LMFDB - Embedded newform 4016.2.a.l.1.7

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			1.7
Root			$-0.455245$ 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)

$f(q)$	$=$	$q-0.455245 q^{3} +2.37571 q^{5} +1.84768 q^{7} -2.79275 q^{9} +O(q^{10})$	$q-0.455245 q^{3} +2.37571 q^{5} +1.84768 q^{7} -2.79275 q^{9} +3.06325 q^{11} +0.393593 q^{13} -1.08153 q^{15} +1.07180 q^{17} +7.88022 q^{19} -0.841147 q^{21} -7.82955 q^{23} +0.644013 q^{25} +2.63712 q^{27} +3.82916 q^{29} +10.4086 q^{31} -1.39453 q^{33} +4.38956 q^{35} -1.08555 q^{37} -0.179181 q^{39} -9.35205 q^{41} +8.99795 q^{43} -6.63478 q^{45} -9.78417 q^{47} -3.58608 q^{49} -0.487932 q^{51} -5.33903 q^{53} +7.27740 q^{55} -3.58743 q^{57} +7.04335 q^{59} +10.3656 q^{61} -5.16011 q^{63} +0.935064 q^{65} -1.48024 q^{67} +3.56436 q^{69} +9.11481 q^{71} +4.29262 q^{73} -0.293184 q^{75} +5.65990 q^{77} +11.8294 q^{79} +7.17772 q^{81} -2.37296 q^{83} +2.54629 q^{85} -1.74320 q^{87} +11.2464 q^{89} +0.727234 q^{91} -4.73846 q^{93} +18.7211 q^{95} -1.31832 q^{97} -8.55489 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.455245	−0.262836	−0.131418	−	0.991327i	$-0.541953\pi$
$3$	−0.455245	−0.262836	−0.131418	+	0.991327i	$0.541953\pi$
$4$	0	0
$5$	2.37571	1.06245	0.531226	−	0.847230i	$-0.321732\pi$
$5$	2.37571	1.06245	0.531226	+	0.847230i	$0.321732\pi$
$6$	0	0
$7$	1.84768	0.698358	0.349179	−	0.937056i	$-0.386461\pi$
$7$	1.84768	0.698358	0.349179	+	0.937056i	$0.386461\pi$
$8$	0	0
$9$	−2.79275	−0.930917
$10$	0	0
$11$	3.06325	0.923604	0.461802	−	0.886983i	$-0.347203\pi$
$11$	3.06325	0.923604	0.461802	+	0.886983i	$0.347203\pi$
$12$	0	0
$13$	0.393593	0.109163	0.0545815	−	0.998509i	$-0.482618\pi$
$13$	0.393593	0.109163	0.0545815	+	0.998509i	$0.482618\pi$
$14$	0	0
$15$	−1.08153	−0.279250
$16$	0	0
$17$	1.07180	0.259950	0.129975	−	0.991517i	$-0.458510\pi$
$17$	1.07180	0.259950	0.129975	+	0.991517i	$0.458510\pi$
$18$	0	0
$19$	7.88022	1.80785	0.903923	−	0.427696i	$-0.140675\pi$
$19$	7.88022	1.80785	0.903923	+	0.427696i	$0.140675\pi$
$20$	0	0
$21$	−0.841147	−0.183553
$22$	0	0
$23$	−7.82955	−1.63257	−0.816287	−	0.577646i	$-0.803971\pi$
$23$	−7.82955	−1.63257	−0.816287	+	0.577646i	$0.803971\pi$
$24$	0	0
$25$	0.644013	0.128803
$26$	0	0
$27$	2.63712	0.507514
$28$	0	0
$29$	3.82916	0.711056	0.355528	−	0.934666i	$-0.384301\pi$
$29$	3.82916	0.711056	0.355528	+	0.934666i	$0.384301\pi$
$30$	0	0
$31$	10.4086	1.86944	0.934720	−	0.355386i	$-0.115651\pi$
$31$	10.4086	1.86944	0.934720	+	0.355386i	$0.115651\pi$
$32$	0	0
$33$	−1.39453	−0.242756
$34$	0	0
$35$	4.38956	0.741971
$36$	0	0
$37$	−1.08555	−0.178464	−0.0892319	−	0.996011i	$-0.528441\pi$
$37$	−1.08555	−0.178464	−0.0892319	+	0.996011i	$0.528441\pi$
$38$	0	0
$39$	−0.179181	−0.0286920
$40$	0	0
$41$	−9.35205	−1.46055	−0.730273	−	0.683156i	$-0.760607\pi$
$41$	−9.35205	−1.46055	−0.730273	+	0.683156i	$0.760607\pi$
$42$	0	0
$43$	8.99795	1.37217	0.686087	−	0.727519i	$-0.259327\pi$
$43$	8.99795	1.37217	0.686087	+	0.727519i	$0.259327\pi$
$44$	0	0
$45$	−6.63478	−0.989054
$46$	0	0
$47$	−9.78417	−1.42717	−0.713584	−	0.700570i	$-0.752929\pi$
$47$	−9.78417	−1.42717	−0.713584	+	0.700570i	$0.752929\pi$
$48$	0	0
$49$	−3.58608	−0.512297
$50$	0	0
$51$	−0.487932	−0.0683241
$52$	0	0
$53$	−5.33903	−0.733373	−0.366686	−	0.930345i	$-0.619508\pi$
$53$	−5.33903	−0.733373	−0.366686	+	0.930345i	$0.619508\pi$
$54$	0	0
$55$	7.27740	0.981284
$56$	0	0
$57$	−3.58743	−0.475167
$58$	0	0
$59$	7.04335	0.916967	0.458483	−	0.888703i	$-0.348393\pi$
$59$	7.04335	0.916967	0.458483	+	0.888703i	$0.348393\pi$
$60$	0	0
$61$	10.3656	1.32719	0.663593	−	0.748094i	$-0.269031\pi$
$61$	10.3656	1.32719	0.663593	+	0.748094i	$0.269031\pi$
$62$	0	0
$63$	−5.16011	−0.650113
$64$	0	0
$65$	0.935064	0.115980
$66$	0	0
$67$	−1.48024	−0.180840	−0.0904200	−	0.995904i	$-0.528821\pi$
$67$	−1.48024	−0.180840	−0.0904200	+	0.995904i	$0.528821\pi$
$68$	0	0
$69$	3.56436	0.429099
$70$	0	0
$71$	9.11481	1.08173	0.540865	−	0.841110i	$-0.318097\pi$
$71$	9.11481	1.08173	0.540865	+	0.841110i	$0.318097\pi$
$72$	0	0
$73$	4.29262	0.502413	0.251206	−	0.967934i	$-0.419173\pi$
$73$	4.29262	0.502413	0.251206	+	0.967934i	$0.419173\pi$
$74$	0	0
$75$	−0.293184	−0.0338540
$76$	0	0
$77$	5.65990	0.645006
$78$	0	0
$79$	11.8294	1.33091	0.665457	−	0.746436i	$-0.268237\pi$
$79$	11.8294	1.33091	0.665457	+	0.746436i	$0.268237\pi$
$80$	0	0
$81$	7.17772	0.797524
$82$	0	0
$83$	−2.37296	−0.260466	−0.130233	−	0.991483i	$-0.541573\pi$
