LMFDB - Embedded newform 4016.2.a.f.1.5

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:	$6$
Coefficient field:	6.6.60853001.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 8x^{4} + 15x^{3} + 20x^{2} - 12x - 5 $ x^6 - 3x^5 - 8x^4 + 15x^3 + 20x^2 - 12*x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.75626$ 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)

$f(q)$	$=$	$q+1.68934 q^{3} +0.420498 q^{5} -0.389856 q^{7} -0.146119 q^{9} +O(q^{10})$	$q+1.68934 q^{3} +0.420498 q^{5} -0.389856 q^{7} -0.146119 q^{9} -3.59698 q^{11} +5.09203 q^{13} +0.710366 q^{15} -4.08440 q^{17} -1.64544 q^{19} -0.658601 q^{21} -6.76389 q^{23} -4.82318 q^{25} -5.31487 q^{27} +9.75295 q^{29} -4.10519 q^{31} -6.07654 q^{33} -0.163934 q^{35} -0.0847341 q^{37} +8.60218 q^{39} -9.32682 q^{41} +6.89010 q^{43} -0.0614426 q^{45} -7.43997 q^{47} -6.84801 q^{49} -6.89995 q^{51} -0.986738 q^{53} -1.51252 q^{55} -2.77971 q^{57} +10.7034 q^{59} -13.5970 q^{61} +0.0569652 q^{63} +2.14119 q^{65} -8.89603 q^{67} -11.4265 q^{69} +2.68678 q^{71} +9.46592 q^{73} -8.14801 q^{75} +1.40231 q^{77} -4.71776 q^{79} -8.54029 q^{81} +11.0976 q^{83} -1.71748 q^{85} +16.4761 q^{87} +13.4964 q^{89} -1.98516 q^{91} -6.93507 q^{93} -0.691904 q^{95} -18.2343 q^{97} +0.525586 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * 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
$3$	1.68934	0.975343	0.487671	−	0.873027i	$-0.337846\pi$
$3$	1.68934	0.975343	0.487671	+	0.873027i	$0.337846\pi$
$4$	0	0
$5$	0.420498	0.188053	0.0940263	−	0.995570i	$-0.470026\pi$
$5$	0.420498	0.188053	0.0940263	+	0.995570i	$0.470026\pi$
$6$	0	0
$7$	−0.389856	−0.147352	−0.0736759	−	0.997282i	$-0.523473\pi$
$7$	−0.389856	−0.147352	−0.0736759	+	0.997282i	$0.523473\pi$
$8$	0	0
$9$	−0.146119	−0.0487062
$10$	0	0
$11$	−3.59698	−1.08453	−0.542265	−	0.840207i	$-0.682433\pi$
$11$	−3.59698	−1.08453	−0.542265	+	0.840207i	$0.682433\pi$
$12$	0	0
$13$	5.09203	1.41227	0.706137	−	0.708075i	$-0.250436\pi$
$13$	5.09203	1.41227	0.706137	+	0.708075i	$0.250436\pi$
$14$	0	0
$15$	0.710366	0.183416
$16$	0	0
$17$	−4.08440	−0.990612	−0.495306	−	0.868718i	$-0.664944\pi$
$17$	−4.08440	−0.990612	−0.495306	+	0.868718i	$0.664944\pi$
$18$	0	0
$19$	−1.64544	−0.377490	−0.188745	−	0.982026i	$-0.560442\pi$
$19$	−1.64544	−0.377490	−0.188745	+	0.982026i	$0.560442\pi$
$20$	0	0
$21$	−0.658601	−0.143719
$22$	0	0
$23$	−6.76389	−1.41037	−0.705184	−	0.709024i	$-0.749136\pi$
$23$	−6.76389	−1.41037	−0.705184	+	0.709024i	$0.749136\pi$
$24$	0	0
$25$	−4.82318	−0.964636
$26$	0	0
$27$	−5.31487	−1.02285
$28$	0	0
$29$	9.75295	1.81108	0.905539	−	0.424263i	$-0.139467\pi$
$29$	9.75295	1.81108	0.905539	+	0.424263i	$0.139467\pi$
$30$	0	0
$31$	−4.10519	−0.737314	−0.368657	−	0.929566i	$-0.620182\pi$
$31$	−4.10519	−0.737314	−0.368657	+	0.929566i	$0.620182\pi$
$32$	0	0
$33$	−6.07654	−1.05779
$34$	0	0
$35$	−0.163934	−0.0277099
$36$	0	0
$37$	−0.0847341	−0.0139302	−0.00696510	−	0.999976i	$-0.502217\pi$
$37$	−0.0847341	−0.0139302	−0.00696510	+	0.999976i	$0.502217\pi$
$38$	0	0
$39$	8.60218	1.37745
$40$	0	0
$41$	−9.32682	−1.45660	−0.728302	−	0.685256i	$-0.759690\pi$
$41$	−9.32682	−1.45660	−0.728302	+	0.685256i	$0.759690\pi$
$42$	0	0
$43$	6.89010	1.05073	0.525365	−	0.850877i	$-0.323929\pi$
$43$	6.89010	1.05073	0.525365	+	0.850877i	$0.323929\pi$
$44$	0	0
$45$	−0.0614426	−0.00915932
$46$	0	0
$47$	−7.43997	−1.08523	−0.542616	−	0.839981i	$-0.682566\pi$
$47$	−7.43997	−1.08523	−0.542616	+	0.839981i	$0.682566\pi$
$48$	0	0
$49$	−6.84801	−0.978287
$50$	0	0
$51$	−6.89995	−0.966187
$52$	0	0
$53$	−0.986738	−0.135539	−0.0677694	−	0.997701i	$-0.521588\pi$
$53$	−0.986738	−0.135539	−0.0677694	+	0.997701i	$0.521588\pi$
$54$	0	0
$55$	−1.51252	−0.203949
$56$	0	0
$57$	−2.77971	−0.368182
$58$	0	0
$59$	10.7034	1.39347	0.696733	−	0.717331i	$-0.254636\pi$
$59$	10.7034	1.39347	0.696733	+	0.717331i	$0.254636\pi$
$60$	0	0
$61$	−13.5970	−1.74092	−0.870458	−	0.492243i	$-0.836177\pi$
$61$	−13.5970	−1.74092	−0.870458	+	0.492243i	$0.836177\pi$
$62$	0	0
$63$	0.0569652	0.00717695
$64$	0	0
$65$	2.14119	0.265582
$66$	0	0
$67$	−8.89603	−1.08682	−0.543412	−	0.839466i	$-0.682868\pi$
$67$	−8.89603	−1.08682	−0.543412	+	0.839466i	$0.682868\pi$
$68$	0	0
$69$	−11.4265	−1.37559
$70$	0	0
$71$	2.68678	0.318862	0.159431	−	0.987209i	$-0.449034\pi$
$71$	2.68678	0.318862	0.159431	+	0.987209i	$0.449034\pi$
$72$	0	0
$73$	9.46592	1.10790	0.553951	−	0.832549i	$-0.313120\pi$
$73$	9.46592	1.10790	0.553951	+	0.832549i	$0.313120\pi$
$74$	0	0
$75$	−8.14801	−0.940851
$76$	0	0
$77$	1.40231	0.159808
$78$	0	0
$79$	−4.71776	−0.530789	−0.265395	−	0.964140i	$-0.585502\pi$
