LMFDB - Embedded newform 4016.2.a.h.1.3

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.42772$ 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)

$f(q)$	$=$	$q-1.42772 q^{3} -3.45271 q^{5} -2.96003 q^{7} -0.961621 q^{9} +O(q^{10})$	$q-1.42772 q^{3} -3.45271 q^{5} -2.96003 q^{7} -0.961621 q^{9} +1.17787 q^{11} +1.76008 q^{13} +4.92950 q^{15} -3.62318 q^{17} -0.411025 q^{19} +4.22609 q^{21} +7.25292 q^{23} +6.92120 q^{25} +5.65608 q^{27} +5.51832 q^{29} +3.97237 q^{31} -1.68167 q^{33} +10.2201 q^{35} -5.35081 q^{37} -2.51289 q^{39} +0.721040 q^{41} -1.15691 q^{43} +3.32020 q^{45} +5.14013 q^{47} +1.76177 q^{49} +5.17287 q^{51} -7.17334 q^{53} -4.06684 q^{55} +0.586828 q^{57} +1.70933 q^{59} +6.75252 q^{61} +2.84642 q^{63} -6.07703 q^{65} +9.81958 q^{67} -10.3551 q^{69} -8.20123 q^{71} -7.49535 q^{73} -9.88153 q^{75} -3.48653 q^{77} -5.47903 q^{79} -5.19042 q^{81} +8.03576 q^{83} +12.5098 q^{85} -7.87860 q^{87} -8.41234 q^{89} -5.20988 q^{91} -5.67143 q^{93} +1.41915 q^{95} +8.26536 q^{97} -1.13266 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

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
$3$	−1.42772	−0.824294	−0.412147	−	0.911117i	$-0.635221\pi$
$3$	−1.42772	−0.824294	−0.412147	+	0.911117i	$0.635221\pi$
$4$	0	0
$5$	−3.45271	−1.54410	−0.772049	−	0.635563i	$-0.780768\pi$
$5$	−3.45271	−1.54410	−0.772049	+	0.635563i	$0.780768\pi$
$6$	0	0
$7$	−2.96003	−1.11879	−0.559393	−	0.828903i	$-0.688966\pi$
$7$	−2.96003	−1.11879	−0.559393	+	0.828903i	$0.688966\pi$
$8$	0	0
$9$	−0.961621	−0.320540
$10$	0	0
$11$	1.17787	0.355141	0.177571	−	0.984108i	$-0.443176\pi$
$11$	1.17787	0.355141	0.177571	+	0.984108i	$0.443176\pi$
$12$	0	0
$13$	1.76008	0.488157	0.244079	−	0.969755i	$-0.421514\pi$
$13$	1.76008	0.488157	0.244079	+	0.969755i	$0.421514\pi$
$14$	0	0
$15$	4.92950	1.27279
$16$	0	0
$17$	−3.62318	−0.878749	−0.439375	−	0.898304i	$-0.644800\pi$
$17$	−3.62318	−0.878749	−0.439375	+	0.898304i	$0.644800\pi$
$18$	0	0
$19$	−0.411025	−0.0942956	−0.0471478	−	0.998888i	$-0.515013\pi$
$19$	−0.411025	−0.0942956	−0.0471478	+	0.998888i	$0.515013\pi$
$20$	0	0
$21$	4.22609	0.922208
$22$	0	0
$23$	7.25292	1.51234	0.756170	−	0.654376i	$-0.227068\pi$
$23$	7.25292	1.51234	0.756170	+	0.654376i	$0.227068\pi$
$24$	0	0
$25$	6.92120	1.38424
$26$	0	0
$27$	5.65608	1.08851
$28$	0	0
$29$	5.51832	1.02473	0.512363	−	0.858769i	$-0.328770\pi$
$29$	5.51832	1.02473	0.512363	+	0.858769i	$0.328770\pi$
$30$	0	0
$31$	3.97237	0.713459	0.356729	−	0.934208i	$-0.383892\pi$
$31$	3.97237	0.713459	0.356729	+	0.934208i	$0.383892\pi$
$32$	0	0
$33$	−1.68167	−0.292740
$34$	0	0
$35$	10.2201	1.72752
$36$	0	0
$37$	−5.35081	−0.879667	−0.439834	−	0.898079i	$-0.644963\pi$
$37$	−5.35081	−0.879667	−0.439834	+	0.898079i	$0.644963\pi$
$38$	0	0
$39$	−2.51289	−0.402385
$40$	0	0
$41$	0.721040	0.112608	0.0563038	−	0.998414i	$-0.482068\pi$
$41$	0.721040	0.112608	0.0563038	+	0.998414i	$0.482068\pi$
$42$	0	0
$43$	−1.15691	−0.176427	−0.0882137	−	0.996102i	$-0.528116\pi$
$43$	−1.15691	−0.176427	−0.0882137	+	0.996102i	$0.528116\pi$
$44$	0	0
$45$	3.32020	0.494946
$46$	0	0
$47$	5.14013	0.749764	0.374882	−	0.927072i	$-0.377683\pi$
$47$	5.14013	0.749764	0.374882	+	0.927072i	$0.377683\pi$
$48$	0	0
$49$	1.76177	0.251681
$50$	0	0
$51$	5.17287	0.724347
$52$	0	0
$53$	−7.17334	−0.985334	−0.492667	−	0.870218i	$-0.663978\pi$
$53$	−7.17334	−0.985334	−0.492667	+	0.870218i	$0.663978\pi$
$54$	0	0
$55$	−4.06684	−0.548373
$56$	0	0
$57$	0.586828	0.0777272
$58$	0	0
$59$	1.70933	0.222536	0.111268	−	0.993790i	$-0.464509\pi$
$59$	1.70933	0.222536	0.111268	+	0.993790i	$0.464509\pi$
$60$	0	0
$61$	6.75252	0.864572	0.432286	−	0.901737i	$-0.357707\pi$
$61$	6.75252	0.864572	0.432286	+	0.901737i	$0.357707\pi$
$62$	0	0
$63$	2.84642	0.358616
$64$	0	0
$65$	−6.07703	−0.753763
$66$	0	0
$67$	9.81958	1.19965	0.599826	−	0.800130i	$-0.295236\pi$
$67$	9.81958	1.19965	0.599826	+	0.800130i	$0.295236\pi$
$68$	0	0
$69$	−10.3551	−1.24661
$70$	0	0
$71$	−8.20123	−0.973307	−0.486654	−	0.873595i	$-0.661783\pi$
$71$	−8.20123	−0.973307	−0.486654	+	0.873595i	$0.661783\pi$
$72$	0	0
$73$	−7.49535	−0.877265	−0.438632	−	0.898667i	$-0.644537\pi$
$73$	−7.49535	−0.877265	−0.438632	+	0.898667i	$0.644537\pi$
$74$	0	0
$75$	−9.88153	−1.14102
$76$	0	0
$77$	−3.48653	−0.397327
$78$	0	0
$79$	−5.47903	−0.616439	−0.308220	−	0.951315i	$-0.599733\pi$
$79$	−5.47903	−0.616439	−0.308220	+	0.951315i	$0.599733\pi$
$80$	0	0
$81$	−5.19042	−0.576714
