LMFDB - Embedded newform 264.6.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [264,6,Mod(1,264)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(264, 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("264.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 264 = 2^{3} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	264.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:	$42.3413284306$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-9.61187$ of defining polynomial
Character	$\chi$	$=$	264.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} +45.6712 q^{5} -16.4475 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} +45.6712 q^{5} -16.4475 q^{7} +81.0000 q^{9} -121.000 q^{11} -339.041 q^{13} -411.041 q^{15} -1456.62 q^{17} +1367.22 q^{19} +148.027 q^{21} +3826.76 q^{23} -1039.14 q^{25} -729.000 q^{27} +256.749 q^{29} -779.105 q^{31} +1089.00 q^{33} -751.178 q^{35} -8668.10 q^{37} +3051.37 q^{39} +3135.51 q^{41} -1432.61 q^{43} +3699.37 q^{45} +4851.27 q^{47} -16536.5 q^{49} +13109.6 q^{51} -15091.8 q^{53} -5526.22 q^{55} -12305.0 q^{57} -43077.5 q^{59} -22940.2 q^{61} -1332.25 q^{63} -15484.4 q^{65} +6935.87 q^{67} -34440.8 q^{69} -39085.6 q^{71} -14217.3 q^{73} +9352.24 q^{75} +1990.15 q^{77} -72174.4 q^{79} +6561.00 q^{81} +3929.54 q^{83} -66525.7 q^{85} -2310.74 q^{87} -140514. q^{89} +5576.38 q^{91} +7011.95 q^{93} +62442.8 q^{95} +15247.3 q^{97} -9801.00 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} - 30 q^{5} + 48 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 - 30 * q^5 + 48 * q^7 + 162 * q^9	$ 2 q - 18 q^{3} - 30 q^{5} + 48 q^{7} + 162 q^{9} - 242 q^{11} + 414 q^{13} + 270 q^{15} - 1174 q^{17} + 2694 q^{19} - 432 q^{21} + 2274 q^{23} + 1562 q^{25} - 1458 q^{27} + 1282 q^{29} - 1720 q^{31} + 2178 q^{33} - 5628 q^{35} - 2856 q^{37} - 3726 q^{39} - 7198 q^{41} - 7274 q^{43} - 2430 q^{45} - 3362 q^{47} - 29190 q^{49} + 10566 q^{51} - 44138 q^{53} + 3630 q^{55} - 24246 q^{57} - 27668 q^{59} - 25050 q^{61} + 3888 q^{63} - 72468 q^{65} + 29080 q^{67} - 20466 q^{69} - 3950 q^{71} - 52784 q^{73} - 14058 q^{75} - 5808 q^{77} - 97268 q^{79} + 13122 q^{81} - 15196 q^{83} - 87912 q^{85} - 11538 q^{87} - 227476 q^{89} + 54108 q^{91} + 15480 q^{93} - 37956 q^{95} - 96996 q^{97} - 19602 q^{99}+O(q^{100}) $ 2 * q - 18 * q^3 - 30 * q^5 + 48 * q^7 + 162 * q^9 - 242 * q^11 + 414 * q^13 + 270 * q^15 - 1174 * q^17 + 2694 * q^19 - 432 * q^21 + 2274 * q^23 + 1562 * q^25 - 1458 * q^27 + 1282 * q^29 - 1720 * q^31 + 2178 * q^33 - 5628 * q^35 - 2856 * q^37 - 3726 * q^39 - 7198 * q^41 - 7274 * q^43 - 2430 * q^45 - 3362 * q^47 - 29190 * q^49 + 10566 * q^51 - 44138 * q^53 + 3630 * q^55 - 24246 * q^57 - 27668 * q^59 - 25050 * q^61 + 3888 * q^63 - 72468 * q^65 + 29080 * q^67 - 20466 * q^69 - 3950 * q^71 - 52784 * q^73 - 14058 * q^75 - 5808 * q^77 - 97268 * q^79 + 13122 * q^81 - 15196 * q^83 - 87912 * q^85 - 11538 * q^87 - 227476 * q^89 + 54108 * q^91 + 15480 * q^93 - 37956 * q^95 - 96996 * q^97 - 19602 * 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
$5$	45.6712	0.816992	0.408496	−	0.912760i	$-0.366053\pi$
$5$	45.6712	0.816992	0.408496	+	0.912760i	$0.366053\pi$
$6$	0	0
$7$	−16.4475	−0.126869	−0.0634344	−	0.997986i	$-0.520205\pi$
$7$	−16.4475	−0.126869	−0.0634344	+	0.997986i	$0.520205\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−339.041	−0.556409	−0.278204	−	0.960522i	$-0.589739\pi$
$13$	−339.041	−0.556409	−0.278204	+	0.960522i	$0.589739\pi$
$14$	0	0
$15$	−411.041	−0.471691
$16$	0	0
$17$	−1456.62	−1.22243	−0.611215	−	0.791464i	$-0.709319\pi$
$17$	−1456.62	−1.22243	−0.611215	+	0.791464i	$0.709319\pi$
$18$	0	0
$19$	1367.22	0.868872	0.434436	−	0.900703i	$-0.356948\pi$
$19$	1367.22	0.868872	0.434436	+	0.900703i	$0.356948\pi$
$20$	0	0
$21$	148.027	0.0732477
$22$	0	0
$23$	3826.76	1.50838	0.754191	−	0.656655i	$-0.228029\pi$
$23$	3826.76	1.50838	0.754191	+	0.656655i	$0.228029\pi$
$24$	0	0
$25$	−1039.14	−0.332524
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	256.749	0.0566909	0.0283455	−	0.999598i	$-0.490976\pi$
$29$	256.749	0.0566909	0.0283455	+	0.999598i	$0.490976\pi$
$30$	0	0
$31$	−779.105	−0.145610	−0.0728051	−	0.997346i	$-0.523195\pi$
$31$	−779.105	−0.145610	−0.0728051	+	0.997346i	$0.523195\pi$
$32$	0	0
$33$	1089.00	0.174078
$34$	0	0
$35$	−751.178	−0.103651
$36$	0	0
$37$	−8668.10	−1.04093	−0.520463	−	0.853884i	$-0.674241\pi$
$37$	−8668.10	−1.04093	−0.520463	+	0.853884i	$0.674241\pi$
$38$	0	0
$39$	3051.37	0.321243
$40$	0	0
$41$	3135.51	0.291305	0.145653	−	0.989336i	$-0.453472\pi$
$41$	3135.51	0.291305	0.145653	+	0.989336i	$0.453472\pi$
$42$	0	0
$43$	−1432.61	−0.118156	−0.0590782	−	0.998253i	$-0.518816\pi$
$43$	−1432.61	−0.118156	−0.0590782	+	0.998253i	$0.518816\pi$
$44$	0	0
$45$	3699.37	0.272331
$46$	0	0
$47$	4851.27	0.320340	0.160170	−	0.987089i	$-0.448796\pi$
$47$	4851.27	0.320340	0.160170	+	0.987089i	$0.448796\pi$
$48$	0	0
$49$	−16536.5	−0.983904
$50$	0	0
$51$	13109.6	0.705771
