LMFDB - Embedded newform 6008.2.a.d.1.14

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:	$49$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	$q-1.42908 q^{3} +1.27830 q^{5} -2.61604 q^{7} -0.957738 q^{9} +O(q^{10})$	$q-1.42908 q^{3} +1.27830 q^{5} -2.61604 q^{7} -0.957738 q^{9} -1.29535 q^{11} -5.80973 q^{13} -1.82679 q^{15} +2.58492 q^{17} +6.56267 q^{19} +3.73852 q^{21} +5.85897 q^{23} -3.36594 q^{25} +5.65591 q^{27} -4.06796 q^{29} -8.26281 q^{31} +1.85116 q^{33} -3.34409 q^{35} -2.48571 q^{37} +8.30255 q^{39} +5.50653 q^{41} -3.95838 q^{43} -1.22428 q^{45} -11.8444 q^{47} -0.156345 q^{49} -3.69405 q^{51} -4.84380 q^{53} -1.65585 q^{55} -9.37856 q^{57} +0.978309 q^{59} -2.55449 q^{61} +2.50548 q^{63} -7.42659 q^{65} +0.387301 q^{67} -8.37292 q^{69} -5.33452 q^{71} -1.99447 q^{73} +4.81019 q^{75} +3.38869 q^{77} +0.356967 q^{79} -5.20953 q^{81} +7.47561 q^{83} +3.30431 q^{85} +5.81343 q^{87} +10.4055 q^{89} +15.1985 q^{91} +11.8082 q^{93} +8.38907 q^{95} +1.59881 q^{97} +1.24061 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	$ 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) $ 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.42908	−0.825078	−0.412539	−	0.910940i	$-0.635358\pi$
$3$	−1.42908	−0.825078	−0.412539	+	0.910940i	$0.635358\pi$
$4$	0	0
$5$	1.27830	0.571674	0.285837	−	0.958278i	$-0.407728\pi$
$5$	1.27830	0.571674	0.285837	+	0.958278i	$0.407728\pi$
$6$	0	0
$7$	−2.61604	−0.988769	−0.494385	−	0.869243i	$-0.664607\pi$
$7$	−2.61604	−0.988769	−0.494385	+	0.869243i	$0.664607\pi$
$8$	0	0
$9$	−0.957738	−0.319246
$10$	0	0
$11$	−1.29535	−0.390563	−0.195282	−	0.980747i	$-0.562562\pi$
$11$	−1.29535	−0.390563	−0.195282	+	0.980747i	$0.562562\pi$
$12$	0	0
$13$	−5.80973	−1.61133	−0.805664	−	0.592372i	$-0.798192\pi$
$13$	−5.80973	−1.61133	−0.805664	+	0.592372i	$0.798192\pi$
$14$	0	0
$15$	−1.82679	−0.471676
$16$	0	0
$17$	2.58492	0.626935	0.313468	−	0.949599i	$-0.398509\pi$
$17$	2.58492	0.626935	0.313468	+	0.949599i	$0.398509\pi$
$18$	0	0
$19$	6.56267	1.50558	0.752789	−	0.658261i	$-0.228708\pi$
$19$	6.56267	1.50558	0.752789	+	0.658261i	$0.228708\pi$
$20$	0	0
$21$	3.73852	0.815812
$22$	0	0
$23$	5.85897	1.22168	0.610840	−	0.791754i	$-0.290832\pi$
$23$	5.85897	1.22168	0.610840	+	0.791754i	$0.290832\pi$
$24$	0	0
$25$	−3.36594	−0.673189
$26$	0	0
$27$	5.65591	1.08848
$28$	0	0
$29$	−4.06796	−0.755401	−0.377700	−	0.925928i	$-0.623285\pi$
$29$	−4.06796	−0.755401	−0.377700	+	0.925928i	$0.623285\pi$
$30$	0	0
$31$	−8.26281	−1.48405	−0.742023	−	0.670375i	$-0.766133\pi$
$31$	−8.26281	−1.48405	−0.742023	+	0.670375i	$0.766133\pi$
$32$	0	0
$33$	1.85116	0.322245
$34$	0	0
$35$	−3.34409	−0.565254
$36$	0	0
$37$	−2.48571	−0.408648	−0.204324	−	0.978903i	$-0.565500\pi$
$37$	−2.48571	−0.408648	−0.204324	+	0.978903i	$0.565500\pi$
$38$	0	0
$39$	8.30255	1.32947
$40$	0	0
$41$	5.50653	0.859976	0.429988	−	0.902835i	$-0.358518\pi$
$41$	5.50653	0.859976	0.429988	+	0.902835i	$0.358518\pi$
$42$	0	0
$43$	−3.95838	−0.603647	−0.301824	−	0.953364i	$-0.597595\pi$
$43$	−3.95838	−0.603647	−0.301824	+	0.953364i	$0.597595\pi$
$44$	0	0
$45$	−1.22428	−0.182505
$46$	0	0
$47$	−11.8444	−1.72769	−0.863845	−	0.503758i	$-0.831950\pi$
$47$	−11.8444	−1.72769	−0.863845	+	0.503758i	$0.831950\pi$
$48$	0	0
$49$	−0.156345	−0.0223351
$50$	0	0
$51$	−3.69405	−0.517271
$52$	0	0
$53$	−4.84380	−0.665346	−0.332673	−	0.943042i	$-0.607951\pi$
$53$	−4.84380	−0.665346	−0.332673	+	0.943042i	$0.607951\pi$
$54$	0	0
$55$	−1.65585	−0.223275
$56$	0	0
$57$	−9.37856	−1.24222
$58$	0	0
$59$	0.978309	0.127365	0.0636825	−	0.997970i	$-0.479716\pi$
$59$	0.978309	0.127365	0.0636825	+	0.997970i	$0.479716\pi$
$60$	0	0
$61$	−2.55449	−0.327068	−0.163534	−	0.986538i	$-0.552289\pi$
$61$	−2.55449	−0.327068	−0.163534	+	0.986538i	$0.552289\pi$
$62$	0	0
$63$	2.50548	0.315661
$64$	0	0
$65$	−7.42659	−0.921155
$66$	0	0
$67$	0.387301	0.0473163	0.0236582	−	0.999720i	$-0.492469\pi$
$67$	0.387301	0.0473163	0.0236582	+	0.999720i	$0.492469\pi$
$68$	0	0
$69$	−8.37292	−1.00798
$70$	0	0
$71$	−5.33452	−0.633091	−0.316545	−	0.948577i	$-0.602523\pi$
$71$	−5.33452	−0.633091	−0.316545	+	0.948577i	$0.602523\pi$
$72$	0	0
$73$	−1.99447	−0.233435	−0.116717	−	0.993165i	$-0.537237\pi$
$73$	−1.99447	−0.233435	−0.116717	+	0.993165i	$0.537237\pi$
$74$	0	0
$75$	4.81019	0.555433
$76$	0	0
$77$	3.38869	0.386177
$78$	0	0
$79$	0.356967	0.0401620	0.0200810	−	0.999798i	$-0.493608\pi$
$79$	0.356967	0.0401620	0.0200810	+	0.999798i	$0.493608\pi$
$80$	0	0
$81$	−5.20953	−0.578836
$82$	0	0
$83$	7.47561	0.820555	0.410278	−	0.911961i	$-0.365432\pi$