$82$	0	0
$83$	8.03576	0.882040	0.441020	−	0.897497i	$-0.354617\pi$
$83$	8.03576	0.882040	0.441020	+	0.897497i	$0.354617\pi$
$84$	0	0
$85$	12.5098	1.35688
$86$	0	0
$87$	−7.87860	−0.844675
$88$	0	0
$89$	−8.41234	−0.891706	−0.445853	−	0.895106i	$-0.647100\pi$
$89$	−8.41234	−0.891706	−0.445853	+	0.895106i	$0.647100\pi$
$90$	0	0
$91$	−5.20988	−0.546144
$92$	0	0
$93$	−5.67143	−0.588100
$94$	0	0
$95$	1.41915	0.145602
$96$	0	0
$97$	8.26536	0.839220	0.419610	−	0.907704i	$-0.362167\pi$
$97$	8.26536	0.839220	0.419610	+	0.907704i	$0.362167\pi$
$98$	0	0
$99$	−1.13266	−0.113837
$100$	0	0
$101$	−7.10989	−0.707461	−0.353730	−	0.935347i	$-0.615087\pi$
$101$	−7.10989	−0.707461	−0.353730	+	0.935347i	$0.615087\pi$
$102$	0	0
$103$	2.74098	0.270077	0.135038	−	0.990840i	$-0.456884\pi$
$103$	2.74098	0.270077	0.135038	+	0.990840i	$0.456884\pi$
$104$	0	0
$105$	−14.5915	−1.42398
$106$	0	0
$107$	3.31364	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$107$	3.31364	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$108$	0	0
$109$	−4.36976	−0.418547	−0.209273	−	0.977857i	$-0.567110\pi$
$109$	−4.36976	−0.418547	−0.209273	+	0.977857i	$0.567110\pi$
$110$	0	0
$111$	7.63944	0.725104
$112$	0	0
$113$	−19.3635	−1.82156	−0.910781	−	0.412890i	$-0.864519\pi$
$113$	−19.3635	−1.82156	−0.910781	+	0.412890i	$0.864519\pi$
$114$	0	0
$115$	−25.0422	−2.33520
$116$	0	0
$117$	−1.69253	−0.156474
$118$	0	0
$119$	10.7247	0.983132
$120$	0	0
$121$	−9.61262	−0.873875
$122$	0	0
$123$	−1.02944	−0.0928217
$124$	0	0
$125$	−6.63335	−0.593305
$126$	0	0
$127$	15.5813	1.38262	0.691308	−	0.722560i	$-0.257035\pi$
$127$	15.5813	1.38262	0.691308	+	0.722560i	$0.257035\pi$
$128$	0	0
$129$	1.65174	0.145428
$130$	0	0
$131$	0.748851	0.0654274	0.0327137	−	0.999465i	$-0.489585\pi$
$131$	0.748851	0.0654274	0.0327137	+	0.999465i	$0.489585\pi$
$132$	0	0
$133$	1.21665	0.105497
$134$	0	0
$135$	−19.5288	−1.68077
$136$	0	0
$137$	10.8818	0.929698	0.464849	−	0.885390i	$-0.346109\pi$
$137$	10.8818	0.929698	0.464849	+	0.885390i	$0.346109\pi$
$138$	0	0
$139$	7.15340	0.606744	0.303372	−	0.952872i	$-0.401888\pi$
$139$	7.15340	0.606744	0.303372	+	0.952872i	$0.401888\pi$
$140$	0	0
$141$	−7.33865	−0.618026
$142$	0	0
$143$	2.07314	0.173365
$144$	0	0
$145$	−19.0532	−1.58228
$146$	0	0
$147$	−2.51531	−0.207459
$148$	0	0
$149$	5.22100	0.427721	0.213860	−	0.976864i	$-0.431396\pi$
$149$	5.22100	0.427721	0.213860	+	0.976864i	$0.431396\pi$
$150$	0	0
$151$	−11.9236	−0.970325	−0.485163	−	0.874424i	$-0.661240\pi$
$151$	−11.9236	−0.970325	−0.485163	+	0.874424i	$0.661240\pi$
$152$	0	0
$153$	3.48412	0.281674
$154$	0	0
$155$	−13.7154	−1.10165
$156$	0	0
$157$	17.2773	1.37888	0.689441	−	0.724342i	$-0.257856\pi$
$157$	17.2773	1.37888	0.689441	+	0.724342i	$0.257856\pi$
$158$	0	0
$159$	10.2415	0.812205
$160$	0	0
$161$	−21.4689	−1.69198
$162$	0	0
$163$	−1.68208	−0.131750	−0.0658752	−	0.997828i	$-0.520984\pi$
$163$	−1.68208	−0.131750	−0.0658752	+	0.997828i	$0.520984\pi$
$164$	0	0
$165$	5.80630	0.452020
$166$	0	0
$167$	9.53790	0.738065	0.369032	−	0.929417i	$-0.379689\pi$
$167$	9.53790	0.738065	0.369032	+	0.929417i	$0.379689\pi$
$168$	0	0
$169$	−9.90213	−0.761702
$170$	0	0
$171$	0.395250	0.0302255
$172$	0	0
$173$	17.0689	1.29772	0.648862	−	0.760906i	$-0.275245\pi$
$173$	17.0689	1.29772	0.648862	+	0.760906i	$0.275245\pi$
$174$	0	0
$175$	−20.4870	−1.54867
$176$	0	0
$177$	−2.44044	−0.183435
$178$	0	0
$179$	−14.4507	−1.08009	−0.540047	−	0.841635i	$-0.681594\pi$
$179$	−14.4507	−1.08009	−0.540047	+	0.841635i	$0.681594\pi$
$180$	0	0
$181$	9.49409	0.705690	0.352845	−	0.935682i	$-0.385214\pi$
$181$	9.49409	0.705690	0.352845	+	0.935682i	$0.385214\pi$
$182$	0	0
$183$	−9.64070	−0.712661
$184$	0	0
$185$	18.4748	1.35829
$186$	0	0
$187$	−4.26763	−0.312080
$188$	0	0
$189$	−16.7422	−1.21781
$190$	0	0
$191$	0.388538	0.0281137	0.0140568	−	0.999901i	$-0.495525\pi$
$191$	0.388538	0.0281137	0.0140568	+	0.999901i	$0.495525\pi$
$192$	0	0
$193$	−24.5127	−1.76446	−0.882230	−	0.470818i	$-0.843959\pi$
$193$	−24.5127	−1.76446	−0.882230	+	0.470818i	$0.843959\pi$
$194$	0	0
$195$	8.67629	0.621322
$196$	0	0
$197$	8.11610	0.578248	0.289124	−	0.957292i	$-0.406636\pi$
$197$	8.11610	0.578248	0.289124	+	0.957292i	$0.406636\pi$
$198$	0	0
$199$	−23.6816	−1.67874	−0.839371	−	0.543559i	$-0.817076\pi$
$199$	−23.6816	−1.67874	−0.839371	+	0.543559i	$0.817076\pi$
$200$	0	0
$201$	−14.0196	−0.988866
$202$	0	0
$203$	−16.3344	−1.14645
$204$	0	0
