LMFDB - Embedded newform 4016.2.a.i.1.6

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.571724$ 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)

$f(q)$	$=$	$q-0.571724 q^{3} +0.953980 q^{5} +4.33542 q^{7} -2.67313 q^{9} +O(q^{10})$	$q-0.571724 q^{3} +0.953980 q^{5} +4.33542 q^{7} -2.67313 q^{9} -1.47410 q^{11} +4.03365 q^{13} -0.545414 q^{15} +0.0115340 q^{17} -3.96227 q^{19} -2.47867 q^{21} -7.83651 q^{23} -4.08992 q^{25} +3.24347 q^{27} -6.75701 q^{29} -7.57934 q^{31} +0.842777 q^{33} +4.13590 q^{35} -5.48972 q^{37} -2.30613 q^{39} +2.39192 q^{41} -8.35952 q^{43} -2.55011 q^{45} -3.64903 q^{47} +11.7959 q^{49} -0.00659429 q^{51} +11.5253 q^{53} -1.40626 q^{55} +2.26533 q^{57} -13.7484 q^{59} -0.911085 q^{61} -11.5891 q^{63} +3.84802 q^{65} -3.80620 q^{67} +4.48032 q^{69} +9.50880 q^{71} -1.39689 q^{73} +2.33831 q^{75} -6.39082 q^{77} -12.8951 q^{79} +6.16502 q^{81} +11.2503 q^{83} +0.0110032 q^{85} +3.86315 q^{87} +8.31945 q^{89} +17.4875 q^{91} +4.33330 q^{93} -3.77993 q^{95} +2.26070 q^{97} +3.94045 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	−0.571724	−0.330085	−0.165043	−	0.986286i	$-0.552776\pi$
$3$	−0.571724	−0.330085	−0.165043	+	0.986286i	$0.552776\pi$
$4$	0	0
$5$	0.953980	0.426633	0.213316	−	0.976983i	$-0.431573\pi$
$5$	0.953980	0.426633	0.213316	+	0.976983i	$0.431573\pi$
$6$	0	0
$7$	4.33542	1.63863	0.819317	−	0.573340i	$-0.194353\pi$
$7$	4.33542	1.63863	0.819317	+	0.573340i	$0.194353\pi$
$8$	0	0
$9$	−2.67313	−0.891044
$10$	0	0
$11$	−1.47410	−0.444457	−0.222228	−	0.974995i	$-0.571333\pi$
$11$	−1.47410	−0.444457	−0.222228	+	0.974995i	$0.571333\pi$
$12$	0	0
$13$	4.03365	1.11873	0.559366	−	0.828921i	$-0.311045\pi$
$13$	4.03365	1.11873	0.559366	+	0.828921i	$0.311045\pi$
$14$	0	0
$15$	−0.545414	−0.140825
$16$	0	0
$17$	0.0115340	0.00279741	0.00139871	−	0.999999i	$-0.499555\pi$
$17$	0.0115340	0.00279741	0.00139871	+	0.999999i	$0.499555\pi$
$18$	0	0
$19$	−3.96227	−0.909007	−0.454503	−	0.890745i	$-0.650183\pi$
$19$	−3.96227	−0.909007	−0.454503	+	0.890745i	$0.650183\pi$
$20$	0	0
$21$	−2.47867	−0.540889
$22$	0	0
$23$	−7.83651	−1.63402	−0.817012	−	0.576620i	$-0.804371\pi$
$23$	−7.83651	−1.63402	−0.817012	+	0.576620i	$0.804371\pi$
$24$	0	0
$25$	−4.08992	−0.817984
$26$	0	0
$27$	3.24347	0.624206
$28$	0	0
$29$	−6.75701	−1.25474	−0.627372	−	0.778719i	$-0.715870\pi$
$29$	−6.75701	−1.25474	−0.627372	+	0.778719i	$0.715870\pi$
$30$	0	0
$31$	−7.57934	−1.36129	−0.680645	−	0.732613i	$-0.738300\pi$
$31$	−7.57934	−1.36129	−0.680645	+	0.732613i	$0.738300\pi$
$32$	0	0
$33$	0.842777	0.146709
$34$	0	0
$35$	4.13590	0.699095
$36$	0	0
$37$	−5.48972	−0.902504	−0.451252	−	0.892396i	$-0.649023\pi$
$37$	−5.48972	−0.902504	−0.451252	+	0.892396i	$0.649023\pi$
$38$	0	0
$39$	−2.30613	−0.369277
$40$	0	0
$41$	2.39192	0.373555	0.186778	−	0.982402i	$-0.440196\pi$
$41$	2.39192	0.373555	0.186778	+	0.982402i	$0.440196\pi$
$42$	0	0
$43$	−8.35952	−1.27481	−0.637407	−	0.770527i	$-0.719993\pi$
$43$	−8.35952	−1.27481	−0.637407	+	0.770527i	$0.719993\pi$
$44$	0	0
$45$	−2.55011	−0.380148
$46$	0	0
$47$	−3.64903	−0.532266	−0.266133	−	0.963936i	$-0.585746\pi$
$47$	−3.64903	−0.532266	−0.266133	+	0.963936i	$0.585746\pi$
$48$	0	0
$49$	11.7959	1.68512
$50$	0	0
$51$	−0.00659429	−0.000923385 0
$52$	0	0
$53$	11.5253	1.58313	0.791564	−	0.611087i	$-0.209267\pi$
$53$	11.5253	1.58313	0.791564	+	0.611087i	$0.209267\pi$
$54$	0	0
$55$	−1.40626	−0.189620
$56$	0	0
$57$	2.26533	0.300050
$58$	0	0
$59$	−13.7484	−1.78989	−0.894947	−	0.446173i	$-0.852787\pi$
$59$	−13.7484	−1.78989	−0.894947	+	0.446173i	$0.852787\pi$
$60$	0	0
$61$	−0.911085	−0.116652	−0.0583262	−	0.998298i	$-0.518576\pi$
$61$	−0.911085	−0.116652	−0.0583262	+	0.998298i	$0.518576\pi$
$62$	0	0
$63$	−11.5891	−1.46009
$64$	0	0
$65$	3.84802	0.477288
$66$	0	0
$67$	−3.80620	−0.465001	−0.232501	−	0.972596i	$-0.574691\pi$
$67$	−3.80620	−0.465001	−0.232501	+	0.972596i	$0.574691\pi$
$68$	0	0
$69$	4.48032	0.539367
$70$	0	0
$71$	9.50880	1.12849	0.564243	−	0.825609i	$-0.309168\pi$
$71$	9.50880	1.12849	0.564243	+	0.825609i	$0.309168\pi$
$72$	0	0
$73$	−1.39689	−0.163494	−0.0817469	−	0.996653i	$-0.526050\pi$
$73$	−1.39689	−0.163494	−0.0817469	+	0.996653i	$0.526050\pi$
$74$	0	0
$75$	2.33831	0.270005
$76$	0	0
$77$	−6.39082	−0.728302
$78$	0	0
$79$	−12.8951	−1.45081	−0.725403	−	0.688324i	$-0.758347\pi$
$79$	−12.8951	−1.45081	−0.725403	+	0.688324i	$0.758347\pi$
$80$	0	0
$81$	6.16502	0.685003
$82$	0	0
$83$	11.2503	1.23488	0.617439	−	0.786619i	$-0.288170\pi$
