LMFDB - Embedded newform 4005.2.a.n.1.6

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:	$6$
Coefficient field:	6.6.10407557.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 2x^{3} + 18x^{2} + 7x - 2 $ x^6 - x^5 - 8x^4 + 2x^3 + 18x^2 + 7x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1335)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.72426$ 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)

$f(q)$	$=$	$q+2.69734 q^{2} +5.27566 q^{4} +1.00000 q^{5} -1.40763 q^{7} +8.83558 q^{8} +O(q^{10})$	$q+2.69734 q^{2} +5.27566 q^{4} +1.00000 q^{5} -1.40763 q^{7} +8.83558 q^{8} +2.69734 q^{10} +0.521859 q^{13} -3.79687 q^{14} +13.2813 q^{16} +0.132621 q^{17} +2.00000 q^{19} +5.27566 q^{20} +1.47334 q^{23} +1.00000 q^{25} +1.40763 q^{26} -7.42620 q^{28} +2.38444 q^{29} +4.81527 q^{31} +18.1530 q^{32} +0.357725 q^{34} -1.40763 q^{35} +1.86738 q^{37} +5.39469 q^{38} +8.83558 q^{40} +4.61573 q^{41} -1.85778 q^{43} +3.97411 q^{46} +0.250998 q^{47} -5.01857 q^{49} +2.69734 q^{50} +2.75315 q^{52} -7.68411 q^{53} -12.4373 q^{56} +6.43165 q^{58} -0.00479999 q^{59} -4.54043 q^{61} +12.9884 q^{62} +22.4023 q^{64} +0.521859 q^{65} +8.45217 q^{67} +0.699664 q^{68} -3.79687 q^{70} -11.0917 q^{71} +11.2929 q^{73} +5.03696 q^{74} +10.5513 q^{76} +2.16274 q^{79} +13.2813 q^{80} +12.4502 q^{82} +4.28861 q^{83} +0.132621 q^{85} -5.01107 q^{86} +1.00000 q^{89} -0.734587 q^{91} +7.77285 q^{92} +0.677028 q^{94} +2.00000 q^{95} -9.21900 q^{97} -13.5368 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	$ 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) $ 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * 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 $

$2$	2.69734	1.90731	0.953655	−	0.300903i	$-0.0972879\pi$
$2$	2.69734	1.90731	0.953655	+	0.300903i	$0.0972879\pi$
$3$	0	0
$3$	0	0
$4$	5.27566	2.63783
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.40763	−0.532036	−0.266018	−	0.963968i	$-0.585708\pi$
$7$	−1.40763	−0.532036	−0.266018	+	0.963968i	$0.585708\pi$
$8$	8.83558	3.12385
$9$	0	0
$10$	2.69734	0.852975
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0.521859	0.144738	0.0723689	−	0.997378i	$-0.476944\pi$
$13$	0.521859	0.144738	0.0723689	+	0.997378i	$0.476944\pi$
$14$	−3.79687	−1.01476
$15$	0	0
$16$	13.2813	3.32032
$17$	0.132621	0.0321653	0.0160827	−	0.999871i	$-0.494881\pi$
$17$	0.132621	0.0321653	0.0160827	+	0.999871i	$0.494881\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	5.27566	1.17967
$21$	0	0
$22$	0	0
$23$	1.47334	0.307213	0.153606	−	0.988132i	$-0.450911\pi$
$23$	1.47334	0.307213	0.153606	+	0.988132i	$0.450911\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.40763	0.276060
$27$	0	0
$28$	−7.42620	−1.40342
$29$	2.38444	0.442779	0.221390	−	0.975185i	$-0.428941\pi$
$29$	2.38444	0.442779	0.221390	+	0.975185i	$0.428941\pi$
$30$	0	0
$31$	4.81527	0.864848	0.432424	−	0.901670i	$-0.357658\pi$
$31$	4.81527	0.864848	0.432424	+	0.901670i	$0.357658\pi$
$32$	18.1530	3.20903
$33$	0	0
$34$	0.357725	0.0613493
$35$	−1.40763	−0.237934
$36$	0	0
$37$	1.86738	0.306995	0.153498	−	0.988149i	$-0.450946\pi$
$37$	1.86738	0.306995	0.153498	+	0.988149i	$0.450946\pi$
$38$	5.39469	0.875134
$39$	0	0
$40$	8.83558	1.39703
$41$	4.61573	0.720856	0.360428	−	0.932787i	$-0.382631\pi$
$41$	4.61573	0.720856	0.360428	+	0.932787i	$0.382631\pi$
$42$	0	0
$43$	−1.85778	−0.283309	−0.141654	−	0.989916i	$-0.545242\pi$
$43$	−1.85778	−0.283309	−0.141654	+	0.989916i	$0.545242\pi$
$44$	0	0
$45$	0	0
$46$	3.97411	0.585950
$47$	0.250998	0.0366118	0.0183059	−	0.999832i	$-0.494173\pi$
$47$	0.250998	0.0366118	0.0183059	+	0.999832i	$0.494173\pi$
$48$	0	0
$49$	−5.01857	−0.716938
$50$	2.69734	0.381462
$51$	0	0
$52$	2.75315	0.381794
$53$	−7.68411	−1.05549	−0.527747	−	0.849402i	$-0.676963\pi$
$53$	−7.68411	−1.05549	−0.527747	+	0.849402i	$0.676963\pi$
$54$	0	0
$55$	0	0
$56$	−12.4373	−1.66200
$57$	0	0
$58$	6.43165	0.844517
$59$	−0.00479999	−0.000624905 0	−0.000312452	−	1.00000i	$-0.500099\pi$
$59$	−0.00479999	−0.000624905 0	−0.000312452		1.00000i	$0.500099\pi$
$60$	0	0
$61$	−4.54043	−0.581342	−0.290671	−	0.956823i	$-0.593879\pi$
$61$	−4.54043	−0.581342	−0.290671	+	0.956823i	$0.593879\pi$
$62$	12.9884	1.64953
$63$	0	0
$64$	22.4023	2.80029
$65$	0.521859	0.0647287
$66$	0	0
$67$	8.45217	1.03260	0.516299	−	0.856409i	$-0.327309\pi$
$67$	8.45217	1.03260	0.516299	+	0.856409i	$0.327309\pi$
$68$	0.699664	0.0848467
$69$	0	0
$70$	−3.79687	−0.453813
$71$	−11.0917	−1.31635	−0.658174	−	0.752866i	$-0.728671\pi$
$71$	−11.0917	−1.31635	−0.658174	+	0.752866i	$0.728671\pi$
$72$	0	0
$73$	11.2929	1.32174	0.660868	−	0.750502i	$-0.270188\pi$
$73$	11.2929	1.32174	0.660868	+	0.750502i	$0.270188\pi$
$74$	5.03696	0.585535
$75$	0	0
$76$	10.5513	1.21032
$77$	0	0
$78$	0	0
$79$	2.16274	0.243328	0.121664	−	0.992571i	$-0.461177\pi$
$79$	2.16274	0.243328	0.121664	+	0.992571i	$0.461177\pi$
