LMFDB - Embedded newform 560.6.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(560, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("560.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.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:	$89.8149390953$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 463x - 1890 $ x^3 - 463*x - 1890
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.24759$ of defining polynomial
Character	$\chi$	$=$	560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.24759 q^{3} -25.0000 q^{5} -49.0000 q^{7} -203.968 q^{9} +O(q^{10})$	$q-6.24759 q^{3} -25.0000 q^{5} -49.0000 q^{7} -203.968 q^{9} -89.4628 q^{11} +459.250 q^{13} +156.190 q^{15} +701.108 q^{17} +396.470 q^{19} +306.132 q^{21} +4316.18 q^{23} +625.000 q^{25} +2792.47 q^{27} -1437.19 q^{29} -3669.99 q^{31} +558.927 q^{33} +1225.00 q^{35} -3166.84 q^{37} -2869.21 q^{39} -6353.05 q^{41} +15510.1 q^{43} +5099.19 q^{45} +3995.07 q^{47} +2401.00 q^{49} -4380.24 q^{51} -24472.8 q^{53} +2236.57 q^{55} -2476.98 q^{57} +2395.32 q^{59} -21052.6 q^{61} +9994.41 q^{63} -11481.3 q^{65} -853.274 q^{67} -26965.7 q^{69} +1378.74 q^{71} +12248.5 q^{73} -3904.75 q^{75} +4383.68 q^{77} +51047.4 q^{79} +32117.9 q^{81} -17615.6 q^{83} -17527.7 q^{85} +8978.99 q^{87} -18346.0 q^{89} -22503.3 q^{91} +22928.6 q^{93} -9911.74 q^{95} -103460. q^{97} +18247.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9}+O(q^{10}) $ 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9	$ 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33} + 3675 q^{35} - 4058 q^{37} + 20910 q^{39} - 9602 q^{41} + 12340 q^{43} - 5225 q^{45} + 41026 q^{47} + 7203 q^{49} - 10006 q^{51} - 37610 q^{53} - 14450 q^{55} - 95456 q^{57} + 37664 q^{59} - 8386 q^{61} - 10241 q^{63} - 2000 q^{65} - 69340 q^{67} - 37712 q^{69} + 34016 q^{71} - 45314 q^{73} - 3750 q^{75} - 28322 q^{77} - 1382 q^{79} - 101005 q^{81} + 10128 q^{83} + 31100 q^{85} + 41510 q^{87} - 222810 q^{89} - 3920 q^{91} - 174292 q^{93} + 23600 q^{95} - 159476 q^{97} + 156568 q^{99}+O(q^{100}) $ 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9 + 578 * q^11 + 80 * q^13 + 150 * q^15 - 1244 * q^17 - 944 * q^19 + 294 * q^21 - 1096 * q^23 + 1875 * q^25 + 3006 * q^27 + 1868 * q^29 + 6620 * q^31 + 2662 * q^33 + 3675 * q^35 - 4058 * q^37 + 20910 * q^39 - 9602 * q^41 + 12340 * q^43 - 5225 * q^45 + 41026 * q^47 + 7203 * q^49 - 10006 * q^51 - 37610 * q^53 - 14450 * q^55 - 95456 * q^57 + 37664 * q^59 - 8386 * q^61 - 10241 * q^63 - 2000 * q^65 - 69340 * q^67 - 37712 * q^69 + 34016 * q^71 - 45314 * q^73 - 3750 * q^75 - 28322 * q^77 - 1382 * q^79 - 101005 * q^81 + 10128 * q^83 + 31100 * q^85 + 41510 * q^87 - 222810 * q^89 - 3920 * q^91 - 174292 * q^93 + 23600 * q^95 - 159476 * q^97 + 156568 * 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
$3$	−6.24759	−0.400783	−0.200392	−	0.979716i	$-0.564221\pi$
$3$	−6.24759	−0.400783	−0.200392	+	0.979716i	$0.564221\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−203.968	−0.839373
$10$	0	0
$11$	−89.4628	−0.222926	−0.111463	−	0.993769i	$-0.535554\pi$
$11$	−89.4628	−0.222926	−0.111463	+	0.993769i	$0.535554\pi$
$12$	0	0
$13$	459.250	0.753687	0.376843	−	0.926277i	$-0.377009\pi$
$13$	459.250	0.753687	0.376843	+	0.926277i	$0.377009\pi$
$14$	0	0
$15$	156.190	0.179236
$16$	0	0
$17$	701.108	0.588387	0.294193	−	0.955746i	$-0.404949\pi$
$17$	701.108	0.588387	0.294193	+	0.955746i	$0.404949\pi$
$18$	0	0
$19$	396.470	0.251957	0.125978	−	0.992033i	$-0.459793\pi$
$19$	396.470	0.251957	0.125978	+	0.992033i	$0.459793\pi$
$20$	0	0
$21$	306.132	0.151482
$22$	0	0
$23$	4316.18	1.70130	0.850649	−	0.525735i	$-0.176210\pi$
$23$	4316.18	1.70130	0.850649	+	0.525735i	$0.176210\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2792.47	0.737190
$28$	0	0
$29$	−1437.19	−0.317336	−0.158668	−	0.987332i	$-0.550720\pi$
$29$	−1437.19	−0.317336	−0.158668	+	0.987332i	$0.550720\pi$
$30$	0	0
$31$	−3669.99	−0.685899	−0.342950	−	0.939354i	$-0.611426\pi$
$31$	−3669.99	−0.685899	−0.342950	+	0.939354i	$0.611426\pi$
$32$	0	0
$33$	558.927	0.0893450
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−3166.84	−0.380295	−0.190148	−	0.981755i	$-0.560897\pi$
$37$	−3166.84	−0.380295	−0.190148	+	0.981755i	$0.560897\pi$
$38$	0	0
$39$	−2869.21	−0.302065
$40$	0	0
$41$	−6353.05	−0.590232	−0.295116	−	0.955461i	$-0.595358\pi$
$41$	−6353.05	−0.590232	−0.295116	+	0.955461i	$0.595358\pi$
$42$	0	0
$43$	15510.1	1.27921	0.639606	−	0.768703i	$-0.279098\pi$
$43$	15510.1	1.27921	0.639606	+	0.768703i	$0.279098\pi$
$44$	0	0
$45$	5099.19	0.375379
$46$	0	0
$47$	3995.07	0.263803	0.131902	−	0.991263i	$-0.457892\pi$
$47$	3995.07	0.263803	0.131902	+	0.991263i	$0.457892\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−4380.24	−0.235816
$52$	0	0
$53$	−24472.8	−1.19673	−0.598363	−	0.801225i	$-0.704182\pi$
$53$	−24472.8	−1.19673	−0.598363	+	0.801225i	$0.704182\pi$
$54$	0	0
$55$	2236.57	0.0996955
$56$	0	0
