LMFDB - Embedded newform 688.6.a.h.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,6,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("688.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	688.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:	$110.344068031$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + \cdots - 22734604 $ x^10 - 2x^9 - 256x^8 + 266x^7 + 21986x^6 - 10450x^5 - 719484x^4 + 384582x^3 + 8437093x^2 - 5752252*x - 22734604
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.48720$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+27.5943 q^{3} +101.308 q^{5} -15.7005 q^{7} +518.448 q^{9} +O(q^{10})$	$q+27.5943 q^{3} +101.308 q^{5} -15.7005 q^{7} +518.448 q^{9} -394.739 q^{11} +666.939 q^{13} +2795.53 q^{15} +172.038 q^{17} +1280.86 q^{19} -433.245 q^{21} -569.882 q^{23} +7138.34 q^{25} +7600.80 q^{27} -6328.41 q^{29} -7795.01 q^{31} -10892.6 q^{33} -1590.59 q^{35} +16252.3 q^{37} +18403.7 q^{39} +7454.95 q^{41} -1849.00 q^{43} +52523.0 q^{45} +5628.68 q^{47} -16560.5 q^{49} +4747.27 q^{51} +22460.1 q^{53} -39990.3 q^{55} +35344.4 q^{57} +9061.48 q^{59} -18280.8 q^{61} -8139.89 q^{63} +67566.3 q^{65} +27428.5 q^{67} -15725.5 q^{69} +12860.9 q^{71} -63446.5 q^{73} +196978. q^{75} +6197.60 q^{77} +1911.26 q^{79} +83756.3 q^{81} +52124.9 q^{83} +17428.8 q^{85} -174628. q^{87} +66185.2 q^{89} -10471.3 q^{91} -215098. q^{93} +129761. q^{95} +37497.8 q^{97} -204652. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	$ 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) $ 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * 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
$3$	27.5943	1.77018	0.885089	−	0.465422i	$-0.154097\pi$
$3$	27.5943	1.77018	0.885089	+	0.465422i	$0.154097\pi$
$4$	0	0
$5$	101.308	1.81226	0.906128	−	0.423004i	$-0.139024\pi$
$5$	101.308	1.81226	0.906128	+	0.423004i	$0.139024\pi$
$6$	0	0
$7$	−15.7005	−0.121107	−0.0605534	−	0.998165i	$-0.519287\pi$
$7$	−15.7005	−0.121107	−0.0605534	+	0.998165i	$0.519287\pi$
$8$	0	0
$9$	518.448	2.13353
$10$	0	0
$11$	−394.739	−0.983623	−0.491811	−	0.870702i	$-0.663665\pi$
$11$	−394.739	−0.983623	−0.491811	+	0.870702i	$0.663665\pi$
$12$	0	0
$13$	666.939	1.09453	0.547265	−	0.836959i	$-0.315669\pi$
$13$	666.939	1.09453	0.547265	+	0.836959i	$0.315669\pi$
$14$	0	0
$15$	2795.53	3.20801
$16$	0	0
$17$	172.038	0.144378	0.0721890	−	0.997391i	$-0.477002\pi$
$17$	172.038	0.144378	0.0721890	+	0.997391i	$0.477002\pi$
$18$	0	0
$19$	1280.86	0.813985	0.406992	−	0.913432i	$-0.366578\pi$
$19$	1280.86	0.813985	0.406992	+	0.913432i	$0.366578\pi$
$20$	0	0
$21$	−433.245	−0.214380
$22$	0	0
$23$	−569.882	−0.224629	−0.112314	−	0.993673i	$-0.535826\pi$
$23$	−569.882	−0.224629	−0.112314	+	0.993673i	$0.535826\pi$
$24$	0	0
$25$	7138.34	2.28427
$26$	0	0
$27$	7600.80	2.00655
$28$	0	0
$29$	−6328.41	−1.39733	−0.698666	−	0.715448i	$-0.746223\pi$
$29$	−6328.41	−1.39733	−0.698666	+	0.715448i	$0.746223\pi$
$30$	0	0
$31$	−7795.01	−1.45684	−0.728421	−	0.685129i	$-0.759746\pi$
$31$	−7795.01	−1.45684	−0.728421	+	0.685129i	$0.759746\pi$
$32$	0	0
$33$	−10892.6	−1.74119
$34$	0	0
$35$	−1590.59	−0.219476
$36$	0	0
$37$	16252.3	1.95169	0.975844	−	0.218469i	$-0.0701061\pi$
$37$	16252.3	1.95169	0.975844	+	0.218469i	$0.0701061\pi$
$38$	0	0
$39$	18403.7	1.93751
$40$	0	0
$41$	7454.95	0.692604	0.346302	−	0.938123i	$-0.387437\pi$
$41$	7454.95	0.692604	0.346302	+	0.938123i	$0.387437\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	52523.0	3.86650
$46$	0	0
$47$	5628.68	0.371673	0.185837	−	0.982581i	$-0.440500\pi$
$47$	5628.68	0.371673	0.185837	+	0.982581i	$0.440500\pi$
$48$	0	0
$49$	−16560.5	−0.985333
$50$	0	0
$51$	4747.27	0.255575
$52$	0	0
$53$	22460.1	1.09830	0.549151	−	0.835723i	$-0.314951\pi$
$53$	22460.1	1.09830	0.549151	+	0.835723i	$0.314951\pi$
$54$	0	0
$55$	−39990.3	−1.78258
$56$	0	0
$57$	35344.4	1.44090
$58$	0	0
$59$	9061.48	0.338898	0.169449	−	0.985539i	$-0.445801\pi$
$59$	9061.48	0.338898	0.169449	+	0.985539i	$0.445801\pi$
$60$	0	0
$61$	−18280.8	−0.629029	−0.314514	−	0.949253i	$-0.601842\pi$
$61$	−18280.8	−0.629029	−0.314514	+	0.949253i	$0.601842\pi$
$62$	0	0
$63$	−8139.89	−0.258385
$64$	0	0
$65$	67566.3	1.98357
$66$	0	0
$67$	27428.5	0.746474	0.373237	−	0.927736i	$-0.378248\pi$
$67$	27428.5	0.746474	0.373237	+	0.927736i	$0.378248\pi$
$68$	0	0
$69$	−15725.5	−0.397633
$70$	0	0
$71$	12860.9	0.302780	0.151390	−	0.988474i	$-0.451625\pi$
$71$	12860.9	0.302780	0.151390	+	0.988474i	$0.451625\pi$
$72$	0	0
$73$	−63446.5	−1.39348	−0.696740	−	0.717324i	$-0.745367\pi$
$73$	−63446.5	−1.39348	−0.696740	+	0.717324i	$0.745367\pi$
$74$	0	0
$75$	196978.	4.04356
$76$	0	0
$77$	6197.60	0.119123
$78$	0	0
$79$	1911.26	0.0344549	0.0172275	−	0.999852i	$-0.494516\pi$
$79$	1911.26	0.0344549	0.0172275	+	0.999852i	$0.494516\pi$