$52$	0	0
$53$	−15091.8	−0.737992	−0.368996	−	0.929431i	$-0.620298\pi$
$53$	−15091.8	−0.737992	−0.368996	+	0.929431i	$0.620298\pi$
$54$	0	0
$55$	−5526.22	−0.246332
$56$	0	0
$57$	−12305.0	−0.501643
$58$	0	0
$59$	−43077.5	−1.61109	−0.805547	−	0.592532i	$-0.798128\pi$
$59$	−43077.5	−1.61109	−0.805547	+	0.592532i	$0.798128\pi$
$60$	0	0
$61$	−22940.2	−0.789356	−0.394678	−	0.918819i	$-0.629144\pi$
$61$	−22940.2	−0.789356	−0.394678	+	0.918819i	$0.629144\pi$
$62$	0	0
$63$	−1332.25	−0.0422896
$64$	0	0
$65$	−15484.4	−0.454582
$66$	0	0
$67$	6935.87	0.188762	0.0943809	−	0.995536i	$-0.469913\pi$
$67$	6935.87	0.188762	0.0943809	+	0.995536i	$0.469913\pi$
$68$	0	0
$69$	−34440.8	−0.870865
$70$	0	0
$71$	−39085.6	−0.920176	−0.460088	−	0.887873i	$-0.652182\pi$
$71$	−39085.6	−0.920176	−0.460088	+	0.887873i	$0.652182\pi$
$72$	0	0
$73$	−14217.3	−0.312255	−0.156128	−	0.987737i	$-0.549901\pi$
$73$	−14217.3	−0.312255	−0.156128	+	0.987737i	$0.549901\pi$
$74$	0	0
$75$	9352.24	0.191983
$76$	0	0
$77$	1990.15	0.0382524
$78$	0	0
$79$	−72174.4	−1.30112	−0.650558	−	0.759457i	$-0.725465\pi$
$79$	−72174.4	−1.30112	−0.650558	+	0.759457i	$0.725465\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	3929.54	0.0626103	0.0313052	−	0.999510i	$-0.490034\pi$
$83$	3929.54	0.0626103	0.0313052	+	0.999510i	$0.490034\pi$
$84$	0	0
$85$	−66525.7	−0.998716
$86$	0	0
$87$	−2310.74	−0.0327305
$88$	0	0
$89$	−140514.	−1.88038	−0.940189	−	0.340652i	$-0.889352\pi$
$89$	−140514.	−1.88038	−0.940189	+	0.340652i	$0.889352\pi$
$90$	0	0
$91$	5576.38	0.0705909
$92$	0	0
$93$	7011.95	0.0840681
$94$	0	0
$95$	62442.8	0.709861
$96$	0	0
$97$	15247.3	0.164537	0.0822683	−	0.996610i	$-0.473784\pi$
$97$	15247.3	0.164537	0.0822683	+	0.996610i	$0.473784\pi$
$98$	0	0
$99$	−9801.00	−0.100504
$100$	0	0
$101$	113196.	1.10415	0.552074	−	0.833795i	$-0.313836\pi$
$101$	113196.	1.10415	0.552074	+	0.833795i	$0.313836\pi$
$102$	0	0
$103$	15229.3	0.141445	0.0707224	−	0.997496i	$-0.477470\pi$
$103$	15229.3	0.141445	0.0707224	+	0.997496i	$0.477470\pi$
$104$	0	0
$105$	6760.60	0.0598428
$106$	0	0
$107$	−143007.	−1.20753	−0.603765	−	0.797162i	$-0.706334\pi$
$107$	−143007.	−1.20753	−0.603765	+	0.797162i	$0.706334\pi$
$108$	0	0
$109$	68192.6	0.549757	0.274879	−	0.961479i	$-0.411362\pi$
$109$	68192.6	0.549757	0.274879	+	0.961479i	$0.411362\pi$
$110$	0	0
$111$	78012.9	0.600979
$112$	0	0
$113$	−137498.	−1.01298	−0.506489	−	0.862246i	$-0.669057\pi$
$113$	−137498.	−1.01298	−0.506489	+	0.862246i	$0.669057\pi$
$114$	0	0
$115$	174773.	1.23234
$116$	0	0
$117$	−27462.3	−0.185470
$118$	0	0
$119$	23957.8	0.155088
$120$	0	0
$121$	14641.0	0.0909091
$122$	0	0
$123$	−28219.6	−0.168185
$124$	0	0
$125$	−190181.	−1.08866
$126$	0	0
$127$	295641.	1.62651	0.813253	−	0.581910i	$-0.197694\pi$
$127$	295641.	1.62651	0.813253	+	0.581910i	$0.197694\pi$
$128$	0	0
$129$	12893.5	0.0682176
$130$	0	0
$131$	−143765.	−0.731937	−0.365968	−	0.930627i	$-0.619262\pi$
$131$	−143765.	−0.731937	−0.365968	+	0.930627i	$0.619262\pi$
$132$	0	0
$133$	−22487.4	−0.110233
$134$	0	0
$135$	−33294.3	−0.157230
$136$	0	0
$137$	−118869.	−0.541086	−0.270543	−	0.962708i	$-0.587203\pi$
$137$	−118869.	−0.541086	−0.270543	+	0.962708i	$0.587203\pi$
$138$	0	0
$139$	−127349.	−0.559058	−0.279529	−	0.960137i	$-0.590178\pi$
$139$	−127349.	−0.559058	−0.279529	+	0.960137i	$0.590178\pi$
$140$	0	0
$141$	−43661.4	−0.184948
$142$	0	0
$143$	41024.0	0.167764
$144$	0	0
$145$	11726.0	0.0463160
$146$	0	0
$147$	148828.	0.568057
$148$	0	0
$149$	−28023.4	−0.103408	−0.0517042	−	0.998662i	$-0.516465\pi$
$149$	−28023.4	−0.103408	−0.0517042	+	0.998662i	$0.516465\pi$
$150$	0	0
$151$	−14839.9	−0.0529648	−0.0264824	−	0.999649i	$-0.508431\pi$
$151$	−14839.9	−0.0529648	−0.0264824	+	0.999649i	$0.508431\pi$
$152$	0	0
$153$	−117986.	−0.407477
$154$	0	0
$155$	−35582.7	−0.118962
$156$	0	0
$157$	−100791.	−0.326341	−0.163171	−	0.986598i	$-0.552172\pi$
$157$	−100791.	−0.326341	−0.163171	+	0.986598i	$0.552172\pi$
$158$	0	0
$159$	135826.	0.426080
$160$	0	0
$161$	−62940.6	−0.191367
$162$	0	0
$163$	216673.	0.638757	0.319378	−	0.947627i	$-0.396526\pi$
$163$	216673.	0.638757	0.319378	+	0.947627i	$0.396526\pi$
$164$	0	0
$165$	49736.0	0.142220
$166$	0	0
$167$	476573.	1.32232	0.661162	−	0.750243i	$-0.270064\pi$
$167$	476573.	1.32232	0.661162	+	0.750243i	$0.270064\pi$
$168$	0	0
$169$	−256344.	−0.690409
$170$	0	0
$171$	110745.	0.289624
$172$	0	0
$173$	97502.3	0.247685	0.123843	−	0.992302i	$-0.460478\pi$
$173$	97502.3	0.247685	0.123843	+	0.992302i	$0.460478\pi$
$174$	0	0
$175$	17091.2	0.0421869
$176$	0	0
$177$	387698.	0.930166
$178$	0	0
$179$	204791.	0.477726	0.238863	−	0.971053i	$-0.423225\pi$
$179$	204791.	0.477726	0.238863	+	0.971053i	$0.423225\pi$
$180$	0	0
$181$	−114294.	−0.259315	−0.129658	−	0.991559i	$-0.541388\pi$
$181$	−114294.	−0.259315	−0.129658	+	0.991559i	$0.541388\pi$
$182$	0	0
$183$	206462.	0.455735
$184$	0	0
$185$	−395883.	−0.850428
$186$	0	0
$187$	176251.	0.368577
$188$	0	0
$189$	11990.2	0.0244159
$190$	0	0
