LMFDB - Embedded newform 448.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,4,Mod(1,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, 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("448.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$448 = 2^{6} \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	448.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:	$26.4328556826$
Analytic rank:	$0$
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	$\chi$	$=$	448.1

$q$ -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} -7.00000 q^{7} +9.00000 q^{9} -56.0000 q^{11} +28.0000 q^{13} +48.0000 q^{15} -90.0000 q^{17} -74.0000 q^{19} +42.0000 q^{21} -96.0000 q^{23} -61.0000 q^{25} +108.000 q^{27} +222.000 q^{29} -100.000 q^{31} +336.000 q^{33} +56.0000 q^{35} -58.0000 q^{37} -168.000 q^{39} +422.000 q^{41} -512.000 q^{43} -72.0000 q^{45} +148.000 q^{47} +49.0000 q^{49} +540.000 q^{51} +642.000 q^{53} +448.000 q^{55} +444.000 q^{57} +318.000 q^{59} -720.000 q^{61} -63.0000 q^{63} -224.000 q^{65} +412.000 q^{67} +576.000 q^{69} +448.000 q^{71} +994.000 q^{73} +366.000 q^{75} +392.000 q^{77} -296.000 q^{79} -891.000 q^{81} -386.000 q^{83} +720.000 q^{85} -1332.00 q^{87} -6.00000 q^{89} -196.000 q^{91} +600.000 q^{93} +592.000 q^{95} -138.000 q^{97} -504.000 q^{99} +O(q^{100})

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	$-0.695913\pi$
$3$	−6.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	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
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	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	$-0.721898\pi$
$17$	−90.0000	−1.28401	−0.642006	+	0.766700i	$0.721898\pi$
$18$	0	0
$19$	−74.0000	−0.893514	−0.446757	−	0.894655i	$-0.647421\pi$
$19$	−74.0000	−0.893514	−0.446757	+	0.894655i	$0.647421\pi$
$20$	0	0
$21$	42.0000	0.436436
$22$	0	0
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	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	$-0.248349\pi$
$29$	222.000	1.42153	0.710765	+	0.703430i	$0.248349\pi$
$30$	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
$35$	56.0000	0.270449
$36$	0	0
$37$	−58.0000	−0.257707	−0.128853	−	0.991664i	$-0.541130\pi$
$37$	−58.0000	−0.257707	−0.128853	+	0.991664i	$0.541130\pi$
$38$	0	0
$39$	−168.000	−0.689783
$40$	0	0
$41$	422.000	1.60745	0.803724	−	0.595003i	$-0.202849\pi$
$41$	422.000	1.60745	0.803724	+	0.595003i	$0.202849\pi$
$42$	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
$49$	49.0000	0.142857
$50$	0	0
$51$	540.000	1.48265
$52$	0	0
$53$	642.000	1.66388	0.831939	−	0.554868i	$-0.187231\pi$
$53$	642.000	1.66388	0.831939	+	0.554868i	$0.187231\pi$
$54$	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	$-0.385893\pi$
$59$	318.000	0.701696	0.350848	+	0.936432i	$0.385893\pi$
$60$	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
$63$	−63.0000	−0.125988
$64$	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	$-0.377841\pi$
$71$	448.000	0.748843	0.374421	+	0.927259i	$0.377841\pi$
$72$	0	0
$73$	994.000	1.59368	0.796842	−	0.604188i	$-0.206502\pi$
$73$	994.000	1.59368	0.796842	+	0.604188i	$0.206502\pi$
$74$	0	0
$75$	366.000	0.563494
$76$	0	0
$77$	392.000	0.580163
$78$	0	0
$79$	−296.000	−0.421552	−0.210776	−	0.977534i	$-0.567599\pi$
$79$	−296.000	−0.421552	−0.210776	+	0.977534i	$0.567599\pi$
