LMFDB - Embedded newform 207.6.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,6,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("207.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	207.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:	$33.1994507013$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 251 x^{8} + 296 x^{7} + 21169 x^{6} - 9042 x^{5} - 685949 x^{4} - 270764 x^{3} + \cdots + 5446216 $ x^10 - 2x^9 - 251x^8 + 296x^7 + 21169x^6 - 9042x^5 - 685949x^4 - 270764x^3 + 6230110x^2 + 11573432*x + 5446216
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.817332$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.81733 q^{2} -28.6973 q^{4} +65.9563 q^{5} +148.913 q^{7} -110.307 q^{8} +O(q^{10})$	\(q+1.81733 q^{2} -28.6973 q^{4} +65.9563 q^{5} +148.913 q^{7} -110.307 q^{8} +119.864 q^{10} +47.5480 q^{11} -72.8459 q^{13} +270.625 q^{14} +717.849 q^{16} -1037.32 q^{17} +1430.22 q^{19} -1892.77 q^{20} +86.4105 q^{22} +529.000 q^{23} +1225.23 q^{25} -132.385 q^{26} -4273.41 q^{28} +3359.23 q^{29} -3470.84 q^{31} +4834.40 q^{32} -1885.16 q^{34} +9821.77 q^{35} +2084.17 q^{37} +2599.19 q^{38} -7275.45 q^{40} +16019.5 q^{41} +21209.3 q^{43} -1364.50 q^{44} +961.369 q^{46} +14672.1 q^{47} +5368.21 q^{49} +2226.64 q^{50} +2090.48 q^{52} +8110.02 q^{53} +3136.09 q^{55} -16426.2 q^{56} +6104.84 q^{58} +38628.5 q^{59} +9764.60 q^{61} -6307.66 q^{62} -14185.5 q^{64} -4804.64 q^{65} -5172.01 q^{67} +29768.3 q^{68} +17849.4 q^{70} -29546.1 q^{71} -46611.0 q^{73} +3787.62 q^{74} -41043.5 q^{76} +7080.54 q^{77} -41029.2 q^{79} +47346.6 q^{80} +29112.8 q^{82} +109485. q^{83} -68417.8 q^{85} +38544.4 q^{86} -5244.88 q^{88} -20909.2 q^{89} -10847.7 q^{91} -15180.9 q^{92} +26664.1 q^{94} +94332.0 q^{95} +109289. q^{97} +9755.82 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	$ 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) $ 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * 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 $

$2$	1.81733	0.321262	0.160631	−	0.987015i	$-0.448647\pi$
$2$	1.81733	0.321262	0.160631	+	0.987015i	$0.448647\pi$
$3$	0	0
$3$	0	0
$4$	−28.6973	−0.896791
$5$	65.9563	1.17986	0.589931	−	0.807454i	$-0.299155\pi$
$5$	65.9563	1.17986	0.589931	+	0.807454i	$0.299155\pi$
$6$	0	0
$7$	148.913	1.14865	0.574326	−	0.818626i	$-0.305264\pi$
$7$	148.913	1.14865	0.574326	+	0.818626i	$0.305264\pi$
$8$	−110.307	−0.609367
$9$	0	0
$10$	119.864	0.379044
$11$	47.5480	0.118481	0.0592407	−	0.998244i	$-0.481132\pi$
$11$	47.5480	0.118481	0.0592407	+	0.998244i	$0.481132\pi$
$12$	0	0
$13$	−72.8459	−0.119549	−0.0597746	−	0.998212i	$-0.519038\pi$
$13$	−72.8459	−0.119549	−0.0597746	+	0.998212i	$0.519038\pi$
$14$	270.625	0.369018
$15$	0	0
$16$	717.849	0.701024
$17$	−1037.32	−0.870545	−0.435272	−	0.900299i	$-0.643348\pi$
$17$	−1037.32	−0.870545	−0.435272	+	0.900299i	$0.643348\pi$
$18$	0	0
$19$	1430.22	0.908906	0.454453	−	0.890771i	$-0.349835\pi$
$19$	1430.22	0.908906	0.454453	+	0.890771i	$0.349835\pi$
$20$	−1892.77	−1.05809
$21$	0	0
$22$	86.4105	0.0380636
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	1225.23	0.392073
$26$	−132.385	−0.0384066
$27$	0	0
$28$	−4273.41	−1.03010
$29$	3359.23	0.741729	0.370864	−	0.928687i	$-0.379061\pi$
$29$	3359.23	0.741729	0.370864	+	0.928687i	$0.379061\pi$
$30$	0	0
$31$	−3470.84	−0.648679	−0.324340	−	0.945941i	$-0.605142\pi$
$31$	−3470.84	−0.648679	−0.324340	+	0.945941i	$0.605142\pi$
$32$	4834.40	0.834579
$33$	0	0
$34$	−1885.16	−0.279673
$35$	9821.77	1.35525
$36$	0	0
$37$	2084.17	0.250281	0.125141	−	0.992139i	$-0.460062\pi$
$37$	2084.17	0.250281	0.125141	+	0.992139i	$0.460062\pi$
$38$	2599.19	0.291997
$39$	0	0
$40$	−7275.45	−0.718968
$41$	16019.5	1.48830	0.744149	−	0.668014i	$-0.232855\pi$
$41$	16019.5	1.48830	0.744149	+	0.668014i	$0.232855\pi$
$42$	0	0
$43$	21209.3	1.74927	0.874633	−	0.484785i	$-0.161102\pi$
$43$	21209.3	1.74927	0.874633	+	0.484785i	$0.161102\pi$
$44$	−1364.50	−0.106253
$45$	0	0
$46$	961.369	0.0669877
$47$	14672.1	0.968831	0.484416	−	0.874838i	$-0.339032\pi$
$47$	14672.1	0.968831	0.484416	+	0.874838i	$0.339032\pi$
$48$	0	0
$49$	5368.21	0.319403
$50$	2226.64	0.125958
$51$	0	0
$52$	2090.48	0.107211
$53$	8110.02	0.396581	0.198291	−	0.980143i	$-0.436461\pi$
$53$	8110.02	0.396581	0.198291	+	0.980143i	$0.436461\pi$
$54$	0	0
$55$	3136.09	0.139792
$56$	−16426.2	−0.699951
$57$	0	0
$58$	6104.84	0.238289
$59$	38628.5	1.44470	0.722350	−	0.691528i	$-0.243062\pi$
$59$	38628.5	1.44470	0.722350	+	0.691528i	$0.243062\pi$
$60$	0	0
$61$	9764.60	0.335993	0.167996	−	0.985788i	$-0.446270\pi$
$61$	9764.60	0.335993	0.167996	+	0.985788i	$0.446270\pi$
$62$	−6307.66	−0.208396
$63$	0	0
$64$	−14185.5	−0.432906
$65$	−4804.64	−0.141052
$66$	0	0
$67$	−5172.01	−0.140758	−0.0703788	−	0.997520i	$-0.522421\pi$
$67$	−5172.01	−0.140758	−0.0703788	+	0.997520i	$0.522421\pi$
$68$	29768.3	0.780697
$69$	0	0
$70$	17849.4	0.435391
$71$	−29546.1	−0.695592	−0.347796	−	0.937570i	$-0.613070\pi$
$71$	−29546.1	−0.695592	−0.347796	+	0.937570i	$0.613070\pi$
$72$	0	0
$73$	−46611.0	−1.02372	−0.511861	−	0.859069i	$-0.671044\pi$
$73$	−46611.0	−1.02372	−0.511861	+	0.859069i	$0.671044\pi$
$74$	3787.62	0.0804058
$75$	0	0
$76$	−41043.5	−0.815099
$77$	7080.54	0.136094
