LMFDB - Embedded newform 4005.2.a.s.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4005,2,Mod(1,4005)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4005, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4005 = 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4005.a (trivial)

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:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.69046$ of defining polynomial
Character	$\chi$	$=$	4005.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.69046 q^{2} +0.857665 q^{4} -1.00000 q^{5} +0.437617 q^{7} +1.93108 q^{8} +O(q^{10})$	\(q-1.69046 q^{2} +0.857665 q^{4} -1.00000 q^{5} +0.437617 q^{7} +1.93108 q^{8} +1.69046 q^{10} -1.16360 q^{11} +4.04364 q^{13} -0.739775 q^{14} -4.97974 q^{16} -3.32298 q^{17} -2.06232 q^{19} -0.857665 q^{20} +1.96702 q^{22} -5.13953 q^{23} +1.00000 q^{25} -6.83562 q^{26} +0.375329 q^{28} +5.54534 q^{29} +10.1466 q^{31} +4.55592 q^{32} +5.61738 q^{34} -0.437617 q^{35} +6.16027 q^{37} +3.48628 q^{38} -1.93108 q^{40} -7.42486 q^{41} -3.71584 q^{43} -0.997978 q^{44} +8.68819 q^{46} -7.37281 q^{47} -6.80849 q^{49} -1.69046 q^{50} +3.46809 q^{52} -5.89482 q^{53} +1.16360 q^{55} +0.845071 q^{56} -9.37419 q^{58} +3.32335 q^{59} -3.83784 q^{61} -17.1524 q^{62} +2.25787 q^{64} -4.04364 q^{65} -14.1398 q^{67} -2.85001 q^{68} +0.739775 q^{70} -9.24075 q^{71} +8.67927 q^{73} -10.4137 q^{74} -1.76878 q^{76} -0.509211 q^{77} +15.4583 q^{79} +4.97974 q^{80} +12.5515 q^{82} +18.0513 q^{83} +3.32298 q^{85} +6.28148 q^{86} -2.24700 q^{88} +1.00000 q^{89} +1.76956 q^{91} -4.40800 q^{92} +12.4635 q^{94} +2.06232 q^{95} +9.69829 q^{97} +11.5095 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−1.69046	−1.19534	−0.597669	−	0.801743i	$-0.703906\pi$
$2$	−1.69046	−1.19534	−0.597669	+	0.801743i	$0.703906\pi$
$3$	0	0
$3$	0	0
$4$	0.857665	0.428832
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.437617	0.165404	0.0827018	−	0.996574i	$-0.473645\pi$
$7$	0.437617	0.165404	0.0827018	+	0.996574i	$0.473645\pi$
$8$	1.93108	0.682738
$9$	0	0
$10$	1.69046	0.534571
$11$	−1.16360	−0.350838	−0.175419	−	0.984494i	$-0.556128\pi$
$11$	−1.16360	−0.350838	−0.175419	+	0.984494i	$0.556128\pi$
$12$	0	0
$13$	4.04364	1.12150	0.560752	−	0.827984i	$-0.310512\pi$
$13$	4.04364	1.12150	0.560752	+	0.827984i	$0.310512\pi$
$14$	−0.739775	−0.197713
$15$	0	0
$16$	−4.97974	−1.24494
$17$	−3.32298	−0.805942	−0.402971	−	0.915213i	$-0.632022\pi$
$17$	−3.32298	−0.805942	−0.402971	+	0.915213i	$0.632022\pi$
$18$	0	0
$19$	−2.06232	−0.473130	−0.236565	−	0.971616i	$-0.576022\pi$
$19$	−2.06232	−0.473130	−0.236565	+	0.971616i	$0.576022\pi$
$20$	−0.857665	−0.191780
$21$	0	0
$22$	1.96702	0.419370
$23$	−5.13953	−1.07167	−0.535833	−	0.844324i	$-0.680003\pi$
$23$	−5.13953	−1.07167	−0.535833	+	0.844324i	$0.680003\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−6.83562	−1.34058
$27$	0	0
$28$	0.375329	0.0709304
$29$	5.54534	1.02974	0.514872	−	0.857267i	$-0.327840\pi$
$29$	5.54534	1.02974	0.514872	+	0.857267i	$0.327840\pi$
$30$	0	0
$31$	10.1466	1.82238	0.911190	−	0.411986i	$-0.135165\pi$
$31$	10.1466	1.82238	0.911190	+	0.411986i	$0.135165\pi$
$32$	4.55592	0.805380
$33$	0	0
$34$	5.61738	0.963373
$35$	−0.437617	−0.0739708
$36$	0	0
$37$	6.16027	1.01274	0.506371	−	0.862315i	$-0.330986\pi$
$37$	6.16027	1.01274	0.506371	+	0.862315i	$0.330986\pi$
$38$	3.48628	0.565550
$39$	0	0
$40$	−1.93108	−0.305330
$41$	−7.42486	−1.15957	−0.579784	−	0.814770i	$-0.696863\pi$
$41$	−7.42486	−1.15957	−0.579784	+	0.814770i	$0.696863\pi$
$42$	0	0
$43$	−3.71584	−0.566660	−0.283330	−	0.959023i	$-0.591439\pi$
$43$	−3.71584	−0.566660	−0.283330	+	0.959023i	$0.591439\pi$
$44$	−0.997978	−0.150451
$45$	0	0
$46$	8.68819	1.28100
$47$	−7.37281	−1.07543	−0.537717	−	0.843125i	$-0.680713\pi$
$47$	−7.37281	−1.07543	−0.537717	+	0.843125i	$0.680713\pi$
$48$	0	0
$49$	−6.80849	−0.972642
$50$	−1.69046	−0.239068
$51$	0	0
$52$	3.46809	0.480937
$53$	−5.89482	−0.809715	−0.404858	−	0.914380i	$-0.632679\pi$
$53$	−5.89482	−0.809715	−0.404858	+	0.914380i	$0.632679\pi$
$54$	0	0
$55$	1.16360	0.156900
$56$	0.845071	0.112927
$57$	0	0
$58$	−9.37419	−1.23089
$59$	3.32335	0.432663	0.216331	−	0.976320i	$-0.430591\pi$
$59$	3.32335	0.432663	0.216331	+	0.976320i	$0.430591\pi$
$60$	0	0
$61$	−3.83784	−0.491385	−0.245692	−	0.969348i	$-0.579015\pi$
$61$	−3.83784	−0.491385	−0.245692	+	0.969348i	$0.579015\pi$
$62$	−17.1524	−2.17836
$63$	0	0
$64$	2.25787	0.282234
$65$	−4.04364	−0.501552
$66$	0	0
$67$	−14.1398	−1.72745	−0.863724	−	0.503964i	$-0.831874\pi$
$67$	−14.1398	−1.72745	−0.863724	+	0.503964i	$0.831874\pi$
$68$	−2.85001	−0.345614
$69$	0	0
$70$	0.739775	0.0884200
$71$	−9.24075	−1.09668	−0.548338	−	0.836257i	$-0.684739\pi$
$71$	−9.24075	−1.09668	−0.548338	+	0.836257i	$0.684739\pi$
$72$	0	0
$73$	8.67927	1.01583	0.507916	−	0.861407i	$-0.330416\pi$
