LMFDB - Embedded newform 6008.2.a.e.1.2

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:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.97826 q^{3} -0.794149 q^{5} -3.63896 q^{7} +5.87005 q^{9} +O(q^{10})$	$q-2.97826 q^{3} -0.794149 q^{5} -3.63896 q^{7} +5.87005 q^{9} -0.674085 q^{11} -1.02843 q^{13} +2.36518 q^{15} +3.65416 q^{17} +3.51928 q^{19} +10.8378 q^{21} +4.04783 q^{23} -4.36933 q^{25} -8.54776 q^{27} +2.95252 q^{29} -3.73972 q^{31} +2.00760 q^{33} +2.88987 q^{35} +3.30857 q^{37} +3.06292 q^{39} -12.0974 q^{41} -1.05637 q^{43} -4.66169 q^{45} -6.46297 q^{47} +6.24201 q^{49} -10.8831 q^{51} +2.27933 q^{53} +0.535324 q^{55} -10.4813 q^{57} -14.8230 q^{59} -6.97730 q^{61} -21.3609 q^{63} +0.816724 q^{65} -12.8224 q^{67} -12.0555 q^{69} -3.34117 q^{71} +13.2862 q^{73} +13.0130 q^{75} +2.45297 q^{77} -15.7620 q^{79} +7.84734 q^{81} +8.63077 q^{83} -2.90195 q^{85} -8.79338 q^{87} +8.61968 q^{89} +3.74240 q^{91} +11.1379 q^{93} -2.79483 q^{95} +13.8912 q^{97} -3.95691 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	$ 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) $ 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * 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
$3$	−2.97826	−1.71950	−0.859750	−	0.510714i	$-0.829381\pi$
$3$	−2.97826	−1.71950	−0.859750	+	0.510714i	$0.829381\pi$
$4$	0	0
$5$	−0.794149	−0.355154	−0.177577	−	0.984107i	$-0.556826\pi$
$5$	−0.794149	−0.355154	−0.177577	+	0.984107i	$0.556826\pi$
$6$	0	0
$7$	−3.63896	−1.37540	−0.687698	−	0.725997i	$-0.741379\pi$
$7$	−3.63896	−1.37540	−0.687698	+	0.725997i	$0.741379\pi$
$8$	0	0
$9$	5.87005	1.95668
$10$	0	0
$11$	−0.674085	−0.203244	−0.101622	−	0.994823i	$-0.532403\pi$
$11$	−0.674085	−0.203244	−0.101622	+	0.994823i	$0.532403\pi$
$12$	0	0
$13$	−1.02843	−0.285234	−0.142617	−	0.989778i	$-0.545552\pi$
$13$	−1.02843	−0.285234	−0.142617	+	0.989778i	$0.545552\pi$
$14$	0	0
$15$	2.36518	0.610688
$16$	0	0
$17$	3.65416	0.886264	0.443132	−	0.896456i	$-0.353867\pi$
$17$	3.65416	0.886264	0.443132	+	0.896456i	$0.353867\pi$
$18$	0	0
$19$	3.51928	0.807378	0.403689	−	0.914896i	$-0.367728\pi$
$19$	3.51928	0.807378	0.403689	+	0.914896i	$0.367728\pi$
$20$	0	0
$21$	10.8378	2.36500
$22$	0	0
$23$	4.04783	0.844031	0.422015	−	0.906589i	$-0.361323\pi$
$23$	4.04783	0.844031	0.422015	+	0.906589i	$0.361323\pi$
$24$	0	0
$25$	−4.36933	−0.873866
$26$	0	0
$27$	−8.54776	−1.64502
$28$	0	0
$29$	2.95252	0.548269	0.274135	−	0.961691i	$-0.411609\pi$
$29$	2.95252	0.548269	0.274135	+	0.961691i	$0.411609\pi$
$30$	0	0
$31$	−3.73972	−0.671673	−0.335837	−	0.941920i	$-0.609019\pi$
$31$	−3.73972	−0.671673	−0.335837	+	0.941920i	$0.609019\pi$
$32$	0	0
$33$	2.00760	0.349479
$34$	0	0
$35$	2.88987	0.488478
$36$	0	0
$37$	3.30857	0.543926	0.271963	−	0.962308i	$-0.412327\pi$
$37$	3.30857	0.543926	0.271963	+	0.962308i	$0.412327\pi$
$38$	0	0
$39$	3.06292	0.490460
$40$	0	0
$41$	−12.0974	−1.88929	−0.944646	−	0.328090i	$-0.893595\pi$
$41$	−12.0974	−1.88929	−0.944646	+	0.328090i	$0.893595\pi$
$42$	0	0
$43$	−1.05637	−0.161095	−0.0805475	−	0.996751i	$-0.525667\pi$
$43$	−1.05637	−0.161095	−0.0805475	+	0.996751i	$0.525667\pi$
$44$	0	0
$45$	−4.66169	−0.694924
$46$	0	0
$47$	−6.46297	−0.942721	−0.471361	−	0.881941i	$-0.656237\pi$
$47$	−6.46297	−0.942721	−0.471361	+	0.881941i	$0.656237\pi$
$48$	0	0
$49$	6.24201	0.891715
$50$	0	0
$51$	−10.8831	−1.52393
$52$	0	0
$53$	2.27933	0.313090	0.156545	−	0.987671i	$-0.449964\pi$
$53$	2.27933	0.313090	0.156545	+	0.987671i	$0.449964\pi$
$54$	0	0
$55$	0.535324	0.0721831
$56$	0	0
$57$	−10.4813	−1.38829
$58$	0	0
$59$	−14.8230	−1.92979	−0.964896	−	0.262633i	$-0.915409\pi$
$59$	−14.8230	−1.92979	−0.964896	+	0.262633i	$0.915409\pi$
$60$	0	0
$61$	−6.97730	−0.893351	−0.446676	−	0.894696i	$-0.647392\pi$
$61$	−6.97730	−0.893351	−0.446676	+	0.894696i	$0.647392\pi$
$62$	0	0
$63$	−21.3609	−2.69121
$64$	0	0
$65$	0.816724	0.101302
$66$	0	0
$67$	−12.8224	−1.56650	−0.783250	−	0.621707i	$-0.786440\pi$
$67$	−12.8224	−1.56650	−0.783250	+	0.621707i	$0.786440\pi$
$68$	0	0
$69$	−12.0555	−1.45131
$70$	0	0
$71$	−3.34117	−0.396523	−0.198262	−	0.980149i	$-0.563530\pi$
$71$	−3.34117	−0.396523	−0.198262	+	0.980149i	$0.563530\pi$
$72$	0	0
$73$	13.2862	1.55503	0.777516	−	0.628863i	$-0.216479\pi$
$73$	13.2862	1.55503	0.777516	+	0.628863i	$0.216479\pi$
$74$	0	0
$75$	13.0130	1.50261
$76$	0	0
$77$	2.45297	0.279541
$78$	0	0
$79$	−15.7620	−1.77336	−0.886679	−	0.462385i	$-0.846994\pi$
$79$	−15.7620	−1.77336	−0.886679	+	0.462385i	$0.846994\pi$
$80$	0	0
$81$	7.84734	0.871926
$82$	0	0
