LMFDB - Embedded newform 4008.2.a.k.1.7

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 $ x^11 - x^10 - 33x^9 + 22x^8 + 417x^7 - 151x^6 - 2470x^5 + 272x^4 + 6584x^3 + 292x^2 - 5687*x + 242
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.60189$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.60189 q^{5} -3.58131 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.60189 q^{5} -3.58131 q^{7} +1.00000 q^{9} -4.72536 q^{11} -0.942538 q^{13} -2.60189 q^{15} -3.64824 q^{17} -2.69214 q^{19} +3.58131 q^{21} -0.381441 q^{23} +1.76983 q^{25} -1.00000 q^{27} +4.31070 q^{29} +0.400729 q^{31} +4.72536 q^{33} -9.31818 q^{35} +0.407293 q^{37} +0.942538 q^{39} +0.759495 q^{41} +4.47426 q^{43} +2.60189 q^{45} +13.2113 q^{47} +5.82578 q^{49} +3.64824 q^{51} +4.41253 q^{53} -12.2949 q^{55} +2.69214 q^{57} -5.47954 q^{59} +10.9868 q^{61} -3.58131 q^{63} -2.45238 q^{65} +4.38579 q^{67} +0.381441 q^{69} -1.17148 q^{71} +12.3989 q^{73} -1.76983 q^{75} +16.9230 q^{77} +8.57361 q^{79} +1.00000 q^{81} +0.551394 q^{83} -9.49231 q^{85} -4.31070 q^{87} +0.764327 q^{89} +3.37552 q^{91} -0.400729 q^{93} -7.00466 q^{95} -4.11552 q^{97} -4.72536 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	$ 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) $ 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * 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
$5$	2.60189	1.16360	0.581800	−	0.813332i	$-0.302348\pi$
$5$	2.60189	1.16360	0.581800	+	0.813332i	$0.302348\pi$
$6$	0	0
$7$	−3.58131	−1.35361	−0.676804	−	0.736163i	$-0.736636\pi$
$7$	−3.58131	−1.35361	−0.676804	+	0.736163i	$0.736636\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.72536	−1.42475	−0.712375	−	0.701799i	$-0.752381\pi$
$11$	−4.72536	−1.42475	−0.712375	+	0.701799i	$0.752381\pi$
$12$	0	0
$13$	−0.942538	−0.261413	−0.130707	−	0.991421i	$-0.541725\pi$
$13$	−0.942538	−0.261413	−0.130707	+	0.991421i	$0.541725\pi$
$14$	0	0
$15$	−2.60189	−0.671805
$16$	0	0
$17$	−3.64824	−0.884827	−0.442414	−	0.896811i	$-0.645878\pi$
$17$	−3.64824	−0.884827	−0.442414	+	0.896811i	$0.645878\pi$
$18$	0	0
$19$	−2.69214	−0.617620	−0.308810	−	0.951124i	$-0.599931\pi$
$19$	−2.69214	−0.617620	−0.308810	+	0.951124i	$0.599931\pi$
$20$	0	0
$21$	3.58131	0.781506
$22$	0	0
$23$	−0.381441	−0.0795360	−0.0397680	−	0.999209i	$-0.512662\pi$
$23$	−0.381441	−0.0795360	−0.0397680	+	0.999209i	$0.512662\pi$
$24$	0	0
$25$	1.76983	0.353967
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.31070	0.800477	0.400239	−	0.916411i	$-0.368927\pi$
$29$	4.31070	0.800477	0.400239	+	0.916411i	$0.368927\pi$
$30$	0	0
$31$	0.400729	0.0719731	0.0359865	−	0.999352i	$-0.488543\pi$
$31$	0.400729	0.0719731	0.0359865	+	0.999352i	$0.488543\pi$
$32$	0	0
$33$	4.72536	0.822580
$34$	0	0
$35$	−9.31818	−1.57506
$36$	0	0
$37$	0.407293	0.0669585	0.0334792	−	0.999439i	$-0.489341\pi$
$37$	0.407293	0.0669585	0.0334792	+	0.999439i	$0.489341\pi$
$38$	0	0
$39$	0.942538	0.150927
$40$	0	0
$41$	0.759495	0.118613	0.0593066	−	0.998240i	$-0.481111\pi$
$41$	0.759495	0.118613	0.0593066	+	0.998240i	$0.481111\pi$
$42$	0	0
$43$	4.47426	0.682319	0.341159	−	0.940005i	$-0.389180\pi$
$43$	4.47426	0.682319	0.341159	+	0.940005i	$0.389180\pi$
$44$	0	0
$45$	2.60189	0.387867
$46$	0	0
$47$	13.2113	1.92707	0.963534	−	0.267587i	$-0.0862263\pi$
$47$	13.2113	1.92707	0.963534	+	0.267587i	$0.0862263\pi$
$48$	0	0
$49$	5.82578	0.832255
$50$	0	0
$51$	3.64824	0.510855
$52$	0	0
$53$	4.41253	0.606108	0.303054	−	0.952973i	$-0.401994\pi$
$53$	4.41253	0.606108	0.303054	+	0.952973i	$0.401994\pi$
$54$	0	0
$55$	−12.2949	−1.65784
$56$	0	0
$57$	2.69214	0.356583
$58$	0	0
$59$	−5.47954	−0.713375	−0.356687	−	0.934224i	$-0.616094\pi$
$59$	−5.47954	−0.713375	−0.356687	+	0.934224i	$0.616094\pi$
$60$	0	0
$61$	10.9868	1.40671	0.703356	−	0.710838i	$-0.251684\pi$
$61$	10.9868	1.40671	0.703356	+	0.710838i	$0.251684\pi$
$62$	0	0
$63$	−3.58131	−0.451203
$64$	0	0
$65$	−2.45238	−0.304180
$66$	0	0
$67$	4.38579	0.535809	0.267905	−	0.963445i	$-0.413669\pi$
$67$	4.38579	0.535809	0.267905	+	0.963445i	$0.413669\pi$
$68$	0	0
$69$	0.381441	0.0459201
$70$	0	0
$71$	−1.17148	−0.139029	−0.0695143	−	0.997581i	$-0.522145\pi$
$71$	−1.17148	−0.139029	−0.0695143	+	0.997581i	$0.522145\pi$
$72$	0	0
$73$	12.3989	1.45118	0.725592	−	0.688125i	$-0.241566\pi$
$73$	12.3989	1.45118	0.725592	+	0.688125i	$0.241566\pi$
$74$	0	0
$75$	−1.76983	−0.204363
$76$	0	0
$77$	16.9230	1.92855
$78$	0	0
$79$	8.57361	0.964607	0.482303	−	0.876004i	$-0.339800\pi$
$79$	8.57361	0.964607	0.482303	+	0.876004i	$0.339800\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.551394	0.0605233	0.0302617	−	0.999542i	$-0.490366\pi$
