LMFDB - Embedded newform 6008.2.a.e.1.12

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			1.12
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)

$f(q)$	$=$	$q-1.56448 q^{3} +2.55356 q^{5} -2.27627 q^{7} -0.552416 q^{9} +O(q^{10})$	$q-1.56448 q^{3} +2.55356 q^{5} -2.27627 q^{7} -0.552416 q^{9} -5.49177 q^{11} +6.01599 q^{13} -3.99498 q^{15} -3.09065 q^{17} -4.77783 q^{19} +3.56117 q^{21} +7.81673 q^{23} +1.52067 q^{25} +5.55767 q^{27} +10.1573 q^{29} -9.04284 q^{31} +8.59175 q^{33} -5.81260 q^{35} -7.43029 q^{37} -9.41186 q^{39} +2.58217 q^{41} -8.81318 q^{43} -1.41063 q^{45} -0.566299 q^{47} -1.81858 q^{49} +4.83525 q^{51} -1.06014 q^{53} -14.0236 q^{55} +7.47480 q^{57} +8.80752 q^{59} -3.65742 q^{61} +1.25745 q^{63} +15.3622 q^{65} -9.02674 q^{67} -12.2291 q^{69} +13.2238 q^{71} -4.91442 q^{73} -2.37905 q^{75} +12.5008 q^{77} +8.48091 q^{79} -7.03759 q^{81} +9.36299 q^{83} -7.89217 q^{85} -15.8909 q^{87} +6.37020 q^{89} -13.6940 q^{91} +14.1473 q^{93} -12.2005 q^{95} +2.29917 q^{97} +3.03374 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.56448	−0.903251	−0.451625	−	0.892208i	$-0.649156\pi$
$3$	−1.56448	−0.903251	−0.451625	+	0.892208i	$0.649156\pi$
$4$	0	0
$5$	2.55356	1.14199	0.570993	−	0.820955i	$-0.306558\pi$
$5$	2.55356	1.14199	0.570993	+	0.820955i	$0.306558\pi$
$6$	0	0
$7$	−2.27627	−0.860350	−0.430175	−	0.902745i	$-0.641548\pi$
$7$	−2.27627	−0.860350	−0.430175	+	0.902745i	$0.641548\pi$
$8$	0	0
$9$	−0.552416	−0.184139
$10$	0	0
$11$	−5.49177	−1.65583	−0.827916	−	0.560852i	$-0.810474\pi$
$11$	−5.49177	−1.65583	−0.827916	+	0.560852i	$0.810474\pi$
$12$	0	0
$13$	6.01599	1.66853	0.834267	−	0.551360i	$-0.185891\pi$
$13$	6.01599	1.66853	0.834267	+	0.551360i	$0.185891\pi$
$14$	0	0
$15$	−3.99498	−1.03150
$16$	0	0
$17$	−3.09065	−0.749594	−0.374797	−	0.927107i	$-0.622288\pi$
$17$	−3.09065	−0.749594	−0.374797	+	0.927107i	$0.622288\pi$
$18$	0	0
$19$	−4.77783	−1.09611	−0.548055	−	0.836442i	$-0.684631\pi$
$19$	−4.77783	−1.09611	−0.548055	+	0.836442i	$0.684631\pi$
$20$	0	0
$21$	3.56117	0.777112
$22$	0	0
$23$	7.81673	1.62990	0.814951	−	0.579530i	$-0.196764\pi$
$23$	7.81673	1.62990	0.814951	+	0.579530i	$0.196764\pi$
$24$	0	0
$25$	1.52067	0.304134
$26$	0	0
$27$	5.55767	1.06957
$28$	0	0
$29$	10.1573	1.88617	0.943083	−	0.332559i	$-0.107912\pi$
$29$	10.1573	1.88617	0.943083	+	0.332559i	$0.107912\pi$
$30$	0	0
$31$	−9.04284	−1.62414	−0.812071	−	0.583559i	$-0.801660\pi$
$31$	−9.04284	−1.62414	−0.812071	+	0.583559i	$0.801660\pi$
$32$	0	0
$33$	8.59175	1.49563
$34$	0	0
$35$	−5.81260	−0.982508
$36$	0	0
$37$	−7.43029	−1.22153	−0.610766	−	0.791811i	$-0.709138\pi$
$37$	−7.43029	−1.22153	−0.610766	+	0.791811i	$0.709138\pi$
$38$	0	0
$39$	−9.41186	−1.50710
$40$	0	0
$41$	2.58217	0.403268	0.201634	−	0.979461i	$-0.435375\pi$
$41$	2.58217	0.403268	0.201634	+	0.979461i	$0.435375\pi$
$42$	0	0
$43$	−8.81318	−1.34400	−0.671998	−	0.740553i	$-0.734564\pi$
$43$	−8.81318	−1.34400	−0.671998	+	0.740553i	$0.734564\pi$
$44$	0	0
$45$	−1.41063	−0.210284
$46$	0	0
$47$	−0.566299	−0.0826032	−0.0413016	−	0.999147i	$-0.513150\pi$
$47$	−0.566299	−0.0826032	−0.0413016	+	0.999147i	$0.513150\pi$
$48$	0	0
$49$	−1.81858	−0.259798
$50$	0	0
$51$	4.83525	0.677071
$52$	0	0
$53$	−1.06014	−0.145621	−0.0728107	−	0.997346i	$-0.523197\pi$
$53$	−1.06014	−0.145621	−0.0728107	+	0.997346i	$0.523197\pi$
$54$	0	0
$55$	−14.0236	−1.89094
$56$	0	0
$57$	7.47480	0.990061
$58$	0	0
$59$	8.80752	1.14664	0.573321	−	0.819331i	$-0.305655\pi$
$59$	8.80752	1.14664	0.573321	+	0.819331i	$0.305655\pi$
$60$	0	0
$61$	−3.65742	−0.468284	−0.234142	−	0.972202i	$-0.575228\pi$
$61$	−3.65742	−0.468284	−0.234142	+	0.972202i	$0.575228\pi$
$62$	0	0
$63$	1.25745	0.158424
$64$	0	0
$65$	15.3622	1.90544
$66$	0	0
$67$	−9.02674	−1.10279	−0.551396	−	0.834244i	$-0.685905\pi$
$67$	−9.02674	−1.10279	−0.551396	+	0.834244i	$0.685905\pi$
$68$	0	0
$69$	−12.2291	−1.47221
$70$	0	0
$71$	13.2238	1.56938	0.784690	−	0.619888i	$-0.212822\pi$
$71$	13.2238	1.56938	0.784690	+	0.619888i	$0.212822\pi$
$72$	0	0
$73$	−4.91442	−0.575189	−0.287595	−	0.957752i	$-0.592856\pi$
$73$	−4.91442	−0.575189	−0.287595	+	0.957752i	$0.592856\pi$
$74$	0	0
$75$	−2.37905	−0.274709
$76$	0	0
$77$	12.5008	1.42460
$78$	0	0
$79$	8.48091	0.954176	0.477088	−	0.878855i	$-0.341692\pi$
$79$	8.48091	0.954176	0.477088	+	0.878855i	$0.341692\pi$
$80$	0	0
$81$	−7.03759	−0.781954
$82$	0	0
$83$	9.36299	1.02772	0.513861	−	0.857874i	$-0.328215\pi$
