LMFDB - Embedded newform 1089.6.a.bl.1.7

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:	$10$
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} - x^{9} - 252 x^{8} + 45 x^{7} + 21644 x^{6} + 14121 x^{5} - 727612 x^{4} - 1049829 x^{3} + \cdots - 5072980 $ x^10 - x^9 - 252x^8 + 45x^7 + 21644x^6 + 14121x^5 - 727612x^4 - 1049829x^3 + 8255839x^2 + 16664560x - 5072980
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			1.7
Root			$-3.80353$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.80353 q^{2} -8.92614 q^{4} -59.7896 q^{5} +72.0300 q^{7} -196.590 q^{8} +O(q^{10})$	$q+4.80353 q^{2} -8.92614 q^{4} -59.7896 q^{5} +72.0300 q^{7} -196.590 q^{8} -287.201 q^{10} -540.791 q^{13} +345.998 q^{14} -658.687 q^{16} +1510.71 q^{17} -315.677 q^{19} +533.691 q^{20} +2610.94 q^{23} +449.799 q^{25} -2597.71 q^{26} -642.950 q^{28} +6630.81 q^{29} +7435.70 q^{31} +3126.85 q^{32} +7256.73 q^{34} -4306.65 q^{35} +6440.18 q^{37} -1516.36 q^{38} +11754.0 q^{40} -1194.66 q^{41} -21637.2 q^{43} +12541.7 q^{46} -1352.89 q^{47} -11618.7 q^{49} +2160.62 q^{50} +4827.18 q^{52} -12845.1 q^{53} -14160.4 q^{56} +31851.3 q^{58} +5548.05 q^{59} +16044.8 q^{61} +35717.6 q^{62} +36097.9 q^{64} +32333.7 q^{65} -71743.7 q^{67} -13484.8 q^{68} -20687.1 q^{70} -34182.3 q^{71} -15613.4 q^{73} +30935.6 q^{74} +2817.78 q^{76} +44250.7 q^{79} +39382.7 q^{80} -5738.57 q^{82} +19538.0 q^{83} -90324.8 q^{85} -103935. q^{86} -111152. q^{89} -38953.2 q^{91} -23305.6 q^{92} -6498.62 q^{94} +18874.2 q^{95} +85165.0 q^{97} -55810.6 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 9 q^{2} + 193 q^{4} - 11 q^{5} - 470 q^{7} + 324 q^{8}+O(q^{10}) $ 10 * q + 9 * q^2 + 193 * q^4 - 11 * q^5 - 470 * q^7 + 324 * q^8	$ 10 q + 9 q^{2} + 193 q^{4} - 11 q^{5} - 470 q^{7} + 324 q^{8} - 976 q^{10} - 2308 q^{13} - 540 q^{14} + 2801 q^{16} + 3093 q^{17} - 5305 q^{19} + 2229 q^{20} - 700 q^{23} + 8381 q^{25} + 15559 q^{26} - 24656 q^{28} + 10392 q^{29} - 101 q^{31} + 34557 q^{32} - 4542 q^{34} + 19867 q^{35} - 1284 q^{37} + 35769 q^{38} - 66596 q^{40} + 17944 q^{41} - 31812 q^{43} - 36417 q^{46} - 8787 q^{47} - 23810 q^{49} - 910 q^{50} - 51663 q^{52} + 3261 q^{53} - 84819 q^{56} + 53125 q^{58} - 49375 q^{59} - 63175 q^{61} + 28399 q^{62} + 124764 q^{64} - 14105 q^{65} + 5365 q^{67} - 53313 q^{68} + 40297 q^{70} + 236675 q^{71} - 200912 q^{73} - 180329 q^{74} - 39606 q^{76} - 210802 q^{79} - 270298 q^{80} - 369223 q^{82} - 178968 q^{83} - 107352 q^{85} - 465999 q^{86} - 90816 q^{89} + 7500 q^{91} + 136407 q^{92} - 71890 q^{94} + 335807 q^{95} + 271521 q^{97} - 75285 q^{98}+O(q^{100}) $ 10 * q + 9 * q^2 + 193 * q^4 - 11 * q^5 - 470 * q^7 + 324 * q^8 - 976 * q^10 - 2308 * q^13 - 540 * q^14 + 2801 * q^16 + 3093 * q^17 - 5305 * q^19 + 2229 * q^20 - 700 * q^23 + 8381 * q^25 + 15559 * q^26 - 24656 * q^28 + 10392 * q^29 - 101 * q^31 + 34557 * q^32 - 4542 * q^34 + 19867 * q^35 - 1284 * q^37 + 35769 * q^38 - 66596 * q^40 + 17944 * q^41 - 31812 * q^43 - 36417 * q^46 - 8787 * q^47 - 23810 * q^49 - 910 * q^50 - 51663 * q^52 + 3261 * q^53 - 84819 * q^56 + 53125 * q^58 - 49375 * q^59 - 63175 * q^61 + 28399 * q^62 + 124764 * q^64 - 14105 * q^65 + 5365 * q^67 - 53313 * q^68 + 40297 * q^70 + 236675 * q^71 - 200912 * q^73 - 180329 * q^74 - 39606 * q^76 - 210802 * q^79 - 270298 * q^80 - 369223 * q^82 - 178968 * q^83 - 107352 * q^85 - 465999 * q^86 - 90816 * q^89 + 7500 * q^91 + 136407 * q^92 - 71890 * q^94 + 335807 * q^95 + 271521 * q^97 - 75285 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	4.80353	0.849151	0.424576	−	0.905392i	$-0.360423\pi$
$2$	4.80353	0.849151	0.424576	+	0.905392i	$0.360423\pi$
$3$	0	0
$3$	0	0
$4$	−8.92614	−0.278942
$5$	−59.7896	−1.06955	−0.534775	−	0.844995i	$-0.679603\pi$
$5$	−59.7896	−1.06955	−0.534775	+	0.844995i	$0.679603\pi$
$6$	0	0
$7$	72.0300	0.555608	0.277804	−	0.960638i	$-0.410393\pi$
$7$	72.0300	0.555608	0.277804	+	0.960638i	$0.410393\pi$
$8$	−196.590	−1.08602
$9$	0	0
$10$	−287.201	−0.908209
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−540.791	−0.887506	−0.443753	−	0.896149i	$-0.646353\pi$
$13$	−540.791	−0.887506	−0.443753	+	0.896149i	$0.646353\pi$
$14$	345.998	0.471795
$15$	0	0
$16$	−658.687	−0.643249
$17$	1510.71	1.26782	0.633912	−	0.773405i	$-0.281448\pi$
$17$	1510.71	1.26782	0.633912	+	0.773405i	$0.281448\pi$
$18$	0	0
$19$	−315.677	−0.200613	−0.100306	−	0.994957i	$-0.531982\pi$
$19$	−315.677	−0.200613	−0.100306	+	0.994957i	$0.531982\pi$
$20$	533.691	0.298342
$21$	0	0
$22$	0	0
$23$	2610.94	1.02914	0.514572	−	0.857447i	$-0.327951\pi$
$23$	2610.94	1.02914	0.514572	+	0.857447i	$0.327951\pi$
$24$	0	0
$25$	449.799	0.143936
$26$	−2597.71	−0.753627
$27$	0	0
$28$	−642.950	−0.154982
$29$	6630.81	1.46410	0.732052	−	0.681249i	$-0.238563\pi$
$29$	6630.81	1.46410	0.732052	+	0.681249i	$0.238563\pi$
$30$	0	0
$31$	7435.70	1.38969	0.694845	−	0.719160i	$-0.255473\pi$
$31$	7435.70	1.38969	0.694845	+	0.719160i	$0.255473\pi$
$32$	3126.85	0.539799
$33$	0	0
$34$	7256.73	1.07657
$35$	−4306.65	−0.594250
$36$	0	0
$37$	6440.18	0.773381	0.386691	−	0.922209i	$-0.373618\pi$
$37$	6440.18	0.773381	0.386691	+	0.922209i	$0.373618\pi$
$38$	−1516.36	−0.170351
$39$	0	0
$40$	11754.0	1.16155
$41$	−1194.66	−0.110990	−0.0554950	−	0.998459i	$-0.517674\pi$
$41$	−1194.66	−0.110990	−0.0554950	+	0.998459i	$0.517674\pi$
$42$	0	0
