LMFDB - Embedded newform 1089.6.a.j.1.2

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{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
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			1.2
Root			$-6.15207$ 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)

$f(q)$	$=$	$q+4.15207 q^{2} -14.7603 q^{4} +37.5207 q^{5} +76.4793 q^{7} -194.152 q^{8} +O(q^{10})$	$q+4.15207 q^{2} -14.7603 q^{4} +37.5207 q^{5} +76.4793 q^{7} -194.152 q^{8} +155.788 q^{10} -169.779 q^{13} +317.547 q^{14} -333.802 q^{16} -0.875959 q^{17} +817.825 q^{19} -553.818 q^{20} -749.657 q^{23} -1717.20 q^{25} -704.932 q^{26} -1128.86 q^{28} +6040.65 q^{29} -1475.69 q^{31} +4826.90 q^{32} -3.63704 q^{34} +2869.56 q^{35} -15862.7 q^{37} +3395.66 q^{38} -7284.72 q^{40} -7626.27 q^{41} +18279.5 q^{43} -3112.62 q^{46} +12878.1 q^{47} -10957.9 q^{49} -7129.93 q^{50} +2505.99 q^{52} -21763.5 q^{53} -14848.6 q^{56} +25081.2 q^{58} +1168.45 q^{59} -14026.3 q^{61} -6127.16 q^{62} +30723.3 q^{64} -6370.21 q^{65} +36954.6 q^{67} +12.9295 q^{68} +11914.6 q^{70} +37513.1 q^{71} -80452.0 q^{73} -65862.9 q^{74} -12071.4 q^{76} +62139.9 q^{79} -12524.5 q^{80} -31664.8 q^{82} -11905.8 q^{83} -32.8666 q^{85} +75897.7 q^{86} -146348. q^{89} -12984.5 q^{91} +11065.2 q^{92} +53470.8 q^{94} +30685.3 q^{95} -138129. q^{97} -45498.0 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5 q^{2} + 37 q^{4} - 58 q^{5} + 286 q^{7} - 375 q^{8}+O(q^{10}) $ 2 * q - 5 * q^2 + 37 * q^4 - 58 * q^5 + 286 * q^7 - 375 * q^8	$ 2 q - 5 q^{2} + 37 q^{4} - 58 q^{5} + 286 q^{7} - 375 q^{8} + 1030 q^{10} + 166 q^{13} - 1600 q^{14} - 335 q^{16} - 800 q^{17} + 1476 q^{19} - 5498 q^{20} + 3370 q^{23} + 4282 q^{25} - 3778 q^{26} + 9716 q^{28} + 6600 q^{29} - 7528 q^{31} + 10625 q^{32} + 7310 q^{34} - 17144 q^{35} - 29916 q^{37} - 2628 q^{38} + 9990 q^{40} - 5780 q^{41} + 16656 q^{43} - 40816 q^{46} - 7850 q^{47} + 16134 q^{49} - 62035 q^{50} + 19886 q^{52} - 14178 q^{53} - 52740 q^{56} + 19962 q^{58} - 17300 q^{59} + 2946 q^{61} + 49264 q^{62} - 22303 q^{64} - 38444 q^{65} + 31336 q^{67} - 41350 q^{68} + 195080 q^{70} + 33810 q^{71} - 60644 q^{73} + 62754 q^{74} + 21996 q^{76} - 1870 q^{79} - 12410 q^{80} - 48562 q^{82} - 58296 q^{83} + 76300 q^{85} + 90756 q^{86} - 92388 q^{89} + 57368 q^{91} + 224300 q^{92} + 243176 q^{94} - 32184 q^{95} + 7120 q^{97} - 293445 q^{98}+O(q^{100}) $ 2 * q - 5 * q^2 + 37 * q^4 - 58 * q^5 + 286 * q^7 - 375 * q^8 + 1030 * q^10 + 166 * q^13 - 1600 * q^14 - 335 * q^16 - 800 * q^17 + 1476 * q^19 - 5498 * q^20 + 3370 * q^23 + 4282 * q^25 - 3778 * q^26 + 9716 * q^28 + 6600 * q^29 - 7528 * q^31 + 10625 * q^32 + 7310 * q^34 - 17144 * q^35 - 29916 * q^37 - 2628 * q^38 + 9990 * q^40 - 5780 * q^41 + 16656 * q^43 - 40816 * q^46 - 7850 * q^47 + 16134 * q^49 - 62035 * q^50 + 19886 * q^52 - 14178 * q^53 - 52740 * q^56 + 19962 * q^58 - 17300 * q^59 + 2946 * q^61 + 49264 * q^62 - 22303 * q^64 - 38444 * q^65 + 31336 * q^67 - 41350 * q^68 + 195080 * q^70 + 33810 * q^71 - 60644 * q^73 + 62754 * q^74 + 21996 * q^76 - 1870 * q^79 - 12410 * q^80 - 48562 * q^82 - 58296 * q^83 + 76300 * q^85 + 90756 * q^86 - 92388 * q^89 + 57368 * q^91 + 224300 * q^92 + 243176 * q^94 - 32184 * q^95 + 7120 * q^97 - 293445 * 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 $

$2$	4.15207	0.733989	0.366994	−	0.930223i	$-0.380387\pi$
$2$	4.15207	0.733989	0.366994	+	0.930223i	$0.380387\pi$
$3$	0	0
$3$	0	0
$4$	−14.7603	−0.461261
$5$	37.5207	0.671190	0.335595	−	0.942006i	$-0.391063\pi$
$5$	37.5207	0.671190	0.335595	+	0.942006i	$0.391063\pi$
$6$	0	0
$7$	76.4793	0.589928	0.294964	−	0.955508i	$-0.404692\pi$
$7$	76.4793	0.589928	0.294964	+	0.955508i	$0.404692\pi$
$8$	−194.152	−1.07255
$9$	0	0
$10$	155.788	0.492646
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−169.779	−0.278628	−0.139314	−	0.990248i	$-0.544490\pi$
$13$	−169.779	−0.278628	−0.139314	+	0.990248i	$0.544490\pi$
$14$	317.547	0.433000
$15$	0	0
$16$	−333.802	−0.325978
$17$	−0.875959	−0.000735126 0	−0.000367563	−	1.00000i	$-0.500117\pi$
$17$	−0.875959	−0.000735126 0	−0.000367563		1.00000i	$0.500117\pi$
$18$	0	0
$19$	817.825	0.519728	0.259864	−	0.965645i	$-0.416322\pi$
$19$	817.825	0.519728	0.259864	+	0.965645i	$0.416322\pi$
$20$	−553.818	−0.309594
$21$	0	0
$22$	0	0
$23$	−749.657	−0.295490	−0.147745	−	0.989025i	$-0.547201\pi$
$23$	−749.657	−0.295490	−0.147745	+	0.989025i	$0.547201\pi$
$24$	0	0
$25$	−1717.20	−0.549504
$26$	−704.932	−0.204510
$27$	0	0
$28$	−1128.86	−0.272110
$29$	6040.65	1.33379	0.666897	−	0.745150i	$-0.267622\pi$
$29$	6040.65	1.33379	0.666897	+	0.745150i	$0.267622\pi$
$30$	0	0
$31$	−1475.69	−0.275798	−0.137899	−	0.990446i	$-0.544035\pi$
$31$	−1475.69	−0.275798	−0.137899	+	0.990446i	$0.544035\pi$
$32$	4826.90	0.833284
$33$	0	0
$34$	−3.63704	−0.000539574 0
$35$	2869.56	0.395954
$36$	0	0
$37$	−15862.7	−1.90490	−0.952450	−	0.304694i	$-0.901446\pi$
$37$	−15862.7	−1.90490	−0.952450	+	0.304694i	$0.901446\pi$
$38$	3395.66	0.381475
$39$	0	0
$40$	−7284.72	−0.719884
$41$	−7626.27	−0.708521	−0.354260	−	0.935147i	$-0.615267\pi$
$41$	−7626.27	−0.708521	−0.354260	+	0.935147i	$0.615267\pi$
$42$	0	0
$43$	18279.5	1.50762	0.753812	−	0.657090i	$-0.228213\pi$
$43$	18279.5	1.50762	0.753812	+	0.657090i	$0.228213\pi$
$44$	0	0
$45$	0	0
$46$	−3112.62	−0.216886
