LMFDB - Embedded newform 4005.2.a.m.1.2

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:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$2.36234$ 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)

$f(q)$	$=$	$q-1.21831 q^{2} -0.515722 q^{4} +1.00000 q^{5} +0.362340 q^{7} +3.06493 q^{8} +O(q^{10})$	\(q-1.21831 q^{2} -0.515722 q^{4} +1.00000 q^{5} +0.362340 q^{7} +3.06493 q^{8} -1.21831 q^{10} +2.43662 q^{11} -2.54921 q^{13} -0.441442 q^{14} -2.70259 q^{16} +0.935072 q^{17} -6.72468 q^{19} -0.515722 q^{20} -2.96856 q^{22} +6.60274 q^{23} +1.00000 q^{25} +3.10572 q^{26} -0.186866 q^{28} -5.53781 q^{29} +1.69324 q^{31} -2.83727 q^{32} -1.13921 q^{34} +0.362340 q^{35} -8.38443 q^{37} +8.19274 q^{38} +3.06493 q^{40} -1.51572 q^{41} +5.85249 q^{43} -1.25662 q^{44} -8.04418 q^{46} -7.83040 q^{47} -6.86871 q^{49} -1.21831 q^{50} +1.31468 q^{52} +4.73199 q^{53} +2.43662 q^{55} +1.11055 q^{56} +6.74677 q^{58} -1.10776 q^{59} -4.65771 q^{61} -2.06289 q^{62} +8.86185 q^{64} -2.54921 q^{65} -2.41000 q^{67} -0.482237 q^{68} -0.441442 q^{70} +6.12986 q^{71} -15.0394 q^{73} +10.2148 q^{74} +3.46806 q^{76} +0.882884 q^{77} +1.05075 q^{79} -2.70259 q^{80} +1.84662 q^{82} -6.89080 q^{83} +0.935072 q^{85} -7.13015 q^{86} +7.46806 q^{88} -1.00000 q^{89} -0.923679 q^{91} -3.40518 q^{92} +9.53985 q^{94} -6.72468 q^{95} +5.70115 q^{97} +8.36822 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	$ 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * 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 $

$2$	−1.21831	−0.861475	−0.430738	−	0.902477i	$-0.641747\pi$
$2$	−1.21831	−0.861475	−0.430738	+	0.902477i	$0.641747\pi$
$3$	0	0
$3$	0	0
$4$	−0.515722	−0.257861
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.362340	0.136952	0.0684758	−	0.997653i	$-0.478186\pi$
$7$	0.362340	0.136952	0.0684758	+	0.997653i	$0.478186\pi$
$8$	3.06493	1.08362
$9$	0	0
$10$	−1.21831	−0.385263
$11$	2.43662	0.734668	0.367334	−	0.930089i	$-0.380271\pi$
$11$	2.43662	0.734668	0.367334	+	0.930089i	$0.380271\pi$
$12$	0	0
$13$	−2.54921	−0.707023	−0.353511	−	0.935430i	$-0.615012\pi$
$13$	−2.54921	−0.707023	−0.353511	+	0.935430i	$0.615012\pi$
$14$	−0.441442	−0.117980
$15$	0	0
$16$	−2.70259	−0.675647
$17$	0.935072	0.226788	0.113394	−	0.993550i	$-0.463828\pi$
$17$	0.935072	0.226788	0.113394	+	0.993550i	$0.463828\pi$
$18$	0	0
$19$	−6.72468	−1.54275	−0.771374	−	0.636382i	$-0.780430\pi$
$19$	−6.72468	−1.54275	−0.771374	+	0.636382i	$0.780430\pi$
$20$	−0.515722	−0.115319
$21$	0	0
$22$	−2.96856	−0.632898
$23$	6.60274	1.37677	0.688383	−	0.725347i	$-0.258321\pi$
$23$	6.60274	1.37677	0.688383	+	0.725347i	$0.258321\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.10572	0.609082
$27$	0	0
$28$	−0.186866	−0.0353144
$29$	−5.53781	−1.02835	−0.514173	−	0.857686i	$-0.671901\pi$
$29$	−5.53781	−1.02835	−0.514173	+	0.857686i	$0.671901\pi$
$30$	0	0
$31$	1.69324	0.304114	0.152057	−	0.988372i	$-0.451410\pi$
$31$	1.69324	0.304114	0.152057	+	0.988372i	$0.451410\pi$
$32$	−2.83727	−0.501563
$33$	0	0
$34$	−1.13921	−0.195372
$35$	0.362340	0.0612466
$36$	0	0
$37$	−8.38443	−1.37839	−0.689196	−	0.724575i	$-0.742036\pi$
$37$	−8.38443	−1.37839	−0.689196	+	0.724575i	$0.742036\pi$
$38$	8.19274	1.32904
$39$	0	0
$40$	3.06493	0.484608
$41$	−1.51572	−0.236716	−0.118358	−	0.992971i	$-0.537763\pi$
$41$	−1.51572	−0.236716	−0.118358	+	0.992971i	$0.537763\pi$
$42$	0	0
$43$	5.85249	0.892497	0.446248	−	0.894909i	$-0.352760\pi$
$43$	5.85249	0.892497	0.446248	+	0.894909i	$0.352760\pi$
$44$	−1.25662	−0.189442
$45$	0	0
$46$	−8.04418	−1.18605
$47$	−7.83040	−1.14218	−0.571091	−	0.820887i	$-0.693480\pi$
$47$	−7.83040	−1.14218	−0.571091	+	0.820887i	$0.693480\pi$
$48$	0	0
$49$	−6.86871	−0.981244
$50$	−1.21831	−0.172295
$51$	0	0
$52$	1.31468	0.182313
$53$	4.73199	0.649989	0.324994	−	0.945716i	$-0.394638\pi$
$53$	4.73199	0.649989	0.324994	+	0.945716i	$0.394638\pi$
$54$	0	0
$55$	2.43662	0.328554
$56$	1.11055	0.148403
$57$	0	0
$58$	6.74677	0.885894
$59$	−1.10776	−0.144219	−0.0721093	−	0.997397i	$-0.522973\pi$
$59$	−1.10776	−0.144219	−0.0721093	+	0.997397i	$0.522973\pi$
$60$	0	0
$61$	−4.65771	−0.596359	−0.298179	−	0.954510i	$-0.596379\pi$
$61$	−4.65771	−0.596359	−0.298179	+	0.954510i	$0.596379\pi$
$62$	−2.06289	−0.261987
$63$	0	0
$64$	8.86185	1.10773
$65$	−2.54921	−0.316190
$66$	0	0
$67$	−2.41000	−0.294428	−0.147214	−	0.989105i	$-0.547031\pi$
$67$	−2.41000	−0.294428	−0.147214	+	0.989105i	$0.547031\pi$
$68$	−0.482237	−0.0584798
$69$	0	0
$70$	−0.441442	−0.0527624
$71$	6.12986	0.727480	0.363740	−	0.931501i	$-0.381500\pi$
$71$	6.12986	0.727480	0.363740	+	0.931501i	$0.381500\pi$
$72$	0	0
$73$	−15.0394	−1.76022	−0.880112	−	0.474766i	$-0.842533\pi$
$73$	−15.0394	−1.76022	−0.880112	+	0.474766i	$0.842533\pi$
$74$	10.2148	1.18745
$75$	0	0
$76$	3.46806	0.397814
$77$	0.882884	0.100614
$78$	0	0
$79$	1.05075	0.118219	0.0591095	−	0.998252i	$-0.481174\pi$
$79$	1.05075	0.118219	0.0591095	+	0.998252i	$0.481174\pi$
$80$	−2.70259	−0.302159
$81$	0	0
$82$	1.84662	0.203925