$43$	−21637.2	−1.78456	−0.892278	−	0.451487i	$-0.850894\pi$
$43$	−21637.2	−1.78456	−0.892278	+	0.451487i	$0.850894\pi$
$44$	0	0
$45$	0	0
$46$	12541.7	0.873900
$47$	−1352.89	−0.0893339	−0.0446670	−	0.999002i	$-0.514223\pi$
$47$	−1352.89	−0.0893339	−0.0446670	+	0.999002i	$0.514223\pi$
$48$	0	0
$49$	−11618.7	−0.691300
$50$	2160.62	0.122223
$51$	0	0
$52$	4827.18	0.247563
$53$	−12845.1	−0.628126	−0.314063	−	0.949402i	$-0.601690\pi$
$53$	−12845.1	−0.628126	−0.314063	+	0.949402i	$0.601690\pi$
$54$	0	0
$55$	0	0
$56$	−14160.4	−0.603399
$57$	0	0
$58$	31851.3	1.24325
$59$	5548.05	0.207496	0.103748	−	0.994604i	$-0.466916\pi$
$59$	5548.05	0.207496	0.103748	+	0.994604i	$0.466916\pi$
$60$	0	0
$61$	16044.8	0.552090	0.276045	−	0.961145i	$-0.410976\pi$
$61$	16044.8	0.552090	0.276045	+	0.961145i	$0.410976\pi$
$62$	35717.6	1.18006
$63$	0	0
$64$	36097.9	1.10162
$65$	32333.7	0.949232
$66$	0	0
$67$	−71743.7	−1.95253	−0.976263	−	0.216587i	$-0.930508\pi$
$67$	−71743.7	−1.95253	−0.976263	+	0.216587i	$0.930508\pi$
$68$	−13484.8	−0.353649
$69$	0	0
$70$	−20687.1	−0.504608
$71$	−34182.3	−0.804740	−0.402370	−	0.915477i	$-0.631813\pi$
$71$	−34182.3	−0.804740	−0.402370	+	0.915477i	$0.631813\pi$
$72$	0	0
$73$	−15613.4	−0.342918	−0.171459	−	0.985191i	$-0.554848\pi$
$73$	−15613.4	−0.342918	−0.171459	+	0.985191i	$0.554848\pi$
$74$	30935.6	0.656718
$75$	0	0
$76$	2817.78	0.0559593
$77$	0	0
$78$	0	0
$79$	44250.7	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$79$	44250.7	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$80$	39382.7	0.687987
$81$	0	0
$82$	−5738.57	−0.0942474
$83$	19538.0	0.311304	0.155652	−	0.987812i	$-0.450252\pi$
$83$	19538.0	0.311304	0.155652	+	0.987812i	$0.450252\pi$
$84$	0	0
$85$	−90324.8	−1.35600
$86$	−103935.	−1.51536
$87$	0	0
$88$	0	0
$89$	−111152.	−1.48745	−0.743725	−	0.668486i	$-0.766943\pi$
$89$	−111152.	−1.48745	−0.743725	+	0.668486i	$0.766943\pi$
$90$	0	0
$91$	−38953.2	−0.493105
$92$	−23305.6	−0.287072
$93$	0	0
$94$	−6498.62	−0.0758580
$95$	18874.2	0.214565
$96$	0	0
$97$	85165.0	0.919034	0.459517	−	0.888169i	$-0.348022\pi$
$97$	85165.0	0.919034	0.459517	+	0.888169i	$0.348022\pi$
$98$	−55810.6	−0.587018
$99$	0	0
$100$	−4014.97	−0.0401497
$101$	96798.9	0.944207	0.472103	−	0.881543i	$-0.343495\pi$
$101$	96798.9	0.944207	0.472103	+	0.881543i	$0.343495\pi$
$102$	0	0
$103$	−197258.	−1.83207	−0.916034	−	0.401102i	$-0.868627\pi$
$103$	−197258.	−1.83207	−0.916034	+	0.401102i	$0.868627\pi$
$104$	106314.	0.963845
$105$	0	0
$106$	−61701.7	−0.533374
$107$	209592.	1.76976	0.884881	−	0.465816i	$-0.154239\pi$
$107$	209592.	1.76976	0.884881	+	0.465816i	$0.154239\pi$
$108$	0	0
$109$	−22068.0	−0.177909	−0.0889544	−	0.996036i	$-0.528353\pi$
$109$	−22068.0	−0.177909	−0.0889544	+	0.996036i	$0.528353\pi$
$110$	0	0
$111$	0	0
$112$	−47445.3	−0.357394
$113$	−9246.95	−0.0681243	−0.0340622	−	0.999420i	$-0.510844\pi$
$113$	−9246.95	−0.0681243	−0.0340622	+	0.999420i	$0.510844\pi$
$114$	0	0
$115$	−156107.	−1.10072
$116$	−59187.6	−0.408400
$117$	0	0
$118$	26650.2	0.176196
$119$	108816.	0.704413
$120$	0	0
$121$	0	0
$122$	77071.7	0.468808
$123$	0	0
$124$	−66372.2	−0.387643
$125$	159949.	0.915603
$126$	0	0
$127$	−97644.1	−0.537201	−0.268600	−	0.963252i	$-0.586561\pi$
$127$	−97644.1	−0.537201	−0.268600	+	0.963252i	$0.586561\pi$
$128$	73338.0	0.395643
$129$	0	0
$130$	155316.	0.806041
$131$	57856.1	0.294558	0.147279	−	0.989095i	$-0.452948\pi$
$131$	57856.1	0.294558	0.147279	+	0.989095i	$0.452948\pi$
$132$	0	0
$133$	−22738.2	−0.111462
$134$	−344623.	−1.65799
$135$	0	0
$136$	−296990.	−1.37688
$137$	325451.	1.48144	0.740720	−	0.671814i	$-0.234485\pi$
$137$	325451.	1.48144	0.740720	+	0.671814i	$0.234485\pi$
$138$	0	0
$139$	−212039.	−0.930848	−0.465424	−	0.885088i	$-0.654098\pi$
$139$	−212039.	−0.930848	−0.465424	+	0.885088i	$0.654098\pi$
$140$	38441.8	0.165761
$141$	0	0
$142$	−164196.	−0.683346
$143$	0	0
$144$	0	0
$145$	−396454.	−1.56593
$146$	−74999.4	−0.291189
$147$	0	0
$148$	−57486.0	−0.215729
$149$	−314649.	−1.16108	−0.580539	−	0.814232i	$-0.697158\pi$
$149$	−314649.	−1.16108	−0.580539	+	0.814232i	$0.697158\pi$
$150$	0	0
$151$	225926.	0.806352	0.403176	−	0.915122i	$-0.367906\pi$
$151$	225926.	0.806352	0.403176	+	0.915122i	$0.367906\pi$
$152$	62058.8	0.217869
$153$	0	0
$154$	0	0
$155$	−444578.	−1.48634
$156$	0	0
$157$	351696.	1.13872	0.569362	−	0.822087i	$-0.307190\pi$
$157$	351696.	1.13872	0.569362	+	0.822087i	$0.307190\pi$
$158$	212560.	0.677389
$159$	0	0
$160$	−186953.	−0.577342
$161$	188066.	0.571801
$162$	0	0
$163$	−461915.	−1.36174	−0.680868	−	0.732406i	$-0.738397\pi$
$163$	−461915.	−1.36174	−0.680868	+	0.732406i	$0.738397\pi$
$164$	10663.7	0.0309598
$165$	0	0
$166$	93851.1	0.264344
$167$	−215682.	−0.598444	−0.299222	−	0.954184i	$-0.596727\pi$
$167$	−215682.	−0.598444	−0.299222	+	0.954184i	$0.596727\pi$
$168$	0	0
$169$	−78837.7	−0.212333
$170$	−433877.	−1.15145
$171$	0	0
$172$	193137.	0.497787
$173$	234410.	0.595471	0.297736	−	0.954648i	$-0.403769\pi$
$173$	234410.	0.595471	0.297736	+	0.954648i	$0.403769\pi$
$174$	0	0
$175$	32399.0	0.0799718
$176$	0	0
$177$	0	0
$178$	−533921.	−1.26307
$179$	102032.	0.238015	0.119008	−	0.992893i	$-0.462029\pi$
$179$	102032.	0.238015	0.119008	+	0.992893i	$0.462029\pi$
$180$	0	0
$181$	−524833.	−1.19076	−0.595381	−	0.803443i	$-0.702999\pi$
$181$	−524833.	−1.19076	−0.595381	+	0.803443i	$0.702999\pi$
$182$	−187113.	−0.418721
$183$	0	0
