LMFDB - Embedded newform 1089.6.a.o.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{313}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 78 $ x^2 - x - 78
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			$9.34590$ 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)

$f(q)$	$=$	$q+9.34590 q^{2} +55.3459 q^{4} -69.4590 q^{5} -8.69181 q^{7} +218.189 q^{8} +O(q^{10})$	$q+9.34590 q^{2} +55.3459 q^{4} -69.4590 q^{5} -8.69181 q^{7} +218.189 q^{8} -649.157 q^{10} +970.666 q^{13} -81.2328 q^{14} +268.100 q^{16} -424.616 q^{17} +1430.62 q^{19} -3844.27 q^{20} -2852.99 q^{23} +1699.56 q^{25} +9071.75 q^{26} -481.056 q^{28} -7467.66 q^{29} +10346.3 q^{31} -4476.40 q^{32} -3968.42 q^{34} +603.724 q^{35} +167.311 q^{37} +13370.4 q^{38} -15155.2 q^{40} +5682.18 q^{41} -21148.9 q^{43} -26663.8 q^{46} +9785.11 q^{47} -16731.5 q^{49} +15883.9 q^{50} +53722.4 q^{52} -25639.5 q^{53} -1896.45 q^{56} -69792.0 q^{58} +23411.5 q^{59} -18591.8 q^{61} +96695.6 q^{62} -50415.2 q^{64} -67421.5 q^{65} +39477.7 q^{67} -23500.8 q^{68} +5642.35 q^{70} -3283.01 q^{71} -29536.7 q^{73} +1563.67 q^{74} +79178.8 q^{76} +10280.8 q^{79} -18622.0 q^{80} +53105.1 q^{82} -38360.3 q^{83} +29493.4 q^{85} -197656. q^{86} +2319.99 q^{89} -8436.84 q^{91} -157902. q^{92} +91450.7 q^{94} -99369.2 q^{95} -81868.3 q^{97} -156371. q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 93 q^{4} + 38 q^{5} + 18 q^{7} + 171 q^{8}+O(q^{10}) $ 2 * q + q^2 + 93 * q^4 + 38 * q^5 + 18 * q^7 + 171 * q^8	$ 2 q + q^{2} + 93 q^{4} + 38 q^{5} + 18 q^{7} + 171 q^{8} - 1546 q^{10} + 66 q^{13} - 304 q^{14} - 543 q^{16} - 920 q^{17} + 2932 q^{19} + 202 q^{20} - 5246 q^{23} + 10122 q^{25} + 16622 q^{26} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} + 3803 q^{32} + 166 q^{34} + 3472 q^{35} + 5996 q^{37} + 840 q^{38} - 20226 q^{40} + 24244 q^{41} - 20360 q^{43} - 6692 q^{46} + 5806 q^{47} - 32826 q^{49} - 54409 q^{50} + 19658 q^{52} - 40770 q^{53} - 3156 q^{56} - 26958 q^{58} - 18212 q^{59} + 11398 q^{61} + 100120 q^{62} - 93559 q^{64} - 164636 q^{65} + 65368 q^{67} - 42154 q^{68} - 18296 q^{70} - 61446 q^{71} - 53412 q^{73} - 47082 q^{74} + 135712 q^{76} - 17122 q^{79} - 105782 q^{80} - 101810 q^{82} - 14304 q^{83} - 23740 q^{85} - 204240 q^{86} + 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 124660 q^{94} + 61968 q^{95} - 183056 q^{97} - 22047 q^{98}+O(q^{100}) $ 2 * q + q^2 + 93 * q^4 + 38 * q^5 + 18 * q^7 + 171 * q^8 - 1546 * q^10 + 66 * q^13 - 304 * q^14 - 543 * q^16 - 920 * q^17 + 2932 * q^19 + 202 * q^20 - 5246 * q^23 + 10122 * q^25 + 16622 * q^26 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 + 3803 * q^32 + 166 * q^34 + 3472 * q^35 + 5996 * q^37 + 840 * q^38 - 20226 * q^40 + 24244 * q^41 - 20360 * q^43 - 6692 * q^46 + 5806 * q^47 - 32826 * q^49 - 54409 * q^50 + 19658 * q^52 - 40770 * q^53 - 3156 * q^56 - 26958 * q^58 - 18212 * q^59 + 11398 * q^61 + 100120 * q^62 - 93559 * q^64 - 164636 * q^65 + 65368 * q^67 - 42154 * q^68 - 18296 * q^70 - 61446 * q^71 - 53412 * q^73 - 47082 * q^74 + 135712 * q^76 - 17122 * q^79 - 105782 * q^80 - 101810 * q^82 - 14304 * q^83 - 23740 * q^85 - 204240 * q^86 + 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 124660 * q^94 + 61968 * q^95 - 183056 * q^97 - 22047 * 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 $

$2$	9.34590	1.65214	0.826069	−	0.563569i	$-0.190572\pi$
$2$	9.34590	1.65214	0.826069	+	0.563569i	$0.190572\pi$
$3$	0	0
$3$	0	0
$4$	55.3459	1.72956
$5$	−69.4590	−1.24252	−0.621260	−	0.783604i	$-0.713379\pi$
$5$	−69.4590	−1.24252	−0.621260	+	0.783604i	$0.713379\pi$
$6$	0	0
$7$	−8.69181	−0.0670448	−0.0335224	−	0.999438i	$-0.510673\pi$
$7$	−8.69181	−0.0670448	−0.0335224	+	0.999438i	$0.510673\pi$
$8$	218.189	1.20533
$9$	0	0
$10$	−649.157	−2.05282
$11$	0	0
$11$	0	0
$12$	0	0
$13$	970.666	1.59298	0.796492	−	0.604649i	$-0.206687\pi$
$13$	970.666	1.59298	0.796492	+	0.604649i	$0.206687\pi$
$14$	−81.2328	−0.110767
$15$	0	0
$16$	268.100	0.261816
$17$	−424.616	−0.356348	−0.178174	−	0.983999i	$-0.557019\pi$
$17$	−424.616	−0.356348	−0.178174	+	0.983999i	$0.557019\pi$
$18$	0	0
$19$	1430.62	0.909158	0.454579	−	0.890707i	$-0.349790\pi$
$19$	1430.62	0.909158	0.454579	+	0.890707i	$0.349790\pi$
$20$	−3844.27	−2.14901
$21$	0	0
$22$	0	0
$23$	−2852.99	−1.12456	−0.562278	−	0.826948i	$-0.690075\pi$
$23$	−2852.99	−1.12456	−0.562278	+	0.826948i	$0.690075\pi$
$24$	0	0
$25$	1699.56	0.543858
$26$	9071.75	2.63183
$27$	0	0
$28$	−481.056	−0.115958
$29$	−7467.66	−1.64888	−0.824441	−	0.565948i	$-0.808510\pi$
$29$	−7467.66	−1.64888	−0.824441	+	0.565948i	$0.808510\pi$
$30$	0	0
$31$	10346.3	1.93366	0.966832	−	0.255411i	$-0.0822109\pi$
$31$	10346.3	1.93366	0.966832	+	0.255411i	$0.0822109\pi$
$32$	−4476.40	−0.772776
$33$	0	0
$34$	−3968.42	−0.588736
$35$	603.724	0.0833045
$36$	0	0
$37$	167.311	0.0200919	0.0100459	−	0.999950i	$-0.496802\pi$
$37$	167.311	0.0200919	0.0100459	+	0.999950i	$0.496802\pi$
$38$	13370.4	1.50205
$39$	0	0
$40$	−15155.2	−1.49765
$41$	5682.18	0.527905	0.263952	−	0.964536i	$-0.414974\pi$
$41$	5682.18	0.527905	0.263952	+	0.964536i	$0.414974\pi$
$42$	0	0
$43$	−21148.9	−1.74428	−0.872142	−	0.489253i	$-0.837269\pi$
$43$	−21148.9	−1.74428	−0.872142	+	0.489253i	$0.837269\pi$
$44$	0	0
$45$	0	0
$46$	−26663.8	−1.85792
