LMFDB - Embedded newform 9054.2.a.bi.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9054,2,Mod(1,9054)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9054, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9054.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9054 = 2 \cdot 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9054.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:	$72.2965539901$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 13 x^{10} + 94 x^{9} + 15 x^{8} - 616 x^{7} + 368 x^{6} + 1643 x^{5} - 1463 x^{4} + \cdots - 208 $ x^12 - 5x^11 - 13x^10 + 94x^9 + 15x^8 - 616x^7 + 368x^6 + 1643x^5 - 1463x^4 - 1556x^3 + 1284x^2 + 576*x - 208
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1006)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$1.74625$ of defining polynomial
Character	$\chi$	$=$	9054.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +2.58578 q^{5} +0.635213 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +2.58578 q^{5} +0.635213 q^{7} -1.00000 q^{8} -2.58578 q^{10} -3.51104 q^{11} +5.12735 q^{13} -0.635213 q^{14} +1.00000 q^{16} +0.322021 q^{17} +3.38371 q^{19} +2.58578 q^{20} +3.51104 q^{22} -3.77692 q^{23} +1.68625 q^{25} -5.12735 q^{26} +0.635213 q^{28} -2.46547 q^{29} +1.98771 q^{31} -1.00000 q^{32} -0.322021 q^{34} +1.64252 q^{35} -7.77317 q^{37} -3.38371 q^{38} -2.58578 q^{40} -10.7578 q^{41} -9.96085 q^{43} -3.51104 q^{44} +3.77692 q^{46} +3.93174 q^{47} -6.59651 q^{49} -1.68625 q^{50} +5.12735 q^{52} -8.45466 q^{53} -9.07877 q^{55} -0.635213 q^{56} +2.46547 q^{58} -11.9394 q^{59} +0.586889 q^{61} -1.98771 q^{62} +1.00000 q^{64} +13.2582 q^{65} -12.9100 q^{67} +0.322021 q^{68} -1.64252 q^{70} +0.0667081 q^{71} -2.81224 q^{73} +7.77317 q^{74} +3.38371 q^{76} -2.23026 q^{77} -4.39323 q^{79} +2.58578 q^{80} +10.7578 q^{82} +16.7077 q^{83} +0.832675 q^{85} +9.96085 q^{86} +3.51104 q^{88} +13.6156 q^{89} +3.25696 q^{91} -3.77692 q^{92} -3.93174 q^{94} +8.74951 q^{95} -12.7052 q^{97} +6.59651 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 8 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 8 q^{7} - 12 q^{8} + 5 q^{10} - 18 q^{11} + 4 q^{13} - 8 q^{14} + 12 q^{16} + 2 q^{17} + 6 q^{19} - 5 q^{20} + 18 q^{22} - 13 q^{23} + q^{25} - 4 q^{26} + 8 q^{28} - 20 q^{29} + 7 q^{31} - 12 q^{32} - 2 q^{34} - q^{35} + 10 q^{37} - 6 q^{38} + 5 q^{40} - 2 q^{41} + 8 q^{43} - 18 q^{44} + 13 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} + 4 q^{52} - 12 q^{53} + 8 q^{55} - 8 q^{56} + 20 q^{58} - 6 q^{59} - 10 q^{61} - 7 q^{62} + 12 q^{64} - 4 q^{65} + 7 q^{67} + 2 q^{68} + q^{70} - 22 q^{71} - 23 q^{73} - 10 q^{74} + 6 q^{76} + 19 q^{77} + 13 q^{79} - 5 q^{80} + 2 q^{82} - q^{83} - 28 q^{85} - 8 q^{86} + 18 q^{88} + 3 q^{89} - 21 q^{91} - 13 q^{92} + 12 q^{94} + 2 q^{95} - 70 q^{97} - 4 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 8 * q^7 - 12 * q^8 + 5 * q^10 - 18 * q^11 + 4 * q^13 - 8 * q^14 + 12 * q^16 + 2 * q^17 + 6 * q^19 - 5 * q^20 + 18 * q^22 - 13 * q^23 + q^25 - 4 * q^26 + 8 * q^28 - 20 * q^29 + 7 * q^31 - 12 * q^32 - 2 * q^34 - q^35 + 10 * q^37 - 6 * q^38 + 5 * q^40 - 2 * q^41 + 8 * q^43 - 18 * q^44 + 13 * q^46 - 12 * q^47 + 4 * q^49 - q^50 + 4 * q^52 - 12 * q^53 + 8 * q^55 - 8 * q^56 + 20 * q^58 - 6 * q^59 - 10 * q^61 - 7 * q^62 + 12 * q^64 - 4 * q^65 + 7 * q^67 + 2 * q^68 + q^70 - 22 * q^71 - 23 * q^73 - 10 * q^74 + 6 * q^76 + 19 * q^77 + 13 * q^79 - 5 * q^80 + 2 * q^82 - q^83 - 28 * q^85 - 8 * q^86 + 18 * q^88 + 3 * q^89 - 21 * q^91 - 13 * q^92 + 12 * q^94 + 2 * q^95 - 70 * q^97 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.58578	1.15639	0.578197	−	0.815897i	$-0.303756\pi$
$5$	2.58578	1.15639	0.578197	+	0.815897i	$0.303756\pi$
$6$	0	0
$7$	0.635213	0.240088	0.120044	−	0.992769i	$-0.461696\pi$
$7$	0.635213	0.240088	0.120044	+	0.992769i	$0.461696\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.58578	−0.817695
$11$	−3.51104	−1.05862	−0.529309	−	0.848429i	$-0.677549\pi$
$11$	−3.51104	−1.05862	−0.529309	+	0.848429i	$0.677549\pi$
$12$	0	0
$13$	5.12735	1.42207	0.711035	−	0.703156i	$-0.248227\pi$
$13$	5.12735	1.42207	0.711035	+	0.703156i	$0.248227\pi$
$14$	−0.635213	−0.169768
$15$	0	0
$16$	1.00000	0.250000
$17$	0.322021	0.0781016	0.0390508	−	0.999237i	$-0.487567\pi$
$17$	0.322021	0.0781016	0.0390508	+	0.999237i	$0.487567\pi$
$18$	0	0
$19$	3.38371	0.776276	0.388138	−	0.921601i	$-0.373118\pi$
$19$	3.38371	0.776276	0.388138	+	0.921601i	$0.373118\pi$
$20$	2.58578	0.578197
$21$	0	0
$22$	3.51104	0.748556
$23$	−3.77692	−0.787543	−0.393772	−	0.919208i	$-0.628830\pi$
$23$	−3.77692	−0.787543	−0.393772	+	0.919208i	$0.628830\pi$
$24$	0	0
$25$	1.68625	0.337249
$26$	−5.12735	−1.00556
$27$	0	0
$28$	0.635213	0.120044
$29$	−2.46547	−0.457826	−0.228913	−	0.973447i	$-0.573517\pi$
$29$	−2.46547	−0.457826	−0.228913	+	0.973447i	$0.573517\pi$
$30$	0	0
$31$	1.98771	0.357003	0.178501	−	0.983940i	$-0.442875\pi$
$31$	1.98771	0.357003	0.178501	+	0.983940i	$0.442875\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.322021	−0.0552262
$35$	1.64252	0.277636
$36$	0	0
$37$	−7.77317	−1.27790	−0.638951	−	0.769248i	$-0.720631\pi$
$37$	−7.77317	−1.27790	−0.638951	+	0.769248i	$0.720631\pi$
$38$	−3.38371	−0.548910
$39$	0	0
$40$	−2.58578	−0.408847
$41$	−10.7578	−1.68009	−0.840045	−	0.542516i	$-0.817472\pi$
