LMFDB - Embedded newform 600.2.r.d.593.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,2,Mod(257,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))

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

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

chi := DirichletCharacter("600.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	600.r (of order $4$, degree $2$, 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:	$4.79102412128$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	600.593
Dual form			600.2.r.d.257.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.292893 + 1.70711i) q^{3} +(-3.00000 + 3.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} -1.41421i q^{11} +(-4.24264 - 4.24264i) q^{17} +4.00000i q^{19} +(-6.00000 - 4.24264i) q^{21} +(-2.82843 + 2.82843i) q^{23} +(-2.53553 - 4.53553i) q^{27} +1.41421 q^{29} -2.00000 q^{31} +(2.41421 - 0.414214i) q^{33} +(2.00000 - 2.00000i) q^{37} +5.65685i q^{41} +(2.00000 + 2.00000i) q^{43} +(5.65685 + 5.65685i) q^{47} -11.0000i q^{49} +(6.00000 - 8.48528i) q^{51} +(-8.48528 + 8.48528i) q^{53} +(-6.82843 + 1.17157i) q^{57} +1.41421 q^{59} -6.00000 q^{61} +(5.48528 - 11.4853i) q^{63} +(-4.00000 + 4.00000i) q^{67} +(-5.65685 - 4.00000i) q^{69} +2.82843i q^{71} +(-3.00000 - 3.00000i) q^{73} +(4.24264 + 4.24264i) q^{77} +10.0000i q^{79} +(7.00000 - 5.65685i) q^{81} +(2.82843 - 2.82843i) q^{83} +(0.414214 + 2.41421i) q^{87} -2.82843 q^{89} +(-0.585786 - 3.41421i) q^{93} +(13.0000 - 13.0000i) q^{97} +(1.41421 + 4.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 12 q^{7} - 24 q^{21} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{43} + 24 q^{51} - 16 q^{57} - 24 q^{61} - 12 q^{63} - 16 q^{67} - 12 q^{73} + 28 q^{81} - 4 q^{87} - 8 q^{93} + 52 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^7 - 24 * q^21 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^43 + 24 * q^51 - 16 * q^57 - 24 * q^61 - 12 * q^63 - 16 * q^67 - 12 * q^73 + 28 * q^81 - 4 * q^87 - 8 * q^93 + 52 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.292893	+	1.70711i	0.169102	+	0.985599i
$3$	0.292893	+	1.70711i	0.169102	+	0.985599i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$		0			0
$9$	−2.82843	+	1.00000i	−0.942809	+	0.333333i
$10$		0			0
$11$		−	1.41421i		−	0.426401i	−0.977008	−	0.213201i	$-0.931611\pi$
$11$		−	1.41421i		−	0.426401i	0.977008	−	0.213201i	$-0.0683888\pi$
$12$		0			0
$13$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$13$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.24264	−	4.24264i	−1.02899	−	1.02899i	−0.999567	−	0.0294245i	$-0.990633\pi$
$17$	−4.24264	−	4.24264i	−1.02899	−	1.02899i	−0.0294245	−	0.999567i	$-0.509367\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$	−6.00000	−	4.24264i	−1.30931	−	0.925820i
$22$		0			0
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	−0.937568	−	0.347801i	$-0.886929\pi$
$23$	−2.82843	+	2.82843i	−0.589768	+	0.589768i	0.347801	+	0.937568i	$0.386929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−2.53553	−	4.53553i	−0.487964	−	0.872864i
$28$		0			0
$29$	1.41421			0.262613			0.131306	−	0.991342i	$-0.458083\pi$
$29$	1.41421			0.262613			0.131306	+	0.991342i	$0.458083\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$	2.41421	−	0.414214i	0.420261	−	0.0721053i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	−	2.00000i	0.328798	−	0.328798i	−0.523331	−	0.852129i	$-0.675311\pi$
$37$	2.00000	−	2.00000i	0.328798	−	0.328798i	0.852129	+	0.523331i	$0.175311\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$		0			0
$43$	2.00000	+	2.00000i	0.304997	+	0.304997i	0.842965	−	0.537968i	$-0.180808\pi$
$43$	2.00000	+	2.00000i	0.304997	+	0.304997i	−0.537968	+	0.842965i	$0.680808\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	0.986840	−	0.161703i	$-0.0516985\pi$
$47$	5.65685	+	5.65685i	0.825137	+	0.825137i	−0.161703	+	0.986840i	$0.551699\pi$
$48$		0			0
$49$		−	11.0000i		−	1.57143i
$50$		0			0
$51$	6.00000	−	8.48528i	0.840168	−	1.18818i
$52$		0			0
$53$	−8.48528	+	8.48528i	−1.16554	+	1.16554i	−0.182300	+	0.983243i	$0.558354\pi$
$53$	−8.48528	+	8.48528i	−1.16554	+	1.16554i	−0.983243	+	0.182300i	$0.941646\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.82843	+	1.17157i	−0.904447	+	0.155179i
$58$		0			0
$59$	1.41421			0.184115			0.0920575	−	0.995754i	$-0.470656\pi$
$59$	1.41421			0.184115			0.0920575	+	0.995754i	$0.470656\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$		0			0
$63$	5.48528	−	11.4853i	0.691080	−	1.44701i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	−0.907889	−	0.419211i	$-0.862307\pi$
$67$	−4.00000	+	4.00000i	−0.488678	+	0.488678i	0.419211	+	0.907889i	$0.362307\pi$
$68$		0			0
$69$	−5.65685	−	4.00000i	−0.681005	−	0.481543i
$70$		0			0
$71$			2.82843i			0.335673i	0.985815	+	0.167836i	$0.0536780\pi$
$71$			2.82843i			0.335673i	−0.985815	+	0.167836i	$0.946322\pi$
$72$		0			0
$73$	−3.00000	−	3.00000i	−0.351123	−	0.351123i	0.509404	−	0.860527i	$-0.329866\pi$
$73$	−3.00000	−	3.00000i	−0.351123	−	0.351123i	−0.860527	+	0.509404i	$0.829866\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.24264	+	4.24264i	0.483494	+	0.483494i
$78$		0			0
$79$			10.0000i			1.12509i	0.826767	+	0.562544i	$0.190177\pi$
$79$			10.0000i			1.12509i	−0.826767	+	0.562544i	$0.809823\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	−0.534628	−	0.845088i	$-0.679548\pi$
$83$	2.82843	−	2.82843i	0.310460	−	0.310460i	0.845088	+	0.534628i	$0.179548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.414214	+	2.41421i	0.0444084	+	0.258831i
$88$		0			0
$89$	−2.82843			−0.299813			−0.149906	−	0.988700i	$-0.547897\pi$