$184$	−513283.	−1.11767
$185$	−385056.	−0.827169
$186$	0	0
$187$	0	0
$188$	12076.1	0.0249190
$189$	0	0
$190$	90662.7	0.182198
$191$	−878656.	−1.74275	−0.871375	−	0.490617i	$-0.836771\pi$
$191$	−878656.	−1.74275	−0.871375	+	0.490617i	$0.836771\pi$
$192$	0	0
$193$	−470133.	−0.908506	−0.454253	−	0.890873i	$-0.650094\pi$
$193$	−470133.	−0.908506	−0.454253	+	0.890873i	$0.650094\pi$
$194$	409092.	0.780399
$195$	0	0
$196$	103710.	0.192833
$197$	−439293.	−0.806471	−0.403235	−	0.915096i	$-0.632114\pi$
$197$	−439293.	−0.806471	−0.403235	+	0.915096i	$0.632114\pi$
$198$	0	0
$199$	−802118.	−1.43584	−0.717920	−	0.696126i	$-0.754905\pi$
$199$	−802118.	−1.43584	−0.717920	+	0.696126i	$0.754905\pi$
$200$	−88425.9	−0.156316
$201$	0	0
$202$	464976.	0.801774
$203$	477617.	0.813467
$204$	0	0
$205$	71428.2	0.118709
$206$	−947533.	−1.55570
$207$	0	0
$208$	356212.	0.570888
$209$	0	0
$210$	0	0
$211$	149733.	0.231533	0.115766	−	0.993276i	$-0.463068\pi$
$211$	149733.	0.231533	0.115766	+	0.993276i	$0.463068\pi$
$212$	114657.	0.175211
$213$	0	0
$214$	1.00678e6	1.50280
$215$	1.29368e6	1.90867
$216$	0	0
$217$	535594.	0.772122
$218$	−106004.	−0.151072
$219$	0	0
$220$	0	0
$221$	−816979.	−1.12520
$222$	0	0
$223$	−349490.	−0.470623	−0.235311	−	0.971920i	$-0.575611\pi$
$223$	−349490.	−0.470623	−0.235311	+	0.971920i	$0.575611\pi$
$224$	225227.	0.299917
$225$	0	0
$226$	−44417.9	−0.0578479
$227$	−1.25342e6	−1.61448	−0.807238	−	0.590226i	$-0.799039\pi$
$227$	−1.25342e6	−1.61448	−0.807238	+	0.590226i	$0.799039\pi$
$228$	0	0
$229$	217491.	0.274065	0.137032	−	0.990567i	$-0.456244\pi$
$229$	217491.	0.274065	0.137032	+	0.990567i	$0.456244\pi$
$230$	−749863.	−0.934679
$231$	0	0
$232$	−1.30355e6	−1.59004
$233$	−1.44863e6	−1.74811	−0.874055	−	0.485827i	$-0.838519\pi$
$233$	−1.44863e6	−1.74811	−0.874055	+	0.485827i	$0.838519\pi$
$234$	0	0
$235$	80888.5	0.0955470
$236$	−49522.7	−0.0578795
$237$	0	0
$238$	522703.	0.598153
$239$	−952015.	−1.07808	−0.539038	−	0.842282i	$-0.681212\pi$
$239$	−952015.	−1.07808	−0.539038	+	0.842282i	$0.681212\pi$
$240$	0	0
$241$	−1.69481e6	−1.87965	−0.939826	−	0.341653i	$-0.889013\pi$
$241$	−1.69481e6	−1.87965	−0.939826	+	0.341653i	$0.889013\pi$
$242$	0	0
$243$	0	0
$244$	−143218.	−0.154001
$245$	694676.	0.739379
$246$	0	0
$247$	170715.	0.178045
$248$	−1.46178e6	−1.50922
$249$	0	0
$250$	768320.	0.777486
$251$	1.08988e6	1.09193	0.545963	−	0.837809i	$-0.316164\pi$
$251$	1.08988e6	1.09193	0.545963	+	0.837809i	$0.316164\pi$
$252$	0	0
$253$	0	0
$254$	−469036.	−0.456165
$255$	0	0
$256$	−802852.	−0.765660
$257$	−1.56490e6	−1.47793	−0.738967	−	0.673742i	$-0.764686\pi$
$257$	−1.56490e6	−1.47793	−0.738967	+	0.673742i	$0.764686\pi$
$258$	0	0
$259$	463886.	0.429697
$260$	−288615.	−0.264781
$261$	0	0
$262$	277913.	0.250124
$263$	−892520.	−0.795662	−0.397831	−	0.917459i	$-0.630237\pi$
$263$	−892520.	−0.795662	−0.397831	+	0.917459i	$0.630237\pi$
$264$	0	0
$265$	768002.	0.671812
$266$	−109224.	−0.0946482
$267$	0	0
$268$	640395.	0.544642
$269$	−229826.	−0.193650	−0.0968250	−	0.995301i	$-0.530869\pi$
$269$	−229826.	−0.193650	−0.0968250	+	0.995301i	$0.530869\pi$
$270$	0	0
$271$	1.48098e6	1.22497	0.612487	−	0.790480i	$-0.290169\pi$
$271$	1.48098e6	1.22497	0.612487	+	0.790480i	$0.290169\pi$
$272$	−995086.	−0.815527
$273$	0	0
$274$	1.56331e6	1.25797
$275$	0	0
$276$	0	0
$277$	1.48183e6	1.16038	0.580188	−	0.814482i	$-0.302979\pi$
$277$	1.48183e6	1.16038	0.580188	+	0.814482i	$0.302979\pi$
$278$	−1.01854e6	−0.790431
$279$	0	0
$280$	846643.	0.645365
$281$	−8168.06	−0.00617097	−0.00308548	−	0.999995i	$-0.500982\pi$
$281$	−8168.06	−0.00617097	−0.00308548	+	0.999995i	$0.500982\pi$
$282$	0	0
$283$	1.45081e6	1.07683	0.538413	−	0.842681i	$-0.319024\pi$
$283$	1.45081e6	1.07683	0.538413	+	0.842681i	$0.319024\pi$
$284$	305116.	0.224476
$285$	0	0
$286$	0	0
$287$	−86051.3	−0.0616670
$288$	0	0
$289$	862388.	0.607376
$290$	−1.90438e6	−1.32971
$291$	0	0
$292$	139368.	0.0956543
$293$	2.19477e6	1.49355	0.746776	−	0.665075i	$-0.231601\pi$
$293$	2.19477e6	1.49355	0.746776	+	0.665075i	$0.231601\pi$
$294$	0	0
$295$	−331716.	−0.221928
$296$	−1.26607e6	−0.839904
$297$	0	0
$298$	−1.51143e6	−0.985931
$299$	−1.41197e6	−0.913372
$300$	0	0
$301$	−1.55853e6	−0.991513
$302$	1.08524e6	0.684715
$303$	0	0
$304$	207932.	0.129044
$305$	−959313.	−0.590488
$306$	0	0
$307$	−537035.	−0.325205	−0.162602	−	0.986692i	$-0.551989\pi$
$307$	−537035.	−0.325205	−0.162602	+	0.986692i	$0.551989\pi$
$308$	0	0
$309$	0	0
$310$	−2.13554e6	−1.26213
$311$	1.21907e6	0.714704	0.357352	−	0.933970i	$-0.383680\pi$
$311$	1.21907e6	0.714704	0.357352	+	0.933970i	$0.383680\pi$
$312$	0	0
$313$	547802.	0.316055	0.158028	−	0.987435i	$-0.449486\pi$
$313$	547802.	0.316055	0.158028	+	0.987435i	$0.449486\pi$
$314$	1.68938e6	0.966949
$315$	0	0
$316$	−394989.	−0.222519
$317$	2.94125e6	1.64393	0.821967	−	0.569536i	$-0.192877\pi$
$317$	2.94125e6	1.64393	0.821967	+	0.569536i	$0.192877\pi$
$318$	0	0
$319$	0	0
$320$	−2.15828e6	−1.17824
$321$	0	0
$322$	903378.	0.485546
$323$	−476896.	−0.254342
$324$	0	0
$325$	−243247.	−0.127744
$326$	−2.21882e6	−1.15632
$327$	0	0
$328$	234858.	0.120537
$329$	−97448.4	−0.0496346
$330$	0	0
$331$	−1.11045e6	−0.557092	−0.278546	−	0.960423i	$-0.589853\pi$
$331$	−1.11045e6	−0.557092	−0.278546	+	0.960423i	$0.589853\pi$
$332$	−174399.	−0.0868356
$333$	0	0
$334$	−1.03604e6	−0.508169
$335$	4.28953e6	2.08832