$47$	12878.1	0.850370	0.425185	−	0.905106i	$-0.360209\pi$
$47$	12878.1	0.850370	0.425185	+	0.905106i	$0.360209\pi$
$48$	0	0
$49$	−10957.9	−0.651985
$50$	−7129.93	−0.403330
$51$	0	0
$52$	2505.99	0.128520
$53$	−21763.5	−1.06424	−0.532118	−	0.846670i	$-0.678604\pi$
$53$	−21763.5	−1.06424	−0.532118	+	0.846670i	$0.678604\pi$
$54$	0	0
$55$	0	0
$56$	−14848.6	−0.632726
$57$	0	0
$58$	25081.2	0.978990
$59$	1168.45	0.0436999	0.0218500	−	0.999761i	$-0.493044\pi$
$59$	1168.45	0.0436999	0.0218500	+	0.999761i	$0.493044\pi$
$60$	0	0
$61$	−14026.3	−0.482635	−0.241318	−	0.970446i	$-0.577580\pi$
$61$	−14026.3	−0.482635	−0.241318	+	0.970446i	$0.577580\pi$
$62$	−6127.16	−0.202432
$63$	0	0
$64$	30723.3	0.937600
$65$	−6370.21	−0.187012
$66$	0	0
$67$	36954.6	1.00573	0.502865	−	0.864365i	$-0.332279\pi$
$67$	36954.6	1.00573	0.502865	+	0.864365i	$0.332279\pi$
$68$	12.9295	0.000339084 0
$69$	0	0
$70$	11914.6	0.290626
$71$	37513.1	0.883155	0.441578	−	0.897223i	$-0.354419\pi$
$71$	37513.1	0.883155	0.441578	+	0.897223i	$0.354419\pi$
$72$	0	0
$73$	−80452.0	−1.76697	−0.883486	−	0.468458i	$-0.844810\pi$
$73$	−80452.0	−1.76697	−0.883486	+	0.468458i	$0.844810\pi$
$74$	−65862.9	−1.39818
$75$	0	0
$76$	−12071.4	−0.239730
$77$	0	0
$78$	0	0
$79$	62139.9	1.12022	0.560109	−	0.828419i	$-0.310759\pi$
$79$	62139.9	1.12022	0.560109	+	0.828419i	$0.310759\pi$
$80$	−12524.5	−0.218793
$81$	0	0
$82$	−31664.8	−0.520046
$83$	−11905.8	−0.189699	−0.0948495	−	0.995492i	$-0.530237\pi$
$83$	−11905.8	−0.189699	−0.0948495	+	0.995492i	$0.530237\pi$
$84$	0	0
$85$	−32.8666	−0.000493409 0
$86$	75897.7	1.10658
$87$	0	0
$88$	0	0
$89$	−146348.	−1.95844	−0.979220	−	0.202800i	$-0.934996\pi$
$89$	−146348.	−1.95844	−0.979220	+	0.202800i	$0.934996\pi$
$90$	0	0
$91$	−12984.5	−0.164370
$92$	11065.2	0.136298
$93$	0	0
$94$	53470.8	0.624162
$95$	30685.3	0.348836
$96$	0	0
$97$	−138129.	−1.49058	−0.745291	−	0.666740i	$-0.767689\pi$
$97$	−138129.	−1.49058	−0.745291	+	0.666740i	$0.767689\pi$
$98$	−45498.0	−0.478550
$99$	0	0
$100$	25346.4	0.253464
$101$	36615.1	0.357155	0.178577	−	0.983926i	$-0.442850\pi$
$101$	36615.1	0.357155	0.178577	+	0.983926i	$0.442850\pi$
$102$	0	0
$103$	45198.5	0.419789	0.209895	−	0.977724i	$-0.432688\pi$
$103$	45198.5	0.419789	0.209895	+	0.977724i	$0.432688\pi$
$104$	32962.9	0.298842
$105$	0	0
$106$	−90363.4	−0.781138
$107$	−28086.8	−0.237161	−0.118580	−	0.992944i	$-0.537834\pi$
$107$	−28086.8	−0.237161	−0.118580	+	0.992944i	$0.537834\pi$
$108$	0	0
$109$	33852.9	0.272916	0.136458	−	0.990646i	$-0.456428\pi$
$109$	33852.9	0.272916	0.136458	+	0.990646i	$0.456428\pi$
$110$	0	0
$111$	0	0
$112$	−25528.9	−0.192304
$113$	−168559.	−1.24181	−0.620904	−	0.783886i	$-0.713234\pi$
$113$	−168559.	−1.24181	−0.620904	+	0.783886i	$0.713234\pi$
$114$	0	0
$115$	−28127.6	−0.198330
$116$	−89162.1	−0.615227
$117$	0	0
$118$	4851.49	0.0320752
$119$	−66.9928	−0.000433671 0
$120$	0	0
$121$	0	0
$122$	−58238.2	−0.354249
$123$	0	0
$124$	21781.7	0.127215
$125$	−181683.	−1.04001
$126$	0	0
$127$	336859.	1.85327	0.926636	−	0.375960i	$-0.122687\pi$
$127$	336859.	1.85327	0.926636	+	0.375960i	$0.122687\pi$
$128$	−26895.7	−0.145097
$129$	0	0
$130$	−26449.5	−0.137265
$131$	−83896.5	−0.427135	−0.213568	−	0.976928i	$-0.568508\pi$
$131$	−83896.5	−0.427135	−0.213568	+	0.976928i	$0.568508\pi$
$132$	0	0
$133$	62546.7	0.306602
$134$	153438.	0.738195
$135$	0	0
$136$	170.069	0.000788458 0
$137$	196507.	0.894492	0.447246	−	0.894411i	$-0.352405\pi$
$137$	196507.	0.894492	0.447246	+	0.894411i	$0.352405\pi$
$138$	0	0
$139$	−263232.	−1.15559	−0.577793	−	0.816183i	$-0.696086\pi$
$139$	−263232.	−1.15559	−0.577793	+	0.816183i	$0.696086\pi$
$140$	−42355.6	−0.182638
$141$	0	0
$142$	155757.	0.648226
$143$	0	0
$144$	0	0
$145$	226649.	0.895230
$146$	−334042.	−1.29694
$147$	0	0
$148$	234139.	0.878655
$149$	−337412.	−1.24507	−0.622536	−	0.782591i	$-0.713898\pi$
$149$	−337412.	−1.24507	−0.622536	+	0.782591i	$0.713898\pi$
$150$	0	0
$151$	157583.	0.562428	0.281214	−	0.959645i	$-0.409263\pi$
$151$	157583.	0.562428	0.281214	+	0.959645i	$0.409263\pi$
$152$	−158782.	−0.557434
$153$	0	0
$154$	0	0
$155$	−55368.8	−0.185113
$156$	0	0
$157$	5369.59	0.0173857	0.00869285	−	0.999962i	$-0.497233\pi$
$157$	5369.59	0.0173857	0.00869285	+	0.999962i	$0.497233\pi$
$158$	258009.	0.822228
$159$	0	0
$160$	181109.	0.559292
$161$	−57333.2	−0.174318
$162$	0	0
$163$	−503102.	−1.48316	−0.741579	−	0.670866i	$-0.765923\pi$
$163$	−503102.	−1.48316	−0.741579	+	0.670866i	$0.765923\pi$
$164$	112566.	0.326813
$165$	0	0
$166$	−49433.9	−0.139237
$167$	−584523.	−1.62185	−0.810924	−	0.585151i	$-0.801035\pi$
$167$	−584523.	−1.62185	−0.810924	+	0.585151i	$0.801035\pi$
$168$	0	0
$169$	−342468.	−0.922367
$170$	−136.464	−0.000362157 0
$171$	0	0
$172$	−269811.	−0.695407
$173$	−55544.0	−0.141098	−0.0705491	−	0.997508i	$-0.522475\pi$
$173$	−55544.0	−0.141098	−0.0705491	+	0.997508i	$0.522475\pi$
$174$	0	0
$175$	−131330.	−0.324168
$176$	0	0
$177$	0	0
$178$	−607645.	−1.43747
$179$	−215136.	−0.501858	−0.250929	−	0.968006i	$-0.580736\pi$
$179$	−215136.	−0.501858	−0.250929	+	0.968006i	$0.580736\pi$
$180$	0	0
$181$	682113.	1.54761	0.773803	−	0.633427i	$-0.218352\pi$
$181$	682113.	1.54761	0.773803	+	0.633427i	$0.218352\pi$
$182$	−53912.7	−0.120646
$183$	0	0
$184$	145547.	0.316927
$185$	−595178.	−1.27855