$205$	−2.48954	−0.173877
$206$	0	0
$207$	−6.97456	−0.484766
$208$	0	0
$209$	−0.484134	−0.0334882
$210$	0	0
$211$	−28.9140	−1.99052	−0.995260	−	0.0972522i	$-0.968995\pi$
$211$	−28.9140	−1.99052	−0.995260	+	0.0972522i	$0.968995\pi$
$212$	0	0
$213$	11.7091	0.802291
$214$	0	0
$215$	3.99448	0.272421
$216$	0	0
$217$	−11.7583	−0.798208
$218$	0	0
$219$	10.7013	0.723124
$220$	0	0
$221$	−6.37707	−0.428968
$222$	0	0
$223$	−18.1650	−1.21642	−0.608209	−	0.793777i	$-0.708112\pi$
$223$	−18.1650	−1.21642	−0.608209	+	0.793777i	$0.708112\pi$
$224$	0	0
$225$	−6.65557	−0.443705
$226$	0	0
$227$	12.3144	0.817337	0.408668	−	0.912683i	$-0.365993\pi$
$227$	12.3144	0.817337	0.408668	+	0.912683i	$0.365993\pi$
$228$	0	0
$229$	5.69718	0.376480	0.188240	−	0.982123i	$-0.439722\pi$
$229$	5.69718	0.376480	0.188240	+	0.982123i	$0.439722\pi$
$230$	0	0
$231$	4.97778	0.327514
$232$	0	0
$233$	−7.74937	−0.507678	−0.253839	−	0.967247i	$-0.581693\pi$
$233$	−7.74937	−0.507678	−0.253839	+	0.967247i	$0.581693\pi$
$234$	0	0
$235$	−17.7474	−1.15771
$236$	0	0
$237$	7.82252	0.508127
$238$	0	0
$239$	1.08353	0.0700878	0.0350439	−	0.999386i	$-0.488843\pi$
$239$	1.08353	0.0700878	0.0350439	+	0.999386i	$0.488843\pi$
$240$	0	0
$241$	−14.6293	−0.942357	−0.471178	−	0.882038i	$-0.656171\pi$
$241$	−14.6293	−0.942357	−0.471178	+	0.882038i	$0.656171\pi$
$242$	0	0
$243$	−9.55777	−0.613131
$244$	0	0
$245$	−6.08288	−0.388621
$246$	0	0
$247$	−0.723435	−0.0460311
$248$	0	0
$249$	−11.4728	−0.727059
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	8.54300	0.537094
$254$	0	0
$255$	−17.8604	−1.11846
$256$	0	0
$257$	−7.41084	−0.462276	−0.231138	−	0.972921i	$-0.574245\pi$
$257$	−7.41084	−0.462276	−0.231138	+	0.972921i	$0.574245\pi$
$258$	0	0
$259$	15.8385	0.984159
$260$	0	0
$261$	−5.30653	−0.328466
$262$	0	0
$263$	−6.22388	−0.383781	−0.191890	−	0.981416i	$-0.561462\pi$
$263$	−6.22388	−0.383781	−0.191890	+	0.981416i	$0.561462\pi$
$264$	0	0
$265$	24.7675	1.52145
$266$	0	0
$267$	12.0104	0.735027
$268$	0	0
$269$	−3.67090	−0.223819	−0.111909	−	0.993718i	$-0.535697\pi$
$269$	−3.67090	−0.223819	−0.111909	+	0.993718i	$0.535697\pi$
$270$	0	0
$271$	19.1969	1.16613	0.583066	−	0.812425i	$-0.301853\pi$
$271$	19.1969	1.16613	0.583066	+	0.812425i	$0.301853\pi$
$272$	0	0
$273$	7.43824	0.450183
$274$	0	0
$275$	8.15227	0.491600
$276$	0	0
$277$	10.8174	0.649958	0.324979	−	0.945721i	$-0.394643\pi$
$277$	10.8174	0.649958	0.324979	+	0.945721i	$0.394643\pi$
$278$	0	0
$279$	−3.81991	−0.228692
$280$	0	0
$281$	31.6874	1.89031	0.945155	−	0.326621i	$-0.105910\pi$
$281$	31.6874	1.89031	0.945155	+	0.326621i	$0.105910\pi$
$282$	0	0
$283$	−4.29670	−0.255412	−0.127706	−	0.991812i	$-0.540761\pi$
$283$	−4.29670	−0.255412	−0.127706	+	0.991812i	$0.540761\pi$
$284$	0	0
$285$	−2.02615	−0.120019
$286$	0	0
$287$	−2.13430	−0.125984
$288$	0	0
$289$	−3.87260	−0.227800
$290$	0	0
$291$	−11.8006	−0.691764
$292$	0	0
$293$	29.4307	1.71936	0.859680	−	0.510832i	$-0.170663\pi$
$293$	29.4307	1.71936	0.859680	+	0.510832i	$0.170663\pi$
$294$	0	0
$295$	−5.90182	−0.343617
$296$	0	0
$297$	6.66212	0.386575
$298$	0	0
$299$	12.7657	0.738260
$300$	0	0
$301$	3.42449	0.197385
$302$	0	0
$303$	10.1509	0.583155
$304$	0	0
$305$	−23.3145	−1.33498
$306$	0	0
$307$	−15.1519	−0.864765	−0.432383	−	0.901690i	$-0.642327\pi$
$307$	−15.1519	−0.864765	−0.432383	+	0.901690i	$0.642327\pi$
$308$	0	0
$309$	−3.91334	−0.222622
$310$	0	0
$311$	23.0781	1.30864	0.654319	−	0.756219i	$-0.272955\pi$
$311$	23.0781	1.30864	0.654319	+	0.756219i	$0.272955\pi$
$312$	0	0
$313$	−4.17440	−0.235951	−0.117976	−	0.993016i	$-0.537640\pi$
$313$	−4.17440	−0.235951	−0.117976	+	0.993016i	$0.537640\pi$
$314$	0	0
$315$	−9.82788	−0.553738
$316$	0	0
$317$	−3.52305	−0.197874	−0.0989372	−	0.995094i	$-0.531544\pi$
$317$	−3.52305	−0.197874	−0.0989372	+	0.995094i	$0.531544\pi$
$318$	0	0
$319$	6.49986	0.363922
$320$	0	0
$321$	−4.73095	−0.264056
$322$	0	0
$323$	1.48922	0.0828621
$324$	0	0
$325$	12.1818	0.675727
$326$	0	0
$327$	6.23878	0.345005
$328$	0	0
$329$	−15.2149	−0.838826
$330$	0	0
$331$	−29.2471	−1.60757	−0.803783	−	0.594922i	$-0.797183\pi$
$331$	−29.2471	−1.60757	−0.803783	+	0.594922i	$0.797183\pi$
$332$	0	0
$333$	5.14545	0.281969
$334$	0	0
$335$	−33.9041	−1.85238
$336$	0	0
$337$	−14.0080	−0.763062	−0.381531	−	0.924356i	$-0.624603\pi$
$337$	−14.0080	−0.763062	−0.381531	+	0.924356i	$0.624603\pi$
$338$	0	0
$339$	27.6456	1.50150
$340$	0	0
$341$	4.67893	0.253378
$342$	0	0
