LMFDB - Embedded newform 504.6.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,6,Mod(1,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, 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("504.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	504.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:	$80.8334451857$
Analytic rank:	$1$
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			1.1
Root			$17.3003$ of defining polynomial
Character	$\chi$	$=$	504.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-67.2012 q^{5} -49.0000 q^{7} +17.2012 q^{11} +807.610 q^{13} -777.604 q^{17} +262.379 q^{19} +1950.86 q^{23} +1391.00 q^{25} +1518.04 q^{29} +5474.47 q^{31} +3292.86 q^{35} +6586.95 q^{37} -2930.30 q^{41} -975.338 q^{43} -22985.1 q^{47} +2401.00 q^{49} -2645.71 q^{53} -1155.94 q^{55} -22510.9 q^{59} +9533.02 q^{61} -54272.3 q^{65} -67852.2 q^{67} +10158.5 q^{71} +60184.1 q^{73} -842.858 q^{77} -27760.3 q^{79} +13431.4 q^{83} +52255.9 q^{85} -120473. q^{89} -39572.9 q^{91} -17632.2 q^{95} -28652.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 98 q^{7} - 100 q^{11} + 540 q^{13} - 1152 q^{17} + 2944 q^{19} - 2684 q^{23} + 2782 q^{25} - 996 q^{29} + 2616 q^{31} - 3492 q^{37} - 17016 q^{41} + 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 37548 q^{53}+ \cdots - 11876 q^{97}+O(q^{100}) $ 2 * q - 98 * q^7 - 100 * q^11 + 540 * q^13 - 1152 * q^17 + 2944 * q^19 - 2684 * q^23 + 2782 * q^25 - 996 * q^29 + 2616 * q^31 - 3492 * q^37 - 17016 * q^41 + 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 37548 * q^53 - 9032 * q^55 + 6320 * q^59 + 39764 * q^61 - 72256 * q^65 - 27376 * q^67 - 35460 * q^71 + 64188 * q^73 + 4900 * q^77 - 110088 * q^79 + 5896 * q^83 + 27096 * q^85 - 167160 * q^89 - 26460 * q^91 + 162576 * q^95 - 11876 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−67.2012	−1.20213	−0.601066	−	0.799200i	$-0.705257\pi$
$5$	−67.2012	−1.20213	−0.601066	+	0.799200i	$0.705257\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	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
$15$	0	0
$16$	0	0
$17$	−777.604	−0.652583	−0.326292	−	0.945269i	$-0.605799\pi$
$17$	−777.604	−0.652583	−0.326292	+	0.945269i	$0.605799\pi$
$18$	0	0
$19$	262.379	0.166742	0.0833709	−	0.996519i	$-0.473431\pi$
$19$	262.379	0.166742	0.0833709	+	0.996519i	$0.473431\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	1518.04	0.335187	0.167593	−	0.985856i	$-0.446400\pi$
$29$	1518.04	0.335187	0.167593	+	0.985856i	$0.446400\pi$
$30$	0	0
$31$	5474.47	1.02315	0.511574	−	0.859239i	$-0.329063\pi$
$31$	5474.47	1.02315	0.511574	+	0.859239i	$0.329063\pi$
$32$	0	0
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	−2930.30	−0.272240	−0.136120	−	0.990692i	$-0.543463\pi$
$41$	−2930.30	−0.272240	−0.136120	+	0.990692i	$0.543463\pi$
$42$	0	0
$43$	−975.338	−0.0804422	−0.0402211	−	0.999191i	$-0.512806\pi$
$43$	−975.338	−0.0804422	−0.0402211	+	0.999191i	$0.512806\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	−2645.71	−0.129376	−0.0646879	−	0.997906i	$-0.520605\pi$
$53$	−2645.71	−0.129376	−0.0646879	+	0.997906i	$0.520605\pi$
$54$	0	0
$55$	−1155.94	−0.0515263
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	−54272.3	−1.59329
$66$	0	0
$67$	−67852.2	−1.84662	−0.923308	−	0.384060i	$-0.874526\pi$