$73$	8.67927	1.01583	0.507916	+	0.861407i	$0.330416\pi$
$74$	−10.4137	−1.21057
$75$	0	0
$76$	−1.76878	−0.202893
$77$	−0.509211	−0.0580299
$78$	0	0
$79$	15.4583	1.73919	0.869596	−	0.493763i	$-0.164379\pi$
$79$	15.4583	1.73919	0.869596	+	0.493763i	$0.164379\pi$
$80$	4.97974	0.556752
$81$	0	0
$82$	12.5515	1.38608
$83$	18.0513	1.98139	0.990693	−	0.136118i	$-0.0434626\pi$
$83$	18.0513	1.98139	0.990693	+	0.136118i	$0.0434626\pi$
$84$	0	0
$85$	3.32298	0.360428
$86$	6.28148	0.677350
$87$	0	0
$88$	−2.24700	−0.239531
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	1.76956	0.185501
$92$	−4.40800	−0.459566
$93$	0	0
$94$	12.4635	1.28551
$95$	2.06232	0.211590
$96$	0	0
$97$	9.69829	0.984713	0.492356	−	0.870394i	$-0.336136\pi$
$97$	9.69829	0.984713	0.492356	+	0.870394i	$0.336136\pi$
$98$	11.5095	1.16264
$99$	0	0
$100$	0.857665	0.0857665
$101$	−5.21056	−0.518470	−0.259235	−	0.965814i	$-0.583470\pi$
$101$	−5.21056	−0.518470	−0.259235	+	0.965814i	$0.583470\pi$
$102$	0	0
$103$	−18.4892	−1.82180	−0.910898	−	0.412631i	$-0.864610\pi$
$103$	−18.4892	−1.82180	−0.910898	+	0.412631i	$0.864610\pi$
$104$	7.80857	0.765693
$105$	0	0
$106$	9.96497	0.967883
$107$	18.9000	1.82713	0.913567	−	0.406689i	$-0.133317\pi$
$107$	18.9000	1.82713	0.913567	+	0.406689i	$0.133317\pi$
$108$	0	0
$109$	1.00030	0.0958113	0.0479057	−	0.998852i	$-0.484745\pi$
$109$	1.00030	0.0958113	0.0479057	+	0.998852i	$0.484745\pi$
$110$	−1.96702	−0.187548
$111$	0	0
$112$	−2.17922	−0.205917
$113$	8.38629	0.788916	0.394458	−	0.918914i	$-0.370932\pi$
$113$	8.38629	0.788916	0.394458	+	0.918914i	$0.370932\pi$
$114$	0	0
$115$	5.13953	0.479264
$116$	4.75604	0.441587
$117$	0	0
$118$	−5.61799	−0.517178
$119$	−1.45419	−0.133306
$120$	0	0
$121$	−9.64604	−0.876913
$122$	6.48773	0.587371
$123$	0	0
$124$	8.70237	0.781496
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−3.70275	−0.328566	−0.164283	−	0.986413i	$-0.552531\pi$
$127$	−3.70275	−0.328566	−0.164283	+	0.986413i	$0.552531\pi$
$128$	−12.9287	−1.14275
$129$	0	0
$130$	6.83562	0.599524
$131$	−4.81775	−0.420928	−0.210464	−	0.977602i	$-0.567498\pi$
$131$	−4.81775	−0.420928	−0.210464	+	0.977602i	$0.567498\pi$
$132$	0	0
$133$	−0.902508	−0.0782574
$134$	23.9028	2.06489
$135$	0	0
$136$	−6.41693	−0.550247
$137$	4.40912	0.376696	0.188348	−	0.982102i	$-0.439687\pi$
$137$	4.40912	0.376696	0.188348	+	0.982102i	$0.439687\pi$
$138$	0	0
$139$	−6.87089	−0.582781	−0.291391	−	0.956604i	$-0.594118\pi$
$139$	−6.87089	−0.582781	−0.291391	+	0.956604i	$0.594118\pi$
$140$	−0.375329	−0.0317211
$141$	0	0
$142$	15.6211	1.31090
$143$	−4.70517	−0.393466
$144$	0	0
$145$	−5.54534	−0.460515
$146$	−14.6720	−1.21426
$147$	0	0
$148$	5.28345	0.434297
$149$	−12.3895	−1.01498	−0.507492	−	0.861657i	$-0.669427\pi$
$149$	−12.3895	−1.01498	−0.507492	+	0.861657i	$0.669427\pi$
$150$	0	0
$151$	−18.5272	−1.50772	−0.753861	−	0.657034i	$-0.771811\pi$
$151$	−18.5272	−1.50772	−0.753861	+	0.657034i	$0.771811\pi$
$152$	−3.98250	−0.323024
$153$	0	0
$154$	0.860802	0.0693654
$155$	−10.1466	−0.814993
$156$	0	0
$157$	8.37765	0.668609	0.334305	−	0.942465i	$-0.391499\pi$
$157$	8.37765	0.668609	0.334305	+	0.942465i	$0.391499\pi$
$158$	−26.1317	−2.07892
$159$	0	0
$160$	−4.55592	−0.360177
$161$	−2.24915	−0.177258
$162$	0	0
$163$	−7.19797	−0.563789	−0.281894	−	0.959445i	$-0.590963\pi$
$163$	−7.19797	−0.563789	−0.281894	+	0.959445i	$0.590963\pi$
$164$	−6.36805	−0.497261
$165$	0	0
$166$	−30.5150	−2.36842
$167$	10.5041	0.812835	0.406417	−	0.913687i	$-0.366778\pi$
$167$	10.5041	0.812835	0.406417	+	0.913687i	$0.366778\pi$
$168$	0	0
$169$	3.35102	0.257770
$170$	−5.61738	−0.430833
$171$	0	0
$172$	−3.18694	−0.243002
$173$	20.7624	1.57853	0.789267	−	0.614050i	$-0.210461\pi$
$173$	20.7624	1.57853	0.789267	+	0.614050i	$0.210461\pi$
$174$	0	0
$175$	0.437617	0.0330807
$176$	5.79442	0.436771
$177$	0	0
$178$	−1.69046	−0.126706
$179$	11.1843	0.835951	0.417975	−	0.908458i	$-0.362740\pi$
$179$	11.1843	0.835951	0.417975	+	0.908458i	$0.362740\pi$
$180$	0	0
$181$	21.2772	1.58153	0.790763	−	0.612123i	$-0.209684\pi$
$181$	21.2772	1.58153	0.790763	+	0.612123i	$0.209684\pi$
$182$	−2.99138	−0.221736
$183$	0	0
$184$	−9.92483	−0.731668
$185$	−6.16027	−0.452912
$186$	0	0
$187$	3.86662	0.282755
$188$	−6.32340	−0.461181
$189$	0	0
$190$	−3.48628	−0.252922
$191$	−22.2412	−1.60932	−0.804659	−	0.593738i	$-0.797652\pi$
$191$	−22.2412	−1.60932	−0.804659	+	0.593738i	$0.797652\pi$
$192$	0	0
$193$	13.3562	0.961400	0.480700	−	0.876885i	$-0.340383\pi$
$193$	13.3562	0.961400	0.480700	+	0.876885i	$0.340383\pi$
$194$	−16.3946	−1.17706
$195$	0	0
$196$	−5.83940	−0.417100
$197$	−7.02410	−0.500446	−0.250223	−	0.968188i	$-0.580504\pi$
$197$	−7.02410	−0.500446	−0.250223	+	0.968188i	$0.580504\pi$
$198$	0	0
$199$	−7.77069	−0.550850	−0.275425	−	0.961323i	$-0.588819\pi$
$199$	−7.77069	−0.550850	−0.275425	+	0.961323i	$0.588819\pi$
