LMFDB - Embedded newform 1089.6.a.bg.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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 167x^{6} - 68x^{5} + 7903x^{4} + 8528x^{3} - 88021x^{2} - 132908x - 19844 $ x^8 - 167x^6 - 68x^5 + 7903x^4 + 8528x^3 - 88021x^2 - 132908x - 19844
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 11^{2} $
Twist minimal:	no (minimal twist has level 11)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$8.98519$ 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)

$f(q)$	$=$	$q-7.98519 q^{2} +31.7633 q^{4} +7.25383 q^{5} -168.938 q^{7} +1.89018 q^{8} +O(q^{10})$	$q-7.98519 q^{2} +31.7633 q^{4} +7.25383 q^{5} -168.938 q^{7} +1.89018 q^{8} -57.9232 q^{10} -265.753 q^{13} +1349.01 q^{14} -1031.52 q^{16} +689.133 q^{17} +2705.87 q^{19} +230.406 q^{20} -1409.32 q^{23} -3072.38 q^{25} +2122.09 q^{26} -5366.04 q^{28} +2679.93 q^{29} +4320.20 q^{31} +8176.39 q^{32} -5502.86 q^{34} -1225.45 q^{35} -786.487 q^{37} -21606.9 q^{38} +13.7111 q^{40} +12894.1 q^{41} -20070.0 q^{43} +11253.7 q^{46} -18817.1 q^{47} +11733.2 q^{49} +24533.6 q^{50} -8441.18 q^{52} -29797.1 q^{53} -319.325 q^{56} -21399.7 q^{58} +20582.3 q^{59} +31083.6 q^{61} -34497.7 q^{62} -32281.4 q^{64} -1927.72 q^{65} -764.687 q^{67} +21889.1 q^{68} +9785.46 q^{70} +52215.0 q^{71} -70779.2 q^{73} +6280.25 q^{74} +85947.4 q^{76} -3782.89 q^{79} -7482.46 q^{80} -102962. q^{82} +5860.40 q^{83} +4998.85 q^{85} +160263. q^{86} -62319.7 q^{89} +44895.8 q^{91} -44764.6 q^{92} +150258. q^{94} +19627.9 q^{95} +99792.1 q^{97} -93691.8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 86 q^{4} - 70 q^{5} - 292 q^{7} + 294 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 86 * q^4 - 70 * q^5 - 292 * q^7 + 294 * q^8	$ 8 q + 8 q^{2} + 86 q^{4} - 70 q^{5} - 292 q^{7} + 294 q^{8} - 336 q^{10} - 514 q^{13} + 1020 q^{14} + 722 q^{16} + 3776 q^{17} - 4108 q^{19} - 10978 q^{20} + 5424 q^{23} - 2722 q^{25} - 508 q^{26} - 24736 q^{28} + 8190 q^{29} + 16268 q^{31} + 29406 q^{32} - 13483 q^{34} + 28096 q^{35} - 866 q^{37} - 157 q^{38} - 49118 q^{40} + 43048 q^{41} - 12188 q^{43} - 6814 q^{46} + 22268 q^{47} - 4602 q^{49} + 44588 q^{50} + 13838 q^{52} - 32818 q^{53} - 35316 q^{56} - 46852 q^{58} + 34748 q^{59} - 18758 q^{61} - 20586 q^{62} - 2110 q^{64} + 20642 q^{65} - 32100 q^{67} - 62733 q^{68} + 32146 q^{70} - 6140 q^{71} + 101260 q^{73} - 46654 q^{74} + 87403 q^{76} - 95268 q^{79} - 106178 q^{80} + 139095 q^{82} - 45636 q^{83} + 150442 q^{85} + 354907 q^{86} - 74182 q^{89} + 286656 q^{91} - 34806 q^{92} + 513780 q^{94} + 137168 q^{95} - 167632 q^{97} - 371846 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 86 * q^4 - 70 * q^5 - 292 * q^7 + 294 * q^8 - 336 * q^10 - 514 * q^13 + 1020 * q^14 + 722 * q^16 + 3776 * q^17 - 4108 * q^19 - 10978 * q^20 + 5424 * q^23 - 2722 * q^25 - 508 * q^26 - 24736 * q^28 + 8190 * q^29 + 16268 * q^31 + 29406 * q^32 - 13483 * q^34 + 28096 * q^35 - 866 * q^37 - 157 * q^38 - 49118 * q^40 + 43048 * q^41 - 12188 * q^43 - 6814 * q^46 + 22268 * q^47 - 4602 * q^49 + 44588 * q^50 + 13838 * q^52 - 32818 * q^53 - 35316 * q^56 - 46852 * q^58 + 34748 * q^59 - 18758 * q^61 - 20586 * q^62 - 2110 * q^64 + 20642 * q^65 - 32100 * q^67 - 62733 * q^68 + 32146 * q^70 - 6140 * q^71 + 101260 * q^73 - 46654 * q^74 + 87403 * q^76 - 95268 * q^79 - 106178 * q^80 + 139095 * q^82 - 45636 * q^83 + 150442 * q^85 + 354907 * q^86 - 74182 * q^89 + 286656 * q^91 - 34806 * q^92 + 513780 * q^94 + 137168 * q^95 - 167632 * q^97 - 371846 * 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 $

$2$	−7.98519	−1.41160	−0.705798	−	0.708413i	$-0.749411\pi$
$2$	−7.98519	−1.41160	−0.705798	+	0.708413i	$0.749411\pi$
$3$	0	0
$3$	0	0
$4$	31.7633	0.992603
$5$	7.25383	0.129760	0.0648802	−	0.997893i	$-0.479333\pi$
$5$	7.25383	0.129760	0.0648802	+	0.997893i	$0.479333\pi$
$6$	0	0
$7$	−168.938	−1.30312	−0.651558	−	0.758599i	$-0.725884\pi$
$7$	−168.938	−1.30312	−0.651558	+	0.758599i	$0.725884\pi$
$8$	1.89018	0.0104419
$9$	0	0
$10$	−57.9232	−0.183169
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−265.753	−0.436133	−0.218067	−	0.975934i	$-0.569975\pi$
$13$	−265.753	−0.436133	−0.218067	+	0.975934i	$0.569975\pi$
$14$	1349.01	1.83947
$15$	0	0
$16$	−1031.52	−1.00734
$17$	689.133	0.578337	0.289168	−	0.957278i	$-0.406621\pi$
$17$	689.133	0.578337	0.289168	+	0.957278i	$0.406621\pi$
$18$	0	0
$19$	2705.87	1.71958	0.859792	−	0.510645i	$-0.170593\pi$
$19$	2705.87	1.71958	0.859792	+	0.510645i	$0.170593\pi$
$20$	230.406	0.128801
$21$	0	0
$22$	0	0
$23$	−1409.32	−0.555507	−0.277754	−	0.960652i	$-0.589590\pi$
$23$	−1409.32	−0.555507	−0.277754	+	0.960652i	$0.589590\pi$
$24$	0	0
$25$	−3072.38	−0.983162
$26$	2122.09	0.615644
$27$	0	0
$28$	−5366.04	−1.29348
$29$	2679.93	0.591736	0.295868	−	0.955229i	$-0.404391\pi$
$29$	2679.93	0.591736	0.295868	+	0.955229i	$0.404391\pi$
$30$	0	0
$31$	4320.20	0.807421	0.403710	−	0.914887i	$-0.367720\pi$
$31$	4320.20	0.807421	0.403710	+	0.914887i	$0.367720\pi$
$32$	8176.39	1.41152
$33$	0	0
$34$	−5502.86	−0.816378
$35$	−1225.45	−0.169093
$36$	0	0
$37$	−786.487	−0.0944468	−0.0472234	−	0.998884i	$-0.515037\pi$
$37$	−786.487	−0.0944468	−0.0472234	+	0.998884i	$0.515037\pi$
$38$	−21606.9	−2.42736
$39$	0	0
$40$	13.7111	0.00135494
$41$	12894.1	1.19793	0.598964	−	0.800776i	$-0.295579\pi$
$41$	12894.1	1.19793	0.598964	+	0.800776i	$0.295579\pi$
