LMFDB - Embedded newform 48.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,6,Mod(1,48)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	48.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:	$7.69842335102$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+9.00000 q^{3} +94.0000 q^{5} -144.000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +94.0000 q^{5} -144.000 q^{7} +81.0000 q^{9} +380.000 q^{11} +814.000 q^{13} +846.000 q^{15} -862.000 q^{17} +1156.00 q^{19} -1296.00 q^{21} +488.000 q^{23} +5711.00 q^{25} +729.000 q^{27} -5466.00 q^{29} -9560.00 q^{31} +3420.00 q^{33} -13536.0 q^{35} -10506.0 q^{37} +7326.00 q^{39} -5190.00 q^{41} +17084.0 q^{43} +7614.00 q^{45} -3168.00 q^{47} +3929.00 q^{49} -7758.00 q^{51} -24770.0 q^{53} +35720.0 q^{55} +10404.0 q^{57} -17380.0 q^{59} +4366.00 q^{61} -11664.0 q^{63} +76516.0 q^{65} +5284.00 q^{67} +4392.00 q^{69} -8360.00 q^{71} +39466.0 q^{73} +51399.0 q^{75} -54720.0 q^{77} -42376.0 q^{79} +6561.00 q^{81} +61828.0 q^{83} -81028.0 q^{85} -49194.0 q^{87} -63078.0 q^{89} -117216. q^{91} -86040.0 q^{93} +108664. q^{95} -16318.0 q^{97} +30780.0 q^{99} +O(q^{100})$

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
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	0	0
$5$	94.0000	1.68152	0.840762	−	0.541406i	$-0.182108\pi$
$5$	94.0000	1.68152	0.840762	+	0.541406i	$0.182108\pi$
$6$	0	0
$7$	−144.000	−1.11075	−0.555376	−	0.831599i	$-0.687426\pi$
$7$	−144.000	−1.11075	−0.555376	+	0.831599i	$0.687426\pi$
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	380.000	0.946895	0.473448	−	0.880822i	$-0.343009\pi$
$11$	380.000	0.946895	0.473448	+	0.880822i	$0.343009\pi$
$12$	0	0
$13$	814.000	1.33588	0.667938	−	0.744217i	$-0.267177\pi$
$13$	814.000	1.33588	0.667938	+	0.744217i	$0.267177\pi$
$14$	0	0
$15$	846.000	0.970828
$16$	0	0
$17$	−862.000	−0.723411	−0.361705	−	0.932292i	$-0.617805\pi$
$17$	−862.000	−0.723411	−0.361705	+	0.932292i	$0.617805\pi$
$18$	0	0
$19$	1156.00	0.734639	0.367319	−	0.930095i	$-0.380276\pi$
$19$	1156.00	0.734639	0.367319	+	0.930095i	$0.380276\pi$
$20$	0	0
$21$	−1296.00	−0.641293
$22$	0	0
$23$	488.000	0.192354	0.0961768	−	0.995364i	$-0.469339\pi$
$23$	488.000	0.192354	0.0961768	+	0.995364i	$0.469339\pi$
$24$	0	0
$25$	5711.00	1.82752
$26$	0	0
$27$	729.000	0.192450
$28$	0	0
$29$	−5466.00	−1.20691	−0.603455	−	0.797397i	$-0.706210\pi$
$29$	−5466.00	−1.20691	−0.603455	+	0.797397i	$0.706210\pi$
$30$	0	0
$31$	−9560.00	−1.78671	−0.893354	−	0.449353i	$-0.851654\pi$
$31$	−9560.00	−1.78671	−0.893354	+	0.449353i	$0.851654\pi$
$32$	0	0
$33$	3420.00	0.546690
$34$	0	0
$35$	−13536.0	−1.86776
$36$	0	0
$37$	−10506.0	−1.26163	−0.630817	−	0.775932i	$-0.717280\pi$
$37$	−10506.0	−1.26163	−0.630817	+	0.775932i	$0.717280\pi$
$38$	0	0
$39$	7326.00	0.771268
$40$	0	0
$41$	−5190.00	−0.482178	−0.241089	−	0.970503i	$-0.577505\pi$
$41$	−5190.00	−0.482178	−0.241089	+	0.970503i	$0.577505\pi$
$42$	0	0
$43$	17084.0	1.40902	0.704512	−	0.709692i	$-0.251166\pi$
$43$	17084.0	1.40902	0.704512	+	0.709692i	$0.251166\pi$
$44$	0	0
$45$	7614.00	0.560508
$46$	0	0
$47$	−3168.00	−0.209190	−0.104595	−	0.994515i	$-0.533355\pi$
$47$	−3168.00	−0.209190	−0.104595	+	0.994515i	$0.533355\pi$
$48$	0	0
$49$	3929.00	0.233772
$50$	0	0
$51$	−7758.00	−0.417661
$52$	0	0
$53$	−24770.0	−1.21126	−0.605629	−	0.795747i	$-0.707078\pi$
$53$	−24770.0	−1.21126	−0.605629	+	0.795747i	$0.707078\pi$
$54$	0	0
$55$	35720.0	1.59223
$56$	0	0
$57$	10404.0	0.424144
$58$	0	0
$59$	−17380.0	−0.650010	−0.325005	−	0.945712i	$-0.605366\pi$
$59$	−17380.0	−0.650010	−0.325005	+	0.945712i	$0.605366\pi$
$60$	0	0
$61$	4366.00	0.150231	0.0751154	−	0.997175i	$-0.476067\pi$
$61$	4366.00	0.150231	0.0751154	+	0.997175i	$0.476067\pi$
$62$	0	0
$63$	−11664.0	−0.370251
$64$	0	0
$65$	76516.0	2.24631
$66$	0	0
$67$	5284.00	0.143806	0.0719028	−	0.997412i	$-0.477093\pi$
$67$	5284.00	0.143806	0.0719028	+	0.997412i	$0.477093\pi$
$68$	0	0
$69$	4392.00	0.111055
$70$	0	0
$71$	−8360.00	−0.196816	−0.0984080	−	0.995146i	$-0.531375\pi$
$71$	−8360.00	−0.196816	−0.0984080	+	0.995146i	$0.531375\pi$
$72$	0	0
$73$	39466.0	0.866794	0.433397	−	0.901203i	$-0.357315\pi$
$73$	39466.0	0.866794	0.433397	+	0.901203i	$0.357315\pi$
$74$	0	0
$75$	51399.0	1.05512
$76$	0	0
$77$	−54720.0	−1.05177
$78$	0	0
$79$	−42376.0	−0.763928	−0.381964	−	0.924177i	$-0.624752\pi$
$79$	−42376.0	−0.763928	−0.381964	+	0.924177i	$0.624752\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	61828.0	0.985122	0.492561	−	0.870278i	$-0.336061\pi$