$186$	0	0
$187$	0	0
$188$	−190085.	−0.392242
$189$	0	0
$190$	127408.	0.256042
$191$	89697.4	0.177908	0.0889542	−	0.996036i	$-0.471648\pi$
$191$	89697.4	0.177908	0.0889542	+	0.996036i	$0.471648\pi$
$192$	0	0
$193$	−657062.	−1.26973	−0.634867	−	0.772621i	$-0.718945\pi$
$193$	−657062.	−1.26973	−0.634867	+	0.772621i	$0.718945\pi$
$194$	−573521.	−1.09407
$195$	0	0
$196$	161742.	0.300735
$197$	720419.	1.32257	0.661286	−	0.750134i	$-0.270011\pi$
$197$	720419.	1.32257	0.661286	+	0.750134i	$0.270011\pi$
$198$	0	0
$199$	140848.	0.252127	0.126063	−	0.992022i	$-0.459766\pi$
$199$	140848.	0.252127	0.126063	+	0.992022i	$0.459766\pi$
$200$	333398.	0.589370
$201$	0	0
$202$	152028.	0.262148
$203$	461985.	0.786842
$204$	0	0
$205$	−286143.	−0.475552
$206$	187667.	0.308121
$207$	0	0
$208$	56672.4	0.0908266
$209$	0	0
$210$	0	0
$211$	111929.	0.173075	0.0865377	−	0.996249i	$-0.472420\pi$
$211$	111929.	0.173075	0.0865377	+	0.996249i	$0.472420\pi$
$212$	321236.	0.490890
$213$	0	0
$214$	−116618.	−0.174073
$215$	685859.	1.01190
$216$	0	0
$217$	−112860.	−0.162701
$218$	140559.	0.200317
$219$	0	0
$220$	0	0
$221$	148.719	0.000204826 0
$222$	0	0
$223$	−906826.	−1.22113	−0.610565	−	0.791966i	$-0.709058\pi$
$223$	−906826.	−1.22113	−0.610565	+	0.791966i	$0.709058\pi$
$224$	369158.	0.491578
$225$	0	0
$226$	−699867.	−0.911474
$227$	−1.35715e6	−1.74809	−0.874046	−	0.485844i	$-0.838512\pi$
$227$	−1.35715e6	−1.74809	−0.874046	+	0.485844i	$0.838512\pi$
$228$	0	0
$229$	−344956.	−0.434685	−0.217342	−	0.976095i	$-0.569739\pi$
$229$	−344956.	−0.434685	−0.217342	+	0.976095i	$0.569739\pi$
$230$	−116788.	−0.145572
$231$	0	0
$232$	−1.17281e6	−1.43056
$233$	−1.18163e6	−1.42591	−0.712955	−	0.701210i	$-0.752643\pi$
$233$	−1.18163e6	−1.42591	−0.712955	+	0.701210i	$0.752643\pi$
$234$	0	0
$235$	483196.	0.570760
$236$	−17246.7	−0.0201570
$237$	0	0
$238$	−278.158	−0.000318310 0
$239$	−723612.	−0.819429	−0.409714	−	0.912214i	$-0.634372\pi$
$239$	−723612.	−0.819429	−0.409714	+	0.912214i	$0.634372\pi$
$240$	0	0
$241$	−394322.	−0.437329	−0.218664	−	0.975800i	$-0.570170\pi$
$241$	−394322.	−0.437329	−0.218664	+	0.975800i	$0.570170\pi$
$242$	0	0
$243$	0	0
$244$	207033.	0.222621
$245$	−411148.	−0.437606
$246$	0	0
$247$	−138849.	−0.144811
$248$	286508.	0.295806
$249$	0	0
$250$	−754358.	−0.763357
$251$	−1.49884e6	−1.50165	−0.750827	−	0.660499i	$-0.770345\pi$
$251$	−1.49884e6	−1.50165	−0.750827	+	0.660499i	$0.770345\pi$
$252$	0	0
$253$	0	0
$254$	1.39866e6	1.36028
$255$	0	0
$256$	−1.09482e6	−1.04410
$257$	512138.	0.483676	0.241838	−	0.970317i	$-0.422250\pi$
$257$	512138.	0.483676	0.241838	+	0.970317i	$0.422250\pi$
$258$	0	0
$259$	−1.21317e6	−1.12375
$260$	94026.4	0.0862614
$261$	0	0
$262$	−348344.	−0.313513
$263$	154350.	0.137600	0.0687998	−	0.997630i	$-0.478083\pi$
$263$	154350.	0.137600	0.0687998	+	0.997630i	$0.478083\pi$
$264$	0	0
$265$	−816580.	−0.714305
$266$	259698.	0.225043
$267$	0	0
$268$	−545463.	−0.463904
$269$	−2.27104e6	−1.91357	−0.956783	−	0.290802i	$-0.906078\pi$
$269$	−2.27104e6	−1.91357	−0.956783	+	0.290802i	$0.906078\pi$
$270$	0	0
$271$	−304916.	−0.252207	−0.126103	−	0.992017i	$-0.540247\pi$
$271$	−304916.	−0.252207	−0.126103	+	0.992017i	$0.540247\pi$
$272$	292.397	0.000239635 0
$273$	0	0
$274$	815910.	0.656547
$275$	0	0
$276$	0	0
$277$	884418.	0.692561	0.346280	−	0.938131i	$-0.387445\pi$
$277$	884418.	0.692561	0.346280	+	0.938131i	$0.387445\pi$
$278$	−1.09296e6	−0.848187
$279$	0	0
$280$	−557130.	−0.424680
$281$	2.03032e6	1.53391	0.766955	−	0.641701i	$-0.221771\pi$
$281$	2.03032e6	1.53391	0.766955	+	0.641701i	$0.221771\pi$
$282$	0	0
$283$	−1.32361e6	−0.982415	−0.491207	−	0.871043i	$-0.663444\pi$
$283$	−1.32361e6	−0.982415	−0.491207	+	0.871043i	$0.663444\pi$
$284$	−553706.	−0.407365
$285$	0	0
$286$	0	0
$287$	−583252.	−0.417976
$288$	0	0
$289$	−1.41986e6	−0.999999
$290$	941063.	0.657088
$291$	0	0
$292$	1.18750e6	0.815034
$293$	−2.50642e6	−1.70563	−0.852816	−	0.522211i	$-0.825107\pi$
$293$	−2.50642e6	−1.70563	−0.852816	+	0.522211i	$0.825107\pi$
$294$	0	0
$295$	43841.1	0.0293310
$296$	3.07977e6	2.04310
$297$	0	0
$298$	−1.40096e6	−0.913869
$299$	127276.	0.0823317
$300$	0	0
$301$	1.39800e6	0.889389
$302$	654295.	0.412816
$303$	0	0
$304$	−272991.	−0.169420
$305$	−526277.	−0.323940
$306$	0	0
$307$	−871198.	−0.527559	−0.263779	−	0.964583i	$-0.584969\pi$
$307$	−871198.	−0.527559	−0.263779	+	0.964583i	$0.584969\pi$
$308$	0	0
$309$	0	0
$310$	−229895.	−0.135871
$311$	−1.26077e6	−0.739156	−0.369578	−	0.929200i	$-0.620498\pi$
$311$	−1.26077e6	−0.739156	−0.369578	+	0.929200i	$0.620498\pi$
$312$	0	0
$313$	326948.	0.188633	0.0943164	−	0.995542i	$-0.469933\pi$
$313$	326948.	0.188633	0.0943164	+	0.995542i	$0.469933\pi$
$314$	22294.9	0.0127609
$315$	0	0
$316$	−917206.	−0.516713
$317$	969525.	0.541890	0.270945	−	0.962595i	$-0.412664\pi$
$317$	969525.	0.541890	0.270945	+	0.962595i	$0.412664\pi$
$318$	0	0
$319$	0	0
$320$	1.15276e6	0.629308
$321$	0	0
$322$	−238051.	−0.127947
$323$	−716.381	−0.000382065 0
$324$	0	0
$325$	291544.	0.153107
$326$	−2.08891e6	−1.08862
$327$	0	0
$328$	1.48066e6	0.759923
$329$	984910.	0.501657
$330$	0	0
$331$	2.06283e6	1.03489	0.517444	−	0.855717i	$-0.326884\pi$
$331$	2.06283e6	1.03489	0.517444	+	0.855717i	$0.326884\pi$
$332$	175734.	0.0875006
$333$	0	0
$334$	−2.42698e6	−1.19042
