LMFDB - Embedded newform 8046.2.a.h.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.55687$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.71034 q^{5} +1.67819 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.71034 q^{5} +1.67819 q^{7} +1.00000 q^{8} +2.71034 q^{10} -4.37358 q^{11} -2.34775 q^{13} +1.67819 q^{14} +1.00000 q^{16} -4.09598 q^{17} -7.26207 q^{19} +2.71034 q^{20} -4.37358 q^{22} +0.0981648 q^{23} +2.34593 q^{25} -2.34775 q^{26} +1.67819 q^{28} -5.89425 q^{29} -1.88042 q^{31} +1.00000 q^{32} -4.09598 q^{34} +4.54847 q^{35} -3.13510 q^{37} -7.26207 q^{38} +2.71034 q^{40} -0.419725 q^{41} -0.0900380 q^{43} -4.37358 q^{44} +0.0981648 q^{46} -1.12980 q^{47} -4.18367 q^{49} +2.34593 q^{50} -2.34775 q^{52} +12.5105 q^{53} -11.8539 q^{55} +1.67819 q^{56} -5.89425 q^{58} -7.63266 q^{59} +3.77438 q^{61} -1.88042 q^{62} +1.00000 q^{64} -6.36319 q^{65} +3.12705 q^{67} -4.09598 q^{68} +4.54847 q^{70} -3.20185 q^{71} +1.15506 q^{73} -3.13510 q^{74} -7.26207 q^{76} -7.33971 q^{77} -15.5113 q^{79} +2.71034 q^{80} -0.419725 q^{82} -14.9118 q^{83} -11.1015 q^{85} -0.0900380 q^{86} -4.37358 q^{88} +7.11363 q^{89} -3.93997 q^{91} +0.0981648 q^{92} -1.12980 q^{94} -19.6827 q^{95} -4.78642 q^{97} -4.18367 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * 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
$5$	2.71034	1.21210	0.606050	−	0.795426i	$-0.292753\pi$
$5$	2.71034	1.21210	0.606050	+	0.795426i	$0.292753\pi$
$6$	0	0
$7$	1.67819	0.634297	0.317148	−	0.948376i	$-0.397275\pi$
$7$	1.67819	0.634297	0.317148	+	0.948376i	$0.397275\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.71034	0.857084
$11$	−4.37358	−1.31869	−0.659343	−	0.751843i	$-0.729165\pi$
$11$	−4.37358	−1.31869	−0.659343	+	0.751843i	$0.729165\pi$
$12$	0	0
$13$	−2.34775	−0.651148	−0.325574	−	0.945517i	$-0.605557\pi$
$13$	−2.34775	−0.651148	−0.325574	+	0.945517i	$0.605557\pi$
$14$	1.67819	0.448515
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.09598	−0.993420	−0.496710	−	0.867916i	$-0.665459\pi$
$17$	−4.09598	−0.993420	−0.496710	+	0.867916i	$0.665459\pi$
$18$	0	0
$19$	−7.26207	−1.66603	−0.833016	−	0.553249i	$-0.813388\pi$
$19$	−7.26207	−1.66603	−0.833016	+	0.553249i	$0.813388\pi$
$20$	2.71034	0.606050
$21$	0	0
$22$	−4.37358	−0.932451
$23$	0.0981648	0.0204688	0.0102344	−	0.999948i	$-0.496742\pi$
$23$	0.0981648	0.0204688	0.0102344	+	0.999948i	$0.496742\pi$
$24$	0	0
$25$	2.34593	0.469187
$26$	−2.34775	−0.460431
$27$	0	0
$28$	1.67819	0.317148
$29$	−5.89425	−1.09454	−0.547268	−	0.836958i	$-0.684332\pi$
$29$	−5.89425	−1.09454	−0.547268	+	0.836958i	$0.684332\pi$
$30$	0	0
$31$	−1.88042	−0.337734	−0.168867	−	0.985639i	$-0.554011\pi$
$31$	−1.88042	−0.337734	−0.168867	+	0.985639i	$0.554011\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.09598	−0.702454
$35$	4.54847	0.768831
$36$	0	0
$37$	−3.13510	−0.515407	−0.257704	−	0.966224i	$-0.582966\pi$
$37$	−3.13510	−0.515407	−0.257704	+	0.966224i	$0.582966\pi$
$38$	−7.26207	−1.17806
$39$	0	0
$40$	2.71034	0.428542
$41$	−0.419725	−0.0655501	−0.0327750	−	0.999463i	$-0.510434\pi$
$41$	−0.419725	−0.0655501	−0.0327750	+	0.999463i	$0.510434\pi$
$42$	0	0
$43$	−0.0900380	−0.0137307	−0.00686534	−	0.999976i	$-0.502185\pi$
$43$	−0.0900380	−0.0137307	−0.00686534	+	0.999976i	$0.502185\pi$
$44$	−4.37358	−0.659343
$45$	0	0
$46$	0.0981648	0.0144736
$47$	−1.12980	−0.164799	−0.0823993	−	0.996599i	$-0.526258\pi$
$47$	−1.12980	−0.164799	−0.0823993	+	0.996599i	$0.526258\pi$
$48$	0	0
$49$	−4.18367	−0.597668
$50$	2.34593	0.331765
$51$	0	0
$52$	−2.34775	−0.325574
$53$	12.5105	1.71845	0.859227	−	0.511595i	$-0.170945\pi$
$53$	12.5105	1.71845	0.859227	+	0.511595i	$0.170945\pi$
$54$	0	0
$55$	−11.8539	−1.59838
$56$	1.67819	0.224258
$57$	0	0
$58$	−5.89425	−0.773953
$59$	−7.63266	−0.993687	−0.496844	−	0.867840i	$-0.665508\pi$
$59$	−7.63266	−0.993687	−0.496844	+	0.867840i	$0.665508\pi$
$60$	0	0
$61$	3.77438	0.483260	0.241630	−	0.970368i	$-0.422318\pi$
$61$	3.77438	0.483260	0.241630	+	0.970368i	$0.422318\pi$
$62$	−1.88042	−0.238814
$63$	0	0
$64$	1.00000	0.125000
$65$	−6.36319	−0.789256
$66$	0	0
$67$	3.12705	0.382030	0.191015	−	0.981587i	$-0.438822\pi$
$67$	3.12705	0.382030	0.191015	+	0.981587i	$0.438822\pi$
$68$	−4.09598	−0.496710
$69$	0	0
$70$	4.54847	0.543646
$71$	−3.20185	−0.379990	−0.189995	−	0.981785i	$-0.560847\pi$
$71$	−3.20185	−0.379990	−0.189995	+	0.981785i	$0.560847\pi$
$72$	0	0
$73$	1.15506	0.135190	0.0675951	−	0.997713i	$-0.478467\pi$
$73$	1.15506	0.135190	0.0675951	+	0.997713i	$0.478467\pi$
$74$	−3.13510	−0.364448
$75$	0	0
$76$	−7.26207	−0.833016
$77$	−7.33971	−0.836437
$78$	0	0
$79$	−15.5113	−1.74516	−0.872578	−	0.488475i	$-0.837553\pi$
$79$	−15.5113	−1.74516	−0.872578	+	0.488475i	$0.837553\pi$
$80$	2.71034	0.303025
$81$	0	0
$82$	−0.419725	−0.0463509
