LMFDB - Embedded newform 4005.2.a.v.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("4005.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$31.9800860095$
Analytic rank:	$1$
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			1.10
Root			$2.11454$ 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)

$f(q)$	$=$	$q+2.11454 q^{2} +2.47129 q^{4} -1.00000 q^{5} -2.16289 q^{7} +0.996554 q^{8} +O(q^{10})$	\(q+2.11454 q^{2} +2.47129 q^{4} -1.00000 q^{5} -2.16289 q^{7} +0.996554 q^{8} -2.11454 q^{10} +1.80543 q^{11} -1.29272 q^{13} -4.57353 q^{14} -2.83532 q^{16} -0.350545 q^{17} +3.25941 q^{19} -2.47129 q^{20} +3.81766 q^{22} +0.455773 q^{23} +1.00000 q^{25} -2.73351 q^{26} -5.34513 q^{28} -2.83632 q^{29} -6.90104 q^{31} -7.98850 q^{32} -0.741242 q^{34} +2.16289 q^{35} +2.22421 q^{37} +6.89216 q^{38} -0.996554 q^{40} -9.83272 q^{41} -5.54973 q^{43} +4.46174 q^{44} +0.963751 q^{46} -5.24085 q^{47} -2.32190 q^{49} +2.11454 q^{50} -3.19468 q^{52} -0.434901 q^{53} -1.80543 q^{55} -2.15544 q^{56} -5.99751 q^{58} -10.4564 q^{59} +11.4379 q^{61} -14.5925 q^{62} -11.2214 q^{64} +1.29272 q^{65} -5.58475 q^{67} -0.866297 q^{68} +4.57353 q^{70} +3.93984 q^{71} -8.14300 q^{73} +4.70318 q^{74} +8.05493 q^{76} -3.90495 q^{77} -12.3914 q^{79} +2.83532 q^{80} -20.7917 q^{82} +12.6733 q^{83} +0.350545 q^{85} -11.7351 q^{86} +1.79921 q^{88} -1.00000 q^{89} +2.79601 q^{91} +1.12635 q^{92} -11.0820 q^{94} -3.25941 q^{95} -11.2090 q^{97} -4.90974 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.11454	1.49521	0.747603	−	0.664145i	$-0.231204\pi$
$2$	2.11454	1.49521	0.747603	+	0.664145i	$0.231204\pi$
$3$	0	0
$3$	0	0
$4$	2.47129	1.23564
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.16289	−0.817497	−0.408748	−	0.912647i	$-0.634035\pi$
$7$	−2.16289	−0.817497	−0.408748	+	0.912647i	$0.634035\pi$
$8$	0.996554	0.352335
$9$	0	0
$10$	−2.11454	−0.668677
$11$	1.80543	0.544358	0.272179	−	0.962247i	$-0.412256\pi$
$11$	1.80543	0.544358	0.272179	+	0.962247i	$0.412256\pi$
$12$	0	0
$13$	−1.29272	−0.358535	−0.179268	−	0.983800i	$-0.557373\pi$
$13$	−1.29272	−0.358535	−0.179268	+	0.983800i	$0.557373\pi$
$14$	−4.57353	−1.22233
$15$	0	0
$16$	−2.83532	−0.708829
$17$	−0.350545	−0.0850196	−0.0425098	−	0.999096i	$-0.513535\pi$
$17$	−0.350545	−0.0850196	−0.0425098	+	0.999096i	$0.513535\pi$
$18$	0	0
$19$	3.25941	0.747760	0.373880	−	0.927477i	$-0.378027\pi$
$19$	3.25941	0.747760	0.373880	+	0.927477i	$0.378027\pi$
$20$	−2.47129	−0.552596
$21$	0	0
$22$	3.81766	0.813928
$23$	0.455773	0.0950353	0.0475176	−	0.998870i	$-0.484869\pi$
$23$	0.455773	0.0950353	0.0475176	+	0.998870i	$0.484869\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.73351	−0.536085
$27$	0	0
$28$	−5.34513	−1.01013
$29$	−2.83632	−0.526691	−0.263345	−	0.964702i	$-0.584826\pi$
$29$	−2.83632	−0.526691	−0.263345	+	0.964702i	$0.584826\pi$
$30$	0	0
$31$	−6.90104	−1.23946	−0.619732	−	0.784814i	$-0.712759\pi$
$31$	−6.90104	−1.23946	−0.619732	+	0.784814i	$0.712759\pi$
$32$	−7.98850	−1.41218
$33$	0	0
$34$	−0.741242	−0.127122
$35$	2.16289	0.365596
$36$	0	0
$37$	2.22421	0.365657	0.182829	−	0.983145i	$-0.441475\pi$
$37$	2.22421	0.365657	0.182829	+	0.983145i	$0.441475\pi$
$38$	6.89216	1.11806
$39$	0	0
$40$	−0.996554	−0.157569
$41$	−9.83272	−1.53561	−0.767807	−	0.640682i	$-0.778652\pi$
$41$	−9.83272	−1.53561	−0.767807	+	0.640682i	$0.778652\pi$
$42$	0	0
$43$	−5.54973	−0.846325	−0.423163	−	0.906054i	$-0.639080\pi$
$43$	−5.54973	−0.846325	−0.423163	+	0.906054i	$0.639080\pi$
$44$	4.46174	0.672632
$45$	0	0
$46$	0.963751	0.142097
$47$	−5.24085	−0.764457	−0.382229	−	0.924068i	$-0.624843\pi$
$47$	−5.24085	−0.764457	−0.382229	+	0.924068i	$0.624843\pi$
$48$	0	0
$49$	−2.32190	−0.331699
$50$	2.11454	0.299041
$51$	0	0
$52$	−3.19468	−0.443022
$53$	−0.434901	−0.0597382	−0.0298691	−	0.999554i	$-0.509509\pi$
$53$	−0.434901	−0.0597382	−0.0298691	+	0.999554i	$0.509509\pi$
$54$	0	0
$55$	−1.80543	−0.243444
$56$	−2.15544	−0.288033
$57$	0	0
$58$	−5.99751	−0.787511
$59$	−10.4564	−1.36131	−0.680656	−	0.732603i	$-0.738305\pi$
$59$	−10.4564	−1.36131	−0.680656	+	0.732603i	$0.738305\pi$
$60$	0	0
$61$	11.4379	1.46448	0.732238	−	0.681048i	$-0.238476\pi$
$61$	11.4379	1.46448	0.732238	+	0.681048i	$0.238476\pi$
$62$	−14.5925	−1.85325
$63$	0	0
$64$	−11.2214	−1.40267
$65$	1.29272	0.160342
$66$	0	0
$67$	−5.58475	−0.682286	−0.341143	−	0.940011i	$-0.610814\pi$
$67$	−5.58475	−0.682286	−0.341143	+	0.940011i	$0.610814\pi$
$68$	−0.866297	−0.105054
$69$	0	0
$70$	4.57353	0.546641
$71$	3.93984	0.467573	0.233787	−	0.972288i	$-0.424888\pi$
$71$	3.93984	0.467573	0.233787	+	0.972288i	$0.424888\pi$
$72$	0	0
$73$	−8.14300	−0.953066	−0.476533	−	0.879156i	$-0.658107\pi$
$73$	−8.14300	−0.953066	−0.476533	+	0.879156i	$0.658107\pi$
$74$	4.70318	0.546733
$75$	0	0
$76$	8.05493	0.923964
$77$	−3.90495	−0.445011
$78$	0	0
