LMFDB - Embedded newform 495.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,6,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("495.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	495.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:	$79.3899908074$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.18257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 26x + 8 $ x^3 - x^2 - 26*x + 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 165)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.305203$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.694797 q^{2} -31.5173 q^{4} -25.0000 q^{5} +83.1683 q^{7} +44.1316 q^{8} +O(q^{10})$	\(q-0.694797 q^{2} -31.5173 q^{4} -25.0000 q^{5} +83.1683 q^{7} +44.1316 q^{8} +17.3699 q^{10} -121.000 q^{11} -674.655 q^{13} -57.7851 q^{14} +977.890 q^{16} -1927.66 q^{17} -149.751 q^{19} +787.931 q^{20} +84.0704 q^{22} +1355.59 q^{23} +625.000 q^{25} +468.748 q^{26} -2621.24 q^{28} +7320.99 q^{29} -4215.76 q^{31} -2091.65 q^{32} +1339.33 q^{34} -2079.21 q^{35} -13420.1 q^{37} +104.046 q^{38} -1103.29 q^{40} -2865.39 q^{41} -22078.1 q^{43} +3813.59 q^{44} -941.861 q^{46} +14556.4 q^{47} -9890.04 q^{49} -434.248 q^{50} +21263.3 q^{52} -13349.7 q^{53} +3025.00 q^{55} +3670.35 q^{56} -5086.60 q^{58} -45803.3 q^{59} +18996.5 q^{61} +2929.10 q^{62} -29839.2 q^{64} +16866.4 q^{65} +6651.05 q^{67} +60754.5 q^{68} +1444.63 q^{70} +61028.7 q^{71} -17353.4 q^{73} +9324.27 q^{74} +4719.73 q^{76} -10063.4 q^{77} +61676.5 q^{79} -24447.2 q^{80} +1990.87 q^{82} +65230.8 q^{83} +48191.4 q^{85} +15339.8 q^{86} -5339.92 q^{88} +109563. q^{89} -56109.9 q^{91} -42724.5 q^{92} -10113.8 q^{94} +3743.76 q^{95} +83736.1 q^{97} +6871.57 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8	$ 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * 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 $

$2$	−0.694797	−0.122824	−0.0614120	−	0.998113i	$-0.519560\pi$
$2$	−0.694797	−0.122824	−0.0614120	+	0.998113i	$0.519560\pi$
$3$	0	0
$3$	0	0
$4$	−31.5173	−0.984914
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	83.1683	0.641523	0.320762	−	0.947160i	$-0.396061\pi$
$7$	83.1683	0.641523	0.320762	+	0.947160i	$0.396061\pi$
$8$	44.1316	0.243795
$9$	0	0
$10$	17.3699	0.0549285
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−674.655	−1.10719	−0.553597	−	0.832785i	$-0.686745\pi$
$13$	−674.655	−1.10719	−0.553597	+	0.832785i	$0.686745\pi$
$14$	−57.7851	−0.0787944
$15$	0	0
$16$	977.890	0.954970
$17$	−1927.66	−1.61774	−0.808868	−	0.587991i	$-0.799919\pi$
$17$	−1927.66	−1.61774	−0.808868	+	0.587991i	$0.799919\pi$
$18$	0	0
$19$	−149.751	−0.0951666	−0.0475833	−	0.998867i	$-0.515152\pi$
$19$	−149.751	−0.0951666	−0.0475833	+	0.998867i	$0.515152\pi$
$20$	787.931	0.440467
$21$	0	0
$22$	84.0704	0.0370328
$23$	1355.59	0.534330	0.267165	−	0.963651i	$-0.413913\pi$
$23$	1355.59	0.534330	0.267165	+	0.963651i	$0.413913\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	468.748	0.135990
$27$	0	0
$28$	−2621.24	−0.631846
$29$	7320.99	1.61650	0.808248	−	0.588842i	$-0.200416\pi$
$29$	7320.99	1.61650	0.808248	+	0.588842i	$0.200416\pi$
$30$	0	0
$31$	−4215.76	−0.787901	−0.393951	−	0.919132i	$-0.628892\pi$
$31$	−4215.76	−0.787901	−0.393951	+	0.919132i	$0.628892\pi$
$32$	−2091.65	−0.361088
$33$	0	0
$34$	1339.33	0.198697
$35$	−2079.21	−0.286898
$36$	0	0
$37$	−13420.1	−1.61158	−0.805792	−	0.592199i	$-0.798260\pi$
$37$	−13420.1	−1.61158	−0.805792	+	0.592199i	$0.798260\pi$
$38$	104.046	0.0116887
$39$	0	0
$40$	−1103.29	−0.109028
$41$	−2865.39	−0.266210	−0.133105	−	0.991102i	$-0.542495\pi$
$41$	−2865.39	−0.266210	−0.133105	+	0.991102i	$0.542495\pi$
$42$	0	0
$43$	−22078.1	−1.82092	−0.910458	−	0.413601i	$-0.864271\pi$
$43$	−22078.1	−1.82092	−0.910458	+	0.413601i	$0.864271\pi$
$44$	3813.59	0.296963
$45$	0	0
$46$	−941.861	−0.0656285
$47$	14556.4	0.961192	0.480596	−	0.876942i	$-0.340420\pi$
$47$	14556.4	0.961192	0.480596	+	0.876942i	$0.340420\pi$
$48$	0	0
$49$	−9890.04	−0.588448
$50$	−434.248	−0.0245648
$51$	0	0
$52$	21263.3	1.09049
$53$	−13349.7	−0.652805	−0.326402	−	0.945231i	$-0.605836\pi$
$53$	−13349.7	−0.652805	−0.326402	+	0.945231i	$0.605836\pi$
$54$	0	0
$55$	3025.00	0.134840
$56$	3670.35	0.156400
$57$	0	0
$58$	−5086.60	−0.198544
$59$	−45803.3	−1.71304	−0.856518	−	0.516117i	$-0.827377\pi$
$59$	−45803.3	−1.71304	−0.856518	+	0.516117i	$0.827377\pi$
$60$	0	0
$61$	18996.5	0.653655	0.326827	−	0.945084i	$-0.394020\pi$
$61$	18996.5	0.653655	0.326827	+	0.945084i	$0.394020\pi$
$62$	2929.10	0.0967731
$63$	0	0
$64$	−29839.2	−0.910620
$65$	16866.4	0.495152
$66$	0	0
$67$	6651.05	0.181010	0.0905052	−	0.995896i	$-0.471152\pi$
$67$	6651.05	0.181010	0.0905052	+	0.995896i	$0.471152\pi$
$68$	60754.5	1.59333
$69$	0	0
$70$	1444.63	0.0352379
$71$	61028.7	1.43677	0.718386	−	0.695644i	$-0.244881\pi$
$71$	61028.7	1.43677	0.718386	+	0.695644i	$0.244881\pi$
$72$	0	0
$73$	−17353.4	−0.381134	−0.190567	−	0.981674i	$-0.561033\pi$
$73$	−17353.4	−0.381134	−0.190567	+	0.981674i	$0.561033\pi$
$74$	9324.27	0.197941
$75$	0	0
$76$	4719.73	0.0937309
$77$	−10063.4	−0.193427
$78$	0	0
$79$	61676.5	1.11187	0.555933	−	0.831227i	$-0.312361\pi$
$79$	61676.5	1.11187	0.555933	+	0.831227i	$0.312361\pi$
$80$	−24447.2	−0.427076
