LMFDB - Embedded newform 1200.3.bg.k.193.2

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:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1200.193
Dual form			1200.3.bg.k.1057.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-1.44949 + 1.44949i) q^{7} +3.00000i q^{9} +O(q^{10})$	$q+(1.22474 + 1.22474i) q^{3} +(-1.44949 + 1.44949i) q^{7} +3.00000i q^{9} +3.34847 q^{11} +(10.4495 + 10.4495i) q^{13} +(2.65153 - 2.65153i) q^{17} -20.6969i q^{19} -3.55051 q^{21} +(16.4495 + 16.4495i) q^{23} +(-3.67423 + 3.67423i) q^{27} -0.853572i q^{29} +18.6969 q^{31} +(4.10102 + 4.10102i) q^{33} +(-38.0454 + 38.0454i) q^{37} +25.5959i q^{39} -28.6969 q^{41} +(22.4949 + 22.4949i) q^{43} +(19.7526 - 19.7526i) q^{47} +44.7980i q^{49} +6.49490 q^{51} +(-28.6969 - 28.6969i) q^{53} +(25.3485 - 25.3485i) q^{57} +111.934i q^{59} +94.0908 q^{61} +(-4.34847 - 4.34847i) q^{63} +(-54.8990 + 54.8990i) q^{67} +40.2929i q^{69} +68.0000 q^{71} +(39.7878 + 39.7878i) q^{73} +(-4.85357 + 4.85357i) q^{77} +24.4949i q^{79} -9.00000 q^{81} +(-21.1464 - 21.1464i) q^{83} +(1.04541 - 1.04541i) q^{87} +94.1816i q^{89} -30.2929 q^{91} +(22.8990 + 22.8990i) q^{93} +(-14.5959 + 14.5959i) q^{97} +10.0454i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 16 q^{11} + 32 q^{13} + 40 q^{17} - 24 q^{21} + 56 q^{23} + 16 q^{31} + 36 q^{33} - 64 q^{37} - 56 q^{41} - 8 q^{43} + 128 q^{47} - 72 q^{51} - 56 q^{53} + 72 q^{57} + 200 q^{61} + 12 q^{63} - 200 q^{67} + 272 q^{71} - 76 q^{73} - 88 q^{77} - 36 q^{81} - 16 q^{83} - 84 q^{87} + 16 q^{91} + 72 q^{93} + 20 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 16 * q^11 + 32 * q^13 + 40 * q^17 - 24 * q^21 + 56 * q^23 + 16 * q^31 + 36 * q^33 - 64 * q^37 - 56 * q^41 - 8 * q^43 + 128 * q^47 - 72 * q^51 - 56 * q^53 + 72 * q^57 + 200 * q^61 + 12 * q^63 - 200 * q^67 + 272 * q^71 - 76 * q^73 - 88 * q^77 - 36 * q^81 - 16 * q^83 - 84 * q^87 + 16 * q^91 + 72 * q^93 + 20 * 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
$7$	−1.44949	+	1.44949i	−0.207070	+	0.207070i	−0.803021	−	0.595951i	$-0.796775\pi$
$7$	−1.44949	+	1.44949i	−0.207070	+	0.207070i	0.595951	+	0.803021i	$0.296775\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	3.34847			0.304406			0.152203	−	0.988349i	$-0.451363\pi$
$11$	3.34847			0.304406			0.152203	+	0.988349i	$0.451363\pi$
$12$		0			0
$13$	10.4495	+	10.4495i	0.803807	+	0.803807i	0.983688	−	0.179881i	$-0.0575714\pi$
$13$	10.4495	+	10.4495i	0.803807	+	0.803807i	−0.179881	+	0.983688i	$0.557571\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.65153	−	2.65153i	0.155972	−	0.155972i	−0.624807	−	0.780779i	$-0.714822\pi$
$17$	2.65153	−	2.65153i	0.155972	−	0.155972i	0.780779	+	0.624807i	$0.214822\pi$
$18$		0			0
$19$		−	20.6969i		−	1.08931i	−0.838659	−	0.544656i	$-0.816660\pi$
$19$		−	20.6969i		−	1.08931i	0.838659	−	0.544656i	$-0.183340\pi$
$20$		0			0
$21$	−3.55051			−0.169072
$22$		0			0
$23$	16.4495	+	16.4495i	0.715195	+	0.715195i	0.967617	−	0.252422i	$-0.0812271\pi$
$23$	16.4495	+	16.4495i	0.715195	+	0.715195i	−0.252422	+	0.967617i	$0.581227\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$		−	0.853572i		−	0.0294335i	−0.999892	−	0.0147168i	$-0.995315\pi$
$29$		−	0.853572i		−	0.0294335i	0.999892	−	0.0147168i	$-0.00468466\pi$
$30$		0			0
$31$	18.6969			0.603127			0.301564	−	0.953446i	$-0.402491\pi$
$31$	18.6969			0.603127			0.301564	+	0.953446i	$0.402491\pi$
$32$		0			0
$33$	4.10102	+	4.10102i	0.124273	+	0.124273i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−38.0454	+	38.0454i	−1.02825	+	1.02825i	−0.0286652	+	0.999589i	$0.509126\pi$
$37$	−38.0454	+	38.0454i	−1.02825	+	1.02825i	−0.999589	+	0.0286652i	$0.990874\pi$
$38$		0			0
$39$			25.5959i			0.656306i
$40$		0			0
$41$	−28.6969			−0.699925			−0.349963	−	0.936764i	$-0.613806\pi$
$41$	−28.6969			−0.699925			−0.349963	+	0.936764i	$0.613806\pi$
$42$		0			0
$43$	22.4949	+	22.4949i	0.523137	+	0.523137i	0.918517	−	0.395380i	$-0.129387\pi$
$43$	22.4949	+	22.4949i	0.523137	+	0.523137i	−0.395380	+	0.918517i	$0.629387\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	19.7526	−	19.7526i	0.420267	−	0.420267i	−0.465029	−	0.885296i	$-0.653956\pi$
$47$	19.7526	−	19.7526i	0.420267	−	0.420267i	0.885296	+	0.465029i	$0.153956\pi$
$48$		0			0
$49$			44.7980i			0.914244i
$50$		0			0
$51$	6.49490			0.127351
$52$		0			0
$53$	−28.6969	−	28.6969i	−0.541452	−	0.541452i	0.382503	−	0.923954i	$-0.375062\pi$
$53$	−28.6969	−	28.6969i	−0.541452	−	0.541452i	−0.923954	+	0.382503i	$0.875062\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	25.3485	−	25.3485i	0.444710	−	0.444710i
$58$		0			0
$59$			111.934i			1.89719i	0.316493	+	0.948595i	$0.397495\pi$
$59$			111.934i			1.89719i	−0.316493	+	0.948595i	$0.602505\pi$
$60$		0			0
$61$	94.0908			1.54247			0.771236	−	0.636549i	$-0.219639\pi$
$61$	94.0908			1.54247			0.771236	+	0.636549i	$0.219639\pi$
$62$		0			0
$63$	−4.34847	−	4.34847i	−0.0690233	−	0.0690233i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−54.8990	+	54.8990i	−0.819388	+	0.819388i	−0.986019	−	0.166631i	$-0.946711\pi$
$67$	−54.8990	+	54.8990i	−0.819388	+	0.819388i	0.166631	+	0.986019i	$0.446711\pi$
$68$		0			0
$69$			40.2929i			0.583954i
$70$		0			0
$71$	68.0000			0.957746			0.478873	−	0.877884i	$-0.341045\pi$
$71$	68.0000			0.957746			0.478873	+	0.877884i	$0.341045\pi$
$72$		0			0
$73$	39.7878	+	39.7878i	0.545038	+	0.545038i	0.925001	−	0.379964i	$-0.124064\pi$
$73$	39.7878	+	39.7878i	0.545038	+	0.545038i	−0.379964	+	0.925001i	$0.624064\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.85357	+	4.85357i	−0.0630334	+	0.0630334i
$78$		0			0
$79$			24.4949i			0.310062i	0.987910	+	0.155031i	$0.0495477\pi$
$79$			24.4949i			0.310062i	−0.987910	+	0.155031i	$0.950452\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−21.1464	−	21.1464i	−0.254776	−	0.254776i	0.568149	−	0.822926i	$-0.307660\pi$
