LMFDB - Embedded newform 768.5.b.i.127.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,5,Mod(127,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("768.127");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	768.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$79.3881316484$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 104 x^{14} - 588 x^{13} + 3846 x^{12} - 15796 x^{11} + 66992 x^{10} - 203224 x^{9} + \cdots + 2196513 $ x^16 - 8x^15 + 104x^14 - 588x^13 + 3846x^12 - 15796x^11 + 66992x^10 - 203224x^9 + 604877x^8 - 1351612x^7 + 2878620x^6 - 4608408x^5 + 6889612x^4 - 7344036x^3 + 7202376x^2 - 4122756*x + 2196513
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{70}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.14
Root			$0.500000 - 1.23482i$ of defining polynomial
Character	$\chi$	$=$	768.127
Dual form			768.5.b.i.127.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{3} +8.12430i q^{5} -42.5107i q^{7} +27.0000 q^{9} +O(q^{10})$	$q+5.19615 q^{3} +8.12430i q^{5} -42.5107i q^{7} +27.0000 q^{9} -166.684 q^{11} +38.3529i q^{13} +42.2151i q^{15} +44.5768 q^{17} -191.758 q^{19} -220.892i q^{21} +122.742i q^{23} +558.996 q^{25} +140.296 q^{27} -132.291i q^{29} +1223.70i q^{31} -866.118 q^{33} +345.370 q^{35} +1290.16i q^{37} +199.288i q^{39} +1494.32 q^{41} +1583.01 q^{43} +219.356i q^{45} +2224.71i q^{47} +593.843 q^{49} +231.628 q^{51} -4573.39i q^{53} -1354.20i q^{55} -996.403 q^{57} +4547.36 q^{59} +4998.83i q^{61} -1147.79i q^{63} -311.591 q^{65} -1113.11 q^{67} +637.788i q^{69} -4223.07i q^{71} +6988.79 q^{73} +2904.63 q^{75} +7085.87i q^{77} +4123.08i q^{79} +729.000 q^{81} -6863.28 q^{83} +362.156i q^{85} -687.404i q^{87} +8149.40 q^{89} +1630.41 q^{91} +6358.51i q^{93} -1557.90i q^{95} -11535.2 q^{97} -4500.48 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 432 q^{9}+O(q^{10}) $ 16 * q + 432 * q^9	$ 16 q + 432 q^{9} + 576 q^{11} - 480 q^{17} + 192 q^{19} - 2672 q^{25} + 9024 q^{35} - 1440 q^{41} + 12224 q^{43} - 2480 q^{49} + 6336 q^{51} + 7488 q^{57} + 15360 q^{59} - 1344 q^{65} + 12288 q^{67} + 8480 q^{73} + 21888 q^{75} + 11664 q^{81} + 13248 q^{83} - 18720 q^{89} - 30272 q^{91} + 13088 q^{97} + 15552 q^{99}+O(q^{100}) $ 16 * q + 432 * q^9 + 576 * q^11 - 480 * q^17 + 192 * q^19 - 2672 * q^25 + 9024 * q^35 - 1440 * q^41 + 12224 * q^43 - 2480 * q^49 + 6336 * q^51 + 7488 * q^57 + 15360 * q^59 - 1344 * q^65 + 12288 * q^67 + 8480 * q^73 + 21888 * q^75 + 11664 * q^81 + 13248 * q^83 - 18720 * q^89 - 30272 * q^91 + 13088 * q^97 + 15552 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	0	0
$5$	8.12430i	0.324972i	0.986711	+	0.162486i	$0.0519513\pi$
$5$	8.12430i	0.324972i	−0.986711	+	0.162486i	$0.948049\pi$
$6$	0	0
$7$	− 42.5107i	− 0.867565i	−0.901018	−	0.433782i	$-0.857179\pi$
$7$	− 42.5107i	− 0.867565i	0.901018	−	0.433782i	$-0.142821\pi$
$8$	0	0
$9$	27.0000	0.333333
$10$	0	0
$11$	−166.684	−1.37756	−0.688779	−	0.724971i	$-0.741853\pi$
$11$	−166.684	−1.37756	−0.688779	+	0.724971i	$0.741853\pi$
$12$	0	0
$13$	38.3529i	0.226941i	0.993541	+	0.113470i	$0.0361967\pi$
$13$	38.3529i	0.226941i	−0.993541	+	0.113470i	$0.963803\pi$
$14$	0	0
$15$	42.2151i	0.187623i
$16$	0	0
$17$	44.5768	0.154245	0.0771225	−	0.997022i	$-0.475427\pi$
$17$	44.5768	0.154245	0.0771225	+	0.997022i	$0.475427\pi$
$18$	0	0
$19$	−191.758	−0.531185	−0.265593	−	0.964085i	$-0.585568\pi$
$19$	−191.758	−0.531185	−0.265593	+	0.964085i	$0.585568\pi$
$20$	0	0
$21$	− 220.892i	− 0.500889i
$22$	0	0
$23$	122.742i	0.232027i	0.993248	+	0.116014i	$0.0370116\pi$
$23$	122.742i	0.232027i	−0.993248	+	0.116014i	$0.962988\pi$
$24$	0	0
$25$	558.996	0.894393
$26$	0	0
$27$	140.296	0.192450
$28$	0	0
$29$	− 132.291i	− 0.157302i	−0.996902	−	0.0786510i	$-0.974939\pi$
$29$	− 132.291i	− 0.157302i	0.996902	−	0.0786510i	$-0.0250613\pi$
$30$	0	0
$31$	1223.70i	1.27336i	0.771130	+	0.636678i	$0.219692\pi$
$31$	1223.70i	1.27336i	−0.771130	+	0.636678i	$0.780308\pi$
$32$	0	0
$33$	−866.118	−0.795333
$34$	0	0
$35$	345.370	0.281934
$36$	0	0
$37$	1290.16i	0.942407i	0.882024	+	0.471204i	$0.156180\pi$
$37$	1290.16i	0.942407i	−0.882024	+	0.471204i	$0.843820\pi$
$38$	0	0
$39$	199.288i	0.131024i
$40$	0	0
$41$	1494.32	0.888947	0.444473	−	0.895792i	$-0.353391\pi$
$41$	1494.32	0.888947	0.444473	+	0.895792i	$0.353391\pi$
$42$	0	0
$43$	1583.01	0.856145	0.428073	−	0.903744i	$-0.359193\pi$
$43$	1583.01	0.856145	0.428073	+	0.903744i	$0.359193\pi$
$44$	0	0
$45$	219.356i	0.108324i
$46$	0	0
$47$	2224.71i	1.00711i	0.863962	+	0.503556i	$0.167975\pi$
$47$	2224.71i	1.00711i	−0.863962	+	0.503556i	$0.832025\pi$
$48$	0	0
$49$	593.843	0.247332
$50$	0	0
$51$	231.628	0.0890534
$52$	0	0
$53$	− 4573.39i	− 1.62812i	−0.580780	−	0.814061i	$-0.697252\pi$
$53$	− 4573.39i	− 1.62812i	0.580780	−	0.814061i	$-0.302748\pi$
$54$	0	0
$55$	− 1354.20i	− 0.447668i
$56$	0	0
$57$	−996.403	−0.306680
$58$	0	0
$59$	4547.36	1.30634	0.653168	−	0.757213i	$-0.273439\pi$