$83$	61828.0	0.985122	0.492561	+	0.870278i	$0.336061\pi$
$84$	0	0
$85$	−81028.0	−1.21643
$86$	0	0
$87$	−49194.0	−0.696809
$88$	0	0
$89$	−63078.0	−0.844117	−0.422059	−	0.906568i	$-0.638692\pi$
$89$	−63078.0	−0.844117	−0.422059	+	0.906568i	$0.638692\pi$
$90$	0	0
$91$	−117216.	−1.48383
$92$	0	0
$93$	−86040.0	−1.03156
$94$	0	0
$95$	108664.	1.23531
$96$	0	0
$97$	−16318.0	−0.176091	−0.0880456	−	0.996116i	$-0.528062\pi$
$97$	−16318.0	−0.176091	−0.0880456	+	0.996116i	$0.528062\pi$
$98$	0	0
$99$	30780.0	0.315632
$100$	0	0
$101$	134766.	1.31455	0.657275	−	0.753651i	$-0.271709\pi$
$101$	134766.	1.31455	0.657275	+	0.753651i	$0.271709\pi$
$102$	0	0
$103$	−45184.0	−0.419654	−0.209827	−	0.977739i	$-0.567290\pi$
$103$	−45184.0	−0.419654	−0.209827	+	0.977739i	$0.567290\pi$
$104$	0	0
$105$	−121824.	−1.07835
$106$	0	0
$107$	−143124.	−1.20852	−0.604259	−	0.796788i	$-0.706531\pi$
$107$	−143124.	−1.20852	−0.604259	+	0.796788i	$0.706531\pi$
$108$	0	0
$109$	153070.	1.23402	0.617012	−	0.786953i	$-0.288343\pi$
$109$	153070.	1.23402	0.617012	+	0.786953i	$0.288343\pi$
$110$	0	0
$111$	−94554.0	−0.728404
$112$	0	0
$113$	−64494.0	−0.475142	−0.237571	−	0.971370i	$-0.576351\pi$
$113$	−64494.0	−0.475142	−0.237571	+	0.971370i	$0.576351\pi$
$114$	0	0
$115$	45872.0	0.323447
$116$	0	0
$117$	65934.0	0.445292
$118$	0	0
$119$	124128.	0.803530
$120$	0	0
$121$	−16651.0	−0.103390
$122$	0	0
$123$	−46710.0	−0.278386
$124$	0	0
$125$	243084.	1.39149
$126$	0	0
$127$	−33256.0	−0.182962	−0.0914810	−	0.995807i	$-0.529160\pi$
$127$	−33256.0	−0.182962	−0.0914810	+	0.995807i	$0.529160\pi$
$128$	0	0
$129$	153756.	0.813500
$130$	0	0
$131$	149284.	0.760038	0.380019	−	0.924979i	$-0.375918\pi$
$131$	149284.	0.760038	0.380019	+	0.924979i	$0.375918\pi$
$132$	0	0
$133$	−166464.	−0.816002
$134$	0	0
$135$	68526.0	0.323609
$136$	0	0
$137$	148602.	0.676431	0.338215	−	0.941069i	$-0.390177\pi$
$137$	148602.	0.676431	0.338215	+	0.941069i	$0.390177\pi$
$138$	0	0
$139$	324012.	1.42241	0.711204	−	0.702986i	$-0.248150\pi$
$139$	324012.	1.42241	0.711204	+	0.702986i	$0.248150\pi$
$140$	0	0
$141$	−28512.0	−0.120776
$142$	0	0
$143$	309320.	1.26493
$144$	0	0
$145$	−513804.	−2.02945
$146$	0	0
$147$	35361.0	0.134968
$148$	0	0
$149$	−59986.0	−0.221352	−0.110676	−	0.993857i	$-0.535302\pi$
$149$	−59986.0	−0.221352	−0.110676	+	0.993857i	$0.535302\pi$
$150$	0	0
$151$	−91520.0	−0.326643	−0.163322	−	0.986573i	$-0.552221\pi$
$151$	−91520.0	−0.326643	−0.163322	+	0.986573i	$0.552221\pi$
$152$	0	0
$153$	−69822.0	−0.241137
$154$	0	0
$155$	−898640.	−3.00439
$156$	0	0
$157$	345550.	1.11882	0.559412	−	0.828890i	$-0.311027\pi$
$157$	345550.	1.11882	0.559412	+	0.828890i	$0.311027\pi$
$158$	0	0
$159$	−222930.	−0.699320
$160$	0	0
$161$	−70272.0	−0.213657
$162$	0	0
$163$	634164.	1.86953	0.934765	−	0.355266i	$-0.115610\pi$
$163$	634164.	1.86953	0.934765	+	0.355266i	$0.115610\pi$
$164$	0	0
$165$	321480.	0.919272
$166$	0	0
$167$	415848.	1.15383	0.576917	−	0.816803i	$-0.304256\pi$
$167$	415848.	1.15383	0.576917	+	0.816803i	$0.304256\pi$
$168$	0	0
$169$	291303.	0.784564
$170$	0	0
$171$	93636.0	0.244880
$172$	0	0
$173$	−324282.	−0.823773	−0.411887	−	0.911235i	$-0.635130\pi$
$173$	−324282.	−0.823773	−0.411887	+	0.911235i	$0.635130\pi$
$174$	0	0
$175$	−822384.	−2.02992
$176$	0	0
$177$	−156420.	−0.375283
$178$	0	0
$179$	−606828.	−1.41558	−0.707788	−	0.706425i	$-0.750307\pi$
$179$	−606828.	−1.41558	−0.707788	+	0.706425i	$0.750307\pi$
$180$	0	0
$181$	463254.	1.05105	0.525524	−	0.850779i	$-0.323869\pi$
$181$	463254.	1.05105	0.525524	+	0.850779i	$0.323869\pi$
$182$	0	0
$183$	39294.0	0.0867358
$184$	0	0
$185$	−987564.	−2.12147
$186$	0	0
$187$	−327560.	−0.684994
$188$	0	0
$189$	−104976.	−0.213764
$190$	0	0
$191$	−955008.	−1.89419	−0.947095	−	0.320953i	$-0.895997\pi$
$191$	−955008.	−1.89419	−0.947095	+	0.320953i	$0.895997\pi$
$192$	0	0
$193$	−256126.	−0.494949	−0.247474	−	0.968894i	$-0.579601\pi$
$193$	−256126.	−0.494949	−0.247474	+	0.968894i	$0.579601\pi$
$194$	0	0
$195$	688644.	1.29691
$196$	0	0
$197$	111150.	0.204053	0.102027	−	0.994782i	$-0.467467\pi$
$197$	111150.	0.204053	0.102027	+	0.994782i	$0.467467\pi$
$198$	0	0
$199$	516800.	0.925102	0.462551	−	0.886593i	$-0.346934\pi$
$199$	516800.	0.925102	0.462551	+	0.886593i	$0.346934\pi$
$200$	0	0
$201$	47556.0	0.0830262
$202$	0	0
$203$	787104.	1.34058
$204$	0	0
$205$	−487860.	−0.810794
$206$	0	0
$207$	39528.0	0.0641179
$208$	0	0
$209$	439280.	0.695626
$210$	0	0
$211$	−400972.	−0.620023	−0.310012	−	0.950733i	$-0.600333\pi$
$211$	−400972.	−0.620023	−0.310012	+	0.950733i	$0.600333\pi$
