LMFDB - Embedded newform 207.6.a.h.1.6

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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 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			1.6
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	$-0.551353\pi$
$2$	−1.81733	−0.321262	−0.160631	+	0.987015i	$0.551353\pi$
$3$	0	0
$3$	0	0
$4$	−28.6973	−0.896791
$5$	−65.9563	−1.17986	−0.589931	−	0.807454i	$-0.700845\pi$
$5$	−65.9563	−1.17986	−0.589931	+	0.807454i	$0.700845\pi$
$6$	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	$-0.518868\pi$
$11$	−47.5480	−0.118481	−0.0592407	+	0.998244i	$0.518868\pi$
$12$	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	$-0.356652\pi$
$17$	1037.32	0.870545	0.435272	+	0.900299i	$0.356652\pi$
$18$	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	$-0.620939\pi$
$29$	−3359.23	−0.741729	−0.370864	+	0.928687i	$0.620939\pi$
$30$	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	$-0.767145\pi$
$41$	−16019.5	−1.48830	−0.744149	+	0.668014i	$0.767145\pi$
$42$	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	$-0.660968\pi$
$47$	−14672.1	−0.968831	−0.484416	+	0.874838i	$0.660968\pi$
$48$	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	$-0.563539\pi$
$53$	−8110.02	−0.396581	−0.198291	+	0.980143i	$0.563539\pi$
$54$	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	$-0.756938\pi$
$59$	−38628.5	−1.44470	−0.722350	+	0.691528i	$0.756938\pi$
$60$	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	$-0.386930\pi$
$71$	29546.1	0.695592	0.347796	+	0.937570i	$0.386930\pi$
$72$	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	$-0.837323\pi$
$83$	−109485.	−1.74445	−0.872224	+	0.489107i	$0.837323\pi$
$84$	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	$-0.455320\pi$
$89$	20909.2	0.279810	0.139905	+	0.990165i	$0.455320\pi$
$90$	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	$-0.444244\pi$
$101$	35731.6	0.348537	0.174269	+	0.984698i	$0.444244\pi$
$102$	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	$-0.484253\pi$
$107$	11712.7	0.0989005	0.0494502	+	0.998777i	$0.484253\pi$
$108$	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	$-0.517949\pi$
$113$	−15299.4	−0.112714	−0.0563571	+	0.998411i	$0.517949\pi$
$114$	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	$-0.675658\pi$
$131$	−205947.	−1.04852	−0.524260	+	0.851558i	$0.675658\pi$
$132$	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	$-0.405371\pi$
$137$	128704.	0.585854	0.292927	+	0.956135i	$0.405371\pi$
$138$	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	$-0.616766\pi$
$149$	−194391.	−0.717317	−0.358659	+	0.933469i	$0.616766\pi$
$150$	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	$-0.138303\pi$
$167$	653836.	1.81417	0.907085	+	0.420948i	$0.138303\pi$
$168$	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	$-0.414451\pi$
$173$	209060.	0.531075	0.265538	+	0.964101i	$0.414451\pi$
$174$	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	$-0.675658\pi$
$179$	−449478.	−1.04852	−0.524259	+	0.851559i	$0.675658\pi$
$180$	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	$-0.762108\pi$
$191$	−739615.	−1.46697	−0.733487	+	0.679704i	$0.762108\pi$
$192$	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	$-0.311914\pi$
$197$	606917.	1.11420	0.557101	+	0.830445i	$0.311914\pi$
$198$	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	$-0.339099\pi$
$227$	751881.	0.968466	0.484233	+	0.874939i	$0.339099\pi$
$228$	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	$-0.709896\pi$
$233$	−1.01539e6	−1.22530	−0.612649	+	0.790355i	$0.709896\pi$
$234$	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	$-0.420414\pi$
$239$	436994.	0.494859	0.247429	+	0.968906i	$0.420414\pi$
$240$	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	$-0.645442\pi$
$251$	−880717.	−0.882373	−0.441186	+	0.897416i	$0.645442\pi$
$252$	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	$-0.251500\pi$
$257$	1.49036e6	1.40753	0.703767	+	0.710431i	$0.251500\pi$
$258$	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	$-0.444117\pi$
$263$	391849.	0.349325	0.174662	+	0.984628i	$0.444117\pi$
