LMFDB - Embedded newform 8016.2.a.bg.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.05600$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.05600 q^{5} -2.59989 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.05600 q^{5} -2.59989 q^{7} +1.00000 q^{9} -4.72064 q^{11} +6.74051 q^{13} +3.05600 q^{15} -3.13623 q^{17} -2.68764 q^{19} +2.59989 q^{21} -6.99219 q^{23} +4.33912 q^{25} -1.00000 q^{27} -7.69073 q^{29} -4.05540 q^{31} +4.72064 q^{33} +7.94526 q^{35} +0.127532 q^{37} -6.74051 q^{39} -8.00281 q^{41} +10.7440 q^{43} -3.05600 q^{45} -1.96112 q^{47} -0.240573 q^{49} +3.13623 q^{51} -7.56814 q^{53} +14.4263 q^{55} +2.68764 q^{57} -10.2078 q^{59} +2.45141 q^{61} -2.59989 q^{63} -20.5990 q^{65} -14.1052 q^{67} +6.99219 q^{69} +3.69140 q^{71} +4.93617 q^{73} -4.33912 q^{75} +12.2731 q^{77} -9.15118 q^{79} +1.00000 q^{81} -7.82320 q^{83} +9.58433 q^{85} +7.69073 q^{87} +5.37710 q^{89} -17.5246 q^{91} +4.05540 q^{93} +8.21342 q^{95} -10.3975 q^{97} -4.72064 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.05600	−1.36668	−0.683342	−	0.730098i	$-0.739474\pi$
$5$	−3.05600	−1.36668	−0.683342	+	0.730098i	$0.739474\pi$
$6$	0	0
$7$	−2.59989	−0.982666	−0.491333	−	0.870972i	$-0.663490\pi$
$7$	−2.59989	−0.982666	−0.491333	+	0.870972i	$0.663490\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.72064	−1.42333	−0.711663	−	0.702521i	$-0.752058\pi$
$11$	−4.72064	−1.42333	−0.711663	+	0.702521i	$0.752058\pi$
$12$	0	0
$13$	6.74051	1.86948	0.934740	−	0.355332i	$-0.115632\pi$
$13$	6.74051	1.86948	0.934740	+	0.355332i	$0.115632\pi$
$14$	0	0
$15$	3.05600	0.789055
$16$	0	0
$17$	−3.13623	−0.760649	−0.380324	−	0.924853i	$-0.624188\pi$
$17$	−3.13623	−0.760649	−0.380324	+	0.924853i	$0.624188\pi$
$18$	0	0
$19$	−2.68764	−0.616587	−0.308293	−	0.951291i	$-0.599758\pi$
$19$	−2.68764	−0.616587	−0.308293	+	0.951291i	$0.599758\pi$
$20$	0	0
$21$	2.59989	0.567342
$22$	0	0
$23$	−6.99219	−1.45797	−0.728986	−	0.684528i	$-0.760008\pi$
$23$	−6.99219	−1.45797	−0.728986	+	0.684528i	$0.760008\pi$
$24$	0	0
$25$	4.33912	0.867825
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.69073	−1.42813	−0.714067	−	0.700078i	$-0.753149\pi$
$29$	−7.69073	−1.42813	−0.714067	+	0.700078i	$0.753149\pi$
$30$	0	0
$31$	−4.05540	−0.728372	−0.364186	−	0.931326i	$-0.618653\pi$
$31$	−4.05540	−0.728372	−0.364186	+	0.931326i	$0.618653\pi$
$32$	0	0
$33$	4.72064	0.821757
$34$	0	0
$35$	7.94526	1.34299
$36$	0	0
$37$	0.127532	0.0209662	0.0104831	−	0.999945i	$-0.496663\pi$
$37$	0.127532	0.0209662	0.0104831	+	0.999945i	$0.496663\pi$
$38$	0	0
$39$	−6.74051	−1.07935
$40$	0	0
$41$	−8.00281	−1.24983	−0.624914	−	0.780693i	$-0.714866\pi$
$41$	−8.00281	−1.24983	−0.624914	+	0.780693i	$0.714866\pi$
$42$	0	0
$43$	10.7440	1.63844	0.819221	−	0.573479i	$-0.194406\pi$
$43$	10.7440	1.63844	0.819221	+	0.573479i	$0.194406\pi$
$44$	0	0
$45$	−3.05600	−0.455561
$46$	0	0
$47$	−1.96112	−0.286059	−0.143030	−	0.989718i	$-0.545684\pi$
$47$	−1.96112	−0.286059	−0.143030	+	0.989718i	$0.545684\pi$
$48$	0	0
$49$	−0.240573	−0.0343676
$50$	0	0
$51$	3.13623	0.439161
$52$	0	0
$53$	−7.56814	−1.03956	−0.519782	−	0.854299i	$-0.673987\pi$
$53$	−7.56814	−1.03956	−0.519782	+	0.854299i	$0.673987\pi$
$54$	0	0
$55$	14.4263	1.94524
$56$	0	0
$57$	2.68764	0.355986
$58$	0	0
$59$	−10.2078	−1.32895	−0.664474	−	0.747311i	$-0.731344\pi$
$59$	−10.2078	−1.32895	−0.664474	+	0.747311i	$0.731344\pi$
$60$	0	0
$61$	2.45141	0.313871	0.156936	−	0.987609i	$-0.449838\pi$
$61$	2.45141	0.313871	0.156936	+	0.987609i	$0.449838\pi$
$62$	0	0
$63$	−2.59989	−0.327555
$64$	0	0
$65$	−20.5990	−2.55499
$66$	0	0
$67$	−14.1052	−1.72323	−0.861614	−	0.507564i	$-0.830546\pi$
$67$	−14.1052	−1.72323	−0.861614	+	0.507564i	$0.830546\pi$
$68$	0	0
$69$	6.99219	0.841761
$70$	0	0
$71$	3.69140	0.438088	0.219044	−	0.975715i	$-0.429706\pi$
$71$	3.69140	0.438088	0.219044	+	0.975715i	$0.429706\pi$
$72$	0	0
$73$	4.93617	0.577735	0.288867	−	0.957369i	$-0.406721\pi$
$73$	4.93617	0.577735	0.288867	+	0.957369i	$0.406721\pi$
$74$	0	0
$75$	−4.33912	−0.501039
$76$	0	0
$77$	12.2731	1.39865
$78$	0	0
$79$	−9.15118	−1.02959	−0.514794	−	0.857314i	$-0.672132\pi$
$79$	−9.15118	−1.02959	−0.514794	+	0.857314i	$0.672132\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.82320	−0.858708	−0.429354	−	0.903136i	$-0.641259\pi$
$83$	−7.82320	−0.858708	−0.429354	+	0.903136i	$0.641259\pi$
$84$	0	0
$85$	9.58433	1.03957
$86$	0	0
