LMFDB - Embedded newform 8016.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.284897.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 5x^{2} + 10x + 3 $ x^5 - 2x^4 - 6x^3 + 5x^2 + 10x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.759623$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.473647 q^{5} -0.663349 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.473647 q^{5} -0.663349 q^{7} +1.00000 q^{9} -3.46722 q^{11} -2.43008 q^{13} +0.473647 q^{15} -5.99358 q^{17} +2.40051 q^{19} +0.663349 q^{21} +8.17617 q^{23} -4.77566 q^{25} -1.00000 q^{27} +3.87701 q^{29} +2.70895 q^{31} +3.46722 q^{33} +0.314193 q^{35} +8.48041 q^{37} +2.43008 q^{39} +3.71876 q^{41} +10.5079 q^{43} -0.473647 q^{45} +10.9069 q^{47} -6.55997 q^{49} +5.99358 q^{51} +0.259988 q^{53} +1.64224 q^{55} -2.40051 q^{57} +1.16237 q^{59} -1.55815 q^{61} -0.663349 q^{63} +1.15100 q^{65} +1.52364 q^{67} -8.17617 q^{69} +4.46519 q^{71} -7.44051 q^{73} +4.77566 q^{75} +2.29998 q^{77} -8.68765 q^{79} +1.00000 q^{81} -15.7936 q^{83} +2.83884 q^{85} -3.87701 q^{87} -5.88769 q^{89} +1.61199 q^{91} -2.70895 q^{93} -1.13700 q^{95} -12.9569 q^{97} -3.46722 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.473647	−0.211821	−0.105911	−	0.994376i	$-0.533776\pi$
$5$	−0.473647	−0.211821	−0.105911	+	0.994376i	$0.533776\pi$
$6$	0	0
$7$	−0.663349	−0.250722	−0.125361	−	0.992111i	$-0.540009\pi$
$7$	−0.663349	−0.250722	−0.125361	+	0.992111i	$0.540009\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.46722	−1.04541	−0.522704	−	0.852514i	$-0.675077\pi$
$11$	−3.46722	−1.04541	−0.522704	+	0.852514i	$0.675077\pi$
$12$	0	0
$13$	−2.43008	−0.673983	−0.336991	−	0.941508i	$-0.609409\pi$
$13$	−2.43008	−0.673983	−0.336991	+	0.941508i	$0.609409\pi$
$14$	0	0
$15$	0.473647	0.122295
$16$	0	0
$17$	−5.99358	−1.45366	−0.726828	−	0.686819i	$-0.759006\pi$
$17$	−5.99358	−1.45366	−0.726828	+	0.686819i	$0.759006\pi$
$18$	0	0
$19$	2.40051	0.550716	0.275358	−	0.961342i	$-0.411204\pi$
$19$	2.40051	0.550716	0.275358	+	0.961342i	$0.411204\pi$
$20$	0	0
$21$	0.663349	0.144755
$22$	0	0
$23$	8.17617	1.70485	0.852425	−	0.522850i	$-0.175131\pi$
$23$	8.17617	1.70485	0.852425	+	0.522850i	$0.175131\pi$
$24$	0	0
$25$	−4.77566	−0.955132
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.87701	0.719942	0.359971	−	0.932963i	$-0.382787\pi$
$29$	3.87701	0.719942	0.359971	+	0.932963i	$0.382787\pi$
$30$	0	0
$31$	2.70895	0.486542	0.243271	−	0.969958i	$-0.421780\pi$
$31$	2.70895	0.486542	0.243271	+	0.969958i	$0.421780\pi$
$32$	0	0
$33$	3.46722	0.603566
$34$	0	0
$35$	0.314193	0.0531083
$36$	0	0
$37$	8.48041	1.39417	0.697086	−	0.716988i	$-0.254480\pi$
$37$	8.48041	1.39417	0.697086	+	0.716988i	$0.254480\pi$
$38$	0	0
$39$	2.43008	0.389124
$40$	0	0
$41$	3.71876	0.580772	0.290386	−	0.956910i	$-0.406216\pi$
$41$	3.71876	0.580772	0.290386	+	0.956910i	$0.406216\pi$
$42$	0	0
$43$	10.5079	1.60245	0.801223	−	0.598365i	$-0.204183\pi$
$43$	10.5079	1.60245	0.801223	+	0.598365i	$0.204183\pi$
$44$	0	0
$45$	−0.473647	−0.0706071
$46$	0	0
$47$	10.9069	1.59094	0.795469	−	0.605995i	$-0.207225\pi$
$47$	10.9069	1.59094	0.795469	+	0.605995i	$0.207225\pi$
$48$	0	0
$49$	−6.55997	−0.937138
$50$	0	0
$51$	5.99358	0.839269
$52$	0	0
$53$	0.259988	0.0357121	0.0178561	−	0.999841i	$-0.494316\pi$
$53$	0.259988	0.0357121	0.0178561	+	0.999841i	$0.494316\pi$
$54$	0	0
$55$	1.64224	0.221439
$56$	0	0
$57$	−2.40051	−0.317956
$58$	0	0
$59$	1.16237	0.151327	0.0756636	−	0.997133i	$-0.475892\pi$
$59$	1.16237	0.151327	0.0756636	+	0.997133i	$0.475892\pi$
$60$	0	0
$61$	−1.55815	−0.199500	−0.0997501	−	0.995013i	$-0.531804\pi$
$61$	−1.55815	−0.199500	−0.0997501	+	0.995013i	$0.531804\pi$
$62$	0	0
$63$	−0.663349	−0.0835741
$64$	0	0
$65$	1.15100	0.142764
$66$	0	0
$67$	1.52364	0.186142	0.0930709	−	0.995659i	$-0.470332\pi$
$67$	1.52364	0.186142	0.0930709	+	0.995659i	$0.470332\pi$
$68$	0	0
$69$	−8.17617	−0.984296
$70$	0	0
$71$	4.46519	0.529921	0.264960	−	0.964259i	$-0.414641\pi$
$71$	4.46519	0.529921	0.264960	+	0.964259i	$0.414641\pi$
$72$	0	0
$73$	−7.44051	−0.870845	−0.435423	−	0.900226i	$-0.643401\pi$
$73$	−7.44051	−0.870845	−0.435423	+	0.900226i	$0.643401\pi$
$74$	0	0
$75$	4.77566	0.551446
$76$	0	0
$77$	2.29998	0.262107
$78$	0	0
$79$	−8.68765	−0.977437	−0.488718	−	0.872442i	$-0.662535\pi$
$79$	−8.68765	−0.977437	−0.488718	+	0.872442i	$0.662535\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.7936	−1.73357	−0.866785	−	0.498681i	$-0.833818\pi$
$83$	−15.7936	−1.73357	−0.866785	+	0.498681i	$0.833818\pi$
$84$	0	0
$85$	2.83884	0.307915
$86$	0	0
$87$	−3.87701	−0.415659
$88$	0	0
