LMFDB - Embedded newform 4005.2.a.s.1.5

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:	$10$
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} - x^{9} - 16x^{8} + 15x^{7} + 85x^{6} - 75x^{5} - 163x^{4} + 138x^{3} + 78x^{2} - 67x + 4 $ x^10 - x^9 - 16x^8 + 15x^7 + 85x^6 - 75x^5 - 163x^4 + 138x^3 + 78x^2 - 67x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.5
Root			$0.644507$ 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)

$f(q)$	$=$	$q-0.644507 q^{2} -1.58461 q^{4} -1.00000 q^{5} +2.88508 q^{7} +2.31031 q^{8} +O(q^{10})$	\(q-0.644507 q^{2} -1.58461 q^{4} -1.00000 q^{5} +2.88508 q^{7} +2.31031 q^{8} +0.644507 q^{10} -5.88276 q^{11} -4.54662 q^{13} -1.85945 q^{14} +1.68022 q^{16} +2.13144 q^{17} +1.71518 q^{19} +1.58461 q^{20} +3.79148 q^{22} +7.78484 q^{23} +1.00000 q^{25} +2.93033 q^{26} -4.57173 q^{28} -4.02396 q^{29} +10.2383 q^{31} -5.70352 q^{32} -1.37373 q^{34} -2.88508 q^{35} -2.54782 q^{37} -1.10544 q^{38} -2.31031 q^{40} -0.599157 q^{41} +3.29967 q^{43} +9.32188 q^{44} -5.01738 q^{46} -7.45867 q^{47} +1.32368 q^{49} -0.644507 q^{50} +7.20462 q^{52} -13.3236 q^{53} +5.88276 q^{55} +6.66541 q^{56} +2.59347 q^{58} -3.62830 q^{59} +5.82164 q^{61} -6.59863 q^{62} +0.315526 q^{64} +4.54662 q^{65} +12.5409 q^{67} -3.37750 q^{68} +1.85945 q^{70} +7.41414 q^{71} -3.00317 q^{73} +1.64209 q^{74} -2.71789 q^{76} -16.9722 q^{77} +4.21954 q^{79} -1.68022 q^{80} +0.386160 q^{82} -8.76712 q^{83} -2.13144 q^{85} -2.12666 q^{86} -13.5910 q^{88} +1.00000 q^{89} -13.1174 q^{91} -12.3359 q^{92} +4.80716 q^{94} -1.71518 q^{95} -19.2984 q^{97} -0.853121 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	$ 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) $ 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * 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 $

$2$	−0.644507	−0.455735	−0.227867	−	0.973692i	$-0.573175\pi$
$2$	−0.644507	−0.455735	−0.227867	+	0.973692i	$0.573175\pi$
$3$	0	0
$3$	0	0
$4$	−1.58461	−0.792306
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.88508	1.09046	0.545229	−	0.838287i	$-0.316443\pi$
$7$	2.88508	1.09046	0.545229	+	0.838287i	$0.316443\pi$
$8$	2.31031	0.816816
$9$	0	0
$10$	0.644507	0.203811
$11$	−5.88276	−1.77372	−0.886859	−	0.462040i	$-0.847118\pi$
$11$	−5.88276	−1.77372	−0.886859	+	0.462040i	$0.847118\pi$
$12$	0	0
$13$	−4.54662	−1.26101	−0.630503	−	0.776187i	$-0.717151\pi$
$13$	−4.54662	−1.26101	−0.630503	+	0.776187i	$0.717151\pi$
$14$	−1.85945	−0.496960
$15$	0	0
$16$	1.68022	0.420054
$17$	2.13144	0.516949	0.258475	−	0.966018i	$-0.416780\pi$
$17$	2.13144	0.516949	0.258475	+	0.966018i	$0.416780\pi$
$18$	0	0
$19$	1.71518	0.393488	0.196744	−	0.980455i	$-0.436963\pi$
$19$	1.71518	0.393488	0.196744	+	0.980455i	$0.436963\pi$
$20$	1.58461	0.354330
$21$	0	0
$22$	3.79148	0.808345
$23$	7.78484	1.62325	0.811626	−	0.584178i	$-0.198583\pi$
$23$	7.78484	1.62325	0.811626	+	0.584178i	$0.198583\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.93033	0.574684
$27$	0	0
$28$	−4.57173	−0.863976
$29$	−4.02396	−0.747231	−0.373616	−	0.927584i	$-0.621882\pi$
$29$	−4.02396	−0.747231	−0.373616	+	0.927584i	$0.621882\pi$
$30$	0	0
$31$	10.2383	1.83885	0.919423	−	0.393270i	$-0.128656\pi$
$31$	10.2383	1.83885	0.919423	+	0.393270i	$0.128656\pi$
$32$	−5.70352	−1.00825
$33$	0	0
$34$	−1.37373	−0.235592
$35$	−2.88508	−0.487667
$36$	0	0
$37$	−2.54782	−0.418860	−0.209430	−	0.977824i	$-0.567161\pi$
$37$	−2.54782	−0.418860	−0.209430	+	0.977824i	$0.567161\pi$
$38$	−1.10544	−0.179326
$39$	0	0
$40$	−2.31031	−0.365291
$41$	−0.599157	−0.0935726	−0.0467863	−	0.998905i	$-0.514898\pi$
$41$	−0.599157	−0.0935726	−0.0467863	+	0.998905i	$0.514898\pi$
$42$	0	0
$43$	3.29967	0.503196	0.251598	−	0.967832i	$-0.419044\pi$
$43$	3.29967	0.503196	0.251598	+	0.967832i	$0.419044\pi$
$44$	9.32188	1.40533
$45$	0	0
$46$	−5.01738	−0.739773
$47$	−7.45867	−1.08796	−0.543979	−	0.839099i	$-0.683083\pi$
$47$	−7.45867	−1.08796	−0.543979	+	0.839099i	$0.683083\pi$
$48$	0	0
$49$	1.32368	0.189097
$50$	−0.644507	−0.0911470
$51$	0	0
$52$	7.20462	0.999102
$53$	−13.3236	−1.83013	−0.915065	−	0.403306i	$-0.867861\pi$
$53$	−13.3236	−1.83013	−0.915065	+	0.403306i	$0.867861\pi$
$54$	0	0
$55$	5.88276	0.793231
$56$	6.66541	0.890703
$57$	0	0
$58$	2.59347	0.340539
$59$	−3.62830	−0.472365	−0.236182	−	0.971709i	$-0.575896\pi$
$59$	−3.62830	−0.472365	−0.236182	+	0.971709i	$0.575896\pi$
$60$	0	0
$61$	5.82164	0.745384	0.372692	−	0.927955i	$-0.378435\pi$
$61$	5.82164	0.745384	0.372692	+	0.927955i	$0.378435\pi$
$62$	−6.59863	−0.838027
$63$	0	0
$64$	0.315526	0.0394407
$65$	4.54662	0.563939
$66$	0	0
$67$	12.5409	1.53211	0.766057	−	0.642773i	$-0.222216\pi$
$67$	12.5409	1.53211	0.766057	+	0.642773i	$0.222216\pi$
$68$	−3.37750	−0.409582
$69$	0	0
$70$	1.85945	0.222247
$71$	7.41414	0.879897	0.439949	−	0.898023i	$-0.354997\pi$
$71$	7.41414	0.879897	0.439949	+	0.898023i	$0.354997\pi$
$72$	0	0
$73$	−3.00317	−0.351495	−0.175747	−	0.984435i	$-0.556234\pi$
$73$	−3.00317	−0.351495	−0.175747	+	0.984435i	$0.556234\pi$
$74$	1.64209	0.190889
