LMFDB - Embedded newform 8008.2.a.z.1.10

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
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			1.10
Root			$-1.32542$ 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)

$f(q)$	$=$	$q+1.32542 q^{3} +2.70237 q^{5} -1.00000 q^{7} -1.24326 q^{9} +O(q^{10})$	$q+1.32542 q^{3} +2.70237 q^{5} -1.00000 q^{7} -1.24326 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.58178 q^{15} +5.60719 q^{17} -2.35759 q^{19} -1.32542 q^{21} -1.20340 q^{23} +2.30283 q^{25} -5.62410 q^{27} +6.92256 q^{29} -2.89311 q^{31} -1.32542 q^{33} -2.70237 q^{35} +8.97917 q^{37} +1.32542 q^{39} +5.99010 q^{41} +6.83208 q^{43} -3.35975 q^{45} -12.9171 q^{47} +1.00000 q^{49} +7.43188 q^{51} +11.8321 q^{53} -2.70237 q^{55} -3.12480 q^{57} -2.16360 q^{59} -3.92191 q^{61} +1.24326 q^{63} +2.70237 q^{65} +12.2775 q^{67} -1.59501 q^{69} +9.50738 q^{71} -4.77815 q^{73} +3.05221 q^{75} +1.00000 q^{77} -3.55169 q^{79} -3.72453 q^{81} +4.94977 q^{83} +15.1527 q^{85} +9.17531 q^{87} +3.42138 q^{89} -1.00000 q^{91} -3.83459 q^{93} -6.37109 q^{95} -5.75875 q^{97} +1.24326 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	1.32542	0.765232	0.382616	−	0.923907i	$-0.375023\pi$
$3$	1.32542	0.765232	0.382616	+	0.923907i	$0.375023\pi$
$4$	0	0
$5$	2.70237	1.20854	0.604269	−	0.796780i	$-0.293465\pi$
$5$	2.70237	1.20854	0.604269	+	0.796780i	$0.293465\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.24326	−0.414420
$10$	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
$15$	3.58178	0.924812
$16$	0	0
$17$	5.60719	1.35994	0.679971	−	0.733239i	$-0.261992\pi$
$17$	5.60719	1.35994	0.679971	+	0.733239i	$0.261992\pi$
$18$	0	0
$19$	−2.35759	−0.540868	−0.270434	−	0.962739i	$-0.587167\pi$
$19$	−2.35759	−0.540868	−0.270434	+	0.962739i	$0.587167\pi$
$20$	0	0
$21$	−1.32542	−0.289231
$22$	0	0
$23$	−1.20340	−0.250927	−0.125463	−	0.992098i	$-0.540042\pi$
$23$	−1.20340	−0.250927	−0.125463	+	0.992098i	$0.540042\pi$
$24$	0	0
$25$	2.30283	0.460565
$26$	0	0
$27$	−5.62410	−1.08236
$28$	0	0
$29$	6.92256	1.28549	0.642744	−	0.766081i	$-0.277796\pi$
$29$	6.92256	1.28549	0.642744	+	0.766081i	$0.277796\pi$
$30$	0	0
$31$	−2.89311	−0.519618	−0.259809	−	0.965660i	$-0.583660\pi$
$31$	−2.89311	−0.519618	−0.259809	+	0.965660i	$0.583660\pi$
$32$	0	0
$33$	−1.32542	−0.230726
$34$	0	0
$35$	−2.70237	−0.456785
$36$	0	0
$37$	8.97917	1.47617	0.738084	−	0.674709i	$-0.235731\pi$
$37$	8.97917	1.47617	0.738084	+	0.674709i	$0.235731\pi$
$38$	0	0
$39$	1.32542	0.212237
$40$	0	0
$41$	5.99010	0.935497	0.467748	−	0.883862i	$-0.345065\pi$
$41$	5.99010	0.935497	0.467748	+	0.883862i	$0.345065\pi$
$42$	0	0
$43$	6.83208	1.04188	0.520941	−	0.853592i	$-0.325581\pi$
$43$	6.83208	1.04188	0.520941	+	0.853592i	$0.325581\pi$
$44$	0	0
$45$	−3.35975	−0.500842
$46$	0	0
$47$	−12.9171	−1.88415	−0.942076	−	0.335401i	$-0.891128\pi$
$47$	−12.9171	−1.88415	−0.942076	+	0.335401i	$0.891128\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.43188	1.04067
$52$	0	0
$53$	11.8321	1.62526	0.812632	−	0.582777i	$-0.198034\pi$
$53$	11.8321	1.62526	0.812632	+	0.582777i	$0.198034\pi$
$54$	0	0
$55$	−2.70237	−0.364388
$56$	0	0
$57$	−3.12480	−0.413890
$58$	0	0
$59$	−2.16360	−0.281677	−0.140839	−	0.990033i	$-0.544980\pi$
$59$	−2.16360	−0.281677	−0.140839	+	0.990033i	$0.544980\pi$
$60$	0	0
$61$	−3.92191	−0.502149	−0.251074	−	0.967968i	$-0.580784\pi$
$61$	−3.92191	−0.502149	−0.251074	+	0.967968i	$0.580784\pi$
$62$	0	0
$63$	1.24326	0.156636
$64$	0	0
$65$	2.70237	0.335188
$66$	0	0
$67$	12.2775	1.49994	0.749969	−	0.661473i	$-0.230068\pi$
$67$	12.2775	1.49994	0.749969	+	0.661473i	$0.230068\pi$
$68$	0	0
$69$	−1.59501	−0.192017
$70$	0	0
$71$	9.50738	1.12832	0.564159	−	0.825666i	$-0.309200\pi$
$71$	9.50738	1.12832	0.564159	+	0.825666i	$0.309200\pi$
$72$	0	0
$73$	−4.77815	−0.559240	−0.279620	−	0.960111i	$-0.590209\pi$
$73$	−4.77815	−0.559240	−0.279620	+	0.960111i	$0.590209\pi$
$74$	0	0
$75$	3.05221	0.352439
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−3.55169	−0.399597	−0.199798	−	0.979837i	$-0.564029\pi$
$79$	−3.55169	−0.399597	−0.199798	+	0.979837i	$0.564029\pi$
$80$	0	0
$81$	−3.72453	−0.413836
$82$	0	0
$83$	4.94977	0.543308	0.271654	−	0.962395i	$-0.412429\pi$
$83$	4.94977	0.543308	0.271654	+	0.962395i	$0.412429\pi$
$84$	0	0
$85$	15.1527	1.64354
$86$	0	0
$87$	9.17531	0.983696
$88$	0	0
$89$	3.42138	0.362666	0.181333	−	0.983422i	$-0.441959\pi$
