LMFDB - Embedded newform 784.4.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [784,4,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, 4, names="a")

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

chi := DirichletCharacter("784.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
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:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+6.00000 q^{3} -8.00000 q^{5} +9.00000 q^{9} +O(q^{10})$

$q+6.00000 q^{3} -8.00000 q^{5} +9.00000 q^{9} -56.0000 q^{11} +28.0000 q^{13} -48.0000 q^{15} +90.0000 q^{17} +74.0000 q^{19} +96.0000 q^{23} -61.0000 q^{25} -108.000 q^{27} -222.000 q^{29} -100.000 q^{31} -336.000 q^{33} +58.0000 q^{37} +168.000 q^{39} -422.000 q^{41} -512.000 q^{43} -72.0000 q^{45} +148.000 q^{47} +540.000 q^{51} -642.000 q^{53} +448.000 q^{55} +444.000 q^{57} -318.000 q^{59} -720.000 q^{61} -224.000 q^{65} +412.000 q^{67} +576.000 q^{69} -448.000 q^{71} -994.000 q^{73} -366.000 q^{75} +296.000 q^{79} -891.000 q^{81} +386.000 q^{83} -720.000 q^{85} -1332.00 q^{87} +6.00000 q^{89} -600.000 q^{93} -592.000 q^{95} +138.000 q^{97} -504.000 q^{99} +O(q^{100})$

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
$3$	6.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	6.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	0	0
$5$	−8.00000	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$5$	−8.00000	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−56.0000	−1.53497	−0.767483	−	0.641069i	$-0.778491\pi$
$11$	−56.0000	−1.53497	−0.767483	+	0.641069i	$0.778491\pi$
$12$	0	0
$13$	28.0000	0.597369	0.298685	−	0.954352i	$-0.403452\pi$
$13$	28.0000	0.597369	0.298685	+	0.954352i	$0.403452\pi$
$14$	0	0
$15$	−48.0000	−0.826236
$16$	0	0
$17$	90.0000	1.28401	0.642006	−	0.766700i	$-0.278102\pi$
$17$	90.0000	1.28401	0.642006	+	0.766700i	$0.278102\pi$
$18$	0	0
$19$	74.0000	0.893514	0.446757	−	0.894655i	$-0.352579\pi$
$19$	74.0000	0.893514	0.446757	+	0.894655i	$0.352579\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	0	0
$27$	−108.000	−0.769800
$28$	0	0
$29$	−222.000	−1.42153	−0.710765	−	0.703430i	$-0.751651\pi$
$29$	−222.000	−1.42153	−0.710765	+	0.703430i	$0.751651\pi$
$30$	0	0
$31$	−100.000	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$31$	−100.000	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$32$	0	0
$33$	−336.000	−1.77243
$34$	0	0
$35$	0	0
$36$	0	0
$37$	58.0000	0.257707	0.128853	−	0.991664i	$-0.458870\pi$
$37$	58.0000	0.257707	0.128853	+	0.991664i	$0.458870\pi$
$38$	0	0
$39$	168.000	0.689783
$40$	0	0
$41$	−422.000	−1.60745	−0.803724	−	0.595003i	$-0.797151\pi$
$41$	−422.000	−1.60745	−0.803724	+	0.595003i	$0.797151\pi$
$42$	0	0
$43$	−512.000	−1.81580	−0.907898	−	0.419190i	$-0.862314\pi$
$43$	−512.000	−1.81580	−0.907898	+	0.419190i	$0.862314\pi$
$44$	0	0
$45$	−72.0000	−0.238514
$46$	0	0
$47$	148.000	0.459320	0.229660	−	0.973271i	$-0.426239\pi$
$47$	148.000	0.459320	0.229660	+	0.973271i	$0.426239\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	540.000	1.48265
$52$	0	0
$53$	−642.000	−1.66388	−0.831939	−	0.554868i	$-0.812769\pi$
$53$	−642.000	−1.66388	−0.831939	+	0.554868i	$0.812769\pi$
$54$	0	0
$55$	448.000	1.09833
$56$	0	0
$57$	444.000	1.03174
$58$	0	0
$59$	−318.000	−0.701696	−0.350848	−	0.936432i	$-0.614107\pi$
$59$	−318.000	−0.701696	−0.350848	+	0.936432i	$0.614107\pi$
$60$	0	0
$61$	−720.000	−1.51125	−0.755627	−	0.655002i	$-0.772668\pi$
$61$	−720.000	−1.51125	−0.755627	+	0.655002i	$0.772668\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−224.000	−0.427443
$66$	0	0
$67$	412.000	0.751251	0.375625	−	0.926772i	$-0.377428\pi$
$67$	412.000	0.751251	0.375625	+	0.926772i	$0.377428\pi$
$68$	0	0
$69$	576.000	1.00496
$70$	0	0
$71$	−448.000	−0.748843	−0.374421	−	0.927259i	$-0.622159\pi$
$71$	−448.000	−0.748843	−0.374421	+	0.927259i	$0.622159\pi$
$72$	0	0
$73$	−994.000	−1.59368	−0.796842	−	0.604188i	$-0.793498\pi$
$73$	−994.000	−1.59368	−0.796842	+	0.604188i	$0.793498\pi$
$74$	0	0
$75$	−366.000	−0.563494
$76$	0	0
$77$	0	0
$78$	0	0
$79$	296.000	0.421552	0.210776	−	0.977534i	$-0.432401\pi$
$79$	296.000	0.421552	0.210776	+	0.977534i	$0.432401\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	386.000	0.510470	0.255235	−	0.966879i	$-0.417847\pi$
$83$	386.000	0.510470	0.255235	+	0.966879i	$0.417847\pi$
$84$	0	0
$85$	−720.000	−0.918764
$86$	0	0
$87$	−1332.00	−1.64144
$88$	0	0