$200$	1.93108	0.136548
$201$	0	0
$202$	8.80826	0.619747
$203$	2.42673	0.170323
$204$	0	0
$205$	7.42486	0.518575
$206$	31.2553	2.17766
$207$	0	0
$208$	−20.1363	−1.39620
$209$	2.39972	0.165992
$210$	0	0
$211$	−13.9886	−0.963012	−0.481506	−	0.876443i	$-0.659910\pi$
$211$	−13.9886	−0.963012	−0.481506	+	0.876443i	$0.659910\pi$
$212$	−5.05578	−0.347232
$213$	0	0
$214$	−31.9498	−2.18404
$215$	3.71584	0.253418
$216$	0	0
$217$	4.44032	0.301428
$218$	−1.69097	−0.114527
$219$	0	0
$220$	0.997978	0.0672837
$221$	−13.4369	−0.903867
$222$	0	0
$223$	−24.7376	−1.65655	−0.828275	−	0.560321i	$-0.810678\pi$
$223$	−24.7376	−1.65655	−0.828275	+	0.560321i	$0.810678\pi$
$224$	1.99375	0.133213
$225$	0	0
$226$	−14.1767	−0.943021
$227$	−22.8781	−1.51848	−0.759238	−	0.650814i	$-0.774428\pi$
$227$	−22.8781	−1.51848	−0.759238	+	0.650814i	$0.774428\pi$
$228$	0	0
$229$	−18.0420	−1.19225	−0.596123	−	0.802893i	$-0.703293\pi$
$229$	−18.0420	−1.19225	−0.596123	+	0.802893i	$0.703293\pi$
$230$	−8.68819	−0.572882
$231$	0	0
$232$	10.7085	0.703045
$233$	0.409756	0.0268440	0.0134220	−	0.999910i	$-0.495728\pi$
$233$	0.409756	0.0268440	0.0134220	+	0.999910i	$0.495728\pi$
$234$	0	0
$235$	7.37281	0.480949
$236$	2.85032	0.185540
$237$	0	0
$238$	2.45826	0.159345
$239$	−13.4598	−0.870642	−0.435321	−	0.900275i	$-0.643365\pi$
$239$	−13.4598	−0.870642	−0.435321	+	0.900275i	$0.643365\pi$
$240$	0	0
$241$	−27.3845	−1.76399	−0.881997	−	0.471256i	$-0.843801\pi$
$241$	−27.3845	−1.76399	−0.881997	+	0.471256i	$0.843801\pi$
$242$	16.3063	1.04821
$243$	0	0
$244$	−3.29158	−0.210722
$245$	6.80849	0.434979
$246$	0	0
$247$	−8.33929	−0.530617
$248$	19.5938	1.24421
$249$	0	0
$250$	1.69046	0.106914
$251$	−0.386022	−0.0243655	−0.0121828	−	0.999926i	$-0.503878\pi$
$251$	−0.386022	−0.0243655	−0.0121828	+	0.999926i	$0.503878\pi$
$252$	0	0
$253$	5.98036	0.375982
$254$	6.25937	0.392748
$255$	0	0
$256$	17.3397	1.08373
$257$	16.8803	1.05296	0.526482	−	0.850186i	$-0.323511\pi$
$257$	16.8803	1.05296	0.526482	+	0.850186i	$0.323511\pi$
$258$	0	0
$259$	2.69584	0.167511
$260$	−3.46809	−0.215082
$261$	0	0
$262$	8.14422	0.503152
$263$	16.9081	1.04260	0.521299	−	0.853374i	$-0.325448\pi$
$263$	16.9081	1.04260	0.521299	+	0.853374i	$0.325448\pi$
$264$	0	0
$265$	5.89482	0.362116
$266$	1.52566	0.0935440
$267$	0	0
$268$	−12.1272	−0.740786
$269$	−12.1421	−0.740320	−0.370160	−	0.928968i	$-0.620697\pi$
$269$	−12.1421	−0.740320	−0.370160	+	0.928968i	$0.620697\pi$
$270$	0	0
$271$	8.27071	0.502410	0.251205	−	0.967934i	$-0.419173\pi$
$271$	8.27071	0.502410	0.251205	+	0.967934i	$0.419173\pi$
$272$	16.5476	1.00335
$273$	0	0
$274$	−7.45345	−0.450279
$275$	−1.16360	−0.0701677
$276$	0	0
$277$	−16.1370	−0.969576	−0.484788	−	0.874632i	$-0.661103\pi$
$277$	−16.1370	−0.969576	−0.484788	+	0.874632i	$0.661103\pi$
$278$	11.6150	0.696621
$279$	0	0
$280$	−0.845071	−0.0505027
$281$	−16.5474	−0.987136	−0.493568	−	0.869707i	$-0.664308\pi$
$281$	−16.5474	−0.987136	−0.493568	+	0.869707i	$0.664308\pi$
$282$	0	0
$283$	14.9648	0.889563	0.444781	−	0.895639i	$-0.353281\pi$
$283$	14.9648	0.889563	0.444781	+	0.895639i	$0.353281\pi$
$284$	−7.92547	−0.470290
$285$	0	0
$286$	7.95392	0.470325
$287$	−3.24925	−0.191797
$288$	0	0
$289$	−5.95778	−0.350458
$290$	9.37419	0.550471
$291$	0	0
$292$	7.44391	0.435622
$293$	−17.0971	−0.998825	−0.499412	−	0.866364i	$-0.666451\pi$
$293$	−17.0971	−0.998825	−0.499412	+	0.866364i	$0.666451\pi$
$294$	0	0
$295$	−3.32335	−0.193493
$296$	11.8960	0.691438
$297$	0	0
$298$	20.9439	1.21325
$299$	−20.7824	−1.20188
$300$	0	0
$301$	−1.62611	−0.0937276
$302$	31.3195	1.80224
$303$	0	0
$304$	10.2698	0.589016
$305$	3.83784	0.219754
$306$	0	0
$307$	−5.48213	−0.312881	−0.156441	−	0.987687i	$-0.550002\pi$
$307$	−5.48213	−0.312881	−0.156441	+	0.987687i	$0.550002\pi$
$308$	−0.436732	−0.0248851
$309$	0	0
$310$	17.1524	0.974192
$311$	13.1365	0.744900	0.372450	−	0.928052i	$-0.378518\pi$
$311$	13.1365	0.744900	0.372450	+	0.928052i	$0.378518\pi$
$312$	0	0
$313$	−29.4525	−1.66475	−0.832376	−	0.554211i	$-0.813020\pi$
$313$	−29.4525	−1.66475	−0.832376	+	0.554211i	$0.813020\pi$
$314$	−14.1621	−0.799214
$315$	0	0
$316$	13.2580	0.745822
$317$	−9.63725	−0.541282	−0.270641	−	0.962680i	$-0.587236\pi$
$317$	−9.63725	−0.541282	−0.270641	+	0.962680i	$0.587236\pi$
$318$	0	0
$319$	−6.45255	−0.361273
$320$	−2.25787	−0.126219
$321$	0	0
$322$	3.80210	0.211883
$323$	6.85307	0.381315
$324$	0	0
$325$	4.04364	0.224301
$326$	12.1679	0.673918
$327$	0	0
$328$	−14.3380	−0.791682
$329$	−3.22646	−0.177881
$330$	0	0
$331$	−32.2712	−1.77378	−0.886891	−	0.461978i	$-0.847140\pi$
$331$	−32.2712	−1.77378	−0.886891	+	0.461978i	$0.847140\pi$
$332$	15.4819	0.849682
$333$	0	0
$334$	−17.7569	−0.971612
$335$	14.1398	0.772539
$336$	0	0
$337$	25.0830	1.36636	0.683178	−	0.730252i	$-0.260597\pi$
$337$	25.0830	1.36636	0.683178	+	0.730252i	$0.260597\pi$
