LMFDB - Embedded newform 4016.2.a.j.1.6

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.46572$ 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)

$f(q)$	$=$	$q-1.46572 q^{3} +0.817924 q^{5} -4.79612 q^{7} -0.851657 q^{9} +O(q^{10})$	$q-1.46572 q^{3} +0.817924 q^{5} -4.79612 q^{7} -0.851657 q^{9} +0.0579767 q^{11} +4.02980 q^{13} -1.19885 q^{15} -0.340875 q^{17} +2.32542 q^{19} +7.02979 q^{21} +1.22363 q^{23} -4.33100 q^{25} +5.64546 q^{27} +7.87416 q^{29} -0.0384920 q^{31} -0.0849778 q^{33} -3.92287 q^{35} +3.18806 q^{37} -5.90657 q^{39} -10.9470 q^{41} +1.02038 q^{43} -0.696591 q^{45} +1.95927 q^{47} +16.0028 q^{49} +0.499628 q^{51} +1.09674 q^{53} +0.0474205 q^{55} -3.40842 q^{57} +12.6889 q^{59} +0.587621 q^{61} +4.08466 q^{63} +3.29607 q^{65} -13.7457 q^{67} -1.79351 q^{69} -8.08740 q^{71} +3.49927 q^{73} +6.34804 q^{75} -0.278064 q^{77} +0.113342 q^{79} -5.71971 q^{81} -2.22079 q^{83} -0.278810 q^{85} -11.5413 q^{87} -13.3707 q^{89} -19.3274 q^{91} +0.0564186 q^{93} +1.90202 q^{95} +2.24936 q^{97} -0.0493763 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	−1.46572	−0.846235	−0.423118	−	0.906075i	$-0.639064\pi$
$3$	−1.46572	−0.846235	−0.423118	+	0.906075i	$0.639064\pi$
$4$	0	0
$5$	0.817924	0.365787	0.182893	−	0.983133i	$-0.441454\pi$
$5$	0.817924	0.365787	0.182893	+	0.983133i	$0.441454\pi$
$6$	0	0
$7$	−4.79612	−1.81276	−0.906382	−	0.422458i	$-0.861167\pi$
$7$	−4.79612	−1.81276	−0.906382	+	0.422458i	$0.861167\pi$
$8$	0	0
$9$	−0.851657	−0.283886
$10$	0	0
$11$	0.0579767	0.0174806	0.00874032	−	0.999962i	$-0.497218\pi$
$11$	0.0579767	0.0174806	0.00874032	+	0.999962i	$0.497218\pi$
$12$	0	0
$13$	4.02980	1.11767	0.558833	−	0.829280i	$-0.311249\pi$
$13$	4.02980	1.11767	0.558833	+	0.829280i	$0.311249\pi$
$14$	0	0
$15$	−1.19885	−0.309542
$16$	0	0
$17$	−0.340875	−0.0826742	−0.0413371	−	0.999145i	$-0.513162\pi$
$17$	−0.340875	−0.0826742	−0.0413371	+	0.999145i	$0.513162\pi$
$18$	0	0
$19$	2.32542	0.533489	0.266744	−	0.963767i	$-0.414052\pi$
$19$	2.32542	0.533489	0.266744	+	0.963767i	$0.414052\pi$
$20$	0	0
$21$	7.02979	1.53403
$22$	0	0
$23$	1.22363	0.255145	0.127573	−	0.991829i	$-0.459281\pi$
$23$	1.22363	0.255145	0.127573	+	0.991829i	$0.459281\pi$
$24$	0	0
$25$	−4.33100	−0.866200
$26$	0	0
$27$	5.64546	1.08647
$28$	0	0
$29$	7.87416	1.46219	0.731097	−	0.682273i	$-0.239009\pi$
$29$	7.87416	1.46219	0.731097	+	0.682273i	$0.239009\pi$
$30$	0	0
$31$	−0.0384920	−0.00691337	−0.00345669	−	0.999994i	$-0.501100\pi$
$31$	−0.0384920	−0.00691337	−0.00345669	+	0.999994i	$0.501100\pi$
$32$	0	0
$33$	−0.0849778	−0.0147927
$34$	0	0
$35$	−3.92287	−0.663085
$36$	0	0
$37$	3.18806	0.524115	0.262057	−	0.965052i	$-0.415599\pi$
$37$	3.18806	0.524115	0.262057	+	0.965052i	$0.415599\pi$
$38$	0	0
$39$	−5.90657	−0.945808
$40$	0	0
$41$	−10.9470	−1.70963	−0.854814	−	0.518935i	$-0.826329\pi$
$41$	−10.9470	−1.70963	−0.854814	+	0.518935i	$0.826329\pi$
$42$	0	0
$43$	1.02038	0.155607	0.0778033	−	0.996969i	$-0.475209\pi$
$43$	1.02038	0.155607	0.0778033	+	0.996969i	$0.475209\pi$
$44$	0	0
$45$	−0.696591	−0.103842
$46$	0	0
$47$	1.95927	0.285789	0.142894	−	0.989738i	$-0.454359\pi$
$47$	1.95927	0.285789	0.142894	+	0.989738i	$0.454359\pi$
$48$	0	0
$49$	16.0028	2.28612
$50$	0	0
$51$	0.499628	0.0699618
$52$	0	0
$53$	1.09674	0.150649	0.0753247	−	0.997159i	$-0.476001\pi$
$53$	1.09674	0.150649	0.0753247	+	0.997159i	$0.476001\pi$
$54$	0	0
$55$	0.0474205	0.00639419
$56$	0	0
$57$	−3.40842	−0.451457
$58$	0	0
$59$	12.6889	1.65195	0.825974	−	0.563708i	$-0.190626\pi$
$59$	12.6889	1.65195	0.825974	+	0.563708i	$0.190626\pi$
$60$	0	0
$61$	0.587621	0.0752371	0.0376186	−	0.999292i	$-0.488023\pi$
$61$	0.587621	0.0752371	0.0376186	+	0.999292i	$0.488023\pi$
$62$	0	0
$63$	4.08466	0.514618
$64$	0	0
$65$	3.29607	0.408827
$66$	0	0
$67$	−13.7457	−1.67930	−0.839650	−	0.543128i	$-0.817240\pi$
$67$	−13.7457	−1.67930	−0.839650	+	0.543128i	$0.817240\pi$
$68$	0	0
$69$	−1.79351	−0.215913
$70$	0	0
$71$	−8.08740	−0.959797	−0.479899	−	0.877324i	$-0.659327\pi$
$71$	−8.08740	−0.959797	−0.479899	+	0.877324i	$0.659327\pi$
$72$	0	0
$73$	3.49927	0.409559	0.204779	−	0.978808i	$-0.434352\pi$
$73$	3.49927	0.409559	0.204779	+	0.978808i	$0.434352\pi$
$74$	0	0
$75$	6.34804	0.733009
$76$	0	0
$77$	−0.278064	−0.0316883
$78$	0	0
$79$	0.113342	0.0127520	0.00637598	−	0.999980i	$-0.497970\pi$
$79$	0.113342	0.0127520	0.00637598	+	0.999980i	$0.497970\pi$
$80$	0	0
$81$	−5.71971	−0.635523
$82$	0	0
$83$	−2.22079	−0.243764	−0.121882	−	0.992545i	$-0.538893\pi$
