LMFDB - Embedded newform 8046.2.a.q.1.1

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:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 42 x^{12} + 70 x^{11} + 680 x^{10} - 882 x^{9} - 5559 x^{8} + 5066 x^{7} + \cdots - 7083 $ x^14 - 2x^13 - 42x^12 + 70x^11 + 680x^10 - 882x^9 - 5559x^8 + 5066x^7 + 24455x^6 - 12990x^5 - 55580x^4 + 9808x^3 + 53551x^2 + 6282*x - 7083
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$4.30228$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.30228 q^{5} -2.58072 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.30228 q^{5} -2.58072 q^{7} -1.00000 q^{8} +4.30228 q^{10} +2.36140 q^{11} -4.34794 q^{13} +2.58072 q^{14} +1.00000 q^{16} +4.59971 q^{17} +4.77611 q^{19} -4.30228 q^{20} -2.36140 q^{22} -6.85875 q^{23} +13.5096 q^{25} +4.34794 q^{26} -2.58072 q^{28} -9.68846 q^{29} +4.83005 q^{31} -1.00000 q^{32} -4.59971 q^{34} +11.1030 q^{35} -4.41878 q^{37} -4.77611 q^{38} +4.30228 q^{40} -4.47634 q^{41} -3.56055 q^{43} +2.36140 q^{44} +6.85875 q^{46} -6.92253 q^{47} -0.339891 q^{49} -13.5096 q^{50} -4.34794 q^{52} +2.33828 q^{53} -10.1594 q^{55} +2.58072 q^{56} +9.68846 q^{58} -3.02010 q^{59} -1.66465 q^{61} -4.83005 q^{62} +1.00000 q^{64} +18.7061 q^{65} -14.9222 q^{67} +4.59971 q^{68} -11.1030 q^{70} +1.22986 q^{71} +16.5077 q^{73} +4.41878 q^{74} +4.77611 q^{76} -6.09411 q^{77} +0.926115 q^{79} -4.30228 q^{80} +4.47634 q^{82} -16.1043 q^{83} -19.7893 q^{85} +3.56055 q^{86} -2.36140 q^{88} -2.40888 q^{89} +11.2208 q^{91} -6.85875 q^{92} +6.92253 q^{94} -20.5482 q^{95} -9.91733 q^{97} +0.339891 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	$ 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) $ 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * 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
$5$	−4.30228	−1.92404	−0.962020	−	0.272979i	$-0.911991\pi$
$5$	−4.30228	−1.92404	−0.962020	+	0.272979i	$0.911991\pi$
$6$	0	0
$7$	−2.58072	−0.975420	−0.487710	−	0.873006i	$-0.662168\pi$
$7$	−2.58072	−0.975420	−0.487710	+	0.873006i	$0.662168\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.30228	1.36050
$11$	2.36140	0.711989	0.355994	−	0.934488i	$-0.384142\pi$
$11$	2.36140	0.711989	0.355994	+	0.934488i	$0.384142\pi$
$12$	0	0
$13$	−4.34794	−1.20590	−0.602950	−	0.797779i	$-0.706008\pi$
$13$	−4.34794	−1.20590	−0.602950	+	0.797779i	$0.706008\pi$
$14$	2.58072	0.689726
$15$	0	0
$16$	1.00000	0.250000
$17$	4.59971	1.11559	0.557797	−	0.829977i	$-0.311647\pi$
$17$	4.59971	1.11559	0.557797	+	0.829977i	$0.311647\pi$
$18$	0	0
$19$	4.77611	1.09571	0.547857	−	0.836572i	$-0.315444\pi$
$19$	4.77611	1.09571	0.547857	+	0.836572i	$0.315444\pi$
$20$	−4.30228	−0.962020
$21$	0	0
$22$	−2.36140	−0.503452
$23$	−6.85875	−1.43015	−0.715074	−	0.699049i	$-0.753607\pi$
$23$	−6.85875	−1.43015	−0.715074	+	0.699049i	$0.753607\pi$
$24$	0	0
$25$	13.5096	2.70193
$26$	4.34794	0.852701
$27$	0	0
$28$	−2.58072	−0.487710
$29$	−9.68846	−1.79910	−0.899551	−	0.436817i	$-0.856106\pi$
$29$	−9.68846	−1.79910	−0.899551	+	0.436817i	$0.856106\pi$
$30$	0	0
$31$	4.83005	0.867503	0.433751	−	0.901033i	$-0.357190\pi$
$31$	4.83005	0.867503	0.433751	+	0.901033i	$0.357190\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.59971	−0.788845
$35$	11.1030	1.87675
$36$	0	0
$37$	−4.41878	−0.726443	−0.363222	−	0.931703i	$-0.618323\pi$
$37$	−4.41878	−0.726443	−0.363222	+	0.931703i	$0.618323\pi$
$38$	−4.77611	−0.774787
$39$	0	0
$40$	4.30228	0.680251
$41$	−4.47634	−0.699087	−0.349543	−	0.936920i	$-0.613663\pi$
$41$	−4.47634	−0.699087	−0.349543	+	0.936920i	$0.613663\pi$
$42$	0	0
$43$	−3.56055	−0.542979	−0.271490	−	0.962441i	$-0.587516\pi$
$43$	−3.56055	−0.542979	−0.271490	+	0.962441i	$0.587516\pi$
$44$	2.36140	0.355994
$45$	0	0
$46$	6.85875	1.01127
$47$	−6.92253	−1.00976	−0.504878	−	0.863191i	$-0.668462\pi$
$47$	−6.92253	−1.00976	−0.504878	+	0.863191i	$0.668462\pi$
$48$	0	0
$49$	−0.339891	−0.0485558
$50$	−13.5096	−1.91055
$51$	0	0
$52$	−4.34794	−0.602950
$53$	2.33828	0.321187	0.160594	−	0.987021i	$-0.448659\pi$
$53$	2.33828	0.321187	0.160594	+	0.987021i	$0.448659\pi$
$54$	0	0
$55$	−10.1594	−1.36989
$56$	2.58072	0.344863
$57$	0	0
$58$	9.68846	1.27216
$59$	−3.02010	−0.393184	−0.196592	−	0.980485i	$-0.562987\pi$
$59$	−3.02010	−0.393184	−0.196592	+	0.980485i	$0.562987\pi$
$60$	0	0
$61$	−1.66465	−0.213136	−0.106568	−	0.994305i	$-0.533986\pi$
$61$	−1.66465	−0.213136	−0.106568	+	0.994305i	$0.533986\pi$
$62$	−4.83005	−0.613417
$63$	0	0
$64$	1.00000	0.125000
$65$	18.7061	2.32020
$66$	0	0
$67$	−14.9222	−1.82303	−0.911516	−	0.411264i	$-0.865088\pi$
$67$	−14.9222	−1.82303	−0.911516	+	0.411264i	$0.865088\pi$
$68$	4.59971	0.557797
$69$	0	0
$70$	−11.1030	−1.32706
$71$	1.22986	0.145958	0.0729788	−	0.997333i	$-0.476749\pi$
$71$	1.22986	0.145958	0.0729788	+	0.997333i	$0.476749\pi$
$72$	0	0
$73$	16.5077	1.93208	0.966038	−	0.258399i	$-0.0831950\pi$
$73$	16.5077	1.93208	0.966038	+	0.258399i	$0.0831950\pi$
$74$	4.41878	0.513673
$75$	0	0
$76$	4.77611	0.547857
$77$	−6.09411	−0.694488
$78$	0	0