$335$	1.38656e6	0.675037
$336$	0	0
$337$	2.86366e6	1.37356	0.686778	−	0.726867i	$-0.259024\pi$
$337$	2.86366e6	1.37356	0.686778	+	0.726867i	$0.259024\pi$
$338$	−1.42195e6	−0.677007
$339$	0	0
$340$	485.122	0.000227590 0
$341$	0	0
$342$	0	0
$343$	−2.12344e6	−0.974552
$344$	−3.54900e6	−1.61700
$345$	0	0
$346$	−230622.	−0.103565
$347$	−2.33015e6	−1.03887	−0.519434	−	0.854511i	$-0.673857\pi$
$347$	−2.33015e6	−1.03887	−0.519434	+	0.854511i	$0.673857\pi$
$348$	0	0
$349$	−949067.	−0.417094	−0.208547	−	0.978012i	$-0.566873\pi$
$349$	−949067.	−0.417094	−0.208547	+	0.978012i	$0.566873\pi$
$350$	−545292.	−0.237935
$351$	0	0
$352$	0	0
$353$	2.05584e6	0.878119	0.439059	−	0.898458i	$-0.355312\pi$
$353$	2.05584e6	0.878119	0.439059	+	0.898458i	$0.355312\pi$
$354$	0	0
$355$	1.40752e6	0.592765
$356$	2.16014e6	0.903351
$357$	0	0
$358$	−893259.	−0.368358
$359$	−1.11901e6	−0.458244	−0.229122	−	0.973398i	$-0.573585\pi$
$359$	−1.11901e6	−0.458244	−0.229122	+	0.973398i	$0.573585\pi$
$360$	0	0
$361$	−1.80726e6	−0.729883
$362$	2.83218e6	1.13592
$363$	0	0
$364$	191656.	0.0758175
$365$	−3.01861e6	−1.18597
$366$	0	0
$367$	4.10134e6	1.58950	0.794749	−	0.606938i	$-0.207602\pi$
$367$	4.10134e6	1.58950	0.794749	+	0.606938i	$0.207602\pi$
$368$	250237.	0.0963233
$369$	0	0
$370$	−2.47122e6	−0.938442
$371$	−1.66445e6	−0.627823
$372$	0	0
$373$	3.54538e6	1.31945	0.659723	−	0.751509i	$-0.270674\pi$
$373$	3.54538e6	1.31945	0.659723	+	0.751509i	$0.270674\pi$
$374$	0	0
$375$	0	0
$376$	−2.50031e6	−0.912063
$377$	−1.02557e6	−0.371632
$378$	0	0
$379$	3.23295e6	1.15612	0.578058	−	0.815996i	$-0.303811\pi$
$379$	3.23295e6	1.15612	0.578058	+	0.815996i	$0.303811\pi$
$380$	−452926.	−0.160904
$381$	0	0
$382$	372430.	0.130583
$383$	3.72375e6	1.29713	0.648565	−	0.761159i	$-0.275369\pi$
$383$	3.72375e6	1.29713	0.648565	+	0.761159i	$0.275369\pi$
$384$	0	0
$385$	0	0
$386$	−2.72817e6	−0.931971
$387$	0	0
$388$	2.03883e6	0.687546
$389$	1.00909e6	0.338108	0.169054	−	0.985607i	$-0.445929\pi$
$389$	1.00909e6	0.338108	0.169054	+	0.985607i	$0.445929\pi$
$390$	0	0
$391$	656.669	0.000217222 0
$392$	2.12750e6	0.699286
$393$	0	0
$394$	2.99123e6	0.970753
$395$	2.33153e6	0.751880
$396$	0	0
$397$	−366919.	−0.116841	−0.0584203	−	0.998292i	$-0.518606\pi$
$397$	−366919.	−0.116841	−0.0584203	+	0.998292i	$0.518606\pi$
$398$	584811.	0.185058
$399$	0	0
$400$	573204.	0.179126
$401$	−1.21086e6	−0.376038	−0.188019	−	0.982165i	$-0.560207\pi$
$401$	−1.21086e6	−0.376038	−0.188019	+	0.982165i	$0.560207\pi$
$402$	0	0
$403$	250540.	0.0768449
$404$	−540451.	−0.164741
$405$	0	0
$406$	1.91819e6	0.577533
$407$	0	0
$408$	0	0
$409$	−314741.	−0.0930346	−0.0465173	−	0.998917i	$-0.514812\pi$
$409$	−314741.	−0.0930346	−0.0465173	+	0.998917i	$0.514812\pi$
$410$	−1.18808e6	−0.349050
$411$	0	0
$412$	−667145.	−0.193632
$413$	89362.4	0.0257798
$414$	0	0
$415$	−446715.	−0.127324
$416$	−819504.	−0.232176
$417$	0	0
$418$	0	0
$419$	−5.98987e6	−1.66679	−0.833397	−	0.552674i	$-0.813607\pi$
$419$	−5.98987e6	−1.66679	−0.833397	+	0.552674i	$0.813607\pi$
$420$	0	0
$421$	−3.19815e6	−0.879414	−0.439707	−	0.898141i	$-0.644918\pi$
$421$	−3.19815e6	−0.879414	−0.439707	+	0.898141i	$0.644918\pi$
$422$	464735.	0.127035
$423$	0	0
$424$	4.22542e6	1.14145
$425$	1504.20	0.000403954 0
$426$	0	0
$427$	−1.07272e6	−0.284720
$428$	414571.	0.109393
$429$	0	0
$430$	2.84773e6	0.742725
$431$	−5.24498e6	−1.36004	−0.680019	−	0.733195i	$-0.738028\pi$
$431$	−5.24498e6	−1.36004	−0.680019	+	0.733195i	$0.738028\pi$
$432$	0	0
$433$	5.44629e6	1.39599	0.697993	−	0.716105i	$-0.254077\pi$
$433$	5.44629e6	1.39599	0.697993	+	0.716105i	$0.254077\pi$
$434$	−468601.	−0.119421
$435$	0	0
$436$	−499680.	−0.125885
$437$	−613088.	−0.153574
$438$	0	0
$439$	6.38391e6	1.58098	0.790488	−	0.612477i	$-0.209827\pi$
$439$	6.38391e6	1.58098	0.790488	+	0.612477i	$0.209827\pi$
$440$	0	0
$441$	0	0
$442$	617.492	0.000150340 0
$443$	−2.26340e6	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$443$	−2.26340e6	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$444$	0	0
$445$	−5.49106e6	−1.31449
$446$	−3.76520e6	−0.896296
$447$	0	0
$448$	2.34969e6	0.553116
$449$	−4.05102e6	−0.948306	−0.474153	−	0.880442i	$-0.657246\pi$
$449$	−4.05102e6	−0.948306	−0.474153	+	0.880442i	$0.657246\pi$
$450$	0	0
$451$	0	0
$452$	2.48798e6	0.572797
$453$	0	0
$454$	−5.63499e6	−1.28308
$455$	−487189.	−0.110324
$456$	0	0
$457$	5.07731e6	1.13722	0.568608	−	0.822609i	$-0.307482\pi$
$457$	5.07731e6	1.13722	0.568608	+	0.822609i	$0.307482\pi$
$458$	−1.43228e6	−0.319054
$459$	0	0
$460$	415173.	0.0914818
$461$	−3.81197e6	−0.835405	−0.417702	−	0.908584i	$-0.637165\pi$
$461$	−3.81197e6	−0.835405	−0.417702	+	0.908584i	$0.637165\pi$
$462$	0	0
$463$	6.00414e6	1.30166	0.650832	−	0.759222i	$-0.274420\pi$
$463$	6.00414e6	1.30166	0.650832	+	0.759222i	$0.274420\pi$
$464$	−2.01638e6	−0.434788
$465$	0	0
$466$	−4.90621e6	−1.04660
$467$	3.29877e6	0.699939	0.349969	−	0.936761i	$-0.386192\pi$
$467$	3.29877e6	0.699939	0.349969	+	0.936761i	$0.386192\pi$
$468$	0	0
$469$	2.82626e6	0.593309
$470$	2.00626e6	0.418931
$471$	0	0
$472$	−226857.	−0.0468703
$473$	0	0
$474$	0	0
$475$	−1.40437e6	−0.285593
$476$	988.836	0.000200035 0
$477$	0	0
$478$	−3.00449e6	−0.601452
$479$	5.86911e6	1.16878	0.584391	−	0.811472i	$-0.301333\pi$
