LMFDB - Embedded newform 448.8.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,8,Mod(1,448)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("448.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(448, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	448.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-46,0,160,0,-343] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$139.948491417$
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)

$f(q)$

$=$

$q-46.0000 q^{3} +160.000 q^{5} -343.000 q^{7} -71.0000 q^{9} +6840.00 q^{11} +2900.00 q^{13} -7360.00 q^{15} +16566.0 q^{17} +6718.00 q^{19} +15778.0 q^{21} -976.000 q^{23} -52525.0 q^{25} +103868. q^{27} +61662.0 q^{29} -69236.0 q^{31} -314640. q^{33} -54880.0 q^{35} +533062. q^{37} -133400. q^{39} +183158. q^{41} -966864. q^{43} -11360.0 q^{45} -190268. q^{47} +117649. q^{49} -762036. q^{51} +785010. q^{53} +1.09440e6 q^{55} -309028. q^{57} -2.89359e6 q^{59} +95896.0 q^{61} +24353.0 q^{63} +464000. q^{65} +991644. q^{67} +44896.0 q^{69} +1.06816e6 q^{71} +2.52346e6 q^{73} +2.41615e6 q^{75} -2.34612e6 q^{77} +285848. q^{79} -4.62265e6 q^{81} -7.09494e6 q^{83} +2.65056e6 q^{85} -2.83645e6 q^{87} -252390. q^{89} -994700. q^{91} +3.18486e6 q^{93} +1.07488e6 q^{95} -1.82479e6 q^{97} -485640. q^{99} +O(q^{100})$

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
$3$	−46.0000	−0.983634	−0.491817	−	0.870699i	$-0.663667\pi$
$3$	−46.0000	−0.983634	−0.491817	+	0.870699i	$0.663667\pi$
$4$	0	0
$5$	160.000	0.572433	0.286217	−	0.958165i	$-0.407602\pi$
$5$	160.000	0.572433	0.286217	+	0.958165i	$0.407602\pi$
$6$	0	0
$7$	−343.000	−0.377964
$7$	−343.000	−0.377964
$8$	0	0
$9$	−71.0000	−0.0324646
$10$	0	0
$11$	6840.00	1.54946	0.774732	−	0.632289i	$-0.217885\pi$
$11$	6840.00	1.54946	0.774732	+	0.632289i	$0.217885\pi$
$12$	0	0
$13$	2900.00	0.366097	0.183049	−	0.983104i	$-0.441403\pi$
$13$	2900.00	0.366097	0.183049	+	0.983104i	$0.441403\pi$
$14$	0	0
$15$	−7360.00	−0.563065
$16$	0	0
$17$	16566.0	0.817799	0.408899	−	0.912579i	$-0.365913\pi$
$17$	16566.0	0.817799	0.408899	+	0.912579i	$0.365913\pi$
$18$	0	0
$19$	6718.00	0.224700	0.112350	−	0.993669i	$-0.464162\pi$
$19$	6718.00	0.224700	0.112350	+	0.993669i	$0.464162\pi$
$20$	0	0
$21$	15778.0	0.371779
$22$	0	0
$23$	−976.000	−0.0167264	−0.00836320	−	0.999965i	$-0.502662\pi$
$23$	−976.000	−0.0167264	−0.00836320	+	0.999965i	$0.502662\pi$
$24$	0	0
$25$	−52525.0	−0.672320
$26$	0	0
$27$	103868.	1.01557
$28$	0	0
$29$	61662.0	0.469488	0.234744	−	0.972057i	$-0.424575\pi$
$29$	61662.0	0.469488	0.234744	+	0.972057i	$0.424575\pi$
$30$	0	0
$31$	−69236.0	−0.417413	−0.208707	−	0.977978i	$-0.566925\pi$
$31$	−69236.0	−0.417413	−0.208707	+	0.977978i	$0.566925\pi$
$32$	0	0
$33$	−314640.	−1.52411
$34$	0	0
$35$	−54880.0	−0.216359
$36$	0	0
$37$	533062.	1.73010	0.865051	−	0.501684i	$-0.167286\pi$
$37$	533062.	1.73010	0.865051	+	0.501684i	$0.167286\pi$
$38$	0	0
$39$	−133400.	−0.360105
$40$	0	0
$41$	183158.	0.415033	0.207516	−	0.978232i	$-0.433462\pi$
$41$	183158.	0.415033	0.207516	+	0.978232i	$0.433462\pi$
$42$	0	0
$43$	−966864.	−1.85450	−0.927248	−	0.374448i	$-0.877832\pi$
$43$	−966864.	−1.85450	−0.927248	+	0.374448i	$0.877832\pi$
$44$	0	0
$45$	−11360.0	−0.0185838
$46$	0	0
$47$	−190268.	−0.267315	−0.133657	−	0.991028i	$-0.542672\pi$
$47$	−190268.	−0.267315	−0.133657	+	0.991028i	$0.542672\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	−762036.	−0.804414
$52$	0	0
$53$	785010.	0.724285	0.362143	−	0.932123i	$-0.382045\pi$
$53$	785010.	0.724285	0.362143	+	0.932123i	$0.382045\pi$
$54$	0	0
$55$	1.09440e6	0.886965
$56$	0	0
$57$	−309028.	−0.221022
$58$	0	0
$59$	−2.89359e6	−1.83424	−0.917119	−	0.398614i	$-0.869491\pi$
$59$	−2.89359e6	−1.83424	−0.917119	+	0.398614i	$0.869491\pi$
$60$	0	0
$61$	95896.0	0.0540936	0.0270468	−	0.999634i	$-0.491390\pi$
$61$	95896.0	0.0540936	0.0270468	+	0.999634i	$0.491390\pi$
$62$	0	0
$63$	24353.0	0.0122705
$64$	0	0
$65$	464000.	0.209566
$66$	0	0
$67$	991644.	0.402804	0.201402	−	0.979509i	$-0.435450\pi$
$67$	991644.	0.402804	0.201402	+	0.979509i	$0.435450\pi$
$68$	0	0
$69$	44896.0	0.0164526
$70$	0	0
$71$	1.06816e6	0.354187	0.177093	−	0.984194i	$-0.443331\pi$
$71$	1.06816e6	0.354187	0.177093	+	0.984194i	$0.443331\pi$
$72$	0	0
$73$	2.52346e6	0.759217	0.379609	−	0.925147i	$-0.376059\pi$
$73$	2.52346e6	0.759217	0.379609	+	0.925147i	$0.376059\pi$
$74$	0	0
$75$	2.41615e6	0.661317
$76$	0	0
$77$	−2.34612e6	−0.585643
$78$	0	0
$79$	285848.	0.0652289	0.0326145	−	0.999468i	$-0.489617\pi$
$79$	285848.	0.0652289	0.0326145	+	0.999468i	$0.489617\pi$