$78$	0	0
$79$	−41029.2	−0.739649	−0.369824	−	0.929102i	$-0.620582\pi$
$79$	−41029.2	−0.739649	−0.369824	+	0.929102i	$0.620582\pi$
$80$	47346.6	0.827112
$81$	0	0
$82$	29112.8	0.478133
$83$	109485.	1.74445	0.872224	−	0.489107i	$-0.162677\pi$
$83$	109485.	1.74445	0.872224	+	0.489107i	$0.162677\pi$
$84$	0	0
$85$	−68417.8	−1.02712
$86$	38544.4	0.561973
$87$	0	0
$88$	−5244.88	−0.0721987
$89$	−20909.2	−0.279810	−0.139905	−	0.990165i	$-0.544680\pi$
$89$	−20909.2	−0.279810	−0.139905	+	0.990165i	$0.544680\pi$
$90$	0	0
$91$	−10847.7	−0.137321
$92$	−15180.9	−0.186994
$93$	0	0
$94$	26664.1	0.311249
$95$	94332.0	1.07238
$96$	0	0
$97$	109289.	1.17936	0.589680	−	0.807637i	$-0.299254\pi$
$97$	109289.	1.17936	0.589680	+	0.807637i	$0.299254\pi$
$98$	9755.82	0.102612
$99$	0	0
$100$	−35160.7	−0.351607
$101$	−35731.6	−0.348537	−0.174269	−	0.984698i	$-0.555756\pi$
$101$	−35731.6	−0.348537	−0.174269	+	0.984698i	$0.555756\pi$
$102$	0	0
$103$	−33786.3	−0.313796	−0.156898	−	0.987615i	$-0.550149\pi$
$103$	−33786.3	−0.313796	−0.156898	+	0.987615i	$0.550149\pi$
$104$	8035.42	0.0728493
$105$	0	0
$106$	14738.6	0.127406
$107$	−11712.7	−0.0989005	−0.0494502	−	0.998777i	$-0.515747\pi$
$107$	−11712.7	−0.0989005	−0.0494502	+	0.998777i	$0.515747\pi$
$108$	0	0
$109$	68684.7	0.553724	0.276862	−	0.960910i	$-0.410705\pi$
$109$	68684.7	0.553724	0.276862	+	0.960910i	$0.410705\pi$
$110$	5699.31	0.0449098
$111$	0	0
$112$	106897.	0.805234
$113$	15299.4	0.112714	0.0563571	−	0.998411i	$-0.482051\pi$
$113$	15299.4	0.112714	0.0563571	+	0.998411i	$0.482051\pi$
$114$	0	0
$115$	34890.9	0.246018
$116$	−96400.9	−0.665175
$117$	0	0
$118$	70200.8	0.464127
$119$	−154471.	−0.999954
$120$	0	0
$121$	−158790.	−0.985962
$122$	17745.5	0.107942
$123$	0	0
$124$	99603.6	0.581729
$125$	−125302.	−0.717270
$126$	0	0
$127$	−96066.6	−0.528522	−0.264261	−	0.964451i	$-0.585128\pi$
$127$	−96066.6	−0.528522	−0.264261	+	0.964451i	$0.585128\pi$
$128$	−180480.	−0.973655
$129$	0	0
$130$	−8731.63	−0.0453145
$131$	205947.	1.04852	0.524260	−	0.851558i	$-0.324342\pi$
$131$	205947.	1.04852	0.524260	+	0.851558i	$0.324342\pi$
$132$	0	0
$133$	212979.	1.04402
$134$	−9399.25	−0.0452201
$135$	0	0
$136$	114424.	0.530481
$137$	−128704.	−0.585854	−0.292927	−	0.956135i	$-0.594629\pi$
$137$	−128704.	−0.585854	−0.292927	+	0.956135i	$0.594629\pi$
$138$	0	0
$139$	−350701.	−1.53957	−0.769787	−	0.638301i	$-0.779637\pi$
$139$	−350701.	−1.53957	−0.769787	+	0.638301i	$0.779637\pi$
$140$	−281858.	−1.21538
$141$	0	0
$142$	−53695.1	−0.223467
$143$	−3463.68	−0.0141644
$144$	0	0
$145$	221562.	0.875137
$146$	−84707.7	−0.328883
$147$	0	0
$148$	−59809.9	−0.224450
$149$	194391.	0.717317	0.358659	−	0.933469i	$-0.383234\pi$
$149$	194391.	0.717317	0.358659	+	0.933469i	$0.383234\pi$
$150$	0	0
$151$	−193968.	−0.692290	−0.346145	−	0.938181i	$-0.612510\pi$
$151$	−193968.	−0.692290	−0.346145	+	0.938181i	$0.612510\pi$
$152$	−157764.	−0.553857
$153$	0	0
$154$	12867.7	0.0437218
$155$	−228923.	−0.765351
$156$	0	0
$157$	1506.30	0.00487710	0.00243855	−	0.999997i	$-0.499224\pi$
$157$	1506.30	0.00487710	0.00243855	+	0.999997i	$0.499224\pi$
$158$	−74563.7	−0.237621
$159$	0	0
$160$	318859.	0.984688
$161$	78775.2	0.239511
$162$	0	0
$163$	−53619.3	−0.158071	−0.0790355	−	0.996872i	$-0.525184\pi$
$163$	−53619.3	−0.158071	−0.0790355	+	0.996872i	$0.525184\pi$
$164$	−459717.	−1.33469
$165$	0	0
$166$	198970.	0.560425
$167$	−653836.	−1.81417	−0.907085	−	0.420948i	$-0.861697\pi$
$167$	−653836.	−1.81417	−0.907085	+	0.420948i	$0.861697\pi$
$168$	0	0
$169$	−365986.	−0.985708
$170$	−124338.	−0.329975
$171$	0	0
$172$	−608651.	−1.56873
$173$	−209060.	−0.531075	−0.265538	−	0.964101i	$-0.585549\pi$
$173$	−209060.	−0.531075	−0.265538	+	0.964101i	$0.585549\pi$
$174$	0	0
$175$	182453.	0.450355
$176$	34132.3	0.0830584
$177$	0	0
$178$	−37999.0	−0.0898922
$179$	449478.	1.04852	0.524259	−	0.851559i	$-0.324342\pi$
$179$	449478.	1.04852	0.524259	+	0.851559i	$0.324342\pi$
$180$	0	0
$181$	−22289.9	−0.0505722	−0.0252861	−	0.999680i	$-0.508050\pi$
$181$	−22289.9	−0.0505722	−0.0252861	+	0.999680i	$0.508050\pi$
$182$	−19713.9	−0.0441159
$183$	0	0
$184$	−58352.5	−0.127062
$185$	137464.	0.295297
$186$	0	0
$187$	−49322.6	−0.103143
$188$	−421050.	−0.868839
$189$	0	0
$190$	171433.	0.344516
$191$	739615.	1.46697	0.733487	−	0.679704i	$-0.237892\pi$
$191$	739615.	1.46697	0.733487	+	0.679704i	$0.237892\pi$
$192$	0	0
$193$	800277.	1.54649	0.773245	−	0.634107i	$-0.218632\pi$
$193$	800277.	1.54649	0.773245	+	0.634107i	$0.218632\pi$
$194$	198614.	0.378883
$195$	0	0
$196$	−154053.	−0.286438
$197$	−606917.	−1.11420	−0.557101	−	0.830445i	$-0.688086\pi$
$197$	−606917.	−1.11420	−0.557101	+	0.830445i	$0.688086\pi$
$198$	0	0
$199$	−36604.2	−0.0655236	−0.0327618	−	0.999463i	$-0.510430\pi$
$199$	−36604.2	−0.0655236	−0.0327618	+	0.999463i	$0.510430\pi$
$200$	−135151.	−0.238916
$201$	0	0
$202$	−64936.2	−0.111972
$203$	500235.	0.851989
$204$	0	0
$205$	1.05659e6	1.75599
$206$	−61400.9	−0.100811
$207$	0	0
$208$	−52292.4	−0.0838069
$209$	68004.1	0.107689
$210$	0	0
$211$	−652670.	−1.00922	−0.504612	−	0.863346i	$-0.668365\pi$
$211$	−652670.	−1.00922	−0.504612	+	0.863346i	$0.668365\pi$
$212$	−232736.	−0.355650
$213$	0	0
$214$	−21285.9	−0.0317729
$215$	1.39889e6	2.06389
