LMFDB - Embedded newform 688.6.a.h.1.5

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			1.5
Root			$-6.91219$ 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)

$f(q)$	$=$	$q-12.8799 q^{3} +79.5677 q^{5} +172.354 q^{7} -77.1083 q^{9} +O(q^{10})$	$q-12.8799 q^{3} +79.5677 q^{5} +172.354 q^{7} -77.1083 q^{9} -452.247 q^{11} -22.7429 q^{13} -1024.82 q^{15} -521.824 q^{17} -1558.56 q^{19} -2219.90 q^{21} +3464.36 q^{23} +3206.01 q^{25} +4122.96 q^{27} +4324.79 q^{29} +3987.66 q^{31} +5824.90 q^{33} +13713.8 q^{35} +10080.4 q^{37} +292.926 q^{39} -16408.5 q^{41} -1849.00 q^{43} -6135.33 q^{45} -24153.7 q^{47} +12898.8 q^{49} +6721.04 q^{51} +21214.2 q^{53} -35984.2 q^{55} +20074.0 q^{57} +25849.7 q^{59} +28577.7 q^{61} -13289.9 q^{63} -1809.60 q^{65} -66762.2 q^{67} -44620.6 q^{69} +10031.8 q^{71} +32145.0 q^{73} -41293.1 q^{75} -77946.5 q^{77} +21913.4 q^{79} -34366.0 q^{81} +66782.4 q^{83} -41520.3 q^{85} -55702.8 q^{87} +48984.7 q^{89} -3919.82 q^{91} -51360.7 q^{93} -124011. q^{95} +93075.8 q^{97} +34872.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−12.8799	−0.826246	−0.413123	−	0.910675i	$-0.635562\pi$
$3$	−12.8799	−0.826246	−0.413123	+	0.910675i	$0.635562\pi$
$4$	0	0
$5$	79.5677	1.42335	0.711675	−	0.702509i	$-0.247937\pi$
$5$	79.5677	1.42335	0.711675	+	0.702509i	$0.247937\pi$
$6$	0	0
$7$	172.354	1.32946	0.664730	−	0.747083i	$-0.268546\pi$
$7$	172.354	1.32946	0.664730	+	0.747083i	$0.268546\pi$
$8$	0	0
$9$	−77.1083	−0.317318
$10$	0	0
$11$	−452.247	−1.12692	−0.563461	−	0.826142i	$-0.690531\pi$
$11$	−452.247	−1.12692	−0.563461	+	0.826142i	$0.690531\pi$
$12$	0	0
$13$	−22.7429	−0.0373240	−0.0186620	−	0.999826i	$-0.505941\pi$
$13$	−22.7429	−0.0373240	−0.0186620	+	0.999826i	$0.505941\pi$
$14$	0	0
$15$	−1024.82	−1.17604
$16$	0	0
$17$	−521.824	−0.437927	−0.218964	−	0.975733i	$-0.570268\pi$
$17$	−521.824	−0.437927	−0.218964	+	0.975733i	$0.570268\pi$
$18$	0	0
$19$	−1558.56	−0.990463	−0.495232	−	0.868761i	$-0.664917\pi$
$19$	−1558.56	−0.990463	−0.495232	+	0.868761i	$0.664917\pi$
$20$	0	0
$21$	−2219.90	−1.09846
$22$	0	0
$23$	3464.36	1.36554	0.682769	−	0.730635i	$-0.260776\pi$
$23$	3464.36	1.36554	0.682769	+	0.730635i	$0.260776\pi$
$24$	0	0
$25$	3206.01	1.02592
$26$	0	0
$27$	4122.96	1.08843
$28$	0	0
$29$	4324.79	0.954926	0.477463	−	0.878652i	$-0.341556\pi$
$29$	4324.79	0.954926	0.477463	+	0.878652i	$0.341556\pi$
$30$	0	0
$31$	3987.66	0.745271	0.372635	−	0.927978i	$-0.378454\pi$
$31$	3987.66	0.745271	0.372635	+	0.927978i	$0.378454\pi$
$32$	0	0
$33$	5824.90	0.931115
$34$	0	0
$35$	13713.8	1.89229
$36$	0	0
$37$	10080.4	1.21053	0.605264	−	0.796025i	$-0.293068\pi$
$37$	10080.4	1.21053	0.605264	+	0.796025i	$0.293068\pi$
$38$	0	0
$39$	292.926	0.0308388
$40$	0	0
$41$	−16408.5	−1.52444	−0.762219	−	0.647319i	$-0.775890\pi$
$41$	−16408.5	−1.52444	−0.762219	+	0.647319i	$0.775890\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	−6135.33	−0.451654
$46$	0	0
$47$	−24153.7	−1.59492	−0.797459	−	0.603373i	$-0.793823\pi$
$47$	−24153.7	−1.59492	−0.797459	+	0.603373i	$0.793823\pi$
$48$	0	0
$49$	12898.8	0.767465
$50$	0	0
$51$	6721.04	0.361835
$52$	0	0
$53$	21214.2	1.03738	0.518688	−	0.854963i	$-0.326420\pi$
$53$	21214.2	1.03738	0.518688	+	0.854963i	$0.326420\pi$
$54$	0	0
$55$	−35984.2	−1.60400
$56$	0	0
$57$	20074.0	0.818366
$58$	0	0
$59$	25849.7	0.966775	0.483387	−	0.875407i	$-0.339406\pi$
$59$	25849.7	0.966775	0.483387	+	0.875407i	$0.339406\pi$
$60$	0	0
$61$	28577.7	0.983338	0.491669	−	0.870782i	$-0.336387\pi$
$61$	28577.7	0.983338	0.491669	+	0.870782i	$0.336387\pi$
$62$	0	0
$63$	−13289.9	−0.421862
$64$	0	0
$65$	−1809.60	−0.0531250
$66$	0	0
$67$	−66762.2	−1.81695	−0.908477	−	0.417935i	$-0.862754\pi$
$67$	−66762.2	−1.81695	−0.908477	+	0.417935i	$0.862754\pi$
$68$	0	0
$69$	−44620.6	−1.12827
$70$	0	0
$71$	10031.8	0.236173	0.118087	−	0.993003i	$-0.462324\pi$
$71$	10031.8	0.236173	0.118087	+	0.993003i	$0.462324\pi$
$72$	0	0
$73$	32145.0	0.706003	0.353001	−	0.935623i	$-0.385161\pi$
$73$	32145.0	0.706003	0.353001	+	0.935623i	$0.385161\pi$
$74$	0	0
$75$	−41293.1	−0.847665
$76$	0	0
$77$	−77946.5	−1.49820
$78$	0	0
$79$	21913.4	0.395040	0.197520	−	0.980299i	$-0.436711\pi$
$79$	21913.4	0.395040	0.197520	+	0.980299i	$0.436711\pi$
$80$	0	0
$81$	−34366.0	−0.581991
$82$	0	0
$83$	66782.4	1.06406	0.532031	−	0.846725i	$-0.321429\pi$
$83$	66782.4	1.06406	0.532031	+	0.846725i	$0.321429\pi$
$84$	0	0
$85$	−41520.3	−0.623323
$86$	0	0
$87$	−55702.8	−0.789004
$88$	0	0
$89$	48984.7	0.655520	0.327760	−	0.944761i	$-0.393706\pi$
