LMFDB - Embedded newform 168.6.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,6,Mod(1,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("168.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	168.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:	$26.9444817286$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{249}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 62 $ x^2 - x - 62
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.38987$ of defining polynomial
Character	$\chi$	$=$	168.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -0.440532 q^{5} +49.0000 q^{7} +81.0000 q^{9} -203.392 q^{11} -402.951 q^{13} +3.96479 q^{15} +1761.53 q^{17} +1775.40 q^{19} -441.000 q^{21} -2710.72 q^{23} -3124.81 q^{25} -729.000 q^{27} +781.595 q^{29} -9655.74 q^{31} +1830.53 q^{33} -21.5861 q^{35} +3788.80 q^{37} +3626.56 q^{39} -18034.1 q^{41} -13199.7 q^{43} -35.6831 q^{45} -26648.6 q^{47} +2401.00 q^{49} -15853.8 q^{51} +22760.3 q^{53} +89.6008 q^{55} -15978.6 q^{57} +1579.56 q^{59} -35327.9 q^{61} +3969.00 q^{63} +177.513 q^{65} -48572.1 q^{67} +24396.5 q^{69} +12764.0 q^{71} +10152.3 q^{73} +28123.3 q^{75} -9966.21 q^{77} -8703.14 q^{79} +6561.00 q^{81} -26277.1 q^{83} -776.012 q^{85} -7034.35 q^{87} +33598.5 q^{89} -19744.6 q^{91} +86901.6 q^{93} -782.121 q^{95} +23000.5 q^{97} -16474.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 64 q^{5} + 98 q^{7} + 162 q^{9} + 540 q^{11} + 204 q^{13} + 576 q^{15} + 304 q^{17} - 1120 q^{19} - 882 q^{21} - 940 q^{23} - 2210 q^{25} - 1458 q^{27} + 932 q^{29} - 16408 q^{31} - 4860 q^{33}+ \cdots + 43740 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 64 * q^5 + 98 * q^7 + 162 * q^9 + 540 * q^11 + 204 * q^13 + 576 * q^15 + 304 * q^17 - 1120 * q^19 - 882 * q^21 - 940 * q^23 - 2210 * q^25 - 1458 * q^27 + 932 * q^29 - 16408 * q^31 - 4860 * q^33 - 3136 * q^35 - 1764 * q^37 - 1836 * q^39 - 2552 * q^41 - 24632 * q^43 - 5184 * q^45 - 36760 * q^47 + 4802 * q^49 - 2736 * q^51 + 9164 * q^53 - 47160 * q^55 + 10080 * q^57 - 39888 * q^59 - 25084 * q^61 + 7938 * q^63 - 38400 * q^65 - 2592 * q^67 + 8460 * q^69 - 35508 * q^71 - 17188 * q^73 + 19890 * q^75 + 26460 * q^77 - 95800 * q^79 + 13122 * q^81 + 65352 * q^83 + 91864 * q^85 - 8388 * q^87 - 23000 * q^89 + 9996 * q^91 + 147672 * q^93 + 183248 * q^95 - 108388 * q^97 + 43740 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−0.440532	−0.00788048	−0.00394024	−	0.999992i	$-0.501254\pi$
$5$	−0.440532	−0.00788048	−0.00394024	+	0.999992i	$0.501254\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−203.392	−0.506818	−0.253409	−	0.967359i	$-0.581552\pi$
$11$	−203.392	−0.506818	−0.253409	+	0.967359i	$0.581552\pi$
$12$	0	0
$13$	−402.951	−0.661294	−0.330647	−	0.943755i	$-0.607267\pi$
$13$	−402.951	−0.661294	−0.330647	+	0.943755i	$0.607267\pi$
$14$	0	0
$15$	3.96479	0.00454980
$16$	0	0
$17$	1761.53	1.47832	0.739160	−	0.673530i	$-0.235223\pi$
$17$	1761.53	1.47832	0.739160	+	0.673530i	$0.235223\pi$
$18$	0	0
$19$	1775.40	1.12827	0.564134	−	0.825683i	$-0.309210\pi$
$19$	1775.40	1.12827	0.564134	+	0.825683i	$0.309210\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	−2710.72	−1.06848	−0.534239	−	0.845334i	$-0.679402\pi$
$23$	−2710.72	−1.06848	−0.534239	+	0.845334i	$0.679402\pi$
$24$	0	0
$25$	−3124.81	−0.999938
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	781.595	0.172578	0.0862892	−	0.996270i	$-0.472499\pi$
$29$	781.595	0.172578	0.0862892	+	0.996270i	$0.472499\pi$
$30$	0	0
$31$	−9655.74	−1.80460	−0.902300	−	0.431108i	$-0.858123\pi$
$31$	−9655.74	−1.80460	−0.902300	+	0.431108i	$0.858123\pi$
$32$	0	0
$33$	1830.53	0.292612
$34$	0	0
$35$	−21.5861	−0.00297854
$36$	0	0
$37$	3788.80	0.454985	0.227493	−	0.973780i	$-0.426947\pi$
$37$	3788.80	0.454985	0.227493	+	0.973780i	$0.426947\pi$
$38$	0	0
$39$	3626.56	0.381798
$40$	0	0
$41$	−18034.1	−1.67546	−0.837730	−	0.546084i	$-0.816118\pi$
$41$	−18034.1	−1.67546	−0.837730	+	0.546084i	$0.816118\pi$
$42$	0	0
$43$	−13199.7	−1.08866	−0.544329	−	0.838872i	$-0.683216\pi$
$43$	−13199.7	−1.08866	−0.544329	+	0.838872i	$0.683216\pi$
$44$	0	0
$45$	−35.6831	−0.00262683
$46$	0	0
$47$	−26648.6	−1.75966	−0.879831	−	0.475286i	$-0.842345\pi$
$47$	−26648.6	−1.75966	−0.879831	+	0.475286i	$0.842345\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−15853.8	−0.853508
$52$	0	0
$53$	22760.3	1.11298	0.556490	−	0.830854i	$-0.312148\pi$
$53$	22760.3	1.11298	0.556490	+	0.830854i	$0.312148\pi$
$54$	0	0
$55$	89.6008	0.00399397
$56$	0	0
$57$	−15978.6	−0.651406
$58$	0	0
$59$	1579.56	0.0590752	0.0295376	−	0.999564i	$-0.490597\pi$
$59$	1579.56	0.0590752	0.0295376	+	0.999564i	$0.490597\pi$
$60$	0	0
$61$	−35327.9	−1.21561	−0.607804	−	0.794087i	$-0.707949\pi$
$61$	−35327.9	−1.21561	−0.607804	+	0.794087i	$0.707949\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	177.513	0.00521131