$87$	7.69073	0.824533
$88$	0	0
$89$	5.37710	0.569971	0.284986	−	0.958532i	$-0.408011\pi$
$89$	5.37710	0.569971	0.284986	+	0.958532i	$0.408011\pi$
$90$	0	0
$91$	−17.5246	−1.83707
$92$	0	0
$93$	4.05540	0.420526
$94$	0	0
$95$	8.21342	0.842679
$96$	0	0
$97$	−10.3975	−1.05571	−0.527854	−	0.849335i	$-0.677003\pi$
$97$	−10.3975	−1.05571	−0.527854	+	0.849335i	$0.677003\pi$
$98$	0	0
$99$	−4.72064	−0.474442
$100$	0	0
$101$	−10.9764	−1.09219	−0.546094	−	0.837724i	$-0.683886\pi$
$101$	−10.9764	−1.09219	−0.546094	+	0.837724i	$0.683886\pi$
$102$	0	0
$103$	2.67179	0.263260	0.131630	−	0.991299i	$-0.457979\pi$
$103$	2.67179	0.263260	0.131630	+	0.991299i	$0.457979\pi$
$104$	0	0
$105$	−7.94526	−0.775378
$106$	0	0
$107$	3.57525	0.345633	0.172816	−	0.984954i	$-0.444713\pi$
$107$	3.57525	0.345633	0.172816	+	0.984954i	$0.444713\pi$
$108$	0	0
$109$	−6.87869	−0.658859	−0.329429	−	0.944180i	$-0.606856\pi$
$109$	−6.87869	−0.658859	−0.329429	+	0.944180i	$0.606856\pi$
$110$	0	0
$111$	−0.127532	−0.0121048
$112$	0	0
$113$	8.92908	0.839977	0.419989	−	0.907529i	$-0.362034\pi$
$113$	8.92908	0.839977	0.419989	+	0.907529i	$0.362034\pi$
$114$	0	0
$115$	21.3681	1.99259
$116$	0	0
$117$	6.74051	0.623160
$118$	0	0
$119$	8.15386	0.747463
$120$	0	0
$121$	11.2844	1.02586
$122$	0	0
$123$	8.00281	0.721589
$124$	0	0
$125$	2.01963	0.180642
$126$	0	0
$127$	−3.06592	−0.272057	−0.136028	−	0.990705i	$-0.543434\pi$
$127$	−3.06592	−0.272057	−0.136028	+	0.990705i	$0.543434\pi$
$128$	0	0
$129$	−10.7440	−0.945954
$130$	0	0
$131$	−16.5982	−1.45019	−0.725097	−	0.688647i	$-0.758205\pi$
$131$	−16.5982	−1.45019	−0.725097	+	0.688647i	$0.758205\pi$
$132$	0	0
$133$	6.98756	0.605899
$134$	0	0
$135$	3.05600	0.263018
$136$	0	0
$137$	0.703806	0.0601302	0.0300651	−	0.999548i	$-0.490429\pi$
$137$	0.703806	0.0601302	0.0300651	+	0.999548i	$0.490429\pi$
$138$	0	0
$139$	−14.7166	−1.24825	−0.624124	−	0.781326i	$-0.714544\pi$
$139$	−14.7166	−1.24825	−0.624124	+	0.781326i	$0.714544\pi$
$140$	0	0
$141$	1.96112	0.165156
$142$	0	0
$143$	−31.8195	−2.66088
$144$	0	0
$145$	23.5029	1.95181
$146$	0	0
$147$	0.240573	0.0198421
$148$	0	0
$149$	−10.3086	−0.844509	−0.422255	−	0.906477i	$-0.638761\pi$
$149$	−10.3086	−0.844509	−0.422255	+	0.906477i	$0.638761\pi$
$150$	0	0
$151$	−9.20583	−0.749160	−0.374580	−	0.927195i	$-0.622213\pi$
$151$	−9.20583	−0.749160	−0.374580	+	0.927195i	$0.622213\pi$
$152$	0	0
$153$	−3.13623	−0.253550
$154$	0	0
$155$	12.3933	0.995454
$156$	0	0
$157$	−20.9456	−1.67164	−0.835820	−	0.549004i	$-0.815007\pi$
$157$	−20.9456	−1.67164	−0.835820	+	0.549004i	$0.815007\pi$
$158$	0	0
$159$	7.56814	0.600192
$160$	0	0
$161$	18.1789	1.43270
$162$	0	0
$163$	−16.4063	−1.28504	−0.642521	−	0.766268i	$-0.722111\pi$
$163$	−16.4063	−1.28504	−0.642521	+	0.766268i	$0.722111\pi$
$164$	0	0
$165$	−14.4263	−1.12308
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	32.4344	2.49496
$170$	0	0
$171$	−2.68764	−0.205529
$172$	0	0
$173$	−4.45025	−0.338346	−0.169173	−	0.985586i	$-0.554110\pi$
$173$	−4.45025	−0.338346	−0.169173	+	0.985586i	$0.554110\pi$
$174$	0	0
$175$	−11.2812	−0.852782
$176$	0	0
$177$	10.2078	0.767269
$178$	0	0
$179$	−17.7672	−1.32798	−0.663992	−	0.747740i	$-0.731139\pi$
$179$	−17.7672	−1.32798	−0.663992	+	0.747740i	$0.731139\pi$
$180$	0	0
$181$	−15.0026	−1.11514	−0.557568	−	0.830132i	$-0.688265\pi$
$181$	−15.0026	−1.11514	−0.557568	+	0.830132i	$0.688265\pi$
$182$	0	0
$183$	−2.45141	−0.181214
$184$	0	0
$185$	−0.389739	−0.0286542
$186$	0	0
$187$	14.8050	1.08265
$188$	0	0
$189$	2.59989	0.189114
$190$	0	0
$191$	20.2983	1.46873	0.734367	−	0.678753i	$-0.237479\pi$
$191$	20.2983	1.46873	0.734367	+	0.678753i	$0.237479\pi$
$192$	0	0
$193$	−0.976343	−0.0702787	−0.0351394	−	0.999382i	$-0.511188\pi$
$193$	−0.976343	−0.0702787	−0.0351394	+	0.999382i	$0.511188\pi$
$194$	0	0
$195$	20.5990	1.47512
$196$	0	0
$197$	22.3449	1.59201	0.796003	−	0.605293i	$-0.206944\pi$
$197$	22.3449	1.59201	0.796003	+	0.605293i	$0.206944\pi$
$198$	0	0
$199$	−9.41395	−0.667337	−0.333669	−	0.942690i	$-0.608287\pi$
$199$	−9.41395	−0.667337	−0.333669	+	0.942690i	$0.608287\pi$
$200$	0	0
$201$	14.1052	0.994906
$202$	0	0
$203$	19.9951	1.40338
$204$	0	0
$205$	24.4566	1.70812
$206$	0	0
$207$	−6.99219	−0.485991
$208$	0	0
$209$	12.6874	0.877604
$210$	0	0
$211$	−18.5125	−1.27445	−0.637227	−	0.770676i	$-0.719919\pi$
$211$	−18.5125	−1.27445	−0.637227	+	0.770676i	$0.719919\pi$
$212$	0	0
