LMFDB - Embedded newform 4016.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.64615$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.64615 q^{3} -4.00715 q^{5} -2.94760 q^{7} -0.290201 q^{9} +O(q^{10})$	$q+1.64615 q^{3} -4.00715 q^{5} -2.94760 q^{7} -0.290201 q^{9} +4.00704 q^{11} +2.51643 q^{13} -6.59635 q^{15} -1.26481 q^{17} +7.62365 q^{19} -4.85218 q^{21} +2.11058 q^{23} +11.0572 q^{25} -5.41615 q^{27} +1.18258 q^{29} -5.32238 q^{31} +6.59618 q^{33} +11.8115 q^{35} -0.498716 q^{37} +4.14242 q^{39} +2.30758 q^{41} -12.3629 q^{43} +1.16288 q^{45} -9.43666 q^{47} +1.68834 q^{49} -2.08206 q^{51} +7.82052 q^{53} -16.0568 q^{55} +12.5496 q^{57} -1.36116 q^{59} -10.6863 q^{61} +0.855396 q^{63} -10.0837 q^{65} -4.80774 q^{67} +3.47433 q^{69} -9.78259 q^{71} +5.77485 q^{73} +18.2018 q^{75} -11.8112 q^{77} -3.33597 q^{79} -8.04518 q^{81} +1.29630 q^{83} +5.06828 q^{85} +1.94670 q^{87} -13.9752 q^{89} -7.41744 q^{91} -8.76142 q^{93} -30.5491 q^{95} -1.37411 q^{97} -1.16285 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.64615	0.950403	0.475202	−	0.879877i	$-0.342375\pi$
$3$	1.64615	0.950403	0.475202	+	0.879877i	$0.342375\pi$
$4$	0	0
$5$	−4.00715	−1.79205	−0.896025	−	0.444003i	$-0.853558\pi$
$5$	−4.00715	−1.79205	−0.896025	+	0.444003i	$0.853558\pi$
$6$	0	0
$7$	−2.94760	−1.11409	−0.557044	−	0.830483i	$-0.688064\pi$
$7$	−2.94760	−1.11409	−0.557044	+	0.830483i	$0.688064\pi$
$8$	0	0
$9$	−0.290201	−0.0967337
$10$	0	0
$11$	4.00704	1.20817	0.604084	−	0.796920i	$-0.293539\pi$
$11$	4.00704	1.20817	0.604084	+	0.796920i	$0.293539\pi$
$12$	0	0
$13$	2.51643	0.697933	0.348966	−	0.937135i	$-0.386533\pi$
$13$	2.51643	0.697933	0.348966	+	0.937135i	$0.386533\pi$
$14$	0	0
$15$	−6.59635	−1.70317
$16$	0	0
$17$	−1.26481	−0.306762	−0.153381	−	0.988167i	$-0.549016\pi$
$17$	−1.26481	−0.306762	−0.153381	+	0.988167i	$0.549016\pi$
$18$	0	0
$19$	7.62365	1.74898	0.874492	−	0.485039i	$-0.161195\pi$
$19$	7.62365	1.74898	0.874492	+	0.485039i	$0.161195\pi$
$20$	0	0
$21$	−4.85218	−1.05883
$22$	0	0
$23$	2.11058	0.440087	0.220043	−	0.975490i	$-0.429380\pi$
$23$	2.11058	0.440087	0.220043	+	0.975490i	$0.429380\pi$
$24$	0	0
$25$	11.0572	2.21144
$26$	0	0
$27$	−5.41615	−1.04234
$28$	0	0
$29$	1.18258	0.219600	0.109800	−	0.993954i	$-0.464979\pi$
$29$	1.18258	0.219600	0.109800	+	0.993954i	$0.464979\pi$
$30$	0	0
$31$	−5.32238	−0.955928	−0.477964	−	0.878379i	$-0.658625\pi$
$31$	−5.32238	−0.955928	−0.477964	+	0.878379i	$0.658625\pi$
$32$	0	0
$33$	6.59618	1.14825
$34$	0	0
$35$	11.8115	1.99650
$36$	0	0
$37$	−0.498716	−0.0819884	−0.0409942	−	0.999159i	$-0.513053\pi$
$37$	−0.498716	−0.0819884	−0.0409942	+	0.999159i	$0.513053\pi$
$38$	0	0
$39$	4.14242	0.663318
$40$	0	0
$41$	2.30758	0.360384	0.180192	−	0.983631i	$-0.442328\pi$
$41$	2.30758	0.360384	0.180192	+	0.983631i	$0.442328\pi$
$42$	0	0
$43$	−12.3629	−1.88532	−0.942661	−	0.333753i	$-0.891685\pi$
$43$	−12.3629	−1.88532	−0.942661	+	0.333753i	$0.891685\pi$
$44$	0	0
$45$	1.16288	0.173352
$46$	0	0
$47$	−9.43666	−1.37648	−0.688239	−	0.725484i	$-0.741616\pi$
$47$	−9.43666	−1.37648	−0.688239	+	0.725484i	$0.741616\pi$
$48$	0	0
$49$	1.68834	0.241191
$50$	0	0
$51$	−2.08206	−0.291547
$52$	0	0
$53$	7.82052	1.07423	0.537116	−	0.843509i	$-0.319514\pi$
$53$	7.82052	1.07423	0.537116	+	0.843509i	$0.319514\pi$
$54$	0	0
$55$	−16.0568	−2.16510
$56$	0	0
$57$	12.5496	1.66224
$58$	0	0
$59$	−1.36116	−0.177208	−0.0886040	−	0.996067i	$-0.528241\pi$
$59$	−1.36116	−0.177208	−0.0886040	+	0.996067i	$0.528241\pi$
$60$	0	0
$61$	−10.6863	−1.36824	−0.684122	−	0.729368i	$-0.739814\pi$
$61$	−10.6863	−1.36824	−0.684122	+	0.729368i	$0.739814\pi$
$62$	0	0
$63$	0.855396	0.107770
$64$	0	0
$65$	−10.0837	−1.25073
$66$	0	0
$67$	−4.80774	−0.587359	−0.293680	−	0.955904i	$-0.594880\pi$
$67$	−4.80774	−0.587359	−0.293680	+	0.955904i	$0.594880\pi$
$68$	0	0
$69$	3.47433	0.418260
$70$	0	0
$71$	−9.78259	−1.16098	−0.580490	−	0.814268i	$-0.697139\pi$
$71$	−9.78259	−1.16098	−0.580490	+	0.814268i	$0.697139\pi$
$72$	0	0
$73$	5.77485	0.675895	0.337948	−	0.941165i	$-0.390267\pi$
$73$	5.77485	0.675895	0.337948	+	0.941165i	$0.390267\pi$
$74$	0	0
$75$	18.2018	2.10176
$76$	0	0
$77$	−11.8112	−1.34601
$78$	0	0
$79$	−3.33597	−0.375326	−0.187663	−	0.982233i	$-0.560091\pi$
$79$	−3.33597	−0.375326	−0.187663	+	0.982233i	$0.560091\pi$
$80$	0	0
$81$	−8.04518	−0.893909
$82$	0	0
$83$	1.29630	0.142288	0.0711440	−	0.997466i	$-0.477335\pi$
