LMFDB - Embedded newform 688.6.a.h.1.6

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			1.6
Root			$9.86547$ 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)

$f(q)$	$=$	$q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} +O(q^{10})$	$q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} -590.079 q^{11} +434.774 q^{13} +56.8015 q^{15} +1925.58 q^{17} -654.129 q^{19} -187.332 q^{21} -2805.03 q^{23} -1694.26 q^{25} +726.434 q^{27} -1456.23 q^{29} -4419.85 q^{31} +886.115 q^{33} -4718.58 q^{35} +3753.09 q^{37} -652.894 q^{39} +1972.85 q^{41} -1849.00 q^{43} +9106.20 q^{45} -2204.84 q^{47} -1245.12 q^{49} -2891.62 q^{51} +24984.2 q^{53} +22319.8 q^{55} +982.298 q^{57} +42756.2 q^{59} -21022.8 q^{61} -30032.3 q^{63} -16445.4 q^{65} +25272.1 q^{67} +4212.28 q^{69} -48082.8 q^{71} +58801.2 q^{73} +2544.25 q^{75} -73610.8 q^{77} -92704.7 q^{79} +57410.1 q^{81} +1849.64 q^{83} -72835.2 q^{85} +2186.80 q^{87} -70380.3 q^{89} +54236.8 q^{91} +6637.23 q^{93} +24742.5 q^{95} -88842.1 q^{97} +142059. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.50169	−0.0963333	−0.0481667	−	0.998839i	$-0.515338\pi$
$3$	−1.50169	−0.0963333	−0.0481667	+	0.998839i	$0.515338\pi$
$4$	0	0
$5$	−37.8251	−0.676636	−0.338318	−	0.941032i	$-0.609858\pi$
$5$	−37.8251	−0.676636	−0.338318	+	0.941032i	$0.609858\pi$
$6$	0	0
$7$	124.747	0.962246	0.481123	−	0.876653i	$-0.340229\pi$
$7$	124.747	0.962246	0.481123	+	0.876653i	$0.340229\pi$
$8$	0	0
$9$	−240.745	−0.990720
$10$	0	0
$11$	−590.079	−1.47038	−0.735188	−	0.677863i	$-0.762906\pi$
$11$	−590.079	−1.47038	−0.735188	+	0.677863i	$0.762906\pi$
$12$	0	0
$13$	434.774	0.713518	0.356759	−	0.934196i	$-0.383882\pi$
$13$	434.774	0.713518	0.356759	+	0.934196i	$0.383882\pi$
$14$	0	0
$15$	56.8015	0.0651826
$16$	0	0
$17$	1925.58	1.61599	0.807995	−	0.589189i	$-0.200553\pi$
$17$	1925.58	1.61599	0.807995	+	0.589189i	$0.200553\pi$
$18$	0	0
$19$	−654.129	−0.415700	−0.207850	−	0.978161i	$-0.566647\pi$
$19$	−654.129	−0.415700	−0.207850	+	0.978161i	$0.566647\pi$
$20$	0	0
$21$	−187.332	−0.0926963
$22$	0	0
$23$	−2805.03	−1.10565	−0.552825	−	0.833297i	$-0.686450\pi$
$23$	−2805.03	−1.10565	−0.552825	+	0.833297i	$0.686450\pi$
$24$	0	0
$25$	−1694.26	−0.542163
$26$	0	0
$27$	726.434	0.191773
$28$	0	0
$29$	−1456.23	−0.321540	−0.160770	−	0.986992i	$-0.551398\pi$
$29$	−1456.23	−0.321540	−0.160770	+	0.986992i	$0.551398\pi$
$30$	0	0
$31$	−4419.85	−0.826044	−0.413022	−	0.910721i	$-0.635527\pi$
$31$	−4419.85	−0.826044	−0.413022	+	0.910721i	$0.635527\pi$
$32$	0	0
$33$	886.115	0.141646
$34$	0	0
$35$	−4718.58	−0.651090
$36$	0	0
$37$	3753.09	0.450697	0.225348	−	0.974278i	$-0.427648\pi$
$37$	3753.09	0.450697	0.225348	+	0.974278i	$0.427648\pi$
$38$	0	0
$39$	−652.894	−0.0687356
$40$	0	0
$41$	1972.85	0.183288	0.0916442	−	0.995792i	$-0.470788\pi$
$41$	1972.85	0.183288	0.0916442	+	0.995792i	$0.470788\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	9106.20	0.670357
$46$	0	0
$47$	−2204.84	−0.145590	−0.0727950	−	0.997347i	$-0.523192\pi$
$47$	−2204.84	−0.145590	−0.0727950	+	0.997347i	$0.523192\pi$
$48$	0	0
$49$	−1245.12	−0.0740834
$50$	0	0
$51$	−2891.62	−0.155674
$52$	0	0
$53$	24984.2	1.22173	0.610865	−	0.791735i	$-0.290822\pi$
$53$	24984.2	1.22173	0.610865	+	0.791735i	$0.290822\pi$
$54$	0	0
$55$	22319.8	0.994910
$56$	0	0
$57$	982.298	0.0400457
$58$	0	0
$59$	42756.2	1.59907	0.799537	−	0.600616i	$-0.205078\pi$
$59$	42756.2	1.59907	0.799537	+	0.600616i	$0.205078\pi$
$60$	0	0
$61$	−21022.8	−0.723379	−0.361689	−	0.932299i	$-0.617800\pi$
$61$	−21022.8	−0.723379	−0.361689	+	0.932299i	$0.617800\pi$
$62$	0	0
$63$	−30032.3	−0.953316
$64$	0	0
$65$	−16445.4	−0.482792
$66$	0	0
$67$	25272.1	0.687788	0.343894	−	0.939008i	$-0.388254\pi$
$67$	25272.1	0.687788	0.343894	+	0.939008i	$0.388254\pi$
$68$	0	0
$69$	4212.28	0.106511
$70$	0	0
$71$	−48082.8	−1.13199	−0.565997	−	0.824407i	$-0.691509\pi$
$71$	−48082.8	−1.13199	−0.565997	+	0.824407i	$0.691509\pi$
$72$	0	0
$73$	58801.2	1.29145	0.645727	−	0.763568i	$-0.276554\pi$
$73$	58801.2	1.29145	0.645727	+	0.763568i	$0.276554\pi$
$74$	0	0
$75$	2544.25	0.0522284
$76$	0	0
$77$	−73610.8	−1.41486
$78$	0	0
$79$	−92704.7	−1.67122	−0.835611	−	0.549322i	$-0.814886\pi$
$79$	−92704.7	−1.67122	−0.835611	+	0.549322i	$0.814886\pi$
$80$	0	0
$81$	57410.1	0.972246
$82$	0	0
$83$	1849.64	0.0294708	0.0147354	−	0.999891i	$-0.495309\pi$
$83$	1849.64	0.0294708	0.0147354	+	0.999891i	$0.495309\pi$
$84$	0	0
$85$	−72835.2	−1.09344
$86$	0	0
$87$	2186.80	0.0309750
$88$	0	0
$89$	−70380.3	−0.941838	−0.470919	−	0.882176i	$-0.656078\pi$
