LMFDB - Embedded newform 336.6.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, 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("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1129}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 282 $ x^2 - x - 282
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-16.3003$ of defining polynomial
Character	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +67.2012 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +67.2012 q^{5} +49.0000 q^{7} +81.0000 q^{9} +17.2012 q^{11} +807.610 q^{13} -604.811 q^{15} +777.604 q^{17} -262.379 q^{19} -441.000 q^{21} +1950.86 q^{23} +1391.00 q^{25} -729.000 q^{27} -1518.04 q^{29} -5474.47 q^{31} -154.811 q^{33} +3292.86 q^{35} +6586.95 q^{37} -7268.49 q^{39} +2930.30 q^{41} +975.338 q^{43} +5443.30 q^{45} -22985.1 q^{47} +2401.00 q^{49} -6998.43 q^{51} +2645.71 q^{53} +1155.94 q^{55} +2361.41 q^{57} -22510.9 q^{59} +9533.02 q^{61} +3969.00 q^{63} +54272.3 q^{65} +67852.2 q^{67} -17557.7 q^{69} +10158.5 q^{71} +60184.1 q^{73} -12519.0 q^{75} +842.858 q^{77} +27760.3 q^{79} +6561.00 q^{81} +13431.4 q^{83} +52255.9 q^{85} +13662.3 q^{87} +120473. q^{89} +39572.9 q^{91} +49270.3 q^{93} -17632.2 q^{95} -28652.0 q^{97} +1393.30 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 98 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 + 98 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} + 98 q^{7} + 162 q^{9} - 100 q^{11} + 540 q^{13} + 1152 q^{17} - 2944 q^{19} - 882 q^{21} - 2684 q^{23} + 2782 q^{25} - 1458 q^{27} + 996 q^{29} - 2616 q^{31} + 900 q^{33} - 3492 q^{37} - 4860 q^{39} + 17016 q^{41} - 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 10368 q^{51} + 37548 q^{53} + 9032 q^{55} + 26496 q^{57} + 6320 q^{59} + 39764 q^{61} + 7938 q^{63} + 72256 q^{65} + 27376 q^{67} + 24156 q^{69} - 35460 q^{71} + 64188 q^{73} - 25038 q^{75} - 4900 q^{77} + 110088 q^{79} + 13122 q^{81} + 5896 q^{83} + 27096 q^{85} - 8964 q^{87} + 167160 q^{89} + 26460 q^{91} + 23544 q^{93} + 162576 q^{95} - 11876 q^{97} - 8100 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 + 98 * q^7 + 162 * q^9 - 100 * q^11 + 540 * q^13 + 1152 * q^17 - 2944 * q^19 - 882 * q^21 - 2684 * q^23 + 2782 * q^25 - 1458 * q^27 + 996 * q^29 - 2616 * q^31 + 900 * q^33 - 3492 * q^37 - 4860 * q^39 + 17016 * q^41 - 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 10368 * q^51 + 37548 * q^53 + 9032 * q^55 + 26496 * q^57 + 6320 * q^59 + 39764 * q^61 + 7938 * q^63 + 72256 * q^65 + 27376 * q^67 + 24156 * q^69 - 35460 * q^71 + 64188 * q^73 - 25038 * q^75 - 4900 * q^77 + 110088 * q^79 + 13122 * q^81 + 5896 * q^83 + 27096 * q^85 - 8964 * q^87 + 167160 * q^89 + 26460 * q^91 + 23544 * q^93 + 162576 * q^95 - 11876 * q^97 - 8100 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	67.2012	1.20213	0.601066	−	0.799200i	$-0.294743\pi$
$5$	67.2012	1.20213	0.601066	+	0.799200i	$0.294743\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	17.2012	0.0428624	0.0214312	−	0.999770i	$-0.493178\pi$
$11$	17.2012	0.0428624	0.0214312	+	0.999770i	$0.493178\pi$
$12$	0	0
$13$	807.610	1.32539	0.662694	−	0.748890i	$-0.269413\pi$
$13$	807.610	1.32539	0.662694	+	0.748890i	$0.269413\pi$
$14$	0	0
$15$	−604.811	−0.694051
$16$	0	0
$17$	777.604	0.652583	0.326292	−	0.945269i	$-0.394201\pi$
$17$	777.604	0.652583	0.326292	+	0.945269i	$0.394201\pi$
$18$	0	0
$19$	−262.379	−0.166742	−0.0833709	−	0.996519i	$-0.526569\pi$
$19$	−262.379	−0.166742	−0.0833709	+	0.996519i	$0.526569\pi$
$20$	0	0
$21$	−441.000	−0.218218
$22$	0	0
$23$	1950.86	0.768964	0.384482	−	0.923132i	$-0.374380\pi$
$23$	1950.86	0.768964	0.384482	+	0.923132i	$0.374380\pi$
$24$	0	0
$25$	1391.00	0.445120
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−1518.04	−0.335187	−0.167593	−	0.985856i	$-0.553600\pi$
$29$	−1518.04	−0.335187	−0.167593	+	0.985856i	$0.553600\pi$
$30$	0	0
$31$	−5474.47	−1.02315	−0.511574	−	0.859239i	$-0.670937\pi$
$31$	−5474.47	−1.02315	−0.511574	+	0.859239i	$0.670937\pi$
$32$	0	0
$33$	−154.811	−0.0247466
$34$	0	0
$35$	3292.86	0.454363
$36$	0	0
$37$	6586.95	0.791006	0.395503	−	0.918465i	$-0.370570\pi$
$37$	6586.95	0.791006	0.395503	+	0.918465i	$0.370570\pi$
$38$	0	0
$39$	−7268.49	−0.765213
$40$	0	0
$41$	2930.30	0.272240	0.136120	−	0.990692i	$-0.456537\pi$
$41$	2930.30	0.272240	0.136120	+	0.990692i	$0.456537\pi$
$42$	0	0
$43$	975.338	0.0804422	0.0402211	−	0.999191i	$-0.487194\pi$
$43$	975.338	0.0804422	0.0402211	+	0.999191i	$0.487194\pi$
$44$	0	0
$45$	5443.30	0.400710
$46$	0	0
$47$	−22985.1	−1.51776	−0.758878	−	0.651233i	$-0.774252\pi$
$47$	−22985.1	−1.51776	−0.758878	+	0.651233i	$0.774252\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−6998.43	−0.376769
$52$	0	0
$53$	2645.71	0.129376	0.0646879	−	0.997906i	$-0.479395\pi$
$53$	2645.71	0.129376	0.0646879	+	0.997906i	$0.479395\pi$
$54$	0	0
$55$	1155.94	0.0515263
$56$	0	0
$57$	2361.41	0.0962684
