LMFDB - Embedded newform 207.6.a.i.1.2

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			1.2
Root			$8.57316$ 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)

$f(q)$	$=$	$q-7.57316 q^{2} +25.3528 q^{4} -21.2577 q^{5} -80.9808 q^{7} +50.3403 q^{8} +O(q^{10})$	\(q-7.57316 q^{2} +25.3528 q^{4} -21.2577 q^{5} -80.9808 q^{7} +50.3403 q^{8} +160.988 q^{10} +209.604 q^{11} -933.213 q^{13} +613.281 q^{14} -1192.53 q^{16} +96.4584 q^{17} +1144.10 q^{19} -538.942 q^{20} -1587.36 q^{22} +529.000 q^{23} -2673.11 q^{25} +7067.37 q^{26} -2053.09 q^{28} -4174.79 q^{29} -4153.45 q^{31} +7420.30 q^{32} -730.495 q^{34} +1721.46 q^{35} -8197.88 q^{37} -8664.49 q^{38} -1070.12 q^{40} -15723.0 q^{41} +6326.82 q^{43} +5314.04 q^{44} -4006.20 q^{46} +14508.1 q^{47} -10249.1 q^{49} +20243.9 q^{50} -23659.6 q^{52} +27096.6 q^{53} -4455.69 q^{55} -4076.60 q^{56} +31616.4 q^{58} +44274.3 q^{59} +45964.9 q^{61} +31454.7 q^{62} -18034.3 q^{64} +19838.0 q^{65} +19548.2 q^{67} +2445.49 q^{68} -13036.9 q^{70} -18406.2 q^{71} +31086.4 q^{73} +62083.9 q^{74} +29006.2 q^{76} -16973.9 q^{77} +15105.6 q^{79} +25350.3 q^{80} +119073. q^{82} +54320.7 q^{83} -2050.48 q^{85} -47914.0 q^{86} +10551.5 q^{88} -14646.8 q^{89} +75572.3 q^{91} +13411.6 q^{92} -109872. q^{94} -24321.0 q^{95} -13876.7 q^{97} +77618.2 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−7.57316	−1.33876	−0.669379	−	0.742921i	$-0.733440\pi$
$2$	−7.57316	−1.33876	−0.669379	+	0.742921i	$0.733440\pi$
$3$	0	0
$3$	0	0
$4$	25.3528	0.792275
$5$	−21.2577	−0.380269	−0.190135	−	0.981758i	$-0.560892\pi$
$5$	−21.2577	−0.380269	−0.190135	+	0.981758i	$0.560892\pi$
$6$	0	0
$7$	−80.9808	−0.624650	−0.312325	−	0.949975i	$-0.601108\pi$
$7$	−80.9808	−0.624650	−0.312325	+	0.949975i	$0.601108\pi$
$8$	50.3403	0.278093
$9$	0	0
$10$	160.988	0.509089
$11$	209.604	0.522296	0.261148	−	0.965299i	$-0.415899\pi$
$11$	209.604	0.522296	0.261148	+	0.965299i	$0.415899\pi$
$12$	0	0
$13$	−933.213	−1.53152	−0.765759	−	0.643127i	$-0.777637\pi$
$13$	−933.213	−1.53152	−0.765759	+	0.643127i	$0.777637\pi$
$14$	613.281	0.836256
$15$	0	0
$16$	−1192.53	−1.16458
$17$	96.4584	0.0809502	0.0404751	−	0.999181i	$-0.487113\pi$
$17$	96.4584	0.0809502	0.0404751	+	0.999181i	$0.487113\pi$
$18$	0	0
$19$	1144.10	0.727079	0.363539	−	0.931579i	$-0.381568\pi$
$19$	1144.10	0.727079	0.363539	+	0.931579i	$0.381568\pi$
$20$	−538.942	−0.301278
$21$	0	0
$22$	−1587.36	−0.699229
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−2673.11	−0.855395
$26$	7067.37	2.05033
$27$	0	0
$28$	−2053.09	−0.494895
$29$	−4174.79	−0.921807	−0.460903	−	0.887450i	$-0.652475\pi$
$29$	−4174.79	−0.921807	−0.460903	+	0.887450i	$0.652475\pi$
$30$	0	0
$31$	−4153.45	−0.776255	−0.388128	−	0.921606i	$-0.626878\pi$
$31$	−4153.45	−0.776255	−0.388128	+	0.921606i	$0.626878\pi$
$32$	7420.30	1.28099
$33$	0	0
$34$	−730.495	−0.108373
$35$	1721.46	0.237535
$36$	0	0
$37$	−8197.88	−0.984459	−0.492229	−	0.870466i	$-0.663818\pi$
$37$	−8197.88	−0.984459	−0.492229	+	0.870466i	$0.663818\pi$
$38$	−8664.49	−0.973383
$39$	0	0
$40$	−1070.12	−0.105750
$41$	−15723.0	−1.46075	−0.730373	−	0.683049i	$-0.760654\pi$
$41$	−15723.0	−1.46075	−0.730373	+	0.683049i	$0.760654\pi$
$42$	0	0
$43$	6326.82	0.521812	0.260906	−	0.965364i	$-0.415979\pi$
$43$	6326.82	0.521812	0.260906	+	0.965364i	$0.415979\pi$
$44$	5314.04	0.413802
$45$	0	0
$46$	−4006.20	−0.279151
$47$	14508.1	0.958002	0.479001	−	0.877814i	$-0.340999\pi$
$47$	14508.1	0.958002	0.479001	+	0.877814i	$0.340999\pi$
$48$	0	0
$49$	−10249.1	−0.609812
$50$	20243.9	1.14517
$51$	0	0
$52$	−23659.6	−1.21338
$53$	27096.6	1.32503	0.662514	−	0.749050i	$-0.269490\pi$
$53$	27096.6	1.32503	0.662514	+	0.749050i	$0.269490\pi$
$54$	0	0
$55$	−4455.69	−0.198613
$56$	−4076.60	−0.173711
$57$	0	0
$58$	31616.4	1.23408
$59$	44274.3	1.65585	0.827927	−	0.560836i	$-0.189520\pi$
$59$	44274.3	1.65585	0.827927	+	0.560836i	$0.189520\pi$
$60$	0	0
$61$	45964.9	1.58162	0.790809	−	0.612062i	$-0.209660\pi$
$61$	45964.9	1.58162	0.790809	+	0.612062i	$0.209660\pi$
$62$	31454.7	1.03922
$63$	0	0
$64$	−18034.3	−0.550364
$65$	19838.0	0.582389
$66$	0	0
$67$	19548.2	0.532009	0.266005	−	0.963972i	$-0.414296\pi$
$67$	19548.2	0.532009	0.266005	+	0.963972i	$0.414296\pi$
$68$	2445.49	0.0641348
$69$	0	0
$70$	−13036.9	−0.318002
$71$	−18406.2	−0.433331	−0.216665	−	0.976246i	$-0.569518\pi$
$71$	−18406.2	−0.433331	−0.216665	+	0.976246i	$0.569518\pi$
$72$	0	0
$73$	31086.4	0.682753	0.341376	−	0.939927i	$-0.389107\pi$
$73$	31086.4	0.682753	0.341376	+	0.939927i	$0.389107\pi$
$74$	62083.9	1.31795
$75$	0	0
$76$	29006.2	0.576046
$77$	−16973.9	−0.326252
$78$	0	0
$79$	15105.6	0.272314	0.136157	−	0.990687i	$-0.456525\pi$
$79$	15105.6	0.272314	0.136157	+	0.990687i	$0.456525\pi$
$80$	25350.3	0.442852
$81$	0	0
$82$	119073.	1.95559
$83$	54320.7	0.865506	0.432753	−	0.901512i	$-0.357542\pi$
$83$	54320.7	0.865506	0.432753	+	0.901512i	$0.357542\pi$
$84$	0	0
$85$	−2050.48	−0.0307829