$89$	−70380.3	−0.941838	−0.470919	+	0.882176i	$0.656078\pi$
$90$	0	0
$91$	54236.8	0.686579
$92$	0	0
$93$	6637.23	0.0795756
$94$	0	0
$95$	24742.5	0.281277
$96$	0	0
$97$	−88842.1	−0.958714	−0.479357	−	0.877620i	$-0.659130\pi$
$97$	−88842.1	−0.958714	−0.479357	+	0.877620i	$0.659130\pi$
$98$	0	0
$99$	142059.	1.45673
$100$	0	0
$101$	179221.	1.74818	0.874090	−	0.485763i	$-0.161458\pi$
$101$	179221.	1.74818	0.874090	+	0.485763i	$0.161458\pi$
$102$	0	0
$103$	−123443.	−1.14650	−0.573248	−	0.819382i	$-0.694317\pi$
$103$	−123443.	−1.14650	−0.573248	+	0.819382i	$0.694317\pi$
$104$	0	0
$105$	7085.84	0.0627217
$106$	0	0
$107$	−98991.7	−0.835871	−0.417936	−	0.908477i	$-0.637246\pi$
$107$	−98991.7	−0.835871	−0.417936	+	0.908477i	$0.637246\pi$
$108$	0	0
$109$	46356.0	0.373715	0.186857	−	0.982387i	$-0.440170\pi$
$109$	46356.0	0.373715	0.186857	+	0.982387i	$0.440170\pi$
$110$	0	0
$111$	−5635.97	−0.0434171
$112$	0	0
$113$	185076.	1.36350	0.681750	−	0.731586i	$-0.261219\pi$
$113$	185076.	1.36350	0.681750	+	0.731586i	$0.261219\pi$
$114$	0	0
$115$	106101.	0.748123
$116$	0	0
$117$	−104670.	−0.706896
$118$	0	0
$119$	240211.	1.55498
$120$	0	0
$121$	187142.	1.16201
$122$	0	0
$123$	−2962.61	−0.0176568
$124$	0	0
$125$	182289.	1.04348
$126$	0	0
$127$	237332.	1.30571	0.652856	−	0.757482i	$-0.273571\pi$
$127$	237332.	1.30571	0.652856	+	0.757482i	$0.273571\pi$
$128$	0	0
$129$	2776.62	0.0146907
$130$	0	0
$131$	283496.	1.44334	0.721671	−	0.692236i	$-0.243374\pi$
$131$	283496.	1.44334	0.721671	+	0.692236i	$0.243374\pi$
$132$	0	0
$133$	−81600.9	−0.400005
$134$	0	0
$135$	−27477.5	−0.129760
$136$	0	0
$137$	330992.	1.50666	0.753331	−	0.657642i	$-0.228446\pi$
$137$	330992.	1.50666	0.753331	+	0.657642i	$0.228446\pi$
$138$	0	0
$139$	−105975.	−0.465230	−0.232615	−	0.972569i	$-0.574728\pi$
$139$	−105975.	−0.465230	−0.232615	+	0.972569i	$0.574728\pi$
$140$	0	0
$141$	3310.98	0.0140252
$142$	0	0
$143$	−256551.	−1.04914
$144$	0	0
$145$	55082.0	0.217565
$146$	0	0
$147$	1869.78	0.00713670
$148$	0	0
$149$	−196700.	−0.725836	−0.362918	−	0.931821i	$-0.618220\pi$
$149$	−196700.	−0.725836	−0.362918	+	0.931821i	$0.618220\pi$
$150$	0	0
$151$	412513.	1.47230	0.736148	−	0.676820i	$-0.236643\pi$
$151$	412513.	1.47230	0.736148	+	0.676820i	$0.236643\pi$
$152$	0	0
$153$	−463573.	−1.60099
$154$	0	0
$155$	167181.	0.558931
$156$	0	0
$157$	410971.	1.33064	0.665322	−	0.746557i	$-0.268294\pi$
$157$	410971.	1.33064	0.665322	+	0.746557i	$0.268294\pi$
$158$	0	0
$159$	−37518.4	−0.117693
$160$	0	0
$161$	−349920.	−1.06391
$162$	0	0
$163$	82488.7	0.243179	0.121589	−	0.992580i	$-0.461201\pi$
$163$	82488.7	0.243179	0.121589	+	0.992580i	$0.461201\pi$
$164$	0	0
$165$	−33517.4	−0.0958430
$166$	0	0
$167$	380336.	1.05530	0.527650	−	0.849462i	$-0.323073\pi$
$167$	380336.	1.05530	0.527650	+	0.849462i	$0.323073\pi$
$168$	0	0
$169$	−182265.	−0.490892
$170$	0	0
$171$	157478.	0.411842
$172$	0	0
$173$	91482.2	0.232392	0.116196	−	0.993226i	$-0.462930\pi$
$173$	91482.2	0.232392	0.116196	+	0.993226i	$0.462930\pi$
$174$	0	0
$175$	−211354.	−0.521694
$176$	0	0
$177$	−64206.4	−0.154044
$178$	0	0
$179$	607572.	1.41731	0.708656	−	0.705554i	$-0.249302\pi$
$179$	607572.	1.41731	0.708656	+	0.705554i	$0.249302\pi$
$180$	0	0
$181$	351519.	0.797540	0.398770	−	0.917051i	$-0.369437\pi$
$181$	351519.	0.797540	0.398770	+	0.917051i	$0.369437\pi$
$182$	0	0
$183$	31569.7	0.0696855
$184$	0	0
$185$	−141961.	−0.304958
$186$	0	0
$187$	−1.13624e6	−2.37611
$188$	0	0
$189$	90620.7	0.184532
$190$	0	0
$191$	123308.	0.244573	0.122286	−	0.992495i	$-0.460977\pi$
$191$	123308.	0.244573	0.122286	+	0.992495i	$0.460977\pi$
$192$	0	0
$193$	−163481.	−0.315917	−0.157959	−	0.987446i	$-0.550491\pi$
$193$	−163481.	−0.315917	−0.157959	+	0.987446i	$0.550491\pi$
$194$	0	0
$195$	24695.8	0.0465090
$196$	0	0
$197$	581124.	1.06685	0.533425	−	0.845848i	$-0.320905\pi$
$197$	581124.	1.06685	0.533425	+	0.845848i	$0.320905\pi$
$198$	0	0
$199$	−351526.	−0.629253	−0.314627	−	0.949216i	$-0.601879\pi$
$199$	−351526.	−0.629253	−0.314627	+	0.949216i	$0.601879\pi$
$200$	0	0
$201$	−37950.9	−0.0662570
$202$	0	0
$203$	−181660.	−0.309400
$204$	0	0
$205$	−74623.4	−0.124020
$206$	0	0
$207$	675297.	1.09539
$208$	0	0
$209$	385988.	0.611235
$210$	0	0
$211$	−327480.	−0.506383	−0.253192	−	0.967416i	$-0.581480\pi$
$211$	−327480.	−0.506383	−0.253192	+	0.967416i	$0.581480\pi$
$212$	0	0
$213$	72205.4	0.109049
$214$	0	0
$215$	69938.6	0.103186
$216$	0	0
$217$	−551364.	−0.794857
$218$	0	0
$219$	−88301.0	−0.124410
$220$	0	0
$221$	837191.	1.15304
$222$	0	0