$42$	0	0
$43$	−20070.0	−1.65530	−0.827650	−	0.561245i	$-0.810323\pi$
$43$	−20070.0	−1.65530	−0.827650	+	0.561245i	$0.810323\pi$
$44$	0	0
$45$	0	0
$46$	11253.7	0.784151
$47$	−18817.1	−1.24253	−0.621267	−	0.783599i	$-0.713382\pi$
$47$	−18817.1	−1.24253	−0.621267	+	0.783599i	$0.713382\pi$
$48$	0	0
$49$	11733.2	0.698113
$50$	24533.6	1.38783
$51$	0	0
$52$	−8441.18	−0.432907
$53$	−29797.1	−1.45709	−0.728543	−	0.685000i	$-0.759802\pi$
$53$	−29797.1	−1.45709	−0.728543	+	0.685000i	$0.759802\pi$
$54$	0	0
$55$	0	0
$56$	−319.325	−0.0136070
$57$	0	0
$58$	−21399.7	−0.835292
$59$	20582.3	0.769773	0.384887	−	0.922964i	$-0.374241\pi$
$59$	20582.3	0.769773	0.384887	+	0.922964i	$0.374241\pi$
$60$	0	0
$61$	31083.6	1.06956	0.534782	−	0.844990i	$-0.320394\pi$
$61$	31083.6	1.06956	0.534782	+	0.844990i	$0.320394\pi$
$62$	−34497.7	−1.13975
$63$	0	0
$64$	−32281.4	−0.985151
$65$	−1927.72	−0.0565928
$66$	0	0
$67$	−764.687	−0.0208112	−0.0104056	−	0.999946i	$-0.503312\pi$
$67$	−764.687	−0.0208112	−0.0104056	+	0.999946i	$0.503312\pi$
$68$	21889.1	0.574059
$69$	0	0
$70$	9785.46	0.238691
$71$	52215.0	1.22928	0.614638	−	0.788810i	$-0.289302\pi$
$71$	52215.0	1.22928	0.614638	+	0.788810i	$0.289302\pi$
$72$	0	0
$73$	−70779.2	−1.55453	−0.777264	−	0.629175i	$-0.783393\pi$
$73$	−70779.2	−1.55453	−0.777264	+	0.629175i	$0.783393\pi$
$74$	6280.25	0.133321
$75$	0	0
$76$	85947.4	1.70686
$77$	0	0
$78$	0	0
$79$	−3782.89	−0.0681955	−0.0340978	−	0.999419i	$-0.510856\pi$
$79$	−3782.89	−0.0681955	−0.0340978	+	0.999419i	$0.510856\pi$
$80$	−7482.46	−0.130713
$81$	0	0
$82$	−102962.	−1.69099
$83$	5860.40	0.0933753	0.0466877	−	0.998910i	$-0.485133\pi$
$83$	5860.40	0.0933753	0.0466877	+	0.998910i	$0.485133\pi$
$84$	0	0
$85$	4998.85	0.0750453
$86$	160263.	2.33661
$87$	0	0
$88$	0	0
$89$	−62319.7	−0.833969	−0.416985	−	0.908914i	$-0.636913\pi$
$89$	−62319.7	−0.833969	−0.416985	+	0.908914i	$0.636913\pi$
$90$	0	0
$91$	44895.8	0.568332
$92$	−44764.6	−0.551398
$93$	0	0
$94$	150258.	1.75396
$95$	19627.9	0.223134
$96$	0	0
$97$	99792.1	1.07688	0.538439	−	0.842664i	$-0.319014\pi$
$97$	99792.1	1.07688	0.538439	+	0.842664i	$0.319014\pi$
$98$	−93691.8	−0.985454
$99$	0	0
$100$	−97589.0	−0.975890
$101$	125663.	1.22576	0.612878	−	0.790177i	$-0.290012\pi$
$101$	125663.	1.22576	0.612878	+	0.790177i	$0.290012\pi$
$102$	0	0
$103$	−93746.4	−0.870686	−0.435343	−	0.900265i	$-0.643373\pi$
$103$	−93746.4	−0.870686	−0.435343	+	0.900265i	$0.643373\pi$
$104$	−502.321	−0.00455405
$105$	0	0
$106$	237936.	2.05682
$107$	−181.966	−0.00153650	−0.000768249	−	1.00000i	$-0.500245\pi$
$107$	−181.966	−0.00153650	−0.000768249		1.00000i	$0.500245\pi$
$108$	0	0
$109$	−16703.4	−0.134660	−0.0673300	−	0.997731i	$-0.521448\pi$
$109$	−16703.4	−0.134660	−0.0673300	+	0.997731i	$0.521448\pi$
$110$	0	0
$111$	0	0
$112$	174263.	1.31268
$113$	−148512.	−1.09412	−0.547061	−	0.837093i	$-0.684254\pi$
$113$	−148512.	−1.09412	−0.547061	+	0.837093i	$0.684254\pi$
$114$	0	0
$115$	−10223.0	−0.0720829
$116$	85123.3	0.587359
$117$	0	0
$118$	−164353.	−1.08661
$119$	−116421.	−0.753640
$120$	0	0
$121$	0	0
$122$	−248208.	−1.50979
$123$	0	0
$124$	137224.	0.801448
$125$	−44954.8	−0.257336
$126$	0	0
$127$	39142.4	0.215347	0.107673	−	0.994186i	$-0.465660\pi$
$127$	39142.4	0.215347	0.107673	+	0.994186i	$0.465660\pi$
$128$	−3870.99	−0.0208832
$129$	0	0
$130$	15393.2	0.0798862
$131$	92680.8	0.471858	0.235929	−	0.971770i	$-0.424187\pi$
$131$	92680.8	0.471858	0.235929	+	0.971770i	$0.424187\pi$
$132$	0	0
$133$	−457126.	−2.24082
$134$	6106.17	0.0293770
$135$	0	0
$136$	1302.59	0.00603893
$137$	−47134.4	−0.214554	−0.107277	−	0.994229i	$-0.534213\pi$
$137$	−47134.4	−0.214554	−0.107277	+	0.994229i	$0.534213\pi$
$138$	0	0
$139$	−298790.	−1.31169	−0.655843	−	0.754898i	$-0.727686\pi$
$139$	−298790.	−1.31169	−0.655843	+	0.754898i	$0.727686\pi$
$140$	−38924.3	−0.167842
$141$	0	0
$142$	−416946.	−1.73524
$143$	0	0
$144$	0	0
$145$	19439.7	0.0767839
$146$	565185.	2.19436
$147$	0	0
$148$	−24981.4	−0.0937481
$149$	306971.	1.13274	0.566372	−	0.824150i	$-0.308347\pi$
$149$	306971.	1.13274	0.566372	+	0.824150i	$0.308347\pi$
$150$	0	0
$151$	−277889.	−0.991813	−0.495906	−	0.868376i	$-0.665164\pi$
$151$	−277889.	−0.991813	−0.495906	+	0.868376i	$0.665164\pi$
$152$	5114.59	0.0179557
$153$	0	0
$154$	0	0
$155$	31338.0	0.104771
$156$	0	0
$157$	−251452.	−0.814154	−0.407077	−	0.913394i	$-0.633452\pi$
$157$	−251452.	−0.814154	−0.407077	+	0.913394i	$0.633452\pi$
$158$	30207.1	0.0962645
$159$	0	0
$160$	59310.1	0.183159
$161$	238088.	0.723891
$162$	0	0
$163$	−64412.2	−0.189889	−0.0949443	−	0.995483i	$-0.530267\pi$
$163$	−64412.2	−0.189889	−0.0949443	+	0.995483i	$0.530267\pi$
$164$	409558.	1.18907
$165$	0	0
$166$	−46796.4	−0.131808
$167$	390469.	1.08342	0.541708	−	0.840566i	$-0.317778\pi$
$167$	390469.	1.08342	0.541708	+	0.840566i	$0.317778\pi$
$168$	0	0
$169$	−300669.	−0.809788
$170$	−39916.8	−0.105934
$171$	0	0
$172$	−637490.	−1.64306
$173$	−580684.	−1.47511	−0.737555	−	0.675287i	$-0.764020\pi$
$173$	−580684.	−1.47511	−0.737555	+	0.675287i	$0.764020\pi$
$174$	0	0
$175$	519043.	1.28118
$176$	0	0
$177$	0	0
$178$	497634.	1.17723
$179$	282280.	0.658488	0.329244	−	0.944245i	$-0.393206\pi$
$179$	282280.	0.658488	0.329244	+	0.944245i	$0.393206\pi$
$180$	0	0
$181$	−453221.	−1.02828	−0.514142	−	0.857705i	$-0.671890\pi$
$181$	−453221.	−1.02828	−0.514142	+	0.857705i	$0.671890\pi$
