LMFDB - Embedded newform 1200.2.v.c.257.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,2,Mod(257,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, 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("1200.257");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1200.v (of order $4$, degree $2$, not 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:	$9.58204824255$
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			257.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1200.257
Dual form			1200.2.v.c.593.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 + 0.292893i) q^{3} +(3.00000 + 3.00000i) q^{7} +(2.82843 - 1.00000i) q^{9} +1.41421i q^{11} +(4.24264 - 4.24264i) q^{17} +4.00000i q^{19} +(-6.00000 - 4.24264i) q^{21} +(-2.82843 - 2.82843i) q^{23} +(-4.53553 + 2.53553i) q^{27} -1.41421 q^{29} +2.00000 q^{31} +(-0.414214 - 2.41421i) q^{33} +(2.00000 + 2.00000i) q^{37} +5.65685i q^{41} +(-2.00000 + 2.00000i) q^{43} +(5.65685 - 5.65685i) q^{47} +11.0000i q^{49} +(-6.00000 + 8.48528i) q^{51} +(8.48528 + 8.48528i) q^{53} +(-1.17157 - 6.82843i) q^{57} +1.41421 q^{59} -6.00000 q^{61} +(11.4853 + 5.48528i) q^{63} +(4.00000 + 4.00000i) q^{67} +(5.65685 + 4.00000i) q^{69} -2.82843i q^{71} +(-3.00000 + 3.00000i) q^{73} +(-4.24264 + 4.24264i) q^{77} +10.0000i q^{79} +(7.00000 - 5.65685i) q^{81} +(2.82843 + 2.82843i) q^{83} +(2.41421 - 0.414214i) q^{87} +2.82843 q^{89} +(-3.41421 + 0.585786i) q^{93} +(13.0000 + 13.0000i) q^{97} +(1.41421 + 4.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.70711	+	0.292893i	−0.985599	+	0.169102i
$3$	−1.70711	+	0.292893i	−0.985599	+	0.169102i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.989524	+	0.144370i	$0.0461154\pi$
$7$	3.00000	+	3.00000i	1.13389	+	1.13389i	0.144370	+	0.989524i	$0.453885\pi$
$8$		0			0
$9$	2.82843	−	1.00000i	0.942809	−	0.333333i
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$		0			0
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.0294245	−	0.999567i	$-0.490633\pi$
$17$	4.24264	−	4.24264i	1.02899	−	1.02899i	0.999567	−	0.0294245i	$-0.00936746\pi$
$18$		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	0.347801	−	0.937568i	$-0.386929\pi$
$23$	−2.82843	−	2.82843i	−0.589768	−	0.589768i	−0.937568	+	0.347801i	$0.886929\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.53553	+	2.53553i	−0.872864	+	0.487964i
$28$		0			0
$29$	−1.41421			−0.262613			−0.131306	−	0.991342i	$-0.541917\pi$
$29$	−1.41421			−0.262613			−0.131306	+	0.991342i	$0.541917\pi$
$30$		0			0
$31$	2.00000			0.359211			0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000			0.359211			0.179605	+	0.983739i	$0.442518\pi$
$32$		0			0
$33$	−0.414214	−	2.41421i	−0.0721053	−	0.420261i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.00000	+	2.00000i	0.328798	+	0.328798i	0.852129	−	0.523331i	$-0.175311\pi$
$37$	2.00000	+	2.00000i	0.328798	+	0.328798i	−0.523331	+	0.852129i	$0.675311\pi$
$38$		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	$-0.819192\pi$
$43$	−2.00000	+	2.00000i	−0.304997	+	0.304997i	0.537968	+	0.842965i	$0.319192\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.65685	−	5.65685i	0.825137	−	0.825137i	−0.161703	−	0.986840i	$-0.551699\pi$
$47$	5.65685	−	5.65685i	0.825137	−	0.825137i	0.986840	+	0.161703i	$0.0516985\pi$
$48$		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	0.983243	+	0.182300i	$0.0583542\pi$
$53$	8.48528	+	8.48528i	1.16554	+	1.16554i	0.182300	+	0.983243i	$0.441646\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.17157	−	6.82843i	−0.155179	−	0.904447i
$58$		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
$63$	11.4853	+	5.48528i	1.44701	+	0.691080i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$		0			0