$83$	1.29630	0.142288	0.0711440	+	0.997466i	$0.477335\pi$
$84$	0	0
$85$	5.06828	0.549732
$86$	0	0
$87$	1.94670	0.208709
$88$	0	0
$89$	−13.9752	−1.48137	−0.740686	−	0.671852i	$-0.765499\pi$
$89$	−13.9752	−1.48137	−0.740686	+	0.671852i	$0.765499\pi$
$90$	0	0
$91$	−7.41744	−0.777559
$92$	0	0
$93$	−8.76142	−0.908517
$94$	0	0
$95$	−30.5491	−3.13427
$96$	0	0
$97$	−1.37411	−0.139519	−0.0697597	−	0.997564i	$-0.522223\pi$
$97$	−1.37411	−0.139519	−0.0697597	+	0.997564i	$0.522223\pi$
$98$	0	0
$99$	−1.16285	−0.116871
$100$	0	0
$101$	14.0215	1.39520	0.697598	−	0.716489i	$-0.254252\pi$
$101$	14.0215	1.39520	0.697598	+	0.716489i	$0.254252\pi$
$102$	0	0
$103$	−6.16148	−0.607109	−0.303554	−	0.952814i	$-0.598173\pi$
$103$	−6.16148	−0.607109	−0.303554	+	0.952814i	$0.598173\pi$
$104$	0	0
$105$	19.4434	1.89748
$106$	0	0
$107$	19.0443	1.84108	0.920540	−	0.390648i	$-0.127749\pi$
$107$	19.0443	1.84108	0.920540	+	0.390648i	$0.127749\pi$
$108$	0	0
$109$	5.03123	0.481904	0.240952	−	0.970537i	$-0.422540\pi$
$109$	5.03123	0.481904	0.240952	+	0.970537i	$0.422540\pi$
$110$	0	0
$111$	−0.820960	−0.0779221
$112$	0	0
$113$	−14.2857	−1.34389	−0.671945	−	0.740601i	$-0.734541\pi$
$113$	−14.2857	−1.34389	−0.671945	+	0.740601i	$0.734541\pi$
$114$	0	0
$115$	−8.45741	−0.788658
$116$	0	0
$117$	−0.730271	−0.0675136
$118$	0	0
$119$	3.72815	0.341759
$120$	0	0
$121$	5.05638	0.459671
$122$	0	0
$123$	3.79862	0.342510
$124$	0	0
$125$	−24.2722	−2.17097
$126$	0	0
$127$	−3.43426	−0.304741	−0.152371	−	0.988323i	$-0.548691\pi$
$127$	−3.43426	−0.304741	−0.152371	+	0.988323i	$0.548691\pi$
$128$	0	0
$129$	−20.3511	−1.79182
$130$	0	0
$131$	−6.73505	−0.588444	−0.294222	−	0.955737i	$-0.595061\pi$
$131$	−6.73505	−0.588444	−0.294222	+	0.955737i	$0.595061\pi$
$132$	0	0
$133$	−22.4715	−1.94852
$134$	0	0
$135$	21.7033	1.86792
$136$	0	0
$137$	−0.110622	−0.00945104	−0.00472552	−	0.999989i	$-0.501504\pi$
$137$	−0.110622	−0.00945104	−0.00472552	+	0.999989i	$0.501504\pi$
$138$	0	0
$139$	5.52678	0.468775	0.234388	−	0.972143i	$-0.424692\pi$
$139$	5.52678	0.468775	0.234388	+	0.972143i	$0.424692\pi$
$140$	0	0
$141$	−15.5341	−1.30821
$142$	0	0
$143$	10.0835	0.843221
$144$	0	0
$145$	−4.73878	−0.393534
$146$	0	0
$147$	2.77926	0.229229
$148$	0	0
$149$	7.17645	0.587918	0.293959	−	0.955818i	$-0.405027\pi$
$149$	7.17645	0.587918	0.293959	+	0.955818i	$0.405027\pi$
$150$	0	0
$151$	−13.4763	−1.09668	−0.548341	−	0.836255i	$-0.684740\pi$
$151$	−13.4763	−1.09668	−0.548341	+	0.836255i	$0.684740\pi$
$152$	0	0
$153$	0.367049	0.0296742
$154$	0	0
$155$	21.3276	1.71307
$156$	0	0
$157$	−13.9660	−1.11461	−0.557304	−	0.830309i	$-0.688164\pi$
$157$	−13.9660	−1.11461	−0.557304	+	0.830309i	$0.688164\pi$
$158$	0	0
$159$	12.8737	1.02095
$160$	0	0
$161$	−6.22115	−0.490295
$162$	0	0
$163$	−17.7137	−1.38744	−0.693722	−	0.720243i	$-0.744030\pi$
$163$	−17.7137	−1.38744	−0.693722	+	0.720243i	$0.744030\pi$
$164$	0	0
$165$	−26.4319	−2.05772
$166$	0	0
$167$	14.6357	1.13254	0.566271	−	0.824219i	$-0.308386\pi$
$167$	14.6357	1.13254	0.566271	+	0.824219i	$0.308386\pi$
$168$	0	0
$169$	−6.66756	−0.512890
$170$	0	0
$171$	−2.21239	−0.169186
$172$	0	0
$173$	18.9679	1.44210	0.721050	−	0.692883i	$-0.243660\pi$
$173$	18.9679	1.44210	0.721050	+	0.692883i	$0.243660\pi$
$174$	0	0
$175$	−32.5923	−2.46374
$176$	0	0
$177$	−2.24067	−0.168419
$178$	0	0
$179$	−19.1969	−1.43485	−0.717423	−	0.696638i	$-0.754678\pi$
$179$	−19.1969	−1.43485	−0.717423	+	0.696638i	$0.754678\pi$
$180$	0	0
$181$	−16.2798	−1.21007	−0.605033	−	0.796200i	$-0.706840\pi$
$181$	−16.2798	−1.21007	−0.605033	+	0.796200i	$0.706840\pi$
$182$	0	0
$183$	−17.5913	−1.30038
$184$	0	0
$185$	1.99843	0.146927
$186$	0	0
$187$	−5.06815	−0.370620
$188$	0	0
$189$	15.9646	1.16126
$190$	0	0
$191$	−24.4396	−1.76839	−0.884194	−	0.467119i	$-0.845292\pi$
$191$	−24.4396	−1.76839	−0.884194	+	0.467119i	$0.845292\pi$
$192$	0	0
$193$	9.20612	0.662671	0.331335	−	0.943513i	$-0.392501\pi$
$193$	9.20612	0.662671	0.331335	+	0.943513i	$0.392501\pi$
$194$	0	0
$195$	−16.5993	−1.18870
$196$	0	0
$197$	−10.2336	−0.729117	−0.364558	−	0.931181i	$-0.618780\pi$
$197$	−10.2336	−0.729117	−0.364558	+	0.931181i	$0.618780\pi$
$198$	0	0
$199$	13.5062	0.957433	0.478716	−	0.877970i	$-0.341102\pi$
$199$	13.5062	0.957433	0.478716	+	0.877970i	$0.341102\pi$
$200$	0	0
$201$	−7.91425	−0.558228
$202$	0	0
$203$	−3.48578	−0.244654
$204$	0	0
$205$	−9.24683	−0.645826
$206$	0	0
$207$	−0.612493	−0.0425712
$208$	0	0
$209$	30.5483	2.11307