$67$	−67852.2	−1.84662	−0.923308	+	0.384060i	$0.874526\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−842.858	−0.0162005
$78$	0	0
$79$	−27760.3	−0.500445	−0.250223	−	0.968188i	$-0.580504\pi$
$79$	−27760.3	−0.500445	−0.250223	+	0.968188i	$0.580504\pi$
$80$	0	0
$81$	0	0
$82$	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
$87$	0	0
$88$	0	0
$89$	−120473.	−1.61219	−0.806095	−	0.591786i	$-0.798423\pi$
$89$	−120473.	−1.61219	−0.806095	+	0.591786i	$0.798423\pi$
$90$	0	0
$91$	−39572.9	−0.500950
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	−70761.8	−0.690233	−0.345117	−	0.938560i	$-0.612161\pi$
$101$	−70761.8	−0.690233	−0.345117	+	0.938560i	$0.612161\pi$
$102$	0	0
$103$	−127139.	−1.18082	−0.590411	−	0.807103i	$-0.701034\pi$
$103$	−127139.	−1.18082	−0.590411	+	0.807103i	$0.701034\pi$
$104$	0	0
$105$	0	0
$106$	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
$111$	0	0
$112$	0	0
$113$	13935.4	0.102665	0.0513327	−	0.998682i	$-0.483653\pi$
$113$	13935.4	0.102665	0.0513327	+	0.998682i	$0.483653\pi$
$114$	0	0
$115$	−131100.	−0.924396
$116$	0	0
$117$	0	0
$118$	0	0
$119$	38102.6	0.246653
$120$	0	0
$121$	−160755.	−0.998163
$122$	0	0
$123$	0	0
$124$	0	0
$125$	116527.	0.667039
$126$	0	0
$127$	−171345.	−0.942675	−0.471337	−	0.881953i	$-0.656229\pi$
$127$	−171345.	−0.942675	−0.471337	+	0.881953i	$0.656229\pi$
$128$	0	0
$129$	0	0
$130$	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
$135$	0	0
$136$	0	0
$137$	−53519.0	−0.243617	−0.121808	−	0.992554i	$-0.538869\pi$
$137$	−53519.0	−0.243617	−0.121808	+	0.992554i	$0.538869\pi$
$138$	0	0
$139$	118543.	0.520400	0.260200	−	0.965555i	$-0.416211\pi$
$139$	118543.	0.520400	0.260200	+	0.965555i	$0.416211\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13891.8	0.0568094
$144$	0	0
$145$	−102014.	−0.402939
$146$	0	0
$147$	0	0
$148$	0	0
$149$	106643.	0.393519	0.196759	−	0.980452i	$-0.436958\pi$
$149$	106643.	0.393519	0.196759	+	0.980452i	$0.436958\pi$
$150$	0	0
$151$	73169.2	0.261148	0.130574	−	0.991439i	$-0.458318\pi$
$151$	73169.2	0.261148	0.130574	+	0.991439i	$0.458318\pi$
$152$	0	0
$153$	0	0
$154$	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
$159$	0	0
$160$	0	0
$161$	−95592.1	−0.290641
$162$	0	0
$163$	−59360.9	−0.174997	−0.0874986	−	0.996165i	$-0.527887\pi$
$163$	−59360.9	−0.174997	−0.0874986	+	0.996165i	$0.527887\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	−656493.	−1.66769	−0.833844	−	0.552000i	$-0.813865\pi$
$173$	−656493.	−1.66769	−0.833844	+	0.552000i	$0.813865\pi$
$174$	0	0
$175$	−68159.0	−0.168240
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	−442651.	−0.950893
$186$	0	0
$187$	−13375.7	−0.0279713
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	−786994.	−1.44479	−0.722397	−	0.691479i	$-0.756960\pi$
$197$	−786994.	−1.44479	−0.722397	+	0.691479i	$0.756960\pi$
$198$	0	0
$199$	−872156.	−1.56121	−0.780605	−	0.625025i	$-0.785089\pi$
$199$	−872156.	−1.56121	−0.780605	+	0.625025i	$0.785089\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−74383.7	−0.126689
$204$	0	0
$205$	196920.	0.327269
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4513.22	0.00714696
$210$	0	0
$211$	894531.	1.38321	0.691607	−	0.722274i	$-0.256903\pi$
$211$	894531.	1.38321	0.691607	+	0.722274i	$0.256903\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	65543.9	0.0967021
$216$	0	0
$217$	−268249.	−0.386713
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−628000.	−0.864926
