LMFDB - Embedded newform 4016.2.a.a.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$2.30278$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.30278 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.30278 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -3.60555 q^{13} -4.30278 q^{15} -6.90833 q^{17} +4.90833 q^{19} -1.00000 q^{21} +3.00000 q^{23} +13.5139 q^{25} +5.00000 q^{27} -1.69722 q^{29} -7.21110 q^{31} +3.00000 q^{33} +4.30278 q^{35} +0.302776 q^{37} +3.60555 q^{39} -9.90833 q^{41} -11.9083 q^{43} -8.60555 q^{45} -1.30278 q^{47} -6.00000 q^{49} +6.90833 q^{51} +13.8167 q^{53} -12.9083 q^{55} -4.90833 q^{57} -0.211103 q^{61} -2.00000 q^{63} -15.5139 q^{65} -13.6056 q^{67} -3.00000 q^{69} -6.90833 q^{71} +1.21110 q^{73} -13.5139 q^{75} -3.00000 q^{77} +15.6056 q^{79} +1.00000 q^{81} -5.09167 q^{83} -29.7250 q^{85} +1.69722 q^{87} +4.81665 q^{89} -3.60555 q^{91} +7.21110 q^{93} +21.1194 q^{95} -10.3944 q^{97} +6.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 5 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 - 4 * q^9	$ 2 q - 2 q^{3} + 5 q^{5} + 2 q^{7} - 4 q^{9} - 6 q^{11} - 5 q^{15} - 3 q^{17} - q^{19} - 2 q^{21} + 6 q^{23} + 9 q^{25} + 10 q^{27} - 7 q^{29} + 6 q^{33} + 5 q^{35} - 3 q^{37} - 9 q^{41} - 13 q^{43} - 10 q^{45} + q^{47} - 12 q^{49} + 3 q^{51} + 6 q^{53} - 15 q^{55} + q^{57} + 14 q^{61} - 4 q^{63} - 13 q^{65} - 20 q^{67} - 6 q^{69} - 3 q^{71} - 12 q^{73} - 9 q^{75} - 6 q^{77} + 24 q^{79} + 2 q^{81} - 21 q^{83} - 27 q^{85} + 7 q^{87} - 12 q^{89} + 17 q^{95} - 28 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 - 4 * q^9 - 6 * q^11 - 5 * q^15 - 3 * q^17 - q^19 - 2 * q^21 + 6 * q^23 + 9 * q^25 + 10 * q^27 - 7 * q^29 + 6 * q^33 + 5 * q^35 - 3 * q^37 - 9 * q^41 - 13 * q^43 - 10 * q^45 + q^47 - 12 * q^49 + 3 * q^51 + 6 * q^53 - 15 * q^55 + q^57 + 14 * q^61 - 4 * q^63 - 13 * q^65 - 20 * q^67 - 6 * q^69 - 3 * q^71 - 12 * q^73 - 9 * q^75 - 6 * q^77 + 24 * q^79 + 2 * q^81 - 21 * q^83 - 27 * q^85 + 7 * q^87 - 12 * q^89 + 17 * q^95 - 28 * 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
$3$	−1.00000	−0.577350	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	4.30278	1.92426	0.962130	−	0.272591i	$-0.0878807\pi$
$5$	4.30278	1.92426	0.962130	+	0.272591i	$0.0878807\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−3.60555	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−3.60555	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$14$	0	0
$15$	−4.30278	−1.11097
$16$	0	0
$17$	−6.90833	−1.67552	−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833	−1.67552	−0.837758	+	0.546042i	$0.816134\pi$
$18$	0	0
$19$	4.90833	1.12605	0.563024	−	0.826441i	$-0.309638\pi$
$19$	4.90833	1.12605	0.563024	+	0.826441i	$0.309638\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	13.5139	2.70278
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−1.69722	−0.315167	−0.157583	−	0.987506i	$-0.550370\pi$
$29$	−1.69722	−0.315167	−0.157583	+	0.987506i	$0.550370\pi$
$30$	0	0
$31$	−7.21110	−1.29515	−0.647576	−	0.762001i	$-0.724217\pi$
$31$	−7.21110	−1.29515	−0.647576	+	0.762001i	$0.724217\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	4.30278	0.727302
$36$	0	0
$37$	0.302776	0.0497760	0.0248880	−	0.999690i	$-0.492077\pi$
$37$	0.302776	0.0497760	0.0248880	+	0.999690i	$0.492077\pi$
$38$	0	0
$39$	3.60555	0.577350
$40$	0	0
$41$	−9.90833	−1.54742	−0.773710	−	0.633540i	$-0.781601\pi$
$41$	−9.90833	−1.54742	−0.773710	+	0.633540i	$0.781601\pi$
$42$	0	0
$43$	−11.9083	−1.81600	−0.908001	−	0.418967i	$-0.862392\pi$
$43$	−11.9083	−1.81600	−0.908001	+	0.418967i	$0.862392\pi$
$44$	0	0
$45$	−8.60555	−1.28284
$46$	0	0
$47$	−1.30278	−0.190029	−0.0950147	−	0.995476i	$-0.530290\pi$
$47$	−1.30278	−0.190029	−0.0950147	+	0.995476i	$0.530290\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	6.90833	0.967359
$52$	0	0
$53$	13.8167	1.89786	0.948932	−	0.315482i	$-0.102166\pi$
$53$	13.8167	1.89786	0.948932	+	0.315482i	$0.102166\pi$
$54$	0	0
$55$	−12.9083	−1.74056
$56$	0	0
$57$	−4.90833	−0.650124
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−0.211103	−0.0270289	−0.0135145	−	0.999909i	$-0.504302\pi$
$61$	−0.211103	−0.0270289	−0.0135145	+	0.999909i	$0.504302\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−15.5139	−1.92426
$66$	0	0
$67$	−13.6056	−1.66218	−0.831091	−	0.556136i	$-0.812283\pi$
$67$	−13.6056	−1.66218	−0.831091	+	0.556136i	$0.812283\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−6.90833	−0.819868	−0.409934	−	0.912115i	$-0.634448\pi$
$71$	−6.90833	−0.819868	−0.409934	+	0.912115i	$0.634448\pi$
$72$	0	0