$89$	3.42138	0.362666	0.181333	+	0.983422i	$0.441959\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−3.83459	−0.397628
$94$	0	0
$95$	−6.37109	−0.653660
$96$	0	0
$97$	−5.75875	−0.584712	−0.292356	−	0.956310i	$-0.594439\pi$
$97$	−5.75875	−0.584712	−0.292356	+	0.956310i	$0.594439\pi$
$98$	0	0
$99$	1.24326	0.124952
$100$	0	0
$101$	−10.3546	−1.03032	−0.515162	−	0.857093i	$-0.672268\pi$
$101$	−10.3546	−1.03032	−0.515162	+	0.857093i	$0.672268\pi$
$102$	0	0
$103$	15.7479	1.55169	0.775844	−	0.630925i	$-0.217325\pi$
$103$	15.7479	1.55169	0.775844	+	0.630925i	$0.217325\pi$
$104$	0	0
$105$	−3.58178	−0.349546
$106$	0	0
$107$	−14.2170	−1.37441	−0.687205	−	0.726463i	$-0.741163\pi$
$107$	−14.2170	−1.37441	−0.687205	+	0.726463i	$0.741163\pi$
$108$	0	0
$109$	−0.225530	−0.0216018	−0.0108009	−	0.999942i	$-0.503438\pi$
$109$	−0.225530	−0.0216018	−0.0108009	+	0.999942i	$0.503438\pi$
$110$	0	0
$111$	11.9012	1.12961
$112$	0	0
$113$	19.8544	1.86775	0.933874	−	0.357602i	$-0.116406\pi$
$113$	19.8544	1.86775	0.933874	+	0.357602i	$0.116406\pi$
$114$	0	0
$115$	−3.25204	−0.303255
$116$	0	0
$117$	−1.24326	−0.114939
$118$	0	0
$119$	−5.60719	−0.514010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.93940	0.715872
$124$	0	0
$125$	−7.28877	−0.651928
$126$	0	0
$127$	5.51387	0.489277	0.244638	−	0.969614i	$-0.421331\pi$
$127$	5.51387	0.489277	0.244638	+	0.969614i	$0.421331\pi$
$128$	0	0
$129$	9.05538	0.797282
$130$	0	0
$131$	10.7895	0.942684	0.471342	−	0.881951i	$-0.343770\pi$
$131$	10.7895	0.942684	0.471342	+	0.881951i	$0.343770\pi$
$132$	0	0
$133$	2.35759	0.204429
$134$	0	0
$135$	−15.1984	−1.30807
$136$	0	0
$137$	−6.56380	−0.560784	−0.280392	−	0.959886i	$-0.590464\pi$
$137$	−6.56380	−0.560784	−0.280392	+	0.959886i	$0.590464\pi$
$138$	0	0
$139$	17.9112	1.51921	0.759605	−	0.650385i	$-0.225393\pi$
$139$	17.9112	1.51921	0.759605	+	0.650385i	$0.225393\pi$
$140$	0	0
$141$	−17.1206	−1.44181
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	18.7073	1.55356
$146$	0	0
$147$	1.32542	0.109319
$148$	0	0
$149$	14.0622	1.15202	0.576009	−	0.817443i	$-0.304609\pi$
$149$	14.0622	1.15202	0.576009	+	0.817443i	$0.304609\pi$
$150$	0	0
$151$	−2.33874	−0.190324	−0.0951619	−	0.995462i	$-0.530337\pi$
$151$	−2.33874	−0.190324	−0.0951619	+	0.995462i	$0.530337\pi$
$152$	0	0
$153$	−6.97119	−0.563587
$154$	0	0
$155$	−7.81827	−0.627978
$156$	0	0
$157$	3.66743	0.292693	0.146346	−	0.989233i	$-0.453249\pi$
$157$	3.66743	0.292693	0.146346	+	0.989233i	$0.453249\pi$
$158$	0	0
$159$	15.6825	1.24370
$160$	0	0
$161$	1.20340	0.0948414
$162$	0	0
$163$	15.1275	1.18488	0.592438	−	0.805616i	$-0.298166\pi$
$163$	15.1275	1.18488	0.592438	+	0.805616i	$0.298166\pi$
$164$	0	0
$165$	−3.58178	−0.278841
$166$	0	0
$167$	−18.4269	−1.42592	−0.712958	−	0.701207i	$-0.752645\pi$
$167$	−18.4269	−1.42592	−0.712958	+	0.701207i	$0.752645\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.93110	0.224147
$172$	0	0
$173$	−8.34816	−0.634699	−0.317349	−	0.948309i	$-0.602793\pi$
$173$	−8.34816	−0.634699	−0.317349	+	0.948309i	$0.602793\pi$
$174$	0	0
$175$	−2.30283	−0.174077
$176$	0	0
$177$	−2.86769	−0.215548
$178$	0	0
$179$	3.67661	0.274803	0.137401	−	0.990515i	$-0.456125\pi$
$179$	3.67661	0.274803	0.137401	+	0.990515i	$0.456125\pi$
$180$	0	0
$181$	6.81057	0.506226	0.253113	−	0.967437i	$-0.418546\pi$
$181$	6.81057	0.506226	0.253113	+	0.967437i	$0.418546\pi$
$182$	0	0
$183$	−5.19818	−0.384260
$184$	0	0
$185$	24.2651	1.78400
$186$	0	0
$187$	−5.60719	−0.410038
$188$	0	0
$189$	5.62410	0.409093
$190$	0	0
$191$	−6.08272	−0.440130	−0.220065	−	0.975485i	$-0.570627\pi$
$191$	−6.08272	−0.440130	−0.220065	+	0.975485i	$0.570627\pi$
$192$	0	0
$193$	−18.6505	−1.34249	−0.671247	−	0.741233i	$-0.734241\pi$
$193$	−18.6505	−1.34249	−0.671247	+	0.741233i	$0.734241\pi$
$194$	0	0
$195$	3.58178	0.256497
$196$	0	0
$197$	16.7703	1.19483	0.597417	−	0.801930i	$-0.296194\pi$
$197$	16.7703	1.19483	0.597417	+	0.801930i	$0.296194\pi$
$198$	0	0
$199$	13.6570	0.968118	0.484059	−	0.875035i	$-0.339162\pi$
$199$	13.6570	0.968118	0.484059	+	0.875035i	$0.339162\pi$
$200$	0	0
$201$	16.2729	1.14780
$202$	0	0
$203$	−6.92256	−0.485868
$204$	0	0
$205$	16.1875	1.13058
$206$	0	0
$207$	1.49614	0.103989
$208$	0	0
$209$	2.35759	0.163078
$210$	0	0
$211$	−9.56477	−0.658466	−0.329233	−	0.944249i	$-0.606790\pi$
$211$	−9.56477	−0.658466	−0.329233	+	0.944249i	$0.606790\pi$
$212$	0	0
$213$	12.6013	0.863426