$89$	48984.7	0.655520	0.327760	+	0.944761i	$0.393706\pi$
$90$	0	0
$91$	−3919.82	−0.0496207
$92$	0	0
$93$	−51360.7	−0.615777
$94$	0	0
$95$	−124011.	−1.40978
$96$	0	0
$97$	93075.8	1.00440	0.502201	−	0.864751i	$-0.332524\pi$
$97$	93075.8	1.00440	0.502201	+	0.864751i	$0.332524\pi$
$98$	0	0
$99$	34872.0	0.357593
$100$	0	0
$101$	47004.9	0.458500	0.229250	−	0.973368i	$-0.426373\pi$
$101$	47004.9	0.458500	0.229250	+	0.973368i	$0.426373\pi$
$102$	0	0
$103$	51489.8	0.478220	0.239110	−	0.970992i	$-0.423144\pi$
$103$	51489.8	0.478220	0.239110	+	0.970992i	$0.423144\pi$
$104$	0	0
$105$	−176632.	−1.56349
$106$	0	0
$107$	−74331.3	−0.627642	−0.313821	−	0.949482i	$-0.601609\pi$
$107$	−74331.3	−0.627642	−0.313821	+	0.949482i	$0.601609\pi$
$108$	0	0
$109$	−204775.	−1.65086	−0.825430	−	0.564505i	$-0.809067\pi$
$109$	−204775.	−1.65086	−0.825430	+	0.564505i	$0.809067\pi$
$110$	0	0
$111$	−129835.	−1.00019
$112$	0	0
$113$	125866.	0.927281	0.463640	−	0.886023i	$-0.346543\pi$
$113$	125866.	0.927281	0.463640	+	0.886023i	$0.346543\pi$
$114$	0	0
$115$	275651.	1.94364
$116$	0	0
$117$	1753.67	0.0118436
$118$	0	0
$119$	−89938.3	−0.582207
$120$	0	0
$121$	43476.5	0.269955
$122$	0	0
$123$	211340.	1.25956
$124$	0	0
$125$	6445.88	0.0368984
$126$	0	0
$127$	318214.	1.75069	0.875347	−	0.483495i	$-0.160633\pi$
$127$	318214.	1.75069	0.875347	+	0.483495i	$0.160633\pi$
$128$	0	0
$129$	23814.9	0.126001
$130$	0	0
$131$	261871.	1.33324	0.666620	−	0.745398i	$-0.267740\pi$
$131$	261871.	1.33324	0.666620	+	0.745398i	$0.267740\pi$
$132$	0	0
$133$	−268623.	−1.31678
$134$	0	0
$135$	328054.	1.54921
$136$	0	0
$137$	−20837.0	−0.0948492	−0.0474246	−	0.998875i	$-0.515101\pi$
$137$	−20837.0	−0.0948492	−0.0474246	+	0.998875i	$0.515101\pi$
$138$	0	0
$139$	182520.	0.801260	0.400630	−	0.916240i	$-0.368791\pi$
$139$	182520.	0.801260	0.400630	+	0.916240i	$0.368791\pi$
$140$	0	0
$141$	311097.	1.31779
$142$	0	0
$143$	10285.4	0.0420612
$144$	0	0
$145$	344113.	1.35919
$146$	0	0
$147$	−166135.	−0.634115
$148$	0	0
$149$	478460.	1.76555	0.882775	−	0.469796i	$-0.155673\pi$
$149$	478460.	1.76555	0.882775	+	0.469796i	$0.155673\pi$
$150$	0	0
$151$	−244853.	−0.873901	−0.436951	−	0.899485i	$-0.643942\pi$
$151$	−244853.	−0.873901	−0.436951	+	0.899485i	$0.643942\pi$
$152$	0	0
$153$	40237.0	0.138962
$154$	0	0
$155$	317289.	1.06078
$156$	0	0
$157$	12990.1	0.0420594	0.0210297	−	0.999779i	$-0.493306\pi$
$157$	12990.1	0.0420594	0.0210297	+	0.999779i	$0.493306\pi$
$158$	0	0
$159$	−273236.	−0.857128
$160$	0	0
$161$	597095.	1.81543
$162$	0	0
$163$	−43866.4	−0.129319	−0.0646596	−	0.997907i	$-0.520596\pi$
$163$	−43866.4	−0.129319	−0.0646596	+	0.997907i	$0.520596\pi$
$164$	0	0
$165$	463473.	1.32530
$166$	0	0
$167$	−454884.	−1.26215	−0.631073	−	0.775724i	$-0.717385\pi$
$167$	−454884.	−1.26215	−0.631073	+	0.775724i	$0.717385\pi$
$168$	0	0
$169$	−370776.	−0.998607
$170$	0	0
$171$	120178.	0.314292
$172$	0	0
$173$	−649198.	−1.64916	−0.824579	−	0.565747i	$-0.808588\pi$
$173$	−649198.	−1.64916	−0.824579	+	0.565747i	$0.808588\pi$
$174$	0	0
$175$	552568.	1.36392
$176$	0	0
$177$	−332941.	−0.798793
$178$	0	0
$179$	570828.	1.33160	0.665798	−	0.746132i	$-0.268091\pi$
$179$	570828.	1.33160	0.665798	+	0.746132i	$0.268091\pi$
$180$	0	0
$181$	−90744.3	−0.205884	−0.102942	−	0.994687i	$-0.532826\pi$
$181$	−90744.3	−0.205884	−0.102942	+	0.994687i	$0.532826\pi$
$182$	0	0
$183$	−368078.	−0.812479
$184$	0	0
$185$	802076.	1.72300
$186$	0	0
$187$	235994.	0.493510
$188$	0	0
$189$	710607.	1.44702
$190$	0	0
$191$	546924.	1.08478	0.542392	−	0.840126i	$-0.317519\pi$
$191$	546924.	1.08478	0.542392	+	0.840126i	$0.317519\pi$
$192$	0	0
$193$	25064.4	0.0484356	0.0242178	−	0.999707i	$-0.492290\pi$
$193$	25064.4	0.0484356	0.0242178	+	0.999707i	$0.492290\pi$
$194$	0	0
$195$	23307.5	0.0438943
$196$	0	0
$197$	579350.	1.06359	0.531797	−	0.846872i	$-0.321517\pi$
$197$	579350.	1.06359	0.531797	+	0.846872i	$0.321517\pi$
$198$	0	0
$199$	−411324.	−0.736295	−0.368147	−	0.929767i	$-0.620008\pi$
$199$	−411324.	−0.736295	−0.368147	+	0.929767i	$0.620008\pi$
$200$	0	0
$201$	859891.	1.50125
$202$	0	0
$203$	745393.	1.26954
$204$	0	0
$205$	−1.30559e6	−2.16981
$206$	0	0
$207$	−267131.	−0.433310
$208$	0	0
$209$	704852.	1.11618
$210$	0	0
$211$	385582.	0.596225	0.298113	−	0.954531i	$-0.403643\pi$
$211$	385582.	0.596225	0.298113	+	0.954531i	$0.403643\pi$
$212$	0	0
$213$	−129208.	−0.195137
$214$	0	0
$215$	−147121.	−0.217059
$216$	0	0
$217$	687288.	0.990808
$218$	0	0
$219$	−414024.	−0.583332
$220$	0	0
$221$	11867.8	0.0163452
$222$	0	0
$223$	1.03807e6	1.39786	0.698930	−	0.715190i	$-0.253660\pi$
