LMFDB - Embedded newform 240.3.l.d.161.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [240,3,Mod(161,240)] 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("240.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	240.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$6.53952634465$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.8
Root			$3.22255 + 1.41421i$ of defining polynomial
Character	$\chi$	$=$	240.161
Dual form			240.3.l.d.161.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.87275 + 0.864473i) q^{3} +2.23607i q^{5} -9.02416 q^{7} +(7.50537 + 4.96683i) q^{9} +21.8827i q^{11} +21.6599 q^{13} +(-1.93302 + 6.42366i) q^{15} +12.1078i q^{17} -3.03757 q^{19} +(-25.9241 - 7.80114i) q^{21} -28.5735i q^{23} -5.00000 q^{25} +(17.2674 + 20.7566i) q^{27} -12.0364i q^{29} -2.19085 q^{31} +(-18.9170 + 62.8636i) q^{33} -20.1786i q^{35} +0.839959 q^{37} +(62.2233 + 18.7244i) q^{39} -35.5690i q^{41} +12.7152 q^{43} +(-11.1062 + 16.7825i) q^{45} +22.5481i q^{47} +32.4354 q^{49} +(-10.4668 + 34.7826i) q^{51} -9.13775i q^{53} -48.9312 q^{55} +(-8.72618 - 2.62590i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(-67.7297 - 44.8214i) q^{63} +48.4329i q^{65} +63.0560 q^{67} +(24.7011 - 82.0846i) q^{69} +17.0218i q^{71} +52.1181 q^{73} +(-14.3637 - 4.32237i) q^{75} -197.473i q^{77} +7.46224 q^{79} +(31.6612 + 74.5558i) q^{81} -82.3758i q^{83} -27.0738 q^{85} +(10.4051 - 34.5774i) q^{87} +27.5850i q^{89} -195.462 q^{91} +(-6.29376 - 1.89393i) q^{93} -6.79221i q^{95} +114.989 q^{97} +(-108.688 + 164.238i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} - 8 q^{13} + 8 q^{19} + 28 q^{21} - 40 q^{25} - 20 q^{27} - 120 q^{31} - 112 q^{33} + 8 q^{37} + 72 q^{39} + 328 q^{43} - 60 q^{45} + 64 q^{49} - 64 q^{51} + 40 q^{55}+ \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 - 40 * q^25 - 20 * q^27 - 120 * q^31 - 112 * q^33 + 8 * q^37 + 72 * q^39 + 328 * q^43 - 60 * q^45 + 64 * q^49 - 64 * q^51 + 40 * q^55 + 72 * q^57 + 8 * q^61 - 88 * q^63 - 152 * q^67 + 100 * q^69 + 32 * q^73 - 20 * q^75 - 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 $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		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	$-0.186580\pi$
$13$	21.6599			1.66614			0.833071	+	0.553166i	$0.186580\pi$
$14$		0			0
$15$	−1.93302	+	6.42366i	−0.128868	+	0.428244i
$16$		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	$-0.525472\pi$
$19$	−3.03757			−0.159872			−0.0799361	+	0.996800i	$0.525472\pi$
$20$		0			0
$21$	−25.9241	−	7.80114i	−1.23448	−	0.371483i
$22$		0			0
$23$		−	28.5735i		−	1.24233i	−0.783681	−	0.621164i	$-0.786660\pi$
$23$		−	28.5735i		−	1.24233i	0.783681	−	0.621164i	$-0.213340\pi$
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	17.2674	+	20.7566i	0.639532	+	0.768765i
$28$		0			0
$29$		−	12.0364i		−	0.415047i	−0.978230	−	0.207523i	$-0.933460\pi$
$29$		−	12.0364i		−	0.415047i	0.978230	−	0.207523i	$-0.0665403\pi$
$30$		0			0
$31$	−2.19085			−0.0706725			−0.0353363	−	0.999375i	$-0.511250\pi$
$31$	−2.19085			−0.0706725			−0.0353363	+	0.999375i	$0.511250\pi$
$32$		0			0
$33$	−18.9170	+	62.8636i	−0.573243	+	1.90496i
$34$		0			0
$35$		−	20.1786i		−	0.576532i
$36$		0			0
$37$	0.839959			0.0227016			0.0113508	−	0.999936i	$-0.496387\pi$
$37$	0.839959			0.0227016			0.0113508	+	0.999936i	$0.496387\pi$
$38$		0			0
$39$	62.2233	+	18.7244i	1.59547	+	0.480112i
$40$		0			0
$41$		−	35.5690i		−	0.867537i	−0.901024	−	0.433769i	$-0.857184\pi$
$41$		−	35.5690i		−	0.867537i	0.901024	−	0.433769i	$-0.142816\pi$
$42$		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