$223$	−619602.	−0.834355	−0.417177	−	0.908825i	$-0.636981\pi$
$223$	−619602.	−0.834355	−0.417177	+	0.908825i	$0.636981\pi$
$224$	0	0
$225$	407885.	0.537132
$226$	0	0
$227$	757446.	0.975635	0.487817	−	0.872946i	$-0.337793\pi$
$227$	757446.	0.975635	0.487817	+	0.872946i	$0.337793\pi$
$228$	0	0
$229$	1.21739e6	1.53405	0.767025	−	0.641617i	$-0.221736\pi$
$229$	1.21739e6	1.53405	0.767025	+	0.641617i	$0.221736\pi$
$230$	0	0
$231$	110540.	0.136299
$232$	0	0
$233$	379715.	0.458213	0.229107	−	0.973401i	$-0.426420\pi$
$233$	379715.	0.458213	0.229107	+	0.973401i	$0.426420\pi$
$234$	0	0
$235$	83398.2	0.0985115
$236$	0	0
$237$	139214.	0.160994
$238$	0	0
$239$	−281756.	−0.319064	−0.159532	−	0.987193i	$-0.550999\pi$
$239$	−281756.	−0.319064	−0.159532	+	0.987193i	$0.550999\pi$
$240$	0	0
$241$	−1.12524e6	−1.24796	−0.623981	−	0.781439i	$-0.714486\pi$
$241$	−1.12524e6	−1.24796	−0.623981	+	0.781439i	$0.714486\pi$
$242$	0	0
$243$	−262736.	−0.285432
$244$	0	0
$245$	47096.8	0.0501275
$246$	0	0
$247$	−284398.	−0.296609
$248$	0	0
$249$	−2777.58	−0.00283902
$250$	0	0
$251$	675567.	0.676837	0.338419	−	0.940996i	$-0.390108\pi$
$251$	675567.	0.676837	0.338419	+	0.940996i	$0.390108\pi$
$252$	0	0
$253$	1.65519e6	1.62572
$254$	0	0
$255$	109376.	0.105335
$256$	0	0
$257$	1.85623e6	1.75306	0.876532	−	0.481343i	$-0.159851\pi$
$257$	1.85623e6	1.75306	0.876532	+	0.481343i	$0.159851\pi$
$258$	0	0
$259$	468188.	0.433681
$260$	0	0
$261$	350580.	0.318556
$262$	0	0
$263$	1.54160e6	1.37430	0.687149	−	0.726516i	$-0.258862\pi$
$263$	1.54160e6	1.37430	0.687149	+	0.726516i	$0.258862\pi$
$264$	0	0
$265$	−945029.	−0.826667
$266$	0	0
$267$	105689.	0.0907304
$268$	0	0
$269$	1.16775e6	0.983945	0.491973	−	0.870611i	$-0.336276\pi$
$269$	1.16775e6	0.983945	0.491973	+	0.870611i	$0.336276\pi$
$270$	0	0
$271$	1.20256e6	0.994684	0.497342	−	0.867555i	$-0.334309\pi$
$271$	1.20256e6	0.994684	0.497342	+	0.867555i	$0.334309\pi$
$272$	0	0
$273$	−81446.8	−0.0661405
$274$	0	0
$275$	999748.	0.797184
$276$	0	0
$277$	−504463.	−0.395030	−0.197515	−	0.980300i	$-0.563287\pi$
$277$	−504463.	−0.395030	−0.197515	+	0.980300i	$0.563287\pi$
$278$	0	0
$279$	1.06406e6	0.818378
$280$	0	0
$281$	1.52712e6	1.15374	0.576870	−	0.816836i	$-0.304274\pi$
$281$	1.52712e6	1.15374	0.576870	+	0.816836i	$0.304274\pi$
$282$	0	0
$283$	−1.81916e6	−1.35022	−0.675110	−	0.737717i	$-0.735904\pi$
$283$	−1.81916e6	−1.35022	−0.675110	+	0.737717i	$0.735904\pi$
$284$	0	0
$285$	−37155.5	−0.0270964
$286$	0	0
$287$	246108.	0.176369
$288$	0	0
$289$	2.28799e6	1.61143
$290$	0	0
$291$	133413.	0.0923562
$292$	0	0
$293$	−2.35975e6	−1.60582	−0.802911	−	0.596099i	$-0.796717\pi$
$293$	−2.35975e6	−1.60582	−0.802911	+	0.596099i	$0.796717\pi$
$294$	0	0
$295$	−1.61726e6	−1.08199
$296$	0	0
$297$	−428654.	−0.281978
$298$	0	0
$299$	−1.21955e6	−0.788901
$300$	0	0
$301$	−230658.	−0.146741
$302$	0	0
$303$	−269135.	−0.168408
$304$	0	0
$305$	795189.	0.489464
$306$	0	0
$307$	2.59790e6	1.57317	0.786587	−	0.617480i	$-0.211846\pi$
$307$	2.59790e6	1.57317	0.786587	+	0.617480i	$0.211846\pi$
$308$	0	0
$309$	185373.	0.110446
$310$	0	0
$311$	2.21471e6	1.29842	0.649212	−	0.760607i	$-0.275099\pi$
$311$	2.21471e6	1.29842	0.649212	+	0.760607i	$0.275099\pi$
$312$	0	0
$313$	−2.43630e6	−1.40563	−0.702813	−	0.711374i	$-0.748073\pi$
$313$	−2.43630e6	−1.40563	−0.702813	+	0.711374i	$0.748073\pi$
$314$	0	0
$315$	1.13597e6	0.645048
$316$	0	0
$317$	−2.59108e6	−1.44821	−0.724106	−	0.689689i	$-0.757747\pi$
$317$	−2.59108e6	−1.44821	−0.724106	+	0.689689i	$0.757747\pi$
$318$	0	0
$319$	859290.	0.472784
$320$	0	0
$321$	148655.	0.0805223
$322$	0	0
$323$	−1.25958e6	−0.671767
$324$	0	0
$325$	−736620.	−0.386843
$326$	0	0
$327$	−69612.3	−0.0360012
$328$	0	0
$329$	−275047.	−0.140093
$330$	0	0
$331$	2.49362e6	1.25101	0.625503	−	0.780221i	$-0.284894\pi$
$331$	2.49362e6	1.25101	0.625503	+	0.780221i	$0.284894\pi$
$332$	0	0
$333$	−903537.	−0.446514
$334$	0	0
$335$	−955921.	−0.465383
$336$	0	0
$337$	−159199.	−0.0763600	−0.0381800	−	0.999271i	$-0.512156\pi$
$337$	−159199.	−0.0763600	−0.0381800	+	0.999271i	$0.512156\pi$
$338$	0	0
$339$	−277927.	−0.131350
$340$	0	0
$341$	2.60806e6	1.21460
$342$	0	0
$343$	−2.25195e6	−1.03353
$344$	0	0
$345$	−159330.	−0.0720692
$346$	0	0
$347$	−4.35264e6	−1.94057	−0.970285	−	0.241966i	$-0.922208\pi$
$347$	−4.35264e6	−1.94057	−0.970285	+	0.241966i	$0.922208\pi$
$348$	0	0
$349$	−502730.	−0.220938	−0.110469	−	0.993880i	$-0.535235\pi$
$349$	−502730.	−0.220938	−0.110469	+	0.993880i	$0.535235\pi$
$350$	0	0
$351$	315834.	0.136833
$352$	0	0