$86$	−47914.0	−0.698581
$87$	0	0
$88$	10551.5	0.145247
$89$	−14646.8	−0.196005	−0.0980025	−	0.995186i	$-0.531245\pi$
$89$	−14646.8	−0.196005	−0.0980025	+	0.995186i	$0.531245\pi$
$90$	0	0
$91$	75572.3	0.956663
$92$	13411.6	0.165201
$93$	0	0
$94$	−109872.	−1.28253
$95$	−24321.0	−0.276486
$96$	0	0
$97$	−13876.7	−0.149747	−0.0748735	−	0.997193i	$-0.523855\pi$
$97$	−13876.7	−0.149747	−0.0748735	+	0.997193i	$0.523855\pi$
$98$	77618.2	0.816392
$99$	0	0
$100$	−67770.9	−0.677709
$101$	−52421.5	−0.511336	−0.255668	−	0.966765i	$-0.582295\pi$
$101$	−52421.5	−0.511336	−0.255668	+	0.966765i	$0.582295\pi$
$102$	0	0
$103$	153395.	1.42468	0.712341	−	0.701834i	$-0.247635\pi$
$103$	153395.	1.42468	0.712341	+	0.701834i	$0.247635\pi$
$104$	−46978.2	−0.425905
$105$	0	0
$106$	−205207.	−1.77389
$107$	73593.2	0.621410	0.310705	−	0.950506i	$-0.399435\pi$
$107$	73593.2	0.621410	0.310705	+	0.950506i	$0.399435\pi$
$108$	0	0
$109$	143402.	1.15608	0.578042	−	0.816007i	$-0.303817\pi$
$109$	143402.	1.15608	0.578042	+	0.816007i	$0.303817\pi$
$110$	33743.6	0.265895
$111$	0	0
$112$	96571.6	0.727452
$113$	−5330.48	−0.0392708	−0.0196354	−	0.999807i	$-0.506251\pi$
$113$	−5330.48	−0.0392708	−0.0196354	+	0.999807i	$0.506251\pi$
$114$	0	0
$115$	−11245.3	−0.0792916
$116$	−105843.	−0.730325
$117$	0	0
$118$	−335297.	−2.21679
$119$	−7811.28	−0.0505655
$120$	0	0
$121$	−117117.	−0.727207
$122$	−348100.	−2.11741
$123$	0	0
$124$	−105302.	−0.615008
$125$	123254.	0.705550
$126$	0	0
$127$	51657.8	0.284202	0.142101	−	0.989852i	$-0.454614\pi$
$127$	51657.8	0.284202	0.142101	+	0.989852i	$0.454614\pi$
$128$	−100873.	−0.544187
$129$	0	0
$130$	−150236.	−0.779679
$131$	−129442.	−0.659019	−0.329509	−	0.944152i	$-0.606883\pi$
$131$	−129442.	−0.659019	−0.329509	+	0.944152i	$0.606883\pi$
$132$	0	0
$133$	−92650.4	−0.454170
$134$	−148041.	−0.712232
$135$	0	0
$136$	4855.75	0.0225117
$137$	396785.	1.80615	0.903074	−	0.429485i	$-0.141305\pi$
$137$	396785.	1.80615	0.903074	+	0.429485i	$0.141305\pi$
$138$	0	0
$139$	−53668.6	−0.235604	−0.117802	−	0.993037i	$-0.537585\pi$
$139$	−53668.6	−0.235604	−0.117802	+	0.993037i	$0.537585\pi$
$140$	43643.9	0.188193
$141$	0	0
$142$	139394.	0.580125
$143$	−195605.	−0.799907
$144$	0	0
$145$	88746.5	0.350535
$146$	−235422.	−0.914041
$147$	0	0
$148$	−207839.	−0.779962
$149$	80528.9	0.297157	0.148579	−	0.988901i	$-0.452530\pi$
$149$	80528.9	0.297157	0.148579	+	0.988901i	$0.452530\pi$
$150$	0	0
$151$	1848.48	0.00659741	0.00329871	−	0.999995i	$-0.498950\pi$
$151$	1848.48	0.00659741	0.00329871	+	0.999995i	$0.498950\pi$
$152$	57594.5	0.202196
$153$	0	0
$154$	128546.	0.436773
$155$	88292.7	0.295186
$156$	0	0
$157$	−402136.	−1.30204	−0.651019	−	0.759062i	$-0.725658\pi$
$157$	−402136.	−1.30204	−0.651019	+	0.759062i	$0.725658\pi$
$158$	−114397.	−0.364562
$159$	0	0
$160$	−157738.	−0.487122
$161$	−42838.8	−0.130249
$162$	0	0
$163$	80663.5	0.237798	0.118899	−	0.992906i	$-0.462064\pi$
$163$	80663.5	0.237798	0.118899	+	0.992906i	$0.462064\pi$
$164$	−398621.	−1.15731
$165$	0	0
$166$	−411380.	−1.15870
$167$	−455044.	−1.26259	−0.631294	−	0.775543i	$-0.717476\pi$
$167$	−455044.	−1.26259	−0.631294	+	0.775543i	$0.717476\pi$
$168$	0	0
$169$	499593.	1.34555
$170$	15528.6	0.0412108
$171$	0	0
$172$	160403.	0.413419
$173$	280.033	0.000711367 0	0.000355684	−	1.00000i	$-0.499887\pi$
$173$	280.033	0.000711367 0	0.000355684		1.00000i	$0.499887\pi$
$174$	0	0
$175$	216471.	0.534323
$176$	−249957.	−0.608253
$177$	0	0
$178$	110922.	0.262403
$179$	−269652.	−0.629030	−0.314515	−	0.949252i	$-0.601842\pi$
$179$	−269652.	−0.629030	−0.314515	+	0.949252i	$0.601842\pi$
$180$	0	0
$181$	287898.	0.653193	0.326597	−	0.945164i	$-0.394098\pi$
$181$	287898.	0.653193	0.326597	+	0.945164i	$0.394098\pi$
$182$	−572321.	−1.28074
$183$	0	0
$184$	26630.0	0.0579865
$185$	174268.	0.374359
$186$	0	0
$187$	20218.0	0.0422800
$188$	367821.	0.759001
$189$	0	0
$190$	184187.	0.370148
$191$	315192.	0.625160	0.312580	−	0.949891i	$-0.398807\pi$
$191$	315192.	0.625160	0.312580	+	0.949891i	$0.398807\pi$
$192$	0	0
$193$	−115756.	−0.223691	−0.111846	−	0.993726i	$-0.535676\pi$
$193$	−115756.	−0.223691	−0.111846	+	0.993726i	$0.535676\pi$
$194$	105091.	0.200475
$195$	0	0
$196$	−259844.	−0.483139
$197$	17669.6	0.0324386	0.0162193	−	0.999868i	$-0.494837\pi$
$197$	17669.6	0.0324386	0.0162193	+	0.999868i	$0.494837\pi$
$198$	0	0
$199$	64634.2	0.115699	0.0578495	−	0.998325i	$-0.481576\pi$
$199$	64634.2	0.115699	0.0578495	+	0.998325i	$0.481576\pi$
$200$	−134565.	−0.237880
$201$	0	0
$202$	396997.	0.684556
$203$	338078.	0.575807
$204$	0	0
$205$	334234.	0.555476
$206$	−1.16168e6	−1.90731
$207$	0	0
$208$	1.11288e6	1.78357
$209$	239808.	0.379751
$210$	0	0
$211$	861368.	1.33193	0.665967	−	0.745981i	$-0.268019\pi$
$211$	861368.	1.33193	0.665967	+	0.745981i	$0.268019\pi$
$212$	686974.	1.04979
$213$	0	0
$214$	−557333.	−0.831918
$215$	−134494.	−0.198429
$216$	0	0
$217$	336349.	0.484888
$218$	−1.08601e6	−1.54772
