LMFDB - Embedded newform 1224.2.bq.a.433.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1224,2,Mod(145,1224)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1224, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1224.145"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1224 = 2^{3} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1224.bq (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,-4,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$9.77368920740$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			433.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1224.433
Dual form			1224.2.bq.a.865.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 - 0.707107i) q^{5} +(-0.292893 + 0.121320i) q^{7} +(-0.292893 - 0.707107i) q^{11} +(-1.00000 + 4.00000i) q^{17} +(-1.58579 + 1.58579i) q^{19} +(2.53553 + 6.12132i) q^{23} +(-1.12132 - 1.12132i) q^{25} +(5.12132 + 2.12132i) q^{29} +(0.878680 - 2.12132i) q^{31} +0.585786 q^{35} +(0.292893 - 0.707107i) q^{37} +(2.29289 - 0.949747i) q^{41} +(7.24264 + 7.24264i) q^{43} +12.8284i q^{47} +(-4.87868 + 4.87868i) q^{49} +(-2.17157 + 2.17157i) q^{53} +1.41421i q^{55} +(-5.58579 - 5.58579i) q^{59} +(-9.12132 + 3.77817i) q^{61} +9.65685 q^{67} +(-3.70711 + 8.94975i) q^{71} +(7.36396 + 3.05025i) q^{73} +(0.171573 + 0.171573i) q^{77} +(5.46447 + 13.1924i) q^{79} +(7.24264 - 7.24264i) q^{83} +(4.53553 - 6.12132i) q^{85} +1.65685i q^{89} +(3.82843 - 1.58579i) q^{95} +(-14.7782 - 6.12132i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} - 4 q^{17} - 12 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{29} + 12 q^{31} + 8 q^{35} + 4 q^{37} + 12 q^{41} + 12 q^{43} - 28 q^{49} - 20 q^{53} - 28 q^{59} - 28 q^{61} + 16 q^{67}+ \cdots - 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^11 - 4 * q^17 - 12 * q^19 - 4 * q^23 + 4 * q^25 + 12 * q^29 + 12 * q^31 + 8 * q^35 + 4 * q^37 + 12 * q^41 + 12 * q^43 - 28 * q^49 - 20 * q^53 - 28 * q^59 - 28 * q^61 + 16 * q^67 - 12 * q^71 + 4 * q^73 + 12 * q^77 + 36 * q^79 + 12 * q^83 + 4 * q^85 + 4 * q^95 - 28 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1224\mathbb{Z}\right)^\times$.

$n$	$137$	$613$	$649$	$919$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{8}\right)$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.70711	−	0.707107i	−0.763441	−	0.316228i	−0.0332288	−	0.999448i	$-0.510579\pi$
$5$	−1.70711	−	0.707107i	−0.763441	−	0.316228i	−0.730213	+	0.683220i	$0.760579\pi$
$6$		0			0
$7$	−0.292893	+	0.121320i	−0.110703	+	0.0458548i	−0.437348	−	0.899293i	$-0.644082\pi$
$7$	−0.292893	+	0.121320i	−0.110703	+	0.0458548i	0.326644	+	0.945147i	$0.394082\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.292893	−	0.707107i	−0.0883106	−	0.213201i	0.873554	−	0.486728i	$-0.161810\pi$
$11$	−0.292893	−	0.707107i	−0.0883106	−	0.213201i	−0.961864	+	0.273527i	$0.911810\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$17$	−1.00000	+	4.00000i	−0.242536	+	0.970143i
$18$		0			0
$19$	−1.58579	+	1.58579i	−0.363804	+	0.363804i	−0.865211	−	0.501407i	$-0.832816\pi$
$19$	−1.58579	+	1.58579i	−0.363804	+	0.363804i	0.501407	+	0.865211i	$0.332816\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.53553	+	6.12132i	0.528695	+	1.27638i	0.932378	+	0.361484i	$0.117730\pi$
$23$	2.53553	+	6.12132i	0.528695	+	1.27638i	−0.403683	+	0.914899i	$0.632270\pi$
$24$		0			0
$25$	−1.12132	−	1.12132i	−0.224264	−	0.224264i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.12132	+	2.12132i	0.951005	+	0.393919i	0.803609	−	0.595158i	$-0.202911\pi$
$29$	5.12132	+	2.12132i	0.951005	+	0.393919i	0.147397	+	0.989077i	$0.452911\pi$
$30$		0			0
$31$	0.878680	−	2.12132i	0.157816	−	0.381000i	−0.825118	−	0.564960i	$-0.808892\pi$
$31$	0.878680	−	2.12132i	0.157816	−	0.381000i	0.982934	+	0.183960i	$0.0588916\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.585786			0.0990160
$36$		0			0
$37$	0.292893	−	0.707107i	0.0481513	−	0.116248i	−0.897974	−	0.440049i	$-0.854961\pi$
$37$	0.292893	−	0.707107i	0.0481513	−	0.116248i	0.946125	+	0.323802i	$0.104961\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	2.29289	−	0.949747i	0.358090	−	0.148326i	−0.196384	−	0.980527i	$-0.562920\pi$
$41$	2.29289	−	0.949747i	0.358090	−	0.148326i	0.554473	+	0.832202i	$0.312920\pi$
$42$		0			0
$43$	7.24264	+	7.24264i	1.10449	+	1.10449i	0.993862	+	0.110631i	$0.0352871\pi$
$43$	7.24264	+	7.24264i	1.10449	+	1.10449i	0.110631	+	0.993862i	$0.464713\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			12.8284i			1.87122i	0.353037	+	0.935609i	$0.385149\pi$
$47$			12.8284i			1.87122i	−0.353037	+	0.935609i	$0.614851\pi$
$48$		0			0
$49$	−4.87868	+	4.87868i	−0.696954	+	0.696954i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.17157	+	2.17157i	−0.298288	+	0.298288i	−0.840343	−	0.542055i	$-0.817647\pi$
$53$	−2.17157	+	2.17157i	−0.298288	+	0.298288i	0.542055	+	0.840343i	$0.317647\pi$
$54$		0			0
$55$			1.41421i			0.190693i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.58579	−	5.58579i	−0.727207	−	0.727207i	0.242855	−	0.970063i	$-0.421916\pi$
$59$	−5.58579	−	5.58579i	−0.727207	−	0.727207i	−0.970063	+	0.242855i	$0.921916\pi$
$60$		0			0
$61$	−9.12132	+	3.77817i	−1.16787	+	0.483746i	−0.880486	−	0.474071i	$-0.842784\pi$
$61$	−9.12132	+	3.77817i	−1.16787	+	0.483746i	−0.287379	+	0.957817i	$0.592784\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	9.65685			1.17977			0.589886	−	0.807486i	$-0.299173\pi$
$67$	9.65685			1.17977			0.589886	+	0.807486i	$0.299173\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.70711	+	8.94975i	−0.439953	+	1.06214i	0.536012	+	0.844210i	$0.319930\pi$
$71$	−3.70711	+	8.94975i	−0.439953	+	1.06214i	−0.975965	+	0.217929i	$0.930070\pi$
$72$		0			0
$73$	7.36396	+	3.05025i	0.861886	+	0.357005i	0.769445	−	0.638713i	$-0.220533\pi$