$81$	0	0
$82$	1990.87	0.0326969
$83$	65230.8	1.03934	0.519670	−	0.854367i	$-0.326055\pi$
$83$	65230.8	1.03934	0.519670	+	0.854367i	$0.326055\pi$
$84$	0	0
$85$	48191.4	0.723473
$86$	15339.8	0.223652
$87$	0	0
$88$	−5339.92	−0.0735069
$89$	109563.	1.46618	0.733091	−	0.680131i	$-0.238077\pi$
$89$	109563.	1.46618	0.733091	+	0.680131i	$0.238077\pi$
$90$	0	0
$91$	−56109.9	−0.710291
$92$	−42724.5	−0.526269
$93$	0	0
$94$	−10113.8	−0.118057
$95$	3743.76	0.0425598
$96$	0	0
$97$	83736.1	0.903615	0.451808	−	0.892115i	$-0.350779\pi$
$97$	83736.1	0.903615	0.451808	+	0.892115i	$0.350779\pi$
$98$	6871.57	0.0722754
$99$	0	0
$100$	−19698.3	−0.196983
$101$	129877.	1.26686	0.633428	−	0.773801i	$-0.281647\pi$
$101$	129877.	1.26686	0.633428	+	0.773801i	$0.281647\pi$
$102$	0	0
$103$	201851.	1.87472	0.937362	−	0.348358i	$-0.113261\pi$
$103$	201851.	1.87472	0.937362	+	0.348358i	$0.113261\pi$
$104$	−29773.6	−0.269928
$105$	0	0
$106$	9275.36	0.0801800
$107$	132285.	1.11699	0.558496	−	0.829507i	$-0.311378\pi$
$107$	132285.	1.11699	0.558496	+	0.829507i	$0.311378\pi$
$108$	0	0
$109$	−144044.	−1.16126	−0.580629	−	0.814168i	$-0.697193\pi$
$109$	−144044.	−1.16126	−0.580629	+	0.814168i	$0.697193\pi$
$110$	−2101.76	−0.0165616
$111$	0	0
$112$	81329.4	0.612636
$113$	1095.62	0.00807165	0.00403582	−	0.999992i	$-0.498715\pi$
$113$	1095.62	0.00807165	0.00403582	+	0.999992i	$0.498715\pi$
$114$	0	0
$115$	−33889.8	−0.238960
$116$	−230737.	−1.59211
$117$	0	0
$118$	31824.0	0.210402
$119$	−160320.	−1.03782
$120$	0	0
$121$	14641.0	0.0909091
$122$	−13198.7	−0.0802844
$123$	0	0
$124$	132869.	0.776015
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	315758.	1.73718	0.868591	−	0.495529i	$-0.165026\pi$
$127$	315758.	1.73718	0.868591	+	0.495529i	$0.165026\pi$
$128$	87664.9	0.472934
$129$	0	0
$130$	−11718.7	−0.0608165
$131$	187001.	0.952064	0.476032	−	0.879428i	$-0.342075\pi$
$131$	187001.	0.952064	0.476032	+	0.879428i	$0.342075\pi$
$132$	0	0
$133$	−12454.5	−0.0610516
$134$	−4621.13	−0.0222324
$135$	0	0
$136$	−85070.6	−0.394396
$137$	−232301.	−1.05743	−0.528713	−	0.848801i	$-0.677325\pi$
$137$	−232301.	−1.05743	−0.528713	+	0.848801i	$0.677325\pi$
$138$	0	0
$139$	−311665.	−1.36820	−0.684102	−	0.729386i	$-0.739806\pi$
$139$	−311665.	−1.36820	−0.684102	+	0.729386i	$0.739806\pi$
$140$	65530.9	0.282570
$141$	0	0
$142$	−42402.5	−0.176470
$143$	81633.3	0.333831
$144$	0	0
$145$	−183025.	−0.722919
$146$	12057.1	0.0468124
$147$	0	0
$148$	422966.	1.58727
$149$	−30354.5	−0.112010	−0.0560050	−	0.998430i	$-0.517836\pi$
$149$	−30354.5	−0.112010	−0.0560050	+	0.998430i	$0.517836\pi$
$150$	0	0
$151$	131015.	0.467604	0.233802	−	0.972284i	$-0.424883\pi$
$151$	131015.	0.467604	0.233802	+	0.972284i	$0.424883\pi$
$152$	−6608.73	−0.0232011
$153$	0	0
$154$	6991.99	0.0237574
$155$	105394.	0.352360
$156$	0	0
$157$	326922.	1.05851	0.529255	−	0.848463i	$-0.322472\pi$
$157$	326922.	1.05851	0.529255	+	0.848463i	$0.322472\pi$
$158$	−42852.7	−0.136564
$159$	0	0
$160$	52291.1	0.161484
$161$	112742.	0.342785
$162$	0	0
$163$	−501854.	−1.47948	−0.739739	−	0.672894i	$-0.765051\pi$
$163$	−501854.	−1.47948	−0.739739	+	0.672894i	$0.765051\pi$
$164$	90309.3	0.262194
$165$	0	0
$166$	−45322.2	−0.127656
$167$	590462.	1.63833	0.819164	−	0.573559i	$-0.194438\pi$
$167$	590462.	1.63833	0.819164	+	0.573559i	$0.194438\pi$
$168$	0	0
$169$	83866.7	0.225877
$170$	−33483.3	−0.0888598
$171$	0	0
$172$	695840.	1.79345
$173$	−570934.	−1.45034	−0.725172	−	0.688568i	$-0.758240\pi$
$173$	−570934.	−1.45034	−0.725172	+	0.688568i	$0.758240\pi$
$174$	0	0
$175$	51980.2	0.128305
$176$	−118325.	−0.287934
$177$	0	0
$178$	−76123.9	−0.180082
$179$	464402.	1.08333	0.541666	−	0.840594i	$-0.317794\pi$
$179$	464402.	1.08333	0.541666	+	0.840594i	$0.317794\pi$
$180$	0	0
$181$	−409792.	−0.929751	−0.464876	−	0.885376i	$-0.653901\pi$
$181$	−409792.	−0.929751	−0.464876	+	0.885376i	$0.653901\pi$
$182$	38985.0	0.0872407
$183$	0	0
$184$	59824.4	0.130267
$185$	335504.	0.720722
$186$	0	0
$187$	233247.	0.487766
$188$	−458778.	−0.946691
$189$	0	0
$190$	−2601.16	−0.00522736
$191$	591029.	1.17226	0.586132	−	0.810216i	$-0.300650\pi$
$191$	591029.	1.17226	0.586132	+	0.810216i	$0.300650\pi$
$192$	0	0
$193$	−310907.	−0.600811	−0.300405	−	0.953812i	$-0.597122\pi$
$193$	−310907.	−0.600811	−0.300405	+	0.953812i	$0.597122\pi$
$194$	−58179.6	−0.110986
$195$	0	0
$196$	311707.	0.579571
$197$	−161613.	−0.296695	−0.148348	−	0.988935i	$-0.547395\pi$
$197$	−161613.	−0.296695	−0.148348	+	0.988935i	$0.547395\pi$
$198$	0	0
$199$	435622.	0.779790	0.389895	−	0.920859i	$-0.372511\pi$
$199$	435622.	0.779790	0.389895	+	0.920859i	$0.372511\pi$
$200$	27582.2	0.0487590
$201$	0	0
$202$	−90237.8	−0.155600
$203$	608874.	1.03702
$204$	0	0
$205$	71634.8	0.119053
$206$	−140245.	−0.230261
$207$	0	0
$208$	−659738.	−1.05734
$209$	18119.8	0.0286938
$210$	0	0
$211$	−539434.	−0.834127	−0.417063	−	0.908877i	$-0.636941\pi$
$211$	−539434.	−0.834127	−0.417063	+	0.908877i	$0.636941\pi$
$212$	420747.	0.642957
$213$	0	0
$214$	−91911.1	−0.137193
$215$	551952.	0.814339
$216$	0	0
$217$	−350618.	−0.505457
$218$	100081.	0.142630
$219$	0	0
$220$	−95339.7	−0.132806
$221$	1.30050e6	1.79115
$222$	0	0
$223$	452834.	0.609786	0.304893	−	0.952387i	$-0.401379\pi$
