LMFDB - Embedded newform 448.6.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [448,6,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, 6); 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, 6, names="a")

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

Newform invariants

comment:select newform

sage:traces = [1,0,14,0,64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
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)

$f(q)$

$=$

$q+14.0000 q^{3} +64.0000 q^{5} +49.0000 q^{7} -47.0000 q^{9} -420.000 q^{11} -860.000 q^{13} +896.000 q^{15} +830.000 q^{17} +490.000 q^{19} +686.000 q^{21} -4872.00 q^{23} +971.000 q^{25} -4060.00 q^{27} -8754.00 q^{29} -5628.00 q^{31} -5880.00 q^{33} +3136.00 q^{35} -1434.00 q^{37} -12040.0 q^{39} -9258.00 q^{41} +14756.0 q^{43} -3008.00 q^{45} +10108.0 q^{47} +2401.00 q^{49} +11620.0 q^{51} +23058.0 q^{53} -26880.0 q^{55} +6860.00 q^{57} -13734.0 q^{59} +25352.0 q^{61} -2303.00 q^{63} -55040.0 q^{65} +19768.0 q^{67} -68208.0 q^{69} -1792.00 q^{71} +37914.0 q^{73} +13594.0 q^{75} -20580.0 q^{77} -95984.0 q^{79} -45419.0 q^{81} +88242.0 q^{83} +53120.0 q^{85} -122556. q^{87} +43762.0 q^{89} -42140.0 q^{91} -78792.0 q^{93} +31360.0 q^{95} +65790.0 q^{97} +19740.0 q^{99} +O(q^{100})$

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
$3$	14.0000	0.898100	0.449050	−	0.893507i	$-0.351762\pi$
$3$	14.0000	0.898100	0.449050	+	0.893507i	$0.351762\pi$
$4$	0	0
$5$	64.0000	1.14487	0.572433	−	0.819951i	$-0.306000\pi$
$5$	64.0000	1.14487	0.572433	+	0.819951i	$0.306000\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	−47.0000	−0.193416
$10$	0	0
$11$	−420.000	−1.04657	−0.523284	−	0.852158i	$-0.675293\pi$
$11$	−420.000	−1.04657	−0.523284	+	0.852158i	$0.675293\pi$
$12$	0	0
$13$	−860.000	−1.41137	−0.705684	−	0.708527i	$-0.749360\pi$
$13$	−860.000	−1.41137	−0.705684	+	0.708527i	$0.749360\pi$
$14$	0	0
$15$	896.000	1.02821
$16$	0	0
$17$	830.000	0.696556	0.348278	−	0.937391i	$-0.386767\pi$
$17$	830.000	0.696556	0.348278	+	0.937391i	$0.386767\pi$
$18$	0	0
$19$	490.000	0.311395	0.155698	−	0.987805i	$-0.450237\pi$
$19$	490.000	0.311395	0.155698	+	0.987805i	$0.450237\pi$
$20$	0	0
$21$	686.000	0.339450
$22$	0	0
$23$	−4872.00	−1.92038	−0.960191	−	0.279344i	$-0.909883\pi$
$23$	−4872.00	−1.92038	−0.960191	+	0.279344i	$0.909883\pi$
$24$	0	0
$25$	971.000	0.310720
$26$	0	0
$27$	−4060.00	−1.07181
$28$	0	0
$29$	−8754.00	−1.93291	−0.966455	−	0.256837i	$-0.917320\pi$
$29$	−8754.00	−1.93291	−0.966455	+	0.256837i	$0.917320\pi$
$30$	0	0
$31$	−5628.00	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$31$	−5628.00	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$32$	0	0
$33$	−5880.00	−0.939923
$34$	0	0
$35$	3136.00	0.432719
$36$	0	0
$37$	−1434.00	−0.172205	−0.0861023	−	0.996286i	$-0.527441\pi$
$37$	−1434.00	−0.172205	−0.0861023	+	0.996286i	$0.527441\pi$
$38$	0	0
$39$	−12040.0	−1.26755
$40$	0	0
$41$	−9258.00	−0.860117	−0.430059	−	0.902801i	$-0.641507\pi$
$41$	−9258.00	−0.860117	−0.430059	+	0.902801i	$0.641507\pi$
$42$	0	0
$43$	14756.0	1.21702	0.608510	−	0.793546i	$-0.291768\pi$
$43$	14756.0	1.21702	0.608510	+	0.793546i	$0.291768\pi$
$44$	0	0
$45$	−3008.00	−0.221435
$46$	0	0
$47$	10108.0	0.667453	0.333726	−	0.942670i	$-0.391694\pi$
$47$	10108.0	0.667453	0.333726	+	0.942670i	$0.391694\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	11620.0	0.625577
$52$	0	0
$53$	23058.0	1.12754	0.563770	−	0.825932i	$-0.309350\pi$
$53$	23058.0	1.12754	0.563770	+	0.825932i	$0.309350\pi$
$54$	0	0
$55$	−26880.0	−1.19818
$56$	0	0
$57$	6860.00	0.279664
$58$	0	0
$59$	−13734.0	−0.513650	−0.256825	−	0.966458i	$-0.582676\pi$
$59$	−13734.0	−0.513650	−0.256825	+	0.966458i	$0.582676\pi$
$60$	0	0
$61$	25352.0	0.872344	0.436172	−	0.899863i	$-0.356334\pi$
$61$	25352.0	0.872344	0.436172	+	0.899863i	$0.356334\pi$
$62$	0	0
$63$	−2303.00	−0.0731042
$64$	0	0
$65$	−55040.0	−1.61583
$66$	0	0
$67$	19768.0	0.537992	0.268996	−	0.963141i	$-0.413308\pi$
$67$	19768.0	0.537992	0.268996	+	0.963141i	$0.413308\pi$
$68$	0	0
$69$	−68208.0	−1.72470
$70$	0	0
$71$	−1792.00	−0.0421883	−0.0210942	−	0.999777i	$-0.506715\pi$
$71$	−1792.00	−0.0421883	−0.0210942	+	0.999777i	$0.506715\pi$
$72$	0	0
$73$	37914.0	0.832707	0.416354	−	0.909203i	$-0.363308\pi$
$73$	37914.0	0.832707	0.416354	+	0.909203i	$0.363308\pi$
$74$	0	0
$75$	13594.0	0.279058
$76$	0	0
$77$	−20580.0	−0.395566
$78$	0	0
$79$	−95984.0	−1.73034	−0.865169	−	0.501480i	$-0.832789\pi$