$73$	1.21110	0.141749	0.0708744	−	0.997485i	$-0.477421\pi$
$73$	1.21110	0.141749	0.0708744	+	0.997485i	$0.477421\pi$
$74$	0	0
$75$	−13.5139	−1.56045
$76$	0	0
$77$	−3.00000	−0.341882
$78$	0	0
$79$	15.6056	1.75576	0.877881	−	0.478879i	$-0.158957\pi$
$79$	15.6056	1.75576	0.877881	+	0.478879i	$0.158957\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.09167	−0.558884	−0.279442	−	0.960163i	$-0.590149\pi$
$83$	−5.09167	−0.558884	−0.279442	+	0.960163i	$0.590149\pi$
$84$	0	0
$85$	−29.7250	−3.22413
$86$	0	0
$87$	1.69722	0.181962
$88$	0	0
$89$	4.81665	0.510564	0.255282	−	0.966867i	$-0.417832\pi$
$89$	4.81665	0.510564	0.255282	+	0.966867i	$0.417832\pi$
$90$	0	0
$91$	−3.60555	−0.377964
$92$	0	0
$93$	7.21110	0.747757
$94$	0	0
$95$	21.1194	2.16681
$96$	0	0
$97$	−10.3944	−1.05540	−0.527698	−	0.849432i	$-0.676945\pi$
$97$	−10.3944	−1.05540	−0.527698	+	0.849432i	$0.676945\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−9.11943	−0.907417	−0.453709	−	0.891150i	$-0.649899\pi$
$101$	−9.11943	−0.907417	−0.453709	+	0.891150i	$0.649899\pi$
$102$	0	0
$103$	1.90833	0.188033	0.0940165	−	0.995571i	$-0.470029\pi$
$103$	1.90833	0.188033	0.0940165	+	0.995571i	$0.470029\pi$
$104$	0	0
$105$	−4.30278	−0.419908
$106$	0	0
$107$	−14.2111	−1.37384	−0.686920	−	0.726733i	$-0.741038\pi$
$107$	−14.2111	−1.37384	−0.686920	+	0.726733i	$0.741038\pi$
$108$	0	0
$109$	−3.60555	−0.345349	−0.172675	−	0.984979i	$-0.555241\pi$
$109$	−3.60555	−0.345349	−0.172675	+	0.984979i	$0.555241\pi$
$110$	0	0
$111$	−0.302776	−0.0287382
$112$	0	0
$113$	−19.4222	−1.82709	−0.913544	−	0.406741i	$-0.866665\pi$
$113$	−19.4222	−1.82709	−0.913544	+	0.406741i	$0.866665\pi$
$114$	0	0
$115$	12.9083	1.20371
$116$	0	0
$117$	7.21110	0.666667
$118$	0	0
$119$	−6.90833	−0.633285
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	9.90833	0.893404
$124$	0	0
$125$	36.6333	3.27658
$126$	0	0
$127$	7.00000	0.621150	0.310575	−	0.950549i	$-0.399478\pi$
$127$	7.00000	0.621150	0.310575	+	0.950549i	$0.399478\pi$
$128$	0	0
$129$	11.9083	1.04847
$130$	0	0
$131$	13.8167	1.20717	0.603583	−	0.797300i	$-0.293739\pi$
$131$	13.8167	1.20717	0.603583	+	0.797300i	$0.293739\pi$
$132$	0	0
$133$	4.90833	0.425606
$134$	0	0
$135$	21.5139	1.85162
$136$	0	0
$137$	−3.51388	−0.300211	−0.150105	−	0.988670i	$-0.547961\pi$
$137$	−3.51388	−0.300211	−0.150105	+	0.988670i	$0.547961\pi$
$138$	0	0
$139$	−12.3028	−1.04351	−0.521754	−	0.853096i	$-0.674722\pi$
$139$	−12.3028	−1.04351	−0.521754	+	0.853096i	$0.674722\pi$
$140$	0	0
$141$	1.30278	0.109714
$142$	0	0
$143$	10.8167	0.904534
$144$	0	0
$145$	−7.30278	−0.606463
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	8.60555	0.704994	0.352497	−	0.935813i	$-0.385333\pi$
$149$	8.60555	0.704994	0.352497	+	0.935813i	$0.385333\pi$
$150$	0	0
$151$	−7.60555	−0.618931	−0.309465	−	0.950911i	$-0.600150\pi$
$151$	−7.60555	−0.618931	−0.309465	+	0.950911i	$0.600150\pi$
$152$	0	0
$153$	13.8167	1.11701
$154$	0	0
$155$	−31.0278	−2.49221
$156$	0	0
$157$	17.9083	1.42924	0.714620	−	0.699513i	$-0.246600\pi$
$157$	17.9083	1.42924	0.714620	+	0.699513i	$0.246600\pi$
$158$	0	0
$159$	−13.8167	−1.09573
$160$	0	0
$161$	3.00000	0.236433
$162$	0	0
$163$	4.90833	0.384450	0.192225	−	0.981351i	$-0.438430\pi$
$163$	4.90833	0.384450	0.192225	+	0.981351i	$0.438430\pi$
$164$	0	0
$165$	12.9083	1.00491
$166$	0	0
$167$	9.90833	0.766729	0.383365	−	0.923597i	$-0.374765\pi$
$167$	9.90833	0.766729	0.383365	+	0.923597i	$0.374765\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−9.81665	−0.750698
$172$	0	0
$173$	23.3305	1.77379	0.886894	−	0.461973i	$-0.152858\pi$
$173$	23.3305	1.77379	0.886894	+	0.461973i	$0.152858\pi$
$174$	0	0
$175$	13.5139	1.02155
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.51388	0.262640	0.131320	−	0.991340i	$-0.458079\pi$
$179$	3.51388	0.262640	0.131320	+	0.991340i	$0.458079\pi$
$180$	0	0
$181$	−3.48612	−0.259121	−0.129561	−	0.991571i	$-0.541357\pi$
$181$	−3.48612	−0.259121	−0.129561	+	0.991571i	$0.541357\pi$
$182$	0	0
$183$	0.211103	0.0156051
$184$	0	0
$185$	1.30278	0.0957820
$186$	0	0
$187$	20.7250	1.51556
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	7.42221	0.537052	0.268526	−	0.963272i	$-0.413463\pi$
$191$	7.42221	0.537052	0.268526	+	0.963272i	$0.413463\pi$
$192$	0	0
$193$	−12.2111	−0.878974	−0.439487	−	0.898249i	$-0.644840\pi$
$193$	−12.2111	−0.878974	−0.439487	+	0.898249i	$0.644840\pi$
$194$	0	0
$195$	15.5139	1.11097
$196$	0	0
$197$	3.90833	0.278457	0.139228	−	0.990260i	$-0.455538\pi$
$197$	3.90833	0.278457	0.139228	+	0.990260i	$0.455538\pi$
$198$	0	0
$199$	7.51388	0.532645	0.266322	−	0.963884i	$-0.414191\pi$