$79$	−12.3914	−1.39414	−0.697071	−	0.717002i	$-0.745514\pi$
$79$	−12.3914	−1.39414	−0.697071	+	0.717002i	$0.745514\pi$
$80$	2.83532	0.316998
$81$	0	0
$82$	−20.7917	−2.29606
$83$	12.6733	1.39108	0.695539	−	0.718488i	$-0.255166\pi$
$83$	12.6733	1.39108	0.695539	+	0.718488i	$0.255166\pi$
$84$	0	0
$85$	0.350545	0.0380219
$86$	−11.7351	−1.26543
$87$	0	0
$88$	1.79921	0.191796
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	2.79601	0.293101
$92$	1.12635	0.117430
$93$	0	0
$94$	−11.0820	−1.14302
$95$	−3.25941	−0.334408
$96$	0	0
$97$	−11.2090	−1.13810	−0.569051	−	0.822302i	$-0.692689\pi$
$97$	−11.2090	−1.13810	−0.569051	+	0.822302i	$0.692689\pi$
$98$	−4.90974	−0.495959
$99$	0	0
$100$	2.47129	0.247129
$101$	−10.2843	−1.02333	−0.511665	−	0.859185i	$-0.670971\pi$
$101$	−10.2843	−1.02333	−0.511665	+	0.859185i	$0.670971\pi$
$102$	0	0
$103$	−1.00366	−0.0988931	−0.0494466	−	0.998777i	$-0.515746\pi$
$103$	−1.00366	−0.0988931	−0.0494466	+	0.998777i	$0.515746\pi$
$104$	−1.28826	−0.126325
$105$	0	0
$106$	−0.919616	−0.0893210
$107$	15.5749	1.50568	0.752839	−	0.658205i	$-0.228684\pi$
$107$	15.5749	1.50568	0.752839	+	0.658205i	$0.228684\pi$
$108$	0	0
$109$	5.07962	0.486539	0.243270	−	0.969959i	$-0.421780\pi$
$109$	5.07962	0.486539	0.243270	+	0.969959i	$0.421780\pi$
$110$	−3.81766	−0.363999
$111$	0	0
$112$	6.13249	0.579465
$113$	1.22163	0.114921	0.0574606	−	0.998348i	$-0.481700\pi$
$113$	1.22163	0.114921	0.0574606	+	0.998348i	$0.481700\pi$
$114$	0	0
$115$	−0.455773	−0.0425011
$116$	−7.00935	−0.650802
$117$	0	0
$118$	−22.1106	−2.03544
$119$	0.758191	0.0695033
$120$	0	0
$121$	−7.74042	−0.703674
$122$	24.1860	2.18970
$123$	0	0
$124$	−17.0544	−1.53153
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.46614	−0.662513	−0.331257	−	0.943541i	$-0.607473\pi$
$127$	−7.46614	−0.662513	−0.331257	+	0.943541i	$0.607473\pi$
$128$	−7.75109	−0.685106
$129$	0	0
$130$	2.73351	0.239744
$131$	−17.8640	−1.56079	−0.780394	−	0.625288i	$-0.784982\pi$
$131$	−17.8640	−1.56079	−0.780394	+	0.625288i	$0.784982\pi$
$132$	0	0
$133$	−7.04975	−0.611291
$134$	−11.8092	−1.02016
$135$	0	0
$136$	−0.349337	−0.0299554
$137$	−4.76146	−0.406799	−0.203399	−	0.979096i	$-0.565199\pi$
$137$	−4.76146	−0.406799	−0.203399	+	0.979096i	$0.565199\pi$
$138$	0	0
$139$	15.1825	1.28776	0.643879	−	0.765127i	$-0.277324\pi$
$139$	15.1825	1.28776	0.643879	+	0.765127i	$0.277324\pi$
$140$	5.34513	0.451746
$141$	0	0
$142$	8.33096	0.699119
$143$	−2.33391	−0.195172
$144$	0	0
$145$	2.83632	0.235543
$146$	−17.2187	−1.42503
$147$	0	0
$148$	5.49665	0.451822
$149$	13.2347	1.08423	0.542114	−	0.840305i	$-0.317624\pi$
$149$	13.2347	1.08423	0.542114	+	0.840305i	$0.317624\pi$
$150$	0	0
$151$	−7.18971	−0.585090	−0.292545	−	0.956252i	$-0.594502\pi$
$151$	−7.18971	−0.585090	−0.292545	+	0.956252i	$0.594502\pi$
$152$	3.24818	0.263462
$153$	0	0
$154$	−8.25719	−0.665383
$155$	6.90104	0.554305
$156$	0	0
$157$	18.7926	1.49981	0.749905	−	0.661546i	$-0.230099\pi$
$157$	18.7926	1.49981	0.749905	+	0.661546i	$0.230099\pi$
$158$	−26.2021	−2.08453
$159$	0	0
$160$	7.98850	0.631547
$161$	−0.985788	−0.0776910
$162$	0	0
$163$	5.52476	0.432733	0.216366	−	0.976312i	$-0.430579\pi$
$163$	5.52476	0.432733	0.216366	+	0.976312i	$0.430579\pi$
$164$	−24.2995	−1.89747
$165$	0	0
$166$	26.7983	2.07995
$167$	11.1967	0.866428	0.433214	−	0.901291i	$-0.357379\pi$
$167$	11.1967	0.866428	0.433214	+	0.901291i	$0.357379\pi$
$168$	0	0
$169$	−11.3289	−0.871452
$170$	0.741242	0.0568507
$171$	0	0
$172$	−13.7150	−1.04576
$173$	16.5895	1.26128	0.630639	−	0.776077i	$-0.282793\pi$
$173$	16.5895	1.26128	0.630639	+	0.776077i	$0.282793\pi$
$174$	0	0
$175$	−2.16289	−0.163499
$176$	−5.11897	−0.385857
$177$	0	0
$178$	−2.11454	−0.158492
$179$	1.76181	0.131684	0.0658420	−	0.997830i	$-0.479027\pi$
$179$	1.76181	0.131684	0.0658420	+	0.997830i	$0.479027\pi$
$180$	0	0
$181$	7.96930	0.592353	0.296177	−	0.955133i	$-0.404288\pi$
$181$	7.96930	0.592353	0.296177	+	0.955133i	$0.404288\pi$
$182$	5.91228	0.438247
$183$	0	0
$184$	0.454203	0.0334843
$185$	−2.22421	−0.163527
$186$	0	0
$187$	−0.632885	−0.0462811
$188$	−12.9516	−0.944596
$189$	0	0
$190$	−6.89216	−0.500010
$191$	13.4941	0.976397	0.488198	−	0.872733i	$-0.337654\pi$
$191$	13.4941	0.976397	0.488198	+	0.872733i	$0.337654\pi$
$192$	0	0
$193$	14.3586	1.03355	0.516777	−	0.856120i	$-0.327132\pi$
$193$	14.3586	1.03355	0.516777	+	0.856120i	$0.327132\pi$
$194$	−23.7019	−1.70170
$195$	0	0
$196$	−5.73807	−0.409862
$197$	20.9216	1.49060	0.745301	−	0.666728i	$-0.232306\pi$
$197$	20.9216	1.49060	0.745301	+	0.666728i	$0.232306\pi$
$198$	0	0
$199$	−14.6667	−1.03969	−0.519847	−	0.854259i	$-0.674011\pi$
$199$	−14.6667	−1.03969	−0.519847	+	0.854259i	$0.674011\pi$
$200$	0.996554	0.0704670
$201$	0	0
$202$	−21.7466	−1.53009
$203$	6.13465	0.430568
$204$	0	0
$205$	9.83272	0.686747
$206$	−2.12227	−0.147866
