LMFDB - Embedded newform 1380.3.d.a.781.26

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1380,3,Mod(781,1380)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

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

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

chi := DirichletCharacter("1380.781");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1380.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$37.6022764817$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			781.26
Character	$\chi$	$=$	1380.781
Dual form			1380.3.d.a.781.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{3} +2.23607i q^{5} -4.23458i q^{7} +3.00000 q^{9} +O(q^{10})$	$q+1.73205 q^{3} +2.23607i q^{5} -4.23458i q^{7} +3.00000 q^{9} +3.42048i q^{11} -13.7692 q^{13} +3.87298i q^{15} -5.48070i q^{17} +4.66120i q^{19} -7.33451i q^{21} +(15.0674 + 17.3774i) q^{23} -5.00000 q^{25} +5.19615 q^{27} +51.3592 q^{29} +36.0494 q^{31} +5.92445i q^{33} +9.46881 q^{35} +70.1247i q^{37} -23.8489 q^{39} -29.1052 q^{41} +24.5556i q^{43} +6.70820i q^{45} +58.9027 q^{47} +31.0683 q^{49} -9.49285i q^{51} -26.4800i q^{53} -7.64843 q^{55} +8.07343i q^{57} -8.79966 q^{59} -30.1183i q^{61} -12.7037i q^{63} -30.7888i q^{65} -5.12183i q^{67} +(26.0975 + 30.0985i) q^{69} -45.7745 q^{71} +75.2188 q^{73} -8.66025 q^{75} +14.4843 q^{77} +62.2814i q^{79} +9.00000 q^{81} +137.687i q^{83} +12.2552 q^{85} +88.9568 q^{87} -64.5505i q^{89} +58.3066i q^{91} +62.4395 q^{93} -10.4228 q^{95} +171.496i q^{97} +10.2614i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 96 q^{9}+O(q^{10}) $ 32 * q + 96 * q^9	$ 32 q + 96 q^{9} + 24 q^{13} + 64 q^{23} - 160 q^{25} - 60 q^{29} - 4 q^{31} - 60 q^{35} - 48 q^{39} - 108 q^{41} + 136 q^{47} - 428 q^{49} + 120 q^{55} - 84 q^{59} + 48 q^{69} + 188 q^{71} + 472 q^{73} - 120 q^{77} + 288 q^{81} + 60 q^{85} - 72 q^{87} - 192 q^{93} + 80 q^{95}+O(q^{100}) $ 32 * q + 96 * q^9 + 24 * q^13 + 64 * q^23 - 160 * q^25 - 60 * q^29 - 4 * q^31 - 60 * q^35 - 48 * q^39 - 108 * q^41 + 136 * q^47 - 428 * q^49 + 120 * q^55 - 84 * q^59 + 48 * q^69 + 188 * q^71 + 472 * q^73 - 120 * q^77 + 288 * q^81 + 60 * q^85 - 72 * q^87 - 192 * q^93 + 80 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1380\mathbb{Z}\right)^\times$.

$n$	$277$	$461$	$691$	$1201$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$			2.23607i			0.447214i
$5$			2.23607i			0.447214i
$6$		0			0
$7$		−	4.23458i		−	0.604940i	−0.953159	−	0.302470i	$-0.902189\pi$
$7$		−	4.23458i		−	0.604940i	0.953159	−	0.302470i	$-0.0978112\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$			3.42048i			0.310953i	0.987840	+	0.155476i	$0.0496913\pi$
$11$			3.42048i			0.310953i	−0.987840	+	0.155476i	$0.950309\pi$
$12$		0			0
$13$	−13.7692			−1.05917			−0.529583	−	0.848258i	$-0.677652\pi$
$13$	−13.7692			−1.05917			−0.529583	+	0.848258i	$0.677652\pi$
$14$		0			0
$15$			3.87298i			0.258199i
$16$		0			0
$17$		−	5.48070i		−	0.322394i	−0.986922	−	0.161197i	$-0.948465\pi$
$17$		−	5.48070i		−	0.322394i	0.986922	−	0.161197i	$-0.0515355\pi$
$18$		0			0
$19$			4.66120i			0.245326i	0.992448	+	0.122663i	$0.0391435\pi$
$19$			4.66120i			0.245326i	−0.992448	+	0.122663i	$0.960857\pi$
$20$		0			0
$21$		−	7.33451i		−	0.349262i
$22$		0			0
$23$	15.0674	+	17.3774i	0.655104	+	0.755539i
$23$	15.0674	+	17.3774i	0.655104	+	0.755539i
$24$		0			0
$25$	−5.00000			−0.200000
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	51.3592			1.77101			0.885504	−	0.464633i	$-0.153814\pi$
$29$	51.3592			1.77101			0.885504	+	0.464633i	$0.153814\pi$
$30$		0			0
$31$	36.0494			1.16288			0.581442	−	0.813588i	$-0.302489\pi$
$31$	36.0494			1.16288			0.581442	+	0.813588i	$0.302489\pi$
$32$		0			0
$33$			5.92445i			0.179529i
$34$		0			0
$35$	9.46881			0.270537
$36$		0			0
$37$			70.1247i			1.89526i	0.319369	+	0.947631i	$0.396529\pi$
$37$			70.1247i			1.89526i	−0.319369	+	0.947631i	$0.603471\pi$
$38$		0			0
$39$	−23.8489			−0.611510
$40$		0			0
$41$	−29.1052			−0.709883			−0.354942	−	0.934888i	$-0.615499\pi$
$41$	−29.1052			−0.709883			−0.354942	+	0.934888i	$0.615499\pi$
$42$		0			0
$43$			24.5556i			0.571062i	0.958370	+	0.285531i	$0.0921699\pi$
$43$			24.5556i			0.571062i	−0.958370	+	0.285531i	$0.907830\pi$
$44$		0			0
$45$			6.70820i			0.149071i
$46$		0			0
$47$	58.9027			1.25325			0.626624	−	0.779322i	$-0.284436\pi$
$47$	58.9027			1.25325			0.626624	+	0.779322i	$0.284436\pi$
$48$		0			0
$49$	31.0683			0.634048
$50$		0			0
$51$		−	9.49285i		−	0.186134i
$52$		0			0
$53$		−	26.4800i		−	0.499622i	−0.968295	−	0.249811i	$-0.919632\pi$
$53$		−	26.4800i		−	0.499622i	0.968295	−	0.249811i	$-0.0803685\pi$
$54$		0			0
$55$	−7.64843			−0.139062
$56$		0			0
$57$			8.07343i			0.141639i
$58$		0			0
$59$	−8.79966			−0.149147			−0.0745734	−	0.997216i	$-0.523759\pi$
$59$	−8.79966			−0.149147			−0.0745734	+	0.997216i	$0.523759\pi$
$60$		0			0
$61$		−	30.1183i		−	0.493743i	−0.969048	−	0.246872i	$-0.920597\pi$
$61$		−	30.1183i		−	0.493743i	0.969048	−	0.246872i	$-0.0794026\pi$
$62$		0			0
$63$		−	12.7037i		−	0.201647i
$64$		0			0
$65$		−	30.7888i		−	0.473674i
$66$		0			0
$67$		−	5.12183i		−	0.0764452i	−0.999269	−	0.0382226i	$-0.987830\pi$
$67$		−	5.12183i		−	0.0764452i	0.999269	−	0.0382226i	$-0.0121696\pi$
$68$		0			0
$69$	26.0975	+	30.0985i	0.378225	+	0.436210i
$70$		0			0
