LMFDB - Embedded newform 891.6.a.e.1.9

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			1.9
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)

$f(q)$	$=$	$q-3.39140 q^{2} -20.4984 q^{4} +25.0381 q^{5} -249.837 q^{7} +178.043 q^{8} +O(q^{10})$	\(q-3.39140 q^{2} -20.4984 q^{4} +25.0381 q^{5} -249.837 q^{7} +178.043 q^{8} -84.9142 q^{10} +121.000 q^{11} +985.621 q^{13} +847.297 q^{14} +52.1340 q^{16} -1349.42 q^{17} +63.6150 q^{19} -513.241 q^{20} -410.359 q^{22} -3320.76 q^{23} -2498.09 q^{25} -3342.63 q^{26} +5121.26 q^{28} -1168.51 q^{29} +2586.19 q^{31} -5874.19 q^{32} +4576.41 q^{34} -6255.44 q^{35} -1418.91 q^{37} -215.744 q^{38} +4457.86 q^{40} +16642.1 q^{41} +16764.4 q^{43} -2480.31 q^{44} +11262.0 q^{46} +8033.35 q^{47} +45611.5 q^{49} +8472.03 q^{50} -20203.7 q^{52} +2442.69 q^{53} +3029.61 q^{55} -44481.7 q^{56} +3962.88 q^{58} +16759.8 q^{59} -20937.8 q^{61} -8770.82 q^{62} +18253.4 q^{64} +24678.1 q^{65} +31658.9 q^{67} +27660.9 q^{68} +21214.7 q^{70} +21807.4 q^{71} -36722.9 q^{73} +4812.10 q^{74} -1304.01 q^{76} -30230.3 q^{77} -44563.9 q^{79} +1305.34 q^{80} -56439.9 q^{82} +86704.0 q^{83} -33786.8 q^{85} -56854.9 q^{86} +21543.2 q^{88} +18797.3 q^{89} -246245. q^{91} +68070.4 q^{92} -27244.3 q^{94} +1592.80 q^{95} -108008. q^{97} -154687. q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−3.39140	−0.599520	−0.299760	−	0.954015i	$-0.596907\pi$
$2$	−3.39140	−0.599520	−0.299760	+	0.954015i	$0.596907\pi$
$3$	0	0
$3$	0	0
$4$	−20.4984	−0.640575
$5$	25.0381	0.447895	0.223948	−	0.974601i	$-0.428106\pi$
$5$	25.0381	0.447895	0.223948	+	0.974601i	$0.428106\pi$
$6$	0	0
$7$	−249.837	−1.92713	−0.963566	−	0.267469i	$-0.913813\pi$
$7$	−249.837	−1.92713	−0.963566	+	0.267469i	$0.913813\pi$
$8$	178.043	0.983558
$9$	0	0
$10$	−84.9142	−0.268522
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	0	0
$13$	985.621	1.61753	0.808764	−	0.588134i	$-0.200137\pi$
$13$	985.621	1.61753	0.808764	+	0.588134i	$0.200137\pi$
$14$	847.297	1.15536
$15$	0	0
$16$	52.1340	0.0509121
$17$	−1349.42	−1.13246	−0.566231	−	0.824247i	$-0.691599\pi$
$17$	−1349.42	−1.13246	−0.566231	+	0.824247i	$0.691599\pi$
$18$	0	0
$19$	63.6150	0.0404274	0.0202137	−	0.999796i	$-0.493565\pi$
$19$	63.6150	0.0404274	0.0202137	+	0.999796i	$0.493565\pi$
$20$	−513.241	−0.286911
$21$	0	0
$22$	−410.359	−0.180762
$23$	−3320.76	−1.30894	−0.654468	−	0.756090i	$-0.727107\pi$
$23$	−3320.76	−1.30894	−0.654468	+	0.756090i	$0.727107\pi$
$24$	0	0
$25$	−2498.09	−0.799390
$26$	−3342.63	−0.969741
$27$	0	0
$28$	5121.26	1.23447
$29$	−1168.51	−0.258010	−0.129005	−	0.991644i	$-0.541178\pi$
$29$	−1168.51	−0.258010	−0.129005	+	0.991644i	$0.541178\pi$
$30$	0	0
$31$	2586.19	0.483345	0.241672	−	0.970358i	$-0.422304\pi$
$31$	2586.19	0.483345	0.241672	+	0.970358i	$0.422304\pi$
$32$	−5874.19	−1.01408
$33$	0	0
$34$	4576.41	0.678934
$35$	−6255.44	−0.863154
$36$	0	0
$37$	−1418.91	−0.170393	−0.0851964	−	0.996364i	$-0.527152\pi$
$37$	−1418.91	−0.170393	−0.0851964	+	0.996364i	$0.527152\pi$
$38$	−215.744	−0.0242370
$39$	0	0
$40$	4457.86	0.440531
$41$	16642.1	1.54614	0.773068	−	0.634323i	$-0.218721\pi$
$41$	16642.1	1.54614	0.773068	+	0.634323i	$0.218721\pi$
$42$	0	0
$43$	16764.4	1.38267	0.691334	−	0.722535i	$-0.257023\pi$
$43$	16764.4	1.38267	0.691334	+	0.722535i	$0.257023\pi$
$44$	−2480.31	−0.193141
$45$	0	0
$46$	11262.0	0.784734
$47$	8033.35	0.530459	0.265230	−	0.964185i	$-0.414552\pi$
$47$	8033.35	0.530459	0.265230	+	0.964185i	$0.414552\pi$
$48$	0	0
$49$	45611.5	2.71384
$50$	8472.03	0.479251
$51$	0	0
$52$	−20203.7	−1.03615
$53$	2442.69	0.119448	0.0597240	−	0.998215i	$-0.480978\pi$
$53$	2442.69	0.119448	0.0597240	+	0.998215i	$0.480978\pi$
$54$	0	0
$55$	3029.61	0.135045
$56$	−44481.7	−1.89545
$57$	0	0
$58$	3962.88	0.154682
$59$	16759.8	0.626814	0.313407	−	0.949619i	$-0.398530\pi$
$59$	16759.8	0.626814	0.313407	+	0.949619i	$0.398530\pi$
$60$	0	0
$61$	−20937.8	−0.720455	−0.360228	−	0.932864i	$-0.617301\pi$
$61$	−20937.8	−0.720455	−0.360228	+	0.932864i	$0.617301\pi$
$62$	−8770.82	−0.289775
$63$	0	0
$64$	18253.4	0.557050
$65$	24678.1	0.724483
$66$	0	0
$67$	31658.9	0.861605	0.430803	−	0.902446i	$-0.358231\pi$
$67$	31658.9	0.861605	0.430803	+	0.902446i	$0.358231\pi$
$68$	27660.9	0.725427
$69$	0	0
$70$	21214.7	0.517478
$71$	21807.4	0.513403	0.256701	−	0.966491i	$-0.417364\pi$
$71$	21807.4	0.513403	0.256701	+	0.966491i	$0.417364\pi$
$72$	0	0
$73$	−36722.9	−0.806548	−0.403274	−	0.915079i	$-0.632128\pi$
$73$	−36722.9	−0.806548	−0.403274	+	0.915079i	$0.632128\pi$
$74$	4812.10	0.102154
$75$	0	0
$76$	−1304.01	−0.0258968
$77$	−30230.3	−0.581052
$78$	0	0
$79$	−44563.9	−0.803370	−0.401685	−	0.915778i	$-0.631575\pi$
$79$	−44563.9	−0.803370	−0.401685	+	0.915778i	$0.631575\pi$
$80$	1305.34	0.0228033
$81$	0	0
$82$	−56439.9	−0.926940
$83$	86704.0	1.38148	0.690739	−	0.723104i	$-0.257285\pi$
$83$	86704.0	1.38148	0.690739	+	0.723104i	$0.257285\pi$
$84$	0	0
$85$	−33786.8	−0.507224
$86$	−56854.9	−0.828938
$87$	0	0