$83$	−2.22079	−0.243764	−0.121882	+	0.992545i	$0.538893\pi$
$84$	0	0
$85$	−0.278810	−0.0302411
$86$	0	0
$87$	−11.5413	−1.23736
$88$	0	0
$89$	−13.3707	−1.41730	−0.708648	−	0.705562i	$-0.750695\pi$
$89$	−13.3707	−1.41730	−0.708648	+	0.705562i	$0.750695\pi$
$90$	0	0
$91$	−19.3274	−2.02606
$92$	0	0
$93$	0.0564186	0.00585034
$94$	0	0
$95$	1.90202	0.195143
$96$	0	0
$97$	2.24936	0.228387	0.114194	−	0.993459i	$-0.463572\pi$
$97$	2.24936	0.228387	0.114194	+	0.993459i	$0.463572\pi$
$98$	0	0
$99$	−0.0493763	−0.00496250
$100$	0	0
$101$	−9.21422	−0.916849	−0.458424	−	0.888733i	$-0.651586\pi$
$101$	−9.21422	−0.916849	−0.458424	+	0.888733i	$0.651586\pi$
$102$	0	0
$103$	−11.8813	−1.17070	−0.585351	−	0.810780i	$-0.699043\pi$
$103$	−11.8813	−1.17070	−0.585351	+	0.810780i	$0.699043\pi$
$104$	0	0
$105$	5.74983	0.561126
$106$	0	0
$107$	−3.57574	−0.345679	−0.172840	−	0.984950i	$-0.555294\pi$
$107$	−3.57574	−0.345679	−0.172840	+	0.984950i	$0.555294\pi$
$108$	0	0
$109$	−17.0760	−1.63559	−0.817794	−	0.575511i	$-0.804803\pi$
$109$	−17.0760	−1.63559	−0.817794	+	0.575511i	$0.804803\pi$
$110$	0	0
$111$	−4.67282	−0.443524
$112$	0	0
$113$	2.41592	0.227270	0.113635	−	0.993523i	$-0.463750\pi$
$113$	2.41592	0.227270	0.113635	+	0.993523i	$0.463750\pi$
$114$	0	0
$115$	1.00084	0.0933287
$116$	0	0
$117$	−3.43201	−0.317289
$118$	0	0
$119$	1.63488	0.149869
$120$	0	0
$121$	−10.9966	−0.999694
$122$	0	0
$123$	16.0452	1.44675
$124$	0	0
$125$	−7.63205	−0.682631
$126$	0	0
$127$	0.690251	0.0612499	0.0306249	−	0.999531i	$-0.490250\pi$
$127$	0.690251	0.0612499	0.0306249	+	0.999531i	$0.490250\pi$
$128$	0	0
$129$	−1.49559	−0.131680
$130$	0	0
$131$	−5.18818	−0.453294	−0.226647	−	0.973977i	$-0.572776\pi$
$131$	−5.18818	−0.453294	−0.226647	+	0.973977i	$0.572776\pi$
$132$	0	0
$133$	−11.1530	−0.967089
$134$	0	0
$135$	4.61756	0.397416
$136$	0	0
$137$	16.1778	1.38216	0.691080	−	0.722778i	$-0.257135\pi$
$137$	16.1778	1.38216	0.691080	+	0.722778i	$0.257135\pi$
$138$	0	0
$139$	1.14974	0.0975198	0.0487599	−	0.998811i	$-0.484473\pi$
$139$	1.14974	0.0975198	0.0487599	+	0.998811i	$0.484473\pi$
$140$	0	0
$141$	−2.87174	−0.241844
$142$	0	0
$143$	0.233635	0.0195375
$144$	0	0
$145$	6.44046	0.534851
$146$	0	0
$147$	−23.4557	−1.93459
$148$	0	0
$149$	−15.0431	−1.23237	−0.616187	−	0.787600i	$-0.711324\pi$
$149$	−15.0431	−1.23237	−0.616187	+	0.787600i	$0.711324\pi$
$150$	0	0
$151$	12.1893	0.991952	0.495976	−	0.868336i	$-0.334810\pi$
$151$	12.1893	0.991952	0.495976	+	0.868336i	$0.334810\pi$
$152$	0	0
$153$	0.290308	0.0234700
$154$	0	0
$155$	−0.0314836	−0.00252882
$156$	0	0
$157$	20.3795	1.62646	0.813232	−	0.581939i	$-0.197706\pi$
$157$	20.3795	1.62646	0.813232	+	0.581939i	$0.197706\pi$
$158$	0	0
$159$	−1.60752	−0.127485
$160$	0	0
$161$	−5.86870	−0.462518
$162$	0	0
$163$	17.4484	1.36667	0.683333	−	0.730107i	$-0.260530\pi$
$163$	17.4484	1.36667	0.683333	+	0.730107i	$0.260530\pi$
$164$	0	0
$165$	−0.0695054	−0.00541099
$166$	0	0
$167$	−5.21199	−0.403316	−0.201658	−	0.979456i	$-0.564633\pi$
$167$	−5.21199	−0.403316	−0.201658	+	0.979456i	$0.564633\pi$
$168$	0	0
$169$	3.23930	0.249177
$170$	0	0
$171$	−1.98046	−0.151450
$172$	0	0
$173$	−17.2337	−1.31026	−0.655129	−	0.755517i	$-0.727386\pi$
$173$	−17.2337	−1.31026	−0.655129	+	0.755517i	$0.727386\pi$
$174$	0	0
$175$	20.7720	1.57022
$176$	0	0
$177$	−18.5983	−1.39794
$178$	0	0
$179$	9.83883	0.735389	0.367694	−	0.929947i	$-0.380147\pi$
$179$	9.83883	0.735389	0.367694	+	0.929947i	$0.380147\pi$
$180$	0	0
$181$	−14.9404	−1.11051	−0.555257	−	0.831679i	$-0.687380\pi$
$181$	−14.9404	−1.11051	−0.555257	+	0.831679i	$0.687380\pi$
$182$	0	0
$183$	−0.861289	−0.0636683
$184$	0	0
$185$	2.60759	0.191714
$186$	0	0
$187$	−0.0197628	−0.00144520
$188$	0	0
$189$	−27.0763	−1.96951
$190$	0	0
$191$	−13.9873	−1.01209	−0.506044	−	0.862507i	$-0.668893\pi$
$191$	−13.9873	−1.01209	−0.506044	+	0.862507i	$0.668893\pi$
$192$	0	0
$193$	10.3390	0.744220	0.372110	−	0.928189i	$-0.378634\pi$
$193$	10.3390	0.744220	0.372110	+	0.928189i	$0.378634\pi$
$194$	0	0
$195$	−4.83113	−0.345964
$196$	0	0
$197$	12.1548	0.865997	0.432998	−	0.901395i	$-0.357456\pi$
$197$	12.1548	0.865997	0.432998	+	0.901395i	$0.357456\pi$
$198$	0	0
$199$	−0.799245	−0.0566569	−0.0283285	−	0.999599i	$-0.509018\pi$
$199$	−0.799245	−0.0566569	−0.0283285	+	0.999599i	$0.509018\pi$
$200$	0	0
$201$	20.1473	1.42108
$202$	0	0
$203$	−37.7654	−2.65061
$204$	0	0
$205$	−8.95378	−0.625359
$206$	0	0
$207$	−1.04212	−0.0724321
$208$	0	0