$343$	15.5053	0.837208
$344$	0	0
$345$	35.7533	1.92489
$346$	0	0
$347$	21.0868	1.13200	0.565998	−	0.824406i	$-0.308491\pi$
$347$	21.0868	1.13200	0.565998	+	0.824406i	$0.308491\pi$
$348$	0	0
$349$	0.107606	0.00576000	0.00288000	−	0.999996i	$-0.499083\pi$
$349$	0.107606	0.00576000	0.00288000	+	0.999996i	$0.499083\pi$
$350$	0	0
$351$	9.95513	0.531366
$352$	0	0
$353$	4.99595	0.265907	0.132954	−	0.991122i	$-0.457554\pi$
$353$	4.99595	0.265907	0.132954	+	0.991122i	$0.457554\pi$
$354$	0	0
$355$	28.3165	1.50288
$356$	0	0
$357$	−15.3119	−0.810389
$358$	0	0
$359$	2.21452	0.116878	0.0584390	−	0.998291i	$-0.481388\pi$
$359$	2.21452	0.116878	0.0584390	+	0.998291i	$0.481388\pi$
$360$	0	0
$361$	−18.8311	−0.991108
$362$	0	0
$363$	13.7241	0.720329
$364$	0	0
$365$	25.8793	1.35458
$366$	0	0
$367$	3.55114	0.185368	0.0926841	−	0.995696i	$-0.470455\pi$
$367$	3.55114	0.185368	0.0926841	+	0.995696i	$0.470455\pi$
$368$	0	0
$369$	−0.693367	−0.0360953
$370$	0	0
$371$	21.2333	1.10238
$372$	0	0
$373$	−22.0737	−1.14293	−0.571465	−	0.820626i	$-0.693625\pi$
$373$	−22.0737	−1.14293	−0.571465	+	0.820626i	$0.693625\pi$
$374$	0	0
$375$	9.47056	0.489057
$376$	0	0
$377$	9.71266	0.500228
$378$	0	0
$379$	9.36188	0.480887	0.240444	−	0.970663i	$-0.422707\pi$
$379$	9.36188	0.480887	0.240444	+	0.970663i	$0.422707\pi$
$380$	0	0
$381$	−22.2457	−1.13968
$382$	0	0
$383$	−35.4810	−1.81299	−0.906496	−	0.422214i	$-0.861253\pi$
$383$	−35.4810	−1.81299	−0.906496	+	0.422214i	$0.861253\pi$
$384$	0	0
$385$	12.0380	0.613512
$386$	0	0
$387$	1.11251	0.0565521
$388$	0	0
$389$	16.7585	0.849690	0.424845	−	0.905266i	$-0.360329\pi$
$389$	16.7585	0.849690	0.424845	+	0.905266i	$0.360329\pi$
$390$	0	0
$391$	−26.2786	−1.32897
$392$	0	0
$393$	−1.06915	−0.0539314
$394$	0	0
$395$	18.9175	0.951843
$396$	0	0
$397$	16.5261	0.829419	0.414710	−	0.909954i	$-0.363883\pi$
$397$	16.5261	0.829419	0.414710	+	0.909954i	$0.363883\pi$
$398$	0	0
$399$	−1.73703	−0.0869601
$400$	0	0
$401$	6.16583	0.307907	0.153953	−	0.988078i	$-0.450799\pi$
$401$	6.16583	0.307907	0.153953	+	0.988078i	$0.450799\pi$
$402$	0	0
$403$	6.99168	0.348280
$404$	0	0
$405$	17.9210	0.890503
$406$	0	0
$407$	−6.30255	−0.312406
$408$	0	0
$409$	8.25865	0.408364	0.204182	−	0.978933i	$-0.434547\pi$
$409$	8.25865	0.408364	0.204182	+	0.978933i	$0.434547\pi$
$410$	0	0
$411$	−15.5362	−0.766344
$412$	0	0
$413$	−5.05967	−0.248970
$414$	0	0
$415$	−27.7452	−1.36196
$416$	0	0
$417$	−10.2130	−0.500135
$418$	0	0
$419$	14.5239	0.709539	0.354770	−	0.934954i	$-0.384559\pi$
$419$	14.5239	0.709539	0.354770	+	0.934954i	$0.384559\pi$
$420$	0	0
$421$	−10.2628	−0.500180	−0.250090	−	0.968223i	$-0.580460\pi$
$421$	−10.2628	−0.500180	−0.250090	+	0.968223i	$0.580460\pi$
$422$	0	0
$423$	−4.94285	−0.240330
$424$	0	0
$425$	−25.0767	−1.21640
$426$	0	0
$427$	−19.9877	−0.967270
$428$	0	0
$429$	−2.95986	−0.142903
$430$	0	0
$431$	−17.8646	−0.860509	−0.430255	−	0.902708i	$-0.641576\pi$
$431$	−17.8646	−0.860509	−0.430255	+	0.902708i	$0.641576\pi$
$432$	0	0
$433$	11.4447	0.549999	0.274999	−	0.961444i	$-0.411322\pi$
$433$	11.4447	0.549999	0.274999	+	0.961444i	$0.411322\pi$
$434$	0	0
$435$	27.2025	1.30426
$436$	0	0
$437$	−2.98113	−0.142607
$438$	0	0
$439$	15.2893	0.729717	0.364858	−	0.931063i	$-0.381117\pi$
$439$	15.2893	0.729717	0.364858	+	0.931063i	$0.381117\pi$
$440$	0	0
$441$	−1.69415	−0.0806740
$442$	0	0
$443$	−16.8040	−0.798381	−0.399190	−	0.916868i	$-0.630709\pi$
$443$	−16.8040	−0.798381	−0.399190	+	0.916868i	$0.630709\pi$
$444$	0	0
$445$	29.0453	1.37688
$446$	0	0
$447$	−7.45411	−0.352567
$448$	0	0
$449$	−35.9088	−1.69464	−0.847320	−	0.531082i	$-0.821786\pi$
$449$	−35.9088	−1.69464	−0.847320	+	0.531082i	$0.821786\pi$
$450$	0	0
$451$	0.849291	0.0399916
$452$	0	0
$453$	17.0235	0.799833
$454$	0	0
$455$	17.9882	0.843299
$456$	0	0
$457$	−26.7592	−1.25174	−0.625872	−	0.779926i	$-0.715257\pi$
$457$	−26.7592	−1.25174	−0.625872	+	0.779926i	$0.715257\pi$
$458$	0	0
$459$	−20.4930	−0.956530
$460$	0	0
$461$	−5.53757	−0.257910	−0.128955	−	0.991650i	$-0.541162\pi$
$461$	−5.53757	−0.257910	−0.128955	+	0.991650i	$0.541162\pi$
$462$	0	0
$463$	0.335915	0.0156113	0.00780564	−	0.999970i	$-0.497515\pi$
$463$	0.335915	0.0156113	0.00780564	+	0.999970i	$0.497515\pi$
$464$	0	0
$465$	19.5818	0.908084
$466$	0	0
$467$	17.6283	0.815741	0.407871	−	0.913040i	$-0.366271\pi$
$467$	17.6283	0.815741	0.407871	+	0.913040i	$0.366271\pi$
$468$	0	0
$469$	−29.0662	−1.34215
$470$	0	0