$75$	0	0
$76$	−2.71789	−0.311763
$77$	−16.9722	−1.93416
$78$	0	0
$79$	4.21954	0.474735	0.237367	−	0.971420i	$-0.423715\pi$
$79$	4.21954	0.474735	0.237367	+	0.971420i	$0.423715\pi$
$80$	−1.68022	−0.187854
$81$	0	0
$82$	0.386160	0.0426443
$83$	−8.76712	−0.962316	−0.481158	−	0.876634i	$-0.659784\pi$
$83$	−8.76712	−0.962316	−0.481158	+	0.876634i	$0.659784\pi$
$84$	0	0
$85$	−2.13144	−0.231187
$86$	−2.12666	−0.229324
$87$	0	0
$88$	−13.5910	−1.44880
$89$	1.00000	0.106000
$89$	1.00000	0.106000
$90$	0	0
$91$	−13.1174	−1.37507
$92$	−12.3359	−1.28611
$93$	0	0
$94$	4.80716	0.495821
$95$	−1.71518	−0.175973
$96$	0	0
$97$	−19.2984	−1.95946	−0.979730	−	0.200322i	$-0.935801\pi$
$97$	−19.2984	−1.95946	−0.979730	+	0.200322i	$0.935801\pi$
$98$	−0.853121	−0.0861782
$99$	0	0
$100$	−1.58461	−0.158461
$101$	−11.2727	−1.12167	−0.560837	−	0.827926i	$-0.689521\pi$
$101$	−11.2727	−1.12167	−0.560837	+	0.827926i	$0.689521\pi$
$102$	0	0
$103$	−0.554712	−0.0546574	−0.0273287	−	0.999627i	$-0.508700\pi$
$103$	−0.554712	−0.0546574	−0.0273287	+	0.999627i	$0.508700\pi$
$104$	−10.5041	−1.03001
$105$	0	0
$106$	8.58712	0.834054
$107$	16.7880	1.62296	0.811479	−	0.584382i	$-0.198663\pi$
$107$	16.7880	1.62296	0.811479	+	0.584382i	$0.198663\pi$
$108$	0	0
$109$	1.19794	0.114742	0.0573710	−	0.998353i	$-0.481728\pi$
$109$	1.19794	0.114742	0.0573710	+	0.998353i	$0.481728\pi$
$110$	−3.79148	−0.361503
$111$	0	0
$112$	4.84755	0.458051
$113$	−3.38460	−0.318397	−0.159198	−	0.987247i	$-0.550891\pi$
$113$	−3.38460	−0.318397	−0.159198	+	0.987247i	$0.550891\pi$
$114$	0	0
$115$	−7.78484	−0.725940
$116$	6.37642	0.592035
$117$	0	0
$118$	2.33847	0.215273
$119$	6.14936	0.563711
$120$	0	0
$121$	23.6068	2.14608
$122$	−3.75208	−0.339698
$123$	0	0
$124$	−16.2237	−1.45693
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	15.2779	1.35570	0.677848	−	0.735202i	$-0.262913\pi$
$127$	15.2779	1.35570	0.677848	+	0.735202i	$0.262913\pi$
$128$	11.2037	0.990275
$129$	0	0
$130$	−2.93033	−0.257007
$131$	2.21981	0.193946	0.0969730	−	0.995287i	$-0.469084\pi$
$131$	2.21981	0.193946	0.0969730	+	0.995287i	$0.469084\pi$
$132$	0	0
$133$	4.94842	0.429082
$134$	−8.08269	−0.698238
$135$	0	0
$136$	4.92427	0.422253
$137$	−11.0702	−0.945790	−0.472895	−	0.881119i	$-0.656791\pi$
$137$	−11.0702	−0.945790	−0.472895	+	0.881119i	$0.656791\pi$
$138$	0	0
$139$	−7.47813	−0.634286	−0.317143	−	0.948378i	$-0.602724\pi$
$139$	−7.47813	−0.634286	−0.317143	+	0.948378i	$0.602724\pi$
$140$	4.57173	0.386382
$141$	0	0
$142$	−4.77846	−0.401000
$143$	26.7467	2.23667
$144$	0	0
$145$	4.02396	0.334172
$146$	1.93557	0.160189
$147$	0	0
$148$	4.03731	0.331865
$149$	16.2609	1.33215	0.666073	−	0.745886i	$-0.267974\pi$
$149$	16.2609	1.33215	0.666073	+	0.745886i	$0.267974\pi$
$150$	0	0
$151$	−11.2356	−0.914344	−0.457172	−	0.889378i	$-0.651138\pi$
$151$	−11.2356	−0.914344	−0.457172	+	0.889378i	$0.651138\pi$
$152$	3.96258	0.321408
$153$	0	0
$154$	10.9387	0.881466
$155$	−10.2383	−0.822357
$156$	0	0
$157$	−15.8532	−1.26523	−0.632613	−	0.774468i	$-0.718018\pi$
$157$	−15.8532	−1.26523	−0.632613	+	0.774468i	$0.718018\pi$
$158$	−2.71952	−0.216353
$159$	0	0
$160$	5.70352	0.450903
$161$	22.4599	1.77009
$162$	0	0
$163$	−7.85634	−0.615356	−0.307678	−	0.951491i	$-0.599552\pi$
$163$	−7.85634	−0.615356	−0.307678	+	0.951491i	$0.599552\pi$
$164$	0.949430	0.0741381
$165$	0	0
$166$	5.65047	0.438561
$167$	−4.54787	−0.351925	−0.175962	−	0.984397i	$-0.556304\pi$
$167$	−4.54787	−0.351925	−0.175962	+	0.984397i	$0.556304\pi$
$168$	0	0
$169$	7.67175	0.590134
$170$	1.37373	0.105360
$171$	0	0
$172$	−5.22870	−0.398685
$173$	−12.3370	−0.937964	−0.468982	−	0.883208i	$-0.655379\pi$
$173$	−12.3370	−0.937964	−0.468982	+	0.883208i	$0.655379\pi$
$174$	0	0
$175$	2.88508	0.218091
$176$	−9.88430	−0.745057
$177$	0	0
$178$	−0.644507	−0.0483078
$179$	−21.1648	−1.58193	−0.790966	−	0.611860i	$-0.790422\pi$
$179$	−21.1648	−1.58193	−0.790966	+	0.611860i	$0.790422\pi$
$180$	0	0
$181$	−14.4563	−1.07453	−0.537265	−	0.843413i	$-0.680543\pi$
$181$	−14.4563	−1.07453	−0.537265	+	0.843413i	$0.680543\pi$
$182$	8.45422	0.626669
$183$	0	0
$184$	17.9854	1.32590
$185$	2.54782	0.187320
$186$	0	0
$187$	−12.5387	−0.916923
$188$	11.8191	0.861995
$189$	0	0
$190$	1.10544	0.0801972
$191$	−20.4540	−1.48000	−0.740001	−	0.672606i	$-0.765175\pi$
$191$	−20.4540	−1.48000	−0.740001	+	0.672606i	$0.765175\pi$
$192$	0	0
$193$	6.96246	0.501169	0.250584	−	0.968095i	$-0.419377\pi$
$193$	6.96246	0.501169	0.250584	+	0.968095i	$0.419377\pi$
$194$	12.4380	0.892994
$195$	0	0
$196$	−2.09752	−0.149823
$197$	17.4508	1.24332	0.621660	−	0.783287i	$-0.286458\pi$
$197$	17.4508	1.24332	0.621660	+	0.783287i	$0.286458\pi$
$198$	0	0
$199$	−2.96888	−0.210458	−0.105229	−	0.994448i	$-0.533558\pi$
$199$	−2.96888	−0.210458	−0.105229	+	0.994448i	$0.533558\pi$
$200$	2.31031	0.163363
$201$	0	0
$202$	7.26532	0.511186
$203$	−11.6094	−0.814824
$204$	0	0
$205$	0.599157	0.0418469
$206$	0.357516	0.0249093
$207$	0	0
$208$	−7.63930	−0.529690
$209$	−10.0900	−0.697937
$210$	0	0
$211$	−7.54301	−0.519282	−0.259641	−	0.965705i	$-0.583604\pi$