$212$	0	0
$213$	−75240.0	−0.113632
$214$	0	0
$215$	1.60590e6	2.36931
$216$	0	0
$217$	1.37664e6	1.98459
$218$	0	0
$219$	355194.	0.500444
$220$	0	0
$221$	−701668.	−0.966387
$222$	0	0
$223$	206168.	0.277625	0.138813	−	0.990319i	$-0.455671\pi$
$223$	206168.	0.277625	0.138813	+	0.990319i	$0.455671\pi$
$224$	0	0
$225$	462591.	0.609173
$226$	0	0
$227$	−363468.	−0.468168	−0.234084	−	0.972216i	$-0.575209\pi$
$227$	−363468.	−0.468168	−0.234084	+	0.972216i	$0.575209\pi$
$228$	0	0
$229$	−589274.	−0.742555	−0.371277	−	0.928522i	$-0.621080\pi$
$229$	−589274.	−0.742555	−0.371277	+	0.928522i	$0.621080\pi$
$230$	0	0
$231$	−492480.	−0.607238
$232$	0	0
$233$	730090.	0.881022	0.440511	−	0.897747i	$-0.354797\pi$
$233$	730090.	0.881022	0.440511	+	0.897747i	$0.354797\pi$
$234$	0	0
$235$	−297792.	−0.351758
$236$	0	0
$237$	−381384.	−0.441054
$238$	0	0
$239$	−472496.	−0.535061	−0.267531	−	0.963549i	$-0.586208\pi$
$239$	−472496.	−0.535061	−0.267531	+	0.963549i	$0.586208\pi$
$240$	0	0
$241$	993042.	1.10135	0.550675	−	0.834720i	$-0.314371\pi$
$241$	993042.	1.10135	0.550675	+	0.834720i	$0.314371\pi$
$242$	0	0
$243$	59049.0	0.0641500
$244$	0	0
$245$	369326.	0.393092
$246$	0	0
$247$	940984.	0.981386
$248$	0	0
$249$	556452.	0.568760
$250$	0	0
$251$	−1.93653e6	−1.94017	−0.970086	−	0.242760i	$-0.921947\pi$
$251$	−1.93653e6	−1.94017	−0.970086	+	0.242760i	$0.921947\pi$
$252$	0	0
$253$	185440.	0.182139
$254$	0	0
$255$	−729252.	−0.702307
$256$	0	0
$257$	735746.	0.694856	0.347428	−	0.937707i	$-0.387055\pi$
$257$	735746.	0.694856	0.347428	+	0.937707i	$0.387055\pi$
$258$	0	0
$259$	1.51286e6	1.40136
$260$	0	0
$261$	−442746.	−0.402303
$262$	0	0
$263$	542984.	0.484058	0.242029	−	0.970269i	$-0.422187\pi$
$263$	542984.	0.484058	0.242029	+	0.970269i	$0.422187\pi$
$264$	0	0
$265$	−2.32838e6	−2.03676
$266$	0	0
$267$	−567702.	−0.487351
$268$	0	0
$269$	438614.	0.369574	0.184787	−	0.982779i	$-0.440840\pi$
$269$	438614.	0.369574	0.184787	+	0.982779i	$0.440840\pi$
$270$	0	0
$271$	−644408.	−0.533013	−0.266506	−	0.963833i	$-0.585869\pi$
$271$	−644408.	−0.533013	−0.266506	+	0.963833i	$0.585869\pi$
$272$	0	0
$273$	−1.05494e6	−0.856688
$274$	0	0
$275$	2.17018e6	1.73047
$276$	0	0
$277$	−1.41953e6	−1.11159	−0.555796	−	0.831319i	$-0.687586\pi$
$277$	−1.41953e6	−1.11159	−0.555796	+	0.831319i	$0.687586\pi$
$278$	0	0
$279$	−774360.	−0.595569
$280$	0	0
$281$	645242.	0.487480	0.243740	−	0.969841i	$-0.421626\pi$
$281$	645242.	0.487480	0.243740	+	0.969841i	$0.421626\pi$
$282$	0	0
$283$	−1.93696e6	−1.43766	−0.718829	−	0.695187i	$-0.755321\pi$
$283$	−1.93696e6	−1.43766	−0.718829	+	0.695187i	$0.755321\pi$
$284$	0	0
$285$	977976.	0.713208
$286$	0	0
$287$	747360.	0.535581
$288$	0	0
$289$	−676813.	−0.476677
$290$	0	0
$291$	−146862.	−0.101666
$292$	0	0
$293$	−1.65429e6	−1.12575	−0.562876	−	0.826541i	$-0.690305\pi$
$293$	−1.65429e6	−1.12575	−0.562876	+	0.826541i	$0.690305\pi$
$294$	0	0
$295$	−1.63372e6	−1.09301
$296$	0	0
$297$	277020.	0.182230
$298$	0	0
$299$	397232.	0.256960
$300$	0	0
$301$	−2.46010e6	−1.56508
$302$	0	0
$303$	1.21289e6	0.758955
$304$	0	0
$305$	410404.	0.252617
$306$	0	0
$307$	2.78236e6	1.68487	0.842436	−	0.538797i	$-0.181121\pi$
$307$	2.78236e6	1.68487	0.842436	+	0.538797i	$0.181121\pi$
$308$	0	0
$309$	−406656.	−0.242287
$310$	0	0
$311$	−1.00510e6	−0.589259	−0.294630	−	0.955611i	$-0.595196\pi$
$311$	−1.00510e6	−0.589259	−0.294630	+	0.955611i	$0.595196\pi$
$312$	0	0
$313$	535386.	0.308892	0.154446	−	0.988001i	$-0.450641\pi$
$313$	535386.	0.308892	0.154446	+	0.988001i	$0.450641\pi$
$314$	0	0
$315$	−1.09642e6	−0.622585
$316$	0	0
$317$	2.96266e6	1.65590	0.827950	−	0.560802i	$-0.189507\pi$
$317$	2.96266e6	1.65590	0.827950	+	0.560802i	$0.189507\pi$
$318$	0	0
$319$	−2.07708e6	−1.14282
$320$	0	0
$321$	−1.28812e6	−0.697738
$322$	0	0
$323$	−996472.	−0.531446
$324$	0	0
$325$	4.64875e6	2.44134
$326$	0	0
$327$	1.37763e6	0.712464
$328$	0	0
$329$	456192.	0.232358
$330$	0	0
$331$	−1.55935e6	−0.782300	−0.391150	−	0.920327i	$-0.627923\pi$
$331$	−1.55935e6	−0.782300	−0.391150	+	0.920327i	$0.627923\pi$
$332$	0	0
$333$	−850986.	−0.420544
$334$	0	0
$335$	496696.	0.241812
$336$	0	0
$337$	3.61557e6	1.73421	0.867106	−	0.498124i	$-0.165978\pi$
$337$	3.61557e6	1.73421	0.867106	+	0.498124i	$0.165978\pi$
$338$	0	0
$339$	−580446.	−0.274323
$340$	0	0
$341$	−3.63280e6	−1.69183
$342$	0	0
$343$	1.85443e6	0.851090
$344$	0	0
$345$	412848.	0.186742
$346$	0	0
$347$	1.76488e6	0.786847	0.393424	−	0.919357i	$-0.371291\pi$
$347$	1.76488e6	0.786847	0.393424	+	0.919357i	$0.371291\pi$
$348$	0	0