$45$	−11.1062	+	16.7825i	−0.246804	+	0.372945i
$46$		0			0
$47$			22.5481i			0.479746i	0.970804	+	0.239873i	$0.0771058\pi$
$47$			22.5481i			0.479746i	−0.970804	+	0.239873i	$0.922894\pi$
$48$		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
$55$	−48.9312			−0.889659
$56$		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
$65$			48.4329i			0.745122i
$66$		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	$-0.383809\pi$
$73$	52.1181			0.713947			0.356973	+	0.934115i	$0.383809\pi$
$74$		0			0
$75$	−14.3637	−	4.32237i	−0.191517	−	0.0576316i
$76$		0			0
$77$		−	197.473i		−	2.56459i
$78$		0			0
$79$	7.46224			0.0944588			0.0472294	−	0.998884i	$-0.484961\pi$
$79$	7.46224			0.0944588			0.0472294	+	0.998884i	$0.484961\pi$
$80$		0			0
$81$	31.6612	+	74.5558i	0.390879	+	0.920442i
$82$		0			0
$83$		−	82.3758i		−	0.992480i	−0.868185	−	0.496240i	$-0.834714\pi$
$83$		−	82.3758i		−	0.992480i	0.868185	−	0.496240i	$-0.165286\pi$
$84$		0			0
$85$	−27.0738			−0.318515
$86$		0			0
$87$	10.4051	−	34.5774i	0.119599	−	0.397442i
$88$		0			0
$89$			27.5850i			0.309944i	0.987919	+	0.154972i	$0.0495287\pi$
$89$			27.5850i			0.309944i	−0.987919	+	0.154972i	$0.950471\pi$
$90$		0			0
$91$	−195.462			−2.14793
$92$		0			0
$93$	−6.29376	−	1.89393i	−0.0676748	−	0.0203648i
$94$		0			0
$95$		−	6.79221i		−	0.0714970i
$96$		0			0
$97$	114.989			1.18545			0.592727	−	0.805404i	$-0.298051\pi$
$97$	114.989			1.18545			0.592727	+	0.805404i	$0.298051\pi$
$98$		0			0
$99$	−108.688	+	164.238i	−1.09786	+	1.65897i
$100$		0			0
$101$		−	122.804i		−	1.21588i	−0.793982	−	0.607941i	$-0.791996\pi$
$101$		−	122.804i		−	1.21588i	0.793982	−	0.607941i	$-0.208004\pi$
$102$		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
$105$	17.4439	−	57.9681i	0.166132	−	0.552077i
$106$		0			0
$107$		−	105.086i		−	0.982113i	−0.871128	−	0.491056i	$-0.836611\pi$
$107$		−	105.086i		−	0.982113i	0.871128	−	0.491056i	$-0.163389\pi$
$108$		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
$115$	63.8924			0.555586
$116$		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
$125$		−	11.1803i		−	0.0894427i
$126$		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
$135$	−46.4133	+	38.6110i	−0.343802	+	0.286007i
$136$		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	$-0.298089\pi$
$139$	164.752			1.18526			0.592632	+	0.805473i	$0.298089\pi$
$140$		0			0
$141$	−19.4922	+	64.7749i	−0.138243	+	0.459397i
$142$		0			0
$143$			473.977i			3.31452i
$144$		0			0
$145$	26.9141			0.185615
$146$		0			0
$147$	93.1788	+	28.0396i	0.633869	+	0.190745i
$148$		0			0
$149$			262.935i			1.76467i	0.470626	+	0.882333i	$0.344028\pi$
$149$			262.935i			1.76467i	−0.470626	+	0.882333i	$0.655972\pi$
$150$		0			0
$151$	−15.8171			−0.104749			−0.0523745	−	0.998628i	$-0.516679\pi$
$151$	−15.8171			−0.104749			−0.0523745	+	0.998628i	$0.516679\pi$
$152$		0			0
$153$	−60.1372	+	90.8733i	−0.393054	+	0.593943i
$154$		0			0
$155$		−	4.89889i		−	0.0316057i
$156$		0			0
$157$	−6.11941			−0.0389771			−0.0194886	−	0.999810i	$-0.506204\pi$
$157$	−6.11941			−0.0389771			−0.0194886	+	0.999810i	$0.506204\pi$
$158$		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
$165$	−140.567	−	42.2998i	−0.851922	−	0.256362i
$166$		0			0
$167$		−	61.2668i		−	0.366867i	−0.983032	−	0.183434i	$-0.941279\pi$
$167$		−	61.2668i		−	0.366867i	0.983032	−	0.183434i	$-0.0587212\pi$
$168$		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
$175$	45.1208			0.257833
$176$		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
$185$			1.87821i			0.0101525i
$186$		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	$-0.685656\pi$