$210$	0	0
$211$	−24.2735	−1.67105	−0.835527	−	0.549449i	$-0.814838\pi$
$211$	−24.2735	−1.67105	−0.835527	+	0.549449i	$0.814838\pi$
$212$	0	0
$213$	−16.1036	−1.10340
$214$	0	0
$215$	49.5399	3.37859
$216$	0	0
$217$	15.6882	1.06499
$218$	0	0
$219$	9.50625	0.642373
$220$	0	0
$221$	−3.18281	−0.214099
$222$	0	0
$223$	2.35768	0.157882	0.0789411	−	0.996879i	$-0.474846\pi$
$223$	2.35768	0.157882	0.0789411	+	0.996879i	$0.474846\pi$
$224$	0	0
$225$	−3.20882	−0.213921
$226$	0	0
$227$	−19.2348	−1.27666	−0.638330	−	0.769763i	$-0.720374\pi$
$227$	−19.2348	−1.27666	−0.638330	+	0.769763i	$0.720374\pi$
$228$	0	0
$229$	0.0398429	0.00263289	0.00131645	−	0.999999i	$-0.499581\pi$
$229$	0.0398429	0.00263289	0.00131645	+	0.999999i	$0.499581\pi$
$230$	0	0
$231$	−19.4429	−1.27925
$232$	0	0
$233$	11.4704	0.751448	0.375724	−	0.926732i	$-0.377394\pi$
$233$	11.4704	0.751448	0.375724	+	0.926732i	$0.377394\pi$
$234$	0	0
$235$	37.8141	2.46672
$236$	0	0
$237$	−5.49150	−0.356711
$238$	0	0
$239$	0.388834	0.0251516	0.0125758	−	0.999921i	$-0.495997\pi$
$239$	0.388834	0.0251516	0.0125758	+	0.999921i	$0.495997\pi$
$240$	0	0
$241$	−21.3151	−1.37302	−0.686512	−	0.727119i	$-0.740859\pi$
$241$	−21.3151	−1.37302	−0.686512	+	0.727119i	$0.740859\pi$
$242$	0	0
$243$	3.00491	0.192765
$244$	0	0
$245$	−6.76543	−0.432227
$246$	0	0
$247$	19.1844	1.22067
$248$	0	0
$249$	2.13391	0.135231
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	8.45719	0.531699
$254$	0	0
$255$	8.34314	0.522467
$256$	0	0
$257$	23.6866	1.47753	0.738766	−	0.673962i	$-0.235409\pi$
$257$	23.6866	1.47753	0.738766	+	0.673962i	$0.235409\pi$
$258$	0	0
$259$	1.47001	0.0913423
$260$	0	0
$261$	−0.343187	−0.0212427
$262$	0	0
$263$	−1.78068	−0.109802	−0.0549009	−	0.998492i	$-0.517484\pi$
$263$	−1.78068	−0.109802	−0.0549009	+	0.998492i	$0.517484\pi$
$264$	0	0
$265$	−31.3380	−1.92508
$266$	0	0
$267$	−23.0053	−1.40790
$268$	0	0
$269$	−22.2554	−1.35694	−0.678468	−	0.734630i	$-0.737356\pi$
$269$	−22.2554	−1.35694	−0.678468	+	0.734630i	$0.737356\pi$
$270$	0	0
$271$	−4.13028	−0.250897	−0.125448	−	0.992100i	$-0.540037\pi$
$271$	−4.13028	−0.250897	−0.125448	+	0.992100i	$0.540037\pi$
$272$	0	0
$273$	−12.2102	−0.738994
$274$	0	0
$275$	44.3068	2.67180
$276$	0	0
$277$	−14.1537	−0.850415	−0.425208	−	0.905096i	$-0.639799\pi$
$277$	−14.1537	−0.850415	−0.425208	+	0.905096i	$0.639799\pi$
$278$	0	0
$279$	1.54456	0.0924704
$280$	0	0
$281$	9.35858	0.558286	0.279143	−	0.960250i	$-0.409950\pi$
$281$	9.35858	0.558286	0.279143	+	0.960250i	$0.409950\pi$
$282$	0	0
$283$	10.2588	0.609825	0.304912	−	0.952380i	$-0.401373\pi$
$283$	10.2588	0.609825	0.304912	+	0.952380i	$0.401373\pi$
$284$	0	0
$285$	−50.2883	−2.97882
$286$	0	0
$287$	−6.80183	−0.401499
$288$	0	0
$289$	−15.4003	−0.905897
$290$	0	0
$291$	−2.26198	−0.132600
$292$	0	0
$293$	15.0778	0.880857	0.440429	−	0.897788i	$-0.354826\pi$
$293$	15.0778	0.880857	0.440429	+	0.897788i	$0.354826\pi$
$294$	0	0
$295$	5.45437	0.317566
$296$	0	0
$297$	−21.7028	−1.25932
$298$	0	0
$299$	5.31114	0.307151
$300$	0	0
$301$	36.4408	2.10041
$302$	0	0
$303$	23.0815	1.32600
$304$	0	0
$305$	42.8217	2.45196
$306$	0	0
$307$	26.0145	1.48473	0.742364	−	0.669997i	$-0.233705\pi$
$307$	26.0145	1.48473	0.742364	+	0.669997i	$0.233705\pi$
$308$	0	0
$309$	−10.1427	−0.576998
$310$	0	0
$311$	−16.3555	−0.927435	−0.463717	−	0.885983i	$-0.653485\pi$
$311$	−16.3555	−0.927435	−0.463717	+	0.885983i	$0.653485\pi$
$312$	0	0
$313$	25.8511	1.46119	0.730595	−	0.682811i	$-0.239243\pi$
$313$	25.8511	1.46119	0.730595	+	0.682811i	$0.239243\pi$
$314$	0	0
$315$	−3.42770	−0.193129
$316$	0	0
$317$	−1.43506	−0.0806012	−0.0403006	−	0.999188i	$-0.512832\pi$
$317$	−1.43506	−0.0806012	−0.0403006	+	0.999188i	$0.512832\pi$
$318$	0	0
$319$	4.73866	0.265314
$320$	0	0
$321$	31.3497	1.74977
$322$	0	0
$323$	−9.64247	−0.536521
$324$	0	0
$325$	27.8248	1.54344
$326$	0	0
$327$	8.28214	0.458003
$328$	0	0
$329$	27.8155	1.53352
$330$	0	0
$331$	−7.73059	−0.424911	−0.212456	−	0.977171i	$-0.568146\pi$
$331$	−7.73059	−0.424911	−0.212456	+	0.977171i	$0.568146\pi$
$332$	0	0
$333$	0.144728	0.00793104
$334$	0	0
$335$	19.2653	1.05258
$336$	0	0
$337$	30.1721	1.64358	0.821789	−	0.569793i	$-0.192976\pi$
$337$	30.1721	1.64358	0.821789	+	0.569793i	$0.192976\pi$
$338$	0	0
$339$	−23.5164	−1.27724
$340$	0	0
$341$	−21.3270	−1.15492
$342$	0	0
$343$	15.6566	0.845379
$344$	0	0
$345$	−13.9221	−0.749543
$346$	0	0
$347$	−23.3337	−1.25262	−0.626310	−	0.779574i	$-0.715436\pi$
