LMFDB - Embedded newform 4016.2.a.k.1.1

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.32547$ 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)

$f(q)$	$=$	$q-3.27059 q^{3} -2.03057 q^{5} -1.64874 q^{7} +7.69673 q^{9} +O(q^{10})$	$q-3.27059 q^{3} -2.03057 q^{5} -1.64874 q^{7} +7.69673 q^{9} +3.10982 q^{11} +5.70062 q^{13} +6.64114 q^{15} +3.66658 q^{17} +5.51191 q^{19} +5.39233 q^{21} +2.16257 q^{23} -0.876797 q^{25} -15.3610 q^{27} +8.08572 q^{29} -5.30190 q^{31} -10.1709 q^{33} +3.34787 q^{35} -2.05773 q^{37} -18.6444 q^{39} -2.65933 q^{41} +9.01212 q^{43} -15.6287 q^{45} -2.30292 q^{47} -4.28167 q^{49} -11.9919 q^{51} -8.44037 q^{53} -6.31471 q^{55} -18.0272 q^{57} -1.95895 q^{59} +11.1945 q^{61} -12.6899 q^{63} -11.5755 q^{65} +1.43672 q^{67} -7.07287 q^{69} +15.2173 q^{71} -1.27528 q^{73} +2.86764 q^{75} -5.12728 q^{77} +9.46821 q^{79} +27.1494 q^{81} +0.845328 q^{83} -7.44524 q^{85} -26.4451 q^{87} -6.17322 q^{89} -9.39881 q^{91} +17.3403 q^{93} -11.1923 q^{95} -9.74063 q^{97} +23.9355 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	−3.27059	−1.88827	−0.944137	−	0.329554i	$-0.893101\pi$
$3$	−3.27059	−1.88827	−0.944137	+	0.329554i	$0.893101\pi$
$4$	0	0
$5$	−2.03057	−0.908097	−0.454049	−	0.890977i	$-0.650021\pi$
$5$	−2.03057	−0.908097	−0.454049	+	0.890977i	$0.650021\pi$
$6$	0	0
$7$	−1.64874	−0.623164	−0.311582	−	0.950219i	$-0.600859\pi$
$7$	−1.64874	−0.623164	−0.311582	+	0.950219i	$0.600859\pi$
$8$	0	0
$9$	7.69673	2.56558
$10$	0	0
$11$	3.10982	0.937647	0.468824	−	0.883292i	$-0.344678\pi$
$11$	3.10982	0.937647	0.468824	+	0.883292i	$0.344678\pi$
$12$	0	0
$13$	5.70062	1.58107	0.790533	−	0.612419i	$-0.209803\pi$
$13$	5.70062	1.58107	0.790533	+	0.612419i	$0.209803\pi$
$14$	0	0
$15$	6.64114	1.71474
$16$	0	0
$17$	3.66658	0.889277	0.444638	−	0.895710i	$-0.353332\pi$
$17$	3.66658	0.889277	0.444638	+	0.895710i	$0.353332\pi$
$18$	0	0
$19$	5.51191	1.26452	0.632259	−	0.774757i	$-0.282128\pi$
$19$	5.51191	1.26452	0.632259	+	0.774757i	$0.282128\pi$
$20$	0	0
$21$	5.39233	1.17670
$22$	0	0
$23$	2.16257	0.450927	0.225463	−	0.974252i	$-0.427610\pi$
$23$	2.16257	0.450927	0.225463	+	0.974252i	$0.427610\pi$
$24$	0	0
$25$	−0.876797	−0.175359
$26$	0	0
$27$	−15.3610	−2.95623
$28$	0	0
$29$	8.08572	1.50148	0.750741	−	0.660597i	$-0.229697\pi$
$29$	8.08572	1.50148	0.750741	+	0.660597i	$0.229697\pi$
$30$	0	0
$31$	−5.30190	−0.952249	−0.476125	−	0.879378i	$-0.657959\pi$
$31$	−5.30190	−0.952249	−0.476125	+	0.879378i	$0.657959\pi$
$32$	0	0
$33$	−10.1709	−1.77053
$34$	0	0
$35$	3.34787	0.565893
$36$	0	0
$37$	−2.05773	−0.338289	−0.169145	−	0.985591i	$-0.554100\pi$
$37$	−2.05773	−0.338289	−0.169145	+	0.985591i	$0.554100\pi$
$38$	0	0
$39$	−18.6444	−2.98549
$40$	0	0
$41$	−2.65933	−0.415318	−0.207659	−	0.978201i	$-0.566584\pi$
$41$	−2.65933	−0.415318	−0.207659	+	0.978201i	$0.566584\pi$
$42$	0	0
$43$	9.01212	1.37434	0.687168	−	0.726499i	$-0.258854\pi$
$43$	9.01212	1.37434	0.687168	+	0.726499i	$0.258854\pi$
$44$	0	0
$45$	−15.6287	−2.32979
$46$	0	0
$47$	−2.30292	−0.335915	−0.167958	−	0.985794i	$-0.553717\pi$
$47$	−2.30292	−0.335915	−0.167958	+	0.985794i	$0.553717\pi$
$48$	0	0
$49$	−4.28167	−0.611667
$50$	0	0
$51$	−11.9919	−1.67920
$52$	0	0
$53$	−8.44037	−1.15937	−0.579687	−	0.814839i	$-0.696825\pi$
$53$	−8.44037	−1.15937	−0.579687	+	0.814839i	$0.696825\pi$
$54$	0	0
$55$	−6.31471	−0.851475
$56$	0	0
$57$	−18.0272	−2.38776
$58$	0	0
$59$	−1.95895	−0.255034	−0.127517	−	0.991836i	$-0.540701\pi$
$59$	−1.95895	−0.255034	−0.127517	+	0.991836i	$0.540701\pi$
$60$	0	0
$61$	11.1945	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$61$	11.1945	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$62$	0	0
$63$	−12.6899	−1.59877
$64$	0	0
$65$	−11.5755	−1.43576
$66$	0	0
$67$	1.43672	0.175523	0.0877615	−	0.996142i	$-0.472029\pi$
$67$	1.43672	0.175523	0.0877615	+	0.996142i	$0.472029\pi$
$68$	0	0
$69$	−7.07287	−0.851473
$70$	0	0
$71$	15.2173	1.80597	0.902983	−	0.429677i	$-0.141373\pi$
$71$	15.2173	1.80597	0.902983	+	0.429677i	$0.141373\pi$
$72$	0	0
$73$	−1.27528	−0.149260	−0.0746301	−	0.997211i	$-0.523778\pi$
$73$	−1.27528	−0.149260	−0.0746301	+	0.997211i	$0.523778\pi$
$74$	0	0
$75$	2.86764	0.331127
$76$	0	0
$77$	−5.12728	−0.584308
$78$	0	0
$79$	9.46821	1.06526	0.532628	−	0.846349i	$-0.321204\pi$
$79$	9.46821	1.06526	0.532628	+	0.846349i	$0.321204\pi$
$80$	0	0
$81$	27.1494	3.01660
$82$	0	0
$83$	0.845328	0.0927868	0.0463934	−	0.998923i	$-0.485227\pi$