$41$	−10.7578	−1.68009	−0.840045	+	0.542516i	$0.817472\pi$
$42$	0	0
$43$	−9.96085	−1.51902	−0.759508	−	0.650498i	$-0.774560\pi$
$43$	−9.96085	−1.51902	−0.759508	+	0.650498i	$0.774560\pi$
$44$	−3.51104	−0.529309
$45$	0	0
$46$	3.77692	0.556877
$47$	3.93174	0.573503	0.286752	−	0.958005i	$-0.407425\pi$
$47$	3.93174	0.573503	0.286752	+	0.958005i	$0.407425\pi$
$48$	0	0
$49$	−6.59651	−0.942358
$50$	−1.68625	−0.238471
$51$	0	0
$52$	5.12735	0.711035
$53$	−8.45466	−1.16134	−0.580669	−	0.814140i	$-0.697209\pi$
$53$	−8.45466	−1.16134	−0.580669	+	0.814140i	$0.697209\pi$
$54$	0	0
$55$	−9.07877	−1.22418
$56$	−0.635213	−0.0848838
$57$	0	0
$58$	2.46547	0.323732
$59$	−11.9394	−1.55437	−0.777187	−	0.629270i	$-0.783354\pi$
$59$	−11.9394	−1.55437	−0.777187	+	0.629270i	$0.783354\pi$
$60$	0	0
$61$	0.586889	0.0751434	0.0375717	−	0.999294i	$-0.488038\pi$
$61$	0.586889	0.0751434	0.0375717	+	0.999294i	$0.488038\pi$
$62$	−1.98771	−0.252439
$63$	0	0
$64$	1.00000	0.125000
$65$	13.2582	1.64448
$66$	0	0
$67$	−12.9100	−1.57721	−0.788605	−	0.614900i	$-0.789196\pi$
$67$	−12.9100	−1.57721	−0.788605	+	0.614900i	$0.789196\pi$
$68$	0.322021	0.0390508
$69$	0	0
$70$	−1.64252	−0.196319
$71$	0.0667081	0.00791679	0.00395840	−	0.999992i	$-0.498740\pi$
$71$	0.0667081	0.00791679	0.00395840	+	0.999992i	$0.498740\pi$
$72$	0	0
$73$	−2.81224	−0.329148	−0.164574	−	0.986365i	$-0.552625\pi$
$73$	−2.81224	−0.329148	−0.164574	+	0.986365i	$0.552625\pi$
$74$	7.77317	0.903613
$75$	0	0
$76$	3.38371	0.388138
$77$	−2.23026	−0.254161
$78$	0	0
$79$	−4.39323	−0.494277	−0.247139	−	0.968980i	$-0.579490\pi$
$79$	−4.39323	−0.494277	−0.247139	+	0.968980i	$0.579490\pi$
$80$	2.58578	0.289099
$81$	0	0
$82$	10.7578	1.18800
$83$	16.7077	1.83391	0.916955	−	0.398992i	$-0.130640\pi$
$83$	16.7077	1.83391	0.916955	+	0.398992i	$0.130640\pi$
$84$	0	0
$85$	0.832675	0.0903163
$86$	9.96085	1.07411
$87$	0	0
$88$	3.51104	0.374278
$89$	13.6156	1.44325	0.721623	−	0.692286i	$-0.243396\pi$
$89$	13.6156	1.44325	0.721623	+	0.692286i	$0.243396\pi$
$90$	0	0
$91$	3.25696	0.341422
$92$	−3.77692	−0.393772
$93$	0	0
$94$	−3.93174	−0.405528
$95$	8.74951	0.897681
$96$	0	0
$97$	−12.7052	−1.29002	−0.645008	−	0.764176i	$-0.723146\pi$
$97$	−12.7052	−1.29002	−0.645008	+	0.764176i	$0.723146\pi$
$98$	6.59651	0.666348
$99$	0	0
$100$	1.68625	0.168625
$101$	−9.15646	−0.911101	−0.455551	−	0.890210i	$-0.650558\pi$
$101$	−9.15646	−0.911101	−0.455551	+	0.890210i	$0.650558\pi$
$102$	0	0
$103$	1.18800	0.117057	0.0585285	−	0.998286i	$-0.481359\pi$
$103$	1.18800	0.117057	0.0585285	+	0.998286i	$0.481359\pi$
$104$	−5.12735	−0.502778
$105$	0	0
$106$	8.45466	0.821189
$107$	10.5335	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$107$	10.5335	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$108$	0	0
$109$	−7.51021	−0.719347	−0.359674	−	0.933078i	$-0.617112\pi$
$109$	−7.51021	−0.719347	−0.359674	+	0.933078i	$0.617112\pi$
$110$	9.07877	0.865627
$111$	0	0
$112$	0.635213	0.0600219
$113$	−14.2153	−1.33726	−0.668632	−	0.743594i	$-0.733120\pi$
$113$	−14.2153	−1.33726	−0.668632	+	0.743594i	$0.733120\pi$
$114$	0	0
$115$	−9.76629	−0.910711
$116$	−2.46547	−0.228913
$117$	0	0
$118$	11.9394	1.09911
$119$	0.204552	0.0187512
$120$	0	0
$121$	1.32741	0.120673
$122$	−0.586889	−0.0531344
$123$	0	0
$124$	1.98771	0.178501
$125$	−8.56863	−0.766401
$126$	0	0
$127$	20.3451	1.80533	0.902667	−	0.430340i	$-0.141606\pi$
$127$	20.3451	1.80533	0.902667	+	0.430340i	$0.141606\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−13.2582	−1.16282
$131$	19.8675	1.73583	0.867917	−	0.496710i	$-0.165459\pi$
$131$	19.8675	1.73583	0.867917	+	0.496710i	$0.165459\pi$
$132$	0	0
$133$	2.14937	0.186374
$134$	12.9100	1.11526
$135$	0	0
$136$	−0.322021	−0.0276131
$137$	−18.2376	−1.55814	−0.779072	−	0.626934i	$-0.784310\pi$
$137$	−18.2376	−1.55814	−0.779072	+	0.626934i	$0.784310\pi$
$138$	0	0
$139$	3.79729	0.322082	0.161041	−	0.986948i	$-0.448515\pi$
$139$	3.79729	0.322082	0.161041	+	0.986948i	$0.448515\pi$
$140$	1.64252	0.138818
$141$	0	0
$142$	−0.0667081	−0.00559802
$143$	−18.0023	−1.50543
$144$	0	0
$145$	−6.37515	−0.529428
$146$	2.81224	0.232743
$147$	0	0
$148$	−7.77317	−0.638951
$149$	−6.47248	−0.530246	−0.265123	−	0.964215i	$-0.585413\pi$
$149$	−6.47248	−0.530246	−0.265123	+	0.964215i	$0.585413\pi$
$150$	0	0
$151$	−20.4636	−1.66530	−0.832652	−	0.553797i	$-0.813178\pi$
$151$	−20.4636	−1.66530	−0.832652	+	0.553797i	$0.813178\pi$
$152$	−3.38371	−0.274455
$153$	0	0
$154$	2.23026	0.179719
$155$	5.13977	0.412836
$156$	0	0
$157$	−14.4827	−1.15585	−0.577924	−	0.816090i	$-0.696137\pi$
$157$	−14.4827	−1.15585	−0.577924	+	0.816090i	$0.696137\pi$
$158$	4.39323	0.349507
$159$	0	0
$160$	−2.58578	−0.204424
$161$	−2.39915	−0.189079
$162$	0	0
$163$	−3.01751	−0.236349	−0.118175	−	0.992993i	$-0.537704\pi$
$163$	−3.01751	−0.236349	−0.118175	+	0.992993i	$0.537704\pi$
$164$	−10.7578	−0.840045
$165$	0	0
$166$	−16.7077	−1.29677
$167$	21.4972	1.66351	0.831753	−	0.555146i	$-0.187338\pi$
$167$	21.4972	1.66351	0.831753	+	0.555146i	$0.187338\pi$
$168$	0	0
$169$	13.2897	1.02229
$170$	−0.832675	−0.0638633
$171$	0	0
$172$	−9.96085	−0.759508
$173$	9.82213	0.746763	0.373381	−	0.927678i	$-0.378198\pi$
$173$	9.82213	0.746763	0.373381	+	0.927678i	$0.378198\pi$
$174$	0	0
$175$	1.07113	0.0809695
$176$	−3.51104	−0.264655