$222$	0	0
$223$	232594.	0.313211	0.156605	−	0.987661i	$-0.449945\pi$
$223$	232594.	0.313211	0.156605	+	0.987661i	$0.449945\pi$
$224$	0	0
$225$	0	0
$226$	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
$231$	0	0
$232$	0	0
$233$	458394.	0.553158	0.276579	−	0.960991i	$-0.410799\pi$
$233$	458394.	0.553158	0.276579	+	0.960991i	$0.410799\pi$
$234$	0	0
$235$	1.54463e6	1.82454
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	−161350.	−0.171733
$246$	0	0
$247$	211899.	0.220998
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	602342.	0.568866	0.284433	−	0.958696i	$-0.408195\pi$
$257$	602342.	0.568866	0.284433	+	0.958696i	$0.408195\pi$
$258$	0	0
$259$	−322760.	−0.298972
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	−1.45538e6	−1.22630	−0.613149	−	0.789967i	$-0.710097\pi$
$269$	−1.45538e6	−1.22630	−0.613149	+	0.789967i	$0.710097\pi$
$270$	0	0
$271$	−612560.	−0.506670	−0.253335	−	0.967379i	$-0.581528\pi$
$271$	−612560.	−0.506670	−0.253335	+	0.967379i	$0.581528\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	1.81846e6	1.37385	0.686924	−	0.726729i	$-0.258960\pi$
$281$	1.81846e6	1.37385	0.686924	+	0.726729i	$0.258960\pi$
$282$	0	0
$283$	708354.	0.525756	0.262878	−	0.964829i	$-0.415328\pi$
$283$	708354.	0.525756	0.262878	+	0.964829i	$0.415328\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	143585.	0.102897
$288$	0	0
$289$	−815190.	−0.574135
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.21719e6	1.50881	0.754404	−	0.656410i	$-0.227926\pi$
$293$	2.21719e6	1.50881	0.754404	+	0.656410i	$0.227926\pi$
$294$	0	0
$295$	1.51276e6	1.01208
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.57553e6	1.01918
$300$	0	0
$301$	47791.6	0.0304043
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−640630.	−0.394328
$306$	0	0
$307$	−1.76619e6	−1.06952	−0.534762	−	0.845003i	$-0.679599\pi$
$307$	−1.76619e6	−1.06952	−0.534762	+	0.845003i	$0.679599\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	1.01468e6	0.567127	0.283564	−	0.958953i	$-0.408483\pi$
$317$	1.01468e6	0.567127	0.283564	+	0.958953i	$0.408483\pi$
$318$	0	0
$319$	26112.0	0.0143669
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−204027.	−0.108813
$324$	0	0
$325$	1.12338e6	0.589957
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.12627e6	0.573658
$330$	0	0
$331$	2.95015e6	1.48004	0.740021	−	0.672583i	$-0.234815\pi$
$331$	2.95015e6	1.48004	0.740021	+	0.672583i	$0.234815\pi$
$332$	0	0
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	94167.5	0.0438546
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	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
$351$	0	0
$352$	0	0
$353$	−3.91161e6	−1.67078	−0.835390	−	0.549658i	$-0.814758\pi$
$353$	−3.91161e6	−1.67078	−0.835390	+	0.549658i	$0.814758\pi$
$354$	0	0
$355$	−682663.	−0.287498
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	−4.04444e6	−1.58901
$366$	0	0
$367$	459296.	0.178003	0.0890016	−	0.996031i	$-0.471632\pi$
$367$	459296.	0.178003	0.0890016	+	0.996031i	$0.471632\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	1.22598e6	0.444253
$378$	0	0
$379$	4.80734e6	1.71912	0.859561	−	0.511032i	$-0.170737\pi$
$379$	4.80734e6	1.71912	0.859561	+	0.511032i	$0.170737\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	−5.18272e6	−1.73653	−0.868267	−	0.496097i	$-0.834766\pi$
