LMFDB - Embedded newform 280.6.a.g.1.1

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:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 232x^{2} + 60x + 5808 $ x^4 - 2x^3 - 232x^2 + 60*x + 5808
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.35079$ 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)

$f(q)$	$=$	$q-17.3765 q^{3} +25.0000 q^{5} +49.0000 q^{7} +58.9420 q^{9} +O(q^{10})$	$q-17.3765 q^{3} +25.0000 q^{5} +49.0000 q^{7} +58.9420 q^{9} +12.2951 q^{11} -678.753 q^{13} -434.412 q^{15} +1508.01 q^{17} -1394.96 q^{19} -851.447 q^{21} +4192.02 q^{23} +625.000 q^{25} +3198.28 q^{27} +6736.73 q^{29} -9618.57 q^{31} -213.646 q^{33} +1225.00 q^{35} -4550.79 q^{37} +11794.3 q^{39} +18293.8 q^{41} -18476.0 q^{43} +1473.55 q^{45} -22498.3 q^{47} +2401.00 q^{49} -26203.8 q^{51} -29071.8 q^{53} +307.378 q^{55} +24239.5 q^{57} -6521.68 q^{59} +8736.94 q^{61} +2888.16 q^{63} -16968.8 q^{65} -15869.0 q^{67} -72842.5 q^{69} -21920.7 q^{71} -29740.8 q^{73} -10860.3 q^{75} +602.462 q^{77} -65215.7 q^{79} -69897.7 q^{81} -102969. q^{83} +37700.2 q^{85} -117061. q^{87} +116391. q^{89} -33258.9 q^{91} +167137. q^{93} -34874.1 q^{95} -118685. q^{97} +724.700 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9}+O(q^{10}) $ 4 * q - 13 * q^3 + 100 * q^5 + 196 * q^7 - 193 * q^9	$ 4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9} - 595 q^{11} - 969 q^{13} - 325 q^{15} - 1315 q^{17} - 1090 q^{19} - 637 q^{21} - 1534 q^{23} + 2500 q^{25} + 173 q^{27} + 4099 q^{29} - 4820 q^{31} + 4149 q^{33} + 4900 q^{35} + 7692 q^{37} - 6371 q^{39} - 9722 q^{41} - 20610 q^{43} - 4825 q^{45} - 1661 q^{47} + 9604 q^{49} - 73361 q^{51} - 28898 q^{53} - 14875 q^{55} - 21246 q^{57} - 101872 q^{59} - 24742 q^{61} - 9457 q^{63} - 24225 q^{65} - 82060 q^{67} + 16914 q^{69} - 102784 q^{71} - 80652 q^{73} - 8125 q^{75} - 29155 q^{77} - 117801 q^{79} - 141052 q^{81} - 155440 q^{83} - 32875 q^{85} - 82519 q^{87} + 56426 q^{89} - 47481 q^{91} - 17332 q^{93} - 27250 q^{95} - 261031 q^{97} - 61686 q^{99}+O(q^{100}) $ 4 * q - 13 * q^3 + 100 * q^5 + 196 * q^7 - 193 * q^9 - 595 * q^11 - 969 * q^13 - 325 * q^15 - 1315 * q^17 - 1090 * q^19 - 637 * q^21 - 1534 * q^23 + 2500 * q^25 + 173 * q^27 + 4099 * q^29 - 4820 * q^31 + 4149 * q^33 + 4900 * q^35 + 7692 * q^37 - 6371 * q^39 - 9722 * q^41 - 20610 * q^43 - 4825 * q^45 - 1661 * q^47 + 9604 * q^49 - 73361 * q^51 - 28898 * q^53 - 14875 * q^55 - 21246 * q^57 - 101872 * q^59 - 24742 * q^61 - 9457 * q^63 - 24225 * q^65 - 82060 * q^67 + 16914 * q^69 - 102784 * q^71 - 80652 * q^73 - 8125 * q^75 - 29155 * q^77 - 117801 * q^79 - 141052 * q^81 - 155440 * q^83 - 32875 * q^85 - 82519 * q^87 + 56426 * q^89 - 47481 * q^91 - 17332 * q^93 - 27250 * q^95 - 261031 * q^97 - 61686 * 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
$3$	−17.3765	−1.11470	−0.557351	−	0.830277i	$-0.688182\pi$
$3$	−17.3765	−1.11470	−0.557351	+	0.830277i	$0.688182\pi$
$4$	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
$9$	58.9420	0.242560
$10$	0	0
$11$	12.2951	0.0306374	0.0153187	−	0.999883i	$-0.495124\pi$
$11$	12.2951	0.0306374	0.0153187	+	0.999883i	$0.495124\pi$
$12$	0	0
$13$	−678.753	−1.11392	−0.556959	−	0.830540i	$-0.688032\pi$
$13$	−678.753	−1.11392	−0.556959	+	0.830540i	$0.688032\pi$
$14$	0	0
$15$	−434.412	−0.498510
$16$	0	0
$17$	1508.01	1.26555	0.632777	−	0.774334i	$-0.281915\pi$
$17$	1508.01	1.26555	0.632777	+	0.774334i	$0.281915\pi$
$18$	0	0
$19$	−1394.96	−0.886499	−0.443250	−	0.896398i	$-0.646175\pi$
$19$	−1394.96	−0.886499	−0.443250	+	0.896398i	$0.646175\pi$
$20$	0	0
$21$	−851.447	−0.421318
$22$	0	0
$23$	4192.02	1.65236	0.826178	−	0.563409i	$-0.190511\pi$
$23$	4192.02	1.65236	0.826178	+	0.563409i	$0.190511\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	3198.28	0.844320
$28$	0	0
$29$	6736.73	1.48749	0.743745	−	0.668464i	$-0.233048\pi$
$29$	6736.73	1.48749	0.743745	+	0.668464i	$0.233048\pi$
$30$	0	0
$31$	−9618.57	−1.79766	−0.898828	−	0.438302i	$-0.855580\pi$
$31$	−9618.57	−1.79766	−0.898828	+	0.438302i	$0.855580\pi$
$32$	0	0
$33$	−213.646	−0.0341515
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−4550.79	−0.546491	−0.273245	−	0.961944i	$-0.588097\pi$
$37$	−4550.79	−0.546491	−0.273245	+	0.961944i	$0.588097\pi$
$38$	0	0
$39$	11794.3	1.24169
$40$	0	0
$41$	18293.8	1.69959	0.849795	−	0.527114i	$-0.176726\pi$
$41$	18293.8	1.69959	0.849795	+	0.527114i	$0.176726\pi$
$42$	0	0
$43$	−18476.0	−1.52383	−0.761915	−	0.647677i	$-0.775741\pi$
$43$	−18476.0	−1.52383	−0.761915	+	0.647677i	$0.775741\pi$
$44$	0	0
$45$	1473.55	0.108476
$46$	0	0
$47$	−22498.3	−1.48561	−0.742804	−	0.669509i	$-0.766505\pi$
$47$	−22498.3	−1.48561	−0.742804	+	0.669509i	$0.766505\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−26203.8	−1.41072
$52$	0	0
$53$	−29071.8	−1.42162	−0.710809	−	0.703385i	$-0.751671\pi$
$53$	−29071.8	−1.42162	−0.710809	+	0.703385i	$0.751671\pi$
$54$	0	0
$55$	307.378	0.0137015
$56$	0	0