$219$	0	0
$220$	−112964.	−0.157356
$221$	−90016.3	−0.123977
$222$	0	0
$223$	−53718.9	−0.0723378	−0.0361689	−	0.999346i	$-0.511515\pi$
$223$	−53718.9	−0.0723378	−0.0361689	+	0.999346i	$0.511515\pi$
$224$	−600901.	−0.800172
$225$	0	0
$226$	40368.6	0.0525741
$227$	443031.	0.570650	0.285325	−	0.958431i	$-0.407899\pi$
$227$	443031.	0.570650	0.285325	+	0.958431i	$0.407899\pi$
$228$	0	0
$229$	−1.36735e6	−1.72302	−0.861512	−	0.507737i	$-0.830482\pi$
$229$	−1.36735e6	−1.72302	−0.861512	+	0.507737i	$0.830482\pi$
$230$	85162.6	0.106152
$231$	0	0
$232$	−210160.	−0.256348
$233$	1.20882e6	1.45871	0.729357	−	0.684133i	$-0.239819\pi$
$233$	1.20882e6	1.45871	0.729357	+	0.684133i	$0.239819\pi$
$234$	0	0
$235$	−308409.	−0.364298
$236$	1.12248e6	1.31189
$237$	0	0
$238$	59156.1	0.0676951
$239$	394437.	0.446666	0.223333	−	0.974742i	$-0.428306\pi$
$239$	394437.	0.446666	0.223333	+	0.974742i	$0.428306\pi$
$240$	0	0
$241$	1.06705e6	1.18343	0.591713	−	0.806149i	$-0.298452\pi$
$241$	1.06705e6	1.18343	0.591713	+	0.806149i	$0.298452\pi$
$242$	886949.	0.973554
$243$	0	0
$244$	1.16534e6	1.25308
$245$	217873.	0.231893
$246$	0	0
$247$	−1.06769e6	−1.11353
$248$	−209086.	−0.215871
$249$	0	0
$250$	−933426.	−0.944561
$251$	1.80953e6	1.81294	0.906468	−	0.422275i	$-0.138768\pi$
$251$	1.80953e6	1.81294	0.906468	+	0.422275i	$0.138768\pi$
$252$	0	0
$253$	110880.	0.108906
$254$	−391213.	−0.380478
$255$	0	0
$256$	1.34102e6	1.27890
$257$	1.83184e6	1.73003	0.865015	−	0.501746i	$-0.167309\pi$
$257$	1.83184e6	1.73003	0.865015	+	0.501746i	$0.167309\pi$
$258$	0	0
$259$	663871.	0.614942
$260$	502948.	0.461413
$261$	0	0
$262$	980287.	0.882267
$263$	−141084.	−0.125773	−0.0628866	−	0.998021i	$-0.520031\pi$
$263$	−141084.	−0.125773	−0.0628866	+	0.998021i	$0.520031\pi$
$264$	0	0
$265$	−576011.	−0.503867
$266$	701657.	0.608024
$267$	0	0
$268$	495601.	0.421498
$269$	−246901.	−0.208038	−0.104019	−	0.994575i	$-0.533170\pi$
$269$	−246901.	−0.208038	−0.104019	+	0.994575i	$0.533170\pi$
$270$	0	0
$271$	−862954.	−0.713780	−0.356890	−	0.934146i	$-0.616163\pi$
$271$	−862954.	−0.713780	−0.356890	+	0.934146i	$0.616163\pi$
$272$	−115029.	−0.0942726
$273$	0	0
$274$	−3.00491e6	−2.41800
$275$	−560293.	−0.446770
$276$	0	0
$277$	−1.50275e6	−1.17676	−0.588379	−	0.808585i	$-0.700234\pi$
$277$	−1.50275e6	−1.17676	−0.588379	+	0.808585i	$0.700234\pi$
$278$	406441.	0.315417
$279$	0	0
$280$	86659.0	0.0660570
$281$	−335120.	−0.253183	−0.126592	−	0.991955i	$-0.540404\pi$
$281$	−335120.	−0.253183	−0.126592	+	0.991955i	$0.540404\pi$
$282$	0	0
$283$	−2.42745e6	−1.80171	−0.900855	−	0.434120i	$-0.857059\pi$
$283$	−2.42745e6	−1.80171	−0.900855	+	0.434120i	$0.857059\pi$
$284$	−466650.	−0.343317
$285$	0	0
$286$	1.48135e6	1.07088
$287$	1.27326e6	0.912455
$288$	0	0
$289$	−1.41055e6	−0.993447
$290$	−672091.	−0.469281
$291$	0	0
$292$	788128.	0.540928
$293$	1.03243e6	0.702575	0.351287	−	0.936268i	$-0.385744\pi$
$293$	1.03243e6	0.702575	0.351287	+	0.936268i	$0.385744\pi$
$294$	0	0
$295$	−941170.	−0.629670
$296$	−412684.	−0.273771
$297$	0	0
$298$	−609859.	−0.397822
$299$	−493670.	−0.319344
$300$	0	0
$301$	−512351.	−0.325950
$302$	−13998.9	−0.00883234
$303$	0	0
$304$	−1.36437e6	−0.846738
$305$	−977108.	−0.601441
$306$	0	0
$307$	2.27462e6	1.37741	0.688704	−	0.725043i	$-0.258180\pi$
$307$	2.27462e6	1.37741	0.688704	+	0.725043i	$0.258180\pi$
$308$	−430335.	−0.258482
$309$	0	0
$310$	−668655.	−0.395183
$311$	−878781.	−0.515204	−0.257602	−	0.966251i	$-0.582932\pi$
$311$	−878781.	−0.515204	−0.257602	+	0.966251i	$0.582932\pi$
$312$	0	0
$313$	−1.81549e6	−1.04745	−0.523725	−	0.851887i	$-0.675458\pi$
$313$	−1.81549e6	−1.04745	−0.523725	+	0.851887i	$0.675458\pi$
$314$	3.04544e6	1.74311
$315$	0	0
$316$	382969.	0.215747
$317$	748185.	0.418178	0.209089	−	0.977897i	$-0.432950\pi$
$317$	748185.	0.418178	0.209089	+	0.977897i	$0.432950\pi$
$318$	0	0
$319$	−875051.	−0.481456
$320$	383368.	0.209286
$321$	0	0
$322$	324425.	0.174371
$323$	110358.	0.0588572
$324$	0	0
$325$	2.49458e6	1.31005
$326$	−610878.	−0.318354
$327$	0	0
$328$	−791498.	−0.406224
$329$	−1.17488e6	−0.598416
$330$	0	0
$331$	−3.22107e6	−1.61596	−0.807978	−	0.589212i	$-0.799438\pi$
$331$	−3.22107e6	−1.61596	−0.807978	+	0.589212i	$0.799438\pi$
$332$	1.37718e6	0.685719
$333$	0	0
$334$	3.44612e6	1.69030
$335$	−415549.	−0.202307
$336$	0	0
$337$	30999.0	0.0148687	0.00743435	−	0.999972i	$-0.497634\pi$
$337$	30999.0	0.0148687	0.00743435	+	0.999972i	$0.497634\pi$
$338$	−3.78350e6	−1.80137
$339$	0	0
$340$	−51985.5	−0.0243885
$341$	−870577.	−0.405435
$342$	0	0
$343$	2.19103e6	1.00557
$344$	318494.	0.145113
$345$	0	0
$346$	−2120.74	−0.000952349 0
$347$	3.88343e6	1.73138	0.865690	−	0.500581i	$-0.166880\pi$
$347$	3.88343e6	1.73138	0.865690	+	0.500581i	$0.166880\pi$
$348$	0	0
$349$	1.14758e6	0.504336	0.252168	−	0.967683i	$-0.418856\pi$
$349$	1.14758e6	0.504336	0.252168	+	0.967683i	$0.418856\pi$
$350$	−1.63937e6	−0.715329
$351$	0	0
$352$	1.55532e6	0.669057
