LMFDB - Embedded newform 1028.6.a.b.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1028,6,Mod(1,1028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1028.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1028 = 2^{2} \cdot 257 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1028.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:	$164.874566768$
Analytic rank:	$0$
Dimension:	$57$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	1028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-21.1022 q^{3} +70.8764 q^{5} +79.9641 q^{7} +202.303 q^{9} +O(q^{10})$	$q-21.1022 q^{3} +70.8764 q^{5} +79.9641 q^{7} +202.303 q^{9} -40.1922 q^{11} -971.397 q^{13} -1495.65 q^{15} -700.289 q^{17} +547.366 q^{19} -1687.42 q^{21} -2773.34 q^{23} +1898.47 q^{25} +858.794 q^{27} +4554.35 q^{29} -3070.66 q^{31} +848.144 q^{33} +5667.57 q^{35} -7034.52 q^{37} +20498.6 q^{39} +1070.78 q^{41} -3599.93 q^{43} +14338.5 q^{45} +16353.3 q^{47} -10412.7 q^{49} +14777.6 q^{51} +37265.5 q^{53} -2848.68 q^{55} -11550.6 q^{57} -5153.68 q^{59} +46425.7 q^{61} +16177.0 q^{63} -68849.2 q^{65} -68689.0 q^{67} +58523.6 q^{69} +58665.9 q^{71} +58983.6 q^{73} -40061.9 q^{75} -3213.93 q^{77} -60712.5 q^{79} -67282.1 q^{81} -50411.1 q^{83} -49634.0 q^{85} -96106.9 q^{87} +22879.1 q^{89} -77676.9 q^{91} +64797.8 q^{93} +38795.4 q^{95} -57094.7 q^{97} -8131.01 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	$ 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) $ 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * 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
$3$	−21.1022	−1.35371	−0.676854	−	0.736118i	$-0.736657\pi$
$3$	−21.1022	−1.35371	−0.676854	+	0.736118i	$0.736657\pi$
$4$	0	0
$5$	70.8764	1.26788	0.633938	−	0.773384i	$-0.281437\pi$
$5$	70.8764	1.26788	0.633938	+	0.773384i	$0.281437\pi$
$6$	0	0
$7$	79.9641	0.616808	0.308404	−	0.951256i	$-0.400205\pi$
$7$	79.9641	0.616808	0.308404	+	0.951256i	$0.400205\pi$
$8$	0	0
$9$	202.303	0.832523
$10$	0	0
$11$	−40.1922	−0.100152	−0.0500760	−	0.998745i	$-0.515946\pi$
$11$	−40.1922	−0.100152	−0.0500760	+	0.998745i	$0.515946\pi$
$12$	0	0
$13$	−971.397	−1.59418	−0.797092	−	0.603858i	$-0.793629\pi$
$13$	−971.397	−1.59418	−0.797092	+	0.603858i	$0.793629\pi$
$14$	0	0
$15$	−1495.65	−1.71633
$16$	0	0
$17$	−700.289	−0.587699	−0.293850	−	0.955852i	$-0.594936\pi$
$17$	−700.289	−0.587699	−0.293850	+	0.955852i	$0.594936\pi$
$18$	0	0
$19$	547.366	0.347851	0.173926	−	0.984759i	$-0.444355\pi$
$19$	547.366	0.347851	0.173926	+	0.984759i	$0.444355\pi$
$20$	0	0
$21$	−1687.42	−0.834977
$22$	0	0
$23$	−2773.34	−1.09316	−0.546580	−	0.837407i	$-0.684071\pi$
$23$	−2773.34	−1.09316	−0.546580	+	0.837407i	$0.684071\pi$
$24$	0	0
$25$	1898.47	0.607510
$26$	0	0
$27$	858.794	0.226715
$28$	0	0
$29$	4554.35	1.00561	0.502807	−	0.864399i	$-0.332301\pi$
$29$	4554.35	1.00561	0.502807	+	0.864399i	$0.332301\pi$
$30$	0	0
$31$	−3070.66	−0.573889	−0.286945	−	0.957947i	$-0.592640\pi$
$31$	−3070.66	−0.573889	−0.286945	+	0.957947i	$0.592640\pi$
$32$	0	0
$33$	848.144	0.135577
$34$	0	0
$35$	5667.57	0.782036
$36$	0	0
$37$	−7034.52	−0.844754	−0.422377	−	0.906420i	$-0.638804\pi$
$37$	−7034.52	−0.844754	−0.422377	+	0.906420i	$0.638804\pi$
$38$	0	0
$39$	20498.6	2.15806
$40$	0	0
$41$	1070.78	0.0994808	0.0497404	−	0.998762i	$-0.484161\pi$
$41$	1070.78	0.0994808	0.0497404	+	0.998762i	$0.484161\pi$
$42$	0	0
$43$	−3599.93	−0.296909	−0.148454	−	0.988919i	$-0.547430\pi$
$43$	−3599.93	−0.296909	−0.148454	+	0.988919i	$0.547430\pi$
$44$	0	0
$45$	14338.5	1.05554
$46$	0	0
$47$	16353.3	1.07985	0.539923	−	0.841714i	$-0.318453\pi$
$47$	16353.3	1.07985	0.539923	+	0.841714i	$0.318453\pi$
$48$	0	0
$49$	−10412.7	−0.619548
$50$	0	0
$51$	14777.6	0.795573
$52$	0	0
$53$	37265.5	1.82229	0.911144	−	0.412087i	$-0.135200\pi$
$53$	37265.5	1.82229	0.911144	+	0.412087i	$0.135200\pi$
$54$	0	0
$55$	−2848.68	−0.126980
$56$	0	0
$57$	−11550.6	−0.470889
$58$	0	0
$59$	−5153.68	−0.192747	−0.0963734	−	0.995345i	$-0.530724\pi$
$59$	−5153.68	−0.192747	−0.0963734	+	0.995345i	$0.530724\pi$
$60$	0	0
$61$	46425.7	1.59747	0.798737	−	0.601680i	$-0.205502\pi$
$61$	46425.7	1.59747	0.798737	+	0.601680i	$0.205502\pi$
$62$	0	0
$63$	16177.0	0.513507
$64$	0	0
$65$	−68849.2	−2.02123
$66$	0	0
$67$	−68689.0	−1.86939	−0.934695	−	0.355450i	$-0.884327\pi$
$67$	−68689.0	−1.86939	−0.934695	+	0.355450i	$0.884327\pi$
$68$	0	0
$69$	58523.6	1.47982
$70$	0	0
$71$	58665.9	1.38115	0.690574	−	0.723262i	$-0.257358\pi$
$71$	58665.9	1.38115	0.690574	+	0.723262i	$0.257358\pi$
$72$	0	0
$73$	58983.6	1.29546	0.647731	−	0.761870i	$-0.275718\pi$
$73$	58983.6	1.29546	0.647731	+	0.761870i	$0.275718\pi$
$74$	0	0
$75$	−40061.9	−0.822391
$76$	0	0
$77$	−3213.93	−0.0617746
$78$	0	0
$79$	−60712.5	−1.09449	−0.547243	−	0.836974i	$-0.684323\pi$