$211$	−7.54301	−0.519282	−0.259641	+	0.965705i	$0.583604\pi$
$212$	21.1126	1.45002
$213$	0	0
$214$	−10.8200	−0.739638
$215$	−3.29967	−0.225036
$216$	0	0
$217$	29.5382	2.00518
$218$	−0.772081	−0.0522920
$219$	0	0
$220$	−9.32188	−0.628481
$221$	−9.69083	−0.651876
$222$	0	0
$223$	−26.1077	−1.74830	−0.874149	−	0.485657i	$-0.838580\pi$
$223$	−26.1077	−1.74830	−0.874149	+	0.485657i	$0.838580\pi$
$224$	−16.4551	−1.09945
$225$	0	0
$226$	2.18140	0.145105
$227$	15.6193	1.03669	0.518344	−	0.855172i	$-0.326549\pi$
$227$	15.6193	1.03669	0.518344	+	0.855172i	$0.326549\pi$
$228$	0	0
$229$	5.22512	0.345286	0.172643	−	0.984984i	$-0.444769\pi$
$229$	5.22512	0.345286	0.172643	+	0.984984i	$0.444769\pi$
$230$	5.01738	0.330836
$231$	0	0
$232$	−9.29658	−0.610351
$233$	−22.0174	−1.44241	−0.721204	−	0.692723i	$-0.756411\pi$
$233$	−22.0174	−1.44241	−0.721204	+	0.692723i	$0.756411\pi$
$234$	0	0
$235$	7.45867	0.486550
$236$	5.74945	0.374257
$237$	0	0
$238$	−3.96331	−0.256903
$239$	−7.79187	−0.504014	−0.252007	−	0.967725i	$-0.581091\pi$
$239$	−7.79187	−0.504014	−0.252007	+	0.967725i	$0.581091\pi$
$240$	0	0
$241$	−4.26553	−0.274767	−0.137383	−	0.990518i	$-0.543869\pi$
$241$	−4.26553	−0.274767	−0.137383	+	0.990518i	$0.543869\pi$
$242$	−15.2148	−0.978042
$243$	0	0
$244$	−9.22503	−0.590572
$245$	−1.32368	−0.0845669
$246$	0	0
$247$	−7.79825	−0.496191
$248$	23.6535	1.50200
$249$	0	0
$250$	0.644507	0.0407622
$251$	17.1133	1.08018	0.540090	−	0.841607i	$-0.318390\pi$
$251$	17.1133	1.08018	0.540090	+	0.841607i	$0.318390\pi$
$252$	0	0
$253$	−45.7963	−2.87919
$254$	−9.84672	−0.617838
$255$	0	0
$256$	−7.85190	−0.490744
$257$	−11.6396	−0.726059	−0.363029	−	0.931778i	$-0.618258\pi$
$257$	−11.6396	−0.726059	−0.363029	+	0.931778i	$0.618258\pi$
$258$	0	0
$259$	−7.35067	−0.456749
$260$	−7.20462	−0.446812
$261$	0	0
$262$	−1.43068	−0.0883880
$263$	1.96198	0.120981	0.0604903	−	0.998169i	$-0.480734\pi$
$263$	1.96198	0.120981	0.0604903	+	0.998169i	$0.480734\pi$
$264$	0	0
$265$	13.3236	0.818459
$266$	−3.18929	−0.195548
$267$	0	0
$268$	−19.8724	−1.21390
$269$	−4.00767	−0.244352	−0.122176	−	0.992508i	$-0.538987\pi$
$269$	−4.00767	−0.244352	−0.122176	+	0.992508i	$0.538987\pi$
$270$	0	0
$271$	−23.9496	−1.45483	−0.727417	−	0.686195i	$-0.759280\pi$
$271$	−23.9496	−1.45483	−0.727417	+	0.686195i	$0.759280\pi$
$272$	3.58127	0.217147
$273$	0	0
$274$	7.13481	0.431030
$275$	−5.88276	−0.354744
$276$	0	0
$277$	−26.9648	−1.62016	−0.810078	−	0.586322i	$-0.800575\pi$
$277$	−26.9648	−1.62016	−0.810078	+	0.586322i	$0.800575\pi$
$278$	4.81970	0.289067
$279$	0	0
$280$	−6.66541	−0.398335
$281$	−27.1762	−1.62120	−0.810599	−	0.585601i	$-0.800859\pi$
$281$	−27.1762	−1.62120	−0.810599	+	0.585601i	$0.800859\pi$
$282$	0	0
$283$	−25.1902	−1.49740	−0.748701	−	0.662908i	$-0.769322\pi$
$283$	−25.1902	−1.49740	−0.748701	+	0.662908i	$0.769322\pi$
$284$	−11.7485	−0.697147
$285$	0	0
$286$	−17.2384	−1.01933
$287$	−1.72861	−0.102037
$288$	0	0
$289$	−12.4570	−0.732763
$290$	−2.59347	−0.152294
$291$	0	0
$292$	4.75886	0.278491
$293$	−22.0579	−1.28864	−0.644319	−	0.764757i	$-0.722859\pi$
$293$	−22.0579	−1.28864	−0.644319	+	0.764757i	$0.722859\pi$
$294$	0	0
$295$	3.62830	0.211248
$296$	−5.88625	−0.342131
$297$	0	0
$298$	−10.4803	−0.607106
$299$	−35.3947	−2.04693
$300$	0	0
$301$	9.51982	0.548713
$302$	7.24145	0.416698
$303$	0	0
$304$	2.88186	0.165286
$305$	−5.82164	−0.333346
$306$	0	0
$307$	3.62804	0.207063	0.103532	−	0.994626i	$-0.466986\pi$
$307$	3.62804	0.207063	0.103532	+	0.994626i	$0.466986\pi$
$308$	26.8944	1.53245
$309$	0	0
$310$	6.59863	0.374777
$311$	16.8633	0.956230	0.478115	−	0.878297i	$-0.341320\pi$
$311$	16.8633	0.956230	0.478115	+	0.878297i	$0.341320\pi$
$312$	0	0
$313$	8.12182	0.459072	0.229536	−	0.973300i	$-0.426279\pi$
$313$	8.12182	0.459072	0.229536	+	0.973300i	$0.426279\pi$
$314$	10.2175	0.576608
$315$	0	0
$316$	−6.68633	−0.376135
$317$	2.00281	0.112489	0.0562444	−	0.998417i	$-0.482087\pi$
$317$	2.00281	0.112489	0.0562444	+	0.998417i	$0.482087\pi$
$318$	0	0
$319$	23.6720	1.32538
$320$	−0.315526	−0.0176384
$321$	0	0
$322$	−14.4755	−0.806690
$323$	3.65579	0.203414
$324$	0	0
$325$	−4.54662	−0.252201
$326$	5.06346	0.280439
$327$	0	0
$328$	−1.38423	−0.0764316
$329$	−21.5188	−1.18637
$330$	0	0
$331$	−5.84583	−0.321316	−0.160658	−	0.987010i	$-0.551362\pi$
$331$	−5.84583	−0.321316	−0.160658	+	0.987010i	$0.551362\pi$
$332$	13.8925	0.762449
$333$	0	0
$334$	2.93113	0.160384
$335$	−12.5409	−0.685182
$336$	0	0
$337$	13.2038	0.719258	0.359629	−	0.933095i	$-0.382903\pi$
$337$	13.2038	0.719258	0.359629	+	0.933095i	$0.382903\pi$
$338$	−4.94449	−0.268945
$339$	0	0
$340$	3.37750	0.183171
$341$	−60.2292	−3.26160
$342$	0	0
$343$	−16.3766	−0.884255
$344$	7.62326	0.411018
$345$	0	0
$346$	7.95127	0.427463
$347$	6.83694	0.367026	0.183513	−	0.983017i	$-0.441253\pi$
$347$	6.83694	0.367026	0.183513	+	0.983017i	$0.441253\pi$
$348$	0	0
$349$	−14.1359	−0.756677	−0.378338	−	0.925667i	$-0.623504\pi$
$349$	−14.1359	−0.756677	−0.378338	+	0.925667i	$0.623504\pi$
$350$	−1.85945	−0.0993919
$351$	0	0