$223$	452834.	0.609786	0.304893	+	0.952387i	$0.401379\pi$
$224$	−173959.	−0.231646
$225$	0	0
$226$	−761.230	−0.000991391 0
$227$	40608.6	0.0523062	0.0261531	−	0.999658i	$-0.491674\pi$
$227$	40608.6	0.0523062	0.0261531	+	0.999658i	$0.491674\pi$
$228$	0	0
$229$	988410.	1.24551	0.622757	−	0.782415i	$-0.286013\pi$
$229$	988410.	1.24551	0.622757	+	0.782415i	$0.286013\pi$
$230$	23546.5	0.0293499
$231$	0	0
$232$	323087.	0.394093
$233$	47673.4	0.0575289	0.0287645	−	0.999586i	$-0.490843\pi$
$233$	47673.4	0.0575289	0.0287645	+	0.999586i	$0.490843\pi$
$234$	0	0
$235$	−363911.	−0.429858
$236$	1.44359e6	1.68719
$237$	0	0
$238$	111390.	0.127469
$239$	165174.	0.187045	0.0935224	−	0.995617i	$-0.470187\pi$
$239$	165174.	0.187045	0.0935224	+	0.995617i	$0.470187\pi$
$240$	0	0
$241$	−1.13302e6	−1.25659	−0.628294	−	0.777976i	$-0.716247\pi$
$241$	−1.13302e6	−1.25659	−0.628294	+	0.777976i	$0.716247\pi$
$242$	−10172.5	−0.0111658
$243$	0	0
$244$	−598717.	−0.643794
$245$	247251.	0.263162
$246$	0	0
$247$	101030.	0.105368
$248$	−186048.	−0.192086
$249$	0	0
$250$	10856.2	0.0109857
$251$	1.08030e6	1.08233	0.541165	−	0.840917i	$-0.317984\pi$
$251$	1.08030e6	1.08233	0.541165	+	0.840917i	$0.317984\pi$
$252$	0	0
$253$	−164027.	−0.161106
$254$	−219388.	−0.213368
$255$	0	0
$256$	893945.	0.852533
$257$	−301460.	−0.284707	−0.142353	−	0.989816i	$-0.545467\pi$
$257$	−301460.	−0.284707	−0.142353	+	0.989816i	$0.545467\pi$
$258$	0	0
$259$	−1.11613e6	−1.03387
$260$	−531582.	−0.487682
$261$	0	0
$262$	−129928.	−0.116936
$263$	347755.	0.310016	0.155008	−	0.987913i	$-0.450460\pi$
$263$	347755.	0.310016	0.155008	+	0.987913i	$0.450460\pi$
$264$	0	0
$265$	333744.	0.291943
$266$	8653.34	0.00749860
$267$	0	0
$268$	−209623.	−0.178280
$269$	−980953.	−0.826547	−0.413274	−	0.910607i	$-0.635615\pi$
$269$	−980953.	−0.826547	−0.413274	+	0.910607i	$0.635615\pi$
$270$	0	0
$271$	9630.73	0.00796592	0.00398296	−	0.999992i	$-0.498732\pi$
$271$	9630.73	0.00796592	0.00398296	+	0.999992i	$0.498732\pi$
$272$	−1.88504e6	−1.54489
$273$	0	0
$274$	161402.	0.129877
$275$	−75625.0	−0.0603023
$276$	0	0
$277$	324368.	0.254003	0.127001	−	0.991903i	$-0.459465\pi$
$277$	324368.	0.254003	0.127001	+	0.991903i	$0.459465\pi$
$278$	216544.	0.168048
$279$	0	0
$280$	−91758.7	−0.0699443
$281$	−866856.	−0.654909	−0.327455	−	0.944867i	$-0.606191\pi$
$281$	−866856.	−0.654909	−0.327455	+	0.944867i	$0.606191\pi$
$282$	0	0
$283$	−575157.	−0.426895	−0.213447	−	0.976955i	$-0.568469\pi$
$283$	−575157.	−0.426895	−0.213447	+	0.976955i	$0.568469\pi$
$284$	−1.92346e6	−1.41510
$285$	0	0
$286$	−56718.6	−0.0410025
$287$	−238310.	−0.170780
$288$	0	0
$289$	2.29601e6	1.61707
$290$	127165.	0.0887917
$291$	0	0
$292$	546932.	0.375384
$293$	−1.96073e6	−1.33429	−0.667144	−	0.744929i	$-0.732483\pi$
$293$	−1.96073e6	−1.33429	−0.667144	+	0.744929i	$0.732483\pi$
$294$	0	0
$295$	1.14508e6	0.766093
$296$	−592252.	−0.392896
$297$	0	0
$298$	21090.2	0.0137575
$299$	−914557.	−0.591606
$300$	0	0
$301$	−1.83620e6	−1.16816
$302$	−91028.8	−0.0574330
$303$	0	0
$304$	−146440.	−0.0908813
$305$	−474912.	−0.292323
$306$	0	0
$307$	1.60549e6	0.972215	0.486108	−	0.873899i	$-0.338416\pi$
$307$	1.60549e6	0.972215	0.486108	+	0.873899i	$0.338416\pi$
$308$	317170.	0.190509
$309$	0	0
$310$	−73227.4	−0.0432782
$311$	−6622.52	−0.00388260	−0.00194130	−	0.999998i	$-0.500618\pi$
$311$	−6622.52	−0.00388260	−0.00194130	+	0.999998i	$0.500618\pi$
$312$	0	0
$313$	−1.08764e6	−0.627516	−0.313758	−	0.949503i	$-0.601588\pi$
$313$	−1.08764e6	−0.627516	−0.313758	+	0.949503i	$0.601588\pi$
$314$	−227144.	−0.130010
$315$	0	0
$316$	−1.94387e6	−1.09509
$317$	456183.	0.254971	0.127486	−	0.991840i	$-0.459309\pi$
$317$	456183.	0.254971	0.127486	+	0.991840i	$0.459309\pi$
$318$	0	0
$319$	−885839.	−0.487392
$320$	745980.	0.407242
$321$	0	0
$322$	−78332.9	−0.0421022
$323$	288668.	0.153954
$324$	0	0
$325$	−421660.	−0.221439
$326$	348687.	0.181715
$327$	0	0
$328$	−126454.	−0.0649006
$329$	1.21063e6	0.616627
$330$	0	0
$331$	1.73077e6	0.868297	0.434149	−	0.900841i	$-0.357049\pi$
$331$	1.73077e6	0.868297	0.434149	+	0.900841i	$0.357049\pi$
$332$	−2.05590e6	−1.02366
$333$	0	0
$334$	−410251.	−0.201226
$335$	−166276.	−0.0809503
$336$	0	0
$337$	3.01707e6	1.44714	0.723571	−	0.690250i	$-0.242499\pi$
$337$	3.01707e6	1.44714	0.723571	+	0.690250i	$0.242499\pi$
$338$	−58270.3	−0.0277431
$339$	0	0
$340$	−1.51886e6	−0.712559
$341$	510107.	0.237561
$342$	0	0
$343$	−2.22035e6	−1.01903
$344$	−974341.	−0.443930
$345$	0	0
$346$	396683.	0.178137
$347$	−1.26947e6	−0.565978	−0.282989	−	0.959123i	$-0.591326\pi$
$347$	−1.26947e6	−0.565978	−0.282989	+	0.959123i	$0.591326\pi$
$348$	0	0
$349$	3.26420e6	1.43454	0.717270	−	0.696795i	$-0.245391\pi$
$349$	3.26420e6	1.43454	0.717270	+	0.696795i	$0.245391\pi$
$350$	−36115.7	−0.0157589
$351$	0	0
$352$	253089.	0.108872
$353$	−1.70741e6	−0.729290	−0.364645	−	0.931147i	$-0.618810\pi$
$353$	−1.70741e6	−0.729290	−0.364645	+	0.931147i	$0.618810\pi$
$354$	0	0
$355$	−1.52572e6	−0.642544
$356$	−3.45312e6	−1.44406
$357$	0	0
$358$	−322665.	−0.133059
$359$	−257688.	−0.105526	−0.0527628	−	0.998607i	$-0.516803\pi$
$359$	−257688.	−0.105526	−0.0527628	+	0.998607i	$0.516803\pi$
$360$	0	0
$361$	−2.45367e6	−0.990943
$362$	284722.	0.114196
$363$	0	0
$364$	1.76843e6	0.699575