$66$	0	0
$67$	−48572.1	−1.32190	−0.660952	−	0.750428i	$-0.729847\pi$
$67$	−48572.1	−1.32190	−0.660952	+	0.750428i	$0.729847\pi$
$68$	0	0
$69$	24396.5	0.616886
$70$	0	0
$71$	12764.0	0.300498	0.150249	−	0.988648i	$-0.451993\pi$
$71$	12764.0	0.300498	0.150249	+	0.988648i	$0.451993\pi$
$72$	0	0
$73$	10152.3	0.222976	0.111488	−	0.993766i	$-0.464438\pi$
$73$	10152.3	0.222976	0.111488	+	0.993766i	$0.464438\pi$
$74$	0	0
$75$	28123.3	0.577314
$76$	0	0
$77$	−9966.21	−0.191559
$78$	0	0
$79$	−8703.14	−0.156895	−0.0784474	−	0.996918i	$-0.524996\pi$
$79$	−8703.14	−0.156895	−0.0784474	+	0.996918i	$0.524996\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−26277.1	−0.418680	−0.209340	−	0.977843i	$-0.567132\pi$
$83$	−26277.1	−0.418680	−0.209340	+	0.977843i	$0.567132\pi$
$84$	0	0
$85$	−776.012	−0.0116499
$86$	0	0
$87$	−7034.35	−0.0996382
$88$	0	0
$89$	33598.5	0.449619	0.224809	−	0.974403i	$-0.427824\pi$
$89$	33598.5	0.449619	0.224809	+	0.974403i	$0.427824\pi$
$90$	0	0
$91$	−19744.6	−0.249946
$92$	0	0
$93$	86901.6	1.04189
$94$	0	0
$95$	−782.121	−0.00889130
$96$	0	0
$97$	23000.5	0.248203	0.124102	−	0.992270i	$-0.460395\pi$
$97$	23000.5	0.248203	0.124102	+	0.992270i	$0.460395\pi$
$98$	0	0
$99$	−16474.8	−0.168939
$100$	0	0
$101$	57527.8	0.561144	0.280572	−	0.959833i	$-0.409476\pi$
$101$	57527.8	0.561144	0.280572	+	0.959833i	$0.409476\pi$
$102$	0	0
$103$	−167794.	−1.55842	−0.779210	−	0.626763i	$-0.784379\pi$
$103$	−167794.	−1.55842	−0.779210	+	0.626763i	$0.784379\pi$
$104$	0	0
$105$	194.275	0.00171966
$106$	0	0
$107$	67979.3	0.574007	0.287004	−	0.957929i	$-0.407341\pi$
$107$	67979.3	0.574007	0.287004	+	0.957929i	$0.407341\pi$
$108$	0	0
$109$	91585.8	0.738349	0.369175	−	0.929360i	$-0.379640\pi$
$109$	91585.8	0.738349	0.369175	+	0.929360i	$0.379640\pi$
$110$	0	0
$111$	−34099.2	−0.262686
$112$	0	0
$113$	154355.	1.13717	0.568584	−	0.822625i	$-0.307492\pi$
$113$	154355.	1.13717	0.568584	+	0.822625i	$0.307492\pi$
$114$	0	0
$115$	1194.16	0.00842012
$116$	0	0
$117$	−32639.1	−0.220431
$118$	0	0
$119$	86315.1	0.558752
$120$	0	0
$121$	−119683.	−0.743135
$122$	0	0
$123$	162307.	0.967328
$124$	0	0
$125$	2753.24	0.0157605
$126$	0	0
$127$	−54085.3	−0.297557	−0.148778	−	0.988871i	$-0.547534\pi$
$127$	−54085.3	−0.297557	−0.148778	+	0.988871i	$0.547534\pi$
$128$	0	0
$129$	118797.	0.628537
$130$	0	0
$131$	43981.5	0.223919	0.111960	−	0.993713i	$-0.464287\pi$
$131$	43981.5	0.223919	0.111960	+	0.993713i	$0.464287\pi$
$132$	0	0
$133$	86994.6	0.426445
$134$	0	0
$135$	321.148	0.00151660
$136$	0	0
$137$	79578.6	0.362239	0.181119	−	0.983461i	$-0.442028\pi$
$137$	79578.6	0.362239	0.181119	+	0.983461i	$0.442028\pi$
$138$	0	0
$139$	−207284.	−0.909973	−0.454987	−	0.890498i	$-0.650356\pi$
$139$	−207284.	−0.909973	−0.454987	+	0.890498i	$0.650356\pi$
$140$	0	0
$141$	239837.	1.01594
$142$	0	0
$143$	81957.1	0.335156
$144$	0	0
$145$	−344.318	−0.00136000
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	382884.	1.41287	0.706434	−	0.707779i	$-0.250303\pi$
$149$	382884.	1.41287	0.706434	+	0.707779i	$0.250303\pi$
$150$	0	0
$151$	50287.6	0.179481	0.0897406	−	0.995965i	$-0.471396\pi$
$151$	50287.6	0.179481	0.0897406	+	0.995965i	$0.471396\pi$
$152$	0	0
$153$	142684.	0.492773
$154$	0	0
$155$	4253.66	0.0142211
$156$	0	0
$157$	367616.	1.19027	0.595135	−	0.803626i	$-0.297099\pi$
$157$	367616.	1.19027	0.595135	+	0.803626i	$0.297099\pi$
$158$	0	0
$159$	−204842.	−0.642579
$160$	0	0
$161$	−132825.	−0.403847
$162$	0	0
$163$	−206972.	−0.610158	−0.305079	−	0.952327i	$-0.598683\pi$
$163$	−206972.	−0.610158	−0.305079	+	0.952327i	$0.598683\pi$
$164$	0	0
$165$	−806.407	−0.00230592
$166$	0	0
$167$	257285.	0.713878	0.356939	−	0.934128i	$-0.383820\pi$
$167$	257285.	0.713878	0.356939	+	0.934128i	$0.383820\pi$
$168$	0	0
$169$	−208923.	−0.562691
$170$	0	0
$171$	143807.	0.376089
$172$	0	0
$173$	−269032.	−0.683421	−0.341710	−	0.939805i	$-0.611006\pi$
$173$	−269032.	−0.683421	−0.341710	+	0.939805i	$0.611006\pi$
$174$	0	0
$175$	−153115.	−0.377941
$176$	0	0
$177$	−14216.0	−0.0341071
$178$	0	0
$179$	533998.	1.24568	0.622841	−	0.782348i	$-0.285978\pi$
$179$	533998.	1.24568	0.622841	+	0.782348i	$0.285978\pi$
$180$	0	0
$181$	−629728.	−1.42875	−0.714375	−	0.699763i	$-0.753289\pi$
$181$	−629728.	−1.42875	−0.714375	+	0.699763i	$0.753289\pi$
$182$	0	0
$183$	317951.	0.701832
$184$	0	0
$185$	−1669.09	−0.00358550
$186$	0	0
$187$	−358282.	−0.749239
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	−452199.	−0.896903	−0.448452	−	0.893807i	$-0.648024\pi$
$191$	−452199.	−0.896903	−0.448452	+	0.893807i	$0.648024\pi$
$192$	0	0
$193$	−372356.	−0.719556	−0.359778	−	0.933038i	$-0.617148\pi$
$193$	−372356.	−0.719556	−0.359778	+	0.933038i	$0.617148\pi$
$194$	0	0
$195$	−1597.62	−0.00300875
$196$	0	0
$197$	−250110.	−0.459161	−0.229581	−	0.973290i	$-0.573735\pi$