$88$	21543.2	0.296554
$89$	18797.3	0.251548	0.125774	−	0.992059i	$-0.459859\pi$
$89$	18797.3	0.251548	0.125774	+	0.992059i	$0.459859\pi$
$90$	0	0
$91$	−246245.	−3.11719
$92$	68070.4	0.838472
$93$	0	0
$94$	−27244.3	−0.318021
$95$	1592.80	0.0181072
$96$	0	0
$97$	−108008.	−1.16554	−0.582770	−	0.812637i	$-0.698031\pi$
$97$	−108008.	−1.16554	−0.582770	+	0.812637i	$0.698031\pi$
$98$	−154687.	−1.62700
$99$	0	0
$100$	51206.9	0.512069
$101$	−88246.3	−0.860782	−0.430391	−	0.902643i	$-0.641624\pi$
$101$	−88246.3	−0.860782	−0.430391	+	0.902643i	$0.641624\pi$
$102$	0	0
$103$	79711.6	0.740335	0.370168	−	0.928965i	$-0.379300\pi$
$103$	79711.6	0.740335	0.370168	+	0.928965i	$0.379300\pi$
$104$	175483.	1.59093
$105$	0	0
$106$	−8284.13	−0.0716115
$107$	−77110.5	−0.651110	−0.325555	−	0.945523i	$-0.605551\pi$
$107$	−77110.5	−0.651110	−0.325555	+	0.945523i	$0.605551\pi$
$108$	0	0
$109$	144478.	1.16476	0.582380	−	0.812916i	$-0.302121\pi$
$109$	144478.	1.16476	0.582380	+	0.812916i	$0.302121\pi$
$110$	−10274.6	−0.0809625
$111$	0	0
$112$	−13025.0	−0.0981143
$113$	199732.	1.47147	0.735736	−	0.677268i	$-0.236836\pi$
$113$	199732.	1.47147	0.735736	+	0.677268i	$0.236836\pi$
$114$	0	0
$115$	−83145.6	−0.586266
$116$	23952.6	0.165275
$117$	0	0
$118$	−56839.2	−0.375788
$119$	337134.	2.18240
$120$	0	0
$121$	14641.0	0.0909091
$122$	71008.5	0.431928
$123$	0	0
$124$	−53012.9	−0.309619
$125$	−140792.	−0.805938
$126$	0	0
$127$	270373.	1.48749	0.743744	−	0.668465i	$-0.233048\pi$
$127$	270373.	1.48749	0.743744	+	0.668465i	$0.233048\pi$
$128$	126069.	0.680118
$129$	0	0
$130$	−83693.2	−0.434342
$131$	−33916.6	−0.172677	−0.0863384	−	0.996266i	$-0.527517\pi$
$131$	−33916.6	−0.172677	−0.0863384	+	0.996266i	$0.527517\pi$
$132$	0	0
$133$	−15893.4	−0.0779089
$134$	−107368.	−0.516550
$135$	0	0
$136$	−240254.	−1.11384
$137$	−71504.1	−0.325484	−0.162742	−	0.986669i	$-0.552034\pi$
$137$	−71504.1	−0.325484	−0.162742	+	0.986669i	$0.552034\pi$
$138$	0	0
$139$	−151814.	−0.666460	−0.333230	−	0.942846i	$-0.608139\pi$
$139$	−151814.	−0.666460	−0.333230	+	0.942846i	$0.608139\pi$
$140$	128227.	0.552915
$141$	0	0
$142$	−73957.7	−0.307795
$143$	119260.	0.487703
$144$	0	0
$145$	−29257.2	−0.115561
$146$	124542.	0.483542
$147$	0	0
$148$	29085.4	0.109149
$149$	−467634.	−1.72560	−0.862800	−	0.505545i	$-0.831292\pi$
$149$	−467634.	−1.72560	−0.862800	+	0.505545i	$0.831292\pi$
$150$	0	0
$151$	−49902.7	−0.178107	−0.0890536	−	0.996027i	$-0.528384\pi$
$151$	−49902.7	−0.178107	−0.0890536	+	0.996027i	$0.528384\pi$
$152$	11326.2	0.0397627
$153$	0	0
$154$	102523.	0.348353
$155$	64753.4	0.216488
$156$	0	0
$157$	132832.	0.430084	0.215042	−	0.976605i	$-0.431011\pi$
$157$	132832.	0.430084	0.215042	+	0.976605i	$0.431011\pi$
$158$	151134.	0.481637
$159$	0	0
$160$	−147078.	−0.454202
$161$	829650.	2.52249
$162$	0	0
$163$	249743.	0.736248	0.368124	−	0.929777i	$-0.380000\pi$
$163$	249743.	0.736248	0.368124	+	0.929777i	$0.380000\pi$
$164$	−341136.	−0.990417
$165$	0	0
$166$	−294048.	−0.828224
$167$	−42563.2	−0.118098	−0.0590491	−	0.998255i	$-0.518807\pi$
$167$	−42563.2	−0.118098	−0.0590491	+	0.998255i	$0.518807\pi$
$168$	0	0
$169$	600156.	1.61639
$170$	114585.	0.304091
$171$	0	0
$172$	−343644.	−0.885703
$173$	−687226.	−1.74576	−0.872879	−	0.487937i	$-0.837750\pi$
$173$	−687226.	−1.74576	−0.872879	+	0.487937i	$0.837750\pi$
$174$	0	0
$175$	624116.	1.54053
$176$	6308.21	0.0153506
$177$	0	0
$178$	−63749.2	−0.150808
$179$	−435842.	−1.01671	−0.508355	−	0.861148i	$-0.669746\pi$
$179$	−435842.	−1.01671	−0.508355	+	0.861148i	$0.669746\pi$
$180$	0	0
$181$	−516438.	−1.17171	−0.585857	−	0.810415i	$-0.699242\pi$
$181$	−516438.	−1.17171	−0.585857	+	0.810415i	$0.699242\pi$
$182$	835114.	1.86882
$183$	0	0
$184$	−591239.	−1.28742
$185$	−35526.9	−0.0763181
$186$	0	0
$187$	−163279.	−0.341450
$188$	−164671.	−0.339799
$189$	0	0
$190$	−5401.82	−0.0108556
$191$	−401717.	−0.796777	−0.398389	−	0.917217i	$-0.630430\pi$
$191$	−401717.	−0.796777	−0.398389	+	0.917217i	$0.630430\pi$
$192$	0	0
$193$	−923446.	−1.78451	−0.892253	−	0.451535i	$-0.850877\pi$
$193$	−923446.	−1.78451	−0.892253	+	0.451535i	$0.850877\pi$
$194$	366299.	0.698765
$195$	0	0
$196$	−934964.	−1.73842
$197$	779664.	1.43134	0.715669	−	0.698440i	$-0.246122\pi$
$197$	779664.	1.43134	0.715669	+	0.698440i	$0.246122\pi$
$198$	0	0
$199$	−290274.	−0.519608	−0.259804	−	0.965661i	$-0.583658\pi$
$199$	−290274.	−0.519608	−0.259804	+	0.965661i	$0.583658\pi$
$200$	−444768.	−0.786247
$201$	0	0
$202$	299278.	0.516056
$203$	291936.	0.497220
$204$	0	0
$205$	416686.	0.692507
$206$	−270334.	−0.443846
$207$	0	0
$208$	51384.3	0.0823517
$209$	7697.41	0.0121893
$210$	0	0
$211$	−151203.	−0.233806	−0.116903	−	0.993143i	$-0.537297\pi$
$211$	−151203.	−0.233806	−0.116903	+	0.993143i	$0.537297\pi$
$212$	−50071.2	−0.0765154
$213$	0	0
$214$	261513.	0.390353
$215$	419750.	0.619290
$216$	0	0
$217$	−646127.	−0.931469
$218$	−489984.	−0.698298
$219$	0	0
$220$	−62102.2	−0.0865068
$221$	−1.33001e6	−1.83179
$222$	0	0