$79$	0.926115	0.104196	0.0520980	−	0.998642i	$-0.483409\pi$
$79$	0.926115	0.104196	0.0520980	+	0.998642i	$0.483409\pi$
$80$	−4.30228	−0.481010
$81$	0	0
$82$	4.47634	0.494329
$83$	−16.1043	−1.76767	−0.883837	−	0.467795i	$-0.845049\pi$
$83$	−16.1043	−1.76767	−0.883837	+	0.467795i	$0.845049\pi$
$84$	0	0
$85$	−19.7893	−2.14645
$86$	3.56055	0.383944
$87$	0	0
$88$	−2.36140	−0.251726
$89$	−2.40888	−0.255341	−0.127671	−	0.991817i	$-0.540750\pi$
$89$	−2.40888	−0.255341	−0.127671	+	0.991817i	$0.540750\pi$
$90$	0	0
$91$	11.2208	1.17626
$92$	−6.85875	−0.715074
$93$	0	0
$94$	6.92253	0.714005
$95$	−20.5482	−2.10820
$96$	0	0
$97$	−9.91733	−1.00695	−0.503476	−	0.864009i	$-0.667946\pi$
$97$	−9.91733	−1.00695	−0.503476	+	0.864009i	$0.667946\pi$
$98$	0.339891	0.0343341
$99$	0	0
$100$	13.5096	1.35096
$101$	−12.8905	−1.28266	−0.641328	−	0.767266i	$-0.721616\pi$
$101$	−12.8905	−1.28266	−0.641328	+	0.767266i	$0.721616\pi$
$102$	0	0
$103$	11.0474	1.08854	0.544268	−	0.838911i	$-0.316807\pi$
$103$	11.0474	1.08854	0.544268	+	0.838911i	$0.316807\pi$
$104$	4.34794	0.426350
$105$	0	0
$106$	−2.33828	−0.227114
$107$	−13.6167	−1.31638	−0.658190	−	0.752852i	$-0.728678\pi$
$107$	−13.6167	−1.31638	−0.658190	+	0.752852i	$0.728678\pi$
$108$	0	0
$109$	−20.0796	−1.92327	−0.961637	−	0.274326i	$-0.911545\pi$
$109$	−20.0796	−1.92327	−0.961637	+	0.274326i	$0.911545\pi$
$110$	10.1594	0.968662
$111$	0	0
$112$	−2.58072	−0.243855
$113$	3.00799	0.282968	0.141484	−	0.989941i	$-0.454813\pi$
$113$	3.00799	0.282968	0.141484	+	0.989941i	$0.454813\pi$
$114$	0	0
$115$	29.5083	2.75166
$116$	−9.68846	−0.899551
$117$	0	0
$118$	3.02010	0.278023
$119$	−11.8706	−1.08817
$120$	0	0
$121$	−5.42380	−0.493072
$122$	1.66465	0.150710
$123$	0	0
$124$	4.83005	0.433751
$125$	−36.6109	−3.27458
$126$	0	0
$127$	14.6905	1.30357	0.651787	−	0.758402i	$-0.274020\pi$
$127$	14.6905	1.30357	0.651787	+	0.758402i	$0.274020\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−18.7061	−1.64063
$131$	7.27060	0.635235	0.317618	−	0.948219i	$-0.397117\pi$
$131$	7.27060	0.635235	0.317618	+	0.948219i	$0.397117\pi$
$132$	0	0
$133$	−12.3258	−1.06878
$134$	14.9222	1.28908
$135$	0	0
$136$	−4.59971	−0.394422
$137$	5.25460	0.448931	0.224465	−	0.974482i	$-0.427936\pi$
$137$	5.25460	0.448931	0.224465	+	0.974482i	$0.427936\pi$
$138$	0	0
$139$	15.8385	1.34341	0.671703	−	0.740821i	$-0.265563\pi$
$139$	15.8385	1.34341	0.671703	+	0.740821i	$0.265563\pi$
$140$	11.1030	0.938374
$141$	0	0
$142$	−1.22986	−0.103208
$143$	−10.2672	−0.858587
$144$	0	0
$145$	41.6825	3.46154
$146$	−16.5077	−1.36618
$147$	0	0
$148$	−4.41878	−0.363222
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	6.79221	0.552742	0.276371	−	0.961051i	$-0.410868\pi$
$151$	6.79221	0.552742	0.276371	+	0.961051i	$0.410868\pi$
$152$	−4.77611	−0.387394
$153$	0	0
$154$	6.09411	0.491077
$155$	−20.7802	−1.66911
$156$	0	0
$157$	−6.19817	−0.494668	−0.247334	−	0.968930i	$-0.579554\pi$
$157$	−6.19817	−0.494668	−0.247334	+	0.968930i	$0.579554\pi$
$158$	−0.926115	−0.0736777
$159$	0	0
$160$	4.30228	0.340125
$161$	17.7005	1.39499
$162$	0	0
$163$	3.35527	0.262805	0.131402	−	0.991329i	$-0.458052\pi$
$163$	3.35527	0.262805	0.131402	+	0.991329i	$0.458052\pi$
$164$	−4.47634	−0.349543
$165$	0	0
$166$	16.1043	1.24993
$167$	3.18225	0.246250	0.123125	−	0.992391i	$-0.460708\pi$
$167$	3.18225	0.246250	0.123125	+	0.992391i	$0.460708\pi$
$168$	0	0
$169$	5.90455	0.454196
$170$	19.7893	1.51777
$171$	0	0
$172$	−3.56055	−0.271490
$173$	−1.29626	−0.0985526	−0.0492763	−	0.998785i	$-0.515691\pi$
$173$	−1.29626	−0.0985526	−0.0492763	+	0.998785i	$0.515691\pi$
$174$	0	0
$175$	−34.8646	−2.63552
$176$	2.36140	0.177997
$177$	0	0
$178$	2.40888	0.180553
$179$	−5.70933	−0.426735	−0.213368	−	0.976972i	$-0.568443\pi$
$179$	−5.70933	−0.426735	−0.213368	+	0.976972i	$0.568443\pi$
$180$	0	0
$181$	15.9052	1.18222	0.591112	−	0.806590i	$-0.298689\pi$
$181$	15.9052	1.18222	0.591112	+	0.806590i	$0.298689\pi$
$182$	−11.2208	−0.831741
$183$	0	0
$184$	6.85875	0.505633
$185$	19.0109	1.39771
$186$	0	0
$187$	10.8618	0.794291
$188$	−6.92253	−0.504878
$189$	0	0
$190$	20.5482	1.49072
$191$	−1.86430	−0.134896	−0.0674479	−	0.997723i	$-0.521486\pi$
$191$	−1.86430	−0.134896	−0.0674479	+	0.997723i	$0.521486\pi$
$192$	0	0
$193$	−16.9932	−1.22320	−0.611599	−	0.791168i	$-0.709473\pi$
$193$	−16.9932	−1.22320	−0.611599	+	0.791168i	$0.709473\pi$
$194$	9.91733	0.712023
$195$	0	0
$196$	−0.339891	−0.0242779
$197$	−8.68393	−0.618704	−0.309352	−	0.950948i	$-0.600112\pi$
$197$	−8.68393	−0.618704	−0.309352	+	0.950948i	$0.600112\pi$
$198$	0	0
$199$	−11.0290	−0.781823	−0.390911	−	0.920428i	$-0.627840\pi$
$199$	−11.0290	−0.781823	−0.390911	+	0.920428i	$0.627840\pi$
$200$	−13.5096	−0.955276
$201$	0	0
$202$	12.8905	0.906975
$203$	25.0032	1.75488
$204$	0	0
$205$	19.2585	1.34507
$206$	−11.0474	−0.769711
$207$	0	0
$208$	−4.34794	−0.301475
$209$	11.2783	0.780136
$210$	0	0
$211$	3.71092	0.255470	0.127735	−	0.991808i	$-0.459229\pi$
$211$	3.71092	0.255470	0.127735	+	0.991808i	$0.459229\pi$
$212$	2.33828	0.160594
