LMFDB - Embedded newform 688.6.a.k.1.3

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:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 2180 x^{12} - 1495 x^{11} + 1754296 x^{10} + 3578989 x^{9} - 638233449 x^{8} + \cdots - 542414322507456 $ x^14 - x^13 - 2180x^12 - 1495x^11 + 1754296x^10 + 3578989x^9 - 638233449x^8 - 1708713010x^7 + 104384337816x^6 + 211948793000x^5 - 7512999124548x^4 - 6583022459712x^3 + 197513018624160x^2 - 31344227384256x - 542414322507456
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	no (minimal twist has level 344)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-18.1345$ 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)

$f(q)$	$=$	$q-20.1345 q^{3} -33.4767 q^{5} -246.400 q^{7} +162.398 q^{9} +O(q^{10})$	$q-20.1345 q^{3} -33.4767 q^{5} -246.400 q^{7} +162.398 q^{9} -714.595 q^{11} +563.648 q^{13} +674.037 q^{15} +215.462 q^{17} +781.092 q^{19} +4961.14 q^{21} +680.490 q^{23} -2004.31 q^{25} +1622.88 q^{27} -1781.22 q^{29} +7111.14 q^{31} +14388.0 q^{33} +8248.67 q^{35} +2012.54 q^{37} -11348.8 q^{39} +11580.5 q^{41} -1849.00 q^{43} -5436.55 q^{45} -24181.1 q^{47} +43906.1 q^{49} -4338.22 q^{51} -1757.38 q^{53} +23922.3 q^{55} -15726.9 q^{57} -8289.88 q^{59} -7967.98 q^{61} -40014.9 q^{63} -18869.1 q^{65} +1527.04 q^{67} -13701.3 q^{69} +11921.0 q^{71} +76304.7 q^{73} +40355.7 q^{75} +176076. q^{77} -40040.0 q^{79} -72138.6 q^{81} +67384.0 q^{83} -7212.96 q^{85} +35864.0 q^{87} -102851. q^{89} -138883. q^{91} -143179. q^{93} -26148.4 q^{95} -31947.4 q^{97} -116049. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 27 q^{3} + 25 q^{5} - 160 q^{7} + 1011 q^{9}+O(q^{10}) $ 14 * q - 27 * q^3 + 25 * q^5 - 160 * q^7 + 1011 * q^9	$ 14 q - 27 q^{3} + 25 q^{5} - 160 q^{7} + 1011 q^{9} - 367 q^{11} + 615 q^{13} - 1780 q^{15} - 1718 q^{17} - 3565 q^{19} + 2258 q^{21} + 2530 q^{23} + 6307 q^{25} - 2118 q^{27} - 5105 q^{29} - 11152 q^{31} + 3518 q^{33} - 13426 q^{35} + 8575 q^{37} - 43148 q^{39} - 9026 q^{41} - 25886 q^{43} + 69212 q^{45} - 14379 q^{47} + 86350 q^{49} - 14235 q^{51} + 75113 q^{53} - 39292 q^{55} + 51044 q^{57} - 80680 q^{59} + 89420 q^{61} - 129922 q^{63} + 70614 q^{65} - 87121 q^{67} + 35232 q^{69} - 129934 q^{71} + 95492 q^{73} - 335468 q^{75} + 70212 q^{77} - 222515 q^{79} + 128574 q^{81} - 180349 q^{83} + 76652 q^{85} - 276293 q^{87} + 87752 q^{89} - 395572 q^{91} + 256669 q^{93} - 453329 q^{95} - 66078 q^{97} - 775955 q^{99}+O(q^{100}) $ 14 * q - 27 * q^3 + 25 * q^5 - 160 * q^7 + 1011 * q^9 - 367 * q^11 + 615 * q^13 - 1780 * q^15 - 1718 * q^17 - 3565 * q^19 + 2258 * q^21 + 2530 * q^23 + 6307 * q^25 - 2118 * q^27 - 5105 * q^29 - 11152 * q^31 + 3518 * q^33 - 13426 * q^35 + 8575 * q^37 - 43148 * q^39 - 9026 * q^41 - 25886 * q^43 + 69212 * q^45 - 14379 * q^47 + 86350 * q^49 - 14235 * q^51 + 75113 * q^53 - 39292 * q^55 + 51044 * q^57 - 80680 * q^59 + 89420 * q^61 - 129922 * q^63 + 70614 * q^65 - 87121 * q^67 + 35232 * q^69 - 129934 * q^71 + 95492 * q^73 - 335468 * q^75 + 70212 * q^77 - 222515 * q^79 + 128574 * q^81 - 180349 * q^83 + 76652 * q^85 - 276293 * q^87 + 87752 * q^89 - 395572 * q^91 + 256669 * q^93 - 453329 * q^95 - 66078 * q^97 - 775955 * 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
$3$	−20.1345	−1.29163	−0.645814	−	0.763495i	$-0.723482\pi$
$3$	−20.1345	−1.29163	−0.645814	+	0.763495i	$0.723482\pi$
$4$	0	0
$5$	−33.4767	−0.598850	−0.299425	−	0.954120i	$-0.596795\pi$
$5$	−33.4767	−0.598850	−0.299425	+	0.954120i	$0.596795\pi$
$6$	0	0
$7$	−246.400	−1.90062	−0.950312	−	0.311300i	$-0.899236\pi$
$7$	−246.400	−1.90062	−0.950312	+	0.311300i	$0.899236\pi$
$8$	0	0
$9$	162.398	0.668304
$10$	0	0
$11$	−714.595	−1.78065	−0.890324	−	0.455328i	$-0.849522\pi$
$11$	−714.595	−1.78065	−0.890324	+	0.455328i	$0.849522\pi$
$12$	0	0
$13$	563.648	0.925017	0.462509	−	0.886615i	$-0.346949\pi$
$13$	563.648	0.925017	0.462509	+	0.886615i	$0.346949\pi$
$14$	0	0
$15$	674.037	0.773492
$16$	0	0
$17$	215.462	0.180821	0.0904104	−	0.995905i	$-0.471182\pi$
$17$	215.462	0.180821	0.0904104	+	0.995905i	$0.471182\pi$
$18$	0	0
$19$	781.092	0.496384	0.248192	−	0.968711i	$-0.420164\pi$
$19$	781.092	0.496384	0.248192	+	0.968711i	$0.420164\pi$
$20$	0	0
$21$	4961.14	2.45490
$22$	0	0
$23$	680.490	0.268227	0.134113	−	0.990966i	$-0.457181\pi$
$23$	680.490	0.268227	0.134113	+	0.990966i	$0.457181\pi$
$24$	0	0
$25$	−2004.31	−0.641379
$26$	0	0
$27$	1622.88	0.428427
$28$	0	0
$29$	−1781.22	−0.393300	−0.196650	−	0.980474i	$-0.563006\pi$
$29$	−1781.22	−0.393300	−0.196650	+	0.980474i	$0.563006\pi$
$30$	0	0
$31$	7111.14	1.32903	0.664516	−	0.747274i	$-0.268638\pi$
$31$	7111.14	1.32903	0.664516	+	0.747274i	$0.268638\pi$
$32$	0	0
$33$	14388.0	2.29994
$34$	0	0
$35$	8248.67	1.13819
$36$	0	0
$37$	2012.54	0.241680	0.120840	−	0.992672i	$-0.461441\pi$
$37$	2012.54	0.241680	0.120840	+	0.992672i	$0.461441\pi$
$38$	0	0
$39$	−11348.8	−1.19478
$40$	0	0
$41$	11580.5	1.07588	0.537942	−	0.842982i	$-0.319202\pi$
$41$	11580.5	1.07588	0.537942	+	0.842982i	$0.319202\pi$
