LMFDB - Embedded newform 448.6.a.f.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,-8,0,-10] 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 14)
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)

$f(q)$

$=$

$q-8.00000 q^{3} -10.0000 q^{5} -49.0000 q^{7} -179.000 q^{9} +340.000 q^{11} +294.000 q^{13} +80.0000 q^{15} +1226.00 q^{17} -2432.00 q^{19} +392.000 q^{21} +2000.00 q^{23} -3025.00 q^{25} +3376.00 q^{27} +6746.00 q^{29} +8856.00 q^{31} -2720.00 q^{33} +490.000 q^{35} -9182.00 q^{37} -2352.00 q^{39} -14574.0 q^{41} -8108.00 q^{43} +1790.00 q^{45} -312.000 q^{47} +2401.00 q^{49} -9808.00 q^{51} +14634.0 q^{53} -3400.00 q^{55} +19456.0 q^{57} +27656.0 q^{59} -34338.0 q^{61} +8771.00 q^{63} -2940.00 q^{65} -12316.0 q^{67} -16000.0 q^{69} +36920.0 q^{71} -61718.0 q^{73} +24200.0 q^{75} -16660.0 q^{77} -64752.0 q^{79} +16489.0 q^{81} +77056.0 q^{83} -12260.0 q^{85} -53968.0 q^{87} -8166.00 q^{89} -14406.0 q^{91} -70848.0 q^{93} +24320.0 q^{95} +20650.0 q^{97} -60860.0 q^{99} +O(q^{100})$

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
$3$	−8.00000	−0.513200	−0.256600	−	0.966518i	$-0.582602\pi$
$3$	−8.00000	−0.513200	−0.256600	+	0.966518i	$0.582602\pi$
$4$	0	0
$5$	−10.0000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−10.0000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	−179.000	−0.736626
$10$	0	0
$11$	340.000	0.847222	0.423611	−	0.905844i	$-0.360762\pi$
$11$	340.000	0.847222	0.423611	+	0.905844i	$0.360762\pi$
$12$	0	0
$13$	294.000	0.482491	0.241245	−	0.970464i	$-0.422444\pi$
$13$	294.000	0.482491	0.241245	+	0.970464i	$0.422444\pi$
$14$	0	0
$15$	80.0000	0.0918040
$16$	0	0
$17$	1226.00	1.02889	0.514444	−	0.857524i	$-0.327998\pi$
$17$	1226.00	1.02889	0.514444	+	0.857524i	$0.327998\pi$
$18$	0	0
$19$	−2432.00	−1.54554	−0.772769	−	0.634688i	$-0.781129\pi$
$19$	−2432.00	−1.54554	−0.772769	+	0.634688i	$0.781129\pi$
$20$	0	0
$21$	392.000	0.193971
$22$	0	0
$23$	2000.00	0.788334	0.394167	−	0.919039i	$-0.371033\pi$
$23$	2000.00	0.788334	0.394167	+	0.919039i	$0.371033\pi$
$24$	0	0
$25$	−3025.00	−0.968000
$26$	0	0
$27$	3376.00	0.891237
$28$	0	0
$29$	6746.00	1.48954	0.744769	−	0.667323i	$-0.232560\pi$
$29$	6746.00	1.48954	0.744769	+	0.667323i	$0.232560\pi$
$30$	0	0
$31$	8856.00	1.65513	0.827567	−	0.561366i	$-0.189724\pi$
$31$	8856.00	1.65513	0.827567	+	0.561366i	$0.189724\pi$
$32$	0	0
$33$	−2720.00	−0.434795
$34$	0	0
$35$	490.000	0.0676123
$36$	0	0
$37$	−9182.00	−1.10264	−0.551319	−	0.834295i	$-0.685875\pi$
$37$	−9182.00	−1.10264	−0.551319	+	0.834295i	$0.685875\pi$
$38$	0	0
$39$	−2352.00	−0.247614
$40$	0	0
$41$	−14574.0	−1.35400	−0.677001	−	0.735982i	$-0.736721\pi$
$41$	−14574.0	−1.35400	−0.677001	+	0.735982i	$0.736721\pi$
$42$	0	0
$43$	−8108.00	−0.668717	−0.334359	−	0.942446i	$-0.608520\pi$
$43$	−8108.00	−0.668717	−0.334359	+	0.942446i	$0.608520\pi$
$44$	0	0
$45$	1790.00	0.131772
$46$	0	0
$47$	−312.000	−0.0206020	−0.0103010	−	0.999947i	$-0.503279\pi$
$47$	−312.000	−0.0206020	−0.0103010	+	0.999947i	$0.503279\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−9808.00	−0.528026
$52$	0	0
$53$	14634.0	0.715605	0.357803	−	0.933797i	$-0.383526\pi$
$53$	14634.0	0.715605	0.357803	+	0.933797i	$0.383526\pi$
$54$	0	0
$55$	−3400.00	−0.151556
$56$	0	0
$57$	19456.0	0.793170
$58$	0	0
$59$	27656.0	1.03433	0.517165	−	0.855886i	$-0.326987\pi$
$59$	27656.0	1.03433	0.517165	+	0.855886i	$0.326987\pi$
$60$	0	0
$61$	−34338.0	−1.18155	−0.590773	−	0.806838i	$-0.701177\pi$
$61$	−34338.0	−1.18155	−0.590773	+	0.806838i	$0.701177\pi$
$62$	0	0
$63$	8771.00	0.278418
$64$	0	0
$65$	−2940.00	−0.0863106
$66$	0	0
$67$	−12316.0	−0.335184	−0.167592	−	0.985856i	$-0.553599\pi$
$67$	−12316.0	−0.335184	−0.167592	+	0.985856i	$0.553599\pi$
$68$	0	0
$69$	−16000.0	−0.404573
$70$	0	0
$71$	36920.0	0.869192	0.434596	−	0.900625i	$-0.356891\pi$
$71$	36920.0	0.869192	0.434596	+	0.900625i	$0.356891\pi$
$72$	0	0
$73$	−61718.0	−1.35552	−0.677758	−	0.735285i	$-0.737048\pi$
$73$	−61718.0	−1.35552	−0.677758	+	0.735285i	$0.737048\pi$
$74$	0	0
$75$	24200.0	0.496778
$76$	0	0
$77$	−16660.0	−0.320220
$78$	0	0
$79$	−64752.0	−1.16731	−0.583654	−	0.812002i	$-0.698378\pi$
$79$	−64752.0	−1.16731	−0.583654	+	0.812002i	$0.698378\pi$
$80$	0	0
$81$	16489.0	0.279243
$82$	0	0
$83$	77056.0	1.22775	0.613877	−	0.789402i	$-0.289609\pi$
$83$	77056.0	1.22775	0.613877	+	0.789402i	$0.289609\pi$
$84$	0	0
$85$	−12260.0	−0.184053
$86$	0	0
$87$	−53968.0	−0.764431
$88$	0	0
$89$	−8166.00	−0.109278	−0.0546392	−	0.998506i	$-0.517401\pi$