$57$	−2476.98	−0.100980
$58$	0	0
$59$	2395.32	0.0895848	0.0447924	−	0.998996i	$-0.485737\pi$
$59$	2395.32	0.0895848	0.0447924	+	0.998996i	$0.485737\pi$
$60$	0	0
$61$	−21052.6	−0.724404	−0.362202	−	0.932100i	$-0.617975\pi$
$61$	−21052.6	−0.724404	−0.362202	+	0.932100i	$0.617975\pi$
$62$	0	0
$63$	9994.41	0.317253
$64$	0	0
$65$	−11481.3	−0.337059
$66$	0	0
$67$	−853.274	−0.0232221	−0.0116111	−	0.999933i	$-0.503696\pi$
$67$	−853.274	−0.0232221	−0.0116111	+	0.999933i	$0.503696\pi$
$68$	0	0
$69$	−26965.7	−0.681851
$70$	0	0
$71$	1378.74	0.0324591	0.0162295	−	0.999868i	$-0.494834\pi$
$71$	1378.74	0.0324591	0.0162295	+	0.999868i	$0.494834\pi$
$72$	0	0
$73$	12248.5	0.269014	0.134507	−	0.990913i	$-0.457055\pi$
$73$	12248.5	0.269014	0.134507	+	0.990913i	$0.457055\pi$
$74$	0	0
$75$	−3904.75	−0.0801566
$76$	0	0
$77$	4383.68	0.0842581
$78$	0	0
$79$	51047.4	0.920250	0.460125	−	0.887854i	$-0.347805\pi$
$79$	51047.4	0.920250	0.460125	+	0.887854i	$0.347805\pi$
$80$	0	0
$81$	32117.9	0.543919
$82$	0	0
$83$	−17615.6	−0.280674	−0.140337	−	0.990104i	$-0.544819\pi$
$83$	−17615.6	−0.280674	−0.140337	+	0.990104i	$0.544819\pi$
$84$	0	0
$85$	−17527.7	−0.263135
$86$	0	0
$87$	8978.99	0.127183
$88$	0	0
$89$	−18346.0	−0.245508	−0.122754	−	0.992437i	$-0.539173\pi$
$89$	−18346.0	−0.245508	−0.122754	+	0.992437i	$0.539173\pi$
$90$	0	0
$91$	−22503.3	−0.284867
$92$	0	0
$93$	22928.6	0.274897
$94$	0	0
$95$	−9911.74	−0.112678
$96$	0	0
$97$	−103460.	−1.11646	−0.558231	−	0.829686i	$-0.688520\pi$
$97$	−103460.	−1.11646	−0.558231	+	0.829686i	$0.688520\pi$
$98$	0	0
$99$	18247.5	0.187118
$100$	0	0
$101$	−155074.	−1.51264	−0.756320	−	0.654202i	$-0.773005\pi$
$101$	−155074.	−1.51264	−0.756320	+	0.654202i	$0.773005\pi$
$102$	0	0
$103$	36262.5	0.336794	0.168397	−	0.985719i	$-0.446141\pi$
$103$	36262.5	0.336794	0.168397	+	0.985719i	$0.446141\pi$
$104$	0	0
$105$	−7653.30	−0.0677447
$106$	0	0
$107$	116698.	0.985378	0.492689	−	0.870205i	$-0.336014\pi$
$107$	116698.	0.985378	0.492689	+	0.870205i	$0.336014\pi$
$108$	0	0
$109$	6940.27	0.0559513	0.0279756	−	0.999609i	$-0.491094\pi$
$109$	6940.27	0.0559513	0.0279756	+	0.999609i	$0.491094\pi$
$110$	0	0
$111$	19785.1	0.152416
$112$	0	0
$113$	−244790.	−1.80342	−0.901711	−	0.432339i	$-0.857688\pi$
$113$	−244790.	−1.80342	−0.901711	+	0.432339i	$0.857688\pi$
$114$	0	0
$115$	−107905.	−0.760843
$116$	0	0
$117$	−93672.2	−0.632624
$118$	0	0
$119$	−34354.3	−0.222389
$120$	0	0
$121$	−153047.	−0.950304
$122$	0	0
$123$	39691.3	0.236555
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−173950.	−0.957008	−0.478504	−	0.878085i	$-0.658821\pi$
$127$	−173950.	−0.957008	−0.478504	+	0.878085i	$0.658821\pi$
$128$	0	0
$129$	−96900.6	−0.512687
$130$	0	0
$131$	133105.	0.677667	0.338833	−	0.940846i	$-0.389968\pi$
$131$	133105.	0.677667	0.338833	+	0.940846i	$0.389968\pi$
$132$	0	0
$133$	−19427.0	−0.0952307
$134$	0	0
$135$	−69811.8	−0.329681
$136$	0	0
$137$	154448.	0.703039	0.351520	−	0.936180i	$-0.385665\pi$
$137$	154448.	0.703039	0.351520	+	0.936180i	$0.385665\pi$
$138$	0	0
$139$	−219692.	−0.964443	−0.482221	−	0.876049i	$-0.660170\pi$
$139$	−219692.	−0.964443	−0.482221	+	0.876049i	$0.660170\pi$
$140$	0	0
$141$	−24959.6	−0.105728
$142$	0	0
$143$	−41085.8	−0.168016
$144$	0	0
$145$	35929.8	0.141917
$146$	0	0
$147$	−15000.5	−0.0572547
$148$	0	0
$149$	259928.	0.959152	0.479576	−	0.877500i	$-0.340790\pi$
$149$	259928.	0.959152	0.479576	+	0.877500i	$0.340790\pi$
$150$	0	0
$151$	−125854.	−0.449186	−0.224593	−	0.974453i	$-0.572105\pi$
$151$	−125854.	−0.449186	−0.224593	+	0.974453i	$0.572105\pi$
$152$	0	0
$153$	−143003.	−0.493876
$154$	0	0
$155$	91749.7	0.306743
$156$	0	0
$157$	−19697.3	−0.0637760	−0.0318880	−	0.999491i	$-0.510152\pi$
$157$	−19697.3	−0.0637760	−0.0318880	+	0.999491i	$0.510152\pi$
$158$	0	0
$159$	152896.	0.479628
$160$	0	0
$161$	−211493.	−0.643030
$162$	0	0
$163$	−490838.	−1.44700	−0.723501	−	0.690324i	$-0.757468\pi$
$163$	−490838.	−1.44700	−0.723501	+	0.690324i	$0.757468\pi$
$164$	0	0
$165$	−13973.2	−0.0399563
$166$	0	0
$167$	−593431.	−1.64657	−0.823283	−	0.567632i	$-0.807860\pi$
$167$	−593431.	−1.64657	−0.823283	+	0.567632i	$0.807860\pi$
$168$	0	0
$169$	−160382.	−0.431956
$170$	0	0
$171$	−80866.9	−0.211486
$172$	0	0
$173$	75225.1	0.191094	0.0955472	−	0.995425i	$-0.469540\pi$
$173$	75225.1	0.191094	0.0955472	+	0.995425i	$0.469540\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−14965.0	−0.0359041
$178$	0	0
$179$	−210903.	−0.491984	−0.245992	−	0.969272i	$-0.579114\pi$
$179$	−210903.	−0.491984	−0.245992	+	0.969272i	$0.579114\pi$
$180$	0	0
$181$	347613.	0.788677	0.394339	−	0.918965i	$-0.370974\pi$
$181$	347613.	0.788677	0.394339	+	0.918965i	$0.370974\pi$
$182$	0	0
$183$	131528.	0.290329
$184$	0	0
$185$	79170.9	0.170073
$186$	0	0
$187$	−62723.1	−0.131167
$188$	0	0
$189$	−136831.	−0.278632
$190$	0	0
$191$	−181647.	−0.360284	−0.180142	−	0.983641i	$-0.557656\pi$
$191$	−181647.	−0.360284	−0.180142	+	0.983641i	$0.557656\pi$