$42$	0	0
$43$	−1849.00	−0.152499
$43$	−1849.00	−0.152499
$44$	0	0
$45$	−5436.55	−0.400214
$46$	0	0
$47$	−24181.1	−1.59673	−0.798366	−	0.602173i	$-0.794302\pi$
$47$	−24181.1	−1.59673	−0.798366	+	0.602173i	$0.794302\pi$
$48$	0	0
$49$	43906.1	2.61237
$50$	0	0
$51$	−4338.22	−0.233553
$52$	0	0
$53$	−1757.38	−0.0859360	−0.0429680	−	0.999076i	$-0.513681\pi$
$53$	−1757.38	−0.0859360	−0.0429680	+	0.999076i	$0.513681\pi$
$54$	0	0
$55$	23922.3	1.06634
$56$	0	0
$57$	−15726.9	−0.641144
$58$	0	0
$59$	−8289.88	−0.310040	−0.155020	−	0.987911i	$-0.549544\pi$
$59$	−8289.88	−0.310040	−0.155020	+	0.987911i	$0.549544\pi$
$60$	0	0
$61$	−7967.98	−0.274172	−0.137086	−	0.990559i	$-0.543774\pi$
$61$	−7967.98	−0.274172	−0.137086	+	0.990559i	$0.543774\pi$
$62$	0	0
$63$	−40014.9	−1.27019
$64$	0	0
$65$	−18869.1	−0.553947
$66$	0	0
$67$	1527.04	0.0415587	0.0207794	−	0.999784i	$-0.493385\pi$
$67$	1527.04	0.0415587	0.0207794	+	0.999784i	$0.493385\pi$
$68$	0	0
$69$	−13701.3	−0.346449
$70$	0	0
$71$	11921.0	0.280651	0.140325	−	0.990105i	$-0.455185\pi$
$71$	11921.0	0.280651	0.140325	+	0.990105i	$0.455185\pi$
$72$	0	0
$73$	76304.7	1.67589	0.837943	−	0.545758i	$-0.183758\pi$
$73$	76304.7	1.67589	0.837943	+	0.545758i	$0.183758\pi$
$74$	0	0
$75$	40355.7	0.828423
$76$	0	0
$77$	176076.	3.38434
$78$	0	0
$79$	−40040.0	−0.721815	−0.360908	−	0.932602i	$-0.617533\pi$
$79$	−40040.0	−0.721815	−0.360908	+	0.932602i	$0.617533\pi$
$80$	0	0
$81$	−72138.6	−1.22167
$82$	0	0
$83$	67384.0	1.07365	0.536824	−	0.843694i	$-0.319624\pi$
$83$	67384.0	1.07365	0.536824	+	0.843694i	$0.319624\pi$
$84$	0	0
$85$	−7212.96	−0.108285
$86$	0	0
$87$	35864.0	0.507997
$88$	0	0
$89$	−102851.	−1.37637	−0.688184	−	0.725537i	$-0.741592\pi$
$89$	−102851.	−1.37637	−0.688184	+	0.725537i	$0.741592\pi$
$90$	0	0
$91$	−138883.	−1.75811
$92$	0	0
$93$	−143179.	−1.71662
$94$	0	0
$95$	−26148.4	−0.297260
$96$	0	0
$97$	−31947.4	−0.344752	−0.172376	−	0.985031i	$-0.555144\pi$
$97$	−31947.4	−0.344752	−0.172376	+	0.985031i	$0.555144\pi$
$98$	0	0
$99$	−116049.	−1.19001
$100$	0	0
$101$	66555.2	0.649201	0.324600	−	0.945851i	$-0.394770\pi$
$101$	66555.2	0.649201	0.324600	+	0.945851i	$0.394770\pi$
$102$	0	0
$103$	178865.	1.66124	0.830621	−	0.556838i	$-0.187986\pi$
$103$	178865.	1.66124	0.830621	+	0.556838i	$0.187986\pi$
$104$	0	0
$105$	−166083.	−1.47012
$106$	0	0
$107$	63117.7	0.532957	0.266478	−	0.963841i	$-0.414140\pi$
$107$	63117.7	0.532957	0.266478	+	0.963841i	$0.414140\pi$
$108$	0	0
$109$	159969.	1.28964	0.644822	−	0.764333i	$-0.276931\pi$
$109$	159969.	1.28964	0.644822	+	0.764333i	$0.276931\pi$
$110$	0	0
$111$	−40521.5	−0.312160
$112$	0	0
$113$	−100539.	−0.740692	−0.370346	−	0.928894i	$-0.620761\pi$
$113$	−100539.	−0.740692	−0.370346	+	0.928894i	$0.620761\pi$
$114$	0	0
$115$	−22780.6	−0.160628
$116$	0	0
$117$	91535.3	0.618193
$118$	0	0
$119$	−53089.9	−0.343672
$120$	0	0
$121$	349594.	2.17071
$122$	0	0
$123$	−233167.	−1.38964
$124$	0	0
$125$	171712.	0.982940
$126$	0	0
$127$	−350162.	−1.92646	−0.963230	−	0.268680i	$-0.913413\pi$
$127$	−350162.	−1.92646	−0.963230	+	0.268680i	$0.913413\pi$
$128$	0	0
$129$	37228.7	0.196972
$130$	0	0
$131$	122840.	0.625407	0.312703	−	0.949851i	$-0.398765\pi$
$131$	122840.	0.625407	0.312703	+	0.949851i	$0.398765\pi$
$132$	0	0
$133$	−192461.	−0.943440
$134$	0	0
$135$	−54328.8	−0.256564
$136$	0	0
$137$	−124402.	−0.566271	−0.283136	−	0.959080i	$-0.591375\pi$
$137$	−124402.	−0.566271	−0.283136	+	0.959080i	$0.591375\pi$
$138$	0	0
$139$	105904.	0.464916	0.232458	−	0.972606i	$-0.425323\pi$
$139$	105904.	0.464916	0.232458	+	0.972606i	$0.425323\pi$
$140$	0	0
$141$	486875.	2.06238
$142$	0	0
$143$	−402780.	−1.64713
$144$	0	0
$145$	59629.5	0.235527
$146$	0	0
$147$	−884027.	−3.37421
$148$	0	0
$149$	−418417.	−1.54399	−0.771994	−	0.635630i	$-0.780740\pi$
$149$	−418417.	−1.54399	−0.771994	+	0.635630i	$0.780740\pi$
$150$	0	0
$151$	104231.	0.372009	0.186005	−	0.982549i	$-0.440446\pi$
$151$	104231.	0.372009	0.186005	+	0.982549i	$0.440446\pi$
$152$	0	0
$153$	34990.6	0.120843
$154$	0	0
$155$	−238058.	−0.795891
$156$	0	0
$157$	475743.	1.54036	0.770182	−	0.637824i	$-0.220165\pi$
$157$	475743.	1.54036	0.770182	+	0.637824i	$0.220165\pi$
$158$	0	0
$159$	35383.9	0.110997
$160$	0	0
$161$	−167673.	−0.509798
$162$	0	0
$163$	−353265.	−1.04143	−0.520717	−	0.853729i	$-0.674335\pi$
$163$	−353265.	−1.04143	−0.520717	+	0.853729i	$0.674335\pi$
$164$	0	0
$165$	−481663.	−1.37732
$166$	0	0
$167$	43153.3	0.119735	0.0598677	−	0.998206i	$-0.480932\pi$
$167$	43153.3	0.119735	0.0598677	+	0.998206i	$0.480932\pi$
$168$	0	0
$169$	−53593.6	−0.144343
$170$	0	0
$171$	126848.	0.331736
$172$	0	0
$173$	−723007.	−1.83665	−0.918327	−	0.395822i	$-0.870460\pi$
$173$	−723007.	−1.83665	−0.918327	+	0.395822i	$0.870460\pi$
$174$	0	0
$175$	493862.	1.21902
$176$	0	0
$177$	166913.	0.400457
$178$	0	0
$179$	−119848.	−0.279574	−0.139787	−	0.990182i	$-0.544642\pi$
$179$	−119848.	−0.279574	−0.139787	+	0.990182i	$0.544642\pi$
$180$	0	0
$181$	452981.	1.02774	0.513870	−	0.857868i	$-0.328211\pi$
