LMFDB - Embedded newform 448.6.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

//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);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
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:	$71.8519512762$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
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)

$f(q)$

$=$

$q+14.0000 q^{3} +56.0000 q^{5} -49.0000 q^{7} -47.0000 q^{9} +O(q^{10})$

$q+14.0000 q^{3} +56.0000 q^{5} -49.0000 q^{7} -47.0000 q^{9} -232.000 q^{11} +140.000 q^{13} +784.000 q^{15} -1722.00 q^{17} +98.0000 q^{19} -686.000 q^{21} +1824.00 q^{23} +11.0000 q^{25} -4060.00 q^{27} -3418.00 q^{29} -7644.00 q^{31} -3248.00 q^{33} -2744.00 q^{35} +10398.0 q^{37} +1960.00 q^{39} -17962.0 q^{41} -10880.0 q^{43} -2632.00 q^{45} +9324.00 q^{47} +2401.00 q^{49} -24108.0 q^{51} -2262.00 q^{53} -12992.0 q^{55} +1372.00 q^{57} +2730.00 q^{59} -25648.0 q^{61} +2303.00 q^{63} +7840.00 q^{65} +48404.0 q^{67} +25536.0 q^{69} -58560.0 q^{71} +68082.0 q^{73} +154.000 q^{75} +11368.0 q^{77} +31784.0 q^{79} -45419.0 q^{81} +20538.0 q^{83} -96432.0 q^{85} -47852.0 q^{87} -50582.0 q^{89} -6860.00 q^{91} -107016. q^{93} +5488.00 q^{95} -58506.0 q^{97} +10904.0 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	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
$5$	56.0000	1.00176	0.500879	−	0.865517i	$-0.333010\pi$
$5$	56.0000	1.00176	0.500879	+	0.865517i	$0.333010\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−47.0000	−0.193416
$10$	0	0
$11$	−232.000	−0.578104	−0.289052	−	0.957313i	$-0.593340\pi$
$11$	−232.000	−0.578104	−0.289052	+	0.957313i	$0.593340\pi$
$12$	0	0
$13$	140.000	0.229757	0.114879	−	0.993380i	$-0.463352\pi$
$13$	140.000	0.229757	0.114879	+	0.993380i	$0.463352\pi$
$14$	0	0
$15$	784.000	0.899680
$16$	0	0
$17$	−1722.00	−1.44514	−0.722572	−	0.691296i	$-0.757040\pi$
$17$	−1722.00	−1.44514	−0.722572	+	0.691296i	$0.757040\pi$
$18$	0	0
$19$	98.0000	0.0622791	0.0311395	−	0.999515i	$-0.490086\pi$
$19$	98.0000	0.0622791	0.0311395	+	0.999515i	$0.490086\pi$
$20$	0	0
$21$	−686.000	−0.339450
$22$	0	0
$23$	1824.00	0.718961	0.359480	−	0.933153i	$-0.382954\pi$
$23$	1824.00	0.718961	0.359480	+	0.933153i	$0.382954\pi$
$24$	0	0
$25$	11.0000	0.00352000
$26$	0	0
$27$	−4060.00	−1.07181
$28$	0	0
$29$	−3418.00	−0.754705	−0.377352	−	0.926070i	$-0.623165\pi$
$29$	−3418.00	−0.754705	−0.377352	+	0.926070i	$0.623165\pi$
$30$	0	0
$31$	−7644.00	−1.42862	−0.714310	−	0.699830i	$-0.753259\pi$
$31$	−7644.00	−1.42862	−0.714310	+	0.699830i	$0.753259\pi$
$32$	0	0
$33$	−3248.00	−0.519196
$34$	0	0
$35$	−2744.00	−0.378629
$36$	0	0
$37$	10398.0	1.24866	0.624332	−	0.781159i	$-0.285371\pi$
$37$	10398.0	1.24866	0.624332	+	0.781159i	$0.285371\pi$
$38$	0	0
$39$	1960.00	0.206345
$40$	0	0
$41$	−17962.0	−1.66876	−0.834382	−	0.551186i	$-0.814175\pi$
$41$	−17962.0	−1.66876	−0.834382	+	0.551186i	$0.814175\pi$
$42$	0	0
$43$	−10880.0	−0.897342	−0.448671	−	0.893697i	$-0.648102\pi$
$43$	−10880.0	−0.897342	−0.448671	+	0.893697i	$0.648102\pi$
$44$	0	0
$45$	−2632.00	−0.193756
$46$	0	0
$47$	9324.00	0.615684	0.307842	−	0.951438i	$-0.400393\pi$
$47$	9324.00	0.615684	0.307842	+	0.951438i	$0.400393\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−24108.0	−1.29788
$52$	0	0
$53$	−2262.00	−0.110612	−0.0553061	−	0.998469i	$-0.517613\pi$
$53$	−2262.00	−0.110612	−0.0553061	+	0.998469i	$0.517613\pi$
$54$	0	0
$55$	−12992.0	−0.579121
$56$	0	0
$57$	1372.00	0.0559329
$58$	0	0
$59$	2730.00	0.102102	0.0510508	−	0.998696i	$-0.483743\pi$
$59$	2730.00	0.102102	0.0510508	+	0.998696i	$0.483743\pi$
$60$	0	0
$61$	−25648.0	−0.882529	−0.441264	−	0.897377i	$-0.645470\pi$
$61$	−25648.0	−0.882529	−0.441264	+	0.897377i	$0.645470\pi$
$62$	0	0
$63$	2303.00	0.0731042
$64$	0	0
$65$	7840.00	0.230161
$66$	0	0
$67$	48404.0	1.31733	0.658664	−	0.752437i	$-0.271122\pi$
$67$	48404.0	1.31733	0.658664	+	0.752437i	$0.271122\pi$
$68$	0	0
$69$	25536.0	0.645699
$70$	0	0
$71$	−58560.0	−1.37865	−0.689327	−	0.724450i	$-0.742094\pi$
$71$	−58560.0	−1.37865	−0.689327	+	0.724450i	$0.742094\pi$
$72$	0	0
$73$	68082.0	1.49529	0.747645	−	0.664099i	$-0.231185\pi$
$73$	68082.0	1.49529	0.747645	+	0.664099i	$0.231185\pi$
$74$	0	0
$75$	154.000	0.00316131
$76$	0	0
$77$	11368.0	0.218503
$78$	0	0
$79$	31784.0	0.572982	0.286491	−	0.958083i	$-0.407511\pi$
$79$	31784.0	0.572982	0.286491	+	0.958083i	$0.407511\pi$
$80$	0	0
$81$	−45419.0	−0.769175
$82$	0	0
$83$	20538.0	0.327237	0.163619	−	0.986524i	$-0.447683\pi$
$83$	20538.0	0.327237	0.163619	+	0.986524i	$0.447683\pi$
