LMFDB - Embedded newform 4005.2.a.r.1.8

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			1.8
Root			$2.21375$ 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)

$f(q)$	$=$	$q+1.21375 q^{2} -0.526821 q^{4} -1.00000 q^{5} -4.72892 q^{7} -3.06692 q^{8} +O(q^{10})$	\(q+1.21375 q^{2} -0.526821 q^{4} -1.00000 q^{5} -4.72892 q^{7} -3.06692 q^{8} -1.21375 q^{10} -3.61190 q^{11} +1.64943 q^{13} -5.73970 q^{14} -2.66882 q^{16} -2.90942 q^{17} -3.25622 q^{19} +0.526821 q^{20} -4.38393 q^{22} +4.84531 q^{23} +1.00000 q^{25} +2.00199 q^{26} +2.49129 q^{28} -0.511250 q^{29} +3.29685 q^{31} +2.89457 q^{32} -3.53130 q^{34} +4.72892 q^{35} -9.07195 q^{37} -3.95222 q^{38} +3.06692 q^{40} -7.85893 q^{41} +8.02520 q^{43} +1.90283 q^{44} +5.88098 q^{46} +0.578635 q^{47} +15.3626 q^{49} +1.21375 q^{50} -0.868956 q^{52} -10.0984 q^{53} +3.61190 q^{55} +14.5032 q^{56} -0.620528 q^{58} -9.94517 q^{59} +4.92149 q^{61} +4.00154 q^{62} +8.85091 q^{64} -1.64943 q^{65} +5.49644 q^{67} +1.53275 q^{68} +5.73970 q^{70} +4.32180 q^{71} +10.8834 q^{73} -11.0110 q^{74} +1.71544 q^{76} +17.0804 q^{77} +2.61545 q^{79} +2.66882 q^{80} -9.53874 q^{82} +5.19278 q^{83} +2.90942 q^{85} +9.74055 q^{86} +11.0774 q^{88} -1.00000 q^{89} -7.80003 q^{91} -2.55261 q^{92} +0.702316 q^{94} +3.25622 q^{95} +14.9686 q^{97} +18.6464 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.21375	0.858248	0.429124	−	0.903246i	$-0.358822\pi$
$2$	1.21375	0.858248	0.429124	+	0.903246i	$0.358822\pi$
$3$	0	0
$3$	0	0
$4$	−0.526821	−0.263411
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.72892	−1.78736	−0.893681	−	0.448702i	$-0.851886\pi$
$7$	−4.72892	−1.78736	−0.893681	+	0.448702i	$0.851886\pi$
$8$	−3.06692	−1.08432
$9$	0	0
$10$	−1.21375	−0.383820
$11$	−3.61190	−1.08903	−0.544515	−	0.838751i	$-0.683286\pi$
$11$	−3.61190	−1.08903	−0.544515	+	0.838751i	$0.683286\pi$
$12$	0	0
$13$	1.64943	0.457471	0.228735	−	0.973489i	$-0.426541\pi$
$13$	1.64943	0.457471	0.228735	+	0.973489i	$0.426541\pi$
$14$	−5.73970	−1.53400
$15$	0	0
$16$	−2.66882	−0.667204
$17$	−2.90942	−0.705638	−0.352819	−	0.935692i	$-0.614777\pi$
$17$	−2.90942	−0.705638	−0.352819	+	0.935692i	$0.614777\pi$
$18$	0	0
$19$	−3.25622	−0.747027	−0.373514	−	0.927625i	$-0.621847\pi$
$19$	−3.25622	−0.747027	−0.373514	+	0.927625i	$0.621847\pi$
$20$	0.526821	0.117801
$21$	0	0
$22$	−4.38393	−0.934658
$23$	4.84531	1.01032	0.505159	−	0.863026i	$-0.331434\pi$
$23$	4.84531	1.01032	0.505159	+	0.863026i	$0.331434\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.00199	0.392623
$27$	0	0
$28$	2.49129	0.470810
$29$	−0.511250	−0.0949368	−0.0474684	−	0.998873i	$-0.515115\pi$
$29$	−0.511250	−0.0949368	−0.0474684	+	0.998873i	$0.515115\pi$
$30$	0	0
$31$	3.29685	0.592132	0.296066	−	0.955168i	$-0.404325\pi$
$31$	3.29685	0.592132	0.296066	+	0.955168i	$0.404325\pi$
$32$	2.89457	0.511693
$33$	0	0
$34$	−3.53130	−0.605613
$35$	4.72892	0.799333
$36$	0	0
$37$	−9.07195	−1.49142	−0.745710	−	0.666271i	$-0.767889\pi$
$37$	−9.07195	−1.49142	−0.745710	+	0.666271i	$0.767889\pi$
$38$	−3.95222	−0.641135
$39$	0	0
$40$	3.06692	0.484922
$41$	−7.85893	−1.22736	−0.613679	−	0.789555i	$-0.710311\pi$
$41$	−7.85893	−1.22736	−0.613679	+	0.789555i	$0.710311\pi$
$42$	0	0
$43$	8.02520	1.22383	0.611916	−	0.790923i	$-0.290399\pi$
$43$	8.02520	1.22383	0.611916	+	0.790923i	$0.290399\pi$
$44$	1.90283	0.286862
$45$	0	0
$46$	5.88098	0.867103
$47$	0.578635	0.0844026	0.0422013	−	0.999109i	$-0.486563\pi$
$47$	0.578635	0.0844026	0.0422013	+	0.999109i	$0.486563\pi$
$48$	0	0
$49$	15.3626	2.19466
$50$	1.21375	0.171650
$51$	0	0
$52$	−0.868956	−0.120503
$53$	−10.0984	−1.38712	−0.693561	−	0.720398i	$-0.743959\pi$
$53$	−10.0984	−1.38712	−0.693561	+	0.720398i	$0.743959\pi$
$54$	0	0
$55$	3.61190	0.487029
$56$	14.5032	1.93807
$57$	0	0
$58$	−0.620528	−0.0814793
$59$	−9.94517	−1.29475	−0.647375	−	0.762171i	$-0.724133\pi$
$59$	−9.94517	−1.29475	−0.647375	+	0.762171i	$0.724133\pi$
$60$	0	0
$61$	4.92149	0.630132	0.315066	−	0.949070i	$-0.397973\pi$
$61$	4.92149	0.630132	0.315066	+	0.949070i	$0.397973\pi$
$62$	4.00154	0.508196
$63$	0	0
$64$	8.85091	1.10636
$65$	−1.64943	−0.204587
$66$	0	0
$67$	5.49644	0.671498	0.335749	−	0.941952i	$-0.391011\pi$
$67$	5.49644	0.671498	0.335749	+	0.941952i	$0.391011\pi$
$68$	1.53275	0.185873
$69$	0	0
$70$	5.73970	0.686026
$71$	4.32180	0.512903	0.256452	−	0.966557i	$-0.417447\pi$
$71$	4.32180	0.512903	0.256452	+	0.966557i	$0.417447\pi$
$72$	0	0
$73$	10.8834	1.27381	0.636904	−	0.770943i	$-0.280215\pi$
$73$	10.8834	1.27381	0.636904	+	0.770943i	$0.280215\pi$
$74$	−11.0110	−1.28001
$75$	0	0
$76$	1.71544	0.196775
$77$	17.0804	1.94649
$78$	0	0
$79$	2.61545	0.294261	0.147130	−	0.989117i	$-0.452996\pi$
$79$	2.61545	0.294261	0.147130	+	0.989117i	$0.452996\pi$
$80$	2.66882	0.298383
$81$	0	0