$58$	0	0
$59$	−22510.9	−0.841903	−0.420951	−	0.907083i	$-0.638304\pi$
$59$	−22510.9	−0.841903	−0.420951	+	0.907083i	$0.638304\pi$
$60$	0	0
$61$	9533.02	0.328024	0.164012	−	0.986458i	$-0.447556\pi$
$61$	9533.02	0.328024	0.164012	+	0.986458i	$0.447556\pi$
$62$	0	0
$63$	3969.00	0.125988
$64$	0	0
$65$	54272.3	1.59329
$66$	0	0
$67$	67852.2	1.84662	0.923308	−	0.384060i	$-0.125474\pi$
$67$	67852.2	1.84662	0.923308	+	0.384060i	$0.125474\pi$
$68$	0	0
$69$	−17557.7	−0.443962
$70$	0	0
$71$	10158.5	0.239157	0.119579	−	0.992825i	$-0.461846\pi$
$71$	10158.5	0.239157	0.119579	+	0.992825i	$0.461846\pi$
$72$	0	0
$73$	60184.1	1.32183	0.660913	−	0.750462i	$-0.270169\pi$
$73$	60184.1	1.32183	0.660913	+	0.750462i	$0.270169\pi$
$74$	0	0
$75$	−12519.0	−0.256990
$76$	0	0
$77$	842.858	0.0162005
$78$	0	0
$79$	27760.3	0.500445	0.250223	−	0.968188i	$-0.419496\pi$
$79$	27760.3	0.500445	0.250223	+	0.968188i	$0.419496\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	13431.4	0.214006	0.107003	−	0.994259i	$-0.465875\pi$
$83$	13431.4	0.214006	0.107003	+	0.994259i	$0.465875\pi$
$84$	0	0
$85$	52255.9	0.784491
$86$	0	0
$87$	13662.3	0.193520
$88$	0	0
$89$	120473.	1.61219	0.806095	−	0.591786i	$-0.201577\pi$
$89$	120473.	1.61219	0.806095	+	0.591786i	$0.201577\pi$
$90$	0	0
$91$	39572.9	0.500950
$92$	0	0
$93$	49270.3	0.590714
$94$	0	0
$95$	−17632.2	−0.200446
$96$	0	0
$97$	−28652.0	−0.309190	−0.154595	−	0.987978i	$-0.549407\pi$
$97$	−28652.0	−0.309190	−0.154595	+	0.987978i	$0.549407\pi$
$98$	0	0
$99$	1393.30	0.0142875
$100$	0	0
$101$	70761.8	0.690233	0.345117	−	0.938560i	$-0.387839\pi$
$101$	70761.8	0.690233	0.345117	+	0.938560i	$0.387839\pi$
$102$	0	0
$103$	127139.	1.18082	0.590411	−	0.807103i	$-0.298966\pi$
$103$	127139.	1.18082	0.590411	+	0.807103i	$0.298966\pi$
$104$	0	0
$105$	−29635.7	−0.262327
$106$	0	0
$107$	230347.	1.94501	0.972507	−	0.232872i	$-0.0748123\pi$
$107$	230347.	1.94501	0.972507	+	0.232872i	$0.0748123\pi$
$108$	0	0
$109$	−210266.	−1.69513	−0.847563	−	0.530695i	$-0.821931\pi$
$109$	−210266.	−1.69513	−0.847563	+	0.530695i	$0.821931\pi$
$110$	0	0
$111$	−59282.5	−0.456688
$112$	0	0
$113$	−13935.4	−0.102665	−0.0513327	−	0.998682i	$-0.516347\pi$
$113$	−13935.4	−0.102665	−0.0513327	+	0.998682i	$0.516347\pi$
$114$	0	0
$115$	131100.	0.924396
$116$	0	0
$117$	65416.4	0.441796
$118$	0	0
$119$	38102.6	0.246653
$120$	0	0
$121$	−160755.	−0.998163
$122$	0	0
$123$	−26372.7	−0.157178
$124$	0	0
$125$	−116527.	−0.667039
$126$	0	0
$127$	171345.	0.942675	0.471337	−	0.881953i	$-0.343771\pi$
$127$	171345.	0.942675	0.471337	+	0.881953i	$0.343771\pi$
$128$	0	0
$129$	−8778.04	−0.0464433
$130$	0	0
$131$	237204.	1.20766	0.603828	−	0.797115i	$-0.293641\pi$
$131$	237204.	1.20766	0.603828	+	0.797115i	$0.293641\pi$
$132$	0	0
$133$	−12856.6	−0.0630225
$134$	0	0
$135$	−48989.7	−0.231350
$136$	0	0
$137$	53519.0	0.243617	0.121808	−	0.992554i	$-0.461131\pi$
$137$	53519.0	0.243617	0.121808	+	0.992554i	$0.461131\pi$
$138$	0	0
$139$	−118543.	−0.520400	−0.260200	−	0.965555i	$-0.583789\pi$
$139$	−118543.	−0.520400	−0.260200	+	0.965555i	$0.583789\pi$
$140$	0	0
$141$	206866.	0.876277
$142$	0	0
$143$	13891.8	0.0568094
$144$	0	0
$145$	−102014.	−0.402939
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	−106643.	−0.393519	−0.196759	−	0.980452i	$-0.563042\pi$
$149$	−106643.	−0.393519	−0.196759	+	0.980452i	$0.563042\pi$
$150$	0	0
$151$	−73169.2	−0.261148	−0.130574	−	0.991439i	$-0.541682\pi$
$151$	−73169.2	−0.261148	−0.130574	+	0.991439i	$0.541682\pi$
$152$	0	0
$153$	62985.9	0.217528
$154$	0	0
$155$	−367891.	−1.22996
$156$	0	0
$157$	−496125.	−1.60636	−0.803179	−	0.595738i	$-0.796860\pi$
$157$	−496125.	−1.60636	−0.803179	+	0.595738i	$0.796860\pi$
$158$	0	0
$159$	−23811.4	−0.0746952
$160$	0	0
$161$	95592.1	0.290641
$162$	0	0
$163$	59360.9	0.174997	0.0874986	−	0.996165i	$-0.472113\pi$
$163$	59360.9	0.174997	0.0874986	+	0.996165i	$0.472113\pi$
$164$	0	0
$165$	−10403.5	−0.0297487
$166$	0	0
$167$	92468.0	0.256567	0.128283	−	0.991738i	$-0.459053\pi$
$167$	92468.0	0.256567	0.128283	+	0.991738i	$0.459053\pi$
$168$	0	0
$169$	280940.	0.756653
$170$	0	0
$171$	−21252.7	−0.0555806
$172$	0	0
$173$	656493.	1.66769	0.833844	−	0.552000i	$-0.186135\pi$
$173$	656493.	1.66769	0.833844	+	0.552000i	$0.186135\pi$
$174$	0	0
$175$	68159.0	0.168240
$176$	0	0
$177$	202598.	0.486073
$178$	0	0
$179$	−347739.	−0.811186	−0.405593	−	0.914054i	$-0.632935\pi$
$179$	−347739.	−0.811186	−0.405593	+	0.914054i	$0.632935\pi$
$180$	0	0
$181$	−829484.	−1.88197	−0.940983	−	0.338454i	$-0.890096\pi$
$181$	−829484.	−1.88197	−0.940983	+	0.338454i	$0.890096\pi$
$182$	0	0
$183$	−85797.2	−0.189385
$184$	0	0
$185$	442651.	0.950893
$186$	0	0
$187$	13375.7	0.0279713
$188$	0	0
$189$	−35721.0	−0.0727393
$190$	0	0
$191$	820119.	1.62665	0.813324	−	0.581811i	$-0.197656\pi$
$191$	820119.	1.62665	0.813324	+	0.581811i	$0.197656\pi$
$192$	0	0
