LMFDB - Embedded newform 280.6.a.c.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{109}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 27 $ x^2 - x - 27
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			$5.72015$ 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)

$f(q)$	$=$	$q+21.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +235.761 q^{9} +O(q^{10})$	$q+21.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +235.761 q^{9} -502.851 q^{11} -858.926 q^{13} +547.015 q^{15} -1498.54 q^{17} +257.179 q^{19} -1072.15 q^{21} -4000.17 q^{23} +625.000 q^{25} -158.389 q^{27} -4302.02 q^{29} +7535.06 q^{31} -11002.7 q^{33} -1225.00 q^{35} +9023.51 q^{37} -18793.8 q^{39} +659.958 q^{41} -11338.6 q^{43} +5894.03 q^{45} -14459.4 q^{47} +2401.00 q^{49} -32788.9 q^{51} +25842.6 q^{53} -12571.3 q^{55} +5627.22 q^{57} +49638.9 q^{59} +25098.3 q^{61} -11552.3 q^{63} -21473.1 q^{65} +5305.69 q^{67} -87526.1 q^{69} +72901.1 q^{71} +28668.8 q^{73} +13675.4 q^{75} +24639.7 q^{77} -77979.8 q^{79} -60755.6 q^{81} -76004.4 q^{83} -37463.4 q^{85} -94130.8 q^{87} -21957.3 q^{89} +42087.4 q^{91} +164872. q^{93} +6429.46 q^{95} -153370. q^{97} -118553. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9	$ 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} - 254 q^{11} - 1342 q^{13} + 50 q^{15} - 1786 q^{17} + 1976 q^{19} - 98 q^{21} - 2112 q^{23} + 1250 q^{25} + 1646 q^{27} - 4762 q^{29} + 13692 q^{31} - 15950 q^{33} - 2450 q^{35} + 2136 q^{37} - 9190 q^{39} - 6740 q^{41} - 26728 q^{43} + 9700 q^{45} - 6326 q^{47} + 4802 q^{49} - 27074 q^{51} + 24624 q^{53} - 6350 q^{55} - 28544 q^{57} + 51336 q^{59} + 4468 q^{61} - 19012 q^{63} - 33550 q^{65} - 39168 q^{67} - 125064 q^{69} + 41232 q^{71} - 36124 q^{73} + 1250 q^{75} + 12446 q^{77} - 140842 q^{79} - 133622 q^{81} - 57712 q^{83} - 44650 q^{85} - 84986 q^{87} - 20236 q^{89} + 65758 q^{91} + 42468 q^{93} + 49400 q^{95} - 183586 q^{97} - 80668 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9 - 254 * q^11 - 1342 * q^13 + 50 * q^15 - 1786 * q^17 + 1976 * q^19 - 98 * q^21 - 2112 * q^23 + 1250 * q^25 + 1646 * q^27 - 4762 * q^29 + 13692 * q^31 - 15950 * q^33 - 2450 * q^35 + 2136 * q^37 - 9190 * q^39 - 6740 * q^41 - 26728 * q^43 + 9700 * q^45 - 6326 * q^47 + 4802 * q^49 - 27074 * q^51 + 24624 * q^53 - 6350 * q^55 - 28544 * q^57 + 51336 * q^59 + 4468 * q^61 - 19012 * q^63 - 33550 * q^65 - 39168 * q^67 - 125064 * q^69 + 41232 * q^71 - 36124 * q^73 + 1250 * q^75 + 12446 * q^77 - 140842 * q^79 - 133622 * q^81 - 57712 * q^83 - 44650 * q^85 - 84986 * q^87 - 20236 * q^89 + 65758 * q^91 + 42468 * q^93 + 49400 * q^95 - 183586 * q^97 - 80668 * 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
$3$	21.8806	1.40364	0.701821	−	0.712353i	$-0.252371\pi$
$3$	21.8806	1.40364	0.701821	+	0.712353i	$0.252371\pi$
$4$	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
$9$	235.761	0.970211
$10$	0	0
$11$	−502.851	−1.25302	−0.626509	−	0.779414i	$-0.715517\pi$
$11$	−502.851	−1.25302	−0.626509	+	0.779414i	$0.715517\pi$
$12$	0	0
$13$	−858.926	−1.40960	−0.704802	−	0.709404i	$-0.748964\pi$
$13$	−858.926	−1.40960	−0.704802	+	0.709404i	$0.748964\pi$
$14$	0	0
$15$	547.015	0.627728
$16$	0	0
$17$	−1498.54	−1.25761	−0.628804	−	0.777564i	$-0.716455\pi$
$17$	−1498.54	−1.25761	−0.628804	+	0.777564i	$0.716455\pi$
$18$	0	0
$19$	257.179	0.163437	0.0817186	−	0.996655i	$-0.473959\pi$
$19$	257.179	0.163437	0.0817186	+	0.996655i	$0.473959\pi$
$20$	0	0
$21$	−1072.15	−0.530527
$22$	0	0
$23$	−4000.17	−1.57673	−0.788367	−	0.615205i	$-0.789073\pi$
$23$	−4000.17	−1.57673	−0.788367	+	0.615205i	$0.789073\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−158.389	−0.0418134
$28$	0	0
$29$	−4302.02	−0.949898	−0.474949	−	0.880013i	$-0.657534\pi$
$29$	−4302.02	−0.949898	−0.474949	+	0.880013i	$0.657534\pi$
$30$	0	0
$31$	7535.06	1.40826	0.704129	−	0.710072i	$-0.251337\pi$
$31$	7535.06	1.40826	0.704129	+	0.710072i	$0.251337\pi$
$32$	0	0
$33$	−11002.7	−1.75879
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	9023.51	1.08361	0.541803	−	0.840506i	$-0.317742\pi$
$37$	9023.51	1.08361	0.541803	+	0.840506i	$0.317742\pi$
$38$	0	0
$39$	−18793.8	−1.97858
$40$	0	0
$41$	659.958	0.0613136	0.0306568	−	0.999530i	$-0.490240\pi$
$41$	659.958	0.0613136	0.0306568	+	0.999530i	$0.490240\pi$
$42$	0	0
$43$	−11338.6	−0.935164	−0.467582	−	0.883950i	$-0.654875\pi$
$43$	−11338.6	−0.935164	−0.467582	+	0.883950i	$0.654875\pi$
$44$	0	0
$45$	5894.03	0.433891
$46$	0	0
$47$	−14459.4	−0.954786	−0.477393	−	0.878690i	$-0.658418\pi$
$47$	−14459.4	−0.954786	−0.477393	+	0.878690i	$0.658418\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−32788.9	−1.76523
$52$	0	0
$53$	25842.6	1.26371	0.631855	−	0.775087i	$-0.282294\pi$
$53$	25842.6	1.26371	0.631855	+	0.775087i	$0.282294\pi$
$54$	0	0
$55$	−12571.3	−0.560367
$56$	0	0
$57$	5627.22	0.229407
$58$	0	0
