LMFDB - Embedded newform 8008.2.a.v.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} + \cdots + 64 $ x^11 - 2x^10 - 19x^9 + 33x^8 + 120x^7 - 178x^6 - 296x^5 + 380x^4 + 280x^3 - 295x^2 - 87x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.983398$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.983398 q^{3} -1.20732 q^{5} -1.00000 q^{7} -2.03293 q^{9} +O(q^{10})$	$q-0.983398 q^{3} -1.20732 q^{5} -1.00000 q^{7} -2.03293 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.18728 q^{15} +3.10486 q^{17} -6.13942 q^{19} +0.983398 q^{21} +9.39792 q^{23} -3.54237 q^{25} +4.94937 q^{27} -2.82165 q^{29} +0.543253 q^{31} -0.983398 q^{33} +1.20732 q^{35} -0.252569 q^{37} +0.983398 q^{39} +0.665720 q^{41} -0.327119 q^{43} +2.45440 q^{45} +9.39792 q^{47} +1.00000 q^{49} -3.05332 q^{51} +8.35464 q^{53} -1.20732 q^{55} +6.03749 q^{57} +8.49766 q^{59} -4.81738 q^{61} +2.03293 q^{63} +1.20732 q^{65} -13.9821 q^{67} -9.24189 q^{69} -9.69998 q^{71} +14.8876 q^{73} +3.48356 q^{75} -1.00000 q^{77} +4.53081 q^{79} +1.23159 q^{81} -9.44676 q^{83} -3.74857 q^{85} +2.77480 q^{87} -3.56230 q^{89} +1.00000 q^{91} -0.534234 q^{93} +7.41225 q^{95} +15.7191 q^{97} -2.03293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	$ 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) $ 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * 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
$3$	−0.983398	−0.567765	−0.283882	−	0.958859i	$-0.591623\pi$
$3$	−0.983398	−0.567765	−0.283882	+	0.958859i	$0.591623\pi$
$4$	0	0
$5$	−1.20732	−0.539931	−0.269965	−	0.962870i	$-0.587012\pi$
$5$	−1.20732	−0.539931	−0.269965	+	0.962870i	$0.587012\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.03293	−0.677643
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.18728	0.306554
$16$	0	0
$17$	3.10486	0.753040	0.376520	−	0.926408i	$-0.377121\pi$
$17$	3.10486	0.753040	0.376520	+	0.926408i	$0.377121\pi$
$18$	0	0
$19$	−6.13942	−1.40848	−0.704240	−	0.709962i	$-0.748712\pi$
$19$	−6.13942	−1.40848	−0.704240	+	0.709962i	$0.748712\pi$
$20$	0	0
$21$	0.983398	0.214595
$22$	0	0
$23$	9.39792	1.95960	0.979800	−	0.199978i	$-0.0640870\pi$
$23$	9.39792	1.95960	0.979800	+	0.199978i	$0.0640870\pi$
$24$	0	0
$25$	−3.54237	−0.708475
$26$	0	0
$27$	4.94937	0.952507
$28$	0	0
$29$	−2.82165	−0.523967	−0.261984	−	0.965072i	$-0.584377\pi$
$29$	−2.82165	−0.523967	−0.261984	+	0.965072i	$0.584377\pi$
$30$	0	0
$31$	0.543253	0.0975711	0.0487855	−	0.998809i	$-0.484465\pi$
$31$	0.543253	0.0975711	0.0487855	+	0.998809i	$0.484465\pi$
$32$	0	0
$33$	−0.983398	−0.171188
$34$	0	0
$35$	1.20732	0.204075
$36$	0	0
$37$	−0.252569	−0.0415221	−0.0207611	−	0.999784i	$-0.506609\pi$
$37$	−0.252569	−0.0415221	−0.0207611	+	0.999784i	$0.506609\pi$
$38$	0	0
$39$	0.983398	0.157470
$40$	0	0
$41$	0.665720	0.103968	0.0519840	−	0.998648i	$-0.483446\pi$
$41$	0.665720	0.103968	0.0519840	+	0.998648i	$0.483446\pi$
$42$	0	0
$43$	−0.327119	−0.0498852	−0.0249426	−	0.999689i	$-0.507940\pi$
$43$	−0.327119	−0.0498852	−0.0249426	+	0.999689i	$0.507940\pi$
$44$	0	0
$45$	2.45440	0.365880
$46$	0	0
$47$	9.39792	1.37083	0.685413	−	0.728154i	$-0.259621\pi$
$47$	9.39792	1.37083	0.685413	+	0.728154i	$0.259621\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.05332	−0.427550
$52$	0	0
$53$	8.35464	1.14760	0.573799	−	0.818996i	$-0.305469\pi$
$53$	8.35464	1.14760	0.573799	+	0.818996i	$0.305469\pi$
$54$	0	0
$55$	−1.20732	−0.162795
$56$	0	0
$57$	6.03749	0.799685
$58$	0	0
$59$	8.49766	1.10630	0.553150	−	0.833081i	$-0.313426\pi$
$59$	8.49766	1.10630	0.553150	+	0.833081i	$0.313426\pi$
$60$	0	0
$61$	−4.81738	−0.616802	−0.308401	−	0.951256i	$-0.599794\pi$
$61$	−4.81738	−0.616802	−0.308401	+	0.951256i	$0.599794\pi$
$62$	0	0
$63$	2.03293	0.256125
$64$	0	0
$65$	1.20732	0.149750
$66$	0	0
$67$	−13.9821	−1.70819	−0.854095	−	0.520117i	$-0.825888\pi$
$67$	−13.9821	−1.70819	−0.854095	+	0.520117i	$0.825888\pi$
$68$	0	0
$69$	−9.24189	−1.11259
$70$	0	0
$71$	−9.69998	−1.15118	−0.575588	−	0.817740i	$-0.695227\pi$
$71$	−9.69998	−1.15118	−0.575588	+	0.817740i	$0.695227\pi$
$72$	0	0
$73$	14.8876	1.74246	0.871232	−	0.490872i	$-0.163322\pi$
$73$	14.8876	1.74246	0.871232	+	0.490872i	$0.163322\pi$
$74$	0	0
$75$	3.48356	0.402247
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	4.53081	0.509756	0.254878	−	0.966973i	$-0.417965\pi$
$79$	4.53081	0.509756	0.254878	+	0.966973i	$0.417965\pi$
$80$	0	0
$81$	1.23159	0.136843
$82$	0	0
$83$	−9.44676	−1.03692	−0.518458	−	0.855103i	$-0.673494\pi$
$83$	−9.44676	−1.03692	−0.518458	+	0.855103i	$0.673494\pi$
$84$	0	0
$85$	−3.74857	−0.406589
$86$	0	0
$87$	2.77480	0.297490
