LMFDB - Embedded newform 8019.2.a.i.1.18

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:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-1.04797 q^{2} -0.901755 q^{4} -2.67790 q^{5} +1.71332 q^{7} +3.04096 q^{8} +O(q^{10})$	\(q-1.04797 q^{2} -0.901755 q^{4} -2.67790 q^{5} +1.71332 q^{7} +3.04096 q^{8} +2.80637 q^{10} -1.00000 q^{11} +3.35189 q^{13} -1.79551 q^{14} -1.38333 q^{16} -0.123959 q^{17} +1.74858 q^{19} +2.41481 q^{20} +1.04797 q^{22} +3.85738 q^{23} +2.17116 q^{25} -3.51268 q^{26} -1.54499 q^{28} -2.71328 q^{29} -8.66848 q^{31} -4.63223 q^{32} +0.129906 q^{34} -4.58809 q^{35} +5.07344 q^{37} -1.83246 q^{38} -8.14339 q^{40} -10.4799 q^{41} +2.74179 q^{43} +0.901755 q^{44} -4.04242 q^{46} +1.08334 q^{47} -4.06455 q^{49} -2.27532 q^{50} -3.02258 q^{52} -10.9784 q^{53} +2.67790 q^{55} +5.21012 q^{56} +2.84345 q^{58} -1.85845 q^{59} +13.6403 q^{61} +9.08432 q^{62} +7.62110 q^{64} -8.97602 q^{65} -1.74066 q^{67} +0.111781 q^{68} +4.80819 q^{70} +3.66818 q^{71} +1.48373 q^{73} -5.31682 q^{74} -1.57679 q^{76} -1.71332 q^{77} -6.84139 q^{79} +3.70442 q^{80} +10.9826 q^{82} -3.58772 q^{83} +0.331951 q^{85} -2.87332 q^{86} -3.04096 q^{88} +1.24310 q^{89} +5.74284 q^{91} -3.47841 q^{92} -1.13531 q^{94} -4.68253 q^{95} +8.49140 q^{97} +4.25953 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	$ 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) $ 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * 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 $

$2$	−1.04797	−0.741028	−0.370514	−	0.928827i	$-0.620818\pi$
$2$	−1.04797	−0.741028	−0.370514	+	0.928827i	$0.620818\pi$
$3$	0	0
$3$	0	0
$4$	−0.901755	−0.450877
$5$	−2.67790	−1.19759	−0.598797	−	0.800901i	$-0.704354\pi$
$5$	−2.67790	−1.19759	−0.598797	+	0.800901i	$0.704354\pi$
$6$	0	0
$7$	1.71332	0.647572	0.323786	−	0.946130i	$-0.395044\pi$
$7$	1.71332	0.647572	0.323786	+	0.946130i	$0.395044\pi$
$8$	3.04096	1.07514
$9$	0	0
$10$	2.80637	0.887451
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.35189	0.929646	0.464823	−	0.885404i	$-0.346118\pi$
$13$	3.35189	0.929646	0.464823	+	0.885404i	$0.346118\pi$
$14$	−1.79551	−0.479869
$15$	0	0
$16$	−1.38333	−0.345832
$17$	−0.123959	−0.0300645	−0.0150323	−	0.999887i	$-0.504785\pi$
$17$	−0.123959	−0.0300645	−0.0150323	+	0.999887i	$0.504785\pi$
$18$	0	0
$19$	1.74858	0.401152	0.200576	−	0.979678i	$-0.435719\pi$
$19$	1.74858	0.401152	0.200576	+	0.979678i	$0.435719\pi$
$20$	2.41481	0.539968
$21$	0	0
$22$	1.04797	0.223428
$23$	3.85738	0.804319	0.402159	−	0.915570i	$-0.368260\pi$
$23$	3.85738	0.804319	0.402159	+	0.915570i	$0.368260\pi$
$24$	0	0
$25$	2.17116	0.434232
$26$	−3.51268	−0.688894
$27$	0	0
$28$	−1.54499	−0.291976
$29$	−2.71328	−0.503844	−0.251922	−	0.967748i	$-0.581063\pi$
$29$	−2.71328	−0.503844	−0.251922	+	0.967748i	$0.581063\pi$
$30$	0	0
$31$	−8.66848	−1.55690	−0.778452	−	0.627704i	$-0.783995\pi$
$31$	−8.66848	−1.55690	−0.778452	+	0.627704i	$0.783995\pi$
$32$	−4.63223	−0.818869
$33$	0	0
$34$	0.129906	0.0222787
$35$	−4.58809	−0.775529
$36$	0	0
$37$	5.07344	0.834068	0.417034	−	0.908891i	$-0.363070\pi$
$37$	5.07344	0.834068	0.417034	+	0.908891i	$0.363070\pi$
$38$	−1.83246	−0.297265
$39$	0	0
$40$	−8.14339	−1.28758
$41$	−10.4799	−1.63668	−0.818341	−	0.574733i	$-0.805106\pi$
$41$	−10.4799	−1.63668	−0.818341	+	0.574733i	$0.805106\pi$
$42$	0	0
$43$	2.74179	0.418119	0.209059	−	0.977903i	$-0.432960\pi$
$43$	2.74179	0.418119	0.209059	+	0.977903i	$0.432960\pi$
$44$	0.901755	0.135945
$45$	0	0
$46$	−4.04242	−0.596023
$47$	1.08334	0.158021	0.0790107	−	0.996874i	$-0.474824\pi$
$47$	1.08334	0.158021	0.0790107	+	0.996874i	$0.474824\pi$
$48$	0	0
$49$	−4.06455	−0.580650
$50$	−2.27532	−0.321778
$51$	0	0
$52$	−3.02258	−0.419156
$53$	−10.9784	−1.50801	−0.754003	−	0.656871i	$-0.771880\pi$
$53$	−10.9784	−1.50801	−0.754003	+	0.656871i	$0.771880\pi$
$54$	0	0
$55$	2.67790	0.361088
$56$	5.21012	0.696232
$57$	0	0
$58$	2.84345	0.373363
$59$	−1.85845	−0.241949	−0.120975	−	0.992656i	$-0.538602\pi$
$59$	−1.85845	−0.241949	−0.120975	+	0.992656i	$0.538602\pi$
$60$	0	0
$61$	13.6403	1.74646	0.873230	−	0.487308i	$-0.162021\pi$
$61$	13.6403	1.74646	0.873230	+	0.487308i	$0.162021\pi$
$62$	9.08432	1.15371
$63$	0	0
$64$	7.62110	0.952638
$65$	−8.97602	−1.11334
$66$	0	0
$67$	−1.74066	−0.212656	−0.106328	−	0.994331i	$-0.533909\pi$
$67$	−1.74066	−0.212656	−0.106328	+	0.994331i	$0.533909\pi$
$68$	0.111781	0.0135554
$69$	0	0
$70$	4.80819	0.574689
$71$	3.66818	0.435333	0.217667	−	0.976023i	$-0.430155\pi$
$71$	3.66818	0.435333	0.217667	+	0.976023i	$0.430155\pi$
$72$	0	0
$73$	1.48373	0.173657	0.0868286	−	0.996223i	$-0.472327\pi$
$73$	1.48373	0.173657	0.0868286	+	0.996223i	$0.472327\pi$
$74$	−5.31682	−0.618068
$75$	0	0
$76$	−1.57679	−0.180870
$77$	−1.71332	−0.195250
$78$	0	0
$79$	−6.84139	−0.769717	−0.384858	−	0.922976i	$-0.625750\pi$
$79$	−6.84139	−0.769717	−0.384858	+	0.922976i	$0.625750\pi$
$80$	3.70442	0.414167
$81$	0	0
$82$	10.9826	1.21283
$83$	−3.58772	−0.393803	−0.196902	−	0.980423i	$-0.563088\pi$
