LMFDB - Embedded newform 784.6.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,6,Mod(1,784)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(784, 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("784.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	784.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:	$125.740914733$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 167x^{3} - 387x^{2} + 1720x + 2340 $ x^5 - 167x^3 - 387x^2 + 1720*x + 2340
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 7 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.45227$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-18.5692 q^{3} -57.9194 q^{5} +101.814 q^{9} +O(q^{10})$	$q-18.5692 q^{3} -57.9194 q^{5} +101.814 q^{9} -271.498 q^{11} +25.9805 q^{13} +1075.51 q^{15} +65.8357 q^{17} +1604.36 q^{19} -3825.98 q^{23} +229.658 q^{25} +2621.71 q^{27} -4459.32 q^{29} -8913.17 q^{31} +5041.48 q^{33} -14974.3 q^{37} -482.437 q^{39} -12759.0 q^{41} +3067.65 q^{43} -5896.98 q^{45} +16103.1 q^{47} -1222.51 q^{51} -31531.2 q^{53} +15725.0 q^{55} -29791.5 q^{57} +10786.9 q^{59} -12600.5 q^{61} -1504.78 q^{65} -27354.6 q^{67} +71045.2 q^{69} +6926.50 q^{71} -60678.0 q^{73} -4264.55 q^{75} -86073.7 q^{79} -73423.7 q^{81} +61029.0 q^{83} -3813.16 q^{85} +82805.8 q^{87} -44637.3 q^{89} +165510. q^{93} -92923.3 q^{95} -99259.1 q^{97} -27642.2 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) $ 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9	$ 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39} - 5310 q^{41} - 764 q^{43} - 38978 q^{45} + 6675 q^{47} - 20975 q^{51} - 30753 q^{53} + 28267 q^{55} - 14389 q^{57} + 87989 q^{59} - 19899 q^{61} + 119470 q^{65} + 33067 q^{67} - 100399 q^{69} + 108720 q^{71} - 141659 q^{73} + 108788 q^{75} - 118919 q^{79} - 143851 q^{81} + 211004 q^{83} - 143379 q^{85} + 302154 q^{87} + 55861 q^{89} + 410381 q^{93} + 26279 q^{95} - 135470 q^{97} + 300154 q^{99}+O(q^{100}) $ 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * 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
$3$	−18.5692	−1.19121	−0.595606	−	0.803277i	$-0.703088\pi$
$3$	−18.5692	−1.19121	−0.595606	+	0.803277i	$0.703088\pi$
$4$	0	0
$5$	−57.9194	−1.03609	−0.518047	−	0.855352i	$-0.673341\pi$
$5$	−57.9194	−1.03609	−0.518047	+	0.855352i	$0.673341\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	101.814	0.418986
$10$	0	0
$11$	−271.498	−0.676526	−0.338263	−	0.941052i	$-0.609839\pi$
$11$	−271.498	−0.676526	−0.338263	+	0.941052i	$0.609839\pi$
$12$	0	0
$13$	25.9805	0.0426373	0.0213187	−	0.999773i	$-0.493214\pi$
$13$	25.9805	0.0426373	0.0213187	+	0.999773i	$0.493214\pi$
$14$	0	0
$15$	1075.51	1.23421
$16$	0	0
$17$	65.8357	0.0552509	0.0276254	−	0.999618i	$-0.491205\pi$
$17$	65.8357	0.0552509	0.0276254	+	0.999618i	$0.491205\pi$
$18$	0	0
$19$	1604.36	1.01957	0.509785	−	0.860302i	$-0.329725\pi$
$19$	1604.36	1.01957	0.509785	+	0.860302i	$0.329725\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3825.98	−1.50807	−0.754037	−	0.656832i	$-0.771896\pi$
$23$	−3825.98	−1.50807	−0.754037	+	0.656832i	$0.771896\pi$
$24$	0	0
$25$	229.658	0.0734905
$26$	0	0
$27$	2621.71	0.692111
$28$	0	0
$29$	−4459.32	−0.984631	−0.492315	−	0.870417i	$-0.663849\pi$
$29$	−4459.32	−0.984631	−0.492315	+	0.870417i	$0.663849\pi$
$30$	0	0
$31$	−8913.17	−1.66582	−0.832910	−	0.553409i	$-0.813327\pi$
$31$	−8913.17	−1.66582	−0.832910	+	0.553409i	$0.813327\pi$
$32$	0	0
$33$	5041.48	0.805886
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−14974.3	−1.79822	−0.899112	−	0.437720i	$-0.855786\pi$
$37$	−14974.3	−1.79822	−0.899112	+	0.437720i	$0.855786\pi$
$38$	0	0
$39$	−482.437	−0.0507901
$40$	0	0
$41$	−12759.0	−1.18538	−0.592689	−	0.805431i	$-0.701934\pi$
$41$	−12759.0	−1.18538	−0.592689	+	0.805431i	$0.701934\pi$
$42$	0	0
$43$	3067.65	0.253008	0.126504	−	0.991966i	$-0.459624\pi$
$43$	3067.65	0.253008	0.126504	+	0.991966i	$0.459624\pi$
$44$	0	0
$45$	−5896.98	−0.434109
$46$	0	0
$47$	16103.1	1.06332	0.531661	−	0.846957i	$-0.321568\pi$
$47$	16103.1	1.06332	0.531661	+	0.846957i	$0.321568\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−1222.51	−0.0658155
$52$	0	0
$53$	−31531.2	−1.54188	−0.770941	−	0.636907i	$-0.780214\pi$
$53$	−31531.2	−1.54188	−0.770941	+	0.636907i	$0.780214\pi$
$54$	0	0
$55$	15725.0	0.700944
$56$	0	0
$57$	−29791.5	−1.21452
$58$	0	0
$59$	10786.9	0.403429	0.201714	−	0.979444i	$-0.435349\pi$
$59$	10786.9	0.403429	0.201714	+	0.979444i	$0.435349\pi$
$60$	0	0
$61$	−12600.5	−0.433574	−0.216787	−	0.976219i	$-0.569558\pi$
$61$	−12600.5	−0.433574	−0.216787	+	0.976219i	$0.569558\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1504.78	−0.0441763
$66$	0	0
$67$	−27354.6	−0.744463	−0.372231	−	0.928140i	$-0.621407\pi$
$67$	−27354.6	−0.744463	−0.372231	+	0.928140i	$0.621407\pi$
$68$	0	0
$69$	71045.2	1.79644
$70$	0	0
