LMFDB - Embedded newform 688.6.a.h.1.9

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			1.9
Root			$5.31531$ 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)

$f(q)$	$=$	$q+23.8469 q^{3} -52.5837 q^{5} +174.859 q^{7} +325.673 q^{9} +O(q^{10})$	$q+23.8469 q^{3} -52.5837 q^{5} +174.859 q^{7} +325.673 q^{9} +447.981 q^{11} +669.141 q^{13} -1253.96 q^{15} -849.648 q^{17} +1288.87 q^{19} +4169.83 q^{21} +378.254 q^{23} -359.956 q^{25} +1971.50 q^{27} +765.100 q^{29} +7094.59 q^{31} +10683.0 q^{33} -9194.71 q^{35} -7908.22 q^{37} +15956.9 q^{39} +12855.9 q^{41} -1849.00 q^{43} -17125.1 q^{45} -26785.7 q^{47} +13768.5 q^{49} -20261.5 q^{51} +30000.7 q^{53} -23556.5 q^{55} +30735.5 q^{57} -1247.64 q^{59} -48441.4 q^{61} +56946.8 q^{63} -35185.9 q^{65} -67004.8 q^{67} +9020.17 q^{69} +74553.1 q^{71} -18066.8 q^{73} -8583.82 q^{75} +78333.4 q^{77} +63230.5 q^{79} -32124.6 q^{81} +88066.7 q^{83} +44677.6 q^{85} +18245.2 q^{87} +50373.5 q^{89} +117005. q^{91} +169184. q^{93} -67773.5 q^{95} +17502.1 q^{97} +145895. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	23.8469	1.52978	0.764889	−	0.644163i	$-0.222794\pi$
$3$	23.8469	1.52978	0.764889	+	0.644163i	$0.222794\pi$
$4$	0	0
$5$	−52.5837	−0.940646	−0.470323	−	0.882494i	$-0.655863\pi$
$5$	−52.5837	−0.940646	−0.470323	+	0.882494i	$0.655863\pi$
$6$	0	0
$7$	174.859	1.34878	0.674391	−	0.738374i	$-0.264406\pi$
$7$	174.859	1.34878	0.674391	+	0.738374i	$0.264406\pi$
$8$	0	0
$9$	325.673	1.34022
$10$	0	0
$11$	447.981	1.11629	0.558147	−	0.829742i	$-0.311513\pi$
$11$	447.981	1.11629	0.558147	+	0.829742i	$0.311513\pi$
$12$	0	0
$13$	669.141	1.09814	0.549072	−	0.835775i	$-0.314981\pi$
$13$	669.141	1.09814	0.549072	+	0.835775i	$0.314981\pi$
$14$	0	0
$15$	−1253.96	−1.43898
$16$	0	0
$17$	−849.648	−0.713045	−0.356522	−	0.934287i	$-0.616038\pi$
$17$	−849.648	−0.713045	−0.356522	+	0.934287i	$0.616038\pi$
$18$	0	0
$19$	1288.87	0.819078	0.409539	−	0.912293i	$-0.365690\pi$
$19$	1288.87	0.819078	0.409539	+	0.912293i	$0.365690\pi$
$20$	0	0
$21$	4169.83	2.06334
$22$	0	0
$23$	378.254	0.149095	0.0745476	−	0.997217i	$-0.476249\pi$
$23$	378.254	0.149095	0.0745476	+	0.997217i	$0.476249\pi$
$24$	0	0
$25$	−359.956	−0.115186
$26$	0	0
$27$	1971.50	0.520459
$28$	0	0
$29$	765.100	0.168936	0.0844682	−	0.996426i	$-0.473081\pi$
$29$	765.100	0.168936	0.0844682	+	0.996426i	$0.473081\pi$
$30$	0	0
$31$	7094.59	1.32594	0.662969	−	0.748647i	$-0.269296\pi$
$31$	7094.59	1.32594	0.662969	+	0.748647i	$0.269296\pi$
$32$	0	0
$33$	10683.0	1.70768
$34$	0	0
$35$	−9194.71	−1.26873
$36$	0	0
$37$	−7908.22	−0.949674	−0.474837	−	0.880074i	$-0.657493\pi$
$37$	−7908.22	−0.949674	−0.474837	+	0.880074i	$0.657493\pi$
$38$	0	0
$39$	15956.9	1.67992
$40$	0	0
$41$	12855.9	1.19438	0.597192	−	0.802098i	$-0.296283\pi$
$41$	12855.9	1.19438	0.597192	+	0.802098i	$0.296283\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	−17125.1	−1.26067
$46$	0	0
$47$	−26785.7	−1.76871	−0.884357	−	0.466811i	$-0.845403\pi$
$47$	−26785.7	−1.76871	−0.884357	+	0.466811i	$0.845403\pi$
$48$	0	0
$49$	13768.5	0.819215
$50$	0	0
$51$	−20261.5	−1.09080
$52$	0	0
$53$	30000.7	1.46704	0.733520	−	0.679668i	$-0.237876\pi$
$53$	30000.7	1.46704	0.733520	+	0.679668i	$0.237876\pi$
$54$	0	0
$55$	−23556.5	−1.05004
$56$	0	0
$57$	30735.5	1.25301
$58$	0	0
$59$	−1247.64	−0.0466614	−0.0233307	−	0.999728i	$-0.507427\pi$
$59$	−1247.64	−0.0466614	−0.0233307	+	0.999728i	$0.507427\pi$
$60$	0	0
$61$	−48441.4	−1.66683	−0.833417	−	0.552644i	$-0.813619\pi$
$61$	−48441.4	−1.66683	−0.833417	+	0.552644i	$0.813619\pi$
$62$	0	0
$63$	56946.8	1.80766
$64$	0	0
$65$	−35185.9	−1.03296
$66$	0	0
$67$	−67004.8	−1.82356	−0.911778	−	0.410683i	$-0.865290\pi$
$67$	−67004.8	−1.82356	−0.911778	+	0.410683i	$0.865290\pi$
$68$	0	0
$69$	9020.17	0.228082
$70$	0	0
$71$	74553.1	1.75517	0.877587	−	0.479418i	$-0.159152\pi$
$71$	74553.1	1.75517	0.877587	+	0.479418i	$0.159152\pi$
$72$	0	0
$73$	−18066.8	−0.396802	−0.198401	−	0.980121i	$-0.563575\pi$
$73$	−18066.8	−0.396802	−0.198401	+	0.980121i	$0.563575\pi$
$74$	0	0
$75$	−8583.82	−0.176209
$76$	0	0
$77$	78333.4	1.50564
$78$	0	0
$79$	63230.5	1.13988	0.569939	−	0.821687i	$-0.306967\pi$
$79$	63230.5	1.13988	0.569939	+	0.821687i	$0.306967\pi$
$80$	0	0
$81$	−32124.6	−0.544033
$82$	0	0
$83$	88066.7	1.40319	0.701595	−	0.712576i	$-0.252472\pi$
$83$	88066.7	1.40319	0.701595	+	0.712576i	$0.252472\pi$
$84$	0	0
$85$	44677.6	0.670723
$86$	0	0
$87$	18245.2	0.258435
$88$	0	0
$89$	50373.5	0.674104	0.337052	−	0.941486i	$-0.390570\pi$
$89$	50373.5	0.674104	0.337052	+	0.941486i	$0.390570\pi$