$213$	−3.69140	−0.252930
$214$	0	0
$215$	−32.8336	−2.23923
$216$	0	0
$217$	10.5436	0.715746
$218$	0	0
$219$	−4.93617	−0.333555
$220$	0	0
$221$	−21.1398	−1.42202
$222$	0	0
$223$	20.0821	1.34480	0.672400	−	0.740188i	$-0.265264\pi$
$223$	20.0821	1.34480	0.672400	+	0.740188i	$0.265264\pi$
$224$	0	0
$225$	4.33912	0.289275
$226$	0	0
$227$	−3.07801	−0.204295	−0.102147	−	0.994769i	$-0.532571\pi$
$227$	−3.07801	−0.204295	−0.102147	+	0.994769i	$0.532571\pi$
$228$	0	0
$229$	19.3435	1.27825	0.639127	−	0.769101i	$-0.279296\pi$
$229$	19.3435	1.27825	0.639127	+	0.769101i	$0.279296\pi$
$230$	0	0
$231$	−12.2731	−0.807513
$232$	0	0
$233$	−7.27948	−0.476895	−0.238447	−	0.971155i	$-0.576638\pi$
$233$	−7.27948	−0.476895	−0.238447	+	0.971155i	$0.576638\pi$
$234$	0	0
$235$	5.99319	0.390953
$236$	0	0
$237$	9.15118	0.594433
$238$	0	0
$239$	16.2464	1.05089	0.525446	−	0.850827i	$-0.323898\pi$
$239$	16.2464	1.05089	0.525446	+	0.850827i	$0.323898\pi$
$240$	0	0
$241$	24.4511	1.57504	0.787518	−	0.616292i	$-0.211366\pi$
$241$	24.4511	1.57504	0.787518	+	0.616292i	$0.211366\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.735191	0.0469697
$246$	0	0
$247$	−18.1160	−1.15270
$248$	0	0
$249$	7.82320	0.495775
$250$	0	0
$251$	28.1202	1.77493	0.887465	−	0.460874i	$-0.152464\pi$
$251$	28.1202	1.77493	0.887465	+	0.460874i	$0.152464\pi$
$252$	0	0
$253$	33.0076	2.07517
$254$	0	0
$255$	−9.58433	−0.600194
$256$	0	0
$257$	−18.2270	−1.13697	−0.568485	−	0.822694i	$-0.692470\pi$
$257$	−18.2270	−1.13697	−0.568485	+	0.822694i	$0.692470\pi$
$258$	0	0
$259$	−0.331570	−0.0206028
$260$	0	0
$261$	−7.69073	−0.476044
$262$	0	0
$263$	5.53367	0.341220	0.170610	−	0.985339i	$-0.445426\pi$
$263$	5.53367	0.341220	0.170610	+	0.985339i	$0.445426\pi$
$264$	0	0
$265$	23.1282	1.42076
$266$	0	0
$267$	−5.37710	−0.329073
$268$	0	0
$269$	−18.1791	−1.10840	−0.554199	−	0.832385i	$-0.686975\pi$
$269$	−18.1791	−1.10840	−0.554199	+	0.832385i	$0.686975\pi$
$270$	0	0
$271$	30.0438	1.82503	0.912516	−	0.409040i	$-0.134136\pi$
$271$	30.0438	1.82503	0.912516	+	0.409040i	$0.134136\pi$
$272$	0	0
$273$	17.5246	1.06064
$274$	0	0
$275$	−20.4834	−1.23520
$276$	0	0
$277$	−20.0646	−1.20557	−0.602783	−	0.797905i	$-0.705942\pi$
$277$	−20.0646	−1.20557	−0.602783	+	0.797905i	$0.705942\pi$
$278$	0	0
$279$	−4.05540	−0.242791
$280$	0	0
$281$	20.1320	1.20097	0.600486	−	0.799635i	$-0.294974\pi$
$281$	20.1320	1.20097	0.600486	+	0.799635i	$0.294974\pi$
$282$	0	0
$283$	25.1003	1.49206	0.746028	−	0.665914i	$-0.231958\pi$
$283$	25.1003	1.49206	0.746028	+	0.665914i	$0.231958\pi$
$284$	0	0
$285$	−8.21342	−0.486521
$286$	0	0
$287$	20.8064	1.22816
$288$	0	0
$289$	−7.16403	−0.421414
$290$	0	0
$291$	10.3975	0.609513
$292$	0	0
$293$	29.6784	1.73383	0.866914	−	0.498457i	$-0.166100\pi$
$293$	29.6784	1.73383	0.866914	+	0.498457i	$0.166100\pi$
$294$	0	0
$295$	31.1952	1.81625
$296$	0	0
$297$	4.72064	0.273919
$298$	0	0
$299$	−47.1309	−2.72565
$300$	0	0
$301$	−27.9332	−1.61004
$302$	0	0
$303$	10.9764	0.630575
$304$	0	0
$305$	−7.49151	−0.428963
$306$	0	0
$307$	−22.0239	−1.25697	−0.628487	−	0.777820i	$-0.716325\pi$
$307$	−22.0239	−1.25697	−0.628487	+	0.777820i	$0.716325\pi$
$308$	0	0
$309$	−2.67179	−0.151993
$310$	0	0
$311$	26.7334	1.51591	0.757956	−	0.652305i	$-0.226198\pi$
$311$	26.7334	1.51591	0.757956	+	0.652305i	$0.226198\pi$
$312$	0	0
$313$	24.6867	1.39538	0.697688	−	0.716402i	$-0.254212\pi$
$313$	24.6867	1.39538	0.697688	+	0.716402i	$0.254212\pi$
$314$	0	0
$315$	7.94526	0.447665
$316$	0	0
$317$	29.6092	1.66302	0.831509	−	0.555511i	$-0.187477\pi$
$317$	29.6092	1.66302	0.831509	+	0.555511i	$0.187477\pi$
$318$	0	0
$319$	36.3052	2.03270
$320$	0	0
$321$	−3.57525	−0.199551
$322$	0	0
$323$	8.42906	0.469006
$324$	0	0
$325$	29.2479	1.62238
$326$	0	0
$327$	6.87869	0.380392
$328$	0	0
$329$	5.09871	0.281101
$330$	0	0
$331$	6.19998	0.340781	0.170391	−	0.985377i	$-0.445497\pi$
$331$	6.19998	0.340781	0.170391	+	0.985377i	$0.445497\pi$
$332$	0	0
$333$	0.127532	0.00698874
$334$	0	0
$335$	43.1056	2.35511
$336$	0	0
$337$	−13.1055	−0.713904	−0.356952	−	0.934123i	$-0.616184\pi$
$337$	−13.1055	−0.713904	−0.356952	+	0.934123i	$0.616184\pi$
$338$	0	0
$339$	−8.92908	−0.484961
$340$	0	0
$341$	19.1441	1.03671
$342$	0	0
$343$	18.8247	1.01644
$344$	0	0
$345$	−21.3681	−1.15042
$346$	0	0
$347$	32.6932	1.75506	0.877530	−	0.479521i	$-0.159190\pi$
$347$	32.6932	1.75506	0.877530	+	0.479521i	$0.159190\pi$
$348$	0	0
