LMFDB - Embedded newform 600.3.l.f.401.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(401,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.401"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.681615360000.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 $ x^8 - 4x^7 - 2x^6 + 20x^5 + 49x^4 - 136x^3 + 168x^2 - 96*x + 864
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.7
Root			$3.22255 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	600.401
Dual form			600.3.l.f.401.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.87275 - 0.864473i) q^{3} -9.02416 q^{7} +(7.50537 - 4.96683i) q^{9} +21.8827i q^{11} -21.6599 q^{13} +12.1078i q^{17} +3.03757 q^{19} +(-25.9241 + 7.80114i) q^{21} +28.5735i q^{23} +(17.2674 - 20.7566i) q^{27} +12.0364i q^{29} +2.19085 q^{31} +(18.9170 + 62.8636i) q^{33} -0.839959 q^{37} +(-62.2233 + 18.7244i) q^{39} +35.5690i q^{41} +12.7152 q^{43} -22.5481i q^{47} +32.4354 q^{49} +(10.4668 + 34.7826i) q^{51} -9.13775i q^{53} +(8.72618 - 2.62590i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(-67.7297 + 44.8214i) q^{63} +63.0560 q^{67} +(24.7011 + 82.0846i) q^{69} +17.0218i q^{71} -52.1181 q^{73} -197.473i q^{77} -7.46224 q^{79} +(31.6612 - 74.5558i) q^{81} +82.3758i q^{83} +(10.4051 + 34.5774i) q^{87} -27.5850i q^{89} +195.462 q^{91} +(6.29376 - 1.89393i) q^{93} -114.989 q^{97} +(108.688 + 164.238i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + 8 q^{13} - 8 q^{19} + 28 q^{21} - 20 q^{27} + 120 q^{31} + 112 q^{33} - 8 q^{37} - 72 q^{39} + 328 q^{43} + 64 q^{49} + 64 q^{51} - 72 q^{57} + 8 q^{61} - 88 q^{63}+ \cdots + 32 q^{99}+O(q^{100}) $ 8 * q + 4 * q^3 - 16 * q^7 + 20 * q^9 + 8 * q^13 - 8 * q^19 + 28 * q^21 - 20 * q^27 + 120 * q^31 + 112 * q^33 - 8 * q^37 - 72 * q^39 + 328 * q^43 + 64 * q^49 + 64 * q^51 - 72 * q^57 + 8 * q^61 - 88 * q^63 - 152 * q^67 + 100 * q^69 - 32 * q^73 + 88 * q^79 + 224 * q^81 + 152 * q^87 + 560 * q^91 + 368 * q^93 - 144 * q^97 + 32 * q^99

Character values

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

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$	2.87275	−	0.864473i	0.957583	−	0.288158i
$3$	2.87275	−	0.864473i	0.957583	−	0.288158i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−9.02416			−1.28917			−0.644583	−	0.764535i	$-0.722969\pi$
$7$	−9.02416			−1.28917			−0.644583	+	0.764535i	$0.722969\pi$
$8$		0			0
$9$	7.50537	−	4.96683i	0.833930	−	0.551870i
$10$		0			0
$11$			21.8827i			1.98934i	0.103119	+	0.994669i	$0.467118\pi$
$11$			21.8827i			1.98934i	−0.103119	+	0.994669i	$0.532882\pi$
$12$		0			0
$13$	−21.6599			−1.66614			−0.833071	−	0.553166i	$-0.813420\pi$
$13$	−21.6599			−1.66614			−0.833071	+	0.553166i	$0.813420\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			12.1078i			0.712221i	0.934444	+	0.356111i	$0.115897\pi$
$17$			12.1078i			0.712221i	−0.934444	+	0.356111i	$0.884103\pi$
$18$		0			0
$19$	3.03757			0.159872			0.0799361	−	0.996800i	$-0.474528\pi$
$19$	3.03757			0.159872			0.0799361	+	0.996800i	$0.474528\pi$
$20$		0			0
$21$	−25.9241	+	7.80114i	−1.23448	+	0.371483i
$22$		0			0
$23$			28.5735i			1.24233i	0.783681	+	0.621164i	$0.213340\pi$
$23$			28.5735i			1.24233i	−0.783681	+	0.621164i	$0.786660\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	17.2674	−	20.7566i	0.639532	−	0.768765i
$28$		0			0
$29$			12.0364i			0.415047i	0.978230	+	0.207523i	$0.0665403\pi$
$29$			12.0364i			0.415047i	−0.978230	+	0.207523i	$0.933460\pi$
$30$		0			0
$31$	2.19085			0.0706725			0.0353363	−	0.999375i	$-0.488750\pi$
$31$	2.19085			0.0706725			0.0353363	+	0.999375i	$0.488750\pi$
$32$		0			0
$33$	18.9170	+	62.8636i	0.573243	+	1.90496i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−0.839959			−0.0227016			−0.0113508	−	0.999936i	$-0.503613\pi$
$37$	−0.839959			−0.0227016			−0.0113508	+	0.999936i	$0.503613\pi$
$38$		0			0
$39$	−62.2233	+	18.7244i	−1.59547	+	0.480112i
$40$		0			0
$41$			35.5690i			0.867537i	0.901024	+	0.433769i	$0.142816\pi$
$41$			35.5690i			0.867537i	−0.901024	+	0.433769i	$0.857184\pi$
$42$		0			0
$43$	12.7152			0.295702			0.147851	−	0.989010i	$-0.452764\pi$
$43$	12.7152			0.295702			0.147851	+	0.989010i	$0.452764\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	22.5481i		−	0.479746i	−0.970804	−	0.239873i	$-0.922894\pi$
$47$		−	22.5481i		−	0.479746i	0.970804	−	0.239873i	$-0.0771058\pi$
$48$		0			0
$49$	32.4354			0.661947
$50$		0			0
$51$	10.4668	+	34.7826i	0.205232	+	0.682011i
$52$		0			0
$53$		−	9.13775i		−	0.172410i	−0.996277	−	0.0862052i	$-0.972526\pi$
$53$		−	9.13775i		−	0.172410i	0.996277	−	0.0862052i	$-0.0274741\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.72618	−	2.62590i	0.153091	−	0.0460684i
$58$		0			0
$59$		−	80.4459i		−	1.36349i	−0.731590	−	0.681745i	$-0.761221\pi$
$59$		−	80.4459i		−	1.36349i	0.731590	−	0.681745i	$-0.238779\pi$
$60$		0			0
$61$	−57.8816			−0.948878			−0.474439	−	0.880288i	$-0.657349\pi$
$61$	−57.8816			−0.948878			−0.474439	+	0.880288i	$0.657349\pi$
$62$		0			0
$63$	−67.7297	+	44.8214i	−1.07507	+	0.711452i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	63.0560			0.941135			0.470567	−	0.882364i	$-0.344049\pi$
$67$	63.0560			0.941135			0.470567	+	0.882364i	$0.344049\pi$
$68$		0			0
$69$	24.7011	+	82.0846i	0.357986	+	1.18963i
$70$		0			0
$71$			17.0218i			0.239743i	0.992789	+	0.119872i	$0.0382483\pi$
$71$			17.0218i			0.239743i	−0.992789	+	0.119872i	$0.961752\pi$
$72$		0			0
$73$	−52.1181			−0.713947			−0.356973	−	0.934115i	$-0.616191\pi$