$84$	0	0
$85$	−96432.0	−1.44768
$86$	0	0
$87$	−47852.0	−0.677801
$88$	0	0
$89$	−50582.0	−0.676894	−0.338447	−	0.940985i	$-0.609902\pi$
$89$	−50582.0	−0.676894	−0.338447	+	0.940985i	$0.609902\pi$
$90$	0	0
$91$	−6860.00	−0.0868402
$92$	0	0
$93$	−107016.	−1.28304
$94$	0	0
$95$	5488.00	0.0623886
$96$	0	0
$97$	−58506.0	−0.631351	−0.315676	−	0.948867i	$-0.602231\pi$
$97$	−58506.0	−0.631351	−0.315676	+	0.948867i	$0.602231\pi$
$98$	0	0
$99$	10904.0	0.111814
$100$	0	0
$101$	−38696.0	−0.377453	−0.188726	−	0.982030i	$-0.560436\pi$
$101$	−38696.0	−0.377453	−0.188726	+	0.982030i	$0.560436\pi$
$102$	0	0
$103$	53060.0	0.492804	0.246402	−	0.969168i	$-0.420752\pi$
$103$	53060.0	0.492804	0.246402	+	0.969168i	$0.420752\pi$
$104$	0	0
$105$	−38416.0	−0.340047
$106$	0	0
$107$	146324.	1.23554	0.617769	−	0.786360i	$-0.288037\pi$
$107$	146324.	1.23554	0.617769	+	0.786360i	$0.288037\pi$
$108$	0	0
$109$	−92898.0	−0.748928	−0.374464	−	0.927241i	$-0.622173\pi$
$109$	−92898.0	−0.748928	−0.374464	+	0.927241i	$0.622173\pi$
$110$	0	0
$111$	145572.	1.12143
$112$	0	0
$113$	−83354.0	−0.614088	−0.307044	−	0.951695i	$-0.599340\pi$
$113$	−83354.0	−0.614088	−0.307044	+	0.951695i	$0.599340\pi$
$114$	0	0
$115$	102144.	0.720225
$116$	0	0
$117$	−6580.00	−0.0444387
$118$	0	0
$119$	84378.0	0.546213
$120$	0	0
$121$	−107227.	−0.665795
$122$	0	0
$123$	−251468.	−1.49872
$124$	0	0
$125$	−174384.	−0.998232
$126$	0	0
$127$	60384.0	0.332210	0.166105	−	0.986108i	$-0.446881\pi$
$127$	60384.0	0.332210	0.166105	+	0.986108i	$0.446881\pi$
$128$	0	0
$129$	−152320.	−0.805903
$130$	0	0
$131$	61586.0	0.313548	0.156774	−	0.987635i	$-0.449891\pi$
$131$	61586.0	0.313548	0.156774	+	0.987635i	$0.449891\pi$
$132$	0	0
$133$	−4802.00	−0.0235393
$134$	0	0
$135$	−227360.	−1.07369
$136$	0	0
$137$	−204462.	−0.930703	−0.465352	−	0.885126i	$-0.654072\pi$
$137$	−204462.	−0.930703	−0.465352	+	0.885126i	$0.654072\pi$
$138$	0	0
$139$	35406.0	0.155432	0.0777159	−	0.996976i	$-0.475237\pi$
$139$	35406.0	0.155432	0.0777159	+	0.996976i	$0.475237\pi$
$140$	0	0
$141$	130536.	0.552946
$142$	0	0
$143$	−32480.0	−0.132824
$144$	0	0
$145$	−191408.	−0.756032
$146$	0	0
$147$	33614.0	0.128300
$148$	0	0
$149$	20226.0	0.0746353	0.0373177	−	0.999303i	$-0.488119\pi$
$149$	20226.0	0.0746353	0.0373177	+	0.999303i	$0.488119\pi$
$150$	0	0
$151$	70904.0	0.253063	0.126531	−	0.991963i	$-0.459616\pi$
$151$	70904.0	0.253063	0.126531	+	0.991963i	$0.459616\pi$
$152$	0	0
$153$	80934.0	0.279513
$154$	0	0
$155$	−428064.	−1.43113
$156$	0	0
$157$	−293524.	−0.950374	−0.475187	−	0.879885i	$-0.657620\pi$
$157$	−293524.	−0.950374	−0.475187	+	0.879885i	$0.657620\pi$
$158$	0	0
$159$	−31668.0	−0.0993408
$160$	0	0
$161$	−89376.0	−0.271742
$162$	0	0
$163$	−13192.0	−0.0388903	−0.0194452	−	0.999811i	$-0.506190\pi$
$163$	−13192.0	−0.0388903	−0.0194452	+	0.999811i	$0.506190\pi$
$164$	0	0
$165$	−181888.	−0.520109
$166$	0	0
$167$	493612.	1.36960	0.684801	−	0.728730i	$-0.259889\pi$
$167$	493612.	1.36960	0.684801	+	0.728730i	$0.259889\pi$
$168$	0	0
$169$	−351693.	−0.947212
$170$	0	0
$171$	−4606.00	−0.0120457
$172$	0	0
$173$	−240716.	−0.611490	−0.305745	−	0.952113i	$-0.598906\pi$
$173$	−240716.	−0.611490	−0.305745	+	0.952113i	$0.598906\pi$
$174$	0	0
$175$	−539.000	−0.00133043
$176$	0	0
$177$	38220.0	0.0916975
$178$	0	0
$179$	−294932.	−0.688001	−0.344001	−	0.938969i	$-0.611782\pi$
$179$	−294932.	−0.688001	−0.344001	+	0.938969i	$0.611782\pi$
$180$	0	0
$181$	336980.	0.764553	0.382277	−	0.924048i	$-0.375140\pi$
$181$	336980.	0.764553	0.382277	+	0.924048i	$0.375140\pi$
$182$	0	0
$183$	−359072.	−0.792600
$184$	0	0
$185$	582288.	1.25086
$186$	0	0
$187$	399504.	0.835444
$188$	0	0
$189$	198940.	0.405105
$190$	0	0
$191$	358264.	0.710591	0.355296	−	0.934754i	$-0.384380\pi$
$191$	358264.	0.710591	0.355296	+	0.934754i	$0.384380\pi$
$192$	0	0
$193$	−989554.	−1.91226	−0.956128	−	0.292948i	$-0.905364\pi$
$193$	−989554.	−1.91226	−0.956128	+	0.292948i	$0.905364\pi$
$194$	0	0
$195$	109760.	0.206708
$196$	0	0
$197$	990050.	1.81757	0.908786	−	0.417263i	$-0.137011\pi$
$197$	990050.	1.81757	0.908786	+	0.417263i	$0.137011\pi$
$198$	0	0
$199$	−840756.	−1.50500	−0.752501	−	0.658591i	$-0.771153\pi$
$199$	−840756.	−1.50500	−0.752501	+	0.658591i	$0.771153\pi$
$200$	0	0
$201$	677656.	1.18309
$202$	0	0
$203$	167482.	0.285252
$204$	0	0
$205$	−1.00587e6	−1.67170
$206$	0	0
$207$	−85728.0	−0.139058
$208$	0	0
$209$	−22736.0	−0.0360038
$210$	0	0
$211$	−1.15073e6	−1.77938	−0.889689	−	0.456568i	$-0.849079\pi$
$211$	−1.15073e6	−1.77938	−0.889689	+	0.456568i	$0.849079\pi$
$212$	0	0