$83$	−21.1464	−	21.1464i	−0.254776	−	0.254776i	−0.822926	+	0.568149i	$0.807660\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.04541	−	1.04541i	0.0120162	−	0.0120162i
$88$		0			0
$89$			94.1816i			1.05822i	0.848553	+	0.529110i	$0.177474\pi$
$89$			94.1816i			1.05822i	−0.848553	+	0.529110i	$0.822526\pi$
$90$		0			0
$91$	−30.2929			−0.332889
$92$		0			0
$93$	22.8990	+	22.8990i	0.246226	+	0.246226i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−14.5959	+	14.5959i	−0.150473	+	0.150473i	−0.778329	−	0.627856i	$-0.783933\pi$
$97$	−14.5959	+	14.5959i	−0.150473	+	0.150473i	0.627856	+	0.778329i	$0.283933\pi$
$98$		0			0
$99$			10.0454i			0.101469i
$100$		0			0
$101$	173.621			1.71902			0.859509	−	0.511120i	$-0.170769\pi$
$101$	173.621			1.71902			0.859509	+	0.511120i	$0.170769\pi$
$102$		0			0
$103$	−64.7526	−	64.7526i	−0.628666	−	0.628666i	0.319067	−	0.947732i	$-0.396631\pi$
$103$	−64.7526	−	64.7526i	−0.628666	−	0.628666i	−0.947732	+	0.319067i	$0.896631\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.74235	+	4.74235i	−0.0443210	+	0.0443210i	−0.728920	−	0.684599i	$-0.759977\pi$
$107$	−4.74235	+	4.74235i	−0.0443210	+	0.0443210i	0.684599	+	0.728920i	$0.259977\pi$
$108$		0			0
$109$		−	39.3031i		−	0.360579i	−0.983614	−	0.180289i	$-0.942297\pi$
$109$		−	39.3031i		−	0.360579i	0.983614	−	0.180289i	$-0.0577034\pi$
$110$		0			0
$111$	−93.1918			−0.839566
$112$		0			0
$113$	−14.3587	−	14.3587i	−0.127068	−	0.127068i	0.640713	−	0.767781i	$-0.278639\pi$
$113$	−14.3587	−	14.3587i	−0.127068	−	0.127068i	−0.767781	+	0.640713i	$0.778639\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−31.3485	+	31.3485i	−0.267936	+	0.267936i
$118$		0			0
$119$			7.68673i			0.0645944i
$120$		0			0
$121$	−109.788			−0.907337
$122$		0			0
$123$	−35.1464	−	35.1464i	−0.285743	−	0.285743i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−114.621	+	114.621i	−0.902527	+	0.902527i	−0.995654	−	0.0931273i	$-0.970314\pi$
$127$	−114.621	+	114.621i	−0.902527	+	0.902527i	0.0931273	+	0.995654i	$0.470314\pi$
$128$		0			0
$129$			55.1010i			0.427140i
$130$		0			0
$131$	26.1362			0.199513			0.0997566	−	0.995012i	$-0.468194\pi$
$131$	26.1362			0.199513			0.0997566	+	0.995012i	$0.468194\pi$
$132$		0			0
$133$	30.0000	+	30.0000i	0.225564	+	0.225564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.6311	+	14.6311i	−0.106796	+	0.106796i	−0.758486	−	0.651689i	$-0.774061\pi$
$137$	−14.6311	+	14.6311i	−0.106796	+	0.106796i	0.651689	+	0.758486i	$0.274061\pi$
$138$		0			0
$139$			83.1714i			0.598356i	0.954197	+	0.299178i	$0.0967124\pi$
$139$			83.1714i			0.598356i	−0.954197	+	0.299178i	$0.903288\pi$
$140$		0			0
$141$	48.3837			0.343147
$142$		0			0
$143$	34.9898	+	34.9898i	0.244684	+	0.244684i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−54.8661	+	54.8661i	−0.373239	+	0.373239i
$148$		0			0
$149$		−	119.146i		−	0.799640i	−0.916594	−	0.399820i	$-0.869073\pi$
$149$		−	119.146i		−	0.799640i	0.916594	−	0.399820i	$-0.130927\pi$
$150$		0			0
$151$	144.969			0.960062			0.480031	−	0.877251i	$-0.340625\pi$
$151$	144.969			0.960062			0.480031	+	0.877251i	$0.340625\pi$
$152$		0			0
$153$	7.95459	+	7.95459i	0.0519908	+	0.0519908i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−51.1464	+	51.1464i	−0.325773	+	0.325773i	−0.850977	−	0.525203i	$-0.823989\pi$
$157$	−51.1464	+	51.1464i	−0.325773	+	0.325773i	0.525203	+	0.850977i	$0.323989\pi$
$158$		0			0
$159$		−	70.2929i		−	0.442093i
$160$		0			0
$161$	−47.6867			−0.296191
$162$		0			0
$163$	189.394	+	189.394i	1.16193	+	1.16193i	0.984054	+	0.177872i	$0.0569213\pi$
$163$	189.394	+	189.394i	1.16193	+	1.16193i	0.177872	+	0.984054i	$0.443079\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	97.0352	−	97.0352i	0.581049	−	0.581049i	−0.354142	−	0.935192i	$-0.615227\pi$
$167$	97.0352	−	97.0352i	0.581049	−	0.581049i	0.935192	+	0.354142i	$0.115227\pi$
$168$		0			0
$169$			49.3837i			0.292211i
$170$		0			0
$171$	62.0908			0.363104
$172$		0			0
$173$	34.6311	+	34.6311i	0.200180	+	0.200180i	0.800077	−	0.599897i	$-0.204792\pi$
$173$	34.6311	+	34.6311i	0.200180	+	0.200180i	−0.599897	+	0.800077i	$0.704792\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−137.091	+	137.091i	−0.774524	+	0.774524i
$178$		0			0
$179$		−	183.712i		−	1.02632i	−0.858292	−	0.513161i	$-0.828474\pi$
$179$		−	183.712i		−	1.02632i	0.858292	−	0.513161i	$-0.171526\pi$
$180$		0			0
$181$	−21.7276			−0.120042			−0.0600209	−	0.998197i	$-0.519117\pi$
$181$	−21.7276			−0.120042			−0.0600209	+	0.998197i	$0.519117\pi$
$182$		0			0
$183$	115.237	+	115.237i	0.629712	+	0.629712i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.87857	−	8.87857i	0.0474790	−	0.0474790i
$188$		0			0
$189$		−	10.6515i		−	0.0563573i
$190$		0			0
$191$	40.0908			0.209900			0.104950	−	0.994478i	$-0.466532\pi$
$191$	40.0908			0.209900			0.104950	+	0.994478i	$0.466532\pi$
$192$		0			0
$193$	−77.5653	−	77.5653i	−0.401893	−	0.401893i	0.477007	−	0.878900i	$-0.341722\pi$
$193$	−77.5653	−	77.5653i	−0.401893	−	0.401893i	−0.878900	+	0.477007i	$0.841722\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	67.3031	−	67.3031i	0.341640	−	0.341640i	−0.515344	−	0.856984i	$-0.672336\pi$
$197$	67.3031	−	67.3031i	0.341640	−	0.341640i	0.856984	+	0.515344i	$0.172336\pi$
$198$		0			0
$199$			251.394i			1.26329i	0.775259	+	0.631643i	$0.217619\pi$
$199$			251.394i			1.26329i	−0.775259	+	0.631643i	$0.782381\pi$
$200$		0			0
$201$	−134.474			−0.669027
$202$		0			0
$203$	1.23724	+	1.23724i	0.00609480	+	0.00609480i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−49.3485	+	49.3485i	−0.238398	+	0.238398i
$208$		0			0
$209$		−	69.3031i		−	0.331594i
$210$		0			0
$211$	−264.788			−1.25492			−0.627459	−	0.778649i	$-0.715905\pi$
$211$	−264.788			−1.25492			−0.627459	+	0.778649i	$0.715905\pi$
$212$		0			0
$213$	83.2827	+	83.2827i	0.390998	+	0.390998i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−27.1010	+	27.1010i	−0.124889	+	0.124889i
$218$		0			0
$219$			97.4597i			0.445021i
$220$		0			0
$221$	55.4143			0.250743
