LMFDB - Embedded newform 8008.2.a.x.1.8

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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.39079$ 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)

$f(q)$	$=$	$q+1.39079 q^{3} +0.208204 q^{5} +1.00000 q^{7} -1.06569 q^{9} +O(q^{10})$	$q+1.39079 q^{3} +0.208204 q^{5} +1.00000 q^{7} -1.06569 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.289568 q^{15} +5.42443 q^{17} +6.55310 q^{19} +1.39079 q^{21} +4.74563 q^{23} -4.95665 q^{25} -5.65454 q^{27} +1.74178 q^{29} +7.90117 q^{31} +1.39079 q^{33} +0.208204 q^{35} +5.54773 q^{37} +1.39079 q^{39} +6.83388 q^{41} -10.0581 q^{43} -0.221881 q^{45} -3.89791 q^{47} +1.00000 q^{49} +7.54427 q^{51} -10.5728 q^{53} +0.208204 q^{55} +9.11402 q^{57} -0.484155 q^{59} -14.7490 q^{61} -1.06569 q^{63} +0.208204 q^{65} +15.2552 q^{67} +6.60019 q^{69} +3.98985 q^{71} -9.20307 q^{73} -6.89368 q^{75} +1.00000 q^{77} -3.65906 q^{79} -4.66723 q^{81} -0.791043 q^{83} +1.12939 q^{85} +2.42246 q^{87} -6.00581 q^{89} +1.00000 q^{91} +10.9889 q^{93} +1.36438 q^{95} +9.36633 q^{97} -1.06569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * 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
$3$	1.39079	0.802976	0.401488	−	0.915864i	$-0.368493\pi$
$3$	1.39079	0.802976	0.401488	+	0.915864i	$0.368493\pi$
$4$	0	0
$5$	0.208204	0.0931115	0.0465557	−	0.998916i	$-0.485175\pi$
$5$	0.208204	0.0931115	0.0465557	+	0.998916i	$0.485175\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.06569	−0.355230
$10$	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
$15$	0.289568	0.0747662
$16$	0	0
$17$	5.42443	1.31562	0.657809	−	0.753185i	$-0.271483\pi$
$17$	5.42443	1.31562	0.657809	+	0.753185i	$0.271483\pi$
$18$	0	0
$19$	6.55310	1.50338	0.751692	−	0.659514i	$-0.229238\pi$
$19$	6.55310	1.50338	0.751692	+	0.659514i	$0.229238\pi$
$20$	0	0
$21$	1.39079	0.303496
$22$	0	0
$23$	4.74563	0.989532	0.494766	−	0.869026i	$-0.335254\pi$
$23$	4.74563	0.989532	0.494766	+	0.869026i	$0.335254\pi$
$24$	0	0
$25$	−4.95665	−0.991330
$26$	0	0
$27$	−5.65454	−1.08822
$28$	0	0
$29$	1.74178	0.323441	0.161721	−	0.986837i	$-0.448296\pi$
$29$	1.74178	0.323441	0.161721	+	0.986837i	$0.448296\pi$
$30$	0	0
$31$	7.90117	1.41909	0.709546	−	0.704659i	$-0.248900\pi$
$31$	7.90117	1.41909	0.709546	+	0.704659i	$0.248900\pi$
$32$	0	0
$33$	1.39079	0.242106
$34$	0	0
$35$	0.208204	0.0351928
$36$	0	0
$37$	5.54773	0.912041	0.456020	−	0.889969i	$-0.349274\pi$
$37$	5.54773	0.912041	0.456020	+	0.889969i	$0.349274\pi$
$38$	0	0
$39$	1.39079	0.222705
$40$	0	0
$41$	6.83388	1.06727	0.533636	−	0.845714i	$-0.320825\pi$
$41$	6.83388	1.06727	0.533636	+	0.845714i	$0.320825\pi$
$42$	0	0
$43$	−10.0581	−1.53385	−0.766923	−	0.641739i	$-0.778213\pi$
$43$	−10.0581	−1.53385	−0.766923	+	0.641739i	$0.778213\pi$
$44$	0	0
$45$	−0.221881	−0.0330760
$46$	0	0
$47$	−3.89791	−0.568569	−0.284284	−	0.958740i	$-0.591756\pi$
$47$	−3.89791	−0.568569	−0.284284	+	0.958740i	$0.591756\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.54427	1.05641
$52$	0	0
$53$	−10.5728	−1.45228	−0.726141	−	0.687546i	$-0.758688\pi$
$53$	−10.5728	−1.45228	−0.726141	+	0.687546i	$0.758688\pi$
$54$	0	0
$55$	0.208204	0.0280742
$56$	0	0
$57$	9.11402	1.20718
$58$	0	0
$59$	−0.484155	−0.0630316	−0.0315158	−	0.999503i	$-0.510033\pi$
$59$	−0.484155	−0.0630316	−0.0315158	+	0.999503i	$0.510033\pi$
$60$	0	0
$61$	−14.7490	−1.88841	−0.944206	−	0.329354i	$-0.893169\pi$
$61$	−14.7490	−1.88841	−0.944206	+	0.329354i	$0.893169\pi$
$62$	0	0
$63$	−1.06569	−0.134264
$64$	0	0
$65$	0.208204	0.0258245
$66$	0	0
$67$	15.2552	1.86372	0.931860	−	0.362818i	$-0.118185\pi$
$67$	15.2552	1.86372	0.931860	+	0.362818i	$0.118185\pi$
$68$	0	0
$69$	6.60019	0.794570
$70$	0	0
$71$	3.98985	0.473508	0.236754	−	0.971570i	$-0.423916\pi$
$71$	3.98985	0.473508	0.236754	+	0.971570i	$0.423916\pi$
$72$	0	0
$73$	−9.20307	−1.07714	−0.538569	−	0.842582i	$-0.681035\pi$
$73$	−9.20307	−1.07714	−0.538569	+	0.842582i	$0.681035\pi$
$74$	0	0
$75$	−6.89368	−0.796014
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.65906	−0.411677	−0.205838	−	0.978586i	$-0.565992\pi$
$79$	−3.65906	−0.411677	−0.205838	+	0.978586i	$0.565992\pi$
$80$	0	0
$81$	−4.66723	−0.518581
$82$	0	0
$83$	−0.791043	−0.0868282	−0.0434141	−	0.999057i	$-0.513823\pi$
$83$	−0.791043	−0.0868282	−0.0434141	+	0.999057i	$0.513823\pi$
$84$	0	0
$85$	1.12939	0.122499
$86$	0	0
$87$	2.42246	0.259715
$88$	0	0