$80$	0	0
$81$	83756.3	1.41842
$82$	0	0
$83$	52124.9	0.830520	0.415260	−	0.909703i	$-0.363691\pi$
$83$	52124.9	0.830520	0.415260	+	0.909703i	$0.363691\pi$
$84$	0	0
$85$	17428.8	0.261650
$86$	0	0
$87$	−174628.	−2.47353
$88$	0	0
$89$	66185.2	0.885698	0.442849	−	0.896596i	$-0.353968\pi$
$89$	66185.2	0.885698	0.442849	+	0.896596i	$0.353968\pi$
$90$	0	0
$91$	−10471.3	−0.132555
$92$	0	0
$93$	−215098.	−2.57887
$94$	0	0
$95$	129761.	1.47515
$96$	0	0
$97$	37497.8	0.404648	0.202324	−	0.979319i	$-0.435151\pi$
$97$	37497.8	0.404648	0.202324	+	0.979319i	$0.435151\pi$
$98$	0	0
$99$	−204652.	−2.09859
$100$	0	0
$101$	−35537.7	−0.346646	−0.173323	−	0.984865i	$-0.555450\pi$
$101$	−35537.7	−0.346646	−0.173323	+	0.984865i	$0.555450\pi$
$102$	0	0
$103$	−177955.	−1.65279	−0.826395	−	0.563091i	$-0.809612\pi$
$103$	−177955.	−1.65279	−0.826395	+	0.563091i	$0.809612\pi$
$104$	0	0
$105$	−43891.2	−0.388512
$106$	0	0
$107$	−43589.8	−0.368065	−0.184033	−	0.982920i	$-0.558915\pi$
$107$	−43589.8	−0.368065	−0.184033	+	0.982920i	$0.558915\pi$
$108$	0	0
$109$	−49084.5	−0.395711	−0.197856	−	0.980231i	$-0.563398\pi$
$109$	−49084.5	−0.395711	−0.197856	+	0.980231i	$0.563398\pi$
$110$	0	0
$111$	448471.	3.45484
$112$	0	0
$113$	−34530.0	−0.254390	−0.127195	−	0.991878i	$-0.540597\pi$
$113$	−34530.0	−0.254390	−0.127195	+	0.991878i	$0.540597\pi$
$114$	0	0
$115$	−57733.7	−0.407085
$116$	0	0
$117$	345773.	2.33521
$118$	0	0
$119$	−2701.08	−0.0174852
$120$	0	0
$121$	−5231.89	−0.0324859
$122$	0	0
$123$	205715.	1.22603
$124$	0	0
$125$	406584.	2.32742
$126$	0	0
$127$	10008.6	0.0550637	0.0275318	−	0.999621i	$-0.491235\pi$
$127$	10008.6	0.0550637	0.0275318	+	0.999621i	$0.491235\pi$
$128$	0	0
$129$	−51021.9	−0.269950
$130$	0	0
$131$	−336036.	−1.71083	−0.855415	−	0.517943i	$-0.826698\pi$
$131$	−336036.	−1.71083	−0.855415	+	0.517943i	$0.826698\pi$
$132$	0	0
$133$	−20110.1	−0.0985791
$134$	0	0
$135$	770023.	3.63638
$136$	0	0
$137$	−1217.37	−0.00554141	−0.00277071	−	0.999996i	$-0.500882\pi$
$137$	−1217.37	−0.00554141	−0.00277071	+	0.999996i	$0.500882\pi$
$138$	0	0
$139$	157629.	0.691989	0.345995	−	0.938237i	$-0.387542\pi$
$139$	157629.	0.691989	0.345995	+	0.938237i	$0.387542\pi$
$140$	0	0
$141$	155320.	0.657928
$142$	0	0
$143$	−263267.	−1.07660
$144$	0	0
$145$	−641119.	−2.53232
$146$	0	0
$147$	−456976.	−1.74422
$148$	0	0
$149$	−501641.	−1.85109	−0.925545	−	0.378639i	$-0.876392\pi$
$149$	−501641.	−1.85109	−0.925545	+	0.378639i	$0.876392\pi$
$150$	0	0
$151$	430865.	1.53780	0.768899	−	0.639370i	$-0.220805\pi$
$151$	430865.	1.53780	0.768899	+	0.639370i	$0.220805\pi$
$152$	0	0
$153$	89192.5	0.308035
$154$	0	0
$155$	−789698.	−2.64017
$156$	0	0
$157$	157952.	0.511418	0.255709	−	0.966754i	$-0.417691\pi$
$157$	157952.	0.511418	0.255709	+	0.966754i	$0.417691\pi$
$158$	0	0
$159$	619772.	1.94419
$160$	0	0
$161$	8947.43	0.0272041
$162$	0	0
$163$	−405846.	−1.19645	−0.598223	−	0.801330i	$-0.704126\pi$
$163$	−405846.	−1.19645	−0.598223	+	0.801330i	$0.704126\pi$
$164$	0	0
$165$	−1.10351e6	−3.15548
$166$	0	0
$167$	−166792.	−0.462789	−0.231394	−	0.972860i	$-0.574329\pi$
$167$	−166792.	−0.462789	−0.231394	+	0.972860i	$0.574329\pi$
$168$	0	0
$169$	73514.3	0.197995
$170$	0	0
$171$	664057.	1.73666
$172$	0	0
$173$	−287751.	−0.730973	−0.365487	−	0.930817i	$-0.619097\pi$
$173$	−287751.	−0.730973	−0.365487	+	0.930817i	$0.619097\pi$
$174$	0	0
$175$	−112076.	−0.276640
$176$	0	0
$177$	250045.	0.599910
$178$	0	0
$179$	−522579.	−1.21904	−0.609522	−	0.792769i	$-0.708639\pi$
$179$	−522579.	−1.21904	−0.609522	+	0.792769i	$0.708639\pi$
$180$	0	0
$181$	81268.9	0.184386	0.0921930	−	0.995741i	$-0.470612\pi$
$181$	81268.9	0.184386	0.0921930	+	0.995741i	$0.470612\pi$
$182$	0	0
$183$	−504447.	−1.11349
$184$	0	0
$185$	1.64649e6	3.53696
$186$	0	0
$187$	−67910.0	−0.142014
$188$	0	0
$189$	−119336.	−0.243007
$190$	0	0
$191$	608799.	1.20751	0.603755	−	0.797170i	$-0.293671\pi$
$191$	608799.	1.20751	0.603755	+	0.797170i	$0.293671\pi$
$192$	0	0
$193$	184959.	0.357423	0.178712	−	0.983902i	$-0.442807\pi$
$193$	184959.	0.357423	0.178712	+	0.983902i	$0.442807\pi$
$194$	0	0
$195$	1.86445e6	3.51127
$196$	0	0
$197$	200949.	0.368910	0.184455	−	0.982841i	$-0.440948\pi$
$197$	200949.	0.368910	0.184455	+	0.982841i	$0.440948\pi$
$198$	0	0
$199$	233653.	0.418253	0.209127	−	0.977889i	$-0.432938\pi$
$199$	233653.	0.418253	0.209127	+	0.977889i	$0.432938\pi$
$200$	0	0
$201$	756871.	1.32139
$202$	0	0
$203$	99359.2	0.169226
$204$	0	0
$205$	755248.	1.25518
$206$	0	0
$207$	−295454.	−0.479252
$208$	0	0
$209$	−505604.	−0.800654
$210$	0	0
$211$	−200513.	−0.310054	−0.155027	−	0.987910i	$-0.549546\pi$
$211$	−200513.	−0.310054	−0.155027	+	0.987910i	$0.549546\pi$
$212$	0	0
$213$	354889.	0.535974
$214$	0	0
$215$	−187319.	−0.276366
$216$	0	0
$217$	122386.	0.176433
$218$	0	0