$83$	0.845328	0.0927868	0.0463934	+	0.998923i	$0.485227\pi$
$84$	0	0
$85$	−7.44524	−0.807550
$86$	0	0
$87$	−26.4451	−2.83521
$88$	0	0
$89$	−6.17322	−0.654360	−0.327180	−	0.944962i	$-0.606098\pi$
$89$	−6.17322	−0.654360	−0.327180	+	0.944962i	$0.606098\pi$
$90$	0	0
$91$	−9.39881	−0.985263
$92$	0	0
$93$	17.3403	1.79811
$94$	0	0
$95$	−11.1923	−1.14831
$96$	0	0
$97$	−9.74063	−0.989012	−0.494506	−	0.869174i	$-0.664651\pi$
$97$	−9.74063	−0.989012	−0.494506	+	0.869174i	$0.664651\pi$
$98$	0	0
$99$	23.9355	2.40560
$100$	0	0
$101$	5.83576	0.580680	0.290340	−	0.956924i	$-0.406232\pi$
$101$	5.83576	0.580680	0.290340	+	0.956924i	$0.406232\pi$
$102$	0	0
$103$	9.73512	0.959229	0.479615	−	0.877479i	$-0.340776\pi$
$103$	9.73512	0.959229	0.479615	+	0.877479i	$0.340776\pi$
$104$	0	0
$105$	−10.9495	−1.06856
$106$	0	0
$107$	7.26599	0.702430	0.351215	−	0.936295i	$-0.385769\pi$
$107$	7.26599	0.702430	0.351215	+	0.936295i	$0.385769\pi$
$108$	0	0
$109$	5.18824	0.496943	0.248472	−	0.968639i	$-0.420072\pi$
$109$	5.18824	0.496943	0.248472	+	0.968639i	$0.420072\pi$
$110$	0	0
$111$	6.72999	0.638782
$112$	0	0
$113$	−17.6145	−1.65703	−0.828516	−	0.559966i	$-0.810814\pi$
$113$	−17.6145	−1.65703	−0.828516	+	0.559966i	$0.810814\pi$
$114$	0	0
$115$	−4.39124	−0.409485
$116$	0	0
$117$	43.8761	4.05635
$118$	0	0
$119$	−6.04523	−0.554165
$120$	0	0
$121$	−1.32899	−0.120818
$122$	0	0
$123$	8.69757	0.784233
$124$	0	0
$125$	11.9332	1.06734
$126$	0	0
$127$	5.45277	0.483855	0.241928	−	0.970294i	$-0.422220\pi$
$127$	5.45277	0.483855	0.241928	+	0.970294i	$0.422220\pi$
$128$	0	0
$129$	−29.4749	−2.59512
$130$	0	0
$131$	13.1350	1.14761	0.573805	−	0.818992i	$-0.305467\pi$
$131$	13.1350	1.14761	0.573805	+	0.818992i	$0.305467\pi$
$132$	0	0
$133$	−9.08768	−0.788002
$134$	0	0
$135$	31.1916	2.68455
$136$	0	0
$137$	5.50982	0.470736	0.235368	−	0.971906i	$-0.424370\pi$
$137$	5.50982	0.470736	0.235368	+	0.971906i	$0.424370\pi$
$138$	0	0
$139$	−18.8534	−1.59912	−0.799560	−	0.600586i	$-0.794934\pi$
$139$	−18.8534	−1.59912	−0.799560	+	0.600586i	$0.794934\pi$
$140$	0	0
$141$	7.53189	0.634300
$142$	0	0
$143$	17.7279	1.48248
$144$	0	0
$145$	−16.4186	−1.36349
$146$	0	0
$147$	14.0036	1.15499
$148$	0	0
$149$	10.6735	0.874411	0.437206	−	0.899362i	$-0.355968\pi$
$149$	10.6735	0.874411	0.437206	+	0.899362i	$0.355968\pi$
$150$	0	0
$151$	7.08667	0.576705	0.288352	−	0.957524i	$-0.406893\pi$
$151$	7.08667	0.576705	0.288352	+	0.957524i	$0.406893\pi$
$152$	0	0
$153$	28.2207	2.28151
$154$	0	0
$155$	10.7659	0.864735
$156$	0	0
$157$	−4.11273	−0.328232	−0.164116	−	0.986441i	$-0.552477\pi$
$157$	−4.11273	−0.328232	−0.164116	+	0.986441i	$0.552477\pi$
$158$	0	0
$159$	27.6050	2.18922
$160$	0	0
$161$	−3.56551	−0.281001
$162$	0	0
$163$	−11.7512	−0.920423	−0.460211	−	0.887809i	$-0.652226\pi$
$163$	−11.7512	−0.920423	−0.460211	+	0.887809i	$0.652226\pi$
$164$	0	0
$165$	20.6528	1.60782
$166$	0	0
$167$	−21.6762	−1.67736	−0.838678	−	0.544628i	$-0.816671\pi$
$167$	−21.6762	−1.67736	−0.838678	+	0.544628i	$0.816671\pi$
$168$	0	0
$169$	19.4970	1.49977
$170$	0	0
$171$	42.4236	3.24422
$172$	0	0
$173$	12.4853	0.949241	0.474621	−	0.880190i	$-0.342585\pi$
$173$	12.4853	0.949241	0.474621	+	0.880190i	$0.342585\pi$
$174$	0	0
$175$	1.44561	0.109278
$176$	0	0
$177$	6.40692	0.481574
$178$	0	0
$179$	−20.7357	−1.54986	−0.774929	−	0.632048i	$-0.782214\pi$
$179$	−20.7357	−1.54986	−0.774929	+	0.632048i	$0.782214\pi$
$180$	0	0
$181$	−9.08366	−0.675183	−0.337591	−	0.941293i	$-0.609612\pi$
$181$	−9.08366	−0.675183	−0.337591	+	0.941293i	$0.609612\pi$
$182$	0	0
$183$	−36.6126	−2.70648
$184$	0	0
$185$	4.17836	0.307199
$186$	0	0
$187$	11.4024	0.833828
$188$	0	0
$189$	25.3263	1.84222
$190$	0	0
$191$	−14.0981	−1.02010	−0.510051	−	0.860144i	$-0.670373\pi$
$191$	−14.0981	−1.02010	−0.510051	+	0.860144i	$0.670373\pi$
$192$	0	0
$193$	13.3588	0.961589	0.480794	−	0.876833i	$-0.340348\pi$
$193$	13.3588	0.961589	0.480794	+	0.876833i	$0.340348\pi$
$194$	0	0
$195$	37.8586	2.71111
$196$	0	0
$197$	−9.08734	−0.647446	−0.323723	−	0.946152i	$-0.604935\pi$
$197$	−9.08734	−0.647446	−0.323723	+	0.946152i	$0.604935\pi$
$198$	0	0
$199$	3.85757	0.273456	0.136728	−	0.990609i	$-0.456341\pi$
$199$	3.85757	0.273456	0.136728	+	0.990609i	$0.456341\pi$
$200$	0	0
$201$	−4.69891	−0.331435
$202$	0	0
$203$	−13.3312	−0.935669
$204$	0	0
$205$	5.39995	0.377149
$206$	0	0
$207$	16.6447	1.15689
$208$	0	0
$209$	17.1411	1.18567
$210$	0	0
