LMFDB - Embedded newform 4008.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, 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("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 16x^{7} + 45x^{6} + 67x^{5} - 166x^{4} - 83x^{3} + 152x^{2} + 51x - 10 $ x^9 - 3x^8 - 16x^7 + 45x^6 + 67x^5 - 166x^4 - 83x^3 + 152x^2 + 51x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.141931$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.858069 q^{5} +2.33225 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.858069 q^{5} +2.33225 q^{7} +1.00000 q^{9} +0.664078 q^{11} -2.80424 q^{13} -0.858069 q^{15} -4.73771 q^{17} -7.58466 q^{19} +2.33225 q^{21} +6.58999 q^{23} -4.26372 q^{25} +1.00000 q^{27} -4.75833 q^{29} -2.96038 q^{31} +0.664078 q^{33} -2.00123 q^{35} +4.12480 q^{37} -2.80424 q^{39} -7.76756 q^{41} +4.13938 q^{43} -0.858069 q^{45} -1.39111 q^{47} -1.56060 q^{49} -4.73771 q^{51} -1.74469 q^{53} -0.569824 q^{55} -7.58466 q^{57} -10.2363 q^{59} -0.762183 q^{61} +2.33225 q^{63} +2.40623 q^{65} +0.00130165 q^{67} +6.58999 q^{69} -2.99630 q^{71} -16.6717 q^{73} -4.26372 q^{75} +1.54880 q^{77} -10.5928 q^{79} +1.00000 q^{81} +17.4041 q^{83} +4.06528 q^{85} -4.75833 q^{87} -4.94100 q^{89} -6.54021 q^{91} -2.96038 q^{93} +6.50816 q^{95} +17.4544 q^{97} +0.664078 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	$ 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) $ 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + 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
$5$	−0.858069	−0.383740	−0.191870	−	0.981420i	$-0.561455\pi$
$5$	−0.858069	−0.383740	−0.191870	+	0.981420i	$0.561455\pi$
$6$	0	0
$7$	2.33225	0.881509	0.440754	−	0.897628i	$-0.354711\pi$
$7$	2.33225	0.881509	0.440754	+	0.897628i	$0.354711\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.664078	0.200227	0.100113	−	0.994976i	$-0.468079\pi$
$11$	0.664078	0.200227	0.100113	+	0.994976i	$0.468079\pi$
$12$	0	0
$13$	−2.80424	−0.777757	−0.388879	−	0.921289i	$-0.627137\pi$
$13$	−2.80424	−0.777757	−0.388879	+	0.921289i	$0.627137\pi$
$14$	0	0
$15$	−0.858069	−0.221552
$16$	0	0
$17$	−4.73771	−1.14906	−0.574532	−	0.818482i	$-0.694816\pi$
$17$	−4.73771	−1.14906	−0.574532	+	0.818482i	$0.694816\pi$
$18$	0	0
$19$	−7.58466	−1.74004	−0.870021	−	0.493015i	$-0.835895\pi$
$19$	−7.58466	−1.74004	−0.870021	+	0.493015i	$0.835895\pi$
$20$	0	0
$21$	2.33225	0.508939
$22$	0	0
$23$	6.58999	1.37411	0.687054	−	0.726606i	$-0.258903\pi$
$23$	6.58999	1.37411	0.687054	+	0.726606i	$0.258903\pi$
$24$	0	0
$25$	−4.26372	−0.852744
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.75833	−0.883600	−0.441800	−	0.897114i	$-0.645660\pi$
$29$	−4.75833	−0.883600	−0.441800	+	0.897114i	$0.645660\pi$
$30$	0	0
$31$	−2.96038	−0.531700	−0.265850	−	0.964014i	$-0.585653\pi$
$31$	−2.96038	−0.531700	−0.265850	+	0.964014i	$0.585653\pi$
$32$	0	0
$33$	0.664078	0.115601
$34$	0	0
$35$	−2.00123	−0.338270
$36$	0	0
$37$	4.12480	0.678113	0.339056	−	0.940766i	$-0.389892\pi$
$37$	4.12480	0.678113	0.339056	+	0.940766i	$0.389892\pi$
$38$	0	0
$39$	−2.80424	−0.449038
$40$	0	0
$41$	−7.76756	−1.21309	−0.606544	−	0.795050i	$-0.707445\pi$
$41$	−7.76756	−1.21309	−0.606544	+	0.795050i	$0.707445\pi$
$42$	0	0
$43$	4.13938	0.631249	0.315624	−	0.948884i	$-0.397786\pi$
$43$	4.13938	0.631249	0.315624	+	0.948884i	$0.397786\pi$
$44$	0	0
$45$	−0.858069	−0.127913
$46$	0	0
$47$	−1.39111	−0.202914	−0.101457	−	0.994840i	$-0.532350\pi$
$47$	−1.39111	−0.202914	−0.101457	+	0.994840i	$0.532350\pi$
$48$	0	0
$49$	−1.56060	−0.222942
$50$	0	0
$51$	−4.73771	−0.663412
$52$	0	0
$53$	−1.74469	−0.239651	−0.119826	−	0.992795i	$-0.538234\pi$
$53$	−1.74469	−0.239651	−0.119826	+	0.992795i	$0.538234\pi$
$54$	0	0
$55$	−0.569824	−0.0768351
$56$	0	0
$57$	−7.58466	−1.00461
$58$	0	0
$59$	−10.2363	−1.33265	−0.666325	−	0.745661i	$-0.732134\pi$
$59$	−10.2363	−1.33265	−0.666325	+	0.745661i	$0.732134\pi$
$60$	0	0
$61$	−0.762183	−0.0975875	−0.0487937	−	0.998809i	$-0.515538\pi$
$61$	−0.762183	−0.0975875	−0.0487937	+	0.998809i	$0.515538\pi$
$62$	0	0
$63$	2.33225	0.293836
$64$	0	0
$65$	2.40623	0.298457
$66$	0	0
$67$	0.00130165	0.000159022 0	7.95109e−5	−	1.00000i	$-0.499975\pi$
$67$	0.00130165	0.000159022 0	7.95109e−5		1.00000i	$0.499975\pi$
$68$	0	0
$69$	6.58999	0.793342
$70$	0	0
$71$	−2.99630	−0.355596	−0.177798	−	0.984067i	$-0.556897\pi$
$71$	−2.99630	−0.355596	−0.177798	+	0.984067i	$0.556897\pi$
$72$	0	0
$73$	−16.6717	−1.95128	−0.975640	−	0.219379i	$-0.929597\pi$
$73$	−16.6717	−1.95128	−0.975640	+	0.219379i	$0.929597\pi$
$74$	0	0
$75$	−4.26372	−0.492332
$76$	0	0
$77$	1.54880	0.176502
$78$	0	0
