LMFDB - Embedded newform 6008.2.a.c.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.214146 q^{3} -4.30470 q^{5} -2.13393 q^{7} -2.95414 q^{9} +O(q^{10})$	$q-0.214146 q^{3} -4.30470 q^{5} -2.13393 q^{7} -2.95414 q^{9} -0.0877904 q^{11} +2.38665 q^{13} +0.921835 q^{15} -1.95558 q^{17} +2.91343 q^{19} +0.456973 q^{21} +6.47045 q^{23} +13.5305 q^{25} +1.27506 q^{27} -2.77740 q^{29} -10.3265 q^{31} +0.0188000 q^{33} +9.18594 q^{35} +11.5991 q^{37} -0.511092 q^{39} +6.66620 q^{41} -2.47308 q^{43} +12.7167 q^{45} +5.00972 q^{47} -2.44634 q^{49} +0.418780 q^{51} -3.85490 q^{53} +0.377912 q^{55} -0.623900 q^{57} +12.1843 q^{59} -4.59295 q^{61} +6.30393 q^{63} -10.2738 q^{65} +1.28239 q^{67} -1.38562 q^{69} -0.431795 q^{71} -2.52891 q^{73} -2.89750 q^{75} +0.187339 q^{77} -16.0189 q^{79} +8.58938 q^{81} +0.112750 q^{83} +8.41821 q^{85} +0.594770 q^{87} +3.74889 q^{89} -5.09295 q^{91} +2.21137 q^{93} -12.5415 q^{95} +5.55434 q^{97} +0.259345 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	$ 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) $ 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.214146	−0.123637	−0.0618186	−	0.998087i	$-0.519690\pi$
$3$	−0.214146	−0.123637	−0.0618186	+	0.998087i	$0.519690\pi$
$4$	0	0
$5$	−4.30470	−1.92512	−0.962561	−	0.271065i	$-0.912624\pi$
$5$	−4.30470	−1.92512	−0.962561	+	0.271065i	$0.912624\pi$
$6$	0	0
$7$	−2.13393	−0.806550	−0.403275	−	0.915079i	$-0.632128\pi$
$7$	−2.13393	−0.806550	−0.403275	+	0.915079i	$0.632128\pi$
$8$	0	0
$9$	−2.95414	−0.984714
$10$	0	0
$11$	−0.0877904	−0.0264698	−0.0132349	−	0.999912i	$-0.504213\pi$
$11$	−0.0877904	−0.0264698	−0.0132349	+	0.999912i	$0.504213\pi$
$12$	0	0
$13$	2.38665	0.661938	0.330969	−	0.943642i	$-0.392624\pi$
$13$	2.38665	0.661938	0.330969	+	0.943642i	$0.392624\pi$
$14$	0	0
$15$	0.921835	0.238017
$16$	0	0
$17$	−1.95558	−0.474299	−0.237149	−	0.971473i	$-0.576213\pi$
$17$	−1.95558	−0.474299	−0.237149	+	0.971473i	$0.576213\pi$
$18$	0	0
$19$	2.91343	0.668387	0.334193	−	0.942504i	$-0.391536\pi$
$19$	2.91343	0.668387	0.334193	+	0.942504i	$0.391536\pi$
$20$	0	0
$21$	0.456973	0.0997196
$22$	0	0
$23$	6.47045	1.34918	0.674591	−	0.738191i	$-0.264320\pi$
$23$	6.47045	1.34918	0.674591	+	0.738191i	$0.264320\pi$
$24$	0	0
$25$	13.5305	2.70610
$26$	0	0
$27$	1.27506	0.245385
$28$	0	0
$29$	−2.77740	−0.515751	−0.257875	−	0.966178i	$-0.583022\pi$
$29$	−2.77740	−0.515751	−0.257875	+	0.966178i	$0.583022\pi$
$30$	0	0
$31$	−10.3265	−1.85469	−0.927345	−	0.374208i	$-0.877915\pi$
$31$	−10.3265	−1.85469	−0.927345	+	0.374208i	$0.877915\pi$
$32$	0	0
$33$	0.0188000	0.00327265
$34$	0	0
$35$	9.18594	1.55271
$36$	0	0
$37$	11.5991	1.90688	0.953438	−	0.301588i	$-0.0975167\pi$
$37$	11.5991	1.90688	0.953438	+	0.301588i	$0.0975167\pi$
$38$	0	0
$39$	−0.511092	−0.0818402
$40$	0	0
$41$	6.66620	1.04109	0.520543	−	0.853835i	$-0.325730\pi$
$41$	6.66620	1.04109	0.520543	+	0.853835i	$0.325730\pi$
$42$	0	0
$43$	−2.47308	−0.377142	−0.188571	−	0.982060i	$-0.560385\pi$
$43$	−2.47308	−0.377142	−0.188571	+	0.982060i	$0.560385\pi$
$44$	0	0
$45$	12.7167	1.89569
$46$	0	0
$47$	5.00972	0.730743	0.365371	−	0.930862i	$-0.380942\pi$
$47$	5.00972	0.730743	0.365371	+	0.930862i	$0.380942\pi$
$48$	0	0
$49$	−2.44634	−0.349477
$50$	0	0
$51$	0.418780	0.0586410
$52$	0	0
$53$	−3.85490	−0.529511	−0.264756	−	0.964316i	$-0.585291\pi$
$53$	−3.85490	−0.529511	−0.264756	+	0.964316i	$0.585291\pi$
$54$	0	0
$55$	0.377912	0.0509576
$56$	0	0
$57$	−0.623900	−0.0826375
$58$	0	0
$59$	12.1843	1.58626	0.793131	−	0.609051i	$-0.208449\pi$
$59$	12.1843	1.58626	0.793131	+	0.609051i	$0.208449\pi$
$60$	0	0
$61$	−4.59295	−0.588067	−0.294033	−	0.955795i	$-0.594998\pi$
$61$	−4.59295	−0.588067	−0.294033	+	0.955795i	$0.594998\pi$
$62$	0	0
$63$	6.30393	0.794221
$64$	0	0
$65$	−10.2738	−1.27431
$66$	0	0
$67$	1.28239	0.156669	0.0783343	−	0.996927i	$-0.475040\pi$
$67$	1.28239	0.156669	0.0783343	+	0.996927i	$0.475040\pi$
$68$	0	0
$69$	−1.38562	−0.166809
$70$	0	0
$71$	−0.431795	−0.0512446	−0.0256223	−	0.999672i	$-0.508157\pi$
$71$	−0.431795	−0.0512446	−0.0256223	+	0.999672i	$0.508157\pi$
$72$	0	0
$73$	−2.52891	−0.295986	−0.147993	−	0.988988i	$-0.547281\pi$
$73$	−2.52891	−0.295986	−0.147993	+	0.988988i	$0.547281\pi$
$74$	0	0
$75$	−2.89750	−0.334574
$76$	0	0
$77$	0.187339	0.0213492
$78$	0	0
$79$	−16.0189	−1.80226	−0.901131	−	0.433547i	$-0.857262\pi$
$79$	−16.0189	−1.80226	−0.901131	+	0.433547i	$0.857262\pi$
$80$	0	0
$81$	8.58938	0.954375
$82$	0	0
$83$	0.112750	0.0123759	0.00618794	−	0.999981i	$-0.498030\pi$
$83$	0.112750	0.0123759	0.00618794	+	0.999981i	$0.498030\pi$
$84$	0	0
$85$	8.41821	0.913083