$59$	4547.36	1.30634	0.653168	+	0.757213i	$0.273439\pi$
$60$	0	0
$61$	4998.83i	1.34341i	0.740818	+	0.671706i	$0.234438\pi$
$61$	4998.83i	1.34341i	−0.740818	+	0.671706i	$0.765562\pi$
$62$	0	0
$63$	− 1147.79i	− 0.289188i
$64$	0	0
$65$	−311.591	−0.0737493
$66$	0	0
$67$	−1113.11	−0.247964	−0.123982	−	0.992284i	$-0.539567\pi$
$67$	−1113.11	−0.247964	−0.123982	+	0.992284i	$0.539567\pi$
$68$	0	0
$69$	637.788i	0.133961i
$70$	0	0
$71$	− 4223.07i	− 0.837744i	−0.908045	−	0.418872i	$-0.862426\pi$
$71$	− 4223.07i	− 0.837744i	0.908045	−	0.418872i	$-0.137574\pi$
$72$	0	0
$73$	6988.79	1.31146	0.655732	−	0.754994i	$-0.272360\pi$
$73$	6988.79	1.31146	0.655732	+	0.754994i	$0.272360\pi$
$74$	0	0
$75$	2904.63	0.516378
$76$	0	0
$77$	7085.87i	1.19512i
$78$	0	0
$79$	4123.08i	0.660644i	0.943868	+	0.330322i	$0.107157\pi$
$79$	4123.08i	0.660644i	−0.943868	+	0.330322i	$0.892843\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	−6863.28	−0.996266	−0.498133	−	0.867101i	$-0.665981\pi$
$83$	−6863.28	−0.996266	−0.498133	+	0.867101i	$0.665981\pi$
$84$	0	0
$85$	362.156i	0.0501253i
$86$	0	0
$87$	− 687.404i	− 0.0908183i
$88$	0	0
$89$	8149.40	1.02883	0.514417	−	0.857540i	$-0.328008\pi$
$89$	8149.40	1.02883	0.514417	+	0.857540i	$0.328008\pi$
$90$	0	0
$91$	1630.41	0.196886
$92$	0	0
$93$	6358.51i	0.735173i
$94$	0	0
$95$	− 1557.90i	− 0.172620i
$96$	0	0
$97$	−11535.2	−1.22597	−0.612985	−	0.790094i	$-0.710032\pi$
$97$	−11535.2	−1.22597	−0.612985	+	0.790094i	$0.710032\pi$
$98$	0	0
$99$	−4500.48	−0.459186
$100$	0	0
$101$	4320.59i	0.423545i	0.977319	+	0.211773i	$0.0679237\pi$
$101$	4320.59i	0.423545i	−0.977319	+	0.211773i	$0.932076\pi$
$102$	0	0
$103$	519.264i	0.0489456i	0.999700	+	0.0244728i	$0.00779072\pi$
$103$	519.264i	0.0489456i	−0.999700	+	0.0244728i	$0.992209\pi$
$104$	0	0
$105$	1794.59	0.162775
$106$	0	0
$107$	4004.52	0.349770	0.174885	−	0.984589i	$-0.444045\pi$
$107$	4004.52	0.349770	0.174885	+	0.984589i	$0.444045\pi$
$108$	0	0
$109$	10673.3i	0.898351i	0.893444	+	0.449175i	$0.148282\pi$
$109$	10673.3i	0.898351i	−0.893444	+	0.449175i	$0.851718\pi$
$110$	0	0
$111$	6703.85i	0.544099i
$112$	0	0
$113$	16997.4	1.33114	0.665572	−	0.746334i	$-0.268188\pi$
$113$	16997.4	1.33114	0.665572	+	0.746334i	$0.268188\pi$
$114$	0	0
$115$	−997.196	−0.0754024
$116$	0	0
$117$	1035.53i	0.0756468i
$118$	0	0
$119$	− 1894.99i	− 0.133818i
$120$	0	0
$121$	13142.7	0.897666
$122$	0	0
$123$	7764.71	0.513234
$124$	0	0
$125$	9619.14i	0.615625i
$126$	0	0
$127$	24891.8i	1.54330i	0.636050	+	0.771648i	$0.280567\pi$
$127$	24891.8i	1.54330i	−0.636050	+	0.771648i	$0.719433\pi$
$128$	0	0
$129$	8225.58	0.494296
$130$	0	0
$131$	−6152.72	−0.358529	−0.179265	−	0.983801i	$-0.557372\pi$
$131$	−6152.72	−0.358529	−0.179265	+	0.983801i	$0.557372\pi$
$132$	0	0
$133$	8151.76i	0.460838i
$134$	0	0
$135$	1139.81i	0.0625409i
$136$	0	0
$137$	17882.0	0.952739	0.476370	−	0.879245i	$-0.341952\pi$
$137$	17882.0	0.952739	0.476370	+	0.879245i	$0.341952\pi$
$138$	0	0
$139$	−24088.5	−1.24675	−0.623375	−	0.781923i	$-0.714239\pi$
$139$	−24088.5	−1.24675	−0.623375	+	0.781923i	$0.714239\pi$
$140$	0	0
$141$	11559.9i	0.581457i
$142$	0	0
$143$	− 6392.84i	− 0.312624i
$144$	0	0
$145$	1074.77	0.0511188
$146$	0	0
$147$	3085.70	0.142797
$148$	0	0
$149$	20425.4i	0.920020i	0.887914	+	0.460010i	$0.152154\pi$
$149$	20425.4i	0.920020i	−0.887914	+	0.460010i	$0.847846\pi$
$150$	0	0
$151$	− 35543.8i	− 1.55887i	−0.626483	−	0.779435i	$-0.715506\pi$
$151$	− 35543.8i	− 1.55887i	0.626483	−	0.779435i	$-0.284494\pi$
$152$	0	0
$153$	1203.57	0.0514150
$154$	0	0
$155$	−9941.67	−0.413805
$156$	0	0
$157$	− 33393.1i	− 1.35474i	−0.735641	−	0.677372i	$-0.763119\pi$
$157$	− 33393.1i	− 1.35474i	0.735641	−	0.677372i	$-0.236881\pi$
$158$	0	0
$159$	− 23764.0i	− 0.939996i
$160$	0	0
$161$	5217.86	0.201299
$162$	0	0
$163$	13261.3	0.499128	0.249564	−	0.968358i	$-0.419713\pi$
$163$	13261.3	0.499128	0.249564	+	0.968358i	$0.419713\pi$
$164$	0	0
$165$	− 7036.61i	− 0.258461i
$166$	0	0
$167$	− 1403.40i	− 0.0503210i	−0.999683	−	0.0251605i	$-0.991990\pi$
$167$	− 1403.40i	− 0.0503210i	0.999683	−	0.0251605i	$-0.00800969\pi$
$168$	0	0
$169$	27090.1	0.948498
$170$	0	0
$171$	−5177.46	−0.177062
$172$	0	0
$173$	46915.2i	1.56755i	0.621046	+	0.783774i	$0.286708\pi$
$173$	46915.2i	1.56755i	−0.621046	+	0.783774i	$0.713292\pi$
$174$	0	0
$175$	− 23763.3i	− 0.775944i
$176$	0	0
$177$	23628.8	0.754214
$178$	0	0
$179$	−29301.6	−0.914504	−0.457252	−	0.889337i	$-0.651166\pi$
$179$	−29301.6	−0.914504	−0.457252	+	0.889337i	$0.651166\pi$
$180$	0	0
$181$	59656.8i	1.82097i	0.413542	+	0.910485i	$0.364291\pi$
$181$	59656.8i	1.82097i	−0.413542	+	0.910485i	$0.635709\pi$
$182$	0	0
$183$	25974.7i	0.775619i
$184$	0	0
$185$	−10481.6	−0.306256
$186$	0	0
$187$	−7430.26	−0.212481
$188$	0	0
$189$	− 5964.08i	− 0.166963i
$190$	0	0
$191$	− 65781.9i	− 1.80318i	−0.432590	−	0.901591i	$-0.642400\pi$
$191$	− 65781.9i	− 1.80318i	0.432590	−	0.901591i	$-0.357600\pi$
$192$	0	0