$211$	−1.50996	−0.103950	−0.0519749	−	0.998648i	$-0.516552\pi$
$211$	−1.50996	−0.103950	−0.0519749	+	0.998648i	$0.516552\pi$
$212$	0	0
$213$	−49.7696	−3.41016
$214$	0	0
$215$	−18.2997	−1.24803
$216$	0	0
$217$	8.74143	0.593407
$218$	0	0
$219$	4.17091	0.281844
$220$	0	0
$221$	20.9018	1.40601
$222$	0	0
$223$	14.9321	0.999926	0.499963	−	0.866047i	$-0.333347\pi$
$223$	14.9321	0.999926	0.499963	+	0.866047i	$0.333347\pi$
$224$	0	0
$225$	−6.74847	−0.449898
$226$	0	0
$227$	−17.7846	−1.18041	−0.590204	−	0.807254i	$-0.700953\pi$
$227$	−17.7846	−1.18041	−0.590204	+	0.807254i	$0.700953\pi$
$228$	0	0
$229$	20.6901	1.36724	0.683619	−	0.729839i	$-0.260405\pi$
$229$	20.6901	1.36724	0.683619	+	0.729839i	$0.260405\pi$
$230$	0	0
$231$	16.7692	1.10333
$232$	0	0
$233$	−26.3174	−1.72411	−0.862055	−	0.506815i	$-0.830823\pi$
$233$	−26.3174	−1.72411	−0.862055	+	0.506815i	$0.830823\pi$
$234$	0	0
$235$	4.67623	0.305044
$236$	0	0
$237$	−30.9666	−2.01150
$238$	0	0
$239$	9.56260	0.618553	0.309277	−	0.950972i	$-0.399913\pi$
$239$	9.56260	0.618553	0.309277	+	0.950972i	$0.399913\pi$
$240$	0	0
$241$	3.19451	0.205776	0.102888	−	0.994693i	$-0.467192\pi$
$241$	3.19451	0.205776	0.102888	+	0.994693i	$0.467192\pi$
$242$	0	0
$243$	−42.7114	−2.73993
$244$	0	0
$245$	8.69422	0.555453
$246$	0	0
$247$	31.4213	1.99929
$248$	0	0
$249$	−2.76472	−0.175207
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	6.72521	0.422810
$254$	0	0
$255$	24.3503	1.52487
$256$	0	0
$257$	−3.41197	−0.212833	−0.106416	−	0.994322i	$-0.533938\pi$
$257$	−3.41197	−0.212833	−0.106416	+	0.994322i	$0.533938\pi$
$258$	0	0
$259$	3.39266	0.210809
$260$	0	0
$261$	62.2336	3.85216
$262$	0	0
$263$	1.39234	0.0858553	0.0429276	−	0.999078i	$-0.486332\pi$
$263$	1.39234	0.0858553	0.0429276	+	0.999078i	$0.486332\pi$
$264$	0	0
$265$	17.1387	1.05282
$266$	0	0
$267$	20.1900	1.23561
$268$	0	0
$269$	2.68252	0.163556	0.0817782	−	0.996651i	$-0.473940\pi$
$269$	2.68252	0.163556	0.0817782	+	0.996651i	$0.473940\pi$
$270$	0	0
$271$	−7.75309	−0.470967	−0.235483	−	0.971878i	$-0.575667\pi$
$271$	−7.75309	−0.470967	−0.235483	+	0.971878i	$0.575667\pi$
$272$	0	0
$273$	30.7396	1.86045
$274$	0	0
$275$	−2.72669	−0.164425
$276$	0	0
$277$	12.5977	0.756922	0.378461	−	0.925617i	$-0.376453\pi$
$277$	12.5977	0.756922	0.378461	+	0.925617i	$0.376453\pi$
$278$	0	0
$279$	−40.8073	−2.44307
$280$	0	0
$281$	29.8821	1.78262	0.891308	−	0.453399i	$-0.149789\pi$
$281$	29.8821	1.78262	0.891308	+	0.453399i	$0.149789\pi$
$282$	0	0
$283$	−0.431925	−0.0256753	−0.0128376	−	0.999918i	$-0.504086\pi$
$283$	−0.431925	−0.0256753	−0.0128376	+	0.999918i	$0.504086\pi$
$284$	0	0
$285$	36.6053	2.16831
$286$	0	0
$287$	4.38454	0.258811
$288$	0	0
$289$	−3.55617	−0.209187
$290$	0	0
$291$	31.8576	1.86752
$292$	0	0
$293$	−12.7686	−0.745949	−0.372974	−	0.927842i	$-0.621662\pi$
$293$	−12.7686	−0.745949	−0.372974	+	0.927842i	$0.621662\pi$
$294$	0	0
$295$	3.97779	0.231596
$296$	0	0
$297$	−47.7701	−2.77190
$298$	0	0
$299$	12.3280	0.712945
$300$	0	0
$301$	−14.8586	−0.856436
$302$	0	0
$303$	−19.0863	−1.09648
$304$	0	0
$305$	−22.7312	−1.30158
$306$	0	0
$307$	11.8932	0.678779	0.339390	−	0.940646i	$-0.389780\pi$
$307$	11.8932	0.678779	0.339390	+	0.940646i	$0.389780\pi$
$308$	0	0
$309$	−31.8395	−1.81129
$310$	0	0
$311$	−31.6291	−1.79352	−0.896760	−	0.442517i	$-0.854086\pi$
$311$	−31.6291	−1.79352	−0.896760	+	0.442517i	$0.854086\pi$
$312$	0	0
$313$	13.4546	0.760496	0.380248	−	0.924884i	$-0.375839\pi$
$313$	13.4546	0.760496	0.380248	+	0.924884i	$0.375839\pi$
$314$	0	0
$315$	25.7676	1.45184
$316$	0	0
$317$	−7.42754	−0.417172	−0.208586	−	0.978004i	$-0.566886\pi$
$317$	−7.42754	−0.417172	−0.208586	+	0.978004i	$0.566886\pi$
$318$	0	0
$319$	25.1452	1.40786
$320$	0	0
$321$	−23.7640	−1.32638
$322$	0	0
$323$	20.2099	1.12451
$324$	0	0
$325$	−4.99829	−0.277255
$326$	0	0
$327$	−16.9686	−0.938365
$328$	0	0
$329$	3.79690	0.209330
$330$	0	0
$331$	22.5079	1.23714	0.618572	−	0.785728i	$-0.287711\pi$
$331$	22.5079	1.23714	0.618572	+	0.785728i	$0.287711\pi$
$332$	0	0
$333$	−15.8378	−0.867906
$334$	0	0
$335$	−2.91735	−0.159392
$336$	0	0
$337$	12.2271	0.666055	0.333028	−	0.942917i	$-0.391930\pi$
$337$	12.2271	0.666055	0.333028	+	0.942917i	$0.391930\pi$
$338$	0	0
$339$	57.6097	3.12893
$340$	0	0
$341$	−16.4880	−0.892874
$342$	0	0
$343$	18.6005	1.00433
$344$	0	0
$345$	14.3619	0.773220
$346$	0	0
$347$	20.3626	1.09312	0.546562	−	0.837419i	$-0.315936\pi$
