LMFDB - Embedded newform 1104.6.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1104,6,Mod(1,1104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1104, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1104.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1104 = 2^{4} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1104.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:	$177.063737074$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 69)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-5.17654$ of defining polynomial
Character	$\chi$	$=$	1104.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.00000 q^{3} -37.4928 q^{5} -154.850 q^{7} +81.0000 q^{9} +O(q^{10})$	$q-9.00000 q^{3} -37.4928 q^{5} -154.850 q^{7} +81.0000 q^{9} -521.289 q^{11} -28.8511 q^{13} +337.435 q^{15} -428.492 q^{17} +2563.67 q^{19} +1393.65 q^{21} +529.000 q^{23} -1719.29 q^{25} -729.000 q^{27} -2401.11 q^{29} +2130.21 q^{31} +4691.60 q^{33} +5805.77 q^{35} +3651.58 q^{37} +259.660 q^{39} +16198.0 q^{41} -6503.63 q^{43} -3036.91 q^{45} +20712.3 q^{47} +7171.64 q^{49} +3856.43 q^{51} +34393.6 q^{53} +19544.6 q^{55} -23073.0 q^{57} +18299.4 q^{59} -26509.8 q^{61} -12542.9 q^{63} +1081.71 q^{65} +26313.7 q^{67} -4761.00 q^{69} -39459.2 q^{71} -37484.5 q^{73} +15473.6 q^{75} +80721.9 q^{77} -37368.8 q^{79} +6561.00 q^{81} -76901.5 q^{83} +16065.3 q^{85} +21610.0 q^{87} +62991.4 q^{89} +4467.60 q^{91} -19171.9 q^{93} -96118.9 q^{95} +61484.7 q^{97} -42224.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	$ 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) $ 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	−37.4928	−0.670691	−0.335345	−	0.942095i	$-0.608853\pi$
$5$	−37.4928	−0.670691	−0.335345	+	0.942095i	$0.608853\pi$
$6$	0	0
$7$	−154.850	−1.19445	−0.597224	−	0.802075i	$-0.703730\pi$
$7$	−154.850	−1.19445	−0.597224	+	0.802075i	$0.703730\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−521.289	−1.29896	−0.649482	−	0.760377i	$-0.725014\pi$
$11$	−521.289	−1.29896	−0.649482	+	0.760377i	$0.725014\pi$
$12$	0	0
$13$	−28.8511	−0.0473483	−0.0236741	−	0.999720i	$-0.507536\pi$
$13$	−28.8511	−0.0473483	−0.0236741	+	0.999720i	$0.507536\pi$
$14$	0	0
$15$	337.435	0.387224
$16$	0	0
$17$	−428.492	−0.359601	−0.179800	−	0.983703i	$-0.557545\pi$
$17$	−428.492	−0.359601	−0.179800	+	0.983703i	$0.557545\pi$
$18$	0	0
$19$	2563.67	1.62921	0.814606	−	0.580015i	$-0.196953\pi$
$19$	2563.67	1.62921	0.814606	+	0.580015i	$0.196953\pi$
$20$	0	0
$21$	1393.65	0.689615
$22$	0	0
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	0	0
$25$	−1719.29	−0.550174
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−2401.11	−0.530172	−0.265086	−	0.964225i	$-0.585400\pi$
$29$	−2401.11	−0.530172	−0.265086	+	0.964225i	$0.585400\pi$
$30$	0	0
$31$	2130.21	0.398124	0.199062	−	0.979987i	$-0.436211\pi$
$31$	2130.21	0.398124	0.199062	+	0.979987i	$0.436211\pi$
$32$	0	0
$33$	4691.60	0.749957
$34$	0	0
$35$	5805.77	0.801105
$36$	0	0
$37$	3651.58	0.438507	0.219253	−	0.975668i	$-0.429638\pi$
$37$	3651.58	0.438507	0.219253	+	0.975668i	$0.429638\pi$
$38$	0	0
$39$	259.660	0.0273365
$40$	0	0
$41$	16198.0	1.50488	0.752440	−	0.658661i	$-0.228877\pi$
$41$	16198.0	1.50488	0.752440	+	0.658661i	$0.228877\pi$
$42$	0	0
$43$	−6503.63	−0.536395	−0.268197	−	0.963364i	$-0.586428\pi$
$43$	−6503.63	−0.536395	−0.268197	+	0.963364i	$0.586428\pi$
$44$	0	0
$45$	−3036.91	−0.223564
$46$	0	0
$47$	20712.3	1.36768	0.683838	−	0.729633i	$-0.260309\pi$
$47$	20712.3	1.36768	0.683838	+	0.729633i	$0.260309\pi$
$48$	0	0
$49$	7171.64	0.426706
$50$	0	0
$51$	3856.43	0.207615
$52$	0	0
$53$	34393.6	1.68185	0.840926	−	0.541149i	$-0.182011\pi$
$53$	34393.6	1.68185	0.840926	+	0.541149i	$0.182011\pi$
$54$	0	0
$55$	19544.6	0.871203
$56$	0	0
$57$	−23073.0	−0.940626
$58$	0	0
$59$	18299.4	0.684393	0.342197	−	0.939628i	$-0.388829\pi$
$59$	18299.4	0.684393	0.342197	+	0.939628i	$0.388829\pi$
$60$	0	0
$61$	−26509.8	−0.912183	−0.456092	−	0.889933i	$-0.650751\pi$
$61$	−26509.8	−0.912183	−0.456092	+	0.889933i	$0.650751\pi$
$62$	0	0
$63$	−12542.9	−0.398149
$64$	0	0
$65$	1081.71	0.0317560
$66$	0	0
$67$	26313.7	0.716136	0.358068	−	0.933696i	$-0.383436\pi$
$67$	26313.7	0.716136	0.358068	+	0.933696i	$0.383436\pi$
$68$	0	0
$69$	−4761.00	−0.120386
$70$	0	0
$71$	−39459.2	−0.928971	−0.464486	−	0.885581i	$-0.653761\pi$
$71$	−39459.2	−0.928971	−0.464486	+	0.885581i	$0.653761\pi$
$72$	0	0
$73$	−37484.5	−0.823273	−0.411637	−	0.911348i	$-0.635043\pi$
$73$	−37484.5	−0.823273	−0.411637	+	0.911348i	$0.635043\pi$
$74$	0	0
$75$	15473.6	0.317643
$76$	0	0
$77$	80721.9	1.55154
$78$	0	0
$79$	−37368.8	−0.673660	−0.336830	−	0.941565i	$-0.609355\pi$
$79$	−37368.8	−0.673660	−0.336830	+	0.941565i	$0.609355\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−76901.5	−1.22529	−0.612646	−	0.790357i	$-0.709895\pi$