$353$	−990932.	−0.423260	−0.211630	−	0.977350i	$-0.567877\pi$
$353$	−990932.	−0.423260	−0.211630	+	0.977350i	$0.567877\pi$
$354$	0	0
$355$	1.81874e6	0.765948
$356$	0	0
$357$	−360721.	−0.149796
$358$	0	0
$359$	1.78805e6	0.732223	0.366112	−	0.930571i	$-0.380689\pi$
$359$	1.78805e6	0.732223	0.366112	+	0.930571i	$0.380689\pi$
$360$	0	0
$361$	−2.04821e6	−0.827194
$362$	0	0
$363$	−281029.	−0.111940
$364$	0	0
$365$	−2.22416e6	−0.873844
$366$	0	0
$367$	68021.3	0.0263621	0.0131810	−	0.999913i	$-0.495804\pi$
$367$	68021.3	0.0263621	0.0131810	+	0.999913i	$0.495804\pi$
$368$	0	0
$369$	−474954.	−0.181588
$370$	0	0
$371$	3.11671e6	1.17560
$372$	0	0
$373$	−949477.	−0.353356	−0.176678	−	0.984269i	$-0.556535\pi$
$373$	−949477.	−0.353356	−0.176678	+	0.984269i	$0.556535\pi$
$374$	0	0
$375$	−273741.	−0.100522
$376$	0	0
$377$	−633130.	−0.229424
$378$	0	0
$379$	−2.53069e6	−0.904982	−0.452491	−	0.891769i	$-0.649465\pi$
$379$	−2.53069e6	−0.904982	−0.452491	+	0.891769i	$0.649465\pi$
$380$	0	0
$381$	−356399.	−0.125784
$382$	0	0
$383$	−500631.	−0.174390	−0.0871948	−	0.996191i	$-0.527790\pi$
$383$	−500631.	−0.174390	−0.0871948	+	0.996191i	$0.527790\pi$
$384$	0	0
$385$	2.78434e6	0.957348
$386$	0	0
$387$	445137.	0.151083
$388$	0	0
$389$	−4.08546e6	−1.36889	−0.684443	−	0.729066i	$-0.739955\pi$
$389$	−4.08546e6	−1.36889	−0.684443	+	0.729066i	$0.739955\pi$
$390$	0	0
$391$	−5.40130e6	−1.78672
$392$	0	0
$393$	−425723.	−0.139042
$394$	0	0
$395$	3.50657e6	1.13081
$396$	0	0
$397$	953651.	0.303678	0.151839	−	0.988405i	$-0.451480\pi$
$397$	953651.	0.303678	0.151839	+	0.988405i	$0.451480\pi$
$398$	0	0
$399$	122539.	0.0385338
$400$	0	0
$401$	4.81115e6	1.49413	0.747064	−	0.664752i	$-0.231463\pi$
$401$	4.81115e6	1.49413	0.747064	+	0.664752i	$0.231463\pi$
$402$	0	0
$403$	−1.92163e6	−0.589397
$404$	0	0
$405$	−2.17155e6	−0.657857
$406$	0	0
$407$	−2.21462e6	−0.662694
$408$	0	0
$409$	−1.42551e6	−0.421370	−0.210685	−	0.977554i	$-0.567569\pi$
$409$	−1.42551e6	−0.421370	−0.210685	+	0.977554i	$0.567569\pi$
$410$	0	0
$411$	−497046.	−0.145142
$412$	0	0
$413$	5.33372e6	1.53870
$414$	0	0
$415$	−69962.7	−0.0199410
$416$	0	0
$417$	159142.	0.0448171
$418$	0	0
$419$	−2.91631e6	−0.811520	−0.405760	−	0.913980i	$-0.632993\pi$
$419$	−2.91631e6	−0.811520	−0.405760	+	0.913980i	$0.632993\pi$
$420$	0	0
$421$	6.91223e6	1.90070	0.950349	−	0.311186i	$-0.100726\pi$
$421$	6.91223e6	1.90070	0.950349	+	0.311186i	$0.100726\pi$
$422$	0	0
$423$	530803.	0.144239
$424$	0	0
$425$	−3.26243e6	−0.876131
$426$	0	0
$427$	−2.62254e6	−0.696068
$428$	0	0
$429$	385259.	0.101067
$430$	0	0
$431$	−2.58575e6	−0.670493	−0.335246	−	0.942131i	$-0.608820\pi$
$431$	−2.58575e6	−0.670493	−0.335246	+	0.942131i	$0.608820\pi$
$432$	0	0
$433$	−3.71200e6	−0.951455	−0.475727	−	0.879593i	$-0.657815\pi$
$433$	−3.71200e6	−0.951455	−0.475727	+	0.879593i	$0.657815\pi$
$434$	0	0
$435$	−82716.0	−0.0209588
$436$	0	0
$437$	1.83485e6	0.459619
$438$	0	0
$439$	415772.	0.102966	0.0514830	−	0.998674i	$-0.483605\pi$
$439$	415772.	0.102966	0.0514830	+	0.998674i	$0.483605\pi$
$440$	0	0
$441$	299756.	0.0733959
$442$	0	0
$443$	602736.	0.145921	0.0729605	−	0.997335i	$-0.476755\pi$
$443$	602736.	0.145921	0.0729605	+	0.997335i	$0.476755\pi$
$444$	0	0
$445$	2.66214e6	0.637282
$446$	0	0
$447$	295382.	0.0699223
$448$	0	0
$449$	1.84907e6	0.432849	0.216424	−	0.976299i	$-0.430560\pi$
$449$	1.84907e6	0.432849	0.216424	+	0.976299i	$0.430560\pi$
$450$	0	0
$451$	−1.16414e6	−0.269503
$452$	0	0
$453$	−619466.	−0.141831
$454$	0	0
$455$	−2.05151e6	−0.464564
$456$	0	0
$457$	−7.35472e6	−1.64731	−0.823655	−	0.567091i	$-0.808069\pi$
$457$	−7.35472e6	−1.64731	−0.823655	+	0.567091i	$0.808069\pi$
$458$	0	0
$459$	1.39881e6	0.309903
$460$	0	0
$461$	−3.51032e6	−0.769298	−0.384649	−	0.923063i	$-0.625678\pi$
$461$	−3.51032e6	−0.769298	−0.384649	+	0.923063i	$0.625678\pi$
$462$	0	0
$463$	6.86607e6	1.48852	0.744262	−	0.667888i	$-0.232801\pi$
$463$	6.86607e6	1.48852	0.744262	+	0.667888i	$0.232801\pi$
$464$	0	0
$465$	−251054.	−0.0538437
$466$	0	0
$467$	−5.66490e6	−1.20199	−0.600993	−	0.799254i	$-0.705228\pi$
$467$	−5.66490e6	−1.20199	−0.600993	+	0.799254i	$0.705228\pi$
$468$	0	0
$469$	3.15263e6	0.661821
$470$	0	0
$471$	−617150.	−0.128185
$472$	0	0
$473$	1.09106e6	0.224230
$474$	0	0
$475$	1.10827e6	0.225377
$476$	0	0
$477$	−6.01481e6	−1.21039
$478$	0	0
$479$	−4.55961e6	−0.908006	−0.454003	−	0.891000i	$-0.650004\pi$
$479$	−4.55961e6	−0.908006	−0.454003	+	0.891000i	$0.650004\pi$
$480$	0	0
$481$	1.63174e6	0.321580
$482$	0	0
$483$	525470.	0.102490
$484$	0	0
$485$	3.36046e6	0.648701
$486$	0	0
