LMFDB - Embedded newform 1089.6.a.bk.1.4

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 3 x^{9} - 228 x^{8} + 523 x^{7} + 17396 x^{6} - 31445 x^{5} - 508100 x^{4} + 960757 x^{3} + \cdots + 5059564 $ x^10 - 3x^9 - 228x^8 + 523x^7 + 17396x^6 - 31445x^5 - 508100x^4 + 960757x^3 + 4870759x^2 - 11540360*x + 5059564
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$4.35046$ 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)

$f(q)$	$=$	$q-3.35046 q^{2} -20.7744 q^{4} -106.493 q^{5} +40.5248 q^{7} +176.818 q^{8} +O(q^{10})$	$q-3.35046 q^{2} -20.7744 q^{4} -106.493 q^{5} +40.5248 q^{7} +176.818 q^{8} +356.801 q^{10} -653.866 q^{13} -135.777 q^{14} +72.3598 q^{16} +1945.16 q^{17} -487.930 q^{19} +2212.34 q^{20} -805.987 q^{23} +8215.83 q^{25} +2190.75 q^{26} -841.881 q^{28} -3646.11 q^{29} -1479.42 q^{31} -5900.63 q^{32} -6517.16 q^{34} -4315.63 q^{35} -729.957 q^{37} +1634.79 q^{38} -18830.0 q^{40} -160.618 q^{41} -9144.84 q^{43} +2700.42 q^{46} +22517.7 q^{47} -15164.7 q^{49} -27526.8 q^{50} +13583.7 q^{52} +29794.3 q^{53} +7165.54 q^{56} +12216.1 q^{58} -37910.2 q^{59} -23174.8 q^{61} +4956.74 q^{62} +17454.3 q^{64} +69632.3 q^{65} -27186.5 q^{67} -40409.6 q^{68} +14459.3 q^{70} -3393.45 q^{71} -74488.8 q^{73} +2445.69 q^{74} +10136.5 q^{76} -59705.1 q^{79} -7705.84 q^{80} +538.145 q^{82} -29019.2 q^{83} -207146. q^{85} +30639.4 q^{86} -88381.2 q^{89} -26497.8 q^{91} +16743.9 q^{92} -75444.5 q^{94} +51961.3 q^{95} -23821.0 q^{97} +50808.8 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8}+O(q^{10}) $ 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8	$ 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8} - 212 q^{10} - 1016 q^{13} - 1566 q^{14} + 2361 q^{16} + 1669 q^{17} + 2929 q^{19} + 10189 q^{20} - 4070 q^{23} + 2425 q^{25} - 8481 q^{26} + 3272 q^{28} + 11940 q^{29} - 16085 q^{31} + 2313 q^{32} + 8270 q^{34} - 6987 q^{35} + 16136 q^{37} - 10721 q^{38} + 9332 q^{40} + 16278 q^{41} - 10844 q^{43} - 25995 q^{46} + 22411 q^{47} + 75150 q^{49} - 738 q^{50} - 8677 q^{52} + 27511 q^{53} - 84447 q^{56} + 16853 q^{58} + 39641 q^{59} - 3509 q^{61} + 227845 q^{62} - 22980 q^{64} + 67097 q^{65} + 10089 q^{67} + 273621 q^{68} - 38919 q^{70} - 60681 q^{71} - 133740 q^{73} + 317933 q^{74} + 23434 q^{76} - 12386 q^{79} + 289014 q^{80} + 385033 q^{82} + 187242 q^{83} - 191504 q^{85} + 43793 q^{86} - 102746 q^{89} - 435248 q^{91} - 30867 q^{92} - 404734 q^{94} + 648147 q^{95} - 120631 q^{97} + 1148087 q^{98}+O(q^{100}) $ 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8 - 212 * q^10 - 1016 * q^13 - 1566 * q^14 + 2361 * q^16 + 1669 * q^17 + 2929 * q^19 + 10189 * q^20 - 4070 * q^23 + 2425 * q^25 - 8481 * q^26 + 3272 * q^28 + 11940 * q^29 - 16085 * q^31 + 2313 * q^32 + 8270 * q^34 - 6987 * q^35 + 16136 * q^37 - 10721 * q^38 + 9332 * q^40 + 16278 * q^41 - 10844 * q^43 - 25995 * q^46 + 22411 * q^47 + 75150 * q^49 - 738 * q^50 - 8677 * q^52 + 27511 * q^53 - 84447 * q^56 + 16853 * q^58 + 39641 * q^59 - 3509 * q^61 + 227845 * q^62 - 22980 * q^64 + 67097 * q^65 + 10089 * q^67 + 273621 * q^68 - 38919 * q^70 - 60681 * q^71 - 133740 * q^73 + 317933 * q^74 + 23434 * q^76 - 12386 * q^79 + 289014 * q^80 + 385033 * q^82 + 187242 * q^83 - 191504 * q^85 + 43793 * q^86 - 102746 * q^89 - 435248 * q^91 - 30867 * q^92 - 404734 * q^94 + 648147 * q^95 - 120631 * q^97 + 1148087 * 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 $

$2$	−3.35046	−0.592283	−0.296141	−	0.955144i	$-0.595700\pi$
$2$	−3.35046	−0.592283	−0.296141	+	0.955144i	$0.595700\pi$
$3$	0	0
$3$	0	0
$4$	−20.7744	−0.649201
$5$	−106.493	−1.90501	−0.952505	−	0.304522i	$-0.901503\pi$
$5$	−106.493	−1.90501	−0.952505	+	0.304522i	$0.901503\pi$
$6$	0	0
$7$	40.5248	0.312591	0.156295	−	0.987710i	$-0.450045\pi$
$7$	40.5248	0.312591	0.156295	+	0.987710i	$0.450045\pi$
$8$	176.818	0.976793
$9$	0	0
$10$	356.801	1.12830
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−653.866	−1.07307	−0.536537	−	0.843876i	$-0.680268\pi$
$13$	−653.866	−1.07307	−0.536537	+	0.843876i	$0.680268\pi$
$14$	−135.777	−0.185142
$15$	0	0
$16$	72.3598	0.0706639
$17$	1945.16	1.63242	0.816211	−	0.577754i	$-0.196071\pi$
$17$	1945.16	1.63242	0.816211	+	0.577754i	$0.196071\pi$
$18$	0	0
$19$	−487.930	−0.310080	−0.155040	−	0.987908i	$-0.549551\pi$
$19$	−487.930	−0.310080	−0.155040	+	0.987908i	$0.549551\pi$
$20$	2212.34	1.23674
$21$	0	0
$22$	0	0
$23$	−805.987	−0.317694	−0.158847	−	0.987303i	$-0.550778\pi$
$23$	−805.987	−0.317694	−0.158847	+	0.987303i	$0.550778\pi$
$24$	0	0
$25$	8215.83	2.62907
$26$	2190.75	0.635564
$27$	0	0
$28$	−841.881	−0.202934
$29$	−3646.11	−0.805073	−0.402536	−	0.915404i	$-0.631871\pi$
$29$	−3646.11	−0.805073	−0.402536	+	0.915404i	$0.631871\pi$
$30$	0	0
$31$	−1479.42	−0.276495	−0.138248	−	0.990398i	$-0.544147\pi$
$31$	−1479.42	−0.276495	−0.138248	+	0.990398i	$0.544147\pi$
$32$	−5900.63	−1.01865
$33$	0	0
$34$	−6517.16	−0.966855
$35$	−4315.63	−0.595489
$36$	0	0
$37$	−729.957	−0.0876582	−0.0438291	−	0.999039i	$-0.513956\pi$
$37$	−729.957	−0.0876582	−0.0438291	+	0.999039i	$0.513956\pi$
$38$	1634.79	0.183655
$39$	0	0
$40$	−18830.0	−1.86080
$41$	−160.618	−0.0149223	−0.00746115	−	0.999972i	$-0.502375\pi$
$41$	−160.618	−0.0149223	−0.00746115	+	0.999972i	$0.502375\pi$
$42$	0	0
$43$	−9144.84	−0.754232	−0.377116	−	0.926166i	$-0.623084\pi$