$193$	4163.11	0.00804497	0.00402249	−	0.999992i	$-0.498720\pi$
$193$	4163.11	0.00804497	0.00402249	+	0.999992i	$0.498720\pi$
$194$	0	0
$195$	−488451.	−0.919887
$196$	0	0
$197$	786994.	1.44479	0.722397	−	0.691479i	$-0.243040\pi$
$197$	786994.	1.44479	0.722397	+	0.691479i	$0.243040\pi$
$198$	0	0
$199$	872156.	1.56121	0.780605	−	0.625025i	$-0.214911\pi$
$199$	872156.	1.56121	0.780605	+	0.625025i	$0.214911\pi$
$200$	0	0
$201$	−610669.	−1.06614
$202$	0	0
$203$	−74383.7	−0.126689
$204$	0	0
$205$	196920.	0.327269
$206$	0	0
$207$	158020.	0.256321
$208$	0	0
$209$	−4513.22	−0.00714696
$210$	0	0
$211$	−894531.	−1.38321	−0.691607	−	0.722274i	$-0.743097\pi$
$211$	−894531.	−1.38321	−0.691607	+	0.722274i	$0.743097\pi$
$212$	0	0
$213$	−91426.4	−0.138077
$214$	0	0
$215$	65543.9	0.0967021
$216$	0	0
$217$	−268249.	−0.386713
$218$	0	0
$219$	−541657.	−0.763157
$220$	0	0
$221$	628000.	0.864926
$222$	0	0
$223$	−232594.	−0.313211	−0.156605	−	0.987661i	$-0.550055\pi$
$223$	−232594.	−0.313211	−0.156605	+	0.987661i	$0.550055\pi$
$224$	0	0
$225$	112671.	0.148373
$226$	0	0
$227$	−981411.	−1.26411	−0.632057	−	0.774922i	$-0.717789\pi$
$227$	−981411.	−1.26411	−0.632057	+	0.774922i	$0.717789\pi$
$228$	0	0
$229$	−1.31461e6	−1.65656	−0.828280	−	0.560315i	$-0.810680\pi$
$229$	−1.31461e6	−1.65656	−0.828280	+	0.560315i	$0.810680\pi$
$230$	0	0
$231$	−7585.72	−0.00935335
$232$	0	0
$233$	−458394.	−0.553158	−0.276579	−	0.960991i	$-0.589201\pi$
$233$	−458394.	−0.553158	−0.276579	+	0.960991i	$0.589201\pi$
$234$	0	0
$235$	−1.54463e6	−1.82454
$236$	0	0
$237$	−249843.	−0.288932
$238$	0	0
$239$	−643863.	−0.729119	−0.364559	−	0.931180i	$-0.618780\pi$
$239$	−643863.	−0.729119	−0.364559	+	0.931180i	$0.618780\pi$
$240$	0	0
$241$	1.56858e6	1.73966	0.869829	−	0.493353i	$-0.164229\pi$
$241$	1.56858e6	1.73966	0.869829	+	0.493353i	$0.164229\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	161350.	0.171733
$246$	0	0
$247$	−211899.	−0.220998
$248$	0	0
$249$	−120882.	−0.123556
$250$	0	0
$251$	58945.2	0.0590560	0.0295280	−	0.999564i	$-0.490600\pi$
$251$	58945.2	0.0590560	0.0295280	+	0.999564i	$0.490600\pi$
$252$	0	0
$253$	33557.1	0.0329597
$254$	0	0
$255$	−470303.	−0.452926
$256$	0	0
$257$	−602342.	−0.568866	−0.284433	−	0.958696i	$-0.591805\pi$
$257$	−602342.	−0.568866	−0.284433	+	0.958696i	$0.591805\pi$
$258$	0	0
$259$	322760.	0.298972
$260$	0	0
$261$	−122961.	−0.111729
$262$	0	0
$263$	334651.	0.298334	0.149167	−	0.988812i	$-0.452341\pi$
$263$	334651.	0.298334	0.149167	+	0.988812i	$0.452341\pi$
$264$	0	0
$265$	177795.	0.155527
$266$	0	0
$267$	−1.08426e6	−0.930799
$268$	0	0
$269$	1.45538e6	1.22630	0.613149	−	0.789967i	$-0.289903\pi$
$269$	1.45538e6	1.22630	0.613149	+	0.789967i	$0.289903\pi$
$270$	0	0
$271$	612560.	0.506670	0.253335	−	0.967379i	$-0.418472\pi$
$271$	612560.	0.506670	0.253335	+	0.967379i	$0.418472\pi$
$272$	0	0
$273$	−356156.	−0.289223
$274$	0	0
$275$	23926.9	0.0190789
$276$	0	0
$277$	−724917.	−0.567661	−0.283830	−	0.958875i	$-0.591605\pi$
$277$	−724917.	−0.567661	−0.283830	+	0.958875i	$0.591605\pi$
$278$	0	0
$279$	−443432.	−0.341049
$280$	0	0
$281$	−1.81846e6	−1.37385	−0.686924	−	0.726729i	$-0.741040\pi$
$281$	−1.81846e6	−1.37385	−0.686924	+	0.726729i	$0.741040\pi$
$282$	0	0
$283$	−708354.	−0.525756	−0.262878	−	0.964829i	$-0.584672\pi$
$283$	−708354.	−0.525756	−0.262878	+	0.964829i	$0.584672\pi$
$284$	0	0
$285$	158689.	0.115727
$286$	0	0
$287$	143585.	0.102897
$288$	0	0
$289$	−815190.	−0.574135
$290$	0	0
$291$	257868.	0.178511
$292$	0	0
$293$	−2.21719e6	−1.50881	−0.754404	−	0.656410i	$-0.772074\pi$
$293$	−2.21719e6	−1.50881	−0.754404	+	0.656410i	$0.772074\pi$
$294$	0	0
$295$	−1.51276e6	−1.01208
$296$	0	0
$297$	−12539.7	−0.00824888
$298$	0	0
$299$	1.57553e6	1.01918
$300$	0	0
$301$	47791.6	0.0304043
$302$	0	0
$303$	−636857.	−0.398506
$304$	0	0
$305$	640630.	0.394328
$306$	0	0
$307$	1.76619e6	1.06952	0.534762	−	0.845003i	$-0.320401\pi$
$307$	1.76619e6	1.06952	0.534762	+	0.845003i	$0.320401\pi$
$308$	0	0
$309$	−1.14425e6	−0.681748
$310$	0	0
$311$	377794.	0.221490	0.110745	−	0.993849i	$-0.464676\pi$
$311$	377794.	0.221490	0.110745	+	0.993849i	$0.464676\pi$
$312$	0	0
$313$	−971758.	−0.560657	−0.280328	−	0.959904i	$-0.590443\pi$
$313$	−971758.	−0.560657	−0.280328	+	0.959904i	$0.590443\pi$
$314$	0	0
$315$	266722.	0.151454
$316$	0	0
$317$	−1.01468e6	−0.567127	−0.283564	−	0.958953i	$-0.591517\pi$
$317$	−1.01468e6	−0.567127	−0.283564	+	0.958953i	$0.591517\pi$
$318$	0	0
$319$	−26112.0	−0.0143669
$320$	0	0
$321$	−2.07312e6	−1.12295
$322$	0	0
$323$	−204027.	−0.108813
$324$	0	0
$325$	1.12338e6	0.589957
$326$	0	0
$327$	1.89239e6	0.978682
$328$	0	0
$329$	−1.12627e6	−0.573658
$330$	0	0
$331$	−2.95015e6	−1.48004	−0.740021	−	0.672583i	$-0.765185\pi$
$331$	−2.95015e6	−1.48004	−0.740021	+	0.672583i	$0.765185\pi$
$332$	0	0
$333$	533543.	0.263669
$334$	0	0
$335$	4.55975e6	2.21988
$336$	0	0
$337$	1.41135e6	0.676954	0.338477	−	0.940975i	$-0.390088\pi$