$83$	−76901.5	−1.22529	−0.612646	+	0.790357i	$0.709895\pi$
$84$	0	0
$85$	16065.3	0.241181
$86$	0	0
$87$	21610.0	0.306095
$88$	0	0
$89$	62991.4	0.842959	0.421480	−	0.906838i	$-0.361511\pi$
$89$	62991.4	0.842959	0.421480	+	0.906838i	$0.361511\pi$
$90$	0	0
$91$	4467.60	0.0565550
$92$	0	0
$93$	−19171.9	−0.229857
$94$	0	0
$95$	−96118.9	−1.09270
$96$	0	0
$97$	61484.7	0.663495	0.331748	−	0.943368i	$-0.392362\pi$
$97$	61484.7	0.663495	0.331748	+	0.943368i	$0.392362\pi$
$98$	0	0
$99$	−42224.4	−0.432988
$100$	0	0
$101$	−89228.8	−0.870365	−0.435183	−	0.900342i	$-0.643316\pi$
$101$	−89228.8	−0.870365	−0.435183	+	0.900342i	$0.643316\pi$
$102$	0	0
$103$	−76234.9	−0.708045	−0.354023	−	0.935237i	$-0.615186\pi$
$103$	−76234.9	−0.708045	−0.354023	+	0.935237i	$0.615186\pi$
$104$	0	0
$105$	−52251.9	−0.462518
$106$	0	0
$107$	98757.7	0.833895	0.416947	−	0.908931i	$-0.363100\pi$
$107$	98757.7	0.833895	0.416947	+	0.908931i	$0.363100\pi$
$108$	0	0
$109$	242295.	1.95334	0.976672	−	0.214738i	$-0.0688898\pi$
$109$	242295.	1.95334	0.976672	+	0.214738i	$0.0688898\pi$
$110$	0	0
$111$	−32864.2	−0.253172
$112$	0	0
$113$	218460.	1.60945	0.804723	−	0.593650i	$-0.202314\pi$
$113$	218460.	1.60945	0.804723	+	0.593650i	$0.202314\pi$
$114$	0	0
$115$	−19833.7	−0.139849
$116$	0	0
$117$	−2336.94	−0.0157828
$118$	0	0
$119$	66352.1	0.429524
$120$	0	0
$121$	110692.	0.687307
$122$	0	0
$123$	−145782.	−0.868842
$124$	0	0
$125$	181626.	1.03969
$126$	0	0
$127$	199105.	1.09540	0.547700	−	0.836675i	$-0.315504\pi$
$127$	199105.	1.09540	0.547700	+	0.836675i	$0.315504\pi$
$128$	0	0
$129$	58532.6	0.309688
$130$	0	0
$131$	−236710.	−1.20514	−0.602572	−	0.798064i	$-0.705858\pi$
$131$	−236710.	−1.20514	−0.602572	+	0.798064i	$0.705858\pi$
$132$	0	0
$133$	−396985.	−1.94601
$134$	0	0
$135$	27332.2	0.129075
$136$	0	0
$137$	−308407.	−1.40386	−0.701929	−	0.712247i	$-0.747678\pi$
$137$	−308407.	−1.40386	−0.701929	+	0.712247i	$0.747678\pi$
$138$	0	0
$139$	99198.8	0.435481	0.217741	−	0.976007i	$-0.430131\pi$
$139$	99198.8	0.435481	0.217741	+	0.976007i	$0.430131\pi$
$140$	0	0
$141$	−186411.	−0.789629
$142$	0	0
$143$	15039.8	0.0615037
$144$	0	0
$145$	90024.2	0.355582
$146$	0	0
$147$	−64544.8	−0.246359
$148$	0	0
$149$	−10925.0	−0.0403140	−0.0201570	−	0.999797i	$-0.506417\pi$
$149$	−10925.0	−0.0403140	−0.0201570	+	0.999797i	$0.506417\pi$
$150$	0	0
$151$	488685.	1.74416	0.872081	−	0.489362i	$-0.162770\pi$
$151$	488685.	1.74416	0.872081	+	0.489362i	$0.162770\pi$
$152$	0	0
$153$	−34707.8	−0.119867
$154$	0	0
$155$	−79867.4	−0.267018
$156$	0	0
$157$	−589168.	−1.90761	−0.953806	−	0.300423i	$-0.902872\pi$
$157$	−589168.	−1.90761	−0.953806	+	0.300423i	$0.902872\pi$
$158$	0	0
$159$	−309542.	−0.971018
$160$	0	0
$161$	−81915.9	−0.249060
$162$	0	0
$163$	−598674.	−1.76490	−0.882452	−	0.470402i	$-0.844109\pi$
$163$	−598674.	−1.76490	−0.882452	+	0.470402i	$0.844109\pi$
$164$	0	0
$165$	−175901.	−0.502989
$166$	0	0
$167$	−222576.	−0.617570	−0.308785	−	0.951132i	$-0.599922\pi$
$167$	−222576.	−0.617570	−0.308785	+	0.951132i	$0.599922\pi$
$168$	0	0
$169$	−370461.	−0.997758
$170$	0	0
$171$	207657.	0.543071
$172$	0	0
$173$	438731.	1.11451	0.557253	−	0.830343i	$-0.311855\pi$
$173$	438731.	1.11451	0.557253	+	0.830343i	$0.311855\pi$
$174$	0	0
$175$	266233.	0.657154
$176$	0	0
$177$	−164694.	−0.395135
$178$	0	0
$179$	−159561.	−0.372216	−0.186108	−	0.982529i	$-0.559587\pi$
$179$	−159561.	−0.372216	−0.186108	+	0.982529i	$0.559587\pi$
$180$	0	0
$181$	448958.	1.01861	0.509306	−	0.860585i	$-0.329902\pi$
$181$	448958.	1.01861	0.509306	+	0.860585i	$0.329902\pi$
$182$	0	0
$183$	238588.	0.526649
$184$	0	0
$185$	−136908.	−0.294103
$186$	0	0
$187$	223368.	0.467108
$188$	0	0
$189$	112886.	0.229872
$190$	0	0
$191$	−170763.	−0.338696	−0.169348	−	0.985556i	$-0.554166\pi$
$191$	−170763.	−0.338696	−0.169348	+	0.985556i	$0.554166\pi$
$192$	0	0
$193$	211013.	0.407771	0.203885	−	0.978995i	$-0.434643\pi$
$193$	211013.	0.407771	0.203885	+	0.978995i	$0.434643\pi$
$194$	0	0
$195$	−9735.37	−0.0183344
$196$	0	0
$197$	−429130.	−0.787813	−0.393906	−	0.919151i	$-0.628877\pi$
$197$	−429130.	−0.787813	−0.393906	+	0.919151i	$0.628877\pi$
$198$	0	0
$199$	646856.	1.15791	0.578955	−	0.815359i	$-0.303461\pi$
$199$	646856.	1.15791	0.578955	+	0.815359i	$0.303461\pi$
$200$	0	0
$201$	−236824.	−0.413461
$202$	0	0
$203$	371813.	0.633263
$204$	0	0
$205$	−607308.	−1.00931
$206$	0	0
$207$	42849.0	0.0695048
$208$	0	0
$209$	−1.33641e6	−2.11629
$210$	0	0
$211$	−282510.	−0.436845	−0.218422	−	0.975854i	$-0.570091\pi$
$211$	−282510.	−0.436845	−0.218422	+	0.975854i	$0.570091\pi$
$212$	0	0
$213$	355133.	0.536342
$214$	0	0
$215$	243839.	0.359755
$216$	0	0
$217$	−329864.	−0.475538
$218$	0	0
$219$	337360.	0.475317
$220$	0	0
$221$	12362.5	0.0170265
$222$	0	0