$83$	−2.37296	−0.260466	−0.130233	+	0.991483i	$0.541573\pi$
$84$	0	0
$85$	2.54629	0.276184
$86$	0	0
$87$	−1.74320	−0.186891
$88$	0	0
$89$	11.2464	1.19212	0.596058	−	0.802942i	$-0.296733\pi$
$89$	11.2464	1.19212	0.596058	+	0.802942i	$0.296733\pi$
$90$	0	0
$91$	0.727234	0.0762348
$92$	0	0
$93$	−4.73846	−0.491356
$94$	0	0
$95$	18.7211	1.92075
$96$	0	0
$97$	−1.31832	−0.133855	−0.0669275	−	0.997758i	$-0.521320\pi$
$97$	−1.31832	−0.133855	−0.0669275	+	0.997758i	$0.521320\pi$
$98$	0	0
$99$	−8.55489	−0.859799
$100$	0	0
$101$	0.281620	0.0280223	0.0140111	−	0.999902i	$-0.495540\pi$
$101$	0.281620	0.0280223	0.0140111	+	0.999902i	$0.495540\pi$
$102$	0	0
$103$	−0.850124	−0.0837652	−0.0418826	−	0.999123i	$-0.513336\pi$
$103$	−0.850124	−0.0837652	−0.0418826	+	0.999123i	$0.513336\pi$
$104$	0	0
$105$	−1.99832	−0.195017
$106$	0	0
$107$	−18.5523	−1.79352	−0.896758	−	0.442521i	$-0.854084\pi$
$107$	−18.5523	−1.79352	−0.896758	+	0.442521i	$0.854084\pi$
$108$	0	0
$109$	−11.7916	−1.12943	−0.564715	−	0.825286i	$-0.691014\pi$
$109$	−11.7916	−1.12943	−0.564715	+	0.825286i	$0.691014\pi$
$110$	0	0
$111$	0.494192	0.0469067
$112$	0	0
$113$	−7.57560	−0.712652	−0.356326	−	0.934362i	$-0.615971\pi$
$113$	−7.57560	−0.712652	−0.356326	+	0.934362i	$0.615971\pi$
$114$	0	0
$115$	−18.6008	−1.73453
$116$	0	0
$117$	−1.09921	−0.101622
$118$	0	0
$119$	1.98034	0.181538
$120$	0	0
$121$	−1.61652	−0.146956
$122$	0	0
$123$	4.25748	0.383884
$124$	0	0
$125$	−10.3486	−0.925605
$126$	0	0
$127$	−4.29285	−0.380929	−0.190465	−	0.981694i	$-0.560999\pi$
$127$	−4.29285	−0.380929	−0.190465	+	0.981694i	$0.560999\pi$
$128$	0	0
$129$	−4.09627	−0.360657
$130$	0	0
$131$	5.28676	0.461906	0.230953	−	0.972965i	$-0.425816\pi$
$131$	5.28676	0.461906	0.230953	+	0.972965i	$0.425816\pi$
$132$	0	0
$133$	14.5601	1.26252
$134$	0	0
$135$	6.26505	0.539209
$136$	0	0
$137$	1.22659	0.104794	0.0523971	−	0.998626i	$-0.483314\pi$
$137$	1.22659	0.104794	0.0523971	+	0.998626i	$0.483314\pi$
$138$	0	0
$139$	13.4521	1.14099	0.570496	−	0.821300i	$-0.306751\pi$
$139$	13.4521	1.14099	0.570496	+	0.821300i	$0.306751\pi$
$140$	0	0
$141$	4.45420	0.375111
$142$	0	0
$143$	1.20567	0.100823
$144$	0	0
$145$	9.09698	0.755463
$146$	0	0
$147$	1.63254	0.134650
$148$	0	0
$149$	14.4714	1.18554	0.592771	−	0.805371i	$-0.298034\pi$
$149$	14.4714	1.18554	0.592771	+	0.805371i	$0.298034\pi$
$150$	0	0
$151$	−9.39147	−0.764267	−0.382134	−	0.924107i	$-0.624811\pi$
$151$	−9.39147	−0.764267	−0.382134	+	0.924107i	$0.624811\pi$
$152$	0	0
$153$	−2.99327	−0.241992
$154$	0	0
$155$	24.7278	1.98619
$156$	0	0
$157$	11.1982	0.893716	0.446858	−	0.894605i	$-0.352543\pi$
$157$	11.1982	0.893716	0.446858	+	0.894605i	$0.352543\pi$
$158$	0	0
$159$	2.43057	0.192757
$160$	0	0
$161$	−14.4665	−1.14012
$162$	0	0
$163$	−1.12124	−0.0878226	−0.0439113	−	0.999035i	$-0.513982\pi$
$163$	−1.12124	−0.0878226	−0.0439113	+	0.999035i	$0.513982\pi$
$164$	0	0
$165$	−3.31300	−0.257917
$166$	0	0
$167$	−15.5396	−1.20249	−0.601245	−	0.799065i	$-0.705328\pi$
$167$	−15.5396	−1.20249	−0.601245	+	0.799065i	$0.705328\pi$
$168$	0	0
$169$	−12.8451	−0.988083
$170$	0	0
$171$	−22.0075	−1.68295
$172$	0	0
$173$	5.74443	0.436741	0.218371	−	0.975866i	$-0.429926\pi$
$173$	5.74443	0.436741	0.218371	+	0.975866i	$0.429926\pi$
$174$	0	0
$175$	1.18993	0.0899503
$176$	0	0
$177$	−3.20645	−0.241012
$178$	0	0
$179$	17.4908	1.30733	0.653663	−	0.756786i	$-0.273231\pi$
$179$	17.4908	1.30733	0.653663	+	0.756786i	$0.273231\pi$
$180$	0	0
$181$	3.71364	0.276033	0.138017	−	0.990430i	$-0.455927\pi$
$181$	3.71364	0.276033	0.138017	+	0.990430i	$0.455927\pi$
$182$	0	0
$183$	−4.71891	−0.348832
$184$	0	0
$185$	−2.57896	−0.189609
$186$	0	0
$187$	3.28319	0.240090
$188$	0	0
$189$	4.87256	0.354426
$190$	0	0
$191$	12.3451	0.893260	0.446630	−	0.894719i	$-0.352624\pi$
$191$	12.3451	0.893260	0.446630	+	0.894719i	$0.352624\pi$
$192$	0	0
$193$	6.54244	0.470935	0.235468	−	0.971882i	$-0.424338\pi$
$193$	6.54244	0.470935	0.235468	+	0.971882i	$0.424338\pi$
$194$	0	0
$195$	−0.425683	−0.0304838
$196$	0	0
$197$	−2.19084	−0.156091	−0.0780455	−	0.996950i	$-0.524868\pi$
$197$	−2.19084	−0.156091	−0.0780455	+	0.996950i	$0.524868\pi$
$198$	0	0
$199$	22.1067	1.56710	0.783551	−	0.621327i	$-0.213406\pi$
$199$	22.1067	1.56710	0.783551	+	0.621327i	$0.213406\pi$
$200$	0	0
$201$	0.673872	0.0475312
$202$	0	0
$203$	7.07506	0.496572
$204$	0	0
$205$	−22.2178	−1.55176
$206$	0	0
$207$	21.8660	1.51979
$208$	0	0
$209$	24.1390	1.66973
