LMFDB - Embedded newform 6008.2.a.b.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, 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("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.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:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.486347 q^{3} +3.55566 q^{5} +0.471175 q^{7} -2.76347 q^{9} +O(q^{10})$	$q-0.486347 q^{3} +3.55566 q^{5} +0.471175 q^{7} -2.76347 q^{9} -0.531450 q^{11} +4.46716 q^{13} -1.72928 q^{15} -4.83915 q^{17} -7.53934 q^{19} -0.229155 q^{21} -6.36896 q^{23} +7.64269 q^{25} +2.80305 q^{27} -0.479671 q^{29} +2.79010 q^{31} +0.258469 q^{33} +1.67534 q^{35} +10.7314 q^{37} -2.17259 q^{39} -0.506440 q^{41} -10.2121 q^{43} -9.82593 q^{45} -0.860204 q^{47} -6.77799 q^{49} +2.35351 q^{51} -6.48986 q^{53} -1.88965 q^{55} +3.66674 q^{57} -3.90844 q^{59} -10.7206 q^{61} -1.30208 q^{63} +15.8837 q^{65} -9.72226 q^{67} +3.09753 q^{69} -7.91477 q^{71} +4.56729 q^{73} -3.71700 q^{75} -0.250406 q^{77} +2.55175 q^{79} +6.92714 q^{81} +15.8184 q^{83} -17.2064 q^{85} +0.233287 q^{87} -10.2860 q^{89} +2.10482 q^{91} -1.35696 q^{93} -26.8073 q^{95} -14.1804 q^{97} +1.46864 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.486347	−0.280793	−0.140396	−	0.990095i	$-0.544838\pi$
$3$	−0.486347	−0.280793	−0.140396	+	0.990095i	$0.544838\pi$
$4$	0	0
$5$	3.55566	1.59014	0.795069	−	0.606519i	$-0.207435\pi$
$5$	3.55566	1.59014	0.795069	+	0.606519i	$0.207435\pi$
$6$	0	0
$7$	0.471175	0.178088	0.0890438	−	0.996028i	$-0.471619\pi$
$7$	0.471175	0.178088	0.0890438	+	0.996028i	$0.471619\pi$
$8$	0	0
$9$	−2.76347	−0.921155
$10$	0	0
$11$	−0.531450	−0.160238	−0.0801191	−	0.996785i	$-0.525530\pi$
$11$	−0.531450	−0.160238	−0.0801191	+	0.996785i	$0.525530\pi$
$12$	0	0
$13$	4.46716	1.23897	0.619484	−	0.785010i	$-0.287342\pi$
$13$	4.46716	1.23897	0.619484	+	0.785010i	$0.287342\pi$
$14$	0	0
$15$	−1.72928	−0.446499
$16$	0	0
$17$	−4.83915	−1.17367	−0.586834	−	0.809707i	$-0.699626\pi$
$17$	−4.83915	−1.17367	−0.586834	+	0.809707i	$0.699626\pi$
$18$	0	0
$19$	−7.53934	−1.72964	−0.864822	−	0.502079i	$-0.832569\pi$
$19$	−7.53934	−1.72964	−0.864822	+	0.502079i	$0.832569\pi$
$20$	0	0
$21$	−0.229155	−0.0500057
$22$	0	0
$23$	−6.36896	−1.32802	−0.664010	−	0.747724i	$-0.731147\pi$
$23$	−6.36896	−1.32802	−0.664010	+	0.747724i	$0.731147\pi$
$24$	0	0
$25$	7.64269	1.52854
$26$	0	0
$27$	2.80305	0.539447
$28$	0	0
$29$	−0.479671	−0.0890726	−0.0445363	−	0.999008i	$-0.514181\pi$
$29$	−0.479671	−0.0890726	−0.0445363	+	0.999008i	$0.514181\pi$
$30$	0	0
$31$	2.79010	0.501117	0.250558	−	0.968102i	$-0.419386\pi$
$31$	2.79010	0.501117	0.250558	+	0.968102i	$0.419386\pi$
$32$	0	0
$33$	0.258469	0.0449938
$34$	0	0
$35$	1.67534	0.283184
$36$	0	0
$37$	10.7314	1.76422	0.882112	−	0.471040i	$-0.156121\pi$
$37$	10.7314	1.76422	0.882112	+	0.471040i	$0.156121\pi$
$38$	0	0
$39$	−2.17259	−0.347893
$40$	0	0
$41$	−0.506440	−0.0790926	−0.0395463	−	0.999218i	$-0.512591\pi$
$41$	−0.506440	−0.0790926	−0.0395463	+	0.999218i	$0.512591\pi$
$42$	0	0
$43$	−10.2121	−1.55734	−0.778668	−	0.627436i	$-0.784104\pi$
$43$	−10.2121	−1.55734	−0.778668	+	0.627436i	$0.784104\pi$
$44$	0	0
$45$	−9.82593	−1.46476
$46$	0	0
$47$	−0.860204	−0.125474	−0.0627368	−	0.998030i	$-0.519983\pi$
$47$	−0.860204	−0.125474	−0.0627368	+	0.998030i	$0.519983\pi$
$48$	0	0
$49$	−6.77799	−0.968285
$50$	0	0
$51$	2.35351	0.329557
$52$	0	0
$53$	−6.48986	−0.891451	−0.445726	−	0.895170i	$-0.647054\pi$
$53$	−6.48986	−0.891451	−0.445726	+	0.895170i	$0.647054\pi$
$54$	0	0
$55$	−1.88965	−0.254801
$56$	0	0
$57$	3.66674	0.485671
$58$	0	0
$59$	−3.90844	−0.508835	−0.254418	−	0.967094i	$-0.581884\pi$
$59$	−3.90844	−0.508835	−0.254418	+	0.967094i	$0.581884\pi$
$60$	0	0
$61$	−10.7206	−1.37263	−0.686316	−	0.727303i	$-0.740773\pi$
$61$	−10.7206	−1.37263	−0.686316	+	0.727303i	$0.740773\pi$
$62$	0	0
$63$	−1.30208	−0.164046
$64$	0	0
$65$	15.8837	1.97013
$66$	0	0
$67$	−9.72226	−1.18776	−0.593881	−	0.804553i	$-0.702405\pi$
$67$	−9.72226	−1.18776	−0.593881	+	0.804553i	$0.702405\pi$
$68$	0	0
$69$	3.09753	0.372898
$70$	0	0
$71$	−7.91477	−0.939311	−0.469655	−	0.882850i	$-0.655622\pi$
$71$	−7.91477	−0.939311	−0.469655	+	0.882850i	$0.655622\pi$
$72$	0	0
$73$	4.56729	0.534561	0.267280	−	0.963619i	$-0.413875\pi$
$73$	4.56729	0.534561	0.267280	+	0.963619i	$0.413875\pi$
$74$	0	0
$75$	−3.71700	−0.429202
$76$	0	0
$77$	−0.250406	−0.0285364
$78$	0	0