$487$	9.17550e6	1.75310	0.876552	−	0.481308i	$-0.159838\pi$
$487$	9.17550e6	1.75310	0.876552	+	0.481308i	$0.159838\pi$
$488$	0	0
$489$	−123872.	−0.0234262
$490$	0	0
$491$	−3.41449e6	−0.639179	−0.319589	−	0.947556i	$-0.603545\pi$
$491$	−3.41449e6	−0.639179	−0.319589	+	0.947556i	$0.603545\pi$
$492$	0	0
$493$	−2.80408e6	−0.519605
$494$	0	0
$495$	−5.37338e6	−0.985677
$496$	0	0
$497$	−5.99820e6	−1.08926
$498$	0	0
$499$	1.99364e6	0.358423	0.179211	−	0.983811i	$-0.442645\pi$
$499$	1.99364e6	0.358423	0.179211	+	0.983811i	$0.442645\pi$
$500$	0	0
$501$	−571146.	−0.101661
$502$	0	0
$503$	−9.15685e6	−1.61371	−0.806856	−	0.590748i	$-0.798833\pi$
$503$	−9.15685e6	−1.61371	−0.806856	+	0.590748i	$0.798833\pi$
$504$	0	0
$505$	−6.77907e6	−1.18288
$506$	0	0
$507$	273705.	0.0472893
$508$	0	0
$509$	6.56697e6	1.12349	0.561746	−	0.827309i	$-0.310130\pi$
$509$	6.56697e6	1.12349	0.561746	+	0.827309i	$0.310130\pi$
$510$	0	0
$511$	7.33529e6	1.24270
$512$	0	0
$513$	−475182.	−0.0797199
$514$	0	0
$515$	4.66924e6	0.775761
$516$	0	0
$517$	1.30103e6	0.214072
$518$	0	0
$519$	−137378.	−0.0223871
$520$	0	0
$521$	6.45805e6	1.04234	0.521168	−	0.853454i	$-0.325497\pi$
$521$	6.45805e6	1.04234	0.521168	+	0.853454i	$0.325497\pi$
$522$	0	0
$523$	−184405.	−0.0294794	−0.0147397	−	0.999891i	$-0.504692\pi$
$523$	−184405.	−0.0294794	−0.0147397	+	0.999891i	$0.504692\pi$
$524$	0	0
$525$	317388.	0.0502566
$526$	0	0
$527$	−8.51076e6	−1.33488
$528$	0	0
$529$	1.43185e6	0.222463
$530$	0	0
$531$	−1.02933e7	−1.58424
$532$	0	0
$533$	857745.	0.130780
$534$	0	0
$535$	3.74437e6	0.565581
$536$	0	0
$537$	−912384.	−0.136534
$538$	0	0
$539$	734719.	0.108931
$540$	0	0
$541$	−7.10369e6	−1.04350	−0.521748	−	0.853100i	$-0.674720\pi$
$541$	−7.10369e6	−1.04350	−0.521748	+	0.853100i	$0.674720\pi$
$542$	0	0
$543$	−527872.	−0.0768297
$544$	0	0
$545$	−1.75342e6	−0.252869
$546$	0	0
$547$	9.75210e6	1.39357	0.696787	−	0.717279i	$-0.254612\pi$
$547$	9.75210e6	1.39357	0.696787	+	0.717279i	$0.254612\pi$
$548$	0	0
$549$	5.06113e6	0.716666
$550$	0	0
$551$	952562.	0.133664
$552$	0	0
$553$	−1.15647e7	−1.60813
$554$	0	0
$555$	213181.	0.0293776
$556$	0	0
$557$	−2.28586e6	−0.312185	−0.156092	−	0.987742i	$-0.549890\pi$
$557$	−2.28586e6	−0.312185	−0.156092	+	0.987742i	$0.549890\pi$
$558$	0	0
$559$	−803896.	−0.108810
$560$	0	0
$561$	1.70628e6	0.228899
$562$	0	0
$563$	−2.31099e6	−0.307275	−0.153638	−	0.988127i	$-0.549099\pi$
$563$	−2.31099e6	−0.307275	−0.153638	+	0.988127i	$0.549099\pi$
$564$	0	0
$565$	−7.00054e6	−0.922593
$566$	0	0
$567$	7.16176e6	0.935539
$568$	0	0
$569$	1.17226e7	1.51790	0.758949	−	0.651151i	$-0.225713\pi$
$569$	1.17226e7	1.51790	0.758949	+	0.651151i	$0.225713\pi$
$570$	0	0
$571$	5.10057e6	0.654679	0.327340	−	0.944907i	$-0.393848\pi$
$571$	5.10057e6	0.654679	0.327340	+	0.944907i	$0.393848\pi$
$572$	0	0
$573$	−185170.	−0.0235605
$574$	0	0
$575$	4.75245e6	0.599443
$576$	0	0
$577$	599429.	0.0749546	0.0374773	−	0.999297i	$-0.488068\pi$
$577$	599429.	0.0749546	0.0374773	+	0.999297i	$0.488068\pi$
$578$	0	0
$579$	245497.	0.0304334
$580$	0	0
$581$	230737.	0.0283581
$582$	0	0
$583$	−1.47426e7	−1.79640
$584$	0	0
$585$	3.95914e6	0.478312
$586$	0	0
$587$	−4.32755e6	−0.518379	−0.259189	−	0.965826i	$-0.583455\pi$
$587$	−4.32755e6	−0.518379	−0.259189	+	0.965826i	$0.583455\pi$
$588$	0	0
$589$	2.89115e6	0.343386
$590$	0	0
$591$	−872667.	−0.102773
$592$	0	0
$593$	−1.59099e7	−1.85794	−0.928968	−	0.370160i	$-0.879303\pi$
$593$	−1.59099e7	−1.85794	−0.928968	+	0.370160i	$0.879303\pi$
$594$	0	0
$595$	−9.08599e6	−1.05216
$596$	0	0
$597$	527883.	0.0606181
$598$	0	0
$599$	−73202.1	−0.00833598	−0.00416799	−	0.999991i	$-0.501327\pi$
$599$	−73202.1	−0.00833598	−0.00416799	+	0.999991i	$0.501327\pi$
$600$	0	0
$601$	−3.77328e6	−0.426121	−0.213060	−	0.977039i	$-0.568343\pi$
$601$	−3.77328e6	−0.426121	−0.213060	+	0.977039i	$0.568343\pi$
$602$	0	0
$603$	−6.08414e6	−0.681406
$604$	0	0
$605$	−7.07868e6	−0.786256
$606$	0	0
$607$	−3.39314e6	−0.373792	−0.186896	−	0.982380i	$-0.559843\pi$
$607$	−3.39314e6	−0.373792	−0.186896	+	0.982380i	$0.559843\pi$
$608$	0	0
$609$	272797.	0.0298055
$610$	0	0
$611$	−958605.	−0.103881
$612$	0	0
$613$	5.93491e6	0.637915	0.318957	−	0.947769i	$-0.396667\pi$
$613$	5.93491e6	0.637915	0.318957	+	0.947769i	$0.396667\pi$
$614$	0	0
$615$	112061.	0.0119472
$616$	0	0
$617$	−1.07871e7	−1.14076	−0.570379	−	0.821382i	$-0.693204\pi$
$617$	−1.07871e7	−1.14076	−0.570379	+	0.821382i	$0.693204\pi$
$618$	0	0
$619$	−9.97743e6	−1.04663	−0.523313	−	0.852140i	$-0.675304\pi$
