LMFDB - Embedded newform 8019.2.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, 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("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.67622 q^{2} +5.16217 q^{4} -0.731731 q^{5} -3.74848 q^{7} -8.46268 q^{8} +O(q^{10})$	\(q-2.67622 q^{2} +5.16217 q^{4} -0.731731 q^{5} -3.74848 q^{7} -8.46268 q^{8} +1.95828 q^{10} -1.00000 q^{11} +4.67345 q^{13} +10.0318 q^{14} +12.3237 q^{16} +2.65769 q^{17} +6.00695 q^{19} -3.77732 q^{20} +2.67622 q^{22} +0.697920 q^{23} -4.46457 q^{25} -12.5072 q^{26} -19.3503 q^{28} -9.83586 q^{29} -3.11456 q^{31} -16.0555 q^{32} -7.11257 q^{34} +2.74288 q^{35} +5.05400 q^{37} -16.0759 q^{38} +6.19241 q^{40} +6.20445 q^{41} +2.72261 q^{43} -5.16217 q^{44} -1.86779 q^{46} -7.21213 q^{47} +7.05114 q^{49} +11.9482 q^{50} +24.1252 q^{52} +4.15299 q^{53} +0.731731 q^{55} +31.7222 q^{56} +26.3230 q^{58} +1.07838 q^{59} +13.0216 q^{61} +8.33526 q^{62} +18.3209 q^{64} -3.41971 q^{65} -4.03512 q^{67} +13.7194 q^{68} -7.34057 q^{70} +3.00237 q^{71} -0.538450 q^{73} -13.5256 q^{74} +31.0089 q^{76} +3.74848 q^{77} -0.0485083 q^{79} -9.01761 q^{80} -16.6045 q^{82} -8.27169 q^{83} -1.94472 q^{85} -7.28632 q^{86} +8.46268 q^{88} -3.36118 q^{89} -17.5184 q^{91} +3.60278 q^{92} +19.3013 q^{94} -4.39547 q^{95} +12.1230 q^{97} -18.8704 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * 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 $

$2$	−2.67622	−1.89238	−0.946188	−	0.323618i	$-0.895101\pi$
$2$	−2.67622	−1.89238	−0.946188	+	0.323618i	$0.895101\pi$
$3$	0	0
$3$	0	0
$4$	5.16217	2.58109
$5$	−0.731731	−0.327240	−0.163620	−	0.986523i	$-0.552317\pi$
$5$	−0.731731	−0.327240	−0.163620	+	0.986523i	$0.552317\pi$
$6$	0	0
$7$	−3.74848	−1.41679	−0.708397	−	0.705814i	$-0.750581\pi$
$7$	−3.74848	−1.41679	−0.708397	+	0.705814i	$0.750581\pi$
$8$	−8.46268	−2.99201
$9$	0	0
$10$	1.95828	0.619261
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.67345	1.29618	0.648091	−	0.761562i	$-0.275567\pi$
$13$	4.67345	1.29618	0.648091	+	0.761562i	$0.275567\pi$
$14$	10.0318	2.68111
$15$	0	0
$16$	12.3237	3.08092
$17$	2.65769	0.644584	0.322292	−	0.946640i	$-0.395547\pi$
$17$	2.65769	0.644584	0.322292	+	0.946640i	$0.395547\pi$
$18$	0	0
$19$	6.00695	1.37809	0.689044	−	0.724719i	$-0.258030\pi$
$19$	6.00695	1.37809	0.689044	+	0.724719i	$0.258030\pi$
$20$	−3.77732	−0.844635
$21$	0	0
$22$	2.67622	0.570573
$23$	0.697920	0.145526	0.0727632	−	0.997349i	$-0.476818\pi$
$23$	0.697920	0.145526	0.0727632	+	0.997349i	$0.476818\pi$
$24$	0	0
$25$	−4.46457	−0.892914
$26$	−12.5072	−2.45287
$27$	0	0
$28$	−19.3503	−3.65687
$29$	−9.83586	−1.82647	−0.913237	−	0.407429i	$-0.866425\pi$
$29$	−9.83586	−1.82647	−0.913237	+	0.407429i	$0.866425\pi$
$30$	0	0
$31$	−3.11456	−0.559391	−0.279696	−	0.960089i	$-0.590234\pi$
$31$	−3.11456	−0.559391	−0.279696	+	0.960089i	$0.590234\pi$
$32$	−16.0555	−2.83824
$33$	0	0
$34$	−7.11257	−1.21980
$35$	2.74288	0.463632
$36$	0	0
$37$	5.05400	0.830873	0.415436	−	0.909622i	$-0.363629\pi$
$37$	5.05400	0.830873	0.415436	+	0.909622i	$0.363629\pi$
$38$	−16.0759	−2.60786
$39$	0	0
$40$	6.19241	0.979105
$41$	6.20445	0.968972	0.484486	−	0.874799i	$-0.339007\pi$
$41$	6.20445	0.968972	0.484486	+	0.874799i	$0.339007\pi$
$42$	0	0
$43$	2.72261	0.415194	0.207597	−	0.978214i	$-0.433436\pi$
$43$	2.72261	0.415194	0.207597	+	0.978214i	$0.433436\pi$
$44$	−5.16217	−0.778227
$45$	0	0
$46$	−1.86779	−0.275391
$47$	−7.21213	−1.05200	−0.525998	−	0.850486i	$-0.676308\pi$
$47$	−7.21213	−1.05200	−0.525998	+	0.850486i	$0.676308\pi$
$48$	0	0
$49$	7.05114	1.00731
$50$	11.9482	1.68973
$51$	0	0
$52$	24.1252	3.34556
$53$	4.15299	0.570457	0.285228	−	0.958460i	$-0.407931\pi$
$53$	4.15299	0.570457	0.285228	+	0.958460i	$0.407931\pi$
$54$	0	0
$55$	0.731731	0.0986666
$56$	31.7222	4.23906
$57$	0	0
$58$	26.3230	3.45637
$59$	1.07838	0.140393	0.0701966	−	0.997533i	$-0.477637\pi$
$59$	1.07838	0.140393	0.0701966	+	0.997533i	$0.477637\pi$
$60$	0	0
$61$	13.0216	1.66725	0.833624	−	0.552333i	$-0.186262\pi$
$61$	13.0216	1.66725	0.833624	+	0.552333i	$0.186262\pi$
$62$	8.33526	1.05858
$63$	0	0
$64$	18.3209	2.29011
$65$	−3.41971	−0.424163
$66$	0	0
$67$	−4.03512	−0.492968	−0.246484	−	0.969147i	$-0.579275\pi$
$67$	−4.03512	−0.492968	−0.246484	+	0.969147i	$0.579275\pi$
$68$	13.7194	1.66373
$69$	0	0
$70$	−7.34057	−0.877366
$71$	3.00237	0.356316	0.178158	−	0.984002i	$-0.442986\pi$
$71$	3.00237	0.356316	0.178158	+	0.984002i	$0.442986\pi$
$72$	0	0
$73$	−0.538450	−0.0630209	−0.0315104	−	0.999503i	$-0.510032\pi$
$73$	−0.538450	−0.0630209	−0.0315104	+	0.999503i	$0.510032\pi$
$74$	−13.5256	−1.57232
$75$	0	0
$76$	31.0089	3.55697
$77$	3.74848	0.427179
$78$	0	0
$79$	−0.0485083	−0.00545760	−0.00272880	−	0.999996i	$-0.500869\pi$
$79$	−0.0485083	−0.00545760	−0.00272880	+	0.999996i	$0.500869\pi$
$80$	−9.01761	−1.00820
$81$	0	0
$82$	−16.6045	−1.83366
$83$	−8.27169	−0.907936	−0.453968	−	0.891018i	$-0.649992\pi$
$83$	−8.27169	−0.907936	−0.453968	+	0.891018i	$0.649992\pi$
