LMFDB - Embedded newform 4005.2.a.u.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 13 x^{10} + 41 x^{9} + 58 x^{8} - 202 x^{7} - 95 x^{6} + 432 x^{5} + 4 x^{4} + \cdots - 27 $ x^12 - 3x^11 - 13x^10 + 41x^9 + 58x^8 - 202x^7 - 95x^6 + 432x^5 + 4x^4 - 368x^3 + 94x^2 + 77*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-2.29792$ 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)

$f(q)$	$=$	$q+2.29792 q^{2} +3.28045 q^{4} +1.00000 q^{5} -2.76426 q^{7} +2.94239 q^{8} +O(q^{10})$	\(q+2.29792 q^{2} +3.28045 q^{4} +1.00000 q^{5} -2.76426 q^{7} +2.94239 q^{8} +2.29792 q^{10} +1.67068 q^{11} -6.25040 q^{13} -6.35206 q^{14} +0.200472 q^{16} -4.10410 q^{17} -4.67447 q^{19} +3.28045 q^{20} +3.83910 q^{22} -9.46614 q^{23} +1.00000 q^{25} -14.3630 q^{26} -9.06802 q^{28} +4.18945 q^{29} +3.14623 q^{31} -5.42410 q^{32} -9.43091 q^{34} -2.76426 q^{35} +10.3869 q^{37} -10.7416 q^{38} +2.94239 q^{40} -6.63318 q^{41} +4.38966 q^{43} +5.48059 q^{44} -21.7525 q^{46} -8.10240 q^{47} +0.641127 q^{49} +2.29792 q^{50} -20.5042 q^{52} +2.86943 q^{53} +1.67068 q^{55} -8.13352 q^{56} +9.62703 q^{58} -1.93816 q^{59} -10.8902 q^{61} +7.22980 q^{62} -12.8651 q^{64} -6.25040 q^{65} +10.0204 q^{67} -13.4633 q^{68} -6.35206 q^{70} +11.3901 q^{71} +0.329414 q^{73} +23.8683 q^{74} -15.3344 q^{76} -4.61819 q^{77} +6.42974 q^{79} +0.200472 q^{80} -15.2425 q^{82} -12.3558 q^{83} -4.10410 q^{85} +10.0871 q^{86} +4.91579 q^{88} +1.00000 q^{89} +17.2777 q^{91} -31.0533 q^{92} -18.6187 q^{94} -4.67447 q^{95} -11.2411 q^{97} +1.47326 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * 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 $

$2$	2.29792	1.62488	0.812439	−	0.583046i	$-0.198139\pi$
$2$	2.29792	1.62488	0.812439	+	0.583046i	$0.198139\pi$
$3$	0	0
$3$	0	0
$4$	3.28045	1.64023
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.76426	−1.04479	−0.522396	−	0.852703i	$-0.674962\pi$
$7$	−2.76426	−1.04479	−0.522396	+	0.852703i	$0.674962\pi$
$8$	2.94239	1.04029
$9$	0	0
$10$	2.29792	0.726667
$11$	1.67068	0.503729	0.251864	−	0.967763i	$-0.418956\pi$
$11$	1.67068	0.503729	0.251864	+	0.967763i	$0.418956\pi$
$12$	0	0
$13$	−6.25040	−1.73355	−0.866775	−	0.498699i	$-0.833811\pi$
$13$	−6.25040	−1.73355	−0.866775	+	0.498699i	$0.833811\pi$
$14$	−6.35206	−1.69766
$15$	0	0
$16$	0.200472	0.0501179
$17$	−4.10410	−0.995390	−0.497695	−	0.867352i	$-0.665820\pi$
$17$	−4.10410	−0.995390	−0.497695	+	0.867352i	$0.665820\pi$
$18$	0	0
$19$	−4.67447	−1.07240	−0.536199	−	0.844092i	$-0.680140\pi$
$19$	−4.67447	−1.07240	−0.536199	+	0.844092i	$0.680140\pi$
$20$	3.28045	0.733532
$21$	0	0
$22$	3.83910	0.818498
$23$	−9.46614	−1.97383	−0.986914	−	0.161249i	$-0.948448\pi$
$23$	−9.46614	−1.97383	−0.986914	+	0.161249i	$0.948448\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−14.3630	−2.81681
$27$	0	0
$28$	−9.06802	−1.71370
$29$	4.18945	0.777961	0.388980	−	0.921246i	$-0.372827\pi$
$29$	4.18945	0.777961	0.388980	+	0.921246i	$0.372827\pi$
$30$	0	0
$31$	3.14623	0.565080	0.282540	−	0.959256i	$-0.408823\pi$
$31$	3.14623	0.565080	0.282540	+	0.959256i	$0.408823\pi$
$32$	−5.42410	−0.958855
$33$	0	0
$34$	−9.43091	−1.61739
$35$	−2.76426	−0.467245
$36$	0	0
$37$	10.3869	1.70759	0.853796	−	0.520608i	$-0.174295\pi$
$37$	10.3869	1.70759	0.853796	+	0.520608i	$0.174295\pi$
$38$	−10.7416	−1.74252
$39$	0	0
$40$	2.94239	0.465232
$41$	−6.63318	−1.03593	−0.517964	−	0.855402i	$-0.673310\pi$
$41$	−6.63318	−1.03593	−0.517964	+	0.855402i	$0.673310\pi$
$42$	0	0
$43$	4.38966	0.669417	0.334708	−	0.942322i	$-0.391362\pi$
$43$	4.38966	0.669417	0.334708	+	0.942322i	$0.391362\pi$
$44$	5.48059	0.826230
$45$	0	0
$46$	−21.7525	−3.20723
$47$	−8.10240	−1.18186	−0.590928	−	0.806724i	$-0.701238\pi$
$47$	−8.10240	−1.18186	−0.590928	+	0.806724i	$0.701238\pi$
$48$	0	0
$49$	0.641127	0.0915895
$50$	2.29792	0.324976
$51$	0	0
$52$	−20.5042	−2.84342
$53$	2.86943	0.394147	0.197074	−	0.980389i	$-0.436856\pi$
$53$	2.86943	0.394147	0.197074	+	0.980389i	$0.436856\pi$
$54$	0	0
$55$	1.67068	0.225274
$56$	−8.13352	−1.08689
$57$	0	0
$58$	9.62703	1.26409
$59$	−1.93816	−0.252327	−0.126164	−	0.992009i	$-0.540266\pi$
$59$	−1.93816	−0.252327	−0.126164	+	0.992009i	$0.540266\pi$
$60$	0	0
$61$	−10.8902	−1.39434	−0.697171	−	0.716905i	$-0.745558\pi$
$61$	−10.8902	−1.39434	−0.697171	+	0.716905i	$0.745558\pi$
$62$	7.22980	0.918185
$63$	0	0
$64$	−12.8651	−1.60814
$65$	−6.25040	−0.775267
$66$	0	0
$67$	10.0204	1.22419	0.612095	−	0.790784i	$-0.290327\pi$
$67$	10.0204	1.22419	0.612095	+	0.790784i	$0.290327\pi$
$68$	−13.4633	−1.63267
$69$	0	0
$70$	−6.35206	−0.759216
$71$	11.3901	1.35175	0.675877	−	0.737014i	$-0.263765\pi$
$71$	11.3901	1.35175	0.675877	+	0.737014i	$0.263765\pi$
$72$	0	0
$73$	0.329414	0.0385550	0.0192775	−	0.999814i	$-0.493863\pi$
$73$	0.329414	0.0385550	0.0192775	+	0.999814i	$0.493863\pi$
$74$	23.8683	2.77463
$75$	0	0
$76$	−15.3344	−1.75898
$77$	−4.61819	−0.526292
$78$	0	0
$79$	6.42974	0.723402	0.361701	−	0.932294i	$-0.382196\pi$
