LMFDB - Embedded newform 6008.2.a.e.1.8

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			1.8
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)

$f(q)$	$=$	$q-2.49253 q^{3} -3.92718 q^{5} +2.18622 q^{7} +3.21269 q^{9} +O(q^{10})$	$q-2.49253 q^{3} -3.92718 q^{5} +2.18622 q^{7} +3.21269 q^{9} +3.47715 q^{11} +3.85586 q^{13} +9.78861 q^{15} -2.14678 q^{17} +7.07416 q^{19} -5.44922 q^{21} -0.572069 q^{23} +10.4227 q^{25} -0.530148 q^{27} +5.29014 q^{29} +8.09240 q^{31} -8.66688 q^{33} -8.58569 q^{35} +4.15971 q^{37} -9.61083 q^{39} +0.245191 q^{41} +5.43293 q^{43} -12.6168 q^{45} +1.50042 q^{47} -2.22043 q^{49} +5.35091 q^{51} -7.38122 q^{53} -13.6554 q^{55} -17.6325 q^{57} -10.9107 q^{59} +8.32991 q^{61} +7.02367 q^{63} -15.1426 q^{65} +5.93170 q^{67} +1.42590 q^{69} -6.26670 q^{71} +0.909284 q^{73} -25.9790 q^{75} +7.60182 q^{77} -9.07280 q^{79} -8.31668 q^{81} +12.2441 q^{83} +8.43080 q^{85} -13.1858 q^{87} +10.9607 q^{89} +8.42976 q^{91} -20.1705 q^{93} -27.7815 q^{95} -7.29623 q^{97} +11.1710 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.49253	−1.43906	−0.719531	−	0.694461i	$-0.755643\pi$
$3$	−2.49253	−1.43906	−0.719531	+	0.694461i	$0.755643\pi$
$4$	0	0
$5$	−3.92718	−1.75629	−0.878144	−	0.478396i	$-0.841218\pi$
$5$	−3.92718	−1.75629	−0.878144	+	0.478396i	$0.841218\pi$
$6$	0	0
$7$	2.18622	0.826315	0.413157	−	0.910660i	$-0.364426\pi$
$7$	2.18622	0.826315	0.413157	+	0.910660i	$0.364426\pi$
$8$	0	0
$9$	3.21269	1.07090
$10$	0	0
$11$	3.47715	1.04840	0.524199	−	0.851596i	$-0.324365\pi$
$11$	3.47715	1.04840	0.524199	+	0.851596i	$0.324365\pi$
$12$	0	0
$13$	3.85586	1.06942	0.534711	−	0.845035i	$-0.320421\pi$
$13$	3.85586	1.06942	0.534711	+	0.845035i	$0.320421\pi$
$14$	0	0
$15$	9.78861	2.52741
$16$	0	0
$17$	−2.14678	−0.520671	−0.260336	−	0.965518i	$-0.583833\pi$
$17$	−2.14678	−0.520671	−0.260336	+	0.965518i	$0.583833\pi$
$18$	0	0
$19$	7.07416	1.62292	0.811462	−	0.584406i	$-0.198672\pi$
$19$	7.07416	1.62292	0.811462	+	0.584406i	$0.198672\pi$
$20$	0	0
$21$	−5.44922	−1.18912
$22$	0	0
$23$	−0.572069	−0.119285	−0.0596423	−	0.998220i	$-0.518996\pi$
$23$	−0.572069	−0.119285	−0.0596423	+	0.998220i	$0.518996\pi$
$24$	0	0
$25$	10.4227	2.08455
$26$	0	0
$27$	−0.530148	−0.102027
$28$	0	0
$29$	5.29014	0.982354	0.491177	−	0.871060i	$-0.336567\pi$
$29$	5.29014	0.982354	0.491177	+	0.871060i	$0.336567\pi$
$30$	0	0
$31$	8.09240	1.45344	0.726719	−	0.686935i	$-0.241044\pi$
$31$	8.09240	1.45344	0.726719	+	0.686935i	$0.241044\pi$
$32$	0	0
$33$	−8.66688	−1.50871
$34$	0	0
$35$	−8.58569	−1.45125
$36$	0	0
$37$	4.15971	0.683852	0.341926	−	0.939727i	$-0.388921\pi$
$37$	4.15971	0.683852	0.341926	+	0.939727i	$0.388921\pi$
$38$	0	0
$39$	−9.61083	−1.53896
$40$	0	0
$41$	0.245191	0.0382924	0.0191462	−	0.999817i	$-0.493905\pi$
$41$	0.245191	0.0382924	0.0191462	+	0.999817i	$0.493905\pi$
$42$	0	0
$43$	5.43293	0.828514	0.414257	−	0.910160i	$-0.364042\pi$
$43$	5.43293	0.828514	0.414257	+	0.910160i	$0.364042\pi$
$44$	0	0
$45$	−12.6168	−1.88081
$46$	0	0
$47$	1.50042	0.218858	0.109429	−	0.993995i	$-0.465098\pi$
$47$	1.50042	0.218858	0.109429	+	0.993995i	$0.465098\pi$
$48$	0	0
$49$	−2.22043	−0.317204
$50$	0	0
$51$	5.35091	0.749278
$52$	0	0
$53$	−7.38122	−1.01389	−0.506944	−	0.861979i	$-0.669225\pi$
$53$	−7.38122	−1.01389	−0.506944	+	0.861979i	$0.669225\pi$
$54$	0	0
$55$	−13.6554	−1.84129
$56$	0	0
$57$	−17.6325	−2.33549
$58$	0	0
$59$	−10.9107	−1.42045	−0.710227	−	0.703972i	$-0.751408\pi$
$59$	−10.9107	−1.42045	−0.710227	+	0.703972i	$0.751408\pi$
$60$	0	0
$61$	8.32991	1.06654	0.533268	−	0.845946i	$-0.320964\pi$
$61$	8.32991	1.06654	0.533268	+	0.845946i	$0.320964\pi$
$62$	0	0
$63$	7.02367	0.884899
$64$	0	0
$65$	−15.1426	−1.87821
$66$	0	0
$67$	5.93170	0.724673	0.362336	−	0.932047i	$-0.381979\pi$
$67$	5.93170	0.724673	0.362336	+	0.932047i	$0.381979\pi$
$68$	0	0
$69$	1.42590	0.171658
$70$	0	0
$71$	−6.26670	−0.743721	−0.371861	−	0.928289i	$-0.621280\pi$
$71$	−6.26670	−0.743721	−0.371861	+	0.928289i	$0.621280\pi$
$72$	0	0
$73$	0.909284	0.106424	0.0532118	−	0.998583i	$-0.483054\pi$
$73$	0.909284	0.106424	0.0532118	+	0.998583i	$0.483054\pi$
$74$	0	0
$75$	−25.9790	−2.99980
$76$	0	0
$77$	7.60182	0.866307
$78$	0	0
$79$	−9.07280	−1.02077	−0.510385	−	0.859946i	$-0.670497\pi$
$79$	−9.07280	−1.02077	−0.510385	+	0.859946i	$0.670497\pi$
$80$	0	0
$81$	−8.31668	−0.924075
$82$	0	0