$69$	5.65685	+	4.00000i	0.681005	+	0.481543i
$70$		0			0
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$		0			0
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	−0.860527	−	0.509404i	$-0.829866\pi$
$73$	−3.00000	+	3.00000i	−0.351123	+	0.351123i	0.509404	+	0.860527i	$0.329866\pi$
$74$		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	0.845088	−	0.534628i	$-0.179548\pi$
$83$	2.82843	+	2.82843i	0.310460	+	0.310460i	−0.534628	+	0.845088i	$0.679548\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.41421	−	0.414214i	0.258831	−	0.0444084i
$88$		0			0
$89$	2.82843			0.299813			0.149906	−	0.988700i	$-0.452103\pi$
$89$	2.82843			0.299813			0.149906	+	0.988700i	$0.452103\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−3.41421	+	0.585786i	−0.354037	+	0.0607432i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.913812	+	0.406138i	$0.133125\pi$
$97$	13.0000	+	13.0000i	1.31995	+	1.31995i	0.406138	+	0.913812i	$0.366875\pi$
$98$		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	−0.0877167	+	0.996145i	$0.527957\pi$
$103$	−11.0000	+	11.0000i	−1.08386	+	1.08386i	−0.996145	+	0.0877167i	$0.972043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	−0.830481	−	0.557047i	$-0.811934\pi$
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	0.557047	+	0.830481i	$0.311934\pi$
$108$		0			0
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		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	$-0.341070\pi$
$113$	−4.24264	−	4.24264i	−0.399114	−	0.399114i	−0.877920	+	0.478806i	$0.841070\pi$
$114$		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
$123$	−1.65685	−	9.65685i	−0.149394	−	0.870729i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.00000	−	3.00000i	−0.266207	−	0.266207i	0.561363	−	0.827570i	$-0.310277\pi$
$127$	−3.00000	−	3.00000i	−0.266207	−	0.266207i	−0.827570	+	0.561363i	$0.810277\pi$
$128$		0			0
$129$	2.82843	−	4.00000i	0.249029	−	0.352180i
$130$		0			0
$131$		−	15.5563i		−	1.35916i	−0.733599	−	0.679582i	$-0.762161\pi$
$131$		−	15.5563i		−	1.35916i	0.733599	−	0.679582i	$-0.237839\pi$
$132$		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	−0.0916263	+	0.995793i	$0.529207\pi$
$137$	−12.7279	+	12.7279i	−1.08742	+	1.08742i	−0.995793	+	0.0916263i	$0.970793\pi$
$138$		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
$147$	−3.22183	−	18.7782i	−0.265732	−	1.54880i
$148$		0			0
$149$	−4.24264			−0.347571			−0.173785	−	0.984784i	$-0.555600\pi$
$149$	−4.24264			−0.347571			−0.173785	+	0.984784i	$0.555600\pi$
$150$		0			0
$151$	−8.00000			−0.651031			−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000			−0.651031			−0.325515	+	0.945537i	$0.605538\pi$
$152$		0			0
$153$	7.75736	−	16.2426i	0.627145	−	1.31314i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−8.00000	−	8.00000i	−0.638470	−	0.638470i	0.311708	−	0.950178i	$-0.399099\pi$
$157$	−8.00000	−	8.00000i	−0.638470	−	0.638470i	−0.950178	+	0.311708i	$0.899099\pi$
$158$		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	$-0.518594\pi$
$163$	12.0000	−	12.0000i	0.939913	−	0.939913i	0.998294	+	0.0583818i	$0.0185941\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$167$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$168$		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	$-0.428635\pi$
$173$	−9.89949	−	9.89949i	−0.752645	−	0.752645i	−0.974972	+	0.222327i	$0.928635\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.41421	+	0.414214i	−0.181463	+	0.0311342i
$178$		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
$183$	10.2426	−	1.75736i	0.757158	−	0.129908i
$184$		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	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$		0			0
$193$	7.00000	−	7.00000i	0.503871	−	0.503871i	−0.408768	−	0.912639i	$-0.634041\pi$