$83$	11.2503	1.23488	0.617439	+	0.786619i	$0.288170\pi$
$84$	0	0
$85$	0.0110032	0.00119347
$86$	0	0
$87$	3.86315	0.414173
$88$	0	0
$89$	8.31945	0.881860	0.440930	−	0.897541i	$-0.354649\pi$
$89$	8.31945	0.881860	0.440930	+	0.897541i	$0.354649\pi$
$90$	0	0
$91$	17.4875	1.83319
$92$	0	0
$93$	4.33330	0.449342
$94$	0	0
$95$	−3.77993	−0.387812
$96$	0	0
$97$	2.26070	0.229540	0.114770	−	0.993392i	$-0.463387\pi$
$97$	2.26070	0.229540	0.114770	+	0.993392i	$0.463387\pi$
$98$	0	0
$99$	3.94045	0.396030
$100$	0	0
$101$	3.25674	0.324058	0.162029	−	0.986786i	$-0.448196\pi$
$101$	3.25674	0.324058	0.162029	+	0.986786i	$0.448196\pi$
$102$	0	0
$103$	−19.5145	−1.92282	−0.961412	−	0.275112i	$-0.911285\pi$
$103$	−19.5145	−1.92282	−0.961412	+	0.275112i	$0.911285\pi$
$104$	0	0
$105$	−2.36460	−0.230761
$106$	0	0
$107$	5.46723	0.528537	0.264268	−	0.964449i	$-0.414870\pi$
$107$	5.46723	0.528537	0.264268	+	0.964449i	$0.414870\pi$
$108$	0	0
$109$	−2.23754	−0.214318	−0.107159	−	0.994242i	$-0.534175\pi$
$109$	−2.23754	−0.214318	−0.107159	+	0.994242i	$0.534175\pi$
$110$	0	0
$111$	3.13861	0.297903
$112$	0	0
$113$	−3.61832	−0.340383	−0.170192	−	0.985411i	$-0.554439\pi$
$113$	−3.61832	−0.340383	−0.170192	+	0.985411i	$0.554439\pi$
$114$	0	0
$115$	−7.47587	−0.697128
$116$	0	0
$117$	−10.7825	−0.996839
$118$	0	0
$119$	0.0500049	0.00458394
$120$	0	0
$121$	−8.82704	−0.802458
$122$	0	0
$123$	−1.36752	−0.123305
$124$	0	0
$125$	−8.67160	−0.775612
$126$	0	0
$127$	0.382968	0.0339830	0.0169915	−	0.999856i	$-0.494591\pi$
$127$	0.382968	0.0339830	0.0169915	+	0.999856i	$0.494591\pi$
$128$	0	0
$129$	4.77934	0.420798
$130$	0	0
$131$	18.2685	1.59612	0.798061	−	0.602576i	$-0.205859\pi$
$131$	18.2685	1.59612	0.798061	+	0.602576i	$0.205859\pi$
$132$	0	0
$133$	−17.1781	−1.48953
$134$	0	0
$135$	3.09420	0.266307
$136$	0	0
$137$	11.3362	0.968514	0.484257	−	0.874926i	$-0.339090\pi$
$137$	11.3362	0.968514	0.484257	+	0.874926i	$0.339090\pi$
$138$	0	0
$139$	−9.33206	−0.791535	−0.395768	−	0.918351i	$-0.629521\pi$
$139$	−9.33206	−0.791535	−0.395768	+	0.918351i	$0.629521\pi$
$140$	0	0
$141$	2.08624	0.175693
$142$	0	0
$143$	−5.94598	−0.497228
$144$	0	0
$145$	−6.44605	−0.535315
$146$	0	0
$147$	−6.74398	−0.556234
$148$	0	0
$149$	5.97972	0.489878	0.244939	−	0.969538i	$-0.421232\pi$
$149$	5.97972	0.489878	0.244939	+	0.969538i	$0.421232\pi$
$150$	0	0
$151$	−10.3154	−0.839454	−0.419727	−	0.907651i	$-0.637874\pi$
$151$	−10.3154	−0.839454	−0.419727	+	0.907651i	$0.637874\pi$
$152$	0	0
$153$	−0.0308320	−0.00249262
$154$	0	0
$155$	−7.23054	−0.580771
$156$	0	0
$157$	−6.55531	−0.523171	−0.261586	−	0.965180i	$-0.584245\pi$
$157$	−6.55531	−0.523171	−0.261586	+	0.965180i	$0.584245\pi$
$158$	0	0
$159$	−6.58932	−0.522567
$160$	0	0
$161$	−33.9745	−2.67757
$162$	0	0
$163$	24.1773	1.89372	0.946858	−	0.321652i	$-0.104238\pi$
$163$	24.1773	1.89372	0.946858	+	0.321652i	$0.104238\pi$
$164$	0	0
$165$	0.803992	0.0625907
$166$	0	0
$167$	−4.36208	−0.337548	−0.168774	−	0.985655i	$-0.553981\pi$
$167$	−4.36208	−0.337548	−0.168774	+	0.985655i	$0.553981\pi$
$168$	0	0
$169$	3.27029	0.251561
$170$	0	0
$171$	10.5917	0.809965
$172$	0	0
$173$	3.92870	0.298693	0.149347	−	0.988785i	$-0.452283\pi$
$173$	3.92870	0.298693	0.149347	+	0.988785i	$0.452283\pi$
$174$	0	0
$175$	−17.7315	−1.34038
$176$	0	0
$177$	7.86032	0.590818
$178$	0	0
$179$	23.6901	1.77068	0.885340	−	0.464944i	$-0.153925\pi$
$179$	23.6901	1.77068	0.885340	+	0.464944i	$0.153925\pi$
$180$	0	0
$181$	25.5276	1.89745	0.948726	−	0.316101i	$-0.102374\pi$
$181$	25.5276	1.89745	0.948726	+	0.316101i	$0.102374\pi$
$182$	0	0
$183$	0.520889	0.0385053
$184$	0	0
$185$	−5.23708	−0.385038
$186$	0	0
$187$	−0.0170023	−0.00124333
$188$	0	0
$189$	14.0618	1.02284
$190$	0	0
$191$	0.864872	0.0625799	0.0312900	−	0.999510i	$-0.490038\pi$
$191$	0.864872	0.0625799	0.0312900	+	0.999510i	$0.490038\pi$
$192$	0	0
$193$	−10.8938	−0.784154	−0.392077	−	0.919932i	$-0.628243\pi$
$193$	−10.8938	−0.784154	−0.392077	+	0.919932i	$0.628243\pi$
$194$	0	0
$195$	−2.20001	−0.157546
$196$	0	0
$197$	−3.06368	−0.218278	−0.109139	−	0.994026i	$-0.534809\pi$
$197$	−3.06368	−0.218278	−0.109139	+	0.994026i	$0.534809\pi$
$198$	0	0
$199$	−12.4350	−0.881492	−0.440746	−	0.897632i	$-0.645286\pi$
$199$	−12.4350	−0.881492	−0.440746	+	0.897632i	$0.645286\pi$
$200$	0	0
$201$	2.17610	0.153490
$202$	0	0
$203$	−29.2945	−2.05607
$204$	0	0
$205$	2.28185	0.159371
$206$	0	0
$207$	20.9480	1.45599
$208$	0	0
$209$	5.84077	0.404014
$210$	0	0
