LMFDB - Embedded newform 4016.2.a.j.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:	$14$
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			1.5
Root			$1.90340$ 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)

$f(q)$	$=$	$q-1.90340 q^{3} +3.78264 q^{5} +0.441155 q^{7} +0.622940 q^{9} +O(q^{10})$	$q-1.90340 q^{3} +3.78264 q^{5} +0.441155 q^{7} +0.622940 q^{9} +3.73416 q^{11} +0.731910 q^{13} -7.19988 q^{15} -2.10658 q^{17} -7.92717 q^{19} -0.839696 q^{21} -7.60160 q^{23} +9.30833 q^{25} +4.52450 q^{27} +4.11495 q^{29} -4.88026 q^{31} -7.10762 q^{33} +1.66873 q^{35} -10.6240 q^{37} -1.39312 q^{39} -9.14263 q^{41} -4.80343 q^{43} +2.35636 q^{45} -5.33757 q^{47} -6.80538 q^{49} +4.00967 q^{51} -9.64365 q^{53} +14.1250 q^{55} +15.0886 q^{57} -6.91474 q^{59} +1.47262 q^{61} +0.274813 q^{63} +2.76855 q^{65} +12.5289 q^{67} +14.4689 q^{69} +9.17270 q^{71} +0.0833940 q^{73} -17.7175 q^{75} +1.64735 q^{77} -9.43340 q^{79} -10.4808 q^{81} +10.4971 q^{83} -7.96843 q^{85} -7.83241 q^{87} -16.3958 q^{89} +0.322886 q^{91} +9.28911 q^{93} -29.9856 q^{95} -2.63233 q^{97} +2.32616 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.90340	−1.09893	−0.549465	−	0.835517i	$-0.685168\pi$
$3$	−1.90340	−1.09893	−0.549465	+	0.835517i	$0.685168\pi$
$4$	0	0
$5$	3.78264	1.69165	0.845823	−	0.533464i	$-0.179110\pi$
$5$	3.78264	1.69165	0.845823	+	0.533464i	$0.179110\pi$
$6$	0	0
$7$	0.441155	0.166741	0.0833705	−	0.996519i	$-0.473432\pi$
$7$	0.441155	0.166741	0.0833705	+	0.996519i	$0.473432\pi$
$8$	0	0
$9$	0.622940	0.207647
$10$	0	0
$11$	3.73416	1.12589	0.562946	−	0.826493i	$-0.309668\pi$
$11$	3.73416	1.12589	0.562946	+	0.826493i	$0.309668\pi$
$12$	0	0
$13$	0.731910	0.202995	0.101498	−	0.994836i	$-0.467637\pi$
$13$	0.731910	0.202995	0.101498	+	0.994836i	$0.467637\pi$
$14$	0	0
$15$	−7.19988	−1.85900
$16$	0	0
$17$	−2.10658	−0.510921	−0.255461	−	0.966819i	$-0.582227\pi$
$17$	−2.10658	−0.510921	−0.255461	+	0.966819i	$0.582227\pi$
$18$	0	0
$19$	−7.92717	−1.81862	−0.909309	−	0.416122i	$-0.863389\pi$
$19$	−7.92717	−1.81862	−0.909309	+	0.416122i	$0.863389\pi$
$20$	0	0
$21$	−0.839696	−0.183237
$22$	0	0
$23$	−7.60160	−1.58504	−0.792521	−	0.609844i	$-0.791232\pi$
$23$	−7.60160	−1.58504	−0.792521	+	0.609844i	$0.791232\pi$
$24$	0	0
$25$	9.30833	1.86167
$26$	0	0
$27$	4.52450	0.870741
$28$	0	0
$29$	4.11495	0.764128	0.382064	−	0.924136i	$-0.375213\pi$
$29$	4.11495	0.764128	0.382064	+	0.924136i	$0.375213\pi$
$30$	0	0
$31$	−4.88026	−0.876521	−0.438261	−	0.898848i	$-0.644405\pi$
$31$	−4.88026	−0.876521	−0.438261	+	0.898848i	$0.644405\pi$
$32$	0	0
$33$	−7.10762	−1.23728
$34$	0	0
$35$	1.66873	0.282067
$36$	0	0
$37$	−10.6240	−1.74658	−0.873291	−	0.487200i	$-0.838018\pi$
$37$	−10.6240	−1.74658	−0.873291	+	0.487200i	$0.838018\pi$
$38$	0	0
$39$	−1.39312	−0.223078
$40$	0	0
$41$	−9.14263	−1.42784	−0.713919	−	0.700228i	$-0.753082\pi$
$41$	−9.14263	−1.42784	−0.713919	+	0.700228i	$0.753082\pi$
$42$	0	0
$43$	−4.80343	−0.732516	−0.366258	−	0.930513i	$-0.619361\pi$
$43$	−4.80343	−0.732516	−0.366258	+	0.930513i	$0.619361\pi$
$44$	0	0
$45$	2.35636	0.351265
$46$	0	0
$47$	−5.33757	−0.778565	−0.389282	−	0.921119i	$-0.627277\pi$
$47$	−5.33757	−0.778565	−0.389282	+	0.921119i	$0.627277\pi$
$48$	0	0
$49$	−6.80538	−0.972197
$50$	0	0
$51$	4.00967	0.561467
$52$	0	0
$53$	−9.64365	−1.32466	−0.662329	−	0.749213i	$-0.730432\pi$
$53$	−9.64365	−1.32466	−0.662329	+	0.749213i	$0.730432\pi$
$54$	0	0
$55$	14.1250	1.90461
$56$	0	0
$57$	15.0886	1.99853
$58$	0	0
$59$	−6.91474	−0.900223	−0.450111	−	0.892972i	$-0.648616\pi$
$59$	−6.91474	−0.900223	−0.450111	+	0.892972i	$0.648616\pi$
$60$	0	0
$61$	1.47262	0.188550	0.0942748	−	0.995546i	$-0.469947\pi$
$61$	1.47262	0.188550	0.0942748	+	0.995546i	$0.469947\pi$
$62$	0	0
$63$	0.274813	0.0346232
$64$	0	0
$65$	2.76855	0.343396
$66$	0	0
$67$	12.5289	1.53065	0.765323	−	0.643646i	$-0.222579\pi$
$67$	12.5289	1.53065	0.765323	+	0.643646i	$0.222579\pi$
$68$	0	0
$69$	14.4689	1.74185
$70$	0	0
$71$	9.17270	1.08860	0.544300	−	0.838891i	$-0.316795\pi$
$71$	9.17270	1.08860	0.544300	+	0.838891i	$0.316795\pi$
$72$	0	0
$73$	0.0833940	0.00976052	0.00488026	−	0.999988i	$-0.498447\pi$
$73$	0.0833940	0.00976052	0.00488026	+	0.999988i	$0.498447\pi$
$74$	0	0
$75$	−17.7175	−2.04584
$76$	0	0
$77$	1.64735	0.187732
$78$	0	0
$79$	−9.43340	−1.06134	−0.530670	−	0.847578i	$-0.678060\pi$
$79$	−9.43340	−1.06134	−0.530670	+	0.847578i	$0.678060\pi$
$80$	0	0
