LMFDB - Embedded newform 1200.3.bg.q.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,3,Mod(193,1200)]

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, 0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1200.193");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1200.bg (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:	$32.6976317232$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 $ x^8 - 4x^7 + 8x^6 + 269x^4 - 1116x^3 + 2312x^2 + 680x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.1
Root			$2.66015 - 2.66015i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.q.1057.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 1.22474i) q^{3} +(-2.57407 + 2.57407i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(-1.22474 - 1.22474i) q^{3} +(-2.57407 + 2.57407i) q^{7} +3.00000i q^{9} +21.2513 q^{11} +(8.47305 + 8.47305i) q^{13} +(-19.3720 + 19.3720i) q^{17} -13.1481i q^{19} +6.30517 q^{21} +(-19.2041 - 19.2041i) q^{23} +(3.67423 - 3.67423i) q^{27} -1.00555i q^{29} -23.3107 q^{31} +(-26.0274 - 26.0274i) q^{33} +(-29.8669 + 29.8669i) q^{37} -20.7547i q^{39} -0.555553 q^{41} +(8.85951 + 8.85951i) q^{43} +(9.53992 - 9.53992i) q^{47} +35.7483i q^{49} +47.4516 q^{51} +(34.4213 + 34.4213i) q^{53} +(-16.1031 + 16.1031i) q^{57} +47.0492i q^{59} -64.3553 q^{61} +(-7.72222 - 7.72222i) q^{63} +(12.4554 - 12.4554i) q^{67} +47.0404i q^{69} +91.2861 q^{71} +(77.6540 + 77.6540i) q^{73} +(-54.7024 + 54.7024i) q^{77} +126.193i q^{79} -9.00000 q^{81} +(50.0560 + 50.0560i) q^{83} +(-1.23154 + 1.23154i) q^{87} +163.435i q^{89} -43.6205 q^{91} +(28.5497 + 28.5497i) q^{93} +(100.275 - 100.275i) q^{97} +63.7538i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} - 32 q^{11} + 4 q^{13} - 52 q^{17} - 24 q^{21} - 40 q^{23} - 96 q^{31} - 60 q^{33} + 60 q^{37} - 152 q^{41} - 88 q^{43} - 16 q^{47} + 168 q^{51} - 108 q^{53} + 24 q^{57} + 264 q^{61} + 12 q^{63} - 216 q^{67} + 240 q^{71} + 208 q^{73} - 168 q^{77} - 72 q^{81} + 336 q^{83} + 252 q^{87} - 592 q^{91} - 264 q^{93} + 208 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 - 32 * q^11 + 4 * q^13 - 52 * q^17 - 24 * q^21 - 40 * q^23 - 96 * q^31 - 60 * q^33 + 60 * q^37 - 152 * q^41 - 88 * q^43 - 16 * q^47 + 168 * q^51 - 108 * q^53 + 24 * q^57 + 264 * q^61 + 12 * q^63 - 216 * q^67 + 240 * q^71 + 208 * q^73 - 168 * q^77 - 72 * q^81 + 336 * q^83 + 252 * q^87 - 592 * q^91 - 264 * q^93 + 208 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$3$	−1.22474	−	1.22474i	−0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.57407	+	2.57407i	−0.367725	+	0.367725i	−0.866647	−	0.498922i	$-0.833729\pi$
$7$	−2.57407	+	2.57407i	−0.367725	+	0.367725i	0.498922	+	0.866647i	$0.333729\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	21.2513			1.93193			0.965967	−	0.258665i	$-0.0832825\pi$
$11$	21.2513			1.93193			0.965967	+	0.258665i	$0.0832825\pi$
$12$		0			0
$13$	8.47305	+	8.47305i	0.651773	+	0.651773i	0.953420	−	0.301646i	$-0.0975362\pi$
$13$	8.47305	+	8.47305i	0.651773	+	0.651773i	−0.301646	+	0.953420i	$0.597536\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.3720	+	19.3720i	−1.13953	+	1.13953i	−0.150997	+	0.988534i	$0.548248\pi$
$17$	−19.3720	+	19.3720i	−1.13953	+	1.13953i	−0.988534	+	0.150997i	$0.951752\pi$
$18$		0			0
$19$		−	13.1481i		−	0.692008i	−0.938233	−	0.346004i	$-0.887538\pi$
$19$		−	13.1481i		−	0.692008i	0.938233	−	0.346004i	$-0.112462\pi$
$20$		0			0
$21$	6.30517			0.300246
$22$		0			0
$23$	−19.2041	−	19.2041i	−0.834963	−	0.834963i	0.153228	−	0.988191i	$-0.451033\pi$
$23$	−19.2041	−	19.2041i	−0.834963	−	0.834963i	−0.988191	+	0.153228i	$0.951033\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	3.67423	−	3.67423i	0.136083	−	0.136083i
$28$		0			0
$29$		−	1.00555i		−	0.0346741i	−0.999850	−	0.0173370i	$-0.994481\pi$
$29$		−	1.00555i		−	0.0346741i	0.999850	−	0.0173370i	$-0.00551883\pi$
$30$		0			0
$31$	−23.3107			−0.751959			−0.375979	−	0.926628i	$-0.622694\pi$
$31$	−23.3107			−0.751959			−0.375979	+	0.926628i	$0.622694\pi$
$32$		0			0
$33$	−26.0274	−	26.0274i	−0.788709	−	0.788709i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−29.8669	+	29.8669i	−0.807214	+	0.807214i	−0.984211	−	0.176997i	$-0.943362\pi$
$37$	−29.8669	+	29.8669i	−0.807214	+	0.807214i	0.176997	+	0.984211i	$0.443362\pi$
$38$		0			0
$39$		−	20.7547i		−	0.532171i
$40$		0			0
$41$	−0.555553			−0.0135501			−0.00677504	−	0.999977i	$-0.502157\pi$
$41$	−0.555553			−0.0135501			−0.00677504	+	0.999977i	$0.502157\pi$
$42$		0			0
$43$	8.85951	+	8.85951i	0.206035	+	0.206035i	0.802580	−	0.596545i	$-0.203460\pi$
$43$	8.85951	+	8.85951i	0.206035	+	0.206035i	−0.596545	+	0.802580i	$0.703460\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.53992	−	9.53992i	0.202977	−	0.202977i	−0.598297	−	0.801274i	$-0.704156\pi$
$47$	9.53992	−	9.53992i	0.202977	−	0.202977i	0.801274	+	0.598297i	$0.204156\pi$
$48$		0			0
$49$			35.7483i			0.729557i
$50$		0			0
$51$	47.4516			0.930423
$52$		0			0
$53$	34.4213	+	34.4213i	0.649458	+	0.649458i	0.952862	−	0.303404i	$-0.0981232\pi$
$53$	34.4213	+	34.4213i	0.649458	+	0.649458i	−0.303404	+	0.952862i	$0.598123\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−16.1031	+	16.1031i	−0.282511	+	0.282511i
$58$		0			0
$59$			47.0492i			0.797445i	0.917072	+	0.398722i	$0.130546\pi$
$59$			47.0492i			0.797445i	−0.917072	+	0.398722i	$0.869454\pi$
$60$		0			0
$61$	−64.3553			−1.05500			−0.527502	−	0.849554i	$-0.676871\pi$
$61$	−64.3553			−1.05500			−0.527502	+	0.849554i	$0.676871\pi$
$62$		0			0
$63$	−7.72222	−	7.72222i	−0.122575	−	0.122575i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	12.4554	−	12.4554i	0.185902	−	0.185902i	−0.608020	−	0.793922i	$-0.708036\pi$
$67$	12.4554	−	12.4554i	0.185902	−	0.185902i	0.793922	+	0.608020i	$0.208036\pi$
$68$		0			0
$69$			47.0404i			0.681744i
$70$		0			0
$71$	91.2861			1.28572			0.642860	−	0.765984i	$-0.277748\pi$
$71$	91.2861			1.28572			0.642860	+	0.765984i	$0.277748\pi$
$72$		0			0