$177$	0	0
$178$	−13.6156	−1.02053
$179$	−6.49532	−0.485483	−0.242742	−	0.970091i	$-0.578047\pi$
$179$	−6.49532	−0.485483	−0.242742	+	0.970091i	$0.578047\pi$
$180$	0	0
$181$	4.08174	0.303393	0.151696	−	0.988427i	$-0.451526\pi$
$181$	4.08174	0.303393	0.151696	+	0.988427i	$0.451526\pi$
$182$	−3.25696	−0.241422
$183$	0	0
$184$	3.77692	0.278439
$185$	−20.0997	−1.47776
$186$	0	0
$187$	−1.13063	−0.0826798
$188$	3.93174	0.286752
$189$	0	0
$190$	−8.74951	−0.634757
$191$	−23.4882	−1.69955	−0.849773	−	0.527148i	$-0.823261\pi$
$191$	−23.4882	−1.69955	−0.849773	+	0.527148i	$0.823261\pi$
$192$	0	0
$193$	9.86940	0.710415	0.355207	−	0.934788i	$-0.384410\pi$
$193$	9.86940	0.710415	0.355207	+	0.934788i	$0.384410\pi$
$194$	12.7052	0.912179
$195$	0	0
$196$	−6.59651	−0.471179
$197$	1.66586	0.118687	0.0593437	−	0.998238i	$-0.481099\pi$
$197$	1.66586	0.118687	0.0593437	+	0.998238i	$0.481099\pi$
$198$	0	0
$199$	12.7759	0.905663	0.452831	−	0.891596i	$-0.350414\pi$
$199$	12.7759	0.905663	0.452831	+	0.891596i	$0.350414\pi$
$200$	−1.68625	−0.119236
$201$	0	0
$202$	9.15646	0.644246
$203$	−1.56610	−0.109918
$204$	0	0
$205$	−27.8174	−1.94285
$206$	−1.18800	−0.0827718
$207$	0	0
$208$	5.12735	0.355518
$209$	−11.8803	−0.821780
$210$	0	0
$211$	16.3237	1.12377	0.561885	−	0.827215i	$-0.310076\pi$
$211$	16.3237	1.12377	0.561885	+	0.827215i	$0.310076\pi$
$212$	−8.45466	−0.580669
$213$	0	0
$214$	−10.5335	−0.720054
$215$	−25.7565	−1.75658
$216$	0	0
$217$	1.26262	0.0857120
$218$	7.51021	0.508655
$219$	0	0
$220$	−9.07877	−0.612091
$221$	1.65111	0.111066
$222$	0	0
$223$	8.02490	0.537387	0.268694	−	0.963226i	$-0.413408\pi$
$223$	8.02490	0.537387	0.268694	+	0.963226i	$0.413408\pi$
$224$	−0.635213	−0.0424419
$225$	0	0
$226$	14.2153	0.945588
$227$	−21.8320	−1.44904	−0.724520	−	0.689254i	$-0.757938\pi$
$227$	−21.8320	−1.44904	−0.724520	+	0.689254i	$0.757938\pi$
$228$	0	0
$229$	13.0772	0.864163	0.432082	−	0.901834i	$-0.357779\pi$
$229$	13.0772	0.864163	0.432082	+	0.901834i	$0.357779\pi$
$230$	9.76629	0.643970
$231$	0	0
$232$	2.46547	0.161866
$233$	5.66551	0.371160	0.185580	−	0.982629i	$-0.440584\pi$
$233$	5.66551	0.371160	0.185580	+	0.982629i	$0.440584\pi$
$234$	0	0
$235$	10.1666	0.663196
$236$	−11.9394	−0.777187
$237$	0	0
$238$	−0.204552	−0.0132591
$239$	3.87186	0.250450	0.125225	−	0.992128i	$-0.460035\pi$
$239$	3.87186	0.250450	0.125225	+	0.992128i	$0.460035\pi$
$240$	0	0
$241$	−18.6798	−1.20327	−0.601636	−	0.798771i	$-0.705484\pi$
$241$	−18.6798	−1.20327	−0.601636	+	0.798771i	$0.705484\pi$
$242$	−1.32741	−0.0853289
$243$	0	0
$244$	0.586889	0.0375717
$245$	−17.0571	−1.08974
$246$	0	0
$247$	17.3494	1.10392
$248$	−1.98771	−0.126219
$249$	0	0
$250$	8.56863	0.541928
$251$	18.0304	1.13807	0.569035	−	0.822313i	$-0.307317\pi$
$251$	18.0304	1.13807	0.569035	+	0.822313i	$0.307317\pi$
$252$	0	0
$253$	13.2609	0.833708
$254$	−20.3451	−1.27656
$255$	0	0
$256$	1.00000	0.0625000
$257$	0.890250	0.0555323	0.0277661	−	0.999614i	$-0.491161\pi$
$257$	0.890250	0.0555323	0.0277661	+	0.999614i	$0.491161\pi$
$258$	0	0
$259$	−4.93762	−0.306808
$260$	13.2582	0.822238
$261$	0	0
$262$	−19.8675	−1.22742
$263$	−7.74170	−0.477373	−0.238687	−	0.971097i	$-0.576717\pi$
$263$	−7.74170	−0.477373	−0.238687	+	0.971097i	$0.576717\pi$
$264$	0	0
$265$	−21.8619	−1.34296
$266$	−2.14937	−0.131787
$267$	0	0
$268$	−12.9100	−0.788605
$269$	27.9359	1.70328	0.851640	−	0.524127i	$-0.175608\pi$
$269$	27.9359	1.70328	0.851640	+	0.524127i	$0.175608\pi$
$270$	0	0
$271$	0.123443	0.00749865	0.00374932	−	0.999993i	$-0.498807\pi$
$271$	0.123443	0.00749865	0.00374932	+	0.999993i	$0.498807\pi$
$272$	0.322021	0.0195254
$273$	0	0
$274$	18.2376	1.10177
$275$	−5.92048	−0.357018
$276$	0	0
$277$	25.7507	1.54721	0.773606	−	0.633667i	$-0.218451\pi$
$277$	25.7507	1.54721	0.773606	+	0.633667i	$0.218451\pi$
$278$	−3.79729	−0.227746
$279$	0	0
$280$	−1.64252	−0.0981593
$281$	8.28626	0.494317	0.247158	−	0.968975i	$-0.420503\pi$
$281$	8.28626	0.494317	0.247158	+	0.968975i	$0.420503\pi$
$282$	0	0
$283$	−18.3897	−1.09316	−0.546578	−	0.837408i	$-0.684070\pi$
$283$	−18.3897	−1.09316	−0.546578	+	0.837408i	$0.684070\pi$
$284$	0.0667081	0.00395840
$285$	0	0
$286$	18.0023	1.06450
$287$	−6.83351	−0.403369
$288$	0	0
$289$	−16.8963	−0.993900
$290$	6.37515	0.374362
$291$	0	0
$292$	−2.81224	−0.164574
$293$	3.18472	0.186053	0.0930267	−	0.995664i	$-0.470346\pi$
$293$	3.18472	0.186053	0.0930267	+	0.995664i	$0.470346\pi$
$294$	0	0
$295$	−30.8726	−1.79747
$296$	7.77317	0.451806
$297$	0	0
$298$	6.47248	0.374940
$299$	−19.3656	−1.11994
$300$	0	0
$301$	−6.32726	−0.364697
$302$	20.4636	1.17755
$303$	0	0
$304$	3.38371	0.194069
$305$	1.51756	0.0868955
$306$	0	0
$307$	−1.54810	−0.0883547	−0.0441774	−	0.999024i	$-0.514067\pi$
$307$	−1.54810	−0.0883547	−0.0441774	+	0.999024i	$0.514067\pi$
$308$	−2.23026	−0.127081
$309$	0	0
$310$	−5.13977	−0.291919
$311$	0.471497	0.0267361	0.0133681	−	0.999911i	$-0.495745\pi$
$311$	0.471497	0.0267361	0.0133681	+	0.999911i	$0.495745\pi$
$312$	0	0
$313$	−12.1078	−0.684373	−0.342187	−	0.939632i	$-0.611168\pi$
$313$	−12.1078	−0.684373	−0.342187	+	0.939632i	$0.611168\pi$
$314$	14.4827	0.817309
$315$	0	0
$316$	−4.39323	−0.247139
$317$	−1.31831	−0.0740439	−0.0370219	−	0.999314i	$-0.511787\pi$
$317$	−1.31831	−0.0740439	−0.0370219	+	0.999314i	$0.511787\pi$