$71$	6926.50	0.163068	0.0815338	−	0.996671i	$-0.474018\pi$
$71$	6926.50	0.163068	0.0815338	+	0.996671i	$0.474018\pi$
$72$	0	0
$73$	−60678.0	−1.33268	−0.666338	−	0.745650i	$-0.732139\pi$
$73$	−60678.0	−1.33268	−0.666338	+	0.745650i	$0.732139\pi$
$74$	0	0
$75$	−4264.55	−0.0875428
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−86073.7	−1.55168	−0.775841	−	0.630928i	$-0.782674\pi$
$79$	−86073.7	−1.55168	−0.775841	+	0.630928i	$0.782674\pi$
$80$	0	0
$81$	−73423.7	−1.24344
$82$	0	0
$83$	61029.0	0.972391	0.486196	−	0.873850i	$-0.338384\pi$
$83$	61029.0	0.972391	0.486196	+	0.873850i	$0.338384\pi$
$84$	0	0
$85$	−3813.16	−0.0572451
$86$	0	0
$87$	82805.8	1.17290
$88$	0	0
$89$	−44637.3	−0.597342	−0.298671	−	0.954356i	$-0.596543\pi$
$89$	−44637.3	−0.597342	−0.298671	+	0.954356i	$0.596543\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	165510.	1.98434
$94$	0	0
$95$	−92923.3	−1.05637
$96$	0	0
$97$	−99259.1	−1.07113	−0.535563	−	0.844495i	$-0.679901\pi$
$97$	−99259.1	−1.07113	−0.535563	+	0.844495i	$0.679901\pi$
$98$	0	0
$99$	−27642.2	−0.283455
$100$	0	0
$101$	−103359.	−1.00819	−0.504096	−	0.863648i	$-0.668174\pi$
$101$	−103359.	−1.00819	−0.504096	+	0.863648i	$0.668174\pi$
$102$	0	0
$103$	−189430.	−1.75936	−0.879681	−	0.475564i	$-0.842244\pi$
$103$	−189430.	−1.75936	−0.879681	+	0.475564i	$0.842244\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−135274.	−1.14223	−0.571115	−	0.820870i	$-0.693489\pi$
$107$	−135274.	−1.14223	−0.571115	+	0.820870i	$0.693489\pi$
$108$	0	0
$109$	27649.6	0.222907	0.111453	−	0.993770i	$-0.464449\pi$
$109$	27649.6	0.222907	0.111453	+	0.993770i	$0.464449\pi$
$110$	0	0
$111$	278061.	2.14206
$112$	0	0
$113$	208553.	1.53646	0.768228	−	0.640176i	$-0.221138\pi$
$113$	208553.	1.53646	0.768228	+	0.640176i	$0.221138\pi$
$114$	0	0
$115$	221598.	1.56251
$116$	0	0
$117$	2645.17	0.0178644
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−87340.0	−0.542313
$122$	0	0
$123$	236924.	1.41204
$124$	0	0
$125$	167697.	0.959951
$126$	0	0
$127$	100956.	0.555420	0.277710	−	0.960665i	$-0.410425\pi$
$127$	100956.	0.555420	0.277710	+	0.960665i	$0.410425\pi$
$128$	0	0
$129$	−56963.6	−0.301386
$130$	0	0
$131$	309467.	1.57556	0.787781	−	0.615955i	$-0.211230\pi$
$131$	309467.	1.57556	0.787781	+	0.615955i	$0.211230\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−151848.	−0.717092
$136$	0	0
$137$	−59726.4	−0.271872	−0.135936	−	0.990718i	$-0.543404\pi$
$137$	−59726.4	−0.271872	−0.135936	+	0.990718i	$0.543404\pi$
$138$	0	0
$139$	80588.0	0.353780	0.176890	−	0.984231i	$-0.443396\pi$
$139$	80588.0	0.353780	0.176890	+	0.984231i	$0.443396\pi$
$140$	0	0
$141$	−299021.	−1.26664
$142$	0	0
$143$	−7053.66	−0.0288452
$144$	0	0
$145$	258281.	1.02017
$146$	0	0
$147$	0	0
$148$	0	0
$149$	414281.	1.52872	0.764362	−	0.644787i	$-0.223054\pi$
$149$	414281.	1.52872	0.764362	+	0.644787i	$0.223054\pi$
$150$	0	0
$151$	−28158.3	−0.100499	−0.0502497	−	0.998737i	$-0.516002\pi$
$151$	−28158.3	−0.100499	−0.0502497	+	0.998737i	$0.516002\pi$
$152$	0	0
$153$	6702.97	0.0231493
$154$	0	0
$155$	516246.	1.72595
$156$	0	0
$157$	−398385.	−1.28989	−0.644947	−	0.764228i	$-0.723120\pi$
$157$	−398385.	−1.28989	−0.644947	+	0.764228i	$0.723120\pi$
$158$	0	0
$159$	585508.	1.83671
$160$	0	0
$161$	0	0
$162$	0	0
$163$	518486.	1.52851	0.764254	−	0.644915i	$-0.223107\pi$
$163$	518486.	1.52851	0.764254	+	0.644915i	$0.223107\pi$
$164$	0	0
$165$	−292000.	−0.834973
$166$	0	0
$167$	42242.7	0.117209	0.0586044	−	0.998281i	$-0.481335\pi$
$167$	42242.7	0.117209	0.0586044	+	0.998281i	$0.481335\pi$
$168$	0	0
$169$	−370618.	−0.998182
$170$	0	0
$171$	163345.	0.427185
$172$	0	0
$173$	−180744.	−0.459143	−0.229571	−	0.973292i	$-0.573732\pi$
$173$	−180744.	−0.459143	−0.229571	+	0.973292i	$0.573732\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−200304.	−0.480569
$178$	0	0
$179$	−158047.	−0.368683	−0.184342	−	0.982862i	$-0.559015\pi$
$179$	−158047.	−0.368683	−0.184342	+	0.982862i	$0.559015\pi$
$180$	0	0
$181$	−174668.	−0.396294	−0.198147	−	0.980172i	$-0.563492\pi$
$181$	−174668.	−0.396294	−0.198147	+	0.980172i	$0.563492\pi$
$182$	0	0
$183$	233981.	0.516479
$184$	0	0
$185$	867305.	1.86313
$186$	0	0
$187$	−17874.2	−0.0373786
$188$	0	0
$189$	0	0
$190$	0	0
$191$	84176.1	0.166957	0.0834786	−	0.996510i	$-0.473397\pi$
$191$	84176.1	0.166957	0.0834786	+	0.996510i	$0.473397\pi$
$192$	0	0
$193$	−314781.	−0.608296	−0.304148	−	0.952625i	$-0.598372\pi$
$193$	−314781.	−0.608296	−0.304148	+	0.952625i	$0.598372\pi$
$194$	0	0
$195$	27942.4	0.0526233
$196$	0	0
$197$	700611.	1.28621	0.643104	−	0.765779i	$-0.277646\pi$
$197$	700611.	1.28621	0.643104	+	0.765779i	$0.277646\pi$
$198$	0	0
$199$	−216211.	−0.387030	−0.193515	−	0.981097i	$-0.561989\pi$
$199$	−216211.	−0.387030	−0.193515	+	0.981097i	$0.561989\pi$
$200$	0	0
$201$	507951.	0.886813
$202$	0	0
$203$	0	0
$204$	0	0
$205$	738994.	1.22816
$206$	0	0