$207$	0	0
$208$	3.66526	0.254140
$209$	5.88464	0.407049
$210$	0	0
$211$	4.04428	0.278420	0.139210	−	0.990263i	$-0.455544\pi$
$211$	4.04428	0.278420	0.139210	+	0.990263i	$0.455544\pi$
$212$	−1.07476	−0.0738151
$213$	0	0
$214$	32.9337	2.25130
$215$	5.54973	0.378488
$216$	0	0
$217$	14.9262	1.01326
$218$	10.7411	0.727477
$219$	0	0
$220$	−4.46174	−0.300810
$221$	0.453156	0.0304826
$222$	0	0
$223$	8.23780	0.551644	0.275822	−	0.961209i	$-0.411050\pi$
$223$	8.23780	0.551644	0.275822	+	0.961209i	$0.411050\pi$
$224$	17.2783	1.15445
$225$	0	0
$226$	2.58319	0.171831
$227$	−4.45567	−0.295733	−0.147867	−	0.989007i	$-0.547241\pi$
$227$	−4.45567	−0.295733	−0.147867	+	0.989007i	$0.547241\pi$
$228$	0	0
$229$	−11.6602	−0.770526	−0.385263	−	0.922807i	$-0.625889\pi$
$229$	−11.6602	−0.770526	−0.385263	+	0.922807i	$0.625889\pi$
$230$	−0.963751	−0.0635479
$231$	0	0
$232$	−2.82654	−0.185572
$233$	12.7047	0.832310	0.416155	−	0.909294i	$-0.363377\pi$
$233$	12.7047	0.832310	0.416155	+	0.909294i	$0.363377\pi$
$234$	0	0
$235$	5.24085	0.341876
$236$	−25.8409	−1.68210
$237$	0	0
$238$	1.60323	0.103922
$239$	−8.43428	−0.545568	−0.272784	−	0.962075i	$-0.587945\pi$
$239$	−8.43428	−0.545568	−0.272784	+	0.962075i	$0.587945\pi$
$240$	0	0
$241$	−4.79402	−0.308810	−0.154405	−	0.988008i	$-0.549346\pi$
$241$	−4.79402	−0.308810	−0.154405	+	0.988008i	$0.549346\pi$
$242$	−16.3674	−1.05214
$243$	0	0
$244$	28.2664	1.80957
$245$	2.32190	0.148340
$246$	0	0
$247$	−4.21350	−0.268098
$248$	−6.87726	−0.436707
$249$	0	0
$250$	−2.11454	−0.133735
$251$	10.3347	0.652322	0.326161	−	0.945314i	$-0.394245\pi$
$251$	10.3347	0.652322	0.326161	+	0.945314i	$0.394245\pi$
$252$	0	0
$253$	0.822867	0.0517332
$254$	−15.7875	−0.990594
$255$	0	0
$256$	6.05278	0.378299
$257$	−26.1211	−1.62939	−0.814695	−	0.579890i	$-0.803095\pi$
$257$	−26.1211	−1.62939	−0.814695	+	0.579890i	$0.803095\pi$
$258$	0	0
$259$	−4.81072	−0.298924
$260$	3.19468	0.198125
$261$	0	0
$262$	−37.7743	−2.33370
$263$	3.48276	0.214756	0.107378	−	0.994218i	$-0.465754\pi$
$263$	3.48276	0.214756	0.107378	+	0.994218i	$0.465754\pi$
$264$	0	0
$265$	0.434901	0.0267157
$266$	−14.9070	−0.914006
$267$	0	0
$268$	−13.8015	−0.843062
$269$	−26.9337	−1.64218	−0.821088	−	0.570802i	$-0.806632\pi$
$269$	−26.9337	−1.64218	−0.821088	+	0.570802i	$0.806632\pi$
$270$	0	0
$271$	20.9769	1.27426	0.637129	−	0.770757i	$-0.280122\pi$
$271$	20.9769	1.27426	0.637129	+	0.770757i	$0.280122\pi$
$272$	0.993906	0.0602644
$273$	0	0
$274$	−10.0683	−0.608248
$275$	1.80543	0.108872
$276$	0	0
$277$	−28.7586	−1.72794	−0.863969	−	0.503545i	$-0.832029\pi$
$277$	−28.7586	−1.72794	−0.863969	+	0.503545i	$0.832029\pi$
$278$	32.1039	1.92547
$279$	0	0
$280$	2.15544	0.128812
$281$	11.5505	0.689047	0.344523	−	0.938778i	$-0.388041\pi$
$281$	11.5505	0.689047	0.344523	+	0.938778i	$0.388041\pi$
$282$	0	0
$283$	7.73357	0.459713	0.229856	−	0.973225i	$-0.426174\pi$
$283$	7.73357	0.459713	0.229856	+	0.973225i	$0.426174\pi$
$284$	9.73648	0.577754
$285$	0	0
$286$	−4.93516	−0.291822
$287$	21.2671	1.25536
$288$	0	0
$289$	−16.8771	−0.992772
$290$	5.99751	0.352186
$291$	0	0
$292$	−20.1237	−1.17765
$293$	1.09106	0.0637403	0.0318701	−	0.999492i	$-0.489854\pi$
$293$	1.09106	0.0637403	0.0318701	+	0.999492i	$0.489854\pi$
$294$	0	0
$295$	10.4564	0.608798
$296$	2.21654	0.128834
$297$	0	0
$298$	27.9853	1.62115
$299$	−0.589186	−0.0340735
$300$	0	0
$301$	12.0035	0.691868
$302$	−15.2029	−0.874831
$303$	0	0
$304$	−9.24146	−0.530034
$305$	−11.4379	−0.654934
$306$	0	0
$307$	−16.8073	−0.959245	−0.479623	−	0.877475i	$-0.659226\pi$
$307$	−16.8073	−0.959245	−0.479623	+	0.877475i	$0.659226\pi$
$308$	−9.65026	−0.549874
$309$	0	0
$310$	14.5925	0.828800
$311$	−2.31757	−0.131417	−0.0657087	−	0.997839i	$-0.520931\pi$
$311$	−2.31757	−0.131417	−0.0657087	+	0.997839i	$0.520931\pi$
$312$	0	0
$313$	−27.7090	−1.56620	−0.783102	−	0.621894i	$-0.786364\pi$
$313$	−27.7090	−1.56620	−0.783102	+	0.621894i	$0.786364\pi$
$314$	39.7377	2.24253
$315$	0	0
$316$	−30.6227	−1.72266
$317$	14.1859	0.796757	0.398379	−	0.917221i	$-0.369573\pi$
$317$	14.1859	0.796757	0.398379	+	0.917221i	$0.369573\pi$
$318$	0	0
$319$	−5.12077	−0.286708
$320$	11.2214	0.627295
$321$	0	0
$322$	−2.08449	−0.116164
$323$	−1.14257	−0.0635743
$324$	0	0
$325$	−1.29272	−0.0717071
$326$	11.6823	0.647025
$327$	0	0
$328$	−9.79884	−0.541051
$329$	11.3354	0.624941
$330$	0	0
$331$	6.02289	0.331048	0.165524	−	0.986206i	$-0.447068\pi$
$331$	6.02289	0.331048	0.165524	+	0.986206i	$0.447068\pi$
$332$	31.3194	1.71888
$333$	0	0
$334$	23.6759	1.29549
$335$	5.58475	0.305127
$336$	0	0
$337$	14.6634	0.798768	0.399384	−	0.916784i	$-0.369224\pi$
$337$	14.6634	0.798768	0.399384	+	0.916784i	$0.369224\pi$
$338$	−23.9554	−1.30300
$339$	0	0
$340$	0.866297	0.0469816
$341$	−12.4594	−0.674712
$342$	0	0
$343$	20.1623	1.08866
$344$	−5.53060	−0.298190