$471$	−24.6672	−1.13660
$472$	0	0
$473$	−1.36269	−0.0626566
$474$	0	0
$475$	−2.84479	−0.130528
$476$	0	0
$477$	6.89803	0.315839
$478$	0	0
$479$	−11.1186	−0.508022	−0.254011	−	0.967201i	$-0.581750\pi$
$479$	−11.1186	−0.508022	−0.254011	+	0.967201i	$0.581750\pi$
$480$	0	0
$481$	−9.41783	−0.429416
$482$	0	0
$483$	30.6515	1.39469
$484$	0	0
$485$	−28.5379	−1.29584
$486$	0	0
$487$	−9.01111	−0.408332	−0.204166	−	0.978936i	$-0.565448\pi$
$487$	−9.01111	−0.408332	−0.204166	+	0.978936i	$0.565448\pi$
$488$	0	0
$489$	2.40153	0.108601
$490$	0	0
$491$	9.58989	0.432786	0.216393	−	0.976306i	$-0.430571\pi$
$491$	9.58989	0.432786	0.216393	+	0.976306i	$0.430571\pi$
$492$	0	0
$493$	−19.9938	−0.900477
$494$	0	0
$495$	3.91076	0.175775
$496$	0	0
$497$	24.2759	1.08892
$498$	0	0
$499$	27.2968	1.22197	0.610987	−	0.791641i	$-0.290773\pi$
$499$	27.2968	1.22197	0.610987	+	0.791641i	$0.290773\pi$
$500$	0	0
$501$	−13.6174	−0.608382
$502$	0	0
$503$	−14.6651	−0.653884	−0.326942	−	0.945044i	$-0.606018\pi$
$503$	−14.6651	−0.653884	−0.326942	+	0.945044i	$0.606018\pi$
$504$	0	0
$505$	24.5484	1.09239
$506$	0	0
$507$	14.1375	0.627866
$508$	0	0
$509$	−34.0353	−1.50859	−0.754293	−	0.656538i	$-0.772020\pi$
$509$	−34.0353	−1.50859	−0.754293	+	0.656538i	$0.772020\pi$
$510$	0	0
$511$	22.1865	0.981471
$512$	0	0
$513$	−2.32479	−0.102642
$514$	0	0
$515$	−9.46380	−0.417025
$516$	0	0
$517$	6.05440	0.266272
$518$	0	0
$519$	−24.3696	−1.06971
$520$	0	0
$521$	−1.68454	−0.0738011	−0.0369006	−	0.999319i	$-0.511748\pi$
$521$	−1.68454	−0.0738011	−0.0369006	+	0.999319i	$0.511748\pi$
$522$	0	0
$523$	−11.7112	−0.512097	−0.256048	−	0.966664i	$-0.582421\pi$
$523$	−11.7112	−0.512097	−0.256048	+	0.966664i	$0.582421\pi$
$524$	0	0
$525$	29.2496	1.27656
$526$	0	0
$527$	−14.3926	−0.626951
$528$	0	0
$529$	29.6049	1.28717
$530$	0	0
$531$	−1.64373	−0.0713317
$532$	0	0
$533$	1.26909	0.0549702
$534$	0	0
$535$	−11.4410	−0.494639
$536$	0	0
$537$	20.6315	0.890315
$538$	0	0
$539$	2.07514	0.0893824
$540$	0	0
$541$	24.5002	1.05335	0.526673	−	0.850068i	$-0.323439\pi$
$541$	24.5002	1.05335	0.526673	+	0.850068i	$0.323439\pi$
$542$	0	0
$543$	−13.5549	−0.581696
$544$	0	0
$545$	15.0875	0.646277
$546$	0	0
$547$	−16.6938	−0.713774	−0.356887	−	0.934148i	$-0.616162\pi$
$547$	−16.6938	−0.713774	−0.356887	+	0.934148i	$0.616162\pi$
$548$	0	0
$549$	−6.49336	−0.277130
$550$	0	0
$551$	−2.26817	−0.0966271
$552$	0	0
$553$	16.2181	0.689664
$554$	0	0
$555$	−26.3768	−1.11963
$556$	0	0
$557$	27.5674	1.16807	0.584034	−	0.811729i	$-0.301473\pi$
$557$	27.5674	1.16807	0.584034	+	0.811729i	$0.301473\pi$
$558$	0	0
$559$	−2.03625	−0.0861244
$560$	0	0
$561$	6.09297	0.257245
$562$	0	0
$563$	−21.1020	−0.889343	−0.444671	−	0.895694i	$-0.646680\pi$
$563$	−21.1020	−0.889343	−0.444671	+	0.895694i	$0.646680\pi$
$564$	0	0
$565$	66.8564	2.81267
$566$	0	0
$567$	15.3638	0.645219
$568$	0	0
$569$	−8.27057	−0.346720	−0.173360	−	0.984859i	$-0.555462\pi$
$569$	−8.27057	−0.346720	−0.173360	+	0.984859i	$0.555462\pi$
$570$	0	0
$571$	−18.7496	−0.784648	−0.392324	−	0.919827i	$-0.628329\pi$
$571$	−18.7496	−0.784648	−0.392324	+	0.919827i	$0.628329\pi$
$572$	0	0
$573$	−0.554723	−0.0231739
$574$	0	0
$575$	50.1990	2.09344
$576$	0	0
$577$	−18.2417	−0.759414	−0.379707	−	0.925107i	$-0.623975\pi$
$577$	−18.2417	−0.759414	−0.379707	+	0.925107i	$0.623975\pi$
$578$	0	0
$579$	34.9972	1.45443
$580$	0	0
$581$	−23.7861	−0.986813
$582$	0	0
$583$	−8.44926	−0.349933
$584$	0	0
$585$	5.84380	0.241611
$586$	0	0
$587$	12.3247	0.508697	0.254348	−	0.967113i	$-0.418139\pi$
$587$	12.3247	0.508697	0.254348	+	0.967113i	$0.418139\pi$
$588$	0	0
$589$	−1.63274	−0.0672760
$590$	0	0
$591$	−11.5875	−0.476646
$592$	0	0
$593$	−32.8645	−1.34958	−0.674791	−	0.738009i	$-0.735766\pi$
$593$	−32.8645	−1.34958	−0.674791	+	0.738009i	$0.735766\pi$
$594$	0	0
$595$	−37.0293	−1.51805
$596$	0	0
$597$	33.8106	1.38378
$598$	0	0
$599$	−24.9876	−1.02097	−0.510483	−	0.859888i	$-0.670533\pi$
$599$	−24.9876	−1.02097	−0.510483	+	0.859888i	$0.670533\pi$
$600$	0	0
$601$	15.9854	0.652058	0.326029	−	0.945360i	$-0.394289\pi$
$601$	15.9854	0.652058	0.326029	+	0.945360i	$0.394289\pi$
$602$	0	0
$603$	−9.44271	−0.384537
$604$	0	0
$605$	33.1896	1.34935
$606$	0	0
$607$	−17.9750	−0.729583	−0.364792	−	0.931089i	$-0.618860\pi$
$607$	−17.9750	−0.729583	−0.364792	+	0.931089i	$0.618860\pi$
$608$	0	0
$609$	23.3209	0.945010
$610$	0	0
$611$	9.04702	0.366003
$612$	0	0
$613$	−6.04092	−0.243990	−0.121995	−	0.992531i	$-0.538929\pi$