$86$	0	0
$87$	0.594770	0.0637660
$88$	0	0
$89$	3.74889	0.397382	0.198691	−	0.980062i	$-0.436331\pi$
$89$	3.74889	0.397382	0.198691	+	0.980062i	$0.436331\pi$
$90$	0	0
$91$	−5.09295	−0.533886
$92$	0	0
$93$	2.21137	0.229309
$94$	0	0
$95$	−12.5415	−1.28673
$96$	0	0
$97$	5.55434	0.563958	0.281979	−	0.959421i	$-0.409009\pi$
$97$	5.55434	0.563958	0.281979	+	0.959421i	$0.409009\pi$
$98$	0	0
$99$	0.259345	0.0260652
$100$	0	0
$101$	14.8409	1.47672	0.738362	−	0.674405i	$-0.235600\pi$
$101$	14.8409	1.47672	0.738362	+	0.674405i	$0.235600\pi$
$102$	0	0
$103$	−1.45965	−0.143823	−0.0719117	−	0.997411i	$-0.522910\pi$
$103$	−1.45965	−0.143823	−0.0719117	+	0.997411i	$0.522910\pi$
$104$	0	0
$105$	−1.96713	−0.191972
$106$	0	0
$107$	−12.3245	−1.19146	−0.595729	−	0.803185i	$-0.703137\pi$
$107$	−12.3245	−1.19146	−0.595729	+	0.803185i	$0.703137\pi$
$108$	0	0
$109$	−1.16930	−0.111999	−0.0559995	−	0.998431i	$-0.517835\pi$
$109$	−1.16930	−0.111999	−0.0559995	+	0.998431i	$0.517835\pi$
$110$	0	0
$111$	−2.48390	−0.235761
$112$	0	0
$113$	8.08235	0.760323	0.380162	−	0.924920i	$-0.375868\pi$
$113$	8.08235	0.760323	0.380162	+	0.924920i	$0.375868\pi$
$114$	0	0
$115$	−27.8534	−2.59734
$116$	0	0
$117$	−7.05050	−0.651819
$118$	0	0
$119$	4.17308	0.382546
$120$	0	0
$121$	−10.9923	−0.999299
$122$	0	0
$123$	−1.42754	−0.128717
$124$	0	0
$125$	−36.7212	−3.28444
$126$	0	0
$127$	10.0397	0.890880	0.445440	−	0.895312i	$-0.353047\pi$
$127$	10.0397	0.890880	0.445440	+	0.895312i	$0.353047\pi$
$128$	0	0
$129$	0.529601	0.0466287
$130$	0	0
$131$	4.26560	0.372687	0.186344	−	0.982485i	$-0.440336\pi$
$131$	4.26560	0.372687	0.186344	+	0.982485i	$0.440336\pi$
$132$	0	0
$133$	−6.21706	−0.539087
$134$	0	0
$135$	−5.48874	−0.472395
$136$	0	0
$137$	9.91748	0.847307	0.423654	−	0.905824i	$-0.360747\pi$
$137$	9.91748	0.847307	0.423654	+	0.905824i	$0.360747\pi$
$138$	0	0
$139$	−2.28929	−0.194175	−0.0970873	−	0.995276i	$-0.530953\pi$
$139$	−2.28929	−0.194175	−0.0970873	+	0.995276i	$0.530953\pi$
$140$	0	0
$141$	−1.07281	−0.0903470
$142$	0	0
$143$	−0.209525	−0.0175214
$144$	0	0
$145$	11.9559	0.992883
$146$	0	0
$147$	0.523874	0.0432084
$148$	0	0
$149$	−4.09066	−0.335120	−0.167560	−	0.985862i	$-0.553589\pi$
$149$	−4.09066	−0.335120	−0.167560	+	0.985862i	$0.553589\pi$
$150$	0	0
$151$	−14.0576	−1.14399	−0.571996	−	0.820256i	$-0.693831\pi$
$151$	−14.0576	−1.14399	−0.571996	+	0.820256i	$0.693831\pi$
$152$	0	0
$153$	5.77707	0.467049
$154$	0	0
$155$	44.4524	3.57050
$156$	0	0
$157$	−7.58487	−0.605339	−0.302669	−	0.953096i	$-0.597878\pi$
$157$	−7.58487	−0.605339	−0.302669	+	0.953096i	$0.597878\pi$
$158$	0	0
$159$	0.825511	0.0654673
$160$	0	0
$161$	−13.8075	−1.08818
$162$	0	0
$163$	3.92362	0.307321	0.153661	−	0.988124i	$-0.450894\pi$
$163$	3.92362	0.307321	0.153661	+	0.988124i	$0.450894\pi$
$164$	0	0
$165$	−0.0809283	−0.00630026
$166$	0	0
$167$	−1.98427	−0.153547	−0.0767736	−	0.997049i	$-0.524462\pi$
$167$	−1.98427	−0.153547	−0.0767736	+	0.997049i	$0.524462\pi$
$168$	0	0
$169$	−7.30390	−0.561838
$170$	0	0
$171$	−8.60669	−0.658170
$172$	0	0
$173$	−11.0120	−0.837230	−0.418615	−	0.908164i	$-0.637484\pi$
$173$	−11.0120	−0.837230	−0.418615	+	0.908164i	$0.637484\pi$
$174$	0	0
$175$	−28.8731	−2.18260
$176$	0	0
$177$	−2.60922	−0.196121
$178$	0	0
$179$	−13.7083	−1.02461	−0.512303	−	0.858805i	$-0.671208\pi$
$179$	−13.7083	−1.02461	−0.512303	+	0.858805i	$0.671208\pi$
$180$	0	0
$181$	−22.6144	−1.68091	−0.840457	−	0.541879i	$-0.817713\pi$
$181$	−22.6144	−1.68091	−0.840457	+	0.541879i	$0.817713\pi$
$182$	0	0
$183$	0.983561	0.0727069
$184$	0	0
$185$	−49.9306	−3.67097
$186$	0	0
$187$	0.171681	0.0125546
$188$	0	0
$189$	−2.72088	−0.197915
$190$	0	0
$191$	23.9900	1.73585	0.867927	−	0.496692i	$-0.165452\pi$
$191$	23.9900	1.73585	0.867927	+	0.496692i	$0.165452\pi$
$192$	0	0
$193$	−7.04369	−0.507016	−0.253508	−	0.967333i	$-0.581584\pi$
$193$	−7.04369	−0.507016	−0.253508	+	0.967333i	$0.581584\pi$
$194$	0	0
$195$	2.20010	0.157552
$196$	0	0
$197$	−15.6927	−1.11806	−0.559028	−	0.829149i	$-0.688826\pi$
$197$	−15.6927	−1.11806	−0.559028	+	0.829149i	$0.688826\pi$
$198$	0	0
$199$	−0.243544	−0.0172644	−0.00863219	−	0.999963i	$-0.502748\pi$
$199$	−0.243544	−0.0172644	−0.00863219	+	0.999963i	$0.502748\pi$
$200$	0	0
$201$	−0.274618	−0.0193701
$202$	0	0
$203$	5.92679	0.415979
$204$	0	0
$205$	−28.6960	−2.00422
$206$	0	0
$207$	−19.1146	−1.32856
$208$	0	0
$209$	−0.255771	−0.0176921
$210$	0	0
$211$	−2.72140	−0.187349	−0.0936743	−	0.995603i	$-0.529861\pi$
$211$	−2.72140	−0.187349	−0.0936743	+	0.995603i	$0.529861\pi$
$212$	0	0
$213$	0.0924671	0.00633574
$214$	0	0
$215$	10.6459	0.726043