$223$	1.03807e6	1.39786	0.698930	+	0.715190i	$0.253660\pi$
$224$	0	0
$225$	−247210.	−0.325544
$226$	0	0
$227$	−1.14006e6	−1.46846	−0.734229	−	0.678902i	$-0.762456\pi$
$227$	−1.14006e6	−1.46846	−0.734229	+	0.678902i	$0.762456\pi$
$228$	0	0
$229$	548999.	0.691804	0.345902	−	0.938271i	$-0.387573\pi$
$229$	548999.	0.691804	0.345902	+	0.938271i	$0.387573\pi$
$230$	0	0
$231$	1.00394e6	1.23788
$232$	0	0
$233$	−1.11203e6	−1.34191	−0.670957	−	0.741496i	$-0.734117\pi$
$233$	−1.11203e6	−1.34191	−0.670957	+	0.741496i	$0.734117\pi$
$234$	0	0
$235$	−1.92185e6	−2.27013
$236$	0	0
$237$	−282242.	−0.326400
$238$	0	0
$239$	1.28067e6	1.45025	0.725123	−	0.688620i	$-0.241783\pi$
$239$	1.28067e6	1.45025	0.725123	+	0.688620i	$0.241783\pi$
$240$	0	0
$241$	725022.	0.804097	0.402049	−	0.915618i	$-0.368298\pi$
$241$	725022.	0.804097	0.402049	+	0.915618i	$0.368298\pi$
$242$	0	0
$243$	−559249.	−0.607561
$244$	0	0
$245$	1.02633e6	1.09237
$246$	0	0
$247$	35446.1	0.0369680
$248$	0	0
$249$	−860150.	−0.879176
$250$	0	0
$251$	1.43637e6	1.43907	0.719533	−	0.694458i	$-0.244356\pi$
$251$	1.43637e6	1.43907	0.719533	+	0.694458i	$0.244356\pi$
$252$	0	0
$253$	−1.56675e6	−1.53886
$254$	0	0
$255$	534777.	0.515018
$256$	0	0
$257$	607070.	0.573331	0.286666	−	0.958031i	$-0.407453\pi$
$257$	607070.	0.573331	0.286666	+	0.958031i	$0.407453\pi$
$258$	0	0
$259$	1.73740e6	1.60935
$260$	0	0
$261$	−333477.	−0.303015
$262$	0	0
$263$	−169221.	−0.150856	−0.0754282	−	0.997151i	$-0.524032\pi$
$263$	−169221.	−0.150856	−0.0754282	+	0.997151i	$0.524032\pi$
$264$	0	0
$265$	1.68796e6	1.47655
$266$	0	0
$267$	−630918.	−0.541620
$268$	0	0
$269$	1.41683e6	1.19382	0.596908	−	0.802310i	$-0.296396\pi$
$269$	1.41683e6	1.19382	0.596908	+	0.802310i	$0.296396\pi$
$270$	0	0
$271$	−947582.	−0.783779	−0.391889	−	0.920012i	$-0.628178\pi$
$271$	−947582.	−0.783779	−0.391889	+	0.920012i	$0.628178\pi$
$272$	0	0
$273$	50486.9	0.0409989
$274$	0	0
$275$	−1.44991e6	−1.15614
$276$	0	0
$277$	37913.3	0.0296888	0.0148444	−	0.999890i	$-0.495275\pi$
$277$	37913.3	0.0296888	0.0148444	+	0.999890i	$0.495275\pi$
$278$	0	0
$279$	−307482.	−0.236488
$280$	0	0
$281$	−2.05268e6	−1.55080	−0.775400	−	0.631471i	$-0.782452\pi$
$281$	−2.05268e6	−1.55080	−0.775400	+	0.631471i	$0.782452\pi$
$282$	0	0
$283$	2.35101e6	1.74497	0.872487	−	0.488636i	$-0.162506\pi$
$283$	2.35101e6	1.74497	0.872487	+	0.488636i	$0.162506\pi$
$284$	0	0
$285$	1.59724e6	1.16482
$286$	0	0
$287$	−2.82807e6	−2.02668
$288$	0	0
$289$	−1.14756e6	−0.808220
$290$	0	0
$291$	−1.19881e6	−0.829882
$292$	0	0
$293$	1.11432e6	0.758300	0.379150	−	0.925335i	$-0.376216\pi$
$293$	1.11432e6	0.758300	0.379150	+	0.925335i	$0.376216\pi$
$294$	0	0
$295$	2.05680e6	1.37606
$296$	0	0
$297$	−1.86460e6	−1.22657
$298$	0	0
$299$	−78789.7	−0.0509673
$300$	0	0
$301$	−318682.	−0.202741
$302$	0	0
$303$	−605418.	−0.378834
$304$	0	0
$305$	2.27386e6	1.39963
$306$	0	0
$307$	1.15903e6	0.701854	0.350927	−	0.936403i	$-0.385866\pi$
$307$	1.15903e6	0.701854	0.350927	+	0.936403i	$0.385866\pi$
$308$	0	0
$309$	−663183.	−0.395127
$310$	0	0
$311$	1.18671e6	0.695736	0.347868	−	0.937543i	$-0.386906\pi$
$311$	1.18671e6	0.695736	0.347868	+	0.937543i	$0.386906\pi$
$312$	0	0
$313$	571287.	0.329605	0.164802	−	0.986327i	$-0.447301\pi$
$313$	571287.	0.329605	0.164802	+	0.986327i	$0.447301\pi$
$314$	0	0
$315$	−1.05745e6	−0.600457
$316$	0	0
$317$	−796360.	−0.445104	−0.222552	−	0.974921i	$-0.571439\pi$
$317$	−796360.	−0.445104	−0.222552	+	0.974921i	$0.571439\pi$
$318$	0	0
$319$	−1.95587e6	−1.07613
$320$	0	0
$321$	957379.	0.518587
$322$	0	0
$323$	813292.	0.433751
$324$	0	0
$325$	−72914.0	−0.0382915
$326$	0	0
$327$	2.63748e6	1.36402
$328$	0	0
$329$	−4.16297e6	−2.12038
$330$	0	0
$331$	390295.	0.195805	0.0979023	−	0.995196i	$-0.468787\pi$
$331$	390295.	0.195805	0.0979023	+	0.995196i	$0.468787\pi$
$332$	0	0
$333$	−777285.	−0.384122
$334$	0	0
$335$	−5.31212e6	−2.58616
$336$	0	0
$337$	765533.	0.367189	0.183594	−	0.983002i	$-0.441227\pi$
$337$	765533.	0.367189	0.183594	+	0.983002i	$0.441227\pi$
$338$	0	0
$339$	−1.62114e6	−0.766162
$340$	0	0
$341$	−1.80341e6	−0.839863
$342$	0	0
$343$	−673595.	−0.309146
$344$	0	0
$345$	−3.55036e6	−1.60592
$346$	0	0
$347$	1.47170e6	0.656140	0.328070	−	0.944653i	$-0.393602\pi$
$347$	1.47170e6	0.656140	0.328070	+	0.944653i	$0.393602\pi$
$348$	0	0
$349$	3.61738e6	1.58975	0.794877	−	0.606770i	$-0.207535\pi$
$349$	3.61738e6	1.58975	0.794877	+	0.606770i	$0.207535\pi$
$350$	0	0
$351$	−93768.2	−0.0406245
$352$	0	0
$353$	500946.	0.213971	0.106985	−	0.994261i	$-0.465880\pi$