$338$	−5.66477	−0.308123
$339$	0	0
$340$	2.85001	0.154563
$341$	−11.8066	−0.639361
$342$	0	0
$343$	−6.04283	−0.326282
$344$	−7.17556	−0.386880
$345$	0	0
$346$	−35.0980	−1.88688
$347$	−21.2082	−1.13851	−0.569257	−	0.822160i	$-0.692769\pi$
$347$	−21.2082	−1.13851	−0.569257	+	0.822160i	$0.692769\pi$
$348$	0	0
$349$	4.60462	0.246480	0.123240	−	0.992377i	$-0.460672\pi$
$349$	4.60462	0.246480	0.123240	+	0.992377i	$0.460672\pi$
$350$	−0.739775	−0.0395426
$351$	0	0
$352$	−5.30126	−0.282558
$353$	5.31988	0.283148	0.141574	−	0.989928i	$-0.454784\pi$
$353$	5.31988	0.283148	0.141574	+	0.989928i	$0.454784\pi$
$354$	0	0
$355$	9.24075	0.490448
$356$	0.857665	0.0454562
$357$	0	0
$358$	−18.9066	−0.999243
$359$	−2.80970	−0.148290	−0.0741450	−	0.997247i	$-0.523623\pi$
$359$	−2.80970	−0.148290	−0.0741450	+	0.997247i	$0.523623\pi$
$360$	0	0
$361$	−14.7468	−0.776148
$362$	−35.9684	−1.89046
$363$	0	0
$364$	1.51769	0.0795488
$365$	−8.67927	−0.454294
$366$	0	0
$367$	−30.2370	−1.57836	−0.789178	−	0.614164i	$-0.789493\pi$
$367$	−30.2370	−1.57836	−0.789178	+	0.614164i	$0.789493\pi$
$368$	25.5935	1.33416
$369$	0	0
$370$	10.4137	0.541383
$371$	−2.57967	−0.133930
$372$	0	0
$373$	−11.9458	−0.618529	−0.309265	−	0.950976i	$-0.600083\pi$
$373$	−11.9458	−0.618529	−0.309265	+	0.950976i	$0.600083\pi$
$374$	−6.53638	−0.337988
$375$	0	0
$376$	−14.2374	−0.734240
$377$	22.4233	1.15486
$378$	0	0
$379$	27.6228	1.41889	0.709444	−	0.704761i	$-0.248946\pi$
$379$	27.6228	1.41889	0.709444	+	0.704761i	$0.248946\pi$
$380$	1.76878	0.0907367
$381$	0	0
$382$	37.5979	1.92368
$383$	−31.1697	−1.59270	−0.796348	−	0.604839i	$-0.793238\pi$
$383$	−31.1697	−1.59270	−0.796348	+	0.604839i	$0.793238\pi$
$384$	0	0
$385$	0.509211	0.0259518
$386$	−22.5782	−1.14920
$387$	0	0
$388$	8.31789	0.422277
$389$	−18.4752	−0.936728	−0.468364	−	0.883536i	$-0.655156\pi$
$389$	−18.4752	−0.936728	−0.468364	+	0.883536i	$0.655156\pi$
$390$	0	0
$391$	17.0786	0.863701
$392$	−13.1477	−0.664060
$393$	0	0
$394$	11.8740	0.598202
$395$	−15.4583	−0.777791
$396$	0	0
$397$	−23.2635	−1.16756	−0.583781	−	0.811911i	$-0.698427\pi$
$397$	−23.2635	−1.16756	−0.583781	+	0.811911i	$0.698427\pi$
$398$	13.1361	0.658451
$399$	0	0
$400$	−4.97974	−0.248987
$401$	−28.3878	−1.41762	−0.708810	−	0.705400i	$-0.750767\pi$
$401$	−28.3878	−1.41762	−0.708810	+	0.705400i	$0.750767\pi$
$402$	0	0
$403$	41.0291	2.04381
$404$	−4.46891	−0.222337
$405$	0	0
$406$	−4.10230	−0.203594
$407$	−7.16809	−0.355309
$408$	0	0
$409$	−5.11844	−0.253091	−0.126545	−	0.991961i	$-0.540389\pi$
$409$	−5.11844	−0.253091	−0.126545	+	0.991961i	$0.540389\pi$
$410$	−12.5515	−0.619872
$411$	0	0
$412$	−15.8576	−0.781246
$413$	1.45435	0.0715640
$414$	0	0
$415$	−18.0513	−0.886102
$416$	18.4225	0.903237
$417$	0	0
$418$	−4.05663	−0.198416
$419$	10.6640	0.520969	0.260484	−	0.965478i	$-0.416118\pi$
$419$	10.6640	0.520969	0.260484	+	0.965478i	$0.416118\pi$
$420$	0	0
$421$	−1.11402	−0.0542942	−0.0271471	−	0.999631i	$-0.508642\pi$
$421$	−1.11402	−0.0542942	−0.0271471	+	0.999631i	$0.508642\pi$
$422$	23.6471	1.15112
$423$	0	0
$424$	−11.3833	−0.552824
$425$	−3.32298	−0.161188
$426$	0	0
$427$	−1.67950	−0.0812769
$428$	16.2099	0.783534
$429$	0	0
$430$	−6.28148	−0.302920
$431$	−26.9679	−1.29900	−0.649500	−	0.760362i	$-0.725022\pi$
$431$	−26.9679	−1.29900	−0.649500	+	0.760362i	$0.725022\pi$
$432$	0	0
$433$	16.8587	0.810176	0.405088	−	0.914278i	$-0.367241\pi$
$433$	16.8587	0.810176	0.405088	+	0.914278i	$0.367241\pi$
$434$	−7.50619	−0.360309
$435$	0	0
$436$	0.857922	0.0410870
$437$	10.5994	0.507037
$438$	0	0
$439$	13.6146	0.649791	0.324896	−	0.945750i	$-0.394671\pi$
$439$	13.6146	0.649791	0.324896	+	0.945750i	$0.394671\pi$
$440$	2.24700	0.107121
$441$	0	0
$442$	22.7147	1.08043
$443$	−10.3788	−0.493114	−0.246557	−	0.969128i	$-0.579299\pi$
$443$	−10.3788	−0.493114	−0.246557	+	0.969128i	$0.579299\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	41.8180	1.98014
$447$	0	0
$448$	0.988083	0.0466825
$449$	4.95494	0.233838	0.116919	−	0.993141i	$-0.462698\pi$
$449$	4.95494	0.233838	0.116919	+	0.993141i	$0.462698\pi$
$450$	0	0
$451$	8.63956	0.406821
$452$	7.19263	0.338313
$453$	0	0
$454$	38.6746	1.81509
$455$	−1.76956	−0.0829585
$456$	0	0
$457$	13.1185	0.613658	0.306829	−	0.951765i	$-0.400732\pi$
$457$	13.1185	0.613658	0.306829	+	0.951765i	$0.400732\pi$
$458$	30.4993	1.42514
$459$	0	0
$460$	4.40800	0.205524
$461$	−13.8692	−0.645952	−0.322976	−	0.946407i	$-0.604683\pi$
$461$	−13.8692	−0.645952	−0.322976	+	0.946407i	$0.604683\pi$
$462$	0	0
$463$	−10.8066	−0.502226	−0.251113	−	0.967958i	$-0.580797\pi$
$463$	−10.8066	−0.502226	−0.251113	+	0.967958i	$0.580797\pi$
$464$	−27.6143	−1.28196
$465$	0	0
$466$	−0.692678	−0.0320877
$467$	26.8084	1.24054	0.620272	−	0.784387i	$-0.287022\pi$
$467$	26.8084	1.24054	0.620272	+	0.784387i	$0.287022\pi$
$468$	0	0
$469$	−6.18781	−0.285726