$57$	24239.5	0.988182
$58$	0	0
$59$	−6521.68	−0.243910	−0.121955	−	0.992536i	$-0.538916\pi$
$59$	−6521.68	−0.243910	−0.121955	+	0.992536i	$0.538916\pi$
$60$	0	0
$61$	8736.94	0.300632	0.150316	−	0.988638i	$-0.451971\pi$
$61$	8736.94	0.300632	0.150316	+	0.988638i	$0.451971\pi$
$62$	0	0
$63$	2888.16	0.0916789
$64$	0	0
$65$	−16968.8	−0.498159
$66$	0	0
$67$	−15869.0	−0.431880	−0.215940	−	0.976407i	$-0.569282\pi$
$67$	−15869.0	−0.431880	−0.215940	+	0.976407i	$0.569282\pi$
$68$	0	0
$69$	−72842.5	−1.84188
$70$	0	0
$71$	−21920.7	−0.516069	−0.258034	−	0.966136i	$-0.583075\pi$
$71$	−21920.7	−0.516069	−0.258034	+	0.966136i	$0.583075\pi$
$72$	0	0
$73$	−29740.8	−0.653198	−0.326599	−	0.945163i	$-0.605903\pi$
$73$	−29740.8	−0.653198	−0.326599	+	0.945163i	$0.605903\pi$
$74$	0	0
$75$	−10860.3	−0.222940
$76$	0	0
$77$	602.462	0.0115798
$78$	0	0
$79$	−65215.7	−1.17567	−0.587834	−	0.808982i	$-0.700019\pi$
$79$	−65215.7	−1.17567	−0.587834	+	0.808982i	$0.700019\pi$
$80$	0	0
$81$	−69897.7	−1.18372
$82$	0	0
$83$	−102969.	−1.64063	−0.820313	−	0.571915i	$-0.806201\pi$
$83$	−102969.	−1.64063	−0.820313	+	0.571915i	$0.806201\pi$
$84$	0	0
$85$	37700.2	0.565973
$86$	0	0
$87$	−117061.	−1.65811
$88$	0	0
$89$	116391.	1.55756	0.778779	−	0.627298i	$-0.215839\pi$
$89$	116391.	1.55756	0.778779	+	0.627298i	$0.215839\pi$
$90$	0	0
$91$	−33258.9	−0.421021
$92$	0	0
$93$	167137.	2.00385
$94$	0	0
$95$	−34874.1	−0.396455
$96$	0	0
$97$	−118685.	−1.28076	−0.640380	−	0.768058i	$-0.721223\pi$
$97$	−118685.	−1.28076	−0.640380	+	0.768058i	$0.721223\pi$
$98$	0	0
$99$	724.700	0.00743139
$100$	0	0
$101$	−78470.7	−0.765427	−0.382714	−	0.923867i	$-0.625010\pi$
$101$	−78470.7	−0.765427	−0.382714	+	0.923867i	$0.625010\pi$
$102$	0	0
$103$	57488.8	0.533938	0.266969	−	0.963705i	$-0.413978\pi$
$103$	57488.8	0.533938	0.266969	+	0.963705i	$0.413978\pi$
$104$	0	0
$105$	−21286.2	−0.188419
$106$	0	0
$107$	140884.	1.18960	0.594801	−	0.803873i	$-0.297231\pi$
$107$	140884.	1.18960	0.594801	+	0.803873i	$0.297231\pi$
$108$	0	0
$109$	93968.6	0.757559	0.378780	−	0.925487i	$-0.376344\pi$
$109$	93968.6	0.757559	0.378780	+	0.925487i	$0.376344\pi$
$110$	0	0
$111$	79076.7	0.609174
$112$	0	0
$113$	−207096.	−1.52572	−0.762861	−	0.646562i	$-0.776206\pi$
$113$	−207096.	−1.52572	−0.762861	+	0.646562i	$0.776206\pi$
$114$	0	0
$115$	104800.	0.738956
$116$	0	0
$117$	−40007.0	−0.270191
$118$	0	0
$119$	73892.3	0.478335
$120$	0	0
$121$	−160900.	−0.999061
$122$	0	0
$123$	−317882.	−1.89453
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	155471.	0.855342	0.427671	−	0.903934i	$-0.359334\pi$
$127$	155471.	0.855342	0.427671	+	0.903934i	$0.359334\pi$
$128$	0	0
$129$	321048.	1.69862
$130$	0	0
$131$	−314522.	−1.60130	−0.800651	−	0.599131i	$-0.795513\pi$
$131$	−314522.	−1.60130	−0.800651	+	0.599131i	$0.795513\pi$
$132$	0	0
$133$	−68353.2	−0.335065
$134$	0	0
$135$	79957.0	0.377591
$136$	0	0
$137$	−18334.9	−0.0834598	−0.0417299	−	0.999129i	$-0.513287\pi$
$137$	−18334.9	−0.0834598	−0.0417299	+	0.999129i	$0.513287\pi$
$138$	0	0
$139$	−2632.98	−0.0115588	−0.00577938	−	0.999983i	$-0.501840\pi$
$139$	−2632.98	−0.0115588	−0.00577938	+	0.999983i	$0.501840\pi$
$140$	0	0
$141$	390940.	1.65601
$142$	0	0
$143$	−8345.35	−0.0341275
$144$	0	0
$145$	168418.	0.665226
$146$	0	0
$147$	−41720.9	−0.159243
$148$	0	0
$149$	145001.	0.535063	0.267531	−	0.963549i	$-0.413792\pi$
$149$	145001.	0.535063	0.267531	+	0.963549i	$0.413792\pi$
$150$	0	0
$151$	220741.	0.787843	0.393921	−	0.919144i	$-0.371118\pi$
$151$	220741.	0.787843	0.393921	+	0.919144i	$0.371118\pi$
$152$	0	0
$153$	88884.9	0.306972
$154$	0	0
$155$	−240464.	−0.803936
$156$	0	0
$157$	−441747.	−1.43029	−0.715145	−	0.698976i	$-0.753639\pi$
$157$	−441747.	−1.43029	−0.715145	+	0.698976i	$0.753639\pi$
$158$	0	0
$159$	505166.	1.58468
$160$	0	0
$161$	205409.	0.624532
$162$	0	0
$163$	−511902.	−1.50910	−0.754549	−	0.656243i	$-0.772144\pi$
$163$	−511902.	−1.50910	−0.754549	+	0.656243i	$0.772144\pi$
$164$	0	0
$165$	−5341.15	−0.0152730
$166$	0	0
$167$	−712308.	−1.97641	−0.988204	−	0.153144i	$-0.951060\pi$
$167$	−712308.	−1.97641	−0.988204	+	0.153144i	$0.951060\pi$
$168$	0	0
$169$	89412.1	0.240813
$170$	0	0
$171$	−82221.9	−0.215029
$172$	0	0
$173$	452601.	1.14974	0.574870	−	0.818245i	$-0.305053\pi$
$173$	452601.	1.14974	0.574870	+	0.818245i	$0.305053\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	113324.	0.271887
$178$	0	0
$179$	−48023.2	−0.112026	−0.0560129	−	0.998430i	$-0.517839\pi$
$179$	−48023.2	−0.112026	−0.0560129	+	0.998430i	$0.517839\pi$
$180$	0	0
$181$	−633941.	−1.43831	−0.719154	−	0.694850i	$-0.755471\pi$
$181$	−633941.	−1.43831	−0.719154	+	0.694850i	$0.755471\pi$
$182$	0	0
$183$	−151817.	−0.335115
$184$	0	0
$185$	−113770.	−0.244398
$186$	0	0
$187$	18541.1	0.0387733
$188$	0	0
$189$	156716.	0.319123
$190$	0	0
$191$	403127.	0.799573	0.399787	−	0.916608i	$-0.369084\pi$
$191$	403127.	0.799573	0.399787	+	0.916608i	$0.369084\pi$
$192$	0	0
