LMFDB - Embedded newform 784.6.f.e.783.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$125.740914733$
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} - 232 x^{14} + 23092 x^{12} - 1258560 x^{10} + 40996255 x^{8} - 775625040 x^{6} + \cdots + 20552376321 $ x^16 - 232x^14 + 23092x^12 - 1258560x^10 + 40996255x^8 - 775625040x^6 + 7073588484x^4 - 15906881088*x^2 + 20552376321
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{34}\cdot 7^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			783.6
Root			$-5.40349 + 0.215219i$ of defining polynomial
Character	$\chi$	$=$	784.783
Dual form			784.6.f.e.783.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-16.1465 q^{3} +48.6032i q^{5} +17.7104 q^{9} -400.090i q^{11} +228.826i q^{13} -784.773i q^{15} +519.281i q^{17} -946.748 q^{19} -1240.48i q^{23} +762.732 q^{25} +3637.64 q^{27} +3638.60 q^{29} -2113.97 q^{31} +6460.06i q^{33} -8590.33 q^{37} -3694.75i q^{39} +1440.06i q^{41} +14544.8i q^{43} +860.783i q^{45} +15759.6 q^{47} -8384.58i q^{51} +13329.8 q^{53} +19445.6 q^{55} +15286.7 q^{57} +31168.4 q^{59} +34886.6i q^{61} -11121.7 q^{65} -61883.7i q^{67} +20029.5i q^{69} +49388.6i q^{71} -4237.71i q^{73} -12315.5 q^{75} -55380.0i q^{79} -63039.0 q^{81} +104615. q^{83} -25238.7 q^{85} -58750.8 q^{87} -13040.9i q^{89} +34133.3 q^{93} -46014.9i q^{95} -12607.0i q^{97} -7085.76i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 1712 q^{9} - 496 q^{25} - 17024 q^{29} + 85664 q^{53} - 95424 q^{57} + 233344 q^{65} - 511216 q^{81} + 697952 q^{85} - 857024 q^{93}+O(q^{100}) $ 16 * q + 1712 * q^9 - 496 * q^25 - 17024 * q^29 + 85664 * q^53 - 95424 * q^57 + 233344 * q^65 - 511216 * q^81 + 697952 * q^85 - 857024 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−16.1465	−1.03580	−0.517900	−	0.855441i	$-0.673286\pi$
$3$	−16.1465	−1.03580	−0.517900	+	0.855441i	$0.673286\pi$
$4$	0	0
$5$	48.6032i	0.869440i	0.900566	+	0.434720i	$0.143153\pi$
$5$	48.6032i	0.869440i	−0.900566	+	0.434720i	$0.856847\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	17.7104	0.0728824
$10$	0	0
$11$	− 400.090i	− 0.996955i	−0.866902	−	0.498478i	$-0.833893\pi$
$11$	− 400.090i	− 0.996955i	0.866902	−	0.498478i	$-0.166107\pi$
$12$	0	0
$13$	228.826i	0.375532i	0.982214	+	0.187766i	$0.0601247\pi$
$13$	228.826i	0.375532i	−0.982214	+	0.187766i	$0.939875\pi$
$14$	0	0
$15$	− 784.773i	− 0.900566i
$16$	0	0
$17$	519.281i	0.435793i	0.975972	+	0.217896i	$0.0699195\pi$
$17$	519.281i	0.435793i	−0.975972	+	0.217896i	$0.930081\pi$
$18$	0	0
$19$	−946.748	−0.601659	−0.300829	−	0.953678i	$-0.597263\pi$
$19$	−946.748	−0.601659	−0.300829	+	0.953678i	$0.597263\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1240.48i	− 0.488958i	−0.969655	−	0.244479i	$-0.921383\pi$
$23$	− 1240.48i	− 0.488958i	0.969655	−	0.244479i	$-0.0786170\pi$
$24$	0	0
$25$	762.732	0.244074
$26$	0	0
$27$	3637.64	0.960309
$28$	0	0
$29$	3638.60	0.803414	0.401707	−	0.915768i	$-0.368417\pi$
$29$	3638.60	0.803414	0.401707	+	0.915768i	$0.368417\pi$
$30$	0	0
$31$	−2113.97	−0.395089	−0.197545	−	0.980294i	$-0.563297\pi$
$31$	−2113.97	−0.395089	−0.197545	+	0.980294i	$0.563297\pi$
$32$	0	0
$33$	6460.06i	1.03265i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−8590.33	−1.03159	−0.515793	−	0.856713i	$-0.672503\pi$
$37$	−8590.33	−1.03159	−0.515793	+	0.856713i	$0.672503\pi$
$38$	0	0
$39$	− 3694.75i	− 0.388977i
$40$	0	0
$41$	1440.06i	0.133789i	0.997760	+	0.0668944i	$0.0213091\pi$
$41$	1440.06i	0.133789i	−0.997760	+	0.0668944i	$0.978691\pi$
$42$	0	0
$43$	14544.8i	1.19960i	0.800149	+	0.599801i	$0.204754\pi$
$43$	14544.8i	1.19960i	−0.800149	+	0.599801i	$0.795246\pi$
$44$	0	0
$45$	860.783i	0.0633669i
$46$	0	0
$47$	15759.6	1.04064	0.520320	−	0.853972i	$-0.325813\pi$
$47$	15759.6	1.04064	0.520320	+	0.853972i	$0.325813\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	− 8384.58i	− 0.451394i
$52$	0	0
$53$	13329.8	0.651832	0.325916	−	0.945399i	$-0.394327\pi$
$53$	13329.8	0.651832	0.325916	+	0.945399i	$0.394327\pi$
$54$	0	0
$55$	19445.6	0.866793
$56$	0	0
$57$	15286.7	0.623198
$58$	0	0
$59$	31168.4	1.16569	0.582847	−	0.812582i	$-0.301939\pi$
$59$	31168.4	1.16569	0.582847	+	0.812582i	$0.301939\pi$
$60$	0	0
$61$	34886.6i	1.20042i	0.799842	+	0.600211i	$0.204917\pi$
$61$	34886.6i	1.20042i	−0.799842	+	0.600211i	$0.795083\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−11121.7	−0.326503
$66$	0	0
$67$	− 61883.7i	− 1.68418i	−0.539335	−	0.842091i	$-0.681324\pi$
$67$	− 61883.7i	− 1.68418i	0.539335	−	0.842091i	$-0.318676\pi$
$68$	0	0
$69$	20029.5i	0.506463i
$70$	0	0
$71$	49388.6i	1.16274i	0.813641	+	0.581368i	$0.197482\pi$
$71$	49388.6i	1.16274i	−0.813641	+	0.581368i	$0.802518\pi$
$72$	0	0
$73$	− 4237.71i	− 0.0930731i	−0.998917	−	0.0465366i	$-0.985182\pi$