$83$	−6.89080	−0.756364	−0.378182	−	0.925731i	$-0.623451\pi$
$83$	−6.89080	−0.756364	−0.378182	+	0.925731i	$0.623451\pi$
$84$	0	0
$85$	0.935072	0.101423
$86$	−7.13015	−0.768864
$87$	0	0
$88$	7.46806	0.796098
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−0.923679	−0.0968279
$92$	−3.40518	−0.355014
$93$	0	0
$94$	9.53985	0.983961
$95$	−6.72468	−0.689938
$96$	0	0
$97$	5.70115	0.578864	0.289432	−	0.957199i	$-0.406534\pi$
$97$	5.70115	0.578864	0.289432	+	0.957199i	$0.406534\pi$
$98$	8.36822	0.845317
$99$	0	0
$100$	−0.515722	−0.0515722
$101$	1.35956	0.135281	0.0676406	−	0.997710i	$-0.478453\pi$
$101$	1.35956	0.135281	0.0676406	+	0.997710i	$0.478453\pi$
$102$	0	0
$103$	−13.3368	−1.31411	−0.657056	−	0.753842i	$-0.728198\pi$
$103$	−13.3368	−1.31411	−0.657056	+	0.753842i	$0.728198\pi$
$104$	−7.81313	−0.766141
$105$	0	0
$106$	−5.76503	−0.559949
$107$	2.60961	0.252280	0.126140	−	0.992012i	$-0.459741\pi$
$107$	2.60961	0.252280	0.126140	+	0.992012i	$0.459741\pi$
$108$	0	0
$109$	15.2148	1.45732	0.728658	−	0.684877i	$-0.240144\pi$
$109$	15.2148	1.45732	0.728658	+	0.684877i	$0.240144\pi$
$110$	−2.96856	−0.283041
$111$	0	0
$112$	−0.979255	−0.0925309
$113$	5.52711	0.519947	0.259974	−	0.965616i	$-0.416286\pi$
$113$	5.52711	0.519947	0.259974	+	0.965616i	$0.416286\pi$
$114$	0	0
$115$	6.60274	0.615709
$116$	2.85597	0.265170
$117$	0	0
$118$	1.34960	0.124241
$119$	0.338814	0.0310590
$120$	0	0
$121$	−5.06289	−0.460262
$122$	5.67453	0.513748
$123$	0	0
$124$	−0.873239	−0.0784191
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.3019	1.71276	0.856381	−	0.516344i	$-0.172707\pi$
$127$	19.3019	1.71276	0.856381	+	0.516344i	$0.172707\pi$
$128$	−5.12194	−0.452720
$129$	0	0
$130$	3.10572	0.272390
$131$	6.26936	0.547756	0.273878	−	0.961764i	$-0.411694\pi$
$131$	6.26936	0.547756	0.273878	+	0.961764i	$0.411694\pi$
$132$	0	0
$133$	−2.43662	−0.211282
$134$	2.93612	0.253643
$135$	0	0
$136$	2.86593	0.245751
$137$	0.819998	0.0700571	0.0350286	−	0.999386i	$-0.488848\pi$
$137$	0.819998	0.0700571	0.0350286	+	0.999386i	$0.488848\pi$
$138$	0	0
$139$	15.9395	1.35197	0.675986	−	0.736915i	$-0.263718\pi$
$139$	15.9395	1.35197	0.675986	+	0.736915i	$0.263718\pi$
$140$	−0.186866	−0.0157931
$141$	0	0
$142$	−7.46806	−0.626706
$143$	−6.21145	−0.519427
$144$	0	0
$145$	−5.53781	−0.459890
$146$	18.3226	1.51639
$147$	0	0
$148$	4.32403	0.355433
$149$	−4.09359	−0.335360	−0.167680	−	0.985841i	$-0.553628\pi$
$149$	−4.09359	−0.335360	−0.167680	+	0.985841i	$0.553628\pi$
$150$	0	0
$151$	−9.30080	−0.756888	−0.378444	−	0.925624i	$-0.623541\pi$
$151$	−9.30080	−0.756888	−0.378444	+	0.925624i	$0.623541\pi$
$152$	−20.6107	−1.67175
$153$	0	0
$154$	−1.07563	−0.0866764
$155$	1.69324	0.136004
$156$	0	0
$157$	−1.68532	−0.134503	−0.0672516	−	0.997736i	$-0.521423\pi$
$157$	−1.68532	−0.134503	−0.0672516	+	0.997736i	$0.521423\pi$
$158$	−1.28014	−0.101843
$159$	0	0
$160$	−2.83727	−0.224306
$161$	2.39244	0.188550
$162$	0	0
$163$	4.96712	0.389055	0.194528	−	0.980897i	$-0.437683\pi$
$163$	4.96712	0.389055	0.194528	+	0.980897i	$0.437683\pi$
$164$	0.781690	0.0610398
$165$	0	0
$166$	8.39513	0.651588
$167$	−8.91886	−0.690162	−0.345081	−	0.938573i	$-0.612149\pi$
$167$	−8.91886	−0.690162	−0.345081	+	0.938573i	$0.612149\pi$
$168$	0	0
$169$	−6.50155	−0.500119
$170$	−1.13921	−0.0873732
$171$	0	0
$172$	−3.01826	−0.230140
$173$	−26.2355	−1.99465	−0.997324	−	0.0731122i	$-0.976707\pi$
$173$	−26.2355	−1.99465	−0.997324	+	0.0731122i	$0.976707\pi$
$174$	0	0
$175$	0.362340	0.0273903
$176$	−6.58518	−0.496376
$177$	0	0
$178$	1.21831	0.0913162
$179$	−6.59482	−0.492920	−0.246460	−	0.969153i	$-0.579267\pi$
$179$	−6.59482	−0.492920	−0.246460	+	0.969153i	$0.579267\pi$
$180$	0	0
$181$	10.9016	0.810309	0.405154	−	0.914248i	$-0.367218\pi$
$181$	10.9016	0.810309	0.405154	+	0.914248i	$0.367218\pi$
$182$	1.12533	0.0834148
$183$	0	0
$184$	20.2369	1.49189
$185$	−8.38443	−0.616436
$186$	0	0
$187$	2.27841	0.166614
$188$	4.03831	0.294524
$189$	0	0
$190$	8.19274	0.594364
$191$	0.901293	0.0652153	0.0326076	−	0.999468i	$-0.489619\pi$
$191$	0.901293	0.0652153	0.0326076	+	0.999468i	$0.489619\pi$
$192$	0	0
$193$	7.79300	0.560952	0.280476	−	0.959861i	$-0.409508\pi$
$193$	7.79300	0.560952	0.280476	+	0.959861i	$0.409508\pi$
$194$	−6.94577	−0.498677
$195$	0	0
$196$	3.54234	0.253024
$197$	−19.4169	−1.38340	−0.691699	−	0.722186i	$-0.743138\pi$
$197$	−19.4169	−1.38340	−0.691699	+	0.722186i	$0.743138\pi$
$198$	0	0
$199$	−22.5754	−1.60033	−0.800165	−	0.599779i	$-0.795255\pi$
$199$	−22.5754	−1.60033	−0.800165	+	0.599779i	$0.795255\pi$
$200$	3.06493	0.216723
$201$	0	0
$202$	−1.65636	−0.116541
$203$	−2.00657	−0.140834
$204$	0	0
$205$	−1.51572	−0.105863
$206$	16.2483	1.13207
$207$	0	0
$208$	6.88945	0.477698
$209$	−16.3855	−1.13341
$210$	0	0
$211$	−20.7689	−1.42979	−0.714894	−	0.699233i	$-0.753525\pi$
$211$	−20.7689	−1.42979	−0.714894	+	0.699233i	$0.753525\pi$
$212$	−2.44039	−0.167607
$213$	0	0
$214$	−3.17931	−0.217333
$215$	5.85249	0.399137
$216$	0	0
$217$	0.613527	0.0416489
$218$	−18.5364	−1.25544