$209$	0.134820	0.00932572
$210$	0	0
$211$	18.5006	1.27363	0.636816	−	0.771016i	$-0.280251\pi$
$211$	18.5006	1.27363	0.636816	+	0.771016i	$0.280251\pi$
$212$	0	0
$213$	11.8539	0.812214
$214$	0	0
$215$	0.834593	0.0569188
$216$	0	0
$217$	0.184613	0.0125323
$218$	0	0
$219$	−5.12896	−0.346583
$220$	0	0
$221$	−1.37366	−0.0924021
$222$	0	0
$223$	−24.0200	−1.60850	−0.804248	−	0.594294i	$-0.797431\pi$
$223$	−24.0200	−1.60850	−0.804248	+	0.594294i	$0.797431\pi$
$224$	0	0
$225$	3.68853	0.245902
$226$	0	0
$227$	−2.01808	−0.133945	−0.0669724	−	0.997755i	$-0.521334\pi$
$227$	−2.01808	−0.133945	−0.0669724	+	0.997755i	$0.521334\pi$
$228$	0	0
$229$	−17.9639	−1.18709	−0.593544	−	0.804802i	$-0.702272\pi$
$229$	−17.9639	−1.18709	−0.593544	+	0.804802i	$0.702272\pi$
$230$	0	0
$231$	0.407564	0.0268157
$232$	0	0
$233$	−23.6989	−1.55257	−0.776283	−	0.630384i	$-0.782897\pi$
$233$	−23.6989	−1.55257	−0.776283	+	0.630384i	$0.782897\pi$
$234$	0	0
$235$	1.60253	0.104538
$236$	0	0
$237$	−0.166128	−0.0107912
$238$	0	0
$239$	1.97327	0.127640	0.0638202	−	0.997961i	$-0.479672\pi$
$239$	1.97327	0.127640	0.0638202	+	0.997961i	$0.479672\pi$
$240$	0	0
$241$	3.38373	0.217965	0.108983	−	0.994044i	$-0.465241\pi$
$241$	3.38373	0.217965	0.108983	+	0.994044i	$0.465241\pi$
$242$	0	0
$243$	−8.55288	−0.548667
$244$	0	0
$245$	13.0891	0.836231
$246$	0	0
$247$	9.37099	0.596262
$248$	0	0
$249$	3.25507	0.206282
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	0.0709422	0.00446010
$254$	0	0
$255$	0.408657	0.0255911
$256$	0	0
$257$	−23.3790	−1.45834	−0.729171	−	0.684332i	$-0.760094\pi$
$257$	−23.3790	−1.45834	−0.729171	+	0.684332i	$0.760094\pi$
$258$	0	0
$259$	−15.2904	−0.950097
$260$	0	0
$261$	−6.70609	−0.415096
$262$	0	0
$263$	23.5460	1.45191	0.725954	−	0.687743i	$-0.241399\pi$
$263$	23.5460	1.45191	0.725954	+	0.687743i	$0.241399\pi$
$264$	0	0
$265$	0.897054	0.0551056
$266$	0	0
$267$	19.5978	1.19937
$268$	0	0
$269$	−9.71962	−0.592616	−0.296308	−	0.955092i	$-0.595755\pi$
$269$	−9.71962	−0.592616	−0.296308	+	0.955092i	$0.595755\pi$
$270$	0	0
$271$	−26.4553	−1.60704	−0.803522	−	0.595275i	$-0.797043\pi$
$271$	−26.4553	−1.60704	−0.803522	+	0.595275i	$0.797043\pi$
$272$	0	0
$273$	28.3286	1.71453
$274$	0	0
$275$	−0.251097	−0.0151417
$276$	0	0
$277$	8.10271	0.486845	0.243422	−	0.969920i	$-0.421730\pi$
$277$	8.10271	0.486845	0.243422	+	0.969920i	$0.421730\pi$
$278$	0	0
$279$	0.0327820	0.00196261
$280$	0	0
$281$	−28.1144	−1.67717	−0.838583	−	0.544774i	$-0.816615\pi$
$281$	−28.1144	−1.67717	−0.838583	+	0.544774i	$0.816615\pi$
$282$	0	0
$283$	6.57104	0.390608	0.195304	−	0.980743i	$-0.437431\pi$
$283$	6.57104	0.390608	0.195304	+	0.980743i	$0.437431\pi$
$284$	0	0
$285$	−2.78783	−0.165137
$286$	0	0
$287$	52.5030	3.09915
$288$	0	0
$289$	−16.8838	−0.993165
$290$	0	0
$291$	−3.29693	−0.193270
$292$	0	0
$293$	−7.14103	−0.417184	−0.208592	−	0.978003i	$-0.566888\pi$
$293$	−7.14103	−0.417184	−0.208592	+	0.978003i	$0.566888\pi$
$294$	0	0
$295$	10.3785	0.604261
$296$	0	0
$297$	0.327305	0.0189922
$298$	0	0
$299$	4.93100	0.285167
$300$	0	0
$301$	−4.89387	−0.282078
$302$	0	0
$303$	13.5055	0.775870
$304$	0	0
$305$	0.480629	0.0275207
$306$	0	0
$307$	13.1900	0.752795	0.376397	−	0.926458i	$-0.377163\pi$
$307$	13.1900	0.752795	0.376397	+	0.926458i	$0.377163\pi$
$308$	0	0
$309$	17.4147	0.990689
$310$	0	0
$311$	23.9450	1.35779	0.678897	−	0.734233i	$-0.262458\pi$
$311$	23.9450	1.35779	0.678897	+	0.734233i	$0.262458\pi$
$312$	0	0
$313$	−5.34599	−0.302173	−0.151087	−	0.988521i	$-0.548277\pi$
$313$	−5.34599	−0.302173	−0.151087	+	0.988521i	$0.548277\pi$
$314$	0	0
$315$	3.34094	0.188241
$316$	0	0
$317$	2.16864	0.121803	0.0609014	−	0.998144i	$-0.480602\pi$
$317$	2.16864	0.121803	0.0609014	+	0.998144i	$0.480602\pi$
$318$	0	0
$319$	0.456518	0.0255601
$320$	0	0
$321$	5.24104	0.292526
$322$	0	0
$323$	−0.792677	−0.0441058
$324$	0	0
$325$	−17.4531	−0.968122
$326$	0	0
$327$	25.0287	1.38409
$328$	0	0
$329$	−9.39690	−0.518068
$330$	0	0
$331$	3.30981	0.181923	0.0909617	−	0.995854i	$-0.471006\pi$
$331$	3.30981	0.181923	0.0909617	+	0.995854i	$0.471006\pi$
$332$	0	0
$333$	−2.71514	−0.148789
$334$	0	0
$335$	−11.2429	−0.614265
$336$	0	0
$337$	10.3709	0.564941	0.282471	−	0.959276i	$-0.408846\pi$
$337$	10.3709	0.564941	0.282471	+	0.959276i	$0.408846\pi$
$338$	0	0
$339$	−3.54106	−0.192324
$340$	0	0
$341$	−0.00223164	−0.000120850 0
$342$	0	0
$343$	−43.1786	−2.33143
$344$	0	0
$345$	−1.46695	−0.0789780
$346$	0	0
$347$	−27.2761	−1.46426	−0.732128	−	0.681167i	$-0.761473\pi$