$79$	−10.5928	−1.19178	−0.595891	−	0.803065i	$-0.703201\pi$
$79$	−10.5928	−1.19178	−0.595891	+	0.803065i	$0.703201\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.4041	1.91035	0.955177	−	0.296037i	$-0.0956650\pi$
$83$	17.4041	1.91035	0.955177	+	0.296037i	$0.0956650\pi$
$84$	0	0
$85$	4.06528	0.440941
$86$	0	0
$87$	−4.75833	−0.510147
$88$	0	0
$89$	−4.94100	−0.523745	−0.261872	−	0.965103i	$-0.584340\pi$
$89$	−4.94100	−0.523745	−0.261872	+	0.965103i	$0.584340\pi$
$90$	0	0
$91$	−6.54021	−0.685600
$92$	0	0
$93$	−2.96038	−0.306977
$94$	0	0
$95$	6.50816	0.667723
$96$	0	0
$97$	17.4544	1.77222	0.886112	−	0.463471i	$-0.153396\pi$
$97$	17.4544	1.77222	0.886112	+	0.463471i	$0.153396\pi$
$98$	0	0
$99$	0.664078	0.0667423
$100$	0	0
$101$	2.83624	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$101$	2.83624	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$102$	0	0
$103$	1.68361	0.165891	0.0829454	−	0.996554i	$-0.473567\pi$
$103$	1.68361	0.165891	0.0829454	+	0.996554i	$0.473567\pi$
$104$	0	0
$105$	−2.00123	−0.195300
$106$	0	0
$107$	10.4960	1.01468	0.507341	−	0.861745i	$-0.330628\pi$
$107$	10.4960	1.01468	0.507341	+	0.861745i	$0.330628\pi$
$108$	0	0
$109$	−3.61751	−0.346494	−0.173247	−	0.984878i	$-0.555426\pi$
$109$	−3.61751	−0.346494	−0.173247	+	0.984878i	$0.555426\pi$
$110$	0	0
$111$	4.12480	0.391509
$112$	0	0
$113$	2.15002	0.202257	0.101129	−	0.994873i	$-0.467755\pi$
$113$	2.15002	0.202257	0.101129	+	0.994873i	$0.467755\pi$
$114$	0	0
$115$	−5.65467	−0.527300
$116$	0	0
$117$	−2.80424	−0.259252
$118$	0	0
$119$	−11.0495	−1.01291
$120$	0	0
$121$	−10.5590	−0.959909
$122$	0	0
$123$	−7.76756	−0.700377
$124$	0	0
$125$	7.94891	0.710972
$126$	0	0
$127$	−12.3626	−1.09701	−0.548504	−	0.836148i	$-0.684802\pi$
$127$	−12.3626	−1.09701	−0.548504	+	0.836148i	$0.684802\pi$
$128$	0	0
$129$	4.13938	0.364452
$130$	0	0
$131$	20.5805	1.79812	0.899061	−	0.437822i	$-0.144250\pi$
$131$	20.5805	1.79812	0.899061	+	0.437822i	$0.144250\pi$
$132$	0	0
$133$	−17.6894	−1.53386
$134$	0	0
$135$	−0.858069	−0.0738508
$136$	0	0
$137$	10.1249	0.865031	0.432515	−	0.901627i	$-0.357626\pi$
$137$	10.1249	0.865031	0.432515	+	0.901627i	$0.357626\pi$
$138$	0	0
$139$	1.84695	0.156656	0.0783280	−	0.996928i	$-0.475042\pi$
$139$	1.84695	0.156656	0.0783280	+	0.996928i	$0.475042\pi$
$140$	0	0
$141$	−1.39111	−0.117153
$142$	0	0
$143$	−1.86224	−0.155728
$144$	0	0
$145$	4.08297	0.339073
$146$	0	0
$147$	−1.56060	−0.128716
$148$	0	0
$149$	5.36994	0.439922	0.219961	−	0.975509i	$-0.429407\pi$
$149$	5.36994	0.439922	0.219961	+	0.975509i	$0.429407\pi$
$150$	0	0
$151$	−13.0563	−1.06251	−0.531254	−	0.847213i	$-0.678279\pi$
$151$	−13.0563	−1.06251	−0.531254	+	0.847213i	$0.678279\pi$
$152$	0	0
$153$	−4.73771	−0.383021
$154$	0	0
$155$	2.54021	0.204034
$156$	0	0
$157$	5.47662	0.437082	0.218541	−	0.975828i	$-0.429870\pi$
$157$	5.47662	0.437082	0.218541	+	0.975828i	$0.429870\pi$
$158$	0	0
$159$	−1.74469	−0.138363
$160$	0	0
$161$	15.3695	1.21129
$162$	0	0
$163$	18.0321	1.41239	0.706193	−	0.708019i	$-0.250411\pi$
$163$	18.0321	1.41239	0.706193	+	0.708019i	$0.250411\pi$
$164$	0	0
$165$	−0.569824	−0.0443607
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.13621	−0.395093
$170$	0	0
$171$	−7.58466	−0.580014
$172$	0	0
$173$	11.9069	0.905266	0.452633	−	0.891697i	$-0.350485\pi$
$173$	11.9069	0.905266	0.452633	+	0.891697i	$0.350485\pi$
$174$	0	0
$175$	−9.94407	−0.751701
$176$	0	0
$177$	−10.2363	−0.769406
$178$	0	0
$179$	−9.99610	−0.747144	−0.373572	−	0.927601i	$-0.621867\pi$
$179$	−9.99610	−0.747144	−0.373572	+	0.927601i	$0.621867\pi$
$180$	0	0
$181$	−12.9764	−0.964526	−0.482263	−	0.876027i	$-0.660185\pi$
$181$	−12.9764	−0.964526	−0.482263	+	0.876027i	$0.660185\pi$
$182$	0	0
$183$	−0.762183	−0.0563422
$184$	0	0
$185$	−3.53936	−0.260219
$186$	0	0
$187$	−3.14621	−0.230073
$188$	0	0
$189$	2.33225	0.169646
$190$	0	0
$191$	−0.780931	−0.0565062	−0.0282531	−	0.999601i	$-0.508994\pi$
$191$	−0.780931	−0.0565062	−0.0282531	+	0.999601i	$0.508994\pi$
$192$	0	0
$193$	−3.18460	−0.229232	−0.114616	−	0.993410i	$-0.536564\pi$
$193$	−3.18460	−0.229232	−0.114616	+	0.993410i	$0.536564\pi$
$194$	0	0
$195$	2.40623	0.172314
$196$	0	0
$197$	−7.29302	−0.519606	−0.259803	−	0.965662i	$-0.583658\pi$
$197$	−7.29302	−0.519606	−0.259803	+	0.965662i	$0.583658\pi$
$198$	0	0
$199$	−20.4949	−1.45284	−0.726422	−	0.687249i	$-0.758818\pi$
$199$	−20.4949	−1.45284	−0.726422	+	0.687249i	$0.758818\pi$
$200$	0	0
$201$	0.00130165	9.18113e−5 0
$202$	0	0
$203$	−11.0976	−0.778901
$204$	0	0
$205$	6.66510	0.465511
$206$	0	0
$207$	6.58999	0.458036
$208$	0	0
$209$	−5.03681	−0.348403
$210$	0	0
$211$	2.79418	0.192359	0.0961795	−	0.995364i	$-0.469338\pi$
