LMFDB - Embedded newform 4016.2.a.l.1.14

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:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.01204$ 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)

$f(q)$	$=$	$q+2.01204 q^{3} +3.16436 q^{5} +2.95084 q^{7} +1.04829 q^{9} +O(q^{10})$	$q+2.01204 q^{3} +3.16436 q^{5} +2.95084 q^{7} +1.04829 q^{9} +2.62744 q^{11} +5.88020 q^{13} +6.36681 q^{15} -6.15607 q^{17} -6.44957 q^{19} +5.93719 q^{21} +4.28119 q^{23} +5.01318 q^{25} -3.92692 q^{27} -4.42819 q^{29} +4.31071 q^{31} +5.28650 q^{33} +9.33751 q^{35} +7.32618 q^{37} +11.8312 q^{39} -6.77263 q^{41} +0.439552 q^{43} +3.31716 q^{45} +6.08009 q^{47} +1.70744 q^{49} -12.3862 q^{51} -10.8340 q^{53} +8.31417 q^{55} -12.9768 q^{57} +9.72474 q^{59} +13.3895 q^{61} +3.09332 q^{63} +18.6071 q^{65} -3.57840 q^{67} +8.61391 q^{69} -15.9495 q^{71} -1.54826 q^{73} +10.0867 q^{75} +7.75315 q^{77} -1.70810 q^{79} -11.0460 q^{81} +10.4706 q^{83} -19.4800 q^{85} -8.90967 q^{87} -9.71911 q^{89} +17.3515 q^{91} +8.67329 q^{93} -20.4088 q^{95} -8.18248 q^{97} +2.75431 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * 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
$3$	2.01204	1.16165	0.580825	−	0.814029i	$-0.302730\pi$
$3$	2.01204	1.16165	0.580825	+	0.814029i	$0.302730\pi$
$4$	0	0
$5$	3.16436	1.41515	0.707573	−	0.706641i	$-0.249790\pi$
$5$	3.16436	1.41515	0.707573	+	0.706641i	$0.249790\pi$
$6$	0	0
$7$	2.95084	1.11531	0.557656	−	0.830072i	$-0.311701\pi$
$7$	2.95084	1.11531	0.557656	+	0.830072i	$0.311701\pi$
$8$	0	0
$9$	1.04829	0.349429
$10$	0	0
$11$	2.62744	0.792203	0.396101	−	0.918207i	$-0.370363\pi$
$11$	2.62744	0.792203	0.396101	+	0.918207i	$0.370363\pi$
$12$	0	0
$13$	5.88020	1.63087	0.815437	−	0.578845i	$-0.196496\pi$
$13$	5.88020	1.63087	0.815437	+	0.578845i	$0.196496\pi$
$14$	0	0
$15$	6.36681	1.64390
$16$	0	0
$17$	−6.15607	−1.49307	−0.746533	−	0.665349i	$-0.768283\pi$
$17$	−6.15607	−1.49307	−0.746533	+	0.665349i	$0.768283\pi$
$18$	0	0
$19$	−6.44957	−1.47963	−0.739816	−	0.672809i	$-0.765088\pi$
$19$	−6.44957	−1.47963	−0.739816	+	0.672809i	$0.765088\pi$
$20$	0	0
$21$	5.93719	1.29560
$22$	0	0
$23$	4.28119	0.892690	0.446345	−	0.894861i	$-0.352725\pi$
$23$	4.28119	0.892690	0.446345	+	0.894861i	$0.352725\pi$
$24$	0	0
$25$	5.01318	1.00264
$26$	0	0
$27$	−3.92692	−0.755735
$28$	0	0
$29$	−4.42819	−0.822294	−0.411147	−	0.911569i	$-0.634872\pi$
$29$	−4.42819	−0.822294	−0.411147	+	0.911569i	$0.634872\pi$
$30$	0	0
$31$	4.31071	0.774226	0.387113	−	0.922032i	$-0.373472\pi$
$31$	4.31071	0.774226	0.387113	+	0.922032i	$0.373472\pi$
$32$	0	0
$33$	5.28650	0.920262
$34$	0	0
$35$	9.33751	1.57833
$36$	0	0
$37$	7.32618	1.20442	0.602208	−	0.798339i	$-0.294288\pi$
$37$	7.32618	1.20442	0.602208	+	0.798339i	$0.294288\pi$
$38$	0	0
$39$	11.8312	1.89450
$40$	0	0
$41$	−6.77263	−1.05771	−0.528853	−	0.848713i	$-0.677378\pi$
$41$	−6.77263	−1.05771	−0.528853	+	0.848713i	$0.677378\pi$
$42$	0	0
$43$	0.439552	0.0670311	0.0335156	−	0.999438i	$-0.489330\pi$
$43$	0.439552	0.0670311	0.0335156	+	0.999438i	$0.489330\pi$
$44$	0	0
$45$	3.31716	0.494493
$46$	0	0
$47$	6.08009	0.886873	0.443436	−	0.896306i	$-0.353759\pi$
$47$	6.08009	0.886873	0.443436	+	0.896306i	$0.353759\pi$
$48$	0	0
$49$	1.70744	0.243920
$50$	0	0
$51$	−12.3862	−1.73442
$52$	0	0
$53$	−10.8340	−1.48816	−0.744082	−	0.668088i	$-0.767113\pi$
$53$	−10.8340	−1.48816	−0.744082	+	0.668088i	$0.767113\pi$
$54$	0	0
$55$	8.31417	1.12108
$56$	0	0
$57$	−12.9768	−1.71881
$58$	0	0
$59$	9.72474	1.26605	0.633027	−	0.774130i	$-0.281812\pi$
$59$	9.72474	1.26605	0.633027	+	0.774130i	$0.281812\pi$
$60$	0	0
$61$	13.3895	1.71435	0.857177	−	0.515021i	$-0.172216\pi$
$61$	13.3895	1.71435	0.857177	+	0.515021i	$0.172216\pi$
$62$	0	0
$63$	3.09332	0.389722
$64$	0	0
$65$	18.6071	2.30792
$66$	0	0
$67$	−3.57840	−0.437171	−0.218586	−	0.975818i	$-0.570144\pi$
$67$	−3.57840	−0.437171	−0.218586	+	0.975818i	$0.570144\pi$
$68$	0	0
$69$	8.61391	1.03699
$70$	0	0
$71$	−15.9495	−1.89286	−0.946430	−	0.322910i	$-0.895339\pi$
$71$	−15.9495	−1.89286	−0.946430	+	0.322910i	$0.895339\pi$
$72$	0	0
$73$	−1.54826	−0.181210	−0.0906051	−	0.995887i	$-0.528880\pi$
$73$	−1.54826	−0.181210	−0.0906051	+	0.995887i	$0.528880\pi$
$74$	0	0
$75$	10.0867	1.16471
$76$	0	0
$77$	7.75315	0.883553
$78$	0	0
$79$	−1.70810	−0.192177	−0.0960884	−	0.995373i	$-0.530633\pi$
$79$	−1.70810	−0.192177	−0.0960884	+	0.995373i	$0.530633\pi$
$80$	0	0
$81$	−11.0460	−1.22733
$82$	0	0