$89$	−8166.00	−0.109278	−0.0546392	+	0.998506i	$0.517401\pi$
$90$	0	0
$91$	−14406.0	−0.182364
$92$	0	0
$93$	−70848.0	−0.849416
$94$	0	0
$95$	24320.0	0.276474
$96$	0	0
$97$	20650.0	0.222839	0.111419	−	0.993773i	$-0.464460\pi$
$97$	20650.0	0.222839	0.111419	+	0.993773i	$0.464460\pi$
$98$	0	0
$99$	−60860.0	−0.624085
$100$	0	0
$101$	−186250.	−1.81674	−0.908370	−	0.418167i	$-0.862673\pi$
$101$	−186250.	−1.81674	−0.908370	+	0.418167i	$0.862673\pi$
$102$	0	0
$103$	−60064.0	−0.557855	−0.278927	−	0.960312i	$-0.589979\pi$
$103$	−60064.0	−0.557855	−0.278927	+	0.960312i	$0.589979\pi$
$104$	0	0
$105$	−3920.00	−0.0346987
$106$	0	0
$107$	−47892.0	−0.404393	−0.202196	−	0.979345i	$-0.564808\pi$
$107$	−47892.0	−0.404393	−0.202196	+	0.979345i	$0.564808\pi$
$108$	0	0
$109$	−22102.0	−0.178183	−0.0890913	−	0.996023i	$-0.528396\pi$
$109$	−22102.0	−0.178183	−0.0890913	+	0.996023i	$0.528396\pi$
$110$	0	0
$111$	73456.0	0.565874
$112$	0	0
$113$	−245054.	−1.80537	−0.902684	−	0.430304i	$-0.858406\pi$
$113$	−245054.	−1.80537	−0.902684	+	0.430304i	$0.858406\pi$
$114$	0	0
$115$	−20000.0	−0.141022
$116$	0	0
$117$	−52626.0	−0.355415
$118$	0	0
$119$	−60074.0	−0.388883
$120$	0	0
$121$	−45451.0	−0.282215
$122$	0	0
$123$	116592.	0.694874
$124$	0	0
$125$	61500.0	0.352047
$126$	0	0
$127$	−96696.0	−0.531985	−0.265992	−	0.963975i	$-0.585700\pi$
$127$	−96696.0	−0.531985	−0.265992	+	0.963975i	$0.585700\pi$
$128$	0	0
$129$	64864.0	0.343186
$130$	0	0
$131$	−134368.	−0.684097	−0.342048	−	0.939682i	$-0.611121\pi$
$131$	−134368.	−0.684097	−0.342048	+	0.939682i	$0.611121\pi$
$132$	0	0
$133$	119168.	0.584158
$134$	0	0
$135$	−33760.0	−0.159429
$136$	0	0
$137$	−294662.	−1.34129	−0.670645	−	0.741778i	$-0.733983\pi$
$137$	−294662.	−1.34129	−0.670645	+	0.741778i	$0.733983\pi$
$138$	0	0
$139$	−314944.	−1.38260	−0.691300	−	0.722568i	$-0.742962\pi$
$139$	−314944.	−1.38260	−0.691300	+	0.722568i	$0.742962\pi$
$140$	0	0
$141$	2496.00	0.0105730
$142$	0	0
$143$	99960.0	0.408777
$144$	0	0
$145$	−67460.0	−0.266457
$146$	0	0
$147$	−19208.0	−0.0733143
$148$	0	0
$149$	−113622.	−0.419273	−0.209636	−	0.977779i	$-0.567228\pi$
$149$	−113622.	−0.419273	−0.209636	+	0.977779i	$0.567228\pi$
$150$	0	0
$151$	408208.	1.45693	0.728466	−	0.685082i	$-0.240234\pi$
$151$	408208.	1.45693	0.728466	+	0.685082i	$0.240234\pi$
$152$	0	0
$153$	−219454.	−0.757905
$154$	0	0
$155$	−88560.0	−0.296080
$156$	0	0
$157$	−293546.	−0.950445	−0.475223	−	0.879866i	$-0.657632\pi$
$157$	−293546.	−0.950445	−0.475223	+	0.879866i	$0.657632\pi$
$158$	0	0
$159$	−117072.	−0.367249
$160$	0	0
$161$	−98000.0	−0.297962
$162$	0	0
$163$	317116.	0.934866	0.467433	−	0.884029i	$-0.345179\pi$
$163$	317116.	0.934866	0.467433	+	0.884029i	$0.345179\pi$
$164$	0	0
$165$	27200.0	0.0777784
$166$	0	0
$167$	141568.	0.392802	0.196401	−	0.980524i	$-0.437075\pi$
$167$	141568.	0.392802	0.196401	+	0.980524i	$0.437075\pi$
$168$	0	0
$169$	−284857.	−0.767203
$170$	0	0
$171$	435328.	1.13848
$172$	0	0
$173$	71222.0	0.180925	0.0904626	−	0.995900i	$-0.471165\pi$
$173$	71222.0	0.180925	0.0904626	+	0.995900i	$0.471165\pi$
$174$	0	0
$175$	148225.	0.365870
$176$	0	0
$177$	−221248.	−0.530819
$178$	0	0
$179$	−485628.	−1.13285	−0.566423	−	0.824114i	$-0.691673\pi$
$179$	−485628.	−1.13285	−0.566423	+	0.824114i	$0.691673\pi$
$180$	0	0
$181$	−657090.	−1.49083	−0.745416	−	0.666600i	$-0.767749\pi$
$181$	−657090.	−1.49083	−0.745416	+	0.666600i	$0.767749\pi$
$182$	0	0
$183$	274704.	0.606369
$184$	0	0
$185$	91820.0	0.197246
$186$	0	0
$187$	416840.	0.871697
$188$	0	0
$189$	−165424.	−0.336856
$190$	0	0
$191$	68304.0	0.135476	0.0677381	−	0.997703i	$-0.478422\pi$
$191$	68304.0	0.135476	0.0677381	+	0.997703i	$0.478422\pi$
$192$	0	0
$193$	352754.	0.681677	0.340839	−	0.940122i	$-0.389289\pi$
$193$	352754.	0.681677	0.340839	+	0.940122i	$0.389289\pi$
$194$	0	0
$195$	23520.0	0.0442946
$196$	0	0
$197$	−196982.	−0.361627	−0.180814	−	0.983517i	$-0.557873\pi$
$197$	−196982.	−0.361627	−0.180814	+	0.983517i	$0.557873\pi$
$198$	0	0
$199$	−1.10392e6	−1.97608	−0.988041	−	0.154192i	$-0.950723\pi$
$199$	−1.10392e6	−1.97608	−0.988041	+	0.154192i	$0.950723\pi$
$200$	0	0
$201$	98528.0	0.172016
$202$	0	0
$203$	−330554.	−0.562992
$204$	0	0
$205$	145740.	0.242211
$206$	0	0
$207$	−358000.	−0.580707
$208$	0	0
$209$	−826880.	−1.30941
$210$	0	0
$211$	103444.	0.159955	0.0799777	−	0.996797i	$-0.474515\pi$
$211$	103444.	0.159955	0.0799777	+	0.996797i	$0.474515\pi$
$212$	0	0
$213$	−295360.	−0.446070
$214$	0	0
$215$	81080.0	0.119624
$216$	0	0
$217$	−433944.	−0.625582
$218$	0	0
$219$	493744.	0.695651