$181$	452981.	1.02774	0.513870	+	0.857868i	$0.328211\pi$
$182$	0	0
$183$	160431.	0.354129
$184$	0	0
$185$	−67373.3	−0.144730
$186$	0	0
$187$	−153968.	−0.321978
$188$	0	0
$189$	−399878.	−0.814279
$190$	0	0
$191$	389962.	0.773463	0.386731	−	0.922192i	$-0.373604\pi$
$191$	389962.	0.773463	0.386731	+	0.922192i	$0.373604\pi$
$192$	0	0
$193$	−694107.	−1.34132	−0.670661	−	0.741764i	$-0.733989\pi$
$193$	−694107.	−1.34132	−0.670661	+	0.741764i	$0.733989\pi$
$194$	0	0
$195$	379920.	0.715493
$196$	0	0
$197$	−612611.	−1.12466	−0.562328	−	0.826914i	$-0.690094\pi$
$197$	−612611.	−1.12466	−0.562328	+	0.826914i	$0.690094\pi$
$198$	0	0
$199$	−689009.	−1.23337	−0.616683	−	0.787212i	$-0.711524\pi$
$199$	−689009.	−1.23337	−0.616683	+	0.787212i	$0.711524\pi$
$200$	0	0
$201$	−30746.1	−0.0536784
$202$	0	0
$203$	438894.	0.747514
$204$	0	0
$205$	−387676.	−0.644294
$206$	0	0
$207$	110510.	0.179257
$208$	0	0
$209$	−558164.	−0.883886
$210$	0	0
$211$	−284199.	−0.439458	−0.219729	−	0.975561i	$-0.570517\pi$
$211$	−284199.	−0.439458	−0.219729	+	0.975561i	$0.570517\pi$
$212$	0	0
$213$	−240023.	−0.362496
$214$	0	0
$215$	61898.5	0.0913238
$216$	0	0
$217$	−1.75219e6	−2.52599
$218$	0	0
$219$	−1.53636e6	−2.16462
$220$	0	0
$221$	121445.	0.167262
$222$	0	0
$223$	−586704.	−0.790055	−0.395027	−	0.918669i	$-0.629265\pi$
$223$	−586704.	−0.790055	−0.395027	+	0.918669i	$0.629265\pi$
$224$	0	0
$225$	−325496.	−0.428636
$226$	0	0
$227$	746243.	0.961204	0.480602	−	0.876939i	$-0.340418\pi$
$227$	746243.	0.961204	0.480602	+	0.876939i	$0.340418\pi$
$228$	0	0
$229$	−343064.	−0.432301	−0.216150	−	0.976360i	$-0.569350\pi$
$229$	−343064.	−0.432301	−0.216150	+	0.976360i	$0.569350\pi$
$230$	0	0
$231$	−3.54521e6	−4.37131
$232$	0	0
$233$	304092.	0.366956	0.183478	−	0.983024i	$-0.441264\pi$
$233$	304092.	0.366956	0.183478	+	0.983024i	$0.441264\pi$
$234$	0	0
$235$	809505.	0.956202
$236$	0	0
$237$	806185.	0.932317
$238$	0	0
$239$	715089.	0.809776	0.404888	−	0.914366i	$-0.367310\pi$
$239$	715089.	0.809776	0.404888	+	0.914366i	$0.367310\pi$
$240$	0	0
$241$	−234912.	−0.260532	−0.130266	−	0.991479i	$-0.541583\pi$
$241$	−234912.	−0.260532	−0.130266	+	0.991479i	$0.541583\pi$
$242$	0	0
$243$	1.05811e6	1.14952
$244$	0	0
$245$	−1.46983e6	−1.56442
$246$	0	0
$247$	440261.	0.459164
$248$	0	0
$249$	−1.35674e6	−1.38675
$250$	0	0
$251$	1.54440e6	1.54730	0.773651	−	0.633612i	$-0.218428\pi$
$251$	1.54440e6	1.54730	0.773651	+	0.633612i	$0.218428\pi$
$252$	0	0
$253$	−486274.	−0.477617
$254$	0	0
$255$	145229.	0.139863
$256$	0	0
$257$	−601035.	−0.567632	−0.283816	−	0.958879i	$-0.591600\pi$
$257$	−601035.	−0.567632	−0.283816	+	0.958879i	$0.591600\pi$
$258$	0	0
$259$	−495890.	−0.459342
$260$	0	0
$261$	−289267.	−0.262844
$262$	0	0
$263$	−504411.	−0.449672	−0.224836	−	0.974397i	$-0.572185\pi$
$263$	−504411.	−0.449672	−0.224836	+	0.974397i	$0.572185\pi$
$264$	0	0
$265$	58831.2	0.0514628
$266$	0	0
$267$	2.07086e6	1.77776
$268$	0	0
$269$	348932.	0.294008	0.147004	−	0.989136i	$-0.453037\pi$
$269$	348932.	0.294008	0.147004	+	0.989136i	$0.453037\pi$
$270$	0	0
$271$	−754116.	−0.623757	−0.311878	−	0.950122i	$-0.600958\pi$
$271$	−754116.	−0.623757	−0.311878	+	0.950122i	$0.600958\pi$
$272$	0	0
$273$	2.79634e6	2.27082
$274$	0	0
$275$	1.43227e6	1.14207
$276$	0	0
$277$	2.40658e6	1.88452	0.942259	−	0.334884i	$-0.108697\pi$
$277$	2.40658e6	1.88452	0.942259	+	0.334884i	$0.108697\pi$
$278$	0	0
$279$	1.15484e6	0.888198
$280$	0	0
$281$	−319753.	−0.241573	−0.120787	−	0.992679i	$-0.538542\pi$
$281$	−319753.	−0.241573	−0.120787	+	0.992679i	$0.538542\pi$
$282$	0	0
$283$	2.10625e6	1.56331	0.781653	−	0.623714i	$-0.214377\pi$
$283$	2.10625e6	1.56331	0.781653	+	0.623714i	$0.214377\pi$
$284$	0	0
$285$	526485.	0.383949
$286$	0	0
$287$	−2.85343e6	−2.04485
$288$	0	0
$289$	−1.37343e6	−0.967304
$290$	0	0
$291$	643245.	0.445291
$292$	0	0
$293$	1.53564e6	1.04501	0.522505	−	0.852636i	$-0.324998\pi$
$293$	1.53564e6	1.04501	0.522505	+	0.852636i	$0.324998\pi$
$294$	0	0
$295$	277518.	0.185668
$296$	0	0
$297$	−1.15970e6	−0.762878
$298$	0	0
$299$	383557.	0.248114
$300$	0	0
$301$	455594.	0.289842
$302$	0	0
$303$	−1.34006e6	−0.838526
$304$	0	0
$305$	266742.	0.164188
$306$	0	0
$307$	3.16184e6	1.91467	0.957335	−	0.288980i	$-0.0933162\pi$
$307$	3.16184e6	1.91467	0.957335	+	0.288980i	$0.0933162\pi$
$308$	0	0
$309$	−3.60136e6	−2.14571
$310$	0	0
$311$	2.20077e6	1.29025	0.645125	−	0.764077i	$-0.276805\pi$
$311$	2.20077e6	1.29025	0.645125	+	0.764077i	$0.276805\pi$
$312$	0	0
$313$	1.05252e6	0.607253	0.303626	−	0.952791i	$-0.401803\pi$
$313$	1.05252e6	0.607253	0.303626	+	0.952791i	$0.401803\pi$
$314$	0	0
$315$	1.33957e6	0.760656
$316$	0	0
$317$	1.90515e6	1.06483	0.532416	−	0.846483i	$-0.321284\pi$
$317$	1.90515e6	1.06483	0.532416	+	0.846483i	$0.321284\pi$
$318$	0	0
$319$	1.27285e6	0.700328
$320$	0	0
$321$	−1.27084e6	−0.688382
$322$	0	0
$323$	168296.	0.0897566
$324$	0	0
$325$	−1.12973e6	−0.593286
$326$	0	0
$327$	−3.22090e6	−1.66574
$328$	0	0
$329$	5.95823e6	3.03478