$89$	−6.00581	−0.636615	−0.318307	−	0.947988i	$-0.603114\pi$
$89$	−6.00581	−0.636615	−0.318307	+	0.947988i	$0.603114\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	10.9889	1.13950
$94$	0	0
$95$	1.36438	0.139982
$96$	0	0
$97$	9.36633	0.951007	0.475503	−	0.879714i	$-0.342266\pi$
$97$	9.36633	0.951007	0.475503	+	0.879714i	$0.342266\pi$
$98$	0	0
$99$	−1.06569	−0.107106
$100$	0	0
$101$	−4.07622	−0.405599	−0.202799	−	0.979220i	$-0.565004\pi$
$101$	−4.07622	−0.405599	−0.202799	+	0.979220i	$0.565004\pi$
$102$	0	0
$103$	−3.66041	−0.360671	−0.180336	−	0.983605i	$-0.557718\pi$
$103$	−3.66041	−0.360671	−0.180336	+	0.983605i	$0.557718\pi$
$104$	0	0
$105$	0.289568	0.0282590
$106$	0	0
$107$	−2.60076	−0.251425	−0.125712	−	0.992067i	$-0.540122\pi$
$107$	−2.60076	−0.251425	−0.125712	+	0.992067i	$0.540122\pi$
$108$	0	0
$109$	1.64314	0.157384	0.0786921	−	0.996899i	$-0.474926\pi$
$109$	1.64314	0.157384	0.0786921	+	0.996899i	$0.474926\pi$
$110$	0	0
$111$	7.71575	0.732346
$112$	0	0
$113$	4.41900	0.415705	0.207852	−	0.978160i	$-0.433353\pi$
$113$	4.41900	0.415705	0.207852	+	0.978160i	$0.433353\pi$
$114$	0	0
$115$	0.988056	0.0921367
$116$	0	0
$117$	−1.06569	−0.0985231
$118$	0	0
$119$	5.42443	0.497257
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.50452	0.856993
$124$	0	0
$125$	−2.07301	−0.185416
$126$	0	0
$127$	3.21283	0.285092	0.142546	−	0.989788i	$-0.454471\pi$
$127$	3.21283	0.285092	0.142546	+	0.989788i	$0.454471\pi$
$128$	0	0
$129$	−13.9887	−1.23164
$130$	0	0
$131$	−17.0924	−1.49337	−0.746683	−	0.665180i	$-0.768355\pi$
$131$	−17.0924	−1.49337	−0.746683	+	0.665180i	$0.768355\pi$
$132$	0	0
$133$	6.55310	0.568226
$134$	0	0
$135$	−1.17730	−0.101325
$136$	0	0
$137$	18.0792	1.54461	0.772306	−	0.635251i	$-0.219103\pi$
$137$	18.0792	1.54461	0.772306	+	0.635251i	$0.219103\pi$
$138$	0	0
$139$	16.3427	1.38617	0.693086	−	0.720855i	$-0.256251\pi$
$139$	16.3427	1.38617	0.693086	+	0.720855i	$0.256251\pi$
$140$	0	0
$141$	−5.42120	−0.456547
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.362646	0.0301161
$146$	0	0
$147$	1.39079	0.114711
$148$	0	0
$149$	−3.91699	−0.320892	−0.160446	−	0.987045i	$-0.551293\pi$
$149$	−3.91699	−0.320892	−0.160446	+	0.987045i	$0.551293\pi$
$150$	0	0
$151$	23.4196	1.90586	0.952930	−	0.303189i	$-0.0980514\pi$
$151$	23.4196	1.90586	0.952930	+	0.303189i	$0.0980514\pi$
$152$	0	0
$153$	−5.78077	−0.467347
$154$	0	0
$155$	1.64505	0.132134
$156$	0	0
$157$	12.4827	0.996231	0.498115	−	0.867111i	$-0.334026\pi$
$157$	12.4827	0.996231	0.498115	+	0.867111i	$0.334026\pi$
$158$	0	0
$159$	−14.7046	−1.16615
$160$	0	0
$161$	4.74563	0.374008
$162$	0	0
$163$	−7.40362	−0.579896	−0.289948	−	0.957042i	$-0.593638\pi$
$163$	−7.40362	−0.579896	−0.289948	+	0.957042i	$0.593638\pi$
$164$	0	0
$165$	0.289568	0.0225429
$166$	0	0
$167$	5.97579	0.462421	0.231210	−	0.972904i	$-0.425731\pi$
$167$	5.97579	0.462421	0.231210	+	0.972904i	$0.425731\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.98358	−0.534048
$172$	0	0
$173$	−1.16128	−0.0882906	−0.0441453	−	0.999025i	$-0.514056\pi$
$173$	−1.16128	−0.0882906	−0.0441453	+	0.999025i	$0.514056\pi$
$174$	0	0
$175$	−4.95665	−0.374688
$176$	0	0
$177$	−0.673360	−0.0506129
$178$	0	0
$179$	16.8749	1.26129	0.630646	−	0.776071i	$-0.282790\pi$
$179$	16.8749	1.26129	0.630646	+	0.776071i	$0.282790\pi$
$180$	0	0
$181$	6.93181	0.515238	0.257619	−	0.966247i	$-0.417062\pi$
$181$	6.93181	0.515238	0.257619	+	0.966247i	$0.417062\pi$
$182$	0	0
$183$	−20.5128	−1.51635
$184$	0	0
$185$	1.15506	0.0849214
$186$	0	0
$187$	5.42443	0.396674
$188$	0	0
$189$	−5.65454	−0.411307
$190$	0	0
$191$	21.5709	1.56082	0.780408	−	0.625271i	$-0.215011\pi$
$191$	21.5709	1.56082	0.780408	+	0.625271i	$0.215011\pi$
$192$	0	0
$193$	−4.51773	−0.325193	−0.162597	−	0.986693i	$-0.551987\pi$
$193$	−4.51773	−0.325193	−0.162597	+	0.986693i	$0.551987\pi$
$194$	0	0
$195$	0.289568	0.0207364
$196$	0	0
$197$	16.6583	1.18685	0.593426	−	0.804889i	$-0.297775\pi$
$197$	16.6583	1.18685	0.593426	+	0.804889i	$0.297775\pi$
$198$	0	0
$199$	8.18017	0.579877	0.289938	−	0.957045i	$-0.406365\pi$
$199$	8.18017	0.579877	0.289938	+	0.957045i	$0.406365\pi$
$200$	0	0
$201$	21.2169	1.49652
$202$	0	0
$203$	1.74178	0.122249
$204$	0	0
$205$	1.42284	0.0993753
$206$	0	0
$207$	−5.05737	−0.351512
$208$	0	0
$209$	6.55310	0.453288
$210$	0	0
$211$	17.6246	1.21333	0.606663	−	0.794959i	$-0.292508\pi$
$211$	17.6246	1.21333	0.606663	+	0.794959i	$0.292508\pi$