$347$	−23.3337	−1.25262	−0.626310	+	0.779574i	$0.715436\pi$
$348$	0	0
$349$	−3.89645	−0.208572	−0.104286	−	0.994547i	$-0.533256\pi$
$349$	−3.89645	−0.208572	−0.104286	+	0.994547i	$0.533256\pi$
$350$	0	0
$351$	−13.6294	−0.727483
$352$	0	0
$353$	−33.8799	−1.80325	−0.901624	−	0.432521i	$-0.857624\pi$
$353$	−33.8799	−1.80325	−0.901624	+	0.432521i	$0.857624\pi$
$354$	0	0
$355$	39.2003	2.08053
$356$	0	0
$357$	6.13709	0.324809
$358$	0	0
$359$	23.2536	1.22728	0.613639	−	0.789586i	$-0.289705\pi$
$359$	23.2536	1.22728	0.613639	+	0.789586i	$0.289705\pi$
$360$	0	0
$361$	39.1200	2.05895
$362$	0	0
$363$	8.32355	0.436873
$364$	0	0
$365$	−23.1407	−1.21124
$366$	0	0
$367$	26.0174	1.35810	0.679048	−	0.734094i	$-0.262393\pi$
$367$	26.0174	1.35810	0.679048	+	0.734094i	$0.262393\pi$
$368$	0	0
$369$	−0.669663	−0.0348613
$370$	0	0
$371$	−23.0518	−1.19679
$372$	0	0
$373$	−30.6393	−1.58644	−0.793222	−	0.608933i	$-0.791598\pi$
$373$	−30.6393	−1.58644	−0.793222	+	0.608933i	$0.791598\pi$
$374$	0	0
$375$	−39.9556	−2.06330
$376$	0	0
$377$	2.97589	0.153266
$378$	0	0
$379$	−23.4667	−1.20540	−0.602702	−	0.797967i	$-0.705909\pi$
$379$	−23.4667	−1.20540	−0.602702	+	0.797967i	$0.705909\pi$
$380$	0	0
$381$	−5.65330	−0.289627
$382$	0	0
$383$	26.4316	1.35059	0.675295	−	0.737548i	$-0.264016\pi$
$383$	26.4316	1.35059	0.675295	+	0.737548i	$0.264016\pi$
$384$	0	0
$385$	47.3290	2.41211
$386$	0	0
$387$	3.58772	0.182374
$388$	0	0
$389$	7.36128	0.373232	0.186616	−	0.982433i	$-0.440248\pi$
$389$	7.36128	0.373232	0.186616	+	0.982433i	$0.440248\pi$
$390$	0	0
$391$	−2.66949	−0.135002
$392$	0	0
$393$	−11.0869	−0.559259
$394$	0	0
$395$	13.3677	0.672604
$396$	0	0
$397$	35.8664	1.80008	0.900042	−	0.435803i	$-0.143536\pi$
$397$	35.8664	1.80008	0.900042	+	0.435803i	$0.143536\pi$
$398$	0	0
$399$	−36.9913	−1.85188
$400$	0	0
$401$	30.2193	1.50908	0.754540	−	0.656254i	$-0.227860\pi$
$401$	30.2193	1.50908	0.754540	+	0.656254i	$0.227860\pi$
$402$	0	0
$403$	−13.3934	−0.667174
$404$	0	0
$405$	32.2382	1.60193
$406$	0	0
$407$	−1.99838	−0.0990558
$408$	0	0
$409$	4.72859	0.233814	0.116907	−	0.993143i	$-0.462702\pi$
$409$	4.72859	0.233814	0.116907	+	0.993143i	$0.462702\pi$
$410$	0	0
$411$	−0.182099	−0.00898230
$412$	0	0
$413$	4.01216	0.197425
$414$	0	0
$415$	−5.19448	−0.254987
$416$	0	0
$417$	9.09789	0.445526
$418$	0	0
$419$	−1.38803	−0.0678096	−0.0339048	−	0.999425i	$-0.510794\pi$
$419$	−1.38803	−0.0678096	−0.0339048	+	0.999425i	$0.510794\pi$
$420$	0	0
$421$	−28.4264	−1.38542	−0.692710	−	0.721216i	$-0.743583\pi$
$421$	−28.4264	−1.38542	−0.692710	+	0.721216i	$0.743583\pi$
$422$	0	0
$423$	2.73853	0.133152
$424$	0	0
$425$	−13.9853	−0.678386
$426$	0	0
$427$	31.4990	1.52434
$428$	0	0
$429$	16.5988	0.801400
$430$	0	0
$431$	16.2800	0.784182	0.392091	−	0.919926i	$-0.371752\pi$
$431$	16.2800	0.784182	0.392091	+	0.919926i	$0.371752\pi$
$432$	0	0
$433$	17.6223	0.846873	0.423436	−	0.905926i	$-0.360824\pi$
$433$	17.6223	0.846873	0.423436	+	0.905926i	$0.360824\pi$
$434$	0	0
$435$	−7.80073	−0.374016
$436$	0	0
$437$	16.0903	0.769705
$438$	0	0
$439$	−17.7183	−0.845648	−0.422824	−	0.906212i	$-0.638961\pi$
$439$	−17.7183	−0.845648	−0.422824	+	0.906212i	$0.638961\pi$
$440$	0	0
$441$	−0.489958	−0.0233313
$442$	0	0
$443$	27.8836	1.32479	0.662395	−	0.749155i	$-0.269540\pi$
$443$	27.8836	1.32479	0.662395	+	0.749155i	$0.269540\pi$
$444$	0	0
$445$	56.0008	2.65469
$446$	0	0
$447$	11.8135	0.558759
$448$	0	0
$449$	7.75914	0.366176	0.183088	−	0.983096i	$-0.441391\pi$
$449$	7.75914	0.366176	0.183088	+	0.983096i	$0.441391\pi$
$450$	0	0
$451$	9.24659	0.435405
$452$	0	0
$453$	−22.1839	−1.04229
$454$	0	0
$455$	29.7228	1.39342
$456$	0	0
$457$	10.1166	0.473232	0.236616	−	0.971603i	$-0.423962\pi$
$457$	10.1166	0.473232	0.236616	+	0.971603i	$0.423962\pi$
$458$	0	0
$459$	6.85041	0.319750
$460$	0	0
$461$	22.9150	1.06726	0.533630	−	0.845718i	$-0.320828\pi$
$461$	22.9150	1.06726	0.533630	+	0.845718i	$0.320828\pi$
$462$	0	0
$463$	−39.0040	−1.81267	−0.906335	−	0.422561i	$-0.861131\pi$
$463$	−39.0040	−1.81267	−0.906335	+	0.422561i	$0.861131\pi$
$464$	0	0
$465$	35.1083	1.62811
$466$	0	0
$467$	−32.9689	−1.52562	−0.762809	−	0.646624i	$-0.776180\pi$
$467$	−32.9689	−1.52562	−0.762809	+	0.646624i	$0.776180\pi$
$468$	0	0
$469$	14.1713	0.654370
$470$	0	0
$471$	−22.9901	−1.05933
$472$	0	0
$473$	−49.5386	−2.27779
$474$	0	0
$475$	84.2964	3.86778
$476$	0	0
$477$	−2.26952	−0.103914
$478$	0	0