$365$	433835.	0.170448
$366$	0	0
$367$	−683489.	−0.264891	−0.132445	−	0.991190i	$-0.542283\pi$
$367$	−683489.	−0.264891	−0.132445	+	0.991190i	$0.542283\pi$
$368$	1.32562e6	0.510269
$369$	0	0
$370$	−233107.	−0.0885219
$371$	−1.11027e6	−0.418790
$372$	0	0
$373$	3.58176e6	1.33298	0.666492	−	0.745512i	$-0.267795\pi$
$373$	3.58176e6	1.33298	0.666492	+	0.745512i	$0.267795\pi$
$374$	−162059.	−0.0599093
$375$	0	0
$376$	642398.	0.234334
$377$	−4.93914e6	−1.78977
$378$	0	0
$379$	−4.87955e6	−1.74494	−0.872472	−	0.488665i	$-0.837484\pi$
$379$	−4.87955e6	−1.74494	−0.872472	+	0.488665i	$0.837484\pi$
$380$	−117993.	−0.0419177
$381$	0	0
$382$	−410645.	−0.143982
$383$	−4.97559e6	−1.73319	−0.866597	−	0.499008i	$-0.833698\pi$
$383$	−4.97559e6	−1.73319	−0.866597	+	0.499008i	$0.833698\pi$
$384$	0	0
$385$	251584.	0.0865030
$386$	216017.	0.0737939
$387$	0	0
$388$	−2.63913e6	−0.889983
$389$	4.09866e6	1.37331	0.686654	−	0.726984i	$-0.259079\pi$
$389$	4.09866e6	1.37331	0.686654	+	0.726984i	$0.259079\pi$
$390$	0	0
$391$	−2.61312e6	−0.864404
$392$	−436463.	−0.143461
$393$	0	0
$394$	112288.	0.0364413
$395$	−1.54191e6	−0.497241
$396$	0	0
$397$	1.61215e6	0.513369	0.256684	−	0.966495i	$-0.417370\pi$
$397$	1.61215e6	0.513369	0.256684	+	0.966495i	$0.417370\pi$
$398$	−302669.	−0.0957768
$399$	0	0
$400$	611181.	0.190994
$401$	648362.	0.201352	0.100676	−	0.994919i	$-0.467899\pi$
$401$	648362.	0.201352	0.100676	+	0.994919i	$0.467899\pi$
$402$	0	0
$403$	2.84419e6	0.872359
$404$	−4.09335e6	−1.24775
$405$	0	0
$406$	−423044.	−0.127371
$407$	1.62384e6	0.485911
$408$	0	0
$409$	1.88327e6	0.556677	0.278338	−	0.960483i	$-0.410216\pi$
$409$	1.88327e6	0.556677	0.278338	+	0.960483i	$0.410216\pi$
$410$	−49771.6	−0.0146225
$411$	0	0
$412$	−6.36178e6	−1.84644
$413$	−3.80938e6	−1.09895
$414$	0	0
$415$	−1.63077e6	−0.464807
$416$	1.41114e6	0.399794
$417$	0	0
$418$	−12589.6	−0.00352429
$419$	5.10510e6	1.42059	0.710296	−	0.703903i	$-0.248561\pi$
$419$	5.10510e6	1.42059	0.710296	+	0.703903i	$0.248561\pi$
$420$	0	0
$421$	5.48811e6	1.50910	0.754550	−	0.656243i	$-0.227855\pi$
$421$	5.48811e6	1.50910	0.754550	+	0.656243i	$0.227855\pi$
$422$	374797.	0.102451
$423$	0	0
$424$	−589145.	−0.159150
$425$	−1.20479e6	−0.323547
$426$	0	0
$427$	1.57990e6	0.419335
$428$	−4.16925e6	−1.10014
$429$	0	0
$430$	−383494.	−0.100020
$431$	3.59589e6	0.932424	0.466212	−	0.884673i	$-0.345618\pi$
$431$	3.59589e6	0.932424	0.466212	+	0.884673i	$0.345618\pi$
$432$	0	0
$433$	−5.03603e6	−1.29083	−0.645414	−	0.763833i	$-0.723315\pi$
$433$	−5.03603e6	−1.29083	−0.645414	+	0.763833i	$0.723315\pi$
$434$	243608.	0.0620822
$435$	0	0
$436$	4.53987e6	1.14374
$437$	−203001.	−0.0508503
$438$	0	0
$439$	−3.25961e6	−0.807244	−0.403622	−	0.914926i	$-0.632249\pi$
$439$	−3.25961e6	−0.807244	−0.403622	+	0.914926i	$0.632249\pi$
$440$	133498.	0.0328733
$441$	0	0
$442$	−903586.	−0.219996
$443$	1.35658e6	0.328424	0.164212	−	0.986425i	$-0.447492\pi$
$443$	1.35658e6	0.328424	0.164212	+	0.986425i	$0.447492\pi$
$444$	0	0
$445$	−2.73907e6	−0.655696
$446$	−314628.	−0.0748962
$447$	0	0
$448$	−2.48167e6	−0.584184
$449$	6.80906e6	1.59394	0.796969	−	0.604020i	$-0.206435\pi$
$449$	6.80906e6	1.59394	0.796969	+	0.604020i	$0.206435\pi$
$450$	0	0
$451$	346712.	0.0802653
$452$	−34530.8	−0.00794988
$453$	0	0
$454$	−28214.7	−0.00642445
$455$	1.40275e6	0.317652
$456$	0	0
$457$	−3.03977e6	−0.680848	−0.340424	−	0.940272i	$-0.610571\pi$
$457$	−3.03977e6	−0.680848	−0.340424	+	0.940272i	$0.610571\pi$
$458$	−686744.	−0.152979
$459$	0	0
$460$	1.06811e6	0.235355
$461$	−3.27858e6	−0.718511	−0.359256	−	0.933239i	$-0.616969\pi$
$461$	−3.27858e6	−0.718511	−0.359256	+	0.933239i	$0.616969\pi$
$462$	0	0
$463$	5.51970e6	1.19664	0.598319	−	0.801258i	$-0.295836\pi$
$463$	5.51970e6	1.19664	0.598319	+	0.801258i	$0.295836\pi$
$464$	7.15912e6	1.54371
$465$	0	0
$466$	−33123.3	−0.00706593
$467$	4.07534e6	0.864711	0.432356	−	0.901703i	$-0.357683\pi$
$467$	4.07534e6	0.864711	0.432356	+	0.901703i	$0.357683\pi$
$468$	0	0
$469$	553157.	0.116122
$470$	252844.	0.0527968
$471$	0	0
$472$	−2.02137e6	−0.417630
$473$	2.67145e6	0.549027
$474$	0	0
$475$	−93594.1	−0.0190333
$476$	5.05284e6	1.02216
$477$	0	0
$478$	−114762.	−0.0229736
$479$	−4.77799e6	−0.951495	−0.475748	−	0.879582i	$-0.657822\pi$
$479$	−4.77799e6	−0.951495	−0.475748	+	0.879582i	$0.657822\pi$
$480$	0	0
$481$	9.05397e6	1.78433
$482$	787216.	0.154339
$483$	0	0
$484$	−461444.	−0.0895377
$485$	−2.09340e6	−0.404109
$486$	0	0
$487$	1.76923e6	0.338035	0.169017	−	0.985613i	$-0.445941\pi$
$487$	1.76923e6	0.338035	0.169017	+	0.985613i	$0.445941\pi$
$488$	838345.	0.159358
$489$	0	0
$490$	−171789.	−0.0323226
$491$	1.04139e6	0.194943	0.0974716	−	0.995238i	$-0.468924\pi$
$491$	1.04139e6	0.194943	0.0974716	+	0.995238i	$0.468924\pi$
$492$	0	0
$493$	−1.41124e7	−2.61506
$494$	−70195.3	−0.0129417
$495$	0	0
$496$	−4.12255e6	−0.752422
$497$	5.07565e6	0.921723
$498$	0	0
$499$	2.03068e6	0.365082	0.182541	−	0.983198i	$-0.441568\pi$
$499$	2.03068e6	0.365082	0.182541	+	0.983198i	$0.441568\pi$
$500$	492457.	0.0880934
$501$	0	0
$502$	−750588.	−0.132936
$503$	5.06008e6	0.891739	0.445869	−	0.895098i	$-0.352895\pi$
$503$	5.06008e6	0.891739	0.445869	+	0.895098i	$0.352895\pi$
$504$	0	0
$505$	−3.24691e6	−0.566555
$506$	113965.	0.0197877
