LMFDB - Embedded newform 207.6.a.h.1.3

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			1.3
Root			$-7.45301$ 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)

$f(q)$	$=$	$q-8.45301 q^{2} +39.4533 q^{4} +58.5895 q^{5} +96.1665 q^{7} -63.0030 q^{8} +O(q^{10})$	\(q-8.45301 q^{2} +39.4533 q^{4} +58.5895 q^{5} +96.1665 q^{7} -63.0030 q^{8} -495.257 q^{10} -584.970 q^{11} +73.5972 q^{13} -812.896 q^{14} -729.941 q^{16} -1727.70 q^{17} +1751.19 q^{19} +2311.55 q^{20} +4944.76 q^{22} -529.000 q^{23} +307.724 q^{25} -622.118 q^{26} +3794.09 q^{28} -6000.75 q^{29} -776.772 q^{31} +8186.30 q^{32} +14604.2 q^{34} +5634.34 q^{35} +14754.3 q^{37} -14802.8 q^{38} -3691.31 q^{40} +4563.00 q^{41} -9523.25 q^{43} -23079.0 q^{44} +4471.64 q^{46} -14499.0 q^{47} -7559.00 q^{49} -2601.20 q^{50} +2903.66 q^{52} +27236.6 q^{53} -34273.1 q^{55} -6058.78 q^{56} +50724.4 q^{58} -43055.4 q^{59} -31186.6 q^{61} +6566.06 q^{62} -45840.7 q^{64} +4312.02 q^{65} +6634.15 q^{67} -68163.5 q^{68} -47627.1 q^{70} +2599.87 q^{71} -75077.0 q^{73} -124718. q^{74} +69090.2 q^{76} -56254.5 q^{77} +34173.8 q^{79} -42766.9 q^{80} -38571.0 q^{82} -59132.0 q^{83} -101225. q^{85} +80500.1 q^{86} +36854.9 q^{88} +75676.2 q^{89} +7077.59 q^{91} -20870.8 q^{92} +122560. q^{94} +102601. q^{95} -52906.5 q^{97} +63896.3 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.45301	−1.49429	−0.747147	−	0.664658i	$-0.768577\pi$
$2$	−8.45301	−1.49429	−0.747147	+	0.664658i	$0.768577\pi$
$3$	0	0
$3$	0	0
$4$	39.4533	1.23292
$5$	58.5895	1.04808	0.524040	−	0.851694i	$-0.324424\pi$
$5$	58.5895	1.04808	0.524040	+	0.851694i	$0.324424\pi$
$6$	0	0
$7$	96.1665	0.741786	0.370893	−	0.928676i	$-0.379052\pi$
$7$	96.1665	0.741786	0.370893	+	0.928676i	$0.379052\pi$
$8$	−63.0030	−0.348046
$9$	0	0
$10$	−495.257	−1.56614
$11$	−584.970	−1.45765	−0.728823	−	0.684702i	$-0.759932\pi$
$11$	−584.970	−1.45765	−0.728823	+	0.684702i	$0.759932\pi$
$12$	0	0
$13$	73.5972	0.120782	0.0603911	−	0.998175i	$-0.480765\pi$
$13$	73.5972	0.120782	0.0603911	+	0.998175i	$0.480765\pi$
$14$	−812.896	−1.10845
$15$	0	0
$16$	−729.941	−0.712833
$17$	−1727.70	−1.44993	−0.724963	−	0.688788i	$-0.758143\pi$
$17$	−1727.70	−1.44993	−0.724963	+	0.688788i	$0.758143\pi$
$18$	0	0
$19$	1751.19	1.11288	0.556440	−	0.830888i	$-0.312167\pi$
$19$	1751.19	1.11288	0.556440	+	0.830888i	$0.312167\pi$
$20$	2311.55	1.29220
$21$	0	0
$22$	4944.76	2.17815
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	0	0
$25$	307.724	0.0984718
$26$	−622.118	−0.180484
$27$	0	0
$28$	3794.09	0.914560
$29$	−6000.75	−1.32498	−0.662492	−	0.749069i	$-0.730501\pi$
$29$	−6000.75	−1.32498	−0.662492	+	0.749069i	$0.730501\pi$
$30$	0	0
$31$	−776.772	−0.145174	−0.0725871	−	0.997362i	$-0.523126\pi$
$31$	−776.772	−0.145174	−0.0725871	+	0.997362i	$0.523126\pi$
$32$	8186.30	1.41323
$33$	0	0
$34$	14604.2	2.16662
$35$	5634.34	0.777451
$36$	0	0
$37$	14754.3	1.77180	0.885900	−	0.463877i	$-0.153542\pi$
$37$	14754.3	1.77180	0.885900	+	0.463877i	$0.153542\pi$
$38$	−14802.8	−1.66297
$39$	0	0
$40$	−3691.31	−0.364780
$41$	4563.00	0.423926	0.211963	−	0.977278i	$-0.432014\pi$
$41$	4563.00	0.423926	0.211963	+	0.977278i	$0.432014\pi$
$42$	0	0
$43$	−9523.25	−0.785442	−0.392721	−	0.919658i	$-0.628466\pi$
$43$	−9523.25	−0.785442	−0.392721	+	0.919658i	$0.628466\pi$
$44$	−23079.0	−1.79716
$45$	0	0
$46$	4471.64	0.311582
$47$	−14499.0	−0.957401	−0.478701	−	0.877978i	$-0.658892\pi$
$47$	−14499.0	−0.957401	−0.478701	+	0.877978i	$0.658892\pi$
$48$	0	0
$49$	−7559.00	−0.449753
$50$	−2601.20	−0.147146
$51$	0	0
$52$	2903.66	0.148914
$53$	27236.6	1.33187	0.665936	−	0.746009i	$-0.268032\pi$
$53$	27236.6	1.33187	0.665936	+	0.746009i	$0.268032\pi$
$54$	0	0
$55$	−34273.1	−1.52773
$56$	−6058.78	−0.258176
$57$	0	0
$58$	50724.4	1.97992
$59$	−43055.4	−1.61026	−0.805132	−	0.593095i	$-0.797906\pi$
$59$	−43055.4	−1.61026	−0.805132	+	0.593095i	$0.797906\pi$
$60$	0	0
$61$	−31186.6	−1.07311	−0.536553	−	0.843866i	$-0.680274\pi$
$61$	−31186.6	−1.07311	−0.536553	+	0.843866i	$0.680274\pi$
$62$	6566.06	0.216933
$63$	0	0
$64$	−45840.7	−1.39895
$65$	4312.02	0.126589
$66$	0	0
$67$	6634.15	0.180550	0.0902751	−	0.995917i	$-0.471225\pi$
$67$	6634.15	0.180550	0.0902751	+	0.995917i	$0.471225\pi$
$68$	−68163.5	−1.78764
$69$	0	0
$70$	−47627.1	−1.16174
$71$	2599.87	0.0612077	0.0306038	−	0.999532i	$-0.490257\pi$
$71$	2599.87	0.0612077	0.0306038	+	0.999532i	$0.490257\pi$
$72$	0	0
$73$	−75077.0	−1.64892	−0.824460	−	0.565920i	$-0.808521\pi$
$73$	−75077.0	−1.64892	−0.824460	+	0.565920i	$0.808521\pi$
$74$	−124718.	−2.64759
$75$	0	0
$76$	69090.2	1.37209
$77$	−56254.5	−1.08126
$78$	0	0
$79$	34173.8	0.616063	0.308032	−	0.951376i	$-0.400330\pi$
$79$	34173.8	0.616063	0.308032	+	0.951376i	$0.400330\pi$
$80$	−42766.9	−0.747106
$81$	0	0
$82$	−38571.0	−0.633471
$83$	−59132.0	−0.942165	−0.471083	−	0.882089i	$-0.656137\pi$
$83$	−59132.0	−0.942165	−0.471083	+	0.882089i	$0.656137\pi$
$84$	0	0