$193$	67239.7	1.80514	0.902571	−	0.430542i	$-0.141678\pi$
$193$	67239.7	1.80514	0.902571	+	0.430542i	$0.141678\pi$
$194$	0	0
$195$	−1619.07	−0.0425792
$196$	0	0
$197$	12921.7i	0.332955i	0.986045	+	0.166478i	$0.0532394\pi$
$197$	12921.7i	0.332955i	−0.986045	+	0.166478i	$0.946761\pi$
$198$	0	0
$199$	26240.0i	0.662609i	0.943524	+	0.331305i	$0.107489\pi$
$199$	26240.0i	0.662609i	−0.943524	+	0.331305i	$0.892511\pi$
$200$	0	0
$201$	−5783.90	−0.143162
$202$	0	0
$203$	−5623.78	−0.136470
$204$	0	0
$205$	12140.3i	0.288883i
$206$	0	0
$207$	3314.04i	0.0773424i
$208$	0	0
$209$	31963.1	0.731739
$210$	0	0
$211$	77611.4	1.74325	0.871626	−	0.490171i	$-0.163066\pi$
$211$	77611.4	1.74325	0.871626	+	0.490171i	$0.163066\pi$
$212$	0	0
$213$	− 21943.7i	− 0.483672i
$214$	0	0
$215$	12860.9i	0.278223i
$216$	0	0
$217$	52020.1	1.10472
$218$	0	0
$219$	36314.8	0.757174
$220$	0	0
$221$	1709.65i	0.0350044i
$222$	0	0
$223$	54436.1i	1.09465i	0.836919	+	0.547327i	$0.184355\pi$
$223$	54436.1i	1.09465i	−0.836919	+	0.547327i	$0.815645\pi$
$224$	0	0
$225$	15092.9	0.298131
$226$	0	0
$227$	29438.7	0.571304	0.285652	−	0.958333i	$-0.407790\pi$
$227$	29438.7	0.571304	0.285652	+	0.958333i	$0.407790\pi$
$228$	0	0
$229$	100696.i	1.92018i	0.279686	+	0.960092i	$0.409770\pi$
$229$	100696.i	1.92018i	−0.279686	+	0.960092i	$0.590230\pi$
$230$	0	0
$231$	36819.3i	0.690003i
$232$	0	0
$233$	−87042.2	−1.60331	−0.801656	−	0.597785i	$-0.796047\pi$
$233$	−87042.2	−1.60331	−0.801656	+	0.597785i	$0.796047\pi$
$234$	0	0
$235$	−18074.2	−0.327283
$236$	0	0
$237$	21424.2i	0.381423i
$238$	0	0
$239$	− 14574.0i	− 0.255142i	−0.991829	−	0.127571i	$-0.959282\pi$
$239$	− 14574.0i	− 0.255142i	0.991829	−	0.127571i	$-0.0407181\pi$
$240$	0	0
$241$	−20401.5	−0.351259	−0.175630	−	0.984456i	$-0.556196\pi$
$241$	−20401.5	−0.351259	−0.175630	+	0.984456i	$0.556196\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	0	0
$245$	4824.56i	0.0803759i
$246$	0	0
$247$	− 7354.48i	− 0.120547i
$248$	0	0
$249$	−35662.6	−0.575194
$250$	0	0
$251$	−68166.3	−1.08199	−0.540994	−	0.841027i	$-0.681952\pi$
$251$	−68166.3	−1.08199	−0.540994	+	0.841027i	$0.681952\pi$
$252$	0	0
$253$	− 20459.3i	− 0.319631i
$254$	0	0
$255$	1881.82i	0.0289399i
$256$	0	0
$257$	109454.	1.65716	0.828580	−	0.559871i	$-0.189149\pi$
$257$	109454.	1.65716	0.828580	+	0.559871i	$0.189149\pi$
$258$	0	0
$259$	54845.4	0.817599
$260$	0	0
$261$	− 3571.86i	− 0.0524340i
$262$	0	0
$263$	85208.9i	1.23189i	0.787787	+	0.615947i	$0.211227\pi$
$263$	85208.9i	1.23189i	−0.787787	+	0.615947i	$0.788773\pi$
$264$	0	0
$265$	37155.6	0.529094
$266$	0	0
$267$	42345.5	0.593998
$268$	0	0
$269$	− 21274.6i	− 0.294007i	−0.989136	−	0.147004i	$-0.953037\pi$
$269$	− 21274.6i	− 0.294007i	0.989136	−	0.147004i	$-0.0469629\pi$
$270$	0	0
$271$	30213.8i	0.411402i	0.978615	+	0.205701i	$0.0659474\pi$
$271$	30213.8i	0.411402i	−0.978615	+	0.205701i	$0.934053\pi$
$272$	0	0
$273$	8471.86	0.113672
$274$	0	0
$275$	−93175.9	−1.23208
$276$	0	0
$277$	− 996.316i	− 0.0129849i	−0.999979	−	0.00649243i	$-0.997933\pi$
$277$	− 996.316i	− 0.0129849i	0.999979	−	0.00649243i	$-0.00206662\pi$
$278$	0	0
$279$	33039.8i	0.424452i
$280$	0	0
$281$	−9173.97	−0.116184	−0.0580918	−	0.998311i	$-0.518502\pi$
$281$	−9173.97	−0.116184	−0.0580918	+	0.998311i	$0.518502\pi$
$282$	0	0
$283$	−51839.5	−0.647273	−0.323637	−	0.946181i	$-0.604906\pi$
$283$	−51839.5	−0.647273	−0.323637	+	0.946181i	$0.604906\pi$
$284$	0	0
$285$	− 8095.08i	− 0.0996625i
$286$	0	0
$287$	− 63524.5i	− 0.771219i
$288$	0	0
$289$	−81533.9	−0.976208
$290$	0	0
$291$	−59938.4	−0.707815
$292$	0	0
$293$	68807.2i	0.801491i	0.916189	+	0.400746i	$0.131249\pi$
$293$	68807.2i	0.801491i	−0.916189	+	0.400746i	$0.868751\pi$
$294$	0	0
$295$	36944.1i	0.424523i
$296$	0	0
$297$	−23385.2	−0.265111
$298$	0	0
$299$	−4707.53	−0.0526564
$300$	0	0
$301$	− 67294.9i	− 0.742761i
$302$	0	0
$303$	22450.4i	0.244534i
$304$	0	0
$305$	−40612.0	−0.436571
$306$	0	0
$307$	123729.	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$307$	123729.	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$308$	0	0
$309$	2698.18i	0.0282588i
$310$	0	0
$311$	− 157600.i	− 1.62943i	−0.579864	−	0.814714i	$-0.696894\pi$
$311$	− 157600.i	− 1.62943i	0.579864	−	0.814714i	$-0.303106\pi$
$312$	0	0
$313$	22157.5	0.226169	0.113084	−	0.993585i	$-0.463927\pi$
$313$	22157.5	0.226169	0.113084	+	0.993585i	$0.463927\pi$
$314$	0	0
$315$	9324.98	0.0939781
$316$	0	0
$317$	25511.8i	0.253876i	0.991911	+	0.126938i	$0.0405149\pi$
$317$	25511.8i	0.253876i	−0.991911	+	0.126938i	$0.959485\pi$
$318$	0	0
$319$	22050.9i	0.216693i
$320$	0	0
$321$	20808.1	0.201940
$322$	0	0
$323$	−8547.96	−0.0819327
$324$	0	0
$325$	21439.1i	0.202974i
$326$	0	0
$327$	55460.1i	0.518663i
$328$	0	0
$329$	94574.0	0.873735
$330$	0	0
$331$	−10247.2	−0.0935292	−0.0467646	−	0.998906i	$-0.514891\pi$
$331$	−10247.2	−0.0935292	−0.0467646	+	0.998906i	$0.514891\pi$
$332$	0	0
$333$	34834.2i	0.314136i
$334$	0	0
$335$	− 9043.26i	− 0.0805815i
$336$	0	0