$89$	−2.82843			−0.299813			−0.149906	+	0.988700i	$0.547897\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−0.585786	−	3.41421i	−0.0607432	−	0.354037i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.0000	−	13.0000i	1.31995	−	1.31995i	0.406138	−	0.913812i	$-0.366875\pi$
$97$	13.0000	−	13.0000i	1.31995	−	1.31995i	0.913812	−	0.406138i	$-0.133125\pi$
$98$		0			0
$99$	1.41421	+	4.00000i	0.142134	+	0.402015i
$100$		0			0
$101$			15.5563i			1.54791i	0.633238	+	0.773957i	$0.281726\pi$
$101$			15.5563i			1.54791i	−0.633238	+	0.773957i	$0.718274\pi$
$102$		0			0
$103$	11.0000	+	11.0000i	1.08386	+	1.08386i	0.996145	+	0.0877167i	$0.0279570\pi$
$103$	11.0000	+	11.0000i	1.08386	+	1.08386i	0.0877167	+	0.996145i	$0.472043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	0.557047	−	0.830481i	$-0.311934\pi$
$107$	−2.82843	−	2.82843i	−0.273434	−	0.273434i	−0.830481	+	0.557047i	$0.811934\pi$
$108$		0			0
$109$			10.0000i			0.957826i	0.877862	+	0.478913i	$0.158969\pi$
$109$			10.0000i			0.957826i	−0.877862	+	0.478913i	$0.841031\pi$
$110$		0			0
$111$	4.00000	+	2.82843i	0.379663	+	0.268462i
$112$		0			0
$113$	4.24264	−	4.24264i	0.399114	−	0.399114i	−0.478806	−	0.877920i	$-0.658930\pi$
$113$	4.24264	−	4.24264i	0.399114	−	0.399114i	0.877920	+	0.478806i	$0.158930\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	25.4558			2.33353
$120$		0			0
$121$	9.00000			0.818182
$122$		0			0
$123$	−9.65685	+	1.65685i	−0.870729	+	0.149394i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	3.00000	−	3.00000i	0.266207	−	0.266207i	−0.561363	−	0.827570i	$-0.689723\pi$
$127$	3.00000	−	3.00000i	0.266207	−	0.266207i	0.827570	+	0.561363i	$0.189723\pi$
$128$		0			0
$129$	−2.82843	+	4.00000i	−0.249029	+	0.352180i
$130$		0			0
$131$			15.5563i			1.35916i	0.733599	+	0.679582i	$0.237839\pi$
$131$			15.5563i			1.35916i	−0.733599	+	0.679582i	$0.762161\pi$
$132$		0			0
$133$	−12.0000	−	12.0000i	−1.04053	−	1.04053i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	12.7279	+	12.7279i	1.08742	+	1.08742i	0.995793	+	0.0916263i	$0.0292065\pi$
$137$	12.7279	+	12.7279i	1.08742	+	1.08742i	0.0916263	+	0.995793i	$0.470793\pi$
$138$		0			0
$139$		−	16.0000i		−	1.35710i	−0.734553	−	0.678551i	$-0.762608\pi$
$139$		−	16.0000i		−	1.35710i	0.734553	−	0.678551i	$-0.237392\pi$
$140$		0			0
$141$	−8.00000	+	11.3137i	−0.673722	+	0.952786i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	18.7782	−	3.22183i	1.54880	−	0.265732i
$148$		0			0
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$	8.00000			0.651031			0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000			0.651031			0.325515	+	0.945537i	$0.394462\pi$
$152$		0			0
$153$	16.2426	+	7.75736i	1.31314	+	0.627145i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	−0.950178	−	0.311708i	$-0.899099\pi$
$157$	−8.00000	+	8.00000i	−0.638470	+	0.638470i	0.311708	+	0.950178i	$0.399099\pi$
$158$		0			0
$159$	−16.9706	−	12.0000i	−1.34585	−	0.951662i
$160$		0			0
$161$		−	16.9706i		−	1.33747i
$162$		0			0
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	0.0583818	−	0.998294i	$-0.481406\pi$
$163$	−12.0000	−	12.0000i	−0.939913	−	0.939913i	−0.998294	+	0.0583818i	$0.981406\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$		−	13.0000i		−	1.00000i
$170$		0			0
$171$	−4.00000	−	11.3137i	−0.305888	−	0.865181i
$172$		0			0
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	−0.222327	−	0.974972i	$-0.571365\pi$
$173$	9.89949	−	9.89949i	0.752645	−	0.752645i	0.974972	+	0.222327i	$0.0713654\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	0.414214	+	2.41421i	0.0311342	+	0.181463i
$178$		0			0
$179$	−15.5563			−1.16274			−0.581368	−	0.813641i	$-0.697482\pi$
$179$	−15.5563			−1.16274			−0.581368	+	0.813641i	$0.697482\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	−1.75736	−	10.2426i	−0.129908	−	0.757158i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.00000	+	6.00000i	−0.438763	+	0.438763i
$188$		0			0
$189$	21.2132	+	6.00000i	1.54303	+	0.436436i
$190$		0			0
$191$		−	2.82843i		−	0.204658i	−0.994751	−	0.102329i	$-0.967371\pi$
$191$		−	2.82843i		−	0.204658i	0.994751	−	0.102329i	$-0.0326294\pi$
$192$		0			0
$193$	7.00000	+	7.00000i	0.503871	+	0.503871i	0.912639	−	0.408768i	$-0.134041\pi$
$193$	7.00000	+	7.00000i	0.503871	+	0.503871i	−0.408768	+	0.912639i	$0.634041\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−8.00000	−	5.65685i	−0.564276	−	0.399004i
$202$		0			0
$203$	−4.24264	+	4.24264i	−0.297775	+	0.297775i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	5.17157	−	10.8284i	0.359449	−	0.752628i
$208$		0			0
$209$	5.65685			0.391293
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$	−4.82843	+	0.828427i	−0.330838	+	0.0567629i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	6.00000	−	6.00000i	0.407307	−	0.407307i
$218$		0			0
$219$	4.24264	−	6.00000i	0.286691	−	0.405442i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−3.00000	−	3.00000i	−0.200895	−	0.200895i	0.599489	−	0.800383i	$-0.295371\pi$
$223$	−3.00000	−	3.00000i	−0.200895	−	0.200895i	−0.800383	+	0.599489i	$0.795371\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.24264	+	4.24264i	0.281594	+	0.281594i	0.833744	−	0.552151i	$-0.186193\pi$
$227$	4.24264	+	4.24264i	0.281594	+	0.281594i	−0.552151	+	0.833744i	$0.686193\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$	−6.00000	+	8.48528i	−0.394771	+	0.558291i
$232$		0			0
$233$	−7.07107	+	7.07107i	−0.463241	+	0.463241i	−0.899716	−	0.436475i	$-0.856227\pi$
$233$	−7.07107	+	7.07107i	−0.463241	+	0.463241i	0.436475	+	0.899716i	$0.356227\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−17.0711	+	2.92893i	−1.10889	+	0.190255i
