LMFDB - Embedded newform 280.6.a.d.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{37}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 9 $ x^2 - x - 9
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.54138$ 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)

$f(q)$	$=$	$q+25.1655 q^{3} +25.0000 q^{5} -49.0000 q^{7} +390.304 q^{9} +O(q^{10})$	$q+25.1655 q^{3} +25.0000 q^{5} -49.0000 q^{7} +390.304 q^{9} +260.642 q^{11} +96.1724 q^{13} +629.138 q^{15} -328.476 q^{17} +560.248 q^{19} -1233.11 q^{21} +2859.75 q^{23} +625.000 q^{25} +3706.97 q^{27} +4170.89 q^{29} +573.503 q^{31} +6559.18 q^{33} -1225.00 q^{35} +13652.6 q^{37} +2420.23 q^{39} -4175.14 q^{41} +5024.25 q^{43} +9757.59 q^{45} -12038.8 q^{47} +2401.00 q^{49} -8266.27 q^{51} +16595.9 q^{53} +6516.04 q^{55} +14098.9 q^{57} -33570.5 q^{59} -26619.0 q^{61} -19124.9 q^{63} +2404.31 q^{65} -25855.7 q^{67} +71967.2 q^{69} +10629.6 q^{71} -19764.8 q^{73} +15728.5 q^{75} -12771.4 q^{77} +70391.7 q^{79} -1555.85 q^{81} +13220.5 q^{83} -8211.90 q^{85} +104963. q^{87} +57979.4 q^{89} -4712.45 q^{91} +14432.5 q^{93} +14006.2 q^{95} +12639.5 q^{97} +101729. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9}+O(q^{10}) $ 2 * q + 26 * q^3 + 50 * q^5 - 98 * q^7 + 148 * q^9	$ 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} - 14 q^{11} + 314 q^{13} + 650 q^{15} - 146 q^{17} + 2848 q^{19} - 1274 q^{21} + 3992 q^{23} + 1250 q^{25} + 3302 q^{27} + 6006 q^{29} + 1220 q^{31} + 6330 q^{33} - 2450 q^{35} + 7232 q^{37} + 2602 q^{39} + 6516 q^{41} - 2336 q^{43} + 3700 q^{45} + 12930 q^{47} + 4802 q^{49} - 8114 q^{51} + 3216 q^{53} - 350 q^{55} + 16008 q^{57} + 6728 q^{59} - 756 q^{61} - 7252 q^{63} + 7850 q^{65} + 11744 q^{67} + 72912 q^{69} - 1904 q^{71} + 22612 q^{73} + 16250 q^{75} + 686 q^{77} + 62486 q^{79} + 56986 q^{81} + 15200 q^{83} - 3650 q^{85} + 106494 q^{87} + 42844 q^{89} - 15386 q^{91} + 14972 q^{93} + 71200 q^{95} + 82822 q^{97} + 168276 q^{99}+O(q^{100}) $ 2 * q + 26 * q^3 + 50 * q^5 - 98 * q^7 + 148 * q^9 - 14 * q^11 + 314 * q^13 + 650 * q^15 - 146 * q^17 + 2848 * q^19 - 1274 * q^21 + 3992 * q^23 + 1250 * q^25 + 3302 * q^27 + 6006 * q^29 + 1220 * q^31 + 6330 * q^33 - 2450 * q^35 + 7232 * q^37 + 2602 * q^39 + 6516 * q^41 - 2336 * q^43 + 3700 * q^45 + 12930 * q^47 + 4802 * q^49 - 8114 * q^51 + 3216 * q^53 - 350 * q^55 + 16008 * q^57 + 6728 * q^59 - 756 * q^61 - 7252 * q^63 + 7850 * q^65 + 11744 * q^67 + 72912 * q^69 - 1904 * q^71 + 22612 * q^73 + 16250 * q^75 + 686 * q^77 + 62486 * q^79 + 56986 * q^81 + 15200 * q^83 - 3650 * q^85 + 106494 * q^87 + 42844 * q^89 - 15386 * q^91 + 14972 * q^93 + 71200 * q^95 + 82822 * q^97 + 168276 * 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
$3$	25.1655	1.61437	0.807185	−	0.590299i	$-0.200990\pi$
$3$	25.1655	1.61437	0.807185	+	0.590299i	$0.200990\pi$
$4$	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
$9$	390.304	1.60619
$10$	0	0
$11$	260.642	0.649474	0.324737	−	0.945804i	$-0.394724\pi$
$11$	260.642	0.649474	0.324737	+	0.945804i	$0.394724\pi$
$12$	0	0
$13$	96.1724	0.157831	0.0789154	−	0.996881i	$-0.474854\pi$
$13$	96.1724	0.157831	0.0789154	+	0.996881i	$0.474854\pi$
$14$	0	0
$15$	629.138	0.721968
$16$	0	0
$17$	−328.476	−0.275665	−0.137832	−	0.990456i	$-0.544014\pi$
$17$	−328.476	−0.275665	−0.137832	+	0.990456i	$0.544014\pi$
$18$	0	0
$19$	560.248	0.356038	0.178019	−	0.984027i	$-0.443031\pi$
$19$	560.248	0.356038	0.178019	+	0.984027i	$0.443031\pi$
$20$	0	0
$21$	−1233.11	−0.610174
$22$	0	0
$23$	2859.75	1.12722	0.563610	−	0.826041i	$-0.309412\pi$
$23$	2859.75	1.12722	0.563610	+	0.826041i	$0.309412\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	3706.97	0.978611
$28$	0	0
$29$	4170.89	0.920945	0.460473	−	0.887674i	$-0.347680\pi$
$29$	4170.89	0.920945	0.460473	+	0.887674i	$0.347680\pi$
$30$	0	0
$31$	573.503	0.107184	0.0535922	−	0.998563i	$-0.482933\pi$
$31$	573.503	0.107184	0.0535922	+	0.998563i	$0.482933\pi$
$32$	0	0
$33$	6559.18	1.04849
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	13652.6	1.63949	0.819747	−	0.572726i	$-0.194114\pi$
$37$	13652.6	1.63949	0.819747	+	0.572726i	$0.194114\pi$
$38$	0	0
$39$	2420.23	0.254797
$40$	0	0
$41$	−4175.14	−0.387892	−0.193946	−	0.981012i	$-0.562129\pi$
$41$	−4175.14	−0.387892	−0.193946	+	0.981012i	$0.562129\pi$
$42$	0	0
$43$	5024.25	0.414381	0.207191	−	0.978301i	$-0.433568\pi$
$43$	5024.25	0.414381	0.207191	+	0.978301i	$0.433568\pi$
$44$	0	0
$45$	9757.59	0.718309
$46$	0	0
$47$	−12038.8	−0.794945	−0.397473	−	0.917614i	$-0.630113\pi$
$47$	−12038.8	−0.794945	−0.397473	+	0.917614i	$0.630113\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−8266.27	−0.445025
$52$	0	0
$53$	16595.9	0.811544	0.405772	−	0.913974i	$-0.367003\pi$
$53$	16595.9	0.811544	0.405772	+	0.913974i	$0.367003\pi$
$54$	0	0
$55$	6516.04	0.290454
$56$	0	0
$57$	14098.9	0.574776
$58$	0	0
$59$	−33570.5	−1.25553	−0.627767	−	0.778402i	$-0.716031\pi$