$349$	553134.	0.243090	0.121545	−	0.992586i	$-0.461215\pi$
$349$	553134.	0.243090	0.121545	+	0.992586i	$0.461215\pi$
$350$	0	0
$351$	593406.	0.257089
$352$	0	0
$353$	−1.17408e6	−0.501488	−0.250744	−	0.968053i	$-0.580675\pi$
$353$	−1.17408e6	−0.501488	−0.250744	+	0.968053i	$0.580675\pi$
$354$	0	0
$355$	−785840.	−0.330951
$356$	0	0
$357$	1.11715e6	0.463919
$358$	0	0
$359$	1.18975e6	0.487215	0.243607	−	0.969874i	$-0.421669\pi$
$359$	1.18975e6	0.487215	0.243607	+	0.969874i	$0.421669\pi$
$360$	0	0
$361$	−1.13976e6	−0.460306
$362$	0	0
$363$	−149859.	−0.0596920
$364$	0	0
$365$	3.70980e6	1.45753
$366$	0	0
$367$	1.35327e6	0.524469	0.262235	−	0.965004i	$-0.415541\pi$
$367$	1.35327e6	0.524469	0.262235	+	0.965004i	$0.415541\pi$
$368$	0	0
$369$	−420390.	−0.160726
$370$	0	0
$371$	3.56688e6	1.34541
$372$	0	0
$373$	2.79306e6	1.03946	0.519731	−	0.854330i	$-0.326032\pi$
$373$	2.79306e6	1.03946	0.519731	+	0.854330i	$0.326032\pi$
$374$	0	0
$375$	2.18776e6	0.803379
$376$	0	0
$377$	−4.44932e6	−1.61228
$378$	0	0
$379$	3.37304e6	1.20621	0.603105	−	0.797662i	$-0.293930\pi$
$379$	3.37304e6	1.20621	0.603105	+	0.797662i	$0.293930\pi$
$380$	0	0
$381$	−299304.	−0.105633
$382$	0	0
$383$	−4.72781e6	−1.64688	−0.823442	−	0.567401i	$-0.807949\pi$
$383$	−4.72781e6	−1.64688	−0.823442	+	0.567401i	$0.807949\pi$
$384$	0	0
$385$	−5.14368e6	−1.76857
$386$	0	0
$387$	1.38380e6	0.469675
$388$	0	0
$389$	188254.	0.0630769	0.0315384	−	0.999503i	$-0.489959\pi$
$389$	188254.	0.0630769	0.0315384	+	0.999503i	$0.489959\pi$
$390$	0	0
$391$	−420656.	−0.139151
$392$	0	0
$393$	1.34356e6	0.438808
$394$	0	0
$395$	−3.98334e6	−1.28456
$396$	0	0
$397$	−524578.	−0.167045	−0.0835226	−	0.996506i	$-0.526617\pi$
$397$	−524578.	−0.167045	−0.0835226	+	0.996506i	$0.526617\pi$
$398$	0	0
$399$	−1.49818e6	−0.471119
$400$	0	0
$401$	−3.08653e6	−0.958537	−0.479269	−	0.877668i	$-0.659098\pi$
$401$	−3.08653e6	−0.958537	−0.479269	+	0.877668i	$0.659098\pi$
$402$	0	0
$403$	−7.78184e6	−2.38682
$404$	0	0
$405$	616734.	0.186836
$406$	0	0
$407$	−3.99228e6	−1.19463
$408$	0	0
$409$	−4.60196e6	−1.36030	−0.680150	−	0.733073i	$-0.738085\pi$
$409$	−4.60196e6	−1.36030	−0.680150	+	0.733073i	$0.738085\pi$
$410$	0	0
$411$	1.33742e6	0.390537
$412$	0	0
$413$	2.50272e6	0.722000
$414$	0	0
$415$	5.81183e6	1.65651
$416$	0	0
$417$	2.91611e6	0.821227
$418$	0	0
$419$	−2.09492e6	−0.582953	−0.291476	−	0.956578i	$-0.594146\pi$
$419$	−2.09492e6	−0.582953	−0.291476	+	0.956578i	$0.594146\pi$
$420$	0	0
$421$	987206.	0.271458	0.135729	−	0.990746i	$-0.456662\pi$
$421$	987206.	0.271458	0.135729	+	0.990746i	$0.456662\pi$
$422$	0	0
$423$	−256608.	−0.0697299
$424$	0	0
$425$	−4.92288e6	−1.32205
$426$	0	0
$427$	−628704.	−0.166869
$428$	0	0
$429$	2.78388e6	0.730310
$430$	0	0
$431$	−3.25933e6	−0.845152	−0.422576	−	0.906327i	$-0.638874\pi$
$431$	−3.25933e6	−0.845152	−0.422576	+	0.906327i	$0.638874\pi$
$432$	0	0
$433$	7.31461e6	1.87487	0.937436	−	0.348159i	$-0.113193\pi$
$433$	7.31461e6	1.87487	0.937436	+	0.348159i	$0.113193\pi$
$434$	0	0
$435$	−4.62424e6	−1.17170
$436$	0	0
$437$	564128.	0.141310
$438$	0	0
$439$	7.31419e6	1.81136	0.905681	−	0.423961i	$-0.139361\pi$
$439$	7.31419e6	1.81136	0.905681	+	0.423961i	$0.139361\pi$
$440$	0	0
$441$	318249.	0.0779239
$442$	0	0
$443$	7.07403e6	1.71261	0.856303	−	0.516474i	$-0.172756\pi$
$443$	7.07403e6	1.71261	0.856303	+	0.516474i	$0.172756\pi$
$444$	0	0
$445$	−5.92933e6	−1.41940
$446$	0	0
$447$	−539874.	−0.127798
$448$	0	0
$449$	7.48079e6	1.75118	0.875591	−	0.483053i	$-0.160472\pi$
$449$	7.48079e6	1.75118	0.875591	+	0.483053i	$0.160472\pi$
$450$	0	0
$451$	−1.97220e6	−0.456572
$452$	0	0
$453$	−823680.	−0.188588
$454$	0	0
$455$	−1.10183e7	−2.49509
$456$	0	0
$457$	2.97108e6	0.665463	0.332732	−	0.943022i	$-0.392030\pi$
$457$	2.97108e6	0.665463	0.332732	+	0.943022i	$0.392030\pi$
$458$	0	0
$459$	−628398.	−0.139220
$460$	0	0
$461$	−8.29078e6	−1.81695	−0.908475	−	0.417939i	$-0.862753\pi$
$461$	−8.29078e6	−1.81695	−0.908475	+	0.417939i	$0.862753\pi$
$462$	0	0
$463$	−955064.	−0.207052	−0.103526	−	0.994627i	$-0.533013\pi$
$463$	−955064.	−0.207052	−0.103526	+	0.994627i	$0.533013\pi$
$464$	0	0
$465$	−8.08776e6	−1.73459
$466$	0	0
$467$	8.19018e6	1.73781	0.868903	−	0.494983i	$-0.164826\pi$
$467$	8.19018e6	1.73781	0.868903	+	0.494983i	$0.164826\pi$
$468$	0	0
$469$	−760896.	−0.159732
$470$	0	0
$471$	3.10995e6	0.645953
$472$	0	0
$473$	6.49192e6	1.33420
$474$	0	0
$475$	6.60192e6	1.34257
$476$	0	0
$477$	−2.00637e6	−0.403752
$478$	0	0
$479$	3.39774e6	0.676631	0.338315	−	0.941033i	$-0.390143\pi$