$220$	0	0
$221$	360444.	0.496429
$222$	0	0
$223$	307328.	0.413847	0.206924	−	0.978357i	$-0.433655\pi$
$223$	307328.	0.413847	0.206924	+	0.978357i	$0.433655\pi$
$224$	0	0
$225$	541475.	0.713053
$226$	0	0
$227$	891792.	1.14868	0.574340	−	0.818617i	$-0.305259\pi$
$227$	891792.	1.14868	0.574340	+	0.818617i	$0.305259\pi$
$228$	0	0
$229$	−276706.	−0.348682	−0.174341	−	0.984685i	$-0.555780\pi$
$229$	−276706.	−0.348682	−0.174341	+	0.984685i	$0.555780\pi$
$230$	0	0
$231$	133280.	0.164337
$232$	0	0
$233$	1.47943e6	1.78528	0.892639	−	0.450772i	$-0.148851\pi$
$233$	1.47943e6	1.78528	0.892639	+	0.450772i	$0.148851\pi$
$234$	0	0
$235$	3120.00	0.00368540
$236$	0	0
$237$	518016.	0.599063
$238$	0	0
$239$	1.00034e6	1.13280	0.566402	−	0.824129i	$-0.308335\pi$
$239$	1.00034e6	1.13280	0.566402	+	0.824129i	$0.308335\pi$
$240$	0	0
$241$	1.35833e6	1.50648	0.753239	−	0.657747i	$-0.228490\pi$
$241$	1.35833e6	1.50648	0.753239	+	0.657747i	$0.228490\pi$
$242$	0	0
$243$	−952280.	−1.03454
$244$	0	0
$245$	−24010.0	−0.0255551
$246$	0	0
$247$	−715008.	−0.745708
$248$	0	0
$249$	−616448.	−0.630083
$250$	0	0
$251$	177408.	0.177742	0.0888708	−	0.996043i	$-0.471674\pi$
$251$	177408.	0.177742	0.0888708	+	0.996043i	$0.471674\pi$
$252$	0	0
$253$	680000.	0.667894
$254$	0	0
$255$	98080.0	0.0944561
$256$	0	0
$257$	326658.	0.308504	0.154252	−	0.988032i	$-0.450703\pi$
$257$	326658.	0.308504	0.154252	+	0.988032i	$0.450703\pi$
$258$	0	0
$259$	449918.	0.416758
$260$	0	0
$261$	−1.20753e6	−1.09723
$262$	0	0
$263$	−34920.0	−0.0311304	−0.0155652	−	0.999879i	$-0.504955\pi$
$263$	−34920.0	−0.0311304	−0.0155652	+	0.999879i	$0.504955\pi$
$264$	0	0
$265$	−146340.	−0.128011
$266$	0	0
$267$	65328.0	0.0560817
$268$	0	0
$269$	−716458.	−0.603685	−0.301842	−	0.953358i	$-0.597602\pi$
$269$	−716458.	−0.603685	−0.301842	+	0.953358i	$0.597602\pi$
$270$	0	0
$271$	−953376.	−0.788571	−0.394286	−	0.918988i	$-0.629008\pi$
$271$	−953376.	−0.788571	−0.394286	+	0.918988i	$0.629008\pi$
$272$	0	0
$273$	115248.	0.0935894
$274$	0	0
$275$	−1.02850e6	−0.820111
$276$	0	0
$277$	1.84729e6	1.44656	0.723279	−	0.690556i	$-0.242634\pi$
$277$	1.84729e6	1.44656	0.723279	+	0.690556i	$0.242634\pi$
$278$	0	0
$279$	−1.58522e6	−1.21921
$280$	0	0
$281$	−1.99601e6	−1.50798	−0.753991	−	0.656885i	$-0.771874\pi$
$281$	−1.99601e6	−1.50798	−0.753991	+	0.656885i	$0.771874\pi$
$282$	0	0
$283$	−234088.	−0.173745	−0.0868726	−	0.996219i	$-0.527687\pi$
$283$	−234088.	−0.173745	−0.0868726	+	0.996219i	$0.527687\pi$
$284$	0	0
$285$	−194560.	−0.141887
$286$	0	0
$287$	714126.	0.511764
$288$	0	0
$289$	83219.0	0.0586108
$290$	0	0
$291$	−165200.	−0.114361
$292$	0	0
$293$	2.50081e6	1.70181	0.850905	−	0.525320i	$-0.176054\pi$
$293$	2.50081e6	1.70181	0.850905	+	0.525320i	$0.176054\pi$
$294$	0	0
$295$	−276560.	−0.185027
$296$	0	0
$297$	1.14784e6	0.755075
$298$	0	0
$299$	588000.	0.380364
$300$	0	0
$301$	397292.	0.252751
$302$	0	0
$303$	1.49000e6	0.932352
$304$	0	0
$305$	343380.	0.211361
$306$	0	0
$307$	−2.34203e6	−1.41823	−0.709115	−	0.705092i	$-0.750905\pi$
$307$	−2.34203e6	−1.41823	−0.709115	+	0.705092i	$0.750905\pi$
$308$	0	0
$309$	480512.	0.286291
$310$	0	0
$311$	−163064.	−0.0955998	−0.0477999	−	0.998857i	$-0.515221\pi$
$311$	−163064.	−0.0955998	−0.0477999	+	0.998857i	$0.515221\pi$
$312$	0	0
$313$	1.73965e6	1.00369	0.501847	−	0.864957i	$-0.332654\pi$
$313$	1.73965e6	1.00369	0.501847	+	0.864957i	$0.332654\pi$
$314$	0	0
$315$	−87710.0	−0.0498050
$316$	0	0
$317$	1.79771e6	1.00478	0.502392	−	0.864640i	$-0.332454\pi$
$317$	1.79771e6	1.00478	0.502392	+	0.864640i	$0.332454\pi$
$318$	0	0
$319$	2.29364e6	1.26197
$320$	0	0
$321$	383136.	0.207535
$322$	0	0
$323$	−2.98163e6	−1.59019
$324$	0	0
$325$	−889350.	−0.467051
$326$	0	0
$327$	176816.	0.0914434
$328$	0	0
$329$	15288.0	0.00778683
$330$	0	0
$331$	2.47541e6	1.24187	0.620937	−	0.783861i	$-0.286752\pi$
$331$	2.47541e6	1.24187	0.620937	+	0.783861i	$0.286752\pi$
$332$	0	0
$333$	1.64358e6	0.812231
$334$	0	0
$335$	123160.	0.0599595
$336$	0	0
$337$	89154.0	0.0427628	0.0213814	−	0.999771i	$-0.493194\pi$
$337$	89154.0	0.0427628	0.0213814	+	0.999771i	$0.493194\pi$
$338$	0	0
$339$	1.96043e6	0.926515
$340$	0	0
$341$	3.01104e6	1.40227
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	160000.	0.0723723
$346$	0	0
$347$	−938556.	−0.418443	−0.209222	−	0.977868i	$-0.567093\pi$
$347$	−938556.	−0.418443	−0.209222	+	0.977868i	$0.567093\pi$
$348$	0	0
$349$	−3.34268e6	−1.46903	−0.734516	−	0.678591i	$-0.762591\pi$
$349$	−3.34268e6	−1.46903	−0.734516	+	0.678591i	$0.762591\pi$
$350$	0	0