$222$		0			0
$223$	−33.4291	−	33.4291i	−0.149906	−	0.149906i	0.628170	−	0.778076i	$-0.283804\pi$
$223$	−33.4291	−	33.4291i	−0.149906	−	0.149906i	−0.778076	+	0.628170i	$0.783804\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−21.1714	+	21.1714i	−0.0932662	+	0.0932662i	−0.752200	−	0.658934i	$-0.771008\pi$
$227$	−21.1714	+	21.1714i	−0.0932662	+	0.0932662i	0.658934	+	0.752200i	$0.271008\pi$
$228$		0			0
$229$			243.798i			1.06462i	0.846550	+	0.532310i	$0.178676\pi$
$229$			243.798i			1.06462i	−0.846550	+	0.532310i	$0.821324\pi$
$230$		0			0
$231$	−11.8888			−0.0514666
$232$		0			0
$233$	−161.712	−	161.712i	−0.694042	−	0.694042i	0.269077	−	0.963119i	$-0.413281\pi$
$233$	−161.712	−	161.712i	−0.694042	−	0.694042i	−0.963119	+	0.269077i	$0.913281\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−30.0000	+	30.0000i	−0.126582	+	0.126582i
$238$		0			0
$239$		−	326.202i		−	1.36486i	−0.730950	−	0.682431i	$-0.760923\pi$
$239$		−	326.202i		−	1.36486i	0.730950	−	0.682431i	$-0.239077\pi$
$240$		0			0
$241$	−133.576			−0.554255			−0.277128	−	0.960833i	$-0.589382\pi$
$241$	−133.576			−0.554255			−0.277128	+	0.960833i	$0.589382\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	216.272	−	216.272i	0.875597	−	0.875597i
$248$		0			0
$249$		−	51.7980i		−	0.208024i
$250$		0			0
$251$	404.742			1.61252			0.806260	−	0.591562i	$-0.201488\pi$
$251$	404.742			1.61252			0.806260	+	0.591562i	$0.201488\pi$
$252$		0			0
$253$	55.0806	+	55.0806i	0.217710	+	0.217710i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	89.2372	−	89.2372i	0.347227	−	0.347227i	−0.511849	−	0.859076i	$-0.671039\pi$
$257$	89.2372	−	89.2372i	0.347227	−	0.347227i	0.859076	+	0.511849i	$0.171039\pi$
$258$		0			0
$259$		−	110.293i		−	0.425841i
$260$		0			0
$261$	2.56072			0.00981117
$262$		0			0
$263$	−341.843	−	341.843i	−1.29978	−	1.29978i	−0.928532	−	0.371253i	$-0.878928\pi$
$263$	−341.843	−	341.843i	−1.29978	−	1.29978i	−0.371253	−	0.928532i	$-0.621072\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−115.348	+	115.348i	−0.432017	+	0.432017i
$268$		0			0
$269$			3.50052i			0.0130131i	0.999979	+	0.00650653i	$0.00207111\pi$
$269$			3.50052i			0.0130131i	−0.999979	+	0.00650653i	$0.997929\pi$
$270$		0			0
$271$	103.576			0.382197			0.191099	−	0.981571i	$-0.438795\pi$
$271$	103.576			0.382197			0.191099	+	0.981571i	$0.438795\pi$
$272$		0			0
$273$	−37.1010	−	37.1010i	−0.135901	−	0.135901i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	285.510	−	285.510i	1.03072	−	1.03072i	0.0312080	−	0.999513i	$-0.490065\pi$
$277$	285.510	−	285.510i	1.03072	−	1.03072i	0.999513	−	0.0312080i	$-0.00993542\pi$
$278$		0			0
$279$			56.0908i			0.201042i
$280$		0			0
$281$	372.697			1.32632			0.663162	−	0.748476i	$-0.269214\pi$
$281$	372.697			1.32632			0.663162	+	0.748476i	$0.269214\pi$
$282$		0			0
$283$	−77.1918	−	77.1918i	−0.272763	−	0.272763i	0.557449	−	0.830211i	$-0.311780\pi$
$283$	−77.1918	−	77.1918i	−0.272763	−	0.272763i	−0.830211	+	0.557449i	$0.811780\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	41.5959	−	41.5959i	0.144934	−	0.144934i
$288$		0			0
$289$			274.939i			0.951345i
$290$		0			0
$291$	−35.7526			−0.122861
$292$		0			0
$293$	236.565	+	236.565i	0.807390	+	0.807390i	0.984238	−	0.176848i	$-0.0565901\pi$
$293$	236.565	+	236.565i	0.807390	+	0.807390i	−0.176848	+	0.984238i	$0.556590\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.3031	+	12.3031i	−0.0414244	+	0.0414244i
$298$		0			0
$299$			343.778i			1.14976i
$300$		0			0
$301$	−65.2122			−0.216652
$302$		0			0
$303$	212.641	+	212.641i	0.701787	+	0.701787i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	168.969	−	168.969i	0.550389	−	0.550389i	−0.376164	−	0.926553i	$-0.622757\pi$
$307$	168.969	−	168.969i	0.550389	−	0.550389i	0.926553	+	0.376164i	$0.122757\pi$
$308$		0			0
$309$		−	158.611i		−	0.513303i
$310$		0			0
$311$	−354.302			−1.13923			−0.569617	−	0.821910i	$-0.692909\pi$
$311$	−354.302			−1.13923			−0.569617	+	0.821910i	$0.692909\pi$
$312$		0			0
$313$	−152.373	−	152.373i	−0.486816	−	0.486816i	0.420484	−	0.907300i	$-0.361860\pi$
$313$	−152.373	−	152.373i	−0.486816	−	0.486816i	−0.907300	+	0.420484i	$0.861860\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	427.217	−	427.217i	1.34769	−	1.34769i	0.459519	−	0.888168i	$-0.348022\pi$
$317$	427.217	−	427.217i	1.34769	−	1.34769i	0.888168	−	0.459519i	$-0.151978\pi$
$318$		0			0
$319$		−	2.85816i		−	0.00895975i
$320$		0			0
$321$	−11.6163			−0.0361879
$322$		0			0
$323$	−54.8786	−	54.8786i	−0.169903	−	0.169903i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	48.1362	−	48.1362i	0.147206	−	0.147206i
$328$		0			0
$329$			57.2622i			0.174049i
$330$		0			0
$331$	−489.423			−1.47862			−0.739310	−	0.673365i	$-0.764848\pi$
$331$	−489.423			−1.47862			−0.739310	+	0.673365i	$0.764848\pi$
$332$		0			0
$333$	−114.136	−	114.136i	−0.342751	−	0.342751i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−292.192	+	292.192i	−0.867038	+	0.867038i	−0.992143	−	0.125105i	$-0.960073\pi$
$337$	−292.192	+	292.192i	−0.867038	+	0.867038i	0.125105	+	0.992143i	$0.460073\pi$
$338$		0			0
$339$		−	35.1714i		−	0.103751i
$340$		0			0
$341$	62.6061			0.183596
$342$		0			0
$343$	−135.959	−	135.959i	−0.396382	−	0.396382i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	320.050	−	320.050i	0.922334	−	0.922334i	−0.0748598	−	0.997194i	$-0.523851\pi$
$347$	320.050	−	320.050i	0.922334	−	0.922334i	0.997194	+	0.0748598i	$0.0238509\pi$
$348$		0			0
$349$		−	574.009i		−	1.64473i	−0.568964	−	0.822363i	$-0.692655\pi$
$349$		−	574.009i		−	1.64473i	0.568964	−	0.822363i	$-0.307345\pi$
$350$		0			0
$351$	−76.7878			−0.218769
$352$		0			0
$353$	−266.520	−	266.520i	−0.755014	−	0.755014i	0.220396	−	0.975410i	$-0.429265\pi$
$353$	−266.520	−	266.520i	−0.755014	−	0.755014i	−0.975410	+	0.220396i	$0.929265\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−9.41429	+	9.41429i	−0.0263706	+	0.0263706i
$358$		0			0