$318$	0	0
$319$	8.65636	0.484663
$320$	2.58578	0.144549
$321$	0	0
$322$	2.39915	0.133699
$323$	1.08963	0.0606284
$324$	0	0
$325$	8.64598	0.479592
$326$	3.01751	0.167124
$327$	0	0
$328$	10.7578	0.594002
$329$	2.49749	0.137691
$330$	0	0
$331$	30.5415	1.67871	0.839357	−	0.543580i	$-0.182931\pi$
$331$	30.5415	1.67871	0.839357	+	0.543580i	$0.182931\pi$
$332$	16.7077	0.916955
$333$	0	0
$334$	−21.4972	−1.17628
$335$	−33.3825	−1.82388
$336$	0	0
$337$	−20.9083	−1.13895	−0.569475	−	0.822009i	$-0.692853\pi$
$337$	−20.9083	−1.13895	−0.569475	+	0.822009i	$0.692853\pi$
$338$	−13.2897	−0.722865
$339$	0	0
$340$	0.832675	0.0451582
$341$	−6.97892	−0.377930
$342$	0	0
$343$	−8.63667	−0.466336
$344$	9.96085	0.537053
$345$	0	0
$346$	−9.82213	−0.528041
$347$	12.4915	0.670577	0.335288	−	0.942116i	$-0.391166\pi$
$347$	12.4915	0.670577	0.335288	+	0.942116i	$0.391166\pi$
$348$	0	0
$349$	−22.3844	−1.19821	−0.599106	−	0.800670i	$-0.704477\pi$
$349$	−22.3844	−1.19821	−0.599106	+	0.800670i	$0.704477\pi$
$350$	−1.07113	−0.0572540
$351$	0	0
$352$	3.51104	0.187139
$353$	2.03469	0.108296	0.0541478	−	0.998533i	$-0.482756\pi$
$353$	2.03469	0.108296	0.0541478	+	0.998533i	$0.482756\pi$
$354$	0	0
$355$	0.172492	0.00915494
$356$	13.6156	0.721623
$357$	0	0
$358$	6.49532	0.343288
$359$	13.8523	0.731098	0.365549	−	0.930792i	$-0.380881\pi$
$359$	13.8523	0.731098	0.365549	+	0.930792i	$0.380881\pi$
$360$	0	0
$361$	−7.55053	−0.397396
$362$	−4.08174	−0.214531
$363$	0	0
$364$	3.25696	0.170711
$365$	−7.27183	−0.380625
$366$	0	0
$367$	17.9620	0.937608	0.468804	−	0.883302i	$-0.344685\pi$
$367$	17.9620	0.937608	0.468804	+	0.883302i	$0.344685\pi$
$368$	−3.77692	−0.196886
$369$	0	0
$370$	20.0997	1.04493
$371$	−5.37051	−0.278823
$372$	0	0
$373$	−2.63034	−0.136194	−0.0680969	−	0.997679i	$-0.521693\pi$
$373$	−2.63034	−0.136194	−0.0680969	+	0.997679i	$0.521693\pi$
$374$	1.13063	0.0584635
$375$	0	0
$376$	−3.93174	−0.202764
$377$	−12.6413	−0.651061
$378$	0	0
$379$	34.5832	1.77642	0.888209	−	0.459439i	$-0.151949\pi$
$379$	34.5832	1.77642	0.888209	+	0.459439i	$0.151949\pi$
$380$	8.74951	0.448841
$381$	0	0
$382$	23.4882	1.20176
$383$	1.34423	0.0686872	0.0343436	−	0.999410i	$-0.489066\pi$
$383$	1.34423	0.0686872	0.0343436	+	0.999410i	$0.489066\pi$
$384$	0	0
$385$	−5.76695	−0.293911
$386$	−9.86940	−0.502339
$387$	0	0
$388$	−12.7052	−0.645008
$389$	−10.2656	−0.520485	−0.260242	−	0.965543i	$-0.583802\pi$
$389$	−10.2656	−0.520485	−0.260242	+	0.965543i	$0.583802\pi$
$390$	0	0
$391$	−1.21625	−0.0615084
$392$	6.59651	0.333174
$393$	0	0
$394$	−1.66586	−0.0839246
$395$	−11.3599	−0.571580
$396$	0	0
$397$	−33.8549	−1.69913	−0.849564	−	0.527486i	$-0.823135\pi$
$397$	−33.8549	−1.69913	−0.849564	+	0.527486i	$0.823135\pi$
$398$	−12.7759	−0.640400
$399$	0	0
$400$	1.68625	0.0843123
$401$	10.4030	0.519499	0.259750	−	0.965676i	$-0.416360\pi$
$401$	10.4030	0.519499	0.259750	+	0.965676i	$0.416360\pi$
$402$	0	0
$403$	10.1917	0.507683
$404$	−9.15646	−0.455551
$405$	0	0
$406$	1.56610	0.0777241
$407$	27.2919	1.35281
$408$	0	0
$409$	15.9724	0.789787	0.394893	−	0.918727i	$-0.370782\pi$
$409$	15.9724	0.789787	0.394893	+	0.918727i	$0.370782\pi$
$410$	27.8174	1.37380
$411$	0	0
$412$	1.18800	0.0585285
$413$	−7.58404	−0.373186
$414$	0	0
$415$	43.2024	2.12072
$416$	−5.12735	−0.251389
$417$	0	0
$418$	11.8803	0.581086
$419$	1.14327	0.0558524	0.0279262	−	0.999610i	$-0.491110\pi$
$419$	1.14327	0.0558524	0.0279262	+	0.999610i	$0.491110\pi$
$420$	0	0
$421$	15.0478	0.733386	0.366693	−	0.930342i	$-0.380490\pi$
$421$	15.0478	0.733386	0.366693	+	0.930342i	$0.380490\pi$
$422$	−16.3237	−0.794626
$423$	0	0
$424$	8.45466	0.410595
$425$	0.543007	0.0263397
$426$	0	0
$427$	0.372799	0.0180410
$428$	10.5335	0.509155
$429$	0	0
$430$	25.7565	1.24209
$431$	−26.5403	−1.27840	−0.639201	−	0.769040i	$-0.720735\pi$
$431$	−26.5403	−1.27840	−0.639201	+	0.769040i	$0.720735\pi$
$432$	0	0
$433$	22.3094	1.07212	0.536061	−	0.844179i	$-0.319912\pi$
$433$	22.3094	1.07212	0.536061	+	0.844179i	$0.319912\pi$
$434$	−1.26262	−0.0606075
$435$	0	0
$436$	−7.51021	−0.359674
$437$	−12.7800	−0.611351
$438$	0	0
$439$	6.48397	0.309463	0.154732	−	0.987957i	$-0.450549\pi$
$439$	6.48397	0.309463	0.154732	+	0.987957i	$0.450549\pi$
$440$	9.07877	0.432813
$441$	0	0
$442$	−1.65111	−0.0785355
$443$	−12.0749	−0.573694	−0.286847	−	0.957976i	$-0.592607\pi$
$443$	−12.0749	−0.573694	−0.286847	+	0.957976i	$0.592607\pi$
$444$	0	0
$445$	35.2068	1.66896
$446$	−8.02490	−0.379990
$447$	0	0
$448$	0.635213	0.0300110
$449$	11.4624	0.540943	0.270471	−	0.962728i	$-0.412820\pi$
$449$	11.4624	0.540943	0.270471	+	0.962728i	$0.412820\pi$
$450$	0	0
$451$	37.7712	1.77857
$452$	−14.2153	−0.668632
$453$	0	0
$454$	21.8320	1.02463
$455$	8.42177	0.394818
$456$	0	0
$457$	1.55079	0.0725431	0.0362715	−	0.999342i	$-0.488452\pi$
$457$	1.55079	0.0725431	0.0362715	+	0.999342i	$0.488452\pi$
$458$	−13.0772	−0.611056
$459$	0	0
$460$	−9.76629	−0.455355
$461$	−21.0083	−0.978454	−0.489227	−	0.872157i	$-0.662721\pi$
$461$	−21.0083	−0.978454	−0.489227	+	0.872157i	$0.662721\pi$
$462$	0	0
$463$	−7.65014	−0.355532	−0.177766	−	0.984073i	$-0.556887\pi$
$463$	−7.65014	−0.355532	−0.177766	+	0.984073i	$0.556887\pi$
$464$	−2.46547	−0.114457
$465$	0	0
$466$	−5.66551	−0.262450
$467$	23.9495	1.10825	0.554124	−	0.832434i	$-0.313053\pi$