$83$	9.36299	1.02772	0.513861	+	0.857874i	$0.328215\pi$
$84$	0	0
$85$	−7.89217	−0.856026
$86$	0	0
$87$	−15.8909	−1.70368
$88$	0	0
$89$	6.37020	0.675240	0.337620	−	0.941283i	$-0.390378\pi$
$89$	6.37020	0.675240	0.337620	+	0.941283i	$0.390378\pi$
$90$	0	0
$91$	−13.6940	−1.43552
$92$	0	0
$93$	14.1473	1.46701
$94$	0	0
$95$	−12.2005	−1.25174
$96$	0	0
$97$	2.29917	0.233445	0.116723	−	0.993165i	$-0.462761\pi$
$97$	2.29917	0.233445	0.116723	+	0.993165i	$0.462761\pi$
$98$	0	0
$99$	3.03374	0.304902
$100$	0	0
$101$	2.68211	0.266879	0.133440	−	0.991057i	$-0.457398\pi$
$101$	2.68211	0.266879	0.133440	+	0.991057i	$0.457398\pi$
$102$	0	0
$103$	14.4548	1.42427	0.712136	−	0.702041i	$-0.247728\pi$
$103$	14.4548	1.42427	0.712136	+	0.702041i	$0.247728\pi$
$104$	0	0
$105$	9.09367	0.887451
$106$	0	0
$107$	−8.08142	−0.781261	−0.390630	−	0.920548i	$-0.627743\pi$
$107$	−8.08142	−0.781261	−0.390630	+	0.920548i	$0.627743\pi$
$108$	0	0
$109$	−9.27715	−0.888590	−0.444295	−	0.895881i	$-0.646546\pi$
$109$	−9.27715	−0.888590	−0.444295	+	0.895881i	$0.646546\pi$
$110$	0	0
$111$	11.6245	1.10335
$112$	0	0
$113$	14.2851	1.34383	0.671914	−	0.740629i	$-0.265472\pi$
$113$	14.2851	1.34383	0.671914	+	0.740629i	$0.265472\pi$
$114$	0	0
$115$	19.9605	1.86133
$116$	0	0
$117$	−3.32332	−0.307241
$118$	0	0
$119$	7.03517	0.644913
$120$	0	0
$121$	19.1596	1.74178
$122$	0	0
$123$	−4.03975	−0.364252
$124$	0	0
$125$	−8.88468	−0.794670
$126$	0	0
$127$	−11.2935	−1.00213	−0.501067	−	0.865408i	$-0.667059\pi$
$127$	−11.2935	−1.00213	−0.501067	+	0.865408i	$0.667059\pi$
$128$	0	0
$129$	13.7880	1.21397
$130$	0	0
$131$	5.50771	0.481211	0.240606	−	0.970623i	$-0.422654\pi$
$131$	5.50771	0.481211	0.240606	+	0.970623i	$0.422654\pi$
$132$	0	0
$133$	10.8756	0.943038
$134$	0	0
$135$	14.1918	1.22144
$136$	0	0
$137$	1.08824	0.0929750	0.0464875	−	0.998919i	$-0.485197\pi$
$137$	1.08824	0.0929750	0.0464875	+	0.998919i	$0.485197\pi$
$138$	0	0
$139$	14.2837	1.21153	0.605763	−	0.795645i	$-0.292868\pi$
$139$	14.2837	1.21153	0.605763	+	0.795645i	$0.292868\pi$
$140$	0	0
$141$	0.885961	0.0746114
$142$	0	0
$143$	−33.0384	−2.76281
$144$	0	0
$145$	25.9373	2.15398
$146$	0	0
$147$	2.84513	0.234662
$148$	0	0
$149$	11.8921	0.974236	0.487118	−	0.873336i	$-0.338048\pi$
$149$	11.8921	0.974236	0.487118	+	0.873336i	$0.338048\pi$
$150$	0	0
$151$	21.4887	1.74873	0.874364	−	0.485271i	$-0.161279\pi$
$151$	21.4887	1.74873	0.874364	+	0.485271i	$0.161279\pi$
$152$	0	0
$153$	1.70733	0.138029
$154$	0	0
$155$	−23.0914	−1.85475
$156$	0	0
$157$	−8.32223	−0.664186	−0.332093	−	0.943247i	$-0.607755\pi$
$157$	−8.32223	−0.664186	−0.332093	+	0.943247i	$0.607755\pi$
$158$	0	0
$159$	1.65856	0.131533
$160$	0	0
$161$	−17.7930	−1.40229
$162$	0	0
$163$	17.2778	1.35330	0.676649	−	0.736305i	$-0.263431\pi$
$163$	17.2778	1.35330	0.676649	+	0.736305i	$0.263431\pi$
$164$	0	0
$165$	21.9395	1.70799
$166$	0	0
$167$	−14.0757	−1.08921	−0.544605	−	0.838693i	$-0.683320\pi$
$167$	−14.0757	−1.08921	−0.544605	+	0.838693i	$0.683320\pi$
$168$	0	0
$169$	23.1921	1.78401
$170$	0	0
$171$	2.63935	0.201836
$172$	0	0
$173$	11.8755	0.902878	0.451439	−	0.892302i	$-0.350911\pi$
$173$	11.8755	0.902878	0.451439	+	0.892302i	$0.350911\pi$
$174$	0	0
$175$	−3.46145	−0.261661
$176$	0	0
$177$	−13.7792	−1.03570
$178$	0	0
$179$	−7.48192	−0.559225	−0.279612	−	0.960113i	$-0.590206\pi$
$179$	−7.48192	−0.559225	−0.279612	+	0.960113i	$0.590206\pi$
$180$	0	0
$181$	−17.3704	−1.29113	−0.645566	−	0.763705i	$-0.723378\pi$
$181$	−17.3704	−1.29113	−0.645566	+	0.763705i	$0.723378\pi$
$182$	0	0
$183$	5.72194	0.422978
$184$	0	0
$185$	−18.9737	−1.39497
$186$	0	0
$187$	16.9732	1.24120
$188$	0	0
$189$	−12.6508	−0.920208
$190$	0	0
$191$	−23.6065	−1.70810	−0.854052	−	0.520188i	$-0.825862\pi$
$191$	−23.6065	−1.70810	−0.854052	+	0.520188i	$0.825862\pi$
$192$	0	0
$193$	−8.95160	−0.644350	−0.322175	−	0.946680i	$-0.604414\pi$
$193$	−8.95160	−0.644350	−0.322175	+	0.946680i	$0.604414\pi$
$194$	0	0
$195$	−24.0338	−1.72109
$196$	0	0
$197$	−10.6082	−0.755804	−0.377902	−	0.925846i	$-0.623354\pi$
$197$	−10.6082	−0.755804	−0.377902	+	0.925846i	$0.623354\pi$
$198$	0	0
$199$	0.0364639	0.00258486	0.00129243	−	0.999999i	$-0.499589\pi$
$199$	0.0364639	0.00258486	0.00129243	+	0.999999i	$0.499589\pi$
$200$	0	0
$201$	14.1221	0.996098
$202$	0	0
$203$	−23.1208	−1.62276
$204$	0	0
$205$	6.59373	0.460526
$206$	0	0
$207$	−4.31808	−0.300128
$208$	0	0