$389$	−5.18272e6	−1.73653	−0.868267	+	0.496097i	$0.834766\pi$
$390$	0	0
$391$	−1.51699e6	−0.501813
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	1.07557e6	0.334024	0.167012	−	0.985955i	$-0.446588\pi$
$401$	1.07557e6	0.334024	0.167012	+	0.985955i	$0.446588\pi$
$402$	0	0
$403$	4.42124e6	1.35607
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	1.10303e6	0.318209
$414$	0	0
$415$	−902605.	−0.257263
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	−1.08165e6	−0.290478
$426$	0	0
$427$	−467118.	−0.123981
$428$	0	0
$429$	0	0
$430$	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
$435$	0	0
$436$	0	0
$437$	511863.	0.128218
$438$	0	0
$439$	−964726.	−0.238915	−0.119457	−	0.992839i	$-0.538115\pi$
$439$	−964726.	−0.238915	−0.119457	+	0.992839i	$0.538115\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	−5.52864e6	−1.29420	−0.647101	−	0.762404i	$-0.724019\pi$
$449$	−5.52864e6	−1.29420	−0.647101	+	0.762404i	$0.724019\pi$
$450$	0	0
$451$	−50404.7	−0.0116689
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	−2.89612e6	−0.634693	−0.317346	−	0.948310i	$-0.602792\pi$
$461$	−2.89612e6	−0.634693	−0.317346	+	0.948310i	$0.602792\pi$
$462$	0	0
$463$	3.84498e6	0.833569	0.416785	−	0.909005i	$-0.363157\pi$
$463$	3.84498e6	0.833569	0.416785	+	0.909005i	$0.363157\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	−16777.0	−0.00344795
$474$	0	0
$475$	364969.	0.0742201
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	1.92545e6	0.371687
$486$	0	0
$487$	2.51835e6	0.481164	0.240582	−	0.970629i	$-0.422662\pi$
$487$	2.51835e6	0.481164	0.240582	+	0.970629i	$0.422662\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−497766.	−0.0903929
$498$	0	0
$499$	−576968.	−0.103729	−0.0518646	−	0.998654i	$-0.516516\pi$
$499$	−576968.	−0.103729	−0.0518646	+	0.998654i	$0.516516\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	3.69798e6	0.632660	0.316330	−	0.948649i	$-0.397549\pi$
$509$	3.69798e6	0.632660	0.316330	+	0.948649i	$0.397549\pi$
$510$	0	0
$511$	−2.94902e6	−0.499604
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.54387e6	1.41950
$516$	0	0
$517$	−395371.	−0.0650547
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.29058e6	−0.369702	−0.184851	−	0.982767i	$-0.559180\pi$
$521$	−2.29058e6	−0.369702	−0.184851	+	0.982767i	$0.559180\pi$
$522$	0	0
$523$	3.31070e6	0.529256	0.264628	−	0.964351i	$-0.414751\pi$
$523$	3.31070e6	0.529256	0.264628	+	0.964351i	$0.414751\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.25697e6	−0.667689
$528$	0	0
$529$	−2.63049e6	−0.408694
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.36654e6	−0.360824
$534$	0	0
$535$	−1.54796e7	−2.33816
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	1.41301e7	2.03776
$546$	0	0
$547$	−658312.	−0.0940727	−0.0470364	−	0.998893i	$-0.514978\pi$
$547$	−658312.	−0.0940727	−0.0470364	+	0.998893i	$0.514978\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	398300.	0.0558896
$552$	0	0
$553$	1.36026e6	0.189151
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.28168e6	−0.175042	−0.0875211	−	0.996163i	$-0.527895\pi$
$557$	−1.28168e6	−0.175042	−0.0875211	+	0.996163i	$0.527895\pi$
$558$	0	0
$559$	−787692.	−0.106617
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	−6.49965e6	−0.841607	−0.420804	−	0.907152i	$-0.638252\pi$
$569$	−6.49965e6	−0.841607	−0.420804	+	0.907152i	$0.638252\pi$