$507$	0	0
$508$	−9.95183e6	−1.71098
$509$	−5.64302e6	−0.965421	−0.482711	−	0.875780i	$-0.660348\pi$
$509$	−5.64302e6	−0.965421	−0.482711	+	0.875780i	$0.660348\pi$
$510$	0	0
$511$	−1.44325e6	−0.244506
$512$	−3.42639e6	−0.577645
$513$	0	0
$514$	209454.	0.0349688
$515$	−5.04627e6	−0.838402
$516$	0	0
$517$	−1.76133e6	−0.289810
$518$	775484.	0.126984
$519$	0	0
$520$	744340.	0.120716
$521$	−5.37673e6	−0.867809	−0.433905	−	0.900959i	$-0.642864\pi$
$521$	−5.37673e6	−0.867809	−0.433905	+	0.900959i	$0.642864\pi$
$522$	0	0
$523$	8.42403e6	1.34668	0.673342	−	0.739331i	$-0.264858\pi$
$523$	8.42403e6	1.34668	0.673342	+	0.739331i	$0.264858\pi$
$524$	−5.89376e6	−0.937701
$525$	0	0
$526$	−241619.	−0.0380773
$527$	8.12654e6	1.27462
$528$	0	0
$529$	−4.59871e6	−0.714492
$530$	−231884.	−0.0358576
$531$	0	0
$532$	392532.	0.0601306
$533$	1.93315e6	0.294746
$534$	0	0
$535$	−3.30712e6	−0.499534
$536$	293522.	0.0441294
$537$	0	0
$538$	681563.	0.101520
$539$	1.19669e6	0.177424
$540$	0	0
$541$	8.85130e6	1.30021	0.650106	−	0.759844i	$-0.274725\pi$
$541$	8.85130e6	1.30021	0.650106	+	0.759844i	$0.274725\pi$
$542$	−6691.40	−0.000978405 0
$543$	0	0
$544$	4.03198e6	0.584145
$545$	3.60110e6	0.519330
$546$	0	0
$547$	3.13327e6	0.447744	0.223872	−	0.974619i	$-0.428130\pi$
$547$	3.13327e6	0.447744	0.223872	+	0.974619i	$0.428130\pi$
$548$	7.32150e6	1.04147
$549$	0	0
$550$	52544.0	0.00740656
$551$	−1.09632e6	−0.153836
$552$	0	0
$553$	5.12953e6	0.713288
$554$	−225370.	−0.0311976
$555$	0	0
$556$	9.82283e6	1.34756
$557$	5.46691e6	0.746627	0.373313	−	0.927705i	$-0.378222\pi$
$557$	5.46691e6	0.746627	0.373313	+	0.927705i	$0.378222\pi$
$558$	0	0
$559$	1.48951e7	2.01611
$560$	−2.03323e6	−0.273979
$561$	0	0
$562$	602289.	0.0804385
$563$	−5.19043e6	−0.690132	−0.345066	−	0.938578i	$-0.612143\pi$
$563$	−5.19043e6	−0.690132	−0.345066	+	0.938578i	$0.612143\pi$
$564$	0	0
$565$	−27390.4	−0.00360975
$566$	399618.	0.0524329
$567$	0	0
$568$	2.69329e6	0.350278
$569$	−1.24058e7	−1.60636	−0.803180	−	0.595737i	$-0.796860\pi$
$569$	−1.24058e7	−1.60636	−0.803180	+	0.595737i	$0.796860\pi$
$570$	0	0
$571$	8.79234e6	1.12853	0.564266	−	0.825593i	$-0.309159\pi$
$571$	8.79234e6	1.12853	0.564266	+	0.825593i	$0.309159\pi$
$572$	−2.57286e6	−0.328795
$573$	0	0
$574$	165577.	0.0209759
$575$	847245.	0.106866
$576$	0	0
$577$	−1.14336e7	−1.42969	−0.714845	−	0.699283i	$-0.753503\pi$
$577$	−1.14336e7	−1.42969	−0.714845	+	0.699283i	$0.753503\pi$
$578$	−1.59526e6	−0.198615
$579$	0	0
$580$	5.76843e6	0.712013
$581$	5.42514e6	0.666761
$582$	0	0
$583$	1.61532e6	0.196828
$584$	−765834.	−0.0929185
$585$	0	0
$586$	1.36231e6	0.163882
$587$	2.32244e6	0.278195	0.139097	−	0.990279i	$-0.455580\pi$
$587$	2.32244e6	0.278195	0.139097	+	0.990279i	$0.455580\pi$
$588$	0	0
$589$	631313.	0.0749819
$590$	−795599.	−0.0940945
$591$	0	0
$592$	−1.31234e7	−1.53901
$593$	−7.98435e6	−0.932401	−0.466201	−	0.884679i	$-0.654377\pi$
$593$	−7.98435e6	−0.932401	−0.466201	+	0.884679i	$0.654377\pi$
$594$	0	0
$595$	4.00800e6	0.464125
$596$	956690.	0.110320
$597$	0	0
$598$	635431.	0.0726634
$599$	139225.	0.0158544	0.00792721	−	0.999969i	$-0.497477\pi$
$599$	139225.	0.0158544	0.00792721	+	0.999969i	$0.497477\pi$
$600$	0	0
$601$	1.55602e7	1.75723	0.878616	−	0.477529i	$-0.158467\pi$
$601$	1.55602e7	1.75723	0.878616	+	0.477529i	$0.158467\pi$
$602$	1.27578e6	0.143478
$603$	0	0
$604$	−4.12923e6	−0.460550
$605$	−366025.	−0.0406558
$606$	0	0
$607$	−6.00544e6	−0.661566	−0.330783	−	0.943707i	$-0.607313\pi$
$607$	−6.00544e6	−0.661566	−0.330783	+	0.943707i	$0.607313\pi$
$608$	313225.	0.0343635
$609$	0	0
$610$	329967.	0.0359043
$611$	−9.82057e6	−1.06423
$612$	0	0
$613$	−2.32775e6	−0.250199	−0.125099	−	0.992144i	$-0.539925\pi$
$613$	−2.32775e6	−0.250199	−0.125099	+	0.992144i	$0.539925\pi$
$614$	−1.11549e6	−0.119411
$615$	0	0
$616$	−444112.	−0.0471564
$617$	−1.30284e7	−1.37777	−0.688886	−	0.724870i	$-0.741900\pi$
$617$	−1.30284e7	−1.37777	−0.688886	+	0.724870i	$0.741900\pi$
$618$	0	0
$619$	−2.53159e6	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$619$	−2.53159e6	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$620$	−3.32173e6	−0.347045
$621$	0	0
$622$	4601.30	0.000476876 0
$623$	9.11214e6	0.940590
$624$	0	0
$625$	390625.	0.0400000
$626$	755689.	0.0770739
$627$	0	0
$628$	−1.03037e7	−1.04254
$629$	2.58694e7	2.60712
$630$	0	0
$631$	−2.17733e6	−0.217696	−0.108848	−	0.994058i	$-0.534716\pi$
$631$	−2.17733e6	−0.217696	−0.108848	+	0.994058i	$0.534716\pi$
$632$	2.72188e6	0.271067
$633$	0	0
$634$	−316955.	−0.0313166
$635$	−7.89396e6	−0.776892
$636$	0	0
$637$	6.67237e6	0.651525
$638$	615478.	0.0598634
$639$	0	0
$640$	−2.19162e6	−0.211503
$641$	2.58180e6	0.248187	0.124093	−	0.992271i	$-0.460398\pi$
$641$	2.58180e6	0.248187	0.124093	+	0.992271i	$0.460398\pi$
$642$	0	0
$643$	−1.67813e7	−1.60066	−0.800329	−	0.599561i	$-0.795342\pi$
$643$	−1.67813e7	−1.60066	−0.800329	+	0.599561i	$0.795342\pi$
$644$	−3.55333e6	−0.337614
$645$	0	0
$646$	−200566.	−0.0189093
$647$	1.01691e7	0.955037	0.477518	−	0.878622i	$-0.341536\pi$
$647$	1.01691e7	0.955037	0.477518	+	0.878622i	$0.341536\pi$
$648$	0	0
$649$	5.54220e6	0.516500
$650$	292968.	0.0271980
$651$	0	0
$652$	1.58171e7	1.45716
$653$	8.99282e6	0.825302	0.412651	−	0.910889i	$-0.364603\pi$