$83$	10.4706	1.14930	0.574651	−	0.818399i	$-0.305138\pi$
$83$	10.4706	1.14930	0.574651	+	0.818399i	$0.305138\pi$
$84$	0	0
$85$	−19.4800	−2.11290
$86$	0	0
$87$	−8.90967	−0.955217
$88$	0	0
$89$	−9.71911	−1.03022	−0.515112	−	0.857123i	$-0.672250\pi$
$89$	−9.71911	−1.03022	−0.515112	+	0.857123i	$0.672250\pi$
$90$	0	0
$91$	17.3515	1.81893
$92$	0	0
$93$	8.67329	0.899379
$94$	0	0
$95$	−20.4088	−2.09390
$96$	0	0
$97$	−8.18248	−0.830805	−0.415402	−	0.909638i	$-0.636359\pi$
$97$	−8.18248	−0.830805	−0.415402	+	0.909638i	$0.636359\pi$
$98$	0	0
$99$	2.75431	0.276819
$100$	0	0
$101$	6.25229	0.622126	0.311063	−	0.950389i	$-0.399315\pi$
$101$	6.25229	0.622126	0.311063	+	0.950389i	$0.399315\pi$
$102$	0	0
$103$	4.89164	0.481988	0.240994	−	0.970527i	$-0.422527\pi$
$103$	4.89164	0.481988	0.240994	+	0.970527i	$0.422527\pi$
$104$	0	0
$105$	18.7874	1.83346
$106$	0	0
$107$	−10.9492	−1.05850	−0.529251	−	0.848465i	$-0.677527\pi$
$107$	−10.9492	−1.05850	−0.529251	+	0.848465i	$0.677527\pi$
$108$	0	0
$109$	−18.6950	−1.79066	−0.895329	−	0.445406i	$-0.853060\pi$
$109$	−18.6950	−1.79066	−0.895329	+	0.445406i	$0.853060\pi$
$110$	0	0
$111$	14.7405	1.39911
$112$	0	0
$113$	−11.4149	−1.07383	−0.536913	−	0.843637i	$-0.680410\pi$
$113$	−11.4149	−1.07383	−0.536913	+	0.843637i	$0.680410\pi$
$114$	0	0
$115$	13.5472	1.26329
$116$	0	0
$117$	6.16414	0.569875
$118$	0	0
$119$	−18.1656	−1.66523
$120$	0	0
$121$	−4.09656	−0.372414
$122$	0	0
$123$	−13.6268	−1.22868
$124$	0	0
$125$	0.0417040	0.00373012
$126$	0	0
$127$	13.4439	1.19295	0.596475	−	0.802632i	$-0.296568\pi$
$127$	13.4439	1.19295	0.596475	+	0.802632i	$0.296568\pi$
$128$	0	0
$129$	0.884395	0.0778666
$130$	0	0
$131$	22.1912	1.93885	0.969425	−	0.245387i	$-0.0789149\pi$
$131$	22.1912	1.93885	0.969425	+	0.245387i	$0.0789149\pi$
$132$	0	0
$133$	−19.0316	−1.65025
$134$	0	0
$135$	−12.4262	−1.06948
$136$	0	0
$137$	−11.4989	−0.982416	−0.491208	−	0.871042i	$-0.663445\pi$
$137$	−11.4989	−0.982416	−0.491208	+	0.871042i	$0.663445\pi$
$138$	0	0
$139$	−5.60160	−0.475122	−0.237561	−	0.971373i	$-0.576348\pi$
$139$	−5.60160	−0.475122	−0.237561	+	0.971373i	$0.576348\pi$
$140$	0	0
$141$	12.2334	1.03023
$142$	0	0
$143$	15.4499	1.29198
$144$	0	0
$145$	−14.0124	−1.16367
$146$	0	0
$147$	3.43544	0.283350
$148$	0	0
$149$	−17.8182	−1.45973	−0.729863	−	0.683593i	$-0.760416\pi$
$149$	−17.8182	−1.45973	−0.729863	+	0.683593i	$0.760416\pi$
$150$	0	0
$151$	3.80576	0.309708	0.154854	−	0.987937i	$-0.450509\pi$
$151$	3.80576	0.309708	0.154854	+	0.987937i	$0.450509\pi$
$152$	0	0
$153$	−6.45332	−0.521720
$154$	0	0
$155$	13.6406	1.09564
$156$	0	0
$157$	−3.37788	−0.269584	−0.134792	−	0.990874i	$-0.543037\pi$
$157$	−3.37788	−0.269584	−0.134792	+	0.990874i	$0.543037\pi$
$158$	0	0
$159$	−21.7984	−1.72873
$160$	0	0
$161$	12.6331	0.995628
$162$	0	0
$163$	11.4601	0.897624	0.448812	−	0.893626i	$-0.351847\pi$
$163$	11.4601	0.897624	0.448812	+	0.893626i	$0.351847\pi$
$164$	0	0
$165$	16.7284	1.30230
$166$	0	0
$167$	19.9212	1.54155	0.770773	−	0.637110i	$-0.219870\pi$
$167$	19.9212	1.54155	0.770773	+	0.637110i	$0.219870\pi$
$168$	0	0
$169$	21.5768	1.65975
$170$	0	0
$171$	−6.76100	−0.517026
$172$	0	0
$173$	12.1864	0.926511	0.463256	−	0.886225i	$-0.346681\pi$
$173$	12.1864	0.926511	0.463256	+	0.886225i	$0.346681\pi$
$174$	0	0
$175$	14.7931	1.11825
$176$	0	0
$177$	19.5665	1.47071
$178$	0	0
$179$	−3.38298	−0.252856	−0.126428	−	0.991976i	$-0.540351\pi$
$179$	−3.38298	−0.252856	−0.126428	+	0.991976i	$0.540351\pi$
$180$	0	0
$181$	22.5465	1.67587	0.837933	−	0.545773i	$-0.183764\pi$
$181$	22.5465	1.67587	0.837933	+	0.545773i	$0.183764\pi$
$182$	0	0
$183$	26.9402	1.99148
$184$	0	0
$185$	23.1827	1.70442
$186$	0	0
$187$	−16.1747	−1.18281
$188$	0	0
$189$	−11.5877	−0.842881
$190$	0	0
$191$	−24.4309	−1.76776	−0.883881	−	0.467712i	$-0.845078\pi$
$191$	−24.4309	−1.76776	−0.883881	+	0.467712i	$0.845078\pi$
$192$	0	0
$193$	−17.5240	−1.26140	−0.630702	−	0.776025i	$-0.717233\pi$
$193$	−17.5240	−1.26140	−0.630702	+	0.776025i	$0.717233\pi$
$194$	0	0
$195$	37.4381	2.68100
$196$	0	0
$197$	4.50280	0.320811	0.160406	−	0.987051i	$-0.448720\pi$
$197$	4.50280	0.320811	0.160406	+	0.987051i	$0.448720\pi$
$198$	0	0
$199$	−14.0484	−0.995864	−0.497932	−	0.867216i	$-0.665907\pi$
$199$	−14.0484	−0.995864	−0.497932	+	0.867216i	$0.665907\pi$
$200$	0	0
$201$	−7.19987	−0.507839
$202$	0	0
$203$	−13.0669	−0.917114
$204$	0	0
$205$	−21.4310	−1.49681
$206$	0	0
$207$	4.48791	0.311932
$208$	0	0
$209$	−16.9459	−1.17217
$210$	0	0