$219$	−1.75076e6	−2.46671
$220$	0	0
$221$	114739.	0.158026
$222$	0	0
$223$	−347051.	−0.467338	−0.233669	−	0.972316i	$-0.575073\pi$
$223$	−347051.	−0.467338	−0.233669	+	0.972316i	$0.575073\pi$
$224$	0	0
$225$	3.70086e6	4.87356
$226$	0	0
$227$	212111.	0.273212	0.136606	−	0.990625i	$-0.456381\pi$
$227$	212111.	0.273212	0.136606	+	0.990625i	$0.456381\pi$
$228$	0	0
$229$	−470629.	−0.593049	−0.296524	−	0.955025i	$-0.595828\pi$
$229$	−470629.	−0.593049	−0.296524	+	0.955025i	$0.595828\pi$
$230$	0	0
$231$	171019.	0.210870
$232$	0	0
$233$	−952248.	−1.14911	−0.574553	−	0.818467i	$-0.694824\pi$
$233$	−952248.	−1.14911	−0.574553	+	0.818467i	$0.694824\pi$
$234$	0	0
$235$	570231.	0.673567
$236$	0	0
$237$	52739.9	0.0609914
$238$	0	0
$239$	1.63690e6	1.85365	0.926824	−	0.375497i	$-0.122528\pi$
$239$	1.63690e6	1.85365	0.926824	+	0.375497i	$0.122528\pi$
$240$	0	0
$241$	−695944.	−0.771848	−0.385924	−	0.922531i	$-0.626117\pi$
$241$	−695944.	−0.771848	−0.385924	+	0.922531i	$0.626117\pi$
$242$	0	0
$243$	464206.	0.504307
$244$	0	0
$245$	−1.67771e6	−1.78568
$246$	0	0
$247$	854253.	0.890931
$248$	0	0
$249$	1.43835e6	1.47017
$250$	0	0
$251$	−657886.	−0.659123	−0.329561	−	0.944134i	$-0.606901\pi$
$251$	−657886.	−0.659123	−0.329561	+	0.944134i	$0.606901\pi$
$252$	0	0
$253$	224955.	0.220950
$254$	0	0
$255$	480937.	0.463167
$256$	0	0
$257$	−599821.	−0.566486	−0.283243	−	0.959048i	$-0.591410\pi$
$257$	−599821.	−0.566486	−0.283243	+	0.959048i	$0.591410\pi$
$258$	0	0
$259$	−255169.	−0.236363
$260$	0	0
$261$	−3.28095e6	−2.98125
$262$	0	0
$263$	1.62868e6	1.45194	0.725968	−	0.687728i	$-0.241392\pi$
$263$	1.62868e6	1.45194	0.725968	+	0.687728i	$0.241392\pi$
$264$	0	0
$265$	2.27539e6	1.99040
$266$	0	0
$267$	1.82634e6	1.56784
$268$	0	0
$269$	1.99405e6	1.68017	0.840087	−	0.542451i	$-0.182504\pi$
$269$	1.99405e6	1.68017	0.840087	+	0.542451i	$0.182504\pi$
$270$	0	0
$271$	1.89207e6	1.56500	0.782501	−	0.622650i	$-0.213944\pi$
$271$	1.89207e6	1.56500	0.782501	+	0.622650i	$0.213944\pi$
$272$	0	0
$273$	−288948.	−0.234646
$274$	0	0
$275$	−2.81778e6	−2.24686
$276$	0	0
$277$	−2.11477e6	−1.65601	−0.828007	−	0.560717i	$-0.810525\pi$
$277$	−2.11477e6	−1.65601	−0.828007	+	0.560717i	$0.810525\pi$
$278$	0	0
$279$	−4.04131e6	−3.10822
$280$	0	0
$281$	−80948.7	−0.0611567	−0.0305784	−	0.999532i	$-0.509735\pi$
$281$	−80948.7	−0.0611567	−0.0305784	+	0.999532i	$0.509735\pi$
$282$	0	0
$283$	−1.01158e6	−0.750814	−0.375407	−	0.926860i	$-0.622497\pi$
$283$	−1.01158e6	−0.750814	−0.375407	+	0.926860i	$0.622497\pi$
$284$	0	0
$285$	3.58067e6	2.61128
$286$	0	0
$287$	−117046.	−0.0838791
$288$	0	0
$289$	−1.39026e6	−0.979155
$290$	0	0
$291$	1.03473e6	0.716298
$292$	0	0
$293$	786392.	0.535143	0.267572	−	0.963538i	$-0.413779\pi$
$293$	786392.	0.535143	0.267572	+	0.963538i	$0.413779\pi$
$294$	0	0
$295$	918001.	0.614170
$296$	0	0
$297$	−3.00034e6	−1.97369
$298$	0	0
$299$	−380076.	−0.245863
$300$	0	0
$301$	29030.2	0.0184686
$302$	0	0
$303$	−980641.	−0.613625
$304$	0	0
$305$	−1.85199e6	−1.13996
$306$	0	0
$307$	−106306.	−0.0643744	−0.0321872	−	0.999482i	$-0.510247\pi$
$307$	−106306.	−0.0643744	−0.0321872	+	0.999482i	$0.510247\pi$
$308$	0	0
$309$	−4.91056e6	−2.92573
$310$	0	0
$311$	82617.8	0.0484365	0.0242183	−	0.999707i	$-0.492290\pi$
$311$	82617.8	0.0484365	0.0242183	+	0.999707i	$0.492290\pi$
$312$	0	0
$313$	1.38431e6	0.798681	0.399341	−	0.916803i	$-0.369239\pi$
$313$	1.38431e6	0.798681	0.399341	+	0.916803i	$0.369239\pi$
$314$	0	0
$315$	−824637.	−0.468259
$316$	0	0
$317$	964944.	0.539329	0.269665	−	0.962954i	$-0.413087\pi$
$317$	964944.	0.539329	0.269665	+	0.962954i	$0.413087\pi$
$318$	0	0
$319$	2.49807e6	1.37445
$320$	0	0
$321$	−1.20283e6	−0.651541
$322$	0	0
$323$	220356.	0.117522
$324$	0	0
$325$	4.76084e6	2.50020
$326$	0	0
$327$	−1.35446e6	−0.700479
$328$	0	0
$329$	−88373.0	−0.0450122
$330$	0	0
$331$	3.21355e6	1.61218	0.806092	−	0.591790i	$-0.201578\pi$
$331$	3.21355e6	1.61218	0.806092	+	0.591790i	$0.201578\pi$
$332$	0	0
$333$	8.42597e6	4.16398
$334$	0	0
$335$	2.77873e6	1.35280
$336$	0	0
$337$	835398.	0.400699	0.200350	−	0.979724i	$-0.435792\pi$
$337$	835398.	0.400699	0.200350	+	0.979724i	$0.435792\pi$
$338$	0	0
$339$	−952834.	−0.450316
$340$	0	0
$341$	3.07700e6	1.43298
$342$	0	0
$343$	523886.	0.240437
$344$	0	0
$345$	−1.59312e6	−0.720612
$346$	0	0
$347$	−2.19943e6	−0.980589	−0.490295	−	0.871557i	$-0.663111\pi$
$347$	−2.19943e6	−0.980589	−0.490295	+	0.871557i	$0.663111\pi$
$348$	0	0
$349$	2.90995e6	1.27886	0.639428	−	0.768851i	$-0.279171\pi$
$349$	2.90995e6	1.27886	0.639428	+	0.768851i	$0.279171\pi$
$350$	0	0
$351$	5.06927e6	2.19623
$352$	0	0
$353$	2.63861e6	1.12704	0.563518	−	0.826104i	$-0.309448\pi$
$353$	2.63861e6	1.12704	0.563518	+	0.826104i	$0.309448\pi$
$354$	0	0
$355$	1.30292e6	0.548714
$356$	0	0
$357$	−74534.4	−0.0309518
$358$	0	0