$88$	0	0
$89$	−3.56230	−0.377603	−0.188801	−	0.982015i	$-0.560460\pi$
$89$	−3.56230	−0.377603	−0.188801	+	0.982015i	$0.560460\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−0.534234	−0.0553974
$94$	0	0
$95$	7.41225	0.760481
$96$	0	0
$97$	15.7191	1.59603	0.798015	−	0.602637i	$-0.205883\pi$
$97$	15.7191	1.59603	0.798015	+	0.602637i	$0.205883\pi$
$98$	0	0
$99$	−2.03293	−0.204317
$100$	0	0
$101$	−14.8431	−1.47694	−0.738470	−	0.674286i	$-0.764451\pi$
$101$	−14.8431	−1.47694	−0.738470	+	0.674286i	$0.764451\pi$
$102$	0	0
$103$	−8.48994	−0.836538	−0.418269	−	0.908323i	$-0.637363\pi$
$103$	−8.48994	−0.836538	−0.418269	+	0.908323i	$0.637363\pi$
$104$	0	0
$105$	−1.18728	−0.115866
$106$	0	0
$107$	6.85572	0.662768	0.331384	−	0.943496i	$-0.392485\pi$
$107$	6.85572	0.662768	0.331384	+	0.943496i	$0.392485\pi$
$108$	0	0
$109$	−12.1174	−1.16064	−0.580320	−	0.814389i	$-0.697072\pi$
$109$	−12.1174	−1.16064	−0.580320	+	0.814389i	$0.697072\pi$
$110$	0	0
$111$	0.248376	0.0235748
$112$	0	0
$113$	12.0522	1.13377	0.566886	−	0.823796i	$-0.308148\pi$
$113$	12.0522	1.13377	0.566886	+	0.823796i	$0.308148\pi$
$114$	0	0
$115$	−11.3463	−1.05805
$116$	0	0
$117$	2.03293	0.187944
$118$	0	0
$119$	−3.10486	−0.284622
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.654667	−0.0590294
$124$	0	0
$125$	10.3134	0.922458
$126$	0	0
$127$	−14.3558	−1.27387	−0.636936	−	0.770917i	$-0.719798\pi$
$127$	−14.3558	−1.27387	−0.636936	+	0.770917i	$0.719798\pi$
$128$	0	0
$129$	0.321689	0.0283231
$130$	0	0
$131$	6.34706	0.554545	0.277272	−	0.960791i	$-0.410570\pi$
$131$	6.34706	0.554545	0.277272	+	0.960791i	$0.410570\pi$
$132$	0	0
$133$	6.13942	0.532355
$134$	0	0
$135$	−5.97548	−0.514288
$136$	0	0
$137$	11.3083	0.966131	0.483066	−	0.875584i	$-0.339523\pi$
$137$	11.3083	0.966131	0.483066	+	0.875584i	$0.339523\pi$
$138$	0	0
$139$	9.20021	0.780352	0.390176	−	0.920740i	$-0.372414\pi$
$139$	9.20021	0.780352	0.390176	+	0.920740i	$0.372414\pi$
$140$	0	0
$141$	−9.24189	−0.778307
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	3.40664	0.282906
$146$	0	0
$147$	−0.983398	−0.0811093
$148$	0	0
$149$	7.01224	0.574465	0.287232	−	0.957861i	$-0.407265\pi$
$149$	7.01224	0.574465	0.287232	+	0.957861i	$0.407265\pi$
$150$	0	0
$151$	−15.2909	−1.24436	−0.622178	−	0.782876i	$-0.713752\pi$
$151$	−15.2909	−1.24436	−0.622178	+	0.782876i	$0.713752\pi$
$152$	0	0
$153$	−6.31197	−0.510292
$154$	0	0
$155$	−0.655881	−0.0526816
$156$	0	0
$157$	10.4837	0.836690	0.418345	−	0.908288i	$-0.362610\pi$
$157$	10.4837	0.836690	0.418345	+	0.908288i	$0.362610\pi$
$158$	0	0
$159$	−8.21593	−0.651566
$160$	0	0
$161$	−9.39792	−0.740660
$162$	0	0
$163$	−4.19699	−0.328734	−0.164367	−	0.986399i	$-0.552558\pi$
$163$	−4.19699	−0.328734	−0.164367	+	0.986399i	$0.552558\pi$
$164$	0	0
$165$	1.18728	0.0924294
$166$	0	0
$167$	−19.8468	−1.53579	−0.767896	−	0.640575i	$-0.778696\pi$
$167$	−19.8468	−1.53579	−0.767896	+	0.640575i	$0.778696\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	12.4810	0.954446
$172$	0	0
$173$	8.42778	0.640752	0.320376	−	0.947290i	$-0.396191\pi$
$173$	8.42778	0.640752	0.320376	+	0.947290i	$0.396191\pi$
$174$	0	0
$175$	3.54237	0.267778
$176$	0	0
$177$	−8.35658	−0.628119
$178$	0	0
$179$	3.90430	0.291821	0.145911	−	0.989298i	$-0.453389\pi$
$179$	3.90430	0.291821	0.145911	+	0.989298i	$0.453389\pi$
$180$	0	0
$181$	−10.2429	−0.761347	−0.380674	−	0.924709i	$-0.624308\pi$
$181$	−10.2429	−0.761347	−0.380674	+	0.924709i	$0.624308\pi$
$182$	0	0
$183$	4.73740	0.350198
$184$	0	0
$185$	0.304932	0.0224191
$186$	0	0
$187$	3.10486	0.227050
$188$	0	0
$189$	−4.94937	−0.360014
$190$	0	0
$191$	16.8254	1.21745	0.608723	−	0.793383i	$-0.291682\pi$
$191$	16.8254	1.21745	0.608723	+	0.793383i	$0.291682\pi$
$192$	0	0
$193$	−24.8726	−1.79037	−0.895186	−	0.445693i	$-0.852957\pi$
$193$	−24.8726	−1.79037	−0.895186	+	0.445693i	$0.852957\pi$
$194$	0	0
$195$	−1.18728	−0.0850227
$196$	0	0
$197$	2.51647	0.179291	0.0896456	−	0.995974i	$-0.471427\pi$
$197$	2.51647	0.179291	0.0896456	+	0.995974i	$0.471427\pi$
$198$	0	0
$199$	−13.2409	−0.938624	−0.469312	−	0.883032i	$-0.655498\pi$
$199$	−13.2409	−0.938624	−0.469312	+	0.883032i	$0.655498\pi$
$200$	0	0
$201$	13.7500	0.969850
$202$	0	0
$203$	2.82165	0.198041
$204$	0	0
$205$	−0.803738	−0.0561355
$206$	0	0
$207$	−19.1053	−1.32791
$208$	0	0
$209$	−6.13942	−0.424672
$210$	0	0
$211$	−24.5369	−1.68919	−0.844595	−	0.535406i	$-0.820159\pi$
$211$	−24.5369	−1.68919	−0.844595	+	0.535406i	$0.820159\pi$
$212$	0	0
$213$	9.53894	0.653597
$214$	0	0
$215$	0.394938	0.0269346
$216$	0	0
$217$	−0.543253	−0.0368784
$218$	0	0
$219$	−14.6404	−0.989310
$220$	0	0
