LMFDB - Embedded newform 1089.6.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,6,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1089.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:	$174.657979776$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.62772 q^{2} -18.8397 q^{4} -57.7228 q^{5} -251.081 q^{7} -184.432 q^{8} +O(q^{10})$	$q+3.62772 q^{2} -18.8397 q^{4} -57.7228 q^{5} -251.081 q^{7} -184.432 q^{8} -209.402 q^{10} +277.549 q^{13} -910.853 q^{14} -66.1983 q^{16} +704.489 q^{17} +2861.18 q^{19} +1087.48 q^{20} +1066.85 q^{23} +206.923 q^{25} +1006.87 q^{26} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} +5661.67 q^{32} +2555.69 q^{34} +14493.1 q^{35} -9042.34 q^{37} +10379.6 q^{38} +10645.9 q^{40} +18219.0 q^{41} +4054.54 q^{43} +3870.24 q^{46} -20750.8 q^{47} +46234.9 q^{49} +750.659 q^{50} -5228.92 q^{52} +26485.9 q^{53} +46307.4 q^{56} -14283.9 q^{58} -4293.12 q^{59} +6831.76 q^{61} -2337.52 q^{62} +22657.3 q^{64} -16020.9 q^{65} -56749.5 q^{67} -13272.3 q^{68} +52577.0 q^{70} -3187.09 q^{71} +7397.14 q^{73} -32803.1 q^{74} -53903.7 q^{76} -24393.7 q^{79} +3821.15 q^{80} +66093.5 q^{82} +102795. q^{83} -40665.1 q^{85} +14708.7 q^{86} -49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} -75278.1 q^{94} -165156. q^{95} -92279.5 q^{97} +167727. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10}) $ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8	$ 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100}) $ 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 - 212 * q^10 + 130 * q^13 + 74 * q^14 + 241 * q^16 - 728 * q^17 + 828 * q^19 + 1072 * q^20 + 238 * q^23 - 2918 * q^25 - 376 * q^26 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 + 1391 * q^32 - 10870 * q^34 + 14464 * q^35 - 1908 * q^37 - 8676 * q^38 + 10584 * q^40 + 36484 * q^41 - 9768 * q^43 - 3898 * q^46 - 43742 * q^47 + 40470 * q^49 - 28537 * q^50 - 13468 * q^52 + 12174 * q^53 + 69786 * q^56 + 29142 * q^58 + 2788 * q^59 + 25302 * q^61 - 94520 * q^62 - 27199 * q^64 - 15980 * q^65 - 40520 * q^67 - 93262 * q^68 + 52304 * q^70 - 31386 * q^71 + 46780 * q^73 + 34062 * q^74 - 167436 * q^76 + 16850 * q^79 + 3736 * q^80 + 237278 * q^82 + 79440 * q^83 - 40268 * q^85 - 114840 * q^86 + 54204 * q^89 - 85192 * q^91 - 66382 * q^92 - 290758 * q^94 - 164592 * q^95 - 241568 * q^97 + 113697 * 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 $

$2$	3.62772	0.641296	0.320648	−	0.947198i	$-0.396099\pi$
$2$	3.62772	0.641296	0.320648	+	0.947198i	$0.396099\pi$
$3$	0	0
$3$	0	0
$4$	−18.8397	−0.588739
$5$	−57.7228	−1.03258	−0.516289	−	0.856415i	$-0.672687\pi$
$5$	−57.7228	−1.03258	−0.516289	+	0.856415i	$0.672687\pi$
$6$	0	0
$7$	−251.081	−1.93673	−0.968366	−	0.249534i	$-0.919722\pi$
$7$	−251.081	−1.93673	−0.968366	+	0.249534i	$0.919722\pi$
$8$	−184.432	−1.01885
$9$	0	0
$10$	−209.402	−0.662188
$11$	0	0
$11$	0	0
$12$	0	0
$13$	277.549	0.455492	0.227746	−	0.973721i	$-0.426864\pi$
$13$	277.549	0.455492	0.227746	+	0.973721i	$0.426864\pi$
$14$	−910.853	−1.24202
$15$	0	0
$16$	−66.1983	−0.0646468
$17$	704.489	0.591224	0.295612	−	0.955308i	$-0.404477\pi$
$17$	704.489	0.591224	0.295612	+	0.955308i	$0.404477\pi$
$18$	0	0
$19$	2861.18	1.81828	0.909142	−	0.416486i	$-0.136739\pi$
$19$	2861.18	1.81828	0.909142	+	0.416486i	$0.136739\pi$
$20$	1087.48	0.607919
$21$	0	0
$22$	0	0
$23$	1066.85	0.420518	0.210259	−	0.977646i	$-0.432569\pi$
$23$	1066.85	0.420518	0.210259	+	0.977646i	$0.432569\pi$
$24$	0	0
$25$	206.923	0.0662154
$26$	1006.87	0.292105
$27$	0	0
$28$	4730.29	1.14023
$29$	−3937.44	−0.869399	−0.434700	−	0.900575i	$-0.643145\pi$
$29$	−3937.44	−0.869399	−0.434700	+	0.900575i	$0.643145\pi$
$30$	0	0
$31$	−644.350	−0.120425	−0.0602126	−	0.998186i	$-0.519178\pi$
$31$	−644.350	−0.120425	−0.0602126	+	0.998186i	$0.519178\pi$
$32$	5661.67	0.977395
$33$	0	0
$34$	2555.69	0.379149
$35$	14493.1	1.99982
$36$	0	0
$37$	−9042.34	−1.08587	−0.542934	−	0.839776i	$-0.682686\pi$
$37$	−9042.34	−1.08587	−0.542934	+	0.839776i	$0.682686\pi$
$38$	10379.6	1.16606
$39$	0	0
$40$	10645.9	1.05204
$41$	18219.0	1.69264	0.846322	−	0.532672i	$-0.178812\pi$
$41$	18219.0	1.69264	0.846322	+	0.532672i	$0.178812\pi$
$42$	0	0
$43$	4054.54	0.334403	0.167202	−	0.985923i	$-0.446527\pi$
$43$	4054.54	0.334403	0.167202	+	0.985923i	$0.446527\pi$
$44$	0	0
$45$	0	0
$46$	3870.24	0.269677
$47$	−20750.8	−1.37022	−0.685110	−	0.728439i	$-0.740246\pi$
$47$	−20750.8	−1.37022	−0.685110	+	0.728439i	$0.740246\pi$
$48$	0	0
$49$	46234.9	2.75093
$50$	750.659	0.0424637
$51$	0	0
$52$	−5228.92	−0.268166
$53$	26485.9	1.29517	0.647583	−	0.761995i	$-0.275780\pi$
$53$	26485.9	1.29517	0.647583	+	0.761995i	$0.275780\pi$
$54$	0	0
$55$	0	0
$56$	46307.4	1.97324
$57$	0	0
$58$	−14283.9	−0.557542
$59$	−4293.12	−0.160562	−0.0802810	−	0.996772i	$-0.525582\pi$
$59$	−4293.12	−0.160562	−0.0802810	+	0.996772i	$0.525582\pi$
$60$	0	0
$61$	6831.76	0.235076	0.117538	−	0.993068i	$-0.462500\pi$
$61$	6831.76	0.235076	0.117538	+	0.993068i	$0.462500\pi$
$62$	−2337.52	−0.0772282
$63$	0	0
$64$	22657.3	0.691446
$65$	−16020.9	−0.470331
$66$	0	0
$67$	−56749.5	−1.54445	−0.772227	−	0.635347i	$-0.780857\pi$
$67$	−56749.5	−1.54445	−0.772227	+	0.635347i	$0.780857\pi$