$211$	−1.46319	−0.100730	−0.0503651	−	0.998731i	$-0.516038\pi$
$211$	−1.46319	−0.100730	−0.0503651	+	0.998731i	$0.516038\pi$
$212$	0	0
$213$	−32.0910	−2.19884
$214$	0	0
$215$	1.39090	0.0948587
$216$	0	0
$217$	12.7202	0.863503
$218$	0	0
$219$	−3.11516	−0.210503
$220$	0	0
$221$	−36.1989	−2.43500
$222$	0	0
$223$	13.5196	0.905337	0.452668	−	0.891679i	$-0.350472\pi$
$223$	13.5196	0.905337	0.452668	+	0.891679i	$0.350472\pi$
$224$	0	0
$225$	5.25525	0.350350
$226$	0	0
$227$	6.08153	0.403645	0.201823	−	0.979422i	$-0.435314\pi$
$227$	6.08153	0.403645	0.201823	+	0.979422i	$0.435314\pi$
$228$	0	0
$229$	3.42676	0.226447	0.113223	−	0.993570i	$-0.463882\pi$
$229$	3.42676	0.226447	0.113223	+	0.993570i	$0.463882\pi$
$230$	0	0
$231$	15.5996	1.02638
$232$	0	0
$233$	20.1228	1.31829	0.659145	−	0.752016i	$-0.270918\pi$
$233$	20.1228	1.31829	0.659145	+	0.752016i	$0.270918\pi$
$234$	0	0
$235$	19.2396	1.25505
$236$	0	0
$237$	−3.43677	−0.223242
$238$	0	0
$239$	−17.0968	−1.10590	−0.552950	−	0.833214i	$-0.686498\pi$
$239$	−17.0968	−1.10590	−0.552950	+	0.833214i	$0.686498\pi$
$240$	0	0
$241$	−8.88898	−0.572590	−0.286295	−	0.958142i	$-0.592424\pi$
$241$	−8.88898	−0.572590	−0.286295	+	0.958142i	$0.592424\pi$
$242$	0	0
$243$	−10.4441	−0.669990
$244$	0	0
$245$	5.40296	0.345183
$246$	0	0
$247$	−37.9248	−2.41310
$248$	0	0
$249$	21.0673	1.33509
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	11.2486	0.707192
$254$	0	0
$255$	−39.1945	−2.45445
$256$	0	0
$257$	−17.6296	−1.09971	−0.549853	−	0.835262i	$-0.685316\pi$
$257$	−17.6296	−1.09971	−0.549853	+	0.835262i	$0.685316\pi$
$258$	0	0
$259$	21.6184	1.34330
$260$	0	0
$261$	−4.64201	−0.287333
$262$	0	0
$263$	2.40575	0.148345	0.0741724	−	0.997245i	$-0.476368\pi$
$263$	2.40575	0.148345	0.0741724	+	0.997245i	$0.476368\pi$
$264$	0	0
$265$	−34.2827	−2.10597
$266$	0	0
$267$	−19.5552	−1.19676
$268$	0	0
$269$	4.88329	0.297739	0.148870	−	0.988857i	$-0.452437\pi$
$269$	4.88329	0.297739	0.148870	+	0.988857i	$0.452437\pi$
$270$	0	0
$271$	15.4719	0.939853	0.469926	−	0.882706i	$-0.344280\pi$
$271$	15.4719	0.939853	0.469926	+	0.882706i	$0.344280\pi$
$272$	0	0
$273$	34.9119	2.11296
$274$	0	0
$275$	13.1718	0.794291
$276$	0	0
$277$	−13.2852	−0.798231	−0.399115	−	0.916901i	$-0.630683\pi$
$277$	−13.2852	−0.798231	−0.399115	+	0.916901i	$0.630683\pi$
$278$	0	0
$279$	4.51886	0.270537
$280$	0	0
$281$	2.14736	0.128101	0.0640505	−	0.997947i	$-0.479598\pi$
$281$	2.14736	0.128101	0.0640505	+	0.997947i	$0.479598\pi$
$282$	0	0
$283$	−0.948321	−0.0563718	−0.0281859	−	0.999603i	$-0.508973\pi$
$283$	−0.948321	−0.0563718	−0.0281859	+	0.999603i	$0.508973\pi$
$284$	0	0
$285$	−41.0632	−2.43237
$286$	0	0
$287$	−19.9849	−1.17967
$288$	0	0
$289$	20.8972	1.22925
$290$	0	0
$291$	−16.4634	−0.965104
$292$	0	0
$293$	10.2997	0.601716	0.300858	−	0.953669i	$-0.402727\pi$
$293$	10.2997	0.601716	0.300858	+	0.953669i	$0.402727\pi$
$294$	0	0
$295$	30.7726	1.79165
$296$	0	0
$297$	−10.3177	−0.598696
$298$	0	0
$299$	25.1743	1.45587
$300$	0	0
$301$	1.29705	0.0747606
$302$	0	0
$303$	12.5798	0.722692
$304$	0	0
$305$	42.3693	2.42606
$306$	0	0
$307$	22.9877	1.31198	0.655989	−	0.754770i	$-0.272252\pi$
$307$	22.9877	1.31198	0.655989	+	0.754770i	$0.272252\pi$
$308$	0	0
$309$	9.84216	0.559901
$310$	0	0
$311$	−19.8510	−1.12565	−0.562824	−	0.826577i	$-0.690285\pi$
$311$	−19.8510	−1.12565	−0.562824	+	0.826577i	$0.690285\pi$
$312$	0	0
$313$	−18.7140	−1.05778	−0.528889	−	0.848691i	$-0.677391\pi$
$313$	−18.7140	−1.05778	−0.528889	+	0.848691i	$0.677391\pi$
$314$	0	0
$315$	9.78839	0.551513
$316$	0	0
$317$	17.5512	0.985773	0.492886	−	0.870094i	$-0.335942\pi$
$317$	17.5512	0.985773	0.492886	+	0.870094i	$0.335942\pi$
$318$	0	0
$319$	−11.6348	−0.651424
$320$	0	0
$321$	−22.0302	−1.22961
$322$	0	0
$323$	39.7040	2.20919
$324$	0	0
$325$	29.4785	1.63517
$326$	0	0
$327$	−37.6150	−2.08012
$328$	0	0
$329$	17.9414	0.989140
$330$	0	0
$331$	−18.1573	−0.998016	−0.499008	−	0.866597i	$-0.666302\pi$
$331$	−18.1573	−0.998016	−0.499008	+	0.866597i	$0.666302\pi$
$332$	0	0
$333$	7.67993	0.420858
$334$	0	0
$335$	−11.3233	−0.618660
$336$	0	0
$337$	22.8639	1.24548	0.622738	−	0.782430i	$-0.286020\pi$
$337$	22.8639	1.24548	0.622738	+	0.782430i	$0.286020\pi$
$338$	0	0
$339$	−22.9673	−1.24741
$340$	0	0
$341$	11.3261	0.613344
$342$	0	0
$343$	−15.6175	−0.843265
$344$	0	0
$345$	27.2575	1.46750
$346$	0	0
$347$	−1.34409	−0.0721543	−0.0360772	−	0.999349i	$-0.511486\pi$
$347$	−1.34409	−0.0721543	−0.0360772	+	0.999349i	$0.511486\pi$