$211$	1.96014	0.134942	0.0674709	−	0.997721i	$-0.478507\pi$
$211$	1.96014	0.134942	0.0674709	+	0.997721i	$0.478507\pi$
$212$	0	0
$213$	−5.43641	−0.372497
$214$	0	0
$215$	−7.97481	−0.543878
$216$	0	0
$217$	−32.8596	−2.23066
$218$	0	0
$219$	0.798637	0.0539669
$220$	0	0
$221$	0.0465242	0.00312956
$222$	0	0
$223$	−10.3392	−0.692365	−0.346182	−	0.938167i	$-0.612522\pi$
$223$	−10.3392	−0.692365	−0.346182	+	0.938167i	$0.612522\pi$
$224$	0	0
$225$	10.9329	0.728860
$226$	0	0
$227$	−29.2634	−1.94228	−0.971138	−	0.238516i	$-0.923339\pi$
$227$	−29.2634	−1.94228	−0.971138	+	0.238516i	$0.923339\pi$
$228$	0	0
$229$	−5.05297	−0.333910	−0.166955	−	0.985965i	$-0.553393\pi$
$229$	−5.05297	−0.333910	−0.166955	+	0.985965i	$0.553393\pi$
$230$	0	0
$231$	3.65379	0.240402
$232$	0	0
$233$	−3.57962	−0.234508	−0.117254	−	0.993102i	$-0.537409\pi$
$233$	−3.57962	−0.234508	−0.117254	+	0.993102i	$0.537409\pi$
$234$	0	0
$235$	−3.48110	−0.227082
$236$	0	0
$237$	7.37242	0.478890
$238$	0	0
$239$	17.1303	1.10807	0.554034	−	0.832494i	$-0.313088\pi$
$239$	17.1303	1.10807	0.554034	+	0.832494i	$0.313088\pi$
$240$	0	0
$241$	−7.00367	−0.451146	−0.225573	−	0.974226i	$-0.572425\pi$
$241$	−7.00367	−0.451146	−0.225573	+	0.974226i	$0.572425\pi$
$242$	0	0
$243$	−13.2551	−0.850315
$244$	0	0
$245$	11.2530	0.718929
$246$	0	0
$247$	−15.9824	−1.01693
$248$	0	0
$249$	−6.43206	−0.407615
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	11.5518	0.726253
$254$	0	0
$255$	−0.00629082	−0.000393947 0
$256$	0	0
$257$	0.215400	0.0134363	0.00671814	−	0.999977i	$-0.497862\pi$
$257$	0.215400	0.0134363	0.00671814	+	0.999977i	$0.497862\pi$
$258$	0	0
$259$	−23.8002	−1.47887
$260$	0	0
$261$	18.0624	1.11803
$262$	0	0
$263$	20.2409	1.24811	0.624055	−	0.781380i	$-0.285484\pi$
$263$	20.2409	1.24811	0.624055	+	0.781380i	$0.285484\pi$
$264$	0	0
$265$	10.9949	0.675414
$266$	0	0
$267$	−4.75643	−0.291089
$268$	0	0
$269$	4.09787	0.249851	0.124926	−	0.992166i	$-0.460131\pi$
$269$	4.09787	0.249851	0.124926	+	0.992166i	$0.460131\pi$
$270$	0	0
$271$	14.8006	0.899071	0.449535	−	0.893263i	$-0.351590\pi$
$271$	14.8006	0.899071	0.449535	+	0.893263i	$0.351590\pi$
$272$	0	0
$273$	−9.99806	−0.605110
$274$	0	0
$275$	6.02894	0.363559
$276$	0	0
$277$	−11.2912	−0.678425	−0.339213	−	0.940710i	$-0.610161\pi$
$277$	−11.2912	−0.678425	−0.339213	+	0.940710i	$0.610161\pi$
$278$	0	0
$279$	20.2606	1.21297
$280$	0	0
$281$	12.3671	0.737760	0.368880	−	0.929477i	$-0.379741\pi$
$281$	12.3671	0.737760	0.368880	+	0.929477i	$0.379741\pi$
$282$	0	0
$283$	3.58037	0.212831	0.106416	−	0.994322i	$-0.466063\pi$
$283$	3.58037	0.212831	0.106416	+	0.994322i	$0.466063\pi$
$284$	0	0
$285$	2.16108	0.128011
$286$	0	0
$287$	10.3700	0.612121
$288$	0	0
$289$	−16.9999	−0.999992
$290$	0	0
$291$	−1.29250	−0.0757676
$292$	0	0
$293$	30.6150	1.78854	0.894272	−	0.447523i	$-0.147694\pi$
$293$	30.6150	1.78854	0.894272	+	0.447523i	$0.147694\pi$
$294$	0	0
$295$	−13.1157	−0.763627
$296$	0	0
$297$	−4.78118	−0.277432
$298$	0	0
$299$	−31.6097	−1.82804
$300$	0	0
$301$	−36.2420	−2.08896
$302$	0	0
$303$	−1.86196	−0.106967
$304$	0	0
$305$	−0.869157	−0.0497678
$306$	0	0
$307$	−5.41902	−0.309280	−0.154640	−	0.987971i	$-0.549422\pi$
$307$	−5.41902	−0.309280	−0.154640	+	0.987971i	$0.549422\pi$
$308$	0	0
$309$	11.1569	0.634696
$310$	0	0
$311$	8.29436	0.470330	0.235165	−	0.971955i	$-0.424437\pi$
$311$	8.29436	0.470330	0.235165	+	0.971955i	$0.424437\pi$
$312$	0	0
$313$	−7.38204	−0.417258	−0.208629	−	0.977995i	$-0.566900\pi$
$313$	−7.38204	−0.417258	−0.208629	+	0.977995i	$0.566900\pi$
$314$	0	0
$315$	−11.0558	−0.622924
$316$	0	0
$317$	−11.0930	−0.623047	−0.311524	−	0.950238i	$-0.600839\pi$
$317$	−11.0930	−0.623047	−0.311524	+	0.950238i	$0.600839\pi$
$318$	0	0
$319$	9.96048	0.557680
$320$	0	0
$321$	−3.12575	−0.174462
$322$	0	0
$323$	−0.0457010	−0.00254287
$324$	0	0
$325$	−16.4973	−0.915105
$326$	0	0
$327$	1.27926	0.0707431
$328$	0	0
$329$	−15.8201	−0.872189
$330$	0	0
$331$	−0.0351291	−0.00193087	−0.000965436	−	1.00000i	$-0.500307\pi$
$331$	−0.0351291	−0.00193087	−0.000965436		1.00000i	$0.500307\pi$
$332$	0	0
$333$	14.6747	0.804171
$334$	0	0
$335$	−3.63104	−0.198385
$336$	0	0
$337$	−30.2218	−1.64629	−0.823144	−	0.567833i	$-0.807782\pi$
$337$	−30.2218	−1.64629	−0.823144	+	0.567833i	$0.807782\pi$
$338$	0	0
$339$	2.06868	0.112355
$340$	0	0
$341$	11.1727	0.605035
$342$	0	0
$343$	20.7921	1.12267
$344$	0	0
$345$	4.27414	0.230112
$346$	0	0
$347$	−14.3410	−0.769867	−0.384934	−	0.922944i	$-0.625776\pi$