$212$	0	0
$213$	5.54906	0.380216
$214$	0	0
$215$	−2.09413	−0.142819
$216$	0	0
$217$	7.90117	0.536366
$218$	0	0
$219$	−12.7996	−0.864915
$220$	0	0
$221$	5.42443	0.364887
$222$	0	0
$223$	−23.3131	−1.56116	−0.780582	−	0.625054i	$-0.785077\pi$
$223$	−23.3131	−1.56116	−0.780582	+	0.625054i	$0.785077\pi$
$224$	0	0
$225$	5.28226	0.352150
$226$	0	0
$227$	−3.60090	−0.239000	−0.119500	−	0.992834i	$-0.538129\pi$
$227$	−3.60090	−0.239000	−0.119500	+	0.992834i	$0.538129\pi$
$228$	0	0
$229$	10.6056	0.700838	0.350419	−	0.936593i	$-0.386039\pi$
$229$	10.6056	0.700838	0.350419	+	0.936593i	$0.386039\pi$
$230$	0	0
$231$	1.39079	0.0915076
$232$	0	0
$233$	18.8321	1.23373	0.616866	−	0.787068i	$-0.288402\pi$
$233$	18.8321	1.23373	0.616866	+	0.787068i	$0.288402\pi$
$234$	0	0
$235$	−0.811559	−0.0529403
$236$	0	0
$237$	−5.08901	−0.330567
$238$	0	0
$239$	−19.9920	−1.29318	−0.646589	−	0.762839i	$-0.723805\pi$
$239$	−19.9920	−1.29318	−0.646589	+	0.762839i	$0.723805\pi$
$240$	0	0
$241$	−15.3611	−0.989494	−0.494747	−	0.869037i	$-0.664739\pi$
$241$	−15.3611	−0.989494	−0.494747	+	0.869037i	$0.664739\pi$
$242$	0	0
$243$	10.4725	0.671809
$244$	0	0
$245$	0.208204	0.0133016
$246$	0	0
$247$	6.55310	0.416964
$248$	0	0
$249$	−1.10018	−0.0697209
$250$	0	0
$251$	−19.1124	−1.20636	−0.603180	−	0.797605i	$-0.706100\pi$
$251$	−19.1124	−1.20636	−0.603180	+	0.797605i	$0.706100\pi$
$252$	0	0
$253$	4.74563	0.298355
$254$	0	0
$255$	1.57074	0.0983639
$256$	0	0
$257$	14.2930	0.891571	0.445786	−	0.895140i	$-0.352924\pi$
$257$	14.2930	0.891571	0.445786	+	0.895140i	$0.352924\pi$
$258$	0	0
$259$	5.54773	0.344719
$260$	0	0
$261$	−1.85620	−0.114896
$262$	0	0
$263$	31.4246	1.93773	0.968863	−	0.247597i	$-0.0796410\pi$
$263$	31.4246	1.93773	0.968863	+	0.247597i	$0.0796410\pi$
$264$	0	0
$265$	−2.20129	−0.135224
$266$	0	0
$267$	−8.35285	−0.511186
$268$	0	0
$269$	4.23525	0.258228	0.129114	−	0.991630i	$-0.458787\pi$
$269$	4.23525	0.258228	0.129114	+	0.991630i	$0.458787\pi$
$270$	0	0
$271$	3.06996	0.186487	0.0932433	−	0.995643i	$-0.470277\pi$
$271$	3.06996	0.186487	0.0932433	+	0.995643i	$0.470277\pi$
$272$	0	0
$273$	1.39079	0.0841747
$274$	0	0
$275$	−4.95665	−0.298897
$276$	0	0
$277$	0.0234775	0.00141062	0.000705312	−	1.00000i	$-0.499775\pi$
$277$	0.0234775	0.00141062	0.000705312		1.00000i	$0.499775\pi$
$278$	0	0
$279$	−8.42020	−0.504104
$280$	0	0
$281$	−12.3364	−0.735931	−0.367965	−	0.929840i	$-0.619945\pi$
$281$	−12.3364	−0.735931	−0.367965	+	0.929840i	$0.619945\pi$
$282$	0	0
$283$	24.5235	1.45777	0.728885	−	0.684636i	$-0.240039\pi$
$283$	24.5235	1.45777	0.728885	+	0.684636i	$0.240039\pi$
$284$	0	0
$285$	1.89757	0.112402
$286$	0	0
$287$	6.83388	0.403391
$288$	0	0
$289$	12.4245	0.730852
$290$	0	0
$291$	13.0266	0.763635
$292$	0	0
$293$	28.4678	1.66310	0.831552	−	0.555447i	$-0.187453\pi$
$293$	28.4678	1.66310	0.831552	+	0.555447i	$0.187453\pi$
$294$	0	0
$295$	−0.100803	−0.00586897
$296$	0	0
$297$	−5.65454	−0.328110
$298$	0	0
$299$	4.74563	0.274447
$300$	0	0
$301$	−10.0581	−0.579739
$302$	0	0
$303$	−5.66918	−0.325686
$304$	0	0
$305$	−3.07079	−0.175833
$306$	0	0
$307$	15.5550	0.887772	0.443886	−	0.896083i	$-0.353600\pi$
$307$	15.5550	0.887772	0.443886	+	0.896083i	$0.353600\pi$
$308$	0	0
$309$	−5.09088	−0.289610
$310$	0	0
$311$	−9.11279	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$311$	−9.11279	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$312$	0	0
$313$	5.22731	0.295465	0.147733	−	0.989027i	$-0.452803\pi$
$313$	5.22731	0.295465	0.147733	+	0.989027i	$0.452803\pi$
$314$	0	0
$315$	−0.221881	−0.0125016
$316$	0	0
$317$	5.13806	0.288582	0.144291	−	0.989535i	$-0.453910\pi$
$317$	5.13806	0.288582	0.144291	+	0.989535i	$0.453910\pi$
$318$	0	0
$319$	1.74178	0.0975212
$320$	0	0
$321$	−3.61712	−0.201888
$322$	0	0
$323$	35.5469	1.97788
$324$	0	0
$325$	−4.95665	−0.274946
$326$	0	0
$327$	2.28527	0.126376
$328$	0	0
$329$	−3.89791	−0.214899
$330$	0	0
$331$	−26.5139	−1.45734	−0.728668	−	0.684868i	$-0.759860\pi$
$331$	−26.5139	−1.45734	−0.728668	+	0.684868i	$0.759860\pi$
$332$	0	0
$333$	−5.91216	−0.323984
$334$	0	0
$335$	3.17619	0.173534
$336$	0	0
$337$	−34.4715	−1.87778	−0.938892	−	0.344213i	$-0.888146\pi$
$337$	−34.4715	−1.87778	−0.938892	+	0.344213i	$0.888146\pi$
$338$	0	0
$339$	6.14592	0.333801
$340$	0	0
$341$	7.90117	0.427872
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.37418	0.0739836
$346$	0	0
$347$	−27.6339	−1.48347	−0.741733	−	0.670696i	$-0.765996\pi$