$359$	37197.0	0.0152325	0.00761626	−	0.999971i	$-0.497576\pi$
$359$	37197.0	0.0152325	0.00761626	+	0.999971i	$0.497576\pi$
$360$	0	0
$361$	−835506.	−0.337428
$362$	0	0
$363$	−144371.	−0.0575059
$364$	0	0
$365$	−6.42765e6	−2.52534
$366$	0	0
$367$	−2.27179e6	−0.880447	−0.440224	−	0.897888i	$-0.645101\pi$
$367$	−2.27179e6	−0.880447	−0.440224	+	0.897888i	$0.645101\pi$
$368$	0	0
$369$	3.86500e6	1.47769
$370$	0	0
$371$	−352635.	−0.133012
$372$	0	0
$373$	2.42341e6	0.901892	0.450946	−	0.892551i	$-0.351087\pi$
$373$	2.42341e6	0.901892	0.450946	+	0.892551i	$0.351087\pi$
$374$	0	0
$375$	1.12194e7	4.11996
$376$	0	0
$377$	−4.22066e6	−1.52942
$378$	0	0
$379$	2.23180e6	0.798101	0.399050	−	0.916929i	$-0.369340\pi$
$379$	2.23180e6	0.798101	0.399050	+	0.916929i	$0.369340\pi$
$380$	0	0
$381$	276182.	0.0974725
$382$	0	0
$383$	1.13758e6	0.396263	0.198131	−	0.980175i	$-0.436513\pi$
$383$	1.13758e6	0.396263	0.198131	+	0.980175i	$0.436513\pi$
$384$	0	0
$385$	627868.	0.215882
$386$	0	0
$387$	−958610.	−0.325360
$388$	0	0
$389$	1.97034e6	0.660188	0.330094	−	0.943948i	$-0.392920\pi$
$389$	1.97034e6	0.660188	0.330094	+	0.943948i	$0.392920\pi$
$390$	0	0
$391$	−98041.1	−0.0324315
$392$	0	0
$393$	−9.27268e6	−3.02847
$394$	0	0
$395$	193626.	0.0624412
$396$	0	0
$397$	−5.13128e6	−1.63399	−0.816996	−	0.576644i	$-0.804362\pi$
$397$	−5.13128e6	−1.63399	−0.816996	+	0.576644i	$0.804362\pi$
$398$	0	0
$399$	−554925.	−0.174503
$400$	0	0
$401$	−2.92332e6	−0.907852	−0.453926	−	0.891039i	$-0.649977\pi$
$401$	−2.92332e6	−0.907852	−0.453926	+	0.891039i	$0.649977\pi$
$402$	0	0
$403$	−5.19880e6	−1.59456
$404$	0	0
$405$	8.48520e6	2.57054
$406$	0	0
$407$	−6.41542e6	−1.91973
$408$	0	0
$409$	−2.10623e6	−0.622582	−0.311291	−	0.950315i	$-0.600761\pi$
$409$	−2.10623e6	−0.622582	−0.311291	+	0.950315i	$0.600761\pi$
$410$	0	0
$411$	−33592.5	−0.00980929
$412$	0	0
$413$	−142270.	−0.0410428
$414$	0	0
$415$	5.28068e6	1.50511
$416$	0	0
$417$	4.34967e6	1.22494
$418$	0	0
$419$	787697.	0.219192	0.109596	−	0.993976i	$-0.465044\pi$
$419$	787697.	0.219192	0.109596	+	0.993976i	$0.465044\pi$
$420$	0	0
$421$	−2.73861e6	−0.753051	−0.376526	−	0.926406i	$-0.622881\pi$
$421$	−2.73861e6	−0.753051	−0.376526	+	0.926406i	$0.622881\pi$
$422$	0	0
$423$	2.91817e6	0.792977
$424$	0	0
$425$	1.22806e6	0.329798
$426$	0	0
$427$	287018.	0.0761796
$428$	0	0
$429$	−7.26468e6	−1.90578
$430$	0	0
$431$	−2.23642e6	−0.579911	−0.289955	−	0.957040i	$-0.593640\pi$
$431$	−2.23642e6	−0.579911	−0.289955	+	0.957040i	$0.593640\pi$
$432$	0	0
$433$	−7.32309e6	−1.87704	−0.938522	−	0.345219i	$-0.887804\pi$
$433$	−7.32309e6	−1.87704	−0.938522	+	0.345219i	$0.887804\pi$
$434$	0	0
$435$	−1.76913e7	−4.48266
$436$	0	0
$437$	−729937.	−0.182844
$438$	0	0
$439$	−4.05522e6	−1.00428	−0.502138	−	0.864788i	$-0.667453\pi$
$439$	−4.05522e6	−1.00428	−0.502138	+	0.864788i	$0.667453\pi$
$440$	0	0
$441$	−8.58575e6	−2.10224
$442$	0	0
$443$	−2.23385e6	−0.540810	−0.270405	−	0.962747i	$-0.587158\pi$
$443$	−2.23385e6	−0.540810	−0.270405	+	0.962747i	$0.587158\pi$
$444$	0	0
$445$	6.70510e6	1.60511
$446$	0	0
$447$	−1.38425e7	−3.27676
$448$	0	0
$449$	−3.90186e6	−0.913389	−0.456695	−	0.889623i	$-0.650967\pi$
$449$	−3.90186e6	−0.913389	−0.456695	+	0.889623i	$0.650967\pi$
$450$	0	0
$451$	−2.94276e6	−0.681262
$452$	0	0
$453$	1.18894e7	2.72218
$454$	0	0
$455$	−1.06083e6	−0.240223
$456$	0	0
$457$	−184098.	−0.0412343	−0.0206172	−	0.999787i	$-0.506563\pi$
$457$	−184098.	−0.0412343	−0.0206172	+	0.999787i	$0.506563\pi$
$458$	0	0
$459$	1.30762e6	0.289702
$460$	0	0
$461$	−4.89953e6	−1.07375	−0.536873	−	0.843663i	$-0.680395\pi$
$461$	−4.89953e6	−1.07375	−0.536873	+	0.843663i	$0.680395\pi$
$462$	0	0
$463$	2.72830e6	0.591479	0.295740	−	0.955269i	$-0.404434\pi$
$463$	2.72830e6	0.591479	0.295740	+	0.955269i	$0.404434\pi$
$464$	0	0
$465$	−2.17912e7	−4.67357
$466$	0	0
$467$	−3.18682e6	−0.676185	−0.338093	−	0.941113i	$-0.609782\pi$
$467$	−3.18682e6	−0.676185	−0.338093	+	0.941113i	$0.609782\pi$
$468$	0	0
$469$	−430641.	−0.0904031
$470$	0	0
$471$	4.35859e6	0.905302
$472$	0	0
$473$	729873.	0.150001
$474$	0	0
$475$	9.14319e6	1.85936
$476$	0	0
$477$	1.16444e7	2.34326
$478$	0	0
$479$	−4.93825e6	−0.983410	−0.491705	−	0.870762i	$-0.663626\pi$
$479$	−4.93825e6	−0.983410	−0.491705	+	0.870762i	$0.663626\pi$
$480$	0	0
$481$	1.08393e7	2.13618
$482$	0	0
$483$	246898.	0.0481560
$484$	0	0
$485$	3.79884e6	0.733325
$486$	0	0
$487$	−534531.	−0.102129	−0.0510646	−	0.998695i	$-0.516261\pi$
$487$	−534531.	−0.102129	−0.0510646	+	0.998695i	$0.516261\pi$
$488$	0	0
$489$	−1.11991e7	−2.11792
$490$	0	0
$491$	−4.65073e6	−0.870597	−0.435299	−	0.900286i	$-0.643357\pi$
$491$	−4.65073e6	−0.870597	−0.435299	+	0.900286i	$0.643357\pi$
$492$	0	0
$493$	−1.08872e6	−0.201744
$494$	0	0
$495$	−2.07329e7	−3.80318
$496$	0	0
$497$	−201923.	−0.0366687