$199$	7.51388	0.532645	0.266322	+	0.963884i	$0.414191\pi$
$200$	0	0
$201$	13.6056	0.959662
$202$	0	0
$203$	−1.69722	−0.119122
$204$	0	0
$205$	−42.6333	−2.97764
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−14.7250	−1.01855
$210$	0	0
$211$	−20.1194	−1.38508	−0.692539	−	0.721380i	$-0.743508\pi$
$211$	−20.1194	−1.38508	−0.692539	+	0.721380i	$0.743508\pi$
$212$	0	0
$213$	6.90833	0.473351
$214$	0	0
$215$	−51.2389	−3.49446
$216$	0	0
$217$	−7.21110	−0.489522
$218$	0	0
$219$	−1.21110	−0.0818387
$220$	0	0
$221$	24.9083	1.67552
$222$	0	0
$223$	−12.4222	−0.831852	−0.415926	−	0.909398i	$-0.636543\pi$
$223$	−12.4222	−0.831852	−0.415926	+	0.909398i	$0.636543\pi$
$224$	0	0
$225$	−27.0278	−1.80185
$226$	0	0
$227$	−17.6056	−1.16852	−0.584261	−	0.811566i	$-0.698615\pi$
$227$	−17.6056	−1.16852	−0.584261	+	0.811566i	$0.698615\pi$
$228$	0	0
$229$	−15.2111	−1.00518	−0.502589	−	0.864525i	$-0.667619\pi$
$229$	−15.2111	−1.00518	−0.502589	+	0.864525i	$0.667619\pi$
$230$	0	0
$231$	3.00000	0.197386
$232$	0	0
$233$	11.0917	0.726640	0.363320	−	0.931664i	$-0.381643\pi$
$233$	11.0917	0.726640	0.363320	+	0.931664i	$0.381643\pi$
$234$	0	0
$235$	−5.60555	−0.365666
$236$	0	0
$237$	−15.6056	−1.01369
$238$	0	0
$239$	−24.6333	−1.59340	−0.796698	−	0.604377i	$-0.793422\pi$
$239$	−24.6333	−1.59340	−0.796698	+	0.604377i	$0.793422\pi$
$240$	0	0
$241$	6.42221	0.413691	0.206845	−	0.978374i	$-0.433680\pi$
$241$	6.42221	0.413691	0.206845	+	0.978374i	$0.433680\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	−25.8167	−1.64937
$246$	0	0
$247$	−17.6972	−1.12605
$248$	0	0
$249$	5.09167	0.322672
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	29.7250	1.86145
$256$	0	0
$257$	9.11943	0.568854	0.284427	−	0.958698i	$-0.408197\pi$
$257$	9.11943	0.568854	0.284427	+	0.958698i	$0.408197\pi$
$258$	0	0
$259$	0.302776	0.0188136
$260$	0	0
$261$	3.39445	0.210111
$262$	0	0
$263$	8.72498	0.538005	0.269003	−	0.963139i	$-0.413306\pi$
$263$	8.72498	0.538005	0.269003	+	0.963139i	$0.413306\pi$
$264$	0	0
$265$	59.4500	3.65198
$266$	0	0
$267$	−4.81665	−0.294774
$268$	0	0
$269$	−8.21110	−0.500640	−0.250320	−	0.968163i	$-0.580536\pi$
$269$	−8.21110	−0.500640	−0.250320	+	0.968163i	$0.580536\pi$
$270$	0	0
$271$	7.39445	0.449181	0.224590	−	0.974453i	$-0.427896\pi$
$271$	7.39445	0.449181	0.224590	+	0.974453i	$0.427896\pi$
$272$	0	0
$273$	3.60555	0.218218
$274$	0	0
$275$	−40.5416	−2.44475
$276$	0	0
$277$	−4.39445	−0.264037	−0.132018	−	0.991247i	$-0.542146\pi$
$277$	−4.39445	−0.264037	−0.132018	+	0.991247i	$0.542146\pi$
$278$	0	0
$279$	14.4222	0.863435
$280$	0	0
$281$	−5.72498	−0.341524	−0.170762	−	0.985312i	$-0.554623\pi$
$281$	−5.72498	−0.341524	−0.170762	+	0.985312i	$0.554623\pi$
$282$	0	0
$283$	−11.7889	−0.700777	−0.350389	−	0.936604i	$-0.613950\pi$
$283$	−11.7889	−0.700777	−0.350389	+	0.936604i	$0.613950\pi$
$284$	0	0
$285$	−21.1194	−1.25101
$286$	0	0
$287$	−9.90833	−0.584870
$288$	0	0
$289$	30.7250	1.80735
$290$	0	0
$291$	10.3944	0.609333
$292$	0	0
$293$	−17.7250	−1.03550	−0.517752	−	0.855531i	$-0.673231\pi$
$293$	−17.7250	−1.03550	−0.517752	+	0.855531i	$0.673231\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−15.0000	−0.870388
$298$	0	0
$299$	−10.8167	−0.625543
$300$	0	0
$301$	−11.9083	−0.686385
$302$	0	0
$303$	9.11943	0.523898
$304$	0	0
$305$	−0.908327	−0.0520106
$306$	0	0
$307$	22.6333	1.29175	0.645876	−	0.763443i	$-0.276492\pi$
$307$	22.6333	1.29175	0.645876	+	0.763443i	$0.276492\pi$
$308$	0	0
$309$	−1.90833	−0.108561
$310$	0	0
$311$	0.394449	0.0223671	0.0111836	−	0.999937i	$-0.496440\pi$
$311$	0.394449	0.0223671	0.0111836	+	0.999937i	$0.496440\pi$
$312$	0	0
$313$	1.09167	0.0617050	0.0308525	−	0.999524i	$-0.490178\pi$
$313$	1.09167	0.0617050	0.0308525	+	0.999524i	$0.490178\pi$
$314$	0	0
$315$	−8.60555	−0.484868
$316$	0	0
$317$	−13.6972	−0.769313	−0.384656	−	0.923060i	$-0.625680\pi$
$317$	−13.6972	−0.769313	−0.384656	+	0.923060i	$0.625680\pi$
$318$	0	0
$319$	5.09167	0.285079
$320$	0	0
$321$	14.2111	0.793186
$322$	0	0
$323$	−33.9083	−1.88671
$324$	0	0
$325$	−48.7250	−2.70278
$326$	0	0
$327$	3.60555	0.199387
$328$	0	0
$329$	−1.30278	−0.0718243
$330$	0	0
$331$	22.2389	1.22236	0.611179	−	0.791492i	$-0.290696\pi$
$331$	22.2389	1.22236	0.611179	+	0.791492i	$0.290696\pi$
$332$	0	0
$333$	−0.605551	−0.0331840
$334$	0	0
$335$	−58.5416	−3.19847
$336$	0	0
$337$	4.60555	0.250880	0.125440	−	0.992101i	$-0.459966\pi$
$337$	4.60555	0.250880	0.125440	+	0.992101i	$0.459966\pi$
$338$	0	0
$339$	19.4222	1.05487
$340$	0	0
$341$	21.6333	1.17151
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−12.9083	−0.694961