$347$	−27.6339	−1.48347	−0.741733	+	0.670696i	$0.765996\pi$
$348$	0	0
$349$	2.09224	0.111995	0.0559974	−	0.998431i	$-0.482166\pi$
$349$	2.09224	0.111995	0.0559974	+	0.998431i	$0.482166\pi$
$350$	0	0
$351$	−5.65454	−0.301817
$352$	0	0
$353$	4.44579	0.236625	0.118313	−	0.992976i	$-0.462251\pi$
$353$	4.44579	0.236625	0.118313	+	0.992976i	$0.462251\pi$
$354$	0	0
$355$	0.830701	0.0440890
$356$	0	0
$357$	7.54427	0.399285
$358$	0	0
$359$	27.1857	1.43481	0.717404	−	0.696658i	$-0.245330\pi$
$359$	27.1857	1.43481	0.717404	+	0.696658i	$0.245330\pi$
$360$	0	0
$361$	23.9431	1.26017
$362$	0	0
$363$	1.39079	0.0729978
$364$	0	0
$365$	−1.91611	−0.100294
$366$	0	0
$367$	−36.3712	−1.89856	−0.949282	−	0.314427i	$-0.898188\pi$
$367$	−36.3712	−1.89856	−0.949282	+	0.314427i	$0.898188\pi$
$368$	0	0
$369$	−7.28280	−0.379127
$370$	0	0
$371$	−10.5728	−0.548911
$372$	0	0
$373$	16.1572	0.836586	0.418293	−	0.908312i	$-0.362628\pi$
$373$	16.1572	0.836586	0.418293	+	0.908312i	$0.362628\pi$
$374$	0	0
$375$	−2.88313	−0.148884
$376$	0	0
$377$	1.74178	0.0897064
$378$	0	0
$379$	9.01019	0.462822	0.231411	−	0.972856i	$-0.425666\pi$
$379$	9.01019	0.462822	0.231411	+	0.972856i	$0.425666\pi$
$380$	0	0
$381$	4.46838	0.228922
$382$	0	0
$383$	14.8182	0.757176	0.378588	−	0.925565i	$-0.376410\pi$
$383$	14.8182	0.757176	0.378588	+	0.925565i	$0.376410\pi$
$384$	0	0
$385$	0.208204	0.0106110
$386$	0	0
$387$	10.7188	0.544868
$388$	0	0
$389$	−1.03175	−0.0523116	−0.0261558	−	0.999658i	$-0.508327\pi$
$389$	−1.03175	−0.0523116	−0.0261558	+	0.999658i	$0.508327\pi$
$390$	0	0
$391$	25.7423	1.30185
$392$	0	0
$393$	−23.7720	−1.19914
$394$	0	0
$395$	−0.761830	−0.0383318
$396$	0	0
$397$	−4.35927	−0.218786	−0.109393	−	0.993999i	$-0.534891\pi$
$397$	−4.35927	−0.218786	−0.109393	+	0.993999i	$0.534891\pi$
$398$	0	0
$399$	9.11402	0.456272
$400$	0	0
$401$	−15.2157	−0.759834	−0.379917	−	0.925021i	$-0.624048\pi$
$401$	−15.2157	−0.759834	−0.379917	+	0.925021i	$0.624048\pi$
$402$	0	0
$403$	7.90117	0.393585
$404$	0	0
$405$	−0.971734	−0.0482859
$406$	0	0
$407$	5.54773	0.274991
$408$	0	0
$409$	−12.3414	−0.610245	−0.305122	−	0.952313i	$-0.598697\pi$
$409$	−12.3414	−0.610245	−0.305122	+	0.952313i	$0.598697\pi$
$410$	0	0
$411$	25.1445	1.24029
$412$	0	0
$413$	−0.484155	−0.0238237
$414$	0	0
$415$	−0.164698	−0.00808470
$416$	0	0
$417$	22.7294	1.11306
$418$	0	0
$419$	16.4939	0.805779	0.402890	−	0.915249i	$-0.368006\pi$
$419$	16.4939	0.805779	0.402890	+	0.915249i	$0.368006\pi$
$420$	0	0
$421$	−33.3207	−1.62395	−0.811976	−	0.583691i	$-0.801608\pi$
$421$	−33.3207	−1.62395	−0.811976	+	0.583691i	$0.801608\pi$
$422$	0	0
$423$	4.15397	0.201973
$424$	0	0
$425$	−26.8870	−1.30421
$426$	0	0
$427$	−14.7490	−0.713753
$428$	0	0
$429$	1.39079	0.0671482
$430$	0	0
$431$	−34.6817	−1.67056	−0.835278	−	0.549827i	$-0.814693\pi$
$431$	−34.6817	−1.67056	−0.835278	+	0.549827i	$0.814693\pi$
$432$	0	0
$433$	−38.8918	−1.86902	−0.934510	−	0.355938i	$-0.884161\pi$
$433$	−38.8918	−1.86902	−0.934510	+	0.355938i	$0.884161\pi$
$434$	0	0
$435$	0.504366	0.0241825
$436$	0	0
$437$	31.0986	1.48765
$438$	0	0
$439$	12.3349	0.588711	0.294355	−	0.955696i	$-0.404895\pi$
$439$	12.3349	0.588711	0.294355	+	0.955696i	$0.404895\pi$
$440$	0	0
$441$	−1.06569	−0.0507472
$442$	0	0
$443$	−30.3625	−1.44256	−0.721282	−	0.692642i	$-0.756447\pi$
$443$	−30.3625	−1.44256	−0.721282	+	0.692642i	$0.756447\pi$
$444$	0	0
$445$	−1.25043	−0.0592762
$446$	0	0
$447$	−5.44772	−0.257668
$448$	0	0
$449$	0.668200	0.0315343	0.0157672	−	0.999876i	$-0.494981\pi$
$449$	0.668200	0.0315343	0.0157672	+	0.999876i	$0.494981\pi$
$450$	0	0
$451$	6.83388	0.321795
$452$	0	0
$453$	32.5719	1.53036
$454$	0	0
$455$	0.208204	0.00976073
$456$	0	0
$457$	31.6032	1.47834	0.739168	−	0.673521i	$-0.235219\pi$
$457$	31.6032	1.47834	0.739168	+	0.673521i	$0.235219\pi$
$458$	0	0
$459$	−30.6727	−1.43168
$460$	0	0
$461$	−31.5966	−1.47160	−0.735799	−	0.677200i	$-0.763193\pi$
$461$	−31.5966	−1.47160	−0.735799	+	0.677200i	$0.763193\pi$
$462$	0	0
$463$	−15.3479	−0.713275	−0.356638	−	0.934243i	$-0.616077\pi$
$463$	−15.3479	−0.713275	−0.356638	+	0.934243i	$0.616077\pi$
$464$	0	0
$465$	2.28793	0.106100
$466$	0	0
$467$	−34.1706	−1.58123	−0.790614	−	0.612315i	$-0.790238\pi$
$467$	−34.1706	−1.58123	−0.790614	+	0.612315i	$0.790238\pi$
$468$	0	0
$469$	15.2552	0.704420
$470$	0	0
$471$	17.3609	0.799949
$472$	0	0
$473$	−10.0581	−0.462472
$474$	0	0