$347$	−14.3410	−0.769867	−0.384934	+	0.922944i	$0.625776\pi$
$348$	0	0
$349$	−11.1074	−0.594565	−0.297283	−	0.954790i	$-0.596080\pi$
$349$	−11.1074	−0.594565	−0.297283	+	0.954790i	$0.596080\pi$
$350$	0	0
$351$	13.0830	0.698319
$352$	0	0
$353$	−30.7404	−1.63615	−0.818074	−	0.575112i	$-0.804958\pi$
$353$	−30.7404	−1.63615	−0.818074	+	0.575112i	$0.804958\pi$
$354$	0	0
$355$	9.07120	0.481449
$356$	0	0
$357$	−0.0285890	−0.00151309
$358$	0	0
$359$	−25.4174	−1.34148	−0.670740	−	0.741693i	$-0.734023\pi$
$359$	−25.4174	−1.34148	−0.670740	+	0.741693i	$0.734023\pi$
$360$	0	0
$361$	−3.30042	−0.173706
$362$	0	0
$363$	5.04664	0.264880
$364$	0	0
$365$	−1.33261	−0.0697518
$366$	0	0
$367$	−23.9263	−1.24894	−0.624472	−	0.781047i	$-0.714686\pi$
$367$	−23.9263	−1.24894	−0.624472	+	0.781047i	$0.714686\pi$
$368$	0	0
$369$	−6.39392	−0.332854
$370$	0	0
$371$	49.9672	2.59417
$372$	0	0
$373$	2.50523	0.129716	0.0648579	−	0.997895i	$-0.479341\pi$
$373$	2.50523	0.129716	0.0648579	+	0.997895i	$0.479341\pi$
$374$	0	0
$375$	4.95777	0.256018
$376$	0	0
$377$	−27.2554	−1.40372
$378$	0	0
$379$	30.6962	1.57676	0.788380	−	0.615189i	$-0.210920\pi$
$379$	30.6962	1.57676	0.788380	+	0.615189i	$0.210920\pi$
$380$	0	0
$381$	−0.218952	−0.0112173
$382$	0	0
$383$	−31.0559	−1.58688	−0.793441	−	0.608647i	$-0.791713\pi$
$383$	−31.0559	−1.58688	−0.793441	+	0.608647i	$0.791713\pi$
$384$	0	0
$385$	−6.09672	−0.310718
$386$	0	0
$387$	22.3461	1.13592
$388$	0	0
$389$	−14.6468	−0.742625	−0.371312	−	0.928508i	$-0.621092\pi$
$389$	−14.6468	−0.742625	−0.371312	+	0.928508i	$0.621092\pi$
$390$	0	0
$391$	−0.0903865	−0.00457104
$392$	0	0
$393$	−10.4445	−0.526857
$394$	0	0
$395$	−12.3016	−0.618962
$396$	0	0
$397$	−28.6220	−1.43650	−0.718249	−	0.695786i	$-0.755056\pi$
$397$	−28.6220	−1.43650	−0.718249	+	0.695786i	$0.755056\pi$
$398$	0	0
$399$	9.82114	0.491672
$400$	0	0
$401$	−12.6366	−0.631041	−0.315520	−	0.948919i	$-0.602179\pi$
$401$	−12.6366	−0.631041	−0.315520	+	0.948919i	$0.602179\pi$
$402$	0	0
$403$	−30.5724	−1.52292
$404$	0	0
$405$	5.88131	0.292245
$406$	0	0
$407$	8.09237	0.401124
$408$	0	0
$409$	−31.5521	−1.56015	−0.780075	−	0.625686i	$-0.784819\pi$
$409$	−31.5521	−1.56015	−0.780075	+	0.625686i	$0.784819\pi$
$410$	0	0
$411$	−6.48116	−0.319692
$412$	0	0
$413$	−59.6052	−2.93298
$414$	0	0
$415$	10.7325	0.526840
$416$	0	0
$417$	5.33537	0.261274
$418$	0	0
$419$	−6.97989	−0.340990	−0.170495	−	0.985359i	$-0.554537\pi$
$419$	−6.97989	−0.340990	−0.170495	+	0.985359i	$0.554537\pi$
$420$	0	0
$421$	17.9008	0.872430	0.436215	−	0.899842i	$-0.356319\pi$
$421$	17.9008	0.872430	0.436215	+	0.899842i	$0.356319\pi$
$422$	0	0
$423$	9.75434	0.474272
$424$	0	0
$425$	−0.0471733	−0.00228824
$426$	0	0
$427$	−3.94993	−0.191151
$428$	0	0
$429$	3.39946	0.164128
$430$	0	0
$431$	−0.861039	−0.0414748	−0.0207374	−	0.999785i	$-0.506601\pi$
$431$	−0.861039	−0.0414748	−0.0207374	+	0.999785i	$0.506601\pi$
$432$	0	0
$433$	28.5192	1.37054	0.685272	−	0.728287i	$-0.259683\pi$
$433$	28.5192	1.37054	0.685272	+	0.728287i	$0.259683\pi$
$434$	0	0
$435$	3.68536	0.176700
$436$	0	0
$437$	31.0503	1.48534
$438$	0	0
$439$	6.86832	0.327807	0.163904	−	0.986476i	$-0.447591\pi$
$439$	6.86832	0.327807	0.163904	+	0.986476i	$0.447591\pi$
$440$	0	0
$441$	−31.5319	−1.50152
$442$	0	0
$443$	−9.84299	−0.467655	−0.233827	−	0.972278i	$-0.575125\pi$
$443$	−9.84299	−0.467655	−0.233827	+	0.972278i	$0.575125\pi$
$444$	0	0
$445$	7.93659	0.376230
$446$	0	0
$447$	−3.41875	−0.161702
$448$	0	0
$449$	7.24568	0.341945	0.170972	−	0.985276i	$-0.445309\pi$
$449$	7.24568	0.341945	0.170972	+	0.985276i	$0.445309\pi$
$450$	0	0
$451$	−3.52592	−0.166029
$452$	0	0
$453$	5.89755	0.277091
$454$	0	0
$455$	16.6828	0.782100
$456$	0	0
$457$	39.0284	1.82567	0.912836	−	0.408327i	$-0.133888\pi$
$457$	39.0284	1.82567	0.912836	+	0.408327i	$0.133888\pi$
$458$	0	0
$459$	0.0374103	0.00174616
$460$	0	0
$461$	34.7619	1.61902	0.809512	−	0.587103i	$-0.199732\pi$
$461$	34.7619	1.61902	0.809512	+	0.587103i	$0.199732\pi$
$462$	0	0
$463$	−26.7005	−1.24088	−0.620439	−	0.784254i	$-0.713046\pi$
$463$	−26.7005	−1.24088	−0.620439	+	0.784254i	$0.713046\pi$
$464$	0	0
$465$	4.13388	0.191704
$466$	0	0
$467$	−29.8400	−1.38083	−0.690415	−	0.723414i	$-0.742572\pi$
$467$	−29.8400	−1.38083	−0.690415	+	0.723414i	$0.742572\pi$
$468$	0	0
$469$	−16.5015	−0.761967
$470$	0	0
$471$	3.74783	0.172691
$472$	0	0
$473$	12.3227	0.566600
$474$	0	0
$475$	16.2054	0.743554
$476$	0	0
$477$	−30.8087	−1.41064
$478$	0	0