$216$	0	0
$217$	22.0360	1.49590
$218$	0	0
$219$	0.541555	0.0365949
$220$	0	0
$221$	−4.66729	−0.313956
$222$	0	0
$223$	0.183759	0.0123054	0.00615271	−	0.999981i	$-0.498042\pi$
$223$	0.183759	0.0123054	0.00615271	+	0.999981i	$0.498042\pi$
$224$	0	0
$225$	−39.9709	−2.66473
$226$	0	0
$227$	−9.05830	−0.601221	−0.300610	−	0.953747i	$-0.597190\pi$
$227$	−9.05830	−0.601221	−0.300610	+	0.953747i	$0.597190\pi$
$228$	0	0
$229$	−15.9895	−1.05662	−0.528309	−	0.849052i	$-0.677174\pi$
$229$	−15.9895	−1.05662	−0.528309	+	0.849052i	$0.677174\pi$
$230$	0	0
$231$	−0.0401178	−0.00263956
$232$	0	0
$233$	23.8326	1.56133	0.780663	−	0.624952i	$-0.214881\pi$
$233$	23.8326	1.56133	0.780663	+	0.624952i	$0.214881\pi$
$234$	0	0
$235$	−21.5654	−1.40677
$236$	0	0
$237$	3.43037	0.222827
$238$	0	0
$239$	21.1895	1.37064	0.685318	−	0.728244i	$-0.259663\pi$
$239$	21.1895	1.37064	0.685318	+	0.728244i	$0.259663\pi$
$240$	0	0
$241$	20.8271	1.34159	0.670794	−	0.741643i	$-0.265953\pi$
$241$	20.8271	1.34159	0.670794	+	0.741643i	$0.265953\pi$
$242$	0	0
$243$	−5.66455	−0.363381
$244$	0	0
$245$	10.5308	0.672786
$246$	0	0
$247$	6.95334	0.442431
$248$	0	0
$249$	−0.0241449	−0.00153012
$250$	0	0
$251$	2.20659	0.139279	0.0696394	−	0.997572i	$-0.477815\pi$
$251$	2.20659	0.139279	0.0696394	+	0.997572i	$0.477815\pi$
$252$	0	0
$253$	−0.568044	−0.0357126
$254$	0	0
$255$	−1.80273	−0.112891
$256$	0	0
$257$	9.33713	0.582434	0.291217	−	0.956657i	$-0.405940\pi$
$257$	9.33713	0.582434	0.291217	+	0.956657i	$0.405940\pi$
$258$	0	0
$259$	−24.7516	−1.53799
$260$	0	0
$261$	8.20484	0.507867
$262$	0	0
$263$	16.0307	0.988496	0.494248	−	0.869321i	$-0.335443\pi$
$263$	16.0307	0.988496	0.494248	+	0.869321i	$0.335443\pi$
$264$	0	0
$265$	16.5942	1.01937
$266$	0	0
$267$	−0.802811	−0.0491312
$268$	0	0
$269$	−8.87784	−0.541292	−0.270646	−	0.962679i	$-0.587237\pi$
$269$	−8.87784	−0.541292	−0.270646	+	0.962679i	$0.587237\pi$
$270$	0	0
$271$	−3.92155	−0.238217	−0.119109	−	0.992881i	$-0.538004\pi$
$271$	−3.92155	−0.238217	−0.119109	+	0.992881i	$0.538004\pi$
$272$	0	0
$273$	1.09063	0.0660082
$274$	0	0
$275$	−1.18785	−0.0716298
$276$	0	0
$277$	3.00059	0.180288	0.0901440	−	0.995929i	$-0.471267\pi$
$277$	3.00059	0.180288	0.0901440	+	0.995929i	$0.471267\pi$
$278$	0	0
$279$	30.5059	1.82634
$280$	0	0
$281$	−18.7472	−1.11837	−0.559183	−	0.829044i	$-0.688885\pi$
$281$	−18.7472	−1.11837	−0.559183	+	0.829044i	$0.688885\pi$
$282$	0	0
$283$	−12.0890	−0.718615	−0.359308	−	0.933219i	$-0.616987\pi$
$283$	−12.0890	−0.718615	−0.359308	+	0.933219i	$0.616987\pi$
$284$	0	0
$285$	2.68570	0.159087
$286$	0	0
$287$	−14.2252	−0.839688
$288$	0	0
$289$	−13.1757	−0.775041
$290$	0	0
$291$	−1.18944	−0.0697262
$292$	0	0
$293$	−3.17529	−0.185502	−0.0927511	−	0.995689i	$-0.529566\pi$
$293$	−3.17529	−0.185502	−0.0927511	+	0.995689i	$0.529566\pi$
$294$	0	0
$295$	−52.4499	−3.05375
$296$	0	0
$297$	−0.111938	−0.00649528
$298$	0	0
$299$	15.4427	0.893075
$300$	0	0
$301$	5.27739	0.304183
$302$	0	0
$303$	−3.17812	−0.182578
$304$	0	0
$305$	19.7713	1.13210
$306$	0	0
$307$	−18.8616	−1.07649	−0.538244	−	0.842789i	$-0.680912\pi$
$307$	−18.8616	−1.07649	−0.538244	+	0.842789i	$0.680912\pi$
$308$	0	0
$309$	0.312578	0.0177819
$310$	0	0
$311$	−21.5838	−1.22390	−0.611952	−	0.790895i	$-0.709615\pi$
$311$	−21.5838	−1.22390	−0.611952	+	0.790895i	$0.709615\pi$
$312$	0	0
$313$	−15.9482	−0.901445	−0.450722	−	0.892664i	$-0.648834\pi$
$313$	−15.9482	−0.901445	−0.450722	+	0.892664i	$0.648834\pi$
$314$	0	0
$315$	−27.1366	−1.52897
$316$	0	0
$317$	7.39536	0.415365	0.207682	−	0.978196i	$-0.433408\pi$
$317$	7.39536	0.415365	0.207682	+	0.978196i	$0.433408\pi$
$318$	0	0
$319$	0.243829	0.0136518
$320$	0	0
$321$	2.63925	0.147309
$322$	0	0
$323$	−5.69746	−0.317015
$324$	0	0
$325$	32.2925	1.79127
$326$	0	0
$327$	0.250402	0.0138473
$328$	0	0
$329$	−10.6904	−0.589380
$330$	0	0
$331$	31.5721	1.73536	0.867679	−	0.497124i	$-0.165611\pi$
$331$	31.5721	1.73536	0.867679	+	0.497124i	$0.165611\pi$
$332$	0	0
$333$	−34.2653	−1.87773
$334$	0	0
$335$	−5.52030	−0.301606
$336$	0	0
$337$	20.4000	1.11126	0.555629	−	0.831431i	$-0.312478\pi$
$337$	20.4000	1.11126	0.555629	+	0.831431i	$0.312478\pi$
$338$	0	0
$339$	−1.73080	−0.0940043
$340$	0	0
$341$	0.906565	0.0490933
$342$	0	0
$343$	20.1578	1.08842
$344$	0	0
$345$	5.96469	0.321128
$346$	0	0
$347$	4.43110	0.237874	0.118937	−	0.992902i	$-0.462051\pi$
$347$	4.43110	0.237874	0.118937	+	0.992902i	$0.462051\pi$
$348$	0	0
$349$	−12.5438	−0.671453	−0.335727	−	0.941959i	$-0.608982\pi$
$349$	−12.5438	−0.671453	−0.335727	+	0.941959i	$0.608982\pi$
$350$	0	0
$351$	3.04311	0.162429
$352$	0	0
