LMFDB - Embedded newform 448.6.a.p.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 = [1,0,30,0,-32] 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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	448.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+30.0000 q^{3} -32.0000 q^{5} -49.0000 q^{7} +657.000 q^{9} -624.000 q^{11} +708.000 q^{13} -960.000 q^{15} +934.000 q^{17} +1858.00 q^{19} -1470.00 q^{21} +1120.00 q^{23} -2101.00 q^{25} +12420.0 q^{27} +1174.00 q^{29} -2908.00 q^{31} -18720.0 q^{33} +1568.00 q^{35} +12462.0 q^{37} +21240.0 q^{39} +2662.00 q^{41} -7144.00 q^{43} -21024.0 q^{45} +7468.00 q^{47} +2401.00 q^{49} +28020.0 q^{51} +27274.0 q^{53} +19968.0 q^{55} +55740.0 q^{57} +2490.00 q^{59} +11096.0 q^{61} -32193.0 q^{63} -22656.0 q^{65} +39756.0 q^{67} +33600.0 q^{69} +69888.0 q^{71} +16450.0 q^{73} -63030.0 q^{75} +30576.0 q^{77} -78376.0 q^{79} +212949. q^{81} +109818. q^{83} -29888.0 q^{85} +35220.0 q^{87} -56966.0 q^{89} -34692.0 q^{91} -87240.0 q^{93} -59456.0 q^{95} -115946. q^{97} -409968. q^{99} +O(q^{100})$

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
$3$	30.0000	1.92450	0.962250	−	0.272166i	$-0.0877398\pi$
$3$	30.0000	1.92450	0.962250	+	0.272166i	$0.0877398\pi$
$4$	0	0
$5$	−32.0000	−0.572433	−0.286217	−	0.958165i	$-0.592398\pi$
$5$	−32.0000	−0.572433	−0.286217	+	0.958165i	$0.592398\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	657.000	2.70370
$10$	0	0
$11$	−624.000	−1.55490	−0.777451	−	0.628944i	$-0.783488\pi$
$11$	−624.000	−1.55490	−0.777451	+	0.628944i	$0.783488\pi$
$12$	0	0
$13$	708.000	1.16192	0.580958	−	0.813933i	$-0.302678\pi$
$13$	708.000	1.16192	0.580958	+	0.813933i	$0.302678\pi$
$14$	0	0
$15$	−960.000	−1.10165
$16$	0	0
$17$	934.000	0.783835	0.391917	−	0.920000i	$-0.371812\pi$
$17$	934.000	0.783835	0.391917	+	0.920000i	$0.371812\pi$
$18$	0	0
$19$	1858.00	1.18076	0.590380	−	0.807125i	$-0.298978\pi$
$19$	1858.00	1.18076	0.590380	+	0.807125i	$0.298978\pi$
$20$	0	0
$21$	−1470.00	−0.727393
$22$	0	0
$23$	1120.00	0.441467	0.220734	−	0.975334i	$-0.429155\pi$
$23$	1120.00	0.441467	0.220734	+	0.975334i	$0.429155\pi$
$24$	0	0
$25$	−2101.00	−0.672320
$26$	0	0
$27$	12420.0	3.27878
$28$	0	0
$29$	1174.00	0.259223	0.129611	−	0.991565i	$-0.458627\pi$
$29$	1174.00	0.259223	0.129611	+	0.991565i	$0.458627\pi$
$30$	0	0
$31$	−2908.00	−0.543488	−0.271744	−	0.962370i	$-0.587600\pi$
$31$	−2908.00	−0.543488	−0.271744	+	0.962370i	$0.587600\pi$
$32$	0	0
$33$	−18720.0	−2.99241
$34$	0	0
$35$	1568.00	0.216359
$36$	0	0
$37$	12462.0	1.49652	0.748262	−	0.663404i	$-0.230889\pi$
$37$	12462.0	1.49652	0.748262	+	0.663404i	$0.230889\pi$
$38$	0	0
$39$	21240.0	2.23611
$40$	0	0
$41$	2662.00	0.247314	0.123657	−	0.992325i	$-0.460538\pi$
$41$	2662.00	0.247314	0.123657	+	0.992325i	$0.460538\pi$
$42$	0	0
$43$	−7144.00	−0.589210	−0.294605	−	0.955619i	$-0.595188\pi$
$43$	−7144.00	−0.589210	−0.294605	+	0.955619i	$0.595188\pi$
$44$	0	0
$45$	−21024.0	−1.54769
$46$	0	0
$47$	7468.00	0.493128	0.246564	−	0.969127i	$-0.420698\pi$
$47$	7468.00	0.493128	0.246564	+	0.969127i	$0.420698\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	28020.0	1.50849
$52$	0	0
$53$	27274.0	1.33370	0.666852	−	0.745191i	$-0.267642\pi$
$53$	27274.0	1.33370	0.666852	+	0.745191i	$0.267642\pi$
$54$	0	0
$55$	19968.0	0.890078
$56$	0	0
$57$	55740.0	2.27237
$58$	0	0
$59$	2490.00	0.0931257	0.0465628	−	0.998915i	$-0.485173\pi$
$59$	2490.00	0.0931257	0.0465628	+	0.998915i	$0.485173\pi$
$60$	0	0
$61$	11096.0	0.381805	0.190903	−	0.981609i	$-0.438859\pi$
$61$	11096.0	0.381805	0.190903	+	0.981609i	$0.438859\pi$
$62$	0	0
$63$	−32193.0	−1.02190
$64$	0	0
$65$	−22656.0	−0.665120
$66$	0	0
$67$	39756.0	1.08197	0.540986	−	0.841032i	$-0.318051\pi$
$67$	39756.0	1.08197	0.540986	+	0.841032i	$0.318051\pi$
$68$	0	0
$69$	33600.0	0.849604
$70$	0	0
$71$	69888.0	1.64534	0.822672	−	0.568516i	$-0.192482\pi$
$71$	69888.0	1.64534	0.822672	+	0.568516i	$0.192482\pi$
$72$	0	0
$73$	16450.0	0.361292	0.180646	−	0.983548i	$-0.442181\pi$
$73$	16450.0	0.361292	0.180646	+	0.983548i	$0.442181\pi$
$74$	0	0
$75$	−63030.0	−1.29388
$76$	0	0
$77$	30576.0	0.587698
$78$	0	0
$79$	−78376.0	−1.41291	−0.706456	−	0.707757i	$-0.749707\pi$
$79$	−78376.0	−1.41291	−0.706456	+	0.707757i	$0.749707\pi$
$80$	0	0
$81$	212949.	3.60631
$82$	0	0
$83$	109818.	1.74976	0.874880	−	0.484340i	$-0.160940\pi$
$83$	109818.	1.74976	0.874880	+	0.484340i	$0.160940\pi$
$84$	0	0
$85$	−29888.0	−0.448693
$86$	0	0
$87$	35220.0	0.498874
$88$	0	0