$207$	−389537.	−0.631862
$208$	0	0
$209$	−435579.	−0.689765
$210$	0	0
$211$	−525296.	−0.812265	−0.406132	−	0.913814i	$-0.633123\pi$
$211$	−525296.	−0.812265	−0.406132	+	0.913814i	$0.633123\pi$
$212$	0	0
$213$	−128619.	−0.194248
$214$	0	0
$215$	−177676.	−0.262140
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.12674e6	1.58750
$220$	0	0
$221$	1710.45	0.00235575
$222$	0	0
$223$	1.06767e6	1.43772	0.718858	−	0.695157i	$-0.244665\pi$
$223$	1.06767e6	1.43772	0.718858	+	0.695157i	$0.244665\pi$
$224$	0	0
$225$	23382.3	0.0307915
$226$	0	0
$227$	442374.	0.569803	0.284902	−	0.958557i	$-0.408039\pi$
$227$	442374.	0.569803	0.284902	+	0.958557i	$0.408039\pi$
$228$	0	0
$229$	987296.	1.24411	0.622055	−	0.782974i	$-0.286298\pi$
$229$	987296.	1.24411	0.622055	+	0.782974i	$0.286298\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.07559e6	1.29795	0.648974	−	0.760810i	$-0.275198\pi$
$233$	1.07559e6	1.29795	0.648974	+	0.760810i	$0.275198\pi$
$234$	0	0
$235$	−932681.	−1.10170
$236$	0	0
$237$	1.59832e6	1.84838
$238$	0	0
$239$	−659780.	−0.747144	−0.373572	−	0.927601i	$-0.621867\pi$
$239$	−659780.	−0.747144	−0.373572	+	0.927601i	$0.621867\pi$
$240$	0	0
$241$	737858.	0.818333	0.409167	−	0.912460i	$-0.365820\pi$
$241$	737858.	0.818333	0.409167	+	0.912460i	$0.365820\pi$
$242$	0	0
$243$	726340.	0.789086
$244$	0	0
$245$	0	0
$246$	0	0
$247$	41682.0	0.0434717
$248$	0	0
$249$	−1.13326e6	−1.15832
$250$	0	0
$251$	−656376.	−0.657610	−0.328805	−	0.944398i	$-0.606646\pi$
$251$	−656376.	−0.657610	−0.328805	+	0.944398i	$0.606646\pi$
$252$	0	0
$253$	1.03874e6	1.02025
$254$	0	0
$255$	70807.3	0.0681910
$256$	0	0
$257$	−233900.	−0.220901	−0.110450	−	0.993882i	$-0.535229\pi$
$257$	−233900.	−0.220901	−0.110450	+	0.993882i	$0.535229\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−454019.	−0.412547
$262$	0	0
$263$	−1.97332e6	−1.75917	−0.879586	−	0.475741i	$-0.842180\pi$
$263$	−1.97332e6	−1.75917	−0.879586	+	0.475741i	$0.842180\pi$
$264$	0	0
$265$	1.82627e6	1.59753
$266$	0	0
$267$	828877.	0.711561
$268$	0	0
$269$	944556.	0.795879	0.397940	−	0.917412i	$-0.369725\pi$
$269$	944556.	0.795879	0.397940	+	0.917412i	$0.369725\pi$
$270$	0	0
$271$	203839.	0.168602	0.0843011	−	0.996440i	$-0.473134\pi$
$271$	203839.	0.168602	0.0843011	+	0.996440i	$0.473134\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−62351.6	−0.0497183
$276$	0	0
$277$	911604.	0.713850	0.356925	−	0.934133i	$-0.383825\pi$
$277$	911604.	0.713850	0.356925	+	0.934133i	$0.383825\pi$
$278$	0	0
$279$	−907482.	−0.697955
$280$	0	0
$281$	28748.0	0.0217191	0.0108596	−	0.999941i	$-0.496543\pi$
$281$	28748.0	0.0217191	0.0108596	+	0.999941i	$0.496543\pi$
$282$	0	0
$283$	−192070.	−0.142559	−0.0712793	−	0.997456i	$-0.522708\pi$
$283$	−192070.	−0.142559	−0.0712793	+	0.997456i	$0.522708\pi$
$284$	0	0
$285$	1.72551e6	1.25836
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.41552e6	−0.996947
$290$	0	0
$291$	1.84316e6	1.27594
$292$	0	0
$293$	1.53510e6	1.04465	0.522323	−	0.852748i	$-0.325066\pi$
$293$	1.53510e6	1.04465	0.522323	+	0.852748i	$0.325066\pi$
$294$	0	0
$295$	−624771.	−0.417990
$296$	0	0
$297$	−711789.	−0.468231
$298$	0	0
$299$	−99400.9	−0.0643002
$300$	0	0
$301$	0	0
$302$	0	0
$303$	1.91928e6	1.20097
$304$	0	0
$305$	729814.	0.449224
$306$	0	0
$307$	1.63502e6	0.990095	0.495047	−	0.868866i	$-0.335151\pi$
$307$	1.63502e6	0.990095	0.495047	+	0.868866i	$0.335151\pi$
$308$	0	0
$309$	3.51755e6	2.09577
$310$	0	0
$311$	−1.56216e6	−0.915852	−0.457926	−	0.888990i	$-0.651408\pi$
$311$	−1.56216e6	−0.915852	−0.457926	+	0.888990i	$0.651408\pi$
$312$	0	0
$313$	−2.16060e6	−1.24656	−0.623281	−	0.781998i	$-0.714201\pi$
$313$	−2.16060e6	−1.24656	−0.623281	+	0.781998i	$0.714201\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.90858e6	−1.06675	−0.533374	−	0.845880i	$-0.679076\pi$
$317$	−1.90858e6	−1.06675	−0.533374	+	0.845880i	$0.679076\pi$
$318$	0	0
$319$	1.21069e6	0.666128
$320$	0	0
$321$	2.51192e6	1.36064
$322$	0	0
$323$	105624.	0.0563321
$324$	0	0
$325$	5966.64	0.00313344
$326$	0	0
$327$	−513430.	−0.265529
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−1.74451e6	−0.875194	−0.437597	−	0.899171i	$-0.644170\pi$
$331$	−1.74451e6	−0.875194	−0.437597	+	0.899171i	$0.644170\pi$
$332$	0	0
$333$	−1.52459e6	−0.753430
$334$	0	0
$335$	1.58436e6	0.771333
$336$	0	0
$337$	−3.87616e6	−1.85921	−0.929603	−	0.368563i	$-0.879850\pi$
$337$	−3.87616e6	−1.85921	−0.929603	+	0.368563i	$0.879850\pi$
$338$	0	0
$339$	−3.87265e6	−1.83025
$340$	0	0
$341$	2.41991e6	1.12697
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4.11489e6	−1.86128
$346$	0	0
$347$	−3.76279e6	−1.67759	−0.838795	−	0.544447i	$-0.816740\pi$
$347$	−3.76279e6	−1.67759	−0.838795	+	0.544447i	$0.816740\pi$
$348$	0	0
$349$	−2.15341e6	−0.946377	−0.473188	−	0.880961i	$-0.656897\pi$
$349$	−2.15341e6	−0.946377	−0.473188	+	0.880961i	$0.656897\pi$
$350$	0	0
$351$	68113.5	0.0295097