$221$	−3.10486	−0.208856
$222$	0	0
$223$	−4.25717	−0.285081	−0.142541	−	0.989789i	$-0.545527\pi$
$223$	−4.25717	−0.285081	−0.142541	+	0.989789i	$0.545527\pi$
$224$	0	0
$225$	7.20140	0.480093
$226$	0	0
$227$	11.9967	0.796246	0.398123	−	0.917332i	$-0.369662\pi$
$227$	11.9967	0.796246	0.398123	+	0.917332i	$0.369662\pi$
$228$	0	0
$229$	6.48496	0.428538	0.214269	−	0.976775i	$-0.431263\pi$
$229$	6.48496	0.428538	0.214269	+	0.976775i	$0.431263\pi$
$230$	0	0
$231$	0.983398	0.0647028
$232$	0	0
$233$	−16.4755	−1.07934	−0.539672	−	0.841875i	$-0.681452\pi$
$233$	−16.4755	−1.07934	−0.539672	+	0.841875i	$0.681452\pi$
$234$	0	0
$235$	−11.3463	−0.740151
$236$	0	0
$237$	−4.45559	−0.289422
$238$	0	0
$239$	−24.0623	−1.55646	−0.778232	−	0.627977i	$-0.783883\pi$
$239$	−24.0623	−1.55646	−0.778232	+	0.627977i	$0.783883\pi$
$240$	0	0
$241$	−22.5115	−1.45009	−0.725046	−	0.688701i	$-0.758181\pi$
$241$	−22.5115	−1.45009	−0.725046	+	0.688701i	$0.758181\pi$
$242$	0	0
$243$	−16.0593	−1.03020
$244$	0	0
$245$	−1.20732	−0.0771329
$246$	0	0
$247$	6.13942	0.390642
$248$	0	0
$249$	9.28993	0.588725
$250$	0	0
$251$	14.3686	0.906936	0.453468	−	0.891273i	$-0.350187\pi$
$251$	14.3686	0.906936	0.453468	+	0.891273i	$0.350187\pi$
$252$	0	0
$253$	9.39792	0.590842
$254$	0	0
$255$	3.68633	0.230847
$256$	0	0
$257$	−8.26987	−0.515860	−0.257930	−	0.966164i	$-0.583040\pi$
$257$	−8.26987	−0.515860	−0.257930	+	0.966164i	$0.583040\pi$
$258$	0	0
$259$	0.252569	0.0156939
$260$	0	0
$261$	5.73621	0.355063
$262$	0	0
$263$	11.7923	0.727145	0.363572	−	0.931566i	$-0.381557\pi$
$263$	11.7923	0.727145	0.363572	+	0.931566i	$0.381557\pi$
$264$	0	0
$265$	−10.0867	−0.619623
$266$	0	0
$267$	3.50315	0.214390
$268$	0	0
$269$	−29.4401	−1.79500	−0.897499	−	0.441017i	$-0.854618\pi$
$269$	−29.4401	−1.79500	−0.897499	+	0.441017i	$0.854618\pi$
$270$	0	0
$271$	31.6383	1.92189	0.960946	−	0.276737i	$-0.0892531\pi$
$271$	31.6383	1.92189	0.960946	+	0.276737i	$0.0892531\pi$
$272$	0	0
$273$	−0.983398	−0.0595179
$274$	0	0
$275$	−3.54237	−0.213613
$276$	0	0
$277$	−24.4158	−1.46700	−0.733501	−	0.679688i	$-0.762115\pi$
$277$	−24.4158	−1.46700	−0.733501	+	0.679688i	$0.762115\pi$
$278$	0	0
$279$	−1.10439	−0.0661184
$280$	0	0
$281$	−15.5560	−0.927990	−0.463995	−	0.885838i	$-0.653585\pi$
$281$	−15.5560	−0.927990	−0.463995	+	0.885838i	$0.653585\pi$
$282$	0	0
$283$	−1.37089	−0.0814909	−0.0407454	−	0.999170i	$-0.512973\pi$
$283$	−1.37089	−0.0814909	−0.0407454	+	0.999170i	$0.512973\pi$
$284$	0	0
$285$	−7.28919	−0.431774
$286$	0	0
$287$	−0.665720	−0.0392962
$288$	0	0
$289$	−7.35982	−0.432930
$290$	0	0
$291$	−15.4581	−0.906170
$292$	0	0
$293$	4.61245	0.269462	0.134731	−	0.990882i	$-0.456983\pi$
$293$	4.61245	0.269462	0.134731	+	0.990882i	$0.456983\pi$
$294$	0	0
$295$	−10.2594	−0.597326
$296$	0	0
$297$	4.94937	0.287192
$298$	0	0
$299$	−9.39792	−0.543495
$300$	0	0
$301$	0.327119	0.0188549
$302$	0	0
$303$	14.5966	0.838555
$304$	0	0
$305$	5.81612	0.333030
$306$	0	0
$307$	−10.8025	−0.616534	−0.308267	−	0.951300i	$-0.599749\pi$
$307$	−10.8025	−0.616534	−0.308267	+	0.951300i	$0.599749\pi$
$308$	0	0
$309$	8.34898	0.474957
$310$	0	0
$311$	−2.10376	−0.119293	−0.0596466	−	0.998220i	$-0.518997\pi$
$311$	−2.10376	−0.119293	−0.0596466	+	0.998220i	$0.518997\pi$
$312$	0	0
$313$	30.8507	1.74379	0.871893	−	0.489696i	$-0.162892\pi$
$313$	30.8507	1.74379	0.871893	+	0.489696i	$0.162892\pi$
$314$	0	0
$315$	−2.45440	−0.138290
$316$	0	0
$317$	−0.515547	−0.0289560	−0.0144780	−	0.999895i	$-0.504609\pi$
$317$	−0.515547	−0.0289560	−0.0144780	+	0.999895i	$0.504609\pi$
$318$	0	0
$319$	−2.82165	−0.157982
$320$	0	0
$321$	−6.74190	−0.376296
$322$	0	0
$323$	−19.0621	−1.06064
$324$	0	0
$325$	3.54237	0.196496
$326$	0	0
$327$	11.9163	0.658970
$328$	0	0
$329$	−9.39792	−0.518124
$330$	0	0
$331$	−27.5435	−1.51393	−0.756963	−	0.653458i	$-0.773318\pi$
$331$	−27.5435	−1.51393	−0.756963	+	0.653458i	$0.773318\pi$
$332$	0	0
$333$	0.513455	0.0281372
$334$	0	0
$335$	16.8809	0.922304
$336$	0	0
$337$	29.4616	1.60487	0.802437	−	0.596737i	$-0.203537\pi$
$337$	29.4616	1.60487	0.802437	+	0.596737i	$0.203537\pi$
$338$	0	0
$339$	−11.8521	−0.643716
$340$	0	0
$341$	0.543253	0.0294188
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.1579	0.600723
$346$	0	0
$347$	−10.1374	−0.544205	−0.272102	−	0.962268i	$-0.587719\pi$
$347$	−10.1374	−0.544205	−0.272102	+	0.962268i	$0.587719\pi$
$348$	0	0
$349$	−24.7739	−1.32612	−0.663058	−	0.748568i	$-0.730742\pi$
$349$	−24.7739	−1.32612	−0.663058	+	0.748568i	$0.730742\pi$
$350$	0	0
$351$	−4.94937	−0.264178
$352$	0	0
$353$	28.7554	1.53050	0.765248	−	0.643736i	$-0.222616\pi$