$79$	6.42974	0.723402	0.361701	+	0.932294i	$0.382196\pi$
$80$	0.200472	0.0224134
$81$	0	0
$82$	−15.2425	−1.68326
$83$	−12.3558	−1.35622	−0.678111	−	0.734959i	$-0.737201\pi$
$83$	−12.3558	−1.35622	−0.678111	+	0.734959i	$0.737201\pi$
$84$	0	0
$85$	−4.10410	−0.445152
$86$	10.0871	1.08772
$87$	0	0
$88$	4.91579	0.524025
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	17.2777	1.81120
$92$	−31.0533	−3.23753
$93$	0	0
$94$	−18.6187	−1.92037
$95$	−4.67447	−0.479591
$96$	0	0
$97$	−11.2411	−1.14137	−0.570683	−	0.821171i	$-0.693321\pi$
$97$	−11.2411	−1.14137	−0.570683	+	0.821171i	$0.693321\pi$
$98$	1.47326	0.148822
$99$	0	0
$100$	3.28045	0.328045
$101$	12.9476	1.28833	0.644166	−	0.764886i	$-0.277205\pi$
$101$	12.9476	1.28833	0.644166	+	0.764886i	$0.277205\pi$
$102$	0	0
$103$	5.53544	0.545423	0.272711	−	0.962096i	$-0.412080\pi$
$103$	5.53544	0.545423	0.272711	+	0.962096i	$0.412080\pi$
$104$	−18.3911	−1.80340
$105$	0	0
$106$	6.59374	0.640441
$107$	15.7730	1.52483	0.762416	−	0.647087i	$-0.224013\pi$
$107$	15.7730	1.52483	0.762416	+	0.647087i	$0.224013\pi$
$108$	0	0
$109$	17.4892	1.67517	0.837583	−	0.546310i	$-0.183968\pi$
$109$	17.4892	1.67517	0.837583	+	0.546310i	$0.183968\pi$
$110$	3.83910	0.366043
$111$	0	0
$112$	−0.554156	−0.0523628
$113$	7.03514	0.661810	0.330905	−	0.943664i	$-0.392646\pi$
$113$	7.03514	0.661810	0.330905	+	0.943664i	$0.392646\pi$
$114$	0	0
$115$	−9.46614	−0.882722
$116$	13.7433	1.27603
$117$	0	0
$118$	−4.45375	−0.410001
$119$	11.3448	1.03998
$120$	0	0
$121$	−8.20883	−0.746257
$122$	−25.0248	−2.26564
$123$	0	0
$124$	10.3211	0.926859
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.9874	−1.68486	−0.842429	−	0.538807i	$-0.818875\pi$
$127$	−18.9874	−1.68486	−0.842429	+	0.538807i	$0.818875\pi$
$128$	−18.7149	−1.65418
$129$	0	0
$130$	−14.3630	−1.25971
$131$	9.33559	0.815655	0.407827	−	0.913059i	$-0.366287\pi$
$131$	9.33559	0.815655	0.407827	+	0.913059i	$0.366287\pi$
$132$	0	0
$133$	12.9215	1.12043
$134$	23.0262	1.98916
$135$	0	0
$136$	−12.0758	−1.03550
$137$	10.4811	0.895459	0.447729	−	0.894169i	$-0.352233\pi$
$137$	10.4811	0.895459	0.447729	+	0.894169i	$0.352233\pi$
$138$	0	0
$139$	−6.02207	−0.510785	−0.255393	−	0.966837i	$-0.582205\pi$
$139$	−6.02207	−0.510785	−0.255393	+	0.966837i	$0.582205\pi$
$140$	−9.06802	−0.766388
$141$	0	0
$142$	26.1736	2.19644
$143$	−10.4424	−0.873239
$144$	0	0
$145$	4.18945	0.347915
$146$	0.756969	0.0626472
$147$	0	0
$148$	34.0737	2.80084
$149$	−15.5144	−1.27099	−0.635495	−	0.772105i	$-0.719204\pi$
$149$	−15.5144	−1.27099	−0.635495	+	0.772105i	$0.719204\pi$
$150$	0	0
$151$	−19.6097	−1.59581	−0.797907	−	0.602780i	$-0.794060\pi$
$151$	−19.6097	−1.59581	−0.797907	+	0.602780i	$0.794060\pi$
$152$	−13.7541	−1.11561
$153$	0	0
$154$	−10.6123	−0.855160
$155$	3.14623	0.252711
$156$	0	0
$157$	−0.546086	−0.0435824	−0.0217912	−	0.999763i	$-0.506937\pi$
$157$	−0.546086	−0.0435824	−0.0217912	+	0.999763i	$0.506937\pi$
$158$	14.7750	1.17544
$159$	0	0
$160$	−5.42410	−0.428813
$161$	26.1669	2.06224
$162$	0	0
$163$	−8.80594	−0.689735	−0.344867	−	0.938651i	$-0.612076\pi$
$163$	−8.80594	−0.689735	−0.344867	+	0.938651i	$0.612076\pi$
$164$	−21.7598	−1.69916
$165$	0	0
$166$	−28.3926	−2.20370
$167$	−3.37699	−0.261319	−0.130660	−	0.991427i	$-0.541710\pi$
$167$	−3.37699	−0.261319	−0.130660	+	0.991427i	$0.541710\pi$
$168$	0	0
$169$	26.0675	2.00520
$170$	−9.43091	−0.723318
$171$	0	0
$172$	14.4001	1.09800
$173$	−10.0821	−0.766526	−0.383263	−	0.923639i	$-0.625200\pi$
$173$	−10.0821	−0.766526	−0.383263	+	0.923639i	$0.625200\pi$
$174$	0	0
$175$	−2.76426	−0.208958
$176$	0.334924	0.0252459
$177$	0	0
$178$	2.29792	0.172237
$179$	−14.3526	−1.07277	−0.536383	−	0.843975i	$-0.680210\pi$
$179$	−14.3526	−1.07277	−0.536383	+	0.843975i	$0.680210\pi$
$180$	0	0
$181$	21.1496	1.57203	0.786017	−	0.618205i	$-0.212140\pi$
$181$	21.1496	1.57203	0.786017	+	0.618205i	$0.212140\pi$
$182$	39.7029	2.94298
$183$	0	0
$184$	−27.8531	−2.05335
$185$	10.3869	0.763658
$186$	0	0
$187$	−6.85664	−0.501407
$188$	−26.5795	−1.93851
$189$	0	0
$190$	−10.7416	−0.779277
$191$	−0.393687	−0.0284862	−0.0142431	−	0.999899i	$-0.504534\pi$
$191$	−0.393687	−0.0284862	−0.0142431	+	0.999899i	$0.504534\pi$
$192$	0	0
$193$	−17.4013	−1.25257	−0.626285	−	0.779594i	$-0.715425\pi$
$193$	−17.4013	−1.25257	−0.626285	+	0.779594i	$0.715425\pi$
$194$	−25.8313	−1.85458
$195$	0	0
$196$	2.10319	0.150228
$197$	−3.69571	−0.263309	−0.131654	−	0.991296i	$-0.542029\pi$
$197$	−3.69571	−0.263309	−0.131654	+	0.991296i	$0.542029\pi$
$198$	0	0
$199$	−8.12845	−0.576210	−0.288105	−	0.957599i	$-0.593025\pi$
$199$	−8.12845	−0.576210	−0.288105	+	0.957599i	$0.593025\pi$
$200$	2.94239	0.208058
$201$	0	0
$202$	29.7525	2.09338
$203$	−11.5807	−0.812807
$204$	0	0
$205$	−6.63318	−0.463281
$206$	12.7200	0.886245
$207$	0	0
$208$	−1.25303	−0.0868819
$209$	−7.80955	−0.540198
$210$	0	0
$211$	−9.00805	−0.620140	−0.310070	−	0.950714i	$-0.600352\pi$
$211$	−9.00805	−0.620140	−0.310070	+	0.950714i	$0.600352\pi$
$212$	9.41305	0.646491
$213$	0	0
$214$	36.2451	2.47767