$570$	0	0
$571$	7.67214e6	0.984750	0.492375	−	0.870383i	$-0.336129\pi$
$571$	7.67214e6	0.984750	0.492375	+	0.870383i	$0.336129\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	−658138.	−0.0808866
$582$	0	0
$583$	−45509.4	−0.00554536
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	−5.27217e6	−0.615677	−0.307838	−	0.951439i	$-0.599606\pi$
$593$	−5.27217e6	−0.615677	−0.307838	+	0.951439i	$0.599606\pi$
$594$	0	0
$595$	−2.56054e6	−0.296510
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	1.08029e7	1.19992
$606$	0	0
$607$	−5.55130e6	−0.611537	−0.305769	−	0.952106i	$-0.598913\pi$
$607$	−5.55130e6	−0.611537	−0.305769	+	0.952106i	$0.598913\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	1.00113e7	1.05871	0.529353	−	0.848402i	$-0.322435\pi$
$617$	1.00113e7	1.05871	0.529353	+	0.848402i	$0.322435\pi$
$618$	0	0
$619$	−9.97498e6	−1.04637	−0.523185	−	0.852219i	$-0.675256\pi$
$619$	−9.97498e6	−1.04637	−0.523185	+	0.852219i	$0.675256\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.90320e6	0.609351
$624$	0	0
$625$	−1.21776e7	−1.24699
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.12203e6	−0.516197
$630$	0	0
$631$	−1.10575e7	−1.10557	−0.552783	−	0.833325i	$-0.686434\pi$
$631$	−1.10575e7	−1.10557	−0.552783	+	0.833325i	$0.686434\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.15146e7	1.13322
$636$	0	0
$637$	1.93907e6	0.189341
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.64372e7	1.58009	0.790045	−	0.613049i	$-0.210057\pi$
$641$	1.64372e7	1.58009	0.790045	+	0.613049i	$0.210057\pi$
$642$	0	0
$643$	−1.85762e6	−0.177186	−0.0885931	−	0.996068i	$-0.528237\pi$
$643$	−1.85762e6	−0.177186	−0.0885931	+	0.996068i	$0.528237\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	8.28230e6	0.760095	0.380048	−	0.924967i	$-0.375908\pi$
$653$	8.28230e6	0.760095	0.380048	+	0.924967i	$0.375908\pi$
$654$	0	0
$655$	−1.59404e7	−1.45176
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	863975.	0.0757613
$666$	0	0
$667$	2.96147e6	0.257747
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−1.53559e7	−1.28766	−0.643832	−	0.765167i	$-0.722657\pi$
$677$	−1.53559e7	−1.28766	−0.643832	+	0.765167i	$0.722657\pi$
$678$	0	0
$679$	1.40395e6	0.116863
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	−2.13670e6	−0.171473
$690$	0	0
$691$	−1.89348e7	−1.50857	−0.754285	−	0.656547i	$-0.772016\pi$
$691$	−1.89348e7	−1.50857	−0.754285	+	0.656547i	$0.772016\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.96620e6	−0.625589
$696$	0	0
$697$	2.27861e6	0.177660
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.25756e7	0.966569	0.483285	−	0.875463i	$-0.339444\pi$
$701$	1.25756e7	0.966569	0.483285	+	0.875463i	$0.339444\pi$
$702$	0	0
$703$	1.72827e6	0.131894
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	1.06799e7	0.786764
$714$	0	0
$715$	−933549.	−0.0682923
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	2.11159e6	0.149198
$726$	0	0
$727$	2.70136e7	1.89560	0.947800	−	0.318866i	$-0.103302\pi$
$727$	2.70136e7	1.89560	0.947800	+	0.318866i	$0.103302\pi$
$728$	0	0
$729$	0	0
$730$	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
$735$	0	0
$736$	0	0
$737$	−1.16714e6	−0.0791505
$738$	0	0
$739$	6.30043e6	0.424384	0.212192	−	0.977228i	$-0.431940\pi$
$739$	6.30043e6	0.424384	0.212192	+	0.977228i	$0.431940\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	−1.12870e7	−0.735146