$79$	−95984.0	−1.73034	−0.865169	+	0.501480i	$0.832789\pi$
$80$	0	0
$81$	−45419.0	−0.769175
$82$	0	0
$83$	88242.0	1.40598	0.702992	−	0.711198i	$-0.251847\pi$
$83$	88242.0	1.40598	0.702992	+	0.711198i	$0.251847\pi$
$84$	0	0
$85$	53120.0	0.797463
$86$	0	0
$87$	−122556.	−1.73595
$88$	0	0
$89$	43762.0	0.585628	0.292814	−	0.956169i	$-0.405408\pi$
$89$	43762.0	0.585628	0.292814	+	0.956169i	$0.405408\pi$
$90$	0	0
$91$	−42140.0	−0.533447
$92$	0	0
$93$	−78792.0	−0.944658
$94$	0	0
$95$	31360.0	0.356506
$96$	0	0
$97$	65790.0	0.709955	0.354977	−	0.934875i	$-0.384489\pi$
$97$	65790.0	0.709955	0.354977	+	0.934875i	$0.384489\pi$
$98$	0	0
$99$	19740.0	0.202423
$100$	0	0
$101$	17728.0	0.172924	0.0864622	−	0.996255i	$-0.472444\pi$
$101$	17728.0	0.172924	0.0864622	+	0.996255i	$0.472444\pi$
$102$	0	0
$103$	−121660.	−1.12994	−0.564969	−	0.825112i	$-0.691112\pi$
$103$	−121660.	−1.12994	−0.564969	+	0.825112i	$0.691112\pi$
$104$	0	0
$105$	43904.0	0.388625
$106$	0	0
$107$	−48272.0	−0.407602	−0.203801	−	0.979012i	$-0.565329\pi$
$107$	−48272.0	−0.407602	−0.203801	+	0.979012i	$0.565329\pi$
$108$	0	0
$109$	177606.	1.43183	0.715915	−	0.698188i	$-0.246010\pi$
$109$	177606.	1.43183	0.715915	+	0.698188i	$0.246010\pi$
$110$	0	0
$111$	−20076.0	−0.154657
$112$	0	0
$113$	128198.	0.944463	0.472232	−	0.881474i	$-0.343449\pi$
$113$	128198.	0.944463	0.472232	+	0.881474i	$0.343449\pi$
$114$	0	0
$115$	−311808.	−2.19858
$116$	0	0
$117$	40420.0	0.272981
$118$	0	0
$119$	40670.0	0.263273
$120$	0	0
$121$	15349.0	0.0953052
$122$	0	0
$123$	−129612.	−0.772471
$124$	0	0
$125$	−137856.	−0.789134
$126$	0	0
$127$	−302960.	−1.66677	−0.833386	−	0.552692i	$-0.813601\pi$
$127$	−302960.	−1.66677	−0.833386	+	0.552692i	$0.813601\pi$
$128$	0	0
$129$	206584.	1.09301
$130$	0	0
$131$	−295638.	−1.50516	−0.752579	−	0.658502i	$-0.771190\pi$
$131$	−295638.	−1.50516	−0.752579	+	0.658502i	$0.771190\pi$
$132$	0	0
$133$	24010.0	0.117696
$134$	0	0
$135$	−259840.	−1.22708
$136$	0	0
$137$	−78318.0	−0.356501	−0.178250	−	0.983985i	$-0.557044\pi$
$137$	−78318.0	−0.356501	−0.178250	+	0.983985i	$0.557044\pi$
$138$	0	0
$139$	−234010.	−1.02730	−0.513650	−	0.858000i	$-0.671707\pi$
$139$	−234010.	−1.02730	−0.513650	+	0.858000i	$0.671707\pi$
$140$	0	0
$141$	141512.	0.599440
$142$	0	0
$143$	361200.	1.47709
$144$	0	0
$145$	−560256.	−2.21292
$146$	0	0
$147$	33614.0	0.128300
$148$	0	0
$149$	397642.	1.46733	0.733663	−	0.679514i	$-0.237809\pi$
$149$	397642.	1.46733	0.733663	+	0.679514i	$0.237809\pi$
$150$	0	0
$151$	236432.	0.843847	0.421924	−	0.906631i	$-0.361355\pi$
$151$	236432.	0.843847	0.421924	+	0.906631i	$0.361355\pi$
$152$	0	0
$153$	−39010.0	−0.134725
$154$	0	0
$155$	−360192.	−1.20422
$156$	0	0
$157$	−31852.0	−0.103131	−0.0515653	−	0.998670i	$-0.516421\pi$
$157$	−31852.0	−0.103131	−0.0515653	+	0.998670i	$0.516421\pi$
$158$	0	0
$159$	322812.	1.01264
$160$	0	0
$161$	−238728.	−0.725836
$162$	0	0
$163$	403172.	1.18856	0.594280	−	0.804258i	$-0.297437\pi$
$163$	403172.	1.18856	0.594280	+	0.804258i	$0.297437\pi$
$164$	0	0
$165$	−376320.	−1.07609
$166$	0	0
$167$	−569044.	−1.57890	−0.789450	−	0.613815i	$-0.789634\pi$
$167$	−569044.	−1.57890	−0.789450	+	0.613815i	$0.789634\pi$
$168$	0	0
$169$	368307.	0.991958
$170$	0	0
$171$	−23030.0	−0.0602287
$172$	0	0
$173$	−500116.	−1.27044	−0.635222	−	0.772330i	$-0.719091\pi$
$173$	−500116.	−1.27044	−0.635222	+	0.772330i	$0.719091\pi$
$174$	0	0
$175$	47579.0	0.117441
$176$	0	0
$177$	−192276.	−0.461309
$178$	0	0
$179$	−112896.	−0.263358	−0.131679	−	0.991292i	$-0.542037\pi$
$179$	−112896.	−0.263358	−0.131679	+	0.991292i	$0.542037\pi$
$180$	0	0
$181$	20780.0	0.0471465	0.0235732	−	0.999722i	$-0.492496\pi$
$181$	20780.0	0.0471465	0.0235732	+	0.999722i	$0.492496\pi$
$182$	0	0
$183$	354928.	0.783452
$184$	0	0
$185$	−91776.0	−0.197151
$186$	0	0
$187$	−348600.	−0.728993
$188$	0	0
$189$	−198940.	−0.405105
$190$	0	0
$191$	215040.	0.426516	0.213258	−	0.976996i	$-0.431592\pi$
$191$	215040.	0.426516	0.213258	+	0.976996i	$0.431592\pi$
$192$	0	0
$193$	408590.	0.789577	0.394788	−	0.918772i	$-0.370818\pi$
$193$	408590.	0.789577	0.394788	+	0.918772i	$0.370818\pi$
$194$	0	0
$195$	−770560.	−1.45118
$196$	0	0
$197$	721674.	1.32488	0.662438	−	0.749116i	$-0.269522\pi$
$197$	721674.	1.32488	0.662438	+	0.749116i	$0.269522\pi$
$198$	0	0
$199$	−378308.	−0.677194	−0.338597	−	0.940932i	$-0.609952\pi$
$199$	−378308.	−0.677194	−0.338597	+	0.940932i	$0.609952\pi$
$200$	0	0
$201$	276752.	0.483171
$202$	0	0