$215$	4.38966	0.299372
$216$	0	0
$217$	−8.69700	−0.590391
$218$	40.1889	2.72194
$219$	0	0
$220$	5.48059	0.369501
$221$	25.6523	1.72556
$222$	0	0
$223$	−15.0082	−1.00502	−0.502510	−	0.864571i	$-0.667590\pi$
$223$	−15.0082	−1.00502	−0.502510	+	0.864571i	$0.667590\pi$
$224$	14.9936	1.00180
$225$	0	0
$226$	16.1662	1.07536
$227$	17.7884	1.18066	0.590328	−	0.807163i	$-0.298998\pi$
$227$	17.7884	1.18066	0.590328	+	0.807163i	$0.298998\pi$
$228$	0	0
$229$	18.5164	1.22360	0.611798	−	0.791014i	$-0.290447\pi$
$229$	18.5164	1.22360	0.611798	+	0.791014i	$0.290447\pi$
$230$	−21.7525	−1.43432
$231$	0	0
$232$	12.3270	0.809306
$233$	−12.8048	−0.838867	−0.419434	−	0.907786i	$-0.637771\pi$
$233$	−12.8048	−0.838867	−0.419434	+	0.907786i	$0.637771\pi$
$234$	0	0
$235$	−8.10240	−0.528542
$236$	−6.35806	−0.413874
$237$	0	0
$238$	26.0695	1.68983
$239$	−11.9266	−0.771466	−0.385733	−	0.922611i	$-0.626051\pi$
$239$	−11.9266	−0.771466	−0.385733	+	0.922611i	$0.626051\pi$
$240$	0	0
$241$	10.2100	0.657683	0.328842	−	0.944385i	$-0.393342\pi$
$241$	10.2100	0.657683	0.328842	+	0.944385i	$0.393342\pi$
$242$	−18.8633	−1.21258
$243$	0	0
$244$	−35.7247	−2.28704
$245$	0.641127	0.0409601
$246$	0	0
$247$	29.2173	1.85906
$248$	9.25743	0.587847
$249$	0	0
$250$	2.29792	0.145333
$251$	−1.04147	−0.0657370	−0.0328685	−	0.999460i	$-0.510464\pi$
$251$	−1.04147	−0.0657370	−0.0328685	+	0.999460i	$0.510464\pi$
$252$	0	0
$253$	−15.8149	−0.994274
$254$	−43.6316	−2.73769
$255$	0	0
$256$	−17.2751	−1.07969
$257$	−23.9809	−1.49589	−0.747943	−	0.663763i	$-0.768958\pi$
$257$	−23.9809	−1.49589	−0.747943	+	0.663763i	$0.768958\pi$
$258$	0	0
$259$	−28.7120	−1.78408
$260$	−20.5042	−1.27161
$261$	0	0
$262$	21.4525	1.32534
$263$	−24.5977	−1.51676	−0.758380	−	0.651813i	$-0.774009\pi$
$263$	−24.5977	−1.51676	−0.758380	+	0.651813i	$0.774009\pi$
$264$	0	0
$265$	2.86943	0.176268
$266$	29.6925	1.82057
$267$	0	0
$268$	32.8716	2.00795
$269$	29.4992	1.79860	0.899300	−	0.437331i	$-0.144076\pi$
$269$	29.4992	1.79860	0.899300	+	0.437331i	$0.144076\pi$
$270$	0	0
$271$	−14.1745	−0.861042	−0.430521	−	0.902581i	$-0.641670\pi$
$271$	−14.1745	−0.861042	−0.430521	+	0.902581i	$0.641670\pi$
$272$	−0.822756	−0.0498869
$273$	0	0
$274$	24.0847	1.45501
$275$	1.67068	0.100746
$276$	0	0
$277$	−20.3299	−1.22151	−0.610753	−	0.791821i	$-0.709133\pi$
$277$	−20.3299	−1.22151	−0.610753	+	0.791821i	$0.709133\pi$
$278$	−13.8383	−0.829963
$279$	0	0
$280$	−8.13352	−0.486071
$281$	−18.0096	−1.07436	−0.537182	−	0.843466i	$-0.680511\pi$
$281$	−18.0096	−1.07436	−0.537182	+	0.843466i	$0.680511\pi$
$282$	0	0
$283$	−14.3319	−0.851940	−0.425970	−	0.904737i	$-0.640067\pi$
$283$	−14.3319	−0.851940	−0.425970	+	0.904737i	$0.640067\pi$
$284$	37.3647	2.21718
$285$	0	0
$286$	−23.9959	−1.41891
$287$	18.3358	1.08233
$288$	0	0
$289$	−0.156374	−0.00919849
$290$	9.62703	0.565319
$291$	0	0
$292$	1.08063	0.0632390
$293$	−3.29112	−0.192269	−0.0961345	−	0.995368i	$-0.530648\pi$
$293$	−3.29112	−0.192269	−0.0961345	+	0.995368i	$0.530648\pi$
$294$	0	0
$295$	−1.93816	−0.112844
$296$	30.5622	1.77639
$297$	0	0
$298$	−35.6509	−2.06520
$299$	59.1672	3.42173
$300$	0	0
$301$	−12.1342	−0.699401
$302$	−45.0616	−2.59300
$303$	0	0
$304$	−0.937100	−0.0537464
$305$	−10.8902	−0.623569
$306$	0	0
$307$	−22.3433	−1.27520	−0.637598	−	0.770369i	$-0.720072\pi$
$307$	−22.3433	−1.27520	−0.637598	+	0.770369i	$0.720072\pi$
$308$	−15.1498	−0.863238
$309$	0	0
$310$	7.22980	0.410625
$311$	−10.1703	−0.576707	−0.288354	−	0.957524i	$-0.593108\pi$
$311$	−10.1703	−0.576707	−0.288354	+	0.957524i	$0.593108\pi$
$312$	0	0
$313$	−16.8177	−0.950590	−0.475295	−	0.879826i	$-0.657659\pi$
$313$	−16.8177	−0.950590	−0.475295	+	0.879826i	$0.657659\pi$
$314$	−1.25486	−0.0708161
$315$	0	0
$316$	21.0925	1.18654
$317$	−1.43637	−0.0806748	−0.0403374	−	0.999186i	$-0.512843\pi$
$317$	−1.43637	−0.0806748	−0.0403374	+	0.999186i	$0.512843\pi$
$318$	0	0
$319$	6.99923	0.391881
$320$	−12.8651	−0.719182
$321$	0	0
$322$	60.1295	3.35088
$323$	19.1845	1.06745
$324$	0	0
$325$	−6.25040	−0.346710
$326$	−20.2354	−1.12073
$327$	0	0
$328$	−19.5174	−1.07767
$329$	22.3971	1.23479
$330$	0	0
$331$	1.25612	0.0690425	0.0345213	−	0.999404i	$-0.489009\pi$
$331$	1.25612	0.0690425	0.0345213	+	0.999404i	$0.489009\pi$
$332$	−40.5326	−2.22451
$333$	0	0
$334$	−7.76007	−0.424612
$335$	10.0204	0.547474
$336$	0	0
$337$	−2.50988	−0.136722	−0.0683610	−	0.997661i	$-0.521777\pi$
$337$	−2.50988	−0.136722	−0.0683610	+	0.997661i	$0.521777\pi$
$338$	59.9012	3.25820
$339$	0	0
$340$	−13.4633	−0.730150
$341$	5.25634	0.284647
$342$	0	0
$343$	17.5776	0.949100
$344$	12.9161	0.696388
$345$	0	0
$346$	−23.1678	−1.24551
$347$	30.6633	1.64609	0.823047	−	0.567973i	$-0.192272\pi$
$347$	30.6633	1.64609	0.823047	+	0.567973i	$0.192272\pi$
$348$	0	0
$349$	−3.64087	−0.194891	−0.0974455	−	0.995241i	$-0.531067\pi$
$349$	−3.64087	−0.194891	−0.0974455	+	0.995241i	$0.531067\pi$
$350$	−6.35206	−0.339532
$351$	0	0
$352$	−9.06194	−0.483003
$353$	14.8542	0.790607	0.395303	−	0.918551i	$-0.370639\pi$