$210$	0	0
$211$	−23.3163	−1.60516	−0.802580	−	0.596545i	$-0.796540\pi$
$211$	−23.3163	−1.60516	−0.802580	+	0.596545i	$0.796540\pi$
$212$	0	0
$213$	−4.14947	−0.284317
$214$	0	0
$215$	21.3766	1.45787
$216$	0	0
$217$	19.2318	1.30554
$218$	0	0
$219$	−1.95419	−0.132052
$220$	0	0
$221$	0.421853	0.0283769
$222$	0	0
$223$	10.3280	0.691617	0.345808	−	0.938305i	$-0.387605\pi$
$223$	10.3280	0.691617	0.345808	+	0.938305i	$0.387605\pi$
$224$	0	0
$225$	−1.79857	−0.119905
$226$	0	0
$227$	8.35126	0.554293	0.277146	−	0.960828i	$-0.410611\pi$
$227$	8.35126	0.554293	0.277146	+	0.960828i	$0.410611\pi$
$228$	0	0
$229$	−22.8776	−1.51179	−0.755897	−	0.654691i	$-0.772799\pi$
$229$	−22.8776	−1.51179	−0.755897	+	0.654691i	$0.772799\pi$
$230$	0	0
$231$	−2.57664	−0.169531
$232$	0	0
$233$	−10.6528	−0.697888	−0.348944	−	0.937144i	$-0.613460\pi$
$233$	−10.6528	−0.697888	−0.348944	+	0.937144i	$0.613460\pi$
$234$	0	0
$235$	−23.2444	−1.51630
$236$	0	0
$237$	−5.38529	−0.349812
$238$	0	0
$239$	20.3032	1.31330	0.656651	−	0.754195i	$-0.271973\pi$
$239$	20.3032	1.31330	0.656651	+	0.754195i	$0.271973\pi$
$240$	0	0
$241$	−3.78841	−0.244033	−0.122016	−	0.992528i	$-0.538936\pi$
$241$	−3.78841	−0.244033	−0.122016	+	0.992528i	$0.538936\pi$
$242$	0	0
$243$	−11.1790	−0.717132
$244$	0	0
$245$	−8.51949	−0.544290
$246$	0	0
$247$	3.10160	0.197350
$248$	0	0
$249$	1.08028	0.0684598
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−23.9838	−1.50785
$254$	0	0
$255$	−1.15919	−0.0725910
$256$	0	0
$257$	0.349082	0.0217751	0.0108876	−	0.999941i	$-0.496534\pi$
$257$	0.349082	0.0217751	0.0108876	+	0.999941i	$0.496534\pi$
$258$	0	0
$259$	−2.00575	−0.124631
$260$	0	0
$261$	−10.6939	−0.661935
$262$	0	0
$263$	−14.7369	−0.908716	−0.454358	−	0.890819i	$-0.650131\pi$
$263$	−14.7369	−0.908716	−0.454358	+	0.890819i	$0.650131\pi$
$264$	0	0
$265$	−12.6840	−0.779173
$266$	0	0
$267$	−5.11986	−0.313331
$268$	0	0
$269$	30.3436	1.85008	0.925040	−	0.379870i	$-0.124031\pi$
$269$	30.3436	1.85008	0.925040	+	0.379870i	$0.124031\pi$
$270$	0	0
$271$	5.07898	0.308526	0.154263	−	0.988030i	$-0.450700\pi$
$271$	5.07898	0.308526	0.154263	+	0.988030i	$0.450700\pi$
$272$	0	0
$273$	−0.331070	−0.0200373
$274$	0	0
$275$	1.97277	0.118963
$276$	0	0
$277$	29.9452	1.79923	0.899616	−	0.436682i	$-0.143846\pi$
$277$	29.9452	1.79923	0.899616	+	0.436682i	$0.143846\pi$
$278$	0	0
$279$	−29.0686	−1.74029
$280$	0	0
$281$	12.0363	0.718026	0.359013	−	0.933332i	$-0.383113\pi$
$281$	12.0363	0.718026	0.359013	+	0.933332i	$0.383113\pi$
$282$	0	0
$283$	−4.97686	−0.295843	−0.147922	−	0.988999i	$-0.547258\pi$
$283$	−4.97686	−0.295843	−0.147922	+	0.988999i	$0.547258\pi$
$284$	0	0
$285$	−8.52270	−0.504841
$286$	0	0
$287$	−17.2796	−1.01998
$288$	0	0
$289$	−15.8512	−0.932426
$290$	0	0
$291$	0.600158	0.0351819
$292$	0	0
$293$	7.65403	0.447153	0.223577	−	0.974686i	$-0.428227\pi$
$293$	7.65403	0.447153	0.223577	+	0.974686i	$0.428227\pi$
$294$	0	0
$295$	16.7330	0.974232
$296$	0	0
$297$	8.07815	0.468742
$298$	0	0
$299$	−3.08166	−0.178217
$300$	0	0
$301$	16.6253	0.958269
$302$	0	0
$303$	−0.128206	−0.00736525
$304$	0	0
$305$	24.6258	1.41007
$306$	0	0
$307$	24.0439	1.37226	0.686128	−	0.727481i	$-0.259309\pi$
$307$	24.0439	1.37226	0.686128	+	0.727481i	$0.259309\pi$
$308$	0	0
$309$	0.387015	0.0220165
$310$	0	0
$311$	−10.4446	−0.592261	−0.296131	−	0.955147i	$-0.595696\pi$
$311$	−10.4446	−0.592261	−0.296131	+	0.955147i	$0.595696\pi$
$312$	0	0
$313$	34.4717	1.94846	0.974228	−	0.225565i	$-0.0724229\pi$
$313$	34.4717	1.94846	0.974228	+	0.225565i	$0.0724229\pi$
$314$	0	0
$315$	−12.2589	−0.690713
$316$	0	0
$317$	−13.6848	−0.768615	−0.384307	−	0.923205i	$-0.625560\pi$
$317$	−13.6848	−0.768615	−0.384307	+	0.923205i	$0.625560\pi$
$318$	0	0
$319$	11.7296	0.656734
$320$	0	0
$321$	8.44583	0.471400
$322$	0	0
$323$	8.44601	0.469949
$324$	0	0
$325$	0.253479	0.0140605
$326$	0	0
$327$	5.36807	0.296855
$328$	0	0
$329$	−18.0780	−0.996673
$330$	0	0
$331$	−17.9842	−0.988499	−0.494249	−	0.869320i	$-0.664557\pi$
$331$	−17.9842	−0.988499	−0.494249	+	0.869320i	$0.664557\pi$
$332$	0	0
$333$	3.03168	0.166135
$334$	0	0
$335$	−3.51662	−0.192134
$336$	0	0
$337$	16.9539	0.923537	0.461769	−	0.887000i	$-0.347215\pi$
$337$	16.9539	0.923537	0.461769	+	0.887000i	$0.347215\pi$
$338$	0	0
$339$	3.44875	0.187311
$340$	0	0
$341$	31.8841	1.72662
$342$	0	0
$343$	−19.5597	−1.05612
$344$	0	0
$345$	8.46791	0.455897
$346$	0	0
$347$	5.77880	0.310222	0.155111	−	0.987897i	$-0.450426\pi$