$613$	−6.04092	−0.243990	−0.121995	+	0.992531i	$0.538929\pi$
$614$	0	0
$615$	3.55437	0.143326
$616$	0	0
$617$	45.9257	1.84890	0.924449	−	0.381307i	$-0.124526\pi$
$617$	45.9257	1.84890	0.924449	+	0.381307i	$0.124526\pi$
$618$	0	0
$619$	−32.9027	−1.32247	−0.661236	−	0.750178i	$-0.729968\pi$
$619$	−32.9027	−1.32247	−0.661236	+	0.750178i	$0.729968\pi$
$620$	0	0
$621$	41.0231	1.64620
$622$	0	0
$623$	24.9008	0.997628
$624$	0	0
$625$	−11.7030	−0.468119
$626$	0	0
$627$	0.691206	0.0276041
$628$	0	0
$629$	19.3869	0.773007
$630$	0	0
$631$	13.7802	0.548582	0.274291	−	0.961647i	$-0.411557\pi$
$631$	13.7802	0.548582	0.274291	+	0.961647i	$0.411557\pi$
$632$	0	0
$633$	41.2810	1.64077
$634$	0	0
$635$	−53.7976	−2.13489
$636$	0	0
$637$	3.10085	0.122860
$638$	0	0
$639$	7.88648	0.311984
$640$	0	0
$641$	31.1970	1.23221	0.616104	−	0.787665i	$-0.288710\pi$
$641$	31.1970	1.23221	0.616104	+	0.787665i	$0.288710\pi$
$642$	0	0
$643$	17.4467	0.688031	0.344015	−	0.938964i	$-0.388213\pi$
$643$	17.4467	0.688031	0.344015	+	0.938964i	$0.388213\pi$
$644$	0	0
$645$	−5.70299	−0.224555
$646$	0	0
$647$	−32.7710	−1.28836	−0.644180	−	0.764874i	$-0.722801\pi$
$647$	−32.7710	−1.28836	−0.644180	+	0.764874i	$0.722801\pi$
$648$	0	0
$649$	2.01337	0.0790316
$650$	0	0
$651$	16.7876	0.657957
$652$	0	0
$653$	−2.45674	−0.0961397	−0.0480699	−	0.998844i	$-0.515307\pi$
$653$	−2.45674	−0.0961397	−0.0480699	+	0.998844i	$0.515307\pi$
$654$	0	0
$655$	−2.58556	−0.101026
$656$	0	0
$657$	7.20769	0.281199
$658$	0	0
$659$	−21.9971	−0.856886	−0.428443	−	0.903569i	$-0.640938\pi$
$659$	−21.9971	−0.856886	−0.428443	+	0.903569i	$0.640938\pi$
$660$	0	0
$661$	−46.5182	−1.80935	−0.904675	−	0.426102i	$-0.859886\pi$
$661$	−46.5182	−1.80935	−0.904675	+	0.426102i	$0.859886\pi$
$662$	0	0
$663$	9.10465	0.353595
$664$	0	0
$665$	−4.20072	−0.162897
$666$	0	0
$667$	40.0239	1.54973
$668$	0	0
$669$	25.9345	1.00269
$670$	0	0
$671$	7.95359	0.307045
$672$	0	0
$673$	−16.8656	−0.650122	−0.325061	−	0.945693i	$-0.605385\pi$
$673$	−16.8656	−0.650122	−0.325061	+	0.945693i	$0.605385\pi$
$674$	0	0
$675$	39.1469	1.50676
$676$	0	0
$677$	−11.4013	−0.438186	−0.219093	−	0.975704i	$-0.570310\pi$
$677$	−11.4013	−0.438186	−0.219093	+	0.975704i	$0.570310\pi$
$678$	0	0
$679$	−24.4657	−0.938907
$680$	0	0
$681$	−17.5815	−0.673725
$682$	0	0
$683$	−43.0851	−1.64861	−0.824303	−	0.566149i	$-0.808433\pi$
$683$	−43.0851	−1.64861	−0.824303	+	0.566149i	$0.808433\pi$
$684$	0	0
$685$	−37.5718	−1.43555
$686$	0	0
$687$	−8.13397	−0.310330
$688$	0	0
$689$	−12.6256	−0.480998
$690$	0	0
$691$	−23.0710	−0.877661	−0.438831	−	0.898570i	$-0.644607\pi$
$691$	−23.0710	−0.877661	−0.438831	+	0.898570i	$0.644607\pi$
$692$	0	0
$693$	3.35272	0.127359
$694$	0	0
$695$	−24.6986	−0.936872
$696$	0	0
$697$	−2.61246	−0.0989538
$698$	0	0
$699$	11.0639	0.418476
$700$	0	0
$701$	24.2724	0.916756	0.458378	−	0.888757i	$-0.348431\pi$
$701$	24.2724	0.916756	0.458378	+	0.888757i	$0.348431\pi$
$702$	0	0
$703$	2.19931	0.0829487
$704$	0	0
$705$	25.3382	0.954293
$706$	0	0
$707$	21.0455	0.791497
$708$	0	0
$709$	−9.47627	−0.355889	−0.177944	−	0.984041i	$-0.556945\pi$
$709$	−9.47627	−0.355889	−0.177944	+	0.984041i	$0.556945\pi$
$710$	0	0
$711$	5.26875	0.197594
$712$	0	0
$713$	28.8113	1.07899
$714$	0	0
$715$	−7.15795	−0.267692
$716$	0	0
$717$	−1.54698	−0.0577730
$718$	0	0
$719$	44.5525	1.66153	0.830765	−	0.556624i	$-0.187904\pi$
$719$	44.5525	1.66153	0.830765	+	0.556624i	$0.187904\pi$
$720$	0	0
$721$	−8.11337	−0.302158
$722$	0	0
$723$	20.8865	0.776779
$724$	0	0
$725$	38.1934	1.41847
$726$	0	0
$727$	8.38059	0.310819	0.155409	−	0.987850i	$-0.450330\pi$
$727$	8.38059	0.310819	0.155409	+	0.987850i	$0.450330\pi$
$728$	0	0
$729$	29.2171	1.08211
$730$	0	0
$731$	4.19170	0.155035
$732$	0	0
$733$	−23.4275	−0.865314	−0.432657	−	0.901559i	$-0.642424\pi$
$733$	−23.4275	−0.865314	−0.432657	+	0.901559i	$0.642424\pi$
$734$	0	0
$735$	8.68464	0.320338
$736$	0	0
$737$	11.5662	0.426046
$738$	0	0
$739$	12.4200	0.456877	0.228438	−	0.973558i	$-0.426638\pi$
$739$	12.4200	0.456877	0.228438	+	0.973558i	$0.426638\pi$
$740$	0	0
$741$	1.03286	0.0379431
$742$	0	0
$743$	48.0187	1.76164	0.880818	−	0.473454i	$-0.156993\pi$
$743$	48.0187	1.76164	0.880818	+	0.473454i	$0.156993\pi$
$744$	0	0
$745$	−18.0266	−0.660443
$746$	0	0
$747$	−7.72735	−0.282729
$748$	0	0
$749$	−9.80847	−0.358394
$750$	0	0
$751$	19.6075	0.715489	0.357745	−	0.933819i	$-0.383546\pi$