$203$	−428946.	−0.730571
$204$	0	0
$205$	−592512.	−0.984719
$206$	0	0
$207$	228984.	0.371432
$208$	0	0
$209$	−205800.	−0.325896
$210$	0	0
$211$	−265272.	−0.410190	−0.205095	−	0.978742i	$-0.565750\pi$
$211$	−265272.	−0.410190	−0.205095	+	0.978742i	$0.565750\pi$
$212$	0	0
$213$	−25088.0	−0.0378893
$214$	0	0
$215$	944384.	1.39333
$216$	0	0
$217$	−275772.	−0.397558
$218$	0	0
$219$	530796.	0.747855
$220$	0	0
$221$	−713800.	−0.983096
$222$	0	0
$223$	−419608.	−0.565043	−0.282522	−	0.959261i	$-0.591171\pi$
$223$	−419608.	−0.565043	−0.282522	+	0.959261i	$0.591171\pi$
$224$	0	0
$225$	−45637.0	−0.0600981
$226$	0	0
$227$	−776678.	−1.00041	−0.500203	−	0.865908i	$-0.666741\pi$
$227$	−776678.	−1.00041	−0.500203	+	0.865908i	$0.666741\pi$
$228$	0	0
$229$	71924.0	0.0906327	0.0453164	−	0.998973i	$-0.485570\pi$
$229$	71924.0	0.0906327	0.0453164	+	0.998973i	$0.485570\pi$
$230$	0	0
$231$	−288120.	−0.355258
$232$	0	0
$233$	−356966.	−0.430762	−0.215381	−	0.976530i	$-0.569099\pi$
$233$	−356966.	−0.430762	−0.215381	+	0.976530i	$0.569099\pi$
$234$	0	0
$235$	646912.	0.764145
$236$	0	0
$237$	−1.34378e6	−1.55402
$238$	0	0
$239$	−1.06383e6	−1.20470	−0.602349	−	0.798233i	$-0.705768\pi$
$239$	−1.06383e6	−1.20470	−0.602349	+	0.798233i	$0.705768\pi$
$240$	0	0
$241$	−834866.	−0.925921	−0.462961	−	0.886379i	$-0.653213\pi$
$241$	−834866.	−0.925921	−0.462961	+	0.886379i	$0.653213\pi$
$242$	0	0
$243$	350714.	0.381011
$244$	0	0
$245$	153664.	0.163552
$246$	0	0
$247$	−421400.	−0.439493
$248$	0	0
$249$	1.23539e6	1.26271
$250$	0	0
$251$	1.59069e6	1.59368	0.796842	−	0.604187i	$-0.206502\pi$
$251$	1.59069e6	1.59368	0.796842	+	0.604187i	$0.206502\pi$
$252$	0	0
$253$	2.04624e6	2.00981
$254$	0	0
$255$	743680.	0.716202
$256$	0	0
$257$	74754.0	0.0705995	0.0352998	−	0.999377i	$-0.488761\pi$
$257$	74754.0	0.0705995	0.0352998	+	0.999377i	$0.488761\pi$
$258$	0	0
$259$	−70266.0	−0.0650872
$260$	0	0
$261$	411438.	0.373855
$262$	0	0
$263$	9744.00	0.00868656	0.00434328	−	0.999991i	$-0.498617\pi$
$263$	9744.00	0.00868656	0.00434328	+	0.999991i	$0.498617\pi$
$264$	0	0
$265$	1.47571e6	1.29088
$266$	0	0
$267$	612668.	0.525953
$268$	0	0
$269$	1.24555e6	1.04949	0.524747	−	0.851258i	$-0.324160\pi$
$269$	1.24555e6	1.04949	0.524747	+	0.851258i	$0.324160\pi$
$270$	0	0
$271$	−1.62635e6	−1.34521	−0.672607	−	0.740000i	$-0.734825\pi$
$271$	−1.62635e6	−1.34521	−0.672607	+	0.740000i	$0.734825\pi$
$272$	0	0
$273$	−589960.	−0.479089
$274$	0	0
$275$	−407820.	−0.325190
$276$	0	0
$277$	−99070.0	−0.0775787	−0.0387894	−	0.999247i	$-0.512350\pi$
$277$	−99070.0	−0.0775787	−0.0387894	+	0.999247i	$0.512350\pi$
$278$	0	0
$279$	264516.	0.203442
$280$	0	0
$281$	−1.79165e6	−1.35359	−0.676797	−	0.736170i	$-0.736632\pi$
$281$	−1.79165e6	−1.35359	−0.676797	+	0.736170i	$0.736632\pi$
$282$	0	0
$283$	1.07132e6	0.795159	0.397579	−	0.917568i	$-0.369850\pi$
$283$	1.07132e6	0.795159	0.397579	+	0.917568i	$0.369850\pi$
$284$	0	0
$285$	439040.	0.320178
$286$	0	0
$287$	−453642.	−0.325094
$288$	0	0
$289$	−730957.	−0.514810
$290$	0	0
$291$	921060.	0.637610
$292$	0	0
$293$	396024.	0.269496	0.134748	−	0.990880i	$-0.456977\pi$
$293$	396024.	0.269496	0.134748	+	0.990880i	$0.456977\pi$
$294$	0	0
$295$	−878976.	−0.588060
$296$	0	0
$297$	1.70520e6	1.12172
$298$	0	0
$299$	4.18992e6	2.71036
$300$	0	0
$301$	723044.	0.459990
$302$	0	0
$303$	248192.	0.155303
$304$	0	0
$305$	1.62253e6	0.998717
$306$	0	0
$307$	−1.26209e6	−0.764263	−0.382132	−	0.924108i	$-0.624810\pi$
$307$	−1.26209e6	−0.764263	−0.382132	+	0.924108i	$0.624810\pi$
$308$	0	0
$309$	−1.70324e6	−1.01480
$310$	0	0
$311$	−1.71018e6	−1.00263	−0.501316	−	0.865264i	$-0.667151\pi$
$311$	−1.71018e6	−1.00263	−0.501316	+	0.865264i	$0.667151\pi$
$312$	0	0
$313$	−640634.	−0.369615	−0.184807	−	0.982775i	$-0.559166\pi$
$313$	−640634.	−0.369615	−0.184807	+	0.982775i	$0.559166\pi$
$314$	0	0
$315$	−147392.	−0.0836946
$316$	0	0
$317$	518738.	0.289935	0.144967	−	0.989436i	$-0.453692\pi$
$317$	518738.	0.289935	0.144967	+	0.989436i	$0.453692\pi$
$318$	0	0
$319$	3.67668e6	2.02292
$320$	0	0
$321$	−675808.	−0.366067
$322$	0	0
$323$	406700.	0.216904
$324$	0	0
$325$	−835060.	−0.438540
$326$	0	0
$327$	2.48648e6	1.28593
$328$	0	0
$329$	495292.	0.252273
$330$	0	0
$331$	699412.	0.350884	0.175442	−	0.984490i	$-0.443865\pi$
$331$	699412.	0.350884	0.175442	+	0.984490i	$0.443865\pi$
$332$	0	0
$333$	67398.0	0.0333071
$334$	0	0
$335$	1.26515e6	0.615929
$336$	0	0
$337$	2.51064e6	1.20423	0.602115	−	0.798409i	$-0.294325\pi$
