LMFDB - Embedded newform 2008.2.a.d.1.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2008,2,Mod(1,2008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2008 = 2^{3} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2008.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:	$16.0339607259$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
Character	$\chi$	$=$	2008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.26165 q^{3} -4.10224 q^{5} -1.61221 q^{7} +7.63839 q^{9} +O(q^{10})$	$q+3.26165 q^{3} -4.10224 q^{5} -1.61221 q^{7} +7.63839 q^{9} +5.52272 q^{11} +3.26255 q^{13} -13.3801 q^{15} -4.72190 q^{17} -3.75601 q^{19} -5.25848 q^{21} +4.65709 q^{23} +11.8284 q^{25} +15.1288 q^{27} -3.54583 q^{29} +5.76666 q^{31} +18.0132 q^{33} +6.61368 q^{35} -0.102706 q^{37} +10.6413 q^{39} +10.2306 q^{41} +2.60241 q^{43} -31.3345 q^{45} +6.85910 q^{47} -4.40077 q^{49} -15.4012 q^{51} +3.77385 q^{53} -22.6555 q^{55} -12.2508 q^{57} +5.75365 q^{59} -9.91122 q^{61} -12.3147 q^{63} -13.3838 q^{65} +13.8484 q^{67} +15.1898 q^{69} +15.0515 q^{71} -15.6479 q^{73} +38.5801 q^{75} -8.90380 q^{77} -6.36204 q^{79} +26.4298 q^{81} +10.9919 q^{83} +19.3704 q^{85} -11.5653 q^{87} +1.56890 q^{89} -5.25992 q^{91} +18.8088 q^{93} +15.4081 q^{95} -6.16465 q^{97} +42.1847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	$ 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) $ 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * 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
$3$	3.26165	1.88312	0.941558	−	0.336850i	$-0.109361\pi$
$3$	3.26165	1.88312	0.941558	+	0.336850i	$0.109361\pi$
$4$	0	0
$5$	−4.10224	−1.83458	−0.917289	−	0.398222i	$-0.869627\pi$
$5$	−4.10224	−1.83458	−0.917289	+	0.398222i	$0.869627\pi$
$6$	0	0
$7$	−1.61221	−0.609359	−0.304679	−	0.952455i	$-0.598549\pi$
$7$	−1.61221	−0.609359	−0.304679	+	0.952455i	$0.598549\pi$
$8$	0	0
$9$	7.63839	2.54613
$10$	0	0
$11$	5.52272	1.66516	0.832582	−	0.553902i	$-0.186862\pi$
$11$	5.52272	1.66516	0.832582	+	0.553902i	$0.186862\pi$
$12$	0	0
$13$	3.26255	0.904868	0.452434	−	0.891798i	$-0.350556\pi$
$13$	3.26255	0.904868	0.452434	+	0.891798i	$0.350556\pi$
$14$	0	0
$15$	−13.3801	−3.45473
$16$	0	0
$17$	−4.72190	−1.14523	−0.572614	−	0.819825i	$-0.694071\pi$
$17$	−4.72190	−1.14523	−0.572614	+	0.819825i	$0.694071\pi$
$18$	0	0
$19$	−3.75601	−0.861688	−0.430844	−	0.902426i	$-0.641784\pi$
$19$	−3.75601	−0.861688	−0.430844	+	0.902426i	$0.641784\pi$
$20$	0	0
$21$	−5.25848	−1.14749
$22$	0	0
$23$	4.65709	0.971070	0.485535	−	0.874217i	$-0.338625\pi$
$23$	4.65709	0.971070	0.485535	+	0.874217i	$0.338625\pi$
$24$	0	0
$25$	11.8284	2.36568
$26$	0	0
$27$	15.1288	2.91154
$28$	0	0
$29$	−3.54583	−0.658443	−0.329222	−	0.944253i	$-0.606786\pi$
$29$	−3.54583	−0.658443	−0.329222	+	0.944253i	$0.606786\pi$
$30$	0	0
$31$	5.76666	1.03572	0.517861	−	0.855465i	$-0.326728\pi$
$31$	5.76666	1.03572	0.517861	+	0.855465i	$0.326728\pi$
$32$	0	0
$33$	18.0132	3.13570
$34$	0	0
$35$	6.61368	1.11792
$36$	0	0
$37$	−0.102706	−0.0168847	−0.00844236	−	0.999964i	$-0.502687\pi$
$37$	−0.102706	−0.0168847	−0.00844236	+	0.999964i	$0.502687\pi$
$38$	0	0
$39$	10.6413	1.70397
$40$	0	0
$41$	10.2306	1.59776	0.798878	−	0.601493i	$-0.205427\pi$
$41$	10.2306	1.59776	0.798878	+	0.601493i	$0.205427\pi$
$42$	0	0
$43$	2.60241	0.396864	0.198432	−	0.980115i	$-0.436415\pi$
$43$	2.60241	0.396864	0.198432	+	0.980115i	$0.436415\pi$
$44$	0	0
$45$	−31.3345	−4.67107
$46$	0	0
$47$	6.85910	1.00050	0.500251	−	0.865880i	$-0.333241\pi$
$47$	6.85910	1.00050	0.500251	+	0.865880i	$0.333241\pi$
$48$	0	0
$49$	−4.40077	−0.628682
$50$	0	0
$51$	−15.4012	−2.15660
$52$	0	0
$53$	3.77385	0.518378	0.259189	−	0.965827i	$-0.416545\pi$
$53$	3.77385	0.518378	0.259189	+	0.965827i	$0.416545\pi$
$54$	0	0
$55$	−22.6555	−3.05487
$56$	0	0
$57$	−12.2508	−1.62266
$58$	0	0
$59$	5.75365	0.749061	0.374531	−	0.927215i	$-0.377804\pi$
$59$	5.75365	0.749061	0.374531	+	0.927215i	$0.377804\pi$
$60$	0	0
$61$	−9.91122	−1.26900	−0.634501	−	0.772922i	$-0.718794\pi$
$61$	−9.91122	−1.26900	−0.634501	+	0.772922i	$0.718794\pi$
$62$	0	0
$63$	−12.3147	−1.55151
$64$	0	0
$65$	−13.3838	−1.66005
$66$	0	0
$67$	13.8484	1.69186	0.845928	−	0.533298i	$-0.179047\pi$
$67$	13.8484	1.69186	0.845928	+	0.533298i	$0.179047\pi$
$68$	0	0
$69$	15.1898	1.82864
$70$	0	0
$71$	15.0515	1.78629	0.893145	−	0.449769i	$-0.148494\pi$
$71$	15.0515	1.78629	0.893145	+	0.449769i	$0.148494\pi$
$72$	0	0
$73$	−15.6479	−1.83145	−0.915723	−	0.401811i	$-0.868381\pi$
$73$	−15.6479	−1.83145	−0.915723	+	0.401811i	$0.868381\pi$
$74$	0	0
$75$	38.5801	4.45485
$76$	0	0
$77$	−8.90380	−1.01468
$78$	0	0