$80$	13.2813	1.48489
$81$	0	0
$82$	12.4502	1.37490
$83$	4.28861	0.470736	0.235368	−	0.971906i	$-0.424370\pi$
$83$	4.28861	0.470736	0.235368	+	0.971906i	$0.424370\pi$
$84$	0	0
$85$	0.132621	0.0143848
$86$	−5.01107	−0.540357
$87$	0	0
$88$	0	0
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−0.734587	−0.0770056
$92$	7.77285	0.810375
$93$	0	0
$94$	0.677028	0.0698301
$95$	2.00000	0.205196
$96$	0	0
$97$	−9.21900	−0.936047	−0.468024	−	0.883716i	$-0.655034\pi$
$97$	−9.21900	−0.936047	−0.468024	+	0.883716i	$0.655034\pi$
$98$	−13.5368	−1.36742
$99$	0	0
$100$	5.27566	0.527566
$101$	4.75991	0.473629	0.236814	−	0.971555i	$-0.423897\pi$
$101$	4.75991	0.473629	0.236814	+	0.971555i	$0.423897\pi$
$102$	0	0
$103$	−9.74372	−0.960078	−0.480039	−	0.877247i	$-0.659377\pi$
$103$	−9.74372	−0.960078	−0.480039	+	0.877247i	$0.659377\pi$
$104$	4.61093	0.452139
$105$	0	0
$106$	−20.7267	−2.01315
$107$	15.8121	1.52861	0.764306	−	0.644853i	$-0.223082\pi$
$107$	15.8121	1.52861	0.764306	+	0.644853i	$0.223082\pi$
$108$	0	0
$109$	−17.9118	−1.71564	−0.857820	−	0.513949i	$-0.828182\pi$
$109$	−17.9118	−1.71564	−0.857820	+	0.513949i	$0.828182\pi$
$110$	0	0
$111$	0	0
$112$	−18.6952	−1.76653
$113$	−0.253114	−0.0238110	−0.0119055	−	0.999929i	$-0.503790\pi$
$113$	−0.253114	−0.0238110	−0.0119055	+	0.999929i	$0.503790\pi$
$114$	0	0
$115$	1.47334	0.137390
$116$	12.5795	1.16798
$117$	0	0
$118$	−0.0129472	−0.00119189
$119$	−0.186682	−0.0171131
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−12.2471	−1.10880
$123$	0	0
$124$	25.4037	2.28132
$125$	1.00000	0.0894427
$126$	0	0
$127$	−15.4365	−1.36977	−0.684885	−	0.728651i	$-0.740148\pi$
$127$	−15.4365	−1.36977	−0.684885	+	0.728651i	$0.740148\pi$
$128$	24.1208	2.13200
$129$	0	0
$130$	1.40763	0.123458
$131$	−15.5447	−1.35815	−0.679075	−	0.734069i	$-0.737619\pi$
$131$	−15.5447	−1.35815	−0.679075	+	0.734069i	$0.737619\pi$
$132$	0	0
$133$	−2.81527	−0.244115
$134$	22.7984	1.96948
$135$	0	0
$136$	1.17178	0.100480
$137$	15.5076	1.32490	0.662452	−	0.749104i	$-0.269516\pi$
$137$	15.5076	1.32490	0.662452	+	0.749104i	$0.269516\pi$
$138$	0	0
$139$	−3.01302	−0.255561	−0.127781	−	0.991802i	$-0.540785\pi$
$139$	−3.01302	−0.255561	−0.127781	+	0.991802i	$0.540785\pi$
$140$	−7.42620	−0.627628
$141$	0	0
$142$	−29.9183	−2.51068
$143$	0	0
$144$	0	0
$145$	2.38444	0.198017
$146$	30.4609	2.52096
$147$	0	0
$148$	9.85166	0.809801
$149$	5.79621	0.474844	0.237422	−	0.971407i	$-0.423698\pi$
$149$	5.79621	0.474844	0.237422	+	0.971407i	$0.423698\pi$
$150$	0	0
$151$	−13.3407	−1.08565	−0.542825	−	0.839846i	$-0.682645\pi$
$151$	−13.3407	−1.08565	−0.542825	+	0.839846i	$0.682645\pi$
$152$	17.6712	1.43332
$153$	0	0
$154$	0	0
$155$	4.81527	0.386772
$156$	0	0
$157$	−19.1395	−1.52750	−0.763750	−	0.645512i	$-0.776644\pi$
$157$	−19.1395	−1.52750	−0.763750	+	0.645512i	$0.776644\pi$
$158$	5.83366	0.464101
$159$	0	0
$160$	18.1530	1.43512
$161$	−2.07392	−0.163448
$162$	0	0
$163$	4.02946	0.315612	0.157806	−	0.987470i	$-0.449558\pi$
$163$	4.02946	0.315612	0.157806	+	0.987470i	$0.449558\pi$
$164$	24.3510	1.90150
$165$	0	0
$166$	11.5679	0.897839
$167$	−5.22159	−0.404059	−0.202030	−	0.979379i	$-0.564754\pi$
$167$	−5.22159	−0.404059	−0.202030	+	0.979379i	$0.564754\pi$
$168$	0	0
$169$	−12.7277	−0.979051
$170$	0.357725	0.0274362
$171$	0	0
$172$	−9.80101	−0.747320
$173$	3.24969	0.247069	0.123535	−	0.992340i	$-0.460577\pi$
$173$	3.24969	0.247069	0.123535	+	0.992340i	$0.460577\pi$
$174$	0	0
$175$	−1.40763	−0.106407
$176$	0	0
$177$	0	0
$178$	2.69734	0.202174
$179$	−14.2397	−1.06432	−0.532162	−	0.846642i	$-0.678620\pi$
$179$	−14.2397	−1.06432	−0.532162	+	0.846642i	$0.678620\pi$
$180$	0	0
$181$	−16.4212	−1.22058	−0.610289	−	0.792179i	$-0.708947\pi$
$181$	−16.4212	−1.22058	−0.610289	+	0.792179i	$0.708947\pi$
$182$	−1.98143	−0.146874
$183$	0	0
$184$	13.0178	0.959687
$185$	1.86738	0.137292
$186$	0	0
$187$	0	0
$188$	1.32418	0.0965758
$189$	0	0
$190$	5.39469	0.391372
$191$	25.6111	1.85315	0.926576	−	0.376107i	$-0.122737\pi$
$191$	25.6111	1.85315	0.926576	+	0.376107i	$0.122737\pi$
$192$	0	0
$193$	4.46225	0.321200	0.160600	−	0.987020i	$-0.448657\pi$
$193$	4.46225	0.321200	0.160600	+	0.987020i	$0.448657\pi$
$194$	−24.8668	−1.78533
$195$	0	0
$196$	−26.4763	−1.89116
$197$	−15.5188	−1.10567	−0.552836	−	0.833290i	$-0.686454\pi$
$197$	−15.5188	−1.10567	−0.552836	+	0.833290i	$0.686454\pi$
$198$	0	0
$199$	8.10960	0.574874	0.287437	−	0.957800i	$-0.407197\pi$
$199$	8.10960	0.574874	0.287437	+	0.957800i	$0.407197\pi$
$200$	8.83558	0.624770
$201$	0	0
$202$	12.8391	0.903357
$203$	−3.35642	−0.235574
$204$	0	0
$205$	4.61573	0.322377
$206$	−26.2822	−1.83117
$207$	0	0
$208$	6.93096	0.480576
$209$	0	0
$210$	0	0
$211$	−17.7841	−1.22431	−0.612153	−	0.790739i	$-0.709697\pi$
$211$	−17.7841	−1.22431	−0.612153	+	0.790739i	$0.709697\pi$
$212$	−40.5388	−2.78422
$213$	0	0
$214$	42.6506	2.91554
$215$	−1.85778	−0.126699
$216$	0	0