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	10.2111	0.546588	0.273294	−	0.961931i	$-0.411887\pi$
$349$	10.2111	0.546588	0.273294	+	0.961931i	$0.411887\pi$
$350$	0	0
$351$	−18.0278	−0.962250
$352$	0	0
$353$	−3.11943	−0.166030	−0.0830152	−	0.996548i	$-0.526455\pi$
$353$	−3.11943	−0.166030	−0.0830152	+	0.996548i	$0.526455\pi$
$354$	0	0
$355$	−29.7250	−1.57764
$356$	0	0
$357$	6.90833	0.365627
$358$	0	0
$359$	12.9083	0.681275	0.340638	−	0.940195i	$-0.389357\pi$
$359$	12.9083	0.681275	0.340638	+	0.940195i	$0.389357\pi$
$360$	0	0
$361$	5.09167	0.267983
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	5.21110	0.272762
$366$	0	0
$367$	16.9083	0.882607	0.441304	−	0.897358i	$-0.354516\pi$
$367$	16.9083	0.882607	0.441304	+	0.897358i	$0.354516\pi$
$368$	0	0
$369$	19.8167	1.03161
$370$	0	0
$371$	13.8167	0.717325
$372$	0	0
$373$	8.63331	0.447016	0.223508	−	0.974702i	$-0.428249\pi$
$373$	8.63331	0.447016	0.223508	+	0.974702i	$0.428249\pi$
$374$	0	0
$375$	−36.6333	−1.89174
$376$	0	0
$377$	6.11943	0.315167
$378$	0	0
$379$	−3.30278	−0.169652	−0.0848261	−	0.996396i	$-0.527033\pi$
$379$	−3.30278	−0.169652	−0.0848261	+	0.996396i	$0.527033\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	22.5416	1.15182	0.575912	−	0.817512i	$-0.304647\pi$
$383$	22.5416	1.15182	0.575912	+	0.817512i	$0.304647\pi$
$384$	0	0
$385$	−12.9083	−0.657869
$386$	0	0
$387$	23.8167	1.21067
$388$	0	0
$389$	−20.2111	−1.02474	−0.512372	−	0.858764i	$-0.671233\pi$
$389$	−20.2111	−1.02474	−0.512372	+	0.858764i	$0.671233\pi$
$390$	0	0
$391$	−20.7250	−1.04811
$392$	0	0
$393$	−13.8167	−0.696958
$394$	0	0
$395$	67.1472	3.37854
$396$	0	0
$397$	−25.0000	−1.25471	−0.627357	−	0.778732i	$-0.715863\pi$
$397$	−25.0000	−1.25471	−0.627357	+	0.778732i	$0.715863\pi$
$398$	0	0
$399$	−4.90833	−0.245724
$400$	0	0
$401$	7.18335	0.358719	0.179360	−	0.983784i	$-0.442597\pi$
$401$	7.18335	0.358719	0.179360	+	0.983784i	$0.442597\pi$
$402$	0	0
$403$	26.0000	1.29515
$404$	0	0
$405$	4.30278	0.213807
$406$	0	0
$407$	−0.908327	−0.0450241
$408$	0	0
$409$	−35.3028	−1.74561	−0.872805	−	0.488069i	$-0.837701\pi$
$409$	−35.3028	−1.74561	−0.872805	+	0.488069i	$0.837701\pi$
$410$	0	0
$411$	3.51388	0.173327
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−21.9083	−1.07544
$416$	0	0
$417$	12.3028	0.602470
$418$	0	0
$419$	4.69722	0.229474	0.114737	−	0.993396i	$-0.463397\pi$
$419$	4.69722	0.229474	0.114737	+	0.993396i	$0.463397\pi$
$420$	0	0
$421$	7.60555	0.370672	0.185336	−	0.982675i	$-0.440663\pi$
$421$	7.60555	0.370672	0.185336	+	0.982675i	$0.440663\pi$
$422$	0	0
$423$	2.60555	0.126686
$424$	0	0
$425$	−93.3583	−4.52854
$426$	0	0
$427$	−0.211103	−0.0102160
$428$	0	0
$429$	−10.8167	−0.522233
$430$	0	0
$431$	12.5139	0.602772	0.301386	−	0.953502i	$-0.402551\pi$
$431$	12.5139	0.602772	0.301386	+	0.953502i	$0.402551\pi$
$432$	0	0
$433$	37.3305	1.79399	0.896995	−	0.442040i	$-0.145745\pi$
$433$	37.3305	1.79399	0.896995	+	0.442040i	$0.145745\pi$
$434$	0	0
$435$	7.30278	0.350141
$436$	0	0
$437$	14.7250	0.704391
$438$	0	0
$439$	−5.78890	−0.276289	−0.138145	−	0.990412i	$-0.544114\pi$
$439$	−5.78890	−0.276289	−0.138145	+	0.990412i	$0.544114\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	−38.8444	−1.84555	−0.922777	−	0.385335i	$-0.874086\pi$
$443$	−38.8444	−1.84555	−0.922777	+	0.385335i	$0.874086\pi$
$444$	0	0
$445$	20.7250	0.982458
$446$	0	0
$447$	−8.60555	−0.407029
$448$	0	0
$449$	−8.09167	−0.381870	−0.190935	−	0.981603i	$-0.561152\pi$
$449$	−8.09167	−0.381870	−0.190935	+	0.981603i	$0.561152\pi$
$450$	0	0
$451$	29.7250	1.39969
$452$	0	0
$453$	7.60555	0.357340
$454$	0	0
$455$	−15.5139	−0.727302
$456$	0	0
$457$	24.3028	1.13684	0.568418	−	0.822740i	$-0.307556\pi$
$457$	24.3028	1.13684	0.568418	+	0.822740i	$0.307556\pi$
$458$	0	0
$459$	−34.5416	−1.61227
$460$	0	0
$461$	3.90833	0.182029	0.0910145	−	0.995850i	$-0.470989\pi$
$461$	3.90833	0.182029	0.0910145	+	0.995850i	$0.470989\pi$
$462$	0	0
$463$	30.2111	1.40403	0.702015	−	0.712163i	$-0.252284\pi$
$463$	30.2111	1.40403	0.702015	+	0.712163i	$0.252284\pi$
$464$	0	0
$465$	31.0278	1.43888
$466$	0	0
$467$	−36.5139	−1.68966	−0.844830	−	0.535034i	$-0.820299\pi$
$467$	−36.5139	−1.68966	−0.844830	+	0.535034i	$0.820299\pi$
$468$	0	0
$469$	−13.6056	−0.628246
$470$	0	0
$471$	−17.9083	−0.825172
$472$	0	0
$473$	35.7250	1.64264
$474$	0	0
$475$	66.3305	3.04345
$476$	0	0
$477$	−27.6333	−1.26524
$478$	0	0
$479$	−3.11943	−0.142530	−0.0712652	−	0.997457i	$-0.522704\pi$
$479$	−3.11943	−0.142530	−0.0712652	+	0.997457i	$0.522704\pi$
$480$	0	0
$481$	−1.09167	−0.0497760