$79$	−60712.5	−1.09449	−0.547243	+	0.836974i	$0.684323\pi$
$80$	0	0
$81$	−67282.1	−1.13943
$82$	0	0
$83$	−50411.1	−0.803214	−0.401607	−	0.915812i	$-0.631548\pi$
$83$	−50411.1	−0.803214	−0.401607	+	0.915812i	$0.631548\pi$
$84$	0	0
$85$	−49634.0	−0.745130
$86$	0	0
$87$	−96106.9	−1.36131
$88$	0	0
$89$	22879.1	0.306171	0.153086	−	0.988213i	$-0.451079\pi$
$89$	22879.1	0.306171	0.153086	+	0.988213i	$0.451079\pi$
$90$	0	0
$91$	−77676.9	−0.983305
$92$	0	0
$93$	64797.8	0.776878
$94$	0	0
$95$	38795.4	0.441033
$96$	0	0
$97$	−57094.7	−0.616122	−0.308061	−	0.951367i	$-0.599680\pi$
$97$	−57094.7	−0.616122	−0.308061	+	0.951367i	$0.599680\pi$
$98$	0	0
$99$	−8131.01	−0.0833789
$100$	0	0
$101$	−171838.	−1.67616	−0.838078	−	0.545550i	$-0.816321\pi$
$101$	−171838.	−1.67616	−0.838078	+	0.545550i	$0.816321\pi$
$102$	0	0
$103$	197384.	1.83324	0.916618	−	0.399764i	$-0.130908\pi$
$103$	197384.	1.83324	0.916618	+	0.399764i	$0.130908\pi$
$104$	0	0
$105$	−119598.	−1.05865
$106$	0	0
$107$	−104637.	−0.883540	−0.441770	−	0.897128i	$-0.645649\pi$
$107$	−104637.	−0.883540	−0.441770	+	0.897128i	$0.645649\pi$
$108$	0	0
$109$	−88500.8	−0.713478	−0.356739	−	0.934204i	$-0.616112\pi$
$109$	−88500.8	−0.713478	−0.356739	+	0.934204i	$0.616112\pi$
$110$	0	0
$111$	148444.	1.14355
$112$	0	0
$113$	213741.	1.57467	0.787337	−	0.616522i	$-0.211459\pi$
$113$	213741.	1.57467	0.787337	+	0.616522i	$0.211459\pi$
$114$	0	0
$115$	−196565.	−1.38599
$116$	0	0
$117$	−196517.	−1.32720
$118$	0	0
$119$	−55998.0	−0.362497
$120$	0	0
$121$	−159436.	−0.989970
$122$	0	0
$123$	−22595.8	−0.134668
$124$	0	0
$125$	−86932.1	−0.497628
$126$	0	0
$127$	−98091.3	−0.539661	−0.269831	−	0.962908i	$-0.586968\pi$
$127$	−98091.3	−0.539661	−0.269831	+	0.962908i	$0.586968\pi$
$128$	0	0
$129$	75966.5	0.401928
$130$	0	0
$131$	−107399.	−0.546790	−0.273395	−	0.961902i	$-0.588147\pi$
$131$	−107399.	−0.546790	−0.273395	+	0.961902i	$0.588147\pi$
$132$	0	0
$133$	43769.6	0.214557
$134$	0	0
$135$	60868.3	0.287446
$136$	0	0
$137$	276643.	1.25927	0.629634	−	0.776892i	$-0.283205\pi$
$137$	276643.	1.25927	0.629634	+	0.776892i	$0.283205\pi$
$138$	0	0
$139$	275819.	1.21084	0.605420	−	0.795906i	$-0.293005\pi$
$139$	275819.	1.21084	0.605420	+	0.795906i	$0.293005\pi$
$140$	0	0
$141$	−345092.	−1.46180
$142$	0	0
$143$	39042.6	0.159661
$144$	0	0
$145$	322796.	1.27500
$146$	0	0
$147$	219732.	0.838687
$148$	0	0
$149$	50135.6	0.185004	0.0925018	−	0.995713i	$-0.470514\pi$
$149$	50135.6	0.185004	0.0925018	+	0.995713i	$0.470514\pi$
$150$	0	0
$151$	−240567.	−0.858605	−0.429302	−	0.903161i	$-0.641241\pi$
$151$	−240567.	−0.858605	−0.429302	+	0.903161i	$0.641241\pi$
$152$	0	0
$153$	−141671.	−0.489273
$154$	0	0
$155$	−217638.	−0.727620
$156$	0	0
$157$	539377.	1.74640	0.873199	−	0.487363i	$-0.162041\pi$
$157$	539377.	1.74640	0.873199	+	0.487363i	$0.162041\pi$
$158$	0	0
$159$	−786384.	−2.46685
$160$	0	0
$161$	−221768.	−0.674269
$162$	0	0
$163$	571786.	1.68564	0.842819	−	0.538198i	$-0.180895\pi$
$163$	571786.	1.68564	0.842819	+	0.538198i	$0.180895\pi$
$164$	0	0
$165$	60113.4	0.171894
$166$	0	0
$167$	345559.	0.958806	0.479403	−	0.877595i	$-0.340853\pi$
$167$	345559.	0.958806	0.479403	+	0.877595i	$0.340853\pi$
$168$	0	0
$169$	572320.	1.54142
$170$	0	0
$171$	110734.	0.289594
$172$	0	0
$173$	−84008.5	−0.213407	−0.106703	−	0.994291i	$-0.534030\pi$
$173$	−84008.5	−0.213407	−0.106703	+	0.994291i	$0.534030\pi$
$174$	0	0
$175$	151809.	0.374717
$176$	0	0
$177$	108754.	0.260923
$178$	0	0
$179$	−400378.	−0.933979	−0.466990	−	0.884263i	$-0.654661\pi$
$179$	−400378.	−0.933979	−0.466990	+	0.884263i	$0.654661\pi$
$180$	0	0
$181$	11487.5	0.0260634	0.0130317	−	0.999915i	$-0.495852\pi$
$181$	11487.5	0.0260634	0.0130317	+	0.999915i	$0.495852\pi$
$182$	0	0
$183$	−979685.	−2.16251
$184$	0	0
$185$	−498582.	−1.07104
$186$	0	0
$187$	28146.2	0.0588593
$188$	0	0
$189$	68672.7	0.139839
$190$	0	0
$191$	556265.	1.10331	0.551656	−	0.834072i	$-0.313996\pi$
$191$	556265.	1.10331	0.551656	+	0.834072i	$0.313996\pi$
$192$	0	0
$193$	587579.	1.13546	0.567731	−	0.823214i	$-0.307821\pi$
$193$	587579.	1.13546	0.567731	+	0.823214i	$0.307821\pi$
$194$	0	0
$195$	1.45287e6	2.73615
$196$	0	0
$197$	683756.	1.25527	0.627633	−	0.778510i	$-0.284024\pi$
$197$	683756.	1.25527	0.627633	+	0.778510i	$0.284024\pi$
$198$	0	0
$199$	−956480.	−1.71216	−0.856078	−	0.516847i	$-0.827105\pi$
$199$	−956480.	−1.71216	−0.856078	+	0.516847i	$0.827105\pi$
$200$	0	0
$201$	1.44949e6	2.53061
$202$	0	0
$203$	364184.	0.620271
$204$	0	0
$205$	75892.8	0.126129
$206$	0	0
$207$	−561055.	−0.910081
$208$	0	0
$209$	−21999.8	−0.0348381
$210$	0	0
$211$	−739899.	−1.14411	−0.572053	−	0.820217i	$-0.693853\pi$
$211$	−739899.	−1.14411	−0.572053	+	0.820217i	$0.693853\pi$
$212$	0	0
$213$	−1.23798e6	−1.86967
$214$	0	0
$215$	−255150.	−0.376444
$216$	0	0
$217$	−245543.	−0.353979