$337$	−142661.	−1.25616	−0.628082	−	0.778147i	$-0.716160\pi$
$337$	−142661.	−1.25616	−0.628082	+	0.778147i	$0.716160\pi$
$338$	0	0
$339$	88320.9	0.768536
$340$	0	0
$341$	− 203971.i	− 1.75412i
$342$	0	0
$343$	− 127313.i	− 1.08214i
$344$	0	0
$345$	−5181.58	−0.0435336
$346$	0	0
$347$	51567.2	0.428267	0.214134	−	0.976804i	$-0.431307\pi$
$347$	51567.2	0.428267	0.214134	+	0.976804i	$0.431307\pi$
$348$	0	0
$349$	− 117685.i	− 0.966206i	−0.875563	−	0.483103i	$-0.839510\pi$
$349$	− 117685.i	− 0.966206i	0.875563	−	0.483103i	$-0.160490\pi$
$350$	0	0
$351$	5380.77i	0.0436747i
$352$	0	0
$353$	−72694.9	−0.583384	−0.291692	−	0.956512i	$-0.594218\pi$
$353$	−72694.9	−0.583384	−0.291692	+	0.956512i	$0.594218\pi$
$354$	0	0
$355$	34309.5	0.272243
$356$	0	0
$357$	− 9846.66i	− 0.0772596i
$358$	0	0
$359$	− 35180.0i	− 0.272965i	−0.990642	−	0.136482i	$-0.956420\pi$
$359$	− 35180.0i	− 0.272965i	0.990642	−	0.136482i	$-0.0435797\pi$
$360$	0	0
$361$	−93549.9	−0.717842
$362$	0	0
$363$	68291.6	0.518267
$364$	0	0
$365$	56779.1i	0.426189i
$366$	0	0
$367$	229357.i	1.70286i	0.524467	+	0.851431i	$0.324264\pi$
$367$	229357.i	1.70286i	−0.524467	+	0.851431i	$0.675736\pi$
$368$	0	0
$369$	40346.6	0.296316
$370$	0	0
$371$	−194418.	−1.41250
$372$	0	0
$373$	− 97648.9i	− 0.701859i	−0.936402	−	0.350929i	$-0.885866\pi$
$373$	− 97648.9i	− 0.701859i	0.936402	−	0.350929i	$-0.114134\pi$
$374$	0	0
$375$	49982.5i	0.355431i
$376$	0	0
$377$	5073.75	0.0356982
$378$	0	0
$379$	140431.	0.977654	0.488827	−	0.872381i	$-0.337425\pi$
$379$	140431.	0.977654	0.488827	+	0.872381i	$0.337425\pi$
$380$	0	0
$381$	129342.i	0.891023i
$382$	0	0
$383$	− 121149.i	− 0.825893i	−0.910755	−	0.412946i	$-0.864500\pi$
$383$	− 121149.i	− 0.825893i	0.910755	−	0.412946i	$-0.135500\pi$
$384$	0	0
$385$	−57567.7	−0.388381
$386$	0	0
$387$	42741.4	0.285382
$388$	0	0
$389$	− 204953.i	− 1.35443i	−0.735787	−	0.677213i	$-0.763187\pi$
$389$	− 204953.i	− 1.35443i	0.735787	−	0.677213i	$-0.236813\pi$
$390$	0	0
$391$	5471.46i	0.0357890i
$392$	0	0
$393$	−31970.5	−0.206997
$394$	0	0
$395$	−33497.2	−0.214691
$396$	0	0
$397$	− 1277.26i	− 0.00810395i	−0.999992	−	0.00405197i	$-0.998710\pi$
$397$	− 1277.26i	− 0.00810395i	0.999992	−	0.00405197i	$-0.00128979\pi$
$398$	0	0
$399$	42357.8i	0.266065i
$400$	0	0
$401$	89400.0	0.555967	0.277983	−	0.960586i	$-0.410334\pi$
$401$	89400.0	0.555967	0.277983	+	0.960586i	$0.410334\pi$
$402$	0	0
$403$	−46932.3	−0.288976
$404$	0	0
$405$	5922.62i	0.0361080i
$406$	0	0
$407$	− 215049.i	− 1.29822i
$408$	0	0
$409$	−21624.0	−0.129267	−0.0646337	−	0.997909i	$-0.520588\pi$
$409$	−21624.0	−0.129267	−0.0646337	+	0.997909i	$0.520588\pi$
$410$	0	0
$411$	92917.4	0.550064
$412$	0	0
$413$	− 193311.i	− 1.13333i
$414$	0	0
$415$	− 55759.3i	− 0.323759i
$416$	0	0
$417$	−125167.	−0.719812
$418$	0	0
$419$	−262506.	−1.49524	−0.747620	−	0.664127i	$-0.768803\pi$
$419$	−262506.	−1.49524	−0.747620	+	0.664127i	$0.768803\pi$
$420$	0	0
$421$	− 209255.i	− 1.18063i	−0.807174	−	0.590313i	$-0.799004\pi$
$421$	− 209255.i	− 1.18063i	0.807174	−	0.590313i	$-0.200996\pi$
$422$	0	0
$423$	60067.2i	0.335704i
$424$	0	0
$425$	24918.2	0.137956
$426$	0	0
$427$	212504.	1.16550
$428$	0	0
$429$	− 33218.2i	− 0.180493i
$430$	0	0
$431$	− 174375.i	− 0.938706i	−0.883011	−	0.469353i	$-0.844487\pi$
$431$	− 174375.i	− 0.938706i	0.883011	−	0.469353i	$-0.155513\pi$
$432$	0	0
$433$	−183750.	−0.980059	−0.490029	−	0.871706i	$-0.663014\pi$
$433$	−183750.	−0.980059	−0.490029	+	0.871706i	$0.663014\pi$
$434$	0	0
$435$	5584.68	0.0295134
$436$	0	0
$437$	− 23536.8i	− 0.123249i
$438$	0	0
$439$	− 105117.i	− 0.545436i	−0.962094	−	0.272718i	$-0.912077\pi$
$439$	− 105117.i	− 0.545436i	0.962094	−	0.272718i	$-0.0879226\pi$
$440$	0	0
$441$	16033.8	0.0824439
$442$	0	0
$443$	−304033.	−1.54922	−0.774611	−	0.632438i	$-0.782054\pi$
$443$	−304033.	−1.54922	−0.774611	+	0.632438i	$0.782054\pi$
$444$	0	0
$445$	66208.2i	0.334342i
$446$	0	0
$447$	106133.i	0.531174i
$448$	0	0
$449$	−365988.	−1.81541	−0.907703	−	0.419614i	$-0.862165\pi$
$449$	−365988.	−1.81541	−0.907703	+	0.419614i	$0.862165\pi$
$450$	0	0
$451$	−249080.	−1.22458
$452$	0	0
$453$	− 184691.i	− 0.900014i
$454$	0	0
$455$	13245.9i	0.0639823i
$456$	0	0
$457$	−53631.6	−0.256796	−0.128398	−	0.991723i	$-0.540984\pi$
$457$	−53631.6	−0.256796	−0.128398	+	0.991723i	$0.540984\pi$
$458$	0	0
$459$	6253.95	0.0296845
$460$	0	0
$461$	− 56830.2i	− 0.267410i	−0.991021	−	0.133705i	$-0.957313\pi$
$461$	− 56830.2i	− 0.267410i	0.991021	−	0.133705i	$-0.0426874\pi$
$462$	0	0
$463$	348232.i	1.62445i	0.583345	+	0.812224i	$0.301744\pi$
$463$	348232.i	1.62445i	−0.583345	+	0.812224i	$0.698256\pi$
$464$	0	0
$465$	−51658.5	−0.238911
$466$	0	0
$467$	−156294.	−0.716650	−0.358325	−	0.933597i	$-0.616652\pi$
$467$	−156294.	−0.716650	−0.358325	+	0.933597i	$0.616652\pi$
$468$	0	0
$469$	47319.1i	0.215125i
$470$	0	0
$471$	− 173515.i	− 0.782161i
$472$	0	0
$473$	−263864.	−1.17939
$474$	0	0
$475$	−107192.	−0.475089
$476$	0	0
$477$	− 123482.i	− 0.542707i
