LMFDB - Embedded newform 891.6.a.f.1.15

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			1.15
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	$-0.403093\pi$
$2$	3.39140	0.599520	0.299760	+	0.954015i	$0.403093\pi$
$3$	0	0
$3$	0	0
$4$	−20.4984	−0.640575
$5$	−25.0381	−0.447895	−0.223948	−	0.974601i	$-0.571894\pi$
$5$	−25.0381	−0.447895	−0.223948	+	0.974601i	$0.571894\pi$
$6$	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	$-0.308401\pi$
$17$	1349.42	1.13246	0.566231	+	0.824247i	$0.308401\pi$
$18$	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	$-0.272893\pi$
$23$	3320.76	1.30894	0.654468	+	0.756090i	$0.272893\pi$
$24$	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	$-0.458822\pi$
$29$	1168.51	0.258010	0.129005	+	0.991644i	$0.458822\pi$
$30$	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	$-0.781279\pi$
$41$	−16642.1	−1.54614	−0.773068	+	0.634323i	$0.781279\pi$
$42$	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	$-0.585448\pi$
$47$	−8033.35	−0.530459	−0.265230	+	0.964185i	$0.585448\pi$
$48$	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	$-0.519022\pi$
$53$	−2442.69	−0.119448	−0.0597240	+	0.998215i	$0.519022\pi$
$54$	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	$-0.601470\pi$
$59$	−16759.8	−0.626814	−0.313407	+	0.949619i	$0.601470\pi$
$60$	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	$-0.582636\pi$
$71$	−21807.4	−0.513403	−0.256701	+	0.966491i	$0.582636\pi$
$72$	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	$-0.742715\pi$
$83$	−86704.0	−1.38148	−0.690739	+	0.723104i	$0.742715\pi$
$84$	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	$-0.540141\pi$
$89$	−18797.3	−0.251548	−0.125774	+	0.992059i	$0.540141\pi$
$90$	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	$-0.358376\pi$
$101$	88246.3	0.860782	0.430391	+	0.902643i	$0.358376\pi$
$102$	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	$-0.394449\pi$
$107$	77110.5	0.651110	0.325555	+	0.945523i	$0.394449\pi$
$108$	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	$-0.763164\pi$
$113$	−199732.	−1.47147	−0.735736	+	0.677268i	$0.763164\pi$
$114$	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	$-0.472483\pi$
$131$	33916.6	0.172677	0.0863384	+	0.996266i	$0.472483\pi$
$132$	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	$-0.447966\pi$
$137$	71504.1	0.325484	0.162742	+	0.986669i	$0.447966\pi$
$138$	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	$-0.168708\pi$
$149$	467634.	1.72560	0.862800	+	0.505545i	$0.168708\pi$
$150$	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	$-0.481193\pi$
$167$	42563.2	0.118098	0.0590491	+	0.998255i	$0.481193\pi$
$168$	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	$-0.162250\pi$
$173$	687226.	1.74576	0.872879	+	0.487937i	$0.162250\pi$
$174$	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	$-0.330254\pi$
$179$	435842.	1.01671	0.508355	+	0.861148i	$0.330254\pi$
$180$	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	$-0.369570\pi$
$191$	401717.	0.796777	0.398389	+	0.917217i	$0.369570\pi$
$192$	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	$-0.753878\pi$
$197$	−779664.	−1.43134	−0.715669	+	0.698440i	$0.753878\pi$
$198$	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	$-0.557914\pi$
$227$	−280951.	−0.361881	−0.180940	+	0.983494i	$0.557914\pi$
$228$	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	$-0.496231\pi$
$233$	19623.7	0.0236805	0.0118402	+	0.999930i	$0.496231\pi$
$234$	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	$-0.268832\pi$
$239$	1.17282e6	1.32812	0.664060	+	0.747680i	$0.268832\pi$
$240$	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	$-0.482001\pi$
$251$	112822.	0.113034	0.0565168	+	0.998402i	$0.482001\pi$
$252$	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	$-0.531041\pi$
$257$	−206184.	−0.194725	−0.0973624	+	0.995249i	$0.531041\pi$
$258$	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	$-0.865351\pi$
$263$	−2.04572e6	−1.82371	−0.911857	+	0.410509i	$0.865351\pi$
$264$	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	$-0.690330\pi$