$79$	2.55175	0.287094	0.143547	−	0.989644i	$-0.454149\pi$
$79$	2.55175	0.287094	0.143547	+	0.989644i	$0.454149\pi$
$80$	0	0
$81$	6.92714	0.769683
$82$	0	0
$83$	15.8184	1.73629	0.868145	−	0.496310i	$-0.165312\pi$
$83$	15.8184	1.73629	0.868145	+	0.496310i	$0.165312\pi$
$84$	0	0
$85$	−17.2064	−1.86629
$86$	0	0
$87$	0.233287	0.0250110
$88$	0	0
$89$	−10.2860	−1.09031	−0.545154	−	0.838336i	$-0.683529\pi$
$89$	−10.2860	−1.09031	−0.545154	+	0.838336i	$0.683529\pi$
$90$	0	0
$91$	2.10482	0.220645
$92$	0	0
$93$	−1.35696	−0.140710
$94$	0	0
$95$	−26.8073	−2.75037
$96$	0	0
$97$	−14.1804	−1.43981	−0.719903	−	0.694075i	$-0.755814\pi$
$97$	−14.1804	−1.43981	−0.719903	+	0.694075i	$0.755814\pi$
$98$	0	0
$99$	1.46864	0.147604
$100$	0	0
$101$	8.92361	0.887932	0.443966	−	0.896044i	$-0.353571\pi$
$101$	8.92361	0.887932	0.443966	+	0.896044i	$0.353571\pi$
$102$	0	0
$103$	−4.44511	−0.437989	−0.218995	−	0.975726i	$-0.570278\pi$
$103$	−4.44511	−0.437989	−0.218995	+	0.975726i	$0.570278\pi$
$104$	0	0
$105$	−0.814796	−0.0795160
$106$	0	0
$107$	−1.57696	−0.152450	−0.0762251	−	0.997091i	$-0.524287\pi$
$107$	−1.57696	−0.152450	−0.0762251	+	0.997091i	$0.524287\pi$
$108$	0	0
$109$	−14.6773	−1.40583	−0.702913	−	0.711276i	$-0.748118\pi$
$109$	−14.6773	−1.40583	−0.702913	+	0.711276i	$0.748118\pi$
$110$	0	0
$111$	−5.21916	−0.495381
$112$	0	0
$113$	20.4523	1.92399	0.961995	−	0.273068i	$-0.0880385\pi$
$113$	20.4523	1.92399	0.961995	+	0.273068i	$0.0880385\pi$
$114$	0	0
$115$	−22.6458	−2.11173
$116$	0	0
$117$	−12.3448	−1.14128
$118$	0	0
$119$	−2.28009	−0.209016
$120$	0	0
$121$	−10.7176	−0.974324
$122$	0	0
$123$	0.246306	0.0222086
$124$	0	0
$125$	9.39648	0.840447
$126$	0	0
$127$	8.11004	0.719649	0.359825	−	0.933020i	$-0.382837\pi$
$127$	8.11004	0.719649	0.359825	+	0.933020i	$0.382837\pi$
$128$	0	0
$129$	4.96664	0.437289
$130$	0	0
$131$	7.89824	0.690072	0.345036	−	0.938589i	$-0.387867\pi$
$131$	7.89824	0.690072	0.345036	+	0.938589i	$0.387867\pi$
$132$	0	0
$133$	−3.55235	−0.308028
$134$	0	0
$135$	9.96667	0.857794
$136$	0	0
$137$	6.31103	0.539188	0.269594	−	0.962974i	$-0.413111\pi$
$137$	6.31103	0.539188	0.269594	+	0.962974i	$0.413111\pi$
$138$	0	0
$139$	1.58288	0.134258	0.0671290	−	0.997744i	$-0.478616\pi$
$139$	1.58288	0.134258	0.0671290	+	0.997744i	$0.478616\pi$
$140$	0	0
$141$	0.418358	0.0352321
$142$	0	0
$143$	−2.37407	−0.198530
$144$	0	0
$145$	−1.70554	−0.141638
$146$	0	0
$147$	3.29646	0.271887
$148$	0	0
$149$	−18.1857	−1.48983	−0.744915	−	0.667159i	$-0.767510\pi$
$149$	−18.1857	−1.48983	−0.744915	+	0.667159i	$0.767510\pi$
$150$	0	0
$151$	−6.36421	−0.517912	−0.258956	−	0.965889i	$-0.583378\pi$
$151$	−6.36421	−0.517912	−0.258956	+	0.965889i	$0.583378\pi$
$152$	0	0
$153$	13.3728	1.08113
$154$	0	0
$155$	9.92063	0.796844
$156$	0	0
$157$	−7.76662	−0.619844	−0.309922	−	0.950762i	$-0.600303\pi$
$157$	−7.76662	−0.619844	−0.309922	+	0.950762i	$0.600303\pi$
$158$	0	0
$159$	3.15633	0.250313
$160$	0	0
$161$	−3.00090	−0.236504
$162$	0	0
$163$	−15.3220	−1.20011	−0.600056	−	0.799958i	$-0.704855\pi$
$163$	−15.3220	−1.20011	−0.600056	+	0.799958i	$0.704855\pi$
$164$	0	0
$165$	0.919028	0.0715463
$166$	0	0
$167$	−20.3910	−1.57790	−0.788950	−	0.614457i	$-0.789375\pi$
$167$	−20.3910	−1.57790	−0.788950	+	0.614457i	$0.789375\pi$
$168$	0	0
$169$	6.95552	0.535040
$170$	0	0
$171$	20.8347	1.59327
$172$	0	0
$173$	16.2476	1.23528	0.617640	−	0.786461i	$-0.288089\pi$
$173$	16.2476	1.23528	0.617640	+	0.786461i	$0.288089\pi$
$174$	0	0
$175$	3.60105	0.272213
$176$	0	0
$177$	1.90086	0.142877
$178$	0	0
$179$	12.1874	0.910930	0.455465	−	0.890254i	$-0.349473\pi$
$179$	12.1874	0.910930	0.455465	+	0.890254i	$0.349473\pi$
$180$	0	0
$181$	21.2565	1.57999	0.789993	−	0.613116i	$-0.210084\pi$
$181$	21.2565	1.57999	0.789993	+	0.613116i	$0.210084\pi$
$182$	0	0
$183$	5.21394	0.385425
$184$	0	0
$185$	38.1570	2.80536
$186$	0	0
$187$	2.57177	0.188066
$188$	0	0
$189$	1.32073	0.0960687
$190$	0	0
$191$	4.73713	0.342766	0.171383	−	0.985204i	$-0.445176\pi$
$191$	4.73713	0.342766	0.171383	+	0.985204i	$0.445176\pi$
$192$	0	0
$193$	14.7747	1.06351	0.531754	−	0.846899i	$-0.321533\pi$
$193$	14.7747	1.06351	0.531754	+	0.846899i	$0.321533\pi$
$194$	0	0
$195$	−7.72499	−0.553198
$196$	0	0
$197$	−9.59995	−0.683968	−0.341984	−	0.939706i	$-0.611099\pi$
$197$	−9.59995	−0.683968	−0.341984	+	0.939706i	$0.611099\pi$
$198$	0	0
$199$	27.3050	1.93560	0.967800	−	0.251719i	$-0.0809958\pi$
$199$	27.3050	1.93560	0.967800	+	0.251719i	$0.0809958\pi$