$479$	21.8689	0.999216	0.499608	−	0.866252i	$-0.333477\pi$
$479$	21.8689	0.999216	0.499608	+	0.866252i	$0.333477\pi$
$480$	0	0
$481$	−22.1436	−1.00966
$482$	0	0
$483$	19.4241	0.883826
$484$	0	0
$485$	2.15666	0.0979291
$486$	0	0
$487$	43.5081	1.97154	0.985770	−	0.168098i	$-0.0537626\pi$
$487$	43.5081	1.97154	0.985770	+	0.168098i	$0.0537626\pi$
$488$	0	0
$489$	−13.8228	−0.625088
$490$	0	0
$491$	−10.7683	−0.485968	−0.242984	−	0.970030i	$-0.578126\pi$
$491$	−10.7683	−0.485968	−0.242984	+	0.970030i	$0.578126\pi$
$492$	0	0
$493$	−0.0779356	−0.00351004
$494$	0	0
$495$	3.75911	0.168960
$496$	0	0
$497$	41.2246	1.84918
$498$	0	0
$499$	24.3170	1.08858	0.544289	−	0.838898i	$-0.316799\pi$
$499$	24.3170	1.08858	0.544289	+	0.838898i	$0.316799\pi$
$500$	0	0
$501$	2.49391	0.111420
$502$	0	0
$503$	−0.496406	−0.0221337	−0.0110668	−	0.999939i	$-0.503523\pi$
$503$	−0.496406	−0.0221337	−0.0110668	+	0.999939i	$0.503523\pi$
$504$	0	0
$505$	3.10687	0.138254
$506$	0	0
$507$	−1.86971	−0.0830366
$508$	0	0
$509$	10.8248	0.479802	0.239901	−	0.970797i	$-0.422885\pi$
$509$	10.8248	0.479802	0.239901	+	0.970797i	$0.422885\pi$
$510$	0	0
$511$	−6.05611	−0.267907
$512$	0	0
$513$	−12.8515	−0.567407
$514$	0	0
$515$	−18.6165	−0.820340
$516$	0	0
$517$	5.37902	0.236569
$518$	0	0
$519$	−2.24613	−0.0985943
$520$	0	0
$521$	1.73823	0.0761532	0.0380766	−	0.999275i	$-0.487877\pi$
$521$	1.73823	0.0761532	0.0380766	+	0.999275i	$0.487877\pi$
$522$	0	0
$523$	28.3752	1.24076	0.620380	−	0.784301i	$-0.286978\pi$
$523$	28.3752	1.24076	0.620380	+	0.784301i	$0.286978\pi$
$524$	0	0
$525$	10.1375	0.442439
$526$	0	0
$527$	−0.0874204	−0.00380809
$528$	0	0
$529$	38.4108	1.67004
$530$	0	0
$531$	36.7514	1.59487
$532$	0	0
$533$	9.64816	0.417908
$534$	0	0
$535$	5.21563	0.225491
$536$	0	0
$537$	−13.5442	−0.584476
$538$	0	0
$539$	−17.3882	−0.748964
$540$	0	0
$541$	33.8125	1.45371	0.726857	−	0.686789i	$-0.240980\pi$
$541$	33.8125	1.45371	0.726857	+	0.686789i	$0.240980\pi$
$542$	0	0
$543$	−14.5948	−0.626321
$544$	0	0
$545$	−2.13457	−0.0914350
$546$	0	0
$547$	1.80024	0.0769728	0.0384864	−	0.999259i	$-0.487746\pi$
$547$	1.80024	0.0769728	0.0384864	+	0.999259i	$0.487746\pi$
$548$	0	0
$549$	2.43545	0.103942
$550$	0	0
$551$	26.7731	1.14057
$552$	0	0
$553$	−55.9055	−2.37734
$554$	0	0
$555$	2.99417	0.127095
$556$	0	0
$557$	17.5544	0.743805	0.371903	−	0.928272i	$-0.378706\pi$
$557$	17.5544	0.743805	0.371903	+	0.928272i	$0.378706\pi$
$558$	0	0
$559$	−33.7193	−1.42618
$560$	0	0
$561$	0.00972062	0.000410405 0
$562$	0	0
$563$	−4.63030	−0.195144	−0.0975719	−	0.995228i	$-0.531108\pi$
$563$	−4.63030	−0.195144	−0.0975719	+	0.995228i	$0.531108\pi$
$564$	0	0
$565$	−3.45181	−0.145219
$566$	0	0
$567$	26.7280	1.12247
$568$	0	0
$569$	−46.3054	−1.94123	−0.970613	−	0.240647i	$-0.922640\pi$
$569$	−46.3054	−1.94123	−0.970613	+	0.240647i	$0.922640\pi$
$570$	0	0
$571$	35.9183	1.50314	0.751568	−	0.659656i	$-0.229298\pi$
$571$	35.9183	1.50314	0.751568	+	0.659656i	$0.229298\pi$
$572$	0	0
$573$	−0.494468	−0.0206567
$574$	0	0
$575$	32.0507	1.33661
$576$	0	0
$577$	−0.739921	−0.0308033	−0.0154017	−	0.999881i	$-0.504903\pi$
$577$	−0.739921	−0.0308033	−0.0154017	+	0.999881i	$0.504903\pi$
$578$	0	0
$579$	6.22826	0.258838
$580$	0	0
$581$	48.7747	2.02351
$582$	0	0
$583$	−16.9895	−0.703631
$584$	0	0
$585$	−10.2863	−0.425284
$586$	0	0
$587$	−13.9550	−0.575985	−0.287992	−	0.957633i	$-0.592988\pi$
$587$	−13.9550	−0.575985	−0.287992	+	0.957633i	$0.592988\pi$
$588$	0	0
$589$	30.0314	1.23742
$590$	0	0
$591$	1.75158	0.0720504
$592$	0	0
$593$	47.8969	1.96689	0.983445	−	0.181205i	$-0.0579999\pi$
$593$	47.8969	1.96689	0.983445	+	0.181205i	$0.0579999\pi$
$594$	0	0
$595$	0.0477037	0.00195566
$596$	0	0
$597$	7.10938	0.290967
$598$	0	0
$599$	−13.3939	−0.547262	−0.273631	−	0.961835i	$-0.588225\pi$
$599$	−13.3939	−0.547262	−0.273631	+	0.961835i	$0.588225\pi$
$600$	0	0
$601$	−33.9145	−1.38340	−0.691700	−	0.722185i	$-0.743138\pi$
$601$	−33.9145	−1.38340	−0.691700	+	0.722185i	$0.743138\pi$
$602$	0	0
$603$	10.1745	0.414336
$604$	0	0
$605$	−8.42082	−0.342355
$606$	0	0
$607$	9.00702	0.365584	0.182792	−	0.983152i	$-0.441487\pi$
$607$	9.00702	0.365584	0.182792	+	0.983152i	$0.441487\pi$
$608$	0	0
$609$	16.7484	0.678678
$610$	0	0
$611$	−14.7189	−0.595463
$612$	0	0
$613$	−33.1363	−1.33836	−0.669181	−	0.743099i	$-0.733355\pi$
$613$	−33.1363	−1.33836	−0.669181	+	0.743099i	$0.733355\pi$
$614$	0	0
$615$	−1.30459	−0.0526060
$616$	0	0
$617$	−10.0083	−0.402919	−0.201460	−	0.979497i	$-0.564568\pi$