$337$	1.41135e6	0.676954	0.338477	+	0.940975i	$0.390088\pi$
$338$	0	0
$339$	125419.	0.0592739
$340$	0	0
$341$	−94167.5	−0.0438546
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.17990e6	−0.533700
$346$	0	0
$347$	1.47510e6	0.657656	0.328828	−	0.944390i	$-0.393346\pi$
$347$	1.47510e6	0.657656	0.328828	+	0.944390i	$0.393346\pi$
$348$	0	0
$349$	1.65202e6	0.726023	0.363012	−	0.931785i	$-0.381749\pi$
$349$	1.65202e6	0.726023	0.363012	+	0.931785i	$0.381749\pi$
$350$	0	0
$351$	−588747.	−0.255071
$352$	0	0
$353$	3.91161e6	1.67078	0.835390	−	0.549658i	$-0.185242\pi$
$353$	3.91161e6	1.67078	0.835390	+	0.549658i	$0.185242\pi$
$354$	0	0
$355$	682663.	0.287498
$356$	0	0
$357$	−342923.	−0.142405
$358$	0	0
$359$	576419.	0.236049	0.118025	−	0.993011i	$-0.462344\pi$
$359$	576419.	0.236049	0.118025	+	0.993011i	$0.462344\pi$
$360$	0	0
$361$	−2.40726e6	−0.972197
$362$	0	0
$363$	1.44680e6	0.576290
$364$	0	0
$365$	4.04444e6	1.58901
$366$	0	0
$367$	−459296.	−0.178003	−0.0890016	−	0.996031i	$-0.528368\pi$
$367$	−459296.	−0.178003	−0.0890016	+	0.996031i	$0.528368\pi$
$368$	0	0
$369$	237354.	0.0907468
$370$	0	0
$371$	129640.	0.0488995
$372$	0	0
$373$	−3.90270e6	−1.45242	−0.726211	−	0.687472i	$-0.758720\pi$
$373$	−3.90270e6	−1.45242	−0.726211	+	0.687472i	$0.758720\pi$
$374$	0	0
$375$	1.04874e6	0.385115
$376$	0	0
$377$	−1.22598e6	−0.444253
$378$	0	0
$379$	−4.80734e6	−1.71912	−0.859561	−	0.511032i	$-0.829263\pi$
$379$	−4.80734e6	−1.71912	−0.859561	+	0.511032i	$0.829263\pi$
$380$	0	0
$381$	−1.54210e6	−0.544253
$382$	0	0
$383$	4.36779e6	1.52148	0.760738	−	0.649059i	$-0.224837\pi$
$383$	4.36779e6	1.52148	0.760738	+	0.649059i	$0.224837\pi$
$384$	0	0
$385$	56641.1	0.0194751
$386$	0	0
$387$	79002.4	0.0268141
$388$	0	0
$389$	5.18272e6	1.73653	0.868267	−	0.496097i	$-0.165234\pi$
$389$	5.18272e6	1.73653	0.868267	+	0.496097i	$0.165234\pi$
$390$	0	0
$391$	1.51699e6	0.501813
$392$	0	0
$393$	−2.13483e6	−0.697240
$394$	0	0
$395$	1.86553e6	0.601601
$396$	0	0
$397$	−2.42731e6	−0.772945	−0.386472	−	0.922301i	$-0.626307\pi$
$397$	−2.42731e6	−0.772945	−0.386472	+	0.922301i	$0.626307\pi$
$398$	0	0
$399$	115709.	0.0363860
$400$	0	0
$401$	−1.07557e6	−0.334024	−0.167012	−	0.985955i	$-0.553412\pi$
$401$	−1.07557e6	−0.334024	−0.167012	+	0.985955i	$0.553412\pi$
$402$	0	0
$403$	−4.42124e6	−1.35607
$404$	0	0
$405$	440907.	0.133570
$406$	0	0
$407$	113303.	0.0339044
$408$	0	0
$409$	2.07931e6	0.614626	0.307313	−	0.951609i	$-0.400570\pi$
$409$	2.07931e6	0.614626	0.307313	+	0.951609i	$0.400570\pi$
$410$	0	0
$411$	−481671.	−0.140652
$412$	0	0
$413$	−1.10303e6	−0.318209
$414$	0	0
$415$	902605.	0.257263
$416$	0	0
$417$	1.06688e6	0.300453
$418$	0	0
$419$	−3.35878e6	−0.934645	−0.467323	−	0.884087i	$-0.654781\pi$
$419$	−3.35878e6	−0.934645	−0.467323	+	0.884087i	$0.654781\pi$
$420$	0	0
$421$	5.07754e6	1.39620	0.698101	−	0.715999i	$-0.254029\pi$
$421$	5.07754e6	1.39620	0.698101	+	0.715999i	$0.254029\pi$
$422$	0	0
$423$	−1.86179e6	−0.505918
$424$	0	0
$425$	1.08165e6	0.290478
$426$	0	0
$427$	467118.	0.123981
$428$	0	0
$429$	−125027.	−0.0327989
$430$	0	0
$431$	3.98269e6	1.03272	0.516361	−	0.856371i	$-0.327286\pi$
$431$	3.98269e6	1.03272	0.516361	+	0.856371i	$0.327286\pi$
$432$	0	0
$433$	2.40905e6	0.617486	0.308743	−	0.951146i	$-0.400092\pi$
$433$	2.40905e6	0.617486	0.308743	+	0.951146i	$0.400092\pi$
$434$	0	0
$435$	918124.	0.232637
$436$	0	0
$437$	−511863.	−0.128218
$438$	0	0
$439$	964726.	0.238915	0.119457	−	0.992839i	$-0.461885\pi$
$439$	964726.	0.238915	0.119457	+	0.992839i	$0.461885\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	−2.07046e6	−0.501253	−0.250627	−	0.968084i	$-0.580637\pi$
$443$	−2.07046e6	−0.501253	−0.250627	+	0.968084i	$0.580637\pi$
$444$	0	0
$445$	8.09596e6	1.93806
$446$	0	0
$447$	959784.	0.227198
$448$	0	0
$449$	5.52864e6	1.29420	0.647101	−	0.762404i	$-0.275981\pi$
$449$	5.52864e6	1.29420	0.647101	+	0.762404i	$0.275981\pi$
$450$	0	0
$451$	50404.7	0.0116689
$452$	0	0
$453$	658523.	0.150774
$454$	0	0
$455$	2.65934e6	0.602207
$456$	0	0
$457$	−2.10358e6	−0.471161	−0.235580	−	0.971855i	$-0.575699\pi$
$457$	−2.10358e6	−0.471161	−0.235580	+	0.971855i	$0.575699\pi$
$458$	0	0
$459$	−566873.	−0.125590
$460$	0	0
$461$	2.89612e6	0.634693	0.317346	−	0.948310i	$-0.397208\pi$
$461$	2.89612e6	0.634693	0.317346	+	0.948310i	$0.397208\pi$
$462$	0	0
$463$	−3.84498e6	−0.833569	−0.416785	−	0.909005i	$-0.636843\pi$
$463$	−3.84498e6	−0.833569	−0.416785	+	0.909005i	$0.636843\pi$
$464$	0	0
$465$	3.31102e6	0.710116
$466$	0	0
$467$	1.59340e6	0.338090	0.169045	−	0.985608i	$-0.445932\pi$
$467$	1.59340e6	0.338090	0.169045	+	0.985608i	$0.445932\pi$
$468$	0	0
$469$	3.32476e6	0.697955
$470$	0	0
$471$	4.46513e6	0.927431
$472$	0	0
$473$	16777.0	0.00344795
$474$	0	0
$475$	−364969.	−0.0742201
$476$	0	0
$477$	214303.	0.0431253
$478$	0	0
$479$	7.01607e6	1.39719	0.698595	−	0.715517i	$-0.253809\pi$
