LMFDB - Embedded newform 891.6.a.f.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,6,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("891.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	891.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:	$142.901983453$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53916 q^{2} -29.6310 q^{4} +17.6583 q^{5} +165.539 q^{7} -94.8598 q^{8} +O(q^{10})$	\(q+1.53916 q^{2} -29.6310 q^{4} +17.6583 q^{5} +165.539 q^{7} -94.8598 q^{8} +27.1789 q^{10} -121.000 q^{11} -231.530 q^{13} +254.790 q^{14} +802.188 q^{16} -235.531 q^{17} -1094.47 q^{19} -523.234 q^{20} -186.238 q^{22} +3433.42 q^{23} -2813.18 q^{25} -356.362 q^{26} -4905.08 q^{28} -3381.90 q^{29} +7734.37 q^{31} +4270.21 q^{32} -362.520 q^{34} +2923.14 q^{35} -6905.91 q^{37} -1684.56 q^{38} -1675.07 q^{40} -7579.18 q^{41} -4967.76 q^{43} +3585.35 q^{44} +5284.58 q^{46} +4839.44 q^{47} +10596.1 q^{49} -4329.93 q^{50} +6860.48 q^{52} -1614.76 q^{53} -2136.66 q^{55} -15703.0 q^{56} -5205.28 q^{58} -13434.8 q^{59} +23726.8 q^{61} +11904.4 q^{62} -19097.5 q^{64} -4088.44 q^{65} +14598.3 q^{67} +6979.03 q^{68} +4499.17 q^{70} -13457.3 q^{71} -2565.88 q^{73} -10629.3 q^{74} +32430.3 q^{76} -20030.2 q^{77} +1674.25 q^{79} +14165.3 q^{80} -11665.5 q^{82} -54921.2 q^{83} -4159.09 q^{85} -7646.17 q^{86} +11478.0 q^{88} +135543. q^{89} -38327.3 q^{91} -101736. q^{92} +7448.65 q^{94} -19326.5 q^{95} +179140. q^{97} +16309.0 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	$ 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) $ 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.53916	0.272087	0.136044	−	0.990703i	$-0.456561\pi$
$2$	1.53916	0.272087	0.136044	+	0.990703i	$0.456561\pi$
$3$	0	0
$3$	0	0
$4$	−29.6310	−0.925969
$5$	17.6583	0.315882	0.157941	−	0.987449i	$-0.449514\pi$
$5$	17.6583	0.315882	0.157941	+	0.987449i	$0.449514\pi$
$6$	0	0
$7$	165.539	1.27689	0.638447	−	0.769666i	$-0.279577\pi$
$7$	165.539	1.27689	0.638447	+	0.769666i	$0.279577\pi$
$8$	−94.8598	−0.524031
$9$	0	0
$10$	27.1789	0.0859474
$11$	−121.000	−0.301511
$11$	−121.000	−0.301511
$12$	0	0
$13$	−231.530	−0.379970	−0.189985	−	0.981787i	$-0.560844\pi$
$13$	−231.530	−0.379970	−0.189985	+	0.981787i	$0.560844\pi$
$14$	254.790	0.347426
$15$	0	0
$16$	802.188	0.783387
$17$	−235.531	−0.197664	−0.0988318	−	0.995104i	$-0.531511\pi$
$17$	−235.531	−0.197664	−0.0988318	+	0.995104i	$0.531511\pi$
$18$	0	0
$19$	−1094.47	−0.695538	−0.347769	−	0.937580i	$-0.613061\pi$
$19$	−1094.47	−0.695538	−0.347769	+	0.937580i	$0.613061\pi$
$20$	−523.234	−0.292497
$21$	0	0
$22$	−186.238	−0.0820373
$23$	3433.42	1.35334	0.676672	−	0.736285i	$-0.263422\pi$
$23$	3433.42	1.35334	0.676672	+	0.736285i	$0.263422\pi$
$24$	0	0
$25$	−2813.18	−0.900219
$26$	−356.362	−0.103385
$27$	0	0
$28$	−4905.08	−1.18236
$29$	−3381.90	−0.746734	−0.373367	−	0.927684i	$-0.621797\pi$
$29$	−3381.90	−0.746734	−0.373367	+	0.927684i	$0.621797\pi$
$30$	0	0
$31$	7734.37	1.44551	0.722754	−	0.691105i	$-0.242876\pi$
$31$	7734.37	1.44551	0.722754	+	0.691105i	$0.242876\pi$
$32$	4270.21	0.737181
$33$	0	0
$34$	−362.520	−0.0537817
$35$	2923.14	0.403347
$36$	0	0
$37$	−6905.91	−0.829309	−0.414655	−	0.909979i	$-0.636098\pi$
$37$	−6905.91	−0.829309	−0.414655	+	0.909979i	$0.636098\pi$
$38$	−1684.56	−0.189247
$39$	0	0
$40$	−1675.07	−0.165532
$41$	−7579.18	−0.704146	−0.352073	−	0.935973i	$-0.614523\pi$
$41$	−7579.18	−0.704146	−0.352073	+	0.935973i	$0.614523\pi$
$42$	0	0
$43$	−4967.76	−0.409722	−0.204861	−	0.978791i	$-0.565674\pi$
$43$	−4967.76	−0.409722	−0.204861	+	0.978791i	$0.565674\pi$
$44$	3585.35	0.279190
$45$	0	0
$46$	5284.58	0.368227
$47$	4839.44	0.319558	0.159779	−	0.987153i	$-0.448922\pi$
$47$	4839.44	0.319558	0.159779	+	0.987153i	$0.448922\pi$
$48$	0	0
$49$	10596.1	0.630457
$50$	−4329.93	−0.244938
$51$	0	0
$52$	6860.48	0.351841
$53$	−1614.76	−0.0789620	−0.0394810	−	0.999220i	$-0.512570\pi$
$53$	−1614.76	−0.0789620	−0.0394810	+	0.999220i	$0.512570\pi$
$54$	0	0
$55$	−2136.66	−0.0952420
$56$	−15703.0	−0.669132
$57$	0	0
$58$	−5205.28	−0.203177
$59$	−13434.8	−0.502460	−0.251230	−	0.967927i	$-0.580835\pi$
$59$	−13434.8	−0.502460	−0.251230	+	0.967927i	$0.580835\pi$
$60$	0	0
$61$	23726.8	0.816421	0.408210	−	0.912888i	$-0.366153\pi$
$61$	23726.8	0.816421	0.408210	+	0.912888i	$0.366153\pi$
$62$	11904.4	0.393304
$63$	0	0
$64$	−19097.5	−0.582809
$65$	−4088.44	−0.120026
$66$	0	0
$67$	14598.3	0.397298	0.198649	−	0.980071i	$-0.436345\pi$
$67$	14598.3	0.397298	0.198649	+	0.980071i	$0.436345\pi$
$68$	6979.03	0.183030
$69$	0	0
$70$	4499.17	0.109746
$71$	−13457.3	−0.316821	−0.158410	−	0.987373i	$-0.550637\pi$
$71$	−13457.3	−0.316821	−0.158410	+	0.987373i	$0.550637\pi$
$72$	0	0
$73$	−2565.88	−0.0563545	−0.0281773	−	0.999603i	$-0.508970\pi$
$73$	−2565.88	−0.0563545	−0.0281773	+	0.999603i	$0.508970\pi$
$74$	−10629.3	−0.225644
$75$	0	0
$76$	32430.3	0.644046
$77$	−20030.2	−0.384998
$78$	0	0
$79$	1674.25	0.0301824	0.0150912	−	0.999886i	$-0.495196\pi$
$79$	1674.25	0.0301824	0.0150912	+	0.999886i	$0.495196\pi$
$80$	14165.3	0.247458
$81$	0	0
$82$	−11665.5	−0.191589
$83$	−54921.2	−0.875075	−0.437537	−	0.899200i	$-0.644149\pi$
$83$	−54921.2	−0.875075	−0.437537	+	0.899200i	$0.644149\pi$
$84$	0	0