$213$	−819840.	−1.23817
$214$	0	0
$215$	−609280.	−0.898919
$216$	0	0
$217$	374556.	0.539967
$218$	0	0
$219$	953148.	1.34292
$220$	0	0
$221$	−241080.	−0.332032
$222$	0	0
$223$	−824264.	−1.10995	−0.554976	−	0.831866i	$-0.687273\pi$
$223$	−824264.	−1.10995	−0.554976	+	0.831866i	$0.687273\pi$
$224$	0	0
$225$	−517.000	−0.000680823 0
$226$	0	0
$227$	−74382.0	−0.0958083	−0.0479042	−	0.998852i	$-0.515254\pi$
$227$	−74382.0	−0.0958083	−0.0479042	+	0.998852i	$0.515254\pi$
$228$	0	0
$229$	−1.13196e6	−1.42640	−0.713199	−	0.700961i	$-0.752755\pi$
$229$	−1.13196e6	−1.42640	−0.713199	+	0.700961i	$0.752755\pi$
$230$	0	0
$231$	159152.	0.196238
$232$	0	0
$233$	−198726.	−0.239809	−0.119904	−	0.992785i	$-0.538259\pi$
$233$	−198726.	−0.239809	−0.119904	+	0.992785i	$0.538259\pi$
$234$	0	0
$235$	522144.	0.616766
$236$	0	0
$237$	444976.	0.514595
$238$	0	0
$239$	482904.	0.546847	0.273424	−	0.961894i	$-0.411844\pi$
$239$	482904.	0.546847	0.273424	+	0.961894i	$0.411844\pi$
$240$	0	0
$241$	805910.	0.893807	0.446904	−	0.894582i	$-0.352527\pi$
$241$	805910.	0.893807	0.446904	+	0.894582i	$0.352527\pi$
$242$	0	0
$243$	350714.	0.381011
$244$	0	0
$245$	134456.	0.143108
$246$	0	0
$247$	13720.0	0.0143091
$248$	0	0
$249$	287532.	0.293892
$250$	0	0
$251$	−430738.	−0.431548	−0.215774	−	0.976443i	$-0.569227\pi$
$251$	−430738.	−0.431548	−0.215774	+	0.976443i	$0.569227\pi$
$252$	0	0
$253$	−423168.	−0.415634
$254$	0	0
$255$	−1.35005e6	−1.30017
$256$	0	0
$257$	−1.17691e6	−1.11150	−0.555751	−	0.831349i	$-0.687569\pi$
$257$	−1.17691e6	−1.11150	−0.555751	+	0.831349i	$0.687569\pi$
$258$	0	0
$259$	−509502.	−0.471951
$260$	0	0
$261$	160646.	0.145972
$262$	0	0
$263$	1.29098e6	1.15088	0.575438	−	0.817845i	$-0.304831\pi$
$263$	1.29098e6	1.15088	0.575438	+	0.817845i	$0.304831\pi$
$264$	0	0
$265$	−126672.	−0.110807
$266$	0	0
$267$	−708148.	−0.607919
$268$	0	0
$269$	1.27756e6	1.07646	0.538232	−	0.842797i	$-0.319093\pi$
$269$	1.27756e6	1.07646	0.538232	+	0.842797i	$0.319093\pi$
$270$	0	0
$271$	1.65054e6	1.36522	0.682612	−	0.730781i	$-0.260844\pi$
$271$	1.65054e6	1.36522	0.682612	+	0.730781i	$0.260844\pi$
$272$	0	0
$273$	−96040.0	−0.0779912
$274$	0	0
$275$	−2552.00	−0.00203493
$276$	0	0
$277$	1.06409e6	0.833257	0.416628	−	0.909077i	$-0.363212\pi$
$277$	1.06409e6	0.833257	0.416628	+	0.909077i	$0.363212\pi$
$278$	0	0
$279$	359268.	0.276317
$280$	0	0
$281$	−22342.0	−0.0168794	−0.00843969	−	0.999964i	$-0.502686\pi$
$281$	−22342.0	−0.0168794	−0.00843969	+	0.999964i	$0.502686\pi$
$282$	0	0
$283$	2.49574e6	1.85239	0.926196	−	0.377042i	$-0.123059\pi$
$283$	2.49574e6	1.85239	0.926196	+	0.377042i	$0.123059\pi$
$284$	0	0
$285$	76832.0	0.0560312
$286$	0	0
$287$	880138.	0.630734
$288$	0	0
$289$	1.54543e6	1.08844
$290$	0	0
$291$	−819084.	−0.567017
$292$	0	0
$293$	1.93178e6	1.31458	0.657291	−	0.753637i	$-0.271702\pi$
$293$	1.93178e6	1.31458	0.657291	+	0.753637i	$0.271702\pi$
$294$	0	0
$295$	152880.	0.102281
$296$	0	0
$297$	941920.	0.619616
$298$	0	0
$299$	255360.	0.165187
$300$	0	0
$301$	533120.	0.339163
$302$	0	0
$303$	−541744.	−0.338991
$304$	0	0
$305$	−1.43629e6	−0.884081
$306$	0	0
$307$	459074.	0.277995	0.138997	−	0.990293i	$-0.455612\pi$
$307$	459074.	0.277995	0.138997	+	0.990293i	$0.455612\pi$
$308$	0	0
$309$	742840.	0.442587
$310$	0	0
$311$	667128.	0.391118	0.195559	−	0.980692i	$-0.437348\pi$
$311$	667128.	0.391118	0.195559	+	0.980692i	$0.437348\pi$
$312$	0	0
$313$	−111034.	−0.0640612	−0.0320306	−	0.999487i	$-0.510197\pi$
$313$	−111034.	−0.0640612	−0.0320306	+	0.999487i	$0.510197\pi$
$314$	0	0
$315$	128968.	0.0732328
$316$	0	0
$317$	68778.0	0.0384416	0.0192208	−	0.999815i	$-0.493881\pi$
$317$	68778.0	0.0384416	0.0192208	+	0.999815i	$0.493881\pi$
$318$	0	0
$319$	792976.	0.436298
$320$	0	0
$321$	2.04854e6	1.10964
$322$	0	0
$323$	−168756.	−0.0900022
$324$	0	0
$325$	1540.00	0.000808746 0
$326$	0	0
$327$	−1.30057e6	−0.672613
$328$	0	0
$329$	−456876.	−0.232707
$330$	0	0
$331$	564448.	0.283174	0.141587	−	0.989926i	$-0.454779\pi$
$331$	564448.	0.283174	0.141587	+	0.989926i	$0.454779\pi$
$332$	0	0
$333$	−488706.	−0.241511
$334$	0	0
$335$	2.71062e6	1.31965
$336$	0	0
$337$	2.07729e6	0.996376	0.498188	−	0.867069i	$-0.333999\pi$
$337$	2.07729e6	0.996376	0.498188	+	0.867069i	$0.333999\pi$
$338$	0	0
$339$	−1.16696e6	−0.551512
$340$	0	0
$341$	1.77341e6	0.825891
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	1.43002e6	0.646834
$346$	0	0
$347$	53248.0	0.0237399	0.0118700	−	0.999930i	$-0.496222\pi$
$347$	53248.0	0.0237399	0.0118700	+	0.999930i	$0.496222\pi$
$348$	0	0