$337$	2.51064e6	1.20423	0.602115	+	0.798409i	$0.294325\pi$
$338$	0	0
$339$	1.79477e6	0.848223
$340$	0	0
$341$	2.36376e6	1.10082
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−4.36531e6	−1.97455
$346$	0	0
$347$	4.16604e6	1.85737	0.928687	−	0.370865i	$-0.120939\pi$
$347$	4.16604e6	1.85737	0.928687	+	0.370865i	$0.120939\pi$
$348$	0	0
$349$	−4.12754e6	−1.81396	−0.906980	−	0.421174i	$-0.861618\pi$
$349$	−4.12754e6	−1.81396	−0.906980	+	0.421174i	$0.861618\pi$
$350$	0	0
$351$	3.49160e6	1.51271
$352$	0	0
$353$	3.54063e6	1.51232	0.756159	−	0.654388i	$-0.227074\pi$
$353$	3.54063e6	1.51232	0.756159	+	0.654388i	$0.227074\pi$
$354$	0	0
$355$	−114688.	−0.0483000
$356$	0	0
$357$	569380.	0.236446
$358$	0	0
$359$	3.27029e6	1.33921	0.669607	−	0.742716i	$-0.266463\pi$
$359$	3.27029e6	1.33921	0.669607	+	0.742716i	$0.266463\pi$
$360$	0	0
$361$	−2.23600e6	−0.903033
$362$	0	0
$363$	214886.	0.0855937
$364$	0	0
$365$	2.42650e6	0.953339
$366$	0	0
$367$	−3.34337e6	−1.29574	−0.647872	−	0.761749i	$-0.724341\pi$
$367$	−3.34337e6	−1.29574	−0.647872	+	0.761749i	$0.724341\pi$
$368$	0	0
$369$	435126.	0.166360
$370$	0	0
$371$	1.12984e6	0.426170
$372$	0	0
$373$	127946.	0.0476162	0.0238081	−	0.999717i	$-0.492421\pi$
$373$	127946.	0.0476162	0.0238081	+	0.999717i	$0.492421\pi$
$374$	0	0
$375$	−1.92998e6	−0.708721
$376$	0	0
$377$	7.52844e6	2.72805
$378$	0	0
$379$	1.40339e6	0.501857	0.250928	−	0.968006i	$-0.419264\pi$
$379$	1.40339e6	0.501857	0.250928	+	0.968006i	$0.419264\pi$
$380$	0	0
$381$	−4.24144e6	−1.49693
$382$	0	0
$383$	1.55134e6	0.540393	0.270197	−	0.962805i	$-0.412911\pi$
$383$	1.55134e6	0.540393	0.270197	+	0.962805i	$0.412911\pi$
$384$	0	0
$385$	−1.31712e6	−0.452870
$386$	0	0
$387$	−693532.	−0.235391
$388$	0	0
$389$	−5.23841e6	−1.75520	−0.877598	−	0.479398i	$-0.840855\pi$
$389$	−5.23841e6	−1.75520	−0.877598	+	0.479398i	$0.840855\pi$
$390$	0	0
$391$	−4.04376e6	−1.33765
$392$	0	0
$393$	−4.13893e6	−1.35178
$394$	0	0
$395$	−6.14298e6	−1.98101
$396$	0	0
$397$	5.66579e6	1.80420	0.902099	−	0.431530i	$-0.142026\pi$
$397$	5.66579e6	1.80420	0.902099	+	0.431530i	$0.142026\pi$
$398$	0	0
$399$	336140.	0.105703
$400$	0	0
$401$	−2.22010e6	−0.689463	−0.344732	−	0.938701i	$-0.612030\pi$
$401$	−2.22010e6	−0.689463	−0.344732	+	0.938701i	$0.612030\pi$
$402$	0	0
$403$	4.84008e6	1.48453
$404$	0	0
$405$	−2.90682e6	−0.880603
$406$	0	0
$407$	602280.	0.180224
$408$	0	0
$409$	−3.89847e6	−1.15235	−0.576176	−	0.817325i	$-0.695456\pi$
$409$	−3.89847e6	−1.15235	−0.576176	+	0.817325i	$0.695456\pi$
$410$	0	0
$411$	−1.09645e6	−0.320173
$412$	0	0
$413$	−672966.	−0.194141
$414$	0	0
$415$	5.64749e6	1.60966
$416$	0	0
$417$	−3.27614e6	−0.922619
$418$	0	0
$419$	−5.78551e6	−1.60993	−0.804965	−	0.593323i	$-0.797816\pi$
$419$	−5.78551e6	−1.60993	−0.804965	+	0.593323i	$0.797816\pi$
$420$	0	0
$421$	563554.	0.154964	0.0774819	−	0.996994i	$-0.475312\pi$
$421$	563554.	0.154964	0.0774819	+	0.996994i	$0.475312\pi$
$422$	0	0
$423$	−475076.	−0.129096
$424$	0	0
$425$	805930.	0.216434
$426$	0	0
$427$	1.24225e6	0.329715
$428$	0	0
$429$	5.05680e6	1.32658
$430$	0	0
$431$	5.61036e6	1.45478	0.727390	−	0.686224i	$-0.240733\pi$
$431$	5.61036e6	1.45478	0.727390	+	0.686224i	$0.240733\pi$
$432$	0	0
$433$	7.27497e6	1.86471	0.932355	−	0.361545i	$-0.117750\pi$
$433$	7.27497e6	1.86471	0.932355	+	0.361545i	$0.117750\pi$
$434$	0	0
$435$	−7.84358e6	−1.98743
$436$	0	0
$437$	−2.38728e6	−0.597998
$438$	0	0
$439$	−4.46326e6	−1.10533	−0.552663	−	0.833405i	$-0.686388\pi$
$439$	−4.46326e6	−1.10533	−0.552663	+	0.833405i	$0.686388\pi$
$440$	0	0
$441$	−112847.	−0.0276308
$442$	0	0
$443$	−3.78006e6	−0.915143	−0.457571	−	0.889173i	$-0.651281\pi$
$443$	−3.78006e6	−0.915143	−0.457571	+	0.889173i	$0.651281\pi$
$444$	0	0
$445$	2.80077e6	0.670467
$446$	0	0
$447$	5.56699e6	1.31781
$448$	0	0
$449$	−6.95465e6	−1.62802	−0.814010	−	0.580851i	$-0.802720\pi$
$449$	−6.95465e6	−1.62802	−0.814010	+	0.580851i	$0.802720\pi$
$450$	0	0
$451$	3.88836e6	0.900171
$452$	0	0
$453$	3.31005e6	0.757860
$454$	0	0
$455$	−2.69696e6	−0.610725
$456$	0	0
$457$	2.25765e6	0.505670	0.252835	−	0.967509i	$-0.418637\pi$
$457$	2.25765e6	0.505670	0.252835	+	0.967509i	$0.418637\pi$
$458$	0	0
$459$	−3.36980e6	−0.746573
$460$	0	0
$461$	−3.56697e6	−0.781713	−0.390857	−	0.920452i	$-0.627821\pi$
$461$	−3.56697e6	−0.781713	−0.390857	+	0.920452i	$0.627821\pi$
$462$	0	0
$463$	1.06417e6	0.230705	0.115353	−	0.993325i	$-0.463200\pi$
$463$	1.06417e6	0.230705	0.115353	+	0.993325i	$0.463200\pi$