$209$	26.2388	1.81497
$210$	0	0
$211$	17.4829	1.20357	0.601785	−	0.798658i	$-0.294456\pi$
$211$	17.4829	1.20357	0.601785	+	0.798658i	$0.294456\pi$
$212$	0	0
$213$	−20.6884	−1.41754
$214$	0	0
$215$	−22.5050	−1.53483
$216$	0	0
$217$	20.5840	1.39733
$218$	0	0
$219$	7.68849	0.519540
$220$	0	0
$221$	−18.5933	−1.25072
$222$	0	0
$223$	−4.30580	−0.288338	−0.144169	−	0.989553i	$-0.546051\pi$
$223$	−4.30580	−0.288338	−0.144169	+	0.989553i	$0.546051\pi$
$224$	0	0
$225$	−0.840041	−0.0560027
$226$	0	0
$227$	24.6948	1.63905	0.819527	−	0.573040i	$-0.194236\pi$
$227$	24.6948	1.63905	0.819527	+	0.573040i	$0.194236\pi$
$228$	0	0
$229$	3.17703	0.209944	0.104972	−	0.994475i	$-0.466525\pi$
$229$	3.17703	0.209944	0.104972	+	0.994475i	$0.466525\pi$
$230$	0	0
$231$	−19.5572	−1.28677
$232$	0	0
$233$	−21.6926	−1.42113	−0.710566	−	0.703631i	$-0.751561\pi$
$233$	−21.6926	−1.42113	−0.710566	+	0.703631i	$0.751561\pi$
$234$	0	0
$235$	−1.44608	−0.0943318
$236$	0	0
$237$	−13.2682	−0.861860
$238$	0	0
$239$	23.5169	1.52118	0.760590	−	0.649232i	$-0.224910\pi$
$239$	23.5169	1.52118	0.760590	+	0.649232i	$0.224910\pi$
$240$	0	0
$241$	20.0100	1.28896	0.644479	−	0.764622i	$-0.277074\pi$
$241$	20.0100	1.28896	0.644479	+	0.764622i	$0.277074\pi$
$242$	0	0
$243$	−5.66286	−0.363273
$244$	0	0
$245$	−4.64386	−0.296685
$246$	0	0
$247$	−28.7434	−1.82890
$248$	0	0
$249$	−14.6482	−0.928290
$250$	0	0
$251$	−11.7447	−0.741321	−0.370660	−	0.928769i	$-0.620869\pi$
$251$	−11.7447	−0.741321	−0.370660	+	0.928769i	$0.620869\pi$
$252$	0	0
$253$	−42.9277	−2.69884
$254$	0	0
$255$	12.3471	0.773206
$256$	0	0
$257$	15.4256	0.962223	0.481112	−	0.876659i	$-0.340233\pi$
$257$	15.4256	0.962223	0.481112	+	0.876659i	$0.340233\pi$
$258$	0	0
$259$	16.9134	1.05095
$260$	0	0
$261$	−5.61106	−0.347316
$262$	0	0
$263$	27.0951	1.67076	0.835379	−	0.549675i	$-0.185248\pi$
$263$	27.0951	1.67076	0.835379	+	0.549675i	$0.185248\pi$
$264$	0	0
$265$	−2.70713	−0.166298
$266$	0	0
$267$	−9.96603	−0.609911
$268$	0	0
$269$	16.5457	1.00881	0.504405	−	0.863467i	$-0.331712\pi$
$269$	16.5457	1.00881	0.504405	+	0.863467i	$0.331712\pi$
$270$	0	0
$271$	16.8104	1.02116	0.510578	−	0.859831i	$-0.329431\pi$
$271$	16.8104	1.02116	0.510578	+	0.859831i	$0.329431\pi$
$272$	0	0
$273$	21.4240	1.29664
$274$	0	0
$275$	−8.35116	−0.503594
$276$	0	0
$277$	18.1850	1.09263	0.546317	−	0.837579i	$-0.316030\pi$
$277$	18.1850	1.09263	0.546317	+	0.837579i	$0.316030\pi$
$278$	0	0
$279$	4.99540	0.299067
$280$	0	0
$281$	−5.34514	−0.318865	−0.159432	−	0.987209i	$-0.550966\pi$
$281$	−5.34514	−0.318865	−0.159432	+	0.987209i	$0.550966\pi$
$282$	0	0
$283$	13.2256	0.786181	0.393091	−	0.919500i	$-0.371406\pi$
$283$	13.2256	0.786181	0.393091	+	0.919500i	$0.371406\pi$
$284$	0	0
$285$	19.0873	1.13064
$286$	0	0
$287$	−5.87773	−0.346951
$288$	0	0
$289$	−7.44785	−0.438109
$290$	0	0
$291$	−3.59700	−0.210860
$292$	0	0
$293$	25.4840	1.48879	0.744394	−	0.667740i	$-0.232738\pi$
$293$	25.4840	1.48879	0.744394	+	0.667740i	$0.232738\pi$
$294$	0	0
$295$	22.4905	1.30945
$296$	0	0
$297$	−30.5215	−1.77103
$298$	0	0
$299$	47.0254	2.71955
$300$	0	0
$301$	20.0612	1.15631
$302$	0	0
$303$	−4.19609	−0.241059
$304$	0	0
$305$	−9.33943	−0.534774
$306$	0	0
$307$	18.6615	1.06507	0.532534	−	0.846409i	$-0.321240\pi$
$307$	18.6615	1.06507	0.532534	+	0.846409i	$0.321240\pi$
$308$	0	0
$309$	−22.6142	−1.28647
$310$	0	0
$311$	−7.32284	−0.415240	−0.207620	−	0.978210i	$-0.566572\pi$
$311$	−7.32284	−0.415240	−0.207620	+	0.978210i	$0.566572\pi$
$312$	0	0
$313$	17.7322	1.00228	0.501140	−	0.865366i	$-0.332914\pi$
$313$	17.7322	1.00228	0.501140	+	0.865366i	$0.332914\pi$
$314$	0	0
$315$	3.21097	0.180918
$316$	0	0
$317$	−17.9485	−1.00809	−0.504044	−	0.863678i	$-0.668155\pi$
$317$	−17.9485	−1.00809	−0.504044	+	0.863678i	$0.668155\pi$
$318$	0	0
$319$	−55.7816	−3.12317
$320$	0	0
$321$	12.6432	0.705674
$322$	0	0
$323$	14.7666	0.821637
$324$	0	0
$325$	9.14832	0.507457
$326$	0	0
$327$	14.5139	0.802619
$328$	0	0
$329$	1.28905	0.0710677
$330$	0	0
$331$	12.1753	0.669215	0.334608	−	0.942358i	$-0.391396\pi$
$331$	12.1753	0.669215	0.334608	+	0.942358i	$0.391396\pi$
$332$	0	0
$333$	4.10461	0.224931
$334$	0	0
$335$	−23.0503	−1.25937
$336$	0	0
$337$	−21.9786	−1.19725	−0.598625	−	0.801029i	$-0.704286\pi$
$337$	−21.9786	−1.19725	−0.598625	+	0.801029i	$0.704286\pi$
$338$	0	0
$339$	−22.3487	−1.21381
$340$	0	0
$341$	49.6612	2.68931
$342$	0	0
$343$	20.0735	1.08387
$344$	0	0