$349$	−6.07122	−0.324985	−0.162492	−	0.986710i	$-0.551953\pi$
$349$	−6.07122	−0.324985	−0.162492	+	0.986710i	$0.551953\pi$
$350$	0	0
$351$	−6.74051	−0.359782
$352$	0	0
$353$	20.1820	1.07418	0.537091	−	0.843525i	$-0.319523\pi$
$353$	20.1820	1.07418	0.537091	+	0.843525i	$0.319523\pi$
$354$	0	0
$355$	−11.2809	−0.598728
$356$	0	0
$357$	−8.15386	−0.431548
$358$	0	0
$359$	−25.8863	−1.36623	−0.683113	−	0.730313i	$-0.739374\pi$
$359$	−25.8863	−1.36623	−0.683113	+	0.730313i	$0.739374\pi$
$360$	0	0
$361$	−11.7766	−0.619821
$362$	0	0
$363$	−11.2844	−0.592278
$364$	0	0
$365$	−15.0849	−0.789581
$366$	0	0
$367$	4.18304	0.218353	0.109176	−	0.994022i	$-0.465179\pi$
$367$	4.18304	0.218353	0.109176	+	0.994022i	$0.465179\pi$
$368$	0	0
$369$	−8.00281	−0.416609
$370$	0	0
$371$	19.6763	1.02154
$372$	0	0
$373$	6.51126	0.337140	0.168570	−	0.985690i	$-0.446085\pi$
$373$	6.51126	0.337140	0.168570	+	0.985690i	$0.446085\pi$
$374$	0	0
$375$	−2.01963	−0.104293
$376$	0	0
$377$	−51.8394	−2.66987
$378$	0	0
$379$	−16.9299	−0.869631	−0.434815	−	0.900520i	$-0.643186\pi$
$379$	−16.9299	−0.869631	−0.434815	+	0.900520i	$0.643186\pi$
$380$	0	0
$381$	3.06592	0.157072
$382$	0	0
$383$	26.0508	1.33113	0.665566	−	0.746339i	$-0.268190\pi$
$383$	26.0508	1.33113	0.665566	+	0.746339i	$0.268190\pi$
$384$	0	0
$385$	−37.5067	−1.91152
$386$	0	0
$387$	10.7440	0.546147
$388$	0	0
$389$	32.2295	1.63410	0.817049	−	0.576568i	$-0.195608\pi$
$389$	32.2295	1.63410	0.817049	+	0.576568i	$0.195608\pi$
$390$	0	0
$391$	21.9292	1.10900
$392$	0	0
$393$	16.5982	0.837270
$394$	0	0
$395$	27.9660	1.40712
$396$	0	0
$397$	−18.5553	−0.931263	−0.465632	−	0.884979i	$-0.654173\pi$
$397$	−18.5553	−0.931263	−0.465632	+	0.884979i	$0.654173\pi$
$398$	0	0
$399$	−6.98756	−0.349816
$400$	0	0
$401$	−10.1254	−0.505640	−0.252820	−	0.967513i	$-0.581358\pi$
$401$	−10.1254	−0.505640	−0.252820	+	0.967513i	$0.581358\pi$
$402$	0	0
$403$	−27.3355	−1.36168
$404$	0	0
$405$	−3.05600	−0.151854
$406$	0	0
$407$	−0.602035	−0.0298417
$408$	0	0
$409$	−32.8568	−1.62467	−0.812333	−	0.583194i	$-0.801803\pi$
$409$	−32.8568	−1.62467	−0.812333	+	0.583194i	$0.801803\pi$
$410$	0	0
$411$	−0.703806	−0.0347162
$412$	0	0
$413$	26.5393	1.30591
$414$	0	0
$415$	23.9077	1.17358
$416$	0	0
$417$	14.7166	0.720676
$418$	0	0
$419$	−25.3416	−1.23802	−0.619010	−	0.785383i	$-0.712466\pi$
$419$	−25.3416	−1.23802	−0.619010	+	0.785383i	$0.712466\pi$
$420$	0	0
$421$	1.17825	0.0574242	0.0287121	−	0.999588i	$-0.490859\pi$
$421$	1.17825	0.0574242	0.0287121	+	0.999588i	$0.490859\pi$
$422$	0	0
$423$	−1.96112	−0.0953531
$424$	0	0
$425$	−13.6085	−0.660110
$426$	0	0
$427$	−6.37340	−0.308431
$428$	0	0
$429$	31.8195	1.53626
$430$	0	0
$431$	18.3794	0.885307	0.442653	−	0.896693i	$-0.354037\pi$
$431$	18.3794	0.885307	0.442653	+	0.896693i	$0.354037\pi$
$432$	0	0
$433$	−24.5067	−1.17772	−0.588859	−	0.808236i	$-0.700423\pi$
$433$	−24.5067	−1.17772	−0.588859	+	0.808236i	$0.700423\pi$
$434$	0	0
$435$	−23.5029	−1.12688
$436$	0	0
$437$	18.7925	0.898967
$438$	0	0
$439$	−10.5405	−0.503070	−0.251535	−	0.967848i	$-0.580935\pi$
$439$	−10.5405	−0.503070	−0.251535	+	0.967848i	$0.580935\pi$
$440$	0	0
$441$	−0.240573	−0.0114559
$442$	0	0
$443$	13.1918	0.626760	0.313380	−	0.949628i	$-0.398539\pi$
$443$	13.1918	0.626760	0.313380	+	0.949628i	$0.398539\pi$
$444$	0	0
$445$	−16.4324	−0.778970
$446$	0	0
$447$	10.3086	0.487578
$448$	0	0
$449$	13.6895	0.646046	0.323023	−	0.946391i	$-0.395301\pi$
$449$	13.6895	0.646046	0.323023	+	0.946391i	$0.395301\pi$
$450$	0	0
$451$	37.7783	1.77891
$452$	0	0
$453$	9.20583	0.432528
$454$	0	0
$455$	53.5551	2.51070
$456$	0	0
$457$	14.7356	0.689303	0.344651	−	0.938731i	$-0.387997\pi$
$457$	14.7356	0.689303	0.344651	+	0.938731i	$0.387997\pi$
$458$	0	0
$459$	3.13623	0.146387
$460$	0	0
$461$	−22.8198	−1.06283	−0.531413	−	0.847113i	$-0.678339\pi$
$461$	−22.8198	−1.06283	−0.531413	+	0.847113i	$0.678339\pi$
$462$	0	0
$463$	−8.50925	−0.395458	−0.197729	−	0.980257i	$-0.563357\pi$
$463$	−8.50925	−0.395458	−0.197729	+	0.980257i	$0.563357\pi$
$464$	0	0
$465$	−12.3933	−0.574726
$466$	0	0
$467$	6.22222	0.287930	0.143965	−	0.989583i	$-0.454015\pi$
$467$	6.22222	0.287930	0.143965	+	0.989583i	$0.454015\pi$
$468$	0	0
$469$	36.6720	1.69336
$470$	0	0
$471$	20.9456	0.965122
$472$	0	0
$473$	−50.7184	−2.33204
$474$	0	0
$475$	−11.6620	−0.535089
$476$	0	0
$477$	−7.56814	−0.346521
$478$	0	0
$479$	−40.6592	−1.85777	−0.928884	−	0.370372i	$-0.879230\pi$
