LMFDB - Embedded newform 4016.2.a.k.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.82015$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.95607 q^{3} +2.29008 q^{5} +2.82675 q^{7} +5.73835 q^{9} +O(q^{10})$	$q+2.95607 q^{3} +2.29008 q^{5} +2.82675 q^{7} +5.73835 q^{9} +0.493604 q^{11} -1.57709 q^{13} +6.76963 q^{15} -3.39908 q^{17} +1.17083 q^{19} +8.35608 q^{21} -0.0257713 q^{23} +0.244452 q^{25} +8.09475 q^{27} -4.46947 q^{29} +6.43033 q^{31} +1.45913 q^{33} +6.47348 q^{35} -5.77787 q^{37} -4.66198 q^{39} +5.69569 q^{41} +7.57449 q^{43} +13.1413 q^{45} -3.16060 q^{47} +0.990530 q^{49} -10.0479 q^{51} +5.63900 q^{53} +1.13039 q^{55} +3.46106 q^{57} -7.51980 q^{59} +14.7093 q^{61} +16.2209 q^{63} -3.61165 q^{65} -4.39704 q^{67} -0.0761816 q^{69} +1.95847 q^{71} +0.678369 q^{73} +0.722619 q^{75} +1.39530 q^{77} -9.30630 q^{79} +6.71359 q^{81} -2.38418 q^{83} -7.78416 q^{85} -13.2121 q^{87} -0.754972 q^{89} -4.45804 q^{91} +19.0085 q^{93} +2.68129 q^{95} +0.821691 q^{97} +2.83247 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	2.95607	1.70669	0.853344	−	0.521349i	$-0.174571\pi$
$3$	2.95607	1.70669	0.853344	+	0.521349i	$0.174571\pi$
$4$	0	0
$5$	2.29008	1.02415	0.512077	−	0.858940i	$-0.328876\pi$
$5$	2.29008	1.02415	0.512077	+	0.858940i	$0.328876\pi$
$6$	0	0
$7$	2.82675	1.06841	0.534206	−	0.845354i	$-0.320611\pi$
$7$	2.82675	1.06841	0.534206	+	0.845354i	$0.320611\pi$
$8$	0	0
$9$	5.73835	1.91278
$10$	0	0
$11$	0.493604	0.148827	0.0744136	−	0.997227i	$-0.476291\pi$
$11$	0.493604	0.148827	0.0744136	+	0.997227i	$0.476291\pi$
$12$	0	0
$13$	−1.57709	−0.437406	−0.218703	−	0.975792i	$-0.570183\pi$
$13$	−1.57709	−0.437406	−0.218703	+	0.975792i	$0.570183\pi$
$14$	0	0
$15$	6.76963	1.74791
$16$	0	0
$17$	−3.39908	−0.824399	−0.412199	−	0.911094i	$-0.635239\pi$
$17$	−3.39908	−0.824399	−0.412199	+	0.911094i	$0.635239\pi$
$18$	0	0
$19$	1.17083	0.268607	0.134304	−	0.990940i	$-0.457120\pi$
$19$	1.17083	0.268607	0.134304	+	0.990940i	$0.457120\pi$
$20$	0	0
$21$	8.35608	1.82345
$22$	0	0
$23$	−0.0257713	−0.00537368	−0.00268684	−	0.999996i	$-0.500855\pi$
$23$	−0.0257713	−0.00537368	−0.00268684	+	0.999996i	$0.500855\pi$
$24$	0	0
$25$	0.244452	0.0488905
$26$	0	0
$27$	8.09475	1.55783
$28$	0	0
$29$	−4.46947	−0.829960	−0.414980	−	0.909831i	$-0.636211\pi$
$29$	−4.46947	−0.829960	−0.414980	+	0.909831i	$0.636211\pi$
$30$	0	0
$31$	6.43033	1.15492	0.577460	−	0.816419i	$-0.304044\pi$
$31$	6.43033	1.15492	0.577460	+	0.816419i	$0.304044\pi$
$32$	0	0
$33$	1.45913	0.254002
$34$	0	0
$35$	6.47348	1.09422
$36$	0	0
$37$	−5.77787	−0.949875	−0.474938	−	0.880019i	$-0.657529\pi$
$37$	−5.77787	−0.949875	−0.474938	+	0.880019i	$0.657529\pi$
$38$	0	0
$39$	−4.66198	−0.746515
$40$	0	0
$41$	5.69569	0.889517	0.444759	−	0.895650i	$-0.353289\pi$
$41$	5.69569	0.889517	0.444759	+	0.895650i	$0.353289\pi$
$42$	0	0
$43$	7.57449	1.15510	0.577549	−	0.816356i	$-0.304009\pi$
$43$	7.57449	1.15510	0.577549	+	0.816356i	$0.304009\pi$
$44$	0	0
$45$	13.1413	1.95898
$46$	0	0
$47$	−3.16060	−0.461021	−0.230510	−	0.973070i	$-0.574040\pi$
$47$	−3.16060	−0.461021	−0.230510	+	0.973070i	$0.574040\pi$
$48$	0	0
$49$	0.990530	0.141504
$50$	0	0
$51$	−10.0479	−1.40699
$52$	0	0
$53$	5.63900	0.774576	0.387288	−	0.921959i	$-0.373412\pi$
$53$	5.63900	0.774576	0.387288	+	0.921959i	$0.373412\pi$
$54$	0	0
$55$	1.13039	0.152422
$56$	0	0
$57$	3.46106	0.458428
$58$	0	0
$59$	−7.51980	−0.978995	−0.489498	−	0.872005i	$-0.662820\pi$
$59$	−7.51980	−0.978995	−0.489498	+	0.872005i	$0.662820\pi$
$60$	0	0
$61$	14.7093	1.88333	0.941664	−	0.336555i	$-0.109262\pi$
$61$	14.7093	1.88333	0.941664	+	0.336555i	$0.109262\pi$
$62$	0	0
$63$	16.2209	2.04364
$64$	0	0
$65$	−3.61165	−0.447971
$66$	0	0
$67$	−4.39704	−0.537184	−0.268592	−	0.963254i	$-0.586558\pi$
$67$	−4.39704	−0.537184	−0.268592	+	0.963254i	$0.586558\pi$
$68$	0	0
$69$	−0.0761816	−0.00917119
$70$	0	0
$71$	1.95847	0.232427	0.116214	−	0.993224i	$-0.462924\pi$
$71$	1.95847	0.232427	0.116214	+	0.993224i	$0.462924\pi$
$72$	0	0
$73$	0.678369	0.0793971	0.0396985	−	0.999212i	$-0.487360\pi$
$73$	0.678369	0.0793971	0.0396985	+	0.999212i	$0.487360\pi$
$74$	0	0
$75$	0.722619	0.0834408
$76$	0	0
$77$	1.39530	0.159009
$78$	0	0
$79$	−9.30630	−1.04704	−0.523520	−	0.852013i	$-0.675382\pi$
$79$	−9.30630	−1.04704	−0.523520	+	0.852013i	$0.675382\pi$
$80$	0	0
$81$	6.71359	0.745954
$82$	0	0
$83$	−2.38418	−0.261698	−0.130849	−	0.991402i	$-0.541770\pi$