$211$	2.79418	0.192359	0.0961795	+	0.995364i	$0.469338\pi$
$212$	0	0
$213$	−2.99630	−0.205303
$214$	0	0
$215$	−3.55187	−0.242235
$216$	0	0
$217$	−6.90435	−0.468698
$218$	0	0
$219$	−16.6717	−1.12657
$220$	0	0
$221$	13.2857	0.893692
$222$	0	0
$223$	−1.07046	−0.0716830	−0.0358415	−	0.999357i	$-0.511411\pi$
$223$	−1.07046	−0.0716830	−0.0358415	+	0.999357i	$0.511411\pi$
$224$	0	0
$225$	−4.26372	−0.284248
$226$	0	0
$227$	−3.06628	−0.203516	−0.101758	−	0.994809i	$-0.532447\pi$
$227$	−3.06628	−0.203516	−0.101758	+	0.994809i	$0.532447\pi$
$228$	0	0
$229$	−0.166978	−0.0110342	−0.00551710	−	0.999985i	$-0.501756\pi$
$229$	−0.166978	−0.0110342	−0.00551710	+	0.999985i	$0.501756\pi$
$230$	0	0
$231$	1.54880	0.101903
$232$	0	0
$233$	9.75862	0.639309	0.319654	−	0.947534i	$-0.396433\pi$
$233$	9.75862	0.639309	0.319654	+	0.947534i	$0.396433\pi$
$234$	0	0
$235$	1.19367	0.0778663
$236$	0	0
$237$	−10.5928	−0.688076
$238$	0	0
$239$	−27.8243	−1.79980	−0.899902	−	0.436093i	$-0.856362\pi$
$239$	−27.8243	−1.79980	−0.899902	+	0.436093i	$0.856362\pi$
$240$	0	0
$241$	−6.03419	−0.388696	−0.194348	−	0.980933i	$-0.562259\pi$
$241$	−6.03419	−0.388696	−0.194348	+	0.980933i	$0.562259\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.33910	0.0855519
$246$	0	0
$247$	21.2693	1.35333
$248$	0	0
$249$	17.4041	1.10294
$250$	0	0
$251$	−19.2169	−1.21296	−0.606481	−	0.795098i	$-0.707419\pi$
$251$	−19.2169	−1.21296	−0.606481	+	0.795098i	$0.707419\pi$
$252$	0	0
$253$	4.37627	0.275133
$254$	0	0
$255$	4.06528	0.254578
$256$	0	0
$257$	−8.38942	−0.523318	−0.261659	−	0.965160i	$-0.584270\pi$
$257$	−8.38942	−0.523318	−0.261659	+	0.965160i	$0.584270\pi$
$258$	0	0
$259$	9.62007	0.597762
$260$	0	0
$261$	−4.75833	−0.294533
$262$	0	0
$263$	−16.9678	−1.04628	−0.523139	−	0.852247i	$-0.675239\pi$
$263$	−16.9678	−1.04628	−0.523139	+	0.852247i	$0.675239\pi$
$264$	0	0
$265$	1.49706	0.0919638
$266$	0	0
$267$	−4.94100	−0.302384
$268$	0	0
$269$	−13.6797	−0.834068	−0.417034	−	0.908891i	$-0.636930\pi$
$269$	−13.6797	−0.834068	−0.417034	+	0.908891i	$0.636930\pi$
$270$	0	0
$271$	−16.2857	−0.989284	−0.494642	−	0.869097i	$-0.664701\pi$
$271$	−16.2857	−0.989284	−0.494642	+	0.869097i	$0.664701\pi$
$272$	0	0
$273$	−6.54021	−0.395831
$274$	0	0
$275$	−2.83144	−0.170742
$276$	0	0
$277$	−4.11423	−0.247200	−0.123600	−	0.992332i	$-0.539444\pi$
$277$	−4.11423	−0.247200	−0.123600	+	0.992332i	$0.539444\pi$
$278$	0	0
$279$	−2.96038	−0.177233
$280$	0	0
$281$	29.3388	1.75020	0.875102	−	0.483939i	$-0.160794\pi$
$281$	29.3388	1.75020	0.875102	+	0.483939i	$0.160794\pi$
$282$	0	0
$283$	−16.5471	−0.983621	−0.491810	−	0.870702i	$-0.663665\pi$
$283$	−16.5471	−0.983621	−0.491810	+	0.870702i	$0.663665\pi$
$284$	0	0
$285$	6.50816	0.385510
$286$	0	0
$287$	−18.1159	−1.06935
$288$	0	0
$289$	5.44589	0.320346
$290$	0	0
$291$	17.4544	1.02319
$292$	0	0
$293$	−31.9181	−1.86468	−0.932338	−	0.361588i	$-0.882235\pi$
$293$	−31.9181	−1.86468	−0.932338	+	0.361588i	$0.882235\pi$
$294$	0	0
$295$	8.78344	0.511391
$296$	0	0
$297$	0.664078	0.0385337
$298$	0	0
$299$	−18.4799	−1.06872
$300$	0	0
$301$	9.65407	0.556451
$302$	0	0
$303$	2.83624	0.162938
$304$	0	0
$305$	0.654005	0.0374482
$306$	0	0
$307$	−12.7567	−0.728063	−0.364031	−	0.931387i	$-0.618600\pi$
$307$	−12.7567	−0.728063	−0.364031	+	0.931387i	$0.618600\pi$
$308$	0	0
$309$	1.68361	0.0957771
$310$	0	0
$311$	−4.06532	−0.230523	−0.115262	−	0.993335i	$-0.536771\pi$
$311$	−4.06532	−0.230523	−0.115262	+	0.993335i	$0.536771\pi$
$312$	0	0
$313$	−9.87994	−0.558447	−0.279224	−	0.960226i	$-0.590077\pi$
$313$	−9.87994	−0.558447	−0.279224	+	0.960226i	$0.590077\pi$
$314$	0	0
$315$	−2.00123	−0.112757
$316$	0	0
$317$	11.2409	0.631353	0.315676	−	0.948867i	$-0.397769\pi$
$317$	11.2409	0.631353	0.315676	+	0.948867i	$0.397769\pi$
$318$	0	0
$319$	−3.15990	−0.176920
$320$	0	0
$321$	10.4960	0.585827
$322$	0	0
$323$	35.9339	1.99942
$324$	0	0
$325$	11.9565	0.663228
$326$	0	0
$327$	−3.61751	−0.200049
$328$	0	0
$329$	−3.24442	−0.178871
$330$	0	0
$331$	−14.6160	−0.803370	−0.401685	−	0.915778i	$-0.631575\pi$
$331$	−14.6160	−0.803370	−0.401685	+	0.915778i	$0.631575\pi$
$332$	0	0
$333$	4.12480	0.226038
$334$	0	0
$335$	−0.00111690	−6.10230e−5 0
$336$	0	0
$337$	12.7052	0.692094	0.346047	−	0.938217i	$-0.387524\pi$
$337$	12.7052	0.692094	0.346047	+	0.938217i	$0.387524\pi$
$338$	0	0
$339$	2.15002	0.116773
$340$	0	0
$341$	−1.96592	−0.106461
$342$	0	0
$343$	−19.9655	−1.07803
$344$	0	0
$345$	−5.65467	−0.304437
$346$	0	0
$347$	34.7085	1.86325	0.931625	−	0.363422i	$-0.118392\pi$
$347$	34.7085	1.86325	0.931625	+	0.363422i	$0.118392\pi$
$348$	0	0