$353$	28.7554	1.53050	0.765248	+	0.643736i	$0.222616\pi$
$354$	0	0
$355$	11.7110	0.621555
$356$	0	0
$357$	3.05332	0.161599
$358$	0	0
$359$	−27.2941	−1.44053	−0.720265	−	0.693699i	$-0.755980\pi$
$359$	−27.2941	−1.44053	−0.720265	+	0.693699i	$0.755980\pi$
$360$	0	0
$361$	18.6925	0.983814
$362$	0	0
$363$	−0.983398	−0.0516150
$364$	0	0
$365$	−17.9741	−0.940810
$366$	0	0
$367$	18.1081	0.945234	0.472617	−	0.881268i	$-0.343309\pi$
$367$	18.1081	0.945234	0.472617	+	0.881268i	$0.343309\pi$
$368$	0	0
$369$	−1.35336	−0.0704532
$370$	0	0
$371$	−8.35464	−0.433751
$372$	0	0
$373$	35.1752	1.82130	0.910652	−	0.413174i	$-0.135580\pi$
$373$	35.1752	1.82130	0.910652	+	0.413174i	$0.135580\pi$
$374$	0	0
$375$	−10.1422	−0.523739
$376$	0	0
$377$	2.82165	0.145322
$378$	0	0
$379$	−28.6829	−1.47334	−0.736670	−	0.676252i	$-0.763603\pi$
$379$	−28.6829	−1.47334	−0.736670	+	0.676252i	$0.763603\pi$
$380$	0	0
$381$	14.1175	0.723260
$382$	0	0
$383$	−12.4746	−0.637421	−0.318710	−	0.947852i	$-0.603250\pi$
$383$	−12.4746	−0.637421	−0.318710	+	0.947852i	$0.603250\pi$
$384$	0	0
$385$	1.20732	0.0615308
$386$	0	0
$387$	0.665011	0.0338044
$388$	0	0
$389$	33.5399	1.70054	0.850270	−	0.526348i	$-0.176439\pi$
$389$	33.5399	1.70054	0.850270	+	0.526348i	$0.176439\pi$
$390$	0	0
$391$	29.1793	1.47566
$392$	0	0
$393$	−6.24168	−0.314851
$394$	0	0
$395$	−5.47015	−0.275233
$396$	0	0
$397$	−6.10662	−0.306482	−0.153241	−	0.988189i	$-0.548971\pi$
$397$	−6.10662	−0.306482	−0.153241	+	0.988189i	$0.548971\pi$
$398$	0	0
$399$	−6.03749	−0.302253
$400$	0	0
$401$	−5.60807	−0.280054	−0.140027	−	0.990148i	$-0.544719\pi$
$401$	−5.60807	−0.280054	−0.140027	+	0.990148i	$0.544719\pi$
$402$	0	0
$403$	−0.543253	−0.0270614
$404$	0	0
$405$	−1.48692	−0.0738858
$406$	0	0
$407$	−0.252569	−0.0125194
$408$	0	0
$409$	−25.2843	−1.25023	−0.625115	−	0.780533i	$-0.714948\pi$
$409$	−25.2843	−1.25023	−0.625115	+	0.780533i	$0.714948\pi$
$410$	0	0
$411$	−11.1205	−0.548535
$412$	0	0
$413$	−8.49766	−0.418142
$414$	0	0
$415$	11.4053	0.559863
$416$	0	0
$417$	−9.04747	−0.443056
$418$	0	0
$419$	−14.7084	−0.718553	−0.359276	−	0.933231i	$-0.616976\pi$
$419$	−14.7084	−0.718553	−0.359276	+	0.933231i	$0.616976\pi$
$420$	0	0
$421$	−2.64194	−0.128760	−0.0643801	−	0.997925i	$-0.520507\pi$
$421$	−2.64194	−0.128760	−0.0643801	+	0.997925i	$0.520507\pi$
$422$	0	0
$423$	−19.1053	−0.928931
$424$	0	0
$425$	−10.9986	−0.533510
$426$	0	0
$427$	4.81738	0.233129
$428$	0	0
$429$	0.983398	0.0474789
$430$	0	0
$431$	17.4640	0.841213	0.420606	−	0.907243i	$-0.361817\pi$
$431$	17.4640	0.841213	0.420606	+	0.907243i	$0.361817\pi$
$432$	0	0
$433$	3.23544	0.155486	0.0777428	−	0.996973i	$-0.475229\pi$
$433$	3.23544	0.155486	0.0777428	+	0.996973i	$0.475229\pi$
$434$	0	0
$435$	−3.35008	−0.160624
$436$	0	0
$437$	−57.6977	−2.76006
$438$	0	0
$439$	−20.5487	−0.980738	−0.490369	−	0.871515i	$-0.663138\pi$
$439$	−20.5487	−0.980738	−0.490369	+	0.871515i	$0.663138\pi$
$440$	0	0
$441$	−2.03293	−0.0968061
$442$	0	0
$443$	−9.59221	−0.455739	−0.227870	−	0.973692i	$-0.573176\pi$
$443$	−9.59221	−0.455739	−0.227870	+	0.973692i	$0.573176\pi$
$444$	0	0
$445$	4.30084	0.203879
$446$	0	0
$447$	−6.89582	−0.326161
$448$	0	0
$449$	−21.4237	−1.01105	−0.505524	−	0.862812i	$-0.668701\pi$
$449$	−21.4237	−1.01105	−0.505524	+	0.862812i	$0.668701\pi$
$450$	0	0
$451$	0.665720	0.0313475
$452$	0	0
$453$	15.0370	0.706502
$454$	0	0
$455$	−1.20732	−0.0566001
$456$	0	0
$457$	−8.89787	−0.416225	−0.208112	−	0.978105i	$-0.566732\pi$
$457$	−8.89787	−0.416225	−0.208112	+	0.978105i	$0.566732\pi$
$458$	0	0
$459$	15.3671	0.717276
$460$	0	0
$461$	27.3690	1.27470	0.637352	−	0.770573i	$-0.280030\pi$
$461$	27.3690	1.27470	0.637352	+	0.770573i	$0.280030\pi$
$462$	0	0
$463$	−39.9067	−1.85462	−0.927311	−	0.374293i	$-0.877886\pi$
$463$	−39.9067	−1.85462	−0.927311	+	0.374293i	$0.877886\pi$
$464$	0	0
$465$	0.644992	0.0299108
$466$	0	0
$467$	−32.7796	−1.51686	−0.758430	−	0.651755i	$-0.774033\pi$
$467$	−32.7796	−1.51686	−0.758430	+	0.651755i	$0.774033\pi$
$468$	0	0
$469$	13.9821	0.645635
$470$	0	0
$471$	−10.3096	−0.475043
$472$	0	0
$473$	−0.327119	−0.0150410
$474$	0	0
$475$	21.7481	0.997872
$476$	0	0
$477$	−16.9844	−0.777661
$478$	0	0
$479$	13.1936	0.602831	0.301416	−	0.953493i	$-0.402541\pi$
$479$	13.1936	0.602831	0.301416	+	0.953493i	$0.402541\pi$
$480$	0	0
$481$	0.252569	0.0115162
$482$	0	0
$483$	9.24189	0.420520
$484$	0	0
$485$	−18.9780	−0.861746
$486$	0	0
$487$	−13.3460	−0.604766	−0.302383	−	0.953187i	$-0.597782\pi$
$487$	−13.3460	−0.604766	−0.302383	+	0.953187i	$0.597782\pi$
$488$	0	0
$489$	4.12731	0.186644
$490$	0	0
$491$	−25.8482	−1.16651	−0.583257	−	0.812288i	$-0.698222\pi$