$73$	7.36396	+	3.05025i	0.861886	+	0.357005i	0.0924414	+	0.995718i	$0.470533\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.171573	+	0.171573i	0.0195525	+	0.0195525i
$78$		0			0
$79$	5.46447	+	13.1924i	0.614800	+	1.48426i	0.857670	+	0.514201i	$0.171911\pi$
$79$	5.46447	+	13.1924i	0.614800	+	1.48426i	−0.242869	+	0.970059i	$0.578089\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.24264	−	7.24264i	0.794983	−	0.794983i	−0.187317	−	0.982300i	$-0.559979\pi$
$83$	7.24264	−	7.24264i	0.794983	−	0.794983i	0.982300	+	0.187317i	$0.0599790\pi$
$84$		0			0
$85$	4.53553	−	6.12132i	0.491948	−	0.663950i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			1.65685i			0.175626i	0.996137	+	0.0878131i	$0.0279878\pi$
$89$			1.65685i			0.175626i	−0.996137	+	0.0878131i	$0.972012\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	3.82843	−	1.58579i	0.392788	−	0.162698i
$96$		0			0
$97$	−14.7782	−	6.12132i	−1.50050	−	0.621526i	−0.526926	−	0.849911i	$-0.676655\pi$
$97$	−14.7782	−	6.12132i	−1.50050	−	0.621526i	−0.973571	+	0.228386i	$0.926655\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.0000			−0.995037			−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000			−0.995037			−0.497519	+	0.867453i	$0.665755\pi$
$102$		0			0
$103$	−2.34315			−0.230877			−0.115439	−	0.993315i	$-0.536827\pi$
$103$	−2.34315			−0.230877			−0.115439	+	0.993315i	$0.536827\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.3640	+	5.53553i	1.29194	+	0.535140i	0.919564	−	0.392940i	$-0.128542\pi$
$107$	13.3640	+	5.53553i	1.29194	+	0.535140i	0.372379	+	0.928081i	$0.378542\pi$
$108$		0			0
$109$	−10.7782	+	4.46447i	−1.03236	+	0.427618i	−0.833564	−	0.552422i	$-0.813704\pi$
$109$	−10.7782	+	4.46447i	−1.03236	+	0.427618i	−0.198797	+	0.980041i	$0.563704\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.12132	−	7.53553i	−0.293629	−	0.708883i	−0.999999	−	0.00104094i	$-0.999669\pi$
$113$	−3.12132	−	7.53553i	−0.293629	−	0.708883i	0.706370	−	0.707842i	$-0.250331\pi$
$114$		0			0
$115$		−	12.2426i		−	1.14163i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.192388	−	1.29289i	−0.0176362	−	0.118519i
$120$		0			0
$121$	7.36396	−	7.36396i	0.669451	−	0.669451i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	4.65685	+	11.2426i	0.416522	+	1.00557i
$126$		0			0
$127$	−1.24264	−	1.24264i	−0.110267	−	0.110267i	0.649821	−	0.760087i	$-0.274844\pi$
$127$	−1.24264	−	1.24264i	−0.110267	−	0.110267i	−0.760087	+	0.649821i	$0.774844\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	13.3640	+	5.53553i	1.16761	+	0.483642i	0.880403	−	0.474225i	$-0.157272\pi$
$131$	13.3640	+	5.53553i	1.16761	+	0.483642i	0.287211	+	0.957867i	$0.407272\pi$
$132$		0			0
$133$	0.272078	−	0.656854i	0.0235921	−	0.0569565i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	0.192388	−	0.464466i	0.0163182	−	0.0393955i	−0.915510	−	0.402294i	$-0.868213\pi$
$139$	0.192388	−	0.464466i	0.0163182	−	0.0393955i	0.931829	+	0.362899i	$0.118213\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−7.24264	−	7.24264i	−0.601469	−	0.601469i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	6.34315i		−	0.519651i	−0.965656	−	0.259825i	$-0.916335\pi$
$149$		−	6.34315i		−	0.519651i	0.965656	−	0.259825i	$-0.0836650\pi$
$150$		0			0
$151$	−14.0711	+	14.0711i	−1.14509	+	1.14509i	−0.157581	+	0.987506i	$0.550370\pi$
$151$	−14.0711	+	14.0711i	−1.14509	+	1.14509i	−0.987506	+	0.157581i	$0.949630\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.00000	+	3.00000i	−0.240966	+	0.240966i
$156$		0			0
$157$		−	15.3137i		−	1.22217i	−0.791566	−	0.611083i	$-0.790734\pi$
$157$		−	15.3137i		−	1.22217i	0.791566	−	0.611083i	$-0.209266\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.48528	−	1.48528i	−0.117057	−	0.117057i
$162$		0			0
$163$	−4.29289	+	1.77817i	−0.336245	+	0.139277i	−0.544417	−	0.838815i	$-0.683249\pi$
$163$	−4.29289	+	1.77817i	−0.336245	+	0.139277i	0.208171	+	0.978092i	$0.433249\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.292893	−	0.707107i	0.0226648	−	0.0547176i	−0.912142	−	0.409874i	$-0.865573\pi$
$167$	0.292893	−	0.707107i	0.0226648	−	0.0547176i	0.934807	+	0.355157i	$0.115573\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.22183	−	17.4350i	0.549065	−	1.32556i	−0.369110	−	0.929386i	$-0.620337\pi$
$173$	7.22183	−	17.4350i	0.549065	−	1.32556i	0.918175	−	0.396175i	$-0.129663\pi$
$174$		0			0
$175$	0.464466	+	0.192388i	0.0351103	+	0.0145432i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−12.8995	−	12.8995i	−0.964154	−	0.964154i	0.0352259	−	0.999379i	$-0.488785\pi$
$179$	−12.8995	−	12.8995i	−0.964154	−	0.964154i	−0.999379	+	0.0352259i	$0.988785\pi$
$180$		0			0
$181$	3.12132	+	7.53553i	0.232006	+	0.560112i	0.996413	−	0.0846219i	$-0.0269682\pi$
$181$	3.12132	+	7.53553i	0.232006	+	0.560112i	−0.764407	+	0.644734i	$0.776968\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.00000	+	1.00000i	−0.0735215	+	0.0735215i
$186$		0			0
$187$	3.12132	−	0.464466i	0.228254	−	0.0339651i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	8.82843i		−	0.638803i	−0.947620	−	0.319401i	$-0.896518\pi$
$191$		−	8.82843i		−	0.638803i	0.947620	−	0.319401i	$-0.103482\pi$
$192$		0			0
$193$	0.292893	+	0.707107i	0.0210829	+	0.0508987i	0.934070	−	0.357089i	$-0.116231\pi$
$193$	0.292893	+	0.707107i	0.0210829	+	0.0508987i	−0.912987	+	0.407988i	$0.866231\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−11.3640	+	4.70711i	−0.809649	+	0.335367i	−0.748814	−	0.662780i	$-0.769376\pi$
$197$	−11.3640	+	4.70711i	−0.809649	+	0.335367i	−0.0608349	+	0.998148i	$0.519376\pi$
$198$		0			0
$199$	20.7782	+	8.60660i	1.47293	+	0.610106i	0.967524	−	0.252778i	$-0.0813444\pi$
$199$	20.7782	+	8.60660i	1.47293	+	0.610106i	0.505402	+	0.862884i	$0.331344\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.75736			−0.123342