$73$	77.6540	+	77.6540i	1.06375	+	1.06375i	0.997824	+	0.0659295i	$0.0210012\pi$
$73$	77.6540	+	77.6540i	1.06375	+	1.06375i	0.0659295	+	0.997824i	$0.478999\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−54.7024	+	54.7024i	−0.710420	+	0.710420i
$78$		0			0
$79$			126.193i			1.59738i	0.601745	+	0.798688i	$0.294472\pi$
$79$			126.193i			1.59738i	−0.601745	+	0.798688i	$0.705528\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	50.0560	+	50.0560i	0.603084	+	0.603084i	0.941130	−	0.338045i	$-0.109766\pi$
$83$	50.0560	+	50.0560i	0.603084	+	0.603084i	−0.338045	+	0.941130i	$0.609766\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−1.23154	+	1.23154i	−0.0141556	+	0.0141556i
$88$		0			0
$89$			163.435i			1.83635i	0.396175	+	0.918175i	$0.370337\pi$
$89$			163.435i			1.83635i	−0.396175	+	0.918175i	$0.629663\pi$
$90$		0			0
$91$	−43.6205			−0.479347
$92$		0			0
$93$	28.5497	+	28.5497i	0.306986	+	0.306986i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	100.275	−	100.275i	1.03377	−	1.03377i	0.0343580	−	0.999410i	$-0.489061\pi$
$97$	100.275	−	100.275i	1.03377	−	1.03377i	0.999410	−	0.0343580i	$-0.0109387\pi$
$98$		0			0
$99$			63.7538i			0.643978i
$100$		0			0
$101$	−72.4212			−0.717041			−0.358521	−	0.933522i	$-0.616719\pi$
$101$	−72.4212			−0.717041			−0.358521	+	0.933522i	$0.616719\pi$
$102$		0			0
$103$	−56.8486	−	56.8486i	−0.551929	−	0.551929i	0.375069	−	0.926997i	$-0.377619\pi$
$103$	−56.8486	−	56.8486i	−0.551929	−	0.551929i	−0.926997	+	0.375069i	$0.877619\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−65.1358	+	65.1358i	−0.608746	+	0.608746i	−0.942618	−	0.333872i	$-0.891645\pi$
$107$	−65.1358	+	65.1358i	−0.608746	+	0.608746i	0.333872	+	0.942618i	$0.391645\pi$
$108$		0			0
$109$		−	99.5281i		−	0.913102i	−0.889697	−	0.456551i	$-0.849085\pi$
$109$		−	99.5281i		−	0.913102i	0.889697	−	0.456551i	$-0.150915\pi$
$110$		0			0
$111$	73.1587			0.659088
$112$		0			0
$113$	−63.3479	−	63.3479i	−0.560601	−	0.560601i	0.368877	−	0.929478i	$-0.379742\pi$
$113$	−63.3479	−	63.3479i	−0.560601	−	0.560601i	−0.929478	+	0.368877i	$0.879742\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−25.4192	+	25.4192i	−0.217258	+	0.217258i
$118$		0			0
$119$		−	99.7301i		−	0.838068i
$120$		0			0
$121$	330.617			2.73237
$122$		0			0
$123$	0.680411	+	0.680411i	0.00553180	+	0.00553180i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	72.5457	−	72.5457i	0.571226	−	0.571226i	−0.361245	−	0.932471i	$-0.617648\pi$
$127$	72.5457	−	72.5457i	0.571226	−	0.571226i	0.932471	+	0.361245i	$0.117648\pi$
$128$		0			0
$129$		−	21.7013i		−	0.168227i
$130$		0			0
$131$	−115.047			−0.878220			−0.439110	−	0.898433i	$-0.644706\pi$
$131$	−115.047			−0.878220			−0.439110	+	0.898433i	$0.644706\pi$
$132$		0			0
$133$	33.8443	+	33.8443i	0.254469	+	0.254469i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−103.572	+	103.572i	−0.755997	+	0.755997i	−0.975591	−	0.219594i	$-0.929527\pi$
$137$	−103.572	+	103.572i	−0.755997	+	0.755997i	0.219594	+	0.975591i	$0.429527\pi$
$138$		0			0
$139$		−	2.67622i		−	0.0192534i	−0.999954	−	0.00962669i	$-0.996936\pi$
$139$		−	2.67622i		−	0.0192534i	0.999954	−	0.00962669i	$-0.00306432\pi$
$140$		0			0
$141$	−23.3679			−0.165730
$142$		0			0
$143$	180.063	+	180.063i	1.25918	+	1.25918i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	43.7825	−	43.7825i	0.297840	−	0.297840i
$148$		0			0
$149$			197.963i			1.32861i	0.747463	+	0.664304i	$0.231272\pi$
$149$			197.963i			1.32861i	−0.747463	+	0.664304i	$0.768728\pi$
$150$		0			0
$151$	78.5904			0.520466			0.260233	−	0.965546i	$-0.416201\pi$
$151$	78.5904			0.520466			0.260233	+	0.965546i	$0.416201\pi$
$152$		0			0
$153$	−58.1161	−	58.1161i	−0.379844	−	0.379844i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−18.6946	+	18.6946i	−0.119074	+	0.119074i	−0.764133	−	0.645059i	$-0.776833\pi$
$157$	−18.6946	+	18.6946i	−0.119074	+	0.119074i	0.645059	+	0.764133i	$0.276833\pi$
$158$		0			0
$159$		−	84.3145i		−	0.530280i
$160$		0			0
$161$	98.8658			0.614073
$162$		0			0
$163$	−7.79719	−	7.79719i	−0.0478355	−	0.0478355i	0.682784	−	0.730620i	$-0.260769\pi$
$163$	−7.79719	−	7.79719i	−0.0478355	−	0.0478355i	−0.730620	+	0.682784i	$0.760769\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−180.167	+	180.167i	−1.07884	+	1.07884i	−0.0822303	+	0.996613i	$0.526204\pi$
$167$	−180.167	+	180.167i	−1.07884	+	1.07884i	−0.996613	+	0.0822303i	$0.973796\pi$
$168$		0			0
$169$		−	25.4147i		−	0.150383i
$170$		0			0
$171$	39.4444			0.230669
$172$		0			0
$173$	−10.9391	−	10.9391i	−0.0632319	−	0.0632319i	0.674784	−	0.738016i	$-0.264237\pi$
$173$	−10.9391	−	10.9391i	−0.0632319	−	0.0632319i	−0.738016	+	0.674784i	$0.764237\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	57.6233	−	57.6233i	0.325555	−	0.325555i
$178$		0			0
$179$			347.641i			1.94213i	0.238819	+	0.971064i	$0.423240\pi$
$179$			347.641i			1.94213i	−0.238819	+	0.971064i	$0.576760\pi$
$180$		0			0
$181$	277.420			1.53271			0.766354	−	0.642418i	$-0.222069\pi$
$181$	277.420			1.53271			0.766354	+	0.642418i	$0.222069\pi$
$182$		0			0
$183$	78.8188	+	78.8188i	0.430704	+	0.430704i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−411.680	+	411.680i	−2.20150	+	2.20150i
$188$		0			0
$189$			18.9155i			0.100082i
$190$		0			0
$191$	−41.0075			−0.214699			−0.107350	−	0.994221i	$-0.534236\pi$
$191$	−41.0075			−0.214699			−0.107350	+	0.994221i	$0.534236\pi$
$192$		0			0
$193$	36.7844	+	36.7844i	0.190593	+	0.190593i	0.795952	−	0.605359i	$-0.206971\pi$
$193$	36.7844	+	36.7844i	0.190593	+	0.190593i	−0.605359	+	0.795952i	$0.706971\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	9.03582	−	9.03582i	0.0458671	−	0.0458671i	−0.683801	−	0.729668i	$-0.739674\pi$
$197$	9.03582	−	9.03582i	0.0458671	−	0.0458671i	0.729668	+	0.683801i	$0.239674\pi$
$198$		0			0
$199$			247.616i			1.24430i	0.782898	+	0.622150i	$0.213741\pi$
$199$			247.616i			1.24430i	−0.782898	+	0.622150i	$0.786259\pi$
$200$		0			0
$201$	−30.5094			−0.151788
$202$		0			0