$83$	−2.38418	−0.261698	−0.130849	+	0.991402i	$0.541770\pi$
$84$	0	0
$85$	−7.78416	−0.844311
$86$	0	0
$87$	−13.2121	−1.41648
$88$	0	0
$89$	−0.754972	−0.0800268	−0.0400134	−	0.999199i	$-0.512740\pi$
$89$	−0.754972	−0.0800268	−0.0400134	+	0.999199i	$0.512740\pi$
$90$	0	0
$91$	−4.45804	−0.467329
$92$	0	0
$93$	19.0085	1.97109
$94$	0	0
$95$	2.68129	0.275095
$96$	0	0
$97$	0.821691	0.0834301	0.0417150	−	0.999130i	$-0.486718\pi$
$97$	0.821691	0.0834301	0.0417150	+	0.999130i	$0.486718\pi$
$98$	0	0
$99$	2.83247	0.284674
$100$	0	0
$101$	7.38296	0.734632	0.367316	−	0.930096i	$-0.380277\pi$
$101$	7.38296	0.734632	0.367316	+	0.930096i	$0.380277\pi$
$102$	0	0
$103$	10.2766	1.01259	0.506293	−	0.862362i	$-0.331015\pi$
$103$	10.2766	1.01259	0.506293	+	0.862362i	$0.331015\pi$
$104$	0	0
$105$	19.1361	1.86749
$106$	0	0
$107$	1.71593	0.165885	0.0829425	−	0.996554i	$-0.473568\pi$
$107$	1.71593	0.165885	0.0829425	+	0.996554i	$0.473568\pi$
$108$	0	0
$109$	9.55184	0.914901	0.457450	−	0.889235i	$-0.348763\pi$
$109$	9.55184	0.914901	0.457450	+	0.889235i	$0.348763\pi$
$110$	0	0
$111$	−17.0798	−1.62114
$112$	0	0
$113$	−15.7921	−1.48559	−0.742797	−	0.669517i	$-0.766501\pi$
$113$	−15.7921	−1.48559	−0.742797	+	0.669517i	$0.766501\pi$
$114$	0	0
$115$	−0.0590182	−0.00550347
$116$	0	0
$117$	−9.04988	−0.836662
$118$	0	0
$119$	−9.60836	−0.880797
$120$	0	0
$121$	−10.7564	−0.977850
$122$	0	0
$123$	16.8369	1.51813
$124$	0	0
$125$	−10.8906	−0.974082
$126$	0	0
$127$	−15.1105	−1.34084	−0.670419	−	0.741983i	$-0.733886\pi$
$127$	−15.1105	−1.34084	−0.670419	+	0.741983i	$0.733886\pi$
$128$	0	0
$129$	22.3907	1.97139
$130$	0	0
$131$	−13.7326	−1.19982	−0.599910	−	0.800067i	$-0.704797\pi$
$131$	−13.7326	−1.19982	−0.599910	+	0.800067i	$0.704797\pi$
$132$	0	0
$133$	3.30965	0.286983
$134$	0	0
$135$	18.5376	1.59546
$136$	0	0
$137$	9.51667	0.813064	0.406532	−	0.913636i	$-0.366738\pi$
$137$	9.51667	0.813064	0.406532	+	0.913636i	$0.366738\pi$
$138$	0	0
$139$	18.9287	1.60551	0.802755	−	0.596309i	$-0.203367\pi$
$139$	18.9287	1.60551	0.802755	+	0.596309i	$0.203367\pi$
$140$	0	0
$141$	−9.34296	−0.786819
$142$	0	0
$143$	−0.778457	−0.0650979
$144$	0	0
$145$	−10.2354	−0.850006
$146$	0	0
$147$	2.92808	0.241504
$148$	0	0
$149$	−16.4217	−1.34531	−0.672657	−	0.739954i	$-0.734847\pi$
$149$	−16.4217	−1.34531	−0.672657	+	0.739954i	$0.734847\pi$
$150$	0	0
$151$	−18.4776	−1.50369	−0.751843	−	0.659342i	$-0.770835\pi$
$151$	−18.4776	−1.50369	−0.751843	+	0.659342i	$0.770835\pi$
$152$	0	0
$153$	−19.5051	−1.57690
$154$	0	0
$155$	14.7259	1.18282
$156$	0	0
$157$	−9.56827	−0.763631	−0.381816	−	0.924239i	$-0.624701\pi$
$157$	−9.56827	−0.763631	−0.381816	+	0.924239i	$0.624701\pi$
$158$	0	0
$159$	16.6693	1.32196
$160$	0	0
$161$	−0.0728490	−0.00574130
$162$	0	0
$163$	−22.8094	−1.78657	−0.893283	−	0.449495i	$-0.851604\pi$
$163$	−22.8094	−1.78657	−0.893283	+	0.449495i	$0.851604\pi$
$164$	0	0
$165$	3.34152	0.260137
$166$	0	0
$167$	13.6503	1.05629	0.528145	−	0.849154i	$-0.322888\pi$
$167$	13.6503	1.05629	0.528145	+	0.849154i	$0.322888\pi$
$168$	0	0
$169$	−10.5128	−0.808676
$170$	0	0
$171$	6.71864	0.513787
$172$	0	0
$173$	−10.0005	−0.760323	−0.380162	−	0.924920i	$-0.624132\pi$
$173$	−10.0005	−0.760323	−0.380162	+	0.924920i	$0.624132\pi$
$174$	0	0
$175$	0.691007	0.0522352
$176$	0	0
$177$	−22.2291	−1.67084
$178$	0	0
$179$	−8.05815	−0.602295	−0.301147	−	0.953578i	$-0.597370\pi$
$179$	−8.05815	−0.602295	−0.301147	+	0.953578i	$0.597370\pi$
$180$	0	0
$181$	22.4920	1.67182	0.835908	−	0.548870i	$-0.184942\pi$
$181$	22.4920	1.67182	0.835908	+	0.548870i	$0.184942\pi$
$182$	0	0
$183$	43.4816	3.21425
$184$	0	0
$185$	−13.2318	−0.972818
$186$	0	0
$187$	−1.67780	−0.122693
$188$	0	0
$189$	22.8818	1.66441
$190$	0	0
$191$	23.3048	1.68628	0.843140	−	0.537694i	$-0.180705\pi$
$191$	23.3048	1.68628	0.843140	+	0.537694i	$0.180705\pi$
$192$	0	0
$193$	−7.21669	−0.519469	−0.259734	−	0.965680i	$-0.583635\pi$
$193$	−7.21669	−0.519469	−0.259734	+	0.965680i	$0.583635\pi$
$194$	0	0
$195$	−10.6763	−0.764546
$196$	0	0
$197$	−14.6270	−1.04213	−0.521067	−	0.853516i	$-0.674466\pi$
$197$	−14.6270	−1.04213	−0.521067	+	0.853516i	$0.674466\pi$
$198$	0	0
$199$	3.31741	0.235165	0.117583	−	0.993063i	$-0.462486\pi$
$199$	3.31741	0.235165	0.117583	+	0.993063i	$0.462486\pi$
$200$	0	0
$201$	−12.9980	−0.916806
$202$	0	0
$203$	−12.6341	−0.886739
$204$	0	0
$205$	13.0436	0.911002
$206$	0	0
$207$	−0.147884	−0.0102787
$208$	0	0
$209$	0.577927	0.0399761
$210$	0	0