$73$	− 4237.71i	− 0.0930731i	0.998917	−	0.0465366i	$-0.0148184\pi$
$74$	0	0
$75$	−12315.5	−0.252812
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 55380.0i	− 0.998355i	−0.866500	−	0.499177i	$-0.833636\pi$
$79$	− 55380.0i	− 0.998355i	0.866500	−	0.499177i	$-0.166364\pi$
$80$	0	0
$81$	−63039.0	−1.06757
$82$	0	0
$83$	104615.	1.66686	0.833432	−	0.552622i	$-0.186373\pi$
$83$	104615.	1.66686	0.833432	+	0.552622i	$0.186373\pi$
$84$	0	0
$85$	−25238.7	−0.378896
$86$	0	0
$87$	−58750.8	−0.832177
$88$	0	0
$89$	− 13040.9i	− 0.174515i	−0.996186	−	0.0872574i	$-0.972190\pi$
$89$	− 13040.9i	− 0.174515i	0.996186	−	0.0872574i	$-0.0278103\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	34133.3	0.409234
$94$	0	0
$95$	− 46014.9i	− 0.523106i
$96$	0	0
$97$	− 12607.0i	− 0.136045i	−0.997684	−	0.0680227i	$-0.978331\pi$
$97$	− 12607.0i	− 0.136045i	0.997684	−	0.0680227i	$-0.0216690\pi$
$98$	0	0
$99$	− 7085.76i	− 0.0726605i
$100$	0	0
$101$	− 82190.0i	− 0.801706i	−0.916142	−	0.400853i	$-0.868714\pi$
$101$	− 82190.0i	− 0.801706i	0.916142	−	0.400853i	$-0.131286\pi$
$102$	0	0
$103$	77813.7	0.722708	0.361354	−	0.932429i	$-0.382315\pi$
$103$	77813.7	0.722708	0.361354	+	0.932429i	$0.382315\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 67015.8i	− 0.565871i	−0.959139	−	0.282936i	$-0.908692\pi$
$107$	− 67015.8i	− 0.565871i	0.959139	−	0.282936i	$-0.0913082\pi$
$108$	0	0
$109$	−115464.	−0.930853	−0.465426	−	0.885087i	$-0.654099\pi$
$109$	−115464.	−0.930853	−0.465426	+	0.885087i	$0.654099\pi$
$110$	0	0
$111$	138704.	1.06852
$112$	0	0
$113$	−133573.	−0.984059	−0.492029	−	0.870579i	$-0.663745\pi$
$113$	−133573.	−0.984059	−0.492029	+	0.870579i	$0.663745\pi$
$114$	0	0
$115$	60291.5	0.425120
$116$	0	0
$117$	4052.61i	0.0273697i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	979.152	0.00607976
$122$	0	0
$123$	− 23251.9i	− 0.138578i
$124$	0	0
$125$	188956.i	1.08165i
$126$	0	0
$127$	137119.i	0.754375i	0.926137	+	0.377188i	$0.123109\pi$
$127$	137119.i	0.754375i	−0.926137	+	0.377188i	$0.876891\pi$
$128$	0	0
$129$	− 234848.i	− 1.24255i
$130$	0	0
$131$	−7599.78	−0.0386922	−0.0193461	−	0.999813i	$-0.506158\pi$
$131$	−7599.78	−0.0386922	−0.0193461	+	0.999813i	$0.506158\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	176801.i	0.834931i
$136$	0	0
$137$	−354434.	−1.61337	−0.806684	−	0.590983i	$-0.798740\pi$
$137$	−354434.	−1.61337	−0.806684	+	0.590983i	$0.798740\pi$
$138$	0	0
$139$	−308453.	−1.35410	−0.677052	−	0.735935i	$-0.736743\pi$
$139$	−308453.	−1.35410	−0.677052	+	0.735935i	$0.736743\pi$
$140$	0	0
$141$	−254463.	−1.07789
$142$	0	0
$143$	91551.0	0.374389
$144$	0	0
$145$	176848.i	0.698520i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	111322.	0.410787	0.205393	−	0.978679i	$-0.434153\pi$
$149$	111322.	0.410787	0.205393	+	0.978679i	$0.434153\pi$
$150$	0	0
$151$	10832.9i	0.0386637i	0.999813	+	0.0193318i	$0.00615390\pi$
$151$	10832.9i	0.0386637i	−0.999813	+	0.0193318i	$0.993846\pi$
$152$	0	0
$153$	9196.69i	0.0317616i
$154$	0	0
$155$	− 102746.i	− 0.343506i
$156$	0	0
$157$	− 144815.i	− 0.468882i	−0.972130	−	0.234441i	$-0.924674\pi$
$157$	− 144815.i	− 0.468882i	0.972130	−	0.234441i	$-0.0753259\pi$
$158$	0	0
$159$	−215231.	−0.675167
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 152647.i	− 0.450006i	−0.974358	−	0.225003i	$-0.927761\pi$
$163$	− 152647.i	− 0.450006i	0.974358	−	0.225003i	$-0.0722392\pi$
$164$	0	0
$165$	−313979.	−0.897824
$166$	0	0
$167$	−347479.	−0.964134	−0.482067	−	0.876134i	$-0.660114\pi$
$167$	−347479.	−0.964134	−0.482067	+	0.876134i	$0.660114\pi$
$168$	0	0
$169$	318932.	0.858975
$170$	0	0
$171$	−16767.3	−0.0438503
$172$	0	0
$173$	728965.i	1.85179i	0.377782	+	0.925894i	$0.376687\pi$
$173$	728965.i	1.85179i	−0.377782	+	0.925894i	$0.623313\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−503262.	−1.20743
$178$	0	0
$179$	739166.i	1.72429i	0.506665	+	0.862143i	$0.330878\pi$
$179$	739166.i	1.72429i	−0.506665	+	0.862143i	$0.669122\pi$
$180$	0	0
$181$	614030.i	1.39313i	0.717492	+	0.696567i	$0.245290\pi$
$181$	614030.i	1.39313i	−0.717492	+	0.696567i	$0.754710\pi$
$182$	0	0
$183$	− 563297.i	− 1.24340i
$184$	0	0
$185$	− 417517.i	− 0.896902i
$186$	0	0
$187$	207759.	0.434466
$188$	0	0
$189$	0	0
$190$	0	0
$191$	685222.i	1.35909i	0.733635	+	0.679544i	$0.237822\pi$
$191$	685222.i	1.35909i	−0.733635	+	0.679544i	$0.762178\pi$
$192$	0	0
$193$	−881141.	−1.70275	−0.851377	−	0.524554i	$-0.824232\pi$
$193$	−881141.	−1.70275	−0.851377	+	0.524554i	$0.824232\pi$
$194$	0	0
$195$	179577.	0.338192
$196$	0	0
$197$	−307862.	−0.565185	−0.282592	−	0.959240i	$-0.591194\pi$
$197$	−307862.	−0.565185	−0.282592	+	0.959240i	$0.591194\pi$
$198$	0	0
$199$	−826375.	−1.47926	−0.739630	−	0.673014i	$-0.764999\pi$
$199$	−826375.	−1.47926	−0.739630	+	0.673014i	$0.764999\pi$