$223$	−948910.	−1.27780	−0.638900	−	0.769290i	$-0.720610\pi$
$223$	−948910.	−1.27780	−0.638900	+	0.769290i	$0.720610\pi$
$224$	1.46759e6	1.95427
$225$	0	0
$226$	−677372.	−0.882178
$227$	280951.	0.361881	0.180940	−	0.983494i	$-0.442086\pi$
$227$	280951.	0.361881	0.180940	+	0.983494i	$0.442086\pi$
$228$	0	0
$229$	−913556.	−1.15119	−0.575594	−	0.817736i	$-0.695229\pi$
$229$	−913556.	−1.15119	−0.575594	+	0.817736i	$0.695229\pi$
$230$	281980.	0.351479
$231$	0	0
$232$	−208045.	−0.253768
$233$	−19623.7	−0.0236805	−0.0118402	−	0.999930i	$-0.503769\pi$
$233$	−19623.7	−0.0236805	−0.0118402	+	0.999930i	$0.503769\pi$
$234$	0	0
$235$	201140.	0.237590
$236$	−343549.	−0.401522
$237$	0	0
$238$	−1.14336e6	−1.30840
$239$	−1.17282e6	−1.32812	−0.664060	−	0.747680i	$-0.731168\pi$
$239$	−1.17282e6	−1.32812	−0.664060	+	0.747680i	$0.731168\pi$
$240$	0	0
$241$	575715.	0.638506	0.319253	−	0.947670i	$-0.396568\pi$
$241$	575715.	0.638506	0.319253	+	0.947670i	$0.396568\pi$
$242$	−49653.5	−0.0545019
$243$	0	0
$244$	429192.	0.461506
$245$	1.14203e6	1.21552
$246$	0	0
$247$	62700.3	0.0653924
$248$	460454.	0.475398
$249$	0	0
$250$	477481.	0.483176
$251$	−112822.	−0.113034	−0.0565168	−	0.998402i	$-0.517999\pi$
$251$	−112822.	−0.113034	−0.0565168	+	0.998402i	$0.517999\pi$
$252$	0	0
$253$	−401813.	−0.394659
$254$	−916941.	−0.891779
$255$	0	0
$256$	−1.01166e6	−0.964795
$257$	206184.	0.194725	0.0973624	−	0.995249i	$-0.468959\pi$
$257$	206184.	0.194725	0.0973624	+	0.995249i	$0.468959\pi$
$258$	0	0
$259$	354497.	0.328370
$260$	−505862.	−0.464086
$261$	0	0
$262$	115025.	0.103523
$263$	2.04572e6	1.82371	0.911857	−	0.410509i	$-0.134649\pi$
$263$	2.04572e6	1.82371	0.911857	+	0.410509i	$0.134649\pi$
$264$	0	0
$265$	61160.3	0.0535001
$266$	53900.8	0.0467080
$267$	0	0
$268$	−648956.	−0.551923
$269$	1.33620e6	1.12588	0.562940	−	0.826498i	$-0.309670\pi$
$269$	1.33620e6	1.12588	0.562940	+	0.826498i	$0.309670\pi$
$270$	0	0
$271$	484299.	0.400581	0.200291	−	0.979737i	$-0.435811\pi$
$271$	484299.	0.400581	0.200291	+	0.979737i	$0.435811\pi$
$272$	−70350.4	−0.0576560
$273$	0	0
$274$	242499.	0.195134
$275$	−302269.	−0.241025
$276$	0	0
$277$	−1.00325e6	−0.785617	−0.392809	−	0.919620i	$-0.628496\pi$
$277$	−1.00325e6	−0.785617	−0.392809	+	0.919620i	$0.628496\pi$
$278$	514861.	0.399556
$279$	0	0
$280$	−1.11374e6	−0.848962
$281$	−956707.	−0.722792	−0.361396	−	0.932413i	$-0.617700\pi$
$281$	−956707.	−0.722792	−0.361396	+	0.932413i	$0.617700\pi$
$282$	0	0
$283$	−1.46904e6	−1.09035	−0.545176	−	0.838322i	$-0.683537\pi$
$283$	−1.46904e6	−1.09035	−0.545176	+	0.838322i	$0.683537\pi$
$284$	−447017.	−0.328873
$285$	0	0
$286$	−404459.	−0.292388
$287$	−4.15780e6	−2.97961
$288$	0	0
$289$	401065.	0.282469
$290$	99222.9	0.0692814
$291$	0	0
$292$	752762.	0.516655
$293$	1.81353e6	1.23411	0.617057	−	0.786918i	$-0.288325\pi$
$293$	1.81353e6	1.23411	0.617057	+	0.786918i	$0.288325\pi$
$294$	0	0
$295$	419634.	0.280747
$296$	−252627.	−0.167591
$297$	0	0
$298$	1.58593e6	1.03453
$299$	−3.27302e6	−2.11724
$300$	0	0
$301$	−4.18838e6	−2.66458
$302$	169240.	0.106779
$303$	0	0
$304$	3316.50	0.00205824
$305$	−524243.	−0.322688
$306$	0	0
$307$	1.96389e6	1.18925	0.594624	−	0.804004i	$-0.297301\pi$
$307$	1.96389e6	1.18925	0.594624	+	0.804004i	$0.297301\pi$
$308$	619673.	0.372208
$309$	0	0
$310$	−219605.	−0.129789
$311$	−2.37283e6	−1.39112	−0.695562	−	0.718466i	$-0.744845\pi$
$311$	−2.37283e6	−1.39112	−0.695562	+	0.718466i	$0.744845\pi$
$312$	0	0
$313$	−2.71301e6	−1.56528	−0.782638	−	0.622477i	$-0.786126\pi$
$313$	−2.71301e6	−1.56528	−0.782638	+	0.622477i	$0.786126\pi$
$314$	−450486.	−0.257844
$315$	0	0
$316$	913490.	0.514619
$317$	−1.07281e6	−0.599620	−0.299810	−	0.953999i	$-0.596923\pi$
$317$	−1.07281e6	−0.599620	−0.299810	+	0.953999i	$0.596923\pi$
$318$	0	0
$319$	−141389.	−0.0777930
$320$	457031.	0.249500
$321$	0	0
$322$	−2.81367e6	−1.51229
$323$	−85843.0	−0.0457824
$324$	0	0
$325$	−2.46217e6	−1.29303
$326$	−846978.	−0.441396
$327$	0	0
$328$	2.96301e6	1.52071
$329$	−2.00703e6	−1.02227
$330$	0	0
$331$	1.21762e6	0.610862	0.305431	−	0.952214i	$-0.401199\pi$
$331$	1.21762e6	0.610862	0.305431	+	0.952214i	$0.401199\pi$
$332$	−1.77729e6	−0.884941
$333$	0	0
$334$	144349.	0.0708022
$335$	792678.	0.385909
$336$	0	0
$337$	148600.	0.0712759	0.0356380	−	0.999365i	$-0.488654\pi$
$337$	148600.	0.0712759	0.0356380	+	0.999365i	$0.488654\pi$
$338$	−2.03537e6	−0.969061
$339$	0	0
$340$	692576.	0.324915
$341$	312929.	0.145734
$342$	0	0
$343$	−7.19644e6	−3.30280
$344$	2.98479e6	1.35993
$345$	0	0
$346$	2.33066e6	1.04662
$347$	3.89042e6	1.73450	0.867248	−	0.497877i	$-0.165887\pi$
$347$	3.89042e6	1.73450	0.867248	+	0.497877i	$0.165887\pi$
$348$	0	0
$349$	1.95028e6	0.857104	0.428552	−	0.903517i	$-0.359024\pi$
$349$	1.95028e6	0.857104	0.428552	+	0.903517i	$0.359024\pi$
$350$	−2.11663e6	−0.923579
$351$	0	0
$352$	−710776.	−0.305757
$353$	2.63119e6	1.12387	0.561934	−	0.827182i	$-0.310058\pi$
$353$	2.63119e6	1.12387	0.561934	+	0.827182i	$0.310058\pi$