$491$	−25.8482	−1.16651	−0.583257	+	0.812288i	$0.698222\pi$
$492$	0	0
$493$	−8.76084	−0.394568
$494$	0	0
$495$	2.45440	0.110317
$496$	0	0
$497$	9.69998	0.435103
$498$	0	0
$499$	−13.5125	−0.604901	−0.302451	−	0.953165i	$-0.597805\pi$
$499$	−13.5125	−0.604901	−0.302451	+	0.953165i	$0.597805\pi$
$500$	0	0
$501$	19.5173	0.871969
$502$	0	0
$503$	−5.23981	−0.233632	−0.116816	−	0.993154i	$-0.537269\pi$
$503$	−5.23981	−0.233632	−0.116816	+	0.993154i	$0.537269\pi$
$504$	0	0
$505$	17.9203	0.797445
$506$	0	0
$507$	−0.983398	−0.0436742
$508$	0	0
$509$	−9.99253	−0.442911	−0.221456	−	0.975170i	$-0.571081\pi$
$509$	−9.99253	−0.442911	−0.221456	+	0.975170i	$0.571081\pi$
$510$	0	0
$511$	−14.8876	−0.658589
$512$	0	0
$513$	−30.3863	−1.34159
$514$	0	0
$515$	10.2501	0.451673
$516$	0	0
$517$	9.39792	0.413320
$518$	0	0
$519$	−8.28786	−0.363797
$520$	0	0
$521$	16.7804	0.735161	0.367580	−	0.929992i	$-0.380186\pi$
$521$	16.7804	0.735161	0.367580	+	0.929992i	$0.380186\pi$
$522$	0	0
$523$	−8.57181	−0.374819	−0.187409	−	0.982282i	$-0.560009\pi$
$523$	−8.57181	−0.374819	−0.187409	+	0.982282i	$0.560009\pi$
$524$	0	0
$525$	−3.48356	−0.152035
$526$	0	0
$527$	1.68673	0.0734750
$528$	0	0
$529$	65.3208	2.84004
$530$	0	0
$531$	−17.2751	−0.749677
$532$	0	0
$533$	−0.665720	−0.0288355
$534$	0	0
$535$	−8.27706	−0.357848
$536$	0	0
$537$	−3.83948	−0.165686
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	44.0296	1.89298	0.946491	−	0.322730i	$-0.104601\pi$
$541$	44.0296	1.89298	0.946491	+	0.322730i	$0.104601\pi$
$542$	0	0
$543$	10.0728	0.432266
$544$	0	0
$545$	14.6296	0.626665
$546$	0	0
$547$	14.6835	0.627820	0.313910	−	0.949453i	$-0.398361\pi$
$547$	14.6835	0.627820	0.313910	+	0.949453i	$0.398361\pi$
$548$	0	0
$549$	9.79338	0.417971
$550$	0	0
$551$	17.3233	0.737997
$552$	0	0
$553$	−4.53081	−0.192670
$554$	0	0
$555$	−0.299870	−0.0127288
$556$	0	0
$557$	11.6989	0.495698	0.247849	−	0.968799i	$-0.420276\pi$
$557$	11.6989	0.495698	0.247849	+	0.968799i	$0.420276\pi$
$558$	0	0
$559$	0.327119	0.0138357
$560$	0	0
$561$	−3.05332	−0.128911
$562$	0	0
$563$	−32.0608	−1.35120	−0.675600	−	0.737269i	$-0.736115\pi$
$563$	−32.0608	−1.35120	−0.675600	+	0.737269i	$0.736115\pi$
$564$	0	0
$565$	−14.5508	−0.612158
$566$	0	0
$567$	−1.23159	−0.0517218
$568$	0	0
$569$	−21.0792	−0.883688	−0.441844	−	0.897092i	$-0.645675\pi$
$569$	−21.0792	−0.883688	−0.441844	+	0.897092i	$0.645675\pi$
$570$	0	0
$571$	−35.6616	−1.49239	−0.746195	−	0.665728i	$-0.768121\pi$
$571$	−35.6616	−1.49239	−0.746195	+	0.665728i	$0.768121\pi$
$572$	0	0
$573$	−16.5461	−0.691223
$574$	0	0
$575$	−33.2909	−1.38833
$576$	0	0
$577$	−33.4537	−1.39270	−0.696349	−	0.717703i	$-0.745193\pi$
$577$	−33.4537	−1.39270	−0.696349	+	0.717703i	$0.745193\pi$
$578$	0	0
$579$	24.4597	1.01651
$580$	0	0
$581$	9.44676	0.391918
$582$	0	0
$583$	8.35464	0.346014
$584$	0	0
$585$	−2.45440	−0.101477
$586$	0	0
$587$	−35.7131	−1.47404	−0.737019	−	0.675872i	$-0.763767\pi$
$587$	−35.7131	−1.47404	−0.737019	+	0.675872i	$0.763767\pi$
$588$	0	0
$589$	−3.33526	−0.137427
$590$	0	0
$591$	−2.47469	−0.101795
$592$	0	0
$593$	43.0926	1.76960	0.884801	−	0.465968i	$-0.154294\pi$
$593$	43.0926	1.76960	0.884801	+	0.465968i	$0.154294\pi$
$594$	0	0
$595$	3.74857	0.153676
$596$	0	0
$597$	13.0211	0.532918
$598$	0	0
$599$	−16.1346	−0.659242	−0.329621	−	0.944113i	$-0.606921\pi$
$599$	−16.1346	−0.659242	−0.329621	+	0.944113i	$0.606921\pi$
$600$	0	0
$601$	−8.22210	−0.335386	−0.167693	−	0.985839i	$-0.553632\pi$
$601$	−8.22210	−0.335386	−0.167693	+	0.985839i	$0.553632\pi$
$602$	0	0
$603$	28.4247	1.15754
$604$	0	0
$605$	−1.20732	−0.0490846
$606$	0	0
$607$	29.3796	1.19248	0.596240	−	0.802806i	$-0.296660\pi$
$607$	29.3796	1.19248	0.596240	+	0.802806i	$0.296660\pi$
$608$	0	0
$609$	−2.77480	−0.112441
$610$	0	0
$611$	−9.39792	−0.380199
$612$	0	0
$613$	18.7971	0.759209	0.379605	−	0.925149i	$-0.376060\pi$
$613$	18.7971	0.759209	0.379605	+	0.925149i	$0.376060\pi$
$614$	0	0
$615$	0.790394	0.0318718
$616$	0	0
$617$	1.83888	0.0740307	0.0370153	−	0.999315i	$-0.488215\pi$
$617$	1.83888	0.0740307	0.0370153	+	0.999315i	$0.488215\pi$
$618$	0	0
$619$	−38.9980	−1.56746	−0.783731	−	0.621100i	$-0.786686\pi$
$619$	−38.9980	−1.56746	−0.783731	+	0.621100i	$0.786686\pi$
$620$	0	0
$621$	46.5138	1.86653
$622$	0	0
$623$	3.56230	0.142720
$624$	0	0
$625$	5.26029	0.210412
$626$	0	0
$627$	6.03749	0.241114
$628$	0	0
$629$	−0.784193	−0.0312678
$630$	0	0
$631$	32.9847	1.31310	0.656550	−	0.754282i	$-0.272015\pi$
$631$	32.9847	1.31310	0.656550	+	0.754282i	$0.272015\pi$
$632$	0	0
$633$	24.1295	0.959063
$634$	0	0
$635$	17.3321	0.687803