$213$	0	0
$214$	13.6167	0.930822
$215$	15.3185	1.04471
$216$	0	0
$217$	−12.4650	−0.846179
$218$	20.0796	1.35996
$219$	0	0
$220$	−10.1594	−0.684947
$221$	−19.9993	−1.34530
$222$	0	0
$223$	2.31235	0.154846	0.0774230	−	0.996998i	$-0.475331\pi$
$223$	2.31235	0.154846	0.0774230	+	0.996998i	$0.475331\pi$
$224$	2.58072	0.172432
$225$	0	0
$226$	−3.00799	−0.200088
$227$	−9.48105	−0.629280	−0.314640	−	0.949211i	$-0.601884\pi$
$227$	−9.48105	−0.629280	−0.314640	+	0.949211i	$0.601884\pi$
$228$	0	0
$229$	−17.1183	−1.13121	−0.565605	−	0.824676i	$-0.691357\pi$
$229$	−17.1183	−1.13121	−0.565605	+	0.824676i	$0.691357\pi$
$230$	−29.5083	−1.94572
$231$	0	0
$232$	9.68846	0.636078
$233$	−1.89499	−0.124145	−0.0620725	−	0.998072i	$-0.519771\pi$
$233$	−1.89499	−0.124145	−0.0620725	+	0.998072i	$0.519771\pi$
$234$	0	0
$235$	29.7827	1.94281
$236$	−3.02010	−0.196592
$237$	0	0
$238$	11.8706	0.769455
$239$	10.4414	0.675400	0.337700	−	0.941254i	$-0.390351\pi$
$239$	10.4414	0.675400	0.337700	+	0.941254i	$0.390351\pi$
$240$	0	0
$241$	9.96284	0.641763	0.320882	−	0.947119i	$-0.396021\pi$
$241$	9.96284	0.641763	0.320882	+	0.947119i	$0.396021\pi$
$242$	5.42380	0.348655
$243$	0	0
$244$	−1.66465	−0.106568
$245$	1.46231	0.0934233
$246$	0	0
$247$	−20.7662	−1.32132
$248$	−4.83005	−0.306708
$249$	0	0
$250$	36.6109	2.31548
$251$	−13.6358	−0.860681	−0.430341	−	0.902667i	$-0.641607\pi$
$251$	−13.6358	−0.860681	−0.430341	+	0.902667i	$0.641607\pi$
$252$	0	0
$253$	−16.1962	−1.01825
$254$	−14.6905	−0.921766
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.3175	−0.830724	−0.415362	−	0.909656i	$-0.636345\pi$
$257$	−13.3175	−0.830724	−0.415362	+	0.909656i	$0.636345\pi$
$258$	0	0
$259$	11.4036	0.708587
$260$	18.7061	1.16010
$261$	0	0
$262$	−7.27060	−0.449179
$263$	1.11201	0.0685698	0.0342849	−	0.999412i	$-0.489085\pi$
$263$	1.11201	0.0685698	0.0342849	+	0.999412i	$0.489085\pi$
$264$	0	0
$265$	−10.0599	−0.617977
$266$	12.3258	0.755743
$267$	0	0
$268$	−14.9222	−0.911516
$269$	27.6181	1.68390	0.841952	−	0.539553i	$-0.181407\pi$
$269$	27.6181	1.68390	0.841952	+	0.539553i	$0.181407\pi$
$270$	0	0
$271$	17.0063	1.03306	0.516528	−	0.856270i	$-0.327224\pi$
$271$	17.0063	1.03306	0.516528	+	0.856270i	$0.327224\pi$
$272$	4.59971	0.278899
$273$	0	0
$274$	−5.25460	−0.317442
$275$	31.9017	1.92374
$276$	0	0
$277$	−13.3266	−0.800717	−0.400358	−	0.916359i	$-0.631114\pi$
$277$	−13.3266	−0.800717	−0.400358	+	0.916359i	$0.631114\pi$
$278$	−15.8385	−0.949931
$279$	0	0
$280$	−11.1030	−0.663530
$281$	3.48845	0.208104	0.104052	−	0.994572i	$-0.466819\pi$
$281$	3.48845	0.208104	0.104052	+	0.994572i	$0.466819\pi$
$282$	0	0
$283$	13.7321	0.816291	0.408145	−	0.912917i	$-0.366176\pi$
$283$	13.7321	0.816291	0.408145	+	0.912917i	$0.366176\pi$
$284$	1.22986	0.0729788
$285$	0	0
$286$	10.2672	0.607113
$287$	11.5522	0.681903
$288$	0	0
$289$	4.15737	0.244551
$290$	−41.6825	−2.44768
$291$	0	0
$292$	16.5077	0.966038
$293$	−7.12602	−0.416307	−0.208153	−	0.978096i	$-0.566745\pi$
$293$	−7.12602	−0.416307	−0.208153	+	0.978096i	$0.566745\pi$
$294$	0	0
$295$	12.9933	0.756502
$296$	4.41878	0.256837
$297$	0	0
$298$	1.00000	0.0579284
$299$	29.8214	1.72462
$300$	0	0
$301$	9.18878	0.529633
$302$	−6.79221	−0.390848
$303$	0	0
$304$	4.77611	0.273929
$305$	7.16178	0.410082
$306$	0	0
$307$	−30.7190	−1.75323	−0.876614	−	0.481195i	$-0.840203\pi$
$307$	−30.7190	−1.75323	−0.876614	+	0.481195i	$0.840203\pi$
$308$	−6.09411	−0.347244
$309$	0	0
$310$	20.7802	1.18024
$311$	28.4027	1.61057	0.805283	−	0.592890i	$-0.202013\pi$
$311$	28.4027	1.61057	0.805283	+	0.592890i	$0.202013\pi$
$312$	0	0
$313$	11.4329	0.646225	0.323113	−	0.946361i	$-0.395271\pi$
$313$	11.4329	0.646225	0.323113	+	0.946361i	$0.395271\pi$
$314$	6.19817	0.349783
$315$	0	0
$316$	0.926115	0.0520980
$317$	28.0375	1.57474	0.787370	−	0.616480i	$-0.211442\pi$
$317$	28.0375	1.57474	0.787370	+	0.616480i	$0.211442\pi$
$318$	0	0
$319$	−22.8783	−1.28094
$320$	−4.30228	−0.240505
$321$	0	0
$322$	−17.7005	−0.986410
$323$	21.9687	1.22237
$324$	0	0
$325$	−58.7391	−3.25826
$326$	−3.35527	−0.185831
$327$	0	0
$328$	4.47634	0.247165
$329$	17.8651	0.984935
$330$	0	0
$331$	25.4851	1.40079	0.700395	−	0.713756i	$-0.253007\pi$
$331$	25.4851	1.40079	0.700395	+	0.713756i	$0.253007\pi$
$332$	−16.1043	−0.883837
$333$	0	0
$334$	−3.18225	−0.174125
$335$	64.1994	3.50759
$336$	0	0
$337$	−7.13084	−0.388441	−0.194221	−	0.980958i	$-0.562218\pi$
$337$	−7.13084	−0.388441	−0.194221	+	0.980958i	$0.562218\pi$
$338$	−5.90455	−0.321165
$339$	0	0
$340$	−19.7893	−1.07322
$341$	11.4057	0.617652
$342$	0	0
$343$	18.9422	1.02278
$344$	3.56055	0.191972
$345$	0	0
$346$	1.29626	0.0696872
$347$	21.4199	1.14988	0.574941	−	0.818195i	$-0.305025\pi$
$347$	21.4199	1.14988	0.574941	+	0.818195i	$0.305025\pi$
$348$	0	0
$349$	15.9563	0.854122	0.427061	−	0.904223i	$-0.359549\pi$
$349$	15.9563	0.854122	0.427061	+	0.904223i	$0.359549\pi$
$350$	34.8646	1.86359
$351$	0	0
$352$	−2.36140	−0.125863
$353$	11.7388	0.624792	0.312396	−	0.949952i	$-0.398868\pi$