$348$	0	0
$349$	−23.3875	−1.25190	−0.625952	−	0.779861i	$-0.715290\pi$
$349$	−23.3875	−1.25190	−0.625952	+	0.779861i	$0.715290\pi$
$350$	0	0
$351$	−23.0911	−1.23251
$352$	0	0
$353$	12.5466	0.667787	0.333893	−	0.942611i	$-0.391637\pi$
$353$	12.5466	0.667787	0.333893	+	0.942611i	$0.391637\pi$
$354$	0	0
$355$	−50.4700	−2.67867
$356$	0	0
$357$	−36.5497	−1.93442
$358$	0	0
$359$	0.608654	0.0321235	0.0160618	−	0.999871i	$-0.494887\pi$
$359$	0.608654	0.0321235	0.0160618	+	0.999871i	$0.494887\pi$
$360$	0	0
$361$	22.5969	1.18931
$362$	0	0
$363$	−8.24242	−0.432615
$364$	0	0
$365$	−4.89926	−0.256439
$366$	0	0
$367$	−12.1596	−0.634728	−0.317364	−	0.948304i	$-0.602798\pi$
$367$	−12.1596	−0.634728	−0.317364	+	0.948304i	$0.602798\pi$
$368$	0	0
$369$	−7.09965	−0.369593
$370$	0	0
$371$	−31.9694	−1.65977
$372$	0	0
$373$	35.5980	1.84320	0.921598	−	0.388146i	$-0.126884\pi$
$373$	35.5980	1.84320	0.921598	+	0.388146i	$0.126884\pi$
$374$	0	0
$375$	0.0839099	0.00433309
$376$	0	0
$377$	−26.0387	−1.34106
$378$	0	0
$379$	−32.6835	−1.67884	−0.839419	−	0.543484i	$-0.817105\pi$
$379$	−32.6835	−1.67884	−0.839419	+	0.543484i	$0.817105\pi$
$380$	0	0
$381$	27.0495	1.38579
$382$	0	0
$383$	34.2548	1.75034	0.875170	−	0.483815i	$-0.160749\pi$
$383$	34.2548	1.75034	0.875170	+	0.483815i	$0.160749\pi$
$384$	0	0
$385$	24.5338	1.25036
$386$	0	0
$387$	0.460777	0.0234226
$388$	0	0
$389$	−8.95740	−0.454158	−0.227079	−	0.973876i	$-0.572918\pi$
$389$	−8.95740	−0.454158	−0.227079	+	0.973876i	$0.572918\pi$
$390$	0	0
$391$	−26.3553	−1.33284
$392$	0	0
$393$	44.6494	2.25226
$394$	0	0
$395$	−5.40506	−0.271958
$396$	0	0
$397$	1.84443	0.0925692	0.0462846	−	0.998928i	$-0.485262\pi$
$397$	1.84443	0.0925692	0.0462846	+	0.998928i	$0.485262\pi$
$398$	0	0
$399$	−38.2923	−1.91701
$400$	0	0
$401$	−8.03342	−0.401170	−0.200585	−	0.979676i	$-0.564284\pi$
$401$	−8.03342	−0.401170	−0.200585	+	0.979676i	$0.564284\pi$
$402$	0	0
$403$	25.3478	1.26267
$404$	0	0
$405$	−34.9534	−1.73685
$406$	0	0
$407$	19.2491	0.954142
$408$	0	0
$409$	30.4925	1.50776	0.753878	−	0.657015i	$-0.228181\pi$
$409$	30.4925	1.50776	0.753878	+	0.657015i	$0.228181\pi$
$410$	0	0
$411$	−23.1362	−1.14122
$412$	0	0
$413$	28.6961	1.41204
$414$	0	0
$415$	33.1329	1.62643
$416$	0	0
$417$	−11.2706	−0.551925
$418$	0	0
$419$	−33.2087	−1.62235	−0.811176	−	0.584802i	$-0.801172\pi$
$419$	−33.2087	−1.62235	−0.811176	+	0.584802i	$0.801172\pi$
$420$	0	0
$421$	25.8726	1.26095	0.630477	−	0.776208i	$-0.282859\pi$
$421$	25.8726	1.26095	0.630477	+	0.776208i	$0.282859\pi$
$422$	0	0
$423$	6.37368	0.309899
$424$	0	0
$425$	−30.8615	−1.49700
$426$	0	0
$427$	39.5103	1.91204
$428$	0	0
$429$	31.0857	1.50083
$430$	0	0
$431$	5.35981	0.258173	0.129086	−	0.991633i	$-0.458796\pi$
$431$	5.35981	0.258173	0.129086	+	0.991633i	$0.458796\pi$
$432$	0	0
$433$	−11.2952	−0.542811	−0.271406	−	0.962465i	$-0.587488\pi$
$433$	−11.2952	−0.542811	−0.271406	+	0.962465i	$0.587488\pi$
$434$	0	0
$435$	−28.1934	−1.35177
$436$	0	0
$437$	−27.6118	−1.32085
$438$	0	0
$439$	32.6636	1.55895	0.779474	−	0.626435i	$-0.215487\pi$
$439$	32.6636	1.55895	0.779474	+	0.626435i	$0.215487\pi$
$440$	0	0
$441$	1.78989	0.0852328
$442$	0	0
$443$	−16.5076	−0.784300	−0.392150	−	0.919901i	$-0.628269\pi$
$443$	−16.5076	−0.784300	−0.392150	+	0.919901i	$0.628269\pi$
$444$	0	0
$445$	−30.7548	−1.45792
$446$	0	0
$447$	−35.8509	−1.69569
$448$	0	0
$449$	18.4587	0.871119	0.435560	−	0.900160i	$-0.356551\pi$
$449$	18.4587	0.871119	0.435560	+	0.900160i	$0.356551\pi$
$450$	0	0
$451$	−17.7947	−0.837918
$452$	0	0
$453$	7.65732	0.359772
$454$	0	0
$455$	54.9065	2.57406
$456$	0	0
$457$	−21.7711	−1.01841	−0.509204	−	0.860646i	$-0.670060\pi$
$457$	−21.7711	−1.01841	−0.509204	+	0.860646i	$0.670060\pi$
$458$	0	0
$459$	24.1744	1.12836
$460$	0	0
$461$	34.1065	1.58850	0.794249	−	0.607593i	$-0.207865\pi$
$461$	34.1065	1.58850	0.794249	+	0.607593i	$0.207865\pi$
$462$	0	0
$463$	17.7876	0.826658	0.413329	−	0.910582i	$-0.364366\pi$
$463$	17.7876	0.826658	0.413329	+	0.910582i	$0.364366\pi$
$464$	0	0
$465$	27.4454	1.27275
$466$	0	0
$467$	8.68690	0.401982	0.200991	−	0.979593i	$-0.435584\pi$
$467$	8.68690	0.401982	0.200991	+	0.979593i	$0.435584\pi$
$468$	0	0
$469$	−10.5593	−0.487582
$470$	0	0
$471$	−6.79641	−0.313162
$472$	0	0
$473$	1.15490	0.0531022
$474$	0	0
$475$	−32.3328	−1.48353
$476$	0	0
$477$	−11.3571	−0.520008
$478$	0	0
$479$	35.4566	1.62005	0.810026	−	0.586394i	$-0.199453\pi$
$479$	35.4566	1.62005	0.810026	+	0.586394i	$0.199453\pi$