$347$	20.3626	1.09312	0.546562	+	0.837419i	$0.315936\pi$
$348$	0	0
$349$	−19.3805	−1.03742	−0.518708	−	0.854951i	$-0.673587\pi$
$349$	−19.3805	−1.03742	−0.518708	+	0.854951i	$0.673587\pi$
$350$	0	0
$351$	−87.5674	−4.67400
$352$	0	0
$353$	−16.3547	−0.870472	−0.435236	−	0.900316i	$-0.643335\pi$
$353$	−16.3547	−0.870472	−0.435236	+	0.900316i	$0.643335\pi$
$354$	0	0
$355$	−30.8998	−1.63999
$356$	0	0
$357$	19.7714	1.04642
$358$	0	0
$359$	10.3782	0.547740	0.273870	−	0.961767i	$-0.411696\pi$
$359$	10.3782	0.547740	0.273870	+	0.961767i	$0.411696\pi$
$360$	0	0
$361$	11.3811	0.599005
$362$	0	0
$363$	4.34659	0.228137
$364$	0	0
$365$	2.58954	0.135543
$366$	0	0
$367$	−27.6014	−1.44078	−0.720390	−	0.693569i	$-0.756037\pi$
$367$	−27.6014	−1.44078	−0.720390	+	0.693569i	$0.756037\pi$
$368$	0	0
$369$	−20.4681	−1.06553
$370$	0	0
$371$	13.9159	0.722480
$372$	0	0
$373$	22.3855	1.15908	0.579538	−	0.814945i	$-0.303233\pi$
$373$	22.3855	1.15908	0.579538	+	0.814945i	$0.303233\pi$
$374$	0	0
$375$	−39.0286	−2.01543
$376$	0	0
$377$	46.0936	2.37394
$378$	0	0
$379$	−13.7898	−0.708337	−0.354168	−	0.935182i	$-0.615236\pi$
$379$	−13.7898	−0.708337	−0.354168	+	0.935182i	$0.615236\pi$
$380$	0	0
$381$	−17.8338	−0.913651
$382$	0	0
$383$	14.4938	0.740600	0.370300	−	0.928912i	$-0.379255\pi$
$383$	14.4938	0.740600	0.370300	+	0.928912i	$0.379255\pi$
$384$	0	0
$385$	10.4113	0.530608
$386$	0	0
$387$	69.3639	3.52596
$388$	0	0
$389$	6.42004	0.325509	0.162755	−	0.986667i	$-0.447962\pi$
$389$	6.42004	0.325509	0.162755	+	0.986667i	$0.447962\pi$
$390$	0	0
$391$	7.92924	0.400999
$392$	0	0
$393$	−42.9591	−2.16700
$394$	0	0
$395$	−19.2258	−0.967356
$396$	0	0
$397$	−20.9700	−1.05245	−0.526227	−	0.850344i	$-0.676394\pi$
$397$	−20.9700	−1.05245	−0.526227	+	0.850344i	$0.676394\pi$
$398$	0	0
$399$	29.7220	1.48796
$400$	0	0
$401$	4.95098	0.247240	0.123620	−	0.992330i	$-0.460550\pi$
$401$	4.95098	0.247240	0.123620	+	0.992330i	$0.460550\pi$
$402$	0	0
$403$	−30.2241	−1.50557
$404$	0	0
$405$	−55.1287	−2.73937
$406$	0	0
$407$	−6.39918	−0.317196
$408$	0	0
$409$	27.6111	1.36528	0.682640	−	0.730755i	$-0.260832\pi$
$409$	27.6111	1.36528	0.682640	+	0.730755i	$0.260832\pi$
$410$	0	0
$411$	−18.0203	−0.888878
$412$	0	0
$413$	3.22980	0.158928
$414$	0	0
$415$	−1.71650	−0.0842594
$416$	0	0
$417$	61.6615	3.01958
$418$	0	0
$419$	−18.0724	−0.882894	−0.441447	−	0.897287i	$-0.645535\pi$
$419$	−18.0724	−0.882894	−0.441447	+	0.897287i	$0.645535\pi$
$420$	0	0
$421$	30.9507	1.50844	0.754221	−	0.656620i	$-0.228015\pi$
$421$	30.9507	1.50844	0.754221	+	0.656620i	$0.228015\pi$
$422$	0	0
$423$	−17.7249	−0.861816
$424$	0	0
$425$	−3.21485	−0.155943
$426$	0	0
$427$	−18.4568	−0.893186
$428$	0	0
$429$	−57.9807	−2.79933
$430$	0	0
$431$	11.5470	0.556200	0.278100	−	0.960552i	$-0.410295\pi$
$431$	11.5470	0.556200	0.278100	+	0.960552i	$0.410295\pi$
$432$	0	0
$433$	4.80347	0.230840	0.115420	−	0.993317i	$-0.463179\pi$
$433$	4.80347	0.230840	0.115420	+	0.993317i	$0.463179\pi$
$434$	0	0
$435$	53.6984	2.57464
$436$	0	0
$437$	11.9199	0.570205
$438$	0	0
$439$	0.351852	0.0167930	0.00839648	−	0.999965i	$-0.497327\pi$
$439$	0.351852	0.0167930	0.00839648	+	0.999965i	$0.497327\pi$
$440$	0	0
$441$	−32.9548	−1.56928
$442$	0	0
$443$	−27.4634	−1.30483	−0.652413	−	0.757864i	$-0.726243\pi$
$443$	−27.4634	−1.30483	−0.652413	+	0.757864i	$0.726243\pi$
$444$	0	0
$445$	12.5351	0.594222
$446$	0	0
$447$	−34.9087	−1.65113
$448$	0	0
$449$	−17.3344	−0.818061	−0.409031	−	0.912521i	$-0.634133\pi$
$449$	−17.3344	−0.818061	−0.409031	+	0.912521i	$0.634133\pi$
$450$	0	0
$451$	−8.27005	−0.389422
$452$	0	0
$453$	−23.1776	−1.08898
$454$	0	0
$455$	19.0849	0.894715
$456$	0	0
$457$	26.9250	1.25950	0.629749	−	0.776799i	$-0.283158\pi$
$457$	26.9250	1.25950	0.629749	+	0.776799i	$0.283158\pi$
$458$	0	0
$459$	−56.3225	−2.62891
$460$	0	0
$461$	3.38583	0.157694	0.0788468	−	0.996887i	$-0.474876\pi$
$461$	3.38583	0.157694	0.0788468	+	0.996887i	$0.474876\pi$
$462$	0	0
$463$	8.86389	0.411940	0.205970	−	0.978558i	$-0.433965\pi$
$463$	8.86389	0.411940	0.205970	+	0.978558i	$0.433965\pi$
$464$	0	0
$465$	−35.2107	−1.63286
$466$	0	0
$467$	6.12549	0.283454	0.141727	−	0.989906i	$-0.454735\pi$
$467$	6.12549	0.283454	0.141727	+	0.989906i	$0.454735\pi$
$468$	0	0
$469$	−2.36877	−0.109380
$470$	0	0
$471$	13.4510	0.619791
$472$	0	0
$473$	28.0261	1.28864
$474$	0	0
$475$	−4.83282	−0.221745
$476$	0	0
$477$	−64.9632	−2.97446
$478$	0	0
$479$	2.51387	0.114862	0.0574308	−	0.998349i	$-0.481709\pi$
