LMFDB - Embedded newform 4005.2.a.u.1.3

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			1.3
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	$-0.768796\pi$
$2$	−2.11454	−1.49521	−0.747603	+	0.664145i	$0.768796\pi$
$3$	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	$-0.587744\pi$
$11$	−1.80543	−0.544358	−0.272179	+	0.962247i	$0.587744\pi$
$12$	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	$-0.486465\pi$
$17$	0.350545	0.0850196	0.0425098	+	0.999096i	$0.486465\pi$
$18$	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	$-0.515131\pi$
$23$	−0.455773	−0.0950353	−0.0475176	+	0.998870i	$0.515131\pi$
$24$	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	$-0.415174\pi$
$29$	2.83632	0.526691	0.263345	+	0.964702i	$0.415174\pi$
$30$	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	$-0.221348\pi$
$41$	9.83272	1.53561	0.767807	+	0.640682i	$0.221348\pi$
$42$	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	$-0.375157\pi$
$47$	5.24085	0.764457	0.382229	+	0.924068i	$0.375157\pi$
$48$	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	$-0.490491\pi$
$53$	0.434901	0.0597382	0.0298691	+	0.999554i	$0.490491\pi$
$54$	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	$-0.261695\pi$
$59$	10.4564	1.36131	0.680656	+	0.732603i	$0.261695\pi$
$60$	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	$-0.575112\pi$
$71$	−3.93984	−0.467573	−0.233787	+	0.972288i	$0.575112\pi$
$72$	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	$-0.744834\pi$
$83$	−12.6733	−1.39108	−0.695539	+	0.718488i	$0.744834\pi$
$84$	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	$-0.329029\pi$
$101$	10.2843	1.02333	0.511665	+	0.859185i	$0.329029\pi$
$102$	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	$-0.771316\pi$
$107$	−15.5749	−1.50568	−0.752839	+	0.658205i	$0.771316\pi$
$108$	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	$-0.518300\pi$
$113$	−1.22163	−0.114921	−0.0574606	+	0.998348i	$0.518300\pi$
$114$	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	$-0.215018\pi$
$131$	17.8640	1.56079	0.780394	+	0.625288i	$0.215018\pi$
$132$	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	$-0.434801\pi$
$137$	4.76146	0.406799	0.203399	+	0.979096i	$0.434801\pi$
$138$	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	$-0.682376\pi$
$149$	−13.2347	−1.08423	−0.542114	+	0.840305i	$0.682376\pi$
$150$	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	$-0.642621\pi$
$167$	−11.1967	−0.866428	−0.433214	+	0.901291i	$0.642621\pi$
$168$	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	$-0.717207\pi$
$173$	−16.5895	−1.26128	−0.630639	+	0.776077i	$0.717207\pi$
$174$	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	$-0.520973\pi$
$179$	−1.76181	−0.131684	−0.0658420	+	0.997830i	$0.520973\pi$
$180$	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	$-0.662346\pi$
$191$	−13.4941	−0.976397	−0.488198	+	0.872733i	$0.662346\pi$
$192$	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	$-0.767694\pi$
$197$	−20.9216	−1.49060	−0.745301	+	0.666728i	$0.767694\pi$
$198$	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	$-0.452759\pi$
$227$	4.45567	0.295733	0.147867	+	0.989007i	$0.452759\pi$
$228$	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	$-0.636623\pi$
$233$	−12.7047	−0.832310	−0.416155	+	0.909294i	$0.636623\pi$
$234$	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	$-0.412055\pi$
$239$	8.43428	0.545568	0.272784	+	0.962075i	$0.412055\pi$
$240$	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	$-0.605755\pi$
$251$	−10.3347	−0.652322	−0.326161	+	0.945314i	$0.605755\pi$
$252$	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	$-0.196905\pi$
$257$	26.1211	1.62939	0.814695	+	0.579890i	$0.196905\pi$
$258$	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	$-0.534246\pi$
$263$	−3.48276	−0.214756	−0.107378	+	0.994218i	$0.534246\pi$
$264$	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	$-0.193368\pi$