$353$	−713904.	−0.304932	−0.152466	−	0.988309i	$-0.548721\pi$
$353$	−713904.	−0.304932	−0.152466	+	0.988309i	$0.548721\pi$
$354$	0	0
$355$	391274.	0.164782
$356$	−371337.	−0.155290
$357$	0	0
$358$	2.04212e6	0.842120
$359$	3.08625e6	1.26385	0.631925	−	0.775030i	$-0.282265\pi$
$359$	3.08625e6	1.26385	0.631925	+	0.775030i	$0.282265\pi$
$360$	0	0
$361$	−1.16713e6	−0.471357
$362$	−2.18030e6	−0.874468
$363$	0	0
$364$	1.91597e6	0.757941
$365$	−660825.	−0.259630
$366$	0	0
$367$	1.40740e6	0.545449	0.272724	−	0.962092i	$-0.412075\pi$
$367$	1.40740e6	0.545449	0.272724	+	0.962092i	$0.412075\pi$
$368$	−630846.	−0.242831
$369$	0	0
$370$	−1.31976e6	−0.501177
$371$	−2.19430e6	−0.827678
$372$	0	0
$373$	−2.14111e6	−0.796834	−0.398417	−	0.917204i	$-0.630440\pi$
$373$	−2.14111e6	−0.796834	−0.398417	+	0.917204i	$0.630440\pi$
$374$	−153114.	−0.0566027
$375$	0	0
$376$	730343.	0.266414
$377$	3.89597e6	1.41176
$378$	0	0
$379$	4.20555e6	1.50392	0.751961	−	0.659208i	$-0.229108\pi$
$379$	4.20555e6	1.50392	0.751961	+	0.659208i	$0.229108\pi$
$380$	−616606.	−0.219053
$381$	0	0
$382$	−2.38700e6	−0.836939
$383$	1.56205e6	0.544125	0.272063	−	0.962280i	$-0.412294\pi$
$383$	1.56205e6	0.544125	0.272063	+	0.962280i	$0.412294\pi$
$384$	0	0
$385$	360825.	0.124064
$386$	876637.	0.299469
$387$	0	0
$388$	−351814.	−0.118641
$389$	−5.33789e6	−1.78853	−0.894264	−	0.447539i	$-0.852301\pi$
$389$	−5.33789e6	−1.78853	−0.894264	+	0.447539i	$0.852301\pi$
$390$	0	0
$391$	51026.5	0.0168793
$392$	−515943.	−0.169585
$393$	0	0
$394$	−133815.	−0.0434275
$395$	−321110.	−0.103553
$396$	0	0
$397$	4.11086e6	1.30905	0.654525	−	0.756040i	$-0.272869\pi$
$397$	4.11086e6	1.30905	0.654525	+	0.756040i	$0.272869\pi$
$398$	−489486.	−0.154893
$399$	0	0
$400$	3.18775e6	0.996172
$401$	5.39510e6	1.67548	0.837738	−	0.546072i	$-0.183878\pi$
$401$	5.39510e6	1.67548	0.837738	+	0.546072i	$0.183878\pi$
$402$	0	0
$403$	3.87605e6	1.18885
$404$	−1.32903e6	−0.405119
$405$	0	0
$406$	−2.56032e6	−0.770866
$407$	−1.71831e6	−0.514179
$408$	0	0
$409$	−323949.	−0.0957564	−0.0478782	−	0.998853i	$-0.515246\pi$
$409$	−323949.	−0.0957564	−0.0478782	+	0.998853i	$0.515246\pi$
$410$	−2.53121e6	−0.743649
$411$	0	0
$412$	3.88899e6	1.12874
$413$	−3.58537e6	−1.03433
$414$	0	0
$415$	−1.15473e6	−0.329125
$416$	−6.92472e6	−1.96186
$417$	0	0
$418$	−1.81611e6	−0.508394
$419$	2.49268e6	0.693636	0.346818	−	0.937933i	$-0.387262\pi$
$419$	2.49268e6	0.693636	0.346818	+	0.937933i	$0.387262\pi$
$420$	0	0
$421$	−822418.	−0.226145	−0.113073	−	0.993587i	$-0.536069\pi$
$421$	−822418.	−0.226145	−0.113073	+	0.993587i	$0.536069\pi$
$422$	−6.52328e6	−1.78314
$423$	0	0
$424$	1.36405e6	0.368481
$425$	−257844.	−0.0692444
$426$	0	0
$427$	−3.72227e6	−0.987958
$428$	1.86579e6	0.492328
$429$	0	0
$430$	1.01854e6	0.265649
$431$	−7.22105e6	−1.87244	−0.936218	−	0.351420i	$-0.885699\pi$
$431$	−7.22105e6	−1.87244	−0.936218	+	0.351420i	$0.885699\pi$
$432$	0	0
$433$	−5.40009e6	−1.38414	−0.692071	−	0.721829i	$-0.743302\pi$
$433$	−5.40009e6	−1.38414	−0.692071	+	0.721829i	$0.743302\pi$
$434$	−2.54723e6	−0.649148
$435$	0	0
$436$	3.63565e6	0.915937
$437$	605231.	0.151606
$438$	0	0
$439$	−3.74237e6	−0.926799	−0.463399	−	0.886150i	$-0.653370\pi$
$439$	−3.74237e6	−0.926799	−0.463399	+	0.886150i	$0.653370\pi$
$440$	−224301.	−0.0552330
$441$	0	0
$442$	681708.	0.165975
$443$	−4.86601e6	−1.17805	−0.589025	−	0.808115i	$-0.700488\pi$
$443$	−4.86601e6	−1.17805	−0.589025	+	0.808115i	$0.700488\pi$
$444$	0	0
$445$	311357.	0.0745346
$446$	406822.	0.0968428
$447$	0	0
$448$	1.46043e6	0.343785
$449$	2.31989e6	0.543066	0.271533	−	0.962429i	$-0.412469\pi$
$449$	2.31989e6	0.543066	0.271533	+	0.962429i	$0.412469\pi$
$450$	0	0
$451$	−3.29559e6	−0.762942
$452$	−135143.	−0.0311133
$453$	0	0
$454$	−3.35515e6	−0.763962
$455$	−1.60649e6	−0.363790
$456$	0	0
$457$	4.69807e6	1.05227	0.526137	−	0.850400i	$-0.323640\pi$
$457$	4.69807e6	1.05227	0.526137	+	0.850400i	$0.323640\pi$
$458$	1.03552e7	2.30671
$459$	0	0
$460$	−285100.	−0.0628208
$461$	−6.45960e6	−1.41564	−0.707821	−	0.706392i	$-0.750321\pi$
$461$	−6.45960e6	−1.41564	−0.707821	+	0.706392i	$0.750321\pi$
$462$	0	0
$463$	−1.11879e6	−0.242547	−0.121273	−	0.992619i	$-0.538698\pi$
$463$	−1.11879e6	−0.242547	−0.121273	+	0.992619i	$0.538698\pi$
$464$	4.97854e6	1.07351
$465$	0	0
$466$	−9.15456e6	−1.95287
$467$	5.49372e6	1.16567	0.582833	−	0.812592i	$-0.301944\pi$
$467$	5.49372e6	1.16567	0.582833	+	0.812592i	$0.301944\pi$
$468$	0	0
$469$	−1.58303e6	−0.332320
$470$	2.33563e6	0.487708
$471$	0	0
$472$	2.22878e6	0.460482
$473$	1.32612e6	0.272541
$474$	0	0
$475$	−3.05832e6	−0.621940
$476$	−198038.	−0.0400618
$477$	0	0
$478$	−2.98714e6	−0.597978
$479$	−258489.	−0.0514759	−0.0257379	−	0.999669i	$-0.508194\pi$
$479$	−258489.	−0.0514759	−0.0257379	+	0.999669i	$0.508194\pi$
$480$	0	0
$481$	7.65037e6	1.50772
$482$	−8.08092e6	−1.58432
$483$	0	0
$484$	−2.96925e6	−0.576148
$485$	294987.	0.0569441
$486$	0	0