$238$		0			0
$239$	−25.4558			−1.64660			−0.823301	−	0.567605i	$-0.807870\pi$
$239$	−25.4558			−1.64660			−0.823301	+	0.567605i	$0.807870\pi$
$240$		0			0
$241$	20.0000			1.28831			0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000			1.28831			0.644157	+	0.764894i	$0.277208\pi$
$242$		0			0
$243$	11.7071	+	10.2929i	0.751011	+	0.660289i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	5.65685	+	4.00000i	0.358489	+	0.253490i
$250$		0			0
$251$			12.7279i			0.803379i	0.915776	+	0.401690i	$0.131577\pi$
$251$			12.7279i			0.803379i	−0.915776	+	0.401690i	$0.868423\pi$
$252$		0			0
$253$	4.00000	+	4.00000i	0.251478	+	0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−15.5563	−	15.5563i	−0.970378	−	0.970378i	0.0291953	−	0.999574i	$-0.490706\pi$
$257$	−15.5563	−	15.5563i	−0.970378	−	0.970378i	−0.999574	+	0.0291953i	$0.990706\pi$
$258$		0			0
$259$			12.0000i			0.745644i
$260$		0			0
$261$	−4.00000	+	1.41421i	−0.247594	+	0.0875376i
$262$		0			0
$263$	−11.3137	+	11.3137i	−0.697633	+	0.697633i	−0.963899	−	0.266266i	$-0.914210\pi$
$263$	−11.3137	+	11.3137i	−0.697633	+	0.697633i	0.266266	+	0.963899i	$0.414210\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.828427	−	4.82843i	−0.0506989	−	0.295495i
$268$		0			0
$269$	9.89949			0.603583			0.301791	−	0.953374i	$-0.402415\pi$
$269$	9.89949			0.603583			0.301791	+	0.953374i	$0.402415\pi$
$270$		0			0
$271$	−24.0000			−1.45790			−0.728948	−	0.684569i	$-0.759990\pi$
$271$	−24.0000			−1.45790			−0.728948	+	0.684569i	$0.759990\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.00000	+	2.00000i	−0.120168	+	0.120168i	−0.764634	−	0.644465i	$-0.777080\pi$
$277$	−2.00000	+	2.00000i	−0.120168	+	0.120168i	0.644465	+	0.764634i	$0.277080\pi$
$278$		0			0
$279$	5.65685	−	2.00000i	0.338667	−	0.119737i
$280$		0			0
$281$		−	25.4558i		−	1.51857i	−0.650759	−	0.759284i	$-0.725549\pi$
$281$		−	25.4558i		−	1.51857i	0.650759	−	0.759284i	$-0.274451\pi$
$282$		0			0
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	0.815928	−	0.578153i	$-0.196226\pi$
$283$	4.00000	+	4.00000i	0.237775	+	0.237775i	−0.578153	+	0.815928i	$0.696226\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−16.9706	−	16.9706i	−1.00174	−	1.00174i
$288$		0			0
$289$			19.0000i			1.11765i
$290$		0			0
$291$	26.0000	+	18.3848i	1.52415	+	1.07773i
$292$		0			0
$293$	−5.65685	+	5.65685i	−0.330477	+	0.330477i	−0.852768	−	0.522291i	$-0.825078\pi$
$293$	−5.65685	+	5.65685i	−0.330477	+	0.330477i	0.522291	+	0.852768i	$0.325078\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−6.41421	+	3.58579i	−0.372190	+	0.208068i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$	−26.5563	+	4.55635i	−1.52562	+	0.261755i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10.0000	+	10.0000i	−0.570730	+	0.570730i	−0.932332	−	0.361602i	$-0.882230\pi$
$307$	−10.0000	+	10.0000i	−0.570730	+	0.570730i	0.361602	+	0.932332i	$0.382230\pi$
$308$		0			0
$309$	−15.5563	+	22.0000i	−0.884970	+	1.25154i
$310$		0			0
$311$		−	31.1127i		−	1.76424i	−0.471025	−	0.882120i	$-0.656116\pi$
$311$		−	31.1127i		−	1.76424i	0.471025	−	0.882120i	$-0.343884\pi$
$312$		0			0
$313$	15.0000	+	15.0000i	0.847850	+	0.847850i	0.989865	−	0.142014i	$-0.0453579\pi$
$313$	15.0000	+	15.0000i	0.847850	+	0.847850i	−0.142014	+	0.989865i	$0.545358\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.41421	−	1.41421i	−0.0794301	−	0.0794301i	0.666276	−	0.745706i	$-0.267887\pi$
$317$	−1.41421	−	1.41421i	−0.0794301	−	0.0794301i	−0.745706	+	0.666276i	$0.767887\pi$
$318$		0			0
$319$		−	2.00000i		−	0.111979i
$320$		0			0
$321$	4.00000	−	5.65685i	0.223258	−	0.315735i
$322$		0			0
$323$	16.9706	−	16.9706i	0.944267	−	0.944267i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−17.0711	+	2.92893i	−0.944032	+	0.161970i
$328$		0			0
$329$	−33.9411			−1.87123
$330$		0			0
$331$	−16.0000			−0.879440			−0.439720	−	0.898135i	$-0.644922\pi$
$331$	−16.0000			−0.879440			−0.439720	+	0.898135i	$0.644922\pi$
$332$		0			0
$333$	−3.65685	+	7.65685i	−0.200394	+	0.419593i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	−0.985687	−	0.168585i	$-0.946080\pi$
$337$	−15.0000	+	15.0000i	−0.817102	+	0.817102i	0.168585	+	0.985687i	$0.446080\pi$
$338$		0			0
$339$	8.48528	+	6.00000i	0.460857	+	0.325875i
$340$		0			0
$341$			2.82843i			0.153168i
$342$		0			0
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.7279	−	12.7279i	−0.683271	−	0.683271i	0.277465	−	0.960736i	$-0.410506\pi$
$347$	−12.7279	−	12.7279i	−0.683271	−	0.683271i	−0.960736	+	0.277465i	$0.910506\pi$
$348$		0			0
$349$			2.00000i			0.107058i	0.998566	+	0.0535288i	$0.0170469\pi$
$349$			2.00000i			0.107058i	−0.998566	+	0.0535288i	$0.982953\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	18.3848	−	18.3848i	0.978523	−	0.978523i	−0.0212513	−	0.999774i	$-0.506765\pi$
$353$	18.3848	−	18.3848i	0.978523	−	0.978523i	0.999774	+	0.0212513i	$0.00676499\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	7.45584	+	43.4558i	0.394605	+	2.29993i
$358$		0			0
$359$	−22.6274			−1.19423			−0.597115	−	0.802156i	$-0.703686\pi$
$359$	−22.6274			−1.19423			−0.597115	+	0.802156i	$0.703686\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	2.63604	+	15.3640i	0.138356	+	0.806399i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	7.00000	−	7.00000i	0.365397	−	0.365397i	−0.500398	−	0.865795i	$-0.666813\pi$
$367$	7.00000	−	7.00000i	0.365397	−	0.365397i	0.865795	+	0.500398i	$0.166813\pi$
$368$		0			0
$369$	−5.65685	−	16.0000i	−0.294484	−	0.832927i
$370$		0			0
$371$		−	50.9117i		−	2.64320i
$372$		0			0
$373$	−12.0000	−	12.0000i	−0.621336	−	0.621336i	0.324537	−	0.945873i	$-0.394792\pi$