$89$	−5.88769	−0.624094	−0.312047	−	0.950067i	$-0.601015\pi$
$89$	−5.88769	−0.624094	−0.312047	+	0.950067i	$0.601015\pi$
$90$	0	0
$91$	1.61199	0.168983
$92$	0	0
$93$	−2.70895	−0.280905
$94$	0	0
$95$	−1.13700	−0.116653
$96$	0	0
$97$	−12.9569	−1.31557	−0.657787	−	0.753204i	$-0.728507\pi$
$97$	−12.9569	−1.31557	−0.657787	+	0.753204i	$0.728507\pi$
$98$	0	0
$99$	−3.46722	−0.348469
$100$	0	0
$101$	−3.71030	−0.369188	−0.184594	−	0.982815i	$-0.559097\pi$
$101$	−3.71030	−0.369188	−0.184594	+	0.982815i	$0.559097\pi$
$102$	0	0
$103$	−14.3630	−1.41523	−0.707616	−	0.706598i	$-0.750229\pi$
$103$	−14.3630	−1.41523	−0.707616	+	0.706598i	$0.750229\pi$
$104$	0	0
$105$	−0.314193	−0.0306621
$106$	0	0
$107$	0.312825	0.0302419	0.0151210	−	0.999886i	$-0.495187\pi$
$107$	0.312825	0.0302419	0.0151210	+	0.999886i	$0.495187\pi$
$108$	0	0
$109$	9.69339	0.928458	0.464229	−	0.885715i	$-0.346331\pi$
$109$	9.69339	0.928458	0.464229	+	0.885715i	$0.346331\pi$
$110$	0	0
$111$	−8.48041	−0.804925
$112$	0	0
$113$	7.03077	0.661399	0.330700	−	0.943736i	$-0.392715\pi$
$113$	7.03077	0.661399	0.330700	+	0.943736i	$0.392715\pi$
$114$	0	0
$115$	−3.87262	−0.361123
$116$	0	0
$117$	−2.43008	−0.224661
$118$	0	0
$119$	3.97583	0.364464
$120$	0	0
$121$	1.02164	0.0928767
$122$	0	0
$123$	−3.71876	−0.335309
$124$	0	0
$125$	4.63021	0.414138
$126$	0	0
$127$	−3.30843	−0.293576	−0.146788	−	0.989168i	$-0.546894\pi$
$127$	−3.30843	−0.293576	−0.146788	+	0.989168i	$0.546894\pi$
$128$	0	0
$129$	−10.5079	−0.925173
$130$	0	0
$131$	−8.96333	−0.783130	−0.391565	−	0.920151i	$-0.628066\pi$
$131$	−8.96333	−0.783130	−0.391565	+	0.920151i	$0.628066\pi$
$132$	0	0
$133$	−1.59238	−0.138077
$134$	0	0
$135$	0.473647	0.0407650
$136$	0	0
$137$	−3.77363	−0.322403	−0.161201	−	0.986922i	$-0.551537\pi$
$137$	−3.77363	−0.322403	−0.161201	+	0.986922i	$0.551537\pi$
$138$	0	0
$139$	3.16082	0.268097	0.134049	−	0.990975i	$-0.457202\pi$
$139$	3.16082	0.268097	0.134049	+	0.990975i	$0.457202\pi$
$140$	0	0
$141$	−10.9069	−0.918528
$142$	0	0
$143$	8.42563	0.704586
$144$	0	0
$145$	−1.83633	−0.152499
$146$	0	0
$147$	6.55997	0.541057
$148$	0	0
$149$	17.2900	1.41645	0.708227	−	0.705985i	$-0.249495\pi$
$149$	17.2900	1.41645	0.708227	+	0.705985i	$0.249495\pi$
$150$	0	0
$151$	5.79304	0.471431	0.235716	−	0.971822i	$-0.424257\pi$
$151$	5.79304	0.471431	0.235716	+	0.971822i	$0.424257\pi$
$152$	0	0
$153$	−5.99358	−0.484552
$154$	0	0
$155$	−1.28308	−0.103060
$156$	0	0
$157$	−19.9447	−1.59176	−0.795881	−	0.605453i	$-0.792992\pi$
$157$	−19.9447	−1.59176	−0.795881	+	0.605453i	$0.792992\pi$
$158$	0	0
$159$	−0.259988	−0.0206184
$160$	0	0
$161$	−5.42366	−0.427444
$162$	0	0
$163$	10.2632	0.803874	0.401937	−	0.915667i	$-0.368337\pi$
$163$	10.2632	0.803874	0.401937	+	0.915667i	$0.368337\pi$
$164$	0	0
$165$	−1.64224	−0.127848
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−7.09472	−0.545747
$170$	0	0
$171$	2.40051	0.183572
$172$	0	0
$173$	−9.66098	−0.734510	−0.367255	−	0.930120i	$-0.619702\pi$
$173$	−9.66098	−0.734510	−0.367255	+	0.930120i	$0.619702\pi$
$174$	0	0
$175$	3.16793	0.239473
$176$	0	0
$177$	−1.16237	−0.0873688
$178$	0	0
$179$	21.0225	1.57130	0.785649	−	0.618672i	$-0.212329\pi$
$179$	21.0225	1.57130	0.785649	+	0.618672i	$0.212329\pi$
$180$	0	0
$181$	−10.8532	−0.806711	−0.403355	−	0.915043i	$-0.632156\pi$
$181$	−10.8532	−0.806711	−0.403355	+	0.915043i	$0.632156\pi$
$182$	0	0
$183$	1.55815	0.115181
$184$	0	0
$185$	−4.01672	−0.295315
$186$	0	0
$187$	20.7811	1.51966
$188$	0	0
$189$	0.663349	0.0482515
$190$	0	0
$191$	−12.8612	−0.930601	−0.465301	−	0.885153i	$-0.654054\pi$
$191$	−12.8612	−0.930601	−0.465301	+	0.885153i	$0.654054\pi$
$192$	0	0
$193$	−7.46080	−0.537040	−0.268520	−	0.963274i	$-0.586535\pi$
$193$	−7.46080	−0.537040	−0.268520	+	0.963274i	$0.586535\pi$
$194$	0	0
$195$	−1.15100	−0.0824247
$196$	0	0
$197$	−2.45809	−0.175131	−0.0875657	−	0.996159i	$-0.527909\pi$
$197$	−2.45809	−0.175131	−0.0875657	+	0.996159i	$0.527909\pi$
$198$	0	0
$199$	−18.6699	−1.32348	−0.661738	−	0.749735i	$-0.730181\pi$
$199$	−18.6699	−1.32348	−0.661738	+	0.749735i	$0.730181\pi$
$200$	0	0
$201$	−1.52364	−0.107469
$202$	0	0
$203$	−2.57181	−0.180506
$204$	0	0
$205$	−1.76138	−0.123020
$206$	0	0
$207$	8.17617	0.568283
$208$	0	0
$209$	−8.32312	−0.575722
$210$	0	0
$211$	−4.26671	−0.293732	−0.146866	−	0.989156i	$-0.546919\pi$
$211$	−4.26671	−0.293732	−0.146866	+	0.989156i	$0.546919\pi$
$212$	0	0
$213$	−4.46519	−0.305950
$214$	0	0
$215$	−4.97705	−0.339432
$216$	0	0
$217$	−1.79698	−0.121987
$218$	0	0
$219$	7.44051	0.502783
$220$	0	0
$221$	14.5649	0.979739
$222$	0	0