$90$	0	0
$91$	117005.	1.48116
$92$	0	0
$93$	169184.	2.02839
$94$	0	0
$95$	−67773.5	−0.770462
$96$	0	0
$97$	17502.1	0.188869	0.0944345	−	0.995531i	$-0.469896\pi$
$97$	17502.1	0.188869	0.0944345	+	0.995531i	$0.469896\pi$
$98$	0	0
$99$	145895.	1.49608
$100$	0	0
$101$	182669.	1.78181	0.890906	−	0.454188i	$-0.150071\pi$
$101$	182669.	1.78181	0.890906	+	0.454188i	$0.150071\pi$
$102$	0	0
$103$	110578.	1.02702	0.513508	−	0.858085i	$-0.328346\pi$
$103$	110578.	1.02702	0.513508	+	0.858085i	$0.328346\pi$
$104$	0	0
$105$	−219265.	−1.94087
$106$	0	0
$107$	43935.4	0.370984	0.185492	−	0.982646i	$-0.440612\pi$
$107$	43935.4	0.370984	0.185492	+	0.982646i	$0.440612\pi$
$108$	0	0
$109$	−78170.4	−0.630197	−0.315098	−	0.949059i	$-0.602038\pi$
$109$	−78170.4	−0.630197	−0.315098	+	0.949059i	$0.602038\pi$
$110$	0	0
$111$	−188586.	−1.45279
$112$	0	0
$113$	23097.5	0.170165	0.0850823	−	0.996374i	$-0.472885\pi$
$113$	23097.5	0.170165	0.0850823	+	0.996374i	$0.472885\pi$
$114$	0	0
$115$	−19890.0	−0.140246
$116$	0	0
$117$	217921.	1.47175
$118$	0	0
$119$	−148568.	−0.961743
$120$	0	0
$121$	39636.3	0.246110
$122$	0	0
$123$	306574.	1.82714
$124$	0	0
$125$	183252.	1.04899
$126$	0	0
$127$	−218127.	−1.20005	−0.600027	−	0.799980i	$-0.704844\pi$
$127$	−218127.	−1.20005	−0.600027	+	0.799980i	$0.704844\pi$
$128$	0	0
$129$	−44092.9	−0.233289
$130$	0	0
$131$	−107160.	−0.545575	−0.272787	−	0.962074i	$-0.587946\pi$
$131$	−107160.	−0.545575	−0.272787	+	0.962074i	$0.587946\pi$
$132$	0	0
$133$	225370.	1.10476
$134$	0	0
$135$	−103668.	−0.489567
$136$	0	0
$137$	−168770.	−0.768234	−0.384117	−	0.923284i	$-0.625494\pi$
$137$	−168770.	−0.768234	−0.384117	+	0.923284i	$0.625494\pi$
$138$	0	0
$139$	−294636.	−1.29345	−0.646725	−	0.762723i	$-0.723862\pi$
$139$	−294636.	−1.29345	−0.646725	+	0.762723i	$0.723862\pi$
$140$	0	0
$141$	−638754.	−2.70574
$142$	0	0
$143$	299763.	1.22585
$144$	0	0
$145$	−40231.8	−0.158909
$146$	0	0
$147$	328337.	1.25322
$148$	0	0
$149$	121995.	0.450169	0.225084	−	0.974339i	$-0.427734\pi$
$149$	121995.	0.450169	0.225084	+	0.974339i	$0.427734\pi$
$150$	0	0
$151$	515468.	1.83975	0.919877	−	0.392208i	$-0.128289\pi$
$151$	515468.	1.83975	0.919877	+	0.392208i	$0.128289\pi$
$152$	0	0
$153$	−276708.	−0.955636
$154$	0	0
$155$	−373060.	−1.24724
$156$	0	0
$157$	178197.	0.576969	0.288484	−	0.957485i	$-0.406849\pi$
$157$	178197.	0.576969	0.288484	+	0.957485i	$0.406849\pi$
$158$	0	0
$159$	715423.	2.24424
$160$	0	0
$161$	66140.9	0.201097
$162$	0	0
$163$	111853.	0.329747	0.164873	−	0.986315i	$-0.447278\pi$
$163$	111853.	0.329747	0.164873	+	0.986315i	$0.447278\pi$
$164$	0	0
$165$	−561749.	−1.60632
$166$	0	0
$167$	128666.	0.357003	0.178501	−	0.983940i	$-0.442875\pi$
$167$	128666.	0.357003	0.178501	+	0.983940i	$0.442875\pi$
$168$	0	0
$169$	76457.2	0.205922
$170$	0	0
$171$	419750.	1.09774
$172$	0	0
$173$	553314.	1.40558	0.702792	−	0.711396i	$-0.251937\pi$
$173$	553314.	1.40558	0.702792	+	0.711396i	$0.251937\pi$
$174$	0	0
$175$	−62941.4	−0.155361
$176$	0	0
$177$	−29752.2	−0.0713816
$178$	0	0
$179$	302733.	0.706200	0.353100	−	0.935586i	$-0.385128\pi$
$179$	302733.	0.706200	0.353100	+	0.935586i	$0.385128\pi$
$180$	0	0
$181$	−702706.	−1.59433	−0.797164	−	0.603763i	$-0.793667\pi$
$181$	−702706.	−1.59433	−0.797164	+	0.603763i	$0.793667\pi$
$182$	0	0
$183$	−1.15518e6	−2.54989
$184$	0	0
$185$	415843.	0.893306
$186$	0	0
$187$	−380627.	−0.795967
$188$	0	0
$189$	344733.	0.701986
$190$	0	0
$191$	−105355.	−0.208965	−0.104482	−	0.994527i	$-0.533319\pi$
$191$	−105355.	−0.208965	−0.104482	+	0.994527i	$0.533319\pi$
$192$	0	0
$193$	11272.6	0.0217836	0.0108918	−	0.999941i	$-0.496533\pi$
$193$	11272.6	0.0217836	0.0108918	+	0.999941i	$0.496533\pi$
$194$	0	0
$195$	−839074.	−1.58021
$196$	0	0
$197$	44453.7	0.0816098	0.0408049	−	0.999167i	$-0.487008\pi$
$197$	44453.7	0.0816098	0.0408049	+	0.999167i	$0.487008\pi$
$198$	0	0
$199$	−1.01040e6	−1.80867	−0.904333	−	0.426827i	$-0.859631\pi$
$199$	−1.01040e6	−1.80867	−0.904333	+	0.426827i	$0.859631\pi$
$200$	0	0
$201$	−1.59786e6	−2.78963
$202$	0	0
$203$	133784.	0.227858
$204$	0	0
$205$	−676013.	−1.12349
$206$	0	0
$207$	123187.	0.199820
$208$	0	0
$209$	577390.	0.914331
$210$	0	0
$211$	−758827.	−1.17337	−0.586687	−	0.809814i	$-0.699568\pi$
$211$	−758827.	−1.17337	−0.586687	+	0.809814i	$0.699568\pi$
$212$	0	0
$213$	1.77786e6	2.68502
$214$	0	0
$215$	97227.2	0.143447
$216$	0	0
$217$	1.24055e6	1.78840
$218$	0	0
$219$	−430837.	−0.607019
$220$	0	0
$221$	−568535.	−0.783026
$222$	0	0
$223$	517178.	0.696431	0.348215	−	0.937415i	$-0.386788\pi$