$89$	6.00000	0.00714605	0.00357303	−	0.999994i	$-0.498863\pi$
$89$	6.00000	0.00714605	0.00357303	+	0.999994i	$0.498863\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−600.000	−0.669001
$94$	0	0
$95$	−592.000	−0.639347
$96$	0	0
$97$	138.000	0.144451	0.0722257	−	0.997388i	$-0.476990\pi$
$97$	138.000	0.144451	0.0722257	+	0.997388i	$0.476990\pi$
$98$	0	0
$99$	−504.000	−0.511656
$100$	0	0
$101$	664.000	0.654163	0.327082	−	0.944996i	$-0.393935\pi$
$101$	664.000	0.654163	0.327082	+	0.944996i	$0.393935\pi$
$102$	0	0
$103$	2012.00	1.92474	0.962370	−	0.271742i	$-0.0875997\pi$
$103$	2012.00	1.92474	0.962370	+	0.271742i	$0.0875997\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−900.000	−0.813143	−0.406571	−	0.913619i	$-0.633276\pi$
$107$	−900.000	−0.813143	−0.406571	+	0.913619i	$0.633276\pi$
$108$	0	0
$109$	1706.00	1.49913	0.749565	−	0.661931i	$-0.230263\pi$
$109$	1706.00	1.49913	0.749565	+	0.661931i	$0.230263\pi$
$110$	0	0
$111$	348.000	0.297574
$112$	0	0
$113$	−442.000	−0.367963	−0.183982	−	0.982930i	$-0.558899\pi$
$113$	−442.000	−0.367963	−0.183982	+	0.982930i	$0.558899\pi$
$114$	0	0
$115$	−768.000	−0.622751
$116$	0	0
$117$	252.000	0.199123
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1805.00	1.35612
$122$	0	0
$123$	−2532.00	−1.85612
$124$	0	0
$125$	1488.00	1.06473
$126$	0	0
$127$	1952.00	1.36387	0.681937	−	0.731411i	$-0.261138\pi$
$127$	1952.00	1.36387	0.681937	+	0.731411i	$0.261138\pi$
$128$	0	0
$129$	−3072.00	−2.09670
$130$	0	0
$131$	−742.000	−0.494877	−0.247438	−	0.968904i	$-0.579589\pi$
$131$	−742.000	−0.494877	−0.247438	+	0.968904i	$0.579589\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	864.000	0.550824
$136$	0	0
$137$	−1854.00	−1.15619	−0.578095	−	0.815970i	$-0.696204\pi$
$137$	−1854.00	−1.15619	−0.578095	+	0.815970i	$0.696204\pi$
$138$	0	0
$139$	−74.0000	−0.0451554	−0.0225777	−	0.999745i	$-0.507187\pi$
$139$	−74.0000	−0.0451554	−0.0225777	+	0.999745i	$0.507187\pi$
$140$	0	0
$141$	888.000	0.530377
$142$	0	0
$143$	−1568.00	−0.916942
$144$	0	0
$145$	1776.00	1.01716
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2502.00	1.37565	0.687825	−	0.725877i	$-0.258566\pi$
$149$	2502.00	1.37565	0.687825	+	0.725877i	$0.258566\pi$
$150$	0	0
$151$	888.000	0.478572	0.239286	−	0.970949i	$-0.423087\pi$
$151$	888.000	0.478572	0.239286	+	0.970949i	$0.423087\pi$
$152$	0	0
$153$	810.000	0.428004
$154$	0	0
$155$	800.000	0.414565
$156$	0	0
$157$	−1092.00	−0.555102	−0.277551	−	0.960711i	$-0.589523\pi$
$157$	−1092.00	−0.555102	−0.277551	+	0.960711i	$0.589523\pi$
$158$	0	0
$159$	−3852.00	−1.92128
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1704.00	0.818820	0.409410	−	0.912351i	$-0.365735\pi$
$163$	1704.00	0.818820	0.409410	+	0.912351i	$0.365735\pi$
$164$	0	0
$165$	2688.00	1.26825
$166$	0	0
$167$	436.000	0.202028	0.101014	−	0.994885i	$-0.467791\pi$
$167$	436.000	0.202028	0.101014	+	0.994885i	$0.467791\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	666.000	0.297838
$172$	0	0
$173$	996.000	0.437714	0.218857	−	0.975757i	$-0.429767\pi$
$173$	996.000	0.437714	0.218857	+	0.975757i	$0.429767\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1908.00	−0.810249
$178$	0	0
$179$	−1244.00	−0.519447	−0.259723	−	0.965683i	$-0.583631\pi$
$179$	−1244.00	−0.519447	−0.259723	+	0.965683i	$0.583631\pi$
$180$	0	0
$181$	−412.000	−0.169192	−0.0845959	−	0.996415i	$-0.526960\pi$
$181$	−412.000	−0.169192	−0.0845959	+	0.996415i	$0.526960\pi$
$182$	0	0
$183$	−4320.00	−1.74505
$184$	0	0
$185$	−464.000	−0.184400
$186$	0	0
$187$	−5040.00	−1.97092
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1160.00	−0.439448	−0.219724	−	0.975562i	$-0.570516\pi$
$191$	−1160.00	−0.439448	−0.219724	+	0.975562i	$0.570516\pi$
$192$	0	0
$193$	−3698.00	−1.37921	−0.689606	−	0.724185i	$-0.742216\pi$
$193$	−3698.00	−1.37921	−0.689606	+	0.724185i	$0.742216\pi$
$194$	0	0
$195$	−1344.00	−0.493568
$196$	0	0
$197$	−2874.00	−1.03941	−0.519706	−	0.854345i	$-0.673958\pi$
$197$	−2874.00	−1.03941	−0.519706	+	0.854345i	$0.673958\pi$
$198$	0	0
$199$	3348.00	1.19263	0.596315	−	0.802750i	$-0.296631\pi$
$199$	3348.00	1.19263	0.596315	+	0.802750i	$0.296631\pi$
$200$	0	0
$201$	2472.00	0.867470
$202$	0	0
$203$	0	0
$204$	0	0
$205$	3376.00	1.15020
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	−4144.00	−1.37151
$210$	0	0
$211$	2556.00	0.833945	0.416972	−	0.908919i	$-0.363091\pi$
$211$	2556.00	0.833945	0.416972	+	0.908919i	$0.363091\pi$