$352$	0	0
$353$	3.45391e6	1.47528	0.737639	−	0.675195i	$-0.235940\pi$
$353$	3.45391e6	1.47528	0.737639	+	0.675195i	$0.235940\pi$
$354$	0	0
$355$	−401179.	−0.168953
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.52303e6	0.623693	0.311847	−	0.950132i	$-0.399052\pi$
$359$	1.52303e6	0.623693	0.311847	+	0.950132i	$0.399052\pi$
$360$	0	0
$361$	97858.7	0.0395213
$362$	0	0
$363$	1.62183e6	0.646009
$364$	0	0
$365$	3.51444e6	1.38078
$366$	0	0
$367$	−2.42996e6	−0.941747	−0.470873	−	0.882201i	$-0.656061\pi$
$367$	−2.42996e6	−0.941747	−0.470873	+	0.882201i	$0.656061\pi$
$368$	0	0
$369$	−1.29904e6	−0.496657
$370$	0	0
$371$	0	0
$372$	0	0
$373$	2.48301e6	0.924072	0.462036	−	0.886861i	$-0.347119\pi$
$373$	2.48301e6	0.924072	0.462036	+	0.886861i	$0.347119\pi$
$374$	0	0
$375$	−3.11398e6	−1.14350
$376$	0	0
$377$	−115855.	−0.0419820
$378$	0	0
$379$	1.22556e6	0.438266	0.219133	−	0.975695i	$-0.429677\pi$
$379$	1.22556e6	0.438266	0.219133	+	0.975695i	$0.429677\pi$
$380$	0	0
$381$	−1.87466e6	−0.661623
$382$	0	0
$383$	658469.	0.229371	0.114685	−	0.993402i	$-0.463414\pi$
$383$	658469.	0.229371	0.114685	+	0.993402i	$0.463414\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	312328.	0.106007
$388$	0	0
$389$	−2.38385e6	−0.798738	−0.399369	−	0.916790i	$-0.630771\pi$
$389$	−2.38385e6	−0.798738	−0.399369	+	0.916790i	$0.630771\pi$
$390$	0	0
$391$	−251886.	−0.0833224
$392$	0	0
$393$	−5.74653e6	−1.87683
$394$	0	0
$395$	4.98534e6	1.60769
$396$	0	0
$397$	−2.03048e6	−0.646581	−0.323291	−	0.946300i	$-0.604789\pi$
$397$	−2.03048e6	−0.646581	−0.323291	+	0.946300i	$0.604789\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.43939e6	1.06812	0.534061	−	0.845446i	$-0.320665\pi$
$401$	3.43939e6	1.06812	0.534061	+	0.845446i	$0.320665\pi$
$402$	0	0
$403$	−231569.	−0.0710261
$404$	0	0
$405$	4.25266e6	1.28832
$406$	0	0
$407$	4.06550e6	1.21654
$408$	0	0
$409$	5.96700e6	1.76379	0.881897	−	0.471443i	$-0.156267\pi$
$409$	5.96700e6	1.76379	0.881897	+	0.471443i	$0.156267\pi$
$410$	0	0
$411$	1.10907e6	0.323858
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3.53476e6	−1.00749
$416$	0	0
$417$	−1.49645e6	−0.421427
$418$	0	0
$419$	2.17586e6	0.605474	0.302737	−	0.953074i	$-0.402100\pi$
$419$	2.17586e6	0.605474	0.302737	+	0.953074i	$0.402100\pi$
$420$	0	0
$421$	−1.52580e6	−0.419557	−0.209779	−	0.977749i	$-0.567274\pi$
$421$	−1.52580e6	−0.419557	−0.209779	+	0.977749i	$0.567274\pi$
$422$	0	0
$423$	1.63951e6	0.445517
$424$	0	0
$425$	15119.7	0.00406042
$426$	0	0
$427$	0	0
$428$	0	0
$429$	130980.	0.0343608
$430$	0	0
$431$	4.70930e6	1.22113	0.610567	−	0.791964i	$-0.290941\pi$
$431$	4.70930e6	1.22113	0.610567	+	0.791964i	$0.290941\pi$
$432$	0	0
$433$	−2.24467e6	−0.575351	−0.287676	−	0.957728i	$-0.592882\pi$
$433$	−2.24467e6	−0.575351	−0.287676	+	0.957728i	$0.592882\pi$
$434$	0	0
$435$	−4.79606e6	−1.21524
$436$	0	0
$437$	−6.13823e6	−1.53759
$438$	0	0
$439$	308428.	0.0763824	0.0381912	−	0.999270i	$-0.487840\pi$
$439$	308428.	0.0763824	0.0381912	+	0.999270i	$0.487840\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.56593e6	−0.379109	−0.189554	−	0.981870i	$-0.560704\pi$
$443$	−1.56593e6	−0.379109	−0.189554	+	0.981870i	$0.560704\pi$
$444$	0	0
$445$	2.58537e6	0.618902
$446$	0	0
$447$	−7.69285e6	−1.82104
$448$	0	0
$449$	2.59822e6	0.608220	0.304110	−	0.952637i	$-0.401641\pi$
$449$	2.59822e6	0.608220	0.304110	+	0.952637i	$0.401641\pi$
$450$	0	0
$451$	3.46404e6	0.801940
$452$	0	0
$453$	522876.	0.119716
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.90597e6	−0.426900	−0.213450	−	0.976954i	$-0.568470\pi$
$457$	−1.90597e6	−0.426900	−0.213450	+	0.976954i	$0.568470\pi$
$458$	0	0
$459$	172602.	0.0382397
$460$	0	0
$461$	−6.06896e6	−1.33003	−0.665016	−	0.746829i	$-0.731575\pi$
$461$	−6.06896e6	−1.33003	−0.665016	+	0.746829i	$0.731575\pi$
$462$	0	0
$463$	−7.99741e6	−1.73379	−0.866896	−	0.498490i	$-0.833888\pi$
$463$	−7.99741e6	−1.73379	−0.866896	+	0.498490i	$0.833888\pi$
$464$	0	0
$465$	−9.58625e6	−2.05597
$466$	0	0
$467$	−3.86676e6	−0.820455	−0.410227	−	0.911983i	$-0.634551\pi$
$467$	−3.86676e6	−0.820455	−0.410227	+	0.911983i	$0.634551\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	7.39767e6	1.53654
$472$	0	0
$473$	−832859.	−0.171166
$474$	0	0
$475$	368453.	0.0749287
$476$	0	0
$477$	−3.21031e6	−0.646027
$478$	0	0
$479$	−580053.	−0.115512	−0.0577562	−	0.998331i	$-0.518395\pi$
$479$	−580053.	−0.115512	−0.0577562	+	0.998331i	$0.518395\pi$
$480$	0	0
$481$	−389042.	−0.0766714
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.74903e6	1.10979
$486$	0	0
$487$	493178.	0.0942283	0.0471141	−	0.998890i	$-0.484998\pi$
$487$	493178.	0.0942283	0.0471141	+	0.998890i	$0.484998\pi$
$488$	0	0
$489$	−9.62784e6	−1.82078
$490$	0	0
$491$	2.96699e6	0.555408	0.277704	−	0.960667i	$-0.410427\pi$
$491$	2.96699e6	0.555408	0.277704	+	0.960667i	$0.410427\pi$
$492$	0	0