$204$		0			0
$205$	−4.58579			−0.320285
$206$		0			0
$207$		0			0
$208$		0			0
$209$	1.58579	+	0.656854i	0.109691	+	0.0454356i
$210$		0			0
$211$	24.1924	−	10.0208i	1.66547	−	0.689861i	0.666997	−	0.745060i	$-0.267579\pi$
$211$	24.1924	−	10.0208i	1.66547	−	0.689861i	0.998475	+	0.0551988i	$0.0175792\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−7.24264	−	17.4853i	−0.493944	−	1.19249i
$216$		0			0
$217$			0.727922i			0.0494146i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−12.4142	+	12.4142i	−0.831317	+	0.831317i	−0.987697	−	0.156380i	$-0.950018\pi$
$223$	−12.4142	+	12.4142i	−0.831317	+	0.831317i	0.156380	+	0.987697i	$0.450018\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.36396	+	3.29289i	0.0905293	+	0.218557i	0.962659	−	0.270719i	$-0.0872613\pi$
$227$	1.36396	+	3.29289i	0.0905293	+	0.218557i	−0.872129	+	0.489276i	$0.837261\pi$
$228$		0			0
$229$	−7.00000	−	7.00000i	−0.462573	−	0.462573i	0.436925	−	0.899498i	$-0.356068\pi$
$229$	−7.00000	−	7.00000i	−0.462573	−	0.462573i	−0.899498	+	0.436925i	$0.856068\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.535534	−	0.221825i	−0.0350840	−	0.0145323i	0.365072	−	0.930979i	$-0.381044\pi$
$233$	−0.535534	−	0.221825i	−0.0350840	−	0.0145323i	−0.400156	+	0.916447i	$0.631044\pi$
$234$		0			0
$235$	9.07107	−	21.8995i	0.591731	−	1.42857i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−11.3137			−0.731823			−0.365911	−	0.930650i	$-0.619243\pi$
$239$	−11.3137			−0.731823			−0.365911	+	0.930650i	$0.619243\pi$
$240$		0			0
$241$	−5.36396	+	12.9497i	−0.345523	+	0.834167i	0.651614	+	0.758551i	$0.274092\pi$
$241$	−5.36396	+	12.9497i	−0.345523	+	0.834167i	−0.997137	+	0.0756158i	$0.975908\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.7782	−	4.87868i	0.752480	−	0.311687i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			0.828427i			0.0522899i	0.999658	+	0.0261449i	$0.00832314\pi$
$251$			0.828427i			0.0522899i	−0.999658	+	0.0261449i	$0.991677\pi$
$252$		0			0
$253$	3.58579	−	3.58579i	0.225436	−	0.225436i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.82843	−	5.82843i	0.363567	−	0.363567i	−0.501557	−	0.865124i	$-0.667239\pi$
$257$	5.82843	−	5.82843i	0.363567	−	0.363567i	0.865124	+	0.501557i	$0.167239\pi$
$258$		0			0
$259$			0.242641i			0.0150770i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2.75736	−	2.75736i	−0.170026	−	0.170026i	0.616965	−	0.786991i	$-0.288362\pi$
$263$	−2.75736	−	2.75736i	−0.170026	−	0.170026i	−0.786991	+	0.616965i	$0.788362\pi$
$264$		0			0
$265$	5.24264	−	2.17157i	0.322053	−	0.133399i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.36396	−	12.9497i	0.327046	−	0.789560i	−0.671762	−	0.740767i	$-0.734462\pi$
$269$	5.36396	−	12.9497i	0.327046	−	0.789560i	0.998809	−	0.0487934i	$-0.0155376\pi$
$270$		0			0
$271$	−11.3137			−0.687259			−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137			−0.687259			−0.343629	+	0.939105i	$0.611656\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.464466	+	1.12132i	−0.0280084	+	0.0676182i
$276$		0			0
$277$	8.53553	+	3.53553i	0.512851	+	0.212430i	0.624073	−	0.781366i	$-0.285477\pi$
$277$	8.53553	+	3.53553i	0.512851	+	0.212430i	−0.111223	+	0.993796i	$0.535477\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	2.31371	+	2.31371i	0.138024	+	0.138024i	0.772743	−	0.634719i	$-0.218884\pi$
$281$	2.31371	+	2.31371i	0.138024	+	0.138024i	−0.634719	+	0.772743i	$0.718884\pi$
$282$		0			0
$283$	−10.0503	−	24.2635i	−0.597426	−	1.44231i	−0.876196	−	0.481955i	$-0.839927\pi$
$283$	−10.0503	−	24.2635i	−0.597426	−	1.44231i	0.278770	−	0.960358i	$-0.410073\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.556349	+	0.556349i	−0.0328403	+	0.0328403i
$288$		0			0
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	7.31371i		−	0.427271i	−0.976913	−	0.213636i	$-0.931469\pi$
$293$		−	7.31371i		−	0.427271i	0.976913	−	0.213636i	$-0.0685306\pi$
$294$		0			0
$295$	5.58579	+	13.4853i	0.325217	+	0.785143i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−3.00000	−	1.24264i	−0.172917	−	0.0716246i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	18.2426			1.04457
$306$		0			0
$307$	−26.6274			−1.51971			−0.759853	−	0.650094i	$-0.774729\pi$
$307$	−26.6274			−1.51971			−0.759853	+	0.650094i	$0.774729\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3.70711	+	1.53553i	0.210211	+	0.0870721i	0.485304	−	0.874345i	$-0.338709\pi$
$311$	3.70711	+	1.53553i	0.210211	+	0.0870721i	−0.275094	+	0.961417i	$0.588709\pi$
$312$		0			0
$313$	−3.94975	+	1.63604i	−0.223253	+	0.0924744i	−0.491506	−	0.870874i	$-0.663554\pi$
$313$	−3.94975	+	1.63604i	−0.223253	+	0.0924744i	0.268253	+	0.963348i	$0.413554\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.05025	+	14.6066i	0.339816	+	0.820388i	0.997733	+	0.0672979i	$0.0214378\pi$
$317$	6.05025	+	14.6066i	0.339816	+	0.820388i	−0.657917	+	0.753091i	$0.728562\pi$
$318$		0			0
$319$		−	4.24264i		−	0.237542i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.75736	−	7.92893i	−0.264707	−	0.441178i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.55635	−	3.75736i	−0.0858043	−	0.207150i
$330$		0			0
$331$	−9.72792	−	9.72792i	−0.534695	−	0.534695i	0.387271	−	0.921966i	$-0.373418\pi$
$331$	−9.72792	−	9.72792i	−0.534695	−	0.534695i	−0.921966	+	0.387271i	$0.873418\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.4853	−	6.82843i	−0.900687	−	0.373077i
$336$		0			0
$337$	5.94975	−	14.3640i	0.324103	−	0.782455i	−0.674904	−	0.737906i	$-0.735815\pi$
$337$	5.94975	−	14.3640i	0.324103	−	0.782455i	0.999007	−	0.0445491i	$-0.0141851\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1.75736			−0.0951663
$342$		0			0
$343$	1.68629	−	4.07107i	0.0910512	−	0.219817i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−5.36396	+	2.22183i	−0.287953	+	0.119274i	−0.521984	−	0.852955i	$-0.674808\pi$