$330$	0	0
$331$	−1.18249e6	−0.593235	−0.296617	−	0.954996i	$-0.595859\pi$
$331$	−1.18249e6	−0.593235	−0.296617	+	0.954996i	$0.595859\pi$
$332$	0	0
$333$	326832.	0.161516
$334$	0	0
$335$	−51120.2	−0.0248874
$336$	0	0
$337$	337706.	0.161981	0.0809904	−	0.996715i	$-0.474192\pi$
$337$	337706.	0.161981	0.0809904	+	0.996715i	$0.474192\pi$
$338$	0	0
$339$	2.02430e6	0.956699
$340$	0	0
$341$	−5.08159e6	−2.36654
$342$	0	0
$343$	−6.67722e6	−3.06451
$344$	0	0
$345$	458675.	0.207471
$346$	0	0
$347$	−1.50633e6	−0.671577	−0.335788	−	0.941937i	$-0.609003\pi$
$347$	−1.50633e6	−0.671577	−0.335788	+	0.941937i	$0.609003\pi$
$348$	0	0
$349$	−39675.3	−0.0174364	−0.00871820	−	0.999962i	$-0.502775\pi$
$349$	−39675.3	−0.0174364	−0.00871820	+	0.999962i	$0.502775\pi$
$350$	0	0
$351$	914734.	0.396303
$352$	0	0
$353$	2.54517e6	1.08713	0.543563	−	0.839368i	$-0.317075\pi$
$353$	2.54517e6	1.08713	0.543563	+	0.839368i	$0.317075\pi$
$354$	0	0
$355$	−399076.	−0.168068
$356$	0	0
$357$	1.06894e6	0.443897
$358$	0	0
$359$	−2.48269e6	−1.01669	−0.508343	−	0.861154i	$-0.669742\pi$
$359$	−2.48269e6	−1.01669	−0.508343	+	0.861154i	$0.669742\pi$
$360$	0	0
$361$	−1.86599e6	−0.753603
$362$	0	0
$363$	−7.03891e6	−2.80375
$364$	0	0
$365$	−2.55443e6	−1.00360
$366$	0	0
$367$	−22728.0	−0.00880838	−0.00440419	−	0.999990i	$-0.501402\pi$
$367$	−22728.0	−0.00880838	−0.00440419	+	0.999990i	$0.501402\pi$
$368$	0	0
$369$	1.88064e6	0.719019
$370$	0	0
$371$	433018.	0.163332
$372$	0	0
$373$	−5.29620e6	−1.97103	−0.985514	−	0.169596i	$-0.945754\pi$
$373$	−5.29620e6	−1.97103	−0.985514	+	0.169596i	$0.945754\pi$
$374$	0	0
$375$	−3.45734e6	−1.26959
$376$	0	0
$377$	−1.00398e6	−0.363809
$378$	0	0
$379$	3.14785e6	1.12568	0.562842	−	0.826564i	$-0.309708\pi$
$379$	3.14785e6	1.12568	0.562842	+	0.826564i	$0.309708\pi$
$380$	0	0
$381$	7.05034e6	2.48827
$382$	0	0
$383$	1.38000e6	0.480710	0.240355	−	0.970685i	$-0.422736\pi$
$383$	1.38000e6	0.480710	0.240355	+	0.970685i	$0.422736\pi$
$384$	0	0
$385$	−5.89446e6	−2.02671
$386$	0	0
$387$	−300274.	−0.101915
$388$	0	0
$389$	4.59041e6	1.53807	0.769037	−	0.639205i	$-0.220736\pi$
$389$	4.59041e6	1.53807	0.769037	+	0.639205i	$0.220736\pi$
$390$	0	0
$391$	146620.	0.0485010
$392$	0	0
$393$	−2.47333e6	−0.807793
$394$	0	0
$395$	1.34041e6	0.432259
$396$	0	0
$397$	3.42723e6	1.09136	0.545679	−	0.837995i	$-0.316272\pi$
$397$	3.42723e6	1.09136	0.545679	+	0.837995i	$0.316272\pi$
$398$	0	0
$399$	3.87511e6	1.21857
$400$	0	0
$401$	−1.35819e6	−0.421795	−0.210897	−	0.977508i	$-0.567639\pi$
$401$	−1.35819e6	−0.421795	−0.210897	+	0.977508i	$0.567639\pi$
$402$	0	0
$403$	4.00818e6	1.22938
$404$	0	0
$405$	2.41496e6	0.731599
$406$	0	0
$407$	−1.43815e6	−0.430346
$408$	0	0
$409$	−922252.	−0.272610	−0.136305	−	0.990667i	$-0.543523\pi$
$409$	−922252.	−0.272610	−0.136305	+	0.990667i	$0.543523\pi$
$410$	0	0
$411$	2.50476e6	0.731412
$412$	0	0
$413$	2.04263e6	0.589270
$414$	0	0
$415$	−2.25580e6	−0.642954
$416$	0	0
$417$	−2.13232e6	−0.600499
$418$	0	0
$419$	195692.	0.0544549	0.0272275	−	0.999629i	$-0.491332\pi$
$419$	195692.	0.0544549	0.0272275	+	0.999629i	$0.491332\pi$
$420$	0	0
$421$	−3.03599e6	−0.834823	−0.417412	−	0.908718i	$-0.637063\pi$
$421$	−3.03599e6	−0.834823	−0.417412	+	0.908718i	$0.637063\pi$
$422$	0	0
$423$	−3.92697e6	−1.06710
$424$	0	0
$425$	−431852.	−0.115975
$426$	0	0
$427$	1.96331e6	0.521098
$428$	0	0
$429$	8.10977e6	2.12748
$430$	0	0
$431$	7.38580e6	1.91516	0.957579	−	0.288172i	$-0.0930475\pi$
$431$	7.38580e6	1.91516	0.957579	+	0.288172i	$0.0930475\pi$
$432$	0	0
$433$	−4.54563e6	−1.16513	−0.582565	−	0.812784i	$-0.697951\pi$
$433$	−4.54563e6	−1.16513	−0.582565	+	0.812784i	$0.697951\pi$
$434$	0	0
$435$	−1.20061e6	−0.304214
$436$	0	0
$437$	531525.	0.133144
$438$	0	0
$439$	2.70940e6	0.670983	0.335492	−	0.942043i	$-0.391098\pi$
$439$	2.70940e6	0.670983	0.335492	+	0.942043i	$0.391098\pi$
$440$	0	0
$441$	7.13026e6	1.74586
$442$	0	0
$443$	−6.80016e6	−1.64630	−0.823151	−	0.567822i	$-0.807786\pi$
$443$	−6.80016e6	−1.64630	−0.823151	+	0.567822i	$0.807786\pi$
$444$	0	0
$445$	3.44312e6	0.824237
$446$	0	0
$447$	8.42462e6	1.99426
$448$	0	0
$449$	−6.70400e6	−1.56934	−0.784672	−	0.619911i	$-0.787169\pi$
$449$	−6.70400e6	−1.56934	−0.784672	+	0.619911i	$0.787169\pi$
$450$	0	0
$451$	−8.27533e6	−1.91577
$452$	0	0
$453$	−2.09864e6	−0.480498
$454$	0	0
$455$	4.64935e6	1.05284
$456$	0	0
$457$	4.12552e6	0.924033	0.462017	−	0.886871i	$-0.347126\pi$
$457$	4.12552e6	0.924033	0.462017	+	0.886871i	$0.347126\pi$
$458$	0	0
$459$	349669.	0.0774686
$460$	0	0
$461$	1.13848e6	0.249502	0.124751	−	0.992188i	$-0.460187\pi$
$461$	1.13848e6	0.249502	0.124751	+	0.992188i	$0.460187\pi$
$462$	0	0
$463$	−4.24526e6	−0.920347	−0.460174	−	0.887829i	$-0.652213\pi$
$463$	−4.24526e6	−0.920347	−0.460174	+	0.887829i	$0.652213\pi$
$464$	0	0
$465$	4.79318e6	1.02799
$466$	0	0
$467$	−3.55557e6	−0.754427	−0.377214	−	0.926126i	$-0.623118\pi$
$467$	−3.55557e6	−0.754427	−0.377214	+	0.926126i	$0.623118\pi$
$468$	0	0
$469$	−376262.	−0.0789875
$470$	0	0