$193$	369374.	0.713794	0.356897	−	0.934144i	$-0.383835\pi$
$193$	369374.	0.713794	0.356897	+	0.934144i	$0.383835\pi$
$194$	0	0
$195$	294858.	0.555299
$196$	0	0
$197$	659845.	1.21137	0.605684	−	0.795705i	$-0.292899\pi$
$197$	659845.	1.21137	0.605684	+	0.795705i	$0.292899\pi$
$198$	0	0
$199$	−474610.	−0.849581	−0.424790	−	0.905292i	$-0.639652\pi$
$199$	−474610.	−0.849581	−0.424790	+	0.905292i	$0.639652\pi$
$200$	0	0
$201$	275748.	0.481418
$202$	0	0
$203$	330100.	0.562218
$204$	0	0
$205$	457345.	0.760079
$206$	0	0
$207$	247086.	0.400795
$208$	0	0
$209$	−17151.2	−0.0271600
$210$	0	0
$211$	666444.	1.03052	0.515261	−	0.857033i	$-0.327695\pi$
$211$	666444.	1.03052	0.515261	+	0.857033i	$0.327695\pi$
$212$	0	0
$213$	380904.	0.575263
$214$	0	0
$215$	−461900.	−0.681477
$216$	0	0
$217$	−471310.	−0.679450
$218$	0	0
$219$	516790.	0.728121
$220$	0	0
$221$	−1.02356e6	−1.40972
$222$	0	0
$223$	−1.02172e6	−1.37585	−0.687924	−	0.725783i	$-0.741478\pi$
$223$	−1.02172e6	−1.37585	−0.687924	+	0.725783i	$0.741478\pi$
$224$	0	0
$225$	36838.7	0.0485119
$226$	0	0
$227$	−489354.	−0.630316	−0.315158	−	0.949039i	$-0.602058\pi$
$227$	−489354.	−0.630316	−0.315158	+	0.949039i	$0.602058\pi$
$228$	0	0
$229$	877333.	1.10554	0.552772	−	0.833333i	$-0.313570\pi$
$229$	877333.	1.10554	0.552772	+	0.833333i	$0.313570\pi$
$230$	0	0
$231$	−10468.7	−0.0129081
$232$	0	0
$233$	803477.	0.969580	0.484790	−	0.874631i	$-0.338896\pi$
$233$	803477.	0.969580	0.484790	+	0.874631i	$0.338896\pi$
$234$	0	0
$235$	−562456.	−0.664384
$236$	0	0
$237$	1.13322e6	1.31052
$238$	0	0
$239$	1.63686e6	1.85361	0.926803	−	0.375548i	$-0.122545\pi$
$239$	1.63686e6	1.85361	0.926803	+	0.375548i	$0.122545\pi$
$240$	0	0
$241$	1.16907e6	1.29657	0.648287	−	0.761396i	$-0.275486\pi$
$241$	1.16907e6	1.29657	0.648287	+	0.761396i	$0.275486\pi$
$242$	0	0
$243$	437395.	0.475180
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	946834.	0.987487
$248$	0	0
$249$	1.78923e6	1.82881
$250$	0	0
$251$	−1.19373e6	−1.19597	−0.597985	−	0.801508i	$-0.704032\pi$
$251$	−1.19373e6	−1.19597	−0.597985	+	0.801508i	$0.704032\pi$
$252$	0	0
$253$	51541.4	0.0506238
$254$	0	0
$255$	−655096.	−0.630891
$256$	0	0
$257$	−374511.	−0.353697	−0.176849	−	0.984238i	$-0.556590\pi$
$257$	−374511.	−0.353697	−0.176849	+	0.984238i	$0.556590\pi$
$258$	0	0
$259$	−222989.	−0.206554
$260$	0	0
$261$	397076.	0.360805
$262$	0	0
$263$	1.39145e6	1.24045	0.620224	−	0.784425i	$-0.287042\pi$
$263$	1.39145e6	1.24045	0.620224	+	0.784425i	$0.287042\pi$
$264$	0	0
$265$	−726796.	−0.635767
$266$	0	0
$267$	−2.02247e6	−1.73621
$268$	0	0
$269$	−502448.	−0.423360	−0.211680	−	0.977339i	$-0.567894\pi$
$269$	−502448.	−0.423360	−0.211680	+	0.977339i	$0.567894\pi$
$270$	0	0
$271$	619842.	0.512693	0.256347	−	0.966585i	$-0.417481\pi$
$271$	619842.	0.512693	0.256347	+	0.966585i	$0.417481\pi$
$272$	0	0
$273$	577922.	0.469313
$274$	0	0
$275$	7684.46	0.00612748
$276$	0	0
$277$	−1.28566e6	−1.00677	−0.503383	−	0.864064i	$-0.667911\pi$
$277$	−1.28566e6	−1.00677	−0.503383	+	0.864064i	$0.667911\pi$
$278$	0	0
$279$	−566938.	−0.436039
$280$	0	0
$281$	329027.	0.248579	0.124290	−	0.992246i	$-0.460335\pi$
$281$	329027.	0.248579	0.124290	+	0.992246i	$0.460335\pi$
$282$	0	0
$283$	−740249.	−0.549429	−0.274715	−	0.961526i	$-0.588583\pi$
$283$	−740249.	−0.549429	−0.274715	+	0.961526i	$0.588583\pi$
$284$	0	0
$285$	605988.	0.441929
$286$	0	0
$287$	896396.	0.642384
$288$	0	0
$289$	854226.	0.601628
$290$	0	0
$291$	2.06233e6	1.42767
$292$	0	0
$293$	182685.	0.124318	0.0621588	−	0.998066i	$-0.480201\pi$
$293$	182685.	0.124318	0.0621588	+	0.998066i	$0.480201\pi$
$294$	0	0
$295$	−163042.	−0.109080
$296$	0	0
$297$	39323.3	0.0258677
$298$	0	0
$299$	−2.84534e6	−1.84059
$300$	0	0
$301$	−905323.	−0.575953
$302$	0	0
$303$	1.36354e6	0.853223
$304$	0	0
$305$	218424.	0.134447
$306$	0	0
$307$	−1.56942e6	−0.950368	−0.475184	−	0.879886i	$-0.657619\pi$
$307$	−1.56942e6	−0.950368	−0.475184	+	0.879886i	$0.657619\pi$
$308$	0	0
$309$	−998953.	−0.595181
$310$	0	0
$311$	318418.	0.186680	0.0933399	−	0.995634i	$-0.470246\pi$
$311$	318418.	0.186680	0.0933399	+	0.995634i	$0.470246\pi$
$312$	0	0
$313$	2.63277e6	1.51898	0.759490	−	0.650519i	$-0.225449\pi$
$313$	2.63277e6	1.51898	0.759490	+	0.650519i	$0.225449\pi$
$314$	0	0
$315$	72203.9	0.0410001
$316$	0	0
$317$	−2.67715e6	−1.49632	−0.748161	−	0.663517i	$-0.769063\pi$
$317$	−2.67715e6	−1.49632	−0.748161	+	0.663517i	$0.769063\pi$
$318$	0	0
$319$	82829.0	0.0455728
$320$	0	0
$321$	−2.44806e6	−1.32605
$322$	0	0
$323$	−2.10361e6	−1.12191
$324$	0	0
$325$	−424220.	−0.222784
$326$	0	0
$327$	−1.63284e6	−0.844452
$328$	0	0
$329$	−1.10241e6	−0.561507
$330$	0	0
$331$	1.68153e6	0.843598	0.421799	−	0.906689i	$-0.361399\pi$
$331$	1.68153e6	0.843598	0.421799	+	0.906689i	$0.361399\pi$
$332$	0	0
$333$	−268233.	−0.132557
$334$	0	0
$335$	−396726.	−0.193143
$336$	0	0
$337$	−1.37934e6	−0.661600	−0.330800	−	0.943701i	$-0.607319\pi$
$337$	−1.37934e6	−0.661600	−0.330800	+	0.943701i	$0.607319\pi$