$59$	49638.9	1.85649	0.928245	−	0.371970i	$-0.121318\pi$
$59$	49638.9	1.85649	0.928245	+	0.371970i	$0.121318\pi$
$60$	0	0
$61$	25098.3	0.863613	0.431807	−	0.901966i	$-0.357876\pi$
$61$	25098.3	0.863613	0.431807	+	0.901966i	$0.357876\pi$
$62$	0	0
$63$	−11552.3	−0.366705
$64$	0	0
$65$	−21473.1	−0.630394
$66$	0	0
$67$	5305.69	0.144396	0.0721980	−	0.997390i	$-0.476999\pi$
$67$	5305.69	0.144396	0.0721980	+	0.997390i	$0.476999\pi$
$68$	0	0
$69$	−87526.1	−2.21317
$70$	0	0
$71$	72901.1	1.71628	0.858140	−	0.513416i	$-0.171620\pi$
$71$	72901.1	1.71628	0.858140	+	0.513416i	$0.171620\pi$
$72$	0	0
$73$	28668.8	0.629655	0.314827	−	0.949149i	$-0.398053\pi$
$73$	28668.8	0.629655	0.314827	+	0.949149i	$0.398053\pi$
$74$	0	0
$75$	13675.4	0.280728
$76$	0	0
$77$	24639.7	0.473597
$78$	0	0
$79$	−77979.8	−1.40577	−0.702885	−	0.711303i	$-0.748105\pi$
$79$	−77979.8	−1.40577	−0.702885	+	0.711303i	$0.748105\pi$
$80$	0	0
$81$	−60755.6	−1.02890
$82$	0	0
$83$	−76004.4	−1.21100	−0.605499	−	0.795846i	$-0.707027\pi$
$83$	−76004.4	−1.21100	−0.605499	+	0.795846i	$0.707027\pi$
$84$	0	0
$85$	−37463.4	−0.562420
$86$	0	0
$87$	−94130.8	−1.33332
$88$	0	0
$89$	−21957.3	−0.293835	−0.146918	−	0.989149i	$-0.546935\pi$
$89$	−21957.3	−0.293835	−0.146918	+	0.989149i	$0.546935\pi$
$90$	0	0
$91$	42087.4	0.532780
$92$	0	0
$93$	164872.	1.97669
$94$	0	0
$95$	6429.46	0.0730913
$96$	0	0
$97$	−153370.	−1.65505	−0.827525	−	0.561430i	$-0.810252\pi$
$97$	−153370.	−1.65505	−0.827525	+	0.561430i	$0.810252\pi$
$98$	0	0
$99$	−118553.	−1.21569
$100$	0	0
$101$	−197578.	−1.92724	−0.963620	−	0.267276i	$-0.913876\pi$
$101$	−197578.	−1.92724	−0.963620	+	0.267276i	$0.913876\pi$
$102$	0	0
$103$	51165.9	0.475212	0.237606	−	0.971362i	$-0.423637\pi$
$103$	51165.9	0.475212	0.237606	+	0.971362i	$0.423637\pi$
$104$	0	0
$105$	−26803.8	−0.237259
$106$	0	0
$107$	54653.3	0.461484	0.230742	−	0.973015i	$-0.425885\pi$
$107$	54653.3	0.461484	0.230742	+	0.973015i	$0.425885\pi$
$108$	0	0
$109$	−27531.0	−0.221951	−0.110975	−	0.993823i	$-0.535397\pi$
$109$	−27531.0	−0.221951	−0.110975	+	0.993823i	$0.535397\pi$
$110$	0	0
$111$	197440.	1.52099
$112$	0	0
$113$	54947.3	0.404809	0.202404	−	0.979302i	$-0.435124\pi$
$113$	54947.3	0.404809	0.202404	+	0.979302i	$0.435124\pi$
$114$	0	0
$115$	−100004.	−0.705137
$116$	0	0
$117$	−202501.	−1.36761
$118$	0	0
$119$	73428.4	0.475331
$120$	0	0
$121$	91808.2	0.570056
$122$	0	0
$123$	14440.3	0.0860624
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	86986.9	0.478569	0.239285	−	0.970949i	$-0.423087\pi$
$127$	86986.9	0.478569	0.239285	+	0.970949i	$0.423087\pi$
$128$	0	0
$129$	−248095.	−1.31263
$130$	0	0
$131$	−212679.	−1.08279	−0.541397	−	0.840767i	$-0.682104\pi$
$131$	−212679.	−1.08279	−0.541397	+	0.840767i	$0.682104\pi$
$132$	0	0
$133$	−12601.7	−0.0617734
$134$	0	0
$135$	−3959.72	−0.0186995
$136$	0	0
$137$	−242248.	−1.10270	−0.551351	−	0.834273i	$-0.685888\pi$
$137$	−242248.	−1.10270	−0.551351	+	0.834273i	$0.685888\pi$
$138$	0	0
$139$	−265950.	−1.16752	−0.583759	−	0.811927i	$-0.698419\pi$
$139$	−265950.	−1.16752	−0.583759	+	0.811927i	$0.698419\pi$
$140$	0	0
$141$	−316381.	−1.34018
$142$	0	0
$143$	431912.	1.76626
$144$	0	0
$145$	−107550.	−0.424807
$146$	0	0
$147$	52535.4	0.200520
$148$	0	0
$149$	−331862.	−1.22459	−0.612297	−	0.790628i	$-0.709754\pi$
$149$	−331862.	−1.22459	−0.612297	+	0.790628i	$0.709754\pi$
$150$	0	0
$151$	102268.	0.365004	0.182502	−	0.983205i	$-0.441580\pi$
$151$	102268.	0.365004	0.182502	+	0.983205i	$0.441580\pi$
$152$	0	0
$153$	−353297.	−1.22015
$154$	0	0
$155$	188377.	0.629793
$156$	0	0
$157$	−140480.	−0.454847	−0.227424	−	0.973796i	$-0.573030\pi$
$157$	−140480.	−0.454847	−0.227424	+	0.973796i	$0.573030\pi$
$158$	0	0
$159$	565453.	1.77380
$160$	0	0
$161$	196008.	0.595949
$162$	0	0
$163$	272553.	0.803493	0.401746	−	0.915751i	$-0.368403\pi$
$163$	272553.	0.803493	0.401746	+	0.915751i	$0.368403\pi$
$164$	0	0
$165$	−275067.	−0.786555
$166$	0	0
$167$	538267.	1.49350	0.746752	−	0.665103i	$-0.231612\pi$
$167$	538267.	1.49350	0.746752	+	0.665103i	$0.231612\pi$
$168$	0	0
$169$	366460.	0.986983
$170$	0	0
$171$	60632.7	0.158568
$172$	0	0
$173$	36662.6	0.0931340	0.0465670	−	0.998915i	$-0.485172\pi$
$173$	36662.6	0.0931340	0.0465670	+	0.998915i	$0.485172\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	1.08613e6	2.60585
$178$	0	0
$179$	−364299.	−0.849817	−0.424909	−	0.905236i	$-0.639694\pi$
$179$	−364299.	−0.849817	−0.424909	+	0.905236i	$0.639694\pi$
$180$	0	0
$181$	−115318.	−0.261637	−0.130818	−	0.991406i	$-0.541760\pi$
$181$	−115318.	−0.261637	−0.130818	+	0.991406i	$0.541760\pi$
$182$	0	0
$183$	549166.	1.21220
$184$	0	0
$185$	225588.	0.484603
$186$	0	0
$187$	753541.	1.57581
$188$	0	0
$189$	7761.05	0.0158040
$190$	0	0
$191$	860992.	1.70772	0.853858	−	0.520506i	$-0.174257\pi$
$191$	860992.	1.70772	0.853858	+	0.520506i	$0.174257\pi$
$192$	0	0
$193$	33630.5	0.0649890	0.0324945	−	0.999472i	$-0.489655\pi$