$487$	5.48182e6	1.04737	0.523687	−	0.851911i	$-0.324556\pi$
$487$	5.48182e6	1.04737	0.523687	+	0.851911i	$0.324556\pi$
$488$	2.31389e6	0.439838
$489$	0	0
$490$	−1.64998e6	−0.310448
$491$	−7.75324e6	−1.45137	−0.725687	−	0.688025i	$-0.758478\pi$
$491$	−7.75324e6	−1.45137	−0.725687	+	0.688025i	$0.758478\pi$
$492$	0	0
$493$	−402694.	−0.0746204
$494$	8.08581e6	1.49075
$495$	0	0
$496$	4.95309e6	0.904007
$497$	1.49055e6	0.270680
$498$	0	0
$499$	−1.56839e6	−0.281971	−0.140985	−	0.990012i	$-0.545027\pi$
$499$	−1.56839e6	−0.281971	−0.140985	+	0.990012i	$0.545027\pi$
$500$	3.12485e6	0.558989
$501$	0	0
$502$	−1.37039e7	−2.42708
$503$	−4.45901e6	−0.785812	−0.392906	−	0.919579i	$-0.628530\pi$
$503$	−4.45901e6	−0.785812	−0.392906	+	0.919579i	$0.628530\pi$
$504$	0	0
$505$	1.11436e6	0.194445
$506$	−839714.	−0.145799
$507$	0	0
$508$	1.30967e6	0.225166
$509$	3.57171e6	0.611056	0.305528	−	0.952183i	$-0.401167\pi$
$509$	3.57171e6	0.611056	0.305528	+	0.952183i	$0.401167\pi$
$510$	0	0
$511$	−2.51740e6	−0.426482
$512$	−6.92786e6	−1.16795
$513$	0	0
$514$	−1.38728e7	−2.31609
$515$	−3.26082e6	−0.541762
$516$	0	0
$517$	3.04095e6	0.500361
$518$	−5.02760e6	−0.823259
$519$	0	0
$520$	998648.	0.161959
$521$	1.18934e7	1.91960	0.959801	−	0.280683i	$-0.0905608\pi$
$521$	1.18934e7	1.91960	0.959801	+	0.280683i	$0.0905608\pi$
$522$	0	0
$523$	1.67375e6	0.267569	0.133784	−	0.991010i	$-0.457287\pi$
$523$	1.67375e6	0.267569	0.133784	+	0.991010i	$0.457287\pi$
$524$	−3.28172e6	−0.522124
$525$	0	0
$526$	1.06845e6	0.168380
$527$	−400635.	−0.0628380
$528$	0	0
$529$	279841.	0.0434783
$530$	4.36222e6	0.674556
$531$	0	0
$532$	−2.34895e6	−0.359827
$533$	1.46729e7	2.23716
$534$	0	0
$535$	−1.56442e6	−0.236303
$536$	984060.	0.147948
$537$	0	0
$538$	1.86982e6	0.278512
$539$	−2.14825e6	−0.318503
$540$	0	0
$541$	−7.25610e6	−1.06589	−0.532943	−	0.846151i	$-0.678914\pi$
$541$	−7.25610e6	−1.06589	−0.532943	+	0.846151i	$0.678914\pi$
$542$	6.53529e6	0.955579
$543$	0	0
$544$	715750.	0.103697
$545$	−3.04840e6	−0.439623
$546$	0	0
$547$	1.28186e7	1.83178	0.915889	−	0.401432i	$-0.131487\pi$
$547$	1.28186e7	1.83178	0.915889	+	0.401432i	$0.131487\pi$
$548$	1.00596e7	1.43097
$549$	0	0
$550$	4.24319e6	0.598117
$551$	−4.77640e6	−0.670226
$552$	0	0
$553$	−1.22326e6	−0.170101
$554$	1.13806e7	1.57540
$555$	0	0
$556$	−1.36065e6	−0.186664
$557$	3.82521e6	0.522417	0.261209	−	0.965282i	$-0.415879\pi$
$557$	3.82521e6	0.522417	0.261209	+	0.965282i	$0.415879\pi$
$558$	0	0
$559$	−5.90427e6	−0.799165
$560$	−2.05289e6	−0.276628
$561$	0	0
$562$	2.53792e6	0.338951
$563$	1.09965e6	0.146213	0.0731063	−	0.997324i	$-0.476709\pi$
$563$	1.09965e6	0.146213	0.0731063	+	0.997324i	$0.476709\pi$
$564$	0	0
$565$	113314.	0.0149335
$566$	1.83835e7	2.41206
$567$	0	0
$568$	−926576.	−0.120506
$569$	−8.13877e6	−1.05385	−0.526924	−	0.849912i	$-0.676655\pi$
$569$	−8.13877e6	−1.05385	−0.526924	+	0.849912i	$0.676655\pi$
$570$	0	0
$571$	8.56316e6	1.09912	0.549559	−	0.835455i	$-0.314796\pi$
$571$	8.56316e6	1.09912	0.549559	+	0.835455i	$0.314796\pi$
$572$	−4.95913e6	−0.633746
$573$	0	0
$574$	−9.64258e6	−1.22156
$575$	−1.41408e6	−0.178362
$576$	0	0
$577$	6.15395e6	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$577$	6.15395e6	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$578$	1.06823e7	1.32999
$579$	0	0
$580$	2.24997e6	0.277720
$581$	−4.39893e6	−0.540638
$582$	0	0
$583$	5.67954e6	0.692057
$584$	1.56490e6	0.189869
$585$	0	0
$586$	−7.81878e6	−0.940578
$587$	−1.11707e6	−0.133808	−0.0669042	−	0.997759i	$-0.521312\pi$
$587$	−1.11707e6	−0.133808	−0.0669042	+	0.997759i	$0.521312\pi$
$588$	0	0
$589$	−4.75197e6	−0.564399
$590$	7.12764e6	0.842977
$591$	0	0
$592$	9.77618e6	1.14648
$593$	8.17811e6	0.955028	0.477514	−	0.878624i	$-0.341538\pi$
$593$	8.17811e6	0.955028	0.477514	+	0.878624i	$0.341538\pi$
$594$	0	0
$595$	166050.	0.0192285
$596$	2.04163e6	0.235430
$597$	0	0
$598$	3.73864e6	0.427524
$599$	−1.09051e7	−1.24183	−0.620917	−	0.783877i	$-0.713239\pi$
$599$	−1.09051e7	−1.24183	−0.620917	+	0.783877i	$0.713239\pi$
$600$	0	0
$601$	1.29506e7	1.46252	0.731261	−	0.682098i	$-0.238932\pi$
$601$	1.29506e7	1.46252	0.731261	+	0.682098i	$0.238932\pi$
$602$	3.88011e6	0.436368
$603$	0	0
$604$	46864.3	0.00522696
$605$	2.48964e6	0.276534
$606$	0	0
$607$	1.21012e7	1.33308	0.666542	−	0.745468i	$-0.267774\pi$
$607$	1.21012e7	1.33308	0.666542	+	0.745468i	$0.267774\pi$
$608$	8.48959e6	0.931382
$609$	0	0
$610$	7.39980e6	0.805184
$611$	−1.35392e7	−1.46720
$612$	0	0
$613$	4.64061e6	0.498798	0.249399	−	0.968401i	$-0.419767\pi$
$613$	4.64061e6	0.498798	0.249399	+	0.968401i	$0.419767\pi$
$614$	−1.72261e7	−1.84402
$615$	0	0
$616$	−854469.	−0.0907287
$617$	−7.24099e6	−0.765747	−0.382873	−	0.923801i	$-0.625065\pi$
$617$	−7.24099e6	−0.765747	−0.382873	+	0.923801i	$0.625065\pi$
$618$	0	0
$619$	−296767.	−0.0311307	−0.0155653	−	0.999879i	$-0.504955\pi$
$619$	−296767.	−0.0311307	−0.0155653	+	0.999879i	$0.504955\pi$
$620$	2.23847e6	0.233868