$83$	0.551394	0.0605233	0.0302617	+	0.999542i	$0.490366\pi$
$84$	0	0
$85$	−9.49231	−1.02959
$86$	0	0
$87$	−4.31070	−0.462156
$88$	0	0
$89$	0.764327	0.0810185	0.0405092	−	0.999179i	$-0.487102\pi$
$89$	0.764327	0.0810185	0.0405092	+	0.999179i	$0.487102\pi$
$90$	0	0
$91$	3.37552	0.353851
$92$	0	0
$93$	−0.400729	−0.0415537
$94$	0	0
$95$	−7.00466	−0.718663
$96$	0	0
$97$	−4.11552	−0.417868	−0.208934	−	0.977930i	$-0.566999\pi$
$97$	−4.11552	−0.417868	−0.208934	+	0.977930i	$0.566999\pi$
$98$	0	0
$99$	−4.72536	−0.474917
$100$	0	0
$101$	4.37168	0.434999	0.217499	−	0.976060i	$-0.430210\pi$
$101$	4.37168	0.434999	0.217499	+	0.976060i	$0.430210\pi$
$102$	0	0
$103$	10.2673	1.01167	0.505833	−	0.862632i	$-0.331185\pi$
$103$	10.2673	1.01167	0.505833	+	0.862632i	$0.331185\pi$
$104$	0	0
$105$	9.31818	0.909361
$106$	0	0
$107$	3.33602	0.322505	0.161252	−	0.986913i	$-0.448447\pi$
$107$	3.33602	0.322505	0.161252	+	0.986913i	$0.448447\pi$
$108$	0	0
$109$	6.99888	0.670371	0.335185	−	0.942152i	$-0.391201\pi$
$109$	6.99888	0.670371	0.335185	+	0.942152i	$0.391201\pi$
$110$	0	0
$111$	−0.407293	−0.0386585
$112$	0	0
$113$	−11.1190	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$113$	−11.1190	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$114$	0	0
$115$	−0.992468	−0.0925481
$116$	0	0
$117$	−0.942538	−0.0871377
$118$	0	0
$119$	13.0655	1.19771
$120$	0	0
$121$	11.3290	1.02991
$122$	0	0
$123$	−0.759495	−0.0684813
$124$	0	0
$125$	−8.40454	−0.751725
$126$	0	0
$127$	−6.13214	−0.544139	−0.272070	−	0.962278i	$-0.587708\pi$
$127$	−6.13214	−0.544139	−0.272070	+	0.962278i	$0.587708\pi$
$128$	0	0
$129$	−4.47426	−0.393937
$130$	0	0
$131$	−12.0777	−1.05524	−0.527619	−	0.849481i	$-0.676915\pi$
$131$	−12.0777	−1.05524	−0.527619	+	0.849481i	$0.676915\pi$
$132$	0	0
$133$	9.64140	0.836016
$134$	0	0
$135$	−2.60189	−0.223935
$136$	0	0
$137$	1.52242	0.130069	0.0650345	−	0.997883i	$-0.479284\pi$
$137$	1.52242	0.130069	0.0650345	+	0.997883i	$0.479284\pi$
$138$	0	0
$139$	−0.416070	−0.0352906	−0.0176453	−	0.999844i	$-0.505617\pi$
$139$	−0.416070	−0.0352906	−0.0176453	+	0.999844i	$0.505617\pi$
$140$	0	0
$141$	−13.2113	−1.11259
$142$	0	0
$143$	4.45383	0.372448
$144$	0	0
$145$	11.2160	0.931436
$146$	0	0
$147$	−5.82578	−0.480502
$148$	0	0
$149$	17.3291	1.41965	0.709827	−	0.704376i	$-0.248773\pi$
$149$	17.3291	1.41965	0.709827	+	0.704376i	$0.248773\pi$
$150$	0	0
$151$	−4.06385	−0.330712	−0.165356	−	0.986234i	$-0.552877\pi$
$151$	−4.06385	−0.330712	−0.165356	+	0.986234i	$0.552877\pi$
$152$	0	0
$153$	−3.64824	−0.294942
$154$	0	0
$155$	1.04265	0.0837479
$156$	0	0
$157$	−4.55587	−0.363598	−0.181799	−	0.983336i	$-0.558192\pi$
$157$	−4.55587	−0.363598	−0.181799	+	0.983336i	$0.558192\pi$
$158$	0	0
$159$	−4.41253	−0.349937
$160$	0	0
$161$	1.36606	0.107661
$162$	0	0
$163$	0.730514	0.0572183	0.0286091	−	0.999591i	$-0.490892\pi$
$163$	0.730514	0.0572183	0.0286091	+	0.999591i	$0.490892\pi$
$164$	0	0
$165$	12.2949	0.957155
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.1116	−0.931663
$170$	0	0
$171$	−2.69214	−0.205873
$172$	0	0
$173$	−0.0679638	−0.00516719	−0.00258359	−	0.999997i	$-0.500822\pi$
$173$	−0.0679638	−0.00516719	−0.00258359	+	0.999997i	$0.500822\pi$
$174$	0	0
$175$	−6.33832	−0.479132
$176$	0	0
$177$	5.47954	0.411867
$178$	0	0
$179$	2.86435	0.214092	0.107046	−	0.994254i	$-0.465861\pi$
$179$	2.86435	0.214092	0.107046	+	0.994254i	$0.465861\pi$
$180$	0	0
$181$	−7.39711	−0.549823	−0.274911	−	0.961470i	$-0.588649\pi$
$181$	−7.39711	−0.549823	−0.274911	+	0.961470i	$0.588649\pi$
$182$	0	0
$183$	−10.9868	−0.812165
$184$	0	0
$185$	1.05973	0.0779129
$186$	0	0
$187$	17.2392	1.26066
$188$	0	0
$189$	3.58131	0.260502
$190$	0	0
$191$	1.50228	0.108701	0.0543505	−	0.998522i	$-0.482691\pi$
$191$	1.50228	0.108701	0.0543505	+	0.998522i	$0.482691\pi$
$192$	0	0
$193$	11.6710	0.840097	0.420048	−	0.907502i	$-0.362013\pi$
$193$	11.6710	0.840097	0.420048	+	0.907502i	$0.362013\pi$
$194$	0	0
$195$	2.45238	0.175619
$196$	0	0
$197$	−17.0576	−1.21530	−0.607652	−	0.794203i	$-0.707889\pi$
$197$	−17.0576	−1.21530	−0.607652	+	0.794203i	$0.707889\pi$
$198$	0	0
$199$	−13.9842	−0.991313	−0.495656	−	0.868519i	$-0.665072\pi$
$199$	−13.9842	−0.991313	−0.495656	+	0.868519i	$0.665072\pi$
$200$	0	0
$201$	−4.38579	−0.309350
$202$	0	0
$203$	−15.4380	−1.08353
$204$	0	0
$205$	1.97612	0.138018
$206$	0	0
$207$	−0.381441	−0.0265120
$208$	0	0
$209$	12.7214	0.879955
$210$	0	0
$211$	−2.83616	−0.195249	−0.0976246	−	0.995223i	$-0.531124\pi$
$211$	−2.83616	−0.195249	−0.0976246	+	0.995223i	$0.531124\pi$
$212$	0	0
$213$	1.17148	0.0802682
$214$	0	0