$83$	12.2441	1.34396	0.671980	−	0.740569i	$-0.265444\pi$
$83$	12.2441	1.34396	0.671980	+	0.740569i	$0.265444\pi$
$84$	0	0
$85$	8.43080	0.914449
$86$	0	0
$87$	−13.1858	−1.41367
$88$	0	0
$89$	10.9607	1.16184	0.580918	−	0.813962i	$-0.302694\pi$
$89$	10.9607	1.16184	0.580918	+	0.813962i	$0.302694\pi$
$90$	0	0
$91$	8.42976	0.883679
$92$	0	0
$93$	−20.1705	−2.09159
$94$	0	0
$95$	−27.7815	−2.85032
$96$	0	0
$97$	−7.29623	−0.740820	−0.370410	−	0.928868i	$-0.620783\pi$
$97$	−7.29623	−0.740820	−0.370410	+	0.928868i	$0.620783\pi$
$98$	0	0
$99$	11.1710	1.12273
$100$	0	0
$101$	−2.15477	−0.214408	−0.107204	−	0.994237i	$-0.534190\pi$
$101$	−2.15477	−0.214408	−0.107204	+	0.994237i	$0.534190\pi$
$102$	0	0
$103$	15.8738	1.56410	0.782048	−	0.623218i	$-0.214175\pi$
$103$	15.8738	1.56410	0.782048	+	0.623218i	$0.214175\pi$
$104$	0	0
$105$	21.4001	2.08843
$106$	0	0
$107$	9.70889	0.938594	0.469297	−	0.883040i	$-0.344507\pi$
$107$	9.70889	0.938594	0.469297	+	0.883040i	$0.344507\pi$
$108$	0	0
$109$	−0.266281	−0.0255051	−0.0127525	−	0.999919i	$-0.504059\pi$
$109$	−0.266281	−0.0255051	−0.0127525	+	0.999919i	$0.504059\pi$
$110$	0	0
$111$	−10.3682	−0.984105
$112$	0	0
$113$	−7.96495	−0.749280	−0.374640	−	0.927170i	$-0.622234\pi$
$113$	−7.96495	−0.749280	−0.374640	+	0.927170i	$0.622234\pi$
$114$	0	0
$115$	2.24662	0.209498
$116$	0	0
$117$	12.3877	1.14524
$118$	0	0
$119$	−4.69334	−0.430238
$120$	0	0
$121$	1.09054	0.0991398
$122$	0	0
$123$	−0.611145	−0.0551051
$124$	0	0
$125$	−21.2961	−1.90478
$126$	0	0
$127$	−6.21230	−0.551253	−0.275626	−	0.961265i	$-0.588885\pi$
$127$	−6.21230	−0.551253	−0.275626	+	0.961265i	$0.588885\pi$
$128$	0	0
$129$	−13.5417	−1.19228
$130$	0	0
$131$	−14.7337	−1.28729	−0.643645	−	0.765324i	$-0.722579\pi$
$131$	−14.7337	−1.28729	−0.643645	+	0.765324i	$0.722579\pi$
$132$	0	0
$133$	15.4657	1.34105
$134$	0	0
$135$	2.08199	0.179189
$136$	0	0
$137$	9.14072	0.780945	0.390472	−	0.920615i	$-0.372312\pi$
$137$	9.14072	0.780945	0.390472	+	0.920615i	$0.372312\pi$
$138$	0	0
$139$	1.35070	0.114564	0.0572822	−	0.998358i	$-0.481757\pi$
$139$	1.35070	0.114564	0.0572822	+	0.998358i	$0.481757\pi$
$140$	0	0
$141$	−3.73983	−0.314950
$142$	0	0
$143$	13.4074	1.12118
$144$	0	0
$145$	−20.7753	−1.72530
$146$	0	0
$147$	5.53448	0.456476
$148$	0	0
$149$	12.3909	1.01510	0.507552	−	0.861621i	$-0.330550\pi$
$149$	12.3909	1.01510	0.507552	+	0.861621i	$0.330550\pi$
$150$	0	0
$151$	−17.7936	−1.44802	−0.724012	−	0.689788i	$-0.757704\pi$
$151$	−17.7936	−1.44802	−0.724012	+	0.689788i	$0.757704\pi$
$152$	0	0
$153$	−6.89695	−0.557586
$154$	0	0
$155$	−31.7803	−2.55266
$156$	0	0
$157$	13.1429	1.04892	0.524459	−	0.851435i	$-0.324267\pi$
$157$	13.1429	1.04892	0.524459	+	0.851435i	$0.324267\pi$
$158$	0	0
$159$	18.3979	1.45905
$160$	0	0
$161$	−1.25067	−0.0985667
$162$	0	0
$163$	6.93629	0.543292	0.271646	−	0.962397i	$-0.412432\pi$
$163$	6.93629	0.543292	0.271646	+	0.962397i	$0.412432\pi$
$164$	0	0
$165$	34.0364	2.64973
$166$	0	0
$167$	−12.5044	−0.967622	−0.483811	−	0.875173i	$-0.660748\pi$
$167$	−12.5044	−0.967622	−0.483811	+	0.875173i	$0.660748\pi$
$168$	0	0
$169$	1.86762	0.143663
$170$	0	0
$171$	22.7271	1.73799
$172$	0	0
$173$	21.2106	1.61261	0.806307	−	0.591497i	$-0.201463\pi$
$173$	21.2106	1.61261	0.806307	+	0.591497i	$0.201463\pi$
$174$	0	0
$175$	22.7865	1.72249
$176$	0	0
$177$	27.1953	2.04412
$178$	0	0
$179$	−18.3274	−1.36986	−0.684929	−	0.728610i	$-0.740167\pi$
$179$	−18.3274	−1.36986	−0.684929	+	0.728610i	$0.740167\pi$
$180$	0	0
$181$	11.4582	0.851685	0.425842	−	0.904797i	$-0.359978\pi$
$181$	11.4582	0.851685	0.425842	+	0.904797i	$0.359978\pi$
$182$	0	0
$183$	−20.7625	−1.53481
$184$	0	0
$185$	−16.3359	−1.20104
$186$	0	0
$187$	−7.46467	−0.545871
$188$	0	0
$189$	−1.15902	−0.0843064
$190$	0	0
$191$	−7.23850	−0.523759	−0.261880	−	0.965101i	$-0.584342\pi$
$191$	−7.23850	−0.523759	−0.261880	+	0.965101i	$0.584342\pi$
$192$	0	0
$193$	−6.26334	−0.450845	−0.225423	−	0.974261i	$-0.572376\pi$
$193$	−6.26334	−0.450845	−0.225423	+	0.974261i	$0.572376\pi$
$194$	0	0
$195$	37.7435	2.70286
$196$	0	0
$197$	5.18663	0.369532	0.184766	−	0.982783i	$-0.440847\pi$
$197$	5.18663	0.369532	0.184766	+	0.982783i	$0.440847\pi$
$198$	0	0
$199$	−20.9473	−1.48491	−0.742456	−	0.669895i	$-0.766339\pi$
$199$	−20.9473	−1.48491	−0.742456	+	0.669895i	$0.766339\pi$
$200$	0	0
$201$	−14.7849	−1.04285
$202$	0	0
$203$	11.5654	0.811734
$204$	0	0
$205$	−0.962908	−0.0672524
$206$	0	0