$349$	−0.432938	−0.0231746	−0.0115873	−	0.999933i	$-0.503688\pi$
$349$	−0.432938	−0.0231746	−0.0115873	+	0.999933i	$0.503688\pi$
$350$	0	0
$351$	−2.80424	−0.149679
$352$	0	0
$353$	15.5524	0.827769	0.413884	−	0.910329i	$-0.364172\pi$
$353$	15.5524	0.827769	0.413884	+	0.910329i	$0.364172\pi$
$354$	0	0
$355$	2.57103	0.136456
$356$	0	0
$357$	−11.0495	−0.584803
$358$	0	0
$359$	12.9062	0.681163	0.340581	−	0.940215i	$-0.389376\pi$
$359$	12.9062	0.681163	0.340581	+	0.940215i	$0.389376\pi$
$360$	0	0
$361$	38.5271	2.02774
$362$	0	0
$363$	−10.5590	−0.554204
$364$	0	0
$365$	14.3055	0.748784
$366$	0	0
$367$	−24.5614	−1.28210	−0.641048	−	0.767501i	$-0.721500\pi$
$367$	−24.5614	−1.28210	−0.641048	+	0.767501i	$0.721500\pi$
$368$	0	0
$369$	−7.76756	−0.404363
$370$	0	0
$371$	−4.06905	−0.211255
$372$	0	0
$373$	25.7222	1.33184	0.665922	−	0.746021i	$-0.268038\pi$
$373$	25.7222	1.33184	0.665922	+	0.746021i	$0.268038\pi$
$374$	0	0
$375$	7.94891	0.410480
$376$	0	0
$377$	13.3435	0.687226
$378$	0	0
$379$	34.4480	1.76947	0.884737	−	0.466090i	$-0.154338\pi$
$379$	34.4480	1.76947	0.884737	+	0.466090i	$0.154338\pi$
$380$	0	0
$381$	−12.3626	−0.633358
$382$	0	0
$383$	19.7939	1.01142	0.505711	−	0.862703i	$-0.331230\pi$
$383$	19.7939	1.01142	0.505711	+	0.862703i	$0.331230\pi$
$384$	0	0
$385$	−1.32897	−0.0677308
$386$	0	0
$387$	4.13938	0.210416
$388$	0	0
$389$	18.5357	0.939798	0.469899	−	0.882720i	$-0.344290\pi$
$389$	18.5357	0.939798	0.469899	+	0.882720i	$0.344290\pi$
$390$	0	0
$391$	−31.2215	−1.57894
$392$	0	0
$393$	20.5805	1.03815
$394$	0	0
$395$	9.08935	0.457335
$396$	0	0
$397$	−5.39197	−0.270615	−0.135308	−	0.990804i	$-0.543202\pi$
$397$	−5.39197	−0.270615	−0.135308	+	0.990804i	$0.543202\pi$
$398$	0	0
$399$	−17.6894	−0.885575
$400$	0	0
$401$	−13.8183	−0.690055	−0.345027	−	0.938593i	$-0.612130\pi$
$401$	−13.8183	−0.690055	−0.345027	+	0.938593i	$0.612130\pi$
$402$	0	0
$403$	8.30163	0.413534
$404$	0	0
$405$	−0.858069	−0.0426378
$406$	0	0
$407$	2.73919	0.135776
$408$	0	0
$409$	6.26519	0.309794	0.154897	−	0.987931i	$-0.450495\pi$
$409$	6.26519	0.309794	0.154897	+	0.987931i	$0.450495\pi$
$410$	0	0
$411$	10.1249	0.499426
$412$	0	0
$413$	−23.8736	−1.17474
$414$	0	0
$415$	−14.9340	−0.733079
$416$	0	0
$417$	1.84695	0.0904454
$418$	0	0
$419$	0.914237	0.0446634	0.0223317	−	0.999751i	$-0.492891\pi$
$419$	0.914237	0.0446634	0.0223317	+	0.999751i	$0.492891\pi$
$420$	0	0
$421$	31.5414	1.53723	0.768617	−	0.639709i	$-0.220945\pi$
$421$	31.5414	1.53723	0.768617	+	0.639709i	$0.220945\pi$
$422$	0	0
$423$	−1.39111	−0.0676381
$424$	0	0
$425$	20.2003	0.979856
$426$	0	0
$427$	−1.77760	−0.0860242
$428$	0	0
$429$	−1.86224	−0.0899096
$430$	0	0
$431$	38.5807	1.85837	0.929184	−	0.369619i	$-0.120512\pi$
$431$	38.5807	1.85837	0.929184	+	0.369619i	$0.120512\pi$
$432$	0	0
$433$	8.12866	0.390639	0.195319	−	0.980740i	$-0.437426\pi$
$433$	8.12866	0.390639	0.195319	+	0.980740i	$0.437426\pi$
$434$	0	0
$435$	4.08297	0.195764
$436$	0	0
$437$	−49.9829	−2.39101
$438$	0	0
$439$	−28.3001	−1.35069	−0.675344	−	0.737503i	$-0.736005\pi$
$439$	−28.3001	−1.35069	−0.675344	+	0.737503i	$0.736005\pi$
$440$	0	0
$441$	−1.56060	−0.0743141
$442$	0	0
$443$	−38.5945	−1.83368	−0.916841	−	0.399253i	$-0.869269\pi$
$443$	−38.5945	−1.83368	−0.916841	+	0.399253i	$0.869269\pi$
$444$	0	0
$445$	4.23971	0.200982
$446$	0	0
$447$	5.36994	0.253989
$448$	0	0
$449$	9.70687	0.458096	0.229048	−	0.973415i	$-0.426439\pi$
$449$	9.70687	0.458096	0.229048	+	0.973415i	$0.426439\pi$
$450$	0	0
$451$	−5.15826	−0.242893
$452$	0	0
$453$	−13.0563	−0.613439
$454$	0	0
$455$	5.61195	0.263092
$456$	0	0
$457$	−25.1697	−1.17739	−0.588694	−	0.808356i	$-0.700358\pi$
$457$	−25.1697	−1.17739	−0.588694	+	0.808356i	$0.700358\pi$
$458$	0	0
$459$	−4.73771	−0.221137
$460$	0	0
$461$	33.4704	1.55887	0.779435	−	0.626483i	$-0.215506\pi$
$461$	33.4704	1.55887	0.779435	+	0.626483i	$0.215506\pi$
$462$	0	0
$463$	23.8185	1.10694	0.553469	−	0.832870i	$-0.313304\pi$
$463$	23.8185	1.10694	0.553469	+	0.832870i	$0.313304\pi$
$464$	0	0
$465$	2.54021	0.117799
$466$	0	0
$467$	5.42414	0.250999	0.125500	−	0.992094i	$-0.459947\pi$
$467$	5.42414	0.250999	0.125500	+	0.992094i	$0.459947\pi$
$468$	0	0
$469$	0.00303578	0.000140179 0
$470$	0	0
$471$	5.47662	0.252349
$472$	0	0
$473$	2.74887	0.126393
$474$	0	0
$475$	32.3389	1.48381
$476$	0	0
$477$	−1.74469	−0.0798838
$478$	0	0
$479$	12.8509	0.587172	0.293586	−	0.955933i	$-0.405151\pi$
$479$	12.8509	0.587172	0.293586	+	0.955933i	$0.405151\pi$
$480$	0	0
$481$	−11.5669	−0.527407
$482$	0	0
$483$	15.3695	0.699338
$484$	0	0
$485$	−14.9771	−0.680073
$486$	0	0
$487$	−32.9193	−1.49171	−0.745857	−	0.666106i	$-0.767960\pi$