$215$	11.6415	0.793947
$216$	0	0
$217$	−1.43514	−0.0974234
$218$	0	0
$219$	−12.3989	−0.837841
$220$	0	0
$221$	3.43860	0.231305
$222$	0	0
$223$	27.1850	1.82044	0.910221	−	0.414123i	$-0.135912\pi$
$223$	27.1850	1.82044	0.910221	+	0.414123i	$0.135912\pi$
$224$	0	0
$225$	1.76983	0.117989
$226$	0	0
$227$	−3.04918	−0.202381	−0.101191	−	0.994867i	$-0.532265\pi$
$227$	−3.04918	−0.202381	−0.101191	+	0.994867i	$0.532265\pi$
$228$	0	0
$229$	−18.4941	−1.22213	−0.611063	−	0.791582i	$-0.709258\pi$
$229$	−18.4941	−1.22213	−0.611063	+	0.791582i	$0.709258\pi$
$230$	0	0
$231$	−16.9230	−1.11345
$232$	0	0
$233$	0.657303	0.0430614	0.0215307	−	0.999768i	$-0.493146\pi$
$233$	0.657303	0.0430614	0.0215307	+	0.999768i	$0.493146\pi$
$234$	0	0
$235$	34.3744	2.24234
$236$	0	0
$237$	−8.57361	−0.556916
$238$	0	0
$239$	20.0775	1.29871	0.649353	−	0.760488i	$-0.275040\pi$
$239$	20.0775	1.29871	0.649353	+	0.760488i	$0.275040\pi$
$240$	0	0
$241$	−5.36567	−0.345633	−0.172817	−	0.984954i	$-0.555287\pi$
$241$	−5.36567	−0.345633	−0.172817	+	0.984954i	$0.555287\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	15.1580	0.968412
$246$	0	0
$247$	2.53745	0.161454
$248$	0	0
$249$	−0.551394	−0.0349432
$250$	0	0
$251$	22.9699	1.44985	0.724923	−	0.688830i	$-0.241875\pi$
$251$	22.9699	1.44985	0.724923	+	0.688830i	$0.241875\pi$
$252$	0	0
$253$	1.80245	0.113319
$254$	0	0
$255$	9.49231	0.594432
$256$	0	0
$257$	5.41168	0.337572	0.168786	−	0.985653i	$-0.446015\pi$
$257$	5.41168	0.337572	0.168786	+	0.985653i	$0.446015\pi$
$258$	0	0
$259$	−1.45864	−0.0906355
$260$	0	0
$261$	4.31070	0.266826
$262$	0	0
$263$	27.6992	1.70801	0.854004	−	0.520267i	$-0.174168\pi$
$263$	27.6992	1.70801	0.854004	+	0.520267i	$0.174168\pi$
$264$	0	0
$265$	11.4809	0.705268
$266$	0	0
$267$	−0.764327	−0.0467760
$268$	0	0
$269$	10.7253	0.653932	0.326966	−	0.945036i	$-0.393974\pi$
$269$	10.7253	0.653932	0.326966	+	0.945036i	$0.393974\pi$
$270$	0	0
$271$	8.82097	0.535836	0.267918	−	0.963442i	$-0.413664\pi$
$271$	8.82097	0.535836	0.267918	+	0.963442i	$0.413664\pi$
$272$	0	0
$273$	−3.37552	−0.204296
$274$	0	0
$275$	−8.36310	−0.504314
$276$	0	0
$277$	−18.3165	−1.10053	−0.550265	−	0.834990i	$-0.685473\pi$
$277$	−18.3165	−1.10053	−0.550265	+	0.834990i	$0.685473\pi$
$278$	0	0
$279$	0.400729	0.0239910
$280$	0	0
$281$	22.8776	1.36476	0.682382	−	0.730996i	$-0.260944\pi$
$281$	22.8776	1.36476	0.682382	+	0.730996i	$0.260944\pi$
$282$	0	0
$283$	14.7628	0.877556	0.438778	−	0.898595i	$-0.355411\pi$
$283$	14.7628	0.877556	0.438778	+	0.898595i	$0.355411\pi$
$284$	0	0
$285$	7.00466	0.414920
$286$	0	0
$287$	−2.71999	−0.160556
$288$	0	0
$289$	−3.69037	−0.217081
$290$	0	0
$291$	4.11552	0.241256
$292$	0	0
$293$	1.04558	0.0610834	0.0305417	−	0.999533i	$-0.490277\pi$
$293$	1.04558	0.0610834	0.0305417	+	0.999533i	$0.490277\pi$
$294$	0	0
$295$	−14.2572	−0.830083
$296$	0	0
$297$	4.72536	0.274193
$298$	0	0
$299$	0.359523	0.0207917
$300$	0	0
$301$	−16.0237	−0.923592
$302$	0	0
$303$	−4.37168	−0.251147
$304$	0	0
$305$	28.5864	1.63685
$306$	0	0
$307$	10.8164	0.617326	0.308663	−	0.951171i	$-0.400118\pi$
$307$	10.8164	0.617326	0.308663	+	0.951171i	$0.400118\pi$
$308$	0	0
$309$	−10.2673	−0.584085
$310$	0	0
$311$	−13.1845	−0.747625	−0.373812	−	0.927504i	$-0.621950\pi$
$311$	−13.1845	−0.747625	−0.373812	+	0.927504i	$0.621950\pi$
$312$	0	0
$313$	−7.37178	−0.416678	−0.208339	−	0.978057i	$-0.566806\pi$
$313$	−7.37178	−0.416678	−0.208339	+	0.978057i	$0.566806\pi$
$314$	0	0
$315$	−9.31818	−0.525020
$316$	0	0
$317$	−10.2372	−0.574978	−0.287489	−	0.957784i	$-0.592821\pi$
$317$	−10.2372	−0.574978	−0.287489	+	0.957784i	$0.592821\pi$
$318$	0	0
$319$	−20.3696	−1.14048
$320$	0	0
$321$	−3.33602	−0.186198
$322$	0	0
$323$	9.82158	0.546487
$324$	0	0
$325$	−1.66814	−0.0925315
$326$	0	0
$327$	−6.99888	−0.387039
$328$	0	0
$329$	−47.3138	−2.60849
$330$	0	0
$331$	−34.3313	−1.88702	−0.943509	−	0.331348i	$-0.892496\pi$
$331$	−34.3313	−1.88702	−0.943509	+	0.331348i	$0.892496\pi$
$332$	0	0
$333$	0.407293	0.0223195
$334$	0	0
$335$	11.4113	0.623468
$336$	0	0
$337$	1.75514	0.0956088	0.0478044	−	0.998857i	$-0.484778\pi$
$337$	1.75514	0.0956088	0.0478044	+	0.998857i	$0.484778\pi$
$338$	0	0
$339$	11.1190	0.603901
$340$	0	0
$341$	−1.89359	−0.102544
$342$	0	0
$343$	4.20524	0.227062
$344$	0	0
$345$	0.992468	0.0534327
$346$	0	0
$347$	22.0967	1.18621	0.593106	−	0.805125i	$-0.297902\pi$
$347$	22.0967	1.18621	0.593106	+	0.805125i	$0.297902\pi$
$348$	0	0
$349$	3.71549	0.198886	0.0994428	−	0.995043i	$-0.468294\pi$
$349$	3.71549	0.198886	0.0994428	+	0.995043i	$0.468294\pi$
$350$	0	0
$351$	0.942538	0.0503090
$352$	0	0