$68$	−13272.3	−0.348077
$69$	0	0
$70$	52577.0	1.28248
$71$	−3187.09	−0.0750323	−0.0375161	−	0.999296i	$-0.511945\pi$
$71$	−3187.09	−0.0750323	−0.0375161	+	0.999296i	$0.511945\pi$
$72$	0	0
$73$	7397.14	0.162464	0.0812319	−	0.996695i	$-0.474115\pi$
$73$	7397.14	0.162464	0.0812319	+	0.996695i	$0.474115\pi$
$74$	−32803.1	−0.696362
$75$	0	0
$76$	−53903.7	−1.07050
$77$	0	0
$78$	0	0
$79$	−24393.7	−0.439754	−0.219877	−	0.975528i	$-0.570566\pi$
$79$	−24393.7	−0.439754	−0.219877	+	0.975528i	$0.570566\pi$
$80$	3821.15	0.0667528
$81$	0	0
$82$	66093.5	1.08549
$83$	102795.	1.63786	0.818932	−	0.573890i	$-0.194566\pi$
$83$	102795.	1.63786	0.818932	+	0.573890i	$0.194566\pi$
$84$	0	0
$85$	−40665.1	−0.610484
$86$	14708.7	0.214451
$87$	0	0
$88$	0	0
$89$	−49599.4	−0.663745	−0.331873	−	0.943324i	$-0.607680\pi$
$89$	−49599.4	−0.663745	−0.331873	+	0.943324i	$0.607680\pi$
$90$	0	0
$91$	−69687.4	−0.882166
$92$	−20099.1	−0.247576
$93$	0	0
$94$	−75278.1	−0.878717
$95$	−165156.	−1.87752
$96$	0	0
$97$	−92279.5	−0.995808	−0.497904	−	0.867232i	$-0.665897\pi$
$97$	−92279.5	−0.995808	−0.497904	+	0.867232i	$0.665897\pi$
$98$	167727.	1.76416
$99$	0	0
$100$	−3898.36	−0.0389836
$101$	−35546.1	−0.346728	−0.173364	−	0.984858i	$-0.555464\pi$
$101$	−35546.1	−0.346728	−0.173364	+	0.984858i	$0.555464\pi$
$102$	0	0
$103$	−59876.8	−0.556116	−0.278058	−	0.960564i	$-0.589691\pi$
$103$	−59876.8	−0.556116	−0.278058	+	0.960564i	$0.589691\pi$
$104$	−51188.9	−0.464079
$105$	0	0
$106$	96083.5	0.830586
$107$	89253.8	0.753646	0.376823	−	0.926285i	$-0.377017\pi$
$107$	89253.8	0.753646	0.376823	+	0.926285i	$0.377017\pi$
$108$	0	0
$109$	−22796.0	−0.183777	−0.0918887	−	0.995769i	$-0.529290\pi$
$109$	−22796.0	−0.183777	−0.0918887	+	0.995769i	$0.529290\pi$
$110$	0	0
$111$	0	0
$112$	16621.2	0.125203
$113$	166064.	1.22343	0.611717	−	0.791077i	$-0.290479\pi$
$113$	166064.	1.22343	0.611717	+	0.791077i	$0.290479\pi$
$114$	0	0
$115$	−61581.7	−0.434218
$116$	74180.1	0.511850
$117$	0	0
$118$	−15574.2	−0.102968
$119$	−176884.	−1.14504
$120$	0	0
$121$	0	0
$122$	24783.7	0.150753
$123$	0	0
$124$	12139.3	0.0708991
$125$	168440.	0.964205
$126$	0	0
$127$	−159190.	−0.875802	−0.437901	−	0.899023i	$-0.644278\pi$
$127$	−159190.	−0.875802	−0.437901	+	0.899023i	$0.644278\pi$
$128$	−98979.2	−0.533973
$129$	0	0
$130$	−58119.3	−0.301621
$131$	232174.	1.18205	0.591025	−	0.806654i	$-0.298724\pi$
$131$	232174.	1.18205	0.591025	+	0.806654i	$0.298724\pi$
$132$	0	0
$133$	−718390.	−3.52153
$134$	−205871.	−0.990452
$135$	0	0
$136$	−129930.	−0.602369
$137$	−68205.4	−0.310468	−0.155234	−	0.987878i	$-0.549613\pi$
$137$	−68205.4	−0.310468	−0.155234	+	0.987878i	$0.549613\pi$
$138$	0	0
$139$	−298162.	−1.30893	−0.654464	−	0.756093i	$-0.727106\pi$
$139$	−298162.	−1.30893	−0.654464	+	0.756093i	$0.727106\pi$
$140$	−273046.	−1.17738
$141$	0	0
$142$	−11561.9	−0.0481179
$143$	0	0
$144$	0	0
$145$	227280.	0.897722
$146$	26834.7	0.104187
$147$	0	0
$148$	170355.	0.639293
$149$	83131.5	0.306761	0.153380	−	0.988167i	$-0.450984\pi$
$149$	83131.5	0.306761	0.153380	+	0.988167i	$0.450984\pi$
$150$	0	0
$151$	−167598.	−0.598171	−0.299085	−	0.954226i	$-0.596682\pi$
$151$	−167598.	−0.598171	−0.299085	+	0.954226i	$0.596682\pi$
$152$	−527694.	−1.85256
$153$	0	0
$154$	0	0
$155$	37193.7	0.124348
$156$	0	0
$157$	−63259.7	−0.204823	−0.102411	−	0.994742i	$-0.532656\pi$
$157$	−63259.7	−0.204823	−0.102411	+	0.994742i	$0.532656\pi$
$158$	−88493.4	−0.282012
$159$	0	0
$160$	−326808.	−1.00924
$161$	−267867.	−0.814431
$162$	0	0
$163$	237727.	0.700825	0.350412	−	0.936596i	$-0.386041\pi$
$163$	237727.	0.700825	0.350412	+	0.936596i	$0.386041\pi$
$164$	−343240.	−0.996526
$165$	0	0
$166$	372912.	1.05036
$167$	487880.	1.35370	0.676849	−	0.736122i	$-0.263345\pi$
$167$	487880.	1.35370	0.676849	+	0.736122i	$0.263345\pi$
$168$	0	0
$169$	−294260.	−0.792527
$170$	−147521.	−0.391501
$171$	0	0
$172$	−76386.1	−0.196876
$173$	325942.	0.827991	0.413996	−	0.910279i	$-0.364133\pi$
$173$	325942.	0.827991	0.413996	+	0.910279i	$0.364133\pi$
$174$	0	0
$175$	−51954.6	−0.128242
$176$	0	0
$177$	0	0
$178$	−179933.	−0.425657
$179$	−462906.	−1.07984	−0.539921	−	0.841716i	$-0.681546\pi$
$179$	−462906.	−1.07984	−0.539921	+	0.841716i	$0.681546\pi$
$180$	0	0
$181$	23901.3	0.0542281	0.0271141	−	0.999632i	$-0.491368\pi$
$181$	23901.3	0.0542281	0.0271141	+	0.999632i	$0.491368\pi$
$182$	−252806.	−0.565730
$183$	0	0
$184$	−196762.	−0.428446
$185$	521950.	1.12124
$186$	0	0
$187$	0	0
$188$	390938.	0.806703
$189$	0	0
$190$	−599138.	−1.20405
$191$	565986.	1.12259	0.561297	−	0.827615i	$-0.310303\pi$
$191$	565986.	1.12259	0.561297	+	0.827615i	$0.310303\pi$
$192$	0	0
$193$	−91762.3	−0.177325	−0.0886627	−	0.996062i	$-0.528259\pi$
$193$	−91762.3	−0.177325	−0.0886627	+	0.996062i	$0.528259\pi$
$194$	−334764.	−0.638608
$195$	0	0
$196$	−871049.	−1.61958
$197$	485247.	0.890836	0.445418	−	0.895323i	$-0.353055\pi$
$197$	485247.	0.890836	0.445418	+	0.895323i	$0.353055\pi$
$198$	0	0
$199$	692632.	1.23985	0.619926	−	0.784660i	$-0.287162\pi$
$199$	692632.	1.23985	0.619926	+	0.784660i	$0.287162\pi$
$200$	−38163.2	−0.0674637
$201$	0	0
$202$	−128951.	−0.222355
$203$	988619.	1.68379
$204$	0	0
$205$	−1.05165e6	−1.74778