$89$	−56966.0	−0.762326	−0.381163	−	0.924508i	$-0.624476\pi$
$89$	−56966.0	−0.762326	−0.381163	+	0.924508i	$0.624476\pi$
$90$	0	0
$91$	−34692.0	−0.439163
$92$	0	0
$93$	−87240.0	−1.04594
$94$	0	0
$95$	−59456.0	−0.675907
$96$	0	0
$97$	−115946.	−1.25120	−0.625600	−	0.780144i	$-0.715146\pi$
$97$	−115946.	−1.25120	−0.625600	+	0.780144i	$0.715146\pi$
$98$	0	0
$99$	−409968.	−4.20399
$100$	0	0
$101$	−8352.00	−0.0814680	−0.0407340	−	0.999170i	$-0.512970\pi$
$101$	−8352.00	−0.0814680	−0.0407340	+	0.999170i	$0.512970\pi$
$102$	0	0
$103$	−179484.	−1.66699	−0.833494	−	0.552528i	$-0.813663\pi$
$103$	−179484.	−1.66699	−0.833494	+	0.552528i	$0.813663\pi$
$104$	0	0
$105$	47040.0	0.416384
$106$	0	0
$107$	−53892.0	−0.455056	−0.227528	−	0.973772i	$-0.573064\pi$
$107$	−53892.0	−0.455056	−0.227528	+	0.973772i	$0.573064\pi$
$108$	0	0
$109$	−105970.	−0.854312	−0.427156	−	0.904178i	$-0.640485\pi$
$109$	−105970.	−0.854312	−0.427156	+	0.904178i	$0.640485\pi$
$110$	0	0
$111$	373860.	2.88006
$112$	0	0
$113$	2502.00	0.0184328	0.00921640	−	0.999958i	$-0.497066\pi$
$113$	2502.00	0.0184328	0.00921640	+	0.999958i	$0.497066\pi$
$114$	0	0
$115$	−35840.0	−0.252711
$116$	0	0
$117$	465156.	3.14148
$118$	0	0
$119$	−45766.0	−0.296262
$120$	0	0
$121$	228325.	1.41772
$122$	0	0
$123$	79860.0	0.475956
$124$	0	0
$125$	167232.	0.957292
$126$	0	0
$127$	−287792.	−1.58332	−0.791661	−	0.610960i	$-0.790784\pi$
$127$	−287792.	−1.58332	−0.791661	+	0.610960i	$0.790784\pi$
$128$	0	0
$129$	−214320.	−1.13394
$130$	0	0
$131$	−47662.0	−0.242658	−0.121329	−	0.992612i	$-0.538716\pi$
$131$	−47662.0	−0.242658	−0.121329	+	0.992612i	$0.538716\pi$
$132$	0	0
$133$	−91042.0	−0.446285
$134$	0	0
$135$	−397440.	−1.87688
$136$	0	0
$137$	223154.	1.01579	0.507894	−	0.861419i	$-0.330424\pi$
$137$	223154.	1.01579	0.507894	+	0.861419i	$0.330424\pi$
$138$	0	0
$139$	250542.	1.09988	0.549938	−	0.835206i	$-0.314651\pi$
$139$	250542.	1.09988	0.549938	+	0.835206i	$0.314651\pi$
$140$	0	0
$141$	224040.	0.949025
$142$	0	0
$143$	−441792.	−1.80667
$144$	0	0
$145$	−37568.0	−0.148388
$146$	0	0
$147$	72030.0	0.274929
$148$	0	0
$149$	487394.	1.79852	0.899258	−	0.437418i	$-0.144107\pi$
$149$	487394.	1.79852	0.899258	+	0.437418i	$0.144107\pi$
$150$	0	0
$151$	54680.0	0.195158	0.0975790	−	0.995228i	$-0.468890\pi$
$151$	54680.0	0.195158	0.0975790	+	0.995228i	$0.468890\pi$
$152$	0	0
$153$	613638.	2.11926
$154$	0	0
$155$	93056.0	0.311111
$156$	0	0
$157$	−211068.	−0.683397	−0.341699	−	0.939810i	$-0.611002\pi$
$157$	−211068.	−0.683397	−0.341699	+	0.939810i	$0.611002\pi$
$158$	0	0
$159$	818220.	2.56671
$160$	0	0
$161$	−54880.0	−0.166859
$162$	0	0
$163$	−20192.0	−0.0595265	−0.0297632	−	0.999557i	$-0.509475\pi$
$163$	−20192.0	−0.0595265	−0.0297632	+	0.999557i	$0.509475\pi$
$164$	0	0
$165$	599040.	1.71296
$166$	0	0
$167$	4524.00	0.0125525	0.00627627	−	0.999980i	$-0.498002\pi$
$167$	4524.00	0.0125525	0.00627627	+	0.999980i	$0.498002\pi$
$168$	0	0
$169$	129971.	0.350050
$170$	0	0
$171$	1.22071e6	3.19243
$172$	0	0
$173$	104332.	0.265034	0.132517	−	0.991181i	$-0.457694\pi$
$173$	104332.	0.265034	0.132517	+	0.991181i	$0.457694\pi$
$174$	0	0
$175$	102949.	0.254113
$176$	0	0
$177$	74700.0	0.179220
$178$	0	0
$179$	−201724.	−0.470571	−0.235285	−	0.971926i	$-0.575602\pi$
$179$	−201724.	−0.470571	−0.235285	+	0.971926i	$0.575602\pi$
$180$	0	0
$181$	−655700.	−1.48768	−0.743839	−	0.668359i	$-0.766997\pi$
$181$	−655700.	−1.48768	−0.743839	+	0.668359i	$0.766997\pi$
$182$	0	0
$183$	332880.	0.734784
$184$	0	0
$185$	−398784.	−0.856660
$186$	0	0
$187$	−582816.	−1.21879
$188$	0	0
$189$	−608580.	−1.23926
$190$	0	0
$191$	151496.	0.300482	0.150241	−	0.988649i	$-0.451995\pi$
$191$	151496.	0.300482	0.150241	+	0.988649i	$0.451995\pi$
$192$	0	0
$193$	229326.	0.443159	0.221580	−	0.975142i	$-0.428879\pi$
$193$	229326.	0.443159	0.221580	+	0.975142i	$0.428879\pi$
$194$	0	0
$195$	−679680.	−1.28002
$196$	0	0
$197$	−421086.	−0.773046	−0.386523	−	0.922280i	$-0.626324\pi$
$197$	−421086.	−0.773046	−0.386523	+	0.922280i	$0.626324\pi$
$198$	0	0
$199$	−197300.	−0.353179	−0.176589	−	0.984285i	$-0.556506\pi$
$199$	−197300.	−0.353179	−0.176589	+	0.984285i	$0.556506\pi$
$200$	0	0
$201$	1.19268e6	2.08225
$202$	0	0
$203$	−57526.0	−0.0979770
$204$	0	0
$205$	−85184.0	−0.141571
$206$	0	0
$207$	735840.	1.19360
$208$	0	0
$209$	−1.15939e6	−1.83597
$210$	0	0
$211$	679052.	1.05002	0.525009	−	0.851097i	$-0.324062\pi$
$211$	679052.	1.05002	0.525009	+	0.851097i	$0.324062\pi$
$212$	0	0
$213$	2.09664e6	3.16647
$214$	0	0