$191$	−403294.	−0.799906	−0.399953	−	0.916536i	$-0.630973\pi$
$191$	−403294.	−0.799906	−0.399953	+	0.916536i	$0.630973\pi$
$192$	0	0
$193$	913272.	1.76485	0.882423	−	0.470457i	$-0.155911\pi$
$193$	913272.	1.76485	0.882423	+	0.470457i	$0.155911\pi$
$194$	0	0
$195$	139360.	0.262453
$196$	0	0
$197$	−335788.	−0.616452	−0.308226	−	0.951313i	$-0.599735\pi$
$197$	−335788.	−0.616452	−0.308226	+	0.951313i	$0.599735\pi$
$198$	0	0
$199$	813774.	1.45670	0.728352	−	0.685203i	$-0.240287\pi$
$199$	813774.	1.45670	0.728352	+	0.685203i	$0.240287\pi$
$200$	0	0
$201$	−62422.8	−0.108982
$202$	0	0
$203$	−4222.87	−0.00719231
$204$	0	0
$205$	143203.	0.237994
$206$	0	0
$207$	309967.	0.502794
$208$	0	0
$209$	−165434.	−0.261975
$210$	0	0
$211$	−51490.7	−0.0796200	−0.0398100	−	0.999207i	$-0.512675\pi$
$211$	−51490.7	−0.0796200	−0.0398100	+	0.999207i	$0.512675\pi$
$212$	0	0
$213$	351770.	0.531264
$214$	0	0
$215$	−65429.1	−0.0965328
$216$	0	0
$217$	12814.3	0.0184734
$218$	0	0
$219$	127956.	0.180281
$220$	0	0
$221$	493855.	0.680171
$222$	0	0
$223$	114357.	0.153993	0.0769965	−	0.997031i	$-0.475467\pi$
$223$	114357.	0.153993	0.0769965	+	0.997031i	$0.475467\pi$
$224$	0	0
$225$	−84170.1	−0.110841
$226$	0	0
$227$	1.11695e6	1.43869	0.719347	−	0.694650i	$-0.244441\pi$
$227$	1.11695e6	1.43869	0.719347	+	0.694650i	$0.244441\pi$
$228$	0	0
$229$	1.16688e6	1.47041	0.735203	−	0.677847i	$-0.237087\pi$
$229$	1.16688e6	1.47041	0.735203	+	0.677847i	$0.237087\pi$
$230$	0	0
$231$	−17911.3	−0.0220850
$232$	0	0
$233$	−1.23996e6	−1.49629	−0.748147	−	0.663534i	$-0.769056\pi$
$233$	−1.23996e6	−1.49629	−0.748147	+	0.663534i	$0.769056\pi$
$234$	0	0
$235$	221564.	0.261715
$236$	0	0
$237$	649570.	0.751199
$238$	0	0
$239$	−799712.	−0.905605	−0.452803	−	0.891611i	$-0.649576\pi$
$239$	−799712.	−0.905605	−0.452803	+	0.891611i	$0.649576\pi$
$240$	0	0
$241$	−41211.0	−0.0457058	−0.0228529	−	0.999739i	$-0.507275\pi$
$241$	−41211.0	−0.0457058	−0.0228529	+	0.999739i	$0.507275\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−755242.	−0.803842
$246$	0	0
$247$	−463545.	−0.483448
$248$	0	0
$249$	−35365.8	−0.0361481
$250$	0	0
$251$	1.03303e6	1.03497	0.517485	−	0.855692i	$-0.326868\pi$
$251$	1.03303e6	1.03497	0.517485	+	0.855692i	$0.326868\pi$
$252$	0	0
$253$	−463038.	−0.454794
$254$	0	0
$255$	598731.	0.576609
$256$	0	0
$257$	−1.43627e6	−1.35645	−0.678223	−	0.734856i	$-0.737250\pi$
$257$	−1.43627e6	−1.35645	−0.678223	+	0.734856i	$0.737250\pi$
$258$	0	0
$259$	142569.	0.132061
$260$	0	0
$261$	20796.7	0.0188970
$262$	0	0
$263$	−82594.3	−0.0736310	−0.0368155	−	0.999322i	$-0.511721\pi$
$263$	−82594.3	−0.0736310	−0.0368155	+	0.999322i	$0.511721\pi$
$264$	0	0
$265$	−689262.	−0.602933
$266$	0	0
$267$	1.26463e6	1.08564
$268$	0	0
$269$	672241.	0.566427	0.283214	−	0.959057i	$-0.408599\pi$
$269$	672241.	0.566427	0.283214	+	0.959057i	$0.408599\pi$
$270$	0	0
$271$	−836861.	−0.692197	−0.346099	−	0.938198i	$-0.612494\pi$
$271$	−836861.	−0.692197	−0.346099	+	0.938198i	$0.612494\pi$
$272$	0	0
$273$	−50187.4	−0.0407557
$274$	0	0
$275$	125736.	0.100260
$276$	0	0
$277$	1.81771e6	1.42340	0.711699	−	0.702485i	$-0.247926\pi$
$277$	1.81771e6	1.42340	0.711699	+	0.702485i	$0.247926\pi$
$278$	0	0
$279$	−63107.5	−0.0485367
$280$	0	0
$281$	−107509.	−0.0812228	−0.0406114	−	0.999175i	$-0.512931\pi$
$281$	−107509.	−0.0812228	−0.0406114	+	0.999175i	$0.512931\pi$
$282$	0	0
$283$	1.67391e6	1.24241	0.621205	−	0.783648i	$-0.286643\pi$
$283$	1.67391e6	1.24241	0.621205	+	0.783648i	$0.286643\pi$
$284$	0	0
$285$	−561985.	−0.409839
$286$	0	0
$287$	−51571.3	−0.0369575
$288$	0	0
$289$	701888.	0.494337
$290$	0	0
$291$	−137225.	−0.0949952
$292$	0	0
$293$	−1.77990e6	−1.21123	−0.605613	−	0.795759i	$-0.707072\pi$
$293$	−1.77990e6	−1.21123	−0.605613	+	0.795759i	$0.707072\pi$
$294$	0	0
$295$	−1.96740e6	−1.31625
$296$	0	0
$297$	88209.0	0.0580259
$298$	0	0
$299$	−1.29743e6	−0.839277
$300$	0	0
$301$	23562.9	0.0149904
$302$	0	0
$303$	−1.01876e6	−0.637481
$304$	0	0
$305$	−1.04771e6	−0.644898
$306$	0	0
$307$	−1.95346e6	−1.18293	−0.591463	−	0.806332i	$-0.701450\pi$
$307$	−1.95346e6	−1.18293	−0.591463	+	0.806332i	$0.701450\pi$
$308$	0	0
$309$	−137064.	−0.0816631
$310$	0	0
$311$	−1.74203e6	−1.02130	−0.510652	−	0.859787i	$-0.670596\pi$
$311$	−1.74203e6	−1.02130	−0.510652	+	0.859787i	$0.670596\pi$
$312$	0	0
$313$	2.59514e6	1.49727	0.748635	−	0.662982i	$-0.230709\pi$
$313$	2.59514e6	1.49727	0.748635	+	0.662982i	$0.230709\pi$
$314$	0	0
$315$	−60845.4	−0.0345503
$316$	0	0
$317$	−1.44929e6	−0.810044	−0.405022	−	0.914307i	$-0.632736\pi$
$317$	−1.44929e6	−0.810044	−0.405022	+	0.914307i	$0.632736\pi$
$318$	0	0
$319$	−31066.6	−0.0170930
$320$	0	0
$321$	1.28706e6	0.697168
$322$	0	0
$323$	−1.99153e6	−1.06214
$324$	0	0
$325$	352310.	0.185019
$326$	0	0
$327$	−613733.	−0.317402
$328$	0	0
$329$	−79791.3	−0.0406411
$330$	0	0
$331$	772053.	0.387326	0.193663	−	0.981068i	$-0.437963\pi$
$331$	772053.	0.387326	0.193663	+	0.981068i	$0.437963\pi$
$332$	0	0
$333$	−702116.	−0.346975
