LMFDB - Embedded newform 4005.2.a.r.1.10

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:	$0$
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} - 4x^{9} - 8x^{8} + 35x^{7} + 29x^{6} - 103x^{5} - 57x^{4} + 106x^{3} + 29x^{2} - 39x + 2 $ x^10 - 4x^9 - 8x^8 + 35x^7 + 29x^6 - 103x^5 - 57x^4 + 106x^3 + 29x^2 - 39*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			1.10
Root			$3.44917$ 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)

$f(q)$	$=$	$q+2.44917 q^{2} +3.99846 q^{4} -1.00000 q^{5} +4.85425 q^{7} +4.89457 q^{8} +O(q^{10})$	\(q+2.44917 q^{2} +3.99846 q^{4} -1.00000 q^{5} +4.85425 q^{7} +4.89457 q^{8} -2.44917 q^{10} +6.48840 q^{11} -3.08676 q^{13} +11.8889 q^{14} +3.99074 q^{16} -2.06316 q^{17} +3.08811 q^{19} -3.99846 q^{20} +15.8912 q^{22} +0.433530 q^{23} +1.00000 q^{25} -7.56002 q^{26} +19.4095 q^{28} -8.86624 q^{29} +10.6537 q^{31} -0.0151246 q^{32} -5.05304 q^{34} -4.85425 q^{35} -7.75757 q^{37} +7.56333 q^{38} -4.89457 q^{40} +2.82754 q^{41} -4.99828 q^{43} +25.9436 q^{44} +1.06179 q^{46} -5.56560 q^{47} +16.5638 q^{49} +2.44917 q^{50} -12.3423 q^{52} -9.32640 q^{53} -6.48840 q^{55} +23.7595 q^{56} -21.7150 q^{58} +4.40806 q^{59} +9.44678 q^{61} +26.0929 q^{62} -8.01851 q^{64} +3.08676 q^{65} -0.197229 q^{67} -8.24945 q^{68} -11.8889 q^{70} -12.8267 q^{71} -1.68387 q^{73} -18.9996 q^{74} +12.3477 q^{76} +31.4963 q^{77} +5.54201 q^{79} -3.99074 q^{80} +6.92513 q^{82} -2.81993 q^{83} +2.06316 q^{85} -12.2417 q^{86} +31.7579 q^{88} -1.00000 q^{89} -14.9839 q^{91} +1.73345 q^{92} -13.6311 q^{94} -3.08811 q^{95} -6.95572 q^{97} +40.5675 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	$ 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) $ 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * 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 $

$2$	2.44917	1.73183	0.865914	−	0.500193i	$-0.166738\pi$
$2$	2.44917	1.73183	0.865914	+	0.500193i	$0.166738\pi$
$3$	0	0
$3$	0	0
$4$	3.99846	1.99923
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.85425	1.83473	0.917367	−	0.398042i	$-0.130310\pi$
$7$	4.85425	1.83473	0.917367	+	0.398042i	$0.130310\pi$
$8$	4.89457	1.73049
$9$	0	0
$10$	−2.44917	−0.774497
$11$	6.48840	1.95633	0.978164	−	0.207836i	$-0.0666420\pi$
$11$	6.48840	1.95633	0.978164	+	0.207836i	$0.0666420\pi$
$12$	0	0
$13$	−3.08676	−0.856114	−0.428057	−	0.903752i	$-0.640802\pi$
$13$	−3.08676	−0.856114	−0.428057	+	0.903752i	$0.640802\pi$
$14$	11.8889	3.17744
$15$	0	0
$16$	3.99074	0.997684
$17$	−2.06316	−0.500390	−0.250195	−	0.968196i	$-0.580495\pi$
$17$	−2.06316	−0.500390	−0.250195	+	0.968196i	$0.580495\pi$
$18$	0	0
$19$	3.08811	0.708462	0.354231	−	0.935158i	$-0.384743\pi$
$19$	3.08811	0.708462	0.354231	+	0.935158i	$0.384743\pi$
$20$	−3.99846	−0.894082
$21$	0	0
$22$	15.8912	3.38802
$23$	0.433530	0.0903972	0.0451986	−	0.998978i	$-0.485608\pi$
$23$	0.433530	0.0903972	0.0451986	+	0.998978i	$0.485608\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.56002	−1.48264
$27$	0	0
$28$	19.4095	3.66805
$29$	−8.86624	−1.64642	−0.823210	−	0.567737i	$-0.807819\pi$
$29$	−8.86624	−1.64642	−0.823210	+	0.567737i	$0.807819\pi$
$30$	0	0
$31$	10.6537	1.91347	0.956735	−	0.290962i	$-0.0939752\pi$
$31$	10.6537	1.91347	0.956735	+	0.290962i	$0.0939752\pi$
$32$	−0.0151246	−0.00267368
$33$	0	0
$34$	−5.05304	−0.866589
$35$	−4.85425	−0.820518
$36$	0	0
$37$	−7.75757	−1.27534	−0.637668	−	0.770311i	$-0.720101\pi$
$37$	−7.75757	−1.27534	−0.637668	+	0.770311i	$0.720101\pi$
$38$	7.56333	1.22693
$39$	0	0
$40$	−4.89457	−0.773899
$41$	2.82754	0.441587	0.220794	−	0.975321i	$-0.429135\pi$
$41$	2.82754	0.441587	0.220794	+	0.975321i	$0.429135\pi$
$42$	0	0
$43$	−4.99828	−0.762231	−0.381115	−	0.924528i	$-0.624460\pi$
$43$	−4.99828	−0.762231	−0.381115	+	0.924528i	$0.624460\pi$
$44$	25.9436	3.91114
$45$	0	0
$46$	1.06179	0.156552
$47$	−5.56560	−0.811826	−0.405913	−	0.913912i	$-0.633046\pi$
$47$	−5.56560	−0.811826	−0.405913	+	0.913912i	$0.633046\pi$
$48$	0	0
$49$	16.5638	2.36625
$50$	2.44917	0.346366
$51$	0	0
$52$	−12.3423	−1.71157
$53$	−9.32640	−1.28108	−0.640540	−	0.767925i	$-0.721289\pi$
$53$	−9.32640	−1.28108	−0.640540	+	0.767925i	$0.721289\pi$
$54$	0	0
$55$	−6.48840	−0.874896
$56$	23.7595	3.17499
$57$	0	0
$58$	−21.7150	−2.85132
$59$	4.40806	0.573881	0.286940	−	0.957948i	$-0.407362\pi$
$59$	4.40806	0.573881	0.286940	+	0.957948i	$0.407362\pi$
$60$	0	0
$61$	9.44678	1.20954	0.604768	−	0.796402i	$-0.293266\pi$
$61$	9.44678	1.20954	0.604768	+	0.796402i	$0.293266\pi$
$62$	26.0929	3.31380
$63$	0	0
$64$	−8.01851	−1.00231
$65$	3.08676	0.382866
$66$	0	0
$67$	−0.197229	−0.0240954	−0.0120477	−	0.999927i	$-0.503835\pi$
$67$	−0.197229	−0.0240954	−0.0120477	+	0.999927i	$0.503835\pi$
$68$	−8.24945	−1.00039
$69$	0	0
$70$	−11.8889	−1.42100
$71$	−12.8267	−1.52225	−0.761125	−	0.648605i	$-0.775353\pi$
$71$	−12.8267	−1.52225	−0.761125	+	0.648605i	$0.775353\pi$
$72$	0	0
$73$	−1.68387	−0.197082	−0.0985411	−	0.995133i	$-0.531418\pi$
$73$	−1.68387	−0.197082	−0.0985411	+	0.995133i	$0.531418\pi$