$751$	19.6075	0.715489	0.357745	+	0.933819i	$0.383546\pi$
$752$	0	0
$753$	1.42772	0.0520289
$754$	0	0
$755$	41.1686	1.49828
$756$	0	0
$757$	−34.9740	−1.27115	−0.635575	−	0.772039i	$-0.719237\pi$
$757$	−34.9740	−1.27115	−0.635575	+	0.772039i	$0.719237\pi$
$758$	0	0
$759$	−12.1970	−0.442723
$760$	0	0
$761$	40.2851	1.46033	0.730167	−	0.683269i	$-0.239442\pi$
$761$	40.2851	1.46033	0.730167	+	0.683269i	$0.239442\pi$
$762$	0	0
$763$	12.9346	0.468264
$764$	0	0
$765$	−12.0297	−0.434933
$766$	0	0
$767$	3.00855	0.108633
$768$	0	0
$769$	−17.2057	−0.620452	−0.310226	−	0.950663i	$-0.600405\pi$
$769$	−17.2057	−0.620452	−0.310226	+	0.950663i	$0.600405\pi$
$770$	0	0
$771$	10.5806	0.381051
$772$	0	0
$773$	−15.2599	−0.548860	−0.274430	−	0.961607i	$-0.588489\pi$
$773$	−15.2599	−0.548860	−0.274430	+	0.961607i	$0.588489\pi$
$774$	0	0
$775$	27.4936	0.987599
$776$	0	0
$777$	−22.6130	−0.811236
$778$	0	0
$779$	−0.296366	−0.0106184
$780$	0	0
$781$	−9.65998	−0.345661
$782$	0	0
$783$	31.2120	1.11543
$784$	0	0
$785$	−59.6536	−2.12913
$786$	0	0
$787$	54.0839	1.92788	0.963941	−	0.266116i	$-0.0857405\pi$
$787$	54.0839	1.92788	0.963941	+	0.266116i	$0.0857405\pi$
$788$	0	0
$789$	8.88594	0.316348
$790$	0	0
$791$	57.3164	2.03794
$792$	0	0
$793$	11.8850	0.422047
$794$	0	0
$795$	−35.3610	−1.25412
$796$	0	0
$797$	34.3246	1.21584	0.607920	−	0.793998i	$-0.292004\pi$
$797$	34.3246	1.21584	0.607920	+	0.793998i	$0.292004\pi$
$798$	0	0
$799$	−18.6236	−0.658855
$800$	0	0
$801$	8.08947	0.285828
$802$	0	0
$803$	−8.82855	−0.311553
$804$	0	0
$805$	74.1257	2.61259
$806$	0	0
$807$	5.24101	0.184492
$808$	0	0
$809$	−26.4035	−0.928296	−0.464148	−	0.885758i	$-0.653639\pi$
$809$	−26.4035	−0.928296	−0.464148	+	0.885758i	$0.653639\pi$
$810$	0	0
$811$	−12.0644	−0.423639	−0.211820	−	0.977309i	$-0.567939\pi$
$811$	−12.0644	−0.423639	−0.211820	+	0.977309i	$0.567939\pi$
$812$	0	0
$813$	−27.4078	−0.961235
$814$	0	0
$815$	5.80772	0.203436
$816$	0	0
$817$	0.475520	0.0166363
$818$	0	0
$819$	5.00993	0.175061
$820$	0	0
$821$	−39.0872	−1.36415	−0.682076	−	0.731282i	$-0.738923\pi$
$821$	−39.0872	−1.36415	−0.682076	+	0.731282i	$0.738923\pi$
$822$	0	0
$823$	15.9271	0.555184	0.277592	−	0.960699i	$-0.410464\pi$
$823$	15.9271	0.555184	0.277592	+	0.960699i	$0.410464\pi$
$824$	0	0
$825$	−11.6391	−0.405223
$826$	0	0
$827$	9.14351	0.317951	0.158975	−	0.987283i	$-0.449181\pi$
$827$	9.14351	0.317951	0.158975	+	0.987283i	$0.449181\pi$
$828$	0	0
$829$	−41.9821	−1.45810	−0.729049	−	0.684462i	$-0.760037\pi$
$829$	−41.9821	−1.45810	−0.729049	+	0.684462i	$0.760037\pi$
$830$	0	0
$831$	−15.4443	−0.535756
$832$	0	0
$833$	−6.38320	−0.221165
$834$	0	0
$835$	−32.9316	−1.13964
$836$	0	0
$837$	22.4680	0.776609
$838$	0	0
$839$	26.9855	0.931644	0.465822	−	0.884879i	$-0.345759\pi$
$839$	26.9855	0.931644	0.465822	+	0.884879i	$0.345759\pi$
$840$	0	0
$841$	1.45184	0.0500636
$842$	0	0
$843$	−45.2407	−1.55817
$844$	0	0
$845$	34.1892	1.17614
$846$	0	0
$847$	28.4536	0.977679
$848$	0	0
$849$	6.13448	0.210535
$850$	0	0
$851$	−38.8090	−1.33036
$852$	0	0
$853$	−5.55966	−0.190359	−0.0951796	−	0.995460i	$-0.530343\pi$
$853$	−5.55966	−0.190359	−0.0951796	+	0.995460i	$0.530343\pi$
$854$	0	0
$855$	−1.36468	−0.0466712
$856$	0	0
$857$	−50.5395	−1.72640	−0.863199	−	0.504864i	$-0.831543\pi$
$857$	−50.5395	−1.72640	−0.863199	+	0.504864i	$0.831543\pi$
$858$	0	0
$859$	−15.4386	−0.526758	−0.263379	−	0.964692i	$-0.584837\pi$
$859$	−15.4386	−0.526758	−0.263379	+	0.964692i	$0.584837\pi$
$860$	0	0
$861$	3.04718	0.103848
$862$	0	0
$863$	22.0092	0.749201	0.374601	−	0.927186i	$-0.377780\pi$
$863$	22.0092	0.749201	0.374601	+	0.927186i	$0.377780\pi$
$864$	0	0
$865$	−58.9339	−2.00381
$866$	0	0
$867$	5.52898	0.187774
$868$	0	0
$869$	−6.45359	−0.218923
$870$	0	0
$871$	17.2832	0.585619
$872$	0	0
$873$	−7.94814	−0.269004
$874$	0	0
$875$	19.6349	0.663781
$876$	0	0
$877$	−58.4257	−1.97290	−0.986448	−	0.164074i	$-0.947536\pi$
$877$	−58.4257	−1.97290	−0.986448	+	0.164074i	$0.947536\pi$
$878$	0	0
$879$	−42.0188	−1.41726
$880$	0	0
$881$	−22.1759	−0.747124	−0.373562	−	0.927605i	$-0.621864\pi$
$881$	−22.1759	−0.747124	−0.373562	+	0.927605i	$0.621864\pi$
$882$	0	0
$883$	12.1099	0.407531	0.203766	−	0.979020i	$-0.434682\pi$
$883$	12.1099	0.407531	0.203766	+	0.979020i	$0.434682\pi$
$884$	0	0
$885$	8.42614	0.283242
$886$	0	0
$887$	17.3918	0.583959	0.291979	−	0.956425i	$-0.405686\pi$
$887$	17.3918	0.583959	0.291979	+	0.956425i	$0.405686\pi$
$888$	0	0
$889$	−46.1211	−1.54685