$193$	33630.5	0.0649890	0.0324945	+	0.999472i	$0.489655\pi$
$194$	0	0
$195$	−469845.	−0.884848
$196$	0	0
$197$	−295477.	−0.542448	−0.271224	−	0.962516i	$-0.587428\pi$
$197$	−295477.	−0.542448	−0.271224	+	0.962516i	$0.587428\pi$
$198$	0	0
$199$	−163026.	−0.291827	−0.145913	−	0.989297i	$-0.546612\pi$
$199$	−163026.	−0.291827	−0.145913	+	0.989297i	$0.546612\pi$
$200$	0	0
$201$	116092.	0.202680
$202$	0	0
$203$	210799.	0.359028
$204$	0	0
$205$	16499.0	0.0274203
$206$	0	0
$207$	−943084.	−1.52976
$208$	0	0
$209$	−129322.	−0.204790
$210$	0	0
$211$	1.15718e6	1.78935	0.894676	−	0.446716i	$-0.147406\pi$
$211$	1.15718e6	1.78935	0.894676	+	0.446716i	$0.147406\pi$
$212$	0	0
$213$	1.59512e6	2.40904
$214$	0	0
$215$	−283465.	−0.418218
$216$	0	0
$217$	−369218.	−0.532272
$218$	0	0
$219$	627291.	0.883810
$220$	0	0
$221$	1.28713e6	1.77273
$222$	0	0
$223$	−561470.	−0.756074	−0.378037	−	0.925790i	$-0.623401\pi$
$223$	−561470.	−0.756074	−0.378037	+	0.925790i	$0.623401\pi$
$224$	0	0
$225$	147351.	0.194042
$226$	0	0
$227$	−49178.0	−0.0633441	−0.0316721	−	0.999498i	$-0.510083\pi$
$227$	−49178.0	−0.0633441	−0.0316721	+	0.999498i	$0.510083\pi$
$228$	0	0
$229$	488121.	0.615091	0.307545	−	0.951533i	$-0.400492\pi$
$229$	488121.	0.615091	0.307545	+	0.951533i	$0.400492\pi$
$230$	0	0
$231$	539132.	0.664760
$232$	0	0
$233$	31673.9	0.0382218	0.0191109	−	0.999817i	$-0.493916\pi$
$233$	31673.9	0.0382218	0.0191109	+	0.999817i	$0.493916\pi$
$234$	0	0
$235$	−361485.	−0.426993
$236$	0	0
$237$	−1.70625e6	−1.97320
$238$	0	0
$239$	904367.	1.02412	0.512059	−	0.858950i	$-0.328883\pi$
$239$	904367.	1.02412	0.512059	+	0.858950i	$0.328883\pi$
$240$	0	0
$241$	−859934.	−0.953724	−0.476862	−	0.878978i	$-0.658226\pi$
$241$	−859934.	−0.953724	−0.476862	+	0.878978i	$0.658226\pi$
$242$	0	0
$243$	−1.29088e6	−1.40240
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−220897.	−0.230382
$248$	0	0
$249$	−1.66302e6	−1.69981
$250$	0	0
$251$	−867501.	−0.869132	−0.434566	−	0.900640i	$-0.643098\pi$
$251$	−867501.	−0.869132	−0.434566	+	0.900640i	$0.643098\pi$
$252$	0	0
$253$	2.01149e6	1.97568
$254$	0	0
$255$	−819723.	−0.789436
$256$	0	0
$257$	−1.16027e6	−1.09579	−0.547894	−	0.836548i	$-0.684570\pi$
$257$	−1.16027e6	−1.09579	−0.547894	+	0.836548i	$0.684570\pi$
$258$	0	0
$259$	−442152.	−0.409565
$260$	0	0
$261$	−1.01425e6	−0.921602
$262$	0	0
$263$	477983.	0.426111	0.213056	−	0.977040i	$-0.431658\pi$
$263$	477983.	0.426111	0.213056	+	0.977040i	$0.431658\pi$
$264$	0	0
$265$	646066.	0.565148
$266$	0	0
$267$	−480439.	−0.412440
$268$	0	0
$269$	1.87356e6	1.57865	0.789327	−	0.613973i	$-0.210430\pi$
$269$	1.87356e6	1.57865	0.789327	+	0.613973i	$0.210430\pi$
$270$	0	0
$271$	2.34234e6	1.93743	0.968717	−	0.248168i	$-0.0798284\pi$
$271$	2.34234e6	1.93743	0.968717	+	0.248168i	$0.0798284\pi$
$272$	0	0
$273$	920897.	0.747833
$274$	0	0
$275$	−314282.	−0.250604
$276$	0	0
$277$	−562829.	−0.440735	−0.220367	−	0.975417i	$-0.570726\pi$
$277$	−562829.	−0.440735	−0.220367	+	0.975417i	$0.570726\pi$
$278$	0	0
$279$	1.77648e6	1.36631
$280$	0	0
$281$	−1.63404e6	−1.23452	−0.617260	−	0.786759i	$-0.711757\pi$
$281$	−1.63404e6	−1.23452	−0.617260	+	0.786759i	$0.711757\pi$
$282$	0	0
$283$	−1.36758e6	−1.01505	−0.507525	−	0.861637i	$-0.669439\pi$
$283$	−1.36758e6	−1.01505	−0.507525	+	0.861637i	$0.669439\pi$
$284$	0	0
$285$	140681.	0.102594
$286$	0	0
$287$	−32338.0	−0.0231744
$288$	0	0
$289$	825758.	0.581579
$290$	0	0
$291$	−3.35583e6	−2.32310
$292$	0	0
$293$	−746877.	−0.508253	−0.254126	−	0.967171i	$-0.581788\pi$
$293$	−746877.	−0.508253	−0.254126	+	0.967171i	$0.581788\pi$
$294$	0	0
$295$	1.24097e6	0.830247
$296$	0	0
$297$	79646.0	0.0523929
$298$	0	0
$299$	3.43585e6	2.22257
$300$	0	0
$301$	555590.	0.353459
$302$	0	0
$303$	−4.32313e6	−2.70515
$304$	0	0
$305$	627457.	0.386220
$306$	0	0
$307$	−928192.	−0.562072	−0.281036	−	0.959697i	$-0.590678\pi$
$307$	−928192.	−0.562072	−0.281036	+	0.959697i	$0.590678\pi$
$308$	0	0
$309$	1.11954e6	0.667027
$310$	0	0
$311$	−2.08041e6	−1.21969	−0.609843	−	0.792522i	$-0.708768\pi$
$311$	−2.08041e6	−1.21969	−0.609843	+	0.792522i	$0.708768\pi$
$312$	0	0
$313$	−1.62655e6	−0.938442	−0.469221	−	0.883081i	$-0.655465\pi$
$313$	−1.62655e6	−0.938442	−0.469221	+	0.883081i	$0.655465\pi$
$314$	0	0
$315$	−288808.	−0.163996
$316$	0	0
$317$	−1.06388e6	−0.594626	−0.297313	−	0.954780i	$-0.596090\pi$
$317$	−1.06388e6	−0.594626	−0.297313	+	0.954780i	$0.596090\pi$
$318$	0	0
$319$	2.16327e6	1.19024
$320$	0	0
$321$	1.19585e6	0.647759
$322$	0	0
$323$	−385392.	−0.205540
$324$	0	0
$325$	−536828.	−0.281921
$326$	0	0
$327$	−602396.	−0.311539
$328$	0	0
$329$	708511.	0.360875
$330$	0	0
$331$	2.81224e6	1.41085	0.705427	−	0.708782i	$-0.250755\pi$
$331$	2.81224e6	1.41085	0.705427	+	0.708782i	$0.250755\pi$
$332$	0	0
$333$	2.12739e6	1.05133
$334$	0	0
$335$	132642.	0.0645758
$336$	0	0
$337$	−3.11032e6	−1.49187	−0.745934	−	0.666020i	$-0.767997\pi$