$74$	−18.9996	−2.20866
$75$	0	0
$76$	12.3477	1.41638
$77$	31.4963	3.58934
$78$	0	0
$79$	5.54201	0.623525	0.311763	−	0.950160i	$-0.399081\pi$
$79$	5.54201	0.623525	0.311763	+	0.950160i	$0.399081\pi$
$80$	−3.99074	−0.446178
$81$	0	0
$82$	6.92513	0.764753
$83$	−2.81993	−0.309527	−0.154764	−	0.987952i	$-0.549462\pi$
$83$	−2.81993	−0.309527	−0.154764	+	0.987952i	$0.549462\pi$
$84$	0	0
$85$	2.06316	0.223781
$86$	−12.2417	−1.32005
$87$	0	0
$88$	31.7579	3.38541
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−14.9839	−1.57074
$92$	1.73345	0.180725
$93$	0	0
$94$	−13.6311	−1.40594
$95$	−3.08811	−0.316834
$96$	0	0
$97$	−6.95572	−0.706246	−0.353123	−	0.935577i	$-0.614880\pi$
$97$	−6.95572	−0.706246	−0.353123	+	0.935577i	$0.614880\pi$
$98$	40.5675	4.09794
$99$	0	0
$100$	3.99846	0.399846
$101$	11.6750	1.16171	0.580855	−	0.814007i	$-0.302718\pi$
$101$	11.6750	1.16171	0.580855	+	0.814007i	$0.302718\pi$
$102$	0	0
$103$	−8.46284	−0.833868	−0.416934	−	0.908937i	$-0.636895\pi$
$103$	−8.46284	−0.833868	−0.416934	+	0.908937i	$0.636895\pi$
$104$	−15.1084	−1.48150
$105$	0	0
$106$	−22.8420	−2.21861
$107$	10.1189	0.978234	0.489117	−	0.872218i	$-0.337319\pi$
$107$	10.1189	0.978234	0.489117	+	0.872218i	$0.337319\pi$
$108$	0	0
$109$	11.9862	1.14807	0.574034	−	0.818832i	$-0.305378\pi$
$109$	11.9862	1.14807	0.574034	+	0.818832i	$0.305378\pi$
$110$	−15.8912	−1.51517
$111$	0	0
$112$	19.3720	1.83049
$113$	−2.42958	−0.228556	−0.114278	−	0.993449i	$-0.536455\pi$
$113$	−2.42958	−0.228556	−0.114278	+	0.993449i	$0.536455\pi$
$114$	0	0
$115$	−0.433530	−0.0404269
$116$	−35.4513	−3.29157
$117$	0	0
$118$	10.7961	0.993863
$119$	−10.0151	−0.918082
$120$	0	0
$121$	31.0994	2.82722
$122$	23.1368	2.09471
$123$	0	0
$124$	42.5985	3.82546
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.08900	−0.185369	−0.0926843	−	0.995696i	$-0.529545\pi$
$127$	−2.08900	−0.185369	−0.0926843	+	0.995696i	$0.529545\pi$
$128$	−19.6085	−1.73316
$129$	0	0
$130$	7.56002	0.663058
$131$	−0.664163	−0.0580282	−0.0290141	−	0.999579i	$-0.509237\pi$
$131$	−0.664163	−0.0580282	−0.0290141	+	0.999579i	$0.509237\pi$
$132$	0	0
$133$	14.9905	1.29984
$134$	−0.483049	−0.0417291
$135$	0	0
$136$	−10.0983	−0.865919
$137$	11.0114	0.940764	0.470382	−	0.882463i	$-0.344116\pi$
$137$	11.0114	0.940764	0.470382	+	0.882463i	$0.344116\pi$
$138$	0	0
$139$	9.99158	0.847475	0.423737	−	0.905785i	$-0.360718\pi$
$139$	9.99158	0.847475	0.423737	+	0.905785i	$0.360718\pi$
$140$	−19.4095	−1.64040
$141$	0	0
$142$	−31.4148	−2.63627
$143$	−20.0282	−1.67484
$144$	0	0
$145$	8.86624	0.736301
$146$	−4.12409	−0.341312
$147$	0	0
$148$	−31.0183	−2.54969
$149$	0.841792	0.0689623	0.0344811	−	0.999405i	$-0.489022\pi$
$149$	0.841792	0.0689623	0.0344811	+	0.999405i	$0.489022\pi$
$150$	0	0
$151$	−20.8145	−1.69386	−0.846931	−	0.531703i	$-0.821552\pi$
$151$	−20.8145	−1.69386	−0.846931	+	0.531703i	$0.821552\pi$
$152$	15.1150	1.22599
$153$	0	0
$154$	77.1400	6.21612
$155$	−10.6537	−0.855730
$156$	0	0
$157$	7.39423	0.590124	0.295062	−	0.955478i	$-0.404660\pi$
$157$	7.39423	0.590124	0.295062	+	0.955478i	$0.404660\pi$
$158$	13.5734	1.07984
$159$	0	0
$160$	0.0151246	0.00119571
$161$	2.10446	0.165855
$162$	0	0
$163$	0.849449	0.0665340	0.0332670	−	0.999447i	$-0.489409\pi$
$163$	0.849449	0.0665340	0.0332670	+	0.999447i	$0.489409\pi$
$164$	11.3058	0.882833
$165$	0	0
$166$	−6.90649	−0.536048
$167$	−24.8627	−1.92393	−0.961966	−	0.273168i	$-0.911928\pi$
$167$	−24.8627	−1.92393	−0.961966	+	0.273168i	$0.911928\pi$
$168$	0	0
$169$	−3.47189	−0.267068
$170$	5.05304	0.387550
$171$	0	0
$172$	−19.9854	−1.52387
$173$	−6.50967	−0.494921	−0.247460	−	0.968898i	$-0.579596\pi$
$173$	−6.50967	−0.494921	−0.247460	+	0.968898i	$0.579596\pi$
$174$	0	0
$175$	4.85425	0.366947
$176$	25.8935	1.95180
$177$	0	0
$178$	−2.44917	−0.183573
$179$	3.74231	0.279714	0.139857	−	0.990172i	$-0.455336\pi$
$179$	3.74231	0.279714	0.139857	+	0.990172i	$0.455336\pi$
$180$	0	0
$181$	4.12824	0.306850	0.153425	−	0.988160i	$-0.450970\pi$
$181$	4.12824	0.306850	0.153425	+	0.988160i	$0.450970\pi$
$182$	−36.6983	−2.72026
$183$	0	0
$184$	2.12194	0.156432
$185$	7.75757	0.570348
$186$	0	0
$187$	−13.3866	−0.978926
$188$	−22.2538	−1.62302
$189$	0	0
$190$	−7.56333	−0.548701
$191$	10.7171	0.775465	0.387732	−	0.921772i	$-0.373258\pi$
$191$	10.7171	0.775465	0.387732	+	0.921772i	$0.373258\pi$
$192$	0	0
$193$	−21.5806	−1.55341	−0.776703	−	0.629867i	$-0.783109\pi$
$193$	−21.5806	−1.55341	−0.776703	+	0.629867i	$0.783109\pi$
$194$	−17.0358	−1.22310
$195$	0	0
$196$	66.2295	4.73068
$197$	5.91035	0.421095	0.210548	−	0.977584i	$-0.432475\pi$
$197$	5.91035	0.421095	0.210548	+	0.977584i	$0.432475\pi$
$198$	0	0
$199$	−0.998960	−0.0708144	−0.0354072	−	0.999373i	$-0.511273\pi$
$199$	−0.998960	−0.0708144	−0.0354072	+	0.999373i	$0.511273\pi$
$200$	4.89457	0.346098
$201$	0	0
$202$	28.5942	2.01188
$203$	−43.0390	−3.02074
$204$	0	0
$205$	−2.82754	−0.197484
$206$	−20.7270	−1.44412
$207$	0	0
$208$	−12.3185	−0.854131
$209$	20.0369	1.38598
$210$	0	0
$211$	3.33746	0.229760	0.114880	−	0.993379i	$-0.463352\pi$