$79$	−6.36204	−0.715786	−0.357893	−	0.933763i	$-0.616505\pi$
$79$	−6.36204	−0.715786	−0.357893	+	0.933763i	$0.616505\pi$
$80$	0	0
$81$	26.4298	2.93664
$82$	0	0
$83$	10.9919	1.20652	0.603261	−	0.797544i	$-0.293868\pi$
$83$	10.9919	1.20652	0.603261	+	0.797544i	$0.293868\pi$
$84$	0	0
$85$	19.3704	2.10101
$86$	0	0
$87$	−11.5653	−1.23993
$88$	0	0
$89$	1.56890	0.166303	0.0831517	−	0.996537i	$-0.473501\pi$
$89$	1.56890	0.166303	0.0831517	+	0.996537i	$0.473501\pi$
$90$	0	0
$91$	−5.25992	−0.551390
$92$	0	0
$93$	18.8088	1.95039
$94$	0	0
$95$	15.4081	1.58083
$96$	0	0
$97$	−6.16465	−0.625925	−0.312962	−	0.949765i	$-0.601321\pi$
$97$	−6.16465	−0.625925	−0.312962	+	0.949765i	$0.601321\pi$
$98$	0	0
$99$	42.1847	4.23972
$100$	0	0
$101$	2.60250	0.258958	0.129479	−	0.991582i	$-0.458669\pi$
$101$	2.60250	0.258958	0.129479	+	0.991582i	$0.458669\pi$
$102$	0	0
$103$	−0.121978	−0.0120189	−0.00600944	−	0.999982i	$-0.501913\pi$
$103$	−0.121978	−0.0120189	−0.00600944	+	0.999982i	$0.501913\pi$
$104$	0	0
$105$	21.5715	2.10517
$106$	0	0
$107$	−16.5618	−1.60109	−0.800544	−	0.599274i	$-0.795456\pi$
$107$	−16.5618	−1.60109	−0.800544	+	0.599274i	$0.795456\pi$
$108$	0	0
$109$	−7.91023	−0.757662	−0.378831	−	0.925466i	$-0.623674\pi$
$109$	−7.91023	−0.757662	−0.378831	+	0.925466i	$0.623674\pi$
$110$	0	0
$111$	−0.334990	−0.0317959
$112$	0	0
$113$	−7.99692	−0.752287	−0.376143	−	0.926561i	$-0.622750\pi$
$113$	−7.99692	−0.752287	−0.376143	+	0.926561i	$0.622750\pi$
$114$	0	0
$115$	−19.1045	−1.78150
$116$	0	0
$117$	24.9206	2.30391
$118$	0	0
$119$	7.61270	0.697855
$120$	0	0
$121$	19.5005	1.77277
$122$	0	0
$123$	33.3688	3.00876
$124$	0	0
$125$	−28.0117	−2.50544
$126$	0	0
$127$	−3.88187	−0.344460	−0.172230	−	0.985057i	$-0.555097\pi$
$127$	−3.88187	−0.344460	−0.172230	+	0.985057i	$0.555097\pi$
$128$	0	0
$129$	8.48817	0.747341
$130$	0	0
$131$	−11.7914	−1.03022	−0.515109	−	0.857125i	$-0.672248\pi$
$131$	−11.7914	−1.03022	−0.515109	+	0.857125i	$0.672248\pi$
$132$	0	0
$133$	6.05549	0.525077
$134$	0	0
$135$	−62.0620	−5.34145
$136$	0	0
$137$	1.08131	0.0923824	0.0461912	−	0.998933i	$-0.485292\pi$
$137$	1.08131	0.0923824	0.0461912	+	0.998933i	$0.485292\pi$
$138$	0	0
$139$	7.63350	0.647465	0.323733	−	0.946149i	$-0.395062\pi$
$139$	7.63350	0.647465	0.323733	+	0.946149i	$0.395062\pi$
$140$	0	0
$141$	22.3720	1.88406
$142$	0	0
$143$	18.0182	1.50675
$144$	0	0
$145$	14.5458	1.20797
$146$	0	0
$147$	−14.3538	−1.18388
$148$	0	0
$149$	7.96318	0.652369	0.326184	−	0.945306i	$-0.394237\pi$
$149$	7.96318	0.652369	0.326184	+	0.945306i	$0.394237\pi$
$150$	0	0
$151$	−15.9239	−1.29586	−0.647932	−	0.761698i	$-0.724366\pi$
$151$	−15.9239	−1.29586	−0.647932	+	0.761698i	$0.724366\pi$
$152$	0	0
$153$	−36.0677	−2.91590
$154$	0	0
$155$	−23.6562	−1.90011
$156$	0	0
$157$	16.7099	1.33360	0.666798	−	0.745239i	$-0.267664\pi$
$157$	16.7099	1.33360	0.666798	+	0.745239i	$0.267664\pi$
$158$	0	0
$159$	12.3090	0.976167
$160$	0	0
$161$	−7.50822	−0.591730
$162$	0	0
$163$	−7.33905	−0.574839	−0.287419	−	0.957805i	$-0.592797\pi$
$163$	−7.33905	−0.574839	−0.287419	+	0.957805i	$0.592797\pi$
$164$	0	0
$165$	−73.8945	−5.75268
$166$	0	0
$167$	−10.0999	−0.781554	−0.390777	−	0.920485i	$-0.627794\pi$
$167$	−10.0999	−0.781554	−0.390777	+	0.920485i	$0.627794\pi$
$168$	0	0
$169$	−2.35577	−0.181213
$170$	0	0
$171$	−28.6899	−2.19397
$172$	0	0
$173$	1.69787	0.129087	0.0645434	−	0.997915i	$-0.479441\pi$
$173$	1.69787	0.129087	0.0645434	+	0.997915i	$0.479441\pi$
$174$	0	0
$175$	−19.0699	−1.44155
$176$	0	0
$177$	18.7664	1.41057
$178$	0	0
$179$	15.5898	1.16523	0.582617	−	0.812747i	$-0.302029\pi$
$179$	15.5898	1.16523	0.582617	+	0.812747i	$0.302029\pi$
$180$	0	0
$181$	1.66228	0.123556	0.0617781	−	0.998090i	$-0.480323\pi$
$181$	1.66228	0.123556	0.0617781	+	0.998090i	$0.480323\pi$
$182$	0	0
$183$	−32.3270	−2.38968
$184$	0	0
$185$	0.421324	0.0309763
$186$	0	0
$187$	−26.0777	−1.90699
$188$	0	0
$189$	−24.3909	−1.77417
$190$	0	0
$191$	−11.7530	−0.850420	−0.425210	−	0.905095i	$-0.639800\pi$
$191$	−11.7530	−0.850420	−0.425210	+	0.905095i	$0.639800\pi$
$192$	0	0
$193$	16.8485	1.21278	0.606392	−	0.795166i	$-0.292616\pi$
$193$	16.8485	1.21278	0.606392	+	0.795166i	$0.292616\pi$
$194$	0	0
$195$	−43.6532	−3.12607
$196$	0	0
$197$	−17.7093	−1.26173	−0.630867	−	0.775891i	$-0.717301\pi$
$197$	−17.7093	−1.26173	−0.630867	+	0.775891i	$0.717301\pi$
$198$	0	0
$199$	−3.85488	−0.273265	−0.136632	−	0.990622i	$-0.543628\pi$
$199$	−3.85488	−0.273265	−0.136632	+	0.990622i	$0.543628\pi$
$200$	0	0
$201$	45.1688	3.18596
$202$	0	0
$203$	5.71662	0.401228
$204$	0	0
$205$	−41.9685	−2.93121
$206$	0	0
$207$	35.5726	2.47247
$208$	0	0