$487$	−32.9193	−1.49171	−0.745857	+	0.666106i	$0.767960\pi$
$488$	0	0
$489$	18.0321	0.815442
$490$	0	0
$491$	−23.3929	−1.05571	−0.527854	−	0.849335i	$-0.677003\pi$
$491$	−23.3929	−1.05571	−0.527854	+	0.849335i	$0.677003\pi$
$492$	0	0
$493$	22.5436	1.01531
$494$	0	0
$495$	−0.569824	−0.0256117
$496$	0	0
$497$	−6.98813	−0.313461
$498$	0	0
$499$	7.77468	0.348042	0.174021	−	0.984742i	$-0.444324\pi$
$499$	7.77468	0.348042	0.174021	+	0.984742i	$0.444324\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−11.5995	−0.517195	−0.258597	−	0.965985i	$-0.583260\pi$
$503$	−11.5995	−0.517195	−0.258597	+	0.965985i	$0.583260\pi$
$504$	0	0
$505$	−2.43369	−0.108298
$506$	0	0
$507$	−5.13621	−0.228107
$508$	0	0
$509$	−14.3183	−0.634646	−0.317323	−	0.948317i	$-0.602784\pi$
$509$	−14.3183	−0.634646	−0.317323	+	0.948317i	$0.602784\pi$
$510$	0	0
$511$	−38.8827	−1.72007
$512$	0	0
$513$	−7.58466	−0.334871
$514$	0	0
$515$	−1.44465	−0.0636590
$516$	0	0
$517$	−0.923805	−0.0406289
$518$	0	0
$519$	11.9069	0.522656
$520$	0	0
$521$	24.7954	1.08631	0.543153	−	0.839633i	$-0.317230\pi$
$521$	24.7954	1.08631	0.543153	+	0.839633i	$0.317230\pi$
$522$	0	0
$523$	15.8863	0.694660	0.347330	−	0.937743i	$-0.387088\pi$
$523$	15.8863	0.694660	0.347330	+	0.937743i	$0.387088\pi$
$524$	0	0
$525$	−9.94407	−0.433995
$526$	0	0
$527$	14.0254	0.610957
$528$	0	0
$529$	20.4280	0.888174
$530$	0	0
$531$	−10.2363	−0.444217
$532$	0	0
$533$	21.7821	0.943489
$534$	0	0
$535$	−9.00625	−0.389374
$536$	0	0
$537$	−9.99610	−0.431364
$538$	0	0
$539$	−1.03636	−0.0446391
$540$	0	0
$541$	−34.8029	−1.49629	−0.748147	−	0.663532i	$-0.769056\pi$
$541$	−34.8029	−1.49629	−0.748147	+	0.663532i	$0.769056\pi$
$542$	0	0
$543$	−12.9764	−0.556869
$544$	0	0
$545$	3.10407	0.132964
$546$	0	0
$547$	−16.7384	−0.715682	−0.357841	−	0.933782i	$-0.616487\pi$
$547$	−16.7384	−0.715682	−0.357841	+	0.933782i	$0.616487\pi$
$548$	0	0
$549$	−0.762183	−0.0325292
$550$	0	0
$551$	36.0903	1.53750
$552$	0	0
$553$	−24.7051	−1.05057
$554$	0	0
$555$	−3.53936	−0.150237
$556$	0	0
$557$	−4.14278	−0.175535	−0.0877677	−	0.996141i	$-0.527973\pi$
$557$	−4.14278	−0.175535	−0.0877677	+	0.996141i	$0.527973\pi$
$558$	0	0
$559$	−11.6078	−0.490959
$560$	0	0
$561$	−3.14621	−0.132833
$562$	0	0
$563$	31.1358	1.31222	0.656109	−	0.754666i	$-0.272201\pi$
$563$	31.1358	1.31222	0.656109	+	0.754666i	$0.272201\pi$
$564$	0	0
$565$	−1.84487	−0.0776142
$566$	0	0
$567$	2.33225	0.0979454
$568$	0	0
$569$	−16.2278	−0.680305	−0.340153	−	0.940370i	$-0.610479\pi$
$569$	−16.2278	−0.680305	−0.340153	+	0.940370i	$0.610479\pi$
$570$	0	0
$571$	−15.7948	−0.660991	−0.330495	−	0.943808i	$-0.607216\pi$
$571$	−15.7948	−0.660991	−0.330495	+	0.943808i	$0.607216\pi$
$572$	0	0
$573$	−0.780931	−0.0326239
$574$	0	0
$575$	−28.0979	−1.17176
$576$	0	0
$577$	29.3450	1.22165	0.610825	−	0.791766i	$-0.290838\pi$
$577$	29.3450	1.22165	0.610825	+	0.791766i	$0.290838\pi$
$578$	0	0
$579$	−3.18460	−0.132347
$580$	0	0
$581$	40.5909	1.68399
$582$	0	0
$583$	−1.15861	−0.0479847
$584$	0	0
$585$	2.40623	0.0994855
$586$	0	0
$587$	3.97172	0.163931	0.0819653	−	0.996635i	$-0.473880\pi$
$587$	3.97172	0.163931	0.0819653	+	0.996635i	$0.473880\pi$
$588$	0	0
$589$	22.4535	0.925180
$590$	0	0
$591$	−7.29302	−0.299995
$592$	0	0
$593$	13.0664	0.536573	0.268286	−	0.963339i	$-0.413543\pi$
$593$	13.0664	0.536573	0.268286	+	0.963339i	$0.413543\pi$
$594$	0	0
$595$	9.48126	0.388694
$596$	0	0
$597$	−20.4949	−0.838800
$598$	0	0
$599$	26.1897	1.07008	0.535042	−	0.844826i	$-0.320296\pi$
$599$	26.1897	1.07008	0.535042	+	0.844826i	$0.320296\pi$
$600$	0	0
$601$	31.0894	1.26816	0.634082	−	0.773266i	$-0.281378\pi$
$601$	31.0894	1.26816	0.634082	+	0.773266i	$0.281378\pi$
$602$	0	0
$603$	0.00130165	5.30073e−5 0
$604$	0	0
$605$	9.06035	0.368356
$606$	0	0
$607$	7.46967	0.303185	0.151592	−	0.988443i	$-0.451560\pi$
$607$	7.46967	0.303185	0.151592	+	0.988443i	$0.451560\pi$
$608$	0	0
$609$	−11.0976	−0.449699
$610$	0	0
$611$	3.90101	0.157818
$612$	0	0
$613$	−7.66230	−0.309477	−0.154739	−	0.987955i	$-0.549454\pi$
$613$	−7.66230	−0.309477	−0.154739	+	0.987955i	$0.549454\pi$
$614$	0	0
$615$	6.66510	0.268763
$616$	0	0
$617$	11.5871	0.466479	0.233239	−	0.972419i	$-0.425067\pi$
$617$	11.5871	0.466479	0.233239	+	0.972419i	$0.425067\pi$
$618$	0	0
$619$	4.39728	0.176741	0.0883707	−	0.996088i	$-0.471834\pi$
$619$	4.39728	0.176741	0.0883707	+	0.996088i	$0.471834\pi$
$620$	0	0
$621$	6.58999	0.264447
$622$	0	0
$623$	−11.5237	−0.461685
$624$	0	0
$625$	14.4979	0.579915
$626$	0	0
$627$	−5.03681	−0.201151
$628$	0	0
$629$	−19.5421	−0.779194
$630$	0	0
$631$	−25.3849	−1.01056	−0.505279	−	0.862956i	$-0.668610\pi$