$82$	−9.53874	−1.05338
$83$	5.19278	0.569981	0.284991	−	0.958530i	$-0.408010\pi$
$83$	5.19278	0.569981	0.284991	+	0.958530i	$0.408010\pi$
$84$	0	0
$85$	2.90942	0.315571
$86$	9.74055	1.05035
$87$	0	0
$88$	11.0774	1.18086
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−7.80003	−0.817666
$92$	−2.55261	−0.266128
$93$	0	0
$94$	0.702316	0.0724384
$95$	3.25622	0.334081
$96$	0	0
$97$	14.9686	1.51984	0.759918	−	0.650019i	$-0.225239\pi$
$97$	14.9686	1.51984	0.759918	+	0.650019i	$0.225239\pi$
$98$	18.6464	1.88357
$99$	0	0
$100$	−0.526821	−0.0526821
$101$	3.00541	0.299049	0.149525	−	0.988758i	$-0.452226\pi$
$101$	3.00541	0.299049	0.149525	+	0.988758i	$0.452226\pi$
$102$	0	0
$103$	2.82193	0.278053	0.139027	−	0.990289i	$-0.455603\pi$
$103$	2.82193	0.278053	0.139027	+	0.990289i	$0.455603\pi$
$104$	−5.05868	−0.496044
$105$	0	0
$106$	−12.2569	−1.19049
$107$	10.0361	0.970225	0.485113	−	0.874452i	$-0.338779\pi$
$107$	10.0361	0.970225	0.485113	+	0.874452i	$0.338779\pi$
$108$	0	0
$109$	4.18055	0.400424	0.200212	−	0.979753i	$-0.435837\pi$
$109$	4.18055	0.400424	0.200212	+	0.979753i	$0.435837\pi$
$110$	4.38393	0.417992
$111$	0	0
$112$	12.6206	1.19254
$113$	3.13759	0.295159	0.147580	−	0.989050i	$-0.452852\pi$
$113$	3.13759	0.295159	0.147580	+	0.989050i	$0.452852\pi$
$114$	0	0
$115$	−4.84531	−0.451828
$116$	0.269338	0.0250074
$117$	0	0
$118$	−12.0709	−1.11122
$119$	13.7584	1.26123
$120$	0	0
$121$	2.04585	0.185986
$122$	5.97343	0.540809
$123$	0	0
$124$	−1.73685	−0.155974
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.1674	−1.07968	−0.539839	−	0.841768i	$-0.681515\pi$
$127$	−12.1674	−1.07968	−0.539839	+	0.841768i	$0.681515\pi$
$128$	4.95361	0.437841
$129$	0	0
$130$	−2.00199	−0.175586
$131$	−19.5956	−1.71208	−0.856039	−	0.516911i	$-0.827082\pi$
$131$	−19.5956	−1.71208	−0.856039	+	0.516911i	$0.827082\pi$
$132$	0	0
$133$	15.3984	1.33521
$134$	6.67129	0.576311
$135$	0	0
$136$	8.92296	0.765137
$137$	−17.8203	−1.52249	−0.761247	−	0.648462i	$-0.775412\pi$
$137$	−17.8203	−1.52249	−0.761247	+	0.648462i	$0.775412\pi$
$138$	0	0
$139$	19.0939	1.61953	0.809763	−	0.586757i	$-0.199596\pi$
$139$	19.0939	1.61953	0.809763	+	0.586757i	$0.199596\pi$
$140$	−2.49129	−0.210553
$141$	0	0
$142$	5.24557	0.440198
$143$	−5.95759	−0.498199
$144$	0	0
$145$	0.511250	0.0424570
$146$	13.2097	1.09324
$147$	0	0
$148$	4.77930	0.392856
$149$	0.752469	0.0616446	0.0308223	−	0.999525i	$-0.490187\pi$
$149$	0.752469	0.0616446	0.0308223	+	0.999525i	$0.490187\pi$
$150$	0	0
$151$	6.28238	0.511253	0.255627	−	0.966776i	$-0.417718\pi$
$151$	6.28238	0.511253	0.255627	+	0.966776i	$0.417718\pi$
$152$	9.98655	0.810016
$153$	0	0
$154$	20.7313	1.67057
$155$	−3.29685	−0.264809
$156$	0	0
$157$	−7.46647	−0.595889	−0.297944	−	0.954583i	$-0.596301\pi$
$157$	−7.46647	−0.595889	−0.297944	+	0.954583i	$0.596301\pi$
$158$	3.17449	0.252549
$159$	0	0
$160$	−2.89457	−0.228836
$161$	−22.9131	−1.80580
$162$	0	0
$163$	−24.6314	−1.92928	−0.964642	−	0.263564i	$-0.915102\pi$
$163$	−24.6314	−1.92928	−0.964642	+	0.263564i	$0.915102\pi$
$164$	4.14025	0.323299
$165$	0	0
$166$	6.30271	0.489185
$167$	−16.0961	−1.24556	−0.622778	−	0.782398i	$-0.713996\pi$
$167$	−16.0961	−1.24556	−0.622778	+	0.782398i	$0.713996\pi$
$168$	0	0
$169$	−10.2794	−0.790721
$170$	3.53130	0.270838
$171$	0	0
$172$	−4.22784	−0.322370
$173$	−1.00048	−0.0760653	−0.0380326	−	0.999276i	$-0.512109\pi$
$173$	−1.00048	−0.0760653	−0.0380326	+	0.999276i	$0.512109\pi$
$174$	0	0
$175$	−4.72892	−0.357472
$176$	9.63951	0.726605
$177$	0	0
$178$	−1.21375	−0.0909741
$179$	19.8389	1.48283	0.741416	−	0.671046i	$-0.234155\pi$
$179$	19.8389	1.48283	0.741416	+	0.671046i	$0.234155\pi$
$180$	0	0
$181$	11.2563	0.836674	0.418337	−	0.908292i	$-0.362613\pi$
$181$	11.2563	0.836674	0.418337	+	0.908292i	$0.362613\pi$
$182$	−9.46726	−0.701760
$183$	0	0
$184$	−14.8602	−1.09551
$185$	9.07195	0.666983
$186$	0	0
$187$	10.5086	0.768461
$188$	−0.304837	−0.0222326
$189$	0	0
$190$	3.95222	0.286724
$191$	−1.87771	−0.135867	−0.0679333	−	0.997690i	$-0.521640\pi$
$191$	−1.87771	−0.135867	−0.0679333	+	0.997690i	$0.521640\pi$
$192$	0	0
$193$	−12.3298	−0.887518	−0.443759	−	0.896146i	$-0.646355\pi$
$193$	−12.3298	−0.887518	−0.443759	+	0.896146i	$0.646355\pi$
$194$	18.1681	1.30440
$195$	0	0
$196$	−8.09337	−0.578098
$197$	9.69414	0.690679	0.345339	−	0.938478i	$-0.387764\pi$
$197$	9.69414	0.690679	0.345339	+	0.938478i	$0.387764\pi$
$198$	0	0
$199$	17.6905	1.25405	0.627024	−	0.779000i	$-0.284273\pi$
$199$	17.6905	1.25405	0.627024	+	0.779000i	$0.284273\pi$
$200$	−3.06692	−0.216864
$201$	0	0
$202$	3.64780	0.256659
$203$	2.41766	0.169687
$204$	0	0
$205$	7.85893	0.548892
$206$	3.42511	0.238639
$207$	0	0
$208$	−4.40204	−0.305226
$209$	11.7611	0.813535
$210$	0	0
$211$	16.8316	1.15873	0.579366	−	0.815067i	$-0.303300\pi$