$479$	5.86911e6	1.16878	0.584391	+	0.811472i	$0.301333\pi$
$480$	0	0
$481$	2.69314e6	0.530758
$482$	−1.63725e6	−0.320995
$483$	0	0
$484$	0	0
$485$	−5.18269e6	−1.00046
$486$	0	0
$487$	981038.	0.187441	0.0937203	−	0.995599i	$-0.470124\pi$
$487$	981038.	0.187441	0.0937203	+	0.995599i	$0.470124\pi$
$488$	2.72324e6	0.517650
$489$	0	0
$490$	−1.70712e6	−0.321198
$491$	5.94395e6	1.11268	0.556342	−	0.830954i	$-0.312205\pi$
$491$	5.94395e6	1.11268	0.556342	+	0.830954i	$0.312205\pi$
$492$	0	0
$493$	−5291.36	−0.000980506 0
$494$	−576511.	−0.106289
$495$	0	0
$496$	492587.	0.0899040
$497$	2.86898e6	0.520998
$498$	0	0
$499$	6.59714e6	1.18605	0.593027	−	0.805183i	$-0.297933\pi$
$499$	6.59714e6	1.18605	0.593027	+	0.805183i	$0.297933\pi$
$500$	2.68170e6	0.479716
$501$	0	0
$502$	−6.22327e6	−1.10220
$503$	−7.55854e6	−1.33204	−0.666021	−	0.745933i	$-0.732004\pi$
$503$	−7.55854e6	−1.33204	−0.666021	+	0.745933i	$0.732004\pi$
$504$	0	0
$505$	1.37382e6	0.239719
$506$	0	0
$507$	0	0
$508$	−4.97216e6	−0.854841
$509$	−4.67934e6	−0.800553	−0.400276	−	0.916394i	$-0.631086\pi$
$509$	−4.67934e6	−0.800553	−0.400276	+	0.916394i	$0.631086\pi$
$510$	0	0
$511$	−6.15291e6	−1.04239
$512$	−3.68509e6	−0.621260
$513$	0	0
$514$	2.12643e6	0.355013
$515$	1.69588e6	0.281758
$516$	0	0
$517$	0	0
$518$	−5.03715e6	−0.824823
$519$	0	0
$520$	1.23679e6	0.200580
$521$	−557794.	−0.0900284	−0.0450142	−	0.998986i	$-0.514333\pi$
$521$	−557794.	−0.0900284	−0.0450142	+	0.998986i	$0.514333\pi$
$522$	0	0
$523$	−3.87401e6	−0.619308	−0.309654	−	0.950849i	$-0.600213\pi$
$523$	−3.87401e6	−0.619308	−0.309654	+	0.950849i	$0.600213\pi$
$524$	1.23834e6	0.197021
$525$	0	0
$526$	640871.	0.100997
$527$	1292.64	0.000202746 0
$528$	0	0
$529$	−5.87436e6	−0.912686
$530$	−3.39049e6	−0.524292
$531$	0	0
$532$	−923210.	−0.141423
$533$	1.29478e6	0.197414
$534$	0	0
$535$	−1.05384e6	−0.159180
$536$	−7.17481e6	−1.07870
$537$	0	0
$538$	−9.42950e6	−1.40454
$539$	0	0
$540$	0	0
$541$	5.42985e6	0.797618	0.398809	−	0.917034i	$-0.369424\pi$
$541$	5.42985e6	0.797618	0.398809	+	0.917034i	$0.369424\pi$
$542$	−1.26603e6	−0.185117
$543$	0	0
$544$	−4228.17	−0.000612569 0
$545$	1.27018e6	0.183179
$546$	0	0
$547$	853295.	0.121936	0.0609679	−	0.998140i	$-0.480581\pi$
$547$	853295.	0.121936	0.0609679	+	0.998140i	$0.480581\pi$
$548$	−2.90051e6	−0.412594
$549$	0	0
$550$	0	0
$551$	4.94019e6	0.693210
$552$	0	0
$553$	4.75242e6	0.660848
$554$	3.67216e6	0.508332
$555$	0	0
$556$	3.88540e6	0.533026
$557$	1.08605e6	0.148324	0.0741618	−	0.997246i	$-0.476372\pi$
$557$	1.08605e6	0.148324	0.0741618	+	0.997246i	$0.476372\pi$
$558$	0	0
$559$	−3.10347e6	−0.420066
$560$	−957863.	−0.129072
$561$	0	0
$562$	8.43005e6	1.12587
$563$	6.05610e6	0.805235	0.402617	−	0.915368i	$-0.368101\pi$
$563$	6.05610e6	0.805235	0.402617	+	0.915368i	$0.368101\pi$
$564$	0	0
$565$	−6.32443e6	−0.833490
$566$	−5.49573e6	−0.721081
$567$	0	0
$568$	−7.28325e6	−0.947227
$569$	4.03513e6	0.522489	0.261245	−	0.965273i	$-0.415867\pi$
$569$	4.03513e6	0.522489	0.261245	+	0.965273i	$0.415867\pi$
$570$	0	0
$571$	−597325.	−0.0766691	−0.0383345	−	0.999265i	$-0.512205\pi$
$571$	−597325.	−0.0766691	−0.0383345	+	0.999265i	$0.512205\pi$
$572$	0	0
$573$	0	0
$574$	−2.42170e6	−0.306790
$575$	1.28731e6	0.162373
$576$	0	0
$577$	4.08910e6	0.511315	0.255657	−	0.966767i	$-0.417708\pi$
$577$	4.08910e6	0.511315	0.255657	+	0.966767i	$0.417708\pi$
$578$	−5.89534e6	−0.733988
$579$	0	0
$580$	−3.34542e6	−0.412934
$581$	−910551.	−0.111909
$582$	0	0
$583$	0	0
$584$	1.56199e7	1.89516
$585$	0	0
$586$	−1.04068e7	−1.25191
$587$	−1.56181e6	−0.187082	−0.0935409	−	0.995615i	$-0.529819\pi$
$587$	−1.56181e6	−0.187082	−0.0935409	+	0.995615i	$0.529819\pi$
$588$	0	0
$589$	−1.20685e6	−0.143340
$590$	182031.	0.0215286
$591$	0	0
$592$	5.29499e6	0.620956
$593$	−8.87674e6	−1.03661	−0.518307	−	0.855195i	$-0.673437\pi$
$593$	−8.87674e6	−1.03661	−0.518307	+	0.855195i	$0.673437\pi$
$594$	0	0
$595$	−2513.61	−0.000291076 0
$596$	4.98031e6	0.574303
$597$	0	0
$598$	528457.	0.0604306
$599$	7.17157e6	0.816671	0.408336	−	0.912832i	$-0.366109\pi$
$599$	7.17157e6	0.816671	0.408336	+	0.912832i	$0.366109\pi$
$600$	0	0
$601$	4.79599e6	0.541617	0.270809	−	0.962633i	$-0.412709\pi$
$601$	4.79599e6	0.541617	0.270809	+	0.962633i	$0.412709\pi$
$602$	5.80460e6	0.652802
$603$	0	0
$604$	−2.32598e6	−0.259426
$605$	0	0
$606$	0	0
$607$	8.49023e6	0.935293	0.467646	−	0.883916i	$-0.345102\pi$
$607$	8.49023e6	0.935293	0.467646	+	0.883916i	$0.345102\pi$
$608$	3.94756e6	0.433081
$609$	0	0
$610$	−2.18514e6	−0.237768
$611$	−2.18643e6	−0.236937
$612$	0	0
$613$	−6.52329e6	−0.701158	−0.350579	−	0.936533i	$-0.614015\pi$
$613$	−6.52329e6	−0.701158	−0.350579	+	0.936533i	$0.614015\pi$
$614$	−3.61727e6	−0.387222
$615$	0	0
$616$	0	0
$617$	−1.20215e7	−1.27129	−0.635647	−	0.771980i	$-0.719267\pi$
$617$	−1.20215e7	−1.27129	−0.635647	+	0.771980i	$0.719267\pi$
$618$	0	0
$619$	1.34686e7	1.41285	0.706424	−	0.707789i	$-0.250307\pi$
$619$	1.34686e7	1.41285	0.706424	+	0.707789i	$0.250307\pi$
$620$	817263.	0.0853852
$621$	0	0
$622$	−5.23482e6	−0.542532
$623$	−1.11926e7	−1.15534
$624$	0	0
$625$	−1.45061e6	−0.148542
$626$	1.35751e6	0.138454
$627$	0	0
$628$	−79257.0	−0.00801934
$629$	13895.1	0.00140034
$630$	0	0
$631$	−1.08509e7	−1.08490	−0.542452	−	0.840087i	$-0.682504\pi$