$218$	0	0
$219$	−1.24468e6	−1.75367
$220$	0	0
$221$	680259.	0.936901
$222$	0	0
$223$	−1.14762e6	−1.54538	−0.772689	−	0.634785i	$-0.781089\pi$
$223$	−1.14762e6	−1.54538	−0.772689	+	0.634785i	$0.781089\pi$
$224$	0	0
$225$	384066.	0.505766
$226$	0	0
$227$	899716.	1.15889	0.579443	−	0.815013i	$-0.303270\pi$
$227$	899716.	1.15889	0.579443	+	0.815013i	$0.303270\pi$
$228$	0	0
$229$	161144.	0.203060	0.101530	−	0.994832i	$-0.467626\pi$
$229$	161144.	0.203060	0.101530	+	0.994832i	$0.467626\pi$
$230$	0	0
$231$	67821.0	0.0836247
$232$	0	0
$233$	699900.	0.844590	0.422295	−	0.906458i	$-0.361225\pi$
$233$	699900.	0.844590	0.422295	+	0.906458i	$0.361225\pi$
$234$	0	0
$235$	1.15907e6	1.36911
$236$	0	0
$237$	1.28117e6	1.48161
$238$	0	0
$239$	−304783.	−0.345141	−0.172570	−	0.984997i	$-0.555207\pi$
$239$	−304783.	−0.345141	−0.172570	+	0.984997i	$0.555207\pi$
$240$	0	0
$241$	1.56079e6	1.73101	0.865507	−	0.500897i	$-0.166996\pi$
$241$	1.56079e6	1.73101	0.865507	+	0.500897i	$0.166996\pi$
$242$	0	0
$243$	1.21111e6	1.31574
$244$	0	0
$245$	−738019.	−0.785511
$246$	0	0
$247$	−531710.	−0.554539
$248$	0	0
$249$	1.06379e6	1.08732
$250$	0	0
$251$	1.01528e6	1.01719	0.508596	−	0.861005i	$-0.330165\pi$
$251$	1.01528e6	1.01719	0.508596	+	0.861005i	$0.330165\pi$
$252$	0	0
$253$	111467.	0.109482
$254$	0	0
$255$	1.04739e6	1.00869
$256$	0	0
$257$	−66049.0	−0.0623783
$257$	−66049.0	−0.0623783
$258$	0	0
$259$	−562509.	−0.521051
$260$	0	0
$261$	921360.	0.837197
$262$	0	0
$263$	1.09043e6	0.972096	0.486048	−	0.873932i	$-0.338438\pi$
$263$	1.09043e6	0.972096	0.486048	+	0.873932i	$0.338438\pi$
$264$	0	0
$265$	2.64125e6	2.31044
$266$	0	0
$267$	−482800.	−0.414466
$268$	0	0
$269$	897075.	0.755872	0.377936	−	0.925832i	$-0.376634\pi$
$269$	897075.	0.755872	0.377936	+	0.925832i	$0.376634\pi$
$270$	0	0
$271$	2.19185e6	1.81295	0.906477	−	0.422255i	$-0.138761\pi$
$271$	2.19185e6	1.81295	0.906477	+	0.422255i	$0.138761\pi$
$272$	0	0
$273$	1.63915e6	1.33111
$274$	0	0
$275$	−76303.7	−0.0608434
$276$	0	0
$277$	−111711.	−0.0874777	−0.0437389	−	0.999043i	$-0.513927\pi$
$277$	−111711.	−0.0874777	−0.0437389	+	0.999043i	$0.513927\pi$
$278$	0	0
$279$	−621205.	−0.477776
$280$	0	0
$281$	−511093.	−0.386131	−0.193065	−	0.981186i	$-0.561843\pi$
$281$	−511093.	−0.386131	−0.193065	+	0.981186i	$0.561843\pi$
$282$	0	0
$283$	726078.	0.538911	0.269456	−	0.963013i	$-0.413156\pi$
$283$	726078.	0.538911	0.269456	+	0.963013i	$0.413156\pi$
$284$	0	0
$285$	−818668.	−0.597029
$286$	0	0
$287$	85623.7	0.0613605
$288$	0	0
$289$	−929452.	−0.654610
$290$	0	0
$291$	1.20482e6	0.834048
$292$	0	0
$293$	2.69061e6	1.83097	0.915487	−	0.402348i	$-0.131806\pi$
$293$	2.69061e6	1.83097	0.915487	+	0.402348i	$0.131806\pi$
$294$	0	0
$295$	−365274.	−0.244379
$296$	0	0
$297$	−34516.8	−0.0227060
$298$	0	0
$299$	2.69402e6	1.74270
$300$	0	0
$301$	−287865.	−0.183136
$302$	0	0
$303$	3.62615e6	2.26902
$304$	0	0
$305$	3.29049e6	2.02540
$306$	0	0
$307$	−938923.	−0.568570	−0.284285	−	0.958740i	$-0.591756\pi$
$307$	−938923.	−0.568570	−0.284285	+	0.958740i	$0.591756\pi$
$308$	0	0
$309$	−4.16523e6	−2.48167
$310$	0	0
$311$	624950.	0.366391	0.183195	−	0.983077i	$-0.441356\pi$
$311$	624950.	0.366391	0.183195	+	0.983077i	$0.441356\pi$
$312$	0	0
$313$	−2.50361e6	−1.44446	−0.722230	−	0.691653i	$-0.756883\pi$
$313$	−2.50361e6	−1.44446	−0.722230	+	0.691653i	$0.756883\pi$
$314$	0	0
$315$	1.14657e6	0.651063
$316$	0	0
$317$	3.28805e6	1.83777	0.918884	−	0.394527i	$-0.129092\pi$
$317$	3.28805e6	1.83777	0.918884	+	0.394527i	$0.129092\pi$
$318$	0	0
$319$	−183049.	−0.100714
$320$	0	0
$321$	2.20807e6	1.19605
$322$	0	0
$323$	−383315.	−0.204432
$324$	0	0
$325$	−1.84417e6	−0.968483
$326$	0	0
$327$	1.86756e6	0.965841
$328$	0	0
$329$	1.30768e6	0.666057
$330$	0	0
$331$	−3.19310e6	−1.60193	−0.800964	−	0.598712i	$-0.795679\pi$
$331$	−3.19310e6	−1.60193	−0.800964	+	0.598712i	$0.795679\pi$
$332$	0	0
$333$	−1.42311e6	−0.703277
$334$	0	0
$335$	−4.86843e6	−2.37016
$336$	0	0
$337$	1.85201e6	0.888316	0.444158	−	0.895948i	$-0.353503\pi$
$337$	1.85201e6	0.888316	0.444158	+	0.895948i	$0.353503\pi$
$338$	0	0
$339$	−4.51040e6	−2.13165
$340$	0	0
$341$	123417.	0.0574762
$342$	0	0
$343$	−2.17660e6	−0.998950
$344$	0	0
$345$	4.14795e6	1.87623
$346$	0	0
$347$	−1.29606e6	−0.577834	−0.288917	−	0.957354i	$-0.593295\pi$
$347$	−1.29606e6	−0.577834	−0.288917	+	0.957354i	$0.593295\pi$
$348$	0	0
$349$	−47880.1	−0.0210422	−0.0105211	−	0.999945i	$-0.503349\pi$
$349$	−47880.1	−0.0210422	−0.0105211	+	0.999945i	$0.503349\pi$
$350$	0	0
$351$	−834230.	−0.361425
$352$	0	0
$353$	228203.	0.0974732	0.0487366	−	0.998812i	$-0.484481\pi$
$353$	228203.	0.0974732	0.0487366	+	0.998812i	$0.484481\pi$
$354$	0	0
$355$	4.15803e6	1.75112
$356$	0	0
$357$	1.18168e6	0.490715
$358$	0	0
$359$	−2.60031e6	−1.06485	−0.532425	−	0.846477i	$-0.678719\pi$