$269$	26.9337	1.64218	0.821088	+	0.570802i	$0.193368\pi$
$270$	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	$-0.611959\pi$
$281$	−11.5505	−0.689047	−0.344523	+	0.938778i	$0.611959\pi$
$282$	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	$-0.510146\pi$
$293$	−1.09106	−0.0637403	−0.0318701	+	0.999492i	$0.510146\pi$
$294$	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	$-0.479069\pi$
$311$	2.31757	0.131417	0.0657087	+	0.997839i	$0.479069\pi$
$312$	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	$-0.630427\pi$
$317$	−14.1859	−0.796757	−0.398379	+	0.917221i	$0.630427\pi$
$318$	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	$-0.707955\pi$
$347$	−22.6448	−1.21564	−0.607819	+	0.794076i	$0.707955\pi$
$348$	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	$-0.676518\pi$
$353$	−19.7863	−1.05312	−0.526560	+	0.850138i	$0.676518\pi$
$354$	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	$-0.522883\pi$
$359$	−2.72181	−0.143652	−0.0718258	+	0.997417i	$0.522883\pi$
$360$	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	$-0.707130\pi$
$383$	−23.7099	−1.21152	−0.605759	+	0.795648i	$0.707130\pi$
$384$	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	$-0.487695\pi$
$389$	1.52446	0.0772934	0.0386467	+	0.999253i	$0.487695\pi$
$390$	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	$-0.511587\pi$
$401$	−1.45754	−0.0727860	−0.0363930	+	0.999338i	$0.511587\pi$
$402$	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	$-0.375815\pi$
$419$	15.5698	0.760635	0.380318	+	0.924856i	$0.375815\pi$
$420$	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	$-0.652526\pi$
$431$	−19.1432	−0.922094	−0.461047	+	0.887376i	$0.652526\pi$
$432$	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	$-0.782649\pi$
$443$	−32.6570	−1.55158	−0.775791	+	0.630990i	$0.782649\pi$
$444$	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	$-0.454604\pi$
$449$	6.02343	0.284263	0.142132	+	0.989848i	$0.454604\pi$
$450$	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	$-0.562980\pi$
$461$	−8.44097	−0.393135	−0.196568	+	0.980490i	$0.562980\pi$
$462$	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	$-0.540233\pi$
$467$	−5.44837	−0.252120	−0.126060	+	0.992023i	$0.540233\pi$
$468$	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	$-0.421591\pi$
$479$	10.6737	0.487694	0.243847	+	0.969814i	$0.421591\pi$
$480$	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	$-0.400763\pi$
$491$	13.5937	0.613474	0.306737	+	0.951794i	$0.400763\pi$
$492$	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	$-0.524838\pi$
$503$	−3.49658	−0.155905	−0.0779524	+	0.996957i	$0.524838\pi$
$504$	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	$-0.657605\pi$
$509$	−21.4395	−0.950291	−0.475145	+	0.879907i	$0.657605\pi$
$510$	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	$-0.672130\pi$
$521$	−23.5006	−1.02958	−0.514789	+	0.857317i	$0.672130\pi$
$522$	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	$-0.483586\pi$
$557$	2.43288	0.103084	0.0515422	+	0.998671i	$0.483586\pi$
$558$	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	$-0.426620\pi$
$563$	10.8432	0.456987	0.228493	+	0.973545i	$0.426620\pi$
$564$	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	$-0.862814\pi$
$569$	−43.3449	−1.81711	−0.908556	+	0.417763i	$0.862814\pi$
$570$	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	$-0.265163\pi$
$587$	32.5932	1.34527	0.672633	+	0.739976i	$0.265163\pi$
$588$	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	$-0.571880\pi$
$593$	−10.9048	−0.447808	−0.223904	+	0.974611i	$0.571880\pi$
$594$	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	$-0.468439\pi$
$599$	4.84540	0.197978	0.0989888	+	0.995089i	$0.468439\pi$
$600$	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	$-0.533120\pi$
$617$	−5.15972	−0.207723	−0.103861	+	0.994592i	$0.533120\pi$
$618$	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	$-0.388537\pi$