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	19.7194	0.780086
$640$	0	0
$641$	−38.5731	−1.52354	−0.761772	−	0.647845i	$-0.775671\pi$
$641$	−38.5731	−1.52354	−0.761772	+	0.647845i	$0.775671\pi$
$642$	0	0
$643$	39.4676	1.55645	0.778225	−	0.627985i	$-0.216120\pi$
$643$	39.4676	1.55645	0.778225	+	0.627985i	$0.216120\pi$
$644$	0	0
$645$	−0.388381	−0.0152925
$646$	0	0
$647$	−26.7325	−1.05096	−0.525482	−	0.850805i	$-0.676115\pi$
$647$	−26.7325	−1.05096	−0.525482	+	0.850805i	$0.676115\pi$
$648$	0	0
$649$	8.49766	0.333562
$650$	0	0
$651$	0.534234	0.0209383
$652$	0	0
$653$	−41.8212	−1.63659	−0.818295	−	0.574799i	$-0.805080\pi$
$653$	−41.8212	−1.63659	−0.818295	+	0.574799i	$0.805080\pi$
$654$	0	0
$655$	−7.66294	−0.299416
$656$	0	0
$657$	−30.2655	−1.18077
$658$	0	0
$659$	5.87697	0.228934	0.114467	−	0.993427i	$-0.463484\pi$
$659$	5.87697	0.228934	0.114467	+	0.993427i	$0.463484\pi$
$660$	0	0
$661$	22.3236	0.868286	0.434143	−	0.900844i	$-0.357051\pi$
$661$	22.3236	0.868286	0.434143	+	0.900844i	$0.357051\pi$
$662$	0	0
$663$	3.05332	0.118581
$664$	0	0
$665$	−7.41225	−0.287435
$666$	0	0
$667$	−26.5176	−1.02677
$668$	0	0
$669$	4.18649	0.161859
$670$	0	0
$671$	−4.81738	−0.185973
$672$	0	0
$673$	−21.3322	−0.822296	−0.411148	−	0.911569i	$-0.634872\pi$
$673$	−21.3322	−0.822296	−0.411148	+	0.911569i	$0.634872\pi$
$674$	0	0
$675$	−17.5325	−0.674827
$676$	0	0
$677$	−8.83707	−0.339636	−0.169818	−	0.985475i	$-0.554318\pi$
$677$	−8.83707	−0.339636	−0.169818	+	0.985475i	$0.554318\pi$
$678$	0	0
$679$	−15.7191	−0.603243
$680$	0	0
$681$	−11.7975	−0.452081
$682$	0	0
$683$	−15.2393	−0.583116	−0.291558	−	0.956553i	$-0.594174\pi$
$683$	−15.2393	−0.583116	−0.291558	+	0.956553i	$0.594174\pi$
$684$	0	0
$685$	−13.6527	−0.521644
$686$	0	0
$687$	−6.37729	−0.243309
$688$	0	0
$689$	−8.35464	−0.318286
$690$	0	0
$691$	13.9709	0.531479	0.265739	−	0.964045i	$-0.414384\pi$
$691$	13.9709	0.531479	0.265739	+	0.964045i	$0.414384\pi$
$692$	0	0
$693$	2.03293	0.0772246
$694$	0	0
$695$	−11.1076	−0.421336
$696$	0	0
$697$	2.06697	0.0782921
$698$	0	0
$699$	16.2019	0.612814
$700$	0	0
$701$	50.6077	1.91143	0.955714	−	0.294297i	$-0.0950854\pi$
$701$	50.6077	1.91143	0.955714	+	0.294297i	$0.0950854\pi$
$702$	0	0
$703$	1.55063	0.0584831
$704$	0	0
$705$	11.1579	0.420232
$706$	0	0
$707$	14.8431	0.558231
$708$	0	0
$709$	15.5674	0.584645	0.292322	−	0.956320i	$-0.405572\pi$
$709$	15.5674	0.584645	0.292322	+	0.956320i	$0.405572\pi$
$710$	0	0
$711$	−9.21082	−0.345433
$712$	0	0
$713$	5.10544	0.191200
$714$	0	0
$715$	1.20732	0.0451513
$716$	0	0
$717$	23.6628	0.883705
$718$	0	0
$719$	17.9502	0.669429	0.334715	−	0.942320i	$-0.391360\pi$
$719$	17.9502	0.669429	0.334715	+	0.942320i	$0.391360\pi$
$720$	0	0
$721$	8.48994	0.316182
$722$	0	0
$723$	22.1377	0.823311
$724$	0	0
$725$	9.99534	0.371218
$726$	0	0
$727$	−32.7618	−1.21507	−0.607534	−	0.794293i	$-0.707841\pi$
$727$	−32.7618	−1.21507	−0.607534	+	0.794293i	$0.707841\pi$
$728$	0	0
$729$	12.0979	0.448069
$730$	0	0
$731$	−1.01566	−0.0375656
$732$	0	0
$733$	23.1389	0.854655	0.427327	−	0.904097i	$-0.359455\pi$
$733$	23.1389	0.854655	0.427327	+	0.904097i	$0.359455\pi$
$734$	0	0
$735$	1.18728	0.0437934
$736$	0	0
$737$	−13.9821	−0.515039
$738$	0	0
$739$	−42.5508	−1.56526	−0.782628	−	0.622490i	$-0.786121\pi$
$739$	−42.5508	−1.56526	−0.782628	+	0.622490i	$0.786121\pi$
$740$	0	0
$741$	−6.03749	−0.221793
$742$	0	0
$743$	−5.48758	−0.201320	−0.100660	−	0.994921i	$-0.532095\pi$
$743$	−5.48758	−0.201320	−0.100660	+	0.994921i	$0.532095\pi$
$744$	0	0
$745$	−8.46602	−0.310171
$746$	0	0
$747$	19.2046	0.702660
$748$	0	0
$749$	−6.85572	−0.250503
$750$	0	0
$751$	−4.99964	−0.182440	−0.0912198	−	0.995831i	$-0.529077\pi$
$751$	−4.99964	−0.182440	−0.0912198	+	0.995831i	$0.529077\pi$
$752$	0	0
$753$	−14.1300	−0.514926
$754$	0	0
$755$	18.4610	0.671866
$756$	0	0
$757$	−0.999819	−0.0363391	−0.0181695	−	0.999835i	$-0.505784\pi$
$757$	−0.999819	−0.0363391	−0.0181695	+	0.999835i	$0.505784\pi$
$758$	0	0
$759$	−9.24189	−0.335459
$760$	0	0
$761$	26.7254	0.968794	0.484397	−	0.874848i	$-0.339039\pi$
$761$	26.7254	0.968794	0.484397	+	0.874848i	$0.339039\pi$
$762$	0	0
$763$	12.1174	0.438680
$764$	0	0
$765$	7.62058	0.275523
$766$	0	0
$767$	−8.49766	−0.306833
$768$	0	0
$769$	3.38079	0.121915	0.0609573	−	0.998140i	$-0.480585\pi$
$769$	3.38079	0.121915	0.0609573	+	0.998140i	$0.480585\pi$
$770$	0	0
$771$	8.13257	0.292887
$772$	0	0
$773$	−36.7123	−1.32045	−0.660224	−	0.751069i	$-0.729539\pi$
$773$	−36.7123	−1.32045	−0.660224	+	0.751069i	$0.729539\pi$
$774$	0	0
$775$	−1.92440	−0.0691267
$776$	0	0
$777$	−0.248376	−0.00891044
$778$	0	0
$779$	−4.08713	−0.146437
$780$	0	0