$79$	−4.71776	−0.530789	−0.265395	+	0.964140i	$0.585502\pi$
$80$	0	0
$81$	−8.54029	−0.948922
$82$	0	0
$83$	11.0976	1.21812	0.609060	−	0.793124i	$-0.291547\pi$
$83$	11.0976	1.21812	0.609060	+	0.793124i	$0.291547\pi$
$84$	0	0
$85$	−1.71748	−0.186287
$86$	0	0
$87$	16.4761	1.76642
$88$	0	0
$89$	13.4964	1.43062	0.715309	−	0.698808i	$-0.246286\pi$
$89$	13.4964	1.43062	0.715309	+	0.698808i	$0.246286\pi$
$90$	0	0
$91$	−1.98516	−0.208101
$92$	0	0
$93$	−6.93507	−0.719134
$94$	0	0
$95$	−0.691904	−0.0709879
$96$	0	0
$97$	−18.2343	−1.85141	−0.925705	−	0.378247i	$-0.876527\pi$
$97$	−18.2343	−1.85141	−0.925705	+	0.378247i	$0.876527\pi$
$98$	0	0
$99$	0.525586	0.0528234
$100$	0	0
$101$	−1.35163	−0.134492	−0.0672461	−	0.997736i	$-0.521421\pi$
$101$	−1.35163	−0.134492	−0.0672461	+	0.997736i	$0.521421\pi$
$102$	0	0
$103$	0.347801	0.0342699	0.0171349	−	0.999853i	$-0.494546\pi$
$103$	0.347801	0.0342699	0.0171349	+	0.999853i	$0.494546\pi$
$104$	0	0
$105$	−0.276941	−0.0270266
$106$	0	0
$107$	13.1929	1.27541	0.637704	−	0.770281i	$-0.279884\pi$
$107$	13.1929	1.27541	0.637704	+	0.770281i	$0.279884\pi$
$108$	0	0
$109$	−17.9237	−1.71678	−0.858390	−	0.512998i	$-0.828535\pi$
$109$	−17.9237	−1.71678	−0.858390	+	0.512998i	$0.828535\pi$
$110$	0	0
$111$	−0.143145	−0.0135867
$112$	0	0
$113$	15.5910	1.46668	0.733340	−	0.679863i	$-0.237961\pi$
$113$	15.5910	1.46668	0.733340	+	0.679863i	$0.237961\pi$
$114$	0	0
$115$	−2.84420	−0.265223
$116$	0	0
$117$	−0.744040	−0.0687865
$118$	0	0
$119$	1.59233	0.145969
$120$	0	0
$121$	1.93828	0.176207
$122$	0	0
$123$	−15.7562	−1.42069
$124$	0	0
$125$	−4.13063	−0.369455
$126$	0	0
$127$	−9.23999	−0.819917	−0.409958	−	0.912104i	$-0.634457\pi$
$127$	−9.23999	−0.819917	−0.409958	+	0.912104i	$0.634457\pi$
$128$	0	0
$129$	11.6397	1.02482
$130$	0	0
$131$	−0.811612	−0.0709109	−0.0354554	−	0.999371i	$-0.511288\pi$
$131$	−0.811612	−0.0709109	−0.0354554	+	0.999371i	$0.511288\pi$
$132$	0	0
$133$	0.641485	0.0556238
$134$	0	0
$135$	−2.23490	−0.192349
$136$	0	0
$137$	−14.5376	−1.24203	−0.621014	−	0.783799i	$-0.713279\pi$
$137$	−14.5376	−1.24203	−0.621014	+	0.783799i	$0.713279\pi$
$138$	0	0
$139$	21.0326	1.78396	0.891980	−	0.452075i	$-0.149316\pi$
$139$	21.0326	1.78396	0.891980	+	0.452075i	$0.149316\pi$
$140$	0	0
$141$	−12.5687	−1.05847
$142$	0	0
$143$	−18.3159	−1.53165
$144$	0	0
$145$	4.10110	0.340578
$146$	0	0
$147$	−11.5686	−0.954166
$148$	0	0
$149$	3.83647	0.314296	0.157148	−	0.987575i	$-0.449770\pi$
$149$	3.83647	0.314296	0.157148	+	0.987575i	$0.449770\pi$
$150$	0	0
$151$	−4.20044	−0.341827	−0.170914	−	0.985286i	$-0.554672\pi$
$151$	−4.20044	−0.341827	−0.170914	+	0.985286i	$0.554672\pi$
$152$	0	0
$153$	0.596807	0.0482490
$154$	0	0
$155$	−1.72622	−0.138654
$156$	0	0
$157$	−23.7505	−1.89550	−0.947749	−	0.319016i	$-0.896647\pi$
$157$	−23.7505	−1.89550	−0.947749	+	0.319016i	$0.896647\pi$
$158$	0	0
$159$	−1.66694	−0.132197
$160$	0	0
$161$	2.63694	0.207820
$162$	0	0
$163$	−17.6713	−1.38412	−0.692061	−	0.721839i	$-0.743297\pi$
$163$	−17.6713	−1.38412	−0.692061	+	0.721839i	$0.743297\pi$
$164$	0	0
$165$	−2.55517	−0.198920
$166$	0	0
$167$	−14.7328	−1.14006	−0.570030	−	0.821624i	$-0.693068\pi$
$167$	−14.7328	−1.14006	−0.570030	+	0.821624i	$0.693068\pi$
$168$	0	0
$169$	12.9287	0.994518
$170$	0	0
$171$	0.240429	0.0183861
$172$	0	0
$173$	17.5891	1.33727	0.668636	−	0.743590i	$-0.266879\pi$
$173$	17.5891	1.33727	0.668636	+	0.743590i	$0.266879\pi$
$174$	0	0
$175$	1.88035	0.142141
$176$	0	0
$177$	18.0817	1.35911
$178$	0	0
$179$	−19.6224	−1.46665	−0.733324	−	0.679880i	$-0.762032\pi$
$179$	−19.6224	−1.46665	−0.733324	+	0.679880i	$0.762032\pi$
$180$	0	0
$181$	3.71655	0.276249	0.138124	−	0.990415i	$-0.455893\pi$
$181$	3.71655	0.276249	0.138124	+	0.990415i	$0.455893\pi$
$182$	0	0
$183$	−22.9700	−1.69799
$184$	0	0
$185$	−0.0356306	−0.00261961
$186$	0	0
$187$	14.6915	1.07435
$188$	0	0
$189$	2.07204	0.150719
$190$	0	0
$191$	0.769197	0.0556571	0.0278286	−	0.999613i	$-0.491141\pi$
$191$	0.769197	0.0556571	0.0278286	+	0.999613i	$0.491141\pi$
$192$	0	0
$193$	2.97085	0.213846	0.106923	−	0.994267i	$-0.465900\pi$
$193$	2.97085	0.213846	0.106923	+	0.994267i	$0.465900\pi$
$194$	0	0
$195$	3.61720	0.259033
$196$	0	0
$197$	16.6599	1.18697	0.593484	−	0.804846i	$-0.297752\pi$
$197$	16.6599	1.18697	0.593484	+	0.804846i	$0.297752\pi$
$198$	0	0
$199$	−13.3015	−0.942917	−0.471458	−	0.881888i	$-0.656272\pi$
$199$	−13.3015	−0.942917	−0.471458	+	0.881888i	$0.656272\pi$
$200$	0	0
$201$	−15.0285	−1.06003
$202$	0	0
$203$	−3.80225	−0.266866
$204$	0	0
$205$	−3.92191	−0.273918
$206$	0	0
$207$	0.988330	0.0686937
$208$	0	0
$209$	5.91862	0.409399
$210$	0	0