$482$	0	0
$483$	−3.00000	−0.136505
$484$	0	0
$485$	−44.7250	−2.03086
$486$	0	0
$487$	43.5139	1.97180	0.985901	−	0.167330i	$-0.0535144\pi$
$487$	43.5139	1.97180	0.985901	+	0.167330i	$0.0535144\pi$
$488$	0	0
$489$	−4.90833	−0.221962
$490$	0	0
$491$	−22.8167	−1.02970	−0.514851	−	0.857280i	$-0.672153\pi$
$491$	−22.8167	−1.02970	−0.514851	+	0.857280i	$0.672153\pi$
$492$	0	0
$493$	11.7250	0.528067
$494$	0	0
$495$	25.8167	1.16037
$496$	0	0
$497$	−6.90833	−0.309881
$498$	0	0
$499$	21.8444	0.977890	0.488945	−	0.872315i	$-0.337382\pi$
$499$	21.8444	0.977890	0.488945	+	0.872315i	$0.337382\pi$
$500$	0	0
$501$	−9.90833	−0.442671
$502$	0	0
$503$	6.11943	0.272852	0.136426	−	0.990650i	$-0.456438\pi$
$503$	6.11943	0.272852	0.136426	+	0.990650i	$0.456438\pi$
$504$	0	0
$505$	−39.2389	−1.74611
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.0278	−0.444472	−0.222236	−	0.974993i	$-0.571336\pi$
$509$	−10.0278	−0.444472	−0.222236	+	0.974993i	$0.571336\pi$
$510$	0	0
$511$	1.21110	0.0535760
$512$	0	0
$513$	24.5416	1.08354
$514$	0	0
$515$	8.21110	0.361824
$516$	0	0
$517$	3.90833	0.171888
$518$	0	0
$519$	−23.3305	−1.02410
$520$	0	0
$521$	−19.3028	−0.845670	−0.422835	−	0.906207i	$-0.638965\pi$
$521$	−19.3028	−0.845670	−0.422835	+	0.906207i	$0.638965\pi$
$522$	0	0
$523$	−8.23886	−0.360260	−0.180130	−	0.983643i	$-0.557652\pi$
$523$	−8.23886	−0.360260	−0.180130	+	0.983643i	$0.557652\pi$
$524$	0	0
$525$	−13.5139	−0.589794
$526$	0	0
$527$	49.8167	2.17005
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	35.7250	1.54742
$534$	0	0
$535$	−61.1472	−2.64362
$536$	0	0
$537$	−3.51388	−0.151635
$538$	0	0
$539$	18.0000	0.775315
$540$	0	0
$541$	−11.6972	−0.502903	−0.251451	−	0.967870i	$-0.580908\pi$
$541$	−11.6972	−0.502903	−0.251451	+	0.967870i	$0.580908\pi$
$542$	0	0
$543$	3.48612	0.149604
$544$	0	0
$545$	−15.5139	−0.664542
$546$	0	0
$547$	2.57779	0.110219	0.0551093	−	0.998480i	$-0.482449\pi$
$547$	2.57779	0.110219	0.0551093	+	0.998480i	$0.482449\pi$
$548$	0	0
$549$	0.422205	0.0180193
$550$	0	0
$551$	−8.33053	−0.354893
$552$	0	0
$553$	15.6056	0.663616
$554$	0	0
$555$	−1.30278	−0.0552997
$556$	0	0
$557$	−21.6333	−0.916633	−0.458316	−	0.888789i	$-0.651547\pi$
$557$	−21.6333	−0.916633	−0.458316	+	0.888789i	$0.651547\pi$
$558$	0	0
$559$	42.9361	1.81600
$560$	0	0
$561$	−20.7250	−0.875009
$562$	0	0
$563$	−34.5416	−1.45576	−0.727878	−	0.685706i	$-0.759493\pi$
$563$	−34.5416	−1.45576	−0.727878	+	0.685706i	$0.759493\pi$
$564$	0	0
$565$	−83.5694	−3.51579
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	29.6056	1.24113	0.620565	−	0.784155i	$-0.286903\pi$
$569$	29.6056	1.24113	0.620565	+	0.784155i	$0.286903\pi$
$570$	0	0
$571$	−18.6972	−0.782454	−0.391227	−	0.920294i	$-0.627949\pi$
$571$	−18.6972	−0.782454	−0.391227	+	0.920294i	$0.627949\pi$
$572$	0	0
$573$	−7.42221	−0.310067
$574$	0	0
$575$	40.5416	1.69070
$576$	0	0
$577$	1.60555	0.0668400	0.0334200	−	0.999441i	$-0.489360\pi$
$577$	1.60555	0.0668400	0.0334200	+	0.999441i	$0.489360\pi$
$578$	0	0
$579$	12.2111	0.507476
$580$	0	0
$581$	−5.09167	−0.211238
$582$	0	0
$583$	−41.4500	−1.71668
$584$	0	0
$585$	31.0278	1.28284
$586$	0	0
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	0	0
$589$	−35.3944	−1.45840
$590$	0	0
$591$	−3.90833	−0.160767
$592$	0	0
$593$	9.27502	0.380879	0.190440	−	0.981699i	$-0.439009\pi$
$593$	9.27502	0.380879	0.190440	+	0.981699i	$0.439009\pi$
$594$	0	0
$595$	−29.7250	−1.21861
$596$	0	0
$597$	−7.51388	−0.307523
$598$	0	0
$599$	27.3944	1.11931	0.559653	−	0.828727i	$-0.310934\pi$
$599$	27.3944	1.11931	0.559653	+	0.828727i	$0.310934\pi$
$600$	0	0
$601$	−27.8444	−1.13580	−0.567899	−	0.823099i	$-0.692243\pi$
$601$	−27.8444	−1.13580	−0.567899	+	0.823099i	$0.692243\pi$
$602$	0	0
$603$	27.2111	1.10812
$604$	0	0
$605$	−8.60555	−0.349865
$606$	0	0
$607$	35.4222	1.43774	0.718871	−	0.695143i	$-0.244659\pi$
$607$	35.4222	1.43774	0.718871	+	0.695143i	$0.244659\pi$
$608$	0	0
$609$	1.69722	0.0687750
$610$	0	0
$611$	4.69722	0.190029
$612$	0	0
$613$	12.5778	0.508012	0.254006	−	0.967203i	$-0.418252\pi$
$613$	12.5778	0.508012	0.254006	+	0.967203i	$0.418252\pi$
$614$	0	0
$615$	42.6333	1.71914
$616$	0	0
$617$	37.4222	1.50656	0.753281	−	0.657699i	$-0.228470\pi$
$617$	37.4222	1.50656	0.753281	+	0.657699i	$0.228470\pi$
$618$	0	0
$619$	−16.7250	−0.672234	−0.336117	−	0.941820i	$-0.609114\pi$
$619$	−16.7250	−0.672234	−0.336117	+	0.941820i	$0.609114\pi$
$620$	0	0
$621$	15.0000	0.601929
$622$	0	0
$623$	4.81665	0.192975
$624$	0	0
$625$	90.0555	3.60222
$626$	0	0
$627$	14.7250	0.588059
$628$	0	0
$629$	−2.09167	−0.0834005
$630$	0	0