$631$	−25.3849	−1.01056	−0.505279	+	0.862956i	$0.668610\pi$
$632$	0	0
$633$	2.79418	0.111059
$634$	0	0
$635$	10.6080	0.420966
$636$	0	0
$637$	4.37629	0.173395
$638$	0	0
$639$	−2.99630	−0.118532
$640$	0	0
$641$	23.9833	0.947283	0.473642	−	0.880718i	$-0.342939\pi$
$641$	23.9833	0.947283	0.473642	+	0.880718i	$0.342939\pi$
$642$	0	0
$643$	−27.3549	−1.07877	−0.539386	−	0.842059i	$-0.681344\pi$
$643$	−27.3549	−1.07877	−0.539386	+	0.842059i	$0.681344\pi$
$644$	0	0
$645$	−3.55187	−0.139855
$646$	0	0
$647$	−8.92955	−0.351057	−0.175528	−	0.984474i	$-0.556163\pi$
$647$	−8.92955	−0.351057	−0.175528	+	0.984474i	$0.556163\pi$
$648$	0	0
$649$	−6.79769	−0.266833
$650$	0	0
$651$	−6.90435	−0.270603
$652$	0	0
$653$	1.42815	0.0558877	0.0279438	−	0.999609i	$-0.491104\pi$
$653$	1.42815	0.0558877	0.0279438	+	0.999609i	$0.491104\pi$
$654$	0	0
$655$	−17.6594	−0.690012
$656$	0	0
$657$	−16.6717	−0.650426
$658$	0	0
$659$	−20.7009	−0.806394	−0.403197	−	0.915113i	$-0.632101\pi$
$659$	−20.7009	−0.806394	−0.403197	+	0.915113i	$0.632101\pi$
$660$	0	0
$661$	13.2990	0.517272	0.258636	−	0.965975i	$-0.416727\pi$
$661$	13.2990	0.517272	0.258636	+	0.965975i	$0.416727\pi$
$662$	0	0
$663$	13.2857	0.515974
$664$	0	0
$665$	15.1787	0.588604
$666$	0	0
$667$	−31.3574	−1.21416
$668$	0	0
$669$	−1.07046	−0.0413862
$670$	0	0
$671$	−0.506148	−0.0195396
$672$	0	0
$673$	12.6827	0.488882	0.244441	−	0.969664i	$-0.421396\pi$
$673$	12.6827	0.488882	0.244441	+	0.969664i	$0.421396\pi$
$674$	0	0
$675$	−4.26372	−0.164111
$676$	0	0
$677$	2.60520	0.100126	0.0500629	−	0.998746i	$-0.484058\pi$
$677$	2.60520	0.100126	0.0500629	+	0.998746i	$0.484058\pi$
$678$	0	0
$679$	40.7080	1.56223
$680$	0	0
$681$	−3.06628	−0.117500
$682$	0	0
$683$	24.5138	0.937995	0.468998	−	0.883199i	$-0.344615\pi$
$683$	24.5138	0.937995	0.468998	+	0.883199i	$0.344615\pi$
$684$	0	0
$685$	−8.68788	−0.331947
$686$	0	0
$687$	−0.166978	−0.00637060
$688$	0	0
$689$	4.89253	0.186391
$690$	0	0
$691$	32.0143	1.21788	0.608941	−	0.793215i	$-0.291595\pi$
$691$	32.0143	1.21788	0.608941	+	0.793215i	$0.291595\pi$
$692$	0	0
$693$	1.54880	0.0588339
$694$	0	0
$695$	−1.58481	−0.0601152
$696$	0	0
$697$	36.8004	1.39392
$698$	0	0
$699$	9.75862	0.369105
$700$	0	0
$701$	−4.88152	−0.184372	−0.0921862	−	0.995742i	$-0.529386\pi$
$701$	−4.88152	−0.184372	−0.0921862	+	0.995742i	$0.529386\pi$
$702$	0	0
$703$	−31.2852	−1.17994
$704$	0	0
$705$	1.19367	0.0449561
$706$	0	0
$707$	6.61482	0.248776
$708$	0	0
$709$	42.2523	1.58682	0.793408	−	0.608690i	$-0.208305\pi$
$709$	42.2523	1.58682	0.793408	+	0.608690i	$0.208305\pi$
$710$	0	0
$711$	−10.5928	−0.397261
$712$	0	0
$713$	−19.5089	−0.730613
$714$	0	0
$715$	1.59793	0.0597590
$716$	0	0
$717$	−27.8243	−1.03912
$718$	0	0
$719$	−42.4125	−1.58172	−0.790860	−	0.611997i	$-0.790366\pi$
$719$	−42.4125	−1.58172	−0.790860	+	0.611997i	$0.790366\pi$
$720$	0	0
$721$	3.92660	0.146234
$722$	0	0
$723$	−6.03419	−0.224414
$724$	0	0
$725$	20.2882	0.753484
$726$	0	0
$727$	22.1250	0.820570	0.410285	−	0.911957i	$-0.365429\pi$
$727$	22.1250	0.820570	0.410285	+	0.911957i	$0.365429\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−19.6112	−0.725345
$732$	0	0
$733$	35.9161	1.32659	0.663297	−	0.748357i	$-0.269157\pi$
$733$	35.9161	1.32659	0.663297	+	0.748357i	$0.269157\pi$
$734$	0	0
$735$	1.33910	0.0493934
$736$	0	0
$737$	0.000864396 0	3.18404e−5 0
$738$	0	0
$739$	−20.3434	−0.748344	−0.374172	−	0.927359i	$-0.622073\pi$
$739$	−20.3434	−0.748344	−0.374172	+	0.927359i	$0.622073\pi$
$740$	0	0
$741$	21.2693	0.781345
$742$	0	0
$743$	47.9841	1.76036	0.880182	−	0.474636i	$-0.157420\pi$
$743$	47.9841	1.76036	0.880182	+	0.474636i	$0.157420\pi$
$744$	0	0
$745$	−4.60778	−0.168816
$746$	0	0
$747$	17.4041	0.636784
$748$	0	0
$749$	24.4792	0.894451
$750$	0	0
$751$	40.8830	1.49184	0.745922	−	0.666034i	$-0.232009\pi$
$751$	40.8830	1.49184	0.745922	+	0.666034i	$0.232009\pi$
$752$	0	0
$753$	−19.2169	−0.700304
$754$	0	0
$755$	11.2032	0.407727
$756$	0	0
$757$	52.4727	1.90715	0.953576	−	0.301151i	$-0.0973708\pi$
$757$	52.4727	1.90715	0.953576	+	0.301151i	$0.0973708\pi$
$758$	0	0
$759$	4.37627	0.158848
$760$	0	0
$761$	−30.9663	−1.12253	−0.561264	−	0.827636i	$-0.689685\pi$
$761$	−30.9663	−1.12253	−0.561264	+	0.827636i	$0.689685\pi$
$762$	0	0
$763$	−8.43694	−0.305438
$764$	0	0
$765$	4.06528	0.146980
$766$	0	0
$767$	28.7050	1.03648
$768$	0	0
$769$	−24.2091	−0.873003	−0.436501	−	0.899704i	$-0.643783\pi$
$769$	−24.2091	−0.873003	−0.436501	+	0.899704i	$0.643783\pi$
$770$	0	0
$771$	−8.38942	−0.302138
$772$	0	0
$773$	−13.4368	−0.483289	−0.241644	−	0.970365i	$-0.577687\pi$
$773$	−13.4368	−0.483289	−0.241644	+	0.970365i	$0.577687\pi$
$774$	0	0