$200$	0	0
$201$	999207.i	1.74448i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−69991.3	−0.116321
$206$	0	0
$207$	− 21969.5i	− 0.0356365i
$208$	0	0
$209$	378784.i	0.599827i
$210$	0	0
$211$	− 743853.i	− 1.15022i	−0.818076	−	0.575110i	$-0.804959\pi$
$211$	− 743853.i	− 1.15022i	0.818076	−	0.575110i	$-0.195041\pi$
$212$	0	0
$213$	− 797455.i	− 1.20436i
$214$	0	0
$215$	−706924.	−1.04298
$216$	0	0
$217$	0	0
$218$	0	0
$219$	68424.4i	0.0964052i
$220$	0	0
$221$	−118825.	−0.163654
$222$	0	0
$223$	−1.12243e6	−1.51146	−0.755728	−	0.654885i	$-0.772717\pi$
$223$	−1.12243e6	−1.51146	−0.755728	+	0.654885i	$0.772717\pi$
$224$	0	0
$225$	13508.3	0.0177887
$226$	0	0
$227$	504464.	0.649779	0.324890	−	0.945752i	$-0.394673\pi$
$227$	504464.	0.649779	0.324890	+	0.945752i	$0.394673\pi$
$228$	0	0
$229$	1.34917e6i	1.70012i	0.526689	+	0.850058i	$0.323433\pi$
$229$	1.34917e6i	1.70012i	−0.526689	+	0.850058i	$0.676567\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−868468.	−1.04801	−0.524003	−	0.851716i	$-0.675562\pi$
$233$	−868468.	−1.04801	−0.524003	+	0.851716i	$0.675562\pi$
$234$	0	0
$235$	765966.i	0.904773i
$236$	0	0
$237$	894194.i	1.03410i
$238$	0	0
$239$	552839.i	0.626043i	0.949746	+	0.313021i	$0.101341\pi$
$239$	552839.i	0.626043i	−0.949746	+	0.313021i	$0.898659\pi$
$240$	0	0
$241$	− 446683.i	− 0.495401i	−0.968837	−	0.247700i	$-0.920325\pi$
$241$	− 446683.i	− 0.495401i	0.968837	−	0.247700i	$-0.0796749\pi$
$242$	0	0
$243$	133913.	0.145481
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 216641.i	− 0.225942i
$248$	0	0
$249$	−1.68918e6	−1.72654
$250$	0	0
$251$	−83590.2	−0.0837474	−0.0418737	−	0.999123i	$-0.513333\pi$
$251$	−83590.2	−0.0837474	−0.0418737	+	0.999123i	$0.513333\pi$
$252$	0	0
$253$	−496305.	−0.487470
$254$	0	0
$255$	407517.	0.392460
$256$	0	0
$257$	492968.i	0.465571i	0.972528	+	0.232786i	$0.0747840\pi$
$257$	492968.i	0.465571i	−0.972528	+	0.232786i	$0.925216\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	64441.2	0.0585548
$262$	0	0
$263$	− 1.11123e6i	− 0.990637i	−0.868711	−	0.495319i	$-0.835051\pi$
$263$	− 1.11123e6i	− 0.990637i	0.868711	−	0.495319i	$-0.164949\pi$
$264$	0	0
$265$	647873.i	0.566728i
$266$	0	0
$267$	210565.i	0.180762i
$268$	0	0
$269$	− 1.48558e6i	− 1.25174i	−0.779926	−	0.625871i	$-0.784744\pi$
$269$	− 1.48558e6i	− 1.25174i	0.779926	−	0.625871i	$-0.215256\pi$
$270$	0	0
$271$	429636.	0.355368	0.177684	−	0.984088i	$-0.443140\pi$
$271$	429636.	0.355368	0.177684	+	0.984088i	$0.443140\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 305161.i	− 0.243331i
$276$	0	0
$277$	−2.48618e6	−1.94686	−0.973428	−	0.228992i	$-0.926457\pi$
$277$	−2.48618e6	−1.94686	−0.973428	+	0.228992i	$0.926457\pi$
$278$	0	0
$279$	−37439.4	−0.0287951
$280$	0	0
$281$	−895843.	−0.676809	−0.338405	−	0.941001i	$-0.609887\pi$
$281$	−895843.	−0.676809	−0.338405	+	0.941001i	$0.609887\pi$
$282$	0	0
$283$	1.06303e6	0.789003	0.394501	−	0.918895i	$-0.370917\pi$
$283$	1.06303e6	0.789003	0.394501	+	0.918895i	$0.370917\pi$
$284$	0	0
$285$	742981.i	0.541833i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.15020e6	0.810085
$290$	0	0
$291$	203560.i	0.140916i
$292$	0	0
$293$	286091.i	0.194686i	0.995251	+	0.0973430i	$0.0310344\pi$
$293$	286091.i	0.194686i	−0.995251	+	0.0973430i	$0.968966\pi$
$294$	0	0
$295$	1.51488e6i	1.01350i
$296$	0	0
$297$	− 1.45538e6i	− 0.957385i
$298$	0	0
$299$	283855.	0.183620
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.32708e6i	0.830408i
$304$	0	0
$305$	−1.69560e6	−1.04369
$306$	0	0
$307$	2.54873e6	1.54340	0.771700	−	0.635987i	$-0.219407\pi$
$307$	2.54873e6	1.54340	0.771700	+	0.635987i	$0.219407\pi$
$308$	0	0
$309$	−1.25642e6	−0.748581
$310$	0	0
$311$	−2.76090e6	−1.61864	−0.809319	−	0.587369i	$-0.800164\pi$
$311$	−2.76090e6	−1.61864	−0.809319	+	0.587369i	$0.800164\pi$
$312$	0	0
$313$	− 272811.i	− 0.157399i	−0.996898	−	0.0786994i	$-0.974923\pi$
$313$	− 272811.i	− 0.157399i	0.996898	−	0.0786994i	$-0.0250767\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	206589.	0.115467	0.0577337	−	0.998332i	$-0.481613\pi$
$317$	206589.	0.115467	0.0577337	+	0.998332i	$0.481613\pi$
$318$	0	0
$319$	− 1.45577e6i	− 0.800968i
$320$	0	0
$321$	1.08207e6i	0.586129i
$322$	0	0
$323$	− 491628.i	− 0.262198i
$324$	0	0
$325$	174533.i	0.0916578i
$326$	0	0
$327$	1.86435e6	0.964177
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 3.71327e6i	− 1.86289i	−0.363885	−	0.931444i	$-0.618550\pi$
$331$	− 3.71327e6i	− 1.86289i	0.363885	−	0.931444i	$-0.381450\pi$
$332$	0	0
$333$	−152138.	−0.0751845
$334$	0	0
$335$	3.00774e6	1.46430
$336$	0	0
$337$	−1.35554e6	−0.650184	−0.325092	−	0.945682i	$-0.605395\pi$
$337$	−1.35554e6	−0.650184	−0.325092	+	0.945682i	$0.605395\pi$
$338$	0	0
$339$	2.15673e6	1.01929
$340$	0	0