$336$	0	0
$337$	1.44316e6	0.692213	0.346107	−	0.938195i	$-0.387504\pi$
$337$	1.44316e6	0.692213	0.346107	+	0.938195i	$0.387504\pi$
$338$	−378699.	−0.180303
$339$	0	0
$340$	806252.	0.378245
$341$	0	0
$342$	0	0
$343$	−2.04750e6	−0.939700
$344$	4.25365e6	1.93805
$345$	0	0
$346$	1.12599e6	0.505645
$347$	1.63468e6	0.728800	0.364400	−	0.931243i	$-0.381274\pi$
$347$	1.63468e6	0.728800	0.364400	+	0.931243i	$0.381274\pi$
$348$	0	0
$349$	−3.72087e6	−1.63524	−0.817620	−	0.575758i	$-0.804707\pi$
$349$	−3.72087e6	−1.63524	−0.817620	+	0.575758i	$0.804707\pi$
$350$	155630.	0.0679081
$351$	0	0
$352$	0	0
$353$	991536.	0.423518	0.211759	−	0.977322i	$-0.432081\pi$
$353$	991536.	0.423518	0.211759	+	0.977322i	$0.432081\pi$
$354$	0	0
$355$	2.04375e6	0.860709
$356$	992159.	0.414912
$357$	0	0
$358$	490114.	0.202111
$359$	−838000.	−0.343169	−0.171584	−	0.985169i	$-0.554889\pi$
$359$	−838000.	−0.343169	−0.171584	+	0.985169i	$0.554889\pi$
$360$	0	0
$361$	−2.37645e6	−0.959754
$362$	−2.52105e6	−1.01114
$363$	0	0
$364$	347702.	0.137548
$365$	933520.	0.366768
$366$	0	0
$367$	5.08625e6	1.97121	0.985604	−	0.169071i	$-0.0540766\pi$
$367$	5.08625e6	1.97121	0.985604	+	0.169071i	$0.0540766\pi$
$368$	−1.71979e6	−0.661997
$369$	0	0
$370$	−1.84963e6	−0.702392
$371$	−925231.	−0.348992
$372$	0	0
$373$	−4.08135e6	−1.51891	−0.759455	−	0.650559i	$-0.774534\pi$
$373$	−4.08135e6	−1.51891	−0.759455	+	0.650559i	$0.774534\pi$
$374$	0	0
$375$	0	0
$376$	265963.	0.0970180
$377$	−3.58589e6	−1.29940
$378$	0	0
$379$	34844.5	0.0124605	0.00623026	−	0.999981i	$-0.498017\pi$
$379$	34844.5	0.0124605	0.00623026	+	0.999981i	$0.498017\pi$
$380$	−168474.	−0.0598513
$381$	0	0
$382$	−4.22064e6	−1.47986
$383$	−2.54395e6	−0.886157	−0.443079	−	0.896483i	$-0.646114\pi$
$383$	−2.54395e6	−0.886157	−0.443079	+	0.896483i	$0.646114\pi$
$384$	0	0
$385$	0	0
$386$	−2.25830e6	−0.771459
$387$	0	0
$388$	−760195.	−0.256357
$389$	2.36422e6	0.792163	0.396081	−	0.918215i	$-0.370370\pi$
$389$	2.36422e6	0.792163	0.396081	+	0.918215i	$0.370370\pi$
$390$	0	0
$391$	3.94437e6	1.30477
$392$	2.28411e6	0.750762
$393$	0	0
$394$	−2.11015e6	−0.684816
$395$	−2.64574e6	−0.853206
$396$	0	0
$397$	3.56436e6	1.13502	0.567512	−	0.823365i	$-0.307906\pi$
$397$	3.56436e6	1.13502	0.567512	+	0.823365i	$0.307906\pi$
$398$	−3.85300e6	−1.21924
$399$	0	0
$400$	−296277.	−0.0925865
$401$	6.21116e6	1.92891	0.964454	−	0.264251i	$-0.0851249\pi$
$401$	6.21116e6	1.92891	0.964454	+	0.264251i	$0.0851249\pi$
$402$	0	0
$403$	−4.02116e6	−1.23336
$404$	−864041.	−0.263379
$405$	0	0
$406$	2.29425e6	0.690757
$407$	0	0
$408$	0	0
$409$	799568.	0.236345	0.118173	−	0.992993i	$-0.462296\pi$
$409$	799568.	0.236345	0.118173	+	0.992993i	$0.462296\pi$
$410$	343107.	0.100802
$411$	0	0
$412$	1.76075e6	0.511040
$413$	399626.	0.115287
$414$	0	0
$415$	−1.16817e6	−0.332954
$416$	−1.69097e6	−0.479075
$417$	0	0
$418$	0	0
$419$	1.19844e6	0.333490	0.166745	−	0.986000i	$-0.446674\pi$
$419$	1.19844e6	0.333490	0.166745	+	0.986000i	$0.446674\pi$
$420$	0	0
$421$	1.83223e6	0.503819	0.251910	−	0.967751i	$-0.418941\pi$
$421$	1.83223e6	0.503819	0.251910	+	0.967751i	$0.418941\pi$
$422$	719247.	0.196606
$423$	0	0
$424$	2.52521e6	0.682155
$425$	679516.	0.182485
$426$	0	0
$427$	1.15571e6	0.306746
$428$	−1.87085e6	−0.493661
$429$	0	0
$430$	6.21423e6	1.62075
$431$	−1.52861e6	−0.396373	−0.198187	−	0.980164i	$-0.563505\pi$
$431$	−1.52861e6	−0.396373	−0.198187	+	0.980164i	$0.563505\pi$
$432$	0	0
$433$	−5.29350e6	−1.35682	−0.678412	−	0.734682i	$-0.737331\pi$
$433$	−5.29350e6	−1.35682	−0.678412	+	0.734682i	$0.737331\pi$
$434$	2.57274e6	0.655649
$435$	0	0
$436$	196983.	0.0496263
$437$	−824212.	−0.206460
$438$	0	0
$439$	−3.22914e6	−0.799698	−0.399849	−	0.916581i	$-0.630937\pi$
$439$	−3.22914e6	−0.799698	−0.399849	+	0.916581i	$0.630937\pi$
$440$	0	0
$441$	0	0
$442$	−3.92438e6	−0.955466
$443$	−156436.	−0.0378727	−0.0189363	−	0.999821i	$-0.506028\pi$
$443$	−156436.	−0.0378727	−0.0189363	+	0.999821i	$0.506028\pi$
$444$	0	0
$445$	6.64574e6	1.59090
$446$	−1.67879e6	−0.399630
$447$	0	0
$448$	2.60013e6	0.612069
$449$	−4.08810e6	−0.956986	−0.478493	−	0.878091i	$-0.658817\pi$
$449$	−4.08810e6	−0.956986	−0.478493	+	0.878091i	$0.658817\pi$
$450$	0	0
$451$	0	0
$452$	82539.6	0.0190027
$453$	0	0
$454$	−6.02083e6	−1.37093
$455$	2.32900e6	0.527401
$456$	0	0
$457$	−3.01684e6	−0.675712	−0.337856	−	0.941198i	$-0.609702\pi$
$457$	−3.01684e6	−0.675712	−0.337856	+	0.941198i	$0.609702\pi$
$458$	1.04472e6	0.232722
$459$	0	0
$460$	1.39343e6	0.307037
$461$	798809.	0.175061	0.0875307	−	0.996162i	$-0.472102\pi$
$461$	798809.	0.175061	0.0875307	+	0.996162i	$0.472102\pi$
$462$	0	0
$463$	587910.	0.127455	0.0637277	−	0.997967i	$-0.479701\pi$
$463$	587910.	0.127455	0.0637277	+	0.997967i	$0.479701\pi$
$464$	−4.36763e6	−0.941783
$465$	0	0
$466$	−6.95855e6	−1.48441
$467$	−2.05580e6	−0.436203	−0.218101	−	0.975926i	$-0.569986\pi$
$467$	−2.05580e6	−0.436203	−0.218101	+	0.975926i	$0.569986\pi$
$468$	0	0
$469$	−5.16770e6	−1.08484
$470$	388550.	0.0811339
$471$	0	0
$472$	−1.09069e6	−0.225344
$473$	0	0
$474$	0	0
$475$	−141991.	−0.0288753
$476$	−971312.	−0.196490
$477$	0	0
$478$	−4.57303e6	−0.915449
$479$	2.52635e6	0.503101	0.251551	−	0.967844i	$-0.419059\pi$
$479$	2.52635e6	0.503101	0.251551	+	0.967844i	$0.419059\pi$
$480$	0	0
$481$	−3.48279e6	−0.686381
$482$	−8.14105e6	−1.59611
$483$	0	0
$484$	0	0