$43$	−9144.84	−0.754232	−0.377116	+	0.926166i	$0.623084\pi$
$44$	0	0
$45$	0	0
$46$	2700.42	0.188164
$47$	22517.7	1.48689	0.743446	−	0.668796i	$-0.233190\pi$
$47$	22517.7	1.48689	0.743446	+	0.668796i	$0.233190\pi$
$48$	0	0
$49$	−15164.7	−0.902287
$50$	−27526.8	−1.55715
$51$	0	0
$52$	13583.7	0.696642
$53$	29794.3	1.45695	0.728474	−	0.685074i	$-0.240230\pi$
$53$	29794.3	1.45695	0.728474	+	0.685074i	$0.240230\pi$
$54$	0	0
$55$	0	0
$56$	7165.54	0.305337
$57$	0	0
$58$	12216.1	0.476831
$59$	−37910.2	−1.41783	−0.708917	−	0.705291i	$-0.750816\pi$
$59$	−37910.2	−1.41783	−0.708917	+	0.705291i	$0.750816\pi$
$60$	0	0
$61$	−23174.8	−0.797429	−0.398714	−	0.917075i	$-0.630544\pi$
$61$	−23174.8	−0.797429	−0.398714	+	0.917075i	$0.630544\pi$
$62$	4956.74	0.163763
$63$	0	0
$64$	17454.3	0.532662
$65$	69632.3	2.04422
$66$	0	0
$67$	−27186.5	−0.739888	−0.369944	−	0.929054i	$-0.620623\pi$
$67$	−27186.5	−0.739888	−0.369944	+	0.929054i	$0.620623\pi$
$68$	−40409.6	−1.05977
$69$	0	0
$70$	14459.3	0.352698
$71$	−3393.45	−0.0798907	−0.0399453	−	0.999202i	$-0.512718\pi$
$71$	−3393.45	−0.0798907	−0.0399453	+	0.999202i	$0.512718\pi$
$72$	0	0
$73$	−74488.8	−1.63600	−0.818001	−	0.575217i	$-0.804917\pi$
$73$	−74488.8	−1.63600	−0.818001	+	0.575217i	$0.804917\pi$
$74$	2445.69	0.0519184
$75$	0	0
$76$	10136.5	0.201304
$77$	0	0
$78$	0	0
$79$	−59705.1	−1.07632	−0.538162	−	0.842841i	$-0.680881\pi$
$79$	−59705.1	−1.07632	−0.538162	+	0.842841i	$0.680881\pi$
$80$	−7705.84	−0.134615
$81$	0	0
$82$	538.145	0.00883821
$83$	−29019.2	−0.462370	−0.231185	−	0.972910i	$-0.574260\pi$
$83$	−29019.2	−0.462370	−0.231185	+	0.972910i	$0.574260\pi$
$84$	0	0
$85$	−207146.	−3.10978
$86$	30639.4	0.446718
$87$	0	0
$88$	0	0
$89$	−88381.2	−1.18273	−0.591364	−	0.806405i	$-0.701410\pi$
$89$	−88381.2	−1.18273	−0.591364	+	0.806405i	$0.701410\pi$
$90$	0	0
$91$	−26497.8	−0.335433
$92$	16743.9	0.206247
$93$	0	0
$94$	−75444.5	−0.880660
$95$	51961.3	0.590705
$96$	0	0
$97$	−23821.0	−0.257057	−0.128529	−	0.991706i	$-0.541025\pi$
$97$	−23821.0	−0.257057	−0.128529	+	0.991706i	$0.541025\pi$
$98$	50808.8	0.534409
$99$	0	0
$100$	−170679.	−1.70679
$101$	25702.9	0.250714	0.125357	−	0.992112i	$-0.459992\pi$
$101$	25702.9	0.250714	0.125357	+	0.992112i	$0.459992\pi$
$102$	0	0
$103$	−109070.	−1.01300	−0.506502	−	0.862239i	$-0.669062\pi$
$103$	−109070.	−1.01300	−0.506502	+	0.862239i	$0.669062\pi$
$104$	−115615.	−1.04817
$105$	0	0
$106$	−99824.6	−0.862924
$107$	123163.	1.03997	0.519987	−	0.854174i	$-0.325937\pi$
$107$	123163.	1.03997	0.519987	+	0.854174i	$0.325937\pi$
$108$	0	0
$109$	−121789.	−0.981844	−0.490922	−	0.871204i	$-0.663340\pi$
$109$	−121789.	−0.981844	−0.490922	+	0.871204i	$0.663340\pi$
$110$	0	0
$111$	0	0
$112$	2932.37	0.0220889
$113$	28094.3	0.206977	0.103488	−	0.994631i	$-0.467000\pi$
$113$	28094.3	0.206977	0.103488	+	0.994631i	$0.467000\pi$
$114$	0	0
$115$	85832.3	0.605210
$116$	75746.0	0.522654
$117$	0	0
$118$	127016.	0.839759
$119$	78827.2	0.510280
$120$	0	0
$121$	0	0
$122$	77646.2	0.472303
$123$	0	0
$124$	30734.2	0.179501
$125$	−542140.	−3.10339
$126$	0	0
$127$	309785.	1.70432	0.852160	−	0.523281i	$-0.175292\pi$
$127$	309785.	1.70432	0.852160	+	0.523281i	$0.175292\pi$
$128$	130340.	0.703159
$129$	0	0
$130$	−233300.	−1.21076
$131$	−56573.7	−0.288029	−0.144015	−	0.989576i	$-0.546001\pi$
$131$	−56573.7	−0.288029	−0.144015	+	0.989576i	$0.546001\pi$
$132$	0	0
$133$	−19773.3	−0.0969281
$134$	91087.1	0.438223
$135$	0	0
$136$	343940.	1.59454
$137$	−122548.	−0.557834	−0.278917	−	0.960315i	$-0.589975\pi$
$137$	−122548.	−0.557834	−0.278917	+	0.960315i	$0.589975\pi$
$138$	0	0
$139$	93399.8	0.410024	0.205012	−	0.978759i	$-0.434277\pi$
$139$	93399.8	0.410024	0.205012	+	0.978759i	$0.434277\pi$
$140$	89654.7	0.386592
$141$	0	0
$142$	11369.6	0.0473179
$143$	0	0
$144$	0	0
$145$	388287.	1.53367
$146$	249571.	0.968975
$147$	0	0
$148$	15164.4	0.0569079
$149$	−81459.7	−0.300592	−0.150296	−	0.988641i	$-0.548023\pi$
$149$	−81459.7	−0.300592	−0.150296	+	0.988641i	$0.548023\pi$
$150$	0	0
$151$	−389068.	−1.38862	−0.694310	−	0.719676i	$-0.744290\pi$
$151$	−389068.	−1.38862	−0.694310	+	0.719676i	$0.744290\pi$
$152$	−86275.0	−0.302884
$153$	0	0
$154$	0	0
$155$	157549.	0.526726
$156$	0	0
$157$	−22908.2	−0.0741722	−0.0370861	−	0.999312i	$-0.511808\pi$
$157$	−22908.2	−0.0741722	−0.0370861	+	0.999312i	$0.511808\pi$
$158$	200039.	0.637488
$159$	0	0
$160$	628378.	1.94053
$161$	−32662.5	−0.0993081
$162$	0	0
$163$	−181958.	−0.536416	−0.268208	−	0.963361i	$-0.586431\pi$
$163$	−181958.	−0.536416	−0.268208	+	0.963361i	$0.586431\pi$
$164$	3336.76	0.00968757
$165$	0	0
$166$	97227.4	0.273854
$167$	370089.	1.02687	0.513434	−	0.858129i	$-0.328373\pi$
$167$	370089.	1.02687	0.513434	+	0.858129i	$0.328373\pi$
$168$	0	0
$169$	56247.1	0.151490
$170$	694034.	1.84187
$171$	0	0
$172$	189979.	0.489648
$173$	242655.	0.616416	0.308208	−	0.951319i	$-0.400271\pi$
$173$	242655.	0.616416	0.308208	+	0.951319i	$0.400271\pi$
$174$	0	0
$175$	332945.	0.821822
$176$	0	0
$177$	0	0
$178$	296117.	0.700509
$179$	−364575.	−0.850460	−0.425230	−	0.905085i	$-0.639807\pi$
$179$	−364575.	−0.850460	−0.425230	+	0.905085i	$0.639807\pi$
$180$	0	0
$181$	85273.4	0.193472	0.0967358	−	0.995310i	$-0.469160\pi$
$181$	85273.4	0.193472	0.0967358	+	0.995310i	$0.469160\pi$
$182$	88779.7	0.198671
$183$	0	0
$184$	−142513.	−0.310321
$185$	77735.5	0.166990