$211$	−14.1235	−0.972305	−0.486152	−	0.873874i	$-0.661600\pi$
$211$	−14.1235	−0.972305	−0.486152	+	0.873874i	$0.661600\pi$
$212$	0	0
$213$	5.78937	0.396681
$214$	0	0
$215$	17.3462	1.18300
$216$	0	0
$217$	18.1769	1.23393
$218$	0	0
$219$	2.00531	0.135506
$220$	0	0
$221$	5.36065	0.360597
$222$	0	0
$223$	1.57227	0.105287	0.0526436	−	0.998613i	$-0.483235\pi$
$223$	1.57227	0.105287	0.0526436	+	0.998613i	$0.483235\pi$
$224$	0	0
$225$	1.40275	0.0935169
$226$	0	0
$227$	−3.84013	−0.254879	−0.127439	−	0.991846i	$-0.540676\pi$
$227$	−3.84013	−0.254879	−0.127439	+	0.991846i	$0.540676\pi$
$228$	0	0
$229$	14.4539	0.955140	0.477570	−	0.878594i	$-0.341518\pi$
$229$	14.4539	0.955140	0.477570	+	0.878594i	$0.341518\pi$
$230$	0	0
$231$	4.12459	0.271378
$232$	0	0
$233$	−3.87614	−0.253934	−0.126967	−	0.991907i	$-0.540524\pi$
$233$	−3.87614	−0.253934	−0.126967	+	0.991907i	$0.540524\pi$
$234$	0	0
$235$	−7.23802	−0.472156
$236$	0	0
$237$	−27.5101	−1.78697
$238$	0	0
$239$	−4.43348	−0.286778	−0.143389	−	0.989666i	$-0.545800\pi$
$239$	−4.43348	−0.286778	−0.143389	+	0.989666i	$0.545800\pi$
$240$	0	0
$241$	24.0312	1.54799	0.773993	−	0.633194i	$-0.218256\pi$
$241$	24.0312	1.54799	0.773993	+	0.633194i	$0.218256\pi$
$242$	0	0
$243$	−4.43840	−0.284724
$244$	0	0
$245$	2.26839	0.144922
$246$	0	0
$247$	−1.84650	−0.117490
$248$	0	0
$249$	−7.04781	−0.446637
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−0.0127208	−0.000799750 0
$254$	0	0
$255$	−23.0105	−1.44097
$256$	0	0
$257$	3.94937	0.246355	0.123177	−	0.992385i	$-0.460692\pi$
$257$	3.94937	0.246355	0.123177	+	0.992385i	$0.460692\pi$
$258$	0	0
$259$	−16.3326	−1.01486
$260$	0	0
$261$	−25.6474	−1.58753
$262$	0	0
$263$	10.2782	0.633782	0.316891	−	0.948462i	$-0.397361\pi$
$263$	10.2782	0.633782	0.316891	+	0.948462i	$0.397361\pi$
$264$	0	0
$265$	12.9137	0.793285
$266$	0	0
$267$	−2.23175	−0.136581
$268$	0	0
$269$	−4.93747	−0.301043	−0.150522	−	0.988607i	$-0.548095\pi$
$269$	−4.93747	−0.301043	−0.150522	+	0.988607i	$0.548095\pi$
$270$	0	0
$271$	22.6810	1.37777	0.688886	−	0.724870i	$-0.258100\pi$
$271$	22.6810	1.37777	0.688886	+	0.724870i	$0.258100\pi$
$272$	0	0
$273$	−13.1783	−0.797585
$274$	0	0
$275$	0.120663	0.00727624
$276$	0	0
$277$	3.04249	0.182806	0.0914029	−	0.995814i	$-0.470865\pi$
$277$	3.04249	0.182806	0.0914029	+	0.995814i	$0.470865\pi$
$278$	0	0
$279$	36.8994	2.20911
$280$	0	0
$281$	−13.1636	−0.785274	−0.392637	−	0.919694i	$-0.628437\pi$
$281$	−13.1636	−0.785274	−0.392637	+	0.919694i	$0.628437\pi$
$282$	0	0
$283$	10.4524	0.621331	0.310666	−	0.950519i	$-0.399448\pi$
$283$	10.4524	0.621331	0.310666	+	0.950519i	$0.399448\pi$
$284$	0	0
$285$	7.92609	0.469501
$286$	0	0
$287$	16.1003	0.950371
$288$	0	0
$289$	−5.44624	−0.320367
$290$	0	0
$291$	2.42898	0.142389
$292$	0	0
$293$	−5.40280	−0.315635	−0.157817	−	0.987468i	$-0.550446\pi$
$293$	−5.40280	−0.315635	−0.157817	+	0.987468i	$0.550446\pi$
$294$	0	0
$295$	−17.2209	−1.00264
$296$	0	0
$297$	3.99560	0.231848
$298$	0	0
$299$	0.0406436	0.00235048
$300$	0	0
$301$	21.4112	1.23412
$302$	0	0
$303$	21.8245	1.25379
$304$	0	0
$305$	33.6853	1.92882
$306$	0	0
$307$	−23.7635	−1.35625	−0.678127	−	0.734945i	$-0.737208\pi$
$307$	−23.7635	−1.35625	−0.678127	+	0.734945i	$0.737208\pi$
$308$	0	0
$309$	30.3784	1.72817
$310$	0	0
$311$	−4.04081	−0.229133	−0.114567	−	0.993416i	$-0.536548\pi$
$311$	−4.04081	−0.229133	−0.114567	+	0.993416i	$0.536548\pi$
$312$	0	0
$313$	−29.4064	−1.66215	−0.831075	−	0.556160i	$-0.812274\pi$
$313$	−29.4064	−1.66215	−0.831075	+	0.556160i	$0.812274\pi$
$314$	0	0
$315$	37.1471	2.09300
$316$	0	0
$317$	28.7498	1.61475	0.807374	−	0.590040i	$-0.200888\pi$
$317$	28.7498	1.61475	0.807374	+	0.590040i	$0.200888\pi$
$318$	0	0
$319$	−2.20615	−0.123521
$320$	0	0
$321$	5.07240	0.283114
$322$	0	0
$323$	−3.97975	−0.221439
$324$	0	0
$325$	−0.385523	−0.0213850
$326$	0	0
$327$	28.2359	1.56145
$328$	0	0
$329$	−8.93424	−0.492560
$330$	0	0
$331$	−7.51553	−0.413091	−0.206546	−	0.978437i	$-0.566222\pi$
$331$	−7.51553	−0.413091	−0.206546	+	0.978437i	$0.566222\pi$
$332$	0	0
$333$	−33.1554	−1.81690
$334$	0	0
$335$	−10.0696	−0.550159
$336$	0	0
$337$	−3.38920	−0.184621	−0.0923107	−	0.995730i	$-0.529425\pi$
$337$	−3.38920	−0.184621	−0.0923107	+	0.995730i	$0.529425\pi$
$338$	0	0
$339$	−46.6825	−2.53544
$340$	0	0
$341$	3.17404	0.171884
$342$	0	0
$343$	−16.9873	−0.917227
$344$	0	0
$345$	−0.174462	−0.00939271
$346$	0	0
$347$	−30.4999	−1.63732	−0.818660	−	0.574279i	$-0.805282\pi$