$197$	−250110.	−0.459161	−0.229581	+	0.973290i	$0.573735\pi$
$198$	0	0
$199$	−936025.	−1.67554	−0.837770	−	0.546023i	$-0.816141\pi$
$199$	−936025.	−1.67554	−0.837770	+	0.546023i	$0.816141\pi$
$200$	0	0
$201$	437149.	0.763201
$202$	0	0
$203$	38298.1	0.0652285
$204$	0	0
$205$	7944.59	0.0132034
$206$	0	0
$207$	−219568.	−0.356159
$208$	0	0
$209$	−361102.	−0.571827
$210$	0	0
$211$	961698.	1.48707	0.743537	−	0.668695i	$-0.233147\pi$
$211$	961698.	1.48707	0.743537	+	0.668695i	$0.233147\pi$
$212$	0	0
$213$	−114876.	−0.173492
$214$	0	0
$215$	5814.88	0.00857916
$216$	0	0
$217$	−473131.	−0.682075
$218$	0	0
$219$	−91370.9	−0.128735
$220$	0	0
$221$	−709812.	−0.977604
$222$	0	0
$223$	−624199.	−0.840545	−0.420273	−	0.907398i	$-0.638066\pi$
$223$	−624199.	−0.840545	−0.420273	+	0.907398i	$0.638066\pi$
$224$	0	0
$225$	−253109.	−0.333313
$226$	0	0
$227$	698450.	0.899644	0.449822	−	0.893118i	$-0.351487\pi$
$227$	698450.	0.899644	0.449822	+	0.893118i	$0.351487\pi$
$228$	0	0
$229$	−987151.	−1.24393	−0.621964	−	0.783046i	$-0.713665\pi$
$229$	−987151.	−1.24393	−0.621964	+	0.783046i	$0.713665\pi$
$230$	0	0
$231$	89695.9	0.110597
$232$	0	0
$233$	−1.11183e6	−1.34168	−0.670839	−	0.741603i	$-0.734066\pi$
$233$	−1.11183e6	−1.34168	−0.670839	+	0.741603i	$0.734066\pi$
$234$	0	0
$235$	11739.6	0.0138670
$236$	0	0
$237$	78328.3	0.0905832
$238$	0	0
$239$	1.50111e6	1.69988	0.849941	−	0.526878i	$-0.176638\pi$
$239$	1.50111e6	1.69988	0.849941	+	0.526878i	$0.176638\pi$
$240$	0	0
$241$	−172734.	−0.191574	−0.0957869	−	0.995402i	$-0.530537\pi$
$241$	−172734.	−0.191574	−0.0957869	+	0.995402i	$0.530537\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−1057.72	−0.00112578
$246$	0	0
$247$	−715400.	−0.746117
$248$	0	0
$249$	236494.	0.241725
$250$	0	0
$251$	−136981.	−0.137238	−0.0686192	−	0.997643i	$-0.521859\pi$
$251$	−136981.	−0.137238	−0.0686192	+	0.997643i	$0.521859\pi$
$252$	0	0
$253$	551339.	0.541524
$254$	0	0
$255$	6984.11	0.00672606
$256$	0	0
$257$	−1.97671e6	−1.86686	−0.933428	−	0.358765i	$-0.883198\pi$
$257$	−1.97671e6	−1.86686	−0.933428	+	0.358765i	$0.883198\pi$
$258$	0	0
$259$	185651.	0.171968
$260$	0	0
$261$	63309.2	0.0575262
$262$	0	0
$263$	414337.	0.369372	0.184686	−	0.982798i	$-0.440873\pi$
$263$	414337.	0.369372	0.184686	+	0.982798i	$0.440873\pi$
$264$	0	0
$265$	−10026.6	−0.00877082
$266$	0	0
$267$	−302386.	−0.259588
$268$	0	0
$269$	241748.	0.203696	0.101848	−	0.994800i	$-0.467524\pi$
$269$	241748.	0.203696	0.101848	+	0.994800i	$0.467524\pi$
$270$	0	0
$271$	558814.	0.462215	0.231107	−	0.972928i	$-0.425765\pi$
$271$	558814.	0.462215	0.231107	+	0.972928i	$0.425765\pi$
$272$	0	0
$273$	177702.	0.144306
$274$	0	0
$275$	635561.	0.506787
$276$	0	0
$277$	232158.	0.181796	0.0908981	−	0.995860i	$-0.471026\pi$
$277$	232158.	0.181796	0.0908981	+	0.995860i	$0.471026\pi$
$278$	0	0
$279$	−782115.	−0.601534
$280$	0	0
$281$	1.85817e6	1.40385	0.701923	−	0.712253i	$-0.252325\pi$
$281$	1.85817e6	1.40385	0.701923	+	0.712253i	$0.252325\pi$
$282$	0	0
$283$	2.42916e6	1.80297	0.901487	−	0.432805i	$-0.142476\pi$
$283$	2.42916e6	1.80297	0.901487	+	0.432805i	$0.142476\pi$
$284$	0	0
$285$	7039.09	0.00513339
$286$	0	0
$287$	−883670.	−0.633265
$288$	0	0
$289$	1.68314e6	1.18543
$290$	0	0
$291$	−207004.	−0.143300
$292$	0	0
$293$	−1.61912e6	−1.10182	−0.550908	−	0.834566i	$-0.685719\pi$
$293$	−1.61912e6	−1.10182	−0.550908	+	0.834566i	$0.685719\pi$
$294$	0	0
$295$	−695.846	−0.000465541 0
$296$	0	0
$297$	148273.	0.0975372
$298$	0	0
$299$	1.09229e6	0.706577
$300$	0	0
$301$	−646784.	−0.411474
$302$	0	0
$303$	−517750.	−0.323977
$304$	0	0
$305$	15563.1	0.00957958
$306$	0	0
$307$	936495.	0.567100	0.283550	−	0.958957i	$-0.408488\pi$
$307$	936495.	0.567100	0.283550	+	0.958957i	$0.408488\pi$
$308$	0	0
$309$	1.51015e6	0.899754
$310$	0	0
$311$	48784.9	0.0286012	0.0143006	−	0.999898i	$-0.495448\pi$
$311$	48784.9	0.0286012	0.0143006	+	0.999898i	$0.495448\pi$
$312$	0	0
$313$	1.31634e6	0.759466	0.379733	−	0.925096i	$-0.376016\pi$
$313$	1.31634e6	0.759466	0.379733	+	0.925096i	$0.376016\pi$
$314$	0	0
$315$	−1748.47	−0.000992847 0
$316$	0	0
$317$	−3.08008e6	−1.72153	−0.860763	−	0.509006i	$-0.830013\pi$
$317$	−3.08008e6	−1.72153	−0.860763	+	0.509006i	$0.830013\pi$
$318$	0	0
$319$	−158970.	−0.0874659
$320$	0	0
$321$	−611814.	−0.331403
$322$	0	0
$323$	3.12743e6	1.66794
$324$	0	0
$325$	1.25915e6	0.661253
$326$	0	0
$327$	−824272.	−0.426286
$328$	0	0
$329$	−1.30578e6	−0.665090
$330$	0	0
$331$	−3.20819e6	−1.60950	−0.804749	−	0.593615i	$-0.797700\pi$
$331$	−3.20819e6	−1.60950	−0.804749	+	0.593615i	$0.797700\pi$
$332$	0	0
$333$	306893.	0.151662
$334$	0	0
$335$	21397.6	0.0104172
$336$	0	0
$337$	2.77901e6	1.33295	0.666477	−	0.745525i	$-0.267801\pi$
$337$	2.77901e6	1.33295	0.666477	+	0.745525i	$0.267801\pi$
$338$	0	0
$339$	−1.38919e6	−0.656544
$340$	0	0