$352$	33.5524	1.78835
$353$	−8.89072	−0.473205	−0.236603	−	0.971606i	$-0.576034\pi$
$353$	−8.89072	−0.473205	−0.236603	+	0.971606i	$0.576034\pi$
$354$	0	0
$355$	−7.41414	−0.393502
$356$	−1.58461	−0.0839842
$357$	0	0
$358$	13.6409	0.720942
$359$	−27.8829	−1.47160	−0.735802	−	0.677197i	$-0.763194\pi$
$359$	−27.8829	−1.47160	−0.735802	+	0.677197i	$0.763194\pi$
$360$	0	0
$361$	−16.0582	−0.845167
$362$	9.31720	0.489701
$363$	0	0
$364$	20.7859	1.08948
$365$	3.00317	0.157193
$366$	0	0
$367$	33.8955	1.76933	0.884667	−	0.466224i	$-0.154386\pi$
$367$	33.8955	1.76933	0.884667	+	0.466224i	$0.154386\pi$
$368$	13.0802	0.681853
$369$	0	0
$370$	−1.64209	−0.0853681
$371$	−38.4395	−1.99568
$372$	0	0
$373$	−11.1701	−0.578365	−0.289182	−	0.957274i	$-0.593383\pi$
$373$	−11.1701	−0.578365	−0.289182	+	0.957274i	$0.593383\pi$
$374$	8.08129	0.417874
$375$	0	0
$376$	−17.2318	−0.888662
$377$	18.2954	0.942262
$378$	0	0
$379$	−3.23858	−0.166355	−0.0831774	−	0.996535i	$-0.526507\pi$
$379$	−3.23858	−0.166355	−0.0831774	+	0.996535i	$0.526507\pi$
$380$	2.71789	0.139425
$381$	0	0
$382$	13.1828	0.674488
$383$	7.61086	0.388897	0.194448	−	0.980913i	$-0.437708\pi$
$383$	7.61086	0.388897	0.194448	+	0.980913i	$0.437708\pi$
$384$	0	0
$385$	16.9722	0.864985
$386$	−4.48735	−0.228400
$387$	0	0
$388$	30.5805	1.55249
$389$	14.4094	0.730585	0.365292	−	0.930893i	$-0.380969\pi$
$389$	14.4094	0.730585	0.365292	+	0.930893i	$0.380969\pi$
$390$	0	0
$391$	16.5929	0.839139
$392$	3.05811	0.154458
$393$	0	0
$394$	−11.2472	−0.566625
$395$	−4.21954	−0.212308
$396$	0	0
$397$	8.49423	0.426313	0.213157	−	0.977018i	$-0.431626\pi$
$397$	8.49423	0.426313	0.213157	+	0.977018i	$0.431626\pi$
$398$	1.91346	0.0959132
$399$	0	0
$400$	1.68022	0.0840108
$401$	3.94033	0.196771	0.0983854	−	0.995148i	$-0.468632\pi$
$401$	3.94033	0.196771	0.0983854	+	0.995148i	$0.468632\pi$
$402$	0	0
$403$	−46.5495	−2.31879
$404$	17.8628	0.888708
$405$	0	0
$406$	7.48237	0.371344
$407$	14.9882	0.742939
$408$	0	0
$409$	3.45461	0.170820	0.0854099	−	0.996346i	$-0.472780\pi$
$409$	3.45461	0.170820	0.0854099	+	0.996346i	$0.472780\pi$
$410$	−0.386160	−0.0190711
$411$	0	0
$412$	0.879003	0.0433054
$413$	−10.4679	−0.515094
$414$	0	0
$415$	8.76712	0.430361
$416$	25.9317	1.27141
$417$	0	0
$418$	6.50305	0.318074
$419$	20.8328	1.01775	0.508875	−	0.860840i	$-0.330061\pi$
$419$	20.8328	1.01775	0.508875	+	0.860840i	$0.330061\pi$
$420$	0	0
$421$	15.9771	0.778676	0.389338	−	0.921095i	$-0.372704\pi$
$421$	15.9771	0.778676	0.389338	+	0.921095i	$0.372704\pi$
$422$	4.86152	0.236655
$423$	0	0
$424$	−30.7815	−1.49488
$425$	2.13144	0.103390
$426$	0	0
$427$	16.7959	0.812810
$428$	−26.6025	−1.28588
$429$	0	0
$430$	2.12666	0.102557
$431$	32.2599	1.55391	0.776953	−	0.629558i	$-0.216764\pi$
$431$	32.2599	1.55391	0.776953	+	0.629558i	$0.216764\pi$
$432$	0	0
$433$	19.4534	0.934872	0.467436	−	0.884027i	$-0.345178\pi$
$433$	19.4534	0.934872	0.467436	+	0.884027i	$0.345178\pi$
$434$	−19.0376	−0.913832
$435$	0	0
$436$	−1.89827	−0.0909108
$437$	13.3524	0.638730
$438$	0	0
$439$	−20.5511	−0.980851	−0.490426	−	0.871483i	$-0.663159\pi$
$439$	−20.5511	−0.980851	−0.490426	+	0.871483i	$0.663159\pi$
$440$	13.5910	0.647924
$441$	0	0
$442$	6.24581	0.297083
$443$	−7.60486	−0.361318	−0.180659	−	0.983546i	$-0.557823\pi$
$443$	−7.60486	−0.361318	−0.180659	+	0.983546i	$0.557823\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	16.8266	0.796761
$447$	0	0
$448$	0.910317	0.0430084
$449$	−18.2739	−0.862398	−0.431199	−	0.902257i	$-0.641909\pi$
$449$	−18.2739	−0.862398	−0.431199	+	0.902257i	$0.641909\pi$
$450$	0	0
$451$	3.52469	0.165971
$452$	5.36328	0.252267
$453$	0	0
$454$	−10.0667	−0.472455
$455$	13.1174	0.614951
$456$	0	0
$457$	38.0253	1.77875	0.889375	−	0.457179i	$-0.151140\pi$
$457$	38.0253	1.77875	0.889375	+	0.457179i	$0.151140\pi$
$458$	−3.36762	−0.157359
$459$	0	0
$460$	12.3359	0.575167
$461$	−0.888764	−0.0413939	−0.0206969	−	0.999786i	$-0.506589\pi$
$461$	−0.888764	−0.0413939	−0.0206969	+	0.999786i	$0.506589\pi$
$462$	0	0
$463$	−28.7668	−1.33691	−0.668453	−	0.743754i	$-0.733043\pi$
$463$	−28.7668	−1.33691	−0.668453	+	0.743754i	$0.733043\pi$
$464$	−6.76112	−0.313877
$465$	0	0
$466$	14.1904	0.657356
$467$	39.8800	1.84542	0.922712	−	0.385489i	$-0.125967\pi$
$467$	39.8800	1.84542	0.922712	+	0.385489i	$0.125967\pi$
$468$	0	0
$469$	36.1815	1.67070
$470$	−4.80716	−0.221738
$471$	0	0
$472$	−8.38249	−0.385835
$473$	−19.4112	−0.892527
$474$	0	0
$475$	1.71518	0.0786977
$476$	−9.74435	−0.446632
$477$	0	0
$478$	5.02191	0.229697
$479$	38.4309	1.75595	0.877976	−	0.478705i	$-0.158894\pi$
$479$	38.4309	1.75595	0.877976	+	0.478705i	$0.158894\pi$
$480$	0	0
$481$	11.5840	0.528184
$482$	2.74916	0.125221
$483$	0	0
$484$	−37.4077	−1.70035
$485$	19.2984	0.876297
$486$	0	0
$487$	−42.5507	−1.92816	−0.964078	−	0.265620i	$-0.914423\pi$
$487$	−42.5507	−1.92816	−0.964078	+	0.265620i	$0.914423\pi$
$488$	13.4498	0.608842
$489$	0	0
$490$	0.853121	0.0385401
$491$	10.6636	0.481244	0.240622	−	0.970619i	$-0.422649\pi$
$491$	10.6636	0.481244	0.240622	+	0.970619i	$0.422649\pi$