$219$	0	0
$220$	−1.25662	−0.0847211
$221$	−2.38369	−0.160344
$222$	0	0
$223$	−20.5840	−1.37841	−0.689205	−	0.724567i	$-0.742040\pi$
$223$	−20.5840	−1.37841	−0.689205	+	0.724567i	$0.742040\pi$
$224$	−1.02805	−0.0686898
$225$	0	0
$226$	−6.73374	−0.447922
$227$	−14.6261	−0.970771	−0.485385	−	0.874300i	$-0.661321\pi$
$227$	−14.6261	−0.970771	−0.485385	+	0.874300i	$0.661321\pi$
$228$	0	0
$229$	−6.54777	−0.432689	−0.216344	−	0.976317i	$-0.569413\pi$
$229$	−6.54777	−0.432689	−0.216344	+	0.976317i	$0.569413\pi$
$230$	−8.04418	−0.530418
$231$	0	0
$232$	−16.9730	−1.11433
$233$	22.2811	1.45968	0.729840	−	0.683617i	$-0.239594\pi$
$233$	22.2811	1.45968	0.729840	+	0.683617i	$0.239594\pi$
$234$	0	0
$235$	−7.83040	−0.510799
$236$	0.571298	0.0371883
$237$	0	0
$238$	−0.412780	−0.0267566
$239$	−1.59731	−0.103321	−0.0516607	−	0.998665i	$-0.516451\pi$
$239$	−1.59731	−0.103321	−0.0516607	+	0.998665i	$0.516451\pi$
$240$	0	0
$241$	8.04418	0.518171	0.259086	−	0.965854i	$-0.416579\pi$
$241$	8.04418	0.518171	0.259086	+	0.965854i	$0.416579\pi$
$242$	6.16816	0.396505
$243$	0	0
$244$	2.40208	0.153778
$245$	−6.86871	−0.438826
$246$	0	0
$247$	17.1426	1.09076
$248$	5.18965	0.329543
$249$	0	0
$250$	−1.21831	−0.0770527
$251$	2.05732	0.129857	0.0649286	−	0.997890i	$-0.479318\pi$
$251$	2.05732	0.129857	0.0649286	+	0.997890i	$0.479318\pi$
$252$	0	0
$253$	16.0884	1.01147
$254$	−23.5156	−1.47550
$255$	0	0
$256$	−11.4836	−0.717724
$257$	−18.2574	−1.13886	−0.569432	−	0.822039i	$-0.692837\pi$
$257$	−18.2574	−1.13886	−0.569432	+	0.822039i	$0.692837\pi$
$258$	0	0
$259$	−3.03801	−0.188773
$260$	1.31468	0.0815330
$261$	0	0
$262$	−7.63802	−0.471878
$263$	11.3489	0.699803	0.349902	−	0.936786i	$-0.386215\pi$
$263$	11.3489	0.699803	0.349902	+	0.936786i	$0.386215\pi$
$264$	0	0
$265$	4.73199	0.290684
$266$	2.96856	0.182014
$267$	0	0
$268$	1.24289	0.0759215
$269$	−0.991343	−0.0604433	−0.0302216	−	0.999543i	$-0.509621\pi$
$269$	−0.991343	−0.0604433	−0.0302216	+	0.999543i	$0.509621\pi$
$270$	0	0
$271$	19.6591	1.19420	0.597101	−	0.802166i	$-0.296319\pi$
$271$	19.6591	1.19420	0.597101	+	0.802166i	$0.296319\pi$
$272$	−2.52711	−0.153229
$273$	0	0
$274$	−0.999011	−0.0603525
$275$	2.43662	0.146934
$276$	0	0
$277$	−27.4532	−1.64950	−0.824751	−	0.565496i	$-0.808685\pi$
$277$	−27.4532	−1.64950	−0.824751	+	0.565496i	$0.808685\pi$
$278$	−19.4193	−1.16469
$279$	0	0
$280$	1.11055	0.0663678
$281$	2.86810	0.171097	0.0855483	−	0.996334i	$-0.472736\pi$
$281$	2.86810	0.171097	0.0855483	+	0.996334i	$0.472736\pi$
$282$	0	0
$283$	−28.9844	−1.72294	−0.861472	−	0.507806i	$-0.830457\pi$
$283$	−28.9844	−1.72294	−0.861472	+	0.507806i	$0.830457\pi$
$284$	−3.16130	−0.187589
$285$	0	0
$286$	7.56746	0.447474
$287$	−0.549206	−0.0324186
$288$	0	0
$289$	−16.1256	−0.948567
$290$	6.74677	0.396184
$291$	0	0
$292$	7.75612	0.453893
$293$	25.1838	1.47126	0.735628	−	0.677386i	$-0.236887\pi$
$293$	25.1838	1.47126	0.735628	+	0.677386i	$0.236887\pi$
$294$	0	0
$295$	−1.10776	−0.0644965
$296$	−25.6977	−1.49365
$297$	0	0
$298$	4.98726	0.288904
$299$	−16.8317	−0.973405
$300$	0	0
$301$	2.12059	0.122229
$302$	11.3313	0.652040
$303$	0	0
$304$	18.1740	1.04235
$305$	−4.65771	−0.266700
$306$	0	0
$307$	−32.7540	−1.86937	−0.934685	−	0.355478i	$-0.884318\pi$
$307$	−32.7540	−1.86937	−0.934685	+	0.355478i	$0.884318\pi$
$308$	−0.455322	−0.0259444
$309$	0	0
$310$	−2.06289	−0.117164
$311$	−9.70786	−0.550482	−0.275241	−	0.961375i	$-0.588758\pi$
$311$	−9.70786	−0.550482	−0.275241	+	0.961375i	$0.588758\pi$
$312$	0	0
$313$	−3.43866	−0.194365	−0.0971823	−	0.995267i	$-0.530983\pi$
$313$	−3.43866	−0.194365	−0.0971823	+	0.995267i	$0.530983\pi$
$314$	2.05324	0.115871
$315$	0	0
$316$	−0.541896	−0.0304840
$317$	13.0767	0.734459	0.367230	−	0.930130i	$-0.380306\pi$
$317$	13.0767	0.734459	0.367230	+	0.930130i	$0.380306\pi$
$318$	0	0
$319$	−13.4935	−0.755493
$320$	8.86185	0.495392
$321$	0	0
$322$	−2.91473	−0.162431
$323$	−6.28806	−0.349877
$324$	0	0
$325$	−2.54921	−0.141405
$326$	−6.05149	−0.335161
$327$	0	0
$328$	−4.64558	−0.256509
$329$	−2.83727	−0.156424
$330$	0	0
$331$	1.64762	0.0905613	0.0452807	−	0.998974i	$-0.485582\pi$
$331$	1.64762	0.0905613	0.0452807	+	0.998974i	$0.485582\pi$
$332$	3.55374	0.195037
$333$	0	0
$334$	10.8659	0.594557
$335$	−2.41000	−0.131672
$336$	0	0
$337$	−19.9790	−1.08832	−0.544162	−	0.838980i	$-0.683152\pi$
$337$	−19.9790	−1.08832	−0.544162	+	0.838980i	$0.683152\pi$
$338$	7.92090	0.430840
$339$	0	0
$340$	−0.482237	−0.0261530
$341$	4.12577	0.223423
$342$	0	0
$343$	−5.02519	−0.271335
$344$	17.9375	0.967124
$345$	0	0
$346$	31.9630	1.71834
$347$	−31.2684	−1.67857	−0.839287	−	0.543689i	$-0.817027\pi$
$347$	−31.2684	−1.67857	−0.839287	+	0.543689i	$0.817027\pi$
$348$	0	0
$349$	−9.06598	−0.485291	−0.242645	−	0.970115i	$-0.578015\pi$
$349$	−9.06598	−0.485291	−0.242645	+	0.970115i	$0.578015\pi$
$350$	−0.441442	−0.0235961
$351$	0	0
$352$	−6.91334	−0.368482
$353$	26.2261	1.39588	0.697938	−	0.716158i	$-0.254101\pi$
$353$	26.2261	1.39588	0.697938	+	0.716158i	$0.254101\pi$