$347$	−27.2761	−1.46426	−0.732128	+	0.681167i	$0.761473\pi$
$348$	0	0
$349$	20.4903	1.09682	0.548411	−	0.836209i	$-0.315233\pi$
$349$	20.4903	1.09682	0.548411	+	0.836209i	$0.315233\pi$
$350$	0	0
$351$	22.7501	1.21431
$352$	0	0
$353$	−1.91846	−0.102109	−0.0510546	−	0.998696i	$-0.516258\pi$
$353$	−1.91846	−0.102109	−0.0510546	+	0.998696i	$0.516258\pi$
$354$	0	0
$355$	−6.61488	−0.351081
$356$	0	0
$357$	−2.39628	−0.126824
$358$	0	0
$359$	−19.0017	−1.00287	−0.501437	−	0.865194i	$-0.667195\pi$
$359$	−19.0017	−1.00287	−0.501437	+	0.865194i	$0.667195\pi$
$360$	0	0
$361$	−13.5924	−0.715390
$362$	0	0
$363$	16.1180	0.845977
$364$	0	0
$365$	2.86214	0.149811
$366$	0	0
$367$	−25.3220	−1.32180	−0.660899	−	0.750475i	$-0.729825\pi$
$367$	−25.3220	−1.32180	−0.660899	+	0.750475i	$0.729825\pi$
$368$	0	0
$369$	9.32306	0.485339
$370$	0	0
$371$	−5.26012	−0.273092
$372$	0	0
$373$	−10.4830	−0.542789	−0.271395	−	0.962468i	$-0.587485\pi$
$373$	−10.4830	−0.542789	−0.271395	+	0.962468i	$0.587485\pi$
$374$	0	0
$375$	11.1865	0.577667
$376$	0	0
$377$	31.7313	1.63424
$378$	0	0
$379$	−32.1240	−1.65010	−0.825049	−	0.565061i	$-0.808853\pi$
$379$	−32.1240	−1.65010	−0.825049	+	0.565061i	$0.808853\pi$
$380$	0	0
$381$	−1.01172	−0.0518318
$382$	0	0
$383$	−10.1811	−0.520230	−0.260115	−	0.965578i	$-0.583760\pi$
$383$	−10.1811	−0.520230	−0.260115	+	0.965578i	$0.583760\pi$
$384$	0	0
$385$	−0.227435	−0.0115912
$386$	0	0
$387$	−0.869014	−0.0441745
$388$	0	0
$389$	−6.91136	−0.350420	−0.175210	−	0.984531i	$-0.556060\pi$
$389$	−6.91136	−0.350420	−0.175210	+	0.984531i	$0.556060\pi$
$390$	0	0
$391$	−0.417105	−0.0210939
$392$	0	0
$393$	7.60444	0.383593
$394$	0	0
$395$	0.0927051	0.00466450
$396$	0	0
$397$	−19.4913	−0.978242	−0.489121	−	0.872216i	$-0.662682\pi$
$397$	−19.4913	−0.978242	−0.489121	+	0.872216i	$0.662682\pi$
$398$	0	0
$399$	16.3472	0.818385
$400$	0	0
$401$	25.4798	1.27240	0.636200	−	0.771524i	$-0.280505\pi$
$401$	25.4798	1.27240	0.636200	+	0.771524i	$0.280505\pi$
$402$	0	0
$403$	−0.155115	−0.00772684
$404$	0	0
$405$	−4.67829	−0.232466
$406$	0	0
$407$	0.184834	0.00916186
$408$	0	0
$409$	0.480286	0.0237486	0.0118743	−	0.999929i	$-0.496220\pi$
$409$	0.480286	0.0237486	0.0118743	+	0.999929i	$0.496220\pi$
$410$	0	0
$411$	−23.7121	−1.16963
$412$	0	0
$413$	−60.8574	−2.99459
$414$	0	0
$415$	−1.81644	−0.0891656
$416$	0	0
$417$	−1.68520	−0.0825247
$418$	0	0
$419$	−15.4891	−0.756690	−0.378345	−	0.925665i	$-0.623507\pi$
$419$	−15.4891	−0.756690	−0.378345	+	0.925665i	$0.623507\pi$
$420$	0	0
$421$	13.9413	0.679458	0.339729	−	0.940523i	$-0.389665\pi$
$421$	13.9413	0.679458	0.339729	+	0.940523i	$0.389665\pi$
$422$	0	0
$423$	−1.66863	−0.0811314
$424$	0	0
$425$	1.47633	0.0716124
$426$	0	0
$427$	−2.81830	−0.136387
$428$	0	0
$429$	−0.342444	−0.0165333
$430$	0	0
$431$	−10.5565	−0.508487	−0.254244	−	0.967140i	$-0.581826\pi$
$431$	−10.5565	−0.508487	−0.254244	+	0.967140i	$0.581826\pi$
$432$	0	0
$433$	18.5194	0.889984	0.444992	−	0.895534i	$-0.353206\pi$
$433$	18.5194	0.889984	0.444992	+	0.895534i	$0.353206\pi$
$434$	0	0
$435$	−9.43993	−0.452610
$436$	0	0
$437$	2.84546	0.136117
$438$	0	0
$439$	−16.9747	−0.810159	−0.405079	−	0.914282i	$-0.632756\pi$
$439$	−16.9747	−0.810159	−0.405079	+	0.914282i	$0.632756\pi$
$440$	0	0
$441$	−13.6289	−0.648996
$442$	0	0
$443$	2.02522	0.0962212	0.0481106	−	0.998842i	$-0.484680\pi$
$443$	2.02522	0.0962212	0.0481106	+	0.998842i	$0.484680\pi$
$444$	0	0
$445$	−10.9363	−0.518428
$446$	0	0
$447$	22.0489	1.04288
$448$	0	0
$449$	9.81236	0.463074	0.231537	−	0.972826i	$-0.425625\pi$
$449$	9.81236	0.463074	0.231537	+	0.972826i	$0.425625\pi$
$450$	0	0
$451$	−0.634669	−0.0298854
$452$	0	0
$453$	−17.8661	−0.839424
$454$	0	0
$455$	−15.8084	−0.741108
$456$	0	0
$457$	42.1292	1.97072	0.985360	−	0.170488i	$-0.0545344\pi$
$457$	42.1292	1.97072	0.985360	+	0.170488i	$0.0545344\pi$
$458$	0	0
$459$	−1.92439	−0.0898230
$460$	0	0
$461$	16.4619	0.766708	0.383354	−	0.923602i	$-0.374769\pi$
$461$	16.4619	0.766708	0.383354	+	0.923602i	$0.374769\pi$
$462$	0	0
$463$	−15.5936	−0.724695	−0.362347	−	0.932043i	$-0.618025\pi$
$463$	−15.5936	−0.724695	−0.362347	+	0.932043i	$0.618025\pi$
$464$	0	0
$465$	0.0461462	0.00213998
$466$	0	0
$467$	2.81745	0.130376	0.0651881	−	0.997873i	$-0.479235\pi$
$467$	2.81745	0.130376	0.0651881	+	0.997873i	$0.479235\pi$
$468$	0	0
$469$	65.9259	3.04417
$470$	0	0
$471$	−29.8708	−1.37637
$472$	0	0
$473$	0.0591583	0.00272010
$474$	0	0
$475$	−10.0714	−0.462108