$492$	0	0
$493$	−8.57682	−0.386281
$494$	5.02602	0.226131
$495$	0	0
$496$	17.2025	0.772414
$497$	21.3904	0.959490
$498$	0	0
$499$	−37.1464	−1.66290	−0.831451	−	0.555598i	$-0.812489\pi$
$499$	−37.1464	−1.66290	−0.831451	+	0.555598i	$0.812489\pi$
$500$	1.58461	0.0708660
$501$	0	0
$502$	−11.0296	−0.492276
$503$	−7.15895	−0.319202	−0.159601	−	0.987182i	$-0.551021\pi$
$503$	−7.15895	−0.319202	−0.159601	+	0.987182i	$0.551021\pi$
$504$	0	0
$505$	11.2727	0.501628
$506$	29.5160	1.31215
$507$	0	0
$508$	−24.2096	−1.07413
$509$	2.88098	0.127697	0.0638486	−	0.997960i	$-0.479663\pi$
$509$	2.88098	0.127697	0.0638486	+	0.997960i	$0.479663\pi$
$510$	0	0
$511$	−8.66439	−0.383290
$512$	−17.3468	−0.766626
$513$	0	0
$514$	7.50180	0.330890
$515$	0.554712	0.0244435
$516$	0	0
$517$	43.8775	1.92973
$518$	4.73756	0.208156
$519$	0	0
$520$	10.5041	0.460634
$521$	5.01687	0.219793	0.109897	−	0.993943i	$-0.464948\pi$
$521$	5.01687	0.219793	0.109897	+	0.993943i	$0.464948\pi$
$522$	0	0
$523$	29.3831	1.28483	0.642416	−	0.766356i	$-0.277932\pi$
$523$	29.3831	1.28483	0.642416	+	0.766356i	$0.277932\pi$
$524$	−3.51754	−0.153665
$525$	0	0
$526$	−1.26451	−0.0551351
$527$	21.8222	0.950590
$528$	0	0
$529$	37.6038	1.63495
$530$	−8.58712	−0.373000
$531$	0	0
$532$	−7.84132	−0.339964
$533$	2.72414	0.117995
$534$	0	0
$535$	−16.7880	−0.725809
$536$	28.9733	1.25146
$537$	0	0
$538$	2.58297	0.111360
$539$	−7.78689	−0.335405
$540$	0	0
$541$	1.21844	0.0523849	0.0261925	−	0.999657i	$-0.491662\pi$
$541$	1.21844	0.0523849	0.0261925	+	0.999657i	$0.491662\pi$
$542$	15.4357	0.663019
$543$	0	0
$544$	−12.1567	−0.521214
$545$	−1.19794	−0.0513142
$546$	0	0
$547$	0.879195	0.0375916	0.0187958	−	0.999823i	$-0.494017\pi$
$547$	0.879195	0.0375916	0.0187958	+	0.999823i	$0.494017\pi$
$548$	17.5419	0.749355
$549$	0	0
$550$	3.79148	0.161669
$551$	−6.90180	−0.294027
$552$	0	0
$553$	12.1737	0.517678
$554$	17.3790	0.738362
$555$	0	0
$556$	11.8499	0.502549
$557$	13.5473	0.574017	0.287008	−	0.957928i	$-0.407339\pi$
$557$	13.5473	0.574017	0.287008	+	0.957928i	$0.407339\pi$
$558$	0	0
$559$	−15.0024	−0.634532
$560$	−4.84755	−0.204847
$561$	0	0
$562$	17.5153	0.738837
$563$	−11.5776	−0.487939	−0.243969	−	0.969783i	$-0.578450\pi$
$563$	−11.5776	−0.487939	−0.243969	+	0.969783i	$0.578450\pi$
$564$	0	0
$565$	3.38460	0.142391
$566$	16.2352	0.682418
$567$	0	0
$568$	17.1289	0.718714
$569$	−35.3373	−1.48142	−0.740709	−	0.671826i	$-0.765510\pi$
$569$	−35.3373	−1.48142	−0.740709	+	0.671826i	$0.765510\pi$
$570$	0	0
$571$	−37.5099	−1.56974	−0.784869	−	0.619661i	$-0.787270\pi$
$571$	−37.5099	−1.56974	−0.784869	+	0.619661i	$0.787270\pi$
$572$	−42.3831	−1.77212
$573$	0	0
$574$	1.11410	0.0465018
$575$	7.78484	0.324650
$576$	0	0
$577$	6.36646	0.265039	0.132520	−	0.991180i	$-0.457693\pi$
$577$	6.36646	0.265039	0.132520	+	0.991180i	$0.457693\pi$
$578$	8.02860	0.333946
$579$	0	0
$580$	−6.37642	−0.264766
$581$	−25.2938	−1.04937
$582$	0	0
$583$	78.3792	3.24614
$584$	−6.93825	−0.287107
$585$	0	0
$586$	14.2165	0.587277
$587$	−34.4348	−1.42128	−0.710638	−	0.703558i	$-0.751594\pi$
$587$	−34.4348	−1.42128	−0.710638	+	0.703558i	$0.751594\pi$
$588$	0	0
$589$	17.5604	0.723564
$590$	−2.33847	−0.0962731
$591$	0	0
$592$	−4.28089	−0.175944
$593$	22.8276	0.937416	0.468708	−	0.883353i	$-0.344720\pi$
$593$	22.8276	0.937416	0.468708	+	0.883353i	$0.344720\pi$
$594$	0	0
$595$	−6.14936	−0.252099
$596$	−25.7672	−1.05547
$597$	0	0
$598$	22.8121	0.932857
$599$	10.8616	0.443792	0.221896	−	0.975070i	$-0.428775\pi$
$599$	10.8616	0.443792	0.221896	+	0.975070i	$0.428775\pi$
$600$	0	0
$601$	−37.3151	−1.52212	−0.761058	−	0.648684i	$-0.775320\pi$
$601$	−37.3151	−1.52212	−0.761058	+	0.648684i	$0.775320\pi$
$602$	−6.13559	−0.250068
$603$	0	0
$604$	17.8041	0.724440
$605$	−23.6068	−0.959755
$606$	0	0
$607$	17.5719	0.713222	0.356611	−	0.934253i	$-0.383932\pi$
$607$	17.5719	0.713222	0.356611	+	0.934253i	$0.383932\pi$
$608$	−9.78254	−0.396734
$609$	0	0
$610$	3.75208	0.151917
$611$	33.9117	1.37192
$612$	0	0
$613$	−14.1052	−0.569705	−0.284853	−	0.958571i	$-0.591945\pi$
$613$	−14.1052	−0.569705	−0.284853	+	0.958571i	$0.591945\pi$
$614$	−2.33830	−0.0943660
$615$	0	0
$616$	−39.2110	−1.57986
$617$	42.0400	1.69247	0.846234	−	0.532812i	$-0.178865\pi$
$617$	42.0400	1.69247	0.846234	+	0.532812i	$0.178865\pi$
$618$	0	0
$619$	36.7441	1.47687	0.738435	−	0.674325i	$-0.235565\pi$
$619$	36.7441	1.47687	0.738435	+	0.674325i	$0.235565\pi$
$620$	16.2237	0.651558
$621$	0	0
$622$	−10.8685	−0.435788
$623$	2.88508	0.115588
$624$	0	0
$625$	1.00000	0.0400000
$626$	−5.23457	−0.209215
$627$	0	0
$628$	25.1212	1.00245
$629$	−5.43053	−0.216529
$630$	0	0
$631$	20.4893	0.815666	0.407833	−	0.913056i	$-0.366285\pi$
$631$	20.4893	0.815666	0.407833	+	0.913056i	$0.366285\pi$
$632$	9.74842	0.387771
$633$	0	0
$634$	−1.29082	−0.0512651
$635$	−15.2779	−0.606286
$636$	0	0
$637$	−6.01827	−0.238453
$638$	−15.2568	−0.604021
$639$	0	0
$640$	−11.2037	−0.442864
$641$	−13.1521	−0.519477	−0.259738	−	0.965679i	$-0.583636\pi$
$641$	−13.1521	−0.519477	−0.259738	+	0.965679i	$0.583636\pi$
