LMFDB - Embedded newform 280.6.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(1,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, 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("280.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.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:	$44.9074695476$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.996509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 267x + 1100 $ x^3 - x^2 - 267*x + 1100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$14.2949$ of defining polynomial
Character	$\chi$	$=$	280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.3441 q^{3} -25.0000 q^{5} +49.0000 q^{7} +256.259 q^{9} +O(q^{10})$	$q+22.3441 q^{3} -25.0000 q^{5} +49.0000 q^{7} +256.259 q^{9} +492.372 q^{11} +425.357 q^{13} -558.603 q^{15} -608.168 q^{17} -202.719 q^{19} +1094.86 q^{21} -328.225 q^{23} +625.000 q^{25} +296.272 q^{27} +3545.05 q^{29} +4197.85 q^{31} +11001.6 q^{33} -1225.00 q^{35} +4582.02 q^{37} +9504.23 q^{39} +16236.9 q^{41} -8012.20 q^{43} -6406.49 q^{45} +9797.94 q^{47} +2401.00 q^{49} -13589.0 q^{51} +8838.42 q^{53} -12309.3 q^{55} -4529.57 q^{57} -14869.6 q^{59} +37016.9 q^{61} +12556.7 q^{63} -10633.9 q^{65} +11712.6 q^{67} -7333.89 q^{69} -8994.54 q^{71} -4463.91 q^{73} +13965.1 q^{75} +24126.2 q^{77} -6917.23 q^{79} -55651.1 q^{81} -33848.1 q^{83} +15204.2 q^{85} +79211.1 q^{87} -24070.1 q^{89} +20842.5 q^{91} +93797.2 q^{93} +5067.97 q^{95} -57204.6 q^{97} +126175. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9}+O(q^{10}) $ 3 * q + 14 * q^3 - 75 * q^5 + 147 * q^7 + q^9	$ 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9} - 50 q^{11} + 536 q^{13} - 350 q^{15} + 268 q^{17} - 1048 q^{19} + 686 q^{21} - 576 q^{23} + 1875 q^{25} + 1754 q^{27} + 6172 q^{29} - 2404 q^{31} + 11454 q^{33} - 3675 q^{35} + 7806 q^{37} + 2834 q^{39} + 12150 q^{41} + 16868 q^{43} - 25 q^{45} + 30230 q^{47} + 7203 q^{49} + 16758 q^{51} + 25862 q^{53} + 1250 q^{55} + 25960 q^{57} + 36352 q^{59} + 6534 q^{61} + 49 q^{63} - 13400 q^{65} + 45708 q^{67} + 17528 q^{69} + 57456 q^{71} - 17346 q^{73} + 8750 q^{75} - 2450 q^{77} + 21222 q^{79} - 65501 q^{81} + 128816 q^{83} - 6700 q^{85} + 83250 q^{87} + 132286 q^{89} + 26264 q^{91} + 103716 q^{93} + 26200 q^{95} - 74348 q^{97} + 210504 q^{99}+O(q^{100}) $ 3 * q + 14 * q^3 - 75 * q^5 + 147 * q^7 + q^9 - 50 * q^11 + 536 * q^13 - 350 * q^15 + 268 * q^17 - 1048 * q^19 + 686 * q^21 - 576 * q^23 + 1875 * q^25 + 1754 * q^27 + 6172 * q^29 - 2404 * q^31 + 11454 * q^33 - 3675 * q^35 + 7806 * q^37 + 2834 * q^39 + 12150 * q^41 + 16868 * q^43 - 25 * q^45 + 30230 * q^47 + 7203 * q^49 + 16758 * q^51 + 25862 * q^53 + 1250 * q^55 + 25960 * q^57 + 36352 * q^59 + 6534 * q^61 + 49 * q^63 - 13400 * q^65 + 45708 * q^67 + 17528 * q^69 + 57456 * q^71 - 17346 * q^73 + 8750 * q^75 - 2450 * q^77 + 21222 * q^79 - 65501 * q^81 + 128816 * q^83 - 6700 * q^85 + 83250 * q^87 + 132286 * q^89 + 26264 * q^91 + 103716 * q^93 + 26200 * q^95 - 74348 * q^97 + 210504 * 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
$3$	22.3441	1.43338	0.716688	−	0.697394i	$-0.245657\pi$
$3$	22.3441	1.43338	0.716688	+	0.697394i	$0.245657\pi$
$4$	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
$9$	256.259	1.05457
$10$	0	0
$11$	492.372	1.22691	0.613453	−	0.789731i	$-0.289780\pi$
$11$	492.372	1.22691	0.613453	+	0.789731i	$0.289780\pi$
$12$	0	0
$13$	425.357	0.698064	0.349032	−	0.937111i	$-0.386510\pi$
$13$	425.357	0.698064	0.349032	+	0.937111i	$0.386510\pi$
$14$	0	0
$15$	−558.603	−0.641025
$16$	0	0
$17$	−608.168	−0.510389	−0.255195	−	0.966890i	$-0.582140\pi$
$17$	−608.168	−0.510389	−0.255195	+	0.966890i	$0.582140\pi$
$18$	0	0
$19$	−202.719	−0.128828	−0.0644139	−	0.997923i	$-0.520518\pi$
$19$	−202.719	−0.128828	−0.0644139	+	0.997923i	$0.520518\pi$
$20$	0	0
$21$	1094.86	0.541765
$22$	0	0
$23$	−328.225	−0.129375	−0.0646877	−	0.997906i	$-0.520605\pi$
$23$	−328.225	−0.129375	−0.0646877	+	0.997906i	$0.520605\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	296.272	0.0782133
$28$	0	0
$29$	3545.05	0.782758	0.391379	−	0.920229i	$-0.371998\pi$
$29$	3545.05	0.782758	0.391379	+	0.920229i	$0.371998\pi$
$30$	0	0
$31$	4197.85	0.784554	0.392277	−	0.919847i	$-0.371688\pi$
$31$	4197.85	0.784554	0.392277	+	0.919847i	$0.371688\pi$
$32$	0	0
$33$	11001.6	1.75862
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	4582.02	0.550241	0.275120	−	0.961410i	$-0.411282\pi$
$37$	4582.02	0.550241	0.275120	+	0.961410i	$0.411282\pi$
$38$	0	0
$39$	9504.23	1.00059
$40$	0	0
$41$	16236.9	1.50849	0.754247	−	0.656591i	$-0.228002\pi$
$41$	16236.9	1.50849	0.754247	+	0.656591i	$0.228002\pi$
$42$	0	0
$43$	−8012.20	−0.660816	−0.330408	−	0.943838i	$-0.607186\pi$
$43$	−8012.20	−0.660816	−0.330408	+	0.943838i	$0.607186\pi$
$44$	0	0
$45$	−6406.49	−0.471616
$46$	0	0
$47$	9797.94	0.646979	0.323489	−	0.946232i	$-0.395144\pi$
$47$	9797.94	0.646979	0.323489	+	0.946232i	$0.395144\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−13589.0	−0.731579