$200$	0	0
$201$	4.72839	0.333515
$202$	0	0
$203$	−0.226009	−0.0158627
$204$	0	0
$205$	−1.80072	−0.125768
$206$	0	0
$207$	17.6004	1.22331
$208$	0	0
$209$	4.00678	0.277155
$210$	0	0
$211$	−18.6715	−1.28540	−0.642698	−	0.766120i	$-0.722185\pi$
$211$	−18.6715	−1.28540	−0.642698	+	0.766120i	$0.722185\pi$
$212$	0	0
$213$	3.84933	0.263752
$214$	0	0
$215$	−36.3108	−2.47638
$216$	0	0
$217$	1.31463	0.0892426
$218$	0	0
$219$	−2.22129	−0.150101
$220$	0	0
$221$	−21.6173	−1.45414
$222$	0	0
$223$	20.7481	1.38940	0.694698	−	0.719302i	$-0.255538\pi$
$223$	20.7481	1.38940	0.694698	+	0.719302i	$0.255538\pi$
$224$	0	0
$225$	−21.1203	−1.40802
$226$	0	0
$227$	−28.9848	−1.92379	−0.961894	−	0.273422i	$-0.911845\pi$
$227$	−28.9848	−1.92379	−0.961894	+	0.273422i	$0.911845\pi$
$228$	0	0
$229$	−22.2012	−1.46710	−0.733549	−	0.679636i	$-0.762138\pi$
$229$	−22.2012	−1.46710	−0.733549	+	0.679636i	$0.762138\pi$
$230$	0	0
$231$	0.121784	0.00801283
$232$	0	0
$233$	22.9873	1.50595	0.752973	−	0.658051i	$-0.228619\pi$
$233$	22.9873	1.50595	0.752973	+	0.658051i	$0.228619\pi$
$234$	0	0
$235$	−3.05859	−0.199520
$236$	0	0
$237$	−1.24103	−0.0806139
$238$	0	0
$239$	−19.3670	−1.25275	−0.626374	−	0.779522i	$-0.715462\pi$
$239$	−19.3670	−1.25275	−0.626374	+	0.779522i	$0.715462\pi$
$240$	0	0
$241$	10.7978	0.695548	0.347774	−	0.937578i	$-0.386938\pi$
$241$	10.7978	0.695548	0.347774	+	0.937578i	$0.386938\pi$
$242$	0	0
$243$	−11.7781	−0.755568
$244$	0	0
$245$	−24.1002	−1.53971
$246$	0	0
$247$	−33.6794	−2.14297
$248$	0	0
$249$	−7.69322	−0.487538
$250$	0	0
$251$	−24.4356	−1.54236	−0.771180	−	0.636618i	$-0.780333\pi$
$251$	−24.4356	−1.54236	−0.771180	+	0.636618i	$0.780333\pi$
$252$	0	0
$253$	3.38478	0.212799
$254$	0	0
$255$	8.36827	0.524042
$256$	0	0
$257$	28.3410	1.76786	0.883932	−	0.467616i	$-0.154887\pi$
$257$	28.3410	1.76786	0.883932	+	0.467616i	$0.154887\pi$
$258$	0	0
$259$	5.05635	0.314186
$260$	0	0
$261$	1.32555	0.0820497
$262$	0	0
$263$	10.2206	0.630228	0.315114	−	0.949054i	$-0.397957\pi$
$263$	10.2206	0.630228	0.315114	+	0.949054i	$0.397957\pi$
$264$	0	0
$265$	−23.0757	−1.41753
$266$	0	0
$267$	5.00255	0.306151
$268$	0	0
$269$	14.5474	0.886970	0.443485	−	0.896282i	$-0.353742\pi$
$269$	14.5474	0.886970	0.443485	+	0.896282i	$0.353742\pi$
$270$	0	0
$271$	−3.44221	−0.209099	−0.104550	−	0.994520i	$-0.533340\pi$
$271$	−3.44221	−0.209099	−0.104550	+	0.994520i	$0.533340\pi$
$272$	0	0
$273$	−1.02367	−0.0619554
$274$	0	0
$275$	−4.06171	−0.244930
$276$	0	0
$277$	1.89766	0.114019	0.0570096	−	0.998374i	$-0.481843\pi$
$277$	1.89766	0.114019	0.0570096	+	0.998374i	$0.481843\pi$
$278$	0	0
$279$	−7.71034	−0.461606
$280$	0	0
$281$	19.5571	1.16668	0.583340	−	0.812228i	$-0.301746\pi$
$281$	19.5571	1.16668	0.583340	+	0.812228i	$0.301746\pi$
$282$	0	0
$283$	10.3015	0.612361	0.306181	−	0.951973i	$-0.400949\pi$
$283$	10.3015	0.612361	0.306181	+	0.951973i	$0.400949\pi$
$284$	0	0
$285$	13.0377	0.772284
$286$	0	0
$287$	−0.238622	−0.0140854
$288$	0	0
$289$	6.41742	0.377495
$290$	0	0
$291$	6.89662	0.404287
$292$	0	0
$293$	6.85071	0.400223	0.200111	−	0.979773i	$-0.435870\pi$
$293$	6.85071	0.400223	0.200111	+	0.979773i	$0.435870\pi$
$294$	0	0
$295$	−13.8971	−0.809118
$296$	0	0
$297$	−1.48968	−0.0864400
$298$	0	0
$299$	−28.4511	−1.64537
$300$	0	0
$301$	−4.81171	−0.277342
$302$	0	0
$303$	−4.33997	−0.249325
$304$	0	0
$305$	−38.1188	−2.18267
$306$	0	0
$307$	−3.62604	−0.206949	−0.103474	−	0.994632i	$-0.532996\pi$
$307$	−3.62604	−0.206949	−0.103474	+	0.994632i	$0.532996\pi$
$308$	0	0
$309$	2.16187	0.122984
$310$	0	0
$311$	−30.9186	−1.75323	−0.876616	−	0.481190i	$-0.840205\pi$
$311$	−30.9186	−1.75323	−0.876616	+	0.481190i	$0.840205\pi$
$312$	0	0
$313$	−12.7910	−0.722988	−0.361494	−	0.932374i	$-0.617733\pi$
$313$	−12.7910	−0.722988	−0.361494	+	0.932374i	$0.617733\pi$
$314$	0	0
$315$	−4.62974	−0.260856
$316$	0	0
$317$	−1.16410	−0.0653821	−0.0326911	−	0.999466i	$-0.510408\pi$
$317$	−1.16410	−0.0653821	−0.0326911	+	0.999466i	$0.510408\pi$
$318$	0	0
$319$	0.254921	0.0142728
$320$	0	0
$321$	0.766949	0.0428069
$322$	0	0
$323$	36.4840	2.03003
$324$	0	0
$325$	34.1411	1.89381
$326$	0	0
$327$	7.13824	0.394746
$328$	0	0
$329$	−0.405307	−0.0223453
$330$	0	0
$331$	−9.17768	−0.504451	−0.252225	−	0.967668i	$-0.581162\pi$
$331$	−9.17768	−0.504451	−0.252225	+	0.967668i	$0.581162\pi$
$332$	0	0
$333$	−29.6557	−1.62512
$334$	0	0
$335$	−34.5690	−1.88871
$336$	0	0
$337$	15.6228	0.851026	0.425513	−	0.904952i	$-0.360094\pi$