$83$	8.63077	0.947350	0.473675	−	0.880700i	$-0.342927\pi$
$83$	8.63077	0.947350	0.473675	+	0.880700i	$0.342927\pi$
$84$	0	0
$85$	−2.90195	−0.314760
$86$	0	0
$87$	−8.79338	−0.942749
$88$	0	0
$89$	8.61968	0.913684	0.456842	−	0.889548i	$-0.348980\pi$
$89$	8.61968	0.913684	0.456842	+	0.889548i	$0.348980\pi$
$90$	0	0
$91$	3.74240	0.392310
$92$	0	0
$93$	11.1379	1.15494
$94$	0	0
$95$	−2.79483	−0.286744
$96$	0	0
$97$	13.8912	1.41044	0.705219	−	0.708989i	$-0.250849\pi$
$97$	13.8912	1.41044	0.705219	+	0.708989i	$0.250849\pi$
$98$	0	0
$99$	−3.95691	−0.397685
$100$	0	0
$101$	18.3869	1.82957	0.914784	−	0.403943i	$-0.132361\pi$
$101$	18.3869	1.82957	0.914784	+	0.403943i	$0.132361\pi$
$102$	0	0
$103$	2.57401	0.253624	0.126812	−	0.991927i	$-0.459525\pi$
$103$	2.57401	0.253624	0.126812	+	0.991927i	$0.459525\pi$
$104$	0	0
$105$	−8.60680	−0.839938
$106$	0	0
$107$	1.04842	0.101355	0.0506774	−	0.998715i	$-0.483862\pi$
$107$	1.04842	0.101355	0.0506774	+	0.998715i	$0.483862\pi$
$108$	0	0
$109$	−0.281841	−0.0269955	−0.0134977	−	0.999909i	$-0.504297\pi$
$109$	−0.281841	−0.0269955	−0.0134977	+	0.999909i	$0.504297\pi$
$110$	0	0
$111$	−9.85380	−0.935281
$112$	0	0
$113$	−0.371691	−0.0349658	−0.0174829	−	0.999847i	$-0.505565\pi$
$113$	−0.371691	−0.0349658	−0.0174829	+	0.999847i	$0.505565\pi$
$114$	0	0
$115$	−3.21458	−0.299761
$116$	0	0
$117$	−6.03691	−0.558113
$118$	0	0
$119$	−13.2973	−1.21896
$120$	0	0
$121$	−10.5456	−0.958692
$122$	0	0
$123$	36.0292	3.24864
$124$	0	0
$125$	7.44064	0.665511
$126$	0	0
$127$	−4.08332	−0.362336	−0.181168	−	0.983452i	$-0.557988\pi$
$127$	−4.08332	−0.362336	−0.181168	+	0.983452i	$0.557988\pi$
$128$	0	0
$129$	3.14615	0.277003
$130$	0	0
$131$	−16.6139	−1.45156	−0.725782	−	0.687924i	$-0.758522\pi$
$131$	−16.6139	−1.45156	−0.725782	+	0.687924i	$0.758522\pi$
$132$	0	0
$133$	−12.8065	−1.11046
$134$	0	0
$135$	6.78820	0.584235
$136$	0	0
$137$	−22.5413	−1.92584	−0.962918	−	0.269794i	$-0.913044\pi$
$137$	−22.5413	−1.92584	−0.962918	+	0.269794i	$0.913044\pi$
$138$	0	0
$139$	18.3102	1.55305	0.776526	−	0.630086i	$-0.216980\pi$
$139$	18.3102	1.55305	0.776526	+	0.630086i	$0.216980\pi$
$140$	0	0
$141$	19.2484	1.62101
$142$	0	0
$143$	0.693247	0.0579722
$144$	0	0
$145$	−2.34474	−0.194720
$146$	0	0
$147$	−18.5903	−1.53330
$148$	0	0
$149$	13.7808	1.12897	0.564486	−	0.825443i	$-0.309075\pi$
$149$	13.7808	1.12897	0.564486	+	0.825443i	$0.309075\pi$
$150$	0	0
$151$	−11.7703	−0.957856	−0.478928	−	0.877854i	$-0.658975\pi$
$151$	−11.7703	−0.957856	−0.478928	+	0.877854i	$0.658975\pi$
$152$	0	0
$153$	21.4501	1.73414
$154$	0	0
$155$	2.96989	0.238548
$156$	0	0
$157$	−19.0243	−1.51830	−0.759150	−	0.650915i	$-0.774385\pi$
$157$	−19.0243	−1.51830	−0.759150	+	0.650915i	$0.774385\pi$
$158$	0	0
$159$	−6.78845	−0.538359
$160$	0	0
$161$	−14.7299	−1.16088
$162$	0	0
$163$	−10.1480	−0.794854	−0.397427	−	0.917634i	$-0.630097\pi$
$163$	−10.1480	−0.794854	−0.397427	+	0.917634i	$0.630097\pi$
$164$	0	0
$165$	−1.59434	−0.124119
$166$	0	0
$167$	13.5569	1.04907	0.524534	−	0.851390i	$-0.324240\pi$
$167$	13.5569	1.04907	0.524534	+	0.851390i	$0.324240\pi$
$168$	0	0
$169$	−11.9423	−0.918642
$170$	0	0
$171$	20.6583	1.57978
$172$	0	0
$173$	19.4368	1.47775	0.738875	−	0.673843i	$-0.235357\pi$
$173$	19.4368	1.47775	0.738875	+	0.673843i	$0.235357\pi$
$174$	0	0
$175$	15.8998	1.20191
$176$	0	0
$177$	44.1468	3.31828
$178$	0	0
$179$	−11.3997	−0.852051	−0.426025	−	0.904711i	$-0.640087\pi$
$179$	−11.3997	−0.852051	−0.426025	+	0.904711i	$0.640087\pi$
$180$	0	0
$181$	−23.5225	−1.74842	−0.874208	−	0.485552i	$-0.838619\pi$
$181$	−23.5225	−1.74842	−0.874208	+	0.485552i	$0.838619\pi$
$182$	0	0
$183$	20.7802	1.53612
$184$	0	0
$185$	−2.62750	−0.193178
$186$	0	0
$187$	−2.46322	−0.180128
$188$	0	0
$189$	31.1049	2.26255
$190$	0	0
$191$	−8.92852	−0.646045	−0.323022	−	0.946391i	$-0.604699\pi$
$191$	−8.92852	−0.646045	−0.323022	+	0.946391i	$0.604699\pi$
$192$	0	0
$193$	2.02084	0.145463	0.0727315	−	0.997352i	$-0.476828\pi$
$193$	2.02084	0.145463	0.0727315	+	0.997352i	$0.476828\pi$
$194$	0	0
$195$	−2.43242	−0.174189
$196$	0	0
$197$	9.92643	0.707229	0.353615	−	0.935391i	$-0.384952\pi$
$197$	9.92643	0.707229	0.353615	+	0.935391i	$0.384952\pi$
$198$	0	0
$199$	−17.7531	−1.25849	−0.629244	−	0.777208i	$-0.716635\pi$
$199$	−17.7531	−1.25849	−0.629244	+	0.777208i	$0.716635\pi$
$200$	0	0
$201$	38.1883	2.69360
$202$	0	0
$203$	−10.7441	−0.754087
$204$	0	0
$205$	9.60712	0.670990
$206$	0	0
$207$	23.7610	1.65150
$208$	0	0
$209$	−2.37229	−0.164095
$210$	0	0
$211$	25.3158	1.74281	0.871406	−	0.490563i	$-0.163209\pi$
$211$	25.3158	1.74281	0.871406	+	0.490563i	$0.163209\pi$