$619$	−9.97743e6	−1.04663	−0.523313	+	0.852140i	$0.675304\pi$
$620$	0	0
$621$	−2.03767e6	−0.212034
$622$	0	0
$623$	−8.77976e6	−0.906280
$624$	0	0
$625$	−1.60054e6	−0.163895
$626$	0	0
$627$	−579634.	−0.0588823
$628$	0	0
$629$	7.22687e6	0.728322
$630$	0	0
$631$	3.42994e6	0.342936	0.171468	−	0.985190i	$-0.445149\pi$
$631$	3.42994e6	0.342936	0.171468	+	0.985190i	$0.445149\pi$
$632$	0	0
$633$	491774.	0.0487816
$634$	0	0
$635$	−8.97711e6	−0.883492
$636$	0	0
$637$	−541345.	−0.0528598
$638$	0	0
$639$	1.15757e7	1.12149
$640$	0	0
$641$	−1.41725e7	−1.36239	−0.681193	−	0.732104i	$-0.738538\pi$
$641$	−1.41725e7	−1.36239	−0.681193	+	0.732104i	$0.738538\pi$
$642$	0	0
$643$	−4.26802e6	−0.407098	−0.203549	−	0.979065i	$-0.565248\pi$
$643$	−4.26802e6	−0.407098	−0.203549	+	0.979065i	$0.565248\pi$
$644$	0	0
$645$	−105026.	−0.00994026
$646$	0	0
$647$	−1.26038e7	−1.18370	−0.591849	−	0.806049i	$-0.701602\pi$
$647$	−1.26038e7	−1.18370	−0.591849	+	0.806049i	$0.701602\pi$
$648$	0	0
$649$	−2.52295e7	−2.35124
$650$	0	0
$651$	827977.	0.0765712
$652$	0	0
$653$	1.67724e7	1.53926	0.769631	−	0.638489i	$-0.220440\pi$
$653$	1.67724e7	1.53926	0.769631	+	0.638489i	$0.220440\pi$
$654$	0	0
$655$	−1.07233e7	−0.976618
$656$	0	0
$657$	−1.41561e7	−1.27947
$658$	0	0
$659$	1.72113e7	1.54383	0.771915	−	0.635725i	$-0.219299\pi$
$659$	1.72113e7	1.54383	0.771915	+	0.635725i	$0.219299\pi$
$660$	0	0
$661$	5.67723e6	0.505397	0.252699	−	0.967545i	$-0.418682\pi$
$661$	5.67723e6	0.505397	0.252699	+	0.967545i	$0.418682\pi$
$662$	0	0
$663$	−1.25720e6	−0.111076
$664$	0	0
$665$	3.08656e6	0.270658
$666$	0	0
$667$	4.08476e6	0.355510
$668$	0	0
$669$	930449.	0.0803762
$670$	0	0
$671$	1.24051e7	1.06364
$672$	0	0
$673$	4.92989e6	0.419565	0.209783	−	0.977748i	$-0.432724\pi$
$673$	4.92989e6	0.419565	0.209783	+	0.977748i	$0.432724\pi$
$674$	0	0
$675$	−1.23077e6	−0.103972
$676$	0	0
$677$	−1.25498e7	−1.05236	−0.526180	−	0.850373i	$-0.676376\pi$
$677$	−1.25498e7	−1.05236	−0.526180	+	0.850373i	$0.676376\pi$
$678$	0	0
$679$	−1.10828e7	−0.922519
$680$	0	0
$681$	−1.13745e6	−0.0939861
$682$	0	0
$683$	7.08503e6	0.581152	0.290576	−	0.956852i	$-0.406153\pi$
$683$	7.08503e6	0.581152	0.290576	+	0.956852i	$0.406153\pi$
$684$	0	0
$685$	−1.25198e7	−1.01946
$686$	0	0
$687$	−1.82813e6	−0.147780
$688$	0	0
$689$	1.08625e7	0.871726
$690$	0	0
$691$	9.40053e6	0.748958	0.374479	−	0.927235i	$-0.377822\pi$
$691$	9.40053e6	0.748958	0.374479	+	0.927235i	$0.377822\pi$
$692$	0	0
$693$	1.77214e7	1.40173
$694$	0	0
$695$	4.00853e6	0.314791
$696$	0	0
$697$	3.79888e6	0.296192
$698$	0	0
$699$	−570213.	−0.0441412
$700$	0	0
$701$	2.14131e7	1.64583	0.822914	−	0.568167i	$-0.192347\pi$
$701$	2.14131e7	1.64583	0.822914	+	0.568167i	$0.192347\pi$
$702$	0	0
$703$	−2.45501e6	−0.187355
$704$	0	0
$705$	−125238.	−0.00948995
$706$	0	0
$707$	2.23574e7	1.68218
$708$	0	0
$709$	1.70494e7	1.27377	0.636887	−	0.770957i	$-0.280222\pi$
$709$	1.70494e7	1.27377	0.636887	+	0.770957i	$0.280222\pi$
$710$	0	0
$711$	2.23182e7	1.65571
$712$	0	0
$713$	1.23978e7	0.913316
$714$	0	0
$715$	9.70406e6	0.709886
$716$	0	0
$717$	423110.	0.0307365
$718$	0	0
$719$	1.59376e7	1.14974	0.574872	−	0.818243i	$-0.305052\pi$
$719$	1.59376e7	1.14974	0.574872	+	0.818243i	$0.305052\pi$
$720$	0	0
$721$	−1.53991e7	−1.10321
$722$	0	0
$723$	1.68976e6	0.120220
$724$	0	0
$725$	2.46723e6	0.174327
$726$	0	0
$727$	−8.26202e6	−0.579763	−0.289881	−	0.957063i	$-0.593616\pi$
$727$	−8.26202e6	−0.579763	−0.289881	+	0.957063i	$0.593616\pi$
$728$	0	0
$729$	−1.35561e7	−0.944749
$730$	0	0
$731$	−3.56039e6	−0.246436
$732$	0	0
$733$	6.20168e6	0.426334	0.213167	−	0.977016i	$-0.431622\pi$
$733$	6.20168e6	0.426334	0.213167	+	0.977016i	$0.431622\pi$
$734$	0	0
$735$	−70724.7	−0.00482895
$736$	0	0
$737$	−1.49126e7	−1.01131
$738$	0	0
$739$	1.19003e7	0.801578	0.400789	−	0.916170i	$-0.368736\pi$
$739$	1.19003e7	0.801578	0.400789	+	0.916170i	$0.368736\pi$
$740$	0	0
$741$	427077.	0.0285734
$742$	0	0
$743$	−7.41722e6	−0.492912	−0.246456	−	0.969154i	$-0.579266\pi$
$743$	−7.41722e6	−0.492912	−0.246456	+	0.969154i	$0.579266\pi$
$744$	0	0
$745$	7.44020e6	0.491127
$746$	0	0
$747$	−445291.	−0.0291973
$748$	0	0
$749$	−1.23489e7	−0.804314
$750$	0	0
$751$	−1.46691e7	−0.949084	−0.474542	−	0.880233i	$-0.657386\pi$
$751$	−1.46691e7	−0.949084	−0.474542	+	0.880233i	$0.657386\pi$
$752$	0	0
$753$	−1.01449e6	−0.0652020
$754$	0	0
$755$	−1.56033e7	−0.996209
$756$	0	0
$757$	2.41775e6	0.153346	0.0766729	−	0.997056i	$-0.475570\pi$
$757$	2.41775e6	0.153346	0.0766729	+	0.997056i	$0.475570\pi$
$758$	0	0
$759$	−2.48558e6	−0.156611
$760$	0	0