$347$	5.77880	0.310222	0.155111	+	0.987897i	$0.450426\pi$
$348$	0	0
$349$	−0.606329	−0.0324560	−0.0162280	−	0.999868i	$-0.505166\pi$
$349$	−0.606329	−0.0324560	−0.0162280	+	0.999868i	$0.505166\pi$
$350$	0	0
$351$	1.03795	0.0554018
$352$	0	0
$353$	14.3220	0.762286	0.381143	−	0.924516i	$-0.375531\pi$
$353$	14.3220	0.762286	0.381143	+	0.924516i	$0.375531\pi$
$354$	0	0
$355$	21.6542	1.14928
$356$	0	0
$357$	−0.901542	−0.0477146
$358$	0	0
$359$	−21.7960	−1.15035	−0.575174	−	0.818031i	$-0.695066\pi$
$359$	−21.7960	−1.15035	−0.575174	+	0.818031i	$0.695066\pi$
$360$	0	0
$361$	43.0978	2.26831
$362$	0	0
$363$	0.735913	0.0386254
$364$	0	0
$365$	10.1980	0.533789
$366$	0	0
$367$	24.8530	1.29731	0.648657	−	0.761081i	$-0.275331\pi$
$367$	24.8530	1.29731	0.648657	+	0.761081i	$0.275331\pi$
$368$	0	0
$369$	26.1180	1.35965
$370$	0	0
$371$	−9.86483	−0.512156
$372$	0	0
$373$	−15.4160	−0.798209	−0.399104	−	0.916906i	$-0.630679\pi$
$373$	−15.4160	−0.798209	−0.399104	+	0.916906i	$0.630679\pi$
$374$	0	0
$375$	4.71114	0.243282
$376$	0	0
$377$	1.50713	0.0776211
$378$	0	0
$379$	−2.53089	−0.130003	−0.0650016	−	0.997885i	$-0.520705\pi$
$379$	−2.53089	−0.130003	−0.0650016	+	0.997885i	$0.520705\pi$
$380$	0	0
$381$	1.95430	0.100122
$382$	0	0
$383$	14.6117	0.746621	0.373311	−	0.927706i	$-0.378223\pi$
$383$	14.6117	0.746621	0.373311	+	0.927706i	$0.378223\pi$
$384$	0	0
$385$	13.4463	0.685287
$386$	0	0
$387$	−25.1290	−1.27738
$388$	0	0
$389$	−21.4297	−1.08653	−0.543263	−	0.839562i	$-0.682811\pi$
$389$	−21.4297	−1.08653	−0.543263	+	0.839562i	$0.682811\pi$
$390$	0	0
$391$	−8.39171	−0.424387
$392$	0	0
$393$	−2.40677	−0.121406
$394$	0	0
$395$	28.1033	1.41403
$396$	0	0
$397$	−3.07607	−0.154383	−0.0771917	−	0.997016i	$-0.524595\pi$
$397$	−3.07607	−0.154383	−0.0771917	+	0.997016i	$0.524595\pi$
$398$	0	0
$399$	−6.62842	−0.331836
$400$	0	0
$401$	−18.1921	−0.908468	−0.454234	−	0.890882i	$-0.650087\pi$
$401$	−18.1921	−0.908468	−0.454234	+	0.890882i	$0.650087\pi$
$402$	0	0
$403$	4.09675	0.204074
$404$	0	0
$405$	17.0522	0.847331
$406$	0	0
$407$	−3.32532	−0.164830
$408$	0	0
$409$	21.0422	1.04047	0.520235	−	0.854023i	$-0.325844\pi$
$409$	21.0422	1.04047	0.520235	+	0.854023i	$0.325844\pi$
$410$	0	0
$411$	−0.558397	−0.0275437
$412$	0	0
$413$	13.0139	0.640370
$414$	0	0
$415$	−5.63747	−0.276732
$416$	0	0
$417$	−6.12400	−0.299894
$418$	0	0
$419$	32.4228	1.58396	0.791978	−	0.610549i	$-0.209051\pi$
$419$	32.4228	1.58396	0.791978	+	0.610549i	$0.209051\pi$
$420$	0	0
$421$	−37.0914	−1.80773	−0.903863	−	0.427822i	$-0.859281\pi$
$421$	−37.0914	−1.80773	−0.903863	+	0.427822i	$0.859281\pi$
$422$	0	0
$423$	27.3248	1.32858
$424$	0	0
$425$	0.690253	0.0334822
$426$	0	0
$427$	19.1524	0.926850
$428$	0	0
$429$	−0.548876	−0.0265000
$430$	0	0
$431$	−33.2905	−1.60355	−0.801774	−	0.597627i	$-0.796110\pi$
$431$	−33.2905	−1.60355	−0.801774	+	0.597627i	$0.796110\pi$
$432$	0	0
$433$	−4.15512	−0.199682	−0.0998412	−	0.995003i	$-0.531833\pi$
$433$	−4.15512	−0.199682	−0.0998412	+	0.995003i	$0.531833\pi$
$434$	0	0
$435$	−4.14135	−0.198563
$436$	0	0
$437$	−61.6986	−2.95144
$438$	0	0
$439$	−2.48630	−0.118665	−0.0593323	−	0.998238i	$-0.518897\pi$
$439$	−2.48630	−0.118665	−0.0593323	+	0.998238i	$0.518897\pi$
$440$	0	0
$441$	10.0150	0.476906
$442$	0	0
$443$	28.5586	1.35686	0.678431	−	0.734664i	$-0.262660\pi$
$443$	28.5586	1.35686	0.678431	+	0.734664i	$0.262660\pi$
$444$	0	0
$445$	26.7182	1.26656
$446$	0	0
$447$	−6.58802	−0.311603
$448$	0	0
$449$	6.94720	0.327858	0.163929	−	0.986472i	$-0.447583\pi$
$449$	6.94720	0.327858	0.163929	+	0.986472i	$0.447583\pi$
$450$	0	0
$451$	−28.6477	−1.34897
$452$	0	0
$453$	4.27542	0.200877
$454$	0	0
$455$	1.72770	0.0809958
$456$	0	0
$457$	−5.25832	−0.245974	−0.122987	−	0.992408i	$-0.539247\pi$
$457$	−5.25832	−0.245974	−0.122987	+	0.992408i	$0.539247\pi$
$458$	0	0
$459$	2.82647	0.131928
$460$	0	0
$461$	3.54967	0.165325	0.0826623	−	0.996578i	$-0.473658\pi$
$461$	3.54967	0.165325	0.0826623	+	0.996578i	$0.473658\pi$
$462$	0	0
$463$	−24.2988	−1.12926	−0.564630	−	0.825344i	$-0.690981\pi$
$463$	−24.2988	−1.12926	−0.564630	+	0.825344i	$0.690981\pi$
$464$	0	0
$465$	−11.2572	−0.522041
$466$	0	0
$467$	−33.0846	−1.53097	−0.765486	−	0.643453i	$-0.777501\pi$
$467$	−33.0846	−1.53097	−0.765486	+	0.643453i	$0.777501\pi$
$468$	0	0
$469$	−2.73501	−0.126291
$470$	0	0
$471$	−5.09794	−0.234901
$472$	0	0
$473$	27.5629	1.26735
$474$	0	0
$475$	5.07496	0.232855
$476$	0	0