$217$	−6.77814	−0.460130
$218$	−48.3143	−3.27226
$219$	0	0
$220$	0	0
$221$	0.0692096	0.00465554
$222$	0	0
$223$	5.99789	0.401648	0.200824	−	0.979627i	$-0.435638\pi$
$223$	5.99789	0.401648	0.200824	+	0.979627i	$0.435638\pi$
$224$	−25.5528	−1.70732
$225$	0	0
$226$	−0.682736	−0.0454149
$227$	21.5318	1.42912	0.714558	−	0.699577i	$-0.246628\pi$
$227$	21.5318	1.42912	0.714558	+	0.699577i	$0.246628\pi$
$228$	0	0
$229$	12.7539	0.842800	0.421400	−	0.906875i	$-0.361539\pi$
$229$	12.7539	0.842800	0.421400	+	0.906875i	$0.361539\pi$
$230$	3.97411	0.262045
$231$	0	0
$232$	21.0679	1.38318
$233$	4.97614	0.325998	0.162999	−	0.986626i	$-0.447883\pi$
$233$	4.97614	0.325998	0.162999	+	0.986626i	$0.447883\pi$
$234$	0	0
$235$	0.250998	0.0163733
$236$	−0.0253231	−0.00164839
$237$	0	0
$238$	−0.503545	−0.0326400
$239$	0.357185	0.0231044	0.0115522	−	0.999933i	$-0.496323\pi$
$239$	0.357185	0.0231044	0.0115522	+	0.999933i	$0.496323\pi$
$240$	0	0
$241$	−16.5622	−1.06687	−0.533433	−	0.845842i	$-0.679098\pi$
$241$	−16.5622	−1.06687	−0.533433	+	0.845842i	$0.679098\pi$
$242$	−29.6708	−1.90731
$243$	0	0
$244$	−23.9537	−1.53348
$245$	−5.01857	−0.320624
$246$	0	0
$247$	1.04372	0.0664102
$248$	42.5457	2.70165
$249$	0	0
$250$	2.69734	0.170595
$251$	16.5119	1.04222	0.521111	−	0.853489i	$-0.325518\pi$
$251$	16.5119	1.04222	0.521111	+	0.853489i	$0.325518\pi$
$252$	0	0
$253$	0	0
$254$	−41.6376	−2.61258
$255$	0	0
$256$	20.2573	1.26608
$257$	−9.96985	−0.621902	−0.310951	−	0.950426i	$-0.600648\pi$
$257$	−9.96985	−0.621902	−0.310951	+	0.950426i	$0.600648\pi$
$258$	0	0
$259$	−2.62859	−0.163332
$260$	2.75315	0.170743
$261$	0	0
$262$	−41.9295	−2.59041
$263$	−21.8627	−1.34811	−0.674056	−	0.738680i	$-0.735450\pi$
$263$	−21.8627	−1.34811	−0.674056	+	0.738680i	$0.735450\pi$
$264$	0	0
$265$	−7.68411	−0.472031
$266$	−7.59374	−0.465602
$267$	0	0
$268$	44.5908	2.72382
$269$	8.42839	0.513888	0.256944	−	0.966426i	$-0.417284\pi$
$269$	8.42839	0.513888	0.256944	+	0.966426i	$0.417284\pi$
$270$	0	0
$271$	17.5340	1.06511	0.532557	−	0.846394i	$-0.321231\pi$
$271$	17.5340	1.06511	0.532557	+	0.846394i	$0.321231\pi$
$272$	1.76138	0.106799
$273$	0	0
$274$	41.8293	2.52700
$275$	0	0
$276$	0	0
$277$	−16.7676	−1.00747	−0.503735	−	0.863858i	$-0.668041\pi$
$277$	−16.7676	−1.00747	−0.503735	+	0.863858i	$0.668041\pi$
$278$	−8.12716	−0.487435
$279$	0	0
$280$	−12.4373	−0.743269
$281$	21.3190	1.27178	0.635892	−	0.771778i	$-0.280632\pi$
$281$	21.3190	1.27178	0.635892	+	0.771778i	$0.280632\pi$
$282$	0	0
$283$	13.8394	0.822665	0.411333	−	0.911485i	$-0.365063\pi$
$283$	13.8394	0.822665	0.411333	+	0.911485i	$0.365063\pi$
$284$	−58.5163	−3.47230
$285$	0	0
$286$	0	0
$287$	−6.49726	−0.383521
$288$	0	0
$289$	−16.9824	−0.998965
$290$	6.43165	0.377679
$291$	0	0
$292$	59.5776	3.48652
$293$	−4.54716	−0.265648	−0.132824	−	0.991140i	$-0.542404\pi$
$293$	−4.54716	−0.265648	−0.132824	+	0.991140i	$0.542404\pi$
$294$	0	0
$295$	−0.00479999	−0.000279466 0
$296$	16.4994	0.959007
$297$	0	0
$298$	15.6344	0.905675
$299$	0.768877	0.0444653
$300$	0	0
$301$	2.61507	0.150730
$302$	−35.9844	−2.07067
$303$	0	0
$304$	26.5626	1.52347
$305$	−4.54043	−0.259984
$306$	0	0
$307$	−10.3317	−0.589659	−0.294829	−	0.955550i	$-0.595263\pi$
$307$	−10.3317	−0.589659	−0.294829	+	0.955550i	$0.595263\pi$
$308$	0	0
$309$	0	0
$310$	12.9884	0.737693
$311$	1.90538	0.108044	0.0540220	−	0.998540i	$-0.482796\pi$
$311$	1.90538	0.108044	0.0540220	+	0.998540i	$0.482796\pi$
$312$	0	0
$313$	14.8497	0.839355	0.419678	−	0.907673i	$-0.362143\pi$
$313$	14.8497	0.839355	0.419678	+	0.907673i	$0.362143\pi$
$314$	−51.6259	−2.91342
$315$	0	0
$316$	11.4099	0.641857
$317$	−3.23397	−0.181638	−0.0908189	−	0.995867i	$-0.528948\pi$
$317$	−3.23397	−0.181638	−0.0908189	+	0.995867i	$0.528948\pi$
$318$	0	0
$319$	0	0
$320$	22.4023	1.25233
$321$	0	0
$322$	−5.59409	−0.311746
$323$	0.265242	0.0147585
$324$	0	0
$325$	0.521859	0.0289476
$326$	10.8688	0.601970
$327$	0	0
$328$	40.7827	2.25185
$329$	−0.353313	−0.0194788
$330$	0	0
$331$	20.9581	1.15196	0.575980	−	0.817464i	$-0.304621\pi$
$331$	20.9581	1.15196	0.575980	+	0.817464i	$0.304621\pi$
$332$	22.6252	1.24172
$333$	0	0
$334$	−14.0844	−0.770666
$335$	8.45217	0.461791
$336$	0	0
$337$	−7.10264	−0.386906	−0.193453	−	0.981110i	$-0.561969\pi$
$337$	−7.10264	−0.386906	−0.193453	+	0.981110i	$0.561969\pi$
$338$	−34.3309	−1.86735
$339$	0	0
$340$	0.699664	0.0379446
$341$	0	0
$342$	0	0
$343$	16.9177	0.913472
$344$	−16.4146	−0.885014
$345$	0	0
$346$	8.76553	0.471238
$347$	5.95141	0.319488	0.159744	−	0.987158i	$-0.448933\pi$
$347$	5.95141	0.319488	0.159744	+	0.987158i	$0.448933\pi$
$348$	0	0
$349$	30.2973	1.62178	0.810889	−	0.585200i	$-0.198984\pi$
$349$	30.2973	1.62178	0.810889	+	0.585200i	$0.198984\pi$
$350$	−3.79687	−0.202951
$351$	0	0
$352$	0	0
$353$	16.5366	0.880154	0.440077	−	0.897960i	$-0.354951\pi$
$353$	16.5366	0.880154	0.440077	+	0.897960i	$0.354951\pi$
$354$	0	0