$193$	−212.587			−1.10149			−0.550744	+	0.834674i	$0.685656\pi$
$194$		0			0
$195$	−41.8690	+	139.136i	−0.214713	+	0.713516i
$196$		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	$-0.676919\pi$
$199$	−209.996			−1.05526			−0.527629	+	0.849475i	$0.676919\pi$
$200$		0			0
$201$	181.144	+	54.5102i	0.901214	+	0.271195i
$202$		0			0
$203$			108.618i			0.535064i
$204$		0			0
$205$	79.5348			0.387974
$206$		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	$-0.637288\pi$
$211$	−176.419			−0.836110			−0.418055	+	0.908422i	$0.637288\pi$
$212$		0			0
$213$	−14.7149	+	48.8993i	−0.0690839	+	0.229574i
$214$		0			0
$215$			28.4320i			0.132242i
$216$		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
$225$	−37.5269	−	24.8341i	−0.166786	−	0.110374i
$226$		0			0
$227$			187.624i			0.826537i	0.910609	+	0.413268i	$0.135613\pi$
$227$			187.624i			0.826537i	−0.910609	+	0.413268i	$0.864387\pi$
$228$		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
$235$	−50.4190			−0.214549
$236$		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
$245$			72.5278i			0.296032i
$246$		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
$255$	−77.7762	−	23.4046i	−0.305005	−	0.0917826i
$256$		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	$-0.883176\pi$
$263$		−	188.745i		−	0.717660i	0.933403	−	0.358830i	$-0.116824\pi$
$264$		0			0
$265$	20.4326			0.0771043
$266$		0			0
$267$	−23.8465	+	79.2447i	−0.0893127	+	0.296797i
$268$		0			0
$269$			333.372i			1.23930i	0.784878	+	0.619651i	$0.212726\pi$
$269$			333.372i			1.23930i	−0.784878	+	0.619651i	$0.787274\pi$
$270$		0			0
$271$	−262.047			−0.966964			−0.483482	−	0.875354i	$-0.660628\pi$
$271$	−262.047			−0.966964			−0.483482	+	0.875354i	$0.660628\pi$
$272$		0			0
$273$	−561.513	−	168.972i	−2.05682	−	0.618944i
$274$		0			0
$275$		−	109.414i		−	0.397868i
$276$		0			0
$277$	−400.100			−1.44440			−0.722202	−	0.691682i	$-0.756870\pi$
$277$	−400.100			−1.44440			−0.722202	+	0.691682i	$0.756870\pi$
$278$		0			0
$279$	−16.4431	−	10.8816i	−0.0589360	−	0.0390020i
$280$		0			0
$281$		−	350.698i		−	1.24804i	−0.781410	−	0.624018i	$-0.785499\pi$
$281$		−	350.698i		−	1.24804i	0.781410	−	0.624018i	$-0.214501\pi$
$282$		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
$285$	5.87169	−	19.5123i	0.0206024	−	0.0684643i
$286$		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
$295$	179.883			0.609771
$296$		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
$305$		−	129.427i		−	0.424351i
$306$		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	$-0.450254\pi$
$313$	97.4353			0.311295			0.155647	+	0.987813i	$0.450254\pi$
$314$		0			0
$315$	100.224	−	151.448i	0.318171	−	0.480788i
$316$		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
$325$	−108.299			−0.333229
$326$		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	$-0.348525\pi$
$331$	303.273			0.916231			0.458116	+	0.888893i	$0.348525\pi$
$332$		0			0
$333$	6.30420	+	4.17193i	0.0189315	+	0.0125283i
$334$		0			0
$335$			140.998i			0.420888i
$336$		0			0
$337$	−352.738			−1.04670			−0.523350	−	0.852118i	$-0.675318\pi$
$337$	−352.738			−1.04670			−0.523350	+	0.852118i	$0.675318\pi$
$338$		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
$345$	183.547	+	55.2332i	0.532019	+	0.160096i
$346$		0			0
$347$		−	280.382i		−	0.808018i	−0.914755	−	0.404009i	$-0.867616\pi$
$347$		−	280.382i		−	0.808018i	0.914755	−	0.404009i	$-0.132384\pi$
$348$		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
$355$	−38.0618			−0.107216
$356$		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
$365$			116.540i			0.319287i
$366$		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	$-0.336122\pi$