$359$	−2.60031e6	−1.06485	−0.532425	+	0.846477i	$0.678719\pi$
$360$	0	0
$361$	−2.17649e6	−0.878999
$362$	0	0
$363$	3.36444e6	1.34013
$364$	0	0
$365$	4.18055e6	1.64248
$366$	0	0
$367$	1.98613e6	0.769739	0.384869	−	0.922971i	$-0.374247\pi$
$367$	1.98613e6	0.769739	0.384869	+	0.922971i	$0.374247\pi$
$368$	0	0
$369$	216621.	0.0828201
$370$	0	0
$371$	2.97990e6	1.12400
$372$	0	0
$373$	−2.92503e6	−1.08858	−0.544288	−	0.838898i	$-0.683200\pi$
$373$	−2.92503e6	−1.08858	−0.544288	+	0.838898i	$0.683200\pi$
$374$	0	0
$375$	1.83446e6	0.673643
$376$	0	0
$377$	−4.42409e6	−1.60314
$378$	0	0
$379$	−4.20883e6	−1.50509	−0.752546	−	0.658540i	$-0.771175\pi$
$379$	−4.20883e6	−1.50509	−0.752546	+	0.658540i	$0.771175\pi$
$380$	0	0
$381$	2.06994e6	0.730543
$382$	0	0
$383$	−5.24283e6	−1.82629	−0.913143	−	0.407638i	$-0.866352\pi$
$383$	−5.24283e6	−1.82629	−0.913143	+	0.407638i	$0.866352\pi$
$384$	0	0
$385$	−227792.	−0.0783225
$386$	0	0
$387$	−728278.	−0.247184
$388$	0	0
$389$	5.78141e6	1.93714	0.968568	−	0.248751i	$-0.0800200\pi$
$389$	5.78141e6	1.93714	0.968568	+	0.248751i	$0.0800200\pi$
$390$	0	0
$391$	1.94214e6	0.642449
$392$	0	0
$393$	2.26635e6	0.740194
$394$	0	0
$395$	−4.30309e6	−1.38767
$396$	0	0
$397$	4.64801e6	1.48010	0.740049	−	0.672553i	$-0.234802\pi$
$397$	4.64801e6	1.48010	0.740049	+	0.672553i	$0.234802\pi$
$398$	0	0
$399$	−923635.	−0.290448
$400$	0	0
$401$	3.62963e6	1.12720	0.563600	−	0.826048i	$-0.309416\pi$
$401$	3.62963e6	1.12720	0.563600	+	0.826048i	$0.309416\pi$
$402$	0	0
$403$	2.98283e6	0.914885
$404$	0	0
$405$	−4.76872e6	−1.44465
$406$	0	0
$407$	282733.	0.0846039
$408$	0	0
$409$	3.12956e6	0.925070	0.462535	−	0.886601i	$-0.346940\pi$
$409$	3.12956e6	0.925070	0.462535	+	0.886601i	$0.346940\pi$
$410$	0	0
$411$	−5.83777e6	−1.70468
$412$	0	0
$413$	−412109.	−0.118888
$414$	0	0
$415$	−3.57296e6	−1.01838
$416$	0	0
$417$	−5.82038e6	−1.63912
$418$	0	0
$419$	−2.75696e6	−0.767176	−0.383588	−	0.923504i	$-0.625312\pi$
$419$	−2.75696e6	−0.767176	−0.383588	+	0.923504i	$0.625312\pi$
$420$	0	0
$421$	2.29643e6	0.631462	0.315731	−	0.948849i	$-0.397750\pi$
$421$	2.29643e6	0.631462	0.315731	+	0.948849i	$0.397750\pi$
$422$	0	0
$423$	3.30833e6	0.898997
$424$	0	0
$425$	−1.32948e6	−0.357033
$426$	0	0
$427$	3.71239e6	0.985335
$428$	0	0
$429$	−823885.	−0.216134
$430$	0	0
$431$	3.52285e6	0.913484	0.456742	−	0.889599i	$-0.349016\pi$
$431$	3.52285e6	0.913484	0.456742	+	0.889599i	$0.349016\pi$
$432$	0	0
$433$	−5.48947e6	−1.40705	−0.703527	−	0.710668i	$-0.748393\pi$
$433$	−5.48947e6	−1.40705	−0.703527	+	0.710668i	$0.748393\pi$
$434$	0	0
$435$	−6.81171e6	−1.72597
$436$	0	0
$437$	−1.51803e6	−0.380257
$438$	0	0
$439$	−1.25644e6	−0.311157	−0.155579	−	0.987824i	$-0.549724\pi$
$439$	−1.25644e6	−0.311157	−0.155579	+	0.987824i	$0.549724\pi$
$440$	0	0
$441$	−2.10653e6	−0.515788
$442$	0	0
$443$	1.86585e6	0.451718	0.225859	−	0.974160i	$-0.427481\pi$
$443$	1.86585e6	0.451718	0.225859	+	0.974160i	$0.427481\pi$
$444$	0	0
$445$	1.62159e6	0.388187
$446$	0	0
$447$	−1.05797e6	−0.250441
$448$	0	0
$449$	−108113.	−0.0253082	−0.0126541	−	0.999920i	$-0.504028\pi$
$449$	−108113.	−0.0253082	−0.0126541	+	0.999920i	$0.504028\pi$
$450$	0	0
$451$	−43036.9	−0.00996321
$452$	0	0
$453$	5.07649e6	1.16230
$454$	0	0
$455$	−5.50546e6	−1.24671
$456$	0	0
$457$	3.40589e6	0.762851	0.381425	−	0.924400i	$-0.375433\pi$
$457$	3.40589e6	0.762851	0.381425	+	0.924400i	$0.375433\pi$
$458$	0	0
$459$	−601404.	−0.133240
$460$	0	0
$461$	6.92278e6	1.51715	0.758575	−	0.651586i	$-0.225896\pi$
$461$	6.92278e6	1.51715	0.758575	+	0.651586i	$0.225896\pi$
$462$	0	0
$463$	−2.65274e6	−0.575098	−0.287549	−	0.957766i	$-0.592840\pi$
$463$	−2.65274e6	−0.575098	−0.287549	+	0.957766i	$0.592840\pi$
$464$	0	0
$465$	4.59264e6	0.984985
$466$	0	0
$467$	−1.86302e6	−0.395299	−0.197649	−	0.980273i	$-0.563331\pi$
$467$	−1.86302e6	−0.395299	−0.197649	+	0.980273i	$0.563331\pi$
$468$	0	0
$469$	−5.49265e6	−1.15305
$470$	0	0
$471$	−1.13820e7	−2.36411
$472$	0	0
$473$	144689.	0.0297360
$474$	0	0
$475$	1.03916e6	0.211323
$476$	0	0
$477$	7.53892e6	1.51710
$478$	0	0
$479$	−2.95181e6	−0.587826	−0.293913	−	0.955832i	$-0.594958\pi$
$479$	−2.95181e6	−0.587826	−0.293913	+	0.955832i	$0.594958\pi$
$480$	0	0
$481$	6.83331e6	1.34669
$482$	0	0
$483$	4.67979e6	0.912763
$484$	0	0
$485$	−4.04667e6	−0.781166
$486$	0	0
$487$	9.63684e6	1.84125	0.920624	−	0.390451i	$-0.127681\pi$
$487$	9.63684e6	1.84125	0.920624	+	0.390451i	$0.127681\pi$
$488$	0	0
$489$	−1.20659e7	−2.28186
$490$	0	0
$491$	−7.56617e6	−1.41636	−0.708178	−	0.706034i	$-0.750483\pi$
$491$	−7.56617e6	−1.41636	−0.708178	+	0.706034i	$0.750483\pi$
$492$	0	0
$493$	−3.18936e6	−0.590999
$494$	0	0
$495$	−576297.	−0.105714
$496$	0	0
$497$	4.69117e6	0.851903
$498$	0	0
$499$	4.55401e6	0.818733	0.409367	−	0.912370i	$-0.365750\pi$