$216$	0	0
$217$	−516854.	−0.745107
$218$	124823.	0.177891
$219$	0	0
$220$	−89997.3	−0.125364
$221$	75564.6	0.104073
$222$	0	0
$223$	−864101.	−1.16360	−0.581798	−	0.813333i	$-0.697651\pi$
$223$	−864101.	−1.16360	−0.581798	+	0.813333i	$0.697651\pi$
$224$	719907.	0.958642
$225$	0	0
$226$	27804.1	0.0362108
$227$	−751881.	−0.968466	−0.484233	−	0.874939i	$-0.660901\pi$
$227$	−751881.	−0.968466	−0.484233	+	0.874939i	$0.660901\pi$
$228$	0	0
$229$	1.17103e6	1.47564	0.737821	−	0.674996i	$-0.235855\pi$
$229$	1.17103e6	1.47564	0.737821	+	0.674996i	$0.235855\pi$
$230$	63408.3	0.0790362
$231$	0	0
$232$	−370547.	−0.451985
$233$	1.01539e6	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$233$	1.01539e6	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$234$	0	0
$235$	967718.	1.14309
$236$	−1.10853e6	−1.29559
$237$	0	0
$238$	−280725.	−0.321247
$239$	−436994.	−0.494859	−0.247429	−	0.968906i	$-0.579586\pi$
$239$	−436994.	−0.494859	−0.247429	+	0.968906i	$0.579586\pi$
$240$	0	0
$241$	−1.62912e6	−1.80680	−0.903401	−	0.428796i	$-0.858938\pi$
$241$	−1.62912e6	−1.80680	−0.903401	+	0.428796i	$0.858938\pi$
$242$	−288574.	−0.316752
$243$	0	0
$244$	−280218.	−0.301315
$245$	354067.	0.376851
$246$	0	0
$247$	−104186.	−0.108659
$248$	382858.	0.395283
$249$	0	0
$250$	−227715.	−0.230432
$251$	880717.	0.882373	0.441186	−	0.897416i	$-0.354558\pi$
$251$	880717.	0.882373	0.441186	+	0.897416i	$0.354558\pi$
$252$	0	0
$253$	25152.9	0.0247051
$254$	−174585.	−0.169794
$255$	0	0
$256$	125942.	0.120108
$257$	−1.49036e6	−1.40753	−0.703767	−	0.710431i	$-0.748500\pi$
$257$	−1.49036e6	−1.40753	−0.703767	+	0.710431i	$0.748500\pi$
$258$	0	0
$259$	310360.	0.287486
$260$	137880.	0.126494
$261$	0	0
$262$	374273.	0.336849
$263$	−391849.	−0.349325	−0.174662	−	0.984628i	$-0.555883\pi$
$263$	−391849.	−0.349325	−0.174662	+	0.984628i	$0.555883\pi$
$264$	0	0
$265$	534906.	0.467911
$266$	387054.	0.335403
$267$	0	0
$268$	148423.	0.126230
$269$	1.50144e6	1.26510	0.632552	−	0.774518i	$-0.282007\pi$
$269$	1.50144e6	1.26510	0.632552	+	0.774518i	$0.282007\pi$
$270$	0	0
$271$	−482511.	−0.399102	−0.199551	−	0.979887i	$-0.563948\pi$
$271$	−482511.	−0.399102	−0.199551	+	0.979887i	$0.563948\pi$
$272$	−744640.	−0.610273
$273$	0	0
$274$	−233897.	−0.188213
$275$	58257.1	0.0464534
$276$	0	0
$277$	−886396.	−0.694110	−0.347055	−	0.937845i	$-0.612818\pi$
$277$	−886396.	−0.694110	−0.347055	+	0.937845i	$0.612818\pi$
$278$	−637341.	−0.494606
$279$	0	0
$280$	−1.08341e6	−0.825845
$281$	2.27286e6	1.71714	0.858572	−	0.512693i	$-0.171352\pi$
$281$	2.27286e6	1.71714	0.858572	+	0.512693i	$0.171352\pi$
$282$	0	0
$283$	973238.	0.722359	0.361179	−	0.932496i	$-0.382374\pi$
$283$	973238.	0.722359	0.361179	+	0.932496i	$0.382374\pi$
$284$	847894.	0.623800
$285$	0	0
$286$	−6294.65	−0.00455047
$287$	2.38552e6	1.70954
$288$	0	0
$289$	−343821.	−0.242152
$290$	402652.	0.281148
$291$	0	0
$292$	1.33761e6	0.918064
$293$	2.60507e6	1.77276	0.886382	−	0.462955i	$-0.153211\pi$
$293$	2.60507e6	1.77276	0.886382	+	0.462955i	$0.153211\pi$
$294$	0	0
$295$	2.54779e6	1.70454
$296$	−229898.	−0.152513
$297$	0	0
$298$	353273.	0.230447
$299$	−38535.5	−0.0249277
$300$	0	0
$301$	3.15836e6	2.00930
$302$	−352505.	−0.222406
$303$	0	0
$304$	1.02668e6	0.637165
$305$	644036.	0.396425
$306$	0	0
$307$	−2.58204e6	−1.56357	−0.781784	−	0.623550i	$-0.785690\pi$
$307$	−2.58204e6	−1.56357	−0.781784	+	0.623550i	$0.785690\pi$
$308$	−203192.	−0.122048
$309$	0	0
$310$	−416030.	−0.245878
$311$	1.41291e6	0.828347	0.414173	−	0.910198i	$-0.364071\pi$
$311$	1.41291e6	0.828347	0.414173	+	0.910198i	$0.364071\pi$
$312$	0	0
$313$	545902.	0.314959	0.157480	−	0.987522i	$-0.449663\pi$
$313$	545902.	0.314959	0.157480	+	0.987522i	$0.449663\pi$
$314$	2737.44	0.00156683
$315$	0	0
$316$	1.17743e6	0.663310
$317$	2.47613e6	1.38396	0.691981	−	0.721915i	$-0.256738\pi$
$317$	2.47613e6	1.38396	0.691981	+	0.721915i	$0.256738\pi$
$318$	0	0
$319$	159725.	0.0878811
$320$	−935620.	−0.510769
$321$	0	0
$322$	143161.	0.0769457
$323$	−1.48360e6	−0.791243
$324$	0	0
$325$	−89252.8	−0.0468720
$326$	−97444.1	−0.0507822
$327$	0	0
$328$	−1.76707e6	−0.906919
$329$	2.18488e6	1.11285
$330$	0	0
$331$	−543497.	−0.272664	−0.136332	−	0.990663i	$-0.543531\pi$
$331$	−543497.	−0.272664	−0.136332	+	0.990663i	$0.543531\pi$
$332$	−3.14191e6	−1.56440
$333$	0	0
$334$	−1.18824e6	−0.582824
$335$	−341126.	−0.166074
$336$	0	0
$337$	371347.	0.178117	0.0890584	−	0.996026i	$-0.471614\pi$
$337$	371347.	0.178117	0.0890584	+	0.996026i	$0.471614\pi$
$338$	−665119.	−0.316670
$339$	0	0
$340$	1.96341e6	0.921114
$341$	−165031.	−0.0768565
$342$	0	0
$343$	−1.70339e6	−0.781769
$344$	−2.33954e6	−1.06594
$345$	0	0
$346$	−379932.	−0.170614
$347$	−1.95484e6	−0.871540	−0.435770	−	0.900058i	$-0.643524\pi$
$347$	−1.95484e6	−0.871540	−0.435770	+	0.900058i	$0.643524\pi$
$348$	0	0
$349$	−3.91651e6	−1.72122	−0.860608	−	0.509267i	$-0.829916\pi$
$349$	−3.91651e6	−1.72122	−0.860608	+	0.509267i	$0.829916\pi$
$350$	331577.	0.144682
$351$	0	0
$352$	229866.	0.0988822
$353$	524557.	0.224055	0.112028	−	0.993705i	$-0.464265\pi$
$353$	524557.	0.224055	0.112028	+	0.993705i	$0.464265\pi$
$354$	0	0
$355$	−1.94875e6	−0.820702
$356$	600038.	0.250931
$357$	0	0
$358$	816850.	0.336849