$480$	0	0
$481$	43.0794	1.96425
$482$	0	0
$483$	25.4182	1.15657
$484$	0	0
$485$	−25.8923	−1.17571
$486$	0	0
$487$	−8.39717	−0.380512	−0.190256	−	0.981734i	$-0.560932\pi$
$487$	−8.39717	−0.380512	−0.190256	+	0.981734i	$0.560932\pi$
$488$	0	0
$489$	23.0581	1.04272
$490$	0	0
$491$	−8.64831	−0.390293	−0.195146	−	0.980774i	$-0.562518\pi$
$491$	−8.64831	−0.390293	−0.195146	+	0.980774i	$0.562518\pi$
$492$	0	0
$493$	27.2602	1.22774
$494$	0	0
$495$	8.71563	0.391738
$496$	0	0
$497$	−47.0644	−2.11113
$498$	0	0
$499$	−10.5235	−0.471096	−0.235548	−	0.971863i	$-0.575688\pi$
$499$	−10.5235	−0.471096	−0.235548	+	0.971863i	$0.575688\pi$
$500$	0	0
$501$	40.0821	1.79073
$502$	0	0
$503$	−34.8055	−1.55190	−0.775950	−	0.630795i	$-0.782729\pi$
$503$	−34.8055	−1.55190	−0.775950	+	0.630795i	$0.782729\pi$
$504$	0	0
$505$	19.7845	0.880399
$506$	0	0
$507$	43.4133	1.92805
$508$	0	0
$509$	−2.93607	−0.130139	−0.0650694	−	0.997881i	$-0.520727\pi$
$509$	−2.93607	−0.130139	−0.0650694	+	0.997881i	$0.520727\pi$
$510$	0	0
$511$	−4.56867	−0.202106
$512$	0	0
$513$	25.3269	1.11821
$514$	0	0
$515$	15.4789	0.682083
$516$	0	0
$517$	15.9751	0.702583
$518$	0	0
$519$	24.5194	1.07628
$520$	0	0
$521$	−26.2919	−1.15187	−0.575934	−	0.817496i	$-0.695362\pi$
$521$	−26.2919	−1.15187	−0.575934	+	0.817496i	$0.695362\pi$
$522$	0	0
$523$	18.5515	0.811198	0.405599	−	0.914051i	$-0.367063\pi$
$523$	18.5515	0.811198	0.405599	+	0.914051i	$0.367063\pi$
$524$	0	0
$525$	29.7642	1.29902
$526$	0	0
$527$	−26.5370	−1.15597
$528$	0	0
$529$	−4.67140	−0.203105
$530$	0	0
$531$	10.1943	0.442396
$532$	0	0
$533$	−39.8244	−1.72499
$534$	0	0
$535$	−34.6473	−1.49793
$536$	0	0
$537$	−6.80667	−0.293729
$538$	0	0
$539$	4.48620	0.193234
$540$	0	0
$541$	−1.26785	−0.0545090	−0.0272545	−	0.999629i	$-0.508676\pi$
$541$	−1.26785	−0.0545090	−0.0272545	+	0.999629i	$0.508676\pi$
$542$	0	0
$543$	45.3643	1.94677
$544$	0	0
$545$	−59.1578	−2.53404
$546$	0	0
$547$	2.03480	0.0870019	0.0435009	−	0.999053i	$-0.486149\pi$
$547$	2.03480	0.0870019	0.0435009	+	0.999053i	$0.486149\pi$
$548$	0	0
$549$	14.0361	0.599045
$550$	0	0
$551$	28.5599	1.21669
$552$	0	0
$553$	−5.04034	−0.214337
$554$	0	0
$555$	46.6444	1.97994
$556$	0	0
$557$	30.6299	1.29783	0.648916	−	0.760860i	$-0.275223\pi$
$557$	30.6299	1.29783	0.648916	+	0.760860i	$0.275223\pi$
$558$	0	0
$559$	2.58466	0.109319
$560$	0	0
$561$	−32.5441	−1.37401
$562$	0	0
$563$	−4.22200	−0.177936	−0.0889680	−	0.996034i	$-0.528357\pi$
$563$	−4.22200	−0.177936	−0.0889680	+	0.996034i	$0.528357\pi$
$564$	0	0
$565$	−36.1210	−1.51962
$566$	0	0
$567$	−32.5948	−1.36885
$568$	0	0
$569$	−12.7293	−0.533639	−0.266819	−	0.963747i	$-0.585973\pi$
$569$	−12.7293	−0.533639	−0.266819	+	0.963747i	$0.585973\pi$
$570$	0	0
$571$	9.57600	0.400743	0.200372	−	0.979720i	$-0.435785\pi$
$571$	9.57600	0.400743	0.200372	+	0.979720i	$0.435785\pi$
$572$	0	0
$573$	−49.1559	−2.05352
$574$	0	0
$575$	21.4624	0.895043
$576$	0	0
$577$	36.2390	1.50865	0.754326	−	0.656501i	$-0.227964\pi$
$577$	36.2390	1.50865	0.754326	+	0.656501i	$0.227964\pi$
$578$	0	0
$579$	−35.2589	−1.46531
$580$	0	0
$581$	30.8972	1.28183
$582$	0	0
$583$	−28.4657	−1.17893
$584$	0	0
$585$	19.5056	0.806455
$586$	0	0
$587$	26.2995	1.08550	0.542749	−	0.839895i	$-0.317384\pi$
$587$	26.2995	1.08550	0.542749	+	0.839895i	$0.317384\pi$
$588$	0	0
$589$	−27.8022	−1.14557
$590$	0	0
$591$	9.05979	0.372670
$592$	0	0
$593$	22.9101	0.940803	0.470402	−	0.882452i	$-0.344109\pi$
$593$	22.9101	0.940803	0.470402	+	0.882452i	$0.344109\pi$
$594$	0	0
$595$	−57.4824	−2.35655
$596$	0	0
$597$	−28.2659	−1.15684
$598$	0	0
$599$	−3.69697	−0.151054	−0.0755271	−	0.997144i	$-0.524064\pi$
$599$	−3.69697	−0.151054	−0.0755271	+	0.997144i	$0.524064\pi$
$600$	0	0
$601$	32.3111	1.31800	0.658999	−	0.752144i	$-0.270980\pi$
$601$	32.3111	1.31800	0.658999	+	0.752144i	$0.270980\pi$
$602$	0	0
$603$	−3.75119	−0.152760
$604$	0	0
$605$	−12.9630	−0.527021
$606$	0	0
$607$	−0.170839	−0.00693416	−0.00346708	−	0.999994i	$-0.501104\pi$
$607$	−0.170839	−0.00693416	−0.00346708	+	0.999994i	$0.501104\pi$
$608$	0	0
$609$	−26.2910	−1.06537
$610$	0	0
$611$	35.7522	1.44638
$612$	0	0
$613$	21.4642	0.866930	0.433465	−	0.901170i	$-0.357291\pi$
$613$	21.4642	0.866930	0.433465	+	0.901170i	$0.357291\pi$
$614$	0	0
$615$	−43.1200	−1.73877
$616$	0	0
$617$	−25.0857	−1.00991	−0.504956	−	0.863145i	$-0.668491\pi$
$617$	−25.0857	−1.00991	−0.504956	+	0.863145i	$0.668491\pi$
$618$	0	0
$619$	−23.3984	−0.940461	−0.470230	−	0.882544i	$-0.655829\pi$