$775$	12.6222	0.453404
$776$	0	0
$777$	9.62007	0.345118
$778$	0	0
$779$	58.9143	2.11082
$780$	0	0
$781$	−1.98978	−0.0711998
$782$	0	0
$783$	−4.75833	−0.170049
$784$	0	0
$785$	−4.69932	−0.167726
$786$	0	0
$787$	−38.7500	−1.38129	−0.690644	−	0.723195i	$-0.742673\pi$
$787$	−38.7500	−1.38129	−0.690644	+	0.723195i	$0.742673\pi$
$788$	0	0
$789$	−16.9678	−0.604069
$790$	0	0
$791$	5.01440	0.178292
$792$	0	0
$793$	2.13735	0.0758994
$794$	0	0
$795$	1.49706	0.0530953
$796$	0	0
$797$	−4.12743	−0.146201	−0.0731006	−	0.997325i	$-0.523289\pi$
$797$	−4.12743	−0.146201	−0.0731006	+	0.997325i	$0.523289\pi$
$798$	0	0
$799$	6.59067	0.233161
$800$	0	0
$801$	−4.94100	−0.174582
$802$	0	0
$803$	−11.0713	−0.390699
$804$	0	0
$805$	−13.1881	−0.464820
$806$	0	0
$807$	−13.6797	−0.481549
$808$	0	0
$809$	−43.5841	−1.53233	−0.766167	−	0.642641i	$-0.777839\pi$
$809$	−43.5841	−1.53233	−0.766167	+	0.642641i	$0.777839\pi$
$810$	0	0
$811$	−6.11071	−0.214576	−0.107288	−	0.994228i	$-0.534217\pi$
$811$	−6.11071	−0.214576	−0.107288	+	0.994228i	$0.534217\pi$
$812$	0	0
$813$	−16.2857	−0.571163
$814$	0	0
$815$	−15.4728	−0.541989
$816$	0	0
$817$	−31.3958	−1.09840
$818$	0	0
$819$	−6.54021	−0.228533
$820$	0	0
$821$	−44.6483	−1.55824	−0.779118	−	0.626878i	$-0.784333\pi$
$821$	−44.6483	−1.55824	−0.779118	+	0.626878i	$0.784333\pi$
$822$	0	0
$823$	−52.0320	−1.81372	−0.906861	−	0.421429i	$-0.861529\pi$
$823$	−52.0320	−1.81372	−0.906861	+	0.421429i	$0.861529\pi$
$824$	0	0
$825$	−2.83144	−0.0985781
$826$	0	0
$827$	0.434728	0.0151170	0.00755849	−	0.999971i	$-0.497594\pi$
$827$	0.434728	0.0151170	0.00755849	+	0.999971i	$0.497594\pi$
$828$	0	0
$829$	45.0928	1.56614	0.783069	−	0.621935i	$-0.213653\pi$
$829$	45.0928	1.56614	0.783069	+	0.621935i	$0.213653\pi$
$830$	0	0
$831$	−4.11423	−0.142721
$832$	0	0
$833$	7.39365	0.256175
$834$	0	0
$835$	0.858069	0.0296947
$836$	0	0
$837$	−2.96038	−0.102326
$838$	0	0
$839$	−28.8473	−0.995919	−0.497960	−	0.867200i	$-0.665917\pi$
$839$	−28.8473	−0.995919	−0.497960	+	0.867200i	$0.665917\pi$
$840$	0	0
$841$	−6.35829	−0.219251
$842$	0	0
$843$	29.3388	1.01048
$844$	0	0
$845$	4.40722	0.151613
$846$	0	0
$847$	−24.6263	−0.846168
$848$	0	0
$849$	−16.5471	−0.567894
$850$	0	0
$851$	27.1824	0.931800
$852$	0	0
$853$	−34.0205	−1.16484	−0.582420	−	0.812888i	$-0.697894\pi$
$853$	−34.0205	−1.16484	−0.582420	+	0.812888i	$0.697894\pi$
$854$	0	0
$855$	6.50816	0.222574
$856$	0	0
$857$	−4.41103	−0.150678	−0.0753389	−	0.997158i	$-0.524004\pi$
$857$	−4.41103	−0.150678	−0.0753389	+	0.997158i	$0.524004\pi$
$858$	0	0
$859$	0.717858	0.0244930	0.0122465	−	0.999925i	$-0.496102\pi$
$859$	0.717858	0.0244930	0.0122465	+	0.999925i	$0.496102\pi$
$860$	0	0
$861$	−18.1159	−0.617388
$862$	0	0
$863$	37.7870	1.28628	0.643142	−	0.765747i	$-0.277630\pi$
$863$	37.7870	1.28628	0.643142	+	0.765747i	$0.277630\pi$
$864$	0	0
$865$	−10.2170	−0.347387
$866$	0	0
$867$	5.44589	0.184952
$868$	0	0
$869$	−7.03444	−0.238627
$870$	0	0
$871$	−0.00365014	−0.000123680 0
$872$	0	0
$873$	17.4544	0.590741
$874$	0	0
$875$	18.5389	0.626728
$876$	0	0
$877$	−33.0599	−1.11635	−0.558177	−	0.829722i	$-0.688499\pi$
$877$	−33.0599	−1.11635	−0.558177	+	0.829722i	$0.688499\pi$
$878$	0	0
$879$	−31.9181	−1.07657
$880$	0	0
$881$	−33.8792	−1.14142	−0.570710	−	0.821152i	$-0.693332\pi$
$881$	−33.8792	−1.14142	−0.570710	+	0.821152i	$0.693332\pi$
$882$	0	0
$883$	−36.9683	−1.24408	−0.622041	−	0.782985i	$-0.713696\pi$
$883$	−36.9683	−1.24408	−0.622041	+	0.782985i	$0.713696\pi$
$884$	0	0
$885$	8.78344	0.295252
$886$	0	0
$887$	27.6020	0.926786	0.463393	−	0.886153i	$-0.346632\pi$
$887$	27.6020	0.926786	0.463393	+	0.886153i	$0.346632\pi$
$888$	0	0
$889$	−28.8328	−0.967022
$890$	0	0
$891$	0.664078	0.0222474
$892$	0	0
$893$	10.5511	0.353079
$894$	0	0
$895$	8.57734	0.286709
$896$	0	0
$897$	−18.4799	−0.617027
$898$	0	0
$899$	14.0865	0.469810
$900$	0	0
$901$	8.26583	0.275375
$902$	0	0
$903$	9.65407	0.321267
$904$	0	0
$905$	11.1346	0.370127
$906$	0	0
$907$	35.2051	1.16897	0.584484	−	0.811406i	$-0.301297\pi$
$907$	35.2051	1.16897	0.584484	+	0.811406i	$0.301297\pi$
$908$	0	0
$909$	2.83624	0.0940720
$910$	0	0
$911$	3.23223	0.107088	0.0535442	−	0.998565i	$-0.482948\pi$
$911$	3.23223	0.107088	0.0535442	+	0.998565i	$0.482948\pi$
$912$	0	0
$913$	11.5577	0.382504
$914$	0	0
$915$	0.654005	0.0216207
$916$	0	0
$917$	47.9988	1.58506
$918$	0	0
$919$	11.7314	0.386984	0.193492	−	0.981102i	$-0.438019\pi$
$919$	11.7314	0.386984	0.193492	+	0.981102i	$0.438019\pi$
$920$	0	0
$921$	−12.7567	−0.420347
$922$	0	0
$923$	8.40236	0.276567
$924$	0	0
$925$	−17.5870	−0.578256
$926$	0	0
$927$	1.68361	0.0552970
$928$	0	0