$186$	0	0
$187$	0	0
$188$	−467793.	−0.965292
$189$	0	0
$190$	−174094.	−0.349864
$191$	305012.	0.604969	0.302485	−	0.953154i	$-0.402184\pi$
$191$	305012.	0.604969	0.302485	+	0.953154i	$0.402184\pi$
$192$	0	0
$193$	−620080.	−1.19827	−0.599135	−	0.800648i	$-0.704489\pi$
$193$	−620080.	−1.19827	−0.599135	+	0.800648i	$0.704489\pi$
$194$	79811.1	0.152251
$195$	0	0
$196$	315039.	0.585766
$197$	933167.	1.71314	0.856572	−	0.516027i	$-0.172590\pi$
$197$	933167.	1.71314	0.856572	+	0.516027i	$0.172590\pi$
$198$	0	0
$199$	267117.	0.478154	0.239077	−	0.971001i	$-0.423155\pi$
$199$	267117.	0.478154	0.239077	+	0.971001i	$0.423155\pi$
$200$	1.45271e6	2.56805
$201$	0	0
$202$	−86116.4	−0.148493
$203$	−147758.	−0.251658
$204$	0	0
$205$	17104.8	0.0284271
$206$	365434.	0.599985
$207$	0	0
$208$	−47313.6	−0.0758276
$209$	0	0
$210$	0	0
$211$	714150.	1.10429	0.552145	−	0.833748i	$-0.313809\pi$
$211$	714150.	1.10429	0.552145	+	0.833748i	$0.313809\pi$
$212$	−618960.	−0.945852
$213$	0	0
$214$	−412653.	−0.615958
$215$	973864.	1.43682
$216$	0	0
$217$	−59953.3	−0.0864299
$218$	408049.	0.581529
$219$	0	0
$220$	0	0
$221$	−1.27187e6	−1.75171
$222$	0	0
$223$	483777.	0.651453	0.325727	−	0.945464i	$-0.394391\pi$
$223$	483777.	0.651453	0.325727	+	0.945464i	$0.394391\pi$
$224$	−239122.	−0.318419
$225$	0	0
$226$	−94128.6	−0.122589
$227$	−239254.	−0.308173	−0.154086	−	0.988057i	$-0.549243\pi$
$227$	−239254.	−0.308173	−0.154086	+	0.988057i	$0.549243\pi$
$228$	0	0
$229$	56860.0	0.0716504	0.0358252	−	0.999358i	$-0.488594\pi$
$229$	56860.0	0.0716504	0.0358252	+	0.999358i	$0.488594\pi$
$230$	−287577.	−0.358455
$231$	0	0
$232$	−644700.	−0.786390
$233$	−729653.	−0.880494	−0.440247	−	0.897877i	$-0.645109\pi$
$233$	−729653.	−0.880494	−0.440247	+	0.897877i	$0.645109\pi$
$234$	0	0
$235$	−2.39798e6	−2.83254
$236$	787563.	0.920460
$237$	0	0
$238$	−264107.	−0.302230
$239$	−1.53740e6	−1.74098	−0.870489	−	0.492188i	$-0.836197\pi$
$239$	−1.53740e6	−1.74098	−0.870489	+	0.492188i	$0.836197\pi$
$240$	0	0
$241$	311983.	0.346010	0.173005	−	0.984921i	$-0.444652\pi$
$241$	311983.	0.346010	0.173005	+	0.984921i	$0.444652\pi$
$242$	0	0
$243$	0	0
$244$	481444.	0.517692
$245$	1.61494e6	1.71887
$246$	0	0
$247$	319041.	0.332739
$248$	−261589.	−0.270079
$249$	0	0
$250$	1.81642e6	1.83808
$251$	−1.34126e6	−1.34378	−0.671890	−	0.740651i	$-0.734517\pi$
$251$	−1.34126e6	−1.34378	−0.671890	+	0.740651i	$0.734517\pi$
$252$	0	0
$253$	0	0
$254$	−1.03792e6	−1.00944
$255$	0	0
$256$	−995237.	−0.949132
$257$	1.08114e6	1.02106	0.510528	−	0.859861i	$-0.329450\pi$
$257$	1.08114e6	1.02106	0.510528	+	0.859861i	$0.329450\pi$
$258$	0	0
$259$	−29581.4	−0.0274012
$260$	−1.44657e6	−1.32711
$261$	0	0
$262$	189548.	0.170595
$263$	874227.	0.779354	0.389677	−	0.920952i	$-0.372587\pi$
$263$	874227.	0.779354	0.389677	+	0.920952i	$0.372587\pi$
$264$	0	0
$265$	−3.17290e6	−2.77550
$266$	66249.5	0.0574088
$267$	0	0
$268$	564784.	0.480336
$269$	515759.	0.434576	0.217288	−	0.976107i	$-0.430279\pi$
$269$	515759.	0.434576	0.217288	+	0.976107i	$0.430279\pi$
$270$	0	0
$271$	−5463.46	−0.00451903	−0.00225951	−	0.999997i	$-0.500719\pi$
$271$	−5463.46	−0.00451903	−0.00225951	+	0.999997i	$0.500719\pi$
$272$	140751.	0.115353
$273$	0	0
$274$	410592.	0.330395
$275$	0	0
$276$	0	0
$277$	−1.80815e6	−1.41591	−0.707953	−	0.706260i	$-0.750381\pi$
$277$	−1.80815e6	−1.41591	−0.707953	+	0.706260i	$0.750381\pi$
$278$	−312932.	−0.242850
$279$	0	0
$280$	−763082.	−0.581669
$281$	−1.02094e6	−0.771321	−0.385660	−	0.922641i	$-0.626026\pi$
$281$	−1.02094e6	−0.771321	−0.385660	+	0.922641i	$0.626026\pi$
$282$	0	0
$283$	2.17055e6	1.61103	0.805515	−	0.592575i	$-0.201889\pi$
$283$	2.17055e6	1.61103	0.805515	+	0.592575i	$0.201889\pi$
$284$	70497.1	0.0518651
$285$	0	0
$286$	0	0
$287$	−6509.03	−0.00466457
$288$	0	0
$289$	2.36378e6	1.66480
$290$	−1.30094e6	−0.908367
$291$	0	0
$292$	1.54746e6	1.06209
$293$	−1.24518e6	−0.847352	−0.423676	−	0.905814i	$-0.639261\pi$
$293$	−1.24518e6	−0.847352	−0.423676	+	0.905814i	$0.639261\pi$
$294$	0	0
$295$	4.03718e6	2.70099
$296$	−129070.	−0.0856240
$297$	0	0
$298$	272927.	0.178035
$299$	527007.	0.340909
$300$	0	0
$301$	−370593.	−0.235766
$302$	1.30356e6	0.822455
$303$	0	0
$304$	−35306.5	−0.0219114
$305$	2.46797e6	1.51911
$306$	0	0
$307$	262490.	0.158952	0.0794761	−	0.996837i	$-0.474675\pi$
$307$	262490.	0.158952	0.0794761	+	0.996837i	$0.474675\pi$
$308$	0	0
$309$	0	0
$310$	−527859.	−0.311971
$311$	1.51351e6	0.887328	0.443664	−	0.896193i	$-0.353678\pi$
$311$	1.51351e6	0.887328	0.443664	+	0.896193i	$0.353678\pi$
$312$	0	0
$313$	2.00963e6	1.15946	0.579728	−	0.814810i	$-0.303159\pi$
$313$	2.00963e6	1.15946	0.579728	+	0.814810i	$0.303159\pi$
$314$	76752.8	0.0439309
$315$	0	0
$316$	1.24034e6	0.698752
$317$	2.31072e6	1.29151	0.645756	−	0.763544i	$-0.276542\pi$
$317$	2.31072e6	1.29151	0.645756	+	0.763544i	$0.276542\pi$
$318$	0	0
$319$	0	0
$320$	−1.85877e6	−1.01473
$321$	0	0
$322$	109434.	0.0588185
$323$	−949100.	−0.506181
$324$	0	0
$325$	−5.37205e6	−2.82119
$326$	609641.	0.317710
$327$	0	0
$328$	−28400.3	−0.0145760
$329$	912526.	0.464789
$330$	0	0
$331$	2.84028e6	1.42492	0.712460	−	0.701712i	$-0.247581\pi$
$331$	2.84028e6	1.42492	0.712460	+	0.701712i	$0.247581\pi$
$332$	602857.	0.300171
$333$	0	0
$334$	−1.23997e6	−0.608196
$335$	2.89518e6	1.40949
$336$	0	0
$337$	−136641.	−0.0655401	−0.0327701	−	0.999463i	$-0.510433\pi$