$182$	−358502.	−0.802256
$183$	0	0
$184$	−2663.87	−0.00580054
$185$	−5705.04	−0.0122555
$186$	0	0
$187$	0	0
$188$	−597694.	−1.23334
$189$	0	0
$190$	−156733.	−0.314975
$191$	−614986.	−1.21978	−0.609891	−	0.792486i	$-0.708787\pi$
$191$	−614986.	−1.21978	−0.609891	+	0.792486i	$0.708787\pi$
$192$	0	0
$193$	33753.3	0.0652264	0.0326132	−	0.999468i	$-0.489617\pi$
$193$	33753.3	0.0652264	0.0326132	+	0.999468i	$0.489617\pi$
$194$	−796859.	−1.52012
$195$	0	0
$196$	372685.	0.692949
$197$	411926.	0.756230	0.378115	−	0.925759i	$-0.376572\pi$
$197$	411926.	0.756230	0.378115	+	0.925759i	$0.376572\pi$
$198$	0	0
$199$	−766254.	−1.37164	−0.685820	−	0.727772i	$-0.740556\pi$
$199$	−766254.	−1.37164	−0.685820	+	0.727772i	$0.740556\pi$
$200$	−5807.36	−0.0102661
$201$	0	0
$202$	−1.00344e6	−1.73027
$203$	−452743.	−0.771101
$204$	0	0
$205$	93531.5	0.155444
$206$	748583.	1.22906
$207$	0	0
$208$	274129.	0.439336
$209$	0	0
$210$	0	0
$211$	−427093.	−0.660414	−0.330207	−	0.943908i	$-0.607119\pi$
$211$	−427093.	−0.660414	−0.330207	+	0.943908i	$0.607119\pi$
$212$	−946455.	−1.44631
$213$	0	0
$214$	1453.04	0.00216891
$215$	−145585.	−0.214792
$216$	0	0
$217$	−729848.	−1.05216
$218$	133380.	0.190086
$219$	0	0
$220$	0	0
$221$	−183139.	−0.252232
$222$	0	0
$223$	1.00598e6	1.35465	0.677324	−	0.735685i	$-0.263139\pi$
$223$	1.00598e6	1.35465	0.677324	+	0.735685i	$0.263139\pi$
$224$	−1.38131e6	−1.83937
$225$	0	0
$226$	1.18590e6	1.54446
$227$	−494367.	−0.636774	−0.318387	−	0.947961i	$-0.603141\pi$
$227$	−494367.	−0.636774	−0.318387	+	0.947961i	$0.603141\pi$
$228$	0	0
$229$	−101318.	−0.127672	−0.0638361	−	0.997960i	$-0.520333\pi$
$229$	−101318.	−0.127672	−0.0638361	+	0.997960i	$0.520333\pi$
$230$	81632.3	0.101752
$231$	0	0
$232$	5065.55	0.00617884
$233$	−526492.	−0.635333	−0.317667	−	0.948202i	$-0.602899\pi$
$233$	−526492.	−0.635333	−0.317667	+	0.948202i	$0.602899\pi$
$234$	0	0
$235$	−136496.	−0.161232
$236$	653760.	0.764079
$237$	0	0
$238$	929644.	1.06384
$239$	413546.	0.468305	0.234153	−	0.972200i	$-0.424768\pi$
$239$	413546.	0.468305	0.234153	+	0.972200i	$0.424768\pi$
$240$	0	0
$241$	679718.	0.753852	0.376926	−	0.926243i	$-0.376981\pi$
$241$	679718.	0.753852	0.376926	+	0.926243i	$0.376981\pi$
$242$	0	0
$243$	0	0
$244$	987317.	1.06165
$245$	85110.6	0.0905875
$246$	0	0
$247$	−719092.	−0.749967
$248$	8165.98	0.00843100
$249$	0	0
$250$	358972.	0.363254
$251$	−794130.	−0.795623	−0.397811	−	0.917467i	$-0.630230\pi$
$251$	−794130.	−0.795623	−0.397811	+	0.917467i	$0.630230\pi$
$252$	0	0
$253$	0	0
$254$	−312560.	−0.303983
$255$	0	0
$256$	1.06392e6	1.01463
$257$	617850.	0.583512	0.291756	−	0.956493i	$-0.405760\pi$
$257$	617850.	0.583512	0.291756	+	0.956493i	$0.405760\pi$
$258$	0	0
$259$	132868.	0.123075
$260$	−61230.9	−0.0561742
$261$	0	0
$262$	−740074.	−0.666073
$263$	66349.1	0.0591488	0.0295744	−	0.999563i	$-0.490585\pi$
$263$	66349.1	0.0591488	0.0295744	+	0.999563i	$0.490585\pi$
$264$	0	0
$265$	−216143.	−0.189072
$266$	3.65024e6	3.16313
$267$	0	0
$268$	−24289.0	−0.0206572
$269$	1.70469e6	1.43637	0.718184	−	0.695853i	$-0.244974\pi$
$269$	1.70469e6	1.43637	0.718184	+	0.695853i	$0.244974\pi$
$270$	0	0
$271$	225681.	0.186669	0.0933344	−	0.995635i	$-0.470247\pi$
$271$	225681.	0.186669	0.0933344	+	0.995635i	$0.470247\pi$
$272$	−710854.	−0.582583
$273$	0	0
$274$	376377.	0.302863
$275$	0	0
$276$	0	0
$277$	843509.	0.660527	0.330263	−	0.943889i	$-0.392862\pi$
$277$	843509.	0.660527	0.330263	+	0.943889i	$0.392862\pi$
$278$	2.38590e6	1.85157
$279$	0	0
$280$	−2316.33	−0.00176565
$281$	−1.63585e6	−1.23589	−0.617943	−	0.786223i	$-0.712034\pi$
$281$	−1.63585e6	−1.23589	−0.617943	+	0.786223i	$0.712034\pi$
$282$	0	0
$283$	898431.	0.666835	0.333418	−	0.942779i	$-0.391798\pi$
$283$	898431.	0.666835	0.333418	+	0.942779i	$0.391798\pi$
$284$	1.65852e6	1.22018
$285$	0	0
$286$	0	0
$287$	−2.17831e6	−1.56104
$288$	0	0
$289$	−944953.	−0.665527
$290$	−155230.	−0.108388
$291$	0	0
$292$	−2.24818e6	−1.54303
$293$	1.64748e6	1.12112	0.560558	−	0.828115i	$-0.310587\pi$
$293$	1.64748e6	1.12112	0.560558	+	0.828115i	$0.310587\pi$
$294$	0	0
$295$	149300.	0.0998861
$296$	−1486.60	−0.000986202 0
$297$	0	0
$298$	−2.45122e6	−1.59898
$299$	374530.	0.242275
$300$	0	0
$301$	3.39060e6	2.15705
$302$	2.21900e6	1.40004
$303$	0	0
$304$	−2.79116e6	−1.73221
$305$	225475.	0.138787
$306$	0	0
$307$	−1.28570e6	−0.778561	−0.389281	−	0.921119i	$-0.627276\pi$
$307$	−1.28570e6	−0.778561	−0.389281	+	0.921119i	$0.627276\pi$
$308$	0	0
$309$	0	0
$310$	−250240.	−0.147895
$311$	−212432.	−0.124543	−0.0622713	−	0.998059i	$-0.519834\pi$
$311$	−212432.	−0.124543	−0.0622713	+	0.998059i	$0.519834\pi$
$312$	0	0
$313$	2.91996e6	1.68468	0.842339	−	0.538948i	$-0.181178\pi$
$313$	2.91996e6	1.68468	0.842339	+	0.538948i	$0.181178\pi$
$314$	2.00789e6	1.14926
$315$	0	0
$316$	−120157.	−0.0676911
$317$	1.22860e6	0.686693	0.343347	−	0.939209i	$-0.388439\pi$
$317$	1.22860e6	0.686693	0.343347	+	0.939209i	$0.388439\pi$
$318$	0	0
$319$	0	0
$320$	−234164.	−0.127834
$321$	0	0
$322$	−1.90118e6	−1.02184
$323$	1.86471e6	0.994498
$324$	0	0
$325$	816494.	0.428790
$326$	514344.	0.268046
$327$	0	0
$328$	24372.2	0.0125086
$329$	3.17894e6	1.61917
$330$	0	0
$331$	2.85151e6	1.43055	0.715277	−	0.698841i	$-0.246300\pi$
$331$	2.85151e6	1.43055	0.715277	+	0.698841i	$0.246300\pi$
$332$	186146.	0.0926846
$333$	0	0