$212$	0	0
$213$	−2688.00	−0.864689
$214$	0	0
$215$	4096.00	1.29928
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−5964.00	−1.84023
$220$	0	0
$221$	2520.00	0.767030
$222$	0	0
$223$	−312.000	−0.0936909	−0.0468454	−	0.998902i	$-0.514917\pi$
$223$	−312.000	−0.0936909	−0.0468454	+	0.998902i	$0.514917\pi$
$224$	0	0
$225$	−549.000	−0.162667
$226$	0	0
$227$	826.000	0.241513	0.120757	−	0.992682i	$-0.461468\pi$
$227$	826.000	0.241513	0.120757	+	0.992682i	$0.461468\pi$
$228$	0	0
$229$	−4036.00	−1.16466	−0.582328	−	0.812954i	$-0.697858\pi$
$229$	−4036.00	−1.16466	−0.582328	+	0.812954i	$0.697858\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2902.00	−0.815950	−0.407975	−	0.912993i	$-0.633765\pi$
$233$	−2902.00	−0.815950	−0.407975	+	0.912993i	$0.633765\pi$
$234$	0	0
$235$	−1184.00	−0.328662
$236$	0	0
$237$	1776.00	0.486766
$238$	0	0
$239$	−3176.00	−0.859575	−0.429787	−	0.902930i	$-0.641411\pi$
$239$	−3176.00	−0.859575	−0.429787	+	0.902930i	$0.641411\pi$
$240$	0	0
$241$	4682.00	1.25143	0.625714	−	0.780053i	$-0.284808\pi$
$241$	4682.00	1.25143	0.625714	+	0.780053i	$0.284808\pi$
$242$	0	0
$243$	−2430.00	−0.641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2072.00	0.533758
$248$	0	0
$249$	2316.00	0.589440
$250$	0	0
$251$	630.000	0.158427	0.0792136	−	0.996858i	$-0.474759\pi$
$251$	630.000	0.158427	0.0792136	+	0.996858i	$0.474759\pi$
$252$	0	0
$253$	−5376.00	−1.33591
$254$	0	0
$255$	−4320.00	−1.06090
$256$	0	0
$257$	1870.00	0.453881	0.226940	−	0.973909i	$-0.427128\pi$
$257$	1870.00	0.453881	0.226940	+	0.973909i	$0.427128\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1998.00	−0.473843
$262$	0	0
$263$	−3360.00	−0.787781	−0.393891	−	0.919157i	$-0.628871\pi$
$263$	−3360.00	−0.787781	−0.393891	+	0.919157i	$0.628871\pi$
$264$	0	0
$265$	5136.00	1.19057
$266$	0	0
$267$	36.0000	0.00825155
$268$	0	0
$269$	5924.00	1.34272	0.671362	−	0.741130i	$-0.265710\pi$
$269$	5924.00	1.34272	0.671362	+	0.741130i	$0.265710\pi$
$270$	0	0
$271$	2064.00	0.462653	0.231327	−	0.972876i	$-0.425693\pi$
$271$	2064.00	0.462653	0.231327	+	0.972876i	$0.425693\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3416.00	0.749064
$276$	0	0
$277$	−3314.00	−0.718841	−0.359421	−	0.933176i	$-0.617026\pi$
$277$	−3314.00	−0.718841	−0.359421	+	0.933176i	$0.617026\pi$
$278$	0	0
$279$	−900.000	−0.193124
$280$	0	0
$281$	3018.00	0.640707	0.320354	−	0.947298i	$-0.396198\pi$
$281$	3018.00	0.640707	0.320354	+	0.947298i	$0.396198\pi$
$282$	0	0
$283$	6802.00	1.42875	0.714376	−	0.699762i	$-0.246711\pi$
$283$	6802.00	1.42875	0.714376	+	0.699762i	$0.246711\pi$
$284$	0	0
$285$	−3552.00	−0.738254
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3187.00	0.648687
$290$	0	0
$291$	828.000	0.166798
$292$	0	0
$293$	−2784.00	−0.555096	−0.277548	−	0.960712i	$-0.589522\pi$
$293$	−2784.00	−0.555096	−0.277548	+	0.960712i	$0.589522\pi$
$294$	0	0
$295$	2544.00	0.502093
$296$	0	0
$297$	6048.00	1.18162
$298$	0	0
$299$	2688.00	0.519903
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3984.00	0.755362
$304$	0	0
$305$	5760.00	1.08137
$306$	0	0
$307$	−5654.00	−1.05111	−0.525555	−	0.850760i	$-0.676142\pi$
$307$	−5654.00	−1.05111	−0.525555	+	0.850760i	$0.676142\pi$
$308$	0	0
$309$	12072.0	2.22250
$310$	0	0
$311$	2504.00	0.456556	0.228278	−	0.973596i	$-0.426691\pi$
$311$	2504.00	0.456556	0.228278	+	0.973596i	$0.426691\pi$
$312$	0	0
$313$	−6454.00	−1.16550	−0.582750	−	0.812651i	$-0.698023\pi$
$313$	−6454.00	−1.16550	−0.582750	+	0.812651i	$0.698023\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4146.00	−0.734582	−0.367291	−	0.930106i	$-0.619715\pi$
$317$	−4146.00	−0.734582	−0.367291	+	0.930106i	$0.619715\pi$
$318$	0	0
$319$	12432.0	2.18200
$320$	0	0
$321$	−5400.00	−0.938936
$322$	0	0
$323$	6660.00	1.14728
$324$	0	0
$325$	−1708.00	−0.291516
$326$	0	0
$327$	10236.0	1.73105
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−10208.0	−1.69511	−0.847557	−	0.530705i	$-0.821927\pi$
$331$	−10208.0	−1.69511	−0.847557	+	0.530705i	$0.821927\pi$
$332$	0	0
$333$	522.000	0.0859022
$334$	0	0
$335$	−3296.00	−0.537551
$336$	0	0
$337$	8078.00	1.30575	0.652873	−	0.757467i	$-0.273563\pi$
$337$	8078.00	1.30575	0.652873	+	0.757467i	$0.273563\pi$
$338$	0	0
$339$	−2652.00	−0.424888
$340$	0	0
$341$	5600.00	0.889317
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−4608.00	−0.719091
$346$	0	0
$347$	3456.00	0.534662	0.267331	−	0.963605i	$-0.413858\pi$