$353$	14.8542	0.790607	0.395303	+	0.918551i	$0.370639\pi$
$354$	0	0
$355$	11.3901	0.604523
$356$	3.28045	0.173864
$357$	0	0
$358$	−32.9812	−1.74311
$359$	−11.3609	−0.599606	−0.299803	−	0.954001i	$-0.596921\pi$
$359$	−11.3609	−0.599606	−0.299803	+	0.954001i	$0.596921\pi$
$360$	0	0
$361$	2.85071	0.150037
$362$	48.6001	2.55436
$363$	0	0
$364$	56.6788	2.97078
$365$	0.329414	0.0172423
$366$	0	0
$367$	32.5216	1.69761	0.848807	−	0.528703i	$-0.177321\pi$
$367$	32.5216	1.69761	0.848807	+	0.528703i	$0.177321\pi$
$368$	−1.89769	−0.0989241
$369$	0	0
$370$	23.8683	1.24085
$371$	−7.93186	−0.411802
$372$	0	0
$373$	21.8812	1.13297	0.566483	−	0.824074i	$-0.308304\pi$
$373$	21.8812	1.13297	0.566483	+	0.824074i	$0.308304\pi$
$374$	−15.7560	−0.814725
$375$	0	0
$376$	−23.8404	−1.22947
$377$	−26.1857	−1.34863
$378$	0	0
$379$	19.0162	0.976794	0.488397	−	0.872621i	$-0.337582\pi$
$379$	19.0162	0.976794	0.488397	+	0.872621i	$0.337582\pi$
$380$	−15.3344	−0.786638
$381$	0	0
$382$	−0.904662	−0.0462865
$383$	−17.6189	−0.900285	−0.450143	−	0.892957i	$-0.648627\pi$
$383$	−17.6189	−0.900285	−0.450143	+	0.892957i	$0.648627\pi$
$384$	0	0
$385$	−4.61819	−0.235365
$386$	−39.9868	−2.03527
$387$	0	0
$388$	−36.8761	−1.87210
$389$	5.60016	0.283939	0.141970	−	0.989871i	$-0.454656\pi$
$389$	5.60016	0.283939	0.141970	+	0.989871i	$0.454656\pi$
$390$	0	0
$391$	38.8500	1.96473
$392$	1.88644	0.0952797
$393$	0	0
$394$	−8.49247	−0.427844
$395$	6.42974	0.323515
$396$	0	0
$397$	−1.55324	−0.0779549	−0.0389775	−	0.999240i	$-0.512410\pi$
$397$	−1.55324	−0.0779549	−0.0389775	+	0.999240i	$0.512410\pi$
$398$	−18.6785	−0.936271
$399$	0	0
$400$	0.200472	0.0100236
$401$	9.47750	0.473284	0.236642	−	0.971597i	$-0.423953\pi$
$401$	9.47750	0.473284	0.236642	+	0.971597i	$0.423953\pi$
$402$	0	0
$403$	−19.6652	−0.979594
$404$	42.4739	2.11316
$405$	0	0
$406$	−26.6116	−1.32071
$407$	17.3531	0.860164
$408$	0	0
$409$	31.1537	1.54045	0.770227	−	0.637770i	$-0.220143\pi$
$409$	31.1537	1.54045	0.770227	+	0.637770i	$0.220143\pi$
$410$	−15.2425	−0.752775
$411$	0	0
$412$	18.1587	0.894617
$413$	5.35758	0.263630
$414$	0	0
$415$	−12.3558	−0.606521
$416$	33.9028	1.66222
$417$	0	0
$418$	−17.9458	−0.877755
$419$	16.0863	0.785868	0.392934	−	0.919567i	$-0.371460\pi$
$419$	16.0863	0.785868	0.392934	+	0.919567i	$0.371460\pi$
$420$	0	0
$421$	−27.3244	−1.33171	−0.665855	−	0.746081i	$-0.731933\pi$
$421$	−27.3244	−1.33171	−0.665855	+	0.746081i	$0.731933\pi$
$422$	−20.6998	−1.00765
$423$	0	0
$424$	8.44299	0.410028
$425$	−4.10410	−0.199078
$426$	0	0
$427$	30.1032	1.45680
$428$	51.7426	2.50107
$429$	0	0
$430$	10.0871	0.486443
$431$	−12.7776	−0.615476	−0.307738	−	0.951471i	$-0.599572\pi$
$431$	−12.7776	−0.615476	−0.307738	+	0.951471i	$0.599572\pi$
$432$	0	0
$433$	−8.26500	−0.397190	−0.198595	−	0.980082i	$-0.563638\pi$
$433$	−8.26500	−0.397190	−0.198595	+	0.980082i	$0.563638\pi$
$434$	−19.9850	−0.959312
$435$	0	0
$436$	57.3727	2.74765
$437$	44.2492	2.11673
$438$	0	0
$439$	−27.6182	−1.31814	−0.659072	−	0.752080i	$-0.729051\pi$
$439$	−27.6182	−1.31814	−0.659072	+	0.752080i	$0.729051\pi$
$440$	4.91579	0.234351
$441$	0	0
$442$	58.9470	2.80382
$443$	31.7136	1.50676	0.753380	−	0.657586i	$-0.228422\pi$
$443$	31.7136	1.50676	0.753380	+	0.657586i	$0.228422\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	−34.4876	−1.63304
$447$	0	0
$448$	35.5625	1.68017
$449$	17.0273	0.803569	0.401784	−	0.915734i	$-0.368390\pi$
$449$	17.0273	0.803569	0.401784	+	0.915734i	$0.368390\pi$
$450$	0	0
$451$	−11.0819	−0.521827
$452$	23.0785	1.08552
$453$	0	0
$454$	40.8763	1.91842
$455$	17.2777	0.809993
$456$	0	0
$457$	−9.81406	−0.459082	−0.229541	−	0.973299i	$-0.573723\pi$
$457$	−9.81406	−0.459082	−0.229541	+	0.973299i	$0.573723\pi$
$458$	42.5492	1.98819
$459$	0	0
$460$	−31.0533	−1.44787
$461$	29.6425	1.38059	0.690295	−	0.723528i	$-0.257481\pi$
$461$	29.6425	1.38059	0.690295	+	0.723528i	$0.257481\pi$
$462$	0	0
$463$	−3.12392	−0.145181	−0.0725903	−	0.997362i	$-0.523127\pi$
$463$	−3.12392	−0.145181	−0.0725903	+	0.997362i	$0.523127\pi$
$464$	0.839866	0.0389898
$465$	0	0
$466$	−29.4244	−1.36306
$467$	−3.98797	−0.184541	−0.0922707	−	0.995734i	$-0.529412\pi$
$467$	−3.98797	−0.184541	−0.0922707	+	0.995734i	$0.529412\pi$
$468$	0	0
$469$	−27.6991	−1.27902
$470$	−18.6187	−0.858816
$471$	0	0
$472$	−5.70283	−0.262494
$473$	7.33372	0.337205
$474$	0	0
$475$	−4.67447	−0.214480
$476$	37.2161	1.70580
$477$	0	0
$478$	−27.4063	−1.25354
$479$	41.8493	1.91214	0.956071	−	0.293136i	$-0.0946988\pi$
$479$	41.8493	1.91214	0.956071	+	0.293136i	$0.0946988\pi$
$480$	0	0
$481$	−64.9222	−2.96020
$482$	23.4618	1.06865
$483$	0	0
$484$	−26.9287	−1.22403
$485$	−11.2411	−0.510434
$486$	0	0
$487$	−7.63362	−0.345912	−0.172956	−	0.984930i	$-0.555332\pi$
$487$	−7.63362	−0.345912	−0.172956	+	0.984930i	$0.555332\pi$
$488$	−32.0431	−1.45052
$489$	0	0
$490$	1.47326	0.0665551
$491$	43.1517	1.94741	0.973704	−	0.227818i	$-0.0731590\pi$
$491$	43.1517	1.94741	0.973704	+	0.227818i	$0.0731590\pi$
$492$	0	0