$359$	−1.58882e6	−0.650637	−0.325319	−	0.945604i	$-0.605472\pi$
$359$	−1.58882e6	−0.650637	−0.325319	+	0.945604i	$0.605472\pi$
$360$	0	0
$361$	−430568.	−0.173890
$362$	−40508.2	−0.0162469
$363$	0	0
$364$	311301.	0.123148
$365$	−3.07429e6	−1.20785
$366$	0	0
$367$	−1.86611e6	−0.723221	−0.361611	−	0.932329i	$-0.617773\pi$
$367$	−1.86611e6	−0.723221	−0.361611	+	0.932329i	$0.617773\pi$
$368$	379742.	0.146174
$369$	0	0
$370$	249817.	0.0948677
$371$	1.20769e6	0.455534
$372$	0	0
$373$	1.32676e6	0.493766	0.246883	−	0.969045i	$-0.420594\pi$
$373$	1.32676e6	0.493766	0.246883	+	0.969045i	$0.420594\pi$
$374$	−89635.5	−0.0331361
$375$	0	0
$376$	−1.61844e6	−0.590373
$377$	−244706.	−0.0886731
$378$	0	0
$379$	847234.	0.302974	0.151487	−	0.988459i	$-0.451594\pi$
$379$	847234.	0.302974	0.151487	+	0.988459i	$0.451594\pi$
$380$	−2.70707e6	−0.961703
$381$	0	0
$382$	1.34413e6	0.471283
$383$	−4.68818e6	−1.63308	−0.816540	−	0.577289i	$-0.804111\pi$
$383$	−4.68818e6	−1.63308	−0.816540	+	0.577289i	$0.804111\pi$
$384$	0	0
$385$	467006.	0.160572
$386$	1.45437e6	0.496828
$387$	0	0
$388$	−3.13629e6	−1.05764
$389$	−5.71164e6	−1.91376	−0.956879	−	0.290488i	$-0.906182\pi$
$389$	−5.71164e6	−1.91376	−0.956879	+	0.290488i	$0.906182\pi$
$390$	0	0
$391$	−548743.	−0.181521
$392$	−592152.	−0.194634
$393$	0	0
$394$	−1.10297e6	−0.357950
$395$	−2.70613e6	−0.872683
$396$	0	0
$397$	1.41109e6	0.449343	0.224671	−	0.974435i	$-0.427869\pi$
$397$	1.41109e6	0.449343	0.224671	+	0.974435i	$0.427869\pi$
$398$	−66521.9	−0.0210502
$399$	0	0
$400$	879528.	0.274853
$401$	1.24416e6	0.386382	0.193191	−	0.981161i	$-0.438116\pi$
$401$	1.24416e6	0.386382	0.193191	+	0.981161i	$0.438116\pi$
$402$	0	0
$403$	252836.	0.0775491
$404$	1.02540e6	0.312565
$405$	0	0
$406$	909093.	0.273712
$407$	99097.9	0.0296537
$408$	0	0
$409$	−2.31947e6	−0.685616	−0.342808	−	0.939406i	$-0.611378\pi$
$409$	−2.31947e6	−0.685616	−0.342808	+	0.939406i	$0.611378\pi$
$410$	1.92017e6	0.564131
$411$	0	0
$412$	969575.	0.281409
$413$	5.75230e6	1.65946
$414$	0	0
$415$	7.22120e6	2.05821
$416$	−352166.	−0.0997733
$417$	0	0
$418$	123586.	0.0345962
$419$	−3.26602e6	−0.908831	−0.454416	−	0.890790i	$-0.650152\pi$
$419$	−3.26602e6	−0.908831	−0.454416	+	0.890790i	$0.650152\pi$
$420$	0	0
$421$	−843342.	−0.231899	−0.115949	−	0.993255i	$-0.536991\pi$
$421$	−843342.	−0.231899	−0.115949	+	0.993255i	$0.536991\pi$
$422$	−1.18612e6	−0.324225
$423$	0	0
$424$	−894593.	−0.241663
$425$	−1.27095e6	−0.341317
$426$	0	0
$427$	1.45408e6	0.385939
$428$	336123.	0.0886930
$429$	0	0
$430$	2.54225e6	0.663050
$431$	2.13493e6	0.553592	0.276796	−	0.960929i	$-0.410727\pi$
$431$	2.13493e6	0.553592	0.276796	+	0.960929i	$0.410727\pi$
$432$	0	0
$433$	1.21055e6	0.310285	0.155143	−	0.987892i	$-0.450416\pi$
$433$	1.21055e6	0.310285	0.155143	+	0.987892i	$0.450416\pi$
$434$	−939295.	−0.239375
$435$	0	0
$436$	−1.97106e6	−0.496575
$437$	756587.	0.189520
$438$	0	0
$439$	5.94175e6	1.47148	0.735738	−	0.677266i	$-0.236835\pi$
$439$	5.94175e6	1.47148	0.735738	+	0.677266i	$0.236835\pi$
$440$	−345933.	−0.0851844
$441$	0	0
$442$	137326.	0.0334347
$443$	−5.63467e6	−1.36414	−0.682070	−	0.731286i	$-0.738920\pi$
$443$	−5.63467e6	−1.36414	−0.682070	+	0.731286i	$0.738920\pi$
$444$	0	0
$445$	−1.37909e6	−0.330137
$446$	−1.57036e6	−0.373819
$447$	0	0
$448$	−2.11241e6	−0.497259
$449$	−1.43999e6	−0.337087	−0.168544	−	0.985694i	$-0.553906\pi$
$449$	−1.43999e6	−0.337087	−0.168544	+	0.985694i	$0.553906\pi$
$450$	0	0
$451$	761696.	0.176336
$452$	−439052.	−0.101081
$453$	0	0
$454$	−1.36642e6	−0.311131
$455$	−715476.	−0.162019
$456$	0	0
$457$	−2.66277e6	−0.596407	−0.298203	−	0.954502i	$-0.596387\pi$
$457$	−2.66277e6	−0.596407	−0.298203	+	0.954502i	$0.596387\pi$
$458$	2.12816e6	0.474068
$459$	0	0
$460$	−1.00127e6	−0.220627
$461$	−4.51818e6	−0.990173	−0.495086	−	0.868844i	$-0.664864\pi$
$461$	−4.51818e6	−0.990173	−0.495086	+	0.868844i	$0.664864\pi$
$462$	0	0
$463$	−7.42640e6	−1.61000	−0.805000	−	0.593275i	$-0.797835\pi$
$463$	−7.42640e6	−1.61000	−0.805000	+	0.593275i	$0.797835\pi$
$464$	2.41142e6	0.519970
$465$	0	0
$466$	1.84530e6	0.393642
$467$	−2.24427e6	−0.476193	−0.238097	−	0.971241i	$-0.576523\pi$
$467$	−2.24427e6	−0.476193	−0.238097	+	0.971241i	$0.576523\pi$
$468$	0	0
$469$	−770181.	−0.161682
$470$	1.75866e6	0.367230
$471$	0	0
$472$	−4.26100e6	−0.880352
$473$	1.00846e6	0.207256
$474$	0	0
$475$	1.75234e6	0.356357
$476$	4.43291e6	0.896749
$477$	0	0
$478$	−794164.	−0.158979
$479$	−4.52066e6	−0.900251	−0.450125	−	0.892965i	$-0.648621\pi$
$479$	−4.52066e6	−0.900251	−0.450125	+	0.892965i	$0.648621\pi$
$480$	0	0
$481$	−151823.	−0.0299209
$482$	−2.96065e6	−0.580457
$483$	0	0
$484$	4.55685e6	0.884202
$485$	7.20828e6	1.39148
$486$	0	0
$487$	1.44375e6	0.275848	0.137924	−	0.990443i	$-0.455957\pi$
$487$	1.44375e6	0.275848	0.137924	+	0.990443i	$0.455957\pi$
$488$	−1.07710e6	−0.204743
$489$	0	0
$490$	643457.	0.121068
$491$	497140.	0.0930625	0.0465312	−	0.998917i	$-0.485183\pi$
$491$	497140.	0.0930625	0.0465312	+	0.998917i	$0.485183\pi$
$492$	0	0
$493$	−3.48460e6	−0.645708
$494$	−189340.	−0.0349080
$495$	0	0
$496$	−2.49154e6	−0.454740
$497$	−4.39981e6	−0.798994
$498$	0	0
$499$	1.03922e7	1.86834	0.934169	−	0.356832i	$-0.116143\pi$
$499$	1.03922e7	1.86834	0.934169	+	0.356832i	$0.116143\pi$