$215$	228608.	0.337284
$216$	0	0
$217$	142492.	0.205419
$218$	0	0
$219$	493500.	0.695307
$220$	0	0
$221$	661272.	0.910751
$222$	0	0
$223$	184440.	0.248366	0.124183	−	0.992259i	$-0.460369\pi$
$223$	184440.	0.248366	0.124183	+	0.992259i	$0.460369\pi$
$224$	0	0
$225$	−1.38036e6	−1.81775
$226$	0	0
$227$	−868078.	−1.11813	−0.559067	−	0.829122i	$-0.688841\pi$
$227$	−868078.	−1.11813	−0.559067	+	0.829122i	$0.688841\pi$
$228$	0	0
$229$	593860.	0.748334	0.374167	−	0.927361i	$-0.377929\pi$
$229$	593860.	0.748334	0.374167	+	0.927361i	$0.377929\pi$
$230$	0	0
$231$	917280.	1.13102
$232$	0	0
$233$	48218.0	0.0581861	0.0290931	−	0.999577i	$-0.490738\pi$
$233$	48218.0	0.0581861	0.0290931	+	0.999577i	$0.490738\pi$
$234$	0	0
$235$	−238976.	−0.282283
$236$	0	0
$237$	−2.35128e6	−2.71915
$238$	0	0
$239$	−241688.	−0.273691	−0.136845	−	0.990592i	$-0.543696\pi$
$239$	−241688.	−0.273691	−0.136845	+	0.990592i	$0.543696\pi$
$240$	0	0
$241$	565270.	0.626922	0.313461	−	0.949601i	$-0.398512\pi$
$241$	565270.	0.626922	0.313461	+	0.949601i	$0.398512\pi$
$242$	0	0
$243$	3.37041e6	3.66157
$244$	0	0
$245$	−76832.0	−0.0817762
$246$	0	0
$247$	1.31546e6	1.37194
$248$	0	0
$249$	3.29454e6	3.36741
$250$	0	0
$251$	−1.43775e6	−1.44045	−0.720224	−	0.693741i	$-0.755961\pi$
$251$	−1.43775e6	−1.44045	−0.720224	+	0.693741i	$0.755961\pi$
$252$	0	0
$253$	−698880.	−0.686438
$254$	0	0
$255$	−896640.	−0.863511
$256$	0	0
$257$	494802.	0.467303	0.233652	−	0.972320i	$-0.424933\pi$
$257$	494802.	0.467303	0.233652	+	0.972320i	$0.424933\pi$
$258$	0	0
$259$	−610638.	−0.565633
$260$	0	0
$261$	771318.	0.700861
$262$	0	0
$263$	−1.55654e6	−1.38762	−0.693812	−	0.720156i	$-0.744070\pi$
$263$	−1.55654e6	−1.38762	−0.693812	+	0.720156i	$0.744070\pi$
$264$	0	0
$265$	−872768.	−0.763456
$266$	0	0
$267$	−1.70898e6	−1.46710
$268$	0	0
$269$	−1.36204e6	−1.14765	−0.573823	−	0.818979i	$-0.694540\pi$
$269$	−1.36204e6	−1.14765	−0.573823	+	0.818979i	$0.694540\pi$
$270$	0	0
$271$	−558320.	−0.461806	−0.230903	−	0.972977i	$-0.574168\pi$
$271$	−558320.	−0.461806	−0.230903	+	0.972977i	$0.574168\pi$
$272$	0	0
$273$	−1.04076e6	−0.845170
$274$	0	0
$275$	1.31102e6	1.04539
$276$	0	0
$277$	−586342.	−0.459147	−0.229573	−	0.973291i	$-0.573733\pi$
$277$	−586342.	−0.459147	−0.229573	+	0.973291i	$0.573733\pi$
$278$	0	0
$279$	−1.91056e6	−1.46943
$280$	0	0
$281$	606234.	0.458010	0.229005	−	0.973425i	$-0.426453\pi$
$281$	606234.	0.458010	0.229005	+	0.973425i	$0.426453\pi$
$282$	0	0
$283$	−865174.	−0.642151	−0.321076	−	0.947054i	$-0.604044\pi$
$283$	−865174.	−0.642151	−0.321076	+	0.947054i	$0.604044\pi$
$284$	0	0
$285$	−1.78368e6	−1.30078
$286$	0	0
$287$	−130438.	−0.0934758
$288$	0	0
$289$	−547501.	−0.385603
$290$	0	0
$291$	−3.47838e6	−2.40793
$292$	0	0
$293$	353352.	0.240458	0.120229	−	0.992746i	$-0.461637\pi$
$293$	353352.	0.240458	0.120229	+	0.992746i	$0.461637\pi$
$294$	0	0
$295$	−79680.0	−0.0533082
$296$	0	0
$297$	−7.75008e6	−5.09818
$298$	0	0
$299$	792960.	0.512948
$300$	0	0
$301$	350056.	0.222701
$302$	0	0
$303$	−250560.	−0.156785
$304$	0	0
$305$	−355072.	−0.218558
$306$	0	0
$307$	−1.95904e6	−1.18631	−0.593153	−	0.805090i	$-0.702117\pi$
$307$	−1.95904e6	−1.18631	−0.593153	+	0.805090i	$0.702117\pi$
$308$	0	0
$309$	−5.38452e6	−3.20812
$310$	0	0
$311$	−3.06257e6	−1.79550	−0.897749	−	0.440508i	$-0.854798\pi$
$311$	−3.06257e6	−1.79550	−0.897749	+	0.440508i	$0.854798\pi$
$312$	0	0
$313$	−582634.	−0.336151	−0.168076	−	0.985774i	$-0.553755\pi$
$313$	−582634.	−0.336151	−0.168076	+	0.985774i	$0.553755\pi$
$314$	0	0
$315$	1.03018e6	0.584972
$316$	0	0
$317$	3.09585e6	1.73034	0.865171	−	0.501478i	$-0.167210\pi$
$317$	3.09585e6	1.73034	0.865171	+	0.501478i	$0.167210\pi$
$318$	0	0
$319$	−732576.	−0.403066
$320$	0	0
$321$	−1.61676e6	−0.875756
$322$	0	0
$323$	1.73537e6	0.925521
$324$	0	0
$325$	−1.48751e6	−0.781180
$326$	0	0
$327$	−3.17910e6	−1.64412
$328$	0	0
$329$	−365932.	−0.186385
$330$	0	0
$331$	625496.	0.313801	0.156901	−	0.987614i	$-0.449850\pi$
$331$	625496.	0.313801	0.156901	+	0.987614i	$0.449850\pi$
$332$	0	0
$333$	8.18753e6	4.04616
$334$	0	0
$335$	−1.27219e6	−0.619356
$336$	0	0
$337$	2.32494e6	1.11516	0.557580	−	0.830123i	$-0.311730\pi$
$337$	2.32494e6	1.11516	0.557580	+	0.830123i	$0.311730\pi$
$338$	0	0
$339$	75060.0	0.0354739
$340$	0	0
$341$	1.81459e6	0.845071
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−1.07520e6	−0.486342
$346$	0	0
$347$	−781128.	−0.348256	−0.174128	−	0.984723i	$-0.555711\pi$
$347$	−781128.	−0.348256	−0.174128	+	0.984723i	$0.555711\pi$