$207$	−1.83788	−0.127742
$208$	0	0
$209$	24.5979	1.70147
$210$	0	0
$211$	−17.4373	−1.20043	−0.600217	−	0.799837i	$-0.704919\pi$
$211$	−17.4373	−1.20043	−0.600217	+	0.799837i	$0.704919\pi$
$212$	0	0
$213$	15.6199	1.07026
$214$	0	0
$215$	−21.3361	−1.45511
$216$	0	0
$217$	17.6918	1.20100
$218$	0	0
$219$	−2.26642	−0.153150
$220$	0	0
$221$	−8.27768	−0.556817
$222$	0	0
$223$	7.61026	0.509621	0.254810	−	0.966991i	$-0.417987\pi$
$223$	7.61026	0.509621	0.254810	+	0.966991i	$0.417987\pi$
$224$	0	0
$225$	33.4851	2.23234
$226$	0	0
$227$	26.2180	1.74015	0.870074	−	0.492921i	$-0.164071\pi$
$227$	26.2180	1.74015	0.870074	+	0.492921i	$0.164071\pi$
$228$	0	0
$229$	13.1573	0.869462	0.434731	−	0.900560i	$-0.356843\pi$
$229$	13.1573	0.869462	0.434731	+	0.900560i	$0.356843\pi$
$230$	0	0
$231$	−18.9477	−1.24667
$232$	0	0
$233$	3.87355	0.253764	0.126882	−	0.991918i	$-0.459503\pi$
$233$	3.87355	0.253764	0.126882	+	0.991918i	$0.459503\pi$
$234$	0	0
$235$	−5.89241	−0.384378
$236$	0	0
$237$	22.6142	1.46895
$238$	0	0
$239$	−20.4227	−1.32103	−0.660517	−	0.750812i	$-0.729663\pi$
$239$	−20.4227	−1.32103	−0.660517	+	0.750812i	$0.729663\pi$
$240$	0	0
$241$	9.40360	0.605739	0.302870	−	0.953032i	$-0.402055\pi$
$241$	9.40360	0.605739	0.302870	+	0.953032i	$0.402055\pi$
$242$	0	0
$243$	22.3200	1.43183
$244$	0	0
$245$	8.72002	0.557101
$246$	0	0
$247$	27.2769	1.73559
$248$	0	0
$249$	−30.5186	−1.93404
$250$	0	0
$251$	−26.3902	−1.66573	−0.832866	−	0.553475i	$-0.813301\pi$
$251$	−26.3902	−1.66573	−0.832866	+	0.553475i	$0.813301\pi$
$252$	0	0
$253$	−1.98917	−0.125058
$254$	0	0
$255$	−21.0140	−1.31595
$256$	0	0
$257$	19.8601	1.23884	0.619420	−	0.785060i	$-0.287368\pi$
$257$	19.8601	1.23884	0.619420	+	0.785060i	$0.287368\pi$
$258$	0	0
$259$	9.09405	0.565077
$260$	0	0
$261$	16.9956	1.05200
$262$	0	0
$263$	21.7481	1.34105	0.670523	−	0.741889i	$-0.266070\pi$
$263$	21.7481	1.34105	0.670523	+	0.741889i	$0.266070\pi$
$264$	0	0
$265$	28.9874	1.78068
$266$	0	0
$267$	−27.3200	−1.67195
$268$	0	0
$269$	29.7942	1.81658	0.908291	−	0.418339i	$-0.137388\pi$
$269$	29.7942	1.81658	0.908291	+	0.418339i	$0.137388\pi$
$270$	0	0
$271$	4.81290	0.292363	0.146181	−	0.989258i	$-0.453302\pi$
$271$	4.81290	0.292363	0.146181	+	0.989258i	$0.453302\pi$
$272$	0	0
$273$	−21.0114	−1.27167
$274$	0	0
$275$	36.2414	2.18544
$276$	0	0
$277$	−1.87943	−0.112924	−0.0564620	−	0.998405i	$-0.517982\pi$
$277$	−1.87943	−0.112924	−0.0564620	+	0.998405i	$0.517982\pi$
$278$	0	0
$279$	25.9984	1.55648
$280$	0	0
$281$	13.0500	0.778498	0.389249	−	0.921133i	$-0.372735\pi$
$281$	13.0500	0.778498	0.389249	+	0.921133i	$0.372735\pi$
$282$	0	0
$283$	6.75429	0.401501	0.200751	−	0.979642i	$-0.435662\pi$
$283$	6.75429	0.401501	0.200751	+	0.979642i	$0.435662\pi$
$284$	0	0
$285$	69.2462	4.10179
$286$	0	0
$287$	0.536042	0.0316415
$288$	0	0
$289$	−12.3913	−0.728902
$290$	0	0
$291$	18.1861	1.06609
$292$	0	0
$293$	−20.1411	−1.17666	−0.588329	−	0.808622i	$-0.700214\pi$
$293$	−20.1411	−1.17666	−0.588329	+	0.808622i	$0.700214\pi$
$294$	0	0
$295$	42.8484	2.49473
$296$	0	0
$297$	−1.84340	−0.106965
$298$	0	0
$299$	−2.20582	−0.127566
$300$	0	0
$301$	11.8776	0.684613
$302$	0	0
$303$	5.37083	0.308546
$304$	0	0
$305$	−32.7131	−1.87314
$306$	0	0
$307$	4.70210	0.268363	0.134182	−	0.990957i	$-0.457159\pi$
$307$	4.70210	0.268363	0.134182	+	0.990957i	$0.457159\pi$
$308$	0	0
$309$	−39.5660	−2.25083
$310$	0	0
$311$	32.6084	1.84905	0.924527	−	0.381115i	$-0.124460\pi$
$311$	32.6084	1.84905	0.924527	+	0.381115i	$0.124460\pi$
$312$	0	0
$313$	2.32504	0.131419	0.0657096	−	0.997839i	$-0.479069\pi$
$313$	2.32504	0.131419	0.0657096	+	0.997839i	$0.479069\pi$
$314$	0	0
$315$	−27.5832	−1.55414
$316$	0	0
$317$	−13.3206	−0.748160	−0.374080	−	0.927396i	$-0.622041\pi$
$317$	−13.3206	−0.748160	−0.374080	+	0.927396i	$0.622041\pi$
$318$	0	0
$319$	18.3946	1.02990
$320$	0	0
$321$	−24.1997	−1.35069
$322$	0	0
$323$	−15.1867	−0.845009
$324$	0	0
$325$	40.1886	2.22926
$326$	0	0
$327$	0.663712	0.0367034
$328$	0	0
$329$	3.28025	0.180846
$330$	0	0
$331$	−1.14348	−0.0628516	−0.0314258	−	0.999506i	$-0.510005\pi$
$331$	−1.14348	−0.0628516	−0.0314258	+	0.999506i	$0.510005\pi$
$332$	0	0
$333$	13.3639	0.732336
$334$	0	0
$335$	−23.2949	−1.27273
$336$	0	0
$337$	−6.33761	−0.345232	−0.172616	−	0.984989i	$-0.555222\pi$
$337$	−6.33761	−0.345232	−0.172616	+	0.984989i	$0.555222\pi$
$338$	0	0
$339$	19.8529	1.07826
$340$	0	0