$193$	7.00000	−	7.00000i	0.503871	−	0.503871i	0.912639	+	0.408768i	$0.134041\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		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
$207$	−10.8284	−	5.17157i	−0.752628	−	0.359449i
$208$		0			0
$209$	−5.65685			−0.391293
$210$		0			0
$211$	16.0000			1.10149			0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000			1.10149			0.550743	+	0.834675i	$0.314345\pi$
$212$		0			0
$213$	0.828427	+	4.82843i	0.0567629	+	0.330838i
$214$		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	$-0.704629\pi$
$223$	3.00000	−	3.00000i	0.200895	−	0.200895i	0.800383	+	0.599489i	$0.204629\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.24264	−	4.24264i	0.281594	−	0.281594i	−0.552151	−	0.833744i	$-0.686193\pi$
$227$	4.24264	−	4.24264i	0.281594	−	0.281594i	0.833744	+	0.552151i	$0.186193\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		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	$-0.143773\pi$
$233$	7.07107	+	7.07107i	0.463241	+	0.463241i	−0.436475	+	0.899716i	$0.643773\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−2.92893	−	17.0711i	−0.190255	−	1.10889i
$238$		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
$243$	−10.2929	+	11.7071i	−0.660289	+	0.751011i
$244$		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	$-0.868423\pi$
$251$		−	12.7279i		−	0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$		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	$-0.509294\pi$
$257$	15.5563	−	15.5563i	0.970378	−	0.970378i	0.999574	+	0.0291953i	$0.00929448\pi$
$258$		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	0.266266	−	0.963899i	$-0.414210\pi$
$263$	−11.3137	−	11.3137i	−0.697633	−	0.697633i	−0.963899	+	0.266266i	$0.914210\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.82843	+	0.828427i	−0.295495	+	0.0506989i
$268$		0			0
$269$	−9.89949			−0.603583			−0.301791	−	0.953374i	$-0.597585\pi$
$269$	−9.89949			−0.603583			−0.301791	+	0.953374i	$0.597585\pi$
$270$		0			0
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−2.00000	−	2.00000i	−0.120168	−	0.120168i	0.644465	−	0.764634i	$-0.277080\pi$
$277$	−2.00000	−	2.00000i	−0.120168	−	0.120168i	−0.764634	+	0.644465i	$0.777080\pi$
$278$		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	$-0.803774\pi$
$283$	−4.00000	+	4.00000i	−0.237775	+	0.237775i	0.578153	+	0.815928i	$0.303774\pi$
$284$		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	$-0.174922\pi$
$293$	5.65685	+	5.65685i	0.330477	+	0.330477i	−0.522291	+	0.852768i	$0.674922\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−3.58579	−	6.41421i	−0.208068	−	0.372190i
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$		0			0
$303$	−4.55635	−	26.5563i	−0.261755	−	1.52562i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	10.0000	+	10.0000i	0.570730	+	0.570730i	0.932332	−	0.361602i	$-0.117770\pi$
$307$	10.0000	+	10.0000i	0.570730	+	0.570730i	−0.361602	+	0.932332i	$0.617770\pi$
$308$		0			0
$309$	15.5563	−	22.0000i	0.884970	−	1.25154i
$310$		0			0
$311$			31.1127i			1.76424i	0.471025	+	0.882120i	$0.343884\pi$
$311$			31.1127i			1.76424i	−0.471025	+	0.882120i	$0.656116\pi$
$312$		0			0
$313$	15.0000	−	15.0000i	0.847850	−	0.847850i	−0.142014	−	0.989865i	$-0.545358\pi$
$313$	15.0000	−	15.0000i	0.847850	−	0.847850i	0.989865	+	0.142014i	$0.0453579\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.41421	−	1.41421i	0.0794301	−	0.0794301i	−0.666276	−	0.745706i	$-0.732113\pi$
$317$	1.41421	−	1.41421i	0.0794301	−	0.0794301i	0.745706	+	0.666276i	$0.232113\pi$
$318$		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
$327$	2.92893	+	17.0711i	0.161970	+	0.944032i
$328$		0			0
$329$	33.9411			1.87123
$330$		0			0