$47$	9785.11	0.646132	0.323066	−	0.946376i	$-0.395286\pi$
$47$	9785.11	0.646132	0.323066	+	0.946376i	$0.395286\pi$
$48$	0	0
$49$	−16731.5	−0.995505
$50$	15883.9	0.898529
$51$	0	0
$52$	53722.4	2.75516
$53$	−25639.5	−1.25377	−0.626887	−	0.779110i	$-0.715671\pi$
$53$	−25639.5	−1.25377	−0.626887	+	0.779110i	$0.715671\pi$
$54$	0	0
$55$	0	0
$56$	−1896.45	−0.0808113
$57$	0	0
$58$	−69792.0	−2.72418
$59$	23411.5	0.875588	0.437794	−	0.899075i	$-0.355760\pi$
$59$	23411.5	0.875588	0.437794	+	0.899075i	$0.355760\pi$
$60$	0	0
$61$	−18591.8	−0.639732	−0.319866	−	0.947463i	$-0.603638\pi$
$61$	−18591.8	−0.639732	−0.319866	+	0.947463i	$0.603638\pi$
$62$	96695.6	3.19468
$63$	0	0
$64$	−50415.2	−1.53855
$65$	−67421.5	−1.97932
$66$	0	0
$67$	39477.7	1.07440	0.537198	−	0.843456i	$-0.319483\pi$
$67$	39477.7	1.07440	0.537198	+	0.843456i	$0.319483\pi$
$68$	−23500.8	−0.616325
$69$	0	0
$70$	5642.35	0.137631
$71$	−3283.01	−0.0772905	−0.0386453	−	0.999253i	$-0.512304\pi$
$71$	−3283.01	−0.0772905	−0.0386453	+	0.999253i	$0.512304\pi$
$72$	0	0
$73$	−29536.7	−0.648716	−0.324358	−	0.945934i	$-0.605148\pi$
$73$	−29536.7	−0.648716	−0.324358	+	0.945934i	$0.605148\pi$
$74$	1563.67	0.0331945
$75$	0	0
$76$	79178.8	1.57244
$77$	0	0
$78$	0	0
$79$	10280.8	0.185335	0.0926676	−	0.995697i	$-0.470461\pi$
$79$	10280.8	0.185335	0.0926676	+	0.995697i	$0.470461\pi$
$80$	−18622.0	−0.325312
$81$	0	0
$82$	53105.1	0.872171
$83$	−38360.3	−0.611206	−0.305603	−	0.952159i	$-0.598858\pi$
$83$	−38360.3	−0.611206	−0.305603	+	0.952159i	$0.598858\pi$
$84$	0	0
$85$	29493.4	0.442770
$86$	−197656.	−2.88180
$87$	0	0
$88$	0	0
$89$	2319.99	0.0310464	0.0155232	−	0.999880i	$-0.495059\pi$
$89$	2319.99	0.0310464	0.0155232	+	0.999880i	$0.495059\pi$
$90$	0	0
$91$	−8436.84	−0.106801
$92$	−157902.	−1.94499
$93$	0	0
$94$	91450.7	1.06750
$95$	−99369.2	−1.12965
$96$	0	0
$97$	−81868.3	−0.883459	−0.441729	−	0.897148i	$-0.645635\pi$
$97$	−81868.3	−0.883459	−0.441729	+	0.897148i	$0.645635\pi$
$98$	−156371.	−1.64471
$99$	0	0
$100$	94063.5	0.940635
$101$	−180138.	−1.75712	−0.878560	−	0.477631i	$-0.841496\pi$
$101$	−180138.	−1.75712	−0.878560	+	0.477631i	$0.841496\pi$
$102$	0	0
$103$	−109911.	−1.02082	−0.510410	−	0.859931i	$-0.670506\pi$
$103$	−109911.	−1.02082	−0.510410	+	0.859931i	$0.670506\pi$
$104$	211788.	1.92008
$105$	0	0
$106$	−239624.	−2.07141
$107$	65683.5	0.554621	0.277311	−	0.960780i	$-0.410557\pi$
$107$	65683.5	0.554621	0.277311	+	0.960780i	$0.410557\pi$
$108$	0	0
$109$	−146333.	−1.17971	−0.589854	−	0.807510i	$-0.700815\pi$
$109$	−146333.	−1.17971	−0.589854	+	0.807510i	$0.700815\pi$
$110$	0	0
$111$	0	0
$112$	−2330.27	−0.0175534
$113$	−261872.	−1.92927	−0.964636	−	0.263585i	$-0.915095\pi$
$113$	−261872.	−1.92927	−0.964636	+	0.263585i	$0.915095\pi$
$114$	0	0
$115$	198166.	1.39728
$116$	−413304.	−2.85184
$117$	0	0
$118$	218802.	1.44659
$119$	3690.68	0.0238913
$120$	0	0
$121$	0	0
$122$	−173758.	−1.05693
$123$	0	0
$124$	572626.	3.34439
$125$	99009.9	0.566766
$126$	0	0
$127$	22466.6	0.123603	0.0618015	−	0.998088i	$-0.480315\pi$
$127$	22466.6	0.123603	0.0618015	+	0.998088i	$0.480315\pi$
$128$	−327931.	−1.76912
$129$	0	0
$130$	−630115.	−3.27010
$131$	119595.	0.608886	0.304443	−	0.952531i	$-0.401530\pi$
$131$	119595.	0.608886	0.304443	+	0.952531i	$0.401530\pi$
$132$	0	0
$133$	−12434.6	−0.0609543
$134$	368954.	1.77505
$135$	0	0
$136$	−92646.4	−0.429518
$137$	−176443.	−0.803160	−0.401580	−	0.915824i	$-0.631539\pi$
$137$	−176443.	−0.803160	−0.401580	+	0.915824i	$0.631539\pi$
$138$	0	0
$139$	−288218.	−1.26527	−0.632636	−	0.774450i	$-0.718027\pi$
$139$	−288218.	−1.26527	−0.632636	+	0.774450i	$0.718027\pi$
$140$	33413.7	0.144080
$141$	0	0
$142$	−30682.7	−0.127695
$143$	0	0
$144$	0	0
$145$	518696.	2.04877
$146$	−276047.	−1.07177
$147$	0	0
$148$	9259.98	0.0347501
$149$	−133635.	−0.493123	−0.246561	−	0.969127i	$-0.579301\pi$
$149$	−133635.	−0.493123	−0.246561	+	0.969127i	$0.579301\pi$
$150$	0	0
$151$	−406122.	−1.44949	−0.724744	−	0.689019i	$-0.758042\pi$
$151$	−406122.	−1.44949	−0.724744	+	0.689019i	$0.758042\pi$
$152$	312144.	1.09584
$153$	0	0
$154$	0	0
$155$	−718645.	−2.40262
$156$	0	0
$157$	−158746.	−0.513988	−0.256994	−	0.966413i	$-0.582732\pi$
$157$	−158746.	−0.513988	−0.256994	+	0.966413i	$0.582732\pi$
$158$	96083.1	0.306199
$159$	0	0
$160$	310926.	0.960190
$161$	24797.7	0.0753956
$162$	0	0
$163$	174460.	0.514311	0.257155	−	0.966370i	$-0.417215\pi$
$163$	174460.	0.514311	0.257155	+	0.966370i	$0.417215\pi$
$164$	314486.	0.913043
$165$	0	0
$166$	−358512.	−1.00980
$167$	105879.	0.293777	0.146889	−	0.989153i	$-0.453074\pi$
$167$	105879.	0.293777	0.146889	+	0.989153i	$0.453074\pi$
$168$	0	0
$169$	570899.	1.53760
$170$	275643.	0.731517
$171$	0	0
$172$	−1.17051e6	−3.01684
$173$	−432459.	−1.09857	−0.549287	−	0.835634i	$-0.685101\pi$
$173$	−432459.	−1.09857	−0.549287	+	0.835634i	$0.685101\pi$
$174$	0	0
$175$	−14772.2	−0.0364628
$176$	0	0
$177$	0	0
$178$	21682.4	0.0512929
$179$	624448.	1.45668	0.728340	−	0.685216i	$-0.240292\pi$
$179$	624448.	1.45668	0.728340	+	0.685216i	$0.240292\pi$
$180$	0	0
$181$	10602.4	0.0240551	0.0120275	−	0.999928i	$-0.496171\pi$
$181$	10602.4	0.0240551	0.0120275	+	0.999928i	$0.496171\pi$
$182$	−78849.9	−0.176450
$183$	0	0
$184$	−622490.	−1.35546
$185$	−11621.3	−0.0249646
$186$	0	0
$187$	0	0