$52$	0	0
$53$	8838.42	0.432200	0.216100	−	0.976371i	$-0.430666\pi$
$53$	8838.42	0.432200	0.216100	+	0.976371i	$0.430666\pi$
$54$	0	0
$55$	−12309.3	−0.548689
$56$	0	0
$57$	−4529.57	−0.184659
$58$	0	0
$59$	−14869.6	−0.556120	−0.278060	−	0.960564i	$-0.589691\pi$
$59$	−14869.6	−0.556120	−0.278060	+	0.960564i	$0.589691\pi$
$60$	0	0
$61$	37016.9	1.27372	0.636861	−	0.770978i	$-0.280232\pi$
$61$	37016.9	1.27372	0.636861	+	0.770978i	$0.280232\pi$
$62$	0	0
$63$	12556.7	0.398588
$64$	0	0
$65$	−10633.9	−0.312184
$66$	0	0
$67$	11712.6	0.318763	0.159381	−	0.987217i	$-0.449050\pi$
$67$	11712.6	0.318763	0.159381	+	0.987217i	$0.449050\pi$
$68$	0	0
$69$	−7333.89	−0.185443
$70$	0	0
$71$	−8994.54	−0.211755	−0.105877	−	0.994379i	$-0.533765\pi$
$71$	−8994.54	−0.211755	−0.105877	+	0.994379i	$0.533765\pi$
$72$	0	0
$73$	−4463.91	−0.0980411	−0.0490206	−	0.998798i	$-0.515610\pi$
$73$	−4463.91	−0.0980411	−0.0490206	+	0.998798i	$0.515610\pi$
$74$	0	0
$75$	13965.1	0.286675
$76$	0	0
$77$	24126.2	0.463727
$78$	0	0
$79$	−6917.23	−0.124699	−0.0623497	−	0.998054i	$-0.519859\pi$
$79$	−6917.23	−0.124699	−0.0623497	+	0.998054i	$0.519859\pi$
$80$	0	0
$81$	−55651.1	−0.942457
$82$	0	0
$83$	−33848.1	−0.539311	−0.269656	−	0.962957i	$-0.586910\pi$
$83$	−33848.1	−0.539311	−0.269656	+	0.962957i	$0.586910\pi$
$84$	0	0
$85$	15204.2	0.228253
$86$	0	0
$87$	79211.1	1.12199
$88$	0	0
$89$	−24070.1	−0.322109	−0.161054	−	0.986946i	$-0.551489\pi$
$89$	−24070.1	−0.322109	−0.161054	+	0.986946i	$0.551489\pi$
$90$	0	0
$91$	20842.5	0.263844
$92$	0	0
$93$	93797.2	1.12456
$94$	0	0
$95$	5067.97	0.0576136
$96$	0	0
$97$	−57204.6	−0.617308	−0.308654	−	0.951174i	$-0.599878\pi$
$97$	−57204.6	−0.617308	−0.308654	+	0.951174i	$0.599878\pi$
$98$	0	0
$99$	126175.	1.29385
$100$	0	0
$101$	123407.	1.20375	0.601877	−	0.798589i	$-0.294420\pi$
$101$	123407.	1.20375	0.601877	+	0.798589i	$0.294420\pi$
$102$	0	0
$103$	181127.	1.68225	0.841124	−	0.540843i	$-0.181895\pi$
$103$	181127.	1.68225	0.841124	+	0.540843i	$0.181895\pi$
$104$	0	0
$105$	−27371.5	−0.242285
$106$	0	0
$107$	42691.1	0.360477	0.180239	−	0.983623i	$-0.442313\pi$
$107$	42691.1	0.360477	0.180239	+	0.983623i	$0.442313\pi$
$108$	0	0
$109$	−238423.	−1.92212	−0.961062	−	0.276333i	$-0.910881\pi$
$109$	−238423.	−1.92212	−0.961062	+	0.276333i	$0.910881\pi$
$110$	0	0
$111$	102381.	0.788701
$112$	0	0
$113$	168345.	1.24024	0.620119	−	0.784508i	$-0.287084\pi$
$113$	168345.	1.24024	0.620119	+	0.784508i	$0.287084\pi$
$114$	0	0
$115$	8205.61	0.0578584
$116$	0	0
$117$	109002.	0.736155
$118$	0	0
$119$	−29800.2	−0.192909
$120$	0	0
$121$	81378.9	0.505299
$122$	0	0
$123$	362799.	2.16224
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	181913.	1.00082	0.500409	−	0.865789i	$-0.333183\pi$
$127$	181913.	1.00082	0.500409	+	0.865789i	$0.333183\pi$
$128$	0	0
$129$	−179026.	−0.947198
$130$	0	0
$131$	25188.4	0.128240	0.0641199	−	0.997942i	$-0.479576\pi$
$131$	25188.4	0.128240	0.0641199	+	0.997942i	$0.479576\pi$
$132$	0	0
$133$	−9933.22	−0.0486924
$134$	0	0
$135$	−7406.79	−0.0349781
$136$	0	0
$137$	−218817.	−0.996047	−0.498024	−	0.867163i	$-0.665941\pi$
$137$	−218817.	−0.996047	−0.498024	+	0.867163i	$0.665941\pi$
$138$	0	0
$139$	−214369.	−0.941076	−0.470538	−	0.882380i	$-0.655940\pi$
$139$	−214369.	−0.941076	−0.470538	+	0.882380i	$0.655940\pi$
$140$	0	0
$141$	218926.	0.927364
$142$	0	0
$143$	209434.	0.856459
$144$	0	0
$145$	−88626.3	−0.350060
$146$	0	0
$147$	53648.2	0.204768
$148$	0	0
$149$	105397.	0.388922	0.194461	−	0.980910i	$-0.437704\pi$
$149$	105397.	0.388922	0.194461	+	0.980910i	$0.437704\pi$
$150$	0	0
$151$	−79749.8	−0.284634	−0.142317	−	0.989821i	$-0.545455\pi$
$151$	−79749.8	−0.284634	−0.142317	+	0.989821i	$0.545455\pi$
$152$	0	0
$153$	−155849.	−0.538239
$154$	0	0
$155$	−104946.	−0.350863
$156$	0	0
$157$	−411818.	−1.33339	−0.666693	−	0.745332i	$-0.732291\pi$
$157$	−411818.	−1.33339	−0.666693	+	0.745332i	$0.732291\pi$
$158$	0	0
$159$	197487.	0.619506
$160$	0	0
$161$	−16083.0	−0.0488993
$162$	0	0
$163$	370069.	1.09097	0.545486	−	0.838120i	$-0.316345\pi$
$163$	370069.	1.09097	0.545486	+	0.838120i	$0.316345\pi$
$164$	0	0
$165$	−275040.	−0.786478
$166$	0	0
$167$	−413641.	−1.14771	−0.573855	−	0.818957i	$-0.694553\pi$
$167$	−413641.	−1.14771	−0.573855	+	0.818957i	$0.694553\pi$
$168$	0	0
$169$	−190364.	−0.512706
$170$	0	0
$171$	−51948.6	−0.135857
$172$	0	0
$173$	−646770.	−1.64299	−0.821495	−	0.570216i	$-0.806860\pi$
$173$	−646770.	−1.64299	−0.821495	+	0.570216i	$0.806860\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−332247.	−0.797128
$178$	0	0
$179$	−385862.	−0.900117	−0.450059	−	0.892999i	$-0.648597\pi$
$179$	−385862.	−0.900117	−0.450059	+	0.892999i	$0.648597\pi$
$180$	0	0
$181$	−666927.	−1.51315	−0.756574	−	0.653908i	$-0.773129\pi$
$181$	−666927.	−1.51315	−0.756574	+	0.653908i	$0.773129\pi$
$182$	0	0
$183$	827109.	1.82572
$184$	0	0
$185$	−114550.	−0.246075
$186$	0	0
$187$	−299445.	−0.626200