$349$	2.27200e6	0.998494	0.499247	−	0.866460i	$-0.333610\pi$
$349$	2.27200e6	0.998494	0.499247	+	0.866460i	$0.333610\pi$
$350$	0	0
$351$	−568400.	−0.246256
$352$	0	0
$353$	4.00645e6	1.71129	0.855644	−	0.517565i	$-0.173162\pi$
$353$	4.00645e6	1.71129	0.855644	+	0.517565i	$0.173162\pi$
$354$	0	0
$355$	−3.27936e6	−1.38108
$356$	0	0
$357$	1.18129e6	0.490554
$358$	0	0
$359$	73784.0	0.0302152	0.0151076	−	0.999886i	$-0.495191\pi$
$359$	73784.0	0.0302152	0.0151076	+	0.999886i	$0.495191\pi$
$360$	0	0
$361$	−2.46650e6	−0.996121
$362$	0	0
$363$	−1.50118e6	−0.597951
$364$	0	0
$365$	3.81259e6	1.49792
$366$	0	0
$367$	1.40431e6	0.544250	0.272125	−	0.962262i	$-0.412274\pi$
$367$	1.40431e6	0.544250	0.272125	+	0.962262i	$0.412274\pi$
$368$	0	0
$369$	844214.	0.322765
$370$	0	0
$371$	110838.	0.0418075
$372$	0	0
$373$	1.60323e6	0.596657	0.298329	−	0.954463i	$-0.403571\pi$
$373$	1.60323e6	0.596657	0.298329	+	0.954463i	$0.403571\pi$
$374$	0	0
$375$	−2.44138e6	−0.896513
$376$	0	0
$377$	−478520.	−0.173399
$378$	0	0
$379$	4.77012e6	1.70581	0.852906	−	0.522064i	$-0.174838\pi$
$379$	4.77012e6	1.70581	0.852906	+	0.522064i	$0.174838\pi$
$380$	0	0
$381$	845376.	0.298358
$382$	0	0
$383$	−2.23079e6	−0.777072	−0.388536	−	0.921434i	$-0.627019\pi$
$383$	−2.23079e6	−0.777072	−0.388536	+	0.921434i	$0.627019\pi$
$384$	0	0
$385$	636608.	0.218887
$386$	0	0
$387$	511360.	0.173560
$388$	0	0
$389$	−4.84024e6	−1.62178	−0.810892	−	0.585196i	$-0.801018\pi$
$389$	−4.84024e6	−1.62178	−0.810892	+	0.585196i	$0.801018\pi$
$390$	0	0
$391$	−3.14093e6	−1.03900
$392$	0	0
$393$	862204.	0.281597
$394$	0	0
$395$	1.77990e6	0.573989
$396$	0	0
$397$	−995820.	−0.317106	−0.158553	−	0.987350i	$-0.550683\pi$
$397$	−995820.	−0.317106	−0.158553	+	0.987350i	$0.550683\pi$
$398$	0	0
$399$	−67228.0	−0.0211406
$400$	0	0
$401$	−3.31605e6	−1.02982	−0.514909	−	0.857245i	$-0.672174\pi$
$401$	−3.31605e6	−1.02982	−0.514909	+	0.857245i	$0.672174\pi$
$402$	0	0
$403$	−1.07016e6	−0.328236
$404$	0	0
$405$	−2.54346e6	−0.770527
$406$	0	0
$407$	−2.41234e6	−0.721858
$408$	0	0
$409$	3.07273e6	0.908274	0.454137	−	0.890932i	$-0.349948\pi$
$409$	3.07273e6	0.908274	0.454137	+	0.890932i	$0.349948\pi$
$410$	0	0
$411$	−2.86247e6	−0.835865
$412$	0	0
$413$	−133770.	−0.0385908
$414$	0	0
$415$	1.15013e6	0.327813
$416$	0	0
$417$	495684.	0.139593
$418$	0	0
$419$	−2.81438e6	−0.783154	−0.391577	−	0.920145i	$-0.628070\pi$
$419$	−2.81438e6	−0.783154	−0.391577	+	0.920145i	$0.628070\pi$
$420$	0	0
$421$	−3.05802e6	−0.840883	−0.420441	−	0.907320i	$-0.638125\pi$
$421$	−3.05802e6	−0.840883	−0.420441	+	0.907320i	$0.638125\pi$
$422$	0	0
$423$	−438228.	−0.119083
$424$	0	0
$425$	−18942.0	−0.00508690
$426$	0	0
$427$	1.25675e6	0.333565
$428$	0	0
$429$	−454720.	−0.119289
$430$	0	0
$431$	1.93750e6	0.502398	0.251199	−	0.967936i	$-0.419175\pi$
$431$	1.93750e6	0.502398	0.251199	+	0.967936i	$0.419175\pi$
$432$	0	0
$433$	3.94790e6	1.01192	0.505961	−	0.862557i	$-0.331138\pi$
$433$	3.94790e6	1.01192	0.505961	+	0.862557i	$0.331138\pi$
$434$	0	0
$435$	−2.67971e6	−0.678993
$436$	0	0
$437$	178752.	0.0447762
$438$	0	0
$439$	−7.41770e6	−1.83700	−0.918498	−	0.395426i	$-0.870597\pi$
$439$	−7.41770e6	−1.83700	−0.918498	+	0.395426i	$0.870597\pi$
$440$	0	0
$441$	−112847.	−0.0276308
$442$	0	0
$443$	−1.40269e6	−0.339589	−0.169794	−	0.985480i	$-0.554310\pi$
$443$	−1.40269e6	−0.339589	−0.169794	+	0.985480i	$0.554310\pi$
$444$	0	0
$445$	−2.83259e6	−0.678085
$446$	0	0
$447$	283164.	0.0670300
$448$	0	0
$449$	−590574.	−0.138248	−0.0691239	−	0.997608i	$-0.522020\pi$
$449$	−590574.	−0.138248	−0.0691239	+	0.997608i	$0.522020\pi$
$450$	0	0
$451$	4.16718e6	0.964720
$452$	0	0
$453$	992656.	0.227276
$454$	0	0
$455$	−384160.	−0.0869929
$456$	0	0
$457$	−2.90484e6	−0.650627	−0.325313	−	0.945606i	$-0.605470\pi$
$457$	−2.90484e6	−0.650627	−0.325313	+	0.945606i	$0.605470\pi$
$458$	0	0
$459$	6.99132e6	1.54891
$460$	0	0
$461$	922684.	0.202209	0.101105	−	0.994876i	$-0.467762\pi$
$461$	922684.	0.202209	0.101105	+	0.994876i	$0.467762\pi$
$462$	0	0
$463$	7.18235e6	1.55709	0.778546	−	0.627588i	$-0.215958\pi$
$463$	7.18235e6	1.55709	0.778546	+	0.627588i	$0.215958\pi$
$464$	0	0
$465$	−5.99290e6	−1.28530
$466$	0	0
$467$	612570.	0.129976	0.0649881	−	0.997886i	$-0.479299\pi$
$467$	612570.	0.129976	0.0649881	+	0.997886i	$0.479299\pi$
$468$	0	0
$469$	−2.37180e6	−0.497904
$470$	0	0
$471$	−4.10934e6	−0.853531
$472$	0	0
$473$	2.52416e6	0.518757
$474$	0	0
$475$	1078.00	0.000219222 0
$476$	0	0
$477$	106314.	0.0213941
$478$	0	0
$479$	2.60330e6	0.518424	0.259212	−	0.965820i	$-0.416537\pi$
$479$	2.60330e6	0.518424	0.259212	+	0.965820i	$0.416537\pi$