$479$	7.01607e6	1.39719	0.698595	+	0.715517i	$0.253809\pi$
$480$	0	0
$481$	5.31968e6	1.04839
$482$	0	0
$483$	−860329.	−0.167802
$484$	0	0
$485$	−1.92545e6	−0.371687
$486$	0	0
$487$	−2.51835e6	−0.481164	−0.240582	−	0.970629i	$-0.577338\pi$
$487$	−2.51835e6	−0.481164	−0.240582	+	0.970629i	$0.577338\pi$
$488$	0	0
$489$	−534248.	−0.101035
$490$	0	0
$491$	−7.01534e6	−1.31324	−0.656621	−	0.754220i	$-0.728015\pi$
$491$	−7.01534e6	−1.31324	−0.656621	+	0.754220i	$0.728015\pi$
$492$	0	0
$493$	−1.18043e6	−0.218737
$494$	0	0
$495$	93631.2	0.0171754
$496$	0	0
$497$	497766.	0.0903929
$498$	0	0
$499$	576968.	0.103729	0.0518646	−	0.998654i	$-0.483484\pi$
$499$	576968.	0.103729	0.0518646	+	0.998654i	$0.483484\pi$
$500$	0	0
$501$	−832212.	−0.148129
$502$	0	0
$503$	−2.65556e6	−0.467989	−0.233995	−	0.972238i	$-0.575180\pi$
$503$	−2.65556e6	−0.467989	−0.233995	+	0.972238i	$0.575180\pi$
$504$	0	0
$505$	4.75528e6	0.829751
$506$	0	0
$507$	−2.52846e6	−0.436854
$508$	0	0
$509$	−3.69798e6	−0.632660	−0.316330	−	0.948649i	$-0.602451\pi$
$509$	−3.69798e6	−0.632660	−0.316330	+	0.948649i	$0.602451\pi$
$510$	0	0
$511$	2.94902e6	0.499604
$512$	0	0
$513$	191274.	0.0320895
$514$	0	0
$515$	8.54387e6	1.41950
$516$	0	0
$517$	−395371.	−0.0650547
$518$	0	0
$519$	−5.90844e6	−0.962840
$520$	0	0
$521$	2.29058e6	0.369702	0.184851	−	0.982767i	$-0.440820\pi$
$521$	2.29058e6	0.369702	0.184851	+	0.982767i	$0.440820\pi$
$522$	0	0
$523$	−3.31070e6	−0.529256	−0.264628	−	0.964351i	$-0.585249\pi$
$523$	−3.31070e6	−0.529256	−0.264628	+	0.964351i	$0.585249\pi$
$524$	0	0
$525$	−613431.	−0.0971331
$526$	0	0
$527$	−4.25697e6	−0.667689
$528$	0	0
$529$	−2.63049e6	−0.408694
$530$	0	0
$531$	−1.82338e6	−0.280634
$532$	0	0
$533$	2.36654e6	0.360824
$534$	0	0
$535$	1.54796e7	2.33816
$536$	0	0
$537$	3.12965e6	0.468339
$538$	0	0
$539$	41300.1	0.00612320
$540$	0	0
$541$	−1.28977e7	−1.89461	−0.947306	−	0.320329i	$-0.896207\pi$
$541$	−1.28977e7	−1.89461	−0.947306	+	0.320329i	$0.896207\pi$
$542$	0	0
$543$	7.46536e6	1.08655
$544$	0	0
$545$	−1.41301e7	−2.03776
$546$	0	0
$547$	658312.	0.0940727	0.0470364	−	0.998893i	$-0.485022\pi$
$547$	658312.	0.0940727	0.0470364	+	0.998893i	$0.485022\pi$
$548$	0	0
$549$	772174.	0.109341
$550$	0	0
$551$	398300.	0.0558896
$552$	0	0
$553$	1.36026e6	0.189151
$554$	0	0
$555$	−3.98386e6	−0.548999
$556$	0	0
$557$	1.28168e6	0.175042	0.0875211	−	0.996163i	$-0.472105\pi$
$557$	1.28168e6	0.175042	0.0875211	+	0.996163i	$0.472105\pi$
$558$	0	0
$559$	787692.	0.106617
$560$	0	0
$561$	−120381.	−0.0161492
$562$	0	0
$563$	−1.36647e7	−1.81690	−0.908448	−	0.417997i	$-0.862732\pi$
$563$	−1.36647e7	−1.81690	−0.908448	+	0.417997i	$0.862732\pi$
$564$	0	0
$565$	−936477.	−0.123417
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	6.49965e6	0.841607	0.420804	−	0.907152i	$-0.361748\pi$
$569$	6.49965e6	0.841607	0.420804	+	0.907152i	$0.361748\pi$
$570$	0	0
$571$	−7.67214e6	−0.984750	−0.492375	−	0.870383i	$-0.663871\pi$
$571$	−7.67214e6	−0.984750	−0.492375	+	0.870383i	$0.663871\pi$
$572$	0	0
$573$	−7.38107e6	−0.939146
$574$	0	0
$575$	2.71364e6	0.342281
$576$	0	0
$577$	4.80020e6	0.600233	0.300117	−	0.953903i	$-0.402974\pi$
$577$	4.80020e6	0.600233	0.300117	+	0.953903i	$0.402974\pi$
$578$	0	0
$579$	−37468.0	−0.00464477
$580$	0	0
$581$	658138.	0.0808866
$582$	0	0
$583$	45509.4	0.00554536
$584$	0	0
$585$	4.39606e6	0.531097
$586$	0	0
$587$	−1.44629e7	−1.73244	−0.866222	−	0.499660i	$-0.833458\pi$
$587$	−1.44629e7	−1.73244	−0.866222	+	0.499660i	$0.833458\pi$
$588$	0	0
$589$	1.43638e6	0.170601
$590$	0	0
$591$	−7.08295e6	−0.834152
$592$	0	0
$593$	5.27217e6	0.615677	0.307838	−	0.951439i	$-0.400394\pi$
$593$	5.27217e6	0.615677	0.307838	+	0.951439i	$0.400394\pi$
$594$	0	0
$595$	2.56054e6	0.296510
$596$	0	0
$597$	−7.84940e6	−0.901365
$598$	0	0
$599$	−3.87792e6	−0.441603	−0.220801	−	0.975319i	$-0.570867\pi$
$599$	−3.87792e6	−0.441603	−0.220801	+	0.975319i	$0.570867\pi$
$600$	0	0
$601$	−1.54530e7	−1.74512	−0.872560	−	0.488506i	$-0.837542\pi$
$601$	−1.54530e7	−1.74512	−0.872560	+	0.488506i	$0.837542\pi$
$602$	0	0
$603$	5.49602e6	0.615539
$604$	0	0
$605$	−1.08029e7	−1.19992
$606$	0	0
$607$	5.55130e6	0.611537	0.305769	−	0.952106i	$-0.401087\pi$
$607$	5.55130e6	0.611537	0.305769	+	0.952106i	$0.401087\pi$
$608$	0	0
$609$	669454.	0.0731438
$610$	0	0
$611$	−1.85630e7	−2.01161
$612$	0	0
$613$	−240556.	−0.0258562	−0.0129281	−	0.999916i	$-0.504115\pi$
$613$	−240556.	−0.0258562	−0.0129281	+	0.999916i	$0.504115\pi$
$614$	0	0
$615$	−1.77228e6	−0.188949
$616$	0	0
$617$	−1.00113e7	−1.05871	−0.529353	−	0.848402i	$-0.677565\pi$
$617$	−1.00113e7	−1.05871	−0.529353	+	0.848402i	$0.677565\pi$
$618$	0	0
$619$	9.97498e6	1.04637	0.523185	−	0.852219i	$-0.324744\pi$
$619$	9.97498e6	1.04637	0.523185	+	0.852219i	$0.324744\pi$
$620$	0	0
$621$	−1.42218e6	−0.147987
$622$	0	0
$623$	5.90320e6	0.609351
$624$	0	0
$625$	−1.21776e7	−1.24699
$626$	0	0
$627$	40619.0	0.00412630