$351$	992544.	0.430013
$352$	0	0
$353$	−3.76606e6	−1.60861	−0.804305	−	0.594217i	$-0.797462\pi$
$353$	−3.76606e6	−1.60861	−0.804305	+	0.594217i	$0.797462\pi$
$354$	0	0
$355$	−369200.	−0.155486
$356$	0	0
$357$	480592.	0.199575
$358$	0	0
$359$	−1.53934e6	−0.630376	−0.315188	−	0.949029i	$-0.602068\pi$
$359$	−1.53934e6	−0.630376	−0.315188	+	0.949029i	$0.602068\pi$
$360$	0	0
$361$	3.43852e6	1.38869
$362$	0	0
$363$	363608.	0.144833
$364$	0	0
$365$	617180.	0.242482
$366$	0	0
$367$	−859312.	−0.333032	−0.166516	−	0.986039i	$-0.553252\pi$
$367$	−859312.	−0.333032	−0.166516	+	0.986039i	$0.553252\pi$
$368$	0	0
$369$	2.60875e6	0.997392
$370$	0	0
$371$	−717066.	−0.270473
$372$	0	0
$373$	976586.	0.363445	0.181722	−	0.983350i	$-0.441833\pi$
$373$	976586.	0.363445	0.181722	+	0.983350i	$0.441833\pi$
$374$	0	0
$375$	−492000.	−0.180670
$376$	0	0
$377$	1.98332e6	0.718688
$378$	0	0
$379$	−106444.	−0.0380648	−0.0190324	−	0.999819i	$-0.506059\pi$
$379$	−106444.	−0.0380648	−0.0190324	+	0.999819i	$0.506059\pi$
$380$	0	0
$381$	773568.	0.273015
$382$	0	0
$383$	−2.00634e6	−0.698889	−0.349445	−	0.936957i	$-0.613630\pi$
$383$	−2.00634e6	−0.698889	−0.349445	+	0.936957i	$0.613630\pi$
$384$	0	0
$385$	166600.	0.0572827
$386$	0	0
$387$	1.45133e6	0.492594
$388$	0	0
$389$	684002.	0.229184	0.114592	−	0.993413i	$-0.463444\pi$
$389$	684002.	0.229184	0.114592	+	0.993413i	$0.463444\pi$
$390$	0	0
$391$	2.45200e6	0.811108
$392$	0	0
$393$	1.07494e6	0.351079
$394$	0	0
$395$	647520.	0.208814
$396$	0	0
$397$	222870.	0.0709701	0.0354850	−	0.999370i	$-0.488702\pi$
$397$	222870.	0.0709701	0.0354850	+	0.999370i	$0.488702\pi$
$398$	0	0
$399$	−953344.	−0.299790
$400$	0	0
$401$	1.90072e6	0.590279	0.295140	−	0.955454i	$-0.404634\pi$
$401$	1.90072e6	0.590279	0.295140	+	0.955454i	$0.404634\pi$
$402$	0	0
$403$	2.60366e6	0.798587
$404$	0	0
$405$	−164890.	−0.0499524
$406$	0	0
$407$	−3.12188e6	−0.934179
$408$	0	0
$409$	1.77715e6	0.525311	0.262656	−	0.964890i	$-0.415402\pi$
$409$	1.77715e6	0.525311	0.262656	+	0.964890i	$0.415402\pi$
$410$	0	0
$411$	2.35730e6	0.688350
$412$	0	0
$413$	−1.35514e6	−0.390940
$414$	0	0
$415$	−770560.	−0.219627
$416$	0	0
$417$	2.51955e6	0.709550
$418$	0	0
$419$	−28056.0	−0.00780712	−0.00390356	−	0.999992i	$-0.501243\pi$
$419$	−28056.0	−0.00780712	−0.00390356	+	0.999992i	$0.501243\pi$
$420$	0	0
$421$	2.70897e6	0.744902	0.372451	−	0.928052i	$-0.378518\pi$
$421$	2.70897e6	0.744902	0.372451	+	0.928052i	$0.378518\pi$
$422$	0	0
$423$	55848.0	0.0151760
$424$	0	0
$425$	−3.70865e6	−0.995964
$426$	0	0
$427$	1.68256e6	0.446582
$428$	0	0
$429$	−799680.	−0.209784
$430$	0	0
$431$	5.53898e6	1.43627	0.718136	−	0.695902i	$-0.244995\pi$
$431$	5.53898e6	1.43627	0.718136	+	0.695902i	$0.244995\pi$
$432$	0	0
$433$	−868294.	−0.222560	−0.111280	−	0.993789i	$-0.535495\pi$
$433$	−868294.	−0.222560	−0.111280	+	0.993789i	$0.535495\pi$
$434$	0	0
$435$	539680.	0.136746
$436$	0	0
$437$	−4.86400e6	−1.21840
$438$	0	0
$439$	−1.13767e6	−0.281745	−0.140872	−	0.990028i	$-0.544991\pi$
$439$	−1.13767e6	−0.281745	−0.140872	+	0.990028i	$0.544991\pi$
$440$	0	0
$441$	−429779.	−0.105232
$442$	0	0
$443$	−1.75399e6	−0.424636	−0.212318	−	0.977201i	$-0.568101\pi$
$443$	−1.75399e6	−0.424636	−0.212318	+	0.977201i	$0.568101\pi$
$444$	0	0
$445$	81660.0	0.0195483
$446$	0	0
$447$	908976.	0.215171
$448$	0	0
$449$	2.41674e6	0.565736	0.282868	−	0.959159i	$-0.408714\pi$
$449$	2.41674e6	0.565736	0.282868	+	0.959159i	$0.408714\pi$
$450$	0	0
$451$	−4.95516e6	−1.14714
$452$	0	0
$453$	−3.26566e6	−0.747698
$454$	0	0
$455$	144060.	0.0326223
$456$	0	0
$457$	−127430.	−0.0285418	−0.0142709	−	0.999898i	$-0.504543\pi$
$457$	−127430.	−0.0285418	−0.0142709	+	0.999898i	$0.504543\pi$
$458$	0	0
$459$	4.13898e6	0.916983
$460$	0	0
$461$	128198.	0.0280950	0.0140475	−	0.999901i	$-0.495528\pi$
$461$	128198.	0.0280950	0.0140475	+	0.999901i	$0.495528\pi$
$462$	0	0
$463$	−4.01653e6	−0.870760	−0.435380	−	0.900247i	$-0.643386\pi$
$463$	−4.01653e6	−0.870760	−0.435380	+	0.900247i	$0.643386\pi$
$464$	0	0
$465$	708480.	0.151948
$466$	0	0
$467$	−8.67246e6	−1.84014	−0.920069	−	0.391757i	$-0.871867\pi$
$467$	−8.67246e6	−1.84014	−0.920069	+	0.391757i	$0.871867\pi$
$468$	0	0
$469$	603484.	0.126687
$470$	0	0
$471$	2.34837e6	0.487769
$472$	0	0
$473$	−2.75672e6	−0.566552
$474$	0	0
$475$	7.35680e6	1.49608
$476$	0	0
$477$	−2.61949e6	−0.527133
$478$	0	0
$479$	8.28946e6	1.65077	0.825387	−	0.564567i	$-0.190957\pi$
$479$	8.28946e6	1.65077	0.825387	+	0.564567i	$0.190957\pi$
$480$	0	0
$481$	−2.69951e6	−0.532013
$482$	0	0
$483$	784000.	0.152914