$498$	0	0
$499$	215791.	0.0387955	0.0193977	−	0.999812i	$-0.493825\pi$
$499$	215791.	0.0387955	0.0193977	+	0.999812i	$0.493825\pi$
$500$	0	0
$501$	−4.60250e6	−0.819219
$502$	0	0
$503$	6.45815e6	1.13812	0.569060	−	0.822296i	$-0.307307\pi$
$503$	6.45815e6	1.13812	0.569060	+	0.822296i	$0.307307\pi$
$504$	0	0
$505$	−3.60026e6	−0.628211
$506$	0	0
$507$	2.02858e6	0.350487
$508$	0	0
$509$	3.88028e6	0.663847	0.331924	−	0.943306i	$-0.392302\pi$
$509$	3.88028e6	0.663847	0.331924	+	0.943306i	$0.392302\pi$
$510$	0	0
$511$	996142.	0.168760
$512$	0	0
$513$	9.73553e6	1.63330
$514$	0	0
$515$	−1.80283e7	−2.99528
$516$	0	0
$517$	−2.22186e6	−0.365587
$518$	0	0
$519$	−7.94030e6	−1.29395
$520$	0	0
$521$	2.45313e6	0.395937	0.197969	−	0.980208i	$-0.436566\pi$
$521$	2.45313e6	0.395937	0.197969	+	0.980208i	$0.436566\pi$
$522$	0	0
$523$	1.91833e6	0.306668	0.153334	−	0.988174i	$-0.450999\pi$
$523$	1.91833e6	0.306668	0.153334	+	0.988174i	$0.450999\pi$
$524$	0	0
$525$	−3.09265e6	−0.489703
$526$	0	0
$527$	−1.34104e6	−0.210336
$528$	0	0
$529$	−6.11158e6	−0.949542
$530$	0	0
$531$	4.69790e6	0.723049
$532$	0	0
$533$	4.97200e6	0.758076
$534$	0	0
$535$	−4.41600e6	−0.667029
$536$	0	0
$537$	−1.44202e7	−2.15792
$538$	0	0
$539$	6.53708e6	0.969196
$540$	0	0
$541$	−6.89363e6	−1.01264	−0.506320	−	0.862346i	$-0.668995\pi$
$541$	−6.89363e6	−1.01264	−0.506320	+	0.862346i	$0.668995\pi$
$542$	0	0
$543$	2.24256e6	0.326396
$544$	0	0
$545$	−4.97266e6	−0.717130
$546$	0	0
$547$	−4.78180e6	−0.683319	−0.341659	−	0.939824i	$-0.610989\pi$
$547$	−4.78180e6	−0.683319	−0.341659	+	0.939824i	$0.610989\pi$
$548$	0	0
$549$	−9.47764e6	−1.34205
$550$	0	0
$551$	−8.10578e6	−1.13741
$552$	0	0
$553$	−30007.7	−0.00417273
$554$	0	0
$555$	4.54338e7	6.26104
$556$	0	0
$557$	−1.16813e7	−1.59534	−0.797672	−	0.603091i	$-0.793935\pi$
$557$	−1.16813e7	−1.59534	−0.797672	+	0.603091i	$0.793935\pi$
$558$	0	0
$559$	−1.23317e6	−0.166914
$560$	0	0
$561$	−1.87393e6	−0.251389
$562$	0	0
$563$	−9.15272e6	−1.21697	−0.608484	−	0.793566i	$-0.708222\pi$
$563$	−9.15272e6	−1.21697	−0.608484	+	0.793566i	$0.708222\pi$
$564$	0	0
$565$	−3.49817e6	−0.461021
$566$	0	0
$567$	−1.31502e6	−0.171780
$568$	0	0
$569$	−4.46016e6	−0.577523	−0.288762	−	0.957401i	$-0.593243\pi$
$569$	−4.46016e6	−0.577523	−0.288762	+	0.957401i	$0.593243\pi$
$570$	0	0
$571$	−3.82289e6	−0.490684	−0.245342	−	0.969437i	$-0.578900\pi$
$571$	−3.82289e6	−0.490684	−0.245342	+	0.969437i	$0.578900\pi$
$572$	0	0
$573$	1.67994e7	2.13751
$574$	0	0
$575$	−4.06801e6	−0.513112
$576$	0	0
$577$	1.15777e7	1.44772	0.723858	−	0.689949i	$-0.242367\pi$
$577$	1.15777e7	1.44772	0.723858	+	0.689949i	$0.242367\pi$
$578$	0	0
$579$	5.10383e6	0.632702
$580$	0	0
$581$	−818387.	−0.100582
$582$	0	0
$583$	−8.86588e6	−1.08032
$584$	0	0
$585$	3.50296e7	4.23200
$586$	0	0
$587$	−2.16236e6	−0.259020	−0.129510	−	0.991578i	$-0.541340\pi$
$587$	−2.16236e6	−0.259020	−0.129510	+	0.991578i	$0.541340\pi$
$588$	0	0
$589$	−9.98429e6	−1.18585
$590$	0	0
$591$	5.54505e6	0.653036
$592$	0	0
$593$	1.42792e7	1.66751	0.833753	−	0.552137i	$-0.186188\pi$
$593$	1.42792e7	1.66751	0.833753	+	0.552137i	$0.186188\pi$
$594$	0	0
$595$	−273641.	−0.0316876
$596$	0	0
$597$	6.44751e6	0.740382
$598$	0	0
$599$	−1.13983e7	−1.29799	−0.648997	−	0.760791i	$-0.724811\pi$
$599$	−1.13983e7	−1.29799	−0.648997	+	0.760791i	$0.724811\pi$
$600$	0	0
$601$	−665939.	−0.0752053	−0.0376026	−	0.999293i	$-0.511972\pi$
$601$	−665939.	−0.0752053	−0.0376026	+	0.999293i	$0.511972\pi$
$602$	0	0
$603$	1.42202e7	1.59263
$604$	0	0
$605$	−530033.	−0.0588728
$606$	0	0
$607$	1.35927e7	1.49738	0.748692	−	0.662918i	$-0.230682\pi$
$607$	1.35927e7	1.49738	0.748692	+	0.662918i	$0.230682\pi$
$608$	0	0
$609$	2.74175e6	0.299561
$610$	0	0
$611$	3.75398e6	0.406808
$612$	0	0
$613$	−1.08635e7	−1.16767	−0.583833	−	0.811874i	$-0.698448\pi$
$613$	−1.08635e7	−1.16767	−0.583833	+	0.811874i	$0.698448\pi$
$614$	0	0
$615$	2.08406e7	2.22189
$616$	0	0
$617$	−1.62403e6	−0.171744	−0.0858720	−	0.996306i	$-0.527368\pi$
$617$	−1.62403e6	−0.171744	−0.0858720	+	0.996306i	$0.527368\pi$
$618$	0	0
$619$	−3.82440e6	−0.401177	−0.200589	−	0.979676i	$-0.564285\pi$
$619$	−3.82440e6	−0.401177	−0.200589	+	0.979676i	$0.564285\pi$
$620$	0	0
$621$	−4.33156e6	−0.450729
$622$	0	0
$623$	−1.03914e6	−0.107264
$624$	0	0
$625$	1.88830e7	1.93362
$626$	0	0
$627$	−1.39518e7	−1.41730
$628$	0	0
$629$	2.79601e6	0.281781
$630$	0	0
$631$	4.03571e6	0.403503	0.201751	−	0.979437i	$-0.435337\pi$
$631$	4.03571e6	0.403503	0.201751	+	0.979437i	$0.435337\pi$
$632$	0	0
$633$	−5.53303e6	−0.548850
$634$	0	0
$635$	1.01396e6	0.0997894
$636$	0	0
$637$	−1.10448e7	−1.07848
$638$	0	0
$639$	6.66773e6	0.645990
$640$	0	0
$641$	1.03975e7	0.999505	0.499753	−	0.866168i	$-0.333424\pi$
$641$	1.03975e7	0.999505	0.499753	+	0.866168i	$0.333424\pi$
$642$	0	0