$467$	23.9495	1.10825	0.554124	+	0.832434i	$0.313053\pi$
$468$	0	0
$469$	−8.20061	−0.378669
$470$	−10.1666	−0.468950
$471$	0	0
$472$	11.9394	0.549554
$473$	34.9729	1.60806
$474$	0	0
$475$	5.70577	0.261798
$476$	0.204552	0.00937562
$477$	0	0
$478$	−3.87186	−0.177095
$479$	7.57712	0.346207	0.173104	−	0.984904i	$-0.444620\pi$
$479$	7.57712	0.346207	0.173104	+	0.984904i	$0.444620\pi$
$480$	0	0
$481$	−39.8558	−1.81727
$482$	18.6798	0.850841
$483$	0	0
$484$	1.32741	0.0603366
$485$	−32.8528	−1.49177
$486$	0	0
$487$	9.92582	0.449782	0.224891	−	0.974384i	$-0.427797\pi$
$487$	9.92582	0.449782	0.224891	+	0.974384i	$0.427797\pi$
$488$	−0.586889	−0.0265672
$489$	0	0
$490$	17.0571	0.770561
$491$	−2.77590	−0.125275	−0.0626373	−	0.998036i	$-0.519951\pi$
$491$	−2.77590	−0.125275	−0.0626373	+	0.998036i	$0.519951\pi$
$492$	0	0
$493$	−0.793933	−0.0357570
$494$	−17.3494	−0.780589
$495$	0	0
$496$	1.98771	0.0892506
$497$	0.0423738	0.00190072
$498$	0	0
$499$	−1.66429	−0.0745039	−0.0372519	−	0.999306i	$-0.511860\pi$
$499$	−1.66429	−0.0745039	−0.0372519	+	0.999306i	$0.511860\pi$
$500$	−8.56863	−0.383201
$501$	0	0
$502$	−18.0304	−0.804738
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−23.6766	−1.05359
$506$	−13.2609	−0.589520
$507$	0	0
$508$	20.3451	0.902667
$509$	−24.0760	−1.06715	−0.533574	−	0.845753i	$-0.679152\pi$
$509$	−24.0760	−1.06715	−0.533574	+	0.845753i	$0.679152\pi$
$510$	0	0
$511$	−1.78637	−0.0790243
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−0.890250	−0.0392672
$515$	3.07190	0.135364
$516$	0	0
$517$	−13.8045	−0.607121
$518$	4.93762	0.216946
$519$	0	0
$520$	−13.2582	−0.581410
$521$	21.4959	0.941752	0.470876	−	0.882199i	$-0.343938\pi$
$521$	21.4959	0.941752	0.470876	+	0.882199i	$0.343938\pi$
$522$	0	0
$523$	−1.97588	−0.0863991	−0.0431995	−	0.999066i	$-0.513755\pi$
$523$	−1.97588	−0.0863991	−0.0431995	+	0.999066i	$0.513755\pi$
$524$	19.8675	0.867917
$525$	0	0
$526$	7.74170	0.337554
$527$	0.640084	0.0278825
$528$	0	0
$529$	−8.73484	−0.379776
$530$	21.8619	0.949619
$531$	0	0
$532$	2.14937	0.0931871
$533$	−55.1591	−2.38921
$534$	0	0
$535$	27.2372	1.17757
$536$	12.9100	0.557628
$537$	0	0
$538$	−27.9359	−1.20440
$539$	23.1606	0.997598
$540$	0	0
$541$	24.5700	1.05635	0.528173	−	0.849137i	$-0.322877\pi$
$541$	24.5700	1.05635	0.528173	+	0.849137i	$0.322877\pi$
$542$	−0.123443	−0.00530234
$543$	0	0
$544$	−0.322021	−0.0138065
$545$	−19.4197	−0.831850
$546$	0	0
$547$	28.0251	1.19827	0.599133	−	0.800649i	$-0.295512\pi$
$547$	28.0251	1.19827	0.599133	+	0.800649i	$0.295512\pi$
$548$	−18.2376	−0.779072
$549$	0	0
$550$	5.92048	0.252450
$551$	−8.34242	−0.355399
$552$	0	0
$553$	−2.79064	−0.118670
$554$	−25.7507	−1.09404
$555$	0	0
$556$	3.79729	0.161041
$557$	−19.6663	−0.833288	−0.416644	−	0.909070i	$-0.636794\pi$
$557$	−19.6663	−0.833288	−0.416644	+	0.909070i	$0.636794\pi$
$558$	0	0
$559$	−51.0727	−2.16015
$560$	1.64252	0.0694091
$561$	0	0
$562$	−8.28626	−0.349535
$563$	−28.9537	−1.22025	−0.610126	−	0.792304i	$-0.708881\pi$
$563$	−28.9537	−1.22025	−0.610126	+	0.792304i	$0.708881\pi$
$564$	0	0
$565$	−36.7576	−1.54640
$566$	18.3897	0.772978
$567$	0	0
$568$	−0.0667081	−0.00279901
$569$	−13.9319	−0.584054	−0.292027	−	0.956410i	$-0.594330\pi$
$569$	−13.9319	−0.584054	−0.292027	+	0.956410i	$0.594330\pi$
$570$	0	0
$571$	1.05250	0.0440457	0.0220228	−	0.999757i	$-0.492989\pi$
$571$	1.05250	0.0440457	0.0220228	+	0.999757i	$0.492989\pi$
$572$	−18.0023	−0.752715
$573$	0	0
$574$	6.83351	0.285225
$575$	−6.36883	−0.265598
$576$	0	0
$577$	33.3043	1.38648	0.693239	−	0.720708i	$-0.256183\pi$
$577$	33.3043	1.38648	0.693239	+	0.720708i	$0.256183\pi$
$578$	16.8963	0.702794
$579$	0	0
$580$	−6.37515	−0.264714
$581$	10.6129	0.440299
$582$	0	0
$583$	29.6847	1.22941
$584$	2.81224	0.116371
$585$	0	0
$586$	−3.18472	−0.131560
$587$	−9.86829	−0.407308	−0.203654	−	0.979043i	$-0.565282\pi$
$587$	−9.86829	−0.407308	−0.203654	+	0.979043i	$0.565282\pi$
$588$	0	0
$589$	6.72582	0.277132
$590$	30.8726	1.27100
$591$	0	0
$592$	−7.77317	−0.319475
$593$	19.3262	0.793632	0.396816	−	0.917898i	$-0.370115\pi$
$593$	19.3262	0.793632	0.396816	+	0.917898i	$0.370115\pi$
$594$	0	0
$595$	0.528926	0.0216838
$596$	−6.47248	−0.265123
$597$	0	0
$598$	19.3656	0.791919
$599$	−25.9927	−1.06203	−0.531016	−	0.847361i	$-0.678190\pi$
$599$	−25.9927	−1.06203	−0.531016	+	0.847361i	$0.678190\pi$
$600$	0	0
$601$	21.0730	0.859586	0.429793	−	0.902927i	$-0.358586\pi$
$601$	21.0730	0.859586	0.429793	+	0.902927i	$0.358586\pi$
$602$	6.32726	0.257880
$603$	0	0
$604$	−20.4636	−0.832652
$605$	3.43238	0.139546
$606$	0	0
$607$	23.2863	0.945164	0.472582	−	0.881287i	$-0.343322\pi$
$607$	23.2863	0.945164	0.472582	+	0.881287i	$0.343322\pi$
$608$	−3.38371	−0.137227
$609$	0	0
$610$	−1.51756	−0.0614444
$611$	20.1594	0.815562
$612$	0	0
$613$	17.9773	0.726096	0.363048	−	0.931770i	$-0.381736\pi$
$613$	17.9773	0.726096	0.363048	+	0.931770i	$0.381736\pi$
$614$	1.54810	0.0624762
$615$	0	0
$616$	2.23026	0.0898596
$617$	12.6311	0.508511	0.254255	−	0.967137i	$-0.418170\pi$
$617$	12.6311	0.508511	0.254255	+	0.967137i	$0.418170\pi$
$618$	0	0
$619$	14.9543	0.601064	0.300532	−	0.953772i	$-0.402836\pi$
$619$	14.9543	0.601064	0.300532	+	0.953772i	$0.402836\pi$
$620$	5.13977	0.206418
$621$	0	0
$622$	−0.471497	−0.0189053
$623$	8.64877	0.346506
$624$	0	0
$625$	−30.5878	−1.22351