$479$	2.51387	0.114862	0.0574308	+	0.998349i	$0.481709\pi$
$480$	0	0
$481$	−11.7303	−0.534858
$482$	0	0
$483$	11.6613	0.530607
$484$	0	0
$485$	19.7790	0.898119
$486$	0	0
$487$	−7.32964	−0.332138	−0.166069	−	0.986114i	$-0.553107\pi$
$487$	−7.32964	−0.332138	−0.166069	+	0.986114i	$0.553107\pi$
$488$	0	0
$489$	38.4332	1.73801
$490$	0	0
$491$	10.2215	0.461291	0.230646	−	0.973038i	$-0.425916\pi$
$491$	10.2215	0.461291	0.230646	+	0.973038i	$0.425916\pi$
$492$	0	0
$493$	29.6470	1.33523
$494$	0	0
$495$	−48.6026	−2.18452
$496$	0	0
$497$	−25.0894	−1.12541
$498$	0	0
$499$	26.3537	1.17975	0.589876	−	0.807494i	$-0.299177\pi$
$499$	26.3537	1.17975	0.589876	+	0.807494i	$0.299177\pi$
$500$	0	0
$501$	70.8939	3.16730
$502$	0	0
$503$	−4.42095	−0.197120	−0.0985602	−	0.995131i	$-0.531424\pi$
$503$	−4.42095	−0.197120	−0.0985602	+	0.995131i	$0.531424\pi$
$504$	0	0
$505$	−11.8499	−0.527314
$506$	0	0
$507$	−63.7667	−2.83198
$508$	0	0
$509$	19.3043	0.855647	0.427824	−	0.903862i	$-0.359280\pi$
$509$	19.3043	0.855647	0.427824	+	0.903862i	$0.359280\pi$
$510$	0	0
$511$	2.10260	0.0930135
$512$	0	0
$513$	−84.6686	−3.73821
$514$	0	0
$515$	−19.7678	−0.871074
$516$	0	0
$517$	−7.16167	−0.314970
$518$	0	0
$519$	−40.8343	−1.79243
$520$	0	0
$521$	16.8838	0.739691	0.369846	−	0.929093i	$-0.379411\pi$
$521$	16.8838	0.739691	0.369846	+	0.929093i	$0.379411\pi$
$522$	0	0
$523$	37.7621	1.65122	0.825610	−	0.564241i	$-0.190831\pi$
$523$	37.7621	1.65122	0.825610	+	0.564241i	$0.190831\pi$
$524$	0	0
$525$	−4.72798	−0.206346
$526$	0	0
$527$	−19.4399	−0.846813
$528$	0	0
$529$	−18.3233	−0.796665
$530$	0	0
$531$	−15.0775	−0.654309
$532$	0	0
$533$	−15.1598	−0.656645
$534$	0	0
$535$	−14.7541	−0.637874
$536$	0	0
$537$	67.8178	2.92655
$538$	0	0
$539$	−13.3152	−0.573528
$540$	0	0
$541$	2.51970	0.108330	0.0541652	−	0.998532i	$-0.482750\pi$
$541$	2.51970	0.108330	0.0541652	+	0.998532i	$0.482750\pi$
$542$	0	0
$543$	29.7089	1.27493
$544$	0	0
$545$	−10.5351	−0.451273
$546$	0	0
$547$	14.7422	0.630331	0.315165	−	0.949037i	$-0.397940\pi$
$547$	14.7422	0.630331	0.315165	+	0.949037i	$0.397940\pi$
$548$	0	0
$549$	86.1610	3.67726
$550$	0	0
$551$	44.5678	1.89865
$552$	0	0
$553$	−15.6106	−0.663829
$554$	0	0
$555$	−13.6657	−0.580076
$556$	0	0
$557$	2.32712	0.0986035	0.0493017	−	0.998784i	$-0.484300\pi$
$557$	2.32712	0.0986035	0.0493017	+	0.998784i	$0.484300\pi$
$558$	0	0
$559$	51.3747	2.17292
$560$	0	0
$561$	−37.2926	−1.57450
$562$	0	0
$563$	10.1209	0.426546	0.213273	−	0.976993i	$-0.431588\pi$
$563$	10.1209	0.426546	0.213273	+	0.976993i	$0.431588\pi$
$564$	0	0
$565$	35.7674	1.50475
$566$	0	0
$567$	−44.7622	−1.87984
$568$	0	0
$569$	4.70511	0.197249	0.0986243	−	0.995125i	$-0.468556\pi$
$569$	4.70511	0.197249	0.0986243	+	0.995125i	$0.468556\pi$
$570$	0	0
$571$	19.6052	0.820454	0.410227	−	0.911984i	$-0.365450\pi$
$571$	19.6052	0.820454	0.410227	+	0.911984i	$0.365450\pi$
$572$	0	0
$573$	46.1090	1.92623
$574$	0	0
$575$	−1.89614	−0.0790743
$576$	0	0
$577$	4.76106	0.198205	0.0991027	−	0.995077i	$-0.468403\pi$
$577$	4.76106	0.198205	0.0991027	+	0.995077i	$0.468403\pi$
$578$	0	0
$579$	−43.6911	−1.81574
$580$	0	0
$581$	−1.39372	−0.0578214
$582$	0	0
$583$	−26.2481	−1.08708
$584$	0	0
$585$	−89.0934	−3.68356
$586$	0	0
$587$	−27.5881	−1.13868	−0.569341	−	0.822102i	$-0.692801\pi$
$587$	−27.5881	−1.13868	−0.569341	+	0.822102i	$0.692801\pi$
$588$	0	0
$589$	−29.2236	−1.20414
$590$	0	0
$591$	29.7209	1.22256
$592$	0	0
$593$	−35.9409	−1.47592	−0.737958	−	0.674846i	$-0.764210\pi$
$593$	−35.9409	−1.47592	−0.737958	+	0.674846i	$0.764210\pi$
$594$	0	0
$595$	12.2752	0.503236
$596$	0	0
$597$	−12.6165	−0.516360
$598$	0	0
$599$	45.1866	1.84627	0.923137	−	0.384471i	$-0.125616\pi$
$599$	45.1866	1.84627	0.923137	+	0.384471i	$0.125616\pi$
$600$	0	0
$601$	−0.413871	−0.0168822	−0.00844108	−	0.999964i	$-0.502687\pi$
$601$	−0.413871	−0.0168822	−0.00844108	+	0.999964i	$0.502687\pi$
$602$	0	0
$603$	11.0580	0.450317
$604$	0	0
$605$	2.69861	0.109714
$606$	0	0
$607$	−21.6716	−0.879624	−0.439812	−	0.898090i	$-0.644955\pi$
$607$	−21.6716	−0.879624	−0.439812	+	0.898090i	$0.644955\pi$
$608$	0	0
$609$	43.6009	1.76680
$610$	0	0
$611$	−13.1281	−0.531104
$612$	0	0
$613$	0.169735	0.00685555	0.00342777	−	0.999994i	$-0.498909\pi$
$613$	0.169735	0.00685555	0.00342777	+	0.999994i	$0.498909\pi$
$614$	0	0
$615$	−17.6610	−0.712160
$616$	0	0
$617$	27.3815	1.10234	0.551169	−	0.834394i	$-0.314182\pi$