$269$	−1.33620e6	−1.12588	−0.562940	+	0.826498i	$0.690330\pi$
$270$	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	$-0.382300\pi$
$281$	956707.	0.722792	0.361396	+	0.932413i	$0.382300\pi$
$282$	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	$-0.711675\pi$
$293$	−1.81353e6	−1.23411	−0.617057	+	0.786918i	$0.711675\pi$
$294$	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	$-0.255155\pi$
$311$	2.37283e6	1.39112	0.695562	+	0.718466i	$0.255155\pi$
$312$	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	$-0.403077\pi$
$317$	1.07281e6	0.599620	0.299810	+	0.953999i	$0.403077\pi$
$318$	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	$-0.834113\pi$
$347$	−3.89042e6	−1.73450	−0.867248	+	0.497877i	$0.834113\pi$
$348$	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	$-0.689942\pi$
$353$	−2.63119e6	−1.12387	−0.561934	+	0.827182i	$0.689942\pi$
$354$	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	$-0.746914\pi$
$359$	−3.41979e6	−1.40044	−0.700218	+	0.713929i	$0.746914\pi$
$360$	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	$-0.274934\pi$
$383$	3.72973e6	1.29921	0.649606	+	0.760271i	$0.274934\pi$
$384$	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	$-0.264513\pi$
$389$	4.02398e6	1.34828	0.674142	+	0.738602i	$0.264513\pi$
$390$	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	$-0.652322\pi$
$401$	−2.96551e6	−0.920956	−0.460478	+	0.887671i	$0.652322\pi$
$402$	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	$-0.575583\pi$
$419$	−1.69065e6	−0.470455	−0.235227	+	0.971940i	$0.575583\pi$
$420$	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	$-0.308455\pi$
$431$	4.36626e6	1.13218	0.566092	+	0.824342i	$0.308455\pi$
$432$	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	$-0.529271\pi$
$443$	−758611.	−0.183658	−0.0918290	+	0.995775i	$0.529271\pi$
$444$	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	$-0.495762\pi$
$449$	113738.	0.0266251	0.0133125	+	0.999911i	$0.495762\pi$
$450$	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	$-0.454542\pi$
$461$	1.29888e6	0.284653	0.142327	+	0.989820i	$0.454542\pi$
$462$	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	$-0.456192\pi$
$467$	1.29315e6	0.274383	0.137192	+	0.990545i	$0.456192\pi$
$468$	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	$-0.849182\pi$
$479$	−8.93674e6	−1.77967	−0.889837	+	0.456278i	$0.849182\pi$
$480$	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	$-0.195103\pi$
$491$	8.73913e6	1.63593	0.817964	+	0.575269i	$0.195103\pi$
$492$	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	$-0.778775\pi$
$503$	−8.71649e6	−1.53611	−0.768054	+	0.640385i	$0.778775\pi$
$504$	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	$-0.574963\pi$
$509$	−2.72773e6	−0.466668	−0.233334	+	0.972397i	$0.574963\pi$
$510$	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	$-0.426746\pi$
$521$	2.82659e6	0.456214	0.228107	+	0.973636i	$0.426746\pi$
$522$	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	$-0.515730\pi$
$557$	−723407.	−0.0987972	−0.0493986	+	0.998779i	$0.515730\pi$
$558$	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	$-0.352384\pi$
$563$	6.72827e6	0.894607	0.447304	+	0.894382i	$0.352384\pi$
$564$	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	$-0.284559\pi$
$569$	9.67408e6	1.25265	0.626324	+	0.779563i	$0.284559\pi$
$570$	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	$-0.392348\pi$
$587$	5.53973e6	0.663580	0.331790	+	0.943353i	$0.392348\pi$
$588$	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	$-0.543181\pi$
$593$	−2.31621e6	−0.270484	−0.135242	+	0.990813i	$0.543181\pi$
$594$	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	$-0.690834\pi$
$599$	−9.90987e6	−1.12850	−0.564249	+	0.825605i	$0.690834\pi$
$600$	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	$-0.691775\pi$
$617$	−1.07173e7	−1.13338	−0.566688	+	0.823933i	$0.691775\pi$
$618$	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	$-0.268573\pi$
$641$	1.38286e7	1.32934	0.664668	+	0.747139i	$0.268573\pi$