$484$	0	0
$485$	−206500.	−0.0398626
$486$	0	0
$487$	−8.91770e6	−1.70385	−0.851923	−	0.523667i	$-0.824563\pi$
$487$	−8.91770e6	−1.70385	−0.851923	+	0.523667i	$0.824563\pi$
$488$	0	0
$489$	−2.53693e6	−0.479773
$490$	0	0
$491$	5.71537e6	1.06989	0.534947	−	0.844886i	$-0.320332\pi$
$491$	5.71537e6	1.06989	0.534947	+	0.844886i	$0.320332\pi$
$492$	0	0
$493$	8.27060e6	1.53257
$494$	0	0
$495$	608600.	0.111640
$496$	0	0
$497$	−1.80908e6	−0.328524
$498$	0	0
$499$	−125116.	−0.0224937	−0.0112469	−	0.999937i	$-0.503580\pi$
$499$	−125116.	−0.0224937	−0.0112469	+	0.999937i	$0.503580\pi$
$500$	0	0
$501$	−1.13254e6	−0.201586
$502$	0	0
$503$	−2.77116e6	−0.488362	−0.244181	−	0.969730i	$-0.578519\pi$
$503$	−2.77116e6	−0.488362	−0.244181	+	0.969730i	$0.578519\pi$
$504$	0	0
$505$	1.86250e6	0.324988
$506$	0	0
$507$	2.27886e6	0.393729
$508$	0	0
$509$	138534.	0.0237007	0.0118504	−	0.999930i	$-0.496228\pi$
$509$	138534.	0.0237007	0.0118504	+	0.999930i	$0.496228\pi$
$510$	0	0
$511$	3.02418e6	0.512337
$512$	0	0
$513$	−8.21043e6	−1.37744
$514$	0	0
$515$	600640.	0.0997921
$516$	0	0
$517$	−106080.	−0.0174545
$518$	0	0
$519$	−569776.	−0.0928508
$520$	0	0
$521$	−1.80281e6	−0.290976	−0.145488	−	0.989360i	$-0.546475\pi$
$521$	−1.80281e6	−0.290976	−0.145488	+	0.989360i	$0.546475\pi$
$522$	0	0
$523$	−9.77247e6	−1.56225	−0.781124	−	0.624375i	$-0.785354\pi$
$523$	−9.77247e6	−1.56225	−0.781124	+	0.624375i	$0.785354\pi$
$524$	0	0
$525$	−1.18580e6	−0.187764
$526$	0	0
$527$	1.08575e7	1.70295
$528$	0	0
$529$	−2.43634e6	−0.378529
$530$	0	0
$531$	−4.95042e6	−0.761914
$532$	0	0
$533$	−4.28476e6	−0.653293
$534$	0	0
$535$	478920.	0.0723400
$536$	0	0
$537$	3.88502e6	0.581377
$538$	0	0
$539$	816340.	0.121032
$540$	0	0
$541$	−2.45504e6	−0.360633	−0.180316	−	0.983609i	$-0.557712\pi$
$541$	−2.45504e6	−0.360633	−0.180316	+	0.983609i	$0.557712\pi$
$542$	0	0
$543$	5.25672e6	0.765095
$544$	0	0
$545$	221020.	0.0318743
$546$	0	0
$547$	−1.32081e7	−1.88744	−0.943721	−	0.330743i	$-0.892701\pi$
$547$	−1.32081e7	−1.88744	−0.943721	+	0.330743i	$0.892701\pi$
$548$	0	0
$549$	6.14650e6	0.870356
$550$	0	0
$551$	−1.64063e7	−2.30214
$552$	0	0
$553$	3.17285e6	0.441201
$554$	0	0
$555$	−734560.	−0.101227
$556$	0	0
$557$	−7.83293e6	−1.06976	−0.534880	−	0.844928i	$-0.679643\pi$
$557$	−7.83293e6	−1.06976	−0.534880	+	0.844928i	$0.679643\pi$
$558$	0	0
$559$	−2.38375e6	−0.322650
$560$	0	0
$561$	−3.33472e6	−0.447355
$562$	0	0
$563$	−3.57908e6	−0.475883	−0.237942	−	0.971279i	$-0.576473\pi$
$563$	−3.57908e6	−0.475883	−0.237942	+	0.971279i	$0.576473\pi$
$564$	0	0
$565$	2.45054e6	0.322954
$566$	0	0
$567$	−807961.	−0.105544
$568$	0	0
$569$	−3.39581e6	−0.439707	−0.219853	−	0.975533i	$-0.570558\pi$
$569$	−3.39581e6	−0.439707	−0.219853	+	0.975533i	$0.570558\pi$
$570$	0	0
$571$	1.47695e6	0.189572	0.0947862	−	0.995498i	$-0.469783\pi$
$571$	1.47695e6	0.189572	0.0947862	+	0.995498i	$0.469783\pi$
$572$	0	0
$573$	−546432.	−0.0695264
$574$	0	0
$575$	−6.05000e6	−0.763108
$576$	0	0
$577$	−1.49961e7	−1.87516	−0.937580	−	0.347771i	$-0.886939\pi$
$577$	−1.49961e7	−1.87516	−0.937580	+	0.347771i	$0.886939\pi$
$578$	0	0
$579$	−2.82203e6	−0.349837
$580$	0	0
$581$	−3.77574e6	−0.464047
$582$	0	0
$583$	4.97556e6	0.606276
$584$	0	0
$585$	526260.	0.0635786
$586$	0	0
$587$	3.29291e6	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$587$	3.29291e6	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$588$	0	0
$589$	−2.15378e7	−2.55807
$590$	0	0
$591$	1.57586e6	0.185587
$592$	0	0
$593$	−1.17908e7	−1.37692	−0.688459	−	0.725275i	$-0.741713\pi$
$593$	−1.17908e7	−1.37692	−0.688459	+	0.725275i	$0.741713\pi$
$594$	0	0
$595$	600740.	0.0695655
$596$	0	0
$597$	8.83136e6	1.01413
$598$	0	0
$599$	−1.52642e6	−0.173823	−0.0869117	−	0.996216i	$-0.527700\pi$
$599$	−1.52642e6	−0.173823	−0.0869117	+	0.996216i	$0.527700\pi$
$600$	0	0
$601$	−1.00142e7	−1.13092	−0.565458	−	0.824777i	$-0.691301\pi$
$601$	−1.00142e7	−1.13092	−0.565458	+	0.824777i	$0.691301\pi$
$602$	0	0
$603$	2.20456e6	0.246905
$604$	0	0
$605$	454510.	0.0504841
$606$	0	0
$607$	1.20660e7	1.32920	0.664599	−	0.747200i	$-0.268602\pi$
$607$	1.20660e7	1.32920	0.664599	+	0.747200i	$0.268602\pi$
$608$	0	0
$609$	2.64443e6	0.288928
$610$	0	0
$611$	−91728.0	−0.00994029
$612$	0	0
$613$	−5.81950e6	−0.625511	−0.312755	−	0.949834i	$-0.601252\pi$
$613$	−5.81950e6	−0.625511	−0.312755	+	0.949834i	$0.601252\pi$
$614$	0	0
$615$	−1.16592e6	−0.124303
$616$	0	0
$617$	−4.16589e6	−0.440550	−0.220275	−	0.975438i	$-0.570695\pi$
$617$	−4.16589e6	−0.440550	−0.220275	+	0.975438i	$0.570695\pi$