$347$	−5.36396	+	2.22183i	−0.287953	+	0.119274i	0.234032	+	0.972229i	$0.424808\pi$
$348$		0			0
$349$	−15.4853	−	15.4853i	−0.828908	−	0.828908i	0.158458	−	0.987366i	$-0.449348\pi$
$349$	−15.4853	−	15.4853i	−0.828908	−	0.828908i	−0.987366	+	0.158458i	$0.949348\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	24.9706i		−	1.32905i	−0.747266	−	0.664524i	$-0.768634\pi$
$353$		−	24.9706i		−	1.32905i	0.747266	−	0.664524i	$-0.231366\pi$
$354$		0			0
$355$	12.6569	−	12.6569i	0.671756	−	0.671756i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	15.7279	−	15.7279i	0.830088	−	0.830088i	−0.157440	−	0.987528i	$-0.550324\pi$
$359$	15.7279	−	15.7279i	0.830088	−	0.830088i	0.987528	+	0.157440i	$0.0503242\pi$
$360$		0			0
$361$			13.9706i			0.735293i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−10.4142	−	10.4142i	−0.545105	−	0.545105i
$366$		0			0
$367$	−1.94975	+	0.807612i	−0.101776	+	0.0421570i	−0.432990	−	0.901398i	$-0.642542\pi$
$367$	−1.94975	+	0.807612i	−0.101776	+	0.0421570i	0.331214	+	0.943555i	$0.392542\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.372583	−	0.899495i	0.0193435	−	0.0466995i
$372$		0			0
$373$	35.9411			1.86096			0.930480	−	0.366342i	$-0.119390\pi$
$373$	35.9411			1.86096			0.930480	+	0.366342i	$0.119390\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	13.9497	+	5.77817i	0.716550	+	0.296805i	0.711012	−	0.703180i	$-0.248237\pi$
$379$	13.9497	+	5.77817i	0.716550	+	0.296805i	0.00553829	+	0.999985i	$0.498237\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	7.58579	+	7.58579i	0.387616	+	0.387616i	0.873836	−	0.486221i	$-0.161625\pi$
$383$	7.58579	+	7.58579i	0.387616	+	0.387616i	−0.486221	+	0.873836i	$0.661625\pi$
$384$		0			0
$385$	−0.171573	−	0.414214i	−0.00874416	−	0.0211103i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.3137	+	20.3137i	−1.02995	+	1.02995i	−0.0304083	+	0.999538i	$0.509681\pi$
$389$	−20.3137	+	20.3137i	−1.02995	+	1.02995i	−0.999538	+	0.0304083i	$0.990319\pi$
$390$		0			0
$391$	−27.0208	+	4.02082i	−1.36650	+	0.203341i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	26.3848i		−	1.32756i
$396$		0			0
$397$	11.1213	+	26.8492i	0.558163	+	1.34752i	0.911219	+	0.411923i	$0.135143\pi$
$397$	11.1213	+	26.8492i	0.558163	+	1.34752i	−0.353056	+	0.935602i	$0.614857\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−25.0208	+	10.3640i	−1.24948	+	0.517552i	−0.906665	−	0.421851i	$-0.861380\pi$
$401$	−25.0208	+	10.3640i	−1.24948	+	0.517552i	−0.342815	+	0.939403i	$0.611380\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.585786			−0.0290364
$408$		0			0
$409$	18.0000			0.890043			0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000			0.890043			0.445021	+	0.895520i	$0.353196\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.31371	+	0.958369i	0.113850	+	0.0471583i
$414$		0			0
$415$	−17.4853	+	7.24264i	−0.858319	+	0.355527i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	2.53553	+	6.12132i	0.123869	+	0.299046i	0.973634	−	0.228115i	$-0.0732563\pi$
$419$	2.53553	+	6.12132i	0.123869	+	0.299046i	−0.849765	+	0.527161i	$0.823256\pi$
$420$		0			0
$421$			11.3137i			0.551396i	0.961244	+	0.275698i	$0.0889090\pi$
$421$			11.3137i			0.551396i	−0.961244	+	0.275698i	$0.911091\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5.60660	−	3.36396i	0.271960	−	0.163176i
$426$		0			0
$427$	2.21320	−	2.21320i	0.107104	−	0.107104i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.878680	+	2.12132i	0.0423245	+	0.102180i	0.943628	−	0.331008i	$-0.107389\pi$
$431$	0.878680	+	2.12132i	0.0423245	+	0.102180i	−0.901304	+	0.433188i	$0.857389\pi$
$432$		0			0
$433$	3.82843	+	3.82843i	0.183982	+	0.183982i	0.793089	−	0.609106i	$-0.208472\pi$
$433$	3.82843	+	3.82843i	0.183982	+	0.183982i	−0.609106	+	0.793089i	$0.708472\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−13.7279	−	5.68629i	−0.656696	−	0.272012i
$438$		0			0
$439$	−3.60660	+	8.70711i	−0.172134	+	0.415568i	−0.986278	−	0.165096i	$-0.947207\pi$
$439$	−3.60660	+	8.70711i	−0.172134	+	0.415568i	0.814144	+	0.580663i	$0.197207\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	40.2843			1.91396			0.956982	−	0.290148i	$-0.0937046\pi$
$443$	40.2843			1.91396			0.956982	+	0.290148i	$0.0937046\pi$
$444$		0			0
$445$	1.17157	−	2.82843i	0.0555379	−	0.134080i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−16.5355	+	6.84924i	−0.780360	+	0.323236i	−0.737061	−	0.675826i	$-0.763787\pi$
$449$	−16.5355	+	6.84924i	−0.780360	+	0.323236i	−0.0432993	+	0.999062i	$0.513787\pi$
$450$		0			0
$451$	−1.34315	−	1.34315i	−0.0632463	−	0.0632463i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.48528	+	3.48528i	−0.163035	+	0.163035i	−0.783910	−	0.620875i	$-0.786777\pi$
$457$	−3.48528	+	3.48528i	−0.163035	+	0.163035i	0.620875	+	0.783910i	$0.286777\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	2.51472	−	2.51472i	0.117122	−	0.117122i	−0.646117	−	0.763239i	$-0.723608\pi$
$461$	2.51472	−	2.51472i	0.117122	−	0.117122i	0.763239	+	0.646117i	$0.223608\pi$
$462$		0			0
$463$		−	24.1421i		−	1.12198i	−0.827823	−	0.560990i	$-0.810421\pi$
$463$		−	24.1421i		−	1.12198i	0.827823	−	0.560990i	$-0.189579\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−9.58579	−	9.58579i	−0.443577	−	0.443577i	0.449635	−	0.893212i	$-0.351554\pi$
$467$	−9.58579	−	9.58579i	−0.443577	−	0.443577i	−0.893212	+	0.449635i	$0.851554\pi$
$468$		0			0
$469$	−2.82843	+	1.17157i	−0.130605	+	0.0540982i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.00000	−	7.24264i	0.137940	−	0.333017i
$474$		0			0
$475$	3.55635			0.163176
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.5355	+	35.0919i	−0.664145	+	1.60339i	0.127100	+	0.991890i	$0.459433\pi$