$71$	−45.7745			−0.644711			−0.322355	−	0.946619i	$-0.604475\pi$
$71$	−45.7745			−0.644711			−0.322355	+	0.946619i	$0.604475\pi$
$72$		0			0
$73$	75.2188			1.03039			0.515197	−	0.857072i	$-0.327719\pi$
$73$	75.2188			1.03039			0.515197	+	0.857072i	$0.327719\pi$
$74$		0			0
$75$	−8.66025			−0.115470
$76$		0			0
$77$	14.4843			0.188108
$78$		0			0
$79$			62.2814i			0.788372i	0.919031	+	0.394186i	$0.128973\pi$
$79$			62.2814i			0.788372i	−0.919031	+	0.394186i	$0.871027\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$			137.687i			1.65888i	0.558596	+	0.829440i	$0.311340\pi$
$83$			137.687i			1.65888i	−0.558596	+	0.829440i	$0.688660\pi$
$84$		0			0
$85$	12.2552			0.144179
$86$		0			0
$87$	88.9568			1.02249
$88$		0			0
$89$		−	64.5505i		−	0.725286i	−0.931928	−	0.362643i	$-0.881874\pi$
$89$		−	64.5505i		−	0.725286i	0.931928	−	0.362643i	$-0.118126\pi$
$90$		0			0
$91$			58.3066i			0.640732i
$92$		0			0
$93$	62.4395			0.671392
$94$		0			0
$95$	−10.4228			−0.109713
$96$		0			0
$97$			171.496i			1.76800i	0.467486	+	0.884000i	$0.345160\pi$
$97$			171.496i			1.76800i	−0.467486	+	0.884000i	$0.654840\pi$
$98$		0			0
$99$			10.2614i			0.103651i
$100$		0			0
$101$	52.1005			0.515847			0.257923	−	0.966165i	$-0.416962\pi$
$101$	52.1005			0.515847			0.257923	+	0.966165i	$0.416962\pi$
$102$		0			0
$103$		−	32.9011i		−	0.319429i	−0.987163	−	0.159714i	$-0.948943\pi$
$103$		−	32.9011i		−	0.319429i	0.987163	−	0.159714i	$-0.0510573\pi$
$104$		0			0
$105$	16.4005			0.156195
$106$		0			0
$107$		−	35.3025i		−	0.329930i	−0.986299	−	0.164965i	$-0.947249\pi$
$107$		−	35.3025i		−	0.329930i	0.986299	−	0.164965i	$-0.0527511\pi$
$108$		0			0
$109$		−	56.0630i		−	0.514339i	−0.966366	−	0.257170i	$-0.917210\pi$
$109$		−	56.0630i		−	0.514339i	0.966366	−	0.257170i	$-0.0827899\pi$
$110$		0			0
$111$			121.459i			1.09423i
$112$		0			0
$113$		−	79.6420i		−	0.704796i	−0.935850	−	0.352398i	$-0.885366\pi$
$113$		−	79.6420i		−	0.704796i	0.935850	−	0.352398i	$-0.114634\pi$
$114$		0			0
$115$	−38.8570	+	33.6917i	−0.337887	+	0.292971i
$116$		0			0
$117$	−41.3075			−0.353056
$118$		0			0
$119$	−23.2085			−0.195029
$120$		0			0
$121$	109.300			0.903308
$122$		0			0
$123$	−50.4117			−0.409851
$124$		0			0
$125$		−	11.1803i		−	0.0894427i
$126$		0			0
$127$	−48.5699			−0.382440			−0.191220	−	0.981547i	$-0.561244\pi$
$127$	−48.5699			−0.382440			−0.191220	+	0.981547i	$0.561244\pi$
$128$		0			0
$129$			42.5316i			0.329703i
$130$		0			0
$131$	3.25291			0.0248314			0.0124157	−	0.999923i	$-0.496048\pi$
$131$	3.25291			0.0248314			0.0124157	+	0.999923i	$0.496048\pi$
$132$		0			0
$133$	19.7382			0.148408
$134$		0			0
$135$			11.6190i			0.0860663i
$136$		0			0
$137$		−	0.210618i		−	0.00153735i	−1.00000		0.000768677i	$-0.999755\pi$
$137$		−	0.210618i		−	0.00153735i	1.00000		0.000768677i	$-0.000244678\pi$
$138$		0			0
$139$	34.9353			0.251333			0.125666	−	0.992073i	$-0.459893\pi$
$139$	34.9353			0.251333			0.125666	+	0.992073i	$0.459893\pi$
$140$		0			0
$141$	102.022			0.723563
$142$		0			0
$143$		−	47.0972i		−	0.329351i
$144$		0			0
$145$			114.843i			0.792018i
$146$		0			0
$147$	53.8119			0.366068
$148$		0			0
$149$			294.015i			1.97326i	0.162983	+	0.986629i	$0.447889\pi$
$149$			294.015i			1.97326i	−0.162983	+	0.986629i	$0.552111\pi$
$150$		0			0
$151$	−76.5200			−0.506755			−0.253377	−	0.967368i	$-0.581541\pi$
$151$	−76.5200			−0.506755			−0.253377	+	0.967368i	$0.581541\pi$
$152$		0			0
$153$		−	16.4421i		−	0.107465i
$154$		0			0
$155$			80.6090i			0.520058i
$156$		0			0
$157$			118.942i			0.757592i	0.925480	+	0.378796i	$0.123662\pi$
$157$			118.942i			0.757592i	−0.925480	+	0.378796i	$0.876338\pi$
$158$		0			0
$159$		−	45.8646i		−	0.288457i
$160$		0			0
$161$	73.5859	−	63.8041i	0.457055	−	0.396299i
$162$		0			0
$163$	285.654			1.75248			0.876240	−	0.481874i	$-0.160044\pi$
$163$	285.654			1.75248			0.876240	+	0.481874i	$0.160044\pi$
$164$		0			0
$165$	−13.2475			−0.0802877
$166$		0			0
$167$	−144.508			−0.865316			−0.432658	−	0.901558i	$-0.642424\pi$
$167$	−144.508			−0.865316			−0.432658	+	0.901558i	$0.642424\pi$
$168$		0			0
$169$	20.5900			0.121834
$170$		0			0
$171$			13.9836i			0.0817754i
$172$		0			0
$173$	−118.670			−0.685954			−0.342977	−	0.939344i	$-0.611435\pi$
$173$	−118.670			−0.685954			−0.342977	+	0.939344i	$0.611435\pi$
$174$		0			0
$175$			21.1729i			0.120988i
$176$		0			0
$177$	−15.2415			−0.0861099
$178$		0			0
$179$	−12.2624			−0.0685053			−0.0342526	−	0.999413i	$-0.510905\pi$
$179$	−12.2624			−0.0685053			−0.0342526	+	0.999413i	$0.510905\pi$
$180$		0			0
$181$		−	228.077i		−	1.26009i	−0.776557	−	0.630047i	$-0.783035\pi$
$181$		−	228.077i		−	1.26009i	0.776557	−	0.630047i	$-0.216965\pi$
$182$		0			0
$183$		−	52.1665i		−	0.285063i
$184$		0			0
$185$	−156.804			−0.847587
$186$		0			0
$187$	18.7466			0.100249
$188$		0			0
$189$		−	22.0035i		−	0.116421i
$190$		0			0
$191$		−	266.736i		−	1.39652i	−0.715843	−	0.698261i	$-0.753958\pi$
$191$		−	266.736i		−	1.39652i	0.715843	−	0.698261i	$-0.246042\pi$
$192$		0			0
$193$	133.378			0.691077			0.345538	−	0.938405i	$-0.387696\pi$
$193$	133.378			0.691077			0.345538	+	0.938405i	$0.387696\pi$
$194$		0			0
$195$		−	53.3278i		−	0.273476i
$196$		0			0
$197$	−119.937			−0.608819			−0.304409	−	0.952541i	$-0.598459\pi$
$197$	−119.937			−0.608819			−0.304409	+	0.952541i	$0.598459\pi$