$373$	367.327			0.984790			0.492395	+	0.870372i	$0.336122\pi$
$374$		0			0
$375$	9.66511	−	32.1183i	0.0257736	−	0.0856488i
$376$		0			0
$377$		−	260.706i		−	0.691527i
$378$		0			0
$379$	−611.014			−1.61217			−0.806087	−	0.591798i	$-0.798418\pi$
$379$	−611.014			−1.61217			−0.806087	+	0.591798i	$0.798418\pi$
$380$		0			0
$381$	552.887	+	166.376i	1.45115	+	0.436682i
$382$		0			0
$383$			13.6994i			0.0357687i	0.999840	+	0.0178844i	$0.00569307\pi$
$383$			13.6994i			0.0357687i	−0.999840	+	0.0178844i	$0.994307\pi$
$384$		0			0
$385$	441.563			1.14692
$386$		0			0
$387$	95.4322	+	63.1542i	0.246595	+	0.163189i
$388$		0			0
$389$		−	379.601i		−	0.975838i	−0.872889	−	0.487919i	$-0.837756\pi$
$389$		−	379.601i		−	0.975838i	0.872889	−	0.487919i	$-0.162244\pi$
$390$		0			0
$391$	345.962			0.884812
$392$		0			0
$393$	−42.0445	+	139.719i	−0.106983	+	0.355519i
$394$		0			0
$395$			16.6861i			0.0422432i
$396$		0			0
$397$	450.560			1.13491			0.567456	−	0.823404i	$-0.307928\pi$
$397$	450.560			1.13491			0.567456	+	0.823404i	$0.307928\pi$
$398$		0			0
$399$	78.7464	+	23.6965i	0.197359	+	0.0593898i
$400$		0			0
$401$		−	503.683i		−	1.25607i	−0.778186	−	0.628034i	$-0.783860\pi$
$401$		−	503.683i		−	1.25607i	0.778186	−	0.628034i	$-0.216140\pi$
$402$		0			0
$403$	−47.4535			−0.117751
$404$		0			0
$405$	−166.712	+	70.7966i	−0.411634	+	0.174806i
$406$		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
$415$	184.198			0.443850
$416$		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
$425$		−	60.5388i		−	0.142444i
$426$		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	$-0.646062\pi$
$433$	−383.579			−0.885864			−0.442932	+	0.896555i	$0.646062\pi$
$434$		0			0
$435$	77.3175	+	23.2665i	0.177741	+	0.0534863i
$436$		0			0
$437$			86.7941i			0.198614i
$438$		0			0
$439$	−523.900			−1.19340			−0.596698	−	0.802466i	$-0.703521\pi$
$439$	−523.900			−1.19340			−0.596698	+	0.802466i	$0.703521\pi$
$440$		0			0
$441$	243.440	+	161.101i	0.552018	+	0.365309i
$442$		0			0
$443$		−	391.277i		−	0.883243i	−0.897201	−	0.441622i	$-0.854403\pi$
$443$		−	391.277i		−	0.883243i	0.897201	−	0.441622i	$-0.145597\pi$
$444$		0			0
$445$	−61.6819			−0.138611
$446$		0			0
$447$	−227.301	+	755.347i	−0.508502	+	1.68981i
$448$		0			0
$449$			375.014i			0.835220i	0.908626	+	0.417610i	$0.137132\pi$
$449$			375.014i			0.835220i	−0.908626	+	0.417610i	$0.862868\pi$
$450$		0			0
$451$	778.347			1.72582
$452$		0			0
$453$	−45.4386	−	13.6735i	−0.100306	−	0.0301843i
$454$		0			0
$455$		−	437.066i		−	0.960585i
$456$		0			0
$457$	−734.032			−1.60620			−0.803098	−	0.595847i	$-0.796817\pi$
$457$	−734.032			−1.60620			−0.803098	+	0.595847i	$0.796817\pi$
$458$		0			0
$459$	−251.317	+	209.069i	−0.547531	+	0.455488i
$460$		0			0
$461$		−	775.239i		−	1.68165i	−0.541310	−	0.840823i	$-0.682072\pi$
$461$		−	775.239i		−	1.68165i	0.541310	−	0.840823i	$-0.317928\pi$
$462$		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
$465$	4.23496	−	14.0733i	0.00910743	−	0.0302651i
$466$		0			0
$467$			160.785i			0.344294i	0.985071	+	0.172147i	$0.0550704\pi$
$467$			160.785i			0.344294i	−0.985071	+	0.172147i	$0.944930\pi$
$468$		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
$475$	15.1879			0.0319744
$476$		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
$485$			257.123i			0.530151i
$486$		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
$495$	−367.247	−	243.033i	−0.741914	−	0.490976i
$496$		0			0
$497$		−	153.607i		−	0.309069i
$498$		0			0
$499$	600.897			1.20420			0.602101	−	0.798420i	$-0.294330\pi$
$499$	600.897			1.20420			0.602101	+	0.798420i	$0.294330\pi$