$334$	−3.11797e6	−1.52935
$335$	−5546.91	−0.00270047
$336$	0	0
$337$	3.13884e6	1.50555	0.752775	−	0.658278i	$-0.228715\pi$
$337$	3.13884e6	1.50555	0.752775	+	0.658278i	$0.228715\pi$
$338$	2.40090e6	1.14309
$339$	0	0
$340$	158780.	0.0744901
$341$	0	0
$342$	0	0
$343$	857162.	0.393394
$344$	−37936.0	−0.0172844
$345$	0	0
$346$	4.63687e6	2.08226
$347$	1.90227e6	0.848102	0.424051	−	0.905638i	$-0.360608\pi$
$347$	1.90227e6	0.848102	0.424051	+	0.905638i	$0.360608\pi$
$348$	0	0
$349$	2.15328e6	0.946317	0.473158	−	0.880977i	$-0.343114\pi$
$349$	2.15328e6	0.946317	0.473158	+	0.880977i	$0.343114\pi$
$350$	−4.14466e6	−1.80850
$351$	0	0
$352$	0	0
$353$	−588980.	−0.251573	−0.125786	−	0.992057i	$-0.540145\pi$
$353$	−588980.	−0.251573	−0.125786	+	0.992057i	$0.540145\pi$
$354$	0	0
$355$	378758.	0.159511
$356$	−1.97948e6	−0.827800
$357$	0	0
$358$	−2.25406e6	−0.929518
$359$	−1.59533e6	−0.653304	−0.326652	−	0.945145i	$-0.605920\pi$
$359$	−1.59533e6	−0.653304	−0.326652	+	0.945145i	$0.605920\pi$
$360$	0	0
$361$	4.84564e6	1.95697
$362$	3.61905e6	1.45152
$363$	0	0
$364$	1.42604e6	0.564128
$365$	−513420.	−0.201716
$366$	0	0
$367$	−2.23420e6	−0.865877	−0.432938	−	0.901424i	$-0.642523\pi$
$367$	−2.23420e6	−0.865877	−0.432938	+	0.901424i	$0.642523\pi$
$368$	1.45374e6	0.559586
$369$	0	0
$370$	45555.8	0.0172997
$371$	5.03388e6	1.89875
$372$	0	0
$373$	1.30323e6	0.485008	0.242504	−	0.970150i	$-0.422031\pi$
$373$	1.30323e6	0.485008	0.242504	+	0.970150i	$0.422031\pi$
$374$	0	0
$375$	0	0
$376$	−35567.8	−0.0129744
$377$	−712198.	−0.258076
$378$	0	0
$379$	2.98424e6	1.06718	0.533588	−	0.845744i	$-0.320843\pi$
$379$	2.98424e6	1.06718	0.533588	+	0.845744i	$0.320843\pi$
$380$	623448.	0.221483
$381$	0	0
$382$	4.91078e6	1.72184
$383$	3.43961e6	1.19815	0.599077	−	0.800692i	$-0.295534\pi$
$383$	3.43961e6	1.19815	0.599077	+	0.800692i	$0.295534\pi$
$384$	0	0
$385$	0	0
$386$	−269527.	−0.0920733
$387$	0	0
$388$	3.16972e6	1.06891
$389$	−1.16556e6	−0.390535	−0.195268	−	0.980750i	$-0.562558\pi$
$389$	−1.16556e6	−0.390535	−0.195268	+	0.980750i	$0.562558\pi$
$390$	0	0
$391$	−971208.	−0.321270
$392$	22177.9	0.00728962
$393$	0	0
$394$	−3.28931e6	−1.06749
$395$	−27440.4	−0.00884908
$396$	0	0
$397$	996554.	0.317340	0.158670	−	0.987332i	$-0.449279\pi$
$397$	996554.	0.317340	0.158670	+	0.987332i	$0.449279\pi$
$398$	6.11868e6	1.93620
$399$	0	0
$400$	3.16922e6	0.990381
$401$	230543.	0.0715962	0.0357981	−	0.999359i	$-0.488603\pi$
$401$	230543.	0.0715962	0.0357981	+	0.999359i	$0.488603\pi$
$402$	0	0
$403$	−1.14811e6	−0.352143
$404$	3.99147e6	1.21669
$405$	0	0
$406$	3.61524e6	1.08848
$407$	0	0
$408$	0	0
$409$	449640.	0.132910	0.0664549	−	0.997789i	$-0.478831\pi$
$409$	449640.	0.132910	0.0664549	+	0.997789i	$0.478831\pi$
$410$	−746867.	−0.219424
$411$	0	0
$412$	−2.97769e6	−0.864245
$413$	−3.47713e6	−1.00310
$414$	0	0
$415$	42510.4	0.0121164
$416$	−2.17290e6	−0.615610
$417$	0	0
$418$	0	0
$419$	2.53917e6	0.706573	0.353287	−	0.935515i	$-0.385064\pi$
$419$	2.53917e6	0.706573	0.353287	+	0.935515i	$0.385064\pi$
$420$	0	0
$421$	−6.84578e6	−1.88242	−0.941212	−	0.337816i	$-0.890312\pi$
$421$	−6.84578e6	−1.88242	−0.941212	+	0.337816i	$0.890312\pi$
$422$	3.41042e6	0.932238
$423$	0	0
$424$	−56322.1	−0.0152147
$425$	−2.11728e6	−0.568599
$426$	0	0
$427$	−5.25121e6	−1.39377
$428$	−5779.85	−0.00152513
$429$	0	0
$430$	1.16252e6	0.303200
$431$	3.29991e6	0.855674	0.427837	−	0.903856i	$-0.359276\pi$
$431$	3.29991e6	0.855674	0.427837	+	0.903856i	$0.359276\pi$
$432$	0	0
$433$	5.06183e6	1.29744	0.648721	−	0.761026i	$-0.275304\pi$
$433$	5.06183e6	1.29744	0.648721	+	0.761026i	$0.275304\pi$
$434$	5.82798e6	1.48523
$435$	0	0
$436$	−530555.	−0.133664
$437$	−3.81343e6	−0.955241
$438$	0	0
$439$	2.33340e6	0.577868	0.288934	−	0.957349i	$-0.406699\pi$
$439$	2.33340e6	0.577868	0.288934	+	0.957349i	$0.406699\pi$
$440$	0	0
$441$	0	0
$442$	1.46240e6	0.356049
$443$	−1.14085e6	−0.276197	−0.138098	−	0.990419i	$-0.544099\pi$
$443$	−1.14085e6	−0.276197	−0.138098	+	0.990419i	$0.544099\pi$
$444$	0	0
$445$	−452056.	−0.108216
$446$	−8.03293e6	−1.91222
$447$	0	0
$448$	5.45357e6	1.28377
$449$	5.25341e6	1.22977	0.614887	−	0.788615i	$-0.289202\pi$
$449$	5.25341e6	1.22977	0.614887	+	0.788615i	$0.289202\pi$
$450$	0	0
$451$	0	0
$452$	−4.71723e6	−1.08603
$453$	0	0
$454$	3.94762e6	0.898867
$455$	325667.	0.0737471
$456$	0	0
$457$	−2.19002e6	−0.490520	−0.245260	−	0.969457i	$-0.578873\pi$
$457$	−2.19002e6	−0.490520	−0.245260	+	0.969457i	$0.578873\pi$
$458$	809040.	0.180221
$459$	0	0
$460$	−324715.	−0.0715496
$461$	−5.62999e6	−1.23383	−0.616915	−	0.787030i	$-0.711618\pi$
$461$	−5.62999e6	−1.23383	−0.616915	+	0.787030i	$0.711618\pi$
$462$	0	0
$463$	3.63260e6	0.787526	0.393763	−	0.919212i	$-0.371173\pi$
$463$	3.63260e6	0.787526	0.393763	+	0.919212i	$0.371173\pi$
$464$	−2.76439e6	−0.596081
$465$	0	0
$466$	4.20414e6	0.896834
$467$	4.18731e6	0.888470	0.444235	−	0.895910i	$-0.353476\pi$
$467$	4.18731e6	0.888470	0.444235	+	0.895910i	$0.353476\pi$
$468$	0	0
$469$	129185.	0.0271194
$470$	1.08995e6	0.227594
$471$	0	0
$472$	38904.2	0.00803788
$473$	0	0
$474$	0	0
$475$	−8.31347e6	−1.69063
$476$	−3.69792e6	−0.748065
$477$	0	0
$478$	−3.30224e6	−0.661058
$479$	5.25697e6	1.04688	0.523440	−	0.852063i	$-0.324648\pi$
$479$	5.25697e6	1.04688	0.523440	+	0.852063i	$0.324648\pi$
$480$	0	0
$481$	209011.	0.0411914
$482$	−5.42768e6	−1.06413
$483$	0	0