$211$	3.33746	0.229760	0.114880	+	0.993379i	$0.463352\pi$
$212$	−37.2912	−2.56117
$213$	0	0
$214$	24.7830	1.69413
$215$	4.99828	0.340880
$216$	0	0
$217$	51.7160	3.51071
$218$	29.3562	1.98825
$219$	0	0
$220$	−25.9436	−1.74912
$221$	6.36849	0.428391
$222$	0	0
$223$	−7.20938	−0.482776	−0.241388	−	0.970429i	$-0.577603\pi$
$223$	−7.20938	−0.482776	−0.241388	+	0.970429i	$0.577603\pi$
$224$	−0.0734187	−0.00490549
$225$	0	0
$226$	−5.95046	−0.395819
$227$	25.4790	1.69110	0.845551	−	0.533894i	$-0.179272\pi$
$227$	25.4790	1.69110	0.845551	+	0.533894i	$0.179272\pi$
$228$	0	0
$229$	−4.00264	−0.264502	−0.132251	−	0.991216i	$-0.542220\pi$
$229$	−4.00264	−0.264502	−0.132251	+	0.991216i	$0.542220\pi$
$230$	−1.06179	−0.0700124
$231$	0	0
$232$	−43.3964	−2.84911
$233$	−10.5343	−0.690125	−0.345063	−	0.938580i	$-0.612142\pi$
$233$	−10.5343	−0.690125	−0.345063	+	0.938580i	$0.612142\pi$
$234$	0	0
$235$	5.56560	0.363060
$236$	17.6254	1.14732
$237$	0	0
$238$	−24.5287	−1.58996
$239$	−12.3436	−0.798443	−0.399222	−	0.916854i	$-0.630720\pi$
$239$	−12.3436	−0.798443	−0.399222	+	0.916854i	$0.630720\pi$
$240$	0	0
$241$	−17.8406	−1.14922	−0.574608	−	0.818429i	$-0.694845\pi$
$241$	−17.8406	−1.14922	−0.574608	+	0.818429i	$0.694845\pi$
$242$	76.1678	4.89625
$243$	0	0
$244$	37.7725	2.41814
$245$	−16.5638	−1.05822
$246$	0	0
$247$	−9.53228	−0.606524
$248$	52.1455	3.31124
$249$	0	0
$250$	−2.44917	−0.154899
$251$	26.5414	1.67528	0.837641	−	0.546222i	$-0.183934\pi$
$251$	26.5414	1.67528	0.837641	+	0.546222i	$0.183934\pi$
$252$	0	0
$253$	2.81292	0.176847
$254$	−5.11632	−0.321026
$255$	0	0
$256$	−31.9876	−1.99922
$257$	15.8752	0.990270	0.495135	−	0.868816i	$-0.335119\pi$
$257$	15.8752	0.990270	0.495135	+	0.868816i	$0.335119\pi$
$258$	0	0
$259$	−37.6572	−2.33990
$260$	12.3423	0.765436
$261$	0	0
$262$	−1.62665	−0.100495
$263$	−25.6748	−1.58317	−0.791587	−	0.611057i	$-0.790745\pi$
$263$	−25.6748	−1.58317	−0.791587	+	0.611057i	$0.790745\pi$
$264$	0	0
$265$	9.32640	0.572916
$266$	36.7143	2.25110
$267$	0	0
$268$	−0.788613	−0.0481722
$269$	−27.3513	−1.66764	−0.833818	−	0.552040i	$-0.813850\pi$
$269$	−27.3513	−1.66764	−0.833818	+	0.552040i	$0.813850\pi$
$270$	0	0
$271$	17.6246	1.07062	0.535309	−	0.844657i	$-0.320195\pi$
$271$	17.6246	1.07062	0.535309	+	0.844657i	$0.320195\pi$
$272$	−8.23352	−0.499231
$273$	0	0
$274$	26.9687	1.62924
$275$	6.48840	0.391265
$276$	0	0
$277$	−2.61766	−0.157280	−0.0786399	−	0.996903i	$-0.525058\pi$
$277$	−2.61766	−0.157280	−0.0786399	+	0.996903i	$0.525058\pi$
$278$	24.4711	1.46768
$279$	0	0
$280$	−23.7595	−1.41990
$281$	−13.5569	−0.808736	−0.404368	−	0.914596i	$-0.632508\pi$
$281$	−13.5569	−0.808736	−0.404368	+	0.914596i	$0.632508\pi$
$282$	0	0
$283$	15.7479	0.936114	0.468057	−	0.883698i	$-0.344954\pi$
$283$	15.7479	0.936114	0.468057	+	0.883698i	$0.344954\pi$
$284$	−51.2870	−3.04332
$285$	0	0
$286$	−49.0525	−2.90053
$287$	13.7256	0.810195
$288$	0	0
$289$	−12.7434	−0.749610
$290$	21.7150	1.27515
$291$	0	0
$292$	−6.73288	−0.394012
$293$	−3.82011	−0.223173	−0.111587	−	0.993755i	$-0.535593\pi$
$293$	−3.82011	−0.223173	−0.111587	+	0.993755i	$0.535593\pi$
$294$	0	0
$295$	−4.40806	−0.256647
$296$	−37.9699	−2.20696
$297$	0	0
$298$	2.06169	0.119431
$299$	−1.33820	−0.0773904
$300$	0	0
$301$	−24.2629	−1.39849
$302$	−50.9784	−2.93348
$303$	0	0
$304$	12.3238	0.706821
$305$	−9.44678	−0.540921
$306$	0	0
$307$	−26.2265	−1.49683	−0.748414	−	0.663232i	$-0.769184\pi$
$307$	−26.2265	−1.49683	−0.748414	+	0.663232i	$0.769184\pi$
$308$	125.937	7.17591
$309$	0	0
$310$	−26.0929	−1.48198
$311$	−14.7192	−0.834648	−0.417324	−	0.908758i	$-0.637032\pi$
$311$	−14.7192	−0.834648	−0.417324	+	0.908758i	$0.637032\pi$
$312$	0	0
$313$	−1.71674	−0.0970358	−0.0485179	−	0.998822i	$-0.515450\pi$
$313$	−1.71674	−0.0970358	−0.0485179	+	0.998822i	$0.515450\pi$
$314$	18.1098	1.02199
$315$	0	0
$316$	22.1595	1.24657
$317$	23.0960	1.29720	0.648601	−	0.761128i	$-0.275354\pi$
$317$	23.0960	1.29720	0.648601	+	0.761128i	$0.275354\pi$
$318$	0	0
$319$	−57.5277	−3.22094
$320$	8.01851	0.448249
$321$	0	0
$322$	5.15420	0.287232
$323$	−6.37127	−0.354507
$324$	0	0
$325$	−3.08676	−0.171223
$326$	2.08045	0.115225
$327$	0	0
$328$	13.8396	0.764163
$329$	−27.0168	−1.48949
$330$	0	0
$331$	8.54169	0.469493	0.234747	−	0.972057i	$-0.424574\pi$
$331$	8.54169	0.469493	0.234747	+	0.972057i	$0.424574\pi$
$332$	−11.2753	−0.618815
$333$	0	0
$334$	−60.8931	−3.33192
$335$	0.197229	0.0107758
$336$	0	0
$337$	6.43690	0.350640	0.175320	−	0.984511i	$-0.443904\pi$
$337$	6.43690	0.350640	0.175320	+	0.984511i	$0.443904\pi$
$338$	−8.50326	−0.462516
$339$	0	0
$340$	8.24945	0.447389
$341$	69.1258	3.74337
$342$	0	0
$343$	46.4249	2.50671
$344$	−24.4644	−1.31903
$345$	0	0
$346$	−15.9433	−0.857118
$347$	12.7180	0.682737	0.341368	−	0.939930i	$-0.389110\pi$
$347$	12.7180	0.682737	0.341368	+	0.939930i	$0.389110\pi$
$348$	0	0
$349$	13.4702	0.721042	0.360521	−	0.932751i	$-0.382599\pi$
$349$	13.4702	0.721042	0.360521	+	0.932751i	$0.382599\pi$
$350$	11.8889	0.635489
$351$	0	0