$373$	−12.0000	−	12.0000i	−0.621336	−	0.621336i	−0.945873	+	0.324537i	$0.894792\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	24.0000i		−	1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$		−	24.0000i		−	1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$		0			0
$381$	6.00000	+	4.24264i	0.307389	+	0.217357i
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.65685	−	3.65685i	−0.389220	−	0.185888i
$388$		0			0
$389$	12.7279			0.645331			0.322666	−	0.946513i	$-0.395421\pi$
$389$	12.7279			0.645331			0.322666	+	0.946513i	$0.395421\pi$
$390$		0			0
$391$	24.0000			1.21373
$392$		0			0
$393$	−26.5563	+	4.55635i	−1.33959	+	0.229837i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	10.0000	−	10.0000i	0.501886	−	0.501886i	−0.410138	−	0.912024i	$-0.634519\pi$
$397$	10.0000	−	10.0000i	0.501886	−	0.501886i	0.912024	+	0.410138i	$0.134519\pi$
$398$		0			0
$399$	16.9706	−	24.0000i	0.849591	−	1.20150i
$400$		0			0
$401$			14.1421i			0.706225i	0.935581	+	0.353112i	$0.114877\pi$
$401$			14.1421i			0.706225i	−0.935581	+	0.353112i	$0.885123\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.82843	−	2.82843i	−0.140200	−	0.140200i
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−18.0000	+	25.4558i	−0.887875	+	1.25564i
$412$		0			0
$413$	−4.24264	+	4.24264i	−0.208767	+	0.208767i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	27.3137	−	4.68629i	1.33756	−	0.229489i
$418$		0			0
$419$	24.0416			1.17451			0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416			1.17451			0.587255	+	0.809402i	$0.300208\pi$
$420$		0			0
$421$	−34.0000			−1.65706			−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000			−1.65706			−0.828529	+	0.559946i	$0.810822\pi$
$422$		0			0
$423$	−21.6569	−	10.3431i	−1.05299	−	0.502901i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	18.0000	−	18.0000i	0.871081	−	0.871081i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			22.6274i			1.08992i	0.838461	+	0.544962i	$0.183456\pi$
$431$			22.6274i			1.08992i	−0.838461	+	0.544962i	$0.816544\pi$
$432$		0			0
$433$	1.00000	+	1.00000i	0.0480569	+	0.0480569i	0.730727	−	0.682670i	$-0.239181\pi$
$433$	1.00000	+	1.00000i	0.0480569	+	0.0480569i	−0.682670	+	0.730727i	$0.739181\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−11.3137	−	11.3137i	−0.541208	−	0.541208i
$438$		0			0
$439$			8.00000i			0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$			8.00000i			0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$		0			0
$441$	11.0000	+	31.1127i	0.523810	+	1.48156i
$442$		0			0
$443$	7.07107	−	7.07107i	0.335957	−	0.335957i	−0.518887	−	0.854843i	$-0.673653\pi$
$443$	7.07107	−	7.07107i	0.335957	−	0.335957i	0.854843	+	0.518887i	$0.173653\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	1.24264	+	7.24264i	0.0587749	+	0.342565i
$448$		0			0
$449$	19.7990			0.934372			0.467186	−	0.884159i	$-0.345268\pi$
$449$	19.7990			0.934372			0.467186	+	0.884159i	$0.345268\pi$
$450$		0			0
$451$	8.00000			0.376705
$452$		0			0
$453$	2.34315	+	13.6569i	0.110091	+	0.641655i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000	+	1.00000i	−0.0467780	+	0.0467780i	−0.730109	−	0.683331i	$-0.760531\pi$
$457$	−1.00000	+	1.00000i	−0.0467780	+	0.0467780i	0.683331	+	0.730109i	$0.260531\pi$
$458$		0			0
$459$	−8.48528	+	30.0000i	−0.396059	+	1.40028i
$460$		0			0
$461$		−	32.5269i		−	1.51493i	−0.652876	−	0.757465i	$-0.726438\pi$
$461$		−	32.5269i		−	1.51493i	0.652876	−	0.757465i	$-0.273562\pi$
$462$		0			0
$463$	−1.00000	−	1.00000i	−0.0464739	−	0.0464739i	0.683488	−	0.729962i	$-0.260462\pi$
$463$	−1.00000	−	1.00000i	−0.0464739	−	0.0464739i	−0.729962	+	0.683488i	$0.760462\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.4558	+	25.4558i	1.17796	+	1.17796i	0.980264	+	0.197692i	$0.0633445\pi$
$467$	25.4558	+	25.4558i	1.17796	+	1.17796i	0.197692	+	0.980264i	$0.436655\pi$
$468$		0			0
$469$		−	24.0000i		−	1.10822i
$470$		0			0
$471$	−16.0000	−	11.3137i	−0.737241	−	0.521308i
$472$		0			0
$473$	2.82843	−	2.82843i	0.130051	−	0.130051i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	15.5147	−	32.4853i	0.710370	−	1.48740i
$478$		0			0
$479$	2.82843			0.129234			0.0646171	−	0.997910i	$-0.479417\pi$
$479$	2.82843			0.129234			0.0646171	+	0.997910i	$0.479417\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	28.9706	−	4.97056i	1.31821	−	0.226168i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	3.00000	−	3.00000i	0.135943	−	0.135943i	−0.635861	−	0.771804i	$-0.719355\pi$
$487$	3.00000	−	3.00000i	0.135943	−	0.135943i	0.771804	+	0.635861i	$0.219355\pi$
$488$		0			0
$489$	16.9706	−	24.0000i	0.767435	−	1.08532i
$490$		0			0
$491$			15.5563i			0.702048i	0.936366	+	0.351024i	$0.114166\pi$
$491$			15.5563i			0.702048i	−0.936366	+	0.351024i	$0.885834\pi$
$492$		0			0
$493$	−6.00000	−	6.00000i	−0.270226	−	0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.48528	−	8.48528i	−0.380617	−	0.380617i
$498$		0			0
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	25.4558	−	25.4558i	1.13502	−	1.13502i	0.145690	−	0.989330i	$-0.453460\pi$
$503$	25.4558	−	25.4558i	1.13502	−	1.13502i	0.989330	−	0.145690i	$-0.0465401\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	22.1924	−	3.80761i	0.985599	−	0.169102i
$508$		0			0
$509$	−38.1838			−1.69247			−0.846233	−	0.532813i	$-0.821135\pi$
$509$	−38.1838			−1.69247			−0.846233	+	0.532813i	$0.821135\pi$
$510$		0			0
$511$	18.0000			0.796273
$512$		0			0
$513$	18.1421	−	10.1421i	0.800995	−	0.447786i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	8.00000	−	8.00000i	0.351840	−	0.351840i
$518$		0			0
$519$	19.7990	+	14.0000i	0.869079	+	0.614532i