$464$	0	0
$465$	−5.04269e6	−1.08151
$466$	0	0
$467$	−5.83622e6	−1.23834	−0.619170	−	0.785257i	$-0.712531\pi$
$467$	−5.83622e6	−1.23834	−0.619170	+	0.785257i	$0.712531\pi$
$468$	0	0
$469$	968632.	0.203342
$470$	0	0
$471$	−445928.	−0.0926216
$472$	0	0
$473$	−6.19752e6	−1.27369
$474$	0	0
$475$	475790.	0.0967568
$476$	0	0
$477$	−1.08373e6	−0.218084
$478$	0	0
$479$	3.71199e6	0.739210	0.369605	−	0.929189i	$-0.379493\pi$
$479$	3.71199e6	0.739210	0.369605	+	0.929189i	$0.379493\pi$
$480$	0	0
$481$	1.23324e6	0.243044
$482$	0	0
$483$	−3.34219e6	−0.651874
$484$	0	0
$485$	4.21056e6	0.812803
$486$	0	0
$487$	−1.64175e6	−0.313679	−0.156839	−	0.987624i	$-0.550131\pi$
$487$	−1.64175e6	−0.313679	−0.156839	+	0.987624i	$0.550131\pi$
$488$	0	0
$489$	5.64441e6	1.06745
$490$	0	0
$491$	3.18164e6	0.595590	0.297795	−	0.954630i	$-0.403749\pi$
$491$	3.18164e6	0.595590	0.297795	+	0.954630i	$0.403749\pi$
$492$	0	0
$493$	−7.26582e6	−1.34638
$494$	0	0
$495$	1.26336e6	0.231747
$496$	0	0
$497$	−87808.0	−0.0159457
$498$	0	0
$499$	−274120.	−0.0492821	−0.0246411	−	0.999696i	$-0.507844\pi$
$499$	−274120.	−0.0492821	−0.0246411	+	0.999696i	$0.507844\pi$
$500$	0	0
$501$	−7.96662e6	−1.41801
$502$	0	0
$503$	5.34274e6	0.941551	0.470775	−	0.882253i	$-0.343974\pi$
$503$	5.34274e6	0.941551	0.470775	+	0.882253i	$0.343974\pi$
$504$	0	0
$505$	1.13459e6	0.197975
$506$	0	0
$507$	5.15630e6	0.890878
$508$	0	0
$509$	2.82107e6	0.482635	0.241318	−	0.970446i	$-0.422420\pi$
$509$	2.82107e6	0.482635	0.241318	+	0.970446i	$0.422420\pi$
$510$	0	0
$511$	1.85779e6	0.314734
$512$	0	0
$513$	−1.98940e6	−0.333756
$514$	0	0
$515$	−7.78624e6	−1.29363
$516$	0	0
$517$	−4.24536e6	−0.698535
$518$	0	0
$519$	−7.00162e6	−1.14099
$520$	0	0
$521$	1.67233e6	0.269915	0.134957	−	0.990851i	$-0.456910\pi$
$521$	1.67233e6	0.269915	0.134957	+	0.990851i	$0.456910\pi$
$522$	0	0
$523$	9.85991e6	1.57623	0.788113	−	0.615530i	$-0.211058\pi$
$523$	9.85991e6	1.57623	0.788113	+	0.615530i	$0.211058\pi$
$524$	0	0
$525$	666106.	0.105474
$526$	0	0
$527$	−4.67124e6	−0.732665
$528$	0	0
$529$	1.73000e7	2.68787
$530$	0	0
$531$	645498.	0.0993479
$532$	0	0
$533$	7.96188e6	1.21394
$534$	0	0
$535$	−3.08941e6	−0.466650
$536$	0	0
$537$	−1.58054e6	−0.236522
$538$	0	0
$539$	−1.00842e6	−0.149510
$540$	0	0
$541$	9.36207e6	1.37524	0.687620	−	0.726071i	$-0.258655\pi$
$541$	9.36207e6	1.37524	0.687620	+	0.726071i	$0.258655\pi$
$542$	0	0
$543$	290920.	0.0423423
$544$	0	0
$545$	1.13668e7	1.63925
$546$	0	0
$547$	−539252.	−0.0770590	−0.0385295	−	0.999257i	$-0.512267\pi$
$547$	−539252.	−0.0770590	−0.0385295	+	0.999257i	$0.512267\pi$
$548$	0	0
$549$	−1.19154e6	−0.168725
$550$	0	0
$551$	−4.28946e6	−0.601899
$552$	0	0
$553$	−4.70322e6	−0.654007
$554$	0	0
$555$	−1.28486e6	−0.177062
$556$	0	0
$557$	−5.74443e6	−0.784529	−0.392264	−	0.919852i	$-0.628308\pi$
$557$	−5.74443e6	−0.784529	−0.392264	+	0.919852i	$0.628308\pi$
$558$	0	0
$559$	−1.26902e7	−1.71766
$560$	0	0
$561$	−4.88040e6	−0.654709
$562$	0	0
$563$	−1.14723e7	−1.52538	−0.762690	−	0.646764i	$-0.776122\pi$
$563$	−1.14723e7	−1.52538	−0.762690	+	0.646764i	$0.776122\pi$
$564$	0	0
$565$	8.20467e6	1.08128
$566$	0	0
$567$	−2.22553e6	−0.290721
$568$	0	0
$569$	−7.71119e6	−0.998484	−0.499242	−	0.866463i	$-0.666388\pi$
$569$	−7.71119e6	−0.998484	−0.499242	+	0.866463i	$0.666388\pi$
$570$	0	0
$571$	2.86588e6	0.367848	0.183924	−	0.982940i	$-0.441120\pi$
$571$	2.86588e6	0.367848	0.183924	+	0.982940i	$0.441120\pi$
$572$	0	0
$573$	3.01056e6	0.383055
$574$	0	0
$575$	−4.73071e6	−0.596701
$576$	0	0
$577$	5.12357e6	0.640668	0.320334	−	0.947305i	$-0.396205\pi$
$577$	5.12357e6	0.640668	0.320334	+	0.947305i	$0.396205\pi$
$578$	0	0
$579$	5.72026e6	0.709119
$580$	0	0
$581$	4.32386e6	0.531412
$582$	0	0
$583$	−9.68436e6	−1.18005
$584$	0	0
$585$	2.58688e6	0.312526
$586$	0	0
$587$	2.63952e6	0.316176	0.158088	−	0.987425i	$-0.449467\pi$
$587$	2.63952e6	0.316176	0.158088	+	0.987425i	$0.449467\pi$
$588$	0	0
$589$	−2.75772e6	−0.327538
$590$	0	0
$591$	1.01034e7	1.18987
$592$	0	0
$593$	1.59576e7	1.86350	0.931751	−	0.363098i	$-0.118281\pi$
$593$	1.59576e7	1.86350	0.931751	+	0.363098i	$0.118281\pi$
$594$	0	0
$595$	2.60288e6	0.301413
$596$	0	0
$597$	−5.29631e6	−0.608188
$598$	0	0
$599$	−1.16868e7	−1.33085	−0.665425	−	0.746465i	$-0.731749\pi$
$599$	−1.16868e7	−1.33085	−0.665425	+	0.746465i	$0.731749\pi$
$600$	0	0
$601$	−5.10344e6	−0.576337	−0.288169	−	0.957580i	$-0.593046\pi$
$601$	−5.10344e6	−0.576337	−0.288169	+	0.957580i	$0.593046\pi$