$192$	0	0
$193$	−468479.	−0.905309	−0.452655	−	0.891686i	$-0.649523\pi$
$193$	−468479.	−0.905309	−0.452655	+	0.891686i	$0.649523\pi$
$194$	0	0
$195$	71730.2	0.135088
$196$	0	0
$197$	301806.	0.554066	0.277033	−	0.960860i	$-0.410649\pi$
$197$	301806.	0.554066	0.277033	+	0.960860i	$0.410649\pi$
$198$	0	0
$199$	34553.3	0.0618525	0.0309262	−	0.999522i	$-0.490154\pi$
$199$	34553.3	0.0618525	0.0309262	+	0.999522i	$0.490154\pi$
$200$	0	0
$201$	5330.91	0.00930703
$202$	0	0
$203$	70422.4	0.119942
$204$	0	0
$205$	158826.	0.263960
$206$	0	0
$207$	−880361.	−1.42802
$208$	0	0
$209$	−35469.3	−0.0561677
$210$	0	0
$211$	−538344.	−0.832442	−0.416221	−	0.909263i	$-0.636646\pi$
$211$	−538344.	−0.832442	−0.416221	+	0.909263i	$0.636646\pi$
$212$	0	0
$213$	−8613.79	−0.0130090
$214$	0	0
$215$	−387752.	−0.572081
$216$	0	0
$217$	179829.	0.259245
$218$	0	0
$219$	−76523.5	−0.107816
$220$	0	0
$221$	321984.	0.443459
$222$	0	0
$223$	−243135.	−0.327404	−0.163702	−	0.986510i	$-0.552344\pi$
$223$	−243135.	−0.327404	−0.163702	+	0.986510i	$0.552344\pi$
$224$	0	0
$225$	−127480.	−0.167875
$226$	0	0
$227$	−341415.	−0.439762	−0.219881	−	0.975527i	$-0.570567\pi$
$227$	−341415.	−0.439762	−0.219881	+	0.975527i	$0.570567\pi$
$228$	0	0
$229$	−113641.	−0.143201	−0.0716004	−	0.997433i	$-0.522811\pi$
$229$	−113641.	−0.143201	−0.0716004	+	0.997433i	$0.522811\pi$
$230$	0	0
$231$	−27387.4	−0.0337692
$232$	0	0
$233$	−1.10838e6	−1.33751	−0.668756	−	0.743482i	$-0.733173\pi$
$233$	−1.10838e6	−1.33751	−0.668756	+	0.743482i	$0.733173\pi$
$234$	0	0
$235$	−99876.9	−0.117976
$236$	0	0
$237$	−318923.	−0.368821
$238$	0	0
$239$	−274846.	−0.311240	−0.155620	−	0.987817i	$-0.549737\pi$
$239$	−274846.	−0.311240	−0.155620	+	0.987817i	$0.549737\pi$
$240$	0	0
$241$	1.32322e6	1.46754	0.733770	−	0.679397i	$-0.237759\pi$
$241$	1.32322e6	1.46754	0.733770	+	0.679397i	$0.237759\pi$
$242$	0	0
$243$	−879230.	−0.955184
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	182079.	0.189896
$248$	0	0
$249$	110055.	0.112489
$250$	0	0
$251$	1.56166e6	1.56460	0.782298	−	0.622905i	$-0.214047\pi$
$251$	1.56166e6	1.56460	0.782298	+	0.622905i	$0.214047\pi$
$252$	0	0
$253$	−386138.	−0.379263
$254$	0	0
$255$	109506.	0.105460
$256$	0	0
$257$	1.19597e6	1.12950	0.564750	−	0.825262i	$-0.308972\pi$
$257$	1.19597e6	1.12950	0.564750	+	0.825262i	$0.308972\pi$
$258$	0	0
$259$	155175.	0.143738
$260$	0	0
$261$	293140.	0.266363
$262$	0	0
$263$	−1.54763e6	−1.37968	−0.689839	−	0.723962i	$-0.742319\pi$
$263$	−1.54763e6	−1.37968	−0.689839	+	0.723962i	$0.742319\pi$
$264$	0	0
$265$	611821.	0.535192
$266$	0	0
$267$	114618.	0.0983956
$268$	0	0
$269$	−1.04788e6	−0.882941	−0.441470	−	0.897276i	$-0.645543\pi$
$269$	−1.04788e6	−0.882941	−0.441470	+	0.897276i	$0.645543\pi$
$270$	0	0
$271$	−842735.	−0.697056	−0.348528	−	0.937298i	$-0.613318\pi$
$271$	−842735.	−0.697056	−0.348528	+	0.937298i	$0.613318\pi$
$272$	0	0
$273$	140591.	0.114170
$274$	0	0
$275$	−55914.2	−0.0445852
$276$	0	0
$277$	660555.	0.517261	0.258630	−	0.965976i	$-0.416729\pi$
$277$	660555.	0.517261	0.258630	+	0.965976i	$0.416729\pi$
$278$	0	0
$279$	748558.	0.575725
$280$	0	0
$281$	−1.46200e6	−1.10454	−0.552272	−	0.833664i	$-0.686239\pi$
$281$	−1.46200e6	−1.10454	−0.552272	+	0.833664i	$0.686239\pi$
$282$	0	0
$283$	1.68634e6	1.25164	0.625819	−	0.779968i	$-0.284765\pi$
$283$	1.68634e6	1.25164	0.625819	+	0.779968i	$0.284765\pi$
$284$	0	0
$285$	61924.5	0.0451596
$286$	0	0
$287$	311299.	0.223087
$288$	0	0
$289$	−928304.	−0.653801
$290$	0	0
$291$	646377.	0.447459
$292$	0	0
$293$	1.03667e6	0.705458	0.352729	−	0.935725i	$-0.385254\pi$
$293$	1.03667e6	0.705458	0.352729	+	0.935725i	$0.385254\pi$
$294$	0	0
$295$	−59883.1	−0.0400635
$296$	0	0
$297$	−249822.	−0.164339
$298$	0	0
$299$	1.98221e6	1.28225
$300$	0	0
$301$	−759993.	−0.483497
$302$	0	0
$303$	968839.	0.606241
$304$	0	0
$305$	526315.	0.323963
$306$	0	0
$307$	−1.33764e6	−0.810013	−0.405006	−	0.914314i	$-0.632731\pi$
$307$	−1.33764e6	−0.810013	−0.405006	+	0.914314i	$0.632731\pi$
$308$	0	0
$309$	−226553.	−0.134981
$310$	0	0
$311$	−443660.	−0.260105	−0.130053	−	0.991507i	$-0.541515\pi$
$311$	−443660.	−0.260105	−0.130053	+	0.991507i	$0.541515\pi$
$312$	0	0
$313$	242889.	0.140135	0.0700677	−	0.997542i	$-0.477678\pi$
$313$	242889.	0.140135	0.0700677	+	0.997542i	$0.477678\pi$
$314$	0	0
$315$	−249860.	−0.141880
$316$	0	0
$317$	546897.	0.305673	0.152837	−	0.988251i	$-0.451159\pi$
$317$	546897.	0.305673	0.152837	+	0.988251i	$0.451159\pi$
$318$	0	0
$319$	128575.	0.0707425
$320$	0	0
$321$	−729080.	−0.394923
$322$	0	0
$323$	277968.	0.148248
$324$	0	0
$325$	287031.	0.150737
$326$	0	0
$327$	−43360.0	−0.0224243
$328$	0	0
$329$	−195759.	−0.0997083
$330$	0	0
$331$	−518092.	−0.259918	−0.129959	−	0.991519i	$-0.541485\pi$
$331$	−518092.	−0.259918	−0.129959	+	0.991519i	$0.541485\pi$
$332$	0	0
$333$	645932.	0.319210
$334$	0	0
$335$	21331.9	0.0103852
$336$	0	0
$337$	1.93329e6	0.927303	0.463651	−	0.886018i	$-0.346539\pi$