$337$	−136641.	−0.0655401	−0.0327701	+	0.999463i	$0.510433\pi$
$338$	−188454.	−0.0897248
$339$	0	0
$340$	4.30335e6	2.01887
$341$	0	0
$342$	0	0
$343$	−1.29565e6	−0.594637
$344$	−1.61698e6	−0.736728
$345$	0	0
$346$	−813004.	−0.365092
$347$	−1.35883e6	−0.605816	−0.302908	−	0.953020i	$-0.597957\pi$
$347$	−1.35883e6	−0.605816	−0.302908	+	0.953020i	$0.597957\pi$
$348$	0	0
$349$	−3.21501e6	−1.41292	−0.706462	−	0.707751i	$-0.749710\pi$
$349$	−3.21501e6	−1.41292	−0.706462	+	0.707751i	$0.749710\pi$
$350$	−1.11552e6	−0.486751
$351$	0	0
$352$	0	0
$353$	−1.24606e6	−0.532233	−0.266117	−	0.963941i	$-0.585741\pi$
$353$	−1.24606e6	−0.532233	−0.266117	+	0.963941i	$0.585741\pi$
$354$	0	0
$355$	361380.	0.152193
$356$	1.83607e6	0.767829
$357$	0	0
$358$	1.22149e6	0.503713
$359$	−2.63443e6	−1.07883	−0.539413	−	0.842042i	$-0.681354\pi$
$359$	−2.63443e6	−1.07883	−0.539413	+	0.842042i	$0.681354\pi$
$360$	0	0
$361$	−2.23802e6	−0.903851
$362$	−285705.	−0.114590
$363$	0	0
$364$	550477.	0.217764
$365$	7.93256e6	3.11660
$366$	0	0
$367$	1.36904e6	0.530579	0.265289	−	0.964169i	$-0.414532\pi$
$367$	1.36904e6	0.530579	0.265289	+	0.964169i	$0.414532\pi$
$368$	−58321.1	−0.0224495
$369$	0	0
$370$	−260449.	−0.0989052
$371$	1.20741e6	0.455428
$372$	0	0
$373$	−414377.	−0.154214	−0.0771069	−	0.997023i	$-0.524568\pi$
$373$	−414377.	−0.154214	−0.0771069	+	0.997023i	$0.524568\pi$
$374$	0	0
$375$	0	0
$376$	3.98154e6	1.45239
$377$	2.38407e6	0.863904
$378$	0	0
$379$	882904.	0.315730	0.157865	−	0.987461i	$-0.449539\pi$
$379$	882904.	0.315730	0.157865	+	0.987461i	$0.449539\pi$
$380$	−1.07947e6	−0.383487
$381$	0	0
$382$	−1.02193e6	−0.358313
$383$	−5.26643e6	−1.83451	−0.917254	−	0.398303i	$-0.869599\pi$
$383$	−5.26643e6	−1.83451	−0.917254	+	0.398303i	$0.869599\pi$
$384$	0	0
$385$	0	0
$386$	2.07755e6	0.709714
$387$	0	0
$388$	494868.	0.166882
$389$	−5.57189e6	−1.86693	−0.933467	−	0.358664i	$-0.883232\pi$
$389$	−5.57189e6	−1.86693	−0.933467	+	0.358664i	$0.883232\pi$
$390$	0	0
$391$	−1.56777e6	−0.518610
$392$	−2.68141e6	−0.881348
$393$	0	0
$394$	−3.12654e6	−1.01467
$395$	6.35819e6	2.05041
$396$	0	0
$397$	928630.	0.295710	0.147855	−	0.989009i	$-0.452763\pi$
$397$	928630.	0.295710	0.147855	+	0.989009i	$0.452763\pi$
$398$	−894962.	−0.283203
$399$	0	0
$400$	594496.	0.185780
$401$	1.13125e6	0.351315	0.175657	−	0.984451i	$-0.443795\pi$
$401$	1.13125e6	0.351315	0.175657	+	0.984451i	$0.443795\pi$
$402$	0	0
$403$	967343.	0.296700
$404$	−533963.	−0.162764
$405$	0	0
$406$	495057.	0.149053
$407$	0	0
$408$	0	0
$409$	5.00697e6	1.48002	0.740009	−	0.672597i	$-0.234821\pi$
$409$	5.00697e6	1.48002	0.740009	+	0.672597i	$0.234821\pi$
$410$	−57308.8	−0.0168369
$411$	0	0
$412$	2.26586e6	0.657644
$413$	−1.53630e6	−0.443202
$414$	0	0
$415$	3.09035e6	0.880820
$416$	3.85822e6	1.09308
$417$	0	0
$418$	0	0
$419$	−878392.	−0.244429	−0.122215	−	0.992504i	$-0.539000\pi$
$419$	−878392.	−0.244429	−0.122215	+	0.992504i	$0.539000\pi$
$420$	0	0
$421$	−2.73175e6	−0.751164	−0.375582	−	0.926789i	$-0.622557\pi$
$421$	−2.73175e6	−0.751164	−0.375582	+	0.926789i	$0.622557\pi$
$422$	−2.39273e6	−0.654052
$423$	0	0
$424$	5.26819e6	1.42314
$425$	1.59811e7	4.29175
$426$	0	0
$427$	−939156.	−0.249269
$428$	−2.55865e6	−0.675152
$429$	0	0
$430$	−3.26289e6	−0.851003
$431$	−1.54741e6	−0.401247	−0.200624	−	0.979668i	$-0.564297\pi$
$431$	−1.54741e6	−0.401247	−0.200624	+	0.979668i	$0.564297\pi$
$432$	0	0
$433$	5.16212e6	1.32315	0.661574	−	0.749880i	$-0.269889\pi$
$433$	5.16212e6	1.32315	0.661574	+	0.749880i	$0.269889\pi$
$434$	200871.	0.0511909
$435$	0	0
$436$	2.53010e6	0.637414
$437$	393265.	0.0985104
$438$	0	0
$439$	−26209.3	−0.00649074	−0.00324537	−	0.999995i	$-0.501033\pi$
$439$	−26209.3	−0.00649074	−0.00324537	+	0.999995i	$0.501033\pi$
$440$	0	0
$441$	0	0
$442$	4.26135e6	1.03751
$443$	3.66169e6	0.886487	0.443243	−	0.896401i	$-0.353828\pi$
$443$	3.66169e6	0.886487	0.443243	+	0.896401i	$0.353828\pi$
$444$	0	0
$445$	9.41201e6	2.25311
$446$	−1.62087e6	−0.385844
$447$	0	0
$448$	707332.	0.166505
$449$	895165.	0.209550	0.104775	−	0.994496i	$-0.466588\pi$
$449$	895165.	0.209550	0.104775	+	0.994496i	$0.466588\pi$
$450$	0	0
$451$	0	0
$452$	−583643.	−0.134370
$453$	0	0
$454$	801609.	0.182525
$455$	2.82184e6	0.639004
$456$	0	0
$457$	430433.	0.0964085	0.0482043	−	0.998837i	$-0.484650\pi$
$457$	430433.	0.0964085	0.0482043	+	0.998837i	$0.484650\pi$
$458$	−190507.	−0.0424373
$459$	0	0
$460$	−1.78312e6	−0.392903
$461$	8.86156e6	1.94204	0.971020	−	0.238999i	$-0.0768193\pi$
$461$	8.86156e6	1.94204	0.971020	+	0.238999i	$0.0768193\pi$
$462$	0	0
$463$	4.96972e6	1.07741	0.538703	−	0.842496i	$-0.318914\pi$
$463$	4.96972e6	1.07741	0.538703	+	0.842496i	$0.318914\pi$
$464$	−263832.	−0.0568896
$465$	0	0
$466$	2.44467e6	0.521501
$467$	−2.25040e6	−0.477494	−0.238747	−	0.971082i	$-0.576737\pi$
$467$	−2.25040e6	−0.477494	−0.238747	+	0.971082i	$0.576737\pi$
$468$	0	0
$469$	−1.10173e6	−0.231282
$470$	8.03434e6	1.67767
$471$	0	0
$472$	−6.70322e6	−1.38493
$473$	0	0
$474$	0	0
$475$	−4.00875e6	−0.815220
$476$	−1.63759e6	−0.331274
$477$	0	0
$478$	5.15100e6	1.03115
$479$	5.47470e6	1.09024	0.545119	−	0.838359i	$-0.316484\pi$
$479$	5.47470e6	1.09024	0.545119	+	0.838359i	$0.316484\pi$
$480$	0	0
$481$	477293.	0.0940639
$482$	−1.04529e6	−0.204936
$483$	0	0
$484$	0	0
$485$	2.53678e6	0.489697
$486$	0	0