$85$	−4159.09	−0.0624383
$86$	−7646.17	−0.111480
$87$	0	0
$88$	11478.0	0.158001
$89$	135543.	1.81386	0.906929	−	0.421284i	$-0.138420\pi$
$89$	135543.	1.81386	0.906929	+	0.421284i	$0.138420\pi$
$90$	0	0
$91$	−38327.3	−0.485182
$92$	−101736.	−1.25315
$93$	0	0
$94$	7448.65	0.0869477
$95$	−19326.5	−0.219708
$96$	0	0
$97$	179140.	1.93314	0.966571	−	0.256397i	$-0.0825356\pi$
$97$	179140.	1.93314	0.966571	+	0.256397i	$0.0825356\pi$
$98$	16309.0	0.171539
$99$	0	0
$100$	83357.4	0.833574
$101$	−61300.8	−0.597947	−0.298974	−	0.954261i	$-0.596644\pi$
$101$	−61300.8	−0.597947	−0.298974	+	0.954261i	$0.596644\pi$
$102$	0	0
$103$	61978.7	0.575638	0.287819	−	0.957685i	$-0.407070\pi$
$103$	61978.7	0.575638	0.287819	+	0.957685i	$0.407070\pi$
$104$	21962.9	0.199116
$105$	0	0
$106$	−2485.37	−0.0214845
$107$	−196542.	−1.65957	−0.829784	−	0.558084i	$-0.811537\pi$
$107$	−196542.	−1.65957	−0.829784	+	0.558084i	$0.811537\pi$
$108$	0	0
$109$	−120605.	−0.972294	−0.486147	−	0.873877i	$-0.661598\pi$
$109$	−120605.	−0.972294	−0.486147	+	0.873877i	$0.661598\pi$
$110$	−3288.65	−0.0259141
$111$	0	0
$112$	132793.	1.00030
$113$	230062.	1.69492	0.847459	−	0.530861i	$-0.178131\pi$
$113$	230062.	1.69492	0.847459	+	0.530861i	$0.178131\pi$
$114$	0	0
$115$	60628.5	0.427497
$116$	100209.	0.691452
$117$	0	0
$118$	−20678.3	−0.136713
$119$	−38989.6	−0.252395
$120$	0	0
$121$	14641.0	0.0909091
$122$	36519.2	0.222138
$123$	0	0
$124$	−229177.	−1.33850
$125$	−104858.	−0.600245
$126$	0	0
$127$	−277150.	−1.52478	−0.762388	−	0.647120i	$-0.775973\pi$
$127$	−277150.	−1.52478	−0.762388	+	0.647120i	$0.775973\pi$
$128$	−166041.	−0.895755
$129$	0	0
$130$	−6292.75	−0.0326574
$131$	92963.1	0.473296	0.236648	−	0.971595i	$-0.423951\pi$
$131$	92963.1	0.473296	0.236648	+	0.971595i	$0.423951\pi$
$132$	0	0
$133$	−181178.	−0.888127
$134$	22469.1	0.108100
$135$	0	0
$136$	22342.5	0.103582
$137$	188299.	0.857131	0.428565	−	0.903511i	$-0.359019\pi$
$137$	188299.	0.857131	0.428565	+	0.903511i	$0.359019\pi$
$138$	0	0
$139$	−383909.	−1.68536	−0.842678	−	0.538418i	$-0.819022\pi$
$139$	−383909.	−1.68536	−0.842678	+	0.538418i	$0.819022\pi$
$140$	−86615.5	−0.373487
$141$	0	0
$142$	−20713.0	−0.0862028
$143$	28015.2	0.114565
$144$	0	0
$145$	−59718.8	−0.235880
$146$	−3949.29	−0.0153333
$147$	0	0
$148$	204629.	0.767914
$149$	−12802.2	−0.0472412	−0.0236206	−	0.999721i	$-0.507519\pi$
$149$	−12802.2	−0.0472412	−0.0236206	+	0.999721i	$0.507519\pi$
$150$	0	0
$151$	−539992.	−1.92728	−0.963640	−	0.267203i	$-0.913901\pi$
$151$	−539992.	−1.92728	−0.963640	+	0.267203i	$0.913901\pi$
$152$	103821.	0.364483
$153$	0	0
$154$	−30829.6	−0.104753
$155$	136576.	0.456610
$156$	0	0
$157$	−155769.	−0.504350	−0.252175	−	0.967682i	$-0.581146\pi$
$157$	−155769.	−0.504350	−0.252175	+	0.967682i	$0.581146\pi$
$158$	2576.94	0.00821224
$159$	0	0
$160$	75404.7	0.232862
$161$	568365.	1.72807
$162$	0	0
$163$	−333665.	−0.983653	−0.491826	−	0.870693i	$-0.663671\pi$
$163$	−333665.	−0.983653	−0.491826	+	0.870693i	$0.663671\pi$
$164$	224579.	0.652017
$165$	0	0
$166$	−84532.4	−0.238097
$167$	−503962.	−1.39832	−0.699160	−	0.714965i	$-0.746442\pi$
$167$	−503962.	−1.39832	−0.699160	+	0.714965i	$0.746442\pi$
$168$	0	0
$169$	−317687.	−0.855623
$170$	−6401.50	−0.0169887
$171$	0	0
$172$	147200.	0.379390
$173$	−389453.	−0.989327	−0.494664	−	0.869085i	$-0.664709\pi$
$173$	−389453.	−0.989327	−0.494664	+	0.869085i	$0.664709\pi$
$174$	0	0
$175$	−465691.	−1.14948
$176$	−97064.7	−0.236200
$177$	0	0
$178$	208623.	0.493527
$179$	−792594.	−1.84892	−0.924461	−	0.381277i	$-0.875484\pi$
$179$	−792594.	−1.84892	−0.924461	+	0.381277i	$0.875484\pi$
$180$	0	0
$181$	−191168.	−0.433729	−0.216865	−	0.976202i	$-0.569583\pi$
$181$	−191168.	−0.433729	−0.216865	+	0.976202i	$0.569583\pi$
$182$	−58991.7	−0.132012
$183$	0	0
$184$	−325694.	−0.709194
$185$	−121947.	−0.261964
$186$	0	0
$187$	28499.3	0.0595978
$188$	−143397.	−0.295901
$189$	0	0
$190$	−29746.6	−0.0597796
$191$	317863.	0.630460	0.315230	−	0.949015i	$-0.397918\pi$
$191$	317863.	0.630460	0.315230	+	0.949015i	$0.397918\pi$
$192$	0	0
$193$	−689300.	−1.33203	−0.666017	−	0.745937i	$-0.732002\pi$
$193$	−689300.	−1.33203	−0.666017	+	0.745937i	$0.732002\pi$
$194$	275725.	0.525983
$195$	0	0
$196$	−313973.	−0.583783
$197$	−757681.	−1.39098	−0.695489	−	0.718536i	$-0.744812\pi$
$197$	−757681.	−1.39098	−0.695489	+	0.718536i	$0.744812\pi$
$198$	0	0
$199$	−439273.	−0.786325	−0.393163	−	0.919469i	$-0.628619\pi$
$199$	−439273.	−0.786325	−0.393163	+	0.919469i	$0.628619\pi$
$200$	266858.	0.471743
$201$	0	0
$202$	−94351.6	−0.162694
$203$	−559836.	−0.953500
$204$	0	0
$205$	−133836.	−0.222427
$206$	95395.0	0.156624
$207$	0	0
$208$	−185731.	−0.297664
$209$	132431.	0.209712
$210$	0	0
$211$	−917216.	−1.41829	−0.709146	−	0.705062i	$-0.750919\pi$
$211$	−917216.	−1.41829	−0.709146	+	0.705062i	$0.750919\pi$
$212$	47846.9	0.0731164
$213$	0	0
$214$	−302509.	−0.451547
$215$	−87722.4	−0.129424
$216$	0	0
$217$	1.28034e6	1.84576
$218$	−185629.	−0.264549
$219$	0	0
$220$	63311.3	0.0881911
$221$	54532.7	0.0751063
$222$	0	0
$223$	155260.	0.209072	0.104536	−	0.994521i	$-0.466664\pi$
$223$	155260.	0.209072	0.104536	+	0.994521i	$0.466664\pi$
$224$	706885.	0.941301
$225$	0	0
$226$	354102.	0.461165