$641$	17.3711	0.686116	0.343058	+	0.939314i	$0.388537\pi$
$642$	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	$-0.372431\pi$
$647$	19.8467	0.780256	0.390128	+	0.920761i	$0.372431\pi$
$648$	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	$-0.614525\pi$
$653$	−17.9939	−0.704156	−0.352078	+	0.935971i	$0.614525\pi$
$654$	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	$-0.775551\pi$
$659$	−39.0984	−1.52306	−0.761529	+	0.648131i	$0.775551\pi$
$660$	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	$-0.370466\pi$
$677$	20.5970	0.791608	0.395804	+	0.918335i	$0.370466\pi$
$678$	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	$-0.308310\pi$
$683$	29.6084	1.13293	0.566467	+	0.824085i	$0.308310\pi$
$684$	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	$-0.384132\pi$
$701$	18.8526	0.712051	0.356026	+	0.934476i	$0.384132\pi$
$702$	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	$-0.417656\pi$
$719$	13.7190	0.511634	0.255817	+	0.966725i	$0.417656\pi$
$720$	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	$-0.153999\pi$
$743$	48.2594	1.77047	0.885233	+	0.465149i	$0.153999\pi$
$744$	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	$-0.316194\pi$
$761$	30.1178	1.09177	0.545884	+	0.837861i	$0.316194\pi$
$762$	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	$-0.579381\pi$
$773$	−13.7238	−0.493609	−0.246805	+	0.969065i	$0.579381\pi$
$774$	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	$-0.583342\pi$
$797$	−14.6149	−0.517688	−0.258844	+	0.965919i	$0.583342\pi$
$798$	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	$-0.494515\pi$
$809$	0.980108	0.0344588	0.0172294	+	0.999852i	$0.494515\pi$
$810$	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	$-0.296594\pi$
$821$	34.1779	1.19282	0.596409	+	0.802681i	$0.296594\pi$
$822$	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	$-0.621120\pi$
$827$	−21.3608	−0.742789	−0.371394	+	0.928475i	$0.621120\pi$
$828$	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	$-0.250039\pi$
$839$	40.9584	1.41404	0.707020	+	0.707194i	$0.250039\pi$
$840$	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	$-0.374941\pi$
$857$	22.4159	0.765712	0.382856	+	0.923808i	$0.374941\pi$
$858$	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	$-0.413131\pi$
$863$	15.8361	0.539067	0.269533	+	0.962991i	$0.413131\pi$
$864$	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	$-0.636148\pi$
$881$	−24.6237	−0.829595	−0.414797	+	0.909914i	$0.636148\pi$
$882$	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	$-0.594229\pi$
$887$	−17.3766	−0.583450	−0.291725	+	0.956502i	$0.594229\pi$
$888$	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	$-0.649482\pi$
$911$	−27.3178	−0.905080	−0.452540	+	0.891744i	$0.649482\pi$
$912$	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	$-0.379426\pi$
$929$	22.5427	0.739602	0.369801	+	0.929111i	$0.379426\pi$
$930$	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	$-0.614038\pi$
$941$	−21.5126	−0.701291	−0.350645	+	0.936508i	$0.614038\pi$
$942$	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	$-0.288911\pi$
$947$	37.8885	1.23121	0.615606	+	0.788054i	$0.288911\pi$
$948$	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	$-0.587573\pi$
$953$	−16.7727	−0.543321	−0.271660	+	0.962393i	$0.587573\pi$
$954$	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	$-0.620823\pi$
$971$	−23.0920	−0.741057	−0.370529	+	0.928821i	$0.620823\pi$
$972$	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	$-0.412287\pi$
$977$	17.0091	0.544169	0.272085	+	0.962273i	$0.412287\pi$
$978$	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	$-0.581625\pi$
$983$	−15.9042	−0.507265	−0.253633	+	0.967301i	$0.581625\pi$
$984$	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.u.1.3	✓	12
3.2	odd	2		4005.2.a.v.1.10	yes	12

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

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