$341$	1.96390e6	0.914604
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−10747.4	−0.00486136
$346$	0	0
$347$	3.01762e6	1.34537	0.672684	−	0.739930i	$-0.265141\pi$
$347$	3.01762e6	1.34537	0.672684	+	0.739930i	$0.265141\pi$
$348$	0	0
$349$	1.35064e6	0.593574	0.296787	−	0.954944i	$-0.404085\pi$
$349$	1.35064e6	0.593574	0.296787	+	0.954944i	$0.404085\pi$
$350$	0	0
$351$	293752.	0.127266
$352$	0	0
$353$	158501.	0.0677011	0.0338506	−	0.999427i	$-0.489223\pi$
$353$	158501.	0.0677011	0.0338506	+	0.999427i	$0.489223\pi$
$354$	0	0
$355$	−5622.96	−0.00236807
$356$	0	0
$357$	−776836.	−0.322596
$358$	0	0
$359$	−1.54504e6	−0.632709	−0.316354	−	0.948641i	$-0.602459\pi$
$359$	−1.54504e6	−0.632709	−0.316354	+	0.948641i	$0.602459\pi$
$360$	0	0
$361$	675948.	0.272989
$362$	0	0
$363$	1.07714e6	0.429049
$364$	0	0
$365$	−4472.43	−0.00175716
$366$	0	0
$367$	2.51991e6	0.976607	0.488304	−	0.872674i	$-0.337616\pi$
$367$	2.51991e6	0.976607	0.488304	+	0.872674i	$0.337616\pi$
$368$	0	0
$369$	−1.46076e6	−0.558487
$370$	0	0
$371$	1.11525e6	0.420667
$372$	0	0
$373$	2.94012e6	1.09419	0.547096	−	0.837070i	$-0.315733\pi$
$373$	2.94012e6	1.09419	0.547096	+	0.837070i	$0.315733\pi$
$374$	0	0
$375$	−24779.2	−0.00909931
$376$	0	0
$377$	−314945.	−0.114125
$378$	0	0
$379$	4.62230e6	1.65295	0.826475	−	0.562973i	$-0.190342\pi$
$379$	4.62230e6	1.65295	0.826475	+	0.562973i	$0.190342\pi$
$380$	0	0
$381$	486768.	0.171795
$382$	0	0
$383$	−4.60273e6	−1.60331	−0.801656	−	0.597785i	$-0.796048\pi$
$383$	−4.60273e6	−1.60331	−0.801656	+	0.597785i	$0.796048\pi$
$384$	0	0
$385$	4390.44	0.00150958
$386$	0	0
$387$	−1.06917e6	−0.362886
$388$	0	0
$389$	−3.16928e6	−1.06191	−0.530953	−	0.847401i	$-0.678166\pi$
$389$	−3.16928e6	−1.06191	−0.530953	+	0.847401i	$0.678166\pi$
$390$	0	0
$391$	−4.77503e6	−1.57955
$392$	0	0
$393$	−395834.	−0.129280
$394$	0	0
$395$	3834.01	0.00123641
$396$	0	0
$397$	−3.11443e6	−0.991749	−0.495874	−	0.868394i	$-0.665152\pi$
$397$	−3.11443e6	−0.991749	−0.495874	+	0.868394i	$0.665152\pi$
$398$	0	0
$399$	−782952.	−0.246208
$400$	0	0
$401$	6.35836e6	1.97462	0.987312	−	0.158795i	$-0.0507608\pi$
$401$	6.35836e6	1.97462	0.987312	+	0.158795i	$0.0507608\pi$
$402$	0	0
$403$	3.89079e6	1.19337
$404$	0	0
$405$	−2890.33	−0.000875609 0
$406$	0	0
$407$	−770612.	−0.230595
$408$	0	0
$409$	4.46377e6	1.31945	0.659726	−	0.751507i	$-0.270673\pi$
$409$	4.46377e6	1.31945	0.659726	+	0.751507i	$0.270673\pi$
$410$	0	0
$411$	−716208.	−0.209139
$412$	0	0
$413$	77398.3	0.0223283
$414$	0	0
$415$	11575.9	0.00329940
$416$	0	0
$417$	1.86556e6	0.525373
$418$	0	0
$419$	−2.96754e6	−0.825773	−0.412887	−	0.910782i	$-0.635479\pi$
$419$	−2.96754e6	−0.825773	−0.412887	+	0.910782i	$0.635479\pi$
$420$	0	0
$421$	6.81021e6	1.87264	0.936322	−	0.351143i	$-0.114207\pi$
$421$	6.81021e6	1.87264	0.936322	+	0.351143i	$0.114207\pi$
$422$	0	0
$423$	−2.15854e6	−0.586554
$424$	0	0
$425$	−5.50445e6	−1.47823
$426$	0	0
$427$	−1.73107e6	−0.459457
$428$	0	0
$429$	−737614.	−0.193502
$430$	0	0
$431$	−675836.	−0.175246	−0.0876230	−	0.996154i	$-0.527927\pi$
$431$	−675836.	−0.175246	−0.0876230	+	0.996154i	$0.527927\pi$
$432$	0	0
$433$	−1.24399e6	−0.318859	−0.159429	−	0.987209i	$-0.550965\pi$
$433$	−1.24399e6	−0.318859	−0.159429	+	0.987209i	$0.550965\pi$
$434$	0	0
$435$	3098.86	0.000785197 0
$436$	0	0
$437$	−4.81262e6	−1.20553
$438$	0	0
$439$	313929.	0.0777445	0.0388723	−	0.999244i	$-0.487623\pi$
$439$	313929.	0.0777445	0.0388723	+	0.999244i	$0.487623\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−3.43067e6	−0.830557	−0.415279	−	0.909694i	$-0.636316\pi$
$443$	−3.43067e6	−0.830557	−0.415279	+	0.909694i	$0.636316\pi$
$444$	0	0
$445$	−14801.2	−0.00354321
$446$	0	0
$447$	−3.44595e6	−0.815719
$448$	0	0
$449$	363883.	0.0851816	0.0425908	−	0.999093i	$-0.486439\pi$
$449$	363883.	0.0851816	0.0425908	+	0.999093i	$0.486439\pi$
$450$	0	0
$451$	3.66799e6	0.849154
$452$	0	0
$453$	−452589.	−0.103623
$454$	0	0
$455$	8698.14	0.00196969
$456$	0	0
$457$	−5.52131e6	−1.23666	−0.618332	−	0.785917i	$-0.712191\pi$
$457$	−5.52131e6	−1.23666	−0.618332	+	0.785917i	$0.712191\pi$
$458$	0	0
$459$	−1.28416e6	−0.284503
$460$	0	0
$461$	−300842.	−0.0659306	−0.0329653	−	0.999456i	$-0.510495\pi$
$461$	−300842.	−0.0659306	−0.0329653	+	0.999456i	$0.510495\pi$
$462$	0	0
$463$	−6.80343e6	−1.47494	−0.737472	−	0.675378i	$-0.763981\pi$
$463$	−6.80343e6	−1.47494	−0.737472	+	0.675378i	$0.763981\pi$
$464$	0	0
$465$	−38283.0	−0.00821057
$466$	0	0
$467$	2.40398e6	0.510080	0.255040	−	0.966931i	$-0.417911\pi$
$467$	2.40398e6	0.510080	0.255040	+	0.966931i	$0.417911\pi$
$468$	0	0
$469$	−2.38003e6	−0.499633
$470$	0	0
$471$	−3.30855e6	−0.687203
$472$	0	0
$473$	2.68471e6	0.551752
$474$	0	0
$475$	−5.54778e6	−1.12820
$476$	0	0
$477$	1.84358e6	0.370993
$478$	0	0
$479$	5.09265e6	1.01416	0.507078	−	0.861900i	$-0.330726\pi$