$227$	1.21338e6	1.56290	0.781450	−	0.623968i	$-0.214480\pi$
$227$	1.21338e6	1.56290	0.781450	+	0.623968i	$0.214480\pi$
$228$	0	0
$229$	120513.	0.151860	0.0759302	−	0.997113i	$-0.475807\pi$
$229$	120513.	0.151860	0.0759302	+	0.997113i	$0.475807\pi$
$230$	93316.9	0.116316
$231$	0	0
$232$	320807.	0.391312
$233$	−167099.	−0.201644	−0.100822	−	0.994904i	$-0.532147\pi$
$233$	−167099.	−0.201644	−0.100822	+	0.994904i	$0.532147\pi$
$234$	0	0
$235$	85456.4	0.100943
$236$	398087.	0.465262
$237$	0	0
$238$	−60011.1	−0.0686735
$239$	587795.	0.665627	0.332814	−	0.942993i	$-0.392002\pi$
$239$	587795.	0.665627	0.332814	+	0.942993i	$0.392002\pi$
$240$	0	0
$241$	116544.	0.129255	0.0646273	−	0.997909i	$-0.479414\pi$
$241$	116544.	0.129255	0.0646273	+	0.997909i	$0.479414\pi$
$242$	22534.8	0.0247352
$243$	0	0
$244$	−703048.	−0.755980
$245$	187109.	0.199150
$246$	0	0
$247$	253404.	0.264284
$248$	−733681.	−0.757492
$249$	0	0
$250$	−161394.	−0.163319
$251$	−1.38721e6	−1.38982	−0.694908	−	0.719098i	$-0.744555\pi$
$251$	−1.38721e6	−1.38982	−0.694908	+	0.719098i	$0.744555\pi$
$252$	0	0
$253$	−415444.	−0.408048
$254$	−426578.	−0.414872
$255$	0	0
$256$	355557.	0.339086
$257$	−865393.	−0.817298	−0.408649	−	0.912692i	$-0.634000\pi$
$257$	−865393.	−0.817298	−0.408649	+	0.912692i	$0.634000\pi$
$258$	0	0
$259$	−1.14320e6	−1.05894
$260$	121145.	0.111140
$261$	0	0
$262$	143085.	0.128778
$263$	564396.	0.503147	0.251573	−	0.967838i	$-0.419052\pi$
$263$	564396.	0.503147	0.251573	+	0.967838i	$0.419052\pi$
$264$	0	0
$265$	−28514.0	−0.0249427
$266$	−278861.	−0.241648
$267$	0	0
$268$	−432563.	−0.367885
$269$	1.13051e6	0.952561	0.476280	−	0.879294i	$-0.341985\pi$
$269$	1.13051e6	0.952561	0.476280	+	0.879294i	$0.341985\pi$
$270$	0	0
$271$	1.90471e6	1.57545	0.787725	−	0.616027i	$-0.211259\pi$
$271$	1.90471e6	1.57545	0.787725	+	0.616027i	$0.211259\pi$
$272$	−188940.	−0.154847
$273$	0	0
$274$	289822.	0.233214
$275$	340395.	0.271426
$276$	0	0
$277$	−1.65885e6	−1.29899	−0.649497	−	0.760365i	$-0.725020\pi$
$277$	−1.65885e6	−1.29899	−0.649497	+	0.760365i	$0.725020\pi$
$278$	−590897.	−0.458564
$279$	0	0
$280$	−277288.	−0.211367
$281$	−1.67665e6	−1.26671	−0.633354	−	0.773863i	$-0.718322\pi$
$281$	−1.67665e6	−1.26671	−0.633354	+	0.773863i	$0.718322\pi$
$282$	0	0
$283$	−1.18644e6	−0.880599	−0.440300	−	0.897851i	$-0.645128\pi$
$283$	−1.18644e6	−0.880599	−0.440300	+	0.897851i	$0.645128\pi$
$284$	398754.	0.293366
$285$	0	0
$286$	43119.8	0.0311718
$287$	−1.25465e6	−0.899119
$288$	0	0
$289$	−1.36438e6	−0.960929
$290$	−91916.5	−0.0641798
$291$	0	0
$292$	76029.5	0.0521825
$293$	1.89309e6	1.28826	0.644130	−	0.764916i	$-0.277220\pi$
$293$	1.89309e6	1.28826	0.644130	+	0.764916i	$0.277220\pi$
$294$	0	0
$295$	−237236.	−0.158718
$296$	655093.	0.434584
$297$	0	0
$298$	−19704.7	−0.0128537
$299$	−794942.	−0.514230
$300$	0	0
$301$	−822357.	−0.523172
$302$	−831132.	−0.524388
$303$	0	0
$304$	−877972.	−0.544875
$305$	418975.	0.257892
$306$	0	0
$307$	−760186.	−0.460335	−0.230168	−	0.973151i	$-0.573927\pi$
$307$	−760186.	−0.460335	−0.230168	+	0.973151i	$0.573927\pi$
$308$	593515.	0.356496
$309$	0	0
$310$	210212.	0.124238
$311$	1.83067e6	1.07327	0.536634	−	0.843815i	$-0.319696\pi$
$311$	1.83067e6	1.07327	0.536634	+	0.843815i	$0.319696\pi$
$312$	0	0
$313$	801788.	0.462593	0.231296	−	0.972883i	$-0.425703\pi$
$313$	801788.	0.462593	0.231296	+	0.972883i	$0.425703\pi$
$314$	−239753.	−0.137227
$315$	0	0
$316$	−49609.8	−0.0279480
$317$	2.43628e6	1.36169	0.680847	−	0.732426i	$-0.261612\pi$
$317$	2.43628e6	1.36169	0.680847	+	0.732426i	$0.261612\pi$
$318$	0	0
$319$	409210.	0.225149
$320$	−337230.	−0.184099
$321$	0	0
$322$	874803.	0.470187
$323$	257783.	0.137482
$324$	0	0
$325$	651337.	0.342056
$326$	−513563.	−0.267639
$327$	0	0
$328$	718959.	0.368994
$329$	801114.	0.408042
$330$	0	0
$331$	1.31532e6	0.659873	0.329937	−	0.944003i	$-0.392973\pi$
$331$	1.31532e6	0.659873	0.329937	+	0.944003i	$0.392973\pi$
$332$	1.62737e6	0.810292
$333$	0	0
$334$	−775676.	−0.380465
$335$	257782.	0.125499
$336$	0	0
$337$	−1.93220e6	−0.926783	−0.463391	−	0.886154i	$-0.653368\pi$
$337$	−1.93220e6	−0.926783	−0.463391	+	0.886154i	$0.653368\pi$
$338$	−488970.	−0.232804
$339$	0	0
$340$	123238.	0.0578159
$341$	−935859.	−0.435837
$342$	0	0
$343$	−1.02815e6	−0.471867
$344$	471241.	0.214707
$345$	0	0
$346$	−599429.	−0.269183
$347$	3.54743e6	1.58158	0.790788	−	0.612090i	$-0.209671\pi$
$347$	3.54743e6	1.58158	0.790788	+	0.612090i	$0.209671\pi$
$348$	0	0
$349$	848094.	0.372718	0.186359	−	0.982482i	$-0.440331\pi$
$349$	848094.	0.372718	0.186359	+	0.982482i	$0.440331\pi$
$350$	−716771.	−0.312760
$351$	0	0
$352$	−516695.	−0.222268
$353$	−2.27095e6	−0.969999	−0.484999	−	0.874515i	$-0.661180\pi$
$353$	−2.27095e6	−0.969999	−0.484999	+	0.874515i	$0.661180\pi$
$354$	0	0
$355$	−237634.	−0.100078
$356$	−4.01628e6	−1.67958
$357$	0	0
$358$	−1.21993e6	−0.503068
$359$	4.21351e6	1.72547	0.862736	−	0.505654i	$-0.168749\pi$
$359$	4.21351e6	1.72547	0.862736	+	0.505654i	$0.168749\pi$
$360$	0	0
$361$	−1.27823e6	−0.516227
$362$	−294238.	−0.118012
$363$	0	0
$364$	1.13567e6	0.449263
$365$	−45309.1	−0.0178014
$366$	0	0
$367$	−389458.	−0.150937	−0.0754684	−	0.997148i	$-0.524045\pi$
$367$	−389458.	−0.150937	−0.0754684	+	0.997148i	$0.524045\pi$