$83$	7.47561	0.820555	0.410278	+	0.911961i	$0.365432\pi$
$84$	0	0
$85$	3.30431	0.358403
$86$	0	0
$87$	5.81343	0.623265
$88$	0	0
$89$	10.4055	1.10299	0.551493	−	0.834180i	$-0.314058\pi$
$89$	10.4055	1.10299	0.551493	+	0.834180i	$0.314058\pi$
$90$	0	0
$91$	15.1985	1.59323
$92$	0	0
$93$	11.8082	1.22445
$94$	0	0
$95$	8.38907	0.860700
$96$	0	0
$97$	1.59881	0.162335	0.0811673	−	0.996700i	$-0.474135\pi$
$97$	1.59881	0.162335	0.0811673	+	0.996700i	$0.474135\pi$
$98$	0	0
$99$	1.24061	0.124686
$100$	0	0
$101$	1.43333	0.142621	0.0713107	−	0.997454i	$-0.477282\pi$
$101$	1.43333	0.142621	0.0713107	+	0.997454i	$0.477282\pi$
$102$	0	0
$103$	3.96741	0.390920	0.195460	−	0.980712i	$-0.437380\pi$
$103$	3.96741	0.390920	0.195460	+	0.980712i	$0.437380\pi$
$104$	0	0
$105$	4.77896	0.466379
$106$	0	0
$107$	15.8407	1.53137	0.765687	−	0.643213i	$-0.222399\pi$
$107$	15.8407	1.53137	0.765687	+	0.643213i	$0.222399\pi$
$108$	0	0
$109$	−19.5454	−1.87211	−0.936055	−	0.351854i	$-0.885551\pi$
$109$	−19.5454	−1.87211	−0.936055	+	0.351854i	$0.885551\pi$
$110$	0	0
$111$	3.55227	0.337166
$112$	0	0
$113$	9.51237	0.894849	0.447424	−	0.894322i	$-0.352341\pi$
$113$	9.51237	0.894849	0.447424	+	0.894322i	$0.352341\pi$
$114$	0	0
$115$	7.48954	0.698403
$116$	0	0
$117$	5.56420	0.514410
$118$	0	0
$119$	−6.76225	−0.619895
$120$	0	0
$121$	−9.32206	−0.847460
$122$	0	0
$123$	−7.86926	−0.709548
$124$	0	0
$125$	−10.6942	−0.956519
$126$	0	0
$127$	19.7305	1.75080	0.875400	−	0.483399i	$-0.160598\pi$
$127$	19.7305	1.75080	0.875400	+	0.483399i	$0.160598\pi$
$128$	0	0
$129$	5.65683	0.498056
$130$	0	0
$131$	20.8015	1.81743	0.908717	−	0.417412i	$-0.137063\pi$
$131$	20.8015	1.81743	0.908717	+	0.417412i	$0.137063\pi$
$132$	0	0
$133$	−17.1682	−1.48867
$134$	0	0
$135$	7.22997	0.622256
$136$	0	0
$137$	3.93568	0.336248	0.168124	−	0.985766i	$-0.446229\pi$
$137$	3.93568	0.336248	0.168124	+	0.985766i	$0.446229\pi$
$138$	0	0
$139$	−10.5496	−0.894802	−0.447401	−	0.894333i	$-0.647650\pi$
$139$	−10.5496	−0.894802	−0.447401	+	0.894333i	$0.647650\pi$
$140$	0	0
$141$	16.9266	1.42548
$142$	0	0
$143$	7.52564	0.629326
$144$	0	0
$145$	−5.20008	−0.431843
$146$	0	0
$147$	0.223430	0.0184282
$148$	0	0
$149$	−0.715190	−0.0585907	−0.0292953	−	0.999571i	$-0.509326\pi$
$149$	−0.715190	−0.0585907	−0.0292953	+	0.999571i	$0.509326\pi$
$150$	0	0
$151$	7.42483	0.604224	0.302112	−	0.953272i	$-0.402308\pi$
$151$	7.42483	0.604224	0.302112	+	0.953272i	$0.402308\pi$
$152$	0	0
$153$	−2.47568	−0.200147
$154$	0	0
$155$	−10.5624	−0.848390
$156$	0	0
$157$	−14.8276	−1.18337	−0.591684	−	0.806170i	$-0.701537\pi$
$157$	−14.8276	−1.18337	−0.591684	+	0.806170i	$0.701537\pi$
$158$	0	0
$159$	6.92216	0.548963
$160$	0	0
$161$	−15.3273	−1.20796
$162$	0	0
$163$	19.3776	1.51777	0.758885	−	0.651225i	$-0.225745\pi$
$163$	19.3776	1.51777	0.758885	+	0.651225i	$0.225745\pi$
$164$	0	0
$165$	2.36634	0.184219
$166$	0	0
$167$	8.37146	0.647803	0.323902	−	0.946091i	$-0.395005\pi$
$167$	8.37146	0.647803	0.323902	+	0.946091i	$0.395005\pi$
$168$	0	0
$169$	20.7530	1.59638
$170$	0	0
$171$	−6.28531	−0.480650
$172$	0	0
$173$	5.64425	0.429125	0.214562	−	0.976710i	$-0.431168\pi$
$173$	5.64425	0.429125	0.214562	+	0.976710i	$0.431168\pi$
$174$	0	0
$175$	8.80544	0.665628
$176$	0	0
$177$	−1.39808	−0.105086
$178$	0	0
$179$	17.7744	1.32852	0.664261	−	0.747501i	$-0.268746\pi$
$179$	17.7744	1.32852	0.664261	+	0.747501i	$0.268746\pi$
$180$	0	0
$181$	6.51166	0.484008	0.242004	−	0.970275i	$-0.422195\pi$
$181$	6.51166	0.484008	0.242004	+	0.970275i	$0.422195\pi$
$182$	0	0
$183$	3.65056	0.269857
$184$	0	0
$185$	−3.17748	−0.233613
$186$	0	0
$187$	−3.34838	−0.244858
$188$	0	0
$189$	−14.7961	−1.07626
$190$	0	0
$191$	17.5328	1.26863	0.634314	−	0.773075i	$-0.281283\pi$
$191$	17.5328	1.26863	0.634314	+	0.773075i	$0.281283\pi$
$192$	0	0
$193$	−10.9529	−0.788404	−0.394202	−	0.919024i	$-0.628979\pi$
$193$	−10.9529	−0.788404	−0.394202	+	0.919024i	$0.628979\pi$
$194$	0	0
$195$	10.6132	0.760025
$196$	0	0
$197$	15.6052	1.11182	0.555912	−	0.831241i	$-0.312369\pi$
$197$	15.6052	1.11182	0.555912	+	0.831241i	$0.312369\pi$
$198$	0	0
$199$	−11.3953	−0.807790	−0.403895	−	0.914805i	$-0.632344\pi$
$199$	−11.3953	−0.807790	−0.403895	+	0.914805i	$0.632344\pi$
$200$	0	0
$201$	−0.553483	−0.0390397
$202$	0	0
$203$	10.6419	0.746917
$204$	0	0
$205$	7.03902	0.491626
$206$	0	0
$207$	−5.61136	−0.390016
$208$	0	0
$209$	−8.50096	−0.588024
$210$	0	0
$211$	0.807382	0.0555825	0.0277912	−	0.999614i	$-0.491153\pi$
$211$	0.807382	0.0555825	0.0277912	+	0.999614i	$0.491153\pi$
$212$	0	0
$213$	7.62344	0.522349
$214$	0	0
$215$	−5.06000	−0.345089
$216$	0	0
$217$	21.6158	1.46738