$479$	−14.5355	+	35.0919i	−0.664145	+	1.60339i	−0.791245	+	0.611499i	$0.790567\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	20.8995	+	20.8995i	0.948997	+	0.948997i
$486$		0			0
$487$	−6.73654	−	16.2635i	−0.305262	−	0.736968i	−0.999846	−	0.0175559i	$-0.994412\pi$
$487$	−6.73654	−	16.2635i	−0.305262	−	0.736968i	0.694584	−	0.719412i	$-0.255588\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−14.4142	+	14.4142i	−0.650504	+	0.650504i	−0.953114	−	0.302610i	$-0.902142\pi$
$491$	−14.4142	+	14.4142i	−0.650504	+	0.650504i	0.302610	+	0.953114i	$0.402142\pi$
$492$		0			0
$493$	−13.6066	+	18.3640i	−0.612811	+	0.827071i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	3.07107i		−	0.137756i
$498$		0			0
$499$	6.43503	+	15.5355i	0.288071	+	0.695466i	0.999977	−	0.00680940i	$-0.00216752\pi$
$499$	6.43503	+	15.5355i	0.288071	+	0.695466i	−0.711905	+	0.702275i	$0.752168\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.292893	−	0.121320i	0.0130595	−	0.00540941i	−0.376144	−	0.926561i	$-0.622750\pi$
$503$	0.292893	−	0.121320i	0.0130595	−	0.00540941i	0.389204	+	0.921152i	$0.372750\pi$
$504$		0			0
$505$	17.0711	+	7.07107i	0.759653	+	0.314658i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	38.2843			1.69692			0.848460	−	0.529259i	$-0.177530\pi$
$509$	38.2843			1.69692			0.848460	+	0.529259i	$0.177530\pi$
$510$		0			0
$511$	−2.52691			−0.111784
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.00000	+	1.65685i	0.176261	+	0.0730097i
$516$		0			0
$517$	9.07107	−	3.75736i	0.398945	−	0.165248i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	15.0208	+	36.2635i	0.658074	+	1.58873i	0.800775	+	0.598965i	$0.204421\pi$
$521$	15.0208	+	36.2635i	0.658074	+	1.58873i	−0.142701	+	0.989766i	$0.545579\pi$
$522$		0			0
$523$		−	4.14214i		−	0.181123i	−0.995891	−	0.0905615i	$-0.971134\pi$
$523$		−	4.14214i		−	0.181123i	0.995891	−	0.0905615i	$-0.0288662\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	7.60660	+	5.63604i	0.331349	+	0.245510i
$528$		0			0
$529$	−14.7782	+	14.7782i	−0.642529	+	0.642529i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−18.8995	−	18.8995i	−0.817096	−	0.817096i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4.87868	+	2.02082i	0.210140	+	0.0870427i
$540$		0			0
$541$	9.94975	−	24.0208i	0.427773	−	1.03274i	−0.552219	−	0.833699i	$-0.686219\pi$
$541$	9.94975	−	24.0208i	0.427773	−	1.03274i	0.979992	−	0.199036i	$-0.0637812\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	21.5563			0.923373
$546$		0			0
$547$	−17.2635	+	41.6777i	−0.738132	+	1.78201i	−0.124802	+	0.992182i	$0.539830\pi$
$547$	−17.2635	+	41.6777i	−0.738132	+	1.78201i	−0.613330	+	0.789827i	$0.710170\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−11.4853	+	4.75736i	−0.489289	+	0.202670i
$552$		0			0
$553$	−3.20101	−	3.20101i	−0.136121	−	0.136121i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	16.0000i		−	0.677942i	−0.940797	−	0.338971i	$-0.889921\pi$
$557$		−	16.0000i		−	0.677942i	0.940797	−	0.338971i	$-0.110079\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−18.4142	+	18.4142i	−0.776067	+	0.776067i	−0.979159	−	0.203093i	$-0.934901\pi$
$563$	−18.4142	+	18.4142i	−0.776067	+	0.776067i	0.203093	+	0.979159i	$0.434901\pi$
$564$		0			0
$565$			15.0711i			0.634045i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.3431	+	21.3431i	0.894751	+	0.894751i	0.994966	−	0.100215i	$-0.0319529\pi$
$569$	21.3431	+	21.3431i	0.894751	+	0.894751i	−0.100215	+	0.994966i	$0.531953\pi$
$570$		0			0
$571$	29.8492	−	12.3640i	1.24915	−	0.517416i	0.342589	−	0.939486i	$-0.388696\pi$
$571$	29.8492	−	12.3640i	1.24915	−	0.517416i	0.906563	+	0.422070i	$0.138696\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.02082	−	9.70711i	0.167680	−	0.404814i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.24264	+	3.00000i	−0.0515534	+	0.124461i
$582$		0			0
$583$	2.17157	+	0.899495i	0.0899374	+	0.0372533i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.27208	−	6.27208i	−0.258876	−	0.258876i	0.565721	−	0.824597i	$-0.308598\pi$
$587$	−6.27208	−	6.27208i	−0.258876	−	0.258876i	−0.824597	+	0.565721i	$0.808598\pi$
$588$		0			0
$589$	1.97056	+	4.75736i	0.0811956	+	0.196024i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−15.3431	+	15.3431i	−0.630067	+	0.630067i	−0.948085	−	0.318017i	$-0.896983\pi$
$593$	−15.3431	+	15.3431i	−0.630067	+	0.630067i	0.318017	+	0.948085i	$0.396983\pi$
$594$		0			0
$595$	−0.585786	+	2.34315i	−0.0240149	+	0.0960596i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			44.8284i			1.83164i	0.401589	+	0.915820i	$0.368458\pi$
$599$			44.8284i			1.83164i	−0.401589	+	0.915820i	$0.631542\pi$
$600$		0			0
$601$	11.6066	+	28.0208i	0.473443	+	1.14299i	0.962631	+	0.270815i	$0.0872931\pi$
$601$	11.6066	+	28.0208i	0.473443	+	1.14299i	−0.489188	+	0.872178i	$0.662707\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.7782	+	7.36396i	−0.722786	+	0.299388i
$606$		0			0
$607$	36.0919	+	14.9497i	1.46492	+	0.606792i	0.965695	−	0.259678i	$-0.0836166\pi$
$607$	36.0919	+	14.9497i	1.46492	+	0.606792i	0.499229	+	0.866470i	$0.333617\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−29.3137			−1.18397			−0.591985	−	0.805949i	$-0.701655\pi$
$613$	−29.3137			−1.18397			−0.591985	+	0.805949i	$0.701655\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	22.7782	+	9.43503i	0.917015	+	0.379840i	0.790738	−	0.612155i	$-0.209697\pi$
$617$	22.7782	+	9.43503i	0.917015	+	0.379840i	0.126277	+	0.991995i	$0.459697\pi$
$618$		0			0
$619$	−1.94975	+	0.807612i	−0.0783670	+	0.0324607i	−0.421523	−	0.906818i	$-0.638504\pi$
$619$	−1.94975	+	0.807612i	−0.0783670	+	0.0324607i	0.343156	+	0.939279i	$0.388504\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−0.201010	−	0.485281i	−0.00805330	−	0.0194424i
$624$		0			0