$471$	−9.57885e6	−1.98958
$472$	0	0
$473$	1.32129e6	0.271546
$474$	0	0
$475$	−1.56555e6	−0.318370
$476$	0	0
$477$	−285395.	−0.0574314
$478$	0	0
$479$	9.60433e6	1.91262	0.956309	−	0.292357i	$-0.0944396\pi$
$479$	9.60433e6	1.91262	0.956309	+	0.292357i	$0.0944396\pi$
$480$	0	0
$481$	1.13436e6	0.223558
$482$	0	0
$483$	3.37601e6	0.658470
$484$	0	0
$485$	1.06950e6	0.206455
$486$	0	0
$487$	6.72843e6	1.28556	0.642779	−	0.766052i	$-0.277782\pi$
$487$	6.72843e6	1.28556	0.642779	+	0.766052i	$0.277782\pi$
$488$	0	0
$489$	7.11282e6	1.34515
$490$	0	0
$491$	945647.	0.177021	0.0885106	−	0.996075i	$-0.471789\pi$
$491$	945647.	0.177021	0.0885106	+	0.996075i	$0.471789\pi$
$492$	0	0
$493$	−383786.	−0.0711167
$494$	0	0
$495$	3.88493e6	0.712640
$496$	0	0
$497$	−2.93733e6	−0.533411
$498$	0	0
$499$	9.25726e6	1.66430	0.832149	−	0.554552i	$-0.187110\pi$
$499$	9.25726e6	1.66430	0.832149	+	0.554552i	$0.187110\pi$
$500$	0	0
$501$	−868870.	−0.154654
$502$	0	0
$503$	2.96119e6	0.521851	0.260926	−	0.965359i	$-0.415972\pi$
$503$	2.96119e6	0.521851	0.260926	+	0.965359i	$0.415972\pi$
$504$	0	0
$505$	−2.22805e6	−0.388774
$506$	0	0
$507$	1.07908e6	0.186438
$508$	0	0
$509$	7.66195e6	1.31083	0.655413	−	0.755271i	$-0.272495\pi$
$509$	7.66195e6	1.31083	0.655413	+	0.755271i	$0.272495\pi$
$510$	0	0
$511$	−1.88015e7	−3.18523
$512$	0	0
$513$	1.26762e6	0.212665
$514$	0	0
$515$	−5.98783e6	−0.994835
$516$	0	0
$517$	1.72797e7	2.84322
$518$	0	0
$519$	1.45574e7	2.37228
$520$	0	0
$521$	555968.	0.0897336	0.0448668	−	0.998993i	$-0.485714\pi$
$521$	555968.	0.0897336	0.0448668	+	0.998993i	$0.485714\pi$
$522$	0	0
$523$	−4.16674e6	−0.666104	−0.333052	−	0.942908i	$-0.608079\pi$
$523$	−4.16674e6	−0.666104	−0.333052	+	0.942908i	$0.608079\pi$
$524$	0	0
$525$	−9.94366e6	−1.57452
$526$	0	0
$527$	1.53218e6	0.240317
$528$	0	0
$529$	−5.97328e6	−0.928054
$530$	0	0
$531$	−1.34626e6	−0.207201
$532$	0	0
$533$	6.52730e6	0.995212
$534$	0	0
$535$	−2.11298e6	−0.319161
$536$	0	0
$537$	2.41307e6	0.361106
$538$	0	0
$539$	−3.13750e7	−4.65171
$540$	0	0
$541$	−631639.	−0.0927846	−0.0463923	−	0.998923i	$-0.514772\pi$
$541$	−631639.	−0.0927846	−0.0463923	+	0.998923i	$0.514772\pi$
$542$	0	0
$543$	−9.12054e6	−1.32746
$544$	0	0
$545$	−5.35524e6	−0.772303
$546$	0	0
$547$	−9.65450e6	−1.37963	−0.689813	−	0.723988i	$-0.742307\pi$
$547$	−9.65450e6	−1.37963	−0.689813	+	0.723988i	$0.742307\pi$
$548$	0	0
$549$	−1.29398e6	−0.183231
$550$	0	0
$551$	−1.39130e6	−0.195228
$552$	0	0
$553$	9.86586e6	1.37190
$554$	0	0
$555$	1.35653e6	0.186937
$556$	0	0
$557$	−1.36715e6	−0.186714	−0.0933570	−	0.995633i	$-0.529760\pi$
$557$	−1.36715e6	−0.186714	−0.0933570	+	0.995633i	$0.529760\pi$
$558$	0	0
$559$	−1.04219e6	−0.141064
$560$	0	0
$561$	3.10007e6	0.415876
$562$	0	0
$563$	−1.24214e7	−1.65158	−0.825792	−	0.563975i	$-0.809271\pi$
$563$	−1.24214e7	−1.65158	−0.825792	+	0.563975i	$0.809271\pi$
$564$	0	0
$565$	3.36571e6	0.443563
$566$	0	0
$567$	1.77750e7	2.32194
$568$	0	0
$569$	−1.17545e7	−1.52203	−0.761014	−	0.648736i	$-0.775298\pi$
$569$	−1.17545e7	−1.52203	−0.761014	+	0.648736i	$0.775298\pi$
$570$	0	0
$571$	8.36928e6	1.07423	0.537115	−	0.843509i	$-0.319514\pi$
$571$	8.36928e6	1.07423	0.537115	+	0.843509i	$0.319514\pi$
$572$	0	0
$573$	−7.85170e6	−0.999027
$574$	0	0
$575$	−1.36391e6	−0.172035
$576$	0	0
$577$	9.62196e6	1.20316	0.601581	−	0.798812i	$-0.294538\pi$
$577$	9.62196e6	1.20316	0.601581	+	0.798812i	$0.294538\pi$
$578$	0	0
$579$	1.39755e7	1.73249
$580$	0	0
$581$	−1.66034e7	−2.04060
$582$	0	0
$583$	1.25581e6	0.153022
$584$	0	0
$585$	−3.06430e6	−0.370205
$586$	0	0
$587$	1.14938e7	1.37679	0.688396	−	0.725335i	$-0.258315\pi$
$587$	1.14938e7	1.37679	0.688396	+	0.725335i	$0.258315\pi$
$588$	0	0
$589$	5.55446e6	0.659711
$590$	0	0
$591$	1.23346e7	1.45264
$592$	0	0
$593$	−1.30120e7	−1.51953	−0.759763	−	0.650200i	$-0.774685\pi$
$593$	−1.30120e7	−1.51953	−0.759763	+	0.650200i	$0.774685\pi$
$594$	0	0
$595$	1.77728e6	0.205808
$596$	0	0
$597$	1.38728e7	1.59305
$598$	0	0
$599$	6.55495e6	0.746453	0.373226	−	0.927740i	$-0.378251\pi$
$599$	6.55495e6	0.746453	0.373226	+	0.927740i	$0.378251\pi$
$600$	0	0
$601$	1.55377e7	1.75469	0.877343	−	0.479863i	$-0.159314\pi$
$601$	1.55377e7	1.75469	0.877343	+	0.479863i	$0.159314\pi$
$602$	0	0
$603$	247988.	0.0277739
$604$	0	0
$605$	−1.17033e7	−1.29993
$606$	0	0
$607$	−1.16095e7	−1.27892	−0.639459	−	0.768825i	$-0.720841\pi$
$607$	−1.16095e7	−1.27892	−0.639459	+	0.768825i	$0.720841\pi$
$608$	0	0
$609$	−8.83691e6	−0.965511
$610$	0	0
$611$	−1.36296e7	−1.47700
$612$	0	0
$613$	8.14363e6	0.875320	0.437660	−	0.899141i	$-0.355807\pi$
$613$	8.14363e6	0.875320	0.437660	+	0.899141i	$0.355807\pi$
$614$	0	0
$615$	7.80565e6	0.832188
$616$	0	0
$617$	−9.10441e6	−0.962806	−0.481403	−	0.876499i	$-0.659873\pi$
$617$	−9.10441e6	−0.962806	−0.481403	+	0.876499i	$0.659873\pi$
$618$	0	0
$619$	−1.44152e7	−1.51215	−0.756073	−	0.654487i	$-0.772885\pi$
$619$	−1.44152e7	−1.51215	−0.756073	+	0.654487i	$0.772885\pi$
$620$	0	0
$621$	1.10435e6	0.114916
$622$	0	0
$623$	2.53426e7	2.61596