$476$	0	0
$477$	−0.934050	−0.0427672
$478$	0	0
$479$	18.5553	0.847812	0.423906	−	0.905706i	$-0.360659\pi$
$479$	18.5553	0.847812	0.423906	+	0.905706i	$0.360659\pi$
$480$	0	0
$481$	12.8473	0.585785
$482$	0	0
$483$	8.60188	0.391399
$484$	0	0
$485$	1.83980	0.0835411
$486$	0	0
$487$	2.32983	0.105574	0.0527872	−	0.998606i	$-0.483189\pi$
$487$	2.32983	0.105574	0.0527872	+	0.998606i	$0.483189\pi$
$488$	0	0
$489$	−25.5746	−1.15652
$490$	0	0
$491$	−22.9807	−1.03710	−0.518552	−	0.855046i	$-0.673529\pi$
$491$	−22.9807	−1.03710	−0.518552	+	0.855046i	$0.673529\pi$
$492$	0	0
$493$	−2.68410	−0.120886
$494$	0	0
$495$	−0.0403861	−0.00181522
$496$	0	0
$497$	38.7882	1.73989
$498$	0	0
$499$	4.85251	0.217228	0.108614	−	0.994084i	$-0.465359\pi$
$499$	4.85251	0.217228	0.108614	+	0.994084i	$0.465359\pi$
$500$	0	0
$501$	7.63933	0.341300
$502$	0	0
$503$	10.3618	0.462008	0.231004	−	0.972953i	$-0.425799\pi$
$503$	10.3618	0.462008	0.231004	+	0.972953i	$0.425799\pi$
$504$	0	0
$505$	−7.53653	−0.335371
$506$	0	0
$507$	−4.74791	−0.210862
$508$	0	0
$509$	20.4586	0.906811	0.453405	−	0.891304i	$-0.350209\pi$
$509$	20.4586	0.906811	0.453405	+	0.891304i	$0.350209\pi$
$510$	0	0
$511$	−16.7829	−0.742434
$512$	0	0
$513$	13.1281	0.579619
$514$	0	0
$515$	−9.71802	−0.428227
$516$	0	0
$517$	0.113592	0.00499577
$518$	0	0
$519$	25.2599	1.10879
$520$	0	0
$521$	22.2361	0.974181	0.487090	−	0.873352i	$-0.338058\pi$
$521$	22.2361	0.974181	0.487090	+	0.873352i	$0.338058\pi$
$522$	0	0
$523$	0.0985361	0.00430868	0.00215434	−	0.999998i	$-0.499314\pi$
$523$	0.0985361	0.00430868	0.00215434	+	0.999998i	$0.499314\pi$
$524$	0	0
$525$	−30.4460	−1.32877
$526$	0	0
$527$	0.0131210	0.000571558 0
$528$	0	0
$529$	−21.5027	−0.934901
$530$	0	0
$531$	−10.8066	−0.468965
$532$	0	0
$533$	−44.1141	−1.91079
$534$	0	0
$535$	−2.92468	−0.126445
$536$	0	0
$537$	−14.4210	−0.622312
$538$	0	0
$539$	0.927791	0.0399628
$540$	0	0
$541$	11.9447	0.513543	0.256772	−	0.966472i	$-0.417341\pi$
$541$	11.9447	0.513543	0.256772	+	0.966472i	$0.417341\pi$
$542$	0	0
$543$	21.8985	0.939756
$544$	0	0
$545$	−13.9669	−0.598276
$546$	0	0
$547$	−41.5823	−1.77793	−0.888966	−	0.457973i	$-0.848576\pi$
$547$	−41.5823	−1.77793	−0.888966	+	0.457973i	$0.848576\pi$
$548$	0	0
$549$	−0.500452	−0.0213588
$550$	0	0
$551$	18.3107	0.780064
$552$	0	0
$553$	−0.543602	−0.0231163
$554$	0	0
$555$	−3.82201	−0.162235
$556$	0	0
$557$	−28.7352	−1.21755	−0.608775	−	0.793343i	$-0.708339\pi$
$557$	−28.7352	−1.21755	−0.608775	+	0.793343i	$0.708339\pi$
$558$	0	0
$559$	4.11193	0.173916
$560$	0	0
$561$	0.0289668	0.00122298
$562$	0	0
$563$	−12.8019	−0.539536	−0.269768	−	0.962925i	$-0.586947\pi$
$563$	−12.8019	−0.539536	−0.269768	+	0.962925i	$0.586947\pi$
$564$	0	0
$565$	1.97604	0.0831325
$566$	0	0
$567$	27.4324	1.15205
$568$	0	0
$569$	14.8997	0.624627	0.312314	−	0.949979i	$-0.398896\pi$
$569$	14.8997	0.624627	0.312314	+	0.949979i	$0.398896\pi$
$570$	0	0
$571$	17.2993	0.723954	0.361977	−	0.932187i	$-0.382102\pi$
$571$	17.2993	0.723954	0.361977	+	0.932187i	$0.382102\pi$
$572$	0	0
$573$	20.5016	0.856465
$574$	0	0
$575$	−5.29955	−0.221007
$576$	0	0
$577$	33.3717	1.38928	0.694641	−	0.719357i	$-0.255563\pi$
$577$	33.3717	1.38928	0.694641	+	0.719357i	$0.255563\pi$
$578$	0	0
$579$	−15.1542	−0.629786
$580$	0	0
$581$	10.6512	0.441886
$582$	0	0
$583$	0.0635856	0.00263345
$584$	0	0
$585$	−2.80712	−0.116060
$586$	0	0
$587$	7.40266	0.305541	0.152770	−	0.988262i	$-0.451181\pi$
$587$	7.40266	0.305541	0.152770	+	0.988262i	$0.451181\pi$
$588$	0	0
$589$	−0.0895102	−0.00368820
$590$	0	0
$591$	−17.8156	−0.732837
$592$	0	0
$593$	−5.84946	−0.240209	−0.120104	−	0.992761i	$-0.538323\pi$
$593$	−5.84946	−0.240209	−0.120104	+	0.992761i	$0.538323\pi$
$594$	0	0
$595$	1.33721	0.0548201
$596$	0	0
$597$	1.17147	0.0479451
$598$	0	0
$599$	−29.6239	−1.21040	−0.605201	−	0.796073i	$-0.706907\pi$
$599$	−29.6239	−1.21040	−0.605201	+	0.796073i	$0.706907\pi$
$600$	0	0
$601$	−41.5890	−1.69645	−0.848226	−	0.529635i	$-0.822329\pi$
$601$	−41.5890	−1.69645	−0.848226	+	0.529635i	$0.822329\pi$
$602$	0	0
$603$	11.7066	0.476729
$604$	0	0
$605$	−8.99442	−0.365675
$606$	0	0
$607$	−35.3359	−1.43424	−0.717119	−	0.696951i	$-0.754540\pi$
$607$	−35.3359	−1.43424	−0.717119	+	0.696951i	$0.754540\pi$
$608$	0	0
$609$	55.3537	2.24304
$610$	0	0
$611$	7.89546	0.319416
$612$	0	0
$613$	−23.1440	−0.934778	−0.467389	−	0.884052i	$-0.654805\pi$
$613$	−23.1440	−0.934778	−0.467389	+	0.884052i	$0.654805\pi$
$614$	0	0
$615$	13.1238	0.529201
$616$	0	0