$345$	−31.2277	−1.68124
$346$	0	0
$347$	9.90242	0.531590	0.265795	−	0.964030i	$-0.414366\pi$
$347$	9.90242	0.531590	0.265795	+	0.964030i	$0.414366\pi$
$348$	0	0
$349$	−16.2225	−0.868372	−0.434186	−	0.900823i	$-0.642964\pi$
$349$	−16.2225	−0.868372	−0.434186	+	0.900823i	$0.642964\pi$
$350$	0	0
$351$	33.4349	1.78462
$352$	0	0
$353$	5.91781	0.314973	0.157487	−	0.987521i	$-0.449661\pi$
$353$	5.91781	0.314973	0.157487	+	0.987521i	$0.449661\pi$
$354$	0	0
$355$	33.7678	1.79221
$356$	0	0
$357$	−11.0064	−0.582518
$358$	0	0
$359$	8.38475	0.442530	0.221265	−	0.975214i	$-0.428981\pi$
$359$	8.38475	0.442530	0.221265	+	0.975214i	$0.428981\pi$
$360$	0	0
$361$	3.82766	0.201456
$362$	0	0
$363$	−29.9747	−1.57326
$364$	0	0
$365$	−12.5493	−0.656858
$366$	0	0
$367$	−8.25170	−0.430735	−0.215368	−	0.976533i	$-0.569095\pi$
$367$	−8.25170	−0.430735	−0.215368	+	0.976533i	$0.569095\pi$
$368$	0	0
$369$	−1.42643	−0.0742571
$370$	0	0
$371$	2.41317	0.125285
$372$	0	0
$373$	36.1508	1.87182	0.935910	−	0.352240i	$-0.114580\pi$
$373$	36.1508	1.87182	0.935910	+	0.352240i	$0.114580\pi$
$374$	0	0
$375$	13.8999	0.717786
$376$	0	0
$377$	61.1062	3.14713
$378$	0	0
$379$	−2.69390	−0.138376	−0.0691881	−	0.997604i	$-0.522041\pi$
$379$	−2.69390	−0.138376	−0.0691881	+	0.997604i	$0.522041\pi$
$380$	0	0
$381$	17.6684	0.905179
$382$	0	0
$383$	5.32147	0.271914	0.135957	−	0.990715i	$-0.456589\pi$
$383$	5.32147	0.271914	0.135957	+	0.990715i	$0.456589\pi$
$384$	0	0
$385$	31.9215	1.62687
$386$	0	0
$387$	4.86854	0.247482
$388$	0	0
$389$	14.5183	0.736109	0.368054	−	0.929804i	$-0.380024\pi$
$389$	14.5183	0.736109	0.368054	+	0.929804i	$0.380024\pi$
$390$	0	0
$391$	−24.1588	−1.22176
$392$	0	0
$393$	−8.61669	−0.434654
$394$	0	0
$395$	21.6565	1.08966
$396$	0	0
$397$	−10.6123	−0.532615	−0.266307	−	0.963888i	$-0.585804\pi$
$397$	−10.6123	−0.532615	−0.266307	+	0.963888i	$0.585804\pi$
$398$	0	0
$399$	−17.0147	−0.851800
$400$	0	0
$401$	29.0551	1.45094	0.725472	−	0.688252i	$-0.241622\pi$
$401$	29.0551	1.45094	0.725472	+	0.688252i	$0.241622\pi$
$402$	0	0
$403$	−54.4016	−2.70994
$404$	0	0
$405$	−17.9709	−0.892982
$406$	0	0
$407$	40.8055	2.02265
$408$	0	0
$409$	−16.4481	−0.813306	−0.406653	−	0.913583i	$-0.633304\pi$
$409$	−16.4481	−0.813306	−0.406653	+	0.913583i	$0.633304\pi$
$410$	0	0
$411$	−1.70253	−0.0839797
$412$	0	0
$413$	−20.0483	−0.986513
$414$	0	0
$415$	23.9090	1.17364
$416$	0	0
$417$	−22.3465	−1.09431
$418$	0	0
$419$	−24.1151	−1.17810	−0.589050	−	0.808097i	$-0.700498\pi$
$419$	−24.1151	−1.17810	−0.589050	+	0.808097i	$0.700498\pi$
$420$	0	0
$421$	16.6029	0.809178	0.404589	−	0.914499i	$-0.367415\pi$
$421$	16.6029	0.809178	0.404589	+	0.914499i	$0.367415\pi$
$422$	0	0
$423$	0.312832	0.0152104
$424$	0	0
$425$	−4.69986	−0.227977
$426$	0	0
$427$	8.32528	0.402888
$428$	0	0
$429$	51.6878	2.49551
$430$	0	0
$431$	−23.7608	−1.14452	−0.572259	−	0.820073i	$-0.693933\pi$
$431$	−23.7608	−1.14452	−0.572259	+	0.820073i	$0.693933\pi$
$432$	0	0
$433$	38.1369	1.83274	0.916371	−	0.400331i	$-0.131105\pi$
$433$	38.1369	1.83274	0.916371	+	0.400331i	$0.131105\pi$
$434$	0	0
$435$	−40.5783	−1.94558
$436$	0	0
$437$	−37.3470	−1.78655
$438$	0	0
$439$	−25.9263	−1.23740	−0.618698	−	0.785629i	$-0.712340\pi$
$439$	−25.9263	−1.23740	−0.618698	+	0.785629i	$0.712340\pi$
$440$	0	0
$441$	1.00461	0.0478387
$442$	0	0
$443$	36.6056	1.73919	0.869593	−	0.493770i	$-0.164381\pi$
$443$	36.6056	1.73919	0.869593	+	0.493770i	$0.164381\pi$
$444$	0	0
$445$	16.2667	0.771115
$446$	0	0
$447$	−18.6049	−0.879979
$448$	0	0
$449$	−15.3170	−0.722853	−0.361427	−	0.932401i	$-0.617710\pi$
$449$	−15.3170	−0.722853	−0.361427	+	0.932401i	$0.617710\pi$
$450$	0	0
$451$	−14.1807	−0.667743
$452$	0	0
$453$	−33.6186	−1.57954
$454$	0	0
$455$	−34.9685	−1.63935
$456$	0	0
$457$	−22.8122	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$457$	−22.8122	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$458$	0	0
$459$	−17.1768	−0.801746
$460$	0	0
$461$	−2.66668	−0.124200	−0.0620998	−	0.998070i	$-0.519780\pi$
$461$	−2.66668	−0.124200	−0.0620998	+	0.998070i	$0.519780\pi$
$462$	0	0
$463$	29.1229	1.35346	0.676728	−	0.736233i	$-0.263397\pi$
$463$	29.1229	1.35346	0.676728	+	0.736233i	$0.263397\pi$
$464$	0	0
$465$	36.1260	1.67530
$466$	0	0
$467$	−17.7065	−0.819360	−0.409680	−	0.912229i	$-0.634360\pi$
$467$	−17.7065	−0.819360	−0.409680	+	0.912229i	$0.634360\pi$
$468$	0	0
$469$	20.5473	0.948788
$470$	0	0
$471$	13.0199	0.599926