$331$	16.0000			0.879440			0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000			0.879440			0.439720	+	0.898135i	$0.355078\pi$
$332$		0			0
$333$	7.65685	+	3.65685i	0.419593	+	0.200394i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	0.168585	−	0.985687i	$-0.446080\pi$
$337$	−15.0000	−	15.0000i	−0.817102	−	0.817102i	−0.985687	+	0.168585i	$0.946080\pi$
$338$		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	−0.960736	−	0.277465i	$-0.910506\pi$
$347$	−12.7279	+	12.7279i	−0.683271	+	0.683271i	0.277465	+	0.960736i	$0.410506\pi$
$348$		0			0
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−18.3848	−	18.3848i	−0.978523	−	0.978523i	0.0212513	−	0.999774i	$-0.493235\pi$
$353$	−18.3848	−	18.3848i	−0.978523	−	0.978523i	−0.999774	+	0.0212513i	$0.993235\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−43.4558	+	7.45584i	−2.29993	+	0.394605i
$358$		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
$363$	−15.3640	+	2.63604i	−0.806399	+	0.138356i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−7.00000	−	7.00000i	−0.365397	−	0.365397i	0.500398	−	0.865795i	$-0.333187\pi$
$367$	−7.00000	−	7.00000i	−0.365397	−	0.365397i	−0.865795	+	0.500398i	$0.833187\pi$
$368$		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	−0.945873	−	0.324537i	$-0.894792\pi$
$373$	−12.0000	+	12.0000i	−0.621336	+	0.621336i	0.324537	+	0.945873i	$0.394792\pi$
$374$		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	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.65685	+	7.65685i	−0.185888	+	0.389220i
$388$		0			0
$389$	−12.7279			−0.645331			−0.322666	−	0.946513i	$-0.604579\pi$
$389$	−12.7279			−0.645331			−0.322666	+	0.946513i	$0.604579\pi$
$390$		0			0
$391$	−24.0000			−1.21373
$392$		0			0
$393$	4.55635	+	26.5563i	0.229837	+	1.33959i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	10.0000	+	10.0000i	0.501886	+	0.501886i	0.912024	−	0.410138i	$-0.134519\pi$
$397$	10.0000	+	10.0000i	0.501886	+	0.501886i	−0.410138	+	0.912024i	$0.634519\pi$
$398$		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
$417$	4.68629	+	27.3137i	0.229489	+	1.33756i
$418$		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
$423$	10.3431	−	21.6569i	0.502901	−	1.05299i
$424$		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	$-0.816544\pi$
$431$		−	22.6274i		−	1.08992i	0.838461	−	0.544962i	$-0.183456\pi$
$432$		0			0
$433$	1.00000	−	1.00000i	0.0480569	−	0.0480569i	−0.682670	−	0.730727i	$-0.739181\pi$
$433$	1.00000	−	1.00000i	0.0480569	−	0.0480569i	0.730727	+	0.682670i	$0.239181\pi$
$434$		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	0.854843	−	0.518887i	$-0.173653\pi$
$443$	7.07107	+	7.07107i	0.335957	+	0.335957i	−0.518887	+	0.854843i	$0.673653\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	7.24264	−	1.24264i	0.342565	−	0.0587749i
$448$		0			0
$449$	−19.7990			−0.934372			−0.467186	−	0.884159i	$-0.654732\pi$
$449$	−19.7990			−0.934372			−0.467186	+	0.884159i	$0.654732\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$		0			0
$453$	13.6569	−	2.34315i	0.641655	−	0.110091i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000	−	1.00000i	−0.0467780	−	0.0467780i	0.683331	−	0.730109i	$-0.260531\pi$
$457$	−1.00000	−	1.00000i	−0.0467780	−	0.0467780i	−0.730109	+	0.683331i	$0.760531\pi$
$458$		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	$-0.739538\pi$
$463$	1.00000	−	1.00000i	0.0464739	−	0.0464739i	0.729962	+	0.683488i	$0.239538\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	25.4558	−	25.4558i	1.17796	−	1.17796i	0.197692	−	0.980264i	$-0.436655\pi$
$467$	25.4558	−	25.4558i	1.17796	−	1.17796i	0.980264	−	0.197692i	$-0.0633445\pi$
$468$		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
$477$	32.4853	+	15.5147i	1.48740	+	0.710370i
$478$		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