$83$	−14.9118	−1.63678	−0.818389	−	0.574665i	$-0.805132\pi$
$83$	−14.9118	−1.63678	−0.818389	+	0.574665i	$0.805132\pi$
$84$	0	0
$85$	−11.1015	−1.20412
$86$	−0.0900380	−0.00970905
$87$	0	0
$88$	−4.37358	−0.466226
$89$	7.11363	0.754043	0.377022	−	0.926204i	$-0.376948\pi$
$89$	7.11363	0.754043	0.377022	+	0.926204i	$0.376948\pi$
$90$	0	0
$91$	−3.93997	−0.413021
$92$	0.0981648	0.0102344
$93$	0	0
$94$	−1.12980	−0.116530
$95$	−19.6827	−2.01940
$96$	0	0
$97$	−4.78642	−0.485987	−0.242994	−	0.970028i	$-0.578129\pi$
$97$	−4.78642	−0.485987	−0.242994	+	0.970028i	$0.578129\pi$
$98$	−4.18367	−0.422615
$99$	0	0
$100$	2.34593	0.234593
$101$	−3.25163	−0.323550	−0.161775	−	0.986828i	$-0.551722\pi$
$101$	−3.25163	−0.323550	−0.161775	+	0.986828i	$0.551722\pi$
$102$	0	0
$103$	−6.91855	−0.681705	−0.340852	−	0.940117i	$-0.610716\pi$
$103$	−6.91855	−0.681705	−0.340852	+	0.940117i	$0.610716\pi$
$104$	−2.34775	−0.230216
$105$	0	0
$106$	12.5105	1.21513
$107$	2.61340	0.252647	0.126323	−	0.991989i	$-0.459682\pi$
$107$	2.61340	0.252647	0.126323	+	0.991989i	$0.459682\pi$
$108$	0	0
$109$	−8.13251	−0.778953	−0.389477	−	0.921036i	$-0.627344\pi$
$109$	−8.13251	−0.778953	−0.389477	+	0.921036i	$0.627344\pi$
$110$	−11.8539	−1.13022
$111$	0	0
$112$	1.67819	0.158574
$113$	6.09305	0.573186	0.286593	−	0.958052i	$-0.407477\pi$
$113$	6.09305	0.573186	0.286593	+	0.958052i	$0.407477\pi$
$114$	0	0
$115$	0.266060	0.0248102
$116$	−5.89425	−0.547268
$117$	0	0
$118$	−7.63266	−0.702643
$119$	−6.87383	−0.630123
$120$	0	0
$121$	8.12823	0.738930
$122$	3.77438	0.341717
$123$	0	0
$124$	−1.88042	−0.168867
$125$	−7.19342	−0.643399
$126$	0	0
$127$	3.03062	0.268924	0.134462	−	0.990919i	$-0.457069\pi$
$127$	3.03062	0.268924	0.134462	+	0.990919i	$0.457069\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−6.36319	−0.558089
$131$	22.8099	1.99291	0.996454	−	0.0841386i	$-0.0268138\pi$
$131$	22.8099	1.99291	0.996454	+	0.0841386i	$0.0268138\pi$
$132$	0	0
$133$	−12.1871	−1.05676
$134$	3.12705	0.270136
$135$	0	0
$136$	−4.09598	−0.351227
$137$	12.4026	1.05962	0.529812	−	0.848115i	$-0.322262\pi$
$137$	12.4026	1.05962	0.529812	+	0.848115i	$0.322262\pi$
$138$	0	0
$139$	16.1737	1.37184	0.685918	−	0.727679i	$-0.259401\pi$
$139$	16.1737	1.37184	0.685918	+	0.727679i	$0.259401\pi$
$140$	4.54847	0.384415
$141$	0	0
$142$	−3.20185	−0.268693
$143$	10.2681	0.858659
$144$	0	0
$145$	−15.9754	−1.32669
$146$	1.15506	0.0955938
$147$	0	0
$148$	−3.13510	−0.257704
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	10.5180	0.855946	0.427973	−	0.903791i	$-0.359228\pi$
$151$	10.5180	0.855946	0.427973	+	0.903791i	$0.359228\pi$
$152$	−7.26207	−0.589031
$153$	0	0
$154$	−7.33971	−0.591451
$155$	−5.09658	−0.409367
$156$	0	0
$157$	19.1463	1.52804	0.764020	−	0.645193i	$-0.223223\pi$
$157$	19.1463	1.52804	0.764020	+	0.645193i	$0.223223\pi$
$158$	−15.5113	−1.23401
$159$	0	0
$160$	2.71034	0.214271
$161$	0.164739	0.0129833
$162$	0	0
$163$	−6.53078	−0.511530	−0.255765	−	0.966739i	$-0.582327\pi$
$163$	−6.53078	−0.511530	−0.255765	+	0.966739i	$0.582327\pi$
$164$	−0.419725	−0.0327750
$165$	0	0
$166$	−14.9118	−1.15738
$167$	−11.7228	−0.907134	−0.453567	−	0.891222i	$-0.649849\pi$
$167$	−11.7228	−0.907134	−0.453567	+	0.891222i	$0.649849\pi$
$168$	0	0
$169$	−7.48808	−0.576006
$170$	−11.1015	−0.851445
$171$	0	0
$172$	−0.0900380	−0.00686534
$173$	−1.93624	−0.147210	−0.0736049	−	0.997287i	$-0.523450\pi$
$173$	−1.93624	−0.147210	−0.0736049	+	0.997287i	$0.523450\pi$
$174$	0	0
$175$	3.93692	0.297604
$176$	−4.37358	−0.329671
$177$	0	0
$178$	7.11363	0.533189
$179$	13.5559	1.01321	0.506606	−	0.862178i	$-0.330900\pi$
$179$	13.5559	1.01321	0.506606	+	0.862178i	$0.330900\pi$
$180$	0	0
$181$	14.9431	1.11071	0.555354	−	0.831614i	$-0.312583\pi$
$181$	14.9431	1.11071	0.555354	+	0.831614i	$0.312583\pi$
$182$	−3.93997	−0.292050
$183$	0	0
$184$	0.0981648	0.00723681
$185$	−8.49718	−0.624725
$186$	0	0
$187$	17.9141	1.31001
$188$	−1.12980	−0.0823993
$189$	0	0
$190$	−19.6827	−1.42793
$191$	−3.69535	−0.267386	−0.133693	−	0.991023i	$-0.542684\pi$
$191$	−3.69535	−0.267386	−0.133693	+	0.991023i	$0.542684\pi$
$192$	0	0
$193$	7.65239	0.550831	0.275416	−	0.961325i	$-0.411185\pi$
$193$	7.65239	0.550831	0.275416	+	0.961325i	$0.411185\pi$
$194$	−4.78642	−0.343645
$195$	0	0
$196$	−4.18367	−0.298834
$197$	−18.2992	−1.30376	−0.651882	−	0.758321i	$-0.726020\pi$
$197$	−18.2992	−1.30376	−0.651882	+	0.758321i	$0.726020\pi$
$198$	0	0
$199$	12.9256	0.916269	0.458134	−	0.888883i	$-0.348518\pi$
$199$	12.9256	0.916269	0.458134	+	0.888883i	$0.348518\pi$
$200$	2.34593	0.165883
$201$	0	0
$202$	−3.25163	−0.228784
$203$	−9.89168	−0.694260
$204$	0	0
$205$	−1.13760	−0.0794532
$206$	−6.91855	−0.482038
$207$	0	0
$208$	−2.34775	−0.162787
$209$	31.7613	2.19697
$210$	0	0
$211$	−12.7410	−0.877126	−0.438563	−	0.898700i	$-0.644512\pi$