$626$	12.1078	0.483925
$627$	0	0
$628$	−14.4827	−0.577924
$629$	−2.50313	−0.0998061
$630$	0	0
$631$	−19.1430	−0.762071	−0.381035	−	0.924561i	$-0.624432\pi$
$631$	−19.1430	−0.762071	−0.381035	+	0.924561i	$0.624432\pi$
$632$	4.39323	0.174753
$633$	0	0
$634$	1.31831	0.0523569
$635$	52.6078	2.08768
$636$	0	0
$637$	−33.8226	−1.34010
$638$	−8.65636	−0.342709
$639$	0	0
$640$	−2.58578	−0.102212
$641$	−12.4900	−0.493326	−0.246663	−	0.969101i	$-0.579334\pi$
$641$	−12.4900	−0.493326	−0.246663	+	0.969101i	$0.579334\pi$
$642$	0	0
$643$	−10.7889	−0.425473	−0.212737	−	0.977110i	$-0.568238\pi$
$643$	−10.7889	−0.425473	−0.212737	+	0.977110i	$0.568238\pi$
$644$	−2.39915	−0.0945397
$645$	0	0
$646$	−1.08963	−0.0428707
$647$	37.4908	1.47392	0.736958	−	0.675939i	$-0.236262\pi$
$647$	37.4908	1.47392	0.736958	+	0.675939i	$0.236262\pi$
$648$	0	0
$649$	41.9196	1.64549
$650$	−8.64598	−0.339123
$651$	0	0
$652$	−3.01751	−0.118175
$653$	−41.5290	−1.62516	−0.812578	−	0.582853i	$-0.801936\pi$
$653$	−41.5290	−1.62516	−0.812578	+	0.582853i	$0.801936\pi$
$654$	0	0
$655$	51.3730	2.00731
$656$	−10.7578	−0.420023
$657$	0	0
$658$	−2.49749	−0.0973623
$659$	17.9805	0.700422	0.350211	−	0.936671i	$-0.386110\pi$
$659$	17.9805	0.700422	0.350211	+	0.936671i	$0.386110\pi$
$660$	0	0
$661$	−15.5956	−0.606599	−0.303300	−	0.952895i	$-0.598088\pi$
$661$	−15.5956	−0.606599	−0.303300	+	0.952895i	$0.598088\pi$
$662$	−30.5415	−1.18703
$663$	0	0
$664$	−16.7077	−0.648385
$665$	5.55780	0.215522
$666$	0	0
$667$	9.31189	0.360558
$668$	21.4972	0.831753
$669$	0	0
$670$	33.3825	1.28968
$671$	−2.06059	−0.0795482
$672$	0	0
$673$	−44.7748	−1.72594	−0.862972	−	0.505253i	$-0.831399\pi$
$673$	−44.7748	−1.72594	−0.862972	+	0.505253i	$0.831399\pi$
$674$	20.9083	0.805359
$675$	0	0
$676$	13.2897	0.511143
$677$	4.40068	0.169132	0.0845659	−	0.996418i	$-0.473050\pi$
$677$	4.40068	0.169132	0.0845659	+	0.996418i	$0.473050\pi$
$678$	0	0
$679$	−8.07049	−0.309717
$680$	−0.832675	−0.0319316
$681$	0	0
$682$	6.97892	0.267237
$683$	−47.9219	−1.83368	−0.916840	−	0.399255i	$-0.869269\pi$
$683$	−47.9219	−1.83368	−0.916840	+	0.399255i	$0.869269\pi$
$684$	0	0
$685$	−47.1584	−1.80183
$686$	8.63667	0.329750
$687$	0	0
$688$	−9.96085	−0.379754
$689$	−43.3500	−1.65150
$690$	0	0
$691$	3.83809	0.146008	0.0730039	−	0.997332i	$-0.476741\pi$
$691$	3.83809	0.146008	0.0730039	+	0.997332i	$0.476741\pi$
$692$	9.82213	0.373381
$693$	0	0
$694$	−12.4915	−0.474169
$695$	9.81895	0.372454
$696$	0	0
$697$	−3.46425	−0.131218
$698$	22.3844	0.847264
$699$	0	0
$700$	1.07113	0.0404847
$701$	32.9235	1.24351	0.621753	−	0.783214i	$-0.286421\pi$
$701$	32.9235	1.24351	0.621753	+	0.783214i	$0.286421\pi$
$702$	0	0
$703$	−26.3021	−0.992004
$704$	−3.51104	−0.132327
$705$	0	0
$706$	−2.03469	−0.0765766
$707$	−5.81630	−0.218744
$708$	0	0
$709$	5.77126	0.216744	0.108372	−	0.994110i	$-0.465436\pi$
$709$	5.77126	0.216744	0.108372	+	0.994110i	$0.465436\pi$
$710$	−0.172492	−0.00647352
$711$	0	0
$712$	−13.6156	−0.510265
$713$	−7.50742	−0.281155
$714$	0	0
$715$	−46.5500	−1.74087
$716$	−6.49532	−0.242742
$717$	0	0
$718$	−13.8523	−0.516964
$719$	12.1263	0.452234	0.226117	−	0.974100i	$-0.427397\pi$
$719$	12.1263	0.452234	0.226117	+	0.974100i	$0.427397\pi$
$720$	0	0
$721$	0.754632	0.0281040
$722$	7.55053	0.281001
$723$	0	0
$724$	4.08174	0.151696
$725$	−4.15739	−0.154402
$726$	0	0
$727$	1.41480	0.0524720	0.0262360	−	0.999656i	$-0.491648\pi$
$727$	1.41480	0.0524720	0.0262360	+	0.999656i	$0.491648\pi$
$728$	−3.25696	−0.120711
$729$	0	0
$730$	7.27183	0.269142
$731$	−3.20760	−0.118638
$732$	0	0
$733$	−45.8355	−1.69297	−0.846486	−	0.532411i	$-0.821286\pi$
$733$	−45.8355	−1.69297	−0.846486	+	0.532411i	$0.821286\pi$
$734$	−17.9620	−0.662989
$735$	0	0
$736$	3.77692	0.139219
$737$	45.3276	1.66966
$738$	0	0
$739$	24.1465	0.888245	0.444123	−	0.895966i	$-0.353515\pi$
$739$	24.1465	0.888245	0.444123	+	0.895966i	$0.353515\pi$
$740$	−20.0997	−0.738879
$741$	0	0
$742$	5.37051	0.197158
$743$	2.04962	0.0751931	0.0375966	−	0.999293i	$-0.488030\pi$
$743$	2.04962	0.0751931	0.0375966	+	0.999293i	$0.488030\pi$
$744$	0	0
$745$	−16.7364	−0.613174
$746$	2.63034	0.0963036
$747$	0	0
$748$	−1.13063	−0.0413399
$749$	6.69100	0.244484
$750$	0	0
$751$	−4.28315	−0.156294	−0.0781471	−	0.996942i	$-0.524900\pi$
$751$	−4.28315	−0.156294	−0.0781471	+	0.996942i	$0.524900\pi$
$752$	3.93174	0.143376
$753$	0	0
$754$	12.6413	0.460370
$755$	−52.9143	−1.92575
$756$	0	0
$757$	−37.8734	−1.37653	−0.688266	−	0.725458i	$-0.741628\pi$
$757$	−37.8734	−1.37653	−0.688266	+	0.725458i	$0.741628\pi$
$758$	−34.5832	−1.25612
$759$	0	0
$760$	−8.74951	−0.317378
$761$	34.5346	1.25188	0.625940	−	0.779872i	$-0.284716\pi$
$761$	34.5346	1.25188	0.625940	+	0.779872i	$0.284716\pi$
$762$	0	0
$763$	−4.77058	−0.172706
$764$	−23.4882	−0.849773
$765$	0	0
$766$	−1.34423	−0.0485692
$767$	−61.2173	−2.21043
$768$	0	0
$769$	−45.2000	−1.62995	−0.814977	−	0.579493i	$-0.803251\pi$
$769$	−45.2000	−1.62995	−0.814977	+	0.579493i	$0.803251\pi$
$770$	5.76695	0.207826
$771$	0	0
$772$	9.86940	0.355207
$773$	−3.76139	−0.135288	−0.0676439	−	0.997710i	$-0.521548\pi$
$773$	−3.76139	−0.135288	−0.0676439	+	0.997710i	$0.521548\pi$
$774$	0	0
$775$	3.35176	0.120399
$776$	12.7052	0.456089
$777$	0	0
$778$	10.2656	0.368038
$779$	−36.4013	−1.30421
$780$	0	0