$500$		0			0
$501$	52.9635	−	176.004i	0.105716	−	0.351306i
$502$		0			0
$503$		−	688.332i		−	1.36845i	−0.729270	−	0.684226i	$-0.760140\pi$
$503$		−	688.332i		−	1.36845i	0.729270	−	0.684226i	$-0.239860\pi$
$504$		0			0
$505$	274.598			0.543759
$506$		0			0
$507$	862.254	+	259.471i	1.70070	+	0.511777i
$508$		0			0
$509$		−	823.791i		−	1.61845i	−0.587498	−	0.809225i	$-0.699887\pi$
$509$		−	823.791i		−	1.61845i	0.587498	−	0.809225i	$-0.300113\pi$
$510$		0			0
$511$	−470.322			−0.920395
$512$		0			0
$513$	−52.4508	−	63.0498i	−0.102243	−	0.122904i
$514$		0			0
$515$			104.710i			0.203320i
$516$		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	$0.376479\pi$
$521$			964.525i			1.85130i	−0.378386	+	0.925648i	$0.623521\pi$
$522$		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
$525$	129.621	+	39.0057i	0.246897	+	0.0742966i
$526$		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
$535$	234.980			0.439214
$536$		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
$545$		−	261.120i		−	0.479120i
$546$		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
$555$	−1.62366	+	5.39561i	−0.00292551	+	0.00972182i
$556$		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	$-0.886195\pi$
$563$		−	394.056i		−	0.699922i	0.936764	−	0.349961i	$-0.113805\pi$
$564$		0			0
$565$	24.1755			0.0427885
$566$		0			0
$567$	−285.716	−	672.803i	−0.503908	−	1.18660i
$568$		0			0
$569$		−	537.452i		−	0.944556i	−0.881450	−	0.472278i	$-0.843432\pi$
$569$		−	537.452i		−	0.944556i	0.881450	−	0.472278i	$-0.156568\pi$
$570$		0			0
$571$	−710.555			−1.24440			−0.622202	−	0.782856i	$-0.713762\pi$
$571$	−710.555			−1.24440			−0.622202	+	0.782856i	$0.713762\pi$
$572$		0			0
$573$	−65.0521	+	216.176i	−0.113529	+	0.377271i
$574$		0			0
$575$			142.868i			0.248466i
$576$		0			0
$577$	452.846			0.784828			0.392414	−	0.919789i	$-0.371640\pi$
$577$	452.846			0.784828			0.392414	+	0.919789i	$0.371640\pi$
$578$		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
$585$	−240.558	+	363.507i	−0.411210	+	0.621379i
$586$		0			0
$587$		−	1103.34i		−	1.87963i	−0.341690	−	0.939813i	$-0.610999\pi$
$587$		−	1103.34i		−	1.87963i	0.341690	−	0.939813i	$-0.389001\pi$
$588$		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
$595$	244.318			0.410619
$596$		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
$605$		−	800.185i		−	1.32262i
$606$		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	$-0.376039\pi$
$613$	465.472			0.759334			0.379667	+	0.925123i	$0.376039\pi$
$614$		0			0
$615$	228.483	+	68.7557i	0.371518	+	0.111798i
$616$		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	$-0.444877\pi$
$619$	213.318			0.344617			0.172309	+	0.985043i	$0.444877\pi$
$620$		0			0
$621$	593.091	−	493.389i	0.955058	−	0.794508i
$622$		0			0
$623$		−	248.931i		−	0.399569i
$624$		0			0
$625$	25.0000			0.0400000
$626$		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	$-0.652710\pi$
$631$	−582.489			−0.923121			−0.461560	+	0.887109i	$0.652710\pi$
$632$		0			0
$633$	−506.808	−	152.510i	−0.800644	−	0.240931i
$634$		0			0
$635$			430.352i			0.677720i
$636$		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	$0.0802088\pi$
$641$			319.635i			0.498650i	−0.968420	+	0.249325i	$0.919791\pi$
$642$		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
$645$	−24.5787	+	81.6781i	−0.0381066	+	0.126633i
$646$		0			0
$647$			507.599i			0.784543i	0.919850	+	0.392271i	$0.128311\pi$
$647$			507.599i			0.784543i	−0.919850	+	0.392271i	$0.871689\pi$
$648$		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
$655$	−108.753			−0.166036
$656$		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