$209$	−20.7434	−1.43485
$210$	0	0
$211$	−14.4082	−0.991899	−0.495949	−	0.868351i	$-0.665180\pi$
$211$	−14.4082	−0.991899	−0.495949	+	0.868351i	$0.665180\pi$
$212$	0	0
$213$	49.0929	3.36379
$214$	0	0
$215$	−10.6757	−0.728078
$216$	0	0
$217$	−9.29708	−0.631127
$218$	0	0
$219$	−51.0380	−3.44883
$220$	0	0
$221$	−15.4054	−1.03628
$222$	0	0
$223$	16.5318	1.10705	0.553525	−	0.832832i	$-0.313282\pi$
$223$	16.5318	1.10705	0.553525	+	0.832832i	$0.313282\pi$
$224$	0	0
$225$	90.3498	6.02332
$226$	0	0
$227$	5.18928	0.344425	0.172212	−	0.985060i	$-0.444908\pi$
$227$	5.18928	0.344425	0.172212	+	0.985060i	$0.444908\pi$
$228$	0	0
$229$	−16.4771	−1.08884	−0.544420	−	0.838813i	$-0.683250\pi$
$229$	−16.4771	−1.08884	−0.544420	+	0.838813i	$0.683250\pi$
$230$	0	0
$231$	−29.0411	−1.91076
$232$	0	0
$233$	16.8186	1.10182	0.550911	−	0.834564i	$-0.314280\pi$
$233$	16.8186	1.10182	0.550911	+	0.834564i	$0.314280\pi$
$234$	0	0
$235$	−28.1377	−1.83550
$236$	0	0
$237$	−20.7508	−1.34791
$238$	0	0
$239$	−10.9090	−0.705646	−0.352823	−	0.935690i	$-0.614778\pi$
$239$	−10.9090	−0.705646	−0.352823	+	0.935690i	$0.614778\pi$
$240$	0	0
$241$	−3.58735	−0.231081	−0.115541	−	0.993303i	$-0.536860\pi$
$241$	−3.58735	−0.231081	−0.115541	+	0.993303i	$0.536860\pi$
$242$	0	0
$243$	40.8184	2.61850
$244$	0	0
$245$	18.0530	1.15337
$246$	0	0
$247$	−12.2542	−0.779714
$248$	0	0
$249$	35.8519	2.27202
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	25.7198	1.61699
$254$	0	0
$255$	63.1794	3.95645
$256$	0	0
$257$	−17.9081	−1.11708	−0.558539	−	0.829478i	$-0.688638\pi$
$257$	−17.9081	−1.11708	−0.558539	+	0.829478i	$0.688638\pi$
$258$	0	0
$259$	0.165583	0.0102888
$260$	0	0
$261$	−27.0844	−1.67648
$262$	0	0
$263$	−5.39226	−0.332501	−0.166251	−	0.986084i	$-0.553166\pi$
$263$	−5.39226	−0.332501	−0.166251	+	0.986084i	$0.553166\pi$
$264$	0	0
$265$	−15.4812	−0.951005
$266$	0	0
$267$	5.11722	0.313169
$268$	0	0
$269$	−20.7985	−1.26811	−0.634054	−	0.773289i	$-0.718610\pi$
$269$	−20.7985	−1.26811	−0.634054	+	0.773289i	$0.718610\pi$
$270$	0	0
$271$	23.8707	1.45004	0.725020	−	0.688728i	$-0.241831\pi$
$271$	23.8707	1.45004	0.725020	+	0.688728i	$0.241831\pi$
$272$	0	0
$273$	−17.1560	−1.03833
$274$	0	0
$275$	65.3249	3.93924
$276$	0	0
$277$	−8.62585	−0.518277	−0.259139	−	0.965840i	$-0.583439\pi$
$277$	−8.62585	−0.518277	−0.259139	+	0.965840i	$0.583439\pi$
$278$	0	0
$279$	44.0480	2.63708
$280$	0	0
$281$	3.66452	0.218607	0.109303	−	0.994008i	$-0.465138\pi$
$281$	3.66452	0.218607	0.109303	+	0.994008i	$0.465138\pi$
$282$	0	0
$283$	−7.51120	−0.446495	−0.223247	−	0.974762i	$-0.571666\pi$
$283$	−7.51120	−0.446495	−0.223247	+	0.974762i	$0.571666\pi$
$284$	0	0
$285$	50.2558	2.97690
$286$	0	0
$287$	−16.4939	−0.973607
$288$	0	0
$289$	5.29630	0.311547
$290$	0	0
$291$	−20.1069	−1.17869
$292$	0	0
$293$	−15.1600	−0.885659	−0.442830	−	0.896606i	$-0.646025\pi$
$293$	−15.1600	−0.885659	−0.442830	+	0.896606i	$0.646025\pi$
$294$	0	0
$295$	−23.6029	−1.37421
$296$	0	0
$297$	83.5522	4.84819
$298$	0	0
$299$	15.1940	0.878691
$300$	0	0
$301$	−4.19564	−0.241833
$302$	0	0
$303$	8.48845	0.487649
$304$	0	0
$305$	40.6582	2.32808
$306$	0	0
$307$	−14.7844	−0.843789	−0.421895	−	0.906645i	$-0.638635\pi$
$307$	−14.7844	−0.843789	−0.421895	+	0.906645i	$0.638635\pi$
$308$	0	0
$309$	−0.397851	−0.0226330
$310$	0	0
$311$	−26.9521	−1.52831	−0.764156	−	0.645032i	$-0.776844\pi$
$311$	−26.9521	−1.52831	−0.764156	+	0.645032i	$0.776844\pi$
$312$	0	0
$313$	−25.9965	−1.46941	−0.734704	−	0.678388i	$-0.762679\pi$
$313$	−25.9965	−1.46941	−0.734704	+	0.678388i	$0.762679\pi$
$314$	0	0
$315$	50.5179	2.84636
$316$	0	0
$317$	23.4660	1.31798	0.658991	−	0.752150i	$-0.270983\pi$
$317$	23.4660	1.31798	0.658991	+	0.752150i	$0.270983\pi$
$318$	0	0
$319$	−19.5826	−1.09642
$320$	0	0
$321$	−54.0188	−3.01504
$322$	0	0
$323$	17.7355	0.986829
$324$	0	0
$325$	38.5907	2.14063
$326$	0	0
$327$	−25.8004	−1.42677
$328$	0	0
$329$	−11.0583	−0.609665
$330$	0	0
$331$	−12.0296	−0.661204	−0.330602	−	0.943770i	$-0.607252\pi$
$331$	−12.0296	−0.661204	−0.330602	+	0.943770i	$0.607252\pi$
$332$	0	0
$333$	−0.784506	−0.0429907
$334$	0	0
$335$	−56.8096	−3.10384
$336$	0	0
$337$	−8.67913	−0.472782	−0.236391	−	0.971658i	$-0.575965\pi$
$337$	−8.67913	−0.472782	−0.236391	+	0.971658i	$0.575965\pi$
$338$	0	0
$339$	−26.0832	−1.41664
$340$	0	0
$341$	31.8477	1.72465
$342$	0	0
$343$	18.3805	0.992452
$344$	0	0
$345$	−62.3123	−3.35478
$346$	0	0
$347$	−11.8869	−0.638125	−0.319062	−	0.947734i	$-0.603368\pi$
$347$	−11.8869	−0.638125	−0.319062	+	0.947734i	$0.603368\pi$
$348$	0	0
$349$	−20.3590	−1.08979	−0.544896	−	0.838503i	$-0.683431\pi$