$188$	0	0
$189$	14517.3	0.0295619
$190$	0	0
$191$	894336.	1.77385	0.886925	−	0.461913i	$-0.152837\pi$
$191$	894336.	1.77385	0.886925	+	0.461913i	$0.152837\pi$
$192$	0	0
$193$	−884790.	−1.70981	−0.854903	−	0.518788i	$-0.826383\pi$
$193$	−884790.	−1.70981	−0.854903	+	0.518788i	$0.826383\pi$
$194$	0	0
$195$	−237606.	−0.447477
$196$	0	0
$197$	−23094.4	−0.0423975	−0.0211988	−	0.999775i	$-0.506748\pi$
$197$	−23094.4	−0.0423975	−0.0211988	+	0.999775i	$0.506748\pi$
$198$	0	0
$199$	−938873.	−1.68064	−0.840319	−	0.542092i	$-0.817632\pi$
$199$	−938873.	−1.68064	−0.840319	+	0.542092i	$0.817632\pi$
$200$	0	0
$201$	261708.	0.456907
$202$	0	0
$203$	173708.	0.295855
$204$	0	0
$205$	−405923.	−0.674619
$206$	0	0
$207$	−84110.7	−0.136435
$208$	0	0
$209$	−99812.9	−0.158060
$210$	0	0
$211$	−798819.	−1.23521	−0.617607	−	0.786487i	$-0.711898\pi$
$211$	−798819.	−1.23521	−0.617607	+	0.786487i	$0.711898\pi$
$212$	0	0
$213$	−200975.	−0.303524
$214$	0	0
$215$	200305.	0.295526
$216$	0	0
$217$	205695.	0.296533
$218$	0	0
$219$	−99742.1	−0.140530
$220$	0	0
$221$	−258689.	−0.356284
$222$	0	0
$223$	123401.	0.166172	0.0830859	−	0.996542i	$-0.473522\pi$
$223$	123401.	0.166172	0.0830859	+	0.996542i	$0.473522\pi$
$224$	0	0
$225$	160162.	0.210913
$226$	0	0
$227$	−996824.	−1.28397	−0.641983	−	0.766719i	$-0.721888\pi$
$227$	−996824.	−1.28397	−0.641983	+	0.766719i	$0.721888\pi$
$228$	0	0
$229$	1.41776e6	1.78654	0.893270	−	0.449520i	$-0.148405\pi$
$229$	1.41776e6	1.78654	0.893270	+	0.449520i	$0.148405\pi$
$230$	0	0
$231$	539079.	0.664695
$232$	0	0
$233$	140310.	0.169316	0.0846581	−	0.996410i	$-0.473020\pi$
$233$	140310.	0.169316	0.0846581	+	0.996410i	$0.473020\pi$
$234$	0	0
$235$	−244949.	−0.289338
$236$	0	0
$237$	−154559.	−0.178741
$238$	0	0
$239$	841648.	0.953094	0.476547	−	0.879149i	$-0.341888\pi$
$239$	841648.	0.953094	0.476547	+	0.879149i	$0.341888\pi$
$240$	0	0
$241$	731589.	0.811380	0.405690	−	0.914011i	$-0.367031\pi$
$241$	731589.	0.811380	0.405690	+	0.914011i	$0.367031\pi$
$242$	0	0
$243$	−1.31547e6	−1.42911
$244$	0	0
$245$	−60025.0	−0.0638877
$246$	0	0
$247$	−86227.9	−0.0899301
$248$	0	0
$249$	−756306.	−0.773036
$250$	0	0
$251$	−1.20502e6	−1.20729	−0.603644	−	0.797254i	$-0.706285\pi$
$251$	−1.20502e6	−1.20729	−0.603644	+	0.797254i	$0.706285\pi$
$252$	0	0
$253$	−161608.	−0.158731
$254$	0	0
$255$	339725.	0.327172
$256$	0	0
$257$	905674.	0.855341	0.427670	−	0.903935i	$-0.359334\pi$
$257$	905674.	0.855341	0.427670	+	0.903935i	$0.359334\pi$
$258$	0	0
$259$	224519.	0.207971
$260$	0	0
$261$	908453.	0.825470
$262$	0	0
$263$	−285878.	−0.254854	−0.127427	−	0.991848i	$-0.540672\pi$
$263$	−285878.	−0.254854	−0.127427	+	0.991848i	$0.540672\pi$
$264$	0	0
$265$	−220961.	−0.193286
$266$	0	0
$267$	−537824.	−0.461703
$268$	0	0
$269$	853757.	0.719372	0.359686	−	0.933073i	$-0.382884\pi$
$269$	853757.	0.719372	0.359686	+	0.933073i	$0.382884\pi$
$270$	0	0
$271$	−372845.	−0.308393	−0.154197	−	0.988040i	$-0.549279\pi$
$271$	−372845.	−0.308393	−0.154197	+	0.988040i	$0.549279\pi$
$272$	0	0
$273$	465707.	0.378187
$274$	0	0
$275$	307732.	0.245381
$276$	0	0
$277$	−34474.4	−0.0269959	−0.0134979	−	0.999909i	$-0.504297\pi$
$277$	−34474.4	−0.0269959	−0.0134979	+	0.999909i	$0.504297\pi$
$278$	0	0
$279$	1.07574e6	0.827363
$280$	0	0
$281$	−1.15911e6	−0.875706	−0.437853	−	0.899047i	$-0.644261\pi$
$281$	−1.15911e6	−0.875706	−0.437853	+	0.899047i	$0.644261\pi$
$282$	0	0
$283$	2.27549e6	1.68892	0.844460	−	0.535619i	$-0.179922\pi$
$283$	2.27549e6	1.68892	0.844460	+	0.535619i	$0.179922\pi$
$284$	0	0
$285$	113239.	0.0825819
$286$	0	0
$287$	795608.	0.570157
$288$	0	0
$289$	−1.04999e6	−0.739503
$290$	0	0
$291$	−1.27819e6	−0.884834
$292$	0	0
$293$	−1.44972e6	−0.986541	−0.493270	−	0.869876i	$-0.664199\pi$
$293$	−1.44972e6	−0.986541	−0.493270	+	0.869876i	$0.664199\pi$
$294$	0	0
$295$	371739.	0.248704
$296$	0	0
$297$	145876.	0.0959604
$298$	0	0
$299$	−139613.	−0.0903123
$300$	0	0
$301$	−392598.	−0.249765
$302$	0	0
$303$	2.75743e6	1.72543
$304$	0	0
$305$	−925421.	−0.569626
$306$	0	0
$307$	−777947.	−0.471090	−0.235545	−	0.971863i	$-0.575688\pi$
$307$	−777947.	−0.471090	−0.235545	+	0.971863i	$0.575688\pi$
$308$	0	0
$309$	4.04712e6	2.41129
$310$	0	0
$311$	1.22692e6	0.719309	0.359655	−	0.933085i	$-0.382894\pi$
$311$	1.22692e6	0.719309	0.359655	+	0.933085i	$0.382894\pi$
$312$	0	0
$313$	194520.	0.112229	0.0561143	−	0.998424i	$-0.482129\pi$
$313$	194520.	0.112229	0.0561143	+	0.998424i	$0.482129\pi$
$314$	0	0
$315$	−313918.	−0.178254
$316$	0	0
$317$	−1.96510e6	−1.09834	−0.549169	−	0.835711i	$-0.685056\pi$
$317$	−1.96510e6	−1.09834	−0.549169	+	0.835711i	$0.685056\pi$
$318$	0	0
$319$	1.74548e6	0.960371
$320$	0	0
$321$	953895.	0.516700
$322$	0	0
$323$	123287.	0.0657523
$324$	0	0
$325$	265848.	0.139613
$326$	0	0
$327$	−5.32735e6	−2.75513
$328$	0	0
$329$	480099.	0.244535
$330$	0	0
$331$	−2.75257e6	−1.38092	−0.690460	−	0.723370i	$-0.742592\pi$
$331$	−2.75257e6	−1.38092	−0.690460	+	0.723370i	$0.742592\pi$