$479$	−40.6592	−1.85777	−0.928884	+	0.370372i	$0.879230\pi$
$480$	0	0
$481$	0.859634	0.0391959
$482$	0	0
$483$	−18.1789	−0.827170
$484$	0	0
$485$	31.7748	1.44282
$486$	0	0
$487$	25.3758	1.14989	0.574944	−	0.818193i	$-0.305024\pi$
$487$	25.3758	1.14989	0.574944	+	0.818193i	$0.305024\pi$
$488$	0	0
$489$	16.4063	0.741919
$490$	0	0
$491$	−2.83989	−0.128162	−0.0640812	−	0.997945i	$-0.520412\pi$
$491$	−2.83989	−0.128162	−0.0640812	+	0.997945i	$0.520412\pi$
$492$	0	0
$493$	24.1199	1.08631
$494$	0	0
$495$	14.4263	0.648412
$496$	0	0
$497$	−9.59722	−0.430494
$498$	0	0
$499$	−11.3425	−0.507759	−0.253879	−	0.967236i	$-0.581707\pi$
$499$	−11.3425	−0.507759	−0.253879	+	0.967236i	$0.581707\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−23.7022	−1.05683	−0.528415	−	0.848986i	$-0.677214\pi$
$503$	−23.7022	−1.05683	−0.528415	+	0.848986i	$0.677214\pi$
$504$	0	0
$505$	33.5437	1.49268
$506$	0	0
$507$	−32.4344	−1.44046
$508$	0	0
$509$	1.62782	0.0721517	0.0360758	−	0.999349i	$-0.488514\pi$
$509$	1.62782	0.0721517	0.0360758	+	0.999349i	$0.488514\pi$
$510$	0	0
$511$	−12.8335	−0.567720
$512$	0	0
$513$	2.68764	0.118662
$514$	0	0
$515$	−8.16500	−0.359793
$516$	0	0
$517$	9.25775	0.407156
$518$	0	0
$519$	4.45025	0.195344
$520$	0	0
$521$	5.03728	0.220687	0.110344	−	0.993893i	$-0.464805\pi$
$521$	5.03728	0.220687	0.110344	+	0.993893i	$0.464805\pi$
$522$	0	0
$523$	−24.7066	−1.08034	−0.540172	−	0.841555i	$-0.681641\pi$
$523$	−24.7066	−1.08034	−0.540172	+	0.841555i	$0.681641\pi$
$524$	0	0
$525$	11.2812	0.492354
$526$	0	0
$527$	12.7187	0.554035
$528$	0	0
$529$	25.8908	1.12568
$530$	0	0
$531$	−10.2078	−0.442983
$532$	0	0
$533$	−53.9430	−2.33653
$534$	0	0
$535$	−10.9260	−0.472370
$536$	0	0
$537$	17.7672	0.766712
$538$	0	0
$539$	1.13566	0.0489163
$540$	0	0
$541$	2.18767	0.0940552	0.0470276	−	0.998894i	$-0.485025\pi$
$541$	2.18767	0.0940552	0.0470276	+	0.998894i	$0.485025\pi$
$542$	0	0
$543$	15.0026	0.643824
$544$	0	0
$545$	21.0213	0.900451
$546$	0	0
$547$	−15.8845	−0.679172	−0.339586	−	0.940575i	$-0.610287\pi$
$547$	−15.8845	−0.679172	−0.339586	+	0.940575i	$0.610287\pi$
$548$	0	0
$549$	2.45141	0.104624
$550$	0	0
$551$	20.6699	0.880568
$552$	0	0
$553$	23.7921	1.01174
$554$	0	0
$555$	0.389739	0.0165435
$556$	0	0
$557$	1.88069	0.0796873	0.0398436	−	0.999206i	$-0.487314\pi$
$557$	1.88069	0.0796873	0.0398436	+	0.999206i	$0.487314\pi$
$558$	0	0
$559$	72.4199	3.06303
$560$	0	0
$561$	−14.8050	−0.625069
$562$	0	0
$563$	−19.8631	−0.837131	−0.418565	−	0.908187i	$-0.637467\pi$
$563$	−19.8631	−0.837131	−0.418565	+	0.908187i	$0.637467\pi$
$564$	0	0
$565$	−27.2873	−1.14798
$566$	0	0
$567$	−2.59989	−0.109185
$568$	0	0
$569$	−21.6129	−0.906058	−0.453029	−	0.891496i	$-0.649657\pi$
$569$	−21.6129	−0.906058	−0.453029	+	0.891496i	$0.649657\pi$
$570$	0	0
$571$	−8.15758	−0.341384	−0.170692	−	0.985324i	$-0.554600\pi$
$571$	−8.15758	−0.341384	−0.170692	+	0.985324i	$0.554600\pi$
$572$	0	0
$573$	−20.2983	−0.847974
$574$	0	0
$575$	−30.3400	−1.26527
$576$	0	0
$577$	−14.3812	−0.598698	−0.299349	−	0.954144i	$-0.596769\pi$
$577$	−14.3812	−0.598698	−0.299349	+	0.954144i	$0.596769\pi$
$578$	0	0
$579$	0.976343	0.0405754
$580$	0	0
$581$	20.3395	0.843823
$582$	0	0
$583$	35.7264	1.47964
$584$	0	0
$585$	−20.5990	−0.851663
$586$	0	0
$587$	−16.4163	−0.677573	−0.338786	−	0.940863i	$-0.610016\pi$
$587$	−16.4163	−0.677573	−0.338786	+	0.940863i	$0.610016\pi$
$588$	0	0
$589$	10.8995	0.449104
$590$	0	0
$591$	−22.3449	−0.919145
$592$	0	0
$593$	−26.4055	−1.08434	−0.542171	−	0.840268i	$-0.682398\pi$
$593$	−26.4055	−1.08434	−0.542171	+	0.840268i	$0.682398\pi$
$594$	0	0
$595$	−24.9182	−1.02155
$596$	0	0
$597$	9.41395	0.385287
$598$	0	0
$599$	4.34234	0.177423	0.0887116	−	0.996057i	$-0.471725\pi$
$599$	4.34234	0.177423	0.0887116	+	0.996057i	$0.471725\pi$
$600$	0	0
$601$	21.5387	0.878583	0.439292	−	0.898344i	$-0.355229\pi$
$601$	21.5387	0.878583	0.439292	+	0.898344i	$0.355229\pi$
$602$	0	0
$603$	−14.1052	−0.574409
$604$	0	0
$605$	−34.4852	−1.40202
$606$	0	0
$607$	6.25626	0.253934	0.126967	−	0.991907i	$-0.459476\pi$
$607$	6.25626	0.253934	0.126967	+	0.991907i	$0.459476\pi$
$608$	0	0
$609$	−19.9951	−0.810241
$610$	0	0
$611$	−13.2190	−0.534782
$612$	0	0
$613$	−32.0598	−1.29488	−0.647441	−	0.762116i	$-0.724161\pi$
$613$	−32.0598	−1.29488	−0.647441	+	0.762116i	$0.724161\pi$
$614$	0	0
$615$	−24.4566	−0.986184
$616$	0	0