$84$	0	0
$85$	−1.94472	−0.210934
$86$	−7.28632	−0.785704
$87$	0	0
$88$	8.46268	0.902124
$89$	−3.36118	−0.356285	−0.178142	−	0.984005i	$-0.557009\pi$
$89$	−3.36118	−0.356285	−0.178142	+	0.984005i	$0.557009\pi$
$90$	0	0
$91$	−17.5184	−1.83642
$92$	3.60278	0.375616
$93$	0	0
$94$	19.3013	1.99077
$95$	−4.39547	−0.450966
$96$	0	0
$97$	12.1230	1.23090	0.615451	−	0.788175i	$-0.288974\pi$
$97$	12.1230	1.23090	0.615451	+	0.788175i	$0.288974\pi$
$98$	−18.8704	−1.90620
$99$	0	0
$100$	−23.0469	−2.30469
$101$	12.1572	1.20968	0.604842	−	0.796346i	$-0.293236\pi$
$101$	12.1572	1.20968	0.604842	+	0.796346i	$0.293236\pi$
$102$	0	0
$103$	−16.0329	−1.57977	−0.789886	−	0.613254i	$-0.789860\pi$
$103$	−16.0329	−1.57977	−0.789886	+	0.613254i	$0.789860\pi$
$104$	−39.5499	−3.87819
$105$	0	0
$106$	−11.1143	−1.07952
$107$	−6.14959	−0.594503	−0.297252	−	0.954799i	$-0.596070\pi$
$107$	−6.14959	−0.594503	−0.297252	+	0.954799i	$0.596070\pi$
$108$	0	0
$109$	19.8183	1.89825	0.949126	−	0.314898i	$-0.101970\pi$
$109$	19.8183	1.89825	0.949126	+	0.314898i	$0.101970\pi$
$110$	−1.95828	−0.186714
$111$	0	0
$112$	−46.1951	−4.36502
$113$	−4.99228	−0.469634	−0.234817	−	0.972040i	$-0.575449\pi$
$113$	−4.99228	−0.469634	−0.234817	+	0.972040i	$0.575449\pi$
$114$	0	0
$115$	−0.510690	−0.0476221
$116$	−50.7744	−4.71429
$117$	0	0
$118$	−2.88599	−0.265677
$119$	−9.96231	−0.913243
$120$	0	0
$121$	1.00000	0.0909091
$122$	−34.8488	−3.15506
$123$	0	0
$124$	−16.0779	−1.44384
$125$	6.92552	0.619438
$126$	0	0
$127$	−17.1640	−1.52306	−0.761530	−	0.648130i	$-0.775552\pi$
$127$	−17.1640	−1.52306	−0.761530	+	0.648130i	$0.775552\pi$
$128$	−16.9196	−1.49550
$129$	0	0
$130$	9.15192	0.802676
$131$	−15.6005	−1.36302	−0.681512	−	0.731807i	$-0.738677\pi$
$131$	−15.6005	−1.36302	−0.681512	+	0.731807i	$0.738677\pi$
$132$	0	0
$133$	−22.5170	−1.95247
$134$	10.7989	0.932881
$135$	0	0
$136$	−22.4912	−1.92860
$137$	−1.36318	−0.116465	−0.0582323	−	0.998303i	$-0.518546\pi$
$137$	−1.36318	−0.116465	−0.0582323	+	0.998303i	$0.518546\pi$
$138$	0	0
$139$	13.0538	1.10721	0.553605	−	0.832779i	$-0.313252\pi$
$139$	13.0538	1.10721	0.553605	+	0.832779i	$0.313252\pi$
$140$	14.1592	1.19667
$141$	0	0
$142$	−8.03501	−0.674283
$143$	−4.67345	−0.390814
$144$	0	0
$145$	7.19721	0.597696
$146$	1.44101	0.119259
$147$	0	0
$148$	26.0896	2.14455
$149$	0.954673	0.0782098	0.0391049	−	0.999235i	$-0.487549\pi$
$149$	0.954673	0.0782098	0.0391049	+	0.999235i	$0.487549\pi$
$150$	0	0
$151$	0.917278	0.0746470	0.0373235	−	0.999303i	$-0.488117\pi$
$151$	0.917278	0.0746470	0.0373235	+	0.999303i	$0.488117\pi$
$152$	−50.8349	−4.12325
$153$	0	0
$154$	−10.0318	−0.808384
$155$	2.27902	0.183055
$156$	0	0
$157$	−7.30756	−0.583207	−0.291604	−	0.956539i	$-0.594189\pi$
$157$	−7.30756	−0.583207	−0.291604	+	0.956539i	$0.594189\pi$
$158$	0.129819	0.0103278
$159$	0	0
$160$	11.7483	0.928788
$161$	−2.61614	−0.206181
$162$	0	0
$163$	−10.6767	−0.836261	−0.418130	−	0.908387i	$-0.637315\pi$
$163$	−10.6767	−0.836261	−0.418130	+	0.908387i	$0.637315\pi$
$164$	32.0284	2.50100
$165$	0	0
$166$	22.1369	1.71816
$167$	24.2747	1.87843	0.939217	−	0.343324i	$-0.111553\pi$
$167$	24.2747	1.87843	0.939217	+	0.343324i	$0.111553\pi$
$168$	0	0
$169$	8.84117	0.680090
$170$	5.20449	0.399166
$171$	0	0
$172$	14.0546	1.07165
$173$	−12.6559	−0.962207	−0.481104	−	0.876664i	$-0.659764\pi$
$173$	−12.6559	−0.962207	−0.481104	+	0.876664i	$0.659764\pi$
$174$	0	0
$175$	16.7354	1.26507
$176$	−12.3237	−0.928931
$177$	0	0
$178$	8.99528	0.674224
$179$	−5.06085	−0.378266	−0.189133	−	0.981951i	$-0.560568\pi$
$179$	−5.06085	−0.378266	−0.189133	+	0.981951i	$0.560568\pi$
$180$	0	0
$181$	1.41012	0.104813	0.0524066	−	0.998626i	$-0.483311\pi$
$181$	1.41012	0.104813	0.0524066	+	0.998626i	$0.483311\pi$
$182$	46.8831	3.47520
$183$	0	0
$184$	−5.90627	−0.435416
$185$	−3.69817	−0.271895
$186$	0	0
$187$	−2.65769	−0.194350
$188$	−37.2302	−2.71529
$189$	0	0
$190$	11.7633	0.853397
$191$	8.74974	0.633109	0.316554	−	0.948574i	$-0.397474\pi$
$191$	8.74974	0.633109	0.316554	+	0.948574i	$0.397474\pi$
$192$	0	0
$193$	−23.6070	−1.69927	−0.849633	−	0.527374i	$-0.823177\pi$
$193$	−23.6070	−1.69927	−0.849633	+	0.527374i	$0.823177\pi$
$194$	−32.4438	−2.32933
$195$	0	0
$196$	36.3992	2.59994
$197$	−4.40579	−0.313900	−0.156950	−	0.987607i	$-0.550166\pi$
$197$	−4.40579	−0.313900	−0.156950	+	0.987607i	$0.550166\pi$
$198$	0	0
$199$	−1.26608	−0.0897503	−0.0448751	−	0.998993i	$-0.514289\pi$
$199$	−1.26608	−0.0897503	−0.0448751	+	0.998993i	$0.514289\pi$
$200$	37.7822	2.67160
$201$	0	0
$202$	−32.5353	−2.28918
$203$	36.8696	2.58774
$204$	0	0
$205$	−4.53999	−0.317087
$206$	42.9077	2.98952
$207$	0	0
$208$	57.5941	3.99343
$209$	−6.00695	−0.415509
$210$	0	0
$211$	−3.61363	−0.248773	−0.124386	−	0.992234i	$-0.539696\pi$
$211$	−3.61363	−0.248773	−0.124386	+	0.992234i	$0.539696\pi$
$212$	21.4384	1.47240
$213$	0	0
$214$	16.4577	1.12502
$215$	−1.99222	−0.135868
$216$	0	0
$217$	11.6749	0.792542
$218$	−53.0383	−3.59220
$219$	0	0
$220$	3.77732	0.254667