$479$	3.39774e6	0.676631	0.338315	+	0.941033i	$0.390143\pi$
$480$	0	0
$481$	−8.55188e6	−1.68538
$482$	0	0
$483$	−632448.	−0.123355
$484$	0	0
$485$	−1.53389e6	−0.296101
$486$	0	0
$487$	−1.94814e6	−0.372219	−0.186110	−	0.982529i	$-0.559588\pi$
$487$	−1.94814e6	−0.372219	−0.186110	+	0.982529i	$0.559588\pi$
$488$	0	0
$489$	5.70748e6	1.07937
$490$	0	0
$491$	3.92574e6	0.734882	0.367441	−	0.930047i	$-0.380234\pi$
$491$	3.92574e6	0.734882	0.367441	+	0.930047i	$0.380234\pi$
$492$	0	0
$493$	4.71169e6	0.873091
$494$	0	0
$495$	2.89332e6	0.530742
$496$	0	0
$497$	1.20384e6	0.218614
$498$	0	0
$499$	−2.07102e6	−0.372334	−0.186167	−	0.982518i	$-0.559607\pi$
$499$	−2.07102e6	−0.372334	−0.186167	+	0.982518i	$0.559607\pi$
$500$	0	0
$501$	3.74263e6	0.666167
$502$	0	0
$503$	9.16500e6	1.61515	0.807574	−	0.589766i	$-0.200780\pi$
$503$	9.16500e6	1.61515	0.807574	+	0.589766i	$0.200780\pi$
$504$	0	0
$505$	1.26680e7	2.21045
$506$	0	0
$507$	2.62173e6	0.452968
$508$	0	0
$509$	6.91517e6	1.18307	0.591533	−	0.806281i	$-0.298523\pi$
$509$	6.91517e6	1.18307	0.591533	+	0.806281i	$0.298523\pi$
$510$	0	0
$511$	−5.68310e6	−0.962794
$512$	0	0
$513$	842724.	0.141381
$514$	0	0
$515$	−4.24730e6	−0.705658
$516$	0	0
$517$	−1.20384e6	−0.198081
$518$	0	0
$519$	−2.91854e6	−0.475606
$520$	0	0
$521$	−2.26077e6	−0.364891	−0.182445	−	0.983216i	$-0.558401\pi$
$521$	−2.26077e6	−0.364891	−0.182445	+	0.983216i	$0.558401\pi$
$522$	0	0
$523$	−8.25524e6	−1.31970	−0.659850	−	0.751397i	$-0.729380\pi$
$523$	−8.25524e6	−1.31970	−0.659850	+	0.751397i	$0.729380\pi$
$524$	0	0
$525$	−7.40146e6	−1.17198
$526$	0	0
$527$	8.24072e6	1.29252
$528$	0	0
$529$	−6.19820e6	−0.963000
$530$	0	0
$531$	−1.40778e6	−0.216670
$532$	0	0
$533$	−4.22466e6	−0.644130
$534$	0	0
$535$	−1.34537e7	−2.03215
$536$	0	0
$537$	−5.46145e6	−0.817283
$538$	0	0
$539$	1.49302e6	0.221357
$540$	0	0
$541$	1.90467e6	0.279786	0.139893	−	0.990167i	$-0.455324\pi$
$541$	1.90467e6	0.279786	0.139893	+	0.990167i	$0.455324\pi$
$542$	0	0
$543$	4.16929e6	0.606823
$544$	0	0
$545$	1.43886e7	2.07504
$546$	0	0
$547$	−1.57060e6	−0.224439	−0.112220	−	0.993683i	$-0.535796\pi$
$547$	−1.57060e6	−0.224439	−0.112220	+	0.993683i	$0.535796\pi$
$548$	0	0
$549$	353646.	0.0500770
$550$	0	0
$551$	−6.31870e6	−0.886642
$552$	0	0
$553$	6.10214e6	0.848535
$554$	0	0
$555$	−8.88808e6	−1.22483
$556$	0	0
$557$	2.36849e6	0.323469	0.161735	−	0.986834i	$-0.448291\pi$
$557$	2.36849e6	0.323469	0.161735	+	0.986834i	$0.448291\pi$
$558$	0	0
$559$	1.39064e7	1.88228
$560$	0	0
$561$	−2.94804e6	−0.395482
$562$	0	0
$563$	−6.64976e6	−0.884168	−0.442084	−	0.896974i	$-0.645761\pi$
$563$	−6.64976e6	−0.884168	−0.442084	+	0.896974i	$0.645761\pi$
$564$	0	0
$565$	−6.06244e6	−0.798962
$566$	0	0
$567$	−944784.	−0.123417
$568$	0	0
$569$	−8.94565e6	−1.15833	−0.579164	−	0.815211i	$-0.696621\pi$
$569$	−8.94565e6	−1.15833	−0.579164	+	0.815211i	$0.696621\pi$
$570$	0	0
$571$	4.19526e6	0.538479	0.269239	−	0.963073i	$-0.413228\pi$
$571$	4.19526e6	0.538479	0.269239	+	0.963073i	$0.413228\pi$
$572$	0	0
$573$	−8.59507e6	−1.09361
$574$	0	0
$575$	2.78697e6	0.351530
$576$	0	0
$577$	−6.24845e6	−0.781326	−0.390663	−	0.920534i	$-0.627754\pi$
$577$	−6.24845e6	−0.781326	−0.390663	+	0.920534i	$0.627754\pi$
$578$	0	0
$579$	−2.30513e6	−0.285759
$580$	0	0
$581$	−8.90323e6	−1.09423
$582$	0	0
$583$	−9.41260e6	−1.14693
$584$	0	0
$585$	6.19780e6	0.748769
$586$	0	0
$587$	4.55284e6	0.545365	0.272683	−	0.962104i	$-0.412089\pi$
$587$	4.55284e6	0.545365	0.272683	+	0.962104i	$0.412089\pi$
$588$	0	0
$589$	−1.10514e7	−1.31259
$590$	0	0
$591$	1.00035e6	0.117810
$592$	0	0
$593$	210802.	0.0246172	0.0123086	−	0.999924i	$-0.496082\pi$
$593$	210802.	0.0246172	0.0123086	+	0.999924i	$0.496082\pi$
$594$	0	0
$595$	1.16680e7	1.35116
$596$	0	0
$597$	4.65120e6	0.534108
$598$	0	0
$599$	−3.05945e6	−0.348398	−0.174199	−	0.984710i	$-0.555734\pi$
$599$	−3.05945e6	−0.348398	−0.174199	+	0.984710i	$0.555734\pi$
$600$	0	0
$601$	7.37094e6	0.832409	0.416204	−	0.909271i	$-0.363360\pi$
$601$	7.37094e6	0.832409	0.416204	+	0.909271i	$0.363360\pi$
$602$	0	0
$603$	428004.	0.0479352
$604$	0	0
$605$	−1.56519e6	−0.173852
$606$	0	0
$607$	−2.12257e6	−0.233824	−0.116912	−	0.993142i	$-0.537300\pi$
$607$	−2.12257e6	−0.233824	−0.116912	+	0.993142i	$0.537300\pi$
$608$	0	0
$609$	7.08394e6	0.773983
$610$	0	0
$611$	−2.57875e6	−0.279452
$612$	0	0
$613$	−1.09689e7	−1.17899	−0.589496	−	0.807771i	$-0.700674\pi$
$613$	−1.09689e7	−1.17899	−0.589496	+	0.807771i	$0.700674\pi$
$614$	0	0
$615$	−4.39074e6	−0.468112
$616$	0	0
$617$	−7.28953e6	−0.770879	−0.385440	−	0.922733i	$-0.625950\pi$