$621$	0	0
$622$	6.65515e6	0.689734
$623$	1.18611e6	0.122435
$624$	0	0
$625$	5.73337e6	0.587097
$626$	1.37490e7	1.40228
$627$	0	0
$628$	−1.01953e7	−1.03157
$629$	−790755.	−0.0796921
$630$	0	0
$631$	−1.94372e7	−1.94339	−0.971697	−	0.236232i	$-0.924087\pi$
$631$	−1.94372e7	−1.94339	−0.971697	+	0.236232i	$0.924087\pi$
$632$	760419.	0.0757287
$633$	0	0
$634$	−5.66613e6	−0.559839
$635$	−1.09813e6	−0.108073
$636$	0	0
$637$	9.56461e6	0.933939
$638$	6.62691e6	0.644554
$639$	0	0
$640$	2.14432e6	0.206938
$641$	−1.49736e7	−1.43940	−0.719702	−	0.694283i	$-0.755721\pi$
$641$	−1.49736e7	−1.43940	−0.719702	+	0.694283i	$0.755721\pi$
$642$	0	0
$643$	−2.52305e6	−0.240657	−0.120329	−	0.992734i	$-0.538395\pi$
$643$	−2.52305e6	−0.240657	−0.120329	+	0.992734i	$0.538395\pi$
$644$	−1.08608e6	−0.103193
$645$	0	0
$646$	−835763.	−0.0787955
$647$	−1.42486e7	−1.33817	−0.669087	−	0.743184i	$-0.733315\pi$
$647$	−1.42486e7	−1.33817	−0.669087	+	0.743184i	$0.733315\pi$
$648$	0	0
$649$	9.28006e6	0.864847
$650$	−1.88919e7	−1.75385
$651$	0	0
$652$	2.04505e6	0.188401
$653$	2.57158e6	0.236003	0.118002	−	0.993013i	$-0.462351\pi$
$653$	2.57158e6	0.236003	0.118002	+	0.993013i	$0.462351\pi$
$654$	0	0
$655$	2.75164e6	0.250604
$656$	1.87500e7	1.70115
$657$	0	0
$658$	8.89754e6	0.801134
$659$	1.92179e7	1.72382	0.861911	−	0.507060i	$-0.169268\pi$
$659$	1.92179e7	1.72382	0.861911	+	0.507060i	$0.169268\pi$
$660$	0	0
$661$	2.65004e6	0.235911	0.117956	−	0.993019i	$-0.462366\pi$
$661$	2.65004e6	0.235911	0.117956	+	0.993019i	$0.462366\pi$
$662$	2.43937e7	2.16338
$663$	0	0
$664$	2.73452e6	0.240692
$665$	1.96953e6	0.172707
$666$	0	0
$667$	−2.20847e6	−0.192210
$668$	−1.15366e7	−1.00032
$669$	0	0
$670$	3.14702e6	0.270840
$671$	9.63441e6	0.826074
$672$	0	0
$673$	−538482.	−0.0458283	−0.0229142	−	0.999737i	$-0.507294\pi$
$673$	−538482.	−0.0458283	−0.0229142	+	0.999737i	$0.507294\pi$
$674$	−234760.	−0.0199056
$675$	0	0
$676$	1.26661e7	1.06605
$677$	1.98499e7	1.66451	0.832255	−	0.554393i	$-0.187050\pi$
$677$	1.98499e7	1.66451	0.832255	+	0.554393i	$0.187050\pi$
$678$	0	0
$679$	1.12375e6	0.0935394
$680$	−103222.	−0.00856051
$681$	0	0
$682$	6.59302e6	0.542780
$683$	−1.73706e7	−1.42483	−0.712416	−	0.701757i	$-0.752399\pi$
$683$	−1.73706e7	−1.42483	−0.712416	+	0.701757i	$0.752399\pi$
$684$	0	0
$685$	−8.43472e6	−0.686822
$686$	−1.65930e7	−1.34621
$687$	0	0
$688$	−7.54489e6	−0.607690
$689$	−2.52869e7	−2.02930
$690$	0	0
$691$	−6.20965e6	−0.494734	−0.247367	−	0.968922i	$-0.579565\pi$
$691$	−6.20965e6	−0.494734	−0.247367	+	0.968922i	$0.579565\pi$
$692$	7099.62	0.000563599 0
$693$	0	0
$694$	−2.94099e7	−2.31790
$695$	1.14087e6	0.0895931
$696$	0	0
$697$	−1.51661e6	−0.118248
$698$	−8.69083e6	−0.675185
$699$	0	0
$700$	5.48814e6	0.423331
$701$	1.14791e7	0.882291	0.441145	−	0.897436i	$-0.354572\pi$
$701$	1.14791e7	0.882291	0.441145	+	0.897436i	$0.354572\pi$
$702$	0	0
$703$	−9.37923e6	−0.715779
$704$	−3.78006e6	−0.287453
$705$	0	0
$706$	5.40651e6	0.408230
$707$	4.24514e6	0.319406
$708$	0	0
$709$	−5.52237e6	−0.412582	−0.206291	−	0.978491i	$-0.566139\pi$
$709$	−5.52237e6	−0.412582	−0.206291	+	0.978491i	$0.566139\pi$
$710$	−2.96318e6	−0.220604
$711$	0	0
$712$	−737323.	−0.0545077
$713$	−2.19717e6	−0.161860
$714$	0	0
$715$	4.15810e6	0.304180
$716$	−6.83644e6	−0.498365
$717$	0	0
$718$	−2.33727e7	−1.69199
$719$	−1.15081e7	−0.830195	−0.415097	−	0.909777i	$-0.636252\pi$
$719$	−1.15081e7	−0.830195	−0.415097	+	0.909777i	$0.636252\pi$
$720$	0	0
$721$	−1.24220e7	−0.889928
$722$	8.83883e6	0.631033
$723$	0	0
$724$	7.29901e6	0.517509
$725$	1.11597e7	0.788509
$726$	0	0
$727$	1.08042e7	0.758154	0.379077	−	0.925365i	$-0.376242\pi$
$727$	1.08042e7	0.758154	0.379077	+	0.925365i	$0.376242\pi$
$728$	3.80433e6	0.266042
$729$	0	0
$730$	5.00454e6	0.347582
$731$	610275.	0.0422408
$732$	0	0
$733$	−2.25707e7	−1.55162	−0.775810	−	0.630967i	$-0.782659\pi$
$733$	−2.25707e7	−1.55162	−0.775810	+	0.630967i	$0.782659\pi$
$734$	−1.06585e7	−0.730224
$735$	0	0
$736$	3.92534e6	0.267105
$737$	4.09737e6	0.277866
$738$	0	0
$739$	−1.66951e7	−1.12455	−0.562275	−	0.826950i	$-0.690074\pi$
$739$	−1.66951e7	−1.12455	−0.562275	+	0.826950i	$0.690074\pi$
$740$	4.41818e6	0.296596
$741$	0	0
$742$	1.66178e7	1.10806
$743$	2.36393e6	0.157095	0.0785477	−	0.996910i	$-0.474972\pi$
$743$	2.36393e6	0.157095	0.0785477	+	0.996910i	$0.474972\pi$
$744$	0	0
$745$	−1.71186e6	−0.113000
$746$	1.62150e7	1.06677
$747$	0	0
$748$	512584.	0.0334974
$749$	−5.95963e6	−0.388164
$750$	0	0
$751$	6.79787e6	0.439818	0.219909	−	0.975520i	$-0.429424\pi$
$751$	6.79787e6	0.439818	0.219909	+	0.975520i	$0.429424\pi$
$752$	−1.73013e7	−1.11566
$753$	0	0
$754$	−2.95048e7	−1.89001
$755$	−39294.5	−0.00250879
$756$	0	0
$757$	−1.83531e6	−0.116404	−0.0582021	−	0.998305i	$-0.518537\pi$
$757$	−1.83531e6	−0.116404	−0.0582021	+	0.998305i	$0.518537\pi$
$758$	−3.18494e7	−2.01339
$759$	0	0
$760$	−1.22433e6	−0.0768888
$761$	1.58462e6	0.0991891	0.0495945	−	0.998769i	$-0.484207\pi$