$499$	4.55401e6	0.818733	0.409367	+	0.912370i	$0.365750\pi$
$500$	0	0
$501$	−7.29205e6	−1.29794
$502$	0	0
$503$	−2.47656e6	−0.436445	−0.218223	−	0.975899i	$-0.570026\pi$
$503$	−2.47656e6	−0.436445	−0.218223	+	0.975899i	$0.570026\pi$
$504$	0	0
$505$	−1.21792e7	−2.12516
$506$	0	0
$507$	−1.20772e7	−2.08664
$508$	0	0
$509$	653027.	0.111722	0.0558608	−	0.998439i	$-0.482210\pi$
$509$	653027.	0.111722	0.0558608	+	0.998439i	$0.482210\pi$
$510$	0	0
$511$	4.71657e6	0.799050
$512$	0	0
$513$	470075.	0.0788630
$514$	0	0
$515$	1.39899e7	2.32432
$516$	0	0
$517$	−657277.	−0.108149
$518$	0	0
$519$	1.77276e6	0.288890
$520$	0	0
$521$	7.36119e6	1.18810	0.594051	−	0.804428i	$-0.297528\pi$
$521$	7.36119e6	1.18810	0.594051	+	0.804428i	$0.297528\pi$
$522$	0	0
$523$	−1.85490e6	−0.296528	−0.148264	−	0.988948i	$-0.547368\pi$
$523$	−1.85490e6	−0.296528	−0.148264	+	0.988948i	$0.547368\pi$
$524$	0	0
$525$	−3.20351e6	−0.507257
$526$	0	0
$527$	2.15035e6	0.337274
$528$	0	0
$529$	1.25508e6	0.194998
$530$	0	0
$531$	−1.04261e6	−0.160466
$532$	0	0
$533$	−1.04015e6	−0.158591
$534$	0	0
$535$	−7.41630e6	−1.12022
$536$	0	0
$537$	8.44885e6	1.26433
$538$	0	0
$539$	418511.	0.0620491
$540$	0	0
$541$	676830.	0.0994229	0.0497114	−	0.998764i	$-0.484170\pi$
$541$	676830.	0.0994229	0.0497114	+	0.998764i	$0.484170\pi$
$542$	0	0
$543$	−242412.	−0.0352821
$544$	0	0
$545$	−6.27262e6	−0.904602
$546$	0	0
$547$	9.99363e6	1.42809	0.714044	−	0.700101i	$-0.246862\pi$
$547$	9.99363e6	1.42809	0.714044	+	0.700101i	$0.246862\pi$
$548$	0	0
$549$	9.39207e6	1.32993
$550$	0	0
$551$	2.49290e6	0.349805
$552$	0	0
$553$	−4.85482e6	−0.675088
$554$	0	0
$555$	1.05212e7	1.44988
$556$	0	0
$557$	2.23874e6	0.305749	0.152875	−	0.988246i	$-0.451147\pi$
$557$	2.23874e6	0.305749	0.152875	+	0.988246i	$0.451147\pi$
$558$	0	0
$559$	3.49696e6	0.473327
$560$	0	0
$561$	−593946.	−0.0796783
$562$	0	0
$563$	8.21104e6	1.09176	0.545880	−	0.837863i	$-0.316196\pi$
$563$	8.21104e6	1.09176	0.545880	+	0.837863i	$0.316196\pi$
$564$	0	0
$565$	1.51492e7	1.99649
$566$	0	0
$567$	−5.38015e6	−0.702808
$568$	0	0
$569$	7.39514e6	0.957559	0.478780	−	0.877935i	$-0.341079\pi$
$569$	7.39514e6	0.957559	0.478780	+	0.877935i	$0.341079\pi$
$570$	0	0
$571$	7.60599e6	0.976259	0.488130	−	0.872771i	$-0.337679\pi$
$571$	7.60599e6	0.976259	0.488130	+	0.872771i	$0.337679\pi$
$572$	0	0
$573$	−1.17384e7	−1.49356
$574$	0	0
$575$	−5.26510e6	−0.664106
$576$	0	0
$577$	−8.52470e6	−1.06596	−0.532978	−	0.846129i	$-0.678927\pi$
$577$	−8.52470e6	−1.06596	−0.532978	+	0.846129i	$0.678927\pi$
$578$	0	0
$579$	−1.23992e7	−1.53708
$580$	0	0
$581$	−4.03108e6	−0.495429
$582$	0	0
$583$	−1.49778e6	−0.182506
$584$	0	0
$585$	−1.39284e7	−1.68272
$586$	0	0
$587$	1.19067e7	1.42625	0.713127	−	0.701035i	$-0.247278\pi$
$587$	1.19067e7	1.42625	0.713127	+	0.701035i	$0.247278\pi$
$588$	0	0
$589$	−1.68078e6	−0.199628
$590$	0	0
$591$	−1.44288e7	−1.69926
$592$	0	0
$593$	5.28713e6	0.617424	0.308712	−	0.951156i	$-0.400102\pi$
$593$	5.28713e6	0.617424	0.308712	+	0.951156i	$0.400102\pi$
$594$	0	0
$595$	−3.96894e6	−0.459602
$596$	0	0
$597$	2.01838e7	2.31776
$598$	0	0
$599$	1.11555e7	1.27034	0.635172	−	0.772370i	$-0.280929\pi$
$599$	1.11555e7	1.27034	0.635172	+	0.772370i	$0.280929\pi$
$600$	0	0
$601$	1.40696e7	1.58890	0.794450	−	0.607330i	$-0.207759\pi$
$601$	1.40696e7	1.58890	0.794450	+	0.607330i	$0.207759\pi$
$602$	0	0
$603$	−1.38960e7	−1.55631
$604$	0	0
$605$	−1.13002e7	−1.25516
$606$	0	0
$607$	457367.	0.0503841	0.0251920	−	0.999683i	$-0.491980\pi$
$607$	457367.	0.0503841	0.0251920	+	0.999683i	$0.491980\pi$
$608$	0	0
$609$	−7.68510e6	−0.839665
$610$	0	0
$611$	−1.58856e7	−1.72147
$612$	0	0
$613$	1.76472e7	1.89682	0.948408	−	0.317053i	$-0.102693\pi$
$613$	1.76472e7	1.89682	0.948408	+	0.317053i	$0.102693\pi$
$614$	0	0
$615$	−1.60151e6	−0.170742
$616$	0	0
$617$	−1.77618e6	−0.187834	−0.0939171	−	0.995580i	$-0.529939\pi$
$617$	−1.77618e6	−0.187834	−0.0939171	+	0.995580i	$0.529939\pi$
$618$	0	0
$619$	4.46946e6	0.468844	0.234422	−	0.972135i	$-0.424680\pi$
$619$	4.46946e6	0.468844	0.234422	+	0.972135i	$0.424680\pi$
$620$	0	0
$621$	−2.38173e6	−0.247835
$622$	0	0
$623$	1.82951e6	0.188849
$624$	0	0
$625$	−1.20942e7	−1.23844
$626$	0	0
$627$	464245.	0.0471605
$628$	0	0
$629$	4.92620e6	0.496461
$630$	0	0
$631$	−319072.	−0.0319018	−0.0159509	−	0.999873i	$-0.505078\pi$
$631$	−319072.	−0.0319018	−0.0159509	+	0.999873i	$0.505078\pi$
$632$	0	0
$633$	1.56135e7	1.54878
$634$	0	0
$635$	−6.95236e6	−0.684224
$636$	0	0
$637$	1.01149e7	0.987674
$638$	0	0
$639$	1.18683e7	1.14984
$640$	0	0
$641$	−6.84061e6	−0.657582	−0.328791	−	0.944403i	$-0.606641\pi$
$641$	−6.84061e6	−0.657582	−0.328791	+	0.944403i	$0.606641\pi$
$642$	0	0
$643$	1.39098e7	1.32676	0.663381	−	0.748282i	$-0.269121\pi$
$643$	1.39098e7	1.32676	0.663381	+	0.748282i	$0.269121\pi$