$617$	−7.28953e6	−0.770879	−0.385440	+	0.922733i	$0.625950\pi$
$618$	0	0
$619$	2.23675e6	0.234634	0.117317	−	0.993095i	$-0.462571\pi$
$619$	2.23675e6	0.234634	0.117317	+	0.993095i	$0.462571\pi$
$620$	0	0
$621$	355752.	0.0370185
$622$	0	0
$623$	9.08323e6	0.937606
$624$	0	0
$625$	5.00302e6	0.512309
$626$	0	0
$627$	3.95352e6	0.401620
$628$	0	0
$629$	9.05617e6	0.912679
$630$	0	0
$631$	7.06160e6	0.706041	0.353020	−	0.935616i	$-0.385155\pi$
$631$	7.06160e6	0.706041	0.353020	+	0.935616i	$0.385155\pi$
$632$	0	0
$633$	−3.60875e6	−0.357971
$634$	0	0
$635$	−3.12606e6	−0.307655
$636$	0	0
$637$	3.19821e6	0.312290
$638$	0	0
$639$	−677160.	−0.0656053
$640$	0	0
$641$	1.80189e7	1.73214	0.866070	−	0.499922i	$-0.166638\pi$
$641$	1.80189e7	1.73214	0.866070	+	0.499922i	$0.166638\pi$
$642$	0	0
$643$	−1.07252e6	−0.102301	−0.0511505	−	0.998691i	$-0.516289\pi$
$643$	−1.07252e6	−0.102301	−0.0511505	+	0.998691i	$0.516289\pi$
$644$	0	0
$645$	1.44531e7	1.36792
$646$	0	0
$647$	−2.03054e7	−1.90701	−0.953503	−	0.301385i	$-0.902551\pi$
$647$	−2.03054e7	−1.90701	−0.953503	+	0.301385i	$0.902551\pi$
$648$	0	0
$649$	−6.60440e6	−0.615491
$650$	0	0
$651$	1.23898e7	1.14580
$652$	0	0
$653$	1.97292e7	1.81061	0.905306	−	0.424759i	$-0.139641\pi$
$653$	1.97292e7	1.81061	0.905306	+	0.424759i	$0.139641\pi$
$654$	0	0
$655$	1.40327e7	1.27802
$656$	0	0
$657$	3.19675e6	0.288931
$658$	0	0
$659$	−1.29511e7	−1.16170	−0.580848	−	0.814012i	$-0.697279\pi$
$659$	−1.29511e7	−1.16170	−0.580848	+	0.814012i	$0.697279\pi$
$660$	0	0
$661$	2.01888e7	1.79725	0.898623	−	0.438721i	$-0.144568\pi$
$661$	2.01888e7	1.79725	0.898623	+	0.438721i	$0.144568\pi$
$662$	0	0
$663$	−6.31501e6	−0.557944
$664$	0	0
$665$	−1.56476e7	−1.37213
$666$	0	0
$667$	−2.66741e6	−0.232153
$668$	0	0
$669$	1.85551e6	0.160287
$670$	0	0
$671$	1.65908e6	0.142253
$672$	0	0
$673$	−1.83989e7	−1.56587	−0.782934	−	0.622105i	$-0.786278\pi$
$673$	−1.83989e7	−1.56587	−0.782934	+	0.622105i	$0.786278\pi$
$674$	0	0
$675$	4.16332e6	0.351706
$676$	0	0
$677$	−4.72791e6	−0.396458	−0.198229	−	0.980156i	$-0.563519\pi$
$677$	−4.72791e6	−0.396458	−0.198229	+	0.980156i	$0.563519\pi$
$678$	0	0
$679$	2.34979e6	0.195594
$680$	0	0
$681$	−3.27121e6	−0.270297
$682$	0	0
$683$	1.38084e7	1.13264	0.566321	−	0.824184i	$-0.308366\pi$
$683$	1.38084e7	1.13264	0.566321	+	0.824184i	$0.308366\pi$
$684$	0	0
$685$	1.39686e7	1.13743
$686$	0	0
$687$	−5.30347e6	−0.428714
$688$	0	0
$689$	−2.01628e7	−1.61809
$690$	0	0
$691$	−1.28964e7	−1.02748	−0.513742	−	0.857945i	$-0.671741\pi$
$691$	−1.28964e7	−1.02748	−0.513742	+	0.857945i	$0.671741\pi$
$692$	0	0
$693$	−4.43232e6	−0.350589
$694$	0	0
$695$	3.04571e7	2.39181
$696$	0	0
$697$	4.47378e6	0.348813
$698$	0	0
$699$	6.57081e6	0.508658
$700$	0	0
$701$	−8.81078e6	−0.677204	−0.338602	−	0.940930i	$-0.609954\pi$
$701$	−8.81078e6	−0.677204	−0.338602	+	0.940930i	$0.609954\pi$
$702$	0	0
$703$	−1.21449e7	−0.926845
$704$	0	0
$705$	−2.68013e6	−0.203087
$706$	0	0
$707$	−1.94063e7	−1.46014
$708$	0	0
$709$	1.83369e7	1.36997	0.684984	−	0.728558i	$-0.259809\pi$
$709$	1.83369e7	1.36997	0.684984	+	0.728558i	$0.259809\pi$
$710$	0	0
$711$	−3.43246e6	−0.254643
$712$	0	0
$713$	−4.66528e6	−0.343680
$714$	0	0
$715$	2.90761e7	2.12702
$716$	0	0
$717$	−4.25246e6	−0.308918
$718$	0	0
$719$	−2.26238e7	−1.63209	−0.816044	−	0.577990i	$-0.803837\pi$
$719$	−2.26238e7	−1.63209	−0.816044	+	0.577990i	$0.803837\pi$
$720$	0	0
$721$	6.50650e6	0.466132
$722$	0	0
$723$	8.93738e6	0.635864
$724$	0	0
$725$	−3.12163e7	−2.20565
$726$	0	0
$727$	1.25658e7	0.881769	0.440885	−	0.897564i	$-0.354665\pi$
$727$	1.25658e7	0.881769	0.440885	+	0.897564i	$0.354665\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−1.47264e7	−1.01930
$732$	0	0
$733$	833534.	0.0573012	0.0286506	−	0.999589i	$-0.490879\pi$
$733$	833534.	0.0573012	0.0286506	+	0.999589i	$0.490879\pi$
$734$	0	0
$735$	3.32393e6	0.226952
$736$	0	0
$737$	2.00792e6	0.136169
$738$	0	0
$739$	−8.84188e6	−0.595571	−0.297786	−	0.954633i	$-0.596248\pi$
$739$	−8.84188e6	−0.595571	−0.297786	+	0.954633i	$0.596248\pi$
$740$	0	0
$741$	8.46886e6	0.566603
$742$	0	0
$743$	2.62140e7	1.74205	0.871025	−	0.491239i	$-0.163456\pi$
$743$	2.62140e7	1.74205	0.871025	+	0.491239i	$0.163456\pi$
$744$	0	0
$745$	−5.63868e6	−0.372209
$746$	0	0
$747$	5.00807e6	0.328374
$748$	0	0
$749$	2.06099e7	1.34236
$750$	0	0
$751$	−6.17143e6	−0.399288	−0.199644	−	0.979869i	$-0.563979\pi$
$751$	−6.17143e6	−0.399288	−0.199644	+	0.979869i	$0.563979\pi$
$752$	0	0
$753$	−1.74288e7	−1.12016
$754$	0	0
$755$	−8.60288e6	−0.549258