$211$	−12.7410	−0.877126	−0.438563	+	0.898700i	$0.644512\pi$
$212$	12.5105	0.859227
$213$	0	0
$214$	2.61340	0.178648
$215$	−0.244034	−0.0166429
$216$	0	0
$217$	−3.15571	−0.214223
$218$	−8.13251	−0.550803
$219$	0	0
$220$	−11.8539	−0.799189
$221$	9.61632	0.646864
$222$	0	0
$223$	−10.0497	−0.672980	−0.336490	−	0.941687i	$-0.609240\pi$
$223$	−10.0497	−0.672980	−0.336490	+	0.941687i	$0.609240\pi$
$224$	1.67819	0.112129
$225$	0	0
$226$	6.09305	0.405303
$227$	9.51471	0.631514	0.315757	−	0.948840i	$-0.397742\pi$
$227$	9.51471	0.631514	0.315757	+	0.948840i	$0.397742\pi$
$228$	0	0
$229$	17.3081	1.14375	0.571875	−	0.820341i	$-0.306216\pi$
$229$	17.3081	1.14375	0.571875	+	0.820341i	$0.306216\pi$
$230$	0.266060	0.0175435
$231$	0	0
$232$	−5.89425	−0.386977
$233$	−21.8741	−1.43302	−0.716511	−	0.697576i	$-0.754262\pi$
$233$	−21.8741	−1.43302	−0.716511	+	0.697576i	$0.754262\pi$
$234$	0	0
$235$	−3.06215	−0.199752
$236$	−7.63266	−0.496844
$237$	0	0
$238$	−6.87383	−0.445564
$239$	25.8220	1.67029	0.835143	−	0.550032i	$-0.185385\pi$
$239$	25.8220	1.67029	0.835143	+	0.550032i	$0.185385\pi$
$240$	0	0
$241$	−12.3157	−0.793327	−0.396663	−	0.917964i	$-0.629832\pi$
$241$	−12.3157	−0.793327	−0.396663	+	0.917964i	$0.629832\pi$
$242$	8.12823	0.522503
$243$	0	0
$244$	3.77438	0.241630
$245$	−11.3392	−0.724433
$246$	0	0
$247$	17.0495	1.08483
$248$	−1.88042	−0.119407
$249$	0	0
$250$	−7.19342	−0.454952
$251$	24.1336	1.52330	0.761649	−	0.647990i	$-0.224390\pi$
$251$	24.1336	1.52330	0.761649	+	0.647990i	$0.224390\pi$
$252$	0	0
$253$	−0.429332	−0.0269919
$254$	3.03062	0.190158
$255$	0	0
$256$	1.00000	0.0625000
$257$	−1.94139	−0.121101	−0.0605504	−	0.998165i	$-0.519286\pi$
$257$	−1.94139	−0.121101	−0.0605504	+	0.998165i	$0.519286\pi$
$258$	0	0
$259$	−5.26129	−0.326921
$260$	−6.36319	−0.394628
$261$	0	0
$262$	22.8099	1.40920
$263$	−12.2957	−0.758187	−0.379093	−	0.925358i	$-0.623764\pi$
$263$	−12.2957	−0.758187	−0.379093	+	0.925358i	$0.623764\pi$
$264$	0	0
$265$	33.9078	2.08294
$266$	−12.1871	−0.747241
$267$	0	0
$268$	3.12705	0.191015
$269$	−5.20845	−0.317565	−0.158782	−	0.987314i	$-0.550757\pi$
$269$	−5.20845	−0.317565	−0.158782	+	0.987314i	$0.550757\pi$
$270$	0	0
$271$	9.53266	0.579068	0.289534	−	0.957168i	$-0.406500\pi$
$271$	9.53266	0.579068	0.289534	+	0.957168i	$0.406500\pi$
$272$	−4.09598	−0.248355
$273$	0	0
$274$	12.4026	0.749268
$275$	−10.2601	−0.618710
$276$	0	0
$277$	12.2562	0.736404	0.368202	−	0.929746i	$-0.379973\pi$
$277$	12.2562	0.736404	0.368202	+	0.929746i	$0.379973\pi$
$278$	16.1737	0.970034
$279$	0	0
$280$	4.54847	0.271823
$281$	1.20525	0.0718992	0.0359496	−	0.999354i	$-0.488554\pi$
$281$	1.20525	0.0718992	0.0359496	+	0.999354i	$0.488554\pi$
$282$	0	0
$283$	−24.3087	−1.44500	−0.722500	−	0.691371i	$-0.757007\pi$
$283$	−24.3087	−1.44500	−0.722500	+	0.691371i	$0.757007\pi$
$284$	−3.20185	−0.189995
$285$	0	0
$286$	10.2681	0.607164
$287$	−0.704379	−0.0415782
$288$	0	0
$289$	−0.222971	−0.0131159
$290$	−15.9754	−0.938109
$291$	0	0
$292$	1.15506	0.0675951
$293$	9.58867	0.560176	0.280088	−	0.959974i	$-0.409636\pi$
$293$	9.58867	0.560176	0.280088	+	0.959974i	$0.409636\pi$
$294$	0	0
$295$	−20.6871	−1.20445
$296$	−3.13510	−0.182224
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−0.230466	−0.0133282
$300$	0	0
$301$	−0.151101	−0.00870932
$302$	10.5180	0.605246
$303$	0	0
$304$	−7.26207	−0.416508
$305$	10.2299	0.585760
$306$	0	0
$307$	−20.9118	−1.19350	−0.596750	−	0.802427i	$-0.703542\pi$
$307$	−20.9118	−1.19350	−0.596750	+	0.802427i	$0.703542\pi$
$308$	−7.33971	−0.418219
$309$	0	0
$310$	−5.09658	−0.289466
$311$	13.7307	0.778598	0.389299	−	0.921111i	$-0.372717\pi$
$311$	13.7307	0.778598	0.389299	+	0.921111i	$0.372717\pi$
$312$	0	0
$313$	−27.3384	−1.54526	−0.772628	−	0.634859i	$-0.781058\pi$
$313$	−27.3384	−1.54526	−0.772628	+	0.634859i	$0.781058\pi$
$314$	19.1463	1.08049
$315$	0	0
$316$	−15.5113	−0.872578
$317$	19.0031	1.06732	0.533659	−	0.845700i	$-0.320817\pi$
$317$	19.0031	1.06732	0.533659	+	0.845700i	$0.320817\pi$
$318$	0	0
$319$	25.7790	1.44335
$320$	2.71034	0.151513
$321$	0	0
$322$	0.164739	0.00918056
$323$	29.7453	1.65507
$324$	0	0
$325$	−5.50766	−0.305510
$326$	−6.53078	−0.361707
$327$	0	0
$328$	−0.419725	−0.0231754
$329$	−1.89602	−0.104531
$330$	0	0
$331$	−24.2079	−1.33059	−0.665294	−	0.746581i	$-0.731694\pi$
$331$	−24.2079	−1.33059	−0.665294	+	0.746581i	$0.731694\pi$
$332$	−14.9118	−0.818389
$333$	0	0
$334$	−11.7228	−0.641440
$335$	8.47536	0.463059
$336$	0	0
$337$	−21.7597	−1.18533	−0.592663	−	0.805450i	$-0.701924\pi$
$337$	−21.7597	−1.18533	−0.592663	+	0.805450i	$0.701924\pi$
$338$	−7.48808	−0.407298
$339$	0	0
$340$	−11.1015	−0.602062
$341$	8.22418	0.445364
$342$	0	0
$343$	−18.7683	−1.01340
$344$	−0.0900380	−0.00485453
$345$	0	0
$346$	−1.93624	−0.104093
