q^{34} +(4.24264 - 1.41421i) q^{35} +0.585786i q^{37} +6.24264i q^{38} +(-2.12132 + 0.707107i) q^{40} +6.82843 q^{41} -10.2426i q^{43} -4.24264 q^{44} -1.00000 q^{46} +0.828427i q^{47} +3.00000 q^{49} +(3.00000 + 4.00000i) q^{50} -0.828427i q^{52} +10.5858i q^{53} +(3.00000 + 9.00000i) q^{55} +2.00000 q^{56} -3.65685i q^{58} +8.48528 q^{59} -0.585786 q^{61} -6.00000i q^{62} -1.00000 q^{64} +(-1.75736 + 0.585786i) q^{65} -3.41421i q^{67} +6.82843i q^{68} +(-1.41421 - 4.24264i) q^{70} +5.65685 q^{71} -7.65685i q^{73} +0.585786 q^{74} +6.24264 q^{76} -8.48528i q^{77} +3.65685 q^{79} +(0.707107 + 2.12132i) q^{80} -6.82843i q^{82} +1.41421i q^{83} +(14.4853 - 4.82843i) q^{85} -10.2426 q^{86} +4.24264i q^{88} +9.17157 q^{89} +1.65685 q^{91} +1.00000i q^{92} +0.828427 q^{94} +(-4.41421 - 13.2426i) q^{95} -6.00000i q^{97} -3.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 8 q^{14} + 4 q^{16} - 8 q^{19} - 16 q^{25} - 8 q^{26} - 8 q^{29} + 24 q^{31} - 16 q^{34} + 16 q^{41} - 4 q^{46} + 12 q^{49} + 12 q^{50} + 12 q^{55} + 8 q^{56} - 8 q^{61} - 4 q^{64} - 24 q^{65} + 8 q^{74} + 8 q^{76} - 8 q^{79} + 24 q^{85} - 24 q^{86} + 48 q^{89} - 16 q^{91} - 8 q^{94} - 12 q^{95}+O(q^{100}) \) 4 * q - 4 * q^4 - 8 * q^14 + 4 * q^16 - 8 * q^19 - 16 * q^25 - 8 * q^26 - 8 * q^29 + 24 * q^31 - 16 * q^34 + 16 * q^41 - 4 * q^46 + 12 * q^49 + 12 * q^50 + 12 * q^55 + 8 * q^56 - 8 * q^61 - 4 * q^64 - 24 * q^65 + 8 * q^74 + 8 * q^76 - 8 * q^79 + 24 * q^85 - 24 * q^86 + 48 * q^89 - 16 * q^91 - 8 * q^94 - 12 * q^95

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