$617$	22.7573	0.916176	0.458088	−	0.888907i	$-0.348534\pi$
$617$	22.7573	0.916176	0.458088	+	0.888907i	$0.348534\pi$
$618$	0	0
$619$	−9.44784	−0.379741	−0.189870	−	0.981809i	$-0.560807\pi$
$619$	−9.44784	−0.379741	−0.189870	+	0.981809i	$0.560807\pi$
$620$	0	0
$621$	6.90797	0.277207
$622$	0	0
$623$	64.1278	2.56923
$624$	0	0
$625$	15.4126	0.616503
$626$	0	0
$627$	−0.197609	−0.00789175
$628$	0	0
$629$	−1.08673	−0.0433308
$630$	0	0
$631$	−25.2392	−1.00476	−0.502379	−	0.864647i	$-0.667542\pi$
$631$	−25.2392	−1.00476	−0.502379	+	0.864647i	$0.667542\pi$
$632$	0	0
$633$	−27.1167	−1.07779
$634$	0	0
$635$	0.564573	0.0224044
$636$	0	0
$637$	64.4882	2.55511
$638$	0	0
$639$	6.88769	0.272473
$640$	0	0
$641$	42.7727	1.68942	0.844711	−	0.535223i	$-0.179772\pi$
$641$	42.7727	1.68942	0.844711	+	0.535223i	$0.179772\pi$
$642$	0	0
$643$	25.0816	0.989120	0.494560	−	0.869143i	$-0.335329\pi$
$643$	25.0816	0.989120	0.494560	+	0.869143i	$0.335329\pi$
$644$	0	0
$645$	−1.22328	−0.0481667
$646$	0	0
$647$	−10.9478	−0.430401	−0.215201	−	0.976570i	$-0.569041\pi$
$647$	−10.9478	−0.430401	−0.215201	+	0.976570i	$0.569041\pi$
$648$	0	0
$649$	0.735658	0.0288771
$650$	0	0
$651$	−0.270591	−0.0106053
$652$	0	0
$653$	−48.4416	−1.89567	−0.947833	−	0.318768i	$-0.896731\pi$
$653$	−48.4416	−1.89567	−0.947833	+	0.318768i	$0.896731\pi$
$654$	0	0
$655$	−4.24354	−0.165809
$656$	0	0
$657$	−2.98018	−0.116268
$658$	0	0
$659$	−16.1039	−0.627317	−0.313659	−	0.949536i	$-0.601555\pi$
$659$	−16.1039	−0.627317	−0.313659	+	0.949536i	$0.601555\pi$
$660$	0	0
$661$	−35.8484	−1.39434	−0.697170	−	0.716906i	$-0.745558\pi$
$661$	−35.8484	−1.39434	−0.697170	+	0.716906i	$0.745558\pi$
$662$	0	0
$663$	2.01340	0.0781940
$664$	0	0
$665$	−9.12232	−0.353748
$666$	0	0
$667$	9.63508	0.373072
$668$	0	0
$669$	35.2066	1.36117
$670$	0	0
$671$	0.0340683	0.00131519
$672$	0	0
$673$	−1.45698	−0.0561626	−0.0280813	−	0.999606i	$-0.508940\pi$
$673$	−1.45698	−0.0561626	−0.0280813	+	0.999606i	$0.508940\pi$
$674$	0	0
$675$	−24.4505	−0.941100
$676$	0	0
$677$	43.9879	1.69059	0.845296	−	0.534299i	$-0.179424\pi$
$677$	43.9879	1.69059	0.845296	+	0.534299i	$0.179424\pi$
$678$	0	0
$679$	−10.7882	−0.414013
$680$	0	0
$681$	2.95795	0.113349
$682$	0	0
$683$	40.1670	1.53695	0.768474	−	0.639881i	$-0.221016\pi$
$683$	40.1670	1.53695	0.768474	+	0.639881i	$0.221016\pi$
$684$	0	0
$685$	13.2322	0.505576
$686$	0	0
$687$	26.3301	1.00456
$688$	0	0
$689$	4.41966	0.168376
$690$	0	0
$691$	−9.58519	−0.364638	−0.182319	−	0.983239i	$-0.558360\pi$
$691$	−9.58519	−0.364638	−0.182319	+	0.983239i	$0.558360\pi$
$692$	0	0
$693$	0.236815	0.00899585
$694$	0	0
$695$	0.940401	0.0356714
$696$	0	0
$697$	3.73154	0.141342
$698$	0	0
$699$	34.7360	1.31384
$700$	0	0
$701$	−12.2125	−0.461260	−0.230630	−	0.973042i	$-0.574079\pi$
$701$	−12.2125	−0.461260	−0.230630	+	0.973042i	$0.574079\pi$
$702$	0	0
$703$	7.41360	0.279609
$704$	0	0
$705$	−2.34887	−0.0884635
$706$	0	0
$707$	44.1925	1.66203
$708$	0	0
$709$	−26.8645	−1.00892	−0.504458	−	0.863436i	$-0.668308\pi$
$709$	−26.8645	−1.00892	−0.504458	+	0.863436i	$0.668308\pi$
$710$	0	0
$711$	−0.0965285	−0.00362010
$712$	0	0
$713$	−0.0471001	−0.00176391
$714$	0	0
$715$	0.191095	0.00714656
$716$	0	0
$717$	−2.89227	−0.108014
$718$	0	0
$719$	22.2113	0.828343	0.414172	−	0.910199i	$-0.364071\pi$
$719$	22.2113	0.828343	0.414172	+	0.910199i	$0.364071\pi$
$720$	0	0
$721$	56.9843	2.12221
$722$	0	0
$723$	−4.95962	−0.184450
$724$	0	0
$725$	−34.1030	−1.26655
$726$	0	0
$727$	22.6978	0.841816	0.420908	−	0.907103i	$-0.361712\pi$
$727$	22.6978	0.841816	0.420908	+	0.907103i	$0.361712\pi$
$728$	0	0
$729$	29.6953	1.09982
$730$	0	0
$731$	−0.347822	−0.0128646
$732$	0	0
$733$	43.4250	1.60394	0.801969	−	0.597366i	$-0.203786\pi$
$733$	43.4250	1.60394	0.801969	+	0.597366i	$0.203786\pi$
$734$	0	0
$735$	−19.1850	−0.707648
$736$	0	0
$737$	−0.796928	−0.0293552
$738$	0	0
$739$	−27.7047	−1.01913	−0.509567	−	0.860431i	$-0.670194\pi$
$739$	−27.7047	−1.01913	−0.509567	+	0.860431i	$0.670194\pi$
$740$	0	0
$741$	−13.7353	−0.504578
$742$	0	0
$743$	−12.9106	−0.473644	−0.236822	−	0.971553i	$-0.576106\pi$
$743$	−12.9106	−0.473644	−0.236822	+	0.971553i	$0.576106\pi$
$744$	0	0
$745$	−12.3041	−0.450786
$746$	0	0
$747$	1.89136	0.0692011
$748$	0	0
$749$	17.1497	0.626636
$750$	0	0
$751$	10.7814	0.393419	0.196710	−	0.980462i	$-0.436974\pi$
$751$	10.7814	0.393419	0.196710	+	0.980462i	$0.436974\pi$
$752$	0	0
$753$	−1.46572	−0.0534139
$754$	0	0
$755$	9.96993	0.362843
$756$	0	0