$59$	−33570.5	−1.25553	−0.627767	+	0.778402i	$0.716031\pi$
$60$	0	0
$61$	−26619.0	−0.915942	−0.457971	−	0.888967i	$-0.651424\pi$
$61$	−26619.0	−0.915942	−0.457971	+	0.888967i	$0.651424\pi$
$62$	0	0
$63$	−19124.9	−0.607082
$64$	0	0
$65$	2404.31	0.0705841
$66$	0	0
$67$	−25855.7	−0.703670	−0.351835	−	0.936062i	$-0.614442\pi$
$67$	−25855.7	−0.703670	−0.351835	+	0.936062i	$0.614442\pi$
$68$	0	0
$69$	71967.2	1.81975
$70$	0	0
$71$	10629.6	0.250248	0.125124	−	0.992141i	$-0.460067\pi$
$71$	10629.6	0.250248	0.125124	+	0.992141i	$0.460067\pi$
$72$	0	0
$73$	−19764.8	−0.434094	−0.217047	−	0.976161i	$-0.569643\pi$
$73$	−19764.8	−0.434094	−0.217047	+	0.976161i	$0.569643\pi$
$74$	0	0
$75$	15728.5	0.322874
$76$	0	0
$77$	−12771.4	−0.245478
$78$	0	0
$79$	70391.7	1.26898	0.634488	−	0.772933i	$-0.281211\pi$
$79$	70391.7	1.26898	0.634488	+	0.772933i	$0.281211\pi$
$80$	0	0
$81$	−1555.85	−0.0263484
$82$	0	0
$83$	13220.5	0.210645	0.105323	−	0.994438i	$-0.466412\pi$
$83$	13220.5	0.210645	0.105323	+	0.994438i	$0.466412\pi$
$84$	0	0
$85$	−8211.90	−0.123281
$86$	0	0
$87$	104963.	1.48675
$88$	0	0
$89$	57979.4	0.775887	0.387944	−	0.921683i	$-0.373185\pi$
$89$	57979.4	0.775887	0.387944	+	0.921683i	$0.373185\pi$
$90$	0	0
$91$	−4712.45	−0.0596545
$92$	0	0
$93$	14432.5	0.173035
$94$	0	0
$95$	14006.2	0.159225
$96$	0	0
$97$	12639.5	0.136396	0.0681980	−	0.997672i	$-0.478275\pi$
$97$	12639.5	0.136396	0.0681980	+	0.997672i	$0.478275\pi$
$98$	0	0
$99$	101729.	1.04318
$100$	0	0
$101$	−43944.0	−0.428643	−0.214322	−	0.976763i	$-0.568754\pi$
$101$	−43944.0	−0.428643	−0.214322	+	0.976763i	$0.568754\pi$
$102$	0	0
$103$	−63108.5	−0.586131	−0.293066	−	0.956092i	$-0.594675\pi$
$103$	−63108.5	−0.586131	−0.293066	+	0.956092i	$0.594675\pi$
$104$	0	0
$105$	−30827.8	−0.272878
$106$	0	0
$107$	−68108.9	−0.575102	−0.287551	−	0.957765i	$-0.592841\pi$
$107$	−68108.9	−0.575102	−0.287551	+	0.957765i	$0.592841\pi$
$108$	0	0
$109$	156415.	1.26099	0.630496	−	0.776193i	$-0.282852\pi$
$109$	156415.	1.26099	0.630496	+	0.776193i	$0.282852\pi$
$110$	0	0
$111$	343574.	2.64675
$112$	0	0
$113$	−225985.	−1.66488	−0.832440	−	0.554116i	$-0.813057\pi$
$113$	−225985.	−1.66488	−0.832440	+	0.554116i	$0.813057\pi$
$114$	0	0
$115$	71493.8	0.504108
$116$	0	0
$117$	37536.4	0.253506
$118$	0	0
$119$	16095.3	0.104192
$120$	0	0
$121$	−93117.0	−0.578183
$122$	0	0
$123$	−105069.	−0.626201
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	240733.	1.32442	0.662210	−	0.749318i	$-0.269618\pi$
$127$	240733.	1.32442	0.662210	+	0.749318i	$0.269618\pi$
$128$	0	0
$129$	126438.	0.668965
$130$	0	0
$131$	104008.	0.529525	0.264763	−	0.964314i	$-0.414706\pi$
$131$	104008.	0.529525	0.264763	+	0.964314i	$0.414706\pi$
$132$	0	0
$133$	−27452.1	−0.134570
$134$	0	0
$135$	92674.3	0.437648
$136$	0	0
$137$	213352.	0.971169	0.485585	−	0.874190i	$-0.338607\pi$
$137$	213352.	0.971169	0.485585	+	0.874190i	$0.338607\pi$
$138$	0	0
$139$	398724.	1.75039	0.875195	−	0.483770i	$-0.160733\pi$
$139$	398724.	1.75039	0.875195	+	0.483770i	$0.160733\pi$
$140$	0	0
$141$	−302962.	−1.28334
$142$	0	0
$143$	25066.5	0.102507
$144$	0	0
$145$	104272.	0.411859
$146$	0	0
$147$	60422.4	0.230624
$148$	0	0
$149$	−283153.	−1.04485	−0.522427	−	0.852684i	$-0.674973\pi$
$149$	−283153.	−1.04485	−0.522427	+	0.852684i	$0.674973\pi$
$150$	0	0
$151$	214574.	0.765835	0.382917	−	0.923783i	$-0.374919\pi$
$151$	214574.	0.765835	0.382917	+	0.923783i	$0.374919\pi$
$152$	0	0
$153$	−128205.	−0.442770
$154$	0	0
$155$	14337.6	0.0479343
$156$	0	0
$157$	−370991.	−1.20120	−0.600599	−	0.799550i	$-0.705071\pi$
$157$	−370991.	−1.20120	−0.600599	+	0.799550i	$0.705071\pi$
$158$	0	0
$159$	417645.	1.31013
$160$	0	0
$161$	−140128.	−0.426049
$162$	0	0
$163$	−591372.	−1.74338	−0.871689	−	0.490059i	$-0.836975\pi$
$163$	−591372.	−1.74338	−0.871689	+	0.490059i	$0.836975\pi$
$164$	0	0
$165$	163980.	0.468900
$166$	0	0
$167$	312521.	0.867138	0.433569	−	0.901120i	$-0.357254\pi$
$167$	312521.	0.867138	0.433569	+	0.901120i	$0.357254\pi$
$168$	0	0
$169$	−362044.	−0.975089
$170$	0	0
$171$	218667.	0.571864
$172$	0	0
$173$	−352761.	−0.896118	−0.448059	−	0.894004i	$-0.647884\pi$
$173$	−352761.	−0.896118	−0.448059	+	0.894004i	$0.647884\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−844820.	−2.02689
$178$	0	0
$179$	−127748.	−0.298003	−0.149002	−	0.988837i	$-0.547606\pi$
$179$	−127748.	−0.298003	−0.149002	+	0.988837i	$0.547606\pi$
$180$	0	0
$181$	−783880.	−1.77850	−0.889249	−	0.457424i	$-0.848772\pi$
$181$	−783880.	−1.77850	−0.889249	+	0.457424i	$0.848772\pi$
$182$	0	0
$183$	−669882.	−1.47867
$184$	0	0
$185$	341314.	0.733204
$186$	0	0
$187$	−85614.5	−0.179037
$188$	0	0
$189$	−181642.	−0.369880
$190$	0	0
$191$	−642784.	−1.27492	−0.637458	−	0.770485i	$-0.720014\pi$
$191$	−642784.	−1.27492	−0.637458	+	0.770485i	$0.720014\pi$
$192$	0	0
$193$	123342.	0.238352	0.119176	−	0.992873i	$-0.461975\pi$