$631$	−1.08509e7	−1.08490	−0.542452	+	0.840087i	$0.682504\pi$
$632$	−1.20646e7	−1.20149
$633$	0	0
$634$	4.02554e6	0.397741
$635$	1.26392e7	1.24390
$636$	0	0
$637$	1.86042e6	0.181661
$638$	0	0
$639$	0	0
$640$	−1.00915e6	−0.0973876
$641$	9.31338e6	0.895287	0.447643	−	0.894212i	$-0.352263\pi$
$641$	9.31338e6	0.895287	0.447643	+	0.894212i	$0.352263\pi$
$642$	0	0
$643$	1.35760e7	1.29493	0.647464	−	0.762096i	$-0.275830\pi$
$643$	1.35760e7	1.29493	0.647464	+	0.762096i	$0.275830\pi$
$644$	846258.	0.0804059
$645$	0	0
$646$	−2974.46	−0.000280432 0
$647$	1.18628e7	1.11411	0.557053	−	0.830477i	$-0.311932\pi$
$647$	1.18628e7	1.11411	0.557053	+	0.830477i	$0.311932\pi$
$648$	0	0
$649$	0	0
$650$	1.21051e6	0.112379
$651$	0	0
$652$	7.42596e6	0.684122
$653$	−1.22670e7	−1.12579	−0.562893	−	0.826530i	$-0.690312\pi$
$653$	−1.22670e7	−1.12579	−0.562893	+	0.826530i	$0.690312\pi$
$654$	0	0
$655$	−3.14785e6	−0.286689
$656$	2.54566e6	0.230962
$657$	0	0
$658$	4.08941e6	0.368211
$659$	4.60998e6	0.413509	0.206755	−	0.978393i	$-0.433710\pi$
$659$	4.60998e6	0.413509	0.206755	+	0.978393i	$0.433710\pi$
$660$	0	0
$661$	−1.04708e7	−0.932130	−0.466065	−	0.884750i	$-0.654329\pi$
$661$	−1.04708e7	−0.932130	−0.466065	+	0.884750i	$0.654329\pi$
$662$	8.56500e6	0.759596
$663$	0	0
$664$	2.31154e6	0.203461
$665$	2.34679e6	0.205788
$666$	0	0
$667$	−4.52841e6	−0.394123
$668$	8.62775e6	0.748094
$669$	0	0
$670$	5.75710e6	0.495469
$671$	0	0
$672$	0	0
$673$	5.32923e6	0.453552	0.226776	−	0.973947i	$-0.427181\pi$
$673$	5.32923e6	0.453552	0.226776	+	0.973947i	$0.427181\pi$
$674$	1.18901e7	1.00818
$675$	0	0
$676$	5.05495e6	0.425451
$677$	1.75693e6	0.147327	0.0736634	−	0.997283i	$-0.476531\pi$
$677$	1.75693e6	0.147327	0.0736634	+	0.997283i	$0.476531\pi$
$678$	0	0
$679$	−1.05640e7	−0.879335
$680$	6381.11	0.000529205 0
$681$	0	0
$682$	0	0
$683$	1.42170e7	1.16615	0.583077	−	0.812417i	$-0.301848\pi$
$683$	1.42170e7	1.16615	0.583077	+	0.812417i	$0.301848\pi$
$684$	0	0
$685$	7.37307e6	0.600374
$686$	−8.81667e6	−0.715310
$687$	0	0
$688$	−6.10173e6	−0.491453
$689$	3.69497e6	0.296526
$690$	0	0
$691$	−2.40296e7	−1.91448	−0.957240	−	0.289294i	$-0.906579\pi$
$691$	−2.40296e7	−1.91448	−0.957240	+	0.289294i	$0.906579\pi$
$692$	819848.	0.0650831
$693$	0	0
$694$	−9.67494e6	−0.762517
$695$	−9.87665e6	−0.775618
$696$	0	0
$697$	6680.30	0.000520852 0
$698$	−3.94059e6	−0.306142
$699$	0	0
$700$	1.93848e6	0.149526
$701$	−5.04194e6	−0.387528	−0.193764	−	0.981048i	$-0.562070\pi$
$701$	−5.04194e6	−0.387528	−0.193764	+	0.981048i	$0.562070\pi$
$702$	0	0
$703$	−1.29729e7	−0.990030
$704$	0	0
$705$	0	0
$706$	8.53600e6	0.644529
$707$	2.80030e6	0.210696
$708$	0	0
$709$	1.87351e7	1.39972	0.699860	−	0.714280i	$-0.253246\pi$
$709$	1.87351e7	1.39972	0.699860	+	0.714280i	$0.253246\pi$
$710$	5.84411e6	0.435083
$711$	0	0
$712$	2.84137e7	2.10052
$713$	1.10626e6	0.0814955
$714$	0	0
$715$	0	0
$716$	3.17548e6	0.231487
$717$	0	0
$718$	−4.64619e6	−0.336346
$719$	−9.84863e6	−0.710483	−0.355242	−	0.934775i	$-0.615601\pi$
$719$	−9.84863e6	−0.710483	−0.355242	+	0.934775i	$0.615601\pi$
$720$	0	0
$721$	3.45675e6	0.247645
$722$	−7.50387e6	−0.535726
$723$	0	0
$724$	−1.00682e7	−0.713849
$725$	−1.03730e7	−0.732925
$726$	0	0
$727$	−1.76700e7	−1.23994	−0.619968	−	0.784627i	$-0.712854\pi$
$727$	−1.76700e7	−1.23994	−0.619968	+	0.784627i	$0.712854\pi$
$728$	2.52098e6	0.176295
$729$	0	0
$730$	−1.25335e7	−0.870492
$731$	−16012.1	−0.00110829
$732$	0	0
$733$	1.72881e7	1.18847	0.594235	−	0.804291i	$-0.297455\pi$
$733$	1.72881e7	1.18847	0.594235	+	0.804291i	$0.297455\pi$
$734$	1.70290e7	1.16667
$735$	0	0
$736$	−3.61852e6	−0.246227
$737$	0	0
$738$	0	0
$739$	5.26063e6	0.354345	0.177172	−	0.984180i	$-0.443305\pi$
$739$	5.26063e6	0.354345	0.177172	+	0.984180i	$0.443305\pi$
$740$	8.78503e6	0.589745
$741$	0	0
$742$	−6.91093e6	−0.460815
$743$	5.40803e6	0.359391	0.179695	−	0.983722i	$-0.442489\pi$
$743$	5.40803e6	0.359391	0.179695	+	0.983722i	$0.442489\pi$
$744$	0	0
$745$	−1.26599e7	−0.835681
$746$	1.47207e7	0.968458
$747$	0	0
$748$	0	0
$749$	−2.14806e6	−0.139908
$750$	0	0
$751$	−1.50076e7	−0.970980	−0.485490	−	0.874242i	$-0.661359\pi$
$751$	−1.50076e7	−0.970980	−0.485490	+	0.874242i	$0.661359\pi$
$752$	−4.29874e6	−0.277202
$753$	0	0
$754$	−4.25825e6	−0.272774
$755$	5.91262e6	0.377496
$756$	0	0
$757$	5.65482e6	0.358657	0.179328	−	0.983789i	$-0.442608\pi$
$757$	5.65482e6	0.358657	0.179328	+	0.983789i	$0.442608\pi$
$758$	1.34234e7	0.848576
$759$	0	0
$760$	−5.95762e6	−0.374144
$761$	1.90760e7	1.19406	0.597030	−	0.802219i	$-0.296347\pi$
$761$	1.90760e7	1.19406	0.597030	+	0.802219i	$0.296347\pi$
$762$	0	0
$763$	2.58904e6	0.161001
$764$	−1.32396e6	−0.0820621
$765$	0	0
$766$	1.54613e7	0.952079
$767$	−198378.	−0.0121760
$768$	0	0
$769$	−2.03557e7	−1.24128	−0.620639	−	0.784096i	$-0.713127\pi$
$769$	−2.03557e7	−1.24128	−0.620639	+	0.784096i	$0.713127\pi$
$770$	0	0
$771$	0	0
$772$	9.69845e6	0.585678
$773$	9.29224e6	0.559334	0.279667	−	0.960097i	$-0.409776\pi$
$773$	9.29224e6	0.559334	0.279667	+	0.960097i	$0.409776\pi$
$774$	0	0
$775$	2.53405e6	0.151552
$776$	2.68180e7	1.59872
$777$	0	0
$778$	4.18981e6	0.248168
$779$	−6.23695e6	−0.368238
$780$	0	0
$781$	0	0
$782$	2726.53	0.000159439 0
$783$	0	0
$784$	3.65777e6	0.212533
$785$	201471.	0.0116691
$786$	0	0
$787$	−1.28703e7	−0.740716	−0.370358	−	0.928889i	$-0.620765\pi$