$337$	1.93329e6	0.927303	0.463651	+	0.886018i	$0.346539\pi$
$338$	0	0
$339$	1.52935e6	0.722782
$340$	0	0
$341$	328327.	0.152905
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	674144.	0.304933
$346$	0	0
$347$	−2.20916e6	−0.984925	−0.492463	−	0.870334i	$-0.663903\pi$
$347$	−2.20916e6	−0.984925	−0.492463	+	0.870334i	$0.663903\pi$
$348$	0	0
$349$	−2.05917e6	−0.904957	−0.452479	−	0.891775i	$-0.649460\pi$
$349$	−2.05917e6	−0.904957	−0.452479	+	0.891775i	$0.649460\pi$
$350$	0	0
$351$	1.28244e6	0.555610
$352$	0	0
$353$	−3.62255e6	−1.54731	−0.773655	−	0.633607i	$-0.781574\pi$
$353$	−3.62255e6	−1.54731	−0.773655	+	0.633607i	$0.781574\pi$
$354$	0	0
$355$	−34468.4	−0.0145161
$356$	0	0
$357$	214632.	0.0891299
$358$	0	0
$359$	3.42481e6	1.40249	0.701247	−	0.712919i	$-0.252627\pi$
$359$	3.42481e6	1.40249	0.701247	+	0.712919i	$0.252627\pi$
$360$	0	0
$361$	−2.31891e6	−0.936518
$362$	0	0
$363$	956178.	0.380866
$364$	0	0
$365$	−306212.	−0.120307
$366$	0	0
$367$	2.57705e6	0.998751	0.499375	−	0.866386i	$-0.333563\pi$
$367$	2.57705e6	0.998751	0.499375	+	0.866386i	$0.333563\pi$
$368$	0	0
$369$	1.29582e6	0.495424
$370$	0	0
$371$	1.19917e6	0.452320
$372$	0	0
$373$	−647873.	−0.241111	−0.120556	−	0.992707i	$-0.538468\pi$
$373$	−647873.	−0.241111	−0.120556	+	0.992707i	$0.538468\pi$
$374$	0	0
$375$	97618.6	0.0358471
$376$	0	0
$377$	−660030.	−0.239172
$378$	0	0
$379$	3.07292e6	1.09889	0.549443	−	0.835531i	$-0.314840\pi$
$379$	3.07292e6	1.09889	0.549443	+	0.835531i	$0.314840\pi$
$380$	0	0
$381$	1.08677e6	0.383553
$382$	0	0
$383$	1.50940e6	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$383$	1.50940e6	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$384$	0	0
$385$	−109592.	−0.0376814
$386$	0	0
$387$	−3.16355e6	−1.07374
$388$	0	0
$389$	1.06421e6	0.356576	0.178288	−	0.983978i	$-0.442944\pi$
$389$	1.06421e6	0.356576	0.178288	+	0.983978i	$0.442944\pi$
$390$	0	0
$391$	3.02611e6	1.00102
$392$	0	0
$393$	−831586.	−0.271597
$394$	0	0
$395$	−1.27619e6	−0.411548
$396$	0	0
$397$	4.99307e6	1.58998	0.794990	−	0.606622i	$-0.207476\pi$
$397$	4.99307e6	1.58998	0.794990	+	0.606622i	$0.207476\pi$
$398$	0	0
$399$	121372.	0.0381669
$400$	0	0
$401$	2.28925e6	0.710939	0.355470	−	0.934688i	$-0.384321\pi$
$401$	2.28925e6	0.710939	0.355470	+	0.934688i	$0.384321\pi$
$402$	0	0
$403$	−1.68544e6	−0.516953
$404$	0	0
$405$	−802948.	−0.243248
$406$	0	0
$407$	283314.	0.0847777
$408$	0	0
$409$	−2.34946e6	−0.694479	−0.347239	−	0.937777i	$-0.612881\pi$
$409$	−2.34946e6	−0.694479	−0.347239	+	0.937777i	$0.612881\pi$
$410$	0	0
$411$	−964925.	−0.281766
$412$	0	0
$413$	−117371.	−0.0338599
$414$	0	0
$415$	440390.	0.125521
$416$	0	0
$417$	1.37254e6	0.386533
$418$	0	0
$419$	6.91433e6	1.92405	0.962023	−	0.272969i	$-0.0880058\pi$
$419$	6.91433e6	1.92405	0.962023	+	0.272969i	$0.0880058\pi$
$420$	0	0
$421$	−910368.	−0.250329	−0.125165	−	0.992136i	$-0.539946\pi$
$421$	−910368.	−0.250329	−0.125165	+	0.992136i	$0.539946\pi$
$422$	0	0
$423$	−814866.	−0.221429
$424$	0	0
$425$	438193.	0.117677
$426$	0	0
$427$	1.03158e6	0.273799
$428$	0	0
$429$	256687.	0.0673381
$430$	0	0
$431$	−2.71383e6	−0.703704	−0.351852	−	0.936056i	$-0.614448\pi$
$431$	−2.71383e6	−0.703704	−0.351852	+	0.936056i	$0.614448\pi$
$432$	0	0
$433$	−5.02281e6	−1.28744	−0.643720	−	0.765261i	$-0.722610\pi$
$433$	−5.02281e6	−1.28744	−0.643720	+	0.765261i	$0.722610\pi$
$434$	0	0
$435$	−224475.	−0.0568780
$436$	0	0
$437$	1.71124e6	0.428653
$438$	0	0
$439$	1.26405e6	0.313041	0.156521	−	0.987675i	$-0.449972\pi$
$439$	1.26405e6	0.313041	0.156521	+	0.987675i	$0.449972\pi$
$440$	0	0
$441$	−489726.	−0.119910
$442$	0	0
$443$	−1.63750e6	−0.396435	−0.198217	−	0.980158i	$-0.563515\pi$
$443$	−1.63750e6	−0.396435	−0.198217	+	0.980158i	$0.563515\pi$
$444$	0	0
$445$	458650.	0.109795
$446$	0	0
$447$	−1.62392e6	−0.384412
$448$	0	0
$449$	−4.63732e6	−1.08555	−0.542777	−	0.839877i	$-0.682627\pi$
$449$	−4.63732e6	−1.08555	−0.542777	+	0.839877i	$0.682627\pi$
$450$	0	0
$451$	568361.	0.131578
$452$	0	0
$453$	786287.	0.180026
$454$	0	0
$455$	562581.	0.127396
$456$	0	0
$457$	5.25976e6	1.17808	0.589040	−	0.808104i	$-0.299506\pi$
$457$	5.25976e6	1.17808	0.589040	+	0.808104i	$0.299506\pi$
$458$	0	0
$459$	1.95782e6	0.433753
$460$	0	0
$461$	3.52812e6	0.773199	0.386599	−	0.922248i	$-0.373650\pi$
$461$	3.52812e6	0.773199	0.386599	+	0.922248i	$0.373650\pi$
$462$	0	0
$463$	−402711.	−0.0873054	−0.0436527	−	0.999047i	$-0.513899\pi$
$463$	−402711.	−0.0873054	−0.0436527	+	0.999047i	$0.513899\pi$
$464$	0	0
$465$	−573214.	−0.122938
$466$	0	0
$467$	4.65025e6	0.986697	0.493348	−	0.869832i	$-0.335773\pi$
$467$	4.65025e6	0.986697	0.493348	+	0.869832i	$0.335773\pi$
$468$	0	0
$469$	41810.4	0.00877713
$470$	0	0
$471$	123061.	0.0255604
$472$	0	0
$473$	−1.38757e6	−0.285170
$474$	0	0
$475$	247793.	0.0503913
$476$	0	0
$477$	4.99167e6	1.00450
$478$	0	0
$479$	7.54178e6	1.50188	0.750940	−	0.660371i	$-0.229601\pi$
$479$	7.54178e6	1.50188	0.750940	+	0.660371i	$0.229601\pi$