$223$	637440.	0.858375	0.429187	−	0.903215i	$-0.358800\pi$
$223$	637440.	0.858375	0.429187	+	0.903215i	$0.358800\pi$
$224$	0	0
$225$	−139263.	−0.183391
$226$	0	0
$227$	−687712.	−0.885813	−0.442906	−	0.896568i	$-0.646053\pi$
$227$	−687712.	−0.885813	−0.442906	+	0.896568i	$0.646053\pi$
$228$	0	0
$229$	−606450.	−0.764199	−0.382100	−	0.924121i	$-0.624799\pi$
$229$	−606450.	−0.764199	−0.382100	+	0.924121i	$0.624799\pi$
$230$	0	0
$231$	−726497.	−0.895785
$232$	0	0
$233$	1.44933e6	1.74895	0.874475	−	0.485070i	$-0.161206\pi$
$233$	1.44933e6	1.74895	0.874475	+	0.485070i	$0.161206\pi$
$234$	0	0
$235$	−776561.	−0.917289
$236$	0	0
$237$	336319.	0.388938
$238$	0	0
$239$	−1.10668e6	−1.25322	−0.626610	−	0.779333i	$-0.715558\pi$
$239$	−1.10668e6	−1.25322	−0.626610	+	0.779333i	$0.715558\pi$
$240$	0	0
$241$	−1.02594e6	−1.13784	−0.568919	−	0.822394i	$-0.692638\pi$
$241$	−1.02594e6	−1.13784	−0.568919	+	0.822394i	$0.692638\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	−268885.	−0.286188
$246$	0	0
$247$	−73964.6	−0.0771403
$248$	0	0
$249$	692114.	0.707423
$250$	0	0
$251$	1.07943e6	1.08146	0.540730	−	0.841196i	$-0.318148\pi$
$251$	1.07943e6	1.08146	0.540730	+	0.841196i	$0.318148\pi$
$252$	0	0
$253$	−275762.	−0.270853
$254$	0	0
$255$	−144588.	−0.139246
$256$	0	0
$257$	1.24866e6	1.17926	0.589631	−	0.807673i	$-0.299273\pi$
$257$	1.24866e6	1.17926	0.589631	+	0.807673i	$0.299273\pi$
$258$	0	0
$259$	−565448.	−0.523774
$260$	0	0
$261$	−194490.	−0.176724
$262$	0	0
$263$	−53460.0	−0.0476584	−0.0238292	−	0.999716i	$-0.507586\pi$
$263$	−53460.0	−0.0476584	−0.0238292	+	0.999716i	$0.507586\pi$
$264$	0	0
$265$	−1.28951e6	−1.12800
$266$	0	0
$267$	−566923.	−0.486683
$268$	0	0
$269$	−1.19910e6	−1.01035	−0.505177	−	0.863016i	$-0.668573\pi$
$269$	−1.19910e6	−1.01035	−0.505177	+	0.863016i	$0.668573\pi$
$270$	0	0
$271$	−595615.	−0.492654	−0.246327	−	0.969187i	$-0.579224\pi$
$271$	−595615.	−0.492654	−0.246327	+	0.969187i	$0.579224\pi$
$272$	0	0
$273$	−40208.4	−0.0326521
$274$	0	0
$275$	896249.	0.714656
$276$	0	0
$277$	2.14463e6	1.67940	0.839698	−	0.543053i	$-0.182732\pi$
$277$	2.14463e6	1.67940	0.839698	+	0.543053i	$0.182732\pi$
$278$	0	0
$279$	172547.	0.132708
$280$	0	0
$281$	−1.27455e6	−0.962921	−0.481461	−	0.876468i	$-0.659894\pi$
$281$	−1.27455e6	−0.962921	−0.481461	+	0.876468i	$0.659894\pi$
$282$	0	0
$283$	−1.33116e6	−0.988014	−0.494007	−	0.869458i	$-0.664468\pi$
$283$	−1.33116e6	−0.988014	−0.494007	+	0.869458i	$0.664468\pi$
$284$	0	0
$285$	865070.	0.630869
$286$	0	0
$287$	−2.50827e6	−1.79750
$288$	0	0
$289$	−1.23625e6	−0.870687
$290$	0	0
$291$	−553362.	−0.383069
$292$	0	0
$293$	−543720.	−0.370004	−0.185002	−	0.982738i	$-0.559229\pi$
$293$	−543720.	−0.370004	−0.185002	+	0.982738i	$0.559229\pi$
$294$	0	0
$295$	−686093.	−0.459016
$296$	0	0
$297$	380020.	0.249986
$298$	0	0
$299$	−15262.2	−0.00987279
$300$	0	0
$301$	1.00709e6	0.640695
$302$	0	0
$303$	803059.	0.502506
$304$	0	0
$305$	993926.	0.611793
$306$	0	0
$307$	−2.16140e6	−1.30885	−0.654424	−	0.756127i	$-0.727089\pi$
$307$	−2.16140e6	−1.30885	−0.654424	+	0.756127i	$0.727089\pi$
$308$	0	0
$309$	686114.	0.408790
$310$	0	0
$311$	2.27827e6	1.33569	0.667843	−	0.744302i	$-0.267218\pi$
$311$	2.27827e6	1.33569	0.667843	+	0.744302i	$0.267218\pi$
$312$	0	0
$313$	−2.34170e6	−1.35105	−0.675524	−	0.737338i	$-0.736083\pi$
$313$	−2.34170e6	−1.35105	−0.675524	+	0.737338i	$0.736083\pi$
$314$	0	0
$315$	470267.	0.267035
$316$	0	0
$317$	1.70927e6	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$317$	1.70927e6	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$318$	0	0
$319$	1.25167e6	0.688675
$320$	0	0
$321$	−888819.	−0.481449
$322$	0	0
$323$	−1.09851e6	−0.585865
$324$	0	0
$325$	49603.5	0.0260498
$326$	0	0
$327$	−2.18066e6	−1.12776
$328$	0	0
$329$	−3.20731e6	−1.63362
$330$	0	0
$331$	−3.09295e6	−1.55168	−0.775840	−	0.630930i	$-0.782674\pi$
$331$	−3.09295e6	−1.55168	−0.775840	+	0.630930i	$0.782674\pi$
$332$	0	0
$333$	295778.	0.146169
$334$	0	0
$335$	−986575.	−0.480306
$336$	0	0
$337$	−691524.	−0.331690	−0.165845	−	0.986152i	$-0.553035\pi$
$337$	−691524.	−0.331690	−0.165845	+	0.986152i	$0.553035\pi$
$338$	0	0
$339$	−1.96614e6	−0.929214
$340$	0	0
$341$	−1.11045e6	−0.517148
$342$	0	0
$343$	1.49204e6	0.684770
$344$	0	0
$345$	178503.	0.0807417
$346$	0	0
$347$	4.15931e6	1.85437	0.927187	−	0.374599i	$-0.122220\pi$
$347$	4.15931e6	1.85437	0.927187	+	0.374599i	$0.122220\pi$
$348$	0	0
$349$	−3.44531e6	−1.51414	−0.757068	−	0.653337i	$-0.773369\pi$
$349$	−3.44531e6	−1.51414	−0.757068	+	0.653337i	$0.773369\pi$
$350$	0	0
$351$	21032.5	0.00911218
$352$	0	0
$353$	−2.27651e6	−0.972373	−0.486186	−	0.873855i	$-0.661612\pi$
$353$	−2.27651e6	−0.972373	−0.486186	+	0.873855i	$0.661612\pi$
$354$	0	0
$355$	1.47943e6	0.623053
$356$	0	0
$357$	−597169.	−0.247986
$358$	0	0
$359$	602616.	0.246777	0.123388	−	0.992358i	$-0.460624\pi$