$198$		0			0
$199$		−	37.3932i		−	0.187905i	−0.995577	−	0.0939526i	$-0.970050\pi$
$199$		−	37.3932i		−	0.187905i	0.995577	−	0.0939526i	$-0.0299502\pi$
$200$		0			0
$201$		−	8.87126i		−	0.0441356i
$202$		0			0
$203$		−	217.485i		−	1.07135i
$204$		0			0
$205$		−	65.0812i		−	0.317469i
$206$		0			0
$207$	45.2022	+	52.1322i	0.218368	+	0.251846i
$208$		0			0
$209$	−15.9435			−0.0762849
$210$		0			0
$211$	145.771			0.690858			0.345429	−	0.938445i	$-0.387733\pi$
$211$	145.771			0.690858			0.345429	+	0.938445i	$0.387733\pi$
$212$		0			0
$213$	−79.2837			−0.372224
$214$		0			0
$215$	−54.9081			−0.255387
$216$		0			0
$217$		−	152.654i		−	0.703476i
$218$		0			0
$219$	130.283			0.594899
$220$		0			0
$221$			75.4647i			0.341469i
$222$		0			0
$223$	202.014			0.905894			0.452947	−	0.891538i	$-0.350373\pi$
$223$	202.014			0.905894			0.452947	+	0.891538i	$0.350373\pi$
$224$		0			0
$225$	−15.0000			−0.0666667
$226$		0			0
$227$			222.735i			0.981213i	0.871381	+	0.490607i	$0.163225\pi$
$227$			222.735i			0.981213i	−0.871381	+	0.490607i	$0.836775\pi$
$228$		0			0
$229$		−	10.7736i		−	0.0470461i	−0.999723	−	0.0235231i	$-0.992512\pi$
$229$		−	10.7736i		−	0.0470461i	0.999723	−	0.0235231i	$-0.00748832\pi$
$230$		0			0
$231$	25.0875			0.108604
$232$		0			0
$233$	306.711			1.31636			0.658179	−	0.752862i	$-0.271327\pi$
$233$	306.711			1.31636			0.658179	+	0.752862i	$0.271327\pi$
$234$		0			0
$235$			131.710i			0.560470i
$236$		0			0
$237$			107.875i			0.455167i
$238$		0			0
$239$	−133.517			−0.558647			−0.279323	−	0.960197i	$-0.590110\pi$
$239$	−133.517			−0.558647			−0.279323	+	0.960197i	$0.590110\pi$
$240$		0			0
$241$			164.609i			0.683023i	0.939878	+	0.341511i	$0.110939\pi$
$241$			164.609i			0.683023i	−0.939878	+	0.341511i	$0.889061\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$			69.4709i			0.283555i
$246$		0			0
$247$		−	64.1808i		−	0.259841i
$248$		0			0
$249$			238.481i			0.957754i
$250$		0			0
$251$			214.404i			0.854200i	0.904205	+	0.427100i	$0.140465\pi$
$251$			214.404i			0.854200i	−0.904205	+	0.427100i	$0.859535\pi$
$252$		0			0
$253$	−59.4390	+	51.5378i	−0.234937	+	0.203707i
$254$		0			0
$255$	21.2267			0.0832418
$256$		0			0
$257$	155.697			0.605827			0.302913	−	0.953018i	$-0.402041\pi$
$257$	155.697			0.605827			0.302913	+	0.953018i	$0.402041\pi$
$258$		0			0
$259$	296.948			1.14652
$260$		0			0
$261$	154.078			0.590336
$262$		0			0
$263$		−	508.041i		−	1.93171i	−0.259077	−	0.965857i	$-0.583418\pi$
$263$		−	508.041i		−	1.93171i	0.259077	−	0.965857i	$-0.416582\pi$
$264$		0			0
$265$	59.2110			0.223438
$266$		0			0
$267$		−	111.805i		−	0.418744i
$268$		0			0
$269$	−137.722			−0.511977			−0.255988	−	0.966680i	$-0.582401\pi$
$269$	−137.722			−0.511977			−0.255988	+	0.966680i	$0.582401\pi$
$270$		0			0
$271$	−449.193			−1.65754			−0.828769	−	0.559590i	$-0.810958\pi$
$271$	−449.193			−1.65754			−0.828769	+	0.559590i	$0.810958\pi$
$272$		0			0
$273$			100.990i			0.369927i
$274$		0			0
$275$		−	17.1024i		−	0.0621906i
$276$		0			0
$277$	−265.137			−0.957173			−0.478586	−	0.878041i	$-0.658851\pi$
$277$	−265.137			−0.957173			−0.478586	+	0.878041i	$0.658851\pi$
$278$		0			0
$279$	108.148			0.387628
$280$		0			0
$281$		−	81.2479i		−	0.289138i	−0.989495	−	0.144569i	$-0.953820\pi$
$281$		−	81.2479i		−	0.289138i	0.989495	−	0.144569i	$-0.0461796\pi$
$282$		0			0
$283$			279.323i			0.987009i	0.869743	+	0.493504i	$0.164284\pi$
$283$			279.323i			0.987009i	−0.869743	+	0.493504i	$0.835716\pi$
$284$		0			0
$285$	−18.0527			−0.0633430
$286$		0			0
$287$			123.248i			0.429437i
$288$		0			0
$289$	258.962			0.896062
$290$		0			0
$291$			297.040i			1.02076i
$292$		0			0
$293$		−	144.258i		−	0.492348i	−0.969226	−	0.246174i	$-0.920827\pi$
$293$		−	144.258i		−	0.492348i	0.969226	−	0.246174i	$-0.0791734\pi$
$294$		0			0
$295$		−	19.6766i		−	0.0667005i
$296$		0			0
$297$			17.7733i			0.0598429i
$298$		0			0
$299$	−207.466	−	239.272i	−0.693865	−	0.800241i
$300$		0			0
$301$	103.983			0.345458
$302$		0			0
$303$	90.2408			0.297824
$304$		0			0
$305$	67.3466			0.220809
$306$		0			0
$307$	−463.385			−1.50940			−0.754698	−	0.656072i	$-0.772217\pi$
$307$	−463.385			−1.50940			−0.754698	+	0.656072i	$0.772217\pi$
$308$		0			0
$309$		−	56.9865i		−	0.184422i
$310$		0			0
$311$	−365.413			−1.17496			−0.587480	−	0.809238i	$-0.699880\pi$
$311$	−365.413			−1.17496			−0.587480	+	0.809238i	$0.699880\pi$
$312$		0			0
$313$		−	224.561i		−	0.717449i	−0.933444	−	0.358724i	$-0.883212\pi$
$313$		−	224.561i		−	0.717449i	0.933444	−	0.358724i	$-0.116788\pi$
$314$		0			0
$315$	28.4064			0.0901791
$316$		0			0
$317$	112.230			0.354037			0.177018	−	0.984208i	$-0.443355\pi$
$317$	112.230			0.354037			0.177018	+	0.984208i	$0.443355\pi$
$318$		0			0
$319$			175.673i			0.550700i
$320$		0			0
$321$		−	61.1457i		−	0.190485i
$322$		0			0
$323$	25.5466			0.0790918
$324$		0			0
$325$	68.8458			0.211833
$326$		0			0
$327$		−	97.1039i		−	0.296954i
$328$		0			0
$329$		−	249.428i		−	0.758140i
$330$		0			0
$331$	446.817			1.34990			0.674950	−	0.737863i	$-0.264165\pi$
$331$	446.817			1.34990			0.674950	+	0.737863i	$0.264165\pi$
$332$		0			0
$333$			210.374i			0.631754i
$334$		0			0
$335$	11.4528			0.0341873
$336$		0			0
$337$			615.666i			1.82690i	0.406949	+	0.913451i	$0.366593\pi$