$73$	−52.1181			−0.713947			−0.356973	+	0.934115i	$0.616191\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	197.473i		−	2.56459i
$78$		0			0
$79$	−7.46224			−0.0944588			−0.0472294	−	0.998884i	$-0.515039\pi$
$79$	−7.46224			−0.0944588			−0.0472294	+	0.998884i	$0.515039\pi$
$80$		0			0
$81$	31.6612	−	74.5558i	0.390879	−	0.920442i
$82$		0			0
$83$			82.3758i			0.992480i	0.868185	+	0.496240i	$0.165286\pi$
$83$			82.3758i			0.992480i	−0.868185	+	0.496240i	$0.834714\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	10.4051	+	34.5774i	0.119599	+	0.397442i
$88$		0			0
$89$		−	27.5850i		−	0.309944i	−0.987919	−	0.154972i	$-0.950471\pi$
$89$		−	27.5850i		−	0.309944i	0.987919	−	0.154972i	$-0.0495287\pi$
$90$		0			0
$91$	195.462			2.14793
$92$		0			0
$93$	6.29376	−	1.89393i	0.0676748	−	0.0203648i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−114.989			−1.18545			−0.592727	−	0.805404i	$-0.701949\pi$
$97$	−114.989			−1.18545			−0.592727	+	0.805404i	$0.701949\pi$
$98$		0			0
$99$	108.688	+	164.238i	1.09786	+	1.65897i
$100$		0			0
$101$			122.804i			1.21588i	0.793982	+	0.607941i	$0.208004\pi$
$101$			122.804i			1.21588i	−0.793982	+	0.607941i	$0.791996\pi$
$102$		0			0
$103$	46.8275			0.454636			0.227318	−	0.973821i	$-0.427004\pi$
$103$	46.8275			0.454636			0.227318	+	0.973821i	$0.427004\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			105.086i			0.982113i	0.871128	+	0.491056i	$0.163389\pi$
$107$			105.086i			0.982113i	−0.871128	+	0.491056i	$0.836611\pi$
$108$		0			0
$109$	−116.777			−1.07135			−0.535673	−	0.844426i	$-0.679942\pi$
$109$	−116.777			−1.07135			−0.535673	+	0.844426i	$0.679942\pi$
$110$		0			0
$111$	−2.41299	+	0.726122i	−0.0217387	+	0.00654164i
$112$		0			0
$113$		−	10.8116i		−	0.0956779i	−0.998855	−	0.0478389i	$-0.984767\pi$
$113$		−	10.8116i		−	0.0956779i	0.998855	−	0.0478389i	$-0.0152334\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−162.565	+	107.581i	−1.38945	+	0.919494i
$118$		0			0
$119$		−	109.262i		−	0.918171i
$120$		0			0
$121$	−357.853			−2.95747
$122$		0			0
$123$	30.7485	+	102.181i	0.249988	+	0.830739i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	192.459			1.51543			0.757714	−	0.652587i	$-0.226316\pi$
$127$	192.459			1.51543			0.757714	+	0.652587i	$0.226316\pi$
$128$		0			0
$129$	36.5276	−	10.9919i	0.283159	−	0.0852089i
$130$		0			0
$131$			48.6360i			0.371267i	0.982619	+	0.185633i	$0.0594337\pi$
$131$			48.6360i			0.371267i	−0.982619	+	0.185633i	$0.940566\pi$
$132$		0			0
$133$	−27.4115			−0.206102
$134$		0			0
$135$		0			0
$136$		0			0
$137$			157.869i			1.15233i	0.817335	+	0.576163i	$0.195451\pi$
$137$			157.869i			1.15233i	−0.817335	+	0.576163i	$0.804549\pi$
$138$		0			0
$139$	−164.752			−1.18526			−0.592632	−	0.805473i	$-0.701911\pi$
$139$	−164.752			−1.18526			−0.592632	+	0.805473i	$0.701911\pi$
$140$		0			0
$141$	−19.4922	−	64.7749i	−0.138243	−	0.459397i
$142$		0			0
$143$		−	473.977i		−	3.31452i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	93.1788	−	28.0396i	0.633869	−	0.190745i
$148$		0			0
$149$		−	262.935i		−	1.76467i	−0.470626	−	0.882333i	$-0.655972\pi$
$149$		−	262.935i		−	1.76467i	0.470626	−	0.882333i	$-0.344028\pi$
$150$		0			0
$151$	15.8171			0.104749			0.0523745	−	0.998628i	$-0.483321\pi$
$151$	15.8171			0.104749			0.0523745	+	0.998628i	$0.483321\pi$
$152$		0			0
$153$	60.1372	+	90.8733i	0.393054	+	0.593943i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	6.11941			0.0389771			0.0194886	−	0.999810i	$-0.493796\pi$
$157$	6.11941			0.0389771			0.0194886	+	0.999810i	$0.493796\pi$
$158$		0			0
$159$	−7.89934	−	26.2505i	−0.0496814	−	0.165097i
$160$		0			0
$161$		−	257.852i		−	1.60157i
$162$		0			0
$163$	170.444			1.04567			0.522833	−	0.852435i	$-0.324875\pi$
$163$	170.444			1.04567			0.522833	+	0.852435i	$0.324875\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			61.2668i			0.366867i	0.983032	+	0.183434i	$0.0587212\pi$
$167$			61.2668i			0.366867i	−0.983032	+	0.183434i	$0.941279\pi$
$168$		0			0
$169$	300.149			1.77603
$170$		0			0
$171$	22.7981	−	15.0871i	0.133322	−	0.0882286i
$172$		0			0
$173$		−	262.548i		−	1.51762i	−0.651312	−	0.758810i	$-0.725781\pi$
$173$		−	262.548i		−	1.51762i	0.651312	−	0.758810i	$-0.274219\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−69.5434	−	231.101i	−0.392900	−	1.30566i
$178$		0			0
$179$		−	6.88752i		−	0.0384778i	−0.999815	−	0.0192389i	$-0.993876\pi$
$179$		−	6.88752i		−	0.0384778i	0.999815	−	0.0192389i	$-0.00612431\pi$
$180$		0			0
$181$	218.536			1.20738			0.603691	−	0.797218i	$-0.293696\pi$
$181$	218.536			1.20738			0.603691	+	0.797218i	$0.293696\pi$
$182$		0			0
$183$	−166.279	+	50.0371i	−0.908630	+	0.273427i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−264.951			−1.41685
$188$		0			0
$189$	−155.823	+	187.311i	−0.824462	+	0.991065i
$190$		0			0
$191$			75.2506i			0.393982i	0.980405	+	0.196991i	$0.0631170\pi$
$191$			75.2506i			0.393982i	−0.980405	+	0.196991i	$0.936883\pi$
$192$		0			0
$193$	212.587			1.10149			0.550744	−	0.834674i	$-0.314344\pi$
$193$	212.587			1.10149			0.550744	+	0.834674i	$0.314344\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			190.640i			0.967718i	0.875146	+	0.483859i	$0.160765\pi$
$197$			190.640i			0.967718i	−0.875146	+	0.483859i	$0.839235\pi$
$198$		0			0
$199$	209.996			1.05526			0.527629	−	0.849475i	$-0.323081\pi$
$199$	209.996			1.05526			0.527629	+	0.849475i	$0.323081\pi$
$200$		0			0
$201$	181.144	−	54.5102i	0.901214	−	0.271195i
$202$		0			0
$203$		−	108.618i		−	0.535064i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	141.920	+	214.455i	0.685603	+	1.03601i