$223$	19.9512	1.33603	0.668015	−	0.744148i	$-0.267144\pi$
$223$	19.9512	1.33603	0.668015	+	0.744148i	$0.267144\pi$
$224$	0	0
$225$	−4.77566	−0.318377
$226$	0	0
$227$	22.5716	1.49813	0.749065	−	0.662497i	$-0.230503\pi$
$227$	22.5716	1.49813	0.749065	+	0.662497i	$0.230503\pi$
$228$	0	0
$229$	−9.25689	−0.611712	−0.305856	−	0.952078i	$-0.598943\pi$
$229$	−9.25689	−0.611712	−0.305856	+	0.952078i	$0.598943\pi$
$230$	0	0
$231$	−2.29998	−0.151328
$232$	0	0
$233$	1.24492	0.0815571	0.0407786	−	0.999168i	$-0.487016\pi$
$233$	1.24492	0.0815571	0.0407786	+	0.999168i	$0.487016\pi$
$234$	0	0
$235$	−5.16602	−0.336994
$236$	0	0
$237$	8.68765	0.564323
$238$	0	0
$239$	−6.50808	−0.420972	−0.210486	−	0.977597i	$-0.567505\pi$
$239$	−6.50808	−0.420972	−0.210486	+	0.977597i	$0.567505\pi$
$240$	0	0
$241$	13.9256	0.897026	0.448513	−	0.893776i	$-0.351954\pi$
$241$	13.9256	0.897026	0.448513	+	0.893776i	$0.351954\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	3.10711	0.198506
$246$	0	0
$247$	−5.83344	−0.371173
$248$	0	0
$249$	15.7936	1.00088
$250$	0	0
$251$	−17.3433	−1.09470	−0.547349	−	0.836905i	$-0.684363\pi$
$251$	−17.3433	−1.09470	−0.547349	+	0.836905i	$0.684363\pi$
$252$	0	0
$253$	−28.3486	−1.78226
$254$	0	0
$255$	−2.83884	−0.177775
$256$	0	0
$257$	−0.261552	−0.0163152	−0.00815759	−	0.999967i	$-0.502597\pi$
$257$	−0.261552	−0.0163152	−0.00815759	+	0.999967i	$0.502597\pi$
$258$	0	0
$259$	−5.62547	−0.349550
$260$	0	0
$261$	3.87701	0.239981
$262$	0	0
$263$	−25.1235	−1.54918	−0.774590	−	0.632464i	$-0.782044\pi$
$263$	−25.1235	−1.54918	−0.774590	+	0.632464i	$0.782044\pi$
$264$	0	0
$265$	−0.123143	−0.00756459
$266$	0	0
$267$	5.88769	0.360321
$268$	0	0
$269$	−8.26812	−0.504116	−0.252058	−	0.967712i	$-0.581107\pi$
$269$	−8.26812	−0.504116	−0.252058	+	0.967712i	$0.581107\pi$
$270$	0	0
$271$	−9.04656	−0.549540	−0.274770	−	0.961510i	$-0.588602\pi$
$271$	−9.04656	−0.549540	−0.274770	+	0.961510i	$0.588602\pi$
$272$	0	0
$273$	−1.61199	−0.0975621
$274$	0	0
$275$	16.5583	0.998502
$276$	0	0
$277$	−25.4147	−1.52702	−0.763511	−	0.645795i	$-0.776526\pi$
$277$	−25.4147	−1.52702	−0.763511	+	0.645795i	$0.776526\pi$
$278$	0	0
$279$	2.70895	0.162181
$280$	0	0
$281$	−5.13820	−0.306519	−0.153260	−	0.988186i	$-0.548977\pi$
$281$	−5.13820	−0.306519	−0.153260	+	0.988186i	$0.548977\pi$
$282$	0	0
$283$	−13.5596	−0.806036	−0.403018	−	0.915192i	$-0.632039\pi$
$283$	−13.5596	−0.806036	−0.403018	+	0.915192i	$0.632039\pi$
$284$	0	0
$285$	1.13700	0.0673498
$286$	0	0
$287$	−2.46683	−0.145613
$288$	0	0
$289$	18.9230	1.11312
$290$	0	0
$291$	12.9569	0.759547
$292$	0	0
$293$	21.0838	1.23173	0.615864	−	0.787853i	$-0.288807\pi$
$293$	21.0838	1.23173	0.615864	+	0.787853i	$0.288807\pi$
$294$	0	0
$295$	−0.550551	−0.0320543
$296$	0	0
$297$	3.46722	0.201189
$298$	0	0
$299$	−19.8687	−1.14904
$300$	0	0
$301$	−6.97044	−0.401769
$302$	0	0
$303$	3.71030	0.213151
$304$	0	0
$305$	0.738011	0.0422584
$306$	0	0
$307$	29.4875	1.68294	0.841471	−	0.540302i	$-0.181690\pi$
$307$	29.4875	1.68294	0.841471	+	0.540302i	$0.181690\pi$
$308$	0	0
$309$	14.3630	0.817084
$310$	0	0
$311$	−19.2057	−1.08905	−0.544527	−	0.838743i	$-0.683291\pi$
$311$	−19.2057	−1.08905	−0.544527	+	0.838743i	$0.683291\pi$
$312$	0	0
$313$	−7.13192	−0.403120	−0.201560	−	0.979476i	$-0.564601\pi$
$313$	−7.13192	−0.403120	−0.201560	+	0.979476i	$0.564601\pi$
$314$	0	0
$315$	0.314193	0.0177028
$316$	0	0
$317$	7.54451	0.423742	0.211871	−	0.977298i	$-0.432044\pi$
$317$	7.54451	0.423742	0.211871	+	0.977298i	$0.432044\pi$
$318$	0	0
$319$	−13.4425	−0.752633
$320$	0	0
$321$	−0.312825	−0.0174602
$322$	0	0
$323$	−14.3877	−0.800551
$324$	0	0
$325$	11.6052	0.643742
$326$	0	0
$327$	−9.69339	−0.536046
$328$	0	0
$329$	−7.23509	−0.398884
$330$	0	0
$331$	21.5151	1.18258	0.591289	−	0.806459i	$-0.298619\pi$
$331$	21.5151	1.18258	0.591289	+	0.806459i	$0.298619\pi$
$332$	0	0
$333$	8.48041	0.464724
$334$	0	0
$335$	−0.721666	−0.0394288
$336$	0	0
$337$	−33.0750	−1.80171	−0.900856	−	0.434118i	$-0.857060\pi$
$337$	−33.0750	−1.80171	−0.900856	+	0.434118i	$0.857060\pi$
$338$	0	0
$339$	−7.03077	−0.381859
$340$	0	0
$341$	−9.39253	−0.508634
$342$	0	0
$343$	8.99499	0.485684
$344$	0	0
$345$	3.87262	0.208495
$346$	0	0
$347$	4.38501	0.235400	0.117700	−	0.993049i	$-0.462448\pi$
$347$	4.38501	0.235400	0.117700	+	0.993049i	$0.462448\pi$
$348$	0	0
$349$	−28.5821	−1.52996	−0.764982	−	0.644052i	$-0.777252\pi$
$349$	−28.5821	−1.52996	−0.764982	+	0.644052i	$0.777252\pi$
$350$	0	0
$351$	2.43008	0.129708
$352$	0	0
$353$	−10.2891	−0.547636	−0.273818	−	0.961782i	$-0.588287\pi$
$353$	−10.2891	−0.547636	−0.273818	+	0.961782i	$0.588287\pi$