$218$	0	0
$219$	2.85025	0.192602
$220$	0	0
$221$	−15.0177	−1.01020
$222$	0	0
$223$	18.8075	1.25944	0.629722	−	0.776821i	$-0.283169\pi$
$223$	18.8075	1.25944	0.629722	+	0.776821i	$0.283169\pi$
$224$	0	0
$225$	3.22369	0.214913
$226$	0	0
$227$	−10.2774	−0.682133	−0.341067	−	0.940039i	$-0.610788\pi$
$227$	−10.2774	−0.682133	−0.341067	+	0.940039i	$0.610788\pi$
$228$	0	0
$229$	−15.7971	−1.04390	−0.521952	−	0.852975i	$-0.674796\pi$
$229$	−15.7971	−1.04390	−0.521952	+	0.852975i	$0.674796\pi$
$230$	0	0
$231$	−4.84270	−0.318626
$232$	0	0
$233$	23.8087	1.55976	0.779879	−	0.625931i	$-0.215281\pi$
$233$	23.8087	1.55976	0.779879	+	0.625931i	$0.215281\pi$
$234$	0	0
$235$	−15.1408	−0.987675
$236$	0	0
$237$	−0.510134	−0.0331368
$238$	0	0
$239$	1.66017	0.107387	0.0536936	−	0.998557i	$-0.482901\pi$
$239$	1.66017	0.107387	0.0536936	+	0.998557i	$0.482901\pi$
$240$	0	0
$241$	−5.87209	−0.378255	−0.189127	−	0.981953i	$-0.560566\pi$
$241$	−5.87209	−0.378255	−0.189127	+	0.981953i	$0.560566\pi$
$242$	0	0
$243$	−9.52292	−0.610896
$244$	0	0
$245$	−0.199857	−0.0127684
$246$	0	0
$247$	−38.1273	−2.42598
$248$	0	0
$249$	−10.6832	−0.677022
$250$	0	0
$251$	28.5898	1.80457	0.902286	−	0.431139i	$-0.141888\pi$
$251$	28.5898	1.80457	0.902286	+	0.431139i	$0.141888\pi$
$252$	0	0
$253$	−7.58943	−0.477143
$254$	0	0
$255$	−4.72212	−0.295710
$256$	0	0
$257$	−4.29490	−0.267909	−0.133954	−	0.990988i	$-0.542768\pi$
$257$	−4.29490	−0.267909	−0.133954	+	0.990988i	$0.542768\pi$
$258$	0	0
$259$	6.50270	0.404058
$260$	0	0
$261$	3.89604	0.241159
$262$	0	0
$263$	7.12914	0.439602	0.219801	−	0.975545i	$-0.429459\pi$
$263$	7.12914	0.439602	0.219801	+	0.975545i	$0.429459\pi$
$264$	0	0
$265$	−6.19183	−0.380361
$266$	0	0
$267$	−14.8703	−0.910050
$268$	0	0
$269$	−20.8590	−1.27180	−0.635898	−	0.771773i	$-0.719370\pi$
$269$	−20.8590	−1.27180	−0.635898	+	0.771773i	$0.719370\pi$
$270$	0	0
$271$	−26.0052	−1.57970	−0.789851	−	0.613299i	$-0.789842\pi$
$271$	−26.0052	−1.57970	−0.789851	+	0.613299i	$0.789842\pi$
$272$	0	0
$273$	−21.7198	−1.31454
$274$	0	0
$275$	4.36008	0.262923
$276$	0	0
$277$	23.4834	1.41098	0.705491	−	0.708719i	$-0.250727\pi$
$277$	23.4834	1.41098	0.705491	+	0.708719i	$0.250727\pi$
$278$	0	0
$279$	7.91361	0.473775
$280$	0	0
$281$	30.0996	1.79559	0.897796	−	0.440413i	$-0.145168\pi$
$281$	30.0996	1.79559	0.897796	+	0.440413i	$0.145168\pi$
$282$	0	0
$283$	7.12709	0.423662	0.211831	−	0.977306i	$-0.432057\pi$
$283$	7.12709	0.423662	0.211831	+	0.977306i	$0.432057\pi$
$284$	0	0
$285$	−11.9886	−0.710145
$286$	0	0
$287$	−14.4053	−0.850318
$288$	0	0
$289$	−10.3182	−0.606952
$290$	0	0
$291$	−2.28482	−0.133939
$292$	0	0
$293$	−0.581453	−0.0339688	−0.0169844	−	0.999856i	$-0.505407\pi$
$293$	−0.581453	−0.0339688	−0.0169844	+	0.999856i	$0.505407\pi$
$294$	0	0
$295$	1.25057	0.0728113
$296$	0	0
$297$	−7.32640	−0.425121
$298$	0	0
$299$	−34.0390	−1.96853
$300$	0	0
$301$	10.3553	0.596868
$302$	0	0
$303$	−2.04834	−0.117674
$304$	0	0
$305$	−3.26541	−0.186977
$306$	0	0
$307$	10.4237	0.594911	0.297455	−	0.954736i	$-0.403862\pi$
$307$	10.4237	0.594911	0.297455	+	0.954736i	$0.403862\pi$
$308$	0	0
$309$	−5.66973	−0.322540
$310$	0	0
$311$	−12.4576	−0.706405	−0.353203	−	0.935547i	$-0.614907\pi$
$311$	−12.4576	−0.706405	−0.353203	+	0.935547i	$0.614907\pi$
$312$	0	0
$313$	9.36713	0.529462	0.264731	−	0.964322i	$-0.414717\pi$
$313$	9.36713	0.529462	0.264731	+	0.964322i	$0.414717\pi$
$314$	0	0
$315$	3.20276	0.180455
$316$	0	0
$317$	8.18149	0.459518	0.229759	−	0.973248i	$-0.426206\pi$
$317$	8.18149	0.459518	0.229759	+	0.973248i	$0.426206\pi$
$318$	0	0
$319$	5.26944	0.295032
$320$	0	0
$321$	−22.6375	−1.26350
$322$	0	0
$323$	16.9640	0.943901
$324$	0	0
$325$	19.5552	1.08473
$326$	0	0
$327$	27.9319	1.54464
$328$	0	0
$329$	30.9855	1.70829
$330$	0	0
$331$	−21.3672	−1.17445	−0.587225	−	0.809424i	$-0.699779\pi$
$331$	−21.3672	−1.17445	−0.587225	+	0.809424i	$0.699779\pi$
$332$	0	0
$333$	2.38065	0.130459
$334$	0	0
$335$	0.495087	0.0270495
$336$	0	0
$337$	−25.7981	−1.40531	−0.702657	−	0.711528i	$-0.748003\pi$
$337$	−25.7981	−1.40531	−0.702657	+	0.711528i	$0.748003\pi$
$338$	0	0
$339$	−13.5939	−0.738320
$340$	0	0
$341$	10.7033	0.579614
$342$	0	0
$343$	18.7213	1.01085
$344$	0	0
$345$	−10.7031	−0.576237
$346$	0	0
$347$	5.71238	0.306657	0.153328	−	0.988175i	$-0.451001\pi$
$347$	5.71238	0.306657	0.153328	+	0.988175i	$0.451001\pi$
$348$	0	0
$349$	8.17056	0.437360	0.218680	−	0.975797i	$-0.429825\pi$
$349$	8.17056	0.437360	0.218680	+	0.975797i	$0.429825\pi$
$350$	0	0
$351$	−32.8593	−1.75390
$352$	0	0
$353$	4.03012	0.214502	0.107251	−	0.994232i	$-0.465795\pi$