$264$	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	$-0.717993\pi$
$269$	−1.50144e6	−1.26510	−0.632552	+	0.774518i	$0.717993\pi$
$270$	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	$-0.828648\pi$
$281$	−2.27286e6	−1.71714	−0.858572	+	0.512693i	$0.828648\pi$
$282$	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	$-0.846789\pi$
$293$	−2.60507e6	−1.77276	−0.886382	+	0.462955i	$0.846789\pi$
$294$	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	$-0.635929\pi$
$311$	−1.41291e6	−0.828347	−0.414173	+	0.910198i	$0.635929\pi$
$312$	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	$-0.743262\pi$
$317$	−2.47613e6	−1.38396	−0.691981	+	0.721915i	$0.743262\pi$
$318$	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	$-0.356476\pi$
$347$	1.95484e6	0.871540	0.435770	+	0.900058i	$0.356476\pi$
$348$	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	$-0.535735\pi$
$353$	−524557.	−0.224055	−0.112028	+	0.993705i	$0.535735\pi$
$354$	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	$-0.394528\pi$
$359$	1.58882e6	0.650637	0.325319	+	0.945604i	$0.394528\pi$
$360$	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	$-0.195889\pi$
$383$	4.68818e6	1.63308	0.816540	+	0.577289i	$0.195889\pi$
$384$	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	$-0.0938177\pi$
$389$	5.71164e6	1.91376	0.956879	+	0.290488i	$0.0938177\pi$
$390$	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	$-0.561884\pi$
$401$	−1.24416e6	−0.386382	−0.193191	+	0.981161i	$0.561884\pi$
$402$	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	$-0.349848\pi$
$419$	3.26602e6	0.908831	0.454416	+	0.890790i	$0.349848\pi$
$420$	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	$-0.589273\pi$
$431$	−2.13493e6	−0.553592	−0.276796	+	0.960929i	$0.589273\pi$
$432$	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	$-0.261080\pi$
$443$	5.63467e6	1.36414	0.682070	+	0.731286i	$0.261080\pi$
$444$	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	$-0.446094\pi$
$449$	1.43999e6	0.337087	0.168544	+	0.985694i	$0.446094\pi$
$450$	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	$-0.335136\pi$
$461$	4.51818e6	0.990173	0.495086	+	0.868844i	$0.335136\pi$
$462$	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	$-0.423477\pi$
$467$	2.24427e6	0.476193	0.238097	+	0.971241i	$0.423477\pi$
$468$	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	$-0.351379\pi$
$479$	4.52066e6	0.900251	0.450125	+	0.892965i	$0.351379\pi$
$480$	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	$-0.514817\pi$
$491$	−497140.	−0.0930625	−0.0465312	+	0.998917i	$0.514817\pi$
$492$	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	$-0.711415\pi$
$503$	−6.99554e6	−1.23282	−0.616412	+	0.787423i	$0.711415\pi$
$504$	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	$-0.378068\pi$
$509$	4.36936e6	0.747521	0.373760	+	0.927525i	$0.378068\pi$
$510$	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	$-0.538028\pi$
$521$	−1.47687e6	−0.238368	−0.119184	+	0.992872i	$0.538028\pi$
$522$	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	$-0.716526\pi$
$557$	−9.21091e6	−1.25795	−0.628977	+	0.777424i	$0.716526\pi$
$558$	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	$-0.376714\pi$
$563$	5.68136e6	0.755407	0.377704	+	0.925927i	$0.376714\pi$
$564$	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	$-0.294582\pi$
$569$	9.29020e6	1.20294	0.601471	+	0.798895i	$0.294582\pi$
$570$	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	$-0.465795\pi$
$587$	1.79072e6	0.214502	0.107251	+	0.994232i	$0.465795\pi$
$588$	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	$-0.415031\pi$
$593$	4.51759e6	0.527558	0.263779	+	0.964583i	$0.415031\pi$
$594$	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	$-0.770909\pi$
$599$	−1.32073e7	−1.50399	−0.751997	+	0.659166i	$0.770909\pi$
$600$	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	$-0.610079\pi$
$617$	−6.41070e6	−0.677942	−0.338971	+	0.940797i	$0.610079\pi$
$618$	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	$-0.719909\pi$