$493$	−17.1939	−0.774375
$494$	67.1392	3.02074
$495$	0	0
$496$	0.630730	0.0283206
$497$	−31.4851	−1.41230
$498$	0	0
$499$	12.1230	0.542700	0.271350	−	0.962481i	$-0.412530\pi$
$499$	12.1230	0.542700	0.271350	+	0.962481i	$0.412530\pi$
$500$	3.28045	0.146706
$501$	0	0
$502$	−2.39322	−0.106815
$503$	−24.0532	−1.07248	−0.536240	−	0.844066i	$-0.680156\pi$
$503$	−24.0532	−1.07248	−0.536240	+	0.844066i	$0.680156\pi$
$504$	0	0
$505$	12.9476	0.576159
$506$	−36.3414	−1.61557
$507$	0	0
$508$	−62.2872	−2.76355
$509$	−34.1519	−1.51376	−0.756878	−	0.653556i	$-0.773276\pi$
$509$	−34.1519	−1.51376	−0.756878	+	0.653556i	$0.773276\pi$
$510$	0	0
$511$	−0.910587	−0.0402820
$512$	−2.26711	−0.100193
$513$	0	0
$514$	−55.1062	−2.43063
$515$	5.53544	0.243920
$516$	0	0
$517$	−13.5365	−0.595335
$518$	−65.9780	−2.89891
$519$	0	0
$520$	−18.3911	−0.806503
$521$	14.0598	0.615973	0.307986	−	0.951391i	$-0.400345\pi$
$521$	14.0598	0.615973	0.307986	+	0.951391i	$0.400345\pi$
$522$	0	0
$523$	−32.1744	−1.40689	−0.703444	−	0.710751i	$-0.748355\pi$
$523$	−32.1744	−1.40689	−0.703444	+	0.710751i	$0.748355\pi$
$524$	30.6250	1.33786
$525$	0	0
$526$	−56.5237	−2.46455
$527$	−12.9124	−0.562475
$528$	0	0
$529$	66.6079	2.89599
$530$	6.59374	0.286414
$531$	0	0
$532$	42.3882	1.83776
$533$	41.4600	1.79583
$534$	0	0
$535$	15.7730	0.681926
$536$	29.4840	1.27351
$537$	0	0
$538$	67.7870	2.92251
$539$	1.07112	0.0461363
$540$	0	0
$541$	−39.4599	−1.69651	−0.848256	−	0.529587i	$-0.822347\pi$
$541$	−39.4599	−1.69651	−0.848256	+	0.529587i	$0.822347\pi$
$542$	−32.5720	−1.39909
$543$	0	0
$544$	22.2611	0.954435
$545$	17.4892	0.749157
$546$	0	0
$547$	−25.2883	−1.08125	−0.540625	−	0.841263i	$-0.681812\pi$
$547$	−25.2883	−1.08125	−0.540625	+	0.841263i	$0.681812\pi$
$548$	34.3827	1.46876
$549$	0	0
$550$	3.83910	0.163700
$551$	−19.5835	−0.834284
$552$	0	0
$553$	−17.7735	−0.755804
$554$	−46.7166	−1.98480
$555$	0	0
$556$	−19.7551	−0.837804
$557$	34.9103	1.47920	0.739599	−	0.673048i	$-0.235015\pi$
$557$	34.9103	1.47920	0.739599	+	0.673048i	$0.235015\pi$
$558$	0	0
$559$	−27.4371	−1.16047
$560$	−0.554156	−0.0234174
$561$	0	0
$562$	−41.3847	−1.74571
$563$	12.4859	0.526219	0.263109	−	0.964766i	$-0.415252\pi$
$563$	12.4859	0.526219	0.263109	+	0.964766i	$0.415252\pi$
$564$	0	0
$565$	7.03514	0.295971
$566$	−32.9335	−1.38430
$567$	0	0
$568$	33.5140	1.40622
$569$	−11.8067	−0.494962	−0.247481	−	0.968893i	$-0.579603\pi$
$569$	−11.8067	−0.494962	−0.247481	+	0.968893i	$0.579603\pi$
$570$	0	0
$571$	−41.6476	−1.74290	−0.871449	−	0.490486i	$-0.836819\pi$
$571$	−41.6476	−1.74290	−0.871449	+	0.490486i	$0.836819\pi$
$572$	−34.2559	−1.43231
$573$	0	0
$574$	42.1343	1.75865
$575$	−9.46614	−0.394765
$576$	0	0
$577$	14.1333	0.588378	0.294189	−	0.955747i	$-0.404950\pi$
$577$	14.1333	0.588378	0.294189	+	0.955747i	$0.404950\pi$
$578$	−0.359336	−0.0149464
$579$	0	0
$580$	13.7433	0.570659
$581$	34.1546	1.41697
$582$	0	0
$583$	4.79391	0.198543
$584$	0.969264	0.0401084
$585$	0	0
$586$	−7.56273	−0.312414
$587$	−13.6225	−0.562260	−0.281130	−	0.959670i	$-0.590709\pi$
$587$	−13.6225	−0.562260	−0.281130	+	0.959670i	$0.590709\pi$
$588$	0	0
$589$	−14.7070	−0.605990
$590$	−4.45375	−0.183358
$591$	0	0
$592$	2.08228	0.0855810
$593$	7.26420	0.298305	0.149152	−	0.988814i	$-0.452346\pi$
$593$	7.26420	0.298305	0.149152	+	0.988814i	$0.452346\pi$
$594$	0	0
$595$	11.3448	0.465091
$596$	−50.8943	−2.08471
$597$	0	0
$598$	135.962	5.55989
$599$	7.24921	0.296195	0.148097	−	0.988973i	$-0.452685\pi$
$599$	7.24921	0.296195	0.148097	+	0.988973i	$0.452685\pi$
$600$	0	0
$601$	16.6264	0.678206	0.339103	−	0.940749i	$-0.389877\pi$
$601$	16.6264	0.678206	0.339103	+	0.940749i	$0.389877\pi$
$602$	−27.8834	−1.13644
$603$	0	0
$604$	−64.3287	−2.61750
$605$	−8.20883	−0.333736
$606$	0	0
$607$	1.38060	0.0560370	0.0280185	−	0.999607i	$-0.491080\pi$
$607$	1.38060	0.0560370	0.0280185	+	0.999607i	$0.491080\pi$
$608$	25.3548	1.02827
$609$	0	0
$610$	−25.0248	−1.01322
$611$	50.6432	2.04881
$612$	0	0
$613$	33.5725	1.35598	0.677990	−	0.735071i	$-0.262851\pi$
$613$	33.5725	1.35598	0.677990	+	0.735071i	$0.262851\pi$
$614$	−51.3431	−2.07204
$615$	0	0
$616$	−13.5885	−0.547496
$617$	−26.1073	−1.05104	−0.525521	−	0.850781i	$-0.676130\pi$
$617$	−26.1073	−1.05104	−0.525521	+	0.850781i	$0.676130\pi$
$618$	0	0
$619$	−26.6976	−1.07307	−0.536533	−	0.843879i	$-0.680266\pi$
$619$	−26.6976	−1.07307	−0.536533	+	0.843879i	$0.680266\pi$
$620$	10.3211	0.414504
$621$	0	0
$622$	−23.3707	−0.937079
$623$	−2.76426	−0.110748
$624$	0	0
$625$	1.00000	0.0400000
$626$	−38.6457	−1.54459
$627$	0	0
$628$	−1.79141	−0.0714851
$629$	−42.6288	−1.69972
$630$	0	0
$631$	−3.75181	−0.149357	−0.0746787	−	0.997208i	$-0.523793\pi$
$631$	−3.75181	−0.149357	−0.0746787	+	0.997208i	$0.523793\pi$
$632$	18.9188	0.752548
$633$	0	0
$634$	−3.30068	−0.131087
$635$	−18.9874	−0.753491
$636$	0	0
$637$	−4.00730	−0.158775
$638$	16.0837	0.636759
$639$	0	0
$640$	−18.7149	−0.739770
$641$	−39.1848	−1.54771	−0.773853	−	0.633365i	$-0.781673\pi$