$520$		0			0
$521$			8.48528i			0.371747i	0.982574	+	0.185873i	$0.0595115\pi$
$521$			8.48528i			0.371747i	−0.982574	+	0.185873i	$0.940489\pi$
$522$		0			0
$523$	−24.0000	−	24.0000i	−1.04945	−	1.04945i	−0.998712	−	0.0507346i	$-0.983844\pi$
$523$	−24.0000	−	24.0000i	−1.04945	−	1.04945i	−0.0507346	−	0.998712i	$-0.516156\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	8.48528	+	8.48528i	0.369625	+	0.369625i
$528$		0			0
$529$			7.00000i			0.304348i
$530$		0			0
$531$	−4.00000	+	1.41421i	−0.173585	+	0.0613716i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−4.55635	−	26.5563i	−0.196621	−	1.14599i
$538$		0			0
$539$	−15.5563			−0.670059
$540$		0			0
$541$	14.0000			0.601907			0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000			0.601907			0.300954	+	0.953639i	$0.402695\pi$
$542$		0			0
$543$	2.92893	+	17.0711i	0.125693	+	0.732590i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−30.0000	+	30.0000i	−1.28271	+	1.28271i	−0.343586	+	0.939121i	$0.611642\pi$
$547$	−30.0000	+	30.0000i	−1.28271	+	1.28271i	−0.939121	+	0.343586i	$0.888358\pi$
$548$		0			0
$549$	16.9706	−	6.00000i	0.724286	−	0.256074i
$550$		0			0
$551$			5.65685i			0.240990i
$552$		0			0
$553$	−30.0000	−	30.0000i	−1.27573	−	1.27573i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.9706	+	16.9706i	0.719066	+	0.719066i	0.968414	−	0.249348i	$-0.0802163\pi$
$557$	16.9706	+	16.9706i	0.719066	+	0.719066i	−0.249348	+	0.968414i	$0.580216\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−12.0000	−	8.48528i	−0.506640	−	0.358249i
$562$		0			0
$563$	−24.0416	+	24.0416i	−1.01323	+	1.01323i	−0.0133227	+	0.999911i	$0.504241\pi$
$563$	−24.0416	+	24.0416i	−1.01323	+	1.01323i	−0.999911	+	0.0133227i	$0.995759\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−4.02944	+	37.9706i	−0.169220	+	1.59461i
$568$		0			0
$569$	14.1421			0.592869			0.296435	−	0.955053i	$-0.404202\pi$
$569$	14.1421			0.592869			0.296435	+	0.955053i	$0.404202\pi$
$570$		0			0
$571$	32.0000			1.33916			0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000			1.33916			0.669579	+	0.742741i	$0.266474\pi$
$572$		0			0
$573$	4.82843	−	0.828427i	0.201710	−	0.0346080i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−21.0000	+	21.0000i	−0.874241	+	0.874241i	−0.992931	−	0.118690i	$-0.962131\pi$
$577$	−21.0000	+	21.0000i	−0.874241	+	0.874241i	0.118690	+	0.992931i	$0.462131\pi$
$578$		0			0
$579$	−9.89949	+	14.0000i	−0.411409	+	0.581820i
$580$		0			0
$581$			16.9706i			0.704058i
$582$		0			0
$583$	12.0000	+	12.0000i	0.496989	+	0.496989i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.7990	+	19.7990i	0.817192	+	0.817192i	0.985700	−	0.168508i	$-0.0538950\pi$
$587$	19.7990	+	19.7990i	0.817192	+	0.817192i	−0.168508	+	0.985700i	$0.553895\pi$
$588$		0			0
$589$		−	8.00000i		−	0.329634i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.41421	+	1.41421i	−0.0580748	+	0.0580748i	−0.735548	−	0.677473i	$-0.763075\pi$
$593$	−1.41421	+	1.41421i	−0.0580748	+	0.0580748i	0.677473	+	0.735548i	$0.263075\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.3137			−0.462266			−0.231133	−	0.972922i	$-0.574243\pi$
$599$	−11.3137			−0.462266			−0.231133	+	0.972922i	$0.574243\pi$
$600$		0			0
$601$	22.0000			0.897399			0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000			0.897399			0.448699	+	0.893683i	$0.351887\pi$
$602$		0			0
$603$	7.31371	−	15.3137i	0.297837	−	0.623622i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	25.0000	−	25.0000i	1.01472	−	1.01472i	0.0148286	−	0.999890i	$-0.495280\pi$
$607$	25.0000	−	25.0000i	1.01472	−	1.01472i	0.999890	−	0.0148286i	$-0.00472028\pi$
$608$		0			0
$609$	−8.48528	−	6.00000i	−0.343841	−	0.243132i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	30.0000	+	30.0000i	1.21169	+	1.21169i	0.970471	+	0.241218i	$0.0775467\pi$
$613$	30.0000	+	30.0000i	1.21169	+	1.21169i	0.241218	+	0.970471i	$0.422453\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.24264	+	4.24264i	0.170802	+	0.170802i	0.787332	−	0.616530i	$-0.211462\pi$
$617$	4.24264	+	4.24264i	0.170802	+	0.170802i	−0.616530	+	0.787332i	$0.711462\pi$
$618$		0			0
$619$		−	48.0000i		−	1.92928i	−0.263566	−	0.964641i	$-0.584899\pi$
$619$		−	48.0000i		−	1.92928i	0.263566	−	0.964641i	$-0.415101\pi$
$620$		0			0
$621$	20.0000	+	5.65685i	0.802572	+	0.227002i
$622$		0			0
$623$	8.48528	−	8.48528i	0.339956	−	0.339956i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	1.65685	+	9.65685i	0.0661684	+	0.385658i
$628$		0			0
$629$	−16.9706			−0.676661
$630$		0			0
$631$	34.0000			1.35352			0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000			1.35352			0.676759	+	0.736204i	$0.263384\pi$
$632$		0			0
$633$	−4.68629	−	27.3137i	−0.186263	−	1.08562i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.82843	−	8.00000i	−0.111891	−	0.316475i
$640$		0			0
$641$			28.2843i			1.11716i	0.829450	+	0.558581i	$0.188654\pi$
$641$			28.2843i			1.11716i	−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	−8.00000	−	8.00000i	−0.315489	−	0.315489i	0.531542	−	0.847032i	$-0.321613\pi$
$643$	−8.00000	−	8.00000i	−0.315489	−	0.315489i	−0.847032	+	0.531542i	$0.821613\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.82843	+	2.82843i	0.111197	+	0.111197i	0.760516	−	0.649319i	$-0.224946\pi$
$647$	2.82843	+	2.82843i	0.111197	+	0.111197i	−0.649319	+	0.760516i	$0.724946\pi$
$648$		0			0
$649$		−	2.00000i		−	0.0785069i
$650$		0			0
$651$	12.0000	+	8.48528i	0.470317	+	0.332564i
$652$		0			0
$653$	4.24264	−	4.24264i	0.166027	−	0.166027i	−0.619203	−	0.785231i	$-0.712544\pi$