$641$	−1.32572e7	−1.27441	−0.637203	+	0.770696i	$0.719909\pi$
$642$	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	$-0.650075\pi$
$647$	−9.67249e6	−0.908401	−0.454201	+	0.890899i	$0.650075\pi$
$648$	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	$-0.591478\pi$
$653$	−6.17708e6	−0.566892	−0.283446	+	0.958988i	$0.591478\pi$
$654$	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	$-0.242095\pi$
$659$	1.61529e7	1.44889	0.724447	+	0.689330i	$0.242095\pi$
$660$	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	$-0.111071\pi$
$677$	2.24134e7	1.87947	0.939736	+	0.341901i	$0.111071\pi$
$678$	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	$-0.202140\pi$
$683$	1.96292e7	1.61009	0.805047	+	0.593211i	$0.202140\pi$
$684$	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	$-0.0816953\pi$
$701$	2.51687e7	1.93449	0.967245	+	0.253845i	$0.0816953\pi$
$702$	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	$-0.499538\pi$
$719$	40244.1	0.00290322	0.00145161	+	0.999999i	$0.499538\pi$
$720$	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	$-0.358019\pi$
$743$	1.29833e7	0.862803	0.431402	+	0.902160i	$0.358019\pi$
$744$	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	$-0.334570\pi$
$761$	1.58681e7	0.993261	0.496630	+	0.867962i	$0.334570\pi$
$762$	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	$-0.680304\pi$
$773$	−1.78302e7	−1.07327	−0.536634	+	0.843815i	$0.680304\pi$
$774$	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	$-0.550587\pi$
$797$	−5.67591e6	−0.316512	−0.158256	+	0.987398i	$0.550587\pi$
$798$	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	$-0.420931\pi$
$809$	9.15340e6	0.491713	0.245856	+	0.969306i	$0.420931\pi$
$810$	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	$-0.783669\pi$
$821$	−3.00442e7	−1.55562	−0.777808	+	0.628501i	$0.783669\pi$
$822$	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	$-0.636693\pi$
$827$	−1.63780e7	−0.832714	−0.416357	+	0.909201i	$0.636693\pi$
$828$	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	$-0.161887\pi$
$839$	3.56177e7	1.74687	0.873435	+	0.486941i	$0.161887\pi$
$840$	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	$-0.668621\pi$
$857$	−2.17289e7	−1.01061	−0.505307	+	0.862940i	$0.668621\pi$
$858$	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	$-0.545361\pi$
$863$	−6.21472e6	−0.284050	−0.142025	+	0.989863i	$0.545361\pi$
$864$	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	$-0.687608\pi$
$881$	−2.56112e7	−1.11171	−0.555853	+	0.831280i	$0.687608\pi$
$882$	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	$-0.363960\pi$
$887$	1.94247e7	0.828981	0.414490	+	0.910054i	$0.363960\pi$
$888$	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	$-0.393701\pi$
$911$	1.64211e7	0.655551	0.327776	+	0.944756i	$0.393701\pi$
$912$	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	$-0.796603\pi$
$929$	−4.22301e7	−1.60540	−0.802699	+	0.596385i	$0.796603\pi$
$930$	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	$-0.672466\pi$
$941$	−2.80154e7	−1.03139	−0.515694	+	0.856773i	$0.672466\pi$
$942$	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	$-0.437577\pi$
$947$	1.07551e7	0.389707	0.194853	+	0.980832i	$0.437577\pi$
$948$	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	$-0.660261\pi$
$953$	−2.70542e7	−0.964944	−0.482472	+	0.875911i	$0.660261\pi$
$954$	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	$-0.552283\pi$
$971$	−9.60795e6	−0.327026	−0.163513	+	0.986541i	$0.552283\pi$
$972$	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	$-0.771444\pi$
$977$	−4.49388e7	−1.50621	−0.753104	+	0.657902i	$0.771444\pi$
$978$	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	$-0.714405\pi$
$983$	−3.77962e7	−1.24757	−0.623783	+	0.781597i	$0.714405\pi$
$984$	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.h.1.6	✓	10
3.2	odd	2		207.6.a.i.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.6	✓	10	1.1	even	1	trivial
207.6.a.i.1.5	yes	10	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
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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