$641$	−39.1848	−1.54771	−0.773853	+	0.633365i	$0.781673\pi$
$642$	0	0
$643$	−21.2124	−0.836536	−0.418268	−	0.908324i	$-0.637363\pi$
$643$	−21.2124	−0.836536	−0.418268	+	0.908324i	$0.637363\pi$
$644$	85.8392	3.38254
$645$	0	0
$646$	44.0845	1.73448
$647$	5.68895	0.223656	0.111828	−	0.993728i	$-0.464329\pi$
$647$	5.68895	0.223656	0.111828	+	0.993728i	$0.464329\pi$
$648$	0	0
$649$	−3.23805	−0.127105
$650$	−14.3630	−0.563361
$651$	0	0
$652$	−28.8875	−1.13132
$653$	−32.2198	−1.26086	−0.630428	−	0.776248i	$-0.717121\pi$
$653$	−32.2198	−1.26086	−0.630428	+	0.776248i	$0.717121\pi$
$654$	0	0
$655$	9.33559	0.364772
$656$	−1.32976	−0.0519186
$657$	0	0
$658$	51.4669	2.00639
$659$	−24.0521	−0.936936	−0.468468	−	0.883481i	$-0.655194\pi$
$659$	−24.0521	−0.936936	−0.468468	+	0.883481i	$0.655194\pi$
$660$	0	0
$661$	−10.4564	−0.406708	−0.203354	−	0.979105i	$-0.565184\pi$
$661$	−10.4564	−0.406708	−0.203354	+	0.979105i	$0.565184\pi$
$662$	2.88646	0.112186
$663$	0	0
$664$	−36.3555	−1.41087
$665$	12.9215	0.501073
$666$	0	0
$667$	−39.6579	−1.53556
$668$	−11.0781	−0.428623
$669$	0	0
$670$	23.0262	0.889579
$671$	−18.1940	−0.702371
$672$	0	0
$673$	−15.3040	−0.589925	−0.294963	−	0.955509i	$-0.595307\pi$
$673$	−15.3040	−0.589925	−0.294963	+	0.955509i	$0.595307\pi$
$674$	−5.76752	−0.222157
$675$	0	0
$676$	85.5134	3.28898
$677$	−4.43937	−0.170619	−0.0853094	−	0.996355i	$-0.527188\pi$
$677$	−4.43937	−0.170619	−0.0853094	+	0.996355i	$0.527188\pi$
$678$	0	0
$679$	31.0734	1.19249
$680$	−12.0758	−0.463087
$681$	0	0
$682$	12.0787	0.462517
$683$	10.8069	0.413514	0.206757	−	0.978392i	$-0.433709\pi$
$683$	10.8069	0.413514	0.206757	+	0.978392i	$0.433709\pi$
$684$	0	0
$685$	10.4811	0.400461
$686$	40.3919	1.54217
$687$	0	0
$688$	0.880003	0.0335498
$689$	−17.9351	−0.683274
$690$	0	0
$691$	−30.5624	−1.16265	−0.581324	−	0.813672i	$-0.697465\pi$
$691$	−30.5624	−1.16265	−0.581324	+	0.813672i	$0.697465\pi$
$692$	−33.0738	−1.25728
$693$	0	0
$694$	70.4620	2.67470
$695$	−6.02207	−0.228430
$696$	0	0
$697$	27.2232	1.03115
$698$	−8.36643	−0.316674
$699$	0	0
$700$	−9.06802	−0.342739
$701$	−2.93500	−0.110853	−0.0554267	−	0.998463i	$-0.517652\pi$
$701$	−2.93500	−0.110853	−0.0554267	+	0.998463i	$0.517652\pi$
$702$	0	0
$703$	−48.5532	−1.83122
$704$	−21.4935	−0.810067
$705$	0	0
$706$	34.1337	1.28464
$707$	−35.7904	−1.34604
$708$	0	0
$709$	37.7930	1.41935	0.709673	−	0.704532i	$-0.248843\pi$
$709$	37.7930	1.41935	0.709673	+	0.704532i	$0.248843\pi$
$710$	26.1736	0.982276
$711$	0	0
$712$	2.94239	0.110271
$713$	−29.7827	−1.11537
$714$	0	0
$715$	−10.4424	−0.390525
$716$	−47.0831	−1.75958
$717$	0	0
$718$	−26.1065	−0.974287
$719$	42.0546	1.56837	0.784187	−	0.620525i	$-0.213080\pi$
$719$	42.0546	1.56837	0.784187	+	0.620525i	$0.213080\pi$
$720$	0	0
$721$	−15.3014	−0.569853
$722$	6.55071	0.243792
$723$	0	0
$724$	69.3801	2.57849
$725$	4.18945	0.155592
$726$	0	0
$727$	−11.3299	−0.420202	−0.210101	−	0.977680i	$-0.567379\pi$
$727$	−11.3299	−0.420202	−0.210101	+	0.977680i	$0.567379\pi$
$728$	50.8378	1.88417
$729$	0	0
$730$	0.756969	0.0280167
$731$	−18.0156	−0.666331
$732$	0	0
$733$	−19.0442	−0.703413	−0.351707	−	0.936110i	$-0.614399\pi$
$733$	−19.0442	−0.703413	−0.351707	+	0.936110i	$0.614399\pi$
$734$	74.7322	2.75841
$735$	0	0
$736$	51.3453	1.89261
$737$	16.7409	0.616660
$738$	0	0
$739$	41.6836	1.53336	0.766678	−	0.642032i	$-0.221908\pi$
$739$	41.6836	1.53336	0.766678	+	0.642032i	$0.221908\pi$
$740$	34.0737	1.25257
$741$	0	0
$742$	−18.2268	−0.669127
$743$	−34.8399	−1.27815	−0.639076	−	0.769144i	$-0.720683\pi$
$743$	−34.8399	−1.27815	−0.639076	+	0.769144i	$0.720683\pi$
$744$	0	0
$745$	−15.5144	−0.568404
$746$	50.2813	1.84093
$747$	0	0
$748$	−22.4929	−0.822421
$749$	−43.6006	−1.59313
$750$	0	0
$751$	−35.0492	−1.27896	−0.639482	−	0.768806i	$-0.720851\pi$
$751$	−35.0492	−1.27896	−0.639482	+	0.768806i	$0.720851\pi$
$752$	−1.62430	−0.0592322
$753$	0	0
$754$	−60.1728	−2.19137
$755$	−19.6097	−0.713670
$756$	0	0
$757$	38.0039	1.38128	0.690638	−	0.723201i	$-0.257330\pi$
$757$	38.0039	1.38128	0.690638	+	0.723201i	$0.257330\pi$
$758$	43.6977	1.58717
$759$	0	0
$760$	−13.7541	−0.498914
$761$	1.89193	0.0685824	0.0342912	−	0.999412i	$-0.489083\pi$
$761$	1.89193	0.0685824	0.0342912	+	0.999412i	$0.489083\pi$
$762$	0	0
$763$	−48.3448	−1.75020
$764$	−1.29147	−0.0467238
$765$	0	0
$766$	−40.4870	−1.46285
$767$	12.1143	0.437422
$768$	0	0
$769$	31.8756	1.14946	0.574731	−	0.818342i	$-0.305106\pi$
$769$	31.8756	1.14946	0.574731	+	0.818342i	$0.305106\pi$
$770$	−10.6123	−0.382439
$771$	0	0
$772$	−57.0841	−2.05450
$773$	27.1169	0.975327	0.487664	−	0.873032i	$-0.337849\pi$
$773$	27.1169	0.975327	0.487664	+	0.873032i	$0.337849\pi$
$774$	0	0
$775$	3.14623	0.113016
$776$	−33.0758	−1.18735
$777$	0	0
$778$	12.8687	0.461367
$779$	31.0066	1.11093
$780$	0	0
$781$	19.0292	0.680918
$782$	89.2743	3.19244
$783$	0	0
$784$	0.128528	0.00459028
$785$	−0.546086	−0.0194907
$786$	0	0
$787$	19.4628	0.693775	0.346887	−	0.937907i	$-0.387239\pi$
$787$	19.4628	0.693775	0.346887	+	0.937907i	$0.387239\pi$