$348$	0	0
$349$	−1.48586e6	−0.653002	−0.326501	−	0.945197i	$-0.605870\pi$
$349$	−1.48586e6	−0.653002	−0.326501	+	0.945197i	$0.605870\pi$
$350$	0	0
$351$	8.79336e6	3.80967
$352$	0	0
$353$	1.44463e6	0.617048	0.308524	−	0.951217i	$-0.400165\pi$
$353$	1.44463e6	0.617048	0.308524	+	0.951217i	$0.400165\pi$
$354$	0	0
$355$	−2.23642e6	−0.941850
$356$	0	0
$357$	−1.37298e6	−0.570156
$358$	0	0
$359$	−404040.	−0.165458	−0.0827291	−	0.996572i	$-0.526364\pi$
$359$	−404040.	−0.165458	−0.0827291	+	0.996572i	$0.526364\pi$
$360$	0	0
$361$	976065.	0.394195
$362$	0	0
$363$	6.84975e6	2.72840
$364$	0	0
$365$	−526400.	−0.206816
$366$	0	0
$367$	2.71698e6	1.05298	0.526492	−	0.850180i	$-0.323507\pi$
$367$	2.71698e6	1.05298	0.526492	+	0.850180i	$0.323507\pi$
$368$	0	0
$369$	1.74893e6	0.668663
$370$	0	0
$371$	−1.33643e6	−0.504092
$372$	0	0
$373$	1.79399e6	0.667647	0.333824	−	0.942636i	$-0.391661\pi$
$373$	1.79399e6	0.667647	0.333824	+	0.942636i	$0.391661\pi$
$374$	0	0
$375$	5.01696e6	1.84231
$376$	0	0
$377$	831192.	0.301195
$378$	0	0
$379$	−18624.0	−0.00666001	−0.00333001	−	0.999994i	$-0.501060\pi$
$379$	−18624.0	−0.00666001	−0.00333001	+	0.999994i	$0.501060\pi$
$380$	0	0
$381$	−8.63376e6	−3.04711
$382$	0	0
$383$	1.33004e6	0.463307	0.231654	−	0.972798i	$-0.425586\pi$
$383$	1.33004e6	0.463307	0.231654	+	0.972798i	$0.425586\pi$
$384$	0	0
$385$	−978432.	−0.336418
$386$	0	0
$387$	−4.69361e6	−1.59305
$388$	0	0
$389$	−2.26506e6	−0.758936	−0.379468	−	0.925205i	$-0.623893\pi$
$389$	−2.26506e6	−0.758936	−0.379468	+	0.925205i	$0.623893\pi$
$390$	0	0
$391$	1.04608e6	0.346037
$392$	0	0
$393$	−1.42986e6	−0.466995
$394$	0	0
$395$	2.50803e6	0.808798
$396$	0	0
$397$	4.48900e6	1.42947	0.714733	−	0.699398i	$-0.246548\pi$
$397$	4.48900e6	1.42947	0.714733	+	0.699398i	$0.246548\pi$
$398$	0	0
$399$	−2.73126e6	−0.858877
$400$	0	0
$401$	−95442.0	−0.0296400	−0.0148200	−	0.999890i	$-0.504718\pi$
$401$	−95442.0	−0.0296400	−0.0148200	+	0.999890i	$0.504718\pi$
$402$	0	0
$403$	−2.05886e6	−0.631488
$404$	0	0
$405$	−6.81437e6	−2.06437
$406$	0	0
$407$	−7.77629e6	−2.32695
$408$	0	0
$409$	−2.99003e6	−0.883828	−0.441914	−	0.897057i	$-0.645700\pi$
$409$	−2.99003e6	−0.883828	−0.441914	+	0.897057i	$0.645700\pi$
$410$	0	0
$411$	6.69462e6	1.95489
$412$	0	0
$413$	−122010.	−0.0351982
$414$	0	0
$415$	−3.51418e6	−1.00162
$416$	0	0
$417$	7.51626e6	2.11671
$418$	0	0
$419$	3.39037e6	0.943436	0.471718	−	0.881749i	$-0.343634\pi$
$419$	3.39037e6	0.943436	0.471718	+	0.881749i	$0.343634\pi$
$420$	0	0
$421$	−3.38397e6	−0.930512	−0.465256	−	0.885176i	$-0.654038\pi$
$421$	−3.38397e6	−0.930512	−0.465256	+	0.885176i	$0.654038\pi$
$422$	0	0
$423$	4.90648e6	1.33327
$424$	0	0
$425$	−1.96233e6	−0.526988
$426$	0	0
$427$	−543704.	−0.144309
$428$	0	0
$429$	−1.32538e7	−3.47693
$430$	0	0
$431$	1.98353e6	0.514334	0.257167	−	0.966367i	$-0.417211\pi$
$431$	1.98353e6	0.514334	0.257167	+	0.966367i	$0.417211\pi$
$432$	0	0
$433$	−7.17581e6	−1.83929	−0.919647	−	0.392746i	$-0.871525\pi$
$433$	−7.17581e6	−1.83929	−0.919647	+	0.392746i	$0.871525\pi$
$434$	0	0
$435$	−1.12704e6	−0.285572
$436$	0	0
$437$	2.08096e6	0.521267
$438$	0	0
$439$	2.44390e6	0.605231	0.302616	−	0.953113i	$-0.402140\pi$
$439$	2.44390e6	0.605231	0.302616	+	0.953113i	$0.402140\pi$
$440$	0	0
$441$	1.57746e6	0.386243
$442$	0	0
$443$	231716.	0.0560979	0.0280490	−	0.999607i	$-0.491071\pi$
$443$	231716.	0.0560979	0.0280490	+	0.999607i	$0.491071\pi$
$444$	0	0
$445$	1.82291e6	0.436381
$446$	0	0
$447$	1.46218e7	3.46125
$448$	0	0
$449$	−4.73637e6	−1.10874	−0.554370	−	0.832271i	$-0.687041\pi$
$449$	−4.73637e6	−1.10874	−0.554370	+	0.832271i	$0.687041\pi$
$450$	0	0
$451$	−1.66109e6	−0.384549
$452$	0	0
$453$	1.64040e6	0.375582
$454$	0	0
$455$	1.11014e6	0.251392
$456$	0	0
$457$	−7.87486e6	−1.76381	−0.881906	−	0.471426i	$-0.843740\pi$
$457$	−7.87486e6	−1.76381	−0.881906	+	0.471426i	$0.843740\pi$
$458$	0	0
$459$	1.16003e7	2.57002
$460$	0	0
$461$	8.23218e6	1.80411	0.902054	−	0.431623i	$-0.142059\pi$
$461$	8.23218e6	1.80411	0.902054	+	0.431623i	$0.142059\pi$
$462$	0	0
$463$	−2.36038e6	−0.511717	−0.255859	−	0.966714i	$-0.582358\pi$
$463$	−2.36038e6	−0.511717	−0.255859	+	0.966714i	$0.582358\pi$
$464$	0	0
$465$	2.79168e6	0.598733
$466$	0	0
$467$	−6.31700e6	−1.34035	−0.670175	−	0.742203i	$-0.733781\pi$
$467$	−6.31700e6	−1.34035	−0.670175	+	0.742203i	$0.733781\pi$
$468$	0	0
$469$	−1.94804e6	−0.408947
$470$	0	0
$471$	−6.33204e6	−1.31520
$472$	0	0
$473$	4.45786e6	0.916164
$474$	0	0
$475$	−3.90366e6	−0.793849
$476$	0	0