$354$	0	0
$355$	546016.	0.229951
$356$	−385315.	−0.161135
$357$	0	0
$358$	1.47812e6	0.609538
$359$	3.41979e6	1.40044	0.700218	−	0.713929i	$-0.253086\pi$
$359$	3.41979e6	1.40044	0.700218	+	0.713929i	$0.253086\pi$
$360$	0	0
$361$	−2.47205e6	−0.998366
$362$	1.75145e6	0.702466
$363$	0	0
$364$	5.04762e6	1.99680
$365$	−919473.	−0.361249
$366$	0	0
$367$	499776.	0.193692	0.0968458	−	0.995299i	$-0.469125\pi$
$367$	499776.	0.193692	0.0968458	+	0.995299i	$0.469125\pi$
$368$	−173125.	−0.0666407
$369$	0	0
$370$	120486.	0.0457543
$371$	−610274.	−0.230192
$372$	0	0
$373$	−4.01256e6	−1.49331	−0.746655	−	0.665211i	$-0.768341\pi$
$373$	−4.01256e6	−1.49331	−0.746655	+	0.665211i	$0.768341\pi$
$374$	553745.	0.204706
$375$	0	0
$376$	1.43028e6	0.521738
$377$	−1.15171e6	−0.417338
$378$	0	0
$379$	3.39905e6	1.21551	0.607756	−	0.794124i	$-0.292070\pi$
$379$	3.39905e6	1.21551	0.607756	+	0.794124i	$0.292070\pi$
$380$	−32649.8	−0.0115990
$381$	0	0
$382$	1.36238e6	0.477684
$383$	−3.72973e6	−1.29921	−0.649606	−	0.760271i	$-0.725066\pi$
$383$	−3.72973e6	−1.29921	−0.649606	+	0.760271i	$0.725066\pi$
$384$	0	0
$385$	−756909.	−0.260251
$386$	3.13177e6	1.06985
$387$	0	0
$388$	2.21399e6	0.746616
$389$	−4.02398e6	−1.34828	−0.674142	−	0.738602i	$-0.735487\pi$
$389$	−4.02398e6	−1.34828	−0.674142	+	0.738602i	$0.735487\pi$
$390$	0	0
$391$	4.48109e6	1.48232
$392$	8.12082e6	2.66922
$393$	0	0
$394$	−2.64415e6	−0.858116
$395$	−1.11580e6	−0.359826
$396$	0	0
$397$	−4.67876e6	−1.48989	−0.744946	−	0.667124i	$-0.767525\pi$
$397$	−4.67876e6	−1.48989	−0.744946	+	0.667124i	$0.767525\pi$
$398$	984435.	0.311515
$399$	0	0
$400$	−130236.	−0.0406986
$401$	2.96551e6	0.920956	0.460478	−	0.887671i	$-0.347678\pi$
$401$	2.96551e6	0.920956	0.460478	+	0.887671i	$0.347678\pi$
$402$	0	0
$403$	2.54901e6	0.781823
$404$	1.80891e6	0.551396
$405$	0	0
$406$	−990073.	−0.298093
$407$	−171688.	−0.0513754
$408$	0	0
$409$	−2.49097e6	−0.736308	−0.368154	−	0.929765i	$-0.620010\pi$
$409$	−2.49097e6	−0.736308	−0.368154	+	0.929765i	$0.620010\pi$
$410$	−1.41315e6	−0.415172
$411$	0	0
$412$	−1.63396e6	−0.474241
$413$	−4.18722e6	−1.20795
$414$	0	0
$415$	2.17090e6	0.618757
$416$	−5.78972e6	−1.64030
$417$	0	0
$418$	−26105.0	−0.00730774
$419$	1.69065e6	0.470455	0.235227	−	0.971940i	$-0.424417\pi$
$419$	1.69065e6	0.470455	0.235227	+	0.971940i	$0.424417\pi$
$420$	0	0
$421$	−7.17918e6	−1.97410	−0.987051	−	0.160404i	$-0.948720\pi$
$421$	−7.17918e6	−1.97410	−0.987051	+	0.160404i	$0.948720\pi$
$422$	512791.	0.140171
$423$	0	0
$424$	434904.	0.117484
$425$	3.37097e6	0.905278
$426$	0	0
$427$	5.23104e6	1.38841
$428$	1.58064e6	0.417085
$429$	0	0
$430$	−1.42354e6	−0.371277
$431$	−4.36626e6	−1.13218	−0.566092	−	0.824342i	$-0.691545\pi$
$431$	−4.36626e6	−1.13218	−0.566092	+	0.824342i	$0.691545\pi$
$432$	0	0
$433$	−3.28300e6	−0.841495	−0.420748	−	0.907178i	$-0.638232\pi$
$433$	−3.28300e6	−0.841495	−0.420748	+	0.907178i	$0.638232\pi$
$434$	2.19127e6	0.558435
$435$	0	0
$436$	−2.96158e6	−0.746117
$437$	−211250.	−0.0529168
$438$	0	0
$439$	−2.34308e6	−0.580265	−0.290132	−	0.956987i	$-0.593699\pi$
$439$	−2.34308e6	−0.580265	−0.290132	+	0.956987i	$0.593699\pi$
$440$	539401.	0.132825
$441$	0	0
$442$	4.51060e6	1.09819
$443$	758611.	0.183658	0.0918290	−	0.995775i	$-0.470729\pi$
$443$	758611.	0.183658	0.0918290	+	0.995775i	$0.470729\pi$
$444$	0	0
$445$	470649.	0.112667
$446$	3.21813e6	0.766067
$447$	0	0
$448$	−4.56038e6	−1.07351
$449$	−113738.	−0.0266251	−0.0133125	−	0.999911i	$-0.504238\pi$
$449$	−113738.	−0.0266251	−0.0133125	+	0.999911i	$0.504238\pi$
$450$	0	0
$451$	2.01369e6	0.466178
$452$	−4.09419e6	−0.942589
$453$	0	0
$454$	−952816.	−0.216955
$455$	−6.16550e6	−1.39617
$456$	0	0
$457$	−4.06387e6	−0.910226	−0.455113	−	0.890434i	$-0.650401\pi$
$457$	−4.06387e6	−0.910226	−0.455113	+	0.890434i	$0.650401\pi$
$458$	3.09823e6	0.690161
$459$	0	0
$460$	1.70435e6	0.375548
$461$	−1.29888e6	−0.284653	−0.142327	−	0.989820i	$-0.545458\pi$
$461$	−1.29888e6	−0.284653	−0.142327	+	0.989820i	$0.545458\pi$
$462$	0	0
$463$	−5.49072e6	−1.19036	−0.595178	−	0.803594i	$-0.702918\pi$
$463$	−5.49072e6	−1.19036	−0.595178	+	0.803594i	$0.702918\pi$
$464$	−60918.9	−0.0131358
$465$	0	0
$466$	66551.7	0.0141969
$467$	−1.29315e6	−0.274383	−0.137192	−	0.990545i	$-0.543808\pi$
$467$	−1.29315e6	−0.274383	−0.137192	+	0.990545i	$0.543808\pi$
$468$	0	0
$469$	−7.90956e6	−1.66043
$470$	−682146.	−0.142440
$471$	0	0
$472$	2.98397e6	0.616509
$473$	2.02850e6	0.416890
$474$	0	0
$475$	−158916.	−0.0323172
$476$	−6.91071e6	−1.39799
$477$	0	0
$478$	3.97750e6	0.796234
$479$	8.93674e6	1.77967	0.889837	−	0.456278i	$-0.150818\pi$
$479$	8.93674e6	1.77967	0.889837	+	0.456278i	$0.150818\pi$
$480$	0	0
$481$	−1.39851e6	−0.275615
$482$	−1.95248e6	−0.382797
$483$	0	0
$484$	−300117.	−0.0582341
$485$	−2.70432e6	−0.522040
$486$	0	0
$487$	−6.18513e6	−1.18175	−0.590876	−	0.806763i	$-0.701218\pi$
$487$	−6.18513e6	−1.18175	−0.590876	+	0.806763i	$0.701218\pi$
$488$	−3.72783e6	−0.708610