$347$	−11.4110	−0.612574	−0.306287	−	0.951939i	$-0.599087\pi$
$347$	−11.4110	−0.612574	−0.306287	+	0.951939i	$0.599087\pi$
$348$	0	0
$349$	29.2720	1.56689	0.783446	−	0.621460i	$-0.213460\pi$
$349$	29.2720	1.56689	0.783446	+	0.621460i	$0.213460\pi$
$350$	3.93692	0.210437
$351$	0	0
$352$	−4.37358	−0.233113
$353$	25.2789	1.34546	0.672730	−	0.739888i	$-0.265121\pi$
$353$	25.2789	1.34546	0.672730	+	0.739888i	$0.265121\pi$
$354$	0	0
$355$	−8.67810	−0.460586
$356$	7.11363	0.377022
$357$	0	0
$358$	13.5559	0.716449
$359$	3.38548	0.178679	0.0893393	−	0.996001i	$-0.471524\pi$
$359$	3.38548	0.178679	0.0893393	+	0.996001i	$0.471524\pi$
$360$	0	0
$361$	33.7376	1.77566
$362$	14.9431	0.785389
$363$	0	0
$364$	−3.93997	−0.206510
$365$	3.13062	0.163864
$366$	0	0
$367$	7.07969	0.369557	0.184778	−	0.982780i	$-0.440843\pi$
$367$	7.07969	0.369557	0.184778	+	0.982780i	$0.440843\pi$
$368$	0.0981648	0.00511719
$369$	0	0
$370$	−8.49718	−0.441747
$371$	20.9951	1.09001
$372$	0	0
$373$	7.58277	0.392621	0.196311	−	0.980542i	$-0.437104\pi$
$373$	7.58277	0.392621	0.196311	+	0.980542i	$0.437104\pi$
$374$	17.9141	0.926316
$375$	0	0
$376$	−1.12980	−0.0582651
$377$	13.8382	0.712704
$378$	0	0
$379$	−37.0298	−1.90210	−0.951048	−	0.309044i	$-0.899991\pi$
$379$	−37.0298	−1.90210	−0.951048	+	0.309044i	$0.899991\pi$
$380$	−19.6827	−1.00970
$381$	0	0
$382$	−3.69535	−0.189071
$383$	9.47515	0.484158	0.242079	−	0.970257i	$-0.422171\pi$
$383$	9.47515	0.484158	0.242079	+	0.970257i	$0.422171\pi$
$384$	0	0
$385$	−19.8931	−1.01385
$386$	7.65239	0.389496
$387$	0	0
$388$	−4.78642	−0.242994
$389$	8.01018	0.406132	0.203066	−	0.979165i	$-0.434909\pi$
$389$	8.01018	0.406132	0.203066	+	0.979165i	$0.434909\pi$
$390$	0	0
$391$	−0.402081	−0.0203341
$392$	−4.18367	−0.211307
$393$	0	0
$394$	−18.2992	−0.921900
$395$	−42.0408	−2.11530
$396$	0	0
$397$	−16.4377	−0.824984	−0.412492	−	0.910961i	$-0.635342\pi$
$397$	−16.4377	−0.824984	−0.412492	+	0.910961i	$0.635342\pi$
$398$	12.9256	0.647900
$399$	0	0
$400$	2.34593	0.117297
$401$	17.7661	0.887199	0.443599	−	0.896225i	$-0.353701\pi$
$401$	17.7661	0.887199	0.443599	+	0.896225i	$0.353701\pi$
$402$	0	0
$403$	4.41475	0.219914
$404$	−3.25163	−0.161775
$405$	0	0
$406$	−9.89168	−0.490916
$407$	13.7116	0.679660
$408$	0	0
$409$	−24.3199	−1.20254	−0.601270	−	0.799046i	$-0.705338\pi$
$409$	−24.3199	−1.20254	−0.601270	+	0.799046i	$0.705338\pi$
$410$	−1.13760	−0.0561819
$411$	0	0
$412$	−6.91855	−0.340852
$413$	−12.8091	−0.630293
$414$	0	0
$415$	−40.4159	−1.98394
$416$	−2.34775	−0.115108
$417$	0	0
$418$	31.7613	1.55349
$419$	−23.5738	−1.15166	−0.575829	−	0.817570i	$-0.695321\pi$
$419$	−23.5738	−1.15166	−0.575829	+	0.817570i	$0.695321\pi$
$420$	0	0
$421$	−0.218135	−0.0106312	−0.00531562	−	0.999986i	$-0.501692\pi$
$421$	−0.218135	−0.0106312	−0.00531562	+	0.999986i	$0.501692\pi$
$422$	−12.7410	−0.620222
$423$	0	0
$424$	12.5105	0.607565
$425$	−9.60889	−0.466100
$426$	0	0
$427$	6.33414	0.306530
$428$	2.61340	0.126323
$429$	0	0
$430$	−0.244034	−0.0117683
$431$	16.3041	0.785339	0.392670	−	0.919680i	$-0.371552\pi$
$431$	16.3041	0.785339	0.392670	+	0.919680i	$0.371552\pi$
$432$	0	0
$433$	16.4350	0.789815	0.394908	−	0.918721i	$-0.370777\pi$
$433$	16.4350	0.789815	0.394908	+	0.918721i	$0.370777\pi$
$434$	−3.15571	−0.151479
$435$	0	0
$436$	−8.13251	−0.389477
$437$	−0.712879	−0.0341016
$438$	0	0
$439$	−25.1285	−1.19932	−0.599660	−	0.800255i	$-0.704697\pi$
$439$	−25.1285	−1.19932	−0.599660	+	0.800255i	$0.704697\pi$
$440$	−11.8539	−0.565112
$441$	0	0
$442$	9.61632	0.457402
$443$	26.4593	1.25712	0.628560	−	0.777761i	$-0.283645\pi$
$443$	26.4593	1.25712	0.628560	+	0.777761i	$0.283645\pi$
$444$	0	0
$445$	19.2803	0.913976
$446$	−10.0497	−0.475869
$447$	0	0
$448$	1.67819	0.0792871
$449$	11.6779	0.551112	0.275556	−	0.961285i	$-0.411138\pi$
$449$	11.6779	0.551112	0.275556	+	0.961285i	$0.411138\pi$
$450$	0	0
$451$	1.83570	0.0864399
$452$	6.09305	0.286593
$453$	0	0
$454$	9.51471	0.446548
$455$	−10.6786	−0.500623
$456$	0	0
$457$	11.2598	0.526709	0.263355	−	0.964699i	$-0.415171\pi$
$457$	11.2598	0.526709	0.263355	+	0.964699i	$0.415171\pi$
$458$	17.3081	0.808753
$459$	0	0
$460$	0.266060	0.0124051
$461$	−14.7144	−0.685318	−0.342659	−	0.939460i	$-0.611328\pi$
$461$	−14.7144	−0.685318	−0.342659	+	0.939460i	$0.611328\pi$
$462$	0	0
$463$	15.2474	0.708607	0.354303	−	0.935131i	$-0.384718\pi$
$463$	15.2474	0.708607	0.354303	+	0.935131i	$0.384718\pi$
$464$	−5.89425	−0.273634
$465$	0	0
$466$	−21.8741	−1.01330
$467$	−31.9586	−1.47887	−0.739433	−	0.673230i	$-0.764906\pi$
$467$	−31.9586	−1.47887	−0.739433	+	0.673230i	$0.764906\pi$
$468$	0	0
$469$	5.24779	0.242320
$470$	−3.06215	−0.141246
$471$	0	0
$472$	−7.63266	−0.351322
$473$	0.393789	0.0181064
$474$	0	0
$475$	−17.0363	−0.781680
$476$	−6.87383	−0.315062
$477$	0	0
$478$	25.8220	1.18107