$211$	4.53200	0.311996	0.155998	−	0.987757i	$-0.450141\pi$
$211$	4.53200	0.311996	0.155998	+	0.987757i	$0.450141\pi$
$212$	0	0
$213$	4.53890	0.311000
$214$	0	0
$215$	2.89728	0.197593
$216$	0	0
$217$	1.60043	0.108644
$218$	0	0
$219$	15.9912	1.08058
$220$	0	0
$221$	−20.7979	−1.39902
$222$	0	0
$223$	3.36742	0.225499	0.112749	−	0.993623i	$-0.464034\pi$
$223$	3.36742	0.225499	0.112749	+	0.993623i	$0.464034\pi$
$224$	0	0
$225$	0.704756	0.0469838
$226$	0	0
$227$	16.9125	1.12252	0.561262	−	0.827638i	$-0.310316\pi$
$227$	16.9125	1.12252	0.561262	+	0.827638i	$0.310316\pi$
$228$	0	0
$229$	19.7122	1.30262	0.651309	−	0.758813i	$-0.274220\pi$
$229$	19.7122	1.30262	0.651309	+	0.758813i	$0.274220\pi$
$230$	0	0
$231$	2.36898	0.155867
$232$	0	0
$233$	2.18695	0.143272	0.0716359	−	0.997431i	$-0.477178\pi$
$233$	2.18695	0.143272	0.0716359	+	0.997431i	$0.477178\pi$
$234$	0	0
$235$	−3.12849	−0.204080
$236$	0	0
$237$	−7.96992	−0.517702
$238$	0	0
$239$	−8.48291	−0.548714	−0.274357	−	0.961628i	$-0.588465\pi$
$239$	−8.48291	−0.548714	−0.274357	+	0.961628i	$0.588465\pi$
$240$	0	0
$241$	−22.9356	−1.47741	−0.738706	−	0.674028i	$-0.764563\pi$
$241$	−22.9356	−1.47741	−0.738706	+	0.674028i	$0.764563\pi$
$242$	0	0
$243$	1.51714	0.0973243
$244$	0	0
$245$	−2.87958	−0.183969
$246$	0	0
$247$	−8.37862	−0.533119
$248$	0	0
$249$	18.7477	1.18808
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	24.3296	1.52959
$254$	0	0
$255$	−2.90142	−0.181694
$256$	0	0
$257$	−7.22447	−0.450650	−0.225325	−	0.974284i	$-0.572344\pi$
$257$	−7.22447	−0.450650	−0.225325	+	0.974284i	$0.572344\pi$
$258$	0	0
$259$	0.0330341	0.00205264
$260$	0	0
$261$	−1.42509	−0.0882107
$262$	0	0
$263$	7.76982	0.479108	0.239554	−	0.970883i	$-0.422999\pi$
$263$	7.76982	0.479108	0.239554	+	0.970883i	$0.422999\pi$
$264$	0	0
$265$	−0.414922	−0.0254884
$266$	0	0
$267$	22.8001	1.39534
$268$	0	0
$269$	−17.8855	−1.09050	−0.545249	−	0.838274i	$-0.683565\pi$
$269$	−17.8855	−1.09050	−0.545249	+	0.838274i	$0.683565\pi$
$270$	0	0
$271$	27.0367	1.64236	0.821181	−	0.570668i	$-0.193316\pi$
$271$	27.0367	1.64236	0.821181	+	0.570668i	$0.193316\pi$
$272$	0	0
$273$	−3.35361	−0.202970
$274$	0	0
$275$	17.3489	1.04618
$276$	0	0
$277$	−5.40388	−0.324688	−0.162344	−	0.986734i	$-0.551905\pi$
$277$	−5.40388	−0.324688	−0.162344	+	0.986734i	$0.551905\pi$
$278$	0	0
$279$	0.599844	0.0359117
$280$	0	0
$281$	22.3464	1.33308	0.666538	−	0.745471i	$-0.267775\pi$
$281$	22.3464	1.33308	0.666538	+	0.745471i	$0.267775\pi$
$282$	0	0
$283$	2.00875	0.119408	0.0597040	−	0.998216i	$-0.480984\pi$
$283$	2.00875	0.119408	0.0597040	+	0.998216i	$0.480984\pi$
$284$	0	0
$285$	−1.16886	−0.0692375
$286$	0	0
$287$	3.63612	0.214633
$288$	0	0
$289$	−0.317682	−0.0186872
$290$	0	0
$291$	−30.8039	−1.80576
$292$	0	0
$293$	26.5082	1.54862	0.774312	−	0.632804i	$-0.218096\pi$
$293$	26.5082	1.54862	0.774312	+	0.632804i	$0.218096\pi$
$294$	0	0
$295$	4.50077	0.262045
$296$	0	0
$297$	19.1175	1.10931
$298$	0	0
$299$	−34.4419	−1.99183
$300$	0	0
$301$	−2.68615	−0.154827
$302$	0	0
$303$	−2.28337	−0.131176
$304$	0	0
$305$	−5.71751	−0.327384
$306$	0	0
$307$	8.94146	0.510316	0.255158	−	0.966899i	$-0.417872\pi$
$307$	8.94146	0.510316	0.255158	+	0.966899i	$0.417872\pi$
$308$	0	0
$309$	0.587556	0.0334249
$310$	0	0
$311$	10.3987	0.589657	0.294828	−	0.955550i	$-0.404738\pi$
$311$	10.3987	0.589657	0.294828	+	0.955550i	$0.404738\pi$
$312$	0	0
$313$	14.4546	0.817024	0.408512	−	0.912753i	$-0.366048\pi$
$313$	14.4546	0.817024	0.408512	+	0.912753i	$0.366048\pi$
$314$	0	0
$315$	0.0239538	0.00134964
$316$	0	0
$317$	−3.20673	−0.180108	−0.0900540	−	0.995937i	$-0.528704\pi$
$317$	−3.20673	−0.180108	−0.0900540	+	0.995937i	$0.528704\pi$
$318$	0	0
$319$	−35.0812	−1.96417
$320$	0	0
$321$	22.2874	1.24396
$322$	0	0
$323$	6.72063	0.373946
$324$	0	0
$325$	−24.5598	−1.36233
$326$	0	0
$327$	−30.2793	−1.67445
$328$	0	0
$329$	2.90052	0.159911
$330$	0	0
$331$	8.70769	0.478618	0.239309	−	0.970943i	$-0.423079\pi$
$331$	8.70769	0.478618	0.239309	+	0.970943i	$0.423079\pi$
$332$	0	0
$333$	0.0123812	0.000678487 0
$334$	0	0
$335$	−3.74077	−0.204380
$336$	0	0
$337$	−12.2632	−0.668022	−0.334011	−	0.942569i	$-0.608402\pi$
$337$	−12.2632	−0.668022	−0.334011	+	0.942569i	$0.608402\pi$
$338$	0	0
$339$	26.3386	1.43052
$340$	0	0
$341$	14.7663	0.799639
$342$	0	0
$343$	5.39873	0.291504
$344$	0	0
$345$	−4.80484	−0.258684
$346$	0	0
$347$	−19.6809	−1.05653	−0.528264	−	0.849080i	$-0.677157\pi$
$347$	−19.6809	−1.05653	−0.528264	+	0.849080i	$0.677157\pi$
$348$	0	0
$349$	−28.9835	−1.55145	−0.775725	−	0.631071i	$-0.782616\pi$
$349$	−28.9835	−1.55145	−0.775725	+	0.631071i	$0.782616\pi$
$350$	0	0