$470$	−12.4635	−0.574896
$471$	0	0
$472$	6.41763	0.295395
$473$	4.32374	0.198806
$474$	0	0
$475$	−2.06232	−0.0946259
$476$	−1.24721	−0.0571658
$477$	0	0
$478$	22.7533	1.04071
$479$	−35.4256	−1.61864	−0.809319	−	0.587370i	$-0.800164\pi$
$479$	−35.4256	−1.61864	−0.809319	+	0.587370i	$0.800164\pi$
$480$	0	0
$481$	24.9099	1.13579
$482$	46.2925	2.10857
$483$	0	0
$484$	−8.27307	−0.376049
$485$	−9.69829	−0.440377
$486$	0	0
$487$	32.2380	1.46084	0.730421	−	0.682997i	$-0.239324\pi$
$487$	32.2380	1.46084	0.730421	+	0.682997i	$0.239324\pi$
$488$	−7.41116	−0.335487
$489$	0	0
$490$	−11.5095	−0.519946
$491$	−20.9922	−0.947365	−0.473682	−	0.880696i	$-0.657076\pi$
$491$	−20.9922	−0.947365	−0.473682	+	0.880696i	$0.657076\pi$
$492$	0	0
$493$	−18.4271	−0.829913
$494$	14.0973	0.634266
$495$	0	0
$496$	−50.5274	−2.26875
$497$	−4.04391	−0.181394
$498$	0	0
$499$	−13.3111	−0.595888	−0.297944	−	0.954583i	$-0.596301\pi$
$499$	−13.3111	−0.595888	−0.297944	+	0.954583i	$0.596301\pi$
$500$	−0.857665	−0.0383559
$501$	0	0
$502$	0.652556	0.0291250
$503$	17.9462	0.800182	0.400091	−	0.916475i	$-0.368978\pi$
$503$	17.9462	0.800182	0.400091	+	0.916475i	$0.368978\pi$
$504$	0	0
$505$	5.21056	0.231867
$506$	−10.1096	−0.449425
$507$	0	0
$508$	−3.17572	−0.140900
$509$	−17.7790	−0.788041	−0.394020	−	0.919102i	$-0.628916\pi$
$509$	−17.7790	−0.788041	−0.394020	+	0.919102i	$0.628916\pi$
$510$	0	0
$511$	3.79819	0.168022
$512$	−3.45478	−0.152681
$513$	0	0
$514$	−28.5355	−1.25865
$515$	18.4892	0.814732
$516$	0	0
$517$	8.57899	0.377303
$518$	−4.55722	−0.200233
$519$	0	0
$520$	−7.80857	−0.342428
$521$	8.24284	0.361125	0.180563	−	0.983563i	$-0.442208\pi$
$521$	8.24284	0.361125	0.180563	+	0.983563i	$0.442208\pi$
$522$	0	0
$523$	−36.8696	−1.61220	−0.806099	−	0.591781i	$-0.798425\pi$
$523$	−36.8696	−1.61220	−0.806099	+	0.591781i	$0.798425\pi$
$524$	−4.13201	−0.180508
$525$	0	0
$526$	−28.5825	−1.24626
$527$	−33.7169	−1.46873
$528$	0	0
$529$	3.41481	0.148470
$530$	−9.96497	−0.432851
$531$	0	0
$532$	−0.774049	−0.0335593
$533$	−30.0235	−1.30046
$534$	0	0
$535$	−18.9000	−0.817119
$536$	−27.3050	−1.17940
$537$	0	0
$538$	20.5258	0.884932
$539$	7.92235	0.341240
$540$	0	0
$541$	27.6025	1.18672	0.593361	−	0.804936i	$-0.297801\pi$
$541$	27.6025	1.18672	0.593361	+	0.804936i	$0.297801\pi$
$542$	−13.9813	−0.600550
$543$	0	0
$544$	−15.1392	−0.649089
$545$	−1.00030	−0.0428481
$546$	0	0
$547$	21.1074	0.902486	0.451243	−	0.892401i	$-0.350981\pi$
$547$	21.1074	0.902486	0.451243	+	0.892401i	$0.350981\pi$
$548$	3.78154	0.161540
$549$	0	0
$550$	1.96702	0.0838740
$551$	−11.4363	−0.487202
$552$	0	0
$553$	6.76481	0.287669
$554$	27.2789	1.15897
$555$	0	0
$556$	−5.89292	−0.249916
$557$	39.8805	1.68979	0.844896	−	0.534931i	$-0.179662\pi$
$557$	39.8805	1.68979	0.844896	+	0.534931i	$0.179662\pi$
$558$	0	0
$559$	−15.0255	−0.635511
$560$	2.17922	0.0920888
$561$	0	0
$562$	27.9728	1.17996
$563$	−7.67034	−0.323266	−0.161633	−	0.986851i	$-0.551676\pi$
$563$	−7.67034	−0.323266	−0.161633	+	0.986851i	$0.551676\pi$
$564$	0	0
$565$	−8.38629	−0.352814
$566$	−25.2974	−1.06333
$567$	0	0
$568$	−17.8446	−0.748742
$569$	1.45628	0.0610506	0.0305253	−	0.999534i	$-0.490282\pi$
$569$	1.45628	0.0610506	0.0305253	+	0.999534i	$0.490282\pi$
$570$	0	0
$571$	25.0036	1.04637	0.523185	−	0.852219i	$-0.324744\pi$
$571$	25.0036	1.04637	0.523185	+	0.852219i	$0.324744\pi$
$572$	−4.03546	−0.168731
$573$	0	0
$574$	5.49273	0.229262
$575$	−5.13953	−0.214333
$576$	0	0
$577$	38.5293	1.60399	0.801997	−	0.597328i	$-0.203771\pi$
$577$	38.5293	1.60399	0.801997	+	0.597328i	$0.203771\pi$
$578$	10.0714	0.418916
$579$	0	0
$580$	−4.75604	−0.197484
$581$	7.89954	0.327728
$582$	0	0
$583$	6.85920	0.284079
$584$	16.7603	0.693547
$585$	0	0
$586$	28.9021	1.19393
$587$	42.8837	1.77000	0.884999	−	0.465593i	$-0.154159\pi$
$587$	42.8837	1.77000	0.884999	+	0.465593i	$0.154159\pi$
$588$	0	0
$589$	−20.9255	−0.862222
$590$	5.61799	0.231289
$591$	0	0
$592$	−30.6766	−1.26080
$593$	−31.3911	−1.28908	−0.644539	−	0.764572i	$-0.722951\pi$
$593$	−31.3911	−1.28908	−0.644539	+	0.764572i	$0.722951\pi$
$594$	0	0
$595$	1.45419	0.0596161
$596$	−10.6260	−0.435258
$597$	0	0
$598$	35.1319	1.43665
$599$	17.2291	0.703963	0.351981	−	0.936007i	$-0.385508\pi$
$599$	17.2291	0.703963	0.351981	+	0.936007i	$0.385508\pi$
$600$	0	0
$601$	−10.6441	−0.434181	−0.217091	−	0.976151i	$-0.569657\pi$
$601$	−10.6441	−0.434181	−0.217091	+	0.976151i	$0.569657\pi$
$602$	2.74888	0.112036
$603$	0	0
$604$	−15.8901	−0.646560
$605$	9.64604	0.392167
$606$	0	0
$607$	−38.6062	−1.56698	−0.783488	−	0.621407i	$-0.786561\pi$
$607$	−38.6062	−1.56698	−0.783488	+	0.621407i	$0.786561\pi$
$608$	−9.39578	−0.381049
$609$	0	0
$610$	−6.48773	−0.262680
$611$	−29.8130	−1.20610
$612$	0	0
$613$	−1.40969	−0.0569370	−0.0284685	−	0.999595i	$-0.509063\pi$
$613$	−1.40969	−0.0569370	−0.0284685	+	0.999595i	$0.509063\pi$