$359$		−	216.272i		−	0.602430i	−0.953556	−	0.301215i	$-0.902608\pi$
$359$		−	216.272i		−	0.602430i	0.953556	−	0.301215i	$-0.0973922\pi$
$360$		0			0
$361$	−67.3633			−0.186602
$362$		0			0
$363$	−134.462	−	134.462i	−0.370419	−	0.370419i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−240.510	+	240.510i	−0.655340	+	0.655340i	−0.954274	−	0.298934i	$-0.903369\pi$
$367$	−240.510	+	240.510i	−0.655340	+	0.655340i	0.298934	+	0.954274i	$0.403369\pi$
$368$		0			0
$369$		−	86.0908i		−	0.233308i
$370$		0			0
$371$	83.1918			0.224237
$372$		0			0
$373$	330.207	+	330.207i	0.885272	+	0.885272i	0.994065	−	0.108792i	$-0.0346983\pi$
$373$	330.207	+	330.207i	0.885272	+	0.885272i	−0.108792	+	0.994065i	$0.534698\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8.91939	−	8.91939i	0.0236589	−	0.0236589i
$378$		0			0
$379$		−	210.000i		−	0.554090i	−0.960857	−	0.277045i	$-0.910645\pi$
$379$		−	210.000i		−	0.554090i	0.960857	−	0.277045i	$-0.0893551\pi$
$380$		0			0
$381$	−280.763			−0.736910
$382$		0			0
$383$	170.631	+	170.631i	0.445512	+	0.445512i	0.893859	−	0.448347i	$-0.147987\pi$
$383$	170.631	+	170.631i	0.445512	+	0.445512i	−0.448347	+	0.893859i	$0.647987\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−67.4847	+	67.4847i	−0.174379	+	0.174379i
$388$		0			0
$389$		−	547.337i		−	1.40704i	−0.710677	−	0.703518i	$-0.751611\pi$
$389$		−	547.337i		−	1.40704i	0.710677	−	0.703518i	$-0.248389\pi$
$390$		0			0
$391$	87.2327			0.223101
$392$		0			0
$393$	32.0102	+	32.0102i	0.0814509	+	0.0814509i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−45.2577	+	45.2577i	−0.113999	+	0.113999i	−0.761805	−	0.647806i	$-0.775687\pi$
$397$	−45.2577	+	45.2577i	−0.113999	+	0.113999i	0.647806	+	0.761805i	$0.275687\pi$
$398$		0			0
$399$			73.4847i			0.184172i
$400$		0			0
$401$	−520.302			−1.29751			−0.648756	−	0.760997i	$-0.724710\pi$
$401$	−520.302			−1.29751			−0.648756	+	0.760997i	$0.724710\pi$
$402$		0			0
$403$	195.373	+	195.373i	0.484798	+	0.484798i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−127.394	+	127.394i	−0.313007	+	0.313007i
$408$		0			0
$409$		−	347.110i		−	0.848680i	−0.905503	−	0.424340i	$-0.860506\pi$
$409$		−	347.110i		−	0.848680i	0.905503	−	0.424340i	$-0.139494\pi$
$410$		0			0
$411$	−35.8388			−0.0871990
$412$		0			0
$413$	−162.247	−	162.247i	−0.392851	−	0.392851i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−101.864	+	101.864i	−0.244278	+	0.244278i
$418$		0			0
$419$		−	583.398i		−	1.39236i	−0.717868	−	0.696180i	$-0.754882\pi$
$419$		−	583.398i		−	1.39236i	0.717868	−	0.696180i	$-0.245118\pi$
$420$		0			0
$421$	213.151			0.506297			0.253148	−	0.967427i	$-0.418534\pi$
$421$	213.151			0.506297			0.253148	+	0.967427i	$0.418534\pi$
$422$		0			0
$423$	59.2577	+	59.2577i	0.140089	+	0.140089i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−136.384	+	136.384i	−0.319400	+	0.319400i
$428$		0			0
$429$			85.7071i			0.199784i
$430$		0			0
$431$	−187.364			−0.434720			−0.217360	−	0.976092i	$-0.569745\pi$
$431$	−187.364			−0.434720			−0.217360	+	0.976092i	$0.569745\pi$
$432$		0			0
$433$	−154.848	−	154.848i	−0.357617	−	0.357617i	0.505317	−	0.862934i	$-0.331376\pi$
$433$	−154.848	−	154.848i	−0.357617	−	0.357617i	−0.862934	+	0.505317i	$0.831376\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	340.454	−	340.454i	0.779071	−	0.779071i
$438$		0			0
$439$		−	252.929i		−	0.576147i	−0.957608	−	0.288074i	$-0.906985\pi$
$439$		−	252.929i		−	0.576147i	0.957608	−	0.288074i	$-0.0930148\pi$
$440$		0			0
$441$	−134.394			−0.304748
$442$		0			0
$443$	−421.131	−	421.131i	−0.950633	−	0.950633i	0.0482041	−	0.998838i	$-0.484650\pi$
$443$	−421.131	−	421.131i	−0.950633	−	0.950633i	−0.998838	+	0.0482041i	$0.984650\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	145.924	−	145.924i	0.326452	−	0.326452i
$448$		0			0
$449$		−	297.909i		−	0.663495i	−0.943368	−	0.331747i	$-0.892362\pi$
$449$		−	297.909i		−	0.663495i	0.943368	−	0.331747i	$-0.107638\pi$
$450$		0			0
$451$	−96.0908			−0.213062
$452$		0			0
$453$	177.551	+	177.551i	0.391944	+	0.391944i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−285.747	+	285.747i	−0.625267	+	0.625267i	−0.946873	−	0.321607i	$-0.895777\pi$
$457$	−285.747	+	285.747i	−0.625267	+	0.625267i	0.321607	+	0.946873i	$0.395777\pi$
$458$		0			0
$459$			19.4847i			0.0424503i
$460$		0			0
$461$	526.620			1.14234			0.571171	−	0.820831i	$-0.306489\pi$
$461$	526.620			1.14234			0.571171	+	0.820831i	$0.306489\pi$
$462$		0			0
$463$	335.702	+	335.702i	0.725057	+	0.725057i	0.969631	−	0.244573i	$-0.0786479\pi$
$463$	335.702	+	335.702i	0.725057	+	0.725057i	−0.244573	+	0.969631i	$0.578648\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	488.742	−	488.742i	1.04656	−	1.04656i	0.0476956	−	0.998862i	$-0.484812\pi$
$467$	488.742	−	488.742i	1.04656	−	1.04656i	0.998862	−	0.0476956i	$-0.0151877\pi$
$468$		0			0
$469$		−	159.151i		−	0.339341i
$470$		0			0
$471$	−125.283			−0.265993
$472$		0			0
$473$	75.3235	+	75.3235i	0.159246	+	0.159246i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	86.0908	−	86.0908i	0.180484	−	0.180484i
$478$		0			0
$479$		−	184.949i		−	0.386115i	−0.981187	−	0.193057i	$-0.938160\pi$
$479$		−	184.949i		−	0.386115i	0.981187	−	0.193057i	$-0.0618404\pi$
$480$		0			0
$481$	−795.110			−1.65304
$482$		0			0
$483$	−58.4041	−	58.4041i	−0.120919	−	0.120919i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−120.682	+	120.682i	−0.247807	+	0.247807i	−0.820070	−	0.572263i	$-0.806066\pi$
$487$	−120.682	+	120.682i	−0.247807	+	0.247807i	0.572263	+	0.820070i	$0.306066\pi$
$488$		0			0
$489$			463.918i			0.948708i
$490$		0			0
$491$	105.682			0.215239			0.107619	−	0.994192i	$-0.465677\pi$
$491$	105.682			0.215239			0.107619	+	0.994192i	$0.465677\pi$
$492$		0			0
$493$	−2.26327	−	2.26327i	−0.00459082	−	0.00459082i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−98.5653	+	98.5653i	−0.198321	+	0.198321i
$498$		0			0
$499$		−	739.585i		−	1.48213i	−0.671431	−	0.741067i	$-0.734320\pi$