$341$	28.1385	1.52378
$342$	0	0
$343$	−20.1579	−1.08843
$344$	0	0
$345$	−5.59976	−0.301481
$346$	0	0
$347$	4.93567	0.264961	0.132480	−	0.991186i	$-0.457706\pi$
$347$	4.93567	0.264961	0.132480	+	0.991186i	$0.457706\pi$
$348$	0	0
$349$	−8.75307	−0.468541	−0.234270	−	0.972171i	$-0.575270\pi$
$349$	−8.75307	−0.468541	−0.234270	+	0.972171i	$0.575270\pi$
$350$	0	0
$351$	−2.04417	−0.109110
$352$	0	0
$353$	5.01433	0.266886	0.133443	−	0.991056i	$-0.457397\pi$
$353$	5.01433	0.266886	0.133443	+	0.991056i	$0.457397\pi$
$354$	0	0
$355$	24.6105	1.30619
$356$	0	0
$357$	11.6983	0.619139
$358$	0	0
$359$	31.8687	1.68197	0.840984	−	0.541060i	$-0.181977\pi$
$359$	31.8687	1.68197	0.840984	+	0.541060i	$0.181977\pi$
$360$	0	0
$361$	31.0437	1.63388
$362$	0	0
$363$	−2.71820	−0.142668
$364$	0	0
$365$	−3.57092	−0.186911
$366$	0	0
$367$	2.03363	0.106155	0.0530773	−	0.998590i	$-0.483097\pi$
$367$	2.03363	0.106155	0.0530773	+	0.998590i	$0.483097\pi$
$368$	0	0
$369$	0.787723	0.0410072
$370$	0	0
$371$	−16.1370	−0.837791
$372$	0	0
$373$	−23.3372	−1.20835	−0.604176	−	0.796851i	$-0.706498\pi$
$373$	−23.3372	−1.20835	−0.604176	+	0.796851i	$0.706498\pi$
$374$	0	0
$375$	53.0811	2.74110
$376$	0	0
$377$	20.3980	1.05055
$378$	0	0
$379$	−27.0205	−1.38795	−0.693975	−	0.719999i	$-0.744142\pi$
$379$	−27.0205	−1.38795	−0.693975	+	0.719999i	$0.744142\pi$
$380$	0	0
$381$	15.4843	0.793287
$382$	0	0
$383$	−8.47464	−0.433034	−0.216517	−	0.976279i	$-0.569470\pi$
$383$	−8.47464	−0.433034	−0.216517	+	0.976279i	$0.569470\pi$
$384$	0	0
$385$	−29.8537	−1.52149
$386$	0	0
$387$	17.4543	0.887254
$388$	0	0
$389$	5.56427	0.282120	0.141060	−	0.990001i	$-0.454949\pi$
$389$	5.56427	0.282120	0.141060	+	0.990001i	$0.454949\pi$
$390$	0	0
$391$	1.22811	0.0621081
$392$	0	0
$393$	36.7242	1.85249
$394$	0	0
$395$	35.6305	1.79277
$396$	0	0
$397$	−16.8815	−0.847257	−0.423628	−	0.905836i	$-0.639244\pi$
$397$	−16.8815	−0.847257	−0.423628	+	0.905836i	$0.639244\pi$
$398$	0	0
$399$	−38.5487	−1.92985
$400$	0	0
$401$	33.8108	1.68843	0.844215	−	0.536005i	$-0.180067\pi$
$401$	33.8108	1.68843	0.844215	+	0.536005i	$0.180067\pi$
$402$	0	0
$403$	31.2031	1.55434
$404$	0	0
$405$	32.6611	1.62294
$406$	0	0
$407$	14.4639	0.716949
$408$	0	0
$409$	2.34695	0.116049	0.0580247	−	0.998315i	$-0.481520\pi$
$409$	2.34695	0.116049	0.0580247	+	0.998315i	$0.481520\pi$
$410$	0	0
$411$	−22.7835	−1.12383
$412$	0	0
$413$	−23.8533	−1.17374
$414$	0	0
$415$	−48.0846	−2.36038
$416$	0	0
$417$	−3.36665	−0.164865
$418$	0	0
$419$	4.68586	0.228919	0.114460	−	0.993428i	$-0.463486\pi$
$419$	4.68586	0.228919	0.114460	+	0.993428i	$0.463486\pi$
$420$	0	0
$421$	−5.06792	−0.246996	−0.123498	−	0.992345i	$-0.539411\pi$
$421$	−5.06792	−0.246996	−0.123498	+	0.992345i	$0.539411\pi$
$422$	0	0
$423$	4.82038	0.234375
$424$	0	0
$425$	−22.3754	−1.08536
$426$	0	0
$427$	18.2110	0.881294
$428$	0	0
$429$	−33.4182	−1.61345
$430$	0	0
$431$	16.8409	0.811196	0.405598	−	0.914052i	$-0.367063\pi$
$431$	16.8409	0.811196	0.405598	+	0.914052i	$0.367063\pi$
$432$	0	0
$433$	19.0307	0.914556	0.457278	−	0.889324i	$-0.348825\pi$
$433$	19.0307	0.914556	0.457278	+	0.889324i	$0.348825\pi$
$434$	0	0
$435$	51.7831	2.48281
$436$	0	0
$437$	−4.04691	−0.193590
$438$	0	0
$439$	−35.0666	−1.67364	−0.836819	−	0.547479i	$-0.815587\pi$
$439$	−35.0666	−1.67364	−0.836819	+	0.547479i	$0.815587\pi$
$440$	0	0
$441$	−7.13355	−0.339693
$442$	0	0
$443$	−21.7054	−1.03126	−0.515628	−	0.856812i	$-0.672441\pi$
$443$	−21.7054	−1.03126	−0.515628	+	0.856812i	$0.672441\pi$
$444$	0	0
$445$	−43.0448	−2.04052
$446$	0	0
$447$	−30.8847	−1.46080
$448$	0	0
$449$	−27.7769	−1.31087	−0.655436	−	0.755251i	$-0.727515\pi$
$449$	−27.7769	−1.31087	−0.655436	+	0.755251i	$0.727515\pi$
$450$	0	0
$451$	0.852564	0.0401457
$452$	0	0
$453$	44.3511	2.08380
$454$	0	0
$455$	−33.1052	−1.55200
$456$	0	0
$457$	−1.78593	−0.0835421	−0.0417710	−	0.999127i	$-0.513300\pi$
$457$	−1.78593	−0.0835421	−0.0417710	+	0.999127i	$0.513300\pi$
$458$	0	0
$459$	1.13811	0.0531225
$460$	0	0
$461$	13.4279	0.625400	0.312700	−	0.949852i	$-0.398766\pi$
$461$	13.4279	0.625400	0.312700	+	0.949852i	$0.398766\pi$
$462$	0	0
$463$	−38.4010	−1.78465	−0.892323	−	0.451398i	$-0.850926\pi$
$463$	−38.4010	−1.78465	−0.892323	+	0.451398i	$0.850926\pi$
$464$	0	0
$465$	79.2133	3.67343
$466$	0	0
$467$	−34.1437	−1.57998	−0.789991	−	0.613118i	$-0.789915\pi$
$467$	−34.1437	−1.57998	−0.789991	+	0.613118i	$0.789915\pi$