$642$	0	0
$643$	−19.1217	−0.754084	−0.377042	−	0.926196i	$-0.623059\pi$
$643$	−19.1217	−0.754084	−0.377042	+	0.926196i	$0.623059\pi$
$644$	−35.5902	−1.40245
$645$	0	0
$646$	−2.35618	−0.0927027
$647$	20.7566	0.816028	0.408014	−	0.912976i	$-0.366222\pi$
$647$	20.7566	0.816028	0.408014	+	0.912976i	$0.366222\pi$
$648$	0	0
$649$	21.3444	0.837842
$650$	2.93033	0.114937
$651$	0	0
$652$	12.4492	0.487550
$653$	9.08533	0.355537	0.177768	−	0.984072i	$-0.443112\pi$
$653$	9.08533	0.355537	0.177768	+	0.984072i	$0.443112\pi$
$654$	0	0
$655$	−2.21981	−0.0867353
$656$	−1.00671	−0.0393055
$657$	0	0
$658$	13.8690	0.540671
$659$	−21.0299	−0.819207	−0.409603	−	0.912264i	$-0.634333\pi$
$659$	−21.0299	−0.819207	−0.409603	+	0.912264i	$0.634333\pi$
$660$	0	0
$661$	−7.67527	−0.298533	−0.149267	−	0.988797i	$-0.547691\pi$
$661$	−7.67527	−0.298533	−0.149267	+	0.988797i	$0.547691\pi$
$662$	3.76768	0.146435
$663$	0	0
$664$	−20.2547	−0.786036
$665$	−4.94842	−0.191891
$666$	0	0
$667$	−31.3259	−1.21294
$668$	7.20660	0.278832
$669$	0	0
$670$	8.08269	0.312261
$671$	−34.2473	−1.32210
$672$	0	0
$673$	−10.9814	−0.423302	−0.211651	−	0.977345i	$-0.567884\pi$
$673$	−10.9814	−0.423302	−0.211651	+	0.977345i	$0.567884\pi$
$674$	−8.50995	−0.327791
$675$	0	0
$676$	−12.1567	−0.467567
$677$	−23.9558	−0.920697	−0.460348	−	0.887738i	$-0.652275\pi$
$677$	−23.9558	−0.920697	−0.460348	+	0.887738i	$0.652275\pi$
$678$	0	0
$679$	−55.6775	−2.13671
$680$	−4.92427	−0.188837
$681$	0	0
$682$	38.8181	1.48642
$683$	3.14083	0.120181	0.0600903	−	0.998193i	$-0.480861\pi$
$683$	3.14083	0.120181	0.0600903	+	0.998193i	$0.480861\pi$
$684$	0	0
$685$	11.0702	0.422970
$686$	10.5548	0.402986
$687$	0	0
$688$	5.54416	0.211369
$689$	60.5771	2.30780
$690$	0	0
$691$	−9.94070	−0.378162	−0.189081	−	0.981961i	$-0.560551\pi$
$691$	−9.94070	−0.378162	−0.189081	+	0.981961i	$0.560551\pi$
$692$	19.5493	0.743154
$693$	0	0
$694$	−4.40645	−0.167267
$695$	7.47813	0.283662
$696$	0	0
$697$	−1.27706	−0.0483723
$698$	9.11067	0.344844
$699$	0	0
$700$	−4.57173	−0.172795
$701$	−44.2423	−1.67101	−0.835503	−	0.549485i	$-0.814824\pi$
$701$	−44.2423	−1.67101	−0.835503	+	0.549485i	$0.814824\pi$
$702$	0	0
$703$	−4.36996	−0.164816
$704$	−1.85616	−0.0699568
$705$	0	0
$706$	5.73013	0.215656
$707$	−32.5226	−1.22314
$708$	0	0
$709$	−1.64752	−0.0618740	−0.0309370	−	0.999521i	$-0.509849\pi$
$709$	−1.64752	−0.0618740	−0.0309370	+	0.999521i	$0.509849\pi$
$710$	4.77846	0.179333
$711$	0	0
$712$	2.31031	0.0865824
$713$	79.7033	2.98491
$714$	0	0
$715$	−26.7467	−1.00027
$716$	33.5380	1.25337
$717$	0	0
$718$	17.9707	0.670661
$719$	−21.0475	−0.784939	−0.392469	−	0.919765i	$-0.628379\pi$
$719$	−21.0475	−0.784939	−0.392469	+	0.919765i	$0.628379\pi$
$720$	0	0
$721$	−1.60039	−0.0596016
$722$	10.3496	0.385172
$723$	0	0
$724$	22.9077	0.851356
$725$	−4.02396	−0.149446
$726$	0	0
$727$	−29.5400	−1.09558	−0.547788	−	0.836617i	$-0.684530\pi$
$727$	−29.5400	−1.09558	−0.547788	+	0.836617i	$0.684530\pi$
$728$	−30.3051	−1.12318
$729$	0	0
$730$	−1.93557	−0.0716385
$731$	7.03305	0.260127
$732$	0	0
$733$	−15.4471	−0.570552	−0.285276	−	0.958445i	$-0.592085\pi$
$733$	−15.4471	−0.570552	−0.285276	+	0.958445i	$0.592085\pi$
$734$	−21.8459	−0.806347
$735$	0	0
$736$	−44.4010	−1.63664
$737$	−73.7750	−2.71754
$738$	0	0
$739$	42.2176	1.55300	0.776500	−	0.630117i	$-0.216993\pi$
$739$	42.2176	1.55300	0.776500	+	0.630117i	$0.216993\pi$
$740$	−4.03731	−0.148414
$741$	0	0
$742$	24.7745	0.909501
$743$	18.8161	0.690297	0.345148	−	0.938548i	$-0.387829\pi$
$743$	18.8161	0.690297	0.345148	+	0.938548i	$0.387829\pi$
$744$	0	0
$745$	−16.2609	−0.595754
$746$	7.19919	0.263581
$747$	0	0
$748$	19.8690	0.726483
$749$	48.4347	1.76977
$750$	0	0
$751$	27.5824	1.00650	0.503248	−	0.864142i	$-0.332138\pi$
$751$	27.5824	1.00650	0.503248	+	0.864142i	$0.332138\pi$
$752$	−12.5322	−0.457001
$753$	0	0
$754$	−11.7915	−0.429422
$755$	11.2356	0.408907
$756$	0	0
$757$	−0.996633	−0.0362232	−0.0181116	−	0.999836i	$-0.505765\pi$
$757$	−0.996633	−0.0362232	−0.0181116	+	0.999836i	$0.505765\pi$
$758$	2.08729	0.0758137
$759$	0	0
$760$	−3.96258	−0.143738
$761$	−15.8353	−0.574030	−0.287015	−	0.957926i	$-0.592663\pi$
$761$	−15.8353	−0.574030	−0.287015	+	0.957926i	$0.592663\pi$
$762$	0	0
$763$	3.45616	0.125121
$764$	32.4117	1.17261
$765$	0	0
$766$	−4.90525	−0.177234
$767$	16.4965	0.595655
$768$	0	0
$769$	−6.27941	−0.226441	−0.113221	−	0.993570i	$-0.536117\pi$
$769$	−6.27941	−0.226441	−0.113221	+	0.993570i	$0.536117\pi$
$770$	−10.9387	−0.394204
$771$	0	0
$772$	−11.0328	−0.397079
$773$	−28.3270	−1.01885	−0.509426	−	0.860515i	$-0.670142\pi$
$773$	−28.3270	−1.01885	−0.509426	+	0.860515i	$0.670142\pi$
$774$	0	0
$775$	10.2383	0.367769
$776$	−44.5853	−1.60052
$777$	0	0
$778$	−9.28694	−0.332953
$779$	−1.02766	−0.0368197
$780$	0	0
$781$	−43.6156	−1.56069
$782$	−10.6942	−0.382425
$783$	0	0
$784$	2.22407	0.0794310
$785$	15.8532	0.565826
$786$	0	0
$787$	6.61090	0.235653	0.117826	−	0.993034i	$-0.462407\pi$
$787$	6.61090	0.235653	0.117826	+	0.993034i	$0.462407\pi$
$788$	−27.6528	−0.985090