$338$	0	0
$339$	3.59860e6	1.70073
$340$	0	0
$341$	−118262.	−0.0550754
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−1.82106e6	−0.823715
$346$	0	0
$347$	1.05189e6	0.468973	0.234486	−	0.972119i	$-0.424659\pi$
$347$	1.05189e6	0.468973	0.234486	+	0.972119i	$0.424659\pi$
$348$	0	0
$349$	−358733.	−0.157655	−0.0788274	−	0.996888i	$-0.525118\pi$
$349$	−358733.	−0.157655	−0.0788274	+	0.996888i	$0.525118\pi$
$350$	0	0
$351$	−2.17084e6	−0.940503
$352$	0	0
$353$	−2.50034e6	−1.06798	−0.533990	−	0.845491i	$-0.679308\pi$
$353$	−2.50034e6	−1.06798	−0.533990	+	0.845491i	$0.679308\pi$
$354$	0	0
$355$	−548017.	−0.230793
$356$	0	0
$357$	−1.28399e6	−0.533200
$358$	0	0
$359$	−754687.	−0.309051	−0.154526	−	0.987989i	$-0.549385\pi$
$359$	−754687.	−0.309051	−0.154526	+	0.987989i	$0.549385\pi$
$360$	0	0
$361$	−530179.	−0.214119
$362$	0	0
$363$	2.79587e6	1.11366
$364$	0	0
$365$	−743519.	−0.292119
$366$	0	0
$367$	−1.12698e6	−0.436767	−0.218383	−	0.975863i	$-0.570078\pi$
$367$	−1.12698e6	−0.436767	−0.218383	+	0.975863i	$0.570078\pi$
$368$	0	0
$369$	1.07827e6	0.412252
$370$	0	0
$371$	−1.42452e6	−0.537321
$372$	0	0
$373$	−2.83541e6	−1.05522	−0.527611	−	0.849486i	$-0.676912\pi$
$373$	−2.83541e6	−1.05522	−0.527611	+	0.849486i	$0.676912\pi$
$374$	0	0
$375$	−271507.	−0.0997019
$376$	0	0
$377$	−4.57257e6	−1.65694
$378$	0	0
$379$	−1.82276e6	−0.651824	−0.325912	−	0.945400i	$-0.605671\pi$
$379$	−1.82276e6	−0.651824	−0.325912	+	0.945400i	$0.605671\pi$
$380$	0	0
$381$	−2.70154e6	−0.953451
$382$	0	0
$383$	−23039.8	−0.00802569	−0.00401285	−	0.999992i	$-0.501277\pi$
$383$	−23039.8	−0.00802569	−0.00401285	+	0.999992i	$0.501277\pi$
$384$	0	0
$385$	15061.5	0.00517866
$386$	0	0
$387$	−1.08901e6	−0.369620
$388$	0	0
$389$	−1.25000e6	−0.418829	−0.209415	−	0.977827i	$-0.567156\pi$
$389$	−1.25000e6	−0.418829	−0.209415	+	0.977827i	$0.567156\pi$
$390$	0	0
$391$	6.32159e6	2.09115
$392$	0	0
$393$	5.46529e6	1.78497
$394$	0	0
$395$	−1.63039e6	−0.525774
$396$	0	0
$397$	−2.86228e6	−0.911458	−0.455729	−	0.890119i	$-0.650621\pi$
$397$	−2.86228e6	−0.911458	−0.455729	+	0.890119i	$0.650621\pi$
$398$	0	0
$399$	1.18774e6	0.373498
$400$	0	0
$401$	989803.	0.307389	0.153694	−	0.988118i	$-0.450883\pi$
$401$	989803.	0.307389	0.153694	+	0.988118i	$0.450883\pi$
$402$	0	0
$403$	6.52863e6	2.00244
$404$	0	0
$405$	−1.74744e6	−0.529378
$406$	0	0
$407$	−55952.6	−0.0167430
$408$	0	0
$409$	−4.21316e6	−1.24537	−0.622687	−	0.782471i	$-0.713959\pi$
$409$	−4.21316e6	−1.24537	−0.622687	+	0.782471i	$0.713959\pi$
$410$	0	0
$411$	318596.	0.0930328
$412$	0	0
$413$	−319562.	−0.0921893
$414$	0	0
$415$	−2.57421e6	−0.733710
$416$	0	0
$417$	45752.0	0.0128846
$418$	0	0
$419$	−1.11376e6	−0.309926	−0.154963	−	0.987920i	$-0.549526\pi$
$419$	−1.11376e6	−0.309926	−0.154963	+	0.987920i	$0.549526\pi$
$420$	0	0
$421$	−2.98167e6	−0.819888	−0.409944	−	0.912111i	$-0.634452\pi$
$421$	−2.98167e6	−0.819888	−0.409944	+	0.912111i	$0.634452\pi$
$422$	0	0
$423$	−1.32609e6	−0.360349
$424$	0	0
$425$	942504.	0.253111
$426$	0	0
$427$	428110.	0.113628
$428$	0	0
$429$	145013.	0.0380420
$430$	0	0
$431$	3.21821e6	0.834490	0.417245	−	0.908794i	$-0.362996\pi$
$431$	3.21821e6	0.834490	0.417245	+	0.908794i	$0.362996\pi$
$432$	0	0
$433$	2.06658e6	0.529703	0.264851	−	0.964289i	$-0.414677\pi$
$433$	2.06658e6	0.529703	0.264851	+	0.964289i	$0.414677\pi$
$434$	0	0
$435$	−2.92651e6	−0.741528
$436$	0	0
$437$	−5.84771e6	−1.46481
$438$	0	0
$439$	3.72548e6	0.922615	0.461308	−	0.887240i	$-0.347380\pi$
$439$	3.72548e6	0.922615	0.461308	+	0.887240i	$0.347380\pi$
$440$	0	0
$441$	141520.	0.0346514
$442$	0	0
$443$	3.16294e6	0.765740	0.382870	−	0.923802i	$-0.374936\pi$
$443$	3.16294e6	0.765740	0.382870	+	0.923802i	$0.374936\pi$
$444$	0	0
$445$	2.90978e6	0.696562
$446$	0	0
$447$	−2.51960e6	−0.596435
$448$	0	0
$449$	2.95841e6	0.692537	0.346268	−	0.938135i	$-0.387449\pi$
$449$	2.95841e6	0.692537	0.346268	+	0.938135i	$0.387449\pi$
$450$	0	0
$451$	224925.	0.0520710
$452$	0	0
$453$	−3.83569e6	−0.878210
$454$	0	0
$455$	−831472.	−0.188286
$456$	0	0
$457$	−180703.	−0.0404738	−0.0202369	−	0.999795i	$-0.506442\pi$
$457$	−180703.	−0.0404738	−0.0202369	+	0.999795i	$0.506442\pi$
$458$	0	0
$459$	4.82303e6	1.06853
$460$	0	0
$461$	−1.73861e6	−0.381021	−0.190511	−	0.981685i	$-0.561014\pi$
$461$	−1.73861e6	−0.381021	−0.190511	+	0.981685i	$0.561014\pi$
$462$	0	0
$463$	1.25266e6	0.271570	0.135785	−	0.990738i	$-0.456644\pi$
$463$	1.25266e6	0.271570	0.135785	+	0.990738i	$0.456644\pi$
$464$	0	0
$465$	4.17842e6	0.896149
$466$	0	0
$467$	5.40683e6	1.14723	0.573615	−	0.819125i	$-0.305541\pi$
$467$	5.40683e6	1.14723	0.573615	+	0.819125i	$0.305541\pi$
$468$	0	0
$469$	−777582.	−0.163235
$470$	0	0
$471$	7.67600e6	1.59435
$472$	0	0
$473$	−227165.	−0.0466861
$474$	0	0
$475$	−871851.	−0.177300
$476$	0	0
$477$	−1.71355e6	−0.344827
$478$	0	0
$479$	−1.94788e6	−0.387904	−0.193952	−	0.981011i	$-0.562131\pi$
$479$	−1.94788e6	−0.387904	−0.193952	+	0.981011i	$0.562131\pi$