$214$	0	0
$215$	18.4628	1.25916
$216$	0	0
$217$	2.89311	0.196397
$218$	0	0
$219$	−6.33306	−0.427949
$220$	0	0
$221$	5.60719	0.377180
$222$	0	0
$223$	−6.65804	−0.445856	−0.222928	−	0.974835i	$-0.571561\pi$
$223$	−6.65804	−0.445856	−0.222928	+	0.974835i	$0.571561\pi$
$224$	0	0
$225$	−2.86301	−0.190867
$226$	0	0
$227$	−8.63686	−0.573248	−0.286624	−	0.958043i	$-0.592533\pi$
$227$	−8.63686	−0.573248	−0.286624	+	0.958043i	$0.592533\pi$
$228$	0	0
$229$	−16.8636	−1.11438	−0.557188	−	0.830386i	$-0.688120\pi$
$229$	−16.8636	−1.11438	−0.557188	+	0.830386i	$0.688120\pi$
$230$	0	0
$231$	1.32542	0.0872063
$232$	0	0
$233$	−21.3984	−1.40185	−0.700927	−	0.713233i	$-0.747230\pi$
$233$	−21.3984	−1.40185	−0.700927	+	0.713233i	$0.747230\pi$
$234$	0	0
$235$	−34.9068	−2.27707
$236$	0	0
$237$	−4.70749	−0.305784
$238$	0	0
$239$	14.9669	0.968128	0.484064	−	0.875033i	$-0.339160\pi$
$239$	14.9669	0.968128	0.484064	+	0.875033i	$0.339160\pi$
$240$	0	0
$241$	22.8771	1.47365	0.736823	−	0.676085i	$-0.236325\pi$
$241$	22.8771	1.47365	0.736823	+	0.676085i	$0.236325\pi$
$242$	0	0
$243$	11.9358	0.765679
$244$	0	0
$245$	2.70237	0.172648
$246$	0	0
$247$	−2.35759	−0.150010
$248$	0	0
$249$	6.56053	0.415757
$250$	0	0
$251$	−3.11738	−0.196767	−0.0983835	−	0.995149i	$-0.531367\pi$
$251$	−3.11738	−0.196767	−0.0983835	+	0.995149i	$0.531367\pi$
$252$	0	0
$253$	1.20340	0.0756572
$254$	0	0
$255$	20.0837	1.25769
$256$	0	0
$257$	−0.145081	−0.00904989	−0.00452495	−	0.999990i	$-0.501440\pi$
$257$	−0.145081	−0.00904989	−0.00452495	+	0.999990i	$0.501440\pi$
$258$	0	0
$259$	−8.97917	−0.557939
$260$	0	0
$261$	−8.60654	−0.532732
$262$	0	0
$263$	−14.8128	−0.913398	−0.456699	−	0.889621i	$-0.650968\pi$
$263$	−14.8128	−0.913398	−0.456699	+	0.889621i	$0.650968\pi$
$264$	0	0
$265$	31.9748	1.96419
$266$	0	0
$267$	4.53477	0.277524
$268$	0	0
$269$	10.8528	0.661707	0.330853	−	0.943682i	$-0.392663\pi$
$269$	10.8528	0.661707	0.330853	+	0.943682i	$0.392663\pi$
$270$	0	0
$271$	6.75431	0.410295	0.205148	−	0.978731i	$-0.434233\pi$
$271$	6.75431	0.410295	0.205148	+	0.978731i	$0.434233\pi$
$272$	0	0
$273$	−1.32542	−0.0802181
$274$	0	0
$275$	−2.30283	−0.138866
$276$	0	0
$277$	11.9752	0.719519	0.359759	−	0.933045i	$-0.382859\pi$
$277$	11.9752	0.719519	0.359759	+	0.933045i	$0.382859\pi$
$278$	0	0
$279$	3.59689	0.215340
$280$	0	0
$281$	7.22438	0.430971	0.215485	−	0.976507i	$-0.430867\pi$
$281$	7.22438	0.430971	0.215485	+	0.976507i	$0.430867\pi$
$282$	0	0
$283$	26.6738	1.58559	0.792796	−	0.609487i	$-0.208625\pi$
$283$	26.6738	1.58559	0.792796	+	0.609487i	$0.208625\pi$
$284$	0	0
$285$	−8.44437	−0.500201
$286$	0	0
$287$	−5.99010	−0.353584
$288$	0	0
$289$	14.4405	0.849444
$290$	0	0
$291$	−7.63276	−0.447440
$292$	0	0
$293$	7.14676	0.417518	0.208759	−	0.977967i	$-0.433058\pi$
$293$	7.14676	0.417518	0.208759	+	0.977967i	$0.433058\pi$
$294$	0	0
$295$	−5.84687	−0.340418
$296$	0	0
$297$	5.62410	0.326344
$298$	0	0
$299$	−1.20340	−0.0695945
$300$	0	0
$301$	−6.83208	−0.393795
$302$	0	0
$303$	−13.7242	−0.788436
$304$	0	0
$305$	−10.5985	−0.606866
$306$	0	0
$307$	2.58563	0.147570	0.0737848	−	0.997274i	$-0.476492\pi$
$307$	2.58563	0.147570	0.0737848	+	0.997274i	$0.476492\pi$
$308$	0	0
$309$	20.8726	1.18740
$310$	0	0
$311$	31.3742	1.77907	0.889534	−	0.456870i	$-0.151029\pi$
$311$	31.3742	1.77907	0.889534	+	0.456870i	$0.151029\pi$
$312$	0	0
$313$	3.24926	0.183659	0.0918295	−	0.995775i	$-0.470729\pi$
$313$	3.24926	0.183659	0.0918295	+	0.995775i	$0.470729\pi$
$314$	0	0
$315$	3.35975	0.189301
$316$	0	0
$317$	9.61325	0.539934	0.269967	−	0.962870i	$-0.412987\pi$
$317$	9.61325	0.539934	0.269967	+	0.962870i	$0.412987\pi$
$318$	0	0
$319$	−6.92256	−0.387589
$320$	0	0
$321$	−18.8435	−1.05174
$322$	0	0
$323$	−13.2194	−0.735549
$324$	0	0
$325$	2.30283	0.127738
$326$	0	0
$327$	−0.298922	−0.0165304
$328$	0	0
$329$	12.9171	0.712142
$330$	0	0
$331$	−7.67786	−0.422013	−0.211007	−	0.977485i	$-0.567674\pi$
$331$	−7.67786	−0.422013	−0.211007	+	0.977485i	$0.567674\pi$
$332$	0	0
$333$	−11.1634	−0.611753
$334$	0	0
$335$	33.1785	1.81273
$336$	0	0
$337$	25.5154	1.38991	0.694956	−	0.719052i	$-0.255424\pi$
$337$	25.5154	1.38991	0.694956	+	0.719052i	$0.255424\pi$
$338$	0	0
$339$	26.3155	1.42926
$340$	0	0
$341$	2.89311	0.156671
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−4.31032	−0.232060
$346$	0	0
$347$	3.95595	0.212367	0.106183	−	0.994347i	$-0.466137\pi$
$347$	3.95595	0.212367	0.106183	+	0.994347i	$0.466137\pi$