$80$	0	0
$81$	−4.62265e6	−0.966481
$82$	0	0
$83$	−7.09494e6	−1.36199	−0.680997	−	0.732286i	$-0.738453\pi$
$83$	−7.09494e6	−1.36199	−0.680997	+	0.732286i	$0.738453\pi$
$84$	0	0
$85$	2.65056e6	0.468135
$86$	0	0
$87$	−2.83645e6	−0.461804
$88$	0	0
$89$	−252390.	−0.0379496	−0.0189748	−	0.999820i	$-0.506040\pi$
$89$	−252390.	−0.0379496	−0.0189748	+	0.999820i	$0.506040\pi$
$90$	0	0
$91$	−994700.	−0.138372
$92$	0	0
$93$	3.18486e6	0.410582
$94$	0	0
$95$	1.07488e6	0.128626
$96$	0	0
$97$	−1.82479e6	−0.203008	−0.101504	−	0.994835i	$-0.532365\pi$
$97$	−1.82479e6	−0.203008	−0.101504	+	0.994835i	$0.532365\pi$
$98$	0	0
$99$	−485640.	−0.0503027
$100$	0	0
$101$	9.61885e6	0.928963	0.464481	−	0.885583i	$-0.346241\pi$
$101$	9.61885e6	0.928963	0.464481	+	0.885583i	$0.346241\pi$
$102$	0	0
$103$	−1.67753e7	−1.51266	−0.756328	−	0.654192i	$-0.773009\pi$
$103$	−1.67753e7	−1.51266	−0.756328	+	0.654192i	$0.773009\pi$
$104$	0	0
$105$	2.52448e6	0.212819
$106$	0	0
$107$	2.72428e6	0.214985	0.107493	−	0.994206i	$-0.465718\pi$
$107$	2.72428e6	0.214985	0.107493	+	0.994206i	$0.465718\pi$
$108$	0	0
$109$	−5.39825e6	−0.399264	−0.199632	−	0.979871i	$-0.563975\pi$
$109$	−5.39825e6	−0.399264	−0.199632	+	0.979871i	$0.563975\pi$
$110$	0	0
$111$	−2.45209e7	−1.70179
$112$	0	0
$113$	−2.51296e7	−1.63836	−0.819182	−	0.573534i	$-0.805572\pi$
$113$	−2.51296e7	−1.63836	−0.819182	+	0.573534i	$0.805572\pi$
$114$	0	0
$115$	−156160.	−0.00957475
$116$	0	0
$117$	−205900.	−0.0118852
$118$	0	0
$119$	−5.68214e6	−0.309099
$120$	0	0
$121$	2.72984e7	1.40084
$122$	0	0
$123$	−8.42527e6	−0.408240
$124$	0	0
$125$	−2.09040e7	−0.957292
$126$	0	0
$127$	9.64146e6	0.417666	0.208833	−	0.977951i	$-0.433033\pi$
$127$	9.64146e6	0.417666	0.208833	+	0.977951i	$0.433033\pi$
$128$	0	0
$129$	4.44757e7	1.82414
$130$	0	0
$131$	2.20160e7	0.855637	0.427819	−	0.903865i	$-0.359282\pi$
$131$	2.20160e7	0.855637	0.427819	+	0.903865i	$0.359282\pi$
$132$	0	0
$133$	−2.30427e6	−0.0849285
$134$	0	0
$135$	1.66189e7	0.581344
$136$	0	0
$137$	5.16437e7	1.71591	0.857957	−	0.513722i	$-0.171734\pi$
$137$	5.16437e7	1.71591	0.857957	+	0.513722i	$0.171734\pi$
$138$	0	0
$139$	5.55268e7	1.75368	0.876841	−	0.480780i	$-0.159646\pi$
$139$	5.55268e7	1.75368	0.876841	+	0.480780i	$0.159646\pi$
$140$	0	0
$141$	8.75233e6	0.262940
$142$	0	0
$143$	1.98360e7	0.567255
$144$	0	0
$145$	9.86592e6	0.268751
$146$	0	0
$147$	−5.41185e6	−0.140519
$148$	0	0
$149$	−2.04378e7	−0.506153	−0.253077	−	0.967446i	$-0.581442\pi$
$149$	−2.04378e7	−0.506153	−0.253077	+	0.967446i	$0.581442\pi$
$150$	0	0
$151$	7.00223e7	1.65507	0.827536	−	0.561412i	$-0.189742\pi$
$151$	7.00223e7	1.65507	0.827536	+	0.561412i	$0.189742\pi$
$152$	0	0
$153$	−1.17619e6	−0.0265495
$154$	0	0
$155$	−1.10778e7	−0.238941
$156$	0	0
$157$	8.05199e7	1.66056	0.830279	−	0.557347i	$-0.188181\pi$
$157$	8.05199e7	1.66056	0.830279	+	0.557347i	$0.188181\pi$
$158$	0	0
$159$	−3.61105e7	−0.712431
$160$	0	0
$161$	334768.	0.00632198
$162$	0	0
$163$	1.49981e7	0.271256	0.135628	−	0.990760i	$-0.456695\pi$
$163$	1.49981e7	0.271256	0.135628	+	0.990760i	$0.456695\pi$
$164$	0	0
$165$	−5.03424e7	−0.872449
$166$	0	0
$167$	3.71770e7	0.617684	0.308842	−	0.951113i	$-0.400059\pi$
$167$	3.71770e7	0.617684	0.308842	+	0.951113i	$0.400059\pi$
$168$	0	0
$169$	−5.43385e7	−0.865973
$170$	0	0
$171$	−476978.	−0.00729478
$172$	0	0
$173$	1.07405e8	1.57712	0.788558	−	0.614960i	$-0.210828\pi$
$173$	1.07405e8	1.57712	0.788558	+	0.614960i	$0.210828\pi$
$174$	0	0
$175$	1.80161e7	0.254113
$176$	0	0
$177$	1.33105e8	1.80422
$178$	0	0
$179$	−1.13390e8	−1.47771	−0.738854	−	0.673865i	$-0.764633\pi$
$179$	−1.13390e8	−1.47771	−0.738854	+	0.673865i	$0.764633\pi$
$180$	0	0
$181$	1.32258e8	1.65785	0.828925	−	0.559359i	$-0.188953\pi$
$181$	1.32258e8	1.65785	0.828925	+	0.559359i	$0.188953\pi$
$182$	0	0
$183$	−4.41122e6	−0.0532083
$184$	0	0
$185$	8.52899e7	0.990368
$186$	0	0
$187$	1.13311e8	1.26715
$188$	0	0
$189$	−3.56267e7	−0.383848
$190$	0	0
$191$	−3.87753e6	−0.0402660	−0.0201330	−	0.999797i	$-0.506409\pi$
$191$	−3.87753e6	−0.0402660	−0.0201330	+	0.999797i	$0.506409\pi$
$192$	0	0
$193$	−1.47799e8	−1.47986	−0.739932	−	0.672682i	$-0.765142\pi$
$193$	−1.47799e8	−1.47986	−0.739932	+	0.672682i	$0.765142\pi$
$194$	0	0
$195$	−2.13440e7	−0.206136
$196$	0	0
$197$	1.95011e8	1.81730	0.908651	−	0.417557i	$-0.137114\pi$
$197$	1.95011e8	1.81730	0.908651	+	0.417557i	$0.137114\pi$
$198$	0	0
$199$	6.68038e7	0.600919	0.300459	−	0.953795i	$-0.402860\pi$
$199$	6.68038e7	0.600919	0.300459	+	0.953795i	$0.402860\pi$
$200$	0	0