$750$	0	0
$751$	2.14846e7	1.39004	0.695021	−	0.718990i	$-0.255395\pi$
$751$	2.14846e7	1.39004	0.695021	+	0.718990i	$0.255395\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	592021.	0.0370575	0.0185287	−	0.999828i	$-0.494102\pi$
$761$	592021.	0.0370575	0.0185287	+	0.999828i	$0.494102\pi$
$762$	0	0
$763$	1.03030e7	0.640697
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−2.00066e7	−1.20427	−0.602136	−	0.798393i	$-0.705684\pi$
$773$	−2.00066e7	−1.20427	−0.602136	+	0.798393i	$0.705684\pi$
$774$	0	0
$775$	7.61499e6	0.455423
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−768848.	−0.0453938
$780$	0	0
$781$	174738.	0.0102509
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.33402e7	1.93105
$786$	0	0
$787$	−1.57761e7	−0.907950	−0.453975	−	0.891014i	$-0.649995\pi$
$787$	−1.57761e7	−0.907950	−0.453975	+	0.891014i	$0.649995\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−682836.	−0.0388039
$792$	0	0
$793$	7.69896e6	0.434759
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.22357e7	1.23995	0.619977	−	0.784620i	$-0.287142\pi$
$797$	2.22357e7	1.23995	0.619977	+	0.784620i	$0.287142\pi$
$798$	0	0
$799$	1.78733e7	0.990462
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.03524e6	0.0566567
$804$	0	0
$805$	6.42390e6	0.349389
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.20194e7	−0.645669	−0.322834	−	0.946455i	$-0.604636\pi$
$809$	−1.20194e7	−0.645669	−0.322834	+	0.946455i	$0.604636\pi$
$810$	0	0
$811$	521352.	0.0278342	0.0139171	−	0.999903i	$-0.495570\pi$
$811$	521352.	0.0278342	0.0139171	+	0.999903i	$0.495570\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.98912e6	0.210370
$816$	0	0
$817$	−255908.	−0.0134131
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−2.42268e6	−0.125441	−0.0627203	−	0.998031i	$-0.519978\pi$
$821$	−2.42268e6	−0.125441	−0.0627203	+	0.998031i	$0.519978\pi$
$822$	0	0
$823$	−1.65869e7	−0.853620	−0.426810	−	0.904341i	$-0.640363\pi$
$823$	−1.65869e7	−0.853620	−0.426810	+	0.904341i	$0.640363\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	−1.86703e6	−0.0932262
$834$	0	0
$835$	−6.21396e6	−0.308427
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	−1.88795e7	−0.909597
$846$	0	0
$847$	7.87700e6	0.377270
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	−1.17972e7	−0.548689	−0.274345	−	0.961631i	$-0.588461\pi$
$857$	−1.17972e7	−0.548689	−0.274345	+	0.961631i	$0.588461\pi$
$858$	0	0
$859$	−1.87297e6	−0.0866060	−0.0433030	−	0.999062i	$-0.513788\pi$
$859$	−1.87297e6	−0.0866060	−0.0433030	+	0.999062i	$0.513788\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	−477510.	−0.0214503
$870$	0	0
$871$	−5.47981e7	−2.44748
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	−1.37674e7	−0.597603	−0.298801	−	0.954315i	$-0.596587\pi$
$881$	−1.37674e7	−0.597603	−0.298801	+	0.954315i	$0.596587\pi$
$882$	0	0
$883$	−1.41355e7	−0.610110	−0.305055	−	0.952335i	$-0.598675\pi$
$883$	−1.41355e7	−0.610110	−0.305055	+	0.952335i	$0.598675\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	−6.03080e6	−0.253073
$894$	0	0
$895$	2.33685e7	0.975153
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.31045e6	0.342946
$900$	0	0
$901$	2.05732e6	0.0844285
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.57423e7	2.26237
$906$	0	0
$907$	687707.	0.0277578	0.0138789	−	0.999904i	$-0.495582\pi$