$347$	3456.00	0.534662	0.267331	+	0.963605i	$0.413858\pi$
$348$	0	0
$349$	−6348.00	−0.973641	−0.486820	−	0.873502i	$-0.661843\pi$
$349$	−6348.00	−0.973641	−0.486820	+	0.873502i	$0.661843\pi$
$350$	0	0
$351$	−3024.00	−0.459855
$352$	0	0
$353$	−2290.00	−0.345282	−0.172641	−	0.984985i	$-0.555230\pi$
$353$	−2290.00	−0.345282	−0.172641	+	0.984985i	$0.555230\pi$
$354$	0	0
$355$	3584.00	0.535828
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9912.00	1.45720	0.728601	−	0.684939i	$-0.240171\pi$
$359$	9912.00	1.45720	0.728601	+	0.684939i	$0.240171\pi$
$360$	0	0
$361$	−1383.00	−0.201633
$362$	0	0
$363$	10830.0	1.56592
$364$	0	0
$365$	7952.00	1.14035
$366$	0	0
$367$	4840.00	0.688409	0.344204	−	0.938895i	$-0.388149\pi$
$367$	4840.00	0.688409	0.344204	+	0.938895i	$0.388149\pi$
$368$	0	0
$369$	−3798.00	−0.535816
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−8090.00	−1.12301	−0.561507	−	0.827472i	$-0.689778\pi$
$373$	−8090.00	−1.12301	−0.561507	+	0.827472i	$0.689778\pi$
$374$	0	0
$375$	8928.00	1.22944
$376$	0	0
$377$	−6216.00	−0.849178
$378$	0	0
$379$	7000.00	0.948723	0.474361	−	0.880330i	$-0.342679\pi$
$379$	7000.00	0.948723	0.474361	+	0.880330i	$0.342679\pi$
$380$	0	0
$381$	11712.0	1.57487
$382$	0	0
$383$	10308.0	1.37523	0.687616	−	0.726074i	$-0.258657\pi$
$383$	10308.0	1.37523	0.687616	+	0.726074i	$0.258657\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4608.00	−0.605266
$388$	0	0
$389$	−11510.0	−1.50021	−0.750103	−	0.661321i	$-0.769996\pi$
$389$	−11510.0	−1.50021	−0.750103	+	0.661321i	$0.769996\pi$
$390$	0	0
$391$	8640.00	1.11750
$392$	0	0
$393$	−4452.00	−0.571434
$394$	0	0
$395$	−2368.00	−0.301638
$396$	0	0
$397$	8132.00	1.02804	0.514022	−	0.857777i	$-0.328155\pi$
$397$	8132.00	1.02804	0.514022	+	0.857777i	$0.328155\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1074.00	−0.133748	−0.0668741	−	0.997761i	$-0.521303\pi$
$401$	−1074.00	−0.133748	−0.0668741	+	0.997761i	$0.521303\pi$
$402$	0	0
$403$	−2800.00	−0.346099
$404$	0	0
$405$	7128.00	0.874551
$406$	0	0
$407$	−3248.00	−0.395571
$408$	0	0
$409$	−3406.00	−0.411775	−0.205887	−	0.978576i	$-0.566008\pi$
$409$	−3406.00	−0.411775	−0.205887	+	0.978576i	$0.566008\pi$
$410$	0	0
$411$	−11124.0	−1.33505
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−3088.00	−0.365263
$416$	0	0
$417$	−444.000	−0.0521409
$418$	0	0
$419$	−13522.0	−1.57659	−0.788297	−	0.615295i	$-0.789037\pi$
$419$	−13522.0	−1.57659	−0.788297	+	0.615295i	$0.789037\pi$
$420$	0	0
$421$	1198.00	0.138686	0.0693432	−	0.997593i	$-0.477910\pi$
$421$	1198.00	0.138686	0.0693432	+	0.997593i	$0.477910\pi$
$422$	0	0
$423$	1332.00	0.153107
$424$	0	0
$425$	−5490.00	−0.626598
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9408.00	−1.05879
$430$	0	0
$431$	12824.0	1.43320	0.716601	−	0.697483i	$-0.245697\pi$
$431$	12824.0	1.43320	0.716601	+	0.697483i	$0.245697\pi$
$432$	0	0
$433$	12770.0	1.41729	0.708646	−	0.705565i	$-0.249307\pi$
$433$	12770.0	1.41729	0.708646	+	0.705565i	$0.249307\pi$
$434$	0	0
$435$	10656.0	1.17452
$436$	0	0
$437$	7104.00	0.777644
$438$	0	0
$439$	13224.0	1.43769	0.718846	−	0.695169i	$-0.244670\pi$
$439$	13224.0	1.43769	0.718846	+	0.695169i	$0.244670\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−972.000	−0.104246	−0.0521232	−	0.998641i	$-0.516599\pi$
$443$	−972.000	−0.104246	−0.0521232	+	0.998641i	$0.516599\pi$
$444$	0	0
$445$	−48.0000	−0.00511330
$446$	0	0
$447$	15012.0	1.58846
$448$	0	0
$449$	6034.00	0.634214	0.317107	−	0.948390i	$-0.397289\pi$
$449$	6034.00	0.634214	0.317107	+	0.948390i	$0.397289\pi$
$450$	0	0
$451$	23632.0	2.46738
$452$	0	0
$453$	5328.00	0.552608
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12742.0	1.30426	0.652129	−	0.758108i	$-0.273876\pi$
$457$	12742.0	1.30426	0.652129	+	0.758108i	$0.273876\pi$
$458$	0	0
$459$	−9720.00	−0.988433
$460$	0	0
$461$	−6132.00	−0.619513	−0.309757	−	0.950816i	$-0.600248\pi$
$461$	−6132.00	−0.619513	−0.309757	+	0.950816i	$0.600248\pi$
$462$	0	0
$463$	−2672.00	−0.268204	−0.134102	−	0.990968i	$-0.542815\pi$
$463$	−2672.00	−0.268204	−0.134102	+	0.990968i	$0.542815\pi$
$464$	0	0
$465$	4800.00	0.478698
$466$	0	0
$467$	−17246.0	−1.70889	−0.854443	−	0.519545i	$-0.826101\pi$
$467$	−17246.0	−1.70889	−0.854443	+	0.519545i	$0.826101\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6552.00	−0.640977
$472$	0	0
$473$	28672.0	2.78719
$474$	0	0
$475$	−4514.00	−0.436035