$489$	0	0
$490$	−3.87307e6	−0.728727
$491$	−8.73913e6	−1.63593	−0.817964	−	0.575269i	$-0.804897\pi$
$491$	−8.73913e6	−1.63593	−0.817964	+	0.575269i	$0.804897\pi$
$492$	0	0
$493$	1.57680e6	0.292186
$494$	−212642.	−0.0392041
$495$	0	0
$496$	134829.	0.0246081
$497$	−5.44830e6	−0.989396
$498$	0	0
$499$	−3.27503e6	−0.588795	−0.294398	−	0.955683i	$-0.595119\pi$
$499$	−3.27503e6	−0.588795	−0.294398	+	0.955683i	$0.595119\pi$
$500$	2.88600e6	0.516264
$501$	0	0
$502$	382623.	0.0677660
$503$	8.71649e6	1.53611	0.768054	−	0.640385i	$-0.221225\pi$
$503$	8.71649e6	1.53611	0.768054	+	0.640385i	$0.221225\pi$
$504$	0	0
$505$	−2.20952e6	−0.385540
$506$	1.36271e6	0.236606
$507$	0	0
$508$	−5.54221e6	−0.952848
$509$	2.72773e6	0.466668	0.233334	−	0.972397i	$-0.425037\pi$
$509$	2.72773e6	0.466668	0.233334	+	0.972397i	$0.425037\pi$
$510$	0	0
$511$	9.17475e6	1.55433
$512$	−603272.	−0.101704
$513$	0	0
$514$	−699251.	−0.116741
$515$	1.99583e6	0.331593
$516$	0	0
$517$	972036.	0.159940
$518$	−1.20224e6	−0.196864
$519$	0	0
$520$	4.39376e6	0.712571
$521$	−2.82659e6	−0.456214	−0.228107	−	0.973636i	$-0.573254\pi$
$521$	−2.82659e6	−0.456214	−0.228107	+	0.973636i	$0.573254\pi$
$522$	0	0
$523$	3.35264e6	0.535960	0.267980	−	0.963424i	$-0.413644\pi$
$523$	3.35264e6	0.535960	0.267980	+	0.963424i	$0.413644\pi$
$524$	695236.	0.110613
$525$	0	0
$526$	−6.93785e6	−1.09335
$527$	−3.48985e6	−0.547369
$528$	0	0
$529$	4.59113e6	0.713314
$530$	−207419.	−0.0320744
$531$	0	0
$532$	325789.	0.0499065
$533$	1.64028e7	2.50092
$534$	0	0
$535$	−1.93070e6	−0.291629
$536$	5.63664e6	0.847439
$537$	0	0
$538$	−4.53160e6	−0.674987
$539$	5.51899e6	0.818254
$540$	0	0
$541$	−6.15263e6	−0.903790	−0.451895	−	0.892071i	$-0.649252\pi$
$541$	−6.15263e6	−0.903790	−0.451895	+	0.892071i	$0.649252\pi$
$542$	−1.64245e6	−0.240156
$543$	0	0
$544$	7.92672e6	1.14841
$545$	3.61747e6	0.521691
$546$	0	0
$547$	1.45229e6	0.207532	0.103766	−	0.994602i	$-0.466911\pi$
$547$	1.45229e6	0.207532	0.103766	+	0.994602i	$0.466911\pi$
$548$	1.46572e6	0.208497
$549$	0	0
$550$	1.02512e6	0.144499
$551$	−74334.6	−0.0104307
$552$	0	0
$553$	1.11337e7	1.54820
$554$	3.40243e6	0.470993
$555$	0	0
$556$	3.11194e6	0.426918
$557$	723407.	0.0987972	0.0493986	−	0.998779i	$-0.484270\pi$
$557$	723407.	0.0987972	0.0493986	+	0.998779i	$0.484270\pi$
$558$	0	0
$559$	1.65234e7	2.23650
$560$	−326121.	−0.0439449
$561$	0	0
$562$	3.24457e6	0.433328
$563$	−6.72827e6	−0.894607	−0.447304	−	0.894382i	$-0.647616\pi$
$563$	−6.72827e6	−0.894607	−0.447304	+	0.894382i	$0.647616\pi$
$564$	0	0
$565$	5.00092e6	0.659065
$566$	4.98209e6	0.653688
$567$	0	0
$568$	3.88266e6	0.504962
$569$	−9.67408e6	−1.25265	−0.626324	−	0.779563i	$-0.715441\pi$
$569$	−9.67408e6	−1.25265	−0.626324	+	0.779563i	$0.715441\pi$
$570$	0	0
$571$	−1.04188e7	−1.33730	−0.668651	−	0.743577i	$-0.733128\pi$
$571$	−1.04188e7	−1.33730	−0.668651	+	0.743577i	$0.733128\pi$
$572$	−2.44464e6	−0.312410
$573$	0	0
$574$	1.41008e7	1.78634
$575$	8.29558e6	1.04635
$576$	0	0
$577$	6.79081e6	0.849145	0.424573	−	0.905394i	$-0.360424\pi$
$577$	6.79081e6	0.849145	0.424573	+	0.905394i	$0.360424\pi$
$578$	−1.36017e6	−0.169346
$579$	0	0
$580$	599726.	0.0740258
$581$	−2.16619e7	−2.66229
$582$	0	0
$583$	295565.	0.0360149
$584$	−6.53827e6	−0.793287
$585$	0	0
$586$	−6.15040e6	−0.739877
$587$	−5.53973e6	−0.663580	−0.331790	−	0.943353i	$-0.607652\pi$
$587$	−5.53973e6	−0.663580	−0.331790	+	0.943353i	$0.607652\pi$
$588$	0	0
$589$	164521.	0.0195403
$590$	−1.42315e6	−0.168314
$591$	0	0
$592$	−73973.5	−0.00867505
$593$	2.31621e6	0.270484	0.135242	−	0.990813i	$-0.456819\pi$
$593$	2.31621e6	0.270484	0.135242	+	0.990813i	$0.456819\pi$
$594$	0	0
$595$	8.44119e6	0.977488
$596$	9.58575e6	1.10538
$597$	0	0
$598$	1.11001e7	1.26933
$599$	9.90987e6	1.12850	0.564249	−	0.825605i	$-0.309166\pi$
$599$	9.90987e6	1.12850	0.564249	+	0.825605i	$0.309166\pi$
$600$	0	0
$601$	−3.77767e6	−0.426616	−0.213308	−	0.976985i	$-0.568424\pi$
$601$	−3.77767e6	−0.426616	−0.213308	+	0.976985i	$0.568424\pi$
$602$	1.42045e7	1.59747
$603$	0	0
$604$	1.02293e6	0.114091
$605$	366583.	0.0407177
$606$	0	0
$607$	3.89494e6	0.429071	0.214536	−	0.976716i	$-0.431176\pi$
$607$	3.89494e6	0.429071	0.214536	+	0.976716i	$0.431176\pi$
$608$	−373686.	−0.0409966
$609$	0	0
$610$	1.77792e6	0.193458
$611$	7.91784e6	0.858033
$612$	0	0
$613$	1.43817e7	1.54582	0.772909	−	0.634517i	$-0.218801\pi$
$613$	1.43817e7	1.54582	0.772909	+	0.634517i	$0.218801\pi$
$614$	−6.66035e6	−0.712978
$615$	0	0
$616$	−5.38229e6	−0.571499
$617$	1.07173e7	1.13338	0.566688	−	0.823933i	$-0.308225\pi$
$617$	1.07173e7	1.13338	0.566688	+	0.823933i	$0.308225\pi$
$618$	0	0
$619$	9.86482e6	1.03481	0.517407	−	0.855739i	$-0.326897\pi$
$619$	9.86482e6	1.03481	0.517407	+	0.855739i	$0.326897\pi$
$620$	−1.32734e6	−0.138677
$621$	0	0
$622$	8.04722e6	0.834007
$623$	−4.69626e6	−0.484766
$624$	0	0
$625$	4.28139e6	0.438414
$626$	9.20091e6	0.938415
$627$	0	0
$628$	−2.72284e6	−0.275501
$629$	1.91470e6	0.192963
$630$	0	0