$479$	6.10151	0.278785	0.139392	−	0.990237i	$-0.455485\pi$
$479$	6.10151	0.278785	0.139392	+	0.990237i	$0.455485\pi$
$480$	0	0
$481$	−1.25499	−0.0572224
$482$	0	0
$483$	−10.2409	−0.465978
$484$	0	0
$485$	5.50624	0.250026
$486$	0	0
$487$	10.2887	0.466224	0.233112	−	0.972450i	$-0.425109\pi$
$487$	10.2887	0.466224	0.233112	+	0.972450i	$0.425109\pi$
$488$	0	0
$489$	−29.1593	−1.31863
$490$	0	0
$491$	−39.7687	−1.79473	−0.897367	−	0.441284i	$-0.854523\pi$
$491$	−39.7687	−1.79473	−0.897367	+	0.441284i	$0.854523\pi$
$492$	0	0
$493$	−1.49574	−0.0673649
$494$	0	0
$495$	4.65970	0.209438
$496$	0	0
$497$	28.8351	1.29343
$498$	0	0
$499$	−41.7840	−1.87051	−0.935255	−	0.353975i	$-0.884830\pi$
$499$	−41.7840	−1.87051	−0.935255	+	0.353975i	$0.884830\pi$
$500$	0	0
$501$	24.0924	1.07637
$502$	0	0
$503$	−31.1299	−1.38801	−0.694007	−	0.719968i	$-0.744156\pi$
$503$	−31.1299	−1.38801	−0.694007	+	0.719968i	$0.744156\pi$
$504$	0	0
$505$	−56.1864	−2.50026
$506$	0	0
$507$	−10.9758	−0.487452
$508$	0	0
$509$	−11.9533	−0.529819	−0.264909	−	0.964273i	$-0.585342\pi$
$509$	−11.9533	−0.529819	−0.264909	+	0.964273i	$0.585342\pi$
$510$	0	0
$511$	−17.0219	−0.753007
$512$	0	0
$513$	−41.2908	−1.82304
$514$	0	0
$515$	24.6900	1.08797
$516$	0	0
$517$	−37.8131	−1.66302
$518$	0	0
$519$	31.2239	1.37058
$520$	0	0
$521$	37.0829	1.62463	0.812317	−	0.583217i	$-0.198206\pi$
$521$	37.0829	1.62463	0.812317	+	0.583217i	$0.198206\pi$
$522$	0	0
$523$	−32.2972	−1.41226	−0.706128	−	0.708084i	$-0.749560\pi$
$523$	−32.2972	−1.41226	−0.706128	+	0.708084i	$0.749560\pi$
$524$	0	0
$525$	−53.6517	−2.34155
$526$	0	0
$527$	6.73180	0.293242
$528$	0	0
$529$	−18.5454	−0.806324
$530$	0	0
$531$	0.395010	0.0171420
$532$	0	0
$533$	5.80688	0.251524
$534$	0	0
$535$	−76.3132	−3.29931
$536$	0	0
$537$	−31.6010	−1.36368
$538$	0	0
$539$	6.76525	0.291400
$540$	0	0
$541$	11.3496	0.487957	0.243979	−	0.969781i	$-0.421547\pi$
$541$	11.3496	0.487957	0.243979	+	0.969781i	$0.421547\pi$
$542$	0	0
$543$	−26.7989	−1.15005
$544$	0	0
$545$	−20.1609	−0.863597
$546$	0	0
$547$	−11.2669	−0.481737	−0.240868	−	0.970558i	$-0.577432\pi$
$547$	−11.2669	−0.481737	−0.240868	+	0.970558i	$0.577432\pi$
$548$	0	0
$549$	3.10118	0.132355
$550$	0	0
$551$	9.01559	0.384077
$552$	0	0
$553$	9.83311	0.418147
$554$	0	0
$555$	3.28971	0.139640
$556$	0	0
$557$	3.62858	0.153748	0.0768740	−	0.997041i	$-0.475506\pi$
$557$	3.62858	0.153748	0.0768740	+	0.997041i	$0.475506\pi$
$558$	0	0
$559$	−31.1104	−1.31583
$560$	0	0
$561$	−8.34292	−0.352238
$562$	0	0
$563$	5.09947	0.214917	0.107458	−	0.994210i	$-0.465729\pi$
$563$	5.09947	0.214917	0.107458	+	0.994210i	$0.465729\pi$
$564$	0	0
$565$	57.2450	2.40832
$566$	0	0
$567$	23.7140	0.995893
$568$	0	0
$569$	−12.4211	−0.520718	−0.260359	−	0.965512i	$-0.583841\pi$
$569$	−12.4211	−0.520718	−0.260359	+	0.965512i	$0.583841\pi$
$570$	0	0
$571$	−39.4521	−1.65102	−0.825509	−	0.564390i	$-0.809112\pi$
$571$	−39.4521	−1.65102	−0.825509	+	0.564390i	$0.809112\pi$
$572$	0	0
$573$	−40.2312	−1.68068
$574$	0	0
$575$	23.3372	0.973228
$576$	0	0
$577$	10.6957	0.445269	0.222634	−	0.974902i	$-0.428534\pi$
$577$	10.6957	0.445269	0.222634	+	0.974902i	$0.428534\pi$
$578$	0	0
$579$	15.1546	0.629804
$580$	0	0
$581$	−3.82099	−0.158521
$582$	0	0
$583$	31.3371	1.29785
$584$	0	0
$585$	2.92630	0.120988
$586$	0	0
$587$	2.90373	0.119850	0.0599248	−	0.998203i	$-0.480914\pi$
$587$	2.90373	0.119850	0.0599248	+	0.998203i	$0.480914\pi$
$588$	0	0
$589$	−40.5760	−1.67190
$590$	0	0
$591$	−16.8461	−0.692955
$592$	0	0
$593$	−7.79111	−0.319943	−0.159971	−	0.987122i	$-0.551140\pi$
$593$	−7.79111	−0.319943	−0.159971	+	0.987122i	$0.551140\pi$
$594$	0	0
$595$	−14.9393	−0.612450
$596$	0	0
$597$	22.2333	0.909947
$598$	0	0
$599$	−13.6048	−0.555878	−0.277939	−	0.960599i	$-0.589651\pi$
$599$	−13.6048	−0.555878	−0.277939	+	0.960599i	$0.589651\pi$
$600$	0	0
$601$	−11.8000	−0.481333	−0.240666	−	0.970608i	$-0.577366\pi$
$601$	−11.8000	−0.481333	−0.240666	+	0.970608i	$0.577366\pi$
$602$	0	0
$603$	1.39521	0.0568174
$604$	0	0
$605$	−20.2617	−0.823754
$606$	0	0
$607$	33.8716	1.37480	0.687402	−	0.726277i	$-0.258751\pi$
$607$	33.8716	1.37480	0.687402	+	0.726277i	$0.258751\pi$
$608$	0	0
$609$	−5.73810	−0.232520
$610$	0	0
$611$	−23.7467	−0.960690
$612$	0	0
$613$	3.50660	0.141630	0.0708150	−	0.997489i	$-0.477440\pi$
$613$	3.50660	0.141630	0.0708150	+	0.997489i	$0.477440\pi$
$614$	0	0
$615$	−15.2216	−0.613796
$616$	0	0