$223$	517178.	0.696431	0.348215	+	0.937415i	$0.386788\pi$
$224$	0	0
$225$	−117228.	−0.154374
$226$	0	0
$227$	−627619.	−0.808409	−0.404205	−	0.914669i	$-0.632452\pi$
$227$	−627619.	−0.808409	−0.404205	+	0.914669i	$0.632452\pi$
$228$	0	0
$229$	800425.	1.00863	0.504315	−	0.863520i	$-0.331745\pi$
$229$	800425.	1.00863	0.504315	+	0.863520i	$0.331745\pi$
$230$	0	0
$231$	1.86801e6	2.30329
$232$	0	0
$233$	102393.	0.123560	0.0617802	−	0.998090i	$-0.480322\pi$
$233$	102393.	0.123560	0.0617802	+	0.998090i	$0.480322\pi$
$234$	0	0
$235$	1.40849e6	1.66373
$236$	0	0
$237$	1.50785e6	1.74376
$238$	0	0
$239$	879306.	0.995739	0.497869	−	0.867252i	$-0.334116\pi$
$239$	879306.	0.995739	0.497869	+	0.867252i	$0.334116\pi$
$240$	0	0
$241$	196740.	0.218198	0.109099	−	0.994031i	$-0.465203\pi$
$241$	196740.	0.218198	0.109099	+	0.994031i	$0.465203\pi$
$242$	0	0
$243$	−1.24514e6	−1.35271
$244$	0	0
$245$	−724001.	−0.770591
$246$	0	0
$247$	862436.	0.899466
$248$	0	0
$249$	2.10011e6	2.14657
$250$	0	0
$251$	798778.	0.800280	0.400140	−	0.916454i	$-0.368962\pi$
$251$	798778.	0.800280	0.400140	+	0.916454i	$0.368962\pi$
$252$	0	0
$253$	169451.	0.166434
$254$	0	0
$255$	1.06542e6	1.02606
$256$	0	0
$257$	−93933.3	−0.0887129	−0.0443565	−	0.999016i	$-0.514124\pi$
$257$	−93933.3	−0.0887129	−0.0443565	+	0.999016i	$0.514124\pi$
$258$	0	0
$259$	−1.38282e6	−1.28090
$260$	0	0
$261$	249173.	0.226412
$262$	0	0
$263$	898502.	0.800995	0.400497	−	0.916298i	$-0.368837\pi$
$263$	898502.	0.800995	0.400497	+	0.916298i	$0.368837\pi$
$264$	0	0
$265$	−1.57755e6	−1.37996
$266$	0	0
$267$	1.20125e6	1.03123
$268$	0	0
$269$	−921887.	−0.776778	−0.388389	−	0.921495i	$-0.626968\pi$
$269$	−921887.	−0.776778	−0.388389	+	0.921495i	$0.626968\pi$
$270$	0	0
$271$	227722.	0.188357	0.0941784	−	0.995555i	$-0.469978\pi$
$271$	227722.	0.188357	0.0941784	+	0.995555i	$0.469978\pi$
$272$	0	0
$273$	2.79021e6	2.26584
$274$	0	0
$275$	−161253.	−0.128581
$276$	0	0
$277$	1.76180e6	1.37961	0.689806	−	0.723994i	$-0.257696\pi$
$277$	1.76180e6	1.37961	0.689806	+	0.723994i	$0.257696\pi$
$278$	0	0
$279$	2.31052e6	1.77705
$280$	0	0
$281$	−270410.	−0.204295	−0.102147	−	0.994769i	$-0.532571\pi$
$281$	−270410.	−0.204295	−0.102147	+	0.994769i	$0.532571\pi$
$282$	0	0
$283$	−2.04589e6	−1.51851	−0.759254	−	0.650795i	$-0.774436\pi$
$283$	−2.04589e6	−1.51851	−0.759254	+	0.650795i	$0.774436\pi$
$284$	0	0
$285$	−1.61619e6	−1.17863
$286$	0	0
$287$	2.24797e6	1.61096
$288$	0	0
$289$	−697955.	−0.491567
$290$	0	0
$291$	417370.	0.288927
$292$	0	0
$293$	1.15172e6	0.783751	0.391875	−	0.920018i	$-0.371826\pi$
$293$	1.15172e6	0.783751	0.391875	+	0.920018i	$0.371826\pi$
$294$	0	0
$295$	65605.3	0.0438919
$296$	0	0
$297$	883193.	0.580984
$298$	0	0
$299$	253105.	0.163728
$300$	0	0
$301$	−323314.	−0.205687
$302$	0	0
$303$	4.35609e6	2.72578
$304$	0	0
$305$	2.54723e6	1.56790
$306$	0	0
$307$	1.88053e6	1.13877	0.569384	−	0.822072i	$-0.307182\pi$
$307$	1.88053e6	1.13877	0.569384	+	0.822072i	$0.307182\pi$
$308$	0	0
$309$	2.63695e6	1.57111
$310$	0	0
$311$	−1.94805e6	−1.14209	−0.571043	−	0.820920i	$-0.693461\pi$
$311$	−1.94805e6	−1.14209	−0.571043	+	0.820920i	$0.693461\pi$
$312$	0	0
$313$	−944428.	−0.544889	−0.272445	−	0.962171i	$-0.587832\pi$
$313$	−944428.	−0.544889	−0.272445	+	0.962171i	$0.587832\pi$
$314$	0	0
$315$	−2.99447e6	−1.70037
$316$	0	0
$317$	34149.2	0.0190868	0.00954339	−	0.999954i	$-0.496962\pi$
$317$	34149.2	0.0190868	0.00954339	+	0.999954i	$0.496962\pi$
$318$	0	0
$319$	342751.	0.188583
$320$	0	0
$321$	1.04772e6	0.567522
$322$	0	0
$323$	−1.09509e6	−0.584039
$324$	0	0
$325$	−240861.	−0.126491
$326$	0	0
$327$	−1.86412e6	−0.964061
$328$	0	0
$329$	−4.68370e6	−2.38561
$330$	0	0
$331$	−1.73996e6	−0.872908	−0.436454	−	0.899727i	$-0.643766\pi$
$331$	−1.73996e6	−0.872908	−0.436454	+	0.899727i	$0.643766\pi$
$332$	0	0
$333$	−2.57549e6	−1.27277
$334$	0	0
$335$	3.52336e6	1.71532
$336$	0	0
$337$	−1.09453e6	−0.524990	−0.262495	−	0.964933i	$-0.584545\pi$
$337$	−1.09453e6	−0.524990	−0.262495	+	0.964933i	$0.584545\pi$
$338$	0	0
$339$	550804.	0.260314
$340$	0	0
$341$	3.17825e6	1.48014
$342$	0	0
$343$	−531301.	−0.243840
$344$	0	0
$345$	−474314.	−0.214545
$346$	0	0
$347$	561799.	0.250471	0.125236	−	0.992127i	$-0.460031\pi$
$347$	561799.	0.250471	0.125236	+	0.992127i	$0.460031\pi$
$348$	0	0
$349$	2.06827e6	0.908957	0.454478	−	0.890758i	$-0.349826\pi$
$349$	2.06827e6	0.908957	0.454478	+	0.890758i	$0.349826\pi$
$350$	0	0
$351$	1.31921e6	0.571539
$352$	0	0
$353$	568169.	0.242684	0.121342	−	0.992611i	$-0.461280\pi$
$353$	568169.	0.242684	0.121342	+	0.992611i	$0.461280\pi$