$83$	−3.58772	−0.393803	−0.196902	+	0.980423i	$0.563088\pi$
$84$	0	0
$85$	0.331951	0.0360051
$86$	−2.87332	−0.309838
$87$	0	0
$88$	−3.04096	−0.324167
$89$	1.24310	0.131768	0.0658840	−	0.997827i	$-0.479013\pi$
$89$	1.24310	0.131768	0.0658840	+	0.997827i	$0.479013\pi$
$90$	0	0
$91$	5.74284	0.602013
$92$	−3.47841	−0.362649
$93$	0	0
$94$	−1.13531	−0.117098
$95$	−4.68253	−0.480417
$96$	0	0
$97$	8.49140	0.862171	0.431086	−	0.902311i	$-0.358131\pi$
$97$	8.49140	0.862171	0.431086	+	0.902311i	$0.358131\pi$
$98$	4.25953	0.430278
$99$	0	0
$100$	−1.95785	−0.195785
$101$	15.4057	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$101$	15.4057	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$102$	0	0
$103$	9.27263	0.913660	0.456830	−	0.889554i	$-0.348985\pi$
$103$	9.27263	0.913660	0.456830	+	0.889554i	$0.348985\pi$
$104$	10.1929	0.999500
$105$	0	0
$106$	11.5051	1.11747
$107$	18.5514	1.79343	0.896714	−	0.442611i	$-0.145948\pi$
$107$	18.5514	1.79343	0.896714	+	0.442611i	$0.145948\pi$
$108$	0	0
$109$	12.7273	1.21906	0.609528	−	0.792765i	$-0.291359\pi$
$109$	12.7273	1.21906	0.609528	+	0.792765i	$0.291359\pi$
$110$	−2.80637	−0.267577
$111$	0	0
$112$	−2.37008	−0.223951
$113$	−0.0959083	−0.00902230	−0.00451115	−	0.999990i	$-0.501436\pi$
$113$	−0.0959083	−0.00902230	−0.00451115	+	0.999990i	$0.501436\pi$
$114$	0	0
$115$	−10.3297	−0.963247
$116$	2.44672	0.227172
$117$	0	0
$118$	1.94760	0.179291
$119$	−0.212381	−0.0194690
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.2946	−1.29418
$123$	0	0
$124$	7.81684	0.701973
$125$	7.57536	0.677560
$126$	0	0
$127$	−12.6881	−1.12589	−0.562943	−	0.826496i	$-0.690331\pi$
$127$	−12.6881	−1.12589	−0.562943	+	0.826496i	$0.690331\pi$
$128$	1.27775	0.112938
$129$	0	0
$130$	9.40662	0.825015
$131$	−6.70594	−0.585901	−0.292950	−	0.956128i	$-0.594637\pi$
$131$	−6.70594	−0.585901	−0.292950	+	0.956128i	$0.594637\pi$
$132$	0	0
$133$	2.99587	0.259775
$134$	1.82417	0.157584
$135$	0	0
$136$	−0.376955	−0.0323236
$137$	−15.3952	−1.31530	−0.657649	−	0.753325i	$-0.728449\pi$
$137$	−15.3952	−1.31530	−0.657649	+	0.753325i	$0.728449\pi$
$138$	0	0
$139$	3.79613	0.321983	0.160992	−	0.986956i	$-0.448531\pi$
$139$	3.79613	0.321983	0.160992	+	0.986956i	$0.448531\pi$
$140$	4.13733	0.349669
$141$	0	0
$142$	−3.84415	−0.322594
$143$	−3.35189	−0.280299
$144$	0	0
$145$	7.26591	0.603401
$146$	−1.55490	−0.128685
$147$	0	0
$148$	−4.57500	−0.376062
$149$	−7.39952	−0.606192	−0.303096	−	0.952960i	$-0.598020\pi$
$149$	−7.39952	−0.606192	−0.303096	+	0.952960i	$0.598020\pi$
$150$	0	0
$151$	12.4047	1.00948	0.504739	−	0.863272i	$-0.331589\pi$
$151$	12.4047	1.00948	0.504739	+	0.863272i	$0.331589\pi$
$152$	5.31736	0.431295
$153$	0	0
$154$	1.79551	0.144686
$155$	23.2133	1.86454
$156$	0	0
$157$	−2.33762	−0.186563	−0.0932814	−	0.995640i	$-0.529736\pi$
$157$	−2.33762	−0.186563	−0.0932814	+	0.995640i	$0.529736\pi$
$158$	7.16959	0.570382
$159$	0	0
$160$	12.4046	0.980673
$161$	6.60890	0.520855
$162$	0	0
$163$	−17.8920	−1.40141	−0.700706	−	0.713450i	$-0.747132\pi$
$163$	−17.8920	−1.40141	−0.700706	+	0.713450i	$0.747132\pi$
$164$	9.45028	0.737943
$165$	0	0
$166$	3.75983	0.291819
$167$	−7.52787	−0.582524	−0.291262	−	0.956643i	$-0.594075\pi$
$167$	−7.52787	−0.582524	−0.291262	+	0.956643i	$0.594075\pi$
$168$	0	0
$169$	−1.76486	−0.135759
$170$	−0.347875	−0.0266808
$171$	0	0
$172$	−2.47242	−0.188520
$173$	−16.8637	−1.28212	−0.641061	−	0.767490i	$-0.721505\pi$
$173$	−16.8637	−1.28212	−0.641061	+	0.767490i	$0.721505\pi$
$174$	0	0
$175$	3.71988	0.281197
$176$	1.38333	0.104272
$177$	0	0
$178$	−1.30273	−0.0976438
$179$	−3.81548	−0.285182	−0.142591	−	0.989782i	$-0.545543\pi$
$179$	−3.81548	−0.285182	−0.142591	+	0.989782i	$0.545543\pi$
$180$	0	0
$181$	−20.9749	−1.55905	−0.779527	−	0.626369i	$-0.784540\pi$
$181$	−20.9749	−1.55905	−0.779527	+	0.626369i	$0.784540\pi$
$182$	−6.01833	−0.446109
$183$	0	0
$184$	11.7301	0.864756
$185$	−13.5862	−0.998875
$186$	0	0
$187$	0.123959	0.00906480
$188$	−0.976907	−0.0712483
$189$	0	0
$190$	4.90716	0.356002
$191$	6.81491	0.493110	0.246555	−	0.969129i	$-0.420701\pi$
$191$	6.81491	0.493110	0.246555	+	0.969129i	$0.420701\pi$
$192$	0	0
$193$	12.0345	0.866260	0.433130	−	0.901331i	$-0.357409\pi$
$193$	12.0345	0.866260	0.433130	+	0.901331i	$0.357409\pi$
$194$	−8.89875	−0.638893
$195$	0	0
$196$	3.66523	0.261802
$197$	−12.6354	−0.900232	−0.450116	−	0.892970i	$-0.648617\pi$
$197$	−12.6354	−0.900232	−0.450116	+	0.892970i	$0.648617\pi$
$198$	0	0
$199$	5.50111	0.389963	0.194982	−	0.980807i	$-0.437535\pi$
$199$	5.50111	0.389963	0.194982	+	0.980807i	$0.437535\pi$
$200$	6.60241	0.466861
$201$	0	0
$202$	−16.1448	−1.13594
$203$	−4.64871	−0.326276
$204$	0	0
$205$	28.0641	1.96008
$206$	−9.71746	−0.677047
$207$	0	0
$208$	−4.63676	−0.321502
$209$	−1.74858	−0.120952
$210$	0	0
$211$	11.8198	0.813711	0.406856	−	0.913493i	$-0.366625\pi$
$211$	11.8198	0.813711	0.406856	+	0.913493i	$0.366625\pi$
$212$	9.89987	0.679926
$213$	0	0
$214$	−19.4413	−1.32898
$215$	−7.34224	−0.500737
$216$	0	0
$217$	−14.8518	−1.00821