$485$	−5.09198e6	−0.982952
$486$	0	0
$487$	29503.9	0.00563711	0.00281856	−	0.999996i	$-0.499103\pi$
$487$	29503.9	0.00563711	0.00281856	+	0.999996i	$0.499103\pi$
$488$	−3.15425e6	−0.599578
$489$	0	0
$490$	3.33690e6	0.627845
$491$	3.21878e6	0.602542	0.301271	−	0.953539i	$-0.402589\pi$
$491$	3.21878e6	0.602542	0.301271	+	0.953539i	$0.402589\pi$
$492$	0	0
$493$	1.00172e7	1.85622
$494$	820035.	0.151187
$495$	0	0
$496$	−4.89780e6	−0.893917
$497$	−2.46215e6	−0.447120
$498$	0	0
$499$	8.68764e6	1.56189	0.780945	−	0.624599i	$-0.214738\pi$
$499$	8.68764e6	1.56189	0.780945	+	0.624599i	$0.214738\pi$
$500$	−1.42773e6	−0.255400
$501$	0	0
$502$	5.23525e6	0.927211
$503$	−440914.	−0.0777023	−0.0388512	−	0.999245i	$-0.512370\pi$
$503$	−440914.	−0.0777023	−0.0388512	+	0.999245i	$0.512370\pi$
$504$	0	0
$505$	−5.78757e6	−1.00988
$506$	0	0
$507$	0	0
$508$	871585.	0.149848
$509$	−2.70392e6	−0.462594	−0.231297	−	0.972883i	$-0.574297\pi$
$509$	−2.70392e6	−0.462594	−0.231297	+	0.972883i	$0.574297\pi$
$510$	0	0
$511$	−1.12463e6	−0.190528
$512$	−6.20334e6	−1.04580
$513$	0	0
$514$	−7.51705e6	−1.25499
$515$	1.17940e7	1.95949
$516$	0	0
$517$	0	0
$518$	2.22829e6	0.364878
$519$	0	0
$520$	−6.35648e6	−1.03088
$521$	−6.40024e6	−1.03300	−0.516502	−	0.856286i	$-0.672766\pi$
$521$	−6.40024e6	−1.03300	−0.516502	+	0.856286i	$0.672766\pi$
$522$	0	0
$523$	−7.79308e6	−1.24582	−0.622909	−	0.782294i	$-0.714049\pi$
$523$	−7.79308e6	−1.24582	−0.622909	+	0.782294i	$0.714049\pi$
$524$	−516432.	−0.0821646
$525$	0	0
$526$	−4.28724e6	−0.675637
$527$	1.12332e7	1.76188
$528$	0	0
$529$	380641.	0.0591393
$530$	3.68912e6	0.570470
$531$	0	0
$532$	202965.	0.0310915
$533$	646061.	0.0985044
$534$	0	0
$535$	−1.25314e7	−1.89285
$536$	1.41041e7	2.12047
$537$	0	0
$538$	−1.10397e6	−0.164438
$539$	0	0
$540$	0	0
$541$	9.77128e6	1.43535	0.717676	−	0.696378i	$-0.245206\pi$
$541$	9.77128e6	1.43535	0.717676	+	0.696378i	$0.245206\pi$
$542$	7.11395e6	1.04019
$543$	0	0
$544$	4.72377e6	0.684370
$545$	1.31944e6	0.190282
$546$	0	0
$547$	−2.46761e6	−0.352621	−0.176311	−	0.984335i	$-0.556416\pi$
$547$	−2.46761e6	−0.352621	−0.176311	+	0.984335i	$0.556416\pi$
$548$	−2.90502e6	−0.413236
$549$	0	0
$550$	0	0
$551$	−2.09319e6	−0.293718
$552$	0	0
$553$	3.18738e6	0.443222
$554$	7.11801e6	0.985335
$555$	0	0
$556$	1.89269e6	0.259653
$557$	4.18102e6	0.571010	0.285505	−	0.958377i	$-0.407839\pi$
$557$	4.18102e6	0.571010	0.285505	+	0.958377i	$0.407839\pi$
$558$	0	0
$559$	1.17012e7	1.58380
$560$	2.83673e6	0.382251
$561$	0	0
$562$	−39235.5	−0.00524008
$563$	−1.01719e6	−0.135249	−0.0676244	−	0.997711i	$-0.521542\pi$
$563$	−1.01719e6	−0.135249	−0.0676244	+	0.997711i	$0.521542\pi$
$564$	0	0
$565$	552871.	0.0728623
$566$	6.96902e6	0.914388
$567$	0	0
$568$	6.71989e6	0.873960
$569$	−1.02955e7	−1.33311	−0.666555	−	0.745456i	$-0.732232\pi$
$569$	−1.02955e7	−1.33311	−0.666555	+	0.745456i	$0.732232\pi$
$570$	0	0
$571$	3.45391e6	0.443323	0.221662	−	0.975124i	$-0.428852\pi$
$571$	3.45391e6	0.443323	0.221662	+	0.975124i	$0.428852\pi$
$572$	0	0
$573$	0	0
$574$	−413350.	−0.0523646
$575$	1.17440e6	0.148131
$576$	0	0
$577$	−797397.	−0.0997092	−0.0498546	−	0.998756i	$-0.515876\pi$
$577$	−797397.	−0.0997092	−0.0498546	+	0.998756i	$0.515876\pi$
$578$	4.14250e6	0.515755
$579$	0	0
$580$	3.53880e6	0.436804
$581$	1.40732e6	0.172963
$582$	0	0
$583$	0	0
$584$	3.06944e6	0.372414
$585$	0	0
$586$	1.05426e7	1.26825
$587$	−192576.	−0.0230678	−0.0115339	−	0.999933i	$-0.503671\pi$
$587$	−192576.	−0.0230678	−0.0115339	+	0.999933i	$0.503671\pi$
$588$	0	0
$589$	−2.34728e6	−0.278790
$590$	−1.59341e6	−0.188450
$591$	0	0
$592$	−4.24206e6	−0.497477
$593$	−6.70390e6	−0.782872	−0.391436	−	0.920205i	$-0.628022\pi$
$593$	−6.70390e6	−0.782872	−0.391436	+	0.920205i	$0.628022\pi$
$594$	0	0
$595$	−6.50610e6	−0.753404
$596$	2.80861e6	0.323873
$597$	0	0
$598$	−6.78244e6	−0.775591
$599$	−1.47361e6	−0.167809	−0.0839044	−	0.996474i	$-0.526739\pi$
$599$	−1.47361e6	−0.167809	−0.0839044	+	0.996474i	$0.526739\pi$
$600$	0	0
$601$	−2.38731e6	−0.269602	−0.134801	−	0.990873i	$-0.543040\pi$
$601$	−2.38731e6	−0.269602	−0.134801	+	0.990873i	$0.543040\pi$
$602$	−7.48643e6	−0.841944
$603$	0	0
$604$	−2.01665e6	−0.224925
$605$	0	0
$606$	0	0
$607$	5.25721e6	0.579141	0.289570	−	0.957157i	$-0.406488\pi$
$607$	5.25721e6	0.579141	0.289570	+	0.957157i	$0.406488\pi$
$608$	−987075.	−0.108291
$609$	0	0
$610$	−4.60809e6	−0.501413
$611$	731629.	0.0792844
$612$	0	0
$613$	1.22410e7	1.31572	0.657862	−	0.753138i	$-0.271461\pi$
$613$	1.22410e7	1.31572	0.657862	+	0.753138i	$0.271461\pi$
$614$	−2.57966e6	−0.276148
$615$	0	0
$616$	0	0
$617$	−3.32157e6	−0.351261	−0.175631	−	0.984456i	$-0.556196\pi$
$617$	−3.32157e6	−0.351261	−0.175631	+	0.984456i	$0.556196\pi$
$618$	0	0
$619$	1.16769e7	1.22490	0.612449	−	0.790510i	$-0.290185\pi$
$619$	1.16769e7	1.22490	0.612449	+	0.790510i	$0.290185\pi$
$620$	3.96837e6	0.414603
$621$	0	0
$622$	5.85581e6	0.606892
$623$	−8.00628e6	−0.826439
$624$	0	0
$625$	−1.09689e7	−1.12322
$626$	2.63138e6	0.268379
$627$	0	0
$628$	−3.13929e6	−0.317638
$629$	9.72924e6	0.980511
$630$	0	0
$631$	8.25895e6	0.825756	0.412878	−	0.910786i	$-0.364524\pi$
$631$	8.25895e6	0.825756	0.412878	+	0.910786i	$0.364524\pi$
$632$	−8.69924e6	−0.866341
$633$	0	0
$634$	1.41284e7	1.39595
$635$	5.83810e6	0.574563
$636$	0	0
$637$	6.28328e6	0.613533
$638$	0	0
$639$	0	0
$640$	−4.38485e6	−0.423160