$208$		0			0
$209$			66.4703i			0.318040i
$210$		0			0
$211$	176.419			0.836110			0.418055	−	0.908422i	$-0.362712\pi$
$211$	176.419			0.836110			0.418055	+	0.908422i	$0.362712\pi$
$212$		0			0
$213$	14.7149	+	48.8993i	0.0690839	+	0.229574i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−19.7706			−0.0911086
$218$		0			0
$219$	−149.722	+	45.0547i	−0.683663	+	0.205729i
$220$		0			0
$221$		−	262.252i		−	1.18666i
$222$		0			0
$223$	−132.362			−0.593552			−0.296776	−	0.954947i	$-0.595911\pi$
$223$	−132.362			−0.593552			−0.296776	+	0.954947i	$0.595911\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	187.624i		−	0.826537i	−0.910609	−	0.413268i	$-0.864387\pi$
$227$		−	187.624i		−	0.826537i	0.910609	−	0.413268i	$-0.135613\pi$
$228$		0			0
$229$	−178.571			−0.779788			−0.389894	−	0.920860i	$-0.627488\pi$
$229$	−178.571			−0.779788			−0.389894	+	0.920860i	$0.627488\pi$
$230$		0			0
$231$	−170.710	−	567.291i	−0.739005	−	2.45580i
$232$		0			0
$233$		−	296.711i		−	1.27344i	−0.771097	−	0.636718i	$-0.780292\pi$
$233$		−	296.711i		−	1.27344i	0.771097	−	0.636718i	$-0.219708\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−21.4371	+	6.45091i	−0.0904521	+	0.0272190i
$238$		0			0
$239$			137.976i			0.577306i	0.957434	+	0.288653i	$0.0932074\pi$
$239$			137.976i			0.577306i	−0.957434	+	0.288653i	$0.906793\pi$
$240$		0			0
$241$	42.5687			0.176633			0.0883167	−	0.996092i	$-0.471851\pi$
$241$	42.5687			0.176633			0.0883167	+	0.996092i	$0.471851\pi$
$242$		0			0
$243$	26.5032	−	241.550i	0.109067	−	0.994034i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−65.7933			−0.266370
$248$		0			0
$249$	71.2117	+	236.645i	0.285991	+	0.950382i
$250$		0			0
$251$		−	205.885i		−	0.820259i	−0.912027	−	0.410130i	$-0.865483\pi$
$251$		−	205.885i		−	0.820259i	0.912027	−	0.410130i	$-0.134517\pi$
$252$		0			0
$253$	−625.267			−2.47141
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	188.382i		−	0.733004i	−0.930417	−	0.366502i	$-0.880555\pi$
$257$		−	188.382i		−	0.733004i	0.930417	−	0.366502i	$-0.119445\pi$
$258$		0			0
$259$	7.57992			0.0292661
$260$		0			0
$261$	59.7825	+	90.3373i	0.229052	+	0.346120i
$262$		0			0
$263$			188.745i			0.717660i	0.933403	+	0.358830i	$0.116824\pi$
$263$			188.745i			0.717660i	−0.933403	+	0.358830i	$0.883176\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−23.8465	−	79.2447i	−0.0893127	−	0.296797i
$268$		0			0
$269$		−	333.372i		−	1.23930i	−0.784878	−	0.619651i	$-0.787274\pi$
$269$		−	333.372i		−	1.23930i	0.784878	−	0.619651i	$-0.212726\pi$
$270$		0			0
$271$	262.047			0.966964			0.483482	−	0.875354i	$-0.339372\pi$
$271$	262.047			0.966964			0.483482	+	0.875354i	$0.339372\pi$
$272$		0			0
$273$	561.513	−	168.972i	2.05682	−	0.618944i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	400.100			1.44440			0.722202	−	0.691682i	$-0.243130\pi$
$277$	400.100			1.44440			0.722202	+	0.691682i	$0.243130\pi$
$278$		0			0
$279$	16.4431	−	10.8816i	0.0589360	−	0.0390020i
$280$		0			0
$281$			350.698i			1.24804i	0.781410	+	0.624018i	$0.214501\pi$
$281$			350.698i			1.24804i	−0.781410	+	0.624018i	$0.785499\pi$
$282$		0			0
$283$	464.015			1.63963			0.819814	−	0.572630i	$-0.194077\pi$
$283$	464.015			1.63963			0.819814	+	0.572630i	$0.194077\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	320.981i		−	1.11840i
$288$		0			0
$289$	142.402			0.492741
$290$		0			0
$291$	−330.334	+	99.4049i	−1.13517	+	0.341598i
$292$		0			0
$293$			47.4080i			0.161802i	0.996722	+	0.0809009i	$0.0257797\pi$
$293$			47.4080i			0.161802i	−0.996722	+	0.0809009i	$0.974220\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	454.212	+	377.857i	1.52933	+	1.27224i
$298$		0			0
$299$		−	618.899i		−	2.06990i
$300$		0			0
$301$	−114.744			−0.381209
$302$		0			0
$303$	106.161	+	352.785i	0.350366	+	1.16431i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−461.894			−1.50454			−0.752270	−	0.658855i	$-0.771041\pi$
$307$	−461.894			−1.50454			−0.752270	+	0.658855i	$0.771041\pi$
$308$		0			0
$309$	134.524	−	40.4812i	0.435352	−	0.131007i
$310$		0			0
$311$			123.057i			0.395681i	0.980234	+	0.197841i	$0.0633929\pi$
$311$			123.057i			0.395681i	−0.980234	+	0.197841i	$0.936607\pi$
$312$		0			0
$313$	−97.4353			−0.311295			−0.155647	−	0.987813i	$-0.549746\pi$
$313$	−97.4353			−0.311295			−0.155647	+	0.987813i	$0.549746\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			252.388i			0.796175i	0.917347	+	0.398088i	$0.130326\pi$
$317$			252.388i			0.796175i	−0.917347	+	0.398088i	$0.869674\pi$
$318$		0			0
$319$	−263.388			−0.825668
$320$		0			0
$321$	90.8441	+	301.886i	0.283003	+	0.940454i
$322$		0			0
$323$			36.7782i			0.113864i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−335.470	+	100.950i	−1.02590	+	0.308716i
$328$		0			0
$329$			203.477i			0.618472i
$330$		0			0
$331$	−303.273			−0.916231			−0.458116	−	0.888893i	$-0.651475\pi$
$331$	−303.273			−0.916231			−0.458116	+	0.888893i	$0.651475\pi$
$332$		0			0
$333$	−6.30420	+	4.17193i	−0.0189315	+	0.0125283i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	352.738			1.04670			0.523350	−	0.852118i	$-0.324682\pi$
$337$	352.738			1.04670			0.523350	+	0.852118i	$0.324682\pi$
$338$		0			0
$339$	−9.34634	−	31.0590i	−0.0275703	−	0.0916195i
$340$		0			0
$341$			47.9417i			0.140592i
$342$		0			0
$343$	149.481			0.435806
$344$		0			0
$345$		0			0
$346$		0			0
$347$			280.382i			0.808018i	0.914755	+	0.404009i	$0.132384\pi$
$347$			280.382i			0.808018i	−0.914755	+	0.404009i	$0.867616\pi$
$348$		0			0
$349$	−586.721			−1.68115			−0.840575	−	0.541696i	$-0.817783\pi$