$218$	−13.3379	−0.903354
$219$	0	0
$220$	−2.41481	−0.162807
$221$	−0.415497	−0.0279494
$222$	0	0
$223$	−22.7564	−1.52388	−0.761941	−	0.647647i	$-0.775753\pi$
$223$	−22.7564	−1.52388	−0.761941	+	0.647647i	$0.775753\pi$
$224$	−7.93646	−0.530277
$225$	0	0
$226$	0.100509	0.00668578
$227$	3.59127	0.238361	0.119180	−	0.992873i	$-0.461973\pi$
$227$	3.59127	0.238361	0.119180	+	0.992873i	$0.461973\pi$
$228$	0	0
$229$	−3.67658	−0.242955	−0.121478	−	0.992594i	$-0.538763\pi$
$229$	−3.67658	−0.242955	−0.121478	+	0.992594i	$0.538763\pi$
$230$	10.8252	0.713793
$231$	0	0
$232$	−8.25098	−0.541704
$233$	−2.21503	−0.145111	−0.0725556	−	0.997364i	$-0.523115\pi$
$233$	−2.21503	−0.145111	−0.0725556	+	0.997364i	$0.523115\pi$
$234$	0	0
$235$	−2.90108	−0.189246
$236$	1.67586	0.109089
$237$	0	0
$238$	0.222570	0.0144271
$239$	−16.3893	−1.06013	−0.530067	−	0.847956i	$-0.677833\pi$
$239$	−16.3893	−1.06013	−0.530067	+	0.847956i	$0.677833\pi$
$240$	0	0
$241$	−6.29862	−0.405730	−0.202865	−	0.979207i	$-0.565025\pi$
$241$	−6.29862	−0.405730	−0.202865	+	0.979207i	$0.565025\pi$
$242$	−1.04797	−0.0673662
$243$	0	0
$244$	−12.3002	−0.787440
$245$	10.8845	0.695383
$246$	0	0
$247$	5.86104	0.372929
$248$	−26.3605	−1.67389
$249$	0	0
$250$	−7.93876	−0.502091
$251$	−21.1739	−1.33648	−0.668241	−	0.743945i	$-0.732953\pi$
$251$	−21.1739	−1.33648	−0.668241	+	0.743945i	$0.732953\pi$
$252$	0	0
$253$	−3.85738	−0.242511
$254$	13.2968	0.834313
$255$	0	0
$256$	−16.5812	−1.03633
$257$	−0.380146	−0.0237129	−0.0118564	−	0.999930i	$-0.503774\pi$
$257$	−0.380146	−0.0237129	−0.0118564	+	0.999930i	$0.503774\pi$
$258$	0	0
$259$	8.69240	0.540119
$260$	8.09417	0.501979
$261$	0	0
$262$	7.02764	0.434169
$263$	12.5739	0.775339	0.387670	−	0.921798i	$-0.373280\pi$
$263$	12.5739	0.775339	0.387670	+	0.921798i	$0.373280\pi$
$264$	0	0
$265$	29.3992	1.80598
$266$	−3.13959	−0.192500
$267$	0	0
$268$	1.56965	0.0958818
$269$	32.2302	1.96511	0.982554	−	0.185978i	$-0.0595454\pi$
$269$	32.2302	1.96511	0.982554	+	0.185978i	$0.0595454\pi$
$270$	0	0
$271$	24.6663	1.49837	0.749185	−	0.662360i	$-0.230445\pi$
$271$	24.6663	1.49837	0.749185	+	0.662360i	$0.230445\pi$
$272$	0.171476	0.0103973
$273$	0	0
$274$	16.1337	0.974672
$275$	−2.17116	−0.130926
$276$	0	0
$277$	21.5150	1.29271	0.646355	−	0.763037i	$-0.276292\pi$
$277$	21.5150	1.29271	0.646355	+	0.763037i	$0.276292\pi$
$278$	−3.97824	−0.238599
$279$	0	0
$280$	−13.9522	−0.833803
$281$	14.6475	0.873797	0.436899	−	0.899511i	$-0.356077\pi$
$281$	14.6475	0.873797	0.436899	+	0.899511i	$0.356077\pi$
$282$	0	0
$283$	0.870033	0.0517181	0.0258590	−	0.999666i	$-0.491768\pi$
$283$	0.870033	0.0517181	0.0258590	+	0.999666i	$0.491768\pi$
$284$	−3.30780	−0.196282
$285$	0	0
$286$	3.51268	0.207709
$287$	−17.9553	−1.05987
$288$	0	0
$289$	−16.9846	−0.999096
$290$	−7.61447	−0.447137
$291$	0	0
$292$	−1.33796	−0.0782981
$293$	−8.58105	−0.501310	−0.250655	−	0.968076i	$-0.580646\pi$
$293$	−8.58105	−0.501310	−0.250655	+	0.968076i	$0.580646\pi$
$294$	0	0
$295$	4.97674	0.289757
$296$	15.4281	0.896740
$297$	0	0
$298$	7.75449	0.449205
$299$	12.9295	0.747731
$300$	0	0
$301$	4.69755	0.270762
$302$	−12.9997	−0.748051
$303$	0	0
$304$	−2.41886	−0.138731
$305$	−36.5274	−2.09155
$306$	0	0
$307$	20.5058	1.17033	0.585165	−	0.810914i	$-0.301030\pi$
$307$	20.5058	1.17033	0.585165	+	0.810914i	$0.301030\pi$
$308$	1.54499	0.0880340
$309$	0	0
$310$	−24.3269	−1.38168
$311$	−14.5491	−0.825002	−0.412501	−	0.910957i	$-0.635345\pi$
$311$	−14.5491	−0.825002	−0.412501	+	0.910957i	$0.635345\pi$
$312$	0	0
$313$	31.6366	1.78821	0.894103	−	0.447861i	$-0.147814\pi$
$313$	31.6366	1.78821	0.894103	+	0.447861i	$0.147814\pi$
$314$	2.44977	0.138248
$315$	0	0
$316$	6.16926	0.347048
$317$	−17.0191	−0.955891	−0.477945	−	0.878390i	$-0.658618\pi$
$317$	−17.0191	−0.955891	−0.477945	+	0.878390i	$0.658618\pi$
$318$	0	0
$319$	2.71328	0.151915
$320$	−20.4086	−1.14087
$321$	0	0
$322$	−6.92595	−0.385968
$323$	−0.216753	−0.0120604
$324$	0	0
$325$	7.27748	0.403682
$326$	18.7504	1.03849
$327$	0	0
$328$	−31.8689	−1.75966
$329$	1.85610	0.102330
$330$	0	0
$331$	16.1400	0.887134	0.443567	−	0.896241i	$-0.353713\pi$
$331$	16.1400	0.887134	0.443567	+	0.896241i	$0.353713\pi$
$332$	3.23524	0.177557
$333$	0	0
$334$	7.88900	0.431667
$335$	4.66133	0.254676
$336$	0	0
$337$	11.6506	0.634647	0.317323	−	0.948317i	$-0.397216\pi$
$337$	11.6506	0.634647	0.317323	+	0.948317i	$0.397216\pi$
$338$	1.84953	0.100601
$339$	0	0
$340$	−0.299338	−0.0162339
$341$	8.66848	0.469424
$342$	0	0
$343$	−18.9571	−1.02359
$344$	8.33766	0.449537
$345$	0	0
$346$	17.6727	0.950088
$347$	−17.7964	−0.955360	−0.477680	−	0.878534i	$-0.658522\pi$
$347$	−17.7964	−0.955360	−0.477680	+	0.878534i	$0.658522\pi$
$348$	0	0
$349$	12.1228	0.648920	0.324460	−	0.945899i	$-0.394817\pi$
$349$	12.1228	0.648920	0.324460	+	0.945899i	$0.394817\pi$
$350$	−3.89833	−0.208375
$351$	0	0
$352$	4.63223	0.246898
$353$	−29.1460	−1.55128	−0.775642	−	0.631174i	$-0.782574\pi$
$353$	−29.1460	−1.55128	−0.775642	+	0.631174i	$0.782574\pi$
$354$	0	0