$332$	0	0
$333$	1.17419e6	0.580265
$334$	0	0
$335$	−292816.	−0.142555
$336$	0	0
$337$	−1.02935e6	−0.493726	−0.246863	−	0.969050i	$-0.579400\pi$
$337$	−1.02935e6	−0.493726	−0.246863	+	0.969050i	$0.579400\pi$
$338$	0	0
$339$	3.76153e6	1.77773
$340$	0	0
$341$	2.06690e6	0.962574
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	183347.	0.0829328
$346$	0	0
$347$	−1.02803e6	−0.458336	−0.229168	−	0.973387i	$-0.573601\pi$
$347$	−1.02803e6	−0.458336	−0.229168	+	0.973387i	$0.573601\pi$
$348$	0	0
$349$	2.99975e6	1.31832	0.659161	−	0.752002i	$-0.270911\pi$
$349$	2.99975e6	1.31832	0.659161	+	0.752002i	$0.270911\pi$
$350$	0	0
$351$	126021.	0.0545979
$352$	0	0
$353$	−3.55143e6	−1.51693	−0.758467	−	0.651711i	$-0.774051\pi$
$353$	−3.55143e6	−1.51693	−0.758467	+	0.651711i	$0.774051\pi$
$354$	0	0
$355$	224863.	0.0946996
$356$	0	0
$357$	−665860.	−0.276511
$358$	0	0
$359$	1.13496e6	0.464778	0.232389	−	0.972623i	$-0.425346\pi$
$359$	1.13496e6	0.464778	0.232389	+	0.972623i	$0.425346\pi$
$360$	0	0
$361$	−2.43500e6	−0.983403
$362$	0	0
$363$	1.81834e6	0.724283
$364$	0	0
$365$	111598.	0.0438453
$366$	0	0
$367$	−1.97652e6	−0.766015	−0.383007	−	0.923745i	$-0.625112\pi$
$367$	−1.97652e6	−0.766015	−0.383007	+	0.923745i	$0.625112\pi$
$368$	0	0
$369$	4.16086e6	1.59081
$370$	0	0
$371$	433083.	0.163356
$372$	0	0
$373$	−2.60644e6	−0.970010	−0.485005	−	0.874511i	$-0.661182\pi$
$373$	−2.60644e6	−0.970010	−0.485005	+	0.874511i	$0.661182\pi$
$374$	0	0
$375$	−349127.	−0.128205
$376$	0	0
$377$	1.50791e6	0.546416
$378$	0	0
$379$	1.83038e6	0.654550	0.327275	−	0.944929i	$-0.393870\pi$
$379$	1.83038e6	0.654550	0.327275	+	0.944929i	$0.393870\pi$
$380$	0	0
$381$	4.06469e6	1.43455
$382$	0	0
$383$	−3.98294e6	−1.38742	−0.693708	−	0.720256i	$-0.744024\pi$
$383$	−3.98294e6	−1.38742	−0.693708	+	0.720256i	$0.744024\pi$
$384$	0	0
$385$	−603155.	−0.207385
$386$	0	0
$387$	−2.05320e6	−0.696874
$388$	0	0
$389$	935569.	0.313474	0.156737	−	0.987640i	$-0.449902\pi$
$389$	935569.	0.313474	0.156737	+	0.987640i	$0.449902\pi$
$390$	0	0
$391$	199616.	0.0660318
$392$	0	0
$393$	562813.	0.183816
$394$	0	0
$395$	172931.	0.0557673
$396$	0	0
$397$	−1.06805e6	−0.340106	−0.170053	−	0.985435i	$-0.554394\pi$
$397$	−1.06805e6	−0.340106	−0.170053	+	0.985435i	$0.554394\pi$
$398$	0	0
$399$	−221949.	−0.0697944
$400$	0	0
$401$	3.19681e6	0.992786	0.496393	−	0.868098i	$-0.334657\pi$
$401$	3.19681e6	0.992786	0.496393	+	0.868098i	$0.334657\pi$
$402$	0	0
$403$	1.78559e6	0.547669
$404$	0	0
$405$	1.39128e6	0.421479
$406$	0	0
$407$	2.25606e6	0.675094
$408$	0	0
$409$	409311.	0.120989	0.0604944	−	0.998169i	$-0.480732\pi$
$409$	409311.	0.120989	0.0604944	+	0.998169i	$0.480732\pi$
$410$	0	0
$411$	−4.88928e6	−1.42771
$412$	0	0
$413$	−728609.	−0.210193
$414$	0	0
$415$	846203.	0.241187
$416$	0	0
$417$	−4.78988e6	−1.34892
$418$	0	0
$419$	5.80047e6	1.61409	0.807046	−	0.590489i	$-0.201065\pi$
$419$	5.80047e6	1.61409	0.807046	+	0.590489i	$0.201065\pi$
$420$	0	0
$421$	−122524.	−0.0336912	−0.0168456	−	0.999858i	$-0.505362\pi$
$421$	−122524.	−0.0336912	−0.0168456	+	0.999858i	$0.505362\pi$
$422$	0	0
$423$	2.51082e6	0.682282
$424$	0	0
$425$	−380105.	−0.102078
$426$	0	0
$427$	1.81383e6	0.481422
$428$	0	0
$429$	4.67961e6	1.22763
$430$	0	0
$431$	2.65230e6	0.687749	0.343874	−	0.939016i	$-0.388261\pi$
$431$	2.65230e6	0.687749	0.343874	+	0.939016i	$0.388261\pi$
$432$	0	0
$433$	3.83873e6	0.983938	0.491969	−	0.870613i	$-0.336277\pi$
$433$	3.83873e6	0.983938	0.491969	+	0.870613i	$0.336277\pi$
$434$	0	0
$435$	−1.98028e6	−0.501768
$436$	0	0
$437$	66537.3	0.0166671
$438$	0	0
$439$	−2.09582e6	−0.519031	−0.259515	−	0.965739i	$-0.583563\pi$
$439$	−2.09582e6	−0.519031	−0.259515	+	0.965739i	$0.583563\pi$
$440$	0	0
$441$	615279.	0.150652
$442$	0	0
$443$	2.87887e6	0.696969	0.348484	−	0.937315i	$-0.386696\pi$
$443$	2.87887e6	0.696969	0.348484	+	0.937315i	$0.386696\pi$
$444$	0	0
$445$	601752.	0.144051
$446$	0	0
$447$	2.35500e6	0.557472
$448$	0	0
$449$	4.29524e6	1.00548	0.502738	−	0.864439i	$-0.332326\pi$
$449$	4.29524e6	1.00548	0.502738	+	0.864439i	$0.332326\pi$
$450$	0	0
$451$	7.99459e6	1.85078
$452$	0	0
$453$	−1.78194e6	−0.407988
$454$	0	0
$455$	−521063.	−0.117994
$456$	0	0
$457$	−4.43657e6	−0.993704	−0.496852	−	0.867835i	$-0.665511\pi$
$457$	−4.43657e6	−0.993704	−0.496852	+	0.867835i	$0.665511\pi$
$458$	0	0
$459$	−180183.	−0.0399192
$460$	0	0
$461$	8.67471e6	1.90109	0.950545	−	0.310586i	$-0.100525\pi$
$461$	8.67471e6	1.90109	0.950545	+	0.310586i	$0.100525\pi$
$462$	0	0
$463$	−2.01501e6	−0.436843	−0.218422	−	0.975854i	$-0.570091\pi$
$463$	−2.01501e6	−0.436843	−0.218422	+	0.975854i	$0.570091\pi$
$464$	0	0
$465$	−2.34493e6	−0.502918
$466$	0	0
$467$	−419938.	−0.0891031	−0.0445515	−	0.999007i	$-0.514186\pi$
$467$	−419938.	−0.0891031	−0.0445515	+	0.999007i	$0.514186\pi$
$468$	0	0
$469$	573919.	0.120481
$470$	0	0
$471$	−9.20170e6	−1.91124
$472$	0	0
$473$	−3.94498e6	−0.810760
$474$	0	0
$475$	−126699.	−0.0257656
$476$	0	0