$354$	0	0
$355$	−3.92028e6	−1.65100
$356$	0	0
$357$	−3.54289e6	−1.47125
$358$	0	0
$359$	−2.57455e6	−1.05430	−0.527151	−	0.849771i	$-0.676740\pi$
$359$	−2.57455e6	−1.05430	−0.527151	+	0.849771i	$0.676740\pi$
$360$	0	0
$361$	−814914.	−0.329112
$362$	0	0
$363$	945202.	0.376494
$364$	0	0
$365$	950019.	0.373250
$366$	0	0
$367$	−272102.	−0.105455	−0.0527274	−	0.998609i	$-0.516791\pi$
$367$	−272102.	−0.105455	−0.0527274	+	0.998609i	$0.516791\pi$
$368$	0	0
$369$	4.18683e6	1.60074
$370$	0	0
$371$	5.24589e6	1.97872
$372$	0	0
$373$	−268967.	−0.100098	−0.0500491	−	0.998747i	$-0.515938\pi$
$373$	−268967.	−0.100098	−0.0500491	+	0.998747i	$0.515938\pi$
$374$	0	0
$375$	4.36998e6	1.60473
$376$	0	0
$377$	511960.	0.185517
$378$	0	0
$379$	522052.	0.186688	0.0933438	−	0.995634i	$-0.470244\pi$
$379$	522052.	0.186688	0.0933438	+	0.995634i	$0.470244\pi$
$380$	0	0
$381$	−5.20166e6	−1.83582
$382$	0	0
$383$	−2.00529e6	−0.698523	−0.349262	−	0.937025i	$-0.613568\pi$
$383$	−2.00529e6	−0.698523	−0.349262	+	0.937025i	$0.613568\pi$
$384$	0	0
$385$	−4.11906e6	−1.41627
$386$	0	0
$387$	−602170.	−0.204381
$388$	0	0
$389$	−1.10827e6	−0.371340	−0.185670	−	0.982612i	$-0.559446\pi$
$389$	−1.10827e6	−0.371340	−0.185670	+	0.982612i	$0.559446\pi$
$390$	0	0
$391$	−321383.	−0.106312
$392$	0	0
$393$	−2.55543e6	−0.834608
$394$	0	0
$395$	−3.32489e6	−1.07222
$396$	0	0
$397$	−3.77816e6	−1.20311	−0.601554	−	0.798832i	$-0.705452\pi$
$397$	−3.77816e6	−1.20311	−0.601554	+	0.798832i	$0.705452\pi$
$398$	0	0
$399$	5.37437e6	1.69003
$400$	0	0
$401$	1.64922e6	0.512173	0.256086	−	0.966654i	$-0.417567\pi$
$401$	1.64922e6	0.512173	0.256086	+	0.966654i	$0.417567\pi$
$402$	0	0
$403$	4.74729e6	1.45607
$404$	0	0
$405$	1.68923e6	0.511742
$406$	0	0
$407$	−3.54273e6	−1.06011
$408$	0	0
$409$	−2.14199e6	−0.633153	−0.316577	−	0.948567i	$-0.602533\pi$
$409$	−2.14199e6	−0.633153	−0.316577	+	0.948567i	$0.602533\pi$
$410$	0	0
$411$	−4.02463e6	−1.17523
$412$	0	0
$413$	−218160.	−0.0629361
$414$	0	0
$415$	−4.63087e6	−1.31990
$416$	0	0
$417$	−7.02616e6	−1.97869
$418$	0	0
$419$	−6.49479e6	−1.80730	−0.903650	−	0.428272i	$-0.859123\pi$
$419$	−6.49479e6	−1.80730	−0.903650	+	0.428272i	$0.859123\pi$
$420$	0	0
$421$	3.30359e6	0.908408	0.454204	−	0.890898i	$-0.349924\pi$
$421$	3.30359e6	0.908408	0.454204	+	0.890898i	$0.349924\pi$
$422$	0	0
$423$	−8.72337e6	−2.37046
$424$	0	0
$425$	305836.	0.0821327
$426$	0	0
$427$	−8.47040e6	−2.24820
$428$	0	0
$429$	7.14841e6	1.87528
$430$	0	0
$431$	2.23343e6	0.579133	0.289567	−	0.957158i	$-0.406489\pi$
$431$	2.23343e6	0.579133	0.289567	+	0.957158i	$0.406489\pi$
$432$	0	0
$433$	−3.81775e6	−0.978561	−0.489280	−	0.872127i	$-0.662741\pi$
$433$	−3.81775e6	−0.978561	−0.489280	+	0.872127i	$0.662741\pi$
$434$	0	0
$435$	−959402.	−0.243096
$436$	0	0
$437$	487520.	0.122121
$438$	0	0
$439$	−4.94254e6	−1.22402	−0.612011	−	0.790850i	$-0.709639\pi$
$439$	−4.94254e6	−1.22402	−0.612011	+	0.790850i	$0.709639\pi$
$440$	0	0
$441$	4.48404e6	1.09793
$442$	0	0
$443$	740033.	0.179160	0.0895802	−	0.995980i	$-0.471447\pi$
$443$	740033.	0.179160	0.0895802	+	0.995980i	$0.471447\pi$
$444$	0	0
$445$	−2.64882e6	−0.634093
$446$	0	0
$447$	2.90919e6	0.688658
$448$	0	0
$449$	−1.75356e6	−0.410492	−0.205246	−	0.978710i	$-0.565799\pi$
$449$	−1.75356e6	−0.410492	−0.205246	+	0.978710i	$0.565799\pi$
$450$	0	0
$451$	5.75922e6	1.33328
$452$	0	0
$453$	1.22923e7	2.81441
$454$	0	0
$455$	−6.15256e6	−1.39325
$456$	0	0
$457$	1.43140e6	0.320605	0.160302	−	0.987068i	$-0.448753\pi$
$457$	1.43140e6	0.320605	0.160302	+	0.987068i	$0.448753\pi$
$458$	0	0
$459$	−1.67508e6	−0.371110
$460$	0	0
$461$	−8.02421e6	−1.75853	−0.879265	−	0.476332i	$-0.841966\pi$
$461$	−8.02421e6	−1.75853	−0.879265	+	0.476332i	$0.841966\pi$
$462$	0	0
$463$	5.42639e6	1.17641	0.588205	−	0.808712i	$-0.299835\pi$
$463$	5.42639e6	1.17641	0.588205	+	0.808712i	$0.299835\pi$
$464$	0	0
$465$	−8.89631e6	−1.90800
$466$	0	0
$467$	4.68842e6	0.994796	0.497398	−	0.867522i	$-0.334289\pi$
$467$	4.68842e6	0.994796	0.497398	+	0.867522i	$0.334289\pi$
$468$	0	0
$469$	−1.17164e7	−2.45958
$470$	0	0
$471$	4.24945e6	0.882634
$472$	0	0
$473$	−828318.	−0.170233
$474$	0	0
$475$	−463936.	−0.0943461
$476$	0	0
$477$	9.77043e6	1.96615
$478$	0	0
$479$	−2.64873e6	−0.527471	−0.263735	−	0.964595i	$-0.584955\pi$
$479$	−2.64873e6	−0.527471	−0.263735	+	0.964595i	$0.584955\pi$
$480$	0	0
$481$	−5.29172e6	−1.04288
$482$	0	0
$483$	1.57725e6	0.307634
$484$	0	0
$485$	−920324.	−0.177659
$486$	0	0
$487$	−2.23774e6	−0.427550	−0.213775	−	0.976883i	$-0.568576\pi$