$85$	−101225.	−1.51964
$86$	80500.1	1.17368
$87$	0	0
$88$	36854.9	0.507328
$89$	75676.2	1.01271	0.506354	−	0.862326i	$-0.330993\pi$
$89$	75676.2	1.01271	0.506354	+	0.862326i	$0.330993\pi$
$90$	0	0
$91$	7077.59	0.0895946
$92$	−20870.8	−0.257081
$93$	0	0
$94$	122560.	1.43064
$95$	102601.	1.16639
$96$	0	0
$97$	−52906.5	−0.570926	−0.285463	−	0.958390i	$-0.592147\pi$
$97$	−52906.5	−0.570926	−0.285463	+	0.958390i	$0.592147\pi$
$98$	63896.3	0.672064
$99$	0	0
$100$	12140.8	0.121408
$101$	−79822.7	−0.778615	−0.389308	−	0.921108i	$-0.627286\pi$
$101$	−79822.7	−0.778615	−0.389308	+	0.921108i	$0.627286\pi$
$102$	0	0
$103$	35685.9	0.331439	0.165720	−	0.986173i	$-0.447005\pi$
$103$	35685.9	0.331439	0.165720	+	0.986173i	$0.447005\pi$
$104$	−4636.85	−0.0420378
$105$	0	0
$106$	−230231.	−1.99021
$107$	74236.8	0.626844	0.313422	−	0.949614i	$-0.398525\pi$
$107$	74236.8	0.626844	0.313422	+	0.949614i	$0.398525\pi$
$108$	0	0
$109$	−154862.	−1.24847	−0.624235	−	0.781237i	$-0.714589\pi$
$109$	−154862.	−1.24847	−0.624235	+	0.781237i	$0.714589\pi$
$110$	289711.	2.28288
$111$	0	0
$112$	−70195.9	−0.528770
$113$	107935.	0.795184	0.397592	−	0.917562i	$-0.369846\pi$
$113$	107935.	0.795184	0.397592	+	0.917562i	$0.369846\pi$
$114$	0	0
$115$	−30993.8	−0.218540
$116$	−236750.	−1.63359
$117$	0	0
$118$	363947.	2.40621
$119$	−166147.	−1.07553
$120$	0	0
$121$	181139.	1.12473
$122$	263620.	1.60354
$123$	0	0
$124$	−30646.2	−0.178988
$125$	−165063.	−0.944874
$126$	0	0
$127$	239945.	1.32009	0.660043	−	0.751227i	$-0.270538\pi$
$127$	239945.	1.32009	0.660043	+	0.751227i	$0.270538\pi$
$128$	125530.	0.677210
$129$	0	0
$130$	−36449.6	−0.189162
$131$	−98222.9	−0.500074	−0.250037	−	0.968236i	$-0.580443\pi$
$131$	−98222.9	−0.500074	−0.250037	+	0.968236i	$0.580443\pi$
$132$	0	0
$133$	168406.	0.825519
$134$	−56078.5	−0.269795
$135$	0	0
$136$	108850.	0.504641
$137$	−251772.	−1.14606	−0.573028	−	0.819536i	$-0.694231\pi$
$137$	−251772.	−1.14606	−0.573028	+	0.819536i	$0.694231\pi$
$138$	0	0
$139$	70139.3	0.307910	0.153955	−	0.988078i	$-0.450799\pi$
$139$	70139.3	0.307910	0.153955	+	0.988078i	$0.450799\pi$
$140$	222294.	0.958532
$141$	0	0
$142$	−21976.7	−0.0914623
$143$	−43052.2	−0.176058
$144$	0	0
$145$	−351581.	−1.38869
$146$	634626.	2.46397
$147$	0	0
$148$	582106.	2.18448
$149$	197150.	0.727497	0.363748	−	0.931497i	$-0.381497\pi$
$149$	197150.	0.727497	0.363748	+	0.931497i	$0.381497\pi$
$150$	0	0
$151$	112495.	0.401504	0.200752	−	0.979642i	$-0.435661\pi$
$151$	112495.	0.401504	0.200752	+	0.979642i	$0.435661\pi$
$152$	−110330.	−0.387334
$153$	0	0
$154$	475520.	1.61572
$155$	−45510.7	−0.152154
$156$	0	0
$157$	−270700.	−0.876473	−0.438236	−	0.898860i	$-0.644397\pi$
$157$	−270700.	−0.876473	−0.438236	+	0.898860i	$0.644397\pi$
$158$	−288871.	−0.920580
$159$	0	0
$160$	479631.	1.48118
$161$	−50872.1	−0.154673
$162$	0	0
$163$	−559097.	−1.64823	−0.824115	−	0.566422i	$-0.808327\pi$
$163$	−559097.	−1.64823	−0.824115	+	0.566422i	$0.808327\pi$
$164$	180025.	0.522666
$165$	0	0
$166$	499843.	1.40787
$167$	−116556.	−0.323402	−0.161701	−	0.986840i	$-0.551698\pi$
$167$	−116556.	−0.323402	−0.161701	+	0.986840i	$0.551698\pi$
$168$	0	0
$169$	−365876.	−0.985412
$170$	855655.	2.27079
$171$	0	0
$172$	−375724.	−0.968384
$173$	5248.07	0.0133317	0.00666583	−	0.999978i	$-0.497878\pi$
$173$	5248.07	0.0133317	0.00666583	+	0.999978i	$0.497878\pi$
$174$	0	0
$175$	29592.8	0.0730450
$176$	426994.	1.03906
$177$	0	0
$178$	−639692.	−1.51328
$179$	−82974.4	−0.193558	−0.0967791	−	0.995306i	$-0.530854\pi$
$179$	−82974.4	−0.193558	−0.0967791	+	0.995306i	$0.530854\pi$
$180$	0	0
$181$	−412192.	−0.935196	−0.467598	−	0.883941i	$-0.654881\pi$
$181$	−412192.	−0.935196	−0.467598	+	0.883941i	$0.654881\pi$
$182$	−59826.9	−0.133881
$183$	0	0
$184$	33328.6	0.0725726
$185$	864447.	1.85699
$186$	0	0
$187$	1.01065e6	2.11348
$188$	−572035.	−1.18040
$189$	0	0
$190$	−867288.	−1.74293
$191$	488800.	0.969500	0.484750	−	0.874653i	$-0.338911\pi$
$191$	488800.	0.969500	0.484750	+	0.874653i	$0.338911\pi$
$192$	0	0
$193$	−451085.	−0.871696	−0.435848	−	0.900020i	$-0.643552\pi$
$193$	−451085.	−0.871696	−0.435848	+	0.900020i	$0.643552\pi$
$194$	447219.	0.853131
$195$	0	0
$196$	−298228.	−0.554508
$197$	147500.	0.270786	0.135393	−	0.990792i	$-0.456770\pi$
$197$	147500.	0.270786	0.135393	+	0.990792i	$0.456770\pi$
$198$	0	0
$199$	−1.00049e6	−1.79094	−0.895471	−	0.445119i	$-0.853161\pi$
$199$	−1.00049e6	−1.79094	−0.895471	+	0.445119i	$0.853161\pi$
$200$	−19387.6	−0.0342727
$201$	0	0
$202$	674742.	1.16348
$203$	−577071.	−0.982855
$204$	0	0
$205$	267343.	0.444309
$206$	−301653.	−0.495268
$207$	0	0
$208$	−53721.7	−0.0860976
$209$	−1.02439e6	−1.62219
$210$	0	0
$211$	432296.	0.668459	0.334229	−	0.942492i	$-0.391524\pi$
$211$	432296.	0.668459	0.334229	+	0.942492i	$0.391524\pi$
$212$	1.07457e6	1.64209
$213$	0	0
$214$	−627524.	−0.936690
$215$	−557962.	−0.823206
$216$	0	0
$217$	−74699.5	−0.107688
$218$	1.30905e6	1.86558
$219$	0	0