$483$	4.97056	+	28.9706i	0.226168	+	1.31821i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	0.635861	−	0.771804i	$-0.280645\pi$
$487$	−3.00000	−	3.00000i	−0.135943	−	0.135943i	−0.771804	+	0.635861i	$0.780645\pi$
$488$		0			0
$489$	−16.9706	+	24.0000i	−0.767435	+	1.08532i
$490$		0			0
$491$		−	15.5563i		−	0.702048i	−0.936366	−	0.351024i	$-0.885834\pi$
$491$		−	15.5563i		−	0.702048i	0.936366	−	0.351024i	$-0.114166\pi$
$492$		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	0.989330	+	0.145690i	$0.0465401\pi$
$503$	25.4558	+	25.4558i	1.13502	+	1.13502i	0.145690	+	0.989330i	$0.453460\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−3.80761	−	22.1924i	−0.169102	−	0.985599i
$508$		0			0
$509$	38.1838			1.69247			0.846233	−	0.532813i	$-0.178865\pi$
$509$	38.1838			1.69247			0.846233	+	0.532813i	$0.178865\pi$
$510$		0			0
$511$	−18.0000			−0.796273
$512$		0			0
$513$	−10.1421	−	18.1421i	−0.447786	−	0.800995i
$514$		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	0.0507346	−	0.998712i	$-0.483844\pi$
$523$	24.0000	−	24.0000i	1.04945	−	1.04945i	0.998712	−	0.0507346i	$-0.0161562\pi$
$524$		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
$537$	26.5563	−	4.55635i	1.14599	−	0.196621i
$538$		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
$543$	−17.0711	+	2.92893i	−0.732590	+	0.125693i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	30.0000	+	30.0000i	1.28271	+	1.28271i	0.939121	+	0.343586i	$0.111642\pi$
$547$	30.0000	+	30.0000i	1.28271	+	1.28271i	0.343586	+	0.939121i	$0.388358\pi$
$548$		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	$-0.919784\pi$
$557$	−16.9706	+	16.9706i	−0.719066	+	0.719066i	0.249348	+	0.968414i	$0.419784\pi$
$558$		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	−0.999911	−	0.0133227i	$-0.995759\pi$
$563$	−24.0416	−	24.0416i	−1.01323	−	1.01323i	−0.0133227	−	0.999911i	$-0.504241\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	37.9706	+	4.02944i	1.59461	+	0.169220i
$568$		0			0
$569$	−14.1421			−0.592869			−0.296435	−	0.955053i	$-0.595798\pi$
$569$	−14.1421			−0.592869			−0.296435	+	0.955053i	$0.595798\pi$
$570$		0			0
$571$	−32.0000			−1.33916			−0.669579	−	0.742741i	$-0.733526\pi$
$571$	−32.0000			−1.33916			−0.669579	+	0.742741i	$0.733526\pi$
$572$		0			0
$573$	−0.828427	−	4.82843i	−0.0346080	−	0.201710i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−21.0000	−	21.0000i	−0.874241	−	0.874241i	0.118690	−	0.992931i	$-0.462131\pi$
$577$	−21.0000	−	21.0000i	−0.874241	−	0.874241i	−0.992931	+	0.118690i	$0.962131\pi$
$578$		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	−0.168508	−	0.985700i	$-0.553895\pi$
$587$	19.7990	−	19.7990i	0.817192	−	0.817192i	0.985700	+	0.168508i	$0.0538950\pi$
$588$		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	$-0.236925\pi$
$593$	1.41421	+	1.41421i	0.0580748	+	0.0580748i	−0.677473	+	0.735548i	$0.736925\pi$
$594$		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
$603$	15.3137	+	7.31371i	0.623622	+	0.297837i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−25.0000	−	25.0000i	−1.01472	−	1.01472i	−0.999890	−	0.0148286i	$-0.995280\pi$
$607$	−25.0000	−	25.0000i	−1.01472	−	1.01472i	−0.0148286	−	0.999890i	$-0.504720\pi$
$608$		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	0.241218	−	0.970471i	$-0.422453\pi$
$613$	30.0000	−	30.0000i	1.21169	−	1.21169i	0.970471	−	0.241218i	$-0.0775467\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	−0.787332	−	0.616530i	$-0.788538\pi$
$617$	−4.24264	+	4.24264i	−0.170802	+	0.170802i	0.616530	+	0.787332i	$0.288538\pi$
$618$		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
$627$	9.65685	−	1.65685i	0.385658	−	0.0661684i
$628$		0			0
$629$	16.9706			0.676661