$479$	10.9302	0.499413	0.249706	−	0.968322i	$-0.419666\pi$
$479$	10.9302	0.499413	0.249706	+	0.968322i	$0.419666\pi$
$480$	0	0
$481$	7.36042	0.335606
$482$	−12.3157	−0.560967
$483$	0	0
$484$	8.12823	0.369465
$485$	−12.9728	−0.589065
$486$	0	0
$487$	−34.7186	−1.57325	−0.786626	−	0.617430i	$-0.788174\pi$
$487$	−34.7186	−1.57325	−0.786626	+	0.617430i	$0.788174\pi$
$488$	3.77438	0.170858
$489$	0	0
$490$	−11.3392	−0.512252
$491$	−33.3466	−1.50491	−0.752456	−	0.658642i	$-0.771131\pi$
$491$	−33.3466	−1.50491	−0.752456	+	0.658642i	$0.771131\pi$
$492$	0	0
$493$	24.1427	1.08733
$494$	17.0495	0.767093
$495$	0	0
$496$	−1.88042	−0.0844334
$497$	−5.37332	−0.241026
$498$	0	0
$499$	−23.1107	−1.03458	−0.517289	−	0.855811i	$-0.673059\pi$
$499$	−23.1107	−1.03458	−0.517289	+	0.855811i	$0.673059\pi$
$500$	−7.19342	−0.321699
$501$	0	0
$502$	24.1336	1.07713
$503$	31.7411	1.41527	0.707633	−	0.706580i	$-0.249763\pi$
$503$	31.7411	1.41527	0.707633	+	0.706580i	$0.249763\pi$
$504$	0	0
$505$	−8.81303	−0.392175
$506$	−0.429332	−0.0190861
$507$	0	0
$508$	3.03062	0.134462
$509$	15.5438	0.688965	0.344483	−	0.938793i	$-0.388054\pi$
$509$	15.5438	0.688965	0.344483	+	0.938793i	$0.388054\pi$
$510$	0	0
$511$	1.93842	0.0857506
$512$	1.00000	0.0441942
$513$	0	0
$514$	−1.94139	−0.0856311
$515$	−18.7516	−0.826294
$516$	0	0
$517$	4.94129	0.217318
$518$	−5.26129	−0.231168
$519$	0	0
$520$	−6.36319	−0.279044
$521$	18.6011	0.814929	0.407465	−	0.913221i	$-0.366413\pi$
$521$	18.6011	0.814929	0.407465	+	0.913221i	$0.366413\pi$
$522$	0	0
$523$	−16.4760	−0.720444	−0.360222	−	0.932867i	$-0.617299\pi$
$523$	−16.4760	−0.720444	−0.360222	+	0.932867i	$0.617299\pi$
$524$	22.8099	0.996454
$525$	0	0
$526$	−12.2957	−0.536119
$527$	7.70216	0.335511
$528$	0	0
$529$	−22.9904	−0.999581
$530$	33.9078	1.47286
$531$	0	0
$532$	−12.1871	−0.528379
$533$	0.985408	0.0426828
$534$	0	0
$535$	7.08319	0.306233
$536$	3.12705	0.135068
$537$	0	0
$538$	−5.20845	−0.224552
$539$	18.2977	0.788136
$540$	0	0
$541$	17.8765	0.768573	0.384286	−	0.923214i	$-0.374448\pi$
$541$	17.8765	0.768573	0.384286	+	0.923214i	$0.374448\pi$
$542$	9.53266	0.409463
$543$	0	0
$544$	−4.09598	−0.175614
$545$	−22.0419	−0.944169
$546$	0	0
$547$	−27.9324	−1.19430	−0.597152	−	0.802128i	$-0.703701\pi$
$547$	−27.9324	−1.19430	−0.597152	+	0.802128i	$0.703701\pi$
$548$	12.4026	0.529812
$549$	0	0
$550$	−10.2601	−0.437494
$551$	42.8044	1.82353
$552$	0	0
$553$	−26.0309	−1.10695
$554$	12.2562	0.520716
$555$	0	0
$556$	16.1737	0.685918
$557$	−17.0091	−0.720700	−0.360350	−	0.932817i	$-0.617343\pi$
$557$	−17.0091	−0.720700	−0.360350	+	0.932817i	$0.617343\pi$
$558$	0	0
$559$	0.211387	0.00894070
$560$	4.54847	0.192208
$561$	0	0
$562$	1.20525	0.0508404
$563$	−36.1048	−1.52164	−0.760818	−	0.648965i	$-0.775202\pi$
$563$	−36.1048	−1.52164	−0.760818	+	0.648965i	$0.775202\pi$
$564$	0	0
$565$	16.5142	0.694758
$566$	−24.3087	−1.02177
$567$	0	0
$568$	−3.20185	−0.134347
$569$	−6.41843	−0.269074	−0.134537	−	0.990909i	$-0.542955\pi$
$569$	−6.41843	−0.269074	−0.134537	+	0.990909i	$0.542955\pi$
$570$	0	0
$571$	16.2900	0.681716	0.340858	−	0.940115i	$-0.389282\pi$
$571$	16.2900	0.681716	0.340858	+	0.940115i	$0.389282\pi$
$572$	10.2681	0.429329
$573$	0	0
$574$	−0.704379	−0.0294002
$575$	0.230288	0.00960368
$576$	0	0
$577$	14.1764	0.590170	0.295085	−	0.955471i	$-0.404652\pi$
$577$	14.1764	0.590170	0.295085	+	0.955471i	$0.404652\pi$
$578$	−0.222971	−0.00927438
$579$	0	0
$580$	−15.9754	−0.663343
$581$	−25.0248	−1.03820
$582$	0	0
$583$	−54.7159	−2.26610
$584$	1.15506	0.0477969
$585$	0	0
$586$	9.58867	0.396104
$587$	−16.5294	−0.682242	−0.341121	−	0.940019i	$-0.610807\pi$
$587$	−16.5294	−0.682242	−0.341121	+	0.940019i	$0.610807\pi$
$588$	0	0
$589$	13.6557	0.562675
$590$	−20.6871	−0.851674
$591$	0	0
$592$	−3.13510	−0.128852
$593$	32.0924	1.31788	0.658939	−	0.752196i	$-0.271005\pi$
$593$	32.0924	1.31788	0.658939	+	0.752196i	$0.271005\pi$
$594$	0	0
$595$	−18.6304	−0.763772
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−0.230466	−0.00942446
$599$	−22.1189	−0.903754	−0.451877	−	0.892080i	$-0.649245\pi$
$599$	−22.1189	−0.903754	−0.451877	+	0.892080i	$0.649245\pi$
$600$	0	0
$601$	−22.2111	−0.906009	−0.453004	−	0.891508i	$-0.649648\pi$
$601$	−22.2111	−0.906009	−0.453004	+	0.891508i	$0.649648\pi$
$602$	−0.151101	−0.00615842
$603$	0	0
$604$	10.5180	0.427973
$605$	22.0303	0.895658
$606$	0	0
$607$	−5.78678	−0.234878	−0.117439	−	0.993080i	$-0.537468\pi$
$607$	−5.78678	−0.234878	−0.117439	+	0.993080i	$0.537468\pi$
$608$	−7.26207	−0.294516
$609$	0	0
$610$	10.2299	0.414195
$611$	2.65249	0.107308
$612$	0	0
$613$	−28.4581	−1.14941	−0.574706	−	0.818360i	$-0.694884\pi$
$613$	−28.4581	−1.14941	−0.574706	+	0.818360i	$0.694884\pi$
$614$	−20.9118	−0.843932
$615$	0	0
$616$	−7.33971	−0.295725