$220$	−1.35219e6	−1.88356
$221$	−127154.	−0.175125
$222$	0	0
$223$	−594199.	−0.800147	−0.400074	−	0.916483i	$-0.631015\pi$
$223$	−594199.	−0.800147	−0.400074	+	0.916483i	$0.631015\pi$
$224$	787248.	1.04831
$225$	0	0
$226$	−912378.	−1.18824
$227$	−1.17103e6	−1.50836	−0.754178	−	0.656670i	$-0.771964\pi$
$227$	−1.17103e6	−1.50836	−0.754178	+	0.656670i	$0.771964\pi$
$228$	0	0
$229$	−344889.	−0.434601	−0.217300	−	0.976105i	$-0.569725\pi$
$229$	−344889.	−0.434601	−0.217300	+	0.976105i	$0.569725\pi$
$230$	261991.	0.326563
$231$	0	0
$232$	378066.	0.461155
$233$	−391646.	−0.472611	−0.236305	−	0.971679i	$-0.575937\pi$
$233$	−391646.	−0.472611	−0.236305	+	0.971679i	$0.575937\pi$
$234$	0	0
$235$	−849490.	−1.00343
$236$	−1.69868e6	−1.98532
$237$	0	0
$238$	1.40444e6	1.60717
$239$	−1.12027e6	−1.26861	−0.634307	−	0.773081i	$-0.718714\pi$
$239$	−1.12027e6	−1.26861	−0.634307	+	0.773081i	$0.718714\pi$
$240$	0	0
$241$	1.67780e6	1.86079	0.930393	−	0.366563i	$-0.119465\pi$
$241$	1.67780e6	1.86079	0.930393	+	0.366563i	$0.119465\pi$
$242$	−1.53117e6	−1.68068
$243$	0	0
$244$	−1.23041e6	−1.32305
$245$	−442878.	−0.471378
$246$	0	0
$247$	128883.	0.134416
$248$	48939.0	0.0505273
$249$	0	0
$250$	1.39528e6	1.41192
$251$	909849.	0.911559	0.455780	−	0.890093i	$-0.349361\pi$
$251$	909849.	0.911559	0.455780	+	0.890093i	$0.349361\pi$
$252$	0	0
$253$	309449.	0.303940
$254$	−2.02826e6	−1.97260
$255$	0	0
$256$	405794.	0.386995
$257$	−1.06538e6	−1.00617	−0.503087	−	0.864236i	$-0.667803\pi$
$257$	−1.06538e6	−1.00617	−0.503087	+	0.864236i	$0.667803\pi$
$258$	0	0
$259$	1.41887e6	1.31430
$260$	170124.	0.156074
$261$	0	0
$262$	830279.	0.747258
$263$	1.00701e6	0.897731	0.448865	−	0.893599i	$-0.351828\pi$
$263$	1.00701e6	0.897731	0.448865	+	0.893599i	$0.351828\pi$
$264$	0	0
$265$	1.59577e6	1.39591
$266$	−1.42353e6	−1.23357
$267$	0	0
$268$	261739.	0.222603
$269$	−128632.	−0.108385	−0.0541926	−	0.998531i	$-0.517258\pi$
$269$	−128632.	−0.108385	−0.0541926	+	0.998531i	$0.517258\pi$
$270$	0	0
$271$	−77835.5	−0.0643805	−0.0321903	−	0.999482i	$-0.510248\pi$
$271$	−77835.5	−0.0643805	−0.0321903	+	0.999482i	$0.510248\pi$
$272$	1.26112e6	1.03356
$273$	0	0
$274$	2.12823e6	1.71254
$275$	−180010.	−0.143537
$276$	0	0
$277$	−2.12938e6	−1.66746	−0.833729	−	0.552175i	$-0.813798\pi$
$277$	−2.12938e6	−1.66746	−0.833729	+	0.552175i	$0.813798\pi$
$278$	−592888.	−0.460109
$279$	0	0
$280$	−354981.	−0.270589
$281$	854844.	0.645834	0.322917	−	0.946427i	$-0.395337\pi$
$281$	854844.	0.645834	0.322917	+	0.946427i	$0.395337\pi$
$282$	0	0
$283$	1.47330e6	1.09351	0.546757	−	0.837291i	$-0.315862\pi$
$283$	1.47330e6	1.09351	0.546757	+	0.837291i	$0.315862\pi$
$284$	102574.	0.0754639
$285$	0	0
$286$	363920.	0.263082
$287$	438807.	0.314463
$288$	0	0
$289$	1.56508e6	1.10228
$290$	2.97192e6	2.07511
$291$	0	0
$292$	−2.96204e6	−2.03298
$293$	−2.16980e6	−1.47656	−0.738279	−	0.674496i	$-0.764361\pi$
$293$	−2.16980e6	−1.47656	−0.738279	+	0.674496i	$0.764361\pi$
$294$	0	0
$295$	−2.52259e6	−1.68769
$296$	−929566.	−0.616667
$297$	0	0
$298$	−1.66651e6	−1.08709
$299$	−38932.9	−0.0251848
$300$	0	0
$301$	−915817.	−0.582630
$302$	−950919.	−0.599965
$303$	0	0
$304$	−1.27826e6	−0.793299
$305$	−1.82720e6	−1.12470
$306$	0	0
$307$	1.18362e6	0.716745	0.358373	−	0.933579i	$-0.383332\pi$
$307$	1.18362e6	0.716745	0.358373	+	0.933579i	$0.383332\pi$
$308$	−2.21943e6	−1.33310
$309$	0	0
$310$	384702.	0.227363
$311$	506582.	0.296995	0.148497	−	0.988913i	$-0.452556\pi$
$311$	506582.	0.296995	0.148497	+	0.988913i	$0.452556\pi$
$312$	0	0
$313$	2.89243e6	1.66879	0.834395	−	0.551167i	$-0.185817\pi$
$313$	2.89243e6	1.66879	0.834395	+	0.551167i	$0.185817\pi$
$314$	2.28823e6	1.30971
$315$	0	0
$316$	1.34827e6	0.759555
$317$	2.84932e6	1.59255	0.796276	−	0.604934i	$-0.206801\pi$
$317$	2.84932e6	1.59255	0.796276	+	0.604934i	$0.206801\pi$
$318$	0	0
$319$	3.51026e6	1.93136
$320$	−2.68578e6	−1.46621
$321$	0	0
$322$	430022.	0.231127
$323$	−3.02552e6	−1.61359
$324$	0	0
$325$	22647.7	0.0118936
$326$	4.72605e6	2.46294
$327$	0	0
$328$	−287483.	−0.147546
$329$	−1.39432e6	−0.710187
$330$	0	0
$331$	1.33220e6	0.668341	0.334170	−	0.942513i	$-0.391544\pi$
$331$	1.33220e6	0.668341	0.334170	+	0.942513i	$0.391544\pi$
$332$	−2.33295e6	−1.16161
$333$	0	0
$334$	985248.	0.483259
$335$	388691.	0.189231
$336$	0	0
$337$	358217.	0.171819	0.0859094	−	0.996303i	$-0.472620\pi$
$337$	358217.	0.171819	0.0859094	+	0.996303i	$0.472620\pi$
$338$	3.09276e6	1.47250
$339$	0	0
$340$	−3.99366e6	−1.87359
$341$	454389.	0.211613
$342$	0	0
$343$	−2.34319e6	−1.07541
$344$	599993.	0.273370
$345$	0	0
$346$	−44362.0	−0.0199214
$347$	1.50049e6	0.668973	0.334487	−	0.942401i	$-0.391437\pi$
$347$	1.50049e6	0.668973	0.334487	+	0.942401i	$0.391437\pi$
$348$	0	0
$349$	4.26616e6	1.87488	0.937441	−	0.348144i	$-0.113188\pi$
$349$	4.26616e6	1.87488	0.937441	+	0.348144i	$0.113188\pi$
$350$	−250148.	−0.109151
$351$	0	0
$352$	−4.78874e6	−2.05999