$80$	0	0
$81$	−891.000	−1.22222
$82$	0	0
$83$	−386.000	−0.510470	−0.255235	−	0.966879i	$-0.582153\pi$
$83$	−386.000	−0.510470	−0.255235	+	0.966879i	$0.582153\pi$
$84$	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	$-0.501137\pi$
$89$	−6.00000	−0.00714605	−0.00357303	+	0.999994i	$0.501137\pi$
$90$	0	0
$91$	−196.000	−0.225784
$92$	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	$-0.523010\pi$
$97$	−138.000	−0.144451	−0.0722257	+	0.997388i	$0.523010\pi$
$98$	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
$105$	−336.000	−0.312288
$106$	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	$-0.769737\pi$
$109$	−1706.00	−1.49913	−0.749565	+	0.661931i	$0.769737\pi$
$110$	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
$119$	630.000	0.485311
$120$	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	$-0.738862\pi$
$127$	−1952.00	−1.36387	−0.681937	+	0.731411i	$0.738862\pi$
$128$	0	0
$129$	3072.00	2.09670
$130$	0	0
$131$	742.000	0.494877	0.247438	−	0.968904i	$-0.420411\pi$
$131$	742.000	0.494877	0.247438	+	0.968904i	$0.420411\pi$
$132$	0	0
$133$	518.000	0.337717
$134$	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	$-0.492813\pi$
$139$	74.0000	0.0451554	0.0225777	+	0.999745i	$0.492813\pi$
$140$	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
$147$	−294.000	−0.164957
$148$	0	0
$149$	−2502.00	−1.37565	−0.687825	−	0.725877i	$-0.741434\pi$
$149$	−2502.00	−1.37565	−0.687825	+	0.725877i	$0.741434\pi$
$150$	0	0
$151$	−888.000	−0.478572	−0.239286	−	0.970949i	$-0.576913\pi$
$151$	−888.000	−0.478572	−0.239286	+	0.970949i	$0.576913\pi$
$152$	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
$161$	672.000	0.328950
$162$	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
$175$	427.000	0.184447
$176$	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
$189$	−756.000	−0.290957
$190$	0	0
$191$	1160.00	0.439448	0.219724	−	0.975562i	$-0.429484\pi$
$191$	1160.00	0.439448	0.219724	+	0.975562i	$0.429484\pi$
$192$	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	$-0.326042\pi$
$197$	2874.00	1.03941	0.519706	+	0.854345i	$0.326042\pi$
$198$	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
$203$	−1554.00	−0.537288
$204$	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
$217$	700.000	0.218982
$218$	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	$-0.538532\pi$
$227$	−826.000	−0.241513	−0.120757	+	0.992682i	$0.538532\pi$
$228$	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
$231$	−2352.00	−0.669914
$232$	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	$-0.358589\pi$
$239$	3176.00	0.859575	0.429787	+	0.902930i	$0.358589\pi$
$240$	0	0
$241$	−4682.00	−1.25143	−0.625714	−	0.780053i	$-0.715192\pi$
$241$	−4682.00	−1.25143	−0.625714	+	0.780053i	$0.715192\pi$
$242$	0	0
$243$	2430.00	0.641500
$244$	0	0
$245$	−392.000	−0.102220
$246$	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	$-0.525241\pi$
$251$	−630.000	−0.158427	−0.0792136	+	0.996858i	$0.525241\pi$
$252$	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	$-0.572872\pi$
$257$	−1870.00	−0.453881	−0.226940	+	0.973909i	$0.572872\pi$
$258$	0	0
$259$	406.000	0.0974039
$260$	0	0
$261$	1998.00	0.473843
$262$	0	0
$263$	3360.00	0.787781	0.393891	−	0.919157i	$-0.371129\pi$