$193$	123342.	0.238352	0.119176	+	0.992873i	$0.461975\pi$
$194$	0	0
$195$	60505.7	0.113949
$196$	0	0
$197$	−751142.	−1.37898	−0.689488	−	0.724297i	$-0.742164\pi$
$197$	−751142.	−1.37898	−0.689488	+	0.724297i	$0.742164\pi$
$198$	0	0
$199$	−126553.	−0.226537	−0.113268	−	0.993564i	$-0.536132\pi$
$199$	−126553.	−0.226537	−0.113268	+	0.993564i	$0.536132\pi$
$200$	0	0
$201$	−650672.	−1.13598
$202$	0	0
$203$	−204374.	−0.348085
$204$	0	0
$205$	−104378.	−0.173471
$206$	0	0
$207$	1.11617e6	1.81053
$208$	0	0
$209$	146024.	0.231237
$210$	0	0
$211$	−1.08277e6	−1.67429	−0.837143	−	0.546984i	$-0.815776\pi$
$211$	−1.08277e6	−1.67429	−0.837143	+	0.546984i	$0.815776\pi$
$212$	0	0
$213$	267499.	0.403992
$214$	0	0
$215$	125606.	0.185317
$216$	0	0
$217$	−28101.7	−0.0405119
$218$	0	0
$219$	−497390.	−0.700789
$220$	0	0
$221$	−31590.3	−0.0435084
$222$	0	0
$223$	449618.	0.605454	0.302727	−	0.953077i	$-0.402103\pi$
$223$	449618.	0.605454	0.302727	+	0.953077i	$0.402103\pi$
$224$	0	0
$225$	243940.	0.321238
$226$	0	0
$227$	−708245.	−0.912260	−0.456130	−	0.889913i	$-0.650765\pi$
$227$	−708245.	−0.912260	−0.456130	+	0.889913i	$0.650765\pi$
$228$	0	0
$229$	−1.24302e6	−1.56636	−0.783178	−	0.621797i	$-0.786403\pi$
$229$	−1.24302e6	−1.56636	−0.783178	+	0.621797i	$0.786403\pi$
$230$	0	0
$231$	−321400.	−0.396292
$232$	0	0
$233$	828399.	0.999654	0.499827	−	0.866125i	$-0.333397\pi$
$233$	828399.	0.999654	0.499827	+	0.866125i	$0.333397\pi$
$234$	0	0
$235$	−300969.	−0.355510
$236$	0	0
$237$	1.77144e6	2.04860
$238$	0	0
$239$	578151.	0.654707	0.327353	−	0.944902i	$-0.393843\pi$
$239$	578151.	0.654707	0.327353	+	0.944902i	$0.393843\pi$
$240$	0	0
$241$	742369.	0.823336	0.411668	−	0.911334i	$-0.364946\pi$
$241$	742369.	0.823336	0.411668	+	0.911334i	$0.364946\pi$
$242$	0	0
$243$	−939948.	−1.02115
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	53880.4	0.0561938
$248$	0	0
$249$	332700.	0.340059
$250$	0	0
$251$	206564.	0.206952	0.103476	−	0.994632i	$-0.467004\pi$
$251$	206564.	0.206952	0.103476	+	0.994632i	$0.467004\pi$
$252$	0	0
$253$	745370.	0.732101
$254$	0	0
$255$	−206657.	−0.199021
$256$	0	0
$257$	−985444.	−0.930677	−0.465338	−	0.885133i	$-0.654067\pi$
$257$	−985444.	−0.930677	−0.465338	+	0.885133i	$0.654067\pi$
$258$	0	0
$259$	−668975.	−0.619670
$260$	0	0
$261$	1.62791e6	1.47921
$262$	0	0
$263$	−74781.6	−0.0666661	−0.0333331	−	0.999444i	$-0.510612\pi$
$263$	−74781.6	−0.0666661	−0.0333331	+	0.999444i	$0.510612\pi$
$264$	0	0
$265$	414898.	0.362933
$266$	0	0
$267$	1.45908e6	1.25257
$268$	0	0
$269$	−996421.	−0.839580	−0.419790	−	0.907621i	$-0.637896\pi$
$269$	−996421.	−0.839580	−0.419790	+	0.907621i	$0.637896\pi$
$270$	0	0
$271$	1.36997e6	1.13315	0.566575	−	0.824010i	$-0.308268\pi$
$271$	1.36997e6	1.13315	0.566575	+	0.824010i	$0.308268\pi$
$272$	0	0
$273$	−118591.	−0.0963043
$274$	0	0
$275$	162901.	0.129895
$276$	0	0
$277$	1.35956e6	1.06463	0.532314	−	0.846547i	$-0.321323\pi$
$277$	1.35956e6	1.06463	0.532314	+	0.846547i	$0.321323\pi$
$278$	0	0
$279$	223840.	0.172158
$280$	0	0
$281$	−1.70033e6	−1.28460	−0.642300	−	0.766453i	$-0.722020\pi$
$281$	−1.70033e6	−1.28460	−0.642300	+	0.766453i	$0.722020\pi$
$282$	0	0
$283$	−1.66283e6	−1.23419	−0.617096	−	0.786888i	$-0.711691\pi$
$283$	−1.66283e6	−1.23419	−0.617096	+	0.786888i	$0.711691\pi$
$284$	0	0
$285$	352473.	0.257048
$286$	0	0
$287$	204582.	0.146609
$288$	0	0
$289$	−1.31196e6	−0.924009
$290$	0	0
$291$	318080.	0.220194
$292$	0	0
$293$	1.38014e6	0.939194	0.469597	−	0.882881i	$-0.344399\pi$
$293$	1.38014e6	0.939194	0.469597	+	0.882881i	$0.344399\pi$
$294$	0	0
$295$	−839263.	−0.561492
$296$	0	0
$297$	966191.	0.635583
$298$	0	0
$299$	275029.	0.177910
$300$	0	0
$301$	−246188.	−0.156621
$302$	0	0
$303$	−1.10587e6	−0.691989
$304$	0	0
$305$	−665476.	−0.409622
$306$	0	0
$307$	561159.	0.339813	0.169906	−	0.985460i	$-0.445653\pi$
$307$	561159.	0.339813	0.169906	+	0.985460i	$0.445653\pi$
$308$	0	0
$309$	−1.58816e6	−0.946233
$310$	0	0
$311$	2.64984e6	1.55353	0.776765	−	0.629791i	$-0.216859\pi$
$311$	2.64984e6	1.55353	0.776765	+	0.629791i	$0.216859\pi$
$312$	0	0
$313$	−2.55958e6	−1.47675	−0.738377	−	0.674388i	$-0.764408\pi$
$313$	−2.55958e6	−1.47675	−0.738377	+	0.674388i	$0.764408\pi$
$314$	0	0
$315$	−478122.	−0.271495
$316$	0	0
$317$	1.73189e6	0.967994	0.483997	−	0.875070i	$-0.339185\pi$
$317$	1.73189e6	0.967994	0.483997	+	0.875070i	$0.339185\pi$
$318$	0	0
$319$	1.08711e6	0.598130
$320$	0	0
$321$	−1.71400e6	−0.928426
$322$	0	0
$323$	−184028.	−0.0981471
$324$	0	0
$325$	60107.7	0.0315662
$326$	0	0
$327$	3.93626e6	2.03571
$328$	0	0
$329$	589899.	0.300461
$330$	0	0
$331$	2.74279e6	1.37601	0.688006	−	0.725705i	$-0.258486\pi$
$331$	2.74279e6	1.37601	0.688006	+	0.725705i	$0.258486\pi$
$332$	0	0
$333$	5.32864e6	2.63333
$334$	0	0
$335$	−646392.	−0.314691
$336$	0	0
$337$	−508222.	−0.243769	−0.121885	−	0.992544i	$-0.538894\pi$
$337$	−508222.	−0.243769	−0.121885	+	0.992544i	$0.538894\pi$