$618$	0	0
$619$	8.08090e6	0.847683	0.423841	−	0.905736i	$-0.360681\pi$
$619$	8.08090e6	0.847683	0.423841	+	0.905736i	$0.360681\pi$
$620$	0	0
$621$	6.75200e6	0.702592
$622$	0	0
$623$	400134.	0.0413034
$624$	0	0
$625$	8.83812e6	0.905024
$626$	0	0
$627$	6.61504e6	0.671991
$628$	0	0
$629$	−1.12571e7	−1.13449
$630$	0	0
$631$	−8.40878e6	−0.840735	−0.420368	−	0.907354i	$-0.638099\pi$
$631$	−8.40878e6	−0.840735	−0.420368	+	0.907354i	$0.638099\pi$
$632$	0	0
$633$	−827552.	−0.0820892
$634$	0	0
$635$	966960.	0.0951643
$636$	0	0
$637$	705894.	0.0689272
$638$	0	0
$639$	−6.60868e6	−0.640269
$640$	0	0
$641$	6.29760e6	0.605383	0.302691	−	0.953089i	$-0.402115\pi$
$641$	6.29760e6	0.605383	0.302691	+	0.953089i	$0.402115\pi$
$642$	0	0
$643$	−4.39762e6	−0.419460	−0.209730	−	0.977759i	$-0.567259\pi$
$643$	−4.39762e6	−0.419460	−0.209730	+	0.977759i	$0.567259\pi$
$644$	0	0
$645$	−648640.	−0.0613910
$646$	0	0
$647$	6.55397e6	0.615522	0.307761	−	0.951464i	$-0.400420\pi$
$647$	6.55397e6	0.615522	0.307761	+	0.951464i	$0.400420\pi$
$648$	0	0
$649$	9.40304e6	0.876308
$650$	0	0
$651$	3.47155e6	0.321049
$652$	0	0
$653$	−3.79652e6	−0.348420	−0.174210	−	0.984709i	$-0.555737\pi$
$653$	−3.79652e6	−0.348420	−0.174210	+	0.984709i	$0.555737\pi$
$654$	0	0
$655$	1.34368e6	0.122375
$656$	0	0
$657$	1.10475e7	0.998508
$658$	0	0
$659$	8.82684e6	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$659$	8.82684e6	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$660$	0	0
$661$	341270.	0.0303805	0.0151902	−	0.999885i	$-0.495165\pi$
$661$	341270.	0.0303805	0.0151902	+	0.999885i	$0.495165\pi$
$662$	0	0
$663$	−2.88355e6	−0.254767
$664$	0	0
$665$	−1.19168e6	−0.104497
$666$	0	0
$667$	1.34920e7	1.17425
$668$	0	0
$669$	−2.45862e6	−0.212386
$670$	0	0
$671$	−1.16749e7	−1.00103
$672$	0	0
$673$	4.41807e6	0.376006	0.188003	−	0.982168i	$-0.439799\pi$
$673$	4.41807e6	0.376006	0.188003	+	0.982168i	$0.439799\pi$
$674$	0	0
$675$	−1.02124e7	−0.862717
$676$	0	0
$677$	−1.63858e7	−1.37403	−0.687014	−	0.726644i	$-0.741079\pi$
$677$	−1.63858e7	−1.37403	−0.687014	+	0.726644i	$0.741079\pi$
$678$	0	0
$679$	−1.01185e6	−0.0842251
$680$	0	0
$681$	−7.13434e6	−0.589503
$682$	0	0
$683$	1.75399e7	1.43872	0.719360	−	0.694638i	$-0.244435\pi$
$683$	1.75399e7	1.43872	0.719360	+	0.694638i	$0.244435\pi$
$684$	0	0
$685$	2.94662e6	0.239937
$686$	0	0
$687$	2.21365e6	0.178944
$688$	0	0
$689$	4.30240e6	0.345273
$690$	0	0
$691$	−3.14638e6	−0.250678	−0.125339	−	0.992114i	$-0.540002\pi$
$691$	−3.14638e6	−0.250678	−0.125339	+	0.992114i	$0.540002\pi$
$692$	0	0
$693$	2.98214e6	0.235882
$694$	0	0
$695$	3.14944e6	0.247327
$696$	0	0
$697$	−1.78677e7	−1.39312
$698$	0	0
$699$	−1.18355e7	−0.916205
$700$	0	0
$701$	1.90919e7	1.46742	0.733709	−	0.679464i	$-0.237788\pi$
$701$	1.90919e7	1.46742	0.733709	+	0.679464i	$0.237788\pi$
$702$	0	0
$703$	2.23306e7	1.70417
$704$	0	0
$705$	−24960.0	−0.00189135
$706$	0	0
$707$	9.12625e6	0.686663
$708$	0	0
$709$	−990974.	−0.0740366	−0.0370183	−	0.999315i	$-0.511786\pi$
$709$	−990974.	−0.0740366	−0.0370183	+	0.999315i	$0.511786\pi$
$710$	0	0
$711$	1.15906e7	0.859869
$712$	0	0
$713$	1.77120e7	1.30480
$714$	0	0
$715$	−999600.	−0.0731242
$716$	0	0
$717$	−8.00275e6	−0.581355
$718$	0	0
$719$	−1.69014e7	−1.21928	−0.609638	−	0.792680i	$-0.708685\pi$
$719$	−1.69014e7	−1.21928	−0.609638	+	0.792680i	$0.708685\pi$
$720$	0	0
$721$	2.94314e6	0.210849
$722$	0	0
$723$	−1.08666e7	−0.773125
$724$	0	0
$725$	−2.04066e7	−1.44187
$726$	0	0
$727$	−2.34302e7	−1.64414	−0.822071	−	0.569384i	$-0.807182\pi$
$727$	−2.34302e7	−1.64414	−0.822071	+	0.569384i	$0.807182\pi$
$728$	0	0
$729$	3.61141e6	0.251686
$730$	0	0
$731$	−9.94041e6	−0.688035
$732$	0	0
$733$	−975810.	−0.0670819	−0.0335409	−	0.999437i	$-0.510678\pi$
$733$	−975810.	−0.0670819	−0.0335409	+	0.999437i	$0.510678\pi$
$734$	0	0
$735$	192080.	0.0131149
$736$	0	0
$737$	−4.18744e6	−0.283975
$738$	0	0
$739$	6.30208e6	0.424495	0.212247	−	0.977216i	$-0.431922\pi$
$739$	6.30208e6	0.424495	0.212247	+	0.977216i	$0.431922\pi$
$740$	0	0
$741$	5.72006e6	0.382697
$742$	0	0
$743$	−6.95698e6	−0.462326	−0.231163	−	0.972915i	$-0.574253\pi$
$743$	−6.95698e6	−0.462326	−0.231163	+	0.972915i	$0.574253\pi$
$744$	0	0
$745$	1.13622e6	0.0750018
$746$	0	0
$747$	−1.37930e7	−0.904395
$748$	0	0
$749$	2.34671e6	0.152846
$750$	0	0
$751$	2.74535e7	1.77622	0.888112	−	0.459628i	$-0.152017\pi$
$751$	2.74535e7	1.77622	0.888112	+	0.459628i	$0.152017\pi$
$752$	0	0
$753$	−1.41926e6	−0.0912170
$754$	0	0
$755$	−4.08208e6	−0.260624
$756$	0	0