$500$	3.59583e6	0.643241
$501$	0	0
$502$	1.60055e6	0.283473
$503$	6.99554e6	1.23282	0.616412	−	0.787423i	$-0.288585\pi$
$503$	6.99554e6	1.23282	0.616412	+	0.787423i	$0.288585\pi$
$504$	0	0
$505$	−2.35672e6	−0.411226
$506$	45711.1	0.00793681
$507$	0	0
$508$	2.75685e6	0.473974
$509$	−4.36936e6	−0.747521	−0.373760	−	0.927525i	$-0.621932\pi$
$509$	−4.36936e6	−0.747521	−0.373760	+	0.927525i	$0.621932\pi$
$510$	0	0
$511$	−6.94101e6	−1.17590
$512$	6.00425e6	1.01224
$513$	0	0
$514$	−2.70848e6	−0.452187
$515$	−2.22842e6	−0.370236
$516$	0	0
$517$	697630.	0.114789
$518$	564028.	0.0923583
$519$	0	0
$520$	529986.	0.0859521
$521$	1.47687e6	0.238368	0.119184	−	0.992872i	$-0.461972\pi$
$521$	1.47687e6	0.238368	0.119184	+	0.992872i	$0.461972\pi$
$522$	0	0
$523$	9.84627e6	1.57405	0.787023	−	0.616923i	$-0.211621\pi$
$523$	9.84627e6	1.57405	0.787023	+	0.616923i	$0.211621\pi$
$524$	−5.91011e6	−0.940303
$525$	0	0
$526$	−712120.	−0.112225
$527$	3.60037e6	0.564704
$528$	0	0
$529$	279841.	0.0434783
$530$	972102.	0.150322
$531$	0	0
$532$	−6.11192e6	−0.936265
$533$	−1.16696e6	−0.177925
$534$	0	0
$535$	−772527.	−0.116689
$536$	570509.	0.0857730
$537$	0	0
$538$	2.72861e6	0.406430
$539$	255248.	0.0378434
$540$	0	0
$541$	3.42977e6	0.503816	0.251908	−	0.967751i	$-0.418942\pi$
$541$	3.42977e6	0.503816	0.251908	+	0.967751i	$0.418942\pi$
$542$	−876882.	−0.128216
$543$	0	0
$544$	−5.01483e6	−0.726539
$545$	4.53018e6	0.653318
$546$	0	0
$547$	−2.60400e6	−0.372110	−0.186055	−	0.982539i	$-0.559570\pi$
$547$	−2.60400e6	−0.372110	−0.186055	+	0.982539i	$0.559570\pi$
$548$	3.69345e6	0.525388
$549$	0	0
$550$	105872.	0.0149237
$551$	4.80444e6	0.674162
$552$	0	0
$553$	−6.10980e6	−0.849600
$554$	−1.61088e6	−0.222991
$555$	0	0
$556$	1.00642e7	1.38068
$557$	9.21091e6	1.25795	0.628977	−	0.777424i	$-0.283474\pi$
$557$	9.21091e6	1.25795	0.628977	+	0.777424i	$0.283474\pi$
$558$	0	0
$559$	−1.54501e6	−0.209124
$560$	7.05055e6	0.950064
$561$	0	0
$562$	4.13054e6	0.551653
$563$	−5.68136e6	−0.755407	−0.377704	−	0.925927i	$-0.623286\pi$
$563$	−5.68136e6	−0.755407	−0.377704	+	0.925927i	$0.623286\pi$
$564$	0	0
$565$	1.00909e6	0.132987
$566$	1.76870e6	0.232066
$567$	0	0
$568$	3.25915e6	0.423871
$569$	−9.29020e6	−1.20294	−0.601471	−	0.798895i	$-0.705418\pi$
$569$	−9.29020e6	−1.20294	−0.601471	+	0.798895i	$0.705418\pi$
$570$	0	0
$571$	6.03035e6	0.774020	0.387010	−	0.922076i	$-0.373508\pi$
$571$	6.03035e6	0.774020	0.387010	+	0.922076i	$0.373508\pi$
$572$	99398.2	0.0127025
$573$	0	0
$574$	4.33528e6	0.549209
$575$	648145.	0.0817528
$576$	0	0
$577$	−2.54613e6	−0.318376	−0.159188	−	0.987248i	$-0.550888\pi$
$577$	−2.54613e6	−0.318376	−0.159188	+	0.987248i	$0.550888\pi$
$578$	−624836.	−0.0777941
$579$	0	0
$580$	−6.35824e6	−0.784815
$581$	1.63037e7	2.00376
$582$	0	0
$583$	385615.	0.0469875
$584$	5.14153e6	0.623821
$585$	0	0
$586$	4.73428e6	0.569521
$587$	−1.79072e6	−0.214502	−0.107251	−	0.994232i	$-0.534205\pi$
$587$	−1.79072e6	−0.214502	−0.107251	+	0.994232i	$0.534205\pi$
$588$	0	0
$589$	−4.96406e6	−0.589588
$590$	4.63018e6	0.547605
$591$	0	0
$592$	1.49612e6	0.175453
$593$	−4.51759e6	−0.527558	−0.263779	−	0.964583i	$-0.584969\pi$
$593$	−4.51759e6	−0.527558	−0.263779	+	0.964583i	$0.584969\pi$
$594$	0	0
$595$	−1.01883e7	−1.17981
$596$	−5.57851e6	−0.643283
$597$	0	0
$598$	−70031.8	−0.00800833
$599$	1.32073e7	1.50399	0.751997	−	0.659166i	$-0.229091\pi$
$599$	1.32073e7	1.50399	0.751997	+	0.659166i	$0.229091\pi$
$600$	0	0
$601$	−2.70025e6	−0.304942	−0.152471	−	0.988308i	$-0.548723\pi$
$601$	−2.70025e6	−0.304942	−0.152471	+	0.988308i	$0.548723\pi$
$602$	5.73978e6	0.645512
$603$	0	0
$604$	5.56636e6	0.620839
$605$	−1.04732e7	−1.16330
$606$	0	0
$607$	−1.34368e6	−0.148021	−0.0740104	−	0.997257i	$-0.523580\pi$
$607$	−1.34368e6	−0.148021	−0.0740104	+	0.997257i	$0.523580\pi$
$608$	6.91426e6	0.758554
$609$	0	0
$610$	1.17043e6	0.127356
$611$	−1.06880e6	−0.115823
$612$	0	0
$613$	1.16013e7	1.24697	0.623486	−	0.781834i	$-0.285716\pi$
$613$	1.16013e7	1.24697	0.623486	+	0.781834i	$0.285716\pi$
$614$	−4.69242e6	−0.502315
$615$	0	0
$616$	−781034.	−0.0829312
$617$	6.41070e6	0.677942	0.338971	−	0.940797i	$-0.389921\pi$
$617$	6.41070e6	0.677942	0.338971	+	0.940797i	$0.389921\pi$
$618$	0	0
$619$	−6.97091e6	−0.731245	−0.365622	−	0.930763i	$-0.619144\pi$
$619$	−6.97091e6	−0.731245	−0.365622	+	0.930763i	$0.619144\pi$
$620$	6.56948e6	0.686360
$621$	0	0
$622$	2.56772e6	0.266116
$623$	−3.11366e6	−0.321404
$624$	0	0
$625$	−1.20933e7	−1.23835
$626$	992086.	0.101184
$627$	0	0
$628$	−43226.7	−0.00437374
$629$	−2.16195e6	−0.217881
$630$	0	0
$631$	−1.03683e7	−1.03666	−0.518329	−	0.855181i	$-0.673446\pi$
$631$	−1.03683e7	−1.03666	−0.518329	+	0.855181i	$0.673446\pi$
$632$	4.52582e6	0.450717
$633$	0	0
$634$	4.49994e6	0.444615
$635$	−6.33619e6	−0.623583
$636$	0	0
$637$	−391052.	−0.0381844
$638$	290273.	0.0282329
$639$	0	0
$640$	−1.19038e7	−1.14878
$641$	1.32572e7	1.27441	0.637203	−	0.770696i	$-0.280091\pi$
$641$	1.32572e7	1.27441	0.637203	+	0.770696i	$0.280091\pi$
$642$	0	0
$643$	−1.58242e7	−1.50936	−0.754682	−	0.656091i	$-0.772209\pi$
$643$	−1.58242e7	−1.50936	−0.754682	+	0.656091i	$0.772209\pi$
$644$	−2.26064e6	−0.214791
$645$	0	0
$646$	−2.69619e6	−0.254196
$647$	9.67249e6	0.908401	0.454201	−	0.890899i	$-0.349925\pi$