$761$	2.32422e7	1.45484	0.727422	−	0.686191i	$-0.240718\pi$
$761$	2.32422e7	1.45484	0.727422	+	0.686191i	$0.240718\pi$
$762$	0	0
$763$	5.78279e6	0.359605
$764$	0	0
$765$	1.75347e7	1.08329
$766$	0	0
$767$	1.85893e7	1.14097
$768$	0	0
$769$	−2.55432e7	−1.55761	−0.778806	−	0.627265i	$-0.784174\pi$
$769$	−2.55432e7	−1.55761	−0.778806	+	0.627265i	$0.784174\pi$
$770$	0	0
$771$	−2.78747e6	−0.168879
$772$	0	0
$773$	1.72159e7	1.03629	0.518146	−	0.855292i	$-0.326622\pi$
$773$	1.72159e7	1.03629	0.518146	+	0.855292i	$0.326622\pi$
$774$	0	0
$775$	7.48838e6	0.447851
$776$	0	0
$777$	−703072.	−0.0417780
$778$	0	0
$779$	−1.29050e6	−0.0761930
$780$	0	0
$781$	2.83727e7	1.66446
$782$	0	0
$783$	−1.05785e6	−0.0616625
$784$	0	0
$785$	−1.55450e7	−0.900362
$786$	0	0
$787$	1.48812e7	0.856449	0.428225	−	0.903672i	$-0.359139\pi$
$787$	1.48812e7	0.856449	0.428225	+	0.903672i	$0.359139\pi$
$788$	0	0
$789$	−2.31500e6	−0.132391
$790$	0	0
$791$	2.30878e7	1.31202
$792$	0	0
$793$	−9.14016e6	−0.516144
$794$	0	0
$795$	1.41914e6	0.0796356
$796$	0	0
$797$	5.81135e6	0.324064	0.162032	−	0.986785i	$-0.448195\pi$
$797$	5.81135e6	0.324064	0.162032	+	0.986785i	$0.448195\pi$
$798$	0	0
$799$	−4.24559e6	−0.235272
$800$	0	0
$801$	1.69437e7	0.933098
$802$	0	0
$803$	−3.46973e7	−1.89892
$804$	0	0
$805$	1.32358e7	0.719878
$806$	0	0
$807$	−1.75360e6	−0.0947867
$808$	0	0
$809$	−2.41530e7	−1.29748	−0.648740	−	0.761010i	$-0.724704\pi$
$809$	−2.41530e7	−1.29748	−0.648740	+	0.761010i	$0.724704\pi$
$810$	0	0
$811$	−2.14467e7	−1.14501	−0.572505	−	0.819901i	$-0.694028\pi$
$811$	−2.14467e7	−1.14501	−0.572505	+	0.819901i	$0.694028\pi$
$812$	0	0
$813$	−1.80588e6	−0.0958212
$814$	0	0
$815$	−3.12014e6	−0.164543
$816$	0	0
$817$	1.20949e6	0.0633936
$818$	0	0
$819$	−1.30572e7	−0.680208
$820$	0	0
$821$	−1.06313e7	−0.550465	−0.275232	−	0.961378i	$-0.588755\pi$
$821$	−1.06313e7	−0.550465	−0.275232	+	0.961378i	$0.588755\pi$
$822$	0	0
$823$	871518.	0.0448515	0.0224257	−	0.999749i	$-0.492861\pi$
$823$	871518.	0.0448515	0.0224257	+	0.999749i	$0.492861\pi$
$824$	0	0
$825$	−1.50131e6	−0.0767954
$826$	0	0
$827$	1.70354e7	0.866141	0.433071	−	0.901360i	$-0.357430\pi$
$827$	1.70354e7	0.866141	0.433071	+	0.901360i	$0.357430\pi$
$828$	0	0
$829$	5.65731e6	0.285906	0.142953	−	0.989729i	$-0.454340\pi$
$829$	5.65731e6	0.285906	0.142953	+	0.989729i	$0.454340\pi$
$830$	0	0
$831$	757546.	0.0380546
$832$	0	0
$833$	−2.39758e6	−0.119718
$834$	0	0
$835$	−1.43862e7	−0.714054
$836$	0	0
$837$	−3.21073e6	−0.158413
$838$	0	0
$839$	−3.42452e7	−1.67956	−0.839779	−	0.542929i	$-0.817315\pi$
$839$	−3.42452e7	−1.67956	−0.839779	+	0.542929i	$0.817315\pi$
$840$	0	0
$841$	−1.83905e7	−0.896612
$842$	0	0
$843$	−2.29326e6	−0.111144
$844$	0	0
$845$	6.89419e6	0.332155
$846$	0	0
$847$	2.33455e7	1.11814
$848$	0	0
$849$	2.73181e6	0.130071
$850$	0	0
$851$	−1.05275e7	−0.498313
$852$	0	0
$853$	−2.21968e7	−1.04452	−0.522261	−	0.852785i	$-0.674911\pi$
$853$	−2.21968e7	−1.04452	−0.522261	+	0.852785i	$0.674911\pi$
$854$	0	0
$855$	−5.95664e6	−0.278667
$856$	0	0
$857$	3.15803e7	1.46881	0.734403	−	0.678714i	$-0.237462\pi$
$857$	3.15803e7	1.46881	0.734403	+	0.678714i	$0.237462\pi$
$858$	0	0
$859$	1.22037e6	0.0564297	0.0282148	−	0.999602i	$-0.491018\pi$
$859$	1.22037e6	0.0564297	0.0282148	+	0.999602i	$0.491018\pi$
$860$	0	0
$861$	−369578.	−0.0169902
$862$	0	0
$863$	2.43543e7	1.11314	0.556570	−	0.830801i	$-0.312117\pi$
$863$	2.43543e7	1.11314	0.556570	+	0.830801i	$0.312117\pi$
$864$	0	0
$865$	−3.46032e6	−0.157245
$866$	0	0
$867$	−3.43585e6	−0.155234
$868$	0	0
$869$	5.47031e7	2.45732
$870$	0	0
$871$	1.09877e7	0.490749
$872$	0	0
$873$	2.13883e7	0.949817
$874$	0	0
$875$	2.27401e7	1.00409
$876$	0	0
$877$	2.52016e6	0.110644	0.0553222	−	0.998469i	$-0.482381\pi$
$877$	2.52016e6	0.110644	0.0553222	+	0.998469i	$0.482381\pi$
$878$	0	0
$879$	3.54361e6	0.154694
$880$	0	0
$881$	1.04311e6	0.0452785	0.0226393	−	0.999744i	$-0.492793\pi$
$881$	1.04311e6	0.0452785	0.0226393	+	0.999744i	$0.492793\pi$
$882$	0	0
$883$	3.57753e7	1.54412	0.772061	−	0.635548i	$-0.219226\pi$
$883$	3.57753e7	1.54412	0.772061	+	0.635548i	$0.219226\pi$
$884$	0	0
$885$	2.42862e6	0.104232
$886$	0	0
$887$	1.66616e7	0.711061	0.355531	−	0.934665i	$-0.384300\pi$
$887$	1.66616e7	0.711061	0.355531	+	0.934665i	$0.384300\pi$
$888$	0	0
$889$	2.96065e7	1.25642
$890$	0	0
$891$	−3.38765e7	−1.42957
$892$	0	0
$893$	1.44225e6	0.0605218
$894$	0	0
$895$	−2.29815e7	−0.959004
$896$	0	0
$897$	1.83139e6	0.0759975
$898$	0	0
$899$	6.43631e6	0.265606
$900$	0	0
$901$	4.81090e7	1.97430
$902$	0	0
$903$	346376.	0.0141361