$345$	0	0
$346$	35.0792	1.88587
$347$	22.6448	1.21564	0.607819	−	0.794076i	$-0.292045\pi$
$347$	22.6448	1.21564	0.607819	+	0.794076i	$0.292045\pi$
$348$	0	0
$349$	−3.44341	−0.184321	−0.0921606	−	0.995744i	$-0.529377\pi$
$349$	−3.44341	−0.184321	−0.0921606	+	0.995744i	$0.529377\pi$
$350$	−4.57353	−0.244465
$351$	0	0
$352$	−14.4227	−0.768732
$353$	19.7863	1.05312	0.526560	−	0.850138i	$-0.323482\pi$
$353$	19.7863	1.05312	0.526560	+	0.850138i	$0.323482\pi$
$354$	0	0
$355$	−3.93984	−0.209105
$356$	−2.47129	−0.130978
$357$	0	0
$358$	3.72542	0.196895
$359$	2.72181	0.143652	0.0718258	−	0.997417i	$-0.477117\pi$
$359$	2.72181	0.143652	0.0718258	+	0.997417i	$0.477117\pi$
$360$	0	0
$361$	−8.37625	−0.440855
$362$	16.8514	0.885691
$363$	0	0
$364$	6.90974	0.362169
$365$	8.14300	0.426224
$366$	0	0
$367$	−2.76301	−0.144228	−0.0721140	−	0.997396i	$-0.522975\pi$
$367$	−2.76301	−0.144228	−0.0721140	+	0.997396i	$0.522975\pi$
$368$	−1.29226	−0.0673638
$369$	0	0
$370$	−4.70318	−0.244507
$371$	0.940644	0.0488358
$372$	0	0
$373$	−7.39045	−0.382663	−0.191331	−	0.981525i	$-0.561281\pi$
$373$	−7.39045	−0.382663	−0.191331	+	0.981525i	$0.561281\pi$
$374$	−1.33826	−0.0691998
$375$	0	0
$376$	−5.22280	−0.269345
$377$	3.66656	0.188837
$378$	0	0
$379$	−29.4884	−1.51472	−0.757359	−	0.652999i	$-0.773511\pi$
$379$	−29.4884	−1.51472	−0.757359	+	0.652999i	$0.773511\pi$
$380$	−8.05493	−0.413209
$381$	0	0
$382$	28.5338	1.45992
$383$	23.7099	1.21152	0.605759	−	0.795648i	$-0.292870\pi$
$383$	23.7099	1.21152	0.605759	+	0.795648i	$0.292870\pi$
$384$	0	0
$385$	3.90495	0.199015
$386$	30.3619	1.54538
$387$	0	0
$388$	−27.7007	−1.40629
$389$	−1.52446	−0.0772934	−0.0386467	−	0.999253i	$-0.512305\pi$
$389$	−1.52446	−0.0772934	−0.0386467	+	0.999253i	$0.512305\pi$
$390$	0	0
$391$	−0.159769	−0.00807986
$392$	−2.31389	−0.116869
$393$	0	0
$394$	44.2396	2.22876
$395$	12.3914	0.623479
$396$	0	0
$397$	−3.37326	−0.169299	−0.0846495	−	0.996411i	$-0.526977\pi$
$397$	−3.37326	−0.169299	−0.0846495	+	0.996411i	$0.526977\pi$
$398$	−31.0133	−1.55456
$399$	0	0
$400$	−2.83532	−0.141766
$401$	1.45754	0.0727860	0.0363930	−	0.999338i	$-0.488413\pi$
$401$	1.45754	0.0727860	0.0363930	+	0.999338i	$0.488413\pi$
$402$	0	0
$403$	8.92110	0.444391
$404$	−25.4155	−1.26447
$405$	0	0
$406$	12.9720	0.643788
$407$	4.01565	0.199049
$408$	0	0
$409$	−17.6098	−0.870748	−0.435374	−	0.900250i	$-0.643384\pi$
$409$	−17.6098	−0.870748	−0.435374	+	0.900250i	$0.643384\pi$
$410$	20.7917	1.02683
$411$	0	0
$412$	−2.48032	−0.122197
$413$	22.6162	1.11287
$414$	0	0
$415$	−12.6733	−0.622109
$416$	10.3269	0.506317
$417$	0	0
$418$	12.4433	0.608622
$419$	−15.5698	−0.760635	−0.380318	−	0.924856i	$-0.624185\pi$
$419$	−15.5698	−0.760635	−0.380318	+	0.924856i	$0.624185\pi$
$420$	0	0
$421$	−2.11992	−0.103318	−0.0516592	−	0.998665i	$-0.516451\pi$
$421$	−2.11992	−0.103318	−0.0516592	+	0.998665i	$0.516451\pi$
$422$	8.55180	0.416295
$423$	0	0
$424$	−0.433402	−0.0210479
$425$	−0.350545	−0.0170039
$426$	0	0
$427$	−24.7390	−1.19720
$428$	38.4899	1.86048
$429$	0	0
$430$	11.7351	0.565918
$431$	19.1432	0.922094	0.461047	−	0.887376i	$-0.347474\pi$
$431$	19.1432	0.922094	0.461047	+	0.887376i	$0.347474\pi$
$432$	0	0
$433$	28.8788	1.38782	0.693912	−	0.720059i	$-0.255885\pi$
$433$	28.8788	1.38782	0.693912	+	0.720059i	$0.255885\pi$
$434$	31.5621	1.51503
$435$	0	0
$436$	12.5532	0.601189
$437$	1.48555	0.0710635
$438$	0	0
$439$	−22.4166	−1.06989	−0.534944	−	0.844888i	$-0.679667\pi$
$439$	−22.4166	−1.06989	−0.534944	+	0.844888i	$0.679667\pi$
$440$	−1.79921	−0.0857740
$441$	0	0
$442$	0.958217	0.0455777
$443$	32.6570	1.55158	0.775791	−	0.630990i	$-0.217351\pi$
$443$	32.6570	1.55158	0.775791	+	0.630990i	$0.217351\pi$
$444$	0	0
$445$	1.00000	0.0474045
$446$	17.4192	0.824822
$447$	0	0
$448$	24.2707	1.14668
$449$	−6.02343	−0.284263	−0.142132	−	0.989848i	$-0.545396\pi$
$449$	−6.02343	−0.284263	−0.142132	+	0.989848i	$0.545396\pi$
$450$	0	0
$451$	−17.7523	−0.835923
$452$	3.01900	0.142002
$453$	0	0
$454$	−9.42169	−0.442182
$455$	−2.79601	−0.131079
$456$	0	0
$457$	−4.11173	−0.192338	−0.0961692	−	0.995365i	$-0.530659\pi$
$457$	−4.11173	−0.192338	−0.0961692	+	0.995365i	$0.530659\pi$
$458$	−24.6559	−1.15210
$459$	0	0
$460$	−1.12635	−0.0525161
$461$	8.44097	0.393135	0.196568	−	0.980490i	$-0.437020\pi$
$461$	8.44097	0.393135	0.196568	+	0.980490i	$0.437020\pi$
$462$	0	0
$463$	−14.1736	−0.658705	−0.329352	−	0.944207i	$-0.606830\pi$
$463$	−14.1736	−0.658705	−0.329352	+	0.944207i	$0.606830\pi$
$464$	8.04185	0.373334
$465$	0	0
$466$	26.8645	1.24448
$467$	5.44837	0.252120	0.126060	−	0.992023i	$-0.459767\pi$
$467$	5.44837	0.252120	0.126060	+	0.992023i	$0.459767\pi$
$468$	0	0
$469$	12.0792	0.557766
$470$	11.0820	0.511175
$471$	0	0
$472$	−10.4204	−0.479638
$473$	−10.0196	−0.460704
$474$	0	0
$475$	3.25941	0.149552
$476$	1.87371	0.0858812