$212$	0	0
$213$	9.95087	0.681822
$214$	0	0
$215$	0.838916	0.0572136
$216$	0	0
$217$	13.6087	0.923817
$218$	0	0
$219$	−39.5698	−2.67388
$220$	0	0
$221$	−3.75804	−0.252793
$222$	0	0
$223$	18.7341	1.25453	0.627265	−	0.778806i	$-0.284174\pi$
$223$	18.7341	1.25453	0.627265	+	0.778806i	$0.284174\pi$
$224$	0	0
$225$	−25.6482	−1.70988
$226$	0	0
$227$	11.4194	0.757932	0.378966	−	0.925411i	$-0.376280\pi$
$227$	11.4194	0.757932	0.378966	+	0.925411i	$0.376280\pi$
$228$	0	0
$229$	1.67879	0.110938	0.0554689	−	0.998460i	$-0.482335\pi$
$229$	1.67879	0.110938	0.0554689	+	0.998460i	$0.482335\pi$
$230$	0	0
$231$	−7.30558	−0.480672
$232$	0	0
$233$	1.11739	0.0732026	0.0366013	−	0.999330i	$-0.488347\pi$
$233$	1.11739	0.0732026	0.0366013	+	0.999330i	$0.488347\pi$
$234$	0	0
$235$	5.13256	0.334811
$236$	0	0
$237$	46.9433	3.04929
$238$	0	0
$239$	27.3842	1.77134	0.885669	−	0.464318i	$-0.153700\pi$
$239$	27.3842	1.77134	0.885669	+	0.464318i	$0.153700\pi$
$240$	0	0
$241$	−1.17894	−0.0759423	−0.0379712	−	0.999279i	$-0.512090\pi$
$241$	−1.17894	−0.0759423	−0.0379712	+	0.999279i	$0.512090\pi$
$242$	0	0
$243$	2.27186	0.145740
$244$	0	0
$245$	−4.95708	−0.316696
$246$	0	0
$247$	−3.61932	−0.230292
$248$	0	0
$249$	−25.7047	−1.62897
$250$	0	0
$251$	24.8123	1.56614	0.783068	−	0.621936i	$-0.213653\pi$
$251$	24.8123	1.56614	0.783068	+	0.621936i	$0.213653\pi$
$252$	0	0
$253$	−2.72858	−0.171544
$254$	0	0
$255$	8.64277	0.541231
$256$	0	0
$257$	−3.30502	−0.206162	−0.103081	−	0.994673i	$-0.532870\pi$
$257$	−3.30502	−0.206162	−0.103081	+	0.994673i	$0.532870\pi$
$258$	0	0
$259$	−12.0397	−0.748114
$260$	0	0
$261$	17.3314	1.07279
$262$	0	0
$263$	−12.1592	−0.749769	−0.374884	−	0.927072i	$-0.622318\pi$
$263$	−12.1592	−0.749769	−0.374884	+	0.927072i	$0.622318\pi$
$264$	0	0
$265$	−1.81013	−0.111195
$266$	0	0
$267$	−25.6717	−1.57108
$268$	0	0
$269$	13.2356	0.806988	0.403494	−	0.914982i	$-0.367796\pi$
$269$	13.2356	0.806988	0.403494	+	0.914982i	$0.367796\pi$
$270$	0	0
$271$	−3.84027	−0.233280	−0.116640	−	0.993174i	$-0.537212\pi$
$271$	−3.84027	−0.233280	−0.116640	+	0.993174i	$0.537212\pi$
$272$	0	0
$273$	−11.1458	−0.674577
$274$	0	0
$275$	2.94530	0.177608
$276$	0	0
$277$	−6.44068	−0.386983	−0.193491	−	0.981102i	$-0.561981\pi$
$277$	−6.44068	−0.386983	−0.193491	+	0.981102i	$0.561981\pi$
$278$	0	0
$279$	−21.9523	−1.31425
$280$	0	0
$281$	−17.4304	−1.03981	−0.519904	−	0.854224i	$-0.674032\pi$
$281$	−17.4304	−1.03981	−0.519904	+	0.854224i	$0.674032\pi$
$282$	0	0
$283$	2.81014	0.167045	0.0835227	−	0.996506i	$-0.473383\pi$
$283$	2.81014	0.167045	0.0835227	+	0.996506i	$0.473383\pi$
$284$	0	0
$285$	8.32374	0.493056
$286$	0	0
$287$	44.0218	2.59853
$288$	0	0
$289$	−3.64710	−0.214535
$290$	0	0
$291$	−41.3717	−2.42525
$292$	0	0
$293$	24.3123	1.42034	0.710168	−	0.704032i	$-0.248619\pi$
$293$	24.3123	1.42034	0.710168	+	0.704032i	$0.248619\pi$
$294$	0	0
$295$	11.7717	0.685374
$296$	0	0
$297$	5.76192	0.334340
$298$	0	0
$299$	−4.16289	−0.240746
$300$	0	0
$301$	3.84409	0.221570
$302$	0	0
$303$	−54.7611	−3.14594
$304$	0	0
$305$	5.54101	0.317277
$306$	0	0
$307$	9.76945	0.557572	0.278786	−	0.960353i	$-0.410068\pi$
$307$	9.76945	0.557572	0.278786	+	0.960353i	$0.410068\pi$
$308$	0	0
$309$	−7.66607	−0.436107
$310$	0	0
$311$	8.77584	0.497632	0.248816	−	0.968551i	$-0.419959\pi$
$311$	8.77584	0.497632	0.248816	+	0.968551i	$0.419959\pi$
$312$	0	0
$313$	−21.1371	−1.19474	−0.597369	−	0.801967i	$-0.703787\pi$
$313$	−21.1371	−1.19474	−0.597369	+	0.801967i	$0.703787\pi$
$314$	0	0
$315$	16.9637	0.955796
$316$	0	0
$317$	−3.01533	−0.169358	−0.0846788	−	0.996408i	$-0.526986\pi$
$317$	−3.01533	−0.169358	−0.0846788	+	0.996408i	$0.526986\pi$
$318$	0	0
$319$	−1.99025	−0.111433
$320$	0	0
$321$	−3.12248	−0.174280
$322$	0	0
$323$	12.8600	0.715550
$324$	0	0
$325$	4.49353	0.249256
$326$	0	0
$327$	0.839396	0.0464187
$328$	0	0
$329$	23.5185	1.29662
$330$	0	0
$331$	10.6238	0.583938	0.291969	−	0.956428i	$-0.405690\pi$
$331$	10.6238	0.583938	0.291969	+	0.956428i	$0.405690\pi$
$332$	0	0
$333$	19.4215	1.06429
$334$	0	0
$335$	10.1829	0.556349
$336$	0	0
$337$	24.9753	1.36049	0.680246	−	0.732984i	$-0.261873\pi$
$337$	24.9753	1.36049	0.680246	+	0.732984i	$0.261873\pi$
$338$	0	0
$339$	1.10699	0.0601237
$340$	0	0
$341$	2.52089	0.136514
$342$	0	0
$343$	2.75831	0.148935
$344$	0	0
$345$	9.57386	0.515439
$346$	0	0
$347$	4.78427	0.256833	0.128417	−	0.991720i	$-0.459011\pi$
$347$	4.78427	0.256833	0.128417	+	0.991720i	$0.459011\pi$
$348$	0	0
$349$	17.1674	0.918949	0.459474	−	0.888191i	$-0.348038\pi$