$206$	−217216.	−0.356635
$207$	0	0
$208$	−18373.3	−0.0294461
$209$	0	0
$210$	0	0
$211$	441020.	0.681949	0.340975	−	0.940073i	$-0.389243\pi$
$211$	441020.	0.681949	0.340975	+	0.940073i	$0.389243\pi$
$212$	−498986.	−0.762516
$213$	0	0
$214$	323788.	0.483310
$215$	−234039.	−0.345297
$216$	0	0
$217$	161784.	0.233231
$218$	−82697.4	−0.117856
$219$	0	0
$220$	0	0
$221$	195530.	0.269298
$222$	0	0
$223$	1.13133e6	1.52345	0.761726	−	0.647899i	$-0.224352\pi$
$223$	1.13133e6	1.52345	0.761726	+	0.647899i	$0.224352\pi$
$224$	−1.42154e6	−1.89295
$225$	0	0
$226$	602435.	0.784583
$227$	−820354.	−1.05666	−0.528332	−	0.849038i	$-0.677182\pi$
$227$	−820354.	−1.05666	−0.528332	+	0.849038i	$0.677182\pi$
$228$	0	0
$229$	−1.00301e6	−1.26392	−0.631958	−	0.775003i	$-0.717748\pi$
$229$	−1.00301e6	−1.26392	−0.631958	+	0.775003i	$0.717748\pi$
$230$	−223401.	−0.278462
$231$	0	0
$232$	726191.	0.885790
$233$	734.569	0.000886427 0	0.000443214	−	1.00000i	$-0.499859\pi$
$233$	734.569	0.000886427 0	0.000443214		1.00000i	$0.499859\pi$
$234$	0	0
$235$	1.19780e6	1.41486
$236$	80880.9	0.0945291
$237$	0	0
$238$	−641685.	−0.734311
$239$	−1.06403e6	−1.20492	−0.602460	−	0.798149i	$-0.705813\pi$
$239$	−1.06403e6	−1.20492	−0.602460	+	0.798149i	$0.705813\pi$
$240$	0	0
$241$	661636.	0.733798	0.366899	−	0.930261i	$-0.380419\pi$
$241$	661636.	0.733798	0.366899	+	0.930261i	$0.380419\pi$
$242$	0	0
$243$	0	0
$244$	−128708.	−0.138398
$245$	−2.66881e6	−2.84055
$246$	0	0
$247$	794118.	0.828214
$248$	118839.	0.122696
$249$	0	0
$250$	611051.	0.618341
$251$	−1.34864e6	−1.35118	−0.675590	−	0.737277i	$-0.736111\pi$
$251$	−1.34864e6	−1.35118	−0.675590	+	0.737277i	$0.736111\pi$
$252$	0	0
$253$	0	0
$254$	−577496.	−0.561648
$255$	0	0
$256$	−1.08410e6	−1.03388
$257$	1.65015e6	1.55844	0.779220	−	0.626751i	$-0.215616\pi$
$257$	1.65015e6	1.55844	0.779220	+	0.626751i	$0.215616\pi$
$258$	0	0
$259$	2.27036e6	2.10303
$260$	301828.	0.276902
$261$	0	0
$262$	842262.	0.758044
$263$	−868163.	−0.773948	−0.386974	−	0.922091i	$-0.626480\pi$
$263$	−868163.	−0.773948	−0.386974	+	0.922091i	$0.626480\pi$
$264$	0	0
$265$	−1.52884e6	−1.33736
$266$	−2.60612e6	−2.25834
$267$	0	0
$268$	1.06914e6	0.909280
$269$	−271547.	−0.228804	−0.114402	−	0.993435i	$-0.536495\pi$
$269$	−271547.	−0.228804	−0.114402	+	0.993435i	$0.536495\pi$
$270$	0	0
$271$	752770.	0.622643	0.311322	−	0.950305i	$-0.399228\pi$
$271$	752770.	0.622643	0.311322	+	0.950305i	$0.399228\pi$
$272$	−46635.9	−0.0382207
$273$	0	0
$274$	−247430.	−0.199102
$275$	0	0
$276$	0	0
$277$	−811445.	−0.635418	−0.317709	−	0.948188i	$-0.602914\pi$
$277$	−811445.	−0.635418	−0.317709	+	0.948188i	$0.602914\pi$
$278$	−1.08165e6	−0.839411
$279$	0	0
$280$	−2.67300e6	−2.03753
$281$	−1.72395e6	−1.30244	−0.651221	−	0.758888i	$-0.725743\pi$
$281$	−1.72395e6	−1.30244	−0.651221	+	0.758888i	$0.725743\pi$
$282$	0	0
$283$	−272979.	−0.202611	−0.101305	−	0.994855i	$-0.532302\pi$
$283$	−272979.	−0.202611	−0.101305	+	0.994855i	$0.532302\pi$
$284$	60043.6	0.0441744
$285$	0	0
$286$	0	0
$287$	−4.57446e6	−3.27820
$288$	0	0
$289$	−923553.	−0.650455
$290$	824509.	0.575706
$291$	0	0
$292$	−139360.	−0.0956488
$293$	−713441.	−0.485500	−0.242750	−	0.970089i	$-0.578049\pi$
$293$	−713441.	−0.485500	−0.242750	+	0.970089i	$0.578049\pi$
$294$	0	0
$295$	247811.	0.165793
$296$	1.66770e6	1.10634
$297$	0	0
$298$	301578.	0.196725
$299$	296104.	0.191543
$300$	0	0
$301$	−1.01802e6	−0.647649
$302$	−607997.	−0.383605
$303$	0	0
$304$	−189405.	−0.117546
$305$	−394348.	−0.242734
$306$	0	0
$307$	2.67734e6	1.62128	0.810640	−	0.585545i	$-0.199119\pi$
$307$	2.67734e6	1.62128	0.810640	+	0.585545i	$0.199119\pi$
$308$	0	0
$309$	0	0
$310$	134928.	0.0797441
$311$	1.28078e6	0.750886	0.375443	−	0.926846i	$-0.377491\pi$
$311$	1.28078e6	0.750886	0.375443	+	0.926846i	$0.377491\pi$
$312$	0	0
$313$	1.36808e6	0.789313	0.394656	−	0.918829i	$-0.370864\pi$
$313$	1.36808e6	0.789313	0.394656	+	0.918829i	$0.370864\pi$
$314$	−229488.	−0.131352
$315$	0	0
$316$	459569.	0.258900
$317$	7686.28	0.00429604	0.00214802	−	0.999998i	$-0.499316\pi$
$317$	7686.28	0.00429604	0.00214802	+	0.999998i	$0.499316\pi$
$318$	0	0
$319$	0	0
$320$	−1.30784e6	−0.713971
$321$	0	0
$322$	−971746.	−0.522292
$323$	2.01567e6	1.07501
$324$	0	0
$325$	57431.3	0.0301606
$326$	862406.	0.449436
$327$	0	0
$328$	−3.36017e6	−1.72455
$329$	5.21014e6	2.65375
$330$	0	0
$331$	958347.	0.480787	0.240393	−	0.970676i	$-0.422724\pi$
$331$	958347.	0.480787	0.240393	+	0.970676i	$0.422724\pi$
$332$	−1.93663e6	−0.964275
$333$	0	0
$334$	1.76989e6	0.868121
$335$	3.27574e6	1.59477
$336$	0	0
$337$	−4.08768e6	−1.96066	−0.980330	−	0.197365i	$-0.936762\pi$
$337$	−4.08768e6	−1.96066	−0.980330	+	0.197365i	$0.936762\pi$
$338$	−1.06749e6	−0.508244
$339$	0	0
$340$	766116.	0.359416
$341$	0	0
$342$	0	0
$343$	−7.38880e6	−3.39108
$344$	−747787.	−0.340707
$345$	0	0
$346$	1.18243e6	0.530988
$347$	84250.2	0.0375619	0.0187809	−	0.999824i	$-0.494021\pi$
$347$	84250.2	0.0375619	0.0187809	+	0.999824i	$0.494021\pi$
$348$	0	0
$349$	−1.11859e6	−0.491597	−0.245798	−	0.969321i	$-0.579050\pi$
$349$	−1.11859e6	−0.491597	−0.245798	+	0.969321i	$0.579050\pi$
$350$	−188477.	−0.0822408
$351$	0	0
$352$	0	0