$929$	36.8679	1.20960	0.604798	−	0.796379i	$-0.293254\pi$
$929$	36.8679	1.20960	0.604798	+	0.796379i	$0.293254\pi$
$930$	0	0
$931$	11.8366	0.387929
$932$	0	0
$933$	−4.06532	−0.133093
$934$	0	0
$935$	2.69966	0.0882884
$936$	0	0
$937$	−10.7009	−0.349583	−0.174792	−	0.984605i	$-0.555925\pi$
$937$	−10.7009	−0.349583	−0.174792	+	0.984605i	$0.555925\pi$
$938$	0	0
$939$	−9.87994	−0.322420
$940$	0	0
$941$	−4.86481	−0.158588	−0.0792942	−	0.996851i	$-0.525267\pi$
$941$	−4.86481	−0.158588	−0.0792942	+	0.996851i	$0.525267\pi$
$942$	0	0
$943$	−51.1881	−1.66692
$944$	0	0
$945$	−2.00123	−0.0651001
$946$	0	0
$947$	−4.67872	−0.152038	−0.0760190	−	0.997106i	$-0.524221\pi$
$947$	−4.67872	−0.152038	−0.0760190	+	0.997106i	$0.524221\pi$
$948$	0	0
$949$	46.7516	1.51762
$950$	0	0
$951$	11.2409	0.364512
$952$	0	0
$953$	−38.4497	−1.24551	−0.622754	−	0.782418i	$-0.713986\pi$
$953$	−38.4497	−1.24551	−0.622754	+	0.782418i	$0.713986\pi$
$954$	0	0
$955$	0.670092	0.0216837
$956$	0	0
$957$	−3.15990	−0.102145
$958$	0	0
$959$	23.6139	0.762532
$960$	0	0
$961$	−22.2362	−0.717295
$962$	0	0
$963$	10.4960	0.338227
$964$	0	0
$965$	2.73260	0.0879656
$966$	0	0
$967$	30.5693	0.983044	0.491522	−	0.870865i	$-0.336441\pi$
$967$	30.5693	0.983044	0.491522	+	0.870865i	$0.336441\pi$
$968$	0	0
$969$	35.9339	1.15436
$970$	0	0
$971$	0.810886	0.0260226	0.0130113	−	0.999915i	$-0.495858\pi$
$971$	0.810886	0.0260226	0.0130113	+	0.999915i	$0.495858\pi$
$972$	0	0
$973$	4.30755	0.138094
$974$	0	0
$975$	11.9565	0.382915
$976$	0	0
$977$	9.23855	0.295567	0.147784	−	0.989020i	$-0.452786\pi$
$977$	9.23855	0.295567	0.147784	+	0.989020i	$0.452786\pi$
$978$	0	0
$979$	−3.28120	−0.104868
$980$	0	0
$981$	−3.61751	−0.115498
$982$	0	0
$983$	−1.30384	−0.0415861	−0.0207931	−	0.999784i	$-0.506619\pi$
$983$	−1.30384	−0.0415861	−0.0207931	+	0.999784i	$0.506619\pi$
$984$	0	0
$985$	6.25791	0.199394
$986$	0	0
$987$	−3.24442	−0.103271
$988$	0	0
$989$	27.2785	0.867405
$990$	0	0
$991$	53.9125	1.71259	0.856293	−	0.516490i	$-0.172762\pi$
$991$	53.9125	1.71259	0.856293	+	0.516490i	$0.172762\pi$
$992$	0	0
$993$	−14.6160	−0.463826
$994$	0	0
$995$	17.5860	0.557515
$996$	0	0
$997$	−47.6636	−1.50952	−0.754760	−	0.656001i	$-0.772247\pi$
$997$	−47.6636	−1.50952	−0.754760	+	0.656001i	$0.772247\pi$
$998$	0	0
$999$	4.12480	0.130503

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.i.1.5	✓	9
4.3	odd	2		8016.2.a.ba.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.i.1.5	✓	9	1.1	even	1	trivial
8016.2.a.ba.1.5		9	4.3	odd	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.858069 q^{5} +2.33225 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.858069 q^{5} +2.33225 q^{7} +1.00000 q^{9} +0.664078 q^{11} -2.80424 q^{13} -0.858069 q^{15} -4.73771 q^{17} -7.58466 q^{19} +2.33225 q^{21} +6.58999 q^{23} -4.26372 q^{25} +1.00000 q^{27} -4.75833 q^{29} -2.96038 q^{31} +0.664078 q^{33} -2.00123 q^{35} +4.12480 q^{37} -2.80424 q^{39} -7.76756 q^{41} +4.13938 q^{43} -0.858069 q^{45} -1.39111 q^{47} -1.56060 q^{49} -4.73771 q^{51} -1.74469 q^{53} -0.569824 q^{55} -7.58466 q^{57} -10.2363 q^{59} -0.762183 q^{61} +2.33225 q^{63} +2.40623 q^{65} +0.00130165 q^{67} +6.58999 q^{69} -2.99630 q^{71} -16.6717 q^{73} -4.26372 q^{75} +1.54880 q^{77} -10.5928 q^{79} +1.00000 q^{81} +17.4041 q^{83} +4.06528 q^{85} -4.75833 q^{87} -4.94100 q^{89} -6.54021 q^{91} -2.96038 q^{93} +6.50816 q^{95} +17.4544 q^{97} +0.664078 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9	\( 9 q + 9 q^{3} - 6 q^{5} - 11 q^{7} + 9 q^{9} + q^{11} - 4 q^{13} - 6 q^{15} - 9 q^{17} - 8 q^{19} - 11 q^{21} - 7 q^{23} - q^{25} + 9 q^{27} - 9 q^{29} - 25 q^{31} + q^{33} + 5 q^{35} - 6 q^{37} - 4 q^{39} - 4 q^{41} - 24 q^{43} - 6 q^{45} - 16 q^{47} + 4 q^{49} - 9 q^{51} - 26 q^{53} - 29 q^{55} - 8 q^{57} - 4 q^{59} - 20 q^{61} - 11 q^{63} - 8 q^{65} - 25 q^{67} - 7 q^{69} - 15 q^{71} - 10 q^{73} - q^{75} - 20 q^{77} - 34 q^{79} + 9 q^{81} - 4 q^{83} - 13 q^{85} - 9 q^{87} + 13 q^{89} - 21 q^{91} - 25 q^{93} - 7 q^{95} - 4 q^{97} + q^{99}+O(q^{100}) \) 9 * q + 9 * q^3 - 6 * q^5 - 11 * q^7 + 9 * q^9 + q^11 - 4 * q^13 - 6 * q^15 - 9 * q^17 - 8 * q^19 - 11 * q^21 - 7 * q^23 - q^25 + 9 * q^27 - 9 * q^29 - 25 * q^31 + q^33 + 5 * q^35 - 6 * q^37 - 4 * q^39 - 4 * q^41 - 24 * q^43 - 6 * q^45 - 16 * q^47 + 4 * q^49 - 9 * q^51 - 26 * q^53 - 29 * q^55 - 8 * q^57 - 4 * q^59 - 20 * q^61 - 11 * q^63 - 8 * q^65 - 25 * q^67 - 7 * q^69 - 15 * q^71 - 10 * q^73 - q^75 - 20 * q^77 - 34 * q^79 + 9 * q^81 - 4 * q^83 - 13 * q^85 - 9 * q^87 + 13 * q^89 - 21 * q^91 - 25 * q^93 - 7 * q^95 - 4 * q^97 + q^99

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