$221$	12.4206	0.835499
$222$	0	0
$223$	−6.06430	−0.406096	−0.203048	−	0.979169i	$-0.565085\pi$
$223$	−6.06430	−0.406096	−0.203048	+	0.979169i	$0.565085\pi$
$224$	60.1839	4.02121
$225$	0	0
$226$	13.3605	0.888725
$227$	19.0743	1.26600	0.633002	−	0.774150i	$-0.281822\pi$
$227$	19.0743	1.26600	0.633002	+	0.774150i	$0.281822\pi$
$228$	0	0
$229$	−15.6447	−1.03383	−0.516917	−	0.856036i	$-0.672920\pi$
$229$	−15.6447	−1.03383	−0.516917	+	0.856036i	$0.672920\pi$
$230$	1.36672	0.0901189
$231$	0	0
$232$	83.2377	5.46482
$233$	0.253304	0.0165945	0.00829726	−	0.999966i	$-0.497359\pi$
$233$	0.253304	0.0165945	0.00829726	+	0.999966i	$0.497359\pi$
$234$	0	0
$235$	5.27734	0.344256
$236$	5.56678	0.362367
$237$	0	0
$238$	26.6614	1.72820
$239$	−18.3268	−1.18546	−0.592732	−	0.805400i	$-0.701951\pi$
$239$	−18.3268	−1.18546	−0.592732	+	0.805400i	$0.701951\pi$
$240$	0	0
$241$	6.54367	0.421515	0.210757	−	0.977538i	$-0.432407\pi$
$241$	6.54367	0.421515	0.210757	+	0.977538i	$0.432407\pi$
$242$	−2.67622	−0.172034
$243$	0	0
$244$	67.2198	4.30331
$245$	−5.15954	−0.329631
$246$	0	0
$247$	28.0732	1.78626
$248$	26.3575	1.67370
$249$	0	0
$250$	−18.5342	−1.17221
$251$	−15.8615	−1.00117	−0.500584	−	0.865688i	$-0.666881\pi$
$251$	−15.8615	−1.00117	−0.500584	+	0.865688i	$0.666881\pi$
$252$	0	0
$253$	−0.697920	−0.0438779
$254$	45.9347	2.88220
$255$	0	0
$256$	8.63901	0.539938
$257$	11.8986	0.742216	0.371108	−	0.928590i	$-0.378978\pi$
$257$	11.8986	0.742216	0.371108	+	0.928590i	$0.378978\pi$
$258$	0	0
$259$	−18.9448	−1.17718
$260$	−17.6531	−1.09480
$261$	0	0
$262$	41.7505	2.57935
$263$	27.5438	1.69843	0.849213	−	0.528051i	$-0.177077\pi$
$263$	27.5438	1.69843	0.849213	+	0.528051i	$0.177077\pi$
$264$	0	0
$265$	−3.03887	−0.186676
$266$	60.2604	3.69480
$267$	0	0
$268$	−20.8300	−1.27239
$269$	−19.0093	−1.15902	−0.579508	−	0.814967i	$-0.696755\pi$
$269$	−19.0093	−1.15902	−0.579508	+	0.814967i	$0.696755\pi$
$270$	0	0
$271$	29.4696	1.79015	0.895075	−	0.445915i	$-0.147122\pi$
$271$	29.4696	1.79015	0.895075	+	0.445915i	$0.147122\pi$
$272$	32.7525	1.98591
$273$	0	0
$274$	3.64818	0.220395
$275$	4.46457	0.269224
$276$	0	0
$277$	11.3497	0.681937	0.340969	−	0.940075i	$-0.389245\pi$
$277$	11.3497	0.681937	0.340969	+	0.940075i	$0.389245\pi$
$278$	−34.9349	−2.09526
$279$	0	0
$280$	−23.2121	−1.38719
$281$	3.23914	0.193231	0.0966153	−	0.995322i	$-0.469198\pi$
$281$	3.23914	0.193231	0.0966153	+	0.995322i	$0.469198\pi$
$282$	0	0
$283$	−20.4209	−1.21389	−0.606947	−	0.794742i	$-0.707606\pi$
$283$	−20.4209	−1.21389	−0.606947	+	0.794742i	$0.707606\pi$
$284$	15.4987	0.919681
$285$	0	0
$286$	12.5072	0.739567
$287$	−23.2573	−1.37283
$288$	0	0
$289$	−9.93668	−0.584511
$290$	−19.2613	−1.13106
$291$	0	0
$292$	−2.77957	−0.162662
$293$	10.9143	0.637622	0.318811	−	0.947818i	$-0.396716\pi$
$293$	10.9143	0.637622	0.318811	+	0.947818i	$0.396716\pi$
$294$	0	0
$295$	−0.789085	−0.0459423
$296$	−42.7704	−2.48598
$297$	0	0
$298$	−2.55492	−0.148002
$299$	3.26170	0.188629
$300$	0	0
$301$	−10.2057	−0.588245
$302$	−2.45484	−0.141260
$303$	0	0
$304$	74.0276	4.24578
$305$	−9.52833	−0.545590
$306$	0	0
$307$	−10.6588	−0.608332	−0.304166	−	0.952619i	$-0.598378\pi$
$307$	−10.6588	−0.608332	−0.304166	+	0.952619i	$0.598378\pi$
$308$	19.3503	1.10259
$309$	0	0
$310$	−6.09917	−0.346410
$311$	−8.29920	−0.470605	−0.235302	−	0.971922i	$-0.575608\pi$
$311$	−8.29920	−0.470605	−0.235302	+	0.971922i	$0.575608\pi$
$312$	0	0
$313$	−4.31587	−0.243948	−0.121974	−	0.992533i	$-0.538922\pi$
$313$	−4.31587	−0.243948	−0.121974	+	0.992533i	$0.538922\pi$
$314$	19.5567	1.10365
$315$	0	0
$316$	−0.250408	−0.0140865
$317$	−32.7049	−1.83689	−0.918444	−	0.395551i	$-0.870554\pi$
$317$	−32.7049	−1.83689	−0.918444	+	0.395551i	$0.870554\pi$
$318$	0	0
$319$	9.83586	0.550703
$320$	−13.4059	−0.749415
$321$	0	0
$322$	7.00139	0.390172
$323$	15.9646	0.888295
$324$	0	0
$325$	−20.8650	−1.15738
$326$	28.5731	1.58252
$327$	0	0
$328$	−52.5062	−2.89917
$329$	27.0345	1.49046
$330$	0	0
$331$	17.9567	0.986992	0.493496	−	0.869748i	$-0.335719\pi$
$331$	17.9567	0.986992	0.493496	+	0.869748i	$0.335719\pi$
$332$	−42.6999	−2.34346
$333$	0	0
$334$	−64.9646	−3.55470
$335$	2.95262	0.161319
$336$	0	0
$337$	−26.3376	−1.43470	−0.717350	−	0.696713i	$-0.754645\pi$
$337$	−26.3376	−1.43470	−0.717350	+	0.696713i	$0.754645\pi$
$338$	−23.6610	−1.28699
$339$	0	0
$340$	−10.0390	−0.544439
$341$	3.11456	0.168663
$342$	0	0
$343$	−0.191688	−0.0103502
$344$	−23.0406	−1.24226
$345$	0	0
$346$	33.8699	1.82086
$347$	−16.8915	−0.906783	−0.453392	−	0.891311i	$-0.649786\pi$
$347$	−16.8915	−0.906783	−0.453392	+	0.891311i	$0.649786\pi$
$348$	0	0
$349$	−12.0198	−0.643403	−0.321701	−	0.946841i	$-0.604255\pi$
$349$	−12.0198	−0.643403	−0.321701	+	0.946841i	$0.604255\pi$
$350$	−44.7876	−2.39400
$351$	0	0
$352$	16.0555	0.855763
$353$	9.51552	0.506460	0.253230	−	0.967406i	$-0.418507\pi$
$353$	9.51552	0.506460	0.253230	+	0.967406i	$0.418507\pi$
$354$	0	0
$355$	−2.19693	−0.116601
$356$	−17.3510	−0.919601