$355$	−9.82304	−0.521353
$356$	−1.12097	−0.0594112
$357$	0	0
$358$	3.99851	0.211328
$359$	18.0135	0.950719	0.475359	−	0.879792i	$-0.342318\pi$
$359$	18.0135	0.950719	0.475359	+	0.879792i	$0.342318\pi$
$360$	0	0
$361$	−15.9425	−0.839077
$362$	21.9811	1.15530
$363$	0	0
$364$	−5.17863	−0.271434
$365$	−3.97328	−0.207971
$366$	0	0
$367$	−16.4950	−0.861034	−0.430517	−	0.902582i	$-0.641669\pi$
$367$	−16.4950	−0.861034	−0.430517	+	0.902582i	$0.641669\pi$
$368$	−5.33602	−0.278159
$369$	0	0
$370$	14.2379	0.740194
$371$	−18.8095	−0.976543
$372$	0	0
$373$	−13.7233	−0.710567	−0.355284	−	0.934759i	$-0.615616\pi$
$373$	−13.7233	−0.710567	−0.355284	+	0.934759i	$0.615616\pi$
$374$	−0.129906	−0.00671727
$375$	0	0
$376$	3.29439	0.169895
$377$	−9.09462	−0.468397
$378$	0	0
$379$	−38.2439	−1.96446	−0.982229	−	0.187686i	$-0.939901\pi$
$379$	−38.2439	−1.96446	−0.982229	+	0.187686i	$0.939901\pi$
$380$	4.22249	0.216609
$381$	0	0
$382$	−7.14184	−0.365408
$383$	−27.7700	−1.41898	−0.709491	−	0.704715i	$-0.751075\pi$
$383$	−27.7700	−1.41898	−0.709491	+	0.704715i	$0.751075\pi$
$384$	0	0
$385$	4.58809	0.233831
$386$	−12.6118	−0.641923
$387$	0	0
$388$	−7.65716	−0.388734
$389$	−34.9248	−1.77076	−0.885378	−	0.464872i	$-0.846100\pi$
$389$	−34.9248	−1.77076	−0.885378	+	0.464872i	$0.846100\pi$
$390$	0	0
$391$	−0.478157	−0.0241815
$392$	−12.3601	−0.624280
$393$	0	0
$394$	13.2415	0.667097
$395$	18.3206	0.921809
$396$	0	0
$397$	−28.2097	−1.41580	−0.707902	−	0.706310i	$-0.750358\pi$
$397$	−28.2097	−1.41580	−0.707902	+	0.706310i	$0.750358\pi$
$398$	−5.76501	−0.288974
$399$	0	0
$400$	−3.00343	−0.150171
$401$	7.78351	0.388690	0.194345	−	0.980933i	$-0.437742\pi$
$401$	7.78351	0.388690	0.194345	+	0.980933i	$0.437742\pi$
$402$	0	0
$403$	−29.0558	−1.44737
$404$	−13.8922	−0.691162
$405$	0	0
$406$	4.87172	0.241779
$407$	−5.07344	−0.251481
$408$	0	0
$409$	5.33288	0.263694	0.131847	−	0.991270i	$-0.457909\pi$
$409$	5.33288	0.263694	0.131847	+	0.991270i	$0.457909\pi$
$410$	−29.4104	−1.45248
$411$	0	0
$412$	−8.36164	−0.411948
$413$	−3.18411	−0.156680
$414$	0	0
$415$	9.60756	0.471617
$416$	−15.5267	−0.761259
$417$	0	0
$418$	1.83246	0.0896287
$419$	−7.51179	−0.366975	−0.183487	−	0.983022i	$-0.558739\pi$
$419$	−7.51179	−0.366975	−0.183487	+	0.983022i	$0.558739\pi$
$420$	0	0
$421$	−4.89868	−0.238747	−0.119373	−	0.992849i	$-0.538089\pi$
$421$	−4.89868	−0.238747	−0.119373	+	0.992849i	$0.538089\pi$
$422$	−12.3869	−0.602983
$423$	0	0
$424$	−33.3850	−1.62132
$425$	−0.269135	−0.0130550
$426$	0	0
$427$	23.3701	1.13096
$428$	−16.7288	−0.808616
$429$	0	0
$430$	7.69446	0.371060
$431$	−15.0599	−0.725409	−0.362704	−	0.931904i	$-0.618147\pi$
$431$	−15.0599	−0.725409	−0.362704	+	0.931904i	$0.618147\pi$
$432$	0	0
$433$	12.3553	0.593759	0.296879	−	0.954915i	$-0.404054\pi$
$433$	12.3553	0.593759	0.296879	+	0.954915i	$0.404054\pi$
$434$	15.5643	0.747111
$435$	0	0
$436$	−11.4769	−0.549644
$437$	6.74493	0.322654
$438$	0	0
$439$	15.9971	0.763501	0.381751	−	0.924265i	$-0.375321\pi$
$439$	15.9971	0.763501	0.381751	+	0.924265i	$0.375321\pi$
$440$	8.14339	0.388221
$441$	0	0
$442$	0.435429	0.0207113
$443$	−18.8683	−0.896460	−0.448230	−	0.893918i	$-0.647945\pi$
$443$	−18.8683	−0.896460	−0.448230	+	0.893918i	$0.647945\pi$
$444$	0	0
$445$	−3.32889	−0.157805
$446$	23.8481	1.12924
$447$	0	0
$448$	13.0574	0.616902
$449$	−15.8061	−0.745938	−0.372969	−	0.927844i	$-0.621660\pi$
$449$	−15.8061	−0.745938	−0.372969	+	0.927844i	$0.621660\pi$
$450$	0	0
$451$	10.4799	0.493478
$452$	0.0864858	0.00406795
$453$	0	0
$454$	−3.76355	−0.176632
$455$	−15.3788	−0.720967
$456$	0	0
$457$	−26.9792	−1.26203	−0.631016	−	0.775770i	$-0.717362\pi$
$457$	−26.9792	−1.26203	−0.631016	+	0.775770i	$0.717362\pi$
$458$	3.85295	0.180036
$459$	0	0
$460$	9.31483	0.434306
$461$	26.7426	1.24553	0.622764	−	0.782410i	$-0.286010\pi$
$461$	26.7426	1.24553	0.622764	+	0.782410i	$0.286010\pi$
$462$	0	0
$463$	29.8654	1.38796	0.693981	−	0.719993i	$-0.255855\pi$
$463$	29.8654	1.38796	0.693981	+	0.719993i	$0.255855\pi$
$464$	3.75337	0.174246
$465$	0	0
$466$	2.32128	0.107531
$467$	24.7171	1.14377	0.571886	−	0.820333i	$-0.306212\pi$
$467$	24.7171	1.14377	0.571886	+	0.820333i	$0.306212\pi$
$468$	0	0
$469$	−2.98231	−0.137710
$470$	3.04025	0.140236
$471$	0	0
$472$	−5.65146	−0.260130
$473$	−2.74179	−0.126068
$474$	0	0
$475$	3.79645	0.174193
$476$	0.191516	0.00877812
$477$	0	0
$478$	17.1755	0.785589
$479$	−3.73129	−0.170487	−0.0852434	−	0.996360i	$-0.527167\pi$
$479$	−3.73129	−0.170487	−0.0852434	+	0.996360i	$0.527167\pi$
$480$	0	0
$481$	17.0056	0.775388
$482$	6.60078	0.300657
$483$	0	0
$484$	−0.901755	−0.0409889
$485$	−22.7391	−1.03253
$486$	0	0
$487$	−32.2176	−1.45992	−0.729960	−	0.683490i	$-0.760461\pi$
$487$	−32.2176	−1.45992	−0.729960	+	0.683490i	$0.760461\pi$
$488$	41.4796	1.87769
$489$	0	0
$490$	−11.4066	−0.515298
$491$	−4.54097	−0.204931	−0.102466	−	0.994737i	$-0.532673\pi$
$491$	−4.54097	−0.204931	−0.102466	+	0.994737i	$0.532673\pi$
$492$	0	0
$493$	0.336337	0.0151478
$494$	−6.14220	−0.276351
$495$	0	0