$643$	703463.	0.0670986	0.0335493	−	0.999437i	$-0.489319\pi$
$643$	703463.	0.0670986	0.0335493	+	0.999437i	$0.489319\pi$
$644$	0	0
$645$	−5.16894e6	−0.489218
$646$	0	0
$647$	1.39550e7	1.31060	0.655300	−	0.755369i	$-0.272542\pi$
$647$	1.39550e7	1.31060	0.655300	+	0.755369i	$0.272542\pi$
$648$	0	0
$649$	−3.57692e6	−0.333348
$650$	0	0
$651$	3.37715e6	0.312319
$652$	0	0
$653$	4.10881e6	0.377079	0.188540	−	0.982066i	$-0.439625\pi$
$653$	4.10881e6	0.377079	0.188540	+	0.982066i	$0.439625\pi$
$654$	0	0
$655$	−3.40431e7	−3.10046
$656$	0	0
$657$	−3.28937e7	−2.97303
$658$	0	0
$659$	−4.28147e6	−0.384042	−0.192021	−	0.981391i	$-0.561504\pi$
$659$	−4.28147e6	−0.384042	−0.192021	+	0.981391i	$0.561504\pi$
$660$	0	0
$661$	−6.69686e6	−0.596166	−0.298083	−	0.954540i	$-0.596347\pi$
$661$	−6.69686e6	−0.596166	−0.298083	+	0.954540i	$0.596347\pi$
$662$	0	0
$663$	3.16614e6	0.279734
$664$	0	0
$665$	−2.03732e6	−0.178650
$666$	0	0
$667$	3.60645e6	0.313881
$668$	0	0
$669$	−9.57663e6	−0.827271
$670$	0	0
$671$	7.21615e6	0.618727
$672$	0	0
$673$	−1.29636e7	−1.10329	−0.551644	−	0.834080i	$-0.685999\pi$
$673$	−1.29636e7	−1.10329	−0.551644	+	0.834080i	$0.685999\pi$
$674$	0	0
$675$	5.42571e7	4.58350
$676$	0	0
$677$	1.00392e7	0.841833	0.420916	−	0.907099i	$-0.361709\pi$
$677$	1.00392e7	0.841833	0.420916	+	0.907099i	$0.361709\pi$
$678$	0	0
$679$	−588735.	−0.0490055
$680$	0	0
$681$	5.85308e6	0.483633
$682$	0	0
$683$	−1.37778e7	−1.13013	−0.565063	−	0.825048i	$-0.691148\pi$
$683$	−1.37778e7	−1.13013	−0.565063	+	0.825048i	$0.691148\pi$
$684$	0	0
$685$	−123329.	−0.0100425
$686$	0	0
$687$	−1.29867e7	−1.04980
$688$	0	0
$689$	1.49795e7	1.20212
$690$	0	0
$691$	2.12787e7	1.69532	0.847658	−	0.530544i	$-0.178012\pi$
$691$	2.12787e7	1.69532	0.847658	+	0.530544i	$0.178012\pi$
$692$	0	0
$693$	3.21313e6	0.254153
$694$	0	0
$695$	1.59691e7	1.25406
$696$	0	0
$697$	1.28253e6	0.0999969
$698$	0	0
$699$	−2.62767e7	−2.03412
$700$	0	0
$701$	2.07790e7	1.59709	0.798547	−	0.601933i	$-0.205602\pi$
$701$	2.07790e7	1.59709	0.798547	+	0.601933i	$0.205602\pi$
$702$	0	0
$703$	2.08169e7	1.58864
$704$	0	0
$705$	1.57351e7	1.19233
$706$	0	0
$707$	557960.	0.0419812
$708$	0	0
$709$	4.16247e6	0.310982	0.155491	−	0.987837i	$-0.450304\pi$
$709$	4.16247e6	0.310982	0.155491	+	0.987837i	$0.450304\pi$
$710$	0	0
$711$	990888.	0.0735107
$712$	0	0
$713$	4.44224e6	0.327249
$714$	0	0
$715$	−2.66711e7	−1.95108
$716$	0	0
$717$	4.51692e7	3.28129
$718$	0	0
$719$	1.69124e7	1.22007	0.610033	−	0.792376i	$-0.291156\pi$
$719$	1.69124e7	1.22007	0.610033	+	0.792376i	$0.291156\pi$
$720$	0	0
$721$	2.79399e6	0.200164
$722$	0	0
$723$	−1.92041e7	−1.36631
$724$	0	0
$725$	−4.51743e7	−3.19188
$726$	0	0
$727$	−1.27084e7	−0.891776	−0.445888	−	0.895089i	$-0.647112\pi$
$727$	−1.27084e7	−0.891776	−0.445888	+	0.895089i	$0.647112\pi$
$728$	0	0
$729$	−7.54333e6	−0.525708
$730$	0	0
$731$	−318098.	−0.0220174
$732$	0	0
$733$	1.32015e6	0.0907535	0.0453767	−	0.998970i	$-0.485551\pi$
$733$	1.32015e6	0.0907535	0.0453767	+	0.998970i	$0.485551\pi$
$734$	0	0
$735$	−4.62954e7	−3.16096
$736$	0	0
$737$	−1.08271e7	−0.734249
$738$	0	0
$739$	−2.42511e7	−1.63351	−0.816753	−	0.576988i	$-0.804228\pi$
$739$	−2.42511e7	−1.63351	−0.816753	+	0.576988i	$0.804228\pi$
$740$	0	0
$741$	2.35725e7	1.57711
$742$	0	0
$743$	1.31970e7	0.877009	0.438504	−	0.898729i	$-0.355508\pi$
$743$	1.31970e7	0.877009	0.438504	+	0.898729i	$0.355508\pi$
$744$	0	0
$745$	−5.08203e7	−3.35465
$746$	0	0
$747$	2.70240e7	1.77194
$748$	0	0
$749$	684381.	0.0445752
$750$	0	0
$751$	1.06321e7	0.687891	0.343945	−	0.938990i	$-0.388237\pi$
$751$	1.06321e7	0.687891	0.343945	+	0.938990i	$0.388237\pi$
$752$	0	0
$753$	−1.81539e7	−1.16676
$754$	0	0
$755$	4.36502e7	2.78688
$756$	0	0
$757$	2.44667e7	1.55180	0.775900	−	0.630856i	$-0.217296\pi$
$757$	2.44667e7	1.55180	0.775900	+	0.630856i	$0.217296\pi$
$758$	0	0
$759$	6.20748e6	0.391121
$760$	0	0
$761$	2.20641e7	1.38110	0.690550	−	0.723285i	$-0.257369\pi$
$761$	2.20641e7	1.38110	0.690550	+	0.723285i	$0.257369\pi$
$762$	0	0
$763$	770652.	0.0479233
$764$	0	0
$765$	9.03593e6	0.558238
$766$	0	0
$767$	6.04345e6	0.370934
$768$	0	0
$769$	2.58590e7	1.57687	0.788434	−	0.615119i	$-0.210892\pi$
$769$	2.58590e7	1.57687	0.788434	+	0.615119i	$0.210892\pi$
$770$	0	0
$771$	−1.65517e7	−1.00278
$772$	0	0
$773$	−1.57927e7	−0.950620	−0.475310	−	0.879818i	$-0.657664\pi$
$773$	−1.57927e7	−0.950620	−0.475310	+	0.879818i	$0.657664\pi$
$774$	0	0
$775$	−5.56435e7	−3.32782
$776$	0	0
$777$	−7.04122e6	−0.418404
$778$	0	0
$779$	9.54872e6	0.563770
$780$	0	0
$781$	−5.07672e6	−0.297821
$782$	0	0
$783$	−4.81010e7	−2.80382
$784$	0	0
$785$	1.60018e7	0.926821
$786$	0	0
$787$	2.14880e7	1.23669	0.618344	−	0.785908i	$-0.287804\pi$
$787$	2.14880e7	1.23669	0.618344	+	0.785908i	$0.287804\pi$
$788$	0	0
$789$	4.49425e7	2.57019