$353$	3.73570e6	1.59564	0.797820	−	0.602896i	$-0.205987\pi$
$353$	3.73570e6	1.59564	0.797820	+	0.602896i	$0.205987\pi$
$354$	0	0
$355$	183968.	0.0774766
$356$	934436.	0.390773
$357$	0	0
$358$	−1.67929e6	−0.692499
$359$	1.45342e6	0.595189	0.297595	−	0.954692i	$-0.403816\pi$
$359$	1.45342e6	0.595189	0.297595	+	0.954692i	$0.403816\pi$
$360$	0	0
$361$	5.71027e6	2.30616
$362$	86707.1	0.0347763
$363$	0	0
$364$	1.31289e6	0.519366
$365$	−426984.	−0.167756
$366$	0	0
$367$	25363.4	0.00982975	0.00491487	−	0.999988i	$-0.498436\pi$
$367$	25363.4	0.00982975	0.00491487	+	0.999988i	$0.498436\pi$
$368$	−70623.8	−0.0271851
$369$	0	0
$370$	1.89349e6	0.719048
$371$	−6.65013e6	−2.50839
$372$	0	0
$373$	1.72695e6	0.642699	0.321350	−	0.946961i	$-0.395864\pi$
$373$	1.72695e6	0.642699	0.321350	+	0.946961i	$0.395864\pi$
$374$	0	0
$375$	0	0
$376$	3.82711e6	1.39605
$377$	−1.09283e6	−0.396005
$378$	0	0
$379$	−4.29401e6	−1.53555	−0.767777	−	0.640718i	$-0.778637\pi$
$379$	−4.29401e6	−1.53555	−0.767777	+	0.640718i	$0.778637\pi$
$380$	3.11147e6	1.10537
$381$	0	0
$382$	2.05324e6	0.719915
$383$	−1.29297e6	−0.450392	−0.225196	−	0.974313i	$-0.572302\pi$
$383$	−1.29297e6	−0.450392	−0.225196	+	0.974313i	$0.572302\pi$
$384$	0	0
$385$	0	0
$386$	−332888.	−0.113718
$387$	0	0
$388$	1.73851e6	0.586272
$389$	−1.55257e6	−0.520209	−0.260104	−	0.965581i	$-0.583757\pi$
$389$	−1.55257e6	−0.520209	−0.260104	+	0.965581i	$0.583757\pi$
$390$	0	0
$391$	751586.	0.248620
$392$	−8.52719e6	−2.80279
$393$	0	0
$394$	1.76034e6	0.571290
$395$	1.40807e6	0.454080
$396$	0	0
$397$	819569.	0.260981	0.130491	−	0.991450i	$-0.458345\pi$
$397$	819569.	0.260981	0.130491	+	0.991450i	$0.458345\pi$
$398$	2.51268e6	0.795113
$399$	0	0
$400$	−13698.0	−0.00428061
$401$	−1.38956e6	−0.431536	−0.215768	−	0.976445i	$-0.569226\pi$
$401$	−1.38956e6	−0.431536	−0.215768	+	0.976445i	$0.569226\pi$
$402$	0	0
$403$	−178839.	−0.0548528
$404$	669677.	0.204132
$405$	0	0
$406$	3.58643e6	1.07981
$407$	0	0
$408$	0	0
$409$	3.53091e6	1.04371	0.521854	−	0.853035i	$-0.325241\pi$
$409$	3.53091e6	1.04371	0.521854	+	0.853035i	$0.325241\pi$
$410$	−3.81510e6	−1.12085
$411$	0	0
$412$	1.12806e6	0.327407
$413$	1.07792e6	0.310966
$414$	0	0
$415$	−5.93363e6	−1.69122
$416$	1.57139e6	0.445196
$417$	0	0
$418$	0	0
$419$	6.69977e6	1.86434	0.932170	−	0.362021i	$-0.117913\pi$
$419$	6.69977e6	1.86434	0.932170	+	0.362021i	$0.117913\pi$
$420$	0	0
$421$	−5.01124e6	−1.37797	−0.688986	−	0.724775i	$-0.741944\pi$
$421$	−5.01124e6	−1.37797	−0.688986	+	0.724775i	$0.741944\pi$
$422$	1.59990e6	0.437331
$423$	0	0
$424$	−4.88485e6	−1.31958
$425$	145775.	0.0391481
$426$	0	0
$427$	−1.71533e6	−0.455279
$428$	−1.68151e6	−0.443701
$429$	0	0
$430$	−849029.	−0.221438
$431$	−1.14741e6	−0.297526	−0.148763	−	0.988873i	$-0.547529\pi$
$431$	−1.14741e6	−0.297526	−0.148763	+	0.988873i	$0.547529\pi$
$432$	0	0
$433$	−1.13393e6	−0.290649	−0.145324	−	0.989384i	$-0.546423\pi$
$433$	−1.13393e6	−0.290649	−0.145324	+	0.989384i	$0.546423\pi$
$434$	586908.	0.149570
$435$	0	0
$436$	429469.	0.108197
$437$	3.05246e6	0.764622
$438$	0	0
$439$	−7.44692e6	−1.84423	−0.922115	−	0.386915i	$-0.873541\pi$
$439$	−7.44692e6	−1.84423	−0.922115	+	0.386915i	$0.873541\pi$
$440$	0	0
$441$	0	0
$442$	709328.	0.172700
$443$	−6.72521e6	−1.62816	−0.814079	−	0.580754i	$-0.802758\pi$
$443$	−6.72521e6	−1.62816	−0.814079	+	0.580754i	$0.802758\pi$
$444$	0	0
$445$	2.86302e6	0.685368
$446$	4.10416e6	0.976984
$447$	0	0
$448$	−5.68883e6	−1.33915
$449$	−2.17793e6	−0.509834	−0.254917	−	0.966963i	$-0.582048\pi$
$449$	−2.17793e6	−0.509834	−0.254917	+	0.966963i	$0.582048\pi$
$450$	0	0
$451$	0	0
$452$	−3.12860e6	−0.720284
$453$	0	0
$454$	−2.97601e6	−0.677634
$455$	4.02255e6	0.910905
$456$	0	0
$457$	3.72380e6	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$457$	3.72380e6	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$458$	−3.63865e6	−0.810544
$459$	0	0
$460$	1.16018e6	0.255641
$461$	4.74202e6	1.03923	0.519615	−	0.854401i	$-0.326076\pi$
$461$	4.74202e6	1.03923	0.519615	+	0.854401i	$0.326076\pi$
$462$	0	0
$463$	−4.82803e6	−1.04669	−0.523344	−	0.852121i	$-0.675316\pi$
$463$	−4.82803e6	−1.04669	−0.523344	+	0.852121i	$0.675316\pi$
$464$	260652.	0.0562039
$465$	0	0
$466$	2664.81	0.000568462 0
$467$	−7.56254e6	−1.60463	−0.802316	−	0.596900i	$-0.796399\pi$
$467$	−7.56254e6	−1.60463	−0.802316	+	0.596900i	$0.796399\pi$
$468$	0	0
$469$	1.42487e7	2.99119
$470$	4.34526e6	0.907343
$471$	0	0
$472$	791788.	0.163589
$473$	0	0
$474$	0	0
$475$	592045.	0.120398
$476$	3.33243e6	0.674131
$477$	0	0
$478$	−3.85999e6	−0.772710
$479$	−1.71897e6	−0.342319	−0.171159	−	0.985243i	$-0.554751\pi$
$479$	−1.71897e6	−0.342319	−0.171159	+	0.985243i	$0.554751\pi$
$480$	0	0
$481$	−2.50969e6	−0.494604
$482$	2.40023e6	0.470582
$483$	0	0
$484$	0	0
$485$	5.32663e6	1.02825
$486$	0	0
$487$	−3.08125e6	−0.588715	−0.294357	−	0.955695i	$-0.595106\pi$
$487$	−3.08125e6	−0.588715	−0.294357	+	0.955695i	$0.595106\pi$
$488$	−1.25999e6	−0.239508
$489$	0	0
$490$	−9.68168e6	−1.82163
$491$	−1.05909e6	−0.198258	−0.0991290	−	0.995075i	$-0.531606\pi$
$491$	−1.05909e6	−0.198258	−0.0991290	+	0.995075i	$0.531606\pi$
$492$	0	0
$493$	−2.77388e6	−0.514009
$494$	2.88084e6	0.531131
$495$	0	0