$354$	0	0
$355$	6.12986	0.325339
$356$	0.515722	0.0273332
$357$	0	0
$358$	8.03454	0.424639
$359$	15.5354	0.819927	0.409963	−	0.912102i	$-0.365542\pi$
$359$	15.5354	0.819927	0.409963	+	0.912102i	$0.365542\pi$
$360$	0	0
$361$	26.2213	1.38007
$362$	−13.2815	−0.698061
$363$	0	0
$364$	0.476361	0.0249681
$365$	−15.0394	−0.787196
$366$	0	0
$367$	−21.2815	−1.11089	−0.555443	−	0.831555i	$-0.687451\pi$
$367$	−21.2815	−1.11089	−0.555443	+	0.831555i	$0.687451\pi$
$368$	−17.8445	−0.930208
$369$	0	0
$370$	10.2148	0.531044
$371$	1.71459	0.0890170
$372$	0	0
$373$	−18.4957	−0.957670	−0.478835	−	0.877905i	$-0.658941\pi$
$373$	−18.4957	−0.957670	−0.478835	+	0.877905i	$0.658941\pi$
$374$	−2.77581	−0.143534
$375$	0	0
$376$	−23.9996	−1.23769
$377$	14.1170	0.727064
$378$	0	0
$379$	29.9302	1.53741	0.768705	−	0.639604i	$-0.220902\pi$
$379$	29.9302	1.53741	0.768705	+	0.639604i	$0.220902\pi$
$380$	3.46806	0.177908
$381$	0	0
$382$	−1.09805	−0.0561813
$383$	−7.28497	−0.372244	−0.186122	−	0.982527i	$-0.559592\pi$
$383$	−7.28497	−0.372244	−0.186122	+	0.982527i	$0.559592\pi$
$384$	0	0
$385$	0.882884	0.0449959
$386$	−9.49428	−0.483246
$387$	0	0
$388$	−2.94021	−0.149266
$389$	−3.73508	−0.189376	−0.0946881	−	0.995507i	$-0.530185\pi$
$389$	−3.73508	−0.189376	−0.0946881	+	0.995507i	$0.530185\pi$
$390$	0	0
$391$	6.17404	0.312235
$392$	−21.0521	−1.06329
$393$	0	0
$394$	23.6558	1.19176
$395$	1.05075	0.0528691
$396$	0	0
$397$	14.1225	0.708790	0.354395	−	0.935096i	$-0.384687\pi$
$397$	14.1225	0.708790	0.354395	+	0.935096i	$0.384687\pi$
$398$	27.5039	1.37865
$399$	0	0
$400$	−2.70259	−0.135129
$401$	34.6707	1.73137	0.865685	−	0.500588i	$-0.166883\pi$
$401$	34.6707	1.73137	0.865685	+	0.500588i	$0.166883\pi$
$402$	0	0
$403$	−4.31641	−0.215016
$404$	−0.701154	−0.0348837
$405$	0	0
$406$	2.44462	0.121325
$407$	−20.4297	−1.01266
$408$	0	0
$409$	9.97163	0.493065	0.246533	−	0.969134i	$-0.420709\pi$
$409$	9.97163	0.493065	0.246533	+	0.969134i	$0.420709\pi$
$410$	1.84662	0.0911980
$411$	0	0
$412$	6.87806	0.338858
$413$	−0.401387	−0.0197510
$414$	0	0
$415$	−6.89080	−0.338256
$416$	7.23278	0.354616
$417$	0	0
$418$	19.9626	0.976402
$419$	−8.18996	−0.400106	−0.200053	−	0.979785i	$-0.564111\pi$
$419$	−8.18996	−0.400106	−0.200053	+	0.979785i	$0.564111\pi$
$420$	0	0
$421$	11.0888	0.540433	0.270217	−	0.962800i	$-0.412905\pi$
$421$	11.0888	0.540433	0.270217	+	0.962800i	$0.412905\pi$
$422$	25.3029	1.23173
$423$	0	0
$424$	14.5032	0.704338
$425$	0.935072	0.0453577
$426$	0	0
$427$	−1.68767	−0.0816723
$428$	−1.34583	−0.0650531
$429$	0	0
$430$	−7.13015	−0.343846
$431$	−19.2805	−0.928707	−0.464353	−	0.885650i	$-0.653713\pi$
$431$	−19.2805	−0.928707	−0.464353	+	0.885650i	$0.653713\pi$
$432$	0	0
$433$	−28.6999	−1.37923	−0.689614	−	0.724177i	$-0.742220\pi$
$433$	−28.6999	−1.37923	−0.689614	+	0.724177i	$0.742220\pi$
$434$	−0.747466	−0.0358795
$435$	0	0
$436$	−7.84662	−0.375785
$437$	−44.4013	−2.12400
$438$	0	0
$439$	−7.82309	−0.373376	−0.186688	−	0.982419i	$-0.559775\pi$
$439$	−7.82309	−0.373376	−0.186688	+	0.982419i	$0.559775\pi$
$440$	7.46806	0.356026
$441$	0	0
$442$	2.90407	0.138133
$443$	−16.0883	−0.764379	−0.382190	−	0.924084i	$-0.624830\pi$
$443$	−16.0883	−0.764379	−0.382190	+	0.924084i	$0.624830\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	25.0777	1.18747
$447$	0	0
$448$	3.21100	0.151705
$449$	6.75424	0.318752	0.159376	−	0.987218i	$-0.449052\pi$
$449$	6.75424	0.318752	0.159376	+	0.987218i	$0.449052\pi$
$450$	0	0
$451$	−3.69324	−0.173908
$452$	−2.85045	−0.134074
$453$	0	0
$454$	17.8192	0.836295
$455$	−0.923679	−0.0433027
$456$	0	0
$457$	−27.1862	−1.27172	−0.635858	−	0.771806i	$-0.719354\pi$
$457$	−27.1862	−1.27172	−0.635858	+	0.771806i	$0.719354\pi$
$458$	7.97721	0.372751
$459$	0	0
$460$	−3.40518	−0.158767
$461$	−21.9330	−1.02152	−0.510762	−	0.859722i	$-0.670636\pi$
$461$	−21.9330	−1.02152	−0.510762	+	0.859722i	$0.670636\pi$
$462$	0	0
$463$	−1.96201	−0.0911821	−0.0455911	−	0.998960i	$-0.514517\pi$
$463$	−1.96201	−0.0911821	−0.0455911	+	0.998960i	$0.514517\pi$
$464$	14.9664	0.694799
$465$	0	0
$466$	−27.1452	−1.25748
$467$	8.62115	0.398939	0.199470	−	0.979904i	$-0.436078\pi$
$467$	8.62115	0.398939	0.199470	+	0.979904i	$0.436078\pi$
$468$	0	0
$469$	−0.873239	−0.0403224
$470$	9.53985	0.440041
$471$	0	0
$472$	−3.39522	−0.156278
$473$	14.2603	0.655689
$474$	0	0
$475$	−6.72468	−0.308549
$476$	−0.174734	−0.00800890
$477$	0	0
$478$	1.94602	0.0890089
$479$	−14.5761	−0.666000	−0.333000	−	0.942927i	$-0.608061\pi$
$479$	−14.5761	−0.666000	−0.333000	+	0.942927i	$0.608061\pi$
$480$	0	0
$481$	21.3736	0.974554
$482$	−9.80031	−0.446392
$483$	0	0
$484$	2.61104	0.118684
$485$	5.70115	0.258876
$486$	0	0
$487$	17.3473	0.786083	0.393041	−	0.919521i	$-0.371423\pi$
$487$	17.3473	0.786083	0.393041	+	0.919521i	$0.371423\pi$
$488$	−14.2755	−0.646224
$489$	0	0
$490$	8.36822	0.378037
$491$	−35.1810	−1.58769	−0.793847	−	0.608117i	$-0.791925\pi$
$491$	−35.1810	−1.58769	−0.793847	+	0.608117i	$0.791925\pi$
$492$	0	0
$493$	−5.17825	−0.233217