$631$	2.97224	0.118323	0.0591616	−	0.998248i	$-0.481157\pi$
$631$	2.97224	0.118323	0.0591616	+	0.998248i	$0.481157\pi$
$632$	0	0
$633$	20.1194	0.799676
$634$	0	0
$635$	30.1194	1.19525
$636$	0	0
$637$	21.6333	0.857143
$638$	0	0
$639$	13.8167	0.546578
$640$	0	0
$641$	26.6056	1.05086	0.525428	−	0.850838i	$-0.323905\pi$
$641$	26.6056	1.05086	0.525428	+	0.850838i	$0.323905\pi$
$642$	0	0
$643$	26.4222	1.04199	0.520995	−	0.853560i	$-0.325561\pi$
$643$	26.4222	1.04199	0.520995	+	0.853560i	$0.325561\pi$
$644$	0	0
$645$	51.2389	2.01753
$646$	0	0
$647$	17.2111	0.676638	0.338319	−	0.941031i	$-0.390142\pi$
$647$	17.2111	0.676638	0.338319	+	0.941031i	$0.390142\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	7.21110	0.282625
$652$	0	0
$653$	10.8167	0.423288	0.211644	−	0.977347i	$-0.432118\pi$
$653$	10.8167	0.423288	0.211644	+	0.977347i	$0.432118\pi$
$654$	0	0
$655$	59.4500	2.32290
$656$	0	0
$657$	−2.42221	−0.0944992
$658$	0	0
$659$	−3.63331	−0.141534	−0.0707668	−	0.997493i	$-0.522545\pi$
$659$	−3.63331	−0.141534	−0.0707668	+	0.997493i	$0.522545\pi$
$660$	0	0
$661$	−4.39445	−0.170924	−0.0854621	−	0.996341i	$-0.527237\pi$
$661$	−4.39445	−0.170924	−0.0854621	+	0.996341i	$0.527237\pi$
$662$	0	0
$663$	−24.9083	−0.967359
$664$	0	0
$665$	21.1194	0.818976
$666$	0	0
$667$	−5.09167	−0.197150
$668$	0	0
$669$	12.4222	0.480270
$670$	0	0
$671$	0.633308	0.0244486
$672$	0	0
$673$	−10.6333	−0.409884	−0.204942	−	0.978774i	$-0.565701\pi$
$673$	−10.6333	−0.409884	−0.204942	+	0.978774i	$0.565701\pi$
$674$	0	0
$675$	67.5694	2.60075
$676$	0	0
$677$	35.2111	1.35327	0.676636	−	0.736317i	$-0.263437\pi$
$677$	35.2111	1.35327	0.676636	+	0.736317i	$0.263437\pi$
$678$	0	0
$679$	−10.3944	−0.398902
$680$	0	0
$681$	17.6056	0.674646
$682$	0	0
$683$	−15.2750	−0.584482	−0.292241	−	0.956345i	$-0.594401\pi$
$683$	−15.2750	−0.584482	−0.292241	+	0.956345i	$0.594401\pi$
$684$	0	0
$685$	−15.1194	−0.577684
$686$	0	0
$687$	15.2111	0.580340
$688$	0	0
$689$	−49.8167	−1.89786
$690$	0	0
$691$	−14.1194	−0.537128	−0.268564	−	0.963262i	$-0.586549\pi$
$691$	−14.1194	−0.537128	−0.268564	+	0.963262i	$0.586549\pi$
$692$	0	0
$693$	6.00000	0.227921
$694$	0	0
$695$	−52.9361	−2.00798
$696$	0	0
$697$	68.4500	2.59273
$698$	0	0
$699$	−11.0917	−0.419526
$700$	0	0
$701$	44.3305	1.67434	0.837171	−	0.546942i	$-0.184208\pi$
$701$	44.3305	1.67434	0.837171	+	0.546942i	$0.184208\pi$
$702$	0	0
$703$	1.48612	0.0560501
$704$	0	0
$705$	5.60555	0.211117
$706$	0	0
$707$	−9.11943	−0.342971
$708$	0	0
$709$	−0.605551	−0.0227420	−0.0113710	−	0.999935i	$-0.503620\pi$
$709$	−0.605551	−0.0227420	−0.0113710	+	0.999935i	$0.503620\pi$
$710$	0	0
$711$	−31.2111	−1.17051
$712$	0	0
$713$	−21.6333	−0.810174
$714$	0	0
$715$	46.5416	1.74056
$716$	0	0
$717$	24.6333	0.919948
$718$	0	0
$719$	27.3944	1.02164	0.510820	−	0.859688i	$-0.329342\pi$
$719$	27.3944	1.02164	0.510820	+	0.859688i	$0.329342\pi$
$720$	0	0
$721$	1.90833	0.0710698
$722$	0	0
$723$	−6.42221	−0.238844
$724$	0	0
$725$	−22.9361	−0.851825
$726$	0	0
$727$	−30.9361	−1.14736	−0.573678	−	0.819081i	$-0.694484\pi$
$727$	−30.9361	−1.14736	−0.573678	+	0.819081i	$0.694484\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	82.2666	3.04274
$732$	0	0
$733$	−53.2666	−1.96745	−0.983724	−	0.179688i	$-0.942491\pi$
$733$	−53.2666	−1.96745	−0.983724	+	0.179688i	$0.942491\pi$
$734$	0	0
$735$	25.8167	0.952262
$736$	0	0
$737$	40.8167	1.50350
$738$	0	0
$739$	−46.3305	−1.70430	−0.852148	−	0.523301i	$-0.824700\pi$
$739$	−46.3305	−1.70430	−0.852148	+	0.523301i	$0.824700\pi$
$740$	0	0
$741$	17.6972	0.650124
$742$	0	0
$743$	1.69722	0.0622651	0.0311326	−	0.999515i	$-0.490089\pi$
$743$	1.69722	0.0622651	0.0311326	+	0.999515i	$0.490089\pi$
$744$	0	0
$745$	37.0278	1.35659
$746$	0	0
$747$	10.1833	0.372589
$748$	0	0
$749$	−14.2111	−0.519262
$750$	0	0
$751$	−17.2750	−0.630374	−0.315187	−	0.949030i	$-0.602067\pi$
$751$	−17.2750	−0.630374	−0.315187	+	0.949030i	$0.602067\pi$
$752$	0	0
$753$	−1.00000	−0.0364420
$754$	0	0
$755$	−32.7250	−1.19098
$756$	0	0
$757$	−26.8167	−0.974668	−0.487334	−	0.873216i	$-0.662031\pi$
$757$	−26.8167	−0.974668	−0.487334	+	0.873216i	$0.662031\pi$
$758$	0	0
$759$	9.00000	0.326679
$760$	0	0
$761$	14.2111	0.515152	0.257576	−	0.966258i	$-0.417076\pi$
$761$	14.2111	0.515152	0.257576	+	0.966258i	$0.417076\pi$
$762$	0	0
$763$	−3.60555	−0.130530
$764$	0	0
$765$	59.4500	2.14942
$766$	0	0
$767$	0	0
$768$	0	0
$769$	20.1194	0.725525	0.362763	−	0.931882i	$-0.381834\pi$
$769$	20.1194	0.725525	0.362763	+	0.931882i	$0.381834\pi$
$770$	0	0
$771$	−9.11943	−0.328428
$772$	0	0