$625$		−	14.5563i		−	0.582254i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.53553	+	1.87868i	0.101098	+	0.0749079i
$630$		0			0
$631$	−17.3848	+	17.3848i	−0.692077	+	0.692077i	−0.962689	−	0.270612i	$-0.912774\pi$
$631$	−17.3848	+	17.3848i	−0.692077	+	0.692077i	0.270612	+	0.962689i	$0.412774\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.24264	+	3.00000i	0.0493127	+	0.119051i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10.8787	−	4.50610i	−0.429682	−	0.177980i	0.157352	−	0.987543i	$-0.449704\pi$
$641$	−10.8787	−	4.50610i	−0.429682	−	0.177980i	−0.587034	+	0.809563i	$0.699704\pi$
$642$		0			0
$643$	15.0208	−	36.2635i	0.592363	−	1.43009i	−0.288851	−	0.957374i	$-0.593273\pi$
$643$	15.0208	−	36.2635i	0.592363	−	1.43009i	0.881214	−	0.472717i	$-0.156727\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−45.2548			−1.77915			−0.889576	−	0.456788i	$-0.849000\pi$
$647$	−45.2548			−1.77915			−0.889576	+	0.456788i	$0.849000\pi$
$648$		0			0
$649$	−2.31371	+	5.58579i	−0.0908210	+	0.219261i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−23.3640	+	9.67767i	−0.914302	+	0.378716i	−0.789702	−	0.613491i	$-0.789765\pi$
$653$	−23.3640	+	9.67767i	−0.914302	+	0.378716i	−0.124600	+	0.992207i	$0.539765\pi$
$654$		0			0
$655$	−18.8995	−	18.8995i	−0.738464	−	0.738464i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	22.7696i		−	0.886976i	−0.896280	−	0.443488i	$-0.853741\pi$
$659$		−	22.7696i		−	0.886976i	0.896280	−	0.443488i	$-0.146259\pi$
$660$		0			0
$661$	−18.7990	+	18.7990i	−0.731196	+	0.731196i	−0.970857	−	0.239661i	$-0.922964\pi$
$661$	−18.7990	+	18.7990i	−0.731196	+	0.731196i	0.239661	+	0.970857i	$0.422964\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.928932	+	0.928932i	−0.0360224	+	0.0360224i
$666$		0			0
$667$			36.7279i			1.42211i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.34315	+	5.34315i	0.206270	+	0.206270i
$672$		0			0
$673$	14.1924	−	5.87868i	0.547076	−	0.226606i	−0.0919875	−	0.995760i	$-0.529322\pi$
$673$	14.1924	−	5.87868i	0.547076	−	0.226606i	0.639064	+	0.769154i	$0.279322\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	3.90812	−	9.43503i	0.150201	−	0.362618i	−0.830814	−	0.556551i	$-0.812124\pi$
$677$	3.90812	−	9.43503i	0.150201	−	0.362618i	0.981015	+	0.193933i	$0.0621245\pi$
$678$		0			0
$679$	5.07107			0.194610
$680$		0			0
$681$		0			0
$682$		0			0
$683$	12.9792	−	31.3345i	0.496635	−	1.19898i	−0.454650	−	0.890670i	$-0.650236\pi$
$683$	12.9792	−	31.3345i	0.496635	−	1.19898i	0.951285	−	0.308312i	$-0.0997642\pi$
$684$		0			0
$685$	23.8995	+	9.89949i	0.913153	+	0.378240i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−12.3934	−	29.9203i	−0.471467	−	1.13822i	−0.963515	−	0.267654i	$-0.913752\pi$
$691$	−12.3934	−	29.9203i	−0.471467	−	1.13822i	0.492048	−	0.870568i	$-0.336248\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.656854	+	0.656854i	−0.0249159	+	0.0249159i
$696$		0			0
$697$	1.50610	+	10.1213i	0.0570475	+	0.383372i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			28.6863i			1.08347i	0.840551	+	0.541733i	$0.182232\pi$
$701$			28.6863i			1.08347i	−0.840551	+	0.541733i	$0.817768\pi$
$702$		0			0
$703$	0.656854	+	1.58579i	0.0247737	+	0.0598091i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.92893	−	1.21320i	0.110154	−	0.0456272i
$708$		0			0
$709$	34.1924	+	14.1630i	1.28412	+	0.531901i	0.917228	−	0.398362i	$-0.130421\pi$
$709$	34.1924	+	14.1630i	1.28412	+	0.531901i	0.366894	+	0.930263i	$0.380421\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	15.2132			0.569739
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.9497	−	7.43503i	−0.669413	−	0.277280i	0.0219807	−	0.999758i	$-0.493003\pi$
$719$	−17.9497	−	7.43503i	−0.669413	−	0.277280i	−0.691393	+	0.722479i	$0.743003\pi$
$720$		0			0
$721$	0.686292	−	0.284271i	0.0255588	−	0.0105868i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.36396	−	8.12132i	−0.124934	−	0.301618i
$726$		0			0
$727$			17.7990i			0.660128i	0.943959	+	0.330064i	$0.107070\pi$
$727$			17.7990i			0.660128i	−0.943959	+	0.330064i	$0.892930\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−36.2132	+	21.7279i	−1.33939	+	0.803636i
$732$		0			0
$733$	17.4853	−	17.4853i	0.645834	−	0.645834i	−0.306150	−	0.951983i	$-0.599041\pi$
$733$	17.4853	−	17.4853i	0.645834	−	0.645834i	0.951983	+	0.306150i	$0.0990408\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.82843	−	6.82843i	−0.104186	−	0.251528i
$738$		0			0
$739$	−6.41421	−	6.41421i	−0.235951	−	0.235951i	0.579220	−	0.815171i	$-0.303357\pi$
$739$	−6.41421	−	6.41421i	−0.235951	−	0.235951i	−0.815171	+	0.579220i	$0.803357\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	29.8492	+	12.3640i	1.09506	+	0.453590i	0.855769	−	0.517358i	$-0.173084\pi$
$743$	29.8492	+	12.3640i	1.09506	+	0.453590i	0.239293	+	0.970947i	$0.423084\pi$
$744$		0			0
$745$	−4.48528	+	10.8284i	−0.164328	+	0.396723i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.58579			−0.167561
$750$		0			0
$751$	−9.26346	+	22.3640i	−0.338028	+	0.816073i	0.659877	+	0.751374i	$0.270609\pi$
$751$	−9.26346	+	22.3640i	−0.338028	+	0.816073i	−0.997905	+	0.0646985i	$0.979391\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	33.9706	−	14.0711i	1.23632	−	0.512099i
$756$		0			0
$757$	9.97056	+	9.97056i	0.362386	+	0.362386i	0.864691	−	0.502305i	$-0.167514\pi$
$757$	9.97056	+	9.97056i	0.362386	+	0.362386i	−0.502305	+	0.864691i	$0.667514\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	3.02944i		−	0.109817i	−0.998491	−	0.0549085i	$-0.982513\pi$
$761$		−	3.02944i		−	0.109817i	0.998491	−	0.0549085i	$-0.0174867\pi$
$762$		0			0
$763$	2.61522	−	2.61522i	0.0946775	−	0.0946775i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	43.5980i		−	1.57218i	−0.618110	−	0.786092i	$-0.712101\pi$