$353$	−2.55317e6	−1.09054	−0.545272	−	0.838259i	$-0.683574\pi$
$353$	−2.55317e6	−1.09054	−0.545272	+	0.838259i	$0.683574\pi$
$354$	0	0
$355$	152325.	0.0641505
$356$	2.98568e6	1.24858
$357$	0	0
$358$	701383.	0.289233
$359$	1.26231e6	0.516927	0.258463	−	0.966021i	$-0.416784\pi$
$359$	1.26231e6	0.516927	0.258463	+	0.966021i	$0.416784\pi$
$360$	0	0
$361$	590558.	0.238503
$362$	3.48426e6	1.39746
$363$	0	0
$364$	279234.	0.110463
$365$	−4.39872e6	−1.72820
$366$	0	0
$367$	3.25576e6	1.26179	0.630896	−	0.775867i	$-0.282687\pi$
$367$	3.25576e6	1.26179	0.630896	+	0.775867i	$0.282687\pi$
$368$	386139.	0.148636
$369$	0	0
$370$	−7.30717e6	−2.77489
$371$	2.61924e6	0.987964
$372$	0	0
$373$	−5.13033e6	−1.90930	−0.954648	−	0.297737i	$-0.903768\pi$
$373$	−5.13033e6	−1.90930	−0.954648	+	0.297737i	$0.903768\pi$
$374$	−8.54305e6	−3.15816
$375$	0	0
$376$	913482.	0.333220
$377$	−441639.	−0.160035
$378$	0	0
$379$	5.44269e6	1.94632	0.973162	−	0.230120i	$-0.0739120\pi$
$379$	5.44269e6	1.94632	0.973162	+	0.230120i	$0.0739120\pi$
$380$	4.04796e6	1.43806
$381$	0	0
$382$	−4.13183e6	−1.44872
$383$	−964316.	−0.335910	−0.167955	−	0.985795i	$-0.553716\pi$
$383$	−964316.	−0.335910	−0.167955	+	0.985795i	$0.553716\pi$
$384$	0	0
$385$	−3.29592e6	−1.13325
$386$	3.81303e6	1.30257
$387$	0	0
$388$	−2.08734e6	−0.703904
$389$	293083.	0.0982010	0.0491005	−	0.998794i	$-0.484365\pi$
$389$	293083.	0.0982010	0.0491005	+	0.998794i	$0.484365\pi$
$390$	0	0
$391$	913952.	0.302330
$392$	476240.	0.156535
$393$	0	0
$394$	−1.24682e6	−0.404635
$395$	2.00222e6	0.645684
$396$	0	0
$397$	−5.21099e6	−1.65937	−0.829687	−	0.558229i	$-0.811481\pi$
$397$	−5.21099e6	−1.65937	−0.829687	+	0.558229i	$0.811481\pi$
$398$	8.45718e6	2.67620
$399$	0	0
$400$	−224621.	−0.0701940
$401$	2.11203e6	0.655903	0.327951	−	0.944695i	$-0.393642\pi$
$401$	2.11203e6	0.655903	0.327951	+	0.944695i	$0.393642\pi$
$402$	0	0
$403$	−57168.3	−0.0175345
$404$	−3.14927e6	−0.959967
$405$	0	0
$406$	4.87799e6	1.46867
$407$	−8.63083e6	−2.58266
$408$	0	0
$409$	590876.	0.174658	0.0873289	−	0.996180i	$-0.472167\pi$
$409$	590876.	0.174658	0.0873289	+	0.996180i	$0.472167\pi$
$410$	−2.25986e6	−0.663928
$411$	0	0
$412$	1.40793e6	0.408637
$413$	−4.14048e6	−1.19447
$414$	0	0
$415$	−3.46451e6	−0.987465
$416$	602489.	0.170693
$417$	0	0
$418$	8.65920e6	2.42402
$419$	6.56754e6	1.82754	0.913772	−	0.406228i	$-0.133156\pi$
$419$	6.56754e6	1.82754	0.913772	+	0.406228i	$0.133156\pi$
$420$	0	0
$421$	−5.72846e6	−1.57519	−0.787594	−	0.616195i	$-0.788673\pi$
$421$	−5.72846e6	−1.57519	−0.787594	+	0.616195i	$0.788673\pi$
$422$	−3.65420e6	−0.998874
$423$	0	0
$424$	−1.71599e6	−0.463553
$425$	−531655.	−0.142777
$426$	0	0
$427$	−2.99910e6	−0.796016
$428$	2.92889e6	0.772847
$429$	0	0
$430$	4.71646e6	1.23011
$431$	5.31923e6	1.37929	0.689645	−	0.724147i	$-0.257767\pi$
$431$	5.31923e6	1.37929	0.689645	+	0.724147i	$0.257767\pi$
$432$	0	0
$433$	−1.53338e6	−0.393033	−0.196517	−	0.980500i	$-0.562963\pi$
$433$	−1.53338e6	−0.393033	−0.196517	+	0.980500i	$0.562963\pi$
$434$	631435.	0.160918
$435$	0	0
$436$	−6.10982e6	−1.53926
$437$	−926378.	−0.232052
$438$	0	0
$439$	−6.07107e6	−1.50350	−0.751751	−	0.659447i	$-0.770790\pi$
$439$	−6.07107e6	−1.50350	−0.751751	+	0.659447i	$0.770790\pi$
$440$	2.15931e6	0.531720
$441$	0	0
$442$	1.07483e6	0.261689
$443$	7.80797e6	1.89029	0.945145	−	0.326650i	$-0.105920\pi$
$443$	7.80797e6	1.89029	0.945145	+	0.326650i	$0.105920\pi$
$444$	0	0
$445$	4.43383e6	1.06140
$446$	5.02277e6	1.19566
$447$	0	0
$448$	−4.40834e6	−1.03772
$449$	3.89523e6	0.911837	0.455918	−	0.890022i	$-0.349311\pi$
$449$	3.89523e6	0.911837	0.455918	+	0.890022i	$0.349311\pi$
$450$	0	0
$451$	−2.66922e6	−0.617935
$452$	4.25841e6	0.980395
$453$	0	0
$454$	9.89874e6	2.25393
$455$	414672.	0.0939023
$456$	0	0
$457$	1.51928e6	0.340289	0.170145	−	0.985419i	$-0.445576\pi$
$457$	1.51928e6	0.340289	0.170145	+	0.985419i	$0.445576\pi$
$458$	2.91535e6	0.649421
$459$	0	0
$460$	−1.22281e6	−0.269441
$461$	811602.	0.177865	0.0889325	−	0.996038i	$-0.471654\pi$
$461$	811602.	0.177865	0.0889325	+	0.996038i	$0.471654\pi$
$462$	0	0
$463$	2.97891e6	0.645810	0.322905	−	0.946431i	$-0.395340\pi$
$463$	2.97891e6	0.645810	0.322905	+	0.946431i	$0.395340\pi$
$464$	4.38020e6	0.944493
$465$	0	0
$466$	3.31058e6	0.706220
$467$	−5.21736e6	−1.10703	−0.553514	−	0.832840i	$-0.686713\pi$
$467$	−5.21736e6	−1.10703	−0.553514	+	0.832840i	$0.686713\pi$
$468$	0	0
$469$	637983.	0.133930
$470$	7.18074e6	1.49943
$471$	0	0
$472$	2.71262e6	0.560446
$473$	5.57081e6	1.14490
$474$	0	0
$475$	538883.	0.109587
$476$	−6.55504e6	−1.32604
$477$	0	0
$478$	9.46968e6	1.89568
$479$	2.82845e6	0.563260	0.281630	−	0.959523i	$-0.409125\pi$
$479$	2.82845e6	0.563260	0.281630	+	0.959523i	$0.409125\pi$
$480$	0	0
$481$	1.08588e6	0.214002
$482$	−1.41824e7	−2.78056
$483$	0	0
$484$	7.14654e6	1.38670
$485$	−3.09976e6	−0.598376
$486$	0	0
$487$	−4.18928e6	−0.800419	−0.400209	−	0.916424i	$-0.631063\pi$