$480$	0	0
$481$	1.45572e6	0.286890
$482$	0	0
$483$	−1.25126e6	−0.244051
$484$	0	0
$485$	−3.27634e6	−0.632461
$486$	0	0
$487$	5.46309e6	1.04380	0.521898	−	0.853008i	$-0.325224\pi$
$487$	5.46309e6	1.04380	0.521898	+	0.853008i	$0.325224\pi$
$488$	0	0
$489$	−184688.	−0.0349274
$490$	0	0
$491$	−1.64090e6	−0.307170	−0.153585	−	0.988135i	$-0.549082\pi$
$491$	−1.64090e6	−0.307170	−0.153585	+	0.988135i	$0.549082\pi$
$492$	0	0
$493$	5.88580e6	1.09066
$494$	0	0
$495$	610624.	0.112011
$496$	0	0
$497$	2.86944e6	0.521082
$498$	0	0
$499$	−2.99796e6	−0.538983	−0.269491	−	0.963003i	$-0.586856\pi$
$499$	−2.99796e6	−0.538983	−0.269491	+	0.963003i	$0.586856\pi$
$500$	0	0
$501$	6.91057e6	1.23004
$502$	0	0
$503$	−6.89405e6	−1.21494	−0.607469	−	0.794343i	$-0.707815\pi$
$503$	−6.89405e6	−1.21494	−0.607469	+	0.794343i	$0.707815\pi$
$504$	0	0
$505$	−2.16698e6	−0.378117
$506$	0	0
$507$	−4.92370e6	−0.850691
$508$	0	0
$509$	−2.30476e6	−0.394305	−0.197152	−	0.980373i	$-0.563169\pi$
$509$	−2.30476e6	−0.394305	−0.197152	+	0.980373i	$0.563169\pi$
$510$	0	0
$511$	−3.33602e6	−0.565166
$512$	0	0
$513$	−397880.	−0.0667511
$514$	0	0
$515$	2.97136e6	0.493671
$516$	0	0
$517$	−2.16317e6	−0.355929
$518$	0	0
$519$	−3.37002e6	−0.549180
$520$	0	0
$521$	−1.20960e7	−1.95231	−0.976155	−	0.217073i	$-0.930349\pi$
$521$	−1.20960e7	−1.95231	−0.976155	+	0.217073i	$0.930349\pi$
$522$	0	0
$523$	−5.48443e6	−0.876753	−0.438377	−	0.898791i	$-0.644446\pi$
$523$	−5.48443e6	−0.876753	−0.438377	+	0.898791i	$0.644446\pi$
$524$	0	0
$525$	−7546.00	−0.00119486
$526$	0	0
$527$	1.31630e7	2.06456
$528$	0	0
$529$	−3.10937e6	−0.483095
$530$	0	0
$531$	−128310.	−0.0197480
$532$	0	0
$533$	−2.51468e6	−0.383411
$534$	0	0
$535$	8.19414e6	1.23771
$536$	0	0
$537$	−4.12905e6	−0.617894
$538$	0	0
$539$	−557032.	−0.0825863
$540$	0	0
$541$	6.71799e6	0.986839	0.493420	−	0.869791i	$-0.335747\pi$
$541$	6.71799e6	0.986839	0.493420	+	0.869791i	$0.335747\pi$
$542$	0	0
$543$	4.71772e6	0.686646
$544$	0	0
$545$	−5.20229e6	−0.750245
$546$	0	0
$547$	5.00235e6	0.714835	0.357418	−	0.933945i	$-0.383657\pi$
$547$	5.00235e6	0.714835	0.357418	+	0.933945i	$0.383657\pi$
$548$	0	0
$549$	1.20546e6	0.170695
$550$	0	0
$551$	−334964.	−0.0470023
$552$	0	0
$553$	−1.55742e6	−0.216567
$554$	0	0
$555$	8.15203e6	1.12340
$556$	0	0
$557$	−9.01961e6	−1.23183	−0.615913	−	0.787814i	$-0.711213\pi$
$557$	−9.01961e6	−1.23183	−0.615913	+	0.787814i	$0.711213\pi$
$558$	0	0
$559$	−1.52320e6	−0.206171
$560$	0	0
$561$	5.59306e6	0.750312
$562$	0	0
$563$	−1.24051e7	−1.64941	−0.824707	−	0.565561i	$-0.808660\pi$
$563$	−1.24051e7	−1.64941	−0.824707	+	0.565561i	$0.808660\pi$
$564$	0	0
$565$	−4.66782e6	−0.615167
$566$	0	0
$567$	2.22553e6	0.290721
$568$	0	0
$569$	6.48804e6	0.840103	0.420052	−	0.907500i	$-0.362012\pi$
$569$	6.48804e6	0.840103	0.420052	+	0.907500i	$0.362012\pi$
$570$	0	0
$571$	1.02285e7	1.31287	0.656435	−	0.754382i	$-0.272064\pi$
$571$	1.02285e7	1.31287	0.656435	+	0.754382i	$0.272064\pi$
$572$	0	0
$573$	5.01570e6	0.638182
$574$	0	0
$575$	20064.0	0.00253074
$576$	0	0
$577$	2.65338e6	0.331787	0.165894	−	0.986144i	$-0.446949\pi$
$577$	2.65338e6	0.331787	0.165894	+	0.986144i	$0.446949\pi$
$578$	0	0
$579$	−1.38538e7	−1.71740
$580$	0	0
$581$	−1.00636e6	−0.123684
$582$	0	0
$583$	524784.	0.0639454
$584$	0	0
$585$	−368480.	−0.0445168
$586$	0	0
$587$	1.43044e7	1.71346	0.856729	−	0.515766i	$-0.172493\pi$
$587$	1.43044e7	1.71346	0.856729	+	0.515766i	$0.172493\pi$
$588$	0	0
$589$	−749112.	−0.0889731
$590$	0	0
$591$	1.38607e7	1.63236
$592$	0	0
$593$	−1.00265e7	−1.17088	−0.585442	−	0.810714i	$-0.699079\pi$
$593$	−1.00265e7	−1.17088	−0.585442	+	0.810714i	$0.699079\pi$
$594$	0	0
$595$	4.72517e6	0.547173
$596$	0	0
$597$	−1.17706e7	−1.35164
$598$	0	0
$599$	−7.52292e6	−0.856681	−0.428341	−	0.903617i	$-0.640902\pi$
$599$	−7.52292e6	−0.856681	−0.428341	+	0.903617i	$0.640902\pi$
$600$	0	0
$601$	3.38625e6	0.382413	0.191207	−	0.981550i	$-0.438760\pi$
$601$	3.38625e6	0.382413	0.191207	+	0.981550i	$0.438760\pi$
$602$	0	0
$603$	−2.27499e6	−0.254792
$604$	0	0
$605$	−6.00471e6	−0.666966
$606$	0	0
$607$	−6.90861e6	−0.761060	−0.380530	−	0.924769i	$-0.624258\pi$
$607$	−6.90861e6	−0.761060	−0.380530	+	0.924769i	$0.624258\pi$
$608$	0	0
$609$	2.34475e6	0.256185
$610$	0	0
$611$	1.30536e6	0.141458
$612$	0	0
$613$	9.68896e6	1.04142	0.520710	−	0.853734i	$-0.325667\pi$
$613$	9.68896e6	1.04142	0.520710	+	0.853734i	$0.325667\pi$
$614$	0	0
$615$	−1.40822e7	−1.50135
$616$	0	0
$617$	−7.84742e6	−0.829877	−0.414939	−	0.909849i	$-0.636197\pi$