$617$	−32.0392	−1.28985	−0.644925	−	0.764246i	$-0.723111\pi$
$617$	−32.0392	−1.28985	−0.644925	+	0.764246i	$0.723111\pi$
$618$	0	0
$619$	−20.7106	−0.832428	−0.416214	−	0.909267i	$-0.636643\pi$
$619$	−20.7106	−0.832428	−0.416214	+	0.909267i	$0.636643\pi$
$620$	0	0
$621$	6.99219	0.280587
$622$	0	0
$623$	−13.9799	−0.560091
$624$	0	0
$625$	−27.8676	−1.11470
$626$	0	0
$627$	−12.6874	−0.506685
$628$	0	0
$629$	−0.399972	−0.0159479
$630$	0	0
$631$	−26.6970	−1.06279	−0.531395	−	0.847124i	$-0.678332\pi$
$631$	−26.6970	−1.06279	−0.531395	+	0.847124i	$0.678332\pi$
$632$	0	0
$633$	18.5125	0.735806
$634$	0	0
$635$	9.36946	0.371816
$636$	0	0
$637$	−1.62159	−0.0642496
$638$	0	0
$639$	3.69140	0.146029
$640$	0	0
$641$	12.0039	0.474125	0.237062	−	0.971494i	$-0.423815\pi$
$641$	12.0039	0.474125	0.237062	+	0.971494i	$0.423815\pi$
$642$	0	0
$643$	−33.8943	−1.33666	−0.668331	−	0.743864i	$-0.732991\pi$
$643$	−33.8943	−1.33666	−0.668331	+	0.743864i	$0.732991\pi$
$644$	0	0
$645$	32.8336	1.29282
$646$	0	0
$647$	−17.6487	−0.693842	−0.346921	−	0.937894i	$-0.612773\pi$
$647$	−17.6487	−0.693842	−0.346921	+	0.937894i	$0.612773\pi$
$648$	0	0
$649$	48.1875	1.89153
$650$	0	0
$651$	−10.5436	−0.413236
$652$	0	0
$653$	−12.4535	−0.487344	−0.243672	−	0.969858i	$-0.578352\pi$
$653$	−12.4535	−0.487344	−0.243672	+	0.969858i	$0.578352\pi$
$654$	0	0
$655$	50.7242	1.98196
$656$	0	0
$657$	4.93617	0.192578
$658$	0	0
$659$	23.0086	0.896290	0.448145	−	0.893961i	$-0.352085\pi$
$659$	23.0086	0.896290	0.448145	+	0.893961i	$0.352085\pi$
$660$	0	0
$661$	−3.59936	−0.139999	−0.0699994	−	0.997547i	$-0.522300\pi$
$661$	−3.59936	−0.139999	−0.0699994	+	0.997547i	$0.522300\pi$
$662$	0	0
$663$	21.1398	0.821002
$664$	0	0
$665$	−21.3540	−0.828072
$666$	0	0
$667$	53.7751	2.08218
$668$	0	0
$669$	−20.0821	−0.776420
$670$	0	0
$671$	−11.5722	−0.446741
$672$	0	0
$673$	−38.0723	−1.46758	−0.733791	−	0.679376i	$-0.762251\pi$
$673$	−38.0723	−1.46758	−0.733791	+	0.679376i	$0.762251\pi$
$674$	0	0
$675$	−4.33912	−0.167013
$676$	0	0
$677$	27.9008	1.07232	0.536158	−	0.844118i	$-0.319875\pi$
$677$	27.9008	1.07232	0.536158	+	0.844118i	$0.319875\pi$
$678$	0	0
$679$	27.0324	1.03741
$680$	0	0
$681$	3.07801	0.117950
$682$	0	0
$683$	14.7556	0.564608	0.282304	−	0.959325i	$-0.408901\pi$
$683$	14.7556	0.564608	0.282304	+	0.959325i	$0.408901\pi$
$684$	0	0
$685$	−2.15083	−0.0821790
$686$	0	0
$687$	−19.3435	−0.738000
$688$	0	0
$689$	−51.0131	−1.94344
$690$	0	0
$691$	26.6875	1.01524	0.507619	−	0.861581i	$-0.330526\pi$
$691$	26.6875	1.01524	0.507619	+	0.861581i	$0.330526\pi$
$692$	0	0
$693$	12.2731	0.466218
$694$	0	0
$695$	44.9739	1.70596
$696$	0	0
$697$	25.0987	0.950680
$698$	0	0
$699$	7.27948	0.275335
$700$	0	0
$701$	7.51451	0.283819	0.141910	−	0.989880i	$-0.454676\pi$
$701$	7.51451	0.283819	0.141910	+	0.989880i	$0.454676\pi$
$702$	0	0
$703$	−0.342761	−0.0129275
$704$	0	0
$705$	−5.99319	−0.225717
$706$	0	0
$707$	28.5373	1.07326
$708$	0	0
$709$	−42.9604	−1.61341	−0.806706	−	0.590953i	$-0.798752\pi$
$709$	−42.9604	−1.61341	−0.806706	+	0.590953i	$0.798752\pi$
$710$	0	0
$711$	−9.15118	−0.343196
$712$	0	0
$713$	28.3561	1.06195
$714$	0	0
$715$	97.2403	3.63658
$716$	0	0
$717$	−16.2464	−0.606733
$718$	0	0
$719$	−2.20012	−0.0820506	−0.0410253	−	0.999158i	$-0.513062\pi$
$719$	−2.20012	−0.0820506	−0.0410253	+	0.999158i	$0.513062\pi$
$720$	0	0
$721$	−6.94637	−0.258696
$722$	0	0
$723$	−24.4511	−0.909347
$724$	0	0
$725$	−33.3710	−1.23937
$726$	0	0
$727$	10.7783	0.399746	0.199873	−	0.979822i	$-0.435947\pi$
$727$	10.7783	0.399746	0.199873	+	0.979822i	$0.435947\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.6956	−1.24628
$732$	0	0
$733$	46.4940	1.71730	0.858648	−	0.512565i	$-0.171305\pi$
$733$	46.4940	1.71730	0.858648	+	0.512565i	$0.171305\pi$
$734$	0	0
$735$	−0.735191	−0.0271179
$736$	0	0
$737$	66.5857	2.45271
$738$	0	0
$739$	20.9324	0.770011	0.385006	−	0.922914i	$-0.374199\pi$
$739$	20.9324	0.770011	0.385006	+	0.922914i	$0.374199\pi$
$740$	0	0
$741$	18.1160	0.665510
$742$	0	0
$743$	23.1270	0.848446	0.424223	−	0.905558i	$-0.360547\pi$
$743$	23.1270	0.848446	0.424223	+	0.905558i	$0.360547\pi$
$744$	0	0
$745$	31.5029	1.15418
$746$	0	0
$747$	−7.82320	−0.286236
$748$	0	0
$749$	−9.29526	−0.339641
$750$	0	0
$751$	17.0814	0.623311	0.311655	−	0.950195i	$-0.399117\pi$
$751$	17.0814	0.623311	0.311655	+	0.950195i	$0.399117\pi$
$752$	0	0
$753$	−28.1202	−1.02476
$754$	0	0
$755$	28.1330	1.02387