$357$	0	0
$358$	13.5440	0.715821
$359$	29.2152	1.54192	0.770960	−	0.636883i	$-0.219777\pi$
$359$	29.2152	1.54192	0.770960	+	0.636883i	$0.219777\pi$
$360$	0	0
$361$	17.0834	0.899129
$362$	−3.77379	−0.198346
$363$	0	0
$364$	−90.4328	−4.73997
$365$	0.394001	0.0206230
$366$	0	0
$367$	7.66613	0.400169	0.200084	−	0.979779i	$-0.435878\pi$
$367$	7.66613	0.400169	0.200084	+	0.979779i	$0.435878\pi$
$368$	8.60094	0.448355
$369$	0	0
$370$	9.89713	0.514527
$371$	−15.5674	−0.808220
$372$	0	0
$373$	20.3061	1.05141	0.525706	−	0.850666i	$-0.323801\pi$
$373$	20.3061	1.05141	0.525706	+	0.850666i	$0.323801\pi$
$374$	7.11257	0.367782
$375$	0	0
$376$	61.0339	3.14758
$377$	−45.9675	−2.36744
$378$	0	0
$379$	1.41960	0.0729202	0.0364601	−	0.999335i	$-0.488392\pi$
$379$	1.41960	0.0729202	0.0364601	+	0.999335i	$0.488392\pi$
$380$	−22.6902	−1.16398
$381$	0	0
$382$	−23.4163	−1.19808
$383$	17.8960	0.914444	0.457222	−	0.889353i	$-0.348844\pi$
$383$	17.8960	0.914444	0.457222	+	0.889353i	$0.348844\pi$
$384$	0	0
$385$	−2.74288	−0.139790
$386$	63.1775	3.21565
$387$	0	0
$388$	62.5809	3.17706
$389$	16.8314	0.853384	0.426692	−	0.904397i	$-0.359679\pi$
$389$	16.8314	0.853384	0.426692	+	0.904397i	$0.359679\pi$
$390$	0	0
$391$	1.85486	0.0938041
$392$	−59.6715	−3.01387
$393$	0	0
$394$	11.7909	0.594016
$395$	0.0354950	0.00178595
$396$	0	0
$397$	−20.7911	−1.04348	−0.521738	−	0.853106i	$-0.674716\pi$
$397$	−20.7911	−1.04348	−0.521738	+	0.853106i	$0.674716\pi$
$398$	3.38832	0.169841
$399$	0	0
$400$	−55.0199	−2.75099
$401$	11.7150	0.585018	0.292509	−	0.956263i	$-0.405510\pi$
$401$	11.7150	0.585018	0.292509	+	0.956263i	$0.405510\pi$
$402$	0	0
$403$	−14.5557	−0.725074
$404$	62.7574	3.12230
$405$	0	0
$406$	−98.6712	−4.89697
$407$	−5.05400	−0.250518
$408$	0	0
$409$	−2.00914	−0.0993456	−0.0496728	−	0.998766i	$-0.515818\pi$
$409$	−2.00914	−0.0993456	−0.0496728	+	0.998766i	$0.515818\pi$
$410$	12.1500	0.600047
$411$	0	0
$412$	−82.7647	−4.07753
$413$	−4.04229	−0.198908
$414$	0	0
$415$	6.05266	0.297113
$416$	−75.0348	−3.67888
$417$	0	0
$418$	16.0759	0.786300
$419$	21.8033	1.06516	0.532581	−	0.846379i	$-0.321222\pi$
$419$	21.8033	1.06516	0.532581	+	0.846379i	$0.321222\pi$
$420$	0	0
$421$	−16.5019	−0.804254	−0.402127	−	0.915584i	$-0.631729\pi$
$421$	−16.5019	−0.804254	−0.402127	+	0.915584i	$0.631729\pi$
$422$	9.67088	0.470771
$423$	0	0
$424$	−35.1454	−1.70681
$425$	−11.8654	−0.575558
$426$	0	0
$427$	−48.8113	−2.36215
$428$	−31.7452	−1.53446
$429$	0	0
$430$	5.33163	0.257114
$431$	29.9564	1.44295	0.721474	−	0.692442i	$-0.243465\pi$
$431$	29.9564	1.44295	0.721474	+	0.692442i	$0.243465\pi$
$432$	0	0
$433$	32.1300	1.54407	0.772034	−	0.635581i	$-0.219239\pi$
$433$	32.1300	1.54407	0.772034	+	0.635581i	$0.219239\pi$
$434$	−31.2446	−1.49979
$435$	0	0
$436$	102.306	4.89955
$437$	4.19237	0.200548
$438$	0	0
$439$	3.16917	0.151256	0.0756280	−	0.997136i	$-0.475904\pi$
$439$	3.16917	0.151256	0.0756280	+	0.997136i	$0.475904\pi$
$440$	−6.19241	−0.295211
$441$	0	0
$442$	−33.2403	−1.58108
$443$	14.9788	0.711662	0.355831	−	0.934550i	$-0.384198\pi$
$443$	14.9788	0.711662	0.355831	+	0.934550i	$0.384198\pi$
$444$	0	0
$445$	2.45948	0.116591
$446$	16.2294	0.768485
$447$	0	0
$448$	−68.6754	−3.24461
$449$	40.1781	1.89612	0.948060	−	0.318091i	$-0.103042\pi$
$449$	40.1781	1.89612	0.948060	+	0.318091i	$0.103042\pi$
$450$	0	0
$451$	−6.20445	−0.292156
$452$	−25.7710	−1.21217
$453$	0	0
$454$	−51.0470	−2.39576
$455$	12.8187	0.600952
$456$	0	0
$457$	32.8364	1.53602	0.768011	−	0.640437i	$-0.221247\pi$
$457$	32.8364	1.53602	0.768011	+	0.640437i	$0.221247\pi$
$458$	41.8688	1.95640
$459$	0	0
$460$	−2.63627	−0.122917
$461$	17.5697	0.818304	0.409152	−	0.912466i	$-0.365825\pi$
$461$	17.5697	0.818304	0.409152	+	0.912466i	$0.365825\pi$
$462$	0	0
$463$	−20.0790	−0.933150	−0.466575	−	0.884482i	$-0.654512\pi$
$463$	−20.0790	−0.933150	−0.466575	+	0.884482i	$0.654512\pi$
$464$	−121.214	−5.62721
$465$	0	0
$466$	−0.677899	−0.0314030
$467$	11.7666	0.544492	0.272246	−	0.962228i	$-0.412234\pi$
$467$	11.7666	0.544492	0.272246	+	0.962228i	$0.412234\pi$
$468$	0	0
$469$	15.1256	0.698434
$470$	−14.1233	−0.651461
$471$	0	0
$472$	−9.12598	−0.420057
$473$	−2.72261	−0.125186
$474$	0	0
$475$	−26.8184	−1.23051
$476$	−51.4271	−2.35716
$477$	0	0
$478$	49.0467	2.24334
$479$	19.8893	0.908766	0.454383	−	0.890807i	$-0.349860\pi$
$479$	19.8893	0.908766	0.454383	+	0.890807i	$0.349860\pi$
$480$	0	0
$481$	23.6196	1.07696
$482$	−17.5123	−0.797664
$483$	0	0
$484$	5.16217	0.234644
$485$	−8.87077	−0.402801
$486$	0	0
$487$	−8.00764	−0.362861	−0.181430	−	0.983404i	$-0.558073\pi$
$487$	−8.00764	−0.362861	−0.181430	+	0.983404i	$0.558073\pi$
$488$	−110.198	−4.98842
$489$	0	0
$490$	13.8081	0.623785
$491$	28.8379	1.30144	0.650718	−	0.759320i	$-0.274468\pi$
$491$	28.8379	1.30144	0.650718	+	0.759320i	$0.274468\pi$
$492$	0	0
$493$	−26.1407	−1.17732
$494$	−75.1302	−3.38027
$495$	0	0
$496$	−38.3828	−1.72344
$497$	−11.2543	−0.504826
$498$	0	0
$499$	17.1539	0.767916	0.383958	−	0.923351i	$-0.374561\pi$