$478$	0	0
$479$	− 96210.9i	− 0.419328i	−0.977774	−	0.209664i	$-0.932763\pi$
$479$	− 96210.9i	− 0.419328i	0.977774	−	0.209664i	$-0.0672369\pi$
$480$	0	0
$481$	−49481.3	−0.213870
$482$	0	0
$483$	27112.8	0.116220
$484$	0	0
$485$	− 93715.1i	− 0.398406i
$486$	0	0
$487$	306387.i	1.29185i	0.763401	+	0.645925i	$0.223528\pi$
$487$	306387.i	1.29185i	−0.763401	+	0.645925i	$0.776472\pi$
$488$	0	0
$489$	68907.8	0.288171
$490$	0	0
$491$	−38628.3	−0.160229	−0.0801147	−	0.996786i	$-0.525529\pi$
$491$	−38628.3	−0.160229	−0.0801147	+	0.996786i	$0.525529\pi$
$492$	0	0
$493$	− 5897.11i	− 0.0242630i
$494$	0	0
$495$	− 36563.3i	− 0.149223i
$496$	0	0
$497$	−179525.	−0.726797
$498$	0	0
$499$	−16660.1	−0.0669078	−0.0334539	−	0.999440i	$-0.510651\pi$
$499$	−16660.1	−0.0669078	−0.0334539	+	0.999440i	$0.510651\pi$
$500$	0	0
$501$	− 7292.30i	− 0.0290529i
$502$	0	0
$503$	− 470372.i	− 1.85911i	−0.368683	−	0.929555i	$-0.620191\pi$
$503$	− 470372.i	− 1.85911i	0.368683	−	0.929555i	$-0.379809\pi$
$504$	0	0
$505$	−35101.8	−0.137640
$506$	0	0
$507$	140764.	0.547616
$508$	0	0
$509$	190671.i	0.735953i	0.929835	+	0.367977i	$0.119949\pi$
$509$	190671.i	0.735953i	−0.929835	+	0.367977i	$0.880051\pi$
$510$	0	0
$511$	− 297098.i	− 1.13778i
$512$	0	0
$513$	−26902.9	−0.102227
$514$	0	0
$515$	−4218.66	−0.0159060
$516$	0	0
$517$	− 370825.i	− 1.38736i
$518$	0	0
$519$	243778.i	0.905025i
$520$	0	0
$521$	−200021.	−0.736886	−0.368443	−	0.929650i	$-0.620109\pi$
$521$	−200021.	−0.736886	−0.368443	+	0.929650i	$0.620109\pi$
$522$	0	0
$523$	−123166.	−0.450284	−0.225142	−	0.974326i	$-0.572285\pi$
$523$	−123166.	−0.450284	−0.225142	+	0.974326i	$0.572285\pi$
$524$	0	0
$525$	− 123478.i	− 0.447991i
$526$	0	0
$527$	54548.5i	0.196409i
$528$	0	0
$529$	264775.	0.946163
$530$	0	0
$531$	122779.	0.435446
$532$	0	0
$533$	57311.6i	0.201738i
$534$	0	0
$535$	32533.9i	0.113666i
$536$	0	0
$537$	−152256.	−0.527989
$538$	0	0
$539$	−98984.5	−0.340714
$540$	0	0
$541$	125483.i	0.428735i	0.976753	+	0.214368i	$0.0687690\pi$
$541$	125483.i	0.428735i	−0.976753	+	0.214368i	$0.931231\pi$
$542$	0	0
$543$	309986.i	1.05134i
$544$	0	0
$545$	−86713.2	−0.291939
$546$	0	0
$547$	478301.	1.59855	0.799275	−	0.600965i	$-0.205217\pi$
$547$	478301.	1.59855	0.799275	+	0.600965i	$0.205217\pi$
$548$	0	0
$549$	134969.i	0.447804i
$550$	0	0
$551$	25367.8i	0.0835565i
$552$	0	0
$553$	175275.	0.573152
$554$	0	0
$555$	−54464.1	−0.176817
$556$	0	0
$557$	− 413597.i	− 1.33311i	−0.745454	−	0.666557i	$-0.767767\pi$
$557$	− 413597.i	− 1.33311i	0.745454	−	0.666557i	$-0.232233\pi$
$558$	0	0
$559$	60713.2i	0.194294i
$560$	0	0
$561$	−38608.8	−0.122676
$562$	0	0
$563$	184376.	0.581685	0.290842	−	0.956771i	$-0.406064\pi$
$563$	184376.	0.581685	0.290842	+	0.956771i	$0.406064\pi$
$564$	0	0
$565$	138092.i	0.432585i
$566$	0	0
$567$	− 30990.3i	− 0.0963961i
$568$	0	0
$569$	79560.9	0.245740	0.122870	−	0.992423i	$-0.460790\pi$
$569$	79560.9	0.245740	0.122870	+	0.992423i	$0.460790\pi$
$570$	0	0
$571$	450181.	1.38075	0.690375	−	0.723452i	$-0.257446\pi$
$571$	450181.	1.38075	0.690375	+	0.723452i	$0.257446\pi$
$572$	0	0
$573$	− 341813.i	− 1.04107i
$574$	0	0
$575$	68612.5i	0.207524i
$576$	0	0
$577$	−3642.95	−0.0109421	−0.00547106	−	0.999985i	$-0.501742\pi$
$577$	−3642.95	−0.0109421	−0.00547106	+	0.999985i	$0.501742\pi$
$578$	0	0
$579$	349388.	1.04220
$580$	0	0
$581$	291762.i	0.864325i
$582$	0	0
$583$	762314.i	2.24283i
$584$	0	0
$585$	−8412.96	−0.0245831
$586$	0	0
$587$	−282396.	−0.819562	−0.409781	−	0.912184i	$-0.634395\pi$
$587$	−282396.	−0.819562	−0.409781	+	0.912184i	$0.634395\pi$
$588$	0	0
$589$	− 234653.i	− 0.676388i
$590$	0	0
$591$	67142.9i	0.192232i
$592$	0	0
$593$	−131182.	−0.373048	−0.186524	−	0.982450i	$-0.559722\pi$
$593$	−131182.	−0.373048	−0.186524	+	0.982450i	$0.559722\pi$
$594$	0	0
$595$	15395.5	0.0434870
$596$	0	0
$597$	136347.i	0.382558i
$598$	0	0
$599$	− 509679.i	− 1.42051i	−0.703946	−	0.710254i	$-0.748580\pi$
$599$	− 509679.i	− 1.42051i	0.703946	−	0.710254i	$-0.251420\pi$
$600$	0	0
$601$	−403653.	−1.11753	−0.558766	−	0.829326i	$-0.688725\pi$
$601$	−403653.	−1.11753	−0.558766	+	0.829326i	$0.688725\pi$
$602$	0	0
$603$	−30054.0	−0.0826548
$604$	0	0
$605$	106775.i	0.291716i
$606$	0	0
$607$	− 278933.i	− 0.757047i	−0.925592	−	0.378523i	$-0.876432\pi$
$607$	− 278933.i	− 0.757047i	0.925592	−	0.378523i	$-0.123568\pi$
$608$	0	0
$609$	−29222.0	−0.0787908
$610$	0	0
$611$	−85324.2	−0.228555
$612$	0	0
$613$	120872.i	0.321666i	0.986982	+	0.160833i	$0.0514181\pi$
$613$	120872.i	0.321666i	−0.986982	+	0.160833i	$0.948582\pi$
$614$	0	0
$615$	63082.9i	0.166787i
$616$	0	0
$617$	56487.3	0.148382	0.0741909	−	0.997244i	$-0.476363\pi$
$617$	56487.3	0.148382	0.0741909	+	0.997244i	$0.476363\pi$
$618$	0	0
$619$	306312.	0.799435	0.399717	−	0.916638i	$-0.369108\pi$
$619$	306312.	0.799435	0.399717	+	0.916638i	$0.369108\pi$
$620$	0	0
$621$	17220.3i	0.0446537i
$622$	0	0
$623$	− 346436.i	− 0.892580i
$624$	0	0
$625$	271223.	0.694332