$644$	0	0
$645$	5.38424e6	0.509595
$646$	0	0
$647$	−5.98348e6	−0.561944	−0.280972	−	0.959716i	$-0.590657\pi$
$647$	−5.98348e6	−0.561944	−0.280972	+	0.959716i	$0.590657\pi$
$648$	0	0
$649$	207138.	0.0193040
$650$	0	0
$651$	5.18149e6	0.479184
$652$	0	0
$653$	14001.2	0.00128494	0.000642470	−	1.00000i	$-0.499795\pi$
$653$	14001.2	0.00128494	0.000642470		1.00000i	$0.499795\pi$
$654$	0	0
$655$	−7.61204e6	−0.693263
$656$	0	0
$657$	1.19326e7	1.07850
$658$	0	0
$659$	9.78465e6	0.877671	0.438835	−	0.898567i	$-0.355391\pi$
$659$	9.78465e6	0.877671	0.438835	+	0.898567i	$0.355391\pi$
$660$	0	0
$661$	−1.77214e7	−1.57759	−0.788794	−	0.614657i	$-0.789294\pi$
$661$	−1.77214e7	−1.57759	−0.788794	+	0.614657i	$0.789294\pi$
$662$	0	0
$663$	−1.43550e7	−1.26829
$664$	0	0
$665$	3.10223e6	0.272032
$666$	0	0
$667$	−1.26308e7	−1.09930
$668$	0	0
$669$	2.42172e7	2.09199
$670$	0	0
$671$	−1.86595e6	−0.159990
$672$	0	0
$673$	−343782.	−0.0292580	−0.0146290	−	0.999893i	$-0.504657\pi$
$673$	−343782.	−0.0292580	−0.0146290	+	0.999893i	$0.504657\pi$
$674$	0	0
$675$	1.63040e6	0.137732
$676$	0	0
$677$	−1.58869e7	−1.33220	−0.666099	−	0.745863i	$-0.732037\pi$
$677$	−1.58869e7	−1.33220	−0.666099	+	0.745863i	$0.732037\pi$
$678$	0	0
$679$	−4.56552e6	−0.380029
$680$	0	0
$681$	−1.89860e7	−1.56879
$682$	0	0
$683$	1.11009e7	0.910552	0.455276	−	0.890350i	$-0.349541\pi$
$683$	1.11009e7	0.910552	0.455276	+	0.890350i	$0.349541\pi$
$684$	0	0
$685$	1.96075e7	1.59659
$686$	0	0
$687$	−3.40048e6	−0.274884
$688$	0	0
$689$	−3.61996e7	−2.90506
$690$	0	0
$691$	−5.32734e6	−0.424439	−0.212219	−	0.977222i	$-0.568069\pi$
$691$	−5.32734e6	−0.424439	−0.212219	+	0.977222i	$0.568069\pi$
$692$	0	0
$693$	−650188.	−0.0514288
$694$	0	0
$695$	1.95490e7	1.53520
$696$	0	0
$697$	−749853.	−0.0584648
$698$	0	0
$699$	−1.47694e7	−1.14333
$700$	0	0
$701$	2.01484e7	1.54862	0.774310	−	0.632806i	$-0.218097\pi$
$701$	2.01484e7	1.54862	0.774310	+	0.632806i	$0.218097\pi$
$702$	0	0
$703$	−3.85046e6	−0.293849
$704$	0	0
$705$	−2.44589e7	−1.85338
$706$	0	0
$707$	−1.37408e7	−1.03387
$708$	0	0
$709$	5.84423e6	0.436628	0.218314	−	0.975879i	$-0.429944\pi$
$709$	5.84423e6	0.436628	0.218314	+	0.975879i	$0.429944\pi$
$710$	0	0
$711$	−1.22823e7	−0.911185
$712$	0	0
$713$	8.51600e6	0.627353
$714$	0	0
$715$	2.76720e6	0.202430
$716$	0	0
$717$	6.43160e6	0.467219
$718$	0	0
$719$	2.58797e6	0.186697	0.0933486	−	0.995633i	$-0.470243\pi$
$719$	2.58797e6	0.186697	0.0933486	+	0.995633i	$0.470243\pi$
$720$	0	0
$721$	1.57836e7	1.13075
$722$	0	0
$723$	−3.29360e7	−2.34329
$724$	0	0
$725$	8.64630e6	0.610921
$726$	0	0
$727$	2.52317e6	0.177056	0.0885281	−	0.996074i	$-0.471784\pi$
$727$	2.52317e6	0.177056	0.0885281	+	0.996074i	$0.471784\pi$
$728$	0	0
$729$	−9.20762e6	−0.641695
$730$	0	0
$731$	2.52099e6	0.174493
$732$	0	0
$733$	−8.39153e6	−0.576874	−0.288437	−	0.957499i	$-0.593136\pi$
$733$	−8.39153e6	−0.576874	−0.288437	+	0.957499i	$0.593136\pi$
$734$	0	0
$735$	1.55738e7	1.06335
$736$	0	0
$737$	2.76076e6	0.187223
$738$	0	0
$739$	2.84112e7	1.91372	0.956859	−	0.290552i	$-0.0938391\pi$
$739$	2.84112e7	1.91372	0.956859	+	0.290552i	$0.0938391\pi$
$740$	0	0
$741$	1.12203e7	0.750684
$742$	0	0
$743$	9.34348e6	0.620921	0.310461	−	0.950586i	$-0.399517\pi$
$743$	9.34348e6	0.620921	0.310461	+	0.950586i	$0.399517\pi$
$744$	0	0
$745$	3.55343e6	0.234562
$746$	0	0
$747$	−1.01983e7	−0.668694
$748$	0	0
$749$	−8.36720e6	−0.544974
$750$	0	0
$751$	1.91660e7	1.24003	0.620013	−	0.784592i	$-0.287127\pi$
$751$	1.91660e7	1.24003	0.620013	+	0.784592i	$0.287127\pi$
$752$	0	0
$753$	−2.14247e7	−1.37698
$754$	0	0
$755$	−1.70505e7	−1.08860
$756$	0	0
$757$	1.03429e7	0.655999	0.327999	−	0.944678i	$-0.393626\pi$
$757$	1.03429e7	0.655999	0.327999	+	0.944678i	$0.393626\pi$
$758$	0	0
$759$	−2.35219e6	−0.148207
$760$	0	0
$761$	1.37925e7	0.863342	0.431671	−	0.902031i	$-0.357924\pi$
$761$	1.37925e7	0.863342	0.431671	+	0.902031i	$0.357924\pi$
$762$	0	0
$763$	−7.07688e6	−0.440079
$764$	0	0
$765$	−1.00411e7	−0.620338
$766$	0	0
$767$	5.00627e6	0.307274
$768$	0	0
$769$	−1.98790e7	−1.21221	−0.606105	−	0.795384i	$-0.707269\pi$
$769$	−1.98790e7	−1.21221	−0.606105	+	0.795384i	$0.707269\pi$
$770$	0	0
$771$	1.39378e6	0.0844419
$772$	0	0
$773$	−5.08811e6	−0.306273	−0.153136	−	0.988205i	$-0.548937\pi$
$773$	−5.08811e6	−0.306273	−0.153136	+	0.988205i	$0.548937\pi$
$774$	0	0
$775$	−5.82956e6	−0.348644
$776$	0	0
$777$	1.18702e7	0.705350
$778$	0	0
$779$	586107.	0.0346045
$780$	0	0
$781$	−2.35791e6	−0.138325
$782$	0	0
$783$	3.91125e6	0.227988
$784$	0	0
$785$	3.82291e7	2.21422
$786$	0	0
$787$	−1.97998e7	−1.13952	−0.569762	−	0.821810i	$-0.692964\pi$
$787$	−1.97998e7	−1.13952	−0.569762	+	0.821810i	$0.692964\pi$
$788$	0	0
$789$	−2.30105e7	−1.31593
$790$	0	0
$791$	1.70916e7	0.971272
$792$	0	0
$793$	−4.50978e7	−2.54667
$794$	0	0