$341$	845779.i	0.393886i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−973498.	−0.440339
$346$	0	0
$347$	− 1.03682e6i	− 0.462252i	−0.972924	−	0.231126i	$-0.925759\pi$
$347$	− 1.03682e6i	− 0.462252i	0.972924	−	0.231126i	$-0.0742410\pi$
$348$	0	0
$349$	163154.i	0.0717023i	0.999357	+	0.0358511i	$0.0114142\pi$
$349$	163154.i	0.0717023i	−0.999357	+	0.0358511i	$0.988586\pi$
$350$	0	0
$351$	832389.i	0.360627i
$352$	0	0
$353$	− 1.99564e6i	− 0.852406i	−0.904628	−	0.426203i	$-0.859851\pi$
$353$	− 1.99564e6i	− 0.852406i	0.904628	−	0.426203i	$-0.140149\pi$
$354$	0	0
$355$	−2.40044e6	−1.01093
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.48740e6i	− 0.609106i	−0.952495	−	0.304553i	$-0.901493\pi$
$359$	− 1.48740e6i	− 0.609106i	0.952495	−	0.304553i	$-0.0985071\pi$
$360$	0	0
$361$	−1.57977e6	−0.638007
$362$	0	0
$363$	−15809.9	−0.00629742
$364$	0	0
$365$	205966.	0.0809215
$366$	0	0
$367$	−1.50150e6	−0.581916	−0.290958	−	0.956736i	$-0.593974\pi$
$367$	−1.50150e6	−0.581916	−0.290958	+	0.956736i	$0.593974\pi$
$368$	0	0
$369$	25504.0i	0.00975085i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.04712e6	0.761851	0.380926	−	0.924606i	$-0.375605\pi$
$373$	2.04712e6	0.761851	0.380926	+	0.924606i	$0.375605\pi$
$374$	0	0
$375$	− 3.05099e6i	− 1.12037i
$376$	0	0
$377$	832608.i	0.301708i
$378$	0	0
$379$	685344.i	0.245082i	0.992463	+	0.122541i	$0.0391042\pi$
$379$	685344.i	0.245082i	−0.992463	+	0.122541i	$0.960896\pi$
$380$	0	0
$381$	− 2.21399e6i	− 0.781382i
$382$	0	0
$383$	−1.29125e6	−0.449793	−0.224896	−	0.974383i	$-0.572204\pi$
$383$	−1.29125e6	−0.449793	−0.224896	+	0.974383i	$0.572204\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	257595.i	0.0874299i
$388$	0	0
$389$	−148250.	−0.0496731	−0.0248366	−	0.999692i	$-0.507907\pi$
$389$	−148250.	−0.0496731	−0.0248366	+	0.999692i	$0.507907\pi$
$390$	0	0
$391$	644160.	0.213085
$392$	0	0
$393$	122710.	0.0400773
$394$	0	0
$395$	2.69164e6	0.868009
$396$	0	0
$397$	1.54828e6i	0.493031i	0.969139	+	0.246516i	$0.0792856\pi$
$397$	1.54828e6i	0.493031i	−0.969139	+	0.246516i	$0.920714\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.13978e6	−1.28563	−0.642816	−	0.766021i	$-0.722234\pi$
$401$	−4.13978e6	−1.28563	−0.642816	+	0.766021i	$0.722234\pi$
$402$	0	0
$403$	− 483733.i	− 0.148369i
$404$	0	0
$405$	− 3.06389e6i	− 0.928188i
$406$	0	0
$407$	3.43690e6i	1.02845i
$408$	0	0
$409$	6.33427e6i	1.87236i	0.351525	+	0.936178i	$0.385663\pi$
$409$	6.33427e6i	1.87236i	−0.351525	+	0.936178i	$0.614337\pi$
$410$	0	0
$411$	5.72287e6	1.67113
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.08464e6i	1.44924i
$416$	0	0
$417$	4.98045e6	1.40258
$418$	0	0
$419$	−1.79755e6	−0.500204	−0.250102	−	0.968220i	$-0.580464\pi$
$419$	−1.79755e6	−0.500204	−0.250102	+	0.968220i	$0.580464\pi$
$420$	0	0
$421$	6.47827e6	1.78137	0.890684	−	0.454623i	$-0.150226\pi$
$421$	6.47827e6	1.78137	0.890684	+	0.454623i	$0.150226\pi$
$422$	0	0
$423$	279109.	0.0758443
$424$	0	0
$425$	396072.i	0.106366i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−1.47823e6	−0.387792
$430$	0	0
$431$	3.95051e6i	1.02438i	0.858873	+	0.512189i	$0.171165\pi$
$431$	3.95051e6i	1.02438i	−0.858873	+	0.512189i	$0.828835\pi$
$432$	0	0
$433$	4.25945e6i	1.09178i	0.837858	+	0.545888i	$0.183808\pi$
$433$	4.25945e6i	1.09178i	−0.837858	+	0.545888i	$0.816192\pi$
$434$	0	0
$435$	− 2.85547e6i	− 0.723528i
$436$	0	0
$437$	1.17443e6i	0.294186i
$438$	0	0
$439$	−1.88043e6	−0.465690	−0.232845	−	0.972514i	$-0.574804\pi$
$439$	−1.88043e6	−0.465690	−0.232845	+	0.972514i	$0.574804\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 4.38368e6i	− 1.06128i	−0.847598	−	0.530639i	$-0.821952\pi$
$443$	− 4.38368e6i	− 1.06128i	0.847598	−	0.530639i	$-0.178048\pi$
$444$	0	0
$445$	633829.	0.151730
$446$	0	0
$447$	−1.79747e6	−0.425493
$448$	0	0
$449$	−925156.	−0.216570	−0.108285	−	0.994120i	$-0.534536\pi$
$449$	−925156.	−0.216570	−0.108285	+	0.994120i	$0.534536\pi$
$450$	0	0
$451$	576152.	0.133381
$452$	0	0
$453$	− 174914.i	− 0.0400479i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.16812e6	0.709596	0.354798	−	0.934943i	$-0.384550\pi$
$457$	3.16812e6	0.709596	0.354798	+	0.934943i	$0.384550\pi$
$458$	0	0
$459$	1.88896e6i	0.418496i
$460$	0	0
$461$	− 6.38784e6i	− 1.39991i	−0.714185	−	0.699957i	$-0.753202\pi$
$461$	− 6.38784e6i	− 1.39991i	0.714185	−	0.699957i	$-0.246798\pi$
$462$	0	0
$463$	− 219636.i	− 0.0476159i	−0.999717	−	0.0238079i	$-0.992421\pi$
$463$	− 219636.i	− 0.0476159i	0.999717	−	0.0238079i	$-0.00757902\pi$
$464$	0	0
$465$	1.65899e6i	0.355804i
$466$	0	0
$467$	−576006.	−0.122218	−0.0611089	−	0.998131i	$-0.519464\pi$
$467$	−576006.	−0.122218	−0.0611089	+	0.998131i	$0.519464\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.33825e6i	0.485668i
$472$	0	0
$473$	5.81923e6	1.19595
$474$	0	0