$617$	−7.84742e6	−0.829877	−0.414939	+	0.909849i	$0.636197\pi$
$618$	0	0
$619$	1.01972e7	1.06968	0.534840	−	0.844953i	$-0.320372\pi$
$619$	1.01972e7	1.06968	0.534840	+	0.844953i	$0.320372\pi$
$620$	0	0
$621$	−7.40544e6	−0.770587
$622$	0	0
$623$	2.47852e6	0.255842
$624$	0	0
$625$	−9.79988e6	−1.00351
$626$	0	0
$627$	−318304.	−0.0323350
$628$	0	0
$629$	−1.79054e7	−1.80450
$630$	0	0
$631$	−8.36258e6	−0.836116	−0.418058	−	0.908420i	$-0.637289\pi$
$631$	−8.36258e6	−0.836116	−0.418058	+	0.908420i	$0.637289\pi$
$632$	0	0
$633$	−1.61102e7	−1.59806
$634$	0	0
$635$	3.38150e6	0.332794
$636$	0	0
$637$	336140.	0.0328225
$638$	0	0
$639$	2.75232e6	0.266653
$640$	0	0
$641$	1.10283e6	0.106014	0.0530070	−	0.998594i	$-0.483119\pi$
$641$	1.10283e6	0.106014	0.0530070	+	0.998594i	$0.483119\pi$
$642$	0	0
$643$	−1.71354e7	−1.63443	−0.817217	−	0.576330i	$-0.804484\pi$
$643$	−1.71354e7	−1.63443	−0.817217	+	0.576330i	$0.804484\pi$
$644$	0	0
$645$	−8.52992e6	−0.807320
$646$	0	0
$647$	−54964.0	−0.00516200	−0.00258100	−	0.999997i	$-0.500822\pi$
$647$	−54964.0	−0.00516200	−0.00258100	+	0.999997i	$0.500822\pi$
$648$	0	0
$649$	−633360.	−0.0590254
$650$	0	0
$651$	5.24378e6	0.484945
$652$	0	0
$653$	485166.	0.0445254	0.0222627	−	0.999752i	$-0.492913\pi$
$653$	485166.	0.0445254	0.0222627	+	0.999752i	$0.492913\pi$
$654$	0	0
$655$	3.44882e6	0.314099
$656$	0	0
$657$	−3.19985e6	−0.289212
$658$	0	0
$659$	2.72136e6	0.244103	0.122051	−	0.992524i	$-0.461053\pi$
$659$	2.72136e6	0.244103	0.122051	+	0.992524i	$0.461053\pi$
$660$	0	0
$661$	2.14525e6	0.190974	0.0954869	−	0.995431i	$-0.469559\pi$
$661$	2.14525e6	0.190974	0.0954869	+	0.995431i	$0.469559\pi$
$662$	0	0
$663$	−3.37512e6	−0.298198
$664$	0	0
$665$	−268912.	−0.0235807
$666$	0	0
$667$	−6.23443e6	−0.542603
$668$	0	0
$669$	−1.15397e7	−0.996848
$670$	0	0
$671$	5.95034e6	0.510194
$672$	0	0
$673$	2.92796e6	0.249188	0.124594	−	0.992208i	$-0.460237\pi$
$673$	2.92796e6	0.249188	0.124594	+	0.992208i	$0.460237\pi$
$674$	0	0
$675$	−44660.0	−0.00377276
$676$	0	0
$677$	1.34992e7	1.13198	0.565988	−	0.824414i	$-0.308495\pi$
$677$	1.34992e7	1.13198	0.565988	+	0.824414i	$0.308495\pi$
$678$	0	0
$679$	2.86679e6	0.238628
$680$	0	0
$681$	−1.04135e6	−0.0860455
$682$	0	0
$683$	5.42972e6	0.445375	0.222688	−	0.974890i	$-0.428517\pi$
$683$	5.42972e6	0.445375	0.222688	+	0.974890i	$0.428517\pi$
$684$	0	0
$685$	−1.14499e7	−0.932340
$686$	0	0
$687$	−1.58474e7	−1.28105
$688$	0	0
$689$	−316680.	−0.0254140
$690$	0	0
$691$	−2.08280e7	−1.65940	−0.829702	−	0.558207i	$-0.811490\pi$
$691$	−2.08280e7	−1.65940	−0.829702	+	0.558207i	$0.811490\pi$
$692$	0	0
$693$	−534296.	−0.0422619
$694$	0	0
$695$	1.98274e6	0.155705
$696$	0	0
$697$	3.09306e7	2.41160
$698$	0	0
$699$	−2.78216e6	−0.215372
$700$	0	0
$701$	−2.35141e7	−1.80731	−0.903655	−	0.428261i	$-0.859126\pi$
$701$	−2.35141e7	−1.80731	−0.903655	+	0.428261i	$0.859126\pi$
$702$	0	0
$703$	1.01900e6	0.0777656
$704$	0	0
$705$	7.31002e6	0.553918
$706$	0	0
$707$	1.89610e6	0.142664
$708$	0	0
$709$	1.95747e7	1.46244	0.731221	−	0.682140i	$-0.238951\pi$
$709$	1.95747e7	1.46244	0.731221	+	0.682140i	$0.238951\pi$
$710$	0	0
$711$	−1.49385e6	−0.110824
$712$	0	0
$713$	−1.39427e7	−1.02712
$714$	0	0
$715$	−1.81888e6	−0.133057
$716$	0	0
$717$	6.76066e6	0.491124
$718$	0	0
$719$	−2.61152e7	−1.88396	−0.941978	−	0.335674i	$-0.891036\pi$
$719$	−2.61152e7	−1.88396	−0.941978	+	0.335674i	$0.891036\pi$
$720$	0	0
$721$	−2.59994e6	−0.186262
$722$	0	0
$723$	1.12827e7	0.802729
$724$	0	0
$725$	−37598.0	−0.00265656
$726$	0	0
$727$	1.54126e7	1.08154	0.540768	−	0.841172i	$-0.318134\pi$
$727$	1.54126e7	1.08154	0.540768	+	0.841172i	$0.318134\pi$
$728$	0	0
$729$	1.59468e7	1.11136
$730$	0	0
$731$	1.87354e7	1.29679
$732$	0	0
$733$	1.69868e7	1.16776	0.583878	−	0.811841i	$-0.301535\pi$
$733$	1.69868e7	1.16776	0.583878	+	0.811841i	$0.301535\pi$
$734$	0	0
$735$	1.88238e6	0.128526
$736$	0	0
$737$	−1.12297e7	−0.761554
$738$	0	0
$739$	−2.01511e6	−0.135734	−0.0678669	−	0.997694i	$-0.521619\pi$
$739$	−2.01511e6	−0.135734	−0.0678669	+	0.997694i	$0.521619\pi$
$740$	0	0
$741$	192080.	0.0128510
$742$	0	0
$743$	−1.51381e7	−1.00600	−0.503001	−	0.864286i	$-0.667771\pi$
$743$	−1.51381e7	−1.00600	−0.503001	+	0.864286i	$0.667771\pi$
$744$	0	0
$745$	1.13266e6	0.0747666
$746$	0	0
$747$	−965286.	−0.0632928
$748$	0	0
$749$	−7.16988e6	−0.466989
$750$	0	0
$751$	7.21401e6	0.466742	0.233371	−	0.972388i	$-0.425024\pi$
$751$	7.21401e6	0.466742	0.233371	+	0.972388i	$0.425024\pi$
$752$	0	0
$753$	−6.03033e6	−0.387573
$754$	0	0
$755$	3.97062e6	0.253508
$756$	0	0