$496$	42654.9	0.00778510
$497$	800218.	0.145317
$498$	0	0
$499$	2.26415e6	0.407056	0.203528	−	0.979069i	$-0.434759\pi$
$499$	2.26415e6	0.407056	0.203528	+	0.979069i	$0.434759\pi$
$500$	−3.17334e6	−0.567665
$501$	0	0
$502$	−4.89250e6	−0.866507
$503$	−7.65222e6	−1.34855	−0.674276	−	0.738480i	$-0.735544\pi$
$503$	−7.65222e6	−1.34855	−0.674276	+	0.738480i	$0.735544\pi$
$504$	0	0
$505$	2.05182e6	0.358023
$506$	0	0
$507$	0	0
$508$	2.99908e6	0.515619
$509$	−489107.	−0.0836777	−0.0418388	−	0.999124i	$-0.513322\pi$
$509$	−489107.	−0.0836777	−0.0418388	+	0.999124i	$0.513322\pi$
$510$	0	0
$511$	−1.85728e6	−0.314649
$512$	−765484.	−0.129051
$513$	0	0
$514$	5.98627e6	0.999421
$515$	3.45626e6	0.574233
$516$	0	0
$517$	0	0
$518$	8.23624e6	1.34867
$519$	0	0
$520$	2.95477e6	0.479198
$521$	−1.36556e6	−0.220402	−0.110201	−	0.993909i	$-0.535149\pi$
$521$	−1.36556e6	−0.220402	−0.110201	+	0.993909i	$0.535149\pi$
$522$	0	0
$523$	5.63994e6	0.901613	0.450807	−	0.892622i	$-0.351136\pi$
$523$	5.63994e6	0.901613	0.450807	+	0.892622i	$0.351136\pi$
$524$	−4.37408e6	−0.695919
$525$	0	0
$526$	−3.14945e6	−0.496330
$527$	−453937.	−0.0711982
$528$	0	0
$529$	−5.29817e6	−0.823164
$530$	−5.54621e6	−0.857644
$531$	0	0
$532$	1.35342e7	2.07326
$533$	5.05667e6	0.770986
$534$	0	0
$535$	−5.15198e6	−0.778197
$536$	1.04664e7	1.57357
$537$	0	0
$538$	−985095.	−0.146731
$539$	0	0
$540$	0	0
$541$	−8.24934e6	−1.21179	−0.605893	−	0.795546i	$-0.707184\pi$
$541$	−8.24934e6	−1.21179	−0.605893	+	0.795546i	$0.707184\pi$
$542$	2.73084e6	0.399299
$543$	0	0
$544$	3.98859e6	0.577859
$545$	1.31585e6	0.189764
$546$	0	0
$547$	4.74537e6	0.678112	0.339056	−	0.940766i	$-0.389892\pi$
$547$	4.74537e6	0.678112	0.339056	+	0.940766i	$0.389892\pi$
$548$	1.28497e6	0.182785
$549$	0	0
$550$	0	0
$551$	−1.12657e7	−1.58082
$552$	0	0
$553$	6.12480e6	0.851685
$554$	−2.94369e6	−0.407491
$555$	0	0
$556$	5.61728e6	0.770617
$557$	2.99690e6	0.409293	0.204647	−	0.978836i	$-0.434395\pi$
$557$	2.99690e6	0.409293	0.204647	+	0.978836i	$0.434395\pi$
$558$	0	0
$559$	1.12533e6	0.152318
$560$	−959420.	−0.129282
$561$	0	0
$562$	−6.25400e6	−0.835251
$563$	−3.89491e6	−0.517876	−0.258938	−	0.965894i	$-0.583373\pi$
$563$	−3.89491e6	−0.517876	−0.258938	+	0.965894i	$0.583373\pi$
$564$	0	0
$565$	−9.58571e6	−1.26329
$566$	−990290.	−0.129934
$567$	0	0
$568$	587801.	0.0764468
$569$	2.08127e6	0.269493	0.134747	−	0.990880i	$-0.456978\pi$
$569$	2.08127e6	0.269493	0.134747	+	0.990880i	$0.456978\pi$
$570$	0	0
$571$	−1.28958e7	−1.65523	−0.827615	−	0.561297i	$-0.810303\pi$
$571$	−1.28958e7	−1.65523	−0.827615	+	0.561297i	$0.810303\pi$
$572$	0	0
$573$	0	0
$574$	−1.65948e7	−2.10229
$575$	220757.	0.0278448
$576$	0	0
$577$	1.26163e6	0.157759	0.0788795	−	0.996884i	$-0.474866\pi$
$577$	1.26163e6	0.157759	0.0788795	+	0.996884i	$0.474866\pi$
$578$	−3.35039e6	−0.417134
$579$	0	0
$580$	−4.28188e6	−0.528524
$581$	−2.58100e7	−3.17210
$582$	0	0
$583$	0	0
$584$	−1.36427e6	−0.165527
$585$	0	0
$586$	−2.58816e6	−0.311349
$587$	1.25673e7	1.50538	0.752689	−	0.658376i	$-0.228756\pi$
$587$	1.25673e7	1.50538	0.752689	+	0.658376i	$0.228756\pi$
$588$	0	0
$589$	−1.84360e6	−0.218967
$590$	898988.	0.106322
$591$	0	0
$592$	598588.	0.0701978
$593$	−4.32620e6	−0.505207	−0.252604	−	0.967570i	$-0.581287\pi$
$593$	−4.32620e6	−0.505207	−0.252604	+	0.967570i	$0.581287\pi$
$594$	0	0
$595$	1.02102e7	1.18234
$596$	−1.56617e6	−0.180602
$597$	0	0
$598$	1.07418e6	0.122836
$599$	−1.45262e7	−1.65419	−0.827097	−	0.562060i	$-0.810009\pi$
$599$	−1.45262e7	−1.65419	−0.827097	+	0.562060i	$0.810009\pi$
$600$	0	0
$601$	380688.	0.0429915	0.0214958	−	0.999769i	$-0.493157\pi$
$601$	380688.	0.0429915	0.0214958	+	0.999769i	$0.493157\pi$
$602$	−3.69309e6	−0.415335
$603$	0	0
$604$	3.15748e6	0.352167
$605$	0	0
$606$	0	0
$607$	4.99233e6	0.549961	0.274980	−	0.961450i	$-0.411329\pi$
$607$	4.99233e6	0.549961	0.274980	+	0.961450i	$0.411329\pi$
$608$	1.61991e7	1.77718
$609$	0	0
$610$	−1.43058e6	−0.155664
$611$	−5.75936e6	−0.624125
$612$	0	0
$613$	−2.99353e6	−0.321760	−0.160880	−	0.986974i	$-0.551433\pi$
$613$	−2.99353e6	−0.321760	−0.160880	+	0.986974i	$0.551433\pi$
$614$	9.71265e6	1.03972
$615$	0	0
$616$	0	0
$617$	−6.27781e6	−0.663889	−0.331945	−	0.943299i	$-0.607705\pi$
$617$	−6.27781e6	−0.663889	−0.331945	+	0.943299i	$0.607705\pi$
$618$	0	0
$619$	3.96477e6	0.415902	0.207951	−	0.978139i	$-0.433320\pi$
$619$	3.96477e6	0.415902	0.207951	+	0.978139i	$0.433320\pi$
$620$	−700716.	−0.0732088
$621$	0	0
$622$	4.64631e6	0.481540
$623$	1.24535e7	1.28550
$624$	0	0
$625$	−1.03694e7	−1.06183
$626$	4.96299e6	0.506183
$627$	0	0
$628$	1.19179e6	0.120587
$629$	−6.37023e6	−0.641990
$630$	0	0
$631$	−9.62587e6	−0.962425	−0.481212	−	0.876604i	$-0.659803\pi$
$631$	−9.62587e6	−0.962425	−0.481212	+	0.876604i	$0.659803\pi$
$632$	4.49898e6	0.448044
$633$	0	0
$634$	27883.7	0.00275503
$635$	9.18888e6	0.904333
$636$	0	0
$637$	1.28324e7	1.25303
$638$	0	0
$639$	0	0
$640$	5.71336e6	0.551368
$641$	2.20090e6	0.211571	0.105785	−	0.994389i	$-0.466264\pi$
$641$	2.20090e6	0.211571	0.105785	+	0.994389i	$0.466264\pi$
$642$	0	0
$643$	2.00555e7	1.91296	0.956482	−	0.291791i	$-0.0942512\pi$
$643$	2.00555e7	1.91296	0.956482	+	0.291791i	$0.0942512\pi$