$617$	−10.0083	−0.402919	−0.201460	+	0.979497i	$0.564568\pi$
$618$	0	0
$619$	25.1902	1.01248	0.506239	−	0.862393i	$-0.331035\pi$
$619$	25.1902	1.01248	0.506239	+	0.862393i	$0.331035\pi$
$620$	0	0
$621$	−25.4175	−1.01997
$622$	0	0
$623$	36.0683	1.44505
$624$	0	0
$625$	12.1771	0.487083
$626$	0	0
$627$	−3.33931	−0.133359
$628$	0	0
$629$	−0.0633186	−0.00252468
$630$	0	0
$631$	12.1422	0.483373	0.241686	−	0.970354i	$-0.422299\pi$
$631$	12.1422	0.483373	0.241686	+	0.970354i	$0.422299\pi$
$632$	0	0
$633$	−1.12066	−0.0445423
$634$	0	0
$635$	0.365344	0.0144982
$636$	0	0
$637$	47.5803	1.88520
$638$	0	0
$639$	−25.4183	−1.00553
$640$	0	0
$641$	−0.884743	−0.0349453	−0.0174726	−	0.999847i	$-0.505562\pi$
$641$	−0.884743	−0.0349453	−0.0174726	+	0.999847i	$0.505562\pi$
$642$	0	0
$643$	−31.3606	−1.23674	−0.618370	−	0.785887i	$-0.712207\pi$
$643$	−31.3606	−1.23674	−0.618370	+	0.785887i	$0.712207\pi$
$644$	0	0
$645$	4.55940	0.179526
$646$	0	0
$647$	−18.6153	−0.731842	−0.365921	−	0.930646i	$-0.619246\pi$
$647$	−18.6153	−0.731842	−0.365921	+	0.930646i	$0.619246\pi$
$648$	0	0
$649$	20.2665	0.795530
$650$	0	0
$651$	18.7867	0.736307
$652$	0	0
$653$	11.0616	0.432872	0.216436	−	0.976297i	$-0.430557\pi$
$653$	11.0616	0.432872	0.216436	+	0.976297i	$0.430557\pi$
$654$	0	0
$655$	17.4277	0.680958
$656$	0	0
$657$	3.73408	0.145680
$658$	0	0
$659$	−6.22540	−0.242507	−0.121254	−	0.992622i	$-0.538691\pi$
$659$	−6.22540	−0.242507	−0.121254	+	0.992622i	$0.538691\pi$
$660$	0	0
$661$	8.54515	0.332368	0.166184	−	0.986095i	$-0.446855\pi$
$661$	8.54515	0.332368	0.166184	+	0.986095i	$0.446855\pi$
$662$	0	0
$663$	−0.0265990	−0.00103302
$664$	0	0
$665$	−16.3876	−0.635482
$666$	0	0
$667$	52.9513	2.05028
$668$	0	0
$669$	5.91118	0.228539
$670$	0	0
$671$	1.34303	0.0518470
$672$	0	0
$673$	45.8550	1.76758	0.883790	−	0.467884i	$-0.154983\pi$
$673$	45.8550	1.76758	0.883790	+	0.467884i	$0.154983\pi$
$674$	0	0
$675$	−13.2655	−0.510591
$676$	0	0
$677$	45.5854	1.75199	0.875994	−	0.482321i	$-0.160206\pi$
$677$	45.5854	1.75199	0.875994	+	0.482321i	$0.160206\pi$
$678$	0	0
$679$	9.80109	0.376131
$680$	0	0
$681$	16.7306	0.641117
$682$	0	0
$683$	−6.46620	−0.247422	−0.123711	−	0.992318i	$-0.539480\pi$
$683$	−6.46620	−0.247422	−0.123711	+	0.992318i	$0.539480\pi$
$684$	0	0
$685$	10.8145	0.413200
$686$	0	0
$687$	2.88891	0.110219
$688$	0	0
$689$	46.4891	1.77109
$690$	0	0
$691$	24.4978	0.931941	0.465970	−	0.884800i	$-0.345705\pi$
$691$	24.4978	0.931941	0.465970	+	0.884800i	$0.345705\pi$
$692$	0	0
$693$	17.0835	0.648949
$694$	0	0
$695$	−8.90260	−0.337695
$696$	0	0
$697$	0.0275885	0.00104499
$698$	0	0
$699$	2.04655	0.0774078
$700$	0	0
$701$	8.03244	0.303381	0.151691	−	0.988428i	$-0.451528\pi$
$701$	8.03244	0.303381	0.151691	+	0.988428i	$0.451528\pi$
$702$	0	0
$703$	21.7517	0.820383
$704$	0	0
$705$	1.99023	0.0749564
$706$	0	0
$707$	14.1193	0.531012
$708$	0	0
$709$	−24.3761	−0.915465	−0.457733	−	0.889090i	$-0.651338\pi$
$709$	−24.3761	−0.915465	−0.457733	+	0.889090i	$0.651338\pi$
$710$	0	0
$711$	34.4702	1.29273
$712$	0	0
$713$	59.3956	2.22438
$714$	0	0
$715$	−5.67235	−0.212134
$716$	0	0
$717$	−9.79382	−0.365757
$718$	0	0
$719$	−35.2258	−1.31370	−0.656851	−	0.754020i	$-0.728112\pi$
$719$	−35.2258	−1.31370	−0.656851	+	0.754020i	$0.728112\pi$
$720$	0	0
$721$	−84.6037	−3.15081
$722$	0	0
$723$	4.00417	0.148917
$724$	0	0
$725$	27.6356	1.02636
$726$	0	0
$727$	40.5328	1.50328	0.751640	−	0.659574i	$-0.229263\pi$
$727$	40.5328	1.50328	0.751640	+	0.659574i	$0.229263\pi$
$728$	0	0
$729$	−10.9168	−0.404326
$730$	0	0
$731$	−0.0964190	−0.00356619
$732$	0	0
$733$	−39.0072	−1.44076	−0.720382	−	0.693578i	$-0.756033\pi$
$733$	−39.0072	−1.44076	−0.720382	+	0.693578i	$0.756033\pi$
$734$	0	0
$735$	−6.43362	−0.237308
$736$	0	0
$737$	5.61070	0.206673
$738$	0	0
$739$	10.4487	0.384361	0.192180	−	0.981360i	$-0.438444\pi$
$739$	10.4487	0.384361	0.192180	+	0.981360i	$0.438444\pi$
$740$	0	0
$741$	9.13752	0.335675
$742$	0	0
$743$	31.0065	1.13752	0.568759	−	0.822504i	$-0.307424\pi$
$743$	31.0065	1.13752	0.568759	+	0.822504i	$0.307424\pi$
$744$	0	0
$745$	5.70454	0.208998
$746$	0	0
$747$	−30.0735	−1.10033
$748$	0	0
$749$	23.7027	0.866079
$750$	0	0
$751$	38.1076	1.39057	0.695283	−	0.718736i	$-0.255279\pi$
$751$	38.1076	1.39057	0.695283	+	0.718736i	$0.255279\pi$
$752$	0	0
$753$	0.571724	0.0208348
$754$	0	0
$755$	−9.84066	−0.358138
$756$	0	0
$757$	−6.77822	−0.246359	−0.123179	−	0.992384i	$-0.539309\pi$
$757$	−6.77822	−0.246359	−0.123179	+	0.992384i	$0.539309\pi$