$617$	−13.5829	−0.546826	−0.273413	−	0.961897i	$-0.588153\pi$
$617$	−13.5829	−0.546826	−0.273413	+	0.961897i	$0.588153\pi$
$618$	0	0
$619$	23.0851	0.927870	0.463935	−	0.885869i	$-0.346437\pi$
$619$	23.0851	0.927870	0.463935	+	0.885869i	$0.346437\pi$
$620$	−5.09658	−0.204683
$621$	0	0
$622$	13.7307	0.550552
$623$	11.9380	0.478287
$624$	0	0
$625$	−31.2263	−1.24905
$626$	−27.3384	−1.09266
$627$	0	0
$628$	19.1463	0.764020
$629$	12.8413	0.512016
$630$	0	0
$631$	−29.5613	−1.17682	−0.588409	−	0.808563i	$-0.700246\pi$
$631$	−29.5613	−1.17682	−0.588409	+	0.808563i	$0.700246\pi$
$632$	−15.5113	−0.617006
$633$	0	0
$634$	19.0031	0.754708
$635$	8.21400	0.325963
$636$	0	0
$637$	9.82221	0.389170
$638$	25.7790	1.02060
$639$	0	0
$640$	2.71034	0.107136
$641$	3.36162	0.132776	0.0663881	−	0.997794i	$-0.478852\pi$
$641$	3.36162	0.132776	0.0663881	+	0.997794i	$0.478852\pi$
$642$	0	0
$643$	−37.2776	−1.47009	−0.735043	−	0.678020i	$-0.762838\pi$
$643$	−37.2776	−1.47009	−0.735043	+	0.678020i	$0.762838\pi$
$644$	0.164739	0.00649164
$645$	0	0
$646$	29.7453	1.17031
$647$	17.7157	0.696478	0.348239	−	0.937406i	$-0.386780\pi$
$647$	17.7157	0.696478	0.348239	+	0.937406i	$0.386780\pi$
$648$	0	0
$649$	33.3821	1.31036
$650$	−5.50766	−0.216028
$651$	0	0
$652$	−6.53078	−0.255765
$653$	31.2042	1.22111	0.610557	−	0.791972i	$-0.290946\pi$
$653$	31.2042	1.22111	0.610557	+	0.791972i	$0.290946\pi$
$654$	0	0
$655$	61.8225	2.41560
$656$	−0.419725	−0.0163875
$657$	0	0
$658$	−1.89602	−0.0739147
$659$	42.2097	1.64426	0.822128	−	0.569303i	$-0.192787\pi$
$659$	42.2097	1.64426	0.822128	+	0.569303i	$0.192787\pi$
$660$	0	0
$661$	−11.0759	−0.430801	−0.215401	−	0.976526i	$-0.569106\pi$
$661$	−11.0759	−0.430801	−0.215401	+	0.976526i	$0.569106\pi$
$662$	−24.2079	−0.940868
$663$	0	0
$664$	−14.9118	−0.578688
$665$	−33.0313	−1.28090
$666$	0	0
$667$	−0.578608	−0.0224038
$668$	−11.7228	−0.453567
$669$	0	0
$670$	8.47536	0.327432
$671$	−16.5076	−0.637268
$672$	0	0
$673$	17.6065	0.678680	0.339340	−	0.940664i	$-0.389796\pi$
$673$	17.6065	0.678680	0.339340	+	0.940664i	$0.389796\pi$
$674$	−21.7597	−0.838152
$675$	0	0
$676$	−7.48808	−0.288003
$677$	21.4160	0.823083	0.411542	−	0.911391i	$-0.364990\pi$
$677$	21.4160	0.823083	0.411542	+	0.911391i	$0.364990\pi$
$678$	0	0
$679$	−8.03253	−0.308260
$680$	−11.1015	−0.425722
$681$	0	0
$682$	8.22418	0.314920
$683$	−41.4123	−1.58460	−0.792298	−	0.610134i	$-0.791116\pi$
$683$	−41.4123	−1.58460	−0.792298	+	0.610134i	$0.791116\pi$
$684$	0	0
$685$	33.6152	1.28437
$686$	−18.7683	−0.716579
$687$	0	0
$688$	−0.0900380	−0.00343267
$689$	−29.3716	−1.11897
$690$	0	0
$691$	15.2709	0.580932	0.290466	−	0.956885i	$-0.406190\pi$
$691$	15.2709	0.580932	0.290466	+	0.956885i	$0.406190\pi$
$692$	−1.93624	−0.0736049
$693$	0	0
$694$	−11.4110	−0.433155
$695$	43.8362	1.66280
$696$	0	0
$697$	1.71918	0.0651188
$698$	29.2720	1.10796
$699$	0	0
$700$	3.93692	0.148802
$701$	−37.9423	−1.43306	−0.716531	−	0.697555i	$-0.754271\pi$
$701$	−37.9423	−1.43306	−0.716531	+	0.697555i	$0.754271\pi$
$702$	0	0
$703$	22.7673	0.858685
$704$	−4.37358	−0.164836
$705$	0	0
$706$	25.2789	0.951384
$707$	−5.45686	−0.205226
$708$	0	0
$709$	−26.1329	−0.981441	−0.490721	−	0.871317i	$-0.663266\pi$
$709$	−26.1329	−0.981441	−0.490721	+	0.871317i	$0.663266\pi$
$710$	−8.67810	−0.325683
$711$	0	0
$712$	7.11363	0.266595
$713$	−0.184591	−0.00691299
$714$	0	0
$715$	27.8299	1.04078
$716$	13.5559	0.506606
$717$	0	0
$718$	3.38548	0.126345
$719$	−2.05961	−0.0768107	−0.0384053	−	0.999262i	$-0.512228\pi$
$719$	−2.05961	−0.0768107	−0.0384053	+	0.999262i	$0.512228\pi$
$720$	0	0
$721$	−11.6106	−0.432403
$722$	33.7376	1.25558
$723$	0	0
$724$	14.9431	0.555354
$725$	−13.8275	−0.513541
$726$	0	0
$727$	−10.4096	−0.386073	−0.193036	−	0.981192i	$-0.561833\pi$
$727$	−10.4096	−0.386073	−0.193036	+	0.981192i	$0.561833\pi$
$728$	−3.93997	−0.146025
$729$	0	0
$730$	3.13062	0.115869
$731$	0.368794	0.0136403
$732$	0	0
$733$	−18.8723	−0.697065	−0.348533	−	0.937297i	$-0.613320\pi$
$733$	−18.8723	−0.697065	−0.348533	+	0.937297i	$0.613320\pi$
$734$	7.07969	0.261316
$735$	0	0
$736$	0.0981648	0.00361840
$737$	−13.6764	−0.503777
$738$	0	0
$739$	−22.8498	−0.840544	−0.420272	−	0.907398i	$-0.638065\pi$
$739$	−22.8498	−0.840544	−0.420272	+	0.907398i	$0.638065\pi$
$740$	−8.49718	−0.312362
$741$	0	0
$742$	20.9951	0.770753
$743$	−40.5510	−1.48767	−0.743837	−	0.668362i	$-0.766996\pi$
$743$	−40.5510	−1.48767	−0.743837	+	0.668362i	$0.766996\pi$
$744$	0	0
$745$	−2.71034	−0.0992991
$746$	7.58277	0.277625
$747$	0	0
$748$	17.9141	0.655004
$749$	4.38578	0.160253
$750$	0	0
$751$	18.7741	0.685078	0.342539	−	0.939504i	$-0.388713\pi$
$751$	18.7741	0.685078	0.342539	+	0.939504i	$0.388713\pi$
$752$	−1.12980	−0.0411997
$753$	0	0
$754$	13.8382	0.503958
$755$	28.5075	1.03749
$756$	0	0
$757$	−10.1211	−0.367857	−0.183928	−	0.982940i	$-0.558881\pi$