$790$	0	0
$791$	542139.	0.0308084
$792$	0	0
$793$	−1.21922e7	−0.688491
$794$	0	0
$795$	6.27879e7	3.52337
$796$	0	0
$797$	−1.54431e7	−0.861170	−0.430585	−	0.902550i	$-0.641693\pi$
$797$	−1.54431e7	−0.861170	−0.430585	+	0.902550i	$0.641693\pi$
$798$	0	0
$799$	968344.	0.0536615
$800$	0	0
$801$	3.43136e7	1.88966
$802$	0	0
$803$	2.50448e7	1.37066
$804$	0	0
$805$	906448.	0.0493007
$806$	0	0
$807$	5.50244e7	2.97421
$808$	0	0
$809$	−5.05884e6	−0.271756	−0.135878	−	0.990726i	$-0.543386\pi$
$809$	−5.05884e6	−0.271756	−0.135878	+	0.990726i	$0.543386\pi$
$810$	0	0
$811$	−2.95840e7	−1.57944	−0.789722	−	0.613465i	$-0.789775\pi$
$811$	−2.95840e7	−1.57944	−0.789722	+	0.613465i	$0.789775\pi$
$812$	0	0
$813$	5.22105e7	2.77033
$814$	0	0
$815$	−4.11156e7	−2.16826
$816$	0	0
$817$	−2.36830e6	−0.124132
$818$	0	0
$819$	−5.42881e6	−0.282810
$820$	0	0
$821$	4.56831e6	0.236536	0.118268	−	0.992982i	$-0.462266\pi$
$821$	4.56831e6	0.236536	0.118268	+	0.992982i	$0.462266\pi$
$822$	0	0
$823$	−3.99029e6	−0.205355	−0.102677	−	0.994715i	$-0.532741\pi$
$823$	−3.99029e6	−0.205355	−0.102677	+	0.994715i	$0.532741\pi$
$824$	0	0
$825$	−7.77549e7	−3.97734
$826$	0	0
$827$	−7.46365e6	−0.379479	−0.189739	−	0.981835i	$-0.560764\pi$
$827$	−7.46365e6	−0.379479	−0.189739	+	0.981835i	$0.560764\pi$
$828$	0	0
$829$	−1.22180e7	−0.617467	−0.308734	−	0.951149i	$-0.599905\pi$
$829$	−1.22180e7	−0.617467	−0.308734	+	0.951149i	$0.599905\pi$
$830$	0	0
$831$	−5.83557e7	−2.93144
$832$	0	0
$833$	−2.84903e6	−0.142260
$834$	0	0
$835$	−1.68973e7	−0.838692
$836$	0	0
$837$	−5.92484e7	−2.92323
$838$	0	0
$839$	1.83694e7	0.900929	0.450464	−	0.892794i	$-0.351258\pi$
$839$	1.83694e7	0.900929	0.450464	+	0.892794i	$0.351258\pi$
$840$	0	0
$841$	1.95376e7	0.952536
$842$	0	0
$843$	−2.23373e6	−0.108258
$844$	0	0
$845$	7.44760e6	0.358818
$846$	0	0
$847$	82143.3	0.00393426
$848$	0	0
$849$	−2.79138e7	−1.32907
$850$	0	0
$851$	−9.26189e6	−0.438405
$852$	0	0
$853$	−3.51699e7	−1.65500	−0.827501	−	0.561465i	$-0.810238\pi$
$853$	−3.51699e7	−1.65500	−0.827501	+	0.561465i	$0.810238\pi$
$854$	0	0
$855$	6.72744e7	3.14727
$856$	0	0
$857$	−1.23215e7	−0.573076	−0.286538	−	0.958069i	$-0.592504\pi$
$857$	−1.23215e7	−0.573076	−0.286538	+	0.958069i	$0.592504\pi$
$858$	0	0
$859$	−2.32476e6	−0.107497	−0.0537484	−	0.998555i	$-0.517117\pi$
$859$	−2.32476e6	−0.107497	−0.0537484	+	0.998555i	$0.517117\pi$
$860$	0	0
$861$	−3.22982e6	−0.148481
$862$	0	0
$863$	−1.53407e7	−0.701163	−0.350582	−	0.936532i	$-0.614016\pi$
$863$	−1.53407e7	−0.701163	−0.350582	+	0.936532i	$0.614016\pi$
$864$	0	0
$865$	−2.91515e7	−1.32471
$866$	0	0
$867$	−3.83633e7	−1.73328
$868$	0	0
$869$	−754449.	−0.0338907
$870$	0	0
$871$	1.82931e7	0.817038
$872$	0	0
$873$	1.94407e7	0.863328
$874$	0	0
$875$	−6.38358e6	−0.281867
$876$	0	0
$877$	1.21207e7	0.532142	0.266071	−	0.963953i	$-0.414274\pi$
$877$	1.21207e7	0.532142	0.266071	+	0.963953i	$0.414274\pi$
$878$	0	0
$879$	2.17000e7	0.947299
$880$	0	0
$881$	208397.	0.00904589	0.00452294	−	0.999990i	$-0.498560\pi$
$881$	208397.	0.00904589	0.00452294	+	0.999990i	$0.498560\pi$
$882$	0	0
$883$	3.57371e7	1.54247	0.771237	−	0.636548i	$-0.219638\pi$
$883$	3.57371e7	1.54247	0.771237	+	0.636548i	$0.219638\pi$
$884$	0	0
$885$	2.53316e7	1.08719
$886$	0	0
$887$	−4.64203e7	−1.98107	−0.990533	−	0.137276i	$-0.956165\pi$
$887$	−4.64203e7	−1.98107	−0.990533	+	0.137276i	$0.956165\pi$
$888$	0	0
$889$	−157140.	−0.00666858
$890$	0	0
$891$	−3.30619e7	−1.39519
$892$	0	0
$893$	7.20952e6	0.302537
$894$	0	0
$895$	−5.29415e7	−2.20922
$896$	0	0
$897$	−1.04880e7	−0.435221
$898$	0	0
$899$	4.93300e7	2.03569
$900$	0	0
$901$	3.86398e6	0.158571
$902$	0	0
$903$	801070.	0.0326927
$904$	0	0
$905$	8.23320e6	0.334155
$906$	0	0
$907$	2.81640e7	1.13678	0.568390	−	0.822759i	$-0.307566\pi$
$907$	2.81640e7	1.13678	0.568390	+	0.822759i	$0.307566\pi$
$908$	0	0
$909$	−1.84245e7	−0.739580
$910$	0	0
$911$	−8.25253e6	−0.329451	−0.164726	−	0.986339i	$-0.552674\pi$
$911$	−8.25253e6	−0.329451	−0.164726	+	0.986339i	$0.552674\pi$
$912$	0	0
$913$	−2.05758e7	−0.816919
$914$	0	0
$915$	−5.11045e7	−2.01793
$916$	0	0
$917$	5.27593e6	0.207193
$918$	0	0
$919$	1.14244e6	0.0446217	0.0223108	−	0.999751i	$-0.492898\pi$
$919$	1.14244e6	0.0446217	0.0223108	+	0.999751i	$0.492898\pi$
$920$	0	0
$921$	−2.93346e6	−0.113954
$922$	0	0
$923$	8.57746e6	0.331401
$924$	0	0
$925$	1.16014e8	4.45818
$926$	0	0
$927$	−9.22605e7	−3.52628
$928$	0	0
$929$	−3.37164e7	−1.28175	−0.640873	−	0.767647i	$-0.721428\pi$
$929$	−3.37164e7	−1.28175	−0.640873	+	0.767647i	$0.721428\pi$
$930$	0	0
$931$	−2.12116e7	−0.802046
$932$	0	0
$933$	2.27979e6	0.0857413
$934$	0	0
$935$	−6.87984e6	−0.257365
$936$	0	0
$937$	−4.06327e7	−1.51191	−0.755955	−	0.654623i	$-0.772827\pi$
$937$	−4.06327e7	−1.51191	−0.755955	+	0.654623i	$0.772827\pi$
$938$	0	0
