LMFDB - Embedded newform 405.3.d.b.404.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] 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("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.20
Character	$\chi$	$=$	405.404
Dual form			405.3.d.b.404.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.28392 q^{2} +1.21631 q^{4} +(1.08806 + 4.88018i) q^{5} +3.44650i q^{7} -6.35774 q^{8} +(2.48505 + 11.1459i) q^{10} +4.50165i q^{11} +11.8431i q^{13} +7.87153i q^{14} -19.3858 q^{16} -23.3048 q^{17} +11.0739 q^{19} +(1.32342 + 5.93579i) q^{20} +10.2814i q^{22} +29.8365 q^{23} +(-22.6322 + 10.6199i) q^{25} +27.0487i q^{26} +4.19199i q^{28} +35.6319i q^{29} +30.3031 q^{31} -18.8448 q^{32} -53.2264 q^{34} +(-16.8195 + 3.75000i) q^{35} +5.11748i q^{37} +25.2919 q^{38} +(-6.91761 - 31.0269i) q^{40} +22.6476i q^{41} -67.9633i q^{43} +5.47538i q^{44} +68.1442 q^{46} -46.2467 q^{47} +37.1217 q^{49} +(-51.6903 + 24.2550i) q^{50} +14.4048i q^{52} +68.0017 q^{53} +(-21.9688 + 4.89807i) q^{55} -21.9119i q^{56} +81.3806i q^{58} +34.5832i q^{59} +17.3633 q^{61} +69.2099 q^{62} +34.5033 q^{64} +(-57.7963 + 12.8860i) q^{65} -98.2875i q^{67} -28.3458 q^{68} +(-38.4145 + 8.56471i) q^{70} +134.669i q^{71} -134.962i q^{73} +11.6879i q^{74} +13.4693 q^{76} -15.5149 q^{77} +49.3738 q^{79} +(-21.0930 - 94.6062i) q^{80} +51.7254i q^{82} +28.6837 q^{83} +(-25.3571 - 113.732i) q^{85} -155.223i q^{86} -28.6203i q^{88} -65.9204i q^{89} -40.8171 q^{91} +36.2903 q^{92} -105.624 q^{94} +(12.0491 + 54.0426i) q^{95} -71.7451i q^{97} +84.7830 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.28392			1.14196			0.570981	−	0.820963i	$-0.306563\pi$
$2$	2.28392			1.14196			0.570981	+	0.820963i	$0.306563\pi$
$3$		0			0
$3$		0			0
$4$	1.21631			0.304077
$5$	1.08806	+	4.88018i	0.217612	+	0.976035i
$5$	1.08806	+	4.88018i	0.217612	+	0.976035i
$6$		0			0
$7$			3.44650i			0.492357i	0.969225	+	0.246178i	$0.0791749\pi$
$7$			3.44650i			0.492357i	−0.969225	+	0.246178i	$0.920825\pi$
$8$	−6.35774			−0.794718
$9$		0			0
$10$	2.48505	+	11.1459i	0.248505	+	1.11459i
$11$			4.50165i			0.409241i	0.978841	+	0.204620i	$0.0655960\pi$
$11$			4.50165i			0.409241i	−0.978841	+	0.204620i	$0.934404\pi$
$12$		0			0
$13$			11.8431i			0.911005i	0.890234	+	0.455503i	$0.150540\pi$
$13$			11.8431i			0.911005i	−0.890234	+	0.455503i	$0.849460\pi$
$14$			7.87153i			0.562252i
$15$		0			0
$16$	−19.3858			−1.21161
$17$	−23.3048			−1.37087			−0.685436	−	0.728133i	$-0.740388\pi$
$17$	−23.3048			−1.37087			−0.685436	+	0.728133i	$0.740388\pi$
$18$		0			0
$19$	11.0739			0.582837			0.291419	−	0.956596i	$-0.405873\pi$
$19$	11.0739			0.582837			0.291419	+	0.956596i	$0.405873\pi$
$20$	1.32342	+	5.93579i	0.0661708	+	0.296789i
$21$		0			0
$22$			10.2814i			0.467337i
$23$	29.8365			1.29724			0.648619	−	0.761113i	$-0.275347\pi$
$23$	29.8365			1.29724			0.648619	+	0.761113i	$0.275347\pi$
$24$		0			0
$25$	−22.6322	+	10.6199i	−0.905290	+	0.424794i
$26$			27.0487i			1.04033i
$27$		0			0
$28$			4.19199i			0.149714i
$29$			35.6319i			1.22869i	0.789039	+	0.614343i	$0.210579\pi$
$29$			35.6319i			1.22869i	−0.789039	+	0.614343i	$0.789421\pi$
$30$		0			0
$31$	30.3031			0.977519			0.488759	−	0.872419i	$-0.337450\pi$
$31$	30.3031			0.977519			0.488759	+	0.872419i	$0.337450\pi$
$32$	−18.8448			−0.588899
$33$		0			0
$34$	−53.2264			−1.56548
$35$	−16.8195	+	3.75000i	−0.480557	+	0.107143i
$36$		0			0
$37$			5.11748i			0.138310i	0.997606	+	0.0691552i	$0.0220304\pi$
$37$			5.11748i			0.138310i	−0.997606	+	0.0691552i	$0.977970\pi$
$38$	25.2919			0.665578
$39$		0			0
$40$	−6.91761	−	31.0269i	−0.172940	−	0.775673i
$41$			22.6476i			0.552381i	0.961103	+	0.276190i	$0.0890720\pi$
$41$			22.6476i			0.552381i	−0.961103	+	0.276190i	$0.910928\pi$
$42$		0			0
$43$		−	67.9633i		−	1.58054i	−0.612758	−	0.790270i	$-0.709940\pi$
$43$		−	67.9633i		−	1.58054i	0.612758	−	0.790270i	$-0.290060\pi$
$44$			5.47538i			0.124441i
$45$		0			0
$46$	68.1442			1.48140
$47$	−46.2467			−0.983973			−0.491986	−	0.870603i	$-0.663729\pi$
$47$	−46.2467			−0.983973			−0.491986	+	0.870603i	$0.663729\pi$
$48$		0			0
$49$	37.1217			0.757585
$50$	−51.6903	+	24.2550i	−1.03381	+	0.485099i
$51$		0			0
$52$			14.4048i			0.277015i
$53$	68.0017			1.28305			0.641525	−	0.767102i	$-0.278302\pi$
$53$	68.0017			1.28305			0.641525	+	0.767102i	$0.278302\pi$
$54$		0			0
$55$	−21.9688	+	4.89807i	−0.399434	+	0.0890558i
$56$		−	21.9119i		−	0.391285i
$57$		0			0
$58$			81.3806i			1.40311i
$59$			34.5832i			0.586156i	0.956089	+	0.293078i	$0.0946795\pi$
$59$			34.5832i			0.586156i	−0.956089	+	0.293078i	$0.905320\pi$
$60$		0			0
$61$	17.3633			0.284644			0.142322	−	0.989820i	$-0.454543\pi$
$61$	17.3633			0.284644			0.142322	+	0.989820i	$0.454543\pi$
$62$	69.2099			1.11629
$63$		0			0
$64$	34.5033			0.539114
$65$	−57.7963	+	12.8860i	−0.889173	+	0.198246i
$66$		0			0
$67$		−	98.2875i		−	1.46698i	−0.679701	−	0.733489i	$-0.737891\pi$
$67$		−	98.2875i		−	1.46698i	0.679701	−	0.733489i	$-0.262109\pi$
$68$	−28.3458			−0.416850
$69$		0			0
$70$	−38.4145	+	8.56471i	−0.548778	+	0.122353i
$71$			134.669i			1.89674i	0.317164	+	0.948371i	$0.397269\pi$
$71$			134.669i			1.89674i	−0.317164	+	0.948371i	$0.602731\pi$
$72$		0			0
$73$		−	134.962i		−	1.84879i	−0.381436	−	0.924395i	$-0.624570\pi$
$73$		−	134.962i		−	1.84879i	0.381436	−	0.924395i	$-0.375430\pi$
$74$			11.6879i			0.157945i
$75$		0			0
$76$	13.4693			0.177227
$77$	−15.5149			−0.201492
$78$		0			0
$79$	49.3738			0.624985			0.312492	−	0.949920i	$-0.398836\pi$
$79$	49.3738			0.624985			0.312492	+	0.949920i	$0.398836\pi$
$80$	−21.0930	−	94.6062i	−0.263662	−	1.18258i
$81$		0			0
$82$			51.7254i			0.630798i
$83$	28.6837			0.345586			0.172793	−	0.984958i	$-0.444721\pi$
$83$	28.6837			0.345586			0.172793	+	0.984958i	$0.444721\pi$
$84$		0			0
$85$	−25.3571	−	113.732i	−0.298318	−	1.33802i
$86$		−	155.223i		−	1.80492i
$87$		0			0
$88$		−	28.6203i		−	0.325231i