$188$	541566.	1.11752
$189$	0	0
$190$	−928695.	−1.86633
$191$	−441955.	−0.876586	−0.438293	−	0.898832i	$-0.644417\pi$
$191$	−441955.	−0.876586	−0.438293	+	0.898832i	$0.644417\pi$
$192$	0	0
$193$	444354.	0.858690	0.429345	−	0.903141i	$-0.358745\pi$
$193$	444354.	0.858690	0.429345	+	0.903141i	$0.358745\pi$
$194$	−765133.	−1.45960
$195$	0	0
$196$	−926017.	−1.72179
$197$	−416576.	−0.764766	−0.382383	−	0.924004i	$-0.624897\pi$
$197$	−416576.	−0.764766	−0.382383	+	0.924004i	$0.624897\pi$
$198$	0	0
$199$	−502973.	−0.900351	−0.450176	−	0.892940i	$-0.648639\pi$
$199$	−502973.	−0.900351	−0.450176	+	0.892940i	$0.648639\pi$
$200$	370824.	0.655530
$201$	0	0
$202$	−1.68355e6	−2.90301
$203$	64907.4	0.110549
$204$	0	0
$205$	−394679.	−0.655933
$206$	−1.02722e6	−1.68653
$207$	0	0
$208$	260236.	0.417069
$209$	0	0
$210$	0	0
$211$	897392.	1.38764	0.693819	−	0.720149i	$-0.255927\pi$
$211$	897392.	1.38764	0.693819	+	0.720149i	$0.255927\pi$
$212$	−1.41904e6	−2.16848
$213$	0	0
$214$	613871.	0.916311
$215$	1.46898e6	2.16731
$216$	0	0
$217$	−89928.1	−0.129642
$218$	−1.36761e6	−1.94904
$219$	0	0
$220$	0	0
$221$	−412161.	−0.567657
$222$	0	0
$223$	117328.	0.157994	0.0789971	−	0.996875i	$-0.474828\pi$
$223$	117328.	0.157994	0.0789971	+	0.996875i	$0.474828\pi$
$224$	38908.0	0.0518106
$225$	0	0
$226$	−2.44743e6	−3.18742
$227$	75051.3	0.0966704	0.0483352	−	0.998831i	$-0.484608\pi$
$227$	75051.3	0.0966704	0.0483352	+	0.998831i	$0.484608\pi$
$228$	0	0
$229$	634950.	0.800112	0.400056	−	0.916491i	$-0.368991\pi$
$229$	634950.	0.800112	0.400056	+	0.916491i	$0.368991\pi$
$230$	1.85204e6	2.30851
$231$	0	0
$232$	−1.62936e6	−1.98745
$233$	−375281.	−0.452863	−0.226431	−	0.974027i	$-0.572706\pi$
$233$	−375281.	−0.452863	−0.226431	+	0.974027i	$0.572706\pi$
$234$	0	0
$235$	−679664.	−0.802832
$236$	1.29573e6	1.51438
$237$	0	0
$238$	34492.8	0.0394717
$239$	−859529.	−0.973343	−0.486671	−	0.873585i	$-0.661789\pi$
$239$	−859529.	−0.973343	−0.486671	+	0.873585i	$0.661789\pi$
$240$	0	0
$241$	552284.	0.612520	0.306260	−	0.951948i	$-0.400922\pi$
$241$	552284.	0.612520	0.306260	+	0.951948i	$0.400922\pi$
$242$	0	0
$243$	0	0
$244$	−1.02898e6	−1.10645
$245$	1.16215e6	1.23694
$246$	0	0
$247$	1.38865e6	1.44827
$248$	2.25745e6	2.33071
$249$	0	0
$250$	925337.	0.936375
$251$	−198740.	−0.199114	−0.0995569	−	0.995032i	$-0.531743\pi$
$251$	−198740.	−0.199114	−0.0995569	+	0.995032i	$0.531743\pi$
$252$	0	0
$253$	0	0
$254$	209971.	0.204209
$255$	0	0
$256$	−1.45152e6	−1.38428
$257$	76585.7	0.0723294	0.0361647	−	0.999346i	$-0.488486\pi$
$257$	76585.7	0.0723294	0.0361647	+	0.999346i	$0.488486\pi$
$258$	0	0
$259$	−1454.24	−0.00134705
$260$	−3.73150e6	−3.42334
$261$	0	0
$262$	1.11773e6	1.00596
$263$	2.10812e6	1.87935	0.939673	−	0.342073i	$-0.111129\pi$
$263$	2.10812e6	1.87935	0.939673	+	0.342073i	$0.111129\pi$
$264$	0	0
$265$	1.78089e6	1.55784
$266$	−116213.	−0.100705
$267$	0	0
$268$	2.18493e6	1.85823
$269$	1.97320e6	1.66261	0.831306	−	0.555816i	$-0.187594\pi$
$269$	1.97320e6	1.66261	0.831306	+	0.555816i	$0.187594\pi$
$270$	0	0
$271$	926426.	0.766280	0.383140	−	0.923690i	$-0.374843\pi$
$271$	926426.	0.766280	0.383140	+	0.923690i	$0.374843\pi$
$272$	−113840.	−0.0932978
$273$	0	0
$274$	−1.64902e6	−1.32693
$275$	0	0
$276$	0	0
$277$	1.60640e6	1.25792	0.628962	−	0.777436i	$-0.283480\pi$
$277$	1.60640e6	1.25792	0.628962	+	0.777436i	$0.283480\pi$
$278$	−2.69365e6	−2.09040
$279$	0	0
$280$	131726.	0.100410
$281$	−814868.	−0.615632	−0.307816	−	0.951446i	$-0.599598\pi$
$281$	−814868.	−0.615632	−0.307816	+	0.951446i	$0.599598\pi$
$282$	0	0
$283$	61465.6	0.0456211	0.0228106	−	0.999740i	$-0.492739\pi$
$283$	61465.6	0.0456211	0.0228106	+	0.999740i	$0.492739\pi$
$284$	−181701.	−0.133679
$285$	0	0
$286$	0	0
$287$	−49388.4	−0.0353933
$288$	0	0
$289$	−1.23956e6	−0.873016
$290$	4.84769e6	3.38485
$291$	0	0
$292$	−1.63473e6	−1.12199
$293$	−298524.	−0.203147	−0.101573	−	0.994828i	$-0.532388\pi$
$293$	−298524.	−0.203147	−0.101573	+	0.994828i	$0.532388\pi$
$294$	0	0
$295$	−1.62614e6	−1.08794
$296$	36505.4	0.0242174
$297$	0	0
$298$	−1.24894e6	−0.814707
$299$	−2.76930e6	−1.79140
$300$	0	0
$301$	183822.	0.116945
$302$	−3.79558e6	−2.39475
$303$	0	0
$304$	383548.	0.238032
$305$	1.29137e6	0.794880
$306$	0	0
$307$	1.35385e6	0.819834	0.409917	−	0.912123i	$-0.365558\pi$
$307$	1.35385e6	0.819834	0.409917	+	0.912123i	$0.365558\pi$
$308$	0	0
$309$	0	0
$310$	−6.71638e6	−3.96946
$311$	2.68320e6	1.57309	0.786544	−	0.617535i	$-0.211868\pi$
$311$	2.68320e6	1.57309	0.786544	+	0.617535i	$0.211868\pi$
$312$	0	0
$313$	2.85968e6	1.64989	0.824947	−	0.565210i	$-0.191205\pi$
$313$	2.85968e6	1.64989	0.824947	+	0.565210i	$0.191205\pi$
$314$	−1.48362e6	−0.849180
$315$	0	0
$316$	568999.	0.320548
$317$	111110.	0.0621020	0.0310510	−	0.999518i	$-0.490115\pi$
$317$	111110.	0.0621020	0.0310510	+	0.999518i	$0.490115\pi$
$318$	0	0
$319$	0	0
$320$	3.50179e6	1.91168
$321$	0	0
$322$	231757.	0.124564
$323$	−607463.	−0.323977
$324$	0	0
$325$	1.64970e6	0.866357
$326$	1.63048e6	0.849712
$327$	0	0
$328$	1.23979e6	0.636301
$329$	−85050.3	−0.0433198
$330$	0	0
$331$	−1.36346e6	−0.684028	−0.342014	−	0.939695i	$-0.611109\pi$
$331$	−1.36346e6	−0.684028	−0.342014	+	0.939695i	$0.611109\pi$
$332$	−2.12309e6	−1.05712
$333$	0	0
$334$	989534.	0.485361
$335$	−2.74208e6	−1.33496