$338$	0	0
$339$	−5.68702e6	−2.68773
$340$	0	0
$341$	149479.	0.0696135
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.79918e6	0.813817
$346$	0	0
$347$	1.66846e6	0.743862	0.371931	−	0.928260i	$-0.378696\pi$
$347$	1.66846e6	0.743862	0.371931	+	0.928260i	$0.378696\pi$
$348$	0	0
$349$	−4.05821e6	−1.78349	−0.891745	−	0.452538i	$-0.850519\pi$
$349$	−4.05821e6	−1.78349	−0.891745	+	0.452538i	$0.850519\pi$
$350$	0	0
$351$	356508.	0.154455
$352$	0	0
$353$	−988283.	−0.422128	−0.211064	−	0.977472i	$-0.567693\pi$
$353$	−988283.	−0.422128	−0.211064	+	0.977472i	$0.567693\pi$
$354$	0	0
$355$	265739.	0.111914
$356$	0	0
$357$	405047.	0.168204
$358$	0	0
$359$	−3.54797e6	−1.45293	−0.726464	−	0.687204i	$-0.758838\pi$
$359$	−3.54797e6	−1.45293	−0.726464	+	0.687204i	$0.758838\pi$
$360$	0	0
$361$	−2.16222e6	−0.873237
$362$	0	0
$363$	−2.34334e6	−0.933401
$364$	0	0
$365$	−494119.	−0.194133
$366$	0	0
$367$	−1.74201e6	−0.675125	−0.337563	−	0.941303i	$-0.609602\pi$
$367$	−1.74201e6	−0.675125	−0.337563	+	0.941303i	$0.609602\pi$
$368$	0	0
$369$	−1.62957e6	−0.623028
$370$	0	0
$371$	−813200.	−0.306735
$372$	0	0
$373$	4.53753e6	1.68868	0.844339	−	0.535809i	$-0.179993\pi$
$373$	4.53753e6	1.68868	0.844339	+	0.535809i	$0.179993\pi$
$374$	0	0
$375$	393211.	0.144394
$376$	0	0
$377$	401124.	0.145354
$378$	0	0
$379$	−2.43448e6	−0.870580	−0.435290	−	0.900290i	$-0.643354\pi$
$379$	−2.43448e6	−0.870580	−0.435290	+	0.900290i	$0.643354\pi$
$380$	0	0
$381$	6.05817e6	2.13810
$382$	0	0
$383$	3.38210e6	1.17812	0.589061	−	0.808089i	$-0.299498\pi$
$383$	3.38210e6	1.17812	0.589061	+	0.808089i	$0.299498\pi$
$384$	0	0
$385$	−319286.	−0.109781
$386$	0	0
$387$	1.96098e6	0.665574
$388$	0	0
$389$	−2.10107e6	−0.703989	−0.351995	−	0.936002i	$-0.614496\pi$
$389$	−2.10107e6	−0.703989	−0.351995	+	0.936002i	$0.614496\pi$
$390$	0	0
$391$	−939360.	−0.310735
$392$	0	0
$393$	2.61740e6	0.854849
$394$	0	0
$395$	1.75979e6	0.567503
$396$	0	0
$397$	1.16621e6	0.371365	0.185682	−	0.982610i	$-0.440550\pi$
$397$	1.16621e6	0.371365	0.185682	+	0.982610i	$0.440550\pi$
$398$	0	0
$399$	−690847.	−0.217245
$400$	0	0
$401$	−1.05268e6	−0.326917	−0.163458	−	0.986550i	$-0.552265\pi$
$401$	−1.05268e6	−0.326917	−0.163458	+	0.986550i	$0.552265\pi$
$402$	0	0
$403$	55155.2	0.0169170
$404$	0	0
$405$	−38896.2	−0.0117834
$406$	0	0
$407$	3.55842e6	1.06481
$408$	0	0
$409$	−2.23538e6	−0.660759	−0.330379	−	0.943848i	$-0.607177\pi$
$409$	−2.23538e6	−0.660759	−0.330379	+	0.943848i	$0.607177\pi$
$410$	0	0
$411$	5.36911e6	1.56783
$412$	0	0
$413$	1.64496e6	0.474547
$414$	0	0
$415$	330512.	0.0942034
$416$	0	0
$417$	1.00341e7	2.82578
$418$	0	0
$419$	287706.	0.0800598	0.0400299	−	0.999198i	$-0.487255\pi$
$419$	287706.	0.0800598	0.0400299	+	0.999198i	$0.487255\pi$
$420$	0	0
$421$	−2.72578e6	−0.749524	−0.374762	−	0.927121i	$-0.622276\pi$
$421$	−2.72578e6	−0.749524	−0.374762	+	0.927121i	$0.622276\pi$
$422$	0	0
$423$	−4.69877e6	−1.27683
$424$	0	0
$425$	−205298.	−0.0551330
$426$	0	0
$427$	1.30433e6	0.346193
$428$	0	0
$429$	630812.	0.165484
$430$	0	0
$431$	1.79222e6	0.464727	0.232364	−	0.972629i	$-0.425354\pi$
$431$	1.79222e6	0.464727	0.232364	+	0.972629i	$0.425354\pi$
$432$	0	0
$433$	−1.09709e6	−0.281204	−0.140602	−	0.990066i	$-0.544904\pi$
$433$	−1.09709e6	−0.281204	−0.140602	+	0.990066i	$0.544904\pi$
$434$	0	0
$435$	2.62407e6	0.664893
$436$	0	0
$437$	1.60217e6	0.401333
$438$	0	0
$439$	3.13375e6	0.776073	0.388037	−	0.921644i	$-0.373153\pi$
$439$	3.13375e6	0.776073	0.388037	+	0.921644i	$0.373153\pi$
$440$	0	0
$441$	937119.	0.229455
$442$	0	0
$443$	5.23745e6	1.26797	0.633987	−	0.773344i	$-0.281417\pi$
$443$	5.23745e6	1.26797	0.633987	+	0.773344i	$0.281417\pi$
$444$	0	0
$445$	1.44949e6	0.346987
$446$	0	0
$447$	−7.12569e6	−1.68678
$448$	0	0
$449$	4.09755e6	0.959198	0.479599	−	0.877488i	$-0.340782\pi$
$449$	4.09755e6	0.959198	0.479599	+	0.877488i	$0.340782\pi$
$450$	0	0
$451$	−1.08821e6	−0.251926
$452$	0	0
$453$	5.39987e6	1.23634
$454$	0	0
$455$	−117811.	−0.0266783
$456$	0	0
$457$	−85314.3	−0.0191087	−0.00955436	−	0.999954i	$-0.503041\pi$
$457$	−85314.3	−0.0191087	−0.00955436	+	0.999954i	$0.503041\pi$
$458$	0	0
$459$	−1.21765e6	−0.269769
$460$	0	0
$461$	−5.55248e6	−1.21684	−0.608422	−	0.793614i	$-0.708197\pi$
$461$	−5.55248e6	−1.21684	−0.608422	+	0.793614i	$0.708197\pi$
$462$	0	0
$463$	−2.51533e6	−0.545308	−0.272654	−	0.962112i	$-0.587901\pi$
$463$	−2.51533e6	−0.545308	−0.272654	+	0.962112i	$0.587901\pi$
$464$	0	0
$465$	360813.	0.0773837
$466$	0	0
$467$	6.08590e6	1.29132	0.645658	−	0.763627i	$-0.276583\pi$
$467$	6.08590e6	1.29132	0.645658	+	0.763627i	$0.276583\pi$
$468$	0	0
$469$	1.26693e6	0.265962
$470$	0	0
$471$	−9.33619e6	−1.93918
$472$	0	0
$473$	1.30953e6	0.269130
$474$	0	0
$475$	350155.	0.0712076
$476$	0	0
$477$	6.47745e6	1.30349
$478$	0	0
$479$	2.00975e6	0.400224	0.200112	−	0.979773i	$-0.435869\pi$
$479$	2.00975e6	0.400224	0.200112	+	0.979773i	$0.435869\pi$