$477$	14.9106	0.682709
$478$	0	0
$479$	3.55665	0.162508	0.0812538	−	0.996693i	$-0.474108\pi$
$479$	3.55665	0.162508	0.0812538	+	0.996693i	$0.474108\pi$
$480$	0	0
$481$	−0.427266	−0.0194816
$482$	0	0
$483$	6.58581	0.299665
$484$	0	0
$485$	−3.13195	−0.142214
$486$	0	0
$487$	−43.4351	−1.96823	−0.984117	−	0.177521i	$-0.943192\pi$
$487$	−43.4351	−1.96823	−0.984117	+	0.177521i	$0.943192\pi$
$488$	0	0
$489$	0.510441	0.0230829
$490$	0	0
$491$	−28.4394	−1.28345	−0.641726	−	0.766934i	$-0.721782\pi$
$491$	−28.4394	−1.28345	−0.641726	+	0.766934i	$0.721782\pi$
$492$	0	0
$493$	4.10409	0.184839
$494$	0	0
$495$	−20.3240	−0.913494
$496$	0	0
$497$	16.8413	0.755434
$498$	0	0
$499$	33.4001	1.49519	0.747597	−	0.664152i	$-0.231207\pi$
$499$	33.4001	1.49519	0.747597	+	0.664152i	$0.231207\pi$
$500$	0	0
$501$	7.07432	0.316057
$502$	0	0
$503$	−39.2907	−1.75188	−0.875942	−	0.482416i	$-0.839759\pi$
$503$	−39.2907	−1.75188	−0.875942	+	0.482416i	$0.839759\pi$
$504$	0	0
$505$	0.669049	0.0297723
$506$	0	0
$507$	5.84766	0.259704
$508$	0	0
$509$	−3.16225	−0.140164	−0.0700821	−	0.997541i	$-0.522326\pi$
$509$	−3.16225	−0.140164	−0.0700821	+	0.997541i	$0.522326\pi$
$510$	0	0
$511$	7.93139	0.350864
$512$	0	0
$513$	20.7811	0.917508
$514$	0	0
$515$	−2.01965	−0.0889964
$516$	0	0
$517$	−29.9713	−1.31814
$518$	0	0
$519$	−2.61513	−0.114791
$520$	0	0
$521$	15.3491	0.672455	0.336227	−	0.941781i	$-0.390849\pi$
$521$	15.3491	0.672455	0.336227	+	0.941781i	$0.390849\pi$
$522$	0	0
$523$	−0.0591248	−0.00258534	−0.00129267	−	0.999999i	$-0.500411\pi$
$523$	−0.0591248	−0.00258534	−0.00129267	+	0.999999i	$0.500411\pi$
$524$	0	0
$525$	−0.541710	−0.0236422
$526$	0	0
$527$	11.1559	0.485960
$528$	0	0
$529$	38.3019	1.66530
$530$	0	0
$531$	−19.6703	−0.853620
$532$	0	0
$533$	−3.68090	−0.159438
$534$	0	0
$535$	−44.0749	−1.90552
$536$	0	0
$537$	−7.96262	−0.343612
$538$	0	0
$539$	−10.9850	−0.473159
$540$	0	0
$541$	−43.8099	−1.88354	−0.941768	−	0.336263i	$-0.890837\pi$
$541$	−43.8099	−1.88354	−0.941768	+	0.336263i	$0.890837\pi$
$542$	0	0
$543$	−1.69062	−0.0725514
$544$	0	0
$545$	−28.0135	−1.19997
$546$	0	0
$547$	−11.0616	−0.472960	−0.236480	−	0.971636i	$-0.575994\pi$
$547$	−11.0616	−0.472960	−0.236480	+	0.971636i	$0.575994\pi$
$548$	0	0
$549$	−28.9487	−1.23550
$550$	0	0
$551$	30.1746	1.28548
$552$	0	0
$553$	21.8570	0.929454
$554$	0	0
$555$	1.17406	0.0498361
$556$	0	0
$557$	−31.1186	−1.31854	−0.659268	−	0.751908i	$-0.729134\pi$
$557$	−31.1186	−1.31854	−0.659268	+	0.751908i	$0.729134\pi$
$558$	0	0
$559$	3.54153	0.149791
$560$	0	0
$561$	−1.49465	−0.0631044
$562$	0	0
$563$	15.7643	0.664385	0.332192	−	0.943212i	$-0.392212\pi$
$563$	15.7643	0.664385	0.332192	+	0.943212i	$0.392212\pi$
$564$	0	0
$565$	−17.9974	−0.757158
$566$	0	0
$567$	13.2621	0.556957
$568$	0	0
$569$	16.5628	0.694347	0.347174	−	0.937801i	$-0.387141\pi$
$569$	16.5628	0.694347	0.347174	+	0.937801i	$0.387141\pi$
$570$	0	0
$571$	3.38166	0.141518	0.0707591	−	0.997493i	$-0.477458\pi$
$571$	3.38166	0.141518	0.0707591	+	0.997493i	$0.477458\pi$
$572$	0	0
$573$	−5.62005	−0.234781
$574$	0	0
$575$	−5.04234	−0.210280
$576$	0	0
$577$	30.2941	1.26116	0.630580	−	0.776125i	$-0.282817\pi$
$577$	30.2941	1.26116	0.630580	+	0.776125i	$0.282817\pi$
$578$	0	0
$579$	−2.97841	−0.123779
$580$	0	0
$581$	−4.38447	−0.181898
$582$	0	0
$583$	−16.3548	−0.677346
$584$	0	0
$585$	−2.61140	−0.107968
$586$	0	0
$587$	44.8008	1.84913	0.924564	−	0.381027i	$-0.124429\pi$
$587$	44.8008	1.84913	0.924564	+	0.381027i	$0.124429\pi$
$588$	0	0
$589$	82.0220	3.37966
$590$	0	0
$591$	0.997370	0.0410263
$592$	0	0
$593$	−39.7522	−1.63243	−0.816214	−	0.577749i	$-0.803931\pi$
$593$	−39.7522	−1.63243	−0.816214	+	0.577749i	$0.803931\pi$
$594$	0	0
$595$	4.70473	0.192875
$596$	0	0
$597$	−10.0640	−0.411891
$598$	0	0
$599$	−32.9746	−1.34731	−0.673653	−	0.739047i	$-0.735276\pi$
$599$	−32.9746	−1.34731	−0.673653	+	0.739047i	$0.735276\pi$
$600$	0	0
$601$	−0.372771	−0.0152056	−0.00760282	−	0.999971i	$-0.502420\pi$
$601$	−0.372771	−0.0152056	−0.00760282	+	0.999971i	$0.502420\pi$
$602$	0	0
$603$	4.13394	0.168347
$604$	0	0
$605$	−3.84039	−0.156134
$606$	0	0
$607$	28.6391	1.16243	0.581214	−	0.813751i	$-0.302578\pi$
$607$	28.6391	1.16243	0.581214	+	0.813751i	$0.302578\pi$
$608$	0	0
$609$	−3.22088	−0.130517
$610$	0	0
$611$	−3.85098	−0.155794
$612$	0	0
$613$	−44.4233	−1.79424	−0.897120	−	0.441788i	$-0.854344\pi$
$613$	−44.4233	−1.79424	−0.897120	+	0.441788i	$0.854344\pi$
$614$	0	0