$353$	−16.0304	−0.853210	−0.426605	−	0.904438i	$-0.640291\pi$
$353$	−16.0304	−0.853210	−0.426605	+	0.904438i	$0.640291\pi$
$354$	0	0
$355$	1.85875	0.0986521
$356$	0	0
$357$	−0.893648	−0.0472969
$358$	0	0
$359$	12.6710	0.668750	0.334375	−	0.942440i	$-0.391475\pi$
$359$	12.6710	0.668750	0.334375	+	0.942440i	$0.391475\pi$
$360$	0	0
$361$	−10.5119	−0.553259
$362$	0	0
$363$	2.35396	0.123551
$364$	0	0
$365$	10.8862	0.569810
$366$	0	0
$367$	21.6193	1.12852	0.564260	−	0.825597i	$-0.309162\pi$
$367$	21.6193	1.12852	0.564260	+	0.825597i	$0.309162\pi$
$368$	0	0
$369$	−19.6929	−1.02517
$370$	0	0
$371$	8.22609	0.427077
$372$	0	0
$373$	−7.28572	−0.377240	−0.188620	−	0.982050i	$-0.560402\pi$
$373$	−7.28572	−0.377240	−0.188620	+	0.982050i	$0.560402\pi$
$374$	0	0
$375$	7.86369	0.406079
$376$	0	0
$377$	−6.62869	−0.341395
$378$	0	0
$379$	−3.90113	−0.200388	−0.100194	−	0.994968i	$-0.531946\pi$
$379$	−3.90113	−0.200388	−0.100194	+	0.994968i	$0.531946\pi$
$380$	0	0
$381$	−2.14996	−0.110146
$382$	0	0
$383$	11.9429	0.610253	0.305127	−	0.952312i	$-0.401301\pi$
$383$	11.9429	0.610253	0.305127	+	0.952312i	$0.401301\pi$
$384$	0	0
$385$	−0.806437	−0.0410998
$386$	0	0
$387$	7.30583	0.371376
$388$	0	0
$389$	−19.2035	−0.973654	−0.486827	−	0.873498i	$-0.661846\pi$
$389$	−19.2035	−0.973654	−0.486827	+	0.873498i	$0.661846\pi$
$390$	0	0
$391$	−12.6535	−0.639916
$392$	0	0
$393$	−0.913462	−0.0460780
$394$	0	0
$395$	68.9564	3.46957
$396$	0	0
$397$	−9.32344	−0.467930	−0.233965	−	0.972245i	$-0.575170\pi$
$397$	−9.32344	−0.467930	−0.233965	+	0.972245i	$0.575170\pi$
$398$	0	0
$399$	1.33136	0.0666513
$400$	0	0
$401$	−6.25490	−0.312355	−0.156177	−	0.987729i	$-0.549917\pi$
$401$	−6.25490	−0.312355	−0.156177	+	0.987729i	$0.549917\pi$
$402$	0	0
$403$	−24.6457	−1.22769
$404$	0	0
$405$	−36.9747	−1.83729
$406$	0	0
$407$	−1.01829	−0.0504746
$408$	0	0
$409$	−28.0686	−1.38790	−0.693952	−	0.720021i	$-0.744132\pi$
$409$	−28.0686	−1.38790	−0.693952	+	0.720021i	$0.744132\pi$
$410$	0	0
$411$	−2.12379	−0.104759
$412$	0	0
$413$	−26.0005	−1.27940
$414$	0	0
$415$	−0.485354	−0.0238251
$416$	0	0
$417$	0.490241	0.0240072
$418$	0	0
$419$	12.1602	0.594064	0.297032	−	0.954868i	$-0.404003\pi$
$419$	12.1602	0.594064	0.297032	+	0.954868i	$0.404003\pi$
$420$	0	0
$421$	8.92767	0.435108	0.217554	−	0.976048i	$-0.430192\pi$
$421$	8.92767	0.435108	0.217554	+	0.976048i	$0.430192\pi$
$422$	0	0
$423$	−14.7994	−0.719572
$424$	0	0
$425$	−26.4600	−1.28350
$426$	0	0
$427$	9.80103	0.474305
$428$	0	0
$429$	0.0448689	0.00216629
$430$	0	0
$431$	−38.0823	−1.83436	−0.917179	−	0.398475i	$-0.869539\pi$
$431$	−38.0823	−1.83436	−0.917179	+	0.398475i	$0.869539\pi$
$432$	0	0
$433$	−24.3661	−1.17096	−0.585481	−	0.810686i	$-0.699094\pi$
$433$	−24.3661	−1.17096	−0.585481	+	0.810686i	$0.699094\pi$
$434$	0	0
$435$	−2.56031	−0.122757
$436$	0	0
$437$	18.8512	0.901776
$438$	0	0
$439$	−24.2808	−1.15886	−0.579429	−	0.815023i	$-0.696724\pi$
$439$	−24.2808	−1.15886	−0.579429	+	0.815023i	$0.696724\pi$
$440$	0	0
$441$	7.22684	0.344135
$442$	0	0
$443$	12.2006	0.579667	0.289833	−	0.957077i	$-0.406400\pi$
$443$	12.2006	0.579667	0.289833	+	0.957077i	$0.406400\pi$
$444$	0	0
$445$	−16.1379	−0.765009
$446$	0	0
$447$	0.875998	0.0414333
$448$	0	0
$449$	14.1413	0.667369	0.333685	−	0.942685i	$-0.391708\pi$
$449$	14.1413	0.667369	0.333685	+	0.942685i	$0.391708\pi$
$450$	0	0
$451$	−0.585229	−0.0275573
$452$	0	0
$453$	3.01038	0.141440
$454$	0	0
$455$	21.9236	1.02780
$456$	0	0
$457$	33.5373	1.56881	0.784404	−	0.620250i	$-0.212969\pi$
$457$	33.5373	1.56881	0.784404	+	0.620250i	$0.212969\pi$
$458$	0	0
$459$	−2.49348	−0.116386
$460$	0	0
$461$	12.9601	0.603613	0.301807	−	0.953369i	$-0.402410\pi$
$461$	12.9601	0.603613	0.301807	+	0.953369i	$0.402410\pi$
$462$	0	0
$463$	18.5083	0.860155	0.430078	−	0.902792i	$-0.358486\pi$
$463$	18.5083	0.860155	0.430078	+	0.902792i	$0.358486\pi$
$464$	0	0
$465$	−9.51931	−0.441447
$466$	0	0
$467$	1.41209	0.0653436	0.0326718	−	0.999466i	$-0.489598\pi$
$467$	1.41209	0.0653436	0.0326718	+	0.999466i	$0.489598\pi$
$468$	0	0
$469$	−2.73652	−0.126361
$470$	0	0
$471$	1.62427	0.0748424
$472$	0	0
$473$	0.217113	0.00998286
$474$	0	0
$475$	39.4201	1.80872
$476$	0	0
$477$	11.3879	0.521417
$478$	0	0
$479$	−10.5674	−0.482837	−0.241419	−	0.970421i	$-0.577613\pi$
$479$	−10.5674	−0.482837	−0.241419	+	0.970421i	$0.577613\pi$
$480$	0	0
$481$	27.6829	1.26223
$482$	0	0
$483$	2.95682	0.134540
$484$	0	0
$485$	−23.9098	−1.08569
$486$	0	0
$487$	5.77862	0.261854	0.130927	−	0.991392i	$-0.458205\pi$
$487$	5.77862	0.261854	0.130927	+	0.991392i	$0.458205\pi$
$488$	0	0
$489$	−0.840227	−0.0379964