$349$	−20.3590	−1.08979	−0.544896	+	0.838503i	$0.683431\pi$
$350$	0	0
$351$	49.3585	2.63456
$352$	0	0
$353$	32.0911	1.70804	0.854018	−	0.520243i	$-0.174159\pi$
$353$	32.0911	1.70804	0.854018	+	0.520243i	$0.174159\pi$
$354$	0	0
$355$	−61.7451	−3.27709
$356$	0	0
$357$	24.8300	1.31414
$358$	0	0
$359$	1.42583	0.0752525	0.0376262	−	0.999292i	$-0.488020\pi$
$359$	1.42583	0.0752525	0.0376262	+	0.999292i	$0.488020\pi$
$360$	0	0
$361$	−4.89238	−0.257493
$362$	0	0
$363$	63.6038	3.33833
$364$	0	0
$365$	64.1914	3.35993
$366$	0	0
$367$	20.1684	1.05278	0.526392	−	0.850242i	$-0.323545\pi$
$367$	20.1684	1.05278	0.526392	+	0.850242i	$0.323545\pi$
$368$	0	0
$369$	78.1455	4.06809
$370$	0	0
$371$	−6.08425	−0.315878
$372$	0	0
$373$	27.9405	1.44671	0.723353	−	0.690478i	$-0.242600\pi$
$373$	27.9405	1.44671	0.723353	+	0.690478i	$0.242600\pi$
$374$	0	0
$375$	−91.3645	−4.71804
$376$	0	0
$377$	−11.5684	−0.595805
$378$	0	0
$379$	−26.6630	−1.36959	−0.684793	−	0.728738i	$-0.740107\pi$
$379$	−26.6630	−1.36959	−0.684793	+	0.728738i	$0.740107\pi$
$380$	0	0
$381$	−12.6613	−0.648659
$382$	0	0
$383$	26.3023	1.34398	0.671992	−	0.740558i	$-0.265439\pi$
$383$	26.3023	1.34398	0.671992	+	0.740558i	$0.265439\pi$
$384$	0	0
$385$	36.5255	1.86151
$386$	0	0
$387$	19.8782	1.01047
$388$	0	0
$389$	28.0214	1.42074	0.710372	−	0.703827i	$-0.248527\pi$
$389$	28.0214	1.42074	0.710372	+	0.703827i	$0.248527\pi$
$390$	0	0
$391$	−21.9903	−1.11210
$392$	0	0
$393$	−38.4594	−1.94002
$394$	0	0
$395$	26.0986	1.31317
$396$	0	0
$397$	−18.8475	−0.945930	−0.472965	−	0.881081i	$-0.656816\pi$
$397$	−18.8475	−0.945930	−0.472965	+	0.881081i	$0.656816\pi$
$398$	0	0
$399$	19.7509	0.988782
$400$	0	0
$401$	−0.486605	−0.0242999	−0.0121500	−	0.999926i	$-0.503868\pi$
$401$	−0.486605	−0.0242999	−0.0121500	+	0.999926i	$0.503868\pi$
$402$	0	0
$403$	18.8140	0.937193
$404$	0	0
$405$	−108.421	−5.38750
$406$	0	0
$407$	−0.567215	−0.0281158
$408$	0	0
$409$	−30.8030	−1.52311	−0.761555	−	0.648100i	$-0.775564\pi$
$409$	−30.8030	−1.52311	−0.761555	+	0.648100i	$0.775564\pi$
$410$	0	0
$411$	3.52685	0.173967
$412$	0	0
$413$	−9.27610	−0.456447
$414$	0	0
$415$	−45.0916	−2.21346
$416$	0	0
$417$	24.8978	1.21925
$418$	0	0
$419$	−17.4763	−0.853774	−0.426887	−	0.904305i	$-0.640390\pi$
$419$	−17.4763	−0.853774	−0.426887	+	0.904305i	$0.640390\pi$
$420$	0	0
$421$	−15.4448	−0.752732	−0.376366	−	0.926471i	$-0.622826\pi$
$421$	−15.4448	−0.752732	−0.376366	+	0.926471i	$0.622826\pi$
$422$	0	0
$423$	52.3924	2.54741
$424$	0	0
$425$	−55.8524	−2.70924
$426$	0	0
$427$	15.9790	0.773278
$428$	0	0
$429$	58.7690	2.83739
$430$	0	0
$431$	36.2976	1.74839	0.874196	−	0.485573i	$-0.161389\pi$
$431$	36.2976	1.74839	0.874196	+	0.485573i	$0.161389\pi$
$432$	0	0
$433$	−29.7835	−1.43130	−0.715651	−	0.698458i	$-0.753870\pi$
$433$	−29.7835	−1.43130	−0.715651	+	0.698458i	$0.753870\pi$
$434$	0	0
$435$	47.4435	2.27474
$436$	0	0
$437$	−17.4921	−0.836760
$438$	0	0
$439$	−8.17330	−0.390090	−0.195045	−	0.980794i	$-0.562485\pi$
$439$	−8.17330	−0.390090	−0.195045	+	0.980794i	$0.562485\pi$
$440$	0	0
$441$	−33.6148	−1.60070
$442$	0	0
$443$	−15.6197	−0.742116	−0.371058	−	0.928610i	$-0.621005\pi$
$443$	−15.6197	−0.742116	−0.371058	+	0.928610i	$0.621005\pi$
$444$	0	0
$445$	−6.43602	−0.305097
$446$	0	0
$447$	25.9731	1.22849
$448$	0	0
$449$	−9.19163	−0.433780	−0.216890	−	0.976196i	$-0.569591\pi$
$449$	−9.19163	−0.433780	−0.216890	+	0.976196i	$0.569591\pi$
$450$	0	0
$451$	56.5009	2.66053
$452$	0	0
$453$	−51.9381	−2.44027
$454$	0	0
$455$	21.5775	1.01157
$456$	0	0
$457$	16.9137	0.791190	0.395595	−	0.918425i	$-0.370538\pi$
$457$	16.9137	0.791190	0.395595	+	0.918425i	$0.370538\pi$
$458$	0	0
$459$	−71.4367	−3.33438
$460$	0	0
$461$	9.23677	0.430199	0.215100	−	0.976592i	$-0.430992\pi$
$461$	9.23677	0.430199	0.215100	+	0.976592i	$0.430992\pi$
$462$	0	0
$463$	5.60293	0.260390	0.130195	−	0.991488i	$-0.458440\pi$
$463$	5.60293	0.260390	0.130195	+	0.991488i	$0.458440\pi$
$464$	0	0
$465$	−77.1584	−3.57814
$466$	0	0
$467$	−16.0004	−0.740410	−0.370205	−	0.928950i	$-0.620713\pi$
$467$	−16.0004	−0.740410	−0.370205	+	0.928950i	$0.620713\pi$
$468$	0	0
$469$	−22.3266	−1.03095
$470$	0	0
$471$	54.5019	2.51132
$472$	0	0
$473$	14.3724	0.660843
$474$	0	0
$475$	−44.4276	−2.03848
$476$	0	0
$477$	28.8261	1.31986
$478$	0	0
$479$	3.99761	0.182655	0.0913276	−	0.995821i	$-0.470889\pi$
$479$	3.99761	0.182655	0.0913276	+	0.995821i	$0.470889\pi$
$480$	0	0
$481$	−0.335082	−0.0152784
$482$	0	0
$483$	−24.4892	−1.11430
$484$	0	0
$485$	25.2889	1.14831
$486$	0	0
$487$	24.6005	1.11475	0.557377	−	0.830260i	$-0.311808\pi$
$487$	24.6005	1.11475	0.557377	+	0.830260i	$0.311808\pi$
$488$	0	0