$472$	0	0
$473$	48.4000	2.22543
$474$	0	0
$475$	−7.26549	−0.333364
$476$	0	0
$477$	0.585638	0.0268145
$478$	0	0
$479$	6.16540	0.281704	0.140852	−	0.990031i	$-0.455016\pi$
$479$	6.16540	0.281704	0.140852	+	0.990031i	$0.455016\pi$
$480$	0	0
$481$	−44.7005	−2.03817
$482$	0	0
$483$	27.8367	1.26662
$484$	0	0
$485$	5.87107	0.266592
$486$	0	0
$487$	24.3662	1.10414	0.552068	−	0.833799i	$-0.313839\pi$
$487$	24.3662	1.10414	0.552068	+	0.833799i	$0.313839\pi$
$488$	0	0
$489$	−27.0306	−1.22237
$490$	0	0
$491$	−35.3305	−1.59444	−0.797221	−	0.603687i	$-0.793698\pi$
$491$	−35.3305	−1.59444	−0.797221	+	0.603687i	$0.793698\pi$
$492$	0	0
$493$	−31.3927	−1.41386
$494$	0	0
$495$	7.74684	0.348195
$496$	0	0
$497$	−30.1010	−1.35022
$498$	0	0
$499$	41.6013	1.86233	0.931164	−	0.364599i	$-0.118794\pi$
$499$	41.6013	1.86233	0.931164	+	0.364599i	$0.118794\pi$
$500$	0	0
$501$	22.0211	0.983829
$502$	0	0
$503$	−9.93154	−0.442826	−0.221413	−	0.975180i	$-0.571067\pi$
$503$	−9.93154	−0.442826	−0.221413	+	0.975180i	$0.571067\pi$
$504$	0	0
$505$	6.84892	0.304773
$506$	0	0
$507$	−36.2835	−1.61140
$508$	0	0
$509$	3.05956	0.135612	0.0678062	−	0.997699i	$-0.478400\pi$
$509$	3.05956	0.135612	0.0678062	+	0.997699i	$0.478400\pi$
$510$	0	0
$511$	11.1866	0.494864
$512$	0	0
$513$	−26.5536	−1.17237
$514$	0	0
$515$	36.9112	1.62650
$516$	0	0
$517$	3.10999	0.136777
$518$	0	0
$519$	−18.5789	−0.815525
$520$	0	0
$521$	−2.80372	−0.122833	−0.0614165	−	0.998112i	$-0.519562\pi$
$521$	−2.80372	−0.122833	−0.0614165	+	0.998112i	$0.519562\pi$
$522$	0	0
$523$	−37.3390	−1.63272	−0.816361	−	0.577542i	$-0.804012\pi$
$523$	−37.3390	−1.63272	−0.816361	+	0.577542i	$0.804012\pi$
$524$	0	0
$525$	5.41536	0.236346
$526$	0	0
$527$	27.9483	1.21745
$528$	0	0
$529$	38.1013	1.65658
$530$	0	0
$531$	−4.86541	−0.211141
$532$	0	0
$533$	15.5343	0.672866
$534$	0	0
$535$	−20.6364	−0.892189
$536$	0	0
$537$	11.7053	0.505120
$538$	0	0
$539$	9.98724	0.430181
$540$	0	0
$541$	36.3964	1.56480	0.782401	−	0.622775i	$-0.213995\pi$
$541$	36.3964	1.56480	0.782401	+	0.622775i	$0.213995\pi$
$542$	0	0
$543$	27.1756	1.16622
$544$	0	0
$545$	−23.6898	−1.01476
$546$	0	0
$547$	−8.59960	−0.367692	−0.183846	−	0.982955i	$-0.558855\pi$
$547$	−8.59960	−0.367692	−0.183846	+	0.982955i	$0.558855\pi$
$548$	0	0
$549$	2.02041	0.0862292
$550$	0	0
$551$	−48.5299	−2.06744
$552$	0	0
$553$	−19.3049	−0.820926
$554$	0	0
$555$	29.6839	1.26001
$556$	0	0
$557$	3.66173	0.155152	0.0775762	−	0.996986i	$-0.475282\pi$
$557$	3.66173	0.155152	0.0775762	+	0.996986i	$0.475282\pi$
$558$	0	0
$559$	−53.0199	−2.24250
$560$	0	0
$561$	−26.5541	−1.12112
$562$	0	0
$563$	−15.5423	−0.655030	−0.327515	−	0.944846i	$-0.606211\pi$
$563$	−15.5423	−0.655030	−0.327515	+	0.944846i	$0.606211\pi$
$564$	0	0
$565$	36.4778	1.53463
$566$	0	0
$567$	16.0195	0.672755
$568$	0	0
$569$	−2.81949	−0.118199	−0.0590996	−	0.998252i	$-0.518823\pi$
$569$	−2.81949	−0.118199	−0.0590996	+	0.998252i	$0.518823\pi$
$570$	0	0
$571$	−10.7280	−0.448951	−0.224476	−	0.974480i	$-0.572067\pi$
$571$	−10.7280	−0.448951	−0.224476	+	0.974480i	$0.572067\pi$
$572$	0	0
$573$	36.9317	1.54285
$574$	0	0
$575$	11.8867	0.495708
$576$	0	0
$577$	−24.1374	−1.00485	−0.502426	−	0.864620i	$-0.667559\pi$
$577$	−24.1374	−1.00485	−0.502426	+	0.864620i	$0.667559\pi$
$578$	0	0
$579$	14.0046	0.582010
$580$	0	0
$581$	−21.3127	−0.884201
$582$	0	0
$583$	5.82205	0.241125
$584$	0	0
$585$	−8.48631	−0.350866
$586$	0	0
$587$	−17.7835	−0.734003	−0.367001	−	0.930220i	$-0.619616\pi$
$587$	−17.7835	−0.734003	−0.367001	+	0.930220i	$0.619616\pi$
$588$	0	0
$589$	43.2051	1.78024
$590$	0	0
$591$	16.5963	0.682680
$592$	0	0
$593$	25.5882	1.05078	0.525391	−	0.850861i	$-0.323919\pi$
$593$	25.5882	1.05078	0.525391	+	0.850861i	$0.323919\pi$
$594$	0	0
$595$	17.9647	0.736482
$596$	0	0
$597$	−0.0570469	−0.00233478
$598$	0	0
$599$	31.3014	1.27894	0.639470	−	0.768816i	$-0.279154\pi$
$599$	31.3014	1.27894	0.639470	+	0.768816i	$0.279154\pi$
$600$	0	0
$601$	16.0565	0.654959	0.327479	−	0.944858i	$-0.393801\pi$
$601$	16.0565	0.654959	0.327479	+	0.944858i	$0.393801\pi$
$602$	0	0
$603$	4.98651	0.203067
$604$	0	0
$605$	48.9251	1.98909
$606$	0	0
$607$	11.0513	0.448559	0.224280	−	0.974525i	$-0.427997\pi$
$607$	11.0513	0.448559	0.224280	+	0.974525i	$0.427997\pi$
$608$	0	0
$609$	36.1719	1.46576
$610$	0	0
$611$	−3.40685	−0.137826
$612$	0	0
$613$	42.1688	1.70318	0.851592	−	0.524206i	$-0.175638\pi$