$781$	−9.69998	−0.347093
$782$	0	0
$783$	−13.9654	−0.499082
$784$	0	0
$785$	−12.6572	−0.451754
$786$	0	0
$787$	−40.2828	−1.43593	−0.717964	−	0.696080i	$-0.754926\pi$
$787$	−40.2828	−1.43593	−0.717964	+	0.696080i	$0.754926\pi$
$788$	0	0
$789$	−11.5965	−0.412847
$790$	0	0
$791$	−12.0522	−0.428526
$792$	0	0
$793$	4.81738	0.171070
$794$	0	0
$795$	9.91927	0.351800
$796$	0	0
$797$	1.20483	0.0426773	0.0213387	−	0.999772i	$-0.493207\pi$
$797$	1.20483	0.0426773	0.0213387	+	0.999772i	$0.493207\pi$
$798$	0	0
$799$	29.1793	1.03229
$800$	0	0
$801$	7.24190	0.255880
$802$	0	0
$803$	14.8876	0.525373
$804$	0	0
$805$	11.3463	0.399905
$806$	0	0
$807$	28.9514	1.01914
$808$	0	0
$809$	−21.4406	−0.753811	−0.376906	−	0.926252i	$-0.623012\pi$
$809$	−21.4406	−0.753811	−0.376906	+	0.926252i	$0.623012\pi$
$810$	0	0
$811$	18.4290	0.647130	0.323565	−	0.946206i	$-0.395119\pi$
$811$	18.4290	0.647130	0.323565	+	0.946206i	$0.395119\pi$
$812$	0	0
$813$	−31.1131	−1.09118
$814$	0	0
$815$	5.06712	0.177493
$816$	0	0
$817$	2.00832	0.0702623
$818$	0	0
$819$	−2.03293	−0.0710363
$820$	0	0
$821$	−43.9270	−1.53306	−0.766531	−	0.642208i	$-0.778019\pi$
$821$	−43.9270	−1.53306	−0.766531	+	0.642208i	$0.778019\pi$
$822$	0	0
$823$	28.0795	0.978791	0.489396	−	0.872062i	$-0.337217\pi$
$823$	28.0795	0.978791	0.489396	+	0.872062i	$0.337217\pi$
$824$	0	0
$825$	3.48356	0.121282
$826$	0	0
$827$	2.92863	0.101838	0.0509192	−	0.998703i	$-0.483785\pi$
$827$	2.92863	0.101838	0.0509192	+	0.998703i	$0.483785\pi$
$828$	0	0
$829$	−34.8137	−1.20913	−0.604565	−	0.796556i	$-0.706653\pi$
$829$	−34.8137	−1.20913	−0.604565	+	0.796556i	$0.706653\pi$
$830$	0	0
$831$	24.0104	0.832913
$832$	0	0
$833$	3.10486	0.107577
$834$	0	0
$835$	23.9615	0.829221
$836$	0	0
$837$	2.68876	0.0929371
$838$	0	0
$839$	−37.5674	−1.29697	−0.648486	−	0.761227i	$-0.724597\pi$
$839$	−37.5674	−1.29697	−0.648486	+	0.761227i	$0.724597\pi$
$840$	0	0
$841$	−21.0383	−0.725458
$842$	0	0
$843$	15.2977	0.526880
$844$	0	0
$845$	−1.20732	−0.0415331
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	1.34813	0.0462677
$850$	0	0
$851$	−2.37362	−0.0813668
$852$	0	0
$853$	48.4415	1.65861	0.829303	−	0.558800i	$-0.188738\pi$
$853$	48.4415	1.65861	0.829303	+	0.558800i	$0.188738\pi$
$854$	0	0
$855$	−15.0686	−0.515335
$856$	0	0
$857$	−1.95212	−0.0666831	−0.0333416	−	0.999444i	$-0.510615\pi$
$857$	−1.95212	−0.0666831	−0.0333416	+	0.999444i	$0.510615\pi$
$858$	0	0
$859$	−49.5046	−1.68908	−0.844538	−	0.535495i	$-0.820125\pi$
$859$	−49.5046	−1.68908	−0.844538	+	0.535495i	$0.820125\pi$
$860$	0	0
$861$	0.654667	0.0223110
$862$	0	0
$863$	41.5788	1.41536	0.707679	−	0.706534i	$-0.249742\pi$
$863$	41.5788	1.41536	0.707679	+	0.706534i	$0.249742\pi$
$864$	0	0
$865$	−10.1750	−0.345962
$866$	0	0
$867$	7.23763	0.245803
$868$	0	0
$869$	4.53081	0.153697
$870$	0	0
$871$	13.9821	0.473767
$872$	0	0
$873$	−31.9558	−1.08154
$874$	0	0
$875$	−10.3134	−0.348656
$876$	0	0
$877$	−25.2456	−0.852483	−0.426241	−	0.904609i	$-0.640163\pi$
$877$	−25.2456	−0.852483	−0.426241	+	0.904609i	$0.640163\pi$
$878$	0	0
$879$	−4.53588	−0.152991
$880$	0	0
$881$	−43.3232	−1.45960	−0.729798	−	0.683662i	$-0.760386\pi$
$881$	−43.3232	−1.45960	−0.729798	+	0.683662i	$0.760386\pi$
$882$	0	0
$883$	13.0166	0.438044	0.219022	−	0.975720i	$-0.429713\pi$
$883$	13.0166	0.438044	0.219022	+	0.975720i	$0.429713\pi$
$884$	0	0
$885$	10.0891	0.339140
$886$	0	0
$887$	32.2842	1.08400	0.541998	−	0.840380i	$-0.317668\pi$
$887$	32.2842	1.08400	0.541998	+	0.840380i	$0.317668\pi$
$888$	0	0
$889$	14.3558	0.481478
$890$	0	0
$891$	1.23159	0.0412597
$892$	0	0
$893$	−57.6977	−1.93078
$894$	0	0
$895$	−4.71375	−0.157563
$896$	0	0
$897$	9.24189	0.308578
$898$	0	0
$899$	−1.53287	−0.0511240
$900$	0	0
$901$	25.9400	0.864187
$902$	0	0
$903$	−0.321689	−0.0107051
$904$	0	0
$905$	12.3665	0.411075
$906$	0	0
$907$	18.3936	0.610749	0.305375	−	0.952232i	$-0.401218\pi$
$907$	18.3936	0.610749	0.305375	+	0.952232i	$0.401218\pi$
$908$	0	0
$909$	30.1749	1.00084
$910$	0	0
$911$	42.5622	1.41015	0.705074	−	0.709134i	$-0.250914\pi$
$911$	42.5622	1.41015	0.705074	+	0.709134i	$0.250914\pi$
$912$	0	0
$913$	−9.44676	−0.312642
$914$	0	0
$915$	−5.71956	−0.189083
$916$	0	0
$917$	−6.34706	−0.209598
$918$	0	0
$919$	41.0242	1.35326	0.676631	−	0.736322i	$-0.263439\pi$
$919$	41.0242	1.35326	0.676631	+	0.736322i	$0.263439\pi$
$920$	0	0
$921$	10.6232	0.350046
$922$	0	0
$923$	9.69998	0.319279
$924$	0	0
$925$	0.894695	0.0294174
$926$	0	0
$927$	17.2594	0.566874
$928$	0	0
$929$	9.64229	0.316353	0.158177	−	0.987411i	$-0.449438\pi$
$929$	9.64229	0.316353	0.158177	+	0.987411i	$0.449438\pi$
$930$	0	0