$353$	12.0789	0.642893	0.321447	−	0.946928i	$-0.395831\pi$
$353$	12.0789	0.642893	0.321447	+	0.946928i	$0.395831\pi$
$354$	0	0
$355$	−3.04805	−0.161774
$356$	0	0
$357$	−13.0655	−0.691498
$358$	0	0
$359$	−15.0369	−0.793617	−0.396808	−	0.917902i	$-0.629882\pi$
$359$	−15.0369	−0.793617	−0.396808	+	0.917902i	$0.629882\pi$
$360$	0	0
$361$	−11.7524	−0.618545
$362$	0	0
$363$	−11.3290	−0.594621
$364$	0	0
$365$	32.2606	1.68860
$366$	0	0
$367$	1.93985	0.101259	0.0506295	−	0.998718i	$-0.483877\pi$
$367$	1.93985	0.101259	0.0506295	+	0.998718i	$0.483877\pi$
$368$	0	0
$369$	0.759495	0.0395377
$370$	0	0
$371$	−15.8026	−0.820432
$372$	0	0
$373$	24.3710	1.26188	0.630942	−	0.775830i	$-0.282669\pi$
$373$	24.3710	1.26188	0.630942	+	0.775830i	$0.282669\pi$
$374$	0	0
$375$	8.40454	0.434009
$376$	0	0
$377$	−4.06300	−0.209255
$378$	0	0
$379$	−6.87541	−0.353166	−0.176583	−	0.984286i	$-0.556504\pi$
$379$	−6.87541	−0.353166	−0.176583	+	0.984286i	$0.556504\pi$
$380$	0	0
$381$	6.13214	0.314159
$382$	0	0
$383$	−21.2247	−1.08453	−0.542267	−	0.840206i	$-0.682434\pi$
$383$	−21.2247	−1.08453	−0.542267	+	0.840206i	$0.682434\pi$
$384$	0	0
$385$	44.0318	2.24407
$386$	0	0
$387$	4.47426	0.227440
$388$	0	0
$389$	31.3959	1.59184	0.795918	−	0.605405i	$-0.206989\pi$
$389$	31.3959	1.59184	0.795918	+	0.605405i	$0.206989\pi$
$390$	0	0
$391$	1.39159	0.0703756
$392$	0	0
$393$	12.0777	0.609242
$394$	0	0
$395$	22.3076	1.12242
$396$	0	0
$397$	24.8691	1.24815	0.624073	−	0.781366i	$-0.285477\pi$
$397$	24.8691	1.24815	0.624073	+	0.781366i	$0.285477\pi$
$398$	0	0
$399$	−9.64140	−0.482674
$400$	0	0
$401$	−5.28088	−0.263714	−0.131857	−	0.991269i	$-0.542094\pi$
$401$	−5.28088	−0.263714	−0.131857	+	0.991269i	$0.542094\pi$
$402$	0	0
$403$	−0.377703	−0.0188147
$404$	0	0
$405$	2.60189	0.129289
$406$	0	0
$407$	−1.92460	−0.0953991
$408$	0	0
$409$	−21.1368	−1.04515	−0.522575	−	0.852593i	$-0.675028\pi$
$409$	−21.1368	−1.04515	−0.522575	+	0.852593i	$0.675028\pi$
$410$	0	0
$411$	−1.52242	−0.0750954
$412$	0	0
$413$	19.6239	0.965630
$414$	0	0
$415$	1.43467	0.0704250
$416$	0	0
$417$	0.416070	0.0203750
$418$	0	0
$419$	25.0085	1.22175	0.610874	−	0.791728i	$-0.290818\pi$
$419$	25.0085	1.22175	0.610874	+	0.791728i	$0.290818\pi$
$420$	0	0
$421$	1.12666	0.0549102	0.0274551	−	0.999623i	$-0.491260\pi$
$421$	1.12666	0.0549102	0.0274551	+	0.999623i	$0.491260\pi$
$422$	0	0
$423$	13.2113	0.642356
$424$	0	0
$425$	−6.45677	−0.313199
$426$	0	0
$427$	−39.3470	−1.90414
$428$	0	0
$429$	−4.45383	−0.215033
$430$	0	0
$431$	29.5186	1.42186	0.710931	−	0.703262i	$-0.248274\pi$
$431$	29.5186	1.42186	0.710931	+	0.703262i	$0.248274\pi$
$432$	0	0
$433$	2.36702	0.113752	0.0568758	−	0.998381i	$-0.481886\pi$
$433$	2.36702	0.113752	0.0568758	+	0.998381i	$0.481886\pi$
$434$	0	0
$435$	−11.2160	−0.537765
$436$	0	0
$437$	1.02689	0.0491230
$438$	0	0
$439$	−23.5108	−1.12211	−0.561056	−	0.827778i	$-0.689605\pi$
$439$	−23.5108	−1.12211	−0.561056	+	0.827778i	$0.689605\pi$
$440$	0	0
$441$	5.82578	0.277418
$442$	0	0
$443$	31.5813	1.50047	0.750237	−	0.661169i	$-0.229940\pi$
$443$	31.5813	1.50047	0.750237	+	0.661169i	$0.229940\pi$
$444$	0	0
$445$	1.98869	0.0942731
$446$	0	0
$447$	−17.3291	−0.819638
$448$	0	0
$449$	19.1231	0.902475	0.451237	−	0.892404i	$-0.350983\pi$
$449$	19.1231	0.902475	0.451237	+	0.892404i	$0.350983\pi$
$450$	0	0
$451$	−3.58889	−0.168994
$452$	0	0
$453$	4.06385	0.190936
$454$	0	0
$455$	8.78274	0.411741
$456$	0	0
$457$	−15.1813	−0.710153	−0.355076	−	0.934837i	$-0.615545\pi$
$457$	−15.1813	−0.710153	−0.355076	+	0.934837i	$0.615545\pi$
$458$	0	0
$459$	3.64824	0.170285
$460$	0	0
$461$	25.4193	1.18390	0.591948	−	0.805976i	$-0.298359\pi$
$461$	25.4193	1.18390	0.591948	+	0.805976i	$0.298359\pi$
$462$	0	0
$463$	−21.9175	−1.01859	−0.509297	−	0.860591i	$-0.670094\pi$
$463$	−21.9175	−1.01859	−0.509297	+	0.860591i	$0.670094\pi$
$464$	0	0
$465$	−1.04265	−0.0483519
$466$	0	0
$467$	20.3451	0.941457	0.470729	−	0.882278i	$-0.343991\pi$
$467$	20.3451	0.941457	0.470729	+	0.882278i	$0.343991\pi$
$468$	0	0
$469$	−15.7069	−0.725276
$470$	0	0
$471$	4.55587	0.209924
$472$	0	0
$473$	−21.1425	−0.972134
$474$	0	0
$475$	−4.76465	−0.218617
$476$	0	0
$477$	4.41253	0.202036
$478$	0	0
$479$	−24.7792	−1.13219	−0.566095	−	0.824340i	$-0.691546\pi$
$479$	−24.7792	−1.13219	−0.566095	+	0.824340i	$0.691546\pi$
$480$	0	0
$481$	−0.383889	−0.0175038
$482$	0	0
$483$	−1.36606	−0.0621579
$484$	0	0
$485$	−10.7081	−0.486232
$486$	0	0
$487$	18.5285	0.839605	0.419802	−	0.907616i	$-0.362099\pi$
$487$	18.5285	0.839605	0.419802	+	0.907616i	$0.362099\pi$
$488$	0	0
$489$	−0.730514	−0.0330350
$490$	0	0
$491$	−15.4680	−0.698063	−0.349031	−	0.937111i	$-0.613489\pi$