$499$		−	739.585i		−	1.48213i	0.671431	−	0.741067i	$-0.265680\pi$
$500$		0			0
$501$	237.687			0.474425
$502$		0			0
$503$	−406.409	−	406.409i	−0.807970	−	0.807970i	0.176357	−	0.984326i	$-0.443569\pi$
$503$	−406.409	−	406.409i	−0.807970	−	0.807970i	−0.984326	+	0.176357i	$0.943569\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−60.4824	+	60.4824i	−0.119295	+	0.119295i
$508$		0			0
$509$		−	194.511i		−	0.382143i	−0.981576	−	0.191071i	$-0.938804\pi$
$509$		−	194.511i		−	0.382143i	0.981576	−	0.191071i	$-0.0611962\pi$
$510$		0			0
$511$	−115.344			−0.225722
$512$		0			0
$513$	76.0454	+	76.0454i	0.148237	+	0.148237i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	66.1408	−	66.1408i	0.127932	−	0.127932i
$518$		0			0
$519$			84.8286i			0.163446i
$520$		0			0
$521$	−589.605			−1.13168			−0.565840	−	0.824515i	$-0.691448\pi$
$521$	−589.605			−1.13168			−0.565840	+	0.824515i	$0.691448\pi$
$522$		0			0
$523$	−141.546	−	141.546i	−0.270642	−	0.270642i	0.558716	−	0.829359i	$-0.311294\pi$
$523$	−141.546	−	141.546i	−0.270642	−	0.270642i	−0.829359	+	0.558716i	$0.811294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	49.5755	−	49.5755i	0.0940712	−	0.0940712i
$528$		0			0
$529$			12.1714i			0.0230084i
$530$		0			0
$531$	−335.803			−0.632397
$532$		0			0
$533$	−299.868	−	299.868i	−0.562605	−	0.562605i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	225.000	−	225.000i	0.418994	−	0.418994i
$538$		0			0
$539$			150.005i			0.278302i
$540$		0			0
$541$	431.303			0.797233			0.398617	−	0.917118i	$-0.369490\pi$
$541$	431.303			0.797233			0.398617	+	0.917118i	$0.369490\pi$
$542$		0			0
$543$	−26.6107	−	26.6107i	−0.0490068	−	0.0490068i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−446.222	+	446.222i	−0.815763	+	0.815763i	−0.985491	−	0.169728i	$-0.945711\pi$
$547$	−446.222	+	446.222i	−0.815763	+	0.815763i	0.169728	+	0.985491i	$0.445711\pi$
$548$		0			0
$549$			282.272i			0.514157i
$550$		0			0
$551$	−17.6663			−0.0320623
$552$		0			0
$553$	−35.5051	−	35.5051i	−0.0642045	−	0.0642045i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−214.091	+	214.091i	−0.384364	+	0.384364i	−0.872672	−	0.488308i	$-0.837614\pi$
$557$	−214.091	+	214.091i	−0.384364	+	0.384364i	0.488308	+	0.872672i	$0.337614\pi$
$558$		0			0
$559$			470.120i			0.841003i
$560$		0			0
$561$	21.7480			0.0387664
$562$		0			0
$563$	672.009	+	672.009i	1.19362	+	1.19362i	0.976043	+	0.217579i	$0.0698161\pi$
$563$	672.009	+	672.009i	1.19362	+	1.19362i	0.217579	+	0.976043i	$0.430184\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	13.0454	−	13.0454i	0.0230078	−	0.0230078i
$568$		0			0
$569$			972.161i			1.70854i	0.519827	+	0.854272i	$0.325997\pi$
$569$			972.161i			1.70854i	−0.519827	+	0.854272i	$0.674003\pi$
$570$		0			0
$571$	924.030			1.61827			0.809133	−	0.587626i	$-0.199937\pi$
$571$	924.030			1.61827			0.809133	+	0.587626i	$0.199937\pi$
$572$		0			0
$573$	49.1010	+	49.1010i	0.0856911	+	0.0856911i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	497.879	−	497.879i	0.862874	−	0.862874i	−0.128797	−	0.991671i	$-0.541111\pi$
$577$	497.879	−	497.879i	0.862874	−	0.862874i	0.991671	+	0.128797i	$0.0411114\pi$
$578$		0			0
$579$		−	189.995i		−	0.328144i
$580$		0			0
$581$	61.3031			0.105513
$582$		0			0
$583$	−96.0908	−	96.0908i	−0.164821	−	0.164821i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	292.783	−	292.783i	0.498779	−	0.498779i	−0.412279	−	0.911058i	$-0.635267\pi$
$587$	292.783	−	292.783i	0.498779	−	0.498779i	0.911058	+	0.412279i	$0.135267\pi$
$588$		0			0
$589$		−	386.969i		−	0.656994i
$590$		0			0
$591$	164.858			0.278948
$592$		0			0
$593$	−451.258	−	451.258i	−0.760974	−	0.760974i	0.215524	−	0.976498i	$-0.430854\pi$
$593$	−451.258	−	451.258i	−0.760974	−	0.760974i	−0.976498	+	0.215524i	$0.930854\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−307.893	+	307.893i	−0.515734	+	0.515734i
$598$		0			0
$599$			32.8582i			0.0548550i	0.999624	+	0.0274275i	$0.00873154\pi$
$599$			32.8582i			0.0548550i	−0.999624	+	0.0274275i	$0.991268\pi$
$600$		0			0
$601$	−184.484			−0.306961			−0.153481	−	0.988152i	$-0.549048\pi$
$601$	−184.484			−0.306961			−0.153481	+	0.988152i	$0.549048\pi$
$602$		0			0
$603$	−164.697	−	164.697i	−0.273129	−	0.273129i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	136.389	−	136.389i	0.224694	−	0.224694i	−0.585778	−	0.810472i	$-0.699211\pi$
$607$	136.389	−	136.389i	0.224694	−	0.224694i	0.810472	+	0.585778i	$0.199211\pi$
$608$		0			0
$609$			3.03062i			0.00497638i
$610$		0			0
$611$	412.808			0.675627
$612$		0			0
$613$	12.7128	+	12.7128i	0.0207386	+	0.0207386i	0.717400	−	0.696661i	$-0.245332\pi$
$613$	12.7128	+	12.7128i	0.0207386	+	0.0207386i	−0.696661	+	0.717400i	$0.745332\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	398.752	−	398.752i	0.646275	−	0.646275i	−0.305816	−	0.952091i	$-0.598929\pi$
$617$	398.752	−	398.752i	0.646275	−	0.646275i	0.952091	+	0.305816i	$0.0989292\pi$
$618$		0			0
$619$			819.131i			1.32331i	0.749807	+	0.661656i	$0.230146\pi$
$619$			819.131i			1.32331i	−0.749807	+	0.661656i	$0.769854\pi$
$620$		0			0
$621$	−120.879			−0.194651
$622$		0			0
$623$	−136.515	−	136.515i	−0.219126	−	0.219126i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	84.8786	−	84.8786i	0.135373	−	0.135373i
$628$		0			0
$629$			201.757i			0.320759i
$630$		0			0
$631$	−105.485			−0.167171			−0.0835853	−	0.996501i	$-0.526637\pi$
$631$	−105.485			−0.167171			−0.0835853	+	0.996501i	$0.526637\pi$
$632$		0			0
$633$	−324.297	−	324.297i	−0.512318	−	0.512318i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−468.116	+	468.116i	−0.734876	+	0.734876i
$638$		0			0
$639$			204.000i			0.319249i
$640$		0			0
$641$	164.788			0.257079			0.128540	−	0.991704i	$-0.458971\pi$
$641$	164.788			0.257079			0.128540	+	0.991704i	$0.458971\pi$
$642$		0			0
$643$	−764.372	−	764.372i	−1.18876	−	1.18876i	−0.977411	−	0.211349i	$-0.932214\pi$