$619$	−23.3984	−0.940461	−0.470230	+	0.882544i	$0.655829\pi$
$620$	0	0
$621$	−16.8119	−0.674637
$622$	0	0
$623$	−28.6795	−1.14902
$624$	0	0
$625$	−24.9339	−0.997357
$626$	0	0
$627$	−34.0957	−1.36165
$628$	0	0
$629$	−45.1005	−1.79827
$630$	0	0
$631$	−20.3274	−0.809221	−0.404610	−	0.914489i	$-0.632593\pi$
$631$	−20.3274	−0.809221	−0.404610	+	0.914489i	$0.632593\pi$
$632$	0	0
$633$	−2.94399	−0.117013
$634$	0	0
$635$	42.5412	1.68820
$636$	0	0
$637$	10.0401	0.397804
$638$	0	0
$639$	−16.7197	−0.661420
$640$	0	0
$641$	−4.73536	−0.187036	−0.0935178	−	0.995618i	$-0.529811\pi$
$641$	−4.73536	−0.187036	−0.0935178	+	0.995618i	$0.529811\pi$
$642$	0	0
$643$	−18.4239	−0.726567	−0.363284	−	0.931679i	$-0.618344\pi$
$643$	−18.4239	−0.726567	−0.363284	+	0.931679i	$0.618344\pi$
$644$	0	0
$645$	2.79854	0.110193
$646$	0	0
$647$	10.9124	0.429012	0.214506	−	0.976723i	$-0.431186\pi$
$647$	10.9124	0.429012	0.214506	+	0.976723i	$0.431186\pi$
$648$	0	0
$649$	25.5512	1.00297
$650$	0	0
$651$	25.5935	1.00309
$652$	0	0
$653$	10.4053	0.407190	0.203595	−	0.979055i	$-0.434737\pi$
$653$	10.4053	0.407190	0.203595	+	0.979055i	$0.434737\pi$
$654$	0	0
$655$	70.2208	2.74376
$656$	0	0
$657$	−1.62302	−0.0633201
$658$	0	0
$659$	−6.32547	−0.246405	−0.123203	−	0.992382i	$-0.539317\pi$
$659$	−6.32547	−0.246405	−0.123203	+	0.992382i	$0.539317\pi$
$660$	0	0
$661$	−12.4410	−0.483898	−0.241949	−	0.970289i	$-0.577787\pi$
$661$	−12.4410	−0.483898	−0.241949	+	0.970289i	$0.577787\pi$
$662$	0	0
$663$	−72.8335	−2.82862
$664$	0	0
$665$	−60.2230	−2.33535
$666$	0	0
$667$	−18.9579	−0.734054
$668$	0	0
$669$	27.2018	1.05168
$670$	0	0
$671$	35.1802	1.35812
$672$	0	0
$673$	−32.1374	−1.23881	−0.619403	−	0.785073i	$-0.712625\pi$
$673$	−32.1374	−1.23881	−0.619403	+	0.785073i	$0.712625\pi$
$674$	0	0
$675$	−19.6863	−0.757727
$676$	0	0
$677$	26.4846	1.01789	0.508943	−	0.860800i	$-0.330037\pi$
$677$	26.4846	1.01789	0.508943	+	0.860800i	$0.330037\pi$
$678$	0	0
$679$	−24.1452	−0.926607
$680$	0	0
$681$	12.2362	0.468894
$682$	0	0
$683$	15.6606	0.599236	0.299618	−	0.954059i	$-0.403141\pi$
$683$	15.6606	0.599236	0.299618	+	0.954059i	$0.403141\pi$
$684$	0	0
$685$	−36.3866	−1.39026
$686$	0	0
$687$	6.89477	0.263052
$688$	0	0
$689$	−63.7061	−2.42701
$690$	0	0
$691$	−21.4216	−0.814916	−0.407458	−	0.913224i	$-0.633585\pi$
$691$	−21.4216	−0.814916	−0.407458	+	0.913224i	$0.633585\pi$
$692$	0	0
$693$	8.12752	0.308739
$694$	0	0
$695$	−17.7255	−0.672366
$696$	0	0
$697$	41.6928	1.57923
$698$	0	0
$699$	40.4878	1.53139
$700$	0	0
$701$	−50.4639	−1.90600	−0.952998	−	0.302976i	$-0.902020\pi$
$701$	−50.4639	−1.90600	−0.952998	+	0.302976i	$0.902020\pi$
$702$	0	0
$703$	−47.2507	−1.78209
$704$	0	0
$705$	38.7108	1.45793
$706$	0	0
$707$	18.4495	0.693865
$708$	0	0
$709$	−31.2660	−1.17422	−0.587109	−	0.809508i	$-0.699734\pi$
$709$	−31.2660	−1.17422	−0.587109	+	0.809508i	$0.699734\pi$
$710$	0	0
$711$	−1.79058	−0.0671521
$712$	0	0
$713$	18.4550	0.691144
$714$	0	0
$715$	48.8890	1.82834
$716$	0	0
$717$	−34.3994	−1.28467
$718$	0	0
$719$	−31.6914	−1.18189	−0.590945	−	0.806712i	$-0.701245\pi$
$719$	−31.6914	−1.18189	−0.590945	+	0.806712i	$0.701245\pi$
$720$	0	0
$721$	14.4344	0.537567
$722$	0	0
$723$	−17.8849	−0.665148
$724$	0	0
$725$	−22.1993	−0.824462
$726$	0	0
$727$	25.4229	0.942882	0.471441	−	0.881898i	$-0.343734\pi$
$727$	25.4229	0.942882	0.471441	+	0.881898i	$0.343734\pi$
$728$	0	0
$729$	12.1240	0.449036
$730$	0	0
$731$	−2.70591	−0.100082
$732$	0	0
$733$	−31.2979	−1.15601	−0.578007	−	0.816032i	$-0.696169\pi$
$733$	−31.2979	−1.15601	−0.578007	+	0.816032i	$0.696169\pi$
$734$	0	0
$735$	10.8710	0.400981
$736$	0	0
$737$	−9.40203	−0.346328
$738$	0	0
$739$	−9.17900	−0.337655	−0.168828	−	0.985646i	$-0.553998\pi$
$739$	−9.17900	−0.337655	−0.168828	+	0.985646i	$0.553998\pi$
$740$	0	0
$741$	−76.3060	−2.80317
$742$	0	0
$743$	−17.5865	−0.645187	−0.322593	−	0.946538i	$-0.604555\pi$
$743$	−17.5865	−0.645187	−0.322593	+	0.946538i	$0.604555\pi$
$744$	0	0
$745$	−56.3833	−2.06573
$746$	0	0
$747$	10.9762	0.401599
$748$	0	0
$749$	−32.3094	−1.18056
$750$	0	0
$751$	29.7578	1.08588	0.542939	−	0.839772i	$-0.317312\pi$
$751$	29.7578	1.08588	0.542939	+	0.839772i	$0.317312\pi$
$752$	0	0
$753$	2.01204	0.0733226
$754$	0	0
$755$	12.0428	0.438282
$756$	0	0
$757$	−46.8606	−1.70318	−0.851589	−	0.524211i	$-0.824360\pi$
$757$	−46.8606	−1.70318	−0.851589	+	0.524211i	$0.824360\pi$
$758$	0	0
$759$	22.6325	0.821509
$760$	0	0
$761$	24.4010	0.884535	0.442267	−	0.896883i	$-0.354174\pi$