$352$	−0.0981346	−0.00523059
$353$	13.4543	0.716101	0.358051	−	0.933702i	$-0.383442\pi$
$353$	13.4543	0.716101	0.358051	+	0.933702i	$0.383442\pi$
$354$	0	0
$355$	12.8267	0.680771
$356$	−3.99846	−0.211918
$357$	0	0
$358$	9.16557	0.484416
$359$	32.2297	1.70102	0.850510	−	0.525959i	$-0.176293\pi$
$359$	32.2297	1.70102	0.850510	+	0.525959i	$0.176293\pi$
$360$	0	0
$361$	−9.46356	−0.498082
$362$	10.1108	0.531411
$363$	0	0
$364$	−59.9126	−3.14027
$365$	1.68387	0.0881378
$366$	0	0
$367$	−1.63233	−0.0852071	−0.0426035	−	0.999092i	$-0.513565\pi$
$367$	−1.63233	−0.0852071	−0.0426035	+	0.999092i	$0.513565\pi$
$368$	1.73010	0.0901879
$369$	0	0
$370$	18.9996	0.987744
$371$	−45.2727	−2.35044
$372$	0	0
$373$	−9.58384	−0.496233	−0.248116	−	0.968730i	$-0.579812\pi$
$373$	−9.58384	−0.496233	−0.248116	+	0.968730i	$0.579812\pi$
$374$	−32.7861	−1.69533
$375$	0	0
$376$	−27.2412	−1.40486
$377$	27.3680	1.40952
$378$	0	0
$379$	−33.8221	−1.73732	−0.868662	−	0.495405i	$-0.835020\pi$
$379$	−33.8221	−1.73732	−0.868662	+	0.495405i	$0.835020\pi$
$380$	−12.3477	−0.633423
$381$	0	0
$382$	26.2481	1.34297
$383$	−12.6271	−0.645217	−0.322608	−	0.946533i	$-0.604560\pi$
$383$	−12.6271	−0.645217	−0.322608	+	0.946533i	$0.604560\pi$
$384$	0	0
$385$	−31.4963	−1.60520
$386$	−52.8546	−2.69023
$387$	0	0
$388$	−27.8121	−1.41195
$389$	27.9514	1.41719	0.708596	−	0.705614i	$-0.249329\pi$
$389$	27.9514	1.41719	0.708596	+	0.705614i	$0.249329\pi$
$390$	0	0
$391$	−0.894441	−0.0452338
$392$	81.0724	4.09478
$393$	0	0
$394$	14.4755	0.729264
$395$	−5.54201	−0.278849
$396$	0	0
$397$	12.0898	0.606770	0.303385	−	0.952868i	$-0.401883\pi$
$397$	12.0898	0.606770	0.303385	+	0.952868i	$0.401883\pi$
$398$	−2.44663	−0.122638
$399$	0	0
$400$	3.99074	0.199537
$401$	−37.3038	−1.86286	−0.931432	−	0.363915i	$-0.881440\pi$
$401$	−37.3038	−1.86286	−0.931432	+	0.363915i	$0.881440\pi$
$402$	0	0
$403$	−32.8856	−1.63815
$404$	46.6822	2.32252
$405$	0	0
$406$	−105.410	−5.23141
$407$	−50.3342	−2.49498
$408$	0	0
$409$	22.0365	1.08963	0.544816	−	0.838555i	$-0.316599\pi$
$409$	22.0365	1.08963	0.544816	+	0.838555i	$0.316599\pi$
$410$	−6.92513	−0.342008
$411$	0	0
$412$	−33.8383	−1.66709
$413$	21.3978	1.05292
$414$	0	0
$415$	2.81993	0.138425
$416$	0.0466861	0.00228897
$417$	0	0
$418$	49.0739	2.40028
$419$	−1.64093	−0.0801646	−0.0400823	−	0.999196i	$-0.512762\pi$
$419$	−1.64093	−0.0801646	−0.0400823	+	0.999196i	$0.512762\pi$
$420$	0	0
$421$	−11.1537	−0.543597	−0.271798	−	0.962354i	$-0.587618\pi$
$421$	−11.1537	−0.543597	−0.271798	+	0.962354i	$0.587618\pi$
$422$	8.17402	0.397905
$423$	0	0
$424$	−45.6487	−2.21690
$425$	−2.06316	−0.100078
$426$	0	0
$427$	45.8571	2.21918
$428$	40.4601	1.95571
$429$	0	0
$430$	12.2417	0.590345
$431$	−26.0338	−1.25400	−0.627001	−	0.779019i	$-0.715718\pi$
$431$	−26.0338	−1.25400	−0.627001	+	0.779019i	$0.715718\pi$
$432$	0	0
$433$	−20.9949	−1.00895	−0.504475	−	0.863426i	$-0.668314\pi$
$433$	−20.9949	−1.00895	−0.504475	+	0.863426i	$0.668314\pi$
$434$	126.661	6.07994
$435$	0	0
$436$	47.9262	2.29525
$437$	1.33879	0.0640430
$438$	0	0
$439$	12.5778	0.600308	0.300154	−	0.953891i	$-0.402962\pi$
$439$	12.5778	0.600308	0.300154	+	0.953891i	$0.402962\pi$
$440$	−31.7579	−1.51400
$441$	0	0
$442$	15.5975	0.741899
$443$	33.3236	1.58325	0.791625	−	0.611007i	$-0.209235\pi$
$443$	33.3236	1.58325	0.791625	+	0.611007i	$0.209235\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−17.6570	−0.836085
$447$	0	0
$448$	−38.9239	−1.83898
$449$	−3.48062	−0.164261	−0.0821304	−	0.996622i	$-0.526172\pi$
$449$	−3.48062	−0.164261	−0.0821304	+	0.996622i	$0.526172\pi$
$450$	0	0
$451$	18.3462	0.863889
$452$	−9.71456	−0.456935
$453$	0	0
$454$	62.4026	2.92870
$455$	14.9839	0.702457
$456$	0	0
$457$	−30.4818	−1.42588	−0.712939	−	0.701226i	$-0.752636\pi$
$457$	−30.4818	−1.42588	−0.712939	+	0.701226i	$0.752636\pi$
$458$	−9.80316	−0.458072
$459$	0	0
$460$	−1.73345	−0.0808225
$461$	−23.8790	−1.11216	−0.556078	−	0.831130i	$-0.687695\pi$
$461$	−23.8790	−1.11216	−0.556078	+	0.831130i	$0.687695\pi$
$462$	0	0
$463$	−16.6175	−0.772281	−0.386140	−	0.922440i	$-0.626192\pi$
$463$	−16.6175	−0.772281	−0.386140	+	0.922440i	$0.626192\pi$
$464$	−35.3828	−1.64261
$465$	0	0
$466$	−25.8004	−1.19518
$467$	39.0591	1.80744	0.903719	−	0.428127i	$-0.140826\pi$
$467$	39.0591	1.80744	0.903719	+	0.428127i	$0.140826\pi$
$468$	0	0
$469$	−0.957402	−0.0442087
$470$	13.6311	0.628757
$471$	0	0
$472$	21.5756	0.993095
$473$	−32.4309	−1.49117
$474$	0	0
$475$	3.08811	0.141692
$476$	−40.0449	−1.83546
$477$	0	0
$478$	−30.2317	−1.38277
$479$	−11.6787	−0.533615	−0.266807	−	0.963750i	$-0.585969\pi$
$479$	−11.6787	−0.533615	−0.266807	+	0.963750i	$0.585969\pi$
$480$	0	0
$481$	23.9458	1.09183
$482$	−43.6948	−1.99024
$483$	0	0
$484$	124.350	5.65225
$485$	6.95572	0.315843
$486$	0	0
$487$	4.78947	0.217031	0.108516	−	0.994095i	$-0.465390\pi$
$487$	4.78947	0.217031	0.108516	+	0.994095i	$0.465390\pi$
$488$	46.2379	2.09309
$489$	0	0
$490$	−40.5675	−1.83265
$491$	2.55327	0.115228	0.0576138	−	0.998339i	$-0.481651\pi$