$349$	−586.721			−1.68115			−0.840575	+	0.541696i	$0.817783\pi$
$350$		0			0
$351$	−374.008	+	449.586i	−1.06555	+	1.28087i
$352$		0			0
$353$			558.927i			1.58336i	0.610935	+	0.791681i	$0.290794\pi$
$353$			558.927i			1.58336i	−0.610935	+	0.791681i	$0.709206\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−94.4544	−	313.883i	−0.264578	−	0.879225i
$358$		0			0
$359$			323.554i			0.901264i	0.892710	+	0.450632i	$0.148801\pi$
$359$			323.554i			0.901264i	−0.892710	+	0.450632i	$0.851199\pi$
$360$		0			0
$361$	−351.773			−0.974441
$362$		0			0
$363$	−1028.02	+	309.355i	−2.83202	+	0.852217i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−176.636			−0.481296			−0.240648	−	0.970612i	$-0.577360\pi$
$367$	−176.636			−0.481296			−0.240648	+	0.970612i	$0.577360\pi$
$368$		0			0
$369$	176.665	+	266.959i	0.478768	+	0.723466i
$370$		0			0
$371$			82.4605i			0.222265i
$372$		0			0
$373$	−367.327			−0.984790			−0.492395	−	0.870372i	$-0.663878\pi$
$373$	−367.327			−0.984790			−0.492395	+	0.870372i	$0.663878\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	260.706i		−	0.691527i
$378$		0			0
$379$	611.014			1.61217			0.806087	−	0.591798i	$-0.201582\pi$
$379$	611.014			1.61217			0.806087	+	0.591798i	$0.201582\pi$
$380$		0			0
$381$	552.887	−	166.376i	1.45115	−	0.436682i
$382$		0			0
$383$		−	13.6994i		−	0.0357687i	−0.999840	−	0.0178844i	$-0.994307\pi$
$383$		−	13.6994i		−	0.0357687i	0.999840	−	0.0178844i	$-0.00569307\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	95.4322	−	63.1542i	0.246595	−	0.163189i
$388$		0			0
$389$			379.601i			0.975838i	0.872889	+	0.487919i	$0.162244\pi$
$389$			379.601i			0.975838i	−0.872889	+	0.487919i	$0.837756\pi$
$390$		0			0
$391$	−345.962			−0.884812
$392$		0			0
$393$	42.0445	+	139.719i	0.106983	+	0.355519i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−450.560			−1.13491			−0.567456	−	0.823404i	$-0.692072\pi$
$397$	−450.560			−1.13491			−0.567456	+	0.823404i	$0.692072\pi$
$398$		0			0
$399$	−78.7464	+	23.6965i	−0.197359	+	0.0593898i
$400$		0			0
$401$			503.683i			1.25607i	0.778186	+	0.628034i	$0.216140\pi$
$401$			503.683i			1.25607i	−0.778186	+	0.628034i	$0.783860\pi$
$402$		0			0
$403$	−47.4535			−0.117751
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	18.3806i		−	0.0451611i
$408$		0			0
$409$	240.726			0.588573			0.294286	−	0.955717i	$-0.404918\pi$
$409$	240.726			0.588573			0.294286	+	0.955717i	$0.404918\pi$
$410$		0			0
$411$	136.473	+	453.517i	0.332051	+	1.10345i
$412$		0			0
$413$			725.957i			1.75776i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−473.290	+	142.423i	−1.13499	+	0.341543i
$418$		0			0
$419$			676.873i			1.61545i	0.589561	+	0.807724i	$0.299301\pi$
$419$			676.873i			1.61545i	−0.589561	+	0.807724i	$0.700699\pi$
$420$		0			0
$421$	683.755			1.62412			0.812061	−	0.583572i	$-0.198346\pi$
$421$	683.755			1.62412			0.812061	+	0.583572i	$0.198346\pi$
$422$		0			0
$423$	−111.992	−	169.232i	−0.264757	−	0.400075i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	522.332			1.22326
$428$		0			0
$429$	−409.740	−	1361.62i	−0.955105	−	3.17393i
$430$		0			0
$431$			213.608i			0.495610i	0.968810	+	0.247805i	$0.0797092\pi$
$431$			213.608i			0.495610i	−0.968810	+	0.247805i	$0.920291\pi$
$432$		0			0
$433$	383.579			0.885864			0.442932	−	0.896555i	$-0.353938\pi$
$433$	383.579			0.885864			0.442932	+	0.896555i	$0.353938\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			86.7941i			0.198614i
$438$		0			0
$439$	523.900			1.19340			0.596698	−	0.802466i	$-0.296479\pi$
$439$	523.900			1.19340			0.596698	+	0.802466i	$0.296479\pi$
$440$		0			0
$441$	243.440	−	161.101i	0.552018	−	0.365309i
$442$		0			0
$443$			391.277i			0.883243i	0.897201	+	0.441622i	$0.145597\pi$
$443$			391.277i			0.883243i	−0.897201	+	0.441622i	$0.854403\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−227.301	−	755.347i	−0.508502	−	1.68981i
$448$		0			0
$449$		−	375.014i		−	0.835220i	−0.908626	−	0.417610i	$-0.862868\pi$
$449$		−	375.014i		−	0.835220i	0.908626	−	0.417610i	$-0.137132\pi$
$450$		0			0
$451$	−778.347			−1.72582
$452$		0			0
$453$	45.4386	−	13.6735i	0.100306	−	0.0301843i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	734.032			1.60620			0.803098	−	0.595847i	$-0.203183\pi$
$457$	734.032			1.60620			0.803098	+	0.595847i	$0.203183\pi$
$458$		0			0
$459$	251.317	+	209.069i	0.547531	+	0.455488i
$460$		0			0
$461$			775.239i			1.68165i	0.541310	+	0.840823i	$0.317928\pi$
$461$			775.239i			1.68165i	−0.541310	+	0.840823i	$0.682072\pi$
$462$		0			0
$463$	323.967			0.699714			0.349857	−	0.936803i	$-0.386230\pi$
$463$	323.967			0.699714			0.349857	+	0.936803i	$0.386230\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	160.785i		−	0.344294i	−0.985071	−	0.172147i	$-0.944930\pi$
$467$		−	160.785i		−	0.344294i	0.985071	−	0.172147i	$-0.0550704\pi$
$468$		0			0
$469$	−569.027			−1.21328
$470$		0			0
$471$	17.5795	−	5.29007i	0.0373239	−	0.0112316i
$472$		0			0
$473$			278.243i			0.588252i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−45.3856	−	68.5822i	−0.0951481	−	0.143778i
$478$		0			0
$479$			61.7565i			0.128928i	0.997920	+	0.0644640i	$0.0205338\pi$
$479$			61.7565i			0.128928i	−0.997920	+	0.0644640i	$0.979466\pi$
$480$		0			0
$481$	18.1934			0.0378241
$482$		0			0
$483$	−222.906	−	740.744i	−0.461504	−	1.53363i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−129.969			−0.266876			−0.133438	−	0.991057i	$-0.542602\pi$
$487$	−129.969			−0.266876			−0.133438	+	0.991057i	$0.542602\pi$
$488$		0			0