$789$	0	0
$790$	2.71952	0.0967561
$791$	−9.76485	−0.347198
$792$	0	0
$793$	−26.4688	−0.939933
$794$	−5.47459	−0.194286
$795$	0	0
$796$	4.70452	0.166747
$797$	49.5173	1.75399	0.876996	−	0.480497i	$-0.159544\pi$
$797$	49.5173	1.75399	0.876996	+	0.480497i	$0.159544\pi$
$798$	0	0
$799$	−15.8977	−0.562419
$800$	−5.70352	−0.201650
$801$	0	0
$802$	−2.53957	−0.0896754
$803$	17.6669	0.623453
$804$	0	0
$805$	−22.4599	−0.791607
$806$	30.0014	1.05676
$807$	0	0
$808$	−26.0433	−0.916201
$809$	−46.7898	−1.64504	−0.822521	−	0.568734i	$-0.807433\pi$
$809$	−46.7898	−1.64504	−0.822521	+	0.568734i	$0.807433\pi$
$810$	0	0
$811$	51.3182	1.80202	0.901012	−	0.433794i	$-0.142825\pi$
$811$	51.3182	1.80202	0.901012	+	0.433794i	$0.142825\pi$
$812$	18.3965	0.645589
$813$	0	0
$814$	−9.66001	−0.338583
$815$	7.85634	0.275195
$816$	0	0
$817$	5.65952	0.198002
$818$	−2.22652	−0.0778485
$819$	0	0
$820$	−0.949430	−0.0331555
$821$	18.8495	0.657851	0.328926	−	0.944356i	$-0.393313\pi$
$821$	18.8495	0.657851	0.328926	+	0.944356i	$0.393313\pi$
$822$	0	0
$823$	38.8882	1.35556	0.677778	−	0.735266i	$-0.262943\pi$
$823$	38.8882	1.35556	0.677778	+	0.735266i	$0.262943\pi$
$824$	−1.28155	−0.0446451
$825$	0	0
$826$	6.74666	0.234746
$827$	1.46437	0.0509212	0.0254606	−	0.999676i	$-0.491895\pi$
$827$	1.46437	0.0509212	0.0254606	+	0.999676i	$0.491895\pi$
$828$	0	0
$829$	−25.0506	−0.870045	−0.435022	−	0.900420i	$-0.643260\pi$
$829$	−25.0506	−0.870045	−0.435022	+	0.900420i	$0.643260\pi$
$830$	−5.65047	−0.196131
$831$	0	0
$832$	−1.43458	−0.0497350
$833$	2.82134	0.0977537
$834$	0	0
$835$	4.54787	0.157385
$836$	15.9887	0.552980
$837$	0	0
$838$	−13.4269	−0.463825
$839$	−31.7096	−1.09474	−0.547368	−	0.836892i	$-0.684370\pi$
$839$	−31.7096	−1.09474	−0.547368	+	0.836892i	$0.684370\pi$
$840$	0	0
$841$	−12.8077	−0.441646
$842$	−10.2973	−0.354870
$843$	0	0
$844$	11.9527	0.411430
$845$	−7.67175	−0.263916
$846$	0	0
$847$	68.1076	2.34020
$848$	−22.3864	−0.768753
$849$	0	0
$850$	−1.37373	−0.0471184
$851$	−19.8344	−0.679914
$852$	0	0
$853$	26.1967	0.896958	0.448479	−	0.893793i	$-0.351966\pi$
$853$	26.1967	0.896958	0.448479	+	0.893793i	$0.351966\pi$
$854$	−10.8251	−0.370426
$855$	0	0
$856$	38.7854	1.32566
$857$	58.1449	1.98619	0.993096	−	0.117307i	$-0.0374261\pi$
$857$	58.1449	1.98619	0.993096	+	0.117307i	$0.0374261\pi$
$858$	0	0
$859$	−55.2854	−1.88631	−0.943156	−	0.332351i	$-0.892158\pi$
$859$	−55.2854	−1.88631	−0.943156	+	0.332351i	$0.892158\pi$
$860$	5.22870	0.178297
$861$	0	0
$862$	−20.7917	−0.708169
$863$	19.6934	0.670371	0.335185	−	0.942152i	$-0.391201\pi$
$863$	19.6934	0.670371	0.335185	+	0.942152i	$0.391201\pi$
$864$	0	0
$865$	12.3370	0.419470
$866$	−12.5379	−0.426054
$867$	0	0
$868$	−46.8066	−1.58872
$869$	−24.8225	−0.842046
$870$	0	0
$871$	−57.0187	−1.93200
$872$	2.76761	0.0937232
$873$	0	0
$874$	−8.60569	−0.291092
$875$	−2.88508	−0.0975335
$876$	0	0
$877$	23.9917	0.810143	0.405072	−	0.914285i	$-0.367247\pi$
$877$	23.9917	0.810143	0.405072	+	0.914285i	$0.367247\pi$
$878$	13.2453	0.447008
$879$	0	0
$880$	9.88430	0.333200
$881$	−47.0612	−1.58553	−0.792767	−	0.609525i	$-0.791360\pi$
$881$	−47.0612	−1.58553	−0.792767	+	0.609525i	$0.791360\pi$
$882$	0	0
$883$	24.0981	0.810965	0.405482	−	0.914103i	$-0.367104\pi$
$883$	24.0981	0.810965	0.405482	+	0.914103i	$0.367104\pi$
$884$	15.3562	0.516485
$885$	0	0
$886$	4.90138	0.164665
$887$	−18.7201	−0.628559	−0.314279	−	0.949331i	$-0.601763\pi$
$887$	−18.7201	−0.628559	−0.314279	+	0.949331i	$0.601763\pi$
$888$	0	0
$889$	44.0780	1.47833
$890$	0.644507	0.0216039
$891$	0	0
$892$	41.3705	1.38519
$893$	−12.7929	−0.428099
$894$	0	0
$895$	21.1648	0.707462
$896$	32.3235	1.07985
$897$	0	0
$898$	11.7776	0.393025
$899$	−41.1984	−1.37404
$900$	0	0
$901$	−28.3983	−0.946085
$902$	−2.27169	−0.0756389
$903$	0	0
$904$	−7.81947	−0.260072
$905$	14.4563	0.480545
$906$	0	0
$907$	25.1874	0.836335	0.418167	−	0.908370i	$-0.362673\pi$
$907$	25.1874	0.836335	0.418167	+	0.908370i	$0.362673\pi$
$908$	−24.7505	−0.821374
$909$	0	0
$910$	−8.45422	−0.280255
$911$	17.2705	0.572198	0.286099	−	0.958200i	$-0.407641\pi$
$911$	17.2705	0.572198	0.286099	+	0.958200i	$0.407641\pi$
$912$	0	0
$913$	51.5748	1.70688
$914$	−24.5076	−0.810638
$915$	0	0
$916$	−8.27979	−0.273572
$917$	6.40434	0.211490
$918$	0	0
$919$	−6.37155	−0.210178	−0.105089	−	0.994463i	$-0.533513\pi$
$919$	−6.37155	−0.210178	−0.105089	+	0.994463i	$0.533513\pi$
$920$	−17.9854	−0.592960
$921$	0	0
$922$	0.572814	0.0188646
$923$	−33.7093	−1.10955
$924$	0	0
$925$	−2.54782	−0.0837719
$926$	18.5404	0.609275
$927$	0	0
$928$	22.9508	0.753395
$929$	46.7786	1.53476	0.767379	−	0.641194i	$-0.221561\pi$
$929$	46.7786	1.53476	0.767379	+	0.641194i	$0.221561\pi$
$930$	0	0
$931$	2.27035	0.0744076
$932$	34.8890	1.14283
$933$	0	0
$934$	−25.7029	−0.841025
$935$	12.5387	0.410060
$936$	0	0
$937$	11.0113	0.359724	0.179862	−	0.983692i	$-0.442435\pi$
$937$	11.0113	0.359724	0.179862	+	0.983692i	$0.442435\pi$
$938$	−23.3192	−0.761399
$939$	0	0
$940$	−11.8191	−0.385496
$941$	−0.607068	−0.0197899	−0.00989493	−	0.999951i	$-0.503150\pi$