$602$	0	0
$603$	−929096.	−0.104056
$604$	0	0
$605$	982336.	0.109112
$606$	0	0
$607$	9.91995e6	1.09279	0.546396	−	0.837527i	$-0.315999\pi$
$607$	9.91995e6	1.09279	0.546396	+	0.837527i	$0.315999\pi$
$608$	0	0
$609$	−6.00524e6	−0.656126
$610$	0	0
$611$	−8.69288e6	−0.942021
$612$	0	0
$613$	−1.70881e7	−1.83672	−0.918360	−	0.395745i	$-0.870486\pi$
$613$	−1.70881e7	−1.83672	−0.918360	+	0.395745i	$0.870486\pi$
$614$	0	0
$615$	−8.29517e6	−0.884377
$616$	0	0
$617$	6.80959e6	0.720125	0.360063	−	0.932928i	$-0.382755\pi$
$617$	6.80959e6	0.720125	0.360063	+	0.932928i	$0.382755\pi$
$618$	0	0
$619$	5.56275e6	0.583529	0.291765	−	0.956490i	$-0.405758\pi$
$619$	5.56275e6	0.583529	0.291765	+	0.956490i	$0.405758\pi$
$620$	0	0
$621$	1.97803e7	2.05828
$622$	0	0
$623$	2.14434e6	0.221347
$624$	0	0
$625$	−1.18572e7	−1.21417
$626$	0	0
$627$	−2.88120e6	−0.292688
$628$	0	0
$629$	−1.19022e6	−0.119950
$630$	0	0
$631$	−1.11026e6	−0.111007	−0.0555034	−	0.998458i	$-0.517676\pi$
$631$	−1.11026e6	−0.111007	−0.0555034	+	0.998458i	$0.517676\pi$
$632$	0	0
$633$	−3.71381e6	−0.368392
$634$	0	0
$635$	−1.93894e7	−1.90823
$636$	0	0
$637$	−2.06486e6	−0.201624
$638$	0	0
$639$	84224.0	0.00815988
$640$	0	0
$641$	−1.00885e7	−0.969797	−0.484898	−	0.874570i	$-0.661143\pi$
$641$	−1.00885e7	−0.969797	−0.484898	+	0.874570i	$0.661143\pi$
$642$	0	0
$643$	3.87096e6	0.369225	0.184612	−	0.982811i	$-0.440897\pi$
$643$	3.87096e6	0.369225	0.184612	+	0.982811i	$0.440897\pi$
$644$	0	0
$645$	1.32214e7	1.25135
$646$	0	0
$647$	−1.51508e7	−1.42290	−0.711449	−	0.702737i	$-0.751961\pi$
$647$	−1.51508e7	−1.42290	−0.711449	+	0.702737i	$0.751961\pi$
$648$	0	0
$649$	5.76828e6	0.537569
$650$	0	0
$651$	−3.86081e6	−0.357047
$652$	0	0
$653$	−1.86286e7	−1.70961	−0.854806	−	0.518947i	$-0.826324\pi$
$653$	−1.86286e7	−1.70961	−0.854806	+	0.518947i	$0.826324\pi$
$654$	0	0
$655$	−1.89208e7	−1.72321
$656$	0	0
$657$	−1.78196e6	−0.161059
$658$	0	0
$659$	−4.23368e6	−0.379756	−0.189878	−	0.981808i	$-0.560809\pi$
$659$	−4.23368e6	−0.379756	−0.189878	+	0.981808i	$0.560809\pi$
$660$	0	0
$661$	7.91690e6	0.704777	0.352389	−	0.935854i	$-0.385370\pi$
$661$	7.91690e6	0.704777	0.352389	+	0.935854i	$0.385370\pi$
$662$	0	0
$663$	−9.99320e6	−0.882919
$664$	0	0
$665$	1.53664e6	0.134747
$666$	0	0
$667$	4.26495e7	3.71193
$668$	0	0
$669$	−5.87451e6	−0.507465
$670$	0	0
$671$	−1.06478e7	−0.912967
$672$	0	0
$673$	−1.50142e7	−1.27780	−0.638900	−	0.769290i	$-0.720610\pi$
$673$	−1.50142e7	−1.27780	−0.638900	+	0.769290i	$0.720610\pi$
$674$	0	0
$675$	−3.94226e6	−0.333032
$676$	0	0
$677$	−4.52404e6	−0.379362	−0.189681	−	0.981846i	$-0.560745\pi$
$677$	−4.52404e6	−0.379362	−0.189681	+	0.981846i	$0.560745\pi$
$678$	0	0
$679$	3.22371e6	0.268338
$680$	0	0
$681$	−1.08735e7	−0.898465
$682$	0	0
$683$	−1.79046e7	−1.46863	−0.734316	−	0.678808i	$-0.762497\pi$
$683$	−1.79046e7	−1.46863	−0.734316	+	0.678808i	$0.762497\pi$
$684$	0	0
$685$	−5.01235e6	−0.408146
$686$	0	0
$687$	1.00694e6	0.0813973
$688$	0	0
$689$	−1.98299e7	−1.59137
$690$	0	0
$691$	−7.15754e6	−0.570255	−0.285127	−	0.958490i	$-0.592036\pi$
$691$	−7.15754e6	−0.570255	−0.285127	+	0.958490i	$0.592036\pi$
$692$	0	0
$693$	967260.	0.0765086
$694$	0	0
$695$	−1.49766e7	−1.17612
$696$	0	0
$697$	−7.68414e6	−0.599119
$698$	0	0
$699$	−4.99752e6	−0.386867
$700$	0	0
$701$	1.95052e7	1.49918	0.749592	−	0.661900i	$-0.230250\pi$
$701$	1.95052e7	1.49918	0.749592	+	0.661900i	$0.230250\pi$
$702$	0	0
$703$	−702660.	−0.0536237
$704$	0	0
$705$	9.05677e6	0.686279
$706$	0	0
$707$	868672.	0.0653593
$708$	0	0
$709$	−9.18187e6	−0.685986	−0.342993	−	0.939338i	$-0.611441\pi$
$709$	−9.18187e6	−0.685986	−0.342993	+	0.939338i	$0.611441\pi$
$710$	0	0
$711$	4.51125e6	0.334675
$712$	0	0
$713$	2.74196e7	2.01994
$714$	0	0
$715$	2.31168e7	1.69107
$716$	0	0
$717$	−1.48936e7	−1.08194
$718$	0	0
$719$	4.64528e6	0.335112	0.167556	−	0.985863i	$-0.446412\pi$
$719$	4.64528e6	0.335112	0.167556	+	0.985863i	$0.446412\pi$
$720$	0	0
$721$	−5.96134e6	−0.427077
$722$	0	0
$723$	−1.16881e7	−0.831570
$724$	0	0
$725$	−8.50013e6	−0.600594
$726$	0	0
$727$	1.67153e7	1.17295	0.586473	−	0.809969i	$-0.300516\pi$
$727$	1.67153e7	1.17295	0.586473	+	0.809969i	$0.300516\pi$
$728$	0	0
$729$	1.59468e7	1.11136
$730$	0	0
$731$	1.22475e7	0.847722
$732$	0	0
$733$	3.55366e6	0.244295	0.122148	−	0.992512i	$-0.461022\pi$
$733$	3.55366e6	0.244295	0.122148	+	0.992512i	$0.461022\pi$
$734$	0	0
$735$	2.15130e6	0.146886
$736$	0	0
$737$	−8.30256e6	−0.563045