$487$	−4.18928e6	−0.800419	−0.400209	+	0.916424i	$0.631063\pi$
$488$	1.96485e6	0.373490
$489$	0	0
$490$	3.74365e6	0.704377
$491$	1.41765e6	0.265378	0.132689	−	0.991158i	$-0.457639\pi$
$491$	1.41765e6	0.265378	0.132689	+	0.991158i	$0.457639\pi$
$492$	0	0
$493$	1.03675e7	1.92113
$494$	−1.08945e6	−0.200857
$495$	0	0
$496$	566998.	0.103485
$497$	250020.	0.0454030
$498$	0	0
$499$	4.23295e6	0.761013	0.380506	−	0.924778i	$-0.375750\pi$
$499$	4.23295e6	0.761013	0.380506	+	0.924778i	$0.375750\pi$
$500$	−6.51227e6	−1.16495
$501$	0	0
$502$	−7.69096e6	−1.36214
$503$	−8.76497e6	−1.54465	−0.772326	−	0.635227i	$-0.780907\pi$
$503$	−8.76497e6	−1.54465	−0.772326	+	0.635227i	$0.780907\pi$
$504$	0	0
$505$	−4.67677e6	−0.816051
$506$	−2.61578e6	−0.454176
$507$	0	0
$508$	9.46663e6	1.62756
$509$	1.64353e6	0.281179	0.140589	−	0.990068i	$-0.455100\pi$
$509$	1.64353e6	0.281179	0.140589	+	0.990068i	$0.455100\pi$
$510$	0	0
$511$	−7.21989e6	−1.22315
$512$	−7.44715e6	−1.25550
$513$	0	0
$514$	9.00570e6	1.50352
$515$	2.09082e6	0.347375
$516$	0	0
$517$	8.48149e6	1.39555
$518$	−1.19937e7	−1.96395
$519$	0	0
$520$	−271671.	−0.0440590
$521$	8.45825e6	1.36517	0.682584	−	0.730807i	$-0.260856\pi$
$521$	8.45825e6	1.36517	0.682584	+	0.730807i	$0.260856\pi$
$522$	0	0
$523$	−3.54598e6	−0.566867	−0.283434	−	0.958992i	$-0.591474\pi$
$523$	−3.54598e6	−0.566867	−0.283434	+	0.958992i	$0.591474\pi$
$524$	−3.87522e6	−0.616550
$525$	0	0
$526$	−8.51229e6	−1.34147
$527$	1.34203e6	0.210492
$528$	0	0
$529$	279841.	0.0434783
$530$	−1.34891e7	−2.08590
$531$	0	0
$532$	6.64416e6	1.01780
$533$	335824.	0.0512028
$534$	0	0
$535$	4.34949e6	0.656983
$536$	−417971.	−0.0628398
$537$	0	0
$538$	1.08733e6	0.161959
$539$	4.42179e6	0.655581
$540$	0	0
$541$	−2.63550e6	−0.387141	−0.193571	−	0.981086i	$-0.562007\pi$
$541$	−2.63550e6	−0.387141	−0.193571	+	0.981086i	$0.562007\pi$
$542$	657944.	0.0962034
$543$	0	0
$544$	−1.41435e7	−2.04908
$545$	−9.07327e6	−1.30850
$546$	0	0
$547$	−1.18178e7	−1.68876	−0.844378	−	0.535748i	$-0.820030\pi$
$547$	−1.18178e7	−1.68876	−0.844378	+	0.535748i	$0.820030\pi$
$548$	−9.93324e6	−1.41299
$549$	0	0
$550$	1.52162e6	0.214487
$551$	−1.05084e7	−1.47455
$552$	0	0
$553$	3.28637e6	0.456987
$554$	1.79997e7	2.49167
$555$	0	0
$556$	2.76723e6	0.379628
$557$	6.87155e6	0.938462	0.469231	−	0.883075i	$-0.344531\pi$
$557$	6.87155e6	0.938462	0.469231	+	0.883075i	$0.344531\pi$
$558$	0	0
$559$	−700885.	−0.0948674
$560$	−4.11274e6	−0.554193
$561$	0	0
$562$	−7.22600e6	−0.965066
$563$	9.61090e6	1.27789	0.638945	−	0.769253i	$-0.279371\pi$
$563$	9.61090e6	1.27789	0.638945	+	0.769253i	$0.279371\pi$
$564$	0	0
$565$	6.32387e6	0.833416
$566$	−1.24538e7	−1.63403
$567$	0	0
$568$	−163800.	−0.0213031
$569$	−6.67177e6	−0.863894	−0.431947	−	0.901899i	$-0.642173\pi$
$569$	−6.67177e6	−0.863894	−0.431947	+	0.901899i	$0.642173\pi$
$570$	0	0
$571$	−4.42943e6	−0.568535	−0.284268	−	0.958745i	$-0.591750\pi$
$571$	−4.42943e6	−0.568535	−0.284268	+	0.958745i	$0.591750\pi$
$572$	−1.69855e6	−0.217065
$573$	0	0
$574$	−3.70924e6	−0.469900
$575$	−162786.	−0.0205328
$576$	0	0
$577$	5.45140e6	0.681661	0.340831	−	0.940125i	$-0.389292\pi$
$577$	5.45140e6	0.681661	0.340831	+	0.940125i	$0.389292\pi$
$578$	−1.32297e7	−1.64714
$579$	0	0
$580$	−1.38710e7	−1.71214
$581$	−5.68651e6	−0.698885
$582$	0	0
$583$	−1.59326e7	−1.94140
$584$	4.73008e6	0.573900
$585$	0	0
$586$	1.83413e7	2.20641
$587$	−3.28978e6	−0.394068	−0.197034	−	0.980397i	$-0.563131\pi$
$587$	−3.28978e6	−0.394068	−0.197034	+	0.980397i	$0.563131\pi$
$588$	0	0
$589$	−1.36027e6	−0.161562
$590$	2.13235e7	2.52190
$591$	0	0
$592$	−1.07698e7	−1.26300
$593$	171511.	0.0200288	0.0100144	−	0.999950i	$-0.496812\pi$
$593$	171511.	0.0200288	0.0100144	+	0.999950i	$0.496812\pi$
$594$	0	0
$595$	−9.73445e6	−1.12725
$596$	7.77823e6	0.896943
$597$	0	0
$598$	329100.	0.0376336
$599$	224601.	0.0255767	0.0127883	−	0.999918i	$-0.495929\pi$
$599$	224601.	0.0255767	0.0127883	+	0.999918i	$0.495929\pi$
$600$	0	0
$601$	−7.68816e6	−0.868232	−0.434116	−	0.900857i	$-0.642939\pi$
$601$	−7.68816e6	−0.868232	−0.434116	+	0.900857i	$0.642939\pi$
$602$	7.74141e6	0.870620
$603$	0	0
$604$	4.43829e6	0.495021
$605$	1.06128e7	1.17881
$606$	0	0
$607$	−7.16040e6	−0.788798	−0.394399	−	0.918939i	$-0.629047\pi$
$607$	−7.16040e6	−0.788798	−0.394399	+	0.918939i	$0.629047\pi$
$608$	1.43357e7	1.57276
$609$	0	0
$610$	1.54454e7	1.68064
$611$	−1.06709e6	−0.115637
$612$	0	0
$613$	−3.12318e6	−0.335695	−0.167848	−	0.985813i	$-0.553682\pi$
$613$	−3.12318e6	−0.335695	−0.167848	+	0.985813i	$0.553682\pi$
$614$	−1.00051e7	−1.07103
$615$	0	0
$616$	3.54421e6	0.376329
$617$	−1.57456e7	−1.66512	−0.832562	−	0.553932i	$-0.813127\pi$
$617$	−1.57456e7	−1.66512	−0.832562	+	0.553932i	$0.813127\pi$
$618$	0	0
$619$	2.69416e6	0.282616	0.141308	−	0.989966i	$-0.454869\pi$
$619$	2.69416e6	0.282616	0.141308	+	0.989966i	$0.454869\pi$
$620$	−1.79555e6	−0.187593
$621$	0	0
$622$	−4.28214e6	−0.443798
$623$	7.27752e6	0.751213