$337$	15.6228	0.851026	0.425513	+	0.904952i	$0.360094\pi$
$338$	0	0
$339$	−9.94692	−0.540242
$340$	0	0
$341$	−1.48280	−0.0802981
$342$	0	0
$343$	−6.49185	−0.350527
$344$	0	0
$345$	11.0137	0.592959
$346$	0	0
$347$	−4.28442	−0.230000	−0.115000	−	0.993366i	$-0.536687\pi$
$347$	−4.28442	−0.230000	−0.115000	+	0.993366i	$0.536687\pi$
$348$	0	0
$349$	−10.2557	−0.548976	−0.274488	−	0.961591i	$-0.588508\pi$
$349$	−10.2557	−0.548976	−0.274488	+	0.961591i	$0.588508\pi$
$350$	0	0
$351$	12.5217	0.668357
$352$	0	0
$353$	−26.0089	−1.38432	−0.692158	−	0.721746i	$-0.743340\pi$
$353$	−26.0089	−1.38432	−0.692158	+	0.721746i	$0.743340\pi$
$354$	0	0
$355$	−28.1422	−1.49363
$356$	0	0
$357$	1.10892	0.0586901
$358$	0	0
$359$	6.93158	0.365835	0.182917	−	0.983128i	$-0.441446\pi$
$359$	6.93158	0.365835	0.182917	+	0.983128i	$0.441446\pi$
$360$	0	0
$361$	37.8416	1.99167
$362$	0	0
$363$	5.21246	0.273583
$364$	0	0
$365$	16.2397	0.850025
$366$	0	0
$367$	5.61288	0.292990	0.146495	−	0.989211i	$-0.453201\pi$
$367$	5.61288	0.292990	0.146495	+	0.989211i	$0.453201\pi$
$368$	0	0
$369$	1.39953	0.0728566
$370$	0	0
$371$	−3.05786	−0.158756
$372$	0	0
$373$	18.9073	0.978981	0.489490	−	0.872009i	$-0.337183\pi$
$373$	18.9073	0.978981	0.489490	+	0.872009i	$0.337183\pi$
$374$	0	0
$375$	−4.56995	−0.235991
$376$	0	0
$377$	−2.14277	−0.110358
$378$	0	0
$379$	24.0839	1.23711	0.618555	−	0.785742i	$-0.287719\pi$
$379$	24.0839	1.23711	0.618555	+	0.785742i	$0.287719\pi$
$380$	0	0
$381$	−3.94429	−0.202072
$382$	0	0
$383$	−23.2970	−1.19042	−0.595211	−	0.803569i	$-0.702932\pi$
$383$	−23.2970	−1.19042	−0.595211	+	0.803569i	$0.702932\pi$
$384$	0	0
$385$	−0.890359	−0.0453769
$386$	0	0
$387$	28.2209	1.43455
$388$	0	0
$389$	4.68326	0.237451	0.118725	−	0.992927i	$-0.462119\pi$
$389$	4.68326	0.237451	0.118725	+	0.992927i	$0.462119\pi$
$390$	0	0
$391$	30.8204	1.55865
$392$	0	0
$393$	−3.84129	−0.193767
$394$	0	0
$395$	9.07313	0.456519
$396$	0	0
$397$	−30.5325	−1.53238	−0.766192	−	0.642612i	$-0.777851\pi$
$397$	−30.5325	−1.53238	−0.766192	+	0.642612i	$0.777851\pi$
$398$	0	0
$399$	1.72768	0.0864920
$400$	0	0
$401$	−8.38688	−0.418821	−0.209410	−	0.977828i	$-0.567154\pi$
$401$	−8.38688	−0.418821	−0.209410	+	0.977828i	$0.567154\pi$
$402$	0	0
$403$	12.4638	0.620867
$404$	0	0
$405$	24.6305	1.22390
$406$	0	0
$407$	−5.70318	−0.282696
$408$	0	0
$409$	−34.5383	−1.70781	−0.853905	−	0.520430i	$-0.825772\pi$
$409$	−34.5383	−1.70781	−0.853905	+	0.520430i	$0.825772\pi$
$410$	0	0
$411$	−3.06935	−0.151400
$412$	0	0
$413$	−1.84156	−0.0906172
$414$	0	0
$415$	56.2446	2.76094
$416$	0	0
$417$	−0.769829	−0.0376987
$418$	0	0
$419$	19.3497	0.945295	0.472647	−	0.881252i	$-0.343298\pi$
$419$	19.3497	0.945295	0.472647	+	0.881252i	$0.343298\pi$
$420$	0	0
$421$	−27.7411	−1.35202	−0.676009	−	0.736893i	$-0.736292\pi$
$421$	−27.7411	−1.35202	−0.676009	+	0.736893i	$0.736292\pi$
$422$	0	0
$423$	2.37714	0.115581
$424$	0	0
$425$	−36.9841	−1.79399
$426$	0	0
$427$	−5.05129	−0.244449
$428$	0	0
$429$	1.15462	0.0557458
$430$	0	0
$431$	18.9533	0.912947	0.456474	−	0.889737i	$-0.349112\pi$
$431$	18.9533	0.912947	0.456474	+	0.889737i	$0.349112\pi$
$432$	0	0
$433$	11.9612	0.574821	0.287410	−	0.957808i	$-0.407206\pi$
$433$	11.9612	0.574821	0.287410	+	0.957808i	$0.407206\pi$
$434$	0	0
$435$	0.829487	0.0397709
$436$	0	0
$437$	48.0177	2.29700
$438$	0	0
$439$	−41.3205	−1.97212	−0.986060	−	0.166391i	$-0.946789\pi$
$439$	−41.3205	−1.97212	−0.986060	+	0.166391i	$0.946789\pi$
$440$	0	0
$441$	18.7308	0.891941
$442$	0	0
$443$	16.9817	0.806827	0.403413	−	0.915018i	$-0.367824\pi$
$443$	16.9817	0.806827	0.403413	+	0.915018i	$0.367824\pi$
$444$	0	0
$445$	−36.5733	−1.73374
$446$	0	0
$447$	8.84457	0.418334
$448$	0	0
$449$	15.7016	0.741005	0.370502	−	0.928832i	$-0.379186\pi$
$449$	15.7016	0.741005	0.370502	+	0.928832i	$0.379186\pi$
$450$	0	0
$451$	0.269147	0.0126737
$452$	0	0
$453$	3.09522	0.145426
$454$	0	0
$455$	7.48400	0.350855
$456$	0	0
$457$	−1.80809	−0.0845789	−0.0422894	−	0.999105i	$-0.513465\pi$
$457$	−1.80809	−0.0845789	−0.0422894	+	0.999105i	$0.513465\pi$
$458$	0	0
$459$	−13.5644	−0.633131
$460$	0	0
$461$	12.7436	0.593530	0.296765	−	0.954951i	$-0.404092\pi$
$461$	12.7436	0.593530	0.296765	+	0.954951i	$0.404092\pi$
$462$	0	0
$463$	−21.6426	−1.00582	−0.502908	−	0.864340i	$-0.667737\pi$
$463$	−21.6426	−1.00582	−0.502908	+	0.864340i	$0.667737\pi$
$464$	0	0
$465$	−4.82487	−0.223748
$466$	0	0