$738$	0	0
$739$	−1.34812e7	−0.908069	−0.454035	−	0.890984i	$-0.650016\pi$
$739$	−1.34812e7	−0.908069	−0.454035	+	0.890984i	$0.650016\pi$
$740$	0	0
$741$	−5.89960e6	−0.394709
$742$	0	0
$743$	1.04869e7	0.696907	0.348454	−	0.937326i	$-0.386707\pi$
$743$	1.04869e7	0.696907	0.348454	+	0.937326i	$0.386707\pi$
$744$	0	0
$745$	2.54491e7	1.67989
$746$	0	0
$747$	−4.14737e6	−0.271939
$748$	0	0
$749$	−2.36533e6	−0.154059
$750$	0	0
$751$	−1.15094e7	−0.744651	−0.372326	−	0.928102i	$-0.621440\pi$
$751$	−1.15094e7	−0.744651	−0.372326	+	0.928102i	$0.621440\pi$
$752$	0	0
$753$	2.22697e7	1.43129
$754$	0	0
$755$	1.51316e7	0.966093
$756$	0	0
$757$	1.45928e7	0.925548	0.462774	−	0.886476i	$-0.346854\pi$
$757$	1.45928e7	0.925548	0.462774	+	0.886476i	$0.346854\pi$
$758$	0	0
$759$	2.86474e7	1.80501
$760$	0	0
$761$	−3.33690e6	−0.208873	−0.104436	−	0.994532i	$-0.533304\pi$
$761$	−3.33690e6	−0.208873	−0.104436	+	0.994532i	$0.533304\pi$
$762$	0	0
$763$	8.70269e6	0.541181
$764$	0	0
$765$	−2.49664e6	−0.154242
$766$	0	0
$767$	1.18112e7	0.724948
$768$	0	0
$769$	−1.49953e7	−0.914408	−0.457204	−	0.889362i	$-0.651149\pi$
$769$	−1.49953e7	−0.914408	−0.457204	+	0.889362i	$0.651149\pi$
$770$	0	0
$771$	1.04656e6	0.0634054
$772$	0	0
$773$	1.89084e7	1.13817	0.569085	−	0.822279i	$-0.307298\pi$
$773$	1.89084e7	1.13817	0.569085	+	0.822279i	$0.307298\pi$
$774$	0	0
$775$	−5.46479e6	−0.326828
$776$	0	0
$777$	−983724.	−0.0584549
$778$	0	0
$779$	−4.53642e6	−0.267836
$780$	0	0
$781$	752640.	0.0441529
$782$	0	0
$783$	3.55412e7	2.07171
$784$	0	0
$785$	−2.03853e6	−0.118071
$786$	0	0
$787$	−1.28586e7	−0.740044	−0.370022	−	0.929023i	$-0.620650\pi$
$787$	−1.28586e7	−0.740044	−0.370022	+	0.929023i	$0.620650\pi$
$788$	0	0
$789$	136416.	0.00780140
$790$	0	0
$791$	6.28170e6	0.356974
$792$	0	0
$793$	−2.18027e7	−1.23120
$794$	0	0
$795$	2.06600e7	1.15934
$796$	0	0
$797$	−2.38505e7	−1.33000	−0.665001	−	0.746842i	$-0.731569\pi$
$797$	−2.38505e7	−1.33000	−0.665001	+	0.746842i	$0.731569\pi$
$798$	0	0
$799$	8.38964e6	0.464918
$800$	0	0
$801$	−2.05681e6	−0.113270
$802$	0	0
$803$	−1.59239e7	−0.871485
$804$	0	0
$805$	−1.52786e7	−0.830986
$806$	0	0
$807$	1.74377e7	0.942551
$808$	0	0
$809$	−2.95899e7	−1.58955	−0.794773	−	0.606907i	$-0.792410\pi$
$809$	−2.95899e7	−1.58955	−0.794773	+	0.606907i	$0.792410\pi$
$810$	0	0
$811$	2.73826e7	1.46192	0.730959	−	0.682421i	$-0.239073\pi$
$811$	2.73826e7	1.46192	0.730959	+	0.682421i	$0.239073\pi$
$812$	0	0
$813$	−2.27689e7	−1.20814
$814$	0	0
$815$	2.58030e7	1.36074
$816$	0	0
$817$	7.23044e6	0.378974
$818$	0	0
$819$	1.98058e6	0.103177
$820$	0	0
$821$	−1.62427e7	−0.841007	−0.420503	−	0.907291i	$-0.638147\pi$
$821$	−1.62427e7	−0.841007	−0.420503	+	0.907291i	$0.638147\pi$
$822$	0	0
$823$	2.13459e7	1.09854	0.549269	−	0.835646i	$-0.314906\pi$
$823$	2.13459e7	1.09854	0.549269	+	0.835646i	$0.314906\pi$
$824$	0	0
$825$	−5.70948e6	−0.292053
$826$	0	0
$827$	−1.88590e7	−0.958860	−0.479430	−	0.877580i	$-0.659157\pi$
$827$	−1.88590e7	−0.958860	−0.479430	+	0.877580i	$0.659157\pi$
$828$	0	0
$829$	−2.50128e7	−1.26408	−0.632041	−	0.774935i	$-0.717783\pi$
$829$	−2.50128e7	−1.26408	−0.632041	+	0.774935i	$0.717783\pi$
$830$	0	0
$831$	−1.38698e6	−0.0696735
$832$	0	0
$833$	1.99283e6	0.0995079
$834$	0	0
$835$	−3.64188e7	−1.80763
$836$	0	0
$837$	2.28497e7	1.12737
$838$	0	0
$839$	2.34469e6	0.114996	0.0574978	−	0.998346i	$-0.481688\pi$
$839$	2.34469e6	0.114996	0.0574978	+	0.998346i	$0.481688\pi$
$840$	0	0
$841$	5.61214e7	2.73614
$842$	0	0
$843$	−2.50832e7	−1.21566
$844$	0	0
$845$	2.35716e7	1.13566
$846$	0	0
$847$	752101.	0.0360220
$848$	0	0
$849$	1.49985e7	0.714133
$850$	0	0
$851$	6.98645e6	0.330699
$852$	0	0
$853$	3.96869e7	1.86756	0.933780	−	0.357846i	$-0.116489\pi$
$853$	3.96869e7	1.86756	0.933780	+	0.357846i	$0.116489\pi$
$854$	0	0
$855$	−1.47392e6	−0.0689539
$856$	0	0
$857$	4.61557e6	0.214671	0.107335	−	0.994223i	$-0.465768\pi$
$857$	4.61557e6	0.214671	0.107335	+	0.994223i	$0.465768\pi$
$858$	0	0
$859$	5.95701e6	0.275452	0.137726	−	0.990470i	$-0.456021\pi$
$859$	5.95701e6	0.275452	0.137726	+	0.990470i	$0.456021\pi$
$860$	0	0
$861$	−6.35099e6	−0.291967
$862$	0	0
$863$	−5.36222e6	−0.245086	−0.122543	−	0.992463i	$-0.539105\pi$
$863$	−5.36222e6	−0.245086	−0.122543	+	0.992463i	$0.539105\pi$
$864$	0	0
$865$	−3.20074e7	−1.45449
$866$	0	0
$867$	−1.02334e7	−0.462351
$868$	0	0
$869$	4.03133e7	1.81092
$870$	0	0
$871$	−1.70005e7	−0.759304
$872$	0	0
$873$	−3.09213e6	−0.137316
$874$	0	0