$795$	−5.57361e7	−3.12765
$796$	0	0
$797$	1.93440e7	1.07870	0.539349	−	0.842082i	$-0.318670\pi$
$797$	1.93440e7	1.07870	0.539349	+	0.842082i	$0.318670\pi$
$798$	0	0
$799$	−1.14521e7	−0.634625
$800$	0	0
$801$	4.62852e6	0.254895
$802$	0	0
$803$	−2.37068e6	−0.129743
$804$	0	0
$805$	−1.57181e7	−0.854890
$806$	0	0
$807$	−1.89303e7	−1.02323
$808$	0	0
$809$	3.37028e6	0.181048	0.0905242	−	0.995894i	$-0.471146\pi$
$809$	3.37028e6	0.181048	0.0905242	+	0.995894i	$0.471146\pi$
$810$	0	0
$811$	−2.31261e7	−1.23467	−0.617335	−	0.786700i	$-0.711788\pi$
$811$	−2.31261e7	−1.23467	−0.617335	+	0.786700i	$0.711788\pi$
$812$	0	0
$813$	−4.62528e7	−2.45421
$814$	0	0
$815$	4.05261e7	2.13718
$816$	0	0
$817$	−1.97048e6	−0.103280
$818$	0	0
$819$	−1.57143e7	−0.818624
$820$	0	0
$821$	−2.22659e7	−1.15288	−0.576438	−	0.817141i	$-0.695558\pi$
$821$	−2.22659e7	−1.15288	−0.576438	+	0.817141i	$0.695558\pi$
$822$	0	0
$823$	−2.04372e7	−1.05177	−0.525885	−	0.850555i	$-0.676266\pi$
$823$	−2.04372e7	−1.05177	−0.525885	+	0.850555i	$0.676266\pi$
$824$	0	0
$825$	1.61018e6	0.0823642
$826$	0	0
$827$	−3.99284e6	−0.203010	−0.101505	−	0.994835i	$-0.532366\pi$
$827$	−3.99284e6	−0.203010	−0.101505	+	0.994835i	$0.532366\pi$
$828$	0	0
$829$	−2.29564e7	−1.16016	−0.580080	−	0.814560i	$-0.696979\pi$
$829$	−2.29564e7	−1.16016	−0.580080	+	0.814560i	$0.696979\pi$
$830$	0	0
$831$	2.35735e6	0.118419
$832$	0	0
$833$	7.29194e6	0.364108
$834$	0	0
$835$	2.44920e7	1.21565
$836$	0	0
$837$	−2.63707e6	−0.130109
$838$	0	0
$839$	−2.41640e7	−1.18512	−0.592562	−	0.805525i	$-0.701884\pi$
$839$	−2.41640e7	−1.18512	−0.592562	+	0.805525i	$0.701884\pi$
$840$	0	0
$841$	230974.	0.0112609
$842$	0	0
$843$	1.07852e7	0.522708
$844$	0	0
$845$	4.05640e7	1.95433
$846$	0	0
$847$	−1.27491e7	−0.610621
$848$	0	0
$849$	−1.53219e7	−0.729528
$850$	0	0
$851$	1.95091e7	0.923451
$852$	0	0
$853$	1.98103e7	0.932218	0.466109	−	0.884727i	$-0.345655\pi$
$853$	1.98103e7	0.932218	0.466109	+	0.884727i	$0.345655\pi$
$854$	0	0
$855$	7.84842e6	0.367170
$856$	0	0
$857$	−3.76883e6	−0.175289	−0.0876444	−	0.996152i	$-0.527934\pi$
$857$	−3.76883e6	−0.175289	−0.0876444	+	0.996152i	$0.527934\pi$
$858$	0	0
$859$	−2.66938e6	−0.123432	−0.0617160	−	0.998094i	$-0.519657\pi$
$859$	−2.66938e6	−0.123432	−0.0617160	+	0.998094i	$0.519657\pi$
$860$	0	0
$861$	−1.80685e6	−0.0830642
$862$	0	0
$863$	8.07926e6	0.369270	0.184635	−	0.982807i	$-0.440890\pi$
$863$	8.07926e6	0.369270	0.184635	+	0.982807i	$0.440890\pi$
$864$	0	0
$865$	−5.95422e6	−0.270573
$866$	0	0
$867$	1.96135e7	0.886150
$868$	0	0
$869$	2.44017e6	0.109615
$870$	0	0
$871$	6.67243e7	2.98015
$872$	0	0
$873$	−1.15504e7	−0.512935
$874$	0	0
$875$	−6.95144e6	−0.306941
$876$	0	0
$877$	−1.73452e6	−0.0761518	−0.0380759	−	0.999275i	$-0.512123\pi$
$877$	−1.73452e6	−0.0761518	−0.0380759	+	0.999275i	$0.512123\pi$
$878$	0	0
$879$	−5.67779e7	−2.47860
$880$	0	0
$881$	−4.68380e6	−0.203310	−0.101655	−	0.994820i	$-0.532414\pi$
$881$	−4.68380e6	−0.203310	−0.101655	+	0.994820i	$0.532414\pi$
$882$	0	0
$883$	3.46662e7	1.49625	0.748125	−	0.663558i	$-0.230955\pi$
$883$	3.46662e7	1.49625	0.748125	+	0.663558i	$0.230955\pi$
$884$	0	0
$885$	7.70810e6	0.330818
$886$	0	0
$887$	4.18105e7	1.78433	0.892167	−	0.451707i	$-0.149185\pi$
$887$	4.18105e7	1.78433	0.892167	+	0.451707i	$0.149185\pi$
$888$	0	0
$889$	−7.84378e6	−0.332867
$890$	0	0
$891$	2.70422e6	0.114116
$892$	0	0
$893$	8.95126e6	0.375626
$894$	0	0
$895$	−2.83773e7	−1.18417
$896$	0	0
$897$	−5.68497e7	−2.35910
$898$	0	0
$899$	−1.39849e7	−0.577111
$900$	0	0
$901$	−2.60966e7	−1.07096
$902$	0	0
$903$	6.07459e6	0.247912
$904$	0	0
$905$	814195.	0.0330451
$906$	0	0
$907$	−1.19694e7	−0.483118	−0.241559	−	0.970386i	$-0.577659\pi$
$907$	−1.19694e7	−0.483118	−0.241559	+	0.970386i	$0.577659\pi$
$908$	0	0
$909$	−3.47633e7	−1.39544
$910$	0	0
$911$	−2.51941e7	−1.00578	−0.502891	−	0.864350i	$-0.667730\pi$
$911$	−2.51941e7	−1.00578	−0.502891	+	0.864350i	$0.667730\pi$
$912$	0	0
$913$	2.02613e6	0.0804436
$914$	0	0
$915$	−6.94366e7	−2.74180
$916$	0	0
$917$	−8.58804e6	−0.337265
$918$	0	0
$919$	−1.04550e7	−0.408352	−0.204176	−	0.978934i	$-0.565451\pi$
$919$	−1.04550e7	−0.408352	−0.204176	+	0.978934i	$0.565451\pi$
$920$	0	0
$921$	1.98133e7	0.769677
$922$	0	0
$923$	−5.69879e7	−2.20180
$924$	0	0
$925$	−1.33548e7	−0.513197
$926$	0	0
$927$	3.99314e7	1.52621
$928$	0	0
$929$	−3.66684e6	−0.139397	−0.0696983	−	0.997568i	$-0.522204\pi$
$929$	−3.66684e6	−0.139397	−0.0696983	+	0.997568i	$0.522204\pi$
$930$	0	0
$931$	−5.69958e6	−0.215511
$932$	0	0
$933$	−1.31878e7	−0.495986
$934$	0	0
$935$	1.99490e6	0.0746263
$936$	0	0
$937$	1.50261e7	0.559109	0.279555	−	0.960130i	$-0.409813\pi$
$937$	1.50261e7	0.559109	0.279555	+	0.960130i	$0.409813\pi$
$938$	0	0
$939$	5.28316e7	1.95538
$940$	0	0
$941$	1.42229e7	0.523616	0.261808	−	0.965120i	$-0.415681\pi$