$489$	489.642	−	147.344i	1.00131	−	0.301317i
$490$		0			0
$491$		−	810.511i		−	1.65074i	−0.564595	−	0.825368i	$-0.690968\pi$
$491$		−	810.511i		−	1.65074i	0.564595	−	0.825368i	$-0.309032\pi$
$492$		0			0
$493$	−145.733			−0.295605
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	153.607i		−	0.309069i
$498$		0			0
$499$	−600.897			−1.20420			−0.602101	−	0.798420i	$-0.705670\pi$
$499$	−600.897			−1.20420			−0.602101	+	0.798420i	$0.705670\pi$
$500$		0			0
$501$	52.9635	+	176.004i	0.105716	+	0.351306i
$502$		0			0
$503$			688.332i			1.36845i	0.729270	+	0.684226i	$0.239860\pi$
$503$			688.332i			1.36845i	−0.729270	+	0.684226i	$0.760140\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	862.254	−	259.471i	1.70070	−	0.511777i
$508$		0			0
$509$			823.791i			1.61845i	0.587498	+	0.809225i	$0.300113\pi$
$509$			823.791i			1.61845i	−0.587498	+	0.809225i	$0.699887\pi$
$510$		0			0
$511$	470.322			0.920395
$512$		0			0
$513$	52.4508	−	63.0498i	0.102243	−	0.122904i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	493.413			0.954377
$518$		0			0
$519$	−226.966	−	754.235i	−0.437314	−	1.45325i
$520$		0			0
$521$		−	964.525i		−	1.85130i	−0.378386	−	0.925648i	$-0.623521\pi$
$521$		−	964.525i		−	1.85130i	0.378386	−	0.925648i	$-0.376479\pi$
$522$		0			0
$523$	−462.679			−0.884664			−0.442332	−	0.896851i	$-0.645849\pi$
$523$	−462.679			−0.884664			−0.442332	+	0.896851i	$0.645849\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			26.5263i			0.0503345i
$528$		0			0
$529$	−287.447			−0.543378
$530$		0			0
$531$	−399.561	−	603.777i	−0.752469	−	1.13706i
$532$		0			0
$533$		−	770.420i		−	1.44544i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.95408	−	19.7861i	−0.0110877	−	0.0368457i
$538$		0			0
$539$			709.775i			1.31684i
$540$		0			0
$541$	−516.752			−0.955180			−0.477590	−	0.878583i	$-0.658490\pi$
$541$	−516.752			−0.955180			−0.477590	+	0.878583i	$0.658490\pi$
$542$		0			0
$543$	627.800	−	188.919i	1.15617	−	0.347917i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−478.953			−0.875599			−0.437799	−	0.899073i	$-0.644242\pi$
$547$	−478.953			−0.875599			−0.437799	+	0.899073i	$0.644242\pi$
$548$		0			0
$549$	−434.423	+	287.488i	−0.791298	+	0.523657i
$550$		0			0
$551$			36.5613i			0.0663544i
$552$		0			0
$553$	67.3404			0.121773
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	343.982i		−	0.617561i	−0.951133	−	0.308781i	$-0.900079\pi$
$557$		−	343.982i		−	0.617561i	0.951133	−	0.308781i	$-0.0999209\pi$
$558$		0			0
$559$	−275.409			−0.492682
$560$		0			0
$561$	−761.137	+	229.043i	−1.35675	+	0.408276i
$562$		0			0
$563$			394.056i			0.699922i	0.936764	+	0.349961i	$0.113805\pi$
$563$			394.056i			0.699922i	−0.936764	+	0.349961i	$0.886195\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−285.716	+	672.803i	−0.503908	+	1.18660i
$568$		0			0
$569$			537.452i			0.944556i	0.881450	+	0.472278i	$0.156568\pi$
$569$			537.452i			0.944556i	−0.881450	+	0.472278i	$0.843432\pi$
$570$		0			0
$571$	710.555			1.24440			0.622202	−	0.782856i	$-0.286238\pi$
$571$	710.555			1.24440			0.622202	+	0.782856i	$0.286238\pi$
$572$		0			0
$573$	65.0521	+	216.176i	0.113529	+	0.377271i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−452.846			−0.784828			−0.392414	−	0.919789i	$-0.628360\pi$
$577$	−452.846			−0.784828			−0.392414	+	0.919789i	$0.628360\pi$
$578$		0			0
$579$	610.709	−	183.776i	1.05477	−	0.317402i
$580$		0			0
$581$		−	743.372i		−	1.27947i
$582$		0			0
$583$	199.959			0.342983
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1103.34i			1.87963i	0.341690	+	0.939813i	$0.389001\pi$
$587$			1103.34i			1.87963i	−0.341690	+	0.939813i	$0.610999\pi$
$588$		0			0
$589$	6.65486			0.0112986
$590$		0			0
$591$	164.804	+	547.662i	0.278855	+	0.926670i
$592$		0			0
$593$			249.474i			0.420698i	0.977626	+	0.210349i	$0.0674601\pi$
$593$			249.474i			0.420698i	−0.977626	+	0.210349i	$0.932540\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	603.267	−	181.536i	1.01050	−	0.304081i
$598$		0			0
$599$		−	596.120i		−	0.995191i	−0.867409	−	0.497596i	$-0.834216\pi$
$599$		−	596.120i		−	0.995191i	0.867409	−	0.497596i	$-0.165784\pi$
$600$		0			0
$601$	476.515			0.792871			0.396435	−	0.918063i	$-0.370247\pi$
$601$	476.515			0.792871			0.396435	+	0.918063i	$0.370247\pi$
$602$		0			0
$603$	473.259	−	313.188i	0.784841	−	0.519384i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−27.8813			−0.0459329			−0.0229664	−	0.999736i	$-0.507311\pi$
$607$	−27.8813			−0.0459329			−0.0229664	+	0.999736i	$0.507311\pi$
$608$		0			0
$609$	−93.8973	−	312.032i	−0.154183	−	0.512368i
$610$		0			0
$611$			488.388i			0.799325i
$612$		0			0
$613$	−465.472			−0.759334			−0.379667	−	0.925123i	$-0.623961\pi$
$613$	−465.472			−0.759334			−0.379667	+	0.925123i	$0.623961\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	48.6709i		−	0.0788832i	−0.999222	−	0.0394416i	$-0.987442\pi$
$617$		−	48.6709i		−	0.0788832i	0.999222	−	0.0394416i	$-0.0125579\pi$
$618$		0			0
$619$	−213.318			−0.344617			−0.172309	−	0.985043i	$-0.555123\pi$
$619$	−213.318			−0.344617			−0.172309	+	0.985043i	$0.555123\pi$
$620$		0			0
$621$	593.091	+	493.389i	0.955058	+	0.794508i
$622$		0			0
$623$			248.931i			0.399569i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	57.4618	+	190.953i	0.0916456	+	0.304549i
$628$		0			0
$629$		−	10.1700i		−	0.0161686i
$630$		0			0
$631$	582.489			0.923121			0.461560	−	0.887109i	$-0.347290\pi$
$631$	582.489			0.923121			0.461560	+	0.887109i	$0.347290\pi$
$632$		0			0
$633$	506.808	−	152.510i	0.800644	−	0.240931i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−702.546			−1.10290