$347$	−30.4999	−1.63732	−0.818660	+	0.574279i	$0.805282\pi$
$348$	0	0
$349$	13.3455	0.714369	0.357185	−	0.934034i	$-0.383737\pi$
$349$	13.3455	0.714369	0.357185	+	0.934034i	$0.383737\pi$
$350$	0	0
$351$	−12.7661	−0.681406
$352$	0	0
$353$	−11.6145	−0.618179	−0.309090	−	0.951033i	$-0.600024\pi$
$353$	−11.6145	−0.618179	−0.309090	+	0.951033i	$0.600024\pi$
$354$	0	0
$355$	4.48504	0.238041
$356$	0	0
$357$	−28.4030	−1.50325
$358$	0	0
$359$	3.40131	0.179514	0.0897571	−	0.995964i	$-0.471391\pi$
$359$	3.40131	0.179514	0.0897571	+	0.995964i	$0.471391\pi$
$360$	0	0
$361$	−17.6292	−0.927850
$362$	0	0
$363$	−31.7965	−1.66889
$364$	0	0
$365$	1.55352	0.0813148
$366$	0	0
$367$	−12.5610	−0.655681	−0.327840	−	0.944733i	$-0.606321\pi$
$367$	−12.5610	−0.655681	−0.327840	+	0.944733i	$0.606321\pi$
$368$	0	0
$369$	32.6838	1.70145
$370$	0	0
$371$	15.9400	0.827566
$372$	0	0
$373$	−0.303011	−0.0156893	−0.00784466	−	0.999969i	$-0.502497\pi$
$373$	−0.303011	−0.0156893	−0.00784466	+	0.999969i	$0.502497\pi$
$374$	0	0
$375$	−32.1933	−1.66245
$376$	0	0
$377$	7.04875	0.363029
$378$	0	0
$379$	18.2500	0.937438	0.468719	−	0.883347i	$-0.344716\pi$
$379$	18.2500	0.937438	0.468719	+	0.883347i	$0.344716\pi$
$380$	0	0
$381$	−44.6676	−2.28839
$382$	0	0
$383$	23.3955	1.19545	0.597727	−	0.801700i	$-0.296071\pi$
$383$	23.3955	1.19545	0.597727	+	0.801700i	$0.296071\pi$
$384$	0	0
$385$	3.19534	0.162849
$386$	0	0
$387$	43.4651	2.20945
$388$	0	0
$389$	−23.5328	−1.19316	−0.596580	−	0.802553i	$-0.703474\pi$
$389$	−23.5328	−1.19316	−0.596580	+	0.802553i	$0.703474\pi$
$390$	0	0
$391$	0.0875986	0.00443005
$392$	0	0
$393$	−40.5944	−2.04772
$394$	0	0
$395$	−21.3121	−1.07233
$396$	0	0
$397$	28.3940	1.42505	0.712526	−	0.701646i	$-0.247551\pi$
$397$	28.3940	1.42505	0.712526	+	0.701646i	$0.247551\pi$
$398$	0	0
$399$	9.78356	0.489791
$400$	0	0
$401$	2.46931	0.123312	0.0616558	−	0.998097i	$-0.480362\pi$
$401$	2.46931	0.123312	0.0616558	+	0.998097i	$0.480362\pi$
$402$	0	0
$403$	−10.1412	−0.505169
$404$	0	0
$405$	15.3746	0.763972
$406$	0	0
$407$	−2.85198	−0.141367
$408$	0	0
$409$	29.7341	1.47025	0.735127	−	0.677929i	$-0.237122\pi$
$409$	29.7341	1.47025	0.735127	+	0.677929i	$0.237122\pi$
$410$	0	0
$411$	28.1320	1.38765
$412$	0	0
$413$	−21.2566	−1.04597
$414$	0	0
$415$	−5.45996	−0.268019
$416$	0	0
$417$	55.9545	2.74010
$418$	0	0
$419$	24.0923	1.17699	0.588493	−	0.808502i	$-0.299721\pi$
$419$	24.0923	1.17699	0.588493	+	0.808502i	$0.299721\pi$
$420$	0	0
$421$	−29.6570	−1.44539	−0.722697	−	0.691165i	$-0.757098\pi$
$421$	−29.6570	−1.44539	−0.722697	+	0.691165i	$0.757098\pi$
$422$	0	0
$423$	−18.1366	−0.881833
$424$	0	0
$425$	−0.830914	−0.0403053
$426$	0	0
$427$	41.5794	2.01217
$428$	0	0
$429$	−2.30117	−0.111102
$430$	0	0
$431$	37.3141	1.79736	0.898678	−	0.438608i	$-0.144528\pi$
$431$	37.3141	1.79736	0.898678	+	0.438608i	$0.144528\pi$
$432$	0	0
$433$	17.7978	0.855305	0.427653	−	0.903943i	$-0.359341\pi$
$433$	17.7978	0.855305	0.427653	+	0.903943i	$0.359341\pi$
$434$	0	0
$435$	−30.2566	−1.45070
$436$	0	0
$437$	−0.0301738	−0.00144341
$438$	0	0
$439$	6.13512	0.292813	0.146407	−	0.989224i	$-0.453229\pi$
$439$	6.13512	0.292813	0.146407	+	0.989224i	$0.453229\pi$
$440$	0	0
$441$	5.68401	0.270667
$442$	0	0
$443$	−9.63283	−0.457670	−0.228835	−	0.973465i	$-0.573492\pi$
$443$	−9.63283	−0.457670	−0.228835	+	0.973465i	$0.573492\pi$
$444$	0	0
$445$	−1.72894	−0.0819598
$446$	0	0
$447$	−48.5435	−2.29603
$448$	0	0
$449$	−31.3153	−1.47786	−0.738931	−	0.673781i	$-0.764669\pi$
$449$	−31.3153	−1.47786	−0.738931	+	0.673781i	$0.764669\pi$
$450$	0	0
$451$	2.81142	0.132384
$452$	0	0
$453$	−54.6211	−2.56632
$454$	0	0
$455$	−10.2093	−0.478617
$456$	0	0
$457$	7.78229	0.364040	0.182020	−	0.983295i	$-0.441736\pi$
$457$	7.78229	0.364040	0.182020	+	0.983295i	$0.441736\pi$
$458$	0	0
$459$	−27.5147	−1.28428
$460$	0	0
$461$	24.6754	1.14925	0.574623	−	0.818418i	$-0.305149\pi$
$461$	24.6754	1.14925	0.574623	+	0.818418i	$0.305149\pi$
$462$	0	0
$463$	−9.56666	−0.444600	−0.222300	−	0.974978i	$-0.571356\pi$
$463$	−9.56666	−0.444600	−0.222300	+	0.974978i	$0.571356\pi$
$464$	0	0
$465$	43.5309	2.01870
$466$	0	0
$467$	−13.2857	−0.614787	−0.307394	−	0.951582i	$-0.599457\pi$
$467$	−13.2857	−0.614787	−0.307394	+	0.951582i	$0.599457\pi$
$468$	0	0
$469$	−12.4293	−0.573934
$470$	0	0
$471$	−28.2845	−1.30328
$472$	0	0
$473$	3.73880	0.171910
$474$	0	0
$475$	0.286213	0.0131323
$476$	0	0
$477$	32.3585	1.48159