$653$	8.99282e6	0.825302	0.412651	+	0.910889i	$0.364603\pi$
$654$	0	0
$655$	−4.67503e6	−0.425776
$656$	−2.80204e6	−0.254223
$657$	0	0
$658$	−841144.	−0.0757365
$659$	1.03480e7	0.928203	0.464102	−	0.885782i	$-0.346377\pi$
$659$	1.03480e7	0.928203	0.464102	+	0.885782i	$0.346377\pi$
$660$	0	0
$661$	1.48595e7	1.32282	0.661409	−	0.750025i	$-0.269959\pi$
$661$	1.48595e7	1.32282	0.661409	+	0.750025i	$0.269959\pi$
$662$	−1.20253e6	−0.106648
$663$	0	0
$664$	2.87874e6	0.253386
$665$	311362.	0.0273031
$666$	0	0
$667$	9.92427e6	0.863742
$668$	−1.86098e7	−1.61361
$669$	0	0
$670$	115528.	0.00994263
$671$	−2.29857e6	−0.197084
$672$	0	0
$673$	217545.	0.0185145	0.00925725	−	0.999957i	$-0.497053\pi$
$673$	217545.	0.0185145	0.00925725	+	0.999957i	$0.497053\pi$
$674$	−2.09625e6	−0.177744
$675$	0	0
$676$	−2.64325e6	−0.222470
$677$	2.64314e6	0.221640	0.110820	−	0.993840i	$-0.464652\pi$
$677$	2.64314e6	0.221640	0.110820	+	0.993840i	$0.464652\pi$
$678$	0	0
$679$	6.96419e6	0.579690
$680$	2.12676e6	0.176379
$681$	0	0
$682$	−354421.	−0.0291782
$683$	−4.66785e6	−0.382882	−0.191441	−	0.981504i	$-0.561316\pi$
$683$	−4.66785e6	−0.382882	−0.191441	+	0.981504i	$0.561316\pi$
$684$	0	0
$685$	5.80753e6	0.472895
$686$	1.54269e6	0.125161
$687$	0	0
$688$	−2.15899e7	−1.73892
$689$	9.00647e6	0.722781
$690$	0	0
$691$	3.11273e6	0.247997	0.123998	−	0.992282i	$-0.460428\pi$
$691$	3.11273e6	0.247997	0.123998	+	0.992282i	$0.460428\pi$
$692$	1.79943e7	1.42846
$693$	0	0
$694$	882026.	0.0695156
$695$	7.79163e6	0.611880
$696$	0	0
$697$	5.52349e6	0.430657
$698$	−2.26795e6	−0.176196
$699$	0	0
$700$	−1.63827e6	−0.126369
$701$	−1.16786e7	−0.897629	−0.448815	−	0.893625i	$-0.648154\pi$
$701$	−1.16786e7	−0.897629	−0.448815	+	0.893625i	$0.648154\pi$
$702$	0	0
$703$	2.00967e6	0.153369
$704$	3.61054e6	0.274562
$705$	0	0
$706$	1.18630e6	0.0895743
$707$	1.08016e7	0.812718
$708$	0	0
$709$	−1.96571e7	−1.46860	−0.734302	−	0.678823i	$-0.762490\pi$
$709$	−1.96571e7	−1.46860	−0.734302	+	0.678823i	$0.762490\pi$
$710$	1.06006e6	0.0789198
$711$	0	0
$712$	4.83518e6	0.357448
$713$	−5.71485e6	−0.420999
$714$	0	0
$715$	−2.04083e6	−0.149294
$716$	−1.46367e7	−1.06699
$717$	0	0
$718$	179041.	0.0129611
$719$	1.89013e7	1.36354	0.681772	−	0.731565i	$-0.261210\pi$
$719$	1.89013e7	1.36354	0.681772	+	0.731565i	$0.261210\pi$
$720$	0	0
$721$	1.67876e7	1.20268
$722$	1.70480e6	0.121712
$723$	0	0
$724$	1.29155e7	0.915725
$725$	4.57562e6	0.323299
$726$	0	0
$727$	6.78994e6	0.476463	0.238232	−	0.971208i	$-0.423432\pi$
$727$	6.78994e6	0.476463	0.238232	+	0.971208i	$0.423432\pi$
$728$	−2.47622e6	−0.173165
$729$	0	0
$730$	−301427.	−0.0209351
$731$	4.25590e7	2.94576
$732$	0	0
$733$	−3.60300e6	−0.247688	−0.123844	−	0.992302i	$-0.539522\pi$
$733$	−3.60300e6	−0.247688	−0.123844	+	0.992302i	$0.539522\pi$
$734$	474886.	0.0325349
$735$	0	0
$736$	−2.83542e6	−0.192940
$737$	−804778.	−0.0545767
$738$	0	0
$739$	−7.58466e6	−0.510887	−0.255443	−	0.966824i	$-0.582221\pi$
$739$	−7.58466e6	−0.510887	−0.255443	+	0.966824i	$0.582221\pi$
$740$	−1.05742e7	−0.709849
$741$	0	0
$742$	771416.	0.0514374
$743$	−1.17473e7	−0.780669	−0.390335	−	0.920673i	$-0.627641\pi$
$743$	−1.17473e7	−0.780669	−0.390335	+	0.920673i	$0.627641\pi$
$744$	0	0
$745$	758862.	0.0500924
$746$	−2.48860e6	−0.163722
$747$	0	0
$748$	−7.35129e6	−0.480407
$749$	1.10019e7	0.716577
$750$	0	0
$751$	4.05057e6	0.262070	0.131035	−	0.991378i	$-0.458170\pi$
$751$	4.05057e6	0.262070	0.131035	+	0.991378i	$0.458170\pi$
$752$	1.42346e7	0.917910
$753$	0	0
$754$	3.43170e6	0.219827
$755$	−3.27537e6	−0.209119
$756$	0	0
$757$	2.12290e7	1.34645	0.673225	−	0.739438i	$-0.264909\pi$
$757$	2.12290e7	1.34645	0.673225	+	0.739438i	$0.264909\pi$
$758$	3.39029e6	0.214321
$759$	0	0
$760$	165218.	0.0103759
$761$	1.06438e7	0.666247	0.333123	−	0.942883i	$-0.391897\pi$
$761$	1.06438e7	0.666247	0.333123	+	0.942883i	$0.391897\pi$
$762$	0	0
$763$	−1.19799e7	−0.744974
$764$	−1.86276e7	−1.15458
$765$	0	0
$766$	3.45702e6	0.212878
$767$	3.09014e7	1.89666
$768$	0	0
$769$	3.17109e6	0.193372	0.0966859	−	0.995315i	$-0.469176\pi$
$769$	3.17109e6	0.193372	0.0966859	+	0.995315i	$0.469176\pi$
$770$	−174800.	−0.0106246
$771$	0	0
$772$	9.79894e6	0.591747
$773$	−1.80501e7	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$773$	−1.80501e7	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$774$	0	0
$775$	−2.63485e6	−0.157580
$776$	3.69541e6	0.220297
$777$	0	0
$778$	−2.84774e6	−0.168675
$779$	429094.	0.0253343
$780$	0	0
$781$	−7.38447e6	−0.433203
$782$	1.81559e6	0.106169
$783$	0	0
$784$	−9.67137e6	−0.561950
$785$	−8.17304e6	−0.473380
$786$	0	0
$787$	−2.58223e7	−1.48614	−0.743068	−	0.669216i	$-0.766630\pi$
$787$	−2.58223e7	−1.48614	−0.743068	+	0.669216i	$0.766630\pi$
$788$	5.09360e6	0.292219
$789$	0	0
$790$	1.07132e6	0.0610731
$791$	91120.5	0.00517815
$792$	0	0
$793$	−1.28161e7	−0.723722
$794$	−1.12012e6	−0.0630540
$795$	0	0
$796$	−1.37296e7	−0.768026
$797$	−1.50658e7	−0.840129	−0.420065	−	0.907494i	$-0.637993\pi$
$797$	−1.50658e7	−0.840129	−0.420065	+	0.907494i	$0.637993\pi$
$798$	0	0
$799$	−2.80598e7	−1.55495
$800$	−1.30728e6	−0.0722176
$801$	0	0
$802$	−450480.	−0.0247309
$803$	2.09976e6	0.114916
$804$	0	0
$805$	−2.81856e6	−0.153298
$806$	−1.97613e6	−0.107147
$807$	0	0