$480$	0	0
$481$	−1.45437e6	−0.286624
$482$	0	0
$483$	1.32132e6	0.257716
$484$	0	0
$485$	2.58650e6	0.499297
$486$	0	0
$487$	−8.68725e6	−1.65982	−0.829908	−	0.557900i	$-0.811607\pi$
$487$	−8.68725e6	−1.65982	−0.829908	+	0.557900i	$0.811607\pi$
$488$	0	0
$489$	3.06655e6	0.579934
$490$	0	0
$491$	4.24852e6	0.795306	0.397653	−	0.917536i	$-0.369825\pi$
$491$	4.24852e6	0.795306	0.397653	+	0.917536i	$0.369825\pi$
$492$	0	0
$493$	−1.00763e6	−0.186716
$494$	0	0
$495$	−456188.	−0.0836817
$496$	0	0
$497$	−67558.2	−0.0122684
$498$	0	0
$499$	−5.49814e6	−0.988473	−0.494236	−	0.869328i	$-0.664552\pi$
$499$	−5.49814e6	−0.988473	−0.494236	+	0.869328i	$0.664552\pi$
$500$	0	0
$501$	3.70751e6	0.659916
$502$	0	0
$503$	9.46196e6	1.66748	0.833741	−	0.552156i	$-0.186195\pi$
$503$	9.46196e6	1.66748	0.833741	+	0.552156i	$0.186195\pi$
$504$	0	0
$505$	3.87685e6	0.676473
$506$	0	0
$507$	1.00200e6	0.173121
$508$	0	0
$509$	5.87612e6	1.00530	0.502651	−	0.864489i	$-0.332358\pi$
$509$	5.87612e6	1.00530	0.502651	+	0.864489i	$0.332358\pi$
$510$	0	0
$511$	−600175.	−0.101678
$512$	0	0
$513$	1.10713e6	0.185740
$514$	0	0
$515$	−906562.	−0.150619
$516$	0	0
$517$	−357410.	−0.0588086
$518$	0	0
$519$	−469976.	−0.0765874
$520$	0	0
$521$	−1.08233e7	−1.74689	−0.873443	−	0.486926i	$-0.838118\pi$
$521$	−1.08233e7	−1.74689	−0.873443	+	0.486926i	$0.838118\pi$
$522$	0	0
$523$	−3.75209e6	−0.599818	−0.299909	−	0.953968i	$-0.596956\pi$
$523$	−3.75209e6	−0.599818	−0.299909	+	0.953968i	$0.596956\pi$
$524$	0	0
$525$	191333.	0.0302964
$526$	0	0
$527$	−2.57306e6	−0.403574
$528$	0	0
$529$	1.21931e7	1.89441
$530$	0	0
$531$	−488568.	−0.0751950
$532$	0	0
$533$	−2.91764e6	−0.444850
$534$	0	0
$535$	−2.91744e6	−0.440674
$536$	0	0
$537$	1.31764e6	0.197179
$538$	0	0
$539$	−214800.	−0.0318466
$540$	0	0
$541$	5.78772e6	0.850187	0.425093	−	0.905150i	$-0.360241\pi$
$541$	5.78772e6	0.850187	0.425093	+	0.905150i	$0.360241\pi$
$542$	0	0
$543$	−2.17174e6	−0.316089
$544$	0	0
$545$	−173507.	−0.0250222
$546$	0	0
$547$	−8.05012e6	−1.15036	−0.575180	−	0.818027i	$-0.695068\pi$
$547$	−8.05012e6	−1.15036	−0.575180	+	0.818027i	$0.695068\pi$
$548$	0	0
$549$	4.29405e6	0.608045
$550$	0	0
$551$	−569803.	−0.0799550
$552$	0	0
$553$	−2.50132e6	−0.347822
$554$	0	0
$555$	−494627.	−0.0681625
$556$	0	0
$557$	1.13985e7	1.55671	0.778356	−	0.627824i	$-0.216054\pi$
$557$	1.13985e7	1.55671	0.778356	+	0.627824i	$0.216054\pi$
$558$	0	0
$559$	7.12300e6	0.964125
$560$	0	0
$561$	391868.	0.0525694
$562$	0	0
$563$	−9.73011e6	−1.29374	−0.646870	−	0.762601i	$-0.723922\pi$
$563$	−9.73011e6	−1.29374	−0.646870	+	0.762601i	$0.723922\pi$
$564$	0	0
$565$	6.11975e6	0.806515
$566$	0	0
$567$	−1.57378e6	−0.205582
$568$	0	0
$569$	−5.09214e6	−0.659355	−0.329678	−	0.944094i	$-0.606940\pi$
$569$	−5.09214e6	−0.659355	−0.329678	+	0.944094i	$0.606940\pi$
$570$	0	0
$571$	−1.25025e7	−1.60475	−0.802376	−	0.596818i	$-0.796431\pi$
$571$	−1.25025e7	−1.60475	−0.802376	+	0.596818i	$0.796431\pi$
$572$	0	0
$573$	1.13486e6	0.144396
$574$	0	0
$575$	2.69761e6	0.340259
$576$	0	0
$577$	−1.02876e7	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$577$	−1.02876e7	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$578$	0	0
$579$	2.92687e6	0.362833
$580$	0	0
$581$	863164.	0.106085
$582$	0	0
$583$	2.18941e6	0.266781
$584$	0	0
$585$	2.34180e6	0.282918
$586$	0	0
$587$	−411635.	−0.0493079	−0.0246540	−	0.999696i	$-0.507848\pi$
$587$	−411635.	−0.0493079	−0.0246540	+	0.999696i	$0.507848\pi$
$588$	0	0
$589$	−1.45504e6	−0.172817
$590$	0	0
$591$	−1.88556e6	−0.222061
$592$	0	0
$593$	1.85606e6	0.216748	0.108374	−	0.994110i	$-0.465436\pi$
$593$	1.85606e6	0.216748	0.108374	+	0.994110i	$0.465436\pi$
$594$	0	0
$595$	858858.	0.0994555
$596$	0	0
$597$	−215875.	−0.0247894
$598$	0	0
$599$	7.01077e6	0.798359	0.399180	−	0.916873i	$-0.369295\pi$
$599$	7.01077e6	0.798359	0.399180	+	0.916873i	$0.369295\pi$
$600$	0	0
$601$	−8.82536e6	−0.996658	−0.498329	−	0.866988i	$-0.666053\pi$
$601$	−8.82536e6	−0.996658	−0.498329	+	0.866988i	$0.666053\pi$
$602$	0	0
$603$	174040.	0.0194920
$604$	0	0
$605$	3.82619e6	0.424989
$606$	0	0
$607$	−1.49962e7	−1.65200	−0.826000	−	0.563671i	$-0.809389\pi$
$607$	−1.49962e7	−1.65200	−0.826000	+	0.563671i	$0.809389\pi$
$608$	0	0
$609$	−439970.	−0.0480707
$610$	0	0
$611$	1.83474e6	0.198825
$612$	0	0
$613$	1.68954e7	1.81601	0.908004	−	0.418961i	$-0.137606\pi$
$613$	1.68954e7	1.81601	0.908004	+	0.418961i	$0.137606\pi$
$614$	0	0
$615$	−992281.	−0.105791
$616$	0	0
$617$	−1.61632e7	−1.70928	−0.854642	−	0.519218i	$-0.826223\pi$
$617$	−1.61632e7	−1.70928	−0.854642	+	0.519218i	$0.826223\pi$
$618$	0	0
$619$	3.57039e6	0.374532	0.187266	−	0.982309i	$-0.440037\pi$
$619$	3.57039e6	0.374532	0.187266	+	0.982309i	$0.440037\pi$
$620$	0	0
$621$	1.20528e7	1.25418
$622$	0	0
$623$	898954.	0.0927934
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	221598.	0.0225111
$628$	0	0
$629$	−2.22029e6	−0.223761
$630$	0	0