$81$	−10.4808	−1.16453
$82$	0	0
$83$	10.4971	1.15220	0.576102	−	0.817378i	$-0.304573\pi$
$83$	10.4971	1.15220	0.576102	+	0.817378i	$0.304573\pi$
$84$	0	0
$85$	−7.96843	−0.864298
$86$	0	0
$87$	−7.83241	−0.839723
$88$	0	0
$89$	−16.3958	−1.73795	−0.868976	−	0.494854i	$-0.835222\pi$
$89$	−16.3958	−1.73795	−0.868976	+	0.494854i	$0.835222\pi$
$90$	0	0
$91$	0.322886	0.0338476
$92$	0	0
$93$	9.28911	0.963236
$94$	0	0
$95$	−29.9856	−3.07646
$96$	0	0
$97$	−2.63233	−0.267272	−0.133636	−	0.991030i	$-0.542665\pi$
$97$	−2.63233	−0.267272	−0.133636	+	0.991030i	$0.542665\pi$
$98$	0	0
$99$	2.32616	0.233788
$100$	0	0
$101$	8.59417	0.855152	0.427576	−	0.903979i	$-0.359368\pi$
$101$	8.59417	0.855152	0.427576	+	0.903979i	$0.359368\pi$
$102$	0	0
$103$	15.1166	1.48948	0.744740	−	0.667355i	$-0.232573\pi$
$103$	15.1166	1.48948	0.744740	+	0.667355i	$0.232573\pi$
$104$	0	0
$105$	−3.17626	−0.309971
$106$	0	0
$107$	−13.4156	−1.29693	−0.648466	−	0.761243i	$-0.724589\pi$
$107$	−13.4156	−1.29693	−0.648466	+	0.761243i	$0.724589\pi$
$108$	0	0
$109$	4.50511	0.431511	0.215755	−	0.976447i	$-0.430779\pi$
$109$	4.50511	0.431511	0.215755	+	0.976447i	$0.430779\pi$
$110$	0	0
$111$	20.2218	1.91937
$112$	0	0
$113$	16.5759	1.55933	0.779664	−	0.626198i	$-0.215390\pi$
$113$	16.5759	1.55933	0.779664	+	0.626198i	$0.215390\pi$
$114$	0	0
$115$	−28.7541	−2.68133
$116$	0	0
$117$	0.455936	0.0421513
$118$	0	0
$119$	−0.929330	−0.0851915
$120$	0	0
$121$	2.94398	0.267634
$122$	0	0
$123$	17.4021	1.56909
$124$	0	0
$125$	16.2968	1.45763
$126$	0	0
$127$	19.0042	1.68635	0.843175	−	0.537639i	$-0.180684\pi$
$127$	19.0042	1.68635	0.843175	+	0.537639i	$0.180684\pi$
$128$	0	0
$129$	9.14286	0.804984
$130$	0	0
$131$	16.8991	1.47648	0.738241	−	0.674538i	$-0.235657\pi$
$131$	16.8991	1.47648	0.738241	+	0.674538i	$0.235657\pi$
$132$	0	0
$133$	−3.49711	−0.303238
$134$	0	0
$135$	17.1145	1.47298
$136$	0	0
$137$	−3.83602	−0.327733	−0.163867	−	0.986483i	$-0.552397\pi$
$137$	−3.83602	−0.327733	−0.163867	+	0.986483i	$0.552397\pi$
$138$	0	0
$139$	4.94128	0.419114	0.209557	−	0.977796i	$-0.432798\pi$
$139$	4.94128	0.419114	0.209557	+	0.977796i	$0.432798\pi$
$140$	0	0
$141$	10.1595	0.855588
$142$	0	0
$143$	2.73307	0.228551
$144$	0	0
$145$	15.5654	1.29263
$146$	0	0
$147$	12.9534	1.06838
$148$	0	0
$149$	−19.5806	−1.60411	−0.802055	−	0.597251i	$-0.796260\pi$
$149$	−19.5806	−1.60411	−0.802055	+	0.597251i	$0.796260\pi$
$150$	0	0
$151$	−2.89370	−0.235486	−0.117743	−	0.993044i	$-0.537566\pi$
$151$	−2.89370	−0.235486	−0.117743	+	0.993044i	$0.537566\pi$
$152$	0	0
$153$	−1.31228	−0.106091
$154$	0	0
$155$	−18.4603	−1.48276
$156$	0	0
$157$	−1.05263	−0.0840090	−0.0420045	−	0.999117i	$-0.513374\pi$
$157$	−1.05263	−0.0840090	−0.0420045	+	0.999117i	$0.513374\pi$
$158$	0	0
$159$	18.3557	1.45571
$160$	0	0
$161$	−3.35348	−0.264292
$162$	0	0
$163$	−16.7361	−1.31087	−0.655435	−	0.755252i	$-0.727515\pi$
$163$	−16.7361	−1.31087	−0.655435	+	0.755252i	$0.727515\pi$
$164$	0	0
$165$	−26.8855	−2.09303
$166$	0	0
$167$	−1.24869	−0.0966265	−0.0483133	−	0.998832i	$-0.515385\pi$
$167$	−1.24869	−0.0966265	−0.0483133	+	0.998832i	$0.515385\pi$
$168$	0	0
$169$	−12.4643	−0.958793
$170$	0	0
$171$	−4.93815	−0.377630
$172$	0	0
$173$	6.48340	0.492924	0.246462	−	0.969152i	$-0.420732\pi$
$173$	6.48340	0.492924	0.246462	+	0.969152i	$0.420732\pi$
$174$	0	0
$175$	4.10642	0.310416
$176$	0	0
$177$	13.1615	0.989282
$178$	0	0
$179$	−16.4977	−1.23309	−0.616547	−	0.787318i	$-0.711469\pi$
$179$	−16.4977	−1.23309	−0.616547	+	0.787318i	$0.711469\pi$
$180$	0	0
$181$	10.0693	0.748447	0.374223	−	0.927339i	$-0.377909\pi$
$181$	10.0693	0.748447	0.374223	+	0.927339i	$0.377909\pi$
$182$	0	0
$183$	−2.80299	−0.207203
$184$	0	0
$185$	−40.1869	−2.95460
$186$	0	0
$187$	−7.86632	−0.575243
$188$	0	0
$189$	1.99601	0.145188
$190$	0	0
$191$	5.70317	0.412667	0.206333	−	0.978482i	$-0.433847\pi$
$191$	5.70317	0.412667	0.206333	+	0.978482i	$0.433847\pi$
$192$	0	0
$193$	25.6893	1.84916	0.924580	−	0.380989i	$-0.124416\pi$
$193$	25.6893	1.84916	0.924580	+	0.380989i	$0.124416\pi$
$194$	0	0
$195$	−5.26966	−0.377368
$196$	0	0
$197$	−8.18992	−0.583508	−0.291754	−	0.956493i	$-0.594239\pi$
$197$	−8.18992	−0.583508	−0.291754	+	0.956493i	$0.594239\pi$
$198$	0	0
$199$	−5.45297	−0.386551	−0.193275	−	0.981145i	$-0.561911\pi$
$199$	−5.45297	−0.386551	−0.193275	+	0.981145i	$0.561911\pi$
$200$	0	0
$201$	−23.8475	−1.68207
$202$	0	0
$203$	1.81533	0.127411
$204$	0	0
$205$	−34.5832	−2.41540
$206$	0	0