$477$	2.26493e6	0.455784
$478$	0	0
$479$	−3.00621e6	−0.598661	−0.299330	−	0.954150i	$-0.596763\pi$
$479$	−3.00621e6	−0.598661	−0.299330	+	0.954150i	$0.596763\pi$
$480$	0	0
$481$	1.94899e6	0.384103
$482$	0	0
$483$	−359360.	−0.0700910
$484$	0	0
$485$	1.43012e6	0.276068
$486$	0	0
$487$	−6.51495e6	−1.24477	−0.622385	−	0.782711i	$-0.713836\pi$
$487$	−6.51495e6	−1.24477	−0.622385	+	0.782711i	$0.713836\pi$
$488$	0	0
$489$	8.26887e6	1.56377
$490$	0	0
$491$	−619484.	−0.115965	−0.0579824	−	0.998318i	$-0.518467\pi$
$491$	−619484.	−0.115965	−0.0579824	+	0.998318i	$0.518467\pi$
$492$	0	0
$493$	−2.15599e6	−0.399511
$494$	0	0
$495$	−3.15437e6	−0.578629
$496$	0	0
$497$	−440732.	−0.0800357
$498$	0	0
$499$	−7.86867e6	−1.41465	−0.707327	−	0.706887i	$-0.750099\pi$
$499$	−7.86867e6	−1.41465	−0.707327	+	0.706887i	$0.750099\pi$
$500$	0	0
$501$	−9.24243e6	−1.64510
$502$	0	0
$503$	7.76259e6	1.36800	0.684001	−	0.729481i	$-0.260238\pi$
$503$	7.76259e6	1.36800	0.684001	+	0.729481i	$0.260238\pi$
$504$	0	0
$505$	−3.08518e6	−0.538335
$506$	0	0
$507$	−4.25352e6	−0.734901
$508$	0	0
$509$	4.84603e6	0.829071	0.414536	−	0.910033i	$-0.363944\pi$
$509$	4.84603e6	0.829071	0.414536	+	0.910033i	$0.363944\pi$
$510$	0	0
$511$	−218732.	−0.0370561
$512$	0	0
$513$	−60059.8	−0.0100761
$514$	0	0
$515$	−4.52817e6	−0.752324
$516$	0	0
$517$	4.82423e6	0.793782
$518$	0	0
$519$	−1.44515e7	−2.35502
$520$	0	0
$521$	3.30315e6	0.533132	0.266566	−	0.963817i	$-0.414111\pi$
$521$	3.30315e6	0.533132	0.266566	+	0.963817i	$0.414111\pi$
$522$	0	0
$523$	1.50521e6	0.240626	0.120313	−	0.992736i	$-0.461610\pi$
$523$	1.50521e6	0.240626	0.120313	+	0.992736i	$0.461610\pi$
$524$	0	0
$525$	684289.	0.108353
$526$	0	0
$527$	−2.55300e6	−0.400428
$528$	0	0
$529$	−6.32861e6	−0.983262
$530$	0	0
$531$	−3.81047e6	−0.586465
$532$	0	0
$533$	6.90649e6	1.05303
$534$	0	0
$535$	−1.06728e6	−0.161210
$536$	0	0
$537$	−8.62174e6	−1.29021
$538$	0	0
$539$	1.18218e6	0.175272
$540$	0	0
$541$	−1.08960e7	−1.60056	−0.800282	−	0.599624i	$-0.795317\pi$
$541$	−1.08960e7	−1.60056	−0.800282	+	0.599624i	$0.795317\pi$
$542$	0	0
$543$	−1.49019e7	−2.16891
$544$	0	0
$545$	5.96057e6	0.859600
$546$	0	0
$547$	−6.82607e6	−0.975445	−0.487722	−	0.872999i	$-0.662172\pi$
$547$	−6.82607e6	−0.975445	−0.487722	+	0.872999i	$0.662172\pi$
$548$	0	0
$549$	9.48592e6	1.34322
$550$	0	0
$551$	−718649.	−0.100841
$552$	0	0
$553$	−338944.	−0.0471320
$554$	0	0
$555$	−2.55953e6	−0.352718
$556$	0	0
$557$	3.17478e6	0.433587	0.216793	−	0.976217i	$-0.430440\pi$
$557$	3.17478e6	0.433587	0.216793	+	0.976217i	$0.430440\pi$
$558$	0	0
$559$	−3.40805e6	−0.461292
$560$	0	0
$561$	−6.69083e6	−0.897579
$562$	0	0
$563$	1.11114e7	1.47740	0.738700	−	0.674034i	$-0.235440\pi$
$563$	1.11114e7	1.47740	0.738700	+	0.674034i	$0.235440\pi$
$564$	0	0
$565$	−4.20863e6	−0.554651
$566$	0	0
$567$	−2.72691e6	−0.356215
$568$	0	0
$569$	−9.11736e6	−1.18056	−0.590281	−	0.807198i	$-0.700983\pi$
$569$	−9.11736e6	−1.18056	−0.590281	+	0.807198i	$0.700983\pi$
$570$	0	0
$571$	1.36545e7	1.75261	0.876307	−	0.481754i	$-0.160000\pi$
$571$	1.36545e7	1.75261	0.876307	+	0.481754i	$0.160000\pi$
$572$	0	0
$573$	1.99831e7	2.54259
$574$	0	0
$575$	−205140.	−0.0258751
$576$	0	0
$577$	1.19848e6	0.149861	0.0749307	−	0.997189i	$-0.476126\pi$
$577$	1.19848e6	0.149861	0.0749307	+	0.997189i	$0.476126\pi$
$578$	0	0
$579$	−1.97698e7	−2.45079
$580$	0	0
$581$	−1.65856e6	−0.203840
$582$	0	0
$583$	4.35179e6	0.530269
$584$	0	0
$585$	−2.72505e6	−0.329218
$586$	0	0
$587$	−8.85763e6	−1.06102	−0.530509	−	0.847680i	$-0.677999\pi$
$587$	−8.85763e6	−1.06102	−0.530509	+	0.847680i	$0.677999\pi$
$588$	0	0
$589$	−850982.	−0.101072
$590$	0	0
$591$	−516023.	−0.0607716
$592$	0	0
$593$	6.28066e6	0.733447	0.366723	−	0.930330i	$-0.380480\pi$
$593$	6.28066e6	0.733447	0.366723	+	0.930330i	$0.380480\pi$
$594$	0	0
$595$	745006.	0.0862715
$596$	0	0
$597$	−2.09783e7	−2.40899
$598$	0	0
$599$	1.51272e6	0.172263	0.0861316	−	0.996284i	$-0.472549\pi$
$599$	1.51272e6	0.172263	0.0861316	+	0.996284i	$0.472549\pi$
$600$	0	0
$601$	−1.24383e7	−1.40467	−0.702335	−	0.711847i	$-0.747859\pi$
$601$	−1.24383e7	−1.40467	−0.702335	+	0.711847i	$0.747859\pi$
$602$	0	0
$603$	3.00147e6	0.336156
$604$	0	0
$605$	−2.03447e6	−0.225977
$606$	0	0
$607$	−268905.	−0.0296228	−0.0148114	−	0.999890i	$-0.504715\pi$
$607$	−268905.	−0.0296228	−0.0148114	+	0.999890i	$0.504715\pi$
$608$	0	0
$609$	3.88134e6	0.424071
$610$	0	0
$611$	4.16762e6	0.451633
$612$	0	0
$613$	−9.77162e6	−1.05030	−0.525152	−	0.851008i	$-0.675992\pi$
$613$	−9.77162e6	−1.05030	−0.525152	+	0.851008i	$0.675992\pi$
$614$	0	0
$615$	−9.06998e6	−0.966983
$616$	0	0
$617$	−1.60775e7	−1.70023	−0.850113	−	0.526601i	$-0.823466\pi$
$617$	−1.60775e7	−1.70023	−0.850113	+	0.526601i	$0.823466\pi$
$618$	0	0
$619$	1.39133e7	1.45950	0.729751	−	0.683713i	$-0.239636\pi$
$619$	1.39133e7	1.45950	0.729751	+	0.683713i	$0.239636\pi$
$620$	0	0
$621$	−97243.6	−0.0101189
$622$	0	0
$623$	−1.17943e6	−0.121746