$631$	5.01522e6	0.501437	0.250718	−	0.968060i	$-0.419333\pi$
$631$	5.01522e6	0.501437	0.250718	+	0.968060i	$0.419333\pi$
$632$	−7.93430e6	−0.790162
$633$	0	0
$634$	3.63834e6	0.359485
$635$	6.76962e6	0.666239
$636$	0	0
$637$	4.49557e7	4.38971
$638$	479508.	0.0466385
$639$	0	0
$640$	3.15654e6	0.304622
$641$	−1.38286e7	−1.32934	−0.664668	−	0.747139i	$-0.731427\pi$
$641$	−1.38286e7	−1.32934	−0.664668	+	0.747139i	$0.731427\pi$
$642$	0	0
$643$	8.16298e6	0.778612	0.389306	−	0.921108i	$-0.372715\pi$
$643$	8.16298e6	0.778612	0.389306	+	0.921108i	$0.372715\pi$
$644$	−1.70065e7	−1.61585
$645$	0	0
$646$	291128.	0.0274475
$647$	2.75199e6	0.258455	0.129228	−	0.991615i	$-0.458750\pi$
$647$	2.75199e6	0.258455	0.129228	+	0.991615i	$0.458750\pi$
$648$	0	0
$649$	2.02794e6	0.188992
$650$	8.35021e6	0.775201
$651$	0	0
$652$	−5.11934e6	−0.471623
$653$	2.83857e6	0.260505	0.130253	−	0.991481i	$-0.458421\pi$
$653$	2.83857e6	0.260505	0.130253	+	0.991481i	$0.458421\pi$
$654$	0	0
$655$	−849207.	−0.0773411
$656$	867617.	0.0787170
$657$	0	0
$658$	6.80664e6	0.612869
$659$	1.09673e7	0.983756	0.491878	−	0.870664i	$-0.336311\pi$
$659$	1.09673e7	0.983756	0.491878	+	0.870664i	$0.336311\pi$
$660$	0	0
$661$	−2.15566e6	−0.191901	−0.0959504	−	0.995386i	$-0.530589\pi$
$661$	−2.15566e6	−0.191901	−0.0959504	+	0.995386i	$0.530589\pi$
$662$	−4.12945e6	−0.366224
$663$	0	0
$664$	1.54371e7	1.35876
$665$	−397940.	−0.0348950
$666$	0	0
$667$	3.88034e6	0.337719
$668$	872478.	0.0756507
$669$	0	0
$670$	−2.68829e6	−0.231360
$671$	−2.53348e6	−0.217225
$672$	0	0
$673$	−1.71120e7	−1.45634	−0.728170	−	0.685397i	$-0.759629\pi$
$673$	−1.71120e7	−1.45634	−0.728170	+	0.685397i	$0.759629\pi$
$674$	−503960.	−0.0427314
$675$	0	0
$676$	−1.23022e7	−1.03542
$677$	5.25274e6	0.440467	0.220234	−	0.975447i	$-0.429318\pi$
$677$	5.25274e6	0.440467	0.220234	+	0.975447i	$0.429318\pi$
$678$	0	0
$679$	2.69844e7	2.24615
$680$	−6.01551e6	−0.498884
$681$	0	0
$682$	−1.06127e6	−0.0873704
$683$	3.61797e6	0.296765	0.148383	−	0.988930i	$-0.452593\pi$
$683$	3.61797e6	0.296765	0.148383	+	0.988930i	$0.452593\pi$
$684$	0	0
$685$	−1.79033e6	−0.145783
$686$	2.44060e7	1.98010
$687$	0	0
$688$	873997.	0.0703945
$689$	2.40757e6	0.193210
$690$	0	0
$691$	−1.79101e7	−1.42693	−0.713464	−	0.700692i	$-0.752875\pi$
$691$	−1.79101e7	−1.42693	−0.713464	+	0.700692i	$0.752875\pi$
$692$	1.40870e7	1.11829
$693$	0	0
$694$	−1.31940e7	−1.03987
$695$	−3.80113e6	−0.298504
$696$	0	0
$697$	−2.24571e7	−1.75094
$698$	−6.61418e6	−0.513851
$699$	0	0
$700$	−1.27934e7	−0.986826
$701$	319236.	0.0245368	0.0122684	−	0.999925i	$-0.496095\pi$
$701$	319236.	0.0245368	0.0122684	+	0.999925i	$0.496095\pi$
$702$	0	0
$703$	−90264.1	−0.00688853
$704$	2.20866e6	0.167957
$705$	0	0
$706$	−8.92341e6	−0.673781
$707$	2.20472e7	1.65884
$708$	0	0
$709$	−1.10348e7	−0.824424	−0.412212	−	0.911088i	$-0.635244\pi$
$709$	−1.10348e7	−0.824424	−0.412212	+	0.911088i	$0.635244\pi$
$710$	−1.85176e6	−0.137860
$711$	0	0
$712$	3.34673e6	0.247412
$713$	−8.58814e6	−0.632667
$714$	0	0
$715$	2.98605e6	0.218440
$716$	8.93407e6	0.651279
$717$	0	0
$718$	−1.15979e7	−0.839590
$719$	593552.	0.0428190	0.0214095	−	0.999771i	$-0.493185\pi$
$719$	593552.	0.0428190	0.0214095	+	0.999771i	$0.493185\pi$
$720$	0	0
$721$	−1.99149e7	−1.42672
$722$	8.38372e6	0.598541
$723$	0	0
$724$	1.05862e7	0.750571
$725$	2.91904e6	0.206251
$726$	0	0
$727$	3.99256e6	0.280166	0.140083	−	0.990140i	$-0.455263\pi$
$727$	3.99256e6	0.280166	0.140083	+	0.990140i	$0.455263\pi$
$728$	−4.38421e7	−3.06594
$729$	0	0
$730$	3.11830e6	0.216576
$731$	−2.26222e7	−1.56582
$732$	0	0
$733$	−1.22959e6	−0.0845278	−0.0422639	−	0.999106i	$-0.513457\pi$
$733$	−1.22959e6	−0.0845278	−0.0422639	+	0.999106i	$0.513457\pi$
$734$	−1.69494e6	−0.116122
$735$	0	0
$736$	1.95068e7	1.32737
$737$	3.83072e6	0.259784
$738$	0	0
$739$	1.87395e7	1.26225	0.631126	−	0.775680i	$-0.282593\pi$
$739$	1.87395e7	1.26225	0.631126	+	0.775680i	$0.282593\pi$
$740$	728244.	0.0488875
$741$	0	0
$742$	2.06968e6	0.138005
$743$	1.76443e7	1.17256	0.586278	−	0.810110i	$-0.300593\pi$
$743$	1.76443e7	1.17256	0.586278	+	0.810110i	$0.300593\pi$
$744$	0	0
$745$	−1.17087e7	−0.772888
$746$	1.36082e7	0.895270
$747$	0	0
$748$	3.34697e6	0.218724
$749$	1.92651e7	1.25477
$750$	0	0
$751$	5.63962e6	0.364880	0.182440	−	0.983217i	$-0.441600\pi$
$751$	5.63962e6	0.364880	0.182440	+	0.983217i	$0.441600\pi$
$752$	418811.	0.0270068
$753$	0	0
$754$	3.90589e6	0.250203
$755$	−1.24947e6	−0.0797734
$756$	0	0
$757$	5.01554e6	0.318111	0.159055	−	0.987270i	$-0.449155\pi$
$757$	5.01554e6	0.318111	0.159055	+	0.987270i	$0.449155\pi$
$758$	−1.15275e7	−0.728724
$759$	0	0
$760$	283587.	0.0178095
$761$	1.57648e7	0.986794	0.493397	−	0.869804i	$-0.335755\pi$
$761$	1.57648e7	0.986794	0.493397	+	0.869804i	$0.335755\pi$
$762$	0	0
$763$	−3.60961e7	−2.24465
$764$	8.23456e6	0.510396
$765$	0	0
$766$	1.26490e7	0.778904
$767$	1.65188e7	1.01389
$768$	0	0
$769$	−5.15278e6	−0.314214	−0.157107	−	0.987582i	$-0.550217\pi$