$875$	−6.75494e6	−0.298265
$876$	0	0
$877$	4.19646e7	1.84240	0.921200	−	0.389090i	$-0.127210\pi$
$877$	4.19646e7	1.84240	0.921200	+	0.389090i	$0.127210\pi$
$878$	0	0
$879$	5.54434e6	0.242035
$880$	0	0
$881$	−1.03589e7	−0.449649	−0.224824	−	0.974399i	$-0.572181\pi$
$881$	−1.03589e7	−0.449649	−0.224824	+	0.974399i	$0.572181\pi$
$882$	0	0
$883$	3.46323e7	1.49479	0.747394	−	0.664381i	$-0.231305\pi$
$883$	3.46323e7	1.49479	0.747394	+	0.664381i	$0.231305\pi$
$884$	0	0
$885$	−1.23057e7	−0.528137
$886$	0	0
$887$	−2.43735e7	−1.04018	−0.520090	−	0.854112i	$-0.674102\pi$
$887$	−2.43735e7	−1.04018	−0.520090	+	0.854112i	$0.674102\pi$
$888$	0	0
$889$	−1.48450e7	−0.629980
$890$	0	0
$891$	1.90760e7	0.804994
$892$	0	0
$893$	4.95292e6	0.207842
$894$	0	0
$895$	−7.22534e6	−0.301509
$896$	0	0
$897$	5.86589e7	2.43418
$898$	0	0
$899$	4.92675e7	2.03311
$900$	0	0
$901$	1.91381e7	0.785394
$902$	0	0
$903$	1.01226e7	0.413117
$904$	0	0
$905$	1.32992e6	0.0539764
$906$	0	0
$907$	−2.84124e7	−1.14680	−0.573402	−	0.819274i	$-0.694377\pi$
$907$	−2.84124e7	−1.14680	−0.573402	+	0.819274i	$0.694377\pi$
$908$	0	0
$909$	−833216.	−0.0334463
$910$	0	0
$911$	1.72159e7	0.687278	0.343639	−	0.939102i	$-0.388340\pi$
$911$	1.72159e7	0.687278	0.343639	+	0.939102i	$0.388340\pi$
$912$	0	0
$913$	−3.70616e7	−1.47146
$914$	0	0
$915$	2.27154e7	0.896948
$916$	0	0
$917$	−1.44863e7	−0.568896
$918$	0	0
$919$	4.23398e7	1.65371	0.826855	−	0.562415i	$-0.190128\pi$
$919$	4.23398e7	1.65371	0.826855	+	0.562415i	$0.190128\pi$
$920$	0	0
$921$	−1.76692e7	−0.686385
$922$	0	0
$923$	1.54112e6	0.0595432
$924$	0	0
$925$	−1.39241e6	−0.0535074
$926$	0	0
$927$	5.71802e6	0.218548
$928$	0	0
$929$	2.05248e7	0.780260	0.390130	−	0.920760i	$-0.372430\pi$
$929$	2.05248e7	0.780260	0.390130	+	0.920760i	$0.372430\pi$
$930$	0	0
$931$	1.17649e6	0.0444850
$932$	0	0
$933$	−2.39426e7	−0.900465
$934$	0	0
$935$	−2.23104e7	−0.834600
$936$	0	0
$937$	1.86579e7	0.694248	0.347124	−	0.937819i	$-0.387158\pi$
$937$	1.86579e7	0.694248	0.347124	+	0.937819i	$0.387158\pi$
$938$	0	0
$939$	−8.96888e6	−0.331951
$940$	0	0
$941$	−4.42313e6	−0.162838	−0.0814189	−	0.996680i	$-0.525945\pi$
$941$	−4.42313e6	−0.162838	−0.0814189	+	0.996680i	$0.525945\pi$
$942$	0	0
$943$	4.51050e7	1.65175
$944$	0	0
$945$	−1.27322e7	−0.463791
$946$	0	0
$947$	−4.46116e6	−0.161649	−0.0808244	−	0.996728i	$-0.525755\pi$
$947$	−4.46116e6	−0.161649	−0.0808244	+	0.996728i	$0.525755\pi$
$948$	0	0
$949$	−3.26060e7	−1.17526
$950$	0	0
$951$	7.26233e6	0.260390
$952$	0	0
$953$	−1.95591e7	−0.697616	−0.348808	−	0.937194i	$-0.613413\pi$
$953$	−1.95591e7	−0.697616	−0.348808	+	0.937194i	$0.613413\pi$
$954$	0	0
$955$	1.37626e7	0.488305
$956$	0	0
$957$	5.14735e7	1.81679
$958$	0	0
$959$	−3.83758e6	−0.134745
$960$	0	0
$961$	3.04523e6	0.106368
$962$	0	0
$963$	2.26878e6	0.0788365
$964$	0	0
$965$	2.61498e7	0.903960
$966$	0	0
$967$	1.70289e7	0.585625	0.292813	−	0.956170i	$-0.405409\pi$
$967$	1.70289e7	0.585625	0.292813	+	0.956170i	$0.405409\pi$
$968$	0	0
$969$	5.69380e6	0.194802
$970$	0	0
$971$	−1.10092e7	−0.374721	−0.187361	−	0.982291i	$-0.559993\pi$
$971$	−1.10092e7	−0.374721	−0.187361	+	0.982291i	$0.559993\pi$
$972$	0	0
$973$	−1.14665e7	−0.388283
$974$	0	0
$975$	−1.16908e7	−0.393853
$976$	0	0
$977$	−1.84999e7	−0.620058	−0.310029	−	0.950727i	$-0.600339\pi$
$977$	−1.84999e7	−0.620058	−0.310029	+	0.950727i	$0.600339\pi$
$978$	0	0
$979$	−1.83800e7	−0.612900
$980$	0	0
$981$	−8.34748e6	−0.276938
$982$	0	0
$983$	−3.19756e7	−1.05544	−0.527721	−	0.849418i	$-0.676953\pi$
$983$	−3.19756e7	−1.05544	−0.527721	+	0.849418i	$0.676953\pi$
$984$	0	0
$985$	4.61871e7	1.51681
$986$	0	0
$987$	6.93409e6	0.226567
$988$	0	0
$989$	−7.18912e7	−2.33714
$990$	0	0
$991$	−5.23128e7	−1.69209	−0.846046	−	0.533111i	$-0.821023\pi$
$991$	−5.23128e7	−1.69209	−0.846046	+	0.533111i	$0.821023\pi$
$992$	0	0
$993$	9.79177e6	0.315129
$994$	0	0
$995$	−2.42117e7	−0.775296
$996$	0	0
$997$	−2.60757e7	−0.830802	−0.415401	−	0.909638i	$-0.636359\pi$
$997$	−2.60757e7	−0.830802	−0.415401	+	0.909638i	$0.636359\pi$
$998$	0	0
$999$	5.82204e6	0.184570

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.n.1.1		1
4.3	odd	2		448.6.a.d.1.1		1
8.3	odd	2		224.6.a.b.1.1	yes	1
8.5	even	2		224.6.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.6.a.a.1.1	✓	1	8.5	even	2
224.6.a.b.1.1	yes	1	8.3	odd	2
448.6.a.d.1.1		1	4.3	odd	2
448.6.a.n.1.1		1	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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