$489$	−23.9374	−1.08249
$490$	0	0
$491$	−10.6307	−0.479757	−0.239879	−	0.970803i	$-0.577108\pi$
$491$	−10.6307	−0.479757	−0.239879	+	0.970803i	$0.577108\pi$
$492$	0	0
$493$	16.7430	0.754068
$494$	0	0
$495$	−173.052	−7.77810
$496$	0	0
$497$	−24.2663	−1.08849
$498$	0	0
$499$	−10.4414	−0.467422	−0.233711	−	0.972306i	$-0.575087\pi$
$499$	−10.4414	−0.467422	−0.233711	+	0.972306i	$0.575087\pi$
$500$	0	0
$501$	−32.9424	−1.47176
$502$	0	0
$503$	13.2307	0.589927	0.294963	−	0.955509i	$-0.404693\pi$
$503$	13.2307	0.589927	0.294963	+	0.955509i	$0.404693\pi$
$504$	0	0
$505$	−10.6761	−0.475079
$506$	0	0
$507$	−7.68371	−0.341245
$508$	0	0
$509$	30.4125	1.34801	0.674006	−	0.738726i	$-0.264572\pi$
$509$	30.4125	1.34801	0.674006	+	0.738726i	$0.264572\pi$
$510$	0	0
$511$	25.2277	1.11601
$512$	0	0
$513$	−56.8240	−2.50884
$514$	0	0
$515$	0.500385	0.0220496
$516$	0	0
$517$	37.8809	1.66600
$518$	0	0
$519$	5.53787	0.243086
$520$	0	0
$521$	9.59122	0.420199	0.210099	−	0.977680i	$-0.432621\pi$
$521$	9.59122	0.420199	0.210099	+	0.977680i	$0.432621\pi$
$522$	0	0
$523$	41.7426	1.82528	0.912639	−	0.408767i	$-0.134041\pi$
$523$	41.7426	1.82528	0.912639	+	0.408767i	$0.134041\pi$
$524$	0	0
$525$	−62.1993	−2.71460
$526$	0	0
$527$	−27.2296	−1.18614
$528$	0	0
$529$	−1.31152	−0.0570225
$530$	0	0
$531$	43.9486	1.90721
$532$	0	0
$533$	33.3779	1.44576
$534$	0	0
$535$	67.9405	2.93732
$536$	0	0
$537$	50.8484	2.19427
$538$	0	0
$539$	−24.3042	−1.04686
$540$	0	0
$541$	11.7164	0.503726	0.251863	−	0.967763i	$-0.418957\pi$
$541$	11.7164	0.503726	0.251863	+	0.967763i	$0.418957\pi$
$542$	0	0
$543$	5.42178	0.232671
$544$	0	0
$545$	32.4497	1.38999
$546$	0	0
$547$	17.9143	0.765959	0.382979	−	0.923757i	$-0.374898\pi$
$547$	17.9143	0.765959	0.382979	+	0.923757i	$0.374898\pi$
$548$	0	0
$549$	−75.7058	−3.23104
$550$	0	0
$551$	13.3182	0.567373
$552$	0	0
$553$	10.2570	0.436170
$554$	0	0
$555$	1.37421	0.0583320
$556$	0	0
$557$	−8.50127	−0.360210	−0.180105	−	0.983647i	$-0.557644\pi$
$557$	−8.50127	−0.360210	−0.180105	+	0.983647i	$0.557644\pi$
$558$	0	0
$559$	8.49050	0.359110
$560$	0	0
$561$	−85.0565	−3.59109
$562$	0	0
$563$	−14.5512	−0.613261	−0.306631	−	0.951829i	$-0.599202\pi$
$563$	−14.5512	−0.613261	−0.306631	+	0.951829i	$0.599202\pi$
$564$	0	0
$565$	32.8053	1.38013
$566$	0	0
$567$	−42.6104	−1.78947
$568$	0	0
$569$	25.2947	1.06041	0.530205	−	0.847870i	$-0.322115\pi$
$569$	25.2947	1.06041	0.530205	+	0.847870i	$0.322115\pi$
$570$	0	0
$571$	−9.03101	−0.377936	−0.188968	−	0.981983i	$-0.560514\pi$
$571$	−9.03101	−0.377936	−0.188968	+	0.981983i	$0.560514\pi$
$572$	0	0
$573$	−38.3344	−1.60144
$574$	0	0
$575$	55.0859	2.29724
$576$	0	0
$577$	−4.75197	−0.197827	−0.0989135	−	0.995096i	$-0.531537\pi$
$577$	−4.75197	−0.197827	−0.0989135	+	0.995096i	$0.531537\pi$
$578$	0	0
$579$	54.9541	2.28381
$580$	0	0
$581$	−17.7213	−0.735205
$582$	0	0
$583$	20.8419	0.863184
$584$	0	0
$585$	−102.230	−4.22671
$586$	0	0
$587$	−21.8558	−0.902087	−0.451044	−	0.892502i	$-0.648948\pi$
$587$	−21.8558	−0.902087	−0.451044	+	0.892502i	$0.648948\pi$
$588$	0	0
$589$	−21.6596	−0.892470
$590$	0	0
$591$	−57.7615	−2.37599
$592$	0	0
$593$	−4.62303	−0.189845	−0.0949225	−	0.995485i	$-0.530260\pi$
$593$	−4.62303	−0.189845	−0.0949225	+	0.995485i	$0.530260\pi$
$594$	0	0
$595$	−31.2291	−1.28027
$596$	0	0
$597$	−12.5733	−0.514590
$598$	0	0
$599$	31.1976	1.27470	0.637349	−	0.770575i	$-0.280031\pi$
$599$	31.1976	1.27470	0.637349	+	0.770575i	$0.280031\pi$
$600$	0	0
$601$	25.2015	1.02799	0.513996	−	0.857792i	$-0.328165\pi$
$601$	25.2015	1.02799	0.513996	+	0.857792i	$0.328165\pi$
$602$	0	0
$603$	105.780	4.30768
$604$	0	0
$605$	−79.9956	−3.25228
$606$	0	0
$607$	−25.2707	−1.02570	−0.512852	−	0.858477i	$-0.671411\pi$
$607$	−25.2707	−1.02570	−0.512852	+	0.858477i	$0.671411\pi$
$608$	0	0
$609$	18.6456	0.755560
$610$	0	0
$611$	22.3781	0.905323
$612$	0	0
$613$	10.4271	0.421148	0.210574	−	0.977578i	$-0.432467\pi$
$613$	10.4271	0.421148	0.210574	+	0.977578i	$0.432467\pi$
$614$	0	0
$615$	−136.887	−5.51981
$616$	0	0
$617$	1.66477	0.0670210	0.0335105	−	0.999438i	$-0.489331\pi$
$617$	1.66477	0.0670210	0.0335105	+	0.999438i	$0.489331\pi$
$618$	0	0
$619$	−46.7622	−1.87953	−0.939766	−	0.341818i	$-0.888957\pi$
$619$	−46.7622	−1.87953	−0.939766	+	0.341818i	$0.888957\pi$
$620$	0	0
$621$	70.4562	2.82731
$622$	0	0
$623$	−2.52941	−0.101338
$624$	0	0
$625$	55.7688	2.23075
$626$	0	0
$627$	−67.6578	−2.70199
$628$	0	0
$629$	0.484966	0.0193368
$630$	0	0
$631$	−12.8965	−0.513403	−0.256701	−	0.966491i	$-0.582636\pi$
$631$	−12.8965	−0.513403	−0.256701	+	0.966491i	$0.582636\pi$
$632$	0	0
$633$	−46.9944	−1.86786
$634$	0	0