$207$	−4.73534	−0.329129
$208$	0	0
$209$	−29.6013	−2.04757
$210$	0	0
$211$	0.843316	0.0580563	0.0290281	−	0.999579i	$-0.490759\pi$
$211$	0.843316	0.0580563	0.0290281	+	0.999579i	$0.490759\pi$
$212$	0	0
$213$	−17.4593	−1.19629
$214$	0	0
$215$	−18.1696	−1.23916
$216$	0	0
$217$	−2.15295	−0.146152
$218$	0	0
$219$	−0.158732	−0.0107261
$220$	0	0
$221$	−1.54183	−0.103715
$222$	0	0
$223$	8.33005	0.557821	0.278911	−	0.960317i	$-0.410027\pi$
$223$	8.33005	0.557821	0.278911	+	0.960317i	$0.410027\pi$
$224$	0	0
$225$	5.79853	0.386569
$226$	0	0
$227$	−22.7525	−1.51014	−0.755070	−	0.655645i	$-0.772397\pi$
$227$	−22.7525	−1.51014	−0.755070	+	0.655645i	$0.772397\pi$
$228$	0	0
$229$	11.5027	0.760120	0.380060	−	0.924962i	$-0.375903\pi$
$229$	11.5027	0.760120	0.380060	+	0.924962i	$0.375903\pi$
$230$	0	0
$231$	−3.13556	−0.206305
$232$	0	0
$233$	3.91149	0.256250	0.128125	−	0.991758i	$-0.459104\pi$
$233$	3.91149	0.256250	0.128125	+	0.991758i	$0.459104\pi$
$234$	0	0
$235$	−20.1901	−1.31706
$236$	0	0
$237$	17.9556	1.16634
$238$	0	0
$239$	−0.499336	−0.0322994	−0.0161497	−	0.999870i	$-0.505141\pi$
$239$	−0.499336	−0.0322994	−0.0161497	+	0.999870i	$0.505141\pi$
$240$	0	0
$241$	8.75039	0.563662	0.281831	−	0.959464i	$-0.409058\pi$
$241$	8.75039	0.563662	0.281831	+	0.959464i	$0.409058\pi$
$242$	0	0
$243$	6.37561	0.408996
$244$	0	0
$245$	−25.7423	−1.64461
$246$	0	0
$247$	−5.80197	−0.369171
$248$	0	0
$249$	−19.9802	−1.26619
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−28.3856	−1.78459
$254$	0	0
$255$	15.1671	0.949803
$256$	0	0
$257$	3.54293	0.221002	0.110501	−	0.993876i	$-0.464754\pi$
$257$	3.54293	0.221002	0.110501	+	0.993876i	$0.464754\pi$
$258$	0	0
$259$	−4.68685	−0.291227
$260$	0	0
$261$	2.56337	0.158669
$262$	0	0
$263$	26.5453	1.63685	0.818427	−	0.574611i	$-0.194847\pi$
$263$	26.5453	1.63685	0.818427	+	0.574611i	$0.194847\pi$
$264$	0	0
$265$	−36.4784	−2.24085
$266$	0	0
$267$	31.2078	1.90989
$268$	0	0
$269$	6.48096	0.395151	0.197576	−	0.980288i	$-0.436693\pi$
$269$	6.48096	0.395151	0.197576	+	0.980288i	$0.436693\pi$
$270$	0	0
$271$	−14.9486	−0.908064	−0.454032	−	0.890985i	$-0.650015\pi$
$271$	−14.9486	−0.908064	−0.454032	+	0.890985i	$0.650015\pi$
$272$	0	0
$273$	−0.614581	−0.0371962
$274$	0	0
$275$	34.7588	2.09604
$276$	0	0
$277$	−4.67738	−0.281037	−0.140518	−	0.990078i	$-0.544877\pi$
$277$	−4.67738	−0.281037	−0.140518	+	0.990078i	$0.544877\pi$
$278$	0	0
$279$	−3.04011	−0.182007
$280$	0	0
$281$	−18.1672	−1.08376	−0.541881	−	0.840455i	$-0.682288\pi$
$281$	−18.1672	−1.08376	−0.541881	+	0.840455i	$0.682288\pi$
$282$	0	0
$283$	−11.3234	−0.673104	−0.336552	−	0.941665i	$-0.609261\pi$
$283$	−11.3234	−0.673104	−0.336552	+	0.941665i	$0.609261\pi$
$284$	0	0
$285$	57.0746	3.38081
$286$	0	0
$287$	−4.03332	−0.238079
$288$	0	0
$289$	−12.5623	−0.738959
$290$	0	0
$291$	5.01038	0.293714
$292$	0	0
$293$	−7.31435	−0.427309	−0.213654	−	0.976909i	$-0.568537\pi$
$293$	−7.31435	−0.427309	−0.213654	+	0.976909i	$0.568537\pi$
$294$	0	0
$295$	−26.1559	−1.52286
$296$	0	0
$297$	16.8952	0.980360
$298$	0	0
$299$	−5.56368	−0.321756
$300$	0	0
$301$	−2.11906	−0.122140
$302$	0	0
$303$	−16.3582	−0.939752
$304$	0	0
$305$	5.57038	0.318959
$306$	0	0
$307$	−23.3607	−1.33327	−0.666633	−	0.745386i	$-0.732265\pi$
$307$	−23.3607	−1.33327	−0.666633	+	0.745386i	$0.732265\pi$
$308$	0	0
$309$	−28.7729	−1.63683
$310$	0	0
$311$	−5.22814	−0.296461	−0.148230	−	0.988953i	$-0.547358\pi$
$311$	−5.22814	−0.296461	−0.148230	+	0.988953i	$0.547358\pi$
$312$	0	0
$313$	12.7122	0.718534	0.359267	−	0.933235i	$-0.383027\pi$
$313$	12.7122	0.718534	0.359267	+	0.933235i	$0.383027\pi$
$314$	0	0
$315$	1.03952	0.0585702
$316$	0	0
$317$	−21.2693	−1.19460	−0.597300	−	0.802018i	$-0.703760\pi$
$317$	−21.2693	−1.19460	−0.597300	+	0.802018i	$0.703760\pi$
$318$	0	0
$319$	15.3659	0.860326
$320$	0	0
$321$	25.5352	1.42524
$322$	0	0
$323$	16.6992	0.929170
$324$	0	0
$325$	6.81286	0.377909
$326$	0	0
$327$	−8.57503	−0.474200
$328$	0	0
$329$	−2.35470	−0.129819
$330$	0	0
$331$	−14.7572	−0.811132	−0.405566	−	0.914066i	$-0.632925\pi$
$331$	−14.7572	−0.811132	−0.405566	+	0.914066i	$0.632925\pi$
$332$	0	0
$333$	−6.61814	−0.362672
$334$	0	0
$335$	47.3922	2.58931
$336$	0	0
$337$	15.6072	0.850178	0.425089	−	0.905152i	$-0.360243\pi$
$337$	15.6072	0.850178	0.425089	+	0.905152i	$0.360243\pi$
$338$	0	0
$339$	−31.5506	−1.71359
$340$	0	0
$341$	−18.2237	−0.986869
$342$	0	0
$343$	−6.09031	−0.328846
$344$	0	0
$345$	54.7306	2.94659