$761$	1.58462e6	0.0991891	0.0495945	+	0.998769i	$0.484207\pi$
$762$	0	0
$763$	−1.16128e7	−0.722148
$764$	7.99099e6	0.495299
$765$	0	0
$766$	−1.18297e7	−0.728453
$767$	−4.13174e7	−2.53597
$768$	0	0
$769$	−7.89421e6	−0.481385	−0.240693	−	0.970601i	$-0.577375\pi$
$769$	−7.89421e6	−0.481385	−0.240693	+	0.970601i	$0.577375\pi$
$770$	−2.73259e6	−0.166091
$771$	0	0
$772$	−2.93473e6	−0.177225
$773$	1.50786e7	0.907634	0.453817	−	0.891095i	$-0.350062\pi$
$773$	1.50786e7	0.907634	0.453817	+	0.891095i	$0.350062\pi$
$774$	0	0
$775$	1.11026e7	0.664005
$776$	−698559.	−0.0416436
$777$	0	0
$778$	4.04247e7	2.39441
$779$	−1.79887e7	−1.06208
$780$	0	0
$781$	−3.85802e6	−0.226327
$782$	−386432.	−0.0225973
$783$	0	0
$784$	1.22223e7	0.710172
$785$	8.54848e6	0.495125
$786$	0	0
$787$	−1.78357e7	−1.02649	−0.513244	−	0.858243i	$-0.671556\pi$
$787$	−1.78357e7	−1.02649	−0.513244	+	0.858243i	$0.671556\pi$
$788$	447975.	0.0257003
$789$	0	0
$790$	2.43182e6	0.138632
$791$	431666.	0.0245305
$792$	0	0
$793$	−4.28950e7	−2.42228
$794$	−3.11322e7	−1.75250
$795$	0	0
$796$	1.63866e6	0.0916655
$797$	−52333.1	−0.00291831	−0.00145915	−	0.999999i	$-0.500464\pi$
$797$	−52333.1	−0.00291831	−0.00145915	+	0.999999i	$0.500464\pi$
$798$	0	0
$799$	1.39943e6	0.0775504
$800$	−1.98353e7	−1.09575
$801$	0	0
$802$	−4.08579e7	−2.24306
$803$	6.51582e6	0.356599
$804$	0	0
$805$	910655.	0.0495295
$806$	−2.93540e7	−1.59158
$807$	0	0
$808$	−2.63892e6	−0.142199
$809$	1.21835e7	0.654486	0.327243	−	0.944940i	$-0.393880\pi$
$809$	1.21835e7	0.654486	0.327243	+	0.944940i	$0.393880\pi$
$810$	0	0
$811$	7.62814e6	0.407255	0.203628	−	0.979048i	$-0.434727\pi$
$811$	7.62814e6	0.407255	0.203628	+	0.979048i	$0.434727\pi$
$812$	8.57122e6	0.456197
$813$	0	0
$814$	1.30130e7	0.688362
$815$	−1.71472e6	−0.0904272
$816$	0	0
$817$	7.23854e6	0.379399
$818$	2.45332e6	0.128195
$819$	0	0
$820$	8.47376e6	0.440090
$821$	−5.98967e6	−0.310131	−0.155065	−	0.987904i	$-0.549559\pi$
$821$	−5.98967e6	−0.310131	−0.155065	+	0.987904i	$0.549559\pi$
$822$	0	0
$823$	7.73297e6	0.397966	0.198983	−	0.980003i	$-0.436236\pi$
$823$	7.73297e6	0.397966	0.198983	+	0.980003i	$0.436236\pi$
$824$	7.72194e6	0.396195
$825$	0	0
$826$	2.71526e7	1.38472
$827$	−2.64371e7	−1.34416	−0.672079	−	0.740479i	$-0.734599\pi$
$827$	−2.64371e7	−1.34416	−0.672079	+	0.740479i	$0.734599\pi$
$828$	0	0
$829$	1.04323e7	0.527223	0.263612	−	0.964629i	$-0.415086\pi$
$829$	1.04323e7	0.527223	0.263612	+	0.964629i	$0.415086\pi$
$830$	8.74498e6	0.440619
$831$	0	0
$832$	1.68299e7	0.842893
$833$	−988614.	−0.0493644
$834$	0	0
$835$	9.67318e6	0.480123
$836$	6.07981e6	0.300867
$837$	0	0
$838$	−1.88775e7	−0.928611
$839$	1.42644e7	0.699600	0.349800	−	0.936824i	$-0.386249\pi$
$839$	1.42644e7	0.699600	0.349800	+	0.936824i	$0.386249\pi$
$840$	0	0
$841$	−3.08225e6	−0.150272
$842$	6.22831e6	0.302754
$843$	0	0
$844$	2.18381e7	1.05526
$845$	−1.06202e7	−0.511671
$846$	0	0
$847$	9.48425e6	0.454250
$848$	−3.23134e7	−1.54309
$849$	0	0
$850$	1.95270e6	0.0927016
$851$	−4.33668e6	−0.205274
$852$	0	0
$853$	1.41943e7	0.667945	0.333972	−	0.942583i	$-0.391611\pi$
$853$	1.41943e7	0.667945	0.333972	+	0.942583i	$0.391611\pi$
$854$	2.81894e7	1.32264
$855$	0	0
$856$	3.70470e6	0.172810
$857$	−4.24744e7	−1.97549	−0.987746	−	0.156069i	$-0.950118\pi$
$857$	−4.24744e7	−1.97549	−0.987746	+	0.156069i	$0.950118\pi$
$858$	0	0
$859$	−4.10825e7	−1.89965	−0.949825	−	0.312781i	$-0.898740\pi$
$859$	−4.10825e7	−1.89965	−0.949825	+	0.312781i	$0.898740\pi$
$860$	−3.40979e6	−0.157210
$861$	0	0
$862$	5.46862e7	2.50674
$863$	−3.84453e7	−1.75718	−0.878590	−	0.477577i	$-0.841515\pi$
$863$	−3.84453e7	−1.75718	−0.878590	+	0.477577i	$0.841515\pi$
$864$	0	0
$865$	−5952.85	−0.000270511 0
$866$	4.08957e7	1.85303
$867$	0	0
$868$	8.52740e6	0.384165
$869$	3.16618e6	0.142228
$870$	0	0
$871$	−1.82426e7	−0.814782
$872$	7.21891e6	0.321500
$873$	0	0
$874$	−4.58351e6	−0.202964
$875$	−9.98124e6	−0.440722
$876$	0	0
$877$	−2.69102e7	−1.18146	−0.590730	−	0.806870i	$-0.701160\pi$
$877$	−2.69102e7	−1.18146	−0.590730	+	0.806870i	$0.701160\pi$
$878$	2.83416e7	1.24076
$879$	0	0
$880$	5.31352e6	0.231300
$881$	3.50491e7	1.52138	0.760689	−	0.649116i	$-0.224861\pi$
$881$	3.50491e7	1.52138	0.760689	+	0.649116i	$0.224861\pi$
$882$	0	0
$883$	2.36808e7	1.02210	0.511052	−	0.859550i	$-0.329256\pi$
$883$	2.36808e7	1.02210	0.511052	+	0.859550i	$0.329256\pi$
$884$	−2.28216e6	−0.0982237
$885$	0	0
$886$	3.68511e7	1.57713
$887$	2.48690e7	1.06133	0.530663	−	0.847583i	$-0.321943\pi$
$887$	2.48690e7	1.06133	0.530663	+	0.847583i	$0.321943\pi$
$888$	0	0
$889$	−4.18329e6	−0.177527
$890$	−2.35796e6	−0.0997839
$891$	0	0
$892$	−1.36193e6	−0.0573114
$893$	1.65988e7	0.696543
$894$	0	0
$895$	5.73218e6	0.239201
$896$	8.16874e6	0.339927
$897$	0	0
$898$	−1.75689e7	−0.727034
$899$	1.73398e7	0.715557
$900$	0	0
$901$	2.61369e6	0.107261
$902$	2.49580e7	1.02140
$903$	0	0
$904$	−268338.	−0.0109210
$905$	−6.12004e6	−0.248389