$203$	2.58836	+	2.58836i	0.0127505	+	0.0127505i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	57.6124	−	57.6124i	0.278321	−	0.278321i
$208$		0			0
$209$		−	279.415i		−	1.33691i
$210$		0			0
$211$	−70.4586			−0.333927			−0.166964	−	0.985963i	$-0.553396\pi$
$211$	−70.4586			−0.333927			−0.166964	+	0.985963i	$0.553396\pi$
$212$		0			0
$213$	−111.802	−	111.802i	−0.524893	−	0.524893i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	60.0035	−	60.0035i	0.276514	−	0.276514i
$218$		0			0
$219$		−	190.213i		−	0.868551i
$220$		0			0
$221$	−328.281			−1.48543
$222$		0			0
$223$	−141.733	−	141.733i	−0.635575	−	0.635575i	0.313885	−	0.949461i	$-0.398369\pi$
$223$	−141.733	−	141.733i	−0.635575	−	0.635575i	−0.949461	+	0.313885i	$0.898369\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.4031	+	12.4031i	−0.0546391	+	0.0546391i	−0.733898	−	0.679259i	$-0.762301\pi$
$227$	−12.4031	+	12.4031i	−0.0546391	+	0.0546391i	0.679259	+	0.733898i	$0.262301\pi$
$228$		0			0
$229$			57.5227i			0.251191i	0.992082	+	0.125596i	$0.0400842\pi$
$229$			57.5227i			0.251191i	−0.992082	+	0.125596i	$0.959916\pi$
$230$		0			0
$231$	133.993			0.580056
$232$		0			0
$233$	286.748	+	286.748i	1.23068	+	1.23068i	0.963704	+	0.266973i	$0.0860235\pi$
$233$	286.748	+	286.748i	1.23068	+	1.23068i	0.266973	+	0.963704i	$0.413977\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	154.554	−	154.554i	0.652126	−	0.652126i
$238$		0			0
$239$		−	8.38189i		−	0.0350707i	−0.999846	−	0.0175353i	$-0.994418\pi$
$239$		−	8.38189i		−	0.0350707i	0.999846	−	0.0175353i	$-0.00558195\pi$
$240$		0			0
$241$	8.41737			0.0349269			0.0174634	−	0.999848i	$-0.494441\pi$
$241$	8.41737			0.0349269			0.0174634	+	0.999848i	$0.494441\pi$
$242$		0			0
$243$	11.0227	+	11.0227i	0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	111.405	−	111.405i	0.451032	−	0.451032i
$248$		0			0
$249$		−	122.612i		−	0.492416i
$250$		0			0
$251$	38.3930			0.152960			0.0764801	−	0.997071i	$-0.475632\pi$
$251$	38.3930			0.152960			0.0764801	+	0.997071i	$0.475632\pi$
$252$		0			0
$253$	−408.113	−	408.113i	−1.61309	−	1.61309i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	70.3678	−	70.3678i	0.273805	−	0.273805i	−0.556825	−	0.830630i	$-0.687981\pi$
$257$	70.3678	−	70.3678i	0.273805	−	0.273805i	0.830630	+	0.556825i	$0.187981\pi$
$258$		0			0
$259$		−	153.759i		−	0.593666i
$260$		0			0
$261$	3.01665			0.0115580
$262$		0			0
$263$	−32.2620	−	32.2620i	−0.122669	−	0.122669i	0.643107	−	0.765776i	$-0.277645\pi$
$263$	−32.2620	−	32.2620i	−0.122669	−	0.122669i	−0.765776	+	0.643107i	$0.777645\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	200.166	−	200.166i	0.749687	−	0.749687i
$268$		0			0
$269$		−	20.9534i		−	0.0778938i	−0.999241	−	0.0389469i	$-0.987600\pi$
$269$		−	20.9534i		−	0.0778938i	0.999241	−	0.0389469i	$-0.0124003\pi$
$270$		0			0
$271$	−464.292			−1.71325			−0.856627	−	0.515936i	$-0.827444\pi$
$271$	−464.292			−1.71325			−0.856627	+	0.515936i	$0.827444\pi$
$272$		0			0
$273$	53.4240	+	53.4240i	0.195692	+	0.195692i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	191.525	−	191.525i	0.691425	−	0.691425i	−0.271120	−	0.962545i	$-0.587394\pi$
$277$	191.525	−	191.525i	0.691425	−	0.691425i	0.962545	+	0.271120i	$0.0873941\pi$
$278$		0			0
$279$		−	69.9322i		−	0.250653i
$280$		0			0
$281$	−205.135			−0.730019			−0.365009	−	0.931004i	$-0.618934\pi$
$281$	−205.135			−0.730019			−0.365009	+	0.931004i	$0.618934\pi$
$282$		0			0
$283$	−9.92225	−	9.92225i	−0.0350609	−	0.0350609i	0.689359	−	0.724420i	$-0.257892\pi$
$283$	−9.92225	−	9.92225i	−0.0350609	−	0.0350609i	−0.724420	+	0.689359i	$0.757892\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.43004	−	1.43004i	0.00498270	−	0.00498270i
$288$		0			0
$289$		−	461.551i		−	1.59706i
$290$		0			0
$291$	−245.624			−0.844068
$292$		0			0
$293$	−243.520	−	243.520i	−0.831125	−	0.831125i	0.156545	−	0.987671i	$-0.449964\pi$
$293$	−243.520	−	243.520i	−0.831125	−	0.831125i	−0.987671	+	0.156545i	$0.949964\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	78.0822	−	78.0822i	0.262903	−	0.262903i
$298$		0			0
$299$		−	325.436i		−	1.08841i
$300$		0			0
$301$	−45.6101			−0.151528
$302$		0			0
$303$	88.6974	+	88.6974i	0.292731	+	0.292731i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	207.842	−	207.842i	0.677009	−	0.677009i	−0.282313	−	0.959322i	$-0.591102\pi$
$307$	207.842	−	207.842i	0.677009	−	0.677009i	0.959322	+	0.282313i	$0.0911017\pi$
$308$		0			0
$309$			139.250i			0.450648i
$310$		0			0
$311$	−432.256			−1.38989			−0.694945	−	0.719063i	$-0.744571\pi$
$311$	−432.256			−1.38989			−0.694945	+	0.719063i	$0.744571\pi$
$312$		0			0
$313$	45.2058	+	45.2058i	0.144427	+	0.144427i	0.775623	−	0.631196i	$-0.217436\pi$
$313$	45.2058	+	45.2058i	0.144427	+	0.144427i	−0.631196	+	0.775623i	$0.717436\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−264.821	+	264.821i	−0.835397	+	0.835397i	−0.988249	−	0.152853i	$-0.951154\pi$
$317$	−264.821	+	264.821i	−0.835397	+	0.835397i	0.152853	+	0.988249i	$0.451154\pi$
$318$		0			0
$319$		−	21.3692i		−	0.0669881i
$320$		0			0
$321$	159.550			0.497039
$322$		0			0
$323$	254.706	+	254.706i	0.788565	+	0.788565i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−121.896	+	121.896i	−0.372772	+	0.372772i
$328$		0			0
$329$			49.1129i			0.149279i
$330$		0			0
$331$	411.758			1.24398			0.621991	−	0.783025i	$-0.286324\pi$
$331$	411.758			1.24398			0.621991	+	0.783025i	$0.286324\pi$
$332$		0			0
$333$	−89.6008	−	89.6008i	−0.269071	−	0.269071i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.27434	+	1.27434i	−0.00378143	+	0.00378143i	−0.708995	−	0.705214i	$-0.750851\pi$
$337$	−1.27434	+	1.27434i	−0.00378143	+	0.00378143i	0.705214	+	0.708995i	$0.250851\pi$
$338$		0			0
$339$			155.170i			0.457729i
$340$		0			0
$341$	−495.383			−1.45273
$342$		0			0
$343$	−218.148	−	218.148i	−0.636001	−	0.636001i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	345.985	−	345.985i	0.997074	−	0.997074i	−0.00292202	−	0.999996i	$-0.500930\pi$