$354$	0	0
$355$	−2.11492	−0.112248
$356$	0	0
$357$	−3.97583	−0.210423
$358$	0	0
$359$	23.2335	1.22622	0.613108	−	0.789999i	$-0.289919\pi$
$359$	23.2335	1.22622	0.613108	+	0.789999i	$0.289919\pi$
$360$	0	0
$361$	−13.2375	−0.696712
$362$	0	0
$363$	−1.02164	−0.0536224
$364$	0	0
$365$	3.52417	0.184464
$366$	0	0
$367$	−16.4276	−0.857512	−0.428756	−	0.903420i	$-0.641048\pi$
$367$	−16.4276	−0.857512	−0.428756	+	0.903420i	$0.641048\pi$
$368$	0	0
$369$	3.71876	0.193591
$370$	0	0
$371$	−0.172463	−0.00895383
$372$	0	0
$373$	−3.00351	−0.155516	−0.0777579	−	0.996972i	$-0.524776\pi$
$373$	−3.00351	−0.155516	−0.0777579	+	0.996972i	$0.524776\pi$
$374$	0	0
$375$	−4.63021	−0.239103
$376$	0	0
$377$	−9.42143	−0.485229
$378$	0	0
$379$	2.71775	0.139601	0.0698007	−	0.997561i	$-0.477764\pi$
$379$	2.71775	0.139601	0.0698007	+	0.997561i	$0.477764\pi$
$380$	0	0
$381$	3.30843	0.169496
$382$	0	0
$383$	2.85264	0.145763	0.0728816	−	0.997341i	$-0.476780\pi$
$383$	2.85264	0.145763	0.0728816	+	0.997341i	$0.476780\pi$
$384$	0	0
$385$	−1.08938	−0.0555198
$386$	0	0
$387$	10.5079	0.534149
$388$	0	0
$389$	−2.22588	−0.112857	−0.0564284	−	0.998407i	$-0.517971\pi$
$389$	−2.22588	−0.112857	−0.0564284	+	0.998407i	$0.517971\pi$
$390$	0	0
$391$	−49.0045	−2.47827
$392$	0	0
$393$	8.96333	0.452140
$394$	0	0
$395$	4.11488	0.207042
$396$	0	0
$397$	12.2045	0.612527	0.306264	−	0.951947i	$-0.400921\pi$
$397$	12.2045	0.612527	0.306264	+	0.951947i	$0.400921\pi$
$398$	0	0
$399$	1.59238	0.0797187
$400$	0	0
$401$	36.7612	1.83577	0.917883	−	0.396852i	$-0.129897\pi$
$401$	36.7612	1.83577	0.917883	+	0.396852i	$0.129897\pi$
$402$	0	0
$403$	−6.58296	−0.327921
$404$	0	0
$405$	−0.473647	−0.0235357
$406$	0	0
$407$	−29.4035	−1.45748
$408$	0	0
$409$	14.4370	0.713863	0.356931	−	0.934131i	$-0.383823\pi$
$409$	14.4370	0.713863	0.356931	+	0.934131i	$0.383823\pi$
$410$	0	0
$411$	3.77363	0.186139
$412$	0	0
$413$	−0.771054	−0.0379411
$414$	0	0
$415$	7.48058	0.367207
$416$	0	0
$417$	−3.16082	−0.154786
$418$	0	0
$419$	2.34434	0.114529	0.0572643	−	0.998359i	$-0.481762\pi$
$419$	2.34434	0.114529	0.0572643	+	0.998359i	$0.481762\pi$
$420$	0	0
$421$	18.3292	0.893311	0.446655	−	0.894706i	$-0.352615\pi$
$421$	18.3292	0.893311	0.446655	+	0.894706i	$0.352615\pi$
$422$	0	0
$423$	10.9069	0.530312
$424$	0	0
$425$	28.6233	1.38843
$426$	0	0
$427$	1.03359	0.0500191
$428$	0	0
$429$	−8.42563	−0.406793
$430$	0	0
$431$	−5.77531	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$431$	−5.77531	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$432$	0	0
$433$	−3.96341	−0.190469	−0.0952346	−	0.995455i	$-0.530360\pi$
$433$	−3.96341	−0.190469	−0.0952346	+	0.995455i	$0.530360\pi$
$434$	0	0
$435$	1.83633	0.0880454
$436$	0	0
$437$	19.6270	0.938888
$438$	0	0
$439$	0.234378	0.0111863	0.00559313	−	0.999984i	$-0.498220\pi$
$439$	0.234378	0.0111863	0.00559313	+	0.999984i	$0.498220\pi$
$440$	0	0
$441$	−6.55997	−0.312379
$442$	0	0
$443$	−26.2502	−1.24718	−0.623592	−	0.781750i	$-0.714327\pi$
$443$	−26.2502	−1.24718	−0.623592	+	0.781750i	$0.714327\pi$
$444$	0	0
$445$	2.78868	0.132196
$446$	0	0
$447$	−17.2900	−0.817790
$448$	0	0
$449$	−21.7460	−1.02626	−0.513128	−	0.858312i	$-0.671513\pi$
$449$	−21.7460	−1.02626	−0.513128	+	0.858312i	$0.671513\pi$
$450$	0	0
$451$	−12.8938	−0.607144
$452$	0	0
$453$	−5.79304	−0.272181
$454$	0	0
$455$	−0.763514	−0.0357941
$456$	0	0
$457$	25.0107	1.16995	0.584975	−	0.811051i	$-0.301104\pi$
$457$	25.0107	1.16995	0.584975	+	0.811051i	$0.301104\pi$
$458$	0	0
$459$	5.99358	0.279756
$460$	0	0
$461$	−2.70398	−0.125937	−0.0629684	−	0.998016i	$-0.520057\pi$
$461$	−2.70398	−0.125937	−0.0629684	+	0.998016i	$0.520057\pi$
$462$	0	0
$463$	18.2411	0.847734	0.423867	−	0.905725i	$-0.360672\pi$
$463$	18.2411	0.847734	0.423867	+	0.905725i	$0.360672\pi$
$464$	0	0
$465$	1.28308	0.0595016
$466$	0	0
$467$	−14.8934	−0.689183	−0.344591	−	0.938753i	$-0.611983\pi$
$467$	−14.8934	−0.689183	−0.344591	+	0.938753i	$0.611983\pi$
$468$	0	0
$469$	−1.01070	−0.0466699
$470$	0	0
$471$	19.9447	0.919004
$472$	0	0
$473$	−36.4334	−1.67521
$474$	0	0
$475$	−11.4640	−0.526006
$476$	0	0
$477$	0.259988	0.0119040
$478$	0	0
$479$	11.1271	0.508412	0.254206	−	0.967150i	$-0.418186\pi$
$479$	11.1271	0.508412	0.254206	+	0.967150i	$0.418186\pi$
$480$	0	0
$481$	−20.6081	−0.939647
$482$	0	0
$483$	5.42366	0.246785
$484$	0	0
$485$	6.13700	0.278667
$486$	0	0
$487$	−29.0109	−1.31461	−0.657304	−	0.753625i	$-0.728303\pi$
$487$	−29.0109	−1.31461	−0.657304	+	0.753625i	$0.728303\pi$
$488$	0	0
$489$	−10.2632	−0.464117
$490$	0	0
$491$	−42.0429	−1.89737	−0.948685	−	0.316223i	$-0.897585\pi$