$491$	−15.4680	−0.698063	−0.349031	+	0.937111i	$0.613489\pi$
$492$	0	0
$493$	−15.7265	−0.708284
$494$	0	0
$495$	−12.2949	−0.552614
$496$	0	0
$497$	4.19542	0.188190
$498$	0	0
$499$	9.82883	0.439999	0.219999	−	0.975500i	$-0.429394\pi$
$499$	9.82883	0.439999	0.219999	+	0.975500i	$0.429394\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	20.7097	0.923401	0.461701	−	0.887036i	$-0.347239\pi$
$503$	20.7097	0.923401	0.461701	+	0.887036i	$0.347239\pi$
$504$	0	0
$505$	11.3746	0.506165
$506$	0	0
$507$	12.1116	0.537896
$508$	0	0
$509$	19.6701	0.871863	0.435931	−	0.899980i	$-0.356419\pi$
$509$	19.6701	0.871863	0.435931	+	0.899980i	$0.356419\pi$
$510$	0	0
$511$	−44.4044	−1.96433
$512$	0	0
$513$	2.69214	0.118861
$514$	0	0
$515$	26.7143	1.17717
$516$	0	0
$517$	−62.4282	−2.74559
$518$	0	0
$519$	0.0679638	0.00298328
$520$	0	0
$521$	25.0759	1.09859	0.549297	−	0.835627i	$-0.314895\pi$
$521$	25.0759	1.09859	0.549297	+	0.835627i	$0.314895\pi$
$522$	0	0
$523$	27.8230	1.21661	0.608307	−	0.793702i	$-0.291849\pi$
$523$	27.8230	1.21661	0.608307	+	0.793702i	$0.291849\pi$
$524$	0	0
$525$	6.33832	0.276627
$526$	0	0
$527$	−1.46195	−0.0636838
$528$	0	0
$529$	−22.8545	−0.993674
$530$	0	0
$531$	−5.47954	−0.237792
$532$	0	0
$533$	−0.715853	−0.0310070
$534$	0	0
$535$	8.67995	0.375267
$536$	0	0
$537$	−2.86435	−0.123606
$538$	0	0
$539$	−27.5289	−1.18575
$540$	0	0
$541$	−11.5442	−0.496325	−0.248163	−	0.968718i	$-0.579827\pi$
$541$	−11.5442	−0.496325	−0.248163	+	0.968718i	$0.579827\pi$
$542$	0	0
$543$	7.39711	0.317440
$544$	0	0
$545$	18.2103	0.780044
$546$	0	0
$547$	−11.9053	−0.509032	−0.254516	−	0.967069i	$-0.581916\pi$
$547$	−11.9053	−0.509032	−0.254516	+	0.967069i	$0.581916\pi$
$548$	0	0
$549$	10.9868	0.468904
$550$	0	0
$551$	−11.6050	−0.494391
$552$	0	0
$553$	−30.7048	−1.30570
$554$	0	0
$555$	−1.05973	−0.0449831
$556$	0	0
$557$	41.1551	1.74380	0.871899	−	0.489686i	$-0.162888\pi$
$557$	41.1551	1.74380	0.871899	+	0.489686i	$0.162888\pi$
$558$	0	0
$559$	−4.21717	−0.178367
$560$	0	0
$561$	−17.2392	−0.727841
$562$	0	0
$563$	−1.33224	−0.0561472	−0.0280736	−	0.999606i	$-0.508937\pi$
$563$	−1.33224	−0.0561472	−0.0280736	+	0.999606i	$0.508937\pi$
$564$	0	0
$565$	−28.9304	−1.21711
$566$	0	0
$567$	−3.58131	−0.150401
$568$	0	0
$569$	1.31639	0.0551861	0.0275931	−	0.999619i	$-0.491216\pi$
$569$	1.31639	0.0551861	0.0275931	+	0.999619i	$0.491216\pi$
$570$	0	0
$571$	−15.0075	−0.628043	−0.314022	−	0.949416i	$-0.601676\pi$
$571$	−15.0075	−0.628043	−0.314022	+	0.949416i	$0.601676\pi$
$572$	0	0
$573$	−1.50228	−0.0627586
$574$	0	0
$575$	−0.675087	−0.0281531
$576$	0	0
$577$	16.9944	0.707487	0.353743	−	0.935343i	$-0.384909\pi$
$577$	16.9944	0.707487	0.353743	+	0.935343i	$0.384909\pi$
$578$	0	0
$579$	−11.6710	−0.485030
$580$	0	0
$581$	−1.97471	−0.0819249
$582$	0	0
$583$	−20.8508	−0.863552
$584$	0	0
$585$	−2.45238	−0.101393
$586$	0	0
$587$	−25.0492	−1.03389	−0.516945	−	0.856018i	$-0.672931\pi$
$587$	−25.0492	−1.03389	−0.516945	+	0.856018i	$0.672931\pi$
$588$	0	0
$589$	−1.07882	−0.0444520
$590$	0	0
$591$	17.0576	0.701656
$592$	0	0
$593$	20.3612	0.836134	0.418067	−	0.908416i	$-0.362708\pi$
$593$	20.3612	0.836134	0.418067	+	0.908416i	$0.362708\pi$
$594$	0	0
$595$	33.9949	1.39366
$596$	0	0
$597$	13.9842	0.572335
$598$	0	0
$599$	−13.7622	−0.562309	−0.281155	−	0.959663i	$-0.590717\pi$
$599$	−13.7622	−0.562309	−0.281155	+	0.959663i	$0.590717\pi$
$600$	0	0
$601$	−32.7714	−1.33677	−0.668387	−	0.743814i	$-0.733015\pi$
$601$	−32.7714	−1.33677	−0.668387	+	0.743814i	$0.733015\pi$
$602$	0	0
$603$	4.38579	0.178603
$604$	0	0
$605$	29.4769	1.19841
$606$	0	0
$607$	35.7405	1.45066	0.725331	−	0.688401i	$-0.241687\pi$
$607$	35.7405	1.45066	0.725331	+	0.688401i	$0.241687\pi$
$608$	0	0
$609$	15.4380	0.625578
$610$	0	0
$611$	−12.4522	−0.503761
$612$	0	0
$613$	−5.12576	−0.207028	−0.103514	−	0.994628i	$-0.533009\pi$
$613$	−5.12576	−0.207028	−0.103514	+	0.994628i	$0.533009\pi$
$614$	0	0
$615$	−1.97612	−0.0796849
$616$	0	0
$617$	−5.86405	−0.236078	−0.118039	−	0.993009i	$-0.537661\pi$
$617$	−5.86405	−0.236078	−0.118039	+	0.993009i	$0.537661\pi$
$618$	0	0
$619$	11.2491	0.452141	0.226070	−	0.974111i	$-0.427412\pi$
$619$	11.2491	0.452141	0.226070	+	0.974111i	$0.427412\pi$
$620$	0	0
$621$	0.381441	0.0153067
$622$	0	0
$623$	−2.73729	−0.109667
$624$	0	0
$625$	−30.7169	−1.22867
$626$	0	0
$627$	−12.7214	−0.508042
$628$	0	0
$629$	−1.48590	−0.0592467
$630$	0	0
$631$	1.17955	0.0469573	0.0234786	−	0.999724i	$-0.492526\pi$
$631$	1.17955	0.0469573	0.0234786	+	0.999724i	$0.492526\pi$
$632$	0	0
$633$	2.83616	0.112727
$634$	0	0
$635$	−15.9552	−0.633161