$368$	2.75425e6	1.06019
$369$	0	0
$370$	−187695.	−0.0712770
$371$	−267305.	−0.100826
$372$	0	0
$373$	−356789.	−0.132782	−0.0663911	−	0.997794i	$-0.521148\pi$
$373$	−356789.	−0.132782	−0.0663911	+	0.997794i	$0.521148\pi$
$374$	43864.9	0.0162158
$375$	0	0
$376$	−459068.	−0.167459
$377$	783013.	0.283737
$378$	0	0
$379$	−5.45416e6	−1.95043	−0.975214	−	0.221263i	$-0.928982\pi$
$379$	−5.45416e6	−1.95043	−0.975214	+	0.221263i	$0.928982\pi$
$380$	572665.	0.203442
$381$	0	0
$382$	489242.	0.171540
$383$	−1.54285e6	−0.537435	−0.268718	−	0.963219i	$-0.586600\pi$
$383$	−1.54285e6	−0.537435	−0.268718	+	0.963219i	$0.586600\pi$
$384$	0	0
$385$	−353700.	−0.121614
$386$	−1.06094e6	−0.362429
$387$	0	0
$388$	−5.30811e6	−1.79003
$389$	−4.03781e6	−1.35292	−0.676459	−	0.736480i	$-0.736486\pi$
$389$	−4.03781e6	−1.35292	−0.676459	+	0.736480i	$0.736486\pi$
$390$	0	0
$391$	−808680.	−0.267507
$392$	−1.00514e6	−0.330379
$393$	0	0
$394$	−1.16619e6	−0.378467
$395$	29564.5	0.00953407
$396$	0	0
$397$	−192895.	−0.0614248	−0.0307124	−	0.999528i	$-0.509778\pi$
$397$	−192895.	−0.0614248	−0.0307124	+	0.999528i	$0.509778\pi$
$398$	−676111.	−0.213949
$399$	0	0
$400$	−2.25670e6	−0.705219
$401$	2.94148e6	0.913493	0.456747	−	0.889597i	$-0.349015\pi$
$401$	2.94148e6	0.913493	0.456747	+	0.889597i	$0.349015\pi$
$402$	0	0
$403$	−1.79074e6	−0.549250
$404$	1.81640e6	0.553680
$405$	0	0
$406$	−861675.	−0.259435
$407$	835615.	0.250046
$408$	0	0
$409$	−5.46972e6	−1.61680	−0.808401	−	0.588632i	$-0.799667\pi$
$409$	−5.46972e6	−1.61680	−0.808401	+	0.588632i	$0.799667\pi$
$410$	−205994.	−0.0605195
$411$	0	0
$412$	−1.83649e6	−0.533023
$413$	−2.22398e6	−0.641588
$414$	0	0
$415$	−969817.	−0.276420
$416$	−988682.	−0.280107
$417$	0	0
$418$	203832.	0.0570601
$419$	−1.50283e6	−0.418190	−0.209095	−	0.977895i	$-0.567052\pi$
$419$	−1.50283e6	−0.418190	−0.209095	+	0.977895i	$0.567052\pi$
$420$	0	0
$421$	713377.	0.196161	0.0980807	−	0.995178i	$-0.468730\pi$
$421$	713377.	0.196161	0.0980807	+	0.995178i	$0.468730\pi$
$422$	−1.41174e6	−0.385899
$423$	0	0
$424$	153176.	0.0413786
$425$	662593.	0.177940
$426$	0	0
$427$	3.92770e6	1.04248
$428$	5.82373e6	1.53671
$429$	0	0
$430$	−135019.	−0.0352146
$431$	3.71229e6	0.962607	0.481304	−	0.876554i	$-0.340163\pi$
$431$	3.71229e6	0.962607	0.481304	+	0.876554i	$0.340163\pi$
$432$	0	0
$433$	2.58545e6	0.662700	0.331350	−	0.943508i	$-0.392496\pi$
$433$	2.58545e6	0.662700	0.331350	+	0.943508i	$0.392496\pi$
$434$	1.97064e6	0.502208
$435$	0	0
$436$	3.57363e6	0.900313
$437$	−3.75779e6	−0.941301
$438$	0	0
$439$	3.60788e6	0.893493	0.446747	−	0.894660i	$-0.352583\pi$
$439$	3.60788e6	0.893493	0.446747	+	0.894660i	$0.352583\pi$
$440$	202683.	0.0499098
$441$	0	0
$442$	83934.4	0.0204355
$443$	504362.	0.122105	0.0610525	−	0.998135i	$-0.480554\pi$
$443$	504362.	0.122105	0.0610525	+	0.998135i	$0.480554\pi$
$444$	0	0
$445$	2.39347e6	0.572965
$446$	238969.	0.0568858
$447$	0	0
$448$	−3.16138e6	−0.744185
$449$	−4.35927e6	−1.02046	−0.510232	−	0.860037i	$-0.670440\pi$
$449$	−4.35927e6	−1.02046	−0.510232	+	0.860037i	$0.670440\pi$
$450$	0	0
$451$	917081.	0.212308
$452$	−6.81697e6	−1.56944
$453$	0	0
$454$	1.86758e6	0.425245
$455$	−676795.	−0.153260
$456$	0	0
$457$	15918.1	0.00356534	0.00178267	−	0.999998i	$-0.499433\pi$
$457$	15918.1	0.00356534	0.00178267	+	0.999998i	$0.499433\pi$
$458$	185488.	0.0413192
$459$	0	0
$460$	−1.79648e6	−0.395848
$461$	3.27654e6	0.718065	0.359032	−	0.933325i	$-0.383107\pi$
$461$	3.27654e6	0.718065	0.359032	+	0.933325i	$0.383107\pi$
$462$	0	0
$463$	−963849.	−0.208957	−0.104478	−	0.994527i	$-0.533317\pi$
$463$	−963849.	−0.208957	−0.104478	+	0.994527i	$0.533317\pi$
$464$	−2.71292e6	−0.584982
$465$	0	0
$466$	−257192.	−0.0548647
$467$	−700864.	−0.148711	−0.0743553	−	0.997232i	$-0.523690\pi$
$467$	−700864.	−0.148711	−0.0743553	+	0.997232i	$0.523690\pi$
$468$	0	0
$469$	2.41659e6	0.507307
$470$	131531.	0.0274652
$471$	0	0
$472$	1.27442e6	0.263305
$473$	601099.	0.123536
$474$	0	0
$475$	3.07895e6	0.626136
$476$	1.15530e6	0.233710
$477$	0	0
$478$	904709.	0.181109
$479$	674709.	0.134362	0.0671812	−	0.997741i	$-0.478599\pi$
$479$	674709.	0.134362	0.0671812	+	0.997741i	$0.478599\pi$
$480$	0	0
$481$	1.59893e6	0.315113
$482$	179379.	0.0351685
$483$	0	0
$484$	−433827.	−0.0841790
$485$	3.16332e6	0.610645
$486$	0	0
$487$	450197.	0.0860162	0.0430081	−	0.999075i	$-0.486306\pi$
$487$	450197.	0.0860162	0.0430081	+	0.999075i	$0.486306\pi$
$488$	−2.25072e6	−0.427830
$489$	0	0
$490$	287990.	0.0541861
$491$	1.74254e6	0.326197	0.163098	−	0.986610i	$-0.447851\pi$
$491$	1.74254e6	0.326197	0.163098	+	0.986610i	$0.447851\pi$
$492$	0	0
$493$	796545.	0.147602
$494$	390028.	0.0719082
$495$	0	0
$496$	6.20442e6	1.13239
$497$	−2.22771e6	−0.404546
$498$	0	0
$499$	−1.08273e6	−0.194656	−0.0973281	−	0.995252i	$-0.531030\pi$
$499$	−1.08273e6	−0.194656	−0.0973281	+	0.995252i	$0.531030\pi$
$500$	3.10706e6	0.555808
$501$	0	0
$502$	−2.13513e6	−0.378151
$503$	−6.91388e6	−1.21843	−0.609217	−	0.793004i	$-0.708516\pi$
$503$	−6.91388e6	−1.21843	−0.609217	+	0.793004i	$0.708516\pi$
$504$	0	0
$505$	−1.08247e6	−0.188881
$506$	−639434.	−0.111025
$507$	0	0
$508$	8.21224e6	1.41190
$509$	1.36958e6	0.234311	0.117155	−	0.993114i	$-0.462622\pi$