$624$	0	0
$625$	515091.	0.0527453
$626$	0	0
$627$	1.12384e7	1.14165
$628$	0	0
$629$	433626.	0.0437007
$630$	0	0
$631$	1.00641e7	1.00624	0.503120	−	0.864216i	$-0.332185\pi$
$631$	1.00641e7	1.00624	0.503120	+	0.864216i	$0.332185\pi$
$632$	0	0
$633$	5.72221e6	0.567616
$634$	0	0
$635$	1.17223e7	1.15366
$636$	0	0
$637$	2.47476e7	2.41649
$638$	0	0
$639$	1.93594e6	0.187560
$640$	0	0
$641$	2.77855e6	0.267100	0.133550	−	0.991042i	$-0.457362\pi$
$641$	2.77855e6	0.267100	0.133550	+	0.991042i	$0.457362\pi$
$642$	0	0
$643$	1.09347e7	1.04299	0.521494	−	0.853255i	$-0.325375\pi$
$643$	1.09347e7	1.04299	0.521494	+	0.853255i	$0.325375\pi$
$644$	0	0
$645$	−1.24629e6	−0.117956
$646$	0	0
$647$	−7.56717e6	−0.710678	−0.355339	−	0.934738i	$-0.615635\pi$
$647$	−7.56717e6	−0.710678	−0.355339	+	0.934738i	$0.615635\pi$
$648$	0	0
$649$	5.92390e6	0.552072
$650$	0	0
$651$	3.52794e7	3.26264
$652$	0	0
$653$	−1.98013e7	−1.81724	−0.908618	−	0.417627i	$-0.862862\pi$
$653$	−1.98013e7	−1.81724	−0.908618	+	0.417627i	$0.862862\pi$
$654$	0	0
$655$	−4.11229e6	−0.374525
$656$	0	0
$657$	1.23917e7	1.12000
$658$	0	0
$659$	6.64687e6	0.596216	0.298108	−	0.954532i	$-0.403644\pi$
$659$	6.64687e6	0.596216	0.298108	+	0.954532i	$0.403644\pi$
$660$	0	0
$661$	−1.67541e6	−0.149148	−0.0745739	−	0.997215i	$-0.523760\pi$
$661$	−1.67541e6	−0.149148	−0.0745739	+	0.997215i	$0.523760\pi$
$662$	0	0
$663$	−2.44523e6	−0.216041
$664$	0	0
$665$	6.44297e6	0.564979
$666$	0	0
$667$	−1.21210e6	−0.105493
$668$	0	0
$669$	1.18130e7	1.02046
$670$	0	0
$671$	5.69387e6	0.488204
$672$	0	0
$673$	−5.08683e6	−0.432922	−0.216461	−	0.976291i	$-0.569451\pi$
$673$	−5.08683e6	−0.432922	−0.216461	+	0.976291i	$0.569451\pi$
$674$	0	0
$675$	−3.25275e6	−0.274784
$676$	0	0
$677$	−3.61418e6	−0.303067	−0.151533	−	0.988452i	$-0.548421\pi$
$677$	−3.61418e6	−0.303067	−0.151533	+	0.988452i	$0.548421\pi$
$678$	0	0
$679$	7.87185e6	0.655243
$680$	0	0
$681$	−1.50252e7	−1.24152
$682$	0	0
$683$	1.13597e7	0.931783	0.465892	−	0.884842i	$-0.345734\pi$
$683$	1.13597e7	0.931783	0.465892	+	0.884842i	$0.345734\pi$
$684$	0	0
$685$	4.16456e6	0.339111
$686$	0	0
$687$	6.90741e6	0.558372
$688$	0	0
$689$	−990543.	−0.0794923
$690$	0	0
$691$	1.04843e7	0.835300	0.417650	−	0.908608i	$-0.362854\pi$
$691$	1.04843e7	0.835300	0.417650	+	0.908608i	$0.362854\pi$
$692$	0	0
$693$	2.85944e7	2.26177
$694$	0	0
$695$	−3.54531e6	−0.278415
$696$	0	0
$697$	2.49515e6	0.194542
$698$	0	0
$699$	−6.12273e6	−0.473971
$700$	0	0
$701$	−521953.	−0.0401177	−0.0200588	−	0.999799i	$-0.506385\pi$
$701$	−521953.	−0.0401177	−0.0200588	+	0.999799i	$0.506385\pi$
$702$	0	0
$703$	1.57198e6	0.119966
$704$	0	0
$705$	−1.62990e7	−1.23506
$706$	0	0
$707$	−1.63992e7	−1.23389
$708$	0	0
$709$	1.76561e7	1.31911	0.659553	−	0.751658i	$-0.270746\pi$
$709$	1.76561e7	1.31911	0.659553	+	0.751658i	$0.270746\pi$
$710$	0	0
$711$	−6.50241e6	−0.482392
$712$	0	0
$713$	4.83906e6	0.356482
$714$	0	0
$715$	1.34838e7	0.986384
$716$	0	0
$717$	−1.43980e7	−1.04593
$718$	0	0
$719$	−1.79096e7	−1.29201	−0.646003	−	0.763335i	$-0.723561\pi$
$719$	−1.79096e7	−1.29201	−0.646003	+	0.763335i	$0.723561\pi$
$720$	0	0
$721$	−4.40725e7	−3.15740
$722$	0	0
$723$	4.72983e6	0.336511
$724$	0	0
$725$	3.57012e6	0.252254
$726$	0	0
$727$	−1.22126e7	−0.856982	−0.428491	−	0.903546i	$-0.640955\pi$
$727$	−1.22126e7	−0.856982	−0.428491	+	0.903546i	$0.640955\pi$
$728$	0	0
$729$	−3.77492e6	−0.263081
$730$	0	0
$731$	−398389.	−0.0275749
$732$	0	0
$733$	−1.08986e7	−0.749225	−0.374613	−	0.927181i	$-0.622224\pi$
$733$	−1.08986e7	−0.749225	−0.374613	+	0.927181i	$0.622224\pi$
$734$	0	0
$735$	2.95943e7	2.02065
$736$	0	0
$737$	−1.09121e6	−0.0740014
$738$	0	0
$739$	−5.39639e6	−0.363490	−0.181745	−	0.983346i	$-0.558174\pi$
$739$	−5.39639e6	−0.363490	−0.181745	+	0.983346i	$0.558174\pi$
$740$	0	0
$741$	−8.86444e6	−0.593070
$742$	0	0
$743$	−6.90481e6	−0.458859	−0.229430	−	0.973325i	$-0.573686\pi$
$743$	−6.90481e6	−0.458859	−0.229430	+	0.973325i	$0.573686\pi$
$744$	0	0
$745$	1.40072e7	0.924617
$746$	0	0
$747$	1.09430e7	0.717523
$748$	0	0
$749$	−1.55522e7	−1.01295
$750$	0	0
$751$	2.28988e7	1.48154	0.740771	−	0.671758i	$-0.234461\pi$
$751$	2.28988e7	1.48154	0.740771	+	0.671758i	$0.234461\pi$
$752$	0	0
$753$	−3.10957e7	−1.99854
$754$	0	0
$755$	−3.48931e6	−0.222778
$756$	0	0
$757$	8.58710e6	0.544637	0.272318	−	0.962207i	$-0.412210\pi$
$757$	8.58710e6	0.544637	0.272318	+	0.962207i	$0.412210\pi$
$758$	0	0
$759$	9.79089e6	0.616904
$760$	0	0
$761$	−2.97428e7	−1.86175	−0.930873	−	0.365342i	$-0.880952\pi$
$761$	−2.97428e7	−1.86175	−0.930873	+	0.365342i	$0.880952\pi$
$762$	0	0
$763$	−3.94164e7	−2.45113
$764$	0	0
$765$	−1.17137e6	−0.0723670
$766$	0	0
$767$	−4.67258e6	−0.286793
$768$	0	0
$769$	−2.52840e7	−1.54180	−0.770902	−	0.636953i	$-0.780194\pi$
$769$	−2.52840e7	−1.54180	−0.770902	+	0.636953i	$0.780194\pi$
$770$	0	0
$771$	1.21015e7	0.733170
$772$	0	0
$773$	1.40822e7	0.847659	0.423830	−	0.905742i	$-0.360686\pi$
$773$	1.40822e7	0.847659	0.423830	+	0.905742i	$0.360686\pi$
$774$	0	0