$353$	500946.	0.213971	0.106985	+	0.994261i	$0.465880\pi$
$354$	0	0
$355$	798203.	0.336157
$356$	0	0
$357$	1.15840e6	0.481046
$358$	0	0
$359$	3.84739e6	1.57554	0.787770	−	0.615969i	$-0.211235\pi$
$359$	3.84739e6	1.57554	0.787770	+	0.615969i	$0.211235\pi$
$360$	0	0
$361$	−47002.6	−0.0189825
$362$	0	0
$363$	−559973.	−0.223049
$364$	0	0
$365$	2.55770e6	1.00489
$366$	0	0
$367$	−2.28572e6	−0.885844	−0.442922	−	0.896560i	$-0.646058\pi$
$367$	−2.28572e6	−0.885844	−0.442922	+	0.896560i	$0.646058\pi$
$368$	0	0
$369$	1.26523e6	0.483732
$370$	0	0
$371$	3.65634e6	1.37915
$372$	0	0
$373$	−753658.	−0.280480	−0.140240	−	0.990118i	$-0.544787\pi$
$373$	−753658.	−0.280480	−0.140240	+	0.990118i	$0.544787\pi$
$374$	0	0
$375$	−83022.3	−0.0304871
$376$	0	0
$377$	−98358.3	−0.0356416
$378$	0	0
$379$	3.54911e6	1.26917	0.634587	−	0.772851i	$-0.281170\pi$
$379$	3.54911e6	1.26917	0.634587	+	0.772851i	$0.281170\pi$
$380$	0	0
$381$	−4.09857e6	−1.44650
$382$	0	0
$383$	−2.29101e6	−0.798049	−0.399025	−	0.916940i	$-0.630651\pi$
$383$	−2.29101e6	−0.798049	−0.399025	+	0.916940i	$0.630651\pi$
$384$	0	0
$385$	−6.20202e6	−2.13246
$386$	0	0
$387$	142573.	0.0483906
$388$	0	0
$389$	3.75998e6	1.25983	0.629914	−	0.776665i	$-0.283090\pi$
$389$	3.75998e6	1.25983	0.629914	+	0.776665i	$0.283090\pi$
$390$	0	0
$391$	−1.80779e6	−0.598006
$392$	0	0
$393$	−3.37287e6	−1.10158
$394$	0	0
$395$	1.74359e6	0.562280
$396$	0	0
$397$	5.66320e6	1.80337	0.901686	−	0.432392i	$-0.142330\pi$
$397$	5.66320e6	1.80337	0.901686	+	0.432392i	$0.142330\pi$
$398$	0	0
$399$	3.45983e6	1.08799
$400$	0	0
$401$	3.53734e6	1.09854	0.549270	−	0.835645i	$-0.314906\pi$
$401$	3.53734e6	1.09854	0.549270	+	0.835645i	$0.314906\pi$
$402$	0	0
$403$	−90691.1	−0.0278165
$404$	0	0
$405$	−2.73442e6	−0.828377
$406$	0	0
$407$	−4.55885e6	−1.36417
$408$	0	0
$409$	−2.11034e6	−0.623798	−0.311899	−	0.950115i	$-0.600965\pi$
$409$	−2.11034e6	−0.623798	−0.311899	+	0.950115i	$0.600965\pi$
$410$	0	0
$411$	268378.	0.0783688
$412$	0	0
$413$	4.45529e6	1.28529
$414$	0	0
$415$	5.31372e6	1.51453
$416$	0	0
$417$	−2.35084e6	−0.662038
$418$	0	0
$419$	−896357.	−0.249428	−0.124714	−	0.992193i	$-0.539801\pi$
$419$	−896357.	−0.249428	−0.124714	+	0.992193i	$0.539801\pi$
$420$	0	0
$421$	4.09292e6	1.12545	0.562727	−	0.826643i	$-0.309752\pi$
$421$	4.09292e6	1.12545	0.562727	+	0.826643i	$0.309752\pi$
$422$	0	0
$423$	1.86245e6	0.506096
$424$	0	0
$425$	−1.67297e6	−0.449280
$426$	0	0
$427$	4.92547e6	1.30731
$428$	0	0
$429$	−132475.	−0.0347529
$430$	0	0
$431$	1.18319e6	0.306804	0.153402	−	0.988164i	$-0.450977\pi$
$431$	1.18319e6	0.306804	0.153402	+	0.988164i	$0.450977\pi$
$432$	0	0
$433$	1.63492e6	0.419061	0.209530	−	0.977802i	$-0.432806\pi$
$433$	1.63492e6	0.419061	0.209530	+	0.977802i	$0.432806\pi$
$434$	0	0
$435$	−4.43214e6	−1.12303
$436$	0	0
$437$	−5.39940e6	−1.35251
$438$	0	0
$439$	2.23900e6	0.554488	0.277244	−	0.960800i	$-0.410579\pi$
$439$	2.23900e6	0.554488	0.277244	+	0.960800i	$0.410579\pi$
$440$	0	0
$441$	−994603.	−0.243531
$442$	0	0
$443$	−4.42567e6	−1.07144	−0.535722	−	0.844394i	$-0.679961\pi$
$443$	−4.42567e6	−1.07144	−0.535722	+	0.844394i	$0.679961\pi$
$444$	0	0
$445$	3.89760e6	0.933034
$446$	0	0
$447$	−6.16252e6	−1.45878
$448$	0	0
$449$	824623.	0.193037	0.0965183	−	0.995331i	$-0.469229\pi$
$449$	824623.	0.193037	0.0965183	+	0.995331i	$0.469229\pi$
$450$	0	0
$451$	7.42071e6	1.71792
$452$	0	0
$453$	3.15368e6	0.722057
$454$	0	0
$455$	−311891.	−0.0706276
$456$	0	0
$457$	4.78062e6	1.07076	0.535382	−	0.844610i	$-0.320168\pi$
$457$	4.78062e6	1.07076	0.535382	+	0.844610i	$0.320168\pi$
$458$	0	0
$459$	−2.15146e6	−0.476652
$460$	0	0
$461$	177573.	0.0389157	0.0194578	−	0.999811i	$-0.493806\pi$
$461$	177573.	0.0389157	0.0194578	+	0.999811i	$0.493806\pi$
$462$	0	0
$463$	−8.24149e6	−1.78671	−0.893353	−	0.449355i	$-0.851654\pi$
$463$	−8.24149e6	−1.78671	−0.893353	+	0.449355i	$0.851654\pi$
$464$	0	0
$465$	−4.08665e6	−0.876466
$466$	0	0
$467$	−2.93483e6	−0.622718	−0.311359	−	0.950292i	$-0.600784\pi$
$467$	−2.93483e6	−0.622718	−0.311359	+	0.950292i	$0.600784\pi$
$468$	0	0
$469$	−1.15067e7	−2.41557
$470$	0	0
$471$	−167311.	−0.0347514
$472$	0	0
$473$	836205.	0.171854
$474$	0	0
$475$	−4.99675e6	−1.01614
$476$	0	0
$477$	−1.63579e6	−0.329178
$478$	0	0
$479$	−7.53163e6	−1.49986	−0.749929	−	0.661518i	$-0.769912\pi$
$479$	−7.53163e6	−1.49986	−0.749929	+	0.661518i	$0.769912\pi$
$480$	0	0
$481$	−229258.	−0.0451817
$482$	0	0
$483$	−7.69053e6	−1.49999
$484$	0	0
$485$	7.40582e6	1.42961
$486$	0	0
$487$	2.30284e6	0.439988	0.219994	−	0.975501i	$-0.429396\pi$