$337$			615.666i			1.82690i	−0.406949	+	0.913451i	$0.633407\pi$
$338$		0			0
$339$		−	137.944i		−	0.406914i
$340$		0			0
$341$			123.306i			0.361602i
$342$		0			0
$343$		−	339.056i		−	0.988501i
$344$		0			0
$345$	−67.3023	+	58.3558i	−0.195079	+	0.169147i
$346$		0			0
$347$	−499.046			−1.43817			−0.719087	−	0.694920i	$-0.755440\pi$
$347$	−499.046			−1.43817			−0.719087	+	0.694920i	$0.755440\pi$
$348$		0			0
$349$	−380.579			−1.09048			−0.545242	−	0.838279i	$-0.683562\pi$
$349$	−380.579			−1.09048			−0.545242	+	0.838279i	$0.683562\pi$
$350$		0			0
$351$	−71.5467			−0.203837
$352$		0			0
$353$	−419.035			−1.18707			−0.593535	−	0.804809i	$-0.702268\pi$
$353$	−419.035			−1.18707			−0.593535	+	0.804809i	$0.702268\pi$
$354$		0			0
$355$		−	102.355i		−	0.288323i
$356$		0			0
$357$	−40.1982			−0.112600
$358$		0			0
$359$		−	319.353i		−	0.889561i	−0.895640	−	0.444781i	$-0.853282\pi$
$359$		−	319.353i		−	0.889561i	0.895640	−	0.444781i	$-0.146718\pi$
$360$		0			0
$361$	339.273			0.939815
$362$		0			0
$363$	189.314			0.521525
$364$		0			0
$365$			168.194i			0.460807i
$366$		0			0
$367$			109.262i			0.297716i	0.988859	+	0.148858i	$0.0475597\pi$
$367$			109.262i			0.297716i	−0.988859	+	0.148858i	$0.952440\pi$
$368$		0			0
$369$	−87.3156			−0.236628
$370$		0			0
$371$	−112.131			−0.302241
$372$		0			0
$373$			597.530i			1.60196i	0.598694	+	0.800978i	$0.295687\pi$
$373$			597.530i			1.60196i	−0.598694	+	0.800978i	$0.704313\pi$
$374$		0			0
$375$		−	19.3649i		−	0.0516398i
$376$		0			0
$377$	−707.174			−1.87579
$378$		0			0
$379$		−	518.720i		−	1.36865i	−0.729175	−	0.684327i	$-0.760096\pi$
$379$		−	518.720i		−	1.36865i	0.729175	−	0.684327i	$-0.239904\pi$
$380$		0			0
$381$	−84.1255			−0.220802
$382$		0			0
$383$		−	130.657i		−	0.341142i	−0.985345	−	0.170571i	$-0.945439\pi$
$383$		−	130.657i		−	0.341142i	0.985345	−	0.170571i	$-0.0545612\pi$
$384$		0			0
$385$			32.3879i			0.0841244i
$386$		0			0
$387$			73.6669i			0.190354i
$388$		0			0
$389$		−	641.056i		−	1.64796i	−0.566619	−	0.823980i	$-0.691749\pi$
$389$		−	641.056i		−	1.64796i	0.566619	−	0.823980i	$-0.308251\pi$
$390$		0			0
$391$	95.2403	−	82.5799i	0.243581	−	0.211202i
$392$		0			0
$393$	5.63420			0.0143364
$394$		0			0
$395$	−139.265			−0.352571
$396$		0			0
$397$	66.6553			0.167897			0.0839487	−	0.996470i	$-0.473247\pi$
$397$	66.6553			0.167897			0.0839487	+	0.996470i	$0.473247\pi$
$398$		0			0
$399$	34.1876			0.0856832
$400$		0			0
$401$			121.218i			0.302290i	0.988512	+	0.151145i	$0.0482960\pi$
$401$			121.218i			0.302290i	−0.988512	+	0.151145i	$0.951704\pi$
$402$		0			0
$403$	−496.371			−1.23169
$404$		0			0
$405$			20.1246i			0.0496904i
$406$		0			0
$407$	−239.860			−0.589337
$408$		0			0
$409$	534.724			1.30739			0.653697	−	0.756756i	$-0.273217\pi$
$409$	534.724			1.30739			0.653697	+	0.756756i	$0.273217\pi$
$410$		0			0
$411$		−	0.364800i		−	0.000887592i
$412$		0			0
$413$			37.2628i			0.0902248i
$414$		0			0
$415$	−307.877			−0.741873
$416$		0			0
$417$	60.5096			0.145107
$418$		0			0
$419$		−	431.194i		−	1.02910i	−0.857460	−	0.514551i	$-0.827959\pi$
$419$		−	431.194i		−	1.02910i	0.857460	−	0.514551i	$-0.172041\pi$
$420$		0			0
$421$			111.757i			0.265456i	0.991153	+	0.132728i	$0.0423737\pi$
$421$			111.757i			0.265456i	−0.991153	+	0.132728i	$0.957626\pi$
$422$		0			0
$423$	176.708			0.417749
$424$		0			0
$425$			27.4035i			0.0644788i
$426$		0			0
$427$	−127.538			−0.298685
$428$		0			0
$429$		−	81.5747i		−	0.190151i
$430$		0			0
$431$		−	811.648i		−	1.88317i	−0.336769	−	0.941587i	$-0.609334\pi$
$431$		−	811.648i		−	1.88317i	0.336769	−	0.941587i	$-0.390666\pi$
$432$		0			0
$433$		−	583.128i		−	1.34672i	−0.739317	−	0.673358i	$-0.764851\pi$
$433$		−	583.128i		−	1.34672i	0.739317	−	0.673358i	$-0.235149\pi$
$434$		0			0
$435$			198.913i			0.457272i
$436$		0			0
$437$	−80.9994	+	70.2321i	−0.185353	+	0.160714i
$438$		0			0
$439$	−107.124			−0.244017			−0.122009	−	0.992529i	$-0.538934\pi$
$439$	−107.124			−0.244017			−0.122009	+	0.992529i	$0.538934\pi$
$440$		0			0
$441$	93.2050			0.211349
$442$		0			0
$443$	−704.580			−1.59047			−0.795237	−	0.606298i	$-0.792654\pi$
$443$	−704.580			−1.59047			−0.795237	+	0.606298i	$0.792654\pi$
$444$		0			0
$445$	144.339			0.324358
$446$		0			0
$447$			509.250i			1.13926i
$448$		0			0
$449$	744.078			1.65719			0.828595	−	0.559849i	$-0.189141\pi$
$449$	744.078			1.65719			0.828595	+	0.559849i	$0.189141\pi$
$450$		0			0
$451$		−	99.5539i		−	0.220740i
$452$		0			0
$453$	−132.536			−0.292575
$454$		0			0
$455$	−130.378			−0.286544
$456$		0			0
$457$			315.363i			0.690073i	0.938589	+	0.345037i	$0.112133\pi$
$457$			315.363i			0.690073i	−0.938589	+	0.345037i	$0.887867\pi$
$458$		0			0
$459$		−	28.4786i		−	0.0620448i
$460$		0			0
$461$	−135.783			−0.294541			−0.147270	−	0.989096i	$-0.547049\pi$
$461$	−135.783			−0.294541			−0.147270	+	0.989096i	$0.547049\pi$
$462$		0			0
$463$	182.888			0.395006			0.197503	−	0.980302i	$-0.436717\pi$
$463$	182.888			0.395006			0.197503	+	0.980302i	$0.436717\pi$
$464$		0			0
$465$			139.619i			0.300256i
$466$		0			0
$467$		−	590.693i		−	1.26487i	−0.774614	−	0.632434i	$-0.782056\pi$
$467$		−	590.693i		−	1.26487i	0.774614	−	0.632434i	$-0.217944\pi$
$468$		0			0
$469$	−21.6888			−0.0462447
$470$		0			0
$471$			206.013i			0.437396i
$472$		0			0
$473$	−83.9922			−0.177573
$474$		0			0