$89$		−	65.9204i		−	0.740679i	−0.928897	−	0.370339i	$-0.879241\pi$
$89$		−	65.9204i		−	0.740679i	0.928897	−	0.370339i	$-0.120759\pi$
$90$		0			0
$91$	−40.8171			−0.448540
$92$	36.2903			0.394460
$93$		0			0
$94$	−105.624			−1.12366
$95$	12.0491	+	54.0426i	0.126832	+	0.568869i
$96$		0			0
$97$		−	71.7451i		−	0.739640i	−0.929103	−	0.369820i	$-0.879419\pi$
$97$		−	71.7451i		−	0.739640i	0.929103	−	0.369820i	$-0.120581\pi$
$98$	84.7830			0.865133
$99$		0			0
$100$	−27.5277	+	12.9170i	−0.275277	+	0.129170i
$101$		−	50.5696i		−	0.500689i	−0.968157	−	0.250344i	$-0.919456\pi$
$101$		−	50.5696i		−	0.500689i	0.968157	−	0.250344i	$-0.0805438\pi$
$102$		0			0
$103$			193.417i			1.87783i	0.344144	+	0.938917i	$0.388169\pi$
$103$			193.417i			1.87783i	−0.344144	+	0.938917i	$0.611831\pi$
$104$		−	75.2952i		−	0.723992i
$105$		0			0
$106$	155.311			1.46519
$107$	−129.069			−1.20626			−0.603128	−	0.797644i	$-0.706079\pi$
$107$	−129.069			−1.20626			−0.603128	+	0.797644i	$0.706079\pi$
$108$		0			0
$109$	−189.029			−1.73421			−0.867107	−	0.498122i	$-0.834023\pi$
$109$	−189.029			−1.73421			−0.867107	+	0.498122i	$0.834023\pi$
$110$	−50.1752	+	11.1868i	−0.456138	+	0.101698i
$111$		0			0
$112$		−	66.8132i		−	0.596546i
$113$	140.231			1.24098			0.620491	−	0.784213i	$-0.286933\pi$
$113$	140.231			1.24098			0.620491	+	0.784213i	$0.286933\pi$
$114$		0			0
$115$	32.4639	+	145.607i	0.282295	+	1.26615i
$116$			43.3393i			0.373615i
$117$		0			0
$118$			78.9854i			0.669368i
$119$		−	80.3199i		−	0.674957i
$120$		0			0
$121$	100.735			0.832522
$122$	39.6565			0.325053
$123$		0			0
$124$	36.8578			0.297241
$125$	−76.4521	−	98.8943i	−0.611617	−	0.791154i
$126$		0			0
$127$		−	12.6839i		−	0.0998729i	−0.998752	−	0.0499364i	$-0.984098\pi$
$127$		−	12.6839i		−	0.0998729i	0.998752	−	0.0499364i	$-0.0159019\pi$
$128$	154.182			1.20455
$129$		0			0
$130$	−132.002	+	29.4306i	−1.01540	+	0.226389i
$131$		−	114.351i		−	0.872909i	−0.899726	−	0.436455i	$-0.856234\pi$
$131$		−	114.351i		−	0.872909i	0.899726	−	0.436455i	$-0.143766\pi$
$132$		0			0
$133$			38.1662i			0.286964i
$134$		−	224.481i		−	1.67523i
$135$		0			0
$136$	148.166			1.08946
$137$	19.2247			0.140326			0.0701631	−	0.997536i	$-0.477648\pi$
$137$	19.2247			0.140326			0.0701631	+	0.997536i	$0.477648\pi$
$138$		0			0
$139$	−56.1230			−0.403763			−0.201881	−	0.979410i	$-0.564706\pi$
$139$	−56.1230			−0.403763			−0.201881	+	0.979410i	$0.564706\pi$
$140$	−20.4577	+	4.56115i	−0.146126	+	0.0325796i
$141$		0			0
$142$			307.573i			2.16601i
$143$	−53.3134			−0.372821
$144$		0			0
$145$	−173.890	+	38.7697i	−1.19924	+	0.267377i
$146$		−	308.242i		−	2.11125i
$147$		0			0
$148$			6.22442i			0.0420569i
$149$			177.733i			1.19284i	0.802672	+	0.596420i	$0.203411\pi$
$149$			177.733i			1.19284i	−0.802672	+	0.596420i	$0.796589\pi$
$150$		0			0
$151$	282.208			1.86893			0.934465	−	0.356056i	$-0.115879\pi$
$151$	282.208			1.86893			0.934465	+	0.356056i	$0.115879\pi$
$152$	−70.4050			−0.463191
$153$		0			0
$154$	−35.4349			−0.230097
$155$	32.9716	+	147.884i	0.212720	+	0.954093i
$156$		0			0
$157$			186.737i			1.18940i	0.803946	+	0.594702i	$0.202730\pi$
$157$			186.737i			1.18940i	−0.803946	+	0.594702i	$0.797270\pi$
$158$	112.766			0.713708
$159$		0			0
$160$	−20.5043	−	91.9658i	−0.128152	−	0.574786i
$161$			102.831i			0.638704i
$162$		0			0
$163$			33.6033i			0.206155i	0.994673	+	0.103078i	$0.0328690\pi$
$163$			33.6033i			0.206155i	−0.994673	+	0.103078i	$0.967131\pi$
$164$			27.5464i			0.167966i
$165$		0			0
$166$	65.5113			0.394646
$167$	−50.0505			−0.299704			−0.149852	−	0.988708i	$-0.547880\pi$
$167$	−50.0505			−0.299704			−0.149852	+	0.988708i	$0.547880\pi$
$168$		0			0
$169$	28.7417			0.170069
$170$	−57.9136	−	259.754i	−0.340668	−	1.52797i
$171$		0			0
$172$		−	82.6641i		−	0.480605i
$173$	75.5605			0.436766			0.218383	−	0.975863i	$-0.429922\pi$
$173$	75.5605			0.436766			0.218383	+	0.975863i	$0.429922\pi$
$174$		0			0
$175$	−36.6013	−	78.0020i	−0.209150	−	0.445725i
$176$		−	87.2682i		−	0.495842i
$177$		0			0
$178$		−	150.557i		−	0.845827i
$179$			116.454i			0.650583i	0.945614	+	0.325291i	$0.105462\pi$
$179$			116.454i			0.650583i	−0.945614	+	0.325291i	$0.894538\pi$
$180$		0			0
$181$	−45.4016			−0.250838			−0.125419	−	0.992104i	$-0.540027\pi$
$181$	−45.4016			−0.250838			−0.125419	+	0.992104i	$0.540027\pi$
$182$	−93.2231			−0.512215
$183$		0			0
$184$	−189.693			−1.03094
$185$	−24.9742	+	5.56813i	−0.134996	+	0.0300980i
$186$		0			0
$187$		−	104.910i		−	0.561016i
$188$	−56.2502			−0.299203
$189$		0			0
$190$	27.5192	+	123.429i	0.144838	+	0.649627i
$191$			149.606i			0.783278i	0.920119	+	0.391639i	$0.128092\pi$
$191$			149.606i			0.783278i	−0.920119	+	0.391639i	$0.871908\pi$
$192$		0			0
$193$		−	141.802i		−	0.734727i	−0.930077	−	0.367363i	$-0.880261\pi$
$193$		−	141.802i		−	0.734727i	0.930077	−	0.367363i	$-0.119739\pi$
$194$		−	163.860i		−	0.844641i
$195$		0			0
$196$	45.1513			0.230364
$197$	121.128			0.614864			0.307432	−	0.951570i	$-0.400530\pi$
$197$	121.128			0.614864			0.307432	+	0.951570i	$0.400530\pi$
$198$		0			0
$199$	−25.7205			−0.129249			−0.0646243	−	0.997910i	$-0.520585\pi$
$199$	−25.7205			−0.129249			−0.0646243	+	0.997910i	$0.520585\pi$
$200$	143.890	−	67.5184i	0.719450	−	0.337592i
$201$		0			0
$202$		−	115.497i		−	0.571767i
$203$	−122.805			−0.604952
$204$		0			0
$205$	−110.524	+	24.6420i	−0.539143	+	0.120205i
$206$			441.749i			2.14441i
$207$		0			0
$208$		−	229.588i		−	1.10379i
$209$			49.8508i			0.238521i
$210$		0			0
$211$	76.8965			0.364439			0.182219	−	0.983258i	$-0.441672\pi$
$211$	76.8965			0.364439			0.182219	+	0.983258i	$0.441672\pi$
$212$	82.7108			0.390145
$213$		0			0
$214$	−294.785			−1.37750
$215$	331.673	−	73.9482i	1.54266	−	0.343945i
$216$		0			0
$217$			104.439i			0.481288i
$218$	−431.729			−1.98041
$219$		0			0
$220$	−26.7208	+	5.95755i	−0.121458	+	0.0270798i
$221$		−	276.000i		−	1.24887i