$626$	0	0
$627$	166085.	0.422469
$628$	0	0
$629$	57511.0i	0.145362i
$630$	0	0
$631$	100904.i	0.253426i	0.991939	+	0.126713i	$0.0404427\pi$
$631$	100904.i	0.253426i	−0.991939	+	0.126713i	$0.959557\pi$
$632$	0	0
$633$	403280.	1.00647
$634$	0	0
$635$	−202229.	−0.501528
$636$	0	0
$637$	22775.6i	0.0561296i
$638$	0	0
$639$	− 114023.i	− 0.279248i
$640$	0	0
$641$	425801.	1.03631	0.518156	−	0.855286i	$-0.326619\pi$
$641$	425801.	1.03631	0.518156	+	0.855286i	$0.326619\pi$
$642$	0	0
$643$	−168805.	−0.408285	−0.204143	−	0.978941i	$-0.565441\pi$
$643$	−168805.	−0.408285	−0.204143	+	0.978941i	$0.565441\pi$
$644$	0	0
$645$	66827.1i	0.160632i
$646$	0	0
$647$	290432.i	0.693802i	0.937902	+	0.346901i	$0.112766\pi$
$647$	290432.i	0.693802i	−0.937902	+	0.346901i	$0.887234\pi$
$648$	0	0
$649$	−757974.	−1.79955
$650$	0	0
$651$	270304.	0.637810
$652$	0	0
$653$	− 361551.i	− 0.847897i	−0.905686	−	0.423949i	$-0.860644\pi$
$653$	− 361551.i	− 0.847897i	0.905686	−	0.423949i	$-0.139356\pi$
$654$	0	0
$655$	− 49986.6i	− 0.116512i
$656$	0	0
$657$	188697.	0.437155
$658$	0	0
$659$	49982.1	0.115092	0.0575458	−	0.998343i	$-0.481672\pi$
$659$	49982.1	0.115092	0.0575458	+	0.998343i	$0.481672\pi$
$660$	0	0
$661$	− 468306.i	− 1.07183i	−0.844271	−	0.535916i	$-0.819966\pi$
$661$	− 468306.i	− 1.07183i	0.844271	−	0.535916i	$-0.180034\pi$
$662$	0	0
$663$	8883.61i	0.0202098i
$664$	0	0
$665$	−66227.3	−0.149759
$666$	0	0
$667$	16237.7	0.0364983
$668$	0	0
$669$	282858.i	0.631999i
$670$	0	0
$671$	− 833228.i	− 1.85063i
$672$	0	0
$673$	683819.	1.50977	0.754885	−	0.655857i	$-0.227692\pi$
$673$	683819.	1.50977	0.754885	+	0.655857i	$0.227692\pi$
$674$	0	0
$675$	78424.9	0.172126
$676$	0	0
$677$	546313.i	1.19197i	0.802997	+	0.595983i	$0.203238\pi$
$677$	546313.i	1.19197i	−0.802997	+	0.595983i	$0.796762\pi$
$678$	0	0
$679$	490367.i	1.06361i
$680$	0	0
$681$	152968.	0.329842
$682$	0	0
$683$	−861010.	−1.84572	−0.922862	−	0.385130i	$-0.874156\pi$
$683$	−861010.	−1.84572	−0.922862	+	0.385130i	$0.874156\pi$
$684$	0	0
$685$	145278.i	0.309614i
$686$	0	0
$687$	523233.i	1.10862i
$688$	0	0
$689$	175403.	0.369487
$690$	0	0
$691$	−112356.	−0.235310	−0.117655	−	0.993055i	$-0.537538\pi$
$691$	−112356.	−0.235310	−0.117655	+	0.993055i	$0.537538\pi$
$692$	0	0
$693$	191318.i	0.398373i
$694$	0	0
$695$	− 195702.i	− 0.405159i
$696$	0	0
$697$	66612.0	0.137116
$698$	0	0
$699$	−452285.	−0.925673
$700$	0	0
$701$	159939.i	0.325475i	0.986669	+	0.162738i	$0.0520324\pi$
$701$	159939.i	0.325475i	−0.986669	+	0.162738i	$0.947968\pi$
$702$	0	0
$703$	− 247398.i	− 0.500593i
$704$	0	0
$705$	−93916.4	−0.188957
$706$	0	0
$707$	183671.	0.367453
$708$	0	0
$709$	− 68345.2i	− 0.135961i	−0.997687	−	0.0679807i	$-0.978344\pi$
$709$	− 68345.2i	− 0.135961i	0.997687	−	0.0679807i	$-0.0216556\pi$
$710$	0	0
$711$	111323.i	0.220215i
$712$	0	0
$713$	−150199.	−0.295453
$714$	0	0
$715$	51937.4	0.101594
$716$	0	0
$717$	− 75728.6i	− 0.147306i
$718$	0	0
$719$	494631.i	0.956806i	0.878140	+	0.478403i	$0.158784\pi$
$719$	494631.i	0.956806i	−0.878140	+	0.478403i	$0.841216\pi$
$720$	0	0
$721$	22074.3	0.0424635
$722$	0	0
$723$	−106009.	−0.202800
$724$	0	0
$725$	− 73950.1i	− 0.140690i
$726$	0	0
$727$	− 288026.i	− 0.544957i	−0.962162	−	0.272478i	$-0.912157\pi$
$727$	− 288026.i	− 0.544957i	0.962162	−	0.272478i	$-0.0878433\pi$
$728$	0	0
$729$	19683.0	0.0370370
$730$	0	0
$731$	70565.7	0.132056
$732$	0	0
$733$	152590.i	0.284000i	0.989867	+	0.142000i	$0.0453532\pi$
$733$	152590.i	0.284000i	−0.989867	+	0.142000i	$0.954647\pi$
$734$	0	0
$735$	25069.2i	0.0464051i
$736$	0	0
$737$	185539.	0.341585
$738$	0	0
$739$	35159.3	0.0643800	0.0321900	−	0.999482i	$-0.489752\pi$
$739$	35159.3	0.0643800	0.0321900	+	0.999482i	$0.489752\pi$
$740$	0	0
$741$	− 38215.0i	− 0.0695981i
$742$	0	0
$743$	894968.i	1.62118i	0.585617	+	0.810588i	$0.300852\pi$
$743$	894968.i	1.62118i	−0.585617	+	0.810588i	$0.699148\pi$
$744$	0	0
$745$	−165942.	−0.298981
$746$	0	0
$747$	−185308.	−0.332089
$748$	0	0
$749$	− 170235.i	− 0.303448i
$750$	0	0
$751$	− 58904.9i	− 0.104441i	−0.998636	−	0.0522205i	$-0.983370\pi$
$751$	− 58904.9i	− 0.104441i	0.998636	−	0.0522205i	$-0.0166299\pi$
$752$	0	0
$753$	−354202.	−0.624686
$754$	0	0
$755$	288768.	0.506589
$756$	0	0
$757$	313925.i	0.547816i	0.961756	+	0.273908i	$0.0883163\pi$
$757$	313925.i	0.547816i	−0.961756	+	0.273908i	$0.911684\pi$
$758$	0	0
$759$	− 106309.i	− 0.184539i
$760$	0	0
$761$	961807.	1.66080	0.830402	−	0.557164i	$-0.188111\pi$
$761$	961807.	1.66080	0.830402	+	0.557164i	$0.188111\pi$
$762$	0	0
$763$	453729.	0.779377
$764$	0	0
$765$	9778.20i	0.0167084i
$766$	0	0
$767$	174405.i	0.296461i
$768$	0	0
$769$	763172.	1.29053	0.645267	−	0.763957i	$-0.276746\pi$
$769$	763172.	1.29053	0.645267	+	0.763957i	$0.276746\pi$
$770$	0	0
$771$	568738.	0.956762
$772$	0	0
$773$	844678.i	1.41362i	0.707404	+	0.706809i	$0.249866\pi$
$773$	844678.i	1.41362i	−0.707404	+	0.706809i	$0.750134\pi$
$774$	0	0
$775$	684041.i	1.13888i
$776$	0	0