$775$	−1.42529e7	−0.852413
$776$	0	0
$777$	9.98450e6	0.593299
$778$	0	0
$779$	9.04540e6	0.534052
$780$	0	0
$781$	−8.51867e6	−0.499740
$782$	0	0
$783$	−2.89071e6	−0.168500
$784$	0	0
$785$	−1.59263e7	−0.922447
$786$	0	0
$787$	7.99248e6	0.459986	0.229993	−	0.973192i	$-0.426130\pi$
$787$	7.99248e6	0.459986	0.229993	+	0.973192i	$0.426130\pi$
$788$	0	0
$789$	1.01561e7	0.580809
$790$	0	0
$791$	2.47728e7	1.40778
$792$	0	0
$793$	−4.49114e6	−0.253614
$794$	0	0
$795$	−1.18454e6	−0.0664708
$796$	0	0
$797$	−3.30133e6	−0.184096	−0.0920478	−	0.995755i	$-0.529341\pi$
$797$	−3.30133e6	−0.184096	−0.0920478	+	0.995755i	$0.529341\pi$
$798$	0	0
$799$	−5.21011e6	−0.288722
$800$	0	0
$801$	−1.67028e7	−0.919832
$802$	0	0
$803$	−5.45269e7	−2.98416
$804$	0	0
$805$	5.61314e6	0.305293
$806$	0	0
$807$	−7.02556e6	−0.379749
$808$	0	0
$809$	1.47021e7	0.789785	0.394892	−	0.918727i	$-0.370782\pi$
$809$	1.47021e7	0.789785	0.394892	+	0.918727i	$0.370782\pi$
$810$	0	0
$811$	9.42343e6	0.503103	0.251551	−	0.967844i	$-0.419059\pi$
$811$	9.42343e6	0.503103	0.251551	+	0.967844i	$0.419059\pi$
$812$	0	0
$813$	1.51838e7	0.805662
$814$	0	0
$815$	1.18262e7	0.623663
$816$	0	0
$817$	−1.44424e6	−0.0756979
$818$	0	0
$819$	−2.25543e7	−1.17495
$820$	0	0
$821$	1.95310e7	1.01127	0.505634	−	0.862748i	$-0.331259\pi$
$821$	1.95310e7	1.01127	0.505634	+	0.862748i	$0.331259\pi$
$822$	0	0
$823$	−9.06750e6	−0.466647	−0.233323	−	0.972399i	$-0.574960\pi$
$823$	−9.06750e6	−0.466647	−0.233323	+	0.972399i	$0.574960\pi$
$824$	0	0
$825$	−2.88380e7	−1.47513
$826$	0	0
$827$	−2.40317e7	−1.22186	−0.610929	−	0.791686i	$-0.709204\pi$
$827$	−2.40317e7	−1.22186	−0.610929	+	0.791686i	$0.709204\pi$
$828$	0	0
$829$	2.67838e7	1.35359	0.676793	−	0.736173i	$-0.263369\pi$
$829$	2.67838e7	1.35359	0.676793	+	0.736173i	$0.263369\pi$
$830$	0	0
$831$	−4.84552e7	−2.43410
$832$	0	0
$833$	9.46009e6	0.472371
$834$	0	0
$835$	−1.44463e6	−0.0717036
$836$	0	0
$837$	1.15405e7	0.569394
$838$	0	0
$839$	−1.77543e7	−0.870762	−0.435381	−	0.900246i	$-0.643386\pi$
$839$	−1.77543e7	−0.870762	−0.435381	+	0.900246i	$0.643386\pi$
$840$	0	0
$841$	−1.73384e7	−0.845315
$842$	0	0
$843$	6.43806e6	0.312023
$844$	0	0
$845$	1.79414e6	0.0864398
$846$	0	0
$847$	−8.61401e7	−4.12569
$848$	0	0
$849$	−4.24083e7	−2.01921
$850$	0	0
$851$	1.36951e6	0.0648250
$852$	0	0
$853$	−3.50585e7	−1.64976	−0.824880	−	0.565308i	$-0.808757\pi$
$853$	−3.50585e7	−1.64976	−0.824880	+	0.565308i	$0.808757\pi$
$854$	0	0
$855$	−4.24645e6	−0.198660
$856$	0	0
$857$	−2.63218e7	−1.22423	−0.612116	−	0.790768i	$-0.709682\pi$
$857$	−2.63218e7	−1.22423	−0.612116	+	0.790768i	$0.709682\pi$
$858$	0	0
$859$	−2.85578e7	−1.32051	−0.660255	−	0.751041i	$-0.729552\pi$
$859$	−2.85578e7	−1.32051	−0.660255	+	0.751041i	$0.729552\pi$
$860$	0	0
$861$	5.74523e7	2.64119
$862$	0	0
$863$	−3.73040e7	−1.70501	−0.852507	−	0.522716i	$-0.824919\pi$
$863$	−3.73040e7	−1.70501	−0.852507	+	0.522716i	$0.824919\pi$
$864$	0	0
$865$	2.42039e7	1.09988
$866$	0	0
$867$	2.76534e7	1.24940
$868$	0	0
$869$	2.86123e7	1.28530
$870$	0	0
$871$	860711.	0.0384425
$872$	0	0
$873$	−5.18820e6	−0.230399
$874$	0	0
$875$	−4.23100e7	−1.86820
$876$	0	0
$877$	−3.24729e6	−0.142568	−0.0712839	−	0.997456i	$-0.522710\pi$
$877$	−3.24729e6	−0.142568	−0.0712839	+	0.997456i	$0.522710\pi$
$878$	0	0
$879$	−3.09193e7	−1.34976
$880$	0	0
$881$	3.29055e7	1.42833	0.714166	−	0.699977i	$-0.246806\pi$
$881$	3.29055e7	1.42833	0.714166	+	0.699977i	$0.246806\pi$
$882$	0	0
$883$	2.67765e6	0.115572	0.0577858	−	0.998329i	$-0.481596\pi$
$883$	2.67765e6	0.115572	0.0577858	+	0.998329i	$0.481596\pi$
$884$	0	0
$885$	−5.58769e6	−0.239814
$886$	0	0
$887$	−1.24008e7	−0.529227	−0.264614	−	0.964355i	$-0.585244\pi$
$887$	−1.24008e7	−0.529227	−0.264614	+	0.964355i	$0.585244\pi$
$888$	0	0
$889$	8.62800e7	3.66147
$890$	0	0
$891$	5.15499e7	2.17537
$892$	0	0
$893$	−1.88877e7	−0.792593
$894$	0	0
$895$	4.01211e6	0.167423
$896$	0	0
$897$	−7.72273e6	−0.320472
$898$	0	0
$899$	−1.26665e7	−0.522708
$900$	0	0
$901$	−378648.	−0.0155390
$902$	0	0
$903$	−9.17316e6	−0.374369
$904$	0	0
$905$	−1.51643e7	−0.615462
$906$	0	0
$907$	−1.39520e7	−0.563141	−0.281571	−	0.959541i	$-0.590855\pi$
$907$	−1.39520e7	−0.563141	−0.281571	+	0.959541i	$0.590855\pi$
$908$	0	0
$909$	1.08084e7	0.433864
$910$	0	0
$911$	2.68463e7	1.07174	0.535868	−	0.844301i	$-0.319984\pi$
$911$	2.68463e7	1.07174	0.535868	+	0.844301i	$0.319984\pi$
$912$	0	0
$913$	−4.81522e7	−1.91179
$914$	0	0
$915$	−5.37071e6	−0.212070
$916$	0	0
$917$	−3.02679e7	−1.18866
$918$	0	0
$919$	4.01968e7	1.57001	0.785006	−	0.619488i	$-0.212660\pi$
$919$	4.01968e7	1.57001	0.785006	+	0.619488i	$0.212660\pi$
$920$	0	0
$921$	−6.36621e7	−2.47304
$922$	0	0
$923$	6.71924e6	0.259607
$924$	0	0
$925$	−4.03375e6	−0.155008
$926$	0	0
$927$	2.90474e7	1.11022
$928$	0	0
$929$	−2.79804e7	−1.06369	−0.531844	−	0.846842i	$-0.678501\pi$
$929$	−2.79804e7	−1.06369	−0.531844	+	0.846842i	$0.678501\pi$
$930$	0	0
$931$	3.42947e7	1.29674
$932$	0	0
$933$	−4.43114e7	−1.66652
$934$	0	0