$222$		0			0
$223$			194.407i			0.871781i	0.900000	+	0.435890i	$0.143567\pi$
$223$			194.407i			0.871781i	−0.900000	+	0.435890i	$0.856433\pi$
$224$		−	64.9484i		−	0.289948i
$225$		0			0
$226$	320.277			1.41715
$227$	−152.365			−0.671210			−0.335605	−	0.942003i	$-0.608941\pi$
$227$	−152.365			−0.671210			−0.335605	+	0.942003i	$0.608941\pi$
$228$		0			0
$229$	266.415			1.16338			0.581691	−	0.813410i	$-0.302391\pi$
$229$	266.415			1.16338			0.581691	+	0.813410i	$0.302391\pi$
$230$	74.1451	+	332.556i	0.322370	+	1.44589i
$231$		0			0
$232$		−	226.539i		−	0.976459i
$233$	107.497			0.461361			0.230681	−	0.973029i	$-0.425905\pi$
$233$	107.497			0.461361			0.230681	+	0.973029i	$0.425905\pi$
$234$		0			0
$235$	−50.3193	−	225.692i	−0.214124	−	0.960392i
$236$			42.0638i			0.178236i
$237$		0			0
$238$		−	183.445i		−	0.770776i
$239$			311.707i			1.30421i	0.758127	+	0.652106i	$0.226114\pi$
$239$			311.707i			1.30421i	−0.758127	+	0.652106i	$0.773886\pi$
$240$		0			0
$241$	−160.321			−0.665232			−0.332616	−	0.943062i	$-0.607931\pi$
$241$	−160.321			−0.665232			−0.332616	+	0.943062i	$0.607931\pi$
$242$	230.071			0.950708
$243$		0			0
$244$	21.1191			0.0865537
$245$	40.3906	+	181.160i	0.164860	+	0.739430i
$246$		0			0
$247$			131.149i			0.530968i
$248$	−192.659			−0.776852
$249$		0			0
$250$	−174.611	−	225.867i	−0.698443	−	0.903468i
$251$			357.272i			1.42339i	0.702487	+	0.711697i	$0.252073\pi$
$251$			357.272i			1.42339i	−0.702487	+	0.711697i	$0.747927\pi$
$252$		0			0
$253$			134.313i			0.530883i
$254$		−	28.9690i		−	0.114051i
$255$		0			0
$256$	214.127			0.836432
$257$	−455.848			−1.77373			−0.886865	−	0.462029i	$-0.847121\pi$
$257$	−455.848			−1.77373			−0.886865	+	0.462029i	$0.847121\pi$
$258$		0			0
$259$	−17.6374			−0.0680980
$260$	−70.2980	+	15.6733i	−0.270377	+	0.0602819i
$261$		0			0
$262$		−	261.169i		−	0.996829i
$263$	−60.8355			−0.231314			−0.115657	−	0.993289i	$-0.536897\pi$
$263$	−60.8355			−0.231314			−0.115657	+	0.993289i	$0.536897\pi$
$264$		0			0
$265$	73.9900	+	331.860i	0.279207	+	1.25230i
$266$			87.1686i			0.327702i
$267$		0			0
$268$		−	119.548i		−	0.446073i
$269$		−	213.484i		−	0.793620i	−0.917901	−	0.396810i	$-0.870117\pi$
$269$		−	213.484i		−	0.793620i	0.917901	−	0.396810i	$-0.129883\pi$
$270$		0			0
$271$	259.475			0.957474			0.478737	−	0.877958i	$-0.341095\pi$
$271$	259.475			0.957474			0.478737	+	0.877958i	$0.341095\pi$
$272$	451.783			1.66097
$273$		0			0
$274$	43.9077			0.160247
$275$	−47.8069	−	101.882i	−0.173843	−	0.370482i
$276$		0			0
$277$		−	254.189i		−	0.917651i	−0.888527	−	0.458825i	$-0.848270\pi$
$277$		−	254.189i		−	0.917651i	0.888527	−	0.458825i	$-0.151730\pi$
$278$	−128.181			−0.461081
$279$		0			0
$280$	106.934	−	23.8415i	0.381908	−	0.0851483i
$281$		−	250.476i		−	0.891374i	−0.895189	−	0.445687i	$-0.852959\pi$
$281$		−	250.476i		−	0.891374i	0.895189	−	0.445687i	$-0.147041\pi$
$282$		0			0
$283$		−	346.284i		−	1.22362i	−0.791005	−	0.611810i	$-0.790442\pi$
$283$		−	346.284i		−	1.22362i	0.791005	−	0.611810i	$-0.209558\pi$
$284$			163.798i			0.576755i
$285$		0			0
$286$	−121.764			−0.425747
$287$	−78.0549			−0.271968
$288$		0			0
$289$	254.114			0.879287
$290$	−397.152	+	88.5470i	−1.36949	+	0.305335i
$291$		0			0
$292$		−	164.155i		−	0.562174i
$293$	−313.036			−1.06838			−0.534192	−	0.845363i	$-0.679384\pi$
$293$	−313.036			−1.06838			−0.534192	+	0.845363i	$0.679384\pi$
$294$		0			0
$295$	−168.772	+	37.6286i	−0.572109	+	0.127555i
$296$		−	32.5356i		−	0.109918i
$297$		0			0
$298$			405.929i			1.36218i
$299$			353.355i			1.18179i
$300$		0			0
$301$	234.235			0.778190
$302$	644.542			2.13425
$303$		0			0
$304$	−214.677			−0.706173
$305$	18.8923	+	84.7360i	0.0619421	+	0.277823i
$306$		0			0
$307$			267.727i			0.872074i	0.899929	+	0.436037i	$0.143618\pi$
$307$			267.727i			0.872074i	−0.899929	+	0.436037i	$0.856382\pi$
$308$	−18.8709			−0.0612691
$309$		0			0
$310$	75.3046	+	337.757i	0.242918	+	1.08954i
$311$			353.124i			1.13545i	0.823220	+	0.567723i	$0.192176\pi$
$311$			353.124i			1.13545i	−0.823220	+	0.567723i	$0.807824\pi$
$312$		0			0
$313$			381.115i			1.21762i	0.793317	+	0.608809i	$0.208353\pi$
$313$			381.115i			1.21762i	−0.793317	+	0.608809i	$0.791647\pi$
$314$			426.492i			1.35825i
$315$		0			0
$316$	60.0536			0.190043
$317$	1.08266			0.00341535			0.00170767	−	0.999999i	$-0.499456\pi$
$317$	1.08266			0.00341535			0.00170767	+	0.999999i	$0.499456\pi$
$318$		0			0
$319$	−160.402			−0.502829
$320$	37.5417	+	168.382i	0.117318	+	0.526194i
$321$		0			0
$322$			234.859i			0.729375i
$323$	−258.075			−0.798994
$324$		0			0
$325$	−125.772	−	268.035i	−0.386990	−	0.824724i
$326$			76.7474i			0.235422i
$327$		0			0
$328$		−	143.988i		−	0.438987i
$329$		−	159.389i		−	0.484465i
$330$		0			0
$331$	−134.879			−0.407491			−0.203745	−	0.979024i	$-0.565311\pi$
$331$	−134.879			−0.407491			−0.203745	+	0.979024i	$0.565311\pi$
$332$	34.8881			0.105085
$333$		0			0
$334$	−114.312			−0.342250
$335$	479.660	−	106.943i	1.43182	−	0.319232i
$336$		0			0
$337$			343.619i			1.01964i	0.860280	+	0.509821i	$0.170288\pi$
$337$			343.619i			1.01964i	−0.860280	+	0.509821i	$0.829712\pi$
$338$	65.6438			0.194213
$339$		0			0
$340$	−30.8419	−	138.332i	−0.0907116	−	0.406860i
$341$			136.414i			0.400041i
$342$		0			0
$343$			296.818i			0.865359i
$344$			432.093i			1.25608i
$345$		0			0
$346$	172.574			0.498770
$347$	199.494			0.574911			0.287456	−	0.957794i	$-0.407191\pi$
$347$	199.494			0.574911			0.287456	+	0.957794i	$0.407191\pi$
$348$		0			0
$349$	−207.578			−0.594778			−0.297389	−	0.954756i	$-0.596116\pi$
$349$	−207.578			−0.594778			−0.297389	+	0.954756i	$0.596116\pi$
$350$	−83.5946	−	178.150i	−0.238842	−	0.509001i
$351$		0			0
$352$		−	84.8325i		−	0.241001i
$353$	−79.8281			−0.226142			−0.113071	−	0.993587i	$-0.536069\pi$
$353$	−79.8281			−0.226142			−0.113071	+	0.993587i	$0.536069\pi$
$354$		0			0