$635$	15.9244	0.631939
$636$	0	0
$637$	−14.3577	−0.568874
$638$	0	0
$639$	114.970	4.54812
$640$	0	0
$641$	40.5589	1.60198	0.800990	−	0.598678i	$-0.204307\pi$
$641$	40.5589	1.60198	0.800990	+	0.598678i	$0.204307\pi$
$642$	0	0
$643$	−4.90342	−0.193372	−0.0966861	−	0.995315i	$-0.530824\pi$
$643$	−4.90342	−0.193372	−0.0966861	+	0.995315i	$0.530824\pi$
$644$	0	0
$645$	−34.8205	−1.37106
$646$	0	0
$647$	31.2480	1.22849	0.614243	−	0.789117i	$-0.289462\pi$
$647$	31.2480	1.22849	0.614243	+	0.789117i	$0.289462\pi$
$648$	0	0
$649$	31.7758	1.24731
$650$	0	0
$651$	−30.3239	−1.18849
$652$	0	0
$653$	12.9521	0.506853	0.253426	−	0.967355i	$-0.418442\pi$
$653$	12.9521	0.506853	0.253426	+	0.967355i	$0.418442\pi$
$654$	0	0
$655$	48.3711	1.89001
$656$	0	0
$657$	−119.525	−4.66310
$658$	0	0
$659$	−28.5378	−1.11167	−0.555837	−	0.831291i	$-0.687602\pi$
$659$	−28.5378	−1.11167	−0.555837	+	0.831291i	$0.687602\pi$
$660$	0	0
$661$	1.11255	0.0432734	0.0216367	−	0.999766i	$-0.493112\pi$
$661$	1.11255	0.0432734	0.0216367	+	0.999766i	$0.493112\pi$
$662$	0	0
$663$	−50.2471	−1.95144
$664$	0	0
$665$	−24.8411	−0.963295
$666$	0	0
$667$	−16.5132	−0.639395
$668$	0	0
$669$	53.9210	2.08471
$670$	0	0
$671$	−54.7369	−2.11310
$672$	0	0
$673$	38.1214	1.46947	0.734736	−	0.678353i	$-0.237306\pi$
$673$	38.1214	1.46947	0.734736	+	0.678353i	$0.237306\pi$
$674$	0	0
$675$	178.949	6.88777
$676$	0	0
$677$	−37.1434	−1.42754	−0.713768	−	0.700382i	$-0.753013\pi$
$677$	−37.1434	−1.42754	−0.713768	+	0.700382i	$0.753013\pi$
$678$	0	0
$679$	9.93872	0.381413
$680$	0	0
$681$	16.9256	0.648592
$682$	0	0
$683$	6.33670	0.242467	0.121234	−	0.992624i	$-0.461315\pi$
$683$	6.33670	0.242467	0.121234	+	0.992624i	$0.461315\pi$
$684$	0	0
$685$	−4.43579	−0.169483
$686$	0	0
$687$	−53.7427	−2.05041
$688$	0	0
$689$	12.3124	0.469064
$690$	0	0
$691$	−22.8522	−0.869338	−0.434669	−	0.900590i	$-0.643135\pi$
$691$	−22.8522	−0.869338	−0.434669	+	0.900590i	$0.643135\pi$
$692$	0	0
$693$	−68.0107	−2.58351
$694$	0	0
$695$	−31.3145	−1.18783
$696$	0	0
$697$	−48.3080	−1.82980
$698$	0	0
$699$	54.8564	2.07486
$700$	0	0
$701$	28.2026	1.06520	0.532599	−	0.846367i	$-0.321215\pi$
$701$	28.2026	1.06520	0.532599	+	0.846367i	$0.321215\pi$
$702$	0	0
$703$	0.385764	0.0145494
$704$	0	0
$705$	−91.7754	−3.45646
$706$	0	0
$707$	−4.19578	−0.157799
$708$	0	0
$709$	20.5816	0.772959	0.386480	−	0.922298i	$-0.373691\pi$
$709$	20.5816	0.772959	0.386480	+	0.922298i	$0.373691\pi$
$710$	0	0
$711$	−48.5957	−1.82248
$712$	0	0
$713$	26.8558	1.00576
$714$	0	0
$715$	−73.9148	−2.76426
$716$	0	0
$717$	−35.5814	−1.32881
$718$	0	0
$719$	−44.5493	−1.66141	−0.830704	−	0.556714i	$-0.812062\pi$
$719$	−44.5493	−1.66141	−0.830704	+	0.556714i	$0.812062\pi$
$720$	0	0
$721$	0.196655	0.00732381
$722$	0	0
$723$	−11.7007	−0.435153
$724$	0	0
$725$	−41.9414	−1.55766
$726$	0	0
$727$	−20.2366	−0.750535	−0.375267	−	0.926917i	$-0.622449\pi$
$727$	−20.2366	−0.750535	−0.375267	+	0.926917i	$0.622449\pi$
$728$	0	0
$729$	53.8461	1.99430
$730$	0	0
$731$	−12.2883	−0.454500
$732$	0	0
$733$	−47.6432	−1.75974	−0.879871	−	0.475213i	$-0.842371\pi$
$733$	−47.6432	−1.75974	−0.879871	+	0.475213i	$0.842371\pi$
$734$	0	0
$735$	58.8827	2.17192
$736$	0	0
$737$	76.4811	2.81722
$738$	0	0
$739$	−21.0770	−0.775328	−0.387664	−	0.921801i	$-0.626718\pi$
$739$	−21.0770	−0.775328	−0.387664	+	0.921801i	$0.626718\pi$
$740$	0	0
$741$	−39.9689	−1.46829
$742$	0	0
$743$	40.9802	1.50342	0.751708	−	0.659496i	$-0.229230\pi$
$743$	40.9802	1.50342	0.751708	+	0.659496i	$0.229230\pi$
$744$	0	0
$745$	−32.6669	−1.19682
$746$	0	0
$747$	83.9606	3.07196
$748$	0	0
$749$	26.7011	0.975638
$750$	0	0
$751$	−21.2943	−0.777040	−0.388520	−	0.921440i	$-0.627014\pi$
$751$	−21.2943	−0.777040	−0.388520	+	0.921440i	$0.627014\pi$
$752$	0	0
$753$	−3.26165	−0.118861
$754$	0	0
$755$	65.3235	2.37737
$756$	0	0
$757$	−12.5443	−0.455929	−0.227964	−	0.973669i	$-0.573207\pi$
$757$	−12.5443	−0.455929	−0.227964	+	0.973669i	$0.573207\pi$
$758$	0	0
$759$	83.8891	3.04498
$760$	0	0
$761$	15.3106	0.555008	0.277504	−	0.960725i	$-0.410493\pi$
$761$	15.3106	0.555008	0.277504	+	0.960725i	$0.410493\pi$
$762$	0	0
$763$	12.7530	0.461688
$764$	0	0
$765$	147.958	5.34944
$766$	0	0
$767$	18.7716	0.677802
$768$	0	0
$769$	29.6740	1.07007	0.535037	−	0.844829i	$-0.320298\pi$
$769$	29.6740	1.07007	0.535037	+	0.844829i	$0.320298\pi$
$770$	0	0
$771$	−58.4101	−2.10359
$772$	0	0
$773$	−42.3396	−1.52285	−0.761425	−	0.648253i	$-0.775500\pi$
$773$	−42.3396	−1.52285	−0.761425	+	0.648253i	$0.775500\pi$
$774$	0	0
$775$	68.2103	2.45019
$776$	0	0
$777$	0.540076	0.0193751
$778$	0	0
$779$	−38.4264	−1.37677
$780$	0	0
$781$	83.1255	2.97446