$758$	0	0
$759$	−6.60442	−0.239725
$760$	0	0
$761$	5.75991	0.208797	0.104398	−	0.994536i	$-0.466708\pi$
$761$	5.75991	0.208797	0.104398	+	0.994536i	$0.466708\pi$
$762$	0	0
$763$	−9.70069	−0.351188
$764$	0	0
$765$	−0.0294131	−0.00106343
$766$	0	0
$767$	−55.4563	−2.00241
$768$	0	0
$769$	−48.5049	−1.74913	−0.874567	−	0.484905i	$-0.838854\pi$
$769$	−48.5049	−1.74913	−0.874567	+	0.484905i	$0.838854\pi$
$770$	0	0
$771$	−0.123149	−0.00443512
$772$	0	0
$773$	−35.4012	−1.27329	−0.636647	−	0.771155i	$-0.719679\pi$
$773$	−35.4012	−1.27329	−0.636647	+	0.771155i	$0.719679\pi$
$774$	0	0
$775$	30.9989	1.11351
$776$	0	0
$777$	13.6072	0.488155
$778$	0	0
$779$	−9.47744	−0.339564
$780$	0	0
$781$	−14.0169	−0.501563
$782$	0	0
$783$	−21.9161	−0.783219
$784$	0	0
$785$	−6.25364	−0.223202
$786$	0	0
$787$	31.9623	1.13933	0.569667	−	0.821876i	$-0.307072\pi$
$787$	31.9623	1.13933	0.569667	+	0.821876i	$0.307072\pi$
$788$	0	0
$789$	−11.5722	−0.411983
$790$	0	0
$791$	−15.6869	−0.557764
$792$	0	0
$793$	−3.67499	−0.130503
$794$	0	0
$795$	−6.28608	−0.222944
$796$	0	0
$797$	−24.9148	−0.882526	−0.441263	−	0.897378i	$-0.645469\pi$
$797$	−24.9148	−0.882526	−0.441263	+	0.897378i	$0.645469\pi$
$798$	0	0
$799$	−0.0420880	−0.00148897
$800$	0	0
$801$	−22.2390	−0.785776
$802$	0	0
$803$	2.05915	0.0726659
$804$	0	0
$805$	−32.4110	−1.14234
$806$	0	0
$807$	−2.34285	−0.0824722
$808$	0	0
$809$	0.664767	0.0233720	0.0116860	−	0.999932i	$-0.496280\pi$
$809$	0.664767	0.0233720	0.0116860	+	0.999932i	$0.496280\pi$
$810$	0	0
$811$	−40.1050	−1.40828	−0.704139	−	0.710062i	$-0.748667\pi$
$811$	−40.1050	−1.40828	−0.704139	+	0.710062i	$0.748667\pi$
$812$	0	0
$813$	−8.46185	−0.296770
$814$	0	0
$815$	23.0647	0.807921
$816$	0	0
$817$	33.1227	1.15882
$818$	0	0
$819$	−46.7465	−1.63345
$820$	0	0
$821$	−4.75264	−0.165868	−0.0829341	−	0.996555i	$-0.526429\pi$
$821$	−4.75264	−0.165868	−0.0829341	+	0.996555i	$0.526429\pi$
$822$	0	0
$823$	−12.5486	−0.437418	−0.218709	−	0.975790i	$-0.570185\pi$
$823$	−12.5486	−0.437418	−0.218709	+	0.975790i	$0.570185\pi$
$824$	0	0
$825$	−3.44689	−0.120005
$826$	0	0
$827$	−8.40671	−0.292330	−0.146165	−	0.989260i	$-0.546693\pi$
$827$	−8.40671	−0.292330	−0.146165	+	0.989260i	$0.546693\pi$
$828$	0	0
$829$	39.3884	1.36802	0.684008	−	0.729475i	$-0.260235\pi$
$829$	39.3884	1.36802	0.684008	+	0.729475i	$0.260235\pi$
$830$	0	0
$831$	6.45548	0.223938
$832$	0	0
$833$	0.136054	0.00471399
$834$	0	0
$835$	−4.16134	−0.144009
$836$	0	0
$837$	−24.5834	−0.849725
$838$	0	0
$839$	−7.94451	−0.274275	−0.137138	−	0.990552i	$-0.543790\pi$
$839$	−7.94451	−0.274275	−0.137138	+	0.990552i	$0.543790\pi$
$840$	0	0
$841$	16.6572	0.574385
$842$	0	0
$843$	−7.07058	−0.243524
$844$	0	0
$845$	3.11979	0.107324
$846$	0	0
$847$	−38.2689	−1.31494
$848$	0	0
$849$	−2.04699	−0.0702524
$850$	0	0
$851$	43.0202	1.47471
$852$	0	0
$853$	21.4936	0.735925	0.367963	−	0.929841i	$-0.380055\pi$
$853$	21.4936	0.735925	0.367963	+	0.929841i	$0.380055\pi$
$854$	0	0
$855$	10.1042	0.345558
$856$	0	0
$857$	−23.3083	−0.796195	−0.398098	−	0.917343i	$-0.630329\pi$
$857$	−23.3083	−0.796195	−0.398098	+	0.917343i	$0.630329\pi$
$858$	0	0
$859$	−18.0200	−0.614835	−0.307417	−	0.951575i	$-0.599465\pi$
$859$	−18.0200	−0.614835	−0.307417	+	0.951575i	$0.599465\pi$
$860$	0	0
$861$	−5.92877	−0.202052
$862$	0	0
$863$	−54.0180	−1.83880	−0.919398	−	0.393329i	$-0.871323\pi$
$863$	−54.0180	−1.83880	−0.919398	+	0.393329i	$0.871323\pi$
$864$	0	0
$865$	3.74790	0.127432
$866$	0	0
$867$	9.71924	0.330083
$868$	0	0
$869$	19.0085	0.644821
$870$	0	0
$871$	−15.3529	−0.520212
$872$	0	0
$873$	−6.04315	−0.204530
$874$	0	0
$875$	−37.5950	−1.27094
$876$	0	0
$877$	55.4313	1.87178	0.935891	−	0.352291i	$-0.114597\pi$
$877$	55.4313	1.87178	0.935891	+	0.352291i	$0.114597\pi$
$878$	0	0
$879$	−17.5033	−0.590372
$880$	0	0
$881$	−13.8652	−0.467130	−0.233565	−	0.972341i	$-0.575039\pi$
$881$	−13.8652	−0.467130	−0.233565	+	0.972341i	$0.575039\pi$
$882$	0	0
$883$	14.5523	0.489723	0.244862	−	0.969558i	$-0.421257\pi$
$883$	14.5523	0.489723	0.244862	+	0.969558i	$0.421257\pi$
$884$	0	0
$885$	7.49858	0.252062
$886$	0	0
$887$	−27.1395	−0.911255	−0.455627	−	0.890171i	$-0.650585\pi$
$887$	−27.1395	−0.911255	−0.455627	+	0.890171i	$0.650585\pi$
$888$	0	0
$889$	1.66033	0.0556856
$890$	0	0
$891$	−9.08784	−0.304454
$892$	0	0
$893$	14.4584	0.483833
$894$	0	0
$895$	22.5999	0.755430
$896$	0	0
$897$	18.0720	0.603407
$898$	0	0
$899$	51.2137	1.70807
$900$	0	0
$901$	0.132934	0.00442866
$902$	0	0
$903$	20.7205	0.689533