$349$	17.1674	0.918949	0.459474	+	0.888191i	$0.348038\pi$
$350$	0	0
$351$	8.79074	0.469215
$352$	0	0
$353$	−34.5313	−1.83792	−0.918958	−	0.394356i	$-0.870968\pi$
$353$	−34.5313	−1.83792	−0.918958	+	0.394356i	$0.870968\pi$
$354$	0	0
$355$	2.65338	0.140827
$356$	0	0
$357$	39.6030	2.09601
$358$	0	0
$359$	23.1325	1.22089	0.610444	−	0.792059i	$-0.290991\pi$
$359$	23.1325	1.22089	0.610444	+	0.792059i	$0.290991\pi$
$360$	0	0
$361$	−6.61468	−0.348141
$362$	0	0
$363$	31.4076	1.64847
$364$	0	0
$365$	−10.5512	−0.552276
$366$	0	0
$367$	11.5508	0.602949	0.301474	−	0.953474i	$-0.402521\pi$
$367$	11.5508	0.602949	0.301474	+	0.953474i	$0.402521\pi$
$368$	0	0
$369$	−71.0122	−3.69675
$370$	0	0
$371$	−8.29439	−0.430623
$372$	0	0
$373$	36.2752	1.87826	0.939130	−	0.343563i	$-0.111634\pi$
$373$	36.2752	1.87826	0.939130	+	0.343563i	$0.111634\pi$
$374$	0	0
$375$	−22.1602	−1.14435
$376$	0	0
$377$	−3.03645	−0.156385
$378$	0	0
$379$	−29.6449	−1.52275	−0.761377	−	0.648309i	$-0.775477\pi$
$379$	−29.6449	−1.52275	−0.761377	+	0.648309i	$0.775477\pi$
$380$	0	0
$381$	12.1612	0.623037
$382$	0	0
$383$	−17.8011	−0.909596	−0.454798	−	0.890595i	$-0.650289\pi$
$383$	−17.8011	−0.909596	−0.454798	+	0.890595i	$0.650289\pi$
$384$	0	0
$385$	−1.94802	−0.0992803
$386$	0	0
$387$	−6.20095	−0.315212
$388$	0	0
$389$	−36.2806	−1.83950	−0.919751	−	0.392503i	$-0.871609\pi$
$389$	−36.2806	−1.83950	−0.919751	+	0.392503i	$0.871609\pi$
$390$	0	0
$391$	14.7914	0.748034
$392$	0	0
$393$	49.4806	2.49597
$394$	0	0
$395$	12.5173	0.629816
$396$	0	0
$397$	28.7250	1.44167	0.720833	−	0.693109i	$-0.243759\pi$
$397$	28.7250	1.44167	0.720833	+	0.693109i	$0.243759\pi$
$398$	0	0
$399$	38.1411	1.90944
$400$	0	0
$401$	0.316301	0.0157953	0.00789766	−	0.999969i	$-0.497486\pi$
$401$	0.316301	0.0157953	0.00789766	+	0.999969i	$0.497486\pi$
$402$	0	0
$403$	3.84603	0.191584
$404$	0	0
$405$	−6.23195	−0.309668
$406$	0	0
$407$	−2.23026	−0.110550
$408$	0	0
$409$	15.0906	0.746180	0.373090	−	0.927795i	$-0.378298\pi$
$409$	15.0906	0.746180	0.373090	+	0.927795i	$0.378298\pi$
$410$	0	0
$411$	67.1340	3.31148
$412$	0	0
$413$	53.9403	2.65423
$414$	0	0
$415$	−6.85412	−0.336455
$416$	0	0
$417$	−54.5326	−2.67047
$418$	0	0
$419$	6.77733	0.331094	0.165547	−	0.986202i	$-0.447061\pi$
$419$	6.77733	0.331094	0.165547	+	0.986202i	$0.447061\pi$
$420$	0	0
$421$	9.18722	0.447758	0.223879	−	0.974617i	$-0.428128\pi$
$421$	9.18722	0.447758	0.223879	+	0.974617i	$0.428128\pi$
$422$	0	0
$423$	−37.9380	−1.84461
$424$	0	0
$425$	−15.9662	−0.774476
$426$	0	0
$427$	25.3901	1.22871
$428$	0	0
$429$	−2.06467	−0.0996833
$430$	0	0
$431$	−7.70399	−0.371088	−0.185544	−	0.982636i	$-0.559405\pi$
$431$	−7.70399	−0.371088	−0.185544	+	0.982636i	$0.559405\pi$
$432$	0	0
$433$	−24.8136	−1.19247	−0.596234	−	0.802811i	$-0.703337\pi$
$433$	−24.8136	−1.19247	−0.596234	+	0.802811i	$0.703337\pi$
$434$	0	0
$435$	6.98325	0.334821
$436$	0	0
$437$	14.2454	0.681452
$438$	0	0
$439$	21.4745	1.02492	0.512461	−	0.858710i	$-0.328734\pi$
$439$	21.4745	1.02492	0.512461	+	0.858710i	$0.328734\pi$
$440$	0	0
$441$	36.6409	1.74480
$442$	0	0
$443$	−7.32253	−0.347904	−0.173952	−	0.984754i	$-0.555654\pi$
$443$	−7.32253	−0.347904	−0.173952	+	0.984754i	$0.555654\pi$
$444$	0	0
$445$	−6.84531	−0.324499
$446$	0	0
$447$	−41.0430	−1.94127
$448$	0	0
$449$	−3.94124	−0.185999	−0.0929994	−	0.995666i	$-0.529645\pi$
$449$	−3.94124	−0.185999	−0.0929994	+	0.995666i	$0.529645\pi$
$450$	0	0
$451$	8.15466	0.383988
$452$	0	0
$453$	35.0551	1.64703
$454$	0	0
$455$	−2.97202	−0.139331
$456$	0	0
$457$	−37.2086	−1.74054	−0.870272	−	0.492571i	$-0.836057\pi$
$457$	−37.2086	−1.74054	−0.870272	+	0.492571i	$0.836057\pi$
$458$	0	0
$459$	−31.2349	−1.45792
$460$	0	0
$461$	−14.0239	−0.653159	−0.326579	−	0.945170i	$-0.605896\pi$
$461$	−14.0239	−0.653159	−0.326579	+	0.945170i	$0.605896\pi$
$462$	0	0
$463$	35.6643	1.65746	0.828730	−	0.559649i	$-0.189064\pi$
$463$	35.6643	1.65746	0.828730	+	0.559649i	$0.189064\pi$
$464$	0	0
$465$	−8.84512	−0.410183
$466$	0	0
$467$	4.20947	0.194791	0.0973956	−	0.995246i	$-0.468949\pi$
$467$	4.20947	0.194791	0.0973956	+	0.995246i	$0.468949\pi$
$468$	0	0
$469$	46.6600	2.15456
$470$	0	0
$471$	56.6592	2.61072
$472$	0	0
$473$	0.712084	0.0327417
$474$	0	0
$475$	−15.3769	−0.705540
$476$	0	0
$477$	13.3798	0.612619
$478$	0	0
$479$	17.6113	0.804682	0.402341	−	0.915490i	$-0.368197\pi$
$479$	17.6113	0.804682	0.402341	+	0.915490i	$0.368197\pi$
$480$	0	0
$481$	−3.40262	−0.155146
$482$	0	0
$483$	43.8694	1.99613
$484$	0	0
$485$	−11.0317	−0.500923
$486$	0	0
$487$	19.3114	0.875081	0.437541	−	0.899199i	$-0.355850\pi$