$487$	−9.47736e6	−1.81078	−0.905389	−	0.424583i	$-0.860421\pi$
$487$	−9.47736e6	−1.81078	−0.905389	+	0.424583i	$0.860421\pi$
$488$	−4.09774e6	−0.778923
$489$	0	0
$490$	−5.41080e6	−1.01805
$491$	−5.89593e6	−1.10369	−0.551847	−	0.833945i	$-0.686077\pi$
$491$	−5.89593e6	−1.10369	−0.551847	+	0.833945i	$0.686077\pi$
$492$	0	0
$493$	−7.09226e6	−1.31422
$494$	−1.06893e6	−0.197075
$495$	0	0
$496$	−107051.	−0.0195382
$497$	−137519.	−0.0249731
$498$	0	0
$499$	−5.27619e6	−0.948570	−0.474285	−	0.880371i	$-0.657293\pi$
$499$	−5.27619e6	−0.948570	−0.474285	+	0.880371i	$0.657293\pi$
$500$	1.12627e7	2.01472
$501$	0	0
$502$	4.49383e6	0.795898
$503$	8.17988e6	1.44154	0.720770	−	0.693174i	$-0.243788\pi$
$503$	8.17988e6	1.44154	0.720770	+	0.693174i	$0.243788\pi$
$504$	0	0
$505$	−2.73719e6	−0.477613
$506$	0	0
$507$	0	0
$508$	−6.43561e6	−1.10645
$509$	−3.67867e6	−0.629357	−0.314678	−	0.949198i	$-0.601897\pi$
$509$	−3.67867e6	−0.629357	−0.314678	+	0.949198i	$0.601897\pi$
$510$	0	0
$511$	−3.01864e6	−0.511399
$512$	−836394.	−0.141005
$513$	0	0
$514$	−3.62232e6	−0.604754
$515$	1.16152e7	1.92978
$516$	0	0
$517$	0	0
$518$	99111.1	0.0162292
$519$	0	0
$520$	1.23123e7	1.99678
$521$	6.05732e6	0.977656	0.488828	−	0.872380i	$-0.337425\pi$
$521$	6.05732e6	0.977656	0.488828	+	0.872380i	$0.337425\pi$
$522$	0	0
$523$	4.86245e6	0.777322	0.388661	−	0.921381i	$-0.372938\pi$
$523$	4.86245e6	0.777322	0.388661	+	0.921381i	$0.372938\pi$
$524$	1.17529e6	0.186989
$525$	0	0
$526$	−2.92906e6	−0.461598
$527$	−2.87771e6	−0.451357
$528$	0	0
$529$	−5.78673e6	−0.899071
$530$	1.06307e7	1.64388
$531$	0	0
$532$	410779.	0.0629258
$533$	105023.	0.0160127
$534$	0	0
$535$	−1.31161e7	−1.98116
$536$	−4.80707e6	−0.722717
$537$	0	0
$538$	−1.72803e6	−0.257392
$539$	0	0
$540$	0	0
$541$	4.30776e6	0.632788	0.316394	−	0.948628i	$-0.397528\pi$
$541$	4.30776e6	0.632788	0.316394	+	0.948628i	$0.397528\pi$
$542$	18305.1	0.00267654
$543$	0	0
$544$	−1.14776e7	−1.66286
$545$	1.29697e7	1.87042
$546$	0	0
$547$	−1.53873e6	−0.219884	−0.109942	−	0.993938i	$-0.535067\pi$
$547$	−1.53873e6	−0.219884	−0.109942	+	0.993938i	$0.535067\pi$
$548$	2.54587e6	0.362146
$549$	0	0
$550$	0	0
$551$	1.77905e6	0.249637
$552$	0	0
$553$	−2.41954e6	−0.336449
$554$	6.05811e6	0.838616
$555$	0	0
$556$	−1.94033e6	−0.266188
$557$	6.45560e6	0.881655	0.440828	−	0.897592i	$-0.354685\pi$
$557$	6.45560e6	0.881655	0.440828	+	0.897592i	$0.354685\pi$
$558$	0	0
$559$	5.97949e6	0.809347
$560$	−312278.	−0.0420795
$561$	0	0
$562$	3.42062e6	0.456840
$563$	−1.17063e7	−1.55650	−0.778250	−	0.627955i	$-0.783892\pi$
$563$	−1.17063e7	−1.55650	−0.778250	+	0.627955i	$0.783892\pi$
$564$	0	0
$565$	−2.99185e6	−0.394293
$566$	−7.27233e6	−0.954185
$567$	0	0
$568$	−600025.	−0.0780367
$569$	9.49908e6	1.22999	0.614994	−	0.788532i	$-0.289158\pi$
$569$	9.49908e6	1.22999	0.614994	+	0.788532i	$0.289158\pi$
$570$	0	0
$571$	2.48173e6	0.318541	0.159270	−	0.987235i	$-0.449086\pi$
$571$	2.48173e6	0.318541	0.159270	+	0.987235i	$0.449086\pi$
$572$	0	0
$573$	0	0
$574$	21808.2	0.00276274
$575$	−6.62186e6	−0.835238
$576$	0	0
$577$	−3.01546e6	−0.377063	−0.188532	−	0.982067i	$-0.560373\pi$
$577$	−3.01546e6	−0.377063	−0.188532	+	0.982067i	$0.560373\pi$
$578$	−7.91974e6	−0.986032
$579$	0	0
$580$	−8.06644e6	−0.995662
$581$	−1.17600e6	−0.144533
$582$	0	0
$583$	0	0
$584$	−1.31710e7	−1.59803
$585$	0	0
$586$	4.17193e6	0.501872
$587$	−1.43298e7	−1.71651	−0.858253	−	0.513227i	$-0.828450\pi$
$587$	−1.43298e7	−1.71651	−0.858253	+	0.513227i	$0.828450\pi$
$588$	0	0
$589$	721854.	0.0857356
$590$	−1.35264e7	−1.59975
$591$	0	0
$592$	−52819.5	−0.00619427
$593$	2.61274e6	0.305112	0.152556	−	0.988295i	$-0.451250\pi$
$593$	2.61274e6	0.305112	0.152556	+	0.988295i	$0.451250\pi$
$594$	0	0
$595$	−8.39457e6	−0.972089
$596$	1.69228e6	0.195145
$597$	0	0
$598$	−1.76571e6	−0.201915
$599$	−7.91980e6	−0.901877	−0.450938	−	0.892555i	$-0.648911\pi$
$599$	−7.91980e6	−0.901877	−0.450938	+	0.892555i	$0.648911\pi$
$600$	0	0
$601$	−1.54857e6	−0.174882	−0.0874410	−	0.996170i	$-0.527869\pi$
$601$	−1.54857e6	−0.174882	−0.0874410	+	0.996170i	$0.527869\pi$
$602$	1.24166e6	0.139640
$603$	0	0
$604$	8.08267e6	0.901493
$605$	0	0
$606$	0	0
$607$	1.59268e6	0.175451	0.0877256	−	0.996145i	$-0.472040\pi$
$607$	1.59268e6	0.175451	0.0877256	+	0.996145i	$0.472040\pi$
$608$	2.87909e6	0.315862
$609$	0	0
$610$	−8.26881e6	−0.899743
$611$	−1.47235e7	−1.59555
$612$	0	0
$613$	1.25549e7	1.34947	0.674735	−	0.738061i	$-0.264258\pi$
$613$	1.25549e7	1.34947	0.674735	+	0.738061i	$0.264258\pi$
$614$	−879461.	−0.0941447
$615$	0	0
$616$	0	0
$617$	−5.15506e6	−0.545156	−0.272578	−	0.962134i	$-0.587876\pi$
$617$	−5.15506e6	−0.545156	−0.272578	+	0.962134i	$0.587876\pi$
$618$	0	0
$619$	−1.20795e7	−1.26713	−0.633567	−	0.773688i	$-0.718410\pi$
$619$	−1.20795e7	−1.26713	−0.633567	+	0.773688i	$0.718410\pi$
$620$	−3.27298e6	−0.341952
$621$	0	0
$622$	−5.07095e6	−0.525549
$623$	−3.58163e6	−0.369710
$624$	0	0
$625$	3.20598e7	3.28292
$626$	−6.73316e6	−0.686726
$627$	0	0
$628$	475904.	0.0481527
$629$	−1.41988e6	−0.143095
$630$	0	0
$631$	1.63469e7	1.63441	0.817205	−	0.576347i	$-0.195522\pi$
$631$	1.63469e7	1.63441	0.817205	+	0.576347i	$0.195522\pi$
$632$	−1.05570e7	−1.05135
$633$	0	0
$634$	−7.74195e6	−0.764940
$635$	−3.29901e7	−3.24675
$636$	0	0
$637$	9.91570e6	0.968222
$638$	0	0
$639$	0	0
$640$	−1.38804e7	−1.33953