$490$	0	0
$491$	2.51720	0.113599	0.0567997	−	0.998386i	$-0.481910\pi$
$491$	2.51720	0.113599	0.0567997	+	0.998386i	$0.481910\pi$
$492$	0	0
$493$	5.43145	0.244620
$494$	0	0
$495$	−1.11640	−0.0501787
$496$	0	0
$497$	0.921420	0.0413313
$498$	0	0
$499$	10.7137	0.479612	0.239806	−	0.970821i	$-0.422916\pi$
$499$	10.7137	0.479612	0.239806	+	0.970821i	$0.422916\pi$
$500$	0	0
$501$	0.424923	0.0189842
$502$	0	0
$503$	2.05799	0.0917613	0.0458807	−	0.998947i	$-0.485391\pi$
$503$	2.05799	0.0917613	0.0458807	+	0.998947i	$0.485391\pi$
$504$	0	0
$505$	−63.8856	−2.84287
$506$	0	0
$507$	1.56410	0.0694642
$508$	0	0
$509$	−21.7020	−0.961924	−0.480962	−	0.876741i	$-0.659713\pi$
$509$	−21.7020	−0.961924	−0.480962	+	0.876741i	$0.659713\pi$
$510$	0	0
$511$	5.39651	0.238728
$512$	0	0
$513$	3.71479	0.164012
$514$	0	0
$515$	6.28336	0.276878
$516$	0	0
$517$	−0.439805	−0.0193426
$518$	0	0
$519$	2.35818	0.103513
$520$	0	0
$521$	29.4073	1.28836	0.644178	−	0.764876i	$-0.277200\pi$
$521$	29.4073	1.28836	0.644178	+	0.764876i	$0.277200\pi$
$522$	0	0
$523$	15.5818	0.681343	0.340672	−	0.940182i	$-0.389346\pi$
$523$	15.5818	0.681343	0.340672	+	0.940182i	$0.389346\pi$
$524$	0	0
$525$	6.18306	0.269851
$526$	0	0
$527$	20.1943	0.879677
$528$	0	0
$529$	18.8668	0.820294
$530$	0	0
$531$	−35.9942	−1.56201
$532$	0	0
$533$	15.9099	0.689134
$534$	0	0
$535$	53.0535	2.29370
$536$	0	0
$537$	2.93558	0.126680
$538$	0	0
$539$	0.214765	0.00925059
$540$	0	0
$541$	−17.2086	−0.739853	−0.369927	−	0.929061i	$-0.620617\pi$
$541$	−17.2086	−0.739853	−0.369927	+	0.929061i	$0.620617\pi$
$542$	0	0
$543$	4.84278	0.207823
$544$	0	0
$545$	5.03351	0.215612
$546$	0	0
$547$	3.73089	0.159521	0.0797606	−	0.996814i	$-0.474584\pi$
$547$	3.73089	0.159521	0.0797606	+	0.996814i	$0.474584\pi$
$548$	0	0
$549$	13.5682	0.579077
$550$	0	0
$551$	−8.09177	−0.344721
$552$	0	0
$553$	34.1831	1.45361
$554$	0	0
$555$	10.6924	0.453869
$556$	0	0
$557$	−0.128254	−0.00543431	−0.00271716	−	0.999996i	$-0.500865\pi$
$557$	−0.128254	−0.00543431	−0.00271716	+	0.999996i	$0.500865\pi$
$558$	0	0
$559$	−5.90238	−0.249644
$560$	0	0
$561$	−0.0367649	−0.00155222
$562$	0	0
$563$	23.4800	0.989563	0.494782	−	0.869017i	$-0.335248\pi$
$563$	23.4800	0.989563	0.494782	+	0.869017i	$0.335248\pi$
$564$	0	0
$565$	−34.7921	−1.46372
$566$	0	0
$567$	−18.3291	−0.769751
$568$	0	0
$569$	37.8950	1.58864	0.794320	−	0.607499i	$-0.207827\pi$
$569$	37.8950	1.58864	0.794320	+	0.607499i	$0.207827\pi$
$570$	0	0
$571$	9.89740	0.414193	0.207097	−	0.978321i	$-0.433599\pi$
$571$	9.89740	0.414193	0.207097	+	0.978321i	$0.433599\pi$
$572$	0	0
$573$	−5.13736	−0.214616
$574$	0	0
$575$	87.5483	3.65102
$576$	0	0
$577$	−18.7741	−0.781576	−0.390788	−	0.920481i	$-0.627797\pi$
$577$	−18.7741	−0.781576	−0.390788	+	0.920481i	$0.627797\pi$
$578$	0	0
$579$	1.50838	0.0626860
$580$	0	0
$581$	−0.240600	−0.00998176
$582$	0	0
$583$	0.338423	0.0140161
$584$	0	0
$585$	30.3503	1.25483
$586$	0	0
$587$	−10.3374	−0.426670	−0.213335	−	0.976979i	$-0.568433\pi$
$587$	−10.3374	−0.426670	−0.213335	+	0.976979i	$0.568433\pi$
$588$	0	0
$589$	−30.0855	−1.23965
$590$	0	0
$591$	3.36052	0.138233
$592$	0	0
$593$	−20.5355	−0.843292	−0.421646	−	0.906760i	$-0.638548\pi$
$593$	−20.5355	−0.843292	−0.421646	+	0.906760i	$0.638548\pi$
$594$	0	0
$595$	−17.9639	−0.736447
$596$	0	0
$597$	0.0521540	0.00213452
$598$	0	0
$599$	−24.6258	−1.00618	−0.503090	−	0.864234i	$-0.667804\pi$
$599$	−24.6258	−1.00618	−0.503090	+	0.864234i	$0.667804\pi$
$600$	0	0
$601$	34.1096	1.39136	0.695680	−	0.718352i	$-0.255103\pi$
$601$	34.1096	1.39136	0.695680	+	0.718352i	$0.255103\pi$
$602$	0	0
$603$	−3.78835	−0.154274
$604$	0	0
$605$	47.3186	1.92377
$606$	0	0
$607$	37.6200	1.52695	0.763473	−	0.645839i	$-0.223492\pi$
$607$	37.6200	1.52695	0.763473	+	0.645839i	$0.223492\pi$
$608$	0	0
$609$	−1.26920	−0.0514305
$610$	0	0
$611$	11.9564	0.483706
$612$	0	0
$613$	−21.6279	−0.873544	−0.436772	−	0.899572i	$-0.643878\pi$
$613$	−21.6279	−0.873544	−0.436772	+	0.899572i	$0.643878\pi$
$614$	0	0
$615$	6.14514	0.247796
$616$	0	0
$617$	−31.7365	−1.27766	−0.638831	−	0.769347i	$-0.720582\pi$
$617$	−31.7365	−1.27766	−0.638831	+	0.769347i	$0.720582\pi$
$618$	0	0
$619$	41.6586	1.67440	0.837200	−	0.546897i	$-0.184191\pi$
$619$	41.6586	1.67440	0.837200	+	0.546897i	$0.184191\pi$
$620$	0	0
$621$	8.25019	0.331069
$622$	0	0
$623$	−7.99988	−0.320508
$624$	0	0
$625$	90.4214	3.61686
$626$	0	0
$627$	0.0547724	0.00218740
$628$	0	0
$629$	−22.6830	−0.904429
$630$	0	0
$631$	29.1298	1.15964	0.579820	−	0.814744i	$-0.303123\pi$
$631$	29.1298	1.15964	0.579820	+	0.814744i	$0.303123\pi$
$632$	0	0
$633$	0.582776	0.0231633