$641$	−1.50046e7	−1.44238	−0.721189	−	0.692738i	$-0.756404\pi$
$641$	−1.50046e7	−1.44238	−0.721189	+	0.692738i	$0.756404\pi$
$642$	0	0
$643$	−1.75733e7	−1.67620	−0.838100	−	0.545516i	$-0.816334\pi$
$643$	−1.75733e7	−1.67620	−0.838100	+	0.545516i	$0.816334\pi$
$644$	−1.67870e6	−0.159499
$645$	0	0
$646$	−2.29078e6	−0.215975
$647$	−924911.	−0.0868639	−0.0434319	−	0.999056i	$-0.513829\pi$
$647$	−924911.	−0.0868639	−0.0434319	+	0.999056i	$0.513829\pi$
$648$	0	0
$649$	0	0
$650$	−1.16844e6	−0.108474
$651$	0	0
$652$	4.12312e6	0.379845
$653$	8.03324e6	0.737238	0.368619	−	0.929581i	$-0.379831\pi$
$653$	8.03324e6	0.737238	0.368619	+	0.929581i	$0.379831\pi$
$654$	0	0
$655$	−3.45919e6	−0.315044
$656$	786906.	0.0713943
$657$	0	0
$658$	−468096.	−0.0421473
$659$	−8.78898e6	−0.788360	−0.394180	−	0.919033i	$-0.628971\pi$
$659$	−8.78898e6	−0.788360	−0.394180	+	0.919033i	$0.628971\pi$
$660$	0	0
$661$	−1.05882e7	−0.942579	−0.471290	−	0.881979i	$-0.656211\pi$
$661$	−1.05882e7	−0.942579	−0.471290	+	0.881979i	$0.656211\pi$
$662$	−5.33405e6	−0.473056
$663$	0	0
$664$	−3.84096e6	−0.338080
$665$	1.35951e6	0.119214
$666$	0	0
$667$	1.73126e7	1.50677
$668$	1.92521e6	0.166931
$669$	0	0
$670$	2.06049e7	1.77330
$671$	0	0
$672$	0	0
$673$	−2.09744e7	−1.78505	−0.892526	−	0.450996i	$-0.851069\pi$
$673$	−2.09744e7	−1.78505	−0.892526	+	0.450996i	$0.851069\pi$
$674$	6.93226e6	0.587794
$675$	0	0
$676$	703717.	0.0592285
$677$	−6.29560e6	−0.527916	−0.263958	−	0.964534i	$-0.585028\pi$
$677$	−6.29560e6	−0.527916	−0.263958	+	0.964534i	$0.585028\pi$
$678$	0	0
$679$	6.13443e6	0.510623
$680$	1.77569e7	1.47264
$681$	0	0
$682$	0	0
$683$	5.09791e6	0.418158	0.209079	−	0.977899i	$-0.432953\pi$
$683$	5.09791e6	0.418158	0.209079	+	0.977899i	$0.432953\pi$
$684$	0	0
$685$	−1.94586e7	−1.58447
$686$	−9.83523e6	−0.797947
$687$	0	0
$688$	1.42522e7	1.14791
$689$	6.94651e6	0.557466
$690$	0	0
$691$	−1.27923e7	−1.01918	−0.509592	−	0.860416i	$-0.670204\pi$
$691$	−1.27923e7	−1.01918	−0.509592	+	0.860416i	$0.670204\pi$
$692$	−2.09238e6	−0.166102
$693$	0	0
$694$	7.85221e6	0.618862
$695$	1.26777e7	0.995588
$696$	0	0
$697$	−1.80478e6	−0.140716
$698$	−1.78733e7	−1.38857
$699$	0	0
$700$	−289198.	−0.0223075
$701$	−961596.	−0.0739091	−0.0369545	−	0.999317i	$-0.511766\pi$
$701$	−961596.	−0.0739091	−0.0369545	+	0.999317i	$0.511766\pi$
$702$	0	0
$703$	−2.03302e6	−0.155150
$704$	0	0
$705$	0	0
$706$	4.76287e6	0.359631
$707$	6.97243e6	0.524609
$708$	0	0
$709$	−1.78426e6	−0.133304	−0.0666521	−	0.997776i	$-0.521232\pi$
$709$	−1.78426e6	−0.133304	−0.0666521	+	0.997776i	$0.521232\pi$
$710$	9.81720e6	0.730872
$711$	0	0
$712$	2.18513e7	1.61539
$713$	1.94141e7	1.43019
$714$	0	0
$715$	0	0
$716$	−910754.	−0.0663924
$717$	0	0
$718$	−4.02535e6	−0.291402
$719$	−5.82070e6	−0.419907	−0.209953	−	0.977711i	$-0.567331\pi$
$719$	−5.82070e6	−0.419907	−0.209953	+	0.977711i	$0.567331\pi$
$720$	0	0
$721$	−1.42085e7	−1.01791
$722$	−1.14153e7	−0.814977
$723$	0	0
$724$	4.68474e6	0.332154
$725$	2.98253e6	0.210737
$726$	0	0
$727$	7.27069e6	0.510199	0.255100	−	0.966915i	$-0.417892\pi$
$727$	7.27069e6	0.510199	0.255100	+	0.966915i	$0.417892\pi$
$728$	7.65780e6	0.535520
$729$	0	0
$730$	4.48418e6	0.311441
$731$	−3.26875e7	−2.26250
$732$	0	0
$733$	1.17267e7	0.806147	0.403074	−	0.915168i	$-0.367942\pi$
$733$	1.17267e7	0.806147	0.403074	+	0.915168i	$0.367942\pi$
$734$	2.44319e7	1.67385
$735$	0	0
$736$	8.16401e6	0.555532
$737$	0	0
$738$	0	0
$739$	9.37182e6	0.631267	0.315633	−	0.948881i	$-0.397783\pi$
$739$	9.37182e6	0.631267	0.315633	+	0.948881i	$0.397783\pi$
$740$	3.43706e6	0.230732
$741$	0	0
$742$	−4.44437e6	−0.296347
$743$	−8.19123e6	−0.544348	−0.272174	−	0.962248i	$-0.587743\pi$
$743$	−8.19123e6	−0.544348	−0.272174	+	0.962248i	$0.587743\pi$
$744$	0	0
$745$	1.88128e7	1.24183
$746$	−1.96049e7	−1.28979
$747$	0	0
$748$	0	0
$749$	1.50969e7	0.983294
$750$	0	0
$751$	−2.22457e7	−1.43928	−0.719640	−	0.694347i	$-0.755693\pi$
$751$	−2.22457e7	−1.43928	−0.719640	+	0.694347i	$0.755693\pi$
$752$	891129.	0.0574640
$753$	0	0
$754$	−1.72249e7	−1.10339
$755$	−1.35081e7	−0.862433
$756$	0	0
$757$	−3.46824e6	−0.219973	−0.109987	−	0.993933i	$-0.535081\pi$
$757$	−3.46824e6	−0.219973	−0.109987	+	0.993933i	$0.535081\pi$
$758$	167376.	0.0105809
$759$	0	0
$760$	−3.71047e6	−0.233021
$761$	1.24057e7	0.776531	0.388265	−	0.921548i	$-0.373074\pi$
$761$	1.24057e7	0.776531	0.388265	+	0.921548i	$0.373074\pi$
$762$	0	0
$763$	−1.58956e6	−0.0988476
$764$	7.84301e6	0.486126
$765$	0	0
$766$	−1.22199e7	−0.752482
$767$	−3.00034e6	−0.184154
$768$	0	0
$769$	−1.84783e7	−1.12680	−0.563398	−	0.826185i	$-0.690506\pi$
$769$	−1.84783e7	−1.12680	−0.563398	+	0.826185i	$0.690506\pi$
$770$	0	0
$771$	0	0
$772$	4.19648e6	0.253420
$773$	−1.97874e6	−0.119108	−0.0595540	−	0.998225i	$-0.518968\pi$
$773$	−1.97874e6	−0.119108	−0.0595540	+	0.998225i	$0.518968\pi$
$774$	0	0
$775$	3.34457e6	0.200026
$776$	−1.67426e7	−0.998085
$777$	0	0
$778$	1.13566e7	0.672666
$779$	377126.	0.0222660
$780$	0	0
$781$	0	0
$782$	1.89469e7	1.10795
$783$	0	0
$784$	7.65307e6	0.444678
$785$	−2.10278e7	−1.21792
$786$	0	0
$787$	−2.55012e7	−1.46765	−0.733827	−	0.679337i	$-0.762268\pi$
$787$	−2.55012e7	−1.46765	−0.733827	+	0.679337i	$0.762268\pi$
$788$	3.92119e6	0.224959
$789$	0	0
$790$	−1.27089e7	−0.724501
$791$	−666058.	−0.0378504
$792$	0	0
$793$	−8.67690e6	−0.489983
$794$	1.71215e7	0.963807
$795$	0	0