$491$	2.55327	0.115228	0.0576138	+	0.998339i	$0.481651\pi$
$492$	0	0
$493$	18.2925	0.823851
$494$	−23.3462	−1.05040
$495$	0	0
$496$	42.5163	1.90904
$497$	−62.2641	−2.79292
$498$	0	0
$499$	−36.7400	−1.64471	−0.822354	−	0.568977i	$-0.807339\pi$
$499$	−36.7400	−1.64471	−0.822354	+	0.568977i	$0.807339\pi$
$500$	−3.99846	−0.178816
$501$	0	0
$502$	65.0046	2.90130
$503$	−41.7866	−1.86317	−0.931587	−	0.363518i	$-0.881575\pi$
$503$	−41.7866	−1.86317	−0.931587	+	0.363518i	$0.881575\pi$
$504$	0	0
$505$	−11.6750	−0.519533
$506$	6.88933	0.306268
$507$	0	0
$508$	−8.35276	−0.370594
$509$	−17.5949	−0.779878	−0.389939	−	0.920841i	$-0.627504\pi$
$509$	−17.5949	−0.779878	−0.389939	+	0.920841i	$0.627504\pi$
$510$	0	0
$511$	−8.17393	−0.361594
$512$	−39.1262	−1.72915
$513$	0	0
$514$	38.8812	1.71498
$515$	8.46284	0.372917
$516$	0	0
$517$	−36.1118	−1.58820
$518$	−92.2290	−4.05231
$519$	0	0
$520$	15.1084	0.662546
$521$	−33.7103	−1.47688	−0.738438	−	0.674321i	$-0.764436\pi$
$521$	−33.7103	−1.47688	−0.738438	+	0.674321i	$0.764436\pi$
$522$	0	0
$523$	5.56397	0.243295	0.121648	−	0.992573i	$-0.461182\pi$
$523$	5.56397	0.243295	0.121648	+	0.992573i	$0.461182\pi$
$524$	−2.65563	−0.116012
$525$	0	0
$526$	−62.8820	−2.74178
$527$	−21.9804	−0.957480
$528$	0	0
$529$	−22.8121	−0.991828
$530$	22.8420	0.992192
$531$	0	0
$532$	59.9388	2.59867
$533$	−8.72794	−0.378049
$534$	0	0
$535$	−10.1189	−0.437480
$536$	−0.965353	−0.0416969
$537$	0	0
$538$	−66.9880	−2.88806
$539$	107.472	4.62916
$540$	0	0
$541$	−28.6000	−1.22961	−0.614805	−	0.788679i	$-0.710765\pi$
$541$	−28.6000	−1.22961	−0.614805	+	0.788679i	$0.710765\pi$
$542$	43.1657	1.85412
$543$	0	0
$544$	0.0312045	0.00133788
$545$	−11.9862	−0.513431
$546$	0	0
$547$	−10.6473	−0.455244	−0.227622	−	0.973750i	$-0.573095\pi$
$547$	−10.6473	−0.455244	−0.227622	+	0.973750i	$0.573095\pi$
$548$	44.0284	1.88080
$549$	0	0
$550$	15.8912	0.677604
$551$	−27.3800	−1.16643
$552$	0	0
$553$	26.9023	1.14400
$554$	−6.41110	−0.272382
$555$	0	0
$556$	39.9509	1.69429
$557$	−41.2912	−1.74956	−0.874782	−	0.484517i	$-0.838995\pi$
$557$	−41.2912	−1.74956	−0.874782	+	0.484517i	$0.838995\pi$
$558$	0	0
$559$	15.4285	0.652556
$560$	−19.3720	−0.818618
$561$	0	0
$562$	−33.2032	−1.40059
$563$	−16.8936	−0.711981	−0.355991	−	0.934490i	$-0.615857\pi$
$563$	−16.8936	−0.711981	−0.355991	+	0.934490i	$0.615857\pi$
$564$	0	0
$565$	2.42958	0.102213
$566$	38.5693	1.62119
$567$	0	0
$568$	−62.7812	−2.63424
$569$	−18.9067	−0.792609	−0.396304	−	0.918119i	$-0.629707\pi$
$569$	−18.9067	−0.792609	−0.396304	+	0.918119i	$0.629707\pi$
$570$	0	0
$571$	28.6825	1.20032	0.600162	−	0.799878i	$-0.295103\pi$
$571$	28.6825	1.20032	0.600162	+	0.799878i	$0.295103\pi$
$572$	−80.0818	−3.34839
$573$	0	0
$574$	33.6163	1.40312
$575$	0.433530	0.0180794
$576$	0	0
$577$	14.4724	0.602495	0.301247	−	0.953546i	$-0.402597\pi$
$577$	14.4724	0.602495	0.301247	+	0.953546i	$0.402597\pi$
$578$	−31.2107	−1.29820
$579$	0	0
$580$	35.4513	1.47203
$581$	−13.6886	−0.567900
$582$	0	0
$583$	−60.5134	−2.50621
$584$	−8.24182	−0.341049
$585$	0	0
$586$	−9.35612	−0.386498
$587$	22.7931	0.940772	0.470386	−	0.882461i	$-0.344115\pi$
$587$	22.7931	0.940772	0.470386	+	0.882461i	$0.344115\pi$
$588$	0	0
$589$	32.9000	1.35562
$590$	−10.7961	−0.444469
$591$	0	0
$592$	−30.9584	−1.27238
$593$	41.0548	1.68592	0.842959	−	0.537978i	$-0.180812\pi$
$593$	41.0548	1.68592	0.842959	+	0.537978i	$0.180812\pi$
$594$	0	0
$595$	10.0151	0.410579
$596$	3.36587	0.137871
$597$	0	0
$598$	−3.27750	−0.134027
$599$	22.1290	0.904165	0.452083	−	0.891976i	$-0.350681\pi$
$599$	22.1290	0.904165	0.452083	+	0.891976i	$0.350681\pi$
$600$	0	0
$601$	36.0322	1.46978	0.734892	−	0.678185i	$-0.237233\pi$
$601$	36.0322	1.46978	0.734892	+	0.678185i	$0.237233\pi$
$602$	−59.4241	−2.42195
$603$	0	0
$604$	−83.2259	−3.38642
$605$	−31.0994	−1.26437
$606$	0	0
$607$	−6.13394	−0.248969	−0.124484	−	0.992222i	$-0.539728\pi$
$607$	−6.13394	−0.248969	−0.124484	+	0.992222i	$0.539728\pi$
$608$	−0.0467065	−0.00189420
$609$	0	0
$610$	−23.1368	−0.936782
$611$	17.1797	0.695016
$612$	0	0
$613$	27.5605	1.11316	0.556579	−	0.830794i	$-0.312113\pi$
$613$	27.5605	1.11316	0.556579	+	0.830794i	$0.312113\pi$
$614$	−64.2334	−2.59225
$615$	0	0
$616$	154.161	6.21132
$617$	−1.16645	−0.0469595	−0.0234798	−	0.999724i	$-0.507475\pi$
$617$	−1.16645	−0.0469595	−0.0234798	+	0.999724i	$0.507475\pi$
$618$	0	0
$619$	−39.0459	−1.56939	−0.784694	−	0.619883i	$-0.787180\pi$
$619$	−39.0459	−1.56939	−0.784694	+	0.619883i	$0.787180\pi$
$620$	−42.5985	−1.71080
$621$	0	0
$622$	−36.0498	−1.44547
$623$	−4.85425	−0.194481
$624$	0	0
$625$	1.00000	0.0400000
$626$	−4.20459	−0.168049
$627$	0	0
$628$	29.5655	1.17979
$629$	16.0051	0.638165
$630$	0	0
$631$	−45.2720	−1.80225	−0.901126	−	0.433558i	$-0.857258\pi$
$631$	−45.2720	−1.80225	−0.901126	+	0.433558i	$0.857258\pi$
$632$	27.1258	1.07900
$633$	0	0
$634$	56.5662	2.24653
$635$	2.08900	0.0828993
$636$	0	0
$637$	−51.1284	−2.02578
$638$	−140.895	−5.57811
$639$	0	0
$640$	19.6085	0.775094
$641$	−6.29377	−0.248589	−0.124294	−	0.992245i	$-0.539667\pi$