$355$	−657.207	+	146.528i	−1.85129	+	0.412754i
$356$		−	80.1794i		−	0.225223i
$357$		0			0
$358$			265.973i			0.742940i
$359$		−	234.273i		−	0.652571i	−0.945271	−	0.326285i	$-0.894203\pi$
$359$		−	234.273i		−	0.652571i	0.945271	−	0.326285i	$-0.105797\pi$
$360$		0			0
$361$	−238.369			−0.660301
$362$	−103.694			−0.286447
$363$		0			0
$364$	−49.6461			−0.136390
$365$	658.637	−	146.847i	1.80448	−	0.402319i
$366$		0			0
$367$		−	219.296i		−	0.597536i	−0.954326	−	0.298768i	$-0.903424\pi$
$367$		−	219.296i		−	0.597536i	0.954326	−	0.298768i	$-0.0965757\pi$
$368$	−578.405			−1.57175
$369$		0			0
$370$	−57.0392	+	12.7172i	−0.154160	+	0.0343708i
$371$			234.368i			0.631718i
$372$		0			0
$373$		−	161.434i		−	0.432800i	−0.976305	−	0.216400i	$-0.930569\pi$
$373$		−	161.434i		−	0.432800i	0.976305	−	0.216400i	$-0.0694315\pi$
$374$		−	239.607i		−	0.640659i
$375$		0			0
$376$	294.025			0.781981
$377$	−421.991			−1.11934
$378$		0			0
$379$	227.627			0.600598			0.300299	−	0.953845i	$-0.402914\pi$
$379$	227.627			0.600598			0.300299	+	0.953845i	$0.402914\pi$
$380$	14.6554	+	65.7323i	0.0385668	+	0.172980i
$381$		0			0
$382$			341.689i			0.894473i
$383$	−180.111			−0.470264			−0.235132	−	0.971963i	$-0.575552\pi$
$383$	−180.111			−0.470264			−0.235132	+	0.971963i	$0.575552\pi$
$384$		0			0
$385$	−16.8812	−	75.7155i	−0.0438472	−	0.196664i
$386$		−	323.866i		−	0.839030i
$387$		0			0
$388$		−	87.2640i		−	0.224907i
$389$		−	728.327i		−	1.87231i	−0.351594	−	0.936153i	$-0.614360\pi$
$389$		−	728.327i		−	1.87231i	0.351594	−	0.936153i	$-0.385640\pi$
$390$		0			0
$391$	−695.333			−1.77835
$392$	−236.010			−0.602066
$393$		0			0
$394$	276.648			0.702151
$395$	53.7217	+	240.953i	0.136004	+	0.610007i
$396$		0			0
$397$			417.565i			1.05180i	0.850546	+	0.525901i	$0.176272\pi$
$397$			417.565i			1.05180i	−0.850546	+	0.525901i	$0.823728\pi$
$398$	−58.7436			−0.147597
$399$		0			0
$400$	438.745	−	205.875i	1.09686	−	0.514687i
$401$		−	461.319i		−	1.15042i	−0.818005	−	0.575211i	$-0.804920\pi$
$401$		−	461.319i		−	1.15042i	0.818005	−	0.575211i	$-0.195080\pi$
$402$		0			0
$403$			358.882i			0.890525i
$404$		−	61.5081i		−	0.152248i
$405$		0			0
$406$	−280.478			−0.690832
$407$	−23.0371			−0.0566022
$408$		0			0
$409$	532.027			1.30080			0.650400	−	0.759592i	$-0.274601\pi$
$409$	532.027			1.30080			0.650400	+	0.759592i	$0.274601\pi$
$410$	−252.429	+	56.2804i	−0.615681	+	0.137269i
$411$		0			0
$412$			235.254i			0.571005i
$413$	−119.191			−0.288598
$414$		0			0
$415$	31.2096	+	139.981i	0.0752038	+	0.337304i
$416$		−	223.180i		−	0.536490i
$417$		0			0
$418$			113.855i			0.272382i
$419$			469.258i			1.11995i	0.828510	+	0.559974i	$0.189189\pi$
$419$			469.258i			1.11995i	−0.828510	+	0.559974i	$0.810811\pi$
$420$		0			0
$421$	−220.538			−0.523844			−0.261922	−	0.965089i	$-0.584356\pi$
$421$	−220.538			−0.523844			−0.261922	+	0.965089i	$0.584356\pi$
$422$	175.626			0.416175
$423$		0			0
$424$	−432.337			−1.01966
$425$	527.440	−	247.494i	1.24104	−	0.582338i
$426$		0			0
$427$			59.8426i			0.140147i
$428$	−156.988			−0.366794
$429$		0			0
$430$	757.515	−	168.892i	1.76166	−	0.392772i
$431$		−	487.110i		−	1.13018i	−0.825028	−	0.565092i	$-0.808840\pi$
$431$		−	487.110i		−	1.13018i	0.825028	−	0.565092i	$-0.191160\pi$
$432$		0			0
$433$		−	462.822i		−	1.06887i	−0.845208	−	0.534437i	$-0.820524\pi$
$433$		−	462.822i		−	1.06887i	0.845208	−	0.534437i	$-0.179476\pi$
$434$			238.532i			0.549612i
$435$		0			0
$436$	−229.918			−0.527334
$437$	330.406			0.756078
$438$		0			0
$439$	801.389			1.82549			0.912744	−	0.408532i	$-0.133959\pi$
$439$	801.389			1.82549			0.912744	+	0.408532i	$0.133959\pi$
$440$	139.672	−	31.1407i	0.317437	−	0.0707743i
$441$		0			0
$442$		−	630.364i		−	1.42616i
$443$	74.3205			0.167766			0.0838832	−	0.996476i	$-0.473268\pi$
$443$	74.3205			0.167766			0.0838832	+	0.996476i	$0.473268\pi$
$444$		0			0
$445$	321.703	−	71.7254i	0.722928	−	0.161181i
$446$			444.011i			0.995541i
$447$		0			0
$448$			118.916i			0.265436i
$449$		−	358.472i		−	0.798379i	−0.916868	−	0.399190i	$-0.869291\pi$
$449$		−	358.472i		−	0.798379i	0.916868	−	0.399190i	$-0.130709\pi$
$450$		0			0
$451$	−101.952			−0.226057
$452$	170.564			0.377354
$453$		0			0
$454$	−347.989			−0.766496
$455$	−44.4115	−	199.195i	−0.0976077	−	0.437790i
$456$		0			0
$457$		−	115.782i		−	0.253351i	−0.991944	−	0.126676i	$-0.959569\pi$
$457$		−	115.782i		−	0.253351i	0.991944	−	0.126676i	$-0.0404308\pi$
$458$	608.471			1.32854
$459$		0			0
$460$	39.4861	+	177.103i	0.0858393	+	0.385007i
$461$		−	199.669i		−	0.433120i	−0.976269	−	0.216560i	$-0.930516\pi$
$461$		−	199.669i		−	0.433120i	0.976269	−	0.216560i	$-0.0694838\pi$
$462$		0			0
$463$		−	597.086i		−	1.28960i	−0.764350	−	0.644802i	$-0.776940\pi$
$463$		−	597.086i		−	1.28960i	0.764350	−	0.644802i	$-0.223060\pi$
$464$		−	690.754i		−	1.48869i
$465$		0			0
$466$	245.515			0.526857
$467$	605.150			1.29582			0.647912	−	0.761715i	$-0.275643\pi$
$467$	605.150			1.29582			0.647912	+	0.761715i	$0.275643\pi$
$468$		0			0
$469$	338.748			0.722276
$470$	−114.925	−	515.463i	−0.244522	−	1.09673i
$471$		0			0
$472$		−	219.871i		−	0.465829i
$473$	305.947			0.646822
$474$		0			0
$475$	−250.627	+	117.603i	−0.527636	+	0.247586i
$476$		−	97.6936i		−	0.205239i
$477$		0			0
$478$			711.915i			1.48936i
$479$		−	540.426i		−	1.12824i	−0.825694	−	0.564119i	$-0.809216\pi$
$479$		−	540.426i		−	1.12824i	0.825694	−	0.564119i	$-0.190784\pi$
$480$		0			0
$481$	−60.6067			−0.126001
$482$	−366.161			−0.759669
$483$		0			0
$484$	122.525			0.253150
$485$	350.129	−	78.0630i	0.721915	−	0.160955i
$486$		0			0
$487$		−	565.424i		−	1.16104i	−0.814248	−	0.580518i	$-0.802850\pi$
$487$		−	565.424i		−	1.16104i	0.814248	−	0.580518i	$-0.197150\pi$
$488$	−110.391			−0.226212
$489$		0			0
$490$	92.2491	+	413.756i	0.188264	+	0.844400i
$491$		−	267.021i		−	0.543830i	−0.962321	−	0.271915i	$-0.912343\pi$
$491$		−	267.021i		−	0.543830i	0.962321	−	0.271915i	$-0.0876570\pi$