$334$	0	0
$335$	316770.	0.154217
$336$	0	0
$337$	−258697.	−0.124084	−0.0620421	−	0.998074i	$-0.519761\pi$
$337$	−258697.	−0.124084	−0.0620421	+	0.998074i	$0.519761\pi$
$338$	0	0
$339$	1.23748e6	0.584844
$340$	0	0
$341$	94271.7	0.0439031
$342$	0	0
$343$	548417.	0.251695
$344$	0	0
$345$	−1.57296e6	−0.711490
$346$	0	0
$347$	−826097.	−0.368305	−0.184152	−	0.982898i	$-0.558954\pi$
$347$	−826097.	−0.368305	−0.184152	+	0.982898i	$0.558954\pi$
$348$	0	0
$349$	544651.	0.239362	0.119681	−	0.992812i	$-0.461813\pi$
$349$	544651.	0.239362	0.119681	+	0.992812i	$0.461813\pi$
$350$	0	0
$351$	247161.	0.107081
$352$	0	0
$353$	243494.	0.104005	0.0520023	−	0.998647i	$-0.483440\pi$
$353$	243494.	0.104005	0.0520023	+	0.998647i	$0.483440\pi$
$354$	0	0
$355$	−1.78509e6	−0.751776
$356$	0	0
$357$	−215620.	−0.0895403
$358$	0	0
$359$	−2.07447e6	−0.849517	−0.424758	−	0.905307i	$-0.639641\pi$
$359$	−2.07447e6	−0.849517	−0.424758	+	0.905307i	$0.639641\pi$
$360$	0	0
$361$	−606798.	−0.245062
$362$	0	0
$363$	−131769.	−0.0524864
$364$	0	0
$365$	−649322.	−0.255110
$366$	0	0
$367$	−4.69826e6	−1.82084	−0.910421	−	0.413684i	$-0.864242\pi$
$367$	−4.69826e6	−1.82084	−0.910421	+	0.413684i	$0.864242\pi$
$368$	0	0
$369$	253976.	0.0971017
$370$	0	0
$371$	248222.	0.0936281
$372$	0	0
$373$	954562.	0.355248	0.177624	−	0.984098i	$-0.443159\pi$
$373$	954562.	0.355248	0.177624	+	0.984098i	$0.443159\pi$
$374$	0	0
$375$	1.71163e6	0.628539
$376$	0	0
$377$	−87048.4	−0.0315433
$378$	0	0
$379$	145932.	0.0521857	0.0260928	−	0.999660i	$-0.491693\pi$
$379$	145932.	0.0521857	0.0260928	+	0.999660i	$0.491693\pi$
$380$	0	0
$381$	−2.66077e6	−0.939064
$382$	0	0
$383$	−4.88917e6	−1.70309	−0.851547	−	0.524278i	$-0.824335\pi$
$383$	−4.88917e6	−1.70309	−0.851547	+	0.524278i	$0.824335\pi$
$384$	0	0
$385$	90892.5	0.0312519
$386$	0	0
$387$	−116042.	−0.0393855
$388$	0	0
$389$	1.86483e6	0.624834	0.312417	−	0.949945i	$-0.398861\pi$
$389$	1.86483e6	0.624834	0.312417	+	0.949945i	$0.398861\pi$
$390$	0	0
$391$	−5.57414e6	−1.84389
$392$	0	0
$393$	1.29388e6	0.422584
$394$	0	0
$395$	−3.29630e6	−1.06300
$396$	0	0
$397$	1.34549e6	0.428453	0.214227	−	0.976784i	$-0.431277\pi$
$397$	1.34549e6	0.428453	0.214227	+	0.976784i	$0.431277\pi$
$398$	0	0
$399$	202387.	0.0636429
$400$	0	0
$401$	3.44604e6	1.07019	0.535094	−	0.844793i	$-0.320276\pi$
$401$	3.44604e6	1.07019	0.535094	+	0.844793i	$0.320276\pi$
$402$	0	0
$403$	264149.	0.0810188
$404$	0	0
$405$	299649.	0.0907769
$406$	0	0
$407$	1.04884e6	0.313851
$408$	0	0
$409$	1.36013e6	0.402042	0.201021	−	0.979587i	$-0.435574\pi$
$409$	1.36013e6	0.402042	0.201021	+	0.979587i	$0.435574\pi$
$410$	0	0
$411$	1.06982e6	0.312396
$412$	0	0
$413$	708518.	0.204397
$414$	0	0
$415$	179467.	0.0511522
$416$	0	0
$417$	1.14614e6	0.322772
$418$	0	0
$419$	1.09643e6	0.305102	0.152551	−	0.988296i	$-0.451251\pi$
$419$	1.09643e6	0.305102	0.152551	+	0.988296i	$0.451251\pi$
$420$	0	0
$421$	3.59748e6	0.989220	0.494610	−	0.869115i	$-0.335311\pi$
$421$	3.59748e6	0.989220	0.494610	+	0.869115i	$0.335311\pi$
$422$	0	0
$423$	392953.	0.106780
$424$	0	0
$425$	1.51363e6	0.406488
$426$	0	0
$427$	377309.	0.100145
$428$	0	0
$429$	−369216.	−0.0968584
$430$	0	0
$431$	−1.49515e6	−0.387695	−0.193848	−	0.981032i	$-0.562097\pi$
$431$	−1.49515e6	−0.387695	−0.193848	+	0.981032i	$0.562097\pi$
$432$	0	0
$433$	2.36981e6	0.607425	0.303713	−	0.952764i	$-0.401774\pi$
$433$	2.36981e6	0.607425	0.303713	+	0.952764i	$0.401774\pi$
$434$	0	0
$435$	−105534.	−0.0267406
$436$	0	0
$437$	5.23204e6	1.31059
$438$	0	0
$439$	−5.18154e6	−1.28321	−0.641605	−	0.767035i	$-0.721731\pi$
$439$	−5.18154e6	−1.28321	−0.641605	+	0.767035i	$0.721731\pi$
$440$	0	0
$441$	−1.33945e6	−0.327968
$442$	0	0
$443$	7.30898e6	1.76949	0.884744	−	0.466077i	$-0.154333\pi$
$443$	7.30898e6	1.76949	0.884744	+	0.466077i	$0.154333\pi$
$444$	0	0
$445$	−6.41746e6	−1.53625
$446$	0	0
$447$	252211.	0.0597029
$448$	0	0
$449$	−6.99701e6	−1.63794	−0.818968	−	0.573839i	$-0.805453\pi$
$449$	−6.99701e6	−1.63794	−0.818968	+	0.573839i	$0.805453\pi$
$450$	0	0
$451$	−379396.	−0.0878318
$452$	0	0
$453$	133559.	0.0305792
$454$	0	0
$455$	254680.	0.0576722
$456$	0	0
$457$	4.53017e6	1.01467	0.507334	−	0.861749i	$-0.330631\pi$
$457$	4.53017e6	1.01467	0.507334	+	0.861749i	$0.330631\pi$
$458$	0	0
$459$	1.06188e6	0.235257
$460$	0	0
$461$	5.43190e6	1.19042	0.595209	−	0.803571i	$-0.297069\pi$
$461$	5.43190e6	1.19042	0.595209	+	0.803571i	$0.297069\pi$
$462$	0	0
$463$	2.78073e6	0.602847	0.301423	−	0.953490i	$-0.402538\pi$
$463$	2.78073e6	0.602847	0.301423	+	0.953490i	$0.402538\pi$
$464$	0	0
$465$	320244.	0.0686830
$466$	0	0
$467$	6.10093e6	1.29451	0.647253	−	0.762276i	$-0.275918\pi$
$467$	6.10093e6	1.29451	0.647253	+	0.762276i	$0.275918\pi$
$468$	0	0
$469$	−114078.	−0.0239480
$470$	0	0
$471$	907117.	0.188413
$472$	0	0
$473$	173346.	0.0356255
$474$	0	0
$475$	−1.42073e6	−0.288921
$476$	0	0
$477$	−1.22244e6	−0.245997
$478$	0	0
$479$	2.97287e6	0.592020	0.296010	−	0.955185i	$-0.404344\pi$