$935$	5.15434e6	0.192817
$936$	0	0
$937$	3.28750e7	1.22326	0.611628	−	0.791146i	$-0.290515\pi$
$937$	3.28750e7	1.22326	0.611628	+	0.791146i	$0.290515\pi$
$938$	0	0
$939$	−2.11920e7	−0.784345
$940$	0	0
$941$	−3.41713e7	−1.25802	−0.629010	−	0.777397i	$-0.716540\pi$
$941$	−3.41713e7	−1.25802	−0.629010	+	0.777397i	$0.716540\pi$
$942$	0	0
$943$	7.88038e6	0.288581
$944$	0	0
$945$	1.33866e7	0.487631
$946$	0	0
$947$	−2.24743e7	−0.814349	−0.407175	−	0.913350i	$-0.633486\pi$
$947$	−2.24743e7	−0.814349	−0.407175	+	0.913350i	$0.633486\pi$
$948$	0	0
$949$	4.30090e7	1.55022
$950$	0	0
$951$	−3.83593e7	−1.37537
$952$	0	0
$953$	−4.51847e7	−1.61161	−0.805804	−	0.592183i	$-0.798266\pi$
$953$	−4.51847e7	−1.61161	−0.805804	+	0.592183i	$0.798266\pi$
$954$	0	0
$955$	−1.30547e7	−0.463188
$956$	0	0
$957$	−2.56282e7	−0.904564
$958$	0	0
$959$	3.06526e7	1.07627
$960$	0	0
$961$	2.19392e7	0.766325
$962$	0	0
$963$	1.02502e7	0.356177
$964$	0	0
$965$	2.32364e7	0.803251
$966$	0	0
$967$	−2.81645e7	−0.968579	−0.484290	−	0.874908i	$-0.660922\pi$
$967$	−2.81645e7	−0.968579	−0.484290	+	0.874908i	$0.660922\pi$
$968$	0	0
$969$	−3.38855e6	−0.115932
$970$	0	0
$971$	−2.05671e7	−0.700042	−0.350021	−	0.936742i	$-0.613826\pi$
$971$	−2.05671e7	−0.700042	−0.350021	+	0.936742i	$0.613826\pi$
$972$	0	0
$973$	−2.60947e7	−0.883631
$974$	0	0
$975$	2.27464e7	0.766306
$976$	0	0
$977$	7.19366e6	0.241109	0.120555	−	0.992707i	$-0.461533\pi$
$977$	7.19366e6	0.241109	0.120555	+	0.992707i	$0.461533\pi$
$978$	0	0
$979$	7.34969e7	2.45082
$980$	0	0
$981$	2.59787e7	0.861875
$982$	0	0
$983$	3.07154e7	1.01385	0.506924	−	0.861991i	$-0.330783\pi$
$983$	3.07154e7	1.01385	0.506924	+	0.861991i	$0.330783\pi$
$984$	0	0
$985$	2.05082e7	0.673500
$986$	0	0
$987$	−1.19966e8	−3.91981
$988$	0	0
$989$	−1.25823e6	−0.0409042
$990$	0	0
$991$	−1.03996e6	−0.0336380	−0.0168190	−	0.999859i	$-0.505354\pi$
$991$	−1.03996e6	−0.0336380	−0.0168190	+	0.999859i	$0.505354\pi$
$992$	0	0
$993$	2.38088e7	0.766239
$994$	0	0
$995$	2.30658e7	0.738601
$996$	0	0
$997$	−2.43285e7	−0.775135	−0.387568	−	0.921841i	$-0.626685\pi$
$997$	−2.43285e7	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$998$	0	0
$999$	3.26611e6	0.103542

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.6.a.k.1.3		14
4.3	odd	2		344.6.a.c.1.12	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.6.a.c.1.12	✓	14	4.3	odd	2
688.6.a.k.1.3		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-20.1345 q^{3} -33.4767 q^{5} -246.400 q^{7} +162.398 q^{9} +O(q^{10})\)	\(q-20.1345 q^{3} -33.4767 q^{5} -246.400 q^{7} +162.398 q^{9} -714.595 q^{11} +563.648 q^{13} +674.037 q^{15} +215.462 q^{17} +781.092 q^{19} +4961.14 q^{21} +680.490 q^{23} -2004.31 q^{25} +1622.88 q^{27} -1781.22 q^{29} +7111.14 q^{31} +14388.0 q^{33} +8248.67 q^{35} +2012.54 q^{37} -11348.8 q^{39} +11580.5 q^{41} -1849.00 q^{43} -5436.55 q^{45} -24181.1 q^{47} +43906.1 q^{49} -4338.22 q^{51} -1757.38 q^{53} +23922.3 q^{55} -15726.9 q^{57} -8289.88 q^{59} -7967.98 q^{61} -40014.9 q^{63} -18869.1 q^{65} +1527.04 q^{67} -13701.3 q^{69} +11921.0 q^{71} +76304.7 q^{73} +40355.7 q^{75} +176076. q^{77} -40040.0 q^{79} -72138.6 q^{81} +67384.0 q^{83} -7212.96 q^{85} +35864.0 q^{87} -102851. q^{89} -138883. q^{91} -143179. q^{93} -26148.4 q^{95} -31947.4 q^{97} -116049. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 27 q^{3} + 25 q^{5} - 160 q^{7} + 1011 q^{9}+O(q^{10}) \) 14 * q - 27 * q^3 + 25 * q^5 - 160 * q^7 + 1011 * q^9	\( 14 q - 27 q^{3} + 25 q^{5} - 160 q^{7} + 1011 q^{9} - 367 q^{11} + 615 q^{13} - 1780 q^{15} - 1718 q^{17} - 3565 q^{19} + 2258 q^{21} + 2530 q^{23} + 6307 q^{25} - 2118 q^{27} - 5105 q^{29} - 11152 q^{31} + 3518 q^{33} - 13426 q^{35} + 8575 q^{37} - 43148 q^{39} - 9026 q^{41} - 25886 q^{43} + 69212 q^{45} - 14379 q^{47} + 86350 q^{49} - 14235 q^{51} + 75113 q^{53} - 39292 q^{55} + 51044 q^{57} - 80680 q^{59} + 89420 q^{61} - 129922 q^{63} + 70614 q^{65} - 87121 q^{67} + 35232 q^{69} - 129934 q^{71} + 95492 q^{73} - 335468 q^{75} + 70212 q^{77} - 222515 q^{79} + 128574 q^{81} - 180349 q^{83} + 76652 q^{85} - 276293 q^{87} + 87752 q^{89} - 395572 q^{91} + 256669 q^{93} - 453329 q^{95} - 66078 q^{97} - 775955 q^{99}+O(q^{100}) \) 14 * q - 27 * q^3 + 25 * q^5 - 160 * q^7 + 1011 * q^9 - 367 * q^11 + 615 * q^13 - 1780 * q^15 - 1718 * q^17 - 3565 * q^19 + 2258 * q^21 + 2530 * q^23 + 6307 * q^25 - 2118 * q^27 - 5105 * q^29 - 11152 * q^31 + 3518 * q^33 - 13426 * q^35 + 8575 * q^37 - 43148 * q^39 - 9026 * q^41 - 25886 * q^43 + 69212 * q^45 - 14379 * q^47 + 86350 * q^49 - 14235 * q^51 + 75113 * q^53 - 39292 * q^55 + 51044 * q^57 - 80680 * q^59 + 89420 * q^61 - 129922 * q^63 + 70614 * q^65 - 87121 * q^67 + 35232 * q^69 - 129934 * q^71 + 95492 * q^73 - 335468 * q^75 + 70212 * q^77 - 222515 * q^79 + 128574 * q^81 - 180349 * q^83 + 76652 * q^85 - 276293 * q^87 + 87752 * q^89 - 395572 * q^91 + 256669 * q^93 - 453329 * q^95 - 66078 * q^97 - 775955 * q^99

Introduction

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