$641$	−6.29377	−0.248589	−0.124294	+	0.992245i	$0.539667\pi$
$642$	0	0
$643$	10.6360	0.419442	0.209721	−	0.977761i	$-0.432744\pi$
$643$	10.6360	0.419442	0.209721	+	0.977761i	$0.432744\pi$
$644$	8.41460	0.331582
$645$	0	0
$646$	−15.6043	−0.613945
$647$	32.1597	1.26433	0.632164	−	0.774835i	$-0.282167\pi$
$647$	32.1597	1.26433	0.632164	+	0.774835i	$0.282167\pi$
$648$	0	0
$649$	28.6013	1.12270
$650$	−7.56002	−0.296529
$651$	0	0
$652$	3.39648	0.133017
$653$	15.7012	0.614436	0.307218	−	0.951639i	$-0.400602\pi$
$653$	15.7012	0.614436	0.307218	+	0.951639i	$0.400602\pi$
$654$	0	0
$655$	0.664163	0.0259510
$656$	11.2840	0.440565
$657$	0	0
$658$	−66.1689	−2.57953
$659$	24.6976	0.962082	0.481041	−	0.876698i	$-0.340259\pi$
$659$	24.6976	0.962082	0.481041	+	0.876698i	$0.340259\pi$
$660$	0	0
$661$	23.0140	0.895139	0.447570	−	0.894249i	$-0.352290\pi$
$661$	23.0140	0.895139	0.447570	+	0.894249i	$0.352290\pi$
$662$	20.9201	0.813082
$663$	0	0
$664$	−13.8023	−0.535634
$665$	−14.9905	−0.581306
$666$	0	0
$667$	−3.84378	−0.148832
$668$	−99.4123	−3.84638
$669$	0	0
$670$	0.483049	0.0186618
$671$	61.2945	2.36625
$672$	0	0
$673$	29.3981	1.13321	0.566607	−	0.823988i	$-0.308256\pi$
$673$	29.3981	1.13321	0.566607	+	0.823988i	$0.308256\pi$
$674$	15.7651	0.607249
$675$	0	0
$676$	−13.8822	−0.533930
$677$	9.63434	0.370278	0.185139	−	0.982712i	$-0.440727\pi$
$677$	9.63434	0.370278	0.185139	+	0.982712i	$0.440727\pi$
$678$	0	0
$679$	−33.7648	−1.29577
$680$	10.0983	0.387251
$681$	0	0
$682$	169.301	6.48288
$683$	27.2850	1.04403	0.522016	−	0.852936i	$-0.325180\pi$
$683$	27.2850	1.04403	0.522016	+	0.852936i	$0.325180\pi$
$684$	0	0
$685$	−11.0114	−0.420723
$686$	113.703	4.34119
$687$	0	0
$688$	−19.9468	−0.760465
$689$	28.7884	1.09675
$690$	0	0
$691$	28.7454	1.09353	0.546764	−	0.837287i	$-0.315859\pi$
$691$	28.7454	1.09353	0.546764	+	0.837287i	$0.315859\pi$
$692$	−26.0286	−0.989460
$693$	0	0
$694$	31.1485	1.18238
$695$	−9.99158	−0.379002
$696$	0	0
$697$	−5.83366	−0.220966
$698$	32.9908	1.24872
$699$	0	0
$700$	19.4095	0.733610
$701$	26.1589	0.988009	0.494005	−	0.869459i	$-0.335533\pi$
$701$	26.1589	0.988009	0.494005	+	0.869459i	$0.335533\pi$
$702$	0	0
$703$	−23.9562	−0.903527
$704$	−52.0274	−1.96085
$705$	0	0
$706$	32.9520	1.24016
$707$	56.6736	2.13143
$708$	0	0
$709$	1.88884	0.0709367	0.0354684	−	0.999371i	$-0.488708\pi$
$709$	1.88884	0.0709367	0.0354684	+	0.999371i	$0.488708\pi$
$710$	31.4148	1.17898
$711$	0	0
$712$	−4.89457	−0.183432
$713$	4.61872	0.172972
$714$	0	0
$715$	20.0282	0.749011
$716$	14.9635	0.559211
$717$	0	0
$718$	78.9362	2.94587
$719$	6.30871	0.235275	0.117638	−	0.993057i	$-0.462468\pi$
$719$	6.30871	0.235275	0.117638	+	0.993057i	$0.462468\pi$
$720$	0	0
$721$	−41.0808	−1.52993
$722$	−23.1779	−0.862592
$723$	0	0
$724$	16.5066	0.613463
$725$	−8.86624	−0.329284
$726$	0	0
$727$	−35.1807	−1.30478	−0.652391	−	0.757883i	$-0.726234\pi$
$727$	−35.1807	−1.30478	−0.652391	+	0.757883i	$0.726234\pi$
$728$	−73.3398	−2.71816
$729$	0	0
$730$	4.12409	0.152640
$731$	10.3122	0.381412
$732$	0	0
$733$	−25.2608	−0.933031	−0.466515	−	0.884513i	$-0.654491\pi$
$733$	−25.2608	−0.933031	−0.466515	+	0.884513i	$0.654491\pi$
$734$	−3.99787	−0.147564
$735$	0	0
$736$	−0.00655697	−0.000241693 0
$737$	−1.27970	−0.0471385
$738$	0	0
$739$	−47.0551	−1.73095	−0.865475	−	0.500951i	$-0.832984\pi$
$739$	−47.0551	−1.73095	−0.865475	+	0.500951i	$0.832984\pi$
$740$	31.0183	1.14026
$741$	0	0
$742$	−110.881	−4.07056
$743$	43.7930	1.60661	0.803305	−	0.595568i	$-0.203073\pi$
$743$	43.7930	1.60661	0.803305	+	0.595568i	$0.203073\pi$
$744$	0	0
$745$	−0.841792	−0.0308409
$746$	−23.4725	−0.859389
$747$	0	0
$748$	−53.5258	−1.95710
$749$	49.1199	1.79480
$750$	0	0
$751$	−16.9624	−0.618968	−0.309484	−	0.950905i	$-0.600156\pi$
$751$	−16.9624	−0.618968	−0.309484	+	0.950905i	$0.600156\pi$
$752$	−22.2108	−0.809946
$753$	0	0
$754$	67.0290	2.44105
$755$	20.8145	0.757518
$756$	0	0
$757$	16.3208	0.593189	0.296594	−	0.955004i	$-0.404149\pi$
$757$	16.3208	0.593189	0.296594	+	0.955004i	$0.404149\pi$
$758$	−82.8362	−3.00875
$759$	0	0
$760$	−15.1150	−0.548278
$761$	20.1078	0.728909	0.364454	−	0.931221i	$-0.381256\pi$
$761$	20.1078	0.728909	0.364454	+	0.931221i	$0.381256\pi$
$762$	0	0
$763$	58.1839	2.10640
$764$	42.8520	1.55033
$765$	0	0
$766$	−30.9261	−1.11740
$767$	−13.6066	−0.491308
$768$	0	0
$769$	20.2936	0.731807	0.365903	−	0.930653i	$-0.380760\pi$
$769$	20.2936	0.731807	0.365903	+	0.930653i	$0.380760\pi$
$770$	−77.1400	−2.77993
$771$	0	0
$772$	−86.2891	−3.10561
$773$	6.20908	0.223325	0.111662	−	0.993746i	$-0.464382\pi$
$773$	6.20908	0.223325	0.111662	+	0.993746i	$0.464382\pi$
$774$	0	0
$775$	10.6537	0.382694
$776$	−34.0452	−1.22215
$777$	0	0
$778$	68.4579	2.45433
$779$	8.73176	0.312848
$780$	0	0
$781$	−83.2248	−2.97802
$782$	−2.19064	−0.0783372
$783$	0	0
$784$	66.1016	2.36077
$785$	−7.39423	−0.263911
$786$	0	0
$787$	14.2652	0.508500	0.254250	−	0.967139i	$-0.418171\pi$
$787$	14.2652	0.508500	0.254250	+	0.967139i	$0.418171\pi$
$788$	23.6323	0.841865
$789$	0	0