$355$	−11.0917	−0.588689
$356$	5.27566	0.279609
$357$	0	0
$358$	−38.4093	−2.03000
$359$	−7.59538	−0.400869	−0.200434	−	0.979707i	$-0.564235\pi$
$359$	−7.59538	−0.400869	−0.200434	+	0.979707i	$0.564235\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−44.2936	−2.32802
$363$	0	0
$364$	−3.87543	−0.203128
$365$	11.2929	0.591098
$366$	0	0
$367$	−13.4072	−0.699851	−0.349926	−	0.936777i	$-0.613793\pi$
$367$	−13.4072	−0.699851	−0.349926	+	0.936777i	$0.613793\pi$
$368$	19.5679	1.02004
$369$	0	0
$370$	5.03696	0.261859
$371$	10.8164	0.561561
$372$	0	0
$373$	−17.4833	−0.905250	−0.452625	−	0.891701i	$-0.649512\pi$
$373$	−17.4833	−0.905250	−0.452625	+	0.891701i	$0.649512\pi$
$374$	0	0
$375$	0	0
$376$	2.21771	0.114370
$377$	1.24434	0.0640869
$378$	0	0
$379$	−32.2027	−1.65414	−0.827071	−	0.562098i	$-0.809994\pi$
$379$	−32.2027	−1.65414	−0.827071	+	0.562098i	$0.809994\pi$
$380$	10.5513	0.541271
$381$	0	0
$382$	69.0819	3.53454
$383$	8.78107	0.448692	0.224346	−	0.974510i	$-0.427975\pi$
$383$	8.78107	0.448692	0.224346	+	0.974510i	$0.427975\pi$
$384$	0	0
$385$	0	0
$386$	12.0362	0.612627
$387$	0	0
$388$	−48.6363	−2.46913
$389$	−10.2477	−0.519577	−0.259788	−	0.965666i	$-0.583653\pi$
$389$	−10.2477	−0.519577	−0.259788	+	0.965666i	$0.583653\pi$
$390$	0	0
$391$	0.195396	0.00988160
$392$	−44.3420	−2.23961
$393$	0	0
$394$	−41.8596	−2.10886
$395$	2.16274	0.108819
$396$	0	0
$397$	12.5349	0.629107	0.314554	−	0.949240i	$-0.398145\pi$
$397$	12.5349	0.629107	0.314554	+	0.949240i	$0.398145\pi$
$398$	21.8744	1.09646
$399$	0	0
$400$	13.2813	0.664064
$401$	−6.02753	−0.301001	−0.150500	−	0.988610i	$-0.548088\pi$
$401$	−6.02753	−0.301001	−0.150500	+	0.988610i	$0.548088\pi$
$402$	0	0
$403$	2.51289	0.125176
$404$	25.1117	1.24935
$405$	0	0
$406$	−9.05341	−0.449313
$407$	0	0
$408$	0	0
$409$	−32.5472	−1.60935	−0.804677	−	0.593713i	$-0.797661\pi$
$409$	−32.5472	−1.60935	−0.804677	+	0.593713i	$0.797661\pi$
$410$	12.4502	0.614872
$411$	0	0
$412$	−51.4046	−2.53252
$413$	0.00675662	0.000332472 0
$414$	0	0
$415$	4.28861	0.210520
$416$	9.47332	0.464468
$417$	0	0
$418$	0	0
$419$	−11.7278	−0.572938	−0.286469	−	0.958089i	$-0.592482\pi$
$419$	−11.7278	−0.572938	−0.286469	+	0.958089i	$0.592482\pi$
$420$	0	0
$421$	−7.79334	−0.379824	−0.189912	−	0.981801i	$-0.560820\pi$
$421$	−7.79334	−0.379824	−0.189912	+	0.981801i	$0.560820\pi$
$422$	−47.9698	−2.33513
$423$	0	0
$424$	−67.8936	−3.29721
$425$	0.132621	0.00643307
$426$	0	0
$427$	6.39126	0.309295
$428$	83.4192	4.03222
$429$	0	0
$430$	−5.01107	−0.241655
$431$	20.6701	0.995642	0.497821	−	0.867280i	$-0.334134\pi$
$431$	20.6701	0.995642	0.497821	+	0.867280i	$0.334134\pi$
$432$	0	0
$433$	13.4083	0.644360	0.322180	−	0.946678i	$-0.395584\pi$
$433$	13.4083	0.644360	0.322180	+	0.946678i	$0.395584\pi$
$434$	−18.2830	−0.877610
$435$	0	0
$436$	−94.4967	−4.52557
$437$	2.94668	0.140959
$438$	0	0
$439$	−18.2698	−0.871969	−0.435985	−	0.899954i	$-0.643600\pi$
$439$	−18.2698	−0.871969	−0.435985	+	0.899954i	$0.643600\pi$
$440$	0	0
$441$	0	0
$442$	0.186682	0.00887956
$443$	−5.81281	−0.276175	−0.138088	−	0.990420i	$-0.544096\pi$
$443$	−5.81281	−0.276175	−0.138088	+	0.990420i	$0.544096\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	16.1784	0.766068
$447$	0	0
$448$	−31.5343	−1.48985
$449$	−9.61294	−0.453663	−0.226831	−	0.973934i	$-0.572837\pi$
$449$	−9.61294	−0.453663	−0.226831	+	0.973934i	$0.572837\pi$
$450$	0	0
$451$	0	0
$452$	−1.33535	−0.0628094
$453$	0	0
$454$	58.0786	2.72577
$455$	−0.734587	−0.0344380
$456$	0	0
$457$	0.298275	0.0139527	0.00697635	−	0.999976i	$-0.497779\pi$
$457$	0.298275	0.0139527	0.00697635	+	0.999976i	$0.497779\pi$
$458$	34.4016	1.60748
$459$	0	0
$460$	7.77285	0.362411
$461$	−3.36288	−0.156625	−0.0783124	−	0.996929i	$-0.524953\pi$
$461$	−3.36288	−0.156625	−0.0783124	+	0.996929i	$0.524953\pi$
$462$	0	0
$463$	41.7524	1.94040	0.970200	−	0.242307i	$-0.0779040\pi$
$463$	41.7524	1.94040	0.970200	+	0.242307i	$0.0779040\pi$
$464$	31.6684	1.47017
$465$	0	0
$466$	13.4224	0.621779
$467$	−26.3562	−1.21962	−0.609809	−	0.792548i	$-0.708754\pi$
$467$	−26.3562	−1.21962	−0.609809	+	0.792548i	$0.708754\pi$
$468$	0	0
$469$	−11.8976	−0.549379
$470$	0.677028	0.0312290
$471$	0	0
$472$	−0.0424107	−0.00195211
$473$	0	0
$474$	0	0
$475$	2.00000	0.0917663
$476$	−0.984871	−0.0451415
$477$	0	0
$478$	0.963450	0.0440672
$479$	−6.19296	−0.282963	−0.141482	−	0.989941i	$-0.545187\pi$
$479$	−6.19296	−0.282963	−0.141482	+	0.989941i	$0.545187\pi$
$480$	0	0
$481$	0.974509	0.0444338
$482$	−44.6740	−2.03484
$483$	0	0
$484$	−58.0323	−2.63783
$485$	−9.21900	−0.418613
$486$	0	0
$487$	−29.0883	−1.31812	−0.659059	−	0.752091i	$-0.729045\pi$
$487$	−29.0883	−1.31812	−0.659059	+	0.752091i	$0.729045\pi$
$488$	−40.1173	−1.81603
$489$	0	0
$490$	−13.5368	−0.611530
$491$	−29.4712	−1.33002	−0.665009	−	0.746836i	$-0.731572\pi$
$491$	−29.4712	−1.33002	−0.665009	+	0.746836i	$0.731572\pi$
$492$	0	0