$348$	0	0
$349$	−21.7837	−1.16606	−0.583028	−	0.812452i	$-0.698132\pi$
$349$	−21.7837	−1.16606	−0.583028	+	0.812452i	$0.698132\pi$
$350$	0	0
$351$	−5.62410	−0.300192
$352$	0	0
$353$	−16.7081	−0.889282	−0.444641	−	0.895709i	$-0.646669\pi$
$353$	−16.7081	−0.889282	−0.444641	+	0.895709i	$0.646669\pi$
$354$	0	0
$355$	25.6925	1.36362
$356$	0	0
$357$	−7.43188	−0.393337
$358$	0	0
$359$	−5.56445	−0.293680	−0.146840	−	0.989160i	$-0.546910\pi$
$359$	−5.56445	−0.293680	−0.146840	+	0.989160i	$0.546910\pi$
$360$	0	0
$361$	−13.4418	−0.707462
$362$	0	0
$363$	1.32542	0.0695665
$364$	0	0
$365$	−12.9124	−0.675863
$366$	0	0
$367$	−21.1224	−1.10258	−0.551291	−	0.834313i	$-0.685864\pi$
$367$	−21.1224	−1.10258	−0.551291	+	0.834313i	$0.685864\pi$
$368$	0	0
$369$	−7.44725	−0.387688
$370$	0	0
$371$	−11.8321	−0.614292
$372$	0	0
$373$	−23.0982	−1.19598	−0.597989	−	0.801504i	$-0.704034\pi$
$373$	−23.0982	−1.19598	−0.597989	+	0.801504i	$0.704034\pi$
$374$	0	0
$375$	−9.66069	−0.498876
$376$	0	0
$377$	6.92256	0.356530
$378$	0	0
$379$	−4.39538	−0.225776	−0.112888	−	0.993608i	$-0.536010\pi$
$379$	−4.39538	−0.225776	−0.112888	+	0.993608i	$0.536010\pi$
$380$	0	0
$381$	7.30820	0.374410
$382$	0	0
$383$	−30.2836	−1.54742	−0.773711	−	0.633539i	$-0.781602\pi$
$383$	−30.2836	−1.54742	−0.773711	+	0.633539i	$0.781602\pi$
$384$	0	0
$385$	2.70237	0.137726
$386$	0	0
$387$	−8.49405	−0.431777
$388$	0	0
$389$	−13.0130	−0.659784	−0.329892	−	0.944019i	$-0.607012\pi$
$389$	−13.0130	−0.659784	−0.329892	+	0.944019i	$0.607012\pi$
$390$	0	0
$391$	−6.74770	−0.341246
$392$	0	0
$393$	14.3006	0.721372
$394$	0	0
$395$	−9.59801	−0.482928
$396$	0	0
$397$	26.2019	1.31503	0.657517	−	0.753440i	$-0.271607\pi$
$397$	26.2019	1.31503	0.657517	+	0.753440i	$0.271607\pi$
$398$	0	0
$399$	3.12480	0.156436
$400$	0	0
$401$	24.7328	1.23510	0.617548	−	0.786533i	$-0.288126\pi$
$401$	24.7328	1.23510	0.617548	+	0.786533i	$0.288126\pi$
$402$	0	0
$403$	−2.89311	−0.144116
$404$	0	0
$405$	−10.0651	−0.500137
$406$	0	0
$407$	−8.97917	−0.445081
$408$	0	0
$409$	−25.1556	−1.24387	−0.621933	−	0.783071i	$-0.713652\pi$
$409$	−25.1556	−1.24387	−0.621933	+	0.783071i	$0.713652\pi$
$410$	0	0
$411$	−8.69980	−0.429130
$412$	0	0
$413$	2.16360	0.106464
$414$	0	0
$415$	13.3761	0.656609
$416$	0	0
$417$	23.7399	1.16255
$418$	0	0
$419$	−28.9798	−1.41576	−0.707878	−	0.706335i	$-0.750347\pi$
$419$	−28.9798	−1.41576	−0.707878	+	0.706335i	$0.750347\pi$
$420$	0	0
$421$	3.33357	0.162468	0.0812342	−	0.996695i	$-0.474114\pi$
$421$	3.33357	0.162468	0.0812342	+	0.996695i	$0.474114\pi$
$422$	0	0
$423$	16.0593	0.780830
$424$	0	0
$425$	12.9124	0.626342
$426$	0	0
$427$	3.92191	0.189794
$428$	0	0
$429$	−1.32542	−0.0639919
$430$	0	0
$431$	2.98975	0.144011	0.0720056	−	0.997404i	$-0.477060\pi$
$431$	2.98975	0.144011	0.0720056	+	0.997404i	$0.477060\pi$
$432$	0	0
$433$	4.07261	0.195717	0.0978586	−	0.995200i	$-0.468801\pi$
$433$	4.07261	0.195717	0.0978586	+	0.995200i	$0.468801\pi$
$434$	0	0
$435$	24.7951	1.18883
$436$	0	0
$437$	2.83713	0.135718
$438$	0	0
$439$	−31.0857	−1.48364	−0.741819	−	0.670600i	$-0.766037\pi$
$439$	−31.0857	−1.48364	−0.741819	+	0.670600i	$0.766037\pi$
$440$	0	0
$441$	−1.24326	−0.0592029
$442$	0	0
$443$	19.4567	0.924418	0.462209	−	0.886771i	$-0.347057\pi$
$443$	19.4567	0.924418	0.462209	+	0.886771i	$0.347057\pi$
$444$	0	0
$445$	9.24586	0.438296
$446$	0	0
$447$	18.6383	0.881561
$448$	0	0
$449$	−27.0708	−1.27755	−0.638776	−	0.769393i	$-0.720559\pi$
$449$	−27.0708	−1.27755	−0.638776	+	0.769393i	$0.720559\pi$
$450$	0	0
$451$	−5.99010	−0.282063
$452$	0	0
$453$	−3.09981	−0.145642
$454$	0	0
$455$	−2.70237	−0.126689
$456$	0	0
$457$	−37.4597	−1.75229	−0.876146	−	0.482046i	$-0.839894\pi$
$457$	−37.4597	−1.75229	−0.876146	+	0.482046i	$0.839894\pi$
$458$	0	0
$459$	−31.5354	−1.47195
$460$	0	0
$461$	−29.2260	−1.36119	−0.680595	−	0.732660i	$-0.738278\pi$
$461$	−29.2260	−1.36119	−0.680595	+	0.732660i	$0.738278\pi$
$462$	0	0
$463$	−20.9534	−0.973788	−0.486894	−	0.873461i	$-0.661870\pi$
$463$	−20.9534	−0.973788	−0.486894	+	0.873461i	$0.661870\pi$
$464$	0	0
$465$	−10.3625	−0.480549
$466$	0	0
$467$	−22.2768	−1.03085	−0.515424	−	0.856935i	$-0.672365\pi$
$467$	−22.2768	−1.03085	−0.515424	+	0.856935i	$0.672365\pi$
$468$	0	0
$469$	−12.2775	−0.566923
$470$	0	0
$471$	4.86089	0.223978
$472$	0	0
$473$	−6.83208	−0.314139
$474$	0	0
$475$	−5.42912	−0.249105
$476$	0	0
$477$	−14.7104	−0.673542