$480$	0	0
$481$	1.31300e6	0.258763
$482$	0	0
$483$	−3.52639e6	−0.687801
$484$	0	0
$485$	315988.	0.0609982
$486$	0	0
$487$	−4.88471e6	−0.933289	−0.466645	−	0.884445i	$-0.654537\pi$
$487$	−4.88471e6	−0.933289	−0.466645	+	0.884445i	$0.654537\pi$
$488$	0	0
$489$	−1.48822e7	−2.81446
$490$	0	0
$491$	−2.59520e6	−0.485811	−0.242905	−	0.970050i	$-0.578100\pi$
$491$	−2.59520e6	−0.485811	−0.242905	+	0.970050i	$0.578100\pi$
$492$	0	0
$493$	−1.37004e6	−0.253872
$494$	0	0
$495$	2.54323e6	0.466523
$496$	0	0
$497$	−520849.	−0.0945848
$498$	0	0
$499$	−656741.	−0.118071	−0.0590355	−	0.998256i	$-0.518803\pi$
$499$	−656741.	−0.118071	−0.0590355	+	0.998256i	$0.518803\pi$
$500$	0	0
$501$	7.86476e6	1.39988
$502$	0	0
$503$	−5.99589e6	−1.05666	−0.528328	−	0.849040i	$-0.677181\pi$
$503$	−5.99589e6	−1.05666	−0.528328	+	0.849040i	$0.677181\pi$
$504$	0	0
$505$	−1.09860e6	−0.191695
$506$	0	0
$507$	−9.11102e6	−1.57415
$508$	0	0
$509$	2.15207e6	0.368182	0.184091	−	0.982909i	$-0.441066\pi$
$509$	2.15207e6	0.368182	0.184091	+	0.982909i	$0.441066\pi$
$510$	0	0
$511$	968473.	0.164072
$512$	0	0
$513$	2.07682e6	0.348423
$514$	0	0
$515$	−1.57771e6	−0.262126
$516$	0	0
$517$	−3.13780e6	−0.516297
$518$	0	0
$519$	−8.87741e6	−1.44666
$520$	0	0
$521$	296993.	0.0479350	0.0239675	−	0.999713i	$-0.492370\pi$
$521$	296993.	0.0479350	0.0239675	+	0.999713i	$0.492370\pi$
$522$	0	0
$523$	−4.50030e6	−0.719428	−0.359714	−	0.933063i	$-0.617126\pi$
$523$	−4.50030e6	−0.719428	−0.359714	+	0.933063i	$0.617126\pi$
$524$	0	0
$525$	−770694.	−0.122035
$526$	0	0
$527$	−188382.	−0.0295470
$528$	0	0
$529$	1.74184e6	0.270626
$530$	0	0
$531$	−1.31027e7	−2.01662
$532$	0	0
$533$	−401533.	−0.0612214
$534$	0	0
$535$	−1.70272e6	−0.257193
$536$	0	0
$537$	−3.21484e6	−0.481087
$538$	0	0
$539$	625800.	0.0927820
$540$	0	0
$541$	5.71943e6	0.840155	0.420077	−	0.907488i	$-0.362003\pi$
$541$	5.71943e6	0.840155	0.420077	+	0.907488i	$0.362003\pi$
$542$	0	0
$543$	−1.97268e7	−2.87115
$544$	0	0
$545$	3.91037e6	0.563932
$546$	0	0
$547$	1.16694e7	1.66755	0.833776	−	0.552103i	$-0.186175\pi$
$547$	1.16694e7	1.66755	0.833776	+	0.552103i	$0.186175\pi$
$548$	0	0
$549$	−1.03895e7	−1.47117
$550$	0	0
$551$	2.33673e6	0.327891
$552$	0	0
$553$	−3.44919e6	−0.479628
$554$	0	0
$555$	8.58934e6	1.18366
$556$	0	0
$557$	1.24722e7	1.70335	0.851677	−	0.524067i	$-0.175586\pi$
$557$	1.24722e7	1.70335	0.851677	+	0.524067i	$0.175586\pi$
$558$	0	0
$559$	483194.	0.0654022
$560$	0	0
$561$	−2.15453e6	−0.289032
$562$	0	0
$563$	−7.54634e6	−1.00338	−0.501690	−	0.865047i	$-0.667288\pi$
$563$	−7.54634e6	−1.00338	−0.501690	+	0.865047i	$0.667288\pi$
$564$	0	0
$565$	−5.64962e6	−0.744557
$566$	0	0
$567$	76236.5	0.00995876
$568$	0	0
$569$	7.79233e6	1.00899	0.504495	−	0.863415i	$-0.331679\pi$
$569$	7.79233e6	1.00899	0.504495	+	0.863415i	$0.331679\pi$
$570$	0	0
$571$	−6.70418e6	−0.860509	−0.430254	−	0.902708i	$-0.641576\pi$
$571$	−6.70418e6	−0.860509	−0.430254	+	0.902708i	$0.641576\pi$
$572$	0	0
$573$	−1.61760e7	−2.05819
$574$	0	0
$575$	1.78735e6	0.225444
$576$	0	0
$577$	−971236.	−0.121447	−0.0607233	−	0.998155i	$-0.519341\pi$
$577$	−971236.	−0.121447	−0.0607233	+	0.998155i	$0.519341\pi$
$578$	0	0
$579$	3.10398e6	0.384788
$580$	0	0
$581$	−647803.	−0.0796164
$582$	0	0
$583$	4.32559e6	0.527077
$584$	0	0
$585$	938411.	0.113371
$586$	0	0
$587$	4.12339e6	0.493923	0.246962	−	0.969025i	$-0.420568\pi$
$587$	4.12339e6	0.493923	0.246962	+	0.969025i	$0.420568\pi$
$588$	0	0
$589$	321304.	0.0381617
$590$	0	0
$591$	−1.89029e7	−2.22618
$592$	0	0
$593$	−9.45864e6	−1.10457	−0.552284	−	0.833656i	$-0.686243\pi$
$593$	−9.45864e6	−1.10457	−0.552284	+	0.833656i	$0.686243\pi$
$594$	0	0
$595$	402383.	0.0465959
$596$	0	0
$597$	−3.18477e6	−0.365714
$598$	0	0
$599$	1.44764e7	1.64852	0.824261	−	0.566210i	$-0.191591\pi$
$599$	1.44764e7	1.64852	0.824261	+	0.566210i	$0.191591\pi$
$600$	0	0
$601$	−6.63503e6	−0.749302	−0.374651	−	0.927166i	$-0.622237\pi$
$601$	−6.63503e6	−0.749302	−0.374651	+	0.927166i	$0.622237\pi$
$602$	0	0
$603$	−1.00916e7	−1.13023
$604$	0	0
$605$	−2.32792e6	−0.258571
$606$	0	0
$607$	1.05227e7	1.15919	0.579594	−	0.814906i	$-0.303211\pi$
$607$	1.05227e7	1.15919	0.579594	+	0.814906i	$0.303211\pi$
$608$	0	0
$609$	−5.14317e6	−0.561937
$610$	0	0
$611$	−1.15780e6	−0.125467
$612$	0	0
$613$	−8.02285e6	−0.862338	−0.431169	−	0.902271i	$-0.641899\pi$
$613$	−8.02285e6	−0.862338	−0.431169	+	0.902271i	$0.641899\pi$
$614$	0	0
$615$	−2.62674e6	−0.280046
$616$	0	0
$617$	−9.06937e6	−0.959101	−0.479550	−	0.877514i	$-0.659200\pi$
$617$	−9.06937e6	−0.959101	−0.479550	+	0.877514i	$0.659200\pi$
$618$	0	0
$619$	6.50536e6	0.682409	0.341204	−	0.939989i	$-0.389165\pi$
$619$	6.50536e6	0.682409	0.341204	+	0.939989i	$0.389165\pi$
$620$	0	0
$621$	1.06010e7	1.10311
$622$	0	0
$623$	−2.84099e6	−0.293258
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	3.67477e6	0.373302
$628$	0	0
$629$	−4.48454e6	−0.451951
$630$	0	0