$353$	4.03012	0.214502	0.107251	+	0.994232i	$0.465795\pi$
$354$	0	0
$355$	−6.81913	−0.361922
$356$	0	0
$357$	9.66378	0.511462
$358$	0	0
$359$	27.1111	1.43087	0.715435	−	0.698679i	$-0.246228\pi$
$359$	27.1111	1.43087	0.715435	+	0.698679i	$0.246228\pi$
$360$	0	0
$361$	24.0686	1.26677
$362$	0	0
$363$	13.3220	0.699221
$364$	0	0
$365$	−2.54953	−0.133449
$366$	0	0
$367$	−6.86636	−0.358421	−0.179210	−	0.983811i	$-0.557354\pi$
$367$	−6.86636	−0.358421	−0.179210	+	0.983811i	$0.557354\pi$
$368$	0	0
$369$	−5.27382	−0.274544
$370$	0	0
$371$	12.6716	0.657874
$372$	0	0
$373$	−27.7410	−1.43637	−0.718187	−	0.695850i	$-0.755028\pi$
$373$	−27.7410	−1.43637	−0.718187	+	0.695850i	$0.755028\pi$
$374$	0	0
$375$	15.2828	0.789203
$376$	0	0
$377$	23.6337	1.21720
$378$	0	0
$379$	31.4886	1.61746	0.808730	−	0.588180i	$-0.200155\pi$
$379$	31.4886	1.61746	0.808730	+	0.588180i	$0.200155\pi$
$380$	0	0
$381$	−28.1964	−1.44455
$382$	0	0
$383$	−21.0004	−1.07307	−0.536535	−	0.843878i	$-0.680267\pi$
$383$	−21.0004	−1.07307	−0.536535	+	0.843878i	$0.680267\pi$
$384$	0	0
$385$	4.33177	0.220767
$386$	0	0
$387$	3.79109	0.192712
$388$	0	0
$389$	−9.58688	−0.486074	−0.243037	−	0.970017i	$-0.578144\pi$
$389$	−9.58688	−0.486074	−0.243037	+	0.970017i	$0.578144\pi$
$390$	0	0
$391$	15.1450	0.765914
$392$	0	0
$393$	−29.7269	−1.49953
$394$	0	0
$395$	0.456312	0.0229596
$396$	0	0
$397$	29.9011	1.50069	0.750347	−	0.661044i	$-0.229887\pi$
$397$	29.9011	1.50069	0.750347	+	0.661044i	$0.229887\pi$
$398$	0	0
$399$	24.5347	1.22827
$400$	0	0
$401$	−3.53500	−0.176529	−0.0882646	−	0.996097i	$-0.528132\pi$
$401$	−3.53500	−0.176529	−0.0882646	+	0.996097i	$0.528132\pi$
$402$	0	0
$403$	48.0047	2.39129
$404$	0	0
$405$	−6.65935	−0.330906
$406$	0	0
$407$	3.21986	0.159603
$408$	0	0
$409$	−32.5973	−1.61183	−0.805917	−	0.592029i	$-0.798327\pi$
$409$	−32.5973	−1.61183	−0.805917	+	0.592029i	$0.798327\pi$
$410$	0	0
$411$	−5.62439	−0.277431
$412$	0	0
$413$	−2.55929	−0.125935
$414$	0	0
$415$	9.55609	0.469090
$416$	0	0
$417$	15.0761	0.738282
$418$	0	0
$419$	−24.9930	−1.22099	−0.610493	−	0.792021i	$-0.709029\pi$
$419$	−24.9930	−1.22099	−0.610493	+	0.792021i	$0.709029\pi$
$420$	0	0
$421$	31.7230	1.54608	0.773042	−	0.634355i	$-0.218734\pi$
$421$	31.7230	1.54608	0.773042	+	0.634355i	$0.218734\pi$
$422$	0	0
$423$	11.3439	0.551558
$424$	0	0
$425$	−8.70070	−0.422046
$426$	0	0
$427$	6.68263	0.323395
$428$	0	0
$429$	−10.7547	−0.519243
$430$	0	0
$431$	−1.37017	−0.0659989	−0.0329994	−	0.999455i	$-0.510506\pi$
$431$	−1.37017	−0.0659989	−0.0329994	+	0.999455i	$0.510506\pi$
$432$	0	0
$433$	37.9548	1.82399	0.911996	−	0.410199i	$-0.134541\pi$
$433$	37.9548	1.82399	0.911996	+	0.410199i	$0.134541\pi$
$434$	0	0
$435$	7.43131	0.356304
$436$	0	0
$437$	38.4505	1.83934
$438$	0	0
$439$	10.6515	0.508368	0.254184	−	0.967156i	$-0.418193\pi$
$439$	10.6515	0.508368	0.254184	+	0.967156i	$0.418193\pi$
$440$	0	0
$441$	0.149738	0.00713038
$442$	0	0
$443$	−30.8375	−1.46514	−0.732568	−	0.680694i	$-0.761678\pi$
$443$	−30.8375	−1.46514	−0.732568	+	0.680694i	$0.761678\pi$
$444$	0	0
$445$	13.3014	0.630548
$446$	0	0
$447$	1.02206	0.0483419
$448$	0	0
$449$	−5.34558	−0.252273	−0.126137	−	0.992013i	$-0.540258\pi$
$449$	−5.34558	−0.252273	−0.126137	+	0.992013i	$0.540258\pi$
$450$	0	0
$451$	−7.13290	−0.335875
$452$	0	0
$453$	−10.6107	−0.498532
$454$	0	0
$455$	19.4282	0.910810
$456$	0	0
$457$	−2.46003	−0.115075	−0.0575377	−	0.998343i	$-0.518325\pi$
$457$	−2.46003	−0.115075	−0.0575377	+	0.998343i	$0.518325\pi$
$458$	0	0
$459$	14.6201	0.682407
$460$	0	0
$461$	31.9164	1.48649	0.743246	−	0.669018i	$-0.233285\pi$
$461$	31.9164	1.48649	0.743246	+	0.669018i	$0.233285\pi$
$462$	0	0
$463$	38.0961	1.77048	0.885239	−	0.465137i	$-0.153995\pi$
$463$	38.0961	1.77048	0.885239	+	0.465137i	$0.153995\pi$
$464$	0	0
$465$	15.0945	0.699988
$466$	0	0
$467$	7.11541	0.329262	0.164631	−	0.986355i	$-0.447357\pi$
$467$	7.11541	0.329262	0.164631	+	0.986355i	$0.447357\pi$
$468$	0	0
$469$	−1.01319	−0.0467849
$470$	0	0
$471$	21.1897	0.976372
$472$	0	0
$473$	5.12749	0.235762
$474$	0	0
$475$	−22.0896	−1.01354
$476$	0	0
$477$	4.63908	0.212409
$478$	0	0
$479$	29.1571	1.33222	0.666111	−	0.745853i	$-0.267958\pi$
$479$	29.1571	1.33222	0.666111	+	0.745853i	$0.267958\pi$
$480$	0	0
$481$	14.4413	0.658466
$482$	0	0
$483$	21.9039	0.996661
$484$	0	0
$485$	2.04376	0.0928024
$486$	0	0
$487$	27.3645	1.24000	0.620001	−	0.784601i	$-0.287132\pi$
$487$	27.3645	1.24000	0.620001	+	0.784601i	$0.287132\pi$
$488$	0	0
$489$	−27.6921	−1.25228
$490$	0	0
$491$	−4.31513	−0.194739	−0.0973696	−	0.995248i	$-0.531043\pi$