$492$		0			0
$493$		−	830.395i		−	1.68437i
$494$			299.534i			0.606345i
$495$		0			0
$496$	−587.450			−1.18438
$497$	−464.135			−0.933873
$498$		0			0
$499$	−620.321			−1.24313			−0.621564	−	0.783363i	$-0.713503\pi$
$499$	−620.321			−1.24313			−0.621564	+	0.783363i	$0.713503\pi$
$500$	−92.9891	−	120.286i	−0.185978	−	0.240571i
$501$		0			0
$502$			815.981i			1.62546i
$503$	433.671			0.862168			0.431084	−	0.902312i	$-0.358131\pi$
$503$	433.671			0.862168			0.431084	+	0.902312i	$0.358131\pi$
$504$		0			0
$505$	246.788	−	55.0228i	0.488690	−	0.108956i
$506$			306.761i			0.606248i
$507$		0			0
$508$		−	15.4275i		−	0.0303690i
$509$		−	260.241i		−	0.511280i	−0.966772	−	0.255640i	$-0.917714\pi$
$509$		−	260.241i		−	0.511280i	0.966772	−	0.255640i	$-0.0822861\pi$
$510$		0			0
$511$	465.145			0.910264
$512$	−127.679			−0.249373
$513$		0			0
$514$	−1041.12			−2.02553
$515$	−943.908	+	210.449i	−1.83283	+	0.408640i
$516$		0			0
$517$		−	208.186i		−	0.402682i
$518$	−40.2824			−0.0777653
$519$		0			0
$520$	367.454	−	81.9258i	0.706642	−	0.157550i
$521$		−	658.284i		−	1.26350i	−0.775172	−	0.631750i	$-0.782337\pi$
$521$		−	658.284i		−	1.26350i	0.775172	−	0.631750i	$-0.217663\pi$
$522$		0			0
$523$		−	401.215i		−	0.767142i	−0.923511	−	0.383571i	$-0.874694\pi$
$523$		−	401.215i		−	0.767142i	0.923511	−	0.383571i	$-0.125306\pi$
$524$		−	139.086i		−	0.265431i
$525$		0			0
$526$	−138.944			−0.264151
$527$	−706.208			−1.34005
$528$		0			0
$529$	361.215			0.682827
$530$	168.987	+	757.943i	0.318844	+	1.43008i
$531$		0			0
$532$			46.4217i			0.0872589i
$533$	−268.217			−0.503222
$534$		0			0
$535$	−140.435	−	629.882i	−0.262496	−	1.17735i
$536$			624.887i			1.16583i
$537$		0			0
$538$		−	487.580i		−	0.906283i
$539$			167.109i			0.310035i
$540$		0			0
$541$	131.652			0.243350			0.121675	−	0.992570i	$-0.461173\pi$
$541$	131.652			0.243350			0.121675	+	0.992570i	$0.461173\pi$
$542$	592.622			1.09340
$543$		0			0
$544$	439.174			0.807304
$545$	−205.676	−	922.497i	−0.377386	−	1.69265i
$546$		0			0
$547$		−	989.756i		−	1.80943i	−0.426021	−	0.904713i	$-0.640085\pi$
$547$		−	989.756i		−	1.80943i	0.426021	−	0.904713i	$-0.359915\pi$
$548$	23.3831			0.0426699
$549$		0			0
$550$	−109.187	−	232.692i	−0.198522	−	0.423076i
$551$			394.584i			0.716124i
$552$		0			0
$553$			170.167i			0.307715i
$554$		−	580.549i		−	1.04792i
$555$		0			0
$556$	−68.2628			−0.122775
$557$	−299.018			−0.536836			−0.268418	−	0.963302i	$-0.586501\pi$
$557$	−299.018			−0.536836			−0.268418	+	0.963302i	$0.586501\pi$
$558$		0			0
$559$	804.894			1.43988
$560$	326.060	−	72.6968i	0.582250	−	0.129816i
$561$		0			0
$562$		−	572.068i		−	1.01791i
$563$	758.370			1.34702			0.673508	−	0.739180i	$-0.264787\pi$
$563$	758.370			1.34702			0.673508	+	0.739180i	$0.264787\pi$
$564$		0			0
$565$	152.580	+	684.352i	0.270053	+	1.21124i
$566$		−	790.887i		−	1.39733i
$567$		0			0
$568$		−	856.189i		−	1.50737i
$569$		−	581.256i		−	1.02154i	−0.859718	−	0.510770i	$-0.829361\pi$
$569$		−	581.256i		−	1.02154i	0.859718	−	0.510770i	$-0.170639\pi$
$570$		0			0
$571$	−670.478			−1.17422			−0.587109	−	0.809508i	$-0.699734\pi$
$571$	−670.478			−1.17422			−0.587109	+	0.809508i	$0.699734\pi$
$572$	−64.8454			−0.113366
$573$		0			0
$574$	−178.271			−0.310577
$575$	−675.266	+	316.859i	−1.17438	+	0.551060i
$576$		0			0
$577$			135.394i			0.234652i	0.993093	+	0.117326i	$0.0374322\pi$
$577$			135.394i			0.234652i	−0.993093	+	0.117326i	$0.962568\pi$
$578$	580.377			1.00411
$579$		0			0
$580$	−211.504	+	47.1558i	−0.364661	+	0.0813032i
$581$			98.8581i			0.170152i
$582$		0			0
$583$			306.120i			0.525077i
$584$			858.052i			1.46927i
$585$		0			0
$586$	−714.951			−1.22005
$587$	−359.404			−0.612272			−0.306136	−	0.951988i	$-0.599036\pi$
$587$	−359.404			−0.612272			−0.306136	+	0.951988i	$0.599036\pi$
$588$		0			0
$589$	335.573			0.569734
$590$	−385.463	+	85.9409i	−0.653326	+	0.145663i
$591$		0			0
$592$		−	99.2066i		−	0.167579i
$593$	540.430			0.911349			0.455675	−	0.890146i	$-0.349398\pi$
$593$	540.430			0.911349			0.455675	+	0.890146i	$0.349398\pi$
$594$		0			0
$595$	391.975	−	87.3930i	0.658782	−	0.146879i
$596$			216.178i			0.362715i
$597$		0			0
$598$			807.037i			1.34956i
$599$			576.543i			0.962509i	0.876581	+	0.481254i	$0.159819\pi$
$599$			576.543i			0.962509i	−0.876581	+	0.481254i	$0.840181\pi$
$600$		0			0
$601$	−1026.42			−1.70785			−0.853927	−	0.520393i	$-0.825786\pi$
$601$	−1026.42			−1.70785			−0.853927	+	0.520393i	$0.825786\pi$
$602$	534.975			0.888663
$603$		0			0
$604$	343.252			0.568298
$605$	109.606	+	491.605i	0.181167	+	0.812571i
$606$		0			0
$607$			165.327i			0.272368i	0.990684	+	0.136184i	$0.0434838\pi$
$607$			165.327i			0.272368i	−0.990684	+	0.136184i	$0.956516\pi$
$608$	−208.685			−0.343232
$609$		0			0
$610$	43.1487	+	193.531i	0.0707355	+	0.317263i
$611$		−	547.703i		−	0.896404i
$612$		0			0
$613$			251.007i			0.409473i	0.978817	+	0.204737i	$0.0656338\pi$
$613$			251.007i			0.409473i	−0.978817	+	0.204737i	$0.934366\pi$
$614$			611.468i			0.995875i
$615$		0			0
$616$	98.6399			0.160130
$617$	917.091			1.48637			0.743186	−	0.669085i	$-0.233314\pi$
$617$	917.091			1.48637			0.743186	+	0.669085i	$0.233314\pi$
$618$		0			0
$619$	−277.589			−0.448448			−0.224224	−	0.974538i	$-0.571985\pi$
$619$	−277.589			−0.448448			−0.224224	+	0.974538i	$0.571985\pi$
$620$	40.1036	+	179.873i	0.0646832	+	0.290117i
$621$		0			0
$622$			806.508i			1.29664i
$623$	227.194			0.364678
$624$		0			0
$625$	399.437	−	480.703i	0.639099	−	0.769124i
$626$			870.436i			1.39047i
$627$		0			0
$628$			227.129i			0.361670i
$629$		−	119.262i		−	0.189606i
$630$		0			0
$631$	322.116			0.510485			0.255243	−	0.966877i	$-0.417845\pi$
$631$	322.116			0.510485			0.255243	+	0.966877i	$0.417845\pi$
$632$	−313.906			−0.496686
$633$		0			0
$634$	2.47272			0.00390020
$635$	61.8995	−	13.8008i	0.0974795	−	0.0217336i
$636$		0			0