$641$	4.61760e6	0.443886	0.221943	−	0.975060i	$-0.428760\pi$
$641$	4.61760e6	0.443886	0.221943	+	0.975060i	$0.428760\pi$
$642$	0	0
$643$	9.91263e6	0.945500	0.472750	−	0.881197i	$-0.343261\pi$
$643$	9.91263e6	0.945500	0.472750	+	0.881197i	$0.343261\pi$
$644$	678545.	0.0644710
$645$	0	0
$646$	3.17992e6	0.299802
$647$	4.16757e6	0.391401	0.195701	−	0.980664i	$-0.437302\pi$
$647$	4.16757e6	0.391401	0.195701	+	0.980664i	$0.437302\pi$
$648$	0	0
$649$	0	0
$650$	1.79988e7	1.67094
$651$	0	0
$652$	3.78007e6	0.348242
$653$	1.49762e7	1.37441	0.687207	−	0.726462i	$-0.258837\pi$
$653$	1.49762e7	1.37441	0.687207	+	0.726462i	$0.258837\pi$
$654$	0	0
$655$	6.02473e6	0.548699
$656$	−11622.3	−0.00105447
$657$	0	0
$658$	−3.05738e6	−0.275286
$659$	−2.57761e6	−0.231209	−0.115604	−	0.993295i	$-0.536880\pi$
$659$	−2.57761e6	−0.231209	−0.115604	+	0.993295i	$0.536880\pi$
$660$	0	0
$661$	−1.72855e7	−1.53879	−0.769395	−	0.638773i	$-0.779442\pi$
$661$	−1.72855e7	−1.53879	−0.769395	+	0.638773i	$0.779442\pi$
$662$	−9.51622e6	−0.843955
$663$	0	0
$664$	−5.13112e6	−0.451640
$665$	2.10572e6	0.184649
$666$	0	0
$667$	2.93872e6	0.255767
$668$	−7.68839e6	−0.666644
$669$	0	0
$670$	−9.70017e6	−0.834819
$671$	0	0
$672$	0	0
$673$	−234736.	−0.0199775	−0.00998877	−	0.999950i	$-0.503180\pi$
$673$	−234736.	−0.0199775	−0.00998877	+	0.999950i	$0.503180\pi$
$674$	457811.	0.0388183
$675$	0	0
$676$	−1.16850e6	−0.0983474
$677$	5.09508e6	0.427248	0.213624	−	0.976916i	$-0.431473\pi$
$677$	5.09508e6	0.427248	0.213624	+	0.976916i	$0.431473\pi$
$678$	0	0
$679$	−965341.	−0.0803538
$680$	−3.66273e7	−3.03761
$681$	0	0
$682$	0	0
$683$	−6.21558e6	−0.509835	−0.254917	−	0.966963i	$-0.582048\pi$
$683$	−6.21558e6	−0.509835	−0.254917	+	0.966963i	$0.582048\pi$
$684$	0	0
$685$	1.30505e7	1.06268
$686$	4.34102e6	0.352193
$687$	0	0
$688$	−661718.	−0.0532969
$689$	−1.94815e7	−1.56341
$690$	0	0
$691$	−9.02204e6	−0.718803	−0.359401	−	0.933183i	$-0.617019\pi$
$691$	−9.02204e6	−0.718803	−0.359401	+	0.933183i	$0.617019\pi$
$692$	−5.04102e6	−0.400178
$693$	0	0
$694$	4.55269e6	0.358814
$695$	−9.94646e6	−0.781100
$696$	0	0
$697$	−312428.	−0.0243595
$698$	1.07718e7	0.836850
$699$	0	0
$700$	−6.91675e6	−0.533528
$701$	−2.54112e6	−0.195312	−0.0976562	−	0.995220i	$-0.531135\pi$
$701$	−2.54112e6	−0.195312	−0.0976562	+	0.995220i	$0.531135\pi$
$702$	0	0
$703$	356168.	0.0271810
$704$	0	0
$705$	0	0
$706$	4.17487e6	0.315232
$707$	1.04161e6	0.0783709
$708$	0	0
$709$	−1.50865e7	−1.12713	−0.563565	−	0.826072i	$-0.690570\pi$
$709$	−1.50865e7	−1.12713	−0.563565	+	0.826072i	$0.690570\pi$
$710$	−1.21079e6	−0.0901410
$711$	0	0
$712$	−1.56274e7	−1.15528
$713$	1.19239e6	0.0878408
$714$	0	0
$715$	0	0
$716$	7.57384e6	0.552120
$717$	0	0
$718$	8.82655e6	0.638969
$719$	1.21890e7	0.879318	0.439659	−	0.898165i	$-0.355099\pi$
$719$	1.21890e7	0.879318	0.439659	+	0.898165i	$0.355099\pi$
$720$	0	0
$721$	−4.42004e6	−0.316656
$722$	7.49840e6	0.535335
$723$	0	0
$724$	−1.77151e6	−0.125602
$725$	−2.99559e7	−2.11659
$726$	0	0
$727$	1.81872e7	1.27623	0.638117	−	0.769939i	$-0.279713\pi$
$727$	1.81872e7	1.27623	0.638117	+	0.769939i	$0.279713\pi$
$728$	−4.68530e6	−0.327649
$729$	0	0
$730$	−2.65777e7	−1.84591
$731$	−1.77881e7	−1.23122
$732$	0	0
$733$	2.14753e7	1.47632	0.738158	−	0.674628i	$-0.235696\pi$
$733$	2.14753e7	1.47632	0.738158	+	0.674628i	$0.235696\pi$
$734$	−4.58690e6	−0.314253
$735$	0	0
$736$	4.75583e6	0.323617
$737$	0	0
$738$	0	0
$739$	−2.37346e7	−1.59871	−0.799357	−	0.600857i	$-0.794826\pi$
$739$	−2.37346e7	−1.59871	−0.799357	+	0.600857i	$0.794826\pi$
$740$	−1.61491e6	−0.108410
$741$	0	0
$742$	−4.04537e6	−0.269742
$743$	2.48112e7	1.64883	0.824416	−	0.565984i	$-0.191504\pi$
$743$	2.48112e7	1.64883	0.824416	+	0.565984i	$0.191504\pi$
$744$	0	0
$745$	8.67492e6	0.572631
$746$	1.38835e6	0.0913382
$747$	0	0
$748$	0	0
$749$	4.99118e6	0.325086
$750$	0	0
$751$	−1.24137e7	−0.803156	−0.401578	−	0.915825i	$-0.631538\pi$
$751$	−1.24137e7	−0.803156	−0.401578	+	0.915825i	$0.631538\pi$
$752$	1.62938e6	0.105070
$753$	0	0
$754$	−7.98771e6	−0.511675
$755$	4.14332e7	2.64533
$756$	0	0
$757$	1.25890e7	0.798458	0.399229	−	0.916851i	$-0.369278\pi$
$757$	1.25890e7	0.798458	0.399229	+	0.916851i	$0.369278\pi$
$758$	−2.95813e6	−0.187001
$759$	0	0
$760$	9.18771e6	0.576997
$761$	1.19917e7	0.750621	0.375310	−	0.926899i	$-0.377536\pi$
$761$	1.19917e7	0.750621	0.375310	+	0.926899i	$0.377536\pi$
$762$	0	0
$763$	−4.93549e6	−0.306915
$764$	−6.33645e6	−0.392747
$765$	0	0
$766$	1.76449e7	1.08655
$767$	2.47882e7	1.52144
$768$	0	0
$769$	−1.41477e7	−0.862719	−0.431360	−	0.902180i	$-0.641966\pi$
$769$	−1.41477e7	−0.862719	−0.431360	+	0.902180i	$0.641966\pi$
$770$	0	0
$771$	0	0
$772$	1.28818e7	0.777918
$773$	1.31734e6	0.0792954	0.0396477	−	0.999214i	$-0.487376\pi$
$773$	1.31734e6	0.0792954	0.0396477	+	0.999214i	$0.487376\pi$
$774$	0	0
$775$	−1.21547e7	−0.726924
$776$	−4.21199e6	−0.251092
$777$	0	0
$778$	1.86684e7	1.10575
$779$	78370.5	0.00462710
$780$	0	0
$781$	0	0
$782$	5.25275e6	0.307164
$783$	0	0
$784$	−1.09732e6	−0.0637591
$785$	2.43957e6	0.141299
$786$	0	0
$787$	2.52381e7	1.45251	0.726256	−	0.687424i	$-0.241259\pi$
$787$	2.52381e7	1.45251	0.726256	+	0.687424i	$0.241259\pi$
$788$	−1.93860e7	−1.11218
$789$	0	0
$790$	−2.13028e7	−1.21442
$791$	1.13852e6	0.0646991
$792$	0	0
$793$	1.51532e7	0.855701
$794$	−3.11133e6	−0.175144
$795$	0	0