$359$	602616.	0.246777	0.123388	+	0.992358i	$0.460624\pi$
$360$	0	0
$361$	4.09629e6	1.65433
$362$	0	0
$363$	−996224.	−0.396817
$364$	0	0
$365$	1.40540e6	0.552162
$366$	0	0
$367$	−2.69133e6	−1.04304	−0.521521	−	0.853238i	$-0.674635\pi$
$367$	−2.69133e6	−1.04304	−0.521521	+	0.853238i	$0.674635\pi$
$368$	0	0
$369$	1.31204e6	0.501626
$370$	0	0
$371$	−5.32586e6	−2.00889
$372$	0	0
$373$	−2.93803e6	−1.09341	−0.546707	−	0.837324i	$-0.684119\pi$
$373$	−2.93803e6	−1.09341	−0.546707	+	0.837324i	$0.684119\pi$
$374$	0	0
$375$	−1.63463e6	−0.600264
$376$	0	0
$377$	69274.6	0.0251027
$378$	0	0
$379$	−2.78962e6	−0.997578	−0.498789	−	0.866723i	$-0.666222\pi$
$379$	−2.78962e6	−0.997578	−0.498789	+	0.866723i	$0.666222\pi$
$380$	0	0
$381$	−1.79194e6	−0.632429
$382$	0	0
$383$	−3.28779e6	−1.14527	−0.572634	−	0.819811i	$-0.694078\pi$
$383$	−3.28779e6	−1.14527	−0.572634	+	0.819811i	$0.694078\pi$
$384$	0	0
$385$	−3.02649e6	−1.04061
$386$	0	0
$387$	−526794.	−0.178798
$388$	0	0
$389$	−2.17987e6	−0.730392	−0.365196	−	0.930931i	$-0.618998\pi$
$389$	−2.17987e6	−0.730392	−0.365196	+	0.930931i	$0.618998\pi$
$390$	0	0
$391$	−226672.	−0.0749819
$392$	0	0
$393$	2.13039e6	0.695791
$394$	0	0
$395$	1.40106e6	0.451818
$396$	0	0
$397$	−3.30454e6	−1.05229	−0.526144	−	0.850396i	$-0.676363\pi$
$397$	−3.30454e6	−1.05229	−0.526144	+	0.850396i	$0.676363\pi$
$398$	0	0
$399$	3.57286e6	1.12353
$400$	0	0
$401$	3.56153e6	1.10605	0.553026	−	0.833164i	$-0.313473\pi$
$401$	3.56153e6	1.10605	0.553026	+	0.833164i	$0.313473\pi$
$402$	0	0
$403$	−61458.9	−0.0188505
$404$	0	0
$405$	−245990.	−0.0745212
$406$	0	0
$407$	−1.90353e6	−0.569605
$408$	0	0
$409$	−2.96716e6	−0.877066	−0.438533	−	0.898715i	$-0.644502\pi$
$409$	−2.96716e6	−0.877066	−0.438533	+	0.898715i	$0.644502\pi$
$410$	0	0
$411$	2.77567e6	0.810518
$412$	0	0
$413$	−2.83366e6	−0.817472
$414$	0	0
$415$	2.88325e6	0.821792
$416$	0	0
$417$	−892789.	−0.251425
$418$	0	0
$419$	1.51390e6	0.421272	0.210636	−	0.977565i	$-0.432447\pi$
$419$	1.51390e6	0.421272	0.210636	+	0.977565i	$0.432447\pi$
$420$	0	0
$421$	746530.	0.205278	0.102639	−	0.994719i	$-0.467271\pi$
$421$	746530.	0.205278	0.102639	+	0.994719i	$0.467271\pi$
$422$	0	0
$423$	1.67770e6	0.455892
$424$	0	0
$425$	736703.	0.197843
$426$	0	0
$427$	4.10506e6	1.08956
$428$	0	0
$429$	−135358.	−0.0355092
$430$	0	0
$431$	−2.15954e6	−0.559976	−0.279988	−	0.960004i	$-0.590330\pi$
$431$	−2.15954e6	−0.559976	−0.279988	+	0.960004i	$0.590330\pi$
$432$	0	0
$433$	5.74239e6	1.47188	0.735941	−	0.677045i	$-0.236740\pi$
$433$	5.74239e6	1.47188	0.735941	+	0.677045i	$0.236740\pi$
$434$	0	0
$435$	−810218.	−0.205295
$436$	0	0
$437$	1.35618e6	0.339714
$438$	0	0
$439$	−3.04352e6	−0.753728	−0.376864	−	0.926269i	$-0.622998\pi$
$439$	−3.04352e6	−0.753728	−0.376864	+	0.926269i	$0.622998\pi$
$440$	0	0
$441$	580903.	0.142235
$442$	0	0
$443$	2.95192e6	0.714652	0.357326	−	0.933980i	$-0.383689\pi$
$443$	2.95192e6	0.714652	0.357326	+	0.933980i	$0.383689\pi$
$444$	0	0
$445$	−2.36172e6	−0.565365
$446$	0	0
$447$	98325.1	0.0232753
$448$	0	0
$449$	3.29085e6	0.770356	0.385178	−	0.922842i	$-0.374140\pi$
$449$	3.29085e6	0.770356	0.385178	+	0.922842i	$0.374140\pi$
$450$	0	0
$451$	−8.44384e6	−1.95478
$452$	0	0
$453$	−4.39817e6	−1.00699
$454$	0	0
$455$	−167503.	−0.0379309
$456$	0	0
$457$	5.82436e6	1.30454	0.652271	−	0.757986i	$-0.273816\pi$
$457$	5.82436e6	1.30454	0.652271	+	0.757986i	$0.273816\pi$
$458$	0	0
$459$	312371.	0.0692052
$460$	0	0
$461$	−1.73714e6	−0.380700	−0.190350	−	0.981716i	$-0.560962\pi$
$461$	−1.73714e6	−0.380700	−0.190350	+	0.981716i	$0.560962\pi$
$462$	0	0
$463$	−3.08948e6	−0.669780	−0.334890	−	0.942257i	$-0.608699\pi$
$463$	−3.08948e6	−0.669780	−0.334890	+	0.942257i	$0.608699\pi$
$464$	0	0
$465$	718807.	0.154163
$466$	0	0
$467$	1.75231e6	0.371808	0.185904	−	0.982568i	$-0.440479\pi$
$467$	1.75231e6	0.371808	0.185904	+	0.982568i	$0.440479\pi$
$468$	0	0
$469$	−4.07469e6	−0.855387
$470$	0	0
$471$	5.30251e6	1.10136
$472$	0	0
$473$	3.39027e6	0.696757
$474$	0	0
$475$	−4.40769e6	−0.896349
$476$	0	0
$477$	2.78588e6	0.560618
$478$	0	0
$479$	5.38341e6	1.07206	0.536030	−	0.844199i	$-0.319924\pi$
$479$	5.38341e6	1.07206	0.536030	+	0.844199i	$0.319924\pi$
$480$	0	0
$481$	−105352.	−0.0207625
$482$	0	0
$483$	737243.	0.143795
$484$	0	0
$485$	−2.30523e6	−0.445000
$486$	0	0
$487$	6.11771e6	1.16887	0.584436	−	0.811440i	$-0.301316\pi$
$487$	6.11771e6	1.16887	0.584436	+	0.811440i	$0.301316\pi$
$488$	0	0
$489$	5.38806e6	1.01897
$490$	0	0
$491$	−5.22411e6	−0.977932	−0.488966	−	0.872303i	$-0.662626\pi$
$491$	−5.22411e6	−0.977932	−0.488966	+	0.872303i	$0.662626\pi$
$492$	0	0
$493$	1.02886e6	0.190650
$494$	0	0
$495$	1.58311e6	0.290401
$496$	0	0
$497$	6.11027e6	1.10961
$498$	0	0
$499$	−8.89404e6	−1.59900	−0.799498	−	0.600668i	$-0.794901\pi$