$468$	0	0
$469$	12.9680	0.598808
$470$	0	0
$471$	−32.7591	−1.50946
$472$	0	0
$473$	18.8911	0.868613
$474$	0	0
$475$	73.7322	3.38306
$476$	0	0
$477$	−23.7136	−1.08577
$478$	0	0
$479$	16.7021	0.763139	0.381569	−	0.924340i	$-0.375384\pi$
$479$	16.7021	0.763139	0.381569	+	0.924340i	$0.375384\pi$
$480$	0	0
$481$	16.0392	0.731326
$482$	0	0
$483$	3.11733	0.141844
$484$	0	0
$485$	28.6536	1.30109
$486$	0	0
$487$	34.0061	1.54096	0.770482	−	0.637462i	$-0.220016\pi$
$487$	34.0061	1.54096	0.770482	+	0.637462i	$0.220016\pi$
$488$	0	0
$489$	−17.2889	−0.781831
$490$	0	0
$491$	−36.4410	−1.64456	−0.822280	−	0.569083i	$-0.807298\pi$
$491$	−36.4410	−1.64456	−0.822280	+	0.569083i	$0.807298\pi$
$492$	0	0
$493$	−11.3568	−0.511483
$494$	0	0
$495$	−43.8706	−1.97183
$496$	0	0
$497$	−13.7004	−0.614548
$498$	0	0
$499$	19.7536	0.884294	0.442147	−	0.896943i	$-0.354217\pi$
$499$	19.7536	0.884294	0.442147	+	0.896943i	$0.354217\pi$
$500$	0	0
$501$	31.1676	1.39247
$502$	0	0
$503$	25.7793	1.14944	0.574721	−	0.818350i	$-0.305111\pi$
$503$	25.7793	1.14944	0.574721	+	0.818350i	$0.305111\pi$
$504$	0	0
$505$	8.46218	0.376562
$506$	0	0
$507$	−4.65509	−0.206740
$508$	0	0
$509$	−11.7842	−0.522324	−0.261162	−	0.965295i	$-0.584106\pi$
$509$	−11.7842	−0.522324	−0.261162	+	0.965295i	$0.584106\pi$
$510$	0	0
$511$	1.98790	0.0879395
$512$	0	0
$513$	−3.75035	−0.165582
$514$	0	0
$515$	−62.3395	−2.74700
$516$	0	0
$517$	5.21717	0.229451
$518$	0	0
$519$	−52.8681	−2.32065
$520$	0	0
$521$	−19.0809	−0.835949	−0.417974	−	0.908459i	$-0.637260\pi$
$521$	−19.0809	−0.835949	−0.417974	+	0.908459i	$0.637260\pi$
$522$	0	0
$523$	−4.20785	−0.183997	−0.0919983	−	0.995759i	$-0.529325\pi$
$523$	−4.20785	−0.183997	−0.0919983	+	0.995759i	$0.529325\pi$
$524$	0	0
$525$	−56.7959	−2.47878
$526$	0	0
$527$	−17.3726	−0.756763
$528$	0	0
$529$	−22.6727	−0.985771
$530$	0	0
$531$	−35.0528	−1.52116
$532$	0	0
$533$	0.945420	0.0409507
$534$	0	0
$535$	−38.1286	−1.64844
$536$	0	0
$537$	45.6817	1.97131
$538$	0	0
$539$	−7.72075	−0.332556
$540$	0	0
$541$	1.17651	0.0505820	0.0252910	−	0.999680i	$-0.491949\pi$
$541$	1.17651	0.0505820	0.0252910	+	0.999680i	$0.491949\pi$
$542$	0	0
$543$	−28.5600	−1.22563
$544$	0	0
$545$	1.04573	0.0447943
$546$	0	0
$547$	1.24925	0.0534141	0.0267071	−	0.999643i	$-0.491498\pi$
$547$	1.24925	0.0534141	0.0267071	+	0.999643i	$0.491498\pi$
$548$	0	0
$549$	26.7615	1.14215
$550$	0	0
$551$	37.4233	1.59429
$552$	0	0
$553$	−19.8352	−0.843477
$554$	0	0
$555$	40.7178	1.72837
$556$	0	0
$557$	4.66180	0.197527	0.0987635	−	0.995111i	$-0.468511\pi$
$557$	4.66180	0.197527	0.0987635	+	0.995111i	$0.468511\pi$
$558$	0	0
$559$	20.9486	0.886031
$560$	0	0
$561$	18.6059	0.785542
$562$	0	0
$563$	−9.89653	−0.417089	−0.208545	−	0.978013i	$-0.566873\pi$
$563$	−9.89653	−0.417089	−0.208545	+	0.978013i	$0.566873\pi$
$564$	0	0
$565$	31.2798	1.31595
$566$	0	0
$567$	−18.1821	−0.763577
$568$	0	0
$569$	14.5155	0.608521	0.304260	−	0.952589i	$-0.401591\pi$
$569$	14.5155	0.608521	0.304260	+	0.952589i	$0.401591\pi$
$570$	0	0
$571$	27.2594	1.14077	0.570385	−	0.821377i	$-0.306794\pi$
$571$	27.2594	1.14077	0.570385	+	0.821377i	$0.306794\pi$
$572$	0	0
$573$	18.0422	0.753722
$574$	0	0
$575$	−5.96253	−0.248655
$576$	0	0
$577$	−37.2339	−1.55007	−0.775033	−	0.631920i	$-0.782267\pi$
$577$	−37.2339	−1.55007	−0.775033	+	0.631920i	$0.782267\pi$
$578$	0	0
$579$	15.6116	0.648794
$580$	0	0
$581$	26.7682	1.11053
$582$	0	0
$583$	−25.6656	−1.06296
$584$	0	0
$585$	−48.6487	−2.01138
$586$	0	0
$587$	−19.9918	−0.825152	−0.412576	−	0.910923i	$-0.635371\pi$
$587$	−19.9918	−0.825152	−0.412576	+	0.910923i	$0.635371\pi$
$588$	0	0
$589$	57.2469	2.35882
$590$	0	0
$591$	−12.9278	−0.531780
$592$	0	0
$593$	−36.6630	−1.50557	−0.752785	−	0.658266i	$-0.771290\pi$
$593$	−36.6630	−1.50557	−0.752785	+	0.658266i	$0.771290\pi$
$594$	0	0
$595$	18.4316	0.755622
$596$	0	0
$597$	52.2116	2.13688
$598$	0	0
$599$	21.8396	0.892343	0.446172	−	0.894947i	$-0.352787\pi$
$599$	21.8396	0.892343	0.446172	+	0.894947i	$0.352787\pi$
$600$	0	0
$601$	−0.564099	−0.0230101	−0.0115050	−	0.999934i	$-0.503662\pi$
$601$	−0.564099	−0.0230101	−0.0115050	+	0.999934i	$0.503662\pi$
$602$	0	0
$603$	19.0568	0.776051
$604$	0	0
$605$	−4.28274	−0.174118
$606$	0	0
$607$	42.3256	1.71794	0.858972	−	0.512022i	$-0.171103\pi$
$607$	42.3256	1.71794	0.858972	+	0.512022i	$0.171103\pi$
$608$	0	0
$609$	−28.8271	−1.16814