$790$	−13.5734	−0.482918
$791$	−11.7938	−0.419339
$792$	0	0
$793$	−29.1600	−1.03550
$794$	29.6101	1.05082
$795$	0	0
$796$	−3.99430	−0.141574
$797$	−3.45271	−0.122301	−0.0611506	−	0.998129i	$-0.519477\pi$
$797$	−3.45271	−0.122301	−0.0611506	+	0.998129i	$0.519477\pi$
$798$	0	0
$799$	11.4827	0.406229
$800$	−0.0151246	−0.000534736 0
$801$	0	0
$802$	−91.3636	−3.22616
$803$	−10.9256	−0.385557
$804$	0	0
$805$	−2.10446	−0.0741726
$806$	−80.5426	−2.83699
$807$	0	0
$808$	57.1443	2.01033
$809$	16.5488	0.581826	0.290913	−	0.956750i	$-0.406041\pi$
$809$	16.5488	0.581826	0.290913	+	0.956750i	$0.406041\pi$
$810$	0	0
$811$	−54.3743	−1.90934	−0.954670	−	0.297665i	$-0.903792\pi$
$811$	−54.3743	−1.90934	−0.954670	+	0.297665i	$0.903792\pi$
$812$	−172.089	−6.03915
$813$	0	0
$814$	−123.277	−4.32087
$815$	−0.849449	−0.0297549
$816$	0	0
$817$	−15.4353	−0.540011
$818$	53.9711	1.88706
$819$	0	0
$820$	−11.3058	−0.394815
$821$	−0.950390	−0.0331688	−0.0165844	−	0.999862i	$-0.505279\pi$
$821$	−0.950390	−0.0331688	−0.0165844	+	0.999862i	$0.505279\pi$
$822$	0	0
$823$	28.8435	1.00542	0.502711	−	0.864455i	$-0.332336\pi$
$823$	28.8435	1.00542	0.502711	+	0.864455i	$0.332336\pi$
$824$	−41.4219	−1.44300
$825$	0	0
$826$	52.4071	1.82347
$827$	−14.0405	−0.488235	−0.244117	−	0.969746i	$-0.578498\pi$
$827$	−14.0405	−0.488235	−0.244117	+	0.969746i	$0.578498\pi$
$828$	0	0
$829$	32.7278	1.13668	0.568342	−	0.822792i	$-0.307585\pi$
$829$	32.7278	1.13668	0.568342	+	0.822792i	$0.307585\pi$
$830$	6.90649	0.239728
$831$	0	0
$832$	24.7513	0.858096
$833$	−34.1737	−1.18405
$834$	0	0
$835$	24.8627	0.860409
$836$	80.1168	2.77090
$837$	0	0
$838$	−4.01892	−0.138831
$839$	7.50058	0.258949	0.129474	−	0.991583i	$-0.458671\pi$
$839$	7.50058	0.258949	0.129474	+	0.991583i	$0.458671\pi$
$840$	0	0
$841$	49.6102	1.71070
$842$	−27.3173	−0.941416
$843$	0	0
$844$	13.3447	0.459343
$845$	3.47189	0.119437
$846$	0	0
$847$	150.964	5.18719
$848$	−37.2192	−1.27811
$849$	0	0
$850$	−5.05304	−0.173318
$851$	−3.36314	−0.115287
$852$	0	0
$853$	−41.0302	−1.40485	−0.702424	−	0.711758i	$-0.747899\pi$
$853$	−41.0302	−1.40485	−0.702424	+	0.711758i	$0.747899\pi$
$854$	112.312	3.84323
$855$	0	0
$856$	49.5278	1.69283
$857$	12.6490	0.432083	0.216041	−	0.976384i	$-0.430685\pi$
$857$	12.6490	0.432083	0.216041	+	0.976384i	$0.430685\pi$
$858$	0	0
$859$	−23.7316	−0.809712	−0.404856	−	0.914380i	$-0.632678\pi$
$859$	−23.7316	−0.809712	−0.404856	+	0.914380i	$0.632678\pi$
$860$	19.9854	0.681496
$861$	0	0
$862$	−63.7612	−2.17172
$863$	22.2979	0.759028	0.379514	−	0.925186i	$-0.376091\pi$
$863$	22.2979	0.759028	0.379514	+	0.925186i	$0.376091\pi$
$864$	0	0
$865$	6.50967	0.221335
$866$	−51.4202	−1.74733
$867$	0	0
$868$	206.784	7.01871
$869$	35.9588	1.21982
$870$	0	0
$871$	0.608801	0.0206284
$872$	58.6671	1.98672
$873$	0	0
$874$	3.27893	0.110911
$875$	−4.85425	−0.164104
$876$	0	0
$877$	34.1197	1.15214	0.576070	−	0.817400i	$-0.304586\pi$
$877$	34.1197	1.15214	0.576070	+	0.817400i	$0.304586\pi$
$878$	30.8053	1.03963
$879$	0	0
$880$	−25.8935	−0.872870
$881$	−13.2721	−0.447147	−0.223573	−	0.974687i	$-0.571772\pi$
$881$	−13.2721	−0.447147	−0.223573	+	0.974687i	$0.571772\pi$
$882$	0	0
$883$	58.4635	1.96745	0.983727	−	0.179667i	$-0.0575020\pi$
$883$	58.4635	1.96745	0.983727	+	0.179667i	$0.0575020\pi$
$884$	25.4641	0.856451
$885$	0	0
$886$	81.6152	2.74192
$887$	3.90447	0.131099	0.0655496	−	0.997849i	$-0.479120\pi$
$887$	3.90447	0.131099	0.0655496	+	0.997849i	$0.479120\pi$
$888$	0	0
$889$	−10.1405	−0.340102
$890$	2.44917	0.0820965
$891$	0	0
$892$	−28.8264	−0.965179
$893$	−17.1872	−0.575148
$894$	0	0
$895$	−3.74231	−0.125092
$896$	−95.1845	−3.17989
$897$	0	0
$898$	−8.52465	−0.284471
$899$	−94.4587	−3.15037
$900$	0	0
$901$	19.2418	0.641039
$902$	44.9331	1.49611
$903$	0	0
$904$	−11.8917	−0.395513
$905$	−4.12824	−0.137227
$906$	0	0
$907$	−4.90891	−0.162998	−0.0814988	−	0.996673i	$-0.525971\pi$
$907$	−4.90891	−0.162998	−0.0814988	+	0.996673i	$0.525971\pi$
$908$	101.877	3.38090
$909$	0	0
$910$	36.6983	1.21654
$911$	−0.217731	−0.00721376	−0.00360688	−	0.999993i	$-0.501148\pi$
$911$	−0.217731	−0.00721376	−0.00360688	+	0.999993i	$0.501148\pi$
$912$	0	0
$913$	−18.2968	−0.605536
$914$	−74.6552	−2.46938
$915$	0	0
$916$	−16.0044	−0.528799
$917$	−3.22401	−0.106466
$918$	0	0
$919$	−26.4800	−0.873496	−0.436748	−	0.899584i	$-0.643870\pi$
$919$	−26.4800	−0.873496	−0.436748	+	0.899584i	$0.643870\pi$
$920$	−2.12194	−0.0699583
$921$	0	0
$922$	−58.4839	−1.92606
$923$	39.5930	1.30322
$924$	0	0
$925$	−7.75757	−0.255067
$926$	−40.6992	−1.33746
$927$	0	0
$928$	0.134098	0.00440200
$929$	47.6390	1.56299	0.781493	−	0.623915i	$-0.214459\pi$
$929$	47.6390	1.56299	0.781493	+	0.623915i	$0.214459\pi$
$930$	0	0
$931$	51.1508	1.67640
$932$	−42.1210	−1.37972
$933$	0	0
$934$	95.6624	3.13017
$935$	13.3866	0.437789
$936$	0	0
$937$	19.9037	0.650224	0.325112	−	0.945675i	$-0.394598\pi$
$937$	19.9037	0.650224	0.325112	+	0.945675i	$0.394598\pi$
$938$	−2.34484	−0.0765619
$939$	0	0
$940$	22.2538	0.725839
$941$	−47.7254	−1.55580	−0.777901	−	0.628387i	$-0.783716\pi$