$480$	0	0
$481$	3.08886e6	0.608746
$482$	0	0
$483$	−3.56928e6	−0.696167
$484$	0	0
$485$	−2.96714e6	−0.572774
$486$	0	0
$487$	3.14918e6	0.601693	0.300847	−	0.953673i	$-0.402731\pi$
$487$	3.14918e6	0.601693	0.300847	+	0.953673i	$0.402731\pi$
$488$	0	0
$489$	8.89505e6	1.68219
$490$	0	0
$491$	303464.	0.0568073	0.0284036	−	0.999597i	$-0.490958\pi$
$491$	303464.	0.0568073	0.0284036	+	0.999597i	$0.490958\pi$
$492$	0	0
$493$	1.01590e7	1.88250
$494$	0	0
$495$	18117.5	0.00332342
$496$	0	0
$497$	−1.07411e6	−0.195056
$498$	0	0
$499$	4.34208e6	0.780632	0.390316	−	0.920681i	$-0.372366\pi$
$499$	4.34208e6	0.780632	0.390316	+	0.920681i	$0.372366\pi$
$500$	0	0
$501$	1.23774e7	2.20310
$502$	0	0
$503$	−8.22681e6	−1.44981	−0.724906	−	0.688848i	$-0.758117\pi$
$503$	−8.22681e6	−1.44981	−0.724906	+	0.688848i	$0.758117\pi$
$504$	0	0
$505$	−1.96177e6	−0.342310
$506$	0	0
$507$	−1.55367e6	−0.268434
$508$	0	0
$509$	1.12374e7	1.92252	0.961260	−	0.275645i	$-0.0888914\pi$
$509$	1.12374e7	1.92252	0.961260	+	0.275645i	$0.0888914\pi$
$510$	0	0
$511$	−1.45730e6	−0.246886
$512$	0	0
$513$	−4.46148e6	−0.748489
$514$	0	0
$515$	1.43722e6	0.238784
$516$	0	0
$517$	−276619.	−0.0455151
$518$	0	0
$519$	−7.86460e6	−1.28162
$520$	0	0
$521$	−7.35800e6	−1.18759	−0.593794	−	0.804617i	$-0.702371\pi$
$521$	−7.35800e6	−1.18759	−0.593794	+	0.804617i	$0.702371\pi$
$522$	0	0
$523$	5.06396e6	0.809537	0.404768	−	0.914419i	$-0.367352\pi$
$523$	5.06396e6	0.809537	0.404768	+	0.914419i	$0.367352\pi$
$524$	0	0
$525$	−532155.	−0.0842635
$526$	0	0
$527$	−1.45049e7	−2.27503
$528$	0	0
$529$	1.11367e7	1.73028
$530$	0	0
$531$	−384401.	−0.0591627
$532$	0	0
$533$	−1.24170e7	−1.89320
$534$	0	0
$535$	3.52209e6	0.532006
$536$	0	0
$537$	834474.	0.124875
$538$	0	0
$539$	29520.6	0.00437677
$540$	0	0
$541$	−1.16768e7	−1.71526	−0.857631	−	0.514265i	$-0.828065\pi$
$541$	−1.16768e7	−1.71526	−0.857631	+	0.514265i	$0.828065\pi$
$542$	0	0
$543$	1.10157e7	1.60328
$544$	0	0
$545$	2.34921e6	0.338791
$546$	0	0
$547$	−7.83943e6	−1.12025	−0.560127	−	0.828407i	$-0.689247\pi$
$547$	−7.83943e6	−1.12025	−0.560127	+	0.828407i	$0.689247\pi$
$548$	0	0
$549$	514973.	0.0729211
$550$	0	0
$551$	−9.39748e6	−1.31866
$552$	0	0
$553$	−3.19557e6	−0.444361
$554$	0	0
$555$	1.97692e6	0.272431
$556$	0	0
$557$	−1.90193e6	−0.259750	−0.129875	−	0.991530i	$-0.541458\pi$
$557$	−1.90193e6	−0.259750	−0.129875	+	0.991530i	$0.541458\pi$
$558$	0	0
$559$	1.25406e7	1.69742
$560$	0	0
$561$	−322180.	−0.0432206
$562$	0	0
$563$	1.33896e7	1.78031	0.890157	−	0.455653i	$-0.150594\pi$
$563$	1.33896e7	1.78031	0.890157	+	0.455653i	$0.150594\pi$
$564$	0	0
$565$	−5.17740e6	−0.682324
$566$	0	0
$567$	−3.42499e6	−0.447406
$568$	0	0
$569$	1.36893e7	1.77255	0.886276	−	0.463157i	$-0.153283\pi$
$569$	1.36893e7	1.77255	0.886276	+	0.463157i	$0.153283\pi$
$570$	0	0
$571$	−6.70450e6	−0.860550	−0.430275	−	0.902698i	$-0.641583\pi$
$571$	−6.70450e6	−0.860550	−0.430275	+	0.902698i	$0.641583\pi$
$572$	0	0
$573$	−7.00492e6	−0.891285
$574$	0	0
$575$	2.62001e6	0.330471
$576$	0	0
$577$	649416.	0.0812052	0.0406026	−	0.999175i	$-0.487072\pi$
$577$	649416.	0.0812052	0.0406026	+	0.999175i	$0.487072\pi$
$578$	0	0
$579$	−6.41842e6	−0.795668
$580$	0	0
$581$	−5.04546e6	−0.620098
$582$	0	0
$583$	−357442.	−0.0435546
$584$	0	0
$585$	−1.00018e6	−0.120833
$586$	0	0
$587$	7.55735e6	0.905262	0.452631	−	0.891698i	$-0.350485\pi$
$587$	7.55735e6	0.905262	0.452631	+	0.891698i	$0.350485\pi$
$588$	0	0
$589$	1.34175e7	1.59362
$590$	0	0
$591$	−1.14658e7	−1.35031
$592$	0	0
$593$	9.50328e6	1.10978	0.554890	−	0.831924i	$-0.312760\pi$
$593$	9.50328e6	1.10978	0.554890	+	0.831924i	$0.312760\pi$
$594$	0	0
$595$	1.84731e6	0.213918
$596$	0	0
$597$	8.24706e6	0.947029
$598$	0	0
$599$	8.10647e6	0.923133	0.461567	−	0.887106i	$-0.347287\pi$
$599$	8.10647e6	0.923133	0.461567	+	0.887106i	$0.347287\pi$
$600$	0	0
$601$	1.28377e7	1.44977	0.724886	−	0.688869i	$-0.241892\pi$
$601$	1.28377e7	1.44977	0.724886	+	0.688869i	$0.241892\pi$
$602$	0	0
$603$	−935352.	−0.104757
$604$	0	0
$605$	−4.02250e6	−0.446794
$606$	0	0
$607$	1.43703e6	0.158305	0.0791526	−	0.996863i	$-0.474779\pi$
$607$	1.43703e6	0.158305	0.0791526	+	0.996863i	$0.474779\pi$
$608$	0	0
$609$	−5.73597e6	−0.626706
$610$	0	0
$611$	1.52708e7	1.65485
$612$	0	0
$613$	−954783.	−0.102625	−0.0513125	−	0.998683i	$-0.516340\pi$
$613$	−954783.	−0.102625	−0.0513125	+	0.998683i	$0.516340\pi$
$614$	0	0
$615$	−7.94704e6	−0.847262
$616$	0	0
$617$	2.25972e6	0.238969	0.119484	−	0.992836i	$-0.461876\pi$
$617$	2.25972e6	0.238969	0.119484	+	0.992836i	$0.461876\pi$
$618$	0	0
$619$	−7.75380e6	−0.813370	−0.406685	−	0.913568i	$-0.633315\pi$
$619$	−7.75380e6	−0.813370	−0.406685	+	0.913568i	$0.633315\pi$
$620$	0	0
$621$	1.34072e7	1.39512
$622$	0	0
$623$	5.70316e6	0.588702
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	298028.	0.0302753
$628$	0	0
$629$	−6.86262e6	−0.691614
$630$	0	0
$631$	−5.18704e6	−0.518616	−0.259308	−	0.965795i	$-0.583495\pi$