$496$	11.9914	0.538428
$497$	6.28476	0.281910
$498$	0	0
$499$	−34.7318	−1.55481	−0.777405	−	0.629000i	$-0.783464\pi$
$499$	−34.7318	−1.55481	−0.777405	+	0.629000i	$0.783464\pi$
$500$	−6.83111	−0.305497
$501$	0	0
$502$	22.1896	0.990371
$503$	17.1524	0.764786	0.382393	−	0.924000i	$-0.375100\pi$
$503$	17.1524	0.764786	0.382393	+	0.924000i	$0.375100\pi$
$504$	0	0
$505$	−41.2551	−1.83583
$506$	4.04242	0.179708
$507$	0	0
$508$	11.4415	0.507637
$509$	27.6087	1.22374	0.611868	−	0.790960i	$-0.290419\pi$
$509$	27.6087	1.22374	0.611868	+	0.790960i	$0.290419\pi$
$510$	0	0
$511$	2.54209	0.112456
$512$	14.8212	0.655010
$513$	0	0
$514$	0.398383	0.0175719
$515$	−24.8312	−1.09419
$516$	0	0
$517$	−1.08334	−0.0476452
$518$	−9.10939	−0.400244
$519$	0	0
$520$	−27.2957	−1.19700
$521$	−6.69649	−0.293378	−0.146689	−	0.989183i	$-0.546862\pi$
$521$	−6.69649	−0.293378	−0.146689	+	0.989183i	$0.546862\pi$
$522$	0	0
$523$	−39.9038	−1.74487	−0.872435	−	0.488730i	$-0.837460\pi$
$523$	−39.9038	−1.74487	−0.872435	+	0.488730i	$0.837460\pi$
$524$	6.04711	0.264169
$525$	0	0
$526$	−13.1771	−0.574548
$527$	1.07454	0.0468076
$528$	0	0
$529$	−8.12065	−0.353072
$530$	−30.8095	−1.33828
$531$	0	0
$532$	−2.70154	−0.117127
$533$	−35.1274	−1.52154
$534$	0	0
$535$	−49.6787	−2.14780
$536$	−5.29329	−0.228635
$537$	0	0
$538$	−33.7763	−1.45620
$539$	4.06455	0.175073
$540$	0	0
$541$	23.3178	1.00251	0.501256	−	0.865299i	$-0.332872\pi$
$541$	23.3178	1.00251	0.501256	+	0.865299i	$0.332872\pi$
$542$	−25.8496	−1.11033
$543$	0	0
$544$	0.574207	0.0246189
$545$	−34.0825	−1.45993
$546$	0	0
$547$	−36.5121	−1.56115	−0.780573	−	0.625064i	$-0.785073\pi$
$547$	−36.5121	−1.56115	−0.780573	+	0.625064i	$0.785073\pi$
$548$	13.8827	0.593038
$549$	0	0
$550$	2.27532	0.0970198
$551$	−4.74439	−0.202118
$552$	0	0
$553$	−11.7215	−0.498447
$554$	−22.5471	−0.957935
$555$	0	0
$556$	−3.42318	−0.145175
$557$	21.4420	0.908527	0.454264	−	0.890867i	$-0.349902\pi$
$557$	21.4420	0.908527	0.454264	+	0.890867i	$0.349902\pi$
$558$	0	0
$559$	9.19016	0.388702
$560$	6.34684	0.268203
$561$	0	0
$562$	−15.3502	−0.647508
$563$	−28.5936	−1.20508	−0.602539	−	0.798090i	$-0.705844\pi$
$563$	−28.5936	−1.20508	−0.602539	+	0.798090i	$0.705844\pi$
$564$	0	0
$565$	0.256833	0.0108051
$566$	−0.911770	−0.0383246
$567$	0	0
$568$	11.1548	0.468045
$569$	−31.5314	−1.32187	−0.660933	−	0.750444i	$-0.729839\pi$
$569$	−31.5314	−1.32187	−0.660933	+	0.750444i	$0.729839\pi$
$570$	0	0
$571$	31.8065	1.33106	0.665531	−	0.746370i	$-0.268205\pi$
$571$	31.8065	1.33106	0.665531	+	0.746370i	$0.268205\pi$
$572$	3.02258	0.126380
$573$	0	0
$574$	18.8167	0.785394
$575$	8.37498	0.349261
$576$	0	0
$577$	−10.5951	−0.441081	−0.220541	−	0.975378i	$-0.570782\pi$
$577$	−10.5951	−0.441081	−0.220541	+	0.975378i	$0.570782\pi$
$578$	17.7994	0.740358
$579$	0	0
$580$	−6.55207	−0.272060
$581$	−6.14690	−0.255016
$582$	0	0
$583$	10.9784	0.454681
$584$	4.51195	0.186706
$585$	0	0
$586$	8.99270	0.371485
$587$	−17.4105	−0.718607	−0.359303	−	0.933221i	$-0.616986\pi$
$587$	−17.4105	−0.718607	−0.359303	+	0.933221i	$0.616986\pi$
$588$	0	0
$589$	−15.1575	−0.624555
$590$	−5.21549	−0.214718
$591$	0	0
$592$	−7.01823	−0.288448
$593$	−2.94251	−0.120834	−0.0604171	−	0.998173i	$-0.519243\pi$
$593$	−2.94251	−0.120834	−0.0604171	+	0.998173i	$0.519243\pi$
$594$	0	0
$595$	0.568737	0.0233159
$596$	6.67255	0.273318
$597$	0	0
$598$	−13.5497	−0.554090
$599$	25.7940	1.05392	0.526958	−	0.849892i	$-0.323333\pi$
$599$	25.7940	1.05392	0.526958	+	0.849892i	$0.323333\pi$
$600$	0	0
$601$	−16.3682	−0.667672	−0.333836	−	0.942631i	$-0.608343\pi$
$601$	−16.3682	−0.667672	−0.333836	+	0.942631i	$0.608343\pi$
$602$	−4.92290	−0.200642
$603$	0	0
$604$	−11.1860	−0.455151
$605$	−2.67790	−0.108872
$606$	0	0
$607$	21.5714	0.875557	0.437778	−	0.899083i	$-0.355765\pi$
$607$	21.5714	0.875557	0.437778	+	0.899083i	$0.355765\pi$
$608$	−8.09981	−0.328491
$609$	0	0
$610$	38.2797	1.54990
$611$	3.63123	0.146904
$612$	0	0
$613$	−4.58729	−0.185279	−0.0926395	−	0.995700i	$-0.529530\pi$
$613$	−4.58729	−0.185279	−0.0926395	+	0.995700i	$0.529530\pi$
$614$	−21.4895	−0.867247
$615$	0	0
$616$	−5.21012	−0.209922
$617$	16.0366	0.645609	0.322805	−	0.946466i	$-0.395374\pi$
$617$	16.0366	0.645609	0.322805	+	0.946466i	$0.395374\pi$
$618$	0	0
$619$	−39.8917	−1.60338	−0.801692	−	0.597737i	$-0.796067\pi$
$619$	−39.8917	−1.60338	−0.801692	+	0.597737i	$0.796067\pi$
$620$	−20.9327	−0.840679
$621$	0	0
$622$	15.2470	0.611349
$623$	2.12982	0.0853294
$624$	0	0
$625$	−31.1419	−1.24567
$626$	−33.1543	−1.32511
$627$	0	0
$628$	2.10796	0.0841169
$629$	−0.628899	−0.0250759
$630$	0	0
$631$	44.7301	1.78068	0.890339	−	0.455298i	$-0.150467\pi$
$631$	44.7301	1.78068	0.890339	+	0.455298i	$0.150467\pi$
$632$	−20.8044	−0.827554
$633$	0	0
$634$	17.8356	0.708342
$635$	33.9775	1.34835
$636$	0	0
$637$	−13.6239	−0.539799
$638$	−2.84345	−0.112573
$639$	0	0
$640$	−3.42169	−0.135254
$641$	13.8324	0.546347	0.273173	−	0.961965i	$-0.411927\pi$
$641$	13.8324	0.546347	0.273173	+	0.961965i	$0.411927\pi$
$642$	0	0