$628$	0	0
$629$	5.12203e6	0.516197
$630$	0	0
$631$	1.10575e7	1.10557	0.552783	−	0.833325i	$-0.313566\pi$
$631$	1.10575e7	1.10557	0.552783	+	0.833325i	$0.313566\pi$
$632$	0	0
$633$	8.05078e6	0.798599
$634$	0	0
$635$	1.15146e7	1.13322
$636$	0	0
$637$	1.93907e6	0.189341
$638$	0	0
$639$	822838.	0.0797191
$640$	0	0
$641$	−1.64372e7	−1.58009	−0.790045	−	0.613049i	$-0.789943\pi$
$641$	−1.64372e7	−1.58009	−0.790045	+	0.613049i	$0.789943\pi$
$642$	0	0
$643$	1.85762e6	0.177186	0.0885931	−	0.996068i	$-0.471763\pi$
$643$	1.85762e6	0.177186	0.0885931	+	0.996068i	$0.471763\pi$
$644$	0	0
$645$	−589895.	−0.0558310
$646$	0	0
$647$	−1.09418e7	−1.02761	−0.513805	−	0.857907i	$-0.671764\pi$
$647$	−1.09418e7	−1.02761	−0.513805	+	0.857907i	$0.671764\pi$
$648$	0	0
$649$	−387214.	−0.0360860
$650$	0	0
$651$	2.41424e6	0.223269
$652$	0	0
$653$	−8.28230e6	−0.760095	−0.380048	−	0.924967i	$-0.624092\pi$
$653$	−8.28230e6	−0.760095	−0.380048	+	0.924967i	$0.624092\pi$
$654$	0	0
$655$	1.59404e7	1.45176
$656$	0	0
$657$	4.87491e6	0.440609
$658$	0	0
$659$	1.11810e7	1.00293	0.501463	−	0.865179i	$-0.332795\pi$
$659$	1.11810e7	1.00293	0.501463	+	0.865179i	$0.332795\pi$
$660$	0	0
$661$	−1.03445e7	−0.920882	−0.460441	−	0.887690i	$-0.652309\pi$
$661$	−1.03445e7	−0.920882	−0.460441	+	0.887690i	$0.652309\pi$
$662$	0	0
$663$	−5.65200e6	−0.499365
$664$	0	0
$665$	−863975.	−0.0757613
$666$	0	0
$667$	−2.96147e6	−0.257747
$668$	0	0
$669$	2.09335e6	0.180832
$670$	0	0
$671$	163979.	0.0140599
$672$	0	0
$673$	−1.19137e7	−1.01393	−0.506966	−	0.861966i	$-0.669233\pi$
$673$	−1.19137e7	−1.01393	−0.506966	+	0.861966i	$0.669233\pi$
$674$	0	0
$675$	−1.01404e6	−0.0856634
$676$	0	0
$677$	1.53559e7	1.28766	0.643832	−	0.765167i	$-0.277343\pi$
$677$	1.53559e7	1.28766	0.643832	+	0.765167i	$0.277343\pi$
$678$	0	0
$679$	−1.40395e6	−0.116863
$680$	0	0
$681$	8.83270e6	0.729837
$682$	0	0
$683$	1.08388e6	0.0889054	0.0444527	−	0.999011i	$-0.485846\pi$
$683$	1.08388e6	0.0889054	0.0444527	+	0.999011i	$0.485846\pi$
$684$	0	0
$685$	3.59654e6	0.292859
$686$	0	0
$687$	1.18315e7	0.956415
$688$	0	0
$689$	2.13670e6	0.171473
$690$	0	0
$691$	1.89348e7	1.50857	0.754285	−	0.656547i	$-0.227984\pi$
$691$	1.89348e7	1.50857	0.754285	+	0.656547i	$0.227984\pi$
$692$	0	0
$693$	68271.5	0.00540016
$694$	0	0
$695$	−7.96620e6	−0.625589
$696$	0	0
$697$	2.27861e6	0.177660
$698$	0	0
$699$	4.12555e6	0.319366
$700$	0	0
$701$	−1.25756e7	−0.966569	−0.483285	−	0.875463i	$-0.660556\pi$
$701$	−1.25756e7	−0.966569	−0.483285	+	0.875463i	$0.660556\pi$
$702$	0	0
$703$	−1.72827e6	−0.131894
$704$	0	0
$705$	1.39016e7	1.05340
$706$	0	0
$707$	3.46733e6	0.260884
$708$	0	0
$709$	−1.24208e7	−0.927969	−0.463985	−	0.885843i	$-0.653581\pi$
$709$	−1.24208e7	−0.927969	−0.463985	+	0.885843i	$0.653581\pi$
$710$	0	0
$711$	2.24859e6	0.166815
$712$	0	0
$713$	−1.06799e7	−0.786764
$714$	0	0
$715$	933549.	0.0682923
$716$	0	0
$717$	5.79476e6	0.420957
$718$	0	0
$719$	−1.53700e7	−1.10880	−0.554399	−	0.832251i	$-0.687052\pi$
$719$	−1.53700e7	−1.10880	−0.554399	+	0.832251i	$0.687052\pi$
$720$	0	0
$721$	6.22980e6	0.446309
$722$	0	0
$723$	−1.41172e7	−1.00439
$724$	0	0
$725$	−2.11159e6	−0.149198
$726$	0	0
$727$	−2.70136e7	−1.89560	−0.947800	−	0.318866i	$-0.896698\pi$
$727$	−2.70136e7	−1.89560	−0.947800	+	0.318866i	$0.896698\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	758426.	0.0524952
$732$	0	0
$733$	−397706.	−0.0273403	−0.0136701	−	0.999907i	$-0.504351\pi$
$733$	−397706.	−0.0273403	−0.0136701	+	0.999907i	$0.504351\pi$
$734$	0	0
$735$	−1.45215e6	−0.0991501
$736$	0	0
$737$	1.16714e6	0.0791505
$738$	0	0
$739$	−6.30043e6	−0.424384	−0.212192	−	0.977228i	$-0.568060\pi$
$739$	−6.30043e6	−0.424384	−0.212192	+	0.977228i	$0.568060\pi$
$740$	0	0
$741$	1.90709e6	0.127593
$742$	0	0
$743$	−1.23368e7	−0.819839	−0.409920	−	0.912122i	$-0.634443\pi$
$743$	−1.23368e7	−0.819839	−0.409920	+	0.912122i	$0.634443\pi$
$744$	0	0
$745$	−7.16651e6	−0.473061
$746$	0	0
$747$	1.08794e6	0.0713353
$748$	0	0
$749$	1.12870e7	0.735146
$750$	0	0
$751$	−2.14846e7	−1.39004	−0.695021	−	0.718990i	$-0.744605\pi$
$751$	−2.14846e7	−1.39004	−0.695021	+	0.718990i	$0.744605\pi$
$752$	0	0
$753$	−530507.	−0.0340960
$754$	0	0
$755$	−4.91706e6	−0.313934
$756$	0	0
$757$	1.23188e7	0.781317	0.390659	−	0.920536i	$-0.372247\pi$
$757$	1.23188e7	0.781317	0.390659	+	0.920536i	$0.372247\pi$
$758$	0	0
$759$	−302014.	−0.0190293
$760$	0	0
$761$	−592021.	−0.0370575	−0.0185287	−	0.999828i	$-0.505898\pi$
$761$	−592021.	−0.0370575	−0.0185287	+	0.999828i	$0.505898\pi$
$762$	0	0
$763$	−1.03030e7	−0.640697
$764$	0	0
$765$	4.23273e6	0.261497
$766$	0	0
$767$	−1.81800e7	−1.11585
$768$	0	0
$769$	1.02609e7	0.625704	0.312852	−	0.949802i	$-0.398716\pi$
$769$	1.02609e7	0.625704	0.312852	+	0.949802i	$0.398716\pi$
$770$	0	0
$771$	5.42107e6	0.328435
$772$	0	0
$773$	2.00066e7	1.20427	0.602136	−	0.798393i	$-0.294316\pi$
$773$	2.00066e7	1.20427	0.602136	+	0.798393i	$0.294316\pi$