$336$	0	0
$337$	621625.	0.298163	0.149082	−	0.988825i	$-0.452368\pi$
$337$	621625.	0.298163	0.149082	+	0.988825i	$0.452368\pi$
$338$	5.33557e6	2.54032
$339$	0	0
$340$	1.63234e6	0.765797
$341$	0	0
$342$	0	0
$343$	291510.	0.133788
$344$	−4.61445e6	−2.10244
$345$	0	0
$346$	−4.04172e6	−1.81500
$347$	125580.	0.0559883	0.0279942	−	0.999608i	$-0.491088\pi$
$347$	125580.	0.0559883	0.0279942	+	0.999608i	$0.491088\pi$
$348$	0	0
$349$	−3.24565e6	−1.42639	−0.713195	−	0.700966i	$-0.752753\pi$
$349$	−3.24565e6	−1.42639	−0.713195	+	0.700966i	$0.752753\pi$
$350$	−138060.	−0.0602417
$351$	0	0
$352$	0	0
$353$	−1.67773e6	−0.716614	−0.358307	−	0.933604i	$-0.616646\pi$
$353$	−1.67773e6	−0.716614	−0.358307	+	0.933604i	$0.616646\pi$
$354$	0	0
$355$	228035.	0.0960351
$356$	128402.	0.0536966
$357$	0	0
$358$	5.83603e6	2.40663
$359$	−453973.	−0.185906	−0.0929531	−	0.995670i	$-0.529631\pi$
$359$	−453973.	−0.185906	−0.0929531	+	0.995670i	$0.529631\pi$
$360$	0	0
$361$	−429436.	−0.173432
$362$	99088.7	0.0397423
$363$	0	0
$364$	−466944.	−0.184719
$365$	2.05159e6	0.806043
$366$	0	0
$367$	4.87840e6	1.89066	0.945328	−	0.326121i	$-0.105742\pi$
$367$	4.87840e6	1.89066	0.945328	+	0.326121i	$0.105742\pi$
$368$	−764888.	−0.294427
$369$	0	0
$370$	−108611.	−0.0412449
$371$	222853.	0.0840590
$372$	0	0
$373$	605344.	0.225284	0.112642	−	0.993636i	$-0.464069\pi$
$373$	605344.	0.225284	0.112642	+	0.993636i	$0.464069\pi$
$374$	0	0
$375$	0	0
$376$	2.13500e6	0.778804
$377$	−7.24860e6	−2.62664
$378$	0	0
$379$	−3.90340e6	−1.39587	−0.697936	−	0.716160i	$-0.745898\pi$
$379$	−3.90340e6	−1.39587	−0.697936	+	0.716160i	$0.745898\pi$
$380$	−5.49968e6	−1.95379
$381$	0	0
$382$	−4.13047e6	−1.44824
$383$	−2.80018e6	−0.975413	−0.487707	−	0.873008i	$-0.662166\pi$
$383$	−2.80018e6	−0.975413	−0.487707	+	0.873008i	$0.662166\pi$
$384$	0	0
$385$	0	0
$386$	4.15289e6	1.41867
$387$	0	0
$388$	−4.53107e6	−1.52799
$389$	−3.71874e6	−1.24601	−0.623006	−	0.782217i	$-0.714089\pi$
$389$	−3.71874e6	−1.24601	−0.623006	+	0.782217i	$0.714089\pi$
$390$	0	0
$391$	1.21143e6	0.400733
$392$	−3.65061e6	−1.19991
$393$	0	0
$394$	−3.89328e6	−1.26350
$395$	−714093.	−0.230283
$396$	0	0
$397$	3.85343e6	1.22708	0.613538	−	0.789665i	$-0.289746\pi$
$397$	3.85343e6	1.22708	0.613538	+	0.789665i	$0.289746\pi$
$398$	−4.70074e6	−1.48750
$399$	0	0
$400$	455651.	0.142391
$401$	−1.08196e6	−0.336009	−0.168004	−	0.985786i	$-0.553732\pi$
$401$	−1.08196e6	−0.336009	−0.168004	+	0.985786i	$0.553732\pi$
$402$	0	0
$403$	1.00428e7	3.08030
$404$	−9.96989e6	−3.03905
$405$	0	0
$406$	606619.	0.182642
$407$	0	0
$408$	0	0
$409$	−1.98167e6	−0.585765	−0.292883	−	0.956148i	$-0.594615\pi$
$409$	−1.98167e6	−0.585765	−0.292883	+	0.956148i	$0.594615\pi$
$410$	−3.68863e6	−1.08369
$411$	0	0
$412$	−6.08313e6	−1.76557
$413$	−203489.	−0.0587036
$414$	0	0
$415$	2.66447e6	0.759436
$416$	−4.34508e6	−1.23102
$417$	0	0
$418$	0	0
$419$	997473.	0.277566	0.138783	−	0.990323i	$-0.455681\pi$
$419$	997473.	0.277566	0.138783	+	0.990323i	$0.455681\pi$
$420$	0	0
$421$	5.36616e6	1.47556	0.737782	−	0.675039i	$-0.235873\pi$
$421$	5.36616e6	1.47556	0.737782	+	0.675039i	$0.235873\pi$
$422$	8.38694e6	2.29257
$423$	0	0
$424$	−5.59424e6	−1.51122
$425$	−721660.	−0.193803
$426$	0	0
$427$	161597.	0.0428907
$428$	3.63531e6	0.959250
$429$	0	0
$430$	1.37290e7	3.58069
$431$	−3.37897e6	−0.876175	−0.438088	−	0.898932i	$-0.644344\pi$
$431$	−3.37897e6	−0.876175	−0.438088	+	0.898932i	$0.644344\pi$
$432$	0	0
$433$	3.06239e6	0.784948	0.392474	−	0.919763i	$-0.371619\pi$
$433$	3.06239e6	0.784948	0.392474	+	0.919763i	$0.371619\pi$
$434$	−840459.	−0.214187
$435$	0	0
$436$	−8.09891e6	−2.04038
$437$	−4.08154e6	−1.02240
$438$	0	0
$439$	−2.56020e6	−0.634035	−0.317018	−	0.948420i	$-0.602681\pi$
$439$	−2.56020e6	−0.634035	−0.317018	+	0.948420i	$0.602681\pi$
$440$	0	0
$441$	0	0
$442$	−3.85201e6	−0.937847
$443$	1.10849e6	0.268363	0.134181	−	0.990957i	$-0.457160\pi$
$443$	1.10849e6	0.268363	0.134181	+	0.990957i	$0.457160\pi$
$444$	0	0
$445$	−161144.	−0.0385758
$446$	1.09654e6	0.261028
$447$	0	0
$448$	438199.	0.103152
$449$	5.98494e6	1.40102	0.700509	−	0.713643i	$-0.252956\pi$
$449$	5.98494e6	1.40102	0.700509	+	0.713643i	$0.252956\pi$
$450$	0	0
$451$	0	0
$452$	−1.44936e7	−3.33679
$453$	0	0
$454$	701422.	0.159713
$455$	586015.	0.132703
$456$	0	0
$457$	−3.91747e6	−0.877436	−0.438718	−	0.898625i	$-0.644567\pi$
$457$	−3.91747e6	−0.877436	−0.438718	+	0.898625i	$0.644567\pi$
$458$	5.93418e6	1.32190
$459$	0	0
$460$	1.09677e7	2.41669
$461$	−2.16827e6	−0.475184	−0.237592	−	0.971365i	$-0.576358\pi$
$461$	−2.16827e6	−0.475184	−0.237592	+	0.971365i	$0.576358\pi$
$462$	0	0
$463$	−5.01423e6	−1.08706	−0.543528	−	0.839391i	$-0.682912\pi$
$463$	−5.01423e6	−1.08706	−0.543528	+	0.839391i	$0.682912\pi$
$464$	−2.00208e6	−0.431704
$465$	0	0
$466$	−3.50734e6	−0.748191
$467$	2.57098e6	0.545514	0.272757	−	0.962083i	$-0.412064\pi$
$467$	2.57098e6	0.545514	0.272757	+	0.962083i	$0.412064\pi$
$468$	0	0
$469$	−343132.	−0.0720326
$470$	−6.35208e6	−1.32639
$471$	0	0
$472$	5.10813e6	1.05538
$473$	0	0
$474$	0	0
$475$	2.43141e6	0.494453
$476$	204264.	0.0413214
$477$	0	0
$478$	−8.03307e6	−1.60810
$479$	5.16179e6	1.02793	0.513963	−	0.857813i	$-0.328177\pi$
$479$	5.16179e6	1.02793	0.513963	+	0.857813i	$0.328177\pi$
$480$	0	0