$351$	−27.0635	−1.44454
$352$	0	0
$353$	11.7275	0.624190	0.312095	−	0.950051i	$-0.398969\pi$
$353$	11.7275	0.624190	0.312095	+	0.950051i	$0.398969\pi$
$354$	0	0
$355$	1.12979	0.0599629
$356$	0	0
$357$	2.68999	0.142369
$358$	0	0
$359$	1.86647	0.0985087	0.0492544	−	0.998786i	$-0.484316\pi$
$359$	1.86647	0.0985087	0.0492544	+	0.998786i	$0.484316\pi$
$360$	0	0
$361$	−16.2925	−0.857502
$362$	0	0
$363$	3.27442	0.171863
$364$	0	0
$365$	3.98040	0.208344
$366$	0	0
$367$	1.24263	0.0648646	0.0324323	−	0.999474i	$-0.489675\pi$
$367$	1.24263	0.0648646	0.0324323	+	0.999474i	$0.489675\pi$
$368$	0	0
$369$	1.36282	0.0709457
$370$	0	0
$371$	0.384686	0.0199719
$372$	0	0
$373$	−4.04630	−0.209509	−0.104755	−	0.994498i	$-0.533406\pi$
$373$	−4.04630	−0.209509	−0.104755	+	0.994498i	$0.533406\pi$
$374$	0	0
$375$	−6.97805	−0.360345
$376$	0	0
$377$	49.6623	2.55774
$378$	0	0
$379$	29.4387	1.51217	0.756083	−	0.654475i	$-0.227110\pi$
$379$	29.4387	1.51217	0.756083	+	0.654475i	$0.227110\pi$
$380$	0	0
$381$	−15.6095	−0.799700
$382$	0	0
$383$	−24.0805	−1.23046	−0.615229	−	0.788349i	$-0.710936\pi$
$383$	−24.0805	−1.23046	−0.615229	+	0.788349i	$0.710936\pi$
$384$	0	0
$385$	0.589667	0.0300522
$386$	0	0
$387$	−1.00677	−0.0511771
$388$	0	0
$389$	5.01083	0.254059	0.127030	−	0.991899i	$-0.459456\pi$
$389$	5.01083	0.254059	0.127030	+	0.991899i	$0.459456\pi$
$390$	0	0
$391$	27.6264	1.39713
$392$	0	0
$393$	−1.37109	−0.0691624
$394$	0	0
$395$	−1.98381	−0.0998163
$396$	0	0
$397$	−27.1520	−1.36272	−0.681359	−	0.731949i	$-0.738611\pi$
$397$	−27.1520	−1.36272	−0.681359	+	0.731949i	$0.738611\pi$
$398$	0	0
$399$	1.08369	0.0542523
$400$	0	0
$401$	−31.5918	−1.57762	−0.788808	−	0.614639i	$-0.789302\pi$
$401$	−31.5918	−1.57762	−0.788808	+	0.614639i	$0.789302\pi$
$402$	0	0
$403$	−20.9037	−1.04129
$404$	0	0
$405$	−3.59118	−0.178447
$406$	0	0
$407$	0.304787	0.0151077
$408$	0	0
$409$	−6.54021	−0.323392	−0.161696	−	0.986841i	$-0.551696\pi$
$409$	−6.54021	−0.323392	−0.161696	+	0.986841i	$0.551696\pi$
$410$	0	0
$411$	−24.5590	−1.21140
$412$	0	0
$413$	−4.17279	−0.205330
$414$	0	0
$415$	4.66652	0.229071
$416$	0	0
$417$	35.5312	1.73997
$418$	0	0
$419$	27.8898	1.36251	0.681253	−	0.732048i	$-0.261435\pi$
$419$	27.8898	1.36251	0.681253	+	0.732048i	$0.261435\pi$
$420$	0	0
$421$	−10.5398	−0.513680	−0.256840	−	0.966454i	$-0.582681\pi$
$421$	−10.5398	−0.513680	−0.256840	+	0.966454i	$0.582681\pi$
$422$	0	0
$423$	1.08712	0.0528575
$424$	0	0
$425$	19.6998	0.955581
$426$	0	0
$427$	5.30087	0.256527
$428$	0	0
$429$	−30.9419	−1.49389
$430$	0	0
$431$	37.1487	1.78939	0.894695	−	0.446677i	$-0.147393\pi$
$431$	37.1487	1.78939	0.894695	+	0.446677i	$0.147393\pi$
$432$	0	0
$433$	−9.72432	−0.467321	−0.233661	−	0.972318i	$-0.575070\pi$
$433$	−9.72432	−0.467321	−0.233661	+	0.972318i	$0.575070\pi$
$434$	0	0
$435$	6.92817	0.332180
$436$	0	0
$437$	11.1296	0.532400
$438$	0	0
$439$	4.70326	0.224474	0.112237	−	0.993681i	$-0.464198\pi$
$439$	4.70326	0.224474	0.112237	+	0.993681i	$0.464198\pi$
$440$	0	0
$441$	1.00062	0.0476487
$442$	0	0
$443$	−24.9308	−1.18450	−0.592250	−	0.805755i	$-0.701760\pi$
$443$	−24.9308	−1.18450	−0.592250	+	0.805755i	$0.701760\pi$
$444$	0	0
$445$	5.67522	0.269031
$446$	0	0
$447$	6.48112	0.306547
$448$	0	0
$449$	−29.3218	−1.38378	−0.691890	−	0.722003i	$-0.743222\pi$
$449$	−29.3218	−1.38378	−0.691890	+	0.722003i	$0.743222\pi$
$450$	0	0
$451$	33.5484	1.57973
$452$	0	0
$453$	−7.09599	−0.333399
$454$	0	0
$455$	−0.834756	−0.0391339
$456$	0	0
$457$	13.7604	0.643682	0.321841	−	0.946794i	$-0.395698\pi$
$457$	13.7604	0.643682	0.321841	+	0.946794i	$0.395698\pi$
$458$	0	0
$459$	21.7081	1.01325
$460$	0	0
$461$	−4.71079	−0.219403	−0.109702	−	0.993965i	$-0.534990\pi$
$461$	−4.71079	−0.219403	−0.109702	+	0.993965i	$0.534990\pi$
$462$	0	0
$463$	−17.6690	−0.821146	−0.410573	−	0.911828i	$-0.634671\pi$
$463$	−17.6690	−0.821146	−0.410573	+	0.911828i	$0.634671\pi$
$464$	0	0
$465$	−2.91619	−0.135235
$466$	0	0
$467$	19.3375	0.894832	0.447416	−	0.894326i	$-0.352344\pi$
$467$	19.3375	0.894832	0.447416	+	0.894326i	$0.352344\pi$
$468$	0	0
$469$	3.46817	0.160145
$470$	0	0
$471$	−40.1228	−1.84876
$472$	0	0
$473$	−24.7836	−1.13955
$474$	0	0
$475$	7.93625	0.364140
$476$	0	0
$477$	0.144181	0.00660158
$478$	0	0
$479$	8.16624	0.373125	0.186562	−	0.982443i	$-0.440265\pi$
$479$	8.16624	0.373125	0.186562	+	0.982443i	$0.440265\pi$
$480$	0	0
$481$	−0.431469	−0.0196733
$482$	0	0
$483$	4.45470	0.202696
$484$	0	0
$485$	−7.66748	−0.348162
$486$	0	0
$487$	0.0912394	0.00413445	0.00206723	−	0.999998i	$-0.499342\pi$
$487$	0.0912394	0.00413445	0.00206723	+	0.999998i	$0.499342\pi$
$488$	0	0
$489$	−29.8529	−1.34999