$509$	1.36958e6	0.234311	0.117155	+	0.993114i	$0.462622\pi$
$510$	0	0
$511$	−424752.	−0.0719587
$512$	5.86056e6	0.988016
$513$	0	0
$514$	−1.33198e6	−0.222376
$515$	1.09444e6	0.181834
$516$	0	0
$517$	−585572.	−0.0963504
$518$	−1.75956e6	−0.288124
$519$	0	0
$520$	387829.	0.0628972
$521$	−1.00288e7	−1.61866	−0.809332	−	0.587352i	$-0.800170\pi$
$521$	−1.00288e7	−1.61866	−0.809332	+	0.587352i	$0.800170\pi$
$522$	0	0
$523$	−1.16559e7	−1.86334	−0.931672	−	0.363300i	$-0.881650\pi$
$523$	−1.16559e7	−1.86334	−0.931672	+	0.363300i	$0.881650\pi$
$524$	−2.75459e6	−0.438257
$525$	0	0
$526$	868694.	0.136900
$527$	−1.82169e6	−0.285724
$528$	0	0
$529$	5.35206e6	0.831538
$530$	−43887.5	−0.00678658
$531$	0	0
$532$	5.36847e6	0.822378
$533$	1.75481e6	0.267554
$534$	0	0
$535$	−3.47060e6	−0.524228
$536$	−1.38479e6	−0.208196
$537$	0	0
$538$	1.74003e6	0.259179
$539$	−1.28213e6	−0.190090
$540$	0	0
$541$	1.20961e7	1.77686	0.888429	−	0.459015i	$-0.151797\pi$
$541$	1.20961e7	1.77686	0.888429	+	0.459015i	$0.151797\pi$
$542$	2.93164e6	0.428660
$543$	0	0
$544$	−1.00577e6	−0.145714
$545$	−2.12968e6	−0.307130
$546$	0	0
$547$	−7.88308e6	−1.12649	−0.563245	−	0.826290i	$-0.690447\pi$
$547$	−7.88308e6	−1.12649	−0.563245	+	0.826290i	$0.690447\pi$
$548$	−5.57949e6	−0.793676
$549$	0	0
$550$	523922.	0.0738515
$551$	3.70140e6	0.519382
$552$	0	0
$553$	277154.	0.0385397
$554$	−2.55323e6	−0.353439
$555$	0	0
$556$	1.13756e7	1.56059
$557$	1.01340e7	1.38402	0.692008	−	0.721890i	$-0.256726\pi$
$557$	1.01340e7	1.38402	0.692008	+	0.721890i	$0.256726\pi$
$558$	0	0
$559$	1.15019e6	0.155682
$560$	2.34491e6	0.315977
$561$	0	0
$562$	−2.58063e6	−0.344655
$563$	1.14217e7	1.51865	0.759327	−	0.650710i	$-0.225529\pi$
$563$	1.14217e7	1.51865	0.759327	+	0.650710i	$0.225529\pi$
$564$	0	0
$565$	4.06251e6	0.535394
$566$	−1.82611e6	−0.239600
$567$	0	0
$568$	1.27656e6	0.166024
$569$	1.25992e7	1.63141	0.815705	−	0.578469i	$-0.196350\pi$
$569$	1.25992e7	1.63141	0.815705	+	0.578469i	$0.196350\pi$
$570$	0	0
$571$	−1.26486e7	−1.62350	−0.811751	−	0.584004i	$-0.801485\pi$
$571$	−1.26486e7	−1.62350	−0.811751	+	0.584004i	$0.801485\pi$
$572$	−830118.	−0.106084
$573$	0	0
$574$	−1.93110e6	−0.244639
$575$	−9.65885e6	−1.21830
$576$	0	0
$577$	1.33870e7	1.67396	0.836979	−	0.547236i	$-0.184320\pi$
$577$	1.33870e7	1.67396	0.836979	+	0.547236i	$0.184320\pi$
$578$	−2.10000e6	−0.261456
$579$	0	0
$580$	1.76953e6	0.218417
$581$	−9.09159e6	−1.11738
$582$	0	0
$583$	195386.	0.0238079
$584$	243399.	0.0295315
$585$	0	0
$586$	2.91377e6	0.350519
$587$	−3.16215e6	−0.378780	−0.189390	−	0.981902i	$-0.560651\pi$
$587$	−3.16215e6	−0.378780	−0.189390	+	0.981902i	$0.560651\pi$
$588$	0	0
$589$	−8.46505e6	−1.00541
$590$	−365144.	−0.0431851
$591$	0	0
$592$	−5.53984e6	−0.649670
$593$	4.93571e6	0.576385	0.288193	−	0.957572i	$-0.406946\pi$
$593$	4.93571e6	0.576385	0.288193	+	0.957572i	$0.406946\pi$
$594$	0	0
$595$	−688491.	−0.0797271
$596$	379343.	0.0437438
$597$	0	0
$598$	−1.22354e6	−0.139915
$599$	8.12454e6	0.925191	0.462596	−	0.886569i	$-0.346918\pi$
$599$	8.12454e6	0.925191	0.462596	+	0.886569i	$0.346918\pi$
$600$	0	0
$601$	9.66137e6	1.09107	0.545535	−	0.838088i	$-0.316327\pi$
$601$	9.66137e6	1.09107	0.545535	+	0.838088i	$0.316327\pi$
$602$	−1.26574e6	−0.142348
$603$	0	0
$604$	1.60005e7	1.78460
$605$	258536.	0.0287165
$606$	0	0
$607$	−1.23381e7	−1.35918	−0.679590	−	0.733592i	$-0.737842\pi$
$607$	−1.23381e7	−1.35918	−0.679590	+	0.733592i	$0.737842\pi$
$608$	−4.67362e6	−0.512737
$609$	0	0
$610$	644869.	0.0701692
$611$	−1.12048e6	−0.121423
$612$	0	0
$613$	1.55853e6	0.167519	0.0837596	−	0.996486i	$-0.473307\pi$
$613$	1.55853e6	0.167519	0.0837596	+	0.996486i	$0.473307\pi$
$614$	−1.17005e6	−0.125251
$615$	0	0
$616$	1.90006e6	0.201751
$617$	−8.90166e6	−0.941365	−0.470682	−	0.882303i	$-0.655992\pi$
$617$	−8.90166e6	−0.941365	−0.470682	+	0.882303i	$0.655992\pi$
$618$	0	0
$619$	−9.98808e6	−1.04774	−0.523872	−	0.851797i	$-0.675513\pi$
$619$	−9.98808e6	−1.04774	−0.523872	+	0.851797i	$0.675513\pi$
$620$	−4.04689e6	−0.422806
$621$	0	0
$622$	2.81768e6	0.292022
$623$	2.24377e7	2.31610
$624$	0	0
$625$	6.93957e6	0.710612
$626$	1.23408e6	0.125866
$627$	0	0
$628$	4.61560e6	0.467013
$629$	1.62656e6	0.163924
$630$	0	0
$631$	6.12175e6	0.612072	0.306036	−	0.952020i	$-0.400997\pi$
$631$	6.12175e6	0.612072	0.306036	+	0.952020i	$0.400997\pi$
$632$	−158819.	−0.0158165
$633$	0	0
$634$	3.74982e6	0.370499
$635$	−4.89401e6	−0.481649
$636$	0	0
$637$	−2.45332e6	−0.239555
$638$	629839.	0.0612601
$639$	0	0
$640$	−2.93200e6	−0.282953
$641$	−6.90830e6	−0.664089	−0.332044	−	0.943264i	$-0.607738\pi$
$641$	−6.90830e6	−0.664089	−0.332044	+	0.943264i	$0.607738\pi$
$642$	0	0
$643$	1.09176e7	1.04136	0.520680	−	0.853752i	$-0.325679\pi$
$643$	1.09176e7	1.04136	0.520680	+	0.853752i	$0.325679\pi$
$644$	−1.68412e7	−1.60014
$645$	0	0
$646$	396768.	0.0374072
$647$	8.34795e6	0.784005	0.392003	−	0.919964i	$-0.371782\pi$
$647$	8.34795e6	0.784005	0.392003	+	0.919964i	$0.371782\pi$
$648$	0	0
$649$	1.62561e6	0.151497
$650$	1.00251e6	0.0930691
$651$	0	0
$652$	9.88683e6	0.910832
$653$	4.43118e6	0.406665	0.203332	−	0.979110i	$-0.434823\pi$
$653$	4.43118e6	0.406665	0.203332	+	0.979110i	$0.434823\pi$
$654$	0	0
$655$	1.64157e6	0.149505
$656$	−6.07993e6	−0.551618