$476$	0	0
$477$	−5778.00	−0.554626
$478$	0	0
$479$	−19812.0	−1.88984	−0.944920	−	0.327301i	$-0.893861\pi$
$479$	−19812.0	−1.88984	−0.944920	+	0.327301i	$0.893861\pi$
$480$	0	0
$481$	1624.00	0.153946
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1104.00	−0.103361
$486$	0	0
$487$	−14864.0	−1.38306	−0.691532	−	0.722346i	$-0.743064\pi$
$487$	−14864.0	−1.38306	−0.691532	+	0.722346i	$0.743064\pi$
$488$	0	0
$489$	10224.0	0.945491
$490$	0	0
$491$	−15756.0	−1.44818	−0.724092	−	0.689703i	$-0.757741\pi$
$491$	−15756.0	−1.44818	−0.724092	+	0.689703i	$0.757741\pi$
$492$	0	0
$493$	−19980.0	−1.82526
$494$	0	0
$495$	4032.00	0.366111
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1604.00	−0.143898	−0.0719488	−	0.997408i	$-0.522922\pi$
$499$	−1604.00	−0.143898	−0.0719488	+	0.997408i	$0.522922\pi$
$500$	0	0
$501$	2616.00	0.233282
$502$	0	0
$503$	−19232.0	−1.70480	−0.852398	−	0.522893i	$-0.824853\pi$
$503$	−19232.0	−1.70480	−0.852398	+	0.522893i	$0.824853\pi$
$504$	0	0
$505$	−5312.00	−0.468081
$506$	0	0
$507$	−8478.00	−0.742645
$508$	0	0
$509$	−76.0000	−0.00661815	−0.00330908	−	0.999995i	$-0.501053\pi$
$509$	−76.0000	−0.00661815	−0.00330908	+	0.999995i	$0.501053\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−7992.00	−0.687827
$514$	0	0
$515$	−16096.0	−1.37723
$516$	0	0
$517$	−8288.00	−0.705040
$518$	0	0
$519$	5976.00	0.505428
$520$	0	0
$521$	12570.0	1.05701	0.528505	−	0.848930i	$-0.322753\pi$
$521$	12570.0	1.05701	0.528505	+	0.848930i	$0.322753\pi$
$522$	0	0
$523$	−11510.0	−0.962327	−0.481164	−	0.876631i	$-0.659786\pi$
$523$	−11510.0	−0.962327	−0.481164	+	0.876631i	$0.659786\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9000.00	−0.743921
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	−2862.00	−0.233899
$532$	0	0
$533$	−11816.0	−0.960240
$534$	0	0
$535$	7200.00	0.581838
$536$	0	0
$537$	−7464.00	−0.599805
$538$	0	0
$539$	0	0
$540$	0	0
$541$	23438.0	1.86262	0.931311	−	0.364225i	$-0.118666\pi$
$541$	23438.0	1.86262	0.931311	+	0.364225i	$0.118666\pi$
$542$	0	0
$543$	−2472.00	−0.195366
$544$	0	0
$545$	−13648.0	−1.07269
$546$	0	0
$547$	5328.00	0.416470	0.208235	−	0.978079i	$-0.433228\pi$
$547$	5328.00	0.416470	0.208235	+	0.978079i	$0.433228\pi$
$548$	0	0
$549$	−6480.00	−0.503752
$550$	0	0
$551$	−16428.0	−1.27016
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2784.00	−0.212927
$556$	0	0
$557$	−2386.00	−0.181505	−0.0907523	−	0.995873i	$-0.528927\pi$
$557$	−2386.00	−0.181505	−0.0907523	+	0.995873i	$0.528927\pi$
$558$	0	0
$559$	−14336.0	−1.08470
$560$	0	0
$561$	−30240.0	−2.27582
$562$	0	0
$563$	13094.0	0.980189	0.490094	−	0.871669i	$-0.336962\pi$
$563$	13094.0	0.980189	0.490094	+	0.871669i	$0.336962\pi$
$564$	0	0
$565$	3536.00	0.263293
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15638.0	1.15216	0.576080	−	0.817393i	$-0.304582\pi$
$569$	15638.0	1.15216	0.576080	+	0.817393i	$0.304582\pi$
$570$	0	0
$571$	4040.00	0.296092	0.148046	−	0.988980i	$-0.452702\pi$
$571$	4040.00	0.296092	0.148046	+	0.988980i	$0.452702\pi$
$572$	0	0
$573$	−6960.00	−0.507431
$574$	0	0
$575$	−5856.00	−0.424717
$576$	0	0
$577$	−9154.00	−0.660461	−0.330231	−	0.943900i	$-0.607126\pi$
$577$	−9154.00	−0.660461	−0.330231	+	0.943900i	$0.607126\pi$
$578$	0	0
$579$	−22188.0	−1.59258
$580$	0	0
$581$	0	0
$582$	0	0
$583$	35952.0	2.55400
$584$	0	0
$585$	−2016.00	−0.142481
$586$	0	0
$587$	−2754.00	−0.193645	−0.0968226	−	0.995302i	$-0.530868\pi$
$587$	−2754.00	−0.193645	−0.0968226	+	0.995302i	$0.530868\pi$
$588$	0	0
$589$	−7400.00	−0.517677
$590$	0	0
$591$	−17244.0	−1.20021
$592$	0	0
$593$	20678.0	1.43195	0.715973	−	0.698128i	$-0.245983\pi$
$593$	20678.0	1.43195	0.715973	+	0.698128i	$0.245983\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20088.0	1.37713
$598$	0	0
$599$	10200.0	0.695761	0.347880	−	0.937539i	$-0.386902\pi$
$599$	10200.0	0.695761	0.347880	+	0.937539i	$0.386902\pi$
$600$	0	0
$601$	23782.0	1.61412	0.807062	−	0.590467i	$-0.201057\pi$
$601$	23782.0	1.61412	0.807062	+	0.590467i	$0.201057\pi$
$602$	0	0
$603$	3708.00	0.250417
$604$	0	0
$605$	−14440.0	−0.970363
$606$	0	0
$607$	9920.00	0.663328	0.331664	−	0.943397i	$-0.392390\pi$
$607$	9920.00	0.663328	0.331664	+	0.943397i	$0.392390\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4144.00	0.274383
$612$	0	0
$613$	−21030.0	−1.38563	−0.692817	−	0.721113i	$-0.743631\pi$
$613$	−21030.0	−1.38563	−0.692817	+	0.721113i	$0.743631\pi$