$493$	−293582.	−0.0544017
$494$	0	0
$495$	1.60102e6	0.293686
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−9.23086e6	−1.65955	−0.829776	−	0.558097i	$-0.811532\pi$
$499$	−9.23086e6	−1.65955	−0.829776	+	0.558097i	$0.811532\pi$
$500$	0	0
$501$	−784411.	−0.139621
$502$	0	0
$503$	−4.99737e6	−0.880687	−0.440343	−	0.897830i	$-0.645143\pi$
$503$	−4.99737e6	−0.880687	−0.440343	+	0.897830i	$0.645143\pi$
$504$	0	0
$505$	5.98647e6	1.04458
$506$	0	0
$507$	6.88206e6	1.18905
$508$	0	0
$509$	2.78308e6	0.476137	0.238068	−	0.971248i	$-0.423486\pi$
$509$	2.78308e6	0.476137	0.238068	+	0.971248i	$0.423486\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.20616e6	0.705655
$514$	0	0
$515$	1.09717e7	1.82286
$516$	0	0
$517$	−4.37195e6	−0.719365
$518$	0	0
$519$	3.35626e6	0.546936
$520$	0	0
$521$	−3.57172e6	−0.576478	−0.288239	−	0.957558i	$-0.593070\pi$
$521$	−3.57172e6	−0.576478	−0.288239	+	0.957558i	$0.593070\pi$
$522$	0	0
$523$	2.62762e6	0.420057	0.210029	−	0.977695i	$-0.432644\pi$
$523$	2.62762e6	0.420057	0.210029	+	0.977695i	$0.432644\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−586805.	−0.0920380
$528$	0	0
$529$	8.20176e6	1.27429
$530$	0	0
$531$	1.09825e6	0.169031
$532$	0	0
$533$	−331486.	−0.0505414
$534$	0	0
$535$	7.83496e6	1.18346
$536$	0	0
$537$	2.93480e6	0.439180
$538$	0	0
$539$	0	0
$540$	0	0
$541$	1.17259e6	0.172248	0.0861240	−	0.996284i	$-0.472552\pi$
$541$	1.17259e6	0.172248	0.0861240	+	0.996284i	$0.472552\pi$
$542$	0	0
$543$	3.24344e6	0.472070
$544$	0	0
$545$	−1.60145e6	−0.230952
$546$	0	0
$547$	7.59939e6	1.08595	0.542976	−	0.839748i	$-0.317297\pi$
$547$	7.59939e6	1.08595	0.542976	+	0.839748i	$0.317297\pi$
$548$	0	0
$549$	−1.28290e6	−0.181662
$550$	0	0
$551$	−7.15433e6	−1.00390
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.61051e7	−2.21938
$556$	0	0
$557$	1.06811e7	1.45874	0.729371	−	0.684118i	$-0.239813\pi$
$557$	1.06811e7	1.45874	0.729371	+	0.684118i	$0.239813\pi$
$558$	0	0
$559$	79699.1	0.0107876
$560$	0	0
$561$	331910.	0.0445259
$562$	0	0
$563$	−6.11517e6	−0.813088	−0.406544	−	0.913631i	$-0.633266\pi$
$563$	−6.11517e6	−0.813088	−0.406544	+	0.913631i	$0.633266\pi$
$564$	0	0
$565$	−1.20793e7	−1.59191
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.66176e6	0.474143	0.237071	−	0.971492i	$-0.423812\pi$
$569$	3.66176e6	0.474143	0.237071	+	0.971492i	$0.423812\pi$
$570$	0	0
$571$	1.77437e6	0.227748	0.113874	−	0.993495i	$-0.463674\pi$
$571$	1.77437e6	0.227748	0.113874	+	0.993495i	$0.463674\pi$
$572$	0	0
$573$	−1.56308e6	−0.198881
$574$	0	0
$575$	−878666.	−0.110829
$576$	0	0
$577$	−3.64591e6	−0.455897	−0.227948	−	0.973673i	$-0.573202\pi$
$577$	−3.64591e6	−0.455897	−0.227948	+	0.973673i	$0.573202\pi$
$578$	0	0
$579$	5.84522e6	0.724610
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.56066e6	1.04312
$584$	0	0
$585$	−153207.	−0.0185092
$586$	0	0
$587$	−1.81749e6	−0.217709	−0.108854	−	0.994058i	$-0.534718\pi$
$587$	−1.81749e6	−0.217709	−0.108854	+	0.994058i	$0.534718\pi$
$588$	0	0
$589$	−1.42999e7	−1.69842
$590$	0	0
$591$	−1.30098e7	−1.53215
$592$	0	0
$593$	−3.17620e6	−0.370912	−0.185456	−	0.982653i	$-0.559376\pi$
$593$	−3.17620e6	−0.370912	−0.185456	+	0.982653i	$0.559376\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.01485e6	0.461034
$598$	0	0
$599$	1.50122e7	1.70953	0.854766	−	0.519014i	$-0.173701\pi$
$599$	1.50122e7	1.70953	0.854766	+	0.519014i	$0.173701\pi$
$600$	0	0
$601$	−1.23271e7	−1.39211	−0.696057	−	0.717987i	$-0.745064\pi$
$601$	−1.23271e7	−1.39211	−0.696057	+	0.717987i	$0.745064\pi$
$602$	0	0
$603$	−2.78507e6	−0.311919
$604$	0	0
$605$	5.05868e6	0.561887
$606$	0	0
$607$	−4.81660e6	−0.530602	−0.265301	−	0.964166i	$-0.585471\pi$
$607$	−4.81660e6	−0.530602	−0.265301	+	0.964166i	$0.585471\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	418367.	0.0453372
$612$	0	0
$613$	1.43158e7	1.53874	0.769368	−	0.638805i	$-0.220571\pi$
$613$	1.43158e7	1.53874	0.769368	+	0.638805i	$0.220571\pi$
$614$	0	0
$615$	−1.37225e7	−1.46300
$616$	0	0
$617$	−9.12565e6	−0.965052	−0.482526	−	0.875882i	$-0.660281\pi$
$617$	−9.12565e6	−0.965052	−0.482526	+	0.875882i	$0.660281\pi$
$618$	0	0
$619$	1.39353e7	1.46181	0.730905	−	0.682479i	$-0.239098\pi$
$619$	1.39353e7	1.46181	0.730905	+	0.682479i	$0.239098\pi$
$620$	0	0
$621$	−1.00306e7	−1.04375
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.04306e7	−1.06809
$626$	0	0
$627$	8.08833e6	0.821656
$628$	0	0
$629$	−985846.	−0.0993534
$630$	0	0
$631$	5.71493e6	0.571397	0.285698	−	0.958320i	$-0.407774\pi$
$631$	5.71493e6	0.571397	0.285698	+	0.958320i	$0.407774\pi$
$632$	0	0
$633$	9.75430e6	0.967580
$634$	0	0
$635$	−5.84729e6	−0.575467
$636$	0	0
$637$	0	0
$638$	0	0
$639$	705212.	0.0683231
$640$	0	0
$641$	−6.25824e6	−0.601599	−0.300800	−	0.953687i	$-0.597254\pi$
$641$	−6.25824e6	−0.601599	−0.300800	+	0.953687i	$0.597254\pi$
$642$	0	0