$631$	−5.52777e6	−0.552684	−0.276342	−	0.961059i	$-0.589122\pi$
$631$	−5.52777e6	−0.552684	−0.276342	+	0.961059i	$0.589122\pi$
$632$	0	0
$633$	3.36336e6	0.333629
$634$	0	0
$635$	4.34876e6	0.427987
$636$	0	0
$637$	1.10266e6	0.107670
$638$	0	0
$639$	−281218.	−0.0272452
$640$	0	0
$641$	−4.42446e6	−0.425320	−0.212660	−	0.977126i	$-0.568213\pi$
$641$	−4.42446e6	−0.425320	−0.212660	+	0.977126i	$0.568213\pi$
$642$	0	0
$643$	3.13757e6	0.299272	0.149636	−	0.988741i	$-0.452190\pi$
$643$	3.13757e6	0.299272	0.149636	+	0.988741i	$0.452190\pi$
$644$	0	0
$645$	2.42251e6	0.229280
$646$	0	0
$647$	−9.66818e6	−0.907996	−0.453998	−	0.891003i	$-0.650003\pi$
$647$	−9.66818e6	−0.907996	−0.453998	+	0.891003i	$0.650003\pi$
$648$	0	0
$649$	−214292.	−0.0199708
$650$	0	0
$651$	−1.12350e6	−0.103901
$652$	0	0
$653$	−1.38904e7	−1.27477	−0.637386	−	0.770545i	$-0.719984\pi$
$653$	−1.38904e7	−1.27477	−0.637386	+	0.770545i	$0.719984\pi$
$654$	0	0
$655$	−3.32762e6	−0.303062
$656$	0	0
$657$	−2.49829e6	−0.225803
$658$	0	0
$659$	−6.78274e6	−0.608403	−0.304202	−	0.952608i	$-0.598390\pi$
$659$	−6.78274e6	−0.608403	−0.304202	+	0.952608i	$0.598390\pi$
$660$	0	0
$661$	−1.96423e7	−1.74860	−0.874298	−	0.485390i	$-0.838677\pi$
$661$	−1.96423e7	−1.74860	−0.874298	+	0.485390i	$0.838677\pi$
$662$	0	0
$663$	−2.01163e6	−0.177731
$664$	0	0
$665$	485675.	0.0425885
$666$	0	0
$667$	−6.20318e6	−0.539883
$668$	0	0
$669$	1.51901e6	0.131218
$670$	0	0
$671$	1.88342e6	0.161489
$672$	0	0
$673$	−2.14538e7	−1.82586	−0.912930	−	0.408116i	$-0.866186\pi$
$673$	−2.14538e7	−1.82586	−0.912930	+	0.408116i	$0.866186\pi$
$674$	0	0
$675$	1.74529e6	0.147438
$676$	0	0
$677$	6.25982e6	0.524917	0.262458	−	0.964943i	$-0.415467\pi$
$677$	6.25982e6	0.524917	0.262458	+	0.964943i	$0.415467\pi$
$678$	0	0
$679$	5.06955e6	0.421983
$680$	0	0
$681$	2.13302e6	0.176249
$682$	0	0
$683$	−6.24086e6	−0.511909	−0.255954	−	0.966689i	$-0.582390\pi$
$683$	−6.24086e6	−0.511909	−0.255954	+	0.966689i	$0.582390\pi$
$684$	0	0
$685$	−3.86119e6	−0.314409
$686$	0	0
$687$	709982.	0.0573925
$688$	0	0
$689$	−1.12392e7	−0.901957
$690$	0	0
$691$	1.94232e6	0.154748	0.0773740	−	0.997002i	$-0.475346\pi$
$691$	1.94232e6	0.154748	0.0773740	+	0.997002i	$0.475346\pi$
$692$	0	0
$693$	−894128.	−0.0707240
$694$	0	0
$695$	5.49229e6	0.431312
$696$	0	0
$697$	−4.45418e6	−0.347284
$698$	0	0
$699$	6.92469e6	0.536052
$700$	0	0
$701$	−1.06388e7	−0.817707	−0.408854	−	0.912600i	$-0.634071\pi$
$701$	−1.06388e7	−0.817707	−0.408854	+	0.912600i	$0.634071\pi$
$702$	0	0
$703$	−1.25555e6	−0.0958180
$704$	0	0
$705$	623990.	0.0472830
$706$	0	0
$707$	7.59863e6	0.571724
$708$	0	0
$709$	1.81481e7	1.35586	0.677930	−	0.735126i	$-0.262877\pi$
$709$	1.81481e7	1.35586	0.677930	+	0.735126i	$0.262877\pi$
$710$	0	0
$711$	−1.04120e7	−0.772433
$712$	0	0
$713$	−1.58403e7	−1.16692
$714$	0	0
$715$	1.02714e6	0.0751392
$716$	0	0
$717$	1.71713e6	0.124740
$718$	0	0
$719$	−9.37540e6	−0.676344	−0.338172	−	0.941084i	$-0.609809\pi$
$719$	−9.37540e6	−0.676344	−0.338172	+	0.941084i	$0.609809\pi$
$720$	0	0
$721$	−1.77686e6	−0.127296
$722$	0	0
$723$	−8.26695e6	−0.588166
$724$	0	0
$725$	−898245.	−0.0634672
$726$	0	0
$727$	5.67290e6	0.398079	0.199040	−	0.979991i	$-0.436218\pi$
$727$	5.67290e6	0.398079	0.199040	+	0.979991i	$0.436218\pi$
$728$	0	0
$729$	−2.31158e6	−0.161098
$730$	0	0
$731$	1.08742e7	0.752671
$732$	0	0
$733$	−1.16816e7	−0.803051	−0.401526	−	0.915848i	$-0.631520\pi$
$733$	−1.16816e7	−0.803051	−0.401526	+	0.915848i	$0.631520\pi$
$734$	0	0
$735$	375012.	0.0256051
$736$	0	0
$737$	76336.3	0.00517681
$738$	0	0
$739$	−9.25981e6	−0.623722	−0.311861	−	0.950128i	$-0.600952\pi$
$739$	−9.25981e6	−0.623722	−0.311861	+	0.950128i	$0.600952\pi$
$740$	0	0
$741$	−1.13755e6	−0.0761073
$742$	0	0
$743$	−1.02603e7	−0.681850	−0.340925	−	0.940090i	$-0.610740\pi$
$743$	−1.02603e7	−0.681850	−0.340925	+	0.940090i	$0.610740\pi$
$744$	0	0
$745$	−6.49820e6	−0.428946
$746$	0	0
$747$	3.59301e6	0.235590
$748$	0	0
$749$	−5.71819e6	−0.372438
$750$	0	0
$751$	−2.31032e6	−0.149476	−0.0747381	−	0.997203i	$-0.523812\pi$
$751$	−2.31032e6	−0.149476	−0.0747381	+	0.997203i	$0.523812\pi$
$752$	0	0
$753$	−9.75661e6	−0.627064
$754$	0	0
$755$	3.14636e6	0.200882
$756$	0	0
$757$	−1.33223e7	−0.844967	−0.422483	−	0.906371i	$-0.638842\pi$
$757$	−1.33223e7	−0.844967	−0.422483	+	0.906371i	$0.638842\pi$
$758$	0	0
$759$	2.41243e6	0.152002
$760$	0	0
$761$	−1.32578e7	−0.829868	−0.414934	−	0.909852i	$-0.636195\pi$
$761$	−1.32578e7	−0.829868	−0.414934	+	0.909852i	$0.636195\pi$
$762$	0	0
$763$	−340073.	−0.0211476
$764$	0	0
$765$	3.57508e6	0.220868
$766$	0	0
$767$	1.10005e6	0.0675189
$768$	0	0
$769$	1.26945e7	0.774103	0.387051	−	0.922058i	$-0.373494\pi$
$769$	1.26945e7	0.774103	0.387051	+	0.922058i	$0.373494\pi$
$770$	0	0
$771$	−7.47192e6	−0.452685
$772$	0	0
$773$	−2.79823e7	−1.68436	−0.842179	−	0.539198i	$-0.818727\pi$
$773$	−2.79823e7	−1.68436	−0.842179	+	0.539198i	$0.818727\pi$
$774$	0	0
$775$	−2.29374e6	−0.137180
$776$	0	0