$941$	1.42229e7	0.523616	0.261808	+	0.965120i	$0.415681\pi$
$942$	0	0
$943$	−2.96963e6	−0.108748
$944$	0	0
$945$	4.86728e6	0.177299
$946$	0	0
$947$	4.12070e7	1.49312	0.746562	−	0.665316i	$-0.231703\pi$
$947$	4.12070e7	1.49312	0.746562	+	0.665316i	$0.231703\pi$
$948$	0	0
$949$	−5.72965e7	−2.06520
$950$	0	0
$951$	−6.93852e7	−2.48780
$952$	0	0
$953$	−2.39173e6	−0.0853061	−0.0426531	−	0.999090i	$-0.513581\pi$
$953$	−2.39173e6	−0.0853061	−0.0426531	+	0.999090i	$0.513581\pi$
$954$	0	0
$955$	3.94261e7	1.39886
$956$	0	0
$957$	3.86275e6	0.136338
$958$	0	0
$959$	2.21215e7	0.776726
$960$	0	0
$961$	−1.92002e7	−0.670651
$962$	0	0
$963$	−2.11684e7	−0.735567
$964$	0	0
$965$	4.16455e7	1.43963
$966$	0	0
$967$	−1.26512e7	−0.435075	−0.217538	−	0.976052i	$-0.569802\pi$
$967$	−1.26512e7	−0.435075	−0.217538	+	0.976052i	$0.569802\pi$
$968$	0	0
$969$	8.08878e6	0.276741
$970$	0	0
$971$	−2.99820e7	−1.02050	−0.510249	−	0.860027i	$-0.670447\pi$
$971$	−2.99820e7	−1.02050	−0.510249	+	0.860027i	$0.670447\pi$
$972$	0	0
$973$	2.20556e7	0.746855
$974$	0	0
$975$	3.89160e7	1.31104
$976$	0	0
$977$	−2.62397e7	−0.879472	−0.439736	−	0.898127i	$-0.644928\pi$
$977$	−2.62397e7	−0.879472	−0.439736	+	0.898127i	$0.644928\pi$
$978$	0	0
$979$	−919562.	−0.0306637
$980$	0	0
$981$	−1.79040e7	−0.593987
$982$	0	0
$983$	5.91825e7	1.95348	0.976741	−	0.214423i	$-0.0687872\pi$
$983$	5.91825e7	1.95348	0.976741	+	0.214423i	$0.0687872\pi$
$984$	0	0
$985$	4.84622e7	1.59152
$986$	0	0
$987$	−2.75949e7	−0.901646
$988$	0	0
$989$	9.98384e6	0.324569
$990$	0	0
$991$	6.16486e6	0.199406	0.0997032	−	0.995017i	$-0.468211\pi$
$991$	6.16486e6	0.199406	0.0997032	+	0.995017i	$0.468211\pi$
$992$	0	0
$993$	6.73815e7	2.16854
$994$	0	0
$995$	−6.77919e7	−2.17080
$996$	0	0
$997$	−4.02896e7	−1.28368	−0.641838	−	0.766840i	$-0.721828\pi$
$997$	−4.02896e7	−1.28368	−0.641838	+	0.766840i	$0.721828\pi$
$998$	0	0
$999$	−6.04121e6	−0.191518

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1028.6.a.b.1.10	✓	57

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1028.6.a.b.1.10	✓	57	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1028 = 2^{2} \cdot 257 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1028.a (trivial)

\(f(q)\)	\(=\)	\(q-21.1022 q^{3} +70.8764 q^{5} +79.9641 q^{7} +202.303 q^{9} +O(q^{10})\)	\(q-21.1022 q^{3} +70.8764 q^{5} +79.9641 q^{7} +202.303 q^{9} -40.1922 q^{11} -971.397 q^{13} -1495.65 q^{15} -700.289 q^{17} +547.366 q^{19} -1687.42 q^{21} -2773.34 q^{23} +1898.47 q^{25} +858.794 q^{27} +4554.35 q^{29} -3070.66 q^{31} +848.144 q^{33} +5667.57 q^{35} -7034.52 q^{37} +20498.6 q^{39} +1070.78 q^{41} -3599.93 q^{43} +14338.5 q^{45} +16353.3 q^{47} -10412.7 q^{49} +14777.6 q^{51} +37265.5 q^{53} -2848.68 q^{55} -11550.6 q^{57} -5153.68 q^{59} +46425.7 q^{61} +16177.0 q^{63} -68849.2 q^{65} -68689.0 q^{67} +58523.6 q^{69} +58665.9 q^{71} +58983.6 q^{73} -40061.9 q^{75} -3213.93 q^{77} -60712.5 q^{79} -67282.1 q^{81} -50411.1 q^{83} -49634.0 q^{85} -96106.9 q^{87} +22879.1 q^{89} -77676.9 q^{91} +64797.8 q^{93} +38795.4 q^{95} -57094.7 q^{97} -8131.01 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9}+O(q^{10}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9	\( 57 q + 27 q^{3} + 61 q^{5} + 88 q^{7} + 5374 q^{9} + 484 q^{11} + 3162 q^{13} - 244 q^{15} + 3298 q^{17} + 2931 q^{19} + 5009 q^{21} + 3712 q^{23} + 48568 q^{25} + 8682 q^{27} + 10909 q^{29} - 3539 q^{31} + 13277 q^{33} + 13976 q^{35} + 42395 q^{37} - 14986 q^{39} - 8469 q^{41} + 69808 q^{43} + 26347 q^{45} + 12410 q^{47} + 203347 q^{49} + 24220 q^{51} + 49353 q^{53} - 34265 q^{55} + 98112 q^{57} - 9792 q^{59} + 157032 q^{61} + 24981 q^{63} + 16143 q^{65} + 20201 q^{67} - 113743 q^{69} - 229673 q^{71} - 19561 q^{73} - 482061 q^{75} + 21204 q^{77} - 10925 q^{79} + 323569 q^{81} - 62846 q^{83} + 218157 q^{85} - 169628 q^{87} + 69901 q^{89} + 56581 q^{91} + 177641 q^{93} - 192910 q^{95} + 277990 q^{97} + 424882 q^{99}+O(q^{100}) \) 57 * q + 27 * q^3 + 61 * q^5 + 88 * q^7 + 5374 * q^9 + 484 * q^11 + 3162 * q^13 - 244 * q^15 + 3298 * q^17 + 2931 * q^19 + 5009 * q^21 + 3712 * q^23 + 48568 * q^25 + 8682 * q^27 + 10909 * q^29 - 3539 * q^31 + 13277 * q^33 + 13976 * q^35 + 42395 * q^37 - 14986 * q^39 - 8469 * q^41 + 69808 * q^43 + 26347 * q^45 + 12410 * q^47 + 203347 * q^49 + 24220 * q^51 + 49353 * q^53 - 34265 * q^55 + 98112 * q^57 - 9792 * q^59 + 157032 * q^61 + 24981 * q^63 + 16143 * q^65 + 20201 * q^67 - 113743 * q^69 - 229673 * q^71 - 19561 * q^73 - 482061 * q^75 + 21204 * q^77 - 10925 * q^79 + 323569 * q^81 - 62846 * q^83 + 218157 * q^85 - 169628 * q^87 + 69901 * q^89 + 56581 * q^91 + 177641 * q^93 - 192910 * q^95 + 277990 * q^97 + 424882 * q^99

Introduction

L-functions

Modular forms

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