$756$	0	0
$757$	9.85466	0.358174	0.179087	−	0.983833i	$-0.442686\pi$
$757$	9.85466	0.358174	0.179087	+	0.983833i	$0.442686\pi$
$758$	0	0
$759$	−33.0076	−1.19810
$760$	0	0
$761$	28.6740	1.03943	0.519716	−	0.854339i	$-0.326038\pi$
$761$	28.6740	1.03943	0.519716	+	0.854339i	$0.326038\pi$
$762$	0	0
$763$	17.8838	0.647438
$764$	0	0
$765$	9.58433	0.346522
$766$	0	0
$767$	−68.8061	−2.48444
$768$	0	0
$769$	−34.9620	−1.26076	−0.630381	−	0.776286i	$-0.717101\pi$
$769$	−34.9620	−1.26076	−0.630381	+	0.776286i	$0.717101\pi$
$770$	0	0
$771$	18.2270	0.656430
$772$	0	0
$773$	−8.10284	−0.291439	−0.145720	−	0.989326i	$-0.546550\pi$
$773$	−8.10284	−0.291439	−0.145720	+	0.989326i	$0.546550\pi$
$774$	0	0
$775$	−17.5969	−0.632099
$776$	0	0
$777$	0.331570	0.0118950
$778$	0	0
$779$	21.5086	0.770627
$780$	0	0
$781$	−17.4257	−0.623542
$782$	0	0
$783$	7.69073	0.274844
$784$	0	0
$785$	64.0097	2.28460
$786$	0	0
$787$	−37.6200	−1.34101	−0.670504	−	0.741906i	$-0.733922\pi$
$787$	−37.6200	−1.34101	−0.670504	+	0.741906i	$0.733922\pi$
$788$	0	0
$789$	−5.53367	−0.197004
$790$	0	0
$791$	−23.2146	−0.825417
$792$	0	0
$793$	16.5238	0.586776
$794$	0	0
$795$	−23.1282	−0.820273
$796$	0	0
$797$	−29.2219	−1.03509	−0.517546	−	0.855655i	$-0.673154\pi$
$797$	−29.2219	−1.03509	−0.517546	+	0.855655i	$0.673154\pi$
$798$	0	0
$799$	6.15054	0.217591
$800$	0	0
$801$	5.37710	0.189990
$802$	0	0
$803$	−23.3019	−0.822304
$804$	0	0
$805$	−55.5548	−1.95805
$806$	0	0
$807$	18.1791	0.639933
$808$	0	0
$809$	−27.9431	−0.982427	−0.491213	−	0.871039i	$-0.663446\pi$
$809$	−27.9431	−0.982427	−0.491213	+	0.871039i	$0.663446\pi$
$810$	0	0
$811$	46.9869	1.64993	0.824967	−	0.565181i	$-0.191194\pi$
$811$	46.9869	1.64993	0.824967	+	0.565181i	$0.191194\pi$
$812$	0	0
$813$	−30.0438	−1.05368
$814$	0	0
$815$	50.1377	1.75625
$816$	0	0
$817$	−28.8759	−1.01024
$818$	0	0
$819$	−17.5246	−0.612358
$820$	0	0
$821$	−6.11204	−0.213312	−0.106656	−	0.994296i	$-0.534014\pi$
$821$	−6.11204	−0.213312	−0.106656	+	0.994296i	$0.534014\pi$
$822$	0	0
$823$	−23.3315	−0.813285	−0.406643	−	0.913587i	$-0.633301\pi$
$823$	−23.3315	−0.813285	−0.406643	+	0.913587i	$0.633301\pi$
$824$	0	0
$825$	20.4834	0.713142
$826$	0	0
$827$	−47.3352	−1.64601	−0.823003	−	0.568037i	$-0.807703\pi$
$827$	−47.3352	−1.64601	−0.823003	+	0.568037i	$0.807703\pi$
$828$	0	0
$829$	37.7216	1.31012	0.655062	−	0.755575i	$-0.272642\pi$
$829$	37.7216	1.31012	0.655062	+	0.755575i	$0.272642\pi$
$830$	0	0
$831$	20.0646	0.696034
$832$	0	0
$833$	0.754494	0.0261417
$834$	0	0
$835$	−3.05600	−0.105757
$836$	0	0
$837$	4.05540	0.140175
$838$	0	0
$839$	38.0910	1.31505	0.657524	−	0.753434i	$-0.271604\pi$
$839$	38.0910	1.31505	0.657524	+	0.753434i	$0.271604\pi$
$840$	0	0
$841$	30.1474	1.03956
$842$	0	0
$843$	−20.1320	−0.693382
$844$	0	0
$845$	−99.1196	−3.40982
$846$	0	0
$847$	−29.3382	−1.00807
$848$	0	0
$849$	−25.1003	−0.861439
$850$	0	0
$851$	−0.891732	−0.0305682
$852$	0	0
$853$	51.7150	1.77069	0.885343	−	0.464938i	$-0.153923\pi$
$853$	51.7150	1.77069	0.885343	+	0.464938i	$0.153923\pi$
$854$	0	0
$855$	8.21342	0.280893
$856$	0	0
$857$	−1.25379	−0.0428287	−0.0214143	−	0.999771i	$-0.506817\pi$
$857$	−1.25379	−0.0428287	−0.0214143	+	0.999771i	$0.506817\pi$
$858$	0	0
$859$	−48.8539	−1.66687	−0.833436	−	0.552616i	$-0.813630\pi$
$859$	−48.8539	−1.66687	−0.833436	+	0.552616i	$0.813630\pi$
$860$	0	0
$861$	−20.8064	−0.709081
$862$	0	0
$863$	−17.6303	−0.600143	−0.300072	−	0.953917i	$-0.597011\pi$
$863$	−17.6303	−0.600143	−0.300072	+	0.953917i	$0.597011\pi$
$864$	0	0
$865$	13.5999	0.462412
$866$	0	0
$867$	7.16403	0.243303
$868$	0	0
$869$	43.1994	1.46544
$870$	0	0
$871$	−95.0764	−3.22154
$872$	0	0
$873$	−10.3975	−0.351903
$874$	0	0
$875$	−5.25083	−0.177510
$876$	0	0
$877$	−1.77076	−0.0597945	−0.0298972	−	0.999553i	$-0.509518\pi$
$877$	−1.77076	−0.0597945	−0.0298972	+	0.999553i	$0.509518\pi$
$878$	0	0
$879$	−29.6784	−1.00103
$880$	0	0
$881$	−43.8943	−1.47884	−0.739419	−	0.673246i	$-0.764900\pi$
$881$	−43.8943	−1.47884	−0.739419	+	0.673246i	$0.764900\pi$
$882$	0	0
$883$	−17.5616	−0.590996	−0.295498	−	0.955343i	$-0.595486\pi$
$883$	−17.5616	−0.590996	−0.295498	+	0.955343i	$0.595486\pi$
$884$	0	0
$885$	−31.1952	−1.04861
$886$	0	0
$887$	−14.0970	−0.473332	−0.236666	−	0.971591i	$-0.576055\pi$
$887$	−14.0970	−0.473332	−0.236666	+	0.971591i	$0.576055\pi$
$888$	0	0
$889$	7.97106	0.267341
$890$	0	0
$891$	−4.72064	−0.158147