$487$	2.30284e6	0.439988	0.219994	+	0.975501i	$0.429396\pi$
$488$	0	0
$489$	564995.	0.106849
$490$	0	0
$491$	−4.43974e6	−0.831100	−0.415550	−	0.909570i	$-0.636411\pi$
$491$	−4.43974e6	−0.831100	−0.415550	+	0.909570i	$0.636411\pi$
$492$	0	0
$493$	−2.25678e6	−0.418188
$494$	0	0
$495$	2.77468e6	0.508980
$496$	0	0
$497$	1.72901e6	0.313983
$498$	0	0
$499$	−7.95785e6	−1.43069	−0.715343	−	0.698774i	$-0.753729\pi$
$499$	−7.95785e6	−1.43069	−0.715343	+	0.698774i	$0.753729\pi$
$500$	0	0
$501$	5.85886e6	1.04284
$502$	0	0
$503$	4.75804e6	0.838510	0.419255	−	0.907868i	$-0.362291\pi$
$503$	4.75804e6	0.838510	0.419255	+	0.907868i	$0.362291\pi$
$504$	0	0
$505$	3.74007e6	0.652606
$506$	0	0
$507$	4.77555e6	0.825095
$508$	0	0
$509$	−5.42576e6	−0.928252	−0.464126	−	0.885769i	$-0.653632\pi$
$509$	−5.42576e6	−0.928252	−0.464126	+	0.885769i	$0.653632\pi$
$510$	0	0
$511$	5.54031e6	0.938603
$512$	0	0
$513$	−6.42587e6	−1.07805
$514$	0	0
$515$	4.09692e6	0.680675
$516$	0	0
$517$	1.09234e7	1.79735
$518$	0	0
$519$	8.36161e6	1.36261
$520$	0	0
$521$	−8.88444e6	−1.43396	−0.716978	−	0.697096i	$-0.754475\pi$
$521$	−8.88444e6	−1.43396	−0.716978	+	0.697096i	$0.754475\pi$
$522$	0	0
$523$	5.03938e6	0.805606	0.402803	−	0.915287i	$-0.368036\pi$
$523$	5.03938e6	0.805606	0.402803	+	0.915287i	$0.368036\pi$
$524$	0	0
$525$	−7.11702e6	−1.12694
$526$	0	0
$527$	−2.08086e6	−0.326374
$528$	0	0
$529$	5.56545e6	0.864692
$530$	0	0
$531$	−1.99323e6	−0.306775
$532$	0	0
$533$	373178.	0.0568981
$534$	0	0
$535$	−5.91436e6	−0.893354
$536$	0	0
$537$	−7.35221e6	−1.10023
$538$	0	0
$539$	−5.83344e6	−0.864874
$540$	0	0
$541$	3.49430e6	0.513295	0.256648	−	0.966505i	$-0.417382\pi$
$541$	3.49430e6	0.513295	0.256648	+	0.966505i	$0.417382\pi$
$542$	0	0
$543$	1.16878e6	0.170111
$544$	0	0
$545$	−1.62934e7	−2.34975
$546$	0	0
$547$	2.50382e6	0.357796	0.178898	−	0.983868i	$-0.442747\pi$
$547$	2.50382e6	0.357796	0.178898	+	0.983868i	$0.442747\pi$
$548$	0	0
$549$	−2.20358e6	−0.312031
$550$	0	0
$551$	−6.74042e6	−0.945819
$552$	0	0
$553$	3.77685e6	0.525190
$554$	0	0
$555$	−1.03307e7	−1.42362
$556$	0	0
$557$	−1.53305e6	−0.209371	−0.104686	−	0.994505i	$-0.533384\pi$
$557$	−1.53305e6	−0.209371	−0.104686	+	0.994505i	$0.533384\pi$
$558$	0	0
$559$	42051.7	0.00569185
$560$	0	0
$561$	−3.03957e6	−0.407761
$562$	0	0
$563$	−8.52078e6	−1.13294	−0.566472	−	0.824081i	$-0.691692\pi$
$563$	−8.52078e6	−1.13294	−0.566472	+	0.824081i	$0.691692\pi$
$564$	0	0
$565$	1.00148e7	1.31984
$566$	0	0
$567$	−5.92311e6	−0.773734
$568$	0	0
$569$	−429293.	−0.0555870	−0.0277935	−	0.999614i	$-0.508848\pi$
$569$	−429293.	−0.0555870	−0.0277935	+	0.999614i	$0.508848\pi$
$570$	0	0
$571$	6.30376e6	0.809114	0.404557	−	0.914513i	$-0.367426\pi$
$571$	6.30376e6	0.809114	0.404557	+	0.914513i	$0.367426\pi$
$572$	0	0
$573$	−7.04432e6	−0.896298
$574$	0	0
$575$	1.11068e7	1.40094
$576$	0	0
$577$	−8.10411e6	−1.01337	−0.506683	−	0.862133i	$-0.669128\pi$
$577$	−8.10411e6	−1.01337	−0.506683	+	0.862133i	$0.669128\pi$
$578$	0	0
$579$	−322827.	−0.0400197
$580$	0	0
$581$	1.15102e7	1.41463
$582$	0	0
$583$	−9.59405e6	−1.16904
$584$	0	0
$585$	139535.	0.0168575
$586$	0	0
$587$	−5.93951e6	−0.711468	−0.355734	−	0.934587i	$-0.615769\pi$
$587$	−5.93951e6	−0.711468	−0.355734	+	0.934587i	$0.615769\pi$
$588$	0	0
$589$	−6.21499e6	−0.738163
$590$	0	0
$591$	−7.46197e6	−0.878789
$592$	0	0
$593$	−1.03054e6	−0.120345	−0.0601727	−	0.998188i	$-0.519165\pi$
$593$	−1.03054e6	−0.120345	−0.0601727	+	0.998188i	$0.519165\pi$
$594$	0	0
$595$	−7.15618e6	−0.828684
$596$	0	0
$597$	5.29781e6	0.608360
$598$	0	0
$599$	−8.75551e6	−0.997044	−0.498522	−	0.866877i	$-0.666124\pi$
$599$	−8.75551e6	−0.997044	−0.498522	+	0.866877i	$0.666124\pi$
$600$	0	0
$601$	−6.36534e6	−0.718845	−0.359423	−	0.933175i	$-0.617026\pi$
$601$	−6.36534e6	−0.718845	−0.359423	+	0.933175i	$0.617026\pi$
$602$	0	0
$603$	5.14792e6	0.576552
$604$	0	0
$605$	3.45932e6	0.384240
$606$	0	0
$607$	−1.44657e7	−1.59356	−0.796779	−	0.604270i	$-0.793465\pi$
$607$	−1.44657e7	−1.59356	−0.796779	+	0.604270i	$0.793465\pi$
$608$	0	0
$609$	−9.60058e6	−1.04895
$610$	0	0
$611$	549325.	0.0595287
$612$	0	0
$613$	−1.78097e7	−1.91428	−0.957142	−	0.289619i	$-0.906471\pi$
$613$	−1.78097e7	−1.91428	−0.957142	+	0.289619i	$0.906471\pi$
$614$	0	0
$615$	1.68158e7	1.79279
$616$	0	0
$617$	4.11393e6	0.435054	0.217527	−	0.976054i	$-0.430201\pi$
$617$	4.11393e6	0.435054	0.217527	+	0.976054i	$0.430201\pi$
$618$	0	0
$619$	3.56813e6	0.374295	0.187148	−	0.982332i	$-0.440076\pi$
$619$	3.56813e6	0.374295	0.187148	+	0.982332i	$0.440076\pi$
$620$	0	0
$621$	1.42834e7	1.48629