$263$	3360.00	0.787781	0.393891	+	0.919157i	$0.371129\pi$
$264$	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
$273$	1176.00	0.260713
$274$	0	0
$275$	3416.00	0.749064
$276$	0	0
$277$	3314.00	0.718841	0.359421	−	0.933176i	$-0.382974\pi$
$277$	3314.00	0.718841	0.359421	+	0.933176i	$0.382974\pi$
$278$	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	$-0.753289\pi$
$283$	−6802.00	−1.42875	−0.714376	+	0.699762i	$0.753289\pi$
$284$	0	0
$285$	−3552.00	−0.738254
$286$	0	0
$287$	−2954.00	−0.607558
$288$	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
$301$	3584.00	0.686307
$302$	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	$-0.323858\pi$
$307$	5654.00	1.05111	0.525555	+	0.850760i	$0.323858\pi$
$308$	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	$-0.301977\pi$
$313$	6454.00	1.16550	0.582750	+	0.812651i	$0.301977\pi$
$314$	0	0
$315$	504.000	0.0901498
$316$	0	0
$317$	4146.00	0.734582	0.367291	−	0.930106i	$-0.380285\pi$
$317$	4146.00	0.734582	0.367291	+	0.930106i	$0.380285\pi$
$318$	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
$329$	−1036.00	−0.173606
$330$	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
$343$	−343.000	−0.0539949
$344$	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	$-0.444770\pi$
$353$	2290.00	0.345282	0.172641	+	0.984985i	$0.444770\pi$
$354$	0	0
$355$	−3584.00	−0.535828
$356$	0	0
$357$	−3780.00	−0.560389
$358$	0	0
$359$	−9912.00	−1.45720	−0.728601	−	0.684939i	$-0.759829\pi$
$359$	−9912.00	−1.45720	−0.728601	+	0.684939i	$0.759829\pi$
$360$	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
$371$	−4494.00	−0.628886
$372$	0	0
$373$	8090.00	1.12301	0.561507	−	0.827472i	$-0.310222\pi$
$373$	8090.00	1.12301	0.561507	+	0.827472i	$0.310222\pi$
$374$	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
$385$	−3136.00	−0.415131
$386$	0	0
$387$	−4608.00	−0.605266
$388$	0	0
$389$	11510.0	1.50021	0.750103	−	0.661321i	$-0.230004\pi$
$389$	11510.0	1.50021	0.750103	+	0.661321i	$0.230004\pi$
$390$	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
$399$	−3108.00	−0.389961
$400$	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	$-0.433992\pi$
$409$	3406.00	0.411775	0.205887	+	0.978576i	$0.433992\pi$
$410$	0	0
$411$	11124.0	1.33505
$412$	0	0
$413$	−2226.00	−0.265216
$414$	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	$-0.210963\pi$
$419$	13522.0	1.57659	0.788297	+	0.615295i	$0.210963\pi$
$420$	0	0
$421$	−1198.00	−0.138686	−0.0693432	−	0.997593i	$-0.522090\pi$
$421$	−1198.00	−0.138686	−0.0693432	+	0.997593i	$0.522090\pi$
$422$	0	0
$423$	1332.00	0.153107
$424$	0	0
$425$	5490.00	0.626598
$426$	0	0
$427$	5040.00	0.571201
$428$	0	0
$429$	9408.00	1.05879
$430$	0	0
$431$	−12824.0	−1.43320	−0.716601	−	0.697483i	$-0.754303\pi$
$431$	−12824.0	−1.43320	−0.716601	+	0.697483i	$0.754303\pi$
$432$	0	0
$433$	−12770.0	−1.41729	−0.708646	−	0.705565i	$-0.750693\pi$
$433$	−12770.0	−1.41729	−0.708646	+	0.705565i	$0.750693\pi$
$434$	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
$441$	441.000	0.0476190
$442$	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
$455$	1568.00	0.161558
$456$	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	$-0.457185\pi$
$463$	2672.00	0.268204	0.134102	+	0.990968i	$0.457185\pi$
$464$	0	0
$465$	−4800.00	−0.478698
$466$	0	0