$644$	5.04652e6	0.479488
$645$	0	0
$646$	7.31229e6	0.689401
$647$	−128900.	−0.0121057	−0.00605287	−	0.999982i	$-0.501927\pi$
$647$	−128900.	−0.0121057	−0.00605287	+	0.999982i	$0.501927\pi$
$648$	0	0
$649$	0	0
$650$	208345.	0.0193419
$651$	0	0
$652$	−4.47869e6	−0.412603
$653$	−3.03250e6	−0.278303	−0.139151	−	0.990271i	$-0.544437\pi$
$653$	−3.03250e6	−0.278303	−0.139151	+	0.990271i	$0.544437\pi$
$654$	0	0
$655$	−1.34017e7	−1.22056
$656$	−1.20607e6	−0.109424
$657$	0	0
$658$	1.89009e7	1.70184
$659$	−1.59132e7	−1.42740	−0.713698	−	0.700454i	$-0.752981\pi$
$659$	−1.59132e7	−1.42740	−0.713698	+	0.700454i	$0.752981\pi$
$660$	0	0
$661$	6.50892e6	0.579436	0.289718	−	0.957112i	$-0.406439\pi$
$661$	6.50892e6	0.579436	0.289718	+	0.957112i	$0.406439\pi$
$662$	3.47661e6	0.308327
$663$	0	0
$664$	−1.89587e7	−1.66874
$665$	4.14675e7	3.63625
$666$	0	0
$667$	−4.20067e6	−0.365598
$668$	−9.19149e6	−0.796975
$669$	0	0
$670$	1.18835e7	1.02272
$671$	0	0
$672$	0	0
$673$	5.66035e6	0.481732	0.240866	−	0.970558i	$-0.422569\pi$
$673$	5.66035e6	0.481732	0.240866	+	0.970558i	$0.422569\pi$
$674$	−1.48290e7	−1.25736
$675$	0	0
$676$	5.54375e6	0.466592
$677$	−1.53601e7	−1.28802	−0.644012	−	0.765016i	$-0.722731\pi$
$677$	−1.53601e7	−1.28802	−0.644012	+	0.765016i	$0.722731\pi$
$678$	0	0
$679$	2.31697e7	1.92861
$680$	7.49994e6	0.621993
$681$	0	0
$682$	0	0
$683$	−1.47069e6	−0.120634	−0.0603170	−	0.998179i	$-0.519211\pi$
$683$	−1.47069e6	−0.120634	−0.0603170	+	0.998179i	$0.519211\pi$
$684$	0	0
$685$	3.93701e6	0.320583
$686$	−2.68045e7	−2.17469
$687$	0	0
$688$	−268404.	−0.0216181
$689$	7.35114e6	0.589939
$690$	0	0
$691$	−3.65617e6	−0.291294	−0.145647	−	0.989337i	$-0.546526\pi$
$691$	−3.65617e6	−0.291294	−0.145647	+	0.989337i	$0.546526\pi$
$692$	−6.14064e6	−0.487471
$693$	0	0
$694$	305636.	0.0240883
$695$	1.72108e7	1.35157
$696$	0	0
$697$	1.28351e7	1.00073
$698$	−4.05794e6	−0.315259
$699$	0	0
$700$	978806.	0.0755008
$701$	−5.21232e6	−0.400623	−0.200312	−	0.979732i	$-0.564195\pi$
$701$	−5.21232e6	−0.400623	−0.200312	+	0.979732i	$0.564195\pi$
$702$	0	0
$703$	−2.58718e7	−1.97442
$704$	0	0
$705$	0	0
$706$	1.35521e7	1.02328
$707$	8.92498e6	0.671519
$708$	0	0
$709$	−1.48388e7	−1.10862	−0.554311	−	0.832310i	$-0.687018\pi$
$709$	−1.48388e7	−1.10862	−0.554311	+	0.832310i	$0.687018\pi$
$710$	667383.	0.0496854
$711$	0	0
$712$	9.14772e6	0.676258
$713$	−687427.	−0.0506410
$714$	0	0
$715$	0	0
$716$	8.72099e6	0.635745
$717$	0	0
$718$	5.27260e6	0.381693
$719$	−1.37497e7	−0.991906	−0.495953	−	0.868349i	$-0.665181\pi$
$719$	−1.37497e7	−0.991906	−0.495953	+	0.868349i	$0.665181\pi$
$720$	0	0
$721$	1.50339e7	1.07705
$722$	2.07153e7	1.47893
$723$	0	0
$724$	−450292.	−0.0319262
$725$	−814748.	−0.0575676
$726$	0	0
$727$	−8.20549e6	−0.575796	−0.287898	−	0.957661i	$-0.592956\pi$
$727$	−8.20549e6	−0.575796	−0.287898	+	0.957661i	$0.592956\pi$
$728$	1.28526e7	0.898797
$729$	0	0
$730$	−1.54898e6	−0.107582
$731$	2.85638e6	0.197707
$732$	0	0
$733$	−1.20636e7	−0.829312	−0.414656	−	0.909978i	$-0.636098\pi$
$733$	−1.20636e7	−0.829312	−0.414656	+	0.909978i	$0.636098\pi$
$734$	92011.3	0.00630378
$735$	0	0
$736$	6.04017e6	0.411012
$737$	0	0
$738$	0	0
$739$	1.02319e7	0.689203	0.344602	−	0.938749i	$-0.388014\pi$
$739$	1.02319e7	0.689203	0.344602	+	0.938749i	$0.388014\pi$
$740$	−9.83335e6	−0.660119
$741$	0	0
$742$	−2.41248e7	−1.60862
$743$	2.41980e7	1.60808	0.804039	−	0.594577i	$-0.202680\pi$
$743$	2.41980e7	1.60808	0.804039	+	0.594577i	$0.202680\pi$
$744$	0	0
$745$	−4.79858e6	−0.316754
$746$	6.26489e6	0.412161
$747$	0	0
$748$	0	0
$749$	−2.24100e7	−1.45961
$750$	0	0
$751$	1.10121e7	0.712476	0.356238	−	0.934395i	$-0.384059\pi$
$751$	1.10121e7	0.712476	0.356238	+	0.934395i	$0.384059\pi$
$752$	1.37367e6	0.0885803
$753$	0	0
$754$	−3.96449e6	−0.253956
$755$	9.67420e6	0.617657
$756$	0	0
$757$	−2.08747e7	−1.32398	−0.661989	−	0.749513i	$-0.730288\pi$
$757$	−2.08747e7	−1.32398	−0.661989	+	0.749513i	$0.730288\pi$
$758$	−1.55775e7	−0.984744
$759$	0	0
$760$	3.04600e7	1.91291
$761$	2.85723e7	1.78848	0.894240	−	0.447587i	$-0.147716\pi$
$761$	2.85723e7	1.78848	0.894240	+	0.447587i	$0.147716\pi$
$762$	0	0
$763$	5.72365e6	0.355928
$764$	−1.06630e7	−0.660915
$765$	0	0
$766$	−4.69052e6	−0.288835
$767$	−1.19155e6	−0.0731347
$768$	0	0
$769$	3.07246e6	0.187357	0.0936787	−	0.995602i	$-0.470137\pi$
$769$	3.07246e6	0.187357	0.0936787	+	0.995602i	$0.470137\pi$
$770$	0	0
$771$	0	0
$772$	1.72877e6	0.104398
$773$	3.02076e7	1.81831	0.909153	−	0.416461i	$-0.136730\pi$
$773$	3.02076e7	1.81831	0.909153	+	0.416461i	$0.136730\pi$
$774$	0	0
$775$	−133331.	−0.00797401
$776$	1.70193e7	1.01458
$777$	0	0
$778$	−5.63229e6	−0.333608
$779$	5.21280e7	3.07771
$780$	0	0
$781$	0	0
$782$	2.72654e6	0.159439
$783$	0	0
$784$	−3.06067e6	−0.177839
$785$	3.65153e6	0.211495
$786$	0	0
$787$	2.07854e7	1.19625	0.598126	−	0.801402i	$-0.295912\pi$
$787$	2.07854e7	1.19625	0.598126	+	0.801402i	$0.295912\pi$
$788$	−9.14190e6	−0.524470
$789$	0	0
$790$	5.10809e6	0.291200
$791$	−4.16957e7	−2.36946
$792$	0	0
$793$	1.89615e6	0.107075
$794$	2.97317e6	0.167366
$795$	0	0
$796$	−1.30490e7	−0.729950
$797$	7.95535e6	0.443622	0.221811	−	0.975090i	$-0.428803\pi$
$797$	7.95535e6	0.443622	0.221811	+	0.975090i	$0.428803\pi$