$622$	0	0
$623$	8.44270e6	0.871488
$624$	0	0
$625$	−9.50590e6	−0.973404
$626$	0	0
$627$	−9.07843e6	−0.922235
$628$	0	0
$629$	−5.26021e6	−0.530123
$630$	0	0
$631$	−8.10764e6	−0.810627	−0.405314	−	0.914178i	$-0.632838\pi$
$631$	−8.10764e6	−0.810627	−0.405314	+	0.914178i	$0.632838\pi$
$632$	0	0
$633$	−4.96625e6	−0.492629
$634$	0	0
$635$	2.53196e7	2.49185
$636$	0	0
$637$	−293356.	−0.0286448
$638$	0	0
$639$	−773531.	−0.0749421
$640$	0	0
$641$	1.15748e7	1.11267	0.556335	−	0.830958i	$-0.312207\pi$
$641$	1.15748e7	1.11267	0.556335	+	0.830958i	$0.312207\pi$
$642$	0	0
$643$	865667.	0.0825702	0.0412851	−	0.999147i	$-0.486855\pi$
$643$	865667.	0.0825702	0.0412851	+	0.999147i	$0.486855\pi$
$644$	0	0
$645$	1.89490e6	0.179344
$646$	0	0
$647$	3.32273e6	0.312057	0.156029	−	0.987753i	$-0.450131\pi$
$647$	3.32273e6	0.312057	0.156029	+	0.987753i	$0.450131\pi$
$648$	0	0
$649$	−1.16904e7	−1.08948
$650$	0	0
$651$	−8.85220e6	−0.818651
$652$	0	0
$653$	−1.42689e7	−1.30951	−0.654754	−	0.755842i	$-0.727228\pi$
$653$	−1.42689e7	−1.30951	−0.654754	+	0.755842i	$0.727228\pi$
$654$	0	0
$655$	2.08364e7	1.89767
$656$	0	0
$657$	−2.47865e6	−0.224027
$658$	0	0
$659$	1.73298e6	0.155446	0.0777230	−	0.996975i	$-0.475235\pi$
$659$	1.73298e6	0.155446	0.0777230	+	0.996975i	$0.475235\pi$
$660$	0	0
$661$	456270.	0.0406180	0.0203090	−	0.999794i	$-0.493535\pi$
$661$	456270.	0.0406180	0.0203090	+	0.999794i	$0.493535\pi$
$662$	0	0
$663$	−152856.	−0.0135051
$664$	0	0
$665$	−2.13737e7	−1.87424
$666$	0	0
$667$	1.49826e7	1.30399
$668$	0	0
$669$	−1.33702e7	−1.15498
$670$	0	0
$671$	−1.29242e7	−1.10815
$672$	0	0
$673$	−1.34881e7	−1.14793	−0.573963	−	0.818881i	$-0.694595\pi$
$673$	−1.34881e7	−1.14793	−0.573963	+	0.818881i	$0.694595\pi$
$674$	0	0
$675$	1.32183e7	1.11664
$676$	0	0
$677$	1.96810e7	1.65035	0.825174	−	0.564879i	$-0.191077\pi$
$677$	1.96810e7	1.65035	0.825174	+	0.564879i	$0.191077\pi$
$678$	0	0
$679$	1.60420e7	1.33531
$680$	0	0
$681$	1.46838e7	1.21331
$682$	0	0
$683$	−6.25887e6	−0.513386	−0.256693	−	0.966493i	$-0.582633\pi$
$683$	−6.25887e6	−0.513386	−0.256693	+	0.966493i	$0.582633\pi$
$684$	0	0
$685$	−1.65795e6	−0.135004
$686$	0	0
$687$	−7.07106e6	−0.571600
$688$	0	0
$689$	−482472.	−0.0387190
$690$	0	0
$691$	−3.10391e6	−0.247295	−0.123647	−	0.992326i	$-0.539459\pi$
$691$	−3.10391e6	−0.247295	−0.123647	+	0.992326i	$0.539459\pi$
$692$	0	0
$693$	6.01032e6	0.475406
$694$	0	0
$695$	1.45227e7	1.14047
$696$	0	0
$697$	8.56236e6	0.667593
$698$	0	0
$699$	1.43228e7	1.10875
$700$	0	0
$701$	5.25605e6	0.403984	0.201992	−	0.979387i	$-0.435258\pi$
$701$	5.25605e6	0.403984	0.201992	+	0.979387i	$0.435258\pi$
$702$	0	0
$703$	−1.57109e7	−1.19898
$704$	0	0
$705$	2.47532e7	1.87568
$706$	0	0
$707$	8.10147e6	0.609558
$708$	0	0
$709$	8.29295e6	0.619574	0.309787	−	0.950806i	$-0.399742\pi$
$709$	8.29295e6	0.619574	0.309787	+	0.950806i	$0.399742\pi$
$710$	0	0
$711$	−1.68970e6	−0.125353
$712$	0	0
$713$	1.38147e7	1.01770
$714$	0	0
$715$	818387.	0.0598678
$716$	0	0
$717$	−1.64949e7	−1.19826
$718$	0	0
$719$	−3.71146e6	−0.267746	−0.133873	−	0.990999i	$-0.542741\pi$
$719$	−3.71146e6	−0.267746	−0.133873	+	0.990999i	$0.542741\pi$
$720$	0	0
$721$	8.87445e6	0.635775
$722$	0	0
$723$	−9.33821e6	−0.664382
$724$	0	0
$725$	1.38653e7	0.979681
$726$	0	0
$727$	3.10084e6	0.217592	0.108796	−	0.994064i	$-0.465300\pi$
$727$	3.10084e6	0.217592	0.108796	+	0.994064i	$0.465300\pi$
$728$	0	0
$729$	1.55540e7	1.08399
$730$	0	0
$731$	964853.	0.0667833
$732$	0	0
$733$	−1.69858e7	−1.16769	−0.583844	−	0.811866i	$-0.698452\pi$
$733$	−1.69858e7	−1.16769	−0.583844	+	0.811866i	$0.698452\pi$
$734$	0	0
$735$	−1.32190e7	−0.902567
$736$	0	0
$737$	3.01930e7	2.04757
$738$	0	0
$739$	−1.13197e7	−0.762473	−0.381236	−	0.924478i	$-0.624502\pi$
$739$	−1.13197e7	−0.762473	−0.381236	+	0.924478i	$0.624502\pi$
$740$	0	0
$741$	−456542.	−0.0305447
$742$	0	0
$743$	9.42589e6	0.626398	0.313199	−	0.949688i	$-0.398599\pi$
$743$	9.42589e6	0.626398	0.313199	+	0.949688i	$0.398599\pi$
$744$	0	0
$745$	3.80699e7	2.51299
$746$	0	0
$747$	−5.14947e6	−0.337646
$748$	0	0
$749$	−1.28113e7	−0.834425
$750$	0	0
$751$	3.73658e6	0.241754	0.120877	−	0.992667i	$-0.461429\pi$
$751$	3.73658e6	0.241754	0.120877	+	0.992667i	$0.461429\pi$
$752$	0	0
$753$	−1.85002e7	−1.18902
$754$	0	0
$755$	−1.94824e7	−1.24387
$756$	0	0
$757$	1.39033e7	0.881815	0.440908	−	0.897552i	$-0.354657\pi$
$757$	1.39033e7	0.881815	0.440908	+	0.897552i	$0.354657\pi$
$758$	0	0
$759$	2.01795e7	1.27147
$760$	0	0
$761$	−2.95291e7	−1.84837	−0.924185	−	0.381946i	$-0.875254\pi$