$647$	9.67249e6	0.908401	0.454201	+	0.890899i	$0.349925\pi$
$648$	0	0
$649$	1.83671e6	0.171170
$650$	−162202.	−0.0150582
$651$	0	0
$652$	1.53873e6	0.141757
$653$	6.17708e6	0.566892	0.283446	−	0.958988i	$-0.408522\pi$
$653$	6.17708e6	0.566892	0.283446	+	0.958988i	$0.408522\pi$
$654$	0	0
$655$	1.35835e7	1.23711
$656$	1.14996e7	1.04333
$657$	0	0
$658$	3.97064e6	0.357517
$659$	−1.61529e7	−1.44889	−0.724447	−	0.689330i	$-0.757905\pi$
$659$	−1.61529e7	−1.44889	−0.724447	+	0.689330i	$0.757905\pi$
$660$	0	0
$661$	1.12662e7	1.00293	0.501467	−	0.865177i	$-0.332794\pi$
$661$	1.12662e7	1.00293	0.501467	+	0.865177i	$0.332794\pi$
$662$	−987715.	−0.0875965
$663$	0	0
$664$	−1.20769e7	−1.06301
$665$	1.40473e7	1.23180
$666$	0	0
$667$	1.77703e6	0.154661
$668$	1.87633e7	1.62693
$669$	0	0
$670$	−619939.	−0.0533534
$671$	464287.	0.0398089
$672$	0	0
$673$	−1.35668e7	−1.15462	−0.577312	−	0.816524i	$-0.695898\pi$
$673$	−1.35668e7	−1.15462	−0.577312	+	0.816524i	$0.695898\pi$
$674$	674861.	0.0572222
$675$	0	0
$676$	1.05028e7	0.883974
$677$	−2.24134e7	−1.87947	−0.939736	−	0.341901i	$-0.888929\pi$
$677$	−2.24134e7	−1.87947	−0.939736	+	0.341901i	$0.888929\pi$
$678$	0	0
$679$	1.62746e7	1.35467
$680$	7.54698e6	0.625894
$681$	0	0
$682$	−299917.	−0.0246911
$683$	−1.96292e7	−1.61009	−0.805047	−	0.593211i	$-0.797860\pi$
$683$	−1.96292e7	−1.61009	−0.805047	+	0.593211i	$0.797860\pi$
$684$	0	0
$685$	−8.48881e6	−0.691226
$686$	−3.09562e6	−0.251153
$687$	0	0
$688$	1.52251e7	1.22628
$689$	−590782.	−0.0474110
$690$	0	0
$691$	−1.52996e6	−0.121894	−0.0609472	−	0.998141i	$-0.519412\pi$
$691$	−1.52996e6	−0.121894	−0.0609472	+	0.998141i	$0.519412\pi$
$692$	5.99946e6	0.476263
$693$	0	0
$694$	−3.55259e6	−0.279993
$695$	−2.31309e7	−1.81648
$696$	0	0
$697$	−1.66174e7	−1.29563
$698$	−7.11760e6	−0.552961
$699$	0	0
$700$	−5.23590e6	−0.403875
$701$	−2.51687e7	−1.93449	−0.967245	−	0.253845i	$-0.918305\pi$
$701$	−2.51687e7	−1.93449	−0.967245	+	0.253845i	$0.918305\pi$
$702$	0	0
$703$	2.98082e6	0.227482
$704$	−674490.	−0.0512914
$705$	0	0
$706$	953293.	0.0719805
$707$	−5.32092e6	−0.400348
$708$	0	0
$709$	4.11965e6	0.307783	0.153892	−	0.988088i	$-0.450819\pi$
$709$	4.11965e6	0.307783	0.153892	+	0.988088i	$0.450819\pi$
$710$	−3.54153e6	−0.263660
$711$	0	0
$712$	2.30644e6	0.170507
$713$	−1.83607e6	−0.135259
$714$	0	0
$715$	−228451.	−0.0167120
$716$	−1.28988e7	−0.940301
$717$	0	0
$718$	−2.88742e6	−0.209025
$719$	−40244.1	−0.00290322	−0.00145161	−	0.999999i	$-0.500462\pi$
$719$	−40244.1	−0.00290322	−0.00145161	+	0.999999i	$0.500462\pi$
$720$	0	0
$721$	−5.03123e6	−0.360443
$722$	−782485.	−0.0558642
$723$	0	0
$724$	639660.	0.0453527
$725$	4.11582e6	0.290812
$726$	0	0
$727$	−9.85755e6	−0.691724	−0.345862	−	0.938285i	$-0.612413\pi$
$727$	−9.85755e6	−0.691724	−0.345862	+	0.938285i	$0.612413\pi$
$728$	1.19658e6	0.0836786
$729$	0	0
$730$	−5.58700e6	−0.388036
$731$	−2.20009e7	−1.52282
$732$	0	0
$733$	2.26583e7	1.55764	0.778820	−	0.627247i	$-0.215818\pi$
$733$	2.26583e7	1.55764	0.778820	+	0.627247i	$0.215818\pi$
$734$	−3.39133e6	−0.232343
$735$	0	0
$736$	2.55740e6	0.174022
$737$	−245918.	−0.0166772
$738$	0	0
$739$	2.56692e7	1.72902	0.864511	−	0.502614i	$-0.167628\pi$
$739$	2.56692e7	1.72902	0.864511	+	0.502614i	$0.167628\pi$
$740$	−3.94484e6	−0.264820
$741$	0	0
$742$	2.19477e6	0.146346
$743$	−1.29833e7	−0.862803	−0.431402	−	0.902160i	$-0.641981\pi$
$743$	−1.29833e7	−0.862803	−0.431402	+	0.902160i	$0.641981\pi$
$744$	0	0
$745$	1.28213e7	0.846335
$746$	2.41117e6	0.158628
$747$	0	0
$748$	1.41542e6	0.0924981
$749$	−1.74418e6	−0.113602
$750$	0	0
$751$	1.64092e7	1.06167	0.530834	−	0.847476i	$-0.321879\pi$
$751$	1.64092e7	1.06167	0.530834	+	0.847476i	$0.321879\pi$
$752$	1.05324e7	0.679174
$753$	0	0
$754$	−444713.	−0.0284873
$755$	−1.27934e7	−0.816806
$756$	0	0
$757$	2.11499e7	1.34143	0.670716	−	0.741714i	$-0.265987\pi$
$757$	2.11499e7	1.34143	0.670716	+	0.741714i	$0.265987\pi$
$758$	1.53971e6	0.0973341
$759$	0	0
$760$	−1.04055e7	−0.653474
$761$	−1.58681e7	−0.993261	−0.496630	−	0.867962i	$-0.665430\pi$
$761$	−1.58681e7	−0.993261	−0.496630	+	0.867962i	$0.665430\pi$
$762$	0	0
$763$	1.02281e7	0.636037
$764$	−2.12250e7	−1.31557
$765$	0	0
$766$	−8.51998e6	−0.524646
$767$	−2.81393e6	−0.172713
$768$	0	0
$769$	2.22568e7	1.35721	0.678606	−	0.734502i	$-0.262584\pi$
$769$	2.22568e7	1.35721	0.678606	+	0.734502i	$0.262584\pi$
$770$	848704.	0.0515857
$771$	0	0
$772$	−2.29658e7	−1.38688
$773$	1.78302e7	1.07327	0.536634	−	0.843815i	$-0.319696\pi$
$773$	1.78302e7	1.07327	0.536634	+	0.843815i	$0.319696\pi$
$774$	0	0
$775$	−4.25256e6	−0.254329
$776$	−1.20553e7	−0.718662
$777$	0	0
$778$	−1.03799e7	−0.614817
$779$	2.29114e7	1.35272
$780$	0	0
$781$	−1.40486e6	−0.0824148
$782$	−997248.	−0.0583158
$783$	0	0
$784$	3.85356e6	0.223909
$785$	99349.7	0.00575430
$786$	0	0
$787$	−2.67009e7	−1.53670	−0.768349	−	0.640032i	$-0.778921\pi$
$787$	−2.67009e7	−1.53670	−0.768349	+	0.640032i	$0.778921\pi$
$788$	1.74169e7	0.999205
$789$	0	0
$790$	−4.91794e6	−0.280360
$791$	2.27829e6	0.129470
$792$	0	0
$793$	−711311.	−0.0401677
$794$	2.56441e6	0.144357
$795$	0	0
$796$	1.05044e6	0.0587610
$797$	5.67591e6	0.316512	0.158256	−	0.987398i	$-0.449413\pi$
$797$	5.67591e6	0.316512	0.158256	+	0.987398i	$0.449413\pi$