$643$	−764.372	−	764.372i	−1.18876	−	1.18876i	−0.211349	−	0.977411i	$-0.567786\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	321.287	−	321.287i	0.496580	−	0.496580i	−0.413792	−	0.910372i	$-0.635796\pi$
$647$	321.287	−	321.287i	0.496580	−	0.496580i	0.910372	+	0.413792i	$0.135796\pi$
$648$		0			0
$649$			374.808i			0.577516i
$650$		0			0
$651$	−66.3837			−0.101972
$652$		0			0
$653$	169.823	+	169.823i	0.260066	+	0.260066i	0.825081	−	0.565015i	$-0.191130\pi$
$653$	169.823	+	169.823i	0.260066	+	0.260066i	−0.565015	+	0.825081i	$0.691130\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−119.363	+	119.363i	−0.181679	+	0.181679i
$658$		0			0
$659$			958.763i			1.45488i	0.686174	+	0.727438i	$0.259289\pi$
$659$			958.763i			1.45488i	−0.686174	+	0.727438i	$0.740711\pi$
$660$		0			0
$661$	396.393			0.599687			0.299843	−	0.953988i	$-0.403066\pi$
$661$	396.393			0.599687			0.299843	+	0.953988i	$0.403066\pi$
$662$		0			0
$663$	67.8684	+	67.8684i	0.102366	+	0.102366i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	14.0408	−	14.0408i	0.0210507	−	0.0210507i
$668$		0			0
$669$		−	81.8842i		−	0.122398i
$670$		0			0
$671$	315.060			0.469538
$672$		0			0
$673$	−164.707	−	164.707i	−0.244736	−	0.244736i	0.574070	−	0.818806i	$-0.305364\pi$
$673$	−164.707	−	164.707i	−0.244736	−	0.244736i	−0.818806	+	0.574070i	$0.805364\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−544.388	+	544.388i	−0.804119	+	0.804119i	−0.983736	−	0.179618i	$-0.942514\pi$
$677$	−544.388	+	544.388i	−0.804119	+	0.804119i	0.179618	+	0.983736i	$0.442514\pi$
$678$		0			0
$679$		−	42.3133i		−	0.0623170i
$680$		0			0
$681$	−51.8592			−0.0761515
$682$		0			0
$683$	786.590	+	786.590i	1.15167	+	1.15167i	0.986218	+	0.165452i	$0.0529082\pi$
$683$	786.590	+	786.590i	1.15167	+	1.15167i	0.165452	+	0.986218i	$0.447092\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−298.590	+	298.590i	−0.434629	+	0.434629i
$688$		0			0
$689$		−	599.737i		−	0.870445i
$690$		0			0
$691$	−356.879			−0.516467			−0.258233	−	0.966083i	$-0.583140\pi$
$691$	−356.879			−0.516467			−0.258233	+	0.966083i	$0.583140\pi$
$692$		0			0
$693$	−14.5607	−	14.5607i	−0.0210111	−	0.0210111i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−76.0908	+	76.0908i	−0.109169	+	0.109169i
$698$		0			0
$699$		−	396.111i		−	0.566683i
$700$		0			0
$701$	885.680			1.26345			0.631726	−	0.775192i	$-0.282347\pi$
$701$	885.680			1.26345			0.631726	+	0.775192i	$0.282347\pi$
$702$		0			0
$703$	787.423	+	787.423i	1.12009	+	1.12009i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−251.662	+	251.662i	−0.355957	+	0.355957i
$708$		0			0
$709$			731.049i			1.03110i	0.856860	+	0.515549i	$0.172412\pi$
$709$			731.049i			1.03110i	−0.856860	+	0.515549i	$0.827588\pi$
$710$		0			0
$711$	−73.4847			−0.103354
$712$		0			0
$713$	307.555	+	307.555i	0.431354	+	0.431354i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	399.514	−	399.514i	0.557203	−	0.557203i
$718$		0			0
$719$			629.271i			0.875204i	0.899169	+	0.437602i	$0.144172\pi$
$719$			629.271i			0.875204i	−0.899169	+	0.437602i	$0.855828\pi$
$720$		0			0
$721$	187.716			0.260356
$722$		0			0
$723$	−163.596	−	163.596i	−0.226274	−	0.226274i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−15.8740	+	15.8740i	−0.0218349	+	0.0218349i	−0.717940	−	0.696105i	$-0.754915\pi$
$727$	−15.8740	+	15.8740i	−0.0218349	+	0.0218349i	0.696105	+	0.717940i	$0.254915\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	119.292			0.163190
$732$		0			0
$733$	393.237	+	393.237i	0.536476	+	0.536476i	0.922492	−	0.386016i	$-0.126149\pi$
$733$	393.237	+	393.237i	0.536476	+	0.536476i	−0.386016	+	0.922492i	$0.626149\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−183.828	+	183.828i	−0.249427	+	0.249427i
$738$		0			0
$739$		−	192.334i		−	0.260262i	−0.991497	−	0.130131i	$-0.958460\pi$
$739$		−	192.334i		−	0.260262i	0.991497	−	0.130131i	$-0.0415398\pi$
$740$		0			0
$741$	529.757			0.714922
$742$		0			0
$743$	−44.7015	−	44.7015i	−0.0601636	−	0.0601636i	0.676385	−	0.736548i	$-0.263546\pi$
$743$	−44.7015	−	44.7015i	−0.0601636	−	0.0601636i	−0.736548	+	0.676385i	$0.763546\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	63.4393	−	63.4393i	0.0849254	−	0.0849254i
$748$		0			0
$749$		−	13.7480i		−	0.0183551i
$750$		0			0
$751$	−227.787			−0.303311			−0.151656	−	0.988433i	$-0.548460\pi$
$751$	−227.787			−0.303311			−0.151656	+	0.988433i	$0.548460\pi$
$752$		0			0
$753$	495.706	+	495.706i	0.658308	+	0.658308i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	235.925	−	235.925i	0.311658	−	0.311658i	−0.533894	−	0.845552i	$-0.679272\pi$
$757$	235.925	−	235.925i	0.311658	−	0.311658i	0.845552	+	0.533894i	$0.179272\pi$
$758$		0			0
$759$			134.919i			0.177759i
$760$		0			0
$761$	−881.242			−1.15801			−0.579003	−	0.815326i	$-0.696558\pi$
$761$	−881.242			−1.15801			−0.579003	+	0.815326i	$0.696558\pi$
$762$		0			0
$763$	56.9694	+	56.9694i	0.0746650	+	0.0746650i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1169.66	+	1169.66i	−1.52497	+	1.52497i
$768$		0			0
$769$			1208.40i			1.57139i	0.618612	+	0.785697i	$0.287696\pi$
$769$			1208.40i			1.57139i	−0.618612	+	0.785697i	$0.712304\pi$
$770$		0			0
$771$	218.586			0.283509
$772$		0			0
$773$	815.226	+	815.226i	1.05463	+	1.05463i	0.998419	+	0.0562070i	$0.0179007\pi$
$773$	815.226	+	815.226i	1.05463	+	1.05463i	0.0562070	+	0.998419i	$0.482099\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	135.081	−	135.081i	0.173849	−	0.173849i
$778$		0			0
$779$			593.939i			0.762437i
$780$		0			0
$781$	227.696			0.291544
$782$		0			0
$783$	3.13622	+	3.13622i	0.00400539	+	0.00400539i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	813.010	−	813.010i	1.03305	−	1.03305i	0.0336150	−	0.999435i	$-0.489298\pi$
$787$	813.010	−	813.010i	1.03305	−	1.03305i	0.999435	−	0.0336150i	$-0.0107020\pi$
$788$		0			0
$789$		−	837.342i		−	1.06127i
$790$		0			0
$791$	41.6255			0.0526239
$792$		0			0