$477$	1.79190e7	3.60594
$478$	0	0
$479$	−1.45856e6	−0.290459	−0.145229	−	0.989398i	$-0.546392\pi$
$479$	−1.45856e6	−0.290459	−0.145229	+	0.989398i	$0.546392\pi$
$480$	0	0
$481$	8.82310e6	1.73883
$482$	0	0
$483$	−1.64640e6	−0.321120
$484$	0	0
$485$	3.71027e6	0.716228
$486$	0	0
$487$	−9.28782e6	−1.77456	−0.887282	−	0.461228i	$-0.847409\pi$
$487$	−9.28782e6	−1.77456	−0.887282	+	0.461228i	$0.847409\pi$
$488$	0	0
$489$	−605760.	−0.114559
$490$	0	0
$491$	−234972.	−0.0439858	−0.0219929	−	0.999758i	$-0.507001\pi$
$491$	−234972.	−0.0439858	−0.0219929	+	0.999758i	$0.507001\pi$
$492$	0	0
$493$	1.09652e6	0.203188
$494$	0	0
$495$	1.31190e7	2.40651
$496$	0	0
$497$	−3.42451e6	−0.621882
$498$	0	0
$499$	−7.00792e6	−1.25991	−0.629953	−	0.776633i	$-0.716926\pi$
$499$	−7.00792e6	−1.25991	−0.629953	+	0.776633i	$0.716926\pi$
$500$	0	0
$501$	135720.	0.0241574
$502$	0	0
$503$	−4.94752e6	−0.871902	−0.435951	−	0.899970i	$-0.643588\pi$
$503$	−4.94752e6	−0.871902	−0.435951	+	0.899970i	$0.643588\pi$
$504$	0	0
$505$	267264.	0.0466350
$506$	0	0
$507$	3.89913e6	0.673671
$508$	0	0
$509$	−5.50640e6	−0.942049	−0.471025	−	0.882120i	$-0.656116\pi$
$509$	−5.50640e6	−0.942049	−0.471025	+	0.882120i	$0.656116\pi$
$510$	0	0
$511$	−806050.	−0.136556
$512$	0	0
$513$	2.30764e7	3.87145
$514$	0	0
$515$	5.74349e6	0.954240
$516$	0	0
$517$	−4.66003e6	−0.766765
$518$	0	0
$519$	3.12996e6	0.510059
$520$	0	0
$521$	1.63076e6	0.263206	0.131603	−	0.991303i	$-0.457988\pi$
$521$	1.63076e6	0.263206	0.131603	+	0.991303i	$0.457988\pi$
$522$	0	0
$523$	−1.00765e7	−1.61086	−0.805429	−	0.592692i	$-0.798065\pi$
$523$	−1.00765e7	−1.61086	−0.805429	+	0.592692i	$0.798065\pi$
$524$	0	0
$525$	3.08847e6	0.489041
$526$	0	0
$527$	−2.71607e6	−0.426005
$528$	0	0
$529$	−5.18194e6	−0.805107
$530$	0	0
$531$	1.63593e6	0.251784
$532$	0	0
$533$	1.88470e6	0.287358
$534$	0	0
$535$	1.72454e6	0.260489
$536$	0	0
$537$	−6.05172e6	−0.905614
$538$	0	0
$539$	−1.49822e6	−0.222129
$540$	0	0
$541$	1.25225e7	1.83949	0.919746	−	0.392513i	$-0.128394\pi$
$541$	1.25225e7	1.83949	0.919746	+	0.392513i	$0.128394\pi$
$542$	0	0
$543$	−1.96710e7	−2.86304
$544$	0	0
$545$	3.39104e6	0.489037
$546$	0	0
$547$	6.67430e6	0.953756	0.476878	−	0.878970i	$-0.341768\pi$
$547$	6.67430e6	0.953756	0.476878	+	0.878970i	$0.341768\pi$
$548$	0	0
$549$	7.29007e6	1.03229
$550$	0	0
$551$	2.18129e6	0.306080
$552$	0	0
$553$	3.84042e6	0.534031
$554$	0	0
$555$	−1.19635e7	−1.64864
$556$	0	0
$557$	4.61643e6	0.630475	0.315238	−	0.949013i	$-0.397916\pi$
$557$	4.61643e6	0.630475	0.315238	+	0.949013i	$0.397916\pi$
$558$	0	0
$559$	−5.05795e6	−0.684613
$560$	0	0
$561$	−1.74845e7	−2.34555
$562$	0	0
$563$	−8.84218e6	−1.17568	−0.587839	−	0.808978i	$-0.700021\pi$
$563$	−8.84218e6	−1.17568	−0.587839	+	0.808978i	$0.700021\pi$
$564$	0	0
$565$	−80064.0	−0.0105515
$566$	0	0
$567$	−1.04345e7	−1.36306
$568$	0	0
$569$	1.15771e7	1.49906	0.749532	−	0.661968i	$-0.230279\pi$
$569$	1.15771e7	1.49906	0.749532	+	0.661968i	$0.230279\pi$
$570$	0	0
$571$	4.48069e6	0.575115	0.287557	−	0.957763i	$-0.407157\pi$
$571$	4.48069e6	0.575115	0.287557	+	0.957763i	$0.407157\pi$
$572$	0	0
$573$	4.54488e6	0.578277
$574$	0	0
$575$	−2.35312e6	−0.296807
$576$	0	0
$577$	1.32788e7	1.66042	0.830212	−	0.557448i	$-0.188220\pi$
$577$	1.32788e7	1.66042	0.830212	+	0.557448i	$0.188220\pi$
$578$	0	0
$579$	6.87978e6	0.852861
$580$	0	0
$581$	−5.38108e6	−0.661347
$582$	0	0
$583$	−1.70190e7	−2.07378
$584$	0	0
$585$	−1.48850e7	−1.79829
$586$	0	0
$587$	−1.11188e7	−1.33187	−0.665936	−	0.746009i	$-0.731968\pi$
$587$	−1.11188e7	−1.33187	−0.665936	+	0.746009i	$0.731968\pi$
$588$	0	0
$589$	−5.40306e6	−0.641729
$590$	0	0
$591$	−1.26326e7	−1.48773
$592$	0	0
$593$	−3.92737e6	−0.458632	−0.229316	−	0.973352i	$-0.573649\pi$
$593$	−3.92737e6	−0.458632	−0.229316	+	0.973352i	$0.573649\pi$
$594$	0	0
$595$	1.46451e6	0.169590
$596$	0	0
$597$	−5.91900e6	−0.679693
$598$	0	0
$599$	1.74099e7	1.98258	0.991289	−	0.131704i	$-0.0420449\pi$
$599$	1.74099e7	1.98258	0.991289	+	0.131704i	$0.0420449\pi$
$600$	0	0
$601$	7.46243e6	0.842740	0.421370	−	0.906889i	$-0.361549\pi$
$601$	7.46243e6	0.842740	0.421370	+	0.906889i	$0.361549\pi$
$602$	0	0
$603$	2.61197e7	2.92533
$604$	0	0
$605$	−7.30640e6	−0.811549
$606$	0	0
$607$	−701152.	−0.0772397	−0.0386198	−	0.999254i	$-0.512296\pi$
$607$	−701152.	−0.0772397	−0.0386198	+	0.999254i	$0.512296\pi$
$608$	0	0
$609$	−1.72578e6	−0.188557
$610$	0	0
$611$	5.28734e6	0.572974
$612$	0	0
$613$	−1.09575e7	−1.17777	−0.588886	−	0.808216i	$-0.700433\pi$