$643$	−24.6063	−0.970378	−0.485189	−	0.874409i	$-0.661249\pi$
$643$	−24.6063	−0.970378	−0.485189	+	0.874409i	$0.661249\pi$
$644$	−5.95961	−0.234842
$645$	0	0
$646$	0.227151	0.00893712
$647$	−25.2328	−0.992004	−0.496002	−	0.868321i	$-0.665199\pi$
$647$	−25.2328	−0.992004	−0.496002	+	0.868321i	$0.665199\pi$
$648$	0	0
$649$	1.85845	0.0729505
$650$	−7.62660	−0.299140
$651$	0	0
$652$	16.1342	0.631865
$653$	21.1727	0.828554	0.414277	−	0.910151i	$-0.364035\pi$
$653$	21.1727	0.828554	0.414277	+	0.910151i	$0.364035\pi$
$654$	0	0
$655$	17.9578	0.701671
$656$	14.4971	0.566018
$657$	0	0
$658$	−1.94514	−0.0758296
$659$	24.6344	0.959621	0.479811	−	0.877372i	$-0.340705\pi$
$659$	24.6344	0.959621	0.479811	+	0.877372i	$0.340705\pi$
$660$	0	0
$661$	−12.7082	−0.494290	−0.247145	−	0.968978i	$-0.579492\pi$
$661$	−12.7082	−0.494290	−0.247145	+	0.968978i	$0.579492\pi$
$662$	−16.9143	−0.657391
$663$	0	0
$664$	−10.9101	−0.423394
$665$	−8.02264	−0.311105
$666$	0	0
$667$	−10.4662	−0.405251
$668$	6.78829	0.262647
$669$	0	0
$670$	−4.88494	−0.188722
$671$	−13.6403	−0.526578
$672$	0	0
$673$	13.3654	0.515198	0.257599	−	0.966252i	$-0.417069\pi$
$673$	13.3654	0.515198	0.257599	+	0.966252i	$0.417069\pi$
$674$	−12.2095	−0.470291
$675$	0	0
$676$	1.59147	0.0612105
$677$	−3.97779	−0.152879	−0.0764394	−	0.997074i	$-0.524355\pi$
$677$	−3.97779	−0.152879	−0.0764394	+	0.997074i	$0.524355\pi$
$678$	0	0
$679$	14.5485	0.558318
$680$	1.00945	0.0387106
$681$	0	0
$682$	−9.08432	−0.347857
$683$	47.4422	1.81532	0.907662	−	0.419702i	$-0.137865\pi$
$683$	47.4422	1.81532	0.907662	+	0.419702i	$0.137865\pi$
$684$	0	0
$685$	41.2267	1.57519
$686$	19.8665	0.758506
$687$	0	0
$688$	−3.79280	−0.144599
$689$	−36.7985	−1.40191
$690$	0	0
$691$	−23.6199	−0.898542	−0.449271	−	0.893396i	$-0.648316\pi$
$691$	−23.6199	−0.898542	−0.449271	+	0.893396i	$0.648316\pi$
$692$	15.2069	0.578080
$693$	0	0
$694$	18.6501	0.707949
$695$	−10.1657	−0.385605
$696$	0	0
$697$	1.29908	0.0492061
$698$	−12.7044	−0.480868
$699$	0	0
$700$	−3.35442	−0.126785
$701$	21.1265	0.797935	0.398968	−	0.916965i	$-0.369369\pi$
$701$	21.1265	0.797935	0.398968	+	0.916965i	$0.369369\pi$
$702$	0	0
$703$	8.87131	0.334588
$704$	−7.62110	−0.287231
$705$	0	0
$706$	30.5442	1.14954
$707$	26.3949	0.992682
$708$	0	0
$709$	−42.6839	−1.60303	−0.801515	−	0.597975i	$-0.795972\pi$
$709$	−42.6839	−1.60303	−0.801515	+	0.597975i	$0.795972\pi$
$710$	10.2943	0.386337
$711$	0	0
$712$	3.78021	0.141669
$713$	−33.4376	−1.25225
$714$	0	0
$715$	8.97602	0.335684
$716$	3.44063	0.128582
$717$	0	0
$718$	−18.8777	−0.704509
$719$	−14.1985	−0.529513	−0.264757	−	0.964315i	$-0.585292\pi$
$719$	−14.1985	−0.529513	−0.264757	+	0.964315i	$0.585292\pi$
$720$	0	0
$721$	15.8869	0.591661
$722$	16.7073	0.621780
$723$	0	0
$724$	18.9142	0.702942
$725$	−5.89098	−0.218785
$726$	0	0
$727$	−36.2540	−1.34459	−0.672293	−	0.740285i	$-0.734691\pi$
$727$	−36.2540	−1.34459	−0.672293	+	0.740285i	$0.734691\pi$
$728$	17.4637	0.647249
$729$	0	0
$730$	4.16388	0.154112
$731$	−0.339870	−0.0125706
$732$	0	0
$733$	−4.08759	−0.150979	−0.0754894	−	0.997147i	$-0.524052\pi$
$733$	−4.08759	−0.150979	−0.0754894	+	0.997147i	$0.524052\pi$
$734$	17.2863	0.638051
$735$	0	0
$736$	−17.8682	−0.658632
$737$	1.74066	0.0641182
$738$	0	0
$739$	41.2989	1.51920	0.759602	−	0.650388i	$-0.225394\pi$
$739$	41.2989	1.51920	0.759602	+	0.650388i	$0.225394\pi$
$740$	12.2514	0.450370
$741$	0	0
$742$	19.7119	0.723646
$743$	−31.3371	−1.14965	−0.574824	−	0.818277i	$-0.694929\pi$
$743$	−31.3371	−1.14965	−0.574824	+	0.818277i	$0.694929\pi$
$744$	0	0
$745$	19.8152	0.725972
$746$	14.3817	0.526550
$747$	0	0
$748$	−0.111781	−0.00408711
$749$	31.7843	1.16137
$750$	0	0
$751$	41.2787	1.50628	0.753141	−	0.657859i	$-0.228538\pi$
$751$	41.2787	1.50628	0.753141	+	0.657859i	$0.228538\pi$
$752$	−1.49862	−0.0546489
$753$	0	0
$754$	9.53091	0.347095
$755$	−33.2185	−1.20894
$756$	0	0
$757$	−35.6523	−1.29580	−0.647902	−	0.761723i	$-0.724354\pi$
$757$	−35.6523	−1.29580	−0.647902	+	0.761723i	$0.724354\pi$
$758$	40.0786	1.45572
$759$	0	0
$760$	−14.2394	−0.516516
$761$	32.9538	1.19458	0.597288	−	0.802027i	$-0.296245\pi$
$761$	32.9538	1.19458	0.597288	+	0.802027i	$0.296245\pi$
$762$	0	0
$763$	21.8059	0.789427
$764$	−6.14538	−0.222332
$765$	0	0
$766$	29.1022	1.05150
$767$	−6.22930	−0.224927
$768$	0	0
$769$	−46.4481	−1.67496	−0.837480	−	0.546467i	$-0.815972\pi$
$769$	−46.4481	−1.67496	−0.837480	+	0.546467i	$0.815972\pi$
$770$	−4.80819	−0.173275
$771$	0	0
$772$	−10.8521	−0.390577
$773$	5.23651	0.188344	0.0941721	−	0.995556i	$-0.469980\pi$
$773$	5.23651	0.188344	0.0941721	+	0.995556i	$0.469980\pi$
$774$	0	0
$775$	−18.8207	−0.676058
$776$	25.8220	0.926956
$777$	0	0
$778$	36.6002	1.31218
$779$	−18.3249	−0.656558
$780$	0	0
$781$	−3.66818	−0.131258
$782$	0.501096	0.0179191
$783$	0	0
$784$	5.62261	0.200807
$785$	6.25993	0.223427
$786$	0	0
$787$	−2.63693	−0.0939966	−0.0469983	−	0.998895i	$-0.514966\pi$
$787$	−2.63693	−0.0939966	−0.0469983	+	0.998895i	$0.514966\pi$
$788$	11.3940	0.405894
$789$	0	0
$790$	−19.1995	−0.683086