$347$	345.985	−	345.985i	0.997074	−	0.997074i	0.999996	+	0.00292202i	$0.000930109\pi$
$348$		0			0
$349$			495.783i			1.42058i	0.703908	+	0.710291i	$0.251437\pi$
$349$			495.783i			1.42058i	−0.703908	+	0.710291i	$0.748563\pi$
$350$		0			0
$351$	62.2640			0.177390
$352$		0			0
$353$	305.640	+	305.640i	0.865837	+	0.865837i	0.992008	−	0.126172i	$-0.0402690\pi$
$353$	305.640	+	305.640i	0.865837	+	0.865837i	−0.126172	+	0.992008i	$0.540269\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−122.144	+	122.144i	−0.342140	+	0.342140i
$358$		0			0
$359$		−	67.6088i		−	0.188325i	−0.995557	−	0.0941627i	$-0.969983\pi$
$359$		−	67.6088i		−	0.188325i	0.995557	−	0.0941627i	$-0.0300174\pi$
$360$		0			0
$361$	188.126			0.521125
$362$		0			0
$363$	−404.921	−	404.921i	−1.11549	−	1.11549i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	457.051	−	457.051i	1.24537	−	1.24537i	0.287627	−	0.957742i	$-0.407134\pi$
$367$	457.051	−	457.051i	1.24537	−	1.24537i	0.957742	−	0.287627i	$-0.0928664\pi$
$368$		0			0
$369$		−	1.66666i		−	0.00451669i
$370$		0			0
$371$	−177.206			−0.477644
$372$		0			0
$373$	147.406	+	147.406i	0.395191	+	0.395191i	0.876533	−	0.481342i	$-0.159851\pi$
$373$	147.406	+	147.406i	0.395191	+	0.395191i	−0.481342	+	0.876533i	$0.659851\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.52007	−	8.52007i	0.0225997	−	0.0225997i
$378$		0			0
$379$		−	602.616i		−	1.59002i	−0.606599	−	0.795008i	$-0.707466\pi$
$379$		−	602.616i		−	1.59002i	0.606599	−	0.795008i	$-0.292534\pi$
$380$		0			0
$381$	−177.700			−0.466404
$382$		0			0
$383$	−374.041	−	374.041i	−0.976609	−	0.976609i	0.0231233	−	0.999733i	$-0.492639\pi$
$383$	−374.041	−	374.041i	−0.976609	−	0.976609i	−0.999733	+	0.0231233i	$0.992639\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−26.5785	+	26.5785i	−0.0686784	+	0.0686784i
$388$		0			0
$389$			168.693i			0.433659i	0.976209	+	0.216830i	$0.0695716\pi$
$389$			168.693i			0.433659i	−0.976209	+	0.216830i	$0.930428\pi$
$390$		0			0
$391$	744.047			1.90293
$392$		0			0
$393$	140.903	+	140.903i	0.358532	+	0.358532i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−152.238	+	152.238i	−0.383472	+	0.383472i	−0.872351	−	0.488880i	$-0.837406\pi$
$397$	−152.238	+	152.238i	−0.383472	+	0.383472i	0.488880	+	0.872351i	$0.337406\pi$
$398$		0			0
$399$		−	82.9013i		−	0.207773i
$400$		0			0
$401$	304.437			0.759194			0.379597	−	0.925152i	$-0.376063\pi$
$401$	304.437			0.759194			0.379597	+	0.925152i	$0.376063\pi$
$402$		0			0
$403$	−197.513	−	197.513i	−0.490107	−	0.490107i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−634.710	+	634.710i	−1.55949	+	1.55949i
$408$		0			0
$409$		−	715.524i		−	1.74945i	−0.484622	−	0.874724i	$-0.661043\pi$
$409$		−	715.524i		−	1.74945i	0.484622	−	0.874724i	$-0.338957\pi$
$410$		0			0
$411$	253.698			0.617269
$412$		0			0
$413$	−121.108	−	121.108i	−0.293240	−	0.293240i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.27769	+	3.27769i	−0.00786016	+	0.00786016i
$418$		0			0
$419$		−	5.11129i		−	0.0121988i	−0.999981	−	0.00609939i	$-0.998058\pi$
$419$		−	5.11129i		−	0.0121988i	0.999981	−	0.00609939i	$-0.00194151\pi$
$420$		0			0
$421$	−704.328			−1.67299			−0.836494	−	0.547976i	$-0.815398\pi$
$421$	−704.328			−1.67299			−0.836494	+	0.547976i	$0.815398\pi$
$422$		0			0
$423$	28.6198	+	28.6198i	0.0676590	+	0.0676590i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	165.655	−	165.655i	0.387952	−	0.387952i
$428$		0			0
$429$		−	441.063i		−	1.02812i
$430$		0			0
$431$	−48.5871			−0.112731			−0.0563655	−	0.998410i	$-0.517951\pi$
$431$	−48.5871			−0.112731			−0.0563655	+	0.998410i	$0.517951\pi$
$432$		0			0
$433$	26.9117	+	26.9117i	0.0621518	+	0.0621518i	0.737500	−	0.675348i	$-0.236006\pi$
$433$	26.9117	+	26.9117i	0.0621518	+	0.0621518i	−0.675348	+	0.737500i	$0.736006\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−252.499	+	252.499i	−0.577801	+	0.577801i
$438$		0			0
$439$		−	417.501i		−	0.951027i	−0.879708	−	0.475513i	$-0.842262\pi$
$439$		−	417.501i		−	0.951027i	0.879708	−	0.475513i	$-0.157738\pi$
$440$		0			0
$441$	−107.245			−0.243186
$442$		0			0
$443$	163.603	+	163.603i	0.369306	+	0.369306i	0.867224	−	0.497918i	$-0.165902\pi$
$443$	163.603	+	163.603i	0.369306	+	0.369306i	−0.497918	+	0.867224i	$0.665902\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	242.454	−	242.454i	0.542402	−	0.542402i
$448$		0			0
$449$		−	295.233i		−	0.657534i	−0.944411	−	0.328767i	$-0.893367\pi$
$449$		−	295.233i		−	0.657534i	0.944411	−	0.328767i	$-0.106633\pi$
$450$		0			0
$451$	−11.8062			−0.0261779
$452$		0			0
$453$	−96.2532	−	96.2532i	−0.212479	−	0.212479i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	209.231	−	209.231i	0.457836	−	0.457836i	−0.440109	−	0.897944i	$-0.645060\pi$
$457$	209.231	−	209.231i	0.457836	−	0.457836i	0.897944	+	0.440109i	$0.145060\pi$
$458$		0			0
$459$			142.355i			0.310141i
$460$		0			0
$461$	233.787			0.507131			0.253565	−	0.967318i	$-0.418397\pi$
$461$	233.787			0.507131			0.253565	+	0.967318i	$0.418397\pi$
$462$		0			0
$463$	382.073	+	382.073i	0.825212	+	0.825212i	0.986850	−	0.161638i	$-0.0516776\pi$
$463$	382.073	+	382.073i	0.825212	+	0.825212i	−0.161638	+	0.986850i	$0.551678\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−70.6910	+	70.6910i	−0.151373	+	0.151373i	−0.778731	−	0.627358i	$-0.784136\pi$
$467$	−70.6910	+	70.6910i	−0.151373	+	0.151373i	0.627358	+	0.778731i	$0.284136\pi$
$468$		0			0
$469$			64.1224i			0.136721i
$470$		0			0
$471$	45.7923			0.0972235
$472$		0			0
$473$	188.276	+	188.276i	0.398046	+	0.398046i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−103.264	+	103.264i	−0.216486	+	0.216486i
$478$		0			0
$479$		−	457.826i		−	0.955795i	−0.878416	−	0.477897i	$-0.841399\pi$
$479$		−	457.826i		−	0.955795i	0.878416	−	0.477897i	$-0.158601\pi$
$480$		0			0
$481$	−506.128			−1.05224
$482$		0			0
$483$	−121.085	−	121.085i	−0.250694	−	0.250694i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	258.288	−	258.288i	0.530366	−	0.530366i	−0.390315	−	0.920681i	$-0.627634\pi$