$346$	0	0
$347$	2.29790	0.123358	0.0616790	−	0.998096i	$-0.480355\pi$
$347$	2.29790	0.123358	0.0616790	+	0.998096i	$0.480355\pi$
$348$	0	0
$349$	22.0106	1.17820	0.589100	−	0.808060i	$-0.299482\pi$
$349$	22.0106	1.17820	0.589100	+	0.808060i	$0.299482\pi$
$350$	0	0
$351$	3.31153	0.176756
$352$	0	0
$353$	−25.0736	−1.33454	−0.667268	−	0.744818i	$-0.732536\pi$
$353$	−25.0736	−1.33454	−0.667268	+	0.744818i	$0.732536\pi$
$354$	0	0
$355$	34.6970	1.84153
$356$	0	0
$357$	1.76889	0.0936195
$358$	0	0
$359$	25.3297	1.33685	0.668426	−	0.743778i	$-0.266968\pi$
$359$	25.3297	1.33685	0.668426	+	0.743778i	$0.266968\pi$
$360$	0	0
$361$	43.8400	2.30737
$362$	0	0
$363$	−5.60357	−0.294111
$364$	0	0
$365$	0.315449	0.0165114
$366$	0	0
$367$	7.09503	0.370358	0.185179	−	0.982705i	$-0.440714\pi$
$367$	7.09503	0.370358	0.185179	+	0.982705i	$0.440714\pi$
$368$	0	0
$369$	−5.69531	−0.296486
$370$	0	0
$371$	−4.25435	−0.220875
$372$	0	0
$373$	−16.3251	−0.845283	−0.422642	−	0.906297i	$-0.638897\pi$
$373$	−16.3251	−0.845283	−0.422642	+	0.906297i	$0.638897\pi$
$374$	0	0
$375$	−31.0194	−1.60184
$376$	0	0
$377$	3.01177	0.155114
$378$	0	0
$379$	−25.1481	−1.29177	−0.645886	−	0.763434i	$-0.723512\pi$
$379$	−25.1481	−1.29177	−0.645886	+	0.763434i	$0.723512\pi$
$380$	0	0
$381$	−36.1726	−1.85318
$382$	0	0
$383$	8.86458	0.452959	0.226479	−	0.974016i	$-0.427278\pi$
$383$	8.86458	0.452959	0.226479	+	0.974016i	$0.427278\pi$
$384$	0	0
$385$	6.23131	0.317577
$386$	0	0
$387$	−2.99225	−0.152105
$388$	0	0
$389$	−14.8771	−0.754299	−0.377150	−	0.926152i	$-0.623096\pi$
$389$	−14.8771	−0.754299	−0.377150	+	0.926152i	$0.623096\pi$
$390$	0	0
$391$	16.0134	0.809832
$392$	0	0
$393$	−32.1658	−1.62255
$394$	0	0
$395$	−35.6831	−1.79541
$396$	0	0
$397$	24.9231	1.25086	0.625428	−	0.780282i	$-0.284924\pi$
$397$	24.9231	1.25086	0.625428	+	0.780282i	$0.284924\pi$
$398$	0	0
$399$	6.65641	0.333237
$400$	0	0
$401$	−2.84068	−0.141857	−0.0709284	−	0.997481i	$-0.522596\pi$
$401$	−2.84068	−0.141857	−0.0709284	+	0.997481i	$0.522596\pi$
$402$	0	0
$403$	−3.57191	−0.177930
$404$	0	0
$405$	−39.6449	−1.96997
$406$	0	0
$407$	−39.6719	−1.96646
$408$	0	0
$409$	28.6518	1.41674	0.708371	−	0.705841i	$-0.249430\pi$
$409$	28.6518	1.41674	0.708371	+	0.705841i	$0.249430\pi$
$410$	0	0
$411$	7.30149	0.360156
$412$	0	0
$413$	−3.05047	−0.150104
$414$	0	0
$415$	39.7066	1.94912
$416$	0	0
$417$	−9.40525	−0.460577
$418$	0	0
$419$	12.3062	0.601196	0.300598	−	0.953751i	$-0.402814\pi$
$419$	12.3062	0.601196	0.300598	+	0.953751i	$0.402814\pi$
$420$	0	0
$421$	−33.8053	−1.64757	−0.823785	−	0.566902i	$-0.808142\pi$
$421$	−33.8053	−1.64757	−0.823785	+	0.566902i	$0.808142\pi$
$422$	0	0
$423$	−3.32499	−0.161666
$424$	0	0
$425$	−19.6088	−0.951165
$426$	0	0
$427$	0.649654	0.0314389
$428$	0	0
$429$	−5.20213	−0.251161
$430$	0	0
$431$	−15.0215	−0.723558	−0.361779	−	0.932264i	$-0.617831\pi$
$431$	−15.0215	−0.723558	−0.361779	+	0.932264i	$0.617831\pi$
$432$	0	0
$433$	8.45400	0.406273	0.203137	−	0.979150i	$-0.434886\pi$
$433$	8.45400	0.406273	0.203137	+	0.979150i	$0.434886\pi$
$434$	0	0
$435$	−29.6272	−1.42051
$436$	0	0
$437$	60.2592	2.88259
$438$	0	0
$439$	−34.3098	−1.63752	−0.818759	−	0.574137i	$-0.805338\pi$
$439$	−34.3098	−1.63752	−0.818759	+	0.574137i	$0.805338\pi$
$440$	0	0
$441$	−4.23935	−0.201874
$442$	0	0
$443$	17.7598	0.843793	0.421897	−	0.906644i	$-0.361365\pi$
$443$	17.7598	0.843793	0.421897	+	0.906644i	$0.361365\pi$
$444$	0	0
$445$	−62.0194	−2.94000
$446$	0	0
$447$	37.2699	1.76280
$448$	0	0
$449$	−28.2901	−1.33509	−0.667547	−	0.744568i	$-0.732656\pi$
$449$	−28.2901	−1.33509	−0.667547	+	0.744568i	$0.732656\pi$
$450$	0	0
$451$	−34.1401	−1.60759
$452$	0	0
$453$	5.50787	0.258782
$454$	0	0
$455$	1.22136	0.0572582
$456$	0	0
$457$	22.4797	1.05155	0.525777	−	0.850622i	$-0.323775\pi$
$457$	22.4797	1.05155	0.525777	+	0.850622i	$0.323775\pi$
$458$	0	0
$459$	−9.53123	−0.444880
$460$	0	0
$461$	1.32777	0.0618403	0.0309201	−	0.999522i	$-0.490156\pi$
$461$	1.32777	0.0618403	0.0309201	+	0.999522i	$0.490156\pi$
$462$	0	0
$463$	31.8634	1.48082	0.740409	−	0.672156i	$-0.234632\pi$
$463$	31.8634	1.48082	0.740409	+	0.672156i	$0.234632\pi$
$464$	0	0
$465$	35.1373	1.62945
$466$	0	0
$467$	10.9547	0.506923	0.253461	−	0.967346i	$-0.418431\pi$
$467$	10.9547	0.506923	0.253461	+	0.967346i	$0.418431\pi$
$468$	0	0
$469$	5.52718	0.255222
$470$	0	0
$471$	2.00358	0.0923200
$472$	0	0
$473$	−17.9368	−0.824735
$474$	0	0
$475$	−73.7887	−3.38566