$499$	17.1539	0.767916	0.383958	+	0.923351i	$0.374561\pi$
$500$	35.7507	1.59882
$501$	0	0
$502$	42.4489	1.89459
$503$	4.48281	0.199878	0.0999392	−	0.994994i	$-0.468135\pi$
$503$	4.48281	0.199878	0.0999392	+	0.994994i	$0.468135\pi$
$504$	0	0
$505$	−8.89579	−0.395857
$506$	1.86779	0.0830334
$507$	0	0
$508$	−88.6036	−3.93115
$509$	16.8877	0.748533	0.374266	−	0.927321i	$-0.377895\pi$
$509$	16.8877	0.748533	0.374266	+	0.927321i	$0.377895\pi$
$510$	0	0
$511$	2.01837	0.0892876
$512$	10.7194	0.473733
$513$	0	0
$514$	−31.8434	−1.40455
$515$	11.7318	0.516965
$516$	0	0
$517$	7.21213	0.317189
$518$	50.7006	2.22766
$519$	0	0
$520$	28.9399	1.26910
$521$	−12.2366	−0.536097	−0.268048	−	0.963405i	$-0.586379\pi$
$521$	−12.2366	−0.536097	−0.268048	+	0.963405i	$0.586379\pi$
$522$	0	0
$523$	−6.62970	−0.289897	−0.144948	−	0.989439i	$-0.546302\pi$
$523$	−6.62970	−0.289897	−0.144948	+	0.989439i	$0.546302\pi$
$524$	−80.5326	−3.51808
$525$	0	0
$526$	−73.7134	−3.21406
$527$	−8.27753	−0.360575
$528$	0	0
$529$	−22.5129	−0.978822
$530$	8.13270	0.353262
$531$	0	0
$532$	−116.236	−5.03949
$533$	28.9962	1.25597
$534$	0	0
$535$	4.49985	0.194545
$536$	34.1479	1.47496
$537$	0	0
$538$	50.8731	2.19329
$539$	−7.05114	−0.303714
$540$	0	0
$541$	10.8298	0.465610	0.232805	−	0.972523i	$-0.425210\pi$
$541$	10.8298	0.465610	0.232805	+	0.972523i	$0.425210\pi$
$542$	−78.8672	−3.38764
$543$	0	0
$544$	−42.6706	−1.82949
$545$	−14.5017	−0.621184
$546$	0	0
$547$	4.34719	0.185872	0.0929362	−	0.995672i	$-0.470375\pi$
$547$	4.34719	0.185872	0.0929362	+	0.995672i	$0.470375\pi$
$548$	−7.03698	−0.300605
$549$	0	0
$550$	−11.9482	−0.509472
$551$	−59.0835	−2.51704
$552$	0	0
$553$	0.181832	0.00773230
$554$	−30.3743	−1.29048
$555$	0	0
$556$	67.3860	2.85780
$557$	20.2886	0.859655	0.429827	−	0.902911i	$-0.358574\pi$
$557$	20.2886	0.859655	0.429827	+	0.902911i	$0.358574\pi$
$558$	0	0
$559$	12.7240	0.538168
$560$	33.8024	1.42841
$561$	0	0
$562$	−8.66865	−0.365665
$563$	−13.4832	−0.568249	−0.284125	−	0.958787i	$-0.591703\pi$
$563$	−13.4832	−0.568249	−0.284125	+	0.958787i	$0.591703\pi$
$564$	0	0
$565$	3.65301	0.153683
$566$	54.6508	2.29714
$567$	0	0
$568$	−25.4081	−1.06610
$569$	22.3973	0.938944	0.469472	−	0.882947i	$-0.344444\pi$
$569$	22.3973	0.938944	0.469472	+	0.882947i	$0.344444\pi$
$570$	0	0
$571$	−32.9687	−1.37970	−0.689849	−	0.723954i	$-0.742323\pi$
$571$	−32.9687	−1.37970	−0.689849	+	0.723954i	$0.742323\pi$
$572$	−24.1252	−1.00872
$573$	0	0
$574$	62.2417	2.59792
$575$	−3.11591	−0.129943
$576$	0	0
$577$	27.6437	1.15082	0.575412	−	0.817864i	$-0.304842\pi$
$577$	27.6437	1.15082	0.575412	+	0.817864i	$0.304842\pi$
$578$	26.5928	1.10611
$579$	0	0
$580$	37.1532	1.54270
$581$	31.0063	1.28636
$582$	0	0
$583$	−4.15299	−0.171999
$584$	4.55673	0.188559
$585$	0	0
$586$	−29.2092	−1.20662
$587$	−23.6295	−0.975295	−0.487647	−	0.873041i	$-0.662145\pi$
$587$	−23.6295	−0.975295	−0.487647	+	0.873041i	$0.662145\pi$
$588$	0	0
$589$	−18.7090	−0.770891
$590$	2.11177	0.0869401
$591$	0	0
$592$	62.2838	2.55985
$593$	5.41533	0.222381	0.111190	−	0.993799i	$-0.464534\pi$
$593$	5.41533	0.222381	0.111190	+	0.993799i	$0.464534\pi$
$594$	0	0
$595$	7.28974	0.298850
$596$	4.92818	0.201866
$597$	0	0
$598$	−8.72904	−0.356957
$599$	15.2773	0.624215	0.312108	−	0.950047i	$-0.398965\pi$
$599$	15.2773	0.624215	0.312108	+	0.950047i	$0.398965\pi$
$600$	0	0
$601$	5.35652	0.218497	0.109249	−	0.994014i	$-0.465156\pi$
$601$	5.35652	0.218497	0.109249	+	0.994014i	$0.465156\pi$
$602$	27.3126	1.11318
$603$	0	0
$604$	4.73514	0.192670
$605$	−0.731731	−0.0297491
$606$	0	0
$607$	23.2129	0.942183	0.471092	−	0.882084i	$-0.343860\pi$
$607$	23.2129	0.942183	0.471092	+	0.882084i	$0.343860\pi$
$608$	−96.4448	−3.91135
$609$	0	0
$610$	25.4999	1.03246
$611$	−33.7055	−1.36358
$612$	0	0
$613$	1.81854	0.0734502	0.0367251	−	0.999325i	$-0.488307\pi$
$613$	1.81854	0.0734502	0.0367251	+	0.999325i	$0.488307\pi$
$614$	28.5254	1.15119
$615$	0	0
$616$	−31.7222	−1.27812
$617$	1.39855	0.0563036	0.0281518	−	0.999604i	$-0.491038\pi$
$617$	1.39855	0.0563036	0.0281518	+	0.999604i	$0.491038\pi$
$618$	0	0
$619$	−24.3029	−0.976815	−0.488407	−	0.872616i	$-0.662422\pi$
$619$	−24.3029	−0.976815	−0.488407	+	0.872616i	$0.662422\pi$
$620$	11.7647	0.472482
$621$	0	0
$622$	22.2105	0.890561
$623$	12.5993	0.504782
$624$	0	0
$625$	17.2552	0.690209
$626$	11.5502	0.461640
$627$	0	0
$628$	−37.7229	−1.50531
$629$	13.4320	0.535568
$630$	0	0
$631$	7.90849	0.314832	0.157416	−	0.987532i	$-0.449684\pi$
$631$	7.90849	0.314832	0.157416	+	0.987532i	$0.449684\pi$
$632$	0.410510	0.0163292
$633$	0	0
$634$	87.5255	3.47608
$635$	12.5594	0.498407
$636$	0	0
$637$	32.9532	1.30565
$638$	−26.3230	−1.04214
$639$	0	0
$640$	12.3806	0.489387
$641$	29.5968	1.16900	0.584502	−	0.811393i	$-0.301290\pi$
$641$	29.5968	1.16900	0.584502	+	0.811393i	$0.301290\pi$
$642$	0	0
$643$	41.9011	1.65242	0.826209	−	0.563364i	$-0.190493\pi$
$643$	41.9011	1.65242	0.826209	+	0.563364i	$0.190493\pi$
$644$	−13.5050	−0.532171