$475$	−722115.	−0.146849
$476$	0	0
$477$	236077.	0.0475071
$478$	0	0
$479$	−3.61839e6	−0.720571	−0.360285	−	0.932842i	$-0.617321\pi$
$479$	−3.61839e6	−0.720571	−0.360285	+	0.932842i	$0.617321\pi$
$480$	0	0
$481$	− 1.96569e6i	− 0.387394i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	612742.	0.118283
$486$	0	0
$487$	4.82652e6i	0.922172i	0.887356	+	0.461086i	$0.152540\pi$
$487$	4.82652e6i	0.922172i	−0.887356	+	0.461086i	$0.847460\pi$
$488$	0	0
$489$	2.46471e6i	0.466116i
$490$	0	0
$491$	− 2.65250e6i	− 0.496537i	−0.968691	−	0.248269i	$-0.920138\pi$
$491$	− 2.65250e6i	− 0.496537i	0.968691	−	0.248269i	$-0.0798615\pi$
$492$	0	0
$493$	1.88946e6i	0.350122i
$494$	0	0
$495$	344390.	0.0631740
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.37551e6i	0.786643i	0.919401	+	0.393321i	$0.128674\pi$
$499$	4.37551e6i	0.786643i	−0.919401	+	0.393321i	$0.871326\pi$
$500$	0	0
$501$	5.61058e6	0.998651
$502$	0	0
$503$	−8.18256e6	−1.44201	−0.721007	−	0.692928i	$-0.756320\pi$
$503$	−8.18256e6	−1.44201	−0.721007	+	0.692928i	$0.756320\pi$
$504$	0	0
$505$	3.99469e6	0.697035
$506$	0	0
$507$	−5.14964e6	−0.889727
$508$	0	0
$509$	− 5.83251e6i	− 0.997841i	−0.866648	−	0.498920i	$-0.833730\pi$
$509$	− 5.83251e6i	− 0.997841i	0.866648	−	0.498920i	$-0.166270\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.44393e6	−0.577778
$514$	0	0
$515$	3.78199e6i	0.628351i
$516$	0	0
$517$	− 6.30525e6i	− 1.03747i
$518$	0	0
$519$	− 1.17703e7i	− 1.91808i
$520$	0	0
$521$	6.46082e6i	1.04278i	0.853318	+	0.521391i	$0.174587\pi$
$521$	6.46082e6i	1.04278i	−0.853318	+	0.521391i	$0.825413\pi$
$522$	0	0
$523$	−1.18045e7	−1.88710	−0.943550	−	0.331231i	$-0.892536\pi$
$523$	−1.18045e7	−1.88710	−0.943550	+	0.331231i	$0.892536\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.09775e6i	− 0.172177i
$528$	0	0
$529$	4.89754e6	0.760920
$530$	0	0
$531$	552006.	0.0849586
$532$	0	0
$533$	−329523.	−0.0502420
$534$	0	0
$535$	3.25718e6	0.491991
$536$	0	0
$537$	− 1.19350e7i	− 1.78602i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	6.01567e6	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$541$	6.01567e6	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$542$	0	0
$543$	− 9.91445e6i	− 1.44301i
$544$	0	0
$545$	− 5.61192e6i	− 0.809320i
$546$	0	0
$547$	385072.i	0.0550268i	0.999621	+	0.0275134i	$0.00875889\pi$
$547$	385072.i	0.0550268i	−0.999621	+	0.0275134i	$0.991241\pi$
$548$	0	0
$549$	617856.i	0.0874896i
$550$	0	0
$551$	−3.44484e6	−0.483381
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.74145e6i	0.929011i
$556$	0	0
$557$	7.74663e6	1.05797	0.528987	−	0.848630i	$-0.322572\pi$
$557$	7.74663e6	1.05797	0.528987	+	0.848630i	$0.322572\pi$
$558$	0	0
$559$	−3.32824e6	−0.450489
$560$	0	0
$561$	−3.35459e6	−0.450020
$562$	0	0
$563$	953532.	0.126784	0.0633920	−	0.997989i	$-0.479808\pi$
$563$	953532.	0.126784	0.0633920	+	0.997989i	$0.479808\pi$
$564$	0	0
$565$	− 6.49205e6i	− 0.855580i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	2.22245e6	0.287773	0.143887	−	0.989594i	$-0.454040\pi$
$569$	2.22245e6	0.287773	0.143887	+	0.989594i	$0.454040\pi$
$570$	0	0
$571$	1.18157e7i	1.51659i	0.651911	+	0.758295i	$0.273967\pi$
$571$	1.18157e7i	1.51659i	−0.651911	+	0.758295i	$0.726033\pi$
$572$	0	0
$573$	− 1.10640e7i	− 1.40774i
$574$	0	0
$575$	− 946157.i	− 0.119342i
$576$	0	0
$577$	7.60972e6i	0.951544i	0.879569	+	0.475772i	$0.157831\pi$
$577$	7.60972e6i	0.951544i	−0.879569	+	0.475772i	$0.842169\pi$
$578$	0	0
$579$	1.42274e7	1.76371
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 5.33314e6i	− 0.649847i
$584$	0	0
$585$	−196970.	−0.0237963
$586$	0	0
$587$	−4.33875e6	−0.519720	−0.259860	−	0.965646i	$-0.583676\pi$
$587$	−4.33875e6	−0.519720	−0.259860	+	0.965646i	$0.583676\pi$
$588$	0	0
$589$	2.00140e6	0.237709
$590$	0	0
$591$	4.97090e6	0.585419
$592$	0	0
$593$	− 1.37445e7i	− 1.60506i	−0.596612	−	0.802530i	$-0.703487\pi$
$593$	− 1.37445e7i	− 1.60506i	0.596612	−	0.802530i	$-0.296513\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.33431e7	1.53222
$598$	0	0
$599$	5.55539e6i	0.632627i	0.948655	+	0.316314i	$0.102445\pi$
$599$	5.55539e6i	0.632627i	−0.948655	+	0.316314i	$0.897555\pi$
$600$	0	0
$601$	5.23135e6i	0.590782i	0.955376	+	0.295391i	$0.0954499\pi$
$601$	5.23135e6i	0.590782i	−0.955376	+	0.295391i	$0.904550\pi$
$602$	0	0
$603$	− 1.09599e6i	− 0.122747i
$604$	0	0
$605$	47589.9i	0.00528599i
$606$	0	0
$607$	3.82822e6	0.421720	0.210860	−	0.977516i	$-0.432374\pi$
$607$	3.82822e6	0.421720	0.210860	+	0.977516i	$0.432374\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.60621e6i	0.390794i
$612$	0	0
$613$	5.38387e6	0.578687	0.289343	−	0.957225i	$-0.406563\pi$
$613$	5.38387e6	0.578687	0.289343	+	0.957225i	$0.406563\pi$
$614$	0	0
$615$	1.13012e6	0.120486
$616$	0	0
$617$	−5.05258e6	−0.534319	−0.267159	−	0.963652i	$-0.586085\pi$