$487$	258.288	−	258.288i	0.530366	−	0.530366i	0.920681	+	0.390315i	$0.127634\pi$
$488$		0			0
$489$			19.0991i			0.0390575i
$490$		0			0
$491$	202.818			0.413071			0.206536	−	0.978439i	$-0.433781\pi$
$491$	202.818			0.413071			0.206536	+	0.978439i	$0.433781\pi$
$492$		0			0
$493$	19.4795	+	19.4795i	0.0395122	+	0.0395122i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−234.977	+	234.977i	−0.472791	+	0.472791i
$498$		0			0
$499$		−	40.0028i		−	0.0801660i	−0.999196	−	0.0400830i	$-0.987238\pi$
$499$		−	40.0028i		−	0.0801660i	0.999196	−	0.0400830i	$-0.0127622\pi$
$500$		0			0
$501$	441.317			0.880872
$502$		0			0
$503$	146.141	+	146.141i	0.290539	+	0.290539i	0.837293	−	0.546754i	$-0.184137\pi$
$503$	146.141	+	146.141i	0.290539	+	0.290539i	−0.546754	+	0.837293i	$0.684137\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−31.1265	+	31.1265i	−0.0613936	+	0.0613936i
$508$		0			0
$509$			581.320i			1.14208i	0.820921	+	0.571041i	$0.193460\pi$
$509$			581.320i			1.14208i	−0.820921	+	0.571041i	$0.806540\pi$
$510$		0			0
$511$	−399.774			−0.782338
$512$		0			0
$513$	−48.3094	−	48.3094i	−0.0941703	−	0.0941703i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	202.735	−	202.735i	0.392138	−	0.392138i
$518$		0			0
$519$			26.7952i			0.0516286i
$520$		0			0
$521$	151.126			0.290069			0.145035	−	0.989427i	$-0.453671\pi$
$521$	151.126			0.290069			0.145035	+	0.989427i	$0.453671\pi$
$522$		0			0
$523$	−4.40949	−	4.40949i	−0.00843114	−	0.00843114i	0.702879	−	0.711310i	$-0.251898\pi$
$523$	−4.40949	−	4.40949i	−0.00843114	−	0.00843114i	−0.711310	+	0.702879i	$0.751898\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	451.576	−	451.576i	0.856880	−	0.856880i
$528$		0			0
$529$			208.599i			0.394326i
$530$		0			0
$531$	−141.148			−0.265815
$532$		0			0
$533$	−4.70723	−	4.70723i	−0.00883158	−	0.00883158i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	425.771	−	425.771i	0.792871	−	0.792871i
$538$		0			0
$539$			759.697i			1.40946i
$540$		0			0
$541$	−850.097			−1.57134			−0.785672	−	0.618643i	$-0.787683\pi$
$541$	−850.097			−1.57134			−0.785672	+	0.618643i	$0.787683\pi$
$542$		0			0
$543$	−339.769	−	339.769i	−0.625726	−	0.625726i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	318.187	−	318.187i	0.581695	−	0.581695i	−0.353674	−	0.935369i	$-0.615068\pi$
$547$	318.187	−	318.187i	0.581695	−	0.581695i	0.935369	+	0.353674i	$0.115068\pi$
$548$		0			0
$549$		−	193.066i		−	0.351668i
$550$		0			0
$551$	−13.2211			−0.0239947
$552$		0			0
$553$	−324.829	−	324.829i	−0.587395	−	0.587395i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	205.785	−	205.785i	0.369453	−	0.369453i	−0.497825	−	0.867278i	$-0.665868\pi$
$557$	205.785	−	205.785i	0.369453	−	0.369453i	0.867278	+	0.497825i	$0.165868\pi$
$558$		0			0
$559$			150.134i			0.268576i
$560$		0			0
$561$	1008.41			1.79752
$562$		0			0
$563$	−77.4204	−	77.4204i	−0.137514	−	0.137514i	0.634999	−	0.772513i	$-0.281001\pi$
$563$	−77.4204	−	77.4204i	−0.137514	−	0.137514i	−0.772513	+	0.634999i	$0.781001\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	23.1667	−	23.1667i	0.0408583	−	0.0408583i
$568$		0			0
$569$		−	106.138i		−	0.186533i	−0.995641	−	0.0932667i	$-0.970269\pi$
$569$		−	106.138i		−	0.186533i	0.995641	−	0.0932667i	$-0.0297309\pi$
$570$		0			0
$571$	−752.958			−1.31866			−0.659332	−	0.751852i	$-0.729161\pi$
$571$	−752.958			−1.31866			−0.659332	+	0.751852i	$0.729161\pi$
$572$		0			0
$573$	50.2238	+	50.2238i	0.0876506	+	0.0876506i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−189.310	+	189.310i	−0.328093	+	0.328093i	−0.851861	−	0.523768i	$-0.824526\pi$
$577$	−189.310	+	189.310i	−0.328093	+	0.328093i	0.523768	+	0.851861i	$0.324526\pi$
$578$		0			0
$579$		−	90.1031i		−	0.155618i
$580$		0			0
$581$	−257.696			−0.443538
$582$		0			0
$583$	731.496	+	731.496i	1.25471	+	1.25471i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−69.0979	+	69.0979i	−0.117714	+	0.117714i	−0.763510	−	0.645796i	$-0.776526\pi$
$587$	−69.0979	+	69.0979i	−0.117714	+	0.117714i	0.645796	+	0.763510i	$0.276526\pi$
$588$		0			0
$589$			306.493i			0.520361i
$590$		0			0
$591$	−22.1332			−0.0374503
$592$		0			0
$593$	−372.773	−	372.773i	−0.628623	−	0.628623i	0.319099	−	0.947721i	$-0.396620\pi$
$593$	−372.773	−	372.773i	−0.628623	−	0.628623i	−0.947721	+	0.319099i	$0.896620\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	303.266	−	303.266i	0.507983	−	0.507983i
$598$		0			0
$599$			436.398i			0.728543i	0.931293	+	0.364272i	$0.118682\pi$
$599$			436.398i			0.728543i	−0.931293	+	0.364272i	$0.881318\pi$
$600$		0			0
$601$	436.625			0.726498			0.363249	−	0.931692i	$-0.381667\pi$
$601$	436.625			0.726498			0.363249	+	0.931692i	$0.381667\pi$
$602$		0			0
$603$	37.3663	+	37.3663i	0.0619673	+	0.0619673i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−653.239	+	653.239i	−1.07618	+	1.07618i	−0.0793272	+	0.996849i	$0.525277\pi$
$607$	−653.239	+	653.239i	−1.07618	+	1.07618i	−0.996849	+	0.0793272i	$0.974723\pi$
$608$		0			0
$609$		−	6.34015i		−	0.0104108i
$610$		0			0
$611$	161.664			0.264590
$612$		0			0
$613$	129.247	+	129.247i	0.210843	+	0.210843i	0.804626	−	0.593782i	$-0.202366\pi$
$613$	129.247	+	129.247i	0.210843	+	0.210843i	−0.593782	+	0.804626i	$0.702366\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	623.413	−	623.413i	1.01039	−	1.01039i	0.0104486	−	0.999945i	$-0.496674\pi$
$617$	623.413	−	623.413i	1.01039	−	1.01039i	0.999945	−	0.0104486i	$-0.00332596\pi$
$618$		0			0
$619$		−	362.680i		−	0.585913i	−0.956126	−	0.292957i	$-0.905361\pi$
$619$		−	362.680i		−	0.585913i	0.956126	−	0.292957i	$-0.0946391\pi$
$620$		0			0
$621$	−141.121			−0.227248
$622$		0			0
$623$	−420.694	−	420.694i	−0.675272	−	0.675272i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−342.212	+	342.212i	−0.545793	+	0.545793i
$628$		0			0
$629$		−	1157.17i		−	1.83969i
$630$		0			0
$631$	681.881			1.08063			0.540317	−	0.841461i	$-0.318304\pi$