$624$	0	0
$625$	−1.06326e7	−1.08878
$626$	−2.44497e7	−2.49366
$627$	0	0
$628$	−1.06800e7	−1.08062
$629$	−2.54910e7	−2.56898
$630$	0	0
$631$	3.12671e6	0.312619	0.156309	−	0.987708i	$-0.450040\pi$
$631$	3.12671e6	0.312619	0.156309	+	0.987708i	$0.450040\pi$
$632$	−2.15305e6	−0.214418
$633$	0	0
$634$	−2.40853e7	−2.37974
$635$	1.40582e7	1.38356
$636$	0	0
$637$	−556322.	−0.0543222
$638$	−2.96723e7	−2.88602
$639$	0	0
$640$	7.35475e6	0.709771
$641$	−2.92742e6	−0.281410	−0.140705	−	0.990052i	$-0.544937\pi$
$641$	−2.92742e6	−0.281410	−0.140705	+	0.990052i	$0.544937\pi$
$642$	0	0
$643$	9.39031e6	0.895680	0.447840	−	0.894114i	$-0.352194\pi$
$643$	9.39031e6	0.895680	0.447840	+	0.894114i	$0.352194\pi$
$644$	−2.00707e6	−0.190699
$645$	0	0
$646$	2.55748e7	2.41118
$647$	1.81211e7	1.70186	0.850928	−	0.525282i	$-0.176040\pi$
$647$	1.81211e7	1.70186	0.850928	+	0.525282i	$0.176040\pi$
$648$	0	0
$649$	2.51861e7	2.34720
$650$	−191441.	−0.0177726
$651$	0	0
$652$	−2.20582e7	−2.03213
$653$	2.01939e7	1.85326	0.926632	−	0.375969i	$-0.122690\pi$
$653$	2.01939e7	1.85326	0.926632	+	0.375969i	$0.122690\pi$
$654$	0	0
$655$	−5.75483e6	−0.524118
$656$	−3.33072e6	−0.302189
$657$	0	0
$658$	1.17862e7	1.06123
$659$	1.49732e7	1.34308	0.671538	−	0.740970i	$-0.265634\pi$
$659$	1.49732e7	1.34308	0.671538	+	0.740970i	$0.265634\pi$
$660$	0	0
$661$	−1.40017e7	−1.24646	−0.623230	−	0.782039i	$-0.714180\pi$
$661$	−1.40017e7	−1.24646	−0.623230	+	0.782039i	$0.714180\pi$
$662$	−1.12611e7	−0.998698
$663$	0	0
$664$	3.72549e6	0.327917
$665$	9.86679e6	0.865210
$666$	0	0
$667$	3.17440e6	0.276278
$668$	−4.59852e6	−0.398728
$669$	0	0
$670$	−3.28561e6	−0.282767
$671$	1.82432e7	1.56421
$672$	0	0
$673$	2.52167e6	0.214610	0.107305	−	0.994226i	$-0.465778\pi$
$673$	2.52167e6	0.214610	0.107305	+	0.994226i	$0.465778\pi$
$674$	−3.02801e6	−0.256748
$675$	0	0
$676$	−1.44350e7	−1.21493
$677$	1.64931e7	1.38303	0.691515	−	0.722362i	$-0.256944\pi$
$677$	1.64931e7	1.38303	0.691515	+	0.722362i	$0.256944\pi$
$678$	0	0
$679$	−5.08783e6	−0.423505
$680$	6.37748e6	0.528904
$681$	0	0
$682$	−3.84095e6	−0.316211
$683$	−1.87772e7	−1.54021	−0.770105	−	0.637917i	$-0.779796\pi$
$683$	−1.87772e7	−1.54021	−0.770105	+	0.637917i	$0.779796\pi$
$684$	0	0
$685$	−1.47512e7	−1.20116
$686$	1.98070e7	1.60697
$687$	0	0
$688$	6.95141e6	0.559889
$689$	2.00454e6	0.160867
$690$	0	0
$691$	1.58192e7	1.26034	0.630171	−	0.776457i	$-0.282985\pi$
$691$	1.58192e7	1.26034	0.630171	+	0.776457i	$0.282985\pi$
$692$	207054.	0.0164368
$693$	0	0
$694$	−1.26836e7	−0.999643
$695$	4.10942e6	0.322715
$696$	0	0
$697$	−7.88348e6	−0.614662
$698$	−3.60619e7	−2.80163
$699$	0	0
$700$	1.16753e6	0.0900584
$701$	−1.76636e7	−1.35764	−0.678821	−	0.734304i	$-0.737509\pi$
$701$	−1.76636e7	−1.35764	−0.678821	+	0.734304i	$0.737509\pi$
$702$	0	0
$703$	2.58376e7	1.97180
$704$	2.68154e7	2.03917
$705$	0	0
$706$	2.15820e7	1.62959
$707$	−7.67627e6	−0.577566
$708$	0	0
$709$	1.89757e7	1.41770	0.708848	−	0.705361i	$-0.249215\pi$
$709$	1.89757e7	1.41770	0.708848	+	0.705361i	$0.249215\pi$
$710$	−1.28760e6	−0.0958598
$711$	0	0
$712$	−4.76783e6	−0.352469
$713$	410912.	0.0302709
$714$	0	0
$715$	−2.52240e6	−0.184523
$716$	−3.27362e6	−0.238641
$717$	0	0
$718$	−1.06703e7	−0.772441
$719$	−1.66483e7	−1.20101	−0.600507	−	0.799619i	$-0.705035\pi$
$719$	−1.66483e7	−1.20101	−0.600507	+	0.799619i	$0.705035\pi$
$720$	0	0
$721$	3.43179e6	0.245857
$722$	−4.99199e6	−0.356394
$723$	0	0
$724$	−1.62623e7	−1.15302
$725$	−1.84658e6	−0.130474
$726$	0	0
$727$	−4.80326e6	−0.337055	−0.168527	−	0.985697i	$-0.553901\pi$
$727$	−4.80326e6	−0.337055	−0.168527	+	0.985697i	$0.553901\pi$
$728$	−445910.	−0.0311830
$729$	0	0
$730$	3.71824e7	2.58244
$731$	1.64533e7	1.13883
$732$	0	0
$733$	−1.31833e7	−0.906281	−0.453141	−	0.891439i	$-0.649696\pi$
$733$	−1.31833e7	−0.906281	−0.453141	+	0.891439i	$0.649696\pi$
$734$	−2.75210e7	−1.88549
$735$	0	0
$736$	−4.33055e6	−0.294679
$737$	−3.88078e6	−0.263178
$738$	0	0
$739$	1.56118e7	1.05158	0.525789	−	0.850615i	$-0.323770\pi$
$739$	1.56118e7	1.05158	0.525789	+	0.850615i	$0.323770\pi$
$740$	3.41053e7	2.28951
$741$	0	0
$742$	−2.21405e7	−1.47631
$743$	1.09972e7	0.730819	0.365410	−	0.930847i	$-0.380929\pi$
$743$	1.09972e7	0.730819	0.365410	+	0.930847i	$0.380929\pi$
$744$	0	0
$745$	1.15509e7	0.762475
$746$	4.33667e7	2.85305
$747$	0	0
$748$	3.98736e7	2.60574
$749$	7.13909e6	0.464984
$750$	0	0
$751$	2.86453e7	1.85333	0.926665	−	0.375887i	$-0.122662\pi$
$751$	2.86453e7	1.85333	0.926665	+	0.375887i	$0.122662\pi$
$752$	1.05834e7	0.682468
$753$	0	0
$754$	3.73318e6	0.239139
$755$	6.59101e6	0.420808
$756$	0	0
$757$	−4.44332e6	−0.281817	−0.140909	−	0.990023i	$-0.545002\pi$
$757$	−4.44332e6	−0.281817	−0.140909	+	0.990023i	$0.545002\pi$
$758$	−4.60071e7	−2.90838
$759$	0	0
$760$	−6.46418e6	−0.405957
$761$	−2.42137e7	−1.51565	−0.757825	−	0.652458i	$-0.773738\pi$
$761$	−2.42137e7	−1.51565	−0.757825	+	0.652458i	$0.773738\pi$
$762$	0	0
$763$	−1.48925e7	−0.926098