$636$	0	0
$637$	−5.49102	−0.217562
$638$	0	0
$639$	−1.17148	−0.0463428
$640$	0	0
$641$	−33.1443	−1.30912	−0.654561	−	0.756009i	$-0.727147\pi$
$641$	−33.1443	−1.30912	−0.654561	+	0.756009i	$0.727147\pi$
$642$	0	0
$643$	−19.5427	−0.770687	−0.385344	−	0.922773i	$-0.625917\pi$
$643$	−19.5427	−0.770687	−0.385344	+	0.922773i	$0.625917\pi$
$644$	0	0
$645$	−11.6415	−0.458385
$646$	0	0
$647$	3.86419	0.151917	0.0759585	−	0.997111i	$-0.475798\pi$
$647$	3.86419	0.151917	0.0759585	+	0.997111i	$0.475798\pi$
$648$	0	0
$649$	25.8928	1.01638
$650$	0	0
$651$	1.43514	0.0562474
$652$	0	0
$653$	−3.27995	−0.128354	−0.0641772	−	0.997939i	$-0.520442\pi$
$653$	−3.27995	−0.128354	−0.0641772	+	0.997939i	$0.520442\pi$
$654$	0	0
$655$	−31.4250	−1.22787
$656$	0	0
$657$	12.3989	0.483728
$658$	0	0
$659$	−17.6881	−0.689031	−0.344515	−	0.938781i	$-0.611957\pi$
$659$	−17.6881	−0.689031	−0.344515	+	0.938781i	$0.611957\pi$
$660$	0	0
$661$	−24.9186	−0.969220	−0.484610	−	0.874730i	$-0.661038\pi$
$661$	−24.9186	−0.969220	−0.484610	+	0.874730i	$0.661038\pi$
$662$	0	0
$663$	−3.43860	−0.133544
$664$	0	0
$665$	25.0859	0.972788
$666$	0	0
$667$	−1.64428	−0.0636668
$668$	0	0
$669$	−27.1850	−1.05103
$670$	0	0
$671$	−51.9165	−2.00421
$672$	0	0
$673$	15.9326	0.614158	0.307079	−	0.951684i	$-0.400648\pi$
$673$	15.9326	0.614158	0.307079	+	0.951684i	$0.400648\pi$
$674$	0	0
$675$	−1.76983	−0.0681209
$676$	0	0
$677$	21.0904	0.810569	0.405285	−	0.914191i	$-0.367172\pi$
$677$	21.0904	0.810569	0.405285	+	0.914191i	$0.367172\pi$
$678$	0	0
$679$	14.7390	0.565630
$680$	0	0
$681$	3.04918	0.116845
$682$	0	0
$683$	9.92172	0.379644	0.189822	−	0.981819i	$-0.439209\pi$
$683$	9.92172	0.379644	0.189822	+	0.981819i	$0.439209\pi$
$684$	0	0
$685$	3.96117	0.151348
$686$	0	0
$687$	18.4941	0.705595
$688$	0	0
$689$	−4.15898	−0.158445
$690$	0	0
$691$	−9.68186	−0.368315	−0.184158	−	0.982897i	$-0.558956\pi$
$691$	−9.68186	−0.368315	−0.184158	+	0.982897i	$0.558956\pi$
$692$	0	0
$693$	16.9230	0.642851
$694$	0	0
$695$	−1.08257	−0.0410642
$696$	0	0
$697$	−2.77082	−0.104952
$698$	0	0
$699$	−0.657303	−0.0248615
$700$	0	0
$701$	35.5112	1.34124	0.670620	−	0.741801i	$-0.266028\pi$
$701$	35.5112	1.34124	0.670620	+	0.741801i	$0.266028\pi$
$702$	0	0
$703$	−1.09649	−0.0413549
$704$	0	0
$705$	−34.3744	−1.29461
$706$	0	0
$707$	−15.6564	−0.588818
$708$	0	0
$709$	15.2713	0.573525	0.286762	−	0.958002i	$-0.407421\pi$
$709$	15.2713	0.573525	0.286762	+	0.958002i	$0.407421\pi$
$710$	0	0
$711$	8.57361	0.321536
$712$	0	0
$713$	−0.152855	−0.00572445
$714$	0	0
$715$	11.5884	0.433381
$716$	0	0
$717$	−20.0775	−0.749808
$718$	0	0
$719$	−43.4707	−1.62118	−0.810592	−	0.585612i	$-0.800854\pi$
$719$	−43.4707	−1.62118	−0.810592	+	0.585612i	$0.800854\pi$
$720$	0	0
$721$	−36.7703	−1.36940
$722$	0	0
$723$	5.36567	0.199552
$724$	0	0
$725$	7.62923	0.283342
$726$	0	0
$727$	−21.3900	−0.793313	−0.396656	−	0.917967i	$-0.629830\pi$
$727$	−21.3900	−0.793313	−0.396656	+	0.917967i	$0.629830\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.3232	−0.603734
$732$	0	0
$733$	18.5499	0.685156	0.342578	−	0.939489i	$-0.388700\pi$
$733$	18.5499	0.685156	0.342578	+	0.939489i	$0.388700\pi$
$734$	0	0
$735$	−15.1580	−0.559113
$736$	0	0
$737$	−20.7244	−0.763395
$738$	0	0
$739$	0.246619	0.00907202	0.00453601	−	0.999990i	$-0.498556\pi$
$739$	0.246619	0.00907202	0.00453601	+	0.999990i	$0.498556\pi$
$740$	0	0
$741$	−2.53745	−0.0932155
$742$	0	0
$743$	−43.8557	−1.60891	−0.804455	−	0.594013i	$-0.797543\pi$
$743$	−43.8557	−1.60891	−0.804455	+	0.594013i	$0.797543\pi$
$744$	0	0
$745$	45.0884	1.65191
$746$	0	0
$747$	0.551394	0.0201744
$748$	0	0
$749$	−11.9473	−0.436545
$750$	0	0
$751$	31.3326	1.14334	0.571671	−	0.820483i	$-0.306295\pi$
$751$	31.3326	1.14334	0.571671	+	0.820483i	$0.306295\pi$
$752$	0	0
$753$	−22.9699	−0.837069
$754$	0	0
$755$	−10.5737	−0.384816
$756$	0	0
$757$	−47.7102	−1.73406	−0.867029	−	0.498258i	$-0.833973\pi$
$757$	−47.7102	−1.73406	−0.867029	+	0.498258i	$0.833973\pi$
$758$	0	0
$759$	−1.80245	−0.0654247
$760$	0	0
$761$	−33.0321	−1.19741	−0.598706	−	0.800969i	$-0.704318\pi$
$761$	−33.0321	−1.19741	−0.598706	+	0.800969i	$0.704318\pi$
$762$	0	0
$763$	−25.0651	−0.907419
$764$	0	0
$765$	−9.49231	−0.343195
$766$	0	0
$767$	5.16467	0.186485
$768$	0	0
$769$	2.24259	0.0808698	0.0404349	−	0.999182i	$-0.487126\pi$
$769$	2.24259	0.0808698	0.0404349	+	0.999182i	$0.487126\pi$
$770$	0	0
$771$	−5.41168	−0.194897
$772$	0	0
$773$	−33.8407	−1.21716	−0.608582	−	0.793491i	$-0.708262\pi$
$773$	−33.8407	−1.21716	−0.608582	+	0.793491i	$0.708262\pi$
$774$	0	0
$775$	0.709224	0.0254761
$776$	0	0
$777$	1.45864	0.0523284