$493$	0.316227	0.0142421
$494$	2.81527	0.126665
$495$	0	0
$496$	63.9529	2.87157
$497$	15.6131	0.700344
$498$	0	0
$499$	40.7377	1.82367	0.911836	−	0.410556i	$-0.134665\pi$
$499$	40.7377	1.82367	0.911836	+	0.410556i	$0.134665\pi$
$500$	5.27566	0.235935
$501$	0	0
$502$	44.5383	1.98784
$503$	20.7132	0.923555	0.461777	−	0.886996i	$-0.347212\pi$
$503$	20.7132	0.923555	0.461777	+	0.886996i	$0.347212\pi$
$504$	0	0
$505$	4.75991	0.211813
$506$	0	0
$507$	0	0
$508$	−81.4379	−3.61322
$509$	2.21919	0.0983640	0.0491820	−	0.998790i	$-0.484339\pi$
$509$	2.21919	0.0983640	0.0491820	+	0.998790i	$0.484339\pi$
$510$	0	0
$511$	−15.8963	−0.703211
$512$	6.39945	0.282818
$513$	0	0
$514$	−26.8921	−1.18616
$515$	−9.74372	−0.429360
$516$	0	0
$517$	0	0
$518$	−7.09020	−0.311525
$519$	0	0
$520$	4.61093	0.202203
$521$	−0.423043	−0.0185338	−0.00926692	−	0.999957i	$-0.502950\pi$
$521$	−0.423043	−0.0185338	−0.00926692	+	0.999957i	$0.502950\pi$
$522$	0	0
$523$	−3.53734	−0.154677	−0.0773385	−	0.997005i	$-0.524642\pi$
$523$	−3.53734	−0.154677	−0.0773385	+	0.997005i	$0.524642\pi$
$524$	−82.0088	−3.58257
$525$	0	0
$526$	−58.9713	−2.57127
$527$	0.638606	0.0278181
$528$	0	0
$529$	−20.8293	−0.905620
$530$	−20.7267	−0.900310
$531$	0	0
$532$	−14.8524	−0.643933
$533$	2.40876	0.104335
$534$	0	0
$535$	15.8121	0.683616
$536$	74.6799	3.22568
$537$	0	0
$538$	22.7343	0.980144
$539$	0	0
$540$	0	0
$541$	24.0054	1.03207	0.516035	−	0.856567i	$-0.327407\pi$
$541$	24.0054	1.03207	0.516035	+	0.856567i	$0.327407\pi$
$542$	47.2952	2.03150
$543$	0	0
$544$	2.40747	0.103219
$545$	−17.9118	−0.767258
$546$	0	0
$547$	−5.23972	−0.224034	−0.112017	−	0.993706i	$-0.535731\pi$
$547$	−5.23972	−0.224034	−0.112017	+	0.993706i	$0.535731\pi$
$548$	81.8129	3.49487
$549$	0	0
$550$	0	0
$551$	4.76888	0.203161
$552$	0	0
$553$	−3.04435	−0.129459
$554$	−45.2281	−1.92156
$555$	0	0
$556$	−15.8957	−0.674128
$557$	0.735910	0.0311815	0.0155908	−	0.999878i	$-0.495037\pi$
$557$	0.735910	0.0311815	0.0155908	+	0.999878i	$0.495037\pi$
$558$	0	0
$559$	−0.969499	−0.0410055
$560$	−18.6952	−0.790016
$561$	0	0
$562$	57.5046	2.42569
$563$	24.3667	1.02694	0.513468	−	0.858109i	$-0.328361\pi$
$563$	24.3667	1.02694	0.513468	+	0.858109i	$0.328361\pi$
$564$	0	0
$565$	−0.253114	−0.0106486
$566$	37.3295	1.56908
$567$	0	0
$568$	−98.0021	−4.11208
$569$	−46.5854	−1.95296	−0.976482	−	0.215600i	$-0.930829\pi$
$569$	−46.5854	−1.95296	−0.976482	+	0.215600i	$0.930829\pi$
$570$	0	0
$571$	22.4554	0.939729	0.469865	−	0.882738i	$-0.344303\pi$
$571$	22.4554	0.939729	0.469865	+	0.882738i	$0.344303\pi$
$572$	0	0
$573$	0	0
$574$	−17.5253	−0.731494
$575$	1.47334	0.0614426
$576$	0	0
$577$	−9.98244	−0.415574	−0.207787	−	0.978174i	$-0.566626\pi$
$577$	−9.98244	−0.415574	−0.207787	+	0.978174i	$0.566626\pi$
$578$	−45.8074	−1.90534
$579$	0	0
$580$	12.5795	0.522335
$581$	−6.03679	−0.250448
$582$	0	0
$583$	0	0
$584$	99.7795	4.12891
$585$	0	0
$586$	−12.2652	−0.506672
$587$	14.6751	0.605708	0.302854	−	0.953037i	$-0.402061\pi$
$587$	14.6751	0.605708	0.302854	+	0.953037i	$0.402061\pi$
$588$	0	0
$589$	9.63054	0.396819
$590$	−0.0129472	−0.000533028 0
$591$	0	0
$592$	24.8012	1.01932
$593$	20.1533	0.827596	0.413798	−	0.910369i	$-0.364202\pi$
$593$	20.1533	0.827596	0.413798	+	0.910369i	$0.364202\pi$
$594$	0	0
$595$	−0.186682	−0.00765321
$596$	30.5789	1.25256
$597$	0	0
$598$	2.07392	0.0848091
$599$	−15.0616	−0.615401	−0.307701	−	0.951483i	$-0.599560\pi$
$599$	−15.0616	−0.615401	−0.307701	+	0.951483i	$0.599560\pi$
$600$	0	0
$601$	−18.0494	−0.736248	−0.368124	−	0.929777i	$-0.620000\pi$
$601$	−18.0494	−0.736248	−0.368124	+	0.929777i	$0.620000\pi$
$602$	7.05375	0.287489
$603$	0	0
$604$	−70.3810	−2.86376
$605$	−11.0000	−0.447214
$606$	0	0
$607$	29.5143	1.19795	0.598975	−	0.800768i	$-0.295575\pi$
$607$	29.5143	1.19795	0.598975	+	0.800768i	$0.295575\pi$
$608$	36.3060	1.47240
$609$	0	0
$610$	−12.2471	−0.495870
$611$	0.130986	0.00529911
$612$	0	0
$613$	10.0727	0.406832	0.203416	−	0.979092i	$-0.434796\pi$
$613$	10.0727	0.406832	0.203416	+	0.979092i	$0.434796\pi$
$614$	−27.8680	−1.12466
$615$	0	0
$616$	0	0
$617$	15.7509	0.634109	0.317054	−	0.948407i	$-0.397306\pi$
$617$	15.7509	0.634109	0.317054	+	0.948407i	$0.397306\pi$
$618$	0	0
$619$	−13.1219	−0.527415	−0.263707	−	0.964603i	$-0.584945\pi$
$619$	−13.1219	−0.527415	−0.263707	+	0.964603i	$0.584945\pi$
$620$	25.4037	1.02024
$621$	0	0
$622$	5.13946	0.206074
$623$	−1.40763	−0.0563957
$624$	0	0
$625$	1.00000	0.0400000
$626$	40.0548	1.60091
$627$	0	0
$628$	−100.974	−4.02929
$629$	0.247654	0.00987460
$630$	0	0
$631$	14.9568	0.595422	0.297711	−	0.954656i	$-0.403777\pi$
$631$	14.9568	0.595422	0.297711	+	0.954656i	$0.403777\pi$
$632$	19.1091	0.760119
$633$	0	0
$634$	−8.72313	−0.346440
$635$	−15.4365	−0.612580
$636$	0	0
$637$	−2.61899	−0.103768
$638$	0	0
$639$	0	0
$640$	24.1208	0.953457
$641$	30.4543	1.20287	0.601436	−	0.798921i	$-0.294596\pi$