$798$	0	0
$799$	−1.52197e7	−0.843411
$800$	5.92324e6	0.327216
$801$	0	0
$802$	2.26106e6	0.124130
$803$	−2.21626e6	−0.121292
$804$	0	0
$805$	5.19572e6	0.282589
$806$	459487.	0.0249136
$807$	0	0
$808$	3.94145e6	0.212387
$809$	−9.15340e6	−0.491713	−0.245856	−	0.969306i	$-0.579069\pi$
$809$	−9.15340e6	−0.491713	−0.245856	+	0.969306i	$0.579069\pi$
$810$	0	0
$811$	−3.07221e6	−0.164021	−0.0820104	−	0.996631i	$-0.526134\pi$
$811$	−3.07221e6	−0.164021	−0.0820104	+	0.996631i	$0.526134\pi$
$812$	−1.43554e7	−0.764056
$813$	0	0
$814$	180094.	0.00952660
$815$	−3.53653e6	−0.186502
$816$	0	0
$817$	3.03340e7	1.58992
$818$	−4.21525e6	−0.220262
$819$	0	0
$820$	−3.03212e7	−1.57475
$821$	3.00442e7	1.55562	0.777808	−	0.628501i	$-0.216331\pi$
$821$	3.00442e7	1.55562	0.777808	+	0.628501i	$0.216331\pi$
$822$	0	0
$823$	3.97764e6	0.204704	0.102352	−	0.994748i	$-0.467363\pi$
$823$	3.97764e6	0.204704	0.102352	+	0.994748i	$0.467363\pi$
$824$	3.72687e6	0.191217
$825$	0	0
$826$	1.04538e7	0.533121
$827$	1.63780e7	0.832714	0.416357	−	0.909201i	$-0.363307\pi$
$827$	1.63780e7	0.832714	0.416357	+	0.909201i	$0.363307\pi$
$828$	0	0
$829$	−6.73541e6	−0.340391	−0.170195	−	0.985410i	$-0.554440\pi$
$829$	−6.73541e6	−0.340391	−0.170195	+	0.985410i	$0.554440\pi$
$830$	1.31233e7	0.661223
$831$	0	0
$832$	1.03335e6	0.0517536
$833$	−5.56856e6	−0.278055
$834$	0	0
$835$	−4.31246e7	−2.14047
$836$	−1.95154e6	−0.0965741
$837$	0	0
$838$	−5.93543e6	−0.291973
$839$	−3.56177e7	−1.74687	−0.873435	−	0.486941i	$-0.838113\pi$
$839$	−3.56177e7	−1.74687	−0.873435	+	0.486941i	$0.838113\pi$
$840$	0	0
$841$	−9.22671e6	−0.449839
$842$	−1.53263e6	−0.0745002
$843$	0	0
$844$	1.87299e7	0.905063
$845$	−2.41391e7	−1.16300
$846$	0	0
$847$	−2.36460e7	−1.13253
$848$	5.82177e6	0.278013
$849$	0	0
$850$	−2.30975e6	−0.109652
$851$	1.10252e6	0.0521872
$852$	0	0
$853$	6.91398e6	0.325353	0.162677	−	0.986679i	$-0.447987\pi$
$853$	6.91398e6	0.325353	0.162677	+	0.986679i	$0.447987\pi$
$854$	2.64254e6	0.123987
$855$	0	0
$856$	1.29200e6	0.0602666
$857$	2.17289e7	1.01061	0.505307	−	0.862940i	$-0.331379\pi$
$857$	2.17289e7	1.01061	0.505307	+	0.862940i	$0.331379\pi$
$858$	0	0
$859$	−3.40419e7	−1.57409	−0.787047	−	0.616893i	$-0.788391\pi$
$859$	−3.40419e7	−1.57409	−0.787047	+	0.616893i	$0.788391\pi$
$860$	−4.01443e7	−1.85088
$861$	0	0
$862$	3.87987e6	0.177848
$863$	6.21472e6	0.284050	0.142025	−	0.989863i	$-0.454639\pi$
$863$	6.21472e6	0.284050	0.142025	+	0.989863i	$0.454639\pi$
$864$	0	0
$865$	−1.37888e7	−0.626595
$866$	2.19996e6	0.0996829
$867$	0	0
$868$	1.48323e7	0.668205
$869$	−1.95086e6	−0.0876347
$870$	0	0
$871$	376759.	0.0168275
$872$	−7.57641e6	−0.337421
$873$	0	0
$874$	1.37497e6	0.0608856
$875$	−1.86591e7	−0.823894
$876$	0	0
$877$	3.91678e7	1.71961	0.859806	−	0.510621i	$-0.170584\pi$
$877$	3.91678e7	1.71961	0.859806	+	0.510621i	$0.170584\pi$
$878$	1.07981e7	0.472729
$879$	0	0
$880$	2.25124e6	0.0979974
$881$	2.56112e7	1.11171	0.555853	−	0.831280i	$-0.312392\pi$
$881$	2.56112e7	1.11171	0.555853	+	0.831280i	$0.312392\pi$
$882$	0	0
$883$	3.18421e7	1.37436	0.687179	−	0.726488i	$-0.258849\pi$
$883$	3.18421e7	1.37436	0.687179	+	0.726488i	$0.258849\pi$
$884$	−2.16850e6	−0.0933317
$885$	0	0
$886$	−1.02401e7	−0.438247
$887$	−1.94247e7	−0.828981	−0.414490	−	0.910054i	$-0.636040\pi$
$887$	−1.94247e7	−0.828981	−0.414490	+	0.910054i	$0.636040\pi$
$888$	0	0
$889$	−1.43056e7	−0.607088
$890$	−2.50627e6	−0.106060
$891$	0	0
$892$	2.47974e7	1.04350
$893$	2.09844e7	0.880577
$894$	0	0
$895$	2.96459e7	1.23711
$896$	−2.68760e7	−1.11839
$897$	0	0
$898$	−2.61693e6	−0.108293
$899$	−1.16593e7	−0.481144
$900$	0	0
$901$	−8.41270e6	−0.345242
$902$	1.38425e6	0.0566500
$903$	0	0
$904$	−1.68764e6	−0.0686843
$905$	−1.47016e6	−0.0596682
$906$	0	0
$907$	−1.50934e7	−0.609211	−0.304606	−	0.952479i	$-0.598525\pi$
$907$	−1.50934e7	−0.609211	−0.304606	+	0.952479i	$0.598525\pi$
$908$	2.15770e7	0.868512
$909$	0	0
$910$	−1.30026e6	−0.0520506
$911$	−1.64211e7	−0.655551	−0.327776	−	0.944756i	$-0.606299\pi$
$911$	−1.64211e7	−0.655551	−0.327776	+	0.944756i	$0.606299\pi$
$912$	0	0
$913$	5.20578e6	0.206685
$914$	−4.83913e6	−0.191603
$915$	0	0
$916$	−3.36055e7	−1.32334
$917$	3.06682e7	1.20438
$918$	0	0
$919$	1.69013e7	0.660134	0.330067	−	0.943957i	$-0.392929\pi$
$919$	1.69013e7	0.660134	0.330067	+	0.943957i	$0.392929\pi$
$920$	−3.84871e6	−0.149915
$921$	0	0
$922$	−8.21103e6	−0.318105
$923$	2.15231e6	0.0831575
$924$	0	0
$925$	2.55358e6	0.0981284
$926$	−1.34962e7	−0.517232
$927$	0	0
$928$	1.62399e7	0.619031
$929$	4.22301e7	1.60540	0.802699	−	0.596385i	$-0.203397\pi$
$929$	4.22301e7	1.60540	0.802699	+	0.596385i	$0.203397\pi$
$930$	0	0
$931$	7.67772e6	0.290308
$932$	−2.91389e7	−1.09884
$933$	0	0
$934$	−4.07859e6	−0.152983
$935$	−3.25313e6	−0.121695
$936$	0	0
$937$	6.19017e6	0.230332	0.115166	−	0.993346i	$-0.463260\pi$
$937$	6.19017e6	0.230332	0.115166	+	0.993346i	$0.463260\pi$
$938$	−1.39967e6	−0.0519422
$939$	0	0
$940$	−2.77709e7	−1.02511
$941$	2.80154e7	1.03139	0.515694	−	0.856773i	$-0.327534\pi$
$941$	2.80154e7	1.03139	0.515694	+	0.856773i	$0.327534\pi$
$942$	0	0
$943$	8.47433e6	0.310332
$944$	2.77294e7	1.01277
$945$	0	0
$946$	1.83271e6	0.0665834
$947$	−1.07551e7	−0.389707	−0.194853	−	0.980832i	$-0.562423\pi$