$490$	0	0
$491$	−28.4462	−1.28376	−0.641880	−	0.766805i	$-0.721845\pi$
$491$	−28.4462	−1.28376	−0.641880	+	0.766805i	$0.721845\pi$
$492$	0	0
$493$	−39.8350	−1.79408
$494$	0	0
$495$	0.221008	0.00993357
$496$	0	0
$497$	−1.04746	−0.0469850
$498$	0	0
$499$	−11.2803	−0.504974	−0.252487	−	0.967600i	$-0.581249\pi$
$499$	−11.2803	−0.504974	−0.252487	+	0.967600i	$0.581249\pi$
$500$	0	0
$501$	−24.8888	−1.11195
$502$	0	0
$503$	−7.32353	−0.326540	−0.163270	−	0.986581i	$-0.552204\pi$
$503$	−7.32353	−0.326540	−0.163270	+	0.986581i	$0.552204\pi$
$504$	0	0
$505$	−0.568358	−0.0252916
$506$	0	0
$507$	21.8411	0.969996
$508$	0	0
$509$	10.0530	0.445590	0.222795	−	0.974865i	$-0.428482\pi$
$509$	10.0530	0.445590	0.222795	+	0.974865i	$0.428482\pi$
$510$	0	0
$511$	−3.69035	−0.163251
$512$	0	0
$513$	8.74530	0.386115
$514$	0	0
$515$	0.146250	0.00644454
$516$	0	0
$517$	26.7614	1.17697
$518$	0	0
$519$	29.7140	1.30430
$520$	0	0
$521$	−2.70259	−0.118403	−0.0592013	−	0.998246i	$-0.518855\pi$
$521$	−2.70259	−0.118403	−0.0592013	+	0.998246i	$0.518855\pi$
$522$	0	0
$523$	−11.7376	−0.513249	−0.256625	−	0.966511i	$-0.582610\pi$
$523$	−11.7376	−0.513249	−0.256625	+	0.966511i	$0.582610\pi$
$524$	0	0
$525$	3.17655	0.138636
$526$	0	0
$527$	16.7672	0.730392
$528$	0	0
$529$	22.7502	0.989139
$530$	0	0
$531$	−1.56397	−0.0678704
$532$	0	0
$533$	−47.4924	−2.05713
$534$	0	0
$535$	5.54760	0.239844
$536$	0	0
$537$	−33.1490	−1.43048
$538$	0	0
$539$	24.6322	1.06098
$540$	0	0
$541$	40.5595	1.74379	0.871895	−	0.489693i	$-0.162891\pi$
$541$	40.5595	1.74379	0.871895	+	0.489693i	$0.162891\pi$
$542$	0	0
$543$	6.27852	0.269437
$544$	0	0
$545$	−7.53689	−0.322845
$546$	0	0
$547$	−4.49618	−0.192243	−0.0961214	−	0.995370i	$-0.530644\pi$
$547$	−4.49618	−0.192243	−0.0961214	+	0.995370i	$0.530644\pi$
$548$	0	0
$549$	1.98677	0.0847934
$550$	0	0
$551$	−16.0479	−0.683663
$552$	0	0
$553$	1.83925	0.0782128
$554$	0	0
$555$	−0.0601923	−0.00255502
$556$	0	0
$557$	23.1106	0.979227	0.489614	−	0.871939i	$-0.337138\pi$
$557$	23.1106	0.979227	0.489614	+	0.871939i	$0.337138\pi$
$558$	0	0
$559$	35.0846	1.48392
$560$	0	0
$561$	24.8190	1.04786
$562$	0	0
$563$	−32.7399	−1.37982	−0.689912	−	0.723893i	$-0.742351\pi$
$563$	−32.7399	−1.37982	−0.689912	+	0.723893i	$0.742351\pi$
$564$	0	0
$565$	6.55599	0.275813
$566$	0	0
$567$	3.32949	0.139825
$568$	0	0
$569$	18.9034	0.792471	0.396236	−	0.918149i	$-0.370316\pi$
$569$	18.9034	0.792471	0.396236	+	0.918149i	$0.370316\pi$
$570$	0	0
$571$	34.0761	1.42604	0.713021	−	0.701143i	$-0.247326\pi$
$571$	34.0761	1.42604	0.713021	+	0.701143i	$0.247326\pi$
$572$	0	0
$573$	1.29944	0.0542848
$574$	0	0
$575$	32.6235	1.36049
$576$	0	0
$577$	−13.4297	−0.559087	−0.279544	−	0.960133i	$-0.590183\pi$
$577$	−13.4297	−0.559087	−0.279544	+	0.960133i	$0.590183\pi$
$578$	0	0
$579$	5.01879	0.208574
$580$	0	0
$581$	−4.32647	−0.179492
$582$	0	0
$583$	3.54928	0.146996
$584$	0	0
$585$	−0.312867	−0.0129355
$586$	0	0
$587$	18.5311	0.764861	0.382431	−	0.923984i	$-0.375087\pi$
$587$	18.5311	0.764861	0.382431	+	0.923984i	$0.375087\pi$
$588$	0	0
$589$	6.75484	0.278328
$590$	0	0
$591$	28.1443	1.15770
$592$	0	0
$593$	−27.9690	−1.14855	−0.574275	−	0.818663i	$-0.694716\pi$
$593$	−27.9690	−1.14855	−0.574275	+	0.818663i	$0.694716\pi$
$594$	0	0
$595$	0.669571	0.0274497
$596$	0	0
$597$	−22.4708	−0.919667
$598$	0	0
$599$	12.0086	0.490660	0.245330	−	0.969440i	$-0.421104\pi$
$599$	12.0086	0.490660	0.245330	+	0.969440i	$0.421104\pi$
$600$	0	0
$601$	33.1868	1.35372	0.676858	−	0.736113i	$-0.263341\pi$
$601$	33.1868	1.35372	0.676858	+	0.736113i	$0.263341\pi$
$602$	0	0
$603$	1.29988	0.0529350
$604$	0	0
$605$	0.815044	0.0331362
$606$	0	0
$607$	−11.1842	−0.453953	−0.226976	−	0.973900i	$-0.572884\pi$
$607$	−11.1842	−0.453953	−0.226976	+	0.973900i	$0.572884\pi$
$608$	0	0
$609$	−6.42331	−0.260285
$610$	0	0
$611$	−37.8845	−1.53264
$612$	0	0
$613$	9.56803	0.386449	0.193225	−	0.981155i	$-0.438105\pi$
$613$	9.56803	0.386449	0.193225	+	0.981155i	$0.438105\pi$
$614$	0	0
$615$	−6.62546	−0.267164
$616$	0	0
$617$	−21.1891	−0.853041	−0.426520	−	0.904478i	$-0.640261\pi$
$617$	−21.1891	−0.853041	−0.426520	+	0.904478i	$0.640261\pi$
$618$	0	0
$619$	10.2112	0.410423	0.205212	−	0.978718i	$-0.434212\pi$
$619$	10.2112	0.410423	0.205212	+	0.978718i	$0.434212\pi$
$620$	0	0
$621$	35.9492	1.44259
$622$	0	0
$623$	−5.26167	−0.210804
$624$	0	0
$625$	22.3790	0.895159
$626$	0	0
$627$	9.99858	0.399305
$628$	0	0
$629$	0.346088	0.0137994
$630$	0	0
$631$	22.0828	0.879103	0.439551	−	0.898217i	$-0.355137\pi$
$631$	22.0828	0.879103	0.439551	+	0.898217i	$0.355137\pi$
$632$	0	0
$633$	7.65610	0.304303
$634$	0	0
$635$	−3.88540	−0.154187