$761$	−2.95291e7	−1.84837	−0.924185	+	0.381946i	$0.875254\pi$
$762$	0	0
$763$	−3.52937e7	−2.19475
$764$	0	0
$765$	3.20156e6	0.197792
$766$	0	0
$767$	−587897.	−0.0360839
$768$	0	0
$769$	−9.04293e6	−0.551434	−0.275717	−	0.961239i	$-0.588915\pi$
$769$	−9.04293e6	−0.551434	−0.275717	+	0.961239i	$0.588915\pi$
$770$	0	0
$771$	−7.81899e6	−0.473713
$772$	0	0
$773$	−661253.	−0.0398033	−0.0199016	−	0.999802i	$-0.506335\pi$
$773$	−661253.	−0.0398033	−0.0199016	+	0.999802i	$0.506335\pi$
$774$	0	0
$775$	1.27845e7	0.764591
$776$	0	0
$777$	−2.23775e7	−1.32972
$778$	0	0
$779$	2.55736e7	1.50990
$780$	0	0
$781$	−4.53683e6	−0.266149
$782$	0	0
$783$	1.78309e7	1.03937
$784$	0	0
$785$	1.03359e6	0.0598653
$786$	0	0
$787$	6.46523e6	0.372089	0.186045	−	0.982541i	$-0.440433\pi$
$787$	6.46523e6	0.372089	0.186045	+	0.982541i	$0.440433\pi$
$788$	0	0
$789$	2.17954e6	0.124644
$790$	0	0
$791$	2.16934e7	1.23278
$792$	0	0
$793$	−649940.	−0.0367021
$794$	0	0
$795$	−2.17408e7	−1.21999
$796$	0	0
$797$	2.62978e7	1.46647	0.733234	−	0.679976i	$-0.238010\pi$
$797$	2.62978e7	1.46647	0.733234	+	0.679976i	$0.238010\pi$
$798$	0	0
$799$	1.26040e7	0.698458
$800$	0	0
$801$	−3.77713e6	−0.208008
$802$	0	0
$803$	−1.45375e7	−0.795610
$804$	0	0
$805$	4.75095e7	2.58399
$806$	0	0
$807$	−1.82486e7	−0.986385
$808$	0	0
$809$	−9.92533e6	−0.533180	−0.266590	−	0.963810i	$-0.585897\pi$
$809$	−9.92533e6	−0.533180	−0.266590	+	0.963810i	$0.585897\pi$
$810$	0	0
$811$	−1.86128e7	−0.993712	−0.496856	−	0.867833i	$-0.665512\pi$
$811$	−1.86128e7	−0.993712	−0.496856	+	0.867833i	$0.665512\pi$
$812$	0	0
$813$	1.22048e7	0.647594
$814$	0	0
$815$	−3.49035e6	−0.184066
$816$	0	0
$817$	2.88177e6	0.151044
$818$	0	0
$819$	302251.	0.0157456
$820$	0	0
$821$	1.08487e7	0.561718	0.280859	−	0.959749i	$-0.409381\pi$
$821$	1.08487e7	0.561718	0.280859	+	0.959749i	$0.409381\pi$
$822$	0	0
$823$	1.29473e7	0.666316	0.333158	−	0.942871i	$-0.391886\pi$
$823$	1.29473e7	0.666316	0.333158	+	0.942871i	$0.391886\pi$
$824$	0	0
$825$	1.86747e7	0.955253
$826$	0	0
$827$	−2.79589e7	−1.42153	−0.710767	−	0.703428i	$-0.751652\pi$
$827$	−2.79589e7	−1.42153	−0.710767	+	0.703428i	$0.751652\pi$
$828$	0	0
$829$	−1.95464e7	−0.987825	−0.493913	−	0.869512i	$-0.664434\pi$
$829$	−1.95464e7	−0.987825	−0.493913	+	0.869512i	$0.664434\pi$
$830$	0	0
$831$	−488319.	−0.0245302
$832$	0	0
$833$	−6.73090e6	−0.336094
$834$	0	0
$835$	−3.61940e7	−1.79647
$836$	0	0
$837$	1.64410e7	0.811174
$838$	0	0
$839$	−1.66180e7	−0.815031	−0.407516	−	0.913198i	$-0.633605\pi$
$839$	−1.66180e7	−0.815031	−0.407516	+	0.913198i	$0.633605\pi$
$840$	0	0
$841$	−1.80736e6	−0.0881161
$842$	0	0
$843$	2.64383e7	1.28134
$844$	0	0
$845$	−2.95018e7	−1.42137
$846$	0	0
$847$	7.49334e6	0.358894
$848$	0	0
$849$	−3.02808e7	−1.44178
$850$	0	0
$851$	3.49222e7	1.65302
$852$	0	0
$853$	−1.69144e7	−0.795947	−0.397974	−	0.917397i	$-0.630286\pi$
$853$	−1.69144e7	−0.795947	−0.397974	+	0.917397i	$0.630286\pi$
$854$	0	0
$855$	9.56225e6	0.447347
$856$	0	0
$857$	767365.	0.0356903	0.0178451	−	0.999841i	$-0.494319\pi$
$857$	767365.	0.0356903	0.0178451	+	0.999841i	$0.494319\pi$
$858$	0	0
$859$	−1.25709e7	−0.581278	−0.290639	−	0.956833i	$-0.593868\pi$
$859$	−1.25709e7	−0.581278	−0.290639	+	0.956833i	$0.593868\pi$
$860$	0	0
$861$	3.64252e7	1.67454
$862$	0	0
$863$	−4.15225e6	−0.189783	−0.0948913	−	0.995488i	$-0.530250\pi$
$863$	−4.15225e6	−0.189783	−0.0948913	+	0.995488i	$0.530250\pi$
$864$	0	0
$865$	−5.16552e7	−2.34733
$866$	0	0
$867$	1.47804e7	0.667788
$868$	0	0
$869$	−9.91025e6	−0.445180
$870$	0	0
$871$	1.51837e6	0.0678159
$872$	0	0
$873$	−7.17691e6	−0.318715
$874$	0	0
$875$	1.11097e6	0.0490549
$876$	0	0
$877$	−2.05982e6	−0.0904337	−0.0452168	−	0.998977i	$-0.514398\pi$
$877$	−2.05982e6	−0.0904337	−0.0452168	+	0.998977i	$0.514398\pi$
$878$	0	0
$879$	−1.43523e7	−0.626542
$880$	0	0
$881$	3.31902e6	0.144069	0.0720345	−	0.997402i	$-0.477051\pi$
$881$	3.31902e6	0.144069	0.0720345	+	0.997402i	$0.477051\pi$
$882$	0	0
$883$	−3.88804e6	−0.167814	−0.0839071	−	0.996474i	$-0.526740\pi$
$883$	−3.88804e6	−0.167814	−0.0839071	+	0.996474i	$0.526740\pi$
$884$	0	0
$885$	−2.64914e7	−1.13696
$886$	0	0
$887$	4.11225e6	0.175497	0.0877486	−	0.996143i	$-0.472033\pi$
$887$	4.11225e6	0.175497	0.0877486	+	0.996143i	$0.472033\pi$
$888$	0	0
$889$	5.48454e7	2.32748
$890$	0	0
$891$	1.55419e7	0.655859
$892$	0	0
$893$	3.76448e7	1.57971
$894$	0	0
$895$	4.54195e7	1.89533
$896$	0	0
$897$	1.01480e6	0.0421115
$898$	0	0
$899$	1.72458e7	0.711679
$900$	0	0
$901$	−1.10701e7	−0.454295
$902$	0	0
$903$	4.10459e6	0.167514
$904$	0	0
$905$	−7.22031e6	−0.293045