$491$	−42.0429	−1.89737	−0.948685	+	0.316223i	$0.897585\pi$
$492$	0	0
$493$	−23.2371	−1.04655
$494$	0	0
$495$	1.64224	0.0738132
$496$	0	0
$497$	−2.96198	−0.132863
$498$	0	0
$499$	−8.70998	−0.389912	−0.194956	−	0.980812i	$-0.562456\pi$
$499$	−8.70998	−0.389912	−0.194956	+	0.980812i	$0.562456\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−0.952706	−0.0424791	−0.0212395	−	0.999774i	$-0.506761\pi$
$503$	−0.952706	−0.0424791	−0.0212395	+	0.999774i	$0.506761\pi$
$504$	0	0
$505$	1.75737	0.0782019
$506$	0	0
$507$	7.09472	0.315087
$508$	0	0
$509$	−13.0897	−0.580191	−0.290095	−	0.956998i	$-0.593687\pi$
$509$	−13.0897	−0.580191	−0.290095	+	0.956998i	$0.593687\pi$
$510$	0	0
$511$	4.93565	0.218340
$512$	0	0
$513$	−2.40051	−0.105985
$514$	0	0
$515$	6.80300	0.299776
$516$	0	0
$517$	−37.8167	−1.66318
$518$	0	0
$519$	9.66098	0.424070
$520$	0	0
$521$	−23.0754	−1.01095	−0.505475	−	0.862841i	$-0.668683\pi$
$521$	−23.0754	−1.01095	−0.505475	+	0.862841i	$0.668683\pi$
$522$	0	0
$523$	28.3216	1.23842	0.619210	−	0.785226i	$-0.287453\pi$
$523$	28.3216	1.23842	0.619210	+	0.785226i	$0.287453\pi$
$524$	0	0
$525$	−3.16793	−0.138260
$526$	0	0
$527$	−16.2363	−0.707264
$528$	0	0
$529$	43.8498	1.90651
$530$	0	0
$531$	1.16237	0.0504424
$532$	0	0
$533$	−9.03688	−0.391431
$534$	0	0
$535$	−0.148168	−0.00640588
$536$	0	0
$537$	−21.0225	−0.907190
$538$	0	0
$539$	22.7449	0.979691
$540$	0	0
$541$	−20.6244	−0.886710	−0.443355	−	0.896346i	$-0.646212\pi$
$541$	−20.6244	−0.886710	−0.443355	+	0.896346i	$0.646212\pi$
$542$	0	0
$543$	10.8532	0.465755
$544$	0	0
$545$	−4.59124	−0.196667
$546$	0	0
$547$	−32.7328	−1.39956	−0.699778	−	0.714361i	$-0.746718\pi$
$547$	−32.7328	−1.39956	−0.699778	+	0.714361i	$0.746718\pi$
$548$	0	0
$549$	−1.55815	−0.0665000
$550$	0	0
$551$	9.30681	0.396484
$552$	0	0
$553$	5.76294	0.245065
$554$	0	0
$555$	4.01672	0.170500
$556$	0	0
$557$	−0.834402	−0.0353548	−0.0176774	−	0.999844i	$-0.505627\pi$
$557$	−0.834402	−0.0353548	−0.0176774	+	0.999844i	$0.505627\pi$
$558$	0	0
$559$	−25.5351	−1.08002
$560$	0	0
$561$	−20.7811	−0.877378
$562$	0	0
$563$	10.3371	0.435656	0.217828	−	0.975987i	$-0.430103\pi$
$563$	10.3371	0.435656	0.217828	+	0.975987i	$0.430103\pi$
$564$	0	0
$565$	−3.33010	−0.140098
$566$	0	0
$567$	−0.663349	−0.0278580
$568$	0	0
$569$	28.4687	1.19347	0.596735	−	0.802439i	$-0.296465\pi$
$569$	28.4687	1.19347	0.596735	+	0.802439i	$0.296465\pi$
$570$	0	0
$571$	−27.5241	−1.15185	−0.575924	−	0.817503i	$-0.695358\pi$
$571$	−27.5241	−1.15185	−0.575924	+	0.817503i	$0.695358\pi$
$572$	0	0
$573$	12.8612	0.537283
$574$	0	0
$575$	−39.0466	−1.62836
$576$	0	0
$577$	−19.6801	−0.819291	−0.409646	−	0.912245i	$-0.634348\pi$
$577$	−19.6801	−0.819291	−0.409646	+	0.912245i	$0.634348\pi$
$578$	0	0
$579$	7.46080	0.310060
$580$	0	0
$581$	10.4767	0.434645
$582$	0	0
$583$	−0.901438	−0.0373337
$584$	0	0
$585$	1.15100	0.0475879
$586$	0	0
$587$	−3.69629	−0.152562	−0.0762810	−	0.997086i	$-0.524305\pi$
$587$	−3.69629	−0.152562	−0.0762810	+	0.997086i	$0.524305\pi$
$588$	0	0
$589$	6.50287	0.267946
$590$	0	0
$591$	2.45809	0.101112
$592$	0	0
$593$	6.97065	0.286250	0.143125	−	0.989705i	$-0.454285\pi$
$593$	6.97065	0.286250	0.143125	+	0.989705i	$0.454285\pi$
$594$	0	0
$595$	−1.88314	−0.0772012
$596$	0	0
$597$	18.6699	0.764109
$598$	0	0
$599$	−40.9392	−1.67273	−0.836366	−	0.548172i	$-0.815324\pi$
$599$	−40.9392	−1.67273	−0.836366	+	0.548172i	$0.815324\pi$
$600$	0	0
$601$	−21.6168	−0.881769	−0.440884	−	0.897564i	$-0.645335\pi$
$601$	−21.6168	−0.881769	−0.440884	+	0.897564i	$0.645335\pi$
$602$	0	0
$603$	1.52364	0.0620473
$604$	0	0
$605$	−0.483898	−0.0196733
$606$	0	0
$607$	−4.06739	−0.165090	−0.0825451	−	0.996587i	$-0.526305\pi$
$607$	−4.06739	−0.165090	−0.0825451	+	0.996587i	$0.526305\pi$
$608$	0	0
$609$	2.57181	0.104215
$610$	0	0
$611$	−26.5047	−1.07226
$612$	0	0
$613$	−4.06712	−0.164269	−0.0821347	−	0.996621i	$-0.526174\pi$
$613$	−4.06712	−0.164269	−0.0821347	+	0.996621i	$0.526174\pi$
$614$	0	0
$615$	1.76138	0.0710256
$616$	0	0
$617$	−17.9187	−0.721380	−0.360690	−	0.932686i	$-0.617459\pi$
$617$	−17.9187	−0.721380	−0.360690	+	0.932686i	$0.617459\pi$
$618$	0	0
$619$	8.24950	0.331575	0.165788	−	0.986161i	$-0.446983\pi$
$619$	8.24950	0.331575	0.165788	+	0.986161i	$0.446983\pi$
$620$	0	0
$621$	−8.17617	−0.328099
$622$	0	0
$623$	3.90559	0.156474
$624$	0	0
$625$	21.6852	0.867408
$626$	0	0
$627$	8.32312	0.332393
$628$	0	0
$629$	−50.8280	−2.02665
$630$	0	0
$631$	−23.1352	−0.920997	−0.460498	−	0.887661i	$-0.652329\pi$
$631$	−23.1352	−0.920997	−0.460498	+	0.887661i	$0.652329\pi$
$632$	0	0