$201$	−4.56156e7	−0.396212
$202$	0	0
$203$	−2.11501e7	−0.177450
$204$	0	0
$205$	2.93053e7	0.237579
$206$	0	0
$207$	69296.0	0.000543015 0
$208$	0	0
$209$	4.59511e7	0.348164
$210$	0	0
$211$	1.67508e8	1.22757	0.613784	−	0.789474i	$-0.289646\pi$
$211$	1.67508e8	1.22757	0.613784	+	0.789474i	$0.289646\pi$
$212$	0	0
$213$	−4.91354e7	−0.348390
$214$	0	0
$215$	−1.54698e8	−1.06158
$216$	0	0
$217$	2.37479e7	0.157767
$218$	0	0
$219$	−1.16079e8	−0.746792
$220$	0	0
$221$	4.80414e7	0.299394
$222$	0	0
$223$	−7.98486e7	−0.482170	−0.241085	−	0.970504i	$-0.577503\pi$
$223$	−7.98486e7	−0.482170	−0.241085	+	0.970504i	$0.577503\pi$
$224$	0	0
$225$	3.72928e6	0.0218266
$226$	0	0
$227$	−1.28683e7	−0.0730183	−0.0365092	−	0.999333i	$-0.511624\pi$
$227$	−1.28683e7	−0.0730183	−0.0365092	+	0.999333i	$0.511624\pi$
$228$	0	0
$229$	2.86474e8	1.57638	0.788189	−	0.615433i	$-0.211019\pi$
$229$	2.86474e8	1.57638	0.788189	+	0.615433i	$0.211019\pi$
$230$	0	0
$231$	1.07922e8	0.576058
$232$	0	0
$233$	9.85317e7	0.510305	0.255153	−	0.966901i	$-0.417874\pi$
$233$	9.85317e7	0.510305	0.255153	+	0.966901i	$0.417874\pi$
$234$	0	0
$235$	−3.04429e7	−0.153020
$236$	0	0
$237$	−1.31490e7	−0.0641614
$238$	0	0
$239$	2.57316e8	1.21920	0.609599	−	0.792710i	$-0.291330\pi$
$239$	2.57316e8	1.21920	0.609599	+	0.792710i	$0.291330\pi$
$240$	0	0
$241$	2.57739e8	1.18610	0.593049	−	0.805167i	$-0.297924\pi$
$241$	2.57739e8	1.18610	0.593049	+	0.805167i	$0.297924\pi$
$242$	0	0
$243$	−1.45174e7	−0.0649032
$244$	0	0
$245$	1.88238e7	0.0817762
$246$	0	0
$247$	1.94822e7	0.0822619
$248$	0	0
$249$	3.26367e8	1.33970
$250$	0	0
$251$	−3.04338e8	−1.21478	−0.607391	−	0.794403i	$-0.707784\pi$
$251$	−3.04338e8	−1.21478	−0.607391	+	0.794403i	$0.707784\pi$
$252$	0	0
$253$	−6.67584e6	−0.0259170
$254$	0	0
$255$	−1.21926e8	−0.460474
$256$	0	0
$257$	−2.13793e8	−0.785648	−0.392824	−	0.919614i	$-0.628502\pi$
$257$	−2.13793e8	−0.785648	−0.392824	+	0.919614i	$0.628502\pi$
$258$	0	0
$259$	−1.82840e8	−0.653917
$260$	0	0
$261$	−4.37800e6	−0.0152417
$262$	0	0
$263$	7.75259e7	0.262786	0.131393	−	0.991330i	$-0.458055\pi$
$263$	7.75259e7	0.262786	0.131393	+	0.991330i	$0.458055\pi$
$264$	0	0
$265$	1.25602e8	0.414605
$266$	0	0
$267$	1.16099e7	0.0373285
$268$	0	0
$269$	−5.03408e8	−1.57684	−0.788419	−	0.615139i	$-0.789100\pi$
$269$	−5.03408e8	−1.57684	−0.788419	+	0.615139i	$0.789100\pi$
$270$	0	0
$271$	−4.43692e8	−1.35422	−0.677109	−	0.735882i	$-0.736768\pi$
$271$	−4.43692e8	−1.35422	−0.677109	+	0.735882i	$0.736768\pi$
$272$	0	0
$273$	4.57562e7	0.136107
$274$	0	0
$275$	−3.59271e8	−1.04174
$276$	0	0
$277$	8.12666e7	0.229738	0.114869	−	0.993381i	$-0.463355\pi$
$277$	8.12666e7	0.229738	0.114869	+	0.993381i	$0.463355\pi$
$278$	0	0
$279$	4.91576e6	0.0135511
$280$	0	0
$281$	4.55683e8	1.22516	0.612578	−	0.790410i	$-0.290133\pi$
$281$	4.55683e8	1.22516	0.612578	+	0.790410i	$0.290133\pi$
$282$	0	0
$283$	4.31324e8	1.13123	0.565616	−	0.824669i	$-0.308639\pi$
$283$	4.31324e8	1.13123	0.565616	+	0.824669i	$0.308639\pi$
$284$	0	0
$285$	−4.94445e7	−0.126520
$286$	0	0
$287$	−6.28232e7	−0.156868
$288$	0	0
$289$	−1.35906e8	−0.331205
$290$	0	0
$291$	8.39405e7	0.199685
$292$	0	0
$293$	8.35265e8	1.93994	0.969969	−	0.243229i	$-0.0782066\pi$
$293$	8.35265e8	1.93994	0.969969	+	0.243229i	$0.0782066\pi$
$294$	0	0
$295$	−4.62975e8	−1.04998
$296$	0	0
$297$	7.10457e8	1.57359
$298$	0	0
$299$	−2.83040e6	−0.00612348
$300$	0	0
$301$	3.31634e8	0.700933
$302$	0	0
$303$	−4.42467e8	−0.913759
$304$	0	0
$305$	1.53434e7	0.0309650
$306$	0	0
$307$	1.28958e8	0.254369	0.127184	−	0.991879i	$-0.459406\pi$
$307$	1.28958e8	0.254369	0.127184	+	0.991879i	$0.459406\pi$
$308$	0	0
$309$	7.71665e8	1.48790
$310$	0	0
$311$	−5.49015e8	−1.03496	−0.517479	−	0.855696i	$-0.673129\pi$
$311$	−5.49015e8	−1.03496	−0.517479	+	0.855696i	$0.673129\pi$
$312$	0	0
$313$	3.47992e8	0.641452	0.320726	−	0.947172i	$-0.396073\pi$
$313$	3.47992e8	0.641452	0.320726	+	0.947172i	$0.396073\pi$
$314$	0	0
$315$	3.89648e6	0.00702402
$316$	0	0
$317$	−7.73189e8	−1.36326	−0.681629	−	0.731698i	$-0.738728\pi$
$317$	−7.73189e8	−1.36326	−0.681629	+	0.731698i	$0.738728\pi$
$318$	0	0
$319$	4.21768e8	0.727455
$320$	0	0
$321$	−1.25317e8	−0.211467
$322$	0	0
$323$	1.11290e8	0.183759
$324$	0	0
$325$	−1.52322e8	−0.246134
$326$	0	0
$327$	2.48320e8	0.392730
$328$	0	0
$329$	6.52619e7	0.101036
$330$	0	0
$331$	7.55430e8	1.14498	0.572488	−	0.819913i	$-0.305978\pi$
$331$	7.55430e8	1.14498	0.572488	+	0.819913i	$0.305978\pi$
$332$	0	0
$333$	−3.78474e7	−0.0561670
$334$	0	0
$335$	1.58663e8	0.230579
$336$	0	0