$613$	42.1688	1.70318	0.851592	+	0.524206i	$0.175638\pi$
$614$	0	0
$615$	−10.3157	−0.415971
$616$	0	0
$617$	2.95948	0.119144	0.0595722	−	0.998224i	$-0.481026\pi$
$617$	2.95948	0.119144	0.0595722	+	0.998224i	$0.481026\pi$
$618$	0	0
$619$	37.7567	1.51757	0.758785	−	0.651341i	$-0.225793\pi$
$619$	37.7567	1.51757	0.758785	+	0.651341i	$0.225793\pi$
$620$	0	0
$621$	43.4428	1.74330
$622$	0	0
$623$	−14.5003	−0.580943
$624$	0	0
$625$	−30.2909	−1.21164
$626$	0	0
$627$	−41.0499	−1.63938
$628$	0	0
$629$	22.9645	0.915653
$630$	0	0
$631$	−34.0356	−1.35493	−0.677467	−	0.735553i	$-0.736922\pi$
$631$	−34.0356	−1.35493	−0.677467	+	0.735553i	$0.736922\pi$
$632$	0	0
$633$	−27.3515	−1.08713
$634$	0	0
$635$	−28.8386	−1.14442
$636$	0	0
$637$	−10.9406	−0.433481
$638$	0	0
$639$	−7.30505	−0.288983
$640$	0	0
$641$	28.6175	1.13032	0.565162	−	0.824980i	$-0.308814\pi$
$641$	28.6175	1.13032	0.565162	+	0.824980i	$0.308814\pi$
$642$	0	0
$643$	1.34506	0.0530441	0.0265221	−	0.999648i	$-0.491557\pi$
$643$	1.34506	0.0530441	0.0265221	+	0.999648i	$0.491557\pi$
$644$	0	0
$645$	35.2085	1.38633
$646$	0	0
$647$	23.5858	0.927253	0.463627	−	0.886031i	$-0.346548\pi$
$647$	23.5858	0.927253	0.463627	+	0.886031i	$0.346548\pi$
$648$	0	0
$649$	−48.3689	−1.89865
$650$	0	0
$651$	−32.2031	−1.26214
$652$	0	0
$653$	21.5890	0.844843	0.422421	−	0.906400i	$-0.361180\pi$
$653$	21.5890	0.844843	0.422421	+	0.906400i	$0.361180\pi$
$654$	0	0
$655$	14.0643	0.549537
$656$	0	0
$657$	2.71480	0.105915
$658$	0	0
$659$	−34.5852	−1.34725	−0.673624	−	0.739074i	$-0.735263\pi$
$659$	−34.5852	−1.34725	−0.673624	+	0.739074i	$0.735263\pi$
$660$	0	0
$661$	−31.8437	−1.23858	−0.619289	−	0.785163i	$-0.712579\pi$
$661$	−31.8437	−1.23858	−0.619289	+	0.785163i	$0.712579\pi$
$662$	0	0
$663$	29.0888	1.12972
$664$	0	0
$665$	27.7716	1.07694
$666$	0	0
$667$	79.3970	3.07426
$668$	0	0
$669$	6.73632	0.260441
$670$	0	0
$671$	20.0857	0.775400
$672$	0	0
$673$	22.8096	0.879244	0.439622	−	0.898183i	$-0.355112\pi$
$673$	22.8096	0.879244	0.439622	+	0.898183i	$0.355112\pi$
$674$	0	0
$675$	8.45137	0.325293
$676$	0	0
$677$	−50.5440	−1.94256	−0.971282	−	0.237933i	$-0.923530\pi$
$677$	−50.5440	−1.94256	−0.971282	+	0.237933i	$0.923530\pi$
$678$	0	0
$679$	−5.23354	−0.200845
$680$	0	0
$681$	−38.6345	−1.48048
$682$	0	0
$683$	24.4105	0.934042	0.467021	−	0.884246i	$-0.345327\pi$
$683$	24.4105	0.934042	0.467021	+	0.884246i	$0.345327\pi$
$684$	0	0
$685$	2.77890	0.106176
$686$	0	0
$687$	−4.97039	−0.189632
$688$	0	0
$689$	−6.37779	−0.242974
$690$	0	0
$691$	12.2365	0.465499	0.232749	−	0.972537i	$-0.425228\pi$
$691$	12.2365	0.465499	0.232749	+	0.972537i	$0.425228\pi$
$692$	0	0
$693$	−6.90562	−0.262323
$694$	0	0
$695$	36.4743	1.38355
$696$	0	0
$697$	−7.98061	−0.302287
$698$	0	0
$699$	33.9376	1.28364
$700$	0	0
$701$	17.3991	0.657154	0.328577	−	0.944477i	$-0.393431\pi$
$701$	17.3991	0.657154	0.328577	+	0.944477i	$0.393431\pi$
$702$	0	0
$703$	35.5007	1.33893
$704$	0	0
$705$	2.26235	0.0852052
$706$	0	0
$707$	−6.10520	−0.229610
$708$	0	0
$709$	−20.9379	−0.786340	−0.393170	−	0.919466i	$-0.628622\pi$
$709$	−20.9379	−0.786340	−0.393170	+	0.919466i	$0.628622\pi$
$710$	0	0
$711$	−4.68498	−0.175701
$712$	0	0
$713$	−70.6854	−2.64719
$714$	0	0
$715$	−84.3656	−3.15509
$716$	0	0
$717$	−36.7916	−1.37401
$718$	0	0
$719$	−7.43750	−0.277372	−0.138686	−	0.990336i	$-0.544288\pi$
$719$	−7.43750	−0.277372	−0.138686	+	0.990336i	$0.544288\pi$
$720$	0	0
$721$	−32.9030	−1.22537
$722$	0	0
$723$	−31.3052	−1.16425
$724$	0	0
$725$	15.4459	0.573646
$726$	0	0
$727$	−38.7866	−1.43852	−0.719258	−	0.694743i	$-0.755518\pi$
$727$	−38.7866	−1.43852	−0.719258	+	0.694743i	$0.755518\pi$
$728$	0	0
$729$	29.9722	1.11008
$730$	0	0
$731$	27.2385	1.00745
$732$	0	0
$733$	11.3860	0.420551	0.210275	−	0.977642i	$-0.432564\pi$
$733$	11.3860	0.420551	0.210275	+	0.977642i	$0.432564\pi$
$734$	0	0
$735$	7.26521	0.267981
$736$	0	0
$737$	49.5728	1.82604
$738$	0	0
$739$	47.8046	1.75852	0.879261	−	0.476341i	$-0.158037\pi$
$739$	47.8046	1.75852	0.879261	+	0.476341i	$0.158037\pi$
$740$	0	0
$741$	44.9683	1.65195
$742$	0	0
$743$	−21.8896	−0.803050	−0.401525	−	0.915848i	$-0.631520\pi$
$743$	−21.8896	−0.803050	−0.401525	+	0.915848i	$0.631520\pi$
$744$	0	0
$745$	30.3671	1.11256
$746$	0	0
$747$	−5.17226	−0.189243
$748$	0	0
$749$	18.3955	0.672158
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	18.3743	0.669598
$754$	0	0
$755$	54.8727	1.99702
$756$	0	0