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−97.4500	−3.50051
$776$	0	0
$777$	−0.302776	−0.0108620
$778$	0	0
$779$	−48.6333	−1.74247
$780$	0	0
$781$	20.7250	0.741598
$782$	0	0
$783$	−8.48612	−0.303269
$784$	0	0
$785$	77.0555	2.75023
$786$	0	0
$787$	−0.302776	−0.0107928	−0.00539639	−	0.999985i	$-0.501718\pi$
$787$	−0.302776	−0.0107928	−0.00539639	+	0.999985i	$0.501718\pi$
$788$	0	0
$789$	−8.72498	−0.310618
$790$	0	0
$791$	−19.4222	−0.690574
$792$	0	0
$793$	0.761141	0.0270289
$794$	0	0
$795$	−59.4500	−2.10847
$796$	0	0
$797$	−21.6333	−0.766291	−0.383146	−	0.923688i	$-0.625159\pi$
$797$	−21.6333	−0.766291	−0.383146	+	0.923688i	$0.625159\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	−9.63331	−0.340376
$802$	0	0
$803$	−3.63331	−0.128217
$804$	0	0
$805$	12.9083	0.454959
$806$	0	0
$807$	8.21110	0.289044
$808$	0	0
$809$	11.8444	0.416427	0.208214	−	0.978083i	$-0.433235\pi$
$809$	11.8444	0.416427	0.208214	+	0.978083i	$0.433235\pi$
$810$	0	0
$811$	−38.3583	−1.34694	−0.673471	−	0.739214i	$-0.735197\pi$
$811$	−38.3583	−1.34694	−0.673471	+	0.739214i	$0.735197\pi$
$812$	0	0
$813$	−7.39445	−0.259335
$814$	0	0
$815$	21.1194	0.739781
$816$	0	0
$817$	−58.4500	−2.04491
$818$	0	0
$819$	7.21110	0.251976
$820$	0	0
$821$	45.6333	1.59261	0.796307	−	0.604893i	$-0.206784\pi$
$821$	45.6333	1.59261	0.796307	+	0.604893i	$0.206784\pi$
$822$	0	0
$823$	−15.9722	−0.556757	−0.278379	−	0.960471i	$-0.589797\pi$
$823$	−15.9722	−0.556757	−0.278379	+	0.960471i	$0.589797\pi$
$824$	0	0
$825$	40.5416	1.41148
$826$	0	0
$827$	40.9361	1.42349	0.711744	−	0.702439i	$-0.247906\pi$
$827$	40.9361	1.42349	0.711744	+	0.702439i	$0.247906\pi$
$828$	0	0
$829$	−25.6333	−0.890282	−0.445141	−	0.895461i	$-0.646846\pi$
$829$	−25.6333	−0.890282	−0.445141	+	0.895461i	$0.646846\pi$
$830$	0	0
$831$	4.39445	0.152442
$832$	0	0
$833$	41.4500	1.43616
$834$	0	0
$835$	42.6333	1.47539
$836$	0	0
$837$	−36.0555	−1.24626
$838$	0	0
$839$	37.5416	1.29608	0.648041	−	0.761606i	$-0.275589\pi$
$839$	37.5416	1.29608	0.648041	+	0.761606i	$0.275589\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	0	0
$843$	5.72498	0.197179
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	11.7889	0.404594
$850$	0	0
$851$	0.908327	0.0311370
$852$	0	0
$853$	5.66947	0.194119	0.0970594	−	0.995279i	$-0.469056\pi$
$853$	5.66947	0.194119	0.0970594	+	0.995279i	$0.469056\pi$
$854$	0	0
$855$	−42.2389	−1.44454
$856$	0	0
$857$	3.00000	0.102478	0.0512390	−	0.998686i	$-0.483683\pi$
$857$	3.00000	0.102478	0.0512390	+	0.998686i	$0.483683\pi$
$858$	0	0
$859$	14.9361	0.509613	0.254806	−	0.966992i	$-0.417988\pi$
$859$	14.9361	0.509613	0.254806	+	0.966992i	$0.417988\pi$
$860$	0	0
$861$	9.90833	0.337675
$862$	0	0
$863$	−43.0278	−1.46468	−0.732341	−	0.680938i	$-0.761572\pi$
$863$	−43.0278	−1.46468	−0.732341	+	0.680938i	$0.761572\pi$
$864$	0	0
$865$	100.386	3.41323
$866$	0	0
$867$	−30.7250	−1.04348
$868$	0	0
$869$	−46.8167	−1.58815
$870$	0	0
$871$	49.0555	1.66218
$872$	0	0
$873$	20.7889	0.703598
$874$	0	0
$875$	36.6333	1.23843
$876$	0	0
$877$	−36.8444	−1.24415	−0.622074	−	0.782959i	$-0.713710\pi$
$877$	−36.8444	−1.24415	−0.622074	+	0.782959i	$0.713710\pi$
$878$	0	0
$879$	17.7250	0.597849
$880$	0	0
$881$	10.8167	0.364422	0.182211	−	0.983259i	$-0.441675\pi$
$881$	10.8167	0.364422	0.182211	+	0.983259i	$0.441675\pi$
$882$	0	0
$883$	26.8167	0.902452	0.451226	−	0.892410i	$-0.350987\pi$
$883$	26.8167	0.902452	0.451226	+	0.892410i	$0.350987\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−21.9083	−0.735610	−0.367805	−	0.929903i	$-0.619891\pi$
$887$	−21.9083	−0.735610	−0.367805	+	0.929903i	$0.619891\pi$
$888$	0	0
$889$	7.00000	0.234772
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−6.39445	−0.213982
$894$	0	0
$895$	15.1194	0.505387
$896$	0	0
$897$	10.8167	0.361158
$898$	0	0
$899$	12.2389	0.408189
$900$	0	0
$901$	−95.4500	−3.17990
$902$	0	0
$903$	11.9083	0.396284
$904$	0	0
$905$	−15.0000	−0.498617
$906$	0	0
$907$	−23.0000	−0.763702	−0.381851	−	0.924224i	$-0.624713\pi$
$907$	−23.0000	−0.763702	−0.381851	+	0.924224i	$0.624713\pi$
$908$	0	0
$909$	18.2389	0.604945
$910$	0	0
$911$	−1.54163	−0.0510766	−0.0255383	−	0.999674i	$-0.508130\pi$
$911$	−1.54163	−0.0510766	−0.0255383	+	0.999674i	$0.508130\pi$
$912$	0	0
$913$	15.2750	0.505529
$914$	0	0
$915$	0.908327	0.0300284
$916$	0	0
$917$	13.8167	0.456266
$918$	0	0
$919$	−53.4777	−1.76407	−0.882034	−	0.471187i	$-0.843826\pi$
$919$	−53.4777	−1.76407	−0.882034	+	0.471187i	$0.843826\pi$
$920$	0	0
$921$	−22.6333	−0.745793
$922$	0	0
$923$	24.9083	0.819868
$924$	0	0
$925$	4.09167	0.134533
$926$	0	0
$927$	−3.81665	−0.125355