$769$		−	43.5980i		−	1.57218i	0.618110	−	0.786092i	$-0.287899\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	23.2843	+	23.2843i	0.837477	+	0.837477i	0.988526	−	0.151049i	$-0.0482652\pi$
$773$	23.2843	+	23.2843i	0.837477	+	0.837477i	−0.151049	+	0.988526i	$0.548265\pi$
$774$		0			0
$775$	−3.36396	+	1.39340i	−0.120837	+	0.0500523i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.12994	+	5.14214i	−0.0763131	+	0.184236i
$780$		0			0
$781$	7.41421			0.265301
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.8284	+	26.1421i	−0.386483	+	0.933053i
$786$		0			0
$787$	10.4350	+	4.32233i	0.371969	+	0.154074i	0.560833	−	0.827929i	$-0.310481\pi$
$787$	10.4350	+	4.32233i	0.371969	+	0.154074i	−0.188865	+	0.982003i	$0.560481\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.82843	+	1.82843i	0.0650114	+	0.0650114i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.48528	+	9.48528i	−0.335986	+	0.335986i	−0.854854	−	0.518868i	$-0.826354\pi$
$797$	−9.48528	+	9.48528i	−0.335986	+	0.335986i	0.518868	+	0.854854i	$0.326354\pi$
$798$		0			0
$799$	−51.3137	−	12.8284i	−1.81535	−	0.453837i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	6.10051i		−	0.215282i
$804$		0			0
$805$	1.48528	+	3.58579i	0.0523493	+	0.126382i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14.7782	−	6.12132i	0.519573	−	0.215214i	−0.107456	−	0.994210i	$-0.534271\pi$
$809$	14.7782	−	6.12132i	0.519573	−	0.215214i	0.627029	+	0.778996i	$0.284271\pi$
$810$		0			0
$811$	−39.5061	−	16.3640i	−1.38725	−	0.574616i	−0.440838	−	0.897587i	$-0.645319\pi$
$811$	−39.5061	−	16.3640i	−1.38725	−	0.574616i	−0.946409	+	0.322970i	$0.895319\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8.58579			0.300747
$816$		0			0
$817$	−22.9706			−0.803638
$818$		0			0
$819$		0			0
$820$		0			0
$821$	44.4350	+	18.4056i	1.55079	+	0.642360i	0.983459	−	0.181128i	$-0.0579750\pi$
$821$	44.4350	+	18.4056i	1.55079	+	0.642360i	0.567334	+	0.823488i	$0.307975\pi$
$822$		0			0
$823$	−8.77817	+	3.63604i	−0.305988	+	0.126744i	−0.530394	−	0.847751i	$-0.677956\pi$
$823$	−8.77817	+	3.63604i	−0.305988	+	0.126744i	0.224406	+	0.974496i	$0.427956\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.12132	−	7.53553i	−0.108539	−	0.262036i	0.860273	−	0.509834i	$-0.170293\pi$
$827$	−3.12132	−	7.53553i	−0.108539	−	0.262036i	−0.968812	+	0.247798i	$0.920293\pi$
$828$		0			0
$829$		−	53.9411i		−	1.87345i	−0.350062	−	0.936726i	$-0.613840\pi$
$829$		−	53.9411i		−	1.87345i	0.350062	−	0.936726i	$-0.386160\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−14.6360	−	24.3934i	−0.507109	−	0.845181i
$834$		0			0
$835$	−1.00000	+	1.00000i	−0.0346064	+	0.0346064i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	13.3640	+	32.2635i	0.461375	+	1.11386i	0.967833	+	0.251593i	$0.0809546\pi$
$839$	13.3640	+	32.2635i	0.461375	+	1.11386i	−0.506458	+	0.862265i	$0.669045\pi$
$840$		0			0
$841$	1.22183	+	1.22183i	0.0421319	+	0.0421319i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−22.1924	−	9.19239i	−0.763441	−	0.316228i
$846$		0			0
$847$	−1.26346	+	3.05025i	−0.0434129	+	0.104808i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.07107			0.173834
$852$		0			0
$853$	−6.05025	+	14.6066i	−0.207157	+	0.500121i	−0.992973	−	0.118339i	$-0.962243\pi$
$853$	−6.05025	+	14.6066i	−0.207157	+	0.500121i	0.785817	+	0.618460i	$0.212243\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.36396	+	3.05025i	−0.251548	+	0.104195i	−0.504895	−	0.863181i	$-0.668469\pi$
$857$	−7.36396	+	3.05025i	−0.251548	+	0.104195i	0.253347	+	0.967376i	$0.418469\pi$
$858$		0			0
$859$	−29.7279	−	29.7279i	−1.01430	−	1.01430i	−0.999896	−	0.0144074i	$-0.995414\pi$
$859$	−29.7279	−	29.7279i	−1.01430	−	1.01430i	−0.0144074	−	0.999896i	$-0.504586\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	15.8579i		−	0.539808i	−0.962887	−	0.269904i	$-0.913008\pi$
$863$		−	15.8579i		−	0.539808i	0.962887	−	0.269904i	$-0.0869920\pi$
$864$		0			0
$865$	−24.6569	+	24.6569i	−0.838358	+	0.838358i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.72792	−	7.72792i	0.262152	−	0.262152i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.72792	−	2.72792i	−0.0922206	−	0.0922206i
$876$		0			0
$877$	−6.77817	+	2.80761i	−0.228883	+	0.0948063i	−0.494178	−	0.869361i	$-0.664531\pi$
$877$	−6.77817	+	2.80761i	−0.228883	+	0.0948063i	0.265295	+	0.964167i	$0.414531\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.26346	+	22.3640i	−0.312094	+	0.753461i	0.687533	+	0.726153i	$0.258694\pi$
$881$	−9.26346	+	22.3640i	−0.312094	+	0.753461i	−0.999627	+	0.0273081i	$0.991306\pi$
$882$		0			0
$883$	41.2548			1.38834			0.694168	−	0.719813i	$-0.255773\pi$
$883$	41.2548			1.38834			0.694168	+	0.719813i	$0.255773\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.84924	+	14.1213i	−0.196398	+	0.474148i	−0.991143	−	0.132796i	$-0.957605\pi$
$887$	−5.84924	+	14.1213i	−0.196398	+	0.474148i	0.794745	+	0.606943i	$0.207605\pi$
$888$		0			0
$889$	0.514719	+	0.213203i	0.0172631	+	0.00715061i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−20.3431	−	20.3431i	−0.680757	−	0.680757i
$894$		0			0
$895$	12.8995	+	31.1421i	0.431183	+	1.04097i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	9.00000	−	9.00000i	0.300167	−	0.300167i
$900$		0			0
$901$	−6.51472	−	10.8579i	−0.217037	−	0.361728i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	15.0711i		−	0.500979i
$906$		0			0
$907$	8.97918	+	21.6777i	0.298149	+	0.719795i	0.999972	+	0.00744418i	$0.00236958\pi$
$907$	8.97918	+	21.6777i	0.298149	+	0.719795i	−0.701823	+	0.712351i	$0.747630\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.9203	−	11.1508i	0.891910	−	0.369441i	0.110806	−	0.993842i	$-0.464657\pi$
$911$	26.9203	−	11.1508i	0.891910	−	0.369441i	0.781104	+	0.624401i	$0.214657\pi$
$912$		0			0