$757$	1.96889e7	1.24877	0.624384	−	0.781118i	$-0.285350\pi$
$757$	1.96889e7	1.24877	0.624384	+	0.781118i	$0.285350\pi$
$758$	0	0
$759$	−5.44000e6	−0.342763
$760$	0	0
$761$	−2.82079e7	−1.76567	−0.882835	−	0.469684i	$-0.844368\pi$
$761$	−2.82079e7	−1.76567	−0.882835	+	0.469684i	$0.844368\pi$
$762$	0	0
$763$	1.08300e6	0.0673467
$764$	0	0
$765$	2.19454e6	0.135578
$766$	0	0
$767$	8.13086e6	0.499055
$768$	0	0
$769$	−1.38081e6	−0.0842009	−0.0421005	−	0.999113i	$-0.513405\pi$
$769$	−1.38081e6	−0.0842009	−0.0421005	+	0.999113i	$0.513405\pi$
$770$	0	0
$771$	−2.61326e6	−0.158324
$772$	0	0
$773$	1.54347e7	0.929074	0.464537	−	0.885554i	$-0.346221\pi$
$773$	1.54347e7	0.929074	0.464537	+	0.885554i	$0.346221\pi$
$774$	0	0
$775$	−2.67894e7	−1.60217
$776$	0	0
$777$	−3.59934e6	−0.213880
$778$	0	0
$779$	3.54440e7	2.09266
$780$	0	0
$781$	1.25528e7	0.736399
$782$	0	0
$783$	2.27745e7	1.32753
$784$	0	0
$785$	2.93546e6	0.170021
$786$	0	0
$787$	7.10107e6	0.408683	0.204342	−	0.978900i	$-0.434495\pi$
$787$	7.10107e6	0.408683	0.204342	+	0.978900i	$0.434495\pi$
$788$	0	0
$789$	279360.	0.0159761
$790$	0	0
$791$	1.20076e7	0.682365
$792$	0	0
$793$	−1.00954e7	−0.570085
$794$	0	0
$795$	1.17072e6	0.0656954
$796$	0	0
$797$	−6.48182e6	−0.361452	−0.180726	−	0.983533i	$-0.557845\pi$
$797$	−6.48182e6	−0.361452	−0.180726	+	0.983533i	$0.557845\pi$
$798$	0	0
$799$	−382512.	−0.0211972
$800$	0	0
$801$	1.46171e6	0.0804973
$802$	0	0
$803$	−2.09841e7	−1.14842
$804$	0	0
$805$	980000.	0.0533011
$806$	0	0
$807$	5.73166e6	0.309811
$808$	0	0
$809$	1.60578e7	0.862610	0.431305	−	0.902206i	$-0.358053\pi$
$809$	1.60578e7	0.862610	0.431305	+	0.902206i	$0.358053\pi$
$810$	0	0
$811$	−4.84775e6	−0.258814	−0.129407	−	0.991592i	$-0.541307\pi$
$811$	−4.84775e6	−0.258814	−0.129407	+	0.991592i	$0.541307\pi$
$812$	0	0
$813$	7.62701e6	0.404695
$814$	0	0
$815$	−3.17116e6	−0.167234
$816$	0	0
$817$	1.97187e7	1.03353
$818$	0	0
$819$	2.57867e6	0.134334
$820$	0	0
$821$	−2.17976e7	−1.12863	−0.564314	−	0.825560i	$-0.690859\pi$
$821$	−2.17976e7	−1.12863	−0.564314	+	0.825560i	$0.690859\pi$
$822$	0	0
$823$	3.20206e7	1.64790	0.823948	−	0.566665i	$-0.191767\pi$
$823$	3.20206e7	1.64790	0.823948	+	0.566665i	$0.191767\pi$
$824$	0	0
$825$	8.22800e6	0.420881
$826$	0	0
$827$	−2.19008e7	−1.11352	−0.556758	−	0.830675i	$-0.687955\pi$
$827$	−2.19008e7	−1.11352	−0.556758	+	0.830675i	$0.687955\pi$
$828$	0	0
$829$	1.45999e7	0.737844	0.368922	−	0.929460i	$-0.379727\pi$
$829$	1.45999e7	0.737844	0.368922	+	0.929460i	$0.379727\pi$
$830$	0	0
$831$	−1.47783e7	−0.742374
$832$	0	0
$833$	2.94363e6	0.146984
$834$	0	0
$835$	−1.41568e6	−0.0702666
$836$	0	0
$837$	2.98979e7	1.47512
$838$	0	0
$839$	4.60947e6	0.226072	0.113036	−	0.993591i	$-0.463942\pi$
$839$	4.60947e6	0.226072	0.113036	+	0.993591i	$0.463942\pi$
$840$	0	0
$841$	2.49974e7	1.21872
$842$	0	0
$843$	1.59680e7	0.773897
$844$	0	0
$845$	2.84857e6	0.137241
$846$	0	0
$847$	2.22710e6	0.106667
$848$	0	0
$849$	1.87270e6	0.0891661
$850$	0	0
$851$	−1.83640e7	−0.869247
$852$	0	0
$853$	1.98437e7	0.933793	0.466897	−	0.884312i	$-0.345372\pi$
$853$	1.98437e7	0.933793	0.466897	+	0.884312i	$0.345372\pi$
$854$	0	0
$855$	−4.35328e6	−0.203658
$856$	0	0
$857$	−1.22960e6	−0.0571888	−0.0285944	−	0.999591i	$-0.509103\pi$
$857$	−1.22960e6	−0.0571888	−0.0285944	+	0.999591i	$0.509103\pi$
$858$	0	0
$859$	−3.33041e7	−1.53998	−0.769989	−	0.638058i	$-0.779738\pi$
$859$	−3.33041e7	−1.53998	−0.769989	+	0.638058i	$0.779738\pi$
$860$	0	0
$861$	−5.71301e6	−0.262638
$862$	0	0
$863$	−2.36616e7	−1.08148	−0.540738	−	0.841191i	$-0.681855\pi$
$863$	−2.36616e7	−1.08148	−0.540738	+	0.841191i	$0.681855\pi$
$864$	0	0
$865$	−712220.	−0.0323649
$866$	0	0
$867$	−665752.	−0.0300791
$868$	0	0
$869$	−2.20157e7	−0.988969
$870$	0	0
$871$	−3.62090e6	−0.161723
$872$	0	0
$873$	−3.69635e6	−0.164149
$874$	0	0
$875$	−3.01350e6	−0.133061
$876$	0	0
$877$	2.37812e7	1.04408	0.522042	−	0.852920i	$-0.325170\pi$
$877$	2.37812e7	1.04408	0.522042	+	0.852920i	$0.325170\pi$
$878$	0	0
$879$	−2.00064e7	−0.873369
$880$	0	0
$881$	−1.41871e7	−0.615818	−0.307909	−	0.951416i	$-0.599629\pi$
$881$	−1.41871e7	−0.615818	−0.307909	+	0.951416i	$0.599629\pi$
$882$	0	0
$883$	−2.09281e7	−0.903293	−0.451647	−	0.892197i	$-0.649163\pi$
$883$	−2.09281e7	−0.903293	−0.451647	+	0.892197i	$0.649163\pi$
$884$	0	0
$885$	2.21248e6	0.0949557
$886$	0	0
$887$	−7.98586e6	−0.340810	−0.170405	−	0.985374i	$-0.554508\pi$
$887$	−7.98586e6	−0.340810	−0.170405	+	0.985374i	$0.554508\pi$
$888$	0	0
$889$	4.73810e6	0.201071
$890$	0	0
$891$	5.60626e6	0.236581