$761$	24.4010	0.884535	0.442267	+	0.896883i	$0.354174\pi$
$762$	0	0
$763$	−55.1660	−1.99714
$764$	0	0
$765$	−20.4206	−0.738310
$766$	0	0
$767$	57.1835	2.06478
$768$	0	0
$769$	8.31014	0.299671	0.149836	−	0.988711i	$-0.452126\pi$
$769$	8.31014	0.299671	0.149836	+	0.988711i	$0.452126\pi$
$770$	0	0
$771$	−35.4714	−1.27747
$772$	0	0
$773$	51.1344	1.83917	0.919587	−	0.392885i	$-0.128523\pi$
$773$	51.1344	1.83917	0.919587	+	0.392885i	$0.128523\pi$
$774$	0	0
$775$	21.6103	0.776267
$776$	0	0
$777$	43.4969	1.56044
$778$	0	0
$779$	43.6805	1.56502
$780$	0	0
$781$	−41.9064	−1.49953
$782$	0	0
$783$	17.3891	0.621437
$784$	0	0
$785$	−10.6888	−0.381500
$786$	0	0
$787$	23.7122	0.845248	0.422624	−	0.906305i	$-0.361109\pi$
$787$	23.7122	0.845248	0.422624	+	0.906305i	$0.361109\pi$
$788$	0	0
$789$	4.84045	0.172325
$790$	0	0
$791$	−33.6836	−1.19765
$792$	0	0
$793$	78.7332	2.79590
$794$	0	0
$795$	−68.9780	−2.44640
$796$	0	0
$797$	−48.1989	−1.70729	−0.853646	−	0.520853i	$-0.825614\pi$
$797$	−48.1989	−1.70729	−0.853646	+	0.520853i	$0.825614\pi$
$798$	0	0
$799$	−37.4295	−1.32416
$800$	0	0
$801$	−10.1884	−0.359990
$802$	0	0
$803$	−4.06796	−0.143555
$804$	0	0
$805$	39.9757	1.40896
$806$	0	0
$807$	9.82535	0.345869
$808$	0	0
$809$	38.8551	1.36607	0.683036	−	0.730385i	$-0.260659\pi$
$809$	38.8551	1.36607	0.683036	+	0.730385i	$0.260659\pi$
$810$	0	0
$811$	−26.4421	−0.928509	−0.464254	−	0.885702i	$-0.653678\pi$
$811$	−26.4421	−0.928509	−0.464254	+	0.885702i	$0.653678\pi$
$812$	0	0
$813$	31.1301	1.09178
$814$	0	0
$815$	36.2639	1.27027
$816$	0	0
$817$	−2.83492	−0.0991814
$818$	0	0
$819$	18.1894	0.635588
$820$	0	0
$821$	−13.7994	−0.481603	−0.240802	−	0.970574i	$-0.577410\pi$
$821$	−13.7994	−0.481603	−0.240802	+	0.970574i	$0.577410\pi$
$822$	0	0
$823$	−1.94671	−0.0678582	−0.0339291	−	0.999424i	$-0.510802\pi$
$823$	−1.94671	−0.0678582	−0.0339291	+	0.999424i	$0.510802\pi$
$824$	0	0
$825$	26.5022	0.922688
$826$	0	0
$827$	17.4109	0.605437	0.302718	−	0.953080i	$-0.402106\pi$
$827$	17.4109	0.605437	0.302718	+	0.953080i	$0.402106\pi$
$828$	0	0
$829$	−42.5792	−1.47884	−0.739419	−	0.673245i	$-0.764900\pi$
$829$	−42.5792	−1.47884	−0.739419	+	0.673245i	$0.764900\pi$
$830$	0	0
$831$	−26.7303	−0.927264
$832$	0	0
$833$	−10.5111	−0.364189
$834$	0	0
$835$	63.0377	2.18151
$836$	0	0
$837$	−16.9278	−0.585110
$838$	0	0
$839$	5.45966	0.188488	0.0942442	−	0.995549i	$-0.469957\pi$
$839$	5.45966	0.188488	0.0942442	+	0.995549i	$0.469957\pi$
$840$	0	0
$841$	−9.39114	−0.323832
$842$	0	0
$843$	4.32057	0.148808
$844$	0	0
$845$	68.2767	2.34879
$846$	0	0
$847$	−12.0883	−0.415358
$848$	0	0
$849$	−1.90806	−0.0654843
$850$	0	0
$851$	31.3648	1.07517
$852$	0	0
$853$	23.5972	0.807951	0.403976	−	0.914770i	$-0.367628\pi$
$853$	23.5972	0.807951	0.403976	+	0.914770i	$0.367628\pi$
$854$	0	0
$855$	−21.3942	−0.731667
$856$	0	0
$857$	−13.6021	−0.464637	−0.232319	−	0.972640i	$-0.574631\pi$
$857$	−13.6021	−0.464637	−0.232319	+	0.972640i	$0.574631\pi$
$858$	0	0
$859$	−19.9808	−0.681736	−0.340868	−	0.940111i	$-0.610721\pi$
$859$	−19.9808	−0.681736	−0.340868	+	0.940111i	$0.610721\pi$
$860$	0	0
$861$	−40.2104	−1.37037
$862$	0	0
$863$	9.06731	0.308655	0.154327	−	0.988020i	$-0.450679\pi$
$863$	9.06731	0.308655	0.154327	+	0.988020i	$0.450679\pi$
$864$	0	0
$865$	38.5620	1.31115
$866$	0	0
$867$	42.0458	1.42795
$868$	0	0
$869$	−4.48794	−0.152243
$870$	0	0
$871$	−21.0417	−0.712971
$872$	0	0
$873$	−8.57758	−0.290307
$874$	0	0
$875$	0.123062	0.00416025
$876$	0	0
$877$	27.2108	0.918842	0.459421	−	0.888219i	$-0.348057\pi$
$877$	27.2108	0.918842	0.459421	+	0.888219i	$0.348057\pi$
$878$	0	0
$879$	20.7234	0.698983
$880$	0	0
$881$	40.2683	1.35667	0.678336	−	0.734752i	$-0.262701\pi$
$881$	40.2683	1.35667	0.678336	+	0.734752i	$0.262701\pi$
$882$	0	0
$883$	16.8028	0.565458	0.282729	−	0.959200i	$-0.408760\pi$
$883$	16.8028	0.565458	0.282729	+	0.959200i	$0.408760\pi$
$884$	0	0
$885$	61.9156	2.08127
$886$	0	0
$887$	−19.1538	−0.643122	−0.321561	−	0.946889i	$-0.604208\pi$
$887$	−19.1538	−0.643122	−0.321561	+	0.946889i	$0.604208\pi$
$888$	0	0
$889$	39.6706	1.33051
$890$	0	0
$891$	−29.0226	−0.972293
$892$	0	0
$893$	−39.2140	−1.31225
$894$	0	0
$895$	−10.7050	−0.357827
$896$	0	0
$897$	50.6515	1.69121
$898$	0	0
$899$	−19.0886	−0.636641
$900$	0	0
$901$	66.6948	2.22193
$902$	0	0
$903$	2.60971	0.0868456
$904$	0	0
$905$	71.3452	2.37159
$906$	0	0
$907$	46.3178	1.53796	0.768979	−	0.639275i	$-0.220765\pi$
$907$	46.3178	1.53796	0.768979	+	0.639275i	$0.220765\pi$