$781$	−0.234215	−0.00838086
$782$	1.21625	0.0434930
$783$	0	0
$784$	−6.59651	−0.235589
$785$	−37.4492	−1.33662
$786$	0	0
$787$	−10.7287	−0.382438	−0.191219	−	0.981547i	$-0.561244\pi$
$787$	−10.7287	−0.382438	−0.191219	+	0.981547i	$0.561244\pi$
$788$	1.66586	0.0593437
$789$	0	0
$790$	11.3599	0.404168
$791$	−9.02974	−0.321061
$792$	0	0
$793$	3.00918	0.106859
$794$	33.8549	1.20146
$795$	0	0
$796$	12.7759	0.452831
$797$	0.766076	0.0271358	0.0135679	−	0.999908i	$-0.495681\pi$
$797$	0.766076	0.0271358	0.0135679	+	0.999908i	$0.495681\pi$
$798$	0	0
$799$	1.26610	0.0447915
$800$	−1.68625	−0.0596178
$801$	0	0
$802$	−10.4030	−0.367342
$803$	9.87389	0.348442
$804$	0	0
$805$	−6.20367	−0.218651
$806$	−10.1917	−0.358986
$807$	0	0
$808$	9.15646	0.322123
$809$	−30.0576	−1.05677	−0.528384	−	0.849005i	$-0.677202\pi$
$809$	−30.0576	−1.05677	−0.528384	+	0.849005i	$0.677202\pi$
$810$	0	0
$811$	−13.7785	−0.483829	−0.241914	−	0.970298i	$-0.577775\pi$
$811$	−13.7785	−0.483829	−0.241914	+	0.970298i	$0.577775\pi$
$812$	−1.56610	−0.0549592
$813$	0	0
$814$	−27.2919	−0.956581
$815$	−7.80260	−0.273313
$816$	0	0
$817$	−33.7046	−1.17917
$818$	−15.9724	−0.558463
$819$	0	0
$820$	−27.8174	−0.971424
$821$	−21.8806	−0.763639	−0.381819	−	0.924237i	$-0.624702\pi$
$821$	−21.8806	−0.763639	−0.381819	+	0.924237i	$0.624702\pi$
$822$	0	0
$823$	38.1119	1.32850	0.664249	−	0.747511i	$-0.268751\pi$
$823$	38.1119	1.32850	0.664249	+	0.747511i	$0.268751\pi$
$824$	−1.18800	−0.0413859
$825$	0	0
$826$	7.58404	0.263882
$827$	12.0487	0.418975	0.209488	−	0.977811i	$-0.432820\pi$
$827$	12.0487	0.418975	0.209488	+	0.977811i	$0.432820\pi$
$828$	0	0
$829$	−5.44538	−0.189126	−0.0945629	−	0.995519i	$-0.530145\pi$
$829$	−5.44538	−0.189126	−0.0945629	+	0.995519i	$0.530145\pi$
$830$	−43.2024	−1.49958
$831$	0	0
$832$	5.12735	0.177759
$833$	−2.12421	−0.0735997
$834$	0	0
$835$	55.5871	1.92367
$836$	−11.8803	−0.410890
$837$	0	0
$838$	−1.14327	−0.0394936
$839$	−40.8623	−1.41072	−0.705361	−	0.708848i	$-0.749215\pi$
$839$	−40.8623	−1.41072	−0.705361	+	0.708848i	$0.749215\pi$
$840$	0	0
$841$	−22.9215	−0.790395
$842$	−15.0478	−0.518582
$843$	0	0
$844$	16.3237	0.561885
$845$	34.3642	1.18217
$846$	0	0
$847$	0.843185	0.0289722
$848$	−8.45466	−0.290334
$849$	0	0
$850$	−0.543007	−0.0186250
$851$	29.3587	1.00640
$852$	0	0
$853$	33.6213	1.15117	0.575586	−	0.817741i	$-0.304774\pi$
$853$	33.6213	1.15117	0.575586	+	0.817741i	$0.304774\pi$
$854$	−0.372799	−0.0127569
$855$	0	0
$856$	−10.5335	−0.360027
$857$	38.0197	1.29873	0.649365	−	0.760477i	$-0.275035\pi$
$857$	38.0197	1.29873	0.649365	+	0.760477i	$0.275035\pi$
$858$	0	0
$859$	−17.6508	−0.602237	−0.301118	−	0.953587i	$-0.597360\pi$
$859$	−17.6508	−0.602237	−0.301118	+	0.953587i	$0.597360\pi$
$860$	−25.7565	−0.878291
$861$	0	0
$862$	26.5403	0.903966
$863$	−21.0000	−0.714850	−0.357425	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714850	−0.357425	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	25.3978	0.863553
$866$	−22.3094	−0.758105
$867$	0	0
$868$	1.26262	0.0428560
$869$	15.4248	0.523251
$870$	0	0
$871$	−66.1942	−2.24290
$872$	7.51021	0.254328
$873$	0	0
$874$	12.7800	0.432290
$875$	−5.44290	−0.184004
$876$	0	0
$877$	13.3184	0.449731	0.224865	−	0.974390i	$-0.427806\pi$
$877$	13.3184	0.449731	0.224865	+	0.974390i	$0.427806\pi$
$878$	−6.48397	−0.218823
$879$	0	0
$880$	−9.07877	−0.306045
$881$	23.1252	0.779108	0.389554	−	0.921004i	$-0.372629\pi$
$881$	23.1252	0.779108	0.389554	+	0.921004i	$0.372629\pi$
$882$	0	0
$883$	−50.9512	−1.71465	−0.857323	−	0.514780i	$-0.827874\pi$
$883$	−50.9512	−1.71465	−0.857323	+	0.514780i	$0.827874\pi$
$884$	1.65111	0.0555330
$885$	0	0
$886$	12.0749	0.405663
$887$	−24.3607	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$887$	−24.3607	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$888$	0	0
$889$	12.9234	0.433438
$890$	−35.2068	−1.18013
$891$	0	0
$892$	8.02490	0.268694
$893$	13.3039	0.445196
$894$	0	0
$895$	−16.7955	−0.561410
$896$	−0.635213	−0.0212210
$897$	0	0
$898$	−11.4624	−0.382504
$899$	−4.90063	−0.163445
$900$	0	0
$901$	−2.72258	−0.0907023
$902$	−37.7712	−1.25764
$903$	0	0
$904$	14.2153	0.472794
$905$	10.5545	0.350842
$906$	0	0
$907$	−30.6192	−1.01669	−0.508347	−	0.861153i	$-0.669743\pi$
$907$	−30.6192	−1.01669	−0.508347	+	0.861153i	$0.669743\pi$
$908$	−21.8320	−0.724520
$909$	0	0
$910$	−8.42177	−0.279179
$911$	52.5031	1.73951	0.869753	−	0.493487i	$-0.164278\pi$
$911$	52.5031	1.73951	0.869753	+	0.493487i	$0.164278\pi$
$912$	0	0
$913$	−58.6614	−1.94141
$914$	−1.55079	−0.0512957
$915$	0	0
$916$	13.0772	0.432082
$917$	12.6201	0.416752
$918$	0	0
$919$	−21.0907	−0.695719	−0.347860	−	0.937547i	$-0.613091\pi$
$919$	−21.0907	−0.695719	−0.347860	+	0.937547i	$0.613091\pi$
$920$	9.76629	0.321985
$921$	0	0
$922$	21.0083	0.691871
$923$	0.342036	0.0112582
$924$	0	0
$925$	−13.1075	−0.430971
$926$	7.65014	0.251399
$927$	0	0
$928$	2.46547	0.0809330
$929$	−36.4744	−1.19668	−0.598342	−	0.801240i	$-0.704174\pi$
$929$	−36.4744	−1.19668	−0.598342	+	0.801240i	$0.704174\pi$
$930$	0	0
$931$	−22.3206	−0.731529
$932$	5.66551	0.185580
$933$	0	0
$934$	−23.9495	−0.783650
$935$	−2.92356	−0.0956105
$936$	0	0
$937$	35.4964	1.15962	0.579809	−	0.814752i	$-0.303127\pi$
$937$	35.4964	1.15962	0.579809	+	0.814752i	$0.303127\pi$
$938$	8.20061	0.267759
$939$	0	0