$475$	−32.4814	−1.49035
$476$	0	0
$477$	11.2673	0.515894
$478$	0	0
$479$	−24.2760	−1.10920	−0.554600	−	0.832117i	$-0.687129\pi$
$479$	−24.2760	−1.10920	−0.554600	+	0.832117i	$0.687129\pi$
$480$	0	0
$481$	5.54773	0.252955
$482$	0	0
$483$	6.60019	0.300319
$484$	0	0
$485$	1.95010	0.0885496
$486$	0	0
$487$	−33.4028	−1.51363	−0.756813	−	0.653631i	$-0.773245\pi$
$487$	−33.4028	−1.51363	−0.756813	+	0.653631i	$0.773245\pi$
$488$	0	0
$489$	−10.2969	−0.465642
$490$	0	0
$491$	−24.8050	−1.11943	−0.559717	−	0.828684i	$-0.689090\pi$
$491$	−24.8050	−1.11943	−0.559717	+	0.828684i	$0.689090\pi$
$492$	0	0
$493$	9.44819	0.425525
$494$	0	0
$495$	−0.221881	−0.00997279
$496$	0	0
$497$	3.98985	0.178969
$498$	0	0
$499$	−12.9994	−0.581934	−0.290967	−	0.956733i	$-0.593977\pi$
$499$	−12.9994	−0.581934	−0.290967	+	0.956733i	$0.593977\pi$
$500$	0	0
$501$	8.31110	0.371313
$502$	0	0
$503$	−4.04393	−0.180310	−0.0901550	−	0.995928i	$-0.528736\pi$
$503$	−4.04393	−0.180310	−0.0901550	+	0.995928i	$0.528736\pi$
$504$	0	0
$505$	−0.848683	−0.0377659
$506$	0	0
$507$	1.39079	0.0617674
$508$	0	0
$509$	42.6514	1.89049	0.945246	−	0.326359i	$-0.105822\pi$
$509$	42.6514	1.89049	0.945246	+	0.326359i	$0.105822\pi$
$510$	0	0
$511$	−9.20307	−0.407120
$512$	0	0
$513$	−37.0548	−1.63601
$514$	0	0
$515$	−0.762111	−0.0335826
$516$	0	0
$517$	−3.89791	−0.171430
$518$	0	0
$519$	−1.61510	−0.0708952
$520$	0	0
$521$	−6.63349	−0.290618	−0.145309	−	0.989386i	$-0.546418\pi$
$521$	−6.63349	−0.290618	−0.145309	+	0.989386i	$0.546418\pi$
$522$	0	0
$523$	−16.1908	−0.707972	−0.353986	−	0.935251i	$-0.615174\pi$
$523$	−16.1908	−0.707972	−0.353986	+	0.935251i	$0.615174\pi$
$524$	0	0
$525$	−6.89368	−0.300865
$526$	0	0
$527$	42.8594	1.86698
$528$	0	0
$529$	−0.479025	−0.0208272
$530$	0	0
$531$	0.515960	0.0223907
$532$	0	0
$533$	6.83388	0.296008
$534$	0	0
$535$	−0.541487	−0.0234105
$536$	0	0
$537$	23.4696	1.01279
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−17.9655	−0.772396	−0.386198	−	0.922416i	$-0.626212\pi$
$541$	−17.9655	−0.772396	−0.386198	+	0.922416i	$0.626212\pi$
$542$	0	0
$543$	9.64073	0.413723
$544$	0	0
$545$	0.342107	0.0146543
$546$	0	0
$547$	−24.0883	−1.02994	−0.514972	−	0.857207i	$-0.672198\pi$
$547$	−24.0883	−1.02994	−0.514972	+	0.857207i	$0.672198\pi$
$548$	0	0
$549$	15.7178	0.670821
$550$	0	0
$551$	11.4141	0.486256
$552$	0	0
$553$	−3.65906	−0.155599
$554$	0	0
$555$	1.60645	0.0681898
$556$	0	0
$557$	8.06009	0.341517	0.170759	−	0.985313i	$-0.445378\pi$
$557$	8.06009	0.341517	0.170759	+	0.985313i	$0.445378\pi$
$558$	0	0
$559$	−10.0581	−0.425412
$560$	0	0
$561$	7.54427	0.318519
$562$	0	0
$563$	10.1517	0.427844	0.213922	−	0.976851i	$-0.431376\pi$
$563$	10.1517	0.427844	0.213922	+	0.976851i	$0.431376\pi$
$564$	0	0
$565$	0.920052	0.0387069
$566$	0	0
$567$	−4.66723	−0.196005
$568$	0	0
$569$	32.4857	1.36187	0.680935	−	0.732344i	$-0.261574\pi$
$569$	32.4857	1.36187	0.680935	+	0.732344i	$0.261574\pi$
$570$	0	0
$571$	35.3069	1.47755	0.738773	−	0.673954i	$-0.235405\pi$
$571$	35.3069	1.47755	0.738773	+	0.673954i	$0.235405\pi$
$572$	0	0
$573$	30.0007	1.25330
$574$	0	0
$575$	−23.5224	−0.980953
$576$	0	0
$577$	18.2307	0.758952	0.379476	−	0.925202i	$-0.376104\pi$
$577$	18.2307	0.758952	0.379476	+	0.925202i	$0.376104\pi$
$578$	0	0
$579$	−6.28323	−0.261122
$580$	0	0
$581$	−0.791043	−0.0328180
$582$	0	0
$583$	−10.5728	−0.437880
$584$	0	0
$585$	−0.221881	−0.00917363
$586$	0	0
$587$	−15.7814	−0.651369	−0.325685	−	0.945478i	$-0.605595\pi$
$587$	−15.7814	−0.651369	−0.325685	+	0.945478i	$0.605595\pi$
$588$	0	0
$589$	51.7772	2.13344
$590$	0	0
$591$	23.1682	0.953013
$592$	0	0
$593$	6.45766	0.265184	0.132592	−	0.991171i	$-0.457670\pi$
$593$	6.45766	0.265184	0.132592	+	0.991171i	$0.457670\pi$
$594$	0	0
$595$	1.12939	0.0463003
$596$	0	0
$597$	11.3769	0.465627
$598$	0	0
$599$	−30.3081	−1.23836	−0.619178	−	0.785251i	$-0.712534\pi$
$599$	−30.3081	−1.23836	−0.619178	+	0.785251i	$0.712534\pi$
$600$	0	0
$601$	−22.2159	−0.906204	−0.453102	−	0.891459i	$-0.649683\pi$
$601$	−22.2159	−0.906204	−0.453102	+	0.891459i	$0.649683\pi$
$602$	0	0
$603$	−16.2573	−0.662050
$604$	0	0
$605$	0.208204	0.00846468
$606$	0	0
$607$	35.6861	1.44845	0.724227	−	0.689562i	$-0.242197\pi$
$607$	35.6861	1.44845	0.724227	+	0.689562i	$0.242197\pi$
$608$	0	0
$609$	2.42246	0.0981632
$610$	0	0
$611$	−3.89791	−0.157693
$612$	0	0
$613$	36.6085	1.47860	0.739301	−	0.673375i	$-0.235156\pi$