$777$	284985.	0.472041
$778$	0	0
$779$	−286548.	−0.472196
$780$	0	0
$781$	703920.i	1.15404i
$782$	0	0
$783$	− 18559.9i	− 0.0302728i
$784$	0	0
$785$	271295.	0.440254
$786$	0	0
$787$	−276057.	−0.445708	−0.222854	−	0.974852i	$-0.571537\pi$
$787$	−276057.	−0.445708	−0.222854	+	0.974852i	$0.571537\pi$
$788$	0	0
$789$	442759.i	0.711235i
$790$	0	0
$791$	− 722569.i	− 1.15485i
$792$	0	0
$793$	−191720.	−0.304875
$794$	0	0
$795$	193066.	0.305473
$796$	0	0
$797$	447283.i	0.704152i	0.935971	+	0.352076i	$0.114524\pi$
$797$	447283.i	0.704152i	−0.935971	+	0.352076i	$0.885476\pi$
$798$	0	0
$799$	99170.5i	0.155342i
$800$	0	0
$801$	220034.	0.342945
$802$	0	0
$803$	−1.16492e6	−1.80662
$804$	0	0
$805$	42391.5i	0.0654164i
$806$	0	0
$807$	− 110546.i	− 0.169745i
$808$	0	0
$809$	1.24497e6	1.90223	0.951115	−	0.308835i	$-0.0999392\pi$
$809$	1.24497e6	1.90223	0.951115	+	0.308835i	$0.0999392\pi$
$810$	0	0
$811$	699085.	1.06289	0.531445	−	0.847093i	$-0.321649\pi$
$811$	699085.	1.06289	0.531445	+	0.847093i	$0.321649\pi$
$812$	0	0
$813$	156995.i	0.237523i
$814$	0	0
$815$	107739.i	0.162203i
$816$	0	0
$817$	−303555.	−0.454772
$818$	0	0
$819$	44021.1	0.0656285
$820$	0	0
$821$	− 177764.i	− 0.263728i	−0.991268	−	0.131864i	$-0.957904\pi$
$821$	− 177764.i	− 0.263728i	0.991268	−	0.131864i	$-0.0420963\pi$
$822$	0	0
$823$	− 471157.i	− 0.695610i	−0.937567	−	0.347805i	$-0.886927\pi$
$823$	− 471157.i	− 0.695610i	0.937567	−	0.347805i	$-0.113073\pi$
$824$	0	0
$825$	−484156.	−0.711341
$826$	0	0
$827$	93698.7	0.137001	0.0685003	−	0.997651i	$-0.478179\pi$
$827$	93698.7	0.137001	0.0685003	+	0.997651i	$0.478179\pi$
$828$	0	0
$829$	751113.i	1.09294i	0.837479	+	0.546470i	$0.184029\pi$
$829$	751113.i	1.09294i	−0.837479	+	0.546470i	$0.815971\pi$
$830$	0	0
$831$	− 5177.01i	− 0.00749682i
$832$	0	0
$833$	26471.7	0.0381497
$834$	0	0
$835$	11401.7	0.0163529
$836$	0	0
$837$	171680.i	0.245058i
$838$	0	0
$839$	− 705058.i	− 1.00162i	−0.865558	−	0.500808i	$-0.833036\pi$
$839$	− 705058.i	− 1.00162i	0.865558	−	0.500808i	$-0.166964\pi$
$840$	0	0
$841$	689780.	0.975256
$842$	0	0
$843$	−47669.4	−0.0670786
$844$	0	0
$845$	220088.i	0.308235i
$846$	0	0
$847$	− 558706.i	− 0.778783i
$848$	0	0
$849$	−269366.	−0.373703
$850$	0	0
$851$	−158357.	−0.218664
$852$	0	0
$853$	832597.i	1.14429i	0.820152	+	0.572146i	$0.193889\pi$
$853$	832597.i	1.14429i	−0.820152	+	0.572146i	$0.806111\pi$
$854$	0	0
$855$	− 42063.3i	− 0.0575401i
$856$	0	0
$857$	218907.	0.298056	0.149028	−	0.988833i	$-0.452386\pi$
$857$	218907.	0.298056	0.149028	+	0.988833i	$0.452386\pi$
$858$	0	0
$859$	−783329.	−1.06159	−0.530796	−	0.847499i	$-0.678107\pi$
$859$	−783329.	−1.06159	−0.530796	+	0.847499i	$0.678107\pi$
$860$	0	0
$861$	− 330083.i	− 0.445263i
$862$	0	0
$863$	− 360346.i	− 0.483836i	−0.970297	−	0.241918i	$-0.922224\pi$
$863$	− 360346.i	− 0.483836i	0.970297	−	0.241918i	$-0.0777765\pi$
$864$	0	0
$865$	−381153.	−0.509410
$866$	0	0
$867$	−423663.	−0.563614
$868$	0	0
$869$	− 687254.i	− 0.910076i
$870$	0	0
$871$	− 42691.1i	− 0.0562732i
$872$	0	0
$873$	−311449.	−0.408657
$874$	0	0
$875$	408916.	0.534094
$876$	0	0
$877$	453408.i	0.589508i	0.955573	+	0.294754i	$0.0952377\pi$
$877$	453408.i	0.589508i	−0.955573	+	0.294754i	$0.904762\pi$
$878$	0	0
$879$	357533.i	0.462741i
$880$	0	0
$881$	−175535.	−0.226158	−0.113079	−	0.993586i	$-0.536071\pi$
$881$	−175535.	−0.226158	−0.113079	+	0.993586i	$0.536071\pi$
$882$	0	0
$883$	100969.	0.129498	0.0647492	−	0.997902i	$-0.479375\pi$
$883$	100969.	0.129498	0.0647492	+	0.997902i	$0.479375\pi$
$884$	0	0
$885$	191967.i	0.245099i
$886$	0	0
$887$	− 1.42438e6i	− 1.81041i	−0.424974	−	0.905206i	$-0.639717\pi$
$887$	− 1.42438e6i	− 1.81041i	0.424974	−	0.905206i	$-0.360283\pi$
$888$	0	0
$889$	1.05817e6	1.33891
$890$	0	0
$891$	−121513.	−0.153062
$892$	0	0
$893$	− 426606.i	− 0.534963i
$894$	0	0
$895$	− 238055.i	− 0.297188i
$896$	0	0
$897$	−24461.1	−0.0304012
$898$	0	0
$899$	161884.	0.200302
$900$	0	0
$901$	− 203867.i	− 0.251130i
$902$	0	0
$903$	− 349675.i	− 0.428834i
$904$	0	0
$905$	−484670.	−0.591764
$906$	0	0
$907$	−231867.	−0.281854	−0.140927	−	0.990020i	$-0.545008\pi$
$907$	−231867.	−0.281854	−0.140927	+	0.990020i	$0.545008\pi$
$908$	0	0
$909$	116656.i	0.141182i
$910$	0	0
$911$	1.21654e6i	1.46585i	0.680310	+	0.732924i	$0.261845\pi$
$911$	1.21654e6i	1.46585i	−0.680310	+	0.732924i	$0.738155\pi$
$912$	0	0
$913$	1.14400e6	1.37241
$914$	0	0
$915$	−211026.	−0.252055
$916$	0	0
$917$	261556.i	0.311047i
$918$	0	0
$919$	− 1.39879e6i	− 1.65623i	−0.560556	−	0.828117i	$-0.689412\pi$
$919$	− 1.39879e6i	− 1.65623i	0.560556	−	0.828117i	$-0.310588\pi$
$920$	0	0
$921$	642912.	0.757936
$922$	0	0
$923$	161967.	0.190118
$924$	0	0
$925$	721192.i	0.842883i
$926$	0	0
$927$	14020.1i	0.0163152i
$928$	0	0
$929$	−366307.	−0.424437	−0.212219	−	0.977222i	$-0.568069\pi$
$929$	−366307.	−0.424437	−0.212219	+	0.977222i	$0.568069\pi$
$930$	0	0
$931$	−113874.	−0.131379
$932$	0	0
$933$	− 818913.i	− 0.940750i
$934$	0	0
$935$	− 60365.7i	− 0.0690506i