$941$	−47.7254	−1.55580	−0.777901	+	0.628387i	$0.783716\pi$
$942$	0	0
$943$	1.22582	0.0399183
$944$	17.5914	0.572552
$945$	0	0
$946$	−79.4288	−2.58245
$947$	34.6727	1.12671	0.563355	−	0.826215i	$-0.309510\pi$
$947$	34.6727	1.12671	0.563355	+	0.826215i	$0.309510\pi$
$948$	0	0
$949$	5.19771	0.168725
$950$	7.56333	0.245387
$951$	0	0
$952$	−49.0195	−1.58873
$953$	13.0601	0.423058	0.211529	−	0.977372i	$-0.432156\pi$
$953$	13.0601	0.423058	0.211529	+	0.977372i	$0.432156\pi$
$954$	0	0
$955$	−10.7171	−0.346798
$956$	−49.3555	−1.59627
$957$	0	0
$958$	−28.6033	−0.924129
$959$	53.4519	1.72605
$960$	0	0
$961$	82.5023	2.66137
$962$	58.6474	1.89087
$963$	0	0
$964$	−71.3350	−2.29754
$965$	21.5806	0.694704
$966$	0	0
$967$	−57.2687	−1.84164	−0.920818	−	0.389992i	$-0.872478\pi$
$967$	−57.2687	−1.84164	−0.920818	+	0.389992i	$0.872478\pi$
$968$	152.218	4.89247
$969$	0	0
$970$	17.0358	0.546986
$971$	41.8696	1.34366	0.671830	−	0.740705i	$-0.265508\pi$
$971$	41.8696	1.34366	0.671830	+	0.740705i	$0.265508\pi$
$972$	0	0
$973$	48.5016	1.55489
$974$	11.7302	0.375861
$975$	0	0
$976$	37.6996	1.20674
$977$	27.7807	0.888783	0.444392	−	0.895833i	$-0.353420\pi$
$977$	27.7807	0.888783	0.444392	+	0.895833i	$0.353420\pi$
$978$	0	0
$979$	−6.48840	−0.207370
$980$	−66.2295	−2.11562
$981$	0	0
$982$	6.25341	0.199554
$983$	−23.3550	−0.744910	−0.372455	−	0.928050i	$-0.621484\pi$
$983$	−23.3550	−0.744910	−0.372455	+	0.928050i	$0.621484\pi$
$984$	0	0
$985$	−5.91035	−0.188319
$986$	44.8014	1.42677
$987$	0	0
$988$	−38.1144	−1.21258
$989$	−2.16690	−0.0689035
$990$	0	0
$991$	7.67405	0.243774	0.121887	−	0.992544i	$-0.461105\pi$
$991$	7.67405	0.243774	0.121887	+	0.992544i	$0.461105\pi$
$992$	−0.161134	−0.00511600
$993$	0	0
$994$	−152.496	−4.83686
$995$	0.998960	0.0316692
$996$	0	0
$997$	6.87605	0.217767	0.108883	−	0.994055i	$-0.465273\pi$
$997$	6.87605	0.217767	0.108883	+	0.994055i	$0.465273\pi$
$998$	−89.9826	−2.84835
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.r.1.10		10
3.2	odd	2		1335.2.a.k.1.1	✓	10
15.14	odd	2		6675.2.a.y.1.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.k.1.1	✓	10	3.2	odd	2
4005.2.a.r.1.10		10	1.1	even	1	trivial
6675.2.a.y.1.10		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)

\(f(q)\)	\(=\)	\(q+2.44917 q^{2} +3.99846 q^{4} -1.00000 q^{5} +4.85425 q^{7} +4.89457 q^{8} +O(q^{10})\)	\(q+2.44917 q^{2} +3.99846 q^{4} -1.00000 q^{5} +4.85425 q^{7} +4.89457 q^{8} -2.44917 q^{10} +6.48840 q^{11} -3.08676 q^{13} +11.8889 q^{14} +3.99074 q^{16} -2.06316 q^{17} +3.08811 q^{19} -3.99846 q^{20} +15.8912 q^{22} +0.433530 q^{23} +1.00000 q^{25} -7.56002 q^{26} +19.4095 q^{28} -8.86624 q^{29} +10.6537 q^{31} -0.0151246 q^{32} -5.05304 q^{34} -4.85425 q^{35} -7.75757 q^{37} +7.56333 q^{38} -4.89457 q^{40} +2.82754 q^{41} -4.99828 q^{43} +25.9436 q^{44} +1.06179 q^{46} -5.56560 q^{47} +16.5638 q^{49} +2.44917 q^{50} -12.3423 q^{52} -9.32640 q^{53} -6.48840 q^{55} +23.7595 q^{56} -21.7150 q^{58} +4.40806 q^{59} +9.44678 q^{61} +26.0929 q^{62} -8.01851 q^{64} +3.08676 q^{65} -0.197229 q^{67} -8.24945 q^{68} -11.8889 q^{70} -12.8267 q^{71} -1.68387 q^{73} -18.9996 q^{74} +12.3477 q^{76} +31.4963 q^{77} +5.54201 q^{79} -3.99074 q^{80} +6.92513 q^{82} -2.81993 q^{83} +2.06316 q^{85} -12.2417 q^{86} +31.7579 q^{88} -1.00000 q^{89} -14.9839 q^{91} +1.73345 q^{92} -13.6311 q^{94} -3.08811 q^{95} -6.95572 q^{97} +40.5675 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8} + 6 q^{10} + 10 q^{11} + 7 q^{13} + 7 q^{14} + 22 q^{16} - 11 q^{17} + 10 q^{19} - 14 q^{20} + 8 q^{22} - 6 q^{23} + 10 q^{25} + 14 q^{26} + 16 q^{28} + 3 q^{29} + 12 q^{31} - 21 q^{32} - 6 q^{34} - 7 q^{35} - 19 q^{37} - 6 q^{38} + 15 q^{40} - 13 q^{41} + 9 q^{43} + 26 q^{44} + 8 q^{46} - 21 q^{47} + 53 q^{49} - 6 q^{50} - 43 q^{52} - 7 q^{53} - 10 q^{55} + 53 q^{56} - 42 q^{58} + 19 q^{59} + 4 q^{61} + 28 q^{62} + 5 q^{64} - 7 q^{65} - 6 q^{67} - 2 q^{68} - 7 q^{70} + 6 q^{71} + 6 q^{73} - 2 q^{76} + 40 q^{77} + 25 q^{79} - 22 q^{80} + q^{82} + 22 q^{83} + 11 q^{85} + 2 q^{86} + 20 q^{88} - 10 q^{89} - 10 q^{91} - 10 q^{92} + 25 q^{94} - 10 q^{95} + 40 q^{97} + 56 q^{98}+O(q^{100}) \) 10 * q - 6 * q^2 + 14 * q^4 - 10 * q^5 + 7 * q^7 - 15 * q^8 + 6 * q^10 + 10 * q^11 + 7 * q^13 + 7 * q^14 + 22 * q^16 - 11 * q^17 + 10 * q^19 - 14 * q^20 + 8 * q^22 - 6 * q^23 + 10 * q^25 + 14 * q^26 + 16 * q^28 + 3 * q^29 + 12 * q^31 - 21 * q^32 - 6 * q^34 - 7 * q^35 - 19 * q^37 - 6 * q^38 + 15 * q^40 - 13 * q^41 + 9 * q^43 + 26 * q^44 + 8 * q^46 - 21 * q^47 + 53 * q^49 - 6 * q^50 - 43 * q^52 - 7 * q^53 - 10 * q^55 + 53 * q^56 - 42 * q^58 + 19 * q^59 + 4 * q^61 + 28 * q^62 + 5 * q^64 - 7 * q^65 - 6 * q^67 - 2 * q^68 - 7 * q^70 + 6 * q^71 + 6 * q^73 - 2 * q^76 + 40 * q^77 + 25 * q^79 - 22 * q^80 + q^82 + 22 * q^83 + 11 * q^85 + 2 * q^86 + 20 * q^88 - 10 * q^89 - 10 * q^91 - 10 * q^92 + 25 * q^94 - 10 * q^95 + 40 * q^97 + 56 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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