$906$	0	0
$907$	4.04945e7	1.63447	0.817236	−	0.576303i	$-0.195505\pi$
$907$	4.04945e7	1.63447	0.817236	+	0.576303i	$0.195505\pi$
$908$	0	0
$909$	−3.62447e6	−0.145490
$910$	0	0
$911$	3.27575e7	1.30772	0.653860	−	0.756615i	$-0.273148\pi$
$911$	3.27575e7	1.30772	0.653860	+	0.756615i	$0.273148\pi$
$912$	0	0
$913$	−3.02021e7	−1.19911
$914$	0	0
$915$	−2.92871e7	−1.15644
$916$	0	0
$917$	4.51343e7	1.77249
$918$	0	0
$919$	8.68701e6	0.339298	0.169649	−	0.985505i	$-0.445737\pi$
$919$	8.68701e6	0.339298	0.169649	+	0.985505i	$0.445737\pi$
$920$	0	0
$921$	−1.49281e7	−0.579904
$922$	0	0
$923$	−228151.	−0.00881492
$924$	0	0
$925$	3.23180e7	1.24191
$926$	0	0
$927$	−3.97029e6	−0.151748
$928$	0	0
$929$	−6.29057e6	−0.239139	−0.119570	−	0.992826i	$-0.538151\pi$
$929$	−6.29057e6	−0.239139	−0.119570	+	0.992826i	$0.538151\pi$
$930$	0	0
$931$	−2.01035e7	−0.760146
$932$	0	0
$933$	−1.52847e7	−0.574849
$934$	0	0
$935$	1.87775e7	0.702437
$936$	0	0
$937$	−6.97275e6	−0.259451	−0.129725	−	0.991550i	$-0.541410\pi$
$937$	−6.97275e6	−0.259451	−0.129725	+	0.991550i	$0.541410\pi$
$938$	0	0
$939$	−7.35812e6	−0.272335
$940$	0	0
$941$	1.67294e7	0.615894	0.307947	−	0.951404i	$-0.400358\pi$
$941$	1.67294e7	0.615894	0.307947	+	0.951404i	$0.400358\pi$
$942$	0	0
$943$	−5.68450e7	−2.08168
$944$	0	0
$945$	5.65414e7	2.05962
$946$	0	0
$947$	4.20327e7	1.52304	0.761522	−	0.648140i	$-0.224453\pi$
$947$	4.20327e7	1.52304	0.761522	+	0.648140i	$0.224453\pi$
$948$	0	0
$949$	−731071.	−0.0263508
$950$	0	0
$951$	1.02570e7	0.367765
$952$	0	0
$953$	4.38940e7	1.56557	0.782787	−	0.622290i	$-0.213798\pi$
$953$	4.38940e7	1.56557	0.782787	+	0.622290i	$0.213798\pi$
$954$	0	0
$955$	4.35174e7	1.54403
$956$	0	0
$957$	2.51914e7	0.889146
$958$	0	0
$959$	−3.59133e6	−0.126098
$960$	0	0
$961$	−1.27277e7	−0.444571
$962$	0	0
$963$	5.73156e6	0.199162
$964$	0	0
$965$	1.99432e6	0.0689407
$966$	0	0
$967$	−1.42055e6	−0.0488530	−0.0244265	−	0.999702i	$-0.507776\pi$
$967$	−1.42055e6	−0.0488530	−0.0244265	+	0.999702i	$0.507776\pi$
$968$	0	0
$969$	−1.04751e7	−0.358385
$970$	0	0
$971$	4.66877e7	1.58911	0.794556	−	0.607191i	$-0.207704\pi$
$971$	4.66877e7	1.58911	0.794556	+	0.607191i	$0.207704\pi$
$972$	0	0
$973$	3.14580e7	1.06524
$974$	0	0
$975$	939125.	0.0316382
$976$	0	0
$977$	5.11806e6	0.171541	0.0857707	−	0.996315i	$-0.472665\pi$
$977$	5.11806e6	0.171541	0.0857707	+	0.996315i	$0.472665\pi$
$978$	0	0
$979$	−2.21532e7	−0.738720
$980$	0	0
$981$	1.57898e7	0.523848
$982$	0	0
$983$	1.29865e6	0.0428654	0.0214327	−	0.999770i	$-0.493177\pi$
$983$	1.29865e6	0.0428654	0.0214327	+	0.999770i	$0.493177\pi$
$984$	0	0
$985$	4.60975e7	1.51387
$986$	0	0
$987$	5.36187e7	1.75196
$988$	0	0
$989$	−6.40560e6	−0.208242
$990$	0	0
$991$	−2.92905e7	−0.947422	−0.473711	−	0.880680i	$-0.657086\pi$
$991$	−2.92905e7	−0.947422	−0.473711	+	0.880680i	$0.657086\pi$
$992$	0	0
$993$	−5.02696e6	−0.161783
$994$	0	0
$995$	−3.27281e7	−1.04800
$996$	0	0
$997$	−2.66478e7	−0.849031	−0.424515	−	0.905421i	$-0.639555\pi$
$997$	−2.66478e7	−0.849031	−0.424515	+	0.905421i	$0.639555\pi$
$998$	0	0
$999$	4.15612e7	1.31757

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.h.1.5		10
4.3	odd	2		43.6.a.b.1.8	✓	10
12.11	even	2		387.6.a.e.1.3		10
20.19	odd	2		1075.6.a.b.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.6.a.b.1.8	✓	10	4.3	odd	2
387.6.a.e.1.3		10	12.11	even	2
688.6.a.h.1.5		10	1.1	even	1	trivial
1075.6.a.b.1.3		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)

\(f(q)\)	\(=\)	\(q-12.8799 q^{3} +79.5677 q^{5} +172.354 q^{7} -77.1083 q^{9} +O(q^{10})\)	\(q-12.8799 q^{3} +79.5677 q^{5} +172.354 q^{7} -77.1083 q^{9} -452.247 q^{11} -22.7429 q^{13} -1024.82 q^{15} -521.824 q^{17} -1558.56 q^{19} -2219.90 q^{21} +3464.36 q^{23} +3206.01 q^{25} +4122.96 q^{27} +4324.79 q^{29} +3987.66 q^{31} +5824.90 q^{33} +13713.8 q^{35} +10080.4 q^{37} +292.926 q^{39} -16408.5 q^{41} -1849.00 q^{43} -6135.33 q^{45} -24153.7 q^{47} +12898.8 q^{49} +6721.04 q^{51} +21214.2 q^{53} -35984.2 q^{55} +20074.0 q^{57} +25849.7 q^{59} +28577.7 q^{61} -13289.9 q^{63} -1809.60 q^{65} -66762.2 q^{67} -44620.6 q^{69} +10031.8 q^{71} +32145.0 q^{73} -41293.1 q^{75} -77946.5 q^{77} +21913.4 q^{79} -34366.0 q^{81} +66782.4 q^{83} -41520.3 q^{85} -55702.8 q^{87} +48984.7 q^{89} -3919.82 q^{91} -51360.7 q^{93} -124011. q^{95} +93075.8 q^{97} +34872.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 28 q^{3} + 138 q^{5} - 60 q^{7} + 1356 q^{9}+O(q^{10}) \) 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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