$610$	0	0
$611$	5.78539	0.234052
$612$	0	0
$613$	−19.1984	−0.775416	−0.387708	−	0.921782i	$-0.626733\pi$
$613$	−19.1984	−0.775416	−0.387708	+	0.921782i	$0.626733\pi$
$614$	0	0
$615$	2.40008	0.0967804
$616$	0	0
$617$	31.4840	1.26750	0.633750	−	0.773538i	$-0.281515\pi$
$617$	31.4840	1.26750	0.633750	+	0.773538i	$0.281515\pi$
$618$	0	0
$619$	−26.3771	−1.06019	−0.530093	−	0.847939i	$-0.677843\pi$
$619$	−26.3771	−1.06019	−0.530093	+	0.847939i	$0.677843\pi$
$620$	0	0
$621$	0.303281	0.0121703
$622$	0	0
$623$	23.9626	0.960042
$624$	0	0
$625$	31.5199	1.26080
$626$	0	0
$627$	−61.3109	−2.44852
$628$	0	0
$629$	−8.92999	−0.356062
$630$	0	0
$631$	9.85443	0.392298	0.196149	−	0.980574i	$-0.437156\pi$
$631$	9.85443	0.392298	0.196149	+	0.980574i	$0.437156\pi$
$632$	0	0
$633$	43.4630	1.72750
$634$	0	0
$635$	24.3968	0.968159
$636$	0	0
$637$	−8.56164	−0.339225
$638$	0	0
$639$	−20.1330	−0.796450
$640$	0	0
$641$	−0.555236	−0.0219305	−0.0109652	−	0.999940i	$-0.503490\pi$
$641$	−0.555236	−0.0219305	−0.0109652	+	0.999940i	$0.503490\pi$
$642$	0	0
$643$	−15.8519	−0.625136	−0.312568	−	0.949895i	$-0.601189\pi$
$643$	−15.8519	−0.625136	−0.312568	+	0.949895i	$0.601189\pi$
$644$	0	0
$645$	53.1808	2.09399
$646$	0	0
$647$	−32.4741	−1.27669	−0.638345	−	0.769751i	$-0.720380\pi$
$647$	−32.4741	−1.27669	−0.638345	+	0.769751i	$0.720380\pi$
$648$	0	0
$649$	−37.9382	−1.48920
$650$	0	0
$651$	−44.0973	−1.72831
$652$	0	0
$653$	31.1028	1.21715	0.608573	−	0.793498i	$-0.291742\pi$
$653$	31.1028	1.21715	0.608573	+	0.793498i	$0.291742\pi$
$654$	0	0
$655$	57.8619	2.26085
$656$	0	0
$657$	2.92125	0.113969
$658$	0	0
$659$	12.9780	0.505551	0.252776	−	0.967525i	$-0.418657\pi$
$659$	12.9780	0.505551	0.252776	+	0.967525i	$0.418657\pi$
$660$	0	0
$661$	25.6740	0.998603	0.499301	−	0.866428i	$-0.333590\pi$
$661$	25.6740	0.998603	0.499301	+	0.866428i	$0.333590\pi$
$662$	0	0
$663$	20.6323	0.801294
$664$	0	0
$665$	−60.7366	−2.35526
$666$	0	0
$667$	−3.02633	−0.117180
$668$	0	0
$669$	−18.9688	−0.733376
$670$	0	0
$671$	28.9643	1.11815
$672$	0	0
$673$	−9.25271	−0.356666	−0.178333	−	0.983970i	$-0.557070\pi$
$673$	−9.25271	−0.356666	−0.178333	+	0.983970i	$0.557070\pi$
$674$	0	0
$675$	−5.52559	−0.212680
$676$	0	0
$677$	49.2561	1.89307	0.946533	−	0.322607i	$-0.104559\pi$
$677$	49.2561	1.89307	0.946533	+	0.322607i	$0.104559\pi$
$678$	0	0
$679$	−15.9512	−0.612151
$680$	0	0
$681$	−65.3490	−2.50418
$682$	0	0
$683$	−11.1109	−0.425148	−0.212574	−	0.977145i	$-0.568185\pi$
$683$	−11.1109	−0.425148	−0.212574	+	0.977145i	$0.568185\pi$
$684$	0	0
$685$	−35.8973	−1.37156
$686$	0	0
$687$	−32.7950	−1.25121
$688$	0	0
$689$	−28.4609	−1.08427
$690$	0	0
$691$	47.2131	1.79607	0.898036	−	0.439922i	$-0.144994\pi$
$691$	47.2131	1.79607	0.898036	+	0.439922i	$0.144994\pi$
$692$	0	0
$693$	24.4223	0.927727
$694$	0	0
$695$	−5.30442	−0.201208
$696$	0	0
$697$	−0.526371	−0.0199377
$698$	0	0
$699$	−9.65492	−0.365183
$700$	0	0
$701$	−10.3819	−0.392119	−0.196060	−	0.980592i	$-0.562815\pi$
$701$	−10.3819	−0.392119	−0.196060	+	0.980592i	$0.562815\pi$
$702$	0	0
$703$	29.4264	1.10984
$704$	0	0
$705$	14.6870	0.553144
$706$	0	0
$707$	−4.71082	−0.177168
$708$	0	0
$709$	−15.3544	−0.576648	−0.288324	−	0.957533i	$-0.593098\pi$
$709$	−15.3544	−0.576648	−0.288324	+	0.957533i	$0.593098\pi$
$710$	0	0
$711$	−29.1481	−1.09314
$712$	0	0
$713$	−4.62941	−0.173373
$714$	0	0
$715$	−52.6532	−1.96912
$716$	0	0
$717$	50.9041	1.90105
$718$	0	0
$719$	−15.3370	−0.571974	−0.285987	−	0.958233i	$-0.592321\pi$
$719$	−15.3370	−0.571974	−0.285987	+	0.958233i	$0.592321\pi$
$720$	0	0
$721$	34.7038	1.29244
$722$	0	0
$723$	−23.4387	−0.871696
$724$	0	0
$725$	55.1378	2.04777
$726$	0	0
$727$	8.20666	0.304368	0.152184	−	0.988352i	$-0.451369\pi$
$727$	8.20666	0.304368	0.152184	+	0.988352i	$0.451369\pi$
$728$	0	0
$729$	−30.6832	−1.13641
$730$	0	0
$731$	−11.6633	−0.431383
$732$	0	0
$733$	14.9200	0.551082	0.275541	−	0.961289i	$-0.411143\pi$
$733$	14.9200	0.551082	0.275541	+	0.961289i	$0.411143\pi$
$734$	0	0
$735$	−21.7349	−0.801703
$736$	0	0
$737$	20.6254	0.759746
$738$	0	0
$739$	−39.3144	−1.44620	−0.723102	−	0.690742i	$-0.757284\pi$
$739$	−39.3144	−1.44620	−0.723102	+	0.690742i	$0.757284\pi$
$740$	0	0
$741$	−67.9885	−2.49762
$742$	0	0
$743$	−43.4766	−1.59500	−0.797500	−	0.603319i	$-0.793845\pi$
$743$	−43.4766	−1.59500	−0.797500	+	0.603319i	$0.793845\pi$
$744$	0	0
$745$	−48.6614	−1.78282
$746$	0	0