$211$	16.8316	1.15873	0.579366	+	0.815067i	$0.303300\pi$
$212$	5.32005	0.365383
$213$	0	0
$214$	12.1813	0.832694
$215$	−8.02520	−0.547314
$216$	0	0
$217$	−15.5905	−1.05835
$218$	5.07412	0.343663
$219$	0	0
$220$	−1.90283	−0.128289
$221$	−4.79890	−0.322809
$222$	0	0
$223$	−25.0367	−1.67658	−0.838290	−	0.545225i	$-0.816444\pi$
$223$	−25.0367	−1.67658	−0.838290	+	0.545225i	$0.816444\pi$
$224$	−13.6882	−0.914581
$225$	0	0
$226$	3.80823	0.253320
$227$	17.2494	1.14488	0.572440	−	0.819946i	$-0.305997\pi$
$227$	17.2494	1.14488	0.572440	+	0.819946i	$0.305997\pi$
$228$	0	0
$229$	17.0519	1.12682	0.563410	−	0.826177i	$-0.309489\pi$
$229$	17.0519	1.12682	0.563410	+	0.826177i	$0.309489\pi$
$230$	−5.88098	−0.387780
$231$	0	0
$232$	1.56796	0.102942
$233$	−6.78392	−0.444429	−0.222215	−	0.974998i	$-0.571329\pi$
$233$	−6.78392	−0.444429	−0.222215	+	0.974998i	$0.571329\pi$
$234$	0	0
$235$	−0.578635	−0.0377460
$236$	5.23933	0.341051
$237$	0	0
$238$	16.6992	1.08245
$239$	12.3662	0.799905	0.399952	−	0.916536i	$-0.369027\pi$
$239$	12.3662	0.799905	0.399952	+	0.916536i	$0.369027\pi$
$240$	0	0
$241$	2.56833	0.165440	0.0827202	−	0.996573i	$-0.473639\pi$
$241$	2.56833	0.165440	0.0827202	+	0.996573i	$0.473639\pi$
$242$	2.48314	0.159622
$243$	0	0
$244$	−2.59274	−0.165983
$245$	−15.3626	−0.981484
$246$	0	0
$247$	−5.37091	−0.341743
$248$	−10.1112	−0.642060
$249$	0	0
$250$	−1.21375	−0.0767640
$251$	27.8679	1.75900	0.879502	−	0.475896i	$-0.157876\pi$
$251$	27.8679	1.75900	0.879502	+	0.475896i	$0.157876\pi$
$252$	0	0
$253$	−17.5008	−1.10027
$254$	−14.7681	−0.926632
$255$	0	0
$256$	−11.6894	−0.730587
$257$	−17.4296	−1.08723	−0.543614	−	0.839335i	$-0.682944\pi$
$257$	−17.4296	−1.08723	−0.543614	+	0.839335i	$0.682944\pi$
$258$	0	0
$259$	42.9005	2.66571
$260$	0.868956	0.0538904
$261$	0	0
$262$	−23.7841	−1.46939
$263$	6.60294	0.407155	0.203577	−	0.979059i	$-0.434743\pi$
$263$	6.60294	0.407155	0.203577	+	0.979059i	$0.434743\pi$
$264$	0	0
$265$	10.0984	0.620340
$266$	18.6897	1.14594
$267$	0	0
$268$	−2.89564	−0.176880
$269$	−12.3699	−0.754209	−0.377104	−	0.926171i	$-0.623080\pi$
$269$	−12.3699	−0.754209	−0.377104	+	0.926171i	$0.623080\pi$
$270$	0	0
$271$	−11.0353	−0.670348	−0.335174	−	0.942156i	$-0.608795\pi$
$271$	−11.0353	−0.670348	−0.335174	+	0.942156i	$0.608795\pi$
$272$	7.76471	0.470805
$273$	0	0
$274$	−21.6293	−1.30668
$275$	−3.61190	−0.217806
$276$	0	0
$277$	15.2629	0.917061	0.458531	−	0.888679i	$-0.348376\pi$
$277$	15.2629	0.917061	0.458531	+	0.888679i	$0.348376\pi$
$278$	23.1752	1.38996
$279$	0	0
$280$	−14.5032	−0.866732
$281$	−22.6807	−1.35302	−0.676508	−	0.736435i	$-0.736508\pi$
$281$	−22.6807	−1.35302	−0.676508	+	0.736435i	$0.736508\pi$
$282$	0	0
$283$	−9.34813	−0.555689	−0.277844	−	0.960626i	$-0.589620\pi$
$283$	−9.34813	−0.555689	−0.277844	+	0.960626i	$0.589620\pi$
$284$	−2.27682	−0.135104
$285$	0	0
$286$	−7.23100	−0.427578
$287$	37.1642	2.19374
$288$	0	0
$289$	−8.53526	−0.502074
$290$	0.620528	0.0364387
$291$	0	0
$292$	−5.73362	−0.335535
$293$	9.08670	0.530851	0.265425	−	0.964131i	$-0.414488\pi$
$293$	9.08670	0.530851	0.265425	+	0.964131i	$0.414488\pi$
$294$	0	0
$295$	9.94517	0.579030
$296$	27.8229	1.61718
$297$	0	0
$298$	0.913306	0.0529064
$299$	7.99202	0.462190
$300$	0	0
$301$	−37.9505	−2.18743
$302$	7.62522	0.438782
$303$	0	0
$304$	8.69025	0.498420
$305$	−4.92149	−0.281803
$306$	0	0
$307$	−13.7887	−0.786964	−0.393482	−	0.919332i	$-0.628730\pi$
$307$	−13.7887	−0.786964	−0.393482	+	0.919332i	$0.628730\pi$
$308$	−8.99831	−0.512726
$309$	0	0
$310$	−4.00154	−0.227272
$311$	−21.8747	−1.24040	−0.620200	−	0.784444i	$-0.712949\pi$
$311$	−21.8747	−1.24040	−0.620200	+	0.784444i	$0.712949\pi$
$312$	0	0
$313$	−21.4250	−1.21101	−0.605507	−	0.795840i	$-0.707030\pi$
$313$	−21.4250	−1.21101	−0.605507	+	0.795840i	$0.707030\pi$
$314$	−9.06239	−0.511420
$315$	0	0
$316$	−1.37787	−0.0775114
$317$	−5.16611	−0.290157	−0.145079	−	0.989420i	$-0.546344\pi$
$317$	−5.16611	−0.290157	−0.145079	+	0.989420i	$0.546344\pi$
$318$	0	0
$319$	1.84659	0.103389
$320$	−8.85091	−0.494781
$321$	0	0
$322$	−27.8107	−1.54983
$323$	9.47371	0.527131
$324$	0	0
$325$	1.64943	0.0914941
$326$	−29.8963	−1.65580
$327$	0	0
$328$	24.1027	1.33085
$329$	−2.73632	−0.150858
$330$	0	0
$331$	−0.454771	−0.0249965	−0.0124982	−	0.999922i	$-0.503978\pi$
$331$	−0.454771	−0.0249965	−0.0124982	+	0.999922i	$0.503978\pi$
$332$	−2.73566	−0.150139
$333$	0	0
$334$	−19.5366	−1.06900
$335$	−5.49644	−0.300303
$336$	0	0
$337$	−13.5478	−0.737998	−0.368999	−	0.929430i	$-0.620299\pi$
$337$	−13.5478	−0.737998	−0.368999	+	0.929430i	$0.620299\pi$
$338$	−12.4765	−0.678634
$339$	0	0
$340$	−1.53275	−0.0831248
$341$	−11.9079	−0.644849
$342$	0	0
$343$	−39.5463	−2.13530
$344$	−24.6126	−1.32702
$345$	0	0
$346$	−1.21433	−0.0652829
$347$	−0.806099	−0.0432737	−0.0216368	−	0.999766i	$-0.506888\pi$