$353$	11.7388	0.624792	0.312396	+	0.949952i	$0.398868\pi$
$354$	0	0
$355$	−5.29121	−0.280828
$356$	−2.40888	−0.127671
$357$	0	0
$358$	5.70933	0.301747
$359$	31.1311	1.64303	0.821517	−	0.570184i	$-0.193128\pi$
$359$	31.1311	1.64303	0.821517	+	0.570184i	$0.193128\pi$
$360$	0	0
$361$	3.81122	0.200590
$362$	−15.9052	−0.835959
$363$	0	0
$364$	11.2208	0.588130
$365$	−71.0207	−3.71739
$366$	0	0
$367$	−14.0585	−0.733845	−0.366923	−	0.930251i	$-0.619589\pi$
$367$	−14.0585	−0.733845	−0.366923	+	0.930251i	$0.619589\pi$
$368$	−6.85875	−0.357537
$369$	0	0
$370$	−19.0109	−0.988327
$371$	−6.03444	−0.313292
$372$	0	0
$373$	36.1993	1.87433	0.937165	−	0.348886i	$-0.113440\pi$
$373$	36.1993	1.87433	0.937165	+	0.348886i	$0.113440\pi$
$374$	−10.8618	−0.561648
$375$	0	0
$376$	6.92253	0.357002
$377$	42.1248	2.16954
$378$	0	0
$379$	7.83323	0.402366	0.201183	−	0.979554i	$-0.435521\pi$
$379$	7.83323	0.402366	0.201183	+	0.979554i	$0.435521\pi$
$380$	−20.5482	−1.05410
$381$	0	0
$382$	1.86430	0.0953857
$383$	28.6388	1.46338	0.731688	−	0.681639i	$-0.238733\pi$
$383$	28.6388	1.46338	0.731688	+	0.681639i	$0.238733\pi$
$384$	0	0
$385$	26.2186	1.33622
$386$	16.9932	0.864932
$387$	0	0
$388$	−9.91733	−0.503476
$389$	29.8923	1.51560	0.757800	−	0.652487i	$-0.226274\pi$
$389$	29.8923	1.51560	0.757800	+	0.652487i	$0.226274\pi$
$390$	0	0
$391$	−31.5483	−1.59546
$392$	0.339891	0.0171671
$393$	0	0
$394$	8.68393	0.437490
$395$	−3.98441	−0.200477
$396$	0	0
$397$	5.71141	0.286648	0.143324	−	0.989676i	$-0.454221\pi$
$397$	5.71141	0.286648	0.143324	+	0.989676i	$0.454221\pi$
$398$	11.0290	0.552832
$399$	0	0
$400$	13.5096	0.675482
$401$	−0.176944	−0.00883618	−0.00441809	−	0.999990i	$-0.501406\pi$
$401$	−0.176944	−0.00883618	−0.00441809	+	0.999990i	$0.501406\pi$
$402$	0	0
$403$	−21.0008	−1.04612
$404$	−12.8905	−0.641328
$405$	0	0
$406$	−25.0032	−1.24089
$407$	−10.4345	−0.517219
$408$	0	0
$409$	10.5360	0.520973	0.260487	−	0.965477i	$-0.416117\pi$
$409$	10.5360	0.520973	0.260487	+	0.965477i	$0.416117\pi$
$410$	−19.2585	−0.951109
$411$	0	0
$412$	11.0474	0.544268
$413$	7.79404	0.383520
$414$	0	0
$415$	69.2852	3.40108
$416$	4.34794	0.213175
$417$	0	0
$418$	−11.2783	−0.551640
$419$	4.42760	0.216302	0.108151	−	0.994134i	$-0.465507\pi$
$419$	4.42760	0.216302	0.108151	+	0.994134i	$0.465507\pi$
$420$	0	0
$421$	11.8257	0.576350	0.288175	−	0.957578i	$-0.406951\pi$
$421$	11.8257	0.576350	0.288175	+	0.957578i	$0.406951\pi$
$422$	−3.71092	−0.180645
$423$	0	0
$424$	−2.33828	−0.113557
$425$	62.1405	3.01426
$426$	0	0
$427$	4.29598	0.207897
$428$	−13.6167	−0.658190
$429$	0	0
$430$	−15.3185	−0.738724
$431$	−16.9221	−0.815108	−0.407554	−	0.913181i	$-0.633618\pi$
$431$	−16.9221	−0.815108	−0.407554	+	0.913181i	$0.633618\pi$
$432$	0	0
$433$	−20.7977	−0.999475	−0.499737	−	0.866177i	$-0.666570\pi$
$433$	−20.7977	−0.999475	−0.499737	+	0.866177i	$0.666570\pi$
$434$	12.4650	0.598339
$435$	0	0
$436$	−20.0796	−0.961637
$437$	−32.7581	−1.56703
$438$	0	0
$439$	3.52842	0.168402	0.0842011	−	0.996449i	$-0.473166\pi$
$439$	3.52842	0.168402	0.0842011	+	0.996449i	$0.473166\pi$
$440$	10.1594	0.484331
$441$	0	0
$442$	19.9993	0.951268
$443$	−31.0514	−1.47530	−0.737648	−	0.675186i	$-0.764063\pi$
$443$	−31.0514	−1.47530	−0.737648	+	0.675186i	$0.764063\pi$
$444$	0	0
$445$	10.3637	0.491286
$446$	−2.31235	−0.109493
$447$	0	0
$448$	−2.58072	−0.121928
$449$	−23.2911	−1.09918	−0.549588	−	0.835436i	$-0.685215\pi$
$449$	−23.2911	−1.09918	−0.549588	+	0.835436i	$0.685215\pi$
$450$	0	0
$451$	−10.5704	−0.497742
$452$	3.00799	0.141484
$453$	0	0
$454$	9.48105	0.444968
$455$	−48.2751	−2.26317
$456$	0	0
$457$	−35.4856	−1.65995	−0.829973	−	0.557803i	$-0.811645\pi$
$457$	−35.4856	−1.65995	−0.829973	+	0.557803i	$0.811645\pi$
$458$	17.1183	0.799886
$459$	0	0
$460$	29.5083	1.37583
$461$	11.5348	0.537229	0.268614	−	0.963248i	$-0.413434\pi$
$461$	11.5348	0.537229	0.268614	+	0.963248i	$0.413434\pi$
$462$	0	0
$463$	−21.6842	−1.00775	−0.503874	−	0.863777i	$-0.668093\pi$
$463$	−21.6842	−1.00775	−0.503874	+	0.863777i	$0.668093\pi$
$464$	−9.68846	−0.449775
$465$	0	0
$466$	1.89499	0.0877838
$467$	2.36171	0.109287	0.0546434	−	0.998506i	$-0.482598\pi$
$467$	2.36171	0.109287	0.0546434	+	0.998506i	$0.482598\pi$
$468$	0	0
$469$	38.5099	1.77822
$470$	−29.7827	−1.37377
$471$	0	0
$472$	3.02010	0.139012
$473$	−8.40788	−0.386595
$474$	0	0
$475$	64.5236	2.96054
$476$	−11.8706	−0.544087
$477$	0	0
$478$	−10.4414	−0.477580
$479$	−29.8556	−1.36414	−0.682069	−	0.731287i	$-0.738920\pi$
$479$	−29.8556	−1.36414	−0.682069	+	0.731287i	$0.738920\pi$
$480$	0	0
$481$	19.2126	0.876019
$482$	−9.96284	−0.453795
$483$	0	0
$484$	−5.42380	−0.246536
$485$	42.6672	1.93742
$486$	0	0
$487$	−12.0354	−0.545375	−0.272687	−	0.962103i	$-0.587912\pi$
$487$	−12.0354	−0.545375	−0.272687	+	0.962103i	$0.587912\pi$
$488$	1.66465	0.0753550
$489$	0	0
$490$	−1.46231	−0.0660602
$491$	−36.1930	−1.63337	−0.816684	−	0.577085i	$-0.804190\pi$
$491$	−36.1930	−1.63337	−0.816684	+	0.577085i	$0.804190\pi$
$492$	0	0
$493$	−44.5641	−2.00707
$494$	20.7662	0.934316
$495$	0	0