$643$	1.47061e7	1.40272	0.701359	−	0.712809i	$-0.252577\pi$
$643$	1.47061e7	1.40272	0.701359	+	0.712809i	$0.252577\pi$
$644$	0	0
$645$	3.29930e6	0.312264
$646$	0	0
$647$	−1.37019e6	−0.128682	−0.0643412	−	0.997928i	$-0.520495\pi$
$647$	−1.37019e6	−0.128682	−0.0643412	+	0.997928i	$0.520495\pi$
$648$	0	0
$649$	−2.92862e6	−0.272930
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.24684e7	−1.14427	−0.572135	−	0.820159i	$-0.693885\pi$
$653$	−1.24684e7	−1.14427	−0.572135	+	0.820159i	$0.693885\pi$
$654$	0	0
$655$	−1.79241e7	−1.63243
$656$	0	0
$657$	−6.17785e6	−0.558372
$658$	0	0
$659$	3.28182e6	0.294375	0.147188	−	0.989109i	$-0.452978\pi$
$659$	3.28182e6	0.294375	0.147188	+	0.989109i	$0.452978\pi$
$660$	0	0
$661$	3.38691e6	0.301509	0.150754	−	0.988571i	$-0.451830\pi$
$661$	3.38691e6	0.301509	0.150754	+	0.988571i	$0.451830\pi$
$662$	0	0
$663$	−31761.6	−0.00280620
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.70612e7	1.48490
$668$	0	0
$669$	−1.98256e7	−1.71262
$670$	0	0
$671$	3.42101e6	0.293324
$672$	0	0
$673$	1.76015e7	1.49800	0.749000	−	0.662570i	$-0.230534\pi$
$673$	1.76015e7	1.49800	0.749000	+	0.662570i	$0.230534\pi$
$674$	0	0
$675$	602097.	0.0508636
$676$	0	0
$677$	−3.71131e6	−0.311211	−0.155606	−	0.987819i	$-0.549733\pi$
$677$	−3.71131e6	−0.311211	−0.155606	+	0.987819i	$0.549733\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−8.21451e6	−0.678757
$682$	0	0
$683$	−2.73490e6	−0.224331	−0.112166	−	0.993690i	$-0.535779\pi$
$683$	−2.73490e6	−0.224331	−0.112166	+	0.993690i	$0.535779\pi$
$684$	0	0
$685$	3.45932e6	0.281685
$686$	0	0
$687$	−1.83333e7	−1.48200
$688$	0	0
$689$	−819198.	−0.0657417
$690$	0	0
$691$	−2.06048e7	−1.64163	−0.820813	−	0.571197i	$-0.806479\pi$
$691$	−2.06048e7	−1.64163	−0.820813	+	0.571197i	$0.806479\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.66761e6	−0.366550
$696$	0	0
$697$	−839998.	−0.0654932
$698$	0	0
$699$	−1.99728e7	−1.54613
$700$	0	0
$701$	−1.92235e7	−1.47753	−0.738765	−	0.673963i	$-0.764591\pi$
$701$	−1.92235e7	−1.47753	−0.738765	+	0.673963i	$0.764591\pi$
$702$	0	0
$703$	−2.40242e7	−1.83341
$704$	0	0
$705$	1.73191e7	1.31236
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−1.79608e7	−1.34187	−0.670935	−	0.741516i	$-0.734107\pi$
$709$	−1.79608e7	−1.34187	−0.670935	+	0.741516i	$0.734107\pi$
$710$	0	0
$711$	−8.76348e6	−0.650133
$712$	0	0
$713$	3.41016e7	2.51218
$714$	0	0
$715$	408544.	0.0298864
$716$	0	0
$717$	1.22516e7	0.890007
$718$	0	0
$719$	3.57894e6	0.258186	0.129093	−	0.991633i	$-0.458793\pi$
$719$	3.57894e6	0.258186	0.129093	+	0.991633i	$0.458793\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−1.37014e7	−0.974808
$724$	0	0
$725$	−1.02412e6	−0.0723610
$726$	0	0
$727$	−4.53529e6	−0.318250	−0.159125	−	0.987258i	$-0.550867\pi$
$727$	−4.53529e6	−0.318250	−0.159125	+	0.987258i	$0.550867\pi$
$728$	0	0
$729$	4.35443e6	0.303468
$730$	0	0
$731$	201961.	0.0139789
$732$	0	0
$733$	−1.95651e7	−1.34500	−0.672498	−	0.740099i	$-0.734779\pi$
$733$	−1.95651e7	−1.34500	−0.672498	+	0.740099i	$0.734779\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.42670e6	0.503648
$738$	0	0
$739$	789081.	0.0531509	0.0265754	−	0.999647i	$-0.491540\pi$
$739$	789081.	0.0531509	0.0265754	+	0.999647i	$0.491540\pi$
$740$	0	0
$741$	−774000.	−0.0517840
$742$	0	0
$743$	−2.44884e7	−1.62738	−0.813690	−	0.581299i	$-0.802545\pi$
$743$	−2.44884e7	−1.62738	−0.813690	+	0.581299i	$0.802545\pi$
$744$	0	0
$745$	−2.39949e7	−1.58390
$746$	0	0
$747$	6.21358e6	0.407418
$748$	0	0
$749$	0	0
$750$	0	0
$751$	9.19848e6	0.595136	0.297568	−	0.954701i	$-0.403825\pi$
$751$	9.19848e6	0.595136	0.297568	+	0.954701i	$0.403825\pi$
$752$	0	0
$753$	1.21884e7	0.783353
$754$	0	0
$755$	1.63091e6	0.104127
$756$	0	0
$757$	1.60764e6	0.101965	0.0509823	−	0.998700i	$-0.483765\pi$
$757$	1.60764e6	0.101965	0.0509823	+	0.998700i	$0.483765\pi$
$758$	0	0
$759$	−1.92886e7	−1.21534
$760$	0	0
$761$	−6.15590e6	−0.385327	−0.192664	−	0.981265i	$-0.561713\pi$
$761$	−6.15590e6	−0.385327	−0.192664	+	0.981265i	$0.561713\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−388232.	−0.0239849
$766$	0	0
$767$	280249.	0.0172011
$768$	0	0
$769$	−5.31373e6	−0.324029	−0.162014	−	0.986788i	$-0.551799\pi$
$769$	−5.31373e6	−0.324029	−0.162014	+	0.986788i	$0.551799\pi$
$770$	0	0
$771$	4.34332e6	0.263139
$772$	0	0
$773$	−9.04874e6	−0.544678	−0.272339	−	0.962201i	$-0.587797\pi$
$773$	−9.04874e6	−0.544678	−0.272339	+	0.962201i	$0.587797\pi$
$774$	0	0
$775$	−2.04698e6	−0.122422
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.04700e7	−1.20858
$780$	0	0
$781$	−1.88053e6	−0.110319
$782$	0	0
$783$	−1.16910e7	−0.681474
$784$	0	0
$785$	2.30742e7	1.33645
$786$	0	0
$787$	1.39440e7	0.802510	0.401255	−	0.915966i	$-0.368574\pi$
$787$	1.39440e7	0.802510	0.401255	+	0.915966i	$0.368574\pi$
$788$	0	0
$789$	3.66429e7	2.09555
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−327368.	−0.0184864