$484$	0	0
$485$	723875.	0.139736
$486$	0	0
$487$	6.21684e6	1.18781	0.593905	−	0.804535i	$-0.297585\pi$
$487$	6.21684e6	1.18781	0.593905	+	0.804535i	$0.297585\pi$
$488$	58753.7	0.0111683
$489$	0	0
$490$	−679624.	−0.127873
$491$	−5.02966e6	−0.941531	−0.470766	−	0.882258i	$-0.656022\pi$
$491$	−5.02966e6	−0.941531	−0.470766	+	0.882258i	$0.656022\pi$
$492$	0	0
$493$	1.84683e6	0.342223
$494$	5.74209e6	1.05865
$495$	0	0
$496$	−4.45637e6	−0.813349
$497$	−8.82111e6	−1.60189
$498$	0	0
$499$	1.38367e6	0.248759	0.124380	−	0.992235i	$-0.460306\pi$
$499$	1.38367e6	0.248759	0.124380	+	0.992235i	$0.460306\pi$
$500$	−1.42791e6	−0.255432
$501$	0	0
$502$	6.34128e6	1.12310
$503$	6.19975e6	1.09258	0.546291	−	0.837596i	$-0.316039\pi$
$503$	6.19975e6	1.09258	0.546291	+	0.837596i	$0.316039\pi$
$504$	0	0
$505$	911539.	0.159055
$506$	0	0
$507$	0	0
$508$	1.24329e6	0.213754
$509$	8.80085e6	1.50567	0.752836	−	0.658208i	$-0.228685\pi$
$509$	8.80085e6	1.50567	0.752836	+	0.658208i	$0.228685\pi$
$510$	0	0
$511$	1.19573e7	2.02573
$512$	−8.37171e6	−1.41136
$513$	0	0
$514$	−4.93365e6	−0.823683
$515$	−680020.	−0.112981
$516$	0	0
$517$	0	0
$518$	−1.06097e6	−0.173732
$519$	0	0
$520$	−3643.75	−0.000590936 0
$521$	−9.02455e6	−1.45657	−0.728285	−	0.685275i	$-0.759682\pi$
$521$	−9.02455e6	−1.45657	−0.728285	+	0.685275i	$0.759682\pi$
$522$	0	0
$523$	−2.62511e6	−0.419655	−0.209828	−	0.977738i	$-0.567290\pi$
$523$	−2.62511e6	−0.419655	−0.209828	+	0.977738i	$0.567290\pi$
$524$	2.94385e6	0.468368
$525$	0	0
$526$	−529811.	−0.0834942
$527$	2.97720e6	0.466961
$528$	0	0
$529$	−4.45016e6	−0.691412
$530$	1.72595e6	0.266893
$531$	0	0
$532$	−1.45198e7	−2.22424
$533$	−3.42664e6	−0.522456
$534$	0	0
$535$	−1319.95	−0.000199377 0
$536$	−1445.40	−0.000217308 0
$537$	0	0
$538$	−1.36123e7	−2.02757
$539$	0	0
$540$	0	0
$541$	542323.	0.0796645	0.0398322	−	0.999206i	$-0.487318\pi$
$541$	542323.	0.0796645	0.0398322	+	0.999206i	$0.487318\pi$
$542$	−1.80211e6	−0.263501
$543$	0	0
$544$	5.63462e6	0.816333
$545$	−121164.	−0.0174736
$546$	0	0
$547$	−1.81223e6	−0.258967	−0.129483	−	0.991582i	$-0.541332\pi$
$547$	−1.81223e6	−0.258967	−0.129483	+	0.991582i	$0.541332\pi$
$548$	−1.49714e6	−0.212967
$549$	0	0
$550$	0	0
$551$	7.25154e6	1.01754
$552$	0	0
$553$	639075.	0.0888667
$554$	−6.73558e6	−0.932397
$555$	0	0
$556$	−9.49056e6	−1.30198
$557$	1.00923e7	1.37833	0.689166	−	0.724603i	$-0.257977\pi$
$557$	1.00923e7	1.37833	0.689166	+	0.724603i	$0.257977\pi$
$558$	0	0
$559$	5.33366e6	0.721931
$560$	1.26408e6	0.170335
$561$	0	0
$562$	1.30626e7	1.74457
$563$	1.53068e6	0.203523	0.101761	−	0.994809i	$-0.467552\pi$
$563$	1.53068e6	0.203523	0.101761	+	0.994809i	$0.467552\pi$
$564$	0	0
$565$	−1.07728e6	−0.141974
$566$	−7.17414e6	−0.941302
$567$	0	0
$568$	98695.8	0.0128359
$569$	−6.93537e6	−0.898026	−0.449013	−	0.893525i	$-0.648224\pi$
$569$	−6.93537e6	−0.898026	−0.449013	+	0.893525i	$0.648224\pi$
$570$	0	0
$571$	1.50802e7	1.93560	0.967800	−	0.251720i	$-0.0809962\pi$
$571$	1.50802e7	1.93560	0.967800	+	0.251720i	$0.0809962\pi$
$572$	0	0
$573$	0	0
$574$	1.73942e7	2.20356
$575$	4.32996e6	0.546154
$576$	0	0
$577$	−8.05487e6	−1.00721	−0.503604	−	0.863935i	$-0.667993\pi$
$577$	−8.05487e6	−1.00721	−0.503604	+	0.863935i	$0.667993\pi$
$578$	7.54563e6	0.939455
$579$	0	0
$580$	617470.	0.0762159
$581$	−990047.	−0.121679
$582$	0	0
$583$	0	0
$584$	−133786.	−0.0162322
$585$	0	0
$586$	−1.31554e7	−1.58256
$587$	1.52908e7	1.83162	0.915811	−	0.401609i	$-0.131549\pi$
$587$	1.52908e7	1.83162	0.915811	+	0.401609i	$0.131549\pi$
$588$	0	0
$589$	1.16899e7	1.38843
$590$	−1.19219e6	−0.140999
$591$	0	0
$592$	811276.	0.0951402
$593$	1.17854e7	1.37628	0.688141	−	0.725577i	$-0.258427\pi$
$593$	1.17854e7	1.37628	0.688141	+	0.725577i	$0.258427\pi$
$594$	0	0
$595$	−844499.	−0.0977927
$596$	9.75041e6	1.12436
$597$	0	0
$598$	−2.99069e6	−0.341994
$599$	6.04672e6	0.688578	0.344289	−	0.938864i	$-0.388120\pi$
$599$	6.04672e6	0.688578	0.344289	+	0.938864i	$0.388120\pi$
$600$	0	0
$601$	−7.77962e6	−0.878561	−0.439281	−	0.898350i	$-0.644767\pi$
$601$	−7.77962e6	−0.878561	−0.439281	+	0.898350i	$0.644767\pi$
$602$	−2.70746e7	−3.04488
$603$	0	0
$604$	−8.82668e6	−0.984476
$605$	0	0
$606$	0	0
$607$	−1.50112e7	−1.65365	−0.826826	−	0.562458i	$-0.809856\pi$
$607$	−1.50112e7	−1.65365	−0.826826	+	0.562458i	$0.809856\pi$
$608$	2.21243e7	2.42722
$609$	0	0
$610$	−1.80046e6	−0.195911
$611$	5.00070e6	0.541911
$612$	0	0
$613$	320763.	0.0344772	0.0172386	−	0.999851i	$-0.494513\pi$
$613$	320763.	0.0344772	0.0172386	+	0.999851i	$0.494513\pi$
$614$	1.02665e7	1.09901
$615$	0	0
$616$	0	0
$617$	4.56312e6	0.482558	0.241279	−	0.970456i	$-0.422433\pi$
$617$	4.56312e6	0.482558	0.241279	+	0.970456i	$0.422433\pi$
$618$	0	0
$619$	5.24505e6	0.550203	0.275102	−	0.961415i	$-0.411288\pi$
$619$	5.24505e6	0.550203	0.275102	+	0.961415i	$0.411288\pi$
$620$	995399.	0.103996
$621$	0	0
$622$	1.69631e6	0.175804
$623$	1.05282e7	1.08676
$624$	0	0
$625$	9.27510e6	0.949770
$626$	−2.33165e7	−2.37808
$627$	0	0
$628$	−7.98695e6	−0.808131
$629$	−541994.	−0.0546220
$630$	0	0
$631$	−677373.	−0.0677258	−0.0338629	−	0.999426i	$-0.510781\pi$
$631$	−677373.	−0.0677258	−0.0338629	+	0.999426i	$0.510781\pi$
$632$	−7150.35	−0.000712090 0
$633$	0	0
$634$	−9.81062e6	−0.969334
$635$	283933.	0.0279435
$636$	0	0
$637$	−3.11813e6	−0.304470
$638$	0	0
$639$	0	0
$640$	−28079.5	−0.00270981