$617$	−28.8534	−1.16159	−0.580796	−	0.814049i	$-0.697259\pi$
$617$	−28.8534	−1.16159	−0.580796	+	0.814049i	$0.697259\pi$
$618$	0	0
$619$	11.1357	0.447581	0.223791	−	0.974637i	$-0.428157\pi$
$619$	11.1357	0.447581	0.223791	+	0.974637i	$0.428157\pi$
$620$	0	0
$621$	−11.4312	−0.458720
$622$	0	0
$623$	41.1934	1.65038
$624$	0	0
$625$	41.9761	1.67904
$626$	0	0
$627$	50.2869	2.00827
$628$	0	0
$629$	0.630781	0.0251509
$630$	0	0
$631$	−9.89236	−0.393809	−0.196904	−	0.980423i	$-0.563089\pi$
$631$	−9.89236	−0.393809	−0.196904	+	0.980423i	$0.563089\pi$
$632$	0	0
$633$	−39.9577	−1.58818
$634$	0	0
$635$	13.7616	0.546112
$636$	0	0
$637$	4.24860	0.168335
$638$	0	0
$639$	2.83892	0.112306
$640$	0	0
$641$	−22.1693	−0.875636	−0.437818	−	0.899064i	$-0.644249\pi$
$641$	−22.1693	−0.875636	−0.437818	+	0.899064i	$0.644249\pi$
$642$	0	0
$643$	−35.7707	−1.41066	−0.705330	−	0.708879i	$-0.749201\pi$
$643$	−35.7707	−1.41066	−0.705330	+	0.708879i	$0.749201\pi$
$644$	0	0
$645$	81.5499	3.21102
$646$	0	0
$647$	−31.8800	−1.25333	−0.626666	−	0.779288i	$-0.715581\pi$
$647$	−31.8800	−1.25333	−0.626666	+	0.779288i	$0.715581\pi$
$648$	0	0
$649$	−5.45423	−0.214097
$650$	0	0
$651$	25.8252	1.01217
$652$	0	0
$653$	30.0375	1.17546	0.587729	−	0.809058i	$-0.300022\pi$
$653$	30.0375	1.17546	0.587729	+	0.809058i	$0.300022\pi$
$654$	0	0
$655$	26.9883	1.05452
$656$	0	0
$657$	−1.67587	−0.0653818
$658$	0	0
$659$	12.8170	0.499279	0.249640	−	0.968339i	$-0.419688\pi$
$659$	12.8170	0.499279	0.249640	+	0.968339i	$0.419688\pi$
$660$	0	0
$661$	−43.1905	−1.67992	−0.839958	−	0.542651i	$-0.817421\pi$
$661$	−43.1905	−1.67992	−0.839958	+	0.542651i	$0.817421\pi$
$662$	0	0
$663$	−5.23937	−0.203480
$664$	0	0
$665$	90.0464	3.49185
$666$	0	0
$667$	2.49594	0.0966431
$668$	0	0
$669$	3.88109	0.150052
$670$	0	0
$671$	−42.8205	−1.65307
$672$	0	0
$673$	−50.0699	−1.93005	−0.965027	−	0.262152i	$-0.915568\pi$
$673$	−50.0699	−1.93005	−0.965027	+	0.262152i	$0.915568\pi$
$674$	0	0
$675$	−59.8876	−2.30508
$676$	0	0
$677$	4.04693	0.155536	0.0777681	−	0.996971i	$-0.475221\pi$
$677$	4.04693	0.155536	0.0777681	+	0.996971i	$0.475221\pi$
$678$	0	0
$679$	4.05031	0.155437
$680$	0	0
$681$	−31.6633	−1.21334
$682$	0	0
$683$	39.8028	1.52301	0.761505	−	0.648159i	$-0.224461\pi$
$683$	39.8028	1.52301	0.761505	+	0.648159i	$0.224461\pi$
$684$	0	0
$685$	0.443277	0.0169367
$686$	0	0
$687$	0.0655872	0.00250231
$688$	0	0
$689$	19.6798	0.749741
$690$	0	0
$691$	46.2702	1.76020	0.880101	−	0.474786i	$-0.157474\pi$
$691$	46.2702	1.76020	0.880101	+	0.474786i	$0.157474\pi$
$692$	0	0
$693$	3.42761	0.130204
$694$	0	0
$695$	−22.1466	−0.840069
$696$	0	0
$697$	−2.91866	−0.110552
$698$	0	0
$699$	18.8819	0.714179
$700$	0	0
$701$	45.5193	1.71924	0.859619	−	0.510935i	$-0.170701\pi$
$701$	45.5193	1.71924	0.859619	+	0.510935i	$0.170701\pi$
$702$	0	0
$703$	−3.80204	−0.143396
$704$	0	0
$705$	62.2475	2.34438
$706$	0	0
$707$	−41.3299	−1.55437
$708$	0	0
$709$	14.3814	0.540105	0.270052	−	0.962846i	$-0.412959\pi$
$709$	14.3814	0.540105	0.270052	+	0.962846i	$0.412959\pi$
$710$	0	0
$711$	0.968103	0.0363067
$712$	0	0
$713$	−11.2333	−0.420691
$714$	0	0
$715$	−40.4059	−1.51109
$716$	0	0
$717$	0.640078	0.0239041
$718$	0	0
$719$	−16.7714	−0.625468	−0.312734	−	0.949841i	$-0.601245\pi$
$719$	−16.7714	−0.625468	−0.312734	+	0.949841i	$0.601245\pi$
$720$	0	0
$721$	18.1616	0.676372
$722$	0	0
$723$	−35.0877	−1.30493
$724$	0	0
$725$	13.0761	0.485633
$726$	0	0
$727$	−5.93316	−0.220049	−0.110024	−	0.993929i	$-0.535093\pi$
$727$	−5.93316	−0.220049	−0.110024	+	0.993929i	$0.535093\pi$
$728$	0	0
$729$	29.0821	1.07711
$730$	0	0
$731$	15.6367	0.578344
$732$	0	0
$733$	9.99744	0.369264	0.184632	−	0.982808i	$-0.440891\pi$
$733$	9.99744	0.369264	0.184632	+	0.982808i	$0.440891\pi$
$734$	0	0
$735$	−11.1369	−0.410790
$736$	0	0
$737$	−19.2648	−0.709629
$738$	0	0
$739$	−10.3337	−0.380132	−0.190066	−	0.981771i	$-0.560870\pi$
$739$	−10.3337	−0.380132	−0.190066	+	0.981771i	$0.560870\pi$
$740$	0	0
$741$	31.5803	1.16013
$742$	0	0
$743$	47.8387	1.75503	0.877515	−	0.479549i	$-0.159200\pi$
$743$	47.8387	1.75503	0.877515	+	0.479549i	$0.159200\pi$
$744$	0	0
$745$	−28.7571	−1.05358
$746$	0	0
$747$	−0.376189	−0.0137640
$748$	0	0
$749$	−56.1349	−2.05113
$750$	0	0
$751$	−13.4574	−0.491067	−0.245534	−	0.969388i	$-0.578963\pi$
$751$	−13.4574	−0.491067	−0.245534	+	0.969388i	$0.578963\pi$
$752$	0	0
$753$	1.64615	0.0599889
$754$	0	0
$755$	54.0013	1.96531
$756$	0	0