$477$	0	0
$478$	−17.8346	−0.815738
$479$	−10.6737	−0.487694	−0.243847	−	0.969814i	$-0.578409\pi$
$479$	−10.6737	−0.487694	−0.243847	+	0.969814i	$0.578409\pi$
$480$	0	0
$481$	−2.87527	−0.131101
$482$	−10.1371	−0.461735
$483$	0	0
$484$	−19.1288	−0.869491
$485$	11.2090	0.508975
$486$	0	0
$487$	−21.8531	−0.990258	−0.495129	−	0.868819i	$-0.664879\pi$
$487$	−21.8531	−0.990258	−0.495129	+	0.868819i	$0.664879\pi$
$488$	11.3985	0.515987
$489$	0	0
$490$	4.90974	0.221800
$491$	−13.5937	−0.613474	−0.306737	−	0.951794i	$-0.599237\pi$
$491$	−13.5937	−0.613474	−0.306737	+	0.951794i	$0.599237\pi$
$492$	0	0
$493$	0.994256	0.0447790
$494$	−8.90961	−0.400862
$495$	0	0
$496$	19.5666	0.878568
$497$	−8.52146	−0.382240
$498$	0	0
$499$	−23.8359	−1.06704	−0.533522	−	0.845786i	$-0.679132\pi$
$499$	−23.8359	−1.06704	−0.533522	+	0.845786i	$0.679132\pi$
$500$	−2.47129	−0.110519
$501$	0	0
$502$	21.8532	0.975356
$503$	3.49658	0.155905	0.0779524	−	0.996957i	$-0.475162\pi$
$503$	3.49658	0.155905	0.0779524	+	0.996957i	$0.475162\pi$
$504$	0	0
$505$	10.2843	0.457647
$506$	1.73999	0.0773518
$507$	0	0
$508$	−18.4510	−0.818630
$509$	21.4395	0.950291	0.475145	−	0.879907i	$-0.342395\pi$
$509$	21.4395	0.950291	0.475145	+	0.879907i	$0.342395\pi$
$510$	0	0
$511$	17.6124	0.779129
$512$	28.3010	1.25074
$513$	0	0
$514$	−55.2341	−2.43627
$515$	1.00366	0.0442264
$516$	0	0
$517$	−9.46200	−0.416138
$518$	−10.1725	−0.446953
$519$	0	0
$520$	1.28826	0.0564941
$521$	23.5006	1.02958	0.514789	−	0.857317i	$-0.327870\pi$
$521$	23.5006	1.02958	0.514789	+	0.857317i	$0.327870\pi$
$522$	0	0
$523$	−34.1286	−1.49234	−0.746170	−	0.665756i	$-0.768109\pi$
$523$	−34.1286	−1.49234	−0.746170	+	0.665756i	$0.768109\pi$
$524$	−44.1472	−1.92858
$525$	0	0
$526$	7.36444	0.321105
$527$	2.41912	0.105379
$528$	0	0
$529$	−22.7923	−0.990968
$530$	0.919616	0.0399456
$531$	0	0
$532$	−17.4220	−0.755338
$533$	12.7109	0.550572
$534$	0	0
$535$	−15.5749	−0.673360
$536$	−5.56551	−0.240393
$537$	0	0
$538$	−56.9524	−2.45539
$539$	−4.19202	−0.180563
$540$	0	0
$541$	6.62203	0.284703	0.142352	−	0.989816i	$-0.454534\pi$
$541$	6.62203	0.284703	0.142352	+	0.989816i	$0.454534\pi$
$542$	44.3566	1.90528
$543$	0	0
$544$	2.80033	0.120063
$545$	−5.07962	−0.217587
$546$	0	0
$547$	−26.1787	−1.11932	−0.559661	−	0.828722i	$-0.689069\pi$
$547$	−26.1787	−1.11932	−0.559661	+	0.828722i	$0.689069\pi$
$548$	−11.7669	−0.502658
$549$	0	0
$550$	3.81766	0.162786
$551$	−9.24471	−0.393838
$552$	0	0
$553$	26.8013	1.13971
$554$	−60.8113	−2.58362
$555$	0	0
$556$	37.5202	1.59121
$557$	−2.43288	−0.103084	−0.0515422	−	0.998671i	$-0.516414\pi$
$557$	−2.43288	−0.103084	−0.0515422	+	0.998671i	$0.516414\pi$
$558$	0	0
$559$	7.17423	0.303438
$560$	−6.13249	−0.259145
$561$	0	0
$562$	24.4241	1.03027
$563$	−10.8432	−0.456987	−0.228493	−	0.973545i	$-0.573380\pi$
$563$	−10.8432	−0.456987	−0.228493	+	0.973545i	$0.573380\pi$
$564$	0	0
$565$	−1.22163	−0.0513943
$566$	16.3529	0.687366
$567$	0	0
$568$	3.92627	0.164743
$569$	43.3449	1.81711	0.908556	−	0.417763i	$-0.137186\pi$
$569$	43.3449	1.81711	0.908556	+	0.417763i	$0.137186\pi$
$570$	0	0
$571$	14.1175	0.590799	0.295399	−	0.955374i	$-0.404547\pi$
$571$	14.1175	0.590799	0.295399	+	0.955374i	$0.404547\pi$
$572$	−5.76777	−0.241162
$573$	0	0
$574$	44.9702	1.87702
$575$	0.455773	0.0190071
$576$	0	0
$577$	−7.36072	−0.306431	−0.153215	−	0.988193i	$-0.548963\pi$
$577$	−7.36072	−0.306431	−0.153215	+	0.988193i	$0.548963\pi$
$578$	−35.6874	−1.48440
$579$	0	0
$580$	7.00935	0.291047
$581$	−27.4110	−1.13720
$582$	0	0
$583$	−0.785183	−0.0325190
$584$	−8.11495	−0.335799
$585$	0	0
$586$	2.30709	0.0953049
$587$	−32.5932	−1.34527	−0.672633	−	0.739976i	$-0.734837\pi$
$587$	−32.5932	−1.34527	−0.672633	+	0.739976i	$0.734837\pi$
$588$	0	0
$589$	−22.4933	−0.926821
$590$	22.1106	0.910278
$591$	0	0
$592$	−6.30633	−0.259189
$593$	10.9048	0.447808	0.223904	−	0.974611i	$-0.428120\pi$
$593$	10.9048	0.447808	0.223904	+	0.974611i	$0.428120\pi$
$594$	0	0
$595$	−0.758191	−0.0310828
$596$	32.7067	1.33972
$597$	0	0
$598$	−1.24586	−0.0509469
$599$	−4.84540	−0.197978	−0.0989888	−	0.995089i	$-0.531561\pi$
$599$	−4.84540	−0.197978	−0.0989888	+	0.995089i	$0.531561\pi$
$600$	0	0
$601$	5.27870	0.215323	0.107661	−	0.994188i	$-0.465664\pi$
$601$	5.27870	0.215323	0.107661	+	0.994188i	$0.465664\pi$
$602$	25.3818	1.03449
$603$	0	0
$604$	−17.7678	−0.722963
$605$	7.74042	0.314693
$606$	0	0
$607$	−34.5129	−1.40084	−0.700418	−	0.713733i	$-0.747003\pi$
$607$	−34.5129	−1.40084	−0.700418	+	0.713733i	$0.747003\pi$
$608$	−26.0378	−1.05597
$609$	0	0
$610$	−24.1860	−0.979262
$611$	6.77494	0.274085
$612$	0	0
$613$	12.2587	0.495123	0.247561	−	0.968872i	$-0.420371\pi$
$613$	12.2587	0.495123	0.247561	+	0.968872i	$0.420371\pi$
$614$	−35.5398	−1.43427
$615$	0	0
$616$	−3.89150	−0.156793
$617$	5.15972	0.207723	0.103861	−	0.994592i	$-0.466880\pi$