$347$	−0.806099	−0.0432737	−0.0216368	+	0.999766i	$0.506888\pi$
$348$	0	0
$349$	−14.5608	−0.779420	−0.389710	−	0.920938i	$-0.627425\pi$
$349$	−14.5608	−0.779420	−0.389710	+	0.920938i	$0.627425\pi$
$350$	−5.73970	−0.306800
$351$	0	0
$352$	−10.4549	−0.557249
$353$	35.0914	1.86772	0.933862	−	0.357633i	$-0.116416\pi$
$353$	35.0914	1.86772	0.933862	+	0.357633i	$0.116416\pi$
$354$	0	0
$355$	−4.32180	−0.229377
$356$	0.526821	0.0279215
$357$	0	0
$358$	24.0794	1.27264
$359$	7.80571	0.411970	0.205985	−	0.978555i	$-0.433960\pi$
$359$	7.80571	0.411970	0.205985	+	0.978555i	$0.433960\pi$
$360$	0	0
$361$	−8.39705	−0.441950
$362$	13.6623	0.718074
$363$	0	0
$364$	4.10922	0.215382
$365$	−10.8834	−0.569665
$366$	0	0
$367$	−1.87199	−0.0977172	−0.0488586	−	0.998806i	$-0.515558\pi$
$367$	−1.87199	−0.0977172	−0.0488586	+	0.998806i	$0.515558\pi$
$368$	−12.9313	−0.674088
$369$	0	0
$370$	11.0110	0.572437
$371$	47.7545	2.47929
$372$	0	0
$373$	27.4490	1.42126	0.710629	−	0.703567i	$-0.248411\pi$
$373$	27.4490	1.42126	0.710629	+	0.703567i	$0.248411\pi$
$374$	12.7547	0.659530
$375$	0	0
$376$	−1.77463	−0.0915194
$377$	−0.843274	−0.0434308
$378$	0	0
$379$	21.2308	1.09055	0.545276	−	0.838256i	$-0.316425\pi$
$379$	21.2308	1.09055	0.545276	+	0.838256i	$0.316425\pi$
$380$	−1.71544	−0.0880004
$381$	0	0
$382$	−2.27907	−0.116607
$383$	17.9127	0.915298	0.457649	−	0.889133i	$-0.348692\pi$
$383$	17.9127	0.915298	0.457649	+	0.889133i	$0.348692\pi$
$384$	0	0
$385$	−17.0804	−0.870497
$386$	−14.9652	−0.761711
$387$	0	0
$388$	−7.88580	−0.400341
$389$	−1.22591	−0.0621562	−0.0310781	−	0.999517i	$-0.509894\pi$
$389$	−1.22591	−0.0621562	−0.0310781	+	0.999517i	$0.509894\pi$
$390$	0	0
$391$	−14.0971	−0.712919
$392$	−47.1160	−2.37972
$393$	0	0
$394$	11.7662	0.592774
$395$	−2.61545	−0.131597
$396$	0	0
$397$	−28.3452	−1.42261	−0.711303	−	0.702885i	$-0.751895\pi$
$397$	−28.3452	−1.42261	−0.711303	+	0.702885i	$0.751895\pi$
$398$	21.4718	1.07628
$399$	0	0
$400$	−2.66882	−0.133441
$401$	−5.93834	−0.296547	−0.148273	−	0.988946i	$-0.547372\pi$
$401$	−5.93834	−0.296547	−0.148273	+	0.988946i	$0.547372\pi$
$402$	0	0
$403$	5.43793	0.270883
$404$	−1.58331	−0.0787728
$405$	0	0
$406$	2.93443	0.145633
$407$	32.7670	1.62420
$408$	0	0
$409$	−12.1005	−0.598330	−0.299165	−	0.954201i	$-0.596708\pi$
$409$	−12.1005	−0.598330	−0.299165	+	0.954201i	$0.596708\pi$
$410$	9.53874	0.471085
$411$	0	0
$412$	−1.48665	−0.0732422
$413$	47.0299	2.31419
$414$	0	0
$415$	−5.19278	−0.254903
$416$	4.77440	0.234084
$417$	0	0
$418$	14.2750	0.698215
$419$	−16.6219	−0.812035	−0.406017	−	0.913865i	$-0.633083\pi$
$419$	−16.6219	−0.812035	−0.406017	+	0.913865i	$0.633083\pi$
$420$	0	0
$421$	6.53089	0.318296	0.159148	−	0.987255i	$-0.449125\pi$
$421$	6.53089	0.318296	0.159148	+	0.987255i	$0.449125\pi$
$422$	20.4292	0.994480
$423$	0	0
$424$	30.9710	1.50408
$425$	−2.90942	−0.141128
$426$	0	0
$427$	−23.2733	−1.12627
$428$	−5.28722	−0.255568
$429$	0	0
$430$	−9.74055	−0.469731
$431$	14.9261	0.718965	0.359482	−	0.933152i	$-0.382953\pi$
$431$	14.9261	0.718965	0.359482	+	0.933152i	$0.382953\pi$
$432$	0	0
$433$	−4.85821	−0.233471	−0.116735	−	0.993163i	$-0.537243\pi$
$433$	−4.85821	−0.233471	−0.116735	+	0.993163i	$0.537243\pi$
$434$	−18.9229	−0.908330
$435$	0	0
$436$	−2.20240	−0.105476
$437$	−15.7774	−0.754735
$438$	0	0
$439$	27.8037	1.32700	0.663499	−	0.748177i	$-0.269071\pi$
$439$	27.8037	1.32700	0.663499	+	0.748177i	$0.269071\pi$
$440$	−11.0774	−0.528095
$441$	0	0
$442$	−5.82464	−0.277050
$443$	−14.3500	−0.681788	−0.340894	−	0.940102i	$-0.610730\pi$
$443$	−14.3500	−0.681788	−0.340894	+	0.940102i	$0.610730\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−30.3882	−1.43892
$447$	0	0
$448$	−41.8552	−1.97747
$449$	35.6757	1.68364	0.841819	−	0.539759i	$-0.181485\pi$
$449$	35.6757	1.68364	0.841819	+	0.539759i	$0.181485\pi$
$450$	0	0
$451$	28.3857	1.33663
$452$	−1.65295	−0.0777481
$453$	0	0
$454$	20.9363	0.982591
$455$	7.80003	0.365671
$456$	0	0
$457$	7.33145	0.342951	0.171475	−	0.985188i	$-0.445147\pi$
$457$	7.33145	0.342951	0.171475	+	0.985188i	$0.445147\pi$
$458$	20.6966	0.967091
$459$	0	0
$460$	2.55261	0.119016
$461$	−16.4577	−0.766513	−0.383256	−	0.923642i	$-0.625197\pi$
$461$	−16.4577	−0.766513	−0.383256	+	0.923642i	$0.625197\pi$
$462$	0	0
$463$	22.7831	1.05882	0.529410	−	0.848366i	$-0.322413\pi$
$463$	22.7831	1.05882	0.529410	+	0.848366i	$0.322413\pi$
$464$	1.36443	0.0633423
$465$	0	0
$466$	−8.23396	−0.381431
$467$	12.6671	0.586162	0.293081	−	0.956088i	$-0.405319\pi$
$467$	12.6671	0.586162	0.293081	+	0.956088i	$0.405319\pi$
$468$	0	0
$469$	−25.9922	−1.20021
$470$	−0.702316	−0.0323954
$471$	0	0
$472$	30.5010	1.40392
$473$	−28.9862	−1.33279
$474$	0	0
$475$	−3.25622	−0.149405
$476$	−7.24822	−0.332222
$477$	0	0
$478$	15.0095	0.686516