$788$	−12.1236	−0.431886
$789$	0	0
$790$	14.7750	0.525672
$791$	−19.4469	−0.691454
$792$	0	0
$793$	68.0679	2.41716
$794$	−3.56923	−0.126667
$795$	0	0
$796$	−26.6650	−0.945116
$797$	20.5091	0.726470	0.363235	−	0.931698i	$-0.381672\pi$
$797$	20.5091	0.726470	0.363235	+	0.931698i	$0.381672\pi$
$798$	0	0
$799$	33.2530	1.17641
$800$	−5.42410	−0.191771
$801$	0	0
$802$	21.7786	0.769028
$803$	0.550346	0.0194213
$804$	0	0
$805$	26.1669	0.922261
$806$	−45.1892	−1.59172
$807$	0	0
$808$	38.0968	1.34024
$809$	7.06675	0.248454	0.124227	−	0.992254i	$-0.460355\pi$
$809$	7.06675	0.248454	0.124227	+	0.992254i	$0.460355\pi$
$810$	0	0
$811$	10.6174	0.372827	0.186414	−	0.982471i	$-0.440314\pi$
$811$	10.6174	0.372827	0.186414	+	0.982471i	$0.440314\pi$
$812$	−37.9900	−1.33319
$813$	0	0
$814$	39.8762	1.39766
$815$	−8.80594	−0.308459
$816$	0	0
$817$	−20.5194	−0.717881
$818$	71.5889	2.50305
$819$	0	0
$820$	−21.7598	−0.759886
$821$	33.7372	1.17743	0.588717	−	0.808339i	$-0.299633\pi$
$821$	33.7372	1.17743	0.588717	+	0.808339i	$0.299633\pi$
$822$	0	0
$823$	29.3592	1.02340	0.511699	−	0.859164i	$-0.329016\pi$
$823$	29.3592	1.02340	0.511699	+	0.859164i	$0.329016\pi$
$824$	16.2874	0.567398
$825$	0	0
$826$	12.3113	0.428366
$827$	−49.6440	−1.72629	−0.863146	−	0.504955i	$-0.831509\pi$
$827$	−49.6440	−1.72629	−0.863146	+	0.504955i	$0.831509\pi$
$828$	0	0
$829$	2.92242	0.101500	0.0507498	−	0.998711i	$-0.483839\pi$
$829$	2.92242	0.101500	0.0507498	+	0.998711i	$0.483839\pi$
$830$	−28.3926	−0.985523
$831$	0	0
$832$	80.4122	2.78779
$833$	−2.63125	−0.0911673
$834$	0	0
$835$	−3.37699	−0.116866
$836$	−25.6189	−0.886047
$837$	0	0
$838$	36.9651	1.27694
$839$	54.8461	1.89350	0.946749	−	0.321971i	$-0.104345\pi$
$839$	54.8461	1.89350	0.946749	+	0.321971i	$0.104345\pi$
$840$	0	0
$841$	−11.4485	−0.394777
$842$	−62.7894	−2.16387
$843$	0	0
$844$	−29.5505	−1.01717
$845$	26.0675	0.896751
$846$	0	0
$847$	22.6913	0.779683
$848$	0.575241	0.0197538
$849$	0	0
$850$	−9.43091	−0.323477
$851$	−98.3237	−3.37049
$852$	0	0
$853$	38.6230	1.32243	0.661214	−	0.750197i	$-0.270042\pi$
$853$	38.6230	1.32243	0.661214	+	0.750197i	$0.270042\pi$
$854$	69.1749	2.36712
$855$	0	0
$856$	46.4102	1.58627
$857$	4.80167	0.164022	0.0820110	−	0.996631i	$-0.473866\pi$
$857$	4.80167	0.164022	0.0820110	+	0.996631i	$0.473866\pi$
$858$	0	0
$859$	−38.6432	−1.31849	−0.659245	−	0.751928i	$-0.729124\pi$
$859$	−38.6432	−1.31849	−0.659245	+	0.751928i	$0.729124\pi$
$860$	14.4001	0.491039
$861$	0	0
$862$	−29.3620	−1.00007
$863$	6.22230	0.211810	0.105905	−	0.994376i	$-0.466226\pi$
$863$	6.22230	0.211810	0.105905	+	0.994376i	$0.466226\pi$
$864$	0	0
$865$	−10.0821	−0.342801
$866$	−18.9923	−0.645386
$867$	0	0
$868$	−28.5301	−0.968375
$869$	10.7420	0.364398
$870$	0	0
$871$	−62.6317	−2.12219
$872$	51.4601	1.74266
$873$	0	0
$874$	101.681	3.43942
$875$	−2.76426	−0.0934490
$876$	0	0
$877$	50.6442	1.71013	0.855067	−	0.518517i	$-0.173516\pi$
$877$	50.6442	1.71013	0.855067	+	0.518517i	$0.173516\pi$
$878$	−63.4645	−2.14182
$879$	0	0
$880$	0.334924	0.0112903
$881$	−37.9258	−1.27775	−0.638876	−	0.769310i	$-0.720600\pi$
$881$	−37.9258	−1.27775	−0.638876	+	0.769310i	$0.720600\pi$
$882$	0	0
$883$	41.9932	1.41318	0.706592	−	0.707621i	$-0.250231\pi$
$883$	41.9932	1.41318	0.706592	+	0.707621i	$0.250231\pi$
$884$	84.1511	2.83031
$885$	0	0
$886$	72.8755	2.44830
$887$	6.29776	0.211458	0.105729	−	0.994395i	$-0.466282\pi$
$887$	6.29776	0.211458	0.105729	+	0.994395i	$0.466282\pi$
$888$	0	0
$889$	52.4860	1.76033
$890$	2.29792	0.0770266
$891$	0	0
$892$	−49.2336	−1.64846
$893$	37.8744	1.26742
$894$	0	0
$895$	−14.3526	−0.479755
$896$	51.7327	1.72827
$897$	0	0
$898$	39.1275	1.30570
$899$	13.1810	0.439610
$900$	0	0
$901$	−11.7764	−0.392330
$902$	−25.4654	−0.847905
$903$	0	0
$904$	20.7001	0.688475
$905$	21.1496	0.703035
$906$	0	0
$907$	27.1726	0.902252	0.451126	−	0.892460i	$-0.351023\pi$
$907$	27.1726	0.902252	0.451126	+	0.892460i	$0.351023\pi$
$908$	58.3540	1.93654
$909$	0	0
$910$	39.7029	1.31614
$911$	−16.6298	−0.550970	−0.275485	−	0.961305i	$-0.588838\pi$
$911$	−16.6298	−0.550970	−0.275485	+	0.961305i	$0.588838\pi$
$912$	0	0
$913$	−20.6426	−0.683169
$914$	−22.5520	−0.745953
$915$	0	0
$916$	60.7421	2.00698
$917$	−25.8060	−0.852189
$918$	0	0
$919$	−33.6811	−1.11104	−0.555518	−	0.831504i	$-0.687480\pi$
$919$	−33.6811	−1.11104	−0.555518	+	0.831504i	$0.687480\pi$
$920$	−27.8531	−0.918288
$921$	0	0
$922$	68.1163	2.24329
$923$	−71.1926	−2.34333
$924$	0	0
$925$	10.3869	0.341518
$926$	−7.17852	−0.235901
$927$	0	0
$928$	−22.7240	−0.745952
$929$	−18.3051	−0.600570	−0.300285	−	0.953849i	$-0.597082\pi$
$929$	−18.3051	−0.600570	−0.300285	+	0.953849i	$0.597082\pi$
$930$	0	0
$931$	−2.99693	−0.0982204
$932$	−42.0054	−1.37593
$933$	0	0
$934$	−9.16406	−0.299857
$935$	−6.85664	−0.224236
$936$	0	0
$937$	−54.2231	−1.77139	−0.885695	−	0.464267i	$-0.846318\pi$
$937$	−54.2231	−1.77139	−0.885695	+	0.464267i	$0.846318\pi$
$938$	−63.6503	−2.07826
$939$	0	0
$940$	−26.5795	−0.866929