$808$	5.73166e6	0.308853
$809$	2.08560e7	1.12037	0.560183	−	0.828369i	$-0.310731\pi$
$809$	2.08560e7	1.12037	0.560183	+	0.828369i	$0.310731\pi$
$810$	0	0
$811$	8.46233e6	0.451791	0.225896	−	0.974152i	$-0.427469\pi$
$811$	8.46233e6	0.451791	0.225896	+	0.974152i	$0.427469\pi$
$812$	−1.91900e7	−1.02138
$813$	0	0
$814$	−1.12824e6	−0.0596814
$815$	1.25464e7	0.661643
$816$	0	0
$817$	3.30620e6	0.173290
$818$	−1.30849e6	−0.0683732
$819$	0	0
$820$	−2.25773e6	−0.117257
$821$	−8.67359e6	−0.449098	−0.224549	−	0.974463i	$-0.572091\pi$
$821$	−8.67359e6	−0.449098	−0.224549	+	0.974463i	$0.572091\pi$
$822$	0	0
$823$	−1.24194e7	−0.639148	−0.319574	−	0.947561i	$-0.603540\pi$
$823$	−1.24194e7	−0.639148	−0.319574	+	0.947561i	$0.603540\pi$
$824$	8.90799e6	0.457048
$825$	0	0
$826$	2.64675e6	0.134978
$827$	2.38473e7	1.21248	0.606242	−	0.795280i	$-0.292676\pi$
$827$	2.38473e7	1.21248	0.606242	+	0.795280i	$0.292676\pi$
$828$	0	0
$829$	−3.26873e7	−1.65193	−0.825967	−	0.563719i	$-0.809370\pi$
$829$	−3.26873e7	−1.65193	−0.825967	+	0.563719i	$0.809370\pi$
$830$	1.13305e6	0.0570894
$831$	0	0
$832$	2.01312e7	1.00823
$833$	1.90646e7	0.951953
$834$	0	0
$835$	−1.47616e7	−0.732683
$836$	−571087.	−0.0282609
$837$	0	0
$838$	−3.54701e6	−0.174483
$839$	−8.09423e6	−0.396982	−0.198491	−	0.980103i	$-0.563604\pi$
$839$	−8.09423e6	−0.396982	−0.198491	+	0.980103i	$0.563604\pi$
$840$	0	0
$841$	3.30857e7	1.61306
$842$	−3.81313e6	−0.185354
$843$	0	0
$844$	1.70015e7	0.821544
$845$	−2.09667e6	−0.101015
$846$	0	0
$847$	1.21767e6	0.0583203
$848$	−1.30546e7	−0.623409
$849$	0	0
$850$	837082.	0.0397393
$851$	−1.81922e7	−0.861117
$852$	0	0
$853$	3.67970e6	0.173157	0.0865785	−	0.996245i	$-0.472407\pi$
$853$	3.67970e6	0.173157	0.0865785	+	0.996245i	$0.472407\pi$
$854$	−1.09771e6	−0.0515043
$855$	0	0
$856$	5.83794e6	0.272317
$857$	−2.17589e7	−1.01201	−0.506004	−	0.862531i	$-0.668878\pi$
$857$	−2.17589e7	−1.01201	−0.506004	+	0.862531i	$0.668878\pi$
$858$	0	0
$859$	−3.76305e7	−1.74003	−0.870015	−	0.493025i	$-0.835891\pi$
$859$	−3.76305e7	−1.74003	−0.870015	+	0.493025i	$0.835891\pi$
$860$	−1.73960e7	−0.802054
$861$	0	0
$862$	−2.49841e6	−0.114524
$863$	2.50264e7	1.14386	0.571928	−	0.820304i	$-0.306196\pi$
$863$	2.50264e7	1.14386	0.571928	+	0.820304i	$0.306196\pi$
$864$	0	0
$865$	1.42734e7	0.648613
$866$	3.49902e6	0.158545
$867$	0	0
$868$	1.10505e7	0.497832
$869$	−7.46286e6	−0.335240
$870$	0	0
$871$	−4.48717e6	−0.200414
$872$	−6.35689e6	−0.283109
$873$	0	0
$874$	141044.	0.00624564
$875$	−1.29950e6	−0.0573796
$876$	0	0
$877$	1.12677e7	0.494692	0.247346	−	0.968927i	$-0.420442\pi$
$877$	1.12677e7	0.494692	0.247346	+	0.968927i	$0.420442\pi$
$878$	2.26477e6	0.0991488
$879$	0	0
$880$	2.95812e6	0.128768
$881$	4.36223e7	1.89352	0.946758	−	0.321947i	$-0.104337\pi$
$881$	4.36223e7	1.89352	0.946758	+	0.321947i	$0.104337\pi$
$882$	0	0
$883$	−3.14744e7	−1.35849	−0.679244	−	0.733912i	$-0.737692\pi$
$883$	−3.14744e7	−1.35849	−0.679244	+	0.733912i	$0.737692\pi$
$884$	−4.09883e7	−1.76413
$885$	0	0
$886$	−942545.	−0.0403383
$887$	−2.33777e7	−0.997684	−0.498842	−	0.866693i	$-0.666241\pi$
$887$	−2.33777e7	−0.997684	−0.498842	+	0.866693i	$0.666241\pi$
$888$	0	0
$889$	2.62611e7	1.11444
$890$	1.90310e6	0.0805352
$891$	0	0
$892$	−1.42721e7	−0.600587
$893$	−2.17983e6	−0.0914733
$894$	0	0
$895$	−1.16101e7	−0.484481
$896$	7.29093e6	0.303398
$897$	0	0
$898$	−4.73091e6	−0.195774
$899$	−3.08635e7	−1.27364
$900$	0	0
$901$	2.57337e7	1.05607
$902$	−240895.	−0.00985850
$903$	0	0
$904$	48351.3	0.00196783
$905$	1.02448e7	0.415797
$906$	0	0
$907$	2.58345e7	1.04275	0.521377	−	0.853326i	$-0.325418\pi$
$907$	2.58345e7	1.04275	0.521377	+	0.853326i	$0.325418\pi$
$908$	−1.27987e6	−0.0515171
$909$	0	0
$910$	−974625.	−0.0390152
$911$	−1.24995e7	−0.498996	−0.249498	−	0.968375i	$-0.580266\pi$
$911$	−1.24995e7	−0.498996	−0.249498	+	0.968375i	$0.580266\pi$
$912$	0	0
$913$	−7.89293e6	−0.313373
$914$	2.11202e6	0.0836244
$915$	0	0
$916$	−3.11520e7	−1.22672
$917$	1.55526e7	0.610771
$918$	0	0
$919$	1.24766e7	0.487313	0.243657	−	0.969862i	$-0.421653\pi$
$919$	1.24766e7	0.487313	0.243657	+	0.969862i	$0.421653\pi$
$920$	−1.49561e6	−0.0582571
$921$	0	0
$922$	2.27795e6	0.0882503
$923$	−4.11733e7	−1.59079
$924$	0	0
$925$	−8.38759e6	−0.322317
$926$	−3.83507e6	−0.146976
$927$	0	0
$928$	−1.53129e7	−0.583697
$929$	1.63852e6	0.0622891	0.0311445	−	0.999515i	$-0.490085\pi$
$929$	1.63852e6	0.0622891	0.0311445	+	0.999515i	$0.490085\pi$
$930$	0	0
$931$	1.48104e6	0.0560006
$932$	−1.50253e6	−0.0566611
$933$	0	0
$934$	−2.83153e6	−0.106207
$935$	−5.83116e6	−0.218135
$936$	0	0
$937$	1.03137e7	0.383766	0.191883	−	0.981418i	$-0.438541\pi$
$937$	1.03137e7	0.383766	0.191883	+	0.981418i	$0.438541\pi$
$938$	−384331.	−0.0142626
$939$	0	0
$940$	1.14695e7	0.423373
$941$	1.30200e7	0.479332	0.239666	−	0.970855i	$-0.422962\pi$
$941$	1.30200e7	0.479332	0.239666	+	0.970855i	$0.422962\pi$
$942$	0	0
$943$	−3.88430e6	−0.142244
$944$	−4.47906e7	−1.63590
$945$	0	0
$946$	−1.85611e6	−0.0674336
$947$	−4.07950e7	−1.47820	−0.739098	−	0.673598i	$-0.764748\pi$
$947$	−4.07950e7	−1.47820	−0.739098	+	0.673598i	$0.764748\pi$
$948$	0	0
$949$	1.17076e7	0.421989
$950$	65028.9	0.00233775
$951$	0	0
$952$	−7.07517e6	−0.253014
$953$	−4.68439e7	−1.67078	−0.835392	−	0.549654i	$-0.814760\pi$