$657$	0	0
$658$	1.23304e6	0.111023
$659$	−8.07036e6	−0.723902	−0.361951	−	0.932197i	$-0.617889\pi$
$659$	−8.07036e6	−0.723902	−0.361951	+	0.932197i	$0.617889\pi$
$660$	0	0
$661$	5.77054e6	0.513704	0.256852	−	0.966451i	$-0.417315\pi$
$661$	5.77054e6	0.513704	0.256852	+	0.966451i	$0.417315\pi$
$662$	2.02448e6	0.179543
$663$	0	0
$664$	5.20982e6	0.458566
$665$	−3.19929e6	−0.280543
$666$	0	0
$667$	−1.16115e7	−1.01059
$668$	1.49329e7	1.29480
$669$	0	0
$670$	396767.	0.0341467
$671$	−2.87094e6	−0.246160
$672$	0	0
$673$	7.95367e6	0.676908	0.338454	−	0.940983i	$-0.390096\pi$
$673$	7.95367e6	0.676908	0.338454	+	0.940983i	$0.390096\pi$
$674$	−2.97396e6	−0.252166
$675$	0	0
$676$	9.41337e6	0.792280
$677$	−2.24046e7	−1.87873	−0.939366	−	0.342916i	$-0.888585\pi$
$677$	−2.24046e7	−1.87873	−0.939366	+	0.342916i	$0.888585\pi$
$678$	0	0
$679$	2.96547e7	2.46842
$680$	394531.	0.0327196
$681$	0	0
$682$	−1.44043e6	−0.118586
$683$	−4.50123e6	−0.369215	−0.184607	−	0.982812i	$-0.559101\pi$
$683$	−4.50123e6	−0.369215	−0.184607	+	0.982812i	$0.559101\pi$
$684$	0	0
$685$	3.32505e6	0.270752
$686$	−1.58248e6	−0.128389
$687$	0	0
$688$	−3.98508e6	−0.320971
$689$	373866.	0.0300032
$690$	0	0
$691$	−5.34074e6	−0.425507	−0.212753	−	0.977106i	$-0.568243\pi$
$691$	−5.34074e6	−0.425507	−0.212753	+	0.977106i	$0.568243\pi$
$692$	1.15399e7	0.916086
$693$	0	0
$694$	5.46005e6	0.430326
$695$	−6.77920e6	−0.532373
$696$	0	0
$697$	1.78514e6	0.139184
$698$	1.30535e6	0.101412
$699$	0	0
$700$	1.37989e7	1.06439
$701$	−7.02294e6	−0.539789	−0.269894	−	0.962890i	$-0.586989\pi$
$701$	−7.02294e6	−0.539789	−0.269894	+	0.962890i	$0.586989\pi$
$702$	0	0
$703$	7.55832e6	0.576816
$704$	2.31080e6	0.175724
$705$	0	0
$706$	−3.49535e6	−0.263924
$707$	−1.01477e7	−0.763515
$708$	0	0
$709$	−1.00486e7	−0.750742	−0.375371	−	0.926875i	$-0.622485\pi$
$709$	−1.00486e7	−0.750742	−0.375371	+	0.926875i	$0.622485\pi$
$710$	−365756.	−0.0272299
$711$	0	0
$712$	−1.28576e7	−0.950518
$713$	2.65554e7	1.95627
$714$	0	0
$715$	494701.	0.0361891
$716$	2.34854e7	1.71204
$717$	0	0
$718$	6.48525e6	0.469479
$719$	−6.61563e6	−0.477254	−0.238627	−	0.971111i	$-0.576697\pi$
$719$	−6.61563e6	−0.477254	−0.238627	+	0.971111i	$0.576697\pi$
$720$	0	0
$721$	1.02599e7	0.735029
$722$	−1.96740e6	−0.140459
$723$	0	0
$724$	5.66450e6	0.401620
$725$	9.51391e6	0.672224
$726$	0	0
$727$	1.15502e7	0.810501	0.405251	−	0.914206i	$-0.367184\pi$
$727$	1.15502e7	0.810501	0.405251	+	0.914206i	$0.367184\pi$
$728$	3.63572e6	0.254250
$729$	0	0
$730$	−69737.8	−0.00484352
$731$	1.17006e6	0.0809872
$732$	0	0
$733$	1.63607e7	1.12472	0.562358	−	0.826894i	$-0.309894\pi$
$733$	1.63607e7	1.12472	0.562358	+	0.826894i	$0.309894\pi$
$734$	−599436.	−0.0410679
$735$	0	0
$736$	1.46614e7	0.997658
$737$	−1.76640e6	−0.119790
$738$	0	0
$739$	6.15216e6	0.414397	0.207198	−	0.978299i	$-0.433565\pi$
$739$	6.15216e6	0.414397	0.207198	+	0.978299i	$0.433565\pi$
$740$	3.61341e6	0.242570
$741$	0	0
$742$	−411425.	−0.0274335
$743$	−9.49122e6	−0.630739	−0.315370	−	0.948969i	$-0.602129\pi$
$743$	−9.49122e6	−0.630739	−0.315370	+	0.948969i	$0.602129\pi$
$744$	0	0
$745$	−226066.	−0.0149226
$746$	−549155.	−0.0361283
$747$	0	0
$748$	−844463.	−0.0551857
$749$	−3.25353e7	−2.11909
$750$	0	0
$751$	−6.56912e6	−0.425018	−0.212509	−	0.977159i	$-0.568163\pi$
$751$	−6.56912e6	−0.425018	−0.212509	+	0.977159i	$0.568163\pi$
$752$	3.88214e6	0.250338
$753$	0	0
$754$	1.20518e6	0.0772011
$755$	−9.53536e6	−0.608793
$756$	0	0
$757$	−1.94010e7	−1.23051	−0.615254	−	0.788329i	$-0.710946\pi$
$757$	−1.94010e7	−1.23051	−0.615254	+	0.788329i	$0.710946\pi$
$758$	−8.39481e6	−0.530686
$759$	0	0
$760$	1.83331e6	0.115134
$761$	2.22482e7	1.39262	0.696310	−	0.717741i	$-0.254824\pi$
$761$	2.22482e7	1.39262	0.696310	+	0.717741i	$0.254824\pi$
$762$	0	0
$763$	−1.99647e7	−1.24152
$764$	−9.41861e6	−0.583786
$765$	0	0
$766$	−2.37469e6	−0.146229
$767$	3.11057e6	0.190920
$768$	0	0
$769$	−1.50252e7	−0.916227	−0.458114	−	0.888894i	$-0.651475\pi$
$769$	−1.50252e7	−0.916227	−0.458114	+	0.888894i	$0.651475\pi$
$770$	−544399.	−0.0330895
$771$	0	0
$772$	2.04247e7	1.23342
$773$	2.74385e7	1.65163	0.825813	−	0.563944i	$-0.190717\pi$
$773$	2.74385e7	1.65163	0.825813	+	0.563944i	$0.190717\pi$
$774$	0	0
$775$	−2.17582e7	−1.30127
$776$	−1.69932e7	−1.01303
$777$	0	0
$778$	−6.21482e6	−0.368111
$779$	8.29520e6	0.489760
$780$	0	0
$781$	1.62834e6	0.0955250
$782$	−1.24468e6	−0.0727851
$783$	0	0
$784$	8.50005e6	0.493891
$785$	−2.75062e6	−0.159315
$786$	0	0
$787$	9.48553e6	0.545915	0.272957	−	0.962026i	$-0.411998\pi$
$787$	9.48553e6	0.545915	0.272957	+	0.962026i	$0.411998\pi$
$788$	2.24508e7	1.28800
$789$	0	0
$790$	45504.5	0.00259410
$791$	3.80842e7	2.16423
$792$	0	0
$793$	−5.49347e6	−0.310216
$794$	−296895.	−0.0167129
$795$	0	0
$796$	1.30161e7	0.728113
$797$	−6.79546e6	−0.378942	−0.189471	−	0.981886i	$-0.560677\pi$
$797$	−6.79546e6	−0.378942	−0.189471	+	0.981886i	$0.560677\pi$
$798$	0	0
$799$	−1.13984e6	−0.0631650
$800$	−1.20129e7	−0.663624
$801$	0	0
$802$	4.52740e6	0.248550
$803$	310471.	0.0169915
$804$	0	0
$805$	1.00364e7	0.545867
$806$	−2.75623e6	−0.149444
$807$	0	0
$808$	5.81498e6	0.313343
$809$	9.13307e6	0.490620	0.245310	−	0.969445i	$-0.421110\pi$
$809$	9.13307e6	0.490620	0.245310	+	0.969445i	$0.421110\pi$