$613$	36.6085	1.47860	0.739301	+	0.673375i	$0.235156\pi$
$614$	0	0
$615$	1.97887	0.0797959
$616$	0	0
$617$	−24.6155	−0.990984	−0.495492	−	0.868613i	$-0.665012\pi$
$617$	−24.6155	−0.990984	−0.495492	+	0.868613i	$0.665012\pi$
$618$	0	0
$619$	−32.0242	−1.28716	−0.643580	−	0.765379i	$-0.722552\pi$
$619$	−32.0242	−1.28716	−0.643580	+	0.765379i	$0.722552\pi$
$620$	0	0
$621$	−26.8343	−1.07682
$622$	0	0
$623$	−6.00581	−0.240618
$624$	0	0
$625$	24.3516	0.974066
$626$	0	0
$627$	9.11402	0.363979
$628$	0	0
$629$	30.0933	1.19990
$630$	0	0
$631$	4.45031	0.177164	0.0885821	−	0.996069i	$-0.471766\pi$
$631$	4.45031	0.177164	0.0885821	+	0.996069i	$0.471766\pi$
$632$	0	0
$633$	24.5122	0.974272
$634$	0	0
$635$	0.668922	0.0265454
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−4.25195	−0.168204
$640$	0	0
$641$	−0.948699	−0.0374714	−0.0187357	−	0.999824i	$-0.505964\pi$
$641$	−0.948699	−0.0374714	−0.0187357	+	0.999824i	$0.505964\pi$
$642$	0	0
$643$	−24.9786	−0.985061	−0.492531	−	0.870295i	$-0.663928\pi$
$643$	−24.9786	−0.985061	−0.492531	+	0.870295i	$0.663928\pi$
$644$	0	0
$645$	−2.91251	−0.114680
$646$	0	0
$647$	42.7429	1.68040	0.840198	−	0.542280i	$-0.182439\pi$
$647$	42.7429	1.68040	0.840198	+	0.542280i	$0.182439\pi$
$648$	0	0
$649$	−0.484155	−0.0190048
$650$	0	0
$651$	10.9889	0.430689
$652$	0	0
$653$	2.94462	0.115232	0.0576159	−	0.998339i	$-0.481650\pi$
$653$	2.94462	0.115232	0.0576159	+	0.998339i	$0.481650\pi$
$654$	0	0
$655$	−3.55869	−0.139050
$656$	0	0
$657$	9.80762	0.382632
$658$	0	0
$659$	−14.6545	−0.570857	−0.285428	−	0.958400i	$-0.592136\pi$
$659$	−14.6545	−0.570857	−0.285428	+	0.958400i	$0.592136\pi$
$660$	0	0
$661$	9.22255	0.358716	0.179358	−	0.983784i	$-0.442598\pi$
$661$	9.22255	0.358716	0.179358	+	0.983784i	$0.442598\pi$
$662$	0	0
$663$	7.54427	0.292995
$664$	0	0
$665$	1.36438	0.0529084
$666$	0	0
$667$	8.26586	0.320055
$668$	0	0
$669$	−32.4238	−1.25358
$670$	0	0
$671$	−14.7490	−0.569378
$672$	0	0
$673$	−3.64012	−0.140316	−0.0701582	−	0.997536i	$-0.522350\pi$
$673$	−3.64012	−0.140316	−0.0701582	+	0.997536i	$0.522350\pi$
$674$	0	0
$675$	28.0276	1.07878
$676$	0	0
$677$	−13.4791	−0.518044	−0.259022	−	0.965871i	$-0.583400\pi$
$677$	−13.4791	−0.518044	−0.259022	+	0.965871i	$0.583400\pi$
$678$	0	0
$679$	9.36633	0.359447
$680$	0	0
$681$	−5.00811	−0.191911
$682$	0	0
$683$	11.4785	0.439214	0.219607	−	0.975588i	$-0.429522\pi$
$683$	11.4785	0.439214	0.219607	+	0.975588i	$0.429522\pi$
$684$	0	0
$685$	3.76416	0.143821
$686$	0	0
$687$	14.7502	0.562756
$688$	0	0
$689$	−10.5728	−0.402791
$690$	0	0
$691$	−27.9031	−1.06148	−0.530741	−	0.847534i	$-0.678086\pi$
$691$	−27.9031	−1.06148	−0.530741	+	0.847534i	$0.678086\pi$
$692$	0	0
$693$	−1.06569	−0.0404822
$694$	0	0
$695$	3.40261	0.129068
$696$	0	0
$697$	37.0699	1.40412
$698$	0	0
$699$	26.1916	0.990657
$700$	0	0
$701$	34.7465	1.31236	0.656179	−	0.754605i	$-0.272172\pi$
$701$	34.7465	1.31236	0.656179	+	0.754605i	$0.272172\pi$
$702$	0	0
$703$	36.3548	1.37115
$704$	0	0
$705$	−1.12871	−0.0425098
$706$	0	0
$707$	−4.07622	−0.153302
$708$	0	0
$709$	−9.90672	−0.372055	−0.186027	−	0.982545i	$-0.559561\pi$
$709$	−9.90672	−0.372055	−0.186027	+	0.982545i	$0.559561\pi$
$710$	0	0
$711$	3.89943	0.146240
$712$	0	0
$713$	37.4960	1.40424
$714$	0	0
$715$	0.208204	0.00778637
$716$	0	0
$717$	−27.8048	−1.03839
$718$	0	0
$719$	23.4915	0.876085	0.438043	−	0.898954i	$-0.355672\pi$
$719$	23.4915	0.876085	0.438043	+	0.898954i	$0.355672\pi$
$720$	0	0
$721$	−3.66041	−0.136321
$722$	0	0
$723$	−21.3641	−0.794539
$724$	0	0
$725$	−8.63342	−0.320637
$726$	0	0
$727$	18.3589	0.680895	0.340447	−	0.940264i	$-0.389421\pi$
$727$	18.3589	0.680895	0.340447	+	0.940264i	$0.389421\pi$
$728$	0	0
$729$	28.5667	1.05803
$730$	0	0
$731$	−54.5595	−2.01796
$732$	0	0
$733$	11.5726	0.427443	0.213722	−	0.976895i	$-0.431441\pi$
$733$	11.5726	0.427443	0.213722	+	0.976895i	$0.431441\pi$
$734$	0	0
$735$	0.289568	0.0106809
$736$	0	0
$737$	15.2552	0.561933
$738$	0	0
$739$	−42.0160	−1.54558	−0.772792	−	0.634659i	$-0.781141\pi$
$739$	−42.0160	−1.54558	−0.772792	+	0.634659i	$0.781141\pi$
$740$	0	0
$741$	9.11402	0.334812
$742$	0	0
$743$	7.20719	0.264406	0.132203	−	0.991223i	$-0.457795\pi$
$743$	7.20719	0.264406	0.132203	+	0.991223i	$0.457795\pi$
$744$	0	0
$745$	−0.815531	−0.0298787
$746$	0	0
$747$	0.843007	0.0308440
$748$	0	0
$749$	−2.60076	−0.0950296
$750$	0	0
$751$	−28.5524	−1.04189	−0.520946	−	0.853589i	$-0.674421\pi$