$479$	2.97287e6	0.592020	0.296010	+	0.955185i	$0.404344\pi$
$480$	0	0
$481$	2.93884e6	0.579180
$482$	0	0
$483$	566465.	0.110486
$484$	0	0
$485$	696361.	0.134425
$486$	0	0
$487$	−3.99463e6	−0.763227	−0.381613	−	0.924322i	$-0.624631\pi$
$487$	−3.99463e6	−0.763227	−0.381613	+	0.924322i	$0.624631\pi$
$488$	0	0
$489$	−1.95006e6	−0.368786
$490$	0	0
$491$	3.94446e6	0.738386	0.369193	−	0.929353i	$-0.379634\pi$
$491$	3.94446e6	0.738386	0.369193	+	0.929353i	$0.379634\pi$
$492$	0	0
$493$	−373986.	−0.0693007
$494$	0	0
$495$	−447624.	−0.0821108
$496$	0	0
$497$	642860.	0.116742
$498$	0	0
$499$	2.11454e6	0.380158	0.190079	−	0.981769i	$-0.439126\pi$
$499$	2.11454e6	0.380158	0.190079	+	0.981769i	$0.439126\pi$
$500$	0	0
$501$	−4.28915e6	−0.763444
$502$	0	0
$503$	5.07184e6	0.893811	0.446906	−	0.894581i	$-0.352526\pi$
$503$	5.07184e6	0.893811	0.446906	+	0.894581i	$0.352526\pi$
$504$	0	0
$505$	5.16980e6	0.902081
$506$	0	0
$507$	2.30710e6	0.398608
$508$	0	0
$509$	−642011.	−0.109837	−0.0549184	−	0.998491i	$-0.517490\pi$
$509$	−642011.	−0.109837	−0.0549184	+	0.998491i	$0.517490\pi$
$510$	0	0
$511$	233839.	0.0396155
$512$	0	0
$513$	−996706.	−0.167214
$514$	0	0
$515$	695541.	0.115559
$516$	0	0
$517$	−587004.	−0.0965861
$518$	0	0
$519$	−877521.	−0.143001
$520$	0	0
$521$	−7.53918e6	−1.21683	−0.608415	−	0.793619i	$-0.708194\pi$
$521$	−7.53918e6	−1.21683	−0.608415	+	0.793619i	$0.708194\pi$
$522$	0	0
$523$	−5.16739e6	−0.826070	−0.413035	−	0.910715i	$-0.635531\pi$
$523$	−5.16739e6	−0.826070	−0.413035	+	0.910715i	$0.635531\pi$
$524$	0	0
$525$	−153821.	−0.0243566
$526$	0	0
$527$	1.13486e6	0.177998
$528$	0	0
$529$	8.20774e6	1.27522
$530$	0	0
$531$	−3.48928e6	−0.537031
$532$	0	0
$533$	−1.06307e6	−0.162085
$534$	0	0
$535$	−6.53131e6	−0.986543
$536$	0	0
$537$	−1.84312e6	−0.275815
$538$	0	0
$539$	2.00091e6	0.296658
$540$	0	0
$541$	−1.99876e6	−0.293608	−0.146804	−	0.989166i	$-0.546899\pi$
$541$	−1.99876e6	−0.293608	−0.146804	+	0.989166i	$0.546899\pi$
$542$	0	0
$543$	1.02865e6	0.149716
$544$	0	0
$545$	3.11444e6	0.449147
$546$	0	0
$547$	3.47894e6	0.497140	0.248570	−	0.968614i	$-0.420039\pi$
$547$	3.47894e6	0.497140	0.248570	+	0.968614i	$0.420039\pi$
$548$	0	0
$549$	−1.85816e6	−0.263119
$550$	0	0
$551$	351033.	0.0492571
$552$	0	0
$553$	1.18709e6	0.165071
$554$	0	0
$555$	3.56295e6	0.490995
$556$	0	0
$557$	2.07530e6	0.283428	0.141714	−	0.989908i	$-0.454739\pi$
$557$	2.07530e6	0.283428	0.141714	+	0.989908i	$0.454739\pi$
$558$	0	0
$559$	485714.	0.0657433
$560$	0	0
$561$	−1.58626e6	−0.212798
$562$	0	0
$563$	1.10737e7	1.47239	0.736193	−	0.676772i	$-0.236622\pi$
$563$	1.10737e7	1.47239	0.736193	+	0.676772i	$0.236622\pi$
$564$	0	0
$565$	−6.27971e6	−0.827596
$566$	0	0
$567$	−107912.	−0.0140965
$568$	0	0
$569$	7.28513e6	0.943315	0.471658	−	0.881782i	$-0.343656\pi$
$569$	7.28513e6	0.943315	0.471658	+	0.881782i	$0.343656\pi$
$570$	0	0
$571$	3.18116e6	0.408315	0.204157	−	0.978938i	$-0.434555\pi$
$571$	3.18116e6	0.408315	0.204157	+	0.978938i	$0.434555\pi$
$572$	0	0
$573$	3.62965e6	0.461826
$574$	0	0
$575$	−3.97653e6	−0.501573
$576$	0	0
$577$	6.47190e6	0.809268	0.404634	−	0.914479i	$-0.367399\pi$
$577$	6.47190e6	0.809268	0.404634	+	0.914479i	$0.367399\pi$
$578$	0	0
$579$	−8.21945e6	−1.01893
$580$	0	0
$581$	−64631.0	−0.00794330
$582$	0	0
$583$	1.82611e6	0.222513
$584$	0	0
$585$	−1.25424e6	−0.151527
$586$	0	0
$587$	907491.	0.108704	0.0543522	−	0.998522i	$-0.482691\pi$
$587$	907491.	0.108704	0.0543522	+	0.998522i	$0.482691\pi$
$588$	0	0
$589$	−1.06521e6	−0.126517
$590$	0	0
$591$	3.02209e6	0.355909
$592$	0	0
$593$	7.51935e6	0.878099	0.439050	−	0.898463i	$-0.355315\pi$
$593$	7.51935e6	0.878099	0.439050	+	0.898463i	$0.355315\pi$
$594$	0	0
$595$	1.09418e6	0.126706
$596$	0	0
$597$	−7.32397e6	−0.841028
$598$	0	0
$599$	5.07068e6	0.577430	0.288715	−	0.957415i	$-0.406772\pi$
$599$	5.07068e6	0.577430	0.288715	+	0.957415i	$0.406772\pi$
$600$	0	0
$601$	−1.31801e7	−1.48844	−0.744222	−	0.667933i	$-0.767179\pi$
$601$	−1.31801e7	−1.48844	−0.744222	+	0.667933i	$0.767179\pi$
$602$	0	0
$603$	561806.	0.0629206
$604$	0	0
$605$	668673.	0.0742720
$606$	0	0
$607$	8.58376e6	0.945597	0.472798	−	0.881171i	$-0.343244\pi$
$607$	8.58376e6	0.945597	0.472798	+	0.881171i	$0.343244\pi$
$608$	0	0
$609$	38005.9	0.00415248
$610$	0	0
$611$	−1.64478e6	−0.178240
$612$	0	0
$613$	1.00418e7	1.07935	0.539675	−	0.841873i	$-0.318547\pi$
$613$	1.00418e7	1.07935	0.539675	+	0.841873i	$0.318547\pi$
$614$	0	0
$615$	−1.28882e6	−0.137406
$616$	0	0
$617$	2.90447e6	0.307153	0.153576	−	0.988137i	$-0.450921\pi$
$617$	2.90447e6	0.307153	0.153576	+	0.988137i	$0.450921\pi$
$618$	0	0
$619$	−2.16140e6	−0.226730	−0.113365	−	0.993553i	$-0.536163\pi$
$619$	−2.16140e6	−0.226730	−0.113365	+	0.993553i	$0.536163\pi$
$620$	0	0
$621$	−2.78971e6	−0.290288
$622$	0	0
$623$	2.31111e6	0.238561
$624$	0	0
$625$	−5.43851e6	−0.556904
$626$	0	0
$627$	1.48891e6	0.151251
$628$	0	0
$629$	1.26261e7	1.27246
$630$	0	0
$631$	−6.32911e6	−0.632804	−0.316402	−	0.948625i	$-0.602475\pi$