$475$		−	23.3060i		−	0.0490652i
$476$		0			0
$477$		−	79.4399i		−	0.166541i
$478$		0			0
$479$		−	690.631i		−	1.44182i	−0.693030	−	0.720909i	$-0.743725\pi$
$479$		−	690.631i		−	1.44182i	0.693030	−	0.720909i	$-0.256275\pi$
$480$		0			0
$481$		−	965.558i		−	2.00740i
$482$		0			0
$483$	127.455	−	110.512i	0.263881	−	0.228803i
$484$		0			0
$485$	−383.477			−0.790674
$486$		0			0
$487$	415.787			0.853772			0.426886	−	0.904305i	$-0.359611\pi$
$487$	415.787			0.853772			0.426886	+	0.904305i	$0.359611\pi$
$488$		0			0
$489$	494.768			1.01180
$490$		0			0
$491$	−395.472			−0.805443			−0.402721	−	0.915323i	$-0.631936\pi$
$491$	−395.472			−0.805443			−0.402721	+	0.915323i	$0.631936\pi$
$492$		0			0
$493$		−	281.485i		−	0.570963i
$494$		0			0
$495$	−22.9453			−0.0463541
$496$		0			0
$497$			193.836i			0.390011i
$498$		0			0
$499$	−901.606			−1.80683			−0.903413	−	0.428772i	$-0.858946\pi$
$499$	−901.606			−1.80683			−0.903413	+	0.428772i	$0.858946\pi$
$500$		0			0
$501$	−250.295			−0.499590
$502$		0			0
$503$			76.8603i			0.152804i	0.997077	+	0.0764019i	$0.0243432\pi$
$503$			76.8603i			0.152804i	−0.997077	+	0.0764019i	$0.975657\pi$
$504$		0			0
$505$			116.500i			0.230694i
$506$		0			0
$507$	35.6629			0.0703411
$508$		0			0
$509$	530.564			1.04237			0.521183	−	0.853445i	$-0.325491\pi$
$509$	530.564			1.04237			0.521183	+	0.853445i	$0.325491\pi$
$510$		0			0
$511$		−	318.520i		−	0.623327i
$512$		0			0
$513$			24.2203i			0.0472130i
$514$		0			0
$515$	73.5692			0.142853
$516$		0			0
$517$			201.475i			0.389701i
$518$		0			0
$519$	−205.543			−0.396036
$520$		0			0
$521$			194.897i			0.374083i	0.982352	+	0.187042i	$0.0598899\pi$
$521$			194.897i			0.374083i	−0.982352	+	0.187042i	$0.940110\pi$
$522$		0			0
$523$			418.691i			0.800557i	0.916394	+	0.400279i	$0.131087\pi$
$523$			418.691i			0.800557i	−0.916394	+	0.400279i	$0.868913\pi$
$524$		0			0
$525$			36.6725i			0.0698524i
$526$		0			0
$527$		−	197.576i		−	0.374907i
$528$		0			0
$529$	−74.9472	+	523.664i	−0.141677	+	0.989913i
$530$		0			0
$531$	−26.3990			−0.0497156
$532$		0			0
$533$	400.755			0.751885
$534$		0			0
$535$	78.9387			0.147549
$536$		0			0
$537$	−21.2392			−0.0395515
$538$		0			0
$539$			106.269i			0.197159i
$540$		0			0
$541$	−221.165			−0.408808			−0.204404	−	0.978887i	$-0.565526\pi$
$541$	−221.165			−0.408808			−0.204404	+	0.978887i	$0.565526\pi$
$542$		0			0
$543$		−	395.041i		−	0.727516i
$544$		0			0
$545$	125.361			0.230020
$546$		0			0
$547$	−606.362			−1.10852			−0.554262	−	0.832342i	$-0.686999\pi$
$547$	−606.362			−1.10852			−0.554262	+	0.832342i	$0.686999\pi$
$548$		0			0
$549$		−	90.3550i		−	0.164581i
$550$		0			0
$551$			239.395i			0.434474i
$552$		0			0
$553$	263.736			0.476918
$554$		0			0
$555$	−271.592			−0.489354
$556$		0			0
$557$		−	474.639i		−	0.852135i	−0.904691	−	0.426068i	$-0.859899\pi$
$557$		−	474.639i		−	0.852135i	0.904691	−	0.426068i	$-0.140101\pi$
$558$		0			0
$559$		−	338.111i		−	0.604849i
$560$		0			0
$561$	32.4701			0.0578790
$562$		0			0
$563$		−	831.865i		−	1.47756i	−0.673948	−	0.738779i	$-0.735403\pi$
$563$		−	831.865i		−	1.47756i	0.673948	−	0.738779i	$-0.264597\pi$
$564$		0			0
$565$	178.085			0.315194
$566$		0			0
$567$		−	38.1112i		−	0.0672155i
$568$		0			0
$569$		−	58.3947i		−	0.102627i	−0.998683	−	0.0513135i	$-0.983659\pi$
$569$		−	58.3947i		−	0.102627i	0.998683	−	0.0513135i	$-0.0163408\pi$
$570$		0			0
$571$		−	423.510i		−	0.741698i	−0.928693	−	0.370849i	$-0.879067\pi$
$571$		−	423.510i		−	0.741698i	0.928693	−	0.370849i	$-0.120933\pi$
$572$		0			0
$573$		−	462.000i		−	0.806282i
$574$		0			0
$575$	−75.3370	−	86.8869i	−0.131021	−	0.151108i
$576$		0			0
$577$	−434.749			−0.753465			−0.376733	−	0.926322i	$-0.622952\pi$
$577$	−434.749			−0.753465			−0.376733	+	0.926322i	$0.622952\pi$
$578$		0			0
$579$	231.017			0.398993
$580$		0			0
$581$	583.046			1.00352
$582$		0			0
$583$	90.5742			0.155359
$584$		0			0
$585$		−	92.3664i		−	0.157891i
$586$		0			0
$587$	−196.392			−0.334568			−0.167284	−	0.985909i	$-0.553500\pi$
$587$	−196.392			−0.334568			−0.167284	+	0.985909i	$0.553500\pi$
$588$		0			0
$589$			168.034i			0.285286i
$590$		0			0
$591$	−207.737			−0.351502
$592$		0			0
$593$	−779.875			−1.31514			−0.657568	−	0.753395i	$-0.728415\pi$
$593$	−779.875			−1.31514			−0.657568	+	0.753395i	$0.728415\pi$
$594$		0			0
$595$		−	51.8957i		−	0.0872197i
$596$		0			0
$597$		−	64.7668i		−	0.108487i
$598$		0			0
$599$	−729.208			−1.21738			−0.608688	−	0.793410i	$-0.708304\pi$
$599$	−729.208			−1.21738			−0.608688	+	0.793410i	$0.708304\pi$
$600$		0			0
$601$	−98.3505			−0.163645			−0.0818224	−	0.996647i	$-0.526074\pi$
$601$	−98.3505			−0.163645			−0.0818224	+	0.996647i	$0.526074\pi$
$602$		0			0
$603$		−	15.3655i		−	0.0254817i
$604$		0			0
$605$			244.403i			0.403972i
$606$		0			0
$607$	−949.398			−1.56408			−0.782042	−	0.623226i	$-0.785822\pi$
$607$	−949.398			−1.56408			−0.782042	+	0.623226i	$0.785822\pi$
$608$		0			0
$609$		−	376.694i		−	0.618546i
$610$		0			0
$611$	−811.041			−1.32740
$612$		0			0
$613$		−	354.757i		−	0.578722i	−0.957220	−	0.289361i	$-0.906557\pi$
$613$		−	354.757i		−	0.578722i	0.957220	−	0.289361i	$-0.0934428\pi$
$614$		0			0
$615$		−	112.724i		−	0.183291i
$616$		0			0
$617$			1080.07i			1.75052i	0.483654	+	0.875259i	$0.339309\pi$
$617$			1080.07i			1.75052i	−0.483654	+	0.875259i	$0.660691\pi$