$794$	0	0
$795$	−3.39123e7	−1.90300
$796$	0	0
$797$	5.13027e6	0.286085	0.143042	−	0.989717i	$-0.454311\pi$
$797$	5.13027e6	0.286085	0.143042	+	0.989717i	$0.454311\pi$
$798$	0	0
$799$	1.06016e6	0.0587494
$800$	0	0
$801$	−4.54468e6	−0.250278
$802$	0	0
$803$	1.64739e7	0.901590
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.75396e7	−0.948061
$808$	0	0
$809$	2.03948e7	1.09559	0.547794	−	0.836613i	$-0.315468\pi$
$809$	2.03948e7	1.09559	0.547794	+	0.836613i	$0.315468\pi$
$810$	0	0
$811$	1.17222e6	0.0625831	0.0312915	−	0.999510i	$-0.490038\pi$
$811$	1.17222e6	0.0625831	0.0312915	+	0.999510i	$0.490038\pi$
$812$	0	0
$813$	−3.78511e6	−0.200841
$814$	0	0
$815$	−3.00304e7	−1.58368
$816$	0	0
$817$	4.92160e6	0.257959
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.91086e7	−0.989398	−0.494699	−	0.869064i	$-0.664722\pi$
$821$	−1.91086e7	−0.989398	−0.494699	+	0.869064i	$0.664722\pi$
$822$	0	0
$823$	−1.93276e7	−0.994668	−0.497334	−	0.867559i	$-0.665688\pi$
$823$	−1.93276e7	−0.994668	−0.497334	+	0.867559i	$0.665688\pi$
$824$	0	0
$825$	1.15782e6	0.0592250
$826$	0	0
$827$	−1.78980e7	−0.910000	−0.455000	−	0.890491i	$-0.650361\pi$
$827$	−1.78980e7	−0.910000	−0.455000	+	0.890491i	$0.650361\pi$
$828$	0	0
$829$	2.57721e7	1.30246	0.651228	−	0.758882i	$-0.274254\pi$
$829$	2.57721e7	1.30246	0.651228	+	0.758882i	$0.274254\pi$
$830$	0	0
$831$	−1.69277e7	−0.850346
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−2.44667e6	−0.121439
$836$	0	0
$837$	−2.33678e7	−1.15293
$838$	0	0
$839$	−1.18133e7	−0.579384	−0.289692	−	0.957120i	$-0.593553\pi$
$839$	−1.18133e7	−0.579384	−0.289692	+	0.957120i	$0.593553\pi$
$840$	0	0
$841$	−625637.	−0.0305023
$842$	0	0
$843$	−533827.	−0.0258721
$844$	0	0
$845$	2.14660e7	1.03421
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.56658e6	0.169818
$850$	0	0
$851$	5.72915e7	2.71185
$852$	0	0
$853$	−9.75883e6	−0.459224	−0.229612	−	0.973282i	$-0.573746\pi$
$853$	−9.75883e6	−0.459224	−0.229612	+	0.973282i	$0.573746\pi$
$854$	0	0
$855$	−9.46086e6	−0.442604
$856$	0	0
$857$	3.08619e7	1.43539	0.717697	−	0.696355i	$-0.245196\pi$
$857$	3.08619e7	1.43539	0.717697	+	0.696355i	$0.245196\pi$
$858$	0	0
$859$	2.73351e7	1.26398	0.631988	−	0.774979i	$-0.282239\pi$
$859$	2.73351e7	1.26398	0.631988	+	0.774979i	$0.282239\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.64120e7	1.66425	0.832123	−	0.554591i	$-0.187125\pi$
$863$	3.64120e7	1.66425	0.832123	+	0.554591i	$0.187125\pi$
$864$	0	0
$865$	1.04686e7	0.475715
$866$	0	0
$867$	2.62851e7	1.18758
$868$	0	0
$869$	2.33688e7	1.04975
$870$	0	0
$871$	−710686.	−0.0317419
$872$	0	0
$873$	−1.01059e7	−0.448787
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.95863e7	−0.859910	−0.429955	−	0.902850i	$-0.641471\pi$
$877$	−1.95863e7	−0.859910	−0.429955	+	0.902850i	$0.641471\pi$
$878$	0	0
$879$	−2.85056e7	−1.24439
$880$	0	0
$881$	4.32823e7	1.87876	0.939379	−	0.342880i	$-0.111403\pi$
$881$	4.32823e7	1.87876	0.939379	+	0.342880i	$0.111403\pi$
$882$	0	0
$883$	2.52103e7	1.08812	0.544059	−	0.839047i	$-0.316887\pi$
$883$	2.52103e7	1.08812	0.544059	+	0.839047i	$0.316887\pi$
$884$	0	0
$885$	1.16015e7	0.497915
$886$	0	0
$887$	−2.65567e7	−1.13335	−0.566677	−	0.823940i	$-0.691771\pi$
$887$	−2.65567e7	−1.13335	−0.566677	+	0.823940i	$0.691771\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	1.99344e7	0.841217
$892$	0	0
$893$	2.58351e7	1.08413
$894$	0	0
$895$	9.15398e6	0.381990
$896$	0	0
$897$	1.84579e6	0.0765952
$898$	0	0
$899$	3.97467e7	1.64022
$900$	0	0
$901$	−2.07588e6	−0.0851903
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.01167e7	0.410598
$906$	0	0
$907$	3.31130e7	1.33653	0.668267	−	0.743922i	$-0.267037\pi$
$907$	3.31130e7	1.33653	0.668267	+	0.743922i	$0.267037\pi$
$908$	0	0
$909$	−1.05233e7	−0.422418
$910$	0	0
$911$	83307.1	0.00332572	0.00166286	−	0.999999i	$-0.499471\pi$
$911$	83307.1	0.00332572	0.00166286	+	0.999999i	$0.499471\pi$
$912$	0	0
$913$	−1.65692e7	−0.657848
$914$	0	0
$915$	−1.35520e7	−0.535121
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.52919e7	1.76902	0.884508	−	0.466524i	$-0.154494\pi$
$919$	4.52919e7	1.76902	0.884508	+	0.466524i	$0.154494\pi$
$920$	0	0
$921$	−3.03609e7	−1.17941
$922$	0	0
$923$	179954.	0.00695276
$924$	0	0
$925$	−3.43898e6	−0.132152
$926$	0	0
$927$	−1.92865e7	−0.737148
$928$	0	0
$929$	5.88580e6	0.223752	0.111876	−	0.993722i	$-0.464314\pi$
$929$	5.88580e6	0.223752	0.111876	+	0.993722i	$0.464314\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.90081e7	1.09097
$934$	0	0
$935$	1.03527e6	0.0387278
$936$	0	0
$937$	−7.00299e6	−0.260576	−0.130288	−	0.991476i	$-0.541590\pi$
$937$	−7.00299e6	−0.260576	−0.130288	+	0.991476i	$0.541590\pi$
$938$	0	0
$939$	4.01205e7	1.48492
$940$	0	0
$941$	1.33574e7	0.491755	0.245877	−	0.969301i	$-0.420924\pi$
$941$	1.33574e7	0.491755	0.245877	+	0.969301i	$0.420924\pi$
$942$	0	0
$943$	4.88157e7	1.78764