$653$	4.24264	−	4.24264i	0.166027	−	0.166027i	0.785231	+	0.619203i	$0.212544\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	11.4853	+	5.48528i	0.448084	+	0.214001i
$658$		0			0
$659$	46.6690			1.81797			0.908984	−	0.416831i	$-0.136859\pi$
$659$	46.6690			1.81797			0.908984	+	0.416831i	$0.136859\pi$
$660$		0			0
$661$	−14.0000			−0.544537			−0.272268	−	0.962221i	$-0.587774\pi$
$661$	−14.0000			−0.544537			−0.272268	+	0.962221i	$0.587774\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−4.00000	+	4.00000i	−0.154881	+	0.154881i
$668$		0			0
$669$	4.24264	−	6.00000i	0.164030	−	0.231973i
$670$		0			0
$671$			8.48528i			0.327571i
$672$		0			0
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	0.971094	−	0.238698i	$-0.0767205\pi$
$673$	19.0000	+	19.0000i	0.732396	+	0.732396i	−0.238698	+	0.971094i	$0.576721\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	18.3848	+	18.3848i	0.706584	+	0.706584i	0.965815	−	0.259231i	$-0.0834691\pi$
$677$	18.3848	+	18.3848i	0.706584	+	0.706584i	−0.259231	+	0.965815i	$0.583469\pi$
$678$		0			0
$679$			78.0000i			2.99337i
$680$		0			0
$681$	−6.00000	+	8.48528i	−0.229920	+	0.325157i
$682$		0			0
$683$	−25.4558	+	25.4558i	−0.974041	+	0.974041i	−0.999671	−	0.0256307i	$-0.991841\pi$
$683$	−25.4558	+	25.4558i	−0.974041	+	0.974041i	0.0256307	+	0.999671i	$0.491841\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−10.2426	+	1.75736i	−0.390781	+	0.0670474i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.0000			−0.456502			−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000			−0.456502			−0.228251	+	0.973602i	$0.573301\pi$
$692$		0			0
$693$	−16.2426	−	7.75736i	−0.617007	−	0.294678i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	24.0000	−	24.0000i	0.909065	−	0.909065i
$698$		0			0
$699$	−14.1421	−	10.0000i	−0.534905	−	0.378235i
$700$		0			0
$701$			1.41421i			0.0534141i	0.999643	+	0.0267071i	$0.00850213\pi$
$701$			1.41421i			0.0534141i	−0.999643	+	0.0267071i	$0.991498\pi$
$702$		0			0
$703$	8.00000	+	8.00000i	0.301726	+	0.301726i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−46.6690	−	46.6690i	−1.75517	−	1.75517i
$708$		0			0
$709$			6.00000i			0.225335i	0.993633	+	0.112667i	$0.0359394\pi$
$709$			6.00000i			0.225335i	−0.993633	+	0.112667i	$0.964061\pi$
$710$		0			0
$711$	−10.0000	−	28.2843i	−0.375029	−	1.06074i
$712$		0			0
$713$	5.65685	−	5.65685i	0.211851	−	0.211851i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.45584	−	43.4558i	−0.278444	−	1.62289i
$718$		0			0
$719$	45.2548			1.68772			0.843860	−	0.536563i	$-0.180278\pi$
$719$	45.2548			1.68772			0.843860	+	0.536563i	$0.180278\pi$
$720$		0			0
$721$	−66.0000			−2.45797
$722$		0			0
$723$	5.85786	+	34.1421i	0.217856	+	1.26976i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	15.0000	−	15.0000i	0.556319	−	0.556319i	−0.371938	−	0.928257i	$-0.621307\pi$
$727$	15.0000	−	15.0000i	0.556319	−	0.556319i	0.928257	+	0.371938i	$0.121307\pi$
$728$		0			0
$729$	−14.1421	+	23.0000i	−0.523783	+	0.851852i
$730$		0			0
$731$		−	16.9706i		−	0.627679i
$732$		0			0
$733$	−14.0000	−	14.0000i	−0.517102	−	0.517102i	0.399592	−	0.916693i	$-0.369152\pi$
$733$	−14.0000	−	14.0000i	−0.517102	−	0.517102i	−0.916693	+	0.399592i	$0.869152\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.65685	+	5.65685i	0.208373	+	0.208373i
$738$		0			0
$739$		−	16.0000i		−	0.588570i	−0.955718	−	0.294285i	$-0.904919\pi$
$739$		−	16.0000i		−	0.588570i	0.955718	−	0.294285i	$-0.0950814\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	25.4558	−	25.4558i	0.933884	−	0.933884i	−0.0640616	−	0.997946i	$-0.520405\pi$
$743$	25.4558	−	25.4558i	0.933884	−	0.933884i	0.997946	+	0.0640616i	$0.0204054\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−5.17157	+	10.8284i	−0.189218	+	0.396191i
$748$		0			0
$749$	16.9706			0.620091
$750$		0			0
$751$	−30.0000			−1.09472			−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000			−1.09472			−0.547358	+	0.836899i	$0.684366\pi$
$752$		0			0
$753$	−21.7279	+	3.72792i	−0.791809	+	0.135853i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	6.00000	−	6.00000i	0.218074	−	0.218074i	−0.589613	−	0.807686i	$-0.700720\pi$
$757$	6.00000	−	6.00000i	0.218074	−	0.218074i	0.807686	+	0.589613i	$0.200720\pi$
$758$		0			0
$759$	−5.65685	+	8.00000i	−0.205331	+	0.290382i
$760$		0			0
$761$		−	19.7990i		−	0.717713i	−0.933393	−	0.358856i	$-0.883167\pi$
$761$		−	19.7990i		−	0.717713i	0.933393	−	0.358856i	$-0.116833\pi$
$762$		0			0
$763$	−30.0000	−	30.0000i	−1.08607	−	1.08607i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	10.0000i		−	0.360609i	−0.983611	−	0.180305i	$-0.942292\pi$
$769$		−	10.0000i		−	0.360609i	0.983611	−	0.180305i	$-0.0577084\pi$
$770$		0			0
$771$	22.0000	−	31.1127i	0.792311	−	1.12050i
$772$		0			0
$773$	18.3848	−	18.3848i	0.661254	−	0.661254i	−0.294421	−	0.955676i	$-0.595127\pi$
$773$	18.3848	−	18.3848i	0.661254	−	0.661254i	0.955676	+	0.294421i	$0.0951269\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−20.4853	+	3.51472i	−0.734905	+	0.126090i
$778$		0			0
$779$	−22.6274			−0.810711
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$	−3.58579	−	6.41421i	−0.128146	−	0.229225i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	−0.460109	−	0.887863i	$-0.652190\pi$
$787$	12.0000	−	12.0000i	0.427754	−	0.427754i	0.887863	+	0.460109i	$0.152190\pi$
$788$		0			0
$789$	−22.6274	−	16.0000i	−0.805557	−	0.569615i
$790$		0			0
$791$			25.4558i			0.905106i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21.2132	−	21.2132i	−0.751410	−	0.751410i	0.223332	−	0.974742i	$-0.428307\pi$
$797$	−21.2132	−	21.2132i	−0.751410	−	0.751410i	−0.974742	+	0.223332i	$0.928307\pi$
$798$		0			0
$799$		−	48.0000i		−	1.69812i
$800$		0			0