$631$	−6.32911e6	−0.632804	−0.316402	+	0.948625i	$0.602475\pi$
$632$	0	0
$633$	463416.	0.0459686
$634$	0	0
$635$	1.35023e7	1.32884
$636$	0	0
$637$	5.60655e6	0.547453
$638$	0	0
$639$	−3.16593e6	−0.306725
$640$	0	0
$641$	−1.58279e7	−1.52152	−0.760760	−	0.649033i	$-0.775174\pi$
$641$	−1.58279e7	−1.52152	−0.760760	+	0.649033i	$0.775174\pi$
$642$	0	0
$643$	−1.19185e7	−1.13683	−0.568415	−	0.822742i	$-0.692443\pi$
$643$	−1.19185e7	−1.13683	−0.568415	+	0.822742i	$0.692443\pi$
$644$	0	0
$645$	588862.	0.0557333
$646$	0	0
$647$	2.89334e6	0.271731	0.135865	−	0.990727i	$-0.456619\pi$
$647$	2.89334e6	0.271731	0.135865	+	0.990727i	$0.456619\pi$
$648$	0	0
$649$	5.21238e6	0.485763
$650$	0	0
$651$	−115329.	−0.0106656
$652$	0	0
$653$	−1.34330e7	−1.23279	−0.616396	−	0.787436i	$-0.711408\pi$
$653$	−1.34330e7	−1.23279	−0.616396	+	0.787436i	$0.711408\pi$
$654$	0	0
$655$	−6.56590e6	−0.597986
$656$	0	0
$657$	−1.15160e6	−0.104085
$658$	0	0
$659$	1.75666e7	1.57570	0.787849	−	0.615868i	$-0.211195\pi$
$659$	1.75666e7	1.57570	0.787849	+	0.615868i	$0.211195\pi$
$660$	0	0
$661$	1.20193e7	1.06998	0.534990	−	0.844858i	$-0.320315\pi$
$661$	1.20193e7	1.06998	0.534990	+	0.844858i	$0.320315\pi$
$662$	0	0
$663$	−4.44469e6	−0.392697
$664$	0	0
$665$	−1.02703e6	−0.0900592
$666$	0	0
$667$	982516.	0.0855116
$668$	0	0
$669$	−1.02921e6	−0.0889079
$670$	0	0
$671$	2.77577e6	0.238000
$672$	0	0
$673$	5.25023e6	0.446828	0.223414	−	0.974724i	$-0.428280\pi$
$673$	5.25023e6	0.446828	0.223414	+	0.974724i	$0.428280\pi$
$674$	0	0
$675$	757531.	0.0639943
$676$	0	0
$677$	−8.62823e6	−0.723519	−0.361760	−	0.932271i	$-0.617824\pi$
$677$	−8.62823e6	−0.723519	−0.361760	+	0.932271i	$0.617824\pi$
$678$	0	0
$679$	−250779.	−0.0208745
$680$	0	0
$681$	−1.00525e7	−0.830631
$682$	0	0
$683$	8.68783e6	0.712622	0.356311	−	0.934367i	$-0.384034\pi$
$683$	8.68783e6	0.712622	0.356311	+	0.934367i	$0.384034\pi$
$684$	0	0
$685$	−5.42888e6	−0.442063
$686$	0	0
$687$	−1.05019e7	−0.848939
$688$	0	0
$689$	5.11674e6	0.410625
$690$	0	0
$691$	−2.54042e6	−0.202400	−0.101200	−	0.994866i	$-0.532268\pi$
$691$	−2.54042e6	−0.202400	−0.101200	+	0.994866i	$0.532268\pi$
$692$	0	0
$693$	161202.	0.0127508
$694$	0	0
$695$	−5.81617e6	−0.456746
$696$	0	0
$697$	−4.56725e6	−0.356101
$698$	0	0
$699$	1.11596e7	0.863885
$700$	0	0
$701$	−1.27216e7	−0.977789	−0.488895	−	0.872343i	$-0.662600\pi$
$701$	−1.27216e7	−0.977789	−0.488895	+	0.872343i	$0.662600\pi$
$702$	0	0
$703$	−1.18512e7	−0.904431
$704$	0	0
$705$	−1.99407e6	−0.151101
$706$	0	0
$707$	−1.86179e6	−0.140082
$708$	0	0
$709$	−1.92436e7	−1.43770	−0.718852	−	0.695163i	$-0.755332\pi$
$709$	−1.92436e7	−1.43770	−0.718852	+	0.695163i	$0.755332\pi$
$710$	0	0
$711$	−5.84613e6	−0.433705
$712$	0	0
$713$	−2.98145e6	−0.219636
$714$	0	0
$715$	1.87362e6	0.137062
$716$	0	0
$717$	7.19741e6	0.522852
$718$	0	0
$719$	−8.72959e6	−0.629755	−0.314878	−	0.949132i	$-0.601963\pi$
$719$	−8.72959e6	−0.629755	−0.314878	+	0.949132i	$0.601963\pi$
$720$	0	0
$721$	−250484.	−0.0179449
$722$	0	0
$723$	370899.	0.0263882
$724$	0	0
$725$	−266797.	−0.0188511
$726$	0	0
$727$	−2.24040e7	−1.57213	−0.786065	−	0.618143i	$-0.787885\pi$
$727$	−2.24040e7	−1.57213	−0.786065	+	0.618143i	$0.787885\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.08677e6	0.144438
$732$	0	0
$733$	−1.66636e7	−1.14553	−0.572767	−	0.819718i	$-0.694130\pi$
$733$	−1.66636e7	−1.14553	−0.572767	+	0.819718i	$0.694130\pi$
$734$	0	0
$735$	6.79717e6	0.464098
$736$	0	0
$737$	−839240.	−0.0569138
$738$	0	0
$739$	9.34681e6	0.629582	0.314791	−	0.949161i	$-0.398066\pi$
$739$	9.34681e6	0.629582	0.314791	+	0.949161i	$0.398066\pi$
$740$	0	0
$741$	4.17191e6	0.279119
$742$	0	0
$743$	−1.49276e7	−0.992011	−0.496006	−	0.868319i	$-0.665200\pi$
$743$	−1.49276e7	−0.992011	−0.496006	+	0.868319i	$0.665200\pi$
$744$	0	0
$745$	−1.27987e6	−0.0844838
$746$	0	0
$747$	318292.	0.0208701
$748$	0	0
$749$	2.35211e6	0.153198
$750$	0	0
$751$	−2.03700e7	−1.31793	−0.658964	−	0.752175i	$-0.729005\pi$
$751$	−2.03700e7	−1.31793	−0.658964	+	0.752175i	$0.729005\pi$
$752$	0	0
$753$	−9.29725e6	−0.597540
$754$	0	0
$755$	−677755.	−0.0432718
$756$	0	0
$757$	−1.95206e7	−1.23809	−0.619047	−	0.785354i	$-0.712481\pi$
$757$	−1.95206e7	−1.23809	−0.619047	+	0.785354i	$0.712481\pi$
$758$	0	0
$759$	4.16734e6	0.262576
$760$	0	0
$761$	−2.42547e7	−1.51822	−0.759110	−	0.650962i	$-0.774366\pi$
$761$	−2.42547e7	−1.51822	−0.759110	+	0.650962i	$0.774366\pi$
$762$	0	0
$763$	−1.12160e6	−0.0697470
$764$	0	0
$765$	−5.38858e6	−0.332905
$766$	0	0
$767$	1.46051e7	0.896427
$768$	0	0
$769$	−1.59111e7	−0.970252	−0.485126	−	0.874444i	$-0.661226\pi$
$769$	−1.59111e7	−0.970252	−0.485126	+	0.874444i	$0.661226\pi$
$770$	0	0
$771$	1.29264e7	0.783145
$772$	0	0
$773$	−9.11982e6	−0.548956	−0.274478	−	0.961593i	$-0.588505\pi$
$773$	−9.11982e6	−0.548956	−0.274478	+	0.961593i	$0.588505\pi$
$774$	0	0
$775$	809597.	0.0484189
$776$	0	0
$777$	−1.28312e6	−0.0762454
$778$	0	0
$779$	4.28694e6	0.253107
$780$	0	0
$781$	4.72935e6	0.277443