$631$	−547351.	−0.0547258	−0.0273629	−	0.999626i	$-0.508711\pi$
$631$	−547351.	−0.0547258	−0.0273629	+	0.999626i	$0.508711\pi$
$632$	0	0
$633$	−2.72485e7	−2.70292
$634$	0	0
$635$	6.01832e6	0.592299
$636$	0	0
$637$	230910.	0.0225473
$638$	0	0
$639$	4.14876e6	0.401945
$640$	0	0
$641$	1.51178e7	1.45326	0.726629	−	0.687030i	$-0.241086\pi$
$641$	1.51178e7	1.45326	0.726629	+	0.687030i	$0.241086\pi$
$642$	0	0
$643$	1.93741e7	1.84797	0.923984	−	0.382431i	$-0.124913\pi$
$643$	1.93741e7	1.84797	0.923984	+	0.382431i	$0.124913\pi$
$644$	0	0
$645$	3.16095e6	0.299170
$646$	0	0
$647$	8.53091e6	0.801189	0.400594	−	0.916256i	$-0.368804\pi$
$647$	8.53091e6	0.801189	0.400594	+	0.916256i	$0.368804\pi$
$648$	0	0
$649$	−8.74988e6	−0.815437
$650$	0	0
$651$	−707193.	−0.0654012
$652$	0	0
$653$	1.66242e7	1.52566	0.762829	−	0.646600i	$-0.223810\pi$
$653$	1.66242e7	1.52566	0.762829	+	0.646600i	$0.223810\pi$
$654$	0	0
$655$	2.60019e6	0.236811
$656$	0	0
$657$	−7.71425e6	−0.697237
$658$	0	0
$659$	5.30945e6	0.476252	0.238126	−	0.971234i	$-0.423467\pi$
$659$	5.30945e6	0.476252	0.238126	+	0.971234i	$0.423467\pi$
$660$	0	0
$661$	−4.62991e6	−0.412163	−0.206081	−	0.978535i	$-0.566071\pi$
$661$	−4.62991e6	−0.412163	−0.206081	+	0.978535i	$0.566071\pi$
$662$	0	0
$663$	−794987.	−0.0702387
$664$	0	0
$665$	−686303.	−0.0601814
$666$	0	0
$667$	1.19277e7	1.03811
$668$	0	0
$669$	1.13149e7	0.977427
$670$	0	0
$671$	−6.93803e6	−0.594880
$672$	0	0
$673$	4.03005e6	0.342983	0.171491	−	0.985186i	$-0.445141\pi$
$673$	4.03005e6	0.342983	0.171491	+	0.985186i	$0.445141\pi$
$674$	0	0
$675$	2.31686e6	0.195722
$676$	0	0
$677$	1.27163e7	1.06632	0.533161	−	0.846014i	$-0.321004\pi$
$677$	1.27163e7	1.06632	0.533161	+	0.846014i	$0.321004\pi$
$678$	0	0
$679$	−619337.	−0.0515529
$680$	0	0
$681$	−1.78233e7	−1.47272
$682$	0	0
$683$	1.48686e7	1.21960	0.609801	−	0.792555i	$-0.291249\pi$
$683$	1.48686e7	1.21960	0.609801	+	0.792555i	$0.291249\pi$
$684$	0	0
$685$	5.33379e6	0.434320
$686$	0	0
$687$	−3.12813e7	−2.52868
$688$	0	0
$689$	1.59607e6	0.128087
$690$	0	0
$691$	3.72206e6	0.296543	0.148272	−	0.988947i	$-0.452629\pi$
$691$	3.72206e6	0.296543	0.148272	+	0.988947i	$0.452629\pi$
$692$	0	0
$693$	−4.98474e6	−0.394284
$694$	0	0
$695$	9.96809e6	0.782799
$696$	0	0
$697$	1.37143e6	0.106928
$698$	0	0
$699$	2.08471e7	1.61381
$700$	0	0
$701$	2.39713e7	1.84246	0.921228	−	0.389023i	$-0.127187\pi$
$701$	2.39713e7	1.84246	0.921228	+	0.389023i	$0.127187\pi$
$702$	0	0
$703$	7.64881e6	0.583722
$704$	0	0
$705$	−7.57405e6	−0.573925
$706$	0	0
$707$	2.15326e6	0.162012
$708$	0	0
$709$	2.16547e6	0.161784	0.0808922	−	0.996723i	$-0.474223\pi$
$709$	2.16547e6	0.161784	0.0808922	+	0.996723i	$0.474223\pi$
$710$	0	0
$711$	2.74741e7	2.03821
$712$	0	0
$713$	1.64008e6	0.120820
$714$	0	0
$715$	626663.	0.0458426
$716$	0	0
$717$	1.45495e7	1.05694
$718$	0	0
$719$	1.08385e7	0.781894	0.390947	−	0.920413i	$-0.372148\pi$
$719$	1.08385e7	0.781894	0.390947	+	0.920413i	$0.372148\pi$
$720$	0	0
$721$	3.09232e6	0.221537
$722$	0	0
$723$	1.86821e7	1.32917
$724$	0	0
$725$	2.60681e6	0.184189
$726$	0	0
$727$	−1.56366e7	−1.09726	−0.548628	−	0.836067i	$-0.684850\pi$
$727$	−1.56366e7	−1.09726	−0.548628	+	0.836067i	$0.684850\pi$
$728$	0	0
$729$	−2.32762e7	−1.62216
$730$	0	0
$731$	−1.65035e6	−0.114230
$732$	0	0
$733$	−2.91475e6	−0.200374	−0.100187	−	0.994969i	$-0.531944\pi$
$733$	−2.91475e6	−0.200374	−0.100187	+	0.994969i	$0.531944\pi$
$734$	0	0
$735$	1.51056e6	0.103138
$736$	0	0
$737$	−6.73907e6	−0.457016
$738$	0	0
$739$	9.25141e6	0.623156	0.311578	−	0.950221i	$-0.399142\pi$
$739$	9.25141e6	0.623156	0.311578	+	0.950221i	$0.399142\pi$
$740$	0	0
$741$	1.35593e6	0.0907175
$742$	0	0
$743$	8.41963e6	0.559527	0.279764	−	0.960069i	$-0.409744\pi$
$743$	8.41963e6	0.559527	0.279764	+	0.960069i	$0.409744\pi$
$744$	0	0
$745$	−7.07882e6	−0.467273
$746$	0	0
$747$	5.16000e6	0.338336
$748$	0	0
$749$	3.33734e6	0.217368
$750$	0	0
$751$	−2.95913e7	−1.91454	−0.957270	−	0.289195i	$-0.906612\pi$
$751$	−2.95913e7	−1.91454	−0.957270	+	0.289195i	$0.906612\pi$
$752$	0	0
$753$	5.19828e6	0.334097
$754$	0	0
$755$	5.36436e6	0.342492
$756$	0	0
$757$	−2.08396e7	−1.32175	−0.660874	−	0.750497i	$-0.729814\pi$
$757$	−2.08396e7	−1.32175	−0.660874	+	0.750497i	$0.729814\pi$
$758$	0	0
$759$	1.87576e7	1.18188
$760$	0	0
$761$	208580.	0.0130560	0.00652802	−	0.999979i	$-0.497922\pi$
$761$	208580.	0.0130560	0.00652802	+	0.999979i	$0.497922\pi$
$762$	0	0
$763$	−7.66433e6	−0.476610
$764$	0	0
$765$	−3.20513e6	−0.198013
$766$	0	0
$767$	−3.22856e6	−0.198162
$768$	0	0
$769$	−1.46563e7	−0.893736	−0.446868	−	0.894600i	$-0.647461\pi$
$769$	−1.46563e7	−0.893736	−0.446868	+	0.894600i	$0.647461\pi$
$770$	0	0
$771$	−2.47992e7	−1.50246
$772$	0	0
$773$	−4.52964e6	−0.272656	−0.136328	−	0.990664i	$-0.543530\pi$
$773$	−4.52964e6	−0.272656	−0.136328	+	0.990664i	$0.543530\pi$
$774$	0	0
$775$	358440.	0.0214369
$776$	0	0
$777$	−1.68351e7	−1.00038