$481$	162403.	0.0320060
$482$	5.16160e6	1.01197
$483$	0	0
$484$	0	0
$485$	5.68649e6	1.09772
$486$	0	0
$487$	5.85746e6	1.11915	0.559573	−	0.828781i	$-0.310965\pi$
$487$	5.85746e6	1.11915	0.559573	+	0.828781i	$0.310965\pi$
$488$	−4.05653e6	−0.771090
$489$	0	0
$490$	1.08613e7	2.04359
$491$	956557.	0.179064	0.0895318	−	0.995984i	$-0.471463\pi$
$491$	956557.	0.179064	0.0895318	+	0.995984i	$0.471463\pi$
$492$	0	0
$493$	3.17089e6	0.587576
$494$	1.29782e7	2.39275
$495$	0	0
$496$	2.77385e6	0.506265
$497$	28535.3	0.00518192
$498$	0	0
$499$	−1.47579e6	−0.265322	−0.132661	−	0.991161i	$-0.542352\pi$
$499$	−1.47579e6	−0.265322	−0.132661	+	0.991161i	$0.542352\pi$
$500$	5.47979e6	0.980255
$501$	0	0
$502$	−1.85741e6	−0.328963
$503$	991048.	0.174652	0.0873262	−	0.996180i	$-0.472168\pi$
$503$	991048.	0.174652	0.0873262	+	0.996180i	$0.472168\pi$
$504$	0	0
$505$	1.25122e7	2.18326
$506$	0	0
$507$	0	0
$508$	1.24344e6	0.213779
$509$	8.24580e6	1.41071	0.705356	−	0.708854i	$-0.250787\pi$
$509$	8.24580e6	1.41071	0.705356	+	0.708854i	$0.250787\pi$
$510$	0	0
$511$	256727.	0.0434930
$512$	−3.07201e6	−0.517901
$513$	0	0
$514$	715762.	0.119498
$515$	7.63432e6	1.26839
$516$	0	0
$517$	0	0
$518$	−13591.1	−0.00222552
$519$	0	0
$520$	−1.47106e7	−2.38573
$521$	1.18607e6	0.191432	0.0957161	−	0.995409i	$-0.469486\pi$
$521$	1.18607e6	0.191432	0.0957161	+	0.995409i	$0.469486\pi$
$522$	0	0
$523$	−2.80341e6	−0.448159	−0.224079	−	0.974571i	$-0.571938\pi$
$523$	−2.80341e6	−0.448159	−0.224079	+	0.974571i	$0.571938\pi$
$524$	6.61911e6	1.05310
$525$	0	0
$526$	1.97023e7	3.10494
$527$	−4.39321e6	−0.689058
$528$	0	0
$529$	1.70323e6	0.264627
$530$	1.66440e7	2.57377
$531$	0	0
$532$	−688206.	−0.105424
$533$	5.51550e6	0.840944
$534$	0	0
$535$	−4.56231e6	−0.689129
$536$	8.61357e6	1.29500
$537$	0	0
$538$	1.84413e7	2.74686
$539$	0	0
$540$	0	0
$541$	−5.30698e6	−0.779569	−0.389785	−	0.920906i	$-0.627451\pi$
$541$	−5.30698e6	−0.779569	−0.389785	+	0.920906i	$0.627451\pi$
$542$	8.65829e6	1.26600
$543$	0	0
$544$	1.90075e6	0.275377
$545$	1.01641e7	1.46581
$546$	0	0
$547$	−7.62305e6	−1.08933	−0.544666	−	0.838653i	$-0.683344\pi$
$547$	−7.62305e6	−1.08933	−0.544666	+	0.838653i	$0.683344\pi$
$548$	−9.76538e6	−1.38911
$549$	0	0
$550$	0	0
$551$	−1.06834e7	−1.49909
$552$	0	0
$553$	−89358.5	−0.0124258
$554$	1.50133e7	2.07826
$555$	0	0
$556$	−1.59517e7	−2.18836
$557$	7.81748e6	1.06765	0.533825	−	0.845595i	$-0.320754\pi$
$557$	7.81748e6	1.06765	0.533825	+	0.845595i	$0.320754\pi$
$558$	0	0
$559$	−2.05285e7	−2.77861
$560$	161859.	0.0218105
$561$	0	0
$562$	−7.61567e6	−1.01711
$563$	4.46693e6	0.593933	0.296967	−	0.954888i	$-0.404025\pi$
$563$	4.46693e6	0.593933	0.296967	+	0.954888i	$0.404025\pi$
$564$	0	0
$565$	1.81894e7	2.39716
$566$	574452.	0.0753724
$567$	0	0
$568$	−716315.	−0.0931608
$569$	−1.03786e7	−1.34387	−0.671933	−	0.740612i	$-0.734536\pi$
$569$	−1.03786e7	−1.34387	−0.671933	+	0.740612i	$0.734536\pi$
$570$	0	0
$571$	−5.85319e6	−0.751280	−0.375640	−	0.926766i	$-0.622577\pi$
$571$	−5.85319e6	−0.751280	−0.375640	+	0.926766i	$0.622577\pi$
$572$	0	0
$573$	0	0
$574$	−461579.	−0.0584745
$575$	−4.84882e6	−0.611599
$576$	0	0
$577$	1.01944e7	1.27475	0.637373	−	0.770555i	$-0.280021\pi$
$577$	1.01944e7	1.27475	0.637373	+	0.770555i	$0.280021\pi$
$578$	−1.15848e7	−1.44234
$579$	0	0
$580$	2.87077e7	3.54347
$581$	333421.	0.0409781
$582$	0	0
$583$	0	0
$584$	−6.44457e6	−0.781919
$585$	0	0
$586$	−2.78997e6	−0.335626
$587$	3.10970e6	0.372497	0.186249	−	0.982503i	$-0.440367\pi$
$587$	3.10970e6	0.372497	0.186249	+	0.982503i	$0.440367\pi$
$588$	0	0
$589$	1.48016e7	1.75801
$590$	−1.51978e7	−1.79742
$591$	0	0
$592$	44856.1	0.00526038
$593$	−1.59890e7	−1.86717	−0.933587	−	0.358352i	$-0.883339\pi$
$593$	−1.59890e7	−1.86717	−0.933587	+	0.358352i	$0.883339\pi$
$594$	0	0
$595$	−256351.	−0.0296854
$596$	−7.39616e6	−0.852885
$597$	0	0
$598$	−2.58816e7	−2.95964
$599$	4.21384e6	0.479856	0.239928	−	0.970791i	$-0.422876\pi$
$599$	4.21384e6	0.479856	0.239928	+	0.970791i	$0.422876\pi$
$600$	0	0
$601$	−4.68871e6	−0.529502	−0.264751	−	0.964317i	$-0.585290\pi$
$601$	−4.68871e6	−0.529502	−0.264751	+	0.964317i	$0.585290\pi$
$602$	1.71799e6	0.193209
$603$	0	0
$604$	−2.24772e7	−2.50697
$605$	0	0
$606$	0	0
$607$	−4.29265e6	−0.472883	−0.236441	−	0.971646i	$-0.575981\pi$
$607$	−4.29265e6	−0.472883	−0.236441	+	0.971646i	$0.575981\pi$
$608$	−6.40401e6	−0.702575
$609$	0	0
$610$	1.20690e7	1.31325
$611$	9.49807e6	1.02928
$612$	0	0
$613$	−1.67420e7	−1.79951	−0.899757	−	0.436392i	$-0.856256\pi$
$613$	−1.67420e7	−1.79951	−0.899757	+	0.436392i	$0.856256\pi$
$614$	1.26530e7	1.35448
$615$	0	0
$616$	0	0
$617$	2.10611e6	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$617$	2.10611e6	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$618$	0	0
$619$	2.52452e6	0.264821	0.132410	−	0.991195i	$-0.457728\pi$
$619$	2.52452e6	0.264821	0.132410	+	0.991195i	$0.457728\pi$
$620$	−3.97740e7	−4.15547
$621$	0	0
$622$	2.50770e7	2.59896
$623$	−20164.9	−0.00208150
$624$	0	0
$625$	−1.21882e7	−1.24808
$626$	2.67263e7	2.72585
$627$	0	0
$628$	−8.78594e6	−0.888974
$629$	−71043.0	−0.00715970
$630$	0	0
$631$	1.11540e6	0.111521	0.0557607	−	0.998444i	$-0.482242\pi$
$631$	1.11540e6	0.111521	0.0557607	+	0.998444i	$0.482242\pi$
$632$	2.24315e6	0.223391
$633$	0	0