$476$	0	0
$477$	−6.00742	−0.275061
$478$	0	0
$479$	−31.8959	−1.45736	−0.728681	−	0.684853i	$-0.759866\pi$
$479$	−31.8959	−1.45736	−0.728681	+	0.684853i	$0.759866\pi$
$480$	0	0
$481$	−7.77584	−0.354548
$482$	0	0
$483$	6.38303	0.290438
$484$	0	0
$485$	−9.95714	−0.452130
$486$	0	0
$487$	−11.4220	−0.517579	−0.258789	−	0.965934i	$-0.583324\pi$
$487$	−11.4220	−0.517579	−0.258789	+	0.965934i	$0.583324\pi$
$488$	0	0
$489$	31.8555	1.44055
$490$	0	0
$491$	−33.5946	−1.51610	−0.758052	−	0.652195i	$-0.773849\pi$
$491$	−33.5946	−1.51610	−0.758052	+	0.652195i	$0.773849\pi$
$492$	0	0
$493$	−8.66849	−0.390409
$494$	0	0
$495$	8.79902	0.395486
$496$	0	0
$497$	4.04659	0.181514
$498$	0	0
$499$	12.7738	0.571833	0.285917	−	0.958254i	$-0.407702\pi$
$499$	12.7738	0.571833	0.285917	+	0.958254i	$0.407702\pi$
$500$	0	0
$501$	2.37676	0.106186
$502$	0	0
$503$	−3.96822	−0.176934	−0.0884672	−	0.996079i	$-0.528197\pi$
$503$	−3.96822	−0.176934	−0.0884672	+	0.996079i	$0.528197\pi$
$504$	0	0
$505$	32.5086	1.44661
$506$	0	0
$507$	23.7246	1.05365
$508$	0	0
$509$	24.8687	1.10229	0.551144	−	0.834410i	$-0.314192\pi$
$509$	24.8687	1.10229	0.551144	+	0.834410i	$0.314192\pi$
$510$	0	0
$511$	0.0367897	0.00162748
$512$	0	0
$513$	−35.8665	−1.58354
$514$	0	0
$515$	57.1805	2.51967
$516$	0	0
$517$	−19.9314	−0.876580
$518$	0	0
$519$	−12.3405	−0.541689
$520$	0	0
$521$	29.5586	1.29498	0.647492	−	0.762072i	$-0.275818\pi$
$521$	29.5586	1.29498	0.647492	+	0.762072i	$0.275818\pi$
$522$	0	0
$523$	−27.5620	−1.20520	−0.602602	−	0.798042i	$-0.705869\pi$
$523$	−27.5620	−1.20520	−0.602602	+	0.798042i	$0.705869\pi$
$524$	0	0
$525$	−7.81616	−0.341125
$526$	0	0
$527$	10.2807	0.447833
$528$	0	0
$529$	34.7843	1.51236
$530$	0	0
$531$	−4.30747	−0.186928
$532$	0	0
$533$	−6.69158	−0.289844
$534$	0	0
$535$	−50.7462	−2.19395
$536$	0	0
$537$	31.4017	1.35508
$538$	0	0
$539$	−25.4124	−1.09459
$540$	0	0
$541$	25.3582	1.09024	0.545118	−	0.838359i	$-0.316485\pi$
$541$	25.3582	1.09024	0.545118	+	0.838359i	$0.316485\pi$
$542$	0	0
$543$	−19.1660	−0.822491
$544$	0	0
$545$	17.0412	0.729963
$546$	0	0
$547$	−16.7877	−0.717790	−0.358895	−	0.933378i	$-0.616846\pi$
$547$	−16.7877	−0.717790	−0.358895	+	0.933378i	$0.616846\pi$
$548$	0	0
$549$	0.917354	0.0391517
$550$	0	0
$551$	−32.6199	−1.38966
$552$	0	0
$553$	−4.16159	−0.176969
$554$	0	0
$555$	76.4918	3.24689
$556$	0	0
$557$	13.4370	0.569345	0.284673	−	0.958625i	$-0.408115\pi$
$557$	13.4370	0.569345	0.284673	+	0.958625i	$0.408115\pi$
$558$	0	0
$559$	−3.51568	−0.148697
$560$	0	0
$561$	14.9728	0.632151
$562$	0	0
$563$	15.0434	0.634002	0.317001	−	0.948425i	$-0.397324\pi$
$563$	15.0434	0.634002	0.317001	+	0.948425i	$0.397324\pi$
$564$	0	0
$565$	62.7005	2.63783
$566$	0	0
$567$	−4.62364	−0.194175
$568$	0	0
$569$	−6.87759	−0.288323	−0.144162	−	0.989554i	$-0.546049\pi$
$569$	−6.87759	−0.288323	−0.144162	+	0.989554i	$0.546049\pi$
$570$	0	0
$571$	39.3520	1.64683	0.823414	−	0.567440i	$-0.192066\pi$
$571$	39.3520	1.64683	0.823414	+	0.567440i	$0.192066\pi$
$572$	0	0
$573$	−10.8554	−0.453492
$574$	0	0
$575$	−70.7582	−2.95082
$576$	0	0
$577$	−17.5364	−0.730048	−0.365024	−	0.930998i	$-0.618939\pi$
$577$	−17.5364	−0.730048	−0.365024	+	0.930998i	$0.618939\pi$
$578$	0	0
$579$	−48.8971	−2.03210
$580$	0	0
$581$	4.63084	0.192120
$582$	0	0
$583$	−36.0110	−1.49142
$584$	0	0
$585$	1.72464	0.0713051
$586$	0	0
$587$	−2.86907	−0.118419	−0.0592096	−	0.998246i	$-0.518858\pi$
$587$	−2.86907	−0.118419	−0.0592096	+	0.998246i	$0.518858\pi$
$588$	0	0
$589$	38.6867	1.59406
$590$	0	0
$591$	15.5887	0.641234
$592$	0	0
$593$	15.6189	0.641390	0.320695	−	0.947183i	$-0.396084\pi$
$593$	15.6189	0.641390	0.320695	+	0.947183i	$0.396084\pi$
$594$	0	0
$595$	−3.51532	−0.144114
$596$	0	0
$597$	10.3792	0.424792
$598$	0	0
$599$	42.0292	1.71727	0.858634	−	0.512589i	$-0.171314\pi$
$599$	42.0292	1.71727	0.858634	+	0.512589i	$0.171314\pi$
$600$	0	0
$601$	21.2306	0.866016	0.433008	−	0.901390i	$-0.357452\pi$
$601$	21.2306	0.866016	0.433008	+	0.901390i	$0.357452\pi$
$602$	0	0
$603$	7.80475	0.317834
$604$	0	0
$605$	11.1360	0.452742
$606$	0	0
$607$	−16.2187	−0.658296	−0.329148	−	0.944278i	$-0.606761\pi$
$607$	−16.2187	−0.658296	−0.329148	+	0.944278i	$0.606761\pi$
$608$	0	0
$609$	−3.45531	−0.140016
$610$	0	0
$611$	−3.90662	−0.158045
$612$	0	0
$613$	2.96379	0.119706	0.0598531	−	0.998207i	$-0.480937\pi$
$613$	2.96379	0.119706	0.0598531	+	0.998207i	$0.480937\pi$
$614$	0	0
$615$	65.8258	2.65435
$616$	0	0