$487$	19.3114	0.875081	0.437541	+	0.899199i	$0.355850\pi$
$488$	0	0
$489$	30.2235	1.36675
$490$	0	0
$491$	13.5411	0.611099	0.305550	−	0.952176i	$-0.401160\pi$
$491$	13.5411	0.611099	0.305550	+	0.952176i	$0.401160\pi$
$492$	0	0
$493$	10.7890	0.485911
$494$	0	0
$495$	3.14238	0.141239
$496$	0	0
$497$	12.1584	0.545377
$498$	0	0
$499$	16.7939	0.751796	0.375898	−	0.926661i	$-0.377334\pi$
$499$	16.7939	0.751796	0.375898	+	0.926661i	$0.377334\pi$
$500$	0	0
$501$	−40.3761	−1.80387
$502$	0	0
$503$	−9.61359	−0.428649	−0.214324	−	0.976763i	$-0.568755\pi$
$503$	−9.61359	−0.428649	−0.214324	+	0.976763i	$0.568755\pi$
$504$	0	0
$505$	−14.6020	−0.649779
$506$	0	0
$507$	35.5674	1.57960
$508$	0	0
$509$	−22.1395	−0.981315	−0.490657	−	0.871353i	$-0.663243\pi$
$509$	−22.1395	−0.981315	−0.490657	+	0.871353i	$0.663243\pi$
$510$	0	0
$511$	−48.3479	−2.13879
$512$	0	0
$513$	−30.0820	−1.32815
$514$	0	0
$515$	−2.04415	−0.0900758
$516$	0	0
$517$	4.35659	0.191603
$518$	0	0
$519$	−57.8878	−2.54099
$520$	0	0
$521$	23.9418	1.04891	0.524456	−	0.851438i	$-0.324269\pi$
$521$	23.9418	1.04891	0.524456	+	0.851438i	$0.324269\pi$
$522$	0	0
$523$	25.7322	1.12519	0.562595	−	0.826733i	$-0.309803\pi$
$523$	25.7322	1.12519	0.562595	+	0.826733i	$0.309803\pi$
$524$	0	0
$525$	−47.3538	−2.06669
$526$	0	0
$527$	−13.6655	−0.595280
$528$	0	0
$529$	−6.61508	−0.287612
$530$	0	0
$531$	−87.0118	−3.77599
$532$	0	0
$533$	12.4413	0.538891
$534$	0	0
$535$	−0.832603	−0.0359966
$536$	0	0
$537$	33.9512	1.46510
$538$	0	0
$539$	−4.20764	−0.181236
$540$	0	0
$541$	−33.0623	−1.42146	−0.710730	−	0.703465i	$-0.751635\pi$
$541$	−33.0623	−1.42146	−0.710730	+	0.703465i	$0.751635\pi$
$542$	0	0
$543$	70.0563	3.00640
$544$	0	0
$545$	0.223824	0.00958755
$546$	0	0
$547$	34.2049	1.46250	0.731249	−	0.682111i	$-0.238938\pi$
$547$	34.2049	1.46250	0.731249	+	0.682111i	$0.238938\pi$
$548$	0	0
$549$	−40.9571	−1.74801
$550$	0	0
$551$	10.3907	0.442660
$552$	0	0
$553$	57.3571	2.43907
$554$	0	0
$555$	7.82538	0.332169
$556$	0	0
$557$	29.8896	1.26646	0.633231	−	0.773963i	$-0.281728\pi$
$557$	29.8896	1.26646	0.633231	+	0.773963i	$0.281728\pi$
$558$	0	0
$559$	1.08640	0.0459498
$560$	0	0
$561$	7.33610	0.309731
$562$	0	0
$563$	17.6079	0.742087	0.371043	−	0.928616i	$-0.379000\pi$
$563$	17.6079	0.742087	0.371043	+	0.928616i	$0.379000\pi$
$564$	0	0
$565$	0.295178	0.0124182
$566$	0	0
$567$	−28.5561	−1.19924
$568$	0	0
$569$	25.6068	1.07349	0.536746	−	0.843744i	$-0.319653\pi$
$569$	25.6068	1.07349	0.536746	+	0.843744i	$0.319653\pi$
$570$	0	0
$571$	−3.81141	−0.159503	−0.0797514	−	0.996815i	$-0.525413\pi$
$571$	−3.81141	−0.159503	−0.0797514	+	0.996815i	$0.525413\pi$
$572$	0	0
$573$	26.5915	1.11087
$574$	0	0
$575$	−17.6863	−0.737569
$576$	0	0
$577$	22.1253	0.921088	0.460544	−	0.887637i	$-0.347654\pi$
$577$	22.1253	0.921088	0.460544	+	0.887637i	$0.347654\pi$
$578$	0	0
$579$	−6.01858	−0.250124
$580$	0	0
$581$	−31.4070	−1.30298
$582$	0	0
$583$	−1.53646	−0.0636338
$584$	0	0
$585$	4.79421	0.198216
$586$	0	0
$587$	−27.1289	−1.11973	−0.559865	−	0.828584i	$-0.689147\pi$
$587$	−27.1289	−1.11973	−0.559865	+	0.828584i	$0.689147\pi$
$588$	0	0
$589$	−13.1611	−0.542294
$590$	0	0
$591$	−29.5635	−1.21608
$592$	0	0
$593$	11.7915	0.484220	0.242110	−	0.970249i	$-0.422160\pi$
$593$	11.7915	0.484220	0.242110	+	0.970249i	$0.422160\pi$
$594$	0	0
$595$	10.5601	0.432920
$596$	0	0
$597$	52.8735	2.16397
$598$	0	0
$599$	2.26400	0.0925045	0.0462523	−	0.998930i	$-0.485272\pi$
$599$	2.26400	0.0925045	0.0462523	+	0.998930i	$0.485272\pi$
$600$	0	0
$601$	−6.81460	−0.277973	−0.138987	−	0.990294i	$-0.544385\pi$
$601$	−6.81460	−0.277973	−0.138987	+	0.990294i	$0.544385\pi$
$602$	0	0
$603$	−75.2679	−3.06514
$604$	0	0
$605$	8.37478	0.340483
$606$	0	0
$607$	−10.9269	−0.443509	−0.221754	−	0.975103i	$-0.571178\pi$
$607$	−10.9269	−0.443509	−0.221754	+	0.975103i	$0.571178\pi$
$608$	0	0
$609$	31.9987	1.29665
$610$	0	0
$611$	6.64669	0.268896
$612$	0	0
$613$	36.4841	1.47358	0.736790	−	0.676122i	$-0.236341\pi$
$613$	36.4841	1.47358	0.736790	+	0.676122i	$0.236341\pi$
$614$	0	0
$615$	−28.6125	−1.15377
$616$	0	0
$617$	−26.5517	−1.06893	−0.534465	−	0.845190i	$-0.679487\pi$
$617$	−26.5517	−1.06893	−0.534465	+	0.845190i	$0.679487\pi$
$618$	0	0
$619$	−29.2059	−1.17388	−0.586942	−	0.809629i	$-0.699668\pi$
$619$	−29.2059	−1.17388	−0.586942	+	0.809629i	$0.699668\pi$
$620$	0	0
$621$	−34.5999	−1.38845
$622$	0	0
$623$	−31.3666	−1.25668
$624$	0	0
$625$	15.9377	0.637506
$626$	0	0
$627$	7.06531	0.282161
$628$	0	0
$629$	12.0901	0.482062
$630$	0	0
$631$	20.2857	0.807560	0.403780	−	0.914856i	$-0.367696\pi$