$618$		0			0
$619$			207.082i			0.334542i	0.985911	+	0.167271i	$0.0534955\pi$
$619$			207.082i			0.334542i	−0.985911	+	0.167271i	$0.946504\pi$
$620$		0			0
$621$	78.2925	+	90.2956i	0.126075	+	0.145403i
$622$		0			0
$623$	−273.344			−0.438755
$624$		0			0
$625$	25.0000			0.0400000
$626$		0			0
$627$	−27.6150			−0.0440431
$628$		0			0
$629$	384.332			0.611021
$630$		0			0
$631$			899.817i			1.42602i	0.701155	+	0.713009i	$0.252668\pi$
$631$			899.817i			1.42602i	−0.701155	+	0.713009i	$0.747332\pi$
$632$		0			0
$633$	252.483			0.398867
$634$		0			0
$635$		−	108.606i		−	0.171032i
$636$		0			0
$637$	−427.785			−0.671562
$638$		0			0
$639$	−137.323			−0.214904
$640$		0			0
$641$		−	102.776i		−	0.160337i	−0.996781	−	0.0801683i	$-0.974454\pi$
$641$		−	102.776i		−	0.160337i	0.996781	−	0.0801683i	$-0.0255458\pi$
$642$		0			0
$643$		−	120.818i		−	0.187897i	−0.995577	−	0.0939484i	$-0.970051\pi$
$643$		−	120.818i		−	0.187897i	0.995577	−	0.0939484i	$-0.0299489\pi$
$644$		0			0
$645$	−95.1036			−0.147447
$646$		0			0
$647$	−1042.16			−1.61076			−0.805379	−	0.592760i	$-0.798038\pi$
$647$	−1042.16			−1.61076			−0.805379	+	0.592760i	$0.798038\pi$
$648$		0			0
$649$		−	30.0991i		−	0.0463776i
$650$		0			0
$651$		−	264.405i		−	0.406152i
$652$		0			0
$653$	694.904			1.06417			0.532085	−	0.846691i	$-0.321409\pi$
$653$	694.904			1.06417			0.532085	+	0.846691i	$0.321409\pi$
$654$		0			0
$655$			7.27372i			0.0111049i
$656$		0			0
$657$	225.656			0.343465
$658$		0			0
$659$		−	541.740i		−	0.822063i	−0.911621	−	0.411032i	$-0.865169\pi$
$659$		−	541.740i		−	0.822063i	0.911621	−	0.411032i	$-0.134831\pi$
$660$		0			0
$661$		−	243.406i		−	0.368239i	−0.982904	−	0.184120i	$-0.941057\pi$
$661$		−	243.406i		−	0.368239i	0.982904	−	0.184120i	$-0.0589434\pi$
$662$		0			0
$663$			130.709i			0.197147i
$664$		0			0
$665$			44.1360i			0.0663699i
$666$		0			0
$667$	773.850	+	892.489i	1.16019	+	1.33806i
$668$		0			0
$669$	349.899			0.523018
$670$		0			0
$671$	103.019			0.153531
$672$		0			0
$673$	335.090			0.497905			0.248952	−	0.968516i	$-0.419914\pi$
$673$	335.090			0.497905			0.248952	+	0.968516i	$0.419914\pi$
$674$		0			0
$675$	−25.9808			−0.0384900
$676$		0			0
$677$			639.438i			0.944517i	0.881460	+	0.472258i	$0.156561\pi$
$677$			639.438i			0.944517i	−0.881460	+	0.472258i	$0.843439\pi$
$678$		0			0
$679$	726.214			1.06953
$680$		0			0
$681$			385.789i			0.566504i
$682$		0			0
$683$	366.380			0.536428			0.268214	−	0.963359i	$-0.413567\pi$
$683$	366.380			0.536428			0.268214	+	0.963359i	$0.413567\pi$
$684$		0			0
$685$	0.470955			0.000687526
$686$		0			0
$687$		−	18.6604i		−	0.0271621i
$688$		0			0
$689$			364.607i			0.529183i
$690$		0			0
$691$	243.486			0.352367			0.176184	−	0.984357i	$-0.443625\pi$
$691$	243.486			0.352367			0.176184	+	0.984357i	$0.443625\pi$
$692$		0			0
$693$	43.4529			0.0627026
$694$		0			0
$695$			78.1176i			0.112399i
$696$		0			0
$697$			159.517i			0.228862i
$698$		0			0
$699$	531.239			0.759999
$700$		0			0
$701$		−	213.429i		−	0.304463i	−0.988345	−	0.152232i	$-0.951354\pi$
$701$		−	213.429i		−	0.304463i	0.988345	−	0.152232i	$-0.0486460\pi$
$702$		0			0
$703$	−326.865			−0.464957
$704$		0			0
$705$			228.129i			0.323587i
$706$		0			0
$707$		−	220.624i		−	0.312056i
$708$		0			0
$709$		−	804.957i		−	1.13534i	−0.823256	−	0.567671i	$-0.807845\pi$
$709$		−	804.957i		−	1.13534i	0.823256	−	0.567671i	$-0.192155\pi$
$710$		0			0
$711$			186.844i			0.262791i
$712$		0			0
$713$	543.171	+	626.445i	0.761811	+	0.878604i
$714$		0			0
$715$	105.313			0.147290
$716$		0			0
$717$	−231.258			−0.322535
$718$		0			0
$719$	550.092			0.765079			0.382540	−	0.923939i	$-0.375050\pi$
$719$	550.092			0.765079			0.382540	+	0.923939i	$0.375050\pi$
$720$		0			0
$721$	−139.323			−0.193235
$722$		0			0
$723$			285.110i			0.394343i
$724$		0			0
$725$	−256.796			−0.354201
$726$		0			0
$727$			719.138i			0.989185i	0.869125	+	0.494593i	$0.164683\pi$
$727$			719.138i			0.989185i	−0.869125	+	0.494593i	$0.835317\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	134.582			0.184107
$732$		0			0
$733$			1139.86i			1.55506i	0.628844	+	0.777532i	$0.283529\pi$
$733$			1139.86i			1.55506i	−0.628844	+	0.777532i	$0.716471\pi$
$734$		0			0
$735$			120.327i			0.163710i
$736$		0			0
$737$	17.5191			0.0237708
$738$		0			0
$739$	805.483			1.08996			0.544982	−	0.838448i	$-0.316537\pi$
$739$	805.483			1.08996			0.544982	+	0.838448i	$0.316537\pi$
$740$		0			0
$741$		−	111.164i		−	0.150019i
$742$		0			0
$743$		−	75.8211i		−	0.102047i	−0.998697	−	0.0510236i	$-0.983752\pi$
$743$		−	75.8211i		−	0.102047i	0.998697	−	0.0510236i	$-0.0162484\pi$
$744$		0			0
$745$	−657.438			−0.882468
$746$		0			0
$747$			413.061i			0.552960i
$748$		0			0
$749$	−149.491			−0.199588
$750$		0			0
$751$		−	543.632i		−	0.723878i	−0.932202	−	0.361939i	$-0.882115\pi$
$751$		−	543.632i		−	0.723878i	0.932202	−	0.361939i	$-0.117885\pi$
$752$		0			0
$753$			371.359i			0.493172i
$754$		0			0
$755$		−	171.104i		−	0.226628i
$756$		0			0
$757$			650.293i			0.859039i	0.903058	+	0.429520i	$0.141317\pi$
$757$			650.293i			0.859039i	−0.903058	+	0.429520i	$0.858683\pi$
$758$		0			0
$759$	−102.951	+	89.2660i	−0.135641	+	0.117610i
$760$		0			0
$761$	−59.2182			−0.0778162			−0.0389081	−	0.999243i	$-0.512388\pi$