$634$	1.03843e6	0.102601
$635$	−1.56051e6	−0.153579
$636$	0	0
$637$	−1.62406e7	−1.58582
$638$	0	0
$639$	0	0
$640$	2.27777e7	2.19817
$641$	−9.80630e6	−0.942671	−0.471336	−	0.881954i	$-0.656228\pi$
$641$	−9.80630e6	−0.942671	−0.471336	+	0.881954i	$0.656228\pi$
$642$	0	0
$643$	−5.97068e6	−0.569504	−0.284752	−	0.958601i	$-0.591911\pi$
$643$	−5.97068e6	−0.569504	−0.284752	+	0.958601i	$0.591911\pi$
$644$	1.37245e6	0.130401
$645$	0	0
$646$	−5.67729e6	−0.535254
$647$	−8.55002e6	−0.802984	−0.401492	−	0.915863i	$-0.631508\pi$
$647$	−8.55002e6	−0.802984	−0.401492	+	0.915863i	$0.631508\pi$
$648$	0	0
$649$	0	0
$650$	1.54180e7	1.43134
$651$	0	0
$652$	9.65562e6	0.889531
$653$	4.56279e6	0.418743	0.209371	−	0.977836i	$-0.432858\pi$
$653$	4.56279e6	0.418743	0.209371	+	0.977836i	$0.432858\pi$
$654$	0	0
$655$	−8.30697e6	−0.756553
$656$	1.52339e6	0.138214
$657$	0	0
$658$	−794872.	−0.0715702
$659$	−1.97288e6	−0.176965	−0.0884825	−	0.996078i	$-0.528202\pi$
$659$	−1.97288e6	−0.176965	−0.0884825	+	0.996078i	$0.528202\pi$
$660$	0	0
$661$	−1.78964e7	−1.59317	−0.796585	−	0.604527i	$-0.793362\pi$
$661$	−1.78964e7	−1.59317	−0.796585	+	0.604527i	$0.793362\pi$
$662$	−1.27428e7	−1.13011
$663$	0	0
$664$	−8.36979e6	−0.736706
$665$	863698.	0.0757370
$666$	0	0
$667$	2.13052e7	1.85426
$668$	5.85997e6	0.508106
$669$	0	0
$670$	−2.56272e7	−2.20554
$671$	0	0
$672$	0	0
$673$	7.40111e6	0.629882	0.314941	−	0.949111i	$-0.398015\pi$
$673$	7.40111e6	0.629882	0.314941	+	0.949111i	$0.398015\pi$
$674$	5.80965e6	0.492607
$675$	0	0
$676$	3.15969e7	2.65937
$677$	−6.94001e6	−0.581954	−0.290977	−	0.956730i	$-0.593980\pi$
$677$	−6.94001e6	−0.581954	−0.290977	+	0.956730i	$0.593980\pi$
$678$	0	0
$679$	711583.	0.0592313
$680$	6.43513e6	0.533685
$681$	0	0
$682$	0	0
$683$	−5.97233e6	−0.489882	−0.244941	−	0.969538i	$-0.578769\pi$
$683$	−5.97233e6	−0.489882	−0.244941	+	0.969538i	$0.578769\pi$
$684$	0	0
$685$	1.22555e7	0.997943
$686$	2.72442e6	0.221037
$687$	0	0
$688$	−5.67003e6	−0.456682
$689$	−2.48874e7	−1.99724
$690$	0	0
$691$	−2.05711e7	−1.63893	−0.819467	−	0.573126i	$-0.805731\pi$
$691$	−2.05711e7	−1.63893	−0.819467	+	0.573126i	$0.805731\pi$
$692$	−2.39348e7	−1.90005
$693$	0	0
$694$	1.17366e6	0.0925004
$695$	2.00193e7	1.57213
$696$	0	0
$697$	−2.41275e6	−0.188118
$698$	−3.03335e7	−2.35659
$699$	0	0
$700$	−817582.	−0.0630647
$701$	9.61010e6	0.738640	0.369320	−	0.929302i	$-0.379591\pi$
$701$	9.61010e6	0.738640	0.369320	+	0.929302i	$0.379591\pi$
$702$	0	0
$703$	239358.	0.0182667
$704$	0	0
$705$	0	0
$706$	−1.56799e7	−1.18394
$707$	1.56572e6	0.117806
$708$	0	0
$709$	2.13666e6	0.159632	0.0798161	−	0.996810i	$-0.474567\pi$
$709$	2.13666e6	0.159632	0.0798161	+	0.996810i	$0.474567\pi$
$710$	2.13119e6	0.158663
$711$	0	0
$712$	506195.	0.0374212
$713$	−2.95180e7	−2.17451
$714$	0	0
$715$	0	0
$716$	3.45607e7	2.51941
$717$	0	0
$718$	−4.24279e6	−0.307143
$719$	−1.57303e7	−1.13479	−0.567395	−	0.823446i	$-0.692049\pi$
$719$	−1.57303e7	−1.13479	−0.567395	+	0.823446i	$0.692049\pi$
$720$	0	0
$721$	955327.	0.0684406
$722$	−4.01346e6	−0.286534
$723$	0	0
$724$	586797.	0.0416046
$725$	−1.26917e7	−0.896758
$726$	0	0
$727$	1.39695e7	0.980271	0.490136	−	0.871646i	$-0.336947\pi$
$727$	1.39695e7	0.980271	0.490136	+	0.871646i	$0.336947\pi$
$728$	−1.84082e6	−0.128731
$729$	0	0
$730$	1.91740e7	1.33169
$731$	8.98018e6	0.621572
$732$	0	0
$733$	−2.03921e6	−0.140185	−0.0700927	−	0.997540i	$-0.522329\pi$
$733$	−2.03921e6	−0.140185	−0.0700927	+	0.997540i	$0.522329\pi$
$734$	4.55931e7	3.12362
$735$	0	0
$736$	1.27711e7	0.869030
$737$	0	0
$738$	0	0
$739$	1.13456e7	0.764216	0.382108	−	0.924118i	$-0.375198\pi$
$739$	1.13456e7	0.764216	0.382108	+	0.924118i	$0.375198\pi$
$740$	−643189.	−0.0431777
$741$	0	0
$742$	2.08277e6	0.138877
$743$	−1.17404e7	−0.780206	−0.390103	−	0.920771i	$-0.627561\pi$
$743$	−1.17404e7	−0.780206	−0.390103	+	0.920771i	$0.627561\pi$
$744$	0	0
$745$	9.28217e6	0.612715
$746$	5.65749e6	0.372200
$747$	0	0
$748$	0	0
$749$	−570908.	−0.0371845
$750$	0	0
$751$	2.23069e7	1.44324	0.721620	−	0.692289i	$-0.243398\pi$
$751$	2.23069e7	1.44324	0.721620	+	0.692289i	$0.243398\pi$
$752$	2.62339e6	0.169168
$753$	0	0
$754$	−6.77447e7	−4.33957
$755$	2.82089e7	1.80102
$756$	0	0
$757$	−2.86550e7	−1.81744	−0.908721	−	0.417403i	$-0.862940\pi$
$757$	−2.86550e7	−1.81744	−0.908721	+	0.417403i	$0.862940\pi$
$758$	−3.64808e7	−2.30617
$759$	0	0
$760$	−2.16812e7	−1.36160
$761$	−1.61728e7	−1.01233	−0.506167	−	0.862435i	$-0.668938\pi$
$761$	−1.61728e7	−1.01233	−0.506167	+	0.862435i	$0.668938\pi$
$762$	0	0
$763$	1.27189e6	0.0790933
$764$	−2.44604e7	−1.51611
$765$	0	0
$766$	−2.61702e7	−1.61152
$767$	2.27248e7	1.39480
$768$	0	0
$769$	−1.31452e7	−0.801588	−0.400794	−	0.916168i	$-0.631266\pi$
$769$	−1.31452e7	−0.801588	−0.400794	+	0.916168i	$0.631266\pi$
$770$	0	0
$771$	0	0
$772$	2.45932e7	1.48515
$773$	2.45719e7	1.47907	0.739537	−	0.673116i	$-0.235045\pi$
$773$	2.45719e7	1.47907	0.739537	+	0.673116i	$0.235045\pi$
$774$	0	0
$775$	1.75841e7	1.05164
$776$	−1.78627e7	−1.06486
$777$	0	0
$778$	−3.47550e7	−2.05858
$779$	8.12902e6	0.479949
$780$	0	0
$781$	0	0
$782$	1.13219e7	0.662067
$783$	0	0
$784$	−4.48570e6	−0.260640
$785$	1.10263e7	0.638641
$786$	0	0
$787$	2.62366e7	1.50998	0.754989	−	0.655737i	$-0.227642\pi$
$787$	2.62366e7	1.50998	0.754989	+	0.655737i	$0.227642\pi$