$614$	0	0
$615$	20256.0	1.32813
$616$	0	0
$617$	8854.00	0.577713	0.288856	−	0.957372i	$-0.406725\pi$
$617$	8854.00	0.577713	0.288856	+	0.957372i	$0.406725\pi$
$618$	0	0
$619$	−1214.00	−0.0788284	−0.0394142	−	0.999223i	$-0.512549\pi$
$619$	−1214.00	−0.0788284	−0.0394142	+	0.999223i	$0.512549\pi$
$620$	0	0
$621$	−10368.0	−0.669973
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	−24864.0	−1.58369
$628$	0	0
$629$	5220.00	0.330898
$630$	0	0
$631$	1200.00	0.0757072	0.0378536	−	0.999283i	$-0.487948\pi$
$631$	1200.00	0.0757072	0.0378536	+	0.999283i	$0.487948\pi$
$632$	0	0
$633$	15336.0	0.962956
$634$	0	0
$635$	−15616.0	−0.975909
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−4032.00	−0.249614
$640$	0	0
$641$	6894.00	0.424800	0.212400	−	0.977183i	$-0.431872\pi$
$641$	6894.00	0.424800	0.212400	+	0.977183i	$0.431872\pi$
$642$	0	0
$643$	13174.0	0.807981	0.403991	−	0.914763i	$-0.367623\pi$
$643$	13174.0	0.807981	0.403991	+	0.914763i	$0.367623\pi$
$644$	0	0
$645$	24576.0	1.50028
$646$	0	0
$647$	1684.00	0.102326	0.0511630	−	0.998690i	$-0.483707\pi$
$647$	1684.00	0.102326	0.0511630	+	0.998690i	$0.483707\pi$
$648$	0	0
$649$	17808.0	1.07708
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20582.0	−1.23344	−0.616720	−	0.787183i	$-0.711539\pi$
$653$	−20582.0	−1.23344	−0.616720	+	0.787183i	$0.711539\pi$
$654$	0	0
$655$	5936.00	0.354105
$656$	0	0
$657$	−8946.00	−0.531228
$658$	0	0
$659$	−22416.0	−1.32504	−0.662522	−	0.749043i	$-0.730514\pi$
$659$	−22416.0	−1.32504	−0.662522	+	0.749043i	$0.730514\pi$
$660$	0	0
$661$	−20864.0	−1.22771	−0.613854	−	0.789419i	$-0.710382\pi$
$661$	−20864.0	−1.22771	−0.613854	+	0.789419i	$0.710382\pi$
$662$	0	0
$663$	15120.0	0.885690
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−21312.0	−1.23719
$668$	0	0
$669$	−1872.00	−0.108185
$670$	0	0
$671$	40320.0	2.31973
$672$	0	0
$673$	−14438.0	−0.826960	−0.413480	−	0.910513i	$-0.635687\pi$
$673$	−14438.0	−0.826960	−0.413480	+	0.910513i	$0.635687\pi$
$674$	0	0
$675$	6588.00	0.375663
$676$	0	0
$677$	2724.00	0.154641	0.0773204	−	0.997006i	$-0.475364\pi$
$677$	2724.00	0.154641	0.0773204	+	0.997006i	$0.475364\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4956.00	0.278876
$682$	0	0
$683$	−5676.00	−0.317988	−0.158994	−	0.987280i	$-0.550825\pi$
$683$	−5676.00	−0.317988	−0.158994	+	0.987280i	$0.550825\pi$
$684$	0	0
$685$	14832.0	0.827302
$686$	0	0
$687$	−24216.0	−1.34483
$688$	0	0
$689$	−17976.0	−0.993949
$690$	0	0
$691$	−2534.00	−0.139505	−0.0697525	−	0.997564i	$-0.522221\pi$
$691$	−2534.00	−0.139505	−0.0697525	+	0.997564i	$0.522221\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	592.000	0.0323106
$696$	0	0
$697$	−37980.0	−2.06398
$698$	0	0
$699$	−17412.0	−0.942178
$700$	0	0
$701$	−15558.0	−0.838256	−0.419128	−	0.907927i	$-0.637664\pi$
$701$	−15558.0	−0.838256	−0.419128	+	0.907927i	$0.637664\pi$
$702$	0	0
$703$	4292.00	0.230264
$704$	0	0
$705$	−7104.00	−0.379507
$706$	0	0
$707$	0	0
$708$	0	0
$709$	14538.0	0.770079	0.385040	−	0.922900i	$-0.374188\pi$
$709$	14538.0	0.770079	0.385040	+	0.922900i	$0.374188\pi$
$710$	0	0
$711$	2664.00	0.140517
$712$	0	0
$713$	−9600.00	−0.504240
$714$	0	0
$715$	12544.0	0.656110
$716$	0	0
$717$	−19056.0	−0.992551
$718$	0	0
$719$	16364.0	0.848782	0.424391	−	0.905479i	$-0.360488\pi$
$719$	16364.0	0.848782	0.424391	+	0.905479i	$0.360488\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	28092.0	1.44502
$724$	0	0
$725$	13542.0	0.693707
$726$	0	0
$727$	−24884.0	−1.26946	−0.634729	−	0.772735i	$-0.718888\pi$
$727$	−24884.0	−1.26946	−0.634729	+	0.772735i	$0.718888\pi$
$728$	0	0
$729$	9477.00	0.481481
$730$	0	0
$731$	−46080.0	−2.33151
$732$	0	0
$733$	−9248.00	−0.466006	−0.233003	−	0.972476i	$-0.574855\pi$
$733$	−9248.00	−0.466006	−0.233003	+	0.972476i	$0.574855\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−23072.0	−1.15315
$738$	0	0
$739$	−3928.00	−0.195526	−0.0977631	−	0.995210i	$-0.531169\pi$
$739$	−3928.00	−0.195526	−0.0977631	+	0.995210i	$0.531169\pi$
$740$	0	0
$741$	12432.0	0.616331
$742$	0	0
$743$	−2840.00	−0.140228	−0.0701141	−	0.997539i	$-0.522336\pi$
$743$	−2840.00	−0.140228	−0.0701141	+	0.997539i	$0.522336\pi$
$744$	0	0
$745$	−20016.0	−0.984335
$746$	0	0
$747$	3474.00	0.170157
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−904.000	−0.0439247	−0.0219623	−	0.999759i	$-0.506991\pi$
$751$	−904.000	−0.0439247	−0.0219623	+	0.999759i	$0.506991\pi$
$752$	0	0