$796$	7.15983e6	0.400516
$797$	−1.36157e7	−0.759269	−0.379634	−	0.925137i	$-0.623950\pi$
$797$	−1.36157e7	−0.759269	−0.379634	+	0.925137i	$0.623950\pi$
$798$	0	0
$799$	−2.04382e6	−0.113260
$800$	1.40645e6	0.0776964
$801$	0	0
$802$	2.98354e7	1.63793
$803$	0	0
$804$	0	0
$805$	−1.12444e7	−0.611569
$806$	−1.93158e7	−1.04731
$807$	0	0
$808$	−1.90297e7	−1.02542
$809$	−3.79535e6	−0.203883	−0.101941	−	0.994790i	$-0.532505\pi$
$809$	−3.79535e6	−0.203883	−0.101941	+	0.994790i	$0.532505\pi$
$810$	0	0
$811$	−9.11854e6	−0.486825	−0.243413	−	0.969923i	$-0.578267\pi$
$811$	−9.11854e6	−0.486825	−0.243413	+	0.969923i	$0.578267\pi$
$812$	−4.26328e6	−0.226910
$813$	0	0
$814$	0	0
$815$	2.76177e7	1.45644
$816$	0	0
$817$	6.83036e6	0.358005
$818$	3.84075e6	0.200693
$819$	0	0
$820$	−637578.	−0.0331130
$821$	−2.19307e7	−1.13552	−0.567760	−	0.823194i	$-0.692190\pi$
$821$	−2.19307e7	−1.13552	−0.567760	+	0.823194i	$0.692190\pi$
$822$	0	0
$823$	3.45045e7	1.77572	0.887862	−	0.460109i	$-0.152190\pi$
$823$	3.45045e7	1.77572	0.887862	+	0.460109i	$0.152190\pi$
$824$	3.87789e7	1.98965
$825$	0	0
$826$	1.91962e6	0.0978958
$827$	−2.15914e6	−0.109778	−0.0548892	−	0.998492i	$-0.517481\pi$
$827$	−2.15914e6	−0.109778	−0.0548892	+	0.998492i	$0.517481\pi$
$828$	0	0
$829$	8.84351e6	0.446929	0.223464	−	0.974712i	$-0.428263\pi$
$829$	8.84351e6	0.446929	0.223464	+	0.974712i	$0.428263\pi$
$830$	−5.61132e6	−0.282729
$831$	0	0
$832$	−1.95214e7	−0.977695
$833$	−1.75525e7	−0.876446
$834$	0	0
$835$	1.28956e7	0.640065
$836$	0	0
$837$	0	0
$838$	5.75675e6	0.283183
$839$	−1.99568e7	−0.978780	−0.489390	−	0.872065i	$-0.662780\pi$
$839$	−1.99568e7	−0.978780	−0.489390	+	0.872065i	$0.662780\pi$
$840$	0	0
$841$	2.34565e7	1.14360
$842$	8.80116e6	0.427819
$843$	0	0
$844$	−1.33654e6	−0.0645842
$845$	4.71368e6	0.227100
$846$	0	0
$847$	0	0
$848$	8.46089e6	0.404042
$849$	0	0
$850$	3.26407e6	0.154957
$851$	1.68149e7	0.795922
$852$	0	0
$853$	9.98067e6	0.469664	0.234832	−	0.972036i	$-0.424546\pi$
$853$	9.98067e6	0.469664	0.234832	+	0.972036i	$0.424546\pi$
$854$	5.55147e6	0.260474
$855$	0	0
$856$	−4.12036e7	−1.92199
$857$	1.15246e7	0.536012	0.268006	−	0.963417i	$-0.413635\pi$
$857$	1.15246e7	0.536012	0.268006	+	0.963417i	$0.413635\pi$
$858$	0	0
$859$	1.88299e7	0.870693	0.435346	−	0.900263i	$-0.356626\pi$
$859$	1.88299e7	0.870693	0.435346	+	0.900263i	$0.356626\pi$
$860$	−1.15476e7	−0.532408
$861$	0	0
$862$	−7.34273e6	−0.336581
$863$	−1.03363e6	−0.0472432	−0.0236216	−	0.999721i	$-0.507520\pi$
$863$	−1.03363e6	−0.0472432	−0.0236216	+	0.999721i	$0.507520\pi$
$864$	0	0
$865$	−1.40153e7	−0.636886
$866$	−2.54275e7	−1.15215
$867$	0	0
$868$	−4.78079e6	−0.215377
$869$	0	0
$870$	0	0
$871$	3.87984e7	1.73288
$872$	4.33835e6	0.193212
$873$	0	0
$874$	−3.95912e6	−0.175316
$875$	1.15211e7	0.508716
$876$	0	0
$877$	−4.19038e6	−0.183973	−0.0919865	−	0.995760i	$-0.529322\pi$
$877$	−4.19038e6	−0.183973	−0.0919865	+	0.995760i	$0.529322\pi$
$878$	−1.55113e7	−0.679065
$879$	0	0
$880$	0	0
$881$	−4.02320e7	−1.74635	−0.873177	−	0.487403i	$-0.837944\pi$
$881$	−4.02320e7	−1.74635	−0.873177	+	0.487403i	$0.837944\pi$
$882$	0	0
$883$	−1.47547e6	−0.0636839	−0.0318419	−	0.999493i	$-0.510137\pi$
$883$	−1.47547e6	−0.0636839	−0.0318419	+	0.999493i	$0.510137\pi$
$884$	7.29247e6	0.313866
$885$	0	0
$886$	−751442.	−0.0321597
$887$	−1.07269e6	−0.0457791	−0.0228895	−	0.999738i	$-0.507287\pi$
$887$	−1.07269e6	−0.0457791	−0.0228895	+	0.999738i	$0.507287\pi$
$888$	0	0
$889$	−7.03330e6	−0.298473
$890$	3.19230e7	1.35092
$891$	0	0
$892$	3.11960e6	0.131277
$893$	427075.	0.0179215
$894$	0	0
$895$	−6.10047e6	−0.254569
$896$	5.28253e6	0.219823
$897$	0	0
$898$	−1.96373e7	−0.812626
$899$	4.93047e7	2.03465
$900$	0	0
$901$	−1.94052e7	−0.796353
$902$	0	0
$903$	0	0
$904$	1.81786e6	0.0739841
$905$	3.13796e7	1.27358
$906$	0	0
$907$	−1.17276e7	−0.473361	−0.236681	−	0.971588i	$-0.576059\pi$
$907$	−1.17276e7	−0.473361	−0.236681	+	0.971588i	$0.576059\pi$
$908$	1.11882e7	0.450345
$909$	0	0
$910$	1.11874e7	0.447843
$911$	−2.62021e7	−1.04602	−0.523010	−	0.852326i	$-0.675191\pi$
$911$	−2.62021e7	−1.04602	−0.523010	+	0.852326i	$0.675191\pi$
$912$	0	0
$913$	0	0
$914$	−1.44915e7	−0.573781
$915$	0	0
$916$	−1.94136e6	−0.0764482
$917$	4.16738e6	0.163659
$918$	0	0
$919$	2.91878e7	1.14002	0.570010	−	0.821637i	$-0.306939\pi$
$919$	2.91878e7	1.14002	0.570010	+	0.821637i	$0.306939\pi$
$920$	3.06890e7	1.19540
$921$	0	0
$922$	3.83710e6	0.148654
$923$	1.84855e7	0.714212
$924$	0	0
$925$	2.89679e6	0.111317
$926$	2.82404e6	0.108229
$927$	0	0
$928$	2.07336e7	0.790322
$929$	−1.02623e7	−0.390126	−0.195063	−	0.980791i	$-0.562491\pi$
$929$	−1.02623e7	−0.390126	−0.195063	+	0.980791i	$0.562491\pi$
$930$	0	0
$931$	3.66775e6	0.138684
$932$	1.29307e7	0.487621
$933$	0	0
$934$	−9.87508e6	−0.370402
$935$	0	0
$936$	0	0
$937$	2.02516e7	0.753548	0.376774	−	0.926305i	$-0.377033\pi$
$937$	2.02516e7	0.753548	0.376774	+	0.926305i	$0.377033\pi$
$938$	−2.48232e7	−0.921193
$939$	0	0
$940$	−722023.	−0.0266521
$941$	−1.48963e7	−0.548408	−0.274204	−	0.961672i	$-0.588414\pi$
$941$	−1.48963e7	−0.548408	−0.274204	+	0.961672i	$0.588414\pi$
$942$	0	0
$943$	−3.11918e6	−0.114225
$944$	−3.65443e6	−0.133472
$945$	0	0
$946$	0	0
$947$	4.90445e7	1.77711	0.888557	−	0.458767i	$-0.151709\pi$
$947$	4.90445e7	1.77711	0.888557	+	0.458767i	$0.151709\pi$
$948$	0	0
$949$	8.44359e6	0.304342
$950$	−682058.	−0.0245195