$756$	0	0
$757$	−6.70734e6	−0.425413	−0.212706	−	0.977116i	$-0.568228\pi$
$757$	−6.70734e6	−0.425413	−0.212706	+	0.977116i	$0.568228\pi$
$758$	0	0
$759$	1.66896e6	0.105158
$760$	0	0
$761$	−1.08114e7	−0.676734	−0.338367	−	0.941014i	$-0.609875\pi$
$761$	−1.08114e7	−0.676734	−0.338367	+	0.941014i	$0.609875\pi$
$762$	0	0
$763$	−2.20421e7	−1.37070
$764$	0	0
$765$	−6.56327e6	−0.405477
$766$	0	0
$767$	−1.41473e7	−0.868332
$768$	0	0
$769$	−1.33048e7	−0.811322	−0.405661	−	0.914024i	$-0.632959\pi$
$769$	−1.33048e7	−0.811322	−0.405661	+	0.914024i	$0.632959\pi$
$770$	0	0
$771$	6.62171e6	0.401176
$772$	0	0
$773$	−1.29053e7	−0.776818	−0.388409	−	0.921487i	$-0.626975\pi$
$773$	−1.29053e7	−0.776818	−0.388409	+	0.921487i	$0.626975\pi$
$774$	0	0
$775$	−5.45972e7	−3.26525
$776$	0	0
$777$	1.36158e7	0.809077
$778$	0	0
$779$	−5.99964e6	−0.354227
$780$	0	0
$781$	−3.17680e6	−0.186364
$782$	0	0
$783$	−3.98471e6	−0.232270
$784$	0	0
$785$	3.24817e7	1.88133
$786$	0	0
$787$	−6.77705e6	−0.390035	−0.195018	−	0.980800i	$-0.562476\pi$
$787$	−6.77705e6	−0.390035	−0.195018	+	0.980800i	$0.562476\pi$
$788$	0	0
$789$	4.88686e6	0.279471
$790$	0	0
$791$	9.28714e6	0.527765
$792$	0	0
$793$	3.55392e6	0.200690
$794$	0	0
$795$	−2.09554e7	−1.17592
$796$	0	0
$797$	1.49551e7	0.833956	0.416978	−	0.908916i	$-0.363089\pi$
$797$	1.49551e7	0.833956	0.416978	+	0.908916i	$0.363089\pi$
$798$	0	0
$799$	2.73082e6	0.151330
$800$	0	0
$801$	−5.10932e6	−0.281372
$802$	0	0
$803$	1.49971e7	0.820763
$804$	0	0
$805$	−6.60557e6	−0.359270
$806$	0	0
$807$	3.94753e6	0.213374
$808$	0	0
$809$	−8.65841e6	−0.465122	−0.232561	−	0.972582i	$-0.574710\pi$
$809$	−8.65841e6	−0.465122	−0.232561	+	0.972582i	$0.574710\pi$
$810$	0	0
$811$	1.98323e7	1.05881	0.529407	−	0.848368i	$-0.322414\pi$
$811$	1.98323e7	1.05881	0.529407	+	0.848368i	$0.322414\pi$
$812$	0	0
$813$	−5.79967e6	−0.307735
$814$	0	0
$815$	5.96114e7	3.14366
$816$	0	0
$817$	1.97491e7	1.03512
$818$	0	0
$819$	−9.49450e6	−0.494609
$820$	0	0
$821$	3.02228e7	1.56486	0.782431	−	0.622737i	$-0.213979\pi$
$821$	3.02228e7	1.56486	0.782431	+	0.622737i	$0.213979\pi$
$822$	0	0
$823$	2.64692e7	1.36220	0.681100	−	0.732190i	$-0.261502\pi$
$823$	2.64692e7	1.36220	0.681100	+	0.732190i	$0.261502\pi$
$824$	0	0
$825$	1.95316e7	0.999087
$826$	0	0
$827$	−8.81882e6	−0.448380	−0.224190	−	0.974545i	$-0.571974\pi$
$827$	−8.81882e6	−0.448380	−0.224190	+	0.974545i	$0.571974\pi$
$828$	0	0
$829$	−2.31053e7	−1.16768	−0.583841	−	0.811868i	$-0.698451\pi$
$829$	−2.31053e7	−1.16768	−0.583841	+	0.811868i	$0.698451\pi$
$830$	0	0
$831$	−1.27758e7	−0.641778
$832$	0	0
$833$	−3.38680e6	−0.169113
$834$	0	0
$835$	3.90897e7	1.94020
$836$	0	0
$837$	−6.96924e6	−0.343852
$838$	0	0
$839$	−2.65084e7	−1.30010	−0.650052	−	0.759890i	$-0.725253\pi$
$839$	−2.65084e7	−1.30010	−0.650052	+	0.759890i	$0.725253\pi$
$840$	0	0
$841$	9.36601e6	0.456630
$842$	0	0
$843$	5.80718e6	0.281447
$844$	0	0
$845$	2.73825e7	1.31926
$846$	0	0
$847$	2.39774e6	0.114840
$848$	0	0
$849$	−1.74327e7	−0.830032
$850$	0	0
$851$	−5.12693e6	−0.242680
$852$	0	0
$853$	−7.84201e6	−0.369024	−0.184512	−	0.982830i	$-0.559070\pi$
$853$	−7.84201e6	−0.369024	−0.184512	+	0.982830i	$0.559070\pi$
$854$	0	0
$855$	8.80178e6	0.411771
$856$	0	0
$857$	−2.26563e7	−1.05375	−0.526875	−	0.849943i	$-0.676636\pi$
$857$	−2.26563e7	−1.05375	−0.526875	+	0.849943i	$0.676636\pi$
$858$	0	0
$859$	2.67391e7	1.23641	0.618207	−	0.786016i	$-0.287860\pi$
$859$	2.67391e7	1.23641	0.618207	+	0.786016i	$0.287860\pi$
$860$	0	0
$861$	6.72624e6	0.309218
$862$	0	0
$863$	1.29957e7	0.593983	0.296992	−	0.954880i	$-0.404017\pi$
$863$	1.29957e7	0.593983	0.296992	+	0.954880i	$0.404017\pi$
$864$	0	0
$865$	−3.04825e7	−1.38519
$866$	0	0
$867$	−6.09132e6	−0.275210
$868$	0	0
$869$	−1.61029e7	−0.723359
$870$	0	0
$871$	4.30118e6	0.192106
$872$	0	0
$873$	−1.32176e6	−0.0586971
$874$	0	0
$875$	−3.50041e7	−1.54561
$876$	0	0
$877$	−3.16845e6	−0.139107	−0.0695533	−	0.997578i	$-0.522157\pi$
$877$	−3.16845e6	−0.139107	−0.0695533	+	0.997578i	$0.522157\pi$
$878$	0	0
$879$	−1.48886e7	−0.649953
$880$	0	0
$881$	3.85811e7	1.67469	0.837346	−	0.546673i	$-0.184106\pi$
$881$	3.85811e7	1.67469	0.837346	+	0.546673i	$0.184106\pi$
$882$	0	0
$883$	2.31224e7	0.998000	0.499000	−	0.866602i	$-0.333701\pi$
$883$	2.31224e7	0.998000	0.499000	+	0.866602i	$0.333701\pi$
$884$	0	0
$885$	−1.47035e7	−0.631047
$886$	0	0
$887$	2.62602e7	1.12070	0.560350	−	0.828256i	$-0.310667\pi$
$887$	2.62602e7	1.12070	0.560350	+	0.828256i	$0.310667\pi$
$888$	0	0
$889$	4.78886e6	0.203225
$890$	0	0