$633$	4.26671	0.169586
$634$	0	0
$635$	1.56703	0.0621857
$636$	0	0
$637$	15.9412	0.631615
$638$	0	0
$639$	4.46519	0.176640
$640$	0	0
$641$	−37.2103	−1.46972	−0.734859	−	0.678220i	$-0.762751\pi$
$641$	−37.2103	−1.46972	−0.734859	+	0.678220i	$0.762751\pi$
$642$	0	0
$643$	−10.5078	−0.414388	−0.207194	−	0.978300i	$-0.566433\pi$
$643$	−10.5078	−0.414388	−0.207194	+	0.978300i	$0.566433\pi$
$644$	0	0
$645$	4.97705	0.195971
$646$	0	0
$647$	9.81698	0.385945	0.192973	−	0.981204i	$-0.438187\pi$
$647$	9.81698	0.385945	0.192973	+	0.981204i	$0.438187\pi$
$648$	0	0
$649$	−4.03018	−0.158199
$650$	0	0
$651$	1.79698	0.0704291
$652$	0	0
$653$	39.3873	1.54134	0.770671	−	0.637233i	$-0.219921\pi$
$653$	39.3873	1.54134	0.770671	+	0.637233i	$0.219921\pi$
$654$	0	0
$655$	4.24545	0.165883
$656$	0	0
$657$	−7.44051	−0.290282
$658$	0	0
$659$	43.1721	1.68174	0.840872	−	0.541234i	$-0.182043\pi$
$659$	43.1721	1.68174	0.840872	+	0.541234i	$0.182043\pi$
$660$	0	0
$661$	27.4626	1.06817	0.534086	−	0.845430i	$-0.320656\pi$
$661$	27.4626	1.06817	0.534086	+	0.845430i	$0.320656\pi$
$662$	0	0
$663$	−14.5649	−0.565653
$664$	0	0
$665$	0.754225	0.0292476
$666$	0	0
$667$	31.6991	1.22739
$668$	0	0
$669$	−19.9512	−0.771357
$670$	0	0
$671$	5.40244	0.208559
$672$	0	0
$673$	−14.6060	−0.563020	−0.281510	−	0.959558i	$-0.590835\pi$
$673$	−14.6060	−0.563020	−0.281510	+	0.959558i	$0.590835\pi$
$674$	0	0
$675$	4.77566	0.183815
$676$	0	0
$677$	21.4195	0.823217	0.411608	−	0.911361i	$-0.364967\pi$
$677$	21.4195	0.823217	0.411608	+	0.911361i	$0.364967\pi$
$678$	0	0
$679$	8.59495	0.329844
$680$	0	0
$681$	−22.5716	−0.864946
$682$	0	0
$683$	8.83920	0.338223	0.169111	−	0.985597i	$-0.445910\pi$
$683$	8.83920	0.338223	0.169111	+	0.985597i	$0.445910\pi$
$684$	0	0
$685$	1.78737	0.0682917
$686$	0	0
$687$	9.25689	0.353172
$688$	0	0
$689$	−0.631792	−0.0240694
$690$	0	0
$691$	−44.9026	−1.70818	−0.854088	−	0.520129i	$-0.825884\pi$
$691$	−44.9026	−1.70818	−0.854088	+	0.520129i	$0.825884\pi$
$692$	0	0
$693$	2.29998	0.0873690
$694$	0	0
$695$	−1.49711	−0.0567887
$696$	0	0
$697$	−22.2887	−0.844243
$698$	0	0
$699$	−1.24492	−0.0470870
$700$	0	0
$701$	39.2450	1.48226	0.741131	−	0.671360i	$-0.234290\pi$
$701$	39.2450	1.48226	0.741131	+	0.671360i	$0.234290\pi$
$702$	0	0
$703$	20.3574	0.767792
$704$	0	0
$705$	5.16602	0.194564
$706$	0	0
$707$	2.46122	0.0925638
$708$	0	0
$709$	−7.10889	−0.266980	−0.133490	−	0.991050i	$-0.542618\pi$
$709$	−7.10889	−0.266980	−0.133490	+	0.991050i	$0.542618\pi$
$710$	0	0
$711$	−8.68765	−0.325812
$712$	0	0
$713$	22.1488	0.829480
$714$	0	0
$715$	−3.99077	−0.149246
$716$	0	0
$717$	6.50808	0.243049
$718$	0	0
$719$	40.8947	1.52512	0.762558	−	0.646919i	$-0.223943\pi$
$719$	40.8947	1.52512	0.762558	+	0.646919i	$0.223943\pi$
$720$	0	0
$721$	9.52770	0.354830
$722$	0	0
$723$	−13.9256	−0.517898
$724$	0	0
$725$	−18.5153	−0.687640
$726$	0	0
$727$	19.9140	0.738568	0.369284	−	0.929317i	$-0.379603\pi$
$727$	19.9140	0.738568	0.369284	+	0.929317i	$0.379603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−62.9802	−2.32941
$732$	0	0
$733$	14.7061	0.543182	0.271591	−	0.962413i	$-0.412450\pi$
$733$	14.7061	0.543182	0.271591	+	0.962413i	$0.412450\pi$
$734$	0	0
$735$	−3.10711	−0.114607
$736$	0	0
$737$	−5.28279	−0.194594
$738$	0	0
$739$	20.6497	0.759611	0.379806	−	0.925066i	$-0.375991\pi$
$739$	20.6497	0.759611	0.379806	+	0.925066i	$0.375991\pi$
$740$	0	0
$741$	5.83344	0.214297
$742$	0	0
$743$	−7.39332	−0.271235	−0.135617	−	0.990761i	$-0.543302\pi$
$743$	−7.39332	−0.271235	−0.135617	+	0.990761i	$0.543302\pi$
$744$	0	0
$745$	−8.18936	−0.300035
$746$	0	0
$747$	−15.7936	−0.577857
$748$	0	0
$749$	−0.207512	−0.00758232
$750$	0	0
$751$	−39.9001	−1.45598	−0.727988	−	0.685590i	$-0.759544\pi$
$751$	−39.9001	−1.45598	−0.727988	+	0.685590i	$0.759544\pi$
$752$	0	0
$753$	17.3433	0.632024
$754$	0	0
$755$	−2.74386	−0.0998591
$756$	0	0
$757$	−9.09965	−0.330733	−0.165366	−	0.986232i	$-0.552881\pi$
$757$	−9.09965	−0.330733	−0.165366	+	0.986232i	$0.552881\pi$
$758$	0	0
$759$	28.3486	1.02899
$760$	0	0
$761$	24.2613	0.879472	0.439736	−	0.898127i	$-0.355072\pi$
$761$	24.2613	0.879472	0.439736	+	0.898127i	$0.355072\pi$
$762$	0	0
$763$	−6.43010	−0.232785
$764$	0	0
$765$	2.83884	0.102638
$766$	0	0
$767$	−2.82464	−0.101992
$768$	0	0
$769$	−37.0317	−1.33540	−0.667699	−	0.744431i	$-0.732721\pi$
$769$	−37.0317	−1.33540	−0.667699	+	0.744431i	$0.732721\pi$
$770$	0	0
$771$	0.261552	0.00941958
$772$	0	0
$773$	12.1176	0.435841	0.217921	−	0.975966i	$-0.430073\pi$
$773$	12.1176	0.435841	0.217921	+	0.975966i	$0.430073\pi$
$774$	0	0
$775$	−12.9370	−0.464711
$776$	0	0