$478$	0	0
$479$	−14.7151	−0.672351	−0.336176	−	0.941799i	$-0.609134\pi$
$479$	−14.7151	−0.672351	−0.336176	+	0.941799i	$0.609134\pi$
$480$	0	0
$481$	9.11220	0.415481
$482$	0	0
$483$	−0.215347	−0.00979861
$484$	0	0
$485$	1.88174	0.0854452
$486$	0	0
$487$	43.4544	1.96911	0.984553	−	0.175088i	$-0.0560211\pi$
$487$	43.4544	1.96911	0.984553	+	0.175088i	$0.0560211\pi$
$488$	0	0
$489$	−67.4260	−3.04911
$490$	0	0
$491$	3.20875	0.144809	0.0724045	−	0.997375i	$-0.476933\pi$
$491$	3.20875	0.144809	0.0724045	+	0.997375i	$0.476933\pi$
$492$	0	0
$493$	15.1921	0.684218
$494$	0	0
$495$	6.48658	0.291550
$496$	0	0
$497$	5.53610	0.248328
$498$	0	0
$499$	13.7239	0.614364	0.307182	−	0.951651i	$-0.400614\pi$
$499$	13.7239	0.614364	0.307182	+	0.951651i	$0.400614\pi$
$500$	0	0
$501$	40.3511	1.80276
$502$	0	0
$503$	−4.47895	−0.199706	−0.0998532	−	0.995002i	$-0.531837\pi$
$503$	−4.47895	−0.199706	−0.0998532	+	0.995002i	$0.531837\pi$
$504$	0	0
$505$	16.9075	0.752376
$506$	0	0
$507$	−31.0765	−1.38016
$508$	0	0
$509$	−24.0126	−1.06434	−0.532170	−	0.846637i	$-0.678623\pi$
$509$	−24.0126	−1.06434	−0.532170	+	0.846637i	$0.678623\pi$
$510$	0	0
$511$	1.91758	0.0848288
$512$	0	0
$513$	9.47758	0.418445
$514$	0	0
$515$	23.5343	1.03704
$516$	0	0
$517$	−1.56009	−0.0686125
$518$	0	0
$519$	−29.5621	−1.29763
$520$	0	0
$521$	30.6607	1.34327	0.671636	−	0.740881i	$-0.265592\pi$
$521$	30.6607	1.34327	0.671636	+	0.740881i	$0.265592\pi$
$522$	0	0
$523$	−23.7291	−1.03760	−0.518801	−	0.854895i	$-0.673621\pi$
$523$	−23.7291	−1.03760	−0.518801	+	0.854895i	$0.673621\pi$
$524$	0	0
$525$	2.04266	0.0891492
$526$	0	0
$527$	−21.8572	−0.952115
$528$	0	0
$529$	−22.9993	−0.999971
$530$	0	0
$531$	−43.1512	−1.87260
$532$	0	0
$533$	−8.98261	−0.389080
$534$	0	0
$535$	3.92961	0.169892
$536$	0	0
$537$	−23.8205	−1.02793
$538$	0	0
$539$	0.488930	0.0210597
$540$	0	0
$541$	12.5830	0.540987	0.270494	−	0.962722i	$-0.412813\pi$
$541$	12.5830	0.540987	0.270494	+	0.962722i	$0.412813\pi$
$542$	0	0
$543$	66.4879	2.85327
$544$	0	0
$545$	21.8745	0.936999
$546$	0	0
$547$	−29.3704	−1.25579	−0.627894	−	0.778299i	$-0.716083\pi$
$547$	−29.3704	−1.25579	−0.627894	+	0.778299i	$0.716083\pi$
$548$	0	0
$549$	84.4068	3.60240
$550$	0	0
$551$	−5.23300	−0.222933
$552$	0	0
$553$	−26.3066	−1.11867
$554$	0	0
$555$	−39.1140	−1.66030
$556$	0	0
$557$	−8.38061	−0.355098	−0.177549	−	0.984112i	$-0.556817\pi$
$557$	−8.38061	−0.355098	−0.177549	+	0.984112i	$0.556817\pi$
$558$	0	0
$559$	−11.9456	−0.505247
$560$	0	0
$561$	−4.95970	−0.209399
$562$	0	0
$563$	42.4890	1.79070	0.895349	−	0.445365i	$-0.146926\pi$
$563$	42.4890	1.79070	0.895349	+	0.445365i	$0.146926\pi$
$564$	0	0
$565$	−36.1651	−1.52148
$566$	0	0
$567$	18.9777	0.796986
$568$	0	0
$569$	13.7263	0.575435	0.287718	−	0.957715i	$-0.407104\pi$
$569$	13.7263	0.575435	0.287718	+	0.957715i	$0.407104\pi$
$570$	0	0
$571$	25.6251	1.07238	0.536188	−	0.844099i	$-0.319864\pi$
$571$	25.6251	1.07238	0.536188	+	0.844099i	$0.319864\pi$
$572$	0	0
$573$	68.8908	2.87795
$574$	0	0
$575$	−0.00629985	−0.000262722 0
$576$	0	0
$577$	−21.3873	−0.890364	−0.445182	−	0.895440i	$-0.646861\pi$
$577$	−21.3873	−0.890364	−0.445182	+	0.895440i	$0.646861\pi$
$578$	0	0
$579$	−21.3330	−0.886571
$580$	0	0
$581$	−6.73949	−0.279601
$582$	0	0
$583$	2.78343	0.115278
$584$	0	0
$585$	−20.7249	−0.856870
$586$	0	0
$587$	7.49783	0.309469	0.154734	−	0.987956i	$-0.450548\pi$
$587$	7.49783	0.309469	0.154734	+	0.987956i	$0.450548\pi$
$588$	0	0
$589$	7.52883	0.310220
$590$	0	0
$591$	−43.2386	−1.77860
$592$	0	0
$593$	−3.72839	−0.153107	−0.0765534	−	0.997065i	$-0.524392\pi$
$593$	−3.72839	−0.153107	−0.0765534	+	0.997065i	$0.524392\pi$
$594$	0	0
$595$	−22.0039	−0.902072
$596$	0	0
$597$	9.80650	0.401353
$598$	0	0
$599$	−25.2928	−1.03343	−0.516717	−	0.856156i	$-0.672846\pi$
$599$	−25.2928	−1.03343	−0.516717	+	0.856156i	$0.672846\pi$
$600$	0	0
$601$	29.0245	1.18393	0.591967	−	0.805962i	$-0.298351\pi$
$601$	29.0245	1.18393	0.591967	+	0.805962i	$0.298351\pi$
$602$	0	0
$603$	−25.2318	−1.02752
$604$	0	0
$605$	−24.6329	−1.00147
$606$	0	0
$607$	3.84912	0.156231	0.0781156	−	0.996944i	$-0.475110\pi$
$607$	3.84912	0.156231	0.0781156	+	0.996944i	$0.475110\pi$
$608$	0	0
$609$	−37.3472	−1.51339
$610$	0	0
$611$	4.98455	0.201653
$612$	0	0
$613$	24.3746	0.984480	0.492240	−	0.870460i	$-0.336178\pi$
$613$	24.3746	0.984480	0.492240	+	0.870460i	$0.336178\pi$
$614$	0	0
$615$	38.5577	1.55480
$616$	0	0