$757$	1.09697e7	0.695755	0.347877	−	0.937540i	$-0.386903\pi$
$757$	1.09697e7	0.695755	0.347877	+	0.937540i	$0.386903\pi$
$758$	0	0
$759$	−5.92435e6	−0.373281
$760$	0	0
$761$	1.92442e7	1.20459	0.602293	−	0.798275i	$-0.294254\pi$
$761$	1.92442e7	1.20459	0.602293	+	0.798275i	$0.294254\pi$
$762$	0	0
$763$	4.55200e6	0.283068
$764$	0	0
$765$	4.53230e6	0.280005
$766$	0	0
$767$	382200.	0.0234586
$768$	0	0
$769$	8.21185e6	0.500755	0.250378	−	0.968148i	$-0.419445\pi$
$769$	8.21185e6	0.500755	0.250378	+	0.968148i	$0.419445\pi$
$770$	0	0
$771$	−1.64767e7	−0.998241
$772$	0	0
$773$	−1.86187e7	−1.12073	−0.560363	−	0.828247i	$-0.689338\pi$
$773$	−1.86187e7	−1.12073	−0.560363	+	0.828247i	$0.689338\pi$
$774$	0	0
$775$	−84084.0	−0.00502874
$776$	0	0
$777$	−7.13303e6	−0.423859
$778$	0	0
$779$	−1.76028e6	−0.103929
$780$	0	0
$781$	1.35859e7	0.797006
$782$	0	0
$783$	1.38771e7	0.808898
$784$	0	0
$785$	−1.64373e7	−0.952045
$786$	0	0
$787$	−2.62501e7	−1.51075	−0.755377	−	0.655291i	$-0.772546\pi$
$787$	−2.62501e7	−1.51075	−0.755377	+	0.655291i	$0.772546\pi$
$788$	0	0
$789$	1.80737e7	1.03360
$790$	0	0
$791$	4.08435e6	0.232103
$792$	0	0
$793$	−3.59072e6	−0.202768
$794$	0	0
$795$	−1.77341e6	−0.0995155
$796$	0	0
$797$	1.00373e7	0.559720	0.279860	−	0.960041i	$-0.409712\pi$
$797$	1.00373e7	0.559720	0.279860	+	0.960041i	$0.409712\pi$
$798$	0	0
$799$	−1.60559e7	−0.889751
$800$	0	0
$801$	2.37735e6	0.130922
$802$	0	0
$803$	−1.57950e7	−0.864433
$804$	0	0
$805$	−5.00506e6	−0.272220
$806$	0	0
$807$	1.78858e7	0.966772
$808$	0	0
$809$	1.40884e7	0.756816	0.378408	−	0.925639i	$-0.376472\pi$
$809$	1.40884e7	0.756816	0.378408	+	0.925639i	$0.376472\pi$
$810$	0	0
$811$	−1.81433e7	−0.968646	−0.484323	−	0.874889i	$-0.660934\pi$
$811$	−1.81433e7	−0.968646	−0.484323	+	0.874889i	$0.660934\pi$
$812$	0	0
$813$	2.31076e7	1.22611
$814$	0	0
$815$	−738752.	−0.0389587
$816$	0	0
$817$	−1.06624e6	−0.0558856
$818$	0	0
$819$	322420.	0.0167962
$820$	0	0
$821$	2.13669e7	1.10633	0.553164	−	0.833072i	$-0.313420\pi$
$821$	2.13669e7	1.10633	0.553164	+	0.833072i	$0.313420\pi$
$822$	0	0
$823$	1.78017e7	0.916142	0.458071	−	0.888916i	$-0.348541\pi$
$823$	1.78017e7	0.916142	0.458071	+	0.888916i	$0.348541\pi$
$824$	0	0
$825$	−35728.0	−0.00182757
$826$	0	0
$827$	−1.62921e7	−0.828350	−0.414175	−	0.910197i	$-0.635930\pi$
$827$	−1.62921e7	−0.828350	−0.414175	+	0.910197i	$0.635930\pi$
$828$	0	0
$829$	2.08499e6	0.105370	0.0526851	−	0.998611i	$-0.483222\pi$
$829$	2.08499e6	0.105370	0.0526851	+	0.998611i	$0.483222\pi$
$830$	0	0
$831$	1.48973e7	0.748348
$832$	0	0
$833$	−4.13452e6	−0.206449
$834$	0	0
$835$	2.76423e7	1.37201
$836$	0	0
$837$	3.10346e7	1.53120
$838$	0	0
$839$	−2.27850e7	−1.11749	−0.558745	−	0.829340i	$-0.688717\pi$
$839$	−2.27850e7	−1.11749	−0.558745	+	0.829340i	$0.688717\pi$
$840$	0	0
$841$	−8.82842e6	−0.430421
$842$	0	0
$843$	−312788.	−0.0151594
$844$	0	0
$845$	−1.96948e7	−0.948877
$846$	0	0
$847$	5.25412e6	0.251647
$848$	0	0
$849$	3.49403e7	1.66363
$850$	0	0
$851$	1.89660e7	0.897740
$852$	0	0
$853$	2.26975e7	1.06808	0.534042	−	0.845458i	$-0.320672\pi$
$853$	2.26975e7	1.06808	0.534042	+	0.845458i	$0.320672\pi$
$854$	0	0
$855$	−257936.	−0.0120669
$856$	0	0
$857$	2.52900e7	1.17624	0.588120	−	0.808774i	$-0.299868\pi$
$857$	2.52900e7	1.17624	0.588120	+	0.808774i	$0.299868\pi$
$858$	0	0
$859$	1.03947e7	0.480652	0.240326	−	0.970692i	$-0.422746\pi$
$859$	1.03947e7	0.480652	0.240326	+	0.970692i	$0.422746\pi$
$860$	0	0
$861$	1.23219e7	0.566462
$862$	0	0
$863$	4.33399e7	1.98089	0.990447	−	0.137892i	$-0.0440327\pi$
$863$	4.33399e7	1.98089	0.990447	+	0.137892i	$0.0440327\pi$
$864$	0	0
$865$	−1.34801e7	−0.612566
$866$	0	0
$867$	2.16360e7	0.977527
$868$	0	0
$869$	−7.37389e6	−0.331243
$870$	0	0
$871$	6.77656e6	0.302666
$872$	0	0
$873$	2.74978e6	0.122113
$874$	0	0
$875$	8.54482e6	0.377296
$876$	0	0
$877$	−3.71659e7	−1.63172	−0.815861	−	0.578248i	$-0.803736\pi$
$877$	−3.71659e7	−1.63172	−0.815861	+	0.578248i	$0.803736\pi$
$878$	0	0
$879$	2.70449e7	1.18063
$880$	0	0
$881$	9.04785e6	0.392740	0.196370	−	0.980530i	$-0.437085\pi$
$881$	9.04785e6	0.392740	0.196370	+	0.980530i	$0.437085\pi$
$882$	0	0
$883$	−3.29679e7	−1.42295	−0.711474	−	0.702712i	$-0.751972\pi$
$883$	−3.29679e7	−1.42295	−0.711474	+	0.702712i	$0.751972\pi$
$884$	0	0
$885$	2.14032e6	0.0918588
$886$	0	0
$887$	−1.61099e7	−0.687517	−0.343758	−	0.939058i	$-0.611700\pi$
$887$	−1.61099e7	−0.687517	−0.343758	+	0.939058i	$0.611700\pi$
$888$	0	0
$889$	−2.95882e6	−0.125564
$890$	0	0
$891$	1.05372e7	0.444663
$892$	0	0
$893$	913752.	0.0383442
$894$	0	0
$895$	−1.65162e7	−0.689211
$896$	0	0