$778$	0	0
$779$	−2.33911e6	−0.138104
$780$	0	0
$781$	2.77051e6	0.162529
$782$	0	0
$783$	1.54614e7	0.901247
$784$	0	0
$785$	−9.27478e6	−0.537192
$786$	0	0
$787$	2.64851e7	1.52428	0.762139	−	0.647413i	$-0.224149\pi$
$787$	2.64851e7	1.52428	0.762139	+	0.647413i	$0.224149\pi$
$788$	0	0
$789$	−1.88192e6	−0.107624
$790$	0	0
$791$	1.10732e7	0.629265
$792$	0	0
$793$	−2.56002e6	−0.144564
$794$	0	0
$795$	1.04411e7	0.585908
$796$	0	0
$797$	2.06246e7	1.15011	0.575056	−	0.818114i	$-0.304980\pi$
$797$	2.06246e7	1.15011	0.575056	+	0.818114i	$0.304980\pi$
$798$	0	0
$799$	3.95445e6	0.219138
$800$	0	0
$801$	2.26296e7	1.24622
$802$	0	0
$803$	−5.15152e6	−0.281933
$804$	0	0
$805$	−3.50320e6	−0.190535
$806$	0	0
$807$	−2.50754e7	−1.35539
$808$	0	0
$809$	−6.24306e6	−0.335372	−0.167686	−	0.985840i	$-0.553629\pi$
$809$	−6.24306e6	−0.335372	−0.167686	+	0.985840i	$0.553629\pi$
$810$	0	0
$811$	−1.31236e7	−0.700651	−0.350325	−	0.936628i	$-0.613929\pi$
$811$	−1.31236e7	−0.700651	−0.350325	+	0.936628i	$0.613929\pi$
$812$	0	0
$813$	3.44760e7	1.82932
$814$	0	0
$815$	−1.47843e7	−0.779663
$816$	0	0
$817$	2.81483e6	0.147535
$818$	0	0
$819$	−1.83929e6	−0.0958163
$820$	0	0
$821$	−3.64067e7	−1.88505	−0.942527	−	0.334131i	$-0.891557\pi$
$821$	−3.64067e7	−1.88505	−0.942527	+	0.334131i	$0.891557\pi$
$822$	0	0
$823$	3.09385e7	1.59221	0.796103	−	0.605161i	$-0.206891\pi$
$823$	3.09385e7	1.59221	0.796103	+	0.605161i	$0.206891\pi$
$824$	0	0
$825$	4.09949e6	0.209698
$826$	0	0
$827$	−1.11276e7	−0.565769	−0.282884	−	0.959154i	$-0.591291\pi$
$827$	−1.11276e7	−0.565769	−0.282884	+	0.959154i	$0.591291\pi$
$828$	0	0
$829$	1.11566e7	0.563826	0.281913	−	0.959440i	$-0.409031\pi$
$829$	1.11566e7	0.563826	0.281913	+	0.959440i	$0.409031\pi$
$830$	0	0
$831$	3.42139e7	1.71870
$832$	0	0
$833$	−788671.	−0.0393807
$834$	0	0
$835$	7.81303e6	0.387796
$836$	0	0
$837$	2.12596e6	0.104892
$838$	0	0
$839$	1.99106e7	0.976519	0.488259	−	0.872699i	$-0.337632\pi$
$839$	1.99106e7	0.976519	0.488259	+	0.872699i	$0.337632\pi$
$840$	0	0
$841$	−3.11482e6	−0.151860
$842$	0	0
$843$	−4.27898e7	−2.07382
$844$	0	0
$845$	−9.05110e6	−0.436073
$846$	0	0
$847$	4.56273e6	0.218533
$848$	0	0
$849$	−4.18461e7	−1.99244
$850$	0	0
$851$	3.90429e7	1.84807
$852$	0	0
$853$	1.57948e7	0.743262	0.371631	−	0.928381i	$-0.378799\pi$
$853$	1.57948e7	0.743262	0.371631	+	0.928381i	$0.378799\pi$
$854$	0	0
$855$	5.46667e6	0.255745
$856$	0	0
$857$	3.01593e7	1.40272	0.701358	−	0.712810i	$-0.252578\pi$
$857$	3.01593e7	1.40272	0.701358	+	0.712810i	$0.252578\pi$
$858$	0	0
$859$	−3.80634e7	−1.76005	−0.880024	−	0.474930i	$-0.842473\pi$
$859$	−3.80634e7	−1.76005	−0.880024	+	0.474930i	$0.842473\pi$
$860$	0	0
$861$	5.14840e6	0.236682
$862$	0	0
$863$	3.25092e7	1.48586	0.742932	−	0.669367i	$-0.233435\pi$
$863$	3.25092e7	1.48586	0.742932	+	0.669367i	$0.233435\pi$
$864$	0	0
$865$	−8.81902e6	−0.400756
$866$	0	0
$867$	−3.30162e7	−1.49169
$868$	0	0
$869$	1.83470e7	0.824167
$870$	0	0
$871$	−2.48660e6	−0.111061
$872$	0	0
$873$	4.93326e6	0.219078
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	−1.11471e7	−0.489399	−0.244700	−	0.969599i	$-0.578689\pi$
$877$	−1.11471e7	−0.489399	−0.244700	+	0.969599i	$0.578689\pi$
$878$	0	0
$879$	3.47320e7	1.51621
$880$	0	0
$881$	3.47768e7	1.50956	0.754778	−	0.655980i	$-0.227745\pi$
$881$	3.47768e7	1.50956	0.754778	+	0.655980i	$0.227745\pi$
$882$	0	0
$883$	−3.39735e7	−1.46635	−0.733176	−	0.680039i	$-0.761963\pi$
$883$	−3.39735e7	−1.46635	−0.733176	+	0.680039i	$0.761963\pi$
$884$	0	0
$885$	−2.11205e7	−0.906455
$886$	0	0
$887$	−1.99014e7	−0.849324	−0.424662	−	0.905352i	$-0.639607\pi$
$887$	−1.99014e7	−0.849324	−0.424662	+	0.905352i	$0.639607\pi$
$888$	0	0
$889$	−1.17959e7	−0.500584
$890$	0	0
$891$	−405518.	−0.0171126
$892$	0	0
$893$	−6.74469e6	−0.283031
$894$	0	0
$895$	−3.19370e6	−0.133271
$896$	0	0
$897$	6.92125e6	0.287213
$898$	0	0
$899$	2.39202e6	0.0987110
$900$	0	0
$901$	−5.45136e6	−0.223714
$902$	0	0
$903$	−6.19546e6	−0.252845
$904$	0	0
$905$	−1.95970e7	−0.795368
$906$	0	0
$907$	9.25411e6	0.373522	0.186761	−	0.982405i	$-0.440201\pi$
$907$	9.25411e6	0.373522	0.186761	+	0.982405i	$0.440201\pi$
$908$	0	0
$909$	−1.71515e7	−0.688482
$910$	0	0
$911$	−3.59174e7	−1.43387	−0.716934	−	0.697141i	$-0.754455\pi$
$911$	−3.59174e7	−1.43387	−0.716934	+	0.697141i	$0.754455\pi$
$912$	0	0
$913$	3.44580e6	0.136809
$914$	0	0
$915$	−1.67471e7	−0.661280
$916$	0	0
$917$	−5.09637e6	−0.200142
$918$	0	0
$919$	2.34240e7	0.914898	0.457449	−	0.889236i	$-0.348763\pi$
$919$	2.34240e7	0.914898	0.457449	+	0.889236i	$0.348763\pi$
$920$	0	0
$921$	1.41219e7	0.548584
$922$	0	0
$923$	1.02227e6	0.0394968
$924$	0	0
$925$	8.53285e6	0.327899
$926$	0	0
$927$	−2.46315e7	−0.941437
$928$	0	0
$929$	1.63857e7	0.622909	0.311455	−	0.950261i	$-0.399184\pi$
$929$	1.63857e7	0.622909	0.311455	+	0.950261i	$0.399184\pi$
$930$	0	0
$931$	1.34515e6	0.0508625
$932$	0	0
$933$	6.66847e7	2.50797
$934$	0	0