$337$	3.69428e8	0.525805	0.262903	−	0.964822i	$-0.415320\pi$
$337$	3.69428e8	0.525805	0.262903	+	0.964822i	$0.415320\pi$
$338$	0	0
$339$	1.15596e9	1.61155
$340$	0	0
$341$	−4.73574e8	−0.646767
$342$	0	0
$343$	−4.03536e7	−0.0539949
$344$	0	0
$345$	7.18336e6	0.00941805
$346$	0	0
$347$	−3.83443e8	−0.492660	−0.246330	−	0.969186i	$-0.579225\pi$
$347$	−3.83443e8	−0.492660	−0.246330	+	0.969186i	$0.579225\pi$
$348$	0	0
$349$	6.76700e8	0.852133	0.426066	−	0.904692i	$-0.359899\pi$
$349$	6.76700e8	0.852133	0.426066	+	0.904692i	$0.359899\pi$
$350$	0	0
$351$	3.01217e8	0.371796
$352$	0	0
$353$	−8.84640e8	−1.07042	−0.535211	−	0.844718i	$-0.679768\pi$
$353$	−8.84640e8	−1.07042	−0.535211	+	0.844718i	$0.679768\pi$
$354$	0	0
$355$	1.70906e8	0.202748
$356$	0	0
$357$	2.61378e8	0.304040
$358$	0	0
$359$	2.73899e8	0.312435	0.156217	−	0.987723i	$-0.450070\pi$
$359$	2.73899e8	0.312435	0.156217	+	0.987723i	$0.450070\pi$
$360$	0	0
$361$	−8.48740e8	−0.949510
$362$	0	0
$363$	−1.25573e9	−1.37791
$364$	0	0
$365$	4.03753e8	0.434601
$366$	0	0
$367$	−7.18778e8	−0.759038	−0.379519	−	0.925184i	$-0.623910\pi$
$367$	−7.18778e8	−0.759038	−0.379519	+	0.925184i	$0.623910\pi$
$368$	0	0
$369$	−1.30042e7	−0.0134739
$370$	0	0
$371$	−2.69258e8	−0.273754
$372$	0	0
$373$	5.57667e8	0.556409	0.278204	−	0.960522i	$-0.410261\pi$
$373$	5.57667e8	0.556409	0.278204	+	0.960522i	$0.410261\pi$
$374$	0	0
$375$	9.61584e8	0.941625
$376$	0	0
$377$	1.78820e8	0.171878
$378$	0	0
$379$	−3.50444e8	−0.330660	−0.165330	−	0.986238i	$-0.552869\pi$
$379$	−3.50444e8	−0.330660	−0.165330	+	0.986238i	$0.552869\pi$
$380$	0	0
$381$	−4.43507e8	−0.410831
$382$	0	0
$383$	2.09768e9	1.90784	0.953922	−	0.300056i	$-0.0970053\pi$
$383$	2.09768e9	1.90784	0.953922	+	0.300056i	$0.0970053\pi$
$384$	0	0
$385$	−3.75379e8	−0.335241
$386$	0	0
$387$	6.86473e7	0.0602054
$388$	0	0
$389$	3.70723e8	0.319319	0.159660	−	0.987172i	$-0.448960\pi$
$389$	3.70723e8	0.319319	0.159660	+	0.987172i	$0.448960\pi$
$390$	0	0
$391$	−1.61684e7	−0.0136788
$392$	0	0
$393$	−1.01274e9	−0.841634
$394$	0	0
$395$	4.57357e7	0.0373392
$396$	0	0
$397$	−2.15887e9	−1.73165	−0.865823	−	0.500351i	$-0.833204\pi$
$397$	−2.15887e9	−1.73165	−0.865823	+	0.500351i	$0.833204\pi$
$398$	0	0
$399$	1.05997e8	0.0835385
$400$	0	0
$401$	1.99648e8	0.154618	0.0773090	−	0.997007i	$-0.475367\pi$
$401$	1.99648e8	0.154618	0.0773090	+	0.997007i	$0.475367\pi$
$402$	0	0
$403$	−2.00784e8	−0.152814
$404$	0	0
$405$	−7.39624e8	−0.553246
$406$	0	0
$407$	3.64614e9	2.68073
$408$	0	0
$409$	1.95002e9	1.40931	0.704657	−	0.709548i	$-0.251101\pi$
$409$	1.95002e9	1.40931	0.704657	+	0.709548i	$0.251101\pi$
$410$	0	0
$411$	−2.37561e9	−1.68783
$412$	0	0
$413$	9.92503e8	0.693277
$414$	0	0
$415$	−1.13519e9	−0.779651
$416$	0	0
$417$	−2.55423e9	−1.72498
$418$	0	0
$419$	6.45883e8	0.428948	0.214474	−	0.976730i	$-0.431196\pi$
$419$	6.45883e8	0.428948	0.214474	+	0.976730i	$0.431196\pi$
$420$	0	0
$421$	−2.65902e8	−0.173674	−0.0868369	−	0.996223i	$-0.527676\pi$
$421$	−2.65902e8	−0.173674	−0.0868369	+	0.996223i	$0.527676\pi$
$422$	0	0
$423$	1.35090e7	0.00867826
$424$	0	0
$425$	−8.70129e8	−0.549822
$426$	0	0
$427$	−3.28923e7	−0.0204455
$428$	0	0
$429$	−9.12456e8	−0.557971
$430$	0	0
$431$	−1.87449e9	−1.12775	−0.563874	−	0.825861i	$-0.690690\pi$
$431$	−1.87449e9	−1.12775	−0.563874	+	0.825861i	$0.690690\pi$
$432$	0	0
$433$	1.26795e9	0.750573	0.375287	−	0.926909i	$-0.377544\pi$
$433$	1.26795e9	0.750573	0.375287	+	0.926909i	$0.377544\pi$
$434$	0	0
$435$	−4.53832e8	−0.264352
$436$	0	0
$437$	−6.55677e6	−0.00375842
$438$	0	0
$439$	−2.61409e9	−1.47467	−0.737334	−	0.675529i	$-0.763915\pi$
$439$	−2.61409e9	−1.47467	−0.737334	+	0.675529i	$0.763915\pi$
$440$	0	0
$441$	−8.35308e6	−0.00463779
$442$	0	0
$443$	2.21442e9	1.21017	0.605087	−	0.796159i	$-0.293138\pi$
$443$	2.21442e9	1.21017	0.605087	+	0.796159i	$0.293138\pi$
$444$	0	0
$445$	−4.03824e7	−0.0217236
$446$	0	0
$447$	9.40138e8	0.497869
$448$	0	0
$449$	−1.79737e9	−0.937079	−0.468540	−	0.883442i	$-0.655220\pi$
$449$	−1.79737e9	−0.937079	−0.468540	+	0.883442i	$0.655220\pi$
$450$	0	0
$451$	1.25280e9	0.643079
$452$	0	0
$453$	−3.22103e9	−1.62799
$454$	0	0
$455$	−1.59152e8	−0.0792086
$456$	0	0
$457$	−1.14352e9	−0.560449	−0.280224	−	0.959935i	$-0.590409\pi$
$457$	−1.14352e9	−0.560449	−0.280224	+	0.959935i	$0.590409\pi$
$458$	0	0
$459$	1.72068e9	0.830529
$460$	0	0
$461$	3.30189e8	0.156967	0.0784836	−	0.996915i	$-0.474992\pi$
$461$	3.30189e8	0.156967	0.0784836	+	0.996915i	$0.474992\pi$
$462$	0	0
$463$	3.57189e9	1.67249	0.836247	−	0.548352i	$-0.184745\pi$