$614$	9.26733	0.373999
$615$	0	0
$616$	−0.983324	−0.0396192
$617$	−24.2866	−0.977742	−0.488871	−	0.872356i	$-0.662591\pi$
$617$	−24.2866	−0.977742	−0.488871	+	0.872356i	$0.662591\pi$
$618$	0	0
$619$	−32.0699	−1.28900	−0.644499	−	0.764605i	$-0.722934\pi$
$619$	−32.0699	−1.28900	−0.644499	+	0.764605i	$0.722934\pi$
$620$	−8.70237	−0.349496
$621$	0	0
$622$	−22.2067	−0.890407
$623$	0.437617	0.0175328
$624$	0	0
$625$	1.00000	0.0400000
$626$	49.7883	1.98994
$627$	0	0
$628$	7.18521	0.286721
$629$	−20.4705	−0.816212
$630$	0	0
$631$	−0.465730	−0.0185404	−0.00927021	−	0.999957i	$-0.502951\pi$
$631$	−0.465730	−0.0185404	−0.00927021	+	0.999957i	$0.502951\pi$
$632$	29.8511	1.18741
$633$	0	0
$634$	16.2914	0.647015
$635$	3.70275	0.146939
$636$	0	0
$637$	−27.5311	−1.09082
$638$	10.9078	0.431844
$639$	0	0
$640$	12.9287	0.511051
$641$	15.9579	0.630298	0.315149	−	0.949042i	$-0.397946\pi$
$641$	15.9579	0.630298	0.315149	+	0.949042i	$0.397946\pi$
$642$	0	0
$643$	−46.1720	−1.82085	−0.910423	−	0.413679i	$-0.864243\pi$
$643$	−46.1720	−1.82085	−0.910423	+	0.413679i	$0.864243\pi$
$644$	−1.92901	−0.0760138
$645$	0	0
$646$	−11.5849	−0.455800
$647$	−9.83331	−0.386587	−0.193294	−	0.981141i	$-0.561917\pi$
$647$	−9.83331	−0.386587	−0.193294	+	0.981141i	$0.561917\pi$
$648$	0	0
$649$	−3.86704	−0.151795
$650$	−6.83562	−0.268115
$651$	0	0
$652$	−6.17345	−0.241771
$653$	−39.3883	−1.54138	−0.770692	−	0.637207i	$-0.780089\pi$
$653$	−39.3883	−1.54138	−0.770692	+	0.637207i	$0.780089\pi$
$654$	0	0
$655$	4.81775	0.188245
$656$	36.9739	1.44359
$657$	0	0
$658$	5.45422	0.212628
$659$	26.7375	1.04154	0.520772	−	0.853696i	$-0.325644\pi$
$659$	26.7375	1.04154	0.520772	+	0.853696i	$0.325644\pi$
$660$	0	0
$661$	−15.6976	−0.610566	−0.305283	−	0.952262i	$-0.598751\pi$
$661$	−15.6976	−0.610566	−0.305283	+	0.952262i	$0.598751\pi$
$662$	54.5532	2.12027
$663$	0	0
$664$	34.8584	1.35277
$665$	0.902508	0.0349978
$666$	0	0
$667$	−28.5005	−1.10354
$668$	9.00904	0.348570
$669$	0	0
$670$	−23.9028	−0.923445
$671$	4.46571	0.172397
$672$	0	0
$673$	−14.9928	−0.577931	−0.288965	−	0.957340i	$-0.593311\pi$
$673$	−14.9928	−0.577931	−0.288965	+	0.957340i	$0.593311\pi$
$674$	−42.4018	−1.63326
$675$	0	0
$676$	2.87405	0.110540
$677$	30.0654	1.15551	0.577753	−	0.816211i	$-0.303930\pi$
$677$	30.0654	1.15551	0.577753	+	0.816211i	$0.303930\pi$
$678$	0	0
$679$	4.24414	0.162875
$680$	6.41693	0.246078
$681$	0	0
$682$	19.9585	0.764252
$683$	35.1671	1.34563	0.672817	−	0.739809i	$-0.265084\pi$
$683$	35.1671	1.34563	0.672817	+	0.739809i	$0.265084\pi$
$684$	0	0
$685$	−4.40912	−0.168464
$686$	10.2152	0.390017
$687$	0	0
$688$	18.5039	0.705455
$689$	−23.8365	−0.908099
$690$	0	0
$691$	27.6030	1.05007	0.525034	−	0.851081i	$-0.324053\pi$
$691$	27.6030	1.05007	0.525034	+	0.851081i	$0.324053\pi$
$692$	17.8072	0.676927
$693$	0	0
$694$	35.8516	1.36091
$695$	6.87089	0.260628
$696$	0	0
$697$	24.6727	0.934545
$698$	−7.78394	−0.294626
$699$	0	0
$700$	0.375329	0.0141861
$701$	−37.8149	−1.42825	−0.714125	−	0.700018i	$-0.753175\pi$
$701$	−37.8149	−1.42825	−0.714125	+	0.700018i	$0.753175\pi$
$702$	0	0
$703$	−12.7045	−0.479159
$704$	−2.62726	−0.0990185
$705$	0	0
$706$	−8.99305	−0.338458
$707$	−2.28023	−0.0857568
$708$	0	0
$709$	35.1315	1.31939	0.659695	−	0.751533i	$-0.270685\pi$
$709$	35.1315	1.31939	0.659695	+	0.751533i	$0.270685\pi$
$710$	−15.6211	−0.586251
$711$	0	0
$712$	1.93108	0.0723701
$713$	−52.1487	−1.95298
$714$	0	0
$715$	4.70517	0.175964
$716$	9.59234	0.358483
$717$	0	0
$718$	4.74969	0.177257
$719$	5.96425	0.222429	0.111214	−	0.993796i	$-0.464526\pi$
$719$	5.96425	0.222429	0.111214	+	0.993796i	$0.464526\pi$
$720$	0	0
$721$	−8.09119	−0.301332
$722$	24.9290	0.927759
$723$	0	0
$724$	18.2487	0.678209
$725$	5.54534	0.205949
$726$	0	0
$727$	−10.4404	−0.387213	−0.193607	−	0.981079i	$-0.562019\pi$
$727$	−10.4404	−0.387213	−0.193607	+	0.981079i	$0.562019\pi$
$728$	3.41716	0.126648
$729$	0	0
$730$	14.6720	0.543034
$731$	12.3477	0.456695
$732$	0	0
$733$	16.3863	0.605242	0.302621	−	0.953111i	$-0.402138\pi$
$733$	16.3863	0.605242	0.302621	+	0.953111i	$0.402138\pi$
$734$	51.1145	1.88667
$735$	0	0
$736$	−23.4153	−0.863099
$737$	16.4530	0.606055
$738$	0	0
$739$	22.2243	0.817534	0.408767	−	0.912639i	$-0.365959\pi$
$739$	22.2243	0.817534	0.408767	+	0.912639i	$0.365959\pi$
$740$	−5.28345	−0.194224
$741$	0	0
$742$	4.36084	0.160091
$743$	4.05646	0.148817	0.0744085	−	0.997228i	$-0.476293\pi$
$743$	4.05646	0.148817	0.0744085	+	0.997228i	$0.476293\pi$
$744$	0	0
$745$	12.3895	0.453914
$746$	20.1939	0.739351
$747$	0	0
$748$	3.31626	0.121255
$749$	8.27097	0.302215
$750$	0	0
$751$	−31.3543	−1.14414	−0.572068	−	0.820206i	$-0.693859\pi$
$751$	−31.3543	−1.14414	−0.572068	+	0.820206i	$0.693859\pi$
$752$	36.7147	1.33885
$753$	0	0
$754$	−37.9058	−1.38045
$755$	18.5272	0.674273
$756$	0	0