$641$	30.4543	1.20287	0.601436	+	0.798921i	$0.294596\pi$
$642$	0	0
$643$	38.3882	1.51388	0.756941	−	0.653483i	$-0.226693\pi$
$643$	38.3882	1.51388	0.756941	+	0.653483i	$0.226693\pi$
$644$	−10.9413	−0.431149
$645$	0	0
$646$	0.715449	0.0281490
$647$	28.7146	1.12889	0.564443	−	0.825472i	$-0.309091\pi$
$647$	28.7146	1.12889	0.564443	+	0.825472i	$0.309091\pi$
$648$	0	0
$649$	0	0
$650$	1.40763	0.0552119
$651$	0	0
$652$	21.2581	0.832531
$653$	−21.2502	−0.831586	−0.415793	−	0.909459i	$-0.636496\pi$
$653$	−21.2502	−0.831586	−0.415793	+	0.909459i	$0.636496\pi$
$654$	0	0
$655$	−15.5447	−0.607383
$656$	61.3028	2.39347
$657$	0	0
$658$	−0.953007	−0.0371521
$659$	16.5785	0.645806	0.322903	−	0.946432i	$-0.395341\pi$
$659$	16.5785	0.645806	0.322903	+	0.946432i	$0.395341\pi$
$660$	0	0
$661$	8.23673	0.320372	0.160186	−	0.987087i	$-0.448791\pi$
$661$	8.23673	0.320372	0.160186	+	0.987087i	$0.448791\pi$
$662$	56.5311	2.19714
$663$	0	0
$664$	37.8924	1.47051
$665$	−2.81527	−0.109171
$666$	0	0
$667$	3.51309	0.136027
$668$	−27.5474	−1.06584
$669$	0	0
$670$	22.7984	0.880779
$671$	0	0
$672$	0	0
$673$	−25.0513	−0.965657	−0.482829	−	0.875715i	$-0.660391\pi$
$673$	−25.0513	−0.965657	−0.482829	+	0.875715i	$0.660391\pi$
$674$	−19.1583	−0.737949
$675$	0	0
$676$	−67.1468	−2.58257
$677$	15.0796	0.579554	0.289777	−	0.957094i	$-0.406419\pi$
$677$	15.0796	0.579554	0.289777	+	0.957094i	$0.406419\pi$
$678$	0	0
$679$	12.9770	0.498010
$680$	1.17178	0.0449359
$681$	0	0
$682$	0	0
$683$	−2.28604	−0.0874731	−0.0437365	−	0.999043i	$-0.513926\pi$
$683$	−2.28604	−0.0874731	−0.0437365	+	0.999043i	$0.513926\pi$
$684$	0	0
$685$	15.5076	0.592515
$686$	45.6330	1.74227
$687$	0	0
$688$	−24.6737	−0.940675
$689$	−4.01003	−0.152770
$690$	0	0
$691$	49.0588	1.86628	0.933142	−	0.359508i	$-0.117055\pi$
$691$	49.0588	1.86628	0.933142	+	0.359508i	$0.117055\pi$
$692$	17.1443	0.651727
$693$	0	0
$694$	16.0530	0.609363
$695$	−3.01302	−0.114291
$696$	0	0
$697$	0.612143	0.0231866
$698$	81.7223	3.09323
$699$	0	0
$700$	−7.42620	−0.280684
$701$	50.6068	1.91139	0.955697	−	0.294353i	$-0.0951041\pi$
$701$	50.6068	1.91139	0.955697	+	0.294353i	$0.0951041\pi$
$702$	0	0
$703$	3.73476	0.140859
$704$	0	0
$705$	0	0
$706$	44.6049	1.67873
$707$	−6.70021	−0.251987
$708$	0	0
$709$	−30.2120	−1.13463	−0.567317	−	0.823500i	$-0.692019\pi$
$709$	−30.2120	−1.13463	−0.567317	+	0.823500i	$0.692019\pi$
$710$	−29.9183	−1.12281
$711$	0	0
$712$	8.83558	0.331128
$713$	7.09453	0.265692
$714$	0	0
$715$	0	0
$716$	−75.1238	−2.80751
$717$	0	0
$718$	−20.4873	−0.764581
$719$	−18.0305	−0.672425	−0.336213	−	0.941786i	$-0.609146\pi$
$719$	−18.0305	−0.672425	−0.336213	+	0.941786i	$0.609146\pi$
$720$	0	0
$721$	13.7156	0.510796
$722$	−40.4602	−1.50577
$723$	0	0
$724$	−86.6327	−3.21968
$725$	2.38444	0.0885558
$726$	0	0
$727$	48.1142	1.78446	0.892229	−	0.451584i	$-0.149141\pi$
$727$	48.1142	1.78446	0.892229	+	0.451584i	$0.149141\pi$
$728$	−6.49051	−0.240554
$729$	0	0
$730$	30.4609	1.12741
$731$	−0.246381	−0.00911272
$732$	0	0
$733$	4.60628	0.170137	0.0850685	−	0.996375i	$-0.472889\pi$
$733$	4.60628	0.170137	0.0850685	+	0.996375i	$0.472889\pi$
$734$	−36.1639	−1.33483
$735$	0	0
$736$	26.7456	0.985855
$737$	0	0
$738$	0	0
$739$	−20.2584	−0.745217	−0.372609	−	0.927989i	$-0.621537\pi$
$739$	−20.2584	−0.745217	−0.372609	+	0.927989i	$0.621537\pi$
$740$	9.85166	0.362154
$741$	0	0
$742$	29.1756	1.07107
$743$	−9.42951	−0.345935	−0.172968	−	0.984928i	$-0.555336\pi$
$743$	−9.42951	−0.345935	−0.172968	+	0.984928i	$0.555336\pi$
$744$	0	0
$745$	5.79621	0.212357
$746$	−47.1584	−1.72659
$747$	0	0
$748$	0	0
$749$	−22.2576	−0.813276
$750$	0	0
$751$	−0.438102	−0.0159866	−0.00799329	−	0.999968i	$-0.502544\pi$
$751$	−0.438102	−0.0159866	−0.00799329	+	0.999968i	$0.502544\pi$
$752$	3.33357	0.121563
$753$	0	0
$754$	3.35642	0.122233
$755$	−13.3407	−0.485518
$756$	0	0
$757$	−9.72341	−0.353403	−0.176702	−	0.984264i	$-0.556543\pi$
$757$	−9.72341	−0.353403	−0.176702	+	0.984264i	$0.556543\pi$
$758$	−86.8617	−3.15496
$759$	0	0
$760$	17.6712	0.641001
$761$	−30.9455	−1.12177	−0.560887	−	0.827892i	$-0.689540\pi$
$761$	−30.9455	−1.12177	−0.560887	+	0.827892i	$0.689540\pi$
$762$	0	0
$763$	25.2133	0.912782
$764$	135.115	4.88830
$765$	0	0
$766$	23.6856	0.855794
$767$	−0.00250492	−9.04473e−5 0
$768$	0	0
$769$	−12.7164	−0.458566	−0.229283	−	0.973360i	$-0.573638\pi$
$769$	−12.7164	−0.458566	−0.229283	+	0.973360i	$0.573638\pi$
$770$	0	0
$771$	0	0
$772$	23.5413	0.847271
$773$	−27.4367	−0.986829	−0.493415	−	0.869794i	$-0.664251\pi$
$773$	−27.4367	−0.986829	−0.493415	+	0.869794i	$0.664251\pi$
$774$	0	0
$775$	4.81527	0.172970
$776$	−81.4552	−2.92407
$777$	0	0
$778$	−27.6414	−0.990994
$779$	9.23146	0.330752
$780$	0	0
$781$	0	0
$782$	0.527050	0.0188473
$783$	0	0
$784$	−66.6530	−2.38046
$785$	−19.1395	−0.683119
$786$	0	0
$787$	−50.2315	−1.79056	−0.895280	−	0.445503i	$-0.853025\pi$
$787$	−50.2315	−1.79056	−0.895280	+	0.445503i	$0.853025\pi$