$634$	0	0
$635$	−43.2180	−1.71505
$636$	0	0
$637$	−5.83856	−0.231332
$638$	0	0
$639$	1.27558	0.0504613
$640$	0	0
$641$	−7.04916	−0.278425	−0.139213	−	0.990263i	$-0.544457\pi$
$641$	−7.04916	−0.278425	−0.139213	+	0.990263i	$0.544457\pi$
$642$	0	0
$643$	20.2230	0.797515	0.398758	−	0.917056i	$-0.369441\pi$
$643$	20.2230	0.797515	0.398758	+	0.917056i	$0.369441\pi$
$644$	0	0
$645$	−2.27977	−0.0897660
$646$	0	0
$647$	−17.5323	−0.689267	−0.344634	−	0.938737i	$-0.611997\pi$
$647$	−17.5323	−0.689267	−0.344634	+	0.938737i	$0.611997\pi$
$648$	0	0
$649$	−1.06967	−0.0419880
$650$	0	0
$651$	−4.71892	−0.184949
$652$	0	0
$653$	−20.1412	−0.788187	−0.394094	−	0.919070i	$-0.628941\pi$
$653$	−20.1412	−0.788187	−0.394094	+	0.919070i	$0.628941\pi$
$654$	0	0
$655$	−18.3622	−0.717469
$656$	0	0
$657$	7.47075	0.291462
$658$	0	0
$659$	21.3387	0.831239	0.415619	−	0.909539i	$-0.363565\pi$
$659$	21.3387	0.831239	0.415619	+	0.909539i	$0.363565\pi$
$660$	0	0
$661$	−24.4152	−0.949640	−0.474820	−	0.880083i	$-0.657487\pi$
$661$	−24.4152	−0.949640	−0.474820	+	0.880083i	$0.657487\pi$
$662$	0	0
$663$	0.999482	0.0388167
$664$	0	0
$665$	26.7626	1.03781
$666$	0	0
$667$	−17.9711	−0.695842
$668$	0	0
$669$	−0.0393513	−0.00152141
$670$	0	0
$671$	0.403217	0.0155660
$672$	0	0
$673$	−36.4999	−1.40697	−0.703484	−	0.710711i	$-0.748373\pi$
$673$	−36.4999	−1.40697	−0.703484	+	0.710711i	$0.748373\pi$
$674$	0	0
$675$	17.2521	0.664034
$676$	0	0
$677$	−12.4617	−0.478942	−0.239471	−	0.970904i	$-0.576974\pi$
$677$	−12.4617	−0.478942	−0.239471	+	0.970904i	$0.576974\pi$
$678$	0	0
$679$	−11.8526	−0.454860
$680$	0	0
$681$	1.93980	0.0743333
$682$	0	0
$683$	−32.4264	−1.24076	−0.620381	−	0.784300i	$-0.713022\pi$
$683$	−32.4264	−1.24076	−0.620381	+	0.784300i	$0.713022\pi$
$684$	0	0
$685$	−42.6918	−1.63117
$686$	0	0
$687$	3.42409	0.130637
$688$	0	0
$689$	−9.20030	−0.350503
$690$	0	0
$691$	−30.0719	−1.14399	−0.571994	−	0.820258i	$-0.693830\pi$
$691$	−30.0719	−1.14399	−0.571994	+	0.820258i	$0.693830\pi$
$692$	0	0
$693$	−0.553425	−0.0210229
$694$	0	0
$695$	9.85470	0.373810
$696$	0	0
$697$	−13.0363	−0.493786
$698$	0	0
$699$	−5.10366	−0.193038
$700$	0	0
$701$	43.4318	1.64040	0.820198	−	0.572080i	$-0.193863\pi$
$701$	43.4318	1.64040	0.820198	+	0.572080i	$0.193863\pi$
$702$	0	0
$703$	33.7931	1.27453
$704$	0	0
$705$	4.61813	0.173929
$706$	0	0
$707$	−31.6694	−1.19105
$708$	0	0
$709$	−6.71440	−0.252165	−0.126082	−	0.992020i	$-0.540240\pi$
$709$	−6.71440	−0.252165	−0.126082	+	0.992020i	$0.540240\pi$
$710$	0	0
$711$	47.3220	1.77471
$712$	0	0
$713$	−66.8170	−2.50232
$714$	0	0
$715$	0.901943	0.0337308
$716$	0	0
$717$	−4.53765	−0.169462
$718$	0	0
$719$	−6.19906	−0.231186	−0.115593	−	0.993297i	$-0.536877\pi$
$719$	−6.19906	−0.231186	−0.115593	+	0.993297i	$0.536877\pi$
$720$	0	0
$721$	3.11479	0.116001
$722$	0	0
$723$	−4.46003	−0.165870
$724$	0	0
$725$	−37.5796	−1.39567
$726$	0	0
$727$	2.29010	0.0849352	0.0424676	−	0.999098i	$-0.486478\pi$
$727$	2.29010	0.0849352	0.0424676	+	0.999098i	$0.486478\pi$
$728$	0	0
$729$	−24.5551	−0.909448
$730$	0	0
$731$	4.83632	0.178878
$732$	0	0
$733$	−1.57771	−0.0582742	−0.0291371	−	0.999575i	$-0.509276\pi$
$733$	−1.57771	−0.0582742	−0.0291371	+	0.999575i	$0.509276\pi$
$734$	0	0
$735$	−2.25512	−0.0831814
$736$	0	0
$737$	−0.112581	−0.00414698
$738$	0	0
$739$	−16.2081	−0.596223	−0.298111	−	0.954531i	$-0.596357\pi$
$739$	−16.2081	−0.596223	−0.298111	+	0.954531i	$0.596357\pi$
$740$	0	0
$741$	−1.48903	−0.0547009
$742$	0	0
$743$	−24.8518	−0.911722	−0.455861	−	0.890051i	$-0.650669\pi$
$743$	−24.8518	−0.911722	−0.455861	+	0.890051i	$0.650669\pi$
$744$	0	0
$745$	17.6091	0.645146
$746$	0	0
$747$	−0.333078	−0.0121867
$748$	0	0
$749$	26.2997	0.960971
$750$	0	0
$751$	1.00000	0.0364905
$751$	1.00000	0.0364905
$752$	0	0
$753$	−0.472533	−0.0172200
$754$	0	0
$755$	60.5139	2.20233
$756$	0	0
$757$	2.14057	0.0778005	0.0389002	−	0.999243i	$-0.487615\pi$
$757$	2.14057	0.0778005	0.0389002	+	0.999243i	$0.487615\pi$
$758$	0	0
$759$	0.121644	0.00441541
$760$	0	0
$761$	9.67444	0.350698	0.175349	−	0.984506i	$-0.443895\pi$
$761$	9.67444	0.350698	0.175349	+	0.984506i	$0.443895\pi$
$762$	0	0
$763$	2.49521	0.0903328
$764$	0	0
$765$	−24.8686	−0.899126
$766$	0	0
$767$	29.0797	1.05001
$768$	0	0
$769$	−34.8347	−1.25617	−0.628085	−	0.778145i	$-0.716161\pi$
$769$	−34.8347	−1.25617	−0.628085	+	0.778145i	$0.716161\pi$
$770$	0	0
$771$	−1.99951	−0.0720105
$772$	0	0
$773$	−49.5789	−1.78323	−0.891615	−	0.452795i	$-0.850427\pi$
$773$	−49.5789	−1.78323	−0.891615	+	0.452795i	$0.850427\pi$
$774$	0	0
$775$	−139.722	−5.01897
$776$	0	0
$777$	5.30046	0.190153