$788$	−2.30558e7	−1.32271
$789$	0	0
$790$	−6.67384e6	−0.380459
$791$	2.27614e6	0.129348
$792$	0	0
$793$	−1.80465e7	−1.01908
$794$	3.60138e7	2.02730
$795$	0	0
$796$	−2.78375e7	−1.55721
$797$	333892.	0.0186192	0.00930959	−	0.999957i	$-0.497037\pi$
$797$	333892.	0.0186192	0.00930959	+	0.999957i	$0.497037\pi$
$798$	0	0
$799$	−4.15492e6	−0.230248
$800$	−7.60789e6	−0.420280
$801$	0	0
$802$	−1.01119e7	−0.555133
$803$	0	0
$804$	0	0
$805$	−1.72242e6	−0.0936806
$806$	9.38591e7	5.08907
$807$	0	0
$808$	−3.93040e7	−2.11792
$809$	2.96083e6	0.159053	0.0795265	−	0.996833i	$-0.474659\pi$
$809$	2.96083e6	0.159053	0.0795265	+	0.996833i	$0.474659\pi$
$810$	0	0
$811$	2.24739e7	1.19985	0.599925	−	0.800056i	$-0.295197\pi$
$811$	2.24739e7	1.19985	0.599925	+	0.800056i	$0.295197\pi$
$812$	3.59236e6	0.191201
$813$	0	0
$814$	0	0
$815$	−1.21178e7	−0.639042
$816$	0	0
$817$	−3.02560e7	−1.58583
$818$	−1.85205e7	−0.967765
$819$	0	0
$820$	−2.18439e7	−1.13447
$821$	2.07336e7	1.07353	0.536767	−	0.843730i	$-0.319645\pi$
$821$	2.07336e7	1.07353	0.536767	+	0.843730i	$0.319645\pi$
$822$	0	0
$823$	2.99008e7	1.53880	0.769401	−	0.638766i	$-0.220555\pi$
$823$	2.99008e7	1.53880	0.769401	+	0.638766i	$0.220555\pi$
$824$	−2.39814e7	−1.23043
$825$	0	0
$826$	−1.90178e6	−0.0969865
$827$	−3.09628e7	−1.57426	−0.787130	−	0.616787i	$-0.788434\pi$
$827$	−3.09628e7	−1.57426	−0.787130	+	0.616787i	$0.788434\pi$
$828$	0	0
$829$	−2.79498e7	−1.41251	−0.706255	−	0.707957i	$-0.749617\pi$
$829$	−2.79498e7	−1.41251	−0.706255	+	0.707957i	$0.749617\pi$
$830$	2.49019e7	1.25469
$831$	0	0
$832$	−4.89363e7	−2.45088
$833$	7.10445e6	0.354746
$834$	0	0
$835$	−7.35425e6	−0.365025
$836$	0	0
$837$	0	0
$838$	9.32228e6	0.458577
$839$	5.17408e6	0.253763	0.126882	−	0.991918i	$-0.459503\pi$
$839$	5.17408e6	0.253763	0.126882	+	0.991918i	$0.459503\pi$
$840$	0	0
$841$	3.52548e7	1.71881
$842$	5.01516e7	2.43784
$843$	0	0
$844$	4.96670e7	2.40000
$845$	−3.96541e7	−1.91050
$846$	0	0
$847$	0	0
$848$	−6.87394e6	−0.328259
$849$	0	0
$850$	−6.74456e6	−0.320189
$851$	−477337.	−0.0225944
$852$	0	0
$853$	−4.50230e6	−0.211866	−0.105933	−	0.994373i	$-0.533783\pi$
$853$	−4.50230e6	−0.211866	−0.105933	+	0.994373i	$0.533783\pi$
$854$	1.51027e6	0.0708613
$855$	0	0
$856$	1.43314e7	0.668503
$857$	2.97267e7	1.38259	0.691296	−	0.722572i	$-0.257040\pi$
$857$	2.97267e7	1.38259	0.691296	+	0.722572i	$0.257040\pi$
$858$	0	0
$859$	2.65435e7	1.22737	0.613685	−	0.789551i	$-0.289686\pi$
$859$	2.65435e7	1.22737	0.613685	+	0.789551i	$0.289686\pi$
$860$	8.13022e7	3.74849
$861$	0	0
$862$	−3.15795e7	−1.44756
$863$	1.61893e7	0.739949	0.369974	−	0.929042i	$-0.379366\pi$
$863$	1.61893e7	0.739949	0.369974	+	0.929042i	$0.379366\pi$
$864$	0	0
$865$	3.00382e7	1.36500
$866$	2.86208e7	1.29684
$867$	0	0
$868$	−4.97715e6	−0.224224
$869$	0	0
$870$	0	0
$871$	3.83196e7	1.71150
$872$	−3.19281e7	−1.42194
$873$	0	0
$874$	−3.81457e7	−1.68914
$875$	−860575.	−0.0379987
$876$	0	0
$877$	1.71684e6	0.0753757	0.0376879	−	0.999290i	$-0.488001\pi$
$877$	1.71684e6	0.0753757	0.0376879	+	0.999290i	$0.488001\pi$
$878$	−2.39274e7	−1.04751
$879$	0	0
$880$	0	0
$881$	−1.59472e6	−0.0692221	−0.0346110	−	0.999401i	$-0.511019\pi$
$881$	−1.59472e6	−0.0692221	−0.0346110	+	0.999401i	$0.511019\pi$
$882$	0	0
$883$	3.27757e7	1.41465	0.707327	−	0.706887i	$-0.249901\pi$
$883$	3.27757e7	1.41465	0.707327	+	0.706887i	$0.249901\pi$
$884$	−2.28114e7	−0.981796
$885$	0	0
$886$	1.03598e7	0.443372
$887$	2.30536e7	0.983853	0.491926	−	0.870637i	$-0.336293\pi$
$887$	2.30536e7	0.983853	0.491926	+	0.870637i	$0.336293\pi$
$888$	0	0
$889$	−195276.	−0.00828693
$890$	−1.50604e6	−0.0637325
$891$	0	0
$892$	6.49365e6	0.273260
$893$	1.39987e7	0.587436
$894$	0	0
$895$	−4.33736e7	−1.80995
$896$	2.85031e6	0.118610
$897$	0	0
$898$	5.59347e7	2.31468
$899$	−7.72627e7	−3.18838
$900$	0	0
$901$	1.08869e7	0.446780
$902$	0	0
$903$	0	0
$904$	−5.71376e7	−2.32542
$905$	−736430.	−0.0298889
$906$	0	0
$907$	−2.13981e7	−0.863689	−0.431845	−	0.901948i	$-0.642137\pi$
$907$	−2.13981e7	−0.863689	−0.431845	+	0.901948i	$0.642137\pi$
$908$	4.15378e6	0.167197
$909$	0	0
$910$	5.47684e6	0.219243
$911$	2.14014e7	0.854369	0.427184	−	0.904165i	$-0.359506\pi$
$911$	2.14014e7	0.854369	0.427184	+	0.904165i	$0.359506\pi$
$912$	0	0
$913$	0	0
$914$	−3.66123e7	−1.44964
$915$	0	0
$916$	3.51419e7	1.38384
$917$	−1.03950e6	−0.0408226
$918$	0	0
$919$	−5.88304e6	−0.229781	−0.114890	−	0.993378i	$-0.536652\pi$
$919$	−5.88304e6	−0.229781	−0.114890	+	0.993378i	$0.536652\pi$
$920$	4.32376e7	1.68419
$921$	0	0
$922$	−2.02645e7	−0.785069
$923$	−3.18670e6	−0.123123
$924$	0	0
$925$	284355.	0.0109271
$926$	−4.68625e7	−1.79597
$927$	0	0
$928$	3.34282e7	1.27422
$929$	−1.32356e7	−0.503158	−0.251579	−	0.967837i	$-0.580950\pi$
$929$	−1.32356e7	−0.503158	−0.251579	+	0.967837i	$0.580950\pi$
$930$	0	0
$931$	−2.39363e7	−0.905071
$932$	−2.07703e7	−0.783253
$933$	0	0
$934$	2.40281e7	0.901265
$935$	0	0
$936$	0	0
$937$	−4.19890e7	−1.56238	−0.781189	−	0.624294i	$-0.785387\pi$
$937$	−4.19890e7	−1.56238	−0.781189	+	0.624294i	$0.785387\pi$
$938$	−3.20688e6	−0.119008
$939$	0	0
$940$	−3.76166e7	−1.38855
$941$	−4.99313e7	−1.83823	−0.919113	−	0.393993i	$-0.871093\pi$
$941$	−4.99313e7	−1.83823	−0.919113	+	0.393993i	$0.871093\pi$
$942$	0	0
$943$	−1.62112e7	−0.593659