$757$	33.8832	1.23151	0.615754	−	0.787939i	$-0.288852\pi$
$757$	33.8832	1.23151	0.615754	+	0.787939i	$0.288852\pi$
$758$	0	0
$759$	−0.103982	−0.00377429
$760$	0	0
$761$	−47.8712	−1.73533	−0.867665	−	0.497150i	$-0.834380\pi$
$761$	−47.8712	−1.73533	−0.867665	+	0.497150i	$0.834380\pi$
$762$	0	0
$763$	81.8988	2.96494
$764$	0	0
$765$	0.237450	0.00858503
$766$	0	0
$767$	51.1336	1.84633
$768$	0	0
$769$	19.2048	0.692543	0.346272	−	0.938134i	$-0.387448\pi$
$769$	19.2048	0.692543	0.346272	+	0.938134i	$0.387448\pi$
$770$	0	0
$771$	34.2671	1.23410
$772$	0	0
$773$	28.5781	1.02788	0.513942	−	0.857825i	$-0.328185\pi$
$773$	28.5781	1.02788	0.513942	+	0.857825i	$0.328185\pi$
$774$	0	0
$775$	0.166709	0.00598836
$776$	0	0
$777$	22.4114	0.804005
$778$	0	0
$779$	−25.4563	−0.912067
$780$	0	0
$781$	−0.468881	−0.0167779
$782$	0	0
$783$	44.4533	1.58863
$784$	0	0
$785$	16.6689	0.594939
$786$	0	0
$787$	14.7156	0.524554	0.262277	−	0.964993i	$-0.415527\pi$
$787$	14.7156	0.524554	0.262277	+	0.964993i	$0.415527\pi$
$788$	0	0
$789$	−34.5119	−1.22866
$790$	0	0
$791$	−11.5870	−0.411988
$792$	0	0
$793$	2.36799	0.0840900
$794$	0	0
$795$	−1.31483	−0.0466323
$796$	0	0
$797$	25.3408	0.897618	0.448809	−	0.893628i	$-0.351848\pi$
$797$	25.3408	0.897618	0.448809	+	0.893628i	$0.351848\pi$
$798$	0	0
$799$	−0.667865	−0.0236274
$800$	0	0
$801$	11.3873	0.402350
$802$	0	0
$803$	0.202876	0.00715935
$804$	0	0
$805$	−4.80015	−0.169183
$806$	0	0
$807$	14.2463	0.501492
$808$	0	0
$809$	35.8258	1.25957	0.629784	−	0.776771i	$-0.283144\pi$
$809$	35.8258	1.25957	0.629784	+	0.776771i	$0.283144\pi$
$810$	0	0
$811$	56.0401	1.96783	0.983917	−	0.178626i	$-0.0571651\pi$
$811$	56.0401	1.96783	0.983917	+	0.178626i	$0.0571651\pi$
$812$	0	0
$813$	38.7761	1.35994
$814$	0	0
$815$	14.2715	0.499909
$816$	0	0
$817$	2.37282	0.0830143
$818$	0	0
$819$	16.4603	0.575171
$820$	0	0
$821$	−2.29594	−0.0801288	−0.0400644	−	0.999197i	$-0.512756\pi$
$821$	−2.29594	−0.0801288	−0.0400644	+	0.999197i	$0.512756\pi$
$822$	0	0
$823$	−2.58292	−0.0900348	−0.0450174	−	0.998986i	$-0.514334\pi$
$823$	−2.58292	−0.0900348	−0.0450174	+	0.998986i	$0.514334\pi$
$824$	0	0
$825$	0.368039	0.0128135
$826$	0	0
$827$	−35.0286	−1.21807	−0.609033	−	0.793145i	$-0.708442\pi$
$827$	−35.0286	−1.21807	−0.609033	+	0.793145i	$0.708442\pi$
$828$	0	0
$829$	−46.5987	−1.61844	−0.809220	−	0.587505i	$-0.800110\pi$
$829$	−46.5987	−1.61844	−0.809220	+	0.587505i	$0.800110\pi$
$830$	0	0
$831$	−11.8763	−0.411985
$832$	0	0
$833$	−5.45495	−0.189003
$834$	0	0
$835$	−4.26301	−0.147528
$836$	0	0
$837$	−0.217305	−0.00751117
$838$	0	0
$839$	46.9894	1.62225	0.811127	−	0.584871i	$-0.198855\pi$
$839$	46.9894	1.62225	0.811127	+	0.584871i	$0.198855\pi$
$840$	0	0
$841$	33.0024	1.13801
$842$	0	0
$843$	41.2079	1.41928
$844$	0	0
$845$	2.64950	0.0911455
$846$	0	0
$847$	52.7413	1.81221
$848$	0	0
$849$	−9.63132	−0.330546
$850$	0	0
$851$	3.90102	0.133725
$852$	0	0
$853$	−2.29571	−0.0786036	−0.0393018	−	0.999227i	$-0.512513\pi$
$853$	−2.29571	−0.0786036	−0.0393018	+	0.999227i	$0.512513\pi$
$854$	0	0
$855$	−1.61987	−0.0553983
$856$	0	0
$857$	18.7254	0.639646	0.319823	−	0.947477i	$-0.396377\pi$
$857$	18.7254	0.639646	0.319823	+	0.947477i	$0.396377\pi$
$858$	0	0
$859$	−27.2306	−0.929094	−0.464547	−	0.885548i	$-0.653783\pi$
$859$	−27.2306	−0.929094	−0.464547	+	0.885548i	$0.653783\pi$
$860$	0	0
$861$	−76.9548	−2.62261
$862$	0	0
$863$	27.2191	0.926550	0.463275	−	0.886215i	$-0.346674\pi$
$863$	27.2191	0.926550	0.463275	+	0.886215i	$0.346674\pi$
$864$	0	0
$865$	−14.0959	−0.479275
$866$	0	0
$867$	24.7470	0.840451
$868$	0	0
$869$	0.00657119	0.000222912 0
$870$	0	0
$871$	−55.3923	−1.87690
$872$	0	0
$873$	−1.91568	−0.0648360
$874$	0	0
$875$	36.6043	1.23745
$876$	0	0
$877$	−11.4519	−0.386702	−0.193351	−	0.981130i	$-0.561936\pi$
$877$	−11.4519	−0.386702	−0.193351	+	0.981130i	$0.561936\pi$
$878$	0	0
$879$	10.4668	0.353036
$880$	0	0
$881$	−1.67302	−0.0563656	−0.0281828	−	0.999603i	$-0.508972\pi$
$881$	−1.67302	−0.0563656	−0.0281828	+	0.999603i	$0.508972\pi$
$882$	0	0
$883$	−30.1805	−1.01565	−0.507827	−	0.861459i	$-0.669551\pi$
$883$	−30.1805	−1.01565	−0.507827	+	0.861459i	$0.669551\pi$
$884$	0	0
$885$	−15.2120	−0.511347
$886$	0	0
$887$	−31.7142	−1.06486	−0.532430	−	0.846474i	$-0.678721\pi$
$887$	−31.7142	−1.06486	−0.532430	+	0.846474i	$0.678721\pi$
$888$	0	0
$889$	−3.31053	−0.111032
$890$	0	0
$891$	−0.331610	−0.0111093
$892$	0	0
$893$	4.55613	0.152465
$894$	0	0
$895$	8.04742	0.268995
$896$	0	0
$897$	−7.22747	−0.241318