$615$	10.1145	0.407858
$616$	0	0
$617$	−11.3242	−0.455895	−0.227948	−	0.973673i	$-0.573201\pi$
$617$	−11.3242	−0.455895	−0.227948	+	0.973673i	$0.573201\pi$
$618$	0	0
$619$	0.519584	0.0208838	0.0104419	−	0.999945i	$-0.496676\pi$
$619$	0.519584	0.0208838	0.0104419	+	0.999945i	$0.496676\pi$
$620$	0	0
$621$	−20.6475	−0.828555
$622$	0	0
$623$	20.7797	0.832522
$624$	0	0
$625$	−27.8053	−1.11221
$626$	0	0
$627$	−10.9892	−0.438866
$628$	0	0
$629$	−1.16350	−0.0463916
$630$	0	0
$631$	−9.47914	−0.377358	−0.188679	−	0.982039i	$-0.560421\pi$
$631$	−9.47914	−0.377358	−0.188679	+	0.982039i	$0.560421\pi$
$632$	0	0
$633$	10.6146	0.421893
$634$	0	0
$635$	−10.1986	−0.404719
$636$	0	0
$637$	−1.41146	−0.0559239
$638$	0	0
$639$	−25.4554	−1.00700
$640$	0	0
$641$	−23.8855	−0.943421	−0.471710	−	0.881754i	$-0.656363\pi$
$641$	−23.8855	−0.943421	−0.471710	+	0.881754i	$0.656363\pi$
$642$	0	0
$643$	1.87106	0.0737875	0.0368938	−	0.999319i	$-0.488254\pi$
$643$	1.87106	0.0737875	0.0368938	+	0.999319i	$0.488254\pi$
$644$	0	0
$645$	−9.73157	−0.383180
$646$	0	0
$647$	−23.4228	−0.920844	−0.460422	−	0.887700i	$-0.652302\pi$
$647$	−23.4228	−0.920844	−0.460422	+	0.887700i	$0.652302\pi$
$648$	0	0
$649$	21.5755	0.846914
$650$	0	0
$651$	−8.75516	−0.343142
$652$	0	0
$653$	14.3033	0.559730	0.279865	−	0.960039i	$-0.409710\pi$
$653$	14.3033	0.559730	0.279865	+	0.960039i	$0.409710\pi$
$654$	0	0
$655$	12.5598	0.490753
$656$	0	0
$657$	−11.9882	−0.467705
$658$	0	0
$659$	−10.9319	−0.425846	−0.212923	−	0.977069i	$-0.568298\pi$
$659$	−10.9319	−0.425846	−0.212923	+	0.977069i	$0.568298\pi$
$660$	0	0
$661$	1.22320	0.0475769	0.0237885	−	0.999717i	$-0.492427\pi$
$661$	1.22320	0.0475769	0.0237885	+	0.999717i	$0.492427\pi$
$662$	0	0
$663$	−0.192046	−0.00745847
$664$	0	0
$665$	34.5907	1.34137
$666$	0	0
$667$	−29.9806	−1.16085
$668$	0	0
$669$	−4.70179	−0.181782
$670$	0	0
$671$	31.7525	1.22579
$672$	0	0
$673$	25.1159	0.968145	0.484073	−	0.875028i	$-0.339157\pi$
$673$	25.1159	0.968145	0.484073	+	0.875028i	$0.339157\pi$
$674$	0	0
$675$	1.69834	0.0653692
$676$	0	0
$677$	−39.4324	−1.51551	−0.757756	−	0.652538i	$-0.773704\pi$
$677$	−39.4324	−1.51551	−0.757756	+	0.652538i	$0.773704\pi$
$678$	0	0
$679$	−2.43583	−0.0934786
$680$	0	0
$681$	−3.80187	−0.145688
$682$	0	0
$683$	−18.1167	−0.693215	−0.346607	−	0.938010i	$-0.612666\pi$
$683$	−18.1167	−0.693215	−0.346607	+	0.938010i	$0.612666\pi$
$684$	0	0
$685$	2.91402	0.111339
$686$	0	0
$687$	10.4149	0.397354
$688$	0	0
$689$	−2.10141	−0.0800572
$690$	0	0
$691$	3.69277	0.140480	0.0702398	−	0.997530i	$-0.477624\pi$
$691$	3.69277	0.140480	0.0702398	+	0.997530i	$0.477624\pi$
$692$	0	0
$693$	−15.8067	−0.600447
$694$	0	0
$695$	31.9583	1.21225
$696$	0	0
$697$	−10.0235	−0.379668
$698$	0	0
$699$	4.84963	0.183430
$700$	0	0
$701$	35.1297	1.32683	0.663415	−	0.748251i	$-0.269106\pi$
$701$	35.1297	1.32683	0.663415	+	0.748251i	$0.269106\pi$
$702$	0	0
$703$	−8.55439	−0.322635
$704$	0	0
$705$	10.5819	0.398537
$706$	0	0
$707$	0.520344	0.0195696
$708$	0	0
$709$	−3.10292	−0.116532	−0.0582662	−	0.998301i	$-0.518557\pi$
$709$	−3.10292	−0.116532	−0.0582662	+	0.998301i	$0.518557\pi$
$710$	0	0
$711$	−33.0366	−1.23897
$712$	0	0
$713$	−81.4946	−3.05200
$714$	0	0
$715$	2.86433	0.107120
$716$	0	0
$717$	−9.24291	−0.345183
$718$	0	0
$719$	33.0213	1.23149	0.615744	−	0.787946i	$-0.288855\pi$
$719$	33.0213	1.23149	0.615744	+	0.787946i	$0.288855\pi$
$720$	0	0
$721$	−1.57076	−0.0584981
$722$	0	0
$723$	1.72465	0.0641406
$724$	0	0
$725$	2.46603	0.0915860
$726$	0	0
$727$	4.03582	0.149680	0.0748402	−	0.997196i	$-0.476155\pi$
$727$	4.03582	0.149680	0.0748402	+	0.997196i	$0.476155\pi$
$728$	0	0
$729$	−16.4440	−0.609036
$730$	0	0
$731$	9.64400	0.356696
$732$	0	0
$733$	49.7969	1.83929	0.919645	−	0.392750i	$-0.128476\pi$
$733$	49.7969	1.83929	0.919645	+	0.392750i	$0.128476\pi$
$734$	0	0
$735$	3.87846	0.143059
$736$	0	0
$737$	−4.53434	−0.167024
$738$	0	0
$739$	−20.1272	−0.740391	−0.370195	−	0.928954i	$-0.620709\pi$
$739$	−20.1272	−0.740391	−0.370195	+	0.928954i	$0.620709\pi$
$740$	0	0
$741$	−1.41199	−0.0518706
$742$	0	0
$743$	39.1163	1.43504	0.717519	−	0.696539i	$-0.245278\pi$
$743$	39.1163	1.43504	0.717519	+	0.696539i	$0.245278\pi$
$744$	0	0
$745$	34.3798	1.25958
$746$	0	0
$747$	6.62708	0.242472
$748$	0	0
$749$	−34.2787	−1.25252
$750$	0	0
$751$	−42.9269	−1.56643	−0.783213	−	0.621754i	$-0.786420\pi$
$751$	−42.9269	−1.56643	−0.783213	+	0.621754i	$0.786420\pi$
$752$	0	0
$753$	−0.455245	−0.0165901