$665$			61.2940i			0.0921715i
$666$		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	$-0.573150\pi$
$673$	−306.607			−0.455582			−0.227791	+	0.973710i	$0.573150\pi$
$674$		0			0
$675$	−86.3368	−	103.783i	−0.127906	−	0.153753i
$676$		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	$0.202684\pi$
$683$			812.204i			1.18917i	−0.804032	+	0.594586i	$0.797316\pi$
$684$		0			0
$685$	−353.005			−0.515335
$686$		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	$-0.489418\pi$
$691$	45.9358			0.0664773			0.0332387	+	0.999447i	$0.489418\pi$
$692$		0			0
$693$	980.815	−	1482.11i	1.41532	−	2.13869i
$694$		0			0
$695$			368.396i			0.530066i
$696$		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	$-0.840208\pi$
$701$		−	674.615i		−	0.962360i	0.876622	−	0.481180i	$-0.159792\pi$
$702$		0			0
$703$	−2.55143			−0.00362935
$704$		0			0
$705$	−144.841	−	43.5859i	−0.205448	−	0.0618240i
$706$		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
$715$	−1059.84			−1.48230
$716$		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
$725$			60.1818i			0.0830094i
$726$		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	$-0.527775\pi$
$733$	−127.756			−0.174292			−0.0871460	+	0.996196i	$0.527775\pi$
$734$		0			0
$735$	−62.6983	+	208.354i	−0.0853039	+	0.283475i
$736$		0			0
$737$			1379.84i			1.87223i
$738$		0			0
$739$	1401.59			1.89660			0.948302	−	0.317368i	$-0.102799\pi$
$739$	1401.59			1.89660			0.948302	+	0.317368i	$0.102799\pi$
$740$		0			0
$741$	−189.008	−	56.8766i	−0.255071	−	0.0767565i
$742$		0			0
$743$			11.6734i			0.0157112i	0.999969	+	0.00785561i	$0.00250055\pi$
$743$			11.6734i			0.0157112i	−0.999969	+	0.00785561i	$0.997499\pi$
$744$		0			0
$745$	−587.941			−0.789183
$746$		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	$-0.581504\pi$
$751$	−380.403			−0.506529			−0.253264	+	0.967397i	$0.581504\pi$
$752$		0			0
$753$	177.982	−	591.456i	0.236364	−	0.785466i
$754$		0			0
$755$		−	35.3681i		−	0.0468452i
$756$		0			0
$757$	63.6621			0.0840979			0.0420490	−	0.999116i	$-0.486611\pi$
$757$	63.6621			0.0840979			0.0420490	+	0.999116i	$0.486611\pi$
$758$		0			0
$759$	1796.23	+	540.526i	2.36658	+	0.712156i
$760$		0			0
$761$			377.891i			0.496571i	0.968687	+	0.248286i	$0.0798671\pi$
$761$			377.891i			0.496571i	−0.968687	+	0.248286i	$0.920133\pi$
$762$		0			0
$763$	1053.81			1.38114
$764$		0			0
$765$	−203.199	−	134.471i	−0.265619	−	0.175779i
$766$		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
$775$	10.9542			0.0141345
$776$		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
$785$		−	13.6834i		−	0.0174311i
$786$		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
$795$	58.6978	+	17.6635i	0.0738337	+	0.0222182i
$796$		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
$805$	−576.575			−0.716242
$806$		0			0
$807$	−288.191	+	957.694i	−0.357114	+	1.18673i
$808$		0			0
$809$		−	1334.53i		−	1.64961i	−0.565419	−	0.824804i	$-0.691286\pi$
$809$		−	1334.53i		−	1.64961i	0.565419	−	0.824804i	$-0.308714\pi$
$810$		0			0
$811$	−610.590			−0.752885			−0.376443	−	0.926440i	$-0.622853\pi$
$811$	−610.590			−0.752885			−0.376443	+	0.926440i	$0.622853\pi$
$812$		0			0
$813$	−752.796	−	226.533i	−0.925949	−	0.278638i
$814$		0			0
$815$			381.123i			0.467636i
$816$		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	$0.179475\pi$
$821$			877.538i			1.06887i	−0.845211	+	0.534433i	$0.820525\pi$
$822$		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
$825$	94.5851	−	314.318i	0.114649	−	0.380991i
$826$		0			0
$827$			969.434i			1.17223i	0.810228	+	0.586115i	$0.199343\pi$