$467$	−33.0550	−1.52960	−0.764801	−	0.644266i	$-0.777163\pi$
$467$	−33.0550	−1.52960	−0.764801	+	0.644266i	$0.777163\pi$
$468$	0	0
$469$	−4.58089	−0.211526
$470$	0	0
$471$	3.77728	0.174048
$472$	0	0
$473$	5.42724	0.249545
$474$	0	0
$475$	−57.6208	−2.64382
$476$	0	0
$477$	17.9345	0.821165
$478$	0	0
$479$	−4.21102	−0.192406	−0.0962032	−	0.995362i	$-0.530670\pi$
$479$	−4.21102	−0.192406	−0.0962032	+	0.995362i	$0.530670\pi$
$480$	0	0
$481$	47.9387	2.18582
$482$	0	0
$483$	1.45948	0.0664085
$484$	0	0
$485$	−50.4208	−2.28949
$486$	0	0
$487$	−15.7179	−0.712248	−0.356124	−	0.934439i	$-0.615902\pi$
$487$	−15.7179	−0.712248	−0.356124	+	0.934439i	$0.615902\pi$
$488$	0	0
$489$	7.45181	0.336983
$490$	0	0
$491$	12.7693	0.576270	0.288135	−	0.957590i	$-0.406965\pi$
$491$	12.7693	0.576270	0.288135	+	0.957590i	$0.406965\pi$
$492$	0	0
$493$	2.32120	0.104542
$494$	0	0
$495$	5.22199	0.234711
$496$	0	0
$497$	−3.72925	−0.167280
$498$	0	0
$499$	35.9814	1.61075	0.805375	−	0.592766i	$-0.201964\pi$
$499$	35.9814	1.61075	0.805375	+	0.592766i	$0.201964\pi$
$500$	0	0
$501$	9.91709	0.443063
$502$	0	0
$503$	−14.7280	−0.656687	−0.328343	−	0.944558i	$-0.606490\pi$
$503$	−14.7280	−0.656687	−0.328343	+	0.944558i	$0.606490\pi$
$504$	0	0
$505$	31.7293	1.41193
$506$	0	0
$507$	−3.38280	−0.150235
$508$	0	0
$509$	9.10540	0.403590	0.201795	−	0.979428i	$-0.435323\pi$
$509$	9.10540	0.403590	0.201795	+	0.979428i	$0.435323\pi$
$510$	0	0
$511$	2.15199	0.0951986
$512$	0	0
$513$	−21.1331	−0.933050
$514$	0	0
$515$	−15.8053	−0.696463
$516$	0	0
$517$	0.457155	0.0201057
$518$	0	0
$519$	−7.90197	−0.346858
$520$	0	0
$521$	−23.0330	−1.00909	−0.504547	−	0.863384i	$-0.668340\pi$
$521$	−23.0330	−1.00909	−0.504547	+	0.863384i	$0.668340\pi$
$522$	0	0
$523$	4.19303	0.183349	0.0916743	−	0.995789i	$-0.470778\pi$
$523$	4.19303	0.183349	0.0916743	+	0.995789i	$0.470778\pi$
$524$	0	0
$525$	−1.75136	−0.0764356
$526$	0	0
$527$	−13.5017	−0.588144
$528$	0	0
$529$	17.5636	0.763635
$530$	0	0
$531$	10.8008	0.468716
$532$	0	0
$533$	−2.26235	−0.0979931
$534$	0	0
$535$	−5.60712	−0.242417
$536$	0	0
$537$	−5.92731	−0.255782
$538$	0	0
$539$	3.60217	0.155156
$540$	0	0
$541$	−21.4633	−0.922780	−0.461390	−	0.887197i	$-0.652649\pi$
$541$	−21.4633	−0.922780	−0.461390	+	0.887197i	$0.652649\pi$
$542$	0	0
$543$	−10.3381	−0.443649
$544$	0	0
$545$	−52.1873	−2.23546
$546$	0	0
$547$	18.9838	0.811688	0.405844	−	0.913942i	$-0.366978\pi$
$547$	18.9838	0.811688	0.405844	+	0.913942i	$0.366978\pi$
$548$	0	0
$549$	29.6260	1.26441
$550$	0	0
$551$	3.61640	0.154064
$552$	0	0
$553$	1.20232	0.0511278
$554$	0	0
$555$	−18.5576	−0.787724
$556$	0	0
$557$	10.3668	0.439254	0.219627	−	0.975584i	$-0.429516\pi$
$557$	10.3668	0.439254	0.219627	+	0.975584i	$0.429516\pi$
$558$	0	0
$559$	−45.6192	−1.92949
$560$	0	0
$561$	−1.25077	−0.0528077
$562$	0	0
$563$	7.45466	0.314177	0.157088	−	0.987585i	$-0.449789\pi$
$563$	7.45466	0.314177	0.157088	+	0.987585i	$0.449789\pi$
$564$	0	0
$565$	72.7213	3.05941
$566$	0	0
$567$	3.26390	0.137071
$568$	0	0
$569$	−19.0968	−0.800580	−0.400290	−	0.916388i	$-0.631091\pi$
$569$	−19.0968	−0.800580	−0.400290	+	0.916388i	$0.631091\pi$
$570$	0	0
$571$	−11.9090	−0.498377	−0.249189	−	0.968455i	$-0.580164\pi$
$571$	−11.9090	−0.498377	−0.249189	+	0.968455i	$0.580164\pi$
$572$	0	0
$573$	−2.30389	−0.0962464
$574$	0	0
$575$	−48.6759	−2.02993
$576$	0	0
$577$	−1.27577	−0.0531111	−0.0265556	−	0.999647i	$-0.508454\pi$
$577$	−1.27577	−0.0531111	−0.0265556	+	0.999647i	$0.508454\pi$
$578$	0	0
$579$	−7.18565	−0.298626
$580$	0	0
$581$	7.45322	0.309212
$582$	0	0
$583$	3.44904	0.142845
$584$	0	0
$585$	−43.8940	−1.81479
$586$	0	0
$587$	−13.9517	−0.575847	−0.287923	−	0.957653i	$-0.592965\pi$
$587$	−13.9517	−0.575847	−0.287923	+	0.957653i	$0.592965\pi$
$588$	0	0
$589$	−21.0355	−0.866753
$590$	0	0
$591$	4.66891	0.192053
$592$	0	0
$593$	20.1128	0.825932	0.412966	−	0.910746i	$-0.364493\pi$
$593$	20.1128	0.825932	0.412966	+	0.910746i	$0.364493\pi$
$594$	0	0
$595$	−8.10722	−0.332364
$596$	0	0
$597$	−13.2797	−0.543503
$598$	0	0
$599$	−32.0813	−1.31081	−0.655403	−	0.755279i	$-0.727501\pi$
$599$	−32.0813	−1.31081	−0.655403	+	0.755279i	$0.727501\pi$
$600$	0	0
$601$	3.35248	0.136751	0.0683753	−	0.997660i	$-0.478218\pi$
$601$	3.35248	0.136751	0.0683753	+	0.997660i	$0.478218\pi$
$602$	0	0
$603$	26.8671	1.09411
$604$	0	0
$605$	−38.1080	−1.54931
$606$	0	0
$607$	27.5588	1.11858	0.559289	−	0.828973i	$-0.311074\pi$
$607$	27.5588	1.11858	0.559289	+	0.828973i	$0.311074\pi$