$638$		0			0
$639$	84.5442	+	127.755i	0.132307	+	0.199929i
$640$		0			0
$641$		−	319.635i		−	0.498650i	−0.968420	−	0.249325i	$-0.919791\pi$
$641$		−	319.635i		−	0.498650i	0.968420	−	0.249325i	$-0.0802088\pi$
$642$		0			0
$643$	−458.627			−0.713261			−0.356630	−	0.934246i	$-0.616074\pi$
$643$	−458.627			−0.713261			−0.356630	+	0.934246i	$0.616074\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	507.599i		−	0.784543i	−0.919850	−	0.392271i	$-0.871689\pi$
$647$		−	507.599i		−	0.784543i	0.919850	−	0.392271i	$-0.128311\pi$
$648$		0			0
$649$	1760.38			2.71244
$650$		0			0
$651$	−56.7959	+	17.0911i	−0.0872440	+	0.0262536i
$652$		0			0
$653$		−	1197.20i		−	1.83339i	−0.399591	−	0.916694i	$-0.630848\pi$
$653$		−	1197.20i		−	1.83339i	0.399591	−	0.916694i	$-0.369152\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−391.166	+	258.862i	−0.595382	+	0.394006i
$658$		0			0
$659$		−	632.173i		−	0.959291i	−0.877462	−	0.479645i	$-0.840765\pi$
$659$		−	632.173i		−	0.959291i	0.877462	−	0.479645i	$-0.159235\pi$
$660$		0			0
$661$	−565.316			−0.855243			−0.427622	−	0.903958i	$-0.640648\pi$
$661$	−565.316			−0.855243			−0.427622	+	0.903958i	$0.640648\pi$
$662$		0			0
$663$	−226.710	−	753.385i	−0.341946	−	1.13633i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−343.921			−0.515624
$668$		0			0
$669$	−380.243	+	114.423i	−0.568375	+	0.171036i
$670$		0			0
$671$		−	1266.61i		−	1.88764i
$672$		0			0
$673$	306.607			0.455582			0.227791	−	0.973710i	$-0.426850\pi$
$673$	306.607			0.455582			0.227791	+	0.973710i	$0.426850\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			546.672i			0.807491i	0.914871	+	0.403746i	$0.132292\pi$
$677$			546.672i			0.807491i	−0.914871	+	0.403746i	$0.867708\pi$
$678$		0			0
$679$	1037.68			1.52824
$680$		0			0
$681$	−162.196	−	538.996i	−0.238173	−	0.791478i
$682$		0			0
$683$		−	812.204i		−	1.18917i	−0.804032	−	0.594586i	$-0.797316\pi$
$683$		−	812.204i		−	1.18917i	0.804032	−	0.594586i	$-0.202684\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−512.991	+	154.370i	−0.746712	+	0.224702i
$688$		0			0
$689$			197.922i			0.287260i
$690$		0			0
$691$	−45.9358			−0.0664773			−0.0332387	−	0.999447i	$-0.510582\pi$
$691$	−45.9358			−0.0664773			−0.0332387	+	0.999447i	$0.510582\pi$
$692$		0			0
$693$	−980.815	−	1482.11i	−1.41532	−	2.13869i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−430.661			−0.617879
$698$		0			0
$699$	−256.498	−	852.375i	−0.366950	−	1.21942i
$700$		0			0
$701$			674.615i			0.962360i	0.876622	+	0.481180i	$0.159792\pi$
$701$			674.615i			0.962360i	−0.876622	+	0.481180i	$0.840208\pi$
$702$		0			0
$703$	−2.55143			−0.00362935
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	1108.20i		−	1.56747i
$708$		0			0
$709$	656.626			0.926129			0.463065	−	0.886324i	$-0.346750\pi$
$709$	656.626			0.926129			0.463065	+	0.886324i	$0.346750\pi$
$710$		0			0
$711$	−56.0069	+	37.0637i	−0.0787720	+	0.0521290i
$712$		0			0
$713$			62.6003i			0.0877984i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	119.277	+	396.371i	0.166355	+	0.552819i
$718$		0			0
$719$			644.279i			0.896076i	0.894014	+	0.448038i	$0.147877\pi$
$719$			644.279i			0.896076i	−0.894014	+	0.448038i	$0.852123\pi$
$720$		0			0
$721$	−422.579			−0.586101
$722$		0			0
$723$	122.289	−	36.7995i	0.169141	−	0.0508983i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	981.269			1.34975			0.674875	−	0.737932i	$-0.264197\pi$
$727$	981.269			1.34975			0.674875	+	0.737932i	$0.264197\pi$
$728$		0			0
$729$	−132.677	−	716.825i	−0.181998	−	0.983299i
$730$		0			0
$731$			153.953i			0.210605i
$732$		0			0
$733$	127.756			0.174292			0.0871460	−	0.996196i	$-0.472225\pi$
$733$	127.756			0.174292			0.0871460	+	0.996196i	$0.472225\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			1379.84i			1.87223i
$738$		0			0
$739$	−1401.59			−1.89660			−0.948302	−	0.317368i	$-0.897201\pi$
$739$	−1401.59			−1.89660			−0.948302	+	0.317368i	$0.897201\pi$
$740$		0			0
$741$	−189.008	+	56.8766i	−0.255071	+	0.0767565i
$742$		0			0
$743$		−	11.6734i		−	0.0157112i	−0.999969	−	0.00785561i	$-0.997499\pi$
$743$		−	11.6734i		−	0.0157112i	0.999969	−	0.00785561i	$-0.00250055\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	409.147	+	618.261i	0.547720	+	0.827659i
$748$		0			0
$749$		−	948.313i		−	1.26611i
$750$		0			0
$751$	380.403			0.506529			0.253264	−	0.967397i	$-0.418496\pi$
$751$	380.403			0.506529			0.253264	+	0.967397i	$0.418496\pi$
$752$		0			0
$753$	−177.982	−	591.456i	−0.236364	−	0.785466i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−63.6621			−0.0840979			−0.0420490	−	0.999116i	$-0.513389\pi$
$757$	−63.6621			−0.0840979			−0.0420490	+	0.999116i	$0.513389\pi$
$758$		0			0
$759$	−1796.23	+	540.526i	−2.36658	+	0.712156i
$760$		0			0
$761$		−	377.891i		−	0.496571i	−0.968687	−	0.248286i	$-0.920133\pi$
$761$		−	377.891i		−	0.496571i	0.968687	−	0.248286i	$-0.0798671\pi$
$762$		0			0
$763$	1053.81			1.38114
$764$		0			0
$765$		0			0
$766$		0			0
$767$			1742.45i			2.27177i
$768$		0			0
$769$	231.920			0.301586			0.150793	−	0.988565i	$-0.451817\pi$
$769$	231.920			0.301586			0.150793	+	0.988565i	$0.451817\pi$
$770$		0			0
$771$	−162.851	−	541.174i	−0.211221	−	0.701912i
$772$		0			0
$773$			1509.05i			1.95220i	0.217320	+	0.976101i	$0.430269\pi$
$773$			1509.05i			1.95220i	−0.217320	+	0.976101i	$0.569731\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	21.7752	−	6.55264i	0.0280247	−	0.00843326i
$778$		0			0
$779$			108.043i			0.138695i
$780$		0			0