$764$	1.92848e7	1.19531
$765$	0	0
$766$	8.15137e6	0.501948
$767$	−3.16876e6	−0.194491
$768$	0	0
$769$	521625.	0.0318085	0.0159042	−	0.999874i	$-0.494937\pi$
$769$	521625.	0.0318085	0.0159042	+	0.999874i	$0.494937\pi$
$770$	2.78605e7	1.69341
$771$	0	0
$772$	−1.77968e7	−1.07473
$773$	2.66635e7	1.60498	0.802488	−	0.596668i	$-0.203509\pi$
$773$	2.66635e7	1.60498	0.802488	+	0.596668i	$0.203509\pi$
$774$	0	0
$775$	−239032.	−0.0142956
$776$	3.33327e6	0.198708
$777$	0	0
$778$	−2.47743e6	−0.146741
$779$	7.99066e6	0.471780
$780$	0	0
$781$	−1.52085e6	−0.0892191
$782$	−7.72565e6	−0.451771
$783$	0	0
$784$	5.51763e6	0.320599
$785$	−1.58601e7	−0.918614
$786$	0	0
$787$	1.21659e7	0.700178	0.350089	−	0.936716i	$-0.386151\pi$
$787$	1.21659e7	0.700178	0.350089	+	0.936716i	$0.386151\pi$
$788$	5.81937e6	0.333857
$789$	0	0
$790$	−1.69248e7	−0.964842
$791$	1.03798e7	0.589856
$792$	0	0
$793$	−2.29525e6	−0.129612
$794$	4.40486e7	2.47959
$795$	0	0
$796$	−3.94728e7	−2.20808
$797$	1.55120e7	0.865009	0.432505	−	0.901632i	$-0.357630\pi$
$797$	1.55120e7	0.865009	0.432505	+	0.901632i	$0.357630\pi$
$798$	0	0
$799$	2.50499e7	1.38816
$800$	2.51912e6	0.139163
$801$	0	0
$802$	−1.78530e7	−0.980112
$803$	4.39178e7	2.40354
$804$	0	0
$805$	−2.98057e6	−0.162110
$806$	483244.	0.0262017
$807$	0	0
$808$	5.02907e6	0.270994
$809$	1.70698e7	0.916974	0.458487	−	0.888701i	$-0.348392\pi$
$809$	1.70698e7	0.916974	0.458487	+	0.888701i	$0.348392\pi$
$810$	0	0
$811$	−1.04629e7	−0.558599	−0.279299	−	0.960204i	$-0.590102\pi$
$811$	−1.04629e7	−0.558599	−0.279299	+	0.960204i	$0.590102\pi$
$812$	−2.27674e7	−1.21178
$813$	0	0
$814$	7.29565e7	3.85925
$815$	−3.27572e7	−1.72748
$816$	0	0
$817$	−1.66770e7	−0.874103
$818$	−4.99468e6	−0.260990
$819$	0	0
$820$	1.05476e7	0.547796
$821$	−3.21707e7	−1.66572	−0.832862	−	0.553481i	$-0.813299\pi$
$821$	−3.21707e7	−1.66572	−0.832862	+	0.553481i	$0.813299\pi$
$822$	0	0
$823$	3.42407e7	1.76215	0.881076	−	0.472975i	$-0.156820\pi$
$823$	3.42407e7	1.76215	0.881076	+	0.472975i	$0.156820\pi$
$824$	−2.24832e6	−0.115356
$825$	0	0
$826$	3.49995e7	1.78489
$827$	5.25214e6	0.267038	0.133519	−	0.991046i	$-0.457372\pi$
$827$	5.25214e6	0.267038	0.133519	+	0.991046i	$0.457372\pi$
$828$	0	0
$829$	−7.01444e6	−0.354492	−0.177246	−	0.984167i	$-0.556719\pi$
$829$	−7.01444e6	−0.354492	−0.177246	+	0.984167i	$0.556719\pi$
$830$	2.92855e7	1.47556
$831$	0	0
$832$	−3.37375e6	−0.168968
$833$	1.30597e7	0.652109
$834$	0	0
$835$	−6.82895e6	−0.338952
$836$	−4.04157e7	−2.00002
$837$	0	0
$838$	−5.55155e7	−2.73089
$839$	2.04778e7	1.00434	0.502168	−	0.864770i	$-0.332536\pi$
$839$	2.04778e7	1.00434	0.502168	+	0.864770i	$0.332536\pi$
$840$	0	0
$841$	1.54979e7	0.755583
$842$	4.84227e7	2.35379
$843$	0	0
$844$	1.70555e7	0.824154
$845$	−2.14365e7	−1.03279
$846$	0	0
$847$	1.74195e7	0.834310
$848$	−1.98811e7	−0.949403
$849$	0	0
$850$	4.49408e6	0.213351
$851$	−7.80503e6	−0.369446
$852$	0	0
$853$	−3.67137e6	−0.172765	−0.0863824	−	0.996262i	$-0.527531\pi$
$853$	−3.67137e6	−0.172765	−0.0863824	+	0.996262i	$0.527531\pi$
$854$	2.53514e7	1.18948
$855$	0	0
$856$	−4.67714e6	−0.218171
$857$	−1.65715e7	−0.770741	−0.385371	−	0.922762i	$-0.625926\pi$
$857$	−1.65715e7	−0.770741	−0.385371	+	0.922762i	$0.625926\pi$
$858$	0	0
$859$	−3.17360e7	−1.46747	−0.733734	−	0.679437i	$-0.762224\pi$
$859$	−3.17360e7	−1.46747	−0.733734	+	0.679437i	$0.762224\pi$
$860$	−2.20135e7	−1.01494
$861$	0	0
$862$	−4.49635e7	−2.06107
$863$	1.95147e7	0.891941	0.445970	−	0.895048i	$-0.352859\pi$
$863$	1.95147e7	0.891941	0.445970	+	0.895048i	$0.352859\pi$
$864$	0	0
$865$	307482.	0.0139727
$866$	1.29616e7	0.587308
$867$	0	0
$868$	−2.94714e6	−0.132771
$869$	−1.99906e7	−0.898002
$870$	0	0
$871$	488255.	0.0218073
$872$	9.75677e6	0.434525
$873$	0	0
$874$	7.83068e6	0.346754
$875$	−1.58735e7	−0.700894
$876$	0	0
$877$	−3.49789e7	−1.53570	−0.767852	−	0.640627i	$-0.778674\pi$
$877$	−3.49789e7	−1.53570	−0.767852	+	0.640627i	$0.778674\pi$
$878$	5.13188e7	2.24667
$879$	0	0
$880$	2.50173e7	1.08902
$881$	−1.63333e7	−0.708979	−0.354490	−	0.935060i	$-0.615345\pi$
$881$	−1.63333e7	−0.708979	−0.354490	+	0.935060i	$0.615345\pi$
$882$	0	0
$883$	2.31630e7	0.999752	0.499876	−	0.866097i	$-0.333379\pi$
$883$	2.31630e7	0.999752	0.499876	+	0.866097i	$0.333379\pi$
$884$	−5.01664e6	−0.215915
$885$	0	0
$886$	−6.60008e7	−2.82465
$887$	−1.56922e7	−0.669690	−0.334845	−	0.942273i	$-0.608684\pi$
$887$	−1.56922e7	−0.669690	−0.334845	+	0.942273i	$0.608684\pi$
$888$	0	0
$889$	2.30747e7	0.979222
$890$	−3.74792e7	−1.58604
$891$	0	0
$892$	−2.34431e7	−0.986515
$893$	−2.53905e7	−1.06547
$894$	0	0
$895$	−4.86143e6	−0.202865
$896$	1.20718e7	0.502345
$897$	0	0
$898$	−3.29264e7	−1.36255
$899$	4.66122e6	0.192354
$900$	0	0
$901$	−4.70565e7	−1.93111
$902$	2.25629e7	0.923376
$903$	0	0
$904$	−6.80025e6	−0.276760
$905$	−2.41501e7	−0.980161
$906$	0	0
$907$	−3.19642e7	−1.29017	−0.645083	−	0.764112i	$-0.723177\pi$