$496$	4.83005	0.216876
$497$	−3.17393	−0.142370
$498$	0	0
$499$	16.7640	0.750461	0.375231	−	0.926931i	$-0.377563\pi$
$499$	16.7640	0.750461	0.375231	+	0.926931i	$0.377563\pi$
$500$	−36.6109	−1.63729
$501$	0	0
$502$	13.6358	0.608594
$503$	−16.8621	−0.751845	−0.375923	−	0.926651i	$-0.622674\pi$
$503$	−16.8621	−0.751845	−0.375923	+	0.926651i	$0.622674\pi$
$504$	0	0
$505$	55.4588	2.46788
$506$	16.1962	0.720010
$507$	0	0
$508$	14.6905	0.651787
$509$	−0.689284	−0.0305520	−0.0152760	−	0.999883i	$-0.504863\pi$
$509$	−0.689284	−0.0305520	−0.0152760	+	0.999883i	$0.504863\pi$
$510$	0	0
$511$	−42.6016	−1.88459
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.3175	0.587410
$515$	−47.5292	−2.09439
$516$	0	0
$517$	−16.3469	−0.718934
$518$	−11.4036	−0.501047
$519$	0	0
$520$	−18.7061	−0.820315
$521$	−39.6190	−1.73574	−0.867869	−	0.496793i	$-0.834511\pi$
$521$	−39.6190	−1.73574	−0.867869	+	0.496793i	$0.834511\pi$
$522$	0	0
$523$	−17.8746	−0.781600	−0.390800	−	0.920476i	$-0.627802\pi$
$523$	−17.8746	−0.781600	−0.390800	+	0.920476i	$0.627802\pi$
$524$	7.27060	0.317618
$525$	0	0
$526$	−1.11201	−0.0484862
$527$	22.2169	0.967781
$528$	0	0
$529$	24.0424	1.04532
$530$	10.0599	0.436976
$531$	0	0
$532$	−12.3258	−0.534391
$533$	19.4628	0.843029
$534$	0	0
$535$	58.5831	2.53277
$536$	14.9222	0.644539
$537$	0	0
$538$	−27.6181	−1.19070
$539$	−0.802617	−0.0345712
$540$	0	0
$541$	36.3068	1.56095	0.780476	−	0.625186i	$-0.214977\pi$
$541$	36.3068	1.56095	0.780476	+	0.625186i	$0.214977\pi$
$542$	−17.0063	−0.730481
$543$	0	0
$544$	−4.59971	−0.197211
$545$	86.3880	3.70045
$546$	0	0
$547$	26.7836	1.14518	0.572592	−	0.819840i	$-0.305938\pi$
$547$	26.7836	1.14518	0.572592	+	0.819840i	$0.305938\pi$
$548$	5.25460	0.224465
$549$	0	0
$550$	−31.9017	−1.36029
$551$	−46.2731	−1.97130
$552$	0	0
$553$	−2.39004	−0.101635
$554$	13.3266	0.566192
$555$	0	0
$556$	15.8385	0.671703
$557$	28.9109	1.22499	0.612496	−	0.790474i	$-0.290165\pi$
$557$	28.9109	1.22499	0.612496	+	0.790474i	$0.290165\pi$
$558$	0	0
$559$	15.4811	0.654779
$560$	11.1030	0.469187
$561$	0	0
$562$	−3.48845	−0.147151
$563$	29.1613	1.22900	0.614502	−	0.788915i	$-0.289357\pi$
$563$	29.1613	1.22900	0.614502	+	0.788915i	$0.289357\pi$
$564$	0	0
$565$	−12.9412	−0.544441
$566$	−13.7321	−0.577205
$567$	0	0
$568$	−1.22986	−0.0516038
$569$	2.95408	0.123841	0.0619207	−	0.998081i	$-0.480277\pi$
$569$	2.95408	0.123841	0.0619207	+	0.998081i	$0.480277\pi$
$570$	0	0
$571$	10.9821	0.459588	0.229794	−	0.973239i	$-0.426195\pi$
$571$	10.9821	0.459588	0.229794	+	0.973239i	$0.426195\pi$
$572$	−10.2672	−0.429294
$573$	0	0
$574$	−11.5522	−0.482178
$575$	−92.6592	−3.86416
$576$	0	0
$577$	−39.3780	−1.63933	−0.819664	−	0.572845i	$-0.805840\pi$
$577$	−39.3780	−1.63933	−0.819664	+	0.572845i	$0.805840\pi$
$578$	−4.15737	−0.172924
$579$	0	0
$580$	41.6825	1.73077
$581$	41.5606	1.72422
$582$	0	0
$583$	5.52161	0.228682
$584$	−16.5077	−0.683092
$585$	0	0
$586$	7.12602	0.294373
$587$	25.0820	1.03525	0.517623	−	0.855609i	$-0.326817\pi$
$587$	25.0820	1.03525	0.517623	+	0.855609i	$0.326817\pi$
$588$	0	0
$589$	23.0688	0.950535
$590$	−12.9933	−0.534927
$591$	0	0
$592$	−4.41878	−0.181611
$593$	−41.9959	−1.72456	−0.862282	−	0.506428i	$-0.830966\pi$
$593$	−41.9959	−1.72456	−0.862282	+	0.506428i	$0.830966\pi$
$594$	0	0
$595$	51.0706	2.09369
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−29.8214	−1.21949
$599$	−20.7536	−0.847969	−0.423984	−	0.905670i	$-0.639369\pi$
$599$	−20.7536	−0.847969	−0.423984	+	0.905670i	$0.639369\pi$
$600$	0	0
$601$	−10.8460	−0.442416	−0.221208	−	0.975227i	$-0.571000\pi$
$601$	−10.8460	−0.442416	−0.221208	+	0.975227i	$0.571000\pi$
$602$	−9.18878	−0.374507
$603$	0	0
$604$	6.79221	0.276371
$605$	23.3347	0.948691
$606$	0	0
$607$	−3.16725	−0.128555	−0.0642775	−	0.997932i	$-0.520474\pi$
$607$	−3.16725	−0.128555	−0.0642775	+	0.997932i	$0.520474\pi$
$608$	−4.77611	−0.193697
$609$	0	0
$610$	−7.16178	−0.289972
$611$	30.0987	1.21766
$612$	0	0
$613$	−43.5898	−1.76058	−0.880288	−	0.474439i	$-0.842651\pi$
$613$	−43.5898	−1.76058	−0.880288	+	0.474439i	$0.842651\pi$
$614$	30.7190	1.23972
$615$	0	0
$616$	6.09411	0.245539
$617$	37.8700	1.52459	0.762295	−	0.647230i	$-0.224073\pi$
$617$	37.8700	1.52459	0.762295	+	0.647230i	$0.224073\pi$
$618$	0	0
$619$	−6.40979	−0.257631	−0.128816	−	0.991669i	$-0.541118\pi$
$619$	−6.40979	−0.257631	−0.128816	+	0.991669i	$0.541118\pi$
$620$	−20.7802	−0.834555
$621$	0	0
$622$	−28.4027	−1.13884
$623$	6.21665	0.249065
$624$	0	0
$625$	89.9624	3.59849
$626$	−11.4329	−0.456950
$627$	0	0
$628$	−6.19817	−0.247334
$629$	−20.3251	−0.810416
$630$	0	0
$631$	−28.6878	−1.14204	−0.571021	−	0.820935i	$-0.693453\pi$
$631$	−28.6878	−1.14204	−0.571021	+	0.820935i	$0.693453\pi$
$632$	−0.926115	−0.0368389
$633$	0	0
$634$	−28.0375	−1.11351
$635$	−63.2028	−2.50813
$636$	0	0
$637$	1.47782	0.0585535
$638$	22.8783	0.905761
$639$	0	0
$640$	4.30228	0.170063
$641$	17.8921	0.706697	0.353348	−	0.935492i	$-0.385043\pi$
$641$	17.8921	0.706697	0.353348	+	0.935492i	$0.385043\pi$