$613$	−1.09575e7	−1.17777	−0.588886	+	0.808216i	$0.700433\pi$
$614$	0	0
$615$	−2.55552e6	−0.272453
$616$	0	0
$617$	1.90666e6	0.201633	0.100816	−	0.994905i	$-0.467855\pi$
$617$	1.90666e6	0.201633	0.100816	+	0.994905i	$0.467855\pi$
$618$	0	0
$619$	−2.22346e6	−0.233240	−0.116620	−	0.993177i	$-0.537206\pi$
$619$	−2.22346e6	−0.233240	−0.116620	+	0.993177i	$0.537206\pi$
$620$	0	0
$621$	1.39104e7	1.44747
$622$	0	0
$623$	2.79133e6	0.288132
$624$	0	0
$625$	1.21420e6	0.124334
$626$	0	0
$627$	−3.47818e7	−3.53332
$628$	0	0
$629$	1.16395e7	1.17303
$630$	0	0
$631$	−752624.	−0.0752497	−0.0376248	−	0.999292i	$-0.511979\pi$
$631$	−752624.	−0.0752497	−0.0376248	+	0.999292i	$0.511979\pi$
$632$	0	0
$633$	2.03716e7	2.02076
$634$	0	0
$635$	9.20934e6	0.906347
$636$	0	0
$637$	1.69991e6	0.165988
$638$	0	0
$639$	4.59164e7	4.44852
$640$	0	0
$641$	−2.45429e6	−0.235929	−0.117964	−	0.993018i	$-0.537637\pi$
$641$	−2.45429e6	−0.235929	−0.117964	+	0.993018i	$0.537637\pi$
$642$	0	0
$643$	1.58237e7	1.50932	0.754660	−	0.656116i	$-0.227802\pi$
$643$	1.58237e7	1.50932	0.754660	+	0.656116i	$0.227802\pi$
$644$	0	0
$645$	6.85824e6	0.649103
$646$	0	0
$647$	2.65489e6	0.249337	0.124668	−	0.992198i	$-0.460213\pi$
$647$	2.65489e6	0.249337	0.124668	+	0.992198i	$0.460213\pi$
$648$	0	0
$649$	−1.55376e6	−0.144801
$650$	0	0
$651$	4.27476e6	0.395330
$652$	0	0
$653$	−9.26899e6	−0.850648	−0.425324	−	0.905041i	$-0.639840\pi$
$653$	−9.26899e6	−0.850648	−0.425324	+	0.905041i	$0.639840\pi$
$654$	0	0
$655$	1.52518e6	0.138905
$656$	0	0
$657$	1.08076e7	0.976827
$658$	0	0
$659$	−1.68242e7	−1.50911	−0.754556	−	0.656235i	$-0.772148\pi$
$659$	−1.68242e7	−1.50911	−0.754556	+	0.656235i	$0.772148\pi$
$660$	0	0
$661$	6.77217e6	0.602871	0.301435	−	0.953487i	$-0.402534\pi$
$661$	6.77217e6	0.602871	0.301435	+	0.953487i	$0.402534\pi$
$662$	0	0
$663$	1.98382e7	1.75274
$664$	0	0
$665$	2.91334e6	0.255469
$666$	0	0
$667$	1.31488e6	0.114438
$668$	0	0
$669$	5.53320e6	0.477981
$670$	0	0
$671$	−6.92390e6	−0.593669
$672$	0	0
$673$	7.61315e6	0.647928	0.323964	−	0.946069i	$-0.394984\pi$
$673$	7.61315e6	0.647928	0.323964	+	0.946069i	$0.394984\pi$
$674$	0	0
$675$	−2.60944e7	−2.20439
$676$	0	0
$677$	−6.11672e6	−0.512917	−0.256459	−	0.966555i	$-0.582556\pi$
$677$	−6.11672e6	−0.512917	−0.256459	+	0.966555i	$0.582556\pi$
$678$	0	0
$679$	5.68135e6	0.472909
$680$	0	0
$681$	−2.60423e7	−2.15185
$682$	0	0
$683$	−1.12588e7	−0.923511	−0.461755	−	0.887007i	$-0.652780\pi$
$683$	−1.12588e7	−0.923511	−0.461755	+	0.887007i	$0.652780\pi$
$684$	0	0
$685$	−7.14093e6	−0.581471
$686$	0	0
$687$	1.78158e7	1.44017
$688$	0	0
$689$	1.93100e7	1.54965
$690$	0	0
$691$	9.50952e6	0.757641	0.378821	−	0.925470i	$-0.376330\pi$
$691$	9.50952e6	0.757641	0.378821	+	0.925470i	$0.376330\pi$
$692$	0	0
$693$	2.00884e7	1.58896
$694$	0	0
$695$	−8.01734e6	−0.629605
$696$	0	0
$697$	2.48631e6	0.193853
$698$	0	0
$699$	1.44654e6	0.111979
$700$	0	0
$701$	−1.53868e7	−1.18264	−0.591322	−	0.806436i	$-0.701394\pi$
$701$	−1.53868e7	−1.18264	−0.591322	+	0.806436i	$0.701394\pi$
$702$	0	0
$703$	2.31544e7	1.76703
$704$	0	0
$705$	−7.16928e6	−0.543254
$706$	0	0
$707$	409248.	0.0307920
$708$	0	0
$709$	1.21379e6	0.0906834	0.0453417	−	0.998972i	$-0.485562\pi$
$709$	1.21379e6	0.0906834	0.0453417	+	0.998972i	$0.485562\pi$
$710$	0	0
$711$	−5.14930e7	−3.82010
$712$	0	0
$713$	−3.25696e6	−0.239932
$714$	0	0
$715$	1.41373e7	1.03420
$716$	0	0
$717$	−7.25064e6	−0.526718
$718$	0	0
$719$	1.00002e7	0.721418	0.360709	−	0.932678i	$-0.382535\pi$
$719$	1.00002e7	0.721418	0.360709	+	0.932678i	$0.382535\pi$
$720$	0	0
$721$	8.79472e6	0.630063
$722$	0	0
$723$	1.69581e7	1.20651
$724$	0	0
$725$	−2.46657e6	−0.174281
$726$	0	0
$727$	−1.33745e7	−0.938514	−0.469257	−	0.883062i	$-0.655478\pi$
$727$	−1.33745e7	−0.938514	−0.469257	+	0.883062i	$0.655478\pi$
$728$	0	0
$729$	4.93657e7	3.44038
$730$	0	0
$731$	−6.67250e6	−0.461844
$732$	0	0
$733$	−1.61380e7	−1.10940	−0.554701	−	0.832050i	$-0.687167\pi$
$733$	−1.61380e7	−1.10940	−0.554701	+	0.832050i	$0.687167\pi$
$734$	0	0
$735$	−2.30496e6	−0.157378
$736$	0	0
$737$	−2.48077e7	−1.68236
$738$	0	0
$739$	8.61059e6	0.579992	0.289996	−	0.957028i	$-0.406346\pi$
$739$	8.61059e6	0.579992	0.289996	+	0.957028i	$0.406346\pi$
$740$	0	0
$741$	3.94639e7	2.64031
$742$	0	0
$743$	2.85027e7	1.89415	0.947075	−	0.321012i	$-0.104023\pi$
$743$	2.85027e7	1.89415	0.947075	+	0.321012i	$0.104023\pi$
$744$	0	0
$745$	−1.55966e7	−1.02953
$746$	0	0
$747$	7.21504e7	4.73083
$748$	0	0
$749$	2.64071e6	0.171995
$750$	0	0