$479$	32.9371	1.50494	0.752468	−	0.658629i	$-0.228863\pi$
$479$	32.9371	1.50494	0.752468	+	0.658629i	$0.228863\pi$
$480$	0	0
$481$	−14.9636	−0.682280
$482$	3.11729	0.141989
$483$	0	0
$484$	−1.07780	−0.0489907
$485$	−14.9686	−0.679691
$486$	0	0
$487$	−2.60995	−0.118268	−0.0591342	−	0.998250i	$-0.518834\pi$
$487$	−2.60995	−0.118268	−0.0591342	+	0.998250i	$0.518834\pi$
$488$	−15.0938	−0.683264
$489$	0	0
$490$	−18.6464	−0.842356
$491$	12.8005	0.577678	0.288839	−	0.957378i	$-0.406731\pi$
$491$	12.8005	0.577678	0.288839	+	0.957378i	$0.406731\pi$
$492$	0	0
$493$	1.48744	0.0669911
$494$	−6.51892	−0.293300
$495$	0	0
$496$	−8.79869	−0.395073
$497$	−20.4374	−0.916744
$498$	0	0
$499$	40.1956	1.79940	0.899702	−	0.436505i	$-0.143784\pi$
$499$	40.1956	1.79940	0.899702	+	0.436505i	$0.143784\pi$
$500$	0.526821	0.0235602
$501$	0	0
$502$	33.8245	1.50966
$503$	−3.02716	−0.134974	−0.0674871	−	0.997720i	$-0.521498\pi$
$503$	−3.02716	−0.134974	−0.0674871	+	0.997720i	$0.521498\pi$
$504$	0	0
$505$	−3.00541	−0.133739
$506$	−21.2415	−0.944301
$507$	0	0
$508$	6.41002	0.284399
$509$	−15.5270	−0.688221	−0.344110	−	0.938929i	$-0.611819\pi$
$509$	−15.5270	−0.688221	−0.344110	+	0.938929i	$0.611819\pi$
$510$	0	0
$511$	−51.4668	−2.27676
$512$	−24.0952	−1.06487
$513$	0	0
$514$	−21.1551	−0.933111
$515$	−2.82193	−0.124349
$516$	0	0
$517$	−2.08998	−0.0919170
$518$	52.0703	2.28784
$519$	0	0
$520$	5.05868	0.221838
$521$	27.2178	1.19243	0.596217	−	0.802823i	$-0.296670\pi$
$521$	27.2178	1.19243	0.596217	+	0.802823i	$0.296670\pi$
$522$	0	0
$523$	16.9978	0.743261	0.371631	−	0.928381i	$-0.378799\pi$
$523$	16.9978	0.743261	0.371631	+	0.928381i	$0.378799\pi$
$524$	10.3234	0.450979
$525$	0	0
$526$	8.01429	0.349440
$527$	−9.59192	−0.417831
$528$	0	0
$529$	0.477052	0.0207414
$530$	12.2569	0.532405
$531$	0	0
$532$	−8.11219	−0.351708
$533$	−12.9628	−0.561480
$534$	0	0
$535$	−10.0361	−0.433898
$536$	−16.8571	−0.728118
$537$	0	0
$538$	−15.0140	−0.647298
$539$	−55.4884	−2.39006
$540$	0	0
$541$	−27.5938	−1.18635	−0.593174	−	0.805074i	$-0.702126\pi$
$541$	−27.5938	−1.18635	−0.593174	+	0.805074i	$0.702126\pi$
$542$	−13.3941	−0.575325
$543$	0	0
$544$	−8.42153	−0.361070
$545$	−4.18055	−0.179075
$546$	0	0
$547$	18.9492	0.810209	0.405104	−	0.914270i	$-0.367235\pi$
$547$	18.9492	0.810209	0.405104	+	0.914270i	$0.367235\pi$
$548$	9.38813	0.401041
$549$	0	0
$550$	−4.38393	−0.186932
$551$	1.66474	0.0709204
$552$	0	0
$553$	−12.3682	−0.525951
$554$	18.5253	0.787066
$555$	0	0
$556$	−10.0591	−0.426600
$557$	−22.0578	−0.934618	−0.467309	−	0.884094i	$-0.654776\pi$
$557$	−22.0578	−0.934618	−0.467309	+	0.884094i	$0.654776\pi$
$558$	0	0
$559$	13.2370	0.559867
$560$	−12.6206	−0.533318
$561$	0	0
$562$	−27.5286	−1.16122
$563$	−1.39711	−0.0588812	−0.0294406	−	0.999567i	$-0.509373\pi$
$563$	−1.39711	−0.0588812	−0.0294406	+	0.999567i	$0.509373\pi$
$564$	0	0
$565$	−3.13759	−0.131999
$566$	−11.3463	−0.476919
$567$	0	0
$568$	−13.2546	−0.556151
$569$	24.1092	1.01071	0.505356	−	0.862911i	$-0.331361\pi$
$569$	24.1092	1.01071	0.505356	+	0.862911i	$0.331361\pi$
$570$	0	0
$571$	24.5714	1.02828	0.514141	−	0.857706i	$-0.328111\pi$
$571$	24.5714	1.02828	0.514141	+	0.857706i	$0.328111\pi$
$572$	3.13859	0.131231
$573$	0	0
$574$	45.1079	1.88277
$575$	4.84531	0.202063
$576$	0	0
$577$	34.6069	1.44070	0.720351	−	0.693609i	$-0.243981\pi$
$577$	34.6069	1.44070	0.720351	+	0.693609i	$0.243981\pi$
$578$	−10.3596	−0.430904
$579$	0	0
$580$	−0.269338	−0.0111836
$581$	−24.5562	−1.01876
$582$	0	0
$583$	36.4744	1.51062
$584$	−33.3786	−1.38122
$585$	0	0
$586$	11.0289	0.455601
$587$	41.4631	1.71136	0.855682	−	0.517502i	$-0.173138\pi$
$587$	41.4631	1.71136	0.855682	+	0.517502i	$0.173138\pi$
$588$	0	0
$589$	−10.7353	−0.442338
$590$	12.0709	0.496951
$591$	0	0
$592$	24.2114	0.995081
$593$	−39.8769	−1.63755	−0.818774	−	0.574116i	$-0.805346\pi$
$593$	−39.8769	−1.63755	−0.818774	+	0.574116i	$0.805346\pi$
$594$	0	0
$595$	−13.7584	−0.564040
$596$	−0.396416	−0.0162378
$597$	0	0
$598$	9.70028	0.396674
$599$	47.6517	1.94699	0.973497	−	0.228700i	$-0.0734474\pi$
$599$	47.6517	1.94699	0.973497	+	0.228700i	$0.0734474\pi$
$600$	0	0
$601$	33.8623	1.38127	0.690636	−	0.723203i	$-0.257331\pi$
$601$	33.8623	1.38127	0.690636	+	0.723203i	$0.257331\pi$
$602$	−46.0622	−1.87736
$603$	0	0
$604$	−3.30969	−0.134670
$605$	−2.04585	−0.0831755
$606$	0	0
$607$	−25.6735	−1.04205	−0.521027	−	0.853540i	$-0.674451\pi$
$607$	−25.6735	−1.04205	−0.521027	+	0.853540i	$0.674451\pi$
$608$	−9.42535	−0.382249
$609$	0	0
$610$	−5.97343	−0.241857
$611$	0.954421	0.0386117
$612$	0	0
$613$	45.7836	1.84918	0.924590	−	0.380963i	$-0.124408\pi$
$613$	45.7836	1.84918	0.924590	+	0.380963i	$0.124408\pi$
$614$	−16.7360	−0.675410
$615$	0	0
$616$	−52.3842	−2.11062
$617$	20.3509	0.819295	0.409647	−	0.912244i	$-0.365652\pi$