$928$	0	0
$929$	−1.57779	−0.0517658	−0.0258829	−	0.999665i	$-0.508240\pi$
$929$	−1.57779	−0.0517658	−0.0258829	+	0.999665i	$0.508240\pi$
$930$	0	0
$931$	−29.4500	−0.965184
$932$	0	0
$933$	−0.394449	−0.0129137
$934$	0	0
$935$	89.1749	2.91633
$936$	0	0
$937$	−11.9722	−0.391116	−0.195558	−	0.980692i	$-0.562652\pi$
$937$	−11.9722	−0.391116	−0.195558	+	0.980692i	$0.562652\pi$
$938$	0	0
$939$	−1.09167	−0.0356254
$940$	0	0
$941$	−23.4861	−0.765626	−0.382813	−	0.923826i	$-0.625045\pi$
$941$	−23.4861	−0.765626	−0.382813	+	0.923826i	$0.625045\pi$
$942$	0	0
$943$	−29.7250	−0.967979
$944$	0	0
$945$	21.5139	0.699847
$946$	0	0
$947$	17.4500	0.567048	0.283524	−	0.958965i	$-0.408497\pi$
$947$	17.4500	0.567048	0.283524	+	0.958965i	$0.408497\pi$
$948$	0	0
$949$	−4.36669	−0.141749
$950$	0	0
$951$	13.6972	0.444163
$952$	0	0
$953$	−35.0917	−1.13673	−0.568365	−	0.822776i	$-0.692424\pi$
$953$	−35.0917	−1.13673	−0.568365	+	0.822776i	$0.692424\pi$
$954$	0	0
$955$	31.9361	1.03343
$956$	0	0
$957$	−5.09167	−0.164590
$958$	0	0
$959$	−3.51388	−0.113469
$960$	0	0
$961$	21.0000	0.677419
$962$	0	0
$963$	28.4222	0.915893
$964$	0	0
$965$	−52.5416	−1.69138
$966$	0	0
$967$	31.3583	1.00841	0.504207	−	0.863583i	$-0.331785\pi$
$967$	31.3583	1.00841	0.504207	+	0.863583i	$0.331785\pi$
$968$	0	0
$969$	33.9083	1.08929
$970$	0	0
$971$	48.0000	1.54039	0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000	1.54039	0.770197	+	0.637806i	$0.220158\pi$
$972$	0	0
$973$	−12.3028	−0.394409
$974$	0	0
$975$	48.7250	1.56045
$976$	0	0
$977$	44.7250	1.43088	0.715439	−	0.698675i	$-0.246227\pi$
$977$	44.7250	1.43088	0.715439	+	0.698675i	$0.246227\pi$
$978$	0	0
$979$	−14.4500	−0.461823
$980$	0	0
$981$	7.21110	0.230233
$982$	0	0
$983$	−9.63331	−0.307255	−0.153627	−	0.988129i	$-0.549096\pi$
$983$	−9.63331	−0.307255	−0.153627	+	0.988129i	$0.549096\pi$
$984$	0	0
$985$	16.8167	0.535823
$986$	0	0
$987$	1.30278	0.0414678
$988$	0	0
$989$	−35.7250	−1.13599
$990$	0	0
$991$	38.4222	1.22052	0.610261	−	0.792201i	$-0.291065\pi$
$991$	38.4222	1.22052	0.610261	+	0.792201i	$0.291065\pi$
$992$	0	0
$993$	−22.2389	−0.705729
$994$	0	0
$995$	32.3305	1.02495
$996$	0	0
$997$	23.1194	0.732200	0.366100	−	0.930576i	$-0.380693\pi$
$997$	23.1194	0.732200	0.366100	+	0.930576i	$0.380693\pi$
$998$	0	0
$999$	1.51388	0.0478970

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.a.1.2		2
4.3	odd	2		502.2.a.b.1.2	✓	2
12.11	even	2		4518.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.b.1.2	✓	2	4.3	odd	2
4016.2.a.a.1.2		2	1.1	even	1	trivial
4518.2.a.m.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.30278 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.30278 q^{5} +1.00000 q^{7} -2.00000 q^{9} -3.00000 q^{11} -3.60555 q^{13} -4.30278 q^{15} -6.90833 q^{17} +4.90833 q^{19} -1.00000 q^{21} +3.00000 q^{23} +13.5139 q^{25} +5.00000 q^{27} -1.69722 q^{29} -7.21110 q^{31} +3.00000 q^{33} +4.30278 q^{35} +0.302776 q^{37} +3.60555 q^{39} -9.90833 q^{41} -11.9083 q^{43} -8.60555 q^{45} -1.30278 q^{47} -6.00000 q^{49} +6.90833 q^{51} +13.8167 q^{53} -12.9083 q^{55} -4.90833 q^{57} -0.211103 q^{61} -2.00000 q^{63} -15.5139 q^{65} -13.6056 q^{67} -3.00000 q^{69} -6.90833 q^{71} +1.21110 q^{73} -13.5139 q^{75} -3.00000 q^{77} +15.6056 q^{79} +1.00000 q^{81} -5.09167 q^{83} -29.7250 q^{85} +1.69722 q^{87} +4.81665 q^{89} -3.60555 q^{91} +7.21110 q^{93} +21.1194 q^{95} -10.3944 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 5 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 - 4 * q^9	\( 2 q - 2 q^{3} + 5 q^{5} + 2 q^{7} - 4 q^{9} - 6 q^{11} - 5 q^{15} - 3 q^{17} - q^{19} - 2 q^{21} + 6 q^{23} + 9 q^{25} + 10 q^{27} - 7 q^{29} + 6 q^{33} + 5 q^{35} - 3 q^{37} - 9 q^{41} - 13 q^{43} - 10 q^{45} + q^{47} - 12 q^{49} + 3 q^{51} + 6 q^{53} - 15 q^{55} + q^{57} + 14 q^{61} - 4 q^{63} - 13 q^{65} - 20 q^{67} - 6 q^{69} - 3 q^{71} - 12 q^{73} - 9 q^{75} - 6 q^{77} + 24 q^{79} + 2 q^{81} - 21 q^{83} - 27 q^{85} + 7 q^{87} - 12 q^{89} + 17 q^{95} - 28 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 - 4 * q^9 - 6 * q^11 - 5 * q^15 - 3 * q^17 - q^19 - 2 * q^21 + 6 * q^23 + 9 * q^25 + 10 * q^27 - 7 * q^29 + 6 * q^33 + 5 * q^35 - 3 * q^37 - 9 * q^41 - 13 * q^43 - 10 * q^45 + q^47 - 12 * q^49 + 3 * q^51 + 6 * q^53 - 15 * q^55 + q^57 + 14 * q^61 - 4 * q^63 - 13 * q^65 - 20 * q^67 - 6 * q^69 - 3 * q^71 - 12 * q^73 - 9 * q^75 - 6 * q^77 + 24 * q^79 + 2 * q^81 - 21 * q^83 - 27 * q^85 + 7 * q^87 - 12 * q^89 + 17 * q^95 - 28 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

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