$827$			969.434i			1.17223i	−0.810228	+	0.586115i	$0.800657\pi$
$828$		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
$835$	136.997			0.164068
$836$		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
$845$			671.154i			0.794265i
$846$		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	$-0.464343\pi$
$853$	190.704			0.223569			0.111784	+	0.993732i	$0.464343\pi$
$854$		0			0
$855$	33.7358	−	50.9781i	0.0394570	−	0.0596235i
$856$		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	$-0.245002\pi$
$859$	1233.74			1.43625			0.718123	+	0.695916i	$0.245002\pi$
$860$		0			0
$861$	−277.479	+	922.096i	−0.322275	+	1.07096i
$862$		0			0
$863$			463.404i			0.536968i	0.963284	+	0.268484i	$0.0865227\pi$
$863$			463.404i			0.536968i	−0.963284	+	0.268484i	$0.913477\pi$
$864$		0			0
$865$	587.076			0.678700
$866$		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
$875$			100.893i			0.115306i
$876$		0			0
$877$	777.195			0.886197			0.443099	−	0.896473i	$-0.353879\pi$
$877$	777.195			0.886197			0.443099	+	0.896473i	$0.353879\pi$
$878$		0			0
$879$	−40.9829	+	136.191i	−0.0466245	+	0.154939i
$880$		0			0
$881$			1609.50i			1.82690i	0.406951	+	0.913450i	$0.366592\pi$
$881$			1609.50i			1.82690i	−0.406951	+	0.913450i	$0.633408\pi$
$882$		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
$885$	516.757	+	155.504i	0.583907	+	0.175710i
$886$		0			0
$887$			365.778i			0.412377i	0.978512	+	0.206188i	$0.0661060\pi$
$887$			365.778i			0.412377i	−0.978512	+	0.206188i	$0.933894\pi$
$888$		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
$895$	15.4010			0.0172078
$896$		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
$905$			488.662i			0.539958i
$906$		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
$915$	111.886	−	371.812i	0.122280	−	0.406352i
$916$		0			0
$917$		−	438.898i		−	0.478624i
$918$		0			0
$919$	−578.894			−0.629918			−0.314959	−	0.949105i	$-0.601991\pi$
$919$	−578.894			−0.629918			−0.314959	+	0.949105i	$0.601991\pi$
$920$		0			0
$921$	−1326.91	−	399.295i	−1.44072	−	0.433545i
$922$		0			0
$923$			368.689i			0.399447i
$924$		0			0
$925$	−4.19979			−0.00454032
$926$		0			0
$927$	351.458	+	232.584i	0.379135	+	0.250900i
$928$		0			0
$929$			1307.58i			1.40752i	0.710440	+	0.703758i	$0.248496\pi$
$929$			1307.58i			1.40752i	−0.710440	+	0.703758i	$0.751504\pi$
$930$		0			0
$931$	−98.5249			−0.105827
$932$		0			0
$933$	−106.379	+	353.511i	−0.114019	+	0.378898i
$934$		0			0
$935$		−	592.448i		−	0.633634i
$936$		0			0
$937$	−693.314			−0.739929			−0.369965	−	0.929046i	$-0.620630\pi$
$937$	−693.314			−0.739929			−0.369965	+	0.929046i	$0.620630\pi$
$938$		0			0
$939$	279.907	+	84.2302i	0.298091	+	0.0897020i
$940$		0			0
$941$			783.592i			0.832723i	0.909199	+	0.416362i	$0.136695\pi$
$941$			783.592i			0.832723i	−0.909199	+	0.416362i	$0.863305\pi$
$942$		0			0
$943$	−1016.33			−1.07777
$944$		0			0
$945$	418.841	−	348.432i	0.443218	−	0.368711i
$946$		0			0
$947$		−	1621.44i		−	1.71219i	−0.516820	−	0.856094i	$-0.672884\pi$
$947$		−	1621.44i		−	1.71219i	0.516820	−	0.856094i	$-0.327116\pi$
$948$		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
$955$	−168.265			−0.176194
$956$		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
$965$		−	475.359i		−	0.492600i
$966$		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
$975$	−311.117	−	93.6218i	−0.319094	−	0.0960224i
$976$		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	$-0.995515\pi$
$983$		−	27.7025i		−	0.0281815i	0.999901	−	0.0140908i	$-0.00448538\pi$
$984$		0			0
$985$	−426.285			−0.432777
$986$		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	$-0.674012\pi$