$608$	0	0
$609$	0.109919	0.00445414
$610$	0	0
$611$	−3.84267	−0.155458
$612$	0	0
$613$	−22.9924	−0.928655	−0.464328	−	0.885664i	$-0.653704\pi$
$613$	−22.9924	−0.928655	−0.464328	+	0.885664i	$0.653704\pi$
$614$	0	0
$615$	0.875778	0.0353148
$616$	0	0
$617$	−42.9010	−1.72713	−0.863565	−	0.504237i	$-0.831774\pi$
$617$	−42.9010	−1.72713	−0.863565	+	0.504237i	$0.831774\pi$
$618$	0	0
$619$	−4.27994	−0.172025	−0.0860126	−	0.996294i	$-0.527413\pi$
$619$	−4.27994	−0.172025	−0.0860126	+	0.996294i	$0.527413\pi$
$620$	0	0
$621$	−17.8525	−0.716395
$622$	0	0
$623$	−4.84649	−0.194170
$624$	0	0
$625$	−4.80278	−0.192111
$626$	0	0
$627$	−1.94869	−0.0778231
$628$	0	0
$629$	−51.9307	−2.07061
$630$	0	0
$631$	−14.8743	−0.592136	−0.296068	−	0.955167i	$-0.595675\pi$
$631$	−14.8743	−0.592136	−0.296068	+	0.955167i	$0.595675\pi$
$632$	0	0
$633$	9.08081	0.360930
$634$	0	0
$635$	28.8365	1.14434
$636$	0	0
$637$	−30.2784	−1.19967
$638$	0	0
$639$	21.8722	0.865251
$640$	0	0
$641$	8.23306	0.325186	0.162593	−	0.986693i	$-0.448014\pi$
$641$	8.23306	0.325186	0.162593	+	0.986693i	$0.448014\pi$
$642$	0	0
$643$	8.38792	0.330787	0.165394	−	0.986228i	$-0.447111\pi$
$643$	8.38792	0.330787	0.165394	+	0.986228i	$0.447111\pi$
$644$	0	0
$645$	17.6597	0.695349
$646$	0	0
$647$	36.6037	1.43904	0.719520	−	0.694472i	$-0.244362\pi$
$647$	36.6037	1.43904	0.719520	+	0.694472i	$0.244362\pi$
$648$	0	0
$649$	2.07714	0.0815349
$650$	0	0
$651$	−0.639365	−0.0250587
$652$	0	0
$653$	−27.4024	−1.07234	−0.536170	−	0.844110i	$-0.680129\pi$
$653$	−27.4024	−1.07234	−0.536170	+	0.844110i	$0.680129\pi$
$654$	0	0
$655$	28.0834	1.09731
$656$	0	0
$657$	−12.6215	−0.492413
$658$	0	0
$659$	23.7235	0.924138	0.462069	−	0.886844i	$-0.347107\pi$
$659$	23.7235	0.924138	0.462069	+	0.886844i	$0.347107\pi$
$660$	0	0
$661$	33.8612	1.31705	0.658524	−	0.752560i	$-0.271181\pi$
$661$	33.8612	1.31705	0.658524	+	0.752560i	$0.271181\pi$
$662$	0	0
$663$	10.5135	0.408311
$664$	0	0
$665$	−12.6309	−0.489807
$666$	0	0
$667$	3.05500	0.118290
$668$	0	0
$669$	−10.0908	−0.390132
$670$	0	0
$671$	5.69747	0.219948
$672$	0	0
$673$	20.5924	0.793780	0.396890	−	0.917866i	$-0.370089\pi$
$673$	20.5924	0.793780	0.396890	+	0.917866i	$0.370089\pi$
$674$	0	0
$675$	21.4228	0.824564
$676$	0	0
$677$	11.0601	0.425076	0.212538	−	0.977153i	$-0.431827\pi$
$677$	11.0601	0.425076	0.212538	+	0.977153i	$0.431827\pi$
$678$	0	0
$679$	−6.68148	−0.256411
$680$	0	0
$681$	14.0967	0.540186
$682$	0	0
$683$	−23.2209	−0.888523	−0.444261	−	0.895897i	$-0.646534\pi$
$683$	−23.2209	−0.888523	−0.444261	+	0.895897i	$0.646534\pi$
$684$	0	0
$685$	22.4399	0.857383
$686$	0	0
$687$	10.7975	0.411951
$688$	0	0
$689$	−28.9913	−1.10448
$690$	0	0
$691$	−0.393183	−0.0149574	−0.00747870	−	0.999972i	$-0.502381\pi$
$691$	−0.393183	−0.0149574	−0.00747870	+	0.999972i	$0.502381\pi$
$692$	0	0
$693$	0.691989	0.0262865
$694$	0	0
$695$	5.62817	0.213489
$696$	0	0
$697$	2.45074	0.0928284
$698$	0	0
$699$	−11.1798	−0.422859
$700$	0	0
$701$	−39.8171	−1.50387	−0.751935	−	0.659237i	$-0.770879\pi$
$701$	−39.8171	−1.50387	−0.751935	+	0.659237i	$0.770879\pi$
$702$	0	0
$703$	−80.9073	−3.05148
$704$	0	0
$705$	1.48754	0.0560238
$706$	0	0
$707$	4.20459	0.158130
$708$	0	0
$709$	−43.4510	−1.63184	−0.815919	−	0.578166i	$-0.803769\pi$
$709$	−43.4510	−1.63184	−0.815919	+	0.578166i	$0.803769\pi$
$710$	0	0
$711$	−7.05166	−0.264458
$712$	0	0
$713$	−17.7700	−0.665492
$714$	0	0
$715$	−8.44139	−0.315690
$716$	0	0
$717$	9.41910	0.351763
$718$	0	0
$719$	37.2523	1.38928	0.694639	−	0.719358i	$-0.255564\pi$
$719$	37.2523	1.38928	0.694639	+	0.719358i	$0.255564\pi$
$720$	0	0
$721$	−2.09443	−0.0780005
$722$	0	0
$723$	−5.25149	−0.195305
$724$	0	0
$725$	−3.66597	−0.136151
$726$	0	0
$727$	−4.33842	−0.160903	−0.0804515	−	0.996759i	$-0.525636\pi$
$727$	−4.33842	−0.160903	−0.0804515	+	0.996759i	$0.525636\pi$
$728$	0	0
$729$	−15.0532	−0.557525
$730$	0	0
$731$	49.4181	1.82779
$732$	0	0
$733$	14.5444	0.537211	0.268605	−	0.963250i	$-0.413437\pi$
$733$	14.5444	0.537211	0.268605	+	0.963250i	$0.413437\pi$
$734$	0	0
$735$	11.7211	0.432338
$736$	0	0
$737$	5.16690	0.190325
$738$	0	0
$739$	−45.4842	−1.67316	−0.836582	−	0.547841i	$-0.815450\pi$
$739$	−45.4842	−1.67316	−0.836582	+	0.547841i	$0.815450\pi$
$740$	0	0
$741$	16.3799	0.601731
$742$	0	0
$743$	24.4865	0.898324	0.449162	−	0.893450i	$-0.351723\pi$
$743$	24.4865	0.898324	0.449162	+	0.893450i	$0.351723\pi$
$744$	0	0
$745$	−64.6621	−2.36903