$774$	0	0
$775$	−7.61499e6	−0.455423
$776$	0	0
$777$	−2.90484e6	−0.172612
$778$	0	0
$779$	−768848.	−0.0453938
$780$	0	0
$781$	174738.	0.0102509
$782$	0	0
$783$	1.10665e6	0.0645067
$784$	0	0
$785$	−3.33402e7	−1.93105
$786$	0	0
$787$	1.57761e7	0.907950	0.453975	−	0.891014i	$-0.350005\pi$
$787$	1.57761e7	0.907950	0.453975	+	0.891014i	$0.350005\pi$
$788$	0	0
$789$	−3.01186e6	−0.172243
$790$	0	0
$791$	−682836.	−0.0388039
$792$	0	0
$793$	7.69896e6	0.434759
$794$	0	0
$795$	−1.60016e6	−0.0897934
$796$	0	0
$797$	−2.22357e7	−1.23995	−0.619977	−	0.784620i	$-0.712858\pi$
$797$	−2.22357e7	−1.23995	−0.619977	+	0.784620i	$0.712858\pi$
$798$	0	0
$799$	−1.78733e7	−0.990462
$800$	0	0
$801$	9.75835e6	0.537397
$802$	0	0
$803$	1.03524e6	0.0566567
$804$	0	0
$805$	6.42390e6	0.349389
$806$	0	0
$807$	−1.30984e7	−0.708003
$808$	0	0
$809$	1.20194e7	0.645669	0.322834	−	0.946455i	$-0.395364\pi$
$809$	1.20194e7	0.645669	0.322834	+	0.946455i	$0.395364\pi$
$810$	0	0
$811$	−521352.	−0.0278342	−0.0139171	−	0.999903i	$-0.504430\pi$
$811$	−521352.	−0.0278342	−0.0139171	+	0.999903i	$0.504430\pi$
$812$	0	0
$813$	−5.51304e6	−0.292526
$814$	0	0
$815$	3.98912e6	0.210370
$816$	0	0
$817$	−255908.	−0.0134131
$818$	0	0
$819$	3.20540e6	0.166983
$820$	0	0
$821$	2.42268e6	0.125441	0.0627203	−	0.998031i	$-0.480022\pi$
$821$	2.42268e6	0.125441	0.0627203	+	0.998031i	$0.480022\pi$
$822$	0	0
$823$	1.65869e7	0.853620	0.426810	−	0.904341i	$-0.359637\pi$
$823$	1.65869e7	0.853620	0.426810	+	0.904341i	$0.359637\pi$
$824$	0	0
$825$	−215342.	−0.0110152
$826$	0	0
$827$	−9.11159e6	−0.463266	−0.231633	−	0.972803i	$-0.574407\pi$
$827$	−9.11159e6	−0.463266	−0.231633	+	0.972803i	$0.574407\pi$
$828$	0	0
$829$	−2.89737e7	−1.46426	−0.732130	−	0.681165i	$-0.761474\pi$
$829$	−2.89737e7	−1.46426	−0.732130	+	0.681165i	$0.761474\pi$
$830$	0	0
$831$	6.52425e6	0.327739
$832$	0	0
$833$	1.86703e6	0.0932262
$834$	0	0
$835$	6.21396e6	0.308427
$836$	0	0
$837$	3.99089e6	0.196905
$838$	0	0
$839$	−2.06856e7	−1.01452	−0.507262	−	0.861792i	$-0.669342\pi$
$839$	−2.06856e7	−1.01452	−0.507262	+	0.861792i	$0.669342\pi$
$840$	0	0
$841$	−1.82067e7	−0.887650
$842$	0	0
$843$	1.63662e7	0.793192
$844$	0	0
$845$	1.88795e7	0.909597
$846$	0	0
$847$	−7.87700e6	−0.377270
$848$	0	0
$849$	6.37519e6	0.303545
$850$	0	0
$851$	1.28502e7	0.608255
$852$	0	0
$853$	3.02453e7	1.42326	0.711632	−	0.702552i	$-0.247956\pi$
$853$	3.02453e7	1.42326	0.711632	+	0.702552i	$0.247956\pi$
$854$	0	0
$855$	−1.42820e6	−0.0668152
$856$	0	0
$857$	1.17972e7	0.548689	0.274345	−	0.961631i	$-0.411539\pi$
$857$	1.17972e7	0.548689	0.274345	+	0.961631i	$0.411539\pi$
$858$	0	0
$859$	1.87297e6	0.0866060	0.0433030	−	0.999062i	$-0.486212\pi$
$859$	1.87297e6	0.0866060	0.0433030	+	0.999062i	$0.486212\pi$
$860$	0	0
$861$	−1.29226e6	−0.0594077
$862$	0	0
$863$	−2.65653e7	−1.21420	−0.607098	−	0.794627i	$-0.707666\pi$
$863$	−2.65653e7	−1.21420	−0.607098	+	0.794627i	$0.707666\pi$
$864$	0	0
$865$	4.41171e7	2.00478
$866$	0	0
$867$	7.33671e6	0.331477
$868$	0	0
$869$	477510.	0.0214503
$870$	0	0
$871$	5.47981e7	2.44748
$872$	0	0
$873$	−2.32081e6	−0.103063
$874$	0	0
$875$	−5.70982e6	−0.252117
$876$	0	0
$877$	−7.56578e6	−0.332166	−0.166083	−	0.986112i	$-0.553112\pi$
$877$	−7.56578e6	−0.332166	−0.166083	+	0.986112i	$0.553112\pi$
$878$	0	0
$879$	1.99547e7	0.871111
$880$	0	0
$881$	1.37674e7	0.597603	0.298801	−	0.954315i	$-0.403413\pi$
$881$	1.37674e7	0.597603	0.298801	+	0.954315i	$0.403413\pi$
$882$	0	0
$883$	1.41355e7	0.610110	0.305055	−	0.952335i	$-0.401325\pi$
$883$	1.41355e7	0.610110	0.305055	+	0.952335i	$0.401325\pi$
$884$	0	0
$885$	1.36148e7	0.584323
$886$	0	0
$887$	−1.46644e7	−0.625828	−0.312914	−	0.949782i	$-0.601305\pi$
$887$	−1.46644e7	−0.625828	−0.312914	+	0.949782i	$0.601305\pi$
$888$	0	0
$889$	8.39590e6	0.356298
$890$	0	0
$891$	112857.	0.00476249
$892$	0	0
$893$	6.03080e6	0.253073
$894$	0	0
$895$	−2.33685e7	−0.975153
$896$	0	0
$897$	−1.41798e7	−0.588422
$898$	0	0
$899$	8.31045e6	0.342946
$900$	0	0
$901$	2.05732e6	0.0844285
$902$	0	0
$903$	−430124.	−0.0175539
$904$	0	0
$905$	−5.57423e7	−2.26237
$906$	0	0
$907$	−687707.	−0.0277578	−0.0138789	−	0.999904i	$-0.504418\pi$
$907$	−687707.	−0.0277578	−0.0138789	+	0.999904i	$0.504418\pi$
$908$	0	0
$909$	5.73171e6	0.230078
$910$	0	0
$911$	4.91732e7	1.96305	0.981527	−	0.191323i	$-0.0612778\pi$
$911$	4.91732e7	1.96305	0.981527	+	0.191323i	$0.0612778\pi$
$912$	0	0
$913$	231036.	0.00917281
$914$	0	0
$915$	−5.76567e6	−0.227665
$916$	0	0
$917$	1.16230e7	0.456451
$918$	0	0
$919$	5.17193e6	0.202006	0.101003	−	0.994886i	$-0.467795\pi$
$919$	5.17193e6	0.202006	0.101003	+	0.994886i	$0.467795\pi$
$920$	0	0
$921$	−1.58957e7	−0.617490
$922$	0	0
$923$	8.20410e6	0.316976
$924$	0	0
$925$	9.16244e6	0.352093
$926$	0	0
$927$	1.02982e7	0.393608
$928$	0	0
$929$	4.83580e6	0.183835	0.0919177	−	0.995767i	$-0.470700\pi$