$906$	0	0
$907$	3.16625e7	1.27799	0.638994	−	0.769212i	$-0.279351\pi$
$907$	3.16625e7	1.27799	0.638994	+	0.769212i	$0.279351\pi$
$908$	1.12321e7	0.452112
$909$	0	0
$910$	1.21662e7	0.487026
$911$	5.34272e6	0.213288	0.106644	−	0.994297i	$-0.465989\pi$
$911$	5.34272e6	0.213288	0.106644	+	0.994297i	$0.465989\pi$
$912$	0	0
$913$	1.13858e7	0.452051
$914$	−3.55792e7	−1.40874
$915$	0	0
$916$	−3.46662e7	−1.36511
$917$	1.04823e7	0.411656
$918$	0	0
$919$	−3.97879e6	−0.155404	−0.0777021	−	0.996977i	$-0.524758\pi$
$919$	−3.97879e6	−0.155404	−0.0777021	+	0.996977i	$0.524758\pi$
$920$	−566093.	−0.0220505
$921$	0	0
$922$	4.89196e7	1.89520
$923$	1.71769e7	0.663654
$924$	0	0
$925$	2.19139e7	0.842101
$926$	8.47277e6	0.324712
$927$	0	0
$928$	−3.09782e7	−1.18083
$929$	2.19699e7	0.835197	0.417598	−	0.908632i	$-0.362872\pi$
$929$	2.19699e7	0.835197	0.417598	+	0.908632i	$0.362872\pi$
$930$	0	0
$931$	−1.17261e7	−0.443382
$932$	3.06469e7	1.15570
$933$	0	0
$934$	−4.16049e7	−1.56055
$935$	−429789.	−0.0160778
$936$	0	0
$937$	4.84787e7	1.80386	0.901929	−	0.431885i	$-0.142151\pi$
$937$	4.84787e7	1.80386	0.901929	+	0.431885i	$0.142151\pi$
$938$	1.19885e7	0.444896
$939$	0	0
$940$	−7.81903e6	−0.288625
$941$	3.27451e7	1.20551	0.602757	−	0.797925i	$-0.294069\pi$
$941$	3.27451e7	1.20551	0.602757	+	0.797925i	$0.294069\pi$
$942$	0	0
$943$	−8.31744e6	−0.304586
$944$	−5.27983e7	−1.92837
$945$	0	0
$946$	−1.00430e7	−0.364866
$947$	2.15866e7	0.782185	0.391093	−	0.920351i	$-0.372097\pi$
$947$	2.15866e7	0.782185	0.391093	+	0.920351i	$0.372097\pi$
$948$	0	0
$949$	−2.90102e7	−1.04565
$950$	2.31611e7	0.832627
$951$	0	0
$952$	−393222.	−0.0140619
$953$	3.35580e7	1.19692	0.598458	−	0.801154i	$-0.295780\pi$
$953$	3.35580e7	1.19692	0.598458	+	0.801154i	$0.295780\pi$
$954$	0	0
$955$	−6.70025e6	−0.237729
$956$	1.00001e7	0.353882
$957$	0	0
$958$	1.95758e6	0.0689138
$959$	−3.21319e7	−1.12821
$960$	0	0
$961$	−1.13780e7	−0.397428
$962$	−5.79375e7	−2.01847
$963$	0	0
$964$	2.70526e7	0.937599
$965$	2.46070e6	0.0850629
$966$	0	0
$967$	4.58172e7	1.57566	0.787829	−	0.615894i	$-0.211205\pi$
$967$	4.58172e7	1.57566	0.787829	+	0.615894i	$0.211205\pi$
$968$	−5.89572e6	−0.202231
$969$	0	0
$970$	−2.23399e6	−0.0762345
$971$	−133014.	−0.00452740	−0.00226370	−	0.999997i	$-0.500721\pi$
$971$	−133014.	−0.00452740	−0.00226370	+	0.999997i	$0.500721\pi$
$972$	0	0
$973$	4.34613e6	0.147170
$974$	−4.15147e7	−1.40218
$975$	0	0
$976$	−5.48143e7	−1.84191
$977$	3.08041e7	1.03246	0.516229	−	0.856451i	$-0.327335\pi$
$977$	3.08041e7	1.03246	0.516229	+	0.856451i	$0.327335\pi$
$978$	0	0
$979$	−3.07002e6	−0.102373
$980$	5.52368e6	0.183723
$981$	0	0
$982$	5.87165e7	1.94304
$983$	−1.07060e7	−0.353382	−0.176691	−	0.984266i	$-0.556539\pi$
$983$	−1.07060e7	−0.353382	−0.176691	+	0.984266i	$0.556539\pi$
$984$	0	0
$985$	−375616.	−0.0123354
$986$	3.04967e6	0.0998988
$987$	0	0
$988$	−2.70690e7	−0.882226
$989$	3.34689e6	0.108805
$990$	0	0
$991$	2.13829e7	0.691644	0.345822	−	0.938300i	$-0.387600\pi$
$991$	2.13829e7	0.691644	0.345822	+	0.938300i	$0.387600\pi$
$992$	−3.08198e7	−0.994376
$993$	0	0
$994$	−1.12882e7	−0.362375
$995$	−1.37397e6	−0.0439968
$996$	0	0
$997$	−8.42535e6	−0.268442	−0.134221	−	0.990951i	$-0.542853\pi$
$997$	−8.42535e6	−0.268442	−0.134221	+	0.990951i	$0.542853\pi$
$998$	1.18777e7	0.377491
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.6.a.i.1.2	yes	10
3.2	odd	2		207.6.a.h.1.9	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.6.a.h.1.9	✓	10	3.2	odd	2
207.6.a.i.1.2	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)

\(f(q)\)	\(=\)	\(q-7.57316 q^{2} +25.3528 q^{4} -21.2577 q^{5} -80.9808 q^{7} +50.3403 q^{8} +O(q^{10})\)	\(q-7.57316 q^{2} +25.3528 q^{4} -21.2577 q^{5} -80.9808 q^{7} +50.3403 q^{8} +160.988 q^{10} +209.604 q^{11} -933.213 q^{13} +613.281 q^{14} -1192.53 q^{16} +96.4584 q^{17} +1144.10 q^{19} -538.942 q^{20} -1587.36 q^{22} +529.000 q^{23} -2673.11 q^{25} +7067.37 q^{26} -2053.09 q^{28} -4174.79 q^{29} -4153.45 q^{31} +7420.30 q^{32} -730.495 q^{34} +1721.46 q^{35} -8197.88 q^{37} -8664.49 q^{38} -1070.12 q^{40} -15723.0 q^{41} +6326.82 q^{43} +5314.04 q^{44} -4006.20 q^{46} +14508.1 q^{47} -10249.1 q^{49} +20243.9 q^{50} -23659.6 q^{52} +27096.6 q^{53} -4455.69 q^{55} -4076.60 q^{56} +31616.4 q^{58} +44274.3 q^{59} +45964.9 q^{61} +31454.7 q^{62} -18034.3 q^{64} +19838.0 q^{65} +19548.2 q^{67} +2445.49 q^{68} -13036.9 q^{70} -18406.2 q^{71} +31086.4 q^{73} +62083.9 q^{74} +29006.2 q^{76} -16973.9 q^{77} +15105.6 q^{79} +25350.3 q^{80} +119073. q^{82} +54320.7 q^{83} -2050.48 q^{85} -47914.0 q^{86} +10551.5 q^{88} -14646.8 q^{89} +75572.3 q^{91} +13411.6 q^{92} -109872. q^{94} -24321.0 q^{95} -13876.7 q^{97} +77618.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 8 q^{2} + 192 q^{4} + 100 q^{5} + 20 q^{7} + 384 q^{8}+O(q^{10}) \) 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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