$787$	−1.28703e7	−0.740716	−0.370358	+	0.928889i	$0.620765\pi$
$788$	−1.06336e7	−0.610050
$789$	0	0
$790$	9.68067e6	0.551871
$791$	−1.28912e7	−0.732578
$792$	0	0
$793$	2.38137e6	0.134476
$794$	−1.52347e6	−0.0857597
$795$	0	0
$796$	−2.07897e6	−0.116296
$797$	−8.82772e6	−0.492269	−0.246135	−	0.969236i	$-0.579161\pi$
$797$	−8.82772e6	−0.492269	−0.246135	+	0.969236i	$0.579161\pi$
$798$	0	0
$799$	−11280.7	−0.000625129 0
$800$	−8.28875e6	−0.457893
$801$	0	0
$802$	−5.02756e6	−0.276008
$803$	0	0
$804$	0	0
$805$	−2.15118e6	−0.117000
$806$	1.04026e6	0.0564033
$807$	0	0
$808$	−7.10890e6	−0.383066
$809$	640426.	0.0344031	0.0172016	−	0.999852i	$-0.494524\pi$
$809$	640426.	0.0344031	0.0172016	+	0.999852i	$0.494524\pi$
$810$	0	0
$811$	−1.30571e7	−0.697099	−0.348550	−	0.937290i	$-0.613326\pi$
$811$	−1.30571e7	−0.697099	−0.348550	+	0.937290i	$0.613326\pi$
$812$	−6.81905e6	−0.362939
$813$	0	0
$814$	0	0
$815$	−1.88767e7	−0.995481
$816$	0	0
$817$	1.49494e7	0.783555
$818$	−1.30682e6	−0.0682864
$819$	0	0
$820$	4.22356e6	0.219353
$821$	3.01122e7	1.55914	0.779570	−	0.626315i	$-0.215438\pi$
$821$	3.01122e7	1.55914	0.779570	+	0.626315i	$0.215438\pi$
$822$	0	0
$823$	1.95327e7	1.00523	0.502613	−	0.864512i	$-0.332372\pi$
$823$	1.95327e7	1.00523	0.502613	+	0.864512i	$0.332372\pi$
$824$	−8.77539e6	−0.450244
$825$	0	0
$826$	371039.	0.0189221
$827$	6.83832e6	0.347685	0.173842	−	0.984773i	$-0.444382\pi$
$827$	6.83832e6	0.347685	0.173842	+	0.984773i	$0.444382\pi$
$828$	0	0
$829$	7.34516e6	0.371206	0.185603	−	0.982625i	$-0.440576\pi$
$829$	7.34516e6	0.371206	0.185603	+	0.982625i	$0.440576\pi$
$830$	−1.85479e6	−0.0934544
$831$	0	0
$832$	−5.21615e6	−0.261241
$833$	9598.68	0.000479291 0
$834$	0	0
$835$	−2.19317e7	−1.08857
$836$	0	0
$837$	0	0
$838$	−2.48703e7	−1.22341
$839$	3.88085e7	1.90336	0.951682	−	0.307084i	$-0.0993534\pi$
$839$	3.88085e7	1.90336	0.951682	+	0.307084i	$0.0993534\pi$
$840$	0	0
$841$	1.59783e7	0.779007
$842$	−1.32789e7	−0.645480
$843$	0	0
$844$	−1.65211e6	−0.0798328
$845$	−1.28496e7	−0.619083
$846$	0	0
$847$	0	0
$848$	7.26468e6	0.346918
$849$	0	0
$850$	6245.52	0.000296498 0
$851$	1.18916e7	0.562879
$852$	0	0
$853$	1.11803e7	0.526115	0.263058	−	0.964780i	$-0.415269\pi$
$853$	1.11803e7	0.526115	0.263058	+	0.964780i	$0.415269\pi$
$854$	−4.45402e6	−0.208981
$855$	0	0
$856$	5.45311e6	0.254367
$857$	−2.30926e7	−1.07404	−0.537020	−	0.843569i	$-0.680450\pi$
$857$	−2.30926e7	−1.07404	−0.537020	+	0.843569i	$0.680450\pi$
$858$	0	0
$859$	−1.28419e7	−0.593809	−0.296904	−	0.954907i	$-0.595954\pi$
$859$	−1.28419e7	−0.593809	−0.296904	+	0.954907i	$0.595954\pi$
$860$	−1.01235e7	−0.466751
$861$	0	0
$862$	−2.17775e7	−0.998252
$863$	2.98309e7	1.36345	0.681725	−	0.731608i	$-0.261230\pi$
$863$	2.98309e7	1.36345	0.681725	+	0.731608i	$0.261230\pi$
$864$	0	0
$865$	−2.08405e6	−0.0947038
$866$	2.26134e7	1.02464
$867$	0	0
$868$	1.66585e6	0.0750474
$869$	0	0
$870$	0	0
$871$	−6.27410e6	−0.280225
$872$	−6.57260e6	−0.292716
$873$	0	0
$874$	−2.54558e6	−0.112722
$875$	−1.38950e7	−0.613532
$876$	0	0
$877$	3.85241e7	1.69135	0.845674	−	0.533700i	$-0.179199\pi$
$877$	3.85241e7	1.69135	0.845674	+	0.533700i	$0.179199\pi$
$878$	2.65064e7	1.16042
$879$	0	0
$880$	0	0
$881$	3.12023e7	1.35440	0.677200	−	0.735799i	$-0.263193\pi$
$881$	3.12023e7	1.35440	0.677200	+	0.735799i	$0.263193\pi$
$882$	0	0
$883$	−1.08832e7	−0.469738	−0.234869	−	0.972027i	$-0.575466\pi$
$883$	−1.08832e7	−0.469738	−0.234869	+	0.972027i	$0.575466\pi$
$884$	−2195.14	−9.44783e−5 0
$885$	0	0
$886$	−9.39779e6	−0.402199
$887$	4.34842e7	1.85576	0.927882	−	0.372875i	$-0.121628\pi$
$887$	4.34842e7	1.85576	0.927882	+	0.372875i	$0.121628\pi$
$888$	0	0
$889$	2.57628e7	1.09330
$890$	−2.27992e7	−0.964818
$891$	0	0
$892$	1.33851e7	0.563259
$893$	1.05320e7	0.441961
$894$	0	0
$895$	−8.07205e6	−0.336842
$896$	−2.05697e6	−0.0855967
$897$	0	0
$898$	−1.68201e7	−0.696046
$899$	−8.91412e6	−0.367857
$900$	0	0
$901$	19063.9	0.000782348 0
$902$	0	0
$903$	0	0
$904$	3.27260e7	1.33190
$905$	2.55934e7	1.03874
$906$	0	0
$907$	−9.01861e6	−0.364017	−0.182008	−	0.983297i	$-0.558260\pi$
$907$	−9.01861e6	−0.364017	−0.182008	+	0.983297i	$0.558260\pi$
$908$	2.00320e7	0.806326
$909$	0	0
$910$	−2.02284e6	−0.0809764
$911$	6.69826e6	0.267403	0.133701	−	0.991022i	$-0.457314\pi$
$911$	6.69826e6	0.267403	0.133701	+	0.991022i	$0.457314\pi$
$912$	0	0
$913$	0	0
$914$	2.10813e7	0.834704
$915$	0	0
$916$	5.09166e6	0.200503
$917$	−6.41635e6	−0.251979
$918$	0	0
$919$	1.01836e7	0.397753	0.198877	−	0.980025i	$-0.436271\pi$
$919$	1.01836e7	0.397753	0.198877	+	0.980025i	$0.436271\pi$
$920$	5.46104e6	0.212719
$921$	0	0
$922$	−1.58275e7	−0.613178
$923$	−6.36892e6	−0.246072
$924$	0	0
$925$	2.72394e7	1.04675
$926$	2.49296e7	0.955406
$927$	0	0
$928$	2.91576e7	1.11143
$929$	−7.05588e6	−0.268233	−0.134116	−	0.990966i	$-0.542820\pi$
$929$	−7.05588e6	−0.268233	−0.134116	+	0.990966i	$0.542820\pi$
$930$	0	0
$931$	−8.96165e6	−0.338855
$932$	1.74413e7	0.657716
$933$	0	0
$934$	1.36967e7	0.513747
$935$	0	0
$936$	0	0
$937$	−1.62634e6	−0.0605150	−0.0302575	−	0.999542i	$-0.509633\pi$
$937$	−1.62634e6	−0.0605150	−0.0302575	+	0.999542i	$0.509633\pi$
$938$	1.17348e7	0.435482
$939$	0	0
$940$	−7.13213e6	−0.263269
$941$	−2.73619e7	−1.00733	−0.503665	−	0.863899i	$-0.668015\pi$