$467$	17246.0	1.70889	0.854443	−	0.519545i	$-0.173899\pi$
$467$	17246.0	1.70889	0.854443	+	0.519545i	$0.173899\pi$
$468$	0	0
$469$	−2884.00	−0.283946
$470$	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
$483$	−4032.00	−0.379839
$484$	0	0
$485$	1104.00	0.103361
$486$	0	0
$487$	14864.0	1.38306	0.691532	−	0.722346i	$-0.256936\pi$
$487$	14864.0	1.38306	0.691532	+	0.722346i	$0.256936\pi$
$488$	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
$497$	−3136.00	−0.283036
$498$	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
$511$	−6958.00	−0.602356
$512$	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	$-0.677247\pi$
$521$	−12570.0	−1.05701	−0.528505	+	0.848930i	$0.677247\pi$
$522$	0	0
$523$	11510.0	0.962327	0.481164	−	0.876631i	$-0.340214\pi$
$523$	11510.0	0.962327	0.481164	+	0.876631i	$0.340214\pi$
$524$	0	0
$525$	−2562.00	−0.212981
$526$	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
$539$	−2744.00	−0.219281
$540$	0	0
$541$	−23438.0	−1.86262	−0.931311	−	0.364225i	$-0.881334\pi$
$541$	−23438.0	−1.86262	−0.931311	+	0.364225i	$0.881334\pi$
$542$	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
$553$	2072.00	0.159332
$554$	0	0
$555$	−2784.00	−0.212927
$556$	0	0
$557$	2386.00	0.181505	0.0907523	−	0.995873i	$-0.471073\pi$
$557$	2386.00	0.181505	0.0907523	+	0.995873i	$0.471073\pi$
$558$	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	$-0.663038\pi$
$563$	−13094.0	−0.980189	−0.490094	+	0.871669i	$0.663038\pi$
$564$	0	0
$565$	3536.00	0.263293
$566$	0	0
$567$	6237.00	0.461957
$568$	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	$-0.392874\pi$
$577$	9154.00	0.660461	0.330231	+	0.943900i	$0.392874\pi$
$578$	0	0
$579$	22188.0	1.59258
$580$	0	0
$581$	2702.00	0.192939
$582$	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	$-0.469132\pi$
$587$	2754.00	0.193645	0.0968226	+	0.995302i	$0.469132\pi$
$588$	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	$-0.754017\pi$
$593$	−20678.0	−1.43195	−0.715973	+	0.698128i	$0.754017\pi$
$594$	0	0
$595$	−5040.00	−0.347260
$596$	0	0
$597$	−20088.0	−1.37713
$598$	0	0
$599$	−10200.0	−0.695761	−0.347880	−	0.937539i	$-0.613098\pi$
$599$	−10200.0	−0.695761	−0.347880	+	0.937539i	$0.613098\pi$
$600$	0	0
$601$	−23782.0	−1.61412	−0.807062	−	0.590467i	$-0.798943\pi$
$601$	−23782.0	−1.61412	−0.807062	+	0.590467i	$0.798943\pi$
$602$	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
$609$	9324.00	0.620406
$610$	0	0
$611$	4144.00	0.274383
$612$	0	0
$613$	21030.0	1.38563	0.692817	−	0.721113i	$-0.256369\pi$
$613$	21030.0	1.38563	0.692817	+	0.721113i	$0.256369\pi$
$614$	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	$-0.487451\pi$
$619$	1214.00	0.0788284	0.0394142	+	0.999223i	$0.487451\pi$
$620$	0	0
$621$	−10368.0	−0.669973
$622$	0	0
$623$	42.0000	0.00270095
$624$	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	$-0.512052\pi$
$631$	−1200.00	−0.0757072	−0.0378536	+	0.999283i	$0.512052\pi$
$632$	0	0
$633$	−15336.0	−0.962956
$634$	0	0
$635$	15616.0	0.975909
$636$	0	0
$637$	1372.00	0.0853385
$638$	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	$-0.632377\pi$
$643$	−13174.0	−0.807981	−0.403991	+	0.914763i	$0.632377\pi$
$644$	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
$651$	−4200.00	−0.252859
$652$	0	0