$487$	−2.23774e6	−0.427550	−0.213775	+	0.976883i	$0.568576\pi$
$488$	0	0
$489$	2.66736e6	0.504439
$490$	0	0
$491$	−3.10094e6	−0.580484	−0.290242	−	0.956953i	$-0.593736\pi$
$491$	−3.10094e6	−0.580484	−0.290242	+	0.956953i	$0.593736\pi$
$492$	0	0
$493$	−650066.	−0.120459
$494$	0	0
$495$	−7.67172e6	−1.40728
$496$	0	0
$497$	1.30363e7	2.36735
$498$	0	0
$499$	−5.58008e6	−1.00320	−0.501601	−	0.865099i	$-0.667256\pi$
$499$	−5.58008e6	−1.00320	−0.501601	+	0.865099i	$0.667256\pi$
$500$	0	0
$501$	3.06827e6	0.546135
$502$	0	0
$503$	−5.10876e6	−0.900317	−0.450158	−	0.892949i	$-0.648632\pi$
$503$	−5.10876e6	−0.900317	−0.450158	+	0.892949i	$0.648632\pi$
$504$	0	0
$505$	−9.60542e6	−1.67605
$506$	0	0
$507$	1.82327e6	0.315014
$508$	0	0
$509$	2.93415e6	0.501982	0.250991	−	0.967989i	$-0.419244\pi$
$509$	2.93415e6	0.501982	0.250991	+	0.967989i	$0.419244\pi$
$510$	0	0
$511$	−3.15914e6	−0.535200
$512$	0	0
$513$	2.54100e6	0.426296
$514$	0	0
$515$	−5.81462e6	−0.966058
$516$	0	0
$517$	−1.19995e7	−1.97440
$518$	0	0
$519$	1.31948e7	2.15023
$520$	0	0
$521$	−9.22452e6	−1.48885	−0.744423	−	0.667709i	$-0.767275\pi$
$521$	−9.22452e6	−1.48885	−0.744423	+	0.667709i	$0.767275\pi$
$522$	0	0
$523$	4.64359e6	0.742335	0.371168	−	0.928566i	$-0.378958\pi$
$523$	4.64359e6	0.742335	0.371168	+	0.928566i	$0.378958\pi$
$524$	0	0
$525$	−1.50095e6	−0.237667
$526$	0	0
$527$	−6.02791e6	−0.945453
$528$	0	0
$529$	−6.29327e6	−0.977771
$530$	0	0
$531$	−406322.	−0.0625365
$532$	0	0
$533$	8.60244e6	1.31161
$534$	0	0
$535$	−2.31028e6	−0.348964
$536$	0	0
$537$	7.21924e6	1.08033
$538$	0	0
$539$	6.16805e6	0.914484
$540$	0	0
$541$	3.62594e6	0.532632	0.266316	−	0.963886i	$-0.414193\pi$
$541$	3.62594e6	0.532632	0.266316	+	0.963886i	$0.414193\pi$
$542$	0	0
$543$	−1.67573e7	−2.43897
$544$	0	0
$545$	4.11049e6	0.592792
$546$	0	0
$547$	5.56425e6	0.795131	0.397565	−	0.917574i	$-0.369855\pi$
$547$	5.56425e6	0.795131	0.397565	+	0.917574i	$0.369855\pi$
$548$	0	0
$549$	−1.57761e7	−2.23392
$550$	0	0
$551$	986114.	0.138372
$552$	0	0
$553$	1.10564e7	1.53745
$554$	0	0
$555$	9.91656e6	1.36656
$556$	0	0
$557$	6.83963e6	0.934103	0.467052	−	0.884230i	$-0.345316\pi$
$557$	6.83963e6	0.934103	0.467052	+	0.884230i	$0.345316\pi$
$558$	0	0
$559$	−1.23724e6	−0.167465
$560$	0	0
$561$	−9.07675e6	−1.21765
$562$	0	0
$563$	5.62439e6	0.747833	0.373916	−	0.927462i	$-0.378015\pi$
$563$	5.62439e6	0.747833	0.373916	+	0.927462i	$0.378015\pi$
$564$	0	0
$565$	−1.21455e6	−0.160065
$566$	0	0
$567$	−5.61726e6	−0.733782
$568$	0	0
$569$	−4.17021e6	−0.539979	−0.269990	−	0.962863i	$-0.587020\pi$
$569$	−4.17021e6	−0.539979	−0.269990	+	0.962863i	$0.587020\pi$
$570$	0	0
$571$	8.60607e6	1.10462	0.552312	−	0.833637i	$-0.313746\pi$
$571$	8.60607e6	1.10462	0.552312	+	0.833637i	$0.313746\pi$
$572$	0	0
$573$	−2.51239e6	−0.319669
$574$	0	0
$575$	−136155.	−0.0171737
$576$	0	0
$577$	−3.23538e6	−0.404563	−0.202282	−	0.979327i	$-0.564836\pi$
$577$	−3.23538e6	−0.404563	−0.202282	+	0.979327i	$0.564836\pi$
$578$	0	0
$579$	268815.	0.0333240
$580$	0	0
$581$	1.53992e7	1.89260
$582$	0	0
$583$	1.34398e7	1.63765
$584$	0	0
$585$	−1.14591e7	−1.38440
$586$	0	0
$587$	1.12970e7	1.35322	0.676609	−	0.736342i	$-0.263449\pi$
$587$	1.12970e7	1.35322	0.676609	+	0.736342i	$0.263449\pi$
$588$	0	0
$589$	9.14401e6	1.08605
$590$	0	0
$591$	1.06008e6	0.124845
$592$	0	0
$593$	1.61368e7	1.88444	0.942218	−	0.335000i	$-0.108736\pi$
$593$	1.61368e7	1.88444	0.942218	+	0.335000i	$0.108736\pi$
$594$	0	0
$595$	7.81227e6	0.904659
$596$	0	0
$597$	−2.40948e7	−2.76686
$598$	0	0
$599$	−1.14121e7	−1.29957	−0.649785	−	0.760118i	$-0.725141\pi$
$599$	−1.14121e7	−1.29957	−0.649785	+	0.760118i	$0.725141\pi$
$600$	0	0
$601$	1.15192e7	1.30088	0.650438	−	0.759559i	$-0.274585\pi$
$601$	1.15192e7	1.30088	0.650438	+	0.759559i	$0.274585\pi$
$602$	0	0
$603$	−2.18217e7	−2.44396
$604$	0	0
$605$	−2.08422e6	−0.231503
$606$	0	0
$607$	−1.48986e7	−1.64125	−0.820625	−	0.571466i	$-0.806375\pi$
$607$	−1.48986e7	−1.64125	−0.820625	+	0.571466i	$0.806375\pi$
$608$	0	0
$609$	3.19034e6	0.348573
$610$	0	0
$611$	−1.79234e7	−1.94230
$612$	0	0
$613$	−5.39999e6	−0.580419	−0.290210	−	0.956963i	$-0.593725\pi$
$613$	−5.39999e6	−0.580419	−0.290210	+	0.956963i	$0.593725\pi$
$614$	0	0
$615$	−1.61208e7	−1.71869
$616$	0	0
$617$	−1.26228e7	−1.33489	−0.667443	−	0.744660i	$-0.732611\pi$
$617$	−1.26228e7	−1.33489	−0.667443	+	0.744660i	$0.732611\pi$
$618$	0	0
$619$	−1.43557e7	−1.50591	−0.752953	−	0.658074i	$-0.771371\pi$
$619$	−1.43557e7	−1.50591	−0.752953	+	0.658074i	$0.771371\pi$
$620$	0	0
$621$	745725.	0.0775979
$622$	0	0