$617$	20.3509	0.819295	0.409647	+	0.912244i	$0.365652\pi$
$618$	0	0
$619$	−26.6600	−1.07156	−0.535778	−	0.844359i	$-0.679981\pi$
$619$	−26.6600	−1.07156	−0.535778	+	0.844359i	$0.679981\pi$
$620$	1.73685	0.0697536
$621$	0	0
$622$	−26.5503	−1.06457
$623$	4.72892	0.189460
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.0045	−1.03935
$627$	0	0
$628$	3.93349	0.156963
$629$	26.3941	1.05240
$630$	0	0
$631$	7.75223	0.308611	0.154306	−	0.988023i	$-0.450686\pi$
$631$	7.75223	0.308611	0.154306	+	0.988023i	$0.450686\pi$
$632$	−8.02136	−0.319073
$633$	0	0
$634$	−6.27034	−0.249027
$635$	12.1674	0.482847
$636$	0	0
$637$	25.3397	1.00399
$638$	2.24129	0.0887334
$639$	0	0
$640$	−4.95361	−0.195809
$641$	36.7760	1.45256	0.726282	−	0.687397i	$-0.241247\pi$
$641$	36.7760	1.45256	0.726282	+	0.687397i	$0.241247\pi$
$642$	0	0
$643$	−24.8580	−0.980304	−0.490152	−	0.871637i	$-0.663059\pi$
$643$	−24.8580	−0.980304	−0.490152	+	0.871637i	$0.663059\pi$
$644$	12.0711	0.475668
$645$	0	0
$646$	11.4987	0.452409
$647$	9.22307	0.362596	0.181298	−	0.983428i	$-0.441970\pi$
$647$	9.22307	0.362596	0.181298	+	0.983428i	$0.441970\pi$
$648$	0	0
$649$	35.9210	1.41002
$650$	2.00199	0.0785246
$651$	0	0
$652$	12.9764	0.508194
$653$	39.0545	1.52832	0.764160	−	0.645026i	$-0.223154\pi$
$653$	39.0545	1.52832	0.764160	+	0.645026i	$0.223154\pi$
$654$	0	0
$655$	19.5956	0.765664
$656$	20.9740	0.818899
$657$	0	0
$658$	−3.32119	−0.129474
$659$	−14.6394	−0.570270	−0.285135	−	0.958487i	$-0.592038\pi$
$659$	−14.6394	−0.570270	−0.285135	+	0.958487i	$0.592038\pi$
$660$	0	0
$661$	−14.1834	−0.551671	−0.275835	−	0.961205i	$-0.588954\pi$
$661$	−14.1834	−0.551671	−0.275835	+	0.961205i	$0.588954\pi$
$662$	−0.551976	−0.0214532
$663$	0	0
$664$	−15.9258	−0.618042
$665$	−15.3984	−0.597123
$666$	0	0
$667$	−2.47717	−0.0959163
$668$	8.47979	0.328093
$669$	0	0
$670$	−6.67129	−0.257734
$671$	−17.7759	−0.686232
$672$	0	0
$673$	−28.3601	−1.09320	−0.546600	−	0.837394i	$-0.684078\pi$
$673$	−28.3601	−1.09320	−0.546600	+	0.837394i	$0.684078\pi$
$674$	−16.4436	−0.633385
$675$	0	0
$676$	5.41539	0.208284
$677$	−25.7041	−0.987890	−0.493945	−	0.869493i	$-0.664446\pi$
$677$	−25.7041	−0.987890	−0.493945	+	0.869493i	$0.664446\pi$
$678$	0	0
$679$	−70.7855	−2.71650
$680$	−8.92296	−0.342180
$681$	0	0
$682$	−14.4532	−0.553440
$683$	−29.0608	−1.11198	−0.555991	−	0.831188i	$-0.687661\pi$
$683$	−29.0608	−1.11198	−0.555991	+	0.831188i	$0.687661\pi$
$684$	0	0
$685$	17.8203	0.680880
$686$	−47.9991	−1.83261
$687$	0	0
$688$	−21.4178	−0.816545
$689$	−16.6566	−0.634567
$690$	0	0
$691$	43.2863	1.64669	0.823344	−	0.567543i	$-0.192106\pi$
$691$	43.2863	1.64669	0.823344	+	0.567543i	$0.192106\pi$
$692$	0.527075	0.0200364
$693$	0	0
$694$	−0.978400	−0.0371395
$695$	−19.0939	−0.724274
$696$	0	0
$697$	22.8649	0.866072
$698$	−17.6731	−0.668936
$699$	0	0
$700$	2.49129	0.0941620
$701$	−34.0749	−1.28699	−0.643495	−	0.765450i	$-0.722516\pi$
$701$	−34.0749	−1.28699	−0.643495	+	0.765450i	$0.722516\pi$
$702$	0	0
$703$	29.5402	1.11413
$704$	−31.9686	−1.20486
$705$	0	0
$706$	42.5920	1.60297
$707$	−14.2123	−0.534510
$708$	0	0
$709$	5.94934	0.223432	0.111716	−	0.993740i	$-0.464365\pi$
$709$	5.94934	0.223432	0.111716	+	0.993740i	$0.464365\pi$
$710$	−5.24557	−0.196863
$711$	0	0
$712$	3.06692	0.114938
$713$	15.9743	0.598241
$714$	0	0
$715$	5.95759	0.222801
$716$	−10.4516	−0.390594
$717$	0	0
$718$	9.47415	0.353572
$719$	5.26644	0.196405	0.0982025	−	0.995166i	$-0.468691\pi$
$719$	5.26644	0.196405	0.0982025	+	0.995166i	$0.468691\pi$
$720$	0	0
$721$	−13.3447	−0.496982
$722$	−10.1919	−0.379303
$723$	0	0
$724$	−5.93006	−0.220389
$725$	−0.511250	−0.0189874
$726$	0	0
$727$	53.1707	1.97199	0.985996	−	0.166769i	$-0.0533336\pi$
$727$	53.1707	1.97199	0.985996	+	0.166769i	$0.0533336\pi$
$728$	23.9221	0.886611
$729$	0	0
$730$	−13.2097	−0.488913
$731$	−23.3487	−0.863582
$732$	0	0
$733$	25.8482	0.954726	0.477363	−	0.878706i	$-0.341593\pi$
$733$	25.8482	0.954726	0.477363	+	0.878706i	$0.341593\pi$
$734$	−2.27212	−0.0838656
$735$	0	0
$736$	14.0251	0.516972
$737$	−19.8526	−0.731281
$738$	0	0
$739$	28.8334	1.06065	0.530326	−	0.847794i	$-0.322070\pi$
$739$	28.8334	1.06065	0.530326	+	0.847794i	$0.322070\pi$
$740$	−4.77930	−0.175690
$741$	0	0
$742$	57.9618	2.12784
$743$	−50.4944	−1.85246	−0.926230	−	0.376960i	$-0.876969\pi$
$743$	−50.4944	−1.85246	−0.926230	+	0.376960i	$0.876969\pi$
$744$	0	0
$745$	−0.752469	−0.0275683
$746$	33.3161	1.21979
$747$	0	0
$748$	−5.53613	−0.202421
$749$	−47.4598	−1.73414
$750$	0	0
$751$	−44.6027	−1.62757	−0.813787	−	0.581163i	$-0.802598\pi$
$751$	−44.6027	−1.62757	−0.813787	+	0.581163i	$0.802598\pi$
$752$	−1.54427	−0.0563138
$753$	0	0
$754$	−1.02352	−0.0372744
$755$	−6.28238	−0.228639
$756$	0	0
$757$	−9.29110	−0.337691	−0.168845	−	0.985643i	$-0.554004\pi$