$631$	−5.18704e6	−0.518616	−0.259308	+	0.965795i	$0.583495\pi$
$632$	0	0
$633$	−1.15804e7	−1.14872
$634$	0	0
$635$	3.88677e6	0.382521
$636$	0	0
$637$	−1.62968e6	−0.159131
$638$	0	0
$639$	−1.29205e6	−0.125178
$640$	0	0
$641$	−1.14625e6	−0.110188	−0.0550938	−	0.998481i	$-0.517546\pi$
$641$	−1.14625e6	−0.110188	−0.0550938	+	0.998481i	$0.517546\pi$
$642$	0	0
$643$	−7.60562e6	−0.725449	−0.362725	−	0.931896i	$-0.618153\pi$
$643$	−7.60562e6	−0.725449	−0.362725	+	0.931896i	$0.618153\pi$
$644$	0	0
$645$	8.02619e6	0.759644
$646$	0	0
$647$	−9.15232e6	−0.859549	−0.429774	−	0.902936i	$-0.641407\pi$
$647$	−9.15232e6	−0.859549	−0.429774	+	0.902936i	$0.641407\pi$
$648$	0	0
$649$	−80184.9	−0.00747276
$650$	0	0
$651$	8.18971e6	0.757384
$652$	0	0
$653$	−1.86261e7	−1.70938	−0.854689	−	0.519140i	$-0.826252\pi$
$653$	−1.86261e7	−1.70938	−0.854689	+	0.519140i	$0.826252\pi$
$654$	0	0
$655$	−7.86305e6	−0.716124
$656$	0	0
$657$	−1.75298e6	−0.158440
$658$	0	0
$659$	−5.03797e6	−0.451899	−0.225950	−	0.974139i	$-0.572548\pi$
$659$	−5.03797e6	−0.451899	−0.225950	+	0.974139i	$0.572548\pi$
$660$	0	0
$661$	1.45099e6	0.129170	0.0645849	−	0.997912i	$-0.479428\pi$
$661$	1.45099e6	0.129170	0.0645849	+	0.997912i	$0.479428\pi$
$662$	0	0
$663$	1.77859e7	1.57142
$664$	0	0
$665$	−1.70883e6	−0.149846
$666$	0	0
$667$	2.82405e7	2.45786
$668$	0	0
$669$	1.77539e7	1.53366
$670$	0	0
$671$	107422.	0.00921057
$672$	0	0
$673$	1.31819e7	1.12187	0.560934	−	0.827861i	$-0.310442\pi$
$673$	1.31819e7	1.12187	0.560934	+	0.827861i	$0.310442\pi$
$674$	0	0
$675$	1.99892e6	0.168864
$676$	0	0
$677$	−1.50288e7	−1.26024	−0.630121	−	0.776497i	$-0.716995\pi$
$677$	−1.50288e7	−1.26024	−0.630121	+	0.776497i	$0.716995\pi$
$678$	0	0
$679$	−5.81558e6	−0.484082
$680$	0	0
$681$	8.50325e6	0.702615
$682$	0	0
$683$	1.19201e7	0.977751	0.488876	−	0.872354i	$-0.337407\pi$
$683$	1.19201e7	0.977751	0.488876	+	0.872354i	$0.337407\pi$
$684$	0	0
$685$	−458373.	−0.0373243
$686$	0	0
$687$	−1.52450e7	−1.23235
$688$	0	0
$689$	1.97326e7	1.58356
$690$	0	0
$691$	4.49225e6	0.357906	0.178953	−	0.983858i	$-0.442729\pi$
$691$	4.49225e6	0.357906	0.178953	+	0.983858i	$0.442729\pi$
$692$	0	0
$693$	35510.3	0.00280880
$694$	0	0
$695$	−65824.6	−0.00516923
$696$	0	0
$697$	2.75871e7	2.15092
$698$	0	0
$699$	−1.39616e7	−1.08079
$700$	0	0
$701$	−2.04225e6	−0.156969	−0.0784843	−	0.996915i	$-0.525008\pi$
$701$	−2.04225e6	−0.156969	−0.0784843	+	0.996915i	$0.525008\pi$
$702$	0	0
$703$	6.34818e6	0.484464
$704$	0	0
$705$	9.77351e6	0.740590
$706$	0	0
$707$	−3.84506e6	−0.289304
$708$	0	0
$709$	93976.2	0.00702105	0.00351053	−	0.999994i	$-0.498883\pi$
$709$	93976.2	0.00702105	0.00351053	+	0.999994i	$0.498883\pi$
$710$	0	0
$711$	−3.84394e6	−0.285169
$712$	0	0
$713$	−4.03212e7	−2.97037
$714$	0	0
$715$	−208634.	−0.0152623
$716$	0	0
$717$	−2.84429e7	−2.06622
$718$	0	0
$719$	1.20764e7	0.871194	0.435597	−	0.900142i	$-0.356537\pi$
$719$	1.20764e7	0.871194	0.435597	+	0.900142i	$0.356537\pi$
$720$	0	0
$721$	2.81695e6	0.201809
$722$	0	0
$723$	−2.03143e7	−1.44529
$724$	0	0
$725$	4.21045e6	0.297498
$726$	0	0
$727$	−1.67794e7	−1.17744	−0.588721	−	0.808336i	$-0.700368\pi$
$727$	−1.67794e7	−1.17744	−0.588721	+	0.808336i	$0.700368\pi$
$728$	0	0
$729$	9.38477e6	0.654041
$730$	0	0
$731$	−2.78619e7	−1.92849
$732$	0	0
$733$	−7.37863e6	−0.507243	−0.253621	−	0.967304i	$-0.581622\pi$
$733$	−7.37863e6	−0.507243	−0.253621	+	0.967304i	$0.581622\pi$
$734$	0	0
$735$	−1.04302e6	−0.0712157
$736$	0	0
$737$	−195112.	−0.0132317
$738$	0	0
$739$	1.76631e6	0.118975	0.0594874	−	0.998229i	$-0.481053\pi$
$739$	1.76631e6	0.118975	0.0594874	+	0.998229i	$0.481053\pi$
$740$	0	0
$741$	−1.64526e7	−1.10075
$742$	0	0
$743$	−1.46804e7	−0.975587	−0.487794	−	0.872959i	$-0.662198\pi$
$743$	−1.46804e7	−0.975587	−0.487794	+	0.872959i	$0.662198\pi$
$744$	0	0
$745$	3.62502e6	0.239287
$746$	0	0
$747$	−6.06917e6	−0.397949
$748$	0	0
$749$	6.90331e6	0.449627
$750$	0	0
$751$	1.03440e7	0.669252	0.334626	−	0.942351i	$-0.391390\pi$
$751$	1.03440e7	0.669252	0.334626	+	0.942351i	$0.391390\pi$
$752$	0	0
$753$	2.07427e7	1.33315
$754$	0	0
$755$	5.51851e6	0.352334
$756$	0	0
$757$	−6.80621e6	−0.431684	−0.215842	−	0.976428i	$-0.569250\pi$
$757$	−6.80621e6	−0.431684	−0.215842	+	0.976428i	$0.569250\pi$
$758$	0	0
$759$	−895608.	−0.0564305
$760$	0	0
$761$	−2.38947e7	−1.49569	−0.747843	−	0.663876i	$-0.768910\pi$
$761$	−2.38947e7	−1.49569	−0.747843	+	0.663876i	$0.768910\pi$
$762$	0	0
$763$	4.60446e6	0.286330
$764$	0	0
$765$	2.22212e6	0.137282
$766$	0	0
$767$	4.42661e6	0.271696
$768$	0	0
$769$	−6.95378e6	−0.424038	−0.212019	−	0.977266i	$-0.568004\pi$
$769$	−6.95378e6	−0.424038	−0.212019	+	0.977266i	$0.568004\pi$
$770$	0	0
$771$	6.50768e6	0.394267
$772$	0	0
$773$	−6.55112e6	−0.394336	−0.197168	−	0.980370i	$-0.563174\pi$
$773$	−6.55112e6	−0.394336	−0.197168	+	0.980370i	$0.563174\pi$
$774$	0	0
$775$	−6.01161e6	−0.359531
$776$	0	0
$777$	3.87476e6	0.230246