$793$	983.201	+	983.201i	1.23985	+	1.23985i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−311.217	+	311.217i	−0.390485	+	0.390485i	−0.874860	−	0.484375i	$-0.839047\pi$
$797$	−311.217	+	311.217i	−0.390485	+	0.390485i	0.484375	+	0.874860i	$0.339047\pi$
$798$		0			0
$799$		−	104.749i		−	0.131100i
$800$		0			0
$801$	−282.545			−0.352740
$802$		0			0
$803$	133.228	+	133.228i	0.165913	+	0.165913i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−4.28724	+	4.28724i	−0.00531256	+	0.00531256i
$808$		0			0
$809$			150.000i			0.185414i	0.995693	+	0.0927070i	$0.0295520\pi$
$809$			150.000i			0.185414i	−0.995693	+	0.0927070i	$0.970448\pi$
$810$		0			0
$811$	132.847			0.163806			0.0819032	−	0.996640i	$-0.473900\pi$
$811$	132.847			0.163806			0.0819032	+	0.996640i	$0.473900\pi$
$812$		0			0
$813$	126.854	+	126.854i	0.156031	+	0.156031i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	465.576	−	465.576i	0.569860	−	0.569860i
$818$		0			0
$819$		−	90.8786i		−	0.110963i
$820$		0			0
$821$	509.893			0.621064			0.310532	−	0.950563i	$-0.399493\pi$
$821$	509.893			0.621064			0.310532	+	0.950563i	$0.399493\pi$
$822$		0			0
$823$	300.369	+	300.369i	0.364968	+	0.364968i	0.865638	−	0.500670i	$-0.166913\pi$
$823$	300.369	+	300.369i	0.364968	+	0.364968i	−0.500670	+	0.865638i	$0.666913\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1030.76	+	1030.76i	−1.24638	+	1.24638i	−0.289073	+	0.957307i	$0.593347\pi$
$827$	−1030.76	+	1030.76i	−1.24638	+	1.24638i	−0.957307	+	0.289073i	$0.906653\pi$
$828$		0			0
$829$		−	37.4235i		−	0.0451429i	−0.999745	−	0.0225714i	$-0.992815\pi$
$829$		−	37.4235i		−	0.0451429i	0.999745	−	0.0225714i	$-0.00718533\pi$
$830$		0			0
$831$	699.353			0.841580
$832$		0			0
$833$	118.783	+	118.783i	0.142597	+	0.142597i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−68.6969	+	68.6969i	−0.0820752	+	0.0820752i
$838$		0			0
$839$		−	1152.37i		−	1.37351i	−0.726890	−	0.686754i	$-0.759035\pi$
$839$		−	1152.37i		−	1.37351i	0.726890	−	0.686754i	$-0.240965\pi$
$840$		0			0
$841$	840.271			0.999134
$842$		0			0
$843$	456.459	+	456.459i	0.541469	+	0.541469i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	159.136	−	159.136i	0.187882	−	0.187882i
$848$		0			0
$849$		−	189.081i		−	0.222710i
$850$		0			0
$851$	−1251.66			−1.47081
$852$		0			0
$853$	−694.570	−	694.570i	−0.814267	−	0.814267i	0.171003	−	0.985270i	$-0.445299\pi$
$853$	−694.570	−	694.570i	−0.814267	−	0.814267i	−0.985270	+	0.171003i	$0.945299\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−417.176	+	417.176i	−0.486786	+	0.486786i	−0.907291	−	0.420504i	$-0.861853\pi$
$857$	−417.176	+	417.176i	−0.486786	+	0.486786i	0.420504	+	0.907291i	$0.361853\pi$
$858$		0			0
$859$			486.867i			0.566784i	0.959004	+	0.283392i	$0.0914597\pi$
$859$			486.867i			0.566784i	−0.959004	+	0.283392i	$0.908540\pi$
$860$		0			0
$861$	101.889			0.118338
$862$		0			0
$863$	−411.319	−	411.319i	−0.476615	−	0.476615i	0.427432	−	0.904047i	$-0.359418\pi$
$863$	−411.319	−	411.319i	−0.476615	−	0.476615i	−0.904047	+	0.427432i	$0.859418\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−336.730	+	336.730i	−0.388385	+	0.388385i
$868$		0			0
$869$			82.0204i			0.0943848i
$870$		0			0
$871$	−1147.33			−1.31726
$872$		0			0
$873$	−43.7878	−	43.7878i	−0.0501578	−	0.0501578i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−332.540	+	332.540i	−0.379179	+	0.379179i	−0.870806	−	0.491627i	$-0.836402\pi$
$877$	−332.540	+	332.540i	−0.379179	+	0.379179i	0.491627	+	0.870806i	$0.336402\pi$
$878$		0			0
$879$			579.464i			0.659231i
$880$		0			0
$881$	533.151			0.605166			0.302583	−	0.953123i	$-0.402151\pi$
$881$	533.151			0.605166			0.302583	+	0.953123i	$0.402151\pi$
$882$		0			0
$883$	−745.939	−	745.939i	−0.844778	−	0.844778i	0.144698	−	0.989476i	$-0.453779\pi$
$883$	−745.939	−	745.939i	−0.844778	−	0.844778i	−0.989476	+	0.144698i	$0.953779\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−386.207	+	386.207i	−0.435408	+	0.435408i	−0.890463	−	0.455055i	$-0.849619\pi$
$887$	−386.207	+	386.207i	−0.435408	+	0.435408i	0.455055	+	0.890463i	$0.349619\pi$
$888$		0			0
$889$		−	332.284i		−	0.373772i
$890$		0			0
$891$	−30.1362			−0.0338229
$892$		0			0
$893$	−408.817	−	408.817i	−0.457802	−	0.457802i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−421.040	+	421.040i	−0.469387	+	0.469387i
$898$		0			0
$899$		−	15.9592i		−	0.0177521i
$900$		0			0
$901$	−152.182			−0.168903
$902$		0			0
$903$	−79.8684	−	79.8684i	−0.0884478	−	0.0884478i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−947.342	+	947.342i	−1.04448	+	1.04448i	−0.0455146	+	0.998964i	$0.514493\pi$
$907$	−947.342	+	947.342i	−1.04448	+	1.04448i	−0.998964	+	0.0455146i	$0.985507\pi$
$908$		0			0
$909$			520.863i			0.573006i
$910$		0			0
$911$	1149.36			1.26165			0.630824	−	0.775926i	$-0.282717\pi$
$911$	1149.36			1.26165			0.630824	+	0.775926i	$0.282717\pi$
$912$		0			0
$913$	−70.8082	−	70.8082i	−0.0775555	−	0.0775555i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−37.8842	+	37.8842i	−0.0413132	+	0.0413132i
$918$		0			0
$919$			412.577i			0.448941i	0.974481	+	0.224470i	$0.0720652\pi$
$919$			412.577i			0.448941i	−0.974481	+	0.224470i	$0.927935\pi$
$920$		0			0
$921$	413.889			0.449391
$922$		0			0
$923$	710.565	+	710.565i	0.769843	+	0.769843i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	194.258	−	194.258i	0.209555	−	0.209555i
$928$		0			0
$929$			151.707i			0.163302i	0.996661	+	0.0816508i	$0.0260192\pi$
$929$			151.707i			0.163302i	−0.996661	+	0.0816508i	$0.973981\pi$
$930$		0			0
$931$	927.181			0.995898
$932$		0			0
$933$	−433.930	−	433.930i	−0.465091	−	0.465091i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−662.090	+	662.090i	−0.706606	+	0.706606i	−0.965820	−	0.259214i	$-0.916537\pi$
$937$	−662.090	+	662.090i	−0.706606	+	0.706606i	0.259214	+	0.965820i	$0.416537\pi$
$938$		0			0
$939$		−	373.237i		−	0.397484i
$940$		0			0
$941$	−1533.77			−1.62994			−0.814969	−	0.579505i	$-0.803246\pi$