$778$	0	0
$779$	−2.04467	−0.0732579
$780$	0	0
$781$	5.53564	0.198081
$782$	0	0
$783$	−4.31070	−0.154052
$784$	0	0
$785$	−11.8539	−0.423083
$786$	0	0
$787$	22.9925	0.819596	0.409798	−	0.912176i	$-0.365599\pi$
$787$	22.9925	0.819596	0.409798	+	0.912176i	$0.365599\pi$
$788$	0	0
$789$	−27.6992	−0.986119
$790$	0	0
$791$	39.8206	1.41586
$792$	0	0
$793$	−10.3554	−0.367733
$794$	0	0
$795$	−11.4809	−0.407186
$796$	0	0
$797$	−25.2005	−0.892649	−0.446324	−	0.894871i	$-0.647267\pi$
$797$	−25.2005	−0.892649	−0.446324	+	0.894871i	$0.647267\pi$
$798$	0	0
$799$	−48.1980	−1.70512
$800$	0	0
$801$	0.764327	0.0270062
$802$	0	0
$803$	−58.5894	−2.06757
$804$	0	0
$805$	3.55434	0.125274
$806$	0	0
$807$	−10.7253	−0.377548
$808$	0	0
$809$	3.96709	0.139475	0.0697377	−	0.997565i	$-0.477784\pi$
$809$	3.96709	0.139475	0.0697377	+	0.997565i	$0.477784\pi$
$810$	0	0
$811$	8.82118	0.309753	0.154877	−	0.987934i	$-0.450502\pi$
$811$	8.82118	0.309753	0.154877	+	0.987934i	$0.450502\pi$
$812$	0	0
$813$	−8.82097	−0.309365
$814$	0	0
$815$	1.90072	0.0665792
$816$	0	0
$817$	−12.0454	−0.421414
$818$	0	0
$819$	3.37552	0.117950
$820$	0	0
$821$	43.4468	1.51630	0.758151	−	0.652079i	$-0.226103\pi$
$821$	43.4468	1.51630	0.758151	+	0.652079i	$0.226103\pi$
$822$	0	0
$823$	−51.0625	−1.77993	−0.889963	−	0.456033i	$-0.849270\pi$
$823$	−51.0625	−1.77993	−0.889963	+	0.456033i	$0.849270\pi$
$824$	0	0
$825$	8.36310	0.291166
$826$	0	0
$827$	13.7322	0.477516	0.238758	−	0.971079i	$-0.423260\pi$
$827$	13.7322	0.477516	0.238758	+	0.971079i	$0.423260\pi$
$828$	0	0
$829$	17.6105	0.611637	0.305818	−	0.952090i	$-0.401070\pi$
$829$	17.6105	0.611637	0.305818	+	0.952090i	$0.401070\pi$
$830$	0	0
$831$	18.3165	0.635391
$832$	0	0
$833$	−21.2538	−0.736402
$834$	0	0
$835$	2.60189	0.0900421
$836$	0	0
$837$	−0.400729	−0.0138512
$838$	0	0
$839$	−14.8474	−0.512591	−0.256295	−	0.966599i	$-0.582502\pi$
$839$	−14.8474	−0.512591	−0.256295	+	0.966599i	$0.582502\pi$
$840$	0	0
$841$	−10.4178	−0.359236
$842$	0	0
$843$	−22.8776	−0.787946
$844$	0	0
$845$	−31.5131	−1.08408
$846$	0	0
$847$	−40.5728	−1.39410
$848$	0	0
$849$	−14.7628	−0.506657
$850$	0	0
$851$	−0.155358	−0.00532561
$852$	0	0
$853$	−14.9949	−0.513414	−0.256707	−	0.966489i	$-0.582638\pi$
$853$	−14.9949	−0.513414	−0.256707	+	0.966489i	$0.582638\pi$
$854$	0	0
$855$	−7.00466	−0.239554
$856$	0	0
$857$	46.1967	1.57805	0.789024	−	0.614363i	$-0.210587\pi$
$857$	46.1967	1.57805	0.789024	+	0.614363i	$0.210587\pi$
$858$	0	0
$859$	46.7913	1.59650	0.798249	−	0.602327i	$-0.205760\pi$
$859$	46.7913	1.59650	0.798249	+	0.602327i	$0.205760\pi$
$860$	0	0
$861$	2.71999	0.0926969
$862$	0	0
$863$	52.1490	1.77517	0.887587	−	0.460641i	$-0.152380\pi$
$863$	52.1490	1.77517	0.887587	+	0.460641i	$0.152380\pi$
$864$	0	0
$865$	−0.176834	−0.00601254
$866$	0	0
$867$	3.69037	0.125332
$868$	0	0
$869$	−40.5134	−1.37432
$870$	0	0
$871$	−4.13377	−0.140068
$872$	0	0
$873$	−4.11552	−0.139289
$874$	0	0
$875$	30.0993	1.01754
$876$	0	0
$877$	−25.8887	−0.874199	−0.437099	−	0.899413i	$-0.643994\pi$
$877$	−25.8887	−0.874199	−0.437099	+	0.899413i	$0.643994\pi$
$878$	0	0
$879$	−1.04558	−0.0352665
$880$	0	0
$881$	−5.07893	−0.171113	−0.0855567	−	0.996333i	$-0.527267\pi$
$881$	−5.07893	−0.171113	−0.0855567	+	0.996333i	$0.527267\pi$
$882$	0	0
$883$	15.9118	0.535473	0.267737	−	0.963492i	$-0.413724\pi$
$883$	15.9118	0.535473	0.267737	+	0.963492i	$0.413724\pi$
$884$	0	0
$885$	14.2572	0.479249
$886$	0	0
$887$	−4.59352	−0.154235	−0.0771177	−	0.997022i	$-0.524572\pi$
$887$	−4.59352	−0.154235	−0.0771177	+	0.997022i	$0.524572\pi$
$888$	0	0
$889$	21.9611	0.736551
$890$	0	0
$891$	−4.72536	−0.158306
$892$	0	0
$893$	−35.5667	−1.19020
$894$	0	0
$895$	7.45274	0.249118
$896$	0	0
$897$	−0.359523	−0.0120041
$898$	0	0
$899$	1.72742	0.0576128
$900$	0	0
$901$	−16.0980	−0.536301
$902$	0	0
$903$	16.0237	0.533236
$904$	0	0
$905$	−19.2465	−0.639774
$906$	0	0
$907$	21.5834	0.716666	0.358333	−	0.933594i	$-0.383345\pi$
$907$	21.5834	0.716666	0.358333	+	0.933594i	$0.383345\pi$
$908$	0	0
$909$	4.37168	0.145000
$910$	0	0
$911$	11.9986	0.397531	0.198766	−	0.980047i	$-0.436307\pi$
$911$	11.9986	0.397531	0.198766	+	0.980047i	$0.436307\pi$
$912$	0	0
$913$	−2.60554	−0.0862306
$914$	0	0
$915$	−28.5864	−0.945036
$916$	0	0
$917$	43.2541	1.42838
$918$	0	0
$919$	−13.9436	−0.459955	−0.229978	−	0.973196i	$-0.573865\pi$
$919$	−13.9436	−0.459955	−0.229978	+	0.973196i	$0.573865\pi$
$920$	0	0
$921$	−10.8164	−0.356413
$922$	0	0
$923$	1.10416	0.0363439
$924$	0	0
$925$	0.720840	0.0237011
$926$	0	0
$927$	10.2673	0.337222
$928$	0	0
$929$	−1.03719	−0.0340291	−0.0170145	−	0.999855i	$-0.505416\pi$