$491$	−4.31513	−0.194739	−0.0973696	+	0.995248i	$0.531043\pi$
$492$	0	0
$493$	−10.5153	−0.473587
$494$	0	0
$495$	1.58587	0.0712796
$496$	0	0
$497$	13.9553	0.625981
$498$	0	0
$499$	22.5851	1.01105	0.505524	−	0.862813i	$-0.331299\pi$
$499$	22.5851	1.01105	0.505524	+	0.862813i	$0.331299\pi$
$500$	0	0
$501$	−11.9635	−0.534488
$502$	0	0
$503$	−3.32190	−0.148116	−0.0740582	−	0.997254i	$-0.523595\pi$
$503$	−3.32190	−0.148116	−0.0740582	+	0.997254i	$0.523595\pi$
$504$	0	0
$505$	1.83223	0.0815330
$506$	0	0
$507$	−29.6576	−1.31714
$508$	0	0
$509$	27.0772	1.20018	0.600088	−	0.799934i	$-0.295132\pi$
$509$	27.0772	1.20018	0.600088	+	0.799934i	$0.295132\pi$
$510$	0	0
$511$	5.21760	0.230813
$512$	0	0
$513$	37.1179	1.63879
$514$	0	0
$515$	5.07154	0.223479
$516$	0	0
$517$	15.3427	0.674772
$518$	0	0
$519$	−8.06607	−0.354061
$520$	0	0
$521$	25.0963	1.09949	0.549745	−	0.835332i	$-0.314725\pi$
$521$	25.0963	1.09949	0.549745	+	0.835332i	$0.314725\pi$
$522$	0	0
$523$	−34.3388	−1.50153	−0.750765	−	0.660569i	$-0.770315\pi$
$523$	−34.3388	−1.50153	−0.750765	+	0.660569i	$0.770315\pi$
$524$	0	0
$525$	−12.5836	−0.549195
$526$	0	0
$527$	−21.3587	−0.930401
$528$	0	0
$529$	11.3275	0.492502
$530$	0	0
$531$	−0.936963	−0.0406607
$532$	0	0
$533$	−31.9915	−1.38570
$534$	0	0
$535$	20.2492	0.875447
$536$	0	0
$537$	−25.4010	−1.09613
$538$	0	0
$539$	0.202522	0.00872326
$540$	0	0
$541$	−18.2014	−0.782539	−0.391269	−	0.920276i	$-0.627964\pi$
$541$	−18.2014	−0.782539	−0.391269	+	0.920276i	$0.627964\pi$
$542$	0	0
$543$	−9.30566	−0.399344
$544$	0	0
$545$	−24.9849	−1.07024
$546$	0	0
$547$	−22.1414	−0.946697	−0.473348	−	0.880875i	$-0.656955\pi$
$547$	−22.1414	−0.946697	−0.473348	+	0.880875i	$0.656955\pi$
$548$	0	0
$549$	2.44653	0.104415
$550$	0	0
$551$	−26.6966	−1.13732
$552$	0	0
$553$	−0.933840	−0.0397109
$554$	0	0
$555$	4.54087	0.192749
$556$	0	0
$557$	15.3081	0.648625	0.324313	−	0.945950i	$-0.394867\pi$
$557$	15.3081	0.648625	0.324313	+	0.945950i	$0.394867\pi$
$558$	0	0
$559$	22.9971	0.972674
$560$	0	0
$561$	4.78510	0.202027
$562$	0	0
$563$	−29.1968	−1.23050	−0.615249	−	0.788333i	$-0.710944\pi$
$563$	−29.1968	−1.23050	−0.615249	+	0.788333i	$0.710944\pi$
$564$	0	0
$565$	12.1597	0.511562
$566$	0	0
$567$	13.6283	0.572336
$568$	0	0
$569$	−33.4091	−1.40058	−0.700292	−	0.713857i	$-0.746947\pi$
$569$	−33.4091	−1.40058	−0.700292	+	0.713857i	$0.746947\pi$
$570$	0	0
$571$	−33.0857	−1.38459	−0.692296	−	0.721613i	$-0.743401\pi$
$571$	−33.0857	−1.38459	−0.692296	+	0.721613i	$0.743401\pi$
$572$	0	0
$573$	−25.0557	−1.04672
$574$	0	0
$575$	−19.7210	−0.822421
$576$	0	0
$577$	33.4488	1.39249	0.696246	−	0.717803i	$-0.254852\pi$
$577$	33.4488	1.39249	0.696246	+	0.717803i	$0.254852\pi$
$578$	0	0
$579$	15.6525	0.650495
$580$	0	0
$581$	−19.5565	−0.811340
$582$	0	0
$583$	6.27442	0.259860
$584$	0	0
$585$	7.11272	0.294075
$586$	0	0
$587$	14.5543	0.600718	0.300359	−	0.953826i	$-0.402893\pi$
$587$	14.5543	0.600718	0.300359	+	0.953826i	$0.402893\pi$
$588$	0	0
$589$	−54.2261	−2.23435
$590$	0	0
$591$	−22.3010	−0.917341
$592$	0	0
$593$	13.9736	0.573825	0.286913	−	0.957957i	$-0.407371\pi$
$593$	13.9736	0.573825	0.286913	+	0.957957i	$0.407371\pi$
$594$	0	0
$595$	−8.64420	−0.354378
$596$	0	0
$597$	16.2847	0.666490
$598$	0	0
$599$	−35.7950	−1.46254	−0.731272	−	0.682086i	$-0.761073\pi$
$599$	−35.7950	−1.46254	−0.731272	+	0.682086i	$0.761073\pi$
$600$	0	0
$601$	30.1293	1.22900	0.614500	−	0.788917i	$-0.289358\pi$
$601$	30.1293	1.22900	0.614500	+	0.788917i	$0.289358\pi$
$602$	0	0
$603$	−0.370933	−0.0151055
$604$	0	0
$605$	−11.9164	−0.484471
$606$	0	0
$607$	23.3399	0.947337	0.473668	−	0.880703i	$-0.342930\pi$
$607$	23.3399	0.947337	0.473668	+	0.880703i	$0.342930\pi$
$608$	0	0
$609$	−15.2081	−0.616265
$610$	0	0
$611$	68.8130	2.78388
$612$	0	0
$613$	−28.4695	−1.14987	−0.574935	−	0.818199i	$-0.694973\pi$
$613$	−28.4695	−1.14987	−0.574935	+	0.818199i	$0.694973\pi$
$614$	0	0
$615$	−10.0593	−0.405630
$616$	0	0
$617$	−7.47418	−0.300899	−0.150450	−	0.988618i	$-0.548072\pi$
$617$	−7.47418	−0.300899	−0.150450	+	0.988618i	$0.548072\pi$
$618$	0	0
$619$	−29.3172	−1.17836	−0.589180	−	0.808002i	$-0.700549\pi$
$619$	−29.3172	−1.17836	−0.589180	+	0.808002i	$0.700549\pi$
$620$	0	0
$621$	33.1378	1.32978
$622$	0	0
$623$	−27.2213	−1.09060
$624$	0	0
$625$	3.15929	0.126372
$626$	0	0
$627$	12.1485	0.485166
$628$	0	0
$629$	−6.42535	−0.256196
$630$	0	0
$631$	−38.0173	−1.51344	−0.756722	−	0.653737i	$-0.773200\pi$
$631$	−38.0173	−1.51344	−0.756722	+	0.653737i	$0.773200\pi$
$632$	0	0
$633$	−1.15381	−0.0458599
$634$	0	0
$635$	25.2216	1.00089
$636$	0	0
$637$	0.908325	0.0359891
$638$	0	0