$645$	0	0
$646$	−42.7249	−1.68099
$647$	−16.2625	−0.639346	−0.319673	−	0.947528i	$-0.603573\pi$
$647$	−16.2625	−0.639346	−0.319673	+	0.947528i	$0.603573\pi$
$648$	0	0
$649$	−1.07838	−0.0423301
$650$	55.8393	2.19020
$651$	0	0
$652$	−55.1148	−2.15846
$653$	13.1641	0.515149	0.257575	−	0.966258i	$-0.417077\pi$
$653$	13.1641	0.515149	0.257575	+	0.966258i	$0.417077\pi$
$654$	0	0
$655$	11.4154	0.446036
$656$	76.4616	2.98532
$657$	0	0
$658$	−72.3505	−2.82052
$659$	−4.45142	−0.173403	−0.0867013	−	0.996234i	$-0.527633\pi$
$659$	−4.45142	−0.173403	−0.0867013	+	0.996234i	$0.527633\pi$
$660$	0	0
$661$	6.14339	0.238950	0.119475	−	0.992837i	$-0.461879\pi$
$661$	6.14339	0.238950	0.119475	+	0.992837i	$0.461879\pi$
$662$	−48.0563	−1.86776
$663$	0	0
$664$	70.0007	2.71655
$665$	16.4764	0.638926
$666$	0	0
$667$	−6.86465	−0.265800
$668$	125.310	4.84840
$669$	0	0
$670$	−7.90188	−0.305276
$671$	−13.0216	−0.502694
$672$	0	0
$673$	−17.2150	−0.663589	−0.331795	−	0.943352i	$-0.607654\pi$
$673$	−17.2150	−0.663589	−0.331795	+	0.943352i	$0.607654\pi$
$674$	70.4853	2.71499
$675$	0	0
$676$	45.6397	1.75537
$677$	−9.53802	−0.366576	−0.183288	−	0.983059i	$-0.558674\pi$
$677$	−9.53802	−0.366576	−0.183288	+	0.983059i	$0.558674\pi$
$678$	0	0
$679$	−45.4428	−1.74393
$680$	16.4575	0.631116
$681$	0	0
$682$	−8.33526	−0.319173
$683$	29.5286	1.12988	0.564941	−	0.825131i	$-0.308899\pi$
$683$	29.5286	1.12988	0.564941	+	0.825131i	$0.308899\pi$
$684$	0	0
$685$	0.997483	0.0381119
$686$	0.512999	0.0195864
$687$	0	0
$688$	33.5526	1.27918
$689$	19.4088	0.739416
$690$	0	0
$691$	1.67820	0.0638417	0.0319209	−	0.999490i	$-0.489838\pi$
$691$	1.67820	0.0638417	0.0319209	+	0.999490i	$0.489838\pi$
$692$	−65.3317	−2.48354
$693$	0	0
$694$	45.2054	1.71597
$695$	−9.55188	−0.362324
$696$	0	0
$697$	16.4895	0.624585
$698$	32.1675	1.21756
$699$	0	0
$700$	86.3908	3.26527
$701$	45.9169	1.73426	0.867128	−	0.498085i	$-0.165963\pi$
$701$	45.9169	1.73426	0.867128	+	0.498085i	$0.165963\pi$
$702$	0	0
$703$	30.3591	1.14502
$704$	−18.3209	−0.690493
$705$	0	0
$706$	−25.4657	−0.958413
$707$	−45.5710	−1.71387
$708$	0	0
$709$	37.1736	1.39608	0.698042	−	0.716057i	$-0.254055\pi$
$709$	37.1736	1.39608	0.698042	+	0.716057i	$0.254055\pi$
$710$	5.87947	0.220653
$711$	0	0
$712$	28.4446	1.06601
$713$	−2.17371	−0.0814063
$714$	0	0
$715$	3.41971	0.127890
$716$	−26.1250	−0.976337
$717$	0	0
$718$	−78.1865	−2.91789
$719$	5.14202	0.191765	0.0958824	−	0.995393i	$-0.469433\pi$
$719$	5.14202	0.191765	0.0958824	+	0.995393i	$0.469433\pi$
$720$	0	0
$721$	60.0992	2.23821
$722$	−45.7191	−1.70149
$723$	0	0
$724$	7.27927	0.270532
$725$	43.9129	1.63088
$726$	0	0
$727$	−44.5630	−1.65275	−0.826375	−	0.563120i	$-0.809601\pi$
$727$	−44.5630	−1.65275	−0.826375	+	0.563120i	$0.809601\pi$
$728$	148.252	5.49460
$729$	0	0
$730$	−1.05444	−0.0390264
$731$	7.23586	0.267628
$732$	0	0
$733$	−11.0384	−0.407714	−0.203857	−	0.979001i	$-0.565348\pi$
$733$	−11.0384	−0.407714	−0.203857	+	0.979001i	$0.565348\pi$
$734$	−20.5163	−0.757269
$735$	0	0
$736$	−11.2055	−0.413040
$737$	4.03512	0.148635
$738$	0	0
$739$	33.6265	1.23697	0.618485	−	0.785797i	$-0.287747\pi$
$739$	33.6265	1.23697	0.618485	+	0.785797i	$0.287747\pi$
$740$	−19.0906	−0.701784
$741$	0	0
$742$	41.6619	1.52945
$743$	27.6128	1.01302	0.506508	−	0.862235i	$-0.330936\pi$
$743$	27.6128	1.01302	0.506508	+	0.862235i	$0.330936\pi$
$744$	0	0
$745$	−0.698564	−0.0255934
$746$	−54.3437	−1.98967
$747$	0	0
$748$	−13.7194	−0.501633
$749$	23.0516	0.842288
$750$	0	0
$751$	−22.6811	−0.827646	−0.413823	−	0.910357i	$-0.635807\pi$
$751$	−22.6811	−0.827646	−0.413823	+	0.910357i	$0.635807\pi$
$752$	−88.8799	−3.24111
$753$	0	0
$754$	123.019	4.48009
$755$	−0.671201	−0.0244275
$756$	0	0
$757$	−19.3176	−0.702110	−0.351055	−	0.936355i	$-0.614177\pi$
$757$	−19.3176	−0.702110	−0.351055	+	0.936355i	$0.614177\pi$
$758$	−3.79918	−0.137992
$759$	0	0
$760$	37.1975	1.34929
$761$	−45.6174	−1.65363	−0.826815	−	0.562474i	$-0.809850\pi$
$761$	−45.6174	−1.65363	−0.826815	+	0.562474i	$0.809850\pi$
$762$	0	0
$763$	−74.2887	−2.68943
$764$	45.1677	1.63411
$765$	0	0
$766$	−47.8938	−1.73047
$767$	5.03976	0.181975
$768$	0	0
$769$	−0.936460	−0.0337696	−0.0168848	−	0.999857i	$-0.505375\pi$
$769$	−0.936460	−0.0337696	−0.0168848	+	0.999857i	$0.505375\pi$
$770$	7.34057	0.264536
$771$	0	0
$772$	−121.863	−4.38595
$773$	−19.4821	−0.700722	−0.350361	−	0.936615i	$-0.613941\pi$
$773$	−19.4821	−0.700722	−0.350361	+	0.936615i	$0.613941\pi$
$774$	0	0
$775$	13.9052	0.499488
$776$	−102.593	−3.68287
$777$	0	0
$778$	−45.0445	−1.61492
$779$	37.2698	1.33533
$780$	0	0
$781$	−3.00237	−0.107433
$782$	−4.96401	−0.177513
$783$	0	0
$784$	86.8959	3.10342
$785$	5.34717	0.190849
$786$	0	0
$787$	21.2840	0.758694	0.379347	−	0.925254i	$-0.376149\pi$
$787$	21.2840	0.758694	0.379347	+	0.925254i	$0.376149\pi$
$788$	−22.7435	−0.810202
$789$	0	0
$790$	−0.0949926	−0.00337968
$791$	18.7135	0.665375
$792$	0	0
$793$	60.8559	2.16106
$794$	55.6417	1.97465
$795$	0	0
$796$	−6.53574	−0.231653