$337$	−3.11032e6	−1.49187	−0.745934	+	0.666020i	$0.767997\pi$
$338$	0	0
$339$	1.20228e6	0.568207
$340$	0	0
$341$	−3.78901e6	−1.76458
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−2.18815e6	−0.989760
$346$	0	0
$347$	−2.84154e6	−1.26686	−0.633432	−	0.773798i	$-0.718354\pi$
$347$	−2.84154e6	−1.26686	−0.633432	+	0.773798i	$0.718354\pi$
$348$	0	0
$349$	865731.	0.380469	0.190235	−	0.981739i	$-0.439075\pi$
$349$	865731.	0.380469	0.190235	+	0.981739i	$0.439075\pi$
$350$	0	0
$351$	136044.	0.0589403
$352$	0	0
$353$	2.48942e6	1.06332	0.531658	−	0.846959i	$-0.321569\pi$
$353$	2.48942e6	1.06332	0.531658	+	0.846959i	$0.321569\pi$
$354$	0	0
$355$	1.82253e6	0.767543
$356$	0	0
$357$	1.60666e6	0.667195
$358$	0	0
$359$	−2.16715e6	−0.887468	−0.443734	−	0.896159i	$-0.646346\pi$
$359$	−2.16715e6	−0.887468	−0.443734	+	0.896159i	$0.646346\pi$
$360$	0	0
$361$	−2.40996e6	−0.973288
$362$	0	0
$363$	2.00882e6	0.800155
$364$	0	0
$365$	716720.	0.281590
$366$	0	0
$367$	2.43674e6	0.944373	0.472187	−	0.881499i	$-0.343465\pi$
$367$	2.43674e6	0.944373	0.472187	+	0.881499i	$0.343465\pi$
$368$	0	0
$369$	155593.	0.0594871
$370$	0	0
$371$	−1.26629e6	−0.477637
$372$	0	0
$373$	4.62452e6	1.72106	0.860528	−	0.509404i	$-0.170134\pi$
$373$	4.62452e6	1.72106	0.860528	+	0.509404i	$0.170134\pi$
$374$	0	0
$375$	341885.	0.125546
$376$	0	0
$377$	3.69511e6	1.33898
$378$	0	0
$379$	1.16989e6	0.418357	0.209178	−	0.977877i	$-0.432921\pi$
$379$	1.16989e6	0.418357	0.209178	+	0.977877i	$0.432921\pi$
$380$	0	0
$381$	1.90333e6	0.671740
$382$	0	0
$383$	−1.03639e6	−0.361016	−0.180508	−	0.983574i	$-0.557774\pi$
$383$	−1.03639e6	−0.361016	−0.180508	+	0.983574i	$0.557774\pi$
$384$	0	0
$385$	615993.	0.211799
$386$	0	0
$387$	−2.67320e6	−0.907306
$388$	0	0
$389$	−5.04842e6	−1.69154	−0.845768	−	0.533551i	$-0.820857\pi$
$389$	−5.04842e6	−1.69154	−0.845768	+	0.533551i	$0.820857\pi$
$390$	0	0
$391$	5.99440e6	1.98291
$392$	0	0
$393$	−4.65354e6	−1.51986
$394$	0	0
$395$	−1.94949e6	−0.628679
$396$	0	0
$397$	−4.45390e6	−1.41829	−0.709144	−	0.705064i	$-0.750918\pi$
$397$	−4.45390e6	−1.41829	−0.709144	+	0.705064i	$0.750918\pi$
$398$	0	0
$399$	−275734.	−0.0867078
$400$	0	0
$401$	−2.80873e6	−0.872266	−0.436133	−	0.899882i	$-0.643652\pi$
$401$	−2.80873e6	−0.872266	−0.436133	+	0.899882i	$0.643652\pi$
$402$	0	0
$403$	−6.47206e6	−1.98509
$404$	0	0
$405$	−1.51889e6	−0.460139
$406$	0	0
$407$	−4.53748e6	−1.35778
$408$	0	0
$409$	840711.	0.248507	0.124254	−	0.992251i	$-0.460346\pi$
$409$	840711.	0.248507	0.124254	+	0.992251i	$0.460346\pi$
$410$	0	0
$411$	−5.30053e6	−1.54780
$412$	0	0
$413$	−2.43231e6	−0.701687
$414$	0	0
$415$	−1.90011e6	−0.541575
$416$	0	0
$417$	−5.81916e6	−1.63878
$418$	0	0
$419$	5.23517e6	1.45678	0.728392	−	0.685160i	$-0.240268\pi$
$419$	5.23517e6	1.45678	0.728392	+	0.685160i	$0.240268\pi$
$420$	0	0
$421$	4.75280e6	1.30691	0.653453	−	0.756967i	$-0.273320\pi$
$421$	4.75280e6	1.30691	0.653453	+	0.756967i	$0.273320\pi$
$422$	0	0
$423$	−3.40897e6	−0.926344
$424$	0	0
$425$	−936586.	−0.251522
$426$	0	0
$427$	−1.22982e6	−0.326415
$428$	0	0
$429$	9.45049e6	2.47920
$430$	0	0
$431$	−662362.	−0.171752	−0.0858761	−	0.996306i	$-0.527369\pi$
$431$	−662362.	−0.171752	−0.0858761	+	0.996306i	$0.527369\pi$
$432$	0	0
$433$	−1.51928e6	−0.389420	−0.194710	−	0.980861i	$-0.562377\pi$
$433$	−1.51928e6	−0.389420	−0.194710	+	0.980861i	$0.562377\pi$
$434$	0	0
$435$	−2.35327e6	−0.596277
$436$	0	0
$437$	−1.02876e6	−0.257697
$438$	0	0
$439$	1.15686e6	0.286496	0.143248	−	0.989687i	$-0.454245\pi$
$439$	1.15686e6	0.286496	0.143248	+	0.989687i	$0.454245\pi$
$440$	0	0
$441$	566063.	0.138602
$442$	0	0
$443$	−2.30740e6	−0.558617	−0.279308	−	0.960201i	$-0.590105\pi$
$443$	−2.30740e6	−0.558617	−0.279308	+	0.960201i	$0.590105\pi$
$444$	0	0
$445$	−548933.	−0.131407
$446$	0	0
$447$	−7.26135e6	−1.71889
$448$	0	0
$449$	3.53127e6	0.826637	0.413319	−	0.910587i	$-0.364370\pi$
$449$	3.53127e6	0.826637	0.413319	+	0.910587i	$0.364370\pi$
$450$	0	0
$451$	−331861.	−0.0768271
$452$	0	0
$453$	2.23769e6	0.512335
$454$	0	0
$455$	1.05218e6	0.238267
$456$	0	0
$457$	−192471.	−0.0431096	−0.0215548	−	0.999768i	$-0.506862\pi$
$457$	−192471.	−0.0431096	−0.0215548	+	0.999768i	$0.506862\pi$
$458$	0	0
$459$	237352.	0.0525848
$460$	0	0
$461$	1.99454e6	0.437109	0.218555	−	0.975825i	$-0.429866\pi$
$461$	1.99454e6	0.437109	0.218555	+	0.975825i	$0.429866\pi$
$462$	0	0
$463$	517231.	0.112133	0.0560664	−	0.998427i	$-0.482144\pi$
$463$	517231.	0.112133	0.0560664	+	0.998427i	$0.482144\pi$
$464$	0	0
$465$	4.12179e6	0.884003
$466$	0	0
$467$	−612862.	−0.130038	−0.0650190	−	0.997884i	$-0.520711\pi$
$467$	−612862.	−0.130038	−0.0650190	+	0.997884i	$0.520711\pi$
$468$	0	0
$469$	−259979.	−0.0545765
$470$	0	0
$471$	−3.07379e6	−0.638442
$472$	0	0
$473$	5.70162e6	1.17178
$474$	0	0
$475$	160737.	0.0326874
$476$	0	0
$477$	6.09269e6	1.22606
$478$	0	0
$479$	5.97619e6	1.19011	0.595053	−	0.803686i	$-0.297131\pi$