$754$	0	0
$755$	−22.3114	−0.811997
$756$	0	0
$757$	−19.7911	−0.719319	−0.359659	−	0.933084i	$-0.617107\pi$
$757$	−19.7911	−0.719319	−0.359659	+	0.933084i	$0.617107\pi$
$758$	0	0
$759$	10.9185	0.396317
$760$	0	0
$761$	7.36661	0.267039	0.133520	−	0.991046i	$-0.457372\pi$
$761$	7.36661	0.267039	0.133520	+	0.991046i	$0.457372\pi$
$762$	0	0
$763$	−21.7871	−0.788746
$764$	0	0
$765$	−7.11115	−0.257104
$766$	0	0
$767$	2.77221	0.100099
$768$	0	0
$769$	−38.2792	−1.38038	−0.690191	−	0.723627i	$-0.742474\pi$
$769$	−38.2792	−1.38038	−0.690191	+	0.723627i	$0.742474\pi$
$770$	0	0
$771$	−0.158918	−0.00572328
$772$	0	0
$773$	−44.1934	−1.58953	−0.794763	−	0.606920i	$-0.792405\pi$
$773$	−44.1934	−1.58953	−0.794763	+	0.606920i	$0.792405\pi$
$774$	0	0
$775$	6.70328	0.240789
$776$	0	0
$777$	0.913110	0.0327576
$778$	0	0
$779$	−73.6962	−2.64044
$780$	0	0
$781$	27.9209	0.999089
$782$	0	0
$783$	10.0979	0.360871
$784$	0	0
$785$	26.6038	0.949530
$786$	0	0
$787$	−44.9388	−1.60190	−0.800948	−	0.598733i	$-0.795671\pi$
$787$	−44.9388	−1.60190	−0.800948	+	0.598733i	$0.795671\pi$
$788$	0	0
$789$	6.70890	0.238843
$790$	0	0
$791$	−13.9973	−0.497686
$792$	0	0
$793$	4.07985	0.144880
$794$	0	0
$795$	5.77433	0.204795
$796$	0	0
$797$	−29.3815	−1.04075	−0.520373	−	0.853939i	$-0.674207\pi$
$797$	−29.3815	−1.04075	−0.520373	+	0.853939i	$0.674207\pi$
$798$	0	0
$799$	−10.4867	−0.370992
$800$	0	0
$801$	−31.4084	−1.10976
$802$	0	0
$803$	13.1493	0.464030
$804$	0	0
$805$	−34.3683	−1.21132
$806$	0	0
$807$	−13.8138	−0.486267
$808$	0	0
$809$	−41.4861	−1.45857	−0.729286	−	0.684209i	$-0.760148\pi$
$809$	−41.4861	−1.45857	−0.729286	+	0.684209i	$0.760148\pi$
$810$	0	0
$811$	−12.9890	−0.456105	−0.228052	−	0.973649i	$-0.573236\pi$
$811$	−12.9890	−0.456105	−0.228052	+	0.973649i	$0.573236\pi$
$812$	0	0
$813$	−2.31218	−0.0810916
$814$	0	0
$815$	−2.66375	−0.0933072
$816$	0	0
$817$	70.9058	2.48068
$818$	0	0
$819$	−2.03098	−0.0709683
$820$	0	0
$821$	−47.9643	−1.67396	−0.836982	−	0.547230i	$-0.815682\pi$
$821$	−47.9643	−1.67396	−0.836982	+	0.547230i	$0.815682\pi$
$822$	0	0
$823$	−45.4171	−1.58314	−0.791570	−	0.611079i	$-0.790736\pi$
$823$	−45.4171	−1.58314	−0.791570	+	0.611079i	$0.790736\pi$
$824$	0	0
$825$	−0.898095	−0.0312676
$826$	0	0
$827$	−26.2784	−0.913790	−0.456895	−	0.889521i	$-0.651038\pi$
$827$	−26.2784	−0.913790	−0.456895	+	0.889521i	$0.651038\pi$
$828$	0	0
$829$	47.4528	1.64810	0.824052	−	0.566515i	$-0.191709\pi$
$829$	47.4528	1.64810	0.824052	+	0.566515i	$0.191709\pi$
$830$	0	0
$831$	−13.6324	−0.472903
$832$	0	0
$833$	−3.84356	−0.133171
$834$	0	0
$835$	−36.9176	−1.27759
$836$	0	0
$837$	27.4487	0.948767
$838$	0	0
$839$	−19.5245	−0.674060	−0.337030	−	0.941494i	$-0.609422\pi$
$839$	−19.5245	−0.674060	−0.337030	+	0.941494i	$0.609422\pi$
$840$	0	0
$841$	−14.3376	−0.494399
$842$	0	0
$843$	−5.47947	−0.188723
$844$	0	0
$845$	−30.5162	−1.04979
$846$	0	0
$847$	−2.98681	−0.102628
$848$	0	0
$849$	2.26569	0.0777583
$850$	0	0
$851$	8.49939	0.291355
$852$	0	0
$853$	−1.96561	−0.0673012	−0.0336506	−	0.999434i	$-0.510713\pi$
$853$	−1.96561	−0.0673012	−0.0336506	+	0.999434i	$0.510713\pi$
$854$	0	0
$855$	−52.2835	−1.78806
$856$	0	0
$857$	−27.4837	−0.938824	−0.469412	−	0.882979i	$-0.655534\pi$
$857$	−27.4837	−0.938824	−0.469412	+	0.882979i	$0.655534\pi$
$858$	0	0
$859$	17.1576	0.585409	0.292705	−	0.956203i	$-0.405445\pi$
$859$	17.1576	0.585409	0.292705	+	0.956203i	$0.405445\pi$
$860$	0	0
$861$	7.86646	0.268088
$862$	0	0
$863$	12.5487	0.427163	0.213582	−	0.976925i	$-0.431487\pi$
$863$	12.5487	0.427163	0.213582	+	0.976925i	$0.431487\pi$
$864$	0	0
$865$	13.6471	0.464016
$866$	0	0
$867$	7.21620	0.245075
$868$	0	0
$869$	36.2364	1.22924
$870$	0	0
$871$	−0.582612	−0.0197410
$872$	0	0
$873$	3.68174	0.124608
$874$	0	0
$875$	−19.1209	−0.646403
$876$	0	0
$877$	−46.3949	−1.56664	−0.783322	−	0.621616i	$-0.786477\pi$
$877$	−46.3949	−1.56664	−0.783322	+	0.621616i	$0.786477\pi$
$878$	0	0
$879$	−3.48446	−0.117528
$880$	0	0
$881$	−45.5293	−1.53392	−0.766961	−	0.641693i	$-0.778232\pi$
$881$	−45.5293	−1.53392	−0.766961	+	0.641693i	$0.778232\pi$
$882$	0	0
$883$	19.3923	0.652604	0.326302	−	0.945265i	$-0.394197\pi$
$883$	19.3923	0.652604	0.326302	+	0.945265i	$0.394197\pi$
$884$	0	0
$885$	−7.61761	−0.256063
$886$	0	0
$887$	21.4283	0.719492	0.359746	−	0.933050i	$-0.382863\pi$
$887$	21.4283	0.719492	0.359746	+	0.933050i	$0.382863\pi$
$888$	0	0
$889$	−7.93182	−0.266025
$890$	0	0
$891$	21.9871	0.736596