$478$	0	0
$479$	−35.4922	−1.62168	−0.810840	−	0.585267i	$-0.800990\pi$
$479$	−35.4922	−1.62168	−0.810840	+	0.585267i	$0.800990\pi$
$480$	0	0
$481$	8.97917	0.409415
$482$	0	0
$483$	1.59501	0.0725756
$484$	0	0
$485$	−15.5623	−0.706647
$486$	0	0
$487$	13.4698	0.610376	0.305188	−	0.952292i	$-0.401281\pi$
$487$	13.4698	0.610376	0.305188	+	0.952292i	$0.401281\pi$
$488$	0	0
$489$	20.0503	0.906704
$490$	0	0
$491$	1.56770	0.0707493	0.0353747	−	0.999374i	$-0.488738\pi$
$491$	1.56770	0.0707493	0.0353747	+	0.999374i	$0.488738\pi$
$492$	0	0
$493$	38.8161	1.74819
$494$	0	0
$495$	3.35975	0.151010
$496$	0	0
$497$	−9.50738	−0.426464
$498$	0	0
$499$	30.3171	1.35718	0.678590	−	0.734517i	$-0.262591\pi$
$499$	30.3171	1.35718	0.678590	+	0.734517i	$0.262591\pi$
$500$	0	0
$501$	−24.4234	−1.09116
$502$	0	0
$503$	−18.1028	−0.807166	−0.403583	−	0.914943i	$-0.632235\pi$
$503$	−18.1028	−0.807166	−0.403583	+	0.914943i	$0.632235\pi$
$504$	0	0
$505$	−27.9821	−1.24519
$506$	0	0
$507$	1.32542	0.0588640
$508$	0	0
$509$	20.0062	0.886759	0.443380	−	0.896334i	$-0.353779\pi$
$509$	20.0062	0.886759	0.443380	+	0.896334i	$0.353779\pi$
$510$	0	0
$511$	4.77815	0.211373
$512$	0	0
$513$	13.2593	0.585414
$514$	0	0
$515$	42.5567	1.87527
$516$	0	0
$517$	12.9171	0.568093
$518$	0	0
$519$	−11.0648	−0.485692
$520$	0	0
$521$	−20.8501	−0.913460	−0.456730	−	0.889605i	$-0.650979\pi$
$521$	−20.8501	−0.913460	−0.456730	+	0.889605i	$0.650979\pi$
$522$	0	0
$523$	35.0344	1.53195	0.765974	−	0.642872i	$-0.222257\pi$
$523$	35.0344	1.53195	0.765974	+	0.642872i	$0.222257\pi$
$524$	0	0
$525$	−3.05221	−0.133210
$526$	0	0
$527$	−16.2222	−0.706650
$528$	0	0
$529$	−21.5518	−0.937036
$530$	0	0
$531$	2.68992	0.116733
$532$	0	0
$533$	5.99010	0.259460
$534$	0	0
$535$	−38.4197	−1.66103
$536$	0	0
$537$	4.87306	0.210288
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	34.3381	1.47631	0.738154	−	0.674632i	$-0.235698\pi$
$541$	34.3381	1.47631	0.738154	+	0.674632i	$0.235698\pi$
$542$	0	0
$543$	9.02688	0.387380
$544$	0	0
$545$	−0.609465	−0.0261066
$546$	0	0
$547$	−1.05659	−0.0451765	−0.0225883	−	0.999745i	$-0.507191\pi$
$547$	−1.05659	−0.0451765	−0.0225883	+	0.999745i	$0.507191\pi$
$548$	0	0
$549$	4.87595	0.208100
$550$	0	0
$551$	−16.3206	−0.695279
$552$	0	0
$553$	3.55169	0.151033
$554$	0	0
$555$	32.1615	1.36518
$556$	0	0
$557$	27.7910	1.17754	0.588771	−	0.808300i	$-0.299612\pi$
$557$	27.7910	1.17754	0.588771	+	0.808300i	$0.299612\pi$
$558$	0	0
$559$	6.83208	0.288966
$560$	0	0
$561$	−7.43188	−0.313774
$562$	0	0
$563$	−32.0379	−1.35024	−0.675119	−	0.737709i	$-0.735908\pi$
$563$	−32.0379	−1.35024	−0.675119	+	0.737709i	$0.735908\pi$
$564$	0	0
$565$	53.6541	2.25725
$566$	0	0
$567$	3.72453	0.156415
$568$	0	0
$569$	−13.5461	−0.567881	−0.283940	−	0.958842i	$-0.591642\pi$
$569$	−13.5461	−0.567881	−0.283940	+	0.958842i	$0.591642\pi$
$570$	0	0
$571$	−13.4962	−0.564799	−0.282400	−	0.959297i	$-0.591130\pi$
$571$	−13.4962	−0.564799	−0.282400	+	0.959297i	$0.591130\pi$
$572$	0	0
$573$	−8.06217	−0.336802
$574$	0	0
$575$	−2.77123	−0.115568
$576$	0	0
$577$	−11.0256	−0.459002	−0.229501	−	0.973308i	$-0.573709\pi$
$577$	−11.0256	−0.459002	−0.229501	+	0.973308i	$0.573709\pi$
$578$	0	0
$579$	−24.7198	−1.02732
$580$	0	0
$581$	−4.94977	−0.205351
$582$	0	0
$583$	−11.8321	−0.490036
$584$	0	0
$585$	−3.35975	−0.138909
$586$	0	0
$587$	8.16825	0.337140	0.168570	−	0.985690i	$-0.446085\pi$
$587$	8.16825	0.337140	0.168570	+	0.985690i	$0.446085\pi$
$588$	0	0
$589$	6.82076	0.281045
$590$	0	0
$591$	22.2277	0.914326
$592$	0	0
$593$	−18.6962	−0.767760	−0.383880	−	0.923383i	$-0.625412\pi$
$593$	−18.6962	−0.767760	−0.383880	+	0.923383i	$0.625412\pi$
$594$	0	0
$595$	−15.1527	−0.621201
$596$	0	0
$597$	18.1012	0.740835
$598$	0	0
$599$	25.5269	1.04300	0.521499	−	0.853252i	$-0.325373\pi$
$599$	25.5269	1.04300	0.521499	+	0.853252i	$0.325373\pi$
$600$	0	0
$601$	−47.4603	−1.93594	−0.967972	−	0.251057i	$-0.919222\pi$
$601$	−47.4603	−1.93594	−0.967972	+	0.251057i	$0.919222\pi$
$602$	0	0
$603$	−15.2641	−0.621604
$604$	0	0
$605$	2.70237	0.109867
$606$	0	0
$607$	−47.6345	−1.93343	−0.966713	−	0.255863i	$-0.917640\pi$
$607$	−47.6345	−1.93343	−0.966713	+	0.255863i	$0.917640\pi$
$608$	0	0
$609$	−9.17531	−0.371802
$610$	0	0
$611$	−12.9171	−0.522570
$612$	0	0
$613$	−8.84786	−0.357362	−0.178681	−	0.983907i	$-0.557183\pi$
$613$	−8.84786	−0.357362	−0.178681	+	0.983907i	$0.557183\pi$
$614$	0	0