$782$	0	0
$783$	−53.6441	−1.91709
$784$	0	0
$785$	−68.5481	−2.44659
$786$	0	0
$787$	10.6407	0.379301	0.189650	−	0.981852i	$-0.439265\pi$
$787$	10.6407	0.379301	0.189650	+	0.981852i	$0.439265\pi$
$788$	0	0
$789$	−17.5877	−0.626138
$790$	0	0
$791$	12.8927	0.458413
$792$	0	0
$793$	−32.3359	−1.14828
$794$	0	0
$795$	−50.4945	−1.79085
$796$	0	0
$797$	18.2419	0.646162	0.323081	−	0.946371i	$-0.395281\pi$
$797$	18.2419	0.646162	0.323081	+	0.946371i	$0.395281\pi$
$798$	0	0
$799$	−32.3879	−1.14580
$800$	0	0
$801$	11.9839	0.423430
$802$	0	0
$803$	−86.4189	−3.04966
$804$	0	0
$805$	30.8005	1.08558
$806$	0	0
$807$	−67.8375	−2.38799
$808$	0	0
$809$	39.8695	1.40174	0.700868	−	0.713291i	$-0.252796\pi$
$809$	39.8695	1.40174	0.700868	+	0.713291i	$0.252796\pi$
$810$	0	0
$811$	−50.7282	−1.78131	−0.890654	−	0.454682i	$-0.849753\pi$
$811$	−50.7282	−1.78131	−0.890654	+	0.454682i	$0.849753\pi$
$812$	0	0
$813$	77.8579	2.73059
$814$	0	0
$815$	30.1066	1.05459
$816$	0	0
$817$	−9.77469	−0.341973
$818$	0	0
$819$	−40.1773	−1.40391
$820$	0	0
$821$	−7.53704	−0.263045	−0.131522	−	0.991313i	$-0.541986\pi$
$821$	−7.53704	−0.263045	−0.131522	+	0.991313i	$0.541986\pi$
$822$	0	0
$823$	23.9516	0.834901	0.417450	−	0.908700i	$-0.362924\pi$
$823$	23.9516	0.834901	0.417450	+	0.908700i	$0.362924\pi$
$824$	0	0
$825$	213.067	7.41805
$826$	0	0
$827$	34.1475	1.18743	0.593713	−	0.804677i	$-0.297661\pi$
$827$	34.1475	1.18743	0.593713	+	0.804677i	$0.297661\pi$
$828$	0	0
$829$	53.3738	1.85375	0.926874	−	0.375373i	$-0.122485\pi$
$829$	53.3738	1.85375	0.926874	+	0.375373i	$0.122485\pi$
$830$	0	0
$831$	−28.1346	−0.975977
$832$	0	0
$833$	20.7800	0.719984
$834$	0	0
$835$	41.4322	1.43382
$836$	0	0
$837$	87.2427	3.01555
$838$	0	0
$839$	−0.308967	−0.0106667	−0.00533337	−	0.999986i	$-0.501698\pi$
$839$	−0.308967	−0.0106667	−0.00533337	+	0.999986i	$0.501698\pi$
$840$	0	0
$841$	−16.4271	−0.566452
$842$	0	0
$843$	11.9524	0.411662
$844$	0	0
$845$	9.66394	0.332450
$846$	0	0
$847$	−31.4389	−1.08025
$848$	0	0
$849$	−24.4990	−0.840802
$850$	0	0
$851$	−0.478310	−0.0163962
$852$	0	0
$853$	39.2324	1.34329	0.671646	−	0.740872i	$-0.265588\pi$
$853$	39.2324	1.34329	0.671646	+	0.740872i	$0.265588\pi$
$854$	0	0
$855$	117.693	4.02501
$856$	0	0
$857$	−5.22022	−0.178319	−0.0891597	−	0.996017i	$-0.528418\pi$
$857$	−5.22022	−0.178319	−0.0891597	+	0.996017i	$0.528418\pi$
$858$	0	0
$859$	41.5588	1.41797	0.708983	−	0.705225i	$-0.249154\pi$
$859$	41.5588	1.41797	0.708983	+	0.705225i	$0.249154\pi$
$860$	0	0
$861$	−53.7975	−1.83342
$862$	0	0
$863$	14.0049	0.476732	0.238366	−	0.971175i	$-0.423388\pi$
$863$	14.0049	0.476732	0.238366	+	0.971175i	$0.423388\pi$
$864$	0	0
$865$	−6.96508	−0.236820
$866$	0	0
$867$	17.2747	0.586680
$868$	0	0
$869$	−35.1358	−1.19190
$870$	0	0
$871$	45.1812	1.53091
$872$	0	0
$873$	−47.0879	−1.59369
$874$	0	0
$875$	45.1608	1.52671
$876$	0	0
$877$	−3.62374	−0.122365	−0.0611825	−	0.998127i	$-0.519487\pi$
$877$	−3.62374	−0.122365	−0.0611825	+	0.998127i	$0.519487\pi$
$878$	0	0
$879$	−49.4468	−1.66780
$880$	0	0
$881$	1.57415	0.0530344	0.0265172	−	0.999648i	$-0.491558\pi$
$881$	1.57415	0.0530344	0.0265172	+	0.999648i	$0.491558\pi$
$882$	0	0
$883$	−32.9337	−1.10831	−0.554154	−	0.832414i	$-0.686958\pi$
$883$	−32.9337	−1.10831	−0.554154	+	0.832414i	$0.686958\pi$
$884$	0	0
$885$	−76.9843	−2.58780
$886$	0	0
$887$	−7.15176	−0.240132	−0.120066	−	0.992766i	$-0.538311\pi$
$887$	−7.15176	−0.240132	−0.120066	+	0.992766i	$0.538311\pi$
$888$	0	0
$889$	6.25839	0.209900
$890$	0	0
$891$	145.964	4.88999
$892$	0	0
$893$	−25.7629	−0.862121
$894$	0	0
$895$	−63.9530	−2.13771
$896$	0	0
$897$	49.5575	1.65468
$898$	0	0
$899$	−20.4476	−0.681965
$900$	0	0
$901$	−17.8197	−0.593661
$902$	0	0
$903$	−13.6847	−0.455399
$904$	0	0
$905$	−6.81907	−0.226674
$906$	0	0
$907$	11.8811	0.394505	0.197252	−	0.980353i	$-0.436798\pi$
$907$	11.8811	0.394505	0.197252	+	0.980353i	$0.436798\pi$
$908$	0	0
$909$	19.8789	0.659341
$910$	0	0
$911$	−32.0018	−1.06027	−0.530134	−	0.847914i	$-0.677858\pi$
$911$	−32.0018	−1.06027	−0.530134	+	0.847914i	$0.677858\pi$
$912$	0	0
$913$	60.7054	2.00906
$914$	0	0
$915$	132.613	4.38405
$916$	0	0
$917$	19.0102	0.627772
$918$	0	0
$919$	−46.9243	−1.54789	−0.773946	−	0.633252i	$-0.781720\pi$
$919$	−46.9243	−1.54789	−0.773946	+	0.633252i	$0.781720\pi$
$920$	0	0
$921$	−48.2215	−1.58895
$922$	0	0
$923$	49.1064	1.61636
$924$	0	0
$925$	−1.21484	−0.0399438
$926$	0	0
$927$	−0.931718	−0.0306016
$928$	0	0
$929$	37.1512	1.21889	0.609446	−	0.792827i	$-0.291392\pi$
$929$	37.1512	1.21889	0.609446	+	0.792827i	$0.291392\pi$
$930$	0	0
$931$	16.5294	0.541728
$932$	0	0
$933$	−87.9083	−2.87799