$904$	0	0
$905$	−1.32963e7	−0.539645
$906$	0	0
$907$	−3.59470e7	−1.45092	−0.725462	−	0.688262i	$-0.758374\pi$
$907$	−3.59470e7	−1.45092	−0.725462	+	0.688262i	$0.758374\pi$
$908$	0	0
$909$	−4.31466e7	−1.73196
$910$	0	0
$911$	−7.56404e6	−0.301966	−0.150983	−	0.988536i	$-0.548244\pi$
$911$	−7.56404e6	−0.301966	−0.150983	+	0.988536i	$0.548244\pi$
$912$	0	0
$913$	−1.09143e6	−0.0433331
$914$	0	0
$915$	−1.19413e6	−0.0471517
$916$	0	0
$917$	3.53654e7	1.38885
$918$	0	0
$919$	2.22167e7	0.867741	0.433870	−	0.900975i	$-0.357148\pi$
$919$	2.22167e7	0.867741	0.433870	+	0.900975i	$0.357148\pi$
$920$	0	0
$921$	−3.90124e6	−0.151549
$922$	0	0
$923$	−2.09051e7	−0.807698
$924$	0	0
$925$	−6.35871e6	−0.244351
$926$	0	0
$927$	2.97182e7	1.13586
$928$	0	0
$929$	9.18809e6	0.349290	0.174645	−	0.984631i	$-0.444122\pi$
$929$	9.18809e6	0.349290	0.174645	+	0.984631i	$0.444122\pi$
$930$	0	0
$931$	814470.	0.0307965
$932$	0	0
$933$	−3.32581e6	−0.125082
$934$	0	0
$935$	4.29785e7	1.60777
$936$	0	0
$937$	4.06654e7	1.51313	0.756565	−	0.653919i	$-0.226876\pi$
$937$	4.06654e7	1.51313	0.756565	+	0.653919i	$0.226876\pi$
$938$	0	0
$939$	3.65856e6	0.135409
$940$	0	0
$941$	−2.60105e7	−0.957580	−0.478790	−	0.877930i	$-0.658924\pi$
$941$	−2.60105e7	−0.957580	−0.478790	+	0.877930i	$0.658924\pi$
$942$	0	0
$943$	−5.53391e6	−0.202653
$944$	0	0
$945$	−3.42774e6	−0.124861
$946$	0	0
$947$	2.03967e7	0.739067	0.369534	−	0.929217i	$-0.379517\pi$
$947$	2.03967e7	0.739067	0.369534	+	0.929217i	$0.379517\pi$
$948$	0	0
$949$	2.55652e7	0.921475
$950$	0	0
$951$	3.89099e6	0.139511
$952$	0	0
$953$	−4.87462e6	−0.173864	−0.0869318	−	0.996214i	$-0.527706\pi$
$953$	−4.87462e6	−0.173864	−0.0869318	+	0.996214i	$0.527706\pi$
$954$	0	0
$955$	−4.66414e6	−0.165487
$956$	0	0
$957$	−1.29039e6	−0.0455449
$958$	0	0
$959$	4.12903e7	1.44978
$960$	0	0
$961$	−9.09409e6	−0.317651
$962$	0	0
$963$	2.38318e7	0.828114
$964$	0	0
$965$	6.18368e6	0.213761
$966$	0	0
$967$	1.26016e7	0.433371	0.216686	−	0.976241i	$-0.430475\pi$
$967$	1.26016e7	0.433371	0.216686	+	0.976241i	$0.430475\pi$
$968$	0	0
$969$	1.89149e6	0.0647135
$970$	0	0
$971$	1.39978e6	0.0476443	0.0238221	−	0.999716i	$-0.492416\pi$
$971$	1.39978e6	0.0476443	0.0238221	+	0.999716i	$0.492416\pi$
$972$	0	0
$973$	−1.32201e7	−0.447665
$974$	0	0
$975$	1.10617e6	0.0372659
$976$	0	0
$977$	1.43010e6	0.0479325	0.0239662	−	0.999713i	$-0.492371\pi$
$977$	1.43010e6	0.0479325	0.0239662	+	0.999713i	$0.492371\pi$
$978$	0	0
$979$	4.15300e7	1.38486
$980$	0	0
$981$	−1.11600e7	−0.370247
$982$	0	0
$983$	−4.26255e7	−1.40697	−0.703487	−	0.710708i	$-0.748375\pi$
$983$	−4.26255e7	−1.40697	−0.703487	+	0.710708i	$0.748375\pi$
$984$	0	0
$985$	−2.19811e7	−0.721869
$986$	0	0
$987$	413035.	0.0134957
$988$	0	0
$989$	5.18650e6	0.168610
$990$	0	0
$991$	−1.88273e7	−0.608980	−0.304490	−	0.952516i	$-0.598486\pi$
$991$	−1.88273e7	−0.608980	−0.304490	+	0.952516i	$0.598486\pi$
$992$	0	0
$993$	−3.74463e6	−0.120514
$994$	0	0
$995$	1.32965e7	0.425775
$996$	0	0
$997$	2.10574e7	0.670915	0.335458	−	0.942055i	$-0.391109\pi$
$997$	2.10574e7	0.670915	0.335458	+	0.942055i	$0.391109\pi$
$998$	0	0
$999$	2.72637e6	0.0864314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.6		10
4.3	odd	2		43.6.a.b.1.2	✓	10
12.11	even	2		387.6.a.e.1.9		10
20.19	odd	2		1075.6.a.b.1.9		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.2	✓	10	4.3	odd	2
387.6.a.e.1.9		10	12.11	even	2
688.6.a.h.1.6		10	1.1	even	1	trivial
1075.6.a.b.1.9		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)

\(f(q)\)	\(=\)	\(q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} +O(q^{10})\)	\(q-1.50169 q^{3} -37.8251 q^{5} +124.747 q^{7} -240.745 q^{9} -590.079 q^{11} +434.774 q^{13} +56.8015 q^{15} +1925.58 q^{17} -654.129 q^{19} -187.332 q^{21} -2805.03 q^{23} -1694.26 q^{25} +726.434 q^{27} -1456.23 q^{29} -4419.85 q^{31} +886.115 q^{33} -4718.58 q^{35} +3753.09 q^{37} -652.894 q^{39} +1972.85 q^{41} -1849.00 q^{43} +9106.20 q^{45} -2204.84 q^{47} -1245.12 q^{49} -2891.62 q^{51} +24984.2 q^{53} +22319.8 q^{55} +982.298 q^{57} +42756.2 q^{59} -21022.8 q^{61} -30032.3 q^{63} -16445.4 q^{65} +25272.1 q^{67} +4212.28 q^{69} -48082.8 q^{71} +58801.2 q^{73} +2544.25 q^{75} -73610.8 q^{77} -92704.7 q^{79} +57410.1 q^{81} +1849.64 q^{83} -72835.2 q^{85} +2186.80 q^{87} -70380.3 q^{89} +54236.8 q^{91} +6637.23 q^{93} +24742.5 q^{95} -88842.1 q^{97} +142059. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 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

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