$630$		0			0
$631$	−34.0000			−1.35352			−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000			−1.35352			−0.676759	+	0.736204i	$0.736616\pi$
$632$		0			0
$633$	−27.3137	+	4.68629i	−1.08562	+	0.186263i
$634$		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	$-0.678387\pi$
$643$	8.00000	−	8.00000i	0.315489	−	0.315489i	0.847032	+	0.531542i	$0.178387\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.82843	−	2.82843i	0.111197	−	0.111197i	−0.649319	−	0.760516i	$-0.724946\pi$
$647$	2.82843	−	2.82843i	0.111197	−	0.111197i	0.760516	+	0.649319i	$0.224946\pi$
$648$		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	$-0.287456\pi$
$653$	−4.24264	−	4.24264i	−0.166027	−	0.166027i	−0.785231	+	0.619203i	$0.787456\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−5.48528	+	11.4853i	−0.214001	+	0.448084i
$658$		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	−0.238698	−	0.971094i	$-0.576721\pi$
$673$	19.0000	−	19.0000i	0.732396	−	0.732396i	0.971094	+	0.238698i	$0.0767205\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.3848	+	18.3848i	−0.706584	+	0.706584i	−0.965815	−	0.259231i	$-0.916531\pi$
$677$	−18.3848	+	18.3848i	−0.706584	+	0.706584i	0.259231	+	0.965815i	$0.416531\pi$
$678$		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	0.0256307	−	0.999671i	$-0.491841\pi$
$683$	−25.4558	−	25.4558i	−0.974041	−	0.974041i	−0.999671	+	0.0256307i	$0.991841\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.75736	+	10.2426i	0.0670474	+	0.390781i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$		0			0
$693$	−7.75736	+	16.2426i	−0.294678	+	0.617007i
$694$		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	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		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
$717$	43.4558	−	7.45584i	1.62289	−	0.278444i
$718$		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
$723$	−34.1421	+	5.85786i	−1.26976	+	0.217856i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	0.371938	−	0.928257i	$-0.378693\pi$
$727$	−15.0000	−	15.0000i	−0.556319	−	0.556319i	−0.928257	+	0.371938i	$0.878693\pi$
$728$		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	−0.916693	−	0.399592i	$-0.869152\pi$
$733$	−14.0000	+	14.0000i	−0.517102	+	0.517102i	0.399592	+	0.916693i	$0.369152\pi$
$734$		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	0.997946	−	0.0640616i	$-0.0204054\pi$
$743$	25.4558	+	25.4558i	0.933884	+	0.933884i	−0.0640616	+	0.997946i	$0.520405\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	10.8284	+	5.17157i	0.396191	+	0.189218i
$748$		0			0
$749$	−16.9706			−0.620091
$750$		0			0
$751$	30.0000			1.09472			0.547358	−	0.836899i	$-0.315634\pi$
$751$	30.0000			1.09472			0.547358	+	0.836899i	$0.315634\pi$
$752$		0			0
$753$	3.72792	+	21.7279i	0.135853	+	0.791809i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	6.00000	+	6.00000i	0.218074	+	0.218074i	0.807686	−	0.589613i	$-0.200720\pi$
$757$	6.00000	+	6.00000i	0.218074	+	0.218074i	−0.589613	+	0.807686i	$0.700720\pi$
$758$		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	$0.0577084\pi$
$769$			10.0000i			0.360609i	−0.983611	+	0.180305i	$0.942292\pi$
$770$		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	$-0.404873\pi$
$773$	−18.3848	−	18.3848i	−0.661254	−	0.661254i	−0.955676	+	0.294421i	$0.904873\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−3.51472	−	20.4853i	−0.126090	−	0.734905i
$778$		0			0
$779$	−22.6274			−0.810711
$780$		0			0
$781$	4.00000			0.143131
$782$		0			0
$783$	6.41421	−	3.58579i	0.229225	−	0.128146i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−12.0000	−	12.0000i	−0.427754	−	0.427754i	0.460109	−	0.887863i	$-0.347810\pi$
$787$	−12.0000	−	12.0000i	−0.427754	−	0.427754i	−0.887863	+	0.460109i	$0.847810\pi$