$494$	−20.8850	−0.939660
$495$	0	0
$496$	−4.57612	−0.205474
$497$	2.22109	0.0996295
$498$	0	0
$499$	10.3413	0.462940	0.231470	−	0.972842i	$-0.425646\pi$
$499$	10.3413	0.462940	0.231470	+	0.972842i	$0.425646\pi$
$500$	−0.515722	−0.0230638
$501$	0	0
$502$	−2.50646	−0.111869
$503$	22.9479	1.02320	0.511598	−	0.859225i	$-0.329054\pi$
$503$	22.9479	1.02320	0.511598	+	0.859225i	$0.329054\pi$
$504$	0	0
$505$	1.35956	0.0604996
$506$	−19.6006	−0.871353
$507$	0	0
$508$	−9.95438	−0.441654
$509$	−27.8293	−1.23351	−0.616756	−	0.787155i	$-0.711553\pi$
$509$	−27.8293	−1.23351	−0.616756	+	0.787155i	$0.711553\pi$
$510$	0	0
$511$	−5.44936	−0.241065
$512$	24.2344	1.07102
$513$	0	0
$514$	22.2431	0.981103
$515$	−13.3368	−0.587688
$516$	0	0
$517$	−19.0797	−0.839125
$518$	3.70124	0.162623
$519$	0	0
$520$	−7.81313	−0.342629
$521$	22.1372	0.969847	0.484923	−	0.874557i	$-0.338847\pi$
$521$	22.1372	0.969847	0.484923	+	0.874557i	$0.338847\pi$
$522$	0	0
$523$	−11.3099	−0.494546	−0.247273	−	0.968946i	$-0.579534\pi$
$523$	−11.3099	−0.494546	−0.247273	+	0.968946i	$0.579534\pi$
$524$	−3.23324	−0.141245
$525$	0	0
$526$	−13.8265	−0.602863
$527$	1.58330	0.0689696
$528$	0	0
$529$	20.5962	0.895487
$530$	−5.76503	−0.250417
$531$	0	0
$532$	1.25662	0.0544813
$533$	3.86389	0.167364
$534$	0	0
$535$	2.60961	0.112823
$536$	−7.38647	−0.319047
$537$	0	0
$538$	1.20776	0.0520704
$539$	−16.7364	−0.720889
$540$	0	0
$541$	−32.0358	−1.37732	−0.688662	−	0.725082i	$-0.741802\pi$
$541$	−32.0358	−1.37732	−0.688662	+	0.725082i	$0.741802\pi$
$542$	−23.9508	−1.02878
$543$	0	0
$544$	−2.65305	−0.113749
$545$	15.2148	0.651732
$546$	0	0
$547$	−32.6613	−1.39649	−0.698247	−	0.715857i	$-0.746036\pi$
$547$	−32.6613	−1.39649	−0.698247	+	0.715857i	$0.746036\pi$
$548$	−0.422891	−0.0180650
$549$	0	0
$550$	−2.96856	−0.126580
$551$	37.2400	1.58648
$552$	0	0
$553$	0.380730	0.0161903
$554$	33.4465	1.42101
$555$	0	0
$556$	−8.22035	−0.348621
$557$	35.2225	1.49242	0.746212	−	0.665709i	$-0.231871\pi$
$557$	35.2225	1.49242	0.746212	+	0.665709i	$0.231871\pi$
$558$	0	0
$559$	−14.9192	−0.631016
$560$	−0.979255	−0.0413811
$561$	0	0
$562$	−3.49424	−0.147395
$563$	−14.4978	−0.611008	−0.305504	−	0.952191i	$-0.598825\pi$
$563$	−14.4978	−0.611008	−0.305504	+	0.952191i	$0.598825\pi$
$564$	0	0
$565$	5.52711	0.232527
$566$	35.3120	1.48427
$567$	0	0
$568$	18.7876	0.788309
$569$	12.0134	0.503629	0.251815	−	0.967775i	$-0.418973\pi$
$569$	12.0134	0.503629	0.251815	+	0.967775i	$0.418973\pi$
$570$	0	0
$571$	31.9454	1.33687	0.668437	−	0.743769i	$-0.266964\pi$
$571$	31.9454	1.33687	0.668437	+	0.743769i	$0.266964\pi$
$572$	3.20338	0.133940
$573$	0	0
$574$	0.669103	0.0279278
$575$	6.60274	0.275353
$576$	0	0
$577$	−28.8727	−1.20199	−0.600993	−	0.799254i	$-0.705228\pi$
$577$	−28.8727	−1.20199	−0.600993	+	0.799254i	$0.705228\pi$
$578$	19.6460	0.817167
$579$	0	0
$580$	2.85597	0.118588
$581$	−2.49681	−0.103585
$582$	0	0
$583$	11.5301	0.477526
$584$	−46.0946	−1.90741
$585$	0	0
$586$	−30.6817	−1.26745
$587$	14.9520	0.617133	0.308567	−	0.951203i	$-0.400151\pi$
$587$	14.9520	0.617133	0.308567	+	0.951203i	$0.400151\pi$
$588$	0	0
$589$	−11.3865	−0.469171
$590$	1.34960	0.0555621
$591$	0	0
$592$	22.6597	0.931306
$593$	0.0667205	0.00273988	0.00136994	−	0.999999i	$-0.499564\pi$
$593$	0.0667205	0.00273988	0.00136994	+	0.999999i	$0.499564\pi$
$594$	0	0
$595$	0.338814	0.0138900
$596$	2.11115	0.0864762
$597$	0	0
$598$	20.5063	0.838564
$599$	−16.5800	−0.677439	−0.338720	−	0.940887i	$-0.609994\pi$
$599$	−16.5800	−0.677439	−0.338720	+	0.940887i	$0.609994\pi$
$600$	0	0
$601$	−7.87006	−0.321026	−0.160513	−	0.987034i	$-0.551315\pi$
$601$	−7.87006	−0.321026	−0.160513	+	0.987034i	$0.551315\pi$
$602$	−2.58354	−0.105297
$603$	0	0
$604$	4.79662	0.195172
$605$	−5.06289	−0.205836
$606$	0	0
$607$	−6.90045	−0.280081	−0.140040	−	0.990146i	$-0.544723\pi$
$607$	−6.90045	−0.280081	−0.140040	+	0.990146i	$0.544723\pi$
$608$	19.0797	0.773784
$609$	0	0
$610$	5.67453	0.229755
$611$	19.9613	0.807548
$612$	0	0
$613$	13.6459	0.551154	0.275577	−	0.961279i	$-0.411131\pi$
$613$	13.6459	0.551154	0.275577	+	0.961279i	$0.411131\pi$
$614$	39.9045	1.61041
$615$	0	0
$616$	2.70598	0.109027
$617$	−7.74534	−0.311816	−0.155908	−	0.987772i	$-0.549830\pi$
$617$	−7.74534	−0.311816	−0.155908	+	0.987772i	$0.549830\pi$
$618$	0	0
$619$	39.0307	1.56877	0.784387	−	0.620271i	$-0.212977\pi$
$619$	39.0307	1.56877	0.784387	+	0.620271i	$0.212977\pi$
$620$	−0.873239	−0.0350701
$621$	0	0
$622$	11.8272	0.474227
$623$	−0.362340	−0.0145168
$624$	0	0
$625$	1.00000	0.0400000
$626$	4.18935	0.167440
$627$	0	0
$628$	0.869156	0.0346831
$629$	−7.84005	−0.312603
$630$	0	0
$631$	28.2179	1.12334	0.561669	−	0.827362i	$-0.310159\pi$
$631$	28.2179	1.12334	0.561669	+	0.827362i	$0.310159\pi$
$632$	3.22048	0.128104
$633$	0	0
$634$	−15.9314	−0.632718
$635$	19.3019	0.765971
$636$	0	0
$637$	17.5098	0.693762
$638$	16.4393	0.650839
$639$	0	0
$640$	−5.12194	−0.202462
$641$	23.6836	0.935446	0.467723	−	0.883875i	$-0.345075\pi$
$641$	23.6836	0.935446	0.467723	+	0.883875i	$0.345075\pi$