$781$	−372.483			−0.476930
$782$		0			0
$783$	249.834	+	207.836i	0.319073	+	0.265436i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1171.43			−1.48848			−0.744240	−	0.667912i	$-0.767188\pi$
$787$	−1171.43			−1.48848			−0.744240	+	0.667912i	$0.767188\pi$
$788$		0			0
$789$	163.165	+	542.216i	0.206799	+	0.687219i
$790$		0			0
$791$			97.5656i			0.123345i
$792$		0			0
$793$	1253.71			1.58097
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	988.589i		−	1.24039i	−0.784448	−	0.620194i	$-0.787054\pi$
$797$		−	988.589i		−	1.24039i	0.784448	−	0.620194i	$-0.212946\pi$
$798$		0			0
$799$	273.007			0.341685
$800$		0			0
$801$	−137.010	−	207.036i	−0.171049	−	0.258471i
$802$		0			0
$803$		−	1140.49i		−	1.42028i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−288.191	−	957.694i	−0.357114	−	1.18673i
$808$		0			0
$809$			1334.53i			1.64961i	0.565419	+	0.824804i	$0.308714\pi$
$809$			1334.53i			1.64961i	−0.565419	+	0.824804i	$0.691286\pi$
$810$		0			0
$811$	610.590			0.752885			0.376443	−	0.926440i	$-0.377147\pi$
$811$	610.590			0.752885			0.376443	+	0.926440i	$0.377147\pi$
$812$		0			0
$813$	752.796	−	226.533i	0.925949	−	0.278638i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	38.6233			0.0472745
$818$		0			0
$819$	1467.01	−	970.826i	1.79123	−	1.18538i
$820$		0			0
$821$		−	877.538i		−	1.06887i	−0.845211	−	0.534433i	$-0.820525\pi$
$821$		−	877.538i		−	1.06887i	0.845211	−	0.534433i	$-0.179475\pi$
$822$		0			0
$823$	−230.855			−0.280504			−0.140252	−	0.990116i	$-0.544791\pi$
$823$	−230.855			−0.280504			−0.140252	+	0.990116i	$0.544791\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	969.434i		−	1.17223i	−0.810228	−	0.586115i	$-0.800657\pi$
$827$		−	969.434i		−	1.17223i	0.810228	−	0.586115i	$-0.199343\pi$
$828$		0			0
$829$	−1049.09			−1.26549			−0.632746	−	0.774360i	$-0.718072\pi$
$829$	−1049.09			−1.26549			−0.632746	+	0.774360i	$0.718072\pi$
$830$		0			0
$831$	1149.39	−	345.876i	1.38314	−	0.416216i
$832$		0			0
$833$			392.720i			0.471453i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	37.8302	−	45.4747i	0.0451973	−	0.0543305i
$838$		0			0
$839$			1214.62i			1.44770i	0.689959	+	0.723849i	$0.257629\pi$
$839$			1214.62i			1.44770i	−0.689959	+	0.723849i	$0.742371\pi$
$840$		0			0
$841$	696.126			0.827736
$842$		0			0
$843$	303.169	+	1007.47i	0.359631	+	1.19510i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	3229.33			3.81266
$848$		0			0
$849$	1333.00	−	401.128i	1.57008	−	0.472472i
$850$		0			0
$851$		−	24.0006i		−	0.0282028i
$852$		0			0
$853$	−190.704			−0.223569			−0.111784	−	0.993732i	$-0.535657\pi$
$853$	−190.704			−0.223569			−0.111784	+	0.993732i	$0.535657\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			351.238i			0.409846i	0.978778	+	0.204923i	$0.0656944\pi$
$857$			351.238i			0.409846i	−0.978778	+	0.204923i	$0.934306\pi$
$858$		0			0
$859$	−1233.74			−1.43625			−0.718123	−	0.695916i	$-0.754998\pi$
$859$	−1233.74			−1.43625			−0.718123	+	0.695916i	$0.754998\pi$
$860$		0			0
$861$	−277.479	−	922.096i	−0.322275	−	1.07096i
$862$		0			0
$863$		−	463.404i		−	0.536968i	−0.963284	−	0.268484i	$-0.913477\pi$
$863$		−	463.404i		−	0.536968i	0.963284	−	0.268484i	$-0.0865227\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	409.085	−	123.103i	0.471840	−	0.141987i
$868$		0			0
$869$		−	163.294i		−	0.187910i
$870$		0			0
$871$	−1365.78			−1.56806
$872$		0			0
$873$	−863.035	+	571.130i	−0.988585	+	0.654216i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−777.195			−0.886197			−0.443099	−	0.896473i	$-0.646121\pi$
$877$	−777.195			−0.886197			−0.443099	+	0.896473i	$0.646121\pi$
$878$		0			0
$879$	40.9829	+	136.191i	0.0466245	+	0.154939i
$880$		0			0
$881$		−	1609.50i		−	1.82690i	−0.406951	−	0.913450i	$-0.633408\pi$
$881$		−	1609.50i		−	1.82690i	0.406951	−	0.913450i	$-0.366592\pi$
$882$		0			0
$883$	−331.153			−0.375032			−0.187516	−	0.982262i	$-0.560044\pi$
$883$	−331.153			−0.375032			−0.187516	+	0.982262i	$0.560044\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	365.778i		−	0.412377i	−0.978512	−	0.206188i	$-0.933894\pi$
$887$		−	365.778i		−	0.412377i	0.978512	−	0.206188i	$-0.0661060\pi$
$888$		0			0
$889$	−1736.78			−1.95364
$890$		0			0
$891$	1631.48	+	692.833i	1.83107	+	0.777591i
$892$		0			0
$893$		−	68.4913i		−	0.0766980i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−535.021	−	1777.94i	−0.596456	−	1.98210i
$898$		0			0
$899$			26.3698i			0.0293324i
$900$		0			0
$901$	110.638			0.122794
$902$		0			0
$903$	−329.630	+	99.1930i	−0.365039	+	0.109848i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	270.791			0.298556			0.149278	−	0.988795i	$-0.452305\pi$
$907$	270.791			0.298556			0.149278	+	0.988795i	$0.452305\pi$
$908$		0			0
$909$	609.947	+	921.690i	0.671008	+	1.01396i
$910$		0			0
$911$			1238.74i			1.35976i	0.733324	+	0.679879i	$0.237968\pi$
$911$			1238.74i			1.35976i	−0.733324	+	0.679879i	$0.762032\pi$
$912$		0			0
$913$	−1802.61			−1.97438
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	438.898i		−	0.478624i
$918$		0			0
$919$	578.894			0.629918			0.314959	−	0.949105i	$-0.398009\pi$
$919$	578.894			0.629918			0.314959	+	0.949105i	$0.398009\pi$
$920$		0			0
$921$	−1326.91	+	399.295i	−1.44072	+	0.433545i
$922$		0			0
$923$		−	368.689i		−	0.399447i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	351.458	−	232.584i	0.379135	−	0.250900i
$928$		0			0
$929$		−	1307.58i		−	1.40752i	−0.710440	−	0.703758i	$-0.751504\pi$
$929$		−	1307.58i		−	1.40752i	0.710440	−	0.703758i	$-0.248496\pi$
$930$		0			0
$931$	98.5249			0.105827