$747$	39.3364	1.43924
$748$	0	0
$749$	21.2258	0.775574
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	65.7782	2.39709
$754$	0	0
$755$	69.8787	2.54315
$756$	0	0
$757$	20.0740	0.729600	0.364800	−	0.931086i	$-0.381137\pi$
$757$	20.0740	0.729600	0.364800	+	0.931086i	$0.381137\pi$
$758$	0	0
$759$	4.95805	0.179966
$760$	0	0
$761$	16.5998	0.601742	0.300871	−	0.953665i	$-0.402723\pi$
$761$	16.5998	0.601742	0.300871	+	0.953665i	$0.402723\pi$
$762$	0	0
$763$	−0.582149	−0.0210752
$764$	0	0
$765$	27.0856	0.979281
$766$	0	0
$767$	−42.0702	−1.51907
$768$	0	0
$769$	−24.4487	−0.881641	−0.440821	−	0.897595i	$-0.645313\pi$
$769$	−24.4487	−0.881641	−0.440821	+	0.897595i	$0.645313\pi$
$770$	0	0
$771$	−49.5019	−1.78277
$772$	0	0
$773$	−33.8832	−1.21869	−0.609347	−	0.792903i	$-0.708569\pi$
$773$	−33.8832	−1.21869	−0.609347	+	0.792903i	$0.708569\pi$
$774$	0	0
$775$	84.3451	3.02976
$776$	0	0
$777$	−22.6672	−0.813181
$778$	0	0
$779$	1.73452	0.0621455
$780$	0	0
$781$	−21.7902	−0.779716
$782$	0	0
$783$	−2.80455	−0.100227
$784$	0	0
$785$	−51.6146	−1.84220
$786$	0	0
$787$	−8.39663	−0.299307	−0.149654	−	0.988738i	$-0.547816\pi$
$787$	−8.39663	−0.299307	−0.149654	+	0.988738i	$0.547816\pi$
$788$	0	0
$789$	−54.2078	−1.92985
$790$	0	0
$791$	−17.4132	−0.619141
$792$	0	0
$793$	32.1189	1.14058
$794$	0	0
$795$	−72.2519	−2.56251
$796$	0	0
$797$	20.8862	0.739826	0.369913	−	0.929066i	$-0.379387\pi$
$797$	20.8862	0.739826	0.369913	+	0.929066i	$0.379387\pi$
$798$	0	0
$799$	−3.22107	−0.113953
$800$	0	0
$801$	35.2135	1.24421
$802$	0	0
$803$	3.16171	0.111574
$804$	0	0
$805$	4.91161	0.173112
$806$	0	0
$807$	−74.2628	−2.61417
$808$	0	0
$809$	−2.28498	−0.0803355	−0.0401678	−	0.999193i	$-0.512789\pi$
$809$	−2.28498	−0.0803355	−0.0401678	+	0.999193i	$0.512789\pi$
$810$	0	0
$811$	−18.5733	−0.652198	−0.326099	−	0.945336i	$-0.605734\pi$
$811$	−18.5733	−0.652198	−0.326099	+	0.945336i	$0.605734\pi$
$812$	0	0
$813$	−11.9963	−0.420728
$814$	0	0
$815$	−27.2401	−0.954178
$816$	0	0
$817$	38.4334	1.34461
$818$	0	0
$819$	27.0823	0.946330
$820$	0	0
$821$	−13.4261	−0.468575	−0.234287	−	0.972167i	$-0.575276\pi$
$821$	−13.4261	−0.468575	−0.234287	+	0.972167i	$0.575276\pi$
$822$	0	0
$823$	56.5702	1.97191	0.985957	−	0.167001i	$-0.0534082\pi$
$823$	56.5702	1.97191	0.985957	+	0.167001i	$0.0534082\pi$
$824$	0	0
$825$	−90.3327	−3.14498
$826$	0	0
$827$	23.4792	0.816450	0.408225	−	0.912881i	$-0.366148\pi$
$827$	23.4792	0.816450	0.408225	+	0.912881i	$0.366148\pi$
$828$	0	0
$829$	2.60961	0.0906356	0.0453178	−	0.998973i	$-0.485570\pi$
$829$	2.60961	0.0906356	0.0453178	+	0.998973i	$0.485570\pi$
$830$	0	0
$831$	4.68453	0.162504
$832$	0	0
$833$	4.76677	0.165159
$834$	0	0
$835$	49.1072	1.69942
$836$	0	0
$837$	−4.29017	−0.148290
$838$	0	0
$839$	−33.9247	−1.17121	−0.585606	−	0.810596i	$-0.699143\pi$
$839$	−33.9247	−1.17121	−0.585606	+	0.810596i	$0.699143\pi$
$840$	0	0
$841$	−1.01442	−0.0349802
$842$	0	0
$843$	−32.5275	−1.12031
$844$	0	0
$845$	−7.33448	−0.252314
$846$	0	0
$847$	2.38416	0.0819207
$848$	0	0
$849$	−16.8353	−0.577785
$850$	0	0
$851$	−2.37964	−0.0815730
$852$	0	0
$853$	−24.2402	−0.829970	−0.414985	−	0.909828i	$-0.636213\pi$
$853$	−24.2402	−0.829970	−0.414985	+	0.909828i	$0.636213\pi$
$854$	0	0
$855$	−89.2535	−3.05240
$856$	0	0
$857$	−20.8125	−0.710941	−0.355470	−	0.934688i	$-0.615679\pi$
$857$	−20.8125	−0.710941	−0.355470	+	0.934688i	$0.615679\pi$
$858$	0	0
$859$	−30.8237	−1.05169	−0.525846	−	0.850580i	$-0.676251\pi$
$859$	−30.8237	−1.05169	−0.525846	+	0.850580i	$0.676251\pi$
$860$	0	0
$861$	−1.33610	−0.0455341
$862$	0	0
$863$	−43.5436	−1.48224	−0.741121	−	0.671372i	$-0.765705\pi$
$863$	−43.5436	−1.48224	−0.741121	+	0.671372i	$0.765705\pi$
$864$	0	0
$865$	−83.2980	−2.83222
$866$	0	0
$867$	30.8857	1.04893
$868$	0	0
$869$	−31.5474	−1.07017
$870$	0	0
$871$	22.8718	0.774981
$872$	0	0
$873$	−23.4406	−0.793343
$874$	0	0
$875$	−46.5581	−1.57395
$876$	0	0
$877$	−55.6566	−1.87939	−0.939695	−	0.342014i	$-0.888891\pi$
$877$	−55.6566	−1.87939	−0.939695	+	0.342014i	$0.888891\pi$
$878$	0	0
$879$	50.2024	1.69328
$880$	0	0
$881$	2.58474	0.0870821	0.0435411	−	0.999052i	$-0.486136\pi$
$881$	2.58474	0.0870821	0.0435411	+	0.999052i	$0.486136\pi$
$882$	0	0
$883$	−8.56154	−0.288119	−0.144059	−	0.989569i	$-0.546016\pi$
$883$	−8.56154	−0.288119	−0.144059	+	0.989569i	$0.546016\pi$
$884$	0	0
$885$	−106.801	−3.59007
$886$	0	0