$777$	−969470.	−0.0576079
$778$	0	0
$779$	−2.51879e6	−0.148713
$780$	0	0
$781$	−123346.	−0.00723597
$782$	0	0
$783$	−4.01332e6	−0.233937
$784$	0	0
$785$	492432.	0.0285215
$786$	0	0
$787$	−2.95888e6	−0.170290	−0.0851452	−	0.996369i	$-0.527135\pi$
$787$	−2.95888e6	−0.170290	−0.0851452	+	0.996369i	$0.527135\pi$
$788$	0	0
$789$	9.66897e6	0.552952
$790$	0	0
$791$	1.19947e7	0.681630
$792$	0	0
$793$	−9.66841e6	−0.545974
$794$	0	0
$795$	−3.82241e6	−0.214496
$796$	0	0
$797$	−2.52840e6	−0.140994	−0.0704969	−	0.997512i	$-0.522458\pi$
$797$	−2.52840e6	−0.140994	−0.0704969	+	0.997512i	$0.522458\pi$
$798$	0	0
$799$	2.80098e6	0.155218
$800$	0	0
$801$	3.74199e6	0.206073
$802$	0	0
$803$	−1.09578e6	−0.0599702
$804$	0	0
$805$	5.28732e6	0.287572
$806$	0	0
$807$	6.54674e6	0.353868
$808$	0	0
$809$	−1.96696e6	−0.105663	−0.0528316	−	0.998603i	$-0.516825\pi$
$809$	−1.96696e6	−0.105663	−0.0528316	+	0.998603i	$0.516825\pi$
$810$	0	0
$811$	−2.25136e7	−1.20197	−0.600984	−	0.799261i	$-0.705224\pi$
$811$	−2.25136e7	−1.20197	−0.600984	+	0.799261i	$0.705224\pi$
$812$	0	0
$813$	5.26506e6	0.279368
$814$	0	0
$815$	1.22709e7	0.647119
$816$	0	0
$817$	6.14927e6	0.322306
$818$	0	0
$819$	4.58994e6	0.239110
$820$	0	0
$821$	5.29690e6	0.274261	0.137131	−	0.990553i	$-0.456212\pi$
$821$	5.29690e6	0.274261	0.137131	+	0.990553i	$0.456212\pi$
$822$	0	0
$823$	−2.52150e7	−1.29765	−0.648827	−	0.760936i	$-0.724740\pi$
$823$	−2.52150e7	−1.29765	−0.648827	+	0.760936i	$0.724740\pi$
$824$	0	0
$825$	349329.	0.0178690
$826$	0	0
$827$	1.85137e7	0.941305	0.470652	−	0.882319i	$-0.344019\pi$
$827$	1.85137e7	0.941305	0.470652	+	0.882319i	$0.344019\pi$
$828$	0	0
$829$	1.58266e7	0.799837	0.399918	−	0.916551i	$-0.369038\pi$
$829$	1.58266e7	0.799837	0.399918	+	0.916551i	$0.369038\pi$
$830$	0	0
$831$	−4.12688e6	−0.207309
$832$	0	0
$833$	1.68336e6	0.0840552
$834$	0	0
$835$	1.48358e7	0.736366
$836$	0	0
$837$	−1.02483e7	−0.505638
$838$	0	0
$839$	3.93063e7	1.92778	0.963890	−	0.266300i	$-0.0858011\pi$
$839$	3.93063e7	1.92778	0.963890	+	0.266300i	$0.0858011\pi$
$840$	0	0
$841$	−1.84456e7	−0.899298
$842$	0	0
$843$	9.13401e6	0.442683
$844$	0	0
$845$	4.00956e6	0.193177
$846$	0	0
$847$	7.49932e6	0.359181
$848$	0	0
$849$	−1.05356e7	−0.501635
$850$	0	0
$851$	−1.36686e7	−0.646996
$852$	0	0
$853$	2.96226e7	1.39396	0.696979	−	0.717091i	$-0.254527\pi$
$853$	2.96226e7	1.39396	0.696979	+	0.717091i	$0.254527\pi$
$854$	0	0
$855$	2.02167e6	0.0945792
$856$	0	0
$857$	−7.86894e6	−0.365986	−0.182993	−	0.983114i	$-0.558579\pi$
$857$	−7.86894e6	−0.365986	−0.182993	+	0.983114i	$0.558579\pi$
$858$	0	0
$859$	1.38750e7	0.641581	0.320790	−	0.947150i	$-0.396052\pi$
$859$	1.38750e7	0.641581	0.320790	+	0.947150i	$0.396052\pi$
$860$	0	0
$861$	−1.94487e6	−0.0894094
$862$	0	0
$863$	8.53024e6	0.389883	0.194941	−	0.980815i	$-0.437548\pi$
$863$	8.53024e6	0.389883	0.194941	+	0.980815i	$0.437548\pi$
$864$	0	0
$865$	−1.88063e6	−0.0854600
$866$	0	0
$867$	5.79967e6	0.262033
$868$	0	0
$869$	−4.56684e6	−0.205148
$870$	0	0
$871$	−391866.	−0.0175022
$872$	0	0
$873$	2.11025e7	0.937128
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	4.02539e6	0.176729	0.0883647	−	0.996088i	$-0.471836\pi$
$877$	4.02539e6	0.176729	0.0883647	+	0.996088i	$0.471836\pi$
$878$	0	0
$879$	−6.47669e6	−0.282736
$880$	0	0
$881$	−1.20511e7	−0.523102	−0.261551	−	0.965190i	$-0.584234\pi$
$881$	−1.20511e7	−0.523102	−0.261551	+	0.965190i	$0.584234\pi$
$882$	0	0
$883$	−2.00355e7	−0.864765	−0.432383	−	0.901690i	$-0.642327\pi$
$883$	−2.00355e7	−0.864765	−0.432383	+	0.901690i	$0.642327\pi$
$884$	0	0
$885$	374125.	0.0160568
$886$	0	0
$887$	2.87782e7	1.22816	0.614079	−	0.789244i	$-0.289527\pi$
$887$	2.87782e7	1.22816	0.614079	+	0.789244i	$0.289527\pi$
$888$	0	0
$889$	8.52356e6	0.361715
$890$	0	0
$891$	−2.87336e6	−0.121254
$892$	0	0
$893$	1.58393e6	0.0664670
$894$	0	0
$895$	5.27259e6	0.220022
$896$	0	0
$897$	−1.23840e7	−0.513902
$898$	0	0
$899$	5.27447e6	0.217661
$900$	0	0
$901$	−1.71581e7	−0.704138
$902$	0	0
$903$	4.74813e6	0.193777
$904$	0	0
$905$	−8.69032e6	−0.352707
$906$	0	0
$907$	−1.23149e7	−0.497066	−0.248533	−	0.968623i	$-0.579948\pi$
$907$	−1.23149e7	−0.497066	−0.248533	+	0.968623i	$0.579948\pi$
$908$	0	0
$909$	3.16301e7	1.26967
$910$	0	0
$911$	−2.38428e7	−0.951836	−0.475918	−	0.879490i	$-0.657884\pi$
$911$	−2.38428e7	−0.951836	−0.475918	+	0.879490i	$0.657884\pi$
$912$	0	0
$913$	1.57594e6	0.0625695
$914$	0	0
$915$	−3.28820e6	−0.129839
$916$	0	0
$917$	−6.52214e6	−0.256134
$918$	0	0
$919$	6.18286e6	0.241491	0.120745	−	0.992684i	$-0.461472\pi$
$919$	6.18286e6	0.241491	0.120745	+	0.992684i	$0.461472\pi$
$920$	0	0
$921$	8.35700e6	0.324640
$922$	0	0
$923$	633186.	0.0244640
$924$	0	0
$925$	−1.97927e6	−0.0760591
$926$	0	0
$927$	−7.39637e6	−0.282696
$928$	0	0
$929$	2.03934e7	0.775267	0.387633	−	0.921814i	$-0.373293\pi$
$929$	2.03934e7	0.775267	0.387633	+	0.921814i	$0.373293\pi$
$930$	0	0
$931$	951924.	0.0359938
$932$	0	0
$933$	2.77181e6	0.104246