$751$	7.28721e6	0.471478	0.235739	−	0.971816i	$-0.424249\pi$
$751$	7.28721e6	0.471478	0.235739	+	0.971816i	$0.424249\pi$
$752$	0	0
$753$	−4.31324e7	−2.77215
$754$	0	0
$755$	−1.74976e6	−0.111715
$756$	0	0
$757$	2.77165e7	1.75792	0.878958	−	0.476899i	$-0.158239\pi$
$757$	2.77165e7	1.75792	0.878958	+	0.476899i	$0.158239\pi$
$758$	0	0
$759$	−2.09664e7	−1.32105
$760$	0	0
$761$	3.07625e7	1.92557	0.962787	−	0.270262i	$-0.0871103\pi$
$761$	3.07625e7	1.92557	0.962787	+	0.270262i	$0.0871103\pi$
$762$	0	0
$763$	5.19253e6	0.322900
$764$	0	0
$765$	−1.96364e7	−1.21313
$766$	0	0
$767$	1.76292e6	0.108204
$768$	0	0
$769$	−1.83665e7	−1.11998	−0.559990	−	0.828499i	$-0.689195\pi$
$769$	−1.83665e7	−1.11998	−0.559990	+	0.828499i	$0.689195\pi$
$770$	0	0
$771$	1.48441e7	0.899325
$772$	0	0
$773$	1.01397e7	0.610348	0.305174	−	0.952297i	$-0.401285\pi$
$773$	1.01397e7	0.610348	0.305174	+	0.952297i	$0.401285\pi$
$774$	0	0
$775$	6.10971e6	0.365398
$776$	0	0
$777$	−1.83191e7	−1.08856
$778$	0	0
$779$	4.94600e6	0.292018
$780$	0	0
$781$	−4.36101e7	−2.55835
$782$	0	0
$783$	1.45811e7	0.849934
$784$	0	0
$785$	6.75418e6	0.391199
$786$	0	0
$787$	−1.89442e7	−1.09028	−0.545140	−	0.838345i	$-0.683524\pi$
$787$	−1.89442e7	−1.09028	−0.545140	+	0.838345i	$0.683524\pi$
$788$	0	0
$789$	−4.66963e7	−2.67049
$790$	0	0
$791$	−122598.	−0.00696694
$792$	0	0
$793$	7.85597e6	0.443626
$794$	0	0
$795$	−2.61830e7	−1.46927
$796$	0	0
$797$	1.19835e7	0.668248	0.334124	−	0.942529i	$-0.391560\pi$
$797$	1.19835e7	0.668248	0.334124	+	0.942529i	$0.391560\pi$
$798$	0	0
$799$	6.97511e6	0.386531
$800$	0	0
$801$	−3.74267e7	−2.06110
$802$	0	0
$803$	−1.02648e7	−0.561774
$804$	0	0
$805$	1.75616e6	0.0955156
$806$	0	0
$807$	−4.08611e7	−2.20865
$808$	0	0
$809$	−6.31823e6	−0.339410	−0.169705	−	0.985495i	$-0.554281\pi$
$809$	−6.31823e6	−0.339410	−0.169705	+	0.985495i	$0.554281\pi$
$810$	0	0
$811$	1.47079e6	0.0785231	0.0392615	−	0.999229i	$-0.487499\pi$
$811$	1.47079e6	0.0785231	0.0392615	+	0.999229i	$0.487499\pi$
$812$	0	0
$813$	−1.67496e7	−0.888747
$814$	0	0
$815$	646144.	0.0340750
$816$	0	0
$817$	−1.32736e7	−0.695716
$818$	0	0
$819$	−2.27926e7	−1.18737
$820$	0	0
$821$	2.10138e7	1.08804	0.544022	−	0.839071i	$-0.316901\pi$
$821$	2.10138e7	1.08804	0.544022	+	0.839071i	$0.316901\pi$
$822$	0	0
$823$	1.35856e7	0.699163	0.349582	−	0.936906i	$-0.386324\pi$
$823$	1.35856e7	0.699163	0.349582	+	0.936906i	$0.386324\pi$
$824$	0	0
$825$	3.93307e7	2.01186
$826$	0	0
$827$	−2.62070e7	−1.33246	−0.666230	−	0.745747i	$-0.732093\pi$
$827$	−2.62070e7	−1.33246	−0.666230	+	0.745747i	$0.732093\pi$
$828$	0	0
$829$	1.17710e7	0.594876	0.297438	−	0.954741i	$-0.403868\pi$
$829$	1.17710e7	0.594876	0.297438	+	0.954741i	$0.403868\pi$
$830$	0	0
$831$	−1.75903e7	−0.883628
$832$	0	0
$833$	2.24253e6	0.111976
$834$	0	0
$835$	−144768.	−0.00718549
$836$	0	0
$837$	−3.61174e7	−1.78198
$838$	0	0
$839$	−1.44898e7	−0.710651	−0.355326	−	0.934743i	$-0.615630\pi$
$839$	−1.44898e7	−0.710651	−0.355326	+	0.934743i	$0.615630\pi$
$840$	0	0
$841$	−1.91329e7	−0.932804
$842$	0	0
$843$	1.81870e7	0.881440
$844$	0	0
$845$	−4.15907e6	−0.200380
$846$	0	0
$847$	−1.11879e7	−0.535847
$848$	0	0
$849$	−2.59552e7	−1.23582
$850$	0	0
$851$	1.39574e7	0.660666
$852$	0	0
$853$	−1.77865e7	−0.836984	−0.418492	−	0.908220i	$-0.637441\pi$
$853$	−1.77865e7	−0.836984	−0.418492	+	0.908220i	$0.637441\pi$
$854$	0	0
$855$	−3.90626e7	−1.82745
$856$	0	0
$857$	3.79124e7	1.76331	0.881656	−	0.471893i	$-0.156429\pi$
$857$	3.79124e7	1.76331	0.881656	+	0.471893i	$0.156429\pi$
$858$	0	0
$859$	3.32376e7	1.53690	0.768451	−	0.639909i	$-0.221028\pi$
$859$	3.32376e7	1.53690	0.768451	+	0.639909i	$0.221028\pi$
$860$	0	0
$861$	−3.91314e6	−0.179894
$862$	0	0
$863$	−2.33166e6	−0.106571	−0.0532853	−	0.998579i	$-0.516969\pi$
$863$	−2.33166e6	−0.106571	−0.0532853	+	0.998579i	$0.516969\pi$
$864$	0	0
$865$	−3.33862e6	−0.151715
$866$	0	0
$867$	−1.64250e7	−0.742093
$868$	0	0
$869$	4.89066e7	2.19694
$870$	0	0
$871$	2.81472e7	1.25716
$872$	0	0
$873$	−7.61765e7	−3.38287
$874$	0	0
$875$	−8.19437e6	−0.361822
$876$	0	0
$877$	3.38189e7	1.48477	0.742386	−	0.669972i	$-0.233694\pi$
$877$	3.38189e7	1.48477	0.742386	+	0.669972i	$0.233694\pi$
$878$	0	0
$879$	1.06006e7	0.462761
$880$	0	0
$881$	2.65707e7	1.15336	0.576678	−	0.816972i	$-0.304349\pi$
$881$	2.65707e7	1.15336	0.576678	+	0.816972i	$0.304349\pi$
$882$	0	0
$883$	1.74913e7	0.754954	0.377477	−	0.926019i	$-0.376792\pi$
$883$	1.74913e7	0.754954	0.377477	+	0.926019i	$0.376792\pi$
$884$	0	0