$479$	5.97619e6	1.19011	0.595053	+	0.803686i	$0.297131\pi$
$480$	0	0
$481$	−7.75053e6	−1.52746
$482$	0	0
$483$	4.28878e6	0.836500
$484$	0	0
$485$	−3.83425e6	−0.740160
$486$	0	0
$487$	−8.39273e6	−1.60354	−0.801772	−	0.597630i	$-0.796109\pi$
$487$	−8.39273e6	−1.60354	−0.801772	+	0.597630i	$0.796109\pi$
$488$	0	0
$489$	5.96363e6	1.12782
$490$	0	0
$491$	−3.51380e6	−0.657770	−0.328885	−	0.944370i	$-0.606673\pi$
$491$	−3.51380e6	−0.657770	−0.328885	+	0.944370i	$0.606673\pi$
$492$	0	0
$493$	6.44673e6	1.19460
$494$	0	0
$495$	−2.96382e6	−0.543674
$496$	0	0
$497$	−3.57215e6	−0.648693
$498$	0	0
$499$	−2.27527e6	−0.409054	−0.204527	−	0.978861i	$-0.565566\pi$
$499$	−2.27527e6	−0.409054	−0.204527	+	0.978861i	$0.565566\pi$
$500$	0	0
$501$	1.17776e7	2.09634
$502$	0	0
$503$	7.45010e6	1.31293	0.656466	−	0.754355i	$-0.272050\pi$
$503$	7.45010e6	1.31293	0.656466	+	0.754355i	$0.272050\pi$
$504$	0	0
$505$	−4.93946e6	−0.861888
$506$	0	0
$507$	8.01837e6	1.38537
$508$	0	0
$509$	−1.03374e7	−1.76855	−0.884277	−	0.466963i	$-0.845348\pi$
$509$	−1.03374e7	−1.76855	−0.884277	+	0.466963i	$0.845348\pi$
$510$	0	0
$511$	−1.40477e6	−0.237987
$512$	0	0
$513$	−40734.2	−0.00683386
$514$	0	0
$515$	1.27915e6	0.212521
$516$	0	0
$517$	7.27093e6	1.19636
$518$	0	0
$519$	802200.	0.130727
$520$	0	0
$521$	709124.	0.114453	0.0572266	−	0.998361i	$-0.481774\pi$
$521$	709124.	0.114453	0.0572266	+	0.998361i	$0.481774\pi$
$522$	0	0
$523$	2.08425e6	0.333193	0.166596	−	0.986025i	$-0.446722\pi$
$523$	2.08425e6	0.333193	0.166596	+	0.986025i	$0.446722\pi$
$524$	0	0
$525$	−670094.	−0.106105
$526$	0	0
$527$	−1.12916e7	−1.77104
$528$	0	0
$529$	9.56499e6	1.48609
$530$	0	0
$531$	1.17029e7	1.80119
$532$	0	0
$533$	−566855.	−0.0864279
$534$	0	0
$535$	1.36633e6	0.206382
$536$	0	0
$537$	−7.97109e6	−1.19284
$538$	0	0
$539$	−1.20735e6	−0.179003
$540$	0	0
$541$	1.83262e6	0.269203	0.134602	−	0.990900i	$-0.457025\pi$
$541$	1.83262e6	0.269203	0.134602	+	0.990900i	$0.457025\pi$
$542$	0	0
$543$	−2.52322e6	−0.367244
$544$	0	0
$545$	−688276.	−0.0992593
$546$	0	0
$547$	4.58926e6	0.655804	0.327902	−	0.944712i	$-0.393658\pi$
$547$	4.58926e6	0.655804	0.327902	+	0.944712i	$0.393658\pi$
$548$	0	0
$549$	5.91720e6	0.837887
$550$	0	0
$551$	−1.10639e6	−0.155249
$552$	0	0
$553$	3.82101e6	0.531331
$554$	0	0
$555$	4.93600e6	0.680210
$556$	0	0
$557$	9.46004e6	1.29198	0.645989	−	0.763347i	$-0.276445\pi$
$557$	9.46004e6	1.29198	0.645989	+	0.763347i	$0.276445\pi$
$558$	0	0
$559$	9.73900e6	1.31821
$560$	0	0
$561$	1.64879e7	2.21187
$562$	0	0
$563$	1.47230e7	1.95761	0.978806	−	0.204788i	$-0.0656505\pi$
$563$	1.47230e7	1.95761	0.978806	+	0.204788i	$0.0656505\pi$
$564$	0	0
$565$	1.37368e6	0.181036
$566$	0	0
$567$	2.97703e6	0.388888
$568$	0	0
$569$	−1.25917e7	−1.63044	−0.815221	−	0.579151i	$-0.803384\pi$
$569$	−1.25917e7	−1.63044	−0.815221	+	0.579151i	$0.803384\pi$
$570$	0	0
$571$	−5.69802e6	−0.731365	−0.365682	−	0.930740i	$-0.619164\pi$
$571$	−5.69802e6	−0.731365	−0.365682	+	0.930740i	$0.619164\pi$
$572$	0	0
$573$	1.88390e7	2.39702
$574$	0	0
$575$	−2.50010e6	−0.315347
$576$	0	0
$577$	−7.28723e6	−0.911220	−0.455610	−	0.890180i	$-0.650579\pi$
$577$	−7.28723e6	−0.911220	−0.455610	+	0.890180i	$0.650579\pi$
$578$	0	0
$579$	735855.	0.0912213
$580$	0	0
$581$	3.72422e6	0.457714
$582$	0	0
$583$	−1.29950e7	−1.58345
$584$	0	0
$585$	−5.06253e6	−0.611615
$586$	0	0
$587$	5.13918e6	0.615600	0.307800	−	0.951451i	$-0.400407\pi$
$587$	5.13918e6	0.615600	0.307800	+	0.951451i	$0.400407\pi$
$588$	0	0
$589$	1.93786e6	0.230162
$590$	0	0
$591$	−6.46522e6	−0.761403
$592$	0	0
$593$	7.27034e6	0.849021	0.424510	−	0.905423i	$-0.360446\pi$
$593$	7.27034e6	0.849021	0.424510	+	0.905423i	$0.360446\pi$
$594$	0	0
$595$	1.83571e6	0.212575
$596$	0	0
$597$	−3.56711e6	−0.409620
$598$	0	0
$599$	−1.32652e7	−1.51059	−0.755295	−	0.655385i	$-0.772506\pi$
$599$	−1.32652e7	−1.51059	−0.755295	+	0.655385i	$0.772506\pi$
$600$	0	0
$601$	6.95581e6	0.785527	0.392764	−	0.919639i	$-0.371519\pi$
$601$	6.95581e6	0.785527	0.392764	+	0.919639i	$0.371519\pi$
$602$	0	0
$603$	1.25088e6	0.140094
$604$	0	0
$605$	2.29520e6	0.254937
$606$	0	0
$607$	5.15509e6	0.567890	0.283945	−	0.958840i	$-0.408357\pi$
$607$	5.15509e6	0.567890	0.283945	+	0.958840i	$0.408357\pi$
$608$	0	0
$609$	4.61241e6	0.503946
$610$	0	0
$611$	1.24196e7	1.34587
$612$	0	0
$613$	−1.09808e7	−1.18028	−0.590139	−	0.807302i	$-0.700927\pi$
$613$	−1.09808e7	−1.18028	−0.590139	+	0.807302i	$0.700927\pi$
$614$	0	0
$615$	361007.	0.0384883
$616$	0	0
$617$	−1.15104e7	−1.21724	−0.608621	−	0.793461i	$-0.708277\pi$
$617$	−1.15104e7	−1.21724	−0.608621	+	0.793461i	$0.708277\pi$
$618$	0	0
$619$	−3.68302e6	−0.386347	−0.193173	−	0.981165i	$-0.561878\pi$
$619$	−3.68302e6	−0.386347	−0.193173	+	0.981165i	$0.561878\pi$
$620$	0	0
$621$	633582.	0.0659286
$622$	0	0
$623$	1.07591e6	0.111059
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−2.82966e6	−0.287452