$991$	−1030.34			−1.03970			−0.519850	+	0.854258i	$0.674012\pi$
$992$		0			0
$993$	871.226	+	262.171i	0.877367	+	0.264019i
$994$		0			0
$995$		−	469.566i		−	0.471926i
$996$		0			0
$997$	1217.62			1.22128			0.610640	−	0.791909i	$-0.290912\pi$
$997$	1217.62			1.22128			0.610640	+	0.791909i	$0.290912\pi$
$998$		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	240.3.l.d.161.8		8
3.2	odd	2	inner	240.3.l.d.161.7		8
4.3	odd	2		120.3.l.a.41.1	✓	8
5.2	odd	4		1200.3.c.m.449.11		16
5.3	odd	4		1200.3.c.m.449.6		16
5.4	even	2		1200.3.l.x.401.1		8
8.3	odd	2		960.3.l.h.641.8		8
8.5	even	2		960.3.l.g.641.1		8
12.11	even	2		120.3.l.a.41.2	yes	8
15.2	even	4		1200.3.c.m.449.5		16
15.8	even	4		1200.3.c.m.449.12		16
15.14	odd	2		1200.3.l.x.401.2		8
20.3	even	4		600.3.c.d.449.11		16
20.7	even	4		600.3.c.d.449.6		16
20.19	odd	2		600.3.l.f.401.8		8
24.5	odd	2		960.3.l.g.641.2		8
24.11	even	2		960.3.l.h.641.7		8
60.23	odd	4		600.3.c.d.449.5		16
60.47	odd	4		600.3.c.d.449.12		16
60.59	even	2		600.3.l.f.401.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.l.a.41.1	✓	8	4.3	odd	2
120.3.l.a.41.2	yes	8	12.11	even	2
240.3.l.d.161.7		8	3.2	odd	2	inner
240.3.l.d.161.8		8	1.1	even	1	trivial
600.3.c.d.449.5		16	60.23	odd	4
600.3.c.d.449.6		16	20.7	even	4
600.3.c.d.449.11		16	20.3	even	4
600.3.c.d.449.12		16	60.47	odd	4
600.3.l.f.401.7		8	60.59	even	2
600.3.l.f.401.8		8	20.19	odd	2
960.3.l.g.641.1		8	8.5	even	2
960.3.l.g.641.2		8	24.5	odd	2
960.3.l.h.641.7		8	24.11	even	2
960.3.l.h.641.8		8	8.3	odd	2
1200.3.c.m.449.5		16	15.2	even	4
1200.3.c.m.449.6		16	5.3	odd	4
1200.3.c.m.449.11		16	5.2	odd	4
1200.3.c.m.449.12		16	15.8	even	4
1200.3.l.x.401.1		8	5.4	even	2
1200.3.l.x.401.2		8	15.14	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:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	240.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.87275 + 0.864473i) q^{3} +2.23607i q^{5} -9.02416 q^{7} +(7.50537 + 4.96683i) q^{9} +21.8827i q^{11} +21.6599 q^{13} +(-1.93302 + 6.42366i) q^{15} +12.1078i q^{17} -3.03757 q^{19} +(-25.9241 - 7.80114i) q^{21} -28.5735i q^{23} -5.00000 q^{25} +(17.2674 + 20.7566i) q^{27} -12.0364i q^{29} -2.19085 q^{31} +(-18.9170 + 62.8636i) q^{33} -20.1786i q^{35} +0.839959 q^{37} +(62.2233 + 18.7244i) q^{39} -35.5690i q^{41} +12.7152 q^{43} +(-11.1062 + 16.7825i) q^{45} +22.5481i q^{47} +32.4354 q^{49} +(-10.4668 + 34.7826i) q^{51} -9.13775i q^{53} -48.9312 q^{55} +(-8.72618 - 2.62590i) q^{57} -80.4459i q^{59} -57.8816 q^{61} +(-67.7297 - 44.8214i) q^{63} +48.4329i q^{65} +63.0560 q^{67} +(24.7011 - 82.0846i) q^{69} +17.0218i q^{71} +52.1181 q^{73} +(-14.3637 - 4.32237i) q^{75} -197.473i q^{77} +7.46224 q^{79} +(31.6612 + 74.5558i) q^{81} -82.3758i q^{83} -27.0738 q^{85} +(10.4051 - 34.5774i) q^{87} +27.5850i q^{89} -195.462 q^{91} +(-6.29376 - 1.89393i) q^{93} -6.79221i q^{95} +114.989 q^{97} +(-108.688 + 164.238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} - 8 q^{13} + 8 q^{19} + 28 q^{21} - 40 q^{25} - 20 q^{27} - 120 q^{31} - 112 q^{33} + 8 q^{37} + 72 q^{39} + 328 q^{43} - 60 q^{45} + 64 q^{49} - 64 q^{51} + 40 q^{55}+ \cdots - 32 q^{99}+O(q^{100}) \) 8 * q + 4 * q^3 - 16 * q^7 + 20 * q^9 - 8 * q^13 + 8 * q^19 + 28 * q^21 - 40 * q^25 - 20 * q^27 - 120 * q^31 - 112 * q^33 + 8 * q^37 + 72 * q^39 + 328 * q^43 - 60 * q^45 + 64 * q^49 - 64 * q^51 + 40 * q^55 + 72 * q^57 + 8 * q^61 - 88 * q^63 - 152 * q^67 + 100 * q^69 + 32 * q^73 - 20 * q^75 - 88 * q^79 + 224 * q^81 + 152 * q^87 - 560 * q^91 - 368 * q^93 + 144 * q^97 - 32 * q^99

Introduction

L-functions

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