$617$	−5.05258e6	−0.534319	−0.267159	+	0.963652i	$0.586085\pi$
$618$	0	0
$619$	−6.36785e6	−0.667984	−0.333992	−	0.942576i	$-0.608396\pi$
$619$	−6.36785e6	−0.667984	−0.333992	+	0.942576i	$0.608396\pi$
$620$	0	0
$621$	− 4.51244e6i	− 0.469551i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−6.80033e6	−0.696354
$626$	0	0
$627$	− 6.11605e6i	− 0.621301i
$628$	0	0
$629$	− 4.46079e6i	− 0.449558i
$630$	0	0
$631$	1.69140e7i	1.69111i	0.533884	+	0.845557i	$0.320732\pi$
$631$	1.69140e7i	1.69111i	−0.533884	+	0.845557i	$0.679268\pi$
$632$	0	0
$633$	1.20106e7i	1.19140i
$634$	0	0
$635$	−6.66441e6	−0.655884
$636$	0	0
$637$	0	0
$638$	0	0
$639$	874694.i	0.0847430i
$640$	0	0
$641$	1.07281e7	1.03128	0.515641	−	0.856805i	$-0.327554\pi$
$641$	1.07281e7	1.03128	0.515641	+	0.856805i	$0.327554\pi$
$642$	0	0
$643$	−651819.	−0.0621727	−0.0310864	−	0.999517i	$-0.509897\pi$
$643$	−651819.	−0.0621727	−0.0310864	+	0.999517i	$0.509897\pi$
$644$	0	0
$645$	1.14144e7	1.08032
$646$	0	0
$647$	861222.	0.0808825	0.0404412	−	0.999182i	$-0.487124\pi$
$647$	861222.	0.0808825	0.0404412	+	0.999182i	$0.487124\pi$
$648$	0	0
$649$	− 1.24702e7i	− 1.16215i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.19553e6	0.476811	0.238406	−	0.971166i	$-0.423375\pi$
$653$	5.19553e6	0.476811	0.238406	+	0.971166i	$0.423375\pi$
$654$	0	0
$655$	− 369374.i	− 0.0336405i
$656$	0	0
$657$	− 75051.7i	− 0.00678340i
$658$	0	0
$659$	1.50166e7i	1.34697i	0.739201	+	0.673485i	$0.235204\pi$
$659$	1.50166e7i	1.34697i	−0.739201	+	0.673485i	$0.764796\pi$
$660$	0	0
$661$	9.09393e6i	0.809558i	0.914414	+	0.404779i	$0.132652\pi$
$661$	9.09393e6i	0.809558i	−0.914414	+	0.404779i	$0.867348\pi$
$662$	0	0
$663$	1.91861e6	0.169513
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.51363e6i	− 0.392836i
$668$	0	0
$669$	1.81233e7	1.56557
$670$	0	0
$671$	1.39578e7	1.19677
$672$	0	0
$673$	3.36768e6	0.286611	0.143306	−	0.989678i	$-0.454227\pi$
$673$	3.36768e6	0.286611	0.143306	+	0.989678i	$0.454227\pi$
$674$	0	0
$675$	2.77455e6	0.234387
$676$	0	0
$677$	1.03541e7i	0.868244i	0.900854	+	0.434122i	$0.142941\pi$
$677$	1.03541e7i	0.868244i	−0.900854	+	0.434122i	$0.857059\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.14535e6	−0.673041
$682$	0	0
$683$	− 1.58292e7i	− 1.29839i	−0.760621	−	0.649196i	$-0.775105\pi$
$683$	− 1.58292e7i	− 1.29839i	0.760621	−	0.649196i	$-0.224895\pi$
$684$	0	0
$685$	− 1.72266e7i	− 1.40273i
$686$	0	0
$687$	− 2.17844e7i	− 1.76098i
$688$	0	0
$689$	3.05022e6i	0.244784i
$690$	0	0
$691$	2.66927e6	0.212666	0.106333	−	0.994331i	$-0.466089\pi$
$691$	2.66927e6	0.212666	0.106333	+	0.994331i	$0.466089\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.49918e7i	− 1.17731i
$696$	0	0
$697$	−747793.	−0.0583042
$698$	0	0
$699$	1.40227e7	1.08553
$700$	0	0
$701$	−3.67634e6	−0.282566	−0.141283	−	0.989969i	$-0.545123\pi$
$701$	−3.67634e6	−0.282566	−0.141283	+	0.989969i	$0.545123\pi$
$702$	0	0
$703$	8.13287e6	0.620663
$704$	0	0
$705$	− 1.23677e7i	− 0.937165i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	1.02821e7	0.768184	0.384092	−	0.923295i	$-0.374515\pi$
$709$	1.02821e7	0.768184	0.384092	+	0.923295i	$0.374515\pi$
$710$	0	0
$711$	− 980803.i	− 0.0727625i
$712$	0	0
$713$	2.62235e6i	0.193182i
$714$	0	0
$715$	4.44967e6i	0.325509i
$716$	0	0
$717$	− 8.92643e6i	− 0.648455i
$718$	0	0
$719$	−1.72299e6	−0.124297	−0.0621484	−	0.998067i	$-0.519795\pi$
$719$	−1.72299e6	−0.124297	−0.0621484	+	0.998067i	$0.519795\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.21238e6i	0.513136i
$724$	0	0
$725$	2.77528e6	0.196093
$726$	0	0
$727$	1.13322e7	0.795203	0.397601	−	0.917558i	$-0.369843\pi$
$727$	1.13322e7	0.795203	0.397601	+	0.917558i	$0.369843\pi$
$728$	0	0
$729$	1.31562e7	0.916881
$730$	0	0
$731$	−7.55284e6	−0.522778
$732$	0	0
$733$	1.26858e7i	0.872086i	0.899926	+	0.436043i	$0.143620\pi$
$733$	1.26858e7i	0.872086i	−0.899926	+	0.436043i	$0.856380\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2.47590e7	−1.67906
$738$	0	0
$739$	− 2.76738e7i	− 1.86405i	−0.362395	−	0.932024i	$-0.618041\pi$
$739$	− 2.76738e7i	− 1.86405i	0.362395	−	0.932024i	$-0.381959\pi$
$740$	0	0
$741$	3.49800e6i	0.234031i
$742$	0	0
$743$	1.06315e7i	0.706516i	0.935526	+	0.353258i	$0.114926\pi$
$743$	1.06315e7i	0.706516i	−0.935526	+	0.353258i	$0.885074\pi$
$744$	0	0
$745$	5.41062e6i	0.357155i
$746$	0	0
$747$	1.85278e6	0.121485
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 2.17119e7i	− 1.40475i	−0.711809	−	0.702373i	$-0.752124\pi$
$751$	− 2.17119e7i	− 1.40475i	0.711809	−	0.702373i	$-0.247876\pi$
$752$	0	0
$753$	1.34969e6	0.0867456
$754$	0	0
$755$	−526514.	−0.0336158
$756$	0	0
$757$	7.18252e6	0.455551	0.227776	−	0.973714i	$-0.426855\pi$
$757$	7.18252e6	0.455551	0.227776	+	0.973714i	$0.426855\pi$