$892$	0	0
$893$	5.27079	0.176380
$894$	0	0
$895$	54.2966	1.81493
$896$	0	0
$897$	47.1309	1.57366
$898$	0	0
$899$	31.1890	1.04021
$900$	0	0
$901$	23.7355	0.790743
$902$	0	0
$903$	27.9332	0.929557
$904$	0	0
$905$	45.8479	1.52404
$906$	0	0
$907$	−12.8504	−0.426692	−0.213346	−	0.976977i	$-0.568436\pi$
$907$	−12.8504	−0.426692	−0.213346	+	0.976977i	$0.568436\pi$
$908$	0	0
$909$	−10.9764	−0.364063
$910$	0	0
$911$	−18.4133	−0.610060	−0.305030	−	0.952343i	$-0.598667\pi$
$911$	−18.4133	−0.610060	−0.305030	+	0.952343i	$0.598667\pi$
$912$	0	0
$913$	36.9305	1.22222
$914$	0	0
$915$	7.49151	0.247662
$916$	0	0
$917$	43.1536	1.42506
$918$	0	0
$919$	43.6473	1.43979	0.719895	−	0.694082i	$-0.244190\pi$
$919$	43.6473	1.43979	0.719895	+	0.694082i	$0.244190\pi$
$920$	0	0
$921$	22.0239	0.725714
$922$	0	0
$923$	24.8819	0.818997
$924$	0	0
$925$	0.553379	0.0181950
$926$	0	0
$927$	2.67179	0.0877533
$928$	0	0
$929$	−6.48680	−0.212825	−0.106412	−	0.994322i	$-0.533936\pi$
$929$	−6.48680	−0.212825	−0.106412	+	0.994322i	$0.533936\pi$
$930$	0	0
$931$	0.646574	0.0211906
$932$	0	0
$933$	−26.7334	−0.875213
$934$	0	0
$935$	−45.2441	−1.47964
$936$	0	0
$937$	15.5535	0.508110	0.254055	−	0.967190i	$-0.418236\pi$
$937$	15.5535	0.508110	0.254055	+	0.967190i	$0.418236\pi$
$938$	0	0
$939$	−24.6867	−0.805620
$940$	0	0
$941$	47.2420	1.54004	0.770022	−	0.638017i	$-0.220245\pi$
$941$	47.2420	1.54004	0.770022	+	0.638017i	$0.220245\pi$
$942$	0	0
$943$	55.9572	1.82222
$944$	0	0
$945$	−7.94526	−0.258459
$946$	0	0
$947$	31.0655	1.00949	0.504747	−	0.863268i	$-0.331586\pi$
$947$	31.0655	1.00949	0.504747	+	0.863268i	$0.331586\pi$
$948$	0	0
$949$	33.2723	1.08006
$950$	0	0
$951$	−29.6092	−0.960144
$952$	0	0
$953$	−8.77668	−0.284304	−0.142152	−	0.989845i	$-0.545402\pi$
$953$	−8.77668	−0.284304	−0.142152	+	0.989845i	$0.545402\pi$
$954$	0	0
$955$	−62.0316	−2.00730
$956$	0	0
$957$	−36.3052	−1.17358
$958$	0	0
$959$	−1.82982	−0.0590879
$960$	0	0
$961$	−14.5537	−0.469475
$962$	0	0
$963$	3.57525	0.115211
$964$	0	0
$965$	2.98370	0.0960488
$966$	0	0
$967$	23.9661	0.770697	0.385348	−	0.922771i	$-0.374081\pi$
$967$	23.9661	0.770697	0.385348	+	0.922771i	$0.374081\pi$
$968$	0	0
$969$	−8.42906	−0.270781
$970$	0	0
$971$	18.9528	0.608223	0.304112	−	0.952636i	$-0.401640\pi$
$971$	18.9528	0.608223	0.304112	+	0.952636i	$0.401640\pi$
$972$	0	0
$973$	38.2616	1.22661
$974$	0	0
$975$	−29.2479	−0.936683
$976$	0	0
$977$	−46.2775	−1.48055	−0.740274	−	0.672305i	$-0.765304\pi$
$977$	−46.2775	−1.48055	−0.740274	+	0.672305i	$0.765304\pi$
$978$	0	0
$979$	−25.3833	−0.811254
$980$	0	0
$981$	−6.87869	−0.219620
$982$	0	0
$983$	7.63097	0.243390	0.121695	−	0.992568i	$-0.461167\pi$
$983$	7.63097	0.243390	0.121695	+	0.992568i	$0.461167\pi$
$984$	0	0
$985$	−68.2859	−2.17577
$986$	0	0
$987$	−5.09871	−0.162294
$988$	0	0
$989$	−75.1239	−2.38880
$990$	0	0
$991$	−5.91696	−0.187959	−0.0939793	−	0.995574i	$-0.529959\pi$
$991$	−5.91696	−0.187959	−0.0939793	+	0.995574i	$0.529959\pi$
$992$	0	0
$993$	−6.19998	−0.196750
$994$	0	0
$995$	28.7690	0.912039
$996$	0	0
$997$	−3.25299	−0.103023	−0.0515117	−	0.998672i	$-0.516404\pi$
$997$	−3.25299	−0.103023	−0.0515117	+	0.998672i	$0.516404\pi$
$998$	0	0
$999$	−0.127532	−0.00403495

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.3		13
4.3	odd	2		4008.2.a.m.1.3	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.3	✓	13	4.3	odd	2
8016.2.a.bg.1.3		13	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.05600 q^{5} -2.59989 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.05600 q^{5} -2.59989 q^{7} +1.00000 q^{9} -4.72064 q^{11} +6.74051 q^{13} +3.05600 q^{15} -3.13623 q^{17} -2.68764 q^{19} +2.59989 q^{21} -6.99219 q^{23} +4.33912 q^{25} -1.00000 q^{27} -7.69073 q^{29} -4.05540 q^{31} +4.72064 q^{33} +7.94526 q^{35} +0.127532 q^{37} -6.74051 q^{39} -8.00281 q^{41} +10.7440 q^{43} -3.05600 q^{45} -1.96112 q^{47} -0.240573 q^{49} +3.13623 q^{51} -7.56814 q^{53} +14.4263 q^{55} +2.68764 q^{57} -10.2078 q^{59} +2.45141 q^{61} -2.59989 q^{63} -20.5990 q^{65} -14.1052 q^{67} +6.99219 q^{69} +3.69140 q^{71} +4.93617 q^{73} -4.33912 q^{75} +12.2731 q^{77} -9.15118 q^{79} +1.00000 q^{81} -7.82320 q^{83} +9.58433 q^{85} +7.69073 q^{87} +5.37710 q^{89} -17.5246 q^{91} +4.05540 q^{93} +8.21342 q^{95} -10.3975 q^{97} -4.72064 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

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