$653$	20582.0	1.23344	0.616720	−	0.787183i	$-0.288461\pi$
$653$	20582.0	1.23344	0.616720	+	0.787183i	$0.288461\pi$
$654$	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
$665$	−4144.00	−0.241650
$666$	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
$679$	966.000	0.0545975
$680$	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	$-0.477779\pi$
$691$	2534.00	0.139505	0.0697525	+	0.997564i	$0.477779\pi$
$692$	0	0
$693$	3528.00	0.193388
$694$	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	$-0.362336\pi$
$701$	15558.0	0.838256	0.419128	+	0.907927i	$0.362336\pi$
$702$	0	0
$703$	4292.00	0.230264
$704$	0	0
$705$	7104.00	0.379507
$706$	0	0
$707$	−4648.00	−0.247250
$708$	0	0
$709$	−14538.0	−0.770079	−0.385040	−	0.922900i	$-0.625812\pi$
$709$	−14538.0	−0.770079	−0.385040	+	0.922900i	$0.625812\pi$
$710$	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
$721$	−14084.0	−0.727483
$722$	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
$735$	2352.00	0.118034
$736$	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	$-0.477664\pi$
$743$	2840.00	0.140228	0.0701141	+	0.997539i	$0.477664\pi$
$744$	0	0
$745$	20016.0	0.984335
$746$	0	0
$747$	−3474.00	−0.170157
$748$	0	0
$749$	6300.00	0.307339
$750$	0	0
$751$	904.000	0.0439247	0.0219623	−	0.999759i	$-0.493009\pi$
$751$	904.000	0.0439247	0.0219623	+	0.999759i	$0.493009\pi$
$752$	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	$-0.497509\pi$
$757$	326.000	0.0156521	0.00782607	+	0.999969i	$0.497509\pi$
$758$	0	0
$759$	−32256.0	−1.54258
$760$	0	0
$761$	13902.0	0.662217	0.331108	−	0.943593i	$-0.392577\pi$
$761$	13902.0	0.662217	0.331108	+	0.943593i	$0.392577\pi$
$762$	0	0
$763$	11942.0	0.566618
$764$	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	$-0.650128\pi$
$769$	−19378.0	−0.908698	−0.454349	+	0.890824i	$0.650128\pi$
$770$	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
$777$	−2436.00	−0.112472
$778$	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	$-0.231610\pi$
$787$	32974.0	1.49351	0.746757	+	0.665097i	$0.231610\pi$
$788$	0	0
$789$	−20160.0	−0.909651
$790$	0	0
$791$	3094.00	0.139077
$792$	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
$805$	−5376.00	−0.235378
$806$	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	$-0.758413\pi$
$811$	−33514.0	−1.45109	−0.725546	+	0.688174i	$0.758413\pi$
$812$	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
$819$	−1764.00	−0.0752615
$820$	0	0
$821$	10978.0	0.466669	0.233334	−	0.972397i	$-0.425036\pi$
$821$	10978.0	0.466669	0.233334	+	0.972397i	$0.425036\pi$
$822$	0	0
$823$	−2376.00	−0.100634	−0.0503172	−	0.998733i	$-0.516023\pi$
$823$	−2376.00	−0.100634	−0.0503172	+	0.998733i	$0.516023\pi$
$824$	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
$833$	−4410.00	−0.183430
$834$	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
$847$	−12635.0	−0.512566
$848$	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	$-0.185046\pi$
$857$	41934.0	1.67146	0.835728	+	0.549143i	$0.185046\pi$
$858$	0	0
$859$	15946.0	0.633377	0.316688	−	0.948530i	$-0.397429\pi$
$859$	15946.0	0.633377	0.316688	+	0.948530i	$0.397429\pi$
$860$	0	0
$861$	17724.0	0.701547
$862$	0	0
$863$	16984.0	0.669921	0.334961	−	0.942232i	$-0.391277\pi$
$863$	16984.0	0.669921	0.334961	+	0.942232i	$0.391277\pi$