$892$	0	0
$893$	−77.1014	−2.58010
$894$	0	0
$895$	41.5532	1.38897
$896$	0	0
$897$	1.40291	0.0468418
$898$	0	0
$899$	39.8561	1.32928
$900$	0	0
$901$	−5.72238	−0.190640
$902$	0	0
$903$	−7.56860	−0.251867
$904$	0	0
$905$	8.82255	0.293272
$906$	0	0
$907$	36.6874	1.21819	0.609093	−	0.793099i	$-0.291534\pi$
$907$	36.6874	1.21819	0.609093	+	0.793099i	$0.291534\pi$
$908$	0	0
$909$	−0.786495	−0.0260864
$910$	0	0
$911$	14.0306	0.464853	0.232426	−	0.972614i	$-0.425333\pi$
$911$	14.0306	0.464853	0.232426	+	0.972614i	$0.425333\pi$
$912$	0	0
$913$	−7.26896	−0.240567
$914$	0	0
$915$	−11.2108	−0.370617
$916$	0	0
$917$	9.76824	0.322576
$918$	0	0
$919$	−21.7311	−0.716842	−0.358421	−	0.933560i	$-0.616685\pi$
$919$	−21.7311	−0.716842	−0.358421	+	0.933560i	$0.616685\pi$
$920$	0	0
$921$	−10.9458	−0.360678
$922$	0	0
$923$	3.58753	0.118085
$924$	0	0
$925$	−0.699110	−0.0229866
$926$	0	0
$927$	2.37419	0.0779785
$928$	0	0
$929$	−10.2690	−0.336914	−0.168457	−	0.985709i	$-0.553878\pi$
$929$	−10.2690	−0.336914	−0.168457	+	0.985709i	$0.553878\pi$
$930$	0	0
$931$	−28.2591	−0.926153
$932$	0	0
$933$	4.75487	0.155668
$934$	0	0
$935$	7.79991	0.255084
$936$	0	0
$937$	−21.5863	−0.705193	−0.352597	−	0.935775i	$-0.614701\pi$
$937$	−21.5863	−0.705193	−0.352597	+	0.935775i	$0.614701\pi$
$938$	0	0
$939$	−15.6931	−0.512124
$940$	0	0
$941$	−37.7966	−1.23213	−0.616066	−	0.787694i	$-0.711275\pi$
$941$	−37.7966	−1.23213	−0.616066	+	0.787694i	$0.711275\pi$
$942$	0	0
$943$	73.2224	2.38445
$944$	0	0
$945$	11.5758	0.376561
$946$	0	0
$947$	−26.1228	−0.848876	−0.424438	−	0.905457i	$-0.639528\pi$
$947$	−26.1228	−0.848876	−0.424438	+	0.905457i	$0.639528\pi$
$948$	0	0
$949$	1.68954	0.0548449
$950$	0	0
$951$	6.22994	0.202020
$952$	0	0
$953$	56.1976	1.82042	0.910209	−	0.414148i	$-0.135921\pi$
$953$	56.1976	1.82042	0.910209	+	0.414148i	$0.135921\pi$
$954$	0	0
$955$	29.3284	0.949046
$956$	0	0
$957$	−5.33986	−0.172613
$958$	0	0
$959$	2.26634	0.0731839
$960$	0	0
$961$	77.3389	2.49480
$962$	0	0
$963$	51.8119	1.66962
$964$	0	0
$965$	15.5430	0.500346
$966$	0	0
$967$	−42.0958	−1.35371	−0.676854	−	0.736117i	$-0.736657\pi$
$967$	−42.0958	−1.35371	−0.676854	+	0.736117i	$0.736657\pi$
$968$	0	0
$969$	−3.84501	−0.123519
$970$	0	0
$971$	−26.3077	−0.844256	−0.422128	−	0.906536i	$-0.638717\pi$
$971$	−26.3077	−0.844256	−0.422128	+	0.906536i	$0.638717\pi$
$972$	0	0
$973$	24.8552	0.796821
$974$	0	0
$975$	−0.115395	−0.00369560
$976$	0	0
$977$	−6.34556	−0.203013	−0.101506	−	0.994835i	$-0.532366\pi$
$977$	−6.34556	−0.203013	−0.101506	+	0.994835i	$0.532366\pi$
$978$	0	0
$979$	34.4505	1.10104
$980$	0	0
$981$	32.9310	1.05141
$982$	0	0
$983$	44.9220	1.43279	0.716394	−	0.697696i	$-0.245791\pi$
$983$	44.9220	1.43279	0.716394	+	0.697696i	$0.245791\pi$
$984$	0	0
$985$	−5.20481	−0.165839
$986$	0	0
$987$	8.22993	0.261962
$988$	0	0
$989$	−70.4499	−2.24018
$990$	0	0
$991$	44.0073	1.39794	0.698969	−	0.715152i	$-0.253642\pi$
$991$	44.0073	1.39794	0.698969	+	0.715152i	$0.253642\pi$
$992$	0	0
$993$	8.18720	0.259813
$994$	0	0
$995$	52.5192	1.66497
$996$	0	0
$997$	−16.5509	−0.524172	−0.262086	−	0.965045i	$-0.584410\pi$
$997$	−16.5509	−0.524172	−0.262086	+	0.965045i	$0.584410\pi$
$998$	0	0
$999$	−2.86273	−0.0905729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.7		19
4.3	odd	2		2008.2.a.c.1.13	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.13	✓	19	4.3	odd	2
4016.2.a.l.1.7		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)

\(f(q)\)	\(=\)	\(q-0.455245 q^{3} +2.37571 q^{5} +1.84768 q^{7} -2.79275 q^{9} +O(q^{10})\)	\(q-0.455245 q^{3} +2.37571 q^{5} +1.84768 q^{7} -2.79275 q^{9} +3.06325 q^{11} +0.393593 q^{13} -1.08153 q^{15} +1.07180 q^{17} +7.88022 q^{19} -0.841147 q^{21} -7.82955 q^{23} +0.644013 q^{25} +2.63712 q^{27} +3.82916 q^{29} +10.4086 q^{31} -1.39453 q^{33} +4.38956 q^{35} -1.08555 q^{37} -0.179181 q^{39} -9.35205 q^{41} +8.99795 q^{43} -6.63478 q^{45} -9.78417 q^{47} -3.58608 q^{49} -0.487932 q^{51} -5.33903 q^{53} +7.27740 q^{55} -3.58743 q^{57} +7.04335 q^{59} +10.3656 q^{61} -5.16011 q^{63} +0.935064 q^{65} -1.48024 q^{67} +3.56436 q^{69} +9.11481 q^{71} +4.29262 q^{73} -0.293184 q^{75} +5.65990 q^{77} +11.8294 q^{79} +7.17772 q^{81} -2.37296 q^{83} +2.54629 q^{85} -1.74320 q^{87} +11.2464 q^{89} +0.727234 q^{91} -4.73846 q^{93} +18.7211 q^{95} -1.31832 q^{97} -8.55489 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

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