$479$	5.09265e6	1.01416	0.507078	+	0.861900i	$0.330726\pi$
$480$	0	0
$481$	−1.52670e6	−0.300879
$482$	0	0
$483$	1.19543e6	0.233161
$484$	0	0
$485$	−10132.4	−0.00195596
$486$	0	0
$487$	−7.33002e6	−1.40050	−0.700249	−	0.713899i	$-0.746928\pi$
$487$	−7.33002e6	−1.40050	−0.700249	+	0.713899i	$0.746928\pi$
$488$	0	0
$489$	1.86275e6	0.352275
$490$	0	0
$491$	1.36290e6	0.255130	0.127565	−	0.991830i	$-0.459284\pi$
$491$	1.36290e6	0.255130	0.127565	+	0.991830i	$0.459284\pi$
$492$	0	0
$493$	1.37680e6	0.255126
$494$	0	0
$495$	7257.66	0.00133132
$496$	0	0
$497$	625436.	0.113577
$498$	0	0
$499$	5.05087e6	0.908060	0.454030	−	0.890986i	$-0.349986\pi$
$499$	5.05087e6	0.908060	0.454030	+	0.890986i	$0.349986\pi$
$500$	0	0
$501$	−2.31557e6	−0.412157
$502$	0	0
$503$	−5.93004e6	−1.04505	−0.522526	−	0.852623i	$-0.675010\pi$
$503$	−5.93004e6	−1.04505	−0.522526	+	0.852623i	$0.675010\pi$
$504$	0	0
$505$	−25342.9	−0.00442208
$506$	0	0
$507$	1.88031e6	0.324870
$508$	0	0
$509$	−7.01042e6	−1.19936	−0.599680	−	0.800240i	$-0.704706\pi$
$509$	−7.01042e6	−1.19936	−0.599680	+	0.800240i	$0.704706\pi$
$510$	0	0
$511$	497464.	0.0842770
$512$	0	0
$513$	−1.29427e6	−0.217135
$514$	0	0
$515$	73918.9	0.0122811
$516$	0	0
$517$	5.42011e6	0.891829
$518$	0	0
$519$	2.42129e6	0.394573
$520$	0	0
$521$	−4.40461e6	−0.710908	−0.355454	−	0.934694i	$-0.615674\pi$
$521$	−4.40461e6	−0.710908	−0.355454	+	0.934694i	$0.615674\pi$
$522$	0	0
$523$	−7.18270e6	−1.14824	−0.574121	−	0.818770i	$-0.694656\pi$
$523$	−7.18270e6	−1.14824	−0.574121	+	0.818770i	$0.694656\pi$
$524$	0	0
$525$	1.37804e6	0.218204
$526$	0	0
$527$	−1.70089e7	−2.66778
$528$	0	0
$529$	911672.	0.141644
$530$	0	0
$531$	127944.	0.0196917
$532$	0	0
$533$	7.26686e6	1.10797
$534$	0	0
$535$	−29947.1	−0.00452345
$536$	0	0
$537$	−4.80599e6	−0.719195
$538$	0	0
$539$	−488344.	−0.0724026
$540$	0	0
$541$	9.23327e6	1.35632	0.678161	−	0.734914i	$-0.262777\pi$
$541$	9.23327e6	1.35632	0.678161	+	0.734914i	$0.262777\pi$
$542$	0	0
$543$	5.66755e6	0.824889
$544$	0	0
$545$	−40346.5	−0.00581855
$546$	0	0
$547$	−4.61993e6	−0.660187	−0.330093	−	0.943948i	$-0.607080\pi$
$547$	−4.61993e6	−0.660187	−0.330093	+	0.943948i	$0.607080\pi$
$548$	0	0
$549$	−2.86156e6	−0.405203
$550$	0	0
$551$	1.38764e6	0.194715
$552$	0	0
$553$	−426454.	−0.0593006
$554$	0	0
$555$	15021.8	0.00207009
$556$	0	0
$557$	9.71536e6	1.32685	0.663424	−	0.748244i	$-0.269103\pi$
$557$	9.71536e6	1.32685	0.663424	+	0.748244i	$0.269103\pi$
$558$	0	0
$559$	5.31882e6	0.719923
$560$	0	0
$561$	3.22454e6	0.432574
$562$	0	0
$563$	9.70410e6	1.29028	0.645141	−	0.764064i	$-0.276799\pi$
$563$	9.70410e6	1.29028	0.645141	+	0.764064i	$0.276799\pi$
$564$	0	0
$565$	−67998.3	−0.00896143
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	7.91242e6	1.02454	0.512270	−	0.858825i	$-0.328805\pi$
$569$	7.91242e6	1.02454	0.512270	+	0.858825i	$0.328805\pi$
$570$	0	0
$571$	7.91956e6	1.01651	0.508254	−	0.861207i	$-0.330291\pi$
$571$	7.91956e6	1.01651	0.508254	+	0.861207i	$0.330291\pi$
$572$	0	0
$573$	4.06979e6	0.517827
$574$	0	0
$575$	8.47048e6	1.06841
$576$	0	0
$577$	−6.81761e6	−0.852497	−0.426248	−	0.904606i	$-0.640165\pi$
$577$	−6.81761e6	−0.852497	−0.426248	+	0.904606i	$0.640165\pi$
$578$	0	0
$579$	3.35120e6	0.415436
$580$	0	0
$581$	−1.28758e6	−0.158246
$582$	0	0
$583$	−4.62925e6	−0.564079
$584$	0	0
$585$	14378.6	0.00173710
$586$	0	0
$587$	5.75763e6	0.689682	0.344841	−	0.938661i	$-0.387933\pi$
$587$	5.75763e6	0.689682	0.344841	+	0.938661i	$0.387933\pi$
$588$	0	0
$589$	−1.71428e7	−2.03607
$590$	0	0
$591$	2.25099e6	0.265097
$592$	0	0
$593$	1.22350e6	0.142879	0.0714393	−	0.997445i	$-0.477241\pi$
$593$	1.22350e6	0.142879	0.0714393	+	0.997445i	$0.477241\pi$
$594$	0	0
$595$	−38024.6	−0.00440324
$596$	0	0
$597$	8.42422e6	0.967373
$598$	0	0
$599$	−590873.	−0.0672864	−0.0336432	−	0.999434i	$-0.510711\pi$
$599$	−590873.	−0.0672864	−0.0336432	+	0.999434i	$0.510711\pi$
$600$	0	0
$601$	−1.27472e7	−1.43955	−0.719776	−	0.694207i	$-0.755755\pi$
$601$	−1.27472e7	−1.43955	−0.719776	+	0.694207i	$0.755755\pi$
$602$	0	0
$603$	−3.93434e6	−0.440634
$604$	0	0
$605$	52724.1	0.00585626
$606$	0	0
$607$	7.36918e6	0.811797	0.405899	−	0.913918i	$-0.366959\pi$
$607$	7.36918e6	0.811797	0.405899	+	0.913918i	$0.366959\pi$
$608$	0	0
$609$	−344683.	−0.0376597
$610$	0	0
$611$	1.07381e7	1.16365
$612$	0	0
$613$	−5.05204e6	−0.543020	−0.271510	−	0.962436i	$-0.587523\pi$
$613$	−5.05204e6	−0.543020	−0.271510	+	0.962436i	$0.587523\pi$
$614$	0	0
$615$	−71501.3	−0.00762301
$616$	0	0
$617$	1.04265e7	1.10262	0.551309	−	0.834301i	$-0.314129\pi$
$617$	1.04265e7	1.10262	0.551309	+	0.834301i	$0.314129\pi$
$618$	0	0
$619$	1.84513e7	1.93553	0.967765	−	0.251856i	$-0.0810410\pi$
$619$	1.84513e7	1.93553	0.967765	+	0.251856i	$0.0810410\pi$
$620$	0	0
$621$	1.97612e6	0.205629
$622$	0	0
$623$	1.64633e6	0.169940
$624$	0	0