$947$	−1.07551e7	−0.389707	−0.194853	+	0.980832i	$0.562423\pi$
$948$	0	0
$949$	3.39542e6	0.122385
$950$	3.18459e6	0.114484
$951$	0	0
$952$	1.70393e7	0.609338
$953$	2.70542e7	0.964944	0.482472	−	0.875911i	$-0.339739\pi$
$953$	2.70542e7	0.964944	0.482472	+	0.875911i	$0.339739\pi$
$954$	0	0
$955$	4.87822e7	1.73083
$956$	1.25406e7	0.443785
$957$	0	0
$958$	−8.21555e6	−0.289216
$959$	−1.91657e7	−0.672943
$960$	0	0
$961$	−1.65824e7	−0.579215
$962$	−275913.	−0.00961245
$963$	0	0
$964$	4.67514e7	1.62032
$965$	5.27833e7	1.82464
$966$	0	0
$967$	−2.85870e7	−0.983109	−0.491555	−	0.870847i	$-0.663571\pi$
$967$	−2.85870e7	−0.983109	−0.491555	+	0.870847i	$0.663571\pi$
$968$	1.75157e7	0.600812
$969$	0	0
$970$	1.30998e7	0.447030
$971$	9.60795e6	0.327026	0.163513	−	0.986541i	$-0.447717\pi$
$971$	9.60795e6	0.327026	0.163513	+	0.986541i	$0.447717\pi$
$972$	0	0
$973$	−5.22241e7	−1.76844
$974$	2.62378e6	0.0886196
$975$	0	0
$976$	7.00951e6	0.235539
$977$	4.49388e7	1.50621	0.753104	−	0.657902i	$-0.228556\pi$
$977$	4.49388e7	1.50621	0.753104	+	0.657902i	$0.228556\pi$
$978$	0	0
$979$	−994191.	−0.0331523
$980$	−1.01608e7	−0.337957
$981$	0	0
$982$	903468.	0.0298974
$983$	3.77962e7	1.24757	0.623783	−	0.781597i	$-0.285595\pi$
$983$	3.77962e7	1.24757	0.623783	+	0.781597i	$0.285595\pi$
$984$	0	0
$985$	−4.00300e7	−1.31460
$986$	−6.33268e6	−0.207441
$987$	0	0
$988$	2.98985e6	0.0974444
$989$	1.12197e7	0.364747
$990$	0	0
$991$	−7.27751e6	−0.235396	−0.117698	−	0.993049i	$-0.537551\pi$
$991$	−7.27751e6	−0.235396	−0.117698	+	0.993049i	$0.537551\pi$
$992$	−1.67794e7	−0.541374
$993$	0	0
$994$	−7.99592e6	−0.256686
$995$	−2.41427e6	−0.0773087
$996$	0	0
$997$	−1.23992e7	−0.395055	−0.197527	−	0.980297i	$-0.563291\pi$
$997$	−1.23992e7	−0.395055	−0.197527	+	0.980297i	$0.563291\pi$
$998$	1.88860e7	0.600226
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.i.1.5	yes	10
3.2	odd	2		207.6.a.h.1.6	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.6	✓	10	3.2	odd	2
207.6.a.i.1.5	yes	10	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+1.81733 q^{2} -28.6973 q^{4} +65.9563 q^{5} +148.913 q^{7} -110.307 q^{8} +O(q^{10})\)	\(q+1.81733 q^{2} -28.6973 q^{4} +65.9563 q^{5} +148.913 q^{7} -110.307 q^{8} +119.864 q^{10} +47.5480 q^{11} -72.8459 q^{13} +270.625 q^{14} +717.849 q^{16} -1037.32 q^{17} +1430.22 q^{19} -1892.77 q^{20} +86.4105 q^{22} +529.000 q^{23} +1225.23 q^{25} -132.385 q^{26} -4273.41 q^{28} +3359.23 q^{29} -3470.84 q^{31} +4834.40 q^{32} -1885.16 q^{34} +9821.77 q^{35} +2084.17 q^{37} +2599.19 q^{38} -7275.45 q^{40} +16019.5 q^{41} +21209.3 q^{43} -1364.50 q^{44} +961.369 q^{46} +14672.1 q^{47} +5368.21 q^{49} +2226.64 q^{50} +2090.48 q^{52} +8110.02 q^{53} +3136.09 q^{55} -16426.2 q^{56} +6104.84 q^{58} +38628.5 q^{59} +9764.60 q^{61} -6307.66 q^{62} -14185.5 q^{64} -4804.64 q^{65} -5172.01 q^{67} +29768.3 q^{68} +17849.4 q^{70} -29546.1 q^{71} -46611.0 q^{73} +3787.62 q^{74} -41043.5 q^{76} +7080.54 q^{77} -41029.2 q^{79} +47346.6 q^{80} +29112.8 q^{82} +109485. q^{83} -68417.8 q^{85} +38544.4 q^{86} -5244.88 q^{88} -20909.2 q^{89} -10847.7 q^{91} -15180.9 q^{92} +26664.1 q^{94} +94332.0 q^{95} +109289. q^{97} +9755.82 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8} - 250 q^{10} + 460 q^{11} + 464 q^{13} + 3676 q^{14} + 4612 q^{16} + 4756 q^{17} - 1780 q^{19} + 10314 q^{20} - 4214 q^{22} + 5290 q^{23} + 1330 q^{25} - 5152 q^{26} + 7072 q^{28} + 4048 q^{29} + 2816 q^{31} + 27436 q^{32} + 420 q^{34} + 9452 q^{35} + 2872 q^{37} + 31038 q^{38} + 2618 q^{40} + 34056 q^{41} + 7316 q^{43} + 33562 q^{44} + 4232 q^{46} + 49300 q^{47} + 45118 q^{49} + 44764 q^{50} - 25120 q^{52} + 86676 q^{53} - 2120 q^{55} + 290684 q^{56} - 87408 q^{58} + 67100 q^{59} - 40432 q^{61} + 230992 q^{62} + 136776 q^{64} + 184000 q^{65} - 50108 q^{67} + 270592 q^{68} + 117456 q^{70} + 238584 q^{71} - 13804 q^{73} + 150074 q^{74} - 197622 q^{76} + 116248 q^{77} - 9228 q^{79} + 313010 q^{80} - 68604 q^{82} + 155300 q^{83} + 80444 q^{85} - 80914 q^{86} - 237738 q^{88} + 213732 q^{89} - 264352 q^{91} + 101568 q^{92} + 140280 q^{94} - 123612 q^{95} + 42516 q^{97} - 151388 q^{98}+O(q^{100}) \) 10 * q + 8 * q^2 + 192 * q^4 + 100 * q^5 + 20 * q^7 + 384 * q^8 - 250 * q^10 + 460 * q^11 + 464 * q^13 + 3676 * q^14 + 4612 * q^16 + 4756 * q^17 - 1780 * q^19 + 10314 * q^20 - 4214 * q^22 + 5290 * q^23 + 1330 * q^25 - 5152 * q^26 + 7072 * q^28 + 4048 * q^29 + 2816 * q^31 + 27436 * q^32 + 420 * q^34 + 9452 * q^35 + 2872 * q^37 + 31038 * q^38 + 2618 * q^40 + 34056 * q^41 + 7316 * q^43 + 33562 * q^44 + 4232 * q^46 + 49300 * q^47 + 45118 * q^49 + 44764 * q^50 - 25120 * q^52 + 86676 * q^53 - 2120 * q^55 + 290684 * q^56 - 87408 * q^58 + 67100 * q^59 - 40432 * q^61 + 230992 * q^62 + 136776 * q^64 + 184000 * q^65 - 50108 * q^67 + 270592 * q^68 + 117456 * q^70 + 238584 * q^71 - 13804 * q^73 + 150074 * q^74 - 197622 * q^76 + 116248 * q^77 - 9228 * q^79 + 313010 * q^80 - 68604 * q^82 + 155300 * q^83 + 80444 * q^85 - 80914 * q^86 - 237738 * q^88 + 213732 * q^89 - 264352 * q^91 + 101568 * q^92 + 140280 * q^94 - 123612 * q^95 + 42516 * q^97 - 151388 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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