$758$	0	0
$759$	8.01361e6	0.504921
$760$	0	0
$761$	6.84692e6i	0.428582i	0.976770	+	0.214291i	$0.0687440\pi$
$761$	6.84692e6i	0.428582i	−0.976770	+	0.214291i	$0.931256\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−446988.	−0.0276148
$766$	0	0
$767$	7.13215e6i	0.437756i
$768$	0	0
$769$	4.71388e6i	0.287450i	0.989618	+	0.143725i	$0.0459081\pi$
$769$	4.71388e6i	0.287450i	−0.989618	+	0.143725i	$0.954092\pi$
$770$	0	0
$771$	− 7.95972e6i	− 0.482239i
$772$	0	0
$773$	1.36324e7i	0.820588i	0.911953	+	0.410294i	$0.134574\pi$
$773$	1.36324e7i	0.820588i	−0.911953	+	0.410294i	$0.865426\pi$
$774$	0	0
$775$	−1.61239e6	−0.0964311
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.36337e6i	− 0.0804952i
$780$	0	0
$781$	1.97599e7	1.15920
$782$	0	0
$783$	1.32359e7	0.771526
$784$	0	0
$785$	7.03845e6	0.407664
$786$	0	0
$787$	−2.20492e7	−1.26899	−0.634493	−	0.772928i	$-0.718791\pi$
$787$	−2.20492e7	−1.26899	−0.634493	+	0.772928i	$0.718791\pi$
$788$	0	0
$789$	1.79425e7i	1.02610i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−7.98296e6	−0.450797
$794$	0	0
$795$	− 1.04609e7i	− 0.587018i
$796$	0	0
$797$	− 1.07895e6i	− 0.0601664i	−0.999547	−	0.0300832i	$-0.990423\pi$
$797$	− 1.07895e6i	− 0.0601664i	0.999547	−	0.0300832i	$-0.00957723\pi$
$798$	0	0
$799$	8.18365e6i	0.453503i
$800$	0	0
$801$	− 230960.i	− 0.0127191i
$802$	0	0
$803$	−1.69547e6	−0.0927898
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.39869e7i	1.29656i
$808$	0	0
$809$	−1.69245e7	−0.909167	−0.454584	−	0.890704i	$-0.650212\pi$
$809$	−1.69245e7	−0.909167	−0.454584	+	0.890704i	$0.650212\pi$
$810$	0	0
$811$	−6.70607e6	−0.358027	−0.179014	−	0.983847i	$-0.557291\pi$
$811$	−6.70607e6	−0.358027	−0.179014	+	0.983847i	$0.557291\pi$
$812$	0	0
$813$	−6.93714e6	−0.368090
$814$	0	0
$815$	7.41911e6	0.391253
$816$	0	0
$817$	− 1.37703e7i	− 0.721751i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.45166e7	1.78719	0.893593	−	0.448878i	$-0.148176\pi$
$821$	3.45166e7	1.78719	0.893593	+	0.448878i	$0.148176\pi$
$822$	0	0
$823$	2.66931e7i	1.37373i	0.726787	+	0.686863i	$0.241013\pi$
$823$	2.66931e7i	1.37373i	−0.726787	+	0.686863i	$0.758987\pi$
$824$	0	0
$825$	4.92730e6i	0.252042i
$826$	0	0
$827$	3.60205e7i	1.83141i	0.401849	+	0.915706i	$0.368368\pi$
$827$	3.60205e7i	1.83141i	−0.401849	+	0.915706i	$0.631632\pi$
$828$	0	0
$829$	− 3.44066e7i	− 1.73882i	−0.494088	−	0.869412i	$-0.664498\pi$
$829$	− 3.44066e7i	− 1.73882i	0.494088	−	0.869412i	$-0.335502\pi$
$830$	0	0
$831$	4.01433e7	2.01655
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 1.68886e7i	− 0.838257i
$836$	0	0
$837$	−7.68988e6	−0.379408
$838$	0	0
$839$	8.86948e6	0.435004	0.217502	−	0.976060i	$-0.430209\pi$
$839$	8.86948e6	0.435004	0.217502	+	0.976060i	$0.430209\pi$
$840$	0	0
$841$	−7.27172e6	−0.354525
$842$	0	0
$843$	1.44648e7	0.701039
$844$	0	0
$845$	1.55011e7i	0.746828i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−1.71642e7	−0.817249
$850$	0	0
$851$	1.06562e7i	0.504402i
$852$	0	0
$853$	1.01426e7i	0.477284i	0.971108	+	0.238642i	$0.0767022\pi$
$853$	1.01426e7i	0.477284i	−0.971108	+	0.238642i	$0.923298\pi$
$854$	0	0
$855$	− 814944.i	− 0.0381252i
$856$	0	0
$857$	2.80748e7i	1.30576i	0.757460	+	0.652881i	$0.226440\pi$
$857$	2.80748e7i	1.30576i	−0.757460	+	0.652881i	$0.773560\pi$
$858$	0	0
$859$	−2.41550e7	−1.11693	−0.558463	−	0.829529i	$-0.688609\pi$
$859$	−2.41550e7	−1.11693	−0.558463	+	0.829529i	$0.688609\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.41819e6i	0.156232i	0.996944	+	0.0781158i	$0.0248904\pi$
$863$	3.41819e6i	0.156232i	−0.996944	+	0.0781158i	$0.975110\pi$
$864$	0	0
$865$	−3.54300e7	−1.61002
$866$	0	0
$867$	−1.85718e7	−0.839086
$868$	0	0
$869$	−2.21570e7	−0.995315
$870$	0	0
$871$	1.41606e7	0.632465
$872$	0	0
$873$	− 223276.i	− 0.00991532i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.80560e6	0.254887	0.127444	−	0.991846i	$-0.459323\pi$
$877$	5.80560e6	0.254887	0.127444	+	0.991846i	$0.459323\pi$
$878$	0	0
$879$	− 4.61937e6i	− 0.201656i
$880$	0	0
$881$	1.08943e7i	0.472887i	0.971645	+	0.236444i	$0.0759819\pi$
$881$	1.08943e7i	0.472887i	−0.971645	+	0.236444i	$0.924018\pi$
$882$	0	0
$883$	− 2.50812e7i	− 1.08255i	−0.840846	−	0.541274i	$-0.817942\pi$
$883$	− 2.50812e7i	− 1.08255i	0.840846	−	0.541274i	$-0.182058\pi$
$884$	0	0
$885$	− 2.44601e7i	− 1.04978i
$886$	0	0
$887$	−3.08964e7	−1.31856	−0.659278	−	0.751899i	$-0.729138\pi$
$887$	−3.08964e7	−1.31856	−0.659278	+	0.751899i	$0.729138\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.52213e7i	1.06432i
$892$	0	0
$893$	−1.49204e7	−0.626110
$894$	0	0
$895$	−3.59258e7	−1.49916
$896$	0	0
$897$	−4.58328e6	−0.190193
$898$	0	0
$899$	−7.69191e6	−0.317420
$900$	0	0
$901$	6.92193e6i	0.284064i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.98438e7	−1.21125