$642$	0	0
$643$	−25.2265	−0.994838	−0.497419	−	0.867511i	$-0.665719\pi$
$643$	−25.2265	−0.994838	−0.497419	+	0.867511i	$0.665719\pi$
$644$	−1.23383	−0.0486198
$645$	0	0
$646$	7.66080	0.301410
$647$	28.7092	1.12868	0.564338	−	0.825544i	$-0.309131\pi$
$647$	28.7092	1.12868	0.564338	+	0.825544i	$0.309131\pi$
$648$	0	0
$649$	−2.69920	−0.105953
$650$	3.10572	0.121816
$651$	0	0
$652$	−2.56165	−0.100322
$653$	6.22631	0.243655	0.121827	−	0.992551i	$-0.461125\pi$
$653$	6.22631	0.243655	0.121827	+	0.992551i	$0.461125\pi$
$654$	0	0
$655$	6.26936	0.244964
$656$	4.09637	0.159936
$657$	0	0
$658$	3.45667	0.134755
$659$	16.5001	0.642752	0.321376	−	0.946952i	$-0.395855\pi$
$659$	16.5001	0.642752	0.321376	+	0.946952i	$0.395855\pi$
$660$	0	0
$661$	30.8932	1.20160	0.600802	−	0.799398i	$-0.294848\pi$
$661$	30.8932	1.20160	0.600802	+	0.799398i	$0.294848\pi$
$662$	−2.00731	−0.0780163
$663$	0	0
$664$	−21.1198	−0.819608
$665$	−2.43662	−0.0944880
$666$	0	0
$667$	−36.5647	−1.41579
$668$	4.59965	0.177966
$669$	0	0
$670$	2.93612	0.113432
$671$	−11.3491	−0.438126
$672$	0	0
$673$	−9.48849	−0.365755	−0.182877	−	0.983136i	$-0.558541\pi$
$673$	−9.48849	−0.365755	−0.182877	+	0.983136i	$0.558541\pi$
$674$	24.3406	0.937563
$675$	0	0
$676$	3.35299	0.128961
$677$	2.41940	0.0929849	0.0464925	−	0.998919i	$-0.485196\pi$
$677$	2.41940	0.0929849	0.0464925	+	0.998919i	$0.485196\pi$
$678$	0	0
$679$	2.06576	0.0792764
$680$	2.86593	0.109903
$681$	0	0
$682$	−5.02647	−0.192473
$683$	−18.8762	−0.722277	−0.361139	−	0.932512i	$-0.617612\pi$
$683$	−18.8762	−0.722277	−0.361139	+	0.932512i	$0.617612\pi$
$684$	0	0
$685$	0.819998	0.0313305
$686$	6.12223	0.233748
$687$	0	0
$688$	−15.8169	−0.603013
$689$	−12.0628	−0.459557
$690$	0	0
$691$	−10.4930	−0.399173	−0.199587	−	0.979880i	$-0.563960\pi$
$691$	−10.4930	−0.399173	−0.199587	+	0.979880i	$0.563960\pi$
$692$	13.5302	0.514341
$693$	0	0
$694$	38.0946	1.44605
$695$	15.9395	0.604620
$696$	0	0
$697$	−1.41731	−0.0536844
$698$	11.0452	0.418066
$699$	0	0
$700$	−0.186866	−0.00706289
$701$	−11.5661	−0.436845	−0.218422	−	0.975854i	$-0.570091\pi$
$701$	−11.5661	−0.436845	−0.218422	+	0.975854i	$0.570091\pi$
$702$	0	0
$703$	56.3826	2.12651
$704$	21.5929	0.813815
$705$	0	0
$706$	−31.9516	−1.20251
$707$	0.492622	0.0185270
$708$	0	0
$709$	−22.9525	−0.862001	−0.431001	−	0.902352i	$-0.641839\pi$
$709$	−22.9525	−0.862001	−0.431001	+	0.902352i	$0.641839\pi$
$710$	−7.46806	−0.280271
$711$	0	0
$712$	−3.06493	−0.114863
$713$	11.1800	0.418694
$714$	0	0
$715$	−6.21145	−0.232295
$716$	3.40109	0.127105
$717$	0	0
$718$	−18.9269	−0.706346
$719$	33.3840	1.24501	0.622507	−	0.782614i	$-0.286114\pi$
$719$	33.3840	1.24501	0.622507	+	0.782614i	$0.286114\pi$
$720$	0	0
$721$	−4.83244	−0.179970
$722$	−31.9457	−1.18890
$723$	0	0
$724$	−5.62218	−0.208947
$725$	−5.53781	−0.205669
$726$	0	0
$727$	−8.60961	−0.319313	−0.159656	−	0.987173i	$-0.551039\pi$
$727$	−8.60961	−0.319313	−0.159656	+	0.987173i	$0.551039\pi$
$728$	−2.83101	−0.104924
$729$	0	0
$730$	18.3226	0.678150
$731$	5.47250	0.202408
$732$	0	0
$733$	−27.3660	−1.01079	−0.505394	−	0.862889i	$-0.668653\pi$
$733$	−27.3660	−1.01079	−0.505394	+	0.862889i	$0.668653\pi$
$734$	25.9275	0.957000
$735$	0	0
$736$	−18.7337	−0.690535
$737$	−5.87225	−0.216307
$738$	0	0
$739$	−20.1603	−0.741609	−0.370805	−	0.928711i	$-0.620918\pi$
$739$	−20.1603	−0.741609	−0.370805	+	0.928711i	$0.620918\pi$
$740$	4.32403	0.158955
$741$	0	0
$742$	−2.08890	−0.0766859
$743$	−39.7407	−1.45794	−0.728972	−	0.684544i	$-0.760001\pi$
$743$	−39.7407	−1.45794	−0.728972	+	0.684544i	$0.760001\pi$
$744$	0	0
$745$	−4.09359	−0.149978
$746$	22.5335	0.825008
$747$	0	0
$748$	−1.17503	−0.0429633
$749$	0.945564	0.0345502
$750$	0	0
$751$	−4.91081	−0.179198	−0.0895989	−	0.995978i	$-0.528559\pi$
$751$	−4.91081	−0.179198	−0.0895989	+	0.995978i	$0.528559\pi$
$752$	21.1624	0.771712
$753$	0	0
$754$	−17.1989	−0.626347
$755$	−9.30080	−0.338491
$756$	0	0
$757$	14.8908	0.541215	0.270608	−	0.962690i	$-0.412775\pi$
$757$	14.8908	0.541215	0.270608	+	0.962690i	$0.412775\pi$
$758$	−36.4642	−1.32444
$759$	0	0
$760$	−20.6107	−0.747627
$761$	−32.2154	−1.16781	−0.583903	−	0.811824i	$-0.698475\pi$
$761$	−32.2154	−1.16781	−0.583903	+	0.811824i	$0.698475\pi$
$762$	0	0
$763$	5.51294	0.199582
$764$	−0.464816	−0.0168165
$765$	0	0
$766$	8.87534	0.320679
$767$	2.82392	0.101966
$768$	0	0
$769$	31.0741	1.12056	0.560280	−	0.828304i	$-0.310694\pi$
$769$	31.0741	1.12056	0.560280	+	0.828304i	$0.310694\pi$
$770$	−1.07563	−0.0387629
$771$	0	0
$772$	−4.01902	−0.144648
$773$	−2.70573	−0.0973183	−0.0486591	−	0.998815i	$-0.515495\pi$
$773$	−2.70573	−0.0973183	−0.0486591	+	0.998815i	$0.515495\pi$
$774$	0	0
$775$	1.69324	0.0608229
$776$	17.4736	0.627267
$777$	0	0
$778$	4.55049	0.163143
$779$	10.1927	0.365193
$780$	0	0
$781$	14.9361	0.534457
$782$	−7.52189	−0.268982
$783$	0	0
$784$	18.5633	0.662975
$785$	−1.68532	−0.0601516
$786$	0	0
$787$	−42.5306	−1.51605	−0.758027	−	0.652223i	$-0.773836\pi$
$787$	−42.5306	−1.51605	−0.758027	+	0.652223i	$0.773836\pi$