$641$	−4.03615e6	−0.387991	−0.193996	−	0.981002i	$-0.562145\pi$
$641$	−4.03615e6	−0.387991	−0.193996	+	0.981002i	$0.562145\pi$
$642$	0	0
$643$	−1.06199e7	−1.01296	−0.506482	−	0.862251i	$-0.669054\pi$
$643$	−1.06199e7	−1.01296	−0.506482	+	0.862251i	$0.669054\pi$
$644$	7.56246e6	0.718536
$645$	0	0
$646$	−1.48900e7	−1.40383
$647$	−4.12738e6	−0.387627	−0.193813	−	0.981038i	$-0.562086\pi$
$647$	−4.12738e6	−0.387627	−0.193813	+	0.981038i	$0.562086\pi$
$648$	0	0
$649$	0	0
$650$	−6.51986e6	−0.605278
$651$	0	0
$652$	−2.04594e6	−0.188484
$653$	−1.40243e7	−1.28706	−0.643529	−	0.765421i	$-0.722531\pi$
$653$	−1.40243e7	−1.28706	−0.643529	+	0.765421i	$0.722531\pi$
$654$	0	0
$655$	672291.	0.0612285
$656$	−1.33005e7	−1.20672
$657$	0	0
$658$	−2.53844e7	−2.28561
$659$	5.47205e6	0.490836	0.245418	−	0.969417i	$-0.421075\pi$
$659$	5.47205e6	0.490836	0.245418	+	0.969417i	$0.421075\pi$
$660$	0	0
$661$	−7.43446e6	−0.661829	−0.330914	−	0.943661i	$-0.607357\pi$
$661$	−7.43446e6	−0.661829	−0.330914	+	0.943661i	$0.607357\pi$
$662$	−2.27698e7	−2.01937
$663$	0	0
$664$	11077.2	0.000975014 0
$665$	−3.31591e6	−0.290770
$666$	0	0
$667$	−3.77687e6	−0.328713
$668$	1.24026e7	1.07540
$669$	0	0
$670$	44293.1	0.00381197
$671$	0	0
$672$	0	0
$673$	−2.06699e6	−0.175914	−0.0879569	−	0.996124i	$-0.528034\pi$
$673$	−2.06699e6	−0.175914	−0.0879569	+	0.996124i	$0.528034\pi$
$674$	−2.50643e7	−2.12523
$675$	0	0
$676$	−9.55022e6	−0.803798
$677$	−4.59360e6	−0.385196	−0.192598	−	0.981278i	$-0.561691\pi$
$677$	−4.59360e6	−0.385196	−0.192598	+	0.981278i	$0.561691\pi$
$678$	0	0
$679$	−1.68587e7	−1.40330
$680$	9448.75	0.000783614 0
$681$	0	0
$682$	0	0
$683$	−1.18636e7	−0.973120	−0.486560	−	0.873647i	$-0.661748\pi$
$683$	−1.18636e7	−0.973120	−0.486560	+	0.873647i	$0.661748\pi$
$684$	0	0
$685$	−341905.	−0.0278406
$686$	−6.84460e6	−0.555313
$687$	0	0
$688$	2.07026e7	1.66745
$689$	7.91867e6	0.635483
$690$	0	0
$691$	1.48387e7	1.18222	0.591112	−	0.806590i	$-0.298689\pi$
$691$	1.48387e7	1.18222	0.591112	+	0.806590i	$0.298689\pi$
$692$	−1.84444e7	−1.46420
$693$	0	0
$694$	−1.51900e7	−1.19718
$695$	−2.16737e6	−0.170205
$696$	0	0
$697$	8.88574e6	0.692806
$698$	−1.71943e7	−1.33582
$699$	0	0
$700$	1.64865e7	1.27170
$701$	−3.70231e6	−0.284562	−0.142281	−	0.989826i	$-0.545444\pi$
$701$	−3.70231e6	−0.284562	−0.142281	+	0.989826i	$0.545444\pi$
$702$	0	0
$703$	−2.12813e6	−0.162409
$704$	0	0
$705$	0	0
$706$	4.70312e6	0.355119
$707$	−2.12293e7	−1.59730
$708$	0	0
$709$	3.81874e6	0.285302	0.142651	−	0.989773i	$-0.454437\pi$
$709$	3.81874e6	0.285302	0.142651	+	0.989773i	$0.454437\pi$
$710$	−3.02446e6	−0.225166
$711$	0	0
$712$	−117796.	−0.00870821
$713$	−6.08854e6	−0.448528
$714$	0	0
$715$	0	0
$716$	8.96614e6	0.653617
$717$	0	0
$718$	1.27390e7	0.922201
$719$	2.18801e7	1.57843	0.789217	−	0.614115i	$-0.210487\pi$
$719$	2.18801e7	1.57843	0.789217	+	0.614115i	$0.210487\pi$
$720$	0	0
$721$	1.58374e7	1.13461
$722$	−3.86934e7	−2.76245
$723$	0	0
$724$	−1.43958e7	−1.02068
$725$	−8.23376e6	−0.581772
$726$	0	0
$727$	−4.10803e6	−0.288268	−0.144134	−	0.989558i	$-0.546040\pi$
$727$	−4.10803e6	−0.288268	−0.144134	+	0.989558i	$0.546040\pi$
$728$	84861.3	0.00593446
$729$	0	0
$730$	4.09976e6	0.284742
$731$	−1.38309e7	−0.957321
$732$	0	0
$733$	2.39184e7	1.64427	0.822133	−	0.569296i	$-0.192784\pi$
$733$	2.39184e7	1.64427	0.822133	+	0.569296i	$0.192784\pi$
$734$	1.78405e7	1.22227
$735$	0	0
$736$	−1.15231e7	−0.784109
$737$	0	0
$738$	0	0
$739$	2.60451e6	0.175434	0.0877172	−	0.996145i	$-0.472043\pi$
$739$	2.60451e6	0.175434	0.0877172	+	0.996145i	$0.472043\pi$
$740$	−181211.	−0.0121648
$741$	0	0
$742$	−4.01965e7	−2.68027
$743$	1.01391e7	0.673792	0.336896	−	0.941542i	$-0.390623\pi$
$743$	1.01391e7	0.673792	0.336896	+	0.941542i	$0.390623\pi$
$744$	0	0
$745$	2.22672e6	0.146985
$746$	−1.04065e7	−0.684635
$747$	0	0
$748$	0	0
$749$	30741.1	0.00200224
$750$	0	0
$751$	1.85789e7	1.20204	0.601022	−	0.799233i	$-0.294761\pi$
$751$	1.85789e7	1.20204	0.601022	+	0.799233i	$0.294761\pi$
$752$	1.94102e7	1.25166
$753$	0	0
$754$	5.68703e6	0.364299
$755$	−2.01576e6	−0.128698
$756$	0	0
$757$	2.74854e7	1.74326	0.871630	−	0.490164i	$-0.163063\pi$
$757$	2.74854e7	1.74326	0.871630	+	0.490164i	$0.163063\pi$
$758$	−2.38298e7	−1.50642
$759$	0	0
$760$	37100.4	0.00232994
$761$	5.52396e6	0.345771	0.172886	−	0.984942i	$-0.444691\pi$
$761$	5.52396e6	0.345771	0.172886	+	0.984942i	$0.444691\pi$
$762$	0	0
$763$	2.82185e6	0.175478
$764$	−1.95340e7	−1.21076
$765$	0	0
$766$	−2.74660e7	−1.69131
$767$	−5.46979e6	−0.335724
$768$	0	0
$769$	1.46025e7	0.890455	0.445227	−	0.895417i	$-0.353123\pi$
$769$	1.46025e7	0.890455	0.445227	+	0.895417i	$0.353123\pi$
$770$	0	0
$771$	0	0
$772$	1.07212e6	0.0647439
$773$	−2.35744e7	−1.41903	−0.709516	−	0.704689i	$-0.751087\pi$
$773$	−2.35744e7	−1.41903	−0.709516	+	0.704689i	$0.751087\pi$
$774$	0	0
$775$	−1.32733e7	−0.793826
$776$	188625.	0.0112446
$777$	0	0
$778$	9.30721e6	0.551278
$779$	3.48897e7	2.05994
$780$	0	0
$781$	0	0
$782$	7.75528e6	0.453504
$783$	0	0
$784$	−1.21030e7	−0.703239
$785$	−1.82399e6	−0.105645
$786$	0	0
$787$	−7.27486e6	−0.418686	−0.209343	−	0.977842i	$-0.567132\pi$
$787$	−7.27486e6	−0.418686	−0.209343	+	0.977842i	$0.567132\pi$
$788$	1.30841e7	0.750636
$789$	0	0
$790$	219117.	0.0124913
$791$	2.50894e7	1.42577
$792$	0	0
$793$	−8.26055e6	−0.466472
$794$	−7.95767e6	−0.447956
$795$	0	0
$796$	−2.43387e7	−1.36149