$615$	21.4552	0.865159
$616$	0	0
$617$	−33.1179	−1.33328	−0.666638	−	0.745381i	$-0.732267\pi$
$617$	−33.1179	−1.33328	−0.666638	+	0.745381i	$0.732267\pi$
$618$	0	0
$619$	40.5372	1.62933	0.814664	−	0.579934i	$-0.196922\pi$
$619$	40.5372	1.62933	0.814664	+	0.579934i	$0.196922\pi$
$620$	0	0
$621$	6.76806	0.271593
$622$	0	0
$623$	−3.42138	−0.137075
$624$	0	0
$625$	−31.2111	−1.24844
$626$	0	0
$627$	3.12480	0.124792
$628$	0	0
$629$	50.3479	2.00750
$630$	0	0
$631$	32.2390	1.28341	0.641707	−	0.766950i	$-0.278227\pi$
$631$	32.2390	1.28341	0.641707	+	0.766950i	$0.278227\pi$
$632$	0	0
$633$	−12.6773	−0.503879
$634$	0	0
$635$	14.9005	0.591310
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−11.8201	−0.467598
$640$	0	0
$641$	−0.717358	−0.0283339	−0.0141670	−	0.999900i	$-0.504510\pi$
$641$	−0.717358	−0.0283339	−0.0141670	+	0.999900i	$0.504510\pi$
$642$	0	0
$643$	6.32925	0.249601	0.124801	−	0.992182i	$-0.460171\pi$
$643$	6.32925	0.249601	0.124801	+	0.992182i	$0.460171\pi$
$644$	0	0
$645$	24.4710	0.963546
$646$	0	0
$647$	−15.4842	−0.608746	−0.304373	−	0.952553i	$-0.598447\pi$
$647$	−15.4842	−0.608746	−0.304373	+	0.952553i	$0.598447\pi$
$648$	0	0
$649$	2.16360	0.0849289
$650$	0	0
$651$	3.83459	0.150289
$652$	0	0
$653$	33.7697	1.32151	0.660755	−	0.750602i	$-0.270236\pi$
$653$	33.7697	1.32151	0.660755	+	0.750602i	$0.270236\pi$
$654$	0	0
$655$	29.1573	1.13927
$656$	0	0
$657$	5.94048	0.231760
$658$	0	0
$659$	−2.18084	−0.0849535	−0.0424768	−	0.999097i	$-0.513525\pi$
$659$	−2.18084	−0.0849535	−0.0424768	+	0.999097i	$0.513525\pi$
$660$	0	0
$661$	34.0274	1.32351	0.661756	−	0.749719i	$-0.269811\pi$
$661$	34.0274	1.32351	0.661756	+	0.749719i	$0.269811\pi$
$662$	0	0
$663$	7.43188	0.288630
$664$	0	0
$665$	6.37109	0.247060
$666$	0	0
$667$	−8.33062	−0.322563
$668$	0	0
$669$	−8.82471	−0.341183
$670$	0	0
$671$	3.92191	0.151404
$672$	0	0
$673$	6.86399	0.264587	0.132294	−	0.991211i	$-0.457766\pi$
$673$	6.86399	0.264587	0.132294	+	0.991211i	$0.457766\pi$
$674$	0	0
$675$	−12.9513	−0.498497
$676$	0	0
$677$	−29.0443	−1.11626	−0.558132	−	0.829752i	$-0.688482\pi$
$677$	−29.0443	−1.11626	−0.558132	+	0.829752i	$0.688482\pi$
$678$	0	0
$679$	5.75875	0.221000
$680$	0	0
$681$	−11.4475	−0.438668
$682$	0	0
$683$	0.385273	0.0147421	0.00737104	−	0.999973i	$-0.497654\pi$
$683$	0.385273	0.0147421	0.00737104	+	0.999973i	$0.497654\pi$
$684$	0	0
$685$	−17.7379	−0.677729
$686$	0	0
$687$	−22.3513	−0.852757
$688$	0	0
$689$	11.8321	0.450767
$690$	0	0
$691$	24.8777	0.946394	0.473197	−	0.880957i	$-0.343100\pi$
$691$	24.8777	0.946394	0.473197	+	0.880957i	$0.343100\pi$
$692$	0	0
$693$	−1.24326	−0.0472275
$694$	0	0
$695$	48.4028	1.83602
$696$	0	0
$697$	33.5876	1.27222
$698$	0	0
$699$	−28.3619	−1.07274
$700$	0	0
$701$	18.0746	0.682668	0.341334	−	0.939942i	$-0.389121\pi$
$701$	18.0746	0.682668	0.341334	+	0.939942i	$0.389121\pi$
$702$	0	0
$703$	−21.1692	−0.798412
$704$	0	0
$705$	−46.2662	−1.74249
$706$	0	0
$707$	10.3546	0.389426
$708$	0	0
$709$	19.5050	0.732527	0.366263	−	0.930511i	$-0.380637\pi$
$709$	19.5050	0.732527	0.366263	+	0.930511i	$0.380637\pi$
$710$	0	0
$711$	4.41568	0.165601
$712$	0	0
$713$	3.48157	0.130386
$714$	0	0
$715$	−2.70237	−0.101063
$716$	0	0
$717$	19.8374	0.740842
$718$	0	0
$719$	30.9980	1.15603	0.578015	−	0.816026i	$-0.303827\pi$
$719$	30.9980	1.15603	0.578015	+	0.816026i	$0.303827\pi$
$720$	0	0
$721$	−15.7479	−0.586483
$722$	0	0
$723$	30.3218	1.12768
$724$	0	0
$725$	15.9415	0.592051
$726$	0	0
$727$	39.6107	1.46908	0.734540	−	0.678565i	$-0.237398\pi$
$727$	39.6107	1.46908	0.734540	+	0.678565i	$0.237398\pi$
$728$	0	0
$729$	26.9935	0.999758
$730$	0	0
$731$	38.3088	1.41690
$732$	0	0
$733$	−14.4651	−0.534282	−0.267141	−	0.963657i	$-0.586079\pi$
$733$	−14.4651	−0.534282	−0.267141	+	0.963657i	$0.586079\pi$
$734$	0	0
$735$	3.58178	0.132116
$736$	0	0
$737$	−12.2775	−0.452248
$738$	0	0
$739$	−45.0666	−1.65780	−0.828900	−	0.559396i	$-0.811033\pi$
$739$	−45.0666	−1.65780	−0.828900	+	0.559396i	$0.811033\pi$
$740$	0	0
$741$	−3.12480	−0.114792
$742$	0	0
$743$	27.0388	0.991958	0.495979	−	0.868334i	$-0.334809\pi$
$743$	27.0388	0.991958	0.495979	+	0.868334i	$0.334809\pi$
$744$	0	0
$745$	38.0013	1.39226
$746$	0	0
$747$	−6.15385	−0.225158
$748$	0	0
$749$	14.2170	0.519478
$750$	0	0
$751$	−11.4938	−0.419414	−0.209707	−	0.977764i	$-0.567251\pi$
$751$	−11.4938	−0.419414	−0.209707	+	0.977764i	$0.567251\pi$
$752$	0	0
$753$	−4.13184	−0.150572