$757$	−16.1892	−0.588405	−0.294203	−	0.955743i	$-0.595054\pi$
$757$	−16.1892	−0.588405	−0.294203	+	0.955743i	$0.595054\pi$
$758$	0	0
$759$	67.1594	2.43773
$760$	0	0
$761$	−37.8360	−1.37155	−0.685777	−	0.727812i	$-0.740537\pi$
$761$	−37.8360	−1.37155	−0.685777	+	0.727812i	$0.740537\pi$
$762$	0	0
$763$	21.1173	0.764498
$764$	0	0
$765$	4.35976	0.157627
$766$	0	0
$767$	52.9859	1.91321
$768$	0	0
$769$	24.5357	0.884780	0.442390	−	0.896823i	$-0.354131\pi$
$769$	24.5357	0.884780	0.442390	+	0.896823i	$0.354131\pi$
$770$	0	0
$771$	−24.1330	−0.869129
$772$	0	0
$773$	−21.0097	−0.755665	−0.377832	−	0.925874i	$-0.623330\pi$
$773$	−21.0097	−0.755665	−0.377832	+	0.925874i	$0.623330\pi$
$774$	0	0
$775$	−13.7511	−0.493956
$776$	0	0
$777$	−26.4605	−0.949267
$778$	0	0
$779$	−12.3372	−0.442026
$780$	0	0
$781$	−72.6223	−2.59863
$782$	0	0
$783$	56.4510	2.01739
$784$	0	0
$785$	−21.2513	−0.758492
$786$	0	0
$787$	19.5428	0.696627	0.348313	−	0.937378i	$-0.386754\pi$
$787$	19.5428	0.696627	0.348313	+	0.937378i	$0.386754\pi$
$788$	0	0
$789$	−42.3897	−1.50911
$790$	0	0
$791$	−32.5168	−1.15616
$792$	0	0
$793$	−22.0030	−0.781348
$794$	0	0
$795$	4.23524	0.150208
$796$	0	0
$797$	−7.92622	−0.280761	−0.140381	−	0.990098i	$-0.544833\pi$
$797$	−7.92622	−0.280761	−0.140381	+	0.990098i	$0.544833\pi$
$798$	0	0
$799$	1.75023	0.0619189
$800$	0	0
$801$	−3.51900	−0.124338
$802$	0	0
$803$	26.9889	0.952417
$804$	0	0
$805$	−45.4355	−1.60139
$806$	0	0
$807$	−25.8854	−0.911208
$808$	0	0
$809$	−25.9461	−0.912217	−0.456109	−	0.889924i	$-0.650757\pi$
$809$	−25.9461	−0.912217	−0.456109	+	0.889924i	$0.650757\pi$
$810$	0	0
$811$	5.78971	0.203304	0.101652	−	0.994820i	$-0.467587\pi$
$811$	5.78971	0.203304	0.101652	+	0.994820i	$0.467587\pi$
$812$	0	0
$813$	−26.2994	−0.922360
$814$	0	0
$815$	44.1198	1.54545
$816$	0	0
$817$	42.1079	1.47317
$818$	0	0
$819$	7.56479	0.264335
$820$	0	0
$821$	18.1110	0.632078	0.316039	−	0.948746i	$-0.397647\pi$
$821$	18.1110	0.632078	0.316039	+	0.948746i	$0.397647\pi$
$822$	0	0
$823$	4.49904	0.156827	0.0784134	−	0.996921i	$-0.475015\pi$
$823$	4.49904	0.156827	0.0784134	+	0.996921i	$0.475015\pi$
$824$	0	0
$825$	13.0652	0.454872
$826$	0	0
$827$	4.90543	0.170579	0.0852893	−	0.996356i	$-0.472819\pi$
$827$	4.90543	0.170579	0.0852893	+	0.996356i	$0.472819\pi$
$828$	0	0
$829$	7.64569	0.265546	0.132773	−	0.991146i	$-0.457612\pi$
$829$	7.64569	0.265546	0.132773	+	0.991146i	$0.457612\pi$
$830$	0	0
$831$	−28.4501	−0.986921
$832$	0	0
$833$	5.62061	0.194743
$834$	0	0
$835$	−35.9431	−1.24386
$836$	0	0
$837$	−50.2571	−1.73714
$838$	0	0
$839$	8.47301	0.292521	0.146260	−	0.989246i	$-0.453276\pi$
$839$	8.47301	0.292521	0.146260	+	0.989246i	$0.453276\pi$
$840$	0	0
$841$	74.1709	2.55762
$842$	0	0
$843$	8.36235	0.288015
$844$	0	0
$845$	59.2224	2.03731
$846$	0	0
$847$	−43.6124	−1.49854
$848$	0	0
$849$	−20.6912	−0.710118
$850$	0	0
$851$	−58.0806	−1.99098
$852$	0	0
$853$	−29.2668	−1.00208	−0.501038	−	0.865425i	$-0.667048\pi$
$853$	−29.2668	−1.00208	−0.501038	+	0.865425i	$0.667048\pi$
$854$	0	0
$855$	6.73973	0.230494
$856$	0	0
$857$	−52.7586	−1.80220	−0.901100	−	0.433611i	$-0.857239\pi$
$857$	−52.7586	−1.80220	−0.901100	+	0.433611i	$0.857239\pi$
$858$	0	0
$859$	36.8393	1.25694	0.628470	−	0.777834i	$-0.283681\pi$
$859$	36.8393	1.25694	0.628470	+	0.777834i	$0.283681\pi$
$860$	0	0
$861$	9.19557	0.313384
$862$	0	0
$863$	−33.6032	−1.14387	−0.571933	−	0.820300i	$-0.693806\pi$
$863$	−33.6032	−1.14387	−0.571933	+	0.820300i	$0.693806\pi$
$864$	0	0
$865$	30.3248	1.03107
$866$	0	0
$867$	11.6520	0.395722
$868$	0	0
$869$	−46.5752	−1.57996
$870$	0	0
$871$	−54.3048	−1.84005
$872$	0	0
$873$	−1.27010	−0.0429863
$874$	0	0
$875$	20.2240	0.683695
$876$	0	0
$877$	−18.8712	−0.637235	−0.318617	−	0.947883i	$-0.603218\pi$
$877$	−18.8712	−0.637235	−0.318617	+	0.947883i	$0.603218\pi$
$878$	0	0
$879$	−39.8690	−1.34475
$880$	0	0
$881$	4.46039	0.150274	0.0751372	−	0.997173i	$-0.476061\pi$
$881$	4.46039	0.150274	0.0751372	+	0.997173i	$0.476061\pi$
$882$	0	0
$883$	−7.48181	−0.251783	−0.125891	−	0.992044i	$-0.540179\pi$
$883$	−7.48181	−0.251783	−0.125891	+	0.992044i	$0.540179\pi$
$884$	0	0
$885$	−35.1859	−1.18276
$886$	0	0
$887$	−14.1865	−0.476336	−0.238168	−	0.971224i	$-0.576547\pi$
$887$	−14.1865	−0.476336	−0.238168	+	0.971224i	$0.576547\pi$
$888$	0	0
$889$	25.7071	0.862187
$890$	0	0
$891$	38.6488	1.29479
$892$	0	0
$893$	2.70568	0.0905422
$894$	0	0
$895$	−19.1055	−0.638627
$896$	0	0