$944$	6.27664e6	0.229243
$945$	0	0
$946$	0	0
$947$	1.72916e7	0.626557	0.313278	−	0.949661i	$-0.398573\pi$
$947$	1.72916e7	0.626557	0.313278	+	0.949661i	$0.398573\pi$
$948$	0	0
$949$	−2.86703e7	−1.03339
$950$	2.27238e7	0.816904
$951$	0	0
$952$	805265.	0.0287969
$953$	−1.17729e7	−0.419905	−0.209953	−	0.977712i	$-0.567331\pi$
$953$	−1.17729e7	−0.419905	−0.209953	+	0.977712i	$0.567331\pi$
$954$	0	0
$955$	3.06978e7	1.08918
$956$	−4.75714e7	−1.68345
$957$	0	0
$958$	4.82416e7	1.69827
$959$	1.53360e6	0.0538477
$960$	0	0
$961$	7.84170e7	2.73906
$962$	1.51780e6	0.0528783
$963$	0	0
$964$	3.05667e7	1.05939
$965$	−3.08644e7	−1.06694
$966$	0	0
$967$	−4.31614e7	−1.48433	−0.742164	−	0.670218i	$-0.766200\pi$
$967$	−4.31614e7	−1.48433	−0.742164	+	0.670218i	$0.766200\pi$
$968$	0	0
$969$	0	0
$970$	5.31454e7	1.81358
$971$	−1.06735e7	−0.363293	−0.181647	−	0.983364i	$-0.558143\pi$
$971$	−1.06735e7	−0.363293	−0.181647	+	0.983364i	$0.558143\pi$
$972$	0	0
$973$	2.50513e6	0.0848298
$974$	5.47432e7	1.84898
$975$	0	0
$976$	−4.98448e6	−0.167492
$977$	3.55410e7	1.19122	0.595612	−	0.803272i	$-0.296909\pi$
$977$	3.55410e7	1.19122	0.595612	+	0.803272i	$0.296909\pi$
$978$	0	0
$979$	0	0
$980$	6.43203e7	2.13935
$981$	0	0
$982$	8.93989e6	0.295838
$983$	1.30764e6	0.0431623	0.0215811	−	0.999767i	$-0.493130\pi$
$983$	1.30764e6	0.0431623	0.0215811	+	0.999767i	$0.493130\pi$
$984$	0	0
$985$	2.89350e7	0.950238
$986$	2.96348e7	0.970756
$987$	0	0
$988$	7.68561e7	2.50487
$989$	6.03377e7	1.96154
$990$	0	0
$991$	−2.32452e7	−0.751882	−0.375941	−	0.926644i	$-0.622680\pi$
$991$	−2.32452e7	−0.751882	−0.375941	+	0.926644i	$0.622680\pi$
$992$	−4.63142e7	−1.49429
$993$	0	0
$994$	266688.	0.00856125
$995$	3.49360e7	1.11871
$996$	0	0
$997$	−6.20462e6	−0.197686	−0.0988432	−	0.995103i	$-0.531514\pi$
$997$	−6.20462e6	−0.197686	−0.0988432	+	0.995103i	$0.531514\pi$
$998$	−1.37926e7	−0.438348
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.a.d.1.2	✓	2	33.32	even	2
99.6.a.e.1.1		2	11.10	odd	2
363.6.a.g.1.1		2	3.2	odd	2
528.6.a.q.1.2		2	132.131	odd	2
825.6.a.d.1.1		2	165.164	even	2
1089.6.a.o.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)

\(f(q)\)	\(=\)	\(q+9.34590 q^{2} +55.3459 q^{4} -69.4590 q^{5} -8.69181 q^{7} +218.189 q^{8} +O(q^{10})\)	\(q+9.34590 q^{2} +55.3459 q^{4} -69.4590 q^{5} -8.69181 q^{7} +218.189 q^{8} -649.157 q^{10} +970.666 q^{13} -81.2328 q^{14} +268.100 q^{16} -424.616 q^{17} +1430.62 q^{19} -3844.27 q^{20} -2852.99 q^{23} +1699.56 q^{25} +9071.75 q^{26} -481.056 q^{28} -7467.66 q^{29} +10346.3 q^{31} -4476.40 q^{32} -3968.42 q^{34} +603.724 q^{35} +167.311 q^{37} +13370.4 q^{38} -15155.2 q^{40} +5682.18 q^{41} -21148.9 q^{43} -26663.8 q^{46} +9785.11 q^{47} -16731.5 q^{49} +15883.9 q^{50} +53722.4 q^{52} -25639.5 q^{53} -1896.45 q^{56} -69792.0 q^{58} +23411.5 q^{59} -18591.8 q^{61} +96695.6 q^{62} -50415.2 q^{64} -67421.5 q^{65} +39477.7 q^{67} -23500.8 q^{68} +5642.35 q^{70} -3283.01 q^{71} -29536.7 q^{73} +1563.67 q^{74} +79178.8 q^{76} +10280.8 q^{79} -18622.0 q^{80} +53105.1 q^{82} -38360.3 q^{83} +29493.4 q^{85} -197656. q^{86} +2319.99 q^{89} -8436.84 q^{91} -157902. q^{92} +91450.7 q^{94} -99369.2 q^{95} -81868.3 q^{97} -156371. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 93 q^{4} + 38 q^{5} + 18 q^{7} + 171 q^{8}+O(q^{10}) \) 2 * q + q^2 + 93 * q^4 + 38 * q^5 + 18 * q^7 + 171 * q^8	\( 2 q + q^{2} + 93 q^{4} + 38 q^{5} + 18 q^{7} + 171 q^{8} - 1546 q^{10} + 66 q^{13} - 304 q^{14} - 543 q^{16} - 920 q^{17} + 2932 q^{19} + 202 q^{20} - 5246 q^{23} + 10122 q^{25} + 16622 q^{26} + 524 q^{28} - 12600 q^{29} + 9936 q^{31} + 3803 q^{32} + 166 q^{34} + 3472 q^{35} + 5996 q^{37} + 840 q^{38} - 20226 q^{40} + 24244 q^{41} - 20360 q^{43} - 6692 q^{46} + 5806 q^{47} - 32826 q^{49} - 54409 q^{50} + 19658 q^{52} - 40770 q^{53} - 3156 q^{56} - 26958 q^{58} - 18212 q^{59} + 11398 q^{61} + 100120 q^{62} - 93559 q^{64} - 164636 q^{65} + 65368 q^{67} - 42154 q^{68} - 18296 q^{70} - 61446 q^{71} - 53412 q^{73} - 47082 q^{74} + 135712 q^{76} - 17122 q^{79} - 105782 q^{80} - 101810 q^{82} - 14304 q^{83} - 23740 q^{85} - 204240 q^{86} + 58140 q^{89} - 32584 q^{91} - 248008 q^{92} + 124660 q^{94} + 61968 q^{95} - 183056 q^{97} - 22047 q^{98}+O(q^{100}) \) 2 * q + q^2 + 93 * q^4 + 38 * q^5 + 18 * q^7 + 171 * q^8 - 1546 * q^10 + 66 * q^13 - 304 * q^14 - 543 * q^16 - 920 * q^17 + 2932 * q^19 + 202 * q^20 - 5246 * q^23 + 10122 * q^25 + 16622 * q^26 + 524 * q^28 - 12600 * q^29 + 9936 * q^31 + 3803 * q^32 + 166 * q^34 + 3472 * q^35 + 5996 * q^37 + 840 * q^38 - 20226 * q^40 + 24244 * q^41 - 20360 * q^43 - 6692 * q^46 + 5806 * q^47 - 32826 * q^49 - 54409 * q^50 + 19658 * q^52 - 40770 * q^53 - 3156 * q^56 - 26958 * q^58 - 18212 * q^59 + 11398 * q^61 + 100120 * q^62 - 93559 * q^64 - 164636 * q^65 + 65368 * q^67 - 42154 * q^68 - 18296 * q^70 - 61446 * q^71 - 53412 * q^73 - 47082 * q^74 + 135712 * q^76 - 17122 * q^79 - 105782 * q^80 - 101810 * q^82 - 14304 * q^83 - 23740 * q^85 - 204240 * q^86 + 58140 * q^89 - 32584 * q^91 - 248008 * q^92 + 124660 * q^94 + 61968 * q^95 - 183056 * q^97 - 22047 * q^98

Introduction

L-functions

Modular forms

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Database

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