$639$	5.10907	0.202112
$640$	0	0
$641$	9.83649	0.388518	0.194259	−	0.980950i	$-0.437770\pi$
$641$	9.83649	0.388518	0.194259	+	0.980950i	$0.437770\pi$
$642$	0	0
$643$	−3.32351	−0.131066	−0.0655332	−	0.997850i	$-0.520875\pi$
$643$	−3.32351	−0.131066	−0.0655332	+	0.997850i	$0.520875\pi$
$644$	0	0
$645$	7.23114	0.284726
$646$	0	0
$647$	−16.6051	−0.652812	−0.326406	−	0.945230i	$-0.605838\pi$
$647$	−16.6051	−0.652812	−0.326406	+	0.945230i	$0.605838\pi$
$648$	0	0
$649$	−1.26725	−0.0497441
$650$	0	0
$651$	−30.8907	−1.21070
$652$	0	0
$653$	−25.0086	−0.978662	−0.489331	−	0.872098i	$-0.662759\pi$
$653$	−25.0086	−0.978662	−0.489331	+	0.872098i	$0.662759\pi$
$654$	0	0
$655$	26.5906	1.03898
$656$	0	0
$657$	1.91018	0.0745230
$658$	0	0
$659$	22.0858	0.860340	0.430170	−	0.902748i	$-0.358454\pi$
$659$	22.0858	0.860340	0.430170	+	0.902748i	$0.358454\pi$
$660$	0	0
$661$	−34.5855	−1.34522	−0.672611	−	0.739997i	$-0.734827\pi$
$661$	−34.5855	−1.34522	−0.672611	+	0.739997i	$0.734827\pi$
$662$	0	0
$663$	21.4614	0.833493
$664$	0	0
$665$	−21.9461	−0.851034
$666$	0	0
$667$	−23.8340	−0.922858
$668$	0	0
$669$	−26.8774	−1.03914
$670$	0	0
$671$	3.30896	0.127741
$672$	0	0
$673$	2.51310	0.0968727	0.0484364	−	0.998826i	$-0.484576\pi$
$673$	2.51310	0.0968727	0.0484364	+	0.998826i	$0.484576\pi$
$674$	0	0
$675$	−19.0375	−0.732753
$676$	0	0
$677$	31.9791	1.22906	0.614528	−	0.788895i	$-0.289346\pi$
$677$	31.9791	1.22906	0.614528	+	0.788895i	$0.289346\pi$
$678$	0	0
$679$	−4.18255	−0.160511
$680$	0	0
$681$	14.6872	0.562813
$682$	0	0
$683$	−38.0129	−1.45452	−0.727262	−	0.686360i	$-0.759207\pi$
$683$	−38.0129	−1.45452	−0.727262	+	0.686360i	$0.759207\pi$
$684$	0	0
$685$	5.03099	0.192224
$686$	0	0
$687$	22.5753	0.861302
$688$	0	0
$689$	28.1411	1.07209
$690$	0	0
$691$	−15.5119	−0.590100	−0.295050	−	0.955482i	$-0.595336\pi$
$691$	−15.5119	−0.590100	−0.295050	+	0.955482i	$0.595336\pi$
$692$	0	0
$693$	−3.24548	−0.123285
$694$	0	0
$695$	−13.4855	−0.511535
$696$	0	0
$697$	14.2340	0.539150
$698$	0	0
$699$	−34.0244	−1.28692
$700$	0	0
$701$	17.6531	0.666748	0.333374	−	0.942795i	$-0.391813\pi$
$701$	17.6531	0.666748	0.333374	+	0.942795i	$0.391813\pi$
$702$	0	0
$703$	−16.3129	−0.615251
$704$	0	0
$705$	21.6373	0.814909
$706$	0	0
$707$	−3.74964	−0.141020
$708$	0	0
$709$	−6.35072	−0.238506	−0.119253	−	0.992864i	$-0.538050\pi$
$709$	−6.35072	−0.238506	−0.119253	+	0.992864i	$0.538050\pi$
$710$	0	0
$711$	−0.341881	−0.0128215
$712$	0	0
$713$	−48.4116	−1.81303
$714$	0	0
$715$	9.62005	0.359769
$716$	0	0
$717$	−2.37251	−0.0886029
$718$	0	0
$719$	50.0537	1.86669	0.933344	−	0.358984i	$-0.116877\pi$
$719$	50.0537	1.86669	0.933344	+	0.358984i	$0.116877\pi$
$720$	0	0
$721$	−10.3789	−0.386530
$722$	0	0
$723$	8.39167	0.312090
$724$	0	0
$725$	13.6925	0.508527
$726$	0	0
$727$	23.2240	0.861332	0.430666	−	0.902511i	$-0.358279\pi$
$727$	23.2240	0.861332	0.430666	+	0.902511i	$0.358279\pi$
$728$	0	0
$729$	29.2376	1.08287
$730$	0	0
$731$	−10.2321	−0.378448
$732$	0	0
$733$	−24.9916	−0.923086	−0.461543	−	0.887118i	$-0.652704\pi$
$733$	−24.9916	−0.923086	−0.461543	+	0.887118i	$0.652704\pi$
$734$	0	0
$735$	0.285611	0.0105349
$736$	0	0
$737$	−0.501691	−0.0184800
$738$	0	0
$739$	22.1189	0.813659	0.406829	−	0.913504i	$-0.366634\pi$
$739$	22.1189	0.813659	0.406829	+	0.913504i	$0.366634\pi$
$740$	0	0
$741$	54.4869	2.00163
$742$	0	0
$743$	28.2649	1.03694	0.518469	−	0.855096i	$-0.326502\pi$
$743$	28.2649	1.03694	0.518469	+	0.855096i	$0.326502\pi$
$744$	0	0
$745$	−0.914229	−0.0334948
$746$	0	0
$747$	−7.15968	−0.261959
$748$	0	0
$749$	−41.4398	−1.51418
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−40.8570	−1.48891
$754$	0	0
$755$	9.49118	0.345419
$756$	0	0
$757$	−15.0988	−0.548775	−0.274387	−	0.961619i	$-0.588475\pi$
$757$	−15.0988	−0.548775	−0.274387	+	0.961619i	$0.588475\pi$
$758$	0	0
$759$	10.8459	0.393681
$760$	0	0
$761$	−30.0927	−1.09086	−0.545429	−	0.838157i	$-0.683633\pi$
$761$	−30.0927	−1.09086	−0.545429	+	0.838157i	$0.683633\pi$
$762$	0	0
$763$	51.1315	1.85108
$764$	0	0
$765$	−3.16466	−0.114419
$766$	0	0
$767$	−5.68371	−0.205227
$768$	0	0
$769$	33.4658	1.20681	0.603403	−	0.797436i	$-0.293811\pi$
$769$	33.4658	1.20681	0.603403	+	0.797436i	$0.293811\pi$
$770$	0	0
$771$	6.13775	0.221046
$772$	0	0
$773$	23.0717	0.829831	0.414916	−	0.909860i	$-0.363811\pi$
$773$	23.0717	0.829831	0.414916	+	0.909860i	$0.363811\pi$
$774$	0	0
$775$	27.8122	0.999042
$776$	0	0
$777$	−9.29286	−0.333380
$778$	0	0
$779$	36.1375	1.29476
$780$	0	0
$781$	6.91008	0.247262
$782$	0	0
$783$	−23.0080	−0.822239
$784$	0	0
$785$	−18.9541	−0.676501
$786$	0	0