$810$	0	0
$811$	−3.69208e6	−0.197115	−0.0985574	−	0.995131i	$-0.531423\pi$
$811$	−3.69208e6	−0.197115	−0.0985574	+	0.995131i	$0.531423\pi$
$812$	1.65885e7	0.882911
$813$	0	0
$814$	1.28614e6	0.0680343
$815$	−5.89197e6	−0.310718
$816$	0	0
$817$	5.43708e6	0.284977
$818$	−8.41876e6	−0.439911
$819$	0	0
$820$	3.96568e6	0.205960
$821$	6.75134e6	0.349568	0.174784	−	0.984607i	$-0.444077\pi$
$821$	6.75134e6	0.349568	0.174784	+	0.984607i	$0.444077\pi$
$822$	0	0
$823$	2.16819e7	1.11583	0.557915	−	0.829898i	$-0.311602\pi$
$823$	2.16819e7	1.11583	0.557915	+	0.829898i	$0.311602\pi$
$824$	−5.87929e6	−0.301652
$825$	0	0
$826$	−3.42306e6	−0.174568
$827$	1.83885e7	0.934937	0.467468	−	0.884010i	$-0.345166\pi$
$827$	1.83885e7	0.934937	0.467468	+	0.884010i	$0.345166\pi$
$828$	0	0
$829$	−2.95239e7	−1.49207	−0.746033	−	0.665910i	$-0.768044\pi$
$829$	−2.95239e7	−1.49207	−0.746033	+	0.665910i	$0.768044\pi$
$830$	−1.49270e6	−0.0752104
$831$	0	0
$832$	4.42165e6	0.221450
$833$	−2.49571e6	−0.124618
$834$	0	0
$835$	−8.89913e6	−0.441704
$836$	−3.92407e6	−0.194187
$837$	0	0
$838$	−2.31309e6	−0.113784
$839$	−352235.	−0.0172754	−0.00863769	−	0.999963i	$-0.502749\pi$
$839$	−352235.	−0.0172754	−0.00863769	+	0.999963i	$0.502749\pi$
$840$	0	0
$841$	−9.07389e6	−0.442388
$842$	1.09800e6	0.0533730
$843$	0	0
$844$	2.71780e7	1.31329
$845$	−5.60982e6	−0.270276
$846$	0	0
$847$	2.42365e6	0.116081
$848$	−1.29534e6	−0.0618578
$849$	0	0
$850$	1.01984e6	0.0484153
$851$	−2.37109e7	−1.12234
$852$	0	0
$853$	−5.17337e6	−0.243445	−0.121722	−	0.992564i	$-0.538842\pi$
$853$	−5.17337e6	−0.243445	−0.121722	+	0.992564i	$0.538842\pi$
$854$	6.04535e6	0.283646
$855$	0	0
$856$	1.86439e7	0.869666
$857$	−3.81089e7	−1.77245	−0.886226	−	0.463253i	$-0.846682\pi$
$857$	−3.81089e7	−1.77245	−0.886226	+	0.463253i	$0.846682\pi$
$858$	0	0
$859$	−2.17688e7	−1.00659	−0.503293	−	0.864116i	$-0.667878\pi$
$859$	−2.17688e7	−1.00659	−0.503293	+	0.864116i	$0.667878\pi$
$860$	2.59930e6	0.119842
$861$	0	0
$862$	5.71380e6	0.261913
$863$	4.80980e6	0.219837	0.109918	−	0.993941i	$-0.464941\pi$
$863$	4.80980e6	0.219837	0.109918	+	0.993941i	$0.464941\pi$
$864$	0	0
$865$	−6.87709e6	−0.312510
$866$	3.97942e6	0.180312
$867$	0	0
$868$	−3.79377e7	−1.70912
$869$	−202585.	−0.00910034
$870$	0	0
$871$	−3.37996e6	−0.150961
$872$	1.14405e7	0.509512
$873$	0	0
$874$	−5.78382e6	−0.256116
$875$	−1.73581e7	−0.766448
$876$	0	0
$877$	3.23600e7	1.42072	0.710361	−	0.703837i	$-0.248532\pi$
$877$	3.23600e7	1.42072	0.710361	+	0.703837i	$0.248532\pi$
$878$	5.55310e6	0.243108
$879$	0	0
$880$	−1.71400e6	−0.0746113
$881$	4.30302e7	1.86781	0.933907	−	0.357516i	$-0.116376\pi$
$881$	4.30302e7	1.86781	0.933907	+	0.357516i	$0.116376\pi$
$882$	0	0
$883$	1.57650e7	0.680444	0.340222	−	0.940345i	$-0.389498\pi$
$883$	1.57650e7	0.680444	0.340222	+	0.940345i	$0.389498\pi$
$884$	−1.61586e6	−0.0695461
$885$	0	0
$886$	776293.	0.0332232
$887$	−1.83880e7	−0.784738	−0.392369	−	0.919808i	$-0.628344\pi$
$887$	−1.83880e7	−0.784738	−0.392369	+	0.919808i	$0.628344\pi$
$888$	0	0
$889$	−4.58791e7	−1.94698
$890$	3.68393e6	0.155896
$891$	0	0
$892$	−4.60049e6	−0.193594
$893$	−5.29663e6	−0.222265
$894$	0	0
$895$	−1.39959e7	−0.584041
$896$	−2.74862e7	−1.14378
$897$	0	0
$898$	−6.70960e6	−0.277655
$899$	−2.61569e7	−1.07941
$900$	0	0
$901$	380327.	0.0156079
$902$	1.41153e6	0.0577663
$903$	0	0
$904$	−2.18236e7	−0.888190
$905$	−3.37571e6	−0.137007
$906$	0	0
$907$	−1.25279e7	−0.505662	−0.252831	−	0.967510i	$-0.581362\pi$
$907$	−1.25279e7	−0.505662	−0.252831	+	0.967510i	$0.581362\pi$
$908$	−3.59536e7	−1.44720
$909$	0	0
$910$	−1.04169e6	−0.0417001
$911$	4.89593e7	1.95452	0.977258	−	0.212054i	$-0.0680152\pi$
$911$	4.89593e7	1.95452	0.977258	+	0.212054i	$0.0680152\pi$
$912$	0	0
$913$	6.64547e6	0.263845
$914$	24500.5	0.000970084 0
$915$	0	0
$916$	−3.57091e6	−0.140618
$917$	1.53890e7	0.604348
$918$	0	0
$919$	1.27446e7	0.497779	0.248889	−	0.968532i	$-0.419934\pi$
$919$	1.27446e7	0.497779	0.248889	+	0.968532i	$0.419934\pi$
$920$	−5.75121e6	−0.224021
$921$	0	0
$922$	5.04311e6	0.195376
$923$	3.11578e6	0.120382
$924$	0	0
$925$	1.94276e7	0.746560
$926$	−1.48351e6	−0.0568544
$927$	0	0
$928$	−1.44414e7	−0.550478
$929$	1.45762e7	0.554121	0.277060	−	0.960853i	$-0.410640\pi$
$929$	1.45762e7	0.554121	0.277060	+	0.960853i	$0.410640\pi$
$930$	0	0
$931$	−1.15971e7	−0.438506
$932$	4.95132e6	0.186716
$933$	0	0
$934$	−1.07874e6	−0.0404622
$935$	503250.	0.0188259
$936$	0	0
$937$	−3.30324e7	−1.22911	−0.614555	−	0.788874i	$-0.710665\pi$
$937$	−3.30324e7	−1.22911	−0.614555	+	0.788874i	$0.710665\pi$
$938$	3.71951e6	0.138032
$939$	0	0
$940$	−2.53216e6	−0.0934697
$941$	3.10182e7	1.14194	0.570969	−	0.820972i	$-0.306568\pi$
$941$	3.10182e7	1.14194	0.570969	+	0.820972i	$0.306568\pi$
$942$	0	0
$943$	−2.60225e7	−0.952951
$944$	−1.07772e7	−0.393620
$945$	0	0
$946$	925186.	0.0336125
$947$	3.85546e6	0.139701	0.0698507	−	0.997557i	$-0.477748\pi$
$947$	3.85546e6	0.139701	0.0698507	+	0.997557i	$0.477748\pi$
$948$	0	0
$949$	594079.	0.0214130
$950$	4.73899e6	0.170364
$951$	0	0
$952$	3.69854e6	0.132263
$953$	−4.66159e7	−1.66266	−0.831328	−	0.555783i	$-0.812419\pi$
$953$	−4.66159e7	−1.66266	−0.831328	+	0.555783i	$0.812419\pi$
$954$	0	0