$642$	0	0
$643$	−32.6114	−1.28607	−0.643035	−	0.765837i	$-0.722325\pi$
$643$	−32.6114	−1.28607	−0.643035	+	0.765837i	$0.722325\pi$
$644$	17.7005	0.697497
$645$	0	0
$646$	−21.9687	−0.864348
$647$	−4.88499	−0.192049	−0.0960243	−	0.995379i	$-0.530613\pi$
$647$	−4.88499	−0.192049	−0.0960243	+	0.995379i	$0.530613\pi$
$648$	0	0
$649$	−7.13167	−0.279942
$650$	58.7391	2.30394
$651$	0	0
$652$	3.35527	0.131402
$653$	6.49092	0.254009	0.127005	−	0.991902i	$-0.459464\pi$
$653$	6.49092	0.254009	0.127005	+	0.991902i	$0.459464\pi$
$654$	0	0
$655$	−31.2802	−1.22222
$656$	−4.47634	−0.174772
$657$	0	0
$658$	−17.8651	−0.696455
$659$	26.7244	1.04104	0.520518	−	0.853851i	$-0.325739\pi$
$659$	26.7244	1.04104	0.520518	+	0.853851i	$0.325739\pi$
$660$	0	0
$661$	−39.2623	−1.52713	−0.763563	−	0.645733i	$-0.776552\pi$
$661$	−39.2623	−1.52713	−0.763563	+	0.645733i	$0.776552\pi$
$662$	−25.4851	−0.990508
$663$	0	0
$664$	16.1043	0.624967
$665$	53.0291	2.05638
$666$	0	0
$667$	66.4507	2.57298
$668$	3.18225	0.123125
$669$	0	0
$670$	−64.1994	−2.48024
$671$	−3.93089	−0.151750
$672$	0	0
$673$	−15.1320	−0.583295	−0.291647	−	0.956526i	$-0.594203\pi$
$673$	−15.1320	−0.583295	−0.291647	+	0.956526i	$0.594203\pi$
$674$	7.13084	0.274670
$675$	0	0
$676$	5.90455	0.227098
$677$	−24.6721	−0.948225	−0.474113	−	0.880464i	$-0.657231\pi$
$677$	−24.6721	−0.948225	−0.474113	+	0.880464i	$0.657231\pi$
$678$	0	0
$679$	25.5939	0.982202
$680$	19.7893	0.758884
$681$	0	0
$682$	−11.4057	−0.436746
$683$	47.6165	1.82199	0.910997	−	0.412413i	$-0.135314\pi$
$683$	47.6165	1.82199	0.910997	+	0.412413i	$0.135314\pi$
$684$	0	0
$685$	−22.6068	−0.863761
$686$	−18.9422	−0.723216
$687$	0	0
$688$	−3.56055	−0.135745
$689$	−10.1667	−0.387320
$690$	0	0
$691$	−18.4242	−0.700889	−0.350444	−	0.936584i	$-0.613969\pi$
$691$	−18.4242	−0.700889	−0.350444	+	0.936584i	$0.613969\pi$
$692$	−1.29626	−0.0492763
$693$	0	0
$694$	−21.4199	−0.813090
$695$	−68.1418	−2.58477
$696$	0	0
$697$	−20.5899	−0.779898
$698$	−15.9563	−0.603955
$699$	0	0
$700$	−34.8646	−1.31776
$701$	5.10936	0.192978	0.0964890	−	0.995334i	$-0.469239\pi$
$701$	5.10936	0.192978	0.0964890	+	0.995334i	$0.469239\pi$
$702$	0	0
$703$	−21.1046	−0.795975
$704$	2.36140	0.0889986
$705$	0	0
$706$	−11.7388	−0.441794
$707$	33.2669	1.25113
$708$	0	0
$709$	34.6730	1.30217	0.651085	−	0.759005i	$-0.274314\pi$
$709$	34.6730	1.30217	0.651085	+	0.759005i	$0.274314\pi$
$710$	5.29121	0.198576
$711$	0	0
$712$	2.40888	0.0902767
$713$	−33.1281	−1.24066
$714$	0	0
$715$	44.1725	1.65196
$716$	−5.70933	−0.213368
$717$	0	0
$718$	−31.1311	−1.16180
$719$	48.0107	1.79050	0.895248	−	0.445567i	$-0.146998\pi$
$719$	48.0107	1.79050	0.895248	+	0.445567i	$0.146998\pi$
$720$	0	0
$721$	−28.5103	−1.06178
$722$	−3.81122	−0.141839
$723$	0	0
$724$	15.9052	0.591112
$725$	−130.888	−4.86105
$726$	0	0
$727$	30.3222	1.12459	0.562293	−	0.826938i	$-0.309919\pi$
$727$	30.3222	1.12459	0.562293	+	0.826938i	$0.309919\pi$
$728$	−11.2208	−0.415871
$729$	0	0
$730$	71.0207	2.62859
$731$	−16.3775	−0.605745
$732$	0	0
$733$	−4.69123	−0.173275	−0.0866373	−	0.996240i	$-0.527612\pi$
$733$	−4.69123	−0.173275	−0.0866373	+	0.996240i	$0.527612\pi$
$734$	14.0585	0.518907
$735$	0	0
$736$	6.85875	0.252817
$737$	−35.2372	−1.29798
$738$	0	0
$739$	28.8461	1.06112	0.530561	−	0.847647i	$-0.321981\pi$
$739$	28.8461	1.06112	0.530561	+	0.847647i	$0.321981\pi$
$740$	19.0109	0.698853
$741$	0	0
$742$	6.03444	0.221531
$743$	23.7767	0.872281	0.436141	−	0.899878i	$-0.356345\pi$
$743$	23.7767	0.872281	0.436141	+	0.899878i	$0.356345\pi$
$744$	0	0
$745$	4.30228	0.157623
$746$	−36.1993	−1.32535
$747$	0	0
$748$	10.8618	0.397145
$749$	35.1410	1.28402
$750$	0	0
$751$	−16.8466	−0.614743	−0.307371	−	0.951590i	$-0.599449\pi$
$751$	−16.8466	−0.614743	−0.307371	+	0.951590i	$0.599449\pi$
$752$	−6.92253	−0.252439
$753$	0	0
$754$	−42.1248	−1.53409
$755$	−29.2220	−1.06350
$756$	0	0
$757$	12.2260	0.444362	0.222181	−	0.975005i	$-0.428682\pi$
$757$	12.2260	0.444362	0.222181	+	0.975005i	$0.428682\pi$
$758$	−7.83323	−0.284516
$759$	0	0
$760$	20.5482	0.745361
$761$	18.2371	0.661093	0.330546	−	0.943790i	$-0.392767\pi$
$761$	18.2371	0.661093	0.330546	+	0.943790i	$0.392767\pi$
$762$	0	0
$763$	51.8197	1.87600
$764$	−1.86430	−0.0674479
$765$	0	0
$766$	−28.6388	−1.03476
$767$	13.1312	0.474141
$768$	0	0
$769$	−30.5926	−1.10320	−0.551598	−	0.834110i	$-0.685982\pi$
$769$	−30.5926	−1.10320	−0.551598	+	0.834110i	$0.685982\pi$
$770$	−26.2186	−0.944852
$771$	0	0
$772$	−16.9932	−0.611599
$773$	37.6037	1.35251	0.676255	−	0.736667i	$-0.263602\pi$
$773$	37.6037	1.35251	0.676255	+	0.736667i	$0.263602\pi$
$774$	0	0
$775$	65.2523	2.34393
$776$	9.91733	0.356012
$777$	0	0
$778$	−29.8923	−1.07169
$779$	−21.3795	−0.766000
$780$	0	0
$781$	2.90419	0.103920
$782$	31.5483	1.12816
$783$	0	0
$784$	−0.339891	−0.0121389
$785$	26.6663	0.951760
$786$	0	0
$787$	1.42552	0.0508145	0.0254072	−	0.999677i	$-0.491912\pi$
$787$	1.42552	0.0508145	0.0254072	+	0.999677i	$0.491912\pi$
$788$	−8.68393	−0.309352
$789$	0	0
$790$	3.98441	0.141759