$897$	−73.5700	−2.45643
$898$	0	0
$899$	−91.8509	−3.06340
$900$	0	0
$901$	3.27653	0.109157
$902$	0	0
$903$	−31.3852	−1.04444
$904$	0	0
$905$	−44.3563	−1.47445
$906$	0	0
$907$	45.2858	1.50369	0.751846	−	0.659339i	$-0.229164\pi$
$907$	45.2858	1.50369	0.751846	+	0.659339i	$0.229164\pi$
$908$	0	0
$909$	−1.48164	−0.0491428
$910$	0	0
$911$	−16.8228	−0.557364	−0.278682	−	0.960383i	$-0.589898\pi$
$911$	−16.8228	−0.557364	−0.278682	+	0.960383i	$0.589898\pi$
$912$	0	0
$913$	−51.4194	−1.70173
$914$	0	0
$915$	14.6113	0.483035
$916$	0	0
$917$	−12.5371	−0.414010
$918$	0	0
$919$	7.21545	0.238016	0.119008	−	0.992893i	$-0.462029\pi$
$919$	7.21545	0.238016	0.119008	+	0.992893i	$0.462029\pi$
$920$	0	0
$921$	−29.1954	−0.962022
$922$	0	0
$923$	79.5544	2.61856
$924$	0	0
$925$	−11.2990	−0.371509
$926$	0	0
$927$	−7.98505	−0.262263
$928$	0	0
$929$	3.64130	0.119467	0.0597335	−	0.998214i	$-0.480975\pi$
$929$	3.64130	0.119467	0.0597335	+	0.998214i	$0.480975\pi$
$930$	0	0
$931$	8.68888	0.284767
$932$	0	0
$933$	11.4564	0.375066
$934$	0	0
$935$	43.3420	1.41744
$936$	0	0
$937$	1.63208	0.0533176	0.0266588	−	0.999645i	$-0.491513\pi$
$937$	1.63208	0.0533176	0.0266588	+	0.999645i	$0.491513\pi$
$938$	0	0
$939$	−27.7415	−0.905311
$940$	0	0
$941$	40.4765	1.31950	0.659748	−	0.751487i	$-0.270663\pi$
$941$	40.4765	1.31950	0.659748	+	0.751487i	$0.270663\pi$
$942$	0	0
$943$	20.1842	0.657287
$944$	0	0
$945$	−32.3045	−1.05087
$946$	0	0
$947$	−25.1006	−0.815660	−0.407830	−	0.913058i	$-0.633714\pi$
$947$	−25.1006	−0.815660	−0.407830	+	0.913058i	$0.633714\pi$
$948$	0	0
$949$	−29.5651	−0.959723
$950$	0	0
$951$	28.0800	0.910557
$952$	0	0
$953$	52.1992	1.69090	0.845448	−	0.534057i	$-0.179333\pi$
$953$	52.1992	1.69090	0.845448	+	0.534057i	$0.179333\pi$
$954$	0	0
$955$	−60.2805	−1.95063
$956$	0	0
$957$	87.2690	2.82101
$958$	0	0
$959$	−2.47714	−0.0799911
$960$	0	0
$961$	50.7729	1.63784
$962$	0	0
$963$	4.46430	0.143860
$964$	0	0
$965$	−22.8584	−0.735839
$966$	0	0
$967$	−49.1898	−1.58184	−0.790919	−	0.611920i	$-0.790397\pi$
$967$	−49.1898	−1.58184	−0.790919	+	0.611920i	$0.790397\pi$
$968$	0	0
$969$	−23.1020	−0.742144
$970$	0	0
$971$	−40.0516	−1.28532	−0.642658	−	0.766153i	$-0.722169\pi$
$971$	−40.0516	−1.28532	−0.642658	+	0.766153i	$0.722169\pi$
$972$	0	0
$973$	−32.5136	−1.04234
$974$	0	0
$975$	−14.3123	−0.458361
$976$	0	0
$977$	34.2087	1.09443	0.547217	−	0.836991i	$-0.315687\pi$
$977$	34.2087	1.09443	0.547217	+	0.836991i	$0.315687\pi$
$978$	0	0
$979$	−34.9837	−1.11808
$980$	0	0
$981$	5.12484	0.163624
$982$	0	0
$983$	−6.33309	−0.201994	−0.100997	−	0.994887i	$-0.532203\pi$
$983$	−6.33309	−0.201994	−0.100997	+	0.994887i	$0.532203\pi$
$984$	0	0
$985$	−27.0887	−0.863118
$986$	0	0
$987$	−2.01669	−0.0641919
$988$	0	0
$989$	−68.8902	−2.19058
$990$	0	0
$991$	−48.6422	−1.54517	−0.772585	−	0.634911i	$-0.781037\pi$
$991$	−48.6422	−1.54517	−0.772585	+	0.634911i	$0.781037\pi$
$992$	0	0
$993$	−19.0480	−0.604469
$994$	0	0
$995$	0.0931128	0.00295187
$996$	0	0
$997$	8.37412	0.265211	0.132606	−	0.991169i	$-0.457666\pi$
$997$	8.37412	0.265211	0.132606	+	0.991169i	$0.457666\pi$
$998$	0	0
$999$	−41.2951	−1.30652

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.e.1.12	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.e.1.12	✓	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)

\(f(q)\)	\(=\)	\(q-1.56448 q^{3} +2.55356 q^{5} -2.27627 q^{7} -0.552416 q^{9} +O(q^{10})\)	\(q-1.56448 q^{3} +2.55356 q^{5} -2.27627 q^{7} -0.552416 q^{9} -5.49177 q^{11} +6.01599 q^{13} -3.99498 q^{15} -3.09065 q^{17} -4.77783 q^{19} +3.56117 q^{21} +7.81673 q^{23} +1.52067 q^{25} +5.55767 q^{27} +10.1573 q^{29} -9.04284 q^{31} +8.59175 q^{33} -5.81260 q^{35} -7.43029 q^{37} -9.41186 q^{39} +2.58217 q^{41} -8.81318 q^{43} -1.41063 q^{45} -0.566299 q^{47} -1.81858 q^{49} +4.83525 q^{51} -1.06014 q^{53} -14.0236 q^{55} +7.47480 q^{57} +8.80752 q^{59} -3.65742 q^{61} +1.25745 q^{63} +15.3622 q^{65} -9.02674 q^{67} -12.2291 q^{69} +13.2238 q^{71} -4.91442 q^{73} -2.37905 q^{75} +12.5008 q^{77} +8.48091 q^{79} -7.03759 q^{81} +9.36299 q^{83} -7.89217 q^{85} -15.8909 q^{87} +6.37020 q^{89} -13.6940 q^{91} +14.1473 q^{93} -12.2005 q^{95} +2.29917 q^{97} +3.03374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 6 q^{3} + 23 q^{5} + 12 q^{7} + 56 q^{9}+O(q^{10}) \) 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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