$463$	3.57189e9	1.67249	0.836247	+	0.548352i	$0.184745\pi$
$464$	0	0
$465$	5.09577e8	0.235031
$466$	0	0
$467$	2.07936e9	0.944759	0.472380	−	0.881395i	$-0.343395\pi$
$467$	2.07936e9	0.944759	0.472380	+	0.881395i	$0.343395\pi$
$468$	0	0
$469$	−3.40134e8	−0.152246
$470$	0	0
$471$	−3.70391e9	−1.63338
$472$	0	0
$473$	−6.61335e9	−2.87348
$474$	0	0
$475$	−3.52863e8	−0.151070
$476$	0	0
$477$	−5.57357e7	−0.0235136
$478$	0	0
$479$	−2.36832e9	−0.984614	−0.492307	−	0.870422i	$-0.663846\pi$
$479$	−2.36832e9	−0.984614	−0.492307	+	0.870422i	$0.663846\pi$
$480$	0	0
$481$	1.54588e9	0.633385
$482$	0	0
$483$	−1.53993e7	−0.00621852
$484$	0	0
$485$	−2.91967e8	−0.116208
$486$	0	0
$487$	1.50548e9	0.590640	0.295320	−	0.955398i	$-0.404574\pi$
$487$	1.50548e9	0.590640	0.295320	+	0.955398i	$0.404574\pi$
$488$	0	0
$489$	−6.89911e8	−0.266816
$490$	0	0
$491$	−3.41991e9	−1.30385	−0.651927	−	0.758281i	$-0.726039\pi$
$491$	−3.41991e9	−1.30385	−0.651927	+	0.758281i	$0.726039\pi$
$492$	0	0
$493$	1.02149e9	0.383947
$494$	0	0
$495$	−7.77024e7	−0.0287949
$496$	0	0
$497$	−3.66379e8	−0.133870
$498$	0	0
$499$	1.70643e9	0.614802	0.307401	−	0.951580i	$-0.400541\pi$
$499$	1.70643e9	0.614802	0.307401	+	0.951580i	$0.400541\pi$
$500$	0	0
$501$	−1.71014e9	−0.607575
$502$	0	0
$503$	2.40257e9	0.841760	0.420880	−	0.907116i	$-0.361721\pi$
$503$	2.40257e9	0.841760	0.420880	+	0.907116i	$0.361721\pi$
$504$	0	0
$505$	1.53902e9	0.531769
$506$	0	0
$507$	2.49957e9	0.851800
$508$	0	0
$509$	4.76643e8	0.160207	0.0801034	−	0.996787i	$-0.474475\pi$
$509$	4.76643e8	0.160207	0.0801034	+	0.996787i	$0.474475\pi$
$510$	0	0
$511$	−8.65546e8	−0.286957
$512$	0	0
$513$	6.97785e8	0.228198
$514$	0	0
$515$	−2.68405e9	−0.865895
$516$	0	0
$517$	−1.30143e9	−0.414195
$518$	0	0
$519$	−4.94064e9	−1.55130
$520$	0	0
$521$	3.35142e9	1.03824	0.519119	−	0.854702i	$-0.326260\pi$
$521$	3.35142e9	1.03824	0.519119	+	0.854702i	$0.326260\pi$
$522$	0	0
$523$	−3.05061e9	−0.932460	−0.466230	−	0.884663i	$-0.654388\pi$
$523$	−3.05061e9	−0.932460	−0.466230	+	0.884663i	$0.654388\pi$
$524$	0	0
$525$	−8.28739e8	−0.249954
$526$	0	0
$527$	−1.14696e9	−0.341360
$528$	0	0
$529$	−3.40387e9	−0.999720
$530$	0	0
$531$	2.05445e8	0.0595477
$532$	0	0
$533$	5.31158e8	0.151942
$534$	0	0
$535$	4.35885e8	0.123065
$536$	0	0
$537$	5.21593e9	1.45352
$538$	0	0
$539$	8.04719e8	0.221352
$540$	0	0
$541$	−3.71291e9	−1.00815	−0.504073	−	0.863661i	$-0.668166\pi$
$541$	−3.71291e9	−1.00815	−0.504073	+	0.863661i	$0.668166\pi$
$542$	0	0
$543$	−6.08385e9	−1.63072
$544$	0	0
$545$	−8.63720e8	−0.228552
$546$	0	0
$547$	−1.33784e9	−0.349502	−0.174751	−	0.984613i	$-0.555912\pi$
$547$	−1.33784e9	−0.349502	−0.174751	+	0.984613i	$0.555912\pi$
$548$	0	0
$549$	−6.80862e6	−0.00175613
$550$	0	0
$551$	4.14245e8	0.105494
$552$	0	0
$553$	−9.80459e7	−0.0246542
$554$	0	0
$555$	−3.92334e9	−0.974160
$556$	0	0
$557$	−6.96716e9	−1.70829	−0.854147	−	0.520032i	$-0.825920\pi$
$557$	−6.96716e9	−1.70829	−0.854147	+	0.520032i	$0.825920\pi$
$558$	0	0
$559$	−2.80391e9	−0.678925
$560$	0	0
$561$	−5.21233e9	−1.24641
$562$	0	0
$563$	−2.24647e9	−0.530544	−0.265272	−	0.964174i	$-0.585462\pi$
$563$	−2.24647e9	−0.530544	−0.265272	+	0.964174i	$0.585462\pi$
$564$	0	0
$565$	−4.02073e9	−0.937854
$566$	0	0
$567$	1.58557e9	0.365296
$568$	0	0
$569$	−3.52236e9	−0.801570	−0.400785	−	0.916172i	$-0.631263\pi$
$569$	−3.52236e9	−0.801570	−0.400785	+	0.916172i	$0.631263\pi$
$570$	0	0
$571$	−3.33282e9	−0.749180	−0.374590	−	0.927191i	$-0.622217\pi$
$571$	−3.33282e9	−0.749180	−0.374590	+	0.927191i	$0.622217\pi$
$572$	0	0
$573$	1.78366e8	0.0396070
$574$	0	0
$575$	5.12644e7	0.0112455
$576$	0	0
$577$	7.43990e9	1.61232	0.806161	−	0.591697i	$-0.201542\pi$
$577$	7.43990e9	1.61232	0.806161	+	0.591697i	$0.201542\pi$
$578$	0	0
$579$	6.79877e9	1.45564
$580$	0	0
$581$	2.43356e9	0.514785
$582$	0	0
$583$	5.36947e9	1.12225
$584$	0	0
$585$	−3.29440e7	−0.00680348
$586$	0	0
$587$	−2.94111e9	−0.600174	−0.300087	−	0.953912i	$-0.597016\pi$
$587$	−2.94111e9	−0.600174	−0.300087	+	0.953912i	$0.597016\pi$
$588$	0	0
$589$	−4.65127e8	−0.0937926
$590$	0	0
$591$	−8.97050e9	−1.78756
$592$	0	0
$593$	−5.80481e9	−1.14313	−0.571566	−	0.820556i	$-0.693664\pi$
$593$	−5.80481e9	−1.14313	−0.571566	+	0.820556i	$0.693664\pi$
$594$	0	0
$595$	−9.09142e8	−0.176939
$596$	0	0
$597$	−3.07298e9	−0.591084
$598$	0	0
$599$	7.63705e9	1.45188	0.725942	−	0.687756i	$-0.241404\pi$
$599$	7.63705e9	1.45188	0.725942	+	0.687756i	$0.241404\pi$
$600$	0	0
$601$	9.43487e8	0.177286	0.0886432	−	0.996063i	$-0.471747\pi$
$601$	9.43487e8	0.177286	0.0886432	+	0.996063i	$0.471747\pi$