$788$	10.0137	0.356724
$789$	0	0
$790$	−1.28014	−0.0455454
$791$	2.00269	0.0712076
$792$	0	0
$793$	11.8735	0.421639
$794$	−17.2056	−0.610605
$795$	0	0
$796$	11.6426	0.412663
$797$	−26.0828	−0.923902	−0.461951	−	0.886905i	$-0.652850\pi$
$797$	−26.0828	−0.923902	−0.461951	+	0.886905i	$0.652850\pi$
$798$	0	0
$799$	−7.32199	−0.259033
$800$	−2.83727	−0.100313
$801$	0	0
$802$	−42.2396	−1.49153
$803$	−36.6452	−1.29318
$804$	0	0
$805$	2.39244	0.0843223
$806$	5.25872	0.185231
$807$	0	0
$808$	4.16695	0.146593
$809$	49.9510	1.75618	0.878091	−	0.478493i	$-0.158817\pi$
$809$	49.9510	1.75618	0.878091	+	0.478493i	$0.158817\pi$
$810$	0	0
$811$	27.7651	0.974964	0.487482	−	0.873133i	$-0.337915\pi$
$811$	27.7651	0.974964	0.487482	+	0.873133i	$0.337915\pi$
$812$	1.03483	0.0363155
$813$	0	0
$814$	24.8897	0.872382
$815$	4.96712	0.173991
$816$	0	0
$817$	−39.3561	−1.37690
$818$	−12.1485	−0.424764
$819$	0	0
$820$	0.781690	0.0272978
$821$	20.4656	0.714254	0.357127	−	0.934056i	$-0.383756\pi$
$821$	20.4656	0.714254	0.357127	+	0.934056i	$0.383756\pi$
$822$	0	0
$823$	21.4601	0.748054	0.374027	−	0.927418i	$-0.377977\pi$
$823$	21.4601	0.748054	0.374027	+	0.927418i	$0.377977\pi$
$824$	−40.8762	−1.42399
$825$	0	0
$826$	0.489014	0.0170150
$827$	40.7525	1.41710	0.708552	−	0.705659i	$-0.249349\pi$
$827$	40.7525	1.41710	0.708552	+	0.705659i	$0.249349\pi$
$828$	0	0
$829$	39.2359	1.36272	0.681360	−	0.731948i	$-0.261389\pi$
$829$	39.2359	1.36272	0.681360	+	0.731948i	$0.261389\pi$
$830$	8.39513	0.291399
$831$	0	0
$832$	−22.5907	−0.783191
$833$	−6.42274	−0.222535
$834$	0	0
$835$	−8.91886	−0.308650
$836$	8.45035	0.292261
$837$	0	0
$838$	9.97791	0.344681
$839$	43.6965	1.50857	0.754286	−	0.656546i	$-0.227983\pi$
$839$	43.6965	1.50857	0.754286	+	0.656546i	$0.227983\pi$
$840$	0	0
$841$	1.66737	0.0574957
$842$	−13.5096	−0.465570
$843$	0	0
$844$	10.7110	0.368686
$845$	−6.50155	−0.223660
$846$	0	0
$847$	−1.83449	−0.0630337
$848$	−12.7886	−0.439163
$849$	0	0
$850$	−1.13921	−0.0390745
$851$	−55.3602	−1.89772
$852$	0	0
$853$	8.21569	0.281300	0.140650	−	0.990059i	$-0.455081\pi$
$853$	8.21569	0.281300	0.140650	+	0.990059i	$0.455081\pi$
$854$	2.05611	0.0703586
$855$	0	0
$856$	7.99825	0.273375
$857$	43.1418	1.47370	0.736848	−	0.676058i	$-0.236313\pi$
$857$	43.1418	1.47370	0.736848	+	0.676058i	$0.236313\pi$
$858$	0	0
$859$	21.7079	0.740663	0.370331	−	0.928900i	$-0.379244\pi$
$859$	21.7079	0.740663	0.370331	+	0.928900i	$0.379244\pi$
$860$	−3.01826	−0.102922
$861$	0	0
$862$	23.4896	0.800058
$863$	2.25563	0.0767825	0.0383912	−	0.999263i	$-0.487777\pi$
$863$	2.25563	0.0767825	0.0383912	+	0.999263i	$0.487777\pi$
$864$	0	0
$865$	−26.2355	−0.892033
$866$	34.9653	1.18817
$867$	0	0
$868$	−0.316409	−0.0107396
$869$	2.56029	0.0868518
$870$	0	0
$871$	6.14358	0.208167
$872$	46.6324	1.57917
$873$	0	0
$874$	54.0946	1.82978
$875$	0.362340	0.0122493
$876$	0	0
$877$	29.2525	0.987788	0.493894	−	0.869522i	$-0.335573\pi$
$877$	29.2525	0.987788	0.493894	+	0.869522i	$0.335573\pi$
$878$	9.53095	0.321654
$879$	0	0
$880$	−6.58518	−0.221986
$881$	−7.75056	−0.261123	−0.130562	−	0.991440i	$-0.541678\pi$
$881$	−7.75056	−0.261123	−0.130562	+	0.991440i	$0.541678\pi$
$882$	0	0
$883$	−32.2613	−1.08568	−0.542839	−	0.839837i	$-0.682651\pi$
$883$	−32.2613	−1.08568	−0.542839	+	0.839837i	$0.682651\pi$
$884$	1.22932	0.0413465
$885$	0	0
$886$	19.6006	0.658494
$887$	−30.8169	−1.03473	−0.517365	−	0.855765i	$-0.673087\pi$
$887$	−30.8169	−1.03473	−0.517365	+	0.855765i	$0.673087\pi$
$888$	0	0
$889$	6.99383	0.234566
$890$	1.21831	0.0408378
$891$	0	0
$892$	10.6156	0.355438
$893$	52.6569	1.76210
$894$	0	0
$895$	−6.59482	−0.220441
$896$	−1.85588	−0.0620007
$897$	0	0
$898$	−8.22876	−0.274597
$899$	−9.37683	−0.312735
$900$	0	0
$901$	4.42475	0.147410
$902$	4.49951	0.149817
$903$	0	0
$904$	16.9402	0.563423
$905$	10.9016	0.362381
$906$	0	0
$907$	−20.2842	−0.673527	−0.336763	−	0.941589i	$-0.609332\pi$
$907$	−20.2842	−0.673527	−0.336763	+	0.941589i	$0.609332\pi$
$908$	7.54301	0.250324
$909$	0	0
$910$	1.12533	0.0373042
$911$	19.5991	0.649348	0.324674	−	0.945826i	$-0.394745\pi$
$911$	19.5991	0.649348	0.324674	+	0.945826i	$0.394745\pi$
$912$	0	0
$913$	−16.7903	−0.555676
$914$	33.1212	1.09555
$915$	0	0
$916$	3.37683	0.111574
$917$	2.27164	0.0750161
$918$	0	0
$919$	−0.209966	−0.00692613	−0.00346307	−	0.999994i	$-0.501102\pi$
$919$	−0.209966	−0.00692613	−0.00346307	+	0.999994i	$0.501102\pi$
$920$	20.2369	0.667192
$921$	0	0
$922$	26.7212	0.880017
$923$	−15.6263	−0.514345
$924$	0	0
$925$	−8.38443	−0.275678
$926$	2.39033	0.0785511
$927$	0	0
$928$	15.7123	0.515780
$929$	−33.2587	−1.09118	−0.545592	−	0.838051i	$-0.683695\pi$
$929$	−33.2587	−1.09118	−0.545592	+	0.838051i	$0.683695\pi$
$930$	0	0
$931$	46.1899	1.51381
$932$	−11.4908	−0.376395
$933$	0	0
$934$	−10.5032	−0.343676
$935$	2.27841	0.0745121
$936$	0	0
$937$	31.5091	1.02936	0.514679	−	0.857383i	$-0.327911\pi$
$937$	31.5091	1.02936	0.514679	+	0.857383i	$0.327911\pi$
$938$	1.06388	0.0347368
$939$	0	0