$617$	5.15972	0.207723	0.103861	+	0.994592i	$0.466880\pi$
$618$	0	0
$619$	−20.5735	−0.826919	−0.413460	−	0.910522i	$-0.635680\pi$
$619$	−20.5735	−0.826919	−0.413460	+	0.910522i	$0.635680\pi$
$620$	17.0544	0.684923
$621$	0	0
$622$	−4.90060	−0.196496
$623$	2.16289	0.0866545
$624$	0	0
$625$	1.00000	0.0400000
$626$	−58.5918	−2.34180
$627$	0	0
$628$	46.4418	1.85323
$629$	−0.779685	−0.0310881
$630$	0	0
$631$	7.91150	0.314952	0.157476	−	0.987523i	$-0.449664\pi$
$631$	7.91150	0.314952	0.157476	+	0.987523i	$0.449664\pi$
$632$	−12.3487	−0.491205
$633$	0	0
$634$	29.9966	1.19132
$635$	7.46614	0.296285
$636$	0	0
$637$	3.00156	0.118926
$638$	−10.8281	−0.428688
$639$	0	0
$640$	7.75109	0.306389
$641$	−17.3711	−0.686116	−0.343058	−	0.939314i	$-0.611463\pi$
$641$	−17.3711	−0.686116	−0.343058	+	0.939314i	$0.611463\pi$
$642$	0	0
$643$	−16.4698	−0.649506	−0.324753	−	0.945799i	$-0.605281\pi$
$643$	−16.4698	−0.649506	−0.324753	+	0.945799i	$0.605281\pi$
$644$	−2.43617	−0.0959984
$645$	0	0
$646$	−2.41601	−0.0950567
$647$	−19.8467	−0.780256	−0.390128	−	0.920761i	$-0.627569\pi$
$647$	−19.8467	−0.780256	−0.390128	+	0.920761i	$0.627569\pi$
$648$	0	0
$649$	−18.8784	−0.741041
$650$	−2.73351	−0.107217
$651$	0	0
$652$	13.6533	0.534703
$653$	17.9939	0.704156	0.352078	−	0.935971i	$-0.385475\pi$
$653$	17.9939	0.704156	0.352078	+	0.935971i	$0.385475\pi$
$654$	0	0
$655$	17.8640	0.698006
$656$	27.8789	1.08849
$657$	0	0
$658$	23.9692	0.934416
$659$	39.0984	1.52306	0.761529	−	0.648131i	$-0.224449\pi$
$659$	39.0984	1.52306	0.761529	+	0.648131i	$0.224449\pi$
$660$	0	0
$661$	38.8951	1.51284	0.756422	−	0.654084i	$-0.226946\pi$
$661$	38.8951	1.51284	0.756422	+	0.654084i	$0.226946\pi$
$662$	12.7357	0.494985
$663$	0	0
$664$	12.6297	0.490126
$665$	7.04975	0.273378
$666$	0	0
$667$	−1.29272	−0.0500542
$668$	27.6703	1.07060
$669$	0	0
$670$	11.8092	0.456229
$671$	20.6504	0.797200
$672$	0	0
$673$	−1.07312	−0.0413657	−0.0206828	−	0.999786i	$-0.506584\pi$
$673$	−1.07312	−0.0413657	−0.0206828	+	0.999786i	$0.506584\pi$
$674$	31.0064	1.19432
$675$	0	0
$676$	−27.9969	−1.07680
$677$	−20.5970	−0.791608	−0.395804	−	0.918335i	$-0.629534\pi$
$677$	−20.5970	−0.791608	−0.395804	+	0.918335i	$0.629534\pi$
$678$	0	0
$679$	24.2439	0.930395
$680$	0.349337	0.0133965
$681$	0	0
$682$	−26.3458	−1.00883
$683$	−29.6084	−1.13293	−0.566467	−	0.824085i	$-0.691690\pi$
$683$	−29.6084	−1.13293	−0.566467	+	0.824085i	$0.691690\pi$
$684$	0	0
$685$	4.76146	0.181926
$686$	42.6339	1.62777
$687$	0	0
$688$	15.7352	0.599900
$689$	0.562204	0.0214183
$690$	0	0
$691$	−23.9530	−0.911214	−0.455607	−	0.890181i	$-0.650578\pi$
$691$	−23.9530	−0.911214	−0.455607	+	0.890181i	$0.650578\pi$
$692$	40.9974	1.55849
$693$	0	0
$694$	47.8834	1.81763
$695$	−15.1825	−0.575903
$696$	0	0
$697$	3.44681	0.130557
$698$	−7.28122	−0.275598
$699$	0	0
$700$	−5.34513	−0.202027
$701$	−18.8526	−0.712051	−0.356026	−	0.934476i	$-0.615868\pi$
$701$	−18.8526	−0.712051	−0.356026	+	0.934476i	$0.615868\pi$
$702$	0	0
$703$	7.24960	0.273424
$704$	−20.2594	−0.763557
$705$	0	0
$706$	41.8390	1.57463
$707$	22.2439	0.836568
$708$	0	0
$709$	9.40759	0.353309	0.176655	−	0.984273i	$-0.443472\pi$
$709$	9.40759	0.353309	0.176655	+	0.984273i	$0.443472\pi$
$710$	−8.33096	−0.312655
$711$	0	0
$712$	−0.996554	−0.0373475
$713$	−3.14531	−0.117793
$714$	0	0
$715$	2.33391	0.0872834
$716$	4.35394	0.162714
$717$	0	0
$718$	5.75538	0.214789
$719$	−13.7190	−0.511634	−0.255817	−	0.966725i	$-0.582344\pi$
$719$	−13.7190	−0.511634	−0.255817	+	0.966725i	$0.582344\pi$
$720$	0	0
$721$	2.17080	0.0808448
$722$	−17.7119	−0.659170
$723$	0	0
$724$	19.6944	0.731937
$725$	−2.83632	−0.105338
$726$	0	0
$727$	12.8746	0.477493	0.238746	−	0.971082i	$-0.423264\pi$
$727$	12.8746	0.477493	0.238746	+	0.971082i	$0.423264\pi$
$728$	2.78638	0.103270
$729$	0	0
$730$	17.2187	0.637293
$731$	1.94543	0.0719543
$732$	0	0
$733$	12.4235	0.458872	0.229436	−	0.973324i	$-0.426312\pi$
$733$	12.4235	0.458872	0.229436	+	0.973324i	$0.426312\pi$
$734$	−5.84250	−0.215651
$735$	0	0
$736$	−3.64095	−0.134207
$737$	−10.0829	−0.371408
$738$	0	0
$739$	2.47762	0.0911406	0.0455703	−	0.998961i	$-0.485490\pi$
$739$	2.47762	0.0911406	0.0455703	+	0.998961i	$0.485490\pi$
$740$	−5.49665	−0.202061
$741$	0	0
$742$	1.98903	0.0730196
$743$	−48.2594	−1.77047	−0.885233	−	0.465149i	$-0.846001\pi$
$743$	−48.2594	−1.77047	−0.885233	+	0.465149i	$0.846001\pi$
$744$	0	0
$745$	−13.2347	−0.484882
$746$	−15.6274	−0.572160
$747$	0	0
$748$	−1.56404	−0.0571869
$749$	−33.6867	−1.23089
$750$	0	0
$751$	41.7698	1.52420	0.762102	−	0.647458i	$-0.224168\pi$
$751$	41.7698	1.52420	0.762102	+	0.647458i	$0.224168\pi$
$752$	14.8595	0.541870
$753$	0	0
$754$	7.75308	0.282351
$755$	7.18971	0.261660
$756$	0	0
$757$	47.4322	1.72395	0.861975	−	0.506950i	$-0.169227\pi$
$757$	47.4322	1.72395	0.861975	+	0.506950i	$0.169227\pi$