$791$	−0.164321	−0.00584259
$792$	0	0
$793$	45.7207	1.62359
$794$	29.5630	1.04915
$795$	0	0
$796$	−4.96065	−0.175826
$797$	9.16297	0.324569	0.162285	−	0.986744i	$-0.448114\pi$
$797$	9.16297	0.324569	0.162285	+	0.986744i	$0.448114\pi$
$798$	0	0
$799$	−0.134290	−0.00475084
$800$	−10.0573	−0.355579
$801$	0	0
$802$	−8.15690	−0.288030
$803$	−1.48373	−0.0523596
$804$	0	0
$805$	−17.6980	−0.623772
$806$	30.4496	1.07254
$807$	0	0
$808$	46.8482	1.64811
$809$	12.2787	0.431696	0.215848	−	0.976427i	$-0.430748\pi$
$809$	12.2787	0.431696	0.215848	+	0.976427i	$0.430748\pi$
$810$	0	0
$811$	−25.6976	−0.902364	−0.451182	−	0.892432i	$-0.648997\pi$
$811$	−25.6976	−0.902364	−0.451182	+	0.892432i	$0.648997\pi$
$812$	4.19200	0.147110
$813$	0	0
$814$	5.31682	0.186354
$815$	47.9131	1.67832
$816$	0	0
$817$	4.79424	0.167729
$818$	−5.58871	−0.195405
$819$	0	0
$820$	−25.3069	−0.883756
$821$	−38.3536	−1.33855	−0.669276	−	0.743014i	$-0.733395\pi$
$821$	−38.3536	−1.33855	−0.669276	+	0.743014i	$0.733395\pi$
$822$	0	0
$823$	34.2665	1.19445	0.597227	−	0.802072i	$-0.296269\pi$
$823$	34.2665	1.19445	0.597227	+	0.802072i	$0.296269\pi$
$824$	28.1977	0.982313
$825$	0	0
$826$	3.33686	0.116104
$827$	4.30995	0.149872	0.0749358	−	0.997188i	$-0.476125\pi$
$827$	4.30995	0.149872	0.0749358	+	0.997188i	$0.476125\pi$
$828$	0	0
$829$	−16.1836	−0.562078	−0.281039	−	0.959696i	$-0.590679\pi$
$829$	−16.1836	−0.562078	−0.281039	+	0.959696i	$0.590679\pi$
$830$	−10.0685	−0.349481
$831$	0	0
$832$	25.5451	0.885616
$833$	0.503838	0.0174570
$834$	0	0
$835$	20.1589	0.697628
$836$	1.57679	0.0545344
$837$	0	0
$838$	7.87214	0.271939
$839$	−48.8001	−1.68477	−0.842384	−	0.538878i	$-0.818848\pi$
$839$	−48.8001	−1.68477	−0.842384	+	0.538878i	$0.818848\pi$
$840$	0	0
$841$	−21.6381	−0.746141
$842$	5.13368	0.176918
$843$	0	0
$844$	−10.6586	−0.366884
$845$	4.72613	0.162584
$846$	0	0
$847$	1.71332	0.0588702
$848$	15.1868	0.521517
$849$	0	0
$850$	0.282046	0.00967411
$851$	19.5702	0.670856
$852$	0	0
$853$	−49.8446	−1.70665	−0.853324	−	0.521381i	$-0.825417\pi$
$853$	−49.8446	−1.70665	−0.853324	+	0.521381i	$0.825417\pi$
$854$	−24.4912	−0.838073
$855$	0	0
$856$	56.4139	1.92819
$857$	17.8077	0.608300	0.304150	−	0.952624i	$-0.401628\pi$
$857$	17.8077	0.608300	0.304150	+	0.952624i	$0.401628\pi$
$858$	0	0
$859$	−16.3553	−0.558034	−0.279017	−	0.960286i	$-0.590009\pi$
$859$	−16.3553	−0.558034	−0.279017	+	0.960286i	$0.590009\pi$
$860$	6.62090	0.225771
$861$	0	0
$862$	15.7823	0.537548
$863$	−3.13339	−0.106662	−0.0533310	−	0.998577i	$-0.516984\pi$
$863$	−3.13339	−0.106662	−0.0533310	+	0.998577i	$0.516984\pi$
$864$	0	0
$865$	45.1593	1.53546
$866$	−12.9480	−0.439992
$867$	0	0
$868$	13.3927	0.454579
$869$	6.84139	0.232078
$870$	0	0
$871$	−5.83451	−0.197695
$872$	38.7032	1.31066
$873$	0	0
$874$	−7.06850	−0.239095
$875$	12.9790	0.438769
$876$	0	0
$877$	5.33676	0.180210	0.0901048	−	0.995932i	$-0.471280\pi$
$877$	5.33676	0.180210	0.0901048	+	0.995932i	$0.471280\pi$
$878$	−16.7645	−0.565776
$879$	0	0
$880$	−3.70442	−0.124876
$881$	−20.0030	−0.673918	−0.336959	−	0.941519i	$-0.609398\pi$
$881$	−20.0030	−0.673918	−0.336959	+	0.941519i	$0.609398\pi$
$882$	0	0
$883$	26.2438	0.883175	0.441587	−	0.897218i	$-0.354415\pi$
$883$	26.2438	0.883175	0.441587	+	0.897218i	$0.354415\pi$
$884$	0.374677	0.0126017
$885$	0	0
$886$	19.7735	0.664302
$887$	15.1241	0.507816	0.253908	−	0.967228i	$-0.418284\pi$
$887$	15.1241	0.507816	0.253908	+	0.967228i	$0.418284\pi$
$888$	0	0
$889$	−21.7387	−0.729093
$890$	3.48859	0.116938
$891$	0	0
$892$	20.5207	0.687084
$893$	1.89431	0.0633906
$894$	0	0
$895$	10.2175	0.341533
$896$	2.18919	0.0731357
$897$	0	0
$898$	16.5644	0.552761
$899$	23.5201	0.784438
$900$	0	0
$901$	1.36088	0.0453375
$902$	−10.9826	−0.365681
$903$	0	0
$904$	−0.291653	−0.00970024
$905$	56.1688	1.86711
$906$	0	0
$907$	−13.4803	−0.447607	−0.223803	−	0.974634i	$-0.571847\pi$
$907$	−13.4803	−0.447607	−0.223803	+	0.974634i	$0.571847\pi$
$908$	−3.23844	−0.107471
$909$	0	0
$910$	16.1165	0.534257
$911$	−47.9609	−1.58902	−0.794508	−	0.607253i	$-0.792271\pi$
$911$	−47.9609	−1.58902	−0.794508	+	0.607253i	$0.792271\pi$
$912$	0	0
$913$	3.58772	0.118736
$914$	28.2734	0.935201
$915$	0	0
$916$	3.31537	0.109543
$917$	−11.4894	−0.379413
$918$	0	0
$919$	−18.5524	−0.611988	−0.305994	−	0.952033i	$-0.598989\pi$
$919$	−18.5524	−0.611988	−0.305994	+	0.952033i	$0.598989\pi$
$920$	−31.4121	−1.03563
$921$	0	0
$922$	−28.0255	−0.922971
$923$	12.2953	0.404706
$924$	0	0
$925$	11.0152	0.362179
$926$	−31.2981	−1.02852
$927$	0	0
$928$	12.5685	0.412583
$929$	21.1601	0.694239	0.347120	−	0.937821i	$-0.387160\pi$
$929$	21.1601	0.694239	0.347120	+	0.937821i	$0.387160\pi$
$930$	0	0
$931$	−7.10719	−0.232929
$932$	1.99741	0.0654273
$933$	0	0
$934$	−25.9029	−0.847568
$935$	−0.331951	−0.0108560
$936$	0	0
$937$	0.471055	0.0153887	0.00769435	−	0.999970i	$-0.497551\pi$
$937$	0.471055	0.0153887	0.00769435	+	0.999970i	$0.497551\pi$
$938$	3.12538	0.102047
$939$	0	0
$940$	2.61606	0.0853265
$941$	−50.8981	−1.65923	−0.829615	−	0.558336i	$-0.811440\pi$