$907$	687707.	0.0277578	0.0138789	+	0.999904i	$0.495582\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	−1.16230e7	−0.456451
$918$	0	0
$919$	−5.17193e6	−0.202006	−0.101003	−	0.994886i	$-0.532205\pi$
$919$	−5.17193e6	−0.202006	−0.101003	+	0.994886i	$0.532205\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	8.20410e6	0.316976
$924$	0	0
$925$	9.16244e6	0.352093
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.83580e6	−0.183835	−0.0919177	−	0.995767i	$-0.529300\pi$
$929$	−4.83580e6	−0.183835	−0.0919177	+	0.995767i	$0.529300\pi$
$930$	0	0
$931$	629971.	0.0238203
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−1.83528e7	−0.675661	−0.337831	−	0.941207i	$-0.609693\pi$
$941$	−1.83528e7	−0.675661	−0.337831	+	0.941207i	$0.609693\pi$
$942$	0	0
$943$	−5.71660e6	−0.209343
$944$	0	0
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	4.55298e7	1.62391	0.811957	−	0.583717i	$-0.198402\pi$
$953$	4.55298e7	1.62391	0.811957	+	0.583717i	$0.198402\pi$
$954$	0	0
$955$	−5.51130e7	−1.95544
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.62243e6	0.0920784
$960$	0	0
$961$	1.34071e6	0.0468303
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−279766.	−0.00967111
$966$	0	0
$967$	3.67650e7	1.26435	0.632176	−	0.774825i	$-0.282162\pi$
$967$	3.67650e7	1.26435	0.632176	+	0.774825i	$0.282162\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	3.81112e7	1.27737	0.638685	−	0.769468i	$-0.279479\pi$
$977$	3.81112e7	1.27737	0.638685	+	0.769468i	$0.279479\pi$
$978$	0	0
$979$	−2.07229e6	−0.0691024
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	−1.90275e6	−0.0618572
$990$	0	0
$991$	−5.99075e6	−0.193775	−0.0968874	−	0.995295i	$-0.530889\pi$
$991$	−5.99075e6	−0.193775	−0.0968874	+	0.995295i	$0.530889\pi$
$992$	0	0
$993$	0	0
$994$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.j.1.2	✓	2	3.2	odd	2
336.6.a.t.1.2		2	12.11	even	2
504.6.a.o.1.1		2	1.1	even	1	trivial
1008.6.a.bn.1.1		2	4.3	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	504.a (trivial)

\(f(q)\)	\(=\)	\(q-67.2012 q^{5} -49.0000 q^{7} +17.2012 q^{11} +807.610 q^{13} -777.604 q^{17} +262.379 q^{19} +1950.86 q^{23} +1391.00 q^{25} +1518.04 q^{29} +5474.47 q^{31} +3292.86 q^{35} +6586.95 q^{37} -2930.30 q^{41} -975.338 q^{43} -22985.1 q^{47} +2401.00 q^{49} -2645.71 q^{53} -1155.94 q^{55} -22510.9 q^{59} +9533.02 q^{61} -54272.3 q^{65} -67852.2 q^{67} +10158.5 q^{71} +60184.1 q^{73} -842.858 q^{77} -27760.3 q^{79} +13431.4 q^{83} +52255.9 q^{85} -120473. q^{89} -39572.9 q^{91} -17632.2 q^{95} -28652.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 98 q^{7} - 100 q^{11} + 540 q^{13} - 1152 q^{17} + 2944 q^{19} - 2684 q^{23} + 2782 q^{25} - 996 q^{29} + 2616 q^{31} - 3492 q^{37} - 17016 q^{41} + 13640 q^{43} - 11832 q^{47} + 4802 q^{49} - 37548 q^{53}+ \cdots - 11876 q^{97}+O(q^{100}) \) 2 * q - 98 * q^7 - 100 * q^11 + 540 * q^13 - 1152 * q^17 + 2944 * q^19 - 2684 * q^23 + 2782 * q^25 - 996 * q^29 + 2616 * q^31 - 3492 * q^37 - 17016 * q^41 + 13640 * q^43 - 11832 * q^47 + 4802 * q^49 - 37548 * q^53 - 9032 * q^55 + 6320 * q^59 + 39764 * q^61 - 72256 * q^65 - 27376 * q^67 - 35460 * q^71 + 64188 * q^73 + 4900 * q^77 - 110088 * q^79 + 5896 * q^83 + 27096 * q^85 - 167160 * q^89 - 26460 * q^91 + 162576 * q^95 - 11876 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more