$778$	0	0
$779$	−2.55191e7	−1.50669
$780$	0	0
$781$	−269517.	−0.0158110
$782$	0	0
$783$	2.15459e7	1.25592
$784$	0	0
$785$	−1.10437e7	−0.639645
$786$	0	0
$787$	−1.40008e7	−0.805781	−0.402890	−	0.915248i	$-0.631994\pi$
$787$	−1.40008e7	−0.805781	−0.402890	+	0.915248i	$0.631994\pi$
$788$	0	0
$789$	−2.41785e7	−1.38273
$790$	0	0
$791$	−1.01477e7	−0.576669
$792$	0	0
$793$	−5.93022e6	−0.334879
$794$	0	0
$795$	1.26292e7	0.708690
$796$	0	0
$797$	1.53267e7	0.854676	0.427338	−	0.904092i	$-0.359451\pi$
$797$	1.53267e7	0.854676	0.427338	+	0.904092i	$0.359451\pi$
$798$	0	0
$799$	−3.39275e7	−1.88012
$800$	0	0
$801$	6.86032e6	0.377801
$802$	0	0
$803$	−365667.	−0.0200123
$804$	0	0
$805$	5.13522e6	0.279299
$806$	0	0
$807$	8.73077e6	0.471920
$808$	0	0
$809$	−2.48356e7	−1.33415	−0.667074	−	0.744992i	$-0.732453\pi$
$809$	−2.48356e7	−1.33415	−0.667074	+	0.744992i	$0.732453\pi$
$810$	0	0
$811$	−2.15784e7	−1.15204	−0.576020	−	0.817435i	$-0.695395\pi$
$811$	−2.15784e7	−1.15204	−0.576020	+	0.817435i	$0.695395\pi$
$812$	0	0
$813$	−1.07707e7	−0.571500
$814$	0	0
$815$	−1.27975e7	−0.674889
$816$	0	0
$817$	2.57733e7	1.35087
$818$	0	0
$819$	−1.96034e6	−0.102123
$820$	0	0
$821$	2.41038e7	1.24804	0.624019	−	0.781409i	$-0.285499\pi$
$821$	2.41038e7	1.24804	0.624019	+	0.781409i	$0.285499\pi$
$822$	0	0
$823$	−2.17800e7	−1.12088	−0.560439	−	0.828195i	$-0.689368\pi$
$823$	−2.17800e7	−1.12088	−0.560439	+	0.828195i	$0.689368\pi$
$824$	0	0
$825$	−133529.	−0.00683031
$826$	0	0
$827$	−1.23879e6	−0.0629845	−0.0314922	−	0.999504i	$-0.510026\pi$
$827$	−1.23879e6	−0.0629845	−0.0314922	+	0.999504i	$0.510026\pi$
$828$	0	0
$829$	903046.	0.0456377	0.0228188	−	0.999740i	$-0.492736\pi$
$829$	903046.	0.0456377	0.0228188	+	0.999740i	$0.492736\pi$
$830$	0	0
$831$	2.23403e7	1.12224
$832$	0	0
$833$	3.62072e6	0.180794
$834$	0	0
$835$	−1.78077e7	−0.883876
$836$	0	0
$837$	−3.07629e7	−1.51780
$838$	0	0
$839$	−3.15735e7	−1.54852	−0.774261	−	0.632867i	$-0.781878\pi$
$839$	−3.15735e7	−1.54852	−0.774261	+	0.632867i	$0.781878\pi$
$840$	0	0
$841$	2.48723e7	1.21263
$842$	0	0
$843$	−5.71732e6	−0.277092
$844$	0	0
$845$	2.23530e6	0.107695
$846$	0	0
$847$	−7.88409e6	−0.377610
$848$	0	0
$849$	1.28629e7	0.612450
$850$	0	0
$851$	−1.90770e7	−0.902997
$852$	0	0
$853$	−2.95116e7	−1.38874	−0.694369	−	0.719619i	$-0.744317\pi$
$853$	−2.95116e7	−1.38874	−0.694369	+	0.719619i	$0.744317\pi$
$854$	0	0
$855$	−2.05555e6	−0.0961639
$856$	0	0
$857$	3.27248e7	1.52204	0.761019	−	0.648730i	$-0.224699\pi$
$857$	3.27248e7	1.52204	0.761019	+	0.648730i	$0.224699\pi$
$858$	0	0
$859$	1.65551e7	0.765509	0.382754	−	0.923850i	$-0.374976\pi$
$859$	1.65551e7	0.765509	0.382754	+	0.923850i	$0.374976\pi$
$860$	0	0
$861$	−1.55762e7	−0.716067
$862$	0	0
$863$	1.07909e7	0.493209	0.246605	−	0.969116i	$-0.420685\pi$
$863$	1.07909e7	0.493209	0.246605	+	0.969116i	$0.420685\pi$
$864$	0	0
$865$	1.13150e7	0.514180
$866$	0	0
$867$	−1.48434e7	−0.670636
$868$	0	0
$869$	−801836.	−0.0360194
$870$	0	0
$871$	1.07711e7	0.481079
$872$	0	0
$873$	−6.99555e6	−0.310661
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	4.28645e7	1.88191	0.940954	−	0.338534i	$-0.109931\pi$
$877$	4.28645e7	1.88191	0.940954	+	0.338534i	$0.109931\pi$
$878$	0	0
$879$	−3.17441e6	−0.138577
$880$	0	0
$881$	−2.14157e7	−0.929590	−0.464795	−	0.885418i	$-0.653872\pi$
$881$	−2.14157e7	−0.929590	−0.464795	+	0.885418i	$0.653872\pi$
$882$	0	0
$883$	1.45167e7	0.626567	0.313283	−	0.949660i	$-0.398571\pi$
$883$	1.45167e7	0.626567	0.313283	+	0.949660i	$0.398571\pi$
$884$	0	0
$885$	2.83310e6	0.121591
$886$	0	0
$887$	7.06678e6	0.301587	0.150793	−	0.988565i	$-0.451817\pi$
$887$	7.06678e6	0.301587	0.150793	+	0.988565i	$0.451817\pi$
$888$	0	0
$889$	7.61807e6	0.323289
$890$	0	0
$891$	−859402.	−0.0362662
$892$	0	0
$893$	3.13842e7	1.31699
$894$	0	0
$895$	−1.20058e6	−0.0500995
$896$	0	0
$897$	4.94420e7	2.05171
$898$	0	0
$899$	−6.47977e7	−2.67399
$900$	0	0
$901$	−4.38405e7	−1.79913
$902$	0	0
$903$	1.57313e7	0.642016
$904$	0	0
$905$	−1.58485e7	−0.643231
$906$	0	0
$907$	9.96676e6	0.402287	0.201143	−	0.979562i	$-0.435534\pi$
$907$	9.96676e6	0.402287	0.201143	+	0.979562i	$0.435534\pi$
$908$	0	0
$909$	−4.62522e6	−0.185662
$910$	0	0
$911$	3.76798e6	0.150422	0.0752112	−	0.997168i	$-0.476037\pi$
$911$	3.76798e6	0.150422	0.0752112	+	0.997168i	$0.476037\pi$
$912$	0	0
$913$	−1.26601e6	−0.0502644
$914$	0	0
$915$	−3.79543e6	−0.149868
$916$	0	0
$917$	−1.54116e7	−0.605235
$918$	0	0
$919$	−2.88518e6	−0.112690	−0.0563449	−	0.998411i	$-0.517945\pi$
$919$	−2.88518e6	−0.112690	−0.0563449	+	0.998411i	$0.517945\pi$
$920$	0	0
$921$	2.72709e7	1.05938
$922$	0	0
$923$	1.48787e7	0.574858
$924$	0	0
$925$	−2.84425e6	−0.109298
$926$	0	0
$927$	3.38851e6	0.129512
$928$	0	0
$929$	1.55418e7	0.590831	0.295415	−	0.955369i	$-0.404542\pi$
$929$	1.55418e7	0.590831	0.295415	+	0.955369i	$0.404542\pi$
$930$	0	0
$931$	−3.34930e6	−0.126643
$932$	0	0
$933$	−5.53299e6	−0.208092
$934$	0	0
$935$	463529.	0.0173399