$955$	5.61294e6	0.199151
$956$	−1.74169e7	−0.616350
$957$	0	0
$958$	1.03848e6	0.0365583
$959$	3.11708e7	1.09446
$960$	0	0
$961$	3.11913e7	1.08950
$962$	2.46100e6	0.0857382
$963$	0	0
$964$	−3.45330e6	−0.119686
$965$	−1.21719e7	−0.420765
$966$	0	0
$967$	−3.24552e7	−1.11614	−0.558068	−	0.829795i	$-0.688457\pi$
$967$	−3.24552e7	−1.11614	−0.558068	+	0.829795i	$0.688457\pi$
$968$	−1.38884e6	−0.0476392
$969$	0	0
$970$	4.86884e6	0.166149
$971$	2.94347e7	1.00187	0.500935	−	0.865485i	$-0.332989\pi$
$971$	2.94347e7	1.00187	0.500935	+	0.865485i	$0.332989\pi$
$972$	0	0
$973$	−6.35519e7	−2.15202
$974$	692924.	0.0234039
$975$	0	0
$976$	1.90333e7	0.639573
$977$	2.20228e7	0.738136	0.369068	−	0.929402i	$-0.379677\pi$
$977$	2.20228e7	0.738136	0.369068	+	0.929402i	$0.379677\pi$
$978$	0	0
$979$	−1.64007e7	−0.546899
$980$	−5.54423e6	−0.184406
$981$	0	0
$982$	2.68205e6	0.0887539
$983$	−2.56784e7	−0.847588	−0.423794	−	0.905759i	$-0.639302\pi$
$983$	−2.56784e7	−0.847588	−0.423794	+	0.905759i	$0.639302\pi$
$984$	0	0
$985$	−1.33794e7	−0.439385
$986$	1.22601e6	0.0401606
$987$	0	0
$988$	−7.50860e6	−0.244718
$989$	−1.70564e7	−0.554495
$990$	0	0
$991$	4.18895e7	1.35494	0.677472	−	0.735549i	$-0.263075\pi$
$991$	4.18895e7	1.35494	0.677472	+	0.735549i	$0.263075\pi$
$992$	3.30274e7	1.06560
$993$	0	0
$994$	−3.42880e6	−0.110072
$995$	−7.75684e6	−0.248386
$996$	0	0
$997$	−2.50923e7	−0.799472	−0.399736	−	0.916630i	$-0.630898\pi$
$997$	−2.50923e7	−0.799472	−0.399736	+	0.916630i	$0.630898\pi$
$998$	−1.66649e6	−0.0529634
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.f.1.13		23
3.2	odd	2		891.6.a.e.1.11		23
9.2	odd	6		297.6.e.a.199.13		46
9.4	even	3		99.6.e.a.34.11	✓	46
9.5	odd	6		297.6.e.a.100.13		46
9.7	even	3		99.6.e.a.67.11	yes	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.11	✓	46	9.4	even	3
99.6.e.a.67.11	yes	46	9.7	even	3
297.6.e.a.100.13		46	9.5	odd	6
297.6.e.a.199.13		46	9.2	odd	6
891.6.a.e.1.11		23	3.2	odd	2
891.6.a.f.1.13		23	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53916 q^{2} -29.6310 q^{4} +17.6583 q^{5} +165.539 q^{7} -94.8598 q^{8} +O(q^{10})\)	\(q+1.53916 q^{2} -29.6310 q^{4} +17.6583 q^{5} +165.539 q^{7} -94.8598 q^{8} +27.1789 q^{10} -121.000 q^{11} -231.530 q^{13} +254.790 q^{14} +802.188 q^{16} -235.531 q^{17} -1094.47 q^{19} -523.234 q^{20} -186.238 q^{22} +3433.42 q^{23} -2813.18 q^{25} -356.362 q^{26} -4905.08 q^{28} -3381.90 q^{29} +7734.37 q^{31} +4270.21 q^{32} -362.520 q^{34} +2923.14 q^{35} -6905.91 q^{37} -1684.56 q^{38} -1675.07 q^{40} -7579.18 q^{41} -4967.76 q^{43} +3585.35 q^{44} +5284.58 q^{46} +4839.44 q^{47} +10596.1 q^{49} -4329.93 q^{50} +6860.48 q^{52} -1614.76 q^{53} -2136.66 q^{55} -15703.0 q^{56} -5205.28 q^{58} -13434.8 q^{59} +23726.8 q^{61} +11904.4 q^{62} -19097.5 q^{64} -4088.44 q^{65} +14598.3 q^{67} +6979.03 q^{68} +4499.17 q^{70} -13457.3 q^{71} -2565.88 q^{73} -10629.3 q^{74} +32430.3 q^{76} -20030.2 q^{77} +1674.25 q^{79} +14165.3 q^{80} -11665.5 q^{82} -54921.2 q^{83} -4159.09 q^{85} -7646.17 q^{86} +11478.0 q^{88} +135543. q^{89} -38327.3 q^{91} -101736. q^{92} +7448.65 q^{94} -19326.5 q^{95} +179140. q^{97} +16309.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8}+O(q^{10}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8	\( 23 q + 320 q^{4} + 36 q^{5} - 167 q^{7} + 213 q^{8} - 600 q^{10} - 2783 q^{11} - 1871 q^{13} + 1329 q^{14} + 3584 q^{16} + 267 q^{17} - 3641 q^{19} + 1917 q^{20} + 8292 q^{23} + 10049 q^{25} + 9570 q^{26} + 3793 q^{28} + 5970 q^{29} - 9542 q^{31} + 3831 q^{32} - 2982 q^{34} + 3240 q^{35} - 16007 q^{37} - 1221 q^{38} - 40635 q^{40} - 12030 q^{41} - 25943 q^{43} - 38720 q^{44} - 77004 q^{46} - 9756 q^{47} + 6990 q^{49} - 101805 q^{50} - 144446 q^{52} + 53919 q^{53} - 4356 q^{55} + 16602 q^{56} - 95367 q^{58} + 20310 q^{59} - 100247 q^{61} - 15297 q^{62} - 84577 q^{64} - 20931 q^{65} - 84956 q^{67} - 168471 q^{68} - 212292 q^{70} + 36093 q^{71} - 173444 q^{73} - 86619 q^{74} - 340334 q^{76} + 20207 q^{77} - 123113 q^{79} - 15123 q^{80} - 199983 q^{82} - 30672 q^{83} - 268335 q^{85} + 211260 q^{86} - 25773 q^{88} + 32514 q^{89} - 328021 q^{91} + 196731 q^{92} - 230262 q^{94} + 325926 q^{95} - 357002 q^{97} - 214464 q^{98}+O(q^{100}) \) 23 * q + 320 * q^4 + 36 * q^5 - 167 * q^7 + 213 * q^8 - 600 * q^10 - 2783 * q^11 - 1871 * q^13 + 1329 * q^14 + 3584 * q^16 + 267 * q^17 - 3641 * q^19 + 1917 * q^20 + 8292 * q^23 + 10049 * q^25 + 9570 * q^26 + 3793 * q^28 + 5970 * q^29 - 9542 * q^31 + 3831 * q^32 - 2982 * q^34 + 3240 * q^35 - 16007 * q^37 - 1221 * q^38 - 40635 * q^40 - 12030 * q^41 - 25943 * q^43 - 38720 * q^44 - 77004 * q^46 - 9756 * q^47 + 6990 * q^49 - 101805 * q^50 - 144446 * q^52 + 53919 * q^53 - 4356 * q^55 + 16602 * q^56 - 95367 * q^58 + 20310 * q^59 - 100247 * q^61 - 15297 * q^62 - 84577 * q^64 - 20931 * q^65 - 84956 * q^67 - 168471 * q^68 - 212292 * q^70 + 36093 * q^71 - 173444 * q^73 - 86619 * q^74 - 340334 * q^76 + 20207 * q^77 - 123113 * q^79 - 15123 * q^80 - 199983 * q^82 - 30672 * q^83 - 268335 * q^85 + 211260 * q^86 - 25773 * q^88 + 32514 * q^89 - 328021 * q^91 + 196731 * q^92 - 230262 * q^94 + 325926 * q^95 - 357002 * q^97 - 214464 * q^98

Introduction

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