$617$	27.3815	1.10234	0.551169	+	0.834394i	$0.314182\pi$
$618$	0	0
$619$	34.2960	1.37847	0.689237	−	0.724536i	$-0.257946\pi$
$619$	34.2960	1.37847	0.689237	+	0.724536i	$0.257946\pi$
$620$	0	0
$621$	−33.2193	−1.33305
$622$	0	0
$623$	10.1780	0.407773
$624$	0	0
$625$	−19.8472	−0.793890
$626$	0	0
$627$	−56.0613	−2.23887
$628$	0	0
$629$	−7.54484	−0.300833
$630$	0	0
$631$	13.3011	0.529509	0.264755	−	0.964316i	$-0.414709\pi$
$631$	13.3011	0.529509	0.264755	+	0.964316i	$0.414709\pi$
$632$	0	0
$633$	4.93845	0.196286
$634$	0	0
$635$	−11.0722	−0.439388
$636$	0	0
$637$	−24.4082	−0.967086
$638$	0	0
$639$	117.124	4.63334
$640$	0	0
$641$	−29.6261	−1.17016	−0.585080	−	0.810975i	$-0.698937\pi$
$641$	−29.6261	−1.17016	−0.585080	+	0.810975i	$0.698937\pi$
$642$	0	0
$643$	−26.9409	−1.06245	−0.531224	−	0.847232i	$-0.678268\pi$
$643$	−26.9409	−1.06245	−0.531224	+	0.847232i	$0.678268\pi$
$644$	0	0
$645$	59.8508	2.35662
$646$	0	0
$647$	−16.1051	−0.633157	−0.316579	−	0.948566i	$-0.602534\pi$
$647$	−16.1051	−0.633157	−0.316579	+	0.948566i	$0.602534\pi$
$648$	0	0
$649$	−6.09200	−0.239132
$650$	0	0
$651$	−28.5896	−1.12051
$652$	0	0
$653$	−28.4743	−1.11429	−0.557143	−	0.830417i	$-0.688102\pi$
$653$	−28.4743	−1.11429	−0.557143	+	0.830417i	$0.688102\pi$
$654$	0	0
$655$	−26.6715	−1.04214
$656$	0	0
$657$	−9.81548	−0.382938
$658$	0	0
$659$	−5.65381	−0.220241	−0.110121	−	0.993918i	$-0.535124\pi$
$659$	−5.65381	−0.220241	−0.110121	+	0.993918i	$0.535124\pi$
$660$	0	0
$661$	−16.9478	−0.659191	−0.329596	−	0.944122i	$-0.606912\pi$
$661$	−16.9478	−0.659191	−0.329596	+	0.944122i	$0.606912\pi$
$662$	0	0
$663$	−68.3611	−2.65492
$664$	0	0
$665$	18.4531	0.715582
$666$	0	0
$667$	17.4859	0.677058
$668$	0	0
$669$	−48.8367	−1.88813
$670$	0	0
$671$	34.8129	1.34394
$672$	0	0
$673$	18.6054	0.717184	0.358592	−	0.933494i	$-0.383257\pi$
$673$	18.6054	0.717184	0.358592	+	0.933494i	$0.383257\pi$
$674$	0	0
$675$	13.4685	0.518404
$676$	0	0
$677$	31.7825	1.22150	0.610751	−	0.791823i	$-0.290868\pi$
$677$	31.7825	1.22150	0.610751	+	0.791823i	$0.290868\pi$
$678$	0	0
$679$	16.0597	0.616316
$680$	0	0
$681$	58.1661	2.22893
$682$	0	0
$683$	23.0817	0.883196	0.441598	−	0.897213i	$-0.354412\pi$
$683$	23.0817	0.883196	0.441598	+	0.897213i	$0.354412\pi$
$684$	0	0
$685$	−11.1881	−0.427474
$686$	0	0
$687$	−67.6686	−2.58172
$688$	0	0
$689$	−48.1153	−1.83305
$690$	0	0
$691$	−22.4796	−0.855164	−0.427582	−	0.903976i	$-0.640635\pi$
$691$	−22.4796	−0.855164	−0.427582	+	0.903976i	$0.640635\pi$
$692$	0	0
$693$	−39.4633	−1.49909
$694$	0	0
$695$	38.2830	1.45216
$696$	0	0
$697$	−9.75066	−0.369332
$698$	0	0
$699$	86.0733	3.25559
$700$	0	0
$701$	30.5570	1.15412	0.577061	−	0.816701i	$-0.304200\pi$
$701$	30.5570	1.15412	0.577061	+	0.816701i	$0.304200\pi$
$702$	0	0
$703$	−11.3420	−0.427773
$704$	0	0
$705$	−15.2940	−0.576006
$706$	0	0
$707$	−9.62163	−0.361859
$708$	0	0
$709$	44.9477	1.68805	0.844024	−	0.536306i	$-0.180181\pi$
$709$	44.9477	1.68805	0.844024	+	0.536306i	$0.180181\pi$
$710$	0	0
$711$	72.8742	2.73300
$712$	0	0
$713$	−11.4657	−0.429395
$714$	0	0
$715$	−35.9977	−1.34624
$716$	0	0
$717$	−31.2753	−1.16800
$718$	0	0
$719$	28.4208	1.05992	0.529959	−	0.848023i	$-0.322207\pi$
$719$	28.4208	1.05992	0.529959	+	0.848023i	$0.322207\pi$
$720$	0	0
$721$	−16.0506	−0.597757
$722$	0	0
$723$	−10.4479	−0.388562
$724$	0	0
$725$	−7.08954	−0.263299
$726$	0	0
$727$	5.26092	0.195117	0.0975583	−	0.995230i	$-0.468897\pi$
$727$	5.26092	0.195117	0.0975583	+	0.995230i	$0.468897\pi$
$728$	0	0
$729$	58.2429	2.15714
$730$	0	0
$731$	33.0437	1.22217
$732$	0	0
$733$	−43.5916	−1.61009	−0.805046	−	0.593212i	$-0.797860\pi$
$733$	−43.5916	−1.61009	−0.805046	+	0.593212i	$0.797860\pi$
$734$	0	0
$735$	−28.4352	−1.04885
$736$	0	0
$737$	4.46794	0.164579
$738$	0	0
$739$	−42.5955	−1.56690	−0.783450	−	0.621455i	$-0.786542\pi$
$739$	−42.5955	−1.56690	−0.783450	+	0.621455i	$0.786542\pi$
$740$	0	0
$741$	−102.766	−3.77520
$742$	0	0
$743$	9.80687	0.359779	0.179890	−	0.983687i	$-0.442426\pi$
$743$	9.80687	0.359779	0.179890	+	0.983687i	$0.442426\pi$
$744$	0	0
$745$	−21.6734	−0.794050
$746$	0	0
$747$	6.50626	0.238052
$748$	0	0
$749$	−11.9797	−0.437729
$750$	0	0
$751$	−10.5643	−0.385498	−0.192749	−	0.981248i	$-0.561740\pi$
$751$	−10.5643	−0.385498	−0.192749	+	0.981248i	$0.561740\pi$
$752$	0	0
$753$	3.27059	0.119187
$754$	0	0
$755$	−14.3900	−0.523704
$756$	0	0
$757$	20.0840	0.729967	0.364983	−	0.931014i	$-0.381075\pi$