$791$	−7.76277	−0.276012
$792$	0	0
$793$	7.23778	0.257021
$794$	−5.71141	−0.202691
$795$	0	0
$796$	−11.0290	−0.390911
$797$	−6.20525	−0.219801	−0.109901	−	0.993943i	$-0.535053\pi$
$797$	−6.20525	−0.219801	−0.109901	+	0.993943i	$0.535053\pi$
$798$	0	0
$799$	−31.8417	−1.12648
$800$	−13.5096	−0.477638
$801$	0	0
$802$	0.176944	0.00624812
$803$	38.9812	1.37562
$804$	0	0
$805$	−76.1526	−2.68402
$806$	21.0008	0.739720
$807$	0	0
$808$	12.8905	0.453488
$809$	43.4131	1.52632	0.763161	−	0.646208i	$-0.223646\pi$
$809$	43.4131	1.52632	0.763161	+	0.646208i	$0.223646\pi$
$810$	0	0
$811$	−9.38440	−0.329531	−0.164765	−	0.986333i	$-0.552687\pi$
$811$	−9.38440	−0.329531	−0.164765	+	0.986333i	$0.552687\pi$
$812$	25.0032	0.877440
$813$	0	0
$814$	10.4345	0.365729
$815$	−14.4353	−0.505647
$816$	0	0
$817$	−17.0056	−0.594950
$818$	−10.5360	−0.368384
$819$	0	0
$820$	19.2585	0.672535
$821$	17.2247	0.601146	0.300573	−	0.953759i	$-0.402822\pi$
$821$	17.2247	0.601146	0.300573	+	0.953759i	$0.402822\pi$
$822$	0	0
$823$	−39.3874	−1.37296	−0.686478	−	0.727150i	$-0.740844\pi$
$823$	−39.3874	−1.37296	−0.686478	+	0.727150i	$0.740844\pi$
$824$	−11.0474	−0.384856
$825$	0	0
$826$	−7.79404	−0.271189
$827$	10.1788	0.353953	0.176977	−	0.984215i	$-0.443368\pi$
$827$	10.1788	0.353953	0.176977	+	0.984215i	$0.443368\pi$
$828$	0	0
$829$	−11.3518	−0.394264	−0.197132	−	0.980377i	$-0.563163\pi$
$829$	−11.3518	−0.394264	−0.197132	+	0.980377i	$0.563163\pi$
$830$	−69.2852	−2.40492
$831$	0	0
$832$	−4.34794	−0.150738
$833$	−1.56340	−0.0541686
$834$	0	0
$835$	−13.6909	−0.473794
$836$	11.2783	0.390068
$837$	0	0
$838$	−4.42760	−0.152949
$839$	2.73753	0.0945099	0.0472549	−	0.998883i	$-0.484953\pi$
$839$	2.73753	0.0945099	0.0472549	+	0.998883i	$0.484953\pi$
$840$	0	0
$841$	64.8662	2.23677
$842$	−11.8257	−0.407541
$843$	0	0
$844$	3.71092	0.127735
$845$	−25.4031	−0.873892
$846$	0	0
$847$	13.9973	0.480953
$848$	2.33828	0.0802968
$849$	0	0
$850$	−62.1405	−2.13140
$851$	30.3073	1.03892
$852$	0	0
$853$	−15.3394	−0.525211	−0.262605	−	0.964903i	$-0.584582\pi$
$853$	−15.3394	−0.525211	−0.262605	+	0.964903i	$0.584582\pi$
$854$	−4.29598	−0.147006
$855$	0	0
$856$	13.6167	0.465411
$857$	9.87398	0.337289	0.168644	−	0.985677i	$-0.446061\pi$
$857$	9.87398	0.337289	0.168644	+	0.985677i	$0.446061\pi$
$858$	0	0
$859$	−53.1268	−1.81266	−0.906332	−	0.422566i	$-0.861130\pi$
$859$	−53.1268	−1.81266	−0.906332	+	0.422566i	$0.861130\pi$
$860$	15.3185	0.522357
$861$	0	0
$862$	16.9221	0.576369
$863$	−2.81682	−0.0958856	−0.0479428	−	0.998850i	$-0.515267\pi$
$863$	−2.81682	−0.0958856	−0.0479428	+	0.998850i	$0.515267\pi$
$864$	0	0
$865$	5.57687	0.189619
$866$	20.7977	0.706735
$867$	0	0
$868$	−12.4650	−0.423090
$869$	2.18693	0.0741864
$870$	0	0
$871$	64.8806	2.19840
$872$	20.0796	0.679980
$873$	0	0
$874$	32.7581	1.10806
$875$	94.4825	3.19409
$876$	0	0
$877$	−22.8402	−0.771258	−0.385629	−	0.922654i	$-0.626016\pi$
$877$	−22.8402	−0.771258	−0.385629	+	0.922654i	$0.626016\pi$
$878$	−3.52842	−0.119078
$879$	0	0
$880$	−10.1594	−0.342474
$881$	−48.7811	−1.64348	−0.821738	−	0.569865i	$-0.806995\pi$
$881$	−48.7811	−1.64348	−0.821738	+	0.569865i	$0.806995\pi$
$882$	0	0
$883$	34.0519	1.14594	0.572969	−	0.819577i	$-0.305792\pi$
$883$	34.0519	1.14594	0.572969	+	0.819577i	$0.305792\pi$
$884$	−19.9993	−0.672648
$885$	0	0
$886$	31.0514	1.04319
$887$	−18.7194	−0.628535	−0.314268	−	0.949334i	$-0.601759\pi$
$887$	−18.7194	−0.628535	−0.314268	+	0.949334i	$0.601759\pi$
$888$	0	0
$889$	−37.9121	−1.27153
$890$	−10.3637	−0.347392
$891$	0	0
$892$	2.31235	0.0774230
$893$	−33.0628	−1.10640
$894$	0	0
$895$	24.5631	0.821055
$896$	2.58072	0.0862158
$897$	0	0
$898$	23.2911	0.777235
$899$	−46.7957	−1.56072
$900$	0	0
$901$	10.7554	0.358315
$902$	10.5704	0.351957
$903$	0	0
$904$	−3.00799	−0.100044
$905$	−68.4287	−2.27465
$906$	0	0
$907$	46.9204	1.55796	0.778982	−	0.627046i	$-0.215736\pi$
$907$	46.9204	1.55796	0.778982	+	0.627046i	$0.215736\pi$
$908$	−9.48105	−0.314640
$909$	0	0
$910$	48.2751	1.60030
$911$	58.8767	1.95067	0.975337	−	0.220722i	$-0.0708412\pi$
$911$	58.8767	1.95067	0.975337	+	0.220722i	$0.0708412\pi$
$912$	0	0
$913$	−38.0286	−1.25856
$914$	35.4856	1.17376
$915$	0	0
$916$	−17.1183	−0.565605
$917$	−18.7634	−0.619621
$918$	0	0
$919$	46.8315	1.54483	0.772414	−	0.635120i	$-0.219049\pi$
$919$	46.8315	1.54483	0.772414	+	0.635120i	$0.219049\pi$
$920$	−29.5083	−0.972859
$921$	0	0
$922$	−11.5348	−0.379878
$923$	−5.34736	−0.176010
$924$	0	0
$925$	−59.6962	−1.96280
$926$	21.6842	0.712586
$927$	0	0
$928$	9.68846	0.318039
$929$	51.9419	1.70416	0.852080	−	0.523412i	$-0.175341\pi$
$929$	51.9419	1.70416	0.852080	+	0.523412i	$0.175341\pi$
$930$	0	0
$931$	−1.62335	−0.0532033
$932$	−1.89499	−0.0620725
$933$	0	0
$934$	−2.36171	−0.0772774
$935$	−46.7304	−1.52825
$936$	0	0
$937$	−18.2778	−0.597109	−0.298555	−	0.954393i	$-0.596505\pi$
$937$	−18.2778	−0.597109	−0.298555	+	0.954393i	$0.596505\pi$
$938$	−38.5099	−1.25739
$939$	0	0
$940$	29.7827	0.971405
$941$	−28.2186	−0.919900	−0.459950	−	0.887945i	$-0.652133\pi$