$642$	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	$-0.541250\pi$
$647$	−2.75199e6	−0.258455	−0.129228	+	0.991615i	$0.541250\pi$
$648$	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	$-0.541579\pi$
$653$	−2.83857e6	−0.260505	−0.130253	+	0.991481i	$0.541579\pi$
$654$	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	$-0.663689\pi$
$659$	−1.09673e7	−0.983756	−0.491878	+	0.870664i	$0.663689\pi$
$660$	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	$-0.570682\pi$
$677$	−5.25274e6	−0.440467	−0.220234	+	0.975447i	$0.570682\pi$
$678$	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	$-0.547407\pi$
$683$	−3.61797e6	−0.296765	−0.148383	+	0.988930i	$0.547407\pi$
$684$	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	$-0.503905\pi$
$701$	−319236.	−0.0245368	−0.0122684	+	0.999925i	$0.503905\pi$
$702$	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	$-0.506815\pi$
$719$	−593552.	−0.0428190	−0.0214095	+	0.999771i	$0.506815\pi$
$720$	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	$-0.699407\pi$
$743$	−1.76443e7	−1.17256	−0.586278	+	0.810110i	$0.699407\pi$
$744$	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	$-0.664245\pi$
$761$	−1.57648e7	−0.986794	−0.493397	+	0.869804i	$0.664245\pi$
$762$	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	$-0.419105\pi$
$773$	8.35346e6	0.502826	0.251413	+	0.967880i	$0.419105\pi$
$774$	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	$-0.264865\pi$
$797$	2.41491e7	1.34665	0.673326	+	0.739346i	$0.264865\pi$
$798$	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	$-0.356113\pi$
$809$	1.62622e7	0.873590	0.436795	+	0.899561i	$0.356113\pi$
$810$	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	$-0.469521\pi$
$821$	3.69300e6	0.191215	0.0956075	+	0.995419i	$0.469521\pi$
$822$	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	$-0.419999\pi$
$827$	9.78270e6	0.497388	0.248694	+	0.968582i	$0.419999\pi$
$828$	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	$-0.257435\pi$
$839$	2.81537e7	1.38080	0.690399	+	0.723429i	$0.257435\pi$
$840$	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	$-0.179858\pi$
$857$	3.63175e7	1.68913	0.844567	+	0.535450i	$0.179858\pi$
$858$	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	$-0.344038\pi$
$863$	2.05924e7	0.941198	0.470599	+	0.882347i	$0.344038\pi$
$864$	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	$-0.384226\pi$
$881$	1.63913e7	0.711496	0.355748	+	0.934582i	$0.384226\pi$
$882$	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	$-0.459791\pi$
$887$	5.90413e6	0.251969	0.125984	+	0.992032i	$0.459791\pi$
$888$	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	$-0.181071\pi$
$911$	4.22091e7	1.68504	0.842520	+	0.538665i	$0.181071\pi$
$912$	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	$-0.741698\pi$
$929$	−3.62182e7	−1.37685	−0.688426	+	0.725306i	$0.741698\pi$
$930$	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	$-0.703982\pi$
$941$	−3.24791e7	−1.19572	−0.597860	+	0.801600i	$0.703982\pi$
$942$	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	$-0.503094\pi$
$947$	−536431.	−0.0194374	−0.00971872	+	0.999953i	$0.503094\pi$
$948$	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	$-0.375841\pi$
$953$	2.13217e7	0.760484	0.380242	+	0.924887i	$0.375841\pi$
$954$	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	$-0.642557\pi$
$971$	−2.54449e7	−0.866070	−0.433035	+	0.901377i	$0.642557\pi$
$972$	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	$-0.827373\pi$
$977$	−5.11092e7	−1.71302	−0.856511	+	0.516129i	$0.827373\pi$
$978$	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	$-0.474511\pi$
$983$	4.84676e6	0.159981	0.0799903	+	0.996796i	$0.474511\pi$
$984$	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.f.1.15		23
3.2	odd	2		891.6.a.e.1.9		23
9.2	odd	6		297.6.e.a.199.15		46
9.4	even	3		99.6.e.a.34.9	✓	46
9.5	odd	6		297.6.e.a.100.15		46
9.7	even	3		99.6.e.a.67.9	yes	46

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

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

Introduction

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