$940$	4.03831	0.131715
$941$	28.6430	0.933735	0.466868	−	0.884327i	$-0.345382\pi$
$941$	28.6430	0.933735	0.466868	+	0.884327i	$0.345382\pi$
$942$	0	0
$943$	−10.0079	−0.325903
$944$	2.99383	0.0974409
$945$	0	0
$946$	−17.3735	−0.564860
$947$	−44.0053	−1.42998	−0.714991	−	0.699134i	$-0.753569\pi$
$947$	−44.0053	−1.42998	−0.714991	+	0.699134i	$0.753569\pi$
$948$	0	0
$949$	38.3384	1.24452
$950$	8.19274	0.265808
$951$	0	0
$952$	1.03844	0.0336560
$953$	28.5794	0.925776	0.462888	−	0.886417i	$-0.346813\pi$
$953$	28.5794	0.925776	0.462888	+	0.886417i	$0.346813\pi$
$954$	0	0
$955$	0.901293	0.0291652
$956$	0.823768	0.0266426
$957$	0	0
$958$	17.7582	0.573742
$959$	0.297118	0.00959444
$960$	0	0
$961$	−28.1330	−0.907515
$962$	−26.0397	−0.839554
$963$	0	0
$964$	−4.14856	−0.133616
$965$	7.79300	0.250865
$966$	0	0
$967$	−51.6378	−1.66056	−0.830280	−	0.557347i	$-0.811819\pi$
$967$	−51.6378	−1.66056	−0.830280	+	0.557347i	$0.811819\pi$
$968$	−15.5174	−0.498748
$969$	0	0
$970$	−6.94577	−0.223015
$971$	−12.6404	−0.405650	−0.202825	−	0.979215i	$-0.565012\pi$
$971$	−12.6404	−0.405650	−0.202825	+	0.979215i	$0.565012\pi$
$972$	0	0
$973$	5.77552	0.185155
$974$	−21.1344	−0.677191
$975$	0	0
$976$	12.5879	0.402928
$977$	−24.0390	−0.769075	−0.384538	−	0.923109i	$-0.625639\pi$
$977$	−24.0390	−0.769075	−0.384538	+	0.923109i	$0.625639\pi$
$978$	0	0
$979$	−2.43662	−0.0778747
$980$	3.54234	0.113156
$981$	0	0
$982$	42.8613	1.36776
$983$	−30.3043	−0.966558	−0.483279	−	0.875466i	$-0.660554\pi$
$983$	−30.3043	−0.966558	−0.483279	+	0.875466i	$0.660554\pi$
$984$	0	0
$985$	−19.4169	−0.618675
$986$	6.30872	0.200910
$987$	0	0
$988$	−8.84081	−0.281264
$989$	38.6425	1.22876
$990$	0	0
$991$	50.5022	1.60426	0.802128	−	0.597153i	$-0.203701\pi$
$991$	50.5022	1.60426	0.802128	+	0.597153i	$0.203701\pi$
$992$	−4.80416	−0.152532
$993$	0	0
$994$	−2.70598	−0.0858284
$995$	−22.5754	−0.715690
$996$	0	0
$997$	−41.3766	−1.31041	−0.655205	−	0.755451i	$-0.727418\pi$
$997$	−41.3766	−1.31041	−0.655205	+	0.755451i	$0.727418\pi$
$998$	−12.5989	−0.398811
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.m.1.2		4
3.2	odd	2		1335.2.a.f.1.3	✓	4
15.14	odd	2		6675.2.a.r.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.f.1.3	✓	4	3.2	odd	2
4005.2.a.m.1.2		4	1.1	even	1	trivial
6675.2.a.r.1.2		4	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)

\(f(q)\)	\(=\)	\(q-1.21831 q^{2} -0.515722 q^{4} +1.00000 q^{5} +0.362340 q^{7} +3.06493 q^{8} +O(q^{10})\)	\(q-1.21831 q^{2} -0.515722 q^{4} +1.00000 q^{5} +0.362340 q^{7} +3.06493 q^{8} -1.21831 q^{10} +2.43662 q^{11} -2.54921 q^{13} -0.441442 q^{14} -2.70259 q^{16} +0.935072 q^{17} -6.72468 q^{19} -0.515722 q^{20} -2.96856 q^{22} +6.60274 q^{23} +1.00000 q^{25} +3.10572 q^{26} -0.186866 q^{28} -5.53781 q^{29} +1.69324 q^{31} -2.83727 q^{32} -1.13921 q^{34} +0.362340 q^{35} -8.38443 q^{37} +8.19274 q^{38} +3.06493 q^{40} -1.51572 q^{41} +5.85249 q^{43} -1.25662 q^{44} -8.04418 q^{46} -7.83040 q^{47} -6.86871 q^{49} -1.21831 q^{50} +1.31468 q^{52} +4.73199 q^{53} +2.43662 q^{55} +1.11055 q^{56} +6.74677 q^{58} -1.10776 q^{59} -4.65771 q^{61} -2.06289 q^{62} +8.86185 q^{64} -2.54921 q^{65} -2.41000 q^{67} -0.482237 q^{68} -0.441442 q^{70} +6.12986 q^{71} -15.0394 q^{73} +10.2148 q^{74} +3.46806 q^{76} +0.882884 q^{77} +1.05075 q^{79} -2.70259 q^{80} +1.84662 q^{82} -6.89080 q^{83} +0.935072 q^{85} -7.13015 q^{86} +7.46806 q^{88} -1.00000 q^{89} -0.923679 q^{91} -3.40518 q^{92} +9.53985 q^{94} -6.72468 q^{95} +5.70115 q^{97} +8.36822 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8	\( 4 q + 2 q^{4} + 4 q^{5} - 7 q^{7} + 3 q^{8} - 5 q^{13} + q^{14} - 10 q^{16} + 13 q^{17} - 10 q^{19} + 2 q^{20} - 20 q^{22} - 3 q^{23} + 4 q^{25} + 3 q^{26} - 4 q^{28} - 2 q^{29} - 2 q^{31} + q^{32} + 6 q^{34} - 7 q^{35} - 9 q^{37} - 2 q^{38} + 3 q^{40} - 2 q^{41} - 19 q^{43} - 6 q^{44} - 5 q^{47} - 7 q^{49} - 17 q^{52} + 3 q^{53} + 2 q^{56} - 6 q^{58} - 2 q^{59} - 4 q^{61} + 8 q^{62} + q^{64} - 5 q^{65} - 15 q^{67} + q^{68} + q^{70} + 6 q^{71} - 21 q^{73} - 10 q^{74} - 4 q^{76} - 2 q^{77} - 20 q^{79} - 10 q^{80} + 3 q^{82} + 9 q^{83} + 13 q^{85} - 16 q^{86} + 12 q^{88} - 4 q^{89} + 2 q^{91} - 12 q^{92} + 25 q^{94} - 10 q^{95} - 17 q^{97} - 13 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 + 4 * q^5 - 7 * q^7 + 3 * q^8 - 5 * q^13 + q^14 - 10 * q^16 + 13 * q^17 - 10 * q^19 + 2 * q^20 - 20 * q^22 - 3 * q^23 + 4 * q^25 + 3 * q^26 - 4 * q^28 - 2 * q^29 - 2 * q^31 + q^32 + 6 * q^34 - 7 * q^35 - 9 * q^37 - 2 * q^38 + 3 * q^40 - 2 * q^41 - 19 * q^43 - 6 * q^44 - 5 * q^47 - 7 * q^49 - 17 * q^52 + 3 * q^53 + 2 * q^56 - 6 * q^58 - 2 * q^59 - 4 * q^61 + 8 * q^62 + q^64 - 5 * q^65 - 15 * q^67 + q^68 + q^70 + 6 * q^71 - 21 * q^73 - 10 * q^74 - 4 * q^76 - 2 * q^77 - 20 * q^79 - 10 * q^80 + 3 * q^82 + 9 * q^83 + 13 * q^85 - 16 * q^86 + 12 * q^88 - 4 * q^89 + 2 * q^91 - 12 * q^92 + 25 * q^94 - 10 * q^95 - 17 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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