$499$	−8.89404e6	−1.59900	−0.799498	+	0.600668i	$0.794901\pi$
$500$	0	0
$501$	2.00318e6	0.356554
$502$	0	0
$503$	3.62219e6	0.638339	0.319170	−	0.947698i	$-0.396596\pi$
$503$	3.62219e6	0.638339	0.319170	+	0.947698i	$0.396596\pi$
$504$	0	0
$505$	3.34543e6	0.583746
$506$	0	0
$507$	3.33415e6	0.576056
$508$	0	0
$509$	454923.	0.0778294	0.0389147	−	0.999243i	$-0.487610\pi$
$509$	454923.	0.0778294	0.0389147	+	0.999243i	$0.487610\pi$
$510$	0	0
$511$	5.80448e6	0.983357
$512$	0	0
$513$	−1.86891e6	−0.313542
$514$	0	0
$515$	2.85826e6	0.474879
$516$	0	0
$517$	−1.07971e7	−1.77656
$518$	0	0
$519$	−3.94857e6	−0.643461
$520$	0	0
$521$	4.46566e6	0.720761	0.360380	−	0.932805i	$-0.382647\pi$
$521$	4.46566e6	0.720761	0.360380	+	0.932805i	$0.382647\pi$
$522$	0	0
$523$	2.91793e6	0.466467	0.233234	−	0.972421i	$-0.425069\pi$
$523$	2.91793e6	0.466467	0.233234	+	0.972421i	$0.425069\pi$
$524$	0	0
$525$	−2.39610e6	−0.379408
$526$	0	0
$527$	−912777.	−0.143165
$528$	0	0
$529$	279841.	0.0434783
$530$	0	0
$531$	1.48225e6	0.228131
$532$	0	0
$533$	−467330.	−0.0712534
$534$	0	0
$535$	−3.70270e6	−0.559286
$536$	0	0
$537$	1.43605e6	0.214899
$538$	0	0
$539$	−3.73850e6	−0.554275
$540$	0	0
$541$	−2.73195e6	−0.401310	−0.200655	−	0.979662i	$-0.564307\pi$
$541$	−2.73195e6	−0.401310	−0.200655	+	0.979662i	$0.564307\pi$
$542$	0	0
$543$	−4.04062e6	−0.588096
$544$	0	0
$545$	−9.08432e6	−1.31009
$546$	0	0
$547$	5.55388e6	0.793648	0.396824	−	0.917895i	$-0.370112\pi$
$547$	5.55388e6	0.793648	0.396824	+	0.917895i	$0.370112\pi$
$548$	0	0
$549$	−2.14730e6	−0.304061
$550$	0	0
$551$	−6.15564e6	−0.863763
$552$	0	0
$553$	5.78657e6	0.804652
$554$	0	0
$555$	1.23217e6	0.169800
$556$	0	0
$557$	−1.69772e6	−0.231862	−0.115931	−	0.993257i	$-0.536985\pi$
$557$	−1.69772e6	−0.231862	−0.115931	+	0.993257i	$0.536985\pi$
$558$	0	0
$559$	187637.	0.0253974
$560$	0	0
$561$	−2.01031e6	−0.269685
$562$	0	0
$563$	−4.59153e6	−0.610501	−0.305250	−	0.952272i	$-0.598740\pi$
$563$	−4.59153e6	−0.610501	−0.305250	+	0.952272i	$0.598740\pi$
$564$	0	0
$565$	−8.19068e6	−1.07944
$566$	0	0
$567$	−1.01597e6	−0.132716
$568$	0	0
$569$	−1.10517e7	−1.43103	−0.715517	−	0.698595i	$-0.753809\pi$
$569$	−1.10517e7	−1.43103	−0.715517	+	0.698595i	$0.753809\pi$
$570$	0	0
$571$	1.35952e7	1.74500	0.872500	−	0.488613i	$-0.162497\pi$
$571$	1.35952e7	1.74500	0.872500	+	0.488613i	$0.162497\pi$
$572$	0	0
$573$	1.53686e6	0.195546
$574$	0	0
$575$	−909506.	−0.114719
$576$	0	0
$577$	4.14819e6	0.518703	0.259351	−	0.965783i	$-0.416491\pi$
$577$	4.14819e6	0.518703	0.259351	+	0.965783i	$0.416491\pi$
$578$	0	0
$579$	−1.89912e6	−0.235427
$580$	0	0
$581$	1.19082e7	1.46355
$582$	0	0
$583$	−1.79290e7	−2.18467
$584$	0	0
$585$	87618.3	0.0105853
$586$	0	0
$587$	55407.2	0.00663699	0.00331849	−	0.999994i	$-0.498944\pi$
$587$	55407.2	0.00663699	0.00331849	+	0.999994i	$0.498944\pi$
$588$	0	0
$589$	5.46114e6	0.648628
$590$	0	0
$591$	3.86217e6	0.454844
$592$	0	0
$593$	−1.44004e7	−1.68166	−0.840829	−	0.541300i	$-0.817932\pi$
$593$	−1.44004e7	−1.68166	−0.840829	+	0.541300i	$0.817932\pi$
$594$	0	0
$595$	−2.48773e6	−0.288078
$596$	0	0
$597$	−5.82170e6	−0.668520
$598$	0	0
$599$	−1.33225e6	−0.151712	−0.0758559	−	0.997119i	$-0.524169\pi$
$599$	−1.33225e6	−0.151712	−0.0758559	+	0.997119i	$0.524169\pi$
$600$	0	0
$601$	−4.11211e6	−0.464386	−0.232193	−	0.972670i	$-0.574590\pi$
$601$	−4.11211e6	−0.464386	−0.232193	+	0.972670i	$0.574590\pi$
$602$	0	0
$603$	2.13141e6	0.238712
$604$	0	0
$605$	−4.15013e6	−0.460971
$606$	0	0
$607$	5.38575e6	0.593300	0.296650	−	0.954986i	$-0.404131\pi$
$607$	5.38575e6	0.593300	0.296650	+	0.954986i	$0.404131\pi$
$608$	0	0
$609$	−3.34631e6	−0.365615
$610$	0	0
$611$	−597572.	−0.0647571
$612$	0	0
$613$	−2.17889e6	−0.234199	−0.117100	−	0.993120i	$-0.537360\pi$
$613$	−2.17889e6	−0.234199	−0.117100	+	0.993120i	$0.537360\pi$
$614$	0	0
$615$	5.46577e6	0.582725
$616$	0	0
$617$	1.40873e7	1.48976	0.744880	−	0.667199i	$-0.232507\pi$
$617$	1.40873e7	1.48976	0.744880	+	0.667199i	$0.232507\pi$
$618$	0	0
$619$	5.63132e6	0.590722	0.295361	−	0.955386i	$-0.404560\pi$
$619$	5.63132e6	0.590722	0.295361	+	0.955386i	$0.404560\pi$
$620$	0	0
$621$	−385641.	−0.0401286
$622$	0	0
$623$	−9.75425e6	−1.00687
$624$	0	0
$625$	−1.43687e6	−0.147135
$626$	0	0
$627$	1.20277e7	1.22184
$628$	0	0
$629$	−1.56467e6	−0.157687
$630$	0	0
$631$	1.47000e7	1.46976	0.734878	−	0.678200i	$-0.237240\pi$
$631$	1.47000e7	1.46976	0.734878	+	0.678200i	$0.237240\pi$
$632$	0	0
$633$	2.54259e6	0.252212
$634$	0	0
$635$	−7.46499e6	−0.734675
$636$	0	0
$637$	−206910.	−0.0202038
$638$	0	0
$639$	−3.19619e6	−0.309657
$640$	0	0
$641$	−1.20465e7	−1.15802	−0.579012	−	0.815319i	$-0.696561\pi$
$641$	−1.20465e7	−1.15802	−0.579012	+	0.815319i	$0.696561\pi$
$642$	0	0
$643$	−4.72652e6	−0.450831	−0.225416	−	0.974263i	$-0.572374\pi$