$617$	−17.2165	−0.693110	−0.346555	−	0.938030i	$-0.612649\pi$
$617$	−17.2165	−0.693110	−0.346555	+	0.938030i	$0.612649\pi$
$618$	0	0
$619$	9.23998	0.371386	0.185693	−	0.982608i	$-0.440547\pi$
$619$	9.23998	0.371386	0.185693	+	0.982608i	$0.440547\pi$
$620$	0	0
$621$	−0.208612	−0.00837130
$622$	0	0
$623$	−2.13412	−0.0855016
$624$	0	0
$625$	−26.1625	−1.04650
$626$	0	0
$627$	1.70839	0.0682267
$628$	0	0
$629$	19.6394	0.783076
$630$	0	0
$631$	−5.63137	−0.224181	−0.112091	−	0.993698i	$-0.535755\pi$
$631$	−5.63137	−0.224181	−0.112091	+	0.993698i	$0.535755\pi$
$632$	0	0
$633$	−41.7502	−1.65942
$634$	0	0
$635$	−34.6042	−1.37322
$636$	0	0
$637$	−1.56215	−0.0618948
$638$	0	0
$639$	11.2384	0.444583
$640$	0	0
$641$	−8.12365	−0.320865	−0.160432	−	0.987047i	$-0.551289\pi$
$641$	−8.12365	−0.320865	−0.160432	+	0.987047i	$0.551289\pi$
$642$	0	0
$643$	22.6289	0.892396	0.446198	−	0.894934i	$-0.352778\pi$
$643$	22.6289	0.892396	0.446198	+	0.894934i	$0.352778\pi$
$644$	0	0
$645$	51.2765	2.01901
$646$	0	0
$647$	11.0114	0.432905	0.216452	−	0.976293i	$-0.430551\pi$
$647$	11.0114	0.432905	0.216452	+	0.976293i	$0.430551\pi$
$648$	0	0
$649$	−3.71181	−0.145701
$650$	0	0
$651$	53.7323	2.10594
$652$	0	0
$653$	10.8926	0.426261	0.213130	−	0.977024i	$-0.431634\pi$
$653$	10.8926	0.426261	0.213130	+	0.977024i	$0.431634\pi$
$654$	0	0
$655$	−31.4486	−1.22880
$656$	0	0
$657$	3.89272	0.151869
$658$	0	0
$659$	33.9638	1.32304	0.661522	−	0.749926i	$-0.269911\pi$
$659$	33.9638	1.32304	0.661522	+	0.749926i	$0.269911\pi$
$660$	0	0
$661$	0.339588	0.0132084	0.00660422	−	0.999978i	$-0.497898\pi$
$661$	0.339588	0.0132084	0.00660422	+	0.999978i	$0.497898\pi$
$662$	0	0
$663$	15.8465	0.615426
$664$	0	0
$665$	7.57936	0.293915
$666$	0	0
$667$	0.115184	0.00445994
$668$	0	0
$669$	4.64775	0.179692
$670$	0	0
$671$	7.26055	0.280290
$672$	0	0
$673$	−9.84426	−0.379468	−0.189734	−	0.981836i	$-0.560763\pi$
$673$	−9.84426	−0.379468	−0.189734	+	0.981836i	$0.560763\pi$
$674$	0	0
$675$	1.97878	0.0761633
$676$	0	0
$677$	10.5963	0.407247	0.203624	−	0.979049i	$-0.434728\pi$
$677$	10.5963	0.407247	0.203624	+	0.979049i	$0.434728\pi$
$678$	0	0
$679$	2.32272	0.0891377
$680$	0	0
$681$	−11.3517	−0.434998
$682$	0	0
$683$	7.80434	0.298625	0.149312	−	0.988790i	$-0.452294\pi$
$683$	7.80434	0.298625	0.149312	+	0.988790i	$0.452294\pi$
$684$	0	0
$685$	21.7939	0.832703
$686$	0	0
$687$	42.7267	1.63013
$688$	0	0
$689$	−8.89320	−0.338804
$690$	0	0
$691$	31.9084	1.21385	0.606927	−	0.794758i	$-0.292402\pi$
$691$	31.9084	1.21385	0.606927	+	0.794758i	$0.292402\pi$
$692$	0	0
$693$	8.00670	0.304149
$694$	0	0
$695$	43.3481	1.64429
$696$	0	0
$697$	−19.3601	−0.733317
$698$	0	0
$699$	−11.4581	−0.433386
$700$	0	0
$701$	30.9608	1.16937	0.584687	−	0.811259i	$-0.301217\pi$
$701$	30.9608	1.16937	0.584687	+	0.811259i	$0.301217\pi$
$702$	0	0
$703$	−6.76491	−0.255143
$704$	0	0
$705$	−21.3961	−0.805823
$706$	0	0
$707$	20.8698	0.784890
$708$	0	0
$709$	2.33279	0.0876097	0.0438049	−	0.999040i	$-0.486052\pi$
$709$	2.33279	0.0876097	0.0438049	+	0.999040i	$0.486052\pi$
$710$	0	0
$711$	−53.4028	−2.00276
$712$	0	0
$713$	−0.165718	−0.00620617
$714$	0	0
$715$	−1.78273	−0.0666702
$716$	0	0
$717$	−13.1057	−0.489441
$718$	0	0
$719$	8.77184	0.327134	0.163567	−	0.986532i	$-0.447700\pi$
$719$	8.77184	0.327134	0.163567	+	0.986532i	$0.447700\pi$
$720$	0	0
$721$	29.0495	1.08186
$722$	0	0
$723$	71.0379	2.64193
$724$	0	0
$725$	−1.09257	−0.0405771
$726$	0	0
$727$	4.87777	0.180906	0.0904531	−	0.995901i	$-0.471168\pi$
$727$	4.87777	0.180906	0.0904531	+	0.995901i	$0.471168\pi$
$728$	0	0
$729$	−33.2610	−1.23189
$730$	0	0
$731$	−25.7463	−0.952262
$732$	0	0
$733$	6.71871	0.248161	0.124081	−	0.992272i	$-0.460402\pi$
$733$	6.71871	0.248161	0.124081	+	0.992272i	$0.460402\pi$
$734$	0	0
$735$	6.70552	0.247337
$736$	0	0
$737$	−2.17040	−0.0799476
$738$	0	0
$739$	1.50193	0.0552495	0.0276248	−	0.999618i	$-0.491206\pi$
$739$	1.50193	0.0552495	0.0276248	+	0.999618i	$0.491206\pi$
$740$	0	0
$741$	−5.45840	−0.200519
$742$	0	0
$743$	11.3819	0.417561	0.208780	−	0.977963i	$-0.433051\pi$
$743$	11.3819	0.417561	0.208780	+	0.977963i	$0.433051\pi$
$744$	0	0
$745$	−37.6068	−1.37781
$746$	0	0
$747$	−13.6813	−0.500571
$748$	0	0
$749$	4.85051	0.177234
$750$	0	0
$751$	−9.02322	−0.329262	−0.164631	−	0.986355i	$-0.552643\pi$
$751$	−9.02322	−0.329262	−0.164631	+	0.986355i	$0.552643\pi$
$752$	0	0
$753$	−2.95607	−0.107725
$754$	0	0
$755$	−42.3151	−1.54001
$756$	0	0
$757$	26.0960	0.948476	0.474238	−	0.880397i	$-0.342724\pi$