$751$	−28.5524	−1.04189	−0.520946	+	0.853589i	$0.674421\pi$
$752$	0	0
$753$	−26.5814	−0.968678
$754$	0	0
$755$	4.87605	0.177458
$756$	0	0
$757$	−32.0355	−1.16435	−0.582176	−	0.813063i	$-0.697798\pi$
$757$	−32.0355	−1.16435	−0.582176	+	0.813063i	$0.697798\pi$
$758$	0	0
$759$	6.60019	0.239572
$760$	0	0
$761$	7.19483	0.260813	0.130406	−	0.991461i	$-0.458372\pi$
$761$	7.19483	0.260813	0.130406	+	0.991461i	$0.458372\pi$
$762$	0	0
$763$	1.64314	0.0594856
$764$	0	0
$765$	−1.20358	−0.0435154
$766$	0	0
$767$	−0.484155	−0.0174818
$768$	0	0
$769$	19.2371	0.693708	0.346854	−	0.937919i	$-0.387250\pi$
$769$	19.2371	0.693708	0.346854	+	0.937919i	$0.387250\pi$
$770$	0	0
$771$	19.8786	0.715910
$772$	0	0
$773$	12.8115	0.460798	0.230399	−	0.973096i	$-0.425997\pi$
$773$	12.8115	0.460798	0.230399	+	0.973096i	$0.425997\pi$
$774$	0	0
$775$	−39.1633	−1.40679
$776$	0	0
$777$	7.71575	0.276801
$778$	0	0
$779$	44.7831	1.60452
$780$	0	0
$781$	3.98985	0.142768
$782$	0	0
$783$	−9.84899	−0.351974
$784$	0	0
$785$	2.59895	0.0927605
$786$	0	0
$787$	−37.3702	−1.33210	−0.666052	−	0.745905i	$-0.732017\pi$
$787$	−37.3702	−1.33210	−0.666052	+	0.745905i	$0.732017\pi$
$788$	0	0
$789$	43.7052	1.55595
$790$	0	0
$791$	4.41900	0.157122
$792$	0	0
$793$	−14.7490	−0.523752
$794$	0	0
$795$	−3.06154	−0.108582
$796$	0	0
$797$	−29.7191	−1.05270	−0.526352	−	0.850267i	$-0.676441\pi$
$797$	−29.7191	−1.05270	−0.526352	+	0.850267i	$0.676441\pi$
$798$	0	0
$799$	−21.1440	−0.748020
$800$	0	0
$801$	6.40034	0.226145
$802$	0	0
$803$	−9.20307	−0.324769
$804$	0	0
$805$	0.988056	0.0348244
$806$	0	0
$807$	5.89037	0.207351
$808$	0	0
$809$	15.0320	0.528497	0.264249	−	0.964455i	$-0.414876\pi$
$809$	15.0320	0.528497	0.264249	+	0.964455i	$0.414876\pi$
$810$	0	0
$811$	20.1443	0.707361	0.353681	−	0.935366i	$-0.384930\pi$
$811$	20.1443	0.707361	0.353681	+	0.935366i	$0.384930\pi$
$812$	0	0
$813$	4.26968	0.149744
$814$	0	0
$815$	−1.54146	−0.0539950
$816$	0	0
$817$	−65.9117	−2.30596
$818$	0	0
$819$	−1.06569	−0.0372382
$820$	0	0
$821$	27.2104	0.949651	0.474825	−	0.880080i	$-0.342511\pi$
$821$	27.2104	0.949651	0.474825	+	0.880080i	$0.342511\pi$
$822$	0	0
$823$	−2.81900	−0.0982640	−0.0491320	−	0.998792i	$-0.515646\pi$
$823$	−2.81900	−0.0982640	−0.0491320	+	0.998792i	$0.515646\pi$
$824$	0	0
$825$	−6.89368	−0.240007
$826$	0	0
$827$	−25.6686	−0.892583	−0.446291	−	0.894888i	$-0.647256\pi$
$827$	−25.6686	−0.892583	−0.446291	+	0.894888i	$0.647256\pi$
$828$	0	0
$829$	−56.8209	−1.97347	−0.986736	−	0.162335i	$-0.948097\pi$
$829$	−56.8209	−1.97347	−0.986736	+	0.162335i	$0.948097\pi$
$830$	0	0
$831$	0.0326523	0.00113270
$832$	0	0
$833$	5.42443	0.187946
$834$	0	0
$835$	1.24418	0.0430567
$836$	0	0
$837$	−44.6775	−1.54428
$838$	0	0
$839$	11.7617	0.406059	0.203030	−	0.979173i	$-0.434921\pi$
$839$	11.7617	0.406059	0.203030	+	0.979173i	$0.434921\pi$
$840$	0	0
$841$	−25.9662	−0.895386
$842$	0	0
$843$	−17.1575	−0.590934
$844$	0	0
$845$	0.208204	0.00716242
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	34.1071	1.17055
$850$	0	0
$851$	26.3274	0.902493
$852$	0	0
$853$	−6.52472	−0.223402	−0.111701	−	0.993742i	$-0.535630\pi$
$853$	−6.52472	−0.223402	−0.111701	+	0.993742i	$0.535630\pi$
$854$	0	0
$855$	−1.45401	−0.0497260
$856$	0	0
$857$	−45.9601	−1.56997	−0.784984	−	0.619516i	$-0.787329\pi$
$857$	−45.9601	−1.56997	−0.784984	+	0.619516i	$0.787329\pi$
$858$	0	0
$859$	−5.56938	−0.190025	−0.0950125	−	0.995476i	$-0.530289\pi$
$859$	−5.56938	−0.190025	−0.0950125	+	0.995476i	$0.530289\pi$
$860$	0	0
$861$	9.50452	0.323913
$862$	0	0
$863$	29.9813	1.02057	0.510287	−	0.860004i	$-0.329539\pi$
$863$	29.9813	1.02057	0.510287	+	0.860004i	$0.329539\pi$
$864$	0	0
$865$	−0.241783	−0.00822087
$866$	0	0
$867$	17.2799	0.586857
$868$	0	0
$869$	−3.65906	−0.124125
$870$	0	0
$871$	15.2552	0.516903
$872$	0	0
$873$	−9.98161	−0.337826
$874$	0	0
$875$	−2.07301	−0.0700805
$876$	0	0
$877$	38.6138	1.30389	0.651947	−	0.758265i	$-0.273952\pi$
$877$	38.6138	1.30389	0.651947	+	0.758265i	$0.273952\pi$
$878$	0	0
$879$	39.5928	1.33543
$880$	0	0
$881$	17.0991	0.576082	0.288041	−	0.957618i	$-0.406996\pi$
$881$	17.0991	0.576082	0.288041	+	0.957618i	$0.406996\pi$
$882$	0	0
$883$	3.99297	0.134374	0.0671871	−	0.997740i	$-0.478598\pi$
$883$	3.99297	0.134374	0.0671871	+	0.997740i	$0.478598\pi$
$884$	0	0
$885$	−0.140196	−0.00471264
$886$	0	0
$887$	−42.4216	−1.42438	−0.712189	−	0.701988i	$-0.752296\pi$
$887$	−42.4216	−1.42438	−0.712189	+	0.701988i	$0.752296\pi$