$757$	−10.1211	−0.367857	−0.183928	+	0.982940i	$0.558881\pi$
$758$	−37.0298	−1.34498
$759$	0	0
$760$	−19.6827	−0.713965
$761$	20.0772	0.727799	0.363899	−	0.931438i	$-0.381445\pi$
$761$	20.0772	0.727799	0.363899	+	0.931438i	$0.381445\pi$
$762$	0	0
$763$	−13.6479	−0.494087
$764$	−3.69535	−0.133693
$765$	0	0
$766$	9.47515	0.342351
$767$	17.9195	0.647037
$768$	0	0
$769$	6.24889	0.225341	0.112670	−	0.993632i	$-0.464060\pi$
$769$	6.24889	0.225341	0.112670	+	0.993632i	$0.464060\pi$
$770$	−19.8931	−0.716897
$771$	0	0
$772$	7.65239	0.275416
$773$	−1.11866	−0.0402353	−0.0201176	−	0.999798i	$-0.506404\pi$
$773$	−1.11866	−0.0402353	−0.0201176	+	0.999798i	$0.506404\pi$
$774$	0	0
$775$	−4.41134	−0.158460
$776$	−4.78642	−0.171822
$777$	0	0
$778$	8.01018	0.287179
$779$	3.04807	0.109209
$780$	0	0
$781$	14.0036	0.501087
$782$	−0.402081	−0.0143784
$783$	0	0
$784$	−4.18367	−0.149417
$785$	51.8929	1.85214
$786$	0	0
$787$	37.0689	1.32136	0.660682	−	0.750666i	$-0.270267\pi$
$787$	37.0689	1.32136	0.660682	+	0.750666i	$0.270267\pi$
$788$	−18.2992	−0.651882
$789$	0	0
$790$	−42.0408	−1.49575
$791$	10.2253	0.363570
$792$	0	0
$793$	−8.86130	−0.314674
$794$	−16.4377	−0.583352
$795$	0	0
$796$	12.9256	0.458134
$797$	−32.5071	−1.15146	−0.575730	−	0.817640i	$-0.695282\pi$
$797$	−32.5071	−1.15146	−0.575730	+	0.817640i	$0.695282\pi$
$798$	0	0
$799$	4.62765	0.163714
$800$	2.34593	0.0829413
$801$	0	0
$802$	17.7661	0.627344
$803$	−5.05177	−0.178273
$804$	0	0
$805$	0.446499	0.0157370
$806$	4.41475	0.155503
$807$	0	0
$808$	−3.25163	−0.114392
$809$	2.43628	0.0856552	0.0428276	−	0.999082i	$-0.486363\pi$
$809$	2.43628	0.0856552	0.0428276	+	0.999082i	$0.486363\pi$
$810$	0	0
$811$	−38.2685	−1.34379	−0.671894	−	0.740647i	$-0.734519\pi$
$811$	−38.2685	−1.34379	−0.671894	+	0.740647i	$0.734519\pi$
$812$	−9.89168	−0.347130
$813$	0	0
$814$	13.7116	0.480592
$815$	−17.7006	−0.620026
$816$	0	0
$817$	0.653862	0.0228757
$818$	−24.3199	−0.850325
$819$	0	0
$820$	−1.13760	−0.0397266
$821$	0.603988	0.0210793	0.0105397	−	0.999944i	$-0.496645\pi$
$821$	0.603988	0.0210793	0.0105397	+	0.999944i	$0.496645\pi$
$822$	0	0
$823$	−9.46449	−0.329911	−0.164956	−	0.986301i	$-0.552748\pi$
$823$	−9.46449	−0.329911	−0.164956	+	0.986301i	$0.552748\pi$
$824$	−6.91855	−0.241019
$825$	0	0
$826$	−12.8091	−0.445684
$827$	−9.04512	−0.314530	−0.157265	−	0.987556i	$-0.550268\pi$
$827$	−9.04512	−0.314530	−0.157265	+	0.987556i	$0.550268\pi$
$828$	0	0
$829$	−22.2814	−0.773864	−0.386932	−	0.922108i	$-0.626465\pi$
$829$	−22.2814	−0.773864	−0.386932	+	0.922108i	$0.626465\pi$
$830$	−40.4159	−1.40286
$831$	0	0
$832$	−2.34775	−0.0813935
$833$	17.1362	0.593735
$834$	0	0
$835$	−31.7726	−1.09954
$836$	31.7613	1.09849
$837$	0	0
$838$	−23.5738	−0.814345
$839$	3.95317	0.136479	0.0682393	−	0.997669i	$-0.478262\pi$
$839$	3.95317	0.136479	0.0682393	+	0.997669i	$0.478262\pi$
$840$	0	0
$841$	5.74221	0.198007
$842$	−0.218135	−0.00751742
$843$	0	0
$844$	−12.7410	−0.438563
$845$	−20.2952	−0.698178
$846$	0	0
$847$	13.6407	0.468701
$848$	12.5105	0.429613
$849$	0	0
$850$	−9.60889	−0.329582
$851$	−0.307756	−0.0105498
$852$	0	0
$853$	6.31219	0.216125	0.108063	−	0.994144i	$-0.465535\pi$
$853$	6.31219	0.216125	0.108063	+	0.994144i	$0.465535\pi$
$854$	6.33414	0.216750
$855$	0	0
$856$	2.61340	0.0893241
$857$	−22.0163	−0.752064	−0.376032	−	0.926607i	$-0.622712\pi$
$857$	−22.0163	−0.752064	−0.376032	+	0.926607i	$0.622712\pi$
$858$	0	0
$859$	−48.1678	−1.64346	−0.821732	−	0.569875i	$-0.806992\pi$
$859$	−48.1678	−1.64346	−0.821732	+	0.569875i	$0.806992\pi$
$860$	−0.244034	−0.00832147
$861$	0	0
$862$	16.3041	0.555319
$863$	23.0500	0.784631	0.392315	−	0.919831i	$-0.371674\pi$
$863$	23.0500	0.784631	0.392315	+	0.919831i	$0.371674\pi$
$864$	0	0
$865$	−5.24787	−0.178433
$866$	16.4350	0.558484
$867$	0	0
$868$	−3.15571	−0.107112
$869$	67.8399	2.30131
$870$	0	0
$871$	−7.34152	−0.248758
$872$	−8.13251	−0.275402
$873$	0	0
$874$	−0.712879	−0.0241135
$875$	−12.0719	−0.408106
$876$	0	0
$877$	−34.2938	−1.15802	−0.579010	−	0.815320i	$-0.696561\pi$
$877$	−34.2938	−1.15802	−0.579010	+	0.815320i	$0.696561\pi$
$878$	−25.1285	−0.848047
$879$	0	0
$880$	−11.8539	−0.399595
$881$	−30.3973	−1.02411	−0.512056	−	0.858952i	$-0.671116\pi$
$881$	−30.3973	−1.02411	−0.512056	+	0.858952i	$0.671116\pi$
$882$	0	0
$883$	−2.01379	−0.0677695	−0.0338847	−	0.999426i	$-0.510788\pi$
$883$	−2.01379	−0.0677695	−0.0338847	+	0.999426i	$0.510788\pi$
$884$	9.61632	0.323432
$885$	0	0
$886$	26.4593	0.888919
$887$	22.2434	0.746859	0.373429	−	0.927659i	$-0.378182\pi$
$887$	22.2434	0.746859	0.373429	+	0.927659i	$0.378182\pi$
$888$	0	0
$889$	5.08595	0.170577
$890$	19.2803	0.646279
$891$	0	0
$892$	−10.0497	−0.336490
$893$	8.20470	0.274560
$894$	0	0
$895$	36.7410	1.22811
$896$	1.67819	0.0560644
$897$	0	0