$617$	18.2715	0.735584	0.367792	−	0.929908i	$-0.380114\pi$
$617$	18.2715	0.735584	0.367792	+	0.929908i	$0.380114\pi$
$618$	0	0
$619$	24.4339	0.982083	0.491042	−	0.871136i	$-0.336616\pi$
$619$	24.4339	0.982083	0.491042	+	0.871136i	$0.336616\pi$
$620$	0	0
$621$	−34.3934	−1.38016
$622$	0	0
$623$	−7.23310	−0.289788
$624$	0	0
$625$	15.1033	0.604134
$626$	0	0
$627$	56.3433	2.25013
$628$	0	0
$629$	22.3804	0.892365
$630$	0	0
$631$	−9.78202	−0.389416	−0.194708	−	0.980861i	$-0.562376\pi$
$631$	−9.78202	−0.389416	−0.194708	+	0.980861i	$0.562376\pi$
$632$	0	0
$633$	−1.60517	−0.0637998
$634$	0	0
$635$	71.8860	2.85271
$636$	0	0
$637$	−4.98093	−0.197351
$638$	0	0
$639$	5.71405	0.226044
$640$	0	0
$641$	−15.0131	−0.592981	−0.296490	−	0.955036i	$-0.595816\pi$
$641$	−15.0131	−0.592981	−0.296490	+	0.955036i	$0.595816\pi$
$642$	0	0
$643$	−17.5160	−0.690765	−0.345383	−	0.938462i	$-0.612251\pi$
$643$	−17.5160	−0.690765	−0.345383	+	0.938462i	$0.612251\pi$
$644$	0	0
$645$	34.5841	1.36175
$646$	0	0
$647$	0.436342	0.0171544	0.00857719	−	0.999963i	$-0.497270\pi$
$647$	0.436342	0.0171544	0.00857719	+	0.999963i	$0.497270\pi$
$648$	0	0
$649$	−25.8208	−1.01355
$650$	0	0
$651$	4.09794	0.160611
$652$	0	0
$653$	14.2949	0.559403	0.279702	−	0.960087i	$-0.409764\pi$
$653$	14.2949	0.559403	0.279702	+	0.960087i	$0.409764\pi$
$654$	0	0
$655$	63.9231	2.49768
$656$	0	0
$657$	0.0519495	0.00202674
$658$	0	0
$659$	−29.8093	−1.16121	−0.580603	−	0.814187i	$-0.697183\pi$
$659$	−29.8093	−1.16121	−0.580603	+	0.814187i	$0.697183\pi$
$660$	0	0
$661$	−6.38703	−0.248427	−0.124213	−	0.992256i	$-0.539641\pi$
$661$	−6.38703	−0.248427	−0.124213	+	0.992256i	$0.539641\pi$
$662$	0	0
$663$	2.93472	0.113975
$664$	0	0
$665$	−13.2283	−0.512971
$666$	0	0
$667$	−31.2802	−1.21118
$668$	0	0
$669$	−15.8554	−0.613006
$670$	0	0
$671$	5.49900	0.212287
$672$	0	0
$673$	3.80570	0.146699	0.0733495	−	0.997306i	$-0.476631\pi$
$673$	3.80570	0.146699	0.0733495	+	0.997306i	$0.476631\pi$
$674$	0	0
$675$	42.1155	1.62103
$676$	0	0
$677$	−48.8114	−1.87597	−0.937987	−	0.346670i	$-0.887312\pi$
$677$	−48.8114	−1.87597	−0.937987	+	0.346670i	$0.887312\pi$
$678$	0	0
$679$	−1.16127	−0.0445653
$680$	0	0
$681$	43.3072	1.65954
$682$	0	0
$683$	−27.5483	−1.05411	−0.527053	−	0.849832i	$-0.676703\pi$
$683$	−27.5483	−1.05411	−0.527053	+	0.849832i	$0.676703\pi$
$684$	0	0
$685$	−14.5103	−0.554408
$686$	0	0
$687$	−21.8943	−0.835319
$688$	0	0
$689$	−7.05828	−0.268899
$690$	0	0
$691$	24.5169	0.932665	0.466333	−	0.884609i	$-0.345575\pi$
$691$	24.5169	0.932665	0.466333	+	0.884609i	$0.345575\pi$
$692$	0	0
$693$	1.02620	0.0389820
$694$	0	0
$695$	18.6911	0.708993
$696$	0	0
$697$	19.2597	0.729513
$698$	0	0
$699$	−7.44514	−0.281601
$700$	0	0
$701$	−19.6432	−0.741915	−0.370957	−	0.928650i	$-0.620970\pi$
$701$	−19.6432	−0.741915	−0.370957	+	0.928650i	$0.620970\pi$
$702$	0	0
$703$	84.2185	3.17636
$704$	0	0
$705$	38.4298	1.44735
$706$	0	0
$707$	3.79136	0.142589
$708$	0	0
$709$	−8.29490	−0.311521	−0.155761	−	0.987795i	$-0.549783\pi$
$709$	−8.29490	−0.311521	−0.155761	+	0.987795i	$0.549783\pi$
$710$	0	0
$711$	−5.87644	−0.220384
$712$	0	0
$713$	37.0978	1.38932
$714$	0	0
$715$	10.3382	0.386627
$716$	0	0
$717$	0.950438	0.0354948
$718$	0	0
$719$	11.3762	0.424262	0.212131	−	0.977241i	$-0.431960\pi$
$719$	11.3762	0.424262	0.212131	+	0.977241i	$0.431960\pi$
$720$	0	0
$721$	6.66875	0.248357
$722$	0	0
$723$	−16.6555	−0.619425
$724$	0	0
$725$	38.3033	1.42255
$726$	0	0
$727$	−25.6901	−0.952794	−0.476397	−	0.879230i	$-0.658057\pi$
$727$	−25.6901	−0.952794	−0.476397	+	0.879230i	$0.658057\pi$
$728$	0	0
$729$	19.3069	0.715072
$730$	0	0
$731$	10.1188	0.374258
$732$	0	0
$733$	−43.8838	−1.62088	−0.810442	−	0.585819i	$-0.800773\pi$
$733$	−43.8838	−1.62088	−0.810442	+	0.585819i	$0.800773\pi$
$734$	0	0
$735$	48.9979	1.80732
$736$	0	0
$737$	46.7849	1.72334
$738$	0	0
$739$	3.10158	0.114094	0.0570468	−	0.998372i	$-0.481832\pi$
$739$	3.10158	0.114094	0.0570468	+	0.998372i	$0.481832\pi$
$740$	0	0
$741$	11.0435	0.405693
$742$	0	0
$743$	0.494176	0.0181296	0.00906478	−	0.999959i	$-0.497115\pi$
$743$	0.494176	0.0181296	0.00906478	+	0.999959i	$0.497115\pi$
$744$	0	0
$745$	−74.0665	−2.71358
$746$	0	0
$747$	6.53906	0.239252
$748$	0	0
$749$	−5.91835	−0.216252
$750$	0	0
$751$	−30.7682	−1.12275	−0.561374	−	0.827562i	$-0.689727\pi$
$751$	−30.7682	−1.12275	−0.561374	+	0.827562i	$0.689727\pi$
$752$	0	0
$753$	−1.90340	−0.0693638
$754$	0	0
$755$	−10.9458	−0.398358