$936$	0	0
$937$	889237.	1.01283	0.506417	−	0.862289i	$-0.330970\pi$
$937$	889237.	1.01283	0.506417	+	0.862289i	$0.330970\pi$
$938$	0	0
$939$	115134.	0.130579
$940$	0	0
$941$	1.20010e6i	1.35531i	0.735379	+	0.677656i	$0.237004\pi$
$941$	1.20010e6i	1.35531i	−0.735379	+	0.677656i	$0.762996\pi$
$942$	0	0
$943$	183416.i	0.206260i
$944$	0	0
$945$	48454.0	0.0542583
$946$	0	0
$947$	196544.	0.219159	0.109580	−	0.993978i	$-0.465050\pi$
$947$	196544.	0.219159	0.109580	+	0.993978i	$0.465050\pi$
$948$	0	0
$949$	268041.i	0.297624i
$950$	0	0
$951$	132563.i	0.146575i
$952$	0	0
$953$	214377.	0.236044	0.118022	−	0.993011i	$-0.462345\pi$
$953$	214377.	0.236044	0.118022	+	0.993011i	$0.462345\pi$
$954$	0	0
$955$	534432.	0.585984
$956$	0	0
$957$	114580.i	0.125108i
$958$	0	0
$959$	− 760174.i	− 0.826563i
$960$	0	0
$961$	−573910.	−0.621437
$962$	0	0
$963$	108122.	0.116590
$964$	0	0
$965$	546276.i	0.586621i
$966$	0	0
$967$	618682.i	0.661629i	0.943696	+	0.330814i	$0.107323\pi$
$967$	618682.i	0.661629i	−0.943696	+	0.330814i	$0.892677\pi$
$968$	0	0
$969$	−44416.5	−0.0473039
$970$	0	0
$971$	425206.	0.450984	0.225492	−	0.974245i	$-0.427601\pi$
$971$	425206.	0.450984	0.225492	+	0.974245i	$0.427601\pi$
$972$	0	0
$973$	1.02402e6i	1.08164i
$974$	0	0
$975$	111401.i	0.117187i
$976$	0	0
$977$	−165981.	−0.173888	−0.0869438	−	0.996213i	$-0.527710\pi$
$977$	−165981.	−0.173888	−0.0869438	+	0.996213i	$0.527710\pi$
$978$	0	0
$979$	−1.35838e6	−1.41728
$980$	0	0
$981$	288179.i	0.299450i
$982$	0	0
$983$	1.48634e6i	1.53819i	0.639133	+	0.769097i	$0.279293\pi$
$983$	1.48634e6i	1.53819i	−0.639133	+	0.769097i	$0.720707\pi$
$984$	0	0
$985$	−104980.	−0.108201
$986$	0	0
$987$	491421.	0.504451
$988$	0	0
$989$	194303.i	0.198649i
$990$	0	0
$991$	1.30999e6i	1.33390i	0.745104	+	0.666948i	$0.232400\pi$
$991$	1.30999e6i	1.33390i	−0.745104	+	0.666948i	$0.767600\pi$
$992$	0	0
$993$	−53245.8	−0.0539991
$994$	0	0
$995$	−213182.	−0.215329
$996$	0	0
$997$	− 1.51880e6i	− 1.52795i	−0.645246	−	0.763975i	$-0.723245\pi$
$997$	− 1.51880e6i	− 1.52795i	0.645246	−	0.763975i	$-0.276755\pi$
$998$	0	0
$999$	181004.i	0.181366i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.5.b.i.127.14		16
4.3	odd	2		768.5.b.h.127.6		16
8.3	odd	2	inner	768.5.b.i.127.11		16
8.5	even	2		768.5.b.h.127.3		16
16.3	odd	4		384.5.g.b.127.5	yes	16
16.5	even	4		384.5.g.a.127.4	✓	16
16.11	odd	4		384.5.g.a.127.12	yes	16
16.13	even	4		384.5.g.b.127.13	yes	16
48.5	odd	4		1152.5.g.f.127.10		16
48.11	even	4		1152.5.g.f.127.9		16
48.29	odd	4		1152.5.g.c.127.8		16
48.35	even	4		1152.5.g.c.127.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.5.g.a.127.4	✓	16	16.5	even	4
384.5.g.a.127.12	yes	16	16.11	odd	4
384.5.g.b.127.5	yes	16	16.3	odd	4
384.5.g.b.127.13	yes	16	16.13	even	4
768.5.b.h.127.3		16	8.5	even	2
768.5.b.h.127.6		16	4.3	odd	2
768.5.b.i.127.11		16	8.3	odd	2	inner
768.5.b.i.127.14		16	1.1	even	1	trivial
1152.5.g.c.127.7		16	48.35	even	4
1152.5.g.c.127.8		16	48.29	odd	4
1152.5.g.f.127.9		16	48.11	even	4
1152.5.g.f.127.10		16	48.5	odd	4

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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	768.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{3} +8.12430i q^{5} -42.5107i q^{7} +27.0000 q^{9} +O(q^{10})\)	\(q+5.19615 q^{3} +8.12430i q^{5} -42.5107i q^{7} +27.0000 q^{9} -166.684 q^{11} +38.3529i q^{13} +42.2151i q^{15} +44.5768 q^{17} -191.758 q^{19} -220.892i q^{21} +122.742i q^{23} +558.996 q^{25} +140.296 q^{27} -132.291i q^{29} +1223.70i q^{31} -866.118 q^{33} +345.370 q^{35} +1290.16i q^{37} +199.288i q^{39} +1494.32 q^{41} +1583.01 q^{43} +219.356i q^{45} +2224.71i q^{47} +593.843 q^{49} +231.628 q^{51} -4573.39i q^{53} -1354.20i q^{55} -996.403 q^{57} +4547.36 q^{59} +4998.83i q^{61} -1147.79i q^{63} -311.591 q^{65} -1113.11 q^{67} +637.788i q^{69} -4223.07i q^{71} +6988.79 q^{73} +2904.63 q^{75} +7085.87i q^{77} +4123.08i q^{79} +729.000 q^{81} -6863.28 q^{83} +362.156i q^{85} -687.404i q^{87} +8149.40 q^{89} +1630.41 q^{91} +6358.51i q^{93} -1557.90i q^{95} -11535.2 q^{97} -4500.48 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 432 q^{9}+O(q^{10}) \) 16 * q + 432 * q^9	\( 16 q + 432 q^{9} + 576 q^{11} - 480 q^{17} + 192 q^{19} - 2672 q^{25} + 9024 q^{35} - 1440 q^{41} + 12224 q^{43} - 2480 q^{49} + 6336 q^{51} + 7488 q^{57} + 15360 q^{59} - 1344 q^{65} + 12288 q^{67} + 8480 q^{73} + 21888 q^{75} + 11664 q^{81} + 13248 q^{83} - 18720 q^{89} - 30272 q^{91} + 13088 q^{97} + 15552 q^{99}+O(q^{100}) \) 16 * q + 432 * q^9 + 576 * q^11 - 480 * q^17 + 192 * q^19 - 2672 * q^25 + 9024 * q^35 - 1440 * q^41 + 12224 * q^43 - 2480 * q^49 + 6336 * q^51 + 7488 * q^57 + 15360 * q^59 - 1344 * q^65 + 12288 * q^67 + 8480 * q^73 + 21888 * q^75 + 11664 * q^81 + 13248 * q^83 - 18720 * q^89 - 30272 * q^91 + 13088 * q^97 + 15552 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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