$929$	−1.03719	−0.0340291	−0.0170145	+	0.999855i	$0.505416\pi$
$930$	0	0
$931$	−15.6838	−0.514017
$932$	0	0
$933$	13.1845	0.431641
$934$	0	0
$935$	44.8546	1.46690
$936$	0	0
$937$	−8.86334	−0.289553	−0.144776	−	0.989464i	$-0.546246\pi$
$937$	−8.86334	−0.289553	−0.144776	+	0.989464i	$0.546246\pi$
$938$	0	0
$939$	7.37178	0.240569
$940$	0	0
$941$	−26.1467	−0.852357	−0.426178	−	0.904639i	$-0.640140\pi$
$941$	−26.1467	−0.852357	−0.426178	+	0.904639i	$0.640140\pi$
$942$	0	0
$943$	−0.289703	−0.00943401
$944$	0	0
$945$	9.31818	0.303120
$946$	0	0
$947$	−13.7102	−0.445522	−0.222761	−	0.974873i	$-0.571507\pi$
$947$	−13.7102	−0.445522	−0.222761	+	0.974873i	$0.571507\pi$
$948$	0	0
$949$	−11.6865	−0.379358
$950$	0	0
$951$	10.2372	0.331964
$952$	0	0
$953$	−48.4836	−1.57054	−0.785270	−	0.619154i	$-0.787476\pi$
$953$	−48.4836	−1.57054	−0.785270	+	0.619154i	$0.787476\pi$
$954$	0	0
$955$	3.90876	0.126485
$956$	0	0
$957$	20.3696	0.658457
$958$	0	0
$959$	−5.45225	−0.176062
$960$	0	0
$961$	−30.8394	−0.994820
$962$	0	0
$963$	3.33602	0.107502
$964$	0	0
$965$	30.3666	0.977537
$966$	0	0
$967$	1.62702	0.0523214	0.0261607	−	0.999658i	$-0.491672\pi$
$967$	1.62702	0.0523214	0.0261607	+	0.999658i	$0.491672\pi$
$968$	0	0
$969$	−9.82158	−0.315515
$970$	0	0
$971$	−23.8897	−0.766656	−0.383328	−	0.923612i	$-0.625222\pi$
$971$	−23.8897	−0.766656	−0.383328	+	0.923612i	$0.625222\pi$
$972$	0	0
$973$	1.49008	0.0477697
$974$	0	0
$975$	1.66814	0.0534231
$976$	0	0
$977$	37.6398	1.20420	0.602102	−	0.798419i	$-0.294330\pi$
$977$	37.6398	1.20420	0.602102	+	0.798419i	$0.294330\pi$
$978$	0	0
$979$	−3.61172	−0.115431
$980$	0	0
$981$	6.99888	0.223457
$982$	0	0
$983$	−25.8936	−0.825877	−0.412939	−	0.910759i	$-0.635498\pi$
$983$	−25.8936	−0.825877	−0.412939	+	0.910759i	$0.635498\pi$
$984$	0	0
$985$	−44.3820	−1.41413
$986$	0	0
$987$	47.3138	1.50601
$988$	0	0
$989$	−1.70667	−0.0542689
$990$	0	0
$991$	43.5439	1.38322	0.691609	−	0.722273i	$-0.256902\pi$
$991$	43.5439	1.38322	0.691609	+	0.722273i	$0.256902\pi$
$992$	0	0
$993$	34.3313	1.08947
$994$	0	0
$995$	−36.3853	−1.15349
$996$	0	0
$997$	−39.1973	−1.24139	−0.620695	−	0.784052i	$-0.713150\pi$
$997$	−39.1973	−1.24139	−0.620695	+	0.784052i	$0.713150\pi$
$998$	0	0
$999$	−0.407293	−0.0128862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.k.1.7	✓	11
4.3	odd	2		8016.2.a.be.1.7		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.k.1.7	✓	11	1.1	even	1	trivial
8016.2.a.be.1.7		11	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.60189 q^{5} -3.58131 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.60189 q^{5} -3.58131 q^{7} +1.00000 q^{9} -4.72536 q^{11} -0.942538 q^{13} -2.60189 q^{15} -3.64824 q^{17} -2.69214 q^{19} +3.58131 q^{21} -0.381441 q^{23} +1.76983 q^{25} -1.00000 q^{27} +4.31070 q^{29} +0.400729 q^{31} +4.72536 q^{33} -9.31818 q^{35} +0.407293 q^{37} +0.942538 q^{39} +0.759495 q^{41} +4.47426 q^{43} +2.60189 q^{45} +13.2113 q^{47} +5.82578 q^{49} +3.64824 q^{51} +4.41253 q^{53} -12.2949 q^{55} +2.69214 q^{57} -5.47954 q^{59} +10.9868 q^{61} -3.58131 q^{63} -2.45238 q^{65} +4.38579 q^{67} +0.381441 q^{69} -1.17148 q^{71} +12.3989 q^{73} -1.76983 q^{75} +16.9230 q^{77} +8.57361 q^{79} +1.00000 q^{81} +0.551394 q^{83} -9.49231 q^{85} -4.31070 q^{87} +0.764327 q^{89} +3.37552 q^{91} -0.400729 q^{93} -7.00466 q^{95} -4.11552 q^{97} -4.72536 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9	\( 11 q - 11 q^{3} + 10 q^{5} - q^{7} + 11 q^{9} - q^{11} + 10 q^{13} - 10 q^{15} + 17 q^{17} + 2 q^{19} + q^{21} - 3 q^{23} + 21 q^{25} - 11 q^{27} + 17 q^{29} - 15 q^{31} + q^{33} + 11 q^{35} + 4 q^{37} - 10 q^{39} + 16 q^{41} + 10 q^{43} + 10 q^{45} - 16 q^{47} + 22 q^{49} - 17 q^{51} + 42 q^{53} - 5 q^{55} - 2 q^{57} - 2 q^{59} + 12 q^{61} - q^{63} + 10 q^{65} - q^{67} + 3 q^{69} - 9 q^{71} + 24 q^{73} - 21 q^{75} + 22 q^{77} - 30 q^{79} + 11 q^{81} + 16 q^{83} + 25 q^{85} - 17 q^{87} + 37 q^{89} + q^{91} + 15 q^{93} + 5 q^{95} + 4 q^{97} - q^{99}+O(q^{100}) \) 11 * q - 11 * q^3 + 10 * q^5 - q^7 + 11 * q^9 - q^11 + 10 * q^13 - 10 * q^15 + 17 * q^17 + 2 * q^19 + q^21 - 3 * q^23 + 21 * q^25 - 11 * q^27 + 17 * q^29 - 15 * q^31 + q^33 + 11 * q^35 + 4 * q^37 - 10 * q^39 + 16 * q^41 + 10 * q^43 + 10 * q^45 - 16 * q^47 + 22 * q^49 - 17 * q^51 + 42 * q^53 - 5 * q^55 - 2 * q^57 - 2 * q^59 + 12 * q^61 - q^63 + 10 * q^65 - q^67 + 3 * q^69 - 9 * q^71 + 24 * q^73 - 21 * q^75 + 22 * q^77 - 30 * q^79 + 11 * q^81 + 16 * q^83 + 25 * q^85 - 17 * q^87 + 37 * q^89 + q^91 + 15 * q^93 + 5 * q^95 + 4 * q^97 - q^99

Introduction

L-functions

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