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.23023e6	−0.226559
$628$	0	0
$629$	−2.78664e6	−0.280837
$630$	0	0
$631$	1.45615e7	1.45590	0.727952	−	0.685628i	$-0.240472\pi$
$631$	1.45615e7	1.45590	0.727952	+	0.685628i	$0.240472\pi$
$632$	0	0
$633$	−1.78489e7	−1.77053
$634$	0	0
$635$	−4.54784e6	−0.447580
$636$	0	0
$637$	1.02128e6	0.0997235
$638$	0	0
$639$	−2.30494e6	−0.223309
$640$	0	0
$641$	−1.04064e7	−1.00036	−0.500180	−	0.865921i	$-0.666733\pi$
$641$	−1.04064e7	−1.00036	−0.500180	+	0.865921i	$0.666733\pi$
$642$	0	0
$643$	9.12659e6	0.870524	0.435262	−	0.900304i	$-0.356656\pi$
$643$	9.12659e6	0.870524	0.435262	+	0.900304i	$0.356656\pi$
$644$	0	0
$645$	4.47564e6	0.423600
$646$	0	0
$647$	1.56563e7	1.47037	0.735186	−	0.677865i	$-0.237095\pi$
$647$	1.56563e7	1.47037	0.735186	+	0.677865i	$0.237095\pi$
$648$	0	0
$649$	−7.32135e6	−0.682307
$650$	0	0
$651$	4.59606e6	0.425044
$652$	0	0
$653$	1.82151e7	1.67167	0.835833	−	0.548984i	$-0.184985\pi$
$653$	1.82151e7	1.67167	0.835833	+	0.548984i	$0.184985\pi$
$654$	0	0
$655$	−629711.	−0.0573506
$656$	0	0
$657$	−1.14392e6	−0.103391
$658$	0	0
$659$	1.89170e7	1.69683	0.848414	−	0.529333i	$-0.177558\pi$
$659$	1.89170e7	1.69683	0.848414	+	0.529333i	$0.177558\pi$
$660$	0	0
$661$	5.57786e6	0.496551	0.248276	−	0.968689i	$-0.420136\pi$
$661$	5.57786e6	0.496551	0.248276	+	0.968689i	$0.420136\pi$
$662$	0	0
$663$	−5.78017e6	−0.510689
$664$	0	0
$665$	248330.	0.0217759
$666$	0	0
$667$	−1.16357e6	−0.101270
$668$	0	0
$669$	2.75729e6	0.238187
$670$	0	0
$671$	1.82261e7	1.56274
$672$	0	0
$673$	8.39922e6	0.714828	0.357414	−	0.933946i	$-0.383659\pi$
$673$	8.39922e6	0.714828	0.357414	+	0.933946i	$0.383659\pi$
$674$	0	0
$675$	185170.	0.0156427
$676$	0	0
$677$	1.29514e7	1.08604	0.543019	−	0.839721i	$-0.317281\pi$
$677$	1.29514e7	1.08604	0.543019	+	0.839721i	$0.317281\pi$
$678$	0	0
$679$	−2.80303e6	−0.233320
$680$	0	0
$681$	−2.22731e7	−1.84041
$682$	0	0
$683$	3.22727e6	0.264718	0.132359	−	0.991202i	$-0.457745\pi$
$683$	3.22727e6	0.264718	0.132359	+	0.991202i	$0.457745\pi$
$684$	0	0
$685$	5.47043e6	0.445446
$686$	0	0
$687$	3.16785e7	2.56078
$688$	0	0
$689$	3.75949e6	0.301704
$690$	0	0
$691$	2.40047e7	1.91250	0.956249	−	0.292553i	$-0.0945048\pi$
$691$	2.40047e7	1.91250	0.956249	+	0.292553i	$0.0945048\pi$
$692$	0	0
$693$	6.18257e6	0.489031
$694$	0	0
$695$	5.35922e6	0.420862
$696$	0	0
$697$	−9.87477e6	−0.769919
$698$	0	0
$699$	3.13510e6	0.242694
$700$	0	0
$701$	1.19578e7	0.919084	0.459542	−	0.888156i	$-0.348014\pi$
$701$	1.19578e7	0.919084	0.459542	+	0.888156i	$0.348014\pi$
$702$	0	0
$703$	−928861.	−0.0708863
$704$	0	0
$705$	−5.47316e6	−0.414730
$706$	0	0
$707$	6.04696e6	0.454976
$708$	0	0
$709$	1.07851e7	0.805763	0.402882	−	0.915252i	$-0.368009\pi$
$709$	1.07851e7	0.805763	0.402882	+	0.915252i	$0.368009\pi$
$710$	0	0
$711$	−1.77261e6	−0.131504
$712$	0	0
$713$	−1.37784e6	−0.101502
$714$	0	0
$715$	−5.23585e6	−0.383020
$716$	0	0
$717$	1.88059e7	1.36614
$718$	0	0
$719$	−1.22503e7	−0.883741	−0.441871	−	0.897079i	$-0.645685\pi$
$719$	−1.22503e7	−0.883741	−0.441871	+	0.897079i	$0.645685\pi$
$720$	0	0
$721$	8.87522e6	0.635830
$722$	0	0
$723$	1.63467e7	1.16301
$724$	0	0
$725$	2.21566e6	0.156552
$726$	0	0
$727$	1.40579e6	0.0986469	0.0493235	−	0.998783i	$-0.484293\pi$
$727$	1.40579e6	0.0986469	0.0493235	+	0.998783i	$0.484293\pi$
$728$	0	0
$729$	−1.58698e7	−1.10599
$730$	0	0
$731$	4.87277e6	0.337273
$732$	0	0
$733$	−1.04596e7	−0.719043	−0.359521	−	0.933137i	$-0.617060\pi$
$733$	−1.04596e7	−0.719043	−0.359521	+	0.933137i	$0.617060\pi$
$734$	0	0
$735$	−1.34121e6	−0.0915750
$736$	0	0
$737$	5.76697e6	0.391092
$738$	0	0
$739$	−1.02475e7	−0.690252	−0.345126	−	0.938556i	$-0.612164\pi$
$739$	−1.02475e7	−0.690252	−0.345126	+	0.938556i	$0.612164\pi$
$740$	0	0
$741$	−1.92669e6	−0.128904
$742$	0	0
$743$	2.00086e7	1.32967	0.664835	−	0.746991i	$-0.268502\pi$
$743$	2.00086e7	1.32967	0.664835	+	0.746991i	$0.268502\pi$
$744$	0	0
$745$	−2.63493e6	−0.173931
$746$	0	0
$747$	−8.67390e6	−0.568739
$748$	0	0
$749$	2.09187e6	0.136248
$750$	0	0
$751$	−1.33997e7	−0.866955	−0.433477	−	0.901164i	$-0.642714\pi$
$751$	−1.33997e7	−0.866955	−0.433477	+	0.901164i	$0.642714\pi$
$752$	0	0
$753$	−2.69252e7	−1.73050
$754$	0	0
$755$	1.99375e6	0.127292
$756$	0	0
$757$	−2.47008e7	−1.56665	−0.783324	−	0.621613i	$-0.786478\pi$
$757$	−2.47008e7	−1.56665	−0.783324	+	0.621613i	$0.786478\pi$
$758$	0	0
$759$	−3.61100e6	−0.227522
$760$	0	0
$761$	−3.81253e6	−0.238645	−0.119322	−	0.992856i	$-0.538072\pi$
$761$	−3.81253e6	−0.238645	−0.119322	+	0.992856i	$0.538072\pi$
$762$	0	0
$763$	−1.16827e7	−0.726495
$764$	0	0
$765$	3.89622e6	0.240708
$766$	0	0
$767$	−6.32488e6	−0.388207
$768$	0	0
$769$	−1.77407e7	−1.08182	−0.540909	−	0.841081i	$-0.681920\pi$
$769$	−1.77407e7	−1.08182	−0.540909	+	0.841081i	$0.681920\pi$
$770$	0	0
$771$	2.02365e7	1.22602
$772$	0	0
$773$	−1.10218e7	−0.663445	−0.331723	−	0.943377i	$-0.607630\pi$
$773$	−1.10218e7	−0.663445	−0.331723	+	0.943377i	$0.607630\pi$
$774$	0	0