$631$	20.2857	0.807560	0.403780	+	0.914856i	$0.367696\pi$
$632$	0	0
$633$	−75.3971	−2.99677
$634$	0	0
$635$	3.24276	0.128685
$636$	0	0
$637$	−6.41944	−0.254348
$638$	0	0
$639$	−19.6128	−0.775871
$640$	0	0
$641$	2.84368	0.112318	0.0561592	−	0.998422i	$-0.482115\pi$
$641$	2.84368	0.112318	0.0561592	+	0.998422i	$0.482115\pi$
$642$	0	0
$643$	18.3771	0.724722	0.362361	−	0.932038i	$-0.381971\pi$
$643$	18.3771	0.724722	0.362361	+	0.932038i	$0.381971\pi$
$644$	0	0
$645$	−2.49851	−0.0983788
$646$	0	0
$647$	−49.8807	−1.96101	−0.980507	−	0.196483i	$-0.937048\pi$
$647$	−49.8807	−1.96101	−0.980507	+	0.196483i	$0.937048\pi$
$648$	0	0
$649$	9.99197	0.392219
$650$	0	0
$651$	−40.5302	−1.58850
$652$	0	0
$653$	−8.82773	−0.345456	−0.172728	−	0.984970i	$-0.555258\pi$
$653$	−8.82773	−0.345456	−0.172728	+	0.984970i	$0.555258\pi$
$654$	0	0
$655$	13.1939	0.515529
$656$	0	0
$657$	77.9906	3.04270
$658$	0	0
$659$	−18.0733	−0.704037	−0.352018	−	0.935993i	$-0.614505\pi$
$659$	−18.0733	−0.704037	−0.352018	+	0.935993i	$0.614505\pi$
$660$	0	0
$661$	16.5209	0.642589	0.321294	−	0.946979i	$-0.395882\pi$
$661$	16.5209	0.642589	0.321294	+	0.946979i	$0.395882\pi$
$662$	0	0
$663$	11.1924	0.434677
$664$	0	0
$665$	10.1703	0.394386
$666$	0	0
$667$	11.9513	0.462756
$668$	0	0
$669$	−55.7952	−2.15717
$670$	0	0
$671$	4.70329	0.181569
$672$	0	0
$673$	9.45602	0.364503	0.182251	−	0.983252i	$-0.441662\pi$
$673$	9.45602	0.364503	0.182251	+	0.983252i	$0.441662\pi$
$674$	0	0
$675$	37.3480	1.43752
$676$	0	0
$677$	−43.1767	−1.65941	−0.829707	−	0.558199i	$-0.811493\pi$
$677$	−43.1767	−1.65941	−0.829707	+	0.558199i	$0.811493\pi$
$678$	0	0
$679$	−50.5495	−1.93991
$680$	0	0
$681$	−34.0100	−1.30327
$682$	0	0
$683$	−38.7228	−1.48169	−0.740843	−	0.671679i	$-0.765574\pi$
$683$	−38.7228	−1.48169	−0.740843	+	0.671679i	$0.765574\pi$
$684$	0	0
$685$	17.9012	0.683969
$686$	0	0
$687$	−4.99988	−0.190757
$688$	0	0
$689$	−2.34412	−0.0893040
$690$	0	0
$691$	26.4221	1.00514	0.502572	−	0.864535i	$-0.332387\pi$
$691$	26.4221	1.00514	0.502572	+	0.864535i	$0.332387\pi$
$692$	0	0
$693$	14.3990	0.546974
$694$	0	0
$695$	−14.5410	−0.551573
$696$	0	0
$697$	−44.2058	−1.67441
$698$	0	0
$699$	−3.32788	−0.125872
$700$	0	0
$701$	9.17771	0.346637	0.173319	−	0.984866i	$-0.444551\pi$
$701$	9.17771	0.346637	0.173319	+	0.984866i	$0.444551\pi$
$702$	0	0
$703$	11.6438	0.439154
$704$	0	0
$705$	−15.2861	−0.575709
$706$	0	0
$707$	−66.9093	−2.51638
$708$	0	0
$709$	23.1965	0.871164	0.435582	−	0.900149i	$-0.356543\pi$
$709$	23.1965	0.871164	0.435582	+	0.900149i	$0.356543\pi$
$710$	0	0
$711$	−92.5235	−3.46990
$712$	0	0
$713$	−15.1377	−0.566913
$714$	0	0
$715$	−0.550541	−0.0205891
$716$	0	0
$717$	−81.5574	−3.04582
$718$	0	0
$719$	0.742779	0.0277010	0.0138505	−	0.999904i	$-0.495591\pi$
$719$	0.742779	0.0277010	0.0138505	+	0.999904i	$0.495591\pi$
$720$	0	0
$721$	−9.36670	−0.348834
$722$	0	0
$723$	3.51120	0.130583
$724$	0	0
$725$	−12.9005	−0.479113
$726$	0	0
$727$	17.7406	0.657963	0.328981	−	0.944336i	$-0.393295\pi$
$727$	17.7406	0.657963	0.328981	+	0.944336i	$0.393295\pi$
$728$	0	0
$729$	−30.3082	−1.12253
$730$	0	0
$731$	−3.86015	−0.142773
$732$	0	0
$733$	−22.3181	−0.824339	−0.412170	−	0.911107i	$-0.635229\pi$
$733$	−22.3181	−0.824339	−0.412170	+	0.911107i	$0.635229\pi$
$734$	0	0
$735$	14.7635	0.544560
$736$	0	0
$737$	8.64336	0.318382
$738$	0	0
$739$	−20.2234	−0.743931	−0.371965	−	0.928247i	$-0.621316\pi$
$739$	−20.2234	−0.743931	−0.371965	+	0.928247i	$0.621316\pi$
$740$	0	0
$741$	10.7793	0.395987
$742$	0	0
$743$	15.6864	0.575478	0.287739	−	0.957709i	$-0.407096\pi$
$743$	15.6864	0.575478	0.287739	+	0.957709i	$0.407096\pi$
$744$	0	0
$745$	−10.9440	−0.400959
$746$	0	0
$747$	50.6630	1.85366
$748$	0	0
$749$	−3.81516	−0.139403
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−73.8975	−2.69297
$754$	0	0
$755$	9.34740	0.340187
$756$	0	0
$757$	44.3594	1.61227	0.806135	−	0.591731i	$-0.201555\pi$
$757$	44.3594	1.61227	0.806135	+	0.591731i	$0.201555\pi$
$758$	0	0
$759$	8.12643	0.294971
$760$	0	0
$761$	23.1134	0.837859	0.418930	−	0.908019i	$-0.362405\pi$
$761$	23.1134	0.837859	0.418930	+	0.908019i	$0.362405\pi$
$762$	0	0
$763$	1.02561	0.0371295
$764$	0	0
$765$	−17.0346	−0.615887
$766$	0	0
$767$	15.2444	0.550442
$768$	0	0
$769$	−4.01909	−0.144932	−0.0724661	−	0.997371i	$-0.523087\pi$
$769$	−4.01909	−0.144932	−0.0724661	+	0.997371i	$0.523087\pi$
$770$	0	0
$771$	9.84323	0.354495
$772$	0	0
$773$	14.8793	0.535170	0.267585	−	0.963534i	$-0.413774\pi$
$773$	14.8793	0.535170	0.267585	+	0.963534i	$0.413774\pi$
$774$	0	0
$775$	16.3401	0.586952
$776$	0	0