$746$	0	0
$747$	−43.7135	−1.59939
$748$	0	0
$749$	−0.743024	−0.0271495
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	11.8842	0.433083
$754$	0	0
$755$	−22.6289	−0.823551
$756$	0	0
$757$	−2.30651	−0.0838317	−0.0419158	−	0.999121i	$-0.513346\pi$
$757$	−2.30651	−0.0838317	−0.0419158	+	0.999121i	$0.513346\pi$
$758$	0	0
$759$	−1.64618	−0.0597526
$760$	0	0
$761$	39.3856	1.42773	0.713863	−	0.700285i	$-0.246944\pi$
$761$	39.3856	1.42773	0.713863	+	0.700285i	$0.246944\pi$
$762$	0	0
$763$	−6.91556	−0.250360
$764$	0	0
$765$	47.5492	1.71915
$766$	0	0
$767$	−17.4596	−0.630430
$768$	0	0
$769$	−32.1702	−1.16009	−0.580043	−	0.814586i	$-0.696964\pi$
$769$	−32.1702	−1.16009	−0.580043	+	0.814586i	$0.696964\pi$
$770$	0	0
$771$	−13.7836	−0.496403
$772$	0	0
$773$	−4.07712	−0.146644	−0.0733219	−	0.997308i	$-0.523360\pi$
$773$	−4.07712	−0.146644	−0.0733219	+	0.997308i	$0.523360\pi$
$774$	0	0
$775$	21.3239	0.765975
$776$	0	0
$777$	−2.45914	−0.0882212
$778$	0	0
$779$	3.81822	0.136802
$780$	0	0
$781$	4.20631	0.150514
$782$	0	0
$783$	−1.34454	−0.0480499
$784$	0	0
$785$	−27.6154	−0.985637
$786$	0	0
$787$	50.7653	1.80959	0.904794	−	0.425850i	$-0.140025\pi$
$787$	50.7653	1.80959	0.904794	+	0.425850i	$0.140025\pi$
$788$	0	0
$789$	−4.97075	−0.176963
$790$	0	0
$791$	9.63661	0.342639
$792$	0	0
$793$	−47.8907	−1.70065
$794$	0	0
$795$	11.2228	0.398032
$796$	0	0
$797$	16.1180	0.570931	0.285465	−	0.958389i	$-0.407852\pi$
$797$	16.1180	0.570931	0.285465	+	0.958389i	$0.407852\pi$
$798$	0	0
$799$	4.16266	0.147264
$800$	0	0
$801$	28.4249	1.00434
$802$	0	0
$803$	−2.42729	−0.0856571
$804$	0	0
$805$	−10.6702	−0.376073
$806$	0	0
$807$	−7.07509	−0.249055
$808$	0	0
$809$	−33.7460	−1.18645	−0.593223	−	0.805038i	$-0.702145\pi$
$809$	−33.7460	−1.18645	−0.593223	+	0.805038i	$0.702145\pi$
$810$	0	0
$811$	−47.8021	−1.67856	−0.839279	−	0.543701i	$-0.817023\pi$
$811$	−47.8021	−1.67856	−0.839279	+	0.543701i	$0.817023\pi$
$812$	0	0
$813$	1.67411	0.0587135
$814$	0	0
$815$	−54.4797	−1.90834
$816$	0	0
$817$	76.9927	2.69364
$818$	0	0
$819$	−5.81659	−0.203248
$820$	0	0
$821$	−26.0626	−0.909593	−0.454796	−	0.890595i	$-0.650288\pi$
$821$	−26.0626	−0.909593	−0.454796	+	0.890595i	$0.650288\pi$
$822$	0	0
$823$	−28.5055	−0.993638	−0.496819	−	0.867854i	$-0.665499\pi$
$823$	−28.5055	−0.993638	−0.496819	+	0.867854i	$0.665499\pi$
$824$	0	0
$825$	1.97540	0.0687746
$826$	0	0
$827$	22.7240	0.790191	0.395096	−	0.918640i	$-0.370711\pi$
$827$	22.7240	0.790191	0.395096	+	0.918640i	$0.370711\pi$
$828$	0	0
$829$	23.6283	0.820643	0.410322	−	0.911941i	$-0.365416\pi$
$829$	23.6283	0.820643	0.410322	+	0.911941i	$0.365416\pi$
$830$	0	0
$831$	−0.922921	−0.0320158
$832$	0	0
$833$	32.7998	1.13644
$834$	0	0
$835$	−72.5033	−2.50908
$836$	0	0
$837$	7.82078	0.270326
$838$	0	0
$839$	53.4577	1.84557	0.922783	−	0.385320i	$-0.125909\pi$
$839$	53.4577	1.84557	0.922783	+	0.385320i	$0.125909\pi$
$840$	0	0
$841$	−28.7699	−0.992066
$842$	0	0
$843$	−9.51155	−0.327595
$844$	0	0
$845$	24.7314	0.850787
$846$	0	0
$847$	−5.04985	−0.173515
$848$	0	0
$849$	−5.01011	−0.171947
$850$	0	0
$851$	−68.3475	−2.34292
$852$	0	0
$853$	−6.44369	−0.220628	−0.110314	−	0.993897i	$-0.535186\pi$
$853$	−6.44369	−0.220628	−0.110314	+	0.993897i	$0.535186\pi$
$854$	0	0
$855$	74.0811	2.53352
$856$	0	0
$857$	23.0826	0.788485	0.394243	−	0.919006i	$-0.371007\pi$
$857$	23.0826	0.788485	0.394243	+	0.919006i	$0.371007\pi$
$858$	0	0
$859$	18.3323	0.625491	0.312745	−	0.949837i	$-0.398751\pi$
$859$	18.3323	0.625491	0.312745	+	0.949837i	$0.398751\pi$
$860$	0	0
$861$	0.116053	0.00395508
$862$	0	0
$863$	19.4092	0.660695	0.330348	−	0.943859i	$-0.392834\pi$
$863$	19.4092	0.660695	0.330348	+	0.943859i	$0.392834\pi$
$864$	0	0
$865$	57.7708	1.96427
$866$	0	0
$867$	−3.12110	−0.105998
$868$	0	0
$869$	−1.35613	−0.0460034
$870$	0	0
$871$	−43.4309	−1.47160
$872$	0	0
$873$	39.1872	1.32628
$874$	0	0
$875$	4.42739	0.149673
$876$	0	0
$877$	29.4784	0.995414	0.497707	−	0.867345i	$-0.334176\pi$
$877$	29.4784	0.995414	0.497707	+	0.867345i	$0.334176\pi$
$878$	0	0
$879$	−3.33183	−0.112380
$880$	0	0
$881$	38.2282	1.28794	0.643971	−	0.765050i	$-0.277286\pi$
$881$	38.2282	1.28794	0.643971	+	0.765050i	$0.277286\pi$
$882$	0	0
$883$	−16.4754	−0.554441	−0.277221	−	0.960806i	$-0.589413\pi$
$883$	−16.4754	−0.554441	−0.277221	+	0.960806i	$0.589413\pi$
$884$	0	0
$885$	6.75880	0.227194
$886$	0	0
$887$	46.1505	1.54958	0.774790	−	0.632218i	$-0.217855\pi$