$906$	0	0
$907$	− 1.37304e6i	− 0.0554198i	−0.999616	−	0.0277099i	$-0.991179\pi$
$907$	− 1.37304e6i	− 0.0554198i	0.999616	−	0.0277099i	$-0.00882147\pi$
$908$	0	0
$909$	− 1.45562e6i	− 0.0584303i
$910$	0	0
$911$	− 4.47920e7i	− 1.78815i	−0.447916	−	0.894075i	$-0.647834\pi$
$911$	− 4.47920e7i	− 1.78815i	0.447916	−	0.894075i	$-0.352166\pi$
$912$	0	0
$913$	− 4.18555e7i	− 1.66179i
$914$	0	0
$915$	2.73780e7	1.08106
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 1.68926e7i	− 0.659792i	−0.944017	−	0.329896i	$-0.892986\pi$
$919$	− 1.68926e7i	− 0.659792i	0.944017	−	0.329896i	$-0.107014\pi$
$920$	0	0
$921$	−4.11532e7	−1.59865
$922$	0	0
$923$	−1.13014e7	−0.436645
$924$	0	0
$925$	−6.55212e6	−0.251783
$926$	0	0
$927$	1.37811e6	0.0526727
$928$	0	0
$929$	− 1.32849e6i	− 0.0505034i	−0.999681	−	0.0252517i	$-0.991961\pi$
$929$	− 1.32849e6i	− 0.0505034i	0.999681	−	0.0252517i	$-0.00803871\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.45790e7	1.67659
$934$	0	0
$935$	1.00977e7i	0.377742i
$936$	0	0
$937$	3.01493e7i	1.12183i	0.827872	+	0.560917i	$0.189551\pi$
$937$	3.01493e7i	1.12183i	−0.827872	+	0.560917i	$0.810449\pi$
$938$	0	0
$939$	4.40496e6i	0.163034i
$940$	0	0
$941$	− 2.28510e7i	− 0.841263i	−0.907231	−	0.420632i	$-0.861808\pi$
$941$	− 2.28510e7i	− 0.841263i	0.907231	−	0.420632i	$-0.138192\pi$
$942$	0	0
$943$	1.78637e6	0.0654171
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.33819e7i	1.20958i	0.796384	+	0.604792i	$0.206744\pi$
$947$	3.33819e7i	1.20958i	−0.796384	+	0.604792i	$0.793256\pi$
$948$	0	0
$949$	969700.	0.0349520
$950$	0	0
$951$	−3.33570e6	−0.119601
$952$	0	0
$953$	−2.52665e7	−0.901183	−0.450591	−	0.892730i	$-0.648787\pi$
$953$	−2.52665e7	−0.901183	−0.450591	+	0.892730i	$0.648787\pi$
$954$	0	0
$955$	−3.33039e7	−1.18165
$956$	0	0
$957$	2.35056e7i	0.829643i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.41603e7	−0.843904
$962$	0	0
$963$	− 1.18688e6i	− 0.0412421i
$964$	0	0
$965$	− 4.28262e7i	− 1.48044i
$966$	0	0
$967$	− 2.20016e6i	− 0.0756636i	−0.999284	−	0.0378318i	$-0.987955\pi$
$967$	− 2.20016e6i	− 0.0756636i	0.999284	−	0.0378318i	$-0.0120451\pi$
$968$	0	0
$969$	7.93808e6i	0.271585i
$970$	0	0
$971$	−4.05329e7	−1.37962	−0.689810	−	0.723991i	$-0.742306\pi$
$971$	−4.05329e7	−1.37962	−0.689810	+	0.723991i	$0.742306\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 2.81810e6i	− 0.0949392i
$976$	0	0
$977$	−2.72584e7	−0.913617	−0.456808	−	0.889565i	$-0.651008\pi$
$977$	−2.72584e7	−0.913617	−0.456808	+	0.889565i	$0.651008\pi$
$978$	0	0
$979$	−5.21753e6	−0.173983
$980$	0	0
$981$	−2.04492e6	−0.0678428
$982$	0	0
$983$	−2.06387e7	−0.681236	−0.340618	−	0.940202i	$-0.610636\pi$
$983$	−2.06387e7	−0.681236	−0.340618	+	0.940202i	$0.610636\pi$
$984$	0	0
$985$	− 1.49631e7i	− 0.491394i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.80426e7	0.586555
$990$	0	0
$991$	1.59478e7i	0.515841i	0.966166	+	0.257920i	$0.0830372\pi$
$991$	1.59478e7i	0.515841i	−0.966166	+	0.257920i	$0.916963\pi$
$992$	0	0
$993$	5.99564e7i	1.92958i
$994$	0	0
$995$	− 4.01644e7i	− 1.28613i
$996$	0	0
$997$	− 4.70254e7i	− 1.49829i	−0.662408	−	0.749143i	$-0.730466\pi$
$997$	− 4.70254e7i	− 1.49829i	0.662408	−	0.749143i	$-0.269534\pi$
$998$	0	0
$999$	−3.12486e7	−0.990641

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.f.e.783.6	yes	16
4.3	odd	2	inner	784.6.f.e.783.12	yes	16
7.6	odd	2	inner	784.6.f.e.783.11	yes	16
28.27	even	2	inner	784.6.f.e.783.5	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
784.6.f.e.783.5	✓	16	28.27	even	2	inner
784.6.f.e.783.6	yes	16	1.1	even	1	trivial
784.6.f.e.783.11	yes	16	7.6	odd	2	inner
784.6.f.e.783.12	yes	16	4.3	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.1465 q^{3} +48.6032i q^{5} +17.7104 q^{9} -400.090i q^{11} +228.826i q^{13} -784.773i q^{15} +519.281i q^{17} -946.748 q^{19} -1240.48i q^{23} +762.732 q^{25} +3637.64 q^{27} +3638.60 q^{29} -2113.97 q^{31} +6460.06i q^{33} -8590.33 q^{37} -3694.75i q^{39} +1440.06i q^{41} +14544.8i q^{43} +860.783i q^{45} +15759.6 q^{47} -8384.58i q^{51} +13329.8 q^{53} +19445.6 q^{55} +15286.7 q^{57} +31168.4 q^{59} +34886.6i q^{61} -11121.7 q^{65} -61883.7i q^{67} +20029.5i q^{69} +49388.6i q^{71} -4237.71i q^{73} -12315.5 q^{75} -55380.0i q^{79} -63039.0 q^{81} +104615. q^{83} -25238.7 q^{85} -58750.8 q^{87} -13040.9i q^{89} +34133.3 q^{93} -46014.9i q^{95} -12607.0i q^{97} -7085.76i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 1712 q^{9} - 496 q^{25} - 17024 q^{29} + 85664 q^{53} - 95424 q^{57} + 233344 q^{65} - 511216 q^{81} + 697952 q^{85} - 857024 q^{93}+O(q^{100}) \) 16 * q + 1712 * q^9 - 496 * q^25 - 17024 * q^29 + 85664 * q^53 - 95424 * q^57 + 233344 * q^65 - 511216 * q^81 + 697952 * q^85 - 857024 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more