$623$	8.80824e6	0.909220
$624$	0	0
$625$	−8.51120e6	−0.871546
$626$	0	0
$627$	1.37689e7	1.39872
$628$	0	0
$629$	6.71920e6	0.677160
$630$	0	0
$631$	6.76873e6	0.676759	0.338379	−	0.941010i	$-0.390121\pi$
$631$	6.76873e6	0.676759	0.338379	+	0.941010i	$0.390121\pi$
$632$	0	0
$633$	−1.80956e7	−1.79500
$634$	0	0
$635$	1.14699e7	1.12883
$636$	0	0
$637$	9.21310e6	0.899616
$638$	0	0
$639$	2.42799e7	2.35232
$640$	0	0
$641$	−1.51643e6	−0.145773	−0.0728867	−	0.997340i	$-0.523221\pi$
$641$	−1.51643e6	−0.145773	−0.0728867	+	0.997340i	$0.523221\pi$
$642$	0	0
$643$	−6.58876e6	−0.628458	−0.314229	−	0.949347i	$-0.601746\pi$
$643$	−6.58876e6	−0.628458	−0.314229	+	0.949347i	$0.601746\pi$
$644$	0	0
$645$	2.31857e6	0.219442
$646$	0	0
$647$	−9.99129e6	−0.938341	−0.469171	−	0.883108i	$-0.655447\pi$
$647$	−9.99129e6	−0.938341	−0.469171	+	0.883108i	$0.655447\pi$
$648$	0	0
$649$	−558918.	−0.0520878
$650$	0	0
$651$	2.95833e7	2.73586
$652$	0	0
$653$	−1.32714e7	−1.21796	−0.608980	−	0.793186i	$-0.708421\pi$
$653$	−1.32714e7	−1.21796	−0.608980	+	0.793186i	$0.708421\pi$
$654$	0	0
$655$	5.63486e6	0.513192
$656$	0	0
$657$	−5.88387e6	−0.531802
$658$	0	0
$659$	−3.69381e6	−0.331330	−0.165665	−	0.986182i	$-0.552977\pi$
$659$	−3.69381e6	−0.331330	−0.165665	+	0.986182i	$0.552977\pi$
$660$	0	0
$661$	8.97800e6	0.799238	0.399619	−	0.916681i	$-0.369142\pi$
$661$	8.97800e6	0.799238	0.399619	+	0.916681i	$0.369142\pi$
$662$	0	0
$663$	−1.35578e7	−1.19786
$664$	0	0
$665$	−1.18508e7	−1.03919
$666$	0	0
$667$	289402.	0.0251876
$668$	0	0
$669$	1.23331e7	1.06538
$670$	0	0
$671$	−2.17009e7	−1.86068
$672$	0	0
$673$	8.59992e6	0.731909	0.365954	−	0.930633i	$-0.380743\pi$
$673$	8.59992e6	0.731909	0.365954	+	0.930633i	$0.380743\pi$
$674$	0	0
$675$	−709651.	−0.0599495
$676$	0	0
$677$	−8.01663e6	−0.672233	−0.336117	−	0.941820i	$-0.609114\pi$
$677$	−8.01663e6	−0.672233	−0.336117	+	0.941820i	$0.609114\pi$
$678$	0	0
$679$	3.06039e6	0.254743
$680$	0	0
$681$	−1.49667e7	−1.23669
$682$	0	0
$683$	−2.81320e6	−0.230754	−0.115377	−	0.993322i	$-0.536808\pi$
$683$	−2.81320e6	−0.230754	−0.115377	+	0.993322i	$0.536808\pi$
$684$	0	0
$685$	8.87454e6	0.722636
$686$	0	0
$687$	1.90876e7	1.54298
$688$	0	0
$689$	2.00747e7	1.61102
$690$	0	0
$691$	−1.04922e7	−0.835935	−0.417967	−	0.908462i	$-0.637257\pi$
$691$	−1.04922e7	−0.835935	−0.417967	+	0.908462i	$0.637257\pi$
$692$	0	0
$693$	2.55111e7	2.01788
$694$	0	0
$695$	1.54931e7	1.21668
$696$	0	0
$697$	−1.09230e7	−0.851650
$698$	0	0
$699$	2.44175e6	0.189020
$700$	0	0
$701$	−2.37537e7	−1.82573	−0.912865	−	0.408263i	$-0.866135\pi$
$701$	−2.37537e7	−1.82573	−0.912865	+	0.408263i	$0.866135\pi$
$702$	0	0
$703$	−1.01927e7	−0.777856
$704$	0	0
$705$	3.35880e7	2.54514
$706$	0	0
$707$	3.19413e7	2.40328
$708$	0	0
$709$	−1.72000e7	−1.28503	−0.642516	−	0.766273i	$-0.722109\pi$
$709$	−1.72000e7	−1.28503	−0.642516	+	0.766273i	$0.722109\pi$
$710$	0	0
$711$	2.05925e7	1.52769
$712$	0	0
$713$	2.68356e6	0.197691
$714$	0	0
$715$	−1.57626e7	−1.15309
$716$	0	0
$717$	2.09687e7	1.52326
$718$	0	0
$719$	−2.22034e7	−1.60176	−0.800878	−	0.598827i	$-0.795634\pi$
$719$	−2.22034e7	−1.60176	−0.800878	+	0.598827i	$0.795634\pi$
$720$	0	0
$721$	1.93356e7	1.38522
$722$	0	0
$723$	4.69164e6	0.333794
$724$	0	0
$725$	−275402.	−0.0194591
$726$	0	0
$727$	1.95017e7	1.36847	0.684236	−	0.729261i	$-0.260136\pi$
$727$	1.95017e7	1.36847	0.684236	+	0.729261i	$0.260136\pi$
$728$	0	0
$729$	−2.18865e7	−1.52531
$730$	0	0
$731$	1.57100e6	0.108738
$732$	0	0
$733$	1.07906e7	0.741798	0.370899	−	0.928673i	$-0.379050\pi$
$733$	1.07906e7	0.741798	0.370899	+	0.928673i	$0.379050\pi$
$734$	0	0
$735$	−1.72651e7	−1.17883
$736$	0	0
$737$	−3.00169e7	−2.03562
$738$	0	0
$739$	1.44802e7	0.975358	0.487679	−	0.873023i	$-0.337844\pi$
$739$	1.44802e7	0.975358	0.487679	+	0.873023i	$0.337844\pi$
$740$	0	0
$741$	2.05664e7	1.37598
$742$	0	0
$743$	−5.44908e6	−0.362119	−0.181059	−	0.983472i	$-0.557953\pi$
$743$	−5.44908e6	−0.362119	−0.181059	+	0.983472i	$0.557953\pi$
$744$	0	0
$745$	−6.41493e6	−0.423449
$746$	0	0
$747$	2.86809e7	1.88058
$748$	0	0
$749$	7.68248e6	0.500376
$750$	0	0
$751$	2.78004e6	0.179867	0.0899335	−	0.995948i	$-0.471335\pi$
$751$	2.78004e6	0.179867	0.0899335	+	0.995948i	$0.471335\pi$
$752$	0	0
$753$	1.90484e7	1.22425
$754$	0	0
$755$	−2.71052e7	−1.73056
$756$	0	0
$757$	2.46778e7	1.56519	0.782595	−	0.622531i	$-0.213895\pi$
$757$	2.46778e7	1.56519	0.782595	+	0.622531i	$0.213895\pi$
$758$	0	0
$759$	4.04087e6	0.254607
$760$	0	0
$761$	−8.63006e6	−0.540197	−0.270098	−	0.962833i	$-0.587056\pi$