$636$	0	0
$637$	−34.8703	−1.38161
$638$	0	0
$639$	−0.392589	−0.0155306
$640$	0	0
$641$	46.3107	1.82916	0.914581	−	0.404402i	$-0.132520\pi$
$641$	46.3107	1.82916	0.914581	+	0.404402i	$0.132520\pi$
$642$	0	0
$643$	1.47170	0.0580383	0.0290191	−	0.999579i	$-0.490762\pi$
$643$	1.47170	0.0580383	0.0290191	+	0.999579i	$0.490762\pi$
$644$	0	0
$645$	4.89449	0.192720
$646$	0	0
$647$	−40.9789	−1.61105	−0.805524	−	0.592563i	$-0.798116\pi$
$647$	−40.9789	−1.61105	−0.805524	+	0.592563i	$0.798116\pi$
$648$	0	0
$649$	−38.5000	−1.51126
$650$	0	0
$651$	2.70368	0.105966
$652$	0	0
$653$	29.8270	1.16722	0.583610	−	0.812034i	$-0.301640\pi$
$653$	29.8270	1.16722	0.583610	+	0.812034i	$0.301640\pi$
$654$	0	0
$655$	−0.341282	−0.0133350
$656$	0	0
$657$	−1.38315	−0.0539617
$658$	0	0
$659$	−29.7659	−1.15952	−0.579758	−	0.814789i	$-0.696853\pi$
$659$	−29.7659	−1.15952	−0.579758	+	0.814789i	$0.696853\pi$
$660$	0	0
$661$	−11.0597	−0.430174	−0.215087	−	0.976595i	$-0.569004\pi$
$661$	−11.0597	−0.430174	−0.215087	+	0.976595i	$0.569004\pi$
$662$	0	0
$663$	−35.1347	−1.36452
$664$	0	0
$665$	0.269743	0.0104602
$666$	0	0
$667$	−65.9679	−2.55429
$668$	0	0
$669$	5.68872	0.219939
$670$	0	0
$671$	48.9081	1.88808
$672$	0	0
$673$	7.03679	0.271248	0.135624	−	0.990760i	$-0.456696\pi$
$673$	7.03679	0.271248	0.135624	+	0.990760i	$0.456696\pi$
$674$	0	0
$675$	25.6346	0.986676
$676$	0	0
$677$	−10.8703	−0.417779	−0.208889	−	0.977939i	$-0.566985\pi$
$677$	−10.8703	−0.417779	−0.208889	+	0.977939i	$0.566985\pi$
$678$	0	0
$679$	7.10874	0.272809
$680$	0	0
$681$	28.5711	1.09485
$682$	0	0
$683$	−11.0446	−0.422609	−0.211304	−	0.977420i	$-0.567771\pi$
$683$	−11.0446	−0.422609	−0.211304	+	0.977420i	$0.567771\pi$
$684$	0	0
$685$	−6.11302	−0.233567
$686$	0	0
$687$	33.3007	1.27050
$688$	0	0
$689$	−5.02450	−0.191418
$690$	0	0
$691$	−39.1603	−1.48973	−0.744863	−	0.667217i	$-0.767485\pi$
$691$	−39.1603	−1.48973	−0.744863	+	0.667217i	$0.767485\pi$
$692$	0	0
$693$	−0.204903	−0.00778362
$694$	0	0
$695$	8.84416	0.335478
$696$	0	0
$697$	38.0945	1.44293
$698$	0	0
$699$	3.69451	0.139739
$700$	0	0
$701$	−12.8788	−0.486427	−0.243213	−	0.969973i	$-0.578202\pi$
$701$	−12.8788	−0.486427	−0.243213	+	0.969973i	$0.578202\pi$
$702$	0	0
$703$	0.139425	0.00525851
$704$	0	0
$705$	−5.28510	−0.199048
$706$	0	0
$707$	0.526941	0.0198177
$708$	0	0
$709$	46.8186	1.75831	0.879155	−	0.476537i	$-0.158108\pi$
$709$	46.8186	1.75831	0.879155	+	0.476537i	$0.158108\pi$
$710$	0	0
$711$	0.689352	0.0258527
$712$	0	0
$713$	27.7670	1.03988
$714$	0	0
$715$	−7.70182	−0.288032
$716$	0	0
$717$	−14.3305	−0.535184
$718$	0	0
$719$	−14.7716	−0.550889	−0.275445	−	0.961317i	$-0.588825\pi$
$719$	−14.7716	−0.550889	−0.275445	+	0.961317i	$0.588825\pi$
$720$	0	0
$721$	−0.135592	−0.00504973
$722$	0	0
$723$	−38.7461	−1.44098
$724$	0	0
$725$	−47.0403	−1.74703
$726$	0	0
$727$	−42.3426	−1.57040	−0.785201	−	0.619241i	$-0.787440\pi$
$727$	−42.3426	−1.57040	−0.785201	+	0.619241i	$0.787440\pi$
$728$	0	0
$729$	28.1838	1.04385
$730$	0	0
$731$	−28.1419	−1.04087
$732$	0	0
$733$	3.42130	0.126369	0.0631844	−	0.998002i	$-0.479874\pi$
$733$	3.42130	0.126369	0.0631844	+	0.998002i	$0.479874\pi$
$734$	0	0
$735$	−4.86459	−0.179433
$736$	0	0
$737$	31.9989	1.17869
$738$	0	0
$739$	35.3241	1.29942	0.649709	−	0.760183i	$-0.274891\pi$
$739$	35.3241	1.29942	0.649709	+	0.760183i	$0.274891\pi$
$740$	0	0
$741$	−14.1544	−0.519974
$742$	0	0
$743$	−26.1434	−0.959108	−0.479554	−	0.877512i	$-0.659202\pi$
$743$	−26.1434	−0.959108	−0.479554	+	0.877512i	$0.659202\pi$
$744$	0	0
$745$	1.61323	0.0591042
$746$	0	0
$747$	−1.62157	−0.0593300
$748$	0	0
$749$	−5.14334	−0.187934
$750$	0	0
$751$	41.2156	1.50398	0.751990	−	0.659175i	$-0.229094\pi$
$751$	41.2156	1.50398	0.751990	+	0.659175i	$0.229094\pi$
$752$	0	0
$753$	1.68934	0.0615631
$754$	0	0
$755$	−1.76628	−0.0642815
$756$	0	0
$757$	1.29748	0.0471575	0.0235788	−	0.999722i	$-0.492494\pi$
$757$	1.29748	0.0471575	0.0235788	+	0.999722i	$0.492494\pi$
$758$	0	0
$759$	41.1010	1.49187
$760$	0	0
$761$	32.0781	1.16283	0.581415	−	0.813607i	$-0.302499\pi$
$761$	32.0781	1.16283	0.581415	+	0.813607i	$0.302499\pi$
$762$	0	0
$763$	6.98767	0.252971
$764$	0	0
$765$	0.250956	0.00907334
$766$	0	0
$767$	54.5021	1.96796
$768$	0	0
$769$	−19.9507	−0.719440	−0.359720	−	0.933060i	$-0.617128\pi$
$769$	−19.9507	−0.719440	−0.359720	+	0.933060i	$0.617128\pi$
$770$	0	0
$771$	−12.2046	−0.439539
$772$	0	0
$773$	−14.1770	−0.509912	−0.254956	−	0.966953i	$-0.582061\pi$
$773$	−14.1770	−0.509912	−0.254956	+	0.966953i	$0.582061\pi$
$774$	0	0
$775$	19.8001	0.711239
$776$	0	0
$777$	0.0558060	0.00200203
$778$	0	0
$779$	15.3467	0.549853