$801$	8.00000	−	2.82843i	0.282666	−	0.0999376i
$802$		0			0
$803$	−4.24264	+	4.24264i	−0.149720	+	0.149720i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.89949	+	16.8995i	0.102067	+	0.594890i
$808$		0			0
$809$	−45.2548			−1.59108			−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548			−1.59108			−0.795538	+	0.605904i	$0.792811\pi$
$810$		0			0
$811$	−44.0000			−1.54505			−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000			−1.54505			−0.772524	+	0.634985i	$0.781006\pi$
$812$		0			0
$813$	−7.02944	−	40.9706i	−0.246533	−	1.43690i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.00000	+	8.00000i	−0.279885	+	0.279885i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	26.8701i		−	0.937771i	−0.883259	−	0.468886i	$-0.844656\pi$
$821$		−	26.8701i		−	0.937771i	0.883259	−	0.468886i	$-0.155344\pi$
$822$		0			0
$823$	29.0000	+	29.0000i	1.01088	+	1.01088i	0.999940	+	0.0109363i	$0.00348119\pi$
$823$	29.0000	+	29.0000i	1.01088	+	1.01088i	0.0109363	+	0.999940i	$0.496519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.7279	−	12.7279i	−0.442593	−	0.442593i	0.450289	−	0.892883i	$-0.351321\pi$
$827$	−12.7279	−	12.7279i	−0.442593	−	0.442593i	−0.892883	+	0.450289i	$0.851321\pi$
$828$		0			0
$829$		−	14.0000i		−	0.486240i	−0.969996	−	0.243120i	$-0.921829\pi$
$829$		−	14.0000i		−	0.486240i	0.969996	−	0.243120i	$-0.0781709\pi$
$830$		0			0
$831$	−4.00000	−	2.82843i	−0.138758	−	0.0981170i
$832$		0			0
$833$	−46.6690	+	46.6690i	−1.61699	+	1.61699i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5.07107	+	9.07107i	0.175282	+	0.313542i
$838$		0			0
$839$	39.5980			1.36707			0.683537	−	0.729916i	$-0.260441\pi$
$839$	39.5980			1.36707			0.683537	+	0.729916i	$0.260441\pi$
$840$		0			0
$841$	−27.0000			−0.931034
$842$		0			0
$843$	43.4558	−	7.45584i	1.49670	−	0.256793i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−27.0000	+	27.0000i	−0.927731	+	0.927731i
$848$		0			0
$849$	−5.65685	+	8.00000i	−0.194143	+	0.274559i
$850$		0			0
$851$			11.3137i			0.387829i
$852$		0			0
$853$	18.0000	+	18.0000i	0.616308	+	0.616308i	0.944582	−	0.328274i	$-0.106467\pi$
$853$	18.0000	+	18.0000i	0.616308	+	0.616308i	−0.328274	+	0.944582i	$0.606467\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	12.7279	+	12.7279i	0.434778	+	0.434778i	0.890250	−	0.455472i	$-0.150530\pi$
$857$	12.7279	+	12.7279i	0.434778	+	0.434778i	−0.455472	+	0.890250i	$0.650530\pi$
$858$		0			0
$859$			40.0000i			1.36478i	0.730987	+	0.682391i	$0.239060\pi$
$859$			40.0000i			1.36478i	−0.730987	+	0.682391i	$0.760940\pi$
$860$		0			0
$861$	24.0000	−	33.9411i	0.817918	−	1.15671i
$862$		0			0
$863$	−25.4558	+	25.4558i	−0.866527	+	0.866527i	−0.992086	−	0.125559i	$-0.959928\pi$
$863$	−25.4558	+	25.4558i	−0.866527	+	0.866527i	0.125559	+	0.992086i	$0.459928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−32.4350	+	5.56497i	−1.10155	+	0.188996i
$868$		0			0
$869$	14.1421			0.479739
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−23.7696	+	49.7696i	−0.804477	+	1.68444i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36.0000	+	36.0000i	−1.21563	+	1.21563i	−0.246488	+	0.969146i	$0.579276\pi$
$877$	−36.0000	+	36.0000i	−1.21563	+	1.21563i	−0.969146	+	0.246488i	$0.920724\pi$
$878$		0			0
$879$	−11.3137	−	8.00000i	−0.381602	−	0.269833i
$880$		0			0
$881$			19.7990i			0.667045i	0.942742	+	0.333522i	$0.108237\pi$
$881$			19.7990i			0.667045i	−0.942742	+	0.333522i	$0.891763\pi$
$882$		0			0
$883$	12.0000	+	12.0000i	0.403832	+	0.403832i	0.879581	−	0.475749i	$-0.157823\pi$
$883$	12.0000	+	12.0000i	0.403832	+	0.403832i	−0.475749	+	0.879581i	$0.657823\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.7696	+	36.7696i	1.23460	+	1.23460i	0.962178	+	0.272423i	$0.0878251\pi$
$887$	36.7696	+	36.7696i	1.23460	+	1.23460i	0.272423	+	0.962178i	$0.412175\pi$
$888$		0			0
$889$			18.0000i			0.603701i
$890$		0			0
$891$	−8.00000	−	9.89949i	−0.268010	−	0.331646i
$892$		0			0
$893$	−22.6274	+	22.6274i	−0.757198	+	0.757198i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.82843			−0.0943333
$900$		0			0
$901$	72.0000			2.39867
$902$		0			0
$903$	−3.51472	−	20.4853i	−0.116963	−	0.681707i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	26.0000	−	26.0000i	0.863316	−	0.863316i	−0.128406	−	0.991722i	$-0.540986\pi$
$907$	26.0000	−	26.0000i	0.863316	−	0.863316i	0.991722	+	0.128406i	$0.0409860\pi$
$908$		0			0
$909$	−15.5563	−	44.0000i	−0.515972	−	1.45939i
$910$		0			0
$911$			5.65685i			0.187420i	0.995600	+	0.0937100i	$0.0298726\pi$
$911$			5.65685i			0.187420i	−0.995600	+	0.0937100i	$0.970127\pi$
$912$		0			0
$913$	−4.00000	−	4.00000i	−0.132381	−	0.132381i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−46.6690	−	46.6690i	−1.54115	−	1.54115i
$918$		0			0
$919$			2.00000i			0.0659739i	0.999456	+	0.0329870i	$0.0105020\pi$
$919$			2.00000i			0.0659739i	−0.999456	+	0.0329870i	$0.989498\pi$
$920$		0			0
$921$	−20.0000	−	14.1421i	−0.659022	−	0.465999i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−42.1127	−	20.1127i	−1.38316	−	0.660588i
$928$		0			0
$929$	36.7696			1.20637			0.603185	−	0.797601i	$-0.293898\pi$
$929$	36.7696			1.20637			0.603185	+	0.797601i	$0.293898\pi$
$930$		0			0
$931$	44.0000			1.44204
$932$		0			0
$933$	53.1127	−	9.11270i	1.73883	−	0.298336i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	−0.863575	−	0.504221i	$-0.831780\pi$
$937$	−11.0000	+	11.0000i	−0.359354	+	0.359354i	0.504221	+	0.863575i	$0.331780\pi$
$938$		0			0
$939$	−21.2132	+	30.0000i	−0.692267	+	0.979013i
$940$		0			0
$941$			38.1838i			1.24476i	0.782717	+	0.622378i	$0.213833\pi$