$775$	2.62366e6	0.156911
$776$	0	0
$777$	5.01668e6	0.298101
$778$	0	0
$779$	−3.29152e6	−0.194336
$780$	0	0
$781$	−4.42866e6	−0.259803
$782$	0	0
$783$	1.05030e6	0.0612221
$784$	0	0
$785$	1.02954e7	0.596308
$786$	0	0
$787$	2.14388e7	1.23386	0.616928	−	0.787020i	$-0.288377\pi$
$787$	2.14388e7	1.23386	0.616928	+	0.787020i	$0.288377\pi$
$788$	0	0
$789$	−6.38768e6	−0.365301
$790$	0	0
$791$	8.24892e6	0.468766
$792$	0	0
$793$	1.57454e7	0.889141
$794$	0	0
$795$	−4.93717e6	−0.277051
$796$	0	0
$797$	1.11610e7	0.622381	0.311190	−	0.950348i	$-0.399272\pi$
$797$	1.11610e7	0.622381	0.311190	+	0.950348i	$0.399272\pi$
$798$	0	0
$799$	−5.95880e6	−0.330211
$800$	0	0
$801$	−6.16818e6	−0.339685
$802$	0	0
$803$	−2.19790e6	−0.120287
$804$	0	0
$805$	402075.	0.0218684
$806$	0	0
$807$	1.90764e7	1.03113
$808$	0	0
$809$	1.13631e7	0.610418	0.305209	−	0.952285i	$-0.401274\pi$
$809$	1.13631e7	0.610418	0.305209	+	0.952285i	$0.401274\pi$
$810$	0	0
$811$	−5.82633e6	−0.311059	−0.155529	−	0.987831i	$-0.549708\pi$
$811$	−5.82633e6	−0.311059	−0.155529	+	0.987831i	$0.549708\pi$
$812$	0	0
$813$	−8.33088e6	−0.442043
$814$	0	0
$815$	−9.25173e6	−0.487898
$816$	0	0
$817$	1.62422e6	0.0851315
$818$	0	0
$819$	5.34109e6	0.278240
$820$	0	0
$821$	−220571.	−0.0114206	−0.00571032	−	0.999984i	$-0.501818\pi$
$821$	−220571.	−0.0114206	−0.00571032	+	0.999984i	$0.501818\pi$
$822$	0	0
$823$	−2.52105e7	−1.29742	−0.648711	−	0.761035i	$-0.724692\pi$
$823$	−2.52105e7	−1.29742	−0.648711	+	0.761035i	$0.724692\pi$
$824$	0	0
$825$	6.87601e6	0.351724
$826$	0	0
$827$	1.49112e7	0.758137	0.379069	−	0.925369i	$-0.376244\pi$
$827$	1.49112e7	0.758137	0.379069	+	0.925369i	$0.376244\pi$
$828$	0	0
$829$	−648075.	−0.0327521	−0.0163760	−	0.999866i	$-0.505213\pi$
$829$	−648075.	−0.0327521	−0.0163760	+	0.999866i	$0.505213\pi$
$830$	0	0
$831$	−770300.	−0.0386952
$832$	0	0
$833$	−1.46021e6	−0.0729127
$834$	0	0
$835$	1.03410e7	0.513271
$836$	0	0
$837$	1.24370e6	0.0613625
$838$	0	0
$839$	1.27092e7	0.623325	0.311663	−	0.950193i	$-0.399114\pi$
$839$	1.27092e7	0.623325	0.311663	+	0.950193i	$0.399114\pi$
$840$	0	0
$841$	−7.94375e6	−0.387289
$842$	0	0
$843$	−2.58993e7	−1.25522
$844$	0	0
$845$	4.75911e6	0.229289
$846$	0	0
$847$	3.98757e6	0.190985
$848$	0	0
$849$	5.08438e7	2.42086
$850$	0	0
$851$	−1.50393e6	−0.0711875
$852$	0	0
$853$	−9.25234e6	−0.435391	−0.217695	−	0.976017i	$-0.569854\pi$
$853$	−9.25234e6	−0.435391	−0.217695	+	0.976017i	$0.569854\pi$
$854$	0	0
$855$	1.29871e6	0.0607573
$856$	0	0
$857$	−9.97717e6	−0.464040	−0.232020	−	0.972711i	$-0.574533\pi$
$857$	−9.97717e6	−0.464040	−0.232020	+	0.972711i	$0.574533\pi$
$858$	0	0
$859$	1.43941e6	0.0665583	0.0332792	−	0.999446i	$-0.489405\pi$
$859$	1.43941e6	0.0665583	0.0332792	+	0.999446i	$0.489405\pi$
$860$	0	0
$861$	1.77772e7	0.817249
$862$	0	0
$863$	2.26701e7	1.03616	0.518080	−	0.855332i	$-0.326647\pi$
$863$	2.26701e7	1.03616	0.518080	+	0.855332i	$0.326647\pi$
$864$	0	0
$865$	1.61693e7	0.734767
$866$	0	0
$867$	−2.34611e7	−1.05999
$868$	0	0
$869$	−3.40585e6	−0.152995
$870$	0	0
$871$	4.98205e6	0.222517
$872$	0	0
$873$	−1.46592e7	−0.650992
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	1.35959e7	0.596911	0.298455	−	0.954424i	$-0.403529\pi$
$877$	1.35959e7	0.596911	0.298455	+	0.954424i	$0.403529\pi$
$878$	0	0
$879$	−3.23927e7	−1.41408
$880$	0	0
$881$	−2.07616e7	−0.901198	−0.450599	−	0.892726i	$-0.648790\pi$
$881$	−2.07616e7	−0.901198	−0.450599	+	0.892726i	$0.648790\pi$
$882$	0	0
$883$	1.75726e7	0.758464	0.379232	−	0.925302i	$-0.376188\pi$
$883$	1.75726e7	0.758464	0.379232	+	0.925302i	$0.376188\pi$
$884$	0	0
$885$	8.30618e6	0.356487
$886$	0	0
$887$	3.79724e7	1.62054	0.810268	−	0.586059i	$-0.199321\pi$
$887$	3.79724e7	1.62054	0.810268	+	0.586059i	$0.199321\pi$
$888$	0	0
$889$	8.91376e6	0.378274
$890$	0	0
$891$	−2.74010e7	−1.15631
$892$	0	0
$893$	−1.98623e6	−0.0833489
$894$	0	0
$895$	9.64654e6	0.402545
$896$	0	0
$897$	−3.11952e6	−0.129451
$898$	0	0
$899$	1.48816e7	0.614116
$900$	0	0
$901$	−5.37525e6	−0.220590
$902$	0	0
$903$	−8.77225e6	−0.358007
$904$	0	0
$905$	1.66732e7	0.676701
$906$	0	0
$907$	−4.44367e7	−1.79359	−0.896796	−	0.442443i	$-0.854112\pi$
$907$	−4.44367e7	−1.79359	−0.896796	+	0.442443i	$0.854112\pi$
$908$	0	0
$909$	3.16243e7	1.26944
$910$	0	0
$911$	1.19111e6	0.0475506	0.0237753	−	0.999717i	$-0.492431\pi$
$911$	1.19111e6	0.0475506	0.0237753	+	0.999717i	$0.492431\pi$
$912$	0	0
$913$	−1.66659e7	−0.661684
$914$	0	0
$915$	−2.06777e7	−0.816488
$916$	0	0
$917$	1.23423e6	0.0484701
$918$	0	0
$919$	−2.59166e7	−1.01225	−0.506127	−	0.862459i	$-0.668923\pi$
$919$	−2.59166e7	−1.01225	−0.506127	+	0.862459i	$0.668923\pi$
$920$	0	0
$921$	−1.73825e7	−0.675250
$922$	0	0
$923$	−3.82589e6	−0.147818
$924$	0	0
$925$	2.86376e6	0.110048
$926$	0	0
$927$	4.64155e7	1.77404
$928$	0	0
$929$	4.34681e7	1.65246	0.826232	−	0.563330i	$-0.190480\pi$
$929$	4.34681e7	1.65246	0.826232	+	0.563330i	$0.190480\pi$
$930$	0	0
$931$	−486728.	−0.0184040
$932$	0	0
$933$	2.74145e7	1.03104