$798$	0	0
$799$	−1.46187e7	−0.810106
$800$	1.17153e6	0.0647186
$801$	0	0
$802$	−5.04094e6	−0.276742
$803$	0	0
$804$	0	0
$805$	1.54620e7	0.840963
$806$	−648776.	−0.0351769
$807$	0	0
$808$	6.55584e6	0.353265
$809$	−3.04660e7	−1.63661	−0.818303	−	0.574787i	$-0.805085\pi$
$809$	−3.04660e7	−1.63661	−0.818303	+	0.574787i	$0.805085\pi$
$810$	0	0
$811$	−1.28041e7	−0.683594	−0.341797	−	0.939774i	$-0.611036\pi$
$811$	−1.28041e7	−0.683594	−0.341797	+	0.939774i	$0.611036\pi$
$812$	−1.86252e7	−0.991316
$813$	0	0
$814$	0	0
$815$	−1.37223e7	−0.723655
$816$	0	0
$817$	1.16008e7	0.608040
$818$	1.28092e7	0.669325
$819$	0	0
$820$	1.98128e7	1.02899
$821$	1.13635e7	0.588373	0.294186	−	0.955748i	$-0.404951\pi$
$821$	1.13635e7	0.588373	0.294186	+	0.955748i	$0.404951\pi$
$822$	0	0
$823$	8.23741e6	0.423927	0.211964	−	0.977278i	$-0.432014\pi$
$823$	8.23741e6	0.423927	0.211964	+	0.977278i	$0.432014\pi$
$824$	1.10432e7	0.566600
$825$	0	0
$826$	3.91040e6	0.199421
$827$	2.21035e7	1.12382	0.561909	−	0.827199i	$-0.310067\pi$
$827$	2.21035e7	1.12382	0.561909	+	0.827199i	$0.310067\pi$
$828$	0	0
$829$	−5.57875e6	−0.281936	−0.140968	−	0.990014i	$-0.545021\pi$
$829$	−5.57875e6	−0.281936	−0.140968	+	0.990014i	$0.545021\pi$
$830$	−2.15256e7	−1.08457
$831$	0	0
$832$	6.28851e6	0.314948
$833$	3.25720e7	1.62641
$834$	0	0
$835$	−2.81618e7	−1.39780
$836$	0	0
$837$	0	0
$838$	2.43049e7	1.19559
$839$	2.33576e7	1.14558	0.572788	−	0.819704i	$-0.305862\pi$
$839$	2.33576e7	1.14558	0.572788	+	0.819704i	$0.305862\pi$
$840$	0	0
$841$	−5.00769e6	−0.244145
$842$	−1.81794e7	−0.883688
$843$	0	0
$844$	−8.30866e6	−0.401490
$845$	1.69855e7	0.818345
$846$	0	0
$847$	0	0
$848$	−1.75332e6	−0.0837284
$849$	0	0
$850$	528831.	0.0251055
$851$	−9.64685e6	−0.456627
$852$	0	0
$853$	−8.94855e6	−0.421095	−0.210548	−	0.977584i	$-0.567525\pi$
$853$	−8.94855e6	−0.421095	−0.210548	+	0.977584i	$0.567525\pi$
$854$	−6.22273e6	−0.291969
$855$	0	0
$856$	−1.64612e7	−0.767853
$857$	−2.03190e6	−0.0945039	−0.0472520	−	0.998883i	$-0.515046\pi$
$857$	−2.03190e6	−0.0945039	−0.0472520	+	0.998883i	$0.515046\pi$
$858$	0	0
$859$	−1.92359e7	−0.889467	−0.444734	−	0.895663i	$-0.646702\pi$
$859$	−1.92359e7	−0.889467	−0.444734	+	0.895663i	$0.646702\pi$
$860$	4.40922e6	0.203290
$861$	0	0
$862$	−4.16247e6	−0.190802
$863$	−2.00380e7	−0.915856	−0.457928	−	0.888989i	$-0.651408\pi$
$863$	−2.00380e7	−0.915856	−0.457928	+	0.888989i	$0.651408\pi$
$864$	0	0
$865$	−1.88143e7	−0.854965
$866$	−4.11360e6	−0.186392
$867$	0	0
$868$	−3.04796e6	−0.137312
$869$	0	0
$870$	0	0
$871$	−1.57507e7	−0.703486
$872$	4.20431e6	0.187242
$873$	0	0
$874$	1.10735e7	0.490349
$875$	−4.22921e7	−1.86741
$876$	0	0
$877$	−1.98487e7	−0.871430	−0.435715	−	0.900085i	$-0.643504\pi$
$877$	−1.98487e7	−0.871430	−0.435715	+	0.900085i	$0.643504\pi$
$878$	−2.70153e7	−1.18270
$879$	0	0
$880$	0	0
$881$	2.93546e7	1.27420	0.637099	−	0.770782i	$-0.280134\pi$
$881$	2.93546e7	1.27420	0.637099	+	0.770782i	$0.280134\pi$
$882$	0	0
$883$	3.85199e7	1.66258	0.831291	−	0.555838i	$-0.187602\pi$
$883$	3.85199e7	1.66258	0.831291	+	0.555838i	$0.187602\pi$
$884$	−3.68372e6	−0.158546
$885$	0	0
$886$	−2.43972e7	−1.04413
$887$	−2.68797e6	−0.114714	−0.0573569	−	0.998354i	$-0.518267\pi$
$887$	−2.68797e6	−0.114714	−0.0573569	+	0.998354i	$0.518267\pi$
$888$	0	0
$889$	3.99696e7	1.69619
$890$	1.03862e7	0.439524
$891$	0	0
$892$	−2.13139e7	−0.896916
$893$	−5.93719e7	−2.49145
$894$	0	0
$895$	2.67202e7	1.11502
$896$	2.48519e7	1.03416
$897$	0	0
$898$	−7.90093e6	−0.326955
$899$	2.53709e6	0.104698
$900$	0	0
$901$	1.86590e7	0.765733
$902$	0	0
$903$	0	0
$904$	−3.06276e7	−1.24650
$905$	−1.37965e6	−0.0559947
$906$	0	0
$907$	−7.78340e6	−0.314160	−0.157080	−	0.987586i	$-0.550208\pi$
$907$	−7.78340e6	−0.314160	−0.157080	+	0.987586i	$0.550208\pi$
$908$	1.54552e7	0.622099
$909$	0	0
$910$	1.45927e7	0.584160
$911$	−3.15361e7	−1.25896	−0.629481	−	0.777016i	$-0.716732\pi$
$911$	−3.15361e7	−1.25896	−0.629481	+	0.777016i	$0.716732\pi$
$912$	0	0
$913$	0	0
$914$	1.35089e7	0.534878
$915$	0	0
$916$	1.88964e7	0.744117
$917$	−5.82946e7	−2.28931
$918$	0	0
$919$	−4.21184e7	−1.64507	−0.822533	−	0.568717i	$-0.807440\pi$
$919$	−4.21184e7	−1.64507	−0.822533	+	0.568717i	$0.807440\pi$
$920$	1.13576e7	0.442404
$921$	0	0
$922$	1.72027e7	0.666454
$923$	−884572.	−0.0341766
$924$	0	0
$925$	−1.87107e6	−0.0719011
$926$	−1.75147e7	−0.671237
$927$	0	0
$928$	−2.22925e7	−0.849746
$929$	−9.43595e6	−0.358712	−0.179356	−	0.983784i	$-0.557401\pi$
$929$	−9.43595e6	−0.358712	−0.179356	+	0.983784i	$0.557401\pi$
$930$	0	0
$931$	1.32287e8	5.00197
$932$	−13839.0	−0.000521874 0
$933$	0	0
$934$	−2.74348e7	−1.02904
$935$	0	0
$936$	0	0
$937$	−2.17869e7	−0.810674	−0.405337	−	0.914167i	$-0.632846\pi$
$937$	−2.17869e7	−0.810674	−0.405337	+	0.914167i	$0.632846\pi$
$938$	5.16904e7	1.91824
$939$	0	0
$940$	−2.25660e7	−0.832983
$941$	2.18678e7	0.805065	0.402532	−	0.915406i	$-0.368130\pi$
$941$	2.18678e7	0.805065	0.402532	+	0.915406i	$0.368130\pi$
$942$	0	0
$943$	1.94370e7	0.711787
$944$	284197.	0.0103798
$945$	0	0
$946$	0	0
$947$	−2.42335e7	−0.878094	−0.439047	−	0.898464i	$-0.644684\pi$
$947$	−2.42335e7	−0.878094	−0.439047	+	0.898464i	$0.644684\pi$