$892$	0	0
$893$	758784.	0.0318412
$894$	0	0
$895$	4.85628e6	0.202650
$896$	0	0
$897$	−4.70400e6	−0.195203
$898$	0	0
$899$	5.97426e7	2.46538
$900$	0	0
$901$	1.79413e7	0.736278
$902$	0	0
$903$	−3.17834e6	−0.129712
$904$	0	0
$905$	6.57090e6	0.266688
$906$	0	0
$907$	2.31861e7	0.935856	0.467928	−	0.883767i	$-0.345001\pi$
$907$	2.31861e7	0.935856	0.467928	+	0.883767i	$0.345001\pi$
$908$	0	0
$909$	3.33388e7	1.33826
$910$	0	0
$911$	1.65299e7	0.659895	0.329948	−	0.943999i	$-0.392969\pi$
$911$	1.65299e7	0.659895	0.329948	+	0.943999i	$0.392969\pi$
$912$	0	0
$913$	2.61990e7	1.04018
$914$	0	0
$915$	−2.74704e6	−0.108471
$916$	0	0
$917$	6.58403e6	0.258564
$918$	0	0
$919$	1.28087e7	0.500283	0.250142	−	0.968209i	$-0.419523\pi$
$919$	1.28087e7	0.500283	0.250142	+	0.968209i	$0.419523\pi$
$920$	0	0
$921$	1.87363e7	0.727836
$922$	0	0
$923$	1.08545e7	0.419377
$924$	0	0
$925$	2.77756e7	1.06735
$926$	0	0
$927$	1.07515e7	0.410930
$928$	0	0
$929$	2.97319e7	1.13027	0.565136	−	0.824998i	$-0.308824\pi$
$929$	2.97319e7	1.13027	0.565136	+	0.824998i	$0.308824\pi$
$930$	0	0
$931$	−5.83923e6	−0.220791
$932$	0	0
$933$	1.30451e6	0.0490619
$934$	0	0
$935$	−4.16840e6	−0.155934
$936$	0	0
$937$	1.10970e7	0.412911	0.206456	−	0.978456i	$-0.433807\pi$
$937$	1.10970e7	0.412911	0.206456	+	0.978456i	$0.433807\pi$
$938$	0	0
$939$	−1.39172e7	−0.515096
$940$	0	0
$941$	−3.74313e7	−1.37804	−0.689019	−	0.724743i	$-0.741958\pi$
$941$	−3.74313e7	−1.37804	−0.689019	+	0.724743i	$0.741958\pi$
$942$	0	0
$943$	−2.91480e7	−1.06741
$944$	0	0
$945$	1.65424e6	0.0602586
$946$	0	0
$947$	−1.50907e7	−0.546808	−0.273404	−	0.961899i	$-0.588150\pi$
$947$	−1.50907e7	−0.546808	−0.273404	+	0.961899i	$0.588150\pi$
$948$	0	0
$949$	−1.81451e7	−0.654024
$950$	0	0
$951$	−1.43817e7	−0.515655
$952$	0	0
$953$	−2.15741e7	−0.769484	−0.384742	−	0.923024i	$-0.625710\pi$
$953$	−2.15741e7	−0.769484	−0.384742	+	0.923024i	$0.625710\pi$
$954$	0	0
$955$	−683040.	−0.0242347
$956$	0	0
$957$	−1.83491e7	−0.647643
$958$	0	0
$959$	1.44384e7	0.506960
$960$	0	0
$961$	4.97996e7	1.73947
$962$	0	0
$963$	8.57267e6	0.297886
$964$	0	0
$965$	−3.52754e6	−0.121942
$966$	0	0
$967$	−3.29467e7	−1.13304	−0.566520	−	0.824048i	$-0.691711\pi$
$967$	−3.29467e7	−1.13304	−0.566520	+	0.824048i	$0.691711\pi$
$968$	0	0
$969$	2.38531e7	0.816083
$970$	0	0
$971$	−2.24599e7	−0.764470	−0.382235	−	0.924065i	$-0.624846\pi$
$971$	−2.24599e7	−0.764470	−0.382235	+	0.924065i	$0.624846\pi$
$972$	0	0
$973$	1.54323e7	0.522573
$974$	0	0
$975$	7.11480e6	0.239691
$976$	0	0
$977$	−5.16236e7	−1.73026	−0.865132	−	0.501545i	$-0.832765\pi$
$977$	−5.16236e7	−1.73026	−0.865132	+	0.501545i	$0.832765\pi$
$978$	0	0
$979$	−2.77644e6	−0.0925831
$980$	0	0
$981$	3.95626e6	0.131254
$982$	0	0
$983$	−1.10202e7	−0.363751	−0.181876	−	0.983322i	$-0.558217\pi$
$983$	−1.10202e7	−0.363751	−0.181876	+	0.983322i	$0.558217\pi$
$984$	0	0
$985$	1.96982e6	0.0646898
$986$	0	0
$987$	−122304.	−0.00399621
$988$	0	0
$989$	−1.62160e7	−0.527173
$990$	0	0
$991$	3.21029e7	1.03839	0.519194	−	0.854656i	$-0.326232\pi$
$991$	3.21029e7	1.03839	0.519194	+	0.854656i	$0.326232\pi$
$992$	0	0
$993$	−1.98033e7	−0.637330
$994$	0	0
$995$	1.10392e7	0.353492
$996$	0	0
$997$	−2.81772e7	−0.897759	−0.448879	−	0.893592i	$-0.648177\pi$
$997$	−2.81772e7	−0.897759	−0.448879	+	0.893592i	$0.648177\pi$
$998$	0	0
$999$	−3.09984e7	−0.982711

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.f.1.1		1
4.3	odd	2		448.6.a.k.1.1		1
8.3	odd	2		112.6.a.d.1.1		1
8.5	even	2		14.6.a.b.1.1	✓	1
24.5	odd	2		126.6.a.c.1.1		1
24.11	even	2		1008.6.a.n.1.1		1
40.13	odd	4		350.6.c.f.99.1		2
40.29	even	2		350.6.a.b.1.1		1
40.37	odd	4		350.6.c.f.99.2		2
56.5	odd	6		98.6.c.b.67.1		2
56.13	odd	2		98.6.a.b.1.1		1
56.27	even	2		784.6.a.h.1.1		1
56.37	even	6		98.6.c.a.67.1		2
56.45	odd	6		98.6.c.b.79.1		2
56.53	even	6		98.6.c.a.79.1		2
168.125	even	2		882.6.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.a.b.1.1	✓	1	8.5	even	2
98.6.a.b.1.1		1	56.13	odd	2
98.6.c.a.67.1		2	56.37	even	6
98.6.c.a.79.1		2	56.53	even	6
98.6.c.b.67.1		2	56.5	odd	6
98.6.c.b.79.1		2	56.45	odd	6
112.6.a.d.1.1		1	8.3	odd	2
126.6.a.c.1.1		1	24.5	odd	2
350.6.a.b.1.1		1	40.29	even	2
350.6.c.f.99.1		2	40.13	odd	4
350.6.c.f.99.2		2	40.37	odd	4
448.6.a.f.1.1		1	1.1	even	1	trivial
448.6.a.k.1.1		1	4.3	odd	2
784.6.a.h.1.1		1	56.27	even	2
882.6.a.g.1.1		1	168.125	even	2
1008.6.a.n.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 $\F_{q}$
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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