$797$	−20.1721	−0.714534	−0.357267	−	0.934002i	$-0.616291\pi$
$797$	−20.1721	−0.714534	−0.357267	+	0.934002i	$0.616291\pi$
$798$	0	0
$799$	−19.1676	−0.678101
$800$	71.6810	2.53431
$801$	0	0
$802$	−31.3519	−1.10707
$803$	0.538450	0.0190015
$804$	0	0
$805$	1.91431	0.0674707
$806$	38.9544	1.37211
$807$	0	0
$808$	−102.882	−3.61938
$809$	−51.8017	−1.82125	−0.910625	−	0.413234i	$-0.864399\pi$
$809$	−51.8017	−1.82125	−0.910625	+	0.413234i	$0.864399\pi$
$810$	0	0
$811$	−26.4241	−0.927875	−0.463938	−	0.885868i	$-0.653564\pi$
$811$	−26.4241	−0.927875	−0.463938	+	0.885868i	$0.653564\pi$
$812$	190.327	6.67917
$813$	0	0
$814$	13.5256	0.474073
$815$	7.81245	0.273658
$816$	0	0
$817$	16.3546	0.572175
$818$	5.37691	0.187999
$819$	0	0
$820$	−23.4362	−0.818428
$821$	30.1986	1.05394	0.526969	−	0.849884i	$-0.323328\pi$
$821$	30.1986	1.05394	0.526969	+	0.849884i	$0.323328\pi$
$822$	0	0
$823$	−10.4758	−0.365164	−0.182582	−	0.983191i	$-0.558446\pi$
$823$	−10.4758	−0.365164	−0.182582	+	0.983191i	$0.558446\pi$
$824$	135.681	4.72669
$825$	0	0
$826$	10.8181	0.376409
$827$	41.4669	1.44195	0.720973	−	0.692963i	$-0.243695\pi$
$827$	41.4669	1.44195	0.720973	+	0.692963i	$0.243695\pi$
$828$	0	0
$829$	20.0068	0.694865	0.347432	−	0.937705i	$-0.387054\pi$
$829$	20.0068	0.694865	0.347432	+	0.937705i	$0.387054\pi$
$830$	−16.1983	−0.562250
$831$	0	0
$832$	85.6217	2.96840
$833$	18.7397	0.649293
$834$	0	0
$835$	−17.7626	−0.614699
$836$	−31.0089	−1.07247
$837$	0	0
$838$	−58.3506	−2.01569
$839$	32.0255	1.10564	0.552822	−	0.833299i	$-0.313551\pi$
$839$	32.0255	1.10564	0.552822	+	0.833299i	$0.313551\pi$
$840$	0	0
$841$	67.7442	2.33601
$842$	44.1628	1.52195
$843$	0	0
$844$	−18.6542	−0.642103
$845$	−6.46936	−0.222553
$846$	0	0
$847$	−3.74848	−0.128799
$848$	51.1800	1.75753
$849$	0	0
$850$	31.7546	1.08917
$851$	3.52729	0.120914
$852$	0	0
$853$	43.9625	1.50525	0.752624	−	0.658451i	$-0.228788\pi$
$853$	43.9625	1.50525	0.752624	+	0.658451i	$0.228788\pi$
$854$	130.630	4.47007
$855$	0	0
$856$	52.0420	1.77876
$857$	−26.5468	−0.906823	−0.453411	−	0.891301i	$-0.649793\pi$
$857$	−26.5468	−0.906823	−0.453411	+	0.891301i	$0.649793\pi$
$858$	0	0
$859$	−3.65170	−0.124595	−0.0622973	−	0.998058i	$-0.519843\pi$
$859$	−3.65170	−0.124595	−0.0622973	+	0.998058i	$0.519843\pi$
$860$	−10.2842	−0.350688
$861$	0	0
$862$	−80.1699	−2.73060
$863$	−20.3412	−0.692422	−0.346211	−	0.938157i	$-0.612532\pi$
$863$	−20.3412	−0.692422	−0.346211	+	0.938157i	$0.612532\pi$
$864$	0	0
$865$	9.26069	0.314873
$866$	−85.9870	−2.92196
$867$	0	0
$868$	60.2677	2.04562
$869$	0.0485083	0.00164553
$870$	0	0
$871$	−18.8579	−0.638977
$872$	−167.716	−5.67958
$873$	0	0
$874$	−11.2197	−0.379513
$875$	−25.9602	−0.877615
$876$	0	0
$877$	−9.54675	−0.322371	−0.161185	−	0.986924i	$-0.551532\pi$
$877$	−9.54675	−0.322371	−0.161185	+	0.986924i	$0.551532\pi$
$878$	−8.48140	−0.286233
$879$	0	0
$880$	9.01761	0.303984
$881$	−42.7070	−1.43884	−0.719418	−	0.694578i	$-0.755591\pi$
$881$	−42.7070	−1.43884	−0.719418	+	0.694578i	$0.755591\pi$
$882$	0	0
$883$	−25.7393	−0.866197	−0.433099	−	0.901347i	$-0.642580\pi$
$883$	−25.7393	−0.866197	−0.433099	+	0.901347i	$0.642580\pi$
$884$	64.1172	2.15650
$885$	0	0
$886$	−40.0865	−1.34673
$887$	−2.76783	−0.0929345	−0.0464672	−	0.998920i	$-0.514796\pi$
$887$	−2.76783	−0.0929345	−0.0464672	+	0.998920i	$0.514796\pi$
$888$	0	0
$889$	64.3390	2.15786
$890$	−6.58213	−0.220633
$891$	0	0
$892$	−31.3049	−1.04817
$893$	−43.3229	−1.44975
$894$	0	0
$895$	3.70319	0.123784
$896$	63.4230	2.11881
$897$	0	0
$898$	−107.525	−3.58817
$899$	30.6344	1.02171
$900$	0	0
$901$	11.0374	0.367707
$902$	16.6045	0.552869
$903$	0	0
$904$	42.2481	1.40515
$905$	−1.03183	−0.0342991
$906$	0	0
$907$	−13.9244	−0.462354	−0.231177	−	0.972912i	$-0.574258\pi$
$907$	−13.9244	−0.462354	−0.231177	+	0.972912i	$0.574258\pi$
$908$	98.4647	3.26767
$909$	0	0
$910$	−34.3058	−1.13723
$911$	6.34149	0.210103	0.105051	−	0.994467i	$-0.466499\pi$
$911$	6.34149	0.210103	0.105051	+	0.994467i	$0.466499\pi$
$912$	0	0
$913$	8.27169	0.273753
$914$	−87.8775	−2.90673
$915$	0	0
$916$	−80.7608	−2.66841
$917$	58.4783	1.93112
$918$	0	0
$919$	32.7956	1.08183	0.540913	−	0.841078i	$-0.318079\pi$
$919$	32.7956	1.08183	0.540913	+	0.841078i	$0.318079\pi$
$920$	4.32181	0.142486
$921$	0	0
$922$	−47.0205	−1.54854
$923$	14.0314	0.461850
$924$	0	0
$925$	−22.5639	−0.741898
$926$	53.7359	1.76587
$927$	0	0
$928$	157.920	5.18398
$929$	6.77296	0.222214	0.111107	−	0.993808i	$-0.464560\pi$
$929$	6.77296	0.222214	0.111107	+	0.993808i	$0.464560\pi$
$930$	0	0
$931$	42.3558	1.38816
$932$	1.30760	0.0428319
$933$	0	0
$934$	−31.4900	−1.03038
$935$	1.94472	0.0635990
$936$	0	0
$937$	−39.4894	−1.29006	−0.645032	−	0.764156i	$-0.723156\pi$
$937$	−39.4894	−1.29006	−0.645032	+	0.764156i	$0.723156\pi$
$938$	−40.4794	−1.32170
$939$	0	0
$940$	27.2425	0.888553
$941$	39.4793	1.28699	0.643495	−	0.765450i	$-0.277484\pi$
$941$	39.4793	1.28699	0.643495	+	0.765450i	$0.277484\pi$
$942$	0	0
$943$	4.33021	0.141011