$761$	−8.63006e6	−0.540197	−0.270098	+	0.962833i	$0.587056\pi$
$762$	0	0
$763$	−1.36688e7	−0.849999
$764$	0	0
$765$	1.45503e7	0.898915
$766$	0	0
$767$	−834845.	−0.0512410
$768$	0	0
$769$	−1.79104e6	−0.109217	−0.0546083	−	0.998508i	$-0.517391\pi$
$769$	−1.79104e6	−0.109217	−0.0546083	+	0.998508i	$0.517391\pi$
$770$	0	0
$771$	−2.24002e6	−0.135711
$772$	0	0
$773$	−2.64220e7	−1.59044	−0.795219	−	0.606322i	$-0.792644\pi$
$773$	−2.64220e7	−1.59044	−0.795219	+	0.606322i	$0.792644\pi$
$774$	0	0
$775$	−2.55374e6	−0.152729
$776$	0	0
$777$	−3.29759e7	−1.95950
$778$	0	0
$779$	1.65696e7	0.978293
$780$	0	0
$781$	3.33984e7	1.95929
$782$	0	0
$783$	1.50839e6	0.0879244
$784$	0	0
$785$	−9.37028e6	−0.542723
$786$	0	0
$787$	−4.11413e6	−0.236778	−0.118389	−	0.992967i	$-0.537773\pi$
$787$	−4.11413e6	−0.236778	−0.118389	+	0.992967i	$0.537773\pi$
$788$	0	0
$789$	2.14265e7	1.22534
$790$	0	0
$791$	4.03880e6	0.229515
$792$	0	0
$793$	−3.24142e7	−1.83043
$794$	0	0
$795$	−3.76196e7	−2.11104
$796$	0	0
$797$	3.23381e6	0.180330	0.0901651	−	0.995927i	$-0.471261\pi$
$797$	3.23381e6	0.180330	0.0901651	+	0.995927i	$0.471261\pi$
$798$	0	0
$799$	2.27584e7	1.26117
$800$	0	0
$801$	1.64053e7	0.903447
$802$	0	0
$803$	−8.09359e6	−0.442948
$804$	0	0
$805$	−3.47793e6	−0.189161
$806$	0	0
$807$	−2.19841e7	−1.18830
$808$	0	0
$809$	−1.52689e7	−0.820231	−0.410115	−	0.912034i	$-0.634512\pi$
$809$	−1.52689e7	−0.820231	−0.410115	+	0.912034i	$0.634512\pi$
$810$	0	0
$811$	3.04480e7	1.62558	0.812788	−	0.582560i	$-0.197949\pi$
$811$	3.04480e7	1.62558	0.812788	+	0.582560i	$0.197949\pi$
$812$	0	0
$813$	5.43045e6	0.288144
$814$	0	0
$815$	−5.88167e6	−0.310175
$816$	0	0
$817$	−2.38312e6	−0.124908
$818$	0	0
$819$	3.81054e7	1.98508
$820$	0	0
$821$	−2.25035e6	−0.116518	−0.0582590	−	0.998302i	$-0.518555\pi$
$821$	−2.25035e6	−0.116518	−0.0582590	+	0.998302i	$0.518555\pi$
$822$	0	0
$823$	1.66620e7	0.857485	0.428743	−	0.903427i	$-0.358957\pi$
$823$	1.66620e7	0.857485	0.428743	+	0.903427i	$0.358957\pi$
$824$	0	0
$825$	−3.84539e6	−0.196701
$826$	0	0
$827$	−3.70905e7	−1.88581	−0.942907	−	0.333056i	$-0.891920\pi$
$827$	−3.70905e7	−1.88581	−0.942907	+	0.333056i	$0.891920\pi$
$828$	0	0
$829$	2.88276e7	1.45688	0.728438	−	0.685111i	$-0.240246\pi$
$829$	2.88276e7	1.45688	0.728438	+	0.685111i	$0.240246\pi$
$830$	0	0
$831$	4.20134e7	2.11050
$832$	0	0
$833$	−1.16984e7	−0.584137
$834$	0	0
$835$	−6.76572e6	−0.335813
$836$	0	0
$837$	1.39870e7	0.690096
$838$	0	0
$839$	7.10869e6	0.348646	0.174323	−	0.984689i	$-0.444226\pi$
$839$	7.10869e6	0.348646	0.174323	+	0.984689i	$0.444226\pi$
$840$	0	0
$841$	−1.99258e7	−0.971460
$842$	0	0
$843$	−6.44843e6	−0.312525
$844$	0	0
$845$	−4.02040e6	−0.193699
$846$	0	0
$847$	6.93075e6	0.331949
$848$	0	0
$849$	−4.87882e7	−2.32298
$850$	0	0
$851$	−2.99131e6	−0.141592
$852$	0	0
$853$	6.01755e6	0.283170	0.141585	−	0.989926i	$-0.454780\pi$
$853$	6.01755e6	0.283170	0.141585	+	0.989926i	$0.454780\pi$
$854$	0	0
$855$	−2.20720e7	−1.03259
$856$	0	0
$857$	1.40484e7	0.653392	0.326696	−	0.945129i	$-0.394065\pi$
$857$	1.40484e7	0.653392	0.326696	+	0.945129i	$0.394065\pi$
$858$	0	0
$859$	−2.38780e7	−1.10412	−0.552060	−	0.833805i	$-0.686158\pi$
$859$	−2.38780e7	−1.10412	−0.552060	+	0.833805i	$0.686158\pi$
$860$	0	0
$861$	5.36071e7	2.46442
$862$	0	0
$863$	−3.12051e7	−1.42626	−0.713131	−	0.701031i	$-0.752723\pi$
$863$	−3.12051e7	−1.42626	−0.713131	+	0.701031i	$0.752723\pi$
$864$	0	0
$865$	−2.90953e7	−1.32216
$866$	0	0
$867$	−1.66440e7	−0.751988
$868$	0	0
$869$	2.83261e7	1.27244
$870$	0	0
$871$	−4.48357e7	−2.00253
$872$	0	0
$873$	5.69996e6	0.253126
$874$	0	0
$875$	3.20432e7	1.41487
$876$	0	0
$877$	−2.97111e6	−0.130443	−0.0652213	−	0.997871i	$-0.520775\pi$
$877$	−2.97111e6	−0.130443	−0.0652213	+	0.997871i	$0.520775\pi$
$878$	0	0
$879$	2.74649e7	1.19896
$880$	0	0
$881$	3.55211e7	1.54187	0.770934	−	0.636915i	$-0.219790\pi$
$881$	3.55211e7	1.54187	0.770934	+	0.636915i	$0.219790\pi$
$882$	0	0
$883$	−1.87105e7	−0.807576	−0.403788	−	0.914853i	$-0.632307\pi$
$883$	−1.87105e7	−0.807576	−0.403788	+	0.914853i	$0.632307\pi$
$884$	0	0
$885$	1.56448e6	0.0671448
$886$	0	0
$887$	−9.33897e6	−0.398557	−0.199278	−	0.979943i	$-0.563860\pi$
$887$	−9.33897e6	−0.398557	−0.199278	+	0.979943i	$0.563860\pi$
$888$	0	0
$889$	−3.81415e7	−1.61861
$890$	0	0
$891$	−1.43912e7	−0.607300
$892$	0	0
$893$	−3.45232e7	−1.44871
$894$	0	0
$895$	−1.59188e7	−0.664284
$896$	0	0
$897$	6.03577e6	0.250468
$898$	0	0
$899$	5.42807e6	0.223999
$900$	0	0
$901$	−2.54901e7	−1.04607
$902$	0	0
$903$	−7.71002e6	−0.314656
$904$	0	0
$905$	3.69509e7	1.49970
$906$	0	0