$758$	−62.3545	−2.26482
$759$	0	0
$760$	−3.24818	−0.117824
$761$	−30.1178	−1.09177	−0.545884	−	0.837861i	$-0.683806\pi$
$761$	−30.1178	−1.09177	−0.545884	+	0.837861i	$0.683806\pi$
$762$	0	0
$763$	−10.9867	−0.397744
$764$	33.3477	1.20648
$765$	0	0
$766$	50.1355	1.81147
$767$	13.5172	0.488079
$768$	0	0
$769$	−28.2334	−1.01812	−0.509060	−	0.860731i	$-0.670007\pi$
$769$	−28.2334	−1.01812	−0.509060	+	0.860731i	$0.670007\pi$
$770$	8.25719	0.297568
$771$	0	0
$772$	35.4842	1.27710
$773$	13.7238	0.493609	0.246805	−	0.969065i	$-0.420619\pi$
$773$	13.7238	0.493609	0.246805	+	0.969065i	$0.420619\pi$
$774$	0	0
$775$	−6.90104	−0.247893
$776$	−11.1704	−0.400994
$777$	0	0
$778$	−3.22354	−0.115570
$779$	−32.0489	−1.14827
$780$	0	0
$781$	7.11311	0.254527
$782$	−0.337838	−0.0120811
$783$	0	0
$784$	6.58331	0.235118
$785$	−18.7926	−0.670735
$786$	0	0
$787$	−47.6155	−1.69731	−0.848655	−	0.528947i	$-0.822587\pi$
$787$	−47.6155	−1.69731	−0.848655	+	0.528947i	$0.822587\pi$
$788$	51.7033	1.84185
$789$	0	0
$790$	26.2021	0.932230
$791$	−2.64225	−0.0939477
$792$	0	0
$793$	−14.7860	−0.525067
$794$	−7.13289	−0.253137
$795$	0	0
$796$	−36.2456	−1.28469
$797$	14.6149	0.517688	0.258844	−	0.965919i	$-0.416658\pi$
$797$	14.6149	0.517688	0.258844	+	0.965919i	$0.416658\pi$
$798$	0	0
$799$	1.83716	0.0649939
$800$	−7.98850	−0.282436
$801$	0	0
$802$	3.08203	0.108830
$803$	−14.7016	−0.518809
$804$	0	0
$805$	0.985788	0.0347445
$806$	18.8640	0.664457
$807$	0	0
$808$	−10.2489	−0.360555
$809$	−0.980108	−0.0344588	−0.0172294	−	0.999852i	$-0.505485\pi$
$809$	−0.980108	−0.0344588	−0.0172294	+	0.999852i	$0.505485\pi$
$810$	0	0
$811$	−17.7516	−0.623343	−0.311671	−	0.950190i	$-0.600889\pi$
$811$	−17.7516	−0.623343	−0.311671	+	0.950190i	$0.600889\pi$
$812$	15.1605	0.532028
$813$	0	0
$814$	8.49126	0.297619
$815$	−5.52476	−0.193524
$816$	0	0
$817$	−18.0888	−0.632848
$818$	−37.2366	−1.30195
$819$	0	0
$820$	24.2995	0.848574
$821$	−34.1779	−1.19282	−0.596409	−	0.802681i	$-0.703406\pi$
$821$	−34.1779	−1.19282	−0.596409	+	0.802681i	$0.703406\pi$
$822$	0	0
$823$	−43.4595	−1.51490	−0.757452	−	0.652891i	$-0.773556\pi$
$823$	−43.4595	−1.51490	−0.757452	+	0.652891i	$0.773556\pi$
$824$	−1.00020	−0.0348435
$825$	0	0
$826$	47.8228	1.66397
$827$	21.3608	0.742789	0.371394	−	0.928475i	$-0.378880\pi$
$827$	21.3608	0.742789	0.371394	+	0.928475i	$0.378880\pi$
$828$	0	0
$829$	22.5874	0.784493	0.392247	−	0.919860i	$-0.371698\pi$
$829$	22.5874	0.784493	0.392247	+	0.919860i	$0.371698\pi$
$830$	−26.7983	−0.930181
$831$	0	0
$832$	14.5061	0.502908
$833$	0.813929	0.0282010
$834$	0	0
$835$	−11.1967	−0.387478
$836$	14.5426	0.502967
$837$	0	0
$838$	−32.9230	−1.13731
$839$	−40.9584	−1.41404	−0.707020	−	0.707194i	$-0.749961\pi$
$839$	−40.9584	−1.41404	−0.707020	+	0.707194i	$0.749961\pi$
$840$	0	0
$841$	−20.9553	−0.722597
$842$	−4.48265	−0.154482
$843$	0	0
$844$	9.99458	0.344028
$845$	11.3289	0.389725
$846$	0	0
$847$	16.7417	0.575251
$848$	1.23308	0.0423442
$849$	0	0
$850$	−0.741242	−0.0254244
$851$	1.01373	0.0347503
$852$	0	0
$853$	37.2993	1.27710	0.638552	−	0.769578i	$-0.279534\pi$
$853$	37.2993	1.27710	0.638552	+	0.769578i	$0.279534\pi$
$854$	−52.3117	−1.79007
$855$	0	0
$856$	15.5212	0.530503
$857$	−22.4159	−0.765712	−0.382856	−	0.923808i	$-0.625059\pi$
$857$	−22.4159	−0.765712	−0.382856	+	0.923808i	$0.625059\pi$
$858$	0	0
$859$	−30.4452	−1.03878	−0.519389	−	0.854538i	$-0.673841\pi$
$859$	−30.4452	−1.03878	−0.519389	+	0.854538i	$0.673841\pi$
$860$	13.7150	0.467676
$861$	0	0
$862$	40.4790	1.37872
$863$	−15.8361	−0.539067	−0.269533	−	0.962991i	$-0.586869\pi$
$863$	−15.8361	−0.539067	−0.269533	+	0.962991i	$0.586869\pi$
$864$	0	0
$865$	−16.5895	−0.564060
$866$	61.0653	2.07509
$867$	0	0
$868$	36.8869	1.25202
$869$	−22.3718	−0.758912
$870$	0	0
$871$	7.21950	0.244624
$872$	5.06212	0.171425
$873$	0	0
$874$	3.14126	0.106255
$875$	2.16289	0.0731191
$876$	0	0
$877$	5.83896	0.197168	0.0985838	−	0.995129i	$-0.468569\pi$
$877$	5.83896	0.197168	0.0985838	+	0.995129i	$0.468569\pi$
$878$	−47.4009	−1.59970
$879$	0	0
$880$	5.11897	0.172560
$881$	24.6237	0.829595	0.414797	−	0.909914i	$-0.363852\pi$
$881$	24.6237	0.829595	0.414797	+	0.909914i	$0.363852\pi$
$882$	0	0
$883$	33.4217	1.12473	0.562365	−	0.826889i	$-0.309891\pi$
$883$	33.4217	1.12473	0.562365	+	0.826889i	$0.309891\pi$
$884$	1.11988	0.0376656
$885$	0	0
$886$	69.0546	2.31994
$887$	17.3766	0.583450	0.291725	−	0.956502i	$-0.405771\pi$
$887$	17.3766	0.583450	0.291725	+	0.956502i	$0.405771\pi$
$888$	0	0
$889$	16.1485	0.541602
$890$	2.11454	0.0708796
$891$	0	0
$892$	20.3580	0.681635
$893$	−17.0821	−0.571630
$894$	0	0
$895$	−1.76181	−0.0588908
$896$	16.7648	0.560072
$897$	0	0
$898$	−12.7368	−0.425032
$899$	19.5735	0.652814
$900$	0	0
$901$	0.152452	0.00507892
$902$	−37.5380	−1.24988
$903$	0	0
$904$	1.21742	0.0404908