$944$	13.2896	0.432540
$945$	0	0
$946$	7.28632	0.236899
$947$	−41.8682	−1.36053	−0.680266	−	0.732965i	$-0.738136\pi$
$947$	−41.8682	−1.36053	−0.680266	+	0.732965i	$0.738136\pi$
$948$	0	0
$949$	−2.51642	−0.0816866
$950$	71.7721	2.32860
$951$	0	0
$952$	84.3078	2.73243
$953$	15.9067	0.515270	0.257635	−	0.966242i	$-0.417057\pi$
$953$	15.9067	0.515270	0.257635	+	0.966242i	$0.417057\pi$
$954$	0	0
$955$	−6.40246	−0.207179
$956$	−94.6062	−3.05978
$957$	0	0
$958$	−53.2282	−1.71973
$959$	5.10987	0.165006
$960$	0	0
$961$	−21.2995	−0.687081
$962$	−63.2114	−2.03802
$963$	0	0
$964$	33.7795	1.08797
$965$	17.2740	0.556069
$966$	0	0
$967$	7.17898	0.230860	0.115430	−	0.993316i	$-0.463175\pi$
$967$	7.17898	0.230860	0.115430	+	0.993316i	$0.463175\pi$
$968$	−8.46268	−0.272001
$969$	0	0
$970$	23.7402	0.762250
$971$	18.3013	0.587318	0.293659	−	0.955910i	$-0.405127\pi$
$971$	18.3013	0.587318	0.293659	+	0.955910i	$0.405127\pi$
$972$	0	0
$973$	−48.9320	−1.56869
$974$	21.4302	0.686669
$975$	0	0
$976$	160.474	5.13665
$977$	−21.6640	−0.693093	−0.346546	−	0.938033i	$-0.612646\pi$
$977$	−21.6640	−0.693093	−0.346546	+	0.938033i	$0.612646\pi$
$978$	0	0
$979$	3.36118	0.107424
$980$	−26.6344	−0.850805
$981$	0	0
$982$	−77.1766	−2.46280
$983$	−13.1616	−0.419790	−0.209895	−	0.977724i	$-0.567312\pi$
$983$	−13.1616	−0.419790	−0.209895	+	0.977724i	$0.567312\pi$
$984$	0	0
$985$	3.22386	0.102721
$986$	69.9583	2.22793
$987$	0	0
$988$	144.919	4.61048
$989$	1.90017	0.0604218
$990$	0	0
$991$	58.4245	1.85592	0.927958	−	0.372685i	$-0.121563\pi$
$991$	58.4245	1.85592	0.927958	+	0.372685i	$0.121563\pi$
$992$	50.0059	1.58769
$993$	0	0
$994$	30.1191	0.955321
$995$	0.926433	0.0293699
$996$	0	0
$997$	24.6434	0.780464	0.390232	−	0.920717i	$-0.372395\pi$
$997$	24.6434	0.780464	0.390232	+	0.920717i	$0.372395\pi$
$998$	−45.9078	−1.45319
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.k.1.2		51
3.2	odd	2		8019.2.a.l.1.50		51
27.5	odd	18		297.2.j.c.133.17	yes	102
27.11	odd	18		297.2.j.c.67.17	✓	102
27.16	even	9		891.2.j.c.496.1		102
27.22	even	9		891.2.j.c.397.1		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.17	✓	102	27.11	odd	18
297.2.j.c.133.17	yes	102	27.5	odd	18
891.2.j.c.397.1		102	27.22	even	9
891.2.j.c.496.1		102	27.16	even	9
8019.2.a.k.1.2		51	1.1	even	1	trivial
8019.2.a.l.1.50		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67622 q^{2} +5.16217 q^{4} -0.731731 q^{5} -3.74848 q^{7} -8.46268 q^{8} +O(q^{10})\)	\(q-2.67622 q^{2} +5.16217 q^{4} -0.731731 q^{5} -3.74848 q^{7} -8.46268 q^{8} +1.95828 q^{10} -1.00000 q^{11} +4.67345 q^{13} +10.0318 q^{14} +12.3237 q^{16} +2.65769 q^{17} +6.00695 q^{19} -3.77732 q^{20} +2.67622 q^{22} +0.697920 q^{23} -4.46457 q^{25} -12.5072 q^{26} -19.3503 q^{28} -9.83586 q^{29} -3.11456 q^{31} -16.0555 q^{32} -7.11257 q^{34} +2.74288 q^{35} +5.05400 q^{37} -16.0759 q^{38} +6.19241 q^{40} +6.20445 q^{41} +2.72261 q^{43} -5.16217 q^{44} -1.86779 q^{46} -7.21213 q^{47} +7.05114 q^{49} +11.9482 q^{50} +24.1252 q^{52} +4.15299 q^{53} +0.731731 q^{55} +31.7222 q^{56} +26.3230 q^{58} +1.07838 q^{59} +13.0216 q^{61} +8.33526 q^{62} +18.3209 q^{64} -3.41971 q^{65} -4.03512 q^{67} +13.7194 q^{68} -7.34057 q^{70} +3.00237 q^{71} -0.538450 q^{73} -13.5256 q^{74} +31.0089 q^{76} +3.74848 q^{77} -0.0485083 q^{79} -9.01761 q^{80} -16.6045 q^{82} -8.27169 q^{83} -1.94472 q^{85} -7.28632 q^{86} +8.46268 q^{88} -3.36118 q^{89} -17.5184 q^{91} +3.60278 q^{92} +19.3013 q^{94} -4.39547 q^{95} +12.1230 q^{97} -18.8704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} - 6 q^{5} + 12 q^{7} + 18 q^{10} - 51 q^{11} + 30 q^{13} - 12 q^{14} + 78 q^{16} + 30 q^{19} - 18 q^{20} - 3 q^{23} + 75 q^{25} - 9 q^{26} + 36 q^{28} + 42 q^{31} + 15 q^{32} + 42 q^{34} + 9 q^{35} + 48 q^{37} + 3 q^{38} + 54 q^{40} + 18 q^{43} - 60 q^{44} + 42 q^{46} - 30 q^{47} + 99 q^{49} + 30 q^{50} + 60 q^{52} - 18 q^{53} + 6 q^{55} - 21 q^{56} + 30 q^{58} - 24 q^{59} + 99 q^{61} + 114 q^{64} + 15 q^{65} + 39 q^{67} + 39 q^{68} + 48 q^{70} - 30 q^{71} + 69 q^{73} + 90 q^{76} - 12 q^{77} + 48 q^{79} - 42 q^{80} + 42 q^{82} + 21 q^{83} + 84 q^{85} - 24 q^{86} - 15 q^{89} + 69 q^{91} + 66 q^{92} + 66 q^{94} + 12 q^{95} + 72 q^{97} + 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 - 6 * q^5 + 12 * q^7 + 18 * q^10 - 51 * q^11 + 30 * q^13 - 12 * q^14 + 78 * q^16 + 30 * q^19 - 18 * q^20 - 3 * q^23 + 75 * q^25 - 9 * q^26 + 36 * q^28 + 42 * q^31 + 15 * q^32 + 42 * q^34 + 9 * q^35 + 48 * q^37 + 3 * q^38 + 54 * q^40 + 18 * q^43 - 60 * q^44 + 42 * q^46 - 30 * q^47 + 99 * q^49 + 30 * q^50 + 60 * q^52 - 18 * q^53 + 6 * q^55 - 21 * q^56 + 30 * q^58 - 24 * q^59 + 99 * q^61 + 114 * q^64 + 15 * q^65 + 39 * q^67 + 39 * q^68 + 48 * q^70 - 30 * q^71 + 69 * q^73 + 90 * q^76 - 12 * q^77 + 48 * q^79 - 42 * q^80 + 42 * q^82 + 21 * q^83 + 84 * q^85 - 24 * q^86 - 15 * q^89 + 69 * q^91 + 66 * q^92 + 66 * q^94 + 12 * q^95 + 72 * q^97 + 57 * q^98

Introduction

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