$625$	9.76381e6	0.999814
$626$	0	0
$627$	3.24992e6	0.330144
$628$	0	0
$629$	6.67410e6	0.672614
$630$	0	0
$631$	2.28762e6	0.228723	0.114362	−	0.993439i	$-0.463518\pi$
$631$	2.28762e6	0.228723	0.114362	+	0.993439i	$0.463518\pi$
$632$	0	0
$633$	−8.65528e6	−0.858562
$634$	0	0
$635$	23826.3	0.00234489
$636$	0	0
$637$	−967487.	−0.0944705
$638$	0	0
$639$	1.03388e6	0.100166
$640$	0	0
$641$	−6.98845e6	−0.671793	−0.335897	−	0.941899i	$-0.609039\pi$
$641$	−6.98845e6	−0.671793	−0.335897	+	0.941899i	$0.609039\pi$
$642$	0	0
$643$	−1.47461e7	−1.40653	−0.703265	−	0.710928i	$-0.748275\pi$
$643$	−1.47461e7	−1.40653	−0.703265	+	0.710928i	$0.748275\pi$
$644$	0	0
$645$	−52333.9	−0.00495318
$646$	0	0
$647$	−3.20477e6	−0.300979	−0.150490	−	0.988612i	$-0.548085\pi$
$647$	−3.20477e6	−0.300979	−0.150490	+	0.988612i	$0.548085\pi$
$648$	0	0
$649$	−321269.	−0.0299404
$650$	0	0
$651$	4.25818e6	0.393796
$652$	0	0
$653$	−2.10963e6	−0.193608	−0.0968041	−	0.995303i	$-0.530862\pi$
$653$	−2.10963e6	−0.193608	−0.0968041	+	0.995303i	$0.530862\pi$
$654$	0	0
$655$	−19375.3	−0.00176459
$656$	0	0
$657$	822338.	0.0743254
$658$	0	0
$659$	−1.74852e7	−1.56840	−0.784202	−	0.620505i	$-0.786928\pi$
$659$	−1.74852e7	−1.56840	−0.784202	+	0.620505i	$0.786928\pi$
$660$	0	0
$661$	−973131.	−0.0866299	−0.0433149	−	0.999061i	$-0.513792\pi$
$661$	−973131.	−0.0866299	−0.0433149	+	0.999061i	$0.513792\pi$
$662$	0	0
$663$	6.38831e6	0.564420
$664$	0	0
$665$	−38323.9	−0.00336059
$666$	0	0
$667$	−2.11869e6	−0.184396
$668$	0	0
$669$	5.61779e6	0.485289
$670$	0	0
$671$	7.18542e6	0.616092
$672$	0	0
$673$	651051.	0.0554086	0.0277043	−	0.999616i	$-0.491180\pi$
$673$	651051.	0.0554086	0.0277043	+	0.999616i	$0.491180\pi$
$674$	0	0
$675$	2.27798e6	0.192438
$676$	0	0
$677$	−4.62839e6	−0.388113	−0.194057	−	0.980990i	$-0.562165\pi$
$677$	−4.62839e6	−0.388113	−0.194057	+	0.980990i	$0.562165\pi$
$678$	0	0
$679$	1.12702e6	0.0938119
$680$	0	0
$681$	−6.28605e6	−0.519410
$682$	0	0
$683$	1.77222e7	1.45367	0.726836	−	0.686811i	$-0.240990\pi$
$683$	1.77222e7	1.45367	0.726836	+	0.686811i	$0.240990\pi$
$684$	0	0
$685$	−35057.0	−0.00285462
$686$	0	0
$687$	8.88436e6	0.718182
$688$	0	0
$689$	−9.17128e6	−0.736007
$690$	0	0
$691$	1.41125e7	1.12437	0.562186	−	0.827011i	$-0.309960\pi$
$691$	1.41125e7	1.12437	0.562186	+	0.827011i	$0.309960\pi$
$692$	0	0
$693$	−807263.	−0.0638531
$694$	0	0
$695$	91315.3	0.00717103
$696$	0	0
$697$	−3.17676e7	−2.47687
$698$	0	0
$699$	1.00065e7	0.774619
$700$	0	0
$701$	1.02386e6	0.0786947	0.0393474	−	0.999226i	$-0.487472\pi$
$701$	1.02386e6	0.0786947	0.0393474	+	0.999226i	$0.487472\pi$
$702$	0	0
$703$	6.72664e6	0.513346
$704$	0	0
$705$	−105656.	−0.00800611
$706$	0	0
$707$	2.81886e6	0.212092
$708$	0	0
$709$	−7.14313e6	−0.533670	−0.266835	−	0.963742i	$-0.585978\pi$
$709$	−7.14313e6	−0.533670	−0.266835	+	0.963742i	$0.585978\pi$
$710$	0	0
$711$	−704954.	−0.0522982
$712$	0	0
$713$	2.61740e7	1.92818
$714$	0	0
$715$	−36104.8	−0.00264119
$716$	0	0
$717$	−1.35100e7	−0.981427
$718$	0	0
$719$	1.74905e7	1.26177	0.630884	−	0.775878i	$-0.282693\pi$
$719$	1.74905e7	1.26177	0.630884	+	0.775878i	$0.282693\pi$
$720$	0	0
$721$	−8.22193e6	−0.589027
$722$	0	0
$723$	1.55461e6	0.110605
$724$	0	0
$725$	−2.44233e6	−0.172568
$726$	0	0
$727$	−8.07868e6	−0.566897	−0.283449	−	0.958987i	$-0.591479\pi$
$727$	−8.07868e6	−0.566897	−0.283449	+	0.958987i	$0.591479\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−2.32516e7	−1.60939
$732$	0	0
$733$	−6.42261e6	−0.441521	−0.220761	−	0.975328i	$-0.570854\pi$
$733$	−6.42261e6	−0.441521	−0.220761	+	0.975328i	$0.570854\pi$
$734$	0	0
$735$	9519.46	0.000649971 0
$736$	0	0
$737$	9.87917e6	0.669965
$738$	0	0
$739$	−2.45407e7	−1.65301	−0.826506	−	0.562927i	$-0.809675\pi$
$739$	−2.45407e7	−1.65301	−0.826506	+	0.562927i	$0.809675\pi$
$740$	0	0
$741$	6.43860e6	0.430771
$742$	0	0
$743$	−1.53250e7	−1.01842	−0.509212	−	0.860641i	$-0.670063\pi$
$743$	−1.53250e7	−1.01842	−0.509212	+	0.860641i	$0.670063\pi$
$744$	0	0
$745$	−168673.	−0.0111341
$746$	0	0
$747$	−2.12844e6	−0.139560
$748$	0	0
$749$	3.33099e6	0.216954
$750$	0	0
$751$	−1.24971e7	−0.808554	−0.404277	−	0.914637i	$-0.632477\pi$
$751$	−1.24971e7	−0.808554	−0.404277	+	0.914637i	$0.632477\pi$
$752$	0	0
$753$	1.23283e6	0.0792346
$754$	0	0
$755$	−22153.3	−0.00141440
$756$	0	0
$757$	−1.56124e7	−0.990217	−0.495108	−	0.868831i	$-0.664872\pi$
$757$	−1.56124e7	−0.990217	−0.495108	+	0.868831i	$0.664872\pi$
$758$	0	0
$759$	−4.96205e6	−0.312649
$760$	0	0
$761$	1.82820e7	1.14436	0.572179	−	0.820128i	$-0.306098\pi$
$761$	1.82820e7	1.14436	0.572179	+	0.820128i	$0.306098\pi$
$762$	0	0
$763$	4.48770e6	0.279070
$764$	0	0
$765$	−62857.0	−0.00388329
$766$	0	0
$767$	−636485.	−0.0390661
$768$	0	0
$769$	1.32020e7	0.805051	0.402526	−	0.915409i	$-0.368132\pi$
$769$	1.32020e7	0.805051	0.402526	+	0.915409i	$0.368132\pi$