$602$	0	0
$603$	−7.04067e7	−0.0130769
$604$	0	0
$605$	4.36775e9	0.801888
$606$	0	0
$607$	7.51567e9	1.36398	0.681989	−	0.731362i	$-0.261115\pi$
$607$	7.51567e9	1.36398	0.681989	+	0.731362i	$0.261115\pi$
$608$	0	0
$609$	9.72903e8	0.174546
$610$	0	0
$611$	−5.51777e8	−0.0978632
$612$	0	0
$613$	−8.82249e9	−1.54696	−0.773480	−	0.633820i	$-0.781486\pi$
$613$	−8.82249e9	−1.54696	−0.773480	+	0.633820i	$0.781486\pi$
$614$	0	0
$615$	−1.34804e9	−0.233690
$616$	0	0
$617$	−4.02929e8	−0.0690606	−0.0345303	−	0.999404i	$-0.510994\pi$
$617$	−4.02929e8	−0.0690606	−0.0345303	+	0.999404i	$0.510994\pi$
$618$	0	0
$619$	3.23178e9	0.547677	0.273839	−	0.961776i	$-0.411707\pi$
$619$	3.23178e9	0.547677	0.273839	+	0.961776i	$0.411707\pi$
$620$	0	0
$621$	−1.01375e8	−0.0169868
$622$	0	0
$623$	8.65698e7	0.0143436
$624$	0	0
$625$	7.58876e8	0.124334
$626$	0	0
$627$	−2.11375e9	−0.342466
$628$	0	0
$629$	8.83071e9	1.41488
$630$	0	0
$631$	3.34162e9	0.529486	0.264743	−	0.964319i	$-0.414713\pi$
$631$	3.34162e9	0.529486	0.264743	+	0.964319i	$0.414713\pi$
$632$	0	0
$633$	−7.70535e9	−1.20748
$634$	0	0
$635$	1.54263e9	0.239086
$636$	0	0
$637$	3.41182e8	0.0522996
$638$	0	0
$639$	−7.58394e7	−0.0114985
$640$	0	0
$641$	−1.71154e9	−0.256675	−0.128337	−	0.991731i	$-0.540964\pi$
$641$	−1.71154e9	−0.256675	−0.128337	+	0.991731i	$0.540964\pi$
$642$	0	0
$643$	6.40900e9	0.950718	0.475359	−	0.879792i	$-0.342318\pi$
$643$	6.40900e9	0.950718	0.475359	+	0.879792i	$0.342318\pi$
$644$	0	0
$645$	7.11612e9	1.04420
$646$	0	0
$647$	9.49449e9	1.37818	0.689091	−	0.724675i	$-0.258010\pi$
$647$	9.49449e9	1.37818	0.689091	+	0.724675i	$0.258010\pi$
$648$	0	0
$649$	−1.97922e10	−2.84209
$650$	0	0
$651$	−1.09241e9	−0.155185
$652$	0	0
$653$	9.51227e9	1.33687	0.668433	−	0.743772i	$-0.266965\pi$
$653$	9.51227e9	1.33687	0.668433	+	0.743772i	$0.266965\pi$
$654$	0	0
$655$	3.52257e9	0.489795
$656$	0	0
$657$	−1.79166e8	−0.0246477
$658$	0	0
$659$	9.05617e9	1.23267	0.616333	−	0.787486i	$-0.288618\pi$
$659$	9.05617e9	1.23267	0.616333	+	0.787486i	$0.288618\pi$
$660$	0	0
$661$	−2.08227e9	−0.280435	−0.140218	−	0.990121i	$-0.544780\pi$
$661$	−2.08227e9	−0.280435	−0.140218	+	0.990121i	$0.544780\pi$
$662$	0	0
$663$	−2.20990e9	−0.294494
$664$	0	0
$665$	−3.68684e8	−0.0486159
$666$	0	0
$667$	−6.01821e7	−0.00785284
$668$	0	0
$669$	3.67304e9	0.474279
$670$	0	0
$671$	6.55929e8	0.0838162
$672$	0	0
$673$	1.03400e10	1.30758	0.653791	−	0.756675i	$-0.273178\pi$
$673$	1.03400e10	1.30758	0.653791	+	0.756675i	$0.273178\pi$
$674$	0	0
$675$	−5.45567e9	−0.682786
$676$	0	0
$677$	5.87333e9	0.727485	0.363743	−	0.931499i	$-0.381499\pi$
$677$	5.87333e9	0.727485	0.363743	+	0.931499i	$0.381499\pi$
$678$	0	0
$679$	6.25904e8	0.0767297
$680$	0	0
$681$	5.91944e8	0.0718233
$682$	0	0
$683$	3.39405e9	0.407611	0.203805	−	0.979011i	$-0.434669\pi$
$683$	3.39405e9	0.407611	0.203805	+	0.979011i	$0.434669\pi$
$684$	0	0
$685$	8.26299e9	0.982246
$686$	0	0
$687$	−1.31778e10	−1.55058
$688$	0	0
$689$	2.27653e9	0.265159
$690$	0	0
$691$	1.05553e10	1.21702	0.608509	−	0.793547i	$-0.291768\pi$
$691$	1.05553e10	1.21702	0.608509	+	0.793547i	$0.291768\pi$
$692$	0	0
$693$	1.66575e8	0.0190126
$694$	0	0
$695$	8.88429e9	1.00387
$696$	0	0
$697$	3.03420e9	0.339413
$698$	0	0
$699$	−4.53246e9	−0.501954
$700$	0	0
$701$	−6.58559e8	−0.0722074	−0.0361037	−	0.999348i	$-0.511495\pi$
$701$	−6.58559e8	−0.0722074	−0.0361037	+	0.999348i	$0.511495\pi$
$702$	0	0
$703$	3.58111e9	0.388753
$704$	0	0
$705$	1.40037e9	0.150516
$706$	0	0
$707$	−3.29926e9	−0.351115
$708$	0	0
$709$	−1.14257e10	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$709$	−1.14257e10	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$710$	0	0
$711$	−2.02952e7	−0.00211763
$712$	0	0
$713$	6.75743e7	0.00698182
$714$	0	0
$715$	3.17376e9	0.324715
$716$	0	0
$717$	−1.18365e10	−1.19925
$718$	0	0
$719$	−7.62103e8	−0.0764650	−0.0382325	−	0.999269i	$-0.512173\pi$
$719$	−7.62103e8	−0.0764650	−0.0382325	+	0.999269i	$0.512173\pi$
$720$	0	0
$721$	5.75393e9	0.571731
$722$	0	0
$723$	−1.18560e10	−1.16669
$724$	0	0
$725$	−3.23880e9	−0.315646
$726$	0	0
$727$	1.03968e10	1.00353	0.501766	−	0.865004i	$-0.332684\pi$
$727$	1.03968e10	1.00353	0.501766	+	0.865004i	$0.332684\pi$
$728$	0	0
$729$	1.07775e10	1.03032
$730$	0	0
$731$	−1.60171e10	−1.51660
$732$	0	0
$733$	1.89306e10	1.77542	0.887708	−	0.460407i	$-0.152296\pi$
$733$	1.89306e10	1.77542	0.887708	+	0.460407i	$0.152296\pi$
$734$	0	0
$735$	−8.65897e8	−0.0804378
$736$	0	0
$737$	6.78284e9	0.624131
$738$	0	0
$739$	1.98745e9	0.181151	0.0905753	−	0.995890i	$-0.471129\pi$