$907$	−3.89432e7	−1.57186	−0.785930	−	0.618315i	$-0.787816\pi$
$907$	−3.89432e7	−1.57186	−0.785930	+	0.618315i	$0.787816\pi$
$908$	0	0
$909$	5.94904e7	2.38802
$910$	0	0
$911$	−4.28351e7	−1.71003	−0.855015	−	0.518604i	$-0.826452\pi$
$911$	−4.28351e7	−1.71003	−0.855015	+	0.518604i	$0.826452\pi$
$912$	0	0
$913$	3.94522e7	1.56637
$914$	0	0
$915$	6.07435e7	2.39854
$916$	0	0
$917$	−1.87378e7	−0.735862
$918$	0	0
$919$	−881759.	−0.0344398	−0.0172199	−	0.999852i	$-0.505482\pi$
$919$	−881759.	−0.0344398	−0.0172199	+	0.999852i	$0.505482\pi$
$920$	0	0
$921$	4.48448e7	1.74206
$922$	0	0
$923$	4.98866e7	1.92743
$924$	0	0
$925$	2.84661e6	0.109389
$926$	0	0
$927$	3.60124e7	1.37643
$928$	0	0
$929$	−1.81365e7	−0.689468	−0.344734	−	0.938700i	$-0.612031\pi$
$929$	−1.81365e7	−0.689468	−0.344734	+	0.938700i	$0.612031\pi$
$930$	0	0
$931$	1.77459e7	0.671000
$932$	0	0
$933$	−4.64548e7	−1.74714
$934$	0	0
$935$	2.00148e7	0.748723
$936$	0	0
$937$	−1.75363e7	−0.652512	−0.326256	−	0.945281i	$-0.605787\pi$
$937$	−1.75363e7	−0.652512	−0.326256	+	0.945281i	$0.605787\pi$
$938$	0	0
$939$	−2.25217e7	−0.833559
$940$	0	0
$941$	7.69478e6	0.283284	0.141642	−	0.989918i	$-0.454762\pi$
$941$	7.69478e6	0.283284	0.141642	+	0.989918i	$0.454762\pi$
$942$	0	0
$943$	4.86281e6	0.178077
$944$	0	0
$945$	−1.81273e7	−0.660320
$946$	0	0
$947$	3.00889e7	1.09026	0.545131	−	0.838351i	$-0.316480\pi$
$947$	3.00889e7	1.09026	0.545131	+	0.838351i	$0.316480\pi$
$948$	0	0
$949$	−1.20892e7	−0.435746
$950$	0	0
$951$	814352.	0.0291985
$952$	0	0
$953$	2.31755e7	0.826601	0.413301	−	0.910595i	$-0.364376\pi$
$953$	2.31755e7	0.826601	0.413301	+	0.910595i	$0.364376\pi$
$954$	0	0
$955$	5.53997e6	0.196562
$956$	0	0
$957$	8.17353e6	0.288489
$958$	0	0
$959$	−2.95109e7	−1.03618
$960$	0	0
$961$	2.17041e7	0.758112
$962$	0	0
$963$	1.43086e7	0.497199
$964$	0	0
$965$	−592752.	−0.0204906
$966$	0	0
$967$	3.96944e7	1.36510	0.682548	−	0.730841i	$-0.260872\pi$
$967$	3.96944e7	1.36510	0.682548	+	0.730841i	$0.260872\pi$
$968$	0	0
$969$	−2.61144e7	−0.893450
$970$	0	0
$971$	6.90625e6	0.235068	0.117534	−	0.993069i	$-0.462501\pi$
$971$	6.90625e6	0.235068	0.117534	+	0.993069i	$0.462501\pi$
$972$	0	0
$973$	−5.15197e7	−1.74458
$974$	0	0
$975$	−5.74379e6	−0.193503
$976$	0	0
$977$	−1.38767e7	−0.465104	−0.232552	−	0.972584i	$-0.574708\pi$
$977$	−1.38767e7	−0.465104	−0.232552	+	0.972584i	$0.574708\pi$
$978$	0	0
$979$	2.25664e7	0.752498
$980$	0	0
$981$	−2.54580e7	−0.844602
$982$	0	0
$983$	−1.20330e7	−0.397183	−0.198591	−	0.980082i	$-0.563637\pi$
$983$	−1.20330e7	−0.397183	−0.198591	+	0.980082i	$0.563637\pi$
$984$	0	0
$985$	−2.33754e6	−0.0767659
$986$	0	0
$987$	−1.11692e8	−3.64945
$988$	0	0
$989$	−699391.	−0.0227368
$990$	0	0
$991$	−9.67955e6	−0.313091	−0.156546	−	0.987671i	$-0.550036\pi$
$991$	−9.67955e6	−0.313091	−0.156546	+	0.987671i	$0.550036\pi$
$992$	0	0
$993$	−4.14925e7	−1.33535
$994$	0	0
$995$	5.31303e7	1.70131
$996$	0	0
$997$	−273483.	−0.00871351	−0.00435676	−	0.999991i	$-0.501387\pi$
$997$	−273483.	−0.00871351	−0.00435676	+	0.999991i	$0.501387\pi$
$998$	0	0
$999$	−1.55910e7	−0.494266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.9		10
4.3	odd	2		43.6.a.b.1.3	✓	10
12.11	even	2		387.6.a.e.1.8		10
20.19	odd	2		1075.6.a.b.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.3	✓	10	4.3	odd	2
387.6.a.e.1.8		10	12.11	even	2
688.6.a.h.1.9		10	1.1	even	1	trivial
1075.6.a.b.1.8		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)

\(f(q)\)	\(=\)	\(q+23.8469 q^{3} -52.5837 q^{5} +174.859 q^{7} +325.673 q^{9} +O(q^{10})\)	\(q+23.8469 q^{3} -52.5837 q^{5} +174.859 q^{7} +325.673 q^{9} +447.981 q^{11} +669.141 q^{13} -1253.96 q^{15} -849.648 q^{17} +1288.87 q^{19} +4169.83 q^{21} +378.254 q^{23} -359.956 q^{25} +1971.50 q^{27} +765.100 q^{29} +7094.59 q^{31} +10683.0 q^{33} -9194.71 q^{35} -7908.22 q^{37} +15956.9 q^{39} +12855.9 q^{41} -1849.00 q^{43} -17125.1 q^{45} -26785.7 q^{47} +13768.5 q^{49} -20261.5 q^{51} +30000.7 q^{53} -23556.5 q^{55} +30735.5 q^{57} -1247.64 q^{59} -48441.4 q^{61} +56946.8 q^{63} -35185.9 q^{65} -67004.8 q^{67} +9020.17 q^{69} +74553.1 q^{71} -18066.8 q^{73} -8583.82 q^{75} +78333.4 q^{77} +63230.5 q^{79} -32124.6 q^{81} +88066.7 q^{83} +44677.6 q^{85} +18245.2 q^{87} +50373.5 q^{89} +117005. q^{91} +169184. q^{93} -67773.5 q^{95} +17502.1 q^{97} +145895. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 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

Fields

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Database

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