$941$	−28.2186	−0.919900	−0.459950	+	0.887945i	$0.652133\pi$
$942$	0	0
$943$	30.7021	0.999797
$944$	−3.02010	−0.0982960
$945$	0	0
$946$	8.40788	0.273364
$947$	24.9199	0.809788	0.404894	−	0.914364i	$-0.367308\pi$
$947$	24.9199	0.809788	0.404894	+	0.914364i	$0.367308\pi$
$948$	0	0
$949$	−71.7743	−2.32989
$950$	−64.5236	−2.09342
$951$	0	0
$952$	11.8706	0.384727
$953$	−26.5397	−0.859705	−0.429853	−	0.902899i	$-0.641435\pi$
$953$	−26.5397	−0.859705	−0.429853	+	0.902899i	$0.641435\pi$
$954$	0	0
$955$	8.02073	0.259545
$956$	10.4414	0.337700
$957$	0	0
$958$	29.8556	0.964592
$959$	−13.5606	−0.437896
$960$	0	0
$961$	−7.67062	−0.247439
$962$	−19.2126	−0.619439
$963$	0	0
$964$	9.96284	0.320882
$965$	73.1096	2.35348
$966$	0	0
$967$	−5.13455	−0.165116	−0.0825580	−	0.996586i	$-0.526309\pi$
$967$	−5.13455	−0.165116	−0.0825580	+	0.996586i	$0.526309\pi$
$968$	5.42380	0.174327
$969$	0	0
$970$	−42.6672	−1.36996
$971$	−43.7968	−1.40551	−0.702753	−	0.711434i	$-0.748046\pi$
$971$	−43.7968	−1.40551	−0.702753	+	0.711434i	$0.748046\pi$
$972$	0	0
$973$	−40.8748	−1.31039
$974$	12.0354	0.385638
$975$	0	0
$976$	−1.66465	−0.0532840
$977$	24.0519	0.769488	0.384744	−	0.923023i	$-0.374290\pi$
$977$	24.0519	0.769488	0.384744	+	0.923023i	$0.374290\pi$
$978$	0	0
$979$	−5.68833	−0.181800
$980$	1.46231	0.0467116
$981$	0	0
$982$	36.1930	1.15497
$983$	−4.05611	−0.129370	−0.0646849	−	0.997906i	$-0.520604\pi$
$983$	−4.05611	−0.129370	−0.0646849	+	0.997906i	$0.520604\pi$
$984$	0	0
$985$	37.3607	1.19041
$986$	44.5641	1.41921
$987$	0	0
$988$	−20.7662	−0.660661
$989$	24.4209	0.776540
$990$	0	0
$991$	17.1610	0.545136	0.272568	−	0.962136i	$-0.412127\pi$
$991$	17.1610	0.545136	0.272568	+	0.962136i	$0.412127\pi$
$992$	−4.83005	−0.153354
$993$	0	0
$994$	3.17393	0.100671
$995$	47.4497	1.50426
$996$	0	0
$997$	−41.9829	−1.32961	−0.664806	−	0.747016i	$-0.731486\pi$
$997$	−41.9829	−1.32961	−0.664806	+	0.747016i	$0.731486\pi$
$998$	−16.7640	−0.530656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.q.1.1	✓	14
3.2	odd	2		8046.2.a.r.1.14	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.q.1.1	✓	14	1.1	even	1	trivial
8046.2.a.r.1.14	yes	14	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.30228 q^{5} -2.58072 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.30228 q^{5} -2.58072 q^{7} -1.00000 q^{8} +4.30228 q^{10} +2.36140 q^{11} -4.34794 q^{13} +2.58072 q^{14} +1.00000 q^{16} +4.59971 q^{17} +4.77611 q^{19} -4.30228 q^{20} -2.36140 q^{22} -6.85875 q^{23} +13.5096 q^{25} +4.34794 q^{26} -2.58072 q^{28} -9.68846 q^{29} +4.83005 q^{31} -1.00000 q^{32} -4.59971 q^{34} +11.1030 q^{35} -4.41878 q^{37} -4.77611 q^{38} +4.30228 q^{40} -4.47634 q^{41} -3.56055 q^{43} +2.36140 q^{44} +6.85875 q^{46} -6.92253 q^{47} -0.339891 q^{49} -13.5096 q^{50} -4.34794 q^{52} +2.33828 q^{53} -10.1594 q^{55} +2.58072 q^{56} +9.68846 q^{58} -3.02010 q^{59} -1.66465 q^{61} -4.83005 q^{62} +1.00000 q^{64} +18.7061 q^{65} -14.9222 q^{67} +4.59971 q^{68} -11.1030 q^{70} +1.22986 q^{71} +16.5077 q^{73} +4.41878 q^{74} +4.77611 q^{76} -6.09411 q^{77} +0.926115 q^{79} -4.30228 q^{80} +4.47634 q^{82} -16.1043 q^{83} -19.7893 q^{85} +3.56055 q^{86} -2.36140 q^{88} -2.40888 q^{89} +11.2208 q^{91} -6.85875 q^{92} +6.92253 q^{94} -20.5482 q^{95} -9.91733 q^{97} +0.339891 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8}+O(q^{10}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8	\( 14 q - 14 q^{2} + 14 q^{4} - 2 q^{5} + 4 q^{7} - 14 q^{8} + 2 q^{10} - 2 q^{11} + 4 q^{13} - 4 q^{14} + 14 q^{16} - 9 q^{17} + 14 q^{19} - 2 q^{20} + 2 q^{22} - 30 q^{23} + 18 q^{25} - 4 q^{26} + 4 q^{28} - 6 q^{29} + 11 q^{31} - 14 q^{32} + 9 q^{34} + 18 q^{35} + 13 q^{37} - 14 q^{38} + 2 q^{40} + 2 q^{41} + 12 q^{43} - 2 q^{44} + 30 q^{46} - 21 q^{47} + 32 q^{49} - 18 q^{50} + 4 q^{52} - 22 q^{53} - 7 q^{55} - 4 q^{56} + 6 q^{58} - 14 q^{59} + 31 q^{61} - 11 q^{62} + 14 q^{64} + 24 q^{67} - 9 q^{68} - 18 q^{70} - 28 q^{71} + 24 q^{73} - 13 q^{74} + 14 q^{76} - 16 q^{77} + 65 q^{79} - 2 q^{80} - 2 q^{82} + 15 q^{83} - 19 q^{85} - 12 q^{86} + 2 q^{88} + 11 q^{89} + 68 q^{91} - 30 q^{92} + 21 q^{94} - 8 q^{95} + 23 q^{97} - 32 q^{98}+O(q^{100}) \) 14 * q - 14 * q^2 + 14 * q^4 - 2 * q^5 + 4 * q^7 - 14 * q^8 + 2 * q^10 - 2 * q^11 + 4 * q^13 - 4 * q^14 + 14 * q^16 - 9 * q^17 + 14 * q^19 - 2 * q^20 + 2 * q^22 - 30 * q^23 + 18 * q^25 - 4 * q^26 + 4 * q^28 - 6 * q^29 + 11 * q^31 - 14 * q^32 + 9 * q^34 + 18 * q^35 + 13 * q^37 - 14 * q^38 + 2 * q^40 + 2 * q^41 + 12 * q^43 - 2 * q^44 + 30 * q^46 - 21 * q^47 + 32 * q^49 - 18 * q^50 + 4 * q^52 - 22 * q^53 - 7 * q^55 - 4 * q^56 + 6 * q^58 - 14 * q^59 + 31 * q^61 - 11 * q^62 + 14 * q^64 + 24 * q^67 - 9 * q^68 - 18 * q^70 - 28 * q^71 + 24 * q^73 - 13 * q^74 + 14 * q^76 - 16 * q^77 + 65 * q^79 - 2 * q^80 - 2 * q^82 + 15 * q^83 - 19 * q^85 - 12 * q^86 + 2 * q^88 + 11 * q^89 + 68 * q^91 - 30 * q^92 + 21 * q^94 - 8 * q^95 + 23 * q^97 - 32 * q^98

Introduction

L-functions

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