$643$	−4.72652e6	−0.450831	−0.225416	+	0.974263i	$0.572374\pi$
$644$	0	0
$645$	−2.19455e6	−0.207705
$646$	0	0
$647$	1.24621e7	1.17039	0.585196	−	0.810892i	$-0.301018\pi$
$647$	1.24621e7	1.17039	0.585196	+	0.810892i	$0.301018\pi$
$648$	0	0
$649$	−9.53926e6	−0.889002
$650$	0	0
$651$	2.96877e6	0.274552
$652$	0	0
$653$	−7.75118e6	−0.711353	−0.355676	−	0.934609i	$-0.615749\pi$
$653$	−7.75118e6	−0.711353	−0.355676	+	0.934609i	$0.615749\pi$
$654$	0	0
$655$	8.87493e6	0.808280
$656$	0	0
$657$	−3.03624e6	−0.274424
$658$	0	0
$659$	−1.52713e7	−1.36981	−0.684907	−	0.728630i	$-0.740157\pi$
$659$	−1.52713e7	−1.36981	−0.684907	+	0.728630i	$0.740157\pi$
$660$	0	0
$661$	2.21977e7	1.97608	0.988041	−	0.154190i	$-0.0492767\pi$
$661$	2.21977e7	1.97608	0.988041	+	0.154190i	$0.0492767\pi$
$662$	0	0
$663$	−111262.	−0.00983023
$664$	0	0
$665$	1.48841e7	1.30517
$666$	0	0
$667$	−1.27019e6	−0.110549
$668$	0	0
$669$	−5.73696e6	−0.495583
$670$	0	0
$671$	1.38193e7	1.18489
$672$	0	0
$673$	6.38649e6	0.543531	0.271766	−	0.962363i	$-0.412392\pi$
$673$	6.38649e6	0.543531	0.271766	+	0.962363i	$0.412392\pi$
$674$	0	0
$675$	1.25336e6	0.105881
$676$	0	0
$677$	6.89077e6	0.577825	0.288913	−	0.957356i	$-0.406706\pi$
$677$	6.89077e6	0.577825	0.288913	+	0.957356i	$0.406706\pi$
$678$	0	0
$679$	−9.52093e6	−0.792510
$680$	0	0
$681$	6.18941e6	0.511424
$682$	0	0
$683$	2.36161e7	1.93712	0.968558	−	0.248788i	$-0.0800321\pi$
$683$	2.36161e7	1.93712	0.968558	+	0.248788i	$0.0800321\pi$
$684$	0	0
$685$	1.15630e7	0.941555
$686$	0	0
$687$	5.45805e6	0.441211
$688$	0	0
$689$	−992293.	−0.0796328
$690$	0	0
$691$	−1.22060e7	−0.972472	−0.486236	−	0.873828i	$-0.661630\pi$
$691$	−1.22060e7	−0.972472	−0.486236	+	0.873828i	$0.661630\pi$
$692$	0	0
$693$	6.53847e6	0.517182
$694$	0	0
$695$	−3.71924e6	−0.292073
$696$	0	0
$697$	−6.94071e6	−0.541155
$698$	0	0
$699$	−1.30440e7	−1.00976
$700$	0	0
$701$	6.09955e6	0.468816	0.234408	−	0.972138i	$-0.424685\pi$
$701$	6.09955e6	0.468816	0.234408	+	0.972138i	$0.424685\pi$
$702$	0	0
$703$	9.36143e6	0.714420
$704$	0	0
$705$	6.98905e6	0.529597
$706$	0	0
$707$	1.38171e7	1.03961
$708$	0	0
$709$	2.32369e7	1.73605	0.868026	−	0.496518i	$-0.165388\pi$
$709$	2.32369e7	1.73605	0.868026	+	0.496518i	$0.165388\pi$
$710$	0	0
$711$	−3.02687e6	−0.224553
$712$	0	0
$713$	1.12688e6	0.0830145
$714$	0	0
$715$	−563883.	−0.0412500
$716$	0	0
$717$	9.96012e6	0.723547
$718$	0	0
$719$	1.64452e7	1.18636	0.593179	−	0.805071i	$-0.297873\pi$
$719$	1.64452e7	1.18636	0.593179	+	0.805071i	$0.297873\pi$
$720$	0	0
$721$	1.18050e7	0.845723
$722$	0	0
$723$	9.23348e6	0.656931
$724$	0	0
$725$	4.12821e6	0.291687
$726$	0	0
$727$	3.66575e6	0.257233	0.128617	−	0.991694i	$-0.458946\pi$
$727$	3.66575e6	0.257233	0.128617	+	0.991694i	$0.458946\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	2.78675e6	0.192888
$732$	0	0
$733$	−2.15654e7	−1.48251	−0.741256	−	0.671223i	$-0.765769\pi$
$733$	−2.15654e7	−1.48251	−0.741256	+	0.671223i	$0.765769\pi$
$734$	0	0
$735$	2.41996e6	0.165231
$736$	0	0
$737$	−1.37171e7	−0.930235
$738$	0	0
$739$	−2.21913e7	−1.49476	−0.747380	−	0.664397i	$-0.768689\pi$
$739$	−2.21913e7	−1.49476	−0.747380	+	0.664397i	$0.768689\pi$
$740$	0	0
$741$	665681.	0.0445370
$742$	0	0
$743$	6.61440e6	0.439560	0.219780	−	0.975549i	$-0.429466\pi$
$743$	6.61440e6	0.439560	0.219780	+	0.975549i	$0.429466\pi$
$744$	0	0
$745$	409609.	0.0270382
$746$	0	0
$747$	−6.22902e6	−0.408431
$748$	0	0
$749$	−1.52927e7	−0.996044
$750$	0	0
$751$	8.17370e6	0.528834	0.264417	−	0.964409i	$-0.414821\pi$
$751$	8.17370e6	0.528834	0.264417	+	0.964409i	$0.414821\pi$
$752$	0	0
$753$	−9.71487e6	−0.624381
$754$	0	0
$755$	−1.83222e7	−1.16979
$756$	0	0
$757$	−6.07796e6	−0.385495	−0.192747	−	0.981248i	$-0.561740\pi$
$757$	−6.07796e6	−0.385495	−0.192747	+	0.981248i	$0.561740\pi$
$758$	0	0
$759$	2.48186e6	0.156377
$760$	0	0
$761$	2.93696e6	0.183838	0.0919192	−	0.995766i	$-0.470700\pi$
$761$	2.93696e6	0.183838	0.0919192	+	0.995766i	$0.470700\pi$
$762$	0	0
$763$	−3.75195e7	−2.33317
$764$	0	0
$765$	1.30129e6	0.0803936
$766$	0	0
$767$	−527957.	−0.0324048
$768$	0	0
$769$	6.47116e6	0.394608	0.197304	−	0.980342i	$-0.436781\pi$
$769$	6.47116e6	0.394608	0.197304	+	0.980342i	$0.436781\pi$
$770$	0	0
$771$	−1.12379e7	−0.680847
$772$	0	0
$773$	2.52231e7	1.51827	0.759136	−	0.650932i	$-0.225622\pi$
$773$	2.52231e7	1.51827	0.759136	+	0.650932i	$0.225622\pi$
$774$	0	0
$775$	−3.66245e6	−0.219037
$776$	0	0
$777$	5.08904e6	0.302401
$778$	0	0
$779$	4.15263e7	2.45177
$780$	0	0
$781$	2.05697e7	1.20670
$782$	0	0
$783$	1.75041e6	0.102032
$784$	0	0
$785$	2.20895e7	1.27942
$786$	0	0
$787$	1.74674e7	1.00529	0.502644	−	0.864493i	$-0.332361\pi$
$787$	1.74674e7	1.00529	0.502644	+	0.864493i	$0.332361\pi$
$788$	0	0
$789$	481140.	0.0275156
$790$	0	0
$791$	−3.38287e7	−1.92240
$792$	0	0
$793$	764838.	0.0431903
$794$	0	0