$754$	0	0
$755$	−6.32014	−0.230014
$756$	0	0
$757$	12.4011	0.450727	0.225364	−	0.974275i	$-0.427643\pi$
$757$	12.4011	0.450727	0.225364	+	0.974275i	$0.427643\pi$
$758$	0	0
$759$	1.59501	0.0578953
$760$	0	0
$761$	23.2534	0.842933	0.421467	−	0.906844i	$-0.361515\pi$
$761$	23.2534	0.842933	0.421467	+	0.906844i	$0.361515\pi$
$762$	0	0
$763$	0.225530	0.00816472
$764$	0	0
$765$	−18.8388	−0.681117
$766$	0	0
$767$	−2.16360	−0.0781232
$768$	0	0
$769$	6.77407	0.244279	0.122140	−	0.992513i	$-0.461024\pi$
$769$	6.77407	0.244279	0.122140	+	0.992513i	$0.461024\pi$
$770$	0	0
$771$	−0.192293	−0.00692527
$772$	0	0
$773$	26.4827	0.952515	0.476257	−	0.879306i	$-0.341993\pi$
$773$	26.4827	0.952515	0.476257	+	0.879306i	$0.341993\pi$
$774$	0	0
$775$	−6.66233	−0.239318
$776$	0	0
$777$	−11.9012	−0.426953
$778$	0	0
$779$	−14.1222	−0.505980
$780$	0	0
$781$	−9.50738	−0.340201
$782$	0	0
$783$	−38.9332	−1.39136
$784$	0	0
$785$	9.91077	0.353731
$786$	0	0
$787$	−28.5279	−1.01691	−0.508456	−	0.861088i	$-0.669783\pi$
$787$	−28.5279	−1.01691	−0.508456	+	0.861088i	$0.669783\pi$
$788$	0	0
$789$	−19.6332	−0.698962
$790$	0	0
$791$	−19.8544	−0.705942
$792$	0	0
$793$	−3.92191	−0.139271
$794$	0	0
$795$	42.3800	1.50306
$796$	0	0
$797$	−34.2165	−1.21201	−0.606006	−	0.795460i	$-0.707229\pi$
$797$	−34.2165	−1.21201	−0.606006	+	0.795460i	$0.707229\pi$
$798$	0	0
$799$	−72.4285	−2.56234
$800$	0	0
$801$	−4.25367	−0.150296
$802$	0	0
$803$	4.77815	0.168617
$804$	0	0
$805$	3.25204	0.114619
$806$	0	0
$807$	14.3845	0.506359
$808$	0	0
$809$	−0.107374	−0.00377507	−0.00188753	−	0.999998i	$-0.500601\pi$
$809$	−0.107374	−0.00377507	−0.00188753	+	0.999998i	$0.500601\pi$
$810$	0	0
$811$	−7.28175	−0.255697	−0.127848	−	0.991794i	$-0.540807\pi$
$811$	−7.28175	−0.255697	−0.127848	+	0.991794i	$0.540807\pi$
$812$	0	0
$813$	8.95231	0.313971
$814$	0	0
$815$	40.8801	1.43197
$816$	0	0
$817$	−16.1072	−0.563521
$818$	0	0
$819$	1.24326	0.0434430
$820$	0	0
$821$	1.33806	0.0466987	0.0233494	−	0.999727i	$-0.492567\pi$
$821$	1.33806	0.0466987	0.0233494	+	0.999727i	$0.492567\pi$
$822$	0	0
$823$	15.1529	0.528199	0.264099	−	0.964496i	$-0.414925\pi$
$823$	15.1529	0.528199	0.264099	+	0.964496i	$0.414925\pi$
$824$	0	0
$825$	−3.05221	−0.106264
$826$	0	0
$827$	40.1650	1.39667	0.698337	−	0.715769i	$-0.253924\pi$
$827$	40.1650	1.39667	0.698337	+	0.715769i	$0.253924\pi$
$828$	0	0
$829$	55.8976	1.94140	0.970701	−	0.240290i	$-0.0772425\pi$
$829$	55.8976	1.94140	0.970701	+	0.240290i	$0.0772425\pi$
$830$	0	0
$831$	15.8722	0.550599
$832$	0	0
$833$	5.60719	0.194277
$834$	0	0
$835$	−49.7964	−1.72327
$836$	0	0
$837$	16.2712	0.562413
$838$	0	0
$839$	21.2041	0.732048	0.366024	−	0.930605i	$-0.380719\pi$
$839$	21.2041	0.732048	0.366024	+	0.930605i	$0.380719\pi$
$840$	0	0
$841$	18.9218	0.652477
$842$	0	0
$843$	9.57535	0.329792
$844$	0	0
$845$	2.70237	0.0929645
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	35.3540	1.21335
$850$	0	0
$851$	−10.8056	−0.370410
$852$	0	0
$853$	−54.8975	−1.87965	−0.939827	−	0.341650i	$-0.889014\pi$
$853$	−54.8975	−1.87965	−0.939827	+	0.341650i	$0.889014\pi$
$854$	0	0
$855$	7.92092	0.270890
$856$	0	0
$857$	25.1263	0.858298	0.429149	−	0.903234i	$-0.358814\pi$
$857$	25.1263	0.858298	0.429149	+	0.903234i	$0.358814\pi$
$858$	0	0
$859$	−15.4455	−0.526993	−0.263496	−	0.964660i	$-0.584876\pi$
$859$	−15.4455	−0.526993	−0.263496	+	0.964660i	$0.584876\pi$
$860$	0	0
$861$	−7.93940	−0.270574
$862$	0	0
$863$	−39.1868	−1.33393	−0.666967	−	0.745087i	$-0.732408\pi$
$863$	−39.1868	−1.33393	−0.666967	+	0.745087i	$0.732408\pi$
$864$	0	0
$865$	−22.5598	−0.767058
$866$	0	0
$867$	19.1398	0.650021
$868$	0	0
$869$	3.55169	0.120483
$870$	0	0
$871$	12.2775	0.416008
$872$	0	0
$873$	7.15962	0.242316
$874$	0	0
$875$	7.28877	0.246405
$876$	0	0
$877$	0.927049	0.0313042	0.0156521	−	0.999877i	$-0.495018\pi$
$877$	0.927049	0.0313042	0.0156521	+	0.999877i	$0.495018\pi$
$878$	0	0
$879$	9.47246	0.319498
$880$	0	0
$881$	−32.2018	−1.08490	−0.542452	−	0.840087i	$-0.682504\pi$
$881$	−32.2018	−1.08490	−0.542452	+	0.840087i	$0.682504\pi$
$882$	0	0
$883$	−2.31487	−0.0779015	−0.0389508	−	0.999241i	$-0.512402\pi$
$883$	−2.31487	−0.0779015	−0.0389508	+	0.999241i	$0.512402\pi$
$884$	0	0
$885$	−7.74956	−0.260499
$886$	0	0
$887$	−37.0492	−1.24399	−0.621995	−	0.783021i	$-0.713678\pi$
$887$	−37.0492	−1.24399	−0.621995	+	0.783021i	$0.713678\pi$
$888$	0	0
$889$	−5.51387	−0.184929
$890$	0	0