$929$	4.83580e6	0.183835	0.0919177	+	0.995767i	$0.470700\pi$
$930$	0	0
$931$	−629971.	−0.0238203
$932$	0	0
$933$	−3.40014e6	−0.127877
$934$	0	0
$935$	898863.	0.0336252
$936$	0	0
$937$	4.88196e7	1.81654	0.908270	−	0.418385i	$-0.137404\pi$
$937$	4.88196e7	1.81654	0.908270	+	0.418385i	$0.137404\pi$
$938$	0	0
$939$	8.74582e6	0.323695
$940$	0	0
$941$	1.83528e7	0.675661	0.337831	−	0.941207i	$-0.390307\pi$
$941$	1.83528e7	0.675661	0.337831	+	0.941207i	$0.390307\pi$
$942$	0	0
$943$	5.71660e6	0.209343
$944$	0	0
$945$	−2.40049e6	−0.0874422
$946$	0	0
$947$	−1.66145e7	−0.602020	−0.301010	−	0.953621i	$-0.597324\pi$
$947$	−1.66145e7	−0.602020	−0.301010	+	0.953621i	$0.597324\pi$
$948$	0	0
$949$	4.86053e7	1.75193
$950$	0	0
$951$	9.13211e6	0.327431
$952$	0	0
$953$	−4.55298e7	−1.62391	−0.811957	−	0.583717i	$-0.801598\pi$
$953$	−4.55298e7	−1.62391	−0.811957	+	0.583717i	$0.801598\pi$
$954$	0	0
$955$	5.51130e7	1.95544
$956$	0	0
$957$	235008.	0.00829475
$958$	0	0
$959$	2.62243e6	0.0920784
$960$	0	0
$961$	1.34071e6	0.0468303
$962$	0	0
$963$	1.86581e7	0.648338
$964$	0	0
$965$	279766.	0.00967111
$966$	0	0
$967$	−3.67650e7	−1.26435	−0.632176	−	0.774825i	$-0.717838\pi$
$967$	−3.67650e7	−1.26435	−0.632176	+	0.774825i	$0.717838\pi$
$968$	0	0
$969$	1.83624e6	0.0628231
$970$	0	0
$971$	1.33403e7	0.454065	0.227033	−	0.973887i	$-0.427098\pi$
$971$	1.33403e7	0.454065	0.227033	+	0.973887i	$0.427098\pi$
$972$	0	0
$973$	−5.80859e6	−0.196693
$974$	0	0
$975$	−1.01105e7	−0.340612
$976$	0	0
$977$	−3.81112e7	−1.27737	−0.638685	−	0.769468i	$-0.720521\pi$
$977$	−3.81112e7	−1.27737	−0.638685	+	0.769468i	$0.720521\pi$
$978$	0	0
$979$	2.07229e6	0.0691024
$980$	0	0
$981$	−1.70315e7	−0.565042
$982$	0	0
$983$	2.42760e7	0.801296	0.400648	−	0.916232i	$-0.368785\pi$
$983$	2.42760e7	0.801296	0.400648	+	0.916232i	$0.368785\pi$
$984$	0	0
$985$	5.28869e7	1.73683
$986$	0	0
$987$	1.01364e7	0.331201
$988$	0	0
$989$	1.90275e6	0.0618572
$990$	0	0
$991$	5.99075e6	0.193775	0.0968874	−	0.995295i	$-0.469111\pi$
$991$	5.99075e6	0.193775	0.0968874	+	0.995295i	$0.469111\pi$
$992$	0	0
$993$	2.65514e7	0.854503
$994$	0	0
$995$	5.86099e7	1.87678
$996$	0	0
$997$	−1.10752e7	−0.352868	−0.176434	−	0.984312i	$-0.556456\pi$
$997$	−1.10752e7	−0.352868	−0.176434	+	0.984312i	$0.556456\pi$
$998$	0	0
$999$	−4.80188e6	−0.152229

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.j.1.2	✓	2	4.3	odd	2
336.6.a.t.1.2		2	1.1	even	1	trivial
504.6.a.o.1.1		2	12.11	even	2
1008.6.a.bn.1.1		2	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +67.2012 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +67.2012 q^{5} +49.0000 q^{7} +81.0000 q^{9} +17.2012 q^{11} +807.610 q^{13} -604.811 q^{15} +777.604 q^{17} -262.379 q^{19} -441.000 q^{21} +1950.86 q^{23} +1391.00 q^{25} -729.000 q^{27} -1518.04 q^{29} -5474.47 q^{31} -154.811 q^{33} +3292.86 q^{35} +6586.95 q^{37} -7268.49 q^{39} +2930.30 q^{41} +975.338 q^{43} +5443.30 q^{45} -22985.1 q^{47} +2401.00 q^{49} -6998.43 q^{51} +2645.71 q^{53} +1155.94 q^{55} +2361.41 q^{57} -22510.9 q^{59} +9533.02 q^{61} +3969.00 q^{63} +54272.3 q^{65} +67852.2 q^{67} -17557.7 q^{69} +10158.5 q^{71} +60184.1 q^{73} -12519.0 q^{75} +842.858 q^{77} +27760.3 q^{79} +6561.00 q^{81} +13431.4 q^{83} +52255.9 q^{85} +13662.3 q^{87} +120473. q^{89} +39572.9 q^{91} +49270.3 q^{93} -17632.2 q^{95} -28652.0 q^{97} +1393.30 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 98 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 + 98 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 98 q^{7} + 162 q^{9} - 100 q^{11} + 540 q^{13} + 1152 q^{17} - 2944 q^{19} - 882 q^{21} - 2684 q^{23} + 2782 q^{25} - 1458 q^{27} + 996 q^{29} - 2616 q^{31} + 900 q^{33} - 3492 q^{37} - 4860 q^{39} + 17016 q^{41} - 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 10368 q^{51} + 37548 q^{53} + 9032 q^{55} + 26496 q^{57} + 6320 q^{59} + 39764 q^{61} + 7938 q^{63} + 72256 q^{65} + 27376 q^{67} + 24156 q^{69} - 35460 q^{71} + 64188 q^{73} - 25038 q^{75} - 4900 q^{77} + 110088 q^{79} + 13122 q^{81} + 5896 q^{83} + 27096 q^{85} - 8964 q^{87} + 167160 q^{89} + 26460 q^{91} + 23544 q^{93} + 162576 q^{95} - 11876 q^{97} - 8100 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 98 * q^7 + 162 * q^9 - 100 * q^11 + 540 * q^13 + 1152 * q^17 - 2944 * q^19 - 882 * q^21 - 2684 * q^23 + 2782 * q^25 - 1458 * q^27 + 996 * q^29 - 2616 * q^31 + 900 * q^33 - 3492 * q^37 - 4860 * q^39 + 17016 * q^41 - 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 10368 * q^51 + 37548 * q^53 + 9032 * q^55 + 26496 * q^57 + 6320 * q^59 + 39764 * q^61 + 7938 * q^63 + 72256 * q^65 + 27376 * q^67 + 24156 * q^69 - 35460 * q^71 + 64188 * q^73 - 25038 * q^75 - 4900 * q^77 + 110088 * q^79 + 13122 * q^81 + 5896 * q^83 + 27096 * q^85 - 8964 * q^87 + 167160 * q^89 + 26460 * q^91 + 23544 * q^93 + 162576 * q^95 - 11876 * q^97 - 8100 * q^99

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