$795$	1.16056e7	0.651253
$796$	0	0
$797$	2.12790e6	0.118660	0.0593302	−	0.998238i	$-0.481104\pi$
$797$	2.12790e6	0.118660	0.0593302	+	0.998238i	$0.481104\pi$
$798$	0	0
$799$	−8.87505e6	−0.491817
$800$	0	0
$801$	5.10231e6	0.280986
$802$	0	0
$803$	1.95402e7	1.06940
$804$	0	0
$805$	3.07125e6	0.167042
$806$	0	0
$807$	1.07919e7	0.583328
$808$	0	0
$809$	−5.17917e6	−0.278220	−0.139110	−	0.990277i	$-0.544424\pi$
$809$	−5.17917e6	−0.278220	−0.139110	+	0.990277i	$0.544424\pi$
$810$	0	0
$811$	−2.72737e7	−1.45610	−0.728050	−	0.685524i	$-0.759573\pi$
$811$	−2.72737e7	−1.45610	−0.728050	+	0.685524i	$0.759573\pi$
$812$	0	0
$813$	5.36053e6	0.284434
$814$	0	0
$815$	2.24459e7	1.18371
$816$	0	0
$817$	−1.66731e7	−0.873900
$818$	0	0
$819$	361876.	0.0188517
$820$	0	0
$821$	2.28417e7	1.18269	0.591346	−	0.806418i	$-0.298597\pi$
$821$	2.28417e7	1.18269	0.591346	+	0.806418i	$0.298597\pi$
$822$	0	0
$823$	−1.38673e7	−0.713664	−0.356832	−	0.934169i	$-0.616143\pi$
$823$	−1.38673e7	−0.713664	−0.356832	+	0.934169i	$0.616143\pi$
$824$	0	0
$825$	−8.06624e6	−0.412607
$826$	0	0
$827$	7.98777e6	0.406127	0.203064	−	0.979166i	$-0.434910\pi$
$827$	7.98777e6	0.406127	0.203064	+	0.979166i	$0.434910\pi$
$828$	0	0
$829$	−8.88714e6	−0.449134	−0.224567	−	0.974459i	$-0.572097\pi$
$829$	−8.88714e6	−0.449134	−0.224567	+	0.974459i	$0.572097\pi$
$830$	0	0
$831$	−1.93017e7	−0.969600
$832$	0	0
$833$	−3.07299e6	−0.153444
$834$	0	0
$835$	8.34498e6	0.414199
$836$	0	0
$837$	−1.55292e6	−0.0766189
$838$	0	0
$839$	2.25661e7	1.10676	0.553379	−	0.832930i	$-0.313338\pi$
$839$	2.25661e7	1.10676	0.553379	+	0.832930i	$0.313338\pi$
$840$	0	0
$841$	−1.47458e7	−0.718917
$842$	0	0
$843$	1.14709e7	0.555943
$844$	0	0
$845$	1.38896e7	0.669187
$846$	0	0
$847$	−1.71406e7	−0.820953
$848$	0	0
$849$	1.19804e7	0.570430
$850$	0	0
$851$	1.93169e6	0.0914350
$852$	0	0
$853$	−3.35094e7	−1.57686	−0.788431	−	0.615124i	$-0.789106\pi$
$853$	−3.35094e7	−1.57686	−0.788431	+	0.615124i	$0.789106\pi$
$854$	0	0
$855$	−7.78563e6	−0.364232
$856$	0	0
$857$	−1.84965e7	−0.860276	−0.430138	−	0.902763i	$-0.641535\pi$
$857$	−1.84965e7	−0.860276	−0.430138	+	0.902763i	$0.641535\pi$
$858$	0	0
$859$	−3.55324e7	−1.64302	−0.821508	−	0.570198i	$-0.806867\pi$
$859$	−3.55324e7	−1.64302	−0.821508	+	0.570198i	$0.806867\pi$
$860$	0	0
$861$	2.25744e7	1.03779
$862$	0	0
$863$	1.37196e7	0.627067	0.313534	−	0.949577i	$-0.398487\pi$
$863$	1.37196e7	0.627067	0.313534	+	0.949577i	$0.398487\pi$
$864$	0	0
$865$	−1.64492e7	−0.747489
$866$	0	0
$867$	1.11263e7	0.502692
$868$	0	0
$869$	1.94799e7	0.875060
$870$	0	0
$871$	−759180.	−0.0339078
$872$	0	0
$873$	4.98026e6	0.221165
$874$	0	0
$875$	−2.81248e7	−1.24185
$876$	0	0
$877$	−1.74346e7	−0.765443	−0.382721	−	0.923864i	$-0.625013\pi$
$877$	−1.74346e7	−0.765443	−0.382721	+	0.923864i	$0.625013\pi$
$878$	0	0
$879$	4.89348e6	0.213622
$880$	0	0
$881$	3.33087e7	1.44583	0.722916	−	0.690936i	$-0.242801\pi$
$881$	3.33087e7	1.44583	0.722916	+	0.690936i	$0.242801\pi$
$882$	0	0
$883$	−3.36375e6	−0.145185	−0.0725925	−	0.997362i	$-0.523127\pi$
$883$	−3.36375e6	−0.145185	−0.0725925	+	0.997362i	$0.523127\pi$
$884$	0	0
$885$	6.17484e6	0.265013
$886$	0	0
$887$	1.07106e7	0.457095	0.228547	−	0.973533i	$-0.426602\pi$
$887$	1.07106e7	0.457095	0.228547	+	0.973533i	$0.426602\pi$
$888$	0	0
$889$	−3.08315e7	−1.30840
$890$	0	0
$891$	−3.42018e6	−0.144329
$892$	0	0
$893$	5.30994e7	2.22824
$894$	0	0
$895$	5.98239e6	0.249642
$896$	0	0
$897$	137360.	0.00570006
$898$	0	0
$899$	−5.11486e6	−0.211074
$900$	0	0
$901$	−1.47374e7	−0.604795
$902$	0	0
$903$	−9.06380e6	−0.369906
$904$	0	0
$905$	−1.68327e7	−0.683174
$906$	0	0
$907$	−1.90456e7	−0.768735	−0.384367	−	0.923180i	$-0.625580\pi$
$907$	−1.90456e7	−0.768735	−0.384367	+	0.923180i	$0.625580\pi$
$908$	0	0
$909$	−7.22753e6	−0.290122
$910$	0	0
$911$	−1.05019e6	−0.0419249	−0.0209624	−	0.999780i	$-0.506673\pi$
$911$	−1.05019e6	−0.0419249	−0.0209624	+	0.999780i	$0.506673\pi$
$912$	0	0
$913$	4.00879e7	1.59161
$914$	0	0
$915$	−8.94534e6	−0.353219
$916$	0	0
$917$	3.66547e7	1.43948
$918$	0	0
$919$	1.19851e7	0.468114	0.234057	−	0.972223i	$-0.424800\pi$
$919$	1.19851e7	0.468114	0.234057	+	0.972223i	$0.424800\pi$
$920$	0	0
$921$	1.94526e7	0.755664
$922$	0	0
$923$	1.13844e6	0.0439852
$924$	0	0
$925$	−6.27813e6	−0.241255
$926$	0	0
$927$	−6.17503e6	−0.236015
$928$	0	0
$929$	1.29362e7	0.491777	0.245888	−	0.969298i	$-0.420920\pi$
$929$	1.29362e7	0.491777	0.245888	+	0.969298i	$0.420920\pi$
$930$	0	0
$931$	1.83857e7	0.695194
$932$	0	0
$933$	−2.05044e7	−0.771159
$934$	0	0
$935$	−8.37469e6	−0.313285
$936$	0	0
$937$	−1.25390e7	−0.466567	−0.233283	−	0.972409i	$-0.574947\pi$
$937$	−1.25390e7	−0.466567	−0.233283	+	0.972409i	$0.574947\pi$
$938$	0	0
$939$	2.10753e7	0.780028
$940$	0	0
$941$	−2.45328e7	−0.903178	−0.451589	−	0.892226i	$-0.649143\pi$