$770$	0	0
$771$	1.77904e7	1.07783
$772$	0	0
$773$	−1.45645e7	−0.876689	−0.438344	−	0.898807i	$-0.644435\pi$
$773$	−1.45645e7	−0.876689	−0.438344	+	0.898807i	$0.644435\pi$
$774$	0	0
$775$	3.01723e7	1.80449
$776$	0	0
$777$	−1.67086e6	−0.0992860
$778$	0	0
$779$	−3.20177e7	−1.89037
$780$	0	0
$781$	−2.59610e6	−0.152298
$782$	0	0
$783$	−569783.	−0.0332127
$784$	0	0
$785$	−161947.	−0.00937990
$786$	0	0
$787$	1.78163e7	1.02537	0.512685	−	0.858577i	$-0.328651\pi$
$787$	1.78163e7	1.02537	0.512685	+	0.858577i	$0.328651\pi$
$788$	0	0
$789$	−3.72903e6	−0.213257
$790$	0	0
$791$	7.56339e6	0.429809
$792$	0	0
$793$	1.42354e7	0.803874
$794$	0	0
$795$	90239.6	0.00506384
$796$	0	0
$797$	−2.29429e7	−1.27939	−0.639696	−	0.768628i	$-0.720940\pi$
$797$	−2.29429e7	−1.27939	−0.639696	+	0.768628i	$0.720940\pi$
$798$	0	0
$799$	−4.69424e7	−2.60134
$800$	0	0
$801$	2.72148e6	0.149873
$802$	0	0
$803$	−2.06490e6	−0.113008
$804$	0	0
$805$	58513.9	0.00318251
$806$	0	0
$807$	−2.17574e6	−0.117604
$808$	0	0
$809$	2.92525e6	0.157142	0.0785710	−	0.996909i	$-0.474964\pi$
$809$	2.92525e6	0.157142	0.0785710	+	0.996909i	$0.474964\pi$
$810$	0	0
$811$	7.22818e6	0.385902	0.192951	−	0.981208i	$-0.438194\pi$
$811$	7.22818e6	0.385902	0.192951	+	0.981208i	$0.438194\pi$
$812$	0	0
$813$	−5.02933e6	−0.266860
$814$	0	0
$815$	91177.8	0.00480834
$816$	0	0
$817$	−2.34347e7	−1.22830
$818$	0	0
$819$	−1.59931e6	−0.0833152
$820$	0	0
$821$	2.85795e7	1.47978	0.739890	−	0.672728i	$-0.234878\pi$
$821$	2.85795e7	1.47978	0.739890	+	0.672728i	$0.234878\pi$
$822$	0	0
$823$	1.31774e6	0.0678157	0.0339078	−	0.999425i	$-0.489205\pi$
$823$	1.31774e6	0.0678157	0.0339078	+	0.999425i	$0.489205\pi$
$824$	0	0
$825$	−5.72005e6	−0.292593
$826$	0	0
$827$	−3.02202e7	−1.53650	−0.768252	−	0.640147i	$-0.778873\pi$
$827$	−3.02202e7	−1.53650	−0.768252	+	0.640147i	$0.778873\pi$
$828$	0	0
$829$	−8.04061e6	−0.406352	−0.203176	−	0.979142i	$-0.565126\pi$
$829$	−8.04061e6	−0.406352	−0.203176	+	0.979142i	$0.565126\pi$
$830$	0	0
$831$	−2.08942e6	−0.104960
$832$	0	0
$833$	4.22944e6	0.211189
$834$	0	0
$835$	−113342.	−0.00562570
$836$	0	0
$837$	7.03903e6	0.347296
$838$	0	0
$839$	−4.62114e6	−0.226644	−0.113322	−	0.993558i	$-0.536149\pi$
$839$	−4.62114e6	−0.226644	−0.113322	+	0.993558i	$0.536149\pi$
$840$	0	0
$841$	−1.99003e7	−0.970217
$842$	0	0
$843$	−1.67235e7	−0.810511
$844$	0	0
$845$	92037.4	0.00443427
$846$	0	0
$847$	−5.86445e6	−0.280879
$848$	0	0
$849$	−2.18624e7	−1.04095
$850$	0	0
$851$	−1.02704e7	−0.486142
$852$	0	0
$853$	2.20395e7	1.03712	0.518559	−	0.855042i	$-0.326468\pi$
$853$	2.20395e7	1.03712	0.518559	+	0.855042i	$0.326468\pi$
$854$	0	0
$855$	−63351.8	−0.00296377
$856$	0	0
$857$	−1.48477e7	−0.690569	−0.345284	−	0.938498i	$-0.612218\pi$
$857$	−1.48477e7	−0.690569	−0.345284	+	0.938498i	$0.612218\pi$
$858$	0	0
$859$	−668744.	−0.0309227	−0.0154613	−	0.999880i	$-0.504922\pi$
$859$	−668744.	−0.0309227	−0.0154613	+	0.999880i	$0.504922\pi$
$860$	0	0
$861$	7.95303e6	0.365616
$862$	0	0
$863$	−1.09838e7	−0.502024	−0.251012	−	0.967984i	$-0.580763\pi$
$863$	−1.09838e7	−0.502024	−0.251012	+	0.967984i	$0.580763\pi$
$864$	0	0
$865$	118517.	0.00538569
$866$	0	0
$867$	−1.51483e7	−0.684408
$868$	0	0
$869$	1.77015e6	0.0795171
$870$	0	0
$871$	1.95722e7	0.874166
$872$	0	0
$873$	1.86304e6	0.0827344
$874$	0	0
$875$	134909.	0.00595690
$876$	0	0
$877$	4.15400e7	1.82376	0.911879	−	0.410460i	$-0.134632\pi$
$877$	4.15400e7	1.82376	0.911879	+	0.410460i	$0.134632\pi$
$878$	0	0
$879$	1.45721e7	0.636134
$880$	0	0
$881$	4.03497e7	1.75146	0.875730	−	0.482801i	$-0.160381\pi$
$881$	4.03497e7	1.75146	0.875730	+	0.482801i	$0.160381\pi$
$882$	0	0
$883$	−2.10878e7	−0.910183	−0.455092	−	0.890445i	$-0.650394\pi$
$883$	−2.10878e7	−0.910183	−0.455092	+	0.890445i	$0.650394\pi$
$884$	0	0
$885$	6262.61	0.000268780 0
$886$	0	0
$887$	3.52443e6	0.150411	0.0752056	−	0.997168i	$-0.476039\pi$
$887$	3.52443e6	0.150411	0.0752056	+	0.997168i	$0.476039\pi$
$888$	0	0
$889$	−2.65018e6	−0.112466
$890$	0	0
$891$	−1.33446e6	−0.0563131
$892$	0	0
$893$	−4.73119e7	−1.98537
$894$	0	0
$895$	−235244.	−0.00981658
$896$	0	0
$897$	−9.83061e6	−0.407943
$898$	0	0
$899$	−7.54687e6	−0.311435
$900$	0	0
$901$	4.00929e7	1.64534
$902$	0	0
$903$	5.82105e6	0.237565
$904$	0	0
$905$	277415.	0.0112592
$906$	0	0
$907$	−2.95093e7	−1.19108	−0.595540	−	0.803325i	$-0.703062\pi$
$907$	−2.95093e7	−1.19108	−0.595540	+	0.803325i	$0.703062\pi$
$908$	0	0
$909$	4.65975e6	0.187048
$910$	0	0
$911$	1.32643e7	0.529527	0.264763	−	0.964313i	$-0.414706\pi$
$911$	1.32643e7	0.529527	0.264763	+	0.964313i	$0.414706\pi$
$912$	0	0
$913$	5.34455e6	0.212195
$914$	0	0
$915$	−140068.	−0.00553077
$916$	0	0
$917$	2.15509e6	0.0846336
$918$	0	0
$919$	1.89419e7	0.739834	0.369917	−	0.929065i	$-0.379386\pi$
$919$	1.89419e7	0.739834	0.369917	+	0.929065i	$0.379386\pi$