$761$	−59.2182			−0.0778162			−0.0389081	+	0.999243i	$0.512388\pi$
$762$		0			0
$763$	−237.403			−0.311144
$764$		0			0
$765$	36.7657			0.0480597
$766$		0			0
$767$	121.164			0.157971
$768$		0			0
$769$		−	822.423i		−	1.06947i	−0.845019	−	0.534736i	$-0.820411\pi$
$769$		−	822.423i		−	1.06947i	0.845019	−	0.534736i	$-0.179589\pi$
$770$		0			0
$771$	269.676			0.349774
$772$		0			0
$773$		−	62.2749i		−	0.0805626i	−0.999188	−	0.0402813i	$-0.987175\pi$
$773$		−	62.2749i		−	0.0805626i	0.999188	−	0.0402813i	$-0.0128254\pi$
$774$		0			0
$775$	−180.247			−0.232577
$776$		0			0
$777$	514.330			0.661943
$778$		0			0
$779$		−	135.665i		−	0.174153i
$780$		0			0
$781$		−	156.571i		−	0.200475i
$782$		0			0
$783$	266.870			0.340830
$784$		0			0
$785$	−265.962			−0.338805
$786$		0			0
$787$			404.988i			0.514597i	0.966332	+	0.257299i	$0.0828324\pi$
$787$			404.988i			0.514597i	−0.966332	+	0.257299i	$0.917168\pi$
$788$		0			0
$789$		−	879.952i		−	1.11528i
$790$		0			0
$791$	−337.250			−0.426359
$792$		0			0
$793$			414.704i			0.522956i
$794$		0			0
$795$	102.556			0.129002
$796$		0			0
$797$			232.050i			0.291154i	0.989347	+	0.145577i	$0.0465039\pi$
$797$			232.050i			0.291154i	−0.989347	+	0.145577i	$0.953496\pi$
$798$		0			0
$799$		−	322.828i		−	0.404040i
$800$		0			0
$801$		−	193.651i		−	0.241762i
$802$		0			0
$803$			257.285i			0.320404i
$804$		0			0
$805$	142.670	+	164.543i	0.177230	+	0.204401i
$806$		0			0
$807$	−238.541			−0.295590
$808$		0			0
$809$	276.617			0.341925			0.170962	−	0.985278i	$-0.445312\pi$
$809$	276.617			0.341925			0.170962	+	0.985278i	$0.445312\pi$
$810$		0			0
$811$	−53.1869			−0.0655819			−0.0327909	−	0.999462i	$-0.510440\pi$
$811$	−53.1869			−0.0655819			−0.0327909	+	0.999462i	$0.510440\pi$
$812$		0			0
$813$	−778.025			−0.956980
$814$		0			0
$815$			638.743i			0.783733i
$816$		0			0
$817$	−114.459			−0.140096
$818$		0			0
$819$			174.920i			0.213577i
$820$		0			0
$821$	831.053			1.01224			0.506122	−	0.862462i	$-0.331078\pi$
$821$	831.053			1.01224			0.506122	+	0.862462i	$0.331078\pi$
$822$		0			0
$823$	225.779			0.274336			0.137168	−	0.990548i	$-0.456200\pi$
$823$	225.779			0.274336			0.137168	+	0.990548i	$0.456200\pi$
$824$		0			0
$825$		−	29.6222i		−	0.0359057i
$826$		0			0
$827$			454.191i			0.549203i	0.961558	+	0.274602i	$0.0885460\pi$
$827$			454.191i			0.549203i	−0.961558	+	0.274602i	$0.911454\pi$
$828$		0			0
$829$	1391.70			1.67877			0.839386	−	0.543536i	$-0.182915\pi$
$829$	1391.70			1.67877			0.839386	+	0.543536i	$0.182915\pi$
$830$		0			0
$831$	−459.230			−0.552624
$832$		0			0
$833$		−	170.276i		−	0.204413i
$834$		0			0
$835$		−	323.129i		−	0.386981i
$836$		0			0
$837$	187.318			0.223797
$838$		0			0
$839$			132.957i			0.158471i	0.996856	+	0.0792355i	$0.0252479\pi$
$839$			132.957i			0.158471i	−0.996856	+	0.0792355i	$0.974752\pi$
$840$		0			0
$841$	1796.77			2.13647
$842$		0			0
$843$		−	140.726i		−	0.166934i
$844$		0			0
$845$			46.0406i			0.0544860i
$846$		0			0
$847$		−	462.841i		−	0.546447i
$848$		0			0
$849$			483.802i			0.569850i
$850$		0			0
$851$	−1218.58	+	1056.60i	−1.43194	+	1.24159i
$852$		0			0
$853$	104.622			0.122651			0.0613256	−	0.998118i	$-0.480467\pi$
$853$	104.622			0.122651			0.0613256	+	0.998118i	$0.480467\pi$
$854$		0			0
$855$	−31.2683			−0.0365711
$856$		0			0
$857$	31.9854			0.0373226			0.0186613	−	0.999826i	$-0.494060\pi$
$857$	31.9854			0.0373226			0.0186613	+	0.999826i	$0.494060\pi$
$858$		0			0
$859$	984.302			1.14587			0.572935	−	0.819601i	$-0.305805\pi$
$859$	984.302			1.14587			0.572935	+	0.819601i	$0.305805\pi$
$860$		0			0
$861$			213.472i			0.247935i
$862$		0			0
$863$	668.524			0.774651			0.387326	−	0.921943i	$-0.373399\pi$
$863$	668.524			0.774651			0.387326	+	0.921943i	$0.373399\pi$
$864$		0			0
$865$		−	265.354i		−	0.306768i
$866$		0			0
$867$	448.535			0.517342
$868$		0			0
$869$	−213.032			−0.245147
$870$		0			0
$871$			70.5233i			0.0809682i
$872$		0			0
$873$			514.488i			0.589334i
$874$		0			0
$875$	−47.3440			−0.0541075
$876$		0			0
$877$	1103.98			1.25881			0.629405	−	0.777077i	$-0.283299\pi$
$877$	1103.98			1.25881			0.629405	+	0.777077i	$0.283299\pi$
$878$		0			0
$879$		−	249.862i		−	0.284257i
$880$		0			0
$881$		−	607.647i		−	0.689724i	−0.938653	−	0.344862i	$-0.887926\pi$
$881$		−	607.647i		−	0.689724i	0.938653	−	0.344862i	$-0.112074\pi$
$882$		0			0
$883$	85.7974			0.0971659			0.0485829	−	0.998819i	$-0.484529\pi$
$883$	85.7974			0.0971659			0.0485829	+	0.998819i	$0.484529\pi$
$884$		0			0
$885$		−	34.0809i		−	0.0385095i
$886$		0			0
$887$	689.893			0.777782			0.388891	−	0.921284i	$-0.372858\pi$
$887$	689.893			0.777782			0.388891	+	0.921284i	$0.372858\pi$
$888$		0			0
$889$			205.673i			0.231353i
$890$		0			0
$891$			30.7843i			0.0345503i
$892$		0			0
$893$			274.557i			0.307455i
$894$		0			0
$895$		−	27.4197i		−	0.0306365i
$896$		0			0
$897$	−359.341	−	414.432i	−0.400603	−	0.462020i
$898$		0			0
$899$	1851.47			2.05948
$900$		0			0
$901$	−145.129			−0.161075
$902$		0			0
$903$	180.104			0.199450
$904$		0			0
$905$	509.996			0.563531
$906$		0			0
$907$			1182.25i			1.30348i	0.758444	+	0.651738i	$0.225960\pi$
$907$			1182.25i			1.30348i	−0.758444	+	0.651738i	$0.774040\pi$
$908$		0			0
$909$	156.302			0.171949
$910$		0			0
$911$			611.652i			0.671407i	0.941968	+	0.335704i	$0.108974\pi$