$941$	−2.73619e7	−1.00733	−0.503665	+	0.863899i	$0.668015\pi$
$942$	0	0
$943$	5.71709e6	0.209361
$944$	−390031.	−0.0142452
$945$	0	0
$946$	0	0
$947$	−4.28329e7	−1.55204	−0.776019	−	0.630709i	$-0.782764\pi$
$947$	−4.28329e7	−1.55204	−0.776019	+	0.630709i	$0.782764\pi$
$948$	0	0
$949$	1.36590e7	0.492327
$950$	−5.83103e6	−0.209622
$951$	0	0
$952$	13006.8	0.000465133 0
$953$	2.80611e7	1.00086	0.500429	−	0.865777i	$-0.333176\pi$
$953$	2.80611e7	1.00086	0.500429	+	0.865777i	$0.333176\pi$
$954$	0	0
$955$	3.36551e6	0.119410
$956$	1.06808e7	0.377970
$957$	0	0
$958$	2.43690e7	0.857873
$959$	1.50287e7	0.527686
$960$	0	0
$961$	−2.64515e7	−0.923936
$962$	1.11821e7	0.389571
$963$	0	0
$964$	5.82032e6	0.201723
$965$	−2.46534e7	−0.852233
$966$	0	0
$967$	3.20738e7	1.10302	0.551512	−	0.834167i	$-0.314051\pi$
$967$	3.20738e7	1.10302	0.551512	+	0.834167i	$0.314051\pi$
$968$	0	0
$969$	0	0
$970$	−2.15189e7	−0.734329
$971$	−4.92583e7	−1.67661	−0.838304	−	0.545203i	$-0.816452\pi$
$971$	−4.92583e7	−1.67661	−0.838304	+	0.545203i	$0.816452\pi$
$972$	0	0
$973$	−2.01318e7	−0.681712
$974$	4.07334e6	0.137579
$975$	0	0
$976$	4.68201e6	0.157329
$977$	1.26363e7	0.423530	0.211765	−	0.977321i	$-0.432079\pi$
$977$	1.26363e7	0.423530	0.211765	+	0.977321i	$0.432079\pi$
$978$	0	0
$979$	0	0
$980$	6.06869e6	0.201850
$981$	0	0
$982$	2.46797e7	0.816697
$983$	−6.52333e6	−0.215321	−0.107660	−	0.994188i	$-0.534336\pi$
$983$	−6.52333e6	−0.215321	−0.107660	+	0.994188i	$0.534336\pi$
$984$	0	0
$985$	2.70306e7	0.887697
$986$	−21970.1	−0.000719681 0
$987$	0	0
$988$	2.04946e6	0.0667955
$989$	−1.37033e7	−0.445488
$990$	0	0
$991$	2.28897e7	0.740382	0.370191	−	0.928956i	$-0.379292\pi$
$991$	2.28897e7	0.740382	0.370191	+	0.928956i	$0.379292\pi$
$992$	−7.12300e6	−0.229818
$993$	0	0
$994$	1.19122e7	0.382407
$995$	5.28472e6	0.169225
$996$	0	0
$997$	1.82397e7	0.581138	0.290569	−	0.956854i	$-0.406155\pi$
$997$	1.82397e7	0.581138	0.290569	+	0.956854i	$0.406155\pi$
$998$	2.73918e7	0.870550
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.c.1.2	✓	2	33.32	even	2
99.6.a.f.1.1		2	11.10	odd	2
363.6.a.j.1.1		2	3.2	odd	2
528.6.a.s.1.1		2	132.131	odd	2
825.6.a.e.1.1		2	165.164	even	2
1089.6.a.j.1.2		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)

\(f(q)\)	\(=\)	\(q+4.15207 q^{2} -14.7603 q^{4} +37.5207 q^{5} +76.4793 q^{7} -194.152 q^{8} +O(q^{10})\)	\(q+4.15207 q^{2} -14.7603 q^{4} +37.5207 q^{5} +76.4793 q^{7} -194.152 q^{8} +155.788 q^{10} -169.779 q^{13} +317.547 q^{14} -333.802 q^{16} -0.875959 q^{17} +817.825 q^{19} -553.818 q^{20} -749.657 q^{23} -1717.20 q^{25} -704.932 q^{26} -1128.86 q^{28} +6040.65 q^{29} -1475.69 q^{31} +4826.90 q^{32} -3.63704 q^{34} +2869.56 q^{35} -15862.7 q^{37} +3395.66 q^{38} -7284.72 q^{40} -7626.27 q^{41} +18279.5 q^{43} -3112.62 q^{46} +12878.1 q^{47} -10957.9 q^{49} -7129.93 q^{50} +2505.99 q^{52} -21763.5 q^{53} -14848.6 q^{56} +25081.2 q^{58} +1168.45 q^{59} -14026.3 q^{61} -6127.16 q^{62} +30723.3 q^{64} -6370.21 q^{65} +36954.6 q^{67} +12.9295 q^{68} +11914.6 q^{70} +37513.1 q^{71} -80452.0 q^{73} -65862.9 q^{74} -12071.4 q^{76} +62139.9 q^{79} -12524.5 q^{80} -31664.8 q^{82} -11905.8 q^{83} -32.8666 q^{85} +75897.7 q^{86} -146348. q^{89} -12984.5 q^{91} +11065.2 q^{92} +53470.8 q^{94} +30685.3 q^{95} -138129. q^{97} -45498.0 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5 q^{2} + 37 q^{4} - 58 q^{5} + 286 q^{7} - 375 q^{8}+O(q^{10}) \) 2 * q - 5 * q^2 + 37 * q^4 - 58 * q^5 + 286 * q^7 - 375 * q^8	\( 2 q - 5 q^{2} + 37 q^{4} - 58 q^{5} + 286 q^{7} - 375 q^{8} + 1030 q^{10} + 166 q^{13} - 1600 q^{14} - 335 q^{16} - 800 q^{17} + 1476 q^{19} - 5498 q^{20} + 3370 q^{23} + 4282 q^{25} - 3778 q^{26} + 9716 q^{28} + 6600 q^{29} - 7528 q^{31} + 10625 q^{32} + 7310 q^{34} - 17144 q^{35} - 29916 q^{37} - 2628 q^{38} + 9990 q^{40} - 5780 q^{41} + 16656 q^{43} - 40816 q^{46} - 7850 q^{47} + 16134 q^{49} - 62035 q^{50} + 19886 q^{52} - 14178 q^{53} - 52740 q^{56} + 19962 q^{58} - 17300 q^{59} + 2946 q^{61} + 49264 q^{62} - 22303 q^{64} - 38444 q^{65} + 31336 q^{67} - 41350 q^{68} + 195080 q^{70} + 33810 q^{71} - 60644 q^{73} + 62754 q^{74} + 21996 q^{76} - 1870 q^{79} - 12410 q^{80} - 48562 q^{82} - 58296 q^{83} + 76300 q^{85} + 90756 q^{86} - 92388 q^{89} + 57368 q^{91} + 224300 q^{92} + 243176 q^{94} - 32184 q^{95} + 7120 q^{97} - 293445 q^{98}+O(q^{100}) \) 2 * q - 5 * q^2 + 37 * q^4 - 58 * q^5 + 286 * q^7 - 375 * q^8 + 1030 * q^10 + 166 * q^13 - 1600 * q^14 - 335 * q^16 - 800 * q^17 + 1476 * q^19 - 5498 * q^20 + 3370 * q^23 + 4282 * q^25 - 3778 * q^26 + 9716 * q^28 + 6600 * q^29 - 7528 * q^31 + 10625 * q^32 + 7310 * q^34 - 17144 * q^35 - 29916 * q^37 - 2628 * q^38 + 9990 * q^40 - 5780 * q^41 + 16656 * q^43 - 40816 * q^46 - 7850 * q^47 + 16134 * q^49 - 62035 * q^50 + 19886 * q^52 - 14178 * q^53 - 52740 * q^56 + 19962 * q^58 - 17300 * q^59 + 2946 * q^61 + 49264 * q^62 - 22303 * q^64 - 38444 * q^65 + 31336 * q^67 - 41350 * q^68 + 195080 * q^70 + 33810 * q^71 - 60644 * q^73 + 62754 * q^74 + 21996 * q^76 - 1870 * q^79 - 12410 * q^80 - 48562 * q^82 - 58296 * q^83 + 76300 * q^85 + 90756 * q^86 - 92388 * q^89 + 57368 * q^91 + 224300 * q^92 + 243176 * q^94 - 32184 * q^95 + 7120 * q^97 - 293445 * q^98

Introduction

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