$777$	5.62547	0.201813
$778$	0	0
$779$	8.92693	0.319841
$780$	0	0
$781$	−15.4818	−0.553983
$782$	0	0
$783$	−3.87701	−0.138553
$784$	0	0
$785$	9.44675	0.337169
$786$	0	0
$787$	−31.4952	−1.12268	−0.561341	−	0.827585i	$-0.689714\pi$
$787$	−31.4952	−1.12268	−0.561341	+	0.827585i	$0.689714\pi$
$788$	0	0
$789$	25.1235	0.894419
$790$	0	0
$791$	−4.66385	−0.165828
$792$	0	0
$793$	3.78642	0.134460
$794$	0	0
$795$	0.123143	0.00436742
$796$	0	0
$797$	33.7213	1.19447	0.597234	−	0.802067i	$-0.296266\pi$
$797$	33.7213	1.19447	0.597234	+	0.802067i	$0.296266\pi$
$798$	0	0
$799$	−65.3714	−2.31268
$800$	0	0
$801$	−5.88769	−0.208031
$802$	0	0
$803$	25.7979	0.910388
$804$	0	0
$805$	2.56890	0.0905417
$806$	0	0
$807$	8.26812	0.291052
$808$	0	0
$809$	25.2848	0.888968	0.444484	−	0.895787i	$-0.353387\pi$
$809$	25.2848	0.888968	0.444484	+	0.895787i	$0.353387\pi$
$810$	0	0
$811$	23.4610	0.823826	0.411913	−	0.911223i	$-0.364861\pi$
$811$	23.4610	0.823826	0.411913	+	0.911223i	$0.364861\pi$
$812$	0	0
$813$	9.04656	0.317277
$814$	0	0
$815$	−4.86112	−0.170278
$816$	0	0
$817$	25.2245	0.882493
$818$	0	0
$819$	1.61199	0.0563275
$820$	0	0
$821$	−31.9196	−1.11400	−0.557000	−	0.830512i	$-0.688048\pi$
$821$	−31.9196	−1.11400	−0.557000	+	0.830512i	$0.688048\pi$
$822$	0	0
$823$	44.7610	1.56027	0.780136	−	0.625610i	$-0.215150\pi$
$823$	44.7610	1.56027	0.780136	+	0.625610i	$0.215150\pi$
$824$	0	0
$825$	−16.5583	−0.576485
$826$	0	0
$827$	−3.36013	−0.116843	−0.0584216	−	0.998292i	$-0.518607\pi$
$827$	−3.36013	−0.116843	−0.0584216	+	0.998292i	$0.518607\pi$
$828$	0	0
$829$	17.4683	0.606699	0.303350	−	0.952879i	$-0.401895\pi$
$829$	17.4683	0.606699	0.303350	+	0.952879i	$0.401895\pi$
$830$	0	0
$831$	25.4147	0.881627
$832$	0	0
$833$	39.3177	1.36228
$834$	0	0
$835$	0.473647	0.0163912
$836$	0	0
$837$	−2.70895	−0.0936350
$838$	0	0
$839$	−52.9496	−1.82802	−0.914012	−	0.405688i	$-0.867032\pi$
$839$	−52.9496	−1.82802	−0.914012	+	0.405688i	$0.867032\pi$
$840$	0	0
$841$	−13.9688	−0.481683
$842$	0	0
$843$	5.13820	0.176969
$844$	0	0
$845$	3.36039	0.115601
$846$	0	0
$847$	−0.677707	−0.0232863
$848$	0	0
$849$	13.5596	0.465365
$850$	0	0
$851$	69.3373	2.37685
$852$	0	0
$853$	−55.3150	−1.89395	−0.946975	−	0.321308i	$-0.895878\pi$
$853$	−55.3150	−1.89395	−0.946975	+	0.321308i	$0.895878\pi$
$854$	0	0
$855$	−1.13700	−0.0388844
$856$	0	0
$857$	42.6921	1.45834	0.729168	−	0.684335i	$-0.239907\pi$
$857$	42.6921	1.45834	0.729168	+	0.684335i	$0.239907\pi$
$858$	0	0
$859$	35.8971	1.22479	0.612397	−	0.790550i	$-0.290205\pi$
$859$	35.8971	1.22479	0.612397	+	0.790550i	$0.290205\pi$
$860$	0	0
$861$	2.46683	0.0840695
$862$	0	0
$863$	−3.29190	−0.112058	−0.0560289	−	0.998429i	$-0.517844\pi$
$863$	−3.29190	−0.112058	−0.0560289	+	0.998429i	$0.517844\pi$
$864$	0	0
$865$	4.57589	0.155585
$866$	0	0
$867$	−18.9230	−0.642658
$868$	0	0
$869$	30.1220	1.02182
$870$	0	0
$871$	−3.70256	−0.125456
$872$	0	0
$873$	−12.9569	−0.438525
$874$	0	0
$875$	−3.07144	−0.103834
$876$	0	0
$877$	−26.1489	−0.882985	−0.441493	−	0.897265i	$-0.645551\pi$
$877$	−26.1489	−0.882985	−0.441493	+	0.897265i	$0.645551\pi$
$878$	0	0
$879$	−21.0838	−0.711138
$880$	0	0
$881$	52.3331	1.76315	0.881574	−	0.472045i	$-0.156484\pi$
$881$	52.3331	1.76315	0.881574	+	0.472045i	$0.156484\pi$
$882$	0	0
$883$	−12.9420	−0.435531	−0.217766	−	0.976001i	$-0.569877\pi$
$883$	−12.9420	−0.435531	−0.217766	+	0.976001i	$0.569877\pi$
$884$	0	0
$885$	0.550551	0.0185066
$886$	0	0
$887$	34.4393	1.15636	0.578180	−	0.815909i	$-0.303763\pi$
$887$	34.4393	1.15636	0.578180	+	0.815909i	$0.303763\pi$
$888$	0	0
$889$	2.19465	0.0736061
$890$	0	0
$891$	−3.46722	−0.116156
$892$	0	0
$893$	26.1822	0.876154
$894$	0	0
$895$	−9.95726	−0.332834
$896$	0	0
$897$	19.8687	0.663398
$898$	0	0
$899$	10.5026	0.350282
$900$	0	0
$901$	−1.55826	−0.0519132
$902$	0	0
$903$	6.97044	0.231962
$904$	0	0
$905$	5.14057	0.170878
$906$	0	0
$907$	−6.87288	−0.228210	−0.114105	−	0.993469i	$-0.536400\pi$
$907$	−6.87288	−0.228210	−0.114105	+	0.993469i	$0.536400\pi$
$908$	0	0
$909$	−3.71030	−0.123063
$910$	0	0
$911$	−7.84365	−0.259872	−0.129936	−	0.991522i	$-0.541477\pi$
$911$	−7.84365	−0.259872	−0.129936	+	0.991522i	$0.541477\pi$
$912$	0	0
$913$	54.7599	1.81229
$914$	0	0
$915$	−0.738011	−0.0243979
$916$	0	0
$917$	5.94582	0.196348
$918$	0	0
$919$	−3.78026	−0.124699	−0.0623496	−	0.998054i	$-0.519859\pi$
$919$	−3.78026	−0.124699	−0.0623496	+	0.998054i	$0.519859\pi$
$920$	0	0
$921$	−29.4875	−0.971647
$922$	0	0
$923$	−10.8508	−0.357157
$924$	0	0
$925$	−40.4996	−1.33162
$926$	0	0
$927$	−14.3630	−0.471744
$928$	0	0
$929$	−42.9288	−1.40845	−0.704225	−	0.709977i	$-0.748705\pi$