$940$	10.1666	0.331598
$941$	−10.3782	−0.338321	−0.169160	−	0.985589i	$-0.554106\pi$
$941$	−10.3782	−0.338321	−0.169160	+	0.985589i	$0.554106\pi$
$942$	0	0
$943$	40.6315	1.32314
$944$	−11.9394	−0.388593
$945$	0	0
$946$	−34.9729	−1.13707
$947$	−10.9336	−0.355294	−0.177647	−	0.984094i	$-0.556848\pi$
$947$	−10.9336	−0.355294	−0.177647	+	0.984094i	$0.556848\pi$
$948$	0	0
$949$	−14.4193	−0.468071
$950$	−5.70577	−0.185119
$951$	0	0
$952$	−0.204552	−0.00662956
$953$	46.0986	1.49328	0.746640	−	0.665228i	$-0.231666\pi$
$953$	46.0986	1.49328	0.746640	+	0.665228i	$0.231666\pi$
$954$	0	0
$955$	−60.7353	−1.96535
$956$	3.87186	0.125225
$957$	0	0
$958$	−7.57712	−0.244806
$959$	−11.5848	−0.374092
$960$	0	0
$961$	−27.0490	−0.872549
$962$	39.8558	1.28500
$963$	0	0
$964$	−18.6798	−0.601636
$965$	25.5201	0.821520
$966$	0	0
$967$	27.3411	0.879231	0.439615	−	0.898186i	$-0.355115\pi$
$967$	27.3411	0.879231	0.439615	+	0.898186i	$0.355115\pi$
$968$	−1.32741	−0.0426645
$969$	0	0
$970$	32.8528	1.05484
$971$	−18.5972	−0.596812	−0.298406	−	0.954439i	$-0.596455\pi$
$971$	−18.5972	−0.596812	−0.298406	+	0.954439i	$0.596455\pi$
$972$	0	0
$973$	2.41209	0.0773279
$974$	−9.92582	−0.318044
$975$	0	0
$976$	0.586889	0.0187859
$977$	43.2694	1.38431	0.692155	−	0.721749i	$-0.256661\pi$
$977$	43.2694	1.38431	0.692155	+	0.721749i	$0.256661\pi$
$978$	0	0
$979$	−47.8048	−1.52785
$980$	−17.0571	−0.544869
$981$	0	0
$982$	2.77590	0.0885825
$983$	−54.4357	−1.73623	−0.868115	−	0.496363i	$-0.834669\pi$
$983$	−54.4357	−1.73623	−0.868115	+	0.496363i	$0.834669\pi$
$984$	0	0
$985$	4.30753	0.137249
$986$	0.793933	0.0252840
$987$	0	0
$988$	17.3494	0.551959
$989$	37.6214	1.19629
$990$	0	0
$991$	−36.0277	−1.14446	−0.572229	−	0.820094i	$-0.693921\pi$
$991$	−36.0277	−1.14446	−0.572229	+	0.820094i	$0.693921\pi$
$992$	−1.98771	−0.0631097
$993$	0	0
$994$	−0.0423738	−0.00134402
$995$	33.0358	1.04730
$996$	0	0
$997$	−56.7052	−1.79587	−0.897935	−	0.440128i	$-0.854933\pi$
$997$	−56.7052	−1.79587	−0.897935	+	0.440128i	$0.854933\pi$
$998$	1.66429	0.0526822
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9054.2.a.bi.1.11		12
3.2	odd	2		1006.2.a.j.1.8	✓	12
12.11	even	2		8048.2.a.q.1.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1006.2.a.j.1.8	✓	12	3.2	odd	2
8048.2.a.q.1.5		12	12.11	even	2
9054.2.a.bi.1.11		12	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:	\( N \)	\(=\)	\( 9054 = 2 \cdot 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9054.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.58578 q^{5} +0.635213 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +2.58578 q^{5} +0.635213 q^{7} -1.00000 q^{8} -2.58578 q^{10} -3.51104 q^{11} +5.12735 q^{13} -0.635213 q^{14} +1.00000 q^{16} +0.322021 q^{17} +3.38371 q^{19} +2.58578 q^{20} +3.51104 q^{22} -3.77692 q^{23} +1.68625 q^{25} -5.12735 q^{26} +0.635213 q^{28} -2.46547 q^{29} +1.98771 q^{31} -1.00000 q^{32} -0.322021 q^{34} +1.64252 q^{35} -7.77317 q^{37} -3.38371 q^{38} -2.58578 q^{40} -10.7578 q^{41} -9.96085 q^{43} -3.51104 q^{44} +3.77692 q^{46} +3.93174 q^{47} -6.59651 q^{49} -1.68625 q^{50} +5.12735 q^{52} -8.45466 q^{53} -9.07877 q^{55} -0.635213 q^{56} +2.46547 q^{58} -11.9394 q^{59} +0.586889 q^{61} -1.98771 q^{62} +1.00000 q^{64} +13.2582 q^{65} -12.9100 q^{67} +0.322021 q^{68} -1.64252 q^{70} +0.0667081 q^{71} -2.81224 q^{73} +7.77317 q^{74} +3.38371 q^{76} -2.23026 q^{77} -4.39323 q^{79} +2.58578 q^{80} +10.7578 q^{82} +16.7077 q^{83} +0.832675 q^{85} +9.96085 q^{86} +3.51104 q^{88} +13.6156 q^{89} +3.25696 q^{91} -3.77692 q^{92} -3.93174 q^{94} +8.74951 q^{95} -12.7052 q^{97} +6.59651 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 8 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 8 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 8 q^{7} - 12 q^{8} + 5 q^{10} - 18 q^{11} + 4 q^{13} - 8 q^{14} + 12 q^{16} + 2 q^{17} + 6 q^{19} - 5 q^{20} + 18 q^{22} - 13 q^{23} + q^{25} - 4 q^{26} + 8 q^{28} - 20 q^{29} + 7 q^{31} - 12 q^{32} - 2 q^{34} - q^{35} + 10 q^{37} - 6 q^{38} + 5 q^{40} - 2 q^{41} + 8 q^{43} - 18 q^{44} + 13 q^{46} - 12 q^{47} + 4 q^{49} - q^{50} + 4 q^{52} - 12 q^{53} + 8 q^{55} - 8 q^{56} + 20 q^{58} - 6 q^{59} - 10 q^{61} - 7 q^{62} + 12 q^{64} - 4 q^{65} + 7 q^{67} + 2 q^{68} + q^{70} - 22 q^{71} - 23 q^{73} - 10 q^{74} + 6 q^{76} + 19 q^{77} + 13 q^{79} - 5 q^{80} + 2 q^{82} - q^{83} - 28 q^{85} - 8 q^{86} + 18 q^{88} + 3 q^{89} - 21 q^{91} - 13 q^{92} + 12 q^{94} + 2 q^{95} - 70 q^{97} - 4 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 8 * q^7 - 12 * q^8 + 5 * q^10 - 18 * q^11 + 4 * q^13 - 8 * q^14 + 12 * q^16 + 2 * q^17 + 6 * q^19 - 5 * q^20 + 18 * q^22 - 13 * q^23 + q^25 - 4 * q^26 + 8 * q^28 - 20 * q^29 + 7 * q^31 - 12 * q^32 - 2 * q^34 - q^35 + 10 * q^37 - 6 * q^38 + 5 * q^40 - 2 * q^41 + 8 * q^43 - 18 * q^44 + 13 * q^46 - 12 * q^47 + 4 * q^49 - q^50 + 4 * q^52 - 12 * q^53 + 8 * q^55 - 8 * q^56 + 20 * q^58 - 6 * q^59 - 10 * q^61 - 7 * q^62 + 12 * q^64 - 4 * q^65 + 7 * q^67 + 2 * q^68 + q^70 - 22 * q^71 - 23 * q^73 - 10 * q^74 + 6 * q^76 + 19 * q^77 + 13 * q^79 - 5 * q^80 + 2 * q^82 - q^83 - 28 * q^85 - 8 * q^86 + 18 * q^88 + 3 * q^89 - 21 * q^91 - 13 * q^92 + 12 * q^94 + 2 * q^95 - 70 * q^97 - 4 * q^98

Introduction

L-functions

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