$907$	−3.19642e7	−1.29017	−0.645083	+	0.764112i	$0.723177\pi$
$908$	−4.62011e7	−1.85968
$909$	0	0
$910$	−3.50523e6	−0.140318
$911$	−2.11212e7	−0.843186	−0.421593	−	0.906785i	$-0.638529\pi$
$911$	−2.11212e7	−0.843186	−0.421593	+	0.906785i	$0.638529\pi$
$912$	0	0
$913$	3.45904e7	1.37334
$914$	−1.28425e7	−0.508493
$915$	0	0
$916$	−1.36070e7	−0.535826
$917$	−9.44575e6	−0.370948
$918$	0	0
$919$	1.24843e7	0.487613	0.243806	−	0.969824i	$-0.421604\pi$
$919$	1.24843e7	0.487613	0.243806	+	0.969824i	$0.421604\pi$
$920$	1.95270e6	0.0760619
$921$	0	0
$922$	−6.86048e6	−0.265783
$923$	191343.	0.00739280
$924$	0	0
$925$	4.54026e6	0.174472
$926$	−2.51808e7	−0.965031
$927$	0	0
$928$	−4.91239e7	−1.87251
$929$	1.06241e7	0.403880	0.201940	−	0.979398i	$-0.435275\pi$
$929$	1.06241e7	0.403880	0.201940	+	0.979398i	$0.435275\pi$
$930$	0	0
$931$	−1.32372e7	−0.500522
$932$	−1.54517e7	−0.582689
$933$	0	0
$934$	4.41024e7	1.65423
$935$	5.92135e7	2.21509
$936$	0	0
$937$	2.02572e7	0.753754	0.376877	−	0.926263i	$-0.376998\pi$
$937$	2.02572e7	0.753754	0.376877	+	0.926263i	$0.376998\pi$
$938$	−5.39287e6	−0.200130
$939$	0	0
$940$	−3.35152e7	−1.23715
$941$	−3.36641e7	−1.23935	−0.619674	−	0.784859i	$-0.712735\pi$
$941$	−3.36641e7	−1.23935	−0.619674	+	0.784859i	$0.712735\pi$
$942$	0	0
$943$	−2.41383e6	−0.0883948
$944$	3.14279e7	1.14785
$945$	0	0
$946$	−4.70901e7	−1.71081
$947$	1.19381e7	0.432573	0.216287	−	0.976330i	$-0.430605\pi$
$947$	1.19381e7	0.432573	0.216287	+	0.976330i	$0.430605\pi$
$948$	0	0
$949$	−5.52546e6	−0.199160
$950$	−4.55518e6	−0.163756
$951$	0	0
$952$	1.04677e7	0.374335
$953$	−8.76624e6	−0.312666	−0.156333	−	0.987704i	$-0.549967\pi$
$953$	−8.76624e6	−0.312666	−0.156333	+	0.987704i	$0.549967\pi$
$954$	0	0
$955$	2.86385e7	1.01611
$956$	−4.41985e7	−1.56409
$957$	0	0
$958$	−2.39089e7	−0.841677
$959$	−2.42120e7	−0.850128
$960$	0	0
$961$	−2.80258e7	−0.978924
$962$	−9.17892e6	−0.319782
$963$	0	0
$964$	6.61946e7	2.29419
$965$	−2.64288e7	−0.913608
$966$	0	0
$967$	−1.60642e7	−0.552451	−0.276225	−	0.961093i	$-0.589084\pi$
$967$	−1.60642e7	−0.552451	−0.276225	+	0.961093i	$0.589084\pi$
$968$	−1.14123e7	−0.391458
$969$	0	0
$970$	2.62023e7	0.894150
$971$	2.04674e7	0.696650	0.348325	−	0.937374i	$-0.386751\pi$
$971$	2.04674e7	0.696650	0.348325	+	0.937374i	$0.386751\pi$
$972$	0	0
$973$	6.74505e6	0.228404
$974$	3.54120e7	1.19606
$975$	0	0
$976$	2.27644e7	0.764946
$977$	4.90394e6	0.164365	0.0821823	−	0.996617i	$-0.473811\pi$
$977$	4.90394e6	0.164365	0.0821823	+	0.996617i	$0.473811\pi$
$978$	0	0
$979$	−4.42683e7	−1.47617
$980$	−1.74730e7	−0.581169
$981$	0	0
$982$	−1.19834e7	−0.396553
$983$	1.15285e7	0.380531	0.190265	−	0.981733i	$-0.439065\pi$
$983$	1.15285e7	0.380531	0.190265	+	0.981733i	$0.439065\pi$
$984$	0	0
$985$	8.64195e6	0.283806
$986$	−8.76365e7	−2.87073
$987$	0	0
$988$	5.08485e6	0.165724
$989$	5.03780e6	0.163776
$990$	0	0
$991$	4.86112e7	1.57236	0.786180	−	0.617998i	$-0.212056\pi$
$991$	4.86112e7	1.57236	0.786180	+	0.617998i	$0.212056\pi$
$992$	−6.35889e6	−0.205164
$993$	0	0
$994$	−2.11342e6	−0.0678455
$995$	−5.86184e7	−1.87705
$996$	0	0
$997$	−1.56748e7	−0.499416	−0.249708	−	0.968321i	$-0.580335\pi$
$997$	−1.56748e7	−0.499416	−0.249708	+	0.968321i	$0.580335\pi$
$998$	−3.57812e7	−1.13718
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.h.1.3	✓	10
3.2	odd	2		207.6.a.i.1.8	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.3	✓	10	1.1	even	1	trivial
207.6.a.i.1.8	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)

\(f(q)\)	\(=\)	\(q-8.45301 q^{2} +39.4533 q^{4} +58.5895 q^{5} +96.1665 q^{7} -63.0030 q^{8} +O(q^{10})\)	\(q-8.45301 q^{2} +39.4533 q^{4} +58.5895 q^{5} +96.1665 q^{7} -63.0030 q^{8} -495.257 q^{10} -584.970 q^{11} +73.5972 q^{13} -812.896 q^{14} -729.941 q^{16} -1727.70 q^{17} +1751.19 q^{19} +2311.55 q^{20} +4944.76 q^{22} -529.000 q^{23} +307.724 q^{25} -622.118 q^{26} +3794.09 q^{28} -6000.75 q^{29} -776.772 q^{31} +8186.30 q^{32} +14604.2 q^{34} +5634.34 q^{35} +14754.3 q^{37} -14802.8 q^{38} -3691.31 q^{40} +4563.00 q^{41} -9523.25 q^{43} -23079.0 q^{44} +4471.64 q^{46} -14499.0 q^{47} -7559.00 q^{49} -2601.20 q^{50} +2903.66 q^{52} +27236.6 q^{53} -34273.1 q^{55} -6058.78 q^{56} +50724.4 q^{58} -43055.4 q^{59} -31186.6 q^{61} +6566.06 q^{62} -45840.7 q^{64} +4312.02 q^{65} +6634.15 q^{67} -68163.5 q^{68} -47627.1 q^{70} +2599.87 q^{71} -75077.0 q^{73} -124718. q^{74} +69090.2 q^{76} -56254.5 q^{77} +34173.8 q^{79} -42766.9 q^{80} -38571.0 q^{82} -59132.0 q^{83} -101225. q^{85} +80500.1 q^{86} +36854.9 q^{88} +75676.2 q^{89} +7077.59 q^{91} -20870.8 q^{92} +122560. q^{94} +102601. q^{95} -52906.5 q^{97} +63896.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 8 q^{2} + 192 q^{4} - 100 q^{5} + 20 q^{7} - 384 q^{8}+O(q^{10}) \) 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

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Groups

Database

Properties

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