$788$		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	$-0.571693\pi$
$797$	21.2132	−	21.2132i	0.751410	−	0.751410i	0.974742	+	0.223332i	$0.0716935\pi$
$798$		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
$807$	16.8995	−	2.89949i	0.594890	−	0.102067i
$808$		0			0
$809$	45.2548			1.59108			0.795538	−	0.605904i	$-0.207189\pi$
$809$	45.2548			1.59108			0.795538	+	0.605904i	$0.207189\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$	−40.9706	+	7.02944i	−1.43690	+	0.246533i
$814$		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	−0.0109363	+	0.999940i	$0.503481\pi$
$823$	−29.0000	+	29.0000i	−1.01088	+	1.01088i	−0.999940	+	0.0109363i	$0.996519\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−12.7279	+	12.7279i	−0.442593	+	0.442593i	−0.892883	−	0.450289i	$-0.851321\pi$
$827$	−12.7279	+	12.7279i	−0.442593	+	0.442593i	0.450289	+	0.892883i	$0.351321\pi$
$828$		0			0
$829$			14.0000i			0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$			14.0000i			0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$		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
$837$	−9.07107	+	5.07107i	−0.313542	+	0.175282i
$838$		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
$843$	7.45584	+	43.4558i	0.256793	+	1.49670i
$844$		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	−0.328274	−	0.944582i	$-0.606467\pi$
$853$	18.0000	−	18.0000i	0.616308	−	0.616308i	0.944582	+	0.328274i	$0.106467\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−12.7279	+	12.7279i	−0.434778	+	0.434778i	−0.890250	−	0.455472i	$-0.849470\pi$
$857$	−12.7279	+	12.7279i	−0.434778	+	0.434778i	0.455472	+	0.890250i	$0.349470\pi$
$858$		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	0.125559	−	0.992086i	$-0.459928\pi$
$863$	−25.4558	−	25.4558i	−0.866527	−	0.866527i	−0.992086	+	0.125559i	$0.959928\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	5.56497	+	32.4350i	0.188996	+	1.10155i
$868$		0			0
$869$	−14.1421			−0.479739
$870$		0			0
$871$		0			0
$872$		0			0
$873$	49.7696	+	23.7696i	1.68444	+	0.804477i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−36.0000	−	36.0000i	−1.21563	−	1.21563i	−0.969146	−	0.246488i	$-0.920724\pi$
$877$	−36.0000	−	36.0000i	−1.21563	−	1.21563i	−0.246488	−	0.969146i	$-0.579276\pi$
$878$		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	$-0.842177\pi$
$883$	−12.0000	+	12.0000i	−0.403832	+	0.403832i	0.475749	+	0.879581i	$0.342177\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	36.7696	−	36.7696i	1.23460	−	1.23460i	0.272423	−	0.962178i	$-0.412175\pi$
$887$	36.7696	−	36.7696i	1.23460	−	1.23460i	0.962178	−	0.272423i	$-0.0878251\pi$
$888$		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
$903$	20.4853	−	3.51472i	0.681707	−	0.116963i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−26.0000	−	26.0000i	−0.863316	−	0.863316i	0.128406	−	0.991722i	$-0.459014\pi$
$907$	−26.0000	−	26.0000i	−0.863316	−	0.863316i	−0.991722	+	0.128406i	$0.959014\pi$
$908$		0			0
$909$	15.5563	+	44.0000i	0.515972	+	1.45939i
$910$		0			0
$911$		−	5.65685i		−	0.187420i	−0.995600	−	0.0937100i	$-0.970127\pi$
$911$		−	5.65685i		−	0.187420i	0.995600	−	0.0937100i	$-0.0298726\pi$
$912$		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
$927$	−20.1127	+	42.1127i	−0.660588	+	1.38316i
$928$		0			0
$929$	−36.7696			−1.20637			−0.603185	−	0.797601i	$-0.706102\pi$
$929$	−36.7696			−1.20637			−0.603185	+	0.797601i	$0.706102\pi$
$930$		0			0
$931$	−44.0000			−1.44204
$932$		0			0
$933$	−9.11270	−	53.1127i	−0.298336	−	1.73883i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−11.0000	−	11.0000i	−0.359354	−	0.359354i	0.504221	−	0.863575i	$-0.331780\pi$
$937$	−11.0000	−	11.0000i	−0.359354	−	0.359354i	−0.863575	+	0.504221i	$0.831780\pi$