$757$	43.3385	1.57516	0.787581	−	0.616211i	$-0.211333\pi$
$757$	43.3385	1.57516	0.787581	+	0.616211i	$0.211333\pi$
$758$	0	0
$759$	13.9218	0.505328
$760$	0	0
$761$	10.8808	0.394429	0.197214	−	0.980360i	$-0.436811\pi$
$761$	10.8808	0.394429	0.197214	+	0.980360i	$0.436811\pi$
$762$	0	0
$763$	−14.8300	−0.536884
$764$	0	0
$765$	−1.47082	−0.0531776
$766$	0	0
$767$	−3.42527	−0.123679
$768$	0	0
$769$	6.78016	0.244499	0.122249	−	0.992499i	$-0.460989\pi$
$769$	6.78016	0.244499	0.122249	+	0.992499i	$0.460989\pi$
$770$	0	0
$771$	38.9917	1.40425
$772$	0	0
$773$	−11.5175	−0.414255	−0.207128	−	0.978314i	$-0.566412\pi$
$773$	−11.5175	−0.414255	−0.207128	+	0.978314i	$0.566412\pi$
$774$	0	0
$775$	−58.8508	−2.11398
$776$	0	0
$777$	2.41986	0.0868120
$778$	0	0
$779$	17.5922	0.630306
$780$	0	0
$781$	−39.1992	−1.40266
$782$	0	0
$783$	−6.40505	−0.228898
$784$	0	0
$785$	55.9638	1.99743
$786$	0	0
$787$	−8.78796	−0.313257	−0.156628	−	0.987658i	$-0.550063\pi$
$787$	−8.78796	−0.313257	−0.156628	+	0.987658i	$0.550063\pi$
$788$	0	0
$789$	−2.93127	−0.104356
$790$	0	0
$791$	42.1086	1.49721
$792$	0	0
$793$	−26.8914	−0.954942
$794$	0	0
$795$	−51.5869	−1.82960
$796$	0	0
$797$	−19.1814	−0.679441	−0.339720	−	0.940526i	$-0.610332\pi$
$797$	−19.1814	−0.679441	−0.339720	+	0.940526i	$0.610332\pi$
$798$	0	0
$799$	11.9356	0.422251
$800$	0	0
$801$	4.05563	0.143298
$802$	0	0
$803$	23.1401	0.816595
$804$	0	0
$805$	24.9291	0.878634
$806$	0	0
$807$	−36.6357	−1.28964
$808$	0	0
$809$	55.3029	1.94435	0.972173	−	0.234263i	$-0.0752676\pi$
$809$	55.3029	1.94435	0.972173	+	0.234263i	$0.0752676\pi$
$810$	0	0
$811$	9.59127	0.336795	0.168398	−	0.985719i	$-0.446141\pi$
$811$	9.59127	0.336795	0.168398	+	0.985719i	$0.446141\pi$
$812$	0	0
$813$	−6.79905	−0.238453
$814$	0	0
$815$	70.9814	2.48637
$816$	0	0
$817$	−94.2502	−3.29740
$818$	0	0
$819$	2.15255	0.0752161
$820$	0	0
$821$	−47.4468	−1.65590	−0.827952	−	0.560798i	$-0.810494\pi$
$821$	−47.4468	−1.65590	−0.827952	+	0.560798i	$0.810494\pi$
$822$	0	0
$823$	10.2924	0.358771	0.179386	−	0.983779i	$-0.442589\pi$
$823$	10.2924	0.358771	0.179386	+	0.983779i	$0.442589\pi$
$824$	0	0
$825$	72.9354	2.53929
$826$	0	0
$827$	−12.8336	−0.446267	−0.223133	−	0.974788i	$-0.571629\pi$
$827$	−12.8336	−0.446267	−0.223133	+	0.974788i	$0.571629\pi$
$828$	0	0
$829$	11.5004	0.399424	0.199712	−	0.979855i	$-0.435999\pi$
$829$	11.5004	0.399424	0.199712	+	0.979855i	$0.435999\pi$
$830$	0	0
$831$	−23.2991	−0.808238
$832$	0	0
$833$	−2.13543	−0.0739883
$834$	0	0
$835$	−58.6472	−2.02957
$836$	0	0
$837$	28.8268	0.996401
$838$	0	0
$839$	−13.0027	−0.448904	−0.224452	−	0.974485i	$-0.572059\pi$
$839$	−13.0027	−0.448904	−0.224452	+	0.974485i	$0.572059\pi$
$840$	0	0
$841$	−27.6015	−0.951776
$842$	0	0
$843$	15.4056	0.530597
$844$	0	0
$845$	26.7179	0.919124
$846$	0	0
$847$	−14.9042	−0.512114
$848$	0	0
$849$	16.8876	0.579579
$850$	0	0
$851$	−1.05258	−0.0360820
$852$	0	0
$853$	−42.2359	−1.44613	−0.723064	−	0.690781i	$-0.757267\pi$
$853$	−42.2359	−1.44613	−0.723064	+	0.690781i	$0.757267\pi$
$854$	0	0
$855$	8.86537	0.303189
$856$	0	0
$857$	−4.08818	−0.139650	−0.0698248	−	0.997559i	$-0.522244\pi$
$857$	−4.08818	−0.139650	−0.0698248	+	0.997559i	$0.522244\pi$
$858$	0	0
$859$	−4.01949	−0.137143	−0.0685716	−	0.997646i	$-0.521844\pi$
$859$	−4.01949	−0.137143	−0.0685716	+	0.997646i	$0.521844\pi$
$860$	0	0
$861$	−11.1968	−0.381586
$862$	0	0
$863$	15.7841	0.537296	0.268648	−	0.963238i	$-0.413423\pi$
$863$	15.7841	0.537296	0.268648	+	0.963238i	$0.413423\pi$
$864$	0	0
$865$	−76.0070	−2.58432
$866$	0	0
$867$	−25.3511	−0.860968
$868$	0	0
$869$	−13.3674	−0.453458
$870$	0	0
$871$	−12.0984	−0.409937
$872$	0	0
$873$	0.398767	0.0134962
$874$	0	0
$875$	71.5447	2.41865
$876$	0	0
$877$	46.4264	1.56771	0.783853	−	0.620946i	$-0.213251\pi$
$877$	46.4264	1.56771	0.783853	+	0.620946i	$0.213251\pi$
$878$	0	0
$879$	24.8203	0.837170
$880$	0	0
$881$	−1.42346	−0.0479575	−0.0239788	−	0.999712i	$-0.507633\pi$
$881$	−1.42346	−0.0479575	−0.0239788	+	0.999712i	$0.507633\pi$
$882$	0	0
$883$	−15.9966	−0.538329	−0.269164	−	0.963094i	$-0.586748\pi$
$883$	−15.9966	−0.538329	−0.269164	+	0.963094i	$0.586748\pi$
$884$	0	0
$885$	8.97869	0.301815
$886$	0	0
$887$	40.5685	1.36216	0.681078	−	0.732210i	$-0.261511\pi$
$887$	40.5685	1.36216	0.681078	+	0.732210i	$0.261511\pi$
$888$	0	0
$889$	10.1228	0.339509
$890$	0	0
$891$	−32.2374	−1.07999
$892$	0	0
$893$	−71.9418	−2.40744
$894$	0	0
$895$	76.9249	2.57132
$896$	0	0