$941$	−36.6237	−1.19390	−0.596949	−	0.802279i	$-0.703621\pi$
$941$	−36.6237	−1.19390	−0.596949	+	0.802279i	$0.703621\pi$
$942$	0	0
$943$	62.7906	2.04474
$944$	−0.388547	−0.0126461
$945$	0	0
$946$	16.8523	0.547916
$947$	34.9651	1.13621	0.568106	−	0.822955i	$-0.307676\pi$
$947$	34.9651	1.13621	0.568106	+	0.822955i	$0.307676\pi$
$948$	0	0
$949$	−2.05897	−0.0668371
$950$	−10.7416	−0.348503
$951$	0	0
$952$	33.3808	1.08188
$953$	24.1093	0.780978	0.390489	−	0.920608i	$-0.372306\pi$
$953$	24.1093	0.780978	0.390489	+	0.920608i	$0.372306\pi$
$954$	0	0
$955$	−0.393687	−0.0127394
$956$	−39.1246	−1.26538
$957$	0	0
$958$	96.1664	3.10700
$959$	−28.9724	−0.935568
$960$	0	0
$961$	−21.1012	−0.680685
$962$	−149.186	−4.80996
$963$	0	0
$964$	33.4934	1.07875
$965$	−17.4013	−0.560167
$966$	0	0
$967$	−7.70982	−0.247931	−0.123966	−	0.992287i	$-0.539561\pi$
$967$	−7.70982	−0.247931	−0.123966	+	0.992287i	$0.539561\pi$
$968$	−24.1535	−0.776324
$969$	0	0
$970$	−25.8313	−0.829393
$971$	42.5171	1.36444	0.682219	−	0.731148i	$-0.261015\pi$
$971$	42.5171	1.36444	0.682219	+	0.731148i	$0.261015\pi$
$972$	0	0
$973$	16.6466	0.533664
$974$	−17.5415	−0.562065
$975$	0	0
$976$	−2.18317	−0.0698816
$977$	−40.8033	−1.30541	−0.652706	−	0.757611i	$-0.726366\pi$
$977$	−40.8033	−1.30541	−0.652706	+	0.757611i	$0.726366\pi$
$978$	0	0
$979$	1.67068	0.0533952
$980$	2.10319	0.0671838
$981$	0	0
$982$	99.1593	3.16430
$983$	−14.2860	−0.455654	−0.227827	−	0.973702i	$-0.573162\pi$
$983$	−14.2860	−0.455654	−0.227827	+	0.973702i	$0.573162\pi$
$984$	0	0
$985$	−3.69571	−0.117755
$986$	−39.5103	−1.25826
$987$	0	0
$988$	95.8462	3.04927
$989$	−41.5532	−1.32131
$990$	0	0
$991$	18.0018	0.571845	0.285922	−	0.958253i	$-0.407700\pi$
$991$	18.0018	0.571845	0.285922	+	0.958253i	$0.407700\pi$
$992$	−17.0655	−0.541830
$993$	0	0
$994$	−72.3505	−2.29482
$995$	−8.12845	−0.257689
$996$	0	0
$997$	41.6576	1.31931	0.659655	−	0.751568i	$-0.270702\pi$
$997$	41.6576	1.31931	0.659655	+	0.751568i	$0.270702\pi$
$998$	27.8577	0.881821
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.u.1.12	✓	12
3.2	odd	2		4005.2.a.v.1.1	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.12	✓	12	1.1	even	1	trivial
4005.2.a.v.1.1	yes	12	3.2	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)

\(f(q)\)	\(=\)	\(q+2.29792 q^{2} +3.28045 q^{4} +1.00000 q^{5} -2.76426 q^{7} +2.94239 q^{8} +O(q^{10})\)	\(q+2.29792 q^{2} +3.28045 q^{4} +1.00000 q^{5} -2.76426 q^{7} +2.94239 q^{8} +2.29792 q^{10} +1.67068 q^{11} -6.25040 q^{13} -6.35206 q^{14} +0.200472 q^{16} -4.10410 q^{17} -4.67447 q^{19} +3.28045 q^{20} +3.83910 q^{22} -9.46614 q^{23} +1.00000 q^{25} -14.3630 q^{26} -9.06802 q^{28} +4.18945 q^{29} +3.14623 q^{31} -5.42410 q^{32} -9.43091 q^{34} -2.76426 q^{35} +10.3869 q^{37} -10.7416 q^{38} +2.94239 q^{40} -6.63318 q^{41} +4.38966 q^{43} +5.48059 q^{44} -21.7525 q^{46} -8.10240 q^{47} +0.641127 q^{49} +2.29792 q^{50} -20.5042 q^{52} +2.86943 q^{53} +1.67068 q^{55} -8.13352 q^{56} +9.62703 q^{58} -1.93816 q^{59} -10.8902 q^{61} +7.22980 q^{62} -12.8651 q^{64} -6.25040 q^{65} +10.0204 q^{67} -13.4633 q^{68} -6.35206 q^{70} +11.3901 q^{71} +0.329414 q^{73} +23.8683 q^{74} -15.3344 q^{76} -4.61819 q^{77} +6.42974 q^{79} +0.200472 q^{80} -15.2425 q^{82} -12.3558 q^{83} -4.10410 q^{85} +10.0871 q^{86} +4.91579 q^{88} +1.00000 q^{89} +17.2777 q^{91} -31.0533 q^{92} -18.6187 q^{94} -4.67447 q^{95} -11.2411 q^{97} +1.47326 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 11 q^{4} + 12 q^{5} - 8 q^{7} - 9 q^{8} - 3 q^{10} - 12 q^{13} + 4 q^{14} + q^{16} - 24 q^{19} + 11 q^{20} - 16 q^{22} - 24 q^{23} + 12 q^{25} - q^{26} - 44 q^{28} + 8 q^{29} - 12 q^{31} - 31 q^{32} - 18 q^{34} - 8 q^{35} - 10 q^{37} + 2 q^{38} - 9 q^{40} - 10 q^{41} - 42 q^{43} + 42 q^{44} - 24 q^{46} - 22 q^{47} - 4 q^{49} - 3 q^{50} - 30 q^{52} + 8 q^{53} + 27 q^{56} - 12 q^{58} + 4 q^{59} - 52 q^{61} - 14 q^{62} + 7 q^{64} - 12 q^{65} - 40 q^{67} + 23 q^{68} + 4 q^{70} + 2 q^{71} - 8 q^{73} + 26 q^{74} - 46 q^{76} - 12 q^{77} - 26 q^{79} + q^{80} - 26 q^{82} - 14 q^{83} + 32 q^{86} - 60 q^{88} + 12 q^{89} - 24 q^{91} - 38 q^{92} - 26 q^{94} - 24 q^{95} - 6 q^{97} + 30 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 11 * q^4 + 12 * q^5 - 8 * q^7 - 9 * q^8 - 3 * q^10 - 12 * q^13 + 4 * q^14 + q^16 - 24 * q^19 + 11 * q^20 - 16 * q^22 - 24 * q^23 + 12 * q^25 - q^26 - 44 * q^28 + 8 * q^29 - 12 * q^31 - 31 * q^32 - 18 * q^34 - 8 * q^35 - 10 * q^37 + 2 * q^38 - 9 * q^40 - 10 * q^41 - 42 * q^43 + 42 * q^44 - 24 * q^46 - 22 * q^47 - 4 * q^49 - 3 * q^50 - 30 * q^52 + 8 * q^53 + 27 * q^56 - 12 * q^58 + 4 * q^59 - 52 * q^61 - 14 * q^62 + 7 * q^64 - 12 * q^65 - 40 * q^67 + 23 * q^68 + 4 * q^70 + 2 * q^71 - 8 * q^73 + 26 * q^74 - 46 * q^76 - 12 * q^77 - 26 * q^79 + q^80 - 26 * q^82 - 14 * q^83 + 32 * q^86 - 60 * q^88 + 12 * q^89 - 24 * q^91 - 38 * q^92 - 26 * q^94 - 24 * q^95 - 6 * q^97 + 30 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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