$753$	3780.00	0.182936
$754$	0	0
$755$	−7104.00	−0.342438
$756$	0	0
$757$	−326.000	−0.0156521	−0.00782607	−	0.999969i	$-0.502491\pi$
$757$	−326.000	−0.0156521	−0.00782607	+	0.999969i	$0.502491\pi$
$758$	0	0
$759$	−32256.0	−1.54258
$760$	0	0
$761$	−13902.0	−0.662217	−0.331108	−	0.943593i	$-0.607423\pi$
$761$	−13902.0	−0.662217	−0.331108	+	0.943593i	$0.607423\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−6480.00	−0.306255
$766$	0	0
$767$	−8904.00	−0.419172
$768$	0	0
$769$	19378.0	0.908698	0.454349	−	0.890824i	$-0.349872\pi$
$769$	19378.0	0.908698	0.454349	+	0.890824i	$0.349872\pi$
$770$	0	0
$771$	11220.0	0.524097
$772$	0	0
$773$	33280.0	1.54851	0.774255	−	0.632874i	$-0.218125\pi$
$773$	33280.0	1.54851	0.774255	+	0.632874i	$0.218125\pi$
$774$	0	0
$775$	6100.00	0.282734
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31228.0	−1.43628
$780$	0	0
$781$	25088.0	1.14945
$782$	0	0
$783$	23976.0	1.09429
$784$	0	0
$785$	8736.00	0.397199
$786$	0	0
$787$	−32974.0	−1.49351	−0.746757	−	0.665097i	$-0.768390\pi$
$787$	−32974.0	−1.49351	−0.746757	+	0.665097i	$0.768390\pi$
$788$	0	0
$789$	−20160.0	−0.909651
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−20160.0	−0.902778
$794$	0	0
$795$	30816.0	1.37476
$796$	0	0
$797$	−17244.0	−0.766391	−0.383196	−	0.923667i	$-0.625176\pi$
$797$	−17244.0	−0.766391	−0.383196	+	0.923667i	$0.625176\pi$
$798$	0	0
$799$	13320.0	0.589772
$800$	0	0
$801$	54.0000	0.00238202
$802$	0	0
$803$	55664.0	2.44625
$804$	0	0
$805$	0	0
$806$	0	0
$807$	35544.0	1.55044
$808$	0	0
$809$	−27178.0	−1.18112	−0.590561	−	0.806993i	$-0.701093\pi$
$809$	−27178.0	−1.18112	−0.590561	+	0.806993i	$0.701093\pi$
$810$	0	0
$811$	33514.0	1.45109	0.725546	−	0.688174i	$-0.241587\pi$
$811$	33514.0	1.45109	0.725546	+	0.688174i	$0.241587\pi$
$812$	0	0
$813$	12384.0	0.534226
$814$	0	0
$815$	−13632.0	−0.585900
$816$	0	0
$817$	−37888.0	−1.62244
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10978.0	−0.466669	−0.233334	−	0.972397i	$-0.574964\pi$
$821$	−10978.0	−0.466669	−0.233334	+	0.972397i	$0.574964\pi$
$822$	0	0
$823$	2376.00	0.100634	0.0503172	−	0.998733i	$-0.483977\pi$
$823$	2376.00	0.100634	0.0503172	+	0.998733i	$0.483977\pi$
$824$	0	0
$825$	20496.0	0.864944
$826$	0	0
$827$	12844.0	0.540060	0.270030	−	0.962852i	$-0.412966\pi$
$827$	12844.0	0.540060	0.270030	+	0.962852i	$0.412966\pi$
$828$	0	0
$829$	30880.0	1.29374	0.646868	−	0.762602i	$-0.276079\pi$
$829$	30880.0	1.29374	0.646868	+	0.762602i	$0.276079\pi$
$830$	0	0
$831$	−19884.0	−0.830046
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3488.00	−0.144560
$836$	0	0
$837$	10800.0	0.446001
$838$	0	0
$839$	−27924.0	−1.14904	−0.574519	−	0.818491i	$-0.694811\pi$
$839$	−27924.0	−1.14904	−0.574519	+	0.818491i	$0.694811\pi$
$840$	0	0
$841$	24895.0	1.02075
$842$	0	0
$843$	18108.0	0.739825
$844$	0	0
$845$	11304.0	0.460200
$846$	0	0
$847$	0	0
$848$	0	0
$849$	40812.0	1.64978
$850$	0	0
$851$	5568.00	0.224287
$852$	0	0
$853$	10636.0	0.426928	0.213464	−	0.976951i	$-0.431525\pi$
$853$	10636.0	0.426928	0.213464	+	0.976951i	$0.431525\pi$
$854$	0	0
$855$	−5328.00	−0.213116
$856$	0	0
$857$	−41934.0	−1.67146	−0.835728	−	0.549143i	$-0.814954\pi$
$857$	−41934.0	−1.67146	−0.835728	+	0.549143i	$0.814954\pi$
$858$	0	0
$859$	−15946.0	−0.633377	−0.316688	−	0.948530i	$-0.602571\pi$
$859$	−15946.0	−0.633377	−0.316688	+	0.948530i	$0.602571\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16984.0	−0.669921	−0.334961	−	0.942232i	$-0.608723\pi$
$863$	−16984.0	−0.669921	−0.334961	+	0.942232i	$0.608723\pi$
$864$	0	0
$865$	−7968.00	−0.313202
$866$	0	0
$867$	19122.0	0.749039
$868$	0	0
$869$	−16576.0	−0.647068
$870$	0	0
$871$	11536.0	0.448774
$872$	0	0
$873$	1242.00	0.0481504
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6702.00	−0.258051	−0.129025	−	0.991641i	$-0.541185\pi$
$877$	−6702.00	−0.258051	−0.129025	+	0.991641i	$0.541185\pi$
$878$	0	0
$879$	−16704.0	−0.640969
$880$	0	0
$881$	−36810.0	−1.40767	−0.703837	−	0.710362i	$-0.748531\pi$
$881$	−36810.0	−1.40767	−0.703837	+	0.710362i	$0.748531\pi$
$882$	0	0
$883$	5060.00	0.192845	0.0964227	−	0.995340i	$-0.469260\pi$
$883$	5060.00	0.192845	0.0964227	+	0.995340i	$0.469260\pi$
$884$	0	0
$885$	15264.0	0.579767
$886$	0	0
$887$	32156.0	1.21724	0.608621	−	0.793461i	$-0.291723\pi$
$887$	32156.0	1.21724	0.608621	+	0.793461i	$0.291723\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	49896.0	1.87607