$931$	−6.13942	−0.201211
$932$	0	0
$933$	2.06883	0.0677305
$934$	0	0
$935$	−3.74857	−0.122591
$936$	0	0
$937$	56.0178	1.83002	0.915012	−	0.403427i	$-0.132181\pi$
$937$	56.0178	1.83002	0.915012	+	0.403427i	$0.132181\pi$
$938$	0	0
$939$	−30.3385	−0.990061
$940$	0	0
$941$	−13.8445	−0.451317	−0.225658	−	0.974207i	$-0.572453\pi$
$941$	−13.8445	−0.451317	−0.225658	+	0.974207i	$0.572453\pi$
$942$	0	0
$943$	6.25638	0.203736
$944$	0	0
$945$	5.97548	0.194382
$946$	0	0
$947$	−27.8060	−0.903574	−0.451787	−	0.892126i	$-0.649213\pi$
$947$	−27.8060	−0.903574	−0.451787	+	0.892126i	$0.649213\pi$
$948$	0	0
$949$	−14.8876	−0.483273
$950$	0	0
$951$	0.506987	0.0164402
$952$	0	0
$953$	9.93118	0.321702	0.160851	−	0.986979i	$-0.448576\pi$
$953$	9.93118	0.321702	0.160851	+	0.986979i	$0.448576\pi$
$954$	0	0
$955$	−20.3137	−0.657336
$956$	0	0
$957$	2.77480	0.0896967
$958$	0	0
$959$	−11.3083	−0.365163
$960$	0	0
$961$	−30.7049	−0.990480
$962$	0	0
$963$	−13.9372	−0.449120
$964$	0	0
$965$	30.0293	0.966677
$966$	0	0
$967$	−16.4626	−0.529401	−0.264701	−	0.964331i	$-0.585273\pi$
$967$	−16.4626	−0.529401	−0.264701	+	0.964331i	$0.585273\pi$
$968$	0	0
$969$	18.7456	0.602195
$970$	0	0
$971$	1.35918	0.0436182	0.0218091	−	0.999762i	$-0.493057\pi$
$971$	1.35918	0.0436182	0.0218091	+	0.999762i	$0.493057\pi$
$972$	0	0
$973$	−9.20021	−0.294945
$974$	0	0
$975$	−3.48356	−0.111563
$976$	0	0
$977$	−28.3576	−0.907241	−0.453620	−	0.891195i	$-0.649868\pi$
$977$	−28.3576	−0.907241	−0.453620	+	0.891195i	$0.649868\pi$
$978$	0	0
$979$	−3.56230	−0.113852
$980$	0	0
$981$	24.6339	0.786499
$982$	0	0
$983$	50.9697	1.62568	0.812841	−	0.582485i	$-0.197920\pi$
$983$	50.9697	1.62568	0.812841	+	0.582485i	$0.197920\pi$
$984$	0	0
$985$	−3.03819	−0.0968048
$986$	0	0
$987$	9.24189	0.294173
$988$	0	0
$989$	−3.07424	−0.0977552
$990$	0	0
$991$	−58.9843	−1.87370	−0.936848	−	0.349736i	$-0.886271\pi$
$991$	−58.9843	−1.87370	−0.936848	+	0.349736i	$0.886271\pi$
$992$	0	0
$993$	27.0862	0.859554
$994$	0	0
$995$	15.9860	0.506792
$996$	0	0
$997$	−32.0535	−1.01515	−0.507573	−	0.861609i	$-0.669457\pi$
$997$	−32.0535	−1.01515	−0.507573	+	0.861609i	$0.669457\pi$
$998$	0	0
$999$	−1.25006	−0.0395501

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.v.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.v.1.5	✓	11	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.983398 q^{3} -1.20732 q^{5} -1.00000 q^{7} -2.03293 q^{9} +O(q^{10})\)	\(q-0.983398 q^{3} -1.20732 q^{5} -1.00000 q^{7} -2.03293 q^{9} +1.00000 q^{11} -1.00000 q^{13} +1.18728 q^{15} +3.10486 q^{17} -6.13942 q^{19} +0.983398 q^{21} +9.39792 q^{23} -3.54237 q^{25} +4.94937 q^{27} -2.82165 q^{29} +0.543253 q^{31} -0.983398 q^{33} +1.20732 q^{35} -0.252569 q^{37} +0.983398 q^{39} +0.665720 q^{41} -0.327119 q^{43} +2.45440 q^{45} +9.39792 q^{47} +1.00000 q^{49} -3.05332 q^{51} +8.35464 q^{53} -1.20732 q^{55} +6.03749 q^{57} +8.49766 q^{59} -4.81738 q^{61} +2.03293 q^{63} +1.20732 q^{65} -13.9821 q^{67} -9.24189 q^{69} -9.69998 q^{71} +14.8876 q^{73} +3.48356 q^{75} -1.00000 q^{77} +4.53081 q^{79} +1.23159 q^{81} -9.44676 q^{83} -3.74857 q^{85} +2.77480 q^{87} -3.56230 q^{89} +1.00000 q^{91} -0.534234 q^{93} +7.41225 q^{95} +15.7191 q^{97} -2.03293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9	\( 11 q - 2 q^{3} + 2 q^{5} - 11 q^{7} + 9 q^{9} + 11 q^{11} - 11 q^{13} - 7 q^{15} - 4 q^{17} - 16 q^{19} + 2 q^{21} - 3 q^{23} + 11 q^{25} - 11 q^{27} + q^{29} + 14 q^{31} - 2 q^{33} - 2 q^{35} - 8 q^{37} + 2 q^{39} + 4 q^{41} - 30 q^{43} + 13 q^{45} - 3 q^{47} + 11 q^{49} - 14 q^{51} - 5 q^{53} + 2 q^{55} - 22 q^{57} + 11 q^{59} + 15 q^{61} - 9 q^{63} - 2 q^{65} - 41 q^{67} + 12 q^{69} + q^{71} - 8 q^{73} - 24 q^{75} - 11 q^{77} - 26 q^{79} + 19 q^{81} - 31 q^{83} - 27 q^{85} - 25 q^{87} + 2 q^{89} + 11 q^{91} - 37 q^{93} - 6 q^{97} + 9 q^{99}+O(q^{100}) \) 11 * q - 2 * q^3 + 2 * q^5 - 11 * q^7 + 9 * q^9 + 11 * q^11 - 11 * q^13 - 7 * q^15 - 4 * q^17 - 16 * q^19 + 2 * q^21 - 3 * q^23 + 11 * q^25 - 11 * q^27 + q^29 + 14 * q^31 - 2 * q^33 - 2 * q^35 - 8 * q^37 + 2 * q^39 + 4 * q^41 - 30 * q^43 + 13 * q^45 - 3 * q^47 + 11 * q^49 - 14 * q^51 - 5 * q^53 + 2 * q^55 - 22 * q^57 + 11 * q^59 + 15 * q^61 - 9 * q^63 - 2 * q^65 - 41 * q^67 + 12 * q^69 + q^71 - 8 * q^73 - 24 * q^75 - 11 * q^77 - 26 * q^79 + 19 * q^81 - 31 * q^83 - 27 * q^85 - 25 * q^87 + 2 * q^89 + 11 * q^91 - 37 * q^93 - 6 * q^97 + 9 * q^99

Introduction

L-functions

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Varieties

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Groups

Database

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