$891$	2.49318e6	0.105211
$892$	0	0
$893$	−3.66221e6	−0.153679
$894$	0	0
$895$	−5.70418e7	−2.38032
$896$	0	0
$897$	3.57509e6	0.148356
$898$	0	0
$899$	5.22550e7	2.15639
$900$	0	0
$901$	2.13517e7	0.876236
$902$	0	0
$903$	−2.21409e7	−0.903598
$904$	0	0
$905$	4.35459e7	1.76736
$906$	0	0
$907$	−3.13213e7	−1.26422	−0.632109	−	0.774879i	$-0.717811\pi$
$907$	−3.13213e7	−1.26422	−0.632109	+	0.774879i	$0.717811\pi$
$908$	0	0
$909$	1.09160e7	0.438183
$910$	0	0
$911$	−4.49877e6	−0.179596	−0.0897982	−	0.995960i	$-0.528622\pi$
$911$	−4.49877e6	−0.179596	−0.0897982	+	0.995960i	$0.528622\pi$
$912$	0	0
$913$	2.34946e7	0.932807
$914$	0	0
$915$	3.69364e6	0.145848
$916$	0	0
$917$	−2.14969e7	−0.844214
$918$	0	0
$919$	3.03824e6	0.118668	0.0593340	−	0.998238i	$-0.481102\pi$
$919$	3.03824e6	0.118668	0.0593340	+	0.998238i	$0.481102\pi$
$920$	0	0
$921$	2.50412e7	0.972761
$922$	0	0
$923$	−6.80504e6	−0.262922
$924$	0	0
$925$	−5.99998e7	−2.30566
$926$	0	0
$927$	−3.65990e6	−0.139885
$928$	0	0
$929$	−193966.	−0.00737371	−0.00368686	−	0.999993i	$-0.501174\pi$
$929$	−193966.	−0.00737371	−0.00368686	+	0.999993i	$0.501174\pi$
$930$	0	0
$931$	4.54192e6	0.171738
$932$	0	0
$933$	−9.04586e6	−0.340209
$934$	0	0
$935$	−3.07906e7	−1.15183
$936$	0	0
$937$	−3.23213e7	−1.20265	−0.601326	−	0.799003i	$-0.705361\pi$
$937$	−3.23213e7	−1.20265	−0.601326	+	0.799003i	$0.705361\pi$
$938$	0	0
$939$	4.81847e6	0.178339
$940$	0	0
$941$	−4.50288e7	−1.65774	−0.828870	−	0.559442i	$-0.811016\pi$
$941$	−4.50288e7	−1.65774	−0.828870	+	0.559442i	$0.811016\pi$
$942$	0	0
$943$	−2.53272e6	−0.0927487
$944$	0	0
$945$	−9.86774e6	−0.359450
$946$	0	0
$947$	−1.04495e7	−0.378635	−0.189318	−	0.981916i	$-0.560628\pi$
$947$	−1.04495e7	−0.378635	−0.189318	+	0.981916i	$0.560628\pi$
$948$	0	0
$949$	3.21253e7	1.15793
$950$	0	0
$951$	2.66640e7	0.956034
$952$	0	0
$953$	4.31343e7	1.53847	0.769237	−	0.638963i	$-0.220636\pi$
$953$	4.31343e7	1.53847	0.769237	+	0.638963i	$0.220636\pi$
$954$	0	0
$955$	−8.97708e7	−3.18512
$956$	0	0
$957$	−1.86937e7	−0.659805
$958$	0	0
$959$	−2.13987e7	−0.751347
$960$	0	0
$961$	6.27644e7	2.19233
$962$	0	0
$963$	−1.15930e7	−0.402839
$964$	0	0
$965$	−2.40758e7	−0.832268
$966$	0	0
$967$	1.43982e7	0.495157	0.247579	−	0.968868i	$-0.420365\pi$
$967$	1.43982e7	0.495157	0.247579	+	0.968868i	$0.420365\pi$
$968$	0	0
$969$	−8.96825e6	−0.306830
$970$	0	0
$971$	−3.00272e6	−0.102204	−0.0511019	−	0.998693i	$-0.516273\pi$
$971$	−3.00272e6	−0.102204	−0.0511019	+	0.998693i	$0.516273\pi$
$972$	0	0
$973$	−4.66577e7	−1.57994
$974$	0	0
$975$	4.18388e7	1.40951
$976$	0	0
$977$	−3.75164e7	−1.25743	−0.628716	−	0.777635i	$-0.716419\pi$
$977$	−3.75164e7	−1.25743	−0.628716	+	0.777635i	$0.716419\pi$
$978$	0	0
$979$	−2.39696e7	−0.799291
$980$	0	0
$981$	1.23987e7	0.411342
$982$	0	0
$983$	2.57244e6	0.0849105	0.0424553	−	0.999098i	$-0.486482\pi$
$983$	2.57244e6	0.0849105	0.0424553	+	0.999098i	$0.486482\pi$
$984$	0	0
$985$	1.04481e7	0.343121
$986$	0	0
$987$	4.10573e6	0.134152
$988$	0	0
$989$	8.33699e6	0.271031
$990$	0	0
$991$	5.41887e7	1.75277	0.876385	−	0.481611i	$-0.159948\pi$
$991$	5.41887e7	1.75277	0.876385	+	0.481611i	$0.159948\pi$
$992$	0	0
$993$	−1.40341e7	−0.451661
$994$	0	0
$995$	4.85792e7	1.55558
$996$	0	0
$997$	3.52674e7	1.12366	0.561830	−	0.827253i	$-0.310097\pi$
$997$	3.52674e7	1.12366	0.561830	+	0.827253i	$0.310097\pi$
$998$	0	0
$999$	−7.65887e6	−0.242801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.6.a.e.1.1		1
3.2	odd	2		144.6.a.b.1.1		1
4.3	odd	2		24.6.a.b.1.1	✓	1
8.3	odd	2		192.6.a.i.1.1		1
8.5	even	2		192.6.a.a.1.1		1
12.11	even	2		72.6.a.a.1.1		1
16.3	odd	4		768.6.d.d.385.1		2
16.5	even	4		768.6.d.o.385.1		2
16.11	odd	4		768.6.d.d.385.2		2
16.13	even	4		768.6.d.o.385.2		2
20.3	even	4		600.6.f.b.49.1		2
20.7	even	4		600.6.f.b.49.2		2
20.19	odd	2		600.6.a.d.1.1		1
24.5	odd	2		576.6.a.bf.1.1		1
24.11	even	2		576.6.a.bg.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.6.a.b.1.1	✓	1	4.3	odd	2
48.6.a.e.1.1		1	1.1	even	1	trivial
72.6.a.a.1.1		1	12.11	even	2
144.6.a.b.1.1		1	3.2	odd	2
192.6.a.a.1.1		1	8.5	even	2
192.6.a.i.1.1		1	8.3	odd	2
576.6.a.bf.1.1		1	24.5	odd	2
576.6.a.bg.1.1		1	24.11	even	2
600.6.a.d.1.1		1	20.19	odd	2
600.6.f.b.49.1		2	20.3	even	4
600.6.f.b.49.2		2	20.7	even	4
768.6.d.d.385.1		2	16.3	odd	4
768.6.d.d.385.2		2	16.11	odd	4
768.6.d.o.385.1		2	16.5	even	4
768.6.d.o.385.2		2	16.13	even	4

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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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