$887$	46.1505	1.54958	0.774790	+	0.632218i	$0.217855\pi$
$888$	0	0
$889$	3.82125	0.128161
$890$	0	0
$891$	−3.68143	−0.123333
$892$	0	0
$893$	6.48537	0.217025
$894$	0	0
$895$	43.3342	1.44850
$896$	0	0
$897$	13.8371	0.462009
$898$	0	0
$899$	−1.33833	−0.0446358
$900$	0	0
$901$	31.4054	1.04627
$902$	0	0
$903$	2.34016	0.0778757
$904$	0	0
$905$	75.5809	2.51240
$906$	0	0
$907$	31.2290	1.03694	0.518471	−	0.855095i	$-0.326501\pi$
$907$	31.2290	1.03694	0.518471	+	0.855095i	$0.326501\pi$
$908$	0	0
$909$	−24.6601	−0.817924
$910$	0	0
$911$	18.9325	0.627262	0.313631	−	0.949545i	$-0.398454\pi$
$911$	18.9325	0.627262	0.313631	+	0.949545i	$0.398454\pi$
$912$	0	0
$913$	−8.40667	−0.278220
$914$	0	0
$915$	18.5390	0.612879
$916$	0	0
$917$	3.72146	0.122893
$918$	0	0
$919$	−18.0737	−0.596198	−0.298099	−	0.954535i	$-0.596353\pi$
$919$	−18.0737	−0.596198	−0.298099	+	0.954535i	$0.596353\pi$
$920$	0	0
$921$	1.76351	0.0581098
$922$	0	0
$923$	−35.3566	−1.16378
$924$	0	0
$925$	82.0164	2.69668
$926$	0	0
$927$	12.2839	0.403456
$928$	0	0
$929$	−17.3500	−0.569236	−0.284618	−	0.958641i	$-0.591867\pi$
$929$	−17.3500	−0.569236	−0.284618	+	0.958641i	$0.591867\pi$
$930$	0	0
$931$	51.1016	1.67479
$932$	0	0
$933$	15.0372	0.492295
$934$	0	0
$935$	9.14433	0.299051
$936$	0	0
$937$	7.90880	0.258369	0.129185	−	0.991621i	$-0.458764\pi$
$937$	7.90880	0.258369	0.129185	+	0.991621i	$0.458764\pi$
$938$	0	0
$939$	6.22085	0.203010
$940$	0	0
$941$	34.7045	1.13134	0.565668	−	0.824633i	$-0.308618\pi$
$941$	34.7045	1.13134	0.565668	+	0.824633i	$0.308618\pi$
$942$	0	0
$943$	3.22549	0.105036
$944$	0	0
$945$	4.69605	0.152763
$946$	0	0
$947$	−54.3618	−1.76652	−0.883260	−	0.468884i	$-0.844656\pi$
$947$	−54.3618	−1.76652	−0.883260	+	0.468884i	$0.844656\pi$
$948$	0	0
$949$	20.4028	0.662303
$950$	0	0
$951$	0.566155	0.0183588
$952$	0	0
$953$	9.89694	0.320593	0.160297	−	0.987069i	$-0.448755\pi$
$953$	9.89694	0.320593	0.160297	+	0.987069i	$0.448755\pi$
$954$	0	0
$955$	16.8436	0.545046
$956$	0	0
$957$	−0.123980	−0.00400771
$958$	0	0
$959$	2.97360	0.0960227
$960$	0	0
$961$	−23.2153	−0.748882
$962$	0	0
$963$	4.35787	0.140430
$964$	0	0
$965$	52.5339	1.69113
$966$	0	0
$967$	28.1171	0.904185	0.452092	−	0.891971i	$-0.350678\pi$
$967$	28.1171	0.904185	0.452092	+	0.891971i	$0.350678\pi$
$968$	0	0
$969$	−17.7439	−0.570017
$970$	0	0
$971$	15.7537	0.505560	0.252780	−	0.967524i	$-0.418655\pi$
$971$	15.7537	0.505560	0.252780	+	0.967524i	$0.418655\pi$
$972$	0	0
$973$	0.745813	0.0239097
$974$	0	0
$975$	−16.6044	−0.531768
$976$	0	0
$977$	−49.1758	−1.57327	−0.786636	−	0.617417i	$-0.788179\pi$
$977$	−49.1758	−1.57327	−0.786636	+	0.617417i	$0.788179\pi$
$978$	0	0
$979$	5.46647	0.174709
$980$	0	0
$981$	40.5601	1.29498
$982$	0	0
$983$	−44.6943	−1.42553	−0.712763	−	0.701405i	$-0.752556\pi$
$983$	−44.6943	−1.42553	−0.712763	+	0.701405i	$0.752556\pi$
$984$	0	0
$985$	−34.1341	−1.08760
$986$	0	0
$987$	0.197120	0.00627440
$988$	0	0
$989$	65.0406	2.06817
$990$	0	0
$991$	29.5381	0.938307	0.469154	−	0.883117i	$-0.344559\pi$
$991$	29.5381	0.938307	0.469154	+	0.883117i	$0.344559\pi$
$992$	0	0
$993$	4.46354	0.141646
$994$	0	0
$995$	97.0872	3.07787
$996$	0	0
$997$	37.3118	1.18168	0.590838	−	0.806790i	$-0.298797\pi$
$997$	37.3118	1.18168	0.590838	+	0.806790i	$0.298797\pi$
$998$	0	0
$999$	30.0805	0.951704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.21	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.21	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-0.486347 q^{3} +3.55566 q^{5} +0.471175 q^{7} -2.76347 q^{9} +O(q^{10})\)	\(q-0.486347 q^{3} +3.55566 q^{5} +0.471175 q^{7} -2.76347 q^{9} -0.531450 q^{11} +4.46716 q^{13} -1.72928 q^{15} -4.83915 q^{17} -7.53934 q^{19} -0.229155 q^{21} -6.36896 q^{23} +7.64269 q^{25} +2.80305 q^{27} -0.479671 q^{29} +2.79010 q^{31} +0.258469 q^{33} +1.67534 q^{35} +10.7314 q^{37} -2.17259 q^{39} -0.506440 q^{41} -10.2121 q^{43} -9.82593 q^{45} -0.860204 q^{47} -6.77799 q^{49} +2.35351 q^{51} -6.48986 q^{53} -1.88965 q^{55} +3.66674 q^{57} -3.90844 q^{59} -10.7206 q^{61} -1.30208 q^{63} +15.8837 q^{65} -9.72226 q^{67} +3.09753 q^{69} -7.91477 q^{71} +4.56729 q^{73} -3.71700 q^{75} -0.250406 q^{77} +2.55175 q^{79} +6.92714 q^{81} +15.8184 q^{83} -17.2064 q^{85} +0.233287 q^{87} -10.2860 q^{89} +2.10482 q^{91} -1.35696 q^{93} -26.8073 q^{95} -14.1804 q^{97} +1.46864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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