$777$	35.8575	1.28638
$778$	0	0
$779$	−42.5740	−1.52537
$780$	0	0
$781$	2.25223	0.0805911
$782$	0	0
$783$	−25.2374	−0.901912
$784$	0	0
$785$	15.1081	0.539231
$786$	0	0
$787$	12.3080	0.438733	0.219366	−	0.975643i	$-0.429601\pi$
$787$	12.3080	0.438733	0.219366	+	0.975643i	$0.429601\pi$
$788$	0	0
$789$	36.2133	1.28923
$790$	0	0
$791$	1.35257	0.0480918
$792$	0	0
$793$	7.17563	0.254814
$794$	0	0
$795$	5.39104	0.191201
$796$	0	0
$797$	−6.52092	−0.230983	−0.115491	−	0.993308i	$-0.536844\pi$
$797$	−6.52092	−0.230983	−0.115491	+	0.993308i	$0.536844\pi$
$798$	0	0
$799$	−23.6167	−0.835500
$800$	0	0
$801$	50.5980	1.78779
$802$	0	0
$803$	−8.95603	−0.316051
$804$	0	0
$805$	11.6977	0.412290
$806$	0	0
$807$	−39.4191	−1.38762
$808$	0	0
$809$	−20.0644	−0.705426	−0.352713	−	0.935732i	$-0.614741\pi$
$809$	−20.0644	−0.705426	−0.352713	+	0.935732i	$0.614741\pi$
$810$	0	0
$811$	17.2658	0.606284	0.303142	−	0.952945i	$-0.401964\pi$
$811$	17.2658	0.606284	0.303142	+	0.952945i	$0.401964\pi$
$812$	0	0
$813$	11.4373	0.401125
$814$	0	0
$815$	8.05904	0.282296
$816$	0	0
$817$	−3.71766	−0.130065
$818$	0	0
$819$	21.9681	0.767626
$820$	0	0
$821$	35.5425	1.24044	0.620221	−	0.784428i	$-0.287043\pi$
$821$	35.5425	1.24044	0.620221	+	0.784428i	$0.287043\pi$
$822$	0	0
$823$	40.3182	1.40540	0.702702	−	0.711484i	$-0.251977\pi$
$823$	40.3182	1.40540	0.702702	+	0.711484i	$0.251977\pi$
$824$	0	0
$825$	−8.77187	−0.305397
$826$	0	0
$827$	26.1404	0.908990	0.454495	−	0.890749i	$-0.349820\pi$
$827$	26.1404	0.908990	0.454495	+	0.890749i	$0.349820\pi$
$828$	0	0
$829$	51.5006	1.78869	0.894344	−	0.447379i	$-0.147643\pi$
$829$	51.5006	1.78869	0.894344	+	0.447379i	$0.147643\pi$
$830$	0	0
$831$	19.1820	0.665417
$832$	0	0
$833$	22.8093	0.790295
$834$	0	0
$835$	−10.7662	−0.372581
$836$	0	0
$837$	31.9662	1.10491
$838$	0	0
$839$	5.30819	0.183259	0.0916295	−	0.995793i	$-0.470792\pi$
$839$	5.30819	0.183259	0.0916295	+	0.995793i	$0.470792\pi$
$840$	0	0
$841$	−20.2826	−0.699401
$842$	0	0
$843$	51.9122	1.78795
$844$	0	0
$845$	9.48400	0.326259
$846$	0	0
$847$	38.3750	1.31858
$848$	0	0
$849$	−8.36933	−0.287235
$850$	0	0
$851$	13.3925	0.459090
$852$	0	0
$853$	−28.7449	−0.984205	−0.492103	−	0.870537i	$-0.663772\pi$
$853$	−28.7449	−0.984205	−0.492103	+	0.870537i	$0.663772\pi$
$854$	0	0
$855$	−16.4058	−0.561066
$856$	0	0
$857$	46.0432	1.57280	0.786402	−	0.617714i	$-0.211941\pi$
$857$	46.0432	1.57280	0.786402	+	0.617714i	$0.211941\pi$
$858$	0	0
$859$	−12.4321	−0.424177	−0.212089	−	0.977250i	$-0.568027\pi$
$859$	−12.4321	−0.424177	−0.212089	+	0.977250i	$0.568027\pi$
$860$	0	0
$861$	−131.109	−4.46817
$862$	0	0
$863$	2.84583	0.0968732	0.0484366	−	0.998826i	$-0.484576\pi$
$863$	2.84583	0.0968732	0.0484366	+	0.998826i	$0.484576\pi$
$864$	0	0
$865$	−15.4357	−0.524829
$866$	0	0
$867$	10.8620	0.368894
$868$	0	0
$869$	10.6249	0.360425
$870$	0	0
$871$	13.1868	0.446819
$872$	0	0
$873$	81.5421	2.75978
$874$	0	0
$875$	−27.0762	−0.915342
$876$	0	0
$877$	9.77392	0.330042	0.165021	−	0.986290i	$-0.447231\pi$
$877$	9.77392	0.330042	0.165021	+	0.986290i	$0.447231\pi$
$878$	0	0
$879$	−72.4083	−2.44227
$880$	0	0
$881$	35.7683	1.20507	0.602533	−	0.798094i	$-0.294158\pi$
$881$	35.7683	1.20507	0.602533	+	0.798094i	$0.294158\pi$
$882$	0	0
$883$	−15.5981	−0.524919	−0.262460	−	0.964943i	$-0.584534\pi$
$883$	−15.5981	−0.524919	−0.262460	+	0.964943i	$0.584534\pi$
$884$	0	0
$885$	−35.0592	−1.17850
$886$	0	0
$887$	25.9717	0.872045	0.436022	−	0.899936i	$-0.356387\pi$
$887$	25.9717	0.872045	0.436022	+	0.899936i	$0.356387\pi$
$888$	0	0
$889$	14.8590	0.498355
$890$	0	0
$891$	−5.28977	−0.177214
$892$	0	0
$893$	−22.7450	−0.761132
$894$	0	0
$895$	9.05303	0.302609
$896$	0	0
$897$	12.3982	0.413964
$898$	0	0
$899$	−11.0416	−0.368258
$900$	0	0
$901$	8.32905	0.277481
$902$	0	0
$903$	−11.4487	−0.380989
$904$	0	0
$905$	18.6804	0.620957
$906$	0	0
$907$	43.0763	1.43033	0.715163	−	0.698958i	$-0.246353\pi$
$907$	43.0763	1.43033	0.715163	+	0.698958i	$0.246353\pi$
$908$	0	0
$909$	107.932	3.57989
$910$	0	0
$911$	−15.4292	−0.511193	−0.255596	−	0.966784i	$-0.582272\pi$
$911$	−15.4292	−0.511193	−0.255596	+	0.966784i	$0.582272\pi$
$912$	0	0
$913$	−5.81787	−0.192543
$914$	0	0
$915$	−16.5026	−0.545559
$916$	0	0
$917$	60.4573	1.99648
$918$	0	0
$919$	28.8801	0.952667	0.476333	−	0.879265i	$-0.341966\pi$
$919$	28.8801	0.952667	0.476333	+	0.879265i	$0.341966\pi$
$920$	0	0
$921$	−29.0960	−0.958745
$922$	0	0
$923$	3.43614	0.113102
$924$	0	0
$925$	−14.4562	−0.475318
$926$	0	0
$927$	15.1096	0.496263
$928$	0	0
$929$	−14.7303	−0.483286	−0.241643	−	0.970365i	$-0.577686\pi$