$769$	−5.15278e6	−0.314214	−0.157107	+	0.987582i	$0.550217\pi$
$770$	2.56698e6	0.156026
$771$	0	0
$772$	1.89292e7	1.14311
$773$	−8.35346e6	−0.502826	−0.251413	−	0.967880i	$-0.580895\pi$
$773$	−8.35346e6	−0.502826	−0.251413	+	0.967880i	$0.580895\pi$
$774$	0	0
$775$	−6.46055e6	−0.386381
$776$	−1.92301e7	−1.14638
$777$	0	0
$778$	1.36469e7	0.808324
$779$	1.05868e6	0.0625062
$780$	0	0
$781$	2.63870e6	0.154797
$782$	−1.51972e7	−0.888681
$783$	0	0
$784$	2.37791e6	0.138167
$785$	3.32586e6	0.192632
$786$	0	0
$787$	9.51945e6	0.547867	0.273933	−	0.961749i	$-0.411675\pi$
$787$	9.51945e6	0.547867	0.273933	+	0.961749i	$0.411675\pi$
$788$	−1.59819e7	−0.916880
$789$	0	0
$790$	3.78411e6	0.215723
$791$	−4.99005e7	−2.83572
$792$	0	0
$793$	−2.06368e7	−1.16536
$794$	1.58676e7	0.893221
$795$	0	0
$796$	5.95016e6	0.332848
$797$	−2.41491e7	−1.34665	−0.673326	−	0.739346i	$-0.735135\pi$
$797$	−2.41491e7	−1.34665	−0.673326	+	0.739346i	$0.735135\pi$
$798$	0	0
$799$	−1.08403e7	−0.600725
$800$	1.46743e7	0.810646
$801$	0	0
$802$	−1.00572e7	−0.552132
$803$	−4.44348e6	−0.243183
$804$	0	0
$805$	2.07729e7	1.12981
$806$	−8.64470e6	−0.468719
$807$	0	0
$808$	−1.57116e7	−0.846629
$809$	−1.62622e7	−0.873590	−0.436795	−	0.899561i	$-0.643887\pi$
$809$	−1.62622e7	−0.873590	−0.436795	+	0.899561i	$0.643887\pi$
$810$	0	0
$811$	3.19235e6	0.170435	0.0852175	−	0.996362i	$-0.472841\pi$
$811$	3.19235e6	0.170435	0.0852175	+	0.996362i	$0.472841\pi$
$812$	−5.98423e6	−0.318507
$813$	0	0
$814$	582264.	0.0308006
$815$	6.25309e6	0.329762
$816$	0	0
$817$	1.06647e6	0.0558976
$818$	8.44786e6	0.441432
$819$	0	0
$820$	−8.54140e6	−0.443603
$821$	−3.69300e6	−0.191215	−0.0956075	−	0.995419i	$-0.530479\pi$
$821$	−3.69300e6	−0.191215	−0.0956075	+	0.995419i	$0.530479\pi$
$822$	0	0
$823$	−7.63853e6	−0.393107	−0.196553	−	0.980493i	$-0.562975\pi$
$823$	−7.63853e6	−0.393107	−0.196553	+	0.980493i	$0.562975\pi$
$824$	1.41921e7	0.728163
$825$	0	0
$826$	1.42005e7	0.724193
$827$	−9.78270e6	−0.497388	−0.248694	−	0.968582i	$-0.580001\pi$
$827$	−9.78270e6	−0.497388	−0.248694	+	0.968582i	$0.580001\pi$
$828$	0	0
$829$	−2.49371e6	−0.126026	−0.0630129	−	0.998013i	$-0.520071\pi$
$829$	−2.49371e6	−0.126026	−0.0630129	+	0.998013i	$0.520071\pi$
$830$	−7.36240e6	−0.370958
$831$	0	0
$832$	1.79910e7	0.901044
$833$	−6.15489e7	−3.07332
$834$	0	0
$835$	−1.06570e6	−0.0528956
$836$	−157785.	−0.00780817
$837$	0	0
$838$	−5.73366e6	−0.282047
$839$	−2.81537e7	−1.38080	−0.690399	−	0.723429i	$-0.742565\pi$
$839$	−2.81537e7	−1.38080	−0.690399	+	0.723429i	$0.742565\pi$
$840$	0	0
$841$	−1.91457e7	−0.933431
$842$	2.43475e7	1.18351
$843$	0	0
$844$	3.09943e6	0.149770
$845$	1.50268e7	0.723975
$846$	0	0
$847$	−3.65786e6	−0.175194
$848$	127347.	0.00608134
$849$	0	0
$850$	−1.14323e7	−0.542733
$851$	4.71187e6	0.223033
$852$	0	0
$853$	3.32010e7	1.56235	0.781175	−	0.624312i	$-0.214621\pi$
$853$	3.32010e7	1.56235	0.781175	+	0.624312i	$0.214621\pi$
$854$	−1.77406e7	−0.832382
$855$	0	0
$856$	−1.37290e7	−0.640404
$857$	−3.63175e7	−1.68913	−0.844567	−	0.535450i	$-0.820142\pi$
$857$	−3.63175e7	−1.68913	−0.844567	+	0.535450i	$0.820142\pi$
$858$	0	0
$859$	−3.75670e6	−0.173710	−0.0868548	−	0.996221i	$-0.527682\pi$
$859$	−3.75670e6	−0.173710	−0.0868548	+	0.996221i	$0.527682\pi$
$860$	−8.60420e6	−0.396702
$861$	0	0
$862$	1.48077e7	0.678767
$863$	−2.05924e7	−0.941198	−0.470599	−	0.882347i	$-0.655962\pi$
$863$	−2.05924e7	−0.941198	−0.470599	+	0.882347i	$0.655962\pi$
$864$	0	0
$865$	−1.72068e7	−0.781917
$866$	1.11340e7	0.504494
$867$	0	0
$868$	1.32446e7	0.596676
$869$	−5.39224e6	−0.242225
$870$	0	0
$871$	3.12036e7	1.39367
$872$	2.57234e7	1.14561
$873$	0	0
$874$	716434.	0.0317247
$875$	3.51749e7	1.55315
$876$	0	0
$877$	1.16550e7	0.511698	0.255849	−	0.966717i	$-0.417645\pi$
$877$	1.16550e7	0.511698	0.255849	+	0.966717i	$0.417645\pi$
$878$	7.94633e6	0.347881
$879$	0	0
$880$	157946.	0.00687545
$881$	−1.63913e7	−0.711496	−0.355748	−	0.934582i	$-0.615774\pi$
$881$	−1.63913e7	−0.711496	−0.355748	+	0.934582i	$0.615774\pi$
$882$	0	0
$883$	1.53091e7	0.660768	0.330384	−	0.943847i	$-0.392822\pi$
$883$	1.53091e7	0.660768	0.330384	+	0.943847i	$0.392822\pi$
$884$	2.72631e7	1.17340
$885$	0	0
$886$	−2.57275e6	−0.110107
$887$	−5.90413e6	−0.251969	−0.125984	−	0.992032i	$-0.540209\pi$
$887$	−5.90413e6	−0.251969	−0.125984	+	0.992032i	$0.540209\pi$
$888$	0	0
$889$	−6.75491e7	−2.86659
$890$	−1.59616e6	−0.0675462
$891$	0	0
$892$	1.94511e7	0.818527
$893$	511042.	0.0214451
$894$	0	0
$895$	−1.09127e7	−0.455379
$896$	−3.14968e7	−1.31068
$897$	0	0
$898$	385732.	0.0159623
$899$	−3.02199e6	−0.124708
$900$	0	0
$901$	−3.29620e6	−0.135270
$902$	−6.82923e6	−0.279483
$903$	0	0
$904$	3.55609e7	1.44728
$905$	−1.29306e7	−0.524805
$906$	0	0
$907$	−1.76162e7	−0.711039	−0.355520	−	0.934669i	$-0.615696\pi$
$907$	−1.76162e7	−0.711039	−0.355520	+	0.934669i	$0.615696\pi$
$908$	−5.75904e6	−0.231812
$909$	0	0
$910$	2.09097e7	0.837035
$911$	−4.22091e7	−1.68504	−0.842520	−	0.538665i	$-0.818929\pi$