$796$	−5.54920e6	−0.310419
$797$	5.05931e6	0.282128	0.141064	−	0.990001i	$-0.454948\pi$
$797$	5.05931e6	0.282128	0.141064	+	0.990001i	$0.454948\pi$
$798$	0	0
$799$	4.38005e7	2.42723
$800$	−4.84786e7	−2.67809
$801$	0	0
$802$	−3.79019e6	−0.208078
$803$	0	0
$804$	0	0
$805$	3.47834e6	0.189183
$806$	−3.24104e6	−0.175730
$807$	0	0
$808$	4.54474e6	0.244896
$809$	2.43382e6	0.130743	0.0653713	−	0.997861i	$-0.479177\pi$
$809$	2.43382e6	0.130743	0.0653713	+	0.997861i	$0.479177\pi$
$810$	0	0
$811$	−3.33464e7	−1.78032	−0.890158	−	0.455652i	$-0.849406\pi$
$811$	−3.33464e7	−1.78032	−0.890158	+	0.455652i	$0.849406\pi$
$812$	3.06959e6	0.163377
$813$	0	0
$814$	0	0
$815$	1.93773e7	1.02188
$816$	0	0
$817$	4.46204e6	0.233872
$818$	−1.67756e7	−0.876588
$819$	0	0
$820$	−355342.	−0.0184549
$821$	−1.82959e7	−0.947316	−0.473658	−	0.880709i	$-0.657067\pi$
$821$	−1.82959e7	−0.947316	−0.473658	+	0.880709i	$0.657067\pi$
$822$	0	0
$823$	−1.75009e7	−0.900658	−0.450329	−	0.892863i	$-0.648693\pi$
$823$	−1.75009e7	−0.900658	−0.450329	+	0.892863i	$0.648693\pi$
$824$	−1.92856e7	−0.989496
$825$	0	0
$826$	5.14732e6	0.262501
$827$	−2.00385e7	−1.01883	−0.509416	−	0.860521i	$-0.670138\pi$
$827$	−2.00385e7	−1.01883	−0.509416	+	0.860521i	$0.670138\pi$
$828$	0	0
$829$	−5.93667e6	−0.300025	−0.150012	−	0.988684i	$-0.547931\pi$
$829$	−5.93667e6	−0.300025	−0.150012	+	0.988684i	$0.547931\pi$
$830$	−1.03541e7	−0.521694
$831$	0	0
$832$	−1.14128e7	−0.571587
$833$	−2.94978e7	−1.47291
$834$	0	0
$835$	−3.94120e7	−1.95620
$836$	0	0
$837$	0	0
$838$	2.94301e6	0.144771
$839$	−6.60244e6	−0.323817	−0.161908	−	0.986806i	$-0.551765\pi$
$839$	−6.60244e6	−0.323817	−0.161908	+	0.986806i	$0.551765\pi$
$840$	0	0
$841$	−7.21701e6	−0.351858
$842$	9.15259e6	0.444902
$843$	0	0
$844$	−1.48361e7	−0.716907
$845$	−5.98995e6	−0.288590
$846$	0	0
$847$	0	0
$848$	2.15591e6	0.102954
$849$	0	0
$850$	−5.35439e7	−2.54193
$851$	588336.	0.0278485
$852$	0	0
$853$	3.52446e7	1.65852	0.829259	−	0.558865i	$-0.188763\pi$
$853$	3.52446e7	1.65852	0.829259	+	0.558865i	$0.188763\pi$
$854$	3.14660e6	0.147638
$855$	0	0
$856$	2.17776e7	1.01584
$857$	1.76534e7	0.821061	0.410530	−	0.911847i	$-0.365344\pi$
$857$	1.76534e7	0.821061	0.410530	+	0.911847i	$0.365344\pi$
$858$	0	0
$859$	1.80229e7	0.833380	0.416690	−	0.909049i	$-0.363190\pi$
$859$	1.80229e7	0.833380	0.416690	+	0.909049i	$0.363190\pi$
$860$	−2.02315e7	−0.932785
$861$	0	0
$862$	5.18452e6	0.237652
$863$	−1.29297e7	−0.590963	−0.295482	−	0.955348i	$-0.595480\pi$
$863$	−1.29297e7	−0.590963	−0.295482	+	0.955348i	$0.595480\pi$
$864$	0	0
$865$	−2.58411e7	−1.17428
$866$	−1.72955e7	−0.783678
$867$	0	0
$868$	1.24550e6	0.0561104
$869$	0	0
$870$	0	0
$871$	1.77763e7	0.793955
$872$	−2.15346e7	−0.959058
$873$	0	0
$874$	−1.31762e6	−0.0583460
$875$	−2.19701e7	−0.970091
$876$	0	0
$877$	6.32463e6	0.277674	0.138837	−	0.990315i	$-0.455664\pi$
$877$	6.32463e6	0.277674	0.138837	+	0.990315i	$0.455664\pi$
$878$	87813.1	0.00384435
$879$	0	0
$880$	0	0
$881$	2.07506e7	0.900721	0.450360	−	0.892847i	$-0.351296\pi$
$881$	2.07506e7	0.900721	0.450360	+	0.892847i	$0.351296\pi$
$882$	0	0
$883$	−4.75082e6	−0.205053	−0.102527	−	0.994730i	$-0.532693\pi$
$883$	−4.75082e6	−0.205053	−0.102527	+	0.994730i	$0.532693\pi$
$884$	2.64224e7	1.13721
$885$	0	0
$886$	−1.22683e7	−0.525050
$887$	1.82655e7	0.779511	0.389755	−	0.920918i	$-0.372560\pi$
$887$	1.82655e7	0.779511	0.389755	+	0.920918i	$0.372560\pi$
$888$	0	0
$889$	1.25540e7	0.532755
$890$	−3.15345e7	−1.33448
$891$	0	0
$892$	−1.00502e7	−0.422924
$893$	−1.09871e7	−0.461055
$894$	0	0
$895$	3.88248e7	1.62014
$896$	5.28202e6	0.219801
$897$	0	0
$898$	−2.99921e6	−0.124113
$899$	5.39414e6	0.222599
$900$	0	0
$901$	5.79546e7	2.37835
$902$	0	0
$903$	0	0
$904$	4.96759e6	0.202174
$905$	−9.08105e6	−0.368566
$906$	0	0
$907$	−3.48363e7	−1.40609	−0.703047	−	0.711144i	$-0.748178\pi$
$907$	−3.48363e7	−1.40609	−0.703047	+	0.711144i	$0.748178\pi$
$908$	4.97036e6	0.200066
$909$	0	0
$910$	−9.45445e6	−0.378471
$911$	7.83083e6	0.312616	0.156308	−	0.987708i	$-0.450041\pi$
$911$	7.83083e6	0.312616	0.156308	+	0.987708i	$0.450041\pi$
$912$	0	0
$913$	0	0
$914$	−1.44215e6	−0.0571011
$915$	0	0
$916$	−1.18124e6	−0.0465155
$917$	−2.29264e6	−0.0900353
$918$	0	0
$919$	−222249.	−0.00868062	−0.00434031	−	0.999991i	$-0.501382\pi$
$919$	−222249.	−0.00868062	−0.00434031	+	0.999991i	$0.501382\pi$
$920$	1.51767e7	0.591165
$921$	0	0
$922$	−2.96903e7	−1.15024
$923$	2.21886e6	0.0857287
$924$	0	0
$925$	−5.99720e6	−0.230459
$926$	−1.66508e7	−0.638129
$927$	0	0
$928$	2.15144e7	0.820084
$929$	−1.75961e7	−0.668924	−0.334462	−	0.942409i	$-0.608555\pi$
$929$	−1.75961e7	−0.668924	−0.334462	+	0.942409i	$0.608555\pi$
$930$	0	0
$931$	7.39933e6	0.279781
$932$	1.51581e7	0.571618
$933$	0	0
$934$	7.53988e6	0.282812
$935$	0	0
$936$	0	0
$937$	−5.63644e6	−0.209728	−0.104864	−	0.994487i	$-0.533441\pi$
$937$	−5.63644e6	−0.209728	−0.104864	+	0.994487i	$0.533441\pi$
$938$	3.69129e6	0.136984
$939$	0	0
$940$	4.98168e7	1.83889
$941$	−2.35454e7	−0.866825	−0.433413	−	0.901196i	$-0.642691\pi$
$941$	−2.35454e7	−0.866825	−0.433413	+	0.901196i	$0.642691\pi$
$942$	0	0
$943$	129456.	0.00474072
$944$	−2.74317e6	−0.100190
$945$	0	0
$946$	0	0
$947$	3.22218e7	1.16755	0.583775	−	0.811916i	$-0.301575\pi$
$947$	3.22218e7	1.16755	0.583775	+	0.811916i	$0.301575\pi$
$948$	0	0
$949$	4.87056e7	1.75555