$797$	−3.93990e6	−0.219705	−0.109852	−	0.993948i	$-0.535038\pi$
$797$	−3.93990e6	−0.219705	−0.109852	+	0.993948i	$0.535038\pi$
$798$	0	0
$799$	−1.29675e7	−0.718604
$800$	−2.51210e7	−1.38775
$801$	0	0
$802$	−1.84093e6	−0.101065
$803$	0	0
$804$	0	0
$805$	1.72705e6	0.0939324
$806$	9.16784e6	0.497084
$807$	0	0
$808$	237526.	0.0127992
$809$	−1.09467e7	−0.588044	−0.294022	−	0.955799i	$-0.594994\pi$
$809$	−1.09467e7	−0.588044	−0.294022	+	0.955799i	$0.594994\pi$
$810$	0	0
$811$	−5.43387e6	−0.290106	−0.145053	−	0.989424i	$-0.546335\pi$
$811$	−5.43387e6	−0.290106	−0.145053	+	0.989424i	$0.546335\pi$
$812$	−1.43806e7	−0.765397
$813$	0	0
$814$	0	0
$815$	−467235.	−0.0246400
$816$	0	0
$817$	−5.43069e7	−2.84643
$818$	−3.59046e6	−0.187615
$819$	0	0
$820$	2.97087e6	0.154294
$821$	1.82088e7	0.942807	0.471404	−	0.881918i	$-0.343747\pi$
$821$	1.82088e7	0.942807	0.471404	+	0.881918i	$0.343747\pi$
$822$	0	0
$823$	−1.70039e7	−0.875084	−0.437542	−	0.899198i	$-0.644151\pi$
$823$	−1.70039e7	−0.875084	−0.437542	+	0.899198i	$0.644151\pi$
$824$	−177198.	−0.00909160
$825$	0	0
$826$	2.77656e7	1.41598
$827$	3.51645e6	0.178789	0.0893945	−	0.995996i	$-0.471507\pi$
$827$	3.51645e6	0.178789	0.0893945	+	0.995996i	$0.471507\pi$
$828$	0	0
$829$	1.67854e7	0.848293	0.424146	−	0.905594i	$-0.360574\pi$
$829$	1.67854e7	0.848293	0.424146	+	0.905594i	$0.360574\pi$
$830$	−339453.	−0.0171035
$831$	0	0
$832$	8.57888e6	0.429657
$833$	8.08573e6	0.403745
$834$	0	0
$835$	2.83240e6	0.140585
$836$	0	0
$837$	0	0
$838$	−2.02758e7	−0.997396
$839$	2.96212e7	1.45277	0.726387	−	0.687286i	$-0.241198\pi$
$839$	2.96212e7	1.45277	0.726387	+	0.687286i	$0.241198\pi$
$840$	0	0
$841$	−1.33291e7	−0.649849
$842$	5.46648e7	2.65722
$843$	0	0
$844$	−1.35659e7	−0.655529
$845$	−2.18100e6	−0.105078
$846$	0	0
$847$	0	0
$848$	3.07363e7	1.46778
$849$	0	0
$850$	1.69069e7	0.802632
$851$	1.10841e6	0.0524658
$852$	0	0
$853$	−202485.	−0.00952841	−0.00476420	−	0.999989i	$-0.501516\pi$
$853$	−202485.	−0.00952841	−0.00476420	+	0.999989i	$0.501516\pi$
$854$	4.19319e7	1.96743
$855$	0	0
$856$	−343.950	−1.60439e−5 0
$857$	−2.29630e7	−1.06801	−0.534006	−	0.845481i	$-0.679314\pi$
$857$	−2.29630e7	−1.06801	−0.534006	+	0.845481i	$0.679314\pi$
$858$	0	0
$859$	−1.69282e7	−0.782760	−0.391380	−	0.920229i	$-0.628002\pi$
$859$	−1.69282e7	−0.782760	−0.391380	+	0.920229i	$0.628002\pi$
$860$	−4.62424e6	−0.213204
$861$	0	0
$862$	−2.63504e7	−1.20787
$863$	−3.04172e6	−0.139025	−0.0695124	−	0.997581i	$-0.522144\pi$
$863$	−3.04172e6	−0.139025	−0.0695124	+	0.997581i	$0.522144\pi$
$864$	0	0
$865$	−4.21218e6	−0.191411
$866$	−4.04197e7	−1.83146
$867$	0	0
$868$	−2.31824e7	−1.04438
$869$	0	0
$870$	0	0
$871$	203217.	0.00907644
$872$	−31572.5	−0.00140610
$873$	0	0
$874$	3.04510e7	1.34841
$875$	7.59459e6	0.335339
$876$	0	0
$877$	291799.	0.0128111	0.00640553	−	0.999979i	$-0.497961\pi$
$877$	291799.	0.0128111	0.00640553	+	0.999979i	$0.497961\pi$
$878$	−1.86327e7	−0.815716
$879$	0	0
$880$	0	0
$881$	−1.98246e7	−0.860529	−0.430264	−	0.902703i	$-0.641580\pi$
$881$	−1.98246e7	−0.860529	−0.430264	+	0.902703i	$0.641580\pi$
$882$	0	0
$883$	−9.63791e6	−0.415988	−0.207994	−	0.978130i	$-0.566693\pi$
$883$	−9.63791e6	−0.415988	−0.207994	+	0.978130i	$0.566693\pi$
$884$	−5.81709e6	−0.250366
$885$	0	0
$886$	9.10990e6	0.389878
$887$	−3.81684e7	−1.62890	−0.814451	−	0.580233i	$-0.802962\pi$
$887$	−3.81684e7	−1.62890	−0.814451	+	0.580233i	$0.802962\pi$
$888$	0	0
$889$	−6.61266e6	−0.280622
$890$	3.60976e6	0.152758
$891$	0	0
$892$	3.19532e7	1.34463
$893$	−5.09167e7	−2.13664
$894$	0	0
$895$	2.04761e6	0.0854457
$896$	653959.	0.0272132
$897$	0	0
$898$	−4.19495e7	−1.73594
$899$	1.15778e7	0.477780
$900$	0	0
$901$	−2.05342e7	−0.842686
$902$	0	0
$903$	0	0
$904$	−280715.	−0.0114247
$905$	−3.28758e6	−0.133431
$906$	0	0
$907$	3.47894e7	1.40420	0.702100	−	0.712079i	$-0.252246\pi$
$907$	3.47894e7	1.40420	0.702100	+	0.712079i	$0.252246\pi$
$908$	−1.57027e7	−0.632063
$909$	0	0
$910$	−2.60051e6	−0.104101
$911$	1.49429e7	0.596540	0.298270	−	0.954482i	$-0.403590\pi$
$911$	1.49429e7	0.596540	0.298270	+	0.954482i	$0.403590\pi$
$912$	0	0
$913$	0	0
$914$	1.74877e7	0.692416
$915$	0	0
$916$	−3.21818e6	−0.126728
$917$	−1.56573e7	−0.614886
$918$	0	0
$919$	−3.35072e7	−1.30873	−0.654363	−	0.756180i	$-0.727063\pi$
$919$	−3.35072e7	−1.30873	−0.654363	+	0.756180i	$0.727063\pi$
$920$	−19323.3	−0.000752681 0
$921$	0	0
$922$	4.49566e7	1.74167
$923$	−1.38763e7	−0.536128
$924$	0	0
$925$	2.41639e6	0.0928565
$926$	−2.90070e7	−1.11167
$927$	0	0
$928$	2.19121e7	0.835246
$929$	−1.87850e7	−0.714120	−0.357060	−	0.934081i	$-0.616221\pi$
$929$	−1.87850e7	−0.714120	−0.357060	+	0.934081i	$0.616221\pi$
$930$	0	0
$931$	3.17485e7	1.20046
$932$	−1.67231e7	−0.630634
$933$	0	0
$934$	−3.34365e7	−1.25416
$935$	0	0
$936$	0	0
$937$	−1.93571e7	−0.720262	−0.360131	−	0.932902i	$-0.617268\pi$
$937$	−1.93571e7	−0.720262	−0.360131	+	0.932902i	$0.617268\pi$
$938$	−1.03157e6	−0.0382816
$939$	0	0
$940$	−4.33557e6	−0.160039
$941$	1.53228e7	0.564110	0.282055	−	0.959398i	$-0.408984\pi$
$941$	1.53228e7	0.564110	0.282055	+	0.959398i	$0.408984\pi$
$942$	0	0
$943$	−1.81719e7	−0.665458
$944$	−2.12310e7	−0.775425
$945$	0	0
$946$	0	0
$947$	8.60741e6	0.311887	0.155944	−	0.987766i	$-0.450158\pi$
$947$	8.60741e6	0.311887	0.155944	+	0.987766i	$0.450158\pi$
$948$	0	0
$949$	1.88098e7	0.677981