$757$	26.0960	0.948476	0.474238	+	0.880397i	$0.342724\pi$
$758$	0	0
$759$	−0.0376036	−0.00136492
$760$	0	0
$761$	−20.9381	−0.759006	−0.379503	−	0.925190i	$-0.623905\pi$
$761$	−20.9381	−0.759006	−0.379503	+	0.925190i	$0.623905\pi$
$762$	0	0
$763$	27.0007	0.977491
$764$	0	0
$765$	−44.6682	−1.61498
$766$	0	0
$767$	11.8594	0.428218
$768$	0	0
$769$	−12.5096	−0.451108	−0.225554	−	0.974231i	$-0.572419\pi$
$769$	−12.5096	−0.451108	−0.225554	+	0.974231i	$0.572419\pi$
$770$	0	0
$771$	11.6746	0.420451
$772$	0	0
$773$	−48.4373	−1.74217	−0.871084	−	0.491134i	$-0.836583\pi$
$773$	−48.4373	−1.74217	−0.871084	+	0.491134i	$0.836583\pi$
$774$	0	0
$775$	1.57191	0.0564647
$776$	0	0
$777$	−48.2803	−1.73205
$778$	0	0
$779$	6.66869	0.238931
$780$	0	0
$781$	0.966708	0.0345915
$782$	0	0
$783$	−36.1792	−1.29294
$784$	0	0
$785$	−21.9121	−0.782076
$786$	0	0
$787$	2.99058	0.106603	0.0533013	−	0.998578i	$-0.483026\pi$
$787$	2.99058	0.106603	0.0533013	+	0.998578i	$0.483026\pi$
$788$	0	0
$789$	30.3831	1.08167
$790$	0	0
$791$	−44.6403	−1.58723
$792$	0	0
$793$	−23.1978	−0.823778
$794$	0	0
$795$	38.1739	1.35389
$796$	0	0
$797$	−34.6517	−1.22743	−0.613714	−	0.789529i	$-0.710325\pi$
$797$	−34.6517	−1.22743	−0.613714	+	0.789529i	$0.710325\pi$
$798$	0	0
$799$	10.7431	0.380065
$800$	0	0
$801$	−4.33229	−0.153074
$802$	0	0
$803$	0.334846	0.0118164
$804$	0	0
$805$	−0.166830	−0.00587998
$806$	0	0
$807$	−14.5955	−0.513787
$808$	0	0
$809$	−17.4869	−0.614807	−0.307403	−	0.951579i	$-0.599460\pi$
$809$	−17.4869	−0.614807	−0.307403	+	0.951579i	$0.599460\pi$
$810$	0	0
$811$	40.7593	1.43125	0.715626	−	0.698483i	$-0.246141\pi$
$811$	40.7593	1.43125	0.715626	+	0.698483i	$0.246141\pi$
$812$	0	0
$813$	67.0466	2.35143
$814$	0	0
$815$	−52.2352	−1.82972
$816$	0	0
$817$	8.86845	0.310268
$818$	0	0
$819$	−25.5818	−0.893900
$820$	0	0
$821$	31.7359	1.10759	0.553795	−	0.832653i	$-0.313179\pi$
$821$	31.7359	1.10759	0.553795	+	0.832653i	$0.313179\pi$
$822$	0	0
$823$	−35.4568	−1.23595	−0.617973	−	0.786199i	$-0.712046\pi$
$823$	−35.4568	−1.23595	−0.617973	+	0.786199i	$0.712046\pi$
$824$	0	0
$825$	0.356688	0.0124183
$826$	0	0
$827$	5.33426	0.185490	0.0927451	−	0.995690i	$-0.470436\pi$
$827$	5.33426	0.185490	0.0927451	+	0.995690i	$0.470436\pi$
$828$	0	0
$829$	2.57787	0.0895333	0.0447666	−	0.998997i	$-0.485746\pi$
$829$	2.57787	0.0895333	0.0447666	+	0.998997i	$0.485746\pi$
$830$	0	0
$831$	8.99382	0.311992
$832$	0	0
$833$	−3.36689	−0.116656
$834$	0	0
$835$	31.2602	1.08180
$836$	0	0
$837$	52.0519	1.79918
$838$	0	0
$839$	−17.5500	−0.605893	−0.302946	−	0.953008i	$-0.597970\pi$
$839$	−17.5500	−0.605893	−0.302946	+	0.953008i	$0.597970\pi$
$840$	0	0
$841$	−9.02384	−0.311167
$842$	0	0
$843$	−38.9125	−1.34022
$844$	0	0
$845$	−24.0751	−0.828209
$846$	0	0
$847$	−30.4056	−1.04475
$848$	0	0
$849$	30.8981	1.06042
$850$	0	0
$851$	0.148903	0.00510433
$852$	0	0
$853$	33.4519	1.14537	0.572686	−	0.819775i	$-0.305901\pi$
$853$	33.4519	1.14537	0.572686	+	0.819775i	$0.305901\pi$
$854$	0	0
$855$	15.3862	0.526197
$856$	0	0
$857$	0.315009	0.0107605	0.00538025	−	0.999986i	$-0.498287\pi$
$857$	0.315009	0.0107605	0.00538025	+	0.999986i	$0.498287\pi$
$858$	0	0
$859$	14.5354	0.495942	0.247971	−	0.968767i	$-0.420236\pi$
$859$	14.5354	0.495942	0.247971	+	0.968767i	$0.420236\pi$
$860$	0	0
$861$	47.5936	1.62199
$862$	0	0
$863$	−24.2994	−0.827160	−0.413580	−	0.910468i	$-0.635722\pi$
$863$	−24.2994	−0.827160	−0.413580	+	0.910468i	$0.635722\pi$
$864$	0	0
$865$	−22.9019	−0.778688
$866$	0	0
$867$	−16.0995	−0.546766
$868$	0	0
$869$	−4.59363	−0.155828
$870$	0	0
$871$	6.93452	0.234967
$872$	0	0
$873$	4.71515	0.159584
$874$	0	0
$875$	−30.7849	−1.04072
$876$	0	0
$877$	47.1278	1.59139	0.795696	−	0.605697i	$-0.207105\pi$
$877$	47.1278	1.59139	0.795696	+	0.605697i	$0.207105\pi$
$878$	0	0
$879$	−15.9711	−0.538690
$880$	0	0
$881$	34.2629	1.15435	0.577174	−	0.816621i	$-0.304156\pi$
$881$	34.2629	1.15435	0.577174	+	0.816621i	$0.304156\pi$
$882$	0	0
$883$	1.60243	0.0539261	0.0269631	−	0.999636i	$-0.491416\pi$
$883$	1.60243	0.0539261	0.0269631	+	0.999636i	$0.491416\pi$
$884$	0	0
$885$	−50.9063	−1.71120
$886$	0	0
$887$	−29.0994	−0.977062	−0.488531	−	0.872547i	$-0.662467\pi$
$887$	−29.0994	−0.977062	−0.488531	+	0.872547i	$0.662467\pi$
$888$	0	0
$889$	−42.7136	−1.43257
$890$	0	0
$891$	3.31385	0.111018
$892$	0	0
$893$	−3.70053	−0.123834
$894$	0	0
$895$	−18.4538	−0.616842
$896$	0	0
$897$	0.120145	0.00401153