$887$	−45.2711	−1.52006	−0.760028	−	0.649891i	$-0.774815\pi$
$887$	−45.2711	−1.52006	−0.760028	+	0.649891i	$0.774815\pi$
$888$	0	0
$889$	−13.5815	−0.455508
$890$	0	0
$891$	−28.9183	−0.968799
$892$	0	0
$893$	10.6142	0.355190
$894$	0	0
$895$	71.9752	2.40586
$896$	0	0
$897$	5.49806	0.183575
$898$	0	0
$899$	42.8099	1.42779
$900$	0	0
$901$	15.8459	0.527902
$902$	0	0
$903$	−29.6052	−0.985200
$904$	0	0
$905$	−44.9986	−1.49580
$906$	0	0
$907$	35.5777	1.18134	0.590669	−	0.806914i	$-0.298864\pi$
$907$	35.5777	1.18134	0.590669	+	0.806914i	$0.298864\pi$
$908$	0	0
$909$	−6.92263	−0.229609
$910$	0	0
$911$	−13.1500	−0.435679	−0.217839	−	0.975985i	$-0.569901\pi$
$911$	−13.1500	−0.435679	−0.217839	+	0.975985i	$0.569901\pi$
$912$	0	0
$913$	42.5744	1.40901
$914$	0	0
$915$	81.5382	2.69557
$916$	0	0
$917$	−32.2112	−1.06371
$918$	0	0
$919$	−23.1987	−0.765255	−0.382628	−	0.923903i	$-0.624981\pi$
$919$	−23.1987	−0.765255	−0.382628	+	0.923903i	$0.624981\pi$
$920$	0	0
$921$	−11.7201	−0.386191
$922$	0	0
$923$	−24.1635	−0.795351
$924$	0	0
$925$	43.3556	1.42552
$926$	0	0
$927$	50.9978	1.67499
$928$	0	0
$929$	−10.4634	−0.343292	−0.171646	−	0.985159i	$-0.554909\pi$
$929$	−10.4634	−0.343292	−0.171646	+	0.985159i	$0.554909\pi$
$930$	0	0
$931$	−15.7077	−0.514797
$932$	0	0
$933$	−81.2774	−2.66090
$934$	0	0
$935$	29.3151	0.958707
$936$	0	0
$937$	−29.5114	−0.964094	−0.482047	−	0.876145i	$-0.660107\pi$
$937$	−29.5114	−0.964094	−0.482047	+	0.876145i	$0.660107\pi$
$938$	0	0
$939$	−5.79523	−0.189120
$940$	0	0
$941$	44.6333	1.45500	0.727502	−	0.686105i	$-0.240681\pi$
$941$	44.6333	1.45500	0.727502	+	0.686105i	$0.240681\pi$
$942$	0	0
$943$	−0.140266	−0.00456769
$944$	0	0
$945$	4.55169	0.148066
$946$	0	0
$947$	43.7330	1.42113	0.710566	−	0.703631i	$-0.248439\pi$
$947$	43.7330	1.42113	0.710566	+	0.703631i	$0.248439\pi$
$948$	0	0
$949$	3.50607	0.113812
$950$	0	0
$951$	33.2020	1.07665
$952$	0	0
$953$	31.2685	1.01289	0.506444	−	0.862273i	$-0.330960\pi$
$953$	31.2685	1.01289	0.506444	+	0.862273i	$0.330960\pi$
$954$	0	0
$955$	28.4269	0.919872
$956$	0	0
$957$	−45.8490	−1.48209
$958$	0	0
$959$	19.9837	0.645306
$960$	0	0
$961$	34.4870	1.11248
$962$	0	0
$963$	31.1917	1.00514
$964$	0	0
$965$	24.5973	0.791815
$966$	0	0
$967$	3.81997	0.122842	0.0614210	−	0.998112i	$-0.480437\pi$
$967$	3.81997	0.122842	0.0614210	+	0.998112i	$0.480437\pi$
$968$	0	0
$969$	37.8532	1.21602
$970$	0	0
$971$	19.4527	0.624268	0.312134	−	0.950038i	$-0.398956\pi$
$971$	19.4527	0.624268	0.312134	+	0.950038i	$0.398956\pi$
$972$	0	0
$973$	2.95292	0.0946663
$974$	0	0
$975$	−100.171	−3.20805
$976$	0	0
$977$	57.4235	1.83714	0.918571	−	0.395256i	$-0.129344\pi$
$977$	57.4235	1.83714	0.918571	+	0.395256i	$0.129344\pi$
$978$	0	0
$979$	38.1121	1.21807
$980$	0	0
$981$	−0.855479	−0.0273133
$982$	0	0
$983$	−7.00696	−0.223487	−0.111744	−	0.993737i	$-0.535644\pi$
$983$	−7.00696	−0.223487	−0.111744	+	0.993737i	$0.535644\pi$
$984$	0	0
$985$	−20.3688	−0.649005
$986$	0	0
$987$	−8.17610	−0.260248
$988$	0	0
$989$	−3.10801	−0.0988290
$990$	0	0
$991$	−9.87477	−0.313682	−0.156841	−	0.987624i	$-0.550131\pi$
$991$	−9.87477	−0.313682	−0.156841	+	0.987624i	$0.550131\pi$
$992$	0	0
$993$	2.85017	0.0904473
$994$	0	0
$995$	82.2637	2.60793
$996$	0	0
$997$	45.6262	1.44500	0.722498	−	0.691373i	$-0.242994\pi$
$997$	45.6262	1.44500	0.722498	+	0.691373i	$0.242994\pi$
$998$	0	0
$999$	−2.20526	−0.0697713

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.8	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-2.49253 q^{3} -3.92718 q^{5} +2.18622 q^{7} +3.21269 q^{9} +O(q^{10})\)	\(q-2.49253 q^{3} -3.92718 q^{5} +2.18622 q^{7} +3.21269 q^{9} +3.47715 q^{11} +3.85586 q^{13} +9.78861 q^{15} -2.14678 q^{17} +7.07416 q^{19} -5.44922 q^{21} -0.572069 q^{23} +10.4227 q^{25} -0.530148 q^{27} +5.29014 q^{29} +8.09240 q^{31} -8.66688 q^{33} -8.58569 q^{35} +4.15971 q^{37} -9.61083 q^{39} +0.245191 q^{41} +5.43293 q^{43} -12.6168 q^{45} +1.50042 q^{47} -2.22043 q^{49} +5.35091 q^{51} -7.38122 q^{53} -13.6554 q^{55} -17.6325 q^{57} -10.9107 q^{59} +8.32991 q^{61} +7.02367 q^{63} -15.1426 q^{65} +5.93170 q^{67} +1.42590 q^{69} -6.26670 q^{71} +0.909284 q^{73} -25.9790 q^{75} +7.60182 q^{77} -9.07280 q^{79} -8.31668 q^{81} +12.2441 q^{83} +8.43080 q^{85} -13.1858 q^{87} +10.9607 q^{89} +8.42976 q^{91} -20.1705 q^{93} -27.7815 q^{95} -7.29623 q^{97} +11.1710 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more