$908$	0	0
$909$	6.55419	0.217389
$910$	0	0
$911$	16.7506	0.554971	0.277485	−	0.960730i	$-0.410499\pi$
$911$	16.7506	0.554971	0.277485	+	0.960730i	$0.410499\pi$
$912$	0	0
$913$	27.5110	0.910481
$914$	0	0
$915$	85.2486	2.81823
$916$	0	0
$917$	65.4825	2.16242
$918$	0	0
$919$	−13.5484	−0.446922	−0.223461	−	0.974713i	$-0.571735\pi$
$919$	−13.5484	−0.446922	−0.223461	+	0.974713i	$0.571735\pi$
$920$	0	0
$921$	46.2521	1.52406
$922$	0	0
$923$	−93.7864	−3.08702
$924$	0	0
$925$	36.7274	1.20759
$926$	0	0
$927$	5.12784	0.168421
$928$	0	0
$929$	−25.4311	−0.834366	−0.417183	−	0.908822i	$-0.636983\pi$
$929$	−25.4311	−0.834366	−0.417183	+	0.908822i	$0.636983\pi$
$930$	0	0
$931$	−11.0123	−0.360913
$932$	0	0
$933$	−39.9410	−1.30761
$934$	0	0
$935$	−51.1826	−1.67385
$936$	0	0
$937$	45.8163	1.49675	0.748376	−	0.663274i	$-0.230834\pi$
$937$	45.8163	1.49675	0.748376	+	0.663274i	$0.230834\pi$
$938$	0	0
$939$	−37.6532	−1.22877
$940$	0	0
$941$	−19.2037	−0.626022	−0.313011	−	0.949750i	$-0.601338\pi$
$941$	−19.2037	−0.626022	−0.313011	+	0.949750i	$0.601338\pi$
$942$	0	0
$943$	−28.9949	−0.944204
$944$	0	0
$945$	−36.6676	−1.19280
$946$	0	0
$947$	−46.5275	−1.51194	−0.755970	−	0.654606i	$-0.772835\pi$
$947$	−46.5275	−1.51194	−0.755970	+	0.654606i	$0.772835\pi$
$948$	0	0
$949$	−9.10409	−0.295531
$950$	0	0
$951$	35.3136	1.14512
$952$	0	0
$953$	−14.7931	−0.479195	−0.239598	−	0.970872i	$-0.577016\pi$
$953$	−14.7931	−0.479195	−0.239598	+	0.970872i	$0.577016\pi$
$954$	0	0
$955$	−77.3083	−2.50164
$956$	0	0
$957$	−23.4096	−0.756726
$958$	0	0
$959$	−33.9313	−1.09570
$960$	0	0
$961$	−12.4178	−0.400574
$962$	0	0
$963$	−11.4779	−0.369871
$964$	0	0
$965$	−55.4522	−1.78507
$966$	0	0
$967$	56.3967	1.81360	0.906798	−	0.421566i	$-0.138519\pi$
$967$	56.3967	1.81360	0.906798	+	0.421566i	$0.138519\pi$
$968$	0	0
$969$	79.8858	2.56630
$970$	0	0
$971$	−42.2568	−1.35609	−0.678043	−	0.735023i	$-0.737172\pi$
$971$	−42.2568	−1.35609	−0.678043	+	0.735023i	$0.737172\pi$
$972$	0	0
$973$	−16.5294	−0.529909
$974$	0	0
$975$	59.3118	1.89950
$976$	0	0
$977$	−52.3005	−1.67324	−0.836620	−	0.547784i	$-0.815472\pi$
$977$	−52.3005	−1.67324	−0.836620	+	0.547784i	$0.815472\pi$
$978$	0	0
$979$	−25.5364	−0.816146
$980$	0	0
$981$	−19.5977	−0.625708
$982$	0	0
$983$	−6.71073	−0.214039	−0.107020	−	0.994257i	$-0.534131\pi$
$983$	−6.71073	−0.214039	−0.107020	+	0.994257i	$0.534131\pi$
$984$	0	0
$985$	14.2485	0.453994
$986$	0	0
$987$	36.0987	1.14903
$988$	0	0
$989$	1.88181	0.0598380
$990$	0	0
$991$	40.4252	1.28415	0.642074	−	0.766643i	$-0.278074\pi$
$991$	40.4252	1.28415	0.642074	+	0.766643i	$0.278074\pi$
$992$	0	0
$993$	−36.5332	−1.15934
$994$	0	0
$995$	−44.4542	−1.40929
$996$	0	0
$997$	−18.1150	−0.573707	−0.286853	−	0.957974i	$-0.592609\pi$
$997$	−18.1150	−0.573707	−0.286853	+	0.957974i	$0.592609\pi$
$998$	0	0
$999$	−28.7693	−0.910220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.14		19
4.3	odd	2		2008.2.a.c.1.6	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.6	✓	19	4.3	odd	2
4016.2.a.l.1.14		19	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.01204 q^{3} +3.16436 q^{5} +2.95084 q^{7} +1.04829 q^{9} +O(q^{10})\)	\(q+2.01204 q^{3} +3.16436 q^{5} +2.95084 q^{7} +1.04829 q^{9} +2.62744 q^{11} +5.88020 q^{13} +6.36681 q^{15} -6.15607 q^{17} -6.44957 q^{19} +5.93719 q^{21} +4.28119 q^{23} +5.01318 q^{25} -3.92692 q^{27} -4.42819 q^{29} +4.31071 q^{31} +5.28650 q^{33} +9.33751 q^{35} +7.32618 q^{37} +11.8312 q^{39} -6.77263 q^{41} +0.439552 q^{43} +3.31716 q^{45} +6.08009 q^{47} +1.70744 q^{49} -12.3862 q^{51} -10.8340 q^{53} +8.31417 q^{55} -12.9768 q^{57} +9.72474 q^{59} +13.3895 q^{61} +3.09332 q^{63} +18.6071 q^{65} -3.57840 q^{67} +8.61391 q^{69} -15.9495 q^{71} -1.54826 q^{73} +10.0867 q^{75} +7.75315 q^{77} -1.70810 q^{79} -11.0460 q^{81} +10.4706 q^{83} -19.4800 q^{85} -8.90967 q^{87} -9.71911 q^{89} +17.3515 q^{91} +8.67329 q^{93} -20.4088 q^{95} -8.18248 q^{97} +2.75431 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37} + 27 q^{39} + 2 q^{41} + 28 q^{43} - 29 q^{45} + 37 q^{47} + 36 q^{49} + 35 q^{51} - 37 q^{53} + 24 q^{55} - 11 q^{57} + 32 q^{59} - 7 q^{61} + 45 q^{63} + q^{65} + 45 q^{67} - 12 q^{69} + 49 q^{71} + 16 q^{73} + 35 q^{75} - 40 q^{77} + 33 q^{79} + 15 q^{81} + 43 q^{83} - 28 q^{85} + 48 q^{87} + 3 q^{89} + 56 q^{91} - 48 q^{93} + 43 q^{95} + 8 q^{97} + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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