$950$	1.34311e7	0.482841
$951$	0	0
$952$	1.39381e7	0.498438
$953$	1.10921e6	0.0395621	0.0197811	−	0.999804i	$-0.493703\pi$
$953$	1.10921e6	0.0395621	0.0197811	+	0.999804i	$0.493703\pi$
$954$	0	0
$955$	−3.24817e7	−1.15247
$956$	3.19387e7	1.13025
$957$	0	0
$958$	−1.83427e7	−0.645729
$959$	−4.96624e6	−0.174374
$960$	0	0
$961$	−2.64405e7	−0.923550
$962$	−1.59915e6	−0.0557124
$963$	0	0
$964$	−6.48128e6	−0.224630
$965$	6.60344e7	2.28272
$966$	0	0
$967$	−2.40125e7	−0.825792	−0.412896	−	0.910778i	$-0.635483\pi$
$967$	−2.40125e7	−0.825792	−0.412896	+	0.910778i	$0.635483\pi$
$968$	0	0
$969$	0	0
$970$	−8.49936e6	−0.290039
$971$	−8.18827e6	−0.278704	−0.139352	−	0.990243i	$-0.544502\pi$
$971$	−8.18827e6	−0.278704	−0.139352	+	0.990243i	$0.544502\pi$
$972$	0	0
$973$	3.78501e6	0.128170
$974$	3.17535e7	1.07249
$975$	0	0
$976$	−1.67693e6	−0.0563494
$977$	3.61000e7	1.20996	0.604979	−	0.796241i	$-0.293181\pi$
$977$	3.61000e7	1.20996	0.604979	+	0.796241i	$0.293181\pi$
$978$	0	0
$979$	0	0
$980$	−3.35496e7	−1.11589
$981$	0	0
$982$	1.97541e7	0.653699
$983$	−2.57613e7	−0.850323	−0.425162	−	0.905117i	$-0.639783\pi$
$983$	−2.57613e7	−0.850323	−0.425162	+	0.905117i	$0.639783\pi$
$984$	0	0
$985$	−9.93761e7	−3.26356
$986$	2.37623e7	0.778389
$987$	0	0
$988$	−6.62789e6	−0.216015
$989$	7.37062e6	0.239615
$990$	0	0
$991$	−4.90685e6	−0.158715	−0.0793576	−	0.996846i	$-0.525287\pi$
$991$	−4.90685e6	−0.158715	−0.0793576	+	0.996846i	$0.525287\pi$
$992$	8.72952e6	0.281651
$993$	0	0
$994$	460752.	0.0147911
$995$	−2.84461e7	−0.910889
$996$	0	0
$997$	−4.05150e7	−1.29086	−0.645428	−	0.763821i	$-0.723321\pi$
$997$	−4.05150e7	−1.29086	−0.645428	+	0.763821i	$0.723321\pi$
$998$	1.76776e7	0.561821
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bk.1.4		10
3.2	odd	2		363.6.a.r.1.7		10
11.3	even	5		99.6.f.b.64.2		20
11.4	even	5		99.6.f.b.82.2		20
11.10	odd	2		1089.6.a.bi.1.7		10
33.14	odd	10		33.6.e.b.31.4	yes	20
33.26	odd	10		33.6.e.b.16.4	✓	20
33.32	even	2		363.6.a.t.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.b.16.4	✓	20	33.26	odd	10
33.6.e.b.31.4	yes	20	33.14	odd	10
99.6.f.b.64.2		20	11.3	even	5
99.6.f.b.82.2		20	11.4	even	5
363.6.a.r.1.7		10	3.2	odd	2
363.6.a.t.1.4		10	33.32	even	2
1089.6.a.bi.1.7		10	11.10	odd	2
1089.6.a.bk.1.4		10	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)

\(f(q)\)	\(=\)	\(q-3.35046 q^{2} -20.7744 q^{4} -106.493 q^{5} +40.5248 q^{7} +176.818 q^{8} +O(q^{10})\)	\(q-3.35046 q^{2} -20.7744 q^{4} -106.493 q^{5} +40.5248 q^{7} +176.818 q^{8} +356.801 q^{10} -653.866 q^{13} -135.777 q^{14} +72.3598 q^{16} +1945.16 q^{17} -487.930 q^{19} +2212.34 q^{20} -805.987 q^{23} +8215.83 q^{25} +2190.75 q^{26} -841.881 q^{28} -3646.11 q^{29} -1479.42 q^{31} -5900.63 q^{32} -6517.16 q^{34} -4315.63 q^{35} -729.957 q^{37} +1634.79 q^{38} -18830.0 q^{40} -160.618 q^{41} -9144.84 q^{43} +2700.42 q^{46} +22517.7 q^{47} -15164.7 q^{49} -27526.8 q^{50} +13583.7 q^{52} +29794.3 q^{53} +7165.54 q^{56} +12216.1 q^{58} -37910.2 q^{59} -23174.8 q^{61} +4956.74 q^{62} +17454.3 q^{64} +69632.3 q^{65} -27186.5 q^{67} -40409.6 q^{68} +14459.3 q^{70} -3393.45 q^{71} -74488.8 q^{73} +2445.69 q^{74} +10136.5 q^{76} -59705.1 q^{79} -7705.84 q^{80} +538.145 q^{82} -29019.2 q^{83} -207146. q^{85} +30639.4 q^{86} -88381.2 q^{89} -26497.8 q^{91} +16743.9 q^{92} -75444.5 q^{94} +51961.3 q^{95} -23821.0 q^{97} +50808.8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8}+O(q^{10}) \) 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8	\( 10 q + 7 q^{2} + 149 q^{4} + 33 q^{5} + 78 q^{7} + 438 q^{8} - 212 q^{10} - 1016 q^{13} - 1566 q^{14} + 2361 q^{16} + 1669 q^{17} + 2929 q^{19} + 10189 q^{20} - 4070 q^{23} + 2425 q^{25} - 8481 q^{26} + 3272 q^{28} + 11940 q^{29} - 16085 q^{31} + 2313 q^{32} + 8270 q^{34} - 6987 q^{35} + 16136 q^{37} - 10721 q^{38} + 9332 q^{40} + 16278 q^{41} - 10844 q^{43} - 25995 q^{46} + 22411 q^{47} + 75150 q^{49} - 738 q^{50} - 8677 q^{52} + 27511 q^{53} - 84447 q^{56} + 16853 q^{58} + 39641 q^{59} - 3509 q^{61} + 227845 q^{62} - 22980 q^{64} + 67097 q^{65} + 10089 q^{67} + 273621 q^{68} - 38919 q^{70} - 60681 q^{71} - 133740 q^{73} + 317933 q^{74} + 23434 q^{76} - 12386 q^{79} + 289014 q^{80} + 385033 q^{82} + 187242 q^{83} - 191504 q^{85} + 43793 q^{86} - 102746 q^{89} - 435248 q^{91} - 30867 q^{92} - 404734 q^{94} + 648147 q^{95} - 120631 q^{97} + 1148087 q^{98}+O(q^{100}) \) 10 * q + 7 * q^2 + 149 * q^4 + 33 * q^5 + 78 * q^7 + 438 * q^8 - 212 * q^10 - 1016 * q^13 - 1566 * q^14 + 2361 * q^16 + 1669 * q^17 + 2929 * q^19 + 10189 * q^20 - 4070 * q^23 + 2425 * q^25 - 8481 * q^26 + 3272 * q^28 + 11940 * q^29 - 16085 * q^31 + 2313 * q^32 + 8270 * q^34 - 6987 * q^35 + 16136 * q^37 - 10721 * q^38 + 9332 * q^40 + 16278 * q^41 - 10844 * q^43 - 25995 * q^46 + 22411 * q^47 + 75150 * q^49 - 738 * q^50 - 8677 * q^52 + 27511 * q^53 - 84447 * q^56 + 16853 * q^58 + 39641 * q^59 - 3509 * q^61 + 227845 * q^62 - 22980 * q^64 + 67097 * q^65 + 10089 * q^67 + 273621 * q^68 - 38919 * q^70 - 60681 * q^71 - 133740 * q^73 + 317933 * q^74 + 23434 * q^76 - 12386 * q^79 + 289014 * q^80 + 385033 * q^82 + 187242 * q^83 - 191504 * q^85 + 43793 * q^86 - 102746 * q^89 - 435248 * q^91 - 30867 * q^92 - 404734 * q^94 + 648147 * q^95 - 120631 * q^97 + 1148087 * q^98

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