$941$			38.1838i			1.24476i	−0.782717	+	0.622378i	$0.786167\pi$
$942$		0			0
$943$	−16.0000	−	16.0000i	−0.521032	−	0.521032i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15.5563	+	15.5563i	0.505513	+	0.505513i	0.913146	−	0.407633i	$-0.133646\pi$
$947$	15.5563	+	15.5563i	0.505513	+	0.505513i	−0.407633	+	0.913146i	$0.633646\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	2.00000	−	2.82843i	0.0648544	−	0.0917180i
$952$		0			0
$953$	−21.2132	+	21.2132i	−0.687163	+	0.687163i	−0.961604	−	0.274441i	$-0.911507\pi$
$953$	−21.2132	+	21.2132i	−0.687163	+	0.687163i	0.274441	+	0.961604i	$0.411507\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	3.41421	−	0.585786i	0.110366	−	0.0189358i
$958$		0			0
$959$	−76.3675			−2.46604
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	10.8284	+	5.17157i	0.348941	+	0.166652i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−15.0000	+	15.0000i	−0.482367	+	0.482367i	−0.905887	−	0.423520i	$-0.860795\pi$
$967$	−15.0000	+	15.0000i	−0.482367	+	0.482367i	0.423520	+	0.905887i	$0.360795\pi$
$968$		0			0
$969$	33.9411	+	24.0000i	1.09035	+	0.770991i
$970$		0			0
$971$		−	41.0122i		−	1.31614i	−0.752955	−	0.658072i	$-0.771372\pi$
$971$		−	41.0122i		−	1.31614i	0.752955	−	0.658072i	$-0.228628\pi$
$972$		0			0
$973$	48.0000	+	48.0000i	1.53881	+	1.53881i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.2132	+	21.2132i	0.678671	+	0.678671i	0.959699	−	0.281029i	$-0.0906757\pi$
$977$	21.2132	+	21.2132i	0.678671	+	0.678671i	−0.281029	+	0.959699i	$0.590676\pi$
$978$		0			0
$979$			4.00000i			0.127841i
$980$		0			0
$981$	−10.0000	−	28.2843i	−0.319275	−	0.903047i
$982$		0			0
$983$	8.48528	−	8.48528i	0.270638	−	0.270638i	−0.558719	−	0.829357i	$-0.688707\pi$
$983$	8.48528	−	8.48528i	0.270638	−	0.270638i	0.829357	+	0.558719i	$0.188707\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−9.94113	−	57.9411i	−0.316430	−	1.84429i
$988$		0			0
$989$	−11.3137			−0.359755
$990$		0			0
$991$	16.0000			0.508257			0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000			0.508257			0.254128	+	0.967170i	$0.418211\pi$
$992$		0			0
$993$	−4.68629	−	27.3137i	−0.148715	−	0.866774i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.0000	+	24.0000i	−0.760088	+	0.760088i	−0.976338	−	0.216250i	$-0.930617\pi$
$997$	−24.0000	+	24.0000i	−0.760088	+	0.760088i	0.216250	+	0.976338i	$0.430617\pi$
$998$		0			0
$999$	−14.1421	−	4.00000i	−0.447437	−	0.126554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.2.r.d.593.1		4
3.2	odd	2	inner	600.2.r.d.593.2		4
4.3	odd	2		1200.2.v.c.593.2		4
5.2	odd	4	inner	600.2.r.d.257.2		4
5.3	odd	4		120.2.r.a.17.1	✓	4
5.4	even	2		120.2.r.a.113.2	yes	4
12.11	even	2		1200.2.v.c.593.1		4
15.2	even	4	inner	600.2.r.d.257.1		4
15.8	even	4		120.2.r.a.17.2	yes	4
15.14	odd	2		120.2.r.a.113.1	yes	4
20.3	even	4		240.2.v.d.17.2		4
20.7	even	4		1200.2.v.c.257.1		4
20.19	odd	2		240.2.v.d.113.1		4
40.3	even	4		960.2.v.b.257.1		4
40.13	odd	4		960.2.v.l.257.2		4
40.19	odd	2		960.2.v.b.833.2		4
40.29	even	2		960.2.v.l.833.1		4
60.23	odd	4		240.2.v.d.17.1		4
60.47	odd	4		1200.2.v.c.257.2		4
60.59	even	2		240.2.v.d.113.2		4
120.29	odd	2		960.2.v.l.833.2		4
120.53	even	4		960.2.v.l.257.1		4
120.59	even	2		960.2.v.b.833.1		4
120.83	odd	4		960.2.v.b.257.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.r.a.17.1	✓	4	5.3	odd	4
120.2.r.a.17.2	yes	4	15.8	even	4
120.2.r.a.113.1	yes	4	15.14	odd	2
120.2.r.a.113.2	yes	4	5.4	even	2
240.2.v.d.17.1		4	60.23	odd	4
240.2.v.d.17.2		4	20.3	even	4
240.2.v.d.113.1		4	20.19	odd	2
240.2.v.d.113.2		4	60.59	even	2
600.2.r.d.257.1		4	15.2	even	4	inner
600.2.r.d.257.2		4	5.2	odd	4	inner
600.2.r.d.593.1		4	1.1	even	1	trivial
600.2.r.d.593.2		4	3.2	odd	2	inner
960.2.v.b.257.1		4	40.3	even	4
960.2.v.b.257.2		4	120.83	odd	4
960.2.v.b.833.1		4	120.59	even	2
960.2.v.b.833.2		4	40.19	odd	2
960.2.v.l.257.1		4	120.53	even	4
960.2.v.l.257.2		4	40.13	odd	4
960.2.v.l.833.1		4	40.29	even	2
960.2.v.l.833.2		4	120.29	odd	2
1200.2.v.c.257.1		4	20.7	even	4
1200.2.v.c.257.2		4	60.47	odd	4
1200.2.v.c.593.1		4	12.11	even	2
1200.2.v.c.593.2		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.r (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.292893 + 1.70711i) q^{3} +(-3.00000 + 3.00000i) q^{7} +(-2.82843 + 1.00000i) q^{9} -1.41421i q^{11} +(-4.24264 - 4.24264i) q^{17} +4.00000i q^{19} +(-6.00000 - 4.24264i) q^{21} +(-2.82843 + 2.82843i) q^{23} +(-2.53553 - 4.53553i) q^{27} +1.41421 q^{29} -2.00000 q^{31} +(2.41421 - 0.414214i) q^{33} +(2.00000 - 2.00000i) q^{37} +5.65685i q^{41} +(2.00000 + 2.00000i) q^{43} +(5.65685 + 5.65685i) q^{47} -11.0000i q^{49} +(6.00000 - 8.48528i) q^{51} +(-8.48528 + 8.48528i) q^{53} +(-6.82843 + 1.17157i) q^{57} +1.41421 q^{59} -6.00000 q^{61} +(5.48528 - 11.4853i) q^{63} +(-4.00000 + 4.00000i) q^{67} +(-5.65685 - 4.00000i) q^{69} +2.82843i q^{71} +(-3.00000 - 3.00000i) q^{73} +(4.24264 + 4.24264i) q^{77} +10.0000i q^{79} +(7.00000 - 5.65685i) q^{81} +(2.82843 - 2.82843i) q^{83} +(0.414214 + 2.41421i) q^{87} -2.82843 q^{89} +(-0.585786 - 3.41421i) q^{93} +(13.0000 - 13.0000i) q^{97} +(1.41421 + 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 12 q^{7} - 24 q^{21} + 4 q^{27} - 8 q^{31} + 4 q^{33} + 8 q^{37} + 8 q^{43} + 24 q^{51} - 16 q^{57} - 24 q^{61} - 12 q^{63} - 16 q^{67} - 12 q^{73} + 28 q^{81} - 4 q^{87} - 8 q^{93} + 52 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^7 - 24 * q^21 + 4 * q^27 - 8 * q^31 + 4 * q^33 + 8 * q^37 + 8 * q^43 + 24 * q^51 - 16 * q^57 - 24 * q^61 - 12 * q^63 - 16 * q^67 - 12 * q^73 + 28 * q^81 - 4 * q^87 - 8 * q^93 + 52 * q^97

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