$892$	0	0
$893$	10952.0	0.410408
$894$	0	0
$895$	9952.00	0.371686
$896$	0	0
$897$	16128.0	0.600332
$898$	0	0
$899$	22200.0	0.823594
$900$	0	0
$901$	−57780.0	−2.13644
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3296.00	0.121064
$906$	0	0
$907$	2716.00	0.0994303	0.0497152	−	0.998763i	$-0.484169\pi$
$907$	2716.00	0.0994303	0.0497152	+	0.998763i	$0.484169\pi$
$908$	0	0
$909$	5976.00	0.218054
$910$	0	0
$911$	−42048.0	−1.52921	−0.764606	−	0.644498i	$-0.777066\pi$
$911$	−42048.0	−1.52921	−0.764606	+	0.644498i	$0.777066\pi$
$912$	0	0
$913$	−21616.0	−0.783554
$914$	0	0
$915$	34560.0	1.24865
$916$	0	0
$917$	0	0
$918$	0	0
$919$	7816.00	0.280551	0.140275	−	0.990113i	$-0.455201\pi$
$919$	7816.00	0.280551	0.140275	+	0.990113i	$0.455201\pi$
$920$	0	0
$921$	−33924.0	−1.21372
$922$	0	0
$923$	−12544.0	−0.447336
$924$	0	0
$925$	−3538.00	−0.125761
$926$	0	0
$927$	18108.0	0.641580
$928$	0	0
$929$	15578.0	0.550159	0.275079	−	0.961422i	$-0.411296\pi$
$929$	15578.0	0.550159	0.275079	+	0.961422i	$0.411296\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	15024.0	0.527185
$934$	0	0
$935$	40320.0	1.41027
$936$	0	0
$937$	−46210.0	−1.61112	−0.805558	−	0.592518i	$-0.798134\pi$
$937$	−46210.0	−1.61112	−0.805558	+	0.592518i	$0.798134\pi$
$938$	0	0
$939$	−38724.0	−1.34580
$940$	0	0
$941$	24816.0	0.859701	0.429850	−	0.902900i	$-0.358566\pi$
$941$	24816.0	0.859701	0.429850	+	0.902900i	$0.358566\pi$
$942$	0	0
$943$	−40512.0	−1.39899
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12800.0	0.439223	0.219611	−	0.975587i	$-0.429521\pi$
$947$	12800.0	0.439223	0.219611	+	0.975587i	$0.429521\pi$
$948$	0	0
$949$	−27832.0	−0.952018
$950$	0	0
$951$	−24876.0	−0.848222
$952$	0	0
$953$	−16902.0	−0.574512	−0.287256	−	0.957854i	$-0.592743\pi$
$953$	−16902.0	−0.574512	−0.287256	+	0.957854i	$0.592743\pi$
$954$	0	0
$955$	9280.00	0.314444
$956$	0	0
$957$	74592.0	2.51956
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−19791.0	−0.664328
$962$	0	0
$963$	−8100.00	−0.271048
$964$	0	0
$965$	29584.0	0.986884
$966$	0	0
$967$	4000.00	0.133021	0.0665105	−	0.997786i	$-0.478813\pi$
$967$	4000.00	0.133021	0.0665105	+	0.997786i	$0.478813\pi$
$968$	0	0
$969$	39960.0	1.32477
$970$	0	0
$971$	−27486.0	−0.908412	−0.454206	−	0.890897i	$-0.650077\pi$
$971$	−27486.0	−0.908412	−0.454206	+	0.890897i	$0.650077\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10248.0	−0.336614
$976$	0	0
$977$	−1126.00	−0.0368720	−0.0184360	−	0.999830i	$-0.505869\pi$
$977$	−1126.00	−0.0368720	−0.0184360	+	0.999830i	$0.505869\pi$
$978$	0	0
$979$	−336.000	−0.0109690
$980$	0	0
$981$	15354.0	0.499710
$982$	0	0
$983$	4668.00	0.151461	0.0757305	−	0.997128i	$-0.475871\pi$
$983$	4668.00	0.151461	0.0757305	+	0.997128i	$0.475871\pi$
$984$	0	0
$985$	22992.0	0.743742
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−49152.0	−1.58033
$990$	0	0
$991$	−2048.00	−0.0656477	−0.0328238	−	0.999461i	$-0.510450\pi$
$991$	−2048.00	−0.0656477	−0.0328238	+	0.999461i	$0.510450\pi$
$992$	0	0
$993$	−61248.0	−1.95735
$994$	0	0
$995$	−26784.0	−0.853377
$996$	0	0
$997$	26864.0	0.853351	0.426676	−	0.904405i	$-0.359685\pi$
$997$	26864.0	0.853351	0.426676	+	0.904405i	$0.359685\pi$
$998$	0	0
$999$	−6264.00	−0.198383

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.q.1.1		1
4.3	odd	2		392.4.a.a.1.1		1
7.6	odd	2		112.4.a.b.1.1		1
21.20	even	2		1008.4.a.e.1.1		1
28.3	even	6		392.4.i.a.177.1		2
28.11	odd	6		392.4.i.h.177.1		2
28.19	even	6		392.4.i.a.361.1		2
28.23	odd	6		392.4.i.h.361.1		2
28.27	even	2		56.4.a.b.1.1	✓	1
56.13	odd	2		448.4.a.n.1.1		1
56.27	even	2		448.4.a.c.1.1		1
84.83	odd	2		504.4.a.a.1.1		1
140.27	odd	4		1400.4.g.b.449.1		2
140.83	odd	4		1400.4.g.b.449.2		2
140.139	even	2		1400.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.b.1.1	✓	1	28.27	even	2
112.4.a.b.1.1		1	7.6	odd	2
392.4.a.a.1.1		1	4.3	odd	2
392.4.i.a.177.1		2	28.3	even	6
392.4.i.a.361.1		2	28.19	even	6
392.4.i.h.177.1		2	28.11	odd	6
392.4.i.h.361.1		2	28.23	odd	6
448.4.a.c.1.1		1	56.27	even	2
448.4.a.n.1.1		1	56.13	odd	2
504.4.a.a.1.1		1	84.83	odd	2
784.4.a.q.1.1		1	1.1	even	1	trivial
1008.4.a.e.1.1		1	21.20	even	2
1400.4.a.b.1.1		1	140.139	even	2
1400.4.g.b.449.1		2	140.27	odd	4
1400.4.g.b.449.2		2	140.83	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

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L-functions

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