$778$	0	0
$779$	19.4215	0.695848
$780$	0	0
$781$	0.0379074	0.00135643
$782$	0	0
$783$	−3.54134	−0.126557
$784$	0	0
$785$	32.6506	1.16535
$786$	0	0
$787$	44.7626	1.59561	0.797807	−	0.602913i	$-0.205994\pi$
$787$	44.7626	1.59561	0.797807	+	0.602913i	$0.205994\pi$
$788$	0	0
$789$	−3.43291	−0.122215
$790$	0	0
$791$	−17.2472	−0.613239
$792$	0	0
$793$	−10.9618	−0.389263
$794$	0	0
$795$	−3.55358	−0.126033
$796$	0	0
$797$	−32.4277	−1.14865	−0.574325	−	0.818628i	$-0.694735\pi$
$797$	−32.4277	−1.14865	−0.574325	+	0.818628i	$0.694735\pi$
$798$	0	0
$799$	−9.79692	−0.346590
$800$	0	0
$801$	−11.0748	−0.391308
$802$	0	0
$803$	0.222014	0.00783470
$804$	0	0
$805$	59.4372	2.09489
$806$	0	0
$807$	1.90115	0.0669238
$808$	0	0
$809$	14.3624	0.504954	0.252477	−	0.967603i	$-0.418755\pi$
$809$	14.3624	0.504954	0.252477	+	0.967603i	$0.418755\pi$
$810$	0	0
$811$	22.8470	0.802267	0.401134	−	0.916020i	$-0.368616\pi$
$811$	22.8470	0.802267	0.401134	+	0.916020i	$0.368616\pi$
$812$	0	0
$813$	0.839785	0.0294525
$814$	0	0
$815$	−16.8900	−0.591631
$816$	0	0
$817$	−7.20515	−0.252076
$818$	0	0
$819$	15.0453	0.525725
$820$	0	0
$821$	−29.5001	−1.02956	−0.514781	−	0.857322i	$-0.672127\pi$
$821$	−29.5001	−1.02956	−0.514781	+	0.857322i	$0.672127\pi$
$822$	0	0
$823$	−28.9938	−1.01066	−0.505330	−	0.862926i	$-0.668629\pi$
$823$	−28.9938	−1.01066	−0.505330	+	0.862926i	$0.668629\pi$
$824$	0	0
$825$	0.254372	0.00885611
$826$	0	0
$827$	−6.58103	−0.228845	−0.114422	−	0.993432i	$-0.536502\pi$
$827$	−6.58103	−0.228845	−0.114422	+	0.993432i	$0.536502\pi$
$828$	0	0
$829$	−44.5264	−1.54647	−0.773233	−	0.634122i	$-0.781362\pi$
$829$	−44.5264	−1.54647	−0.773233	+	0.634122i	$0.781362\pi$
$830$	0	0
$831$	−0.642564	−0.0222903
$832$	0	0
$833$	4.78402	0.165757
$834$	0	0
$835$	8.54169	0.295597
$836$	0	0
$837$	−13.1668	−0.455112
$838$	0	0
$839$	−30.4902	−1.05264	−0.526320	−	0.850287i	$-0.676429\pi$
$839$	−30.4902	−1.05264	−0.526320	+	0.850287i	$0.676429\pi$
$840$	0	0
$841$	−21.2860	−0.734001
$842$	0	0
$843$	4.01464	0.138272
$844$	0	0
$845$	31.4411	1.08161
$846$	0	0
$847$	23.4568	0.805985
$848$	0	0
$849$	2.58881	0.0888476
$850$	0	0
$851$	75.0513	2.57273
$852$	0	0
$853$	−22.3003	−0.763548	−0.381774	−	0.924256i	$-0.624687\pi$
$853$	−22.3003	−0.763548	−0.381774	+	0.924256i	$0.624687\pi$
$854$	0	0
$855$	37.0492	1.26706
$856$	0	0
$857$	38.5607	1.31721	0.658605	−	0.752489i	$-0.271147\pi$
$857$	38.5607	1.31721	0.658605	+	0.752489i	$0.271147\pi$
$858$	0	0
$859$	−41.0501	−1.40061	−0.700305	−	0.713843i	$-0.746953\pi$
$859$	−41.0501	−1.40061	−0.700305	+	0.713843i	$0.746953\pi$
$860$	0	0
$861$	3.04627	0.103817
$862$	0	0
$863$	−44.8408	−1.52640	−0.763199	−	0.646164i	$-0.776372\pi$
$863$	−44.8408	−1.52640	−0.763199	+	0.646164i	$0.776372\pi$
$864$	0	0
$865$	47.4036	1.61177
$866$	0	0
$867$	2.82152	0.0958239
$868$	0	0
$869$	1.40630	0.0477055
$870$	0	0
$871$	3.06061	0.103705
$872$	0	0
$873$	−16.4083	−0.555337
$874$	0	0
$875$	78.3605	2.64907
$876$	0	0
$877$	−50.5917	−1.70836	−0.854181	−	0.519976i	$-0.825941\pi$
$877$	−50.5917	−1.70836	−0.854181	+	0.519976i	$0.825941\pi$
$878$	0	0
$879$	0.679975	0.0229350
$880$	0	0
$881$	17.6362	0.594179	0.297090	−	0.954850i	$-0.403984\pi$
$881$	17.6362	0.594179	0.297090	+	0.954850i	$0.403984\pi$
$882$	0	0
$883$	20.2157	0.680313	0.340157	−	0.940369i	$-0.389520\pi$
$883$	20.2157	0.680313	0.340157	+	0.940369i	$0.389520\pi$
$884$	0	0
$885$	11.2319	0.377557
$886$	0	0
$887$	−7.07205	−0.237456	−0.118728	−	0.992927i	$-0.537882\pi$
$887$	−7.07205	−0.237456	−0.118728	+	0.992927i	$0.537882\pi$
$888$	0	0
$889$	−21.4240	−0.718539
$890$	0	0
$891$	−0.754065	−0.0252621
$892$	0	0
$893$	14.5955	0.488419
$894$	0	0
$895$	59.0102	1.97249
$896$	0	0
$897$	−3.30699	−0.110417
$898$	0	0
$899$	28.6808	0.956558
$900$	0	0
$901$	7.53858	0.251146
$902$	0	0
$903$	−1.13013	−0.0376084
$904$	0	0
$905$	97.3482	3.23596
$906$	0	0
$907$	−27.1251	−0.900675	−0.450338	−	0.892858i	$-0.648696\pi$
$907$	−27.1251	−0.900675	−0.450338	+	0.892858i	$0.648696\pi$
$908$	0	0
$909$	−43.8421	−1.45415
$910$	0	0
$911$	21.5046	0.712478	0.356239	−	0.934395i	$-0.384059\pi$
$911$	21.5046	0.712478	0.356239	+	0.934395i	$0.384059\pi$
$912$	0	0
$913$	−0.00989833	−0.000327587 0
$914$	0	0
$915$	−4.23394	−0.139970
$916$	0	0
$917$	−9.10250	−0.300591
$918$	0	0
$919$	−16.3760	−0.540195	−0.270098	−	0.962833i	$-0.587056\pi$
$919$	−16.3760	−0.540195	−0.270098	+	0.962833i	$0.587056\pi$
$920$	0	0
$921$	4.03913	0.133094
$922$	0	0
$923$	−1.03054	−0.0339207
$924$	0	0
$925$	156.941	5.16019
$926$	0	0
$927$	4.31201	0.141625
$928$	0	0
$929$	17.3460	0.569103	0.284551	−	0.958661i	$-0.408155\pi$