$938$		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	−0.407633	−	0.913146i	$-0.633646\pi$
$947$	15.5563	−	15.5563i	0.505513	−	0.505513i	0.913146	+	0.407633i	$0.133646\pi$
$948$		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	$-0.0884928\pi$
$953$	21.2132	+	21.2132i	0.687163	+	0.687163i	−0.274441	+	0.961604i	$0.588493\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	0.585786	+	3.41421i	0.0189358	+	0.110366i
$958$		0			0
$959$	−76.3675			−2.46604
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	−5.17157	+	10.8284i	−0.166652	+	0.348941i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	15.0000	+	15.0000i	0.482367	+	0.482367i	0.905887	−	0.423520i	$-0.139205\pi$
$967$	15.0000	+	15.0000i	0.482367	+	0.482367i	−0.423520	+	0.905887i	$0.639205\pi$
$968$		0			0
$969$	−33.9411	−	24.0000i	−1.09035	−	0.770991i
$970$		0			0
$971$			41.0122i			1.31614i	0.752955	+	0.658072i	$0.228628\pi$
$971$			41.0122i			1.31614i	−0.752955	+	0.658072i	$0.771372\pi$
$972$		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	$-0.909324\pi$
$977$	−21.2132	+	21.2132i	−0.678671	+	0.678671i	0.281029	+	0.959699i	$0.409324\pi$
$978$		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	0.829357	−	0.558719i	$-0.188707\pi$
$983$	8.48528	+	8.48528i	0.270638	+	0.270638i	−0.558719	+	0.829357i	$0.688707\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−57.9411	+	9.94113i	−1.84429	+	0.316430i
$988$		0			0
$989$	11.3137			0.359755
$990$		0			0
$991$	−16.0000			−0.508257			−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000			−0.508257			−0.254128	+	0.967170i	$0.581789\pi$
$992$		0			0
$993$	−27.3137	+	4.68629i	−0.866774	+	0.148715i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−24.0000	−	24.0000i	−0.760088	−	0.760088i	0.216250	−	0.976338i	$-0.430617\pi$
$997$	−24.0000	−	24.0000i	−0.760088	−	0.760088i	−0.976338	+	0.216250i	$0.930617\pi$
$998$		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	1200.2.v.c.257.1		4
3.2	odd	2	inner	1200.2.v.c.257.2		4
4.3	odd	2		600.2.r.d.257.2		4
5.2	odd	4		240.2.v.d.113.1		4
5.3	odd	4	inner	1200.2.v.c.593.2		4
5.4	even	2		240.2.v.d.17.2		4
12.11	even	2		600.2.r.d.257.1		4
15.2	even	4		240.2.v.d.113.2		4
15.8	even	4	inner	1200.2.v.c.593.1		4
15.14	odd	2		240.2.v.d.17.1		4
20.3	even	4		600.2.r.d.593.1		4
20.7	even	4		120.2.r.a.113.2	yes	4
20.19	odd	2		120.2.r.a.17.1	✓	4
40.19	odd	2		960.2.v.l.257.2		4
40.27	even	4		960.2.v.l.833.1		4
40.29	even	2		960.2.v.b.257.1		4
40.37	odd	4		960.2.v.b.833.2		4
60.23	odd	4		600.2.r.d.593.2		4
60.47	odd	4		120.2.r.a.113.1	yes	4
60.59	even	2		120.2.r.a.17.2	yes	4
120.29	odd	2		960.2.v.b.257.2		4
120.59	even	2		960.2.v.l.257.1		4
120.77	even	4		960.2.v.b.833.1		4
120.107	odd	4		960.2.v.l.833.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.r.a.17.1	✓	4	20.19	odd	2
120.2.r.a.17.2	yes	4	60.59	even	2
120.2.r.a.113.1	yes	4	60.47	odd	4
120.2.r.a.113.2	yes	4	20.7	even	4
240.2.v.d.17.1		4	15.14	odd	2
240.2.v.d.17.2		4	5.4	even	2
240.2.v.d.113.1		4	5.2	odd	4
240.2.v.d.113.2		4	15.2	even	4
600.2.r.d.257.1		4	12.11	even	2
600.2.r.d.257.2		4	4.3	odd	2
600.2.r.d.593.1		4	20.3	even	4
600.2.r.d.593.2		4	60.23	odd	4
960.2.v.b.257.1		4	40.29	even	2
960.2.v.b.257.2		4	120.29	odd	2
960.2.v.b.833.1		4	120.77	even	4
960.2.v.b.833.2		4	40.37	odd	4
960.2.v.l.257.1		4	120.59	even	2
960.2.v.l.257.2		4	40.19	odd	2
960.2.v.l.833.1		4	40.27	even	4
960.2.v.l.833.2		4	120.107	odd	4
1200.2.v.c.257.1		4	1.1	even	1	trivial
1200.2.v.c.257.2		4	3.2	odd	2	inner
1200.2.v.c.593.1		4	15.8	even	4	inner
1200.2.v.c.593.2		4	5.3	odd	4	inner

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1200.v (of order \(4\), degree \(2\), not minimal)

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