$637$			439.634i			0.690164i
$638$	−366.347			−0.574211
$639$		0			0
$640$	167.759	+	752.435i	0.262124	+	1.17568i
$641$		−	276.966i		−	0.432084i	−0.976384	−	0.216042i	$-0.930685\pi$
$641$		−	276.966i		−	0.432084i	0.976384	−	0.216042i	$-0.0693148\pi$
$642$		0			0
$643$			858.353i			1.33492i	0.744646	+	0.667460i	$0.232618\pi$
$643$			858.353i			1.33492i	−0.744646	+	0.667460i	$0.767382\pi$
$644$			125.074i			0.194215i
$645$		0			0
$646$	−589.424			−0.912421
$647$	−1065.93			−1.64749			−0.823746	−	0.566960i	$-0.808120\pi$
$647$	−1065.93			−1.64749			−0.823746	+	0.566960i	$0.808120\pi$
$648$		0			0
$649$	−155.681			−0.239879
$650$	−287.253	−	612.172i	−0.441928	−	0.941803i
$651$		0			0
$652$			40.8719i			0.0626870i
$653$	356.252			0.545561			0.272781	−	0.962076i	$-0.412057\pi$
$653$	356.252			0.545561			0.272781	+	0.962076i	$0.412057\pi$
$654$		0			0
$655$	558.054	−	124.421i	0.851990	−	0.189956i
$656$		−	439.043i		−	0.669272i
$657$		0			0
$658$		−	364.033i		−	0.553241i
$659$		−	38.8525i		−	0.0589568i	−0.999565	−	0.0294784i	$-0.990615\pi$
$659$		−	38.8525i		−	0.0589568i	0.999565	−	0.0294784i	$-0.00938463\pi$
$660$		0			0
$661$	201.835			0.305349			0.152674	−	0.988277i	$-0.451211\pi$
$661$	201.835			0.305349			0.152674	+	0.988277i	$0.451211\pi$
$662$	−308.054			−0.465339
$663$		0			0
$664$	−182.363			−0.274644
$665$	−186.258	+	41.5271i	−0.280087	+	0.0624468i
$666$		0			0
$667$			1063.13i			1.59390i
$668$	−60.8768			−0.0911329
$669$		0			0
$670$	1095.51	−	244.249i	1.63509	−	0.364551i
$671$			78.1635i			0.116488i
$672$		0			0
$673$		−	449.327i		−	0.667648i	−0.942636	−	0.333824i	$-0.891661\pi$
$673$		−	449.327i		−	0.667648i	0.942636	−	0.333824i	$-0.108339\pi$
$674$			784.801i			1.16439i
$675$		0			0
$676$	34.9587			0.0517141
$677$	702.810			1.03812			0.519062	−	0.854737i	$-0.326281\pi$
$677$	702.810			1.03812			0.519062	+	0.854737i	$0.326281\pi$
$678$		0			0
$679$	247.269			0.364167
$680$	161.214	+	723.076i	0.237079	+	1.06335i
$681$		0			0
$682$			311.559i			0.456831i
$683$	−705.068			−1.03231			−0.516155	−	0.856495i	$-0.672637\pi$
$683$	−705.068			−1.03231			−0.516155	+	0.856495i	$0.672637\pi$
$684$		0			0
$685$	20.9177	+	93.8199i	0.0305367	+	0.136963i
$686$			677.910i			0.988206i
$687$		0			0
$688$			1317.52i			1.91501i
$689$			805.348i			1.16887i
$690$		0			0
$691$	−884.250			−1.27967			−0.639833	−	0.768514i	$-0.720997\pi$
$691$	−884.250			−1.27967			−0.639833	+	0.768514i	$0.720997\pi$
$692$	91.9047			0.132810
$693$		0			0
$694$	455.630			0.656527
$695$	−61.0653	−	273.890i	−0.0878637	−	0.394087i
$696$		0			0
$697$		−	527.798i		−	0.757243i
$698$	−474.091			−0.679214
$699$		0			0
$700$	−44.5184	−	94.8743i	−0.0635977	−	0.135535i
$701$		−	267.537i		−	0.381651i	−0.981624	−	0.190826i	$-0.938884\pi$
$701$		−	267.537i		−	0.381651i	0.981624	−	0.190826i	$-0.0611165\pi$
$702$		0			0
$703$			56.6705i			0.0806124i
$704$			155.322i			0.220628i
$705$		0			0
$706$	−182.321			−0.258246
$707$	174.288			0.246517
$708$		0			0
$709$	−533.247			−0.752111			−0.376056	−	0.926597i	$-0.622720\pi$
$709$	−533.247			−0.752111			−0.376056	+	0.926597i	$0.622720\pi$
$710$	−1501.01	+	334.658i	−2.11410	+	0.471349i
$711$		0			0
$712$			419.105i			0.588631i
$713$	904.137			1.26807
$714$		0			0
$715$	−58.0082	−	260.179i	−0.0811303	−	0.363886i
$716$			141.644i			0.197827i
$717$		0			0
$718$		−	535.061i		−	0.745211i
$719$		−	101.956i		−	0.141803i	−0.997483	−	0.0709015i	$-0.977412\pi$
$719$		−	101.956i		−	0.141803i	0.997483	−	0.0709015i	$-0.0225876\pi$
$720$		0			0
$721$	−666.611			−0.924564
$722$	−544.416			−0.754038
$723$		0			0
$724$	−55.2222			−0.0762738
$725$	−378.406	−	806.430i	−0.521939	−	1.11232i
$726$		0			0
$727$			734.939i			1.01092i	0.862850	+	0.505460i	$0.168677\pi$
$727$			734.939i			1.01092i	−0.862850	+	0.505460i	$0.831323\pi$
$728$	259.505			0.356462
$729$		0			0
$730$	1504.28	−	335.386i	2.06065	−	0.459433i
$731$			1583.87i			2.16672i
$732$		0			0
$733$			1324.21i			1.80656i	0.429047	+	0.903282i	$0.358849\pi$
$733$			1324.21i			1.80656i	−0.429047	+	0.903282i	$0.641151\pi$
$734$		−	500.855i		−	0.682363i
$735$		0			0
$736$	−562.261			−0.763942
$737$	442.456			0.600347
$738$		0			0
$739$	978.711			1.32437			0.662186	−	0.749340i	$-0.269629\pi$
$739$	978.711			1.32437			0.662186	+	0.749340i	$0.269629\pi$
$740$	−30.3763	+	6.77256i	−0.0410490	+	0.00915210i
$741$		0			0
$742$			535.277i			0.721398i
$743$	928.176			1.24923			0.624614	−	0.780934i	$-0.285256\pi$
$743$	928.176			1.24923			0.624614	+	0.780934i	$0.285256\pi$
$744$		0			0
$745$	−867.370	+	193.385i	−1.16425	+	0.259577i
$746$		−	368.704i		−	0.494241i
$747$		0			0
$748$		−	127.603i		−	0.170592i
$749$		−	444.837i		−	0.593908i
$750$		0			0
$751$	−537.997			−0.716374			−0.358187	−	0.933650i	$-0.616605\pi$
$751$	−537.997			−0.716374			−0.358187	+	0.933650i	$0.616605\pi$
$752$	896.531			1.19219
$753$		0			0
$754$	−963.796			−1.27824
$755$	307.060	+	1377.23i	0.406702	+	1.82414i
$756$		0			0
$757$		−	981.098i		−	1.29603i	−0.761626	−	0.648017i	$-0.775599\pi$
$757$		−	981.098i		−	1.29603i	0.761626	−	0.648017i	$-0.224401\pi$
$758$	519.882			0.685860
$759$		0			0
$760$	−76.6050	−	343.589i	−0.100796	−	0.452091i
$761$			572.605i			0.752438i	0.926531	+	0.376219i	$0.122776\pi$
$761$			572.605i			0.752438i	−0.926531	+	0.376219i	$0.877224\pi$
$762$		0			0
$763$		−	651.489i		−	0.853852i
$764$			181.967i			0.238176i
$765$		0			0
$766$	−411.360			−0.537024
$767$	−409.571			−0.533991
$768$		0			0
$769$	932.372			1.21245			0.606223	−	0.795294i	$-0.292684\pi$
$769$	932.372			1.21245			0.606223	+	0.795294i	$0.292684\pi$
$770$	−38.5553	−	172.929i	−0.0500719	−	0.224582i
$771$		0			0
$772$		−	172.475i		−	0.223413i
$773$	280.315			0.362632			0.181316	−	0.983425i	$-0.441964\pi$
$773$	280.315			0.362632			0.181316	+	0.983425i	$0.441964\pi$
$774$		0			0
$775$	−685.827	+	321.815i	−0.884938	+	0.415245i
$776$			456.137i			0.587805i
$777$		0			0