$782$	0	0
$783$	−187170.	−0.0109102
$784$	0	0
$785$	−4.60324e6	−0.266618
$786$	0	0
$787$	1.55665e7	0.895887	0.447944	−	0.894062i	$-0.352157\pi$
$787$	1.55665e7	0.895887	0.447944	+	0.894062i	$0.352157\pi$
$788$	0	0
$789$	743348.	0.0425109
$790$	0	0
$791$	2.26150e6	0.128515
$792$	0	0
$793$	7.77768e6	0.439205
$794$	0	0
$795$	6.20335e6	0.348104
$796$	0	0
$797$	−1.49755e7	−0.835095	−0.417548	−	0.908655i	$-0.637110\pi$
$797$	−1.49755e7	−0.835095	−0.417548	+	0.908655i	$0.637110\pi$
$798$	0	0
$799$	−7.06646e6	−0.391593
$800$	0	0
$801$	−1.13817e7	−0.626793
$802$	0	0
$803$	1.72029e6	0.0941486
$804$	0	0
$805$	−2.87458e6	−0.156345
$806$	0	0
$807$	−6.05017e6	−0.327027
$808$	0	0
$809$	1.63273e7	0.877087	0.438543	−	0.898710i	$-0.355495\pi$
$809$	1.63273e7	0.877087	0.438543	+	0.898710i	$0.355495\pi$
$810$	0	0
$811$	2.87618e7	1.53555	0.767776	−	0.640719i	$-0.221364\pi$
$811$	2.87618e7	1.53555	0.767776	+	0.640719i	$0.221364\pi$
$812$	0	0
$813$	7.53174e6	0.399640
$814$	0	0
$815$	9.89572e6	0.521859
$816$	0	0
$817$	−1.95870e6	−0.102663
$818$	0	0
$819$	451687.	0.0235303
$820$	0	0
$821$	2.00158e7	1.03637	0.518186	−	0.855268i	$-0.326607\pi$
$821$	2.00158e7	1.03637	0.518186	+	0.855268i	$0.326607\pi$
$822$	0	0
$823$	1.33001e7	0.684472	0.342236	−	0.939614i	$-0.388816\pi$
$823$	1.33001e7	0.684472	0.342236	+	0.939614i	$0.388816\pi$
$824$	0	0
$825$	−1.13162e6	−0.0578850
$826$	0	0
$827$	5.22101e6	0.265455	0.132728	−	0.991153i	$-0.457626\pi$
$827$	5.22101e6	0.265455	0.132728	+	0.991153i	$0.457626\pi$
$828$	0	0
$829$	−2.24335e7	−1.13373	−0.566866	−	0.823810i	$-0.691844\pi$
$829$	−2.24335e7	−1.13373	−0.566866	+	0.823810i	$0.691844\pi$
$830$	0	0
$831$	−1.63594e7	−0.821799
$832$	0	0
$833$	2.40874e7	1.20276
$834$	0	0
$835$	2.17657e7	1.08033
$836$	0	0
$837$	567968.	0.0280227
$838$	0	0
$839$	−2.69278e7	−1.32067	−0.660337	−	0.750969i	$-0.729587\pi$
$839$	−2.69278e7	−1.32067	−0.660337	+	0.750969i	$0.729587\pi$
$840$	0	0
$841$	−2.04452e7	−0.996786
$842$	0	0
$843$	967578.	0.0468940
$844$	0	0
$845$	−1.17076e7	−0.564059
$846$	0	0
$847$	−240808.	−0.0115335
$848$	0	0
$849$	−1.50652e7	−0.717306
$850$	0	0
$851$	−3.31707e7	−1.57011
$852$	0	0
$853$	5.53294e6	0.260366	0.130183	−	0.991490i	$-0.458444\pi$
$853$	5.53294e6	0.260366	0.130183	+	0.991490i	$0.458444\pi$
$854$	0	0
$855$	5.05787e6	0.236620
$856$	0	0
$857$	−2.66936e6	−0.124152	−0.0620761	−	0.998071i	$-0.519772\pi$
$857$	−2.66936e6	−0.124152	−0.0620761	+	0.998071i	$0.519772\pi$
$858$	0	0
$859$	1.05572e7	0.488163	0.244081	−	0.969755i	$-0.421514\pi$
$859$	1.05572e7	0.488163	0.244081	+	0.969755i	$0.421514\pi$
$860$	0	0
$861$	464141.	0.0213374
$862$	0	0
$863$	1.55776e7	0.711989	0.355995	−	0.934488i	$-0.384142\pi$
$863$	1.55776e7	0.711989	0.355995	+	0.934488i	$0.384142\pi$
$864$	0	0
$865$	4.45305e6	0.202357
$866$	0	0
$867$	−6.31699e6	−0.285406
$868$	0	0
$869$	8.73311e6	0.392301
$870$	0	0
$871$	−2.35155e6	−0.105029
$872$	0	0
$873$	1.23503e6	0.0548455
$874$	0	0
$875$	3.12801e6	0.138117
$876$	0	0
$877$	3.35577e7	1.47331	0.736653	−	0.676271i	$-0.236405\pi$
$877$	3.35577e7	1.47331	0.736653	+	0.676271i	$0.236405\pi$
$878$	0	0
$879$	1.60191e7	0.699302
$880$	0	0
$881$	3.39756e7	1.47478	0.737390	−	0.675467i	$-0.236058\pi$
$881$	3.39756e7	1.47478	0.737390	+	0.675467i	$0.236058\pi$
$882$	0	0
$883$	−2.56686e7	−1.10790	−0.553950	−	0.832550i	$-0.686880\pi$
$883$	−2.56686e7	−1.10790	−0.553950	+	0.832550i	$0.686880\pi$
$884$	0	0
$885$	1.77066e7	0.759938
$886$	0	0
$887$	−4.12233e7	−1.75927	−0.879637	−	0.475646i	$-0.842214\pi$
$887$	−4.12233e7	−1.75927	−0.879637	+	0.475646i	$0.842214\pi$
$888$	0	0
$889$	−4.86256e6	−0.206353
$890$	0	0
$891$	−793881.	−0.0335013
$892$	0	0
$893$	6.63277e6	0.278334
$894$	0	0
$895$	9.35308e6	0.390299
$896$	0	0
$897$	1.16769e7	0.484557
$898$	0	0
$899$	−200034.	−0.00825477
$900$	0	0
$901$	2.19830e7	0.902144
$902$	0	0
$903$	−212066.	−0.00865469
$904$	0	0
$905$	−5.21997e6	−0.211859
$906$	0	0
$907$	3.57376e7	1.44247	0.721237	−	0.692689i	$-0.243574\pi$
$907$	3.57376e7	1.44247	0.721237	+	0.692689i	$0.243574\pi$
$908$	0	0
$909$	9.16888e6	0.368050
$910$	0	0
$911$	−1.71528e7	−0.684760	−0.342380	−	0.939561i	$-0.611233\pi$
$911$	−1.71528e7	−0.684760	−0.342380	+	0.939561i	$0.611233\pi$
$912$	0	0
$913$	−475474.	−0.0188777
$914$	0	0
$915$	9.42938e6	0.372332
$916$	0	0
$917$	2.36457e6	0.0928599
$918$	0	0
$919$	3.11619e7	1.21712	0.608562	−	0.793506i	$-0.291747\pi$
$919$	3.11619e7	1.21712	0.608562	+	0.793506i	$0.291747\pi$
$920$	0	0
$921$	1.75811e7	0.682963
$922$	0	0
$923$	1.32516e7	0.511994
$924$	0	0
$925$	9.00735e6	0.346133
$926$	0	0
$927$	1.23357e6	0.0471482
$928$	0	0
$929$	−3.80173e7	−1.44525	−0.722623	−	0.691242i	$-0.757064\pi$
$929$	−3.80173e7	−1.44525	−0.722623	+	0.691242i	$0.757064\pi$
$930$	0	0
$931$	−2.26091e7	−0.854886
$932$	0	0
$933$	1.56783e7	0.589650
$934$	0	0
$935$	8.04961e6	0.301124
$936$	0	0
$937$	5.06260e7	1.88376	0.941879	−	0.335953i	$-0.109058\pi$
$937$	5.06260e7	1.88376	0.941879	+	0.335953i	$0.109058\pi$
$938$	0	0