$951$	0	0
$952$	−2.13922e7	−0.765003
$953$	−4.09514e7	−1.46062	−0.730309	−	0.683117i	$-0.760624\pi$
$953$	−4.09514e7	−1.46062	−0.730309	+	0.683117i	$0.760624\pi$
$954$	0	0
$955$	5.25345e7	1.86396
$956$	8.49782e6	0.300720
$957$	0	0
$958$	1.21354e7	0.427209
$959$	2.34422e7	0.823100
$960$	0	0
$961$	2.66605e7	0.931237
$962$	−1.67297e7	−0.582841
$963$	0	0
$964$	1.51281e7	0.524314
$965$	2.81091e7	0.971691
$966$	0	0
$967$	1.37084e7	0.471433	0.235717	−	0.971822i	$-0.424256\pi$
$967$	1.37084e7	0.471433	0.235717	+	0.971822i	$0.424256\pi$
$968$	0	0
$969$	0	0
$970$	−2.44595e7	−0.834675
$971$	9.26748e6	0.315438	0.157719	−	0.987484i	$-0.449586\pi$
$971$	9.26748e6	0.315438	0.157719	+	0.987484i	$0.449586\pi$
$972$	0	0
$973$	−1.52732e7	−0.517187
$974$	141723.	0.00478676
$975$	0	0
$976$	−1.05685e7	−0.355132
$977$	3.31740e7	1.11189	0.555945	−	0.831219i	$-0.312357\pi$
$977$	3.31740e7	1.11189	0.555945	+	0.831219i	$0.312357\pi$
$978$	0	0
$979$	0	0
$980$	−6.20078e6	−0.206244
$981$	0	0
$982$	1.54615e7	0.511649
$983$	3.23287e7	1.06710	0.533549	−	0.845769i	$-0.320858\pi$
$983$	3.23287e7	1.06710	0.533549	+	0.845769i	$0.320858\pi$
$984$	0	0
$985$	2.62652e7	0.862560
$986$	4.81180e7	1.57622
$987$	0	0
$988$	−1.52383e6	−0.0496643
$989$	−5.64933e7	−1.83657
$990$	0	0
$991$	2.51169e7	0.812424	0.406212	−	0.913779i	$-0.366850\pi$
$991$	2.51169e7	0.812424	0.406212	+	0.913779i	$0.366850\pi$
$992$	2.32503e7	0.750153
$993$	0	0
$994$	−1.18270e7	−0.379673
$995$	4.79584e7	1.53570
$996$	0	0
$997$	−8.52150e6	−0.271505	−0.135753	−	0.990743i	$-0.543345\pi$
$997$	−8.52150e6	−0.271505	−0.135753	+	0.990743i	$0.543345\pi$
$998$	4.17313e7	1.32628
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.6.a.bl.1.7		10
3.2	odd	2		363.6.a.q.1.4		10
11.7	odd	10		99.6.f.c.82.2		20
11.8	odd	10		99.6.f.c.64.2		20
11.10	odd	2		1089.6.a.bh.1.4		10
33.8	even	10		33.6.e.a.31.4	yes	20
33.29	even	10		33.6.e.a.16.4	✓	20
33.32	even	2		363.6.a.u.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.6.e.a.16.4	✓	20	33.29	even	10
33.6.e.a.31.4	yes	20	33.8	even	10
99.6.f.c.64.2		20	11.8	odd	10
99.6.f.c.82.2		20	11.7	odd	10
363.6.a.q.1.4		10	3.2	odd	2
363.6.a.u.1.7		10	33.32	even	2
1089.6.a.bh.1.4		10	11.10	odd	2
1089.6.a.bl.1.7		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)

\(f(q)\)	\(=\)	\(q+4.80353 q^{2} -8.92614 q^{4} -59.7896 q^{5} +72.0300 q^{7} -196.590 q^{8} +O(q^{10})\)	\(q+4.80353 q^{2} -8.92614 q^{4} -59.7896 q^{5} +72.0300 q^{7} -196.590 q^{8} -287.201 q^{10} -540.791 q^{13} +345.998 q^{14} -658.687 q^{16} +1510.71 q^{17} -315.677 q^{19} +533.691 q^{20} +2610.94 q^{23} +449.799 q^{25} -2597.71 q^{26} -642.950 q^{28} +6630.81 q^{29} +7435.70 q^{31} +3126.85 q^{32} +7256.73 q^{34} -4306.65 q^{35} +6440.18 q^{37} -1516.36 q^{38} +11754.0 q^{40} -1194.66 q^{41} -21637.2 q^{43} +12541.7 q^{46} -1352.89 q^{47} -11618.7 q^{49} +2160.62 q^{50} +4827.18 q^{52} -12845.1 q^{53} -14160.4 q^{56} +31851.3 q^{58} +5548.05 q^{59} +16044.8 q^{61} +35717.6 q^{62} +36097.9 q^{64} +32333.7 q^{65} -71743.7 q^{67} -13484.8 q^{68} -20687.1 q^{70} -34182.3 q^{71} -15613.4 q^{73} +30935.6 q^{74} +2817.78 q^{76} +44250.7 q^{79} +39382.7 q^{80} -5738.57 q^{82} +19538.0 q^{83} -90324.8 q^{85} -103935. q^{86} -111152. q^{89} -38953.2 q^{91} -23305.6 q^{92} -6498.62 q^{94} +18874.2 q^{95} +85165.0 q^{97} -55810.6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 9 q^{2} + 193 q^{4} - 11 q^{5} - 470 q^{7} + 324 q^{8}+O(q^{10}) \) 10 * q + 9 * q^2 + 193 * q^4 - 11 * q^5 - 470 * q^7 + 324 * q^8	\( 10 q + 9 q^{2} + 193 q^{4} - 11 q^{5} - 470 q^{7} + 324 q^{8} - 976 q^{10} - 2308 q^{13} - 540 q^{14} + 2801 q^{16} + 3093 q^{17} - 5305 q^{19} + 2229 q^{20} - 700 q^{23} + 8381 q^{25} + 15559 q^{26} - 24656 q^{28} + 10392 q^{29} - 101 q^{31} + 34557 q^{32} - 4542 q^{34} + 19867 q^{35} - 1284 q^{37} + 35769 q^{38} - 66596 q^{40} + 17944 q^{41} - 31812 q^{43} - 36417 q^{46} - 8787 q^{47} - 23810 q^{49} - 910 q^{50} - 51663 q^{52} + 3261 q^{53} - 84819 q^{56} + 53125 q^{58} - 49375 q^{59} - 63175 q^{61} + 28399 q^{62} + 124764 q^{64} - 14105 q^{65} + 5365 q^{67} - 53313 q^{68} + 40297 q^{70} + 236675 q^{71} - 200912 q^{73} - 180329 q^{74} - 39606 q^{76} - 210802 q^{79} - 270298 q^{80} - 369223 q^{82} - 178968 q^{83} - 107352 q^{85} - 465999 q^{86} - 90816 q^{89} + 7500 q^{91} + 136407 q^{92} - 71890 q^{94} + 335807 q^{95} + 271521 q^{97} - 75285 q^{98}+O(q^{100}) \) 10 * q + 9 * q^2 + 193 * q^4 - 11 * q^5 - 470 * q^7 + 324 * q^8 - 976 * q^10 - 2308 * q^13 - 540 * q^14 + 2801 * q^16 + 3093 * q^17 - 5305 * q^19 + 2229 * q^20 - 700 * q^23 + 8381 * q^25 + 15559 * q^26 - 24656 * q^28 + 10392 * q^29 - 101 * q^31 + 34557 * q^32 - 4542 * q^34 + 19867 * q^35 - 1284 * q^37 + 35769 * q^38 - 66596 * q^40 + 17944 * q^41 - 31812 * q^43 - 36417 * q^46 - 8787 * q^47 - 23810 * q^49 - 910 * q^50 - 51663 * q^52 + 3261 * q^53 - 84819 * q^56 + 53125 * q^58 - 49375 * q^59 - 63175 * q^61 + 28399 * q^62 + 124764 * q^64 - 14105 * q^65 + 5365 * q^67 - 53313 * q^68 + 40297 * q^70 + 236675 * q^71 - 200912 * q^73 - 180329 * q^74 - 39606 * q^76 - 210802 * q^79 - 270298 * q^80 - 369223 * q^82 - 178968 * q^83 - 107352 * q^85 - 465999 * q^86 - 90816 * q^89 + 7500 * q^91 + 136407 * q^92 - 71890 * q^94 + 335807 * q^95 + 271521 * q^97 - 75285 * q^98

Introduction

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