$631$	681.881			1.08063			0.540317	+	0.841461i	$0.318304\pi$
$632$		0			0
$633$	86.2938	+	86.2938i	0.136325	+	0.136325i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−302.897	+	302.897i	−0.475506	+	0.475506i
$638$		0			0
$639$			273.858i			0.428573i
$640$		0			0
$641$	315.606			0.492365			0.246182	−	0.969224i	$-0.420824\pi$
$641$	315.606			0.492365			0.246182	+	0.969224i	$0.420824\pi$
$642$		0			0
$643$	133.415	+	133.415i	0.207489	+	0.207489i	0.803199	−	0.595710i	$-0.203129\pi$
$643$	133.415	+	133.415i	0.207489	+	0.207489i	−0.595710	+	0.803199i	$0.703129\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−603.672	+	603.672i	−0.933032	+	0.933032i	−0.997894	−	0.0648623i	$-0.979339\pi$
$647$	−603.672	+	603.672i	−0.933032	+	0.933032i	0.0648623	+	0.997894i	$0.479339\pi$
$648$		0			0
$649$			999.856i			1.54061i
$650$		0			0
$651$	−146.978			−0.225773
$652$		0			0
$653$	499.989	+	499.989i	0.765679	+	0.765679i	0.977343	−	0.211663i	$-0.0678880\pi$
$653$	499.989	+	499.989i	0.765679	+	0.765679i	−0.211663	+	0.977343i	$0.567888\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−232.962	+	232.962i	−0.354585	+	0.354585i
$658$		0			0
$659$			786.178i			1.19299i	0.802618	+	0.596493i	$0.203440\pi$
$659$			786.178i			1.19299i	−0.802618	+	0.596493i	$0.796560\pi$
$660$		0			0
$661$	−456.609			−0.690785			−0.345393	−	0.938458i	$-0.612254\pi$
$661$	−456.609			−0.690785			−0.345393	+	0.938458i	$0.612254\pi$
$662$		0			0
$663$	402.060	+	402.060i	0.606425	+	0.606425i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19.3107	+	19.3107i	−0.0289516	+	0.0289516i
$668$		0			0
$669$			347.174i			0.518945i
$670$		0			0
$671$	−1367.63			−2.03820
$672$		0			0
$673$	96.6598	+	96.6598i	0.143625	+	0.143625i	0.775263	−	0.631638i	$-0.217617\pi$
$673$	96.6598	+	96.6598i	0.143625	+	0.143625i	−0.631638	+	0.775263i	$0.717617\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−153.165	+	153.165i	−0.226240	+	0.226240i	−0.811120	−	0.584880i	$-0.801142\pi$
$677$	−153.165	+	153.165i	−0.226240	+	0.226240i	0.584880	+	0.811120i	$0.301142\pi$
$678$		0			0
$679$			516.233i			0.760284i
$680$		0			0
$681$	30.3812			0.0446126
$682$		0			0
$683$	−553.298	−	553.298i	−0.810099	−	0.810099i	0.174549	−	0.984648i	$-0.444153\pi$
$683$	−553.298	−	553.298i	−0.810099	−	0.810099i	−0.984648	+	0.174549i	$0.944153\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	70.4507	−	70.4507i	0.102548	−	0.102548i
$688$		0			0
$689$			583.307i			0.846599i
$690$		0			0
$691$	1059.97			1.53397			0.766984	−	0.641666i	$-0.221756\pi$
$691$	1059.97			1.53397			0.766984	+	0.641666i	$0.221756\pi$
$692$		0			0
$693$	−164.107	−	164.107i	−0.236807	−	0.236807i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	10.7622	−	10.7622i	0.0154407	−	0.0154407i
$698$		0			0
$699$		−	702.386i		−	1.00484i
$700$		0			0
$701$	−552.040			−0.787503			−0.393751	−	0.919217i	$-0.628823\pi$
$701$	−552.040			−0.787503			−0.393751	+	0.919217i	$0.628823\pi$
$702$		0			0
$703$	392.695	+	392.695i	0.558599	+	0.558599i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	186.417	−	186.417i	0.263674	−	0.263674i
$708$		0			0
$709$		−	515.679i		−	0.727333i	−0.931529	−	0.363667i	$-0.881525\pi$
$709$		−	515.679i		−	0.727333i	0.931529	−	0.363667i	$-0.118475\pi$
$710$		0			0
$711$	−378.578			−0.532459
$712$		0			0
$713$	447.662	+	447.662i	0.627858	+	0.627858i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−10.2657	+	10.2657i	−0.0143175	+	0.0143175i
$718$		0			0
$719$		−	636.241i		−	0.884897i	−0.896794	−	0.442448i	$-0.854110\pi$
$719$		−	636.241i		−	0.884897i	0.896794	−	0.442448i	$-0.145890\pi$
$720$		0			0
$721$	292.665			0.405916
$722$		0			0
$723$	−10.3091	−	10.3091i	−0.0142588	−	0.0142588i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−194.095	+	194.095i	−0.266981	+	0.266981i	−0.827883	−	0.560901i	$-0.810455\pi$
$727$	−194.095	+	194.095i	−0.266981	+	0.266981i	0.560901	+	0.827883i	$0.310455\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−343.253			−0.469567
$732$		0			0
$733$	−984.703	−	984.703i	−1.34339	−	1.34339i	−0.892663	−	0.450725i	$-0.851165\pi$
$733$	−984.703	−	984.703i	−1.34339	−	1.34339i	−0.450725	−	0.892663i	$-0.648835\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	264.694	−	264.694i	0.359150	−	0.359150i
$738$		0			0
$739$		−	287.295i		−	0.388762i	−0.980926	−	0.194381i	$-0.937730\pi$
$739$		−	287.295i		−	0.388762i	0.980926	−	0.194381i	$-0.0622699\pi$
$740$		0			0
$741$	−272.885			−0.368266
$742$		0			0
$743$	−425.807	−	425.807i	−0.573091	−	0.573091i	0.359900	−	0.932991i	$-0.382811\pi$
$743$	−425.807	−	425.807i	−0.573091	−	0.573091i	−0.932991	+	0.359900i	$0.882811\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−150.168	+	150.168i	−0.201028	+	0.201028i
$748$		0			0
$749$		−	335.329i		−	0.447702i
$750$		0			0
$751$	287.035			0.382204			0.191102	−	0.981570i	$-0.438794\pi$
$751$	287.035			0.382204			0.191102	+	0.981570i	$0.438794\pi$
$752$		0			0
$753$	−47.0216	−	47.0216i	−0.0624457	−	0.0624457i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	20.8120	−	20.8120i	0.0274927	−	0.0274927i	−0.693227	−	0.720719i	$-0.743812\pi$
$757$	20.8120	−	20.8120i	0.0274927	−	0.0274927i	0.720719	+	0.693227i	$0.243812\pi$
$758$		0			0
$759$			999.668i			1.31709i
$760$		0			0
$761$	1132.90			1.48869			0.744347	−	0.667793i	$-0.232761\pi$
$761$	1132.90			1.48869			0.744347	+	0.667793i	$0.232761\pi$
$762$		0			0
$763$	256.193	+	256.193i	0.335770	+	0.335770i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−398.651	+	398.651i	−0.519753	+	0.519753i
$768$		0			0
$769$			653.519i			0.849829i	0.905233	+	0.424915i	$0.139696\pi$
$769$			653.519i			0.849829i	−0.905233	+	0.424915i	$0.860304\pi$
$770$		0			0
$771$	−172.365			−0.223561
$772$		0			0
$773$	−184.764	−	184.764i	−0.239022	−	0.239022i	0.577423	−	0.816445i	$-0.304058\pi$
$773$	−184.764	−	184.764i	−0.239022	−	0.239022i	−0.816445	+	0.577423i	$0.804058\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−188.316	+	188.316i	−0.242363	+	0.242363i