$739$	1.98745e9	0.181151	0.0905753	+	0.995890i	$0.471129\pi$
$740$	0	0
$741$	−8.96181e8	−0.0809156
$742$	0	0
$743$	−8.47019e9	−0.757587	−0.378793	−	0.925481i	$-0.623661\pi$
$743$	−8.47019e9	−0.757587	−0.378793	+	0.925481i	$0.623661\pi$
$744$	0	0
$745$	−3.27005e9	−0.289739
$746$	0	0
$747$	5.03741e8	0.0442165
$748$	0	0
$749$	−9.34429e8	−0.0812569
$750$	0	0
$751$	−1.70283e10	−1.46700	−0.733502	−	0.679688i	$-0.762115\pi$
$751$	−1.70283e10	−1.46700	−0.733502	+	0.679688i	$0.762115\pi$
$752$	0	0
$753$	1.39996e10	1.19490
$754$	0	0
$755$	1.12036e10	0.947419
$756$	0	0
$757$	−8.18236e9	−0.685556	−0.342778	−	0.939416i	$-0.611368\pi$
$757$	−8.18236e9	−0.685556	−0.342778	+	0.939416i	$0.611368\pi$
$758$	0	0
$759$	3.07089e8	0.0254928
$760$	0	0
$761$	2.89321e8	0.0237977	0.0118988	−	0.999929i	$-0.496212\pi$
$761$	2.89321e8	0.0237977	0.0118988	+	0.999929i	$0.496212\pi$
$762$	0	0
$763$	1.85160e9	0.150908
$764$	0	0
$765$	−1.88190e8	−0.0151978
$766$	0	0
$767$	−8.39142e9	−0.671509
$768$	0	0
$769$	2.09242e10	1.65923	0.829614	−	0.558338i	$-0.188561\pi$
$769$	2.09242e10	1.65923	0.829614	+	0.558338i	$0.188561\pi$
$770$	0	0
$771$	9.83449e9	0.772790
$772$	0	0
$773$	1.21139e10	0.943310	0.471655	−	0.881783i	$-0.343657\pi$
$773$	1.21139e10	0.943310	0.471655	+	0.881783i	$0.343657\pi$
$774$	0	0
$775$	3.63662e9	0.280635
$776$	0	0
$777$	8.41065e9	0.643215
$778$	0	0
$779$	1.23046e9	0.0932577
$780$	0	0
$781$	7.30621e9	0.548800
$782$	0	0
$783$	6.40471e9	0.476797
$784$	0	0
$785$	1.28832e10	0.950559
$786$	0	0
$787$	−5.56243e8	−0.0406774	−0.0203387	−	0.999793i	$-0.506474\pi$
$787$	−5.56243e8	−0.0406774	−0.0203387	+	0.999793i	$0.506474\pi$
$788$	0	0
$789$	−3.56619e9	−0.258485
$790$	0	0
$791$	8.61944e9	0.619243
$792$	0	0
$793$	2.78098e8	0.0198035
$794$	0	0
$795$	−5.77767e9	−0.407820
$796$	0	0
$797$	−4.66707e9	−0.326543	−0.163271	−	0.986581i	$-0.552205\pi$
$797$	−4.66707e9	−0.326543	−0.163271	+	0.986581i	$0.552205\pi$
$798$	0	0
$799$	−3.15198e9	−0.218610
$800$	0	0
$801$	1.79197e7	0.00123202
$802$	0	0
$803$	1.72605e10	1.17638
$804$	0	0
$805$	5.35629e7	0.00361891
$806$	0	0
$807$	2.31568e10	1.55103
$808$	0	0
$809$	6.08320e9	0.403936	0.201968	−	0.979392i	$-0.435266\pi$
$809$	6.08320e9	0.403936	0.201968	+	0.979392i	$0.435266\pi$
$810$	0	0
$811$	8.27584e9	0.544803	0.272401	−	0.962184i	$-0.412182\pi$
$811$	8.27584e9	0.544803	0.272401	+	0.962184i	$0.412182\pi$
$812$	0	0
$813$	2.04098e10	1.33206
$814$	0	0
$815$	2.39969e9	0.155276
$816$	0	0
$817$	−6.49539e9	−0.416705
$818$	0	0
$819$	7.06237e7	0.00449218
$820$	0	0
$821$	3.65230e9	0.230338	0.115169	−	0.993346i	$-0.463259\pi$
$821$	3.65230e9	0.230338	0.115169	+	0.993346i	$0.463259\pi$
$822$	0	0
$823$	−1.65099e10	−1.03239	−0.516197	−	0.856470i	$-0.672653\pi$
$823$	−1.65099e10	−1.03239	−0.516197	+	0.856470i	$0.672653\pi$
$824$	0	0
$825$	1.65265e10	1.02469
$826$	0	0
$827$	−1.76127e10	−1.08282	−0.541409	−	0.840759i	$-0.682109\pi$
$827$	−1.76127e10	−1.08282	−0.541409	+	0.840759i	$0.682109\pi$
$828$	0	0
$829$	1.38475e10	0.844172	0.422086	−	0.906556i	$-0.361298\pi$
$829$	1.38475e10	0.844172	0.422086	+	0.906556i	$0.361298\pi$
$830$	0	0
$831$	−3.73826e9	−0.225978
$832$	0	0
$833$	1.94897e9	0.116828
$834$	0	0
$835$	5.94831e9	0.353583
$836$	0	0
$837$	−7.19140e9	−0.423911
$838$	0	0
$839$	−2.40027e10	−1.40311	−0.701556	−	0.712614i	$-0.747511\pi$
$839$	−2.40027e10	−1.40311	−0.701556	+	0.712614i	$0.747511\pi$
$840$	0	0
$841$	−1.34477e10	−0.779581
$842$	0	0
$843$	−2.09614e10	−1.20510
$844$	0	0
$845$	−8.69416e9	−0.495712
$846$	0	0
$847$	−9.36336e9	−0.529468
$848$	0	0
$849$	−1.98409e10	−1.11272
$850$	0	0
$851$	−5.20269e8	−0.0289384
$852$	0	0
$853$	3.92270e9	0.216403	0.108201	−	0.994129i	$-0.465491\pi$
$853$	3.92270e9	0.216403	0.108201	+	0.994129i	$0.465491\pi$
$854$	0	0
$855$	−7.63165e7	−0.00417577
$856$	0	0
$857$	8.87029e9	0.481399	0.240699	−	0.970600i	$-0.422623\pi$
$857$	8.87029e9	0.481399	0.240699	+	0.970600i	$0.422623\pi$
$858$	0	0
$859$	−2.65180e9	−0.142746	−0.0713731	−	0.997450i	$-0.522738\pi$
$859$	−2.65180e9	−0.142746	−0.0713731	+	0.997450i	$0.522738\pi$
$860$	0	0
$861$	2.88987e9	0.154300
$862$	0	0
$863$	1.97508e9	0.104603	0.0523017	−	0.998631i	$-0.483344\pi$
$863$	1.97508e9	0.104603	0.0523017	+	0.998631i	$0.483344\pi$
$864$	0	0
$865$	1.71848e10	0.902794
$866$	0	0
$867$	6.25169e9	0.325785
$868$	0	0
$869$	1.95520e9	0.101070
$870$	0	0
$871$	2.87577e9	0.147465
$872$	0	0
$873$	1.29560e8	0.00659056
$874$	0	0
$875$	7.17007e9	0.361822
$876$	0	0
$877$	1.59757e10	0.799763	0.399882	−	0.916567i	$-0.369051\pi$
$877$	1.59757e10	0.799763	0.399882	+	0.916567i	$0.369051\pi$