$941$	−2.45328e7	−0.903178	−0.451589	+	0.892226i	$0.649143\pi$
$942$	0	0
$943$	8.56874e6	0.313789
$944$	0	0
$945$	−4.23241e6	−0.154173
$946$	0	0
$947$	2.08008e7	0.753713	0.376856	−	0.926272i	$-0.377005\pi$
$947$	2.08008e7	0.753713	0.376856	+	0.926272i	$0.377005\pi$
$948$	0	0
$949$	1.08147e6	0.0389806
$950$	0	0
$951$	−1.53834e7	−0.551570
$952$	0	0
$953$	−9.76608e6	−0.348328	−0.174164	−	0.984717i	$-0.555722\pi$
$953$	−9.76608e6	−0.348328	−0.174164	+	0.984717i	$0.555722\pi$
$954$	0	0
$955$	6.40236e6	0.227160
$956$	0	0
$957$	−1.12651e7	−0.397606
$958$	0	0
$959$	4.77570e7	1.67684
$960$	0	0
$961$	−2.40914e7	−0.841498
$962$	0	0
$963$	7.99937e6	0.277965
$964$	0	0
$965$	−7.91147e6	−0.273488
$966$	0	0
$967$	2.35516e7	0.809942	0.404971	−	0.914329i	$-0.367282\pi$
$967$	2.35516e7	0.809942	0.404971	+	0.914329i	$0.367282\pi$
$968$	0	0
$969$	9.88659e6	0.338250
$970$	0	0
$971$	−2.32199e7	−0.790336	−0.395168	−	0.918609i	$-0.629314\pi$
$971$	−2.32199e7	−0.790336	−0.395168	+	0.918609i	$0.629314\pi$
$972$	0	0
$973$	−1.53610e7	−0.520160
$974$	0	0
$975$	−446431.	−0.0150398
$976$	0	0
$977$	−3.29920e7	−1.10579	−0.552895	−	0.833251i	$-0.686477\pi$
$977$	−3.29920e7	−1.10579	−0.552895	+	0.833251i	$0.686477\pi$
$978$	0	0
$979$	−3.28368e7	−1.09497
$980$	0	0
$981$	1.96259e7	0.651114
$982$	0	0
$983$	2.82144e7	0.931294	0.465647	−	0.884970i	$-0.345822\pi$
$983$	2.82144e7	0.931294	0.465647	+	0.884970i	$0.345822\pi$
$984$	0	0
$985$	1.60893e7	0.528379
$986$	0	0
$987$	2.88658e7	0.943170
$988$	0	0
$989$	−3.44042e6	−0.111846
$990$	0	0
$991$	3.77541e7	1.22118	0.610590	−	0.791947i	$-0.290932\pi$
$991$	3.77541e7	1.22118	0.610590	+	0.791947i	$0.290932\pi$
$992$	0	0
$993$	2.78365e7	0.895863
$994$	0	0
$995$	−2.42524e7	−0.776600
$996$	0	0
$997$	3.33776e7	1.06345	0.531726	−	0.846917i	$-0.321544\pi$
$997$	3.33776e7	1.06345	0.531726	+	0.846917i	$0.321544\pi$
$998$	0	0
$999$	−2.66200e6	−0.0843907

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1104.6.a.r.1.2		5
4.3	odd	2		69.6.a.e.1.4	✓	5
12.11	even	2		207.6.a.f.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.4	✓	5	4.3	odd	2
207.6.a.f.1.2		5	12.11	even	2
1104.6.a.r.1.2		5	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:	\( N \)	\(=\)	\( 1104 = 2^{4} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1104.a (trivial)

\(f(q)\)	\(=\)	\(q-9.00000 q^{3} -37.4928 q^{5} -154.850 q^{7} +81.0000 q^{9} +O(q^{10})\)	\(q-9.00000 q^{3} -37.4928 q^{5} -154.850 q^{7} +81.0000 q^{9} -521.289 q^{11} -28.8511 q^{13} +337.435 q^{15} -428.492 q^{17} +2563.67 q^{19} +1393.65 q^{21} +529.000 q^{23} -1719.29 q^{25} -729.000 q^{27} -2401.11 q^{29} +2130.21 q^{31} +4691.60 q^{33} +5805.77 q^{35} +3651.58 q^{37} +259.660 q^{39} +16198.0 q^{41} -6503.63 q^{43} -3036.91 q^{45} +20712.3 q^{47} +7171.64 q^{49} +3856.43 q^{51} +34393.6 q^{53} +19544.6 q^{55} -23073.0 q^{57} +18299.4 q^{59} -26509.8 q^{61} -12542.9 q^{63} +1081.71 q^{65} +26313.7 q^{67} -4761.00 q^{69} -39459.2 q^{71} -37484.5 q^{73} +15473.6 q^{75} +80721.9 q^{77} -37368.8 q^{79} +6561.00 q^{81} -76901.5 q^{83} +16065.3 q^{85} +21610.0 q^{87} +62991.4 q^{89} +4467.60 q^{91} -19171.9 q^{93} -96118.9 q^{95} +61484.7 q^{97} -42224.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9}+O(q^{10}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9	\( 5 q - 45 q^{3} + 94 q^{5} - 272 q^{7} + 405 q^{9} - 1100 q^{11} - 978 q^{13} - 846 q^{15} + 2522 q^{17} - 2060 q^{19} + 2448 q^{21} + 2645 q^{23} + 12035 q^{25} - 3645 q^{27} + 1526 q^{29} + 7392 q^{31} + 9900 q^{33} - 6056 q^{35} - 8210 q^{37} + 8802 q^{39} + 21250 q^{41} + 4548 q^{43} + 7614 q^{45} - 536 q^{47} - 27979 q^{49} - 22698 q^{51} - 11482 q^{53} + 77064 q^{55} + 18540 q^{57} - 74676 q^{59} - 44618 q^{61} - 22032 q^{63} - 24388 q^{65} + 1412 q^{67} - 23805 q^{69} - 37912 q^{71} + 46546 q^{73} - 108315 q^{75} + 157008 q^{77} - 50544 q^{79} + 32805 q^{81} - 89588 q^{83} + 147892 q^{85} - 13734 q^{87} + 280410 q^{89} + 27416 q^{91} - 66528 q^{93} - 203120 q^{95} + 90074 q^{97} - 89100 q^{99}+O(q^{100}) \) 5 * q - 45 * q^3 + 94 * q^5 - 272 * q^7 + 405 * q^9 - 1100 * q^11 - 978 * q^13 - 846 * q^15 + 2522 * q^17 - 2060 * q^19 + 2448 * q^21 + 2645 * q^23 + 12035 * q^25 - 3645 * q^27 + 1526 * q^29 + 7392 * q^31 + 9900 * q^33 - 6056 * q^35 - 8210 * q^37 + 8802 * q^39 + 21250 * q^41 + 4548 * q^43 + 7614 * q^45 - 536 * q^47 - 27979 * q^49 - 22698 * q^51 - 11482 * q^53 + 77064 * q^55 + 18540 * q^57 - 74676 * q^59 - 44618 * q^61 - 22032 * q^63 - 24388 * q^65 + 1412 * q^67 - 23805 * q^69 - 37912 * q^71 + 46546 * q^73 - 108315 * q^75 + 157008 * q^77 - 50544 * q^79 + 32805 * q^81 - 89588 * q^83 + 147892 * q^85 - 13734 * q^87 + 280410 * q^89 + 27416 * q^91 - 66528 * q^93 - 203120 * q^95 + 90074 * q^97 - 89100 * q^99

Introduction

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