$911$	−4.22091e7	−1.68504	−0.842520	+	0.538665i	$0.818929\pi$
$912$	0	0
$913$	1.04912e7	0.416531
$914$	1.37822e7	0.545699
$915$	0	0
$916$	1.87264e7	0.737423
$917$	8.47362e6	0.332771
$918$	0	0
$919$	1.40792e7	0.549908	0.274954	−	0.961457i	$-0.411337\pi$
$919$	1.40792e7	0.549908	0.274954	+	0.961457i	$0.411337\pi$
$920$	−1.48035e7	−0.576627
$921$	0	0
$922$	4.40502e6	0.170655
$923$	2.14938e7	0.830443
$924$	0	0
$925$	3.54457e6	0.136210
$926$	1.86212e7	0.713642
$927$	0	0
$928$	6.86403e6	0.261643
$929$	3.62182e7	1.37685	0.688426	−	0.725306i	$-0.258302\pi$
$929$	3.62182e7	1.37685	0.688426	+	0.725306i	$0.258302\pi$
$930$	0	0
$931$	2.90158e6	0.109713
$932$	402254.	0.0151691
$933$	0	0
$934$	4.38560e6	0.164498
$935$	−4.08820e6	−0.152934
$936$	0	0
$937$	−1.01365e7	−0.377172	−0.188586	−	0.982057i	$-0.560391\pi$
$937$	−1.01365e7	−0.377172	−0.188586	+	0.982057i	$0.560391\pi$
$938$	2.68245e7	0.995460
$939$	0	0
$940$	−4.12305e6	−0.152194
$941$	3.24791e7	1.19572	0.597860	−	0.801600i	$-0.296018\pi$
$941$	3.24791e7	1.19572	0.597860	+	0.801600i	$0.296018\pi$
$942$	0	0
$943$	−5.52644e7	−2.02379
$944$	873755.	0.0319124
$945$	0	0
$946$	−6.87944e6	−0.249934
$947$	536431.	0.0194374	0.00971872	−	0.999953i	$-0.496906\pi$
$947$	536431.	0.0194374	0.00971872	+	0.999953i	$0.496906\pi$
$948$	0	0
$949$	−3.61949e7	−1.30461
$950$	538948.	0.0193748
$951$	0	0
$952$	6.00244e7	2.14652
$953$	−2.13217e7	−0.760484	−0.380242	−	0.924887i	$-0.624159\pi$
$953$	−2.13217e7	−0.760484	−0.380242	+	0.924887i	$0.624159\pi$
$954$	0	0
$955$	−1.00582e7	−0.356873
$956$	2.40410e7	0.850760
$957$	0	0
$958$	−3.03081e7	−1.06695
$959$	1.78644e7	0.627250
$960$	0	0
$961$	−2.19408e7	−0.766378
$962$	4.74291e6	0.165237
$963$	0	0
$964$	−1.18012e7	−0.409011
$965$	−2.31213e7	−0.799272
$966$	0	0
$967$	119556.	0.00411156	0.00205578	−	0.999998i	$-0.499346\pi$
$967$	119556.	0.00411156	0.00205578	+	0.999998i	$0.499346\pi$
$968$	2.60673e6	0.0894144
$969$	0	0
$970$	9.17142e6	0.312973
$971$	2.54449e7	0.866070	0.433035	−	0.901377i	$-0.357443\pi$
$971$	2.54449e7	0.866070	0.433035	+	0.901377i	$0.357443\pi$
$972$	0	0
$973$	3.79287e7	1.28436
$974$	2.09762e7	0.708484
$975$	0	0
$976$	−1.09157e6	−0.0366799
$977$	5.11092e7	1.71302	0.856511	−	0.516129i	$-0.172627\pi$
$977$	5.11092e7	1.71302	0.856511	+	0.516129i	$0.172627\pi$
$978$	0	0
$979$	2.27447e6	0.0758445
$980$	−2.34097e7	−0.778630
$981$	0	0
$982$	2.96379e7	0.980772
$983$	−4.84676e6	−0.159981	−0.0799903	−	0.996796i	$-0.525489\pi$
$983$	−4.84676e6	−0.159981	−0.0799903	+	0.996796i	$0.525489\pi$
$984$	0	0
$985$	1.95213e7	0.641089
$986$	−5.34757e6	−0.175172
$987$	0	0
$988$	−1.28526e6	−0.0418887
$989$	−5.56708e7	−1.80982
$990$	0	0
$991$	−2.77987e7	−0.899167	−0.449584	−	0.893238i	$-0.648428\pi$
$991$	−2.77987e7	−0.899167	−0.449584	+	0.893238i	$0.648428\pi$
$992$	−1.51918e7	−0.490151
$993$	0	0
$994$	1.84774e7	0.593163
$995$	−7.26791e6	−0.232730
$996$	0	0
$997$	−3.06185e6	−0.0975541	−0.0487771	−	0.998810i	$-0.515532\pi$
$997$	−3.06185e6	−0.0975541	−0.0487771	+	0.998810i	$0.515532\pi$
$998$	1.11069e7	0.352995
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.6.a.e.1.9		23
3.2	odd	2		891.6.a.f.1.15		23
9.2	odd	6		99.6.e.a.67.9	yes	46
9.4	even	3		297.6.e.a.100.15		46
9.5	odd	6		99.6.e.a.34.9	✓	46
9.7	even	3		297.6.e.a.199.15		46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.6.e.a.34.9	✓	46	9.5	odd	6
99.6.e.a.67.9	yes	46	9.2	odd	6
297.6.e.a.100.15		46	9.4	even	3
297.6.e.a.199.15		46	9.7	even	3
891.6.a.e.1.9		23	1.1	even	1	trivial
891.6.a.f.1.15		23	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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q-3.39140 q^{2} -20.4984 q^{4} +25.0381 q^{5} -249.837 q^{7} +178.043 q^{8} +O(q^{10})\)	\(q-3.39140 q^{2} -20.4984 q^{4} +25.0381 q^{5} -249.837 q^{7} +178.043 q^{8} -84.9142 q^{10} +121.000 q^{11} +985.621 q^{13} +847.297 q^{14} +52.1340 q^{16} -1349.42 q^{17} +63.6150 q^{19} -513.241 q^{20} -410.359 q^{22} -3320.76 q^{23} -2498.09 q^{25} -3342.63 q^{26} +5121.26 q^{28} -1168.51 q^{29} +2586.19 q^{31} -5874.19 q^{32} +4576.41 q^{34} -6255.44 q^{35} -1418.91 q^{37} -215.744 q^{38} +4457.86 q^{40} +16642.1 q^{41} +16764.4 q^{43} -2480.31 q^{44} +11262.0 q^{46} +8033.35 q^{47} +45611.5 q^{49} +8472.03 q^{50} -20203.7 q^{52} +2442.69 q^{53} +3029.61 q^{55} -44481.7 q^{56} +3962.88 q^{58} +16759.8 q^{59} -20937.8 q^{61} -8770.82 q^{62} +18253.4 q^{64} +24678.1 q^{65} +31658.9 q^{67} +27660.9 q^{68} +21214.7 q^{70} +21807.4 q^{71} -36722.9 q^{73} +4812.10 q^{74} -1304.01 q^{76} -30230.3 q^{77} -44563.9 q^{79} +1305.34 q^{80} -56439.9 q^{82} +86704.0 q^{83} -33786.8 q^{85} -56854.9 q^{86} +21543.2 q^{88} +18797.3 q^{89} -246245. q^{91} +68070.4 q^{92} -27244.3 q^{94} +1592.80 q^{95} -108008. q^{97} -154687. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 320 q^{4} - 36 q^{5} - 167 q^{7} - 213 q^{8}+O(q^{10}) \) 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

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