$932$		0			0
$933$	106.379	+	353.511i	0.114019	+	0.378898i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	693.314			0.739929			0.369965	−	0.929046i	$-0.379370\pi$
$937$	693.314			0.739929			0.369965	+	0.929046i	$0.379370\pi$
$938$		0			0
$939$	−279.907	+	84.2302i	−0.298091	+	0.0897020i
$940$		0			0
$941$		−	783.592i		−	0.832723i	−0.909199	−	0.416362i	$-0.863305\pi$
$941$		−	783.592i		−	0.832723i	0.909199	−	0.416362i	$-0.136695\pi$
$942$		0			0
$943$	−1016.33			−1.07777
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1621.44i			1.71219i	0.516820	+	0.856094i	$0.327116\pi$
$947$			1621.44i			1.71219i	−0.516820	+	0.856094i	$0.672884\pi$
$948$		0			0
$949$	1128.87			1.18954
$950$		0			0
$951$	218.182	+	725.046i	0.229424	+	0.762404i
$952$		0			0
$953$		−	1105.90i		−	1.16044i	−0.814461	−	0.580218i	$-0.802967\pi$
$953$		−	1105.90i		−	1.16044i	0.814461	−	0.580218i	$-0.197033\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−756.648	+	227.692i	−0.790646	+	0.237923i
$958$		0			0
$959$		−	1424.63i		−	1.48554i
$960$		0			0
$961$	−956.200			−0.995005
$962$		0			0
$963$	521.945	+	788.710i	0.541998	+	0.819013i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1307.89			−1.35253			−0.676264	−	0.736659i	$-0.736402\pi$
$967$	−1307.89			−1.35253			−0.676264	+	0.736659i	$0.736402\pi$
$968$		0			0
$969$	31.7938	+	105.655i	0.0328109	+	0.109035i
$970$		0			0
$971$			749.988i			0.772387i	0.922418	+	0.386194i	$0.126210\pi$
$971$			749.988i			0.772387i	−0.922418	+	0.386194i	$0.873790\pi$
$972$		0			0
$973$	1486.75			1.52800
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	373.792i		−	0.382591i	−0.981532	−	0.191296i	$-0.938731\pi$
$977$		−	373.792i		−	0.382591i	0.981532	−	0.191296i	$-0.0612690\pi$
$978$		0			0
$979$	603.634			0.616583
$980$		0			0
$981$	−876.452	+	580.009i	−0.893427	+	0.591243i
$982$		0			0
$983$			27.7025i			0.0281815i	0.999901	+	0.0140908i	$0.00448538\pi$
$983$			27.7025i			0.0281815i	−0.999901	+	0.0140908i	$0.995515\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	175.901	+	584.539i	0.178218	+	0.592238i
$988$		0			0
$989$			363.318i			0.367359i
$990$		0			0
$991$	1030.34			1.03970			0.519850	−	0.854258i	$-0.325988\pi$
$991$	1030.34			1.03970			0.519850	+	0.854258i	$0.325988\pi$
$992$		0			0
$993$	−871.226	+	262.171i	−0.877367	+	0.264019i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1217.62			−1.22128			−0.610640	−	0.791909i	$-0.709088\pi$
$997$	−1217.62			−1.22128			−0.610640	+	0.791909i	$0.709088\pi$
$998$		0			0
$999$	−14.5039	+	17.4347i	−0.0145184	+	0.0174522i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.l.f.401.7		8
3.2	odd	2	inner	600.3.l.f.401.8		8
4.3	odd	2		1200.3.l.x.401.2		8
5.2	odd	4		600.3.c.d.449.5		16
5.3	odd	4		600.3.c.d.449.12		16
5.4	even	2		120.3.l.a.41.2	yes	8
12.11	even	2		1200.3.l.x.401.1		8
15.2	even	4		600.3.c.d.449.11		16
15.8	even	4		600.3.c.d.449.6		16
15.14	odd	2		120.3.l.a.41.1	✓	8
20.3	even	4		1200.3.c.m.449.5		16
20.7	even	4		1200.3.c.m.449.12		16
20.19	odd	2		240.3.l.d.161.7		8
40.19	odd	2		960.3.l.g.641.2		8
40.29	even	2		960.3.l.h.641.7		8
60.23	odd	4		1200.3.c.m.449.11		16
60.47	odd	4		1200.3.c.m.449.6		16
60.59	even	2		240.3.l.d.161.8		8
120.29	odd	2		960.3.l.h.641.8		8
120.59	even	2		960.3.l.g.641.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.1	✓	8	15.14	odd	2
120.3.l.a.41.2	yes	8	5.4	even	2
240.3.l.d.161.7		8	20.19	odd	2
240.3.l.d.161.8		8	60.59	even	2
600.3.c.d.449.5		16	5.2	odd	4
600.3.c.d.449.6		16	15.8	even	4
600.3.c.d.449.11		16	15.2	even	4
600.3.c.d.449.12		16	5.3	odd	4
600.3.l.f.401.7		8	1.1	even	1	trivial
600.3.l.f.401.8		8	3.2	odd	2	inner
960.3.l.g.641.1		8	120.59	even	2
960.3.l.g.641.2		8	40.19	odd	2
960.3.l.h.641.7		8	40.29	even	2
960.3.l.h.641.8		8	120.29	odd	2
1200.3.c.m.449.5		16	20.3	even	4
1200.3.c.m.449.6		16	60.47	odd	4
1200.3.c.m.449.11		16	60.23	odd	4
1200.3.c.m.449.12		16	20.7	even	4
1200.3.l.x.401.1		8	12.11	even	2
1200.3.l.x.401.2		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.87275 - 0.864473i) q^{3} -9.02416 q^{7} +(7.50537 - 4.96683i) q^{9} +21.8827i q^{11} -21.6599 q^{13} +12.1078i q^{17} +3.03757 q^{19} +(-25.9241 + 7.80114i) q^{21} +28.5735i q^{23} +(17.2674 - 20.7566i) q^{27} +12.0364i q^{29} +2.19085 q^{31} +(18.9170 + 62.8636i) q^{33} -0.839959 q^{37} +(-62.2233 + 18.7244i) q^{39} +35.5690i q^{41} +12.7152 q^{43} -22.5481i q^{47} +32.4354 q^{49} +(10.4668 + 34.7826i) q^{51} -9.13775i q^{53} +(8.72618 - 2.62590i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(-67.7297 + 44.8214i) q^{63} +63.0560 q^{67} +(24.7011 + 82.0846i) q^{69} +17.0218i q^{71} -52.1181 q^{73} -197.473i q^{77} -7.46224 q^{79} +(31.6612 - 74.5558i) q^{81} +82.3758i q^{83} +(10.4051 + 34.5774i) q^{87} -27.5850i q^{89} +195.462 q^{91} +(6.29376 - 1.89393i) q^{93} -114.989 q^{97} +(108.688 + 164.238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + 8 q^{13} - 8 q^{19} + 28 q^{21} - 20 q^{27} + 120 q^{31} + 112 q^{33} - 8 q^{37} - 72 q^{39} + 328 q^{43} + 64 q^{49} + 64 q^{51} - 72 q^{57} + 8 q^{61} - 88 q^{63}+ \cdots + 32 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 16 * q^7 + 20 * q^9 + 8 * q^13 - 8 * q^19 + 28 * q^21 - 20 * q^27 + 120 * q^31 + 112 * q^33 - 8 * q^37 - 72 * q^39 + 328 * q^43 + 64 * q^49 + 64 * q^51 - 72 * q^57 + 8 * q^61 - 88 * q^63 - 152 * q^67 + 100 * q^69 - 32 * q^73 + 88 * q^79 + 224 * q^81 + 152 * q^87 + 560 * q^91 + 368 * q^93 - 144 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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