$939$	−2.33563e7	−0.864450
$940$	0	0
$941$	−7.27291e6	−0.267753	−0.133876	−	0.990998i	$-0.542743\pi$
$941$	−7.27291e6	−0.267753	−0.133876	+	0.990998i	$0.542743\pi$
$942$	0	0
$943$	1.19988e7	0.439400
$944$	0	0
$945$	547609.	0.0199476
$946$	0	0
$947$	7.13299e6	0.258462	0.129231	−	0.991615i	$-0.458749\pi$
$947$	7.13299e6	0.258462	0.129231	+	0.991615i	$0.458749\pi$
$948$	0	0
$949$	4.82025e6	0.173742
$950$	0	0
$951$	1.30436e7	0.467679
$952$	0	0
$953$	−1.78255e7	−0.635783	−0.317891	−	0.948127i	$-0.602975\pi$
$953$	−1.78255e7	−0.635783	−0.317891	+	0.948127i	$0.602975\pi$
$954$	0	0
$955$	−1.84190e7	−0.653517
$956$	0	0
$957$	279599.	0.00986862
$958$	0	0
$959$	1.95509e6	0.0686469
$960$	0	0
$961$	−2.80221e7	−0.978798
$962$	0	0
$963$	−1.15836e7	−0.402510
$964$	0	0
$965$	4.17103e7	1.44187
$966$	0	0
$967$	−4.33947e7	−1.49235	−0.746174	−	0.665751i	$-0.768111\pi$
$967$	−4.33947e7	−1.49235	−0.746174	+	0.665751i	$0.768111\pi$
$968$	0	0
$969$	1.79237e7	0.613224
$970$	0	0
$971$	−2.89232e7	−0.984459	−0.492230	−	0.870465i	$-0.663818\pi$
$971$	−2.89232e7	−0.984459	−0.492230	+	0.870465i	$0.663818\pi$
$972$	0	0
$973$	2.09456e6	0.0709270
$974$	0	0
$975$	−3.17079e6	−0.106821
$976$	0	0
$977$	−1.51228e7	−0.506870	−0.253435	−	0.967352i	$-0.581560\pi$
$977$	−1.51228e7	−0.506870	−0.253435	+	0.967352i	$0.581560\pi$
$978$	0	0
$979$	1.70022e7	0.566956
$980$	0	0
$981$	5.52360e6	0.183252
$982$	0	0
$983$	−2.55899e7	−0.844666	−0.422333	−	0.906441i	$-0.638789\pi$
$983$	−2.55899e7	−0.844666	−0.422333	+	0.906441i	$0.638789\pi$
$984$	0	0
$985$	−1.53359e7	−0.503637
$986$	0	0
$987$	718121.	0.0234642
$988$	0	0
$989$	−5.48226e6	−0.178225
$990$	0	0
$991$	1.16531e7	0.376928	0.188464	−	0.982080i	$-0.439649\pi$
$991$	1.16531e7	0.376928	0.188464	+	0.982080i	$0.439649\pi$
$992$	0	0
$993$	−6.94848e6	−0.223623
$994$	0	0
$995$	3.71661e7	1.19012
$996$	0	0
$997$	4.08824e7	1.30256	0.651281	−	0.758836i	$-0.274232\pi$
$997$	4.08824e7	1.30256	0.651281	+	0.758836i	$0.274232\pi$
$998$	0	0
$999$	6.31905e6	0.200326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	264.6.a.a.1.2	✓	2
3.2	odd	2		792.6.a.d.1.1		2
4.3	odd	2		528.6.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
264.6.a.a.1.2	✓	2	1.1	even	1	trivial
528.6.a.r.1.2		2	4.3	odd	2
792.6.a.d.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:	\( N \)	\(=\)	\( 264 = 2^{3} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	264.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} +45.6712 q^{5} -16.4475 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} +45.6712 q^{5} -16.4475 q^{7} +81.0000 q^{9} -121.000 q^{11} -339.041 q^{13} -411.041 q^{15} -1456.62 q^{17} +1367.22 q^{19} +148.027 q^{21} +3826.76 q^{23} -1039.14 q^{25} -729.000 q^{27} +256.749 q^{29} -779.105 q^{31} +1089.00 q^{33} -751.178 q^{35} -8668.10 q^{37} +3051.37 q^{39} +3135.51 q^{41} -1432.61 q^{43} +3699.37 q^{45} +4851.27 q^{47} -16536.5 q^{49} +13109.6 q^{51} -15091.8 q^{53} -5526.22 q^{55} -12305.0 q^{57} -43077.5 q^{59} -22940.2 q^{61} -1332.25 q^{63} -15484.4 q^{65} +6935.87 q^{67} -34440.8 q^{69} -39085.6 q^{71} -14217.3 q^{73} +9352.24 q^{75} +1990.15 q^{77} -72174.4 q^{79} +6561.00 q^{81} +3929.54 q^{83} -66525.7 q^{85} -2310.74 q^{87} -140514. q^{89} +5576.38 q^{91} +7011.95 q^{93} +62442.8 q^{95} +15247.3 q^{97} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} - 30 q^{5} + 48 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 - 30 * q^5 + 48 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} - 30 q^{5} + 48 q^{7} + 162 q^{9} - 242 q^{11} + 414 q^{13} + 270 q^{15} - 1174 q^{17} + 2694 q^{19} - 432 q^{21} + 2274 q^{23} + 1562 q^{25} - 1458 q^{27} + 1282 q^{29} - 1720 q^{31} + 2178 q^{33} - 5628 q^{35} - 2856 q^{37} - 3726 q^{39} - 7198 q^{41} - 7274 q^{43} - 2430 q^{45} - 3362 q^{47} - 29190 q^{49} + 10566 q^{51} - 44138 q^{53} + 3630 q^{55} - 24246 q^{57} - 27668 q^{59} - 25050 q^{61} + 3888 q^{63} - 72468 q^{65} + 29080 q^{67} - 20466 q^{69} - 3950 q^{71} - 52784 q^{73} - 14058 q^{75} - 5808 q^{77} - 97268 q^{79} + 13122 q^{81} - 15196 q^{83} - 87912 q^{85} - 11538 q^{87} - 227476 q^{89} + 54108 q^{91} + 15480 q^{93} - 37956 q^{95} - 96996 q^{97} - 19602 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 - 30 * q^5 + 48 * q^7 + 162 * q^9 - 242 * q^11 + 414 * q^13 + 270 * q^15 - 1174 * q^17 + 2694 * q^19 - 432 * q^21 + 2274 * q^23 + 1562 * q^25 - 1458 * q^27 + 1282 * q^29 - 1720 * q^31 + 2178 * q^33 - 5628 * q^35 - 2856 * q^37 - 3726 * q^39 - 7198 * q^41 - 7274 * q^43 - 2430 * q^45 - 3362 * q^47 - 29190 * q^49 + 10566 * q^51 - 44138 * q^53 + 3630 * q^55 - 24246 * q^57 - 27668 * q^59 - 25050 * q^61 + 3888 * q^63 - 72468 * q^65 + 29080 * q^67 - 20466 * q^69 - 3950 * q^71 - 52784 * q^73 - 14058 * q^75 - 5808 * q^77 - 97268 * q^79 + 13122 * q^81 - 15196 * q^83 - 87912 * q^85 - 11538 * q^87 - 227476 * q^89 + 54108 * q^91 + 15480 * q^93 - 37956 * q^95 - 96996 * q^97 - 19602 * q^99

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