$885$	−2.39040e6	−0.102592
$886$	0	0
$887$	1.77452e7	0.757305	0.378652	−	0.925539i	$-0.376388\pi$
$887$	1.77452e7	0.757305	0.378652	+	0.925539i	$0.376388\pi$
$888$	0	0
$889$	1.41018e7	0.598440
$890$	0	0
$891$	−1.32880e8	−5.60746
$892$	0	0
$893$	1.38755e7	0.582266
$894$	0	0
$895$	6.45517e6	0.269370
$896$	0	0
$897$	2.37888e7	0.987169
$898$	0	0
$899$	−3.41399e6	−0.140885
$900$	0	0
$901$	2.54739e7	1.04540
$902$	0	0
$903$	1.05017e7	0.428587
$904$	0	0
$905$	2.09824e7	0.851596
$906$	0	0
$907$	−2.82335e7	−1.13958	−0.569792	−	0.821789i	$-0.692976\pi$
$907$	−2.82335e7	−1.13958	−0.569792	+	0.821789i	$0.692976\pi$
$908$	0	0
$909$	−5.48726e6	−0.220265
$910$	0	0
$911$	−1.71757e7	−0.685674	−0.342837	−	0.939395i	$-0.611388\pi$
$911$	−1.71757e7	−0.685674	−0.342837	+	0.939395i	$0.611388\pi$
$912$	0	0
$913$	−6.85264e7	−2.72070
$914$	0	0
$915$	−1.06522e7	−0.420615
$916$	0	0
$917$	2.33544e6	0.0917160
$918$	0	0
$919$	−2.42273e7	−0.946273	−0.473137	−	0.880989i	$-0.656878\pi$
$919$	−2.42273e7	−0.946273	−0.473137	+	0.880989i	$0.656878\pi$
$920$	0	0
$921$	−5.87711e7	−2.28305
$922$	0	0
$923$	4.94807e7	1.91175
$924$	0	0
$925$	−2.61827e7	−1.00614
$926$	0	0
$927$	−1.17921e8	−4.50704
$928$	0	0
$929$	2.45919e7	0.934874	0.467437	−	0.884026i	$-0.345178\pi$
$929$	2.45919e7	0.934874	0.467437	+	0.884026i	$0.345178\pi$
$930$	0	0
$931$	4.46106e6	0.168680
$932$	0	0
$933$	−9.18770e7	−3.45544
$934$	0	0
$935$	1.86501e7	0.697674
$936$	0	0
$937$	−1.31199e7	−0.488181	−0.244090	−	0.969753i	$-0.578489\pi$
$937$	−1.31199e7	−0.488181	−0.244090	+	0.969753i	$0.578489\pi$
$938$	0	0
$939$	−1.74790e7	−0.646924
$940$	0	0
$941$	−640776.	−0.0235902	−0.0117951	−	0.999930i	$-0.503755\pi$
$941$	−640776.	−0.0235902	−0.0117951	+	0.999930i	$0.503755\pi$
$942$	0	0
$943$	2.98144e6	0.109181
$944$	0	0
$945$	1.94746e7	0.709395
$946$	0	0
$947$	2.01502e7	0.730135	0.365068	−	0.930981i	$-0.381046\pi$
$947$	2.01502e7	0.730135	0.365068	+	0.930981i	$0.381046\pi$
$948$	0	0
$949$	1.16466e7	0.419791
$950$	0	0
$951$	9.28755e7	3.33004
$952$	0	0
$953$	−3.33217e7	−1.18849	−0.594244	−	0.804285i	$-0.702549\pi$
$953$	−3.33217e7	−1.18849	−0.594244	+	0.804285i	$0.702549\pi$
$954$	0	0
$955$	−4.84787e6	−0.172006
$956$	0	0
$957$	−2.19773e7	−0.775701
$958$	0	0
$959$	−1.09345e7	−0.383932
$960$	0	0
$961$	−2.01727e7	−0.704621
$962$	0	0
$963$	−3.54070e7	−1.23034
$964$	0	0
$965$	−7.33843e6	−0.253679
$966$	0	0
$967$	−4.51857e7	−1.55394	−0.776970	−	0.629537i	$-0.783245\pi$
$967$	−4.51857e7	−1.55394	−0.776970	+	0.629537i	$0.783245\pi$
$968$	0	0
$969$	5.20612e7	1.78117
$970$	0	0
$971$	2.62935e7	0.894953	0.447477	−	0.894296i	$-0.352323\pi$
$971$	2.62935e7	0.894953	0.447477	+	0.894296i	$0.352323\pi$
$972$	0	0
$973$	−1.22766e7	−0.415714
$974$	0	0
$975$	−4.46252e7	−1.50338
$976$	0	0
$977$	2.02548e7	0.678879	0.339440	−	0.940628i	$-0.389763\pi$
$977$	2.02548e7	0.678879	0.339440	+	0.940628i	$0.389763\pi$
$978$	0	0
$979$	3.55468e7	1.18534
$980$	0	0
$981$	−6.96223e7	−2.30981
$982$	0	0
$983$	−9.43782e6	−0.311521	−0.155761	−	0.987795i	$-0.549783\pi$
$983$	−9.43782e6	−0.311521	−0.155761	+	0.987795i	$0.549783\pi$
$984$	0	0
$985$	1.34748e7	0.442517
$986$	0	0
$987$	−1.09780e7	−0.358698
$988$	0	0
$989$	−8.00128e6	−0.260117
$990$	0	0
$991$	−1.53265e7	−0.495747	−0.247874	−	0.968792i	$-0.579732\pi$
$991$	−1.53265e7	−0.495747	−0.247874	+	0.968792i	$0.579732\pi$
$992$	0	0
$993$	1.87649e7	0.603911
$994$	0	0
$995$	6.31360e6	0.202171
$996$	0	0
$997$	2.16514e6	0.0689841	0.0344920	−	0.999405i	$-0.489019\pi$
$997$	2.16514e6	0.0689841	0.0344920	+	0.999405i	$0.489019\pi$
$998$	0	0
$999$	1.54778e8	4.90677

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.6.a.p.1.1		1
4.3	odd	2		448.6.a.a.1.1		1
8.3	odd	2		56.6.a.b.1.1	✓	1
8.5	even	2		112.6.a.a.1.1		1
24.5	odd	2		1008.6.a.h.1.1		1
24.11	even	2		504.6.a.b.1.1		1
56.3	even	6		392.6.i.f.177.1		2
56.11	odd	6		392.6.i.a.177.1		2
56.13	odd	2		784.6.a.n.1.1		1
56.19	even	6		392.6.i.f.361.1		2
56.27	even	2		392.6.a.a.1.1		1
56.51	odd	6		392.6.i.a.361.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.b.1.1	✓	1	8.3	odd	2
112.6.a.a.1.1		1	8.5	even	2
392.6.a.a.1.1		1	56.27	even	2
392.6.i.a.177.1		2	56.11	odd	6
392.6.i.a.361.1		2	56.51	odd	6
392.6.i.f.177.1		2	56.3	even	6
392.6.i.f.361.1		2	56.19	even	6
448.6.a.a.1.1		1	4.3	odd	2
448.6.a.p.1.1		1	1.1	even	1	trivial
504.6.a.b.1.1		1	24.11	even	2
784.6.a.n.1.1		1	56.13	odd	2
1008.6.a.h.1.1		1	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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