$757$	20.0840	0.729967	0.364983	+	0.931014i	$0.381075\pi$
$758$	0	0
$759$	−21.9954	−0.798381
$760$	0	0
$761$	3.99080	0.144666	0.0723332	−	0.997381i	$-0.476955\pi$
$761$	3.99080	0.144666	0.0723332	+	0.997381i	$0.476955\pi$
$762$	0	0
$763$	−8.55404	−0.309677
$764$	0	0
$765$	−57.3040	−2.07183
$766$	0	0
$767$	−11.1672	−0.403226
$768$	0	0
$769$	36.1597	1.30395	0.651977	−	0.758239i	$-0.273940\pi$
$769$	36.1597	1.30395	0.651977	+	0.758239i	$0.273940\pi$
$770$	0	0
$771$	11.1591	0.401887
$772$	0	0
$773$	6.56902	0.236271	0.118136	−	0.992997i	$-0.462308\pi$
$773$	6.56902	0.236271	0.118136	+	0.992997i	$0.462308\pi$
$774$	0	0
$775$	4.64869	0.166986
$776$	0	0
$777$	−11.0960	−0.398066
$778$	0	0
$779$	−14.6580	−0.525177
$780$	0	0
$781$	47.3232	1.69336
$782$	0	0
$783$	−124.205	−4.43873
$784$	0	0
$785$	8.35117	0.298066
$786$	0	0
$787$	−29.4650	−1.05032	−0.525158	−	0.851005i	$-0.675994\pi$
$787$	−29.4650	−1.05032	−0.525158	+	0.851005i	$0.675994\pi$
$788$	0	0
$789$	−4.55376	−0.162118
$790$	0	0
$791$	29.0416	1.03260
$792$	0	0
$793$	63.8156	2.26616
$794$	0	0
$795$	−56.0537	−1.98802
$796$	0	0
$797$	6.50783	0.230519	0.115260	−	0.993335i	$-0.463230\pi$
$797$	6.50783	0.230519	0.115260	+	0.993335i	$0.463230\pi$
$798$	0	0
$799$	−8.44384	−0.298722
$800$	0	0
$801$	−47.5136	−1.67881
$802$	0	0
$803$	−3.96590	−0.139953
$804$	0	0
$805$	7.24000	0.255176
$806$	0	0
$807$	−8.77342	−0.308839
$808$	0	0
$809$	−53.0190	−1.86405	−0.932024	−	0.362396i	$-0.881959\pi$
$809$	−53.0190	−1.86405	−0.932024	+	0.362396i	$0.881959\pi$
$810$	0	0
$811$	−48.3467	−1.69768	−0.848841	−	0.528649i	$-0.822699\pi$
$811$	−48.3467	−1.69768	−0.848841	+	0.528649i	$0.822699\pi$
$812$	0	0
$813$	25.3572	0.889314
$814$	0	0
$815$	23.8615	0.835833
$816$	0	0
$817$	49.6740	1.73787
$818$	0	0
$819$	−72.3401	−2.52777
$820$	0	0
$821$	−21.8086	−0.761126	−0.380563	−	0.924755i	$-0.624270\pi$
$821$	−21.8086	−0.761126	−0.380563	+	0.924755i	$0.624270\pi$
$822$	0	0
$823$	−21.5799	−0.752228	−0.376114	−	0.926573i	$-0.622740\pi$
$823$	−21.5799	−0.752228	−0.376114	+	0.926573i	$0.622740\pi$
$824$	0	0
$825$	8.91786	0.310480
$826$	0	0
$827$	14.4586	0.502776	0.251388	−	0.967886i	$-0.419113\pi$
$827$	14.4586	0.502776	0.251388	+	0.967886i	$0.419113\pi$
$828$	0	0
$829$	27.7926	0.965278	0.482639	−	0.875819i	$-0.339678\pi$
$829$	27.7926	0.965278	0.482639	+	0.875819i	$0.339678\pi$
$830$	0	0
$831$	−41.2018	−1.42928
$832$	0	0
$833$	−15.6991	−0.543941
$834$	0	0
$835$	44.0150	1.52320
$836$	0	0
$837$	81.4427	2.81507
$838$	0	0
$839$	3.64173	0.125726	0.0628632	−	0.998022i	$-0.479977\pi$
$839$	3.64173	0.125726	0.0628632	+	0.998022i	$0.479977\pi$
$840$	0	0
$841$	36.3789	1.25445
$842$	0	0
$843$	−97.7319	−3.36606
$844$	0	0
$845$	−39.5900	−1.36194
$846$	0	0
$847$	2.19116	0.0752892
$848$	0	0
$849$	1.41265	0.0484819
$850$	0	0
$851$	−4.44999	−0.152544
$852$	0	0
$853$	−27.1135	−0.928348	−0.464174	−	0.885744i	$-0.653649\pi$
$853$	−27.1135	−0.928348	−0.464174	+	0.885744i	$0.653649\pi$
$854$	0	0
$855$	−86.1440	−2.94606
$856$	0	0
$857$	−27.9166	−0.953611	−0.476806	−	0.879009i	$-0.658205\pi$
$857$	−27.9166	−0.953611	−0.476806	+	0.879009i	$0.658205\pi$
$858$	0	0
$859$	15.0626	0.513930	0.256965	−	0.966421i	$-0.417277\pi$
$859$	15.0626	0.513930	0.256965	+	0.966421i	$0.417277\pi$
$860$	0	0
$861$	−14.3400	−0.488706
$862$	0	0
$863$	−25.5073	−0.868278	−0.434139	−	0.900846i	$-0.642947\pi$
$863$	−25.5073	−0.868278	−0.434139	+	0.900846i	$0.642947\pi$
$864$	0	0
$865$	−25.3523	−0.862003
$866$	0	0
$867$	11.6308	0.395002
$868$	0	0
$869$	29.4445	0.998835
$870$	0	0
$871$	8.19017	0.277513
$872$	0	0
$873$	−74.9710	−2.53738
$874$	0	0
$875$	−19.6748	−0.665128
$876$	0	0
$877$	−1.61233	−0.0544444	−0.0272222	−	0.999629i	$-0.508666\pi$
$877$	−1.61233	−0.0544444	−0.0272222	+	0.999629i	$0.508666\pi$
$878$	0	0
$879$	41.7607	1.40855
$880$	0	0
$881$	45.6924	1.53942	0.769708	−	0.638396i	$-0.220402\pi$
$881$	45.6924	1.53942	0.769708	+	0.638396i	$0.220402\pi$
$882$	0	0
$883$	45.1799	1.52042	0.760212	−	0.649675i	$-0.225095\pi$
$883$	45.1799	1.52042	0.760212	+	0.649675i	$0.225095\pi$
$884$	0	0
$885$	−13.0097	−0.437316
$886$	0	0
$887$	4.42422	0.148551	0.0742754	−	0.997238i	$-0.476336\pi$
$887$	4.42422	0.148551	0.0742754	+	0.997238i	$0.476336\pi$
$888$	0	0
$889$	−8.99018	−0.301521
$890$	0	0
$891$	84.4299	2.82851
$892$	0	0
$893$	−12.6935	−0.424771
$894$	0	0
$895$	42.1052	1.40742
$896$	0	0
$897$	−40.3197	−1.34624
$898$	0	0
$899$	−42.8697	−1.42978
$900$	0	0
$901$	−30.9473	−1.03100