$941$	−0.607068	−0.0197899	−0.00989493	+	0.999951i	$0.503150\pi$
$942$	0	0
$943$	−4.66434	−0.151892
$944$	−6.09633	−0.198419
$945$	0	0
$946$	12.5106	0.406756
$947$	1.90227	0.0618156	0.0309078	−	0.999522i	$-0.490160\pi$
$947$	1.90227	0.0618156	0.0309078	+	0.999522i	$0.490160\pi$
$948$	0	0
$949$	13.6543	0.443237
$950$	−1.10544	−0.0358653
$951$	0	0
$952$	14.2069	0.460449
$953$	15.8767	0.514298	0.257149	−	0.966372i	$-0.417217\pi$
$953$	15.8767	0.514298	0.257149	+	0.966372i	$0.417217\pi$
$954$	0	0
$955$	20.4540	0.661877
$956$	12.3471	0.399333
$957$	0	0
$958$	−24.7689	−0.800248
$959$	−31.9384	−1.03134
$960$	0	0
$961$	73.8220	2.38136
$962$	−7.46595	−0.240712
$963$	0	0
$964$	6.75921	0.217699
$965$	−6.96246	−0.224129
$966$	0	0
$967$	35.8942	1.15428	0.577140	−	0.816645i	$-0.304169\pi$
$967$	35.8942	1.15428	0.577140	+	0.816645i	$0.304169\pi$
$968$	54.5390	1.75295
$969$	0	0
$970$	−12.4380	−0.399359
$971$	−31.0954	−0.997900	−0.498950	−	0.866631i	$-0.666281\pi$
$971$	−31.0954	−0.997900	−0.498950	+	0.866631i	$0.666281\pi$
$972$	0	0
$973$	−21.5750	−0.691662
$974$	27.4242	0.878728
$975$	0	0
$976$	9.78160	0.313101
$977$	−47.3421	−1.51461	−0.757304	−	0.653062i	$-0.773484\pi$
$977$	−47.3421	−1.51461	−0.757304	+	0.653062i	$0.773484\pi$
$978$	0	0
$979$	−5.88276	−0.188014
$980$	2.09752	0.0670028
$981$	0	0
$982$	−6.87279	−0.219320
$983$	−49.5416	−1.58013	−0.790066	−	0.613022i	$-0.789954\pi$
$983$	−49.5416	−1.58013	−0.790066	+	0.613022i	$0.789954\pi$
$984$	0	0
$985$	−17.4508	−0.556030
$986$	5.52782	0.176042
$987$	0	0
$988$	12.3572	0.393135
$989$	25.6874	0.816813
$990$	0	0
$991$	39.9608	1.26940	0.634698	−	0.772760i	$-0.281124\pi$
$991$	39.9608	1.26940	0.634698	+	0.772760i	$0.281124\pi$
$992$	−58.3941	−1.85402
$993$	0	0
$994$	−13.7862	−0.437273
$995$	2.96888	0.0941198
$996$	0	0
$997$	−44.2932	−1.40278	−0.701390	−	0.712778i	$-0.747437\pi$
$997$	−44.2932	−1.40278	−0.701390	+	0.712778i	$0.747437\pi$
$998$	23.9411	0.757842
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.s.1.5		10
3.2	odd	2		1335.2.a.j.1.6	✓	10
15.14	odd	2		6675.2.a.z.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1335.2.a.j.1.6	✓	10	3.2	odd	2
4005.2.a.s.1.5		10	1.1	even	1	trivial
6675.2.a.z.1.5		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)

\(f(q)\)	\(=\)	\(q-0.644507 q^{2} -1.58461 q^{4} -1.00000 q^{5} +2.88508 q^{7} +2.31031 q^{8} +O(q^{10})\)	\(q-0.644507 q^{2} -1.58461 q^{4} -1.00000 q^{5} +2.88508 q^{7} +2.31031 q^{8} +0.644507 q^{10} -5.88276 q^{11} -4.54662 q^{13} -1.85945 q^{14} +1.68022 q^{16} +2.13144 q^{17} +1.71518 q^{19} +1.58461 q^{20} +3.79148 q^{22} +7.78484 q^{23} +1.00000 q^{25} +2.93033 q^{26} -4.57173 q^{28} -4.02396 q^{29} +10.2383 q^{31} -5.70352 q^{32} -1.37373 q^{34} -2.88508 q^{35} -2.54782 q^{37} -1.10544 q^{38} -2.31031 q^{40} -0.599157 q^{41} +3.29967 q^{43} +9.32188 q^{44} -5.01738 q^{46} -7.45867 q^{47} +1.32368 q^{49} -0.644507 q^{50} +7.20462 q^{52} -13.3236 q^{53} +5.88276 q^{55} +6.66541 q^{56} +2.59347 q^{58} -3.62830 q^{59} +5.82164 q^{61} -6.59863 q^{62} +0.315526 q^{64} +4.54662 q^{65} +12.5409 q^{67} -3.37750 q^{68} +1.85945 q^{70} +7.41414 q^{71} -3.00317 q^{73} +1.64209 q^{74} -2.71789 q^{76} -16.9722 q^{77} +4.21954 q^{79} -1.68022 q^{80} +0.386160 q^{82} -8.76712 q^{83} -2.13144 q^{85} -2.12666 q^{86} -13.5910 q^{88} +1.00000 q^{89} -13.1174 q^{91} -12.3359 q^{92} +4.80716 q^{94} -1.71518 q^{95} -19.2984 q^{97} -0.853121 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7}+O(q^{10}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7	\( 10 q - q^{2} + 13 q^{4} - 10 q^{5} - q^{7} + q^{10} - 10 q^{11} + 5 q^{13} - 13 q^{14} + 19 q^{16} - 9 q^{17} - 6 q^{19} - 13 q^{20} + 3 q^{23} + 10 q^{25} - 14 q^{26} - 18 q^{28} - 38 q^{29} + 2 q^{31} - 16 q^{32} - 8 q^{34} + q^{35} + 9 q^{37} - 20 q^{38} - 36 q^{41} - 7 q^{43} - 16 q^{44} + 2 q^{46} + 23 q^{47} + 25 q^{49} - q^{50} + 13 q^{52} - 27 q^{53} + 10 q^{55} - 41 q^{56} - 32 q^{58} - 20 q^{59} + 30 q^{61} + 2 q^{62} - 2 q^{64} - 5 q^{65} - 5 q^{67} + 10 q^{68} + 13 q^{70} - 24 q^{71} - 19 q^{73} - 42 q^{74} - 30 q^{76} - 18 q^{77} + 12 q^{79} - 19 q^{80} + 29 q^{82} - 3 q^{83} + 9 q^{85} - 38 q^{86} - 16 q^{88} + 10 q^{89} - 6 q^{91} - 32 q^{92} + 17 q^{94} + 6 q^{95} + 3 q^{97} + 37 q^{98}+O(q^{100}) \) 10 * q - q^2 + 13 * q^4 - 10 * q^5 - q^7 + q^10 - 10 * q^11 + 5 * q^13 - 13 * q^14 + 19 * q^16 - 9 * q^17 - 6 * q^19 - 13 * q^20 + 3 * q^23 + 10 * q^25 - 14 * q^26 - 18 * q^28 - 38 * q^29 + 2 * q^31 - 16 * q^32 - 8 * q^34 + q^35 + 9 * q^37 - 20 * q^38 - 36 * q^41 - 7 * q^43 - 16 * q^44 + 2 * q^46 + 23 * q^47 + 25 * q^49 - q^50 + 13 * q^52 - 27 * q^53 + 10 * q^55 - 41 * q^56 - 32 * q^58 - 20 * q^59 + 30 * q^61 + 2 * q^62 - 2 * q^64 - 5 * q^65 - 5 * q^67 + 10 * q^68 + 13 * q^70 - 24 * q^71 - 19 * q^73 - 42 * q^74 - 30 * q^76 - 18 * q^77 + 12 * q^79 - 19 * q^80 + 29 * q^82 - 3 * q^83 + 9 * q^85 - 38 * q^86 - 16 * q^88 + 10 * q^89 - 6 * q^91 - 32 * q^92 + 17 * q^94 + 6 * q^95 + 3 * q^97 + 37 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more