$913$	−7.24264	−	3.00000i	−0.239696	−	0.0992855i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4.58579			−0.151436
$918$		0			0
$919$	44.2843			1.46080			0.730402	−	0.683018i	$-0.239333\pi$
$919$	44.2843			1.46080			0.730402	+	0.683018i	$0.239333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.12132	+	0.464466i	−0.0368688	+	0.0152716i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.6360	−	25.6777i	−0.348957	−	0.842457i	−0.996744	−	0.0806351i	$-0.974305\pi$
$929$	−10.6360	−	25.6777i	−0.348957	−	0.842457i	0.647787	−	0.761822i	$-0.275695\pi$
$930$		0			0
$931$		−	15.4731i		−	0.507110i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−5.65685	−	1.41421i	−0.184999	−	0.0462497i
$936$		0			0
$937$	0.313708	−	0.313708i	0.0102484	−	0.0102484i	−0.701964	−	0.712212i	$-0.747693\pi$
$937$	0.313708	−	0.313708i	0.0102484	−	0.0102484i	0.712212	+	0.701964i	$0.247693\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−14.9203	−	36.0208i	−0.486388	−	1.17425i	−0.956525	−	0.291651i	$-0.905795\pi$
$941$	−14.9203	−	36.0208i	−0.486388	−	1.17425i	0.470136	−	0.882594i	$-0.344205\pi$
$942$		0			0
$943$	11.6274	+	11.6274i	0.378641	+	0.378641i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	47.5061	+	19.6777i	1.54374	+	0.639438i	0.982171	−	0.187991i	$-0.0601976\pi$
$947$	47.5061	+	19.6777i	1.54374	+	0.639438i	0.561570	+	0.827429i	$0.310198\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	50.0000			1.61966			0.809829	−	0.586665i	$-0.199560\pi$
$953$	50.0000			1.61966			0.809829	+	0.586665i	$0.199560\pi$
$954$		0			0
$955$	−6.24264	+	15.0711i	−0.202007	+	0.487688i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.10051	−	1.69848i	0.132412	−	0.0548469i
$960$		0			0
$961$	18.1924	+	18.1924i	0.586851	+	0.586851i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.41421i		−	0.0455251i
$966$		0			0
$967$	−24.4142	+	24.4142i	−0.785108	+	0.785108i	−0.980688	−	0.195580i	$-0.937341\pi$
$967$	−24.4142	+	24.4142i	−0.785108	+	0.785108i	0.195580	+	0.980688i	$0.437341\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−38.6985	+	38.6985i	−1.24189	+	1.24189i	−0.282679	+	0.959215i	$0.591223\pi$
$971$	−38.6985	+	38.6985i	−1.24189	+	1.24189i	−0.959215	+	0.282679i	$0.908777\pi$
$972$		0			0
$973$			0.159380i			0.00510947i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.3137	−	24.3137i	−0.777864	−	0.777864i	0.201603	−	0.979467i	$-0.435385\pi$
$977$	−24.3137	−	24.3137i	−0.777864	−	0.777864i	−0.979467	+	0.201603i	$0.935385\pi$
$978$		0			0
$979$	1.17157	−	0.485281i	0.0374436	−	0.0155097i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	14.9203	−	36.0208i	0.475884	−	1.14889i	−0.485639	−	0.874160i	$-0.661413\pi$
$983$	14.9203	−	36.0208i	0.475884	−	1.14889i	0.961523	−	0.274726i	$-0.0885871\pi$
$984$		0			0
$985$	22.7279			0.724172
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−25.9706	+	62.6985i	−0.825816	+	1.99370i
$990$		0			0
$991$	46.9203	+	19.4350i	1.49047	+	0.617374i	0.971420	−	0.237368i	$-0.0762847\pi$
$991$	46.9203	+	19.4350i	1.49047	+	0.617374i	0.519053	+	0.854742i	$0.326285\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−29.3848	−	29.3848i	−0.931560	−	0.931560i
$996$		0			0
$997$	3.32233	+	8.02082i	0.105219	+	0.254022i	0.967717	−	0.252039i	$-0.0811011\pi$
$997$	3.32233	+	8.02082i	0.105219	+	0.254022i	−0.862498	+	0.506061i	$0.831101\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1224.2.bq.a.433.1		4
3.2	odd	2		136.2.n.a.25.1	✓	4
12.11	even	2		272.2.v.e.161.1		4
17.15	even	8	inner	1224.2.bq.a.865.1		4
51.11	even	16		2312.2.b.j.577.2		4
51.23	even	16		2312.2.b.j.577.3		4
51.32	odd	8		136.2.n.a.49.1	yes	4
51.41	even	16		2312.2.a.s.1.3		4
51.44	even	16		2312.2.a.s.1.2		4
204.83	even	8		272.2.v.e.49.1		4
204.95	odd	16		4624.2.a.bm.1.3		4
204.143	odd	16		4624.2.a.bm.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.n.a.25.1	✓	4	3.2	odd	2
136.2.n.a.49.1	yes	4	51.32	odd	8
272.2.v.e.49.1		4	204.83	even	8
272.2.v.e.161.1		4	12.11	even	2
1224.2.bq.a.433.1		4	1.1	even	1	trivial
1224.2.bq.a.865.1		4	17.15	even	8	inner
2312.2.a.s.1.2		4	51.44	even	16
2312.2.a.s.1.3		4	51.41	even	16
2312.2.b.j.577.2		4	51.11	even	16
2312.2.b.j.577.3		4	51.23	even	16
4624.2.a.bm.1.2		4	204.143	odd	16
4624.2.a.bm.1.3		4	204.95	odd	16

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1224.bq (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 - 0.707107i) q^{5} +(-0.292893 + 0.121320i) q^{7} +(-0.292893 - 0.707107i) q^{11} +(-1.00000 + 4.00000i) q^{17} +(-1.58579 + 1.58579i) q^{19} +(2.53553 + 6.12132i) q^{23} +(-1.12132 - 1.12132i) q^{25} +(5.12132 + 2.12132i) q^{29} +(0.878680 - 2.12132i) q^{31} +0.585786 q^{35} +(0.292893 - 0.707107i) q^{37} +(2.29289 - 0.949747i) q^{41} +(7.24264 + 7.24264i) q^{43} +12.8284i q^{47} +(-4.87868 + 4.87868i) q^{49} +(-2.17157 + 2.17157i) q^{53} +1.41421i q^{55} +(-5.58579 - 5.58579i) q^{59} +(-9.12132 + 3.77817i) q^{61} +9.65685 q^{67} +(-3.70711 + 8.94975i) q^{71} +(7.36396 + 3.05025i) q^{73} +(0.171573 + 0.171573i) q^{77} +(5.46447 + 13.1924i) q^{79} +(7.24264 - 7.24264i) q^{83} +(4.53553 - 6.12132i) q^{85} +1.65685i q^{89} +(3.82843 - 1.58579i) q^{95} +(-14.7782 - 6.12132i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 4 q^{7} - 4 q^{11} - 4 q^{17} - 12 q^{19} - 4 q^{23} + 4 q^{25} + 12 q^{29} + 12 q^{31} + 8 q^{35} + 4 q^{37} + 12 q^{41} + 12 q^{43} - 28 q^{49} - 20 q^{53} - 28 q^{59} - 28 q^{61} + 16 q^{67}+ \cdots - 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^7 - 4 * q^11 - 4 * q^17 - 12 * q^19 - 4 * q^23 + 4 * q^25 + 12 * q^29 + 12 * q^31 + 8 * q^35 + 4 * q^37 + 12 * q^41 + 12 * q^43 - 28 * q^49 - 20 * q^53 - 28 * q^59 - 28 * q^61 + 16 * q^67 - 12 * q^71 + 4 * q^73 + 12 * q^77 + 36 * q^79 + 12 * q^83 + 4 * q^85 + 4 * q^95 - 28 * q^97

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