$878$	0	0
$879$	−3.84222e10	−1.90819
$880$	0	0
$881$	−1.93433e10	−0.953049	−0.476525	−	0.879161i	$-0.658104\pi$
$881$	−1.93433e10	−0.953049	−0.476525	+	0.879161i	$0.658104\pi$
$882$	0	0
$883$	2.78010e10	1.35893	0.679466	−	0.733707i	$-0.262212\pi$
$883$	2.78010e10	1.35893	0.679466	+	0.733707i	$0.262212\pi$
$884$	0	0
$885$	2.12969e10	1.03279
$886$	0	0
$887$	2.40826e10	1.15870	0.579350	−	0.815079i	$-0.303306\pi$
$887$	2.40826e10	1.15870	0.579350	+	0.815079i	$0.303306\pi$
$888$	0	0
$889$	−3.30702e9	−0.157863
$890$	0	0
$891$	−3.16189e10	−1.49753
$892$	0	0
$893$	−1.27822e9	−0.0600656
$894$	0	0
$895$	−1.81424e10	−0.845890
$896$	0	0
$897$	1.30198e8	0.00602327
$898$	0	0
$899$	−4.26923e9	−0.195970
$900$	0	0
$901$	1.30045e10	0.592320
$902$	0	0
$903$	−1.52552e10	−0.689462
$904$	0	0
$905$	2.11612e10	0.949009
$906$	0	0
$907$	−3.16564e10	−1.40876	−0.704379	−	0.709824i	$-0.748775\pi$
$907$	−3.16564e10	−1.40876	−0.704379	+	0.709824i	$0.748775\pi$
$908$	0	0
$909$	−6.82938e8	−0.0301584
$910$	0	0
$911$	5.15376e9	0.225845	0.112922	−	0.993604i	$-0.463979\pi$
$911$	5.15376e9	0.225845	0.112922	+	0.993604i	$0.463979\pi$
$912$	0	0
$913$	−4.85294e10	−2.11036
$914$	0	0
$915$	−7.05795e8	−0.0304582
$916$	0	0
$917$	−7.55150e9	−0.323401
$918$	0	0
$919$	−2.39051e10	−1.01598	−0.507991	−	0.861363i	$-0.669612\pi$
$919$	−2.39051e10	−1.01598	−0.507991	+	0.861363i	$0.669612\pi$
$920$	0	0
$921$	−5.93206e9	−0.250206
$922$	0	0
$923$	3.09766e9	0.129667
$924$	0	0
$925$	−2.79991e10	−1.16318
$926$	0	0
$927$	1.19105e9	0.0491077
$928$	0	0
$929$	4.36319e10	1.78546	0.892728	−	0.450596i	$-0.148788\pi$
$929$	4.36319e10	1.78546	0.892728	+	0.450596i	$0.148788\pi$
$930$	0	0
$931$	7.90366e8	0.0321000
$932$	0	0
$933$	2.52547e10	1.01802
$934$	0	0
$935$	1.81298e10	0.725359
$936$	0	0
$937$	4.67554e9	0.185671	0.0928354	−	0.995681i	$-0.470407\pi$
$937$	4.67554e9	0.185671	0.0928354	+	0.995681i	$0.470407\pi$
$938$	0	0
$939$	−1.60076e10	−0.630954
$940$	0	0
$941$	9.72350e9	0.380416	0.190208	−	0.981744i	$-0.439084\pi$
$941$	9.72350e9	0.380416	0.190208	+	0.981744i	$0.439084\pi$
$942$	0	0
$943$	−1.78762e8	−0.00694200
$944$	0	0
$945$	−5.70028e9	−0.219728
$946$	0	0
$947$	1.72888e10	0.661516	0.330758	−	0.943716i	$-0.392696\pi$
$947$	1.72888e10	0.661516	0.330758	+	0.943716i	$0.392696\pi$
$948$	0	0
$949$	7.31803e9	0.277947
$950$	0	0
$951$	3.55667e10	1.34095
$952$	0	0
$953$	2.31448e10	0.866220	0.433110	−	0.901341i	$-0.357416\pi$
$953$	2.31448e10	0.866220	0.433110	+	0.901341i	$0.357416\pi$
$954$	0	0
$955$	−6.20404e8	−0.0230496
$956$	0	0
$957$	−1.94013e10	−0.715550
$958$	0	0
$959$	−1.77138e10	−0.648554
$960$	0	0
$961$	−2.27190e10	−0.825766
$962$	0	0
$963$	−1.93424e8	−0.00697941
$964$	0	0
$965$	−2.36479e10	−0.847124
$966$	0	0
$967$	4.62635e10	1.64530	0.822652	−	0.568546i	$-0.192494\pi$
$967$	4.62635e10	1.64530	0.822652	+	0.568546i	$0.192494\pi$
$968$	0	0
$969$	−5.11936e9	−0.180752
$970$	0	0
$971$	−1.79642e10	−0.629711	−0.314856	−	0.949140i	$-0.601956\pi$
$971$	−1.79642e10	−0.629711	−0.314856	+	0.949140i	$0.601956\pi$
$972$	0	0
$973$	−1.90457e10	−0.662830
$974$	0	0
$975$	7.00684e9	0.242106
$976$	0	0
$977$	7.07040e9	0.242557	0.121278	−	0.992619i	$-0.461301\pi$
$977$	7.07040e9	0.242557	0.121278	+	0.992619i	$0.461301\pi$
$978$	0	0
$979$	−1.72635e9	−0.0588016
$980$	0	0
$981$	3.83276e8	0.0129619
$982$	0	0
$983$	3.91058e10	1.31312	0.656560	−	0.754274i	$-0.272011\pi$
$983$	3.91058e10	1.31312	0.656560	+	0.754274i	$0.272011\pi$
$984$	0	0
$985$	3.12017e10	1.04028
$986$	0	0
$987$	−3.00205e9	−0.0993820
$988$	0	0
$989$	9.43659e8	0.0310190
$990$	0	0
$991$	2.13915e10	0.698206	0.349103	−	0.937084i	$-0.386486\pi$
$991$	2.13915e10	0.698206	0.349103	+	0.937084i	$0.386486\pi$
$992$	0	0
$993$	−3.47498e10	−1.12624
$994$	0	0
$995$	1.06886e10	0.343986
$996$	0	0
$997$	4.26786e10	1.36388	0.681942	−	0.731406i	$-0.261136\pi$
$997$	4.26786e10	1.36388	0.681942	+	0.731406i	$0.261136\pi$
$998$	0	0
$999$	5.53681e10	1.75703

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.8.a.c.1.1		1
4.3	odd	2		448.8.a.h.1.1		1
8.3	odd	2		112.8.a.a.1.1		1
8.5	even	2		56.8.a.b.1.1	✓	1
24.5	odd	2		504.8.a.b.1.1		1
56.13	odd	2		392.8.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.8.a.b.1.1	✓	1	8.5	even	2
112.8.a.a.1.1		1	8.3	odd	2
392.8.a.a.1.1		1	56.13	odd	2
448.8.a.c.1.1		1	1.1	even	1	trivial
448.8.a.h.1.1		1	4.3	odd	2
504.8.a.b.1.1		1	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	448.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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