$757$	−41.1020	−1.49388	−0.746938	−	0.664893i	$-0.768477\pi$
$757$	−41.1020	−1.49388	−0.746938	+	0.664893i	$0.768477\pi$
$758$	−46.6953	−1.69605
$759$	0	0
$760$	3.98250	0.144461
$761$	−4.19536	−0.152082	−0.0760408	−	0.997105i	$-0.524228\pi$
$761$	−4.19536	−0.152082	−0.0760408	+	0.997105i	$0.524228\pi$
$762$	0	0
$763$	0.437748	0.0158475
$764$	−19.0755	−0.690128
$765$	0	0
$766$	52.6912	1.90381
$767$	13.4384	0.485233
$768$	0	0
$769$	2.67684	0.0965293	0.0482646	−	0.998835i	$-0.484631\pi$
$769$	2.67684	0.0965293	0.0482646	+	0.998835i	$0.484631\pi$
$770$	−0.860802	−0.0310211
$771$	0	0
$772$	11.4551	0.412280
$773$	5.63221	0.202577	0.101288	−	0.994857i	$-0.467704\pi$
$773$	5.63221	0.202577	0.101288	+	0.994857i	$0.467704\pi$
$774$	0	0
$775$	10.1466	0.364476
$776$	18.7281	0.672301
$777$	0	0
$778$	31.2316	1.11971
$779$	15.3125	0.548626
$780$	0	0
$781$	10.7525	0.384756
$782$	−28.8707	−1.03241
$783$	0	0
$784$	33.9045	1.21088
$785$	−8.37765	−0.299011
$786$	0	0
$787$	18.6087	0.663330	0.331665	−	0.943397i	$-0.392390\pi$
$787$	18.6087	0.663330	0.331665	+	0.943397i	$0.392390\pi$
$788$	−6.02432	−0.214608
$789$	0	0
$790$	26.1317	0.929723
$791$	3.66998	0.130490
$792$	0	0
$793$	−15.5188	−0.551090
$794$	39.3261	1.39563
$795$	0	0
$796$	−6.66465	−0.236222
$797$	−32.3810	−1.14699	−0.573497	−	0.819207i	$-0.694414\pi$
$797$	−32.3810	−1.14699	−0.573497	+	0.819207i	$0.694414\pi$
$798$	0	0
$799$	24.4997	0.866737
$800$	4.55592	0.161076
$801$	0	0
$802$	47.9885	1.69453
$803$	−10.0992	−0.356393
$804$	0	0
$805$	2.24915	0.0792720
$806$	−69.3582	−2.44304
$807$	0	0
$808$	−10.0620	−0.353979
$809$	21.0267	0.739258	0.369629	−	0.929179i	$-0.379485\pi$
$809$	21.0267	0.739258	0.369629	+	0.929179i	$0.379485\pi$
$810$	0	0
$811$	−19.9211	−0.699524	−0.349762	−	0.936839i	$-0.613738\pi$
$811$	−19.9211	−0.699524	−0.349762	+	0.936839i	$0.613738\pi$
$812$	2.08132	0.0730402
$813$	0	0
$814$	12.1174	0.424714
$815$	7.19797	0.252134
$816$	0	0
$817$	7.66326	0.268104
$818$	8.65254	0.302529
$819$	0	0
$820$	6.36805	0.222382
$821$	22.2161	0.775346	0.387673	−	0.921797i	$-0.373279\pi$
$821$	22.2161	0.775346	0.387673	+	0.921797i	$0.373279\pi$
$822$	0	0
$823$	−13.3803	−0.466407	−0.233204	−	0.972428i	$-0.574921\pi$
$823$	−13.3803	−0.466407	−0.233204	+	0.972428i	$0.574921\pi$
$824$	−35.7041	−1.24381
$825$	0	0
$826$	−2.45853	−0.0855432
$827$	11.6176	0.403984	0.201992	−	0.979387i	$-0.435259\pi$
$827$	11.6176	0.403984	0.201992	+	0.979387i	$0.435259\pi$
$828$	0	0
$829$	20.4059	0.708726	0.354363	−	0.935108i	$-0.384698\pi$
$829$	20.4059	0.708726	0.354363	+	0.935108i	$0.384698\pi$
$830$	30.5150	1.05919
$831$	0	0
$832$	9.13002	0.316527
$833$	22.6245	0.783892
$834$	0	0
$835$	−10.5041	−0.363511
$836$	2.05815	0.0711827
$837$	0	0
$838$	−18.0270	−0.622734
$839$	12.9143	0.445850	0.222925	−	0.974836i	$-0.428439\pi$
$839$	12.9143	0.445850	0.222925	+	0.974836i	$0.428439\pi$
$840$	0	0
$841$	1.75078	0.0603718
$842$	1.88322	0.0649000
$843$	0	0
$844$	−11.9975	−0.412971
$845$	−3.35102	−0.115278
$846$	0	0
$847$	−4.22127	−0.145045
$848$	29.3547	1.00804
$849$	0	0
$850$	5.61738	0.192675
$851$	−31.6609	−1.08532
$852$	0	0
$853$	44.5550	1.52554	0.762768	−	0.646672i	$-0.223840\pi$
$853$	44.5550	1.52554	0.762768	+	0.646672i	$0.223840\pi$
$854$	2.83914	0.0971533
$855$	0	0
$856$	36.4974	1.24745
$857$	12.8473	0.438855	0.219428	−	0.975629i	$-0.429581\pi$
$857$	12.8473	0.438855	0.219428	+	0.975629i	$0.429581\pi$
$858$	0	0
$859$	−12.9416	−0.441562	−0.220781	−	0.975323i	$-0.570861\pi$
$859$	−12.9416	−0.441562	−0.220781	+	0.975323i	$0.570861\pi$
$860$	3.18694	0.108674
$861$	0	0
$862$	45.5883	1.55274
$863$	−23.4478	−0.798171	−0.399085	−	0.916914i	$-0.630672\pi$
$863$	−23.4478	−0.798171	−0.399085	+	0.916914i	$0.630672\pi$
$864$	0	0
$865$	−20.7624	−0.705942
$866$	−28.4990	−0.968434
$867$	0	0
$868$	3.80830	0.129262
$869$	−17.9872	−0.610175
$870$	0	0
$871$	−57.1762	−1.93734
$872$	1.93165	0.0654140
$873$	0	0
$874$	−17.9179	−0.606081
$875$	−0.437617	−0.0147942
$876$	0	0
$877$	−19.7747	−0.667744	−0.333872	−	0.942618i	$-0.608355\pi$
$877$	−19.7747	−0.667744	−0.333872	+	0.942618i	$0.608355\pi$
$878$	−23.0150	−0.776720
$879$	0	0
$880$	−5.79442	−0.195330
$881$	12.8797	0.433930	0.216965	−	0.976179i	$-0.430384\pi$
$881$	12.8797	0.433930	0.216965	+	0.976179i	$0.430384\pi$
$882$	0	0
$883$	−18.1721	−0.611539	−0.305769	−	0.952106i	$-0.598914\pi$
$883$	−18.1721	−0.611539	−0.305769	+	0.952106i	$0.598914\pi$
$884$	−11.5244	−0.387607
$885$	0	0
$886$	17.5451	0.589438
$887$	31.4765	1.05688	0.528439	−	0.848971i	$-0.322778\pi$
$887$	31.4765	1.05688	0.528439	+	0.848971i	$0.322778\pi$
$888$	0	0
$889$	−1.62039	−0.0543461
$890$	1.69046	0.0566644
$891$	0	0
$892$	−21.2166	−0.710383
$893$	15.2051	0.508820
$894$	0	0
$895$	−11.1843	−0.373848
$896$	−5.65781	−0.189014
$897$	0	0
$898$	−8.37614	−0.279515
$899$	56.2662	1.87658
$900$	0	0
$901$	19.5884	0.652583