$953$	−4.68439e7	−1.67078	−0.835392	+	0.549654i	$0.814760\pi$
$954$	0	0
$955$	−1.47757e7	−0.524252
$956$	−5.20582e6	−0.184223
$957$	0	0
$958$	3.31973e6	0.116866
$959$	−1.93201e7	−0.678364
$960$	0	0
$961$	−1.08565e7	−0.379212
$962$	−6.29067e6	−0.219159
$963$	0	0
$964$	3.57095e7	1.23763
$965$	7.77268e6	0.268691
$966$	0	0
$967$	1.63484e7	0.562223	0.281111	−	0.959675i	$-0.409297\pi$
$967$	1.63484e7	0.562223	0.281111	+	0.959675i	$0.409297\pi$
$968$	646131.	0.0221632
$969$	0	0
$970$	1.45449e6	0.0496342
$971$	1.62139e6	0.0551875	0.0275937	−	0.999619i	$-0.491216\pi$
$971$	1.62139e6	0.0551875	0.0275937	+	0.999619i	$0.491216\pi$
$972$	0	0
$973$	−2.59206e7	−0.877735
$974$	−1.22925e6	−0.0415187
$975$	0	0
$976$	1.85765e7	0.624221
$977$	2.33840e7	0.783759	0.391880	−	0.920017i	$-0.371825\pi$
$977$	2.33840e7	0.783759	0.391880	+	0.920017i	$0.371825\pi$
$978$	0	0
$979$	−1.32571e7	−0.442070
$980$	−7.79267e6	−0.259192
$981$	0	0
$982$	−723552.	−0.0239437
$983$	−1.96781e7	−0.649529	−0.324764	−	0.945795i	$-0.605285\pi$
$983$	−1.96781e7	−0.649529	−0.324764	+	0.945795i	$0.605285\pi$
$984$	0	0
$985$	4.04033e6	0.132686
$986$	9.80522e6	0.321192
$987$	0	0
$988$	−3.18419e6	−0.103778
$989$	−2.99289e7	−0.972970
$990$	0	0
$991$	1.56015e7	0.504642	0.252321	−	0.967644i	$-0.418806\pi$
$991$	1.56015e7	0.504642	0.252321	+	0.967644i	$0.418806\pi$
$992$	8.81788e6	0.284502
$993$	0	0
$994$	−3.52654e6	−0.113210
$995$	−1.08906e7	−0.348733
$996$	0	0
$997$	−5.71913e6	−0.182218	−0.0911091	−	0.995841i	$-0.529041\pi$
$997$	−5.71913e6	−0.182218	−0.0911091	+	0.995841i	$0.529041\pi$
$998$	−1.41091e6	−0.0448408
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.6.a.b.1.2		3
3.2	odd	2		165.6.a.c.1.2	✓	3
15.14	odd	2		825.6.a.g.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.6.a.c.1.2	✓	3	3.2	odd	2
495.6.a.b.1.2		3	1.1	even	1	trivial
825.6.a.g.1.2		3	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:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q-0.694797 q^{2} -31.5173 q^{4} -25.0000 q^{5} +83.1683 q^{7} +44.1316 q^{8} +O(q^{10})\)	\(q-0.694797 q^{2} -31.5173 q^{4} -25.0000 q^{5} +83.1683 q^{7} +44.1316 q^{8} +17.3699 q^{10} -121.000 q^{11} -674.655 q^{13} -57.7851 q^{14} +977.890 q^{16} -1927.66 q^{17} -149.751 q^{19} +787.931 q^{20} +84.0704 q^{22} +1355.59 q^{23} +625.000 q^{25} +468.748 q^{26} -2621.24 q^{28} +7320.99 q^{29} -4215.76 q^{31} -2091.65 q^{32} +1339.33 q^{34} -2079.21 q^{35} -13420.1 q^{37} +104.046 q^{38} -1103.29 q^{40} -2865.39 q^{41} -22078.1 q^{43} +3813.59 q^{44} -941.861 q^{46} +14556.4 q^{47} -9890.04 q^{49} -434.248 q^{50} +21263.3 q^{52} -13349.7 q^{53} +3025.00 q^{55} +3670.35 q^{56} -5086.60 q^{58} -45803.3 q^{59} +18996.5 q^{61} +2929.10 q^{62} -29839.2 q^{64} +16866.4 q^{65} +6651.05 q^{67} +60754.5 q^{68} +1444.63 q^{70} +61028.7 q^{71} -17353.4 q^{73} +9324.27 q^{74} +4719.73 q^{76} -10063.4 q^{77} +61676.5 q^{79} -24447.2 q^{80} +1990.87 q^{82} +65230.8 q^{83} +48191.4 q^{85} +15339.8 q^{86} -5339.92 q^{88} +109563. q^{89} -56109.9 q^{91} -42724.5 q^{92} -10113.8 q^{94} +3743.76 q^{95} +83736.1 q^{97} +6871.57 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8	\( 3 q - 2 q^{2} - 42 q^{4} - 75 q^{5} - 68 q^{7} + 24 q^{8} + 50 q^{10} - 363 q^{11} + 290 q^{13} - 916 q^{14} - 590 q^{16} - 434 q^{17} - 2856 q^{19} + 1050 q^{20} + 242 q^{22} + 640 q^{23} + 1875 q^{25} - 2132 q^{26} - 580 q^{28} + 4538 q^{29} - 14968 q^{31} + 2496 q^{32} - 13704 q^{34} + 1700 q^{35} - 6190 q^{37} + 11668 q^{38} - 600 q^{40} + 8926 q^{41} - 33592 q^{43} + 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 14693 q^{49} - 1250 q^{50} + 18780 q^{52} + 22934 q^{53} + 9075 q^{55} + 40012 q^{56} - 32304 q^{58} + 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 35474 q^{64} - 7250 q^{65} + 16868 q^{67} + 71288 q^{68} + 22900 q^{70} - 4856 q^{71} + 1910 q^{73} - 29404 q^{74} + 6116 q^{76} + 8228 q^{77} - 36844 q^{79} + 14750 q^{80} + 84000 q^{82} + 48796 q^{83} + 10850 q^{85} + 83492 q^{86} - 2904 q^{88} + 188978 q^{89} - 93208 q^{91} + 6976 q^{92} + 70472 q^{94} + 71400 q^{95} + 247526 q^{97} + 154654 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 - 42 * q^4 - 75 * q^5 - 68 * q^7 + 24 * q^8 + 50 * q^10 - 363 * q^11 + 290 * q^13 - 916 * q^14 - 590 * q^16 - 434 * q^17 - 2856 * q^19 + 1050 * q^20 + 242 * q^22 + 640 * q^23 + 1875 * q^25 - 2132 * q^26 - 580 * q^28 + 4538 * q^29 - 14968 * q^31 + 2496 * q^32 - 13704 * q^34 + 1700 * q^35 - 6190 * q^37 + 11668 * q^38 - 600 * q^40 + 8926 * q^41 - 33592 * q^43 + 5082 * q^44 - 35680 * q^46 + 24640 * q^47 - 14693 * q^49 - 1250 * q^50 + 18780 * q^52 + 22934 * q^53 + 9075 * q^55 + 40012 * q^56 - 32304 * q^58 + 13756 * q^59 + 24602 * q^61 + 7704 * q^62 + 35474 * q^64 - 7250 * q^65 + 16868 * q^67 + 71288 * q^68 + 22900 * q^70 - 4856 * q^71 + 1910 * q^73 - 29404 * q^74 + 6116 * q^76 + 8228 * q^77 - 36844 * q^79 + 14750 * q^80 + 84000 * q^82 + 48796 * q^83 + 10850 * q^85 + 83492 * q^86 - 2904 * q^88 + 188978 * q^89 - 93208 * q^91 + 6976 * q^92 + 70472 * q^94 + 71400 * q^95 + 247526 * q^97 + 154654 * q^98

Introduction

L-functions

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Database

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