$939$	3.81992e7	1.41381
$940$	0	0
$941$	4.51496e7	1.66219	0.831093	−	0.556134i	$-0.187716\pi$
$941$	4.51496e7	1.66219	0.831093	+	0.556134i	$0.187716\pi$
$942$	0	0
$943$	−4.24844e6	−0.155579
$944$	0	0
$945$	−1.20897e7	−0.440390
$946$	0	0
$947$	1.85178e7	0.670986	0.335493	−	0.942043i	$-0.391097\pi$
$947$	1.85178e7	0.670986	0.335493	+	0.942043i	$0.391097\pi$
$948$	0	0
$949$	−4.23149e7	−1.52520
$950$	0	0
$951$	2.66270e7	0.954709
$952$	0	0
$953$	−3.17478e7	−1.13235	−0.566176	−	0.824284i	$-0.691578\pi$
$953$	−3.17478e7	−1.13235	−0.566176	+	0.824284i	$0.691578\pi$
$954$	0	0
$955$	6.16763e7	2.18832
$956$	0	0
$957$	6.89326e7	2.43302
$958$	0	0
$959$	19113.3	0.000671103 0
$960$	0	0
$961$	3.21331e7	1.12239
$962$	0	0
$963$	−2.25990e7	−0.785279
$964$	0	0
$965$	1.87379e7	0.647742
$966$	0	0
$967$	−4.45707e7	−1.53279	−0.766396	−	0.642368i	$-0.777952\pi$
$967$	−4.45707e7	−1.53279	−0.766396	+	0.642368i	$0.777952\pi$
$968$	0	0
$969$	6.08057e6	0.208034
$970$	0	0
$971$	−2.22017e7	−0.755679	−0.377840	−	0.925871i	$-0.623333\pi$
$971$	−2.22017e7	−0.755679	−0.377840	+	0.925871i	$0.623333\pi$
$972$	0	0
$973$	−2.47485e6	−0.0838045
$974$	0	0
$975$	1.31372e8	4.42580
$976$	0	0
$977$	−3.20865e6	−0.107544	−0.0537719	−	0.998553i	$-0.517124\pi$
$977$	−3.20865e6	−0.107544	−0.0537719	+	0.998553i	$0.517124\pi$
$978$	0	0
$979$	−2.61259e7	−0.871193
$980$	0	0
$981$	−2.54478e7	−0.844262
$982$	0	0
$983$	4.20453e7	1.38782	0.693910	−	0.720061i	$-0.255886\pi$
$983$	4.20453e7	1.38782	0.693910	+	0.720061i	$0.255886\pi$
$984$	0	0
$985$	2.03578e7	0.668558
$986$	0	0
$987$	−2.43860e6	−0.0796795
$988$	0	0
$989$	1.05371e6	0.0342556
$990$	0	0
$991$	−1.08440e7	−0.350755	−0.175377	−	0.984501i	$-0.556115\pi$
$991$	−1.08440e7	−0.350755	−0.175377	+	0.984501i	$0.556115\pi$
$992$	0	0
$993$	8.86757e7	2.85385
$994$	0	0
$995$	2.36710e7	0.757981
$996$	0	0
$997$	3.29287e7	1.04915	0.524574	−	0.851365i	$-0.324225\pi$
$997$	3.29287e7	1.04915	0.524574	+	0.851365i	$0.324225\pi$
$998$	0	0
$999$	1.23530e8	3.91616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.10		10
4.3	odd	2		43.6.a.b.1.6	✓	10
12.11	even	2		387.6.a.e.1.5		10
20.19	odd	2		1075.6.a.b.1.5		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.6	✓	10	4.3	odd	2
387.6.a.e.1.5		10	12.11	even	2
688.6.a.h.1.10		10	1.1	even	1	trivial
1075.6.a.b.1.5		10	20.19	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q+27.5943 q^{3} +101.308 q^{5} -15.7005 q^{7} +518.448 q^{9} +O(q^{10})\)	\(q+27.5943 q^{3} +101.308 q^{5} -15.7005 q^{7} +518.448 q^{9} -394.739 q^{11} +666.939 q^{13} +2795.53 q^{15} +172.038 q^{17} +1280.86 q^{19} -433.245 q^{21} -569.882 q^{23} +7138.34 q^{25} +7600.80 q^{27} -6328.41 q^{29} -7795.01 q^{31} -10892.6 q^{33} -1590.59 q^{35} +16252.3 q^{37} +18403.7 q^{39} +7454.95 q^{41} -1849.00 q^{43} +52523.0 q^{45} +5628.68 q^{47} -16560.5 q^{49} +4747.27 q^{51} +22460.1 q^{53} -39990.3 q^{55} +35344.4 q^{57} +9061.48 q^{59} -18280.8 q^{61} -8139.89 q^{63} +67566.3 q^{65} +27428.5 q^{67} -15725.5 q^{69} +12860.9 q^{71} -63446.5 q^{73} +196978. q^{75} +6197.60 q^{77} +1911.26 q^{79} +83756.3 q^{81} +52124.9 q^{83} +17428.8 q^{85} -174628. q^{87} +66185.2 q^{89} -10471.3 q^{91} -215098. q^{93} +129761. q^{95} +37497.8 q^{97} -204652. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9} - 745 q^{11} + 1917 q^{13} - 1688 q^{15} + 4017 q^{17} + 2404 q^{19} - 228 q^{21} - 1733 q^{23} + 7120 q^{25} + 2324 q^{27} + 6996 q^{29} + 4899 q^{31} - 15734 q^{33} - 7084 q^{35} + 1466 q^{37} + 26542 q^{39} + 10297 q^{41} - 18490 q^{43} + 73822 q^{45} - 48592 q^{47} + 29458 q^{49} - 92972 q^{51} + 127165 q^{53} - 106672 q^{55} + 34060 q^{57} - 99372 q^{59} + 17408 q^{61} - 2244 q^{63} + 54484 q^{65} + 2021 q^{67} + 1654 q^{69} - 11286 q^{71} + 49892 q^{73} + 44662 q^{75} + 98144 q^{77} + 91524 q^{79} - 26450 q^{81} + 105203 q^{83} - 87212 q^{85} - 181200 q^{87} - 62682 q^{89} + 295304 q^{91} - 238430 q^{93} + 305340 q^{95} + 108383 q^{97} + 270499 q^{99}+O(q^{100}) \) 10 * q - 28 * q^3 + 138 * q^5 - 60 * q^7 + 1356 * q^9 - 745 * q^11 + 1917 * q^13 - 1688 * q^15 + 4017 * q^17 + 2404 * q^19 - 228 * q^21 - 1733 * q^23 + 7120 * q^25 + 2324 * q^27 + 6996 * q^29 + 4899 * q^31 - 15734 * q^33 - 7084 * q^35 + 1466 * q^37 + 26542 * q^39 + 10297 * q^41 - 18490 * q^43 + 73822 * q^45 - 48592 * q^47 + 29458 * q^49 - 92972 * q^51 + 127165 * q^53 - 106672 * q^55 + 34060 * q^57 - 99372 * q^59 + 17408 * q^61 - 2244 * q^63 + 54484 * q^65 + 2021 * q^67 + 1654 * q^69 - 11286 * q^71 + 49892 * q^73 + 44662 * q^75 + 98144 * q^77 + 91524 * q^79 - 26450 * q^81 + 105203 * q^83 - 87212 * q^85 - 181200 * q^87 - 62682 * q^89 + 295304 * q^91 - 238430 * q^93 + 305340 * q^95 + 108383 * q^97 + 270499 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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