$941$	−50.8981	−1.65923	−0.829615	+	0.558336i	$0.811440\pi$
$942$	0	0
$943$	−40.4248	−1.31641
$944$	2.57084	0.0836739
$945$	0	0
$946$	2.87332	0.0934196
$947$	−12.7934	−0.415730	−0.207865	−	0.978158i	$-0.566651\pi$
$947$	−12.7934	−0.415730	−0.207865	+	0.978158i	$0.566651\pi$
$948$	0	0
$949$	4.97328	0.161440
$950$	−3.97857	−0.129082
$951$	0	0
$952$	−0.645843	−0.0209319
$953$	43.4960	1.40897	0.704487	−	0.709717i	$-0.251177\pi$
$953$	43.4960	1.40897	0.704487	+	0.709717i	$0.251177\pi$
$954$	0	0
$955$	−18.2497	−0.590545
$956$	14.7791	0.477990
$957$	0	0
$958$	3.91028	0.126336
$959$	−26.3768	−0.851751
$960$	0	0
$961$	44.1425	1.42395
$962$	−17.8214	−0.574584
$963$	0	0
$964$	5.67981	0.182934
$965$	−32.2271	−1.03743
$966$	0	0
$967$	−13.6019	−0.437406	−0.218703	−	0.975791i	$-0.570183\pi$
$967$	−13.6019	−0.437406	−0.218703	+	0.975791i	$0.570183\pi$
$968$	3.04096	0.0977401
$969$	0	0
$970$	23.8300	0.765135
$971$	−23.0575	−0.739949	−0.369975	−	0.929042i	$-0.620634\pi$
$971$	−23.0575	−0.739949	−0.369975	+	0.929042i	$0.620634\pi$
$972$	0	0
$973$	6.50397	0.208508
$974$	33.7632	1.08184
$975$	0	0
$976$	−18.8690	−0.603982
$977$	−50.7066	−1.62225	−0.811124	−	0.584875i	$-0.801144\pi$
$977$	−50.7066	−1.62225	−0.811124	+	0.584875i	$0.801144\pi$
$978$	0	0
$979$	−1.24310	−0.0397296
$980$	−9.81512	−0.313532
$981$	0	0
$982$	4.75881	0.151860
$983$	16.1264	0.514353	0.257177	−	0.966364i	$-0.417208\pi$
$983$	16.1264	0.514353	0.257177	+	0.966364i	$0.417208\pi$
$984$	0	0
$985$	33.8363	1.07811
$986$	−0.352471	−0.0112250
$987$	0	0
$988$	−5.28522	−0.168145
$989$	10.5761	0.336301
$990$	0	0
$991$	12.6439	0.401646	0.200823	−	0.979628i	$-0.435638\pi$
$991$	12.6439	0.401646	0.200823	+	0.979628i	$0.435638\pi$
$992$	40.1544	1.27490
$993$	0	0
$994$	−6.58625	−0.208903
$995$	−14.7314	−0.467018
$996$	0	0
$997$	11.6836	0.370024	0.185012	−	0.982736i	$-0.440768\pi$
$997$	11.6836	0.370024	0.185012	+	0.982736i	$0.440768\pi$
$998$	36.3980	1.15216
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.i.1.18	✓	48
3.2	odd	2		8019.2.a.j.1.31	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8019.2.a.i.1.18	✓	48	1.1	even	1	trivial
8019.2.a.j.1.31	yes	48	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)

\(f(q)\)	\(=\)	\(q-1.04797 q^{2} -0.901755 q^{4} -2.67790 q^{5} +1.71332 q^{7} +3.04096 q^{8} +O(q^{10})\)	\(q-1.04797 q^{2} -0.901755 q^{4} -2.67790 q^{5} +1.71332 q^{7} +3.04096 q^{8} +2.80637 q^{10} -1.00000 q^{11} +3.35189 q^{13} -1.79551 q^{14} -1.38333 q^{16} -0.123959 q^{17} +1.74858 q^{19} +2.41481 q^{20} +1.04797 q^{22} +3.85738 q^{23} +2.17116 q^{25} -3.51268 q^{26} -1.54499 q^{28} -2.71328 q^{29} -8.66848 q^{31} -4.63223 q^{32} +0.129906 q^{34} -4.58809 q^{35} +5.07344 q^{37} -1.83246 q^{38} -8.14339 q^{40} -10.4799 q^{41} +2.74179 q^{43} +0.901755 q^{44} -4.04242 q^{46} +1.08334 q^{47} -4.06455 q^{49} -2.27532 q^{50} -3.02258 q^{52} -10.9784 q^{53} +2.67790 q^{55} +5.21012 q^{56} +2.84345 q^{58} -1.85845 q^{59} +13.6403 q^{61} +9.08432 q^{62} +7.62110 q^{64} -8.97602 q^{65} -1.74066 q^{67} +0.111781 q^{68} +4.80819 q^{70} +3.66818 q^{71} +1.48373 q^{73} -5.31682 q^{74} -1.57679 q^{76} -1.71332 q^{77} -6.84139 q^{79} +3.70442 q^{80} +10.9826 q^{82} -3.58772 q^{83} +0.331951 q^{85} -2.87332 q^{86} -3.04096 q^{88} +1.24310 q^{89} +5.74284 q^{91} -3.47841 q^{92} -1.13531 q^{94} -4.68253 q^{95} +8.49140 q^{97} +4.25953 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8}+O(q^{10}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8	\( 48 q - 6 q^{2} + 54 q^{4} - 24 q^{5} - 18 q^{8} - 48 q^{11} - 24 q^{14} + 66 q^{16} - 24 q^{17} - 48 q^{20} + 6 q^{22} - 12 q^{23} + 60 q^{25} - 36 q^{26} - 18 q^{28} - 60 q^{29} + 36 q^{31} - 42 q^{32} + 12 q^{34} - 24 q^{35} + 6 q^{37} - 24 q^{38} - 72 q^{41} - 12 q^{43} - 54 q^{44} - 30 q^{46} - 36 q^{47} + 60 q^{49} - 42 q^{50} - 48 q^{53} + 24 q^{55} - 72 q^{56} + 12 q^{58} - 60 q^{59} - 24 q^{61} - 36 q^{62} + 90 q^{64} - 48 q^{65} - 60 q^{68} - 30 q^{70} - 60 q^{71} - 18 q^{73} - 36 q^{74} - 42 q^{76} - 12 q^{79} - 96 q^{80} + 12 q^{82} - 36 q^{83} + 18 q^{85} - 48 q^{86} + 18 q^{88} - 96 q^{89} + 30 q^{91} - 36 q^{92} - 48 q^{94} - 48 q^{95} + 30 q^{97} - 54 q^{98}+O(q^{100}) \) 48 * q - 6 * q^2 + 54 * q^4 - 24 * q^5 - 18 * q^8 - 48 * q^11 - 24 * q^14 + 66 * q^16 - 24 * q^17 - 48 * q^20 + 6 * q^22 - 12 * q^23 + 60 * q^25 - 36 * q^26 - 18 * q^28 - 60 * q^29 + 36 * q^31 - 42 * q^32 + 12 * q^34 - 24 * q^35 + 6 * q^37 - 24 * q^38 - 72 * q^41 - 12 * q^43 - 54 * q^44 - 30 * q^46 - 36 * q^47 + 60 * q^49 - 42 * q^50 - 48 * q^53 + 24 * q^55 - 72 * q^56 + 12 * q^58 - 60 * q^59 - 24 * q^61 - 36 * q^62 + 90 * q^64 - 48 * q^65 - 60 * q^68 - 30 * q^70 - 60 * q^71 - 18 * q^73 - 36 * q^74 - 42 * q^76 - 12 * q^79 - 96 * q^80 + 12 * q^82 - 36 * q^83 + 18 * q^85 - 48 * q^86 + 18 * q^88 - 96 * q^89 + 30 * q^91 - 36 * q^92 - 48 * q^94 - 48 * q^95 + 30 * q^97 - 54 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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