$756$	0	0
$757$	−11.0501	−0.401624	−0.200812	−	0.979630i	$-0.564358\pi$
$757$	−11.0501	−0.401624	−0.200812	+	0.979630i	$0.564358\pi$
$758$	0	0
$759$	54.0292	1.96114
$760$	0	0
$761$	49.3951	1.79057	0.895285	−	0.445494i	$-0.146972\pi$
$761$	49.3951	1.79057	0.895285	+	0.445494i	$0.146972\pi$
$762$	0	0
$763$	1.98745	0.0719505
$764$	0	0
$765$	−4.96386	−0.179469
$766$	0	0
$767$	−5.06097	−0.182741
$768$	0	0
$769$	17.0744	0.615719	0.307860	−	0.951432i	$-0.400387\pi$
$769$	17.0744	0.615719	0.307860	+	0.951432i	$0.400387\pi$
$770$	0	0
$771$	−6.74362	−0.242866
$772$	0	0
$773$	−37.2564	−1.34002	−0.670010	−	0.742352i	$-0.733710\pi$
$773$	−37.2564	−1.34002	−0.670010	+	0.742352i	$0.733710\pi$
$774$	0	0
$775$	−45.4271	−1.63179
$776$	0	0
$777$	8.92096	0.320038
$778$	0	0
$779$	72.4751	2.59669
$780$	0	0
$781$	34.2524	1.22565
$782$	0	0
$783$	18.6181	0.665357
$784$	0	0
$785$	−3.98172	−0.142114
$786$	0	0
$787$	31.6280	1.12742	0.563708	−	0.825974i	$-0.309374\pi$
$787$	31.6280	1.12742	0.563708	+	0.825974i	$0.309374\pi$
$788$	0	0
$789$	−50.5264	−1.79879
$790$	0	0
$791$	7.31254	0.260004
$792$	0	0
$793$	1.07782	0.0382747
$794$	0	0
$795$	69.4331	2.46254
$796$	0	0
$797$	45.8027	1.62241	0.811207	−	0.584759i	$-0.198811\pi$
$797$	45.8027	1.62241	0.811207	+	0.584759i	$0.198811\pi$
$798$	0	0
$799$	11.2440	0.397785
$800$	0	0
$801$	−10.2136	−0.360880
$802$	0	0
$803$	0.311407	0.0109893
$804$	0	0
$805$	−12.6850	−0.447088
$806$	0	0
$807$	−12.3359	−0.434243
$808$	0	0
$809$	−20.5951	−0.724086	−0.362043	−	0.932161i	$-0.617921\pi$
$809$	−20.5951	−0.724086	−0.362043	+	0.932161i	$0.617921\pi$
$810$	0	0
$811$	46.0620	1.61746	0.808728	−	0.588183i	$-0.200156\pi$
$811$	46.0620	1.61746	0.808728	+	0.588183i	$0.200156\pi$
$812$	0	0
$813$	28.4532	0.997899
$814$	0	0
$815$	−63.3064	−2.21753
$816$	0	0
$817$	38.0776	1.33217
$818$	0	0
$819$	0.201139	0.00702835
$820$	0	0
$821$	−33.2059	−1.15889	−0.579447	−	0.815010i	$-0.696732\pi$
$821$	−33.2059	−1.15889	−0.579447	+	0.815010i	$0.696732\pi$
$822$	0	0
$823$	10.1112	0.352453	0.176227	−	0.984350i	$-0.443611\pi$
$823$	10.1112	0.352453	0.176227	+	0.984350i	$0.443611\pi$
$824$	0	0
$825$	−66.1600	−2.30340
$826$	0	0
$827$	44.8442	1.55939	0.779693	−	0.626161i	$-0.215375\pi$
$827$	44.8442	1.55939	0.779693	+	0.626161i	$0.215375\pi$
$828$	0	0
$829$	−54.0362	−1.87675	−0.938377	−	0.345614i	$-0.887671\pi$
$829$	−54.0362	−1.87675	−0.938377	+	0.345614i	$0.887671\pi$
$830$	0	0
$831$	8.90294	0.308839
$832$	0	0
$833$	14.3361	0.496716
$834$	0	0
$835$	−4.72334	−0.163458
$836$	0	0
$837$	−22.0808	−0.763223
$838$	0	0
$839$	14.2957	0.493544	0.246772	−	0.969074i	$-0.420630\pi$
$839$	14.2957	0.493544	0.246772	+	0.969074i	$0.420630\pi$
$840$	0	0
$841$	−12.0672	−0.416109
$842$	0	0
$843$	34.5794	1.19098
$844$	0	0
$845$	−47.1479	−1.62194
$846$	0	0
$847$	1.29875	0.0446256
$848$	0	0
$849$	21.5529	0.739694
$850$	0	0
$851$	80.7597	2.76841
$852$	0	0
$853$	−15.6425	−0.535590	−0.267795	−	0.963476i	$-0.586295\pi$
$853$	−15.6425	−0.535590	−0.267795	+	0.963476i	$0.586295\pi$
$854$	0	0
$855$	−18.6792	−0.638816
$856$	0	0
$857$	48.6464	1.66173	0.830864	−	0.556475i	$-0.187847\pi$
$857$	48.6464	1.66173	0.830864	+	0.556475i	$0.187847\pi$
$858$	0	0
$859$	−37.5089	−1.27979	−0.639894	−	0.768463i	$-0.721022\pi$
$859$	−37.5089	−1.27979	−0.639894	+	0.768463i	$0.721022\pi$
$860$	0	0
$861$	7.67702	0.261632
$862$	0	0
$863$	22.3223	0.759861	0.379931	−	0.925015i	$-0.375948\pi$
$863$	22.3223	0.759861	0.379931	+	0.925015i	$0.375948\pi$
$864$	0	0
$865$	24.5243	0.833853
$866$	0	0
$867$	23.9111	0.812065
$868$	0	0
$869$	−35.2259	−1.19496
$870$	0	0
$871$	9.17001	0.310714
$872$	0	0
$873$	−1.63978	−0.0554983
$874$	0	0
$875$	7.18943	0.243047
$876$	0	0
$877$	21.0855	0.712008	0.356004	−	0.934484i	$-0.384139\pi$
$877$	21.0855	0.712008	0.356004	+	0.934484i	$0.384139\pi$
$878$	0	0
$879$	13.9222	0.469582
$880$	0	0
$881$	25.3483	0.854005	0.427002	−	0.904250i	$-0.359570\pi$
$881$	25.3483	0.854005	0.427002	+	0.904250i	$0.359570\pi$
$882$	0	0
$883$	34.8701	1.17347	0.586735	−	0.809779i	$-0.300413\pi$
$883$	34.8701	1.17347	0.586735	+	0.809779i	$0.300413\pi$
$884$	0	0
$885$	49.7853	1.67351
$886$	0	0
$887$	−9.20073	−0.308930	−0.154465	−	0.987998i	$-0.549365\pi$
$887$	−9.20073	−0.308930	−0.154465	+	0.987998i	$0.549365\pi$
$888$	0	0
$889$	8.38380	0.281184
$890$	0	0
$891$	−39.1369	−1.31114
$892$	0	0
$893$	42.3118	1.41591
$894$	0	0
$895$	−62.4047	−2.08596
$896$	0	0
$897$	10.5899	0.353587
$898$	0	0