$934$	0	0
$935$	1.56808e6	0.0586595
$936$	0	0
$937$	−3.22222e7	−1.19896	−0.599481	−	0.800389i	$-0.704626\pi$
$937$	−3.22222e7	−1.19896	−0.599481	+	0.800389i	$0.704626\pi$
$938$	0	0
$939$	−1.51747e6	−0.0561639
$940$	0	0
$941$	−4.01203e6	−0.147703	−0.0738517	−	0.997269i	$-0.523529\pi$
$941$	−4.01203e6	−0.147703	−0.0738517	+	0.997269i	$0.523529\pi$
$942$	0	0
$943$	−2.74209e7	−1.00416
$944$	0	0
$945$	3.42078e6	0.124608
$946$	0	0
$947$	3.07238e7	1.11327	0.556635	−	0.830757i	$-0.312092\pi$
$947$	3.07238e7	1.11327	0.556635	+	0.830757i	$0.312092\pi$
$948$	0	0
$949$	5.62511e6	0.202752
$950$	0	0
$951$	−3.41679e6	−0.122509
$952$	0	0
$953$	−3.10153e7	−1.10622	−0.553112	−	0.833107i	$-0.686560\pi$
$953$	−3.10153e7	−1.10622	−0.553112	+	0.833107i	$0.686560\pi$
$954$	0	0
$955$	4.54118e6	0.161124
$956$	0	0
$957$	−803285.	−0.0283524
$958$	0	0
$959$	−7.56793e6	−0.265724
$960$	0	0
$961$	−1.51604e7	−0.529542
$962$	0	0
$963$	−2.38026e7	−0.827100
$964$	0	0
$965$	1.17120e7	0.404867
$966$	0	0
$967$	−2.39410e7	−0.823335	−0.411668	−	0.911334i	$-0.635054\pi$
$967$	−2.39410e7	−0.823335	−0.411668	+	0.911334i	$0.635054\pi$
$968$	0	0
$969$	−1.73663e6	−0.0594153
$970$	0	0
$971$	7.44188e6	0.253300	0.126650	−	0.991947i	$-0.459578\pi$
$971$	7.44188e6	0.253300	0.126650	+	0.991947i	$0.459578\pi$
$972$	0	0
$973$	1.07649e7	0.364525
$974$	0	0
$975$	−1.79325e6	−0.0604130
$976$	0	0
$977$	−1.26312e7	−0.423359	−0.211680	−	0.977339i	$-0.567893\pi$
$977$	−1.26312e7	−0.423359	−0.211680	+	0.977339i	$0.567893\pi$
$978$	0	0
$979$	1.64128e6	0.0547302
$980$	0	0
$981$	−1.41559e6	−0.0469640
$982$	0	0
$983$	−3.11593e7	−1.02850	−0.514249	−	0.857641i	$-0.671929\pi$
$983$	−3.11593e7	−1.02850	−0.514249	+	0.857641i	$0.671929\pi$
$984$	0	0
$985$	−7.54514e6	−0.247786
$986$	0	0
$987$	1.22302e6	0.0399614
$988$	0	0
$989$	6.69443e7	2.17632
$990$	0	0
$991$	2.08229e7	0.673530	0.336765	−	0.941589i	$-0.390667\pi$
$991$	2.08229e7	0.673530	0.336765	+	0.941589i	$0.390667\pi$
$992$	0	0
$993$	3.23683e6	0.104171
$994$	0	0
$995$	−863833.	−0.0276613
$996$	0	0
$997$	−1.04553e7	−0.333118	−0.166559	−	0.986031i	$-0.553266\pi$
$997$	−1.04553e7	−0.333118	−0.166559	+	0.986031i	$0.553266\pi$
$998$	0	0
$999$	−8.84330e6	−0.280350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.s.1.2		3
4.3	odd	2		280.6.a.e.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.e.1.2	✓	3	4.3	odd	2
560.6.a.s.1.2		3	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

\(f(q)\)	\(=\)	\(q-6.24759 q^{3} -25.0000 q^{5} -49.0000 q^{7} -203.968 q^{9} +O(q^{10})\)	\(q-6.24759 q^{3} -25.0000 q^{5} -49.0000 q^{7} -203.968 q^{9} -89.4628 q^{11} +459.250 q^{13} +156.190 q^{15} +701.108 q^{17} +396.470 q^{19} +306.132 q^{21} +4316.18 q^{23} +625.000 q^{25} +2792.47 q^{27} -1437.19 q^{29} -3669.99 q^{31} +558.927 q^{33} +1225.00 q^{35} -3166.84 q^{37} -2869.21 q^{39} -6353.05 q^{41} +15510.1 q^{43} +5099.19 q^{45} +3995.07 q^{47} +2401.00 q^{49} -4380.24 q^{51} -24472.8 q^{53} +2236.57 q^{55} -2476.98 q^{57} +2395.32 q^{59} -21052.6 q^{61} +9994.41 q^{63} -11481.3 q^{65} -853.274 q^{67} -26965.7 q^{69} +1378.74 q^{71} +12248.5 q^{73} -3904.75 q^{75} +4383.68 q^{77} +51047.4 q^{79} +32117.9 q^{81} -17615.6 q^{83} -17527.7 q^{85} +8978.99 q^{87} -18346.0 q^{89} -22503.3 q^{91} +22928.6 q^{93} -9911.74 q^{95} -103460. q^{97} +18247.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9}+O(q^{10}) \) 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9	\( 3 q - 6 q^{3} - 75 q^{5} - 147 q^{7} + 209 q^{9} + 578 q^{11} + 80 q^{13} + 150 q^{15} - 1244 q^{17} - 944 q^{19} + 294 q^{21} - 1096 q^{23} + 1875 q^{25} + 3006 q^{27} + 1868 q^{29} + 6620 q^{31} + 2662 q^{33} + 3675 q^{35} - 4058 q^{37} + 20910 q^{39} - 9602 q^{41} + 12340 q^{43} - 5225 q^{45} + 41026 q^{47} + 7203 q^{49} - 10006 q^{51} - 37610 q^{53} - 14450 q^{55} - 95456 q^{57} + 37664 q^{59} - 8386 q^{61} - 10241 q^{63} - 2000 q^{65} - 69340 q^{67} - 37712 q^{69} + 34016 q^{71} - 45314 q^{73} - 3750 q^{75} - 28322 q^{77} - 1382 q^{79} - 101005 q^{81} + 10128 q^{83} + 31100 q^{85} + 41510 q^{87} - 222810 q^{89} - 3920 q^{91} - 174292 q^{93} + 23600 q^{95} - 159476 q^{97} + 156568 q^{99}+O(q^{100}) \) 3 * q - 6 * q^3 - 75 * q^5 - 147 * q^7 + 209 * q^9 + 578 * q^11 + 80 * q^13 + 150 * q^15 - 1244 * q^17 - 944 * q^19 + 294 * q^21 - 1096 * q^23 + 1875 * q^25 + 3006 * q^27 + 1868 * q^29 + 6620 * q^31 + 2662 * q^33 + 3675 * q^35 - 4058 * q^37 + 20910 * q^39 - 9602 * q^41 + 12340 * q^43 - 5225 * q^45 + 41026 * q^47 + 7203 * q^49 - 10006 * q^51 - 37610 * q^53 - 14450 * q^55 - 95456 * q^57 + 37664 * q^59 - 8386 * q^61 - 10241 * q^63 - 2000 * q^65 - 69340 * q^67 - 37712 * q^69 + 34016 * q^71 - 45314 * q^73 - 3750 * q^75 - 28322 * q^77 - 1382 * q^79 - 101005 * q^81 + 10128 * q^83 + 31100 * q^85 + 41510 * q^87 - 222810 * q^89 - 3920 * q^91 - 174292 * q^93 + 23600 * q^95 - 159476 * q^97 + 156568 * q^99

Introduction

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