$757$	−9.29110	−0.337691	−0.168845	+	0.985643i	$0.554004\pi$
$758$	25.7688	0.935964
$759$	0	0
$760$	−9.98655	−0.362250
$761$	−22.9912	−0.833432	−0.416716	−	0.909037i	$-0.636819\pi$
$761$	−22.9912	−0.833432	−0.416716	+	0.909037i	$0.636819\pi$
$762$	0	0
$763$	−19.7694	−0.715702
$764$	0.989219	0.0357887
$765$	0	0
$766$	21.7415	0.785553
$767$	−16.4039	−0.592310
$768$	0	0
$769$	1.12510	0.0405723	0.0202862	−	0.999794i	$-0.493542\pi$
$769$	1.12510	0.0405723	0.0202862	+	0.999794i	$0.493542\pi$
$770$	−20.7313	−0.747102
$771$	0	0
$772$	6.49560	0.233782
$773$	22.2413	0.799964	0.399982	−	0.916523i	$-0.369016\pi$
$773$	22.2413	0.799964	0.399982	+	0.916523i	$0.369016\pi$
$774$	0	0
$775$	3.29685	0.118426
$776$	−45.9076	−1.64799
$777$	0	0
$778$	−1.48795	−0.0533454
$779$	25.5904	0.916871
$780$	0	0
$781$	−15.6099	−0.558567
$782$	−17.1102	−0.611861
$783$	0	0
$784$	−41.0001	−1.46429
$785$	7.46647	0.266490
$786$	0	0
$787$	−40.9275	−1.45891	−0.729454	−	0.684030i	$-0.760226\pi$
$787$	−40.9275	−1.45891	−0.729454	+	0.684030i	$0.760226\pi$
$788$	−5.10708	−0.181932
$789$	0	0
$790$	−3.17449	−0.112943
$791$	−14.8374	−0.527557
$792$	0	0
$793$	8.11766	0.288267
$794$	−34.4039	−1.22095
$795$	0	0
$796$	−9.31975	−0.330330
$797$	1.38487	0.0490546	0.0245273	−	0.999699i	$-0.492192\pi$
$797$	1.38487	0.0490546	0.0245273	+	0.999699i	$0.492192\pi$
$798$	0	0
$799$	−1.68349	−0.0595578
$800$	2.89457	0.102339
$801$	0	0
$802$	−7.20764	−0.254510
$803$	−39.3099	−1.38722
$804$	0	0
$805$	22.9131	0.807580
$806$	6.60027	0.232485
$807$	0	0
$808$	−9.21735	−0.324265
$809$	41.6553	1.46452	0.732262	−	0.681023i	$-0.238465\pi$
$809$	41.6553	1.46452	0.732262	+	0.681023i	$0.238465\pi$
$810$	0	0
$811$	−23.9645	−0.841506	−0.420753	−	0.907175i	$-0.638234\pi$
$811$	−23.9645	−0.841506	−0.420753	+	0.907175i	$0.638234\pi$
$812$	−1.27367	−0.0446972
$813$	0	0
$814$	39.7708	1.39397
$815$	24.6314	0.862802
$816$	0	0
$817$	−26.1318	−0.914235
$818$	−14.6869	−0.513516
$819$	0	0
$820$	−4.14025	−0.144584
$821$	−15.9554	−0.556847	−0.278423	−	0.960458i	$-0.589812\pi$
$821$	−15.9554	−0.556847	−0.278423	+	0.960458i	$0.589812\pi$
$822$	0	0
$823$	14.4188	0.502606	0.251303	−	0.967908i	$-0.419141\pi$
$823$	14.4188	0.502606	0.251303	+	0.967908i	$0.419141\pi$
$824$	−8.65464	−0.301499
$825$	0	0
$826$	57.0823	1.98615
$827$	−32.3346	−1.12439	−0.562193	−	0.827006i	$-0.690042\pi$
$827$	−32.3346	−1.12439	−0.562193	+	0.827006i	$0.690042\pi$
$828$	0	0
$829$	−16.9180	−0.587588	−0.293794	−	0.955869i	$-0.594918\pi$
$829$	−16.9180	−0.587588	−0.293794	+	0.955869i	$0.594918\pi$
$830$	−6.30271	−0.218770
$831$	0	0
$832$	14.5990	0.506129
$833$	−44.6964	−1.54864
$834$	0	0
$835$	16.0961	0.557030
$836$	−6.19602	−0.214294
$837$	0	0
$838$	−20.1748	−0.696927
$839$	44.7765	1.54586	0.772928	−	0.634494i	$-0.218791\pi$
$839$	44.7765	1.54586	0.772928	+	0.634494i	$0.218791\pi$
$840$	0	0
$841$	−28.7386	−0.990987
$842$	7.92684	0.273177
$843$	0	0
$844$	−8.86722	−0.305222
$845$	10.2794	0.353621
$846$	0	0
$847$	−9.67464	−0.332425
$848$	26.9508	0.925493
$849$	0	0
$850$	−3.53130	−0.121123
$851$	−43.9564	−1.50681
$852$	0	0
$853$	36.9126	1.26386	0.631932	−	0.775024i	$-0.282262\pi$
$853$	36.9126	1.26386	0.631932	+	0.775024i	$0.282262\pi$
$854$	−28.2479	−0.966622
$855$	0	0
$856$	−30.7799	−1.05203
$857$	38.0376	1.29934	0.649669	−	0.760217i	$-0.274907\pi$
$857$	38.0376	1.29934	0.649669	+	0.760217i	$0.274907\pi$
$858$	0	0
$859$	4.35827	0.148702	0.0743511	−	0.997232i	$-0.476311\pi$
$859$	4.35827	0.148702	0.0743511	+	0.997232i	$0.476311\pi$
$860$	4.22784	0.144168
$861$	0	0
$862$	18.1165	0.617050
$863$	48.8249	1.66202	0.831009	−	0.556260i	$-0.187764\pi$
$863$	48.8249	1.66202	0.831009	+	0.556260i	$0.187764\pi$
$864$	0	0
$865$	1.00048	0.0340174
$866$	−5.89663	−0.200376
$867$	0	0
$868$	8.21342	0.278782
$869$	−9.44674	−0.320459
$870$	0	0
$871$	9.06602	0.307190
$872$	−12.8214	−0.434187
$873$	0	0
$874$	−19.1497	−0.647749
$875$	4.72892	0.159867
$876$	0	0
$877$	−38.1554	−1.28841	−0.644207	−	0.764851i	$-0.722813\pi$
$877$	−38.1554	−1.28841	−0.644207	+	0.764851i	$0.722813\pi$
$878$	33.7466	1.13889
$879$	0	0
$880$	−9.63951	−0.324948
$881$	17.9926	0.606185	0.303092	−	0.952961i	$-0.401981\pi$
$881$	17.9926	0.606185	0.303092	+	0.952961i	$0.401981\pi$
$882$	0	0
$883$	−10.5853	−0.356224	−0.178112	−	0.984010i	$-0.556999\pi$
$883$	−10.5853	−0.356224	−0.178112	+	0.984010i	$0.556999\pi$
$884$	2.52816	0.0850313
$885$	0	0
$886$	−17.4172	−0.585143
$887$	−45.0407	−1.51232	−0.756160	−	0.654387i	$-0.772927\pi$
$887$	−45.0407	−1.51232	−0.756160	+	0.654387i	$0.772927\pi$
$888$	0	0
$889$	57.5384	1.92978
$890$	1.21375	0.0406849
$891$	0	0
$892$	13.1898	0.441629
$893$	−1.88416	−0.0630511
$894$	0	0
$895$	−19.8389	−0.663143
$896$	−23.4252	−0.782581
$897$	0	0
$898$	43.3012	1.44498
$899$	−1.68552	−0.0562151