$778$		0			0
$779$			7.30450i			0.00937676i
$780$		0			0
$781$	1939.95			2.48393
$782$		0			0
$783$	−3.69462	−	3.69462i	−0.00471855	−	0.00471855i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−334.078	+	334.078i	−0.424496	+	0.424496i	−0.886748	−	0.462252i	$-0.847041\pi$
$787$	−334.078	+	334.078i	−0.424496	+	0.424496i	0.462252	+	0.886748i	$0.347041\pi$
$788$		0			0
$789$			79.0255i			0.100159i
$790$		0			0
$791$	326.124			0.412294
$792$		0			0
$793$	−545.286	−	545.286i	−0.687624	−	0.687624i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−235.284	+	235.284i	−0.295212	+	0.295212i	−0.839135	−	0.543923i	$-0.816938\pi$
$797$	−235.284	+	235.284i	−0.295212	+	0.295212i	0.543923	+	0.839135i	$0.316938\pi$
$798$		0			0
$799$			369.615i			0.462597i
$800$		0			0
$801$	−490.305			−0.612117
$802$		0			0
$803$	1650.25	+	1650.25i	2.05510	+	2.05510i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−25.6626	+	25.6626i	−0.0318000	+	0.0318000i
$808$		0			0
$809$		−	1050.82i		−	1.29891i	−0.760400	−	0.649455i	$-0.774997\pi$
$809$		−	1050.82i		−	1.29891i	0.760400	−	0.649455i	$-0.225003\pi$
$810$		0			0
$811$	−50.5273			−0.0623024			−0.0311512	−	0.999515i	$-0.509917\pi$
$811$	−50.5273			−0.0623024			−0.0311512	+	0.999515i	$0.509917\pi$
$812$		0			0
$813$	568.639	+	568.639i	0.699433	+	0.699433i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	116.486	−	116.486i	0.142578	−	0.142578i
$818$		0			0
$819$		−	130.862i		−	0.159782i
$820$		0			0
$821$	1438.48			1.75211			0.876055	−	0.482211i	$-0.160166\pi$
$821$	1438.48			1.75211			0.876055	+	0.482211i	$0.160166\pi$
$822$		0			0
$823$	−617.957	−	617.957i	−0.750859	−	0.750859i	0.223780	−	0.974640i	$-0.428160\pi$
$823$	−617.957	−	617.957i	−0.750859	−	0.750859i	−0.974640	+	0.223780i	$0.928160\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−711.073	+	711.073i	−0.859822	+	0.859822i	−0.991317	−	0.131494i	$-0.958022\pi$
$827$	−711.073	+	711.073i	−0.859822	+	0.859822i	0.131494	+	0.991317i	$0.458022\pi$
$828$		0			0
$829$		−	901.040i		−	1.08690i	−0.839442	−	0.543450i	$-0.817118\pi$
$829$		−	901.040i		−	1.08690i	0.839442	−	0.543450i	$-0.182882\pi$
$830$		0			0
$831$	−469.138			−0.564546
$832$		0			0
$833$	−692.517	−	692.517i	−0.831353	−	0.831353i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−85.6490	+	85.6490i	−0.102329	+	0.102329i
$838$		0			0
$839$			511.590i			0.609762i	0.952390	+	0.304881i	$0.0986167\pi$
$839$			511.590i			0.609762i	−0.952390	+	0.304881i	$0.901383\pi$
$840$		0			0
$841$	839.989			0.998798
$842$		0			0
$843$	251.238	+	251.238i	0.298029	+	0.298029i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−851.032	+	851.032i	−1.00476	+	1.00476i
$848$		0			0
$849$			24.3044i			0.0286271i
$850$		0			0
$851$	1147.14			1.34799
$852$		0			0
$853$	245.561	+	245.561i	0.287879	+	0.287879i	0.836241	−	0.548362i	$-0.184748\pi$
$853$	245.561	+	245.561i	0.287879	+	0.287879i	−0.548362	+	0.836241i	$0.684748\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−609.216	+	609.216i	−0.710871	+	0.710871i	−0.966717	−	0.255847i	$-0.917646\pi$
$857$	−609.216	+	609.216i	−0.710871	+	0.710871i	0.255847	+	0.966717i	$0.417646\pi$
$858$		0			0
$859$			642.894i			0.748422i	0.927344	+	0.374211i	$0.122086\pi$
$859$			642.894i			0.748422i	−0.927344	+	0.374211i	$0.877914\pi$
$860$		0			0
$861$	−3.50286			−0.00406836
$862$		0			0
$863$	339.277	+	339.277i	0.393137	+	0.393137i	0.875804	−	0.482667i	$-0.160332\pi$
$863$	339.277	+	339.277i	0.393137	+	0.393137i	−0.482667	+	0.875804i	$0.660332\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−565.283	+	565.283i	−0.651998	+	0.651998i
$868$		0			0
$869$			2681.76i			3.08603i
$870$		0			0
$871$	211.071			0.242332
$872$		0			0
$873$	300.826	+	300.826i	0.344589	+	0.344589i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−964.971	+	964.971i	−1.10031	+	1.10031i	−0.105935	+	0.994373i	$0.533784\pi$
$877$	−964.971	+	964.971i	−1.10031	+	1.10031i	−0.994373	+	0.105935i	$0.966216\pi$
$878$		0			0
$879$			596.499i			0.678611i
$880$		0			0
$881$	−454.335			−0.515703			−0.257852	−	0.966185i	$-0.583015\pi$
$881$	−454.335			−0.515703			−0.257852	+	0.966185i	$0.583015\pi$
$882$		0			0
$883$	−1165.84	−	1165.84i	−1.32031	−	1.32031i	−0.913521	−	0.406791i	$-0.866648\pi$
$883$	−1165.84	−	1165.84i	−1.32031	−	1.32031i	−0.406791	−	0.913521i	$-0.633352\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	640.792	−	640.792i	0.722426	−	0.722426i	−0.246673	−	0.969099i	$-0.579337\pi$
$887$	640.792	−	640.792i	0.722426	−	0.722426i	0.969099	+	0.246673i	$0.0793373\pi$
$888$		0			0
$889$			373.476i			0.420108i
$890$		0			0
$891$	−191.261			−0.214659
$892$		0			0
$893$	−125.432	−	125.432i	−0.140462	−	0.140462i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−398.576	+	398.576i	−0.444343	+	0.444343i
$898$		0			0
$899$			23.4401i			0.0260735i
$900$		0			0
$901$	−1333.62			−1.48016
$902$		0			0
$903$	55.8607	+	55.8607i	0.0618612	+	0.0618612i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	211.397	−	211.397i	0.233072	−	0.233072i	−0.580901	−	0.813974i	$-0.697300\pi$
$907$	211.397	−	211.397i	0.233072	−	0.233072i	0.813974	+	0.580901i	$0.197300\pi$
$908$		0			0
$909$		−	217.263i		−	0.239014i
$910$		0			0
$911$	1289.87			1.41588			0.707941	−	0.706272i	$-0.249624\pi$
$911$	1289.87			1.41588			0.707941	+	0.706272i	$0.249624\pi$
$912$		0			0
$913$	1063.75	+	1063.75i	1.16512	+	1.16512i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	296.139	−	296.139i	0.322943	−	0.322943i
$918$		0			0
$919$		−	643.370i		−	0.700076i	−0.936736	−	0.350038i	$-0.886169\pi$
$919$		−	643.370i		−	0.700076i	0.936736	−	0.350038i	$-0.113831\pi$
$920$		0			0
$921$	−509.106			−0.552776
$922$		0			0
$923$	773.472	+	773.472i	0.837998	+	0.837998i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	170.546	−	170.546i	0.183976	−	0.183976i
$928$		0			0
$929$		−	82.6632i		−	0.0889808i	−0.999010	−	0.0444904i	$-0.985834\pi$