$888$	0	0
$889$	3.21283	0.107755
$890$	0	0
$891$	−4.66723	−0.156358
$892$	0	0
$893$	−25.5434	−0.854778
$894$	0	0
$895$	3.51342	0.117441
$896$	0	0
$897$	6.60019	0.220374
$898$	0	0
$899$	13.7621	0.458993
$900$	0	0
$901$	−57.3513	−1.91065
$902$	0	0
$903$	−13.9887	−0.465516
$904$	0	0
$905$	1.44323	0.0479745
$906$	0	0
$907$	−43.1236	−1.43190	−0.715948	−	0.698154i	$-0.754005\pi$
$907$	−43.1236	−1.43190	−0.715948	+	0.698154i	$0.754005\pi$
$908$	0	0
$909$	4.34399	0.144081
$910$	0	0
$911$	−9.56379	−0.316862	−0.158431	−	0.987370i	$-0.550644\pi$
$911$	−9.56379	−0.316862	−0.158431	+	0.987370i	$0.550644\pi$
$912$	0	0
$913$	−0.791043	−0.0261797
$914$	0	0
$915$	−4.27084	−0.141190
$916$	0	0
$917$	−17.0924	−0.564440
$918$	0	0
$919$	−30.6362	−1.01060	−0.505298	−	0.862945i	$-0.668617\pi$
$919$	−30.6362	−1.01060	−0.505298	+	0.862945i	$0.668617\pi$
$920$	0	0
$921$	21.6338	0.712859
$922$	0	0
$923$	3.98985	0.131328
$924$	0	0
$925$	−27.4981	−0.904133
$926$	0	0
$927$	3.90087	0.128121
$928$	0	0
$929$	−29.5312	−0.968887	−0.484444	−	0.874823i	$-0.660978\pi$
$929$	−29.5312	−0.968887	−0.484444	+	0.874823i	$0.660978\pi$
$930$	0	0
$931$	6.55310	0.214769
$932$	0	0
$933$	−12.6740	−0.414929
$934$	0	0
$935$	1.12939	0.0369349
$936$	0	0
$937$	8.15965	0.266564	0.133282	−	0.991078i	$-0.457448\pi$
$937$	8.15965	0.266564	0.133282	+	0.991078i	$0.457448\pi$
$938$	0	0
$939$	7.27012	0.237251
$940$	0	0
$941$	24.9055	0.811895	0.405948	−	0.913896i	$-0.366942\pi$
$941$	24.9055	0.811895	0.405948	+	0.913896i	$0.366942\pi$
$942$	0	0
$943$	32.4310	1.05610
$944$	0	0
$945$	−1.17730	−0.0382974
$946$	0	0
$947$	−43.0288	−1.39825	−0.699124	−	0.715000i	$-0.746426\pi$
$947$	−43.0288	−1.39825	−0.699124	+	0.715000i	$0.746426\pi$
$948$	0	0
$949$	−9.20307	−0.298744
$950$	0	0
$951$	7.14599	0.231725
$952$	0	0
$953$	38.5632	1.24919	0.624593	−	0.780951i	$-0.285265\pi$
$953$	38.5632	1.24919	0.624593	+	0.780951i	$0.285265\pi$
$954$	0	0
$955$	4.49114	0.145330
$956$	0	0
$957$	2.42246	0.0783071
$958$	0	0
$959$	18.0792	0.583808
$960$	0	0
$961$	31.4285	1.01382
$962$	0	0
$963$	2.77160	0.0893136
$964$	0	0
$965$	−0.940607	−0.0302792
$966$	0	0
$967$	−57.3770	−1.84512	−0.922559	−	0.385855i	$-0.873906\pi$
$967$	−57.3770	−1.84512	−0.922559	+	0.385855i	$0.873906\pi$
$968$	0	0
$969$	49.4384	1.58819
$970$	0	0
$971$	1.14845	0.0368556	0.0184278	−	0.999830i	$-0.494134\pi$
$971$	1.14845	0.0368556	0.0184278	+	0.999830i	$0.494134\pi$
$972$	0	0
$973$	16.3427	0.523924
$974$	0	0
$975$	−6.89368	−0.220775
$976$	0	0
$977$	−1.29646	−0.0414775	−0.0207387	−	0.999785i	$-0.506602\pi$
$977$	−1.29646	−0.0414775	−0.0207387	+	0.999785i	$0.506602\pi$
$978$	0	0
$979$	−6.00581	−0.191947
$980$	0	0
$981$	−1.75108	−0.0559076
$982$	0	0
$983$	−15.8576	−0.505778	−0.252889	−	0.967495i	$-0.581381\pi$
$983$	−15.8576	−0.505778	−0.252889	+	0.967495i	$0.581381\pi$
$984$	0	0
$985$	3.46831	0.110510
$986$	0	0
$987$	−5.42120	−0.172559
$988$	0	0
$989$	−47.7320	−1.51779
$990$	0	0
$991$	−15.3277	−0.486900	−0.243450	−	0.969913i	$-0.578279\pi$
$991$	−15.3277	−0.486900	−0.243450	+	0.969913i	$0.578279\pi$
$992$	0	0
$993$	−36.8754	−1.17020
$994$	0	0
$995$	1.70314	0.0539932
$996$	0	0
$997$	−2.76193	−0.0874712	−0.0437356	−	0.999043i	$-0.513926\pi$
$997$	−2.76193	−0.0874712	−0.0437356	+	0.999043i	$0.513926\pi$
$998$	0	0
$999$	−31.3698	−0.992498

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.8	✓	12	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.39079 q^{3} +0.208204 q^{5} +1.00000 q^{7} -1.06569 q^{9} +O(q^{10})\)	\(q+1.39079 q^{3} +0.208204 q^{5} +1.00000 q^{7} -1.06569 q^{9} +1.00000 q^{11} +1.00000 q^{13} +0.289568 q^{15} +5.42443 q^{17} +6.55310 q^{19} +1.39079 q^{21} +4.74563 q^{23} -4.95665 q^{25} -5.65454 q^{27} +1.74178 q^{29} +7.90117 q^{31} +1.39079 q^{33} +0.208204 q^{35} +5.54773 q^{37} +1.39079 q^{39} +6.83388 q^{41} -10.0581 q^{43} -0.221881 q^{45} -3.89791 q^{47} +1.00000 q^{49} +7.54427 q^{51} -10.5728 q^{53} +0.208204 q^{55} +9.11402 q^{57} -0.484155 q^{59} -14.7490 q^{61} -1.06569 q^{63} +0.208204 q^{65} +15.2552 q^{67} +6.60019 q^{69} +3.98985 q^{71} -9.20307 q^{73} -6.89368 q^{75} +1.00000 q^{77} -3.65906 q^{79} -4.66723 q^{81} -0.791043 q^{83} +1.12939 q^{85} +2.42246 q^{87} -6.00581 q^{89} +1.00000 q^{91} +10.9889 q^{93} +1.36438 q^{95} +9.36633 q^{97} -1.06569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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