$899$	−20.0821	−0.669774
$900$	0	0
$901$	20.3151	0.676796
$902$	0	0
$903$	4.03342	0.134224
$904$	0	0
$905$	38.0886	1.26611
$906$	0	0
$907$	3.74235	0.124263	0.0621314	−	0.998068i	$-0.480210\pi$
$907$	3.74235	0.124263	0.0621314	+	0.998068i	$0.480210\pi$
$908$	0	0
$909$	5.35366	0.177570
$910$	0	0
$911$	−16.6793	−0.552609	−0.276304	−	0.961070i	$-0.589110\pi$
$911$	−16.6793	−0.552609	−0.276304	+	0.961070i	$0.589110\pi$
$912$	0	0
$913$	39.1978	1.29726
$914$	0	0
$915$	−10.6027	−0.350514
$916$	0	0
$917$	7.45512	0.246190
$918$	0	0
$919$	−43.9044	−1.44827	−0.724137	−	0.689657i	$-0.757761\pi$
$919$	−43.9044	−1.44827	−0.724137	+	0.689657i	$0.757761\pi$
$920$	0	0
$921$	44.4648	1.46517
$922$	0	0
$923$	6.71359	0.220981
$924$	0	0
$925$	−98.8920	−3.25155
$926$	0	0
$927$	9.41672	0.309286
$928$	0	0
$929$	30.7432	1.00865	0.504326	−	0.863513i	$-0.331741\pi$
$929$	30.7432	1.00865	0.504326	+	0.863513i	$0.331741\pi$
$930$	0	0
$931$	53.9474	1.76806
$932$	0	0
$933$	9.95126	0.325790
$934$	0	0
$935$	−29.7554	−0.973107
$936$	0	0
$937$	−51.5672	−1.68463	−0.842314	−	0.538987i	$-0.818807\pi$
$937$	−51.5672	−1.68463	−0.842314	+	0.538987i	$0.818807\pi$
$938$	0	0
$939$	−24.1964	−0.789618
$940$	0	0
$941$	26.4586	0.862526	0.431263	−	0.902226i	$-0.358068\pi$
$941$	26.4586	0.862526	0.431263	+	0.902226i	$0.358068\pi$
$942$	0	0
$943$	69.4986	2.26318
$944$	0	0
$945$	7.55017	0.245607
$946$	0	0
$947$	18.5426	0.602555	0.301277	−	0.953537i	$-0.402587\pi$
$947$	18.5426	0.602555	0.301277	+	0.953537i	$0.402587\pi$
$948$	0	0
$949$	0.0610369	0.00198134
$950$	0	0
$951$	40.4840	1.31278
$952$	0	0
$953$	47.7041	1.54529	0.772643	−	0.634841i	$-0.218934\pi$
$953$	47.7041	1.54529	0.772643	+	0.634841i	$0.218934\pi$
$954$	0	0
$955$	21.5730	0.698086
$956$	0	0
$957$	−29.2475	−0.945438
$958$	0	0
$959$	−1.69228	−0.0546465
$960$	0	0
$961$	−7.18302	−0.231710
$962$	0	0
$963$	−8.35710	−0.269304
$964$	0	0
$965$	97.1734	3.12812
$966$	0	0
$967$	−6.67950	−0.214798	−0.107399	−	0.994216i	$-0.534252\pi$
$967$	−6.67950	−0.214798	−0.107399	+	0.994216i	$0.534252\pi$
$968$	0	0
$969$	−31.7854	−1.02109
$970$	0	0
$971$	5.16353	0.165706	0.0828529	−	0.996562i	$-0.473597\pi$
$971$	5.16353	0.165706	0.0828529	+	0.996562i	$0.473597\pi$
$972$	0	0
$973$	2.17987	0.0698835
$974$	0	0
$975$	−12.9676	−0.415296
$976$	0	0
$977$	−10.4970	−0.335828	−0.167914	−	0.985802i	$-0.553703\pi$
$977$	−10.4970	−0.335828	−0.167914	+	0.985802i	$0.553703\pi$
$978$	0	0
$979$	−61.2246	−1.95675
$980$	0	0
$981$	2.80641	0.0896018
$982$	0	0
$983$	−9.33790	−0.297833	−0.148916	−	0.988850i	$-0.547579\pi$
$983$	−9.33790	−0.297833	−0.148916	+	0.988850i	$0.547579\pi$
$984$	0	0
$985$	−30.9795	−0.987089
$986$	0	0
$987$	4.48193	0.142662
$988$	0	0
$989$	36.5137	1.16107
$990$	0	0
$991$	−32.7764	−1.04118	−0.520588	−	0.853808i	$-0.674287\pi$
$991$	−32.7764	−1.04118	−0.520588	+	0.853808i	$0.674287\pi$
$992$	0	0
$993$	28.0890	0.891377
$994$	0	0
$995$	−20.6266	−0.653907
$996$	0	0
$997$	−19.9322	−0.631259	−0.315630	−	0.948882i	$-0.602216\pi$
$997$	−19.9322	−0.631259	−0.315630	+	0.948882i	$0.602216\pi$
$998$	0	0
$999$	−48.0685	−1.52082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.5		14
4.3	odd	2		1004.2.a.b.1.10	✓	14
12.11	even	2		9036.2.a.m.1.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.10	✓	14	4.3	odd	2
4016.2.a.j.1.5		14	1.1	even	1	trivial
9036.2.a.m.1.1		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)

\(f(q)\)	\(=\)	\(q-1.90340 q^{3} +3.78264 q^{5} +0.441155 q^{7} +0.622940 q^{9} +O(q^{10})\)	\(q-1.90340 q^{3} +3.78264 q^{5} +0.441155 q^{7} +0.622940 q^{9} +3.73416 q^{11} +0.731910 q^{13} -7.19988 q^{15} -2.10658 q^{17} -7.92717 q^{19} -0.839696 q^{21} -7.60160 q^{23} +9.30833 q^{25} +4.52450 q^{27} +4.11495 q^{29} -4.88026 q^{31} -7.10762 q^{33} +1.66873 q^{35} -10.6240 q^{37} -1.39312 q^{39} -9.14263 q^{41} -4.80343 q^{43} +2.35636 q^{45} -5.33757 q^{47} -6.80538 q^{49} +4.00967 q^{51} -9.64365 q^{53} +14.1250 q^{55} +15.0886 q^{57} -6.91474 q^{59} +1.47262 q^{61} +0.274813 q^{63} +2.76855 q^{65} +12.5289 q^{67} +14.4689 q^{69} +9.17270 q^{71} +0.0833940 q^{73} -17.7175 q^{75} +1.64735 q^{77} -9.43340 q^{79} -10.4808 q^{81} +10.4971 q^{83} -7.96843 q^{85} -7.83241 q^{87} -16.3958 q^{89} +0.322886 q^{91} +9.28911 q^{93} -29.9856 q^{95} -2.63233 q^{97} +2.32616 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 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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