$929$	−42.9288	−1.40845	−0.704225	+	0.709977i	$0.748705\pi$
$930$	0	0
$931$	−15.7473	−0.516097
$932$	0	0
$933$	19.2057	0.628766
$934$	0	0
$935$	−9.84289	−0.321897
$936$	0	0
$937$	−17.2737	−0.564307	−0.282154	−	0.959369i	$-0.591049\pi$
$937$	−17.2737	−0.564307	−0.282154	+	0.959369i	$0.591049\pi$
$938$	0	0
$939$	7.13192	0.232741
$940$	0	0
$941$	6.76847	0.220646	0.110323	−	0.993896i	$-0.464812\pi$
$941$	6.76847	0.220646	0.110323	+	0.993896i	$0.464812\pi$
$942$	0	0
$943$	30.4052	0.990130
$944$	0	0
$945$	−0.314193	−0.0102207
$946$	0	0
$947$	−38.9851	−1.26685	−0.633423	−	0.773806i	$-0.718351\pi$
$947$	−38.9851	−1.26685	−0.633423	+	0.773806i	$0.718351\pi$
$948$	0	0
$949$	18.0810	0.586935
$950$	0	0
$951$	−7.54451	−0.244647
$952$	0	0
$953$	−3.88392	−0.125812	−0.0629062	−	0.998019i	$-0.520037\pi$
$953$	−3.88392	−0.125812	−0.0629062	+	0.998019i	$0.520037\pi$
$954$	0	0
$955$	6.09165	0.197121
$956$	0	0
$957$	13.4425	0.434533
$958$	0	0
$959$	2.50323	0.0808336
$960$	0	0
$961$	−23.6616	−0.763277
$962$	0	0
$963$	0.312825	0.0100806
$964$	0	0
$965$	3.53378	0.113757
$966$	0	0
$967$	−11.3477	−0.364917	−0.182459	−	0.983214i	$-0.558406\pi$
$967$	−11.3477	−0.364917	−0.182459	+	0.983214i	$0.558406\pi$
$968$	0	0
$969$	14.3877	0.462199
$970$	0	0
$971$	−24.4384	−0.784265	−0.392132	−	0.919909i	$-0.628262\pi$
$971$	−24.4384	−0.784265	−0.392132	+	0.919909i	$0.628262\pi$
$972$	0	0
$973$	−2.09673	−0.0672180
$974$	0	0
$975$	−11.6052	−0.371665
$976$	0	0
$977$	43.5927	1.39465	0.697326	−	0.716754i	$-0.254373\pi$
$977$	43.5927	1.39465	0.697326	+	0.716754i	$0.254373\pi$
$978$	0	0
$979$	20.4139	0.652432
$980$	0	0
$981$	9.69339	0.309486
$982$	0	0
$983$	−3.69974	−0.118003	−0.0590017	−	0.998258i	$-0.518792\pi$
$983$	−3.69974	−0.118003	−0.0590017	+	0.998258i	$0.518792\pi$
$984$	0	0
$985$	1.16426	0.0370965
$986$	0	0
$987$	7.23509	0.230296
$988$	0	0
$989$	85.9148	2.73193
$990$	0	0
$991$	−54.6092	−1.73472	−0.867360	−	0.497682i	$-0.834185\pi$
$991$	−54.6092	−1.73472	−0.867360	+	0.497682i	$0.834185\pi$
$992$	0	0
$993$	−21.5151	−0.682762
$994$	0	0
$995$	8.84295	0.280340
$996$	0	0
$997$	−21.7888	−0.690057	−0.345029	−	0.938592i	$-0.612131\pi$
$997$	−21.7888	−0.690057	−0.345029	+	0.938592i	$0.612131\pi$
$998$	0	0
$999$	−8.48041	−0.268308

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.s.1.2		5
4.3	odd	2		4008.2.a.f.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.f.1.2	✓	5	4.3	odd	2
8016.2.a.s.1.2		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.473647 q^{5} -0.663349 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.473647 q^{5} -0.663349 q^{7} +1.00000 q^{9} -3.46722 q^{11} -2.43008 q^{13} +0.473647 q^{15} -5.99358 q^{17} +2.40051 q^{19} +0.663349 q^{21} +8.17617 q^{23} -4.77566 q^{25} -1.00000 q^{27} +3.87701 q^{29} +2.70895 q^{31} +3.46722 q^{33} +0.314193 q^{35} +8.48041 q^{37} +2.43008 q^{39} +3.71876 q^{41} +10.5079 q^{43} -0.473647 q^{45} +10.9069 q^{47} -6.55997 q^{49} +5.99358 q^{51} +0.259988 q^{53} +1.64224 q^{55} -2.40051 q^{57} +1.16237 q^{59} -1.55815 q^{61} -0.663349 q^{63} +1.15100 q^{65} +1.52364 q^{67} -8.17617 q^{69} +4.46519 q^{71} -7.44051 q^{73} +4.77566 q^{75} +2.29998 q^{77} -8.68765 q^{79} +1.00000 q^{81} -15.7936 q^{83} +2.83884 q^{85} -3.87701 q^{87} -5.88769 q^{89} +1.61199 q^{91} -2.70895 q^{93} -1.13700 q^{95} -12.9569 q^{97} -3.46722 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 4 q^{7} + 5 q^{9} + 3 q^{11} - 14 q^{13} - q^{15} - 13 q^{17} + 2 q^{19} - 4 q^{21} + 5 q^{23} + 2 q^{25} - 5 q^{27} + 13 q^{29} - 2 q^{31} - 3 q^{33} + 12 q^{35} - 5 q^{37} + 14 q^{39} - 20 q^{41} + 20 q^{43} + q^{45} - q^{47} - 9 q^{49} + 13 q^{51} - 3 q^{53} + 3 q^{55} - 2 q^{57} + q^{59} - 34 q^{61} + 4 q^{63} - 22 q^{65} + 16 q^{67} - 5 q^{69} + 5 q^{71} - 12 q^{73} - 2 q^{75} - 8 q^{77} + 20 q^{79} + 5 q^{81} + 15 q^{83} - 27 q^{85} - 13 q^{87} - 48 q^{89} - 7 q^{91} + 2 q^{93} + 5 q^{95} - 21 q^{97} + 3 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 4 * q^7 + 5 * q^9 + 3 * q^11 - 14 * q^13 - q^15 - 13 * q^17 + 2 * q^19 - 4 * q^21 + 5 * q^23 + 2 * q^25 - 5 * q^27 + 13 * q^29 - 2 * q^31 - 3 * q^33 + 12 * q^35 - 5 * q^37 + 14 * q^39 - 20 * q^41 + 20 * q^43 + q^45 - q^47 - 9 * q^49 + 13 * q^51 - 3 * q^53 + 3 * q^55 - 2 * q^57 + q^59 - 34 * q^61 + 4 * q^63 - 22 * q^65 + 16 * q^67 - 5 * q^69 + 5 * q^71 - 12 * q^73 - 2 * q^75 - 8 * q^77 + 20 * q^79 + 5 * q^81 + 15 * q^83 - 27 * q^85 - 13 * q^87 - 48 * q^89 - 7 * q^91 + 2 * q^93 + 5 * q^95 - 21 * q^97 + 3 * q^99

Introduction

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