$787$	−25.8232	−0.920498	−0.460249	−	0.887790i	$-0.652240\pi$
$787$	−25.8232	−0.920498	−0.460249	+	0.887790i	$0.652240\pi$
$788$	0	0
$789$	−10.1881	−0.362706
$790$	0	0
$791$	−24.8847	−0.884799
$792$	0	0
$793$	14.8409	0.527015
$794$	0	0
$795$	8.84861	0.313828
$796$	0	0
$797$	36.5268	1.29384	0.646922	−	0.762556i	$-0.276056\pi$
$797$	36.5268	1.29384	0.646922	+	0.762556i	$0.276056\pi$
$798$	0	0
$799$	−30.6169	−1.08315
$800$	0	0
$801$	−9.96578	−0.352124
$802$	0	0
$803$	2.58354	0.0911710
$804$	0	0
$805$	−19.5929	−0.690559
$806$	0	0
$807$	29.8091	1.04933
$808$	0	0
$809$	22.4409	0.788979	0.394489	−	0.918901i	$-0.370922\pi$
$809$	22.4409	0.788979	0.394489	+	0.918901i	$0.370922\pi$
$810$	0	0
$811$	−29.7745	−1.04552	−0.522762	−	0.852479i	$-0.675098\pi$
$811$	−29.7745	−1.04552	−0.522762	+	0.852479i	$0.675098\pi$
$812$	0	0
$813$	37.1634	1.30338
$814$	0	0
$815$	24.7704	0.867669
$816$	0	0
$817$	−25.9775	−0.908838
$818$	0	0
$819$	−14.5561	−0.508633
$820$	0	0
$821$	−14.1688	−0.494494	−0.247247	−	0.968953i	$-0.579526\pi$
$821$	−14.1688	−0.494494	−0.247247	+	0.968953i	$0.579526\pi$
$822$	0	0
$823$	−29.0975	−1.01428	−0.507138	−	0.861865i	$-0.669297\pi$
$823$	−29.0975	−1.01428	−0.507138	+	0.861865i	$0.669297\pi$
$824$	0	0
$825$	−6.23089	−0.216932
$826$	0	0
$827$	34.4530	1.19805	0.599024	−	0.800731i	$-0.295555\pi$
$827$	34.4530	1.19805	0.599024	+	0.800731i	$0.295555\pi$
$828$	0	0
$829$	55.7131	1.93499	0.967497	−	0.252882i	$-0.0813783\pi$
$829$	55.7131	1.93499	0.967497	+	0.252882i	$0.0813783\pi$
$830$	0	0
$831$	−33.5596	−1.16417
$832$	0	0
$833$	−0.404141	−0.0140026
$834$	0	0
$835$	10.7013	0.370332
$836$	0	0
$837$	−46.7338	−1.61536
$838$	0	0
$839$	−7.09340	−0.244891	−0.122446	−	0.992475i	$-0.539074\pi$
$839$	−7.09340	−0.244891	−0.122446	+	0.992475i	$0.539074\pi$
$840$	0	0
$841$	−12.4517	−0.429370
$842$	0	0
$843$	−43.0146	−1.48150
$844$	0	0
$845$	26.5285	0.912610
$846$	0	0
$847$	24.3869	0.837943
$848$	0	0
$849$	−10.1852	−0.349554
$850$	0	0
$851$	−14.5637	−0.499236
$852$	0	0
$853$	19.5626	0.669811	0.334905	−	0.942252i	$-0.391296\pi$
$853$	19.5626	0.669811	0.334905	+	0.942252i	$0.391296\pi$
$854$	0	0
$855$	−8.03453	−0.274775
$856$	0	0
$857$	−21.2842	−0.727056	−0.363528	−	0.931583i	$-0.618428\pi$
$857$	−21.2842	−0.727056	−0.363528	+	0.931583i	$0.618428\pi$
$858$	0	0
$859$	−27.3020	−0.931533	−0.465766	−	0.884908i	$-0.654221\pi$
$859$	−27.3020	−0.931533	−0.465766	+	0.884908i	$0.654221\pi$
$860$	0	0
$861$	20.5863	0.701579
$862$	0	0
$863$	−22.0204	−0.749582	−0.374791	−	0.927109i	$-0.622285\pi$
$863$	−22.0204	−0.749582	−0.374791	+	0.927109i	$0.622285\pi$
$864$	0	0
$865$	7.21506	0.245319
$866$	0	0
$867$	14.7455	0.500783
$868$	0	0
$869$	−0.462399	−0.0156858
$870$	0	0
$871$	−2.25011	−0.0762422
$872$	0	0
$873$	−1.53124	−0.0518246
$874$	0	0
$875$	27.9764	0.945776
$876$	0	0
$877$	−16.6470	−0.562129	−0.281064	−	0.959689i	$-0.590687\pi$
$877$	−16.6470	−0.562129	−0.281064	+	0.959689i	$0.590687\pi$
$878$	0	0
$879$	0.830941	0.0280269
$880$	0	0
$881$	−19.9156	−0.670973	−0.335486	−	0.942045i	$-0.608901\pi$
$881$	−19.9156	−0.670973	−0.335486	+	0.942045i	$0.608901\pi$
$882$	0	0
$883$	7.55257	0.254164	0.127082	−	0.991892i	$-0.459439\pi$
$883$	7.55257	0.254164	0.127082	+	0.991892i	$0.459439\pi$
$884$	0	0
$885$	−1.78717	−0.0600750
$886$	0	0
$887$	6.48703	0.217813	0.108907	−	0.994052i	$-0.465265\pi$
$887$	6.48703	0.217813	0.108907	+	0.994052i	$0.465265\pi$
$888$	0	0
$889$	−51.6158	−1.73114
$890$	0	0
$891$	6.74817	0.226072
$892$	0	0
$893$	−77.7311	−2.60117
$894$	0	0
$895$	22.7211	0.759482
$896$	0	0
$897$	48.6444	1.62419
$898$	0	0
$899$	33.6128	1.12105
$900$	0	0
$901$	−12.5208	−0.417129
$902$	0	0
$903$	−14.7985	−0.492463
$904$	0	0
$905$	8.32387	0.276695
$906$	0	0
$907$	−42.7783	−1.42043	−0.710216	−	0.703984i	$-0.751403\pi$
$907$	−42.7783	−1.42043	−0.710216	+	0.703984i	$0.751403\pi$
$908$	0	0
$909$	−1.37275	−0.0455313
$910$	0	0
$911$	−47.1333	−1.56159	−0.780797	−	0.624784i	$-0.785187\pi$
$911$	−47.1333	−1.56159	−0.780797	+	0.624784i	$0.785187\pi$
$912$	0	0
$913$	−9.68355	−0.320479
$914$	0	0
$915$	4.66652	0.154270
$916$	0	0
$917$	−54.4175	−1.79702
$918$	0	0
$919$	37.7261	1.24447	0.622235	−	0.782830i	$-0.286225\pi$
$919$	37.7261	1.24447	0.622235	+	0.782830i	$0.286225\pi$
$920$	0	0
$921$	−14.8962	−0.490848
$922$	0	0
$923$	30.9921	1.02012
$924$	0	0
$925$	8.36675	0.275097
$926$	0	0
$927$	−3.79973	−0.124800
$928$	0	0
$929$	2.50005	0.0820240	0.0410120	−	0.999159i	$-0.486942\pi$
$929$	2.50005	0.0820240	0.0410120	+	0.999159i	$0.486942\pi$
$930$	0	0
$931$	−1.02604	−0.0336272
$932$	0	0
$933$	17.8029	0.582840
$934$	0	0
$935$	−4.28025	−0.139979