$902$	−14.6049	−0.486289
$903$	0	0
$904$	16.1946	0.538623
$905$	−21.2772	−0.707280
$906$	0	0
$907$	−12.4667	−0.413951	−0.206976	−	0.978346i	$-0.566362\pi$
$907$	−12.4667	−0.413951	−0.206976	+	0.978346i	$0.566362\pi$
$908$	−19.6218	−0.651171
$909$	0	0
$910$	2.99138	0.0991634
$911$	−5.05500	−0.167480	−0.0837399	−	0.996488i	$-0.526686\pi$
$911$	−5.05500	−0.167480	−0.0837399	+	0.996488i	$0.526686\pi$
$912$	0	0
$913$	−21.0044	−0.695146
$914$	−22.1763	−0.733528
$915$	0	0
$916$	−15.4740	−0.511274
$917$	−2.10833	−0.0696231
$918$	0	0
$919$	32.6714	1.07773	0.538865	−	0.842392i	$-0.318853\pi$
$919$	32.6714	1.07773	0.538865	+	0.842392i	$0.318853\pi$
$920$	9.92483	0.327212
$921$	0	0
$922$	23.4453	0.772131
$923$	−37.3663	−1.22993
$924$	0	0
$925$	6.16027	0.202549
$926$	18.2682	0.600329
$927$	0	0
$928$	25.2641	0.829335
$929$	−30.2533	−0.992579	−0.496289	−	0.868157i	$-0.665305\pi$
$929$	−30.2533	−0.992579	−0.496289	+	0.868157i	$0.665305\pi$
$930$	0	0
$931$	14.0413	0.460186
$932$	0.351434	0.0115116
$933$	0	0
$934$	−45.3186	−1.48287
$935$	−3.86662	−0.126452
$936$	0	0
$937$	−36.6314	−1.19670	−0.598348	−	0.801236i	$-0.704176\pi$
$937$	−36.6314	−1.19670	−0.598348	+	0.801236i	$0.704176\pi$
$938$	10.4603	0.341539
$939$	0	0
$940$	6.32340	0.206246
$941$	23.3264	0.760419	0.380210	−	0.924900i	$-0.375852\pi$
$941$	23.3264	0.760419	0.380210	+	0.924900i	$0.375852\pi$
$942$	0	0
$943$	38.1603	1.24267
$944$	−16.5494	−0.538637
$945$	0	0
$946$	−7.30913	−0.237640
$947$	38.1854	1.24086	0.620429	−	0.784262i	$-0.286958\pi$
$947$	38.1854	1.24086	0.620429	+	0.784262i	$0.286958\pi$
$948$	0	0
$949$	35.0958	1.13926
$950$	3.48628	0.113110
$951$	0	0
$952$	−2.80816	−0.0910129
$953$	16.7077	0.541215	0.270608	−	0.962690i	$-0.412775\pi$
$953$	16.7077	0.541215	0.270608	+	0.962690i	$0.412775\pi$
$954$	0	0
$955$	22.2412	0.719709
$956$	−11.5440	−0.373360
$957$	0	0
$958$	59.8857	1.93482
$959$	1.92950	0.0623069
$960$	0	0
$961$	71.9532	2.32107
$962$	−42.1093	−1.35766
$963$	0	0
$964$	−23.4868	−0.756457
$965$	−13.3562	−0.429951
$966$	0	0
$967$	36.3350	1.16845	0.584227	−	0.811590i	$-0.301398\pi$
$967$	36.3350	1.16845	0.584227	+	0.811590i	$0.301398\pi$
$968$	−18.6272	−0.598702
$969$	0	0
$970$	16.3946	0.526399
$971$	−55.8004	−1.79072	−0.895360	−	0.445344i	$-0.853082\pi$
$971$	−55.8004	−1.79072	−0.895360	+	0.445344i	$0.853082\pi$
$972$	0	0
$973$	−3.00682	−0.0963941
$974$	−54.4971	−1.74620
$975$	0	0
$976$	19.1114	0.611742
$977$	−18.6043	−0.595203	−0.297601	−	0.954690i	$-0.596187\pi$
$977$	−18.6043	−0.595203	−0.297601	+	0.954690i	$0.596187\pi$
$978$	0	0
$979$	−1.16360	−0.0371888
$980$	5.83940	0.186533
$981$	0	0
$982$	35.4865	1.13242
$983$	−3.20700	−0.102288	−0.0511438	−	0.998691i	$-0.516287\pi$
$983$	−3.20700	−0.102288	−0.0511438	+	0.998691i	$0.516287\pi$
$984$	0	0
$985$	7.02410	0.223806
$986$	31.1503	0.992027
$987$	0	0
$988$	−7.15232	−0.227546
$989$	19.0977	0.607271
$990$	0	0
$991$	−12.8658	−0.408697	−0.204349	−	0.978898i	$-0.565508\pi$
$991$	−12.8658	−0.408697	−0.204349	+	0.978898i	$0.565508\pi$
$992$	46.2270	1.46771
$993$	0	0
$994$	6.83608	0.216827
$995$	7.77069	0.246347
$996$	0	0
$997$	−12.3015	−0.389592	−0.194796	−	0.980844i	$-0.562404\pi$
$997$	−12.3015	−0.389592	−0.194796	+	0.980844i	$0.562404\pi$
$998$	22.5020	0.712288
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.3		10
3.2	odd	2		1335.2.a.j.1.8	✓	10
15.14	odd	2		6675.2.a.z.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.8	✓	10	3.2	odd	2
4005.2.a.s.1.3		10	1.1	even	1	trivial
6675.2.a.z.1.3		10	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4005.a (trivial)

\(f(q)\)	\(=\)	\(q-1.69046 q^{2} +0.857665 q^{4} -1.00000 q^{5} +0.437617 q^{7} +1.93108 q^{8} +O(q^{10})\)	\(q-1.69046 q^{2} +0.857665 q^{4} -1.00000 q^{5} +0.437617 q^{7} +1.93108 q^{8} +1.69046 q^{10} -1.16360 q^{11} +4.04364 q^{13} -0.739775 q^{14} -4.97974 q^{16} -3.32298 q^{17} -2.06232 q^{19} -0.857665 q^{20} +1.96702 q^{22} -5.13953 q^{23} +1.00000 q^{25} -6.83562 q^{26} +0.375329 q^{28} +5.54534 q^{29} +10.1466 q^{31} +4.55592 q^{32} +5.61738 q^{34} -0.437617 q^{35} +6.16027 q^{37} +3.48628 q^{38} -1.93108 q^{40} -7.42486 q^{41} -3.71584 q^{43} -0.997978 q^{44} +8.68819 q^{46} -7.37281 q^{47} -6.80849 q^{49} -1.69046 q^{50} +3.46809 q^{52} -5.89482 q^{53} +1.16360 q^{55} +0.845071 q^{56} -9.37419 q^{58} +3.32335 q^{59} -3.83784 q^{61} -17.1524 q^{62} +2.25787 q^{64} -4.04364 q^{65} -14.1398 q^{67} -2.85001 q^{68} +0.739775 q^{70} -9.24075 q^{71} +8.67927 q^{73} -10.4137 q^{74} -1.76878 q^{76} -0.509211 q^{77} +15.4583 q^{79} +4.97974 q^{80} +12.5515 q^{82} +18.0513 q^{83} +3.32298 q^{85} +6.28148 q^{86} -2.24700 q^{88} +1.00000 q^{89} +1.76956 q^{91} -4.40800 q^{92} +12.4635 q^{94} +2.06232 q^{95} +9.69829 q^{97} +11.5095 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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