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.12125e7	0.768628	0.384314	−	0.923202i	$-0.374438\pi$
$947$	2.12125e7	0.768628	0.384314	+	0.923202i	$0.374438\pi$
$948$	0	0
$949$	−1.57645e6	−0.0568217
$950$	0	0
$951$	3.54407e7	1.27072
$952$	0	0
$953$	−2.61283e7	−0.931920	−0.465960	−	0.884806i	$-0.654291\pi$
$953$	−2.61283e7	−0.931920	−0.465960	+	0.884806i	$0.654291\pi$
$954$	0	0
$955$	−4.87543e6	−0.172983
$956$	0	0
$957$	−2.24816e7	−0.793500
$958$	0	0
$959$	0	0
$960$	0	0
$961$	5.08155e7	1.77496
$962$	0	0
$963$	−1.37727e7	−0.478578
$964$	0	0
$965$	1.82319e7	0.630252
$966$	0	0
$967$	5.49872e7	1.89102	0.945509	−	0.325597i	$-0.105565\pi$
$967$	5.49872e7	1.89102	0.945509	+	0.325597i	$0.105565\pi$
$968$	0	0
$969$	−1.96135e6	−0.0671035
$970$	0	0
$971$	3.87974e7	1.32055	0.660274	−	0.751025i	$-0.270440\pi$
$971$	3.87974e7	1.32055	0.660274	+	0.751025i	$0.270440\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−110795.	−0.00373259
$976$	0	0
$977$	−3.11797e7	−1.04505	−0.522523	−	0.852625i	$-0.675009\pi$
$977$	−3.11797e7	−1.04505	−0.522523	+	0.852625i	$0.675009\pi$
$978$	0	0
$979$	1.21189e7	0.404117
$980$	0	0
$981$	2.81511e6	0.0933948
$982$	0	0
$983$	2.31403e7	0.763810	0.381905	−	0.924202i	$-0.375268\pi$
$983$	2.31403e7	0.763810	0.381905	+	0.924202i	$0.375268\pi$
$984$	0	0
$985$	−4.05790e7	−1.33263
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.17367e7	−0.381555
$990$	0	0
$991$	−1.50192e7	−0.485806	−0.242903	−	0.970051i	$-0.578100\pi$
$991$	−1.50192e7	−0.485806	−0.242903	+	0.970051i	$0.578100\pi$
$992$	0	0
$993$	3.23941e7	1.04254
$994$	0	0
$995$	1.25228e7	0.400999
$996$	0	0
$997$	2.48705e7	0.792403	0.396202	−	0.918163i	$-0.370328\pi$
$997$	2.48705e7	0.792403	0.396202	+	0.918163i	$0.370328\pi$
$998$	0	0
$999$	−3.92584e7	−1.24457

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.6.a.bm.1.1		5
4.3	odd	2		392.6.a.i.1.5		5
7.3	odd	6		112.6.i.g.65.1		10
7.5	odd	6		112.6.i.g.81.1		10
7.6	odd	2		784.6.a.bj.1.5		5
28.3	even	6		56.6.i.a.9.5	✓	10
28.11	odd	6		392.6.i.p.177.1		10
28.19	even	6		56.6.i.a.25.5	yes	10
28.23	odd	6		392.6.i.p.361.1		10
28.27	even	2		392.6.a.l.1.1		5
84.47	odd	6		504.6.s.d.361.4		10
84.59	odd	6		504.6.s.d.289.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.i.a.9.5	✓	10	28.3	even	6
56.6.i.a.25.5	yes	10	28.19	even	6
112.6.i.g.65.1		10	7.3	odd	6
112.6.i.g.81.1		10	7.5	odd	6
392.6.a.i.1.5		5	4.3	odd	2
392.6.a.l.1.1		5	28.27	even	2
392.6.i.p.177.1		10	28.11	odd	6
392.6.i.p.361.1		10	28.23	odd	6
504.6.s.d.289.4		10	84.59	odd	6
504.6.s.d.361.4		10	84.47	odd	6
784.6.a.bj.1.5		5	7.6	odd	2
784.6.a.bm.1.1		5	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q-18.5692 q^{3} -57.9194 q^{5} +101.814 q^{9} +O(q^{10})\)	\(q-18.5692 q^{3} -57.9194 q^{5} +101.814 q^{9} -271.498 q^{11} +25.9805 q^{13} +1075.51 q^{15} +65.8357 q^{17} +1604.36 q^{19} -3825.98 q^{23} +229.658 q^{25} +2621.71 q^{27} -4459.32 q^{29} -8913.17 q^{31} +5041.48 q^{33} -14974.3 q^{37} -482.437 q^{39} -12759.0 q^{41} +3067.65 q^{43} -5896.98 q^{45} +16103.1 q^{47} -1222.51 q^{51} -31531.2 q^{53} +15725.0 q^{55} -29791.5 q^{57} +10786.9 q^{59} -12600.5 q^{61} -1504.78 q^{65} -27354.6 q^{67} +71045.2 q^{69} +6926.50 q^{71} -60678.0 q^{73} -4264.55 q^{75} -86073.7 q^{79} -73423.7 q^{81} +61029.0 q^{83} -3813.16 q^{85} +82805.8 q^{87} -44637.3 q^{89} +165510. q^{93} -92923.3 q^{95} -99259.1 q^{97} -27642.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9}+O(q^{10}) \) 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9	\( 5 q + 13 q^{3} - 31 q^{5} + 230 q^{9} + 351 q^{11} + 54 q^{13} - 607 q^{15} - 111 q^{17} + 1035 q^{19} - 3639 q^{23} + 1540 q^{25} + 3607 q^{27} - 734 q^{29} + 7677 q^{31} + 7439 q^{33} + 13595 q^{37} + 1406 q^{39} - 5310 q^{41} - 764 q^{43} - 38978 q^{45} + 6675 q^{47} - 20975 q^{51} - 30753 q^{53} + 28267 q^{55} - 14389 q^{57} + 87989 q^{59} - 19899 q^{61} + 119470 q^{65} + 33067 q^{67} - 100399 q^{69} + 108720 q^{71} - 141659 q^{73} + 108788 q^{75} - 118919 q^{79} - 143851 q^{81} + 211004 q^{83} - 143379 q^{85} + 302154 q^{87} + 55861 q^{89} + 410381 q^{93} + 26279 q^{95} - 135470 q^{97} + 300154 q^{99}+O(q^{100}) \) 5 * q + 13 * q^3 - 31 * q^5 + 230 * q^9 + 351 * q^11 + 54 * q^13 - 607 * q^15 - 111 * q^17 + 1035 * q^19 - 3639 * q^23 + 1540 * q^25 + 3607 * q^27 - 734 * q^29 + 7677 * q^31 + 7439 * q^33 + 13595 * q^37 + 1406 * q^39 - 5310 * q^41 - 764 * q^43 - 38978 * q^45 + 6675 * q^47 - 20975 * q^51 - 30753 * q^53 + 28267 * q^55 - 14389 * q^57 + 87989 * q^59 - 19899 * q^61 + 119470 * q^65 + 33067 * q^67 - 100399 * q^69 + 108720 * q^71 - 141659 * q^73 + 108788 * q^75 - 118919 * q^79 - 143851 * q^81 + 211004 * q^83 - 143379 * q^85 + 302154 * q^87 + 55861 * q^89 + 410381 * q^93 + 26279 * q^95 - 135470 * q^97 + 300154 * q^99

Introduction

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