$780$	0	0
$781$	−9.66431	−0.345816
$782$	0	0
$783$	−51.8357	−1.85246
$784$	0	0
$785$	−9.98705	−0.356453
$786$	0	0
$787$	20.0106	0.713300	0.356650	−	0.934238i	$-0.383919\pi$
$787$	20.0106	0.713300	0.356650	+	0.934238i	$0.383919\pi$
$788$	0	0
$789$	13.1259	0.467294
$790$	0	0
$791$	−6.07825	−0.216118
$792$	0	0
$793$	−69.2362	−2.45865
$794$	0	0
$795$	−0.700945	−0.0248600
$796$	0	0
$797$	2.82221	0.0999679	0.0499840	−	0.998750i	$-0.484083\pi$
$797$	2.82221	0.0999679	0.0499840	+	0.998750i	$0.484083\pi$
$798$	0	0
$799$	30.3878	1.07504
$800$	0	0
$801$	−1.97208	−0.0696800
$802$	0	0
$803$	−34.0487	−1.20155
$804$	0	0
$805$	1.10883	0.0390811
$806$	0	0
$807$	−30.2147	−1.06361
$808$	0	0
$809$	−27.1741	−0.955391	−0.477696	−	0.878525i	$-0.658528\pi$
$809$	−27.1741	−0.955391	−0.477696	+	0.878525i	$0.658528\pi$
$810$	0	0
$811$	−48.8363	−1.71488	−0.857438	−	0.514587i	$-0.827945\pi$
$811$	−48.8363	−1.71488	−0.857438	+	0.514587i	$0.827945\pi$
$812$	0	0
$813$	45.6743	1.60187
$814$	0	0
$815$	−7.43075	−0.260288
$816$	0	0
$817$	−11.3372	−0.396640
$818$	0	0
$819$	0.290069	0.0101358
$820$	0	0
$821$	−39.6455	−1.38364	−0.691818	−	0.722072i	$-0.743190\pi$
$821$	−39.6455	−1.38364	−0.691818	+	0.722072i	$0.743190\pi$
$822$	0	0
$823$	−6.83378	−0.238211	−0.119105	−	0.992882i	$-0.538003\pi$
$823$	−6.83378	−0.238211	−0.119105	+	0.992882i	$0.538003\pi$
$824$	0	0
$825$	29.3082	1.02038
$826$	0	0
$827$	51.4419	1.78881	0.894405	−	0.447259i	$-0.147600\pi$
$827$	51.4419	1.78881	0.894405	+	0.447259i	$0.147600\pi$
$828$	0	0
$829$	31.9308	1.10900	0.554502	−	0.832182i	$-0.312909\pi$
$829$	31.9308	1.10900	0.554502	+	0.832182i	$0.312909\pi$
$830$	0	0
$831$	−9.12902	−0.316682
$832$	0	0
$833$	27.9700	0.969104
$834$	0	0
$835$	−6.19512	−0.214391
$836$	0	0
$837$	21.8186	0.754160
$838$	0	0
$839$	−10.7646	−0.371635	−0.185817	−	0.982584i	$-0.559493\pi$
$839$	−10.7646	−0.371635	−0.185817	+	0.982584i	$0.559493\pi$
$840$	0	0
$841$	66.1201	2.28000
$842$	0	0
$843$	37.7508	1.30021
$844$	0	0
$845$	5.43651	0.187022
$846$	0	0
$847$	−0.755651	−0.0259645
$848$	0	0
$849$	3.39348	0.116464
$850$	0	0
$851$	0.573132	0.0196467
$852$	0	0
$853$	−28.4363	−0.973640	−0.486820	−	0.873502i	$-0.661843\pi$
$853$	−28.4363	−0.973640	−0.486820	+	0.873502i	$0.661843\pi$
$854$	0	0
$855$	0.101100	0.00345755
$856$	0	0
$857$	26.4119	0.902214	0.451107	−	0.892470i	$-0.351029\pi$
$857$	26.4119	0.902214	0.451107	+	0.892470i	$0.351029\pi$
$858$	0	0
$859$	−25.0628	−0.855133	−0.427566	−	0.903984i	$-0.640629\pi$
$859$	−25.0628	−0.855133	−0.427566	+	0.903984i	$0.640629\pi$
$860$	0	0
$861$	6.14265	0.209341
$862$	0	0
$863$	−25.8964	−0.881524	−0.440762	−	0.897624i	$-0.645292\pi$
$863$	−25.8964	−0.881524	−0.440762	+	0.897624i	$0.645292\pi$
$864$	0	0
$865$	7.39617	0.251477
$866$	0	0
$867$	−0.536675	−0.0182264
$868$	0	0
$869$	16.9697	0.575658
$870$	0	0
$871$	−45.2988	−1.53489
$872$	0	0
$873$	2.66437	0.0901751
$874$	0	0
$875$	1.61035	0.0544398
$876$	0	0
$877$	26.0354	0.879152	0.439576	−	0.898205i	$-0.355129\pi$
$877$	26.0354	0.879152	0.439576	+	0.898205i	$0.355129\pi$
$878$	0	0
$879$	44.7814	1.51044
$880$	0	0
$881$	7.43815	0.250598	0.125299	−	0.992119i	$-0.460011\pi$
$881$	7.43815	0.250598	0.125299	+	0.992119i	$0.460011\pi$
$882$	0	0
$883$	39.5659	1.33150	0.665750	−	0.746175i	$-0.268112\pi$
$883$	39.5659	1.33150	0.665750	+	0.746175i	$0.268112\pi$
$884$	0	0
$885$	7.60334	0.255584
$886$	0	0
$887$	24.2329	0.813662	0.406831	−	0.913504i	$-0.366634\pi$
$887$	24.2329	0.813662	0.406831	+	0.913504i	$0.366634\pi$
$888$	0	0
$889$	3.60227	0.120816
$890$	0	0
$891$	30.7193	1.02913
$892$	0	0
$893$	12.2420	0.409664
$894$	0	0
$895$	−8.25119	−0.275807
$896$	0	0
$897$	−58.1842	−1.94271
$898$	0	0
$899$	−40.0377	−1.33533
$900$	0	0
$901$	4.03023	0.134266
$902$	0	0
$903$	−4.53783	−0.151009
$904$	0	0
$905$	1.56280	0.0519493
$906$	0	0
$907$	−13.0999	−0.434976	−0.217488	−	0.976063i	$-0.569786\pi$
$907$	−13.0999	−0.434976	−0.217488	+	0.976063i	$0.569786\pi$
$908$	0	0
$909$	0.197498	0.00655060
$910$	0	0
$911$	40.0200	1.32592	0.662961	−	0.748654i	$-0.269300\pi$
$911$	40.0200	1.32592	0.662961	+	0.748654i	$0.269300\pi$
$912$	0	0
$913$	−39.9179	−1.32109
$914$	0	0
$915$	−9.65883	−0.319311
$916$	0	0
$917$	0.316412	0.0104488
$918$	0	0
$919$	−23.1724	−0.764386	−0.382193	−	0.924082i	$-0.624831\pi$
$919$	−23.1724	−0.764386	−0.382193	+	0.924082i	$0.624831\pi$
$920$	0	0
$921$	15.1052	0.497733
$922$	0	0
$923$	13.6812	0.450321
$924$	0	0
$925$	0.408688	0.0134376
$926$	0	0
$927$	−0.0508202	−0.00166915
$928$	0	0
$929$	−16.8836	−0.553934	−0.276967	−	0.960879i	$-0.589329\pi$
$929$	−16.8836	−0.553934	−0.276967	+	0.960879i	$0.589329\pi$
$930$	0	0