$788$	−81.8722	−2.91657
$789$	0	0
$790$	5.83366	0.207552
$791$	0.356292	0.0126683
$792$	0	0
$793$	−2.36946	−0.0841421
$794$	33.8109	1.19990
$795$	0	0
$796$	42.7835	1.51642
$797$	11.2952	0.400095	0.200047	−	0.979786i	$-0.435890\pi$
$797$	11.2952	0.400095	0.200047	+	0.979786i	$0.435890\pi$
$798$	0	0
$799$	0.0332876	0.00117763
$800$	18.1530	0.641806
$801$	0	0
$802$	−16.2583	−0.574101
$803$	0	0
$804$	0	0
$805$	−2.07392	−0.0730962
$806$	6.77814	0.238750
$807$	0	0
$808$	42.0566	1.47955
$809$	36.7236	1.29113	0.645566	−	0.763704i	$-0.276621\pi$
$809$	36.7236	1.29113	0.645566	+	0.763704i	$0.276621\pi$
$810$	0	0
$811$	−4.95938	−0.174147	−0.0870737	−	0.996202i	$-0.527752\pi$
$811$	−4.95938	−0.174147	−0.0870737	+	0.996202i	$0.527752\pi$
$812$	−17.7073	−0.621405
$813$	0	0
$814$	0	0
$815$	4.02946	0.141146
$816$	0	0
$817$	−3.71556	−0.129991
$818$	−87.7909	−3.06954
$819$	0	0
$820$	24.3510	0.850375
$821$	−34.8143	−1.21503	−0.607514	−	0.794309i	$-0.707833\pi$
$821$	−34.8143	−1.21503	−0.607514	+	0.794309i	$0.707833\pi$
$822$	0	0
$823$	23.6975	0.826043	0.413021	−	0.910721i	$-0.364473\pi$
$823$	23.6975	0.826043	0.413021	+	0.910721i	$0.364473\pi$
$824$	−86.0915	−2.99914
$825$	0	0
$826$	0.0182249	0.000634127 0
$827$	38.3715	1.33431	0.667154	−	0.744920i	$-0.267512\pi$
$827$	38.3715	1.33431	0.667154	+	0.744920i	$0.267512\pi$
$828$	0	0
$829$	46.1358	1.60236	0.801181	−	0.598422i	$-0.204205\pi$
$829$	46.1358	1.60236	0.801181	+	0.598422i	$0.204205\pi$
$830$	11.5679	0.401526
$831$	0	0
$832$	11.6909	0.405308
$833$	−0.665568	−0.0230606
$834$	0	0
$835$	−5.22159	−0.180701
$836$	0	0
$837$	0	0
$838$	−31.6338	−1.09277
$839$	48.6231	1.67865	0.839327	−	0.543626i	$-0.182949\pi$
$839$	48.6231	1.67865	0.839327	+	0.543626i	$0.182949\pi$
$840$	0	0
$841$	−23.3145	−0.803947
$842$	−21.0213	−0.724443
$843$	0	0
$844$	−93.8228	−3.22951
$845$	−12.7277	−0.437845
$846$	0	0
$847$	15.4840	0.532036
$848$	−102.055	−3.50458
$849$	0	0
$850$	0.357725	0.0122699
$851$	2.75129	0.0943128
$852$	0	0
$853$	47.8962	1.63993	0.819967	−	0.572411i	$-0.193992\pi$
$853$	47.8962	1.63993	0.819967	+	0.572411i	$0.193992\pi$
$854$	17.2394	0.589921
$855$	0	0
$856$	139.709	4.77516
$857$	46.1802	1.57748	0.788742	−	0.614724i	$-0.210733\pi$
$857$	46.1802	1.57748	0.788742	+	0.614724i	$0.210733\pi$
$858$	0	0
$859$	2.99452	0.102172	0.0510859	−	0.998694i	$-0.483732\pi$
$859$	2.99452	0.102172	0.0510859	+	0.998694i	$0.483732\pi$
$860$	−9.80101	−0.334212
$861$	0	0
$862$	55.7543	1.89900
$863$	−18.7276	−0.637496	−0.318748	−	0.947840i	$-0.603262\pi$
$863$	−18.7276	−0.637496	−0.318748	+	0.947840i	$0.603262\pi$
$864$	0	0
$865$	3.24969	0.110493
$866$	36.1667	1.22899
$867$	0	0
$868$	−35.7591	−1.21374
$869$	0	0
$870$	0	0
$871$	4.41085	0.149456
$872$	−158.261	−5.35941
$873$	0	0
$874$	7.94821	0.268852
$875$	−1.40763	−0.0475867
$876$	0	0
$877$	−6.93371	−0.234135	−0.117067	−	0.993124i	$-0.537349\pi$
$877$	−6.93371	−0.234135	−0.117067	+	0.993124i	$0.537349\pi$
$878$	−49.2799	−1.66312
$879$	0	0
$880$	0	0
$881$	8.15596	0.274781	0.137391	−	0.990517i	$-0.456128\pi$
$881$	8.15596	0.274781	0.137391	+	0.990517i	$0.456128\pi$
$882$	0	0
$883$	−28.7180	−0.966439	−0.483219	−	0.875499i	$-0.660533\pi$
$883$	−28.7180	−0.966439	−0.483219	+	0.875499i	$0.660533\pi$
$884$	0.365126	0.0122805
$885$	0	0
$886$	−15.6792	−0.526752
$887$	−28.9430	−0.971811	−0.485905	−	0.874011i	$-0.661510\pi$
$887$	−28.9430	−0.971811	−0.485905	+	0.874011i	$0.661510\pi$
$888$	0	0
$889$	21.7290	0.728767
$890$	2.69734	0.0904152
$891$	0	0
$892$	31.6428	1.05948
$893$	0.501996	0.0167987
$894$	0	0
$895$	−14.2397	−0.475980
$896$	−33.9532	−1.13430
$897$	0	0
$898$	−25.9294	−0.865276
$899$	11.4817	0.382936
$900$	0	0
$901$	−1.01908	−0.0339503
$902$	0	0
$903$	0	0
$904$	−2.23641	−0.0743820
$905$	−16.4212	−0.545859
$906$	0	0
$907$	22.6357	0.751607	0.375803	−	0.926699i	$-0.377367\pi$
$907$	22.6357	0.751607	0.375803	+	0.926699i	$0.377367\pi$
$908$	113.594	3.76976
$909$	0	0
$910$	−1.98143	−0.0656839
$911$	−27.9402	−0.925702	−0.462851	−	0.886436i	$-0.653173\pi$
$911$	−27.9402	−0.925702	−0.462851	+	0.886436i	$0.653173\pi$
$912$	0	0
$913$	0	0
$914$	0.804549	0.0266121
$915$	0	0
$916$	67.2851	2.22316
$917$	21.8813	0.722584
$918$	0	0
$919$	−3.33908	−0.110146	−0.0550730	−	0.998482i	$-0.517539\pi$
$919$	−3.33908	−0.110146	−0.0550730	+	0.998482i	$0.517539\pi$
$920$	13.0178	0.429185
$921$	0	0
$922$	−9.07084	−0.298732
$923$	−5.78833	−0.190525
$924$	0	0
$925$	1.86738	0.0613990
$926$	112.621	3.70094
$927$	0	0
$928$	43.2847	1.42089
$929$	−16.7415	−0.549271	−0.274636	−	0.961548i	$-0.588557\pi$
$929$	−16.7415	−0.549271	−0.274636	+	0.961548i	$0.588557\pi$
$930$	0	0
$931$	−10.0371	−0.328954
$932$	26.2525	0.859928
$933$	0	0
$934$	−71.0917	−2.32619
$935$	0	0
$936$	0	0
$937$	−57.6381	−1.88296	−0.941478	−	0.337075i	$-0.890562\pi$
$937$	−57.6381	−1.88296	−0.941478	+	0.337075i	$0.890562\pi$
$938$	−32.0918	−1.04784