$898$	11.6779	0.389695
$899$	11.0837	0.369661
$900$	0	0
$901$	−51.2429	−1.70715
$902$	1.83570	0.0611222
$903$	0	0
$904$	6.09305	0.202652
$905$	40.5007	1.34629
$906$	0	0
$907$	−1.68674	−0.0560073	−0.0280036	−	0.999608i	$-0.508915\pi$
$907$	−1.68674	−0.0560073	−0.0280036	+	0.999608i	$0.508915\pi$
$908$	9.51471	0.315757
$909$	0	0
$910$	−10.6786	−0.353994
$911$	54.8311	1.81664	0.908318	−	0.418280i	$-0.137367\pi$
$911$	54.8311	1.81664	0.908318	+	0.418280i	$0.137367\pi$
$912$	0	0
$913$	65.2178	2.15839
$914$	11.2598	0.372440
$915$	0	0
$916$	17.3081	0.571875
$917$	38.2793	1.26409
$918$	0	0
$919$	−52.9869	−1.74788	−0.873939	−	0.486036i	$-0.838442\pi$
$919$	−52.9869	−1.74788	−0.873939	+	0.486036i	$0.838442\pi$
$920$	0.266060	0.00877173
$921$	0	0
$922$	−14.7144	−0.484593
$923$	7.51714	0.247430
$924$	0	0
$925$	−7.35473	−0.241822
$926$	15.2474	0.501061
$927$	0	0
$928$	−5.89425	−0.193488
$929$	5.24290	0.172014	0.0860069	−	0.996295i	$-0.472589\pi$
$929$	5.24290	0.172014	0.0860069	+	0.996295i	$0.472589\pi$
$930$	0	0
$931$	30.3821	0.995734
$932$	−21.8741	−0.716511
$933$	0	0
$934$	−31.9586	−1.04572
$935$	48.5533	1.58786
$936$	0	0
$937$	−36.5730	−1.19479	−0.597393	−	0.801948i	$-0.703797\pi$
$937$	−36.5730	−1.19479	−0.597393	+	0.801948i	$0.703797\pi$
$938$	5.24779	0.171346
$939$	0	0
$940$	−3.06215	−0.0998762
$941$	−24.6596	−0.803879	−0.401940	−	0.915666i	$-0.631664\pi$
$941$	−24.6596	−0.803879	−0.401940	+	0.915666i	$0.631664\pi$
$942$	0	0
$943$	−0.0412022	−0.00134173
$944$	−7.63266	−0.248422
$945$	0	0
$946$	0.393789	0.0128032
$947$	−34.3077	−1.11485	−0.557425	−	0.830227i	$-0.688211\pi$
$947$	−34.3077	−1.11485	−0.557425	+	0.830227i	$0.688211\pi$
$948$	0	0
$949$	−2.71180	−0.0880287
$950$	−17.0363	−0.552731
$951$	0	0
$952$	−6.87383	−0.222782
$953$	31.0422	1.00556	0.502778	−	0.864415i	$-0.332311\pi$
$953$	31.0422	1.00556	0.502778	+	0.864415i	$0.332311\pi$
$954$	0	0
$955$	−10.0157	−0.324099
$956$	25.8220	0.835143
$957$	0	0
$958$	10.9302	0.353138
$959$	20.8139	0.672116
$960$	0	0
$961$	−27.4640	−0.885936
$962$	7.36042	0.237309
$963$	0	0
$964$	−12.3157	−0.396663
$965$	20.7406	0.667662
$966$	0	0
$967$	56.2492	1.80885	0.904426	−	0.426631i	$-0.140300\pi$
$967$	56.2492	1.80885	0.904426	+	0.426631i	$0.140300\pi$
$968$	8.12823	0.261251
$969$	0	0
$970$	−12.9728	−0.416532
$971$	−26.1008	−0.837614	−0.418807	−	0.908075i	$-0.637552\pi$
$971$	−26.1008	−0.837614	−0.418807	+	0.908075i	$0.637552\pi$
$972$	0	0
$973$	27.1426	0.870151
$974$	−34.7186	−1.11246
$975$	0	0
$976$	3.77438	0.120815
$977$	13.8161	0.442015	0.221008	−	0.975272i	$-0.429065\pi$
$977$	13.8161	0.442015	0.221008	+	0.975272i	$0.429065\pi$
$978$	0	0
$979$	−31.1121	−0.994346
$980$	−11.3392	−0.362217
$981$	0	0
$982$	−33.3466	−1.06413
$983$	−1.02013	−0.0325370	−0.0162685	−	0.999868i	$-0.505179\pi$
$983$	−1.02013	−0.0325370	−0.0162685	+	0.999868i	$0.505179\pi$
$984$	0	0
$985$	−49.5970	−1.58029
$986$	24.1427	0.768861
$987$	0	0
$988$	17.0495	0.542417
$989$	−0.00883857	−0.000281050 0
$990$	0	0
$991$	−30.5516	−0.970502	−0.485251	−	0.874375i	$-0.661272\pi$
$991$	−30.5516	−0.970502	−0.485251	+	0.874375i	$0.661272\pi$
$992$	−1.88042	−0.0597034
$993$	0	0
$994$	−5.37332	−0.170431
$995$	35.0326	1.11061
$996$	0	0
$997$	14.1051	0.446714	0.223357	−	0.974737i	$-0.428298\pi$
$997$	14.1051	0.446714	0.223357	+	0.974737i	$0.428298\pi$
$998$	−23.1107	−0.731557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.h.1.9	yes	9
3.2	odd	2		8046.2.a.g.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.1	✓	9	3.2	odd	2
8046.2.a.h.1.9	yes	9	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.71034 q^{5} +1.67819 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.71034 q^{5} +1.67819 q^{7} +1.00000 q^{8} +2.71034 q^{10} -4.37358 q^{11} -2.34775 q^{13} +1.67819 q^{14} +1.00000 q^{16} -4.09598 q^{17} -7.26207 q^{19} +2.71034 q^{20} -4.37358 q^{22} +0.0981648 q^{23} +2.34593 q^{25} -2.34775 q^{26} +1.67819 q^{28} -5.89425 q^{29} -1.88042 q^{31} +1.00000 q^{32} -4.09598 q^{34} +4.54847 q^{35} -3.13510 q^{37} -7.26207 q^{38} +2.71034 q^{40} -0.419725 q^{41} -0.0900380 q^{43} -4.37358 q^{44} +0.0981648 q^{46} -1.12980 q^{47} -4.18367 q^{49} +2.34593 q^{50} -2.34775 q^{52} +12.5105 q^{53} -11.8539 q^{55} +1.67819 q^{56} -5.89425 q^{58} -7.63266 q^{59} +3.77438 q^{61} -1.88042 q^{62} +1.00000 q^{64} -6.36319 q^{65} +3.12705 q^{67} -4.09598 q^{68} +4.54847 q^{70} -3.20185 q^{71} +1.15506 q^{73} -3.13510 q^{74} -7.26207 q^{76} -7.33971 q^{77} -15.5113 q^{79} +2.71034 q^{80} -0.419725 q^{82} -14.9118 q^{83} -11.1015 q^{85} -0.0900380 q^{86} -4.37358 q^{88} +7.11363 q^{89} -3.93997 q^{91} +0.0981648 q^{92} -1.12980 q^{94} -19.6827 q^{95} -4.78642 q^{97} -4.18367 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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