$905$	−7.96930	−0.264908
$906$	0	0
$907$	27.0315	0.897568	0.448784	−	0.893640i	$-0.351857\pi$
$907$	27.0315	0.897568	0.448784	+	0.893640i	$0.351857\pi$
$908$	−11.0112	−0.365420
$909$	0	0
$910$	−5.91228	−0.195990
$911$	27.3178	0.905080	0.452540	−	0.891744i	$-0.350518\pi$
$911$	27.3178	0.905080	0.452540	+	0.891744i	$0.350518\pi$
$912$	0	0
$913$	22.8808	0.757244
$914$	−8.69442	−0.287586
$915$	0	0
$916$	−28.8156	−0.952095
$917$	38.6380	1.27594
$918$	0	0
$919$	−12.5082	−0.412608	−0.206304	−	0.978488i	$-0.566144\pi$
$919$	−12.5082	−0.412608	−0.206304	+	0.978488i	$0.566144\pi$
$920$	−0.454203	−0.0149746
$921$	0	0
$922$	17.8488	0.587818
$923$	−5.09310	−0.167642
$924$	0	0
$925$	2.22421	0.0731315
$926$	−29.9707	−0.984900
$927$	0	0
$928$	22.6579	0.743783
$929$	−22.5427	−0.739602	−0.369801	−	0.929111i	$-0.620574\pi$
$929$	−22.5427	−0.739602	−0.369801	+	0.929111i	$0.620574\pi$
$930$	0	0
$931$	−7.56801	−0.248031
$932$	31.3969	1.02844
$933$	0	0
$934$	11.5208	0.376972
$935$	0.632885	0.0206975
$936$	0	0
$937$	49.6922	1.62337	0.811687	−	0.584093i	$-0.198550\pi$
$937$	49.6922	1.62337	0.811687	+	0.584093i	$0.198550\pi$
$938$	25.5420	0.833976
$939$	0	0
$940$	12.9516	0.422436
$941$	21.5126	0.701291	0.350645	−	0.936508i	$-0.385962\pi$
$941$	21.5126	0.701291	0.350645	+	0.936508i	$0.385962\pi$
$942$	0	0
$943$	−4.48149	−0.145937
$944$	29.6473	0.964938
$945$	0	0
$946$	−21.1870	−0.688847
$947$	−37.8885	−1.23121	−0.615606	−	0.788054i	$-0.711089\pi$
$947$	−37.8885	−1.23121	−0.615606	+	0.788054i	$0.711089\pi$
$948$	0	0
$949$	10.5266	0.341708
$950$	6.89216	0.223611
$951$	0	0
$952$	0.755579	0.0244884
$953$	16.7727	0.543321	0.271660	−	0.962393i	$-0.412427\pi$
$953$	16.7727	0.543321	0.271660	+	0.962393i	$0.412427\pi$
$954$	0	0
$955$	−13.4941	−0.436658
$956$	−20.8435	−0.674128
$957$	0	0
$958$	−22.5700	−0.729203
$959$	10.2985	0.332556
$960$	0	0
$961$	16.6243	0.536269
$962$	−6.07988	−0.196023
$963$	0	0
$964$	−11.8474	−0.381579
$965$	−14.3586	−0.462220
$966$	0	0
$967$	−4.20042	−0.135076	−0.0675382	−	0.997717i	$-0.521514\pi$
$967$	−4.20042	−0.135076	−0.0675382	+	0.997717i	$0.521514\pi$
$968$	−7.71375	−0.247929
$969$	0	0
$970$	23.7019	0.761023
$971$	23.0920	0.741057	0.370529	−	0.928821i	$-0.379177\pi$
$971$	23.0920	0.741057	0.370529	+	0.928821i	$0.379177\pi$
$972$	0	0
$973$	−32.8380	−1.05274
$974$	−46.2093	−1.48064
$975$	0	0
$976$	−32.4302	−1.03806
$977$	−17.0091	−0.544169	−0.272085	−	0.962273i	$-0.587713\pi$
$977$	−17.0091	−0.544169	−0.272085	+	0.962273i	$0.587713\pi$
$978$	0	0
$979$	−1.80543	−0.0577018
$980$	5.73807	0.183296
$981$	0	0
$982$	−28.7444	−0.917270
$983$	15.9042	0.507265	0.253633	−	0.967301i	$-0.418375\pi$
$983$	15.9042	0.507265	0.253633	+	0.967301i	$0.418375\pi$
$984$	0	0
$985$	−20.9216	−0.666618
$986$	2.10240	0.0669539
$987$	0	0
$988$	−10.4128	−0.331274
$989$	−2.52942	−0.0804307
$990$	0	0
$991$	−3.40353	−0.108117	−0.0540583	−	0.998538i	$-0.517216\pi$
$991$	−3.40353	−0.108117	−0.0540583	+	0.998538i	$0.517216\pi$
$992$	55.1290	1.75035
$993$	0	0
$994$	−18.0190	−0.571527
$995$	14.6667	0.464965
$996$	0	0
$997$	7.86902	0.249214	0.124607	−	0.992206i	$-0.460233\pi$
$997$	7.86902	0.249214	0.124607	+	0.992206i	$0.460233\pi$
$998$	−50.4021	−1.59545
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4005.2.a.v.1.10	yes	12
3.2	odd	2		4005.2.a.u.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4005.2.a.u.1.3	✓	12	3.2	odd	2
4005.2.a.v.1.10	yes	12	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.11454 q^{2} +2.47129 q^{4} -1.00000 q^{5} -2.16289 q^{7} +0.996554 q^{8} +O(q^{10})\)	\(q+2.11454 q^{2} +2.47129 q^{4} -1.00000 q^{5} -2.16289 q^{7} +0.996554 q^{8} -2.11454 q^{10} +1.80543 q^{11} -1.29272 q^{13} -4.57353 q^{14} -2.83532 q^{16} -0.350545 q^{17} +3.25941 q^{19} -2.47129 q^{20} +3.81766 q^{22} +0.455773 q^{23} +1.00000 q^{25} -2.73351 q^{26} -5.34513 q^{28} -2.83632 q^{29} -6.90104 q^{31} -7.98850 q^{32} -0.741242 q^{34} +2.16289 q^{35} +2.22421 q^{37} +6.89216 q^{38} -0.996554 q^{40} -9.83272 q^{41} -5.54973 q^{43} +4.46174 q^{44} +0.963751 q^{46} -5.24085 q^{47} -2.32190 q^{49} +2.11454 q^{50} -3.19468 q^{52} -0.434901 q^{53} -1.80543 q^{55} -2.15544 q^{56} -5.99751 q^{58} -10.4564 q^{59} +11.4379 q^{61} -14.5925 q^{62} -11.2214 q^{64} +1.29272 q^{65} -5.58475 q^{67} -0.866297 q^{68} +4.57353 q^{70} +3.93984 q^{71} -8.14300 q^{73} +4.70318 q^{74} +8.05493 q^{76} -3.90495 q^{77} -12.3914 q^{79} +2.83532 q^{80} -20.7917 q^{82} +12.6733 q^{83} +0.350545 q^{85} -11.7351 q^{86} +1.79921 q^{88} -1.00000 q^{89} +2.79601 q^{91} +1.12635 q^{92} -11.0820 q^{94} -3.25941 q^{95} -11.2090 q^{97} -4.90974 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 11 q^{4} - 12 q^{5} - 8 q^{7} + 9 q^{8}+O(q^{10}) \) 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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