$890$	0	0
$891$	−6.11364	−0.204815
$892$	0	0
$893$	−2.11272	−0.0706995
$894$	0	0
$895$	49.8940	1.66777
$896$	0	0
$897$	−18.2258	−0.608543
$898$	0	0
$899$	21.9208	0.731100
$900$	0	0
$901$	25.9903	0.865861
$902$	0	0
$903$	−4.88921	−0.162703
$904$	0	0
$905$	−32.7803	−1.08965
$906$	0	0
$907$	−19.1560	−0.636065	−0.318033	−	0.948080i	$-0.603022\pi$
$907$	−19.1560	−0.636065	−0.318033	+	0.948080i	$0.603022\pi$
$908$	0	0
$909$	6.83702	0.226770
$910$	0	0
$911$	8.05313	0.266812	0.133406	−	0.991061i	$-0.457409\pi$
$911$	8.05313	0.266812	0.133406	+	0.991061i	$0.457409\pi$
$912$	0	0
$913$	9.46508	0.313248
$914$	0	0
$915$	33.2865	1.10042
$916$	0	0
$917$	−2.21662	−0.0731992
$918$	0	0
$919$	31.6712	1.04474	0.522368	−	0.852720i	$-0.325049\pi$
$919$	31.6712	1.04474	0.522368	+	0.852720i	$0.325049\pi$
$920$	0	0
$921$	21.6327	0.712820
$922$	0	0
$923$	−14.4348	−0.475127
$924$	0	0
$925$	−37.0340	−1.21767
$926$	0	0
$927$	−2.63578	−0.0865704
$928$	0	0
$929$	−10.7745	−0.353499	−0.176750	−	0.984256i	$-0.556558\pi$
$929$	−10.7745	−0.353499	−0.176750	+	0.984256i	$0.556558\pi$
$930$	0	0
$931$	−0.724131	−0.0237324
$932$	0	0
$933$	−32.9490	−1.07870
$934$	0	0
$935$	14.7349	0.481882
$936$	0	0
$937$	43.5180	1.42167	0.710835	−	0.703358i	$-0.248317\pi$
$937$	43.5180	1.42167	0.710835	+	0.703358i	$0.248317\pi$
$938$	0	0
$939$	5.95987	0.194493
$940$	0	0
$941$	14.5468	0.474213	0.237106	−	0.971484i	$-0.423801\pi$
$941$	14.5468	0.474213	0.237106	+	0.971484i	$0.423801\pi$
$942$	0	0
$943$	5.22965	0.170301
$944$	0	0
$945$	57.8058	1.88042
$946$	0	0
$947$	2.86612	0.0931363	0.0465682	−	0.998915i	$-0.485172\pi$
$947$	2.86612	0.0931363	0.0465682	+	0.998915i	$0.485172\pi$
$948$	0	0
$949$	−13.1924	−0.428243
$950$	0	0
$951$	5.02993	0.163107
$952$	0	0
$953$	11.9853	0.388242	0.194121	−	0.980978i	$-0.437815\pi$
$953$	11.9853	0.388242	0.194121	+	0.980978i	$0.437815\pi$
$954$	0	0
$955$	−1.34151	−0.0434102
$956$	0	0
$957$	−9.27997	−0.299979
$958$	0	0
$959$	−32.2106	−1.04013
$960$	0	0
$961$	−15.2203	−0.490976
$962$	0	0
$963$	−3.18647	−0.102682
$964$	0	0
$965$	84.6351	2.72450
$966$	0	0
$967$	2.35305	0.0756691	0.0378345	−	0.999284i	$-0.487954\pi$
$967$	2.35305	0.0756691	0.0378345	+	0.999284i	$0.487954\pi$
$968$	0	0
$969$	−2.12618	−0.0683027
$970$	0	0
$971$	9.22827	0.296149	0.148075	−	0.988976i	$-0.452692\pi$
$971$	9.22827	0.296149	0.148075	+	0.988976i	$0.452692\pi$
$972$	0	0
$973$	−21.1743	−0.678816
$974$	0	0
$975$	−17.3922	−0.556998
$976$	0	0
$977$	−15.0978	−0.483020	−0.241510	−	0.970398i	$-0.577643\pi$
$977$	−15.0978	−0.483020	−0.241510	+	0.970398i	$0.577643\pi$
$978$	0	0
$979$	−9.90863	−0.316681
$980$	0	0
$981$	4.20205	0.134161
$982$	0	0
$983$	54.8487	1.74940	0.874701	−	0.484663i	$-0.161058\pi$
$983$	54.8487	1.74940	0.874701	+	0.484663i	$0.161058\pi$
$984$	0	0
$985$	−28.0225	−0.892872
$986$	0	0
$987$	21.7226	0.691439
$988$	0	0
$989$	−8.39100	−0.266818
$990$	0	0
$991$	−22.5642	−0.716776	−0.358388	−	0.933573i	$-0.616673\pi$
$991$	−22.5642	−0.716776	−0.358388	+	0.933573i	$0.616673\pi$
$992$	0	0
$993$	41.7566	1.32511
$994$	0	0
$995$	81.7656	2.59214
$996$	0	0
$997$	14.8188	0.469318	0.234659	−	0.972078i	$-0.424603\pi$
$997$	14.8188	0.469318	0.234659	+	0.972078i	$0.424603\pi$
$998$	0	0
$999$	−30.2646	−0.957529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.3		9
4.3	odd	2		2008.2.a.a.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.7	✓	9	4.3	odd	2
4016.2.a.h.1.3		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.42772 q^{3} -3.45271 q^{5} -2.96003 q^{7} -0.961621 q^{9} +O(q^{10})\)	\(q-1.42772 q^{3} -3.45271 q^{5} -2.96003 q^{7} -0.961621 q^{9} +1.17787 q^{11} +1.76008 q^{13} +4.92950 q^{15} -3.62318 q^{17} -0.411025 q^{19} +4.22609 q^{21} +7.25292 q^{23} +6.92120 q^{25} +5.65608 q^{27} +5.51832 q^{29} +3.97237 q^{31} -1.68167 q^{33} +10.2201 q^{35} -5.35081 q^{37} -2.51289 q^{39} +0.721040 q^{41} -1.15691 q^{43} +3.32020 q^{45} +5.14013 q^{47} +1.76177 q^{49} +5.17287 q^{51} -7.17334 q^{53} -4.06684 q^{55} +0.586828 q^{57} +1.70933 q^{59} +6.75252 q^{61} +2.84642 q^{63} -6.07703 q^{65} +9.81958 q^{67} -10.3551 q^{69} -8.20123 q^{71} -7.49535 q^{73} -9.88153 q^{75} -3.48653 q^{77} -5.47903 q^{79} -5.19042 q^{81} +8.03576 q^{83} +12.5098 q^{85} -7.87860 q^{87} -8.41234 q^{89} -5.20988 q^{91} -5.67143 q^{93} +1.41915 q^{95} +8.26536 q^{97} -1.13266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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