$911$			611.652i			0.671407i	−0.941968	+	0.335704i	$0.891026\pi$
$912$		0			0
$913$	−470.956			−0.515833
$914$		0			0
$915$	116.648			0.127484
$916$		0			0
$917$		−	13.7747i		−	0.0150215i
$918$		0			0
$919$		−	1037.12i		−	1.12854i	−0.825592	−	0.564268i	$-0.809159\pi$
$919$		−	1037.12i		−	1.12854i	0.825592	−	0.564268i	$-0.190841\pi$
$920$		0			0
$921$	−802.606			−0.871450
$922$		0			0
$923$	630.276			0.682856
$924$		0			0
$925$		−	350.623i		−	0.379052i
$926$		0			0
$927$		−	98.7034i		−	0.106476i
$928$		0			0
$929$	944.065			1.01622			0.508108	−	0.861293i	$-0.330345\pi$
$929$	944.065			1.01622			0.508108	+	0.861293i	$0.330345\pi$
$930$		0			0
$931$			144.816i			0.155549i
$932$		0			0
$933$	−632.914			−0.678364
$934$		0			0
$935$			41.9188i			0.0448329i
$936$		0			0
$937$			1307.12i			1.39500i	0.716583	+	0.697502i	$0.245705\pi$
$937$			1307.12i			1.39500i	−0.716583	+	0.697502i	$0.754295\pi$
$938$		0			0
$939$		−	388.952i		−	0.414219i
$940$		0			0
$941$			710.928i			0.755502i	0.925907	+	0.377751i	$0.123303\pi$
$941$			710.928i			0.755502i	−0.925907	+	0.377751i	$0.876697\pi$
$942$		0			0
$943$	−438.540	−	505.773i	−0.465048	−	0.536344i
$944$		0			0
$945$	49.2014			0.0520649
$946$		0			0
$947$	674.787			0.712552			0.356276	−	0.934381i	$-0.384046\pi$
$947$	674.787			0.712552			0.356276	+	0.934381i	$0.384046\pi$
$948$		0			0
$949$	−1035.70			−1.09136
$950$		0			0
$951$	194.387			0.204403
$952$		0			0
$953$			671.206i			0.704309i	0.935942	+	0.352154i	$0.114551\pi$
$953$			671.206i			0.704309i	−0.935942	+	0.352154i	$0.885449\pi$
$954$		0			0
$955$	596.439			0.624543
$956$		0			0
$957$			304.275i			0.317947i
$958$		0			0
$959$	−0.891877			−0.000930007
$960$		0			0
$961$	338.562			0.352301
$962$		0			0
$963$		−	105.907i		−	0.109977i
$964$		0			0
$965$			298.242i			0.309059i
$966$		0			0
$967$	−1886.45			−1.95083			−0.975413	−	0.220385i	$-0.929269\pi$
$967$	−1886.45			−1.95083			−0.975413	+	0.220385i	$0.929269\pi$
$968$		0			0
$969$	44.2481			0.0456636
$970$		0			0
$971$			1134.32i			1.16819i	0.811684	+	0.584096i	$0.198551\pi$
$971$			1134.32i			1.16819i	−0.811684	+	0.584096i	$0.801449\pi$
$972$		0			0
$973$		−	147.936i		−	0.152041i
$974$		0			0
$975$	119.244			0.122302
$976$		0			0
$977$			37.6770i			0.0385640i	0.999814	+	0.0192820i	$0.00613803\pi$
$977$			37.6770i			0.0385640i	−0.999814	+	0.0192820i	$0.993862\pi$
$978$		0			0
$979$	220.794			0.225530
$980$		0			0
$981$		−	168.189i		−	0.171446i
$982$		0			0
$983$		−	1653.42i		−	1.68201i	−0.541027	−	0.841005i	$-0.681964\pi$
$983$		−	1653.42i		−	1.68201i	0.541027	−	0.841005i	$-0.318036\pi$
$984$		0			0
$985$		−	268.188i		−	0.272272i
$986$		0			0
$987$		−	432.022i		−	0.437712i
$988$		0			0
$989$	−426.713	+	369.990i	−0.431459	+	0.374105i
$990$		0			0
$991$	1681.88			1.69715			0.848576	−	0.529073i	$-0.177460\pi$
$991$	1681.88			1.69715			0.848576	+	0.529073i	$0.177460\pi$
$992$		0			0
$993$	773.910			0.779365
$994$		0			0
$995$	83.6136			0.0840338
$996$		0			0
$997$	−1474.49			−1.47893			−0.739463	−	0.673198i	$-0.764920\pi$
$997$	−1474.49			−1.47893			−0.739463	+	0.673198i	$0.764920\pi$
$998$		0			0
$999$			364.378i			0.364743i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1380.3.d.a.781.26	yes	32
3.2	odd	2		4140.3.d.c.2161.4		32
23.22	odd	2	inner	1380.3.d.a.781.23	✓	32
69.68	even	2		4140.3.d.c.2161.29		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1380.3.d.a.781.23	✓	32	23.22	odd	2	inner
1380.3.d.a.781.26	yes	32	1.1	even	1	trivial
4140.3.d.c.2161.4		32	3.2	odd	2
4140.3.d.c.2161.29		32	69.68	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1380.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +2.23607i q^{5} -4.23458i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q+1.73205 q^{3} +2.23607i q^{5} -4.23458i q^{7} +3.00000 q^{9} +3.42048i q^{11} -13.7692 q^{13} +3.87298i q^{15} -5.48070i q^{17} +4.66120i q^{19} -7.33451i q^{21} +(15.0674 + 17.3774i) q^{23} -5.00000 q^{25} +5.19615 q^{27} +51.3592 q^{29} +36.0494 q^{31} +5.92445i q^{33} +9.46881 q^{35} +70.1247i q^{37} -23.8489 q^{39} -29.1052 q^{41} +24.5556i q^{43} +6.70820i q^{45} +58.9027 q^{47} +31.0683 q^{49} -9.49285i q^{51} -26.4800i q^{53} -7.64843 q^{55} +8.07343i q^{57} -8.79966 q^{59} -30.1183i q^{61} -12.7037i q^{63} -30.7888i q^{65} -5.12183i q^{67} +(26.0975 + 30.0985i) q^{69} -45.7745 q^{71} +75.2188 q^{73} -8.66025 q^{75} +14.4843 q^{77} +62.2814i q^{79} +9.00000 q^{81} +137.687i q^{83} +12.2552 q^{85} +88.9568 q^{87} -64.5505i q^{89} +58.3066i q^{91} +62.4395 q^{93} -10.4228 q^{95} +171.496i q^{97} +10.2614i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 96 q^{9}+O(q^{10}) \) 32 * q + 96 * q^9	\( 32 q + 96 q^{9} + 24 q^{13} + 64 q^{23} - 160 q^{25} - 60 q^{29} - 4 q^{31} - 60 q^{35} - 48 q^{39} - 108 q^{41} + 136 q^{47} - 428 q^{49} + 120 q^{55} - 84 q^{59} + 48 q^{69} + 188 q^{71} + 472 q^{73} - 120 q^{77} + 288 q^{81} + 60 q^{85} - 72 q^{87} - 192 q^{93} + 80 q^{95}+O(q^{100}) \) 32 * q + 96 * q^9 + 24 * q^13 + 64 * q^23 - 160 * q^25 - 60 * q^29 - 4 * q^31 - 60 * q^35 - 48 * q^39 - 108 * q^41 + 136 * q^47 - 428 * q^49 + 120 * q^55 - 84 * q^59 + 48 * q^69 + 188 * q^71 + 472 * q^73 - 120 * q^77 + 288 * q^81 + 60 * q^85 - 72 * q^87 - 192 * q^93 + 80 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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