$891$	3.72453	0.124776
$892$	0	0
$893$	30.4532	1.01908
$894$	0	0
$895$	9.93558	0.332110
$896$	0	0
$897$	−1.59501	−0.0532560
$898$	0	0
$899$	−20.0277	−0.667962
$900$	0	0
$901$	66.3448	2.21027
$902$	0	0
$903$	−9.05538	−0.301344
$904$	0	0
$905$	18.4047	0.611794
$906$	0	0
$907$	−6.87765	−0.228369	−0.114184	−	0.993460i	$-0.536425\pi$
$907$	−6.87765	−0.228369	−0.114184	+	0.993460i	$0.536425\pi$
$908$	0	0
$909$	12.8735	0.426987
$910$	0	0
$911$	0.727168	0.0240921	0.0120461	−	0.999927i	$-0.496166\pi$
$911$	0.727168	0.0240921	0.0120461	+	0.999927i	$0.496166\pi$
$912$	0	0
$913$	−4.94977	−0.163814
$914$	0	0
$915$	−14.0474	−0.464393
$916$	0	0
$917$	−10.7895	−0.356301
$918$	0	0
$919$	−2.51997	−0.0831260	−0.0415630	−	0.999136i	$-0.513234\pi$
$919$	−2.51997	−0.0831260	−0.0415630	+	0.999136i	$0.513234\pi$
$920$	0	0
$921$	3.42704	0.112925
$922$	0	0
$923$	9.50738	0.312939
$924$	0	0
$925$	20.6775	0.679871
$926$	0	0
$927$	−19.5787	−0.643050
$928$	0	0
$929$	59.1329	1.94009	0.970043	−	0.242933i	$-0.0781096\pi$
$929$	59.1329	1.94009	0.970043	+	0.242933i	$0.0781096\pi$
$930$	0	0
$931$	−2.35759	−0.0772669
$932$	0	0
$933$	41.5840	1.36140
$934$	0	0
$935$	−15.1527	−0.495547
$936$	0	0
$937$	35.2086	1.15022	0.575108	−	0.818077i	$-0.304960\pi$
$937$	35.2086	1.15022	0.575108	+	0.818077i	$0.304960\pi$
$938$	0	0
$939$	4.30664	0.140542
$940$	0	0
$941$	−29.9298	−0.975685	−0.487842	−	0.872932i	$-0.662216\pi$
$941$	−29.9298	−0.975685	−0.487842	+	0.872932i	$0.662216\pi$
$942$	0	0
$943$	−7.20850	−0.234741
$944$	0	0
$945$	15.1984	0.494405
$946$	0	0
$947$	−36.9372	−1.20030	−0.600148	−	0.799889i	$-0.704892\pi$
$947$	−36.9372	−1.20030	−0.600148	+	0.799889i	$0.704892\pi$
$948$	0	0
$949$	−4.77815	−0.155105
$950$	0	0
$951$	12.7416	0.413175
$952$	0	0
$953$	59.7233	1.93463	0.967314	−	0.253582i	$-0.0816088\pi$
$953$	59.7233	1.93463	0.967314	+	0.253582i	$0.0816088\pi$
$954$	0	0
$955$	−16.4378	−0.531914
$956$	0	0
$957$	−9.17531	−0.296595
$958$	0	0
$959$	6.56380	0.211956
$960$	0	0
$961$	−22.6299	−0.729997
$962$	0	0
$963$	17.6754	0.569583
$964$	0	0
$965$	−50.4007	−1.62246
$966$	0	0
$967$	51.6404	1.66064	0.830321	−	0.557285i	$-0.188157\pi$
$967$	51.6404	1.66064	0.830321	+	0.557285i	$0.188157\pi$
$968$	0	0
$969$	−17.5213	−0.562866
$970$	0	0
$971$	35.8018	1.14894	0.574468	−	0.818527i	$-0.305209\pi$
$971$	35.8018	1.14894	0.574468	+	0.818527i	$0.305209\pi$
$972$	0	0
$973$	−17.9112	−0.574207
$974$	0	0
$975$	3.05221	0.0977491
$976$	0	0
$977$	−48.2248	−1.54285	−0.771424	−	0.636322i	$-0.780455\pi$
$977$	−48.2248	−1.54285	−0.771424	+	0.636322i	$0.780455\pi$
$978$	0	0
$979$	−3.42138	−0.109348
$980$	0	0
$981$	0.280392	0.00895223
$982$	0	0
$983$	−12.8222	−0.408965	−0.204482	−	0.978870i	$-0.565551\pi$
$983$	−12.8222	−0.408965	−0.204482	+	0.978870i	$0.565551\pi$
$984$	0	0
$985$	45.3196	1.44400
$986$	0	0
$987$	17.1206	0.544954
$988$	0	0
$989$	−8.22174	−0.261436
$990$	0	0
$991$	−35.0345	−1.11291	−0.556454	−	0.830879i	$-0.687838\pi$
$991$	−35.0345	−1.11291	−0.556454	+	0.830879i	$0.687838\pi$
$992$	0	0
$993$	−10.1764	−0.322938
$994$	0	0
$995$	36.9063	1.17001
$996$	0	0
$997$	35.3937	1.12093	0.560465	−	0.828178i	$-0.310622\pi$
$997$	35.3937	1.12093	0.560465	+	0.828178i	$0.310622\pi$
$998$	0	0
$999$	−50.4998	−1.59774

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.10	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.10	✓	15	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)

\(f(q)\)	\(=\)	\(q+1.32542 q^{3} +2.70237 q^{5} -1.00000 q^{7} -1.24326 q^{9} +O(q^{10})\)	\(q+1.32542 q^{3} +2.70237 q^{5} -1.00000 q^{7} -1.24326 q^{9} -1.00000 q^{11} +1.00000 q^{13} +3.58178 q^{15} +5.60719 q^{17} -2.35759 q^{19} -1.32542 q^{21} -1.20340 q^{23} +2.30283 q^{25} -5.62410 q^{27} +6.92256 q^{29} -2.89311 q^{31} -1.32542 q^{33} -2.70237 q^{35} +8.97917 q^{37} +1.32542 q^{39} +5.99010 q^{41} +6.83208 q^{43} -3.35975 q^{45} -12.9171 q^{47} +1.00000 q^{49} +7.43188 q^{51} +11.8321 q^{53} -2.70237 q^{55} -3.12480 q^{57} -2.16360 q^{59} -3.92191 q^{61} +1.24326 q^{63} +2.70237 q^{65} +12.2775 q^{67} -1.59501 q^{69} +9.50738 q^{71} -4.77815 q^{73} +3.05221 q^{75} +1.00000 q^{77} -3.55169 q^{79} -3.72453 q^{81} +4.94977 q^{83} +15.1527 q^{85} +9.17531 q^{87} +3.42138 q^{89} -1.00000 q^{91} -3.83459 q^{93} -6.37109 q^{95} -5.75875 q^{97} +1.24326 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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