$936$	0	0
$937$	−3.04604e7	−1.13341	−0.566704	−	0.823921i	$-0.691782\pi$
$937$	−3.04604e7	−1.13341	−0.566704	+	0.823921i	$0.691782\pi$
$938$	0	0
$939$	−4.57482e7	−1.69321
$940$	0	0
$941$	−1.72404e7	−0.634709	−0.317354	−	0.948307i	$-0.602794\pi$
$941$	−1.72404e7	−0.634709	−0.317354	+	0.948307i	$0.602794\pi$
$942$	0	0
$943$	7.66879e7	2.80833
$944$	0	0
$945$	3.91789e6	0.142716
$946$	0	0
$947$	4.65509e7	1.68676	0.843380	−	0.537318i	$-0.180563\pi$
$947$	4.65509e7	1.68676	0.843380	+	0.537318i	$0.180563\pi$
$948$	0	0
$949$	2.01866e7	0.727609
$950$	0	0
$951$	4.65195e7	1.66795
$952$	0	0
$953$	−3.93141e6	−0.140222	−0.0701110	−	0.997539i	$-0.522335\pi$
$953$	−3.93141e6	−0.140222	−0.0701110	+	0.997539i	$0.522335\pi$
$954$	0	0
$955$	1.00782e7	0.357580
$956$	0	0
$957$	−1.43928e6	−0.0508000
$958$	0	0
$959$	−898410.	−0.0315448
$960$	0	0
$961$	6.38878e7	2.23156
$962$	0	0
$963$	8.30397e6	0.288549
$964$	0	0
$965$	9.23435e6	0.319219
$966$	0	0
$967$	4.13777e7	1.42298	0.711492	−	0.702694i	$-0.248020\pi$
$967$	4.13777e7	1.42298	0.711492	+	0.702694i	$0.248020\pi$
$968$	0	0
$969$	3.65534e7	1.25060
$970$	0	0
$971$	4.30701e6	0.146598	0.0732990	−	0.997310i	$-0.476647\pi$
$971$	4.30701e6	0.146598	0.0732990	+	0.997310i	$0.476647\pi$
$972$	0	0
$973$	−129016.	−0.00436880
$974$	0	0
$975$	7.37146e6	0.248337
$976$	0	0
$977$	−4.16618e7	−1.39637	−0.698187	−	0.715915i	$-0.746010\pi$
$977$	−4.16618e7	−1.39637	−0.698187	+	0.715915i	$0.746010\pi$
$978$	0	0
$979$	1.43104e6	0.0477195
$980$	0	0
$981$	5.53870e6	0.183753
$982$	0	0
$983$	3.10661e6	0.102542	0.0512712	−	0.998685i	$-0.483673\pi$
$983$	3.10661e6	0.102542	0.0512712	+	0.998685i	$0.483673\pi$
$984$	0	0
$985$	1.64961e7	0.541741
$986$	0	0
$987$	1.91561e7	0.625913
$988$	0	0
$989$	−7.74517e7	−2.51791
$990$	0	0
$991$	−1.85311e7	−0.599400	−0.299700	−	0.954033i	$-0.596887\pi$
$991$	−1.85311e7	−0.599400	−0.299700	+	0.954033i	$0.596887\pi$
$992$	0	0
$993$	−2.92191e7	−0.940360
$994$	0	0
$995$	−1.18653e7	−0.379944
$996$	0	0
$997$	4.46039e7	1.42113	0.710567	−	0.703630i	$-0.248439\pi$
$997$	4.46039e7	1.42113	0.710567	+	0.703630i	$0.248439\pi$
$998$	0	0
$999$	−1.45547e7	−0.461413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.g.1.1	✓	4
4.3	odd	2		560.6.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.g.1.1	✓	4	1.1	even	1	trivial
560.6.a.x.1.4		4	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)

\(f(q)\)	\(=\)	\(q-17.3765 q^{3} +25.0000 q^{5} +49.0000 q^{7} +58.9420 q^{9} +O(q^{10})\)	\(q-17.3765 q^{3} +25.0000 q^{5} +49.0000 q^{7} +58.9420 q^{9} +12.2951 q^{11} -678.753 q^{13} -434.412 q^{15} +1508.01 q^{17} -1394.96 q^{19} -851.447 q^{21} +4192.02 q^{23} +625.000 q^{25} +3198.28 q^{27} +6736.73 q^{29} -9618.57 q^{31} -213.646 q^{33} +1225.00 q^{35} -4550.79 q^{37} +11794.3 q^{39} +18293.8 q^{41} -18476.0 q^{43} +1473.55 q^{45} -22498.3 q^{47} +2401.00 q^{49} -26203.8 q^{51} -29071.8 q^{53} +307.378 q^{55} +24239.5 q^{57} -6521.68 q^{59} +8736.94 q^{61} +2888.16 q^{63} -16968.8 q^{65} -15869.0 q^{67} -72842.5 q^{69} -21920.7 q^{71} -29740.8 q^{73} -10860.3 q^{75} +602.462 q^{77} -65215.7 q^{79} -69897.7 q^{81} -102969. q^{83} +37700.2 q^{85} -117061. q^{87} +116391. q^{89} -33258.9 q^{91} +167137. q^{93} -34874.1 q^{95} -118685. q^{97} +724.700 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9}+O(q^{10}) \) 4 * q - 13 * q^3 + 100 * q^5 + 196 * q^7 - 193 * q^9	\( 4 q - 13 q^{3} + 100 q^{5} + 196 q^{7} - 193 q^{9} - 595 q^{11} - 969 q^{13} - 325 q^{15} - 1315 q^{17} - 1090 q^{19} - 637 q^{21} - 1534 q^{23} + 2500 q^{25} + 173 q^{27} + 4099 q^{29} - 4820 q^{31} + 4149 q^{33} + 4900 q^{35} + 7692 q^{37} - 6371 q^{39} - 9722 q^{41} - 20610 q^{43} - 4825 q^{45} - 1661 q^{47} + 9604 q^{49} - 73361 q^{51} - 28898 q^{53} - 14875 q^{55} - 21246 q^{57} - 101872 q^{59} - 24742 q^{61} - 9457 q^{63} - 24225 q^{65} - 82060 q^{67} + 16914 q^{69} - 102784 q^{71} - 80652 q^{73} - 8125 q^{75} - 29155 q^{77} - 117801 q^{79} - 141052 q^{81} - 155440 q^{83} - 32875 q^{85} - 82519 q^{87} + 56426 q^{89} - 47481 q^{91} - 17332 q^{93} - 27250 q^{95} - 261031 q^{97} - 61686 q^{99}+O(q^{100}) \) 4 * q - 13 * q^3 + 100 * q^5 + 196 * q^7 - 193 * q^9 - 595 * q^11 - 969 * q^13 - 325 * q^15 - 1315 * q^17 - 1090 * q^19 - 637 * q^21 - 1534 * q^23 + 2500 * q^25 + 173 * q^27 + 4099 * q^29 - 4820 * q^31 + 4149 * q^33 + 4900 * q^35 + 7692 * q^37 - 6371 * q^39 - 9722 * q^41 - 20610 * q^43 - 4825 * q^45 - 1661 * q^47 + 9604 * q^49 - 73361 * q^51 - 28898 * q^53 - 14875 * q^55 - 21246 * q^57 - 101872 * q^59 - 24742 * q^61 - 9457 * q^63 - 24225 * q^65 - 82060 * q^67 + 16914 * q^69 - 102784 * q^71 - 80652 * q^73 - 8125 * q^75 - 29155 * q^77 - 117801 * q^79 - 141052 * q^81 - 155440 * q^83 - 32875 * q^85 - 82519 * q^87 + 56426 * q^89 - 47481 * q^91 - 17332 * q^93 - 27250 * q^95 - 261031 * q^97 - 61686 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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