$929$	17.3460	0.569103	0.284551	+	0.958661i	$0.408155\pi$
$930$	0	0
$931$	−7.12724	−0.233586
$932$	0	0
$933$	4.62208	0.151320
$934$	0	0
$935$	−0.739038	−0.0241691
$936$	0	0
$937$	−2.19998	−0.0718701	−0.0359350	−	0.999354i	$-0.511441\pi$
$937$	−2.19998	−0.0718701	−0.0359350	+	0.999354i	$0.511441\pi$
$938$	0	0
$939$	3.41524	0.111452
$940$	0	0
$941$	−34.1463	−1.11314	−0.556569	−	0.830802i	$-0.687882\pi$
$941$	−34.1463	−1.11314	−0.556569	+	0.830802i	$0.687882\pi$
$942$	0	0
$943$	43.1334	1.40462
$944$	0	0
$945$	11.7126	0.381010
$946$	0	0
$947$	−51.3003	−1.66704	−0.833518	−	0.552493i	$-0.813677\pi$
$947$	−51.3003	−1.66704	−0.833518	+	0.552493i	$0.813677\pi$
$948$	0	0
$949$	−6.03562	−0.195924
$950$	0	0
$951$	−1.58369	−0.0513545
$952$	0	0
$953$	18.3829	0.595479	0.297740	−	0.954647i	$-0.403767\pi$
$953$	18.3829	0.595479	0.297740	+	0.954647i	$0.403767\pi$
$954$	0	0
$955$	−103.270	−3.34173
$956$	0	0
$957$	−0.0522151	−0.00168787
$958$	0	0
$959$	−21.1632	−0.683396
$960$	0	0
$961$	75.6361	2.43987
$962$	0	0
$963$	36.4084	1.17325
$964$	0	0
$965$	30.3210	0.976067
$966$	0	0
$967$	50.1458	1.61258	0.806290	−	0.591520i	$-0.201472\pi$
$967$	50.1458	1.61258	0.806290	+	0.591520i	$0.201472\pi$
$968$	0	0
$969$	1.22009	0.0391949
$970$	0	0
$971$	11.7746	0.377864	0.188932	−	0.981990i	$-0.439497\pi$
$971$	11.7746	0.377864	0.188932	+	0.981990i	$0.439497\pi$
$972$	0	0
$973$	4.88518	0.156612
$974$	0	0
$975$	−6.91531	−0.221467
$976$	0	0
$977$	20.0107	0.640199	0.320099	−	0.947384i	$-0.396284\pi$
$977$	20.0107	0.640199	0.320099	+	0.947384i	$0.396284\pi$
$978$	0	0
$979$	−0.329117	−0.0105186
$980$	0	0
$981$	3.45429	0.110287
$982$	0	0
$983$	−50.9481	−1.62499	−0.812496	−	0.582966i	$-0.801892\pi$
$983$	−50.9481	−1.62499	−0.812496	+	0.582966i	$0.801892\pi$
$984$	0	0
$985$	67.5523	2.15240
$986$	0	0
$987$	2.28930	0.0728694
$988$	0	0
$989$	−16.0020	−0.508833
$990$	0	0
$991$	36.3762	1.15553	0.577763	−	0.816204i	$-0.303926\pi$
$991$	36.3762	1.15553	0.577763	+	0.816204i	$0.303926\pi$
$992$	0	0
$993$	−6.76104	−0.214555
$994$	0	0
$995$	1.04838	0.0332360
$996$	0	0
$997$	−45.3884	−1.43747	−0.718733	−	0.695286i	$-0.755278\pi$
$997$	−45.3884	−1.43747	−0.718733	+	0.695286i	$0.755278\pi$
$998$	0	0
$999$	14.7895	0.467918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.c.1.20	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.c.1.20	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.214146 q^{3} -4.30470 q^{5} -2.13393 q^{7} -2.95414 q^{9} +O(q^{10})\)	\(q-0.214146 q^{3} -4.30470 q^{5} -2.13393 q^{7} -2.95414 q^{9} -0.0877904 q^{11} +2.38665 q^{13} +0.921835 q^{15} -1.95558 q^{17} +2.91343 q^{19} +0.456973 q^{21} +6.47045 q^{23} +13.5305 q^{25} +1.27506 q^{27} -2.77740 q^{29} -10.3265 q^{31} +0.0188000 q^{33} +9.18594 q^{35} +11.5991 q^{37} -0.511092 q^{39} +6.66620 q^{41} -2.47308 q^{43} +12.7167 q^{45} +5.00972 q^{47} -2.44634 q^{49} +0.418780 q^{51} -3.85490 q^{53} +0.377912 q^{55} -0.623900 q^{57} +12.1843 q^{59} -4.59295 q^{61} +6.30393 q^{63} -10.2738 q^{65} +1.28239 q^{67} -1.38562 q^{69} -0.431795 q^{71} -2.52891 q^{73} -2.89750 q^{75} +0.187339 q^{77} -16.0189 q^{79} +8.58938 q^{81} +0.112750 q^{83} +8.41821 q^{85} +0.594770 q^{87} +3.74889 q^{89} -5.09295 q^{91} +2.21137 q^{93} -12.5415 q^{95} +5.55434 q^{97} +0.259345 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9	\( 44 q - 4 q^{3} - 21 q^{5} - 10 q^{7} + 38 q^{9} + 11 q^{11} - 36 q^{13} - 5 q^{15} - 10 q^{17} - 7 q^{19} - 42 q^{21} - 5 q^{23} + 29 q^{25} - 16 q^{27} - 57 q^{29} - 21 q^{31} - 32 q^{33} + 17 q^{35} - 52 q^{37} + 8 q^{39} - 16 q^{41} - 9 q^{43} - 84 q^{45} - q^{47} + 28 q^{49} - q^{51} - 52 q^{53} - 39 q^{55} - 15 q^{57} + 7 q^{59} - 85 q^{61} - 25 q^{63} - 9 q^{65} - 36 q^{67} - 72 q^{69} + 12 q^{71} - 60 q^{73} - 5 q^{75} - 81 q^{77} - 13 q^{79} + 20 q^{81} + 5 q^{83} - 72 q^{85} + 9 q^{87} - 37 q^{89} - 23 q^{91} - 60 q^{93} + 24 q^{95} - 79 q^{97} + 42 q^{99}+O(q^{100}) \) 44 * q - 4 * q^3 - 21 * q^5 - 10 * q^7 + 38 * q^9 + 11 * q^11 - 36 * q^13 - 5 * q^15 - 10 * q^17 - 7 * q^19 - 42 * q^21 - 5 * q^23 + 29 * q^25 - 16 * q^27 - 57 * q^29 - 21 * q^31 - 32 * q^33 + 17 * q^35 - 52 * q^37 + 8 * q^39 - 16 * q^41 - 9 * q^43 - 84 * q^45 - q^47 + 28 * q^49 - q^51 - 52 * q^53 - 39 * q^55 - 15 * q^57 + 7 * q^59 - 85 * q^61 - 25 * q^63 - 9 * q^65 - 36 * q^67 - 72 * q^69 + 12 * q^71 - 60 * q^73 - 5 * q^75 - 81 * q^77 - 13 * q^79 + 20 * q^81 + 5 * q^83 - 72 * q^85 + 9 * q^87 - 37 * q^89 - 23 * q^91 - 60 * q^93 + 24 * q^95 - 79 * q^97 + 42 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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