$900$	0	0
$901$	29.3805	0.978806
$902$	34.4530	1.14716
$903$	0	0
$904$	−9.62272	−0.320047
$905$	−11.2563	−0.374172
$906$	0	0
$907$	55.9603	1.85813	0.929065	−	0.369917i	$-0.120614\pi$
$907$	55.9603	1.85813	0.929065	+	0.369917i	$0.120614\pi$
$908$	−9.08733	−0.301574
$909$	0	0
$910$	9.46726	0.313836
$911$	27.8414	0.922425	0.461213	−	0.887290i	$-0.347415\pi$
$911$	27.8414	0.922425	0.461213	+	0.887290i	$0.347415\pi$
$912$	0	0
$913$	−18.7558	−0.620727
$914$	8.89851	0.294337
$915$	0	0
$916$	−8.98329	−0.296816
$917$	92.6661	3.06010
$918$	0	0
$919$	31.9709	1.05462	0.527311	−	0.849672i	$-0.323200\pi$
$919$	31.9709	1.05462	0.527311	+	0.849672i	$0.323200\pi$
$920$	14.8602	0.489926
$921$	0	0
$922$	−19.9755	−0.657858
$923$	7.12852	0.234638
$924$	0	0
$925$	−9.07195	−0.298284
$926$	27.6529	0.908729
$927$	0	0
$928$	−1.47985	−0.0485785
$929$	−25.4843	−0.836111	−0.418056	−	0.908421i	$-0.637288\pi$
$929$	−25.4843	−0.836111	−0.418056	+	0.908421i	$0.637288\pi$
$930$	0	0
$931$	−50.0241	−1.63947
$932$	3.57391	0.117067
$933$	0	0
$934$	15.3746	0.503072
$935$	−10.5086	−0.343666
$936$	0	0
$937$	26.5943	0.868799	0.434400	−	0.900720i	$-0.356961\pi$
$937$	26.5943	0.868799	0.434400	+	0.900720i	$0.356961\pi$
$938$	−31.5480	−1.03008
$939$	0	0
$940$	0.304837	0.00994270
$941$	−10.2341	−0.333622	−0.166811	−	0.985989i	$-0.553347\pi$
$941$	−10.2341	−0.333622	−0.166811	+	0.985989i	$0.553347\pi$
$942$	0	0
$943$	−38.0790	−1.24002
$944$	26.5418	0.863863
$945$	0	0
$946$	−35.1819	−1.14386
$947$	−2.02878	−0.0659265	−0.0329633	−	0.999457i	$-0.510494\pi$
$947$	−2.02878	−0.0659265	−0.0329633	+	0.999457i	$0.510494\pi$
$948$	0	0
$949$	17.9515	0.582730
$950$	−3.95222	−0.128227
$951$	0	0
$952$	−42.1959	−1.36758
$953$	−11.2905	−0.365736	−0.182868	−	0.983137i	$-0.558538\pi$
$953$	−11.2905	−0.365736	−0.182868	+	0.983137i	$0.558538\pi$
$954$	0	0
$955$	1.87771	0.0607613
$956$	−6.51479	−0.210703
$957$	0	0
$958$	39.9773	1.29161
$959$	84.2708	2.72125
$960$	0	0
$961$	−20.1308	−0.649380
$962$	−18.1620	−0.585566
$963$	0	0
$964$	−1.35305	−0.0435788
$965$	12.3298	0.396910
$966$	0	0
$967$	−54.4113	−1.74975	−0.874874	−	0.484350i	$-0.839056\pi$
$967$	−54.4113	−1.74975	−0.874874	+	0.484350i	$0.839056\pi$
$968$	−6.27445	−0.201668
$969$	0	0
$970$	−18.1681	−0.583344
$971$	−43.4681	−1.39496	−0.697478	−	0.716606i	$-0.745695\pi$
$971$	−43.4681	−1.39496	−0.697478	+	0.716606i	$0.745695\pi$
$972$	0	0
$973$	−90.2936	−2.89468
$974$	−3.16782	−0.101504
$975$	0	0
$976$	−13.1345	−0.420426
$977$	−51.9628	−1.66244	−0.831219	−	0.555945i	$-0.812356\pi$
$977$	−51.9628	−1.66244	−0.831219	+	0.555945i	$0.812356\pi$
$978$	0	0
$979$	3.61190	0.115437
$980$	8.09337	0.258533
$981$	0	0
$982$	15.5365	0.495791
$983$	16.0564	0.512118	0.256059	−	0.966661i	$-0.417576\pi$
$983$	16.0564	0.512118	0.256059	+	0.966661i	$0.417576\pi$
$984$	0	0
$985$	−9.69414	−0.308881
$986$	1.80538	0.0574950
$987$	0	0
$988$	2.82951	0.0900187
$989$	38.8846	1.23646
$990$	0	0
$991$	1.35238	0.0429599	0.0214799	−	0.999769i	$-0.493162\pi$
$991$	1.35238	0.0429599	0.0214799	+	0.999769i	$0.493162\pi$
$992$	9.54297	0.302989
$993$	0	0
$994$	−24.8058	−0.786793
$995$	−17.6905	−0.560828
$996$	0	0
$997$	51.8209	1.64118	0.820592	−	0.571515i	$-0.193644\pi$
$997$	51.8209	1.64118	0.820592	+	0.571515i	$0.193644\pi$
$998$	48.7873	1.54433
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.k.1.3	✓	10	3.2	odd	2
4005.2.a.r.1.8		10	1.1	even	1	trivial
6675.2.a.y.1.8		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)

\(f(q)\)	\(=\)	\(q+1.21375 q^{2} -0.526821 q^{4} -1.00000 q^{5} -4.72892 q^{7} -3.06692 q^{8} +O(q^{10})\)	\(q+1.21375 q^{2} -0.526821 q^{4} -1.00000 q^{5} -4.72892 q^{7} -3.06692 q^{8} -1.21375 q^{10} -3.61190 q^{11} +1.64943 q^{13} -5.73970 q^{14} -2.66882 q^{16} -2.90942 q^{17} -3.25622 q^{19} +0.526821 q^{20} -4.38393 q^{22} +4.84531 q^{23} +1.00000 q^{25} +2.00199 q^{26} +2.49129 q^{28} -0.511250 q^{29} +3.29685 q^{31} +2.89457 q^{32} -3.53130 q^{34} +4.72892 q^{35} -9.07195 q^{37} -3.95222 q^{38} +3.06692 q^{40} -7.85893 q^{41} +8.02520 q^{43} +1.90283 q^{44} +5.88098 q^{46} +0.578635 q^{47} +15.3626 q^{49} +1.21375 q^{50} -0.868956 q^{52} -10.0984 q^{53} +3.61190 q^{55} +14.5032 q^{56} -0.620528 q^{58} -9.94517 q^{59} +4.92149 q^{61} +4.00154 q^{62} +8.85091 q^{64} -1.64943 q^{65} +5.49644 q^{67} +1.53275 q^{68} +5.73970 q^{70} +4.32180 q^{71} +10.8834 q^{73} -11.0110 q^{74} +1.71544 q^{76} +17.0804 q^{77} +2.61545 q^{79} +2.66882 q^{80} -9.53874 q^{82} +5.19278 q^{83} +2.90942 q^{85} +9.74055 q^{86} +11.0774 q^{88} -1.00000 q^{89} -7.80003 q^{91} -2.55261 q^{92} +0.702316 q^{94} +3.25622 q^{95} +14.9686 q^{97} +18.6464 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{2} + 14 q^{4} - 10 q^{5} + 7 q^{7} - 15 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more