$941$	−1533.77			−1.62994			−0.814969	+	0.579505i	$0.803246\pi$
$942$		0			0
$943$	−472.050	−	472.050i	−0.500583	−	0.500583i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1173.19	−	1173.19i	1.23885	−	1.23885i	0.278377	−	0.960472i	$-0.410204\pi$
$947$	1173.19	−	1173.19i	1.23885	−	1.23885i	0.960472	−	0.278377i	$-0.0897964\pi$
$948$		0			0
$949$			831.523i			0.876210i
$950$		0			0
$951$	1046.46			1.10038
$952$		0			0
$953$	145.501	+	145.501i	0.152676	+	0.152676i	0.779312	−	0.626636i	$-0.215569\pi$
$953$	145.501	+	145.501i	0.152676	+	0.152676i	−0.626636	+	0.779312i	$0.715569\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	3.50052	−	3.50052i	0.00365780	−	0.00365780i
$958$		0			0
$959$		−	42.4153i		−	0.0442287i
$960$		0			0
$961$	−611.424			−0.636238
$962$		0			0
$963$	−14.2270	−	14.2270i	−0.0147737	−	0.0147737i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1151.69	−	1151.69i	1.19099	−	1.19099i	0.214204	−	0.976789i	$-0.431284\pi$
$967$	1151.69	−	1151.69i	1.19099	−	1.19099i	0.976789	−	0.214204i	$-0.0687158\pi$
$968$		0			0
$969$		−	134.424i		−	0.138725i
$970$		0			0
$971$	−72.4383			−0.0746017			−0.0373009	−	0.999304i	$-0.511876\pi$
$971$	−72.4383			−0.0746017			−0.0373009	+	0.999304i	$0.511876\pi$
$972$		0			0
$973$	−120.556	−	120.556i	−0.123901	−	0.123901i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−706.338	+	706.338i	−0.722966	+	0.722966i	−0.969208	−	0.246242i	$-0.920804\pi$
$977$	−706.338	+	706.338i	−0.722966	+	0.722966i	0.246242	+	0.969208i	$0.420804\pi$
$978$		0			0
$979$			315.364i			0.322129i
$980$		0			0
$981$	117.909			0.120193
$982$		0			0
$983$	−134.663	−	134.663i	−0.136992	−	0.136992i	0.635286	−	0.772277i	$-0.280882\pi$
$983$	−134.663	−	134.663i	−0.136992	−	0.136992i	−0.772277	+	0.635286i	$0.780882\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−70.1316	+	70.1316i	−0.0710554	+	0.0710554i
$988$		0			0
$989$			740.059i			0.748290i
$990$		0			0
$991$	1131.94			1.14222			0.571109	−	0.820874i	$-0.306513\pi$
$991$	1131.94			1.14222			0.571109	+	0.820874i	$0.306513\pi$
$992$		0			0
$993$	−599.419	−	599.419i	−0.603644	−	0.603644i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1115.27	−	1115.27i	1.11862	−	1.11862i	0.126679	−	0.991944i	$-0.459568\pi$
$997$	1115.27	−	1115.27i	1.11862	−	1.11862i	0.991944	−	0.126679i	$-0.0404318\pi$
$998$		0			0
$999$		−	279.576i		−	0.279855i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.3.bg.k.193.2		4
4.3	odd	2		75.3.f.c.43.2		4
5.2	odd	4	inner	1200.3.bg.k.1057.2		4
5.3	odd	4		240.3.bg.a.97.1		4
5.4	even	2		240.3.bg.a.193.1		4
12.11	even	2		225.3.g.a.118.1		4
15.8	even	4		720.3.bh.k.577.1		4
15.14	odd	2		720.3.bh.k.433.1		4
20.3	even	4		15.3.f.a.7.1	✓	4
20.7	even	4		75.3.f.c.7.2		4
20.19	odd	2		15.3.f.a.13.1	yes	4
40.3	even	4		960.3.bg.i.577.1		4
40.13	odd	4		960.3.bg.h.577.2		4
40.19	odd	2		960.3.bg.i.193.1		4
40.29	even	2		960.3.bg.h.193.2		4
60.23	odd	4		45.3.g.b.37.2		4
60.47	odd	4		225.3.g.a.82.1		4
60.59	even	2		45.3.g.b.28.2		4
180.23	odd	12		405.3.l.f.217.1		8
180.43	even	12		405.3.l.h.352.1		8
180.59	even	6		405.3.l.f.298.2		8
180.79	odd	6		405.3.l.h.28.2		8
180.83	odd	12		405.3.l.f.352.2		8
180.103	even	12		405.3.l.h.217.2		8
180.119	even	6		405.3.l.f.28.1		8
180.139	odd	6		405.3.l.h.298.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.3.f.a.7.1	✓	4	20.3	even	4
15.3.f.a.13.1	yes	4	20.19	odd	2
45.3.g.b.28.2		4	60.59	even	2
45.3.g.b.37.2		4	60.23	odd	4
75.3.f.c.7.2		4	20.7	even	4
75.3.f.c.43.2		4	4.3	odd	2
225.3.g.a.82.1		4	60.47	odd	4
225.3.g.a.118.1		4	12.11	even	2
240.3.bg.a.97.1		4	5.3	odd	4
240.3.bg.a.193.1		4	5.4	even	2
405.3.l.f.28.1		8	180.119	even	6
405.3.l.f.217.1		8	180.23	odd	12
405.3.l.f.298.2		8	180.59	even	6
405.3.l.f.352.2		8	180.83	odd	12
405.3.l.h.28.2		8	180.79	odd	6
405.3.l.h.217.2		8	180.103	even	12
405.3.l.h.298.1		8	180.139	odd	6
405.3.l.h.352.1		8	180.43	even	12
720.3.bh.k.433.1		4	15.14	odd	2
720.3.bh.k.577.1		4	15.8	even	4
960.3.bg.h.193.2		4	40.29	even	2
960.3.bg.h.577.2		4	40.13	odd	4
960.3.bg.i.193.1		4	40.19	odd	2
960.3.bg.i.577.1		4	40.3	even	4
1200.3.bg.k.193.2		4	1.1	even	1	trivial
1200.3.bg.k.1057.2		4	5.2	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(-1.44949 + 1.44949i) q^{7} +3.00000i q^{9} +O(q^{10})\)	\(q+(1.22474 + 1.22474i) q^{3} +(-1.44949 + 1.44949i) q^{7} +3.00000i q^{9} +3.34847 q^{11} +(10.4495 + 10.4495i) q^{13} +(2.65153 - 2.65153i) q^{17} -20.6969i q^{19} -3.55051 q^{21} +(16.4495 + 16.4495i) q^{23} +(-3.67423 + 3.67423i) q^{27} -0.853572i q^{29} +18.6969 q^{31} +(4.10102 + 4.10102i) q^{33} +(-38.0454 + 38.0454i) q^{37} +25.5959i q^{39} -28.6969 q^{41} +(22.4949 + 22.4949i) q^{43} +(19.7526 - 19.7526i) q^{47} +44.7980i q^{49} +6.49490 q^{51} +(-28.6969 - 28.6969i) q^{53} +(25.3485 - 25.3485i) q^{57} +111.934i q^{59} +94.0908 q^{61} +(-4.34847 - 4.34847i) q^{63} +(-54.8990 + 54.8990i) q^{67} +40.2929i q^{69} +68.0000 q^{71} +(39.7878 + 39.7878i) q^{73} +(-4.85357 + 4.85357i) q^{77} +24.4949i q^{79} -9.00000 q^{81} +(-21.1464 - 21.1464i) q^{83} +(1.04541 - 1.04541i) q^{87} +94.1816i q^{89} -30.2929 q^{91} +(22.8990 + 22.8990i) q^{93} +(-14.5959 + 14.5959i) q^{97} +10.0454i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 16 q^{11} + 32 q^{13} + 40 q^{17} - 24 q^{21} + 56 q^{23} + 16 q^{31} + 36 q^{33} - 64 q^{37} - 56 q^{41} - 8 q^{43} + 128 q^{47} - 72 q^{51} - 56 q^{53} + 72 q^{57} + 200 q^{61} + 12 q^{63} - 200 q^{67} + 272 q^{71} - 76 q^{73} - 88 q^{77} - 36 q^{81} - 16 q^{83} - 84 q^{87} + 16 q^{91} + 72 q^{93} + 20 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 16 * q^11 + 32 * q^13 + 40 * q^17 - 24 * q^21 + 56 * q^23 + 16 * q^31 + 36 * q^33 - 64 * q^37 - 56 * q^41 - 8 * q^43 + 128 * q^47 - 72 * q^51 - 56 * q^53 + 72 * q^57 + 200 * q^61 + 12 * q^63 - 200 * q^67 + 272 * q^71 - 76 * q^73 - 88 * q^77 - 36 * q^81 - 16 * q^83 - 84 * q^87 + 16 * q^91 + 72 * q^93 + 20 * q^97

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