$778$		−	1663.44i		−	2.13810i
$779$			250.797i			0.321948i
$780$		0			0
$781$	−606.231			−0.776224
$782$	−1588.09			−2.03080
$783$		0			0
$784$	−719.634			−0.917901
$785$	−911.307	+	203.181i	−1.16090	+	0.258829i
$786$		0			0
$787$		−	707.211i		−	0.898617i	−0.893377	−	0.449308i	$-0.851670\pi$
$787$		−	707.211i		−	0.898617i	0.893377	−	0.449308i	$-0.148330\pi$
$788$	147.329			0.186966
$789$		0			0
$790$	122.696	+	550.318i	0.155312	+	0.696605i
$791$			483.306i			0.611006i
$792$		0			0
$793$			205.635i			0.259313i
$794$			953.688i			1.20112i
$795$		0			0
$796$	−31.2840			−0.0393014
$797$	−1346.08			−1.68894			−0.844468	−	0.535606i	$-0.820083\pi$
$797$	−1346.08			−1.68894			−0.844468	+	0.535606i	$0.820083\pi$
$798$		0			0
$799$	1077.77			1.34890
$800$	426.499	−	200.129i	0.533124	−	0.250161i
$801$		0			0
$802$		−	1053.62i		−	1.31374i
$803$	607.550			0.756601
$804$		0			0
$805$	−501.835	+	111.887i	−0.623397	+	0.138990i
$806$			819.658i			1.01695i
$807$		0			0
$808$			321.508i			0.397906i
$809$			63.7611i			0.0788147i	0.999223	+	0.0394073i	$0.0125470\pi$
$809$			63.7611i			0.0788147i	−0.999223	+	0.0394073i	$0.987453\pi$
$810$		0			0
$811$	869.786			1.07249			0.536243	−	0.844064i	$-0.319843\pi$
$811$	869.786			1.07249			0.536243	+	0.844064i	$0.319843\pi$
$812$	−149.369			−0.183952
$813$		0			0
$814$	−52.6150			−0.0646376
$815$	−163.990	+	36.5625i	−0.201215	+	0.0448619i
$816$		0			0
$817$		−	752.619i		−	0.921198i
$818$	1215.11			1.48546
$819$		0			0
$820$	−134.431	+	29.9722i	−0.163941	+	0.0365515i
$821$		−	577.553i		−	0.703475i	−0.936099	−	0.351737i	$-0.885591\pi$
$821$		−	577.553i		−	0.703475i	0.936099	−	0.351737i	$-0.114409\pi$
$822$		0			0
$823$			347.744i			0.422532i	0.977429	+	0.211266i	$0.0677586\pi$
$823$			347.744i			0.422532i	−0.977429	+	0.211266i	$0.932241\pi$
$824$		−	1229.69i		−	1.49235i
$825$		0			0
$826$	−272.223			−0.329568
$827$	19.0847			0.0230770			0.0115385	−	0.999933i	$-0.496327\pi$
$827$	19.0847			0.0230770			0.0115385	+	0.999933i	$0.496327\pi$
$828$		0			0
$829$	−39.6514			−0.0478303			−0.0239152	−	0.999714i	$-0.507613\pi$
$829$	−39.6514			−0.0478303			−0.0239152	+	0.999714i	$0.507613\pi$
$830$	71.2803	+	319.707i	0.0858799	+	0.385189i
$831$		0			0
$832$			408.625i			0.491136i
$833$	−865.113			−1.03855
$834$		0			0
$835$	−54.4581	−	244.255i	−0.0652192	−	0.292522i
$836$			60.6339i			0.0725286i
$837$		0			0
$838$			1071.75i			1.27894i
$839$		−	377.259i		−	0.449653i	−0.974399	−	0.224827i	$-0.927818\pi$
$839$		−	377.259i		−	0.449653i	0.974399	−	0.224827i	$-0.0721816\pi$
$840$		0			0
$841$	−428.633			−0.509671
$842$	−503.693			−0.598210
$843$		0			0
$844$	93.5297			0.110817
$845$	31.2727	+	140.265i	0.0370091	+	0.165994i
$846$		0			0
$847$			347.183i			0.409898i
$848$	−1318.27			−1.55456
$849$		0			0
$850$	1204.63	−	565.257i	1.41722	−	0.665008i
$851$			152.688i			0.179421i
$852$		0			0
$853$		−	164.036i		−	0.192304i	−0.995367	−	0.0961521i	$-0.969346\pi$
$853$		−	164.036i		−	0.192304i	0.995367	−	0.0961521i	$-0.0306535\pi$
$854$			136.676i			0.160042i
$855$		0			0
$856$	820.590			0.958634
$857$	−315.386			−0.368011			−0.184006	−	0.982925i	$-0.558906\pi$
$857$	−315.386			−0.368011			−0.184006	+	0.982925i	$0.558906\pi$
$858$		0			0
$859$	−93.6399			−0.109010			−0.0545052	−	0.998513i	$-0.517358\pi$
$859$	−93.6399			−0.109010			−0.0545052	+	0.998513i	$0.517358\pi$
$860$	403.416	−	89.9436i	0.469088	−	0.104586i
$861$		0			0
$862$		−	1112.52i		−	1.29063i
$863$	1055.96			1.22360			0.611799	−	0.791013i	$-0.290446\pi$
$863$	1055.96			1.22360			0.611799	+	0.791013i	$0.290446\pi$
$864$		0			0
$865$	82.2145	+	368.749i	0.0950457	+	0.426299i
$866$		−	1057.05i		−	1.22061i
$867$		0			0
$868$			127.030i			0.146348i
$869$			222.263i			0.255769i
$870$		0			0
$871$	1164.03			1.33642
$872$	1201.80			1.37821
$873$		0			0
$874$	754.623			0.863413
$875$	340.839	−	263.492i	0.389530	−	0.301133i
$876$		0			0
$877$		−	211.260i		−	0.240890i	−0.992720	−	0.120445i	$-0.961568\pi$
$877$		−	211.260i		−	0.240890i	0.992720	−	0.120445i	$-0.0384321\pi$
$878$	1830.31			2.08464
$879$		0			0
$880$	425.884	−	94.9531i	0.483959	−	0.107901i
$881$		−	824.884i		−	0.936304i	−0.883648	−	0.468152i	$-0.844920\pi$
$881$		−	824.884i		−	0.936304i	0.883648	−	0.468152i	$-0.155080\pi$
$882$		0			0
$883$		−	1155.63i		−	1.30875i	−0.756171	−	0.654375i	$-0.772932\pi$
$883$		−	1155.63i		−	1.30875i	0.756171	−	0.654375i	$-0.227068\pi$
$884$		−	335.701i		−	0.379752i
$885$		0			0
$886$	169.742			0.191583
$887$	−1256.41			−1.41648			−0.708238	−	0.705973i	$-0.750510\pi$
$887$	−1256.41			−1.41648			−0.708238	+	0.705973i	$0.750510\pi$
$888$		0			0
$889$	43.7149			0.0491731
$890$	734.745	−	163.815i	0.825557	−	0.184062i
$891$		0			0
$892$			236.459i			0.265088i
$893$	−512.132			−0.573496
$894$		0			0
$895$	−568.317	+	126.709i	−0.634992	+	0.141575i
$896$			531.388i			0.593066i
$897$		0			0
$898$		−	818.723i		−	0.911719i
$899$			1079.76i			1.20106i
$900$		0			0
$901$	−1584.77			−1.75890
$902$	−232.850			−0.258148
$903$		0			0
$904$	−891.553			−0.986231
$905$	−49.3997	−	221.568i	−0.0545853	−	0.244826i
$906$		0			0
$907$			1154.46i			1.27283i	0.771345	+	0.636417i	$0.219584\pi$
$907$			1154.46i			1.27283i	−0.771345	+	0.636417i	$0.780416\pi$
$908$	−185.322			−0.204099
$909$		0			0
$910$	−101.432	−	454.945i	−0.111464	−	0.499940i
$911$		−	349.820i		−	0.383996i	−0.981395	−	0.191998i	$-0.938503\pi$
$911$		−	349.820i		−	0.383996i	0.981395	−	0.191998i	$-0.0614966\pi$
$912$		0			0
$913$			129.124i			0.141428i
$914$		−	264.436i		−	0.289318i
$915$		0			0
$916$	324.042			0.353757
$917$	394.111			0.429783
$918$		0			0
$919$	−375.011			−0.408064			−0.204032	−	0.978964i	$-0.565405\pi$
$919$	−375.011			−0.408064			−0.204032	+	0.978964i	$0.565405\pi$
$920$	−206.397	−	925.734i	−0.224345	−	1.00623i
$921$		0			0
$922$		−	456.028i		−	0.494607i
$923$	−1594.89			−1.72794
$924$		0			0
$925$	−54.3470	−	115.820i	−0.0587535	−	0.125211i