$864$	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
$875$	−10416.0	−0.402429
$876$	0	0
$877$	6702.00	0.258051	0.129025	−	0.991641i	$-0.458815\pi$
$877$	6702.00	0.258051	0.129025	+	0.991641i	$0.458815\pi$
$878$	0	0
$879$	16704.0	0.640969
$880$	0	0
$881$	36810.0	1.40767	0.703837	−	0.710362i	$-0.251469\pi$
$881$	36810.0	1.40767	0.703837	+	0.710362i	$0.251469\pi$
$882$	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
$889$	13664.0	0.515496
$890$	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
$903$	−21504.0	−0.792479
$904$	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	$-0.222934\pi$
$911$	42048.0	1.52921	0.764606	+	0.644498i	$0.222934\pi$
$912$	0	0
$913$	21616.0	0.783554
$914$	0	0
$915$	−34560.0	−1.24865
$916$	0	0
$917$	−5194.00	−0.187046
$918$	0	0
$919$	−7816.00	−0.280551	−0.140275	−	0.990113i	$-0.544799\pi$
$919$	−7816.00	−0.280551	−0.140275	+	0.990113i	$0.544799\pi$
$920$	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	$-0.588704\pi$
$929$	−15578.0	−0.550159	−0.275079	+	0.961422i	$0.588704\pi$
$930$	0	0
$931$	−3626.00	−0.127645
$932$	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	$-0.201866\pi$
$937$	46210.0	1.61112	0.805558	+	0.592518i	$0.201866\pi$
$938$	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
$945$	6048.00	0.208192
$946$	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
$959$	12978.0	0.436998
$960$	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	$-0.521187\pi$
$967$	−4000.00	−0.133021	−0.0665105	+	0.997786i	$0.521187\pi$
$968$	0	0
$969$	−39960.0	−1.32477
$970$	0	0
$971$	27486.0	0.908412	0.454206	−	0.890897i	$-0.349923\pi$
$971$	27486.0	0.908412	0.454206	+	0.890897i	$0.349923\pi$
$972$	0	0
$973$	−518.000	−0.0170671
$974$	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
$987$	6216.00	0.200463
$988$	0	0
$989$	49152.0	1.58033
$990$	0	0
$991$	2048.00	0.0656477	0.0328238	−	0.999461i	$-0.489550\pi$
$991$	2048.00	0.0656477	0.0328238	+	0.999461i	$0.489550\pi$
$992$	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

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.4.a.c.1.1		1
4.3	odd	2		448.4.a.n.1.1		1
8.3	odd	2		112.4.a.b.1.1		1
8.5	even	2		56.4.a.b.1.1	✓	1
24.5	odd	2		504.4.a.a.1.1		1
24.11	even	2		1008.4.a.e.1.1		1
40.13	odd	4		1400.4.g.b.449.2		2
40.29	even	2		1400.4.a.b.1.1		1
40.37	odd	4		1400.4.g.b.449.1		2
56.5	odd	6		392.4.i.h.361.1		2
56.13	odd	2		392.4.a.a.1.1		1
56.27	even	2		784.4.a.q.1.1		1
56.37	even	6		392.4.i.a.361.1		2
56.45	odd	6		392.4.i.h.177.1		2
56.53	even	6		392.4.i.a.177.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.4.a.b.1.1	✓	1	8.5	even	2
112.4.a.b.1.1		1	8.3	odd	2
392.4.a.a.1.1		1	56.13	odd	2
392.4.i.a.177.1		2	56.53	even	6
392.4.i.a.361.1		2	56.37	even	6
392.4.i.h.177.1		2	56.45	odd	6
392.4.i.h.361.1		2	56.5	odd	6
448.4.a.c.1.1		1	1.1	even	1	trivial
448.4.a.n.1.1		1	4.3	odd	2
504.4.a.a.1.1		1	24.5	odd	2
784.4.a.q.1.1		1	56.27	even	2
1008.4.a.e.1.1		1	24.11	even	2
1400.4.a.b.1.1		1	40.29	even	2
1400.4.g.b.449.1		2	40.37	odd	4
1400.4.g.b.449.2		2	40.13	odd	4

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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	448.4.a.c.1.1

Level	$448$
Weight	$4$
Character	448.1
Self dual	yes
Analytic conductor	$26.433$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$