$920$	0	0
$921$	−8.42845e6	−0.327415
$922$	0	0
$923$	−5.14327e6	−0.198717
$924$	0	0
$925$	−1.18393e7	−0.454957
$926$	0	0
$927$	−1.35913e7	−0.519473
$928$	0	0
$929$	2.92211e7	1.11086	0.555428	−	0.831565i	$-0.312555\pi$
$929$	2.92211e7	1.11086	0.555428	+	0.831565i	$0.312555\pi$
$930$	0	0
$931$	4.26274e6	0.161181
$932$	0	0
$933$	−439064.	−0.0165129
$934$	0	0
$935$	157835.	0.00590437
$936$	0	0
$937$	−6.46944e6	−0.240723	−0.120362	−	0.992730i	$-0.538405\pi$
$937$	−6.46944e6	−0.240723	−0.120362	+	0.992730i	$0.538405\pi$
$938$	0	0
$939$	−1.18471e7	−0.438478
$940$	0	0
$941$	−4.11564e7	−1.51518	−0.757588	−	0.652733i	$-0.773622\pi$
$941$	−4.11564e7	−1.51518	−0.757588	+	0.652733i	$0.773622\pi$
$942$	0	0
$943$	4.88854e7	1.79019
$944$	0	0
$945$	15736.3	0.000573221 0
$946$	0	0
$947$	−3.00657e7	−1.08942	−0.544711	−	0.838624i	$-0.683361\pi$
$947$	−3.00657e7	−1.08942	−0.544711	+	0.838624i	$0.683361\pi$
$948$	0	0
$949$	−4.09089e6	−0.147453
$950$	0	0
$951$	2.77207e7	0.993923
$952$	0	0
$953$	2.95724e7	1.05476	0.527381	−	0.849629i	$-0.323174\pi$
$953$	2.95724e7	1.05476	0.527381	+	0.849629i	$0.323174\pi$
$954$	0	0
$955$	199208.	0.00706803
$956$	0	0
$957$	1.43073e6	0.0504985
$958$	0	0
$959$	3.89935e6	0.136913
$960$	0	0
$961$	6.46041e7	2.25658
$962$	0	0
$963$	5.50633e6	0.191336
$964$	0	0
$965$	164035.	0.00567045
$966$	0	0
$967$	3.49436e7	1.20172	0.600858	−	0.799356i	$-0.294826\pi$
$967$	3.49436e7	1.20172	0.600858	+	0.799356i	$0.294826\pi$
$968$	0	0
$969$	−2.81468e7	−0.962986
$970$	0	0
$971$	−1.13425e7	−0.386065	−0.193032	−	0.981192i	$-0.561832\pi$
$971$	−1.13425e7	−0.386065	−0.193032	+	0.981192i	$0.561832\pi$
$972$	0	0
$973$	−1.01569e7	−0.343938
$974$	0	0
$975$	−1.13323e7	−0.381774
$976$	0	0
$977$	4.24082e7	1.42139	0.710694	−	0.703501i	$-0.248381\pi$
$977$	4.24082e7	1.42139	0.710694	+	0.703501i	$0.248381\pi$
$978$	0	0
$979$	−6.83366e6	−0.227875
$980$	0	0
$981$	7.41845e6	0.246116
$982$	0	0
$983$	−3.98560e7	−1.31556	−0.657779	−	0.753211i	$-0.728504\pi$
$983$	−3.98560e7	−1.31556	−0.657779	+	0.753211i	$0.728504\pi$
$984$	0	0
$985$	110182.	0.00361841
$986$	0	0
$987$	1.17520e7	0.383990
$988$	0	0
$989$	3.57806e7	1.16321
$990$	0	0
$991$	2.81338e7	0.910007	0.455004	−	0.890490i	$-0.349638\pi$
$991$	2.81338e7	0.910007	0.455004	+	0.890490i	$0.349638\pi$
$992$	0	0
$993$	2.88737e7	0.929244
$994$	0	0
$995$	412349.	0.0132041
$996$	0	0
$997$	5.67938e7	1.80952	0.904760	−	0.425922i	$-0.140050\pi$
$997$	5.67938e7	1.80952	0.904760	+	0.425922i	$0.140050\pi$
$998$	0	0
$999$	−2.76204e6	−0.0875620

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.6.a.h.1.2	✓	2
3.2	odd	2		504.6.a.t.1.1		2
4.3	odd	2		336.6.a.w.1.2		2
12.11	even	2		1008.6.a.bu.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.h.1.2	✓	2	1.1	even	1	trivial
336.6.a.w.1.2		2	4.3	odd	2
504.6.a.t.1.1		2	3.2	odd	2
1008.6.a.bu.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	168.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -0.440532 q^{5} +49.0000 q^{7} +81.0000 q^{9} -203.392 q^{11} -402.951 q^{13} +3.96479 q^{15} +1761.53 q^{17} +1775.40 q^{19} -441.000 q^{21} -2710.72 q^{23} -3124.81 q^{25} -729.000 q^{27} +781.595 q^{29} -9655.74 q^{31} +1830.53 q^{33} -21.5861 q^{35} +3788.80 q^{37} +3626.56 q^{39} -18034.1 q^{41} -13199.7 q^{43} -35.6831 q^{45} -26648.6 q^{47} +2401.00 q^{49} -15853.8 q^{51} +22760.3 q^{53} +89.6008 q^{55} -15978.6 q^{57} +1579.56 q^{59} -35327.9 q^{61} +3969.00 q^{63} +177.513 q^{65} -48572.1 q^{67} +24396.5 q^{69} +12764.0 q^{71} +10152.3 q^{73} +28123.3 q^{75} -9966.21 q^{77} -8703.14 q^{79} +6561.00 q^{81} -26277.1 q^{83} -776.012 q^{85} -7034.35 q^{87} +33598.5 q^{89} -19744.6 q^{91} +86901.6 q^{93} -782.121 q^{95} +23000.5 q^{97} -16474.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 64 q^{5} + 98 q^{7} + 162 q^{9} + 540 q^{11} + 204 q^{13} + 576 q^{15} + 304 q^{17} - 1120 q^{19} - 882 q^{21} - 940 q^{23} - 2210 q^{25} - 1458 q^{27} + 932 q^{29} - 16408 q^{31} - 4860 q^{33}+ \cdots + 43740 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 64 * q^5 + 98 * q^7 + 162 * q^9 + 540 * q^11 + 204 * q^13 + 576 * q^15 + 304 * q^17 - 1120 * q^19 - 882 * q^21 - 940 * q^23 - 2210 * q^25 - 1458 * q^27 + 932 * q^29 - 16408 * q^31 - 4860 * q^33 - 3136 * q^35 - 1764 * q^37 - 1836 * q^39 - 2552 * q^41 - 24632 * q^43 - 5184 * q^45 - 36760 * q^47 + 4802 * q^49 - 2736 * q^51 + 9164 * q^53 - 47160 * q^55 + 10080 * q^57 - 39888 * q^59 - 25084 * q^61 + 7938 * q^63 - 38400 * q^65 - 2592 * q^67 + 8460 * q^69 - 35508 * q^71 - 17188 * q^73 + 19890 * q^75 + 26460 * q^77 - 95800 * q^79 + 13122 * q^81 + 65352 * q^83 + 91864 * q^85 - 8388 * q^87 - 23000 * q^89 + 9996 * q^91 + 147672 * q^93 + 183248 * q^95 - 108388 * q^97 + 43740 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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