$897$	3.57504e6	0.148354
$898$	0	0
$899$	2.61272e7	1.07819
$900$	0	0
$901$	3.89516e6	0.159850
$902$	0	0
$903$	7.46368e6	0.304603
$904$	0	0
$905$	1.88709e7	0.765898
$906$	0	0
$907$	4.47286e7	1.80537	0.902686	−	0.430300i	$-0.141592\pi$
$907$	4.47286e7	1.80537	0.902686	+	0.430300i	$0.141592\pi$
$908$	0	0
$909$	1.81871e6	0.0730053
$910$	0	0
$911$	−6.60518e6	−0.263687	−0.131844	−	0.991271i	$-0.542090\pi$
$911$	−6.60518e6	−0.263687	−0.131844	+	0.991271i	$0.542090\pi$
$912$	0	0
$913$	−4.76482e6	−0.189177
$914$	0	0
$915$	−2.01080e7	−0.793993
$916$	0	0
$917$	−3.01771e6	−0.118510
$918$	0	0
$919$	−3.08930e7	−1.20662	−0.603311	−	0.797506i	$-0.706152\pi$
$919$	−3.08930e7	−1.20662	−0.603311	+	0.797506i	$0.706152\pi$
$920$	0	0
$921$	6.42704e6	0.249667
$922$	0	0
$923$	−8.19840e6	−0.316756
$924$	0	0
$925$	114378.	0.00439530
$926$	0	0
$927$	−2.49382e6	−0.0953160
$928$	0	0
$929$	−4.87215e6	−0.185217	−0.0926087	−	0.995703i	$-0.529521\pi$
$929$	−4.87215e6	−0.185217	−0.0926087	+	0.995703i	$0.529521\pi$
$930$	0	0
$931$	235298.	0.00889701
$932$	0	0
$933$	9.33979e6	0.351264
$934$	0	0
$935$	2.23722e7	0.836913
$936$	0	0
$937$	−3.25004e7	−1.20932	−0.604658	−	0.796485i	$-0.706690\pi$
$937$	−3.25004e7	−1.20932	−0.604658	+	0.796485i	$0.706690\pi$
$938$	0	0
$939$	−1.55448e6	−0.0575334
$940$	0	0
$941$	2.64040e6	0.0972066	0.0486033	−	0.998818i	$-0.484523\pi$
$941$	2.64040e6	0.0972066	0.0486033	+	0.998818i	$0.484523\pi$
$942$	0	0
$943$	−3.27627e7	−1.19978
$944$	0	0
$945$	1.11406e7	0.405817
$946$	0	0
$947$	4.08179e7	1.47903	0.739513	−	0.673142i	$-0.235056\pi$
$947$	4.08179e7	1.47903	0.739513	+	0.673142i	$0.235056\pi$
$948$	0	0
$949$	9.53148e6	0.343554
$950$	0	0
$951$	962892.	0.0345244
$952$	0	0
$953$	−6.71983e6	−0.239677	−0.119838	−	0.992793i	$-0.538238\pi$
$953$	−6.71983e6	−0.239677	−0.119838	+	0.992793i	$0.538238\pi$
$954$	0	0
$955$	2.00628e7	0.711841
$956$	0	0
$957$	1.11017e7	0.391840
$958$	0	0
$959$	1.00186e7	0.351773
$960$	0	0
$961$	2.98016e7	1.04095
$962$	0	0
$963$	−6.87723e6	−0.238972
$964$	0	0
$965$	−5.54150e7	−1.91562
$966$	0	0
$967$	−2.78979e6	−0.0959413	−0.0479707	−	0.998849i	$-0.515275\pi$
$967$	−2.78979e6	−0.0959413	−0.0479707	+	0.998849i	$0.515275\pi$
$968$	0	0
$969$	−2.36258e6	−0.0808310
$970$	0	0
$971$	−3.33594e7	−1.13545	−0.567727	−	0.823217i	$-0.692177\pi$
$971$	−3.33594e7	−1.13545	−0.567727	+	0.823217i	$0.692177\pi$
$972$	0	0
$973$	−1.73489e6	−0.0587477
$974$	0	0
$975$	21560.0	0.000726335 0
$976$	0	0
$977$	−7.60033e6	−0.254739	−0.127370	−	0.991855i	$-0.540653\pi$
$977$	−7.60033e6	−0.254739	−0.127370	+	0.991855i	$0.540653\pi$
$978$	0	0
$979$	1.17350e7	0.391316
$980$	0	0
$981$	4.36621e6	0.144854
$982$	0	0
$983$	−5.79760e6	−0.191366	−0.0956829	−	0.995412i	$-0.530503\pi$
$983$	−5.79760e6	−0.191366	−0.0956829	+	0.995412i	$0.530503\pi$
$984$	0	0
$985$	5.54428e7	1.82077
$986$	0	0
$987$	−6.39626e6	−0.208994
$988$	0	0
$989$	−1.98451e7	−0.645153
$990$	0	0
$991$	1.26825e7	0.410224	0.205112	−	0.978739i	$-0.434244\pi$
$991$	1.26825e7	0.410224	0.205112	+	0.978739i	$0.434244\pi$
$992$	0	0
$993$	7.90227e6	0.254319
$994$	0	0
$995$	−4.70823e7	−1.50765
$996$	0	0
$997$	−1.44400e7	−0.460077	−0.230039	−	0.973182i	$-0.573885\pi$
$997$	−1.44400e7	−0.460077	−0.230039	+	0.973182i	$0.573885\pi$
$998$	0	0
$999$	−4.22159e7	−1.33833

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.m.1.1		1
4.3	odd	2		448.6.a.c.1.1		1
8.3	odd	2		112.6.a.g.1.1		1
8.5	even	2		7.6.a.a.1.1	✓	1
24.5	odd	2		63.6.a.e.1.1		1
24.11	even	2		1008.6.a.y.1.1		1
40.13	odd	4		175.6.b.a.99.2		2
40.29	even	2		175.6.a.b.1.1		1
40.37	odd	4		175.6.b.a.99.1		2
56.5	odd	6		49.6.c.b.18.1		2
56.13	odd	2		49.6.a.a.1.1		1
56.27	even	2		784.6.a.c.1.1		1
56.37	even	6		49.6.c.c.18.1		2
56.45	odd	6		49.6.c.b.30.1		2
56.53	even	6		49.6.c.c.30.1		2
88.21	odd	2		847.6.a.b.1.1		1
168.125	even	2		441.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.6.a.a.1.1	✓	1	8.5	even	2
49.6.a.a.1.1		1	56.13	odd	2
49.6.c.b.18.1		2	56.5	odd	6
49.6.c.b.30.1		2	56.45	odd	6
49.6.c.c.18.1		2	56.37	even	6
49.6.c.c.30.1		2	56.53	even	6
63.6.a.e.1.1		1	24.5	odd	2
112.6.a.g.1.1		1	8.3	odd	2
175.6.a.b.1.1		1	40.29	even	2
175.6.b.a.99.1		2	40.37	odd	4
175.6.b.a.99.2		2	40.13	odd	4
441.6.a.k.1.1		1	168.125	even	2
448.6.a.c.1.1		1	4.3	odd	2
448.6.a.m.1.1		1	1.1	even	1	trivial
784.6.a.c.1.1		1	56.27	even	2
847.6.a.b.1.1		1	88.21	odd	2
1008.6.a.y.1.1		1	24.11	even	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 $\F_{q}$

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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