$628$	0	0
$629$	−1.35221e7	−1.36275
$630$	0	0
$631$	7.07723e6	0.707603	0.353801	−	0.935321i	$-0.384889\pi$
$631$	7.07723e6	0.707603	0.353801	+	0.935321i	$0.384889\pi$
$632$	0	0
$633$	2.53199e7	2.51161
$634$	0	0
$635$	2.17467e6	0.214023
$636$	0	0
$637$	−2.06228e6	−0.201372
$638$	0	0
$639$	1.71872e7	1.66515
$640$	0	0
$641$	−1.37562e7	−1.32237	−0.661185	−	0.750223i	$-0.729946\pi$
$641$	−1.37562e7	−1.32237	−0.661185	+	0.750223i	$0.729946\pi$
$642$	0	0
$643$	1.10330e7	1.05236	0.526180	−	0.850373i	$-0.323624\pi$
$643$	1.10330e7	1.05236	0.526180	+	0.850373i	$0.323624\pi$
$644$	0	0
$645$	−6.20238e6	−0.587028
$646$	0	0
$647$	−1.41398e7	−1.32795	−0.663977	−	0.747753i	$-0.731133\pi$
$647$	−1.41398e7	−1.32795	−0.663977	+	0.747753i	$0.731133\pi$
$648$	0	0
$649$	−2.49610e7	−2.32622
$650$	0	0
$651$	−8.07872e6	−0.747119
$652$	0	0
$653$	−1.79284e7	−1.64535	−0.822675	−	0.568512i	$-0.807519\pi$
$653$	−1.79284e7	−1.64535	−0.822675	+	0.568512i	$0.807519\pi$
$654$	0	0
$655$	−5.31697e6	−0.484240
$656$	0	0
$657$	6.75899e6	0.610898
$658$	0	0
$659$	−6.76035e6	−0.606395	−0.303197	−	0.952928i	$-0.598054\pi$
$659$	−6.76035e6	−0.606395	−0.303197	+	0.952928i	$0.598054\pi$
$660$	0	0
$661$	4.66518e6	0.415303	0.207651	−	0.978203i	$-0.433418\pi$
$661$	4.66518e6	0.415303	0.207651	+	0.978203i	$0.433418\pi$
$662$	0	0
$663$	2.81632e7	2.48828
$664$	0	0
$665$	−315044.	−0.0276259
$666$	0	0
$667$	1.72088e7	1.49774
$668$	0	0
$669$	−1.22853e7	−1.06126
$670$	0	0
$671$	−1.26207e7	−1.08212
$672$	0	0
$673$	−9.05510e6	−0.770647	−0.385324	−	0.922782i	$-0.625910\pi$
$673$	−9.05510e6	−0.770647	−0.385324	+	0.922782i	$0.625910\pi$
$674$	0	0
$675$	−98993.0	−0.00836267
$676$	0	0
$677$	−1.81513e7	−1.52208	−0.761038	−	0.648707i	$-0.775310\pi$
$677$	−1.81513e7	−1.52208	−0.761038	+	0.648707i	$0.775310\pi$
$678$	0	0
$679$	7.51513e6	0.625550
$680$	0	0
$681$	−1.07605e6	−0.0889125
$682$	0	0
$683$	−686598.	−0.0563184	−0.0281592	−	0.999603i	$-0.508965\pi$
$683$	−686598.	−0.0563184	−0.0281592	+	0.999603i	$0.508965\pi$
$684$	0	0
$685$	−6.05619e6	−0.493143
$686$	0	0
$687$	1.06804e7	0.863367
$688$	0	0
$689$	−2.21969e7	−1.78133
$690$	0	0
$691$	−7.36416e6	−0.586716	−0.293358	−	0.956003i	$-0.594773\pi$
$691$	−7.36416e6	−0.586716	−0.293358	+	0.956003i	$0.594773\pi$
$692$	0	0
$693$	5.80909e6	0.459489
$694$	0	0
$695$	−6.64876e6	−0.522130
$696$	0	0
$697$	−988972.	−0.0771085
$698$	0	0
$699$	693043.	0.0536497
$700$	0	0
$701$	1.86930e7	1.43676	0.718381	−	0.695650i	$-0.244883\pi$
$701$	1.86930e7	1.43676	0.718381	+	0.695650i	$0.244883\pi$
$702$	0	0
$703$	2.32065e6	0.177101
$704$	0	0
$705$	−7.90952e6	−0.599346
$706$	0	0
$707$	9.68134e6	0.728428
$708$	0	0
$709$	5.27060e6	0.393771	0.196886	−	0.980426i	$-0.436917\pi$
$709$	5.27060e6	0.393771	0.196886	+	0.980426i	$0.436917\pi$
$710$	0	0
$711$	−1.83846e7	−1.36389
$712$	0	0
$713$	−3.01415e7	−2.22045
$714$	0	0
$715$	1.07978e7	0.789896
$716$	0	0
$717$	1.97881e7	1.43749
$718$	0	0
$719$	−3.18584e6	−0.229827	−0.114914	−	0.993375i	$-0.536659\pi$
$719$	−3.18584e6	−0.229827	−0.114914	+	0.993375i	$0.536659\pi$
$720$	0	0
$721$	−2.50713e6	−0.179613
$722$	0	0
$723$	−1.88159e7	−1.33869
$724$	0	0
$725$	−2.68876e6	−0.189980
$726$	0	0
$727$	1.25482e6	0.0880535	0.0440268	−	0.999030i	$-0.485981\pi$
$727$	1.25482e6	0.0880535	0.0440268	+	0.999030i	$0.485981\pi$
$728$	0	0
$729$	−1.34817e7	−0.939561
$730$	0	0
$731$	1.69913e7	1.17607
$732$	0	0
$733$	2.30412e7	1.58396	0.791982	−	0.610544i	$-0.209049\pi$
$733$	2.30412e7	1.58396	0.791982	+	0.610544i	$0.209049\pi$
$734$	0	0
$735$	1.31338e6	0.0896754
$736$	0	0
$737$	−2.66797e6	−0.180931
$738$	0	0
$739$	1.66804e7	1.12356	0.561780	−	0.827286i	$-0.310117\pi$
$739$	1.66804e7	1.12356	0.561780	+	0.827286i	$0.310117\pi$
$740$	0	0
$741$	−4.83337e6	−0.323373
$742$	0	0
$743$	−2.53854e7	−1.68699	−0.843495	−	0.537137i	$-0.819506\pi$
$743$	−2.53854e7	−1.68699	−0.843495	+	0.537137i	$0.819506\pi$
$744$	0	0
$745$	−8.29656e6	−0.547655
$746$	0	0
$747$	−1.79189e7	−1.17492
$748$	0	0
$749$	−2.67801e6	−0.174425
$750$	0	0
$751$	5.35297e6	0.346334	0.173167	−	0.984892i	$-0.444600\pi$
$751$	5.35297e6	0.346334	0.173167	+	0.984892i	$0.444600\pi$
$752$	0	0
$753$	−1.89815e7	−1.21995
$754$	0	0
$755$	2.55670e6	0.163235
$756$	0	0
$757$	1.83168e7	1.16174	0.580870	−	0.813996i	$-0.302712\pi$
$757$	1.83168e7	1.16174	0.580870	+	0.813996i	$0.302712\pi$
$758$	0	0
$759$	4.40126e7	2.77314
$760$	0	0
$761$	2.07348e7	1.29789	0.648946	−	0.760835i	$-0.275210\pi$
$761$	2.07348e7	1.29789	0.648946	+	0.760835i	$0.275210\pi$
$762$	0	0
$763$	1.34902e6	0.0838894
$764$	0	0
$765$	−8.83243e6	−0.545666
$766$	0	0
$767$	−4.26362e7	−2.61692
$768$	0	0
$769$	−1.21310e7	−0.739744	−0.369872	−	0.929083i	$-0.620598\pi$
$769$	−1.21310e7	−0.739744	−0.369872	+	0.929083i	$0.620598\pi$
$770$	0	0
$771$	−2.53875e7	−1.53810
$772$	0	0
$773$	−1.71629e7	−1.03310	−0.516550	−	0.856257i	$-0.672784\pi$
$773$	−1.71629e7	−1.03310	−0.516550	+	0.856257i	$0.672784\pi$