$950$	6.63847e7	2.38649
$951$	0	0
$952$	−220057.	−0.00786943
$953$	1.08192e7	0.385890	0.192945	−	0.981210i	$-0.438196\pi$
$953$	1.08192e7	0.385890	0.192945	+	0.981210i	$0.438196\pi$
$954$	0	0
$955$	−4.46101e6	−0.158279
$956$	1.31356e7	0.464841
$957$	0	0
$958$	−4.19779e7	−1.47777
$959$	7.96280e6	0.279589
$960$	0	0
$961$	−9.96499e6	−0.348071
$962$	−1.66899e6	−0.0581456
$963$	0	0
$964$	2.15901e7	0.748276
$965$	244841.	0.00846381
$966$	0	0
$967$	2.85326e7	0.981241	0.490621	−	0.871373i	$-0.336770\pi$
$967$	2.85326e7	0.981241	0.490621	+	0.871373i	$0.336770\pi$
$968$	0	0
$969$	0	0
$970$	−5.78028e6	−0.197251
$971$	1.97620e7	0.672640	0.336320	−	0.941748i	$-0.390818\pi$
$971$	1.97620e7	0.672640	0.336320	+	0.941748i	$0.390818\pi$
$972$	0	0
$973$	5.04772e7	1.70928
$974$	−4.96426e7	−1.67671
$975$	0	0
$976$	−3.20633e7	−1.07742
$977$	3.64586e7	1.22198	0.610990	−	0.791638i	$-0.290772\pi$
$977$	3.64586e7	1.22198	0.610990	+	0.791638i	$0.290772\pi$
$978$	0	0
$979$	0	0
$980$	2.70339e6	0.0899174
$981$	0	0
$982$	4.01628e7	1.32906
$983$	−2.22458e7	−0.734286	−0.367143	−	0.930165i	$-0.619664\pi$
$983$	−2.22458e7	−0.734286	−0.367143	+	0.930165i	$0.619664\pi$
$984$	0	0
$985$	2.98804e6	0.0981287
$986$	−1.47473e7	−0.483080
$987$	0	0
$988$	−2.28407e7	−0.744420
$989$	2.82850e7	0.919531
$990$	0	0
$991$	5.60917e7	1.81432	0.907161	−	0.420784i	$-0.138245\pi$
$991$	5.60917e7	1.81432	0.907161	+	0.420784i	$0.138245\pi$
$992$	3.53237e7	1.13969
$993$	0	0
$994$	7.04383e7	2.26122
$995$	−5.55827e6	−0.177985
$996$	0	0
$997$	3.56861e7	1.13700	0.568500	−	0.822683i	$-0.307524\pi$
$997$	3.56861e7	1.13700	0.568500	+	0.822683i	$0.307524\pi$
$998$	−1.10488e7	−0.351148
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bg.1.1		8
3.2	odd	2		121.6.a.g.1.8		8
11.3	even	5		99.6.f.a.64.1		16
11.4	even	5		99.6.f.a.82.1		16
11.10	odd	2		1089.6.a.bb.1.8		8
33.14	odd	10		11.6.c.a.9.4	yes	16
33.26	odd	10		11.6.c.a.5.4	✓	16
33.32	even	2		121.6.a.i.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.6.c.a.5.4	✓	16	33.26	odd	10
11.6.c.a.9.4	yes	16	33.14	odd	10
99.6.f.a.64.1		16	11.3	even	5
99.6.f.a.82.1		16	11.4	even	5
121.6.a.g.1.8		8	3.2	odd	2
121.6.a.i.1.1		8	33.32	even	2
1089.6.a.bb.1.8		8	11.10	odd	2
1089.6.a.bg.1.1		8	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)

\(f(q)\)	\(=\)	\(q-7.98519 q^{2} +31.7633 q^{4} +7.25383 q^{5} -168.938 q^{7} +1.89018 q^{8} +O(q^{10})\)	\(q-7.98519 q^{2} +31.7633 q^{4} +7.25383 q^{5} -168.938 q^{7} +1.89018 q^{8} -57.9232 q^{10} -265.753 q^{13} +1349.01 q^{14} -1031.52 q^{16} +689.133 q^{17} +2705.87 q^{19} +230.406 q^{20} -1409.32 q^{23} -3072.38 q^{25} +2122.09 q^{26} -5366.04 q^{28} +2679.93 q^{29} +4320.20 q^{31} +8176.39 q^{32} -5502.86 q^{34} -1225.45 q^{35} -786.487 q^{37} -21606.9 q^{38} +13.7111 q^{40} +12894.1 q^{41} -20070.0 q^{43} +11253.7 q^{46} -18817.1 q^{47} +11733.2 q^{49} +24533.6 q^{50} -8441.18 q^{52} -29797.1 q^{53} -319.325 q^{56} -21399.7 q^{58} +20582.3 q^{59} +31083.6 q^{61} -34497.7 q^{62} -32281.4 q^{64} -1927.72 q^{65} -764.687 q^{67} +21889.1 q^{68} +9785.46 q^{70} +52215.0 q^{71} -70779.2 q^{73} +6280.25 q^{74} +85947.4 q^{76} -3782.89 q^{79} -7482.46 q^{80} -102962. q^{82} +5860.40 q^{83} +4998.85 q^{85} +160263. q^{86} -62319.7 q^{89} +44895.8 q^{91} -44764.6 q^{92} +150258. q^{94} +19627.9 q^{95} +99792.1 q^{97} -93691.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 86 q^{4} - 70 q^{5} - 292 q^{7} + 294 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 86 * q^4 - 70 * q^5 - 292 * q^7 + 294 * q^8	\( 8 q + 8 q^{2} + 86 q^{4} - 70 q^{5} - 292 q^{7} + 294 q^{8} - 336 q^{10} - 514 q^{13} + 1020 q^{14} + 722 q^{16} + 3776 q^{17} - 4108 q^{19} - 10978 q^{20} + 5424 q^{23} - 2722 q^{25} - 508 q^{26} - 24736 q^{28} + 8190 q^{29} + 16268 q^{31} + 29406 q^{32} - 13483 q^{34} + 28096 q^{35} - 866 q^{37} - 157 q^{38} - 49118 q^{40} + 43048 q^{41} - 12188 q^{43} - 6814 q^{46} + 22268 q^{47} - 4602 q^{49} + 44588 q^{50} + 13838 q^{52} - 32818 q^{53} - 35316 q^{56} - 46852 q^{58} + 34748 q^{59} - 18758 q^{61} - 20586 q^{62} - 2110 q^{64} + 20642 q^{65} - 32100 q^{67} - 62733 q^{68} + 32146 q^{70} - 6140 q^{71} + 101260 q^{73} - 46654 q^{74} + 87403 q^{76} - 95268 q^{79} - 106178 q^{80} + 139095 q^{82} - 45636 q^{83} + 150442 q^{85} + 354907 q^{86} - 74182 q^{89} + 286656 q^{91} - 34806 q^{92} + 513780 q^{94} + 137168 q^{95} - 167632 q^{97} - 371846 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 86 * q^4 - 70 * q^5 - 292 * q^7 + 294 * q^8 - 336 * q^10 - 514 * q^13 + 1020 * q^14 + 722 * q^16 + 3776 * q^17 - 4108 * q^19 - 10978 * q^20 + 5424 * q^23 - 2722 * q^25 - 508 * q^26 - 24736 * q^28 + 8190 * q^29 + 16268 * q^31 + 29406 * q^32 - 13483 * q^34 + 28096 * q^35 - 866 * q^37 - 157 * q^38 - 49118 * q^40 + 43048 * q^41 - 12188 * q^43 - 6814 * q^46 + 22268 * q^47 - 4602 * q^49 + 44588 * q^50 + 13838 * q^52 - 32818 * q^53 - 35316 * q^56 - 46852 * q^58 + 34748 * q^59 - 18758 * q^61 - 20586 * q^62 - 2110 * q^64 + 20642 * q^65 - 32100 * q^67 - 62733 * q^68 + 32146 * q^70 - 6140 * q^71 + 101260 * q^73 - 46654 * q^74 + 87403 * q^76 - 95268 * q^79 - 106178 * q^80 + 139095 * q^82 - 45636 * q^83 + 150442 * q^85 + 354907 * q^86 - 74182 * q^89 + 286656 * q^91 - 34806 * q^92 + 513780 * q^94 + 137168 * q^95 - 167632 * q^97 - 371846 * q^98

Introduction

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