$902$	0	0
$903$	48.5964	1.61719
$904$	0	0
$905$	18.4450	0.613132
$906$	0	0
$907$	4.03789	0.134076	0.0670380	−	0.997750i	$-0.478645\pi$
$907$	4.03789	0.134076	0.0670380	+	0.997750i	$0.478645\pi$
$908$	0	0
$909$	44.9162	1.48978
$910$	0	0
$911$	−17.3439	−0.574629	−0.287314	−	0.957836i	$-0.592762\pi$
$911$	−17.3439	−0.574629	−0.287314	+	0.957836i	$0.592762\pi$
$912$	0	0
$913$	2.62882	0.0870013
$914$	0	0
$915$	74.3443	2.45775
$916$	0	0
$917$	−21.6561	−0.715149
$918$	0	0
$919$	52.7790	1.74102	0.870510	−	0.492151i	$-0.163789\pi$
$919$	52.7790	1.74102	0.870510	+	0.492151i	$0.163789\pi$
$920$	0	0
$921$	−38.8976	−1.28172
$922$	0	0
$923$	86.7482	2.85535
$924$	0	0
$925$	1.80421	0.0593222
$926$	0	0
$927$	74.9285	2.46098
$928$	0	0
$929$	3.77741	0.123933	0.0619664	−	0.998078i	$-0.480263\pi$
$929$	3.77741	0.123933	0.0619664	+	0.998078i	$0.480263\pi$
$930$	0	0
$931$	−23.6002	−0.773464
$932$	0	0
$933$	103.446	3.38666
$934$	0	0
$935$	−23.1534	−0.757197
$936$	0	0
$937$	43.9367	1.43535	0.717674	−	0.696379i	$-0.245207\pi$
$937$	43.9367	1.43535	0.717674	+	0.696379i	$0.245207\pi$
$938$	0	0
$939$	−44.0043	−1.43603
$940$	0	0
$941$	−18.4606	−0.601799	−0.300900	−	0.953656i	$-0.597287\pi$
$941$	−18.4606	−0.601799	−0.300900	+	0.953656i	$0.597287\pi$
$942$	0	0
$943$	−5.75099	−0.187278
$944$	0	0
$945$	−51.4268	−1.67291
$946$	0	0
$947$	−25.9437	−0.843058	−0.421529	−	0.906815i	$-0.638506\pi$
$947$	−25.9437	−0.843058	−0.421529	+	0.906815i	$0.638506\pi$
$948$	0	0
$949$	−7.26988	−0.235990
$950$	0	0
$951$	24.2924	0.787736
$952$	0	0
$953$	−17.8588	−0.578504	−0.289252	−	0.957253i	$-0.593407\pi$
$953$	−17.8588	−0.578504	−0.289252	+	0.957253i	$0.593407\pi$
$954$	0	0
$955$	28.6271	0.926351
$956$	0	0
$957$	−82.2395	−2.65842
$958$	0	0
$959$	−9.08425	−0.293346
$960$	0	0
$961$	−2.88986	−0.0932212
$962$	0	0
$963$	55.9243	1.80214
$964$	0	0
$965$	−27.1260	−0.873216
$966$	0	0
$967$	−46.3753	−1.49133	−0.745665	−	0.666321i	$-0.767868\pi$
$967$	−46.3753	−1.49133	−0.745665	+	0.666321i	$0.767868\pi$
$968$	0	0
$969$	−66.0981	−2.12338
$970$	0	0
$971$	−27.0517	−0.868130	−0.434065	−	0.900882i	$-0.642921\pi$
$971$	−27.0517	−0.868130	−0.434065	+	0.900882i	$0.642921\pi$
$972$	0	0
$973$	31.0842	0.996514
$974$	0	0
$975$	16.3473	0.523533
$976$	0	0
$977$	−32.7363	−1.04733	−0.523664	−	0.851925i	$-0.675435\pi$
$977$	−32.7363	−1.04733	−0.523664	+	0.851925i	$0.675435\pi$
$978$	0	0
$979$	−19.1976	−0.613559
$980$	0	0
$981$	39.9325	1.27495
$982$	0	0
$983$	−39.5638	−1.26189	−0.630944	−	0.775828i	$-0.717332\pi$
$983$	−39.5638	−1.26189	−0.630944	+	0.775828i	$0.717332\pi$
$984$	0	0
$985$	18.4525	0.587944
$986$	0	0
$987$	−12.4181	−0.395272
$988$	0	0
$989$	19.4893	0.619725
$990$	0	0
$991$	−9.80219	−0.311377	−0.155688	−	0.987806i	$-0.549760\pi$
$991$	−9.80219	−0.311377	−0.155688	+	0.987806i	$0.549760\pi$
$992$	0	0
$993$	−73.6139	−2.33607
$994$	0	0
$995$	−7.83306	−0.248325
$996$	0	0
$997$	−7.21626	−0.228541	−0.114271	−	0.993450i	$-0.536453\pi$
$997$	−7.21626	−0.228541	−0.114271	+	0.993450i	$0.536453\pi$
$998$	0	0
$999$	31.6089	1.00006

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.1		17
4.3	odd	2		251.2.a.b.1.3	✓	17
12.11	even	2		2259.2.a.k.1.15		17
20.19	odd	2		6275.2.a.e.1.15		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.3	✓	17	4.3	odd	2
2259.2.a.k.1.15		17	12.11	even	2
4016.2.a.k.1.1		17	1.1	even	1	trivial
6275.2.a.e.1.15		17	20.19	odd	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)

\(f(q)\)	\(=\)	\(q-3.27059 q^{3} -2.03057 q^{5} -1.64874 q^{7} +7.69673 q^{9} +O(q^{10})\)	\(q-3.27059 q^{3} -2.03057 q^{5} -1.64874 q^{7} +7.69673 q^{9} +3.10982 q^{11} +5.70062 q^{13} +6.64114 q^{15} +3.66658 q^{17} +5.51191 q^{19} +5.39233 q^{21} +2.16257 q^{23} -0.876797 q^{25} -15.3610 q^{27} +8.08572 q^{29} -5.30190 q^{31} -10.1709 q^{33} +3.34787 q^{35} -2.05773 q^{37} -18.6444 q^{39} -2.65933 q^{41} +9.01212 q^{43} -15.6287 q^{45} -2.30292 q^{47} -4.28167 q^{49} -11.9919 q^{51} -8.44037 q^{53} -6.31471 q^{55} -18.0272 q^{57} -1.95895 q^{59} +11.1945 q^{61} -12.6899 q^{63} -11.5755 q^{65} +1.43672 q^{67} -7.07287 q^{69} +15.2173 q^{71} -1.27528 q^{73} +2.86764 q^{75} -5.12728 q^{77} +9.46821 q^{79} +27.1494 q^{81} +0.845328 q^{83} -7.44524 q^{85} -26.4451 q^{87} -6.17322 q^{89} -9.39881 q^{91} +17.3403 q^{93} -11.1923 q^{95} -9.74063 q^{97} +23.9355 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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