$898$	0	0
$899$	−0.303092	−0.0101087
$900$	0	0
$901$	−0.373852	−0.0124548
$902$	0	0
$903$	7.17306	0.238704
$904$	0	0
$905$	−12.2201	−0.406211
$906$	0	0
$907$	29.7671	0.988401	0.494200	−	0.869348i	$-0.335461\pi$
$907$	29.7671	0.988401	0.494200	+	0.869348i	$0.335461\pi$
$908$	0	0
$909$	7.84736	0.260280
$910$	0	0
$911$	−39.3947	−1.30520	−0.652602	−	0.757701i	$-0.726323\pi$
$911$	−39.3947	−1.30520	−0.652602	+	0.757701i	$0.726323\pi$
$912$	0	0
$913$	−0.128754	−0.00426115
$914$	0	0
$915$	−0.704469	−0.0232890
$916$	0	0
$917$	24.8832	0.821715
$918$	0	0
$919$	30.4197	1.00345	0.501726	−	0.865027i	$-0.332699\pi$
$919$	30.4197	1.00345	0.501726	+	0.865027i	$0.332699\pi$
$920$	0	0
$921$	−19.3329	−0.637041
$922$	0	0
$923$	−32.5906	−1.07273
$924$	0	0
$925$	−13.8075	−0.453988
$926$	0	0
$927$	10.1188	0.332346
$928$	0	0
$929$	7.45438	0.244570	0.122285	−	0.992495i	$-0.460978\pi$
$929$	7.45438	0.244570	0.122285	+	0.992495i	$0.460978\pi$
$930$	0	0
$931$	37.2133	1.21962
$932$	0	0
$933$	−35.0967	−1.14901
$934$	0	0
$935$	−0.0161645	−0.000528634 0
$936$	0	0
$937$	26.0853	0.852168	0.426084	−	0.904684i	$-0.359893\pi$
$937$	26.0853	0.852168	0.426084	+	0.904684i	$0.359893\pi$
$938$	0	0
$939$	7.83574	0.255710
$940$	0	0
$941$	−15.3365	−0.499955	−0.249978	−	0.968252i	$-0.580423\pi$
$941$	−15.3365	−0.499955	−0.249978	+	0.968252i	$0.580423\pi$
$942$	0	0
$943$	−13.3951	−0.436203
$944$	0	0
$945$	−22.1464	−0.720422
$946$	0	0
$947$	10.5973	0.344367	0.172184	−	0.985065i	$-0.444918\pi$
$947$	10.5973	0.344367	0.172184	+	0.985065i	$0.444918\pi$
$948$	0	0
$949$	14.1014	0.457750
$950$	0	0
$951$	−3.17862	−0.103074
$952$	0	0
$953$	−24.6683	−0.799084	−0.399542	−	0.916715i	$-0.630831\pi$
$953$	−24.6683	−0.799084	−0.399542	+	0.916715i	$0.630831\pi$
$954$	0	0
$955$	−11.4406	−0.370209
$956$	0	0
$957$	−0.669128	−0.0216298
$958$	0	0
$959$	−77.5906	−2.50553
$960$	0	0
$961$	−30.9985	−0.999952
$962$	0	0
$963$	3.04530	0.0981335
$964$	0	0
$965$	8.45655	0.272226
$966$	0	0
$967$	−52.4311	−1.68607	−0.843035	−	0.537858i	$-0.819234\pi$
$967$	−52.4311	−1.68607	−0.843035	+	0.537858i	$0.819234\pi$
$968$	0	0
$969$	1.16185	0.0373238
$970$	0	0
$971$	31.0412	0.996158	0.498079	−	0.867132i	$-0.334039\pi$
$971$	31.0412	0.996158	0.498079	+	0.867132i	$0.334039\pi$
$972$	0	0
$973$	−5.51430	−0.176780
$974$	0	0
$975$	25.5814	0.819259
$976$	0	0
$977$	−36.2834	−1.16081	−0.580405	−	0.814328i	$-0.697106\pi$
$977$	−36.2834	−1.16081	−0.580405	+	0.814328i	$0.697106\pi$
$978$	0	0
$979$	−0.775192	−0.0247752
$980$	0	0
$981$	14.5429	0.464320
$982$	0	0
$983$	0.740555	0.0236200	0.0118100	−	0.999930i	$-0.496241\pi$
$983$	0.740555	0.0236200	0.0118100	+	0.999930i	$0.496241\pi$
$984$	0	0
$985$	9.94174	0.316770
$986$	0	0
$987$	13.7732	0.438407
$988$	0	0
$989$	1.24857	0.0397022
$990$	0	0
$991$	14.2191	0.451685	0.225843	−	0.974164i	$-0.427487\pi$
$991$	14.2191	0.451685	0.225843	+	0.974164i	$0.427487\pi$
$992$	0	0
$993$	−4.85126	−0.153950
$994$	0	0
$995$	−0.653721	−0.0207244
$996$	0	0
$997$	−13.8076	−0.437291	−0.218645	−	0.975804i	$-0.570164\pi$
$997$	−13.8076	−0.437291	−0.218645	+	0.975804i	$0.570164\pi$
$998$	0	0
$999$	17.9981	0.569435

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.6		14
4.3	odd	2		1004.2.a.b.1.9	✓	14
12.11	even	2		9036.2.a.m.1.5		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.9	✓	14	4.3	odd	2
4016.2.a.j.1.6		14	1.1	even	1	trivial
9036.2.a.m.1.5		14	12.11	even	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)

\(f(q)\)	\(=\)	\(q-1.46572 q^{3} +0.817924 q^{5} -4.79612 q^{7} -0.851657 q^{9} +O(q^{10})\)	\(q-1.46572 q^{3} +0.817924 q^{5} -4.79612 q^{7} -0.851657 q^{9} +0.0579767 q^{11} +4.02980 q^{13} -1.19885 q^{15} -0.340875 q^{17} +2.32542 q^{19} +7.02979 q^{21} +1.22363 q^{23} -4.33100 q^{25} +5.64546 q^{27} +7.87416 q^{29} -0.0384920 q^{31} -0.0849778 q^{33} -3.92287 q^{35} +3.18806 q^{37} -5.90657 q^{39} -10.9470 q^{41} +1.02038 q^{43} -0.696591 q^{45} +1.95927 q^{47} +16.0028 q^{49} +0.499628 q^{51} +1.09674 q^{53} +0.0474205 q^{55} -3.40842 q^{57} +12.6889 q^{59} +0.587621 q^{61} +4.08466 q^{63} +3.29607 q^{65} -13.7457 q^{67} -1.79351 q^{69} -8.08740 q^{71} +3.49927 q^{73} +6.34804 q^{75} -0.278064 q^{77} +0.113342 q^{79} -5.71971 q^{81} -2.22079 q^{83} -0.278810 q^{85} -11.5413 q^{87} -13.3707 q^{89} -19.3274 q^{91} +0.0564186 q^{93} +1.90202 q^{95} +2.24936 q^{97} -0.0493763 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

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