$936$	0	0
$937$	3.00252	0.0980881	0.0490441	−	0.998797i	$-0.484383\pi$
$937$	3.00252	0.0980881	0.0490441	+	0.998797i	$0.484383\pi$
$938$	0	0
$939$	−13.3864	−0.436847
$940$	0	0
$941$	−8.37912	−0.273152	−0.136576	−	0.990630i	$-0.543610\pi$
$941$	−8.37912	−0.273152	−0.136576	+	0.990630i	$0.543610\pi$
$942$	0	0
$943$	32.2626	1.05062
$944$	0	0
$945$	−18.9139	−0.615268
$946$	0	0
$947$	0.191341	0.00621775	0.00310888	−	0.999995i	$-0.499010\pi$
$947$	0.191341	0.00621775	0.00310888	+	0.999995i	$0.499010\pi$
$948$	0	0
$949$	11.5873	0.376140
$950$	0	0
$951$	−11.6920	−0.379138
$952$	0	0
$953$	−0.113676	−0.00368233	−0.00184117	−	0.999998i	$-0.500586\pi$
$953$	−0.113676	−0.00368233	−0.00184117	+	0.999998i	$0.500586\pi$
$954$	0	0
$955$	22.4122	0.725242
$956$	0	0
$957$	−7.53043	−0.243424
$958$	0	0
$959$	−10.2959	−0.332472
$960$	0	0
$961$	37.2741	1.20239
$962$	0	0
$963$	−15.1712	−0.488885
$964$	0	0
$965$	−14.0011	−0.450710
$966$	0	0
$967$	12.2439	0.393736	0.196868	−	0.980430i	$-0.436923\pi$
$967$	12.2439	0.393736	0.196868	+	0.980430i	$0.436923\pi$
$968$	0	0
$969$	−24.2428	−0.778792
$970$	0	0
$971$	−13.8141	−0.443317	−0.221658	−	0.975124i	$-0.571147\pi$
$971$	−13.8141	−0.443317	−0.221658	+	0.975124i	$0.571147\pi$
$972$	0	0
$973$	27.5981	0.884753
$974$	0	0
$975$	−27.9459	−0.894986
$976$	0	0
$977$	17.1780	0.549573	0.274787	−	0.961505i	$-0.411393\pi$
$977$	17.1780	0.549573	0.274787	+	0.961505i	$0.411393\pi$
$978$	0	0
$979$	−13.4788	−0.430786
$980$	0	0
$981$	18.7194	0.597663
$982$	0	0
$983$	51.3862	1.63897	0.819483	−	0.573104i	$-0.194261\pi$
$983$	51.3862	1.63897	0.819483	+	0.573104i	$0.194261\pi$
$984$	0	0
$985$	19.9481	0.635601
$986$	0	0
$987$	−44.2807	−1.40947
$988$	0	0
$989$	−23.1920	−0.737463
$990$	0	0
$991$	−5.20631	−0.165384	−0.0826919	−	0.996575i	$-0.526352\pi$
$991$	−5.20631	−0.165384	−0.0826919	+	0.996575i	$0.526352\pi$
$992$	0	0
$993$	30.5354	0.969012
$994$	0	0
$995$	−14.5666	−0.461793
$996$	0	0
$997$	54.1674	1.71550	0.857749	−	0.514069i	$-0.171862\pi$
$997$	54.1674	1.71550	0.857749	+	0.514069i	$0.171862\pi$
$998$	0	0
$999$	−14.0589	−0.444805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.d.1.14	✓	49

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.d.1.14	✓	49	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)

\(f(q)\)	\(=\)	\(q-1.42908 q^{3} +1.27830 q^{5} -2.61604 q^{7} -0.957738 q^{9} +O(q^{10})\)	\(q-1.42908 q^{3} +1.27830 q^{5} -2.61604 q^{7} -0.957738 q^{9} -1.29535 q^{11} -5.80973 q^{13} -1.82679 q^{15} +2.58492 q^{17} +6.56267 q^{19} +3.73852 q^{21} +5.85897 q^{23} -3.36594 q^{25} +5.65591 q^{27} -4.06796 q^{29} -8.26281 q^{31} +1.85116 q^{33} -3.34409 q^{35} -2.48571 q^{37} +8.30255 q^{39} +5.50653 q^{41} -3.95838 q^{43} -1.22428 q^{45} -11.8444 q^{47} -0.156345 q^{49} -3.69405 q^{51} -4.84380 q^{53} -1.65585 q^{55} -9.37856 q^{57} +0.978309 q^{59} -2.55449 q^{61} +2.50548 q^{63} -7.42659 q^{65} +0.387301 q^{67} -8.37292 q^{69} -5.33452 q^{71} -1.99447 q^{73} +4.81019 q^{75} +3.38869 q^{77} +0.356967 q^{79} -5.20953 q^{81} +7.47561 q^{83} +3.30431 q^{85} +5.81343 q^{87} +10.4055 q^{89} +15.1985 q^{91} +11.8082 q^{93} +8.38907 q^{95} +1.59881 q^{97} +1.24061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9}+O(q^{10}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9	\( 49 q + 14 q^{3} - 7 q^{5} + 22 q^{7} + 59 q^{9} + 19 q^{11} + 15 q^{13} + 17 q^{15} + 14 q^{17} + 24 q^{19} - 8 q^{21} + 28 q^{23} + 72 q^{25} + 62 q^{27} - 35 q^{29} + 51 q^{31} + 28 q^{33} + 23 q^{35} + 19 q^{37} + 34 q^{39} + 12 q^{41} + 37 q^{43} - 20 q^{45} + 54 q^{47} + 65 q^{49} + 43 q^{51} - 17 q^{53} + 57 q^{55} + 19 q^{57} + 52 q^{59} - 16 q^{61} + 41 q^{63} + 13 q^{65} + 44 q^{67} - 4 q^{69} + 52 q^{71} + 58 q^{73} + 81 q^{75} - 27 q^{77} + 43 q^{79} + 73 q^{81} + 51 q^{83} - 16 q^{85} + 41 q^{87} + 40 q^{89} + 73 q^{91} + 22 q^{93} + 70 q^{95} + 96 q^{97} + 78 q^{99}+O(q^{100}) \) 49 * q + 14 * q^3 - 7 * q^5 + 22 * q^7 + 59 * q^9 + 19 * q^11 + 15 * q^13 + 17 * q^15 + 14 * q^17 + 24 * q^19 - 8 * q^21 + 28 * q^23 + 72 * q^25 + 62 * q^27 - 35 * q^29 + 51 * q^31 + 28 * q^33 + 23 * q^35 + 19 * q^37 + 34 * q^39 + 12 * q^41 + 37 * q^43 - 20 * q^45 + 54 * q^47 + 65 * q^49 + 43 * q^51 - 17 * q^53 + 57 * q^55 + 19 * q^57 + 52 * q^59 - 16 * q^61 + 41 * q^63 + 13 * q^65 + 44 * q^67 - 4 * q^69 + 52 * q^71 + 58 * q^73 + 81 * q^75 - 27 * q^77 + 43 * q^79 + 73 * q^81 + 51 * q^83 - 16 * q^85 + 41 * q^87 + 40 * q^89 + 73 * q^91 + 22 * q^93 + 70 * q^95 + 96 * q^97 + 78 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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