$898$	0	0
$899$	−28.7402	−0.958538
$900$	0	0
$901$	−19.1674	−0.638559
$902$	0	0
$903$	63.2930	2.10626
$904$	0	0
$905$	51.5084	1.71220
$906$	0	0
$907$	−0.844667	−0.0280467	−0.0140234	−	0.999902i	$-0.504464\pi$
$907$	−0.844667	−0.0280467	−0.0140234	+	0.999902i	$0.504464\pi$
$908$	0	0
$909$	42.3660	1.40519
$910$	0	0
$911$	−45.0393	−1.49222	−0.746109	−	0.665824i	$-0.768080\pi$
$911$	−45.0393	−1.49222	−0.746109	+	0.665824i	$0.768080\pi$
$912$	0	0
$913$	−1.17684	−0.0389478
$914$	0	0
$915$	99.5762	3.29189
$916$	0	0
$917$	−38.8186	−1.28190
$918$	0	0
$919$	−0.793907	−0.0261886	−0.0130943	−	0.999914i	$-0.504168\pi$
$919$	−0.793907	−0.0261886	−0.0130943	+	0.999914i	$0.504168\pi$
$920$	0	0
$921$	−70.2465	−2.31470
$922$	0	0
$923$	−3.08868	−0.101665
$924$	0	0
$925$	−1.41241	−0.0464399
$926$	0	0
$927$	58.9708	1.93686
$928$	0	0
$929$	−40.3531	−1.32394	−0.661971	−	0.749529i	$-0.730280\pi$
$929$	−40.3531	−1.32394	−0.661971	+	0.749529i	$0.730280\pi$
$930$	0	0
$931$	1.15974	0.0380091
$932$	0	0
$933$	−11.9449	−0.391059
$934$	0	0
$935$	−3.84229	−0.125656
$936$	0	0
$937$	−37.4603	−1.22378	−0.611888	−	0.790945i	$-0.709590\pi$
$937$	−37.4603	−1.22378	−0.611888	+	0.790945i	$0.709590\pi$
$938$	0	0
$939$	−86.9275	−2.83677
$940$	0	0
$941$	−16.4552	−0.536424	−0.268212	−	0.963360i	$-0.586433\pi$
$941$	−16.4552	−0.536424	−0.268212	+	0.963360i	$0.586433\pi$
$942$	0	0
$943$	−0.146785	−0.00477998
$944$	0	0
$945$	52.4012	1.70461
$946$	0	0
$947$	28.9019	0.939184	0.469592	−	0.882883i	$-0.344401\pi$
$947$	28.9019	0.939184	0.469592	+	0.882883i	$0.344401\pi$
$948$	0	0
$949$	−1.06985	−0.0347287
$950$	0	0
$951$	84.9863	2.75587
$952$	0	0
$953$	−7.97936	−0.258477	−0.129238	−	0.991614i	$-0.541253\pi$
$953$	−7.97936	−0.258477	−0.129238	+	0.991614i	$0.541253\pi$
$954$	0	0
$955$	53.3699	1.72701
$956$	0	0
$957$	−6.52153	−0.210811
$958$	0	0
$959$	26.9013	0.868688
$960$	0	0
$961$	10.3491	0.333842
$962$	0	0
$963$	9.84659	0.317302
$964$	0	0
$965$	−16.5268	−0.532016
$966$	0	0
$967$	6.37745	0.205085	0.102542	−	0.994729i	$-0.467302\pi$
$967$	6.37745	0.205085	0.102542	+	0.994729i	$0.467302\pi$
$968$	0	0
$969$	−11.7644	−0.377928
$970$	0	0
$971$	−27.1525	−0.871365	−0.435682	−	0.900100i	$-0.643493\pi$
$971$	−27.1525	−0.871365	−0.435682	+	0.900100i	$0.643493\pi$
$972$	0	0
$973$	53.5067	1.71535
$974$	0	0
$975$	−1.13963	−0.0364975
$976$	0	0
$977$	13.8156	0.442000	0.221000	−	0.975274i	$-0.429068\pi$
$977$	13.8156	0.442000	0.221000	+	0.975274i	$0.429068\pi$
$978$	0	0
$979$	−0.372657	−0.0119102
$980$	0	0
$981$	54.8118	1.75001
$982$	0	0
$983$	−46.8835	−1.49535	−0.747675	−	0.664065i	$-0.768830\pi$
$983$	−46.8835	−1.49535	−0.747675	+	0.664065i	$0.768830\pi$
$984$	0	0
$985$	−33.4971	−1.06731
$986$	0	0
$987$	−26.4102	−0.840647
$988$	0	0
$989$	−0.195204	−0.00620713
$990$	0	0
$991$	35.7602	1.13596	0.567981	−	0.823042i	$-0.307725\pi$
$991$	35.7602	1.13596	0.567981	+	0.823042i	$0.307725\pi$
$992$	0	0
$993$	−22.2164	−0.705017
$994$	0	0
$995$	7.59713	0.240845
$996$	0	0
$997$	15.2022	0.481457	0.240729	−	0.970592i	$-0.422614\pi$
$997$	15.2022	0.481457	0.240729	+	0.970592i	$0.422614\pi$
$998$	0	0
$999$	−46.7703	−1.47975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.15		17
4.3	odd	2		251.2.a.b.1.17	✓	17
12.11	even	2		2259.2.a.k.1.1		17
20.19	odd	2		6275.2.a.e.1.1		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.17	✓	17	4.3	odd	2
2259.2.a.k.1.1		17	12.11	even	2
4016.2.a.k.1.15		17	1.1	even	1	trivial
6275.2.a.e.1.1		17	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+2.95607 q^{3} +2.29008 q^{5} +2.82675 q^{7} +5.73835 q^{9} +O(q^{10})\)	\(q+2.95607 q^{3} +2.29008 q^{5} +2.82675 q^{7} +5.73835 q^{9} +0.493604 q^{11} -1.57709 q^{13} +6.76963 q^{15} -3.39908 q^{17} +1.17083 q^{19} +8.35608 q^{21} -0.0257713 q^{23} +0.244452 q^{25} +8.09475 q^{27} -4.46947 q^{29} +6.43033 q^{31} +1.45913 q^{33} +6.47348 q^{35} -5.77787 q^{37} -4.66198 q^{39} +5.69569 q^{41} +7.57449 q^{43} +13.1413 q^{45} -3.16060 q^{47} +0.990530 q^{49} -10.0479 q^{51} +5.63900 q^{53} +1.13039 q^{55} +3.46106 q^{57} -7.51980 q^{59} +14.7093 q^{61} +16.2209 q^{63} -3.61165 q^{65} -4.39704 q^{67} -0.0761816 q^{69} +1.95847 q^{71} +0.678369 q^{73} +0.722619 q^{75} +1.39530 q^{77} -9.30630 q^{79} +6.71359 q^{81} -2.38418 q^{83} -7.78416 q^{85} -13.2121 q^{87} -0.754972 q^{89} -4.45804 q^{91} +19.0085 q^{93} +2.68129 q^{95} +0.821691 q^{97} +2.83247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more