$774$	0	0
$775$	4.70941e6	0.281652
$776$	0	0
$777$	−9.67456e6	−0.574882
$778$	0	0
$779$	169727.	0.0100209
$780$	0	0
$781$	−3.66584e7	−2.15053
$782$	0	0
$783$	681391.	0.0397184
$784$	0	0
$785$	−3.51200e6	−0.203414
$786$	0	0
$787$	−3.30464e7	−1.90190	−0.950950	−	0.309344i	$-0.899890\pi$
$787$	−3.30464e7	−1.90190	−0.950950	+	0.309344i	$0.899890\pi$
$788$	0	0
$789$	1.04586e7	0.598108
$790$	0	0
$791$	−2.69242e6	−0.153003
$792$	0	0
$793$	−2.15575e7	−1.21735
$794$	0	0
$795$	1.41363e7	0.793265
$796$	0	0
$797$	1.68991e7	0.942361	0.471181	−	0.882037i	$-0.343828\pi$
$797$	1.68991e7	0.942361	0.471181	+	0.882037i	$0.343828\pi$
$798$	0	0
$799$	2.16680e7	1.20075
$800$	0	0
$801$	−5.17668e6	−0.285082
$802$	0	0
$803$	−1.44161e7	−0.788969
$804$	0	0
$805$	4.90020e6	0.266517
$806$	0	0
$807$	4.09946e7	2.21586
$808$	0	0
$809$	−9.20879e6	−0.494688	−0.247344	−	0.968928i	$-0.579558\pi$
$809$	−9.20879e6	−0.494688	−0.247344	+	0.968928i	$0.579558\pi$
$810$	0	0
$811$	1.67625e7	0.894926	0.447463	−	0.894302i	$-0.352328\pi$
$811$	1.67625e7	0.894926	0.447463	+	0.894302i	$0.352328\pi$
$812$	0	0
$813$	5.12519e7	2.71946
$814$	0	0
$815$	6.81383e6	0.359333
$816$	0	0
$817$	−2.91604e6	−0.152840
$818$	0	0
$819$	9.92257e6	0.516909
$820$	0	0
$821$	3.29155e7	1.70429	0.852144	−	0.523308i	$-0.175302\pi$
$821$	3.29155e7	1.70429	0.852144	+	0.523308i	$0.175302\pi$
$822$	0	0
$823$	3.74797e7	1.92884	0.964421	−	0.264370i	$-0.0851642\pi$
$823$	3.74797e7	1.92884	0.964421	+	0.264370i	$0.0851642\pi$
$824$	0	0
$825$	−6.87668e6	−0.351758
$826$	0	0
$827$	3.05169e6	0.155159	0.0775795	−	0.996986i	$-0.475281\pi$
$827$	3.05169e6	0.155159	0.0775795	+	0.996986i	$0.475281\pi$
$828$	0	0
$829$	−1.08357e6	−0.0547608	−0.0273804	−	0.999625i	$-0.508717\pi$
$829$	−1.08357e6	−0.0547608	−0.0273804	+	0.999625i	$0.508717\pi$
$830$	0	0
$831$	−1.23150e7	−0.618633
$832$	0	0
$833$	−3.59799e6	−0.179658
$834$	0	0
$835$	1.34567e7	0.667915
$836$	0	0
$837$	−1.19347e6	−0.0588840
$838$	0	0
$839$	1.01861e7	0.499578	0.249789	−	0.968300i	$-0.419639\pi$
$839$	1.01861e7	0.499578	0.249789	+	0.968300i	$0.419639\pi$
$840$	0	0
$841$	−2.00380e6	−0.0976934
$842$	0	0
$843$	−3.57539e7	−1.73282
$844$	0	0
$845$	9.16150e6	0.441392
$846$	0	0
$847$	−4.49860e6	−0.215461
$848$	0	0
$849$	−2.99235e7	−1.42477
$850$	0	0
$851$	−3.60956e7	−1.70856
$852$	0	0
$853$	−3.70313e7	−1.74259	−0.871296	−	0.490757i	$-0.836720\pi$
$853$	−3.70313e7	−1.74259	−0.871296	+	0.490757i	$0.836720\pi$
$854$	0	0
$855$	1.51582e6	0.0709140
$856$	0	0
$857$	−2.43472e7	−1.13239	−0.566196	−	0.824271i	$-0.691585\pi$
$857$	−2.43472e7	−1.13239	−0.566196	+	0.824271i	$0.691585\pi$
$858$	0	0
$859$	−1.05001e7	−0.485524	−0.242762	−	0.970086i	$-0.578053\pi$
$859$	−1.05001e7	−0.485524	−0.242762	+	0.970086i	$0.578053\pi$
$860$	0	0
$861$	−707574.	−0.0325285
$862$	0	0
$863$	−4.96667e6	−0.227006	−0.113503	−	0.993538i	$-0.536207\pi$
$863$	−4.96667e6	−0.227006	−0.113503	+	0.993538i	$0.536207\pi$
$864$	0	0
$865$	916565.	0.0416508
$866$	0	0
$867$	1.80681e7	0.816328
$868$	0	0
$869$	3.92122e7	1.76146
$870$	0	0
$871$	−4.55719e6	−0.203541
$872$	0	0
$873$	−3.61587e7	−1.60575
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	1.13373e7	0.497750	0.248875	−	0.968536i	$-0.419939\pi$
$877$	1.13373e7	0.497750	0.248875	+	0.968536i	$0.419939\pi$
$878$	0	0
$879$	−1.63421e7	−0.713405
$880$	0	0
$881$	2.73766e7	1.18834	0.594169	−	0.804340i	$-0.297481\pi$
$881$	2.73766e7	1.18834	0.594169	+	0.804340i	$0.297481\pi$
$882$	0	0
$883$	1.56223e7	0.674285	0.337142	−	0.941454i	$-0.390540\pi$
$883$	1.56223e7	0.674285	0.337142	+	0.941454i	$0.390540\pi$
$884$	0	0
$885$	2.71533e7	1.16537
$886$	0	0
$887$	−3.36613e6	−0.143655	−0.0718276	−	0.997417i	$-0.522883\pi$
$887$	−3.36613e6	−0.143655	−0.0718276	+	0.997417i	$0.522883\pi$
$888$	0	0
$889$	−4.26236e6	−0.180882
$890$	0	0
$891$	3.05510e7	1.28923
$892$	0	0
$893$	−3.71865e6	−0.156047
$894$	0	0
$895$	−9.10748e6	−0.380050
$896$	0	0
$897$	7.51784e7	3.11969
$898$	0	0
$899$	−3.24160e7	−1.33770
$900$	0	0
$901$	−3.87262e7	−1.58925
$902$	0	0
$903$	1.21567e7	0.496129
$904$	0	0
$905$	−2.88294e6	−0.117008
$906$	0	0
$907$	3.01411e6	0.121658	0.0608291	−	0.998148i	$-0.480626\pi$
$907$	3.01411e6	0.121658	0.0608291	+	0.998148i	$0.480626\pi$
$908$	0	0
$909$	−4.65813e7	−1.86983
$910$	0	0
$911$	−2.26103e7	−0.902630	−0.451315	−	0.892365i	$-0.649045\pi$
$911$	−2.26103e7	−0.902630	−0.451315	+	0.892365i	$0.649045\pi$
$912$	0	0
$913$	3.82189e7	1.51740
$914$	0	0
$915$	1.37291e7	0.542114
$916$	0	0
$917$	1.04213e7	0.409258
$918$	0	0
$919$	−5.70895e6	−0.222981	−0.111490	−	0.993766i	$-0.535562\pi$
$919$	−5.70895e6	−0.222981	−0.111490	+	0.993766i	$0.535562\pi$
$920$	0	0
$921$	−2.03094e7	−0.788948
$922$	0	0
$923$	−6.26166e7	−2.41927
$924$	0	0
$925$	5.63970e6	0.216721
$926$	0	0
$927$	1.20629e7	0.461056
$928$	0	0
$929$	3.20537e7	1.21854	0.609269	−	0.792964i	$-0.291463\pi$