$934$	0	0
$935$	106.977	3.49853
$936$	0	0
$937$	−40.6438	−1.32778	−0.663888	−	0.747832i	$-0.731095\pi$
$937$	−40.6438	−1.32778	−0.663888	+	0.747832i	$0.731095\pi$
$938$	0	0
$939$	−84.7915	−2.76707
$940$	0	0
$941$	−16.4220	−0.535341	−0.267670	−	0.963511i	$-0.586254\pi$
$941$	−16.4220	−0.535341	−0.267670	+	0.963511i	$0.586254\pi$
$942$	0	0
$943$	47.6450	1.55153
$944$	0	0
$945$	100.057	3.25486
$946$	0	0
$947$	13.1461	0.427192	0.213596	−	0.976922i	$-0.431482\pi$
$947$	13.1461	0.427192	0.213596	+	0.976922i	$0.431482\pi$
$948$	0	0
$949$	−51.0520	−1.65722
$950$	0	0
$951$	76.5380	2.48192
$952$	0	0
$953$	−4.98372	−0.161439	−0.0807193	−	0.996737i	$-0.525722\pi$
$953$	−4.98372	−0.161439	−0.0807193	+	0.996737i	$0.525722\pi$
$954$	0	0
$955$	48.2138	1.56016
$956$	0	0
$957$	−63.8717	−2.06468
$958$	0	0
$959$	−1.74330	−0.0562941
$960$	0	0
$961$	2.25436	0.0727214
$962$	0	0
$963$	−126.505	−4.07658
$964$	0	0
$965$	−69.1167	−2.22495
$966$	0	0
$967$	35.0081	1.12578	0.562892	−	0.826530i	$-0.309689\pi$
$967$	35.0081	1.12578	0.562892	+	0.826530i	$0.309689\pi$
$968$	0	0
$969$	57.8471	1.85831
$970$	0	0
$971$	−13.4959	−0.433105	−0.216552	−	0.976271i	$-0.569481\pi$
$971$	−13.4959	−0.433105	−0.216552	+	0.976271i	$0.569481\pi$
$972$	0	0
$973$	−12.3068	−0.394539
$974$	0	0
$975$	125.870	4.03105
$976$	0	0
$977$	−5.66777	−0.181328	−0.0906640	−	0.995882i	$-0.528899\pi$
$977$	−5.66777	−0.181328	−0.0906640	+	0.995882i	$0.528899\pi$
$978$	0	0
$979$	8.66462	0.276922
$980$	0	0
$981$	−60.4214	−1.92911
$982$	0	0
$983$	−28.8757	−0.920993	−0.460497	−	0.887661i	$-0.652329\pi$
$983$	−28.8757	−0.920993	−0.460497	+	0.887661i	$0.652329\pi$
$984$	0	0
$985$	72.6477	2.31475
$986$	0	0
$987$	−36.0684	−1.14807
$988$	0	0
$989$	12.1197	0.385383
$990$	0	0
$991$	−15.4635	−0.491214	−0.245607	−	0.969370i	$-0.578987\pi$
$991$	−15.4635	−0.491214	−0.245607	+	0.969370i	$0.578987\pi$
$992$	0	0
$993$	−39.2363	−1.24513
$994$	0	0
$995$	15.8136	0.501326
$996$	0	0
$997$	−27.2654	−0.863503	−0.431751	−	0.901993i	$-0.642104\pi$
$997$	−27.2654	−0.863503	−0.431751	+	0.901993i	$0.642104\pi$
$998$	0	0
$999$	−1.55382	−0.0491605

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2008.2.a.d.1.22	✓	23
4.3	odd	2		4016.2.a.m.1.2		23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.22	✓	23	1.1	even	1	trivial
4016.2.a.m.1.2		23	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:	\( N \)	\(=\)	\( 2008 = 2^{3} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2008.a (trivial)

\(f(q)\)	\(=\)	\(q+3.26165 q^{3} -4.10224 q^{5} -1.61221 q^{7} +7.63839 q^{9} +O(q^{10})\)	\(q+3.26165 q^{3} -4.10224 q^{5} -1.61221 q^{7} +7.63839 q^{9} +5.52272 q^{11} +3.26255 q^{13} -13.3801 q^{15} -4.72190 q^{17} -3.75601 q^{19} -5.25848 q^{21} +4.65709 q^{23} +11.8284 q^{25} +15.1288 q^{27} -3.54583 q^{29} +5.76666 q^{31} +18.0132 q^{33} +6.61368 q^{35} -0.102706 q^{37} +10.6413 q^{39} +10.2306 q^{41} +2.60241 q^{43} -31.3345 q^{45} +6.85910 q^{47} -4.40077 q^{49} -15.4012 q^{51} +3.77385 q^{53} -22.6555 q^{55} -12.2508 q^{57} +5.75365 q^{59} -9.91122 q^{61} -12.3147 q^{63} -13.3838 q^{65} +13.8484 q^{67} +15.1898 q^{69} +15.0515 q^{71} -15.6479 q^{73} +38.5801 q^{75} -8.90380 q^{77} -6.36204 q^{79} +26.4298 q^{81} +10.9919 q^{83} +19.3704 q^{85} -11.5653 q^{87} +1.56890 q^{89} -5.25992 q^{91} +18.8088 q^{93} +15.4081 q^{95} -6.16465 q^{97} +42.1847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9	\( 23 q + 2 q^{3} + 8 q^{5} + 2 q^{7} + 45 q^{9} + 8 q^{11} + 8 q^{13} + 7 q^{15} + 19 q^{17} - 9 q^{19} + 9 q^{21} + 21 q^{23} + 65 q^{25} + 5 q^{27} + 10 q^{29} - 9 q^{31} + 34 q^{33} + 12 q^{35} + 11 q^{37} - 9 q^{39} + 35 q^{41} - 9 q^{43} + 29 q^{45} + 37 q^{47} + 77 q^{49} - 17 q^{51} + 38 q^{53} - 20 q^{55} + 51 q^{57} + 17 q^{59} + 22 q^{63} + 41 q^{65} + 9 q^{67} + 8 q^{69} + 13 q^{71} + 41 q^{73} + 25 q^{75} + 36 q^{77} - 36 q^{79} + 127 q^{81} + 29 q^{83} + 34 q^{85} + 10 q^{87} + 36 q^{89} - 6 q^{91} + 36 q^{93} + 25 q^{95} + 40 q^{97} + 19 q^{99}+O(q^{100}) \) 23 * q + 2 * q^3 + 8 * q^5 + 2 * q^7 + 45 * q^9 + 8 * q^11 + 8 * q^13 + 7 * q^15 + 19 * q^17 - 9 * q^19 + 9 * q^21 + 21 * q^23 + 65 * q^25 + 5 * q^27 + 10 * q^29 - 9 * q^31 + 34 * q^33 + 12 * q^35 + 11 * q^37 - 9 * q^39 + 35 * q^41 - 9 * q^43 + 29 * q^45 + 37 * q^47 + 77 * q^49 - 17 * q^51 + 38 * q^53 - 20 * q^55 + 51 * q^57 + 17 * q^59 + 22 * q^63 + 41 * q^65 + 9 * q^67 + 8 * q^69 + 13 * q^71 + 41 * q^73 + 25 * q^75 + 36 * q^77 - 36 * q^79 + 127 * q^81 + 29 * q^83 + 34 * q^85 + 10 * q^87 + 36 * q^89 - 6 * q^91 + 36 * q^93 + 25 * q^95 + 40 * q^97 + 19 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more