$897$	8.74291	0.291917
$898$	0	0
$899$	−6.29416	−0.209922
$900$	0	0
$901$	−9.89148	−0.329533
$902$	0	0
$903$	59.9869	1.99624
$904$	0	0
$905$	65.2354	2.16850
$906$	0	0
$907$	0.716666	0.0237965	0.0118983	−	0.999929i	$-0.496213\pi$
$907$	0.716666	0.0237965	0.0118983	+	0.999929i	$0.496213\pi$
$908$	0	0
$909$	−4.06907	−0.134962
$910$	0	0
$911$	−25.6532	−0.849928	−0.424964	−	0.905210i	$-0.639713\pi$
$911$	−25.6532	−0.849928	−0.424964	+	0.905210i	$0.639713\pi$
$912$	0	0
$913$	5.19435	0.171908
$914$	0	0
$915$	70.4907	2.33035
$916$	0	0
$917$	19.8522	0.655578
$918$	0	0
$919$	−19.9330	−0.657531	−0.328765	−	0.944412i	$-0.606632\pi$
$919$	−19.9330	−0.657531	−0.328765	+	0.944412i	$0.606632\pi$
$920$	0	0
$921$	42.8237	1.41109
$922$	0	0
$923$	−24.6172	−0.810286
$924$	0	0
$925$	−5.51442	−0.181313
$926$	0	0
$927$	1.78807	0.0587278
$928$	0	0
$929$	−60.6545	−1.99001	−0.995005	−	0.0998220i	$-0.968173\pi$
$929$	−60.6545	−1.99001	−0.995005	+	0.0998220i	$0.968173\pi$
$930$	0	0
$931$	12.8713	0.421840
$932$	0	0
$933$	−26.9235	−0.881437
$934$	0	0
$935$	20.3088	0.664169
$936$	0	0
$937$	18.5007	0.604390	0.302195	−	0.953246i	$-0.402281\pi$
$937$	18.5007	0.604390	0.302195	+	0.953246i	$0.402281\pi$
$938$	0	0
$939$	42.5547	1.38872
$940$	0	0
$941$	−0.339309	−0.0110611	−0.00553057	−	0.999985i	$-0.501760\pi$
$941$	−0.339309	−0.0110611	−0.00553057	+	0.999985i	$0.501760\pi$
$942$	0	0
$943$	4.87035	0.158600
$944$	0	0
$945$	−63.9727	−2.08103
$946$	0	0
$947$	11.1857	0.363487	0.181743	−	0.983346i	$-0.441826\pi$
$947$	11.1857	0.363487	0.181743	+	0.983346i	$0.441826\pi$
$948$	0	0
$949$	14.5320	0.471730
$950$	0	0
$951$	−2.36233	−0.0766037
$952$	0	0
$953$	−35.5040	−1.15009	−0.575044	−	0.818123i	$-0.695015\pi$
$953$	−35.5040	−1.15009	−0.575044	+	0.818123i	$0.695015\pi$
$954$	0	0
$955$	97.9331	3.16904
$956$	0	0
$957$	7.80053	0.252155
$958$	0	0
$959$	0.326068	0.0105293
$960$	0	0
$961$	−2.67225	−0.0862017
$962$	0	0
$963$	−5.52667	−0.178094
$964$	0	0
$965$	−36.8903	−1.18754
$966$	0	0
$967$	45.6415	1.46773	0.733865	−	0.679295i	$-0.237714\pi$
$967$	45.6415	1.46773	0.733865	+	0.679295i	$0.237714\pi$
$968$	0	0
$969$	−15.8729	−0.509912
$970$	0	0
$971$	−14.5138	−0.465770	−0.232885	−	0.972504i	$-0.574817\pi$
$971$	−14.5138	−0.465770	−0.232885	+	0.972504i	$0.574817\pi$
$972$	0	0
$973$	−16.2907	−0.522257
$974$	0	0
$975$	45.8036	1.46689
$976$	0	0
$977$	23.9724	0.766946	0.383473	−	0.923552i	$-0.374728\pi$
$977$	23.9724	0.766946	0.383473	+	0.923552i	$0.374728\pi$
$978$	0	0
$979$	−55.9993	−1.78975
$980$	0	0
$981$	−1.46007	−0.0466163
$982$	0	0
$983$	15.4862	0.493933	0.246966	−	0.969024i	$-0.420566\pi$
$983$	15.4862	0.493933	0.246966	+	0.969024i	$0.420566\pi$
$984$	0	0
$985$	41.0077	1.30661
$986$	0	0
$987$	45.7884	1.45746
$988$	0	0
$989$	−26.0929	−0.829705
$990$	0	0
$991$	23.1189	0.734396	0.367198	−	0.930143i	$-0.380317\pi$
$991$	23.1189	0.734396	0.367198	+	0.930143i	$0.380317\pi$
$992$	0	0
$993$	−12.7257	−0.403837
$994$	0	0
$995$	−54.1215	−1.71577
$996$	0	0
$997$	−17.9176	−0.567456	−0.283728	−	0.958905i	$-0.591571\pi$
$997$	−17.9176	−0.567456	−0.283728	+	0.958905i	$0.591571\pi$
$998$	0	0
$999$	2.70112	0.0854597

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.11		14
4.3	odd	2		1004.2.a.b.1.4	✓	14
12.11	even	2		9036.2.a.m.1.13		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.4	✓	14	4.3	odd	2
4016.2.a.j.1.11		14	1.1	even	1	trivial
9036.2.a.m.1.13		14	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.64615 q^{3} -4.00715 q^{5} -2.94760 q^{7} -0.290201 q^{9} +O(q^{10})\)	\(q+1.64615 q^{3} -4.00715 q^{5} -2.94760 q^{7} -0.290201 q^{9} +4.00704 q^{11} +2.51643 q^{13} -6.59635 q^{15} -1.26481 q^{17} +7.62365 q^{19} -4.85218 q^{21} +2.11058 q^{23} +11.0572 q^{25} -5.41615 q^{27} +1.18258 q^{29} -5.32238 q^{31} +6.59618 q^{33} +11.8115 q^{35} -0.498716 q^{37} +4.14242 q^{39} +2.30758 q^{41} -12.3629 q^{43} +1.16288 q^{45} -9.43666 q^{47} +1.68834 q^{49} -2.08206 q^{51} +7.82052 q^{53} -16.0568 q^{55} +12.5496 q^{57} -1.36116 q^{59} -10.6863 q^{61} +0.855396 q^{63} -10.0837 q^{65} -4.80774 q^{67} +3.47433 q^{69} -9.78259 q^{71} +5.77485 q^{73} +18.2018 q^{75} -11.8112 q^{77} -3.33597 q^{79} -8.04518 q^{81} +1.29630 q^{83} +5.06828 q^{85} +1.94670 q^{87} -13.9752 q^{89} -7.41744 q^{91} -8.76142 q^{93} -30.5491 q^{95} -1.37411 q^{97} -1.16285 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

Introduction

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