$926$		−	1363.70i		−	1.47268i
$927$		0			0
$928$		−	671.475i		−	0.723572i
$929$			1063.34i			1.14461i	0.820040	+	0.572306i	$0.193951\pi$
$929$			1063.34i			1.14461i	−0.820040	+	0.572306i	$0.806049\pi$
$930$		0			0
$931$	411.082			0.441549
$932$	130.750			0.140289
$933$		0			0
$934$	1382.12			1.47978
$935$	511.980	−	114.149i	0.547572	−	0.122084i
$936$		0			0
$937$		−	1021.59i		−	1.09028i	−0.838344	−	0.545141i	$-0.816476\pi$
$937$		−	1021.59i		−	1.09028i	0.838344	−	0.545141i	$-0.183524\pi$
$938$	773.673			0.824812
$939$		0			0
$940$	−61.2036	−	274.511i	−0.0651102	−	0.292033i
$941$		−	437.037i		−	0.464439i	−0.972663	−	0.232220i	$-0.925401\pi$
$941$		−	437.037i		−	0.464439i	0.972663	−	0.232220i	$-0.0745988\pi$
$942$		0			0
$943$			675.725i			0.716569i
$944$		−	670.424i		−	0.710195i
$945$		0			0
$946$	698.759			0.738646
$947$	1658.44			1.75126			0.875631	−	0.482981i	$-0.160446\pi$
$947$	1658.44			1.75126			0.875631	+	0.482981i	$0.160446\pi$
$948$		0			0
$949$	1598.36			1.68426
$950$	−572.414	+	268.597i	−0.602541	+	0.282734i
$951$		0			0
$952$			510.654i			0.536401i
$953$	−413.980			−0.434397			−0.217198	−	0.976127i	$-0.569692\pi$
$953$	−413.980			−0.434397			−0.217198	+	0.976127i	$0.569692\pi$
$954$		0			0
$955$	−730.104	+	162.781i	−0.764507	+	0.170451i
$956$			379.131i			0.396580i
$957$		0			0
$958$		−	1234.29i		−	1.28840i
$959$			66.2579i			0.0690906i
$960$		0			0
$961$	−42.7229			−0.0444567
$962$	−138.421			−0.143889
$963$		0			0
$964$	−194.999			−0.202281
$965$	692.020	−	154.290i	0.717119	−	0.159886i
$966$		0			0
$967$			102.641i			0.106144i	0.998591	+	0.0530720i	$0.0169013\pi$
$967$			102.641i			0.106144i	−0.998591	+	0.0530720i	$0.983099\pi$
$968$	−640.448			−0.661620
$969$		0			0
$970$	799.667	−	178.290i	0.824399	−	0.183804i
$971$			997.502i			1.02729i	0.858002	+	0.513647i	$0.171706\pi$
$971$			997.502i			1.02729i	−0.858002	+	0.513647i	$0.828294\pi$
$972$		0			0
$973$		−	193.428i		−	0.198795i
$974$		−	1291.39i		−	1.32586i
$975$		0			0
$976$	−336.602			−0.344879
$977$	1261.86			1.29157			0.645785	−	0.763520i	$-0.276530\pi$
$977$	1261.86			1.29157			0.645785	+	0.763520i	$0.276530\pi$
$978$		0			0
$979$	296.751			0.303116
$980$	49.1274	+	220.346i	0.0501300	+	0.224843i
$981$		0			0
$982$		−	609.855i		−	0.621033i
$983$	−1036.20			−1.05412			−0.527058	−	0.849829i	$-0.676705\pi$
$983$	−1036.20			−1.05412			−0.527058	+	0.849829i	$0.676705\pi$
$984$		0			0
$985$	131.795	+	591.127i	0.133802	+	0.600129i
$986$		−	1896.56i		−	1.92349i
$987$		0			0
$988$			159.517i			0.161455i
$989$		−	2027.78i		−	2.05034i
$990$		0			0
$991$	−1503.88			−1.51754			−0.758770	−	0.651358i	$-0.774200\pi$
$991$	−1503.88			−1.51754			−0.758770	+	0.651358i	$0.774200\pi$
$992$	−571.054			−0.575660
$993$		0			0
$994$	−1060.05			−1.06645
$995$	−27.9854	−	125.520i	−0.0281261	−	0.126151i
$996$		0			0
$997$			334.836i			0.335843i	0.985800	+	0.167922i	$0.0537056\pi$
$997$			334.836i			0.335843i	−0.985800	+	0.167922i	$0.946294\pi$
$998$	−1416.77			−1.41961
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.b.404.20	yes	24
3.2	odd	2	inner	405.3.d.b.404.5	✓	24
5.4	even	2	inner	405.3.d.b.404.6	yes	24
9.2	odd	6		405.3.h.k.134.20		48
9.4	even	3		405.3.h.k.269.6		48
9.5	odd	6		405.3.h.k.269.19		48
9.7	even	3		405.3.h.k.134.5		48
15.14	odd	2	inner	405.3.d.b.404.19	yes	24
45.4	even	6		405.3.h.k.269.20		48
45.14	odd	6		405.3.h.k.269.5		48
45.29	odd	6		405.3.h.k.134.6		48
45.34	even	6		405.3.h.k.134.19		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.3.d.b.404.5	✓	24	3.2	odd	2	inner
405.3.d.b.404.6	yes	24	5.4	even	2	inner
405.3.d.b.404.19	yes	24	15.14	odd	2	inner
405.3.d.b.404.20	yes	24	1.1	even	1	trivial
405.3.h.k.134.5		48	9.7	even	3
405.3.h.k.134.6		48	45.29	odd	6
405.3.h.k.134.19		48	45.34	even	6
405.3.h.k.134.20		48	9.2	odd	6
405.3.h.k.269.5		48	45.14	odd	6
405.3.h.k.269.6		48	9.4	even	3
405.3.h.k.269.19		48	9.5	odd	6
405.3.h.k.269.20		48	45.4	even	6

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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.28392 q^{2} +1.21631 q^{4} +(1.08806 + 4.88018i) q^{5} +3.44650i q^{7} -6.35774 q^{8} +(2.48505 + 11.1459i) q^{10} +4.50165i q^{11} +11.8431i q^{13} +7.87153i q^{14} -19.3858 q^{16} -23.3048 q^{17} +11.0739 q^{19} +(1.32342 + 5.93579i) q^{20} +10.2814i q^{22} +29.8365 q^{23} +(-22.6322 + 10.6199i) q^{25} +27.0487i q^{26} +4.19199i q^{28} +35.6319i q^{29} +30.3031 q^{31} -18.8448 q^{32} -53.2264 q^{34} +(-16.8195 + 3.75000i) q^{35} +5.11748i q^{37} +25.2919 q^{38} +(-6.91761 - 31.0269i) q^{40} +22.6476i q^{41} -67.9633i q^{43} +5.47538i q^{44} +68.1442 q^{46} -46.2467 q^{47} +37.1217 q^{49} +(-51.6903 + 24.2550i) q^{50} +14.4048i q^{52} +68.0017 q^{53} +(-21.9688 + 4.89807i) q^{55} -21.9119i q^{56} +81.3806i q^{58} +34.5832i q^{59} +17.3633 q^{61} +69.2099 q^{62} +34.5033 q^{64} +(-57.7963 + 12.8860i) q^{65} -98.2875i q^{67} -28.3458 q^{68} +(-38.4145 + 8.56471i) q^{70} +134.669i q^{71} -134.962i q^{73} +11.6879i q^{74} +13.4693 q^{76} -15.5149 q^{77} +49.3738 q^{79} +(-21.0930 - 94.6062i) q^{80} +51.7254i q^{82} +28.6837 q^{83} +(-25.3571 - 113.732i) q^{85} -155.223i q^{86} -28.6203i q^{88} -65.9204i q^{89} -40.8171 q^{91} +36.2903 q^{92} -105.624 q^{94} +(12.0491 + 54.0426i) q^{95} -71.7451i q^{97} +84.7830 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4} + 12 q^{10} + 96 q^{16} + 48 q^{25} + 144 q^{34} + 72 q^{40} - 168 q^{46} - 288 q^{49} - 132 q^{55} - 360 q^{61} - 72 q^{64} - 156 q^{70} + 48 q^{76} - 480 q^{79} - 456 q^{85} - 48 q^{91}+ \cdots + 384 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 + 12 * q^10 + 96 * q^16 + 48 * q^25 + 144 * q^34 + 72 * q^40 - 168 * q^46 - 288 * q^49 - 132 * q^55 - 360 * q^61 - 72 * q^64 - 156 * q^70 + 48 * q^76 - 480 * q^79 - 456 * q^85 - 48 * q^91 + 384 * q^94

Introduction

L-functions

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