$931$	11.2680	0.369293
$932$	0	0
$933$	17.5670	0.575117
$934$	0	0
$935$	6.17775	0.202034
$936$	0	0
$937$	16.4579	0.537655	0.268827	−	0.963188i	$-0.413364\pi$
$937$	16.4579	0.537655	0.268827	+	0.963188i	$0.413364\pi$
$938$	0	0
$939$	24.4188	0.796878
$940$	0	0
$941$	−50.1129	−1.63363	−0.816817	−	0.576897i	$-0.804263\pi$
$941$	−50.1129	−1.63363	−0.816817	+	0.576897i	$0.804263\pi$
$942$	0	0
$943$	63.0856	2.05435
$944$	0	0
$945$	0.871288	0.0283430
$946$	0	0
$947$	29.2422	0.950243	0.475121	−	0.879920i	$-0.342404\pi$
$947$	29.2422	0.950243	0.475121	+	0.879920i	$0.342404\pi$
$948$	0	0
$949$	48.2007	1.56466
$950$	0	0
$951$	−5.41727	−0.175667
$952$	0	0
$953$	−8.21064	−0.265969	−0.132984	−	0.991118i	$-0.542456\pi$
$953$	−8.21064	−0.265969	−0.132984	+	0.991118i	$0.542456\pi$
$954$	0	0
$955$	0.323446	0.0104665
$956$	0	0
$957$	−59.2642	−1.91574
$958$	0	0
$959$	5.66756	0.183015
$960$	0	0
$961$	−14.1474	−0.456369
$962$	0	0
$963$	−1.92773	−0.0621203
$964$	0	0
$965$	1.24924	0.0402144
$966$	0	0
$967$	−0.730530	−0.0234923	−0.0117461	−	0.999931i	$-0.503739\pi$
$967$	−0.730530	−0.0234923	−0.0117461	+	0.999931i	$0.503739\pi$
$968$	0	0
$969$	11.3535	0.364726
$970$	0	0
$971$	−18.3533	−0.588985	−0.294492	−	0.955654i	$-0.595151\pi$
$971$	−18.3533	−0.588985	−0.294492	+	0.955654i	$0.595151\pi$
$972$	0	0
$973$	−8.19968	−0.262870
$974$	0	0
$975$	−41.4899	−1.32874
$976$	0	0
$977$	16.7633	0.536305	0.268153	−	0.963376i	$-0.413587\pi$
$977$	16.7633	0.536305	0.268153	+	0.963376i	$0.413587\pi$
$978$	0	0
$979$	−48.5464	−1.55155
$980$	0	0
$981$	2.61899	0.0836178
$982$	0	0
$983$	−8.00339	−0.255268	−0.127634	−	0.991821i	$-0.540738\pi$
$983$	−8.00339	−0.255268	−0.127634	+	0.991821i	$0.540738\pi$
$984$	0	0
$985$	7.00545	0.223212
$986$	0	0
$987$	4.89997	0.155968
$988$	0	0
$989$	−46.6039	−1.48192
$990$	0	0
$991$	−15.9595	−0.506971	−0.253486	−	0.967339i	$-0.581577\pi$
$991$	−15.9595	−0.506971	−0.253486	+	0.967339i	$0.581577\pi$
$992$	0	0
$993$	14.7103	0.466817
$994$	0	0
$995$	−5.59325	−0.177318
$996$	0	0
$997$	6.37245	0.201817	0.100909	−	0.994896i	$-0.467825\pi$
$997$	6.37245	0.201817	0.100909	+	0.994896i	$0.467825\pi$
$998$	0	0
$999$	0.450351	0.0142485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.f.1.5		6
4.3	odd	2		502.2.a.e.1.2	✓	6
12.11	even	2		4518.2.a.x.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.e.1.2	✓	6	4.3	odd	2
4016.2.a.f.1.5		6	1.1	even	1	trivial
4518.2.a.x.1.4		6	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)

\(f(q)\)	\(=\)	\(q+1.68934 q^{3} +0.420498 q^{5} -0.389856 q^{7} -0.146119 q^{9} +O(q^{10})\)	\(q+1.68934 q^{3} +0.420498 q^{5} -0.389856 q^{7} -0.146119 q^{9} -3.59698 q^{11} +5.09203 q^{13} +0.710366 q^{15} -4.08440 q^{17} -1.64544 q^{19} -0.658601 q^{21} -6.76389 q^{23} -4.82318 q^{25} -5.31487 q^{27} +9.75295 q^{29} -4.10519 q^{31} -6.07654 q^{33} -0.163934 q^{35} -0.0847341 q^{37} +8.60218 q^{39} -9.32682 q^{41} +6.89010 q^{43} -0.0614426 q^{45} -7.43997 q^{47} -6.84801 q^{49} -6.89995 q^{51} -0.986738 q^{53} -1.51252 q^{55} -2.77971 q^{57} +10.7034 q^{59} -13.5970 q^{61} +0.0569652 q^{63} +2.14119 q^{65} -8.89603 q^{67} -11.4265 q^{69} +2.68678 q^{71} +9.46592 q^{73} -8.14801 q^{75} +1.40231 q^{77} -4.71776 q^{79} -8.54029 q^{81} +11.0976 q^{83} -1.71748 q^{85} +16.4761 q^{87} +13.4964 q^{89} -1.98516 q^{91} -6.93507 q^{93} -0.691904 q^{95} -18.2343 q^{97} +0.525586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9}+O(q^{10}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 15 q^{9} - q^{11} - 5 q^{13} - 2 q^{15} + 8 q^{17} - 3 q^{19} - 14 q^{21} - 18 q^{23} - q^{25} - 16 q^{27} + q^{29} - 6 q^{31} - 16 q^{33} - 6 q^{35} - 13 q^{37} + 6 q^{39} + 4 q^{41} + 5 q^{43} - 23 q^{45} - 8 q^{47} + 8 q^{49} + 16 q^{51} - 3 q^{53} + 30 q^{55} - 24 q^{57} + 5 q^{59} - 61 q^{61} + 27 q^{63} - q^{65} + 13 q^{67} - 21 q^{69} - 22 q^{71} + 6 q^{73} + 30 q^{75} + 4 q^{77} - 28 q^{79} + 2 q^{81} - 14 q^{83} - 16 q^{85} + 24 q^{87} + 18 q^{89} + 16 q^{91} + 27 q^{93} - 20 q^{95} + 16 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 15 * q^9 - q^11 - 5 * q^13 - 2 * q^15 + 8 * q^17 - 3 * q^19 - 14 * q^21 - 18 * q^23 - q^25 - 16 * q^27 + q^29 - 6 * q^31 - 16 * q^33 - 6 * q^35 - 13 * q^37 + 6 * q^39 + 4 * q^41 + 5 * q^43 - 23 * q^45 - 8 * q^47 + 8 * q^49 + 16 * q^51 - 3 * q^53 + 30 * q^55 - 24 * q^57 + 5 * q^59 - 61 * q^61 + 27 * q^63 - q^65 + 13 * q^67 - 21 * q^69 - 22 * q^71 + 6 * q^73 + 30 * q^75 + 4 * q^77 - 28 * q^79 + 2 * q^81 - 14 * q^83 - 16 * q^85 + 24 * q^87 + 18 * q^89 + 16 * q^91 + 27 * q^93 - 20 * q^95 + 16 * q^97 - 5 * q^99

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