$948$	0	0
$949$	2.05307e6	0.0740010
$950$	2.14777e6	0.0772110
$951$	0	0
$952$	3.26231e7	1.16663
$953$	5.56092e7	1.98342	0.991710	−	0.128499i	$-0.0410159\pi$
$953$	5.56092e7	1.98342	0.991710	+	0.128499i	$0.0410159\pi$
$954$	0	0
$955$	−3.26703e7	−1.15916
$956$	2.00459e7	0.709383
$957$	0	0
$958$	−6.23595e6	−0.219528
$959$	1.71251e7	0.601294
$960$	0	0
$961$	−2.82140e7	−0.985498
$962$	−9.10446e6	−0.317188
$963$	0	0
$964$	−1.24650e7	−0.432016
$965$	5.29678e6	0.183102
$966$	0	0
$967$	2.09122e7	0.719173	0.359586	−	0.933112i	$-0.382918\pi$
$967$	2.09122e7	0.719173	0.359586	+	0.933112i	$0.382918\pi$
$968$	0	0
$969$	0	0
$970$	1.93235e7	0.659412
$971$	−1.68184e7	−0.572448	−0.286224	−	0.958163i	$-0.592400\pi$
$971$	−1.68184e7	−0.572448	−0.286224	+	0.958163i	$0.592400\pi$
$972$	0	0
$973$	7.48630e7	2.53504
$974$	−1.11779e7	−0.377541
$975$	0	0
$976$	−452251.	−0.0151969
$977$	3.45472e7	1.15792	0.578958	−	0.815358i	$-0.303460\pi$
$977$	3.45472e7	1.15792	0.578958	+	0.815358i	$0.303460\pi$
$978$	0	0
$979$	0	0
$980$	5.02794e7	1.67234
$981$	0	0
$982$	−3.84210e6	−0.127142
$983$	3.40232e7	1.12303	0.561516	−	0.827466i	$-0.310218\pi$
$983$	3.40232e7	1.12303	0.561516	+	0.827466i	$0.310218\pi$
$984$	0	0
$985$	−2.80099e7	−0.919857
$986$	−1.00629e7	−0.329632
$987$	0	0
$988$	−1.49609e7	−0.487602
$989$	4.32560e6	0.140623
$990$	0	0
$991$	−4.47458e7	−1.44733	−0.723665	−	0.690151i	$-0.757544\pi$
$991$	−4.47458e7	−1.44733	−0.723665	+	0.690151i	$0.757544\pi$
$992$	−3.64810e6	−0.117703
$993$	0	0
$994$	2.90297e6	0.0931915
$995$	−3.99807e7	−1.28024
$996$	0	0
$997$	5.86494e6	0.186864	0.0934320	−	0.995626i	$-0.470216\pi$
$997$	5.86494e6	0.186864	0.0934320	+	0.995626i	$0.470216\pi$
$998$	8.21371e6	0.261044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.p.1.1		2
3.2	odd	2		363.6.a.f.1.2		2
11.10	odd	2		99.6.a.d.1.2		2
33.32	even	2		33.6.a.e.1.1	✓	2
132.131	odd	2		528.6.a.o.1.2		2
165.164	even	2		825.6.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.e.1.1	✓	2	33.32	even	2
99.6.a.d.1.2		2	11.10	odd	2
363.6.a.f.1.2		2	3.2	odd	2
528.6.a.o.1.2		2	132.131	odd	2
825.6.a.c.1.2		2	165.164	even	2
1089.6.a.p.1.1		2	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+3.62772 q^{2} -18.8397 q^{4} -57.7228 q^{5} -251.081 q^{7} -184.432 q^{8} +O(q^{10})\)	\(q+3.62772 q^{2} -18.8397 q^{4} -57.7228 q^{5} -251.081 q^{7} -184.432 q^{8} -209.402 q^{10} +277.549 q^{13} -910.853 q^{14} -66.1983 q^{16} +704.489 q^{17} +2861.18 q^{19} +1087.48 q^{20} +1066.85 q^{23} +206.923 q^{25} +1006.87 q^{26} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} +5661.67 q^{32} +2555.69 q^{34} +14493.1 q^{35} -9042.34 q^{37} +10379.6 q^{38} +10645.9 q^{40} +18219.0 q^{41} +4054.54 q^{43} +3870.24 q^{46} -20750.8 q^{47} +46234.9 q^{49} +750.659 q^{50} -5228.92 q^{52} +26485.9 q^{53} +46307.4 q^{56} -14283.9 q^{58} -4293.12 q^{59} +6831.76 q^{61} -2337.52 q^{62} +22657.3 q^{64} -16020.9 q^{65} -56749.5 q^{67} -13272.3 q^{68} +52577.0 q^{70} -3187.09 q^{71} +7397.14 q^{73} -32803.1 q^{74} -53903.7 q^{76} -24393.7 q^{79} +3821.15 q^{80} +66093.5 q^{82} +102795. q^{83} -40665.1 q^{85} +14708.7 q^{86} -49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} -75278.1 q^{94} -165156. q^{95} -92279.5 q^{97} +167727. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8}+O(q^{10}) \) 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8	\( 2 q + 13 q^{2} + 37 q^{4} - 58 q^{5} - 146 q^{7} + 39 q^{8} - 212 q^{10} + 130 q^{13} + 74 q^{14} + 241 q^{16} - 728 q^{17} + 828 q^{19} + 1072 q^{20} + 238 q^{23} - 2918 q^{25} - 376 q^{26} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} + 1391 q^{32} - 10870 q^{34} + 14464 q^{35} - 1908 q^{37} - 8676 q^{38} + 10584 q^{40} + 36484 q^{41} - 9768 q^{43} - 3898 q^{46} - 43742 q^{47} + 40470 q^{49} - 28537 q^{50} - 13468 q^{52} + 12174 q^{53} + 69786 q^{56} + 29142 q^{58} + 2788 q^{59} + 25302 q^{61} - 94520 q^{62} - 27199 q^{64} - 15980 q^{65} - 40520 q^{67} - 93262 q^{68} + 52304 q^{70} - 31386 q^{71} + 46780 q^{73} + 34062 q^{74} - 167436 q^{76} + 16850 q^{79} + 3736 q^{80} + 237278 q^{82} + 79440 q^{83} - 40268 q^{85} - 114840 q^{86} + 54204 q^{89} - 85192 q^{91} - 66382 q^{92} - 290758 q^{94} - 164592 q^{95} - 241568 q^{97} + 113697 q^{98}+O(q^{100}) \) 2 * q + 13 * q^2 + 37 * q^4 - 58 * q^5 - 146 * q^7 + 39 * q^8 - 212 * q^10 + 130 * q^13 + 74 * q^14 + 241 * q^16 - 728 * q^17 + 828 * q^19 + 1072 * q^20 + 238 * q^23 - 2918 * q^25 - 376 * q^26 + 10598 * q^28 + 696 * q^29 - 10480 * q^31 + 1391 * q^32 - 10870 * q^34 + 14464 * q^35 - 1908 * q^37 - 8676 * q^38 + 10584 * q^40 + 36484 * q^41 - 9768 * q^43 - 3898 * q^46 - 43742 * q^47 + 40470 * q^49 - 28537 * q^50 - 13468 * q^52 + 12174 * q^53 + 69786 * q^56 + 29142 * q^58 + 2788 * q^59 + 25302 * q^61 - 94520 * q^62 - 27199 * q^64 - 15980 * q^65 - 40520 * q^67 - 93262 * q^68 + 52304 * q^70 - 31386 * q^71 + 46780 * q^73 + 34062 * q^74 - 167436 * q^76 + 16850 * q^79 + 3736 * q^80 + 237278 * q^82 + 79440 * q^83 - 40268 * q^85 - 114840 * q^86 + 54204 * q^89 - 85192 * q^91 - 66382 * q^92 - 290758 * q^94 - 164592 * q^95 - 241568 * q^97 + 113697 * q^98

Introduction

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