$929$	−14.7303	−0.483286	−0.241643	+	0.970365i	$0.577686\pi$
$930$	0	0
$931$	21.9674	0.719951
$932$	0	0
$933$	−26.1368	−0.855679
$934$	0	0
$935$	1.95616	0.0639733
$936$	0	0
$937$	−25.5827	−0.835751	−0.417876	−	0.908504i	$-0.637225\pi$
$937$	−25.5827	−0.835751	−0.417876	+	0.908504i	$0.637225\pi$
$938$	0	0
$939$	62.9518	2.05435
$940$	0	0
$941$	−17.8800	−0.582872	−0.291436	−	0.956590i	$-0.594133\pi$
$941$	−17.8800	−0.582872	−0.291436	+	0.956590i	$0.594133\pi$
$942$	0	0
$943$	−48.9681	−1.59462
$944$	0	0
$945$	−24.7020	−0.803555
$946$	0	0
$947$	27.0473	0.878921	0.439460	−	0.898262i	$-0.355170\pi$
$947$	27.0473	0.878921	0.439460	+	0.898262i	$0.355170\pi$
$948$	0	0
$949$	−13.6639	−0.443548
$950$	0	0
$951$	8.98044	0.291211
$952$	0	0
$953$	−56.3147	−1.82421	−0.912106	−	0.409955i	$-0.865544\pi$
$953$	−56.3147	−1.82421	−0.912106	+	0.409955i	$0.865544\pi$
$954$	0	0
$955$	7.09057	0.229446
$956$	0	0
$957$	5.92749	0.191608
$958$	0	0
$959$	82.0269	2.64879
$960$	0	0
$961$	−17.0145	−0.548855
$962$	0	0
$963$	6.15429	0.198319
$964$	0	0
$965$	−1.60484	−0.0516618
$966$	0	0
$967$	−43.4123	−1.39605	−0.698023	−	0.716075i	$-0.745937\pi$
$967$	−43.4123	−1.39605	−0.698023	+	0.716075i	$0.745937\pi$
$968$	0	0
$969$	−38.3005	−1.23039
$970$	0	0
$971$	24.6853	0.792188	0.396094	−	0.918210i	$-0.370365\pi$
$971$	24.6853	0.792188	0.396094	+	0.918210i	$0.370365\pi$
$972$	0	0
$973$	−66.6300	−2.13606
$974$	0	0
$975$	−13.3829	−0.428596
$976$	0	0
$977$	6.17037	0.197408	0.0987039	−	0.995117i	$-0.468530\pi$
$977$	6.17037	0.197408	0.0987039	+	0.995117i	$0.468530\pi$
$978$	0	0
$979$	−5.81040	−0.185701
$980$	0	0
$981$	−1.65442	−0.0528216
$982$	0	0
$983$	4.85529	0.154860	0.0774298	−	0.996998i	$-0.475329\pi$
$983$	4.85529	0.154860	0.0774298	+	0.996998i	$0.475329\pi$
$984$	0	0
$985$	−7.88307	−0.251175
$986$	0	0
$987$	−70.0442	−2.22953
$988$	0	0
$989$	−4.27601	−0.135969
$990$	0	0
$991$	−10.7139	−0.340339	−0.170169	−	0.985415i	$-0.554431\pi$
$991$	−10.7139	−0.340339	−0.170169	+	0.985415i	$0.554431\pi$
$992$	0	0
$993$	−31.6405	−1.00408
$994$	0	0
$995$	14.0986	0.446957
$996$	0	0
$997$	−36.1490	−1.14485	−0.572425	−	0.819957i	$-0.693997\pi$
$997$	−36.1490	−1.14485	−0.572425	+	0.819957i	$0.693997\pi$
$998$	0	0
$999$	−28.2809	−0.894768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.2	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.2	✓	50	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)

\(f(q)\)	\(=\)	\(q-2.97826 q^{3} -0.794149 q^{5} -3.63896 q^{7} +5.87005 q^{9} +O(q^{10})\)	\(q-2.97826 q^{3} -0.794149 q^{5} -3.63896 q^{7} +5.87005 q^{9} -0.674085 q^{11} -1.02843 q^{13} +2.36518 q^{15} +3.65416 q^{17} +3.51928 q^{19} +10.8378 q^{21} +4.04783 q^{23} -4.36933 q^{25} -8.54776 q^{27} +2.95252 q^{29} -3.73972 q^{31} +2.00760 q^{33} +2.88987 q^{35} +3.30857 q^{37} +3.06292 q^{39} -12.0974 q^{41} -1.05637 q^{43} -4.66169 q^{45} -6.46297 q^{47} +6.24201 q^{49} -10.8831 q^{51} +2.27933 q^{53} +0.535324 q^{55} -10.4813 q^{57} -14.8230 q^{59} -6.97730 q^{61} -21.3609 q^{63} +0.816724 q^{65} -12.8224 q^{67} -12.0555 q^{69} -3.34117 q^{71} +13.2862 q^{73} +13.0130 q^{75} +2.45297 q^{77} -15.7620 q^{79} +7.84734 q^{81} +8.63077 q^{83} -2.90195 q^{85} -8.79338 q^{87} +8.61968 q^{89} +3.74240 q^{91} +11.1379 q^{93} -2.79483 q^{95} +13.8912 q^{97} -3.95691 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9} - 5 q^{11} + 36 q^{13} + 5 q^{15} + 14 q^{17} + 9 q^{19} + 30 q^{21} + 3 q^{23} + 71 q^{25} + 24 q^{27} + 61 q^{29} + 27 q^{31} + 24 q^{33} - 7 q^{35} + 56 q^{37} - 2 q^{39} + 10 q^{41} + 19 q^{43} + 76 q^{45} + 3 q^{47} + 82 q^{49} - q^{51} + 56 q^{53} + 7 q^{55} + 35 q^{57} - q^{59} + 67 q^{61} + 25 q^{63} + 27 q^{65} + 46 q^{67} + 68 q^{69} + 4 q^{71} + 62 q^{73} + 27 q^{75} + 71 q^{77} + 7 q^{79} + 74 q^{81} - q^{83} + 72 q^{85} + 25 q^{87} + 19 q^{89} + 45 q^{91} + 72 q^{93} - 24 q^{95} + 81 q^{97} + 16 q^{99}+O(q^{100}) \) 50 * q + 6 * q^3 + 23 * q^5 + 12 * q^7 + 56 * q^9 - 5 * q^11 + 36 * q^13 + 5 * q^15 + 14 * q^17 + 9 * q^19 + 30 * q^21 + 3 * q^23 + 71 * q^25 + 24 * q^27 + 61 * q^29 + 27 * q^31 + 24 * q^33 - 7 * q^35 + 56 * q^37 - 2 * q^39 + 10 * q^41 + 19 * q^43 + 76 * q^45 + 3 * q^47 + 82 * q^49 - q^51 + 56 * q^53 + 7 * q^55 + 35 * q^57 - q^59 + 67 * q^61 + 25 * q^63 + 27 * q^65 + 46 * q^67 + 68 * q^69 + 4 * q^71 + 62 * q^73 + 27 * q^75 + 71 * q^77 + 7 * q^79 + 74 * q^81 - q^83 + 72 * q^85 + 25 * q^87 + 19 * q^89 + 45 * q^91 + 72 * q^93 - 24 * q^95 + 81 * q^97 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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