$935$	−2.14036e6	−0.0800679
$936$	0	0
$937$	−7.36467e6	−0.274034	−0.137017	−	0.990569i	$-0.543751\pi$
$937$	−7.36467e6	−0.274034	−0.137017	+	0.990569i	$0.543751\pi$
$938$	0	0
$939$	−6.44132e7	−2.38403
$940$	0	0
$941$	4.46325e7	1.64315	0.821574	−	0.570101i	$-0.193096\pi$
$941$	4.46325e7	1.64315	0.821574	+	0.570101i	$0.193096\pi$
$942$	0	0
$943$	−1.19399e7	−0.437240
$944$	0	0
$945$	−4.54104e6	−0.165415
$946$	0	0
$947$	1.45904e7	0.528680	0.264340	−	0.964430i	$-0.414846\pi$
$947$	1.45904e7	0.528680	0.264340	+	0.964430i	$0.414846\pi$
$948$	0	0
$949$	−1.90082e6	−0.0685135
$950$	0	0
$951$	4.35840e7	1.56270
$952$	0	0
$953$	3.45890e7	1.23369	0.616845	−	0.787085i	$-0.288411\pi$
$953$	3.45890e7	1.23369	0.616845	+	0.787085i	$0.288411\pi$
$954$	0	0
$955$	−1.60696e7	−0.570160
$956$	0	0
$957$	2.73576e7	0.965603
$958$	0	0
$959$	−1.04542e7	−0.367067
$960$	0	0
$961$	−2.83002e7	−0.988511
$962$	0	0
$963$	−2.65832e7	−0.923721
$964$	0	0
$965$	3.08356e6	0.106594
$966$	0	0
$967$	−2.58436e7	−0.888766	−0.444383	−	0.895837i	$-0.646577\pi$
$967$	−2.58436e7	−0.888766	−0.444383	+	0.895837i	$0.646577\pi$
$968$	0	0
$969$	−4.63116e6	−0.158446
$970$	0	0
$971$	−2.22711e7	−0.758043	−0.379022	−	0.925388i	$-0.623739\pi$
$971$	−2.22711e7	−0.758043	−0.379022	+	0.925388i	$0.623739\pi$
$972$	0	0
$973$	−1.95375e7	−0.661586
$974$	0	0
$975$	1.51264e6	0.0509595
$976$	0	0
$977$	4.21805e7	1.41376	0.706880	−	0.707333i	$-0.250102\pi$
$977$	4.21805e7	1.41376	0.706880	+	0.707333i	$0.250102\pi$
$978$	0	0
$979$	1.51118e7	0.503919
$980$	0	0
$981$	6.10493e7	2.02539
$982$	0	0
$983$	−1.60837e6	−0.0530886	−0.0265443	−	0.999648i	$-0.508450\pi$
$983$	−1.60837e6	−0.0530886	−0.0265443	+	0.999648i	$0.508450\pi$
$984$	0	0
$985$	−1.87786e7	−0.616697
$986$	0	0
$987$	1.48451e7	0.485055
$988$	0	0
$989$	1.43681e7	0.467099
$990$	0	0
$991$	2.65982e7	0.860337	0.430168	−	0.902749i	$-0.358454\pi$
$991$	2.65982e7	0.860337	0.430168	+	0.902749i	$0.358454\pi$
$992$	0	0
$993$	6.90237e7	2.22139
$994$	0	0
$995$	−3.16382e6	−0.101310
$996$	0	0
$997$	−8.51064e6	−0.271159	−0.135580	−	0.990766i	$-0.543290\pi$
$997$	−8.51064e6	−0.271159	−0.135580	+	0.990766i	$0.543290\pi$
$998$	0	0
$999$	5.06097e7	1.60443

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.d.1.2	✓	2	1.1	even	1	trivial
560.6.a.j.1.1		2	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)

\(f(q)\)	\(=\)	\(q+25.1655 q^{3} +25.0000 q^{5} -49.0000 q^{7} +390.304 q^{9} +O(q^{10})\)	\(q+25.1655 q^{3} +25.0000 q^{5} -49.0000 q^{7} +390.304 q^{9} +260.642 q^{11} +96.1724 q^{13} +629.138 q^{15} -328.476 q^{17} +560.248 q^{19} -1233.11 q^{21} +2859.75 q^{23} +625.000 q^{25} +3706.97 q^{27} +4170.89 q^{29} +573.503 q^{31} +6559.18 q^{33} -1225.00 q^{35} +13652.6 q^{37} +2420.23 q^{39} -4175.14 q^{41} +5024.25 q^{43} +9757.59 q^{45} -12038.8 q^{47} +2401.00 q^{49} -8266.27 q^{51} +16595.9 q^{53} +6516.04 q^{55} +14098.9 q^{57} -33570.5 q^{59} -26619.0 q^{61} -19124.9 q^{63} +2404.31 q^{65} -25855.7 q^{67} +71967.2 q^{69} +10629.6 q^{71} -19764.8 q^{73} +15728.5 q^{75} -12771.4 q^{77} +70391.7 q^{79} -1555.85 q^{81} +13220.5 q^{83} -8211.90 q^{85} +104963. q^{87} +57979.4 q^{89} -4712.45 q^{91} +14432.5 q^{93} +14006.2 q^{95} +12639.5 q^{97} +101729. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9}+O(q^{10}) \) 2 * q + 26 * q^3 + 50 * q^5 - 98 * q^7 + 148 * q^9	\( 2 q + 26 q^{3} + 50 q^{5} - 98 q^{7} + 148 q^{9} - 14 q^{11} + 314 q^{13} + 650 q^{15} - 146 q^{17} + 2848 q^{19} - 1274 q^{21} + 3992 q^{23} + 1250 q^{25} + 3302 q^{27} + 6006 q^{29} + 1220 q^{31} + 6330 q^{33} - 2450 q^{35} + 7232 q^{37} + 2602 q^{39} + 6516 q^{41} - 2336 q^{43} + 3700 q^{45} + 12930 q^{47} + 4802 q^{49} - 8114 q^{51} + 3216 q^{53} - 350 q^{55} + 16008 q^{57} + 6728 q^{59} - 756 q^{61} - 7252 q^{63} + 7850 q^{65} + 11744 q^{67} + 72912 q^{69} - 1904 q^{71} + 22612 q^{73} + 16250 q^{75} + 686 q^{77} + 62486 q^{79} + 56986 q^{81} + 15200 q^{83} - 3650 q^{85} + 106494 q^{87} + 42844 q^{89} - 15386 q^{91} + 14972 q^{93} + 71200 q^{95} + 82822 q^{97} + 168276 q^{99}+O(q^{100}) \) 2 * q + 26 * q^3 + 50 * q^5 - 98 * q^7 + 148 * q^9 - 14 * q^11 + 314 * q^13 + 650 * q^15 - 146 * q^17 + 2848 * q^19 - 1274 * q^21 + 3992 * q^23 + 1250 * q^25 + 3302 * q^27 + 6006 * q^29 + 1220 * q^31 + 6330 * q^33 - 2450 * q^35 + 7232 * q^37 + 2602 * q^39 + 6516 * q^41 - 2336 * q^43 + 3700 * q^45 + 12930 * q^47 + 4802 * q^49 - 8114 * q^51 + 3216 * q^53 - 350 * q^55 + 16008 * q^57 + 6728 * q^59 - 756 * q^61 - 7252 * q^63 + 7850 * q^65 + 11744 * q^67 + 72912 * q^69 - 1904 * q^71 + 22612 * q^73 + 16250 * q^75 + 686 * q^77 + 62486 * q^79 + 56986 * q^81 + 15200 * q^83 - 3650 * q^85 + 106494 * q^87 + 42844 * q^89 - 15386 * q^91 + 14972 * q^93 + 71200 * q^95 + 82822 * q^97 + 168276 * q^99

Introduction

L-functions

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Varieties

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