$929$	3.20537e7	1.21854	0.609269	+	0.792964i	$0.291463\pi$
$930$	0	0
$931$	617486.	0.0233482
$932$	0	0
$933$	−4.55207e7	−1.71200
$934$	0	0
$935$	1.88385e7	0.704722
$936$	0	0
$937$	−5.69852e6	−0.212038	−0.106019	−	0.994364i	$-0.533810\pi$
$937$	−5.69852e6	−0.212038	−0.106019	+	0.994364i	$0.533810\pi$
$938$	0	0
$939$	−3.55900e7	−1.31724
$940$	0	0
$941$	−1.62837e6	−0.0599485	−0.0299742	−	0.999551i	$-0.509543\pi$
$941$	−1.62837e6	−0.0599485	−0.0299742	+	0.999551i	$0.509543\pi$
$942$	0	0
$943$	−2.63994e6	−0.0966753
$944$	0	0
$945$	194026.	0.00706775
$946$	0	0
$947$	−9.40275e6	−0.340706	−0.170353	−	0.985383i	$-0.554491\pi$
$947$	−9.40275e6	−0.340706	−0.170353	+	0.985383i	$0.554491\pi$
$948$	0	0
$949$	−2.46244e7	−0.887564
$950$	0	0
$951$	−2.32783e7	−0.834642
$952$	0	0
$953$	1.96054e7	0.699267	0.349633	−	0.936887i	$-0.386306\pi$
$953$	1.96054e7	0.699267	0.349633	+	0.936887i	$0.386306\pi$
$954$	0	0
$955$	2.15248e7	0.763714
$956$	0	0
$957$	4.73337e7	1.67067
$958$	0	0
$959$	1.18701e7	0.416782
$960$	0	0
$961$	2.81480e7	0.983193
$962$	0	0
$963$	1.28851e7	0.447737
$964$	0	0
$965$	840762.	0.0290640
$966$	0	0
$967$	−3.32223e7	−1.14252	−0.571259	−	0.820770i	$-0.693545\pi$
$967$	−3.32223e7	−1.14252	−0.571259	+	0.820770i	$0.693545\pi$
$968$	0	0
$969$	−8.43261e6	−0.288504
$970$	0	0
$971$	−1.13564e7	−0.386537	−0.193269	−	0.981146i	$-0.561909\pi$
$971$	−1.13564e7	−0.386537	−0.193269	+	0.981146i	$0.561909\pi$
$972$	0	0
$973$	1.30316e7	0.441280
$974$	0	0
$975$	−1.17461e7	−0.395716
$976$	0	0
$977$	−2.17660e6	−0.0729527	−0.0364763	−	0.999335i	$-0.511613\pi$
$977$	−2.17660e6	−0.0729527	−0.0364763	+	0.999335i	$0.511613\pi$
$978$	0	0
$979$	1.10413e7	0.368181
$980$	0	0
$981$	−6.49075e6	−0.215339
$982$	0	0
$983$	7.70412e6	0.254296	0.127148	−	0.991884i	$-0.459418\pi$
$983$	7.70412e6	0.254296	0.127148	+	0.991884i	$0.459418\pi$
$984$	0	0
$985$	−7.38693e6	−0.242590
$986$	0	0
$987$	1.55027e7	0.506539
$988$	0	0
$989$	4.53562e7	1.47450
$990$	0	0
$991$	5.11224e7	1.65359	0.826794	−	0.562504i	$-0.190162\pi$
$991$	5.11224e7	1.65359	0.826794	+	0.562504i	$0.190162\pi$
$992$	0	0
$993$	6.15335e7	1.98033
$994$	0	0
$995$	−4.07566e6	−0.130509
$996$	0	0
$997$	−9.40075e6	−0.299519	−0.149760	−	0.988722i	$-0.547850\pi$
$997$	−9.40075e6	−0.299519	−0.149760	+	0.988722i	$0.547850\pi$
$998$	0	0
$999$	−1.42922e6	−0.0453092

Twists

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

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

\(f(q)\)	\(=\)	\(q+21.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +235.761 q^{9} +O(q^{10})\)	\(q+21.8806 q^{3} +25.0000 q^{5} -49.0000 q^{7} +235.761 q^{9} -502.851 q^{11} -858.926 q^{13} +547.015 q^{15} -1498.54 q^{17} +257.179 q^{19} -1072.15 q^{21} -4000.17 q^{23} +625.000 q^{25} -158.389 q^{27} -4302.02 q^{29} +7535.06 q^{31} -11002.7 q^{33} -1225.00 q^{35} +9023.51 q^{37} -18793.8 q^{39} +659.958 q^{41} -11338.6 q^{43} +5894.03 q^{45} -14459.4 q^{47} +2401.00 q^{49} -32788.9 q^{51} +25842.6 q^{53} -12571.3 q^{55} +5627.22 q^{57} +49638.9 q^{59} +25098.3 q^{61} -11552.3 q^{63} -21473.1 q^{65} +5305.69 q^{67} -87526.1 q^{69} +72901.1 q^{71} +28668.8 q^{73} +13675.4 q^{75} +24639.7 q^{77} -77979.8 q^{79} -60755.6 q^{81} -76004.4 q^{83} -37463.4 q^{85} -94130.8 q^{87} -21957.3 q^{89} +42087.4 q^{91} +164872. q^{93} +6429.46 q^{95} -153370. q^{97} -118553. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9	\( 2 q + 2 q^{3} + 50 q^{5} - 98 q^{7} + 388 q^{9} - 254 q^{11} - 1342 q^{13} + 50 q^{15} - 1786 q^{17} + 1976 q^{19} - 98 q^{21} - 2112 q^{23} + 1250 q^{25} + 1646 q^{27} - 4762 q^{29} + 13692 q^{31} - 15950 q^{33} - 2450 q^{35} + 2136 q^{37} - 9190 q^{39} - 6740 q^{41} - 26728 q^{43} + 9700 q^{45} - 6326 q^{47} + 4802 q^{49} - 27074 q^{51} + 24624 q^{53} - 6350 q^{55} - 28544 q^{57} + 51336 q^{59} + 4468 q^{61} - 19012 q^{63} - 33550 q^{65} - 39168 q^{67} - 125064 q^{69} + 41232 q^{71} - 36124 q^{73} + 1250 q^{75} + 12446 q^{77} - 140842 q^{79} - 133622 q^{81} - 57712 q^{83} - 44650 q^{85} - 84986 q^{87} - 20236 q^{89} + 65758 q^{91} + 42468 q^{93} + 49400 q^{95} - 183586 q^{97} - 80668 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 50 * q^5 - 98 * q^7 + 388 * q^9 - 254 * q^11 - 1342 * q^13 + 50 * q^15 - 1786 * q^17 + 1976 * q^19 - 98 * q^21 - 2112 * q^23 + 1250 * q^25 + 1646 * q^27 - 4762 * q^29 + 13692 * q^31 - 15950 * q^33 - 2450 * q^35 + 2136 * q^37 - 9190 * q^39 - 6740 * q^41 - 26728 * q^43 + 9700 * q^45 - 6326 * q^47 + 4802 * q^49 - 27074 * q^51 + 24624 * q^53 - 6350 * q^55 - 28544 * q^57 + 51336 * q^59 + 4468 * q^61 - 19012 * q^63 - 33550 * q^65 - 39168 * q^67 - 125064 * q^69 + 41232 * q^71 - 36124 * q^73 + 1250 * q^75 + 12446 * q^77 - 140842 * q^79 - 133622 * q^81 - 57712 * q^83 - 44650 * q^85 - 84986 * q^87 - 20236 * q^89 + 65758 * q^91 + 42468 * q^93 + 49400 * q^95 - 183586 * q^97 - 80668 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more