$929$		−	82.6632i		−	0.0889808i	0.999010	−	0.0444904i	$-0.0141664\pi$
$930$		0			0
$931$	470.024			0.504859
$932$		0			0
$933$	529.403	+	529.403i	0.567420	+	0.567420i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−119.104	+	119.104i	−0.127112	+	0.127112i	−0.767801	−	0.640689i	$-0.778649\pi$
$937$	−119.104	+	119.104i	−0.127112	+	0.127112i	0.640689	+	0.767801i	$0.278649\pi$
$938$		0			0
$939$		−	110.731i		−	0.117925i
$940$		0			0
$941$	−460.144			−0.488995			−0.244498	−	0.969650i	$-0.578623\pi$
$941$	−460.144			−0.488995			−0.244498	+	0.969650i	$0.578623\pi$
$942$		0			0
$943$	10.6689	+	10.6689i	0.0113138	+	0.0113138i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	484.244	−	484.244i	0.511345	−	0.511345i	−0.403594	−	0.914938i	$-0.632239\pi$
$947$	484.244	−	484.244i	0.511345	−	0.511345i	0.914938	+	0.403594i	$0.132239\pi$
$948$		0			0
$949$			1315.93i			1.38665i
$950$		0			0
$951$	648.676			0.682098
$952$		0			0
$953$	−201.628	−	201.628i	−0.211572	−	0.211572i	0.593363	−	0.804935i	$-0.297800\pi$
$953$	−201.628	−	201.628i	−0.211572	−	0.211572i	−0.804935	+	0.593363i	$0.797800\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−26.1718	+	26.1718i	−0.0273478	+	0.0273478i
$958$		0			0
$959$		−	533.202i		−	0.555998i
$960$		0			0
$961$	−417.610			−0.434558
$962$		0			0
$963$	−195.408	−	195.408i	−0.202915	−	0.202915i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	449.704	−	449.704i	0.465050	−	0.465050i	−0.435256	−	0.900307i	$-0.643342\pi$
$967$	449.704	−	449.704i	0.465050	−	0.465050i	0.900307	+	0.435256i	$0.143342\pi$
$968$		0			0
$969$		−	623.901i		−	0.643860i
$970$		0			0
$971$	−278.225			−0.286534			−0.143267	−	0.989684i	$-0.545761\pi$
$971$	−278.225			−0.286534			−0.143267	+	0.989684i	$0.545761\pi$
$972$		0			0
$973$	6.88879	+	6.88879i	0.00707995	+	0.00707995i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	645.478	−	645.478i	0.660674	−	0.660674i	−0.294865	−	0.955539i	$-0.595275\pi$
$977$	645.478	−	645.478i	0.660674	−	0.660674i	0.955539	+	0.294865i	$0.0952748\pi$
$978$		0			0
$979$			3473.21i			3.54771i
$980$		0			0
$981$	298.584			0.304367
$982$		0			0
$983$	−610.333	−	610.333i	−0.620888	−	0.620888i	0.324871	−	0.945758i	$-0.394679\pi$
$983$	−610.333	−	610.333i	−0.620888	−	0.620888i	−0.945758	+	0.324871i	$0.894679\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	60.1508	−	60.1508i	0.0609431	−	0.0609431i
$988$		0			0
$989$		−	340.279i		−	0.344063i
$990$		0			0
$991$	703.508			0.709897			0.354948	−	0.934886i	$-0.384498\pi$
$991$	703.508			0.709897			0.354948	+	0.934886i	$0.384498\pi$
$992$		0			0
$993$	−504.298	−	504.298i	−0.507853	−	0.507853i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−42.1983	+	42.1983i	−0.0423253	+	0.0423253i	−0.727953	−	0.685627i	$-0.759528\pi$
$997$	−42.1983	+	42.1983i	−0.0423253	+	0.0423253i	0.685627	+	0.727953i	$0.259528\pi$
$998$		0			0
$999$			219.476i			0.219696i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.q.193.1		8
4.3	odd	2		600.3.u.h.193.4		8
5.2	odd	4	inner	1200.3.bg.q.1057.1		8
5.3	odd	4		240.3.bg.e.97.3		8
5.4	even	2		240.3.bg.e.193.3		8
12.11	even	2		1800.3.v.t.793.3		8
15.8	even	4		720.3.bh.o.577.4		8
15.14	odd	2		720.3.bh.o.433.4		8
20.3	even	4		120.3.u.b.97.1	yes	8
20.7	even	4		600.3.u.h.457.4		8
20.19	odd	2		120.3.u.b.73.1	✓	8
40.3	even	4		960.3.bg.l.577.4		8
40.13	odd	4		960.3.bg.k.577.2		8
40.19	odd	2		960.3.bg.l.193.4		8
40.29	even	2		960.3.bg.k.193.2		8
60.23	odd	4		360.3.v.f.217.4		8
60.47	odd	4		1800.3.v.t.1657.3		8
60.59	even	2		360.3.v.f.73.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.b.73.1	✓	8	20.19	odd	2
120.3.u.b.97.1	yes	8	20.3	even	4
240.3.bg.e.97.3		8	5.3	odd	4
240.3.bg.e.193.3		8	5.4	even	2
360.3.v.f.73.4		8	60.59	even	2
360.3.v.f.217.4		8	60.23	odd	4
600.3.u.h.193.4		8	4.3	odd	2
600.3.u.h.457.4		8	20.7	even	4
720.3.bh.o.433.4		8	15.14	odd	2
720.3.bh.o.577.4		8	15.8	even	4
960.3.bg.k.193.2		8	40.29	even	2
960.3.bg.k.577.2		8	40.13	odd	4
960.3.bg.l.193.4		8	40.19	odd	2
960.3.bg.l.577.4		8	40.3	even	4
1200.3.bg.q.193.1		8	1.1	even	1	trivial
1200.3.bg.q.1057.1		8	5.2	odd	4	inner
1800.3.v.t.793.3		8	12.11	even	2
1800.3.v.t.1657.3		8	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-2.57407 + 2.57407i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(-1.22474 - 1.22474i) q^{3} +(-2.57407 + 2.57407i) q^{7} +3.00000i q^{9} +21.2513 q^{11} +(8.47305 + 8.47305i) q^{13} +(-19.3720 + 19.3720i) q^{17} -13.1481i q^{19} +6.30517 q^{21} +(-19.2041 - 19.2041i) q^{23} +(3.67423 - 3.67423i) q^{27} -1.00555i q^{29} -23.3107 q^{31} +(-26.0274 - 26.0274i) q^{33} +(-29.8669 + 29.8669i) q^{37} -20.7547i q^{39} -0.555553 q^{41} +(8.85951 + 8.85951i) q^{43} +(9.53992 - 9.53992i) q^{47} +35.7483i q^{49} +47.4516 q^{51} +(34.4213 + 34.4213i) q^{53} +(-16.1031 + 16.1031i) q^{57} +47.0492i q^{59} -64.3553 q^{61} +(-7.72222 - 7.72222i) q^{63} +(12.4554 - 12.4554i) q^{67} +47.0404i q^{69} +91.2861 q^{71} +(77.6540 + 77.6540i) q^{73} +(-54.7024 + 54.7024i) q^{77} +126.193i q^{79} -9.00000 q^{81} +(50.0560 + 50.0560i) q^{83} +(-1.23154 + 1.23154i) q^{87} +163.435i q^{89} -43.6205 q^{91} +(28.5497 + 28.5497i) q^{93} +(100.275 - 100.275i) q^{97} +63.7538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} - 32 q^{11} + 4 q^{13} - 52 q^{17} - 24 q^{21} - 40 q^{23} - 96 q^{31} - 60 q^{33} + 60 q^{37} - 152 q^{41} - 88 q^{43} - 16 q^{47} + 168 q^{51} - 108 q^{53} + 24 q^{57} + 264 q^{61} + 12 q^{63} - 216 q^{67} + 240 q^{71} + 208 q^{73} - 168 q^{77} - 72 q^{81} + 336 q^{83} + 252 q^{87} - 592 q^{91} - 264 q^{93} + 208 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 - 32 * q^11 + 4 * q^13 - 52 * q^17 - 24 * q^21 - 40 * q^23 - 96 * q^31 - 60 * q^33 + 60 * q^37 - 152 * q^41 - 88 * q^43 - 16 * q^47 + 168 * q^51 - 108 * q^53 + 24 * q^57 + 264 * q^61 + 12 * q^63 - 216 * q^67 + 240 * q^71 + 208 * q^73 - 168 * q^77 - 72 * q^81 + 336 * q^83 + 252 * q^87 - 592 * q^91 - 264 * q^93 + 208 * q^97

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