$939$	0	0
$940$	1.32418	0.0431900
$941$	−34.0039	−1.10850	−0.554248	−	0.832352i	$-0.686994\pi$
$941$	−34.0039	−1.10850	−0.554248	+	0.832352i	$0.686994\pi$
$942$	0	0
$943$	6.80055	0.221456
$944$	−0.0637500	−0.00207488
$945$	0	0
$946$	0	0
$947$	−49.0772	−1.59479	−0.797397	−	0.603455i	$-0.793790\pi$
$947$	−49.0772	−1.59479	−0.797397	+	0.603455i	$0.793790\pi$
$948$	0	0
$949$	5.89332	0.191305
$950$	5.39469	0.175027
$951$	0	0
$952$	−1.64944	−0.0534588
$953$	−24.7641	−0.802187	−0.401094	−	0.916037i	$-0.631370\pi$
$953$	−24.7641	−0.802187	−0.401094	+	0.916037i	$0.631370\pi$
$954$	0	0
$955$	25.6111	0.828755
$956$	1.88439	0.0609454
$957$	0	0
$958$	−16.7045	−0.539699
$959$	−21.8290	−0.704896
$960$	0	0
$961$	−7.81319	−0.252039
$962$	2.62859	0.0847490
$963$	0	0
$964$	−87.3767	−2.81421
$965$	4.46225	0.143645
$966$	0	0
$967$	−29.0541	−0.934318	−0.467159	−	0.884173i	$-0.654722\pi$
$967$	−29.0541	−0.934318	−0.467159	+	0.884173i	$0.654722\pi$
$968$	−97.1914	−3.12385
$969$	0	0
$970$	−24.8668	−0.798425
$971$	17.9687	0.576643	0.288322	−	0.957534i	$-0.406903\pi$
$971$	17.9687	0.576643	0.288322	+	0.957534i	$0.406903\pi$
$972$	0	0
$973$	4.24124	0.135968
$974$	−78.4612	−2.51406
$975$	0	0
$976$	−60.3027	−1.93024
$977$	−45.6828	−1.46152	−0.730761	−	0.682634i	$-0.760834\pi$
$977$	−45.6828	−1.46152	−0.730761	+	0.682634i	$0.760834\pi$
$978$	0	0
$979$	0	0
$980$	−26.4763	−0.845753
$981$	0	0
$982$	−79.4940	−2.53676
$983$	14.3270	0.456960	0.228480	−	0.973549i	$-0.426624\pi$
$983$	14.3270	0.456960	0.228480	+	0.973549i	$0.426624\pi$
$984$	0	0
$985$	−15.5188	−0.494471
$986$	0.852972	0.0271642
$987$	0	0
$988$	5.50631	0.175179
$989$	−2.73714	−0.0870360
$990$	0	0
$991$	51.3711	1.63186	0.815929	−	0.578153i	$-0.196226\pi$
$991$	51.3711	1.63186	0.815929	+	0.578153i	$0.196226\pi$
$992$	87.4116	2.77532
$993$	0	0
$994$	42.1140	1.33577
$995$	8.10960	0.257091
$996$	0	0
$997$	−29.4396	−0.932361	−0.466181	−	0.884690i	$-0.654370\pi$
$997$	−29.4396	−0.932361	−0.466181	+	0.884690i	$0.654370\pi$
$998$	109.884	3.47831
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.n.1.6		6
3.2	odd	2		1335.2.a.g.1.1	✓	6
15.14	odd	2		6675.2.a.u.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.g.1.1	✓	6	3.2	odd	2
4005.2.a.n.1.6		6	1.1	even	1	trivial
6675.2.a.u.1.6		6	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)

\(f(q)\)	\(=\)	\(q+2.69734 q^{2} +5.27566 q^{4} +1.00000 q^{5} -1.40763 q^{7} +8.83558 q^{8} +O(q^{10})\)	\(q+2.69734 q^{2} +5.27566 q^{4} +1.00000 q^{5} -1.40763 q^{7} +8.83558 q^{8} +2.69734 q^{10} +0.521859 q^{13} -3.79687 q^{14} +13.2813 q^{16} +0.132621 q^{17} +2.00000 q^{19} +5.27566 q^{20} +1.47334 q^{23} +1.00000 q^{25} +1.40763 q^{26} -7.42620 q^{28} +2.38444 q^{29} +4.81527 q^{31} +18.1530 q^{32} +0.357725 q^{34} -1.40763 q^{35} +1.86738 q^{37} +5.39469 q^{38} +8.83558 q^{40} +4.61573 q^{41} -1.85778 q^{43} +3.97411 q^{46} +0.250998 q^{47} -5.01857 q^{49} +2.69734 q^{50} +2.75315 q^{52} -7.68411 q^{53} -12.4373 q^{56} +6.43165 q^{58} -0.00479999 q^{59} -4.54043 q^{61} +12.9884 q^{62} +22.4023 q^{64} +0.521859 q^{65} +8.45217 q^{67} +0.699664 q^{68} -3.79687 q^{70} -11.0917 q^{71} +11.2929 q^{73} +5.03696 q^{74} +10.5513 q^{76} +2.16274 q^{79} +13.2813 q^{80} +12.4502 q^{82} +4.28861 q^{83} +0.132621 q^{85} -5.01107 q^{86} +1.00000 q^{89} -0.734587 q^{91} +7.77285 q^{92} +0.677028 q^{94} +2.00000 q^{95} -9.21900 q^{97} -13.5368 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8}+O(q^{10}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8	\( 6 q + 4 q^{2} + 8 q^{4} + 6 q^{5} + q^{7} + 9 q^{8} + 4 q^{10} - 3 q^{13} + 5 q^{14} + 12 q^{16} + 13 q^{17} + 12 q^{19} + 8 q^{20} + 19 q^{23} + 6 q^{25} - q^{26} - 6 q^{28} + 10 q^{31} + 17 q^{32} - 2 q^{34} + q^{35} - q^{37} + 8 q^{38} + 9 q^{40} + 4 q^{41} - 7 q^{43} - 6 q^{46} + 15 q^{47} - q^{49} + 4 q^{50} + q^{52} + 27 q^{53} + 14 q^{56} - 6 q^{58} + 4 q^{59} + 8 q^{61} - 2 q^{62} - q^{64} - 3 q^{65} - 11 q^{67} + 47 q^{68} + 5 q^{70} + 16 q^{71} + q^{73} + 10 q^{74} + 16 q^{76} + 12 q^{80} + q^{82} + 17 q^{83} + 13 q^{85} + 20 q^{86} + 6 q^{89} - 18 q^{91} + 36 q^{92} + 17 q^{94} + 12 q^{95} - 29 q^{97} + 23 q^{98}+O(q^{100}) \) 6 * q + 4 * q^2 + 8 * q^4 + 6 * q^5 + q^7 + 9 * q^8 + 4 * q^10 - 3 * q^13 + 5 * q^14 + 12 * q^16 + 13 * q^17 + 12 * q^19 + 8 * q^20 + 19 * q^23 + 6 * q^25 - q^26 - 6 * q^28 + 10 * q^31 + 17 * q^32 - 2 * q^34 + q^35 - q^37 + 8 * q^38 + 9 * q^40 + 4 * q^41 - 7 * q^43 - 6 * q^46 + 15 * q^47 - q^49 + 4 * q^50 + q^52 + 27 * q^53 + 14 * q^56 - 6 * q^58 + 4 * q^59 + 8 * q^61 - 2 * q^62 - q^64 - 3 * q^65 - 11 * q^67 + 47 * q^68 + 5 * q^70 + 16 * q^71 + q^73 + 10 * q^74 + 16 * q^76 + 12 * q^80 + q^82 + 17 * q^83 + 13 * q^85 + 20 * q^86 + 6 * q^89 - 18 * q^91 + 36 * q^92 + 17 * q^94 + 12 * q^95 - 29 * q^97 + 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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