$904$	0	0
$905$	24.3528	0.809515
$906$	0	0
$907$	−21.8933	−0.726954	−0.363477	−	0.931603i	$-0.618411\pi$
$907$	−21.8933	−0.726954	−0.363477	+	0.931603i	$0.618411\pi$
$908$	0	0
$909$	−8.70570	−0.288750
$910$	0	0
$911$	30.1321	0.998319	0.499160	−	0.866510i	$-0.333642\pi$
$911$	30.1321	0.998319	0.499160	+	0.866510i	$0.333642\pi$
$912$	0	0
$913$	−16.5840	−0.548850
$914$	0	0
$915$	0.496918	0.0164276
$916$	0	0
$917$	79.2014	2.61546
$918$	0	0
$919$	24.8532	0.819832	0.409916	−	0.912123i	$-0.365558\pi$
$919$	24.8532	0.819832	0.409916	+	0.912123i	$0.365558\pi$
$920$	0	0
$921$	3.09819	0.102089
$922$	0	0
$923$	38.3551	1.26247
$924$	0	0
$925$	22.4525	0.738235
$926$	0	0
$927$	52.1649	1.71332
$928$	0	0
$929$	−21.0448	−0.690457	−0.345228	−	0.938519i	$-0.612199\pi$
$929$	−21.0448	−0.690457	−0.345228	+	0.938519i	$0.612199\pi$
$930$	0	0
$931$	−46.7384	−1.53179
$932$	0	0
$933$	−4.74209	−0.155249
$934$	0	0
$935$	−0.0162198	−0.000530445 0
$936$	0	0
$937$	43.1939	1.41108	0.705541	−	0.708669i	$-0.250704\pi$
$937$	43.1939	1.41108	0.705541	+	0.708669i	$0.250704\pi$
$938$	0	0
$939$	4.22049	0.137731
$940$	0	0
$941$	−23.4887	−0.765710	−0.382855	−	0.923808i	$-0.625059\pi$
$941$	−23.4887	−0.765710	−0.382855	+	0.923808i	$0.625059\pi$
$942$	0	0
$943$	−18.7443	−0.610399
$944$	0	0
$945$	13.4147	0.436379
$946$	0	0
$947$	−30.4176	−0.988439	−0.494220	−	0.869337i	$-0.664546\pi$
$947$	−30.4176	−0.988439	−0.494220	+	0.869337i	$0.664546\pi$
$948$	0	0
$949$	−5.63457	−0.182906
$950$	0	0
$951$	6.34216	0.205659
$952$	0	0
$953$	−3.08799	−0.100030	−0.0500150	−	0.998748i	$-0.515927\pi$
$953$	−3.08799	−0.100030	−0.0500150	+	0.998748i	$0.515927\pi$
$954$	0	0
$955$	0.825071	0.0266987
$956$	0	0
$957$	−5.69465	−0.184082
$958$	0	0
$959$	49.1470	1.58704
$960$	0	0
$961$	26.4465	0.853111
$962$	0	0
$963$	−14.6146	−0.470950
$964$	0	0
$965$	−10.3925	−0.334546
$966$	0	0
$967$	25.1314	0.808173	0.404086	−	0.914721i	$-0.367590\pi$
$967$	25.1314	0.808173	0.404086	+	0.914721i	$0.367590\pi$
$968$	0	0
$969$	0.0261284	0.000839364 0
$970$	0	0
$971$	−9.95549	−0.319487	−0.159743	−	0.987159i	$-0.551067\pi$
$971$	−9.95549	−0.319487	−0.159743	+	0.987159i	$0.551067\pi$
$972$	0	0
$973$	−40.4584	−1.29704
$974$	0	0
$975$	9.43191	0.302063
$976$	0	0
$977$	−29.0869	−0.930571	−0.465286	−	0.885161i	$-0.654048\pi$
$977$	−29.0869	−0.930571	−0.465286	+	0.885161i	$0.654048\pi$
$978$	0	0
$979$	−12.2637	−0.391949
$980$	0	0
$981$	5.98125	0.190967
$982$	0	0
$983$	29.3161	0.935038	0.467519	−	0.883983i	$-0.345148\pi$
$983$	29.3161	0.935038	0.467519	+	0.883983i	$0.345148\pi$
$984$	0	0
$985$	−2.92269	−0.0931246
$986$	0	0
$987$	9.04472	0.287897
$988$	0	0
$989$	65.5094	2.08308
$990$	0	0
$991$	−33.7133	−1.07094	−0.535469	−	0.844555i	$-0.679865\pi$
$991$	−33.7133	−1.07094	−0.535469	+	0.844555i	$0.679865\pi$
$992$	0	0
$993$	0.0200842	0.000637353 0
$994$	0	0
$995$	−11.8627	−0.376073
$996$	0	0
$997$	−34.8265	−1.10296	−0.551482	−	0.834186i	$-0.685938\pi$
$997$	−34.8265	−1.10296	−0.551482	+	0.834186i	$0.685938\pi$
$998$	0	0
$999$	−17.8057	−0.563348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.6		12
4.3	odd	2		2008.2.a.b.1.7	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.7	✓	12	4.3	odd	2
4016.2.a.i.1.6		12	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)

\(f(q)\)	\(=\)	\(q-0.571724 q^{3} +0.953980 q^{5} +4.33542 q^{7} -2.67313 q^{9} +O(q^{10})\)	\(q-0.571724 q^{3} +0.953980 q^{5} +4.33542 q^{7} -2.67313 q^{9} -1.47410 q^{11} +4.03365 q^{13} -0.545414 q^{15} +0.0115340 q^{17} -3.96227 q^{19} -2.47867 q^{21} -7.83651 q^{23} -4.08992 q^{25} +3.24347 q^{27} -6.75701 q^{29} -7.57934 q^{31} +0.842777 q^{33} +4.13590 q^{35} -5.48972 q^{37} -2.30613 q^{39} +2.39192 q^{41} -8.35952 q^{43} -2.55011 q^{45} -3.64903 q^{47} +11.7959 q^{49} -0.00659429 q^{51} +11.5253 q^{53} -1.40626 q^{55} +2.26533 q^{57} -13.7484 q^{59} -0.911085 q^{61} -11.5891 q^{63} +3.84802 q^{65} -3.80620 q^{67} +4.48032 q^{69} +9.50880 q^{71} -1.39689 q^{73} +2.33831 q^{75} -6.39082 q^{77} -12.8951 q^{79} +6.16502 q^{81} +11.2503 q^{83} +0.0110032 q^{85} +3.86315 q^{87} +8.31945 q^{89} +17.4875 q^{91} +4.33330 q^{93} -3.77993 q^{95} +2.26070 q^{97} +3.94045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

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Varieties

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Representations

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Database

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