$934$	0	0
$935$	7.48612e6	0.280045
$936$	0	0
$937$	−3.01910e7	−1.12338	−0.561692	−	0.827347i	$-0.689849\pi$
$937$	−3.01910e7	−1.12338	−0.561692	+	0.827347i	$0.689849\pi$
$938$	0	0
$939$	4.34638e6	0.160866
$940$	0	0
$941$	1.38315e7	0.509206	0.254603	−	0.967046i	$-0.418055\pi$
$941$	1.38315e7	0.509206	0.254603	+	0.967046i	$0.418055\pi$
$942$	0	0
$943$	−5.32935e6	−0.195162
$944$	0	0
$945$	−362933.	−0.0132205
$946$	0	0
$947$	−5.29155e7	−1.91738	−0.958690	−	0.284453i	$-0.908188\pi$
$947$	−5.29155e7	−1.91738	−0.958690	+	0.284453i	$0.908188\pi$
$948$	0	0
$949$	−1.89876e6	−0.0684390
$950$	0	0
$951$	−4.39084e7	−1.57433
$952$	0	0
$953$	−2.30533e6	−0.0822245	−0.0411123	−	0.999155i	$-0.513090\pi$
$953$	−2.30533e6	−0.0822245	−0.0411123	+	0.999155i	$0.513090\pi$
$954$	0	0
$955$	−2.23584e7	−0.793290
$956$	0	0
$957$	3.90013e7	1.37657
$958$	0	0
$959$	−1.07220e7	−0.376470
$960$	0	0
$961$	−1.10072e7	−0.384476
$962$	0	0
$963$	1.09400e7	0.380147
$964$	0	0
$965$	2.21197e7	0.764649
$966$	0	0
$967$	2.96101e7	1.01829	0.509147	−	0.860679i	$-0.329961\pi$
$967$	2.96101e7	1.01829	0.509147	+	0.860679i	$0.329961\pi$
$968$	0	0
$969$	2.75474e6	0.0942478
$970$	0	0
$971$	2.14233e7	0.729185	0.364592	−	0.931167i	$-0.381208\pi$
$971$	2.14233e7	0.729185	0.364592	+	0.931167i	$0.381208\pi$
$972$	0	0
$973$	−1.05041e7	−0.355693
$974$	0	0
$975$	5.94014e6	0.200118
$976$	0	0
$977$	2.53733e7	0.850435	0.425217	−	0.905091i	$-0.360198\pi$
$977$	2.53733e7	0.850435	0.425217	+	0.905091i	$0.360198\pi$
$978$	0	0
$979$	−1.18514e7	−0.395197
$980$	0	0
$981$	−6.10981e7	−2.02701
$982$	0	0
$983$	5.75355e6	0.189912	0.0949560	−	0.995481i	$-0.469729\pi$
$983$	5.75355e6	0.189912	0.0949560	+	0.995481i	$0.469729\pi$
$984$	0	0
$985$	577359.	0.0189608
$986$	0	0
$987$	1.07274e7	0.350511
$988$	0	0
$989$	2.62980e6	0.0854933
$990$	0	0
$991$	3.20939e7	1.03810	0.519049	−	0.854744i	$-0.326286\pi$
$991$	3.20939e7	1.03810	0.519049	+	0.854744i	$0.326286\pi$
$992$	0	0
$993$	−6.15038e7	−1.97938
$994$	0	0
$995$	2.34718e7	0.751604
$996$	0	0
$997$	−1.94747e6	−0.0620488	−0.0310244	−	0.999519i	$-0.509877\pi$
$997$	−1.94747e6	−0.0620488	−0.0310244	+	0.999519i	$0.509877\pi$
$998$	0	0
$999$	1.35752e6	0.0430361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.f.1.3	✓	3
4.3	odd	2		560.6.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.f.1.3	✓	3	1.1	even	1	trivial
560.6.a.r.1.1		3	4.3	odd	2

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:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.a (trivial)

\(f(q)\)	\(=\)	\(q+22.3441 q^{3} -25.0000 q^{5} +49.0000 q^{7} +256.259 q^{9} +O(q^{10})\)	\(q+22.3441 q^{3} -25.0000 q^{5} +49.0000 q^{7} +256.259 q^{9} +492.372 q^{11} +425.357 q^{13} -558.603 q^{15} -608.168 q^{17} -202.719 q^{19} +1094.86 q^{21} -328.225 q^{23} +625.000 q^{25} +296.272 q^{27} +3545.05 q^{29} +4197.85 q^{31} +11001.6 q^{33} -1225.00 q^{35} +4582.02 q^{37} +9504.23 q^{39} +16236.9 q^{41} -8012.20 q^{43} -6406.49 q^{45} +9797.94 q^{47} +2401.00 q^{49} -13589.0 q^{51} +8838.42 q^{53} -12309.3 q^{55} -4529.57 q^{57} -14869.6 q^{59} +37016.9 q^{61} +12556.7 q^{63} -10633.9 q^{65} +11712.6 q^{67} -7333.89 q^{69} -8994.54 q^{71} -4463.91 q^{73} +13965.1 q^{75} +24126.2 q^{77} -6917.23 q^{79} -55651.1 q^{81} -33848.1 q^{83} +15204.2 q^{85} +79211.1 q^{87} -24070.1 q^{89} +20842.5 q^{91} +93797.2 q^{93} +5067.97 q^{95} -57204.6 q^{97} +126175. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9}+O(q^{10}) \) 3 * q + 14 * q^3 - 75 * q^5 + 147 * q^7 + q^9	\( 3 q + 14 q^{3} - 75 q^{5} + 147 q^{7} + q^{9} - 50 q^{11} + 536 q^{13} - 350 q^{15} + 268 q^{17} - 1048 q^{19} + 686 q^{21} - 576 q^{23} + 1875 q^{25} + 1754 q^{27} + 6172 q^{29} - 2404 q^{31} + 11454 q^{33} - 3675 q^{35} + 7806 q^{37} + 2834 q^{39} + 12150 q^{41} + 16868 q^{43} - 25 q^{45} + 30230 q^{47} + 7203 q^{49} + 16758 q^{51} + 25862 q^{53} + 1250 q^{55} + 25960 q^{57} + 36352 q^{59} + 6534 q^{61} + 49 q^{63} - 13400 q^{65} + 45708 q^{67} + 17528 q^{69} + 57456 q^{71} - 17346 q^{73} + 8750 q^{75} - 2450 q^{77} + 21222 q^{79} - 65501 q^{81} + 128816 q^{83} - 6700 q^{85} + 83250 q^{87} + 132286 q^{89} + 26264 q^{91} + 103716 q^{93} + 26200 q^{95} - 74348 q^{97} + 210504 q^{99}+O(q^{100}) \) 3 * q + 14 * q^3 - 75 * q^5 + 147 * q^7 + q^9 - 50 * q^11 + 536 * q^13 - 350 * q^15 + 268 * q^17 - 1048 * q^19 + 686 * q^21 - 576 * q^23 + 1875 * q^25 + 1754 * q^27 + 6172 * q^29 - 2404 * q^31 + 11454 * q^33 - 3675 * q^35 + 7806 * q^37 + 2834 * q^39 + 12150 * q^41 + 16868 * q^43 - 25 * q^45 + 30230 * q^47 + 7203 * q^49 + 16758 * q^51 + 25862 * q^53 + 1250 * q^55 + 25960 * q^57 + 36352 * q^59 + 6534 * q^61 + 49 * q^63 - 13400 * q^65 + 45708 * q^67 + 17528 * q^69 + 57456 * q^71 - 17346 * q^73 + 8750 * q^75 - 2450 * q^77 + 21222 * q^79 - 65501 * q^81 + 128816 * q^83 - 6700 * q^85 + 83250 * q^87 + 132286 * q^89 + 26264 * q^91 + 103716 * q^93 + 26200 * q^95 - 74348 * q^97 + 210504 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more