LMFDB - Embedded newform 208.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,6,Mod(1,208)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(208, 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("208.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	208.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:	$33.3598345211$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	208.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-14.0000 q^{5} +170.000 q^{7} -243.000 q^{9} +O(q^{10})$

$q-14.0000 q^{5} +170.000 q^{7} -243.000 q^{9} +250.000 q^{11} -169.000 q^{13} +1062.00 q^{17} +78.0000 q^{19} -1576.00 q^{23} -2929.00 q^{25} +2578.00 q^{29} +8654.00 q^{31} -2380.00 q^{35} +10986.0 q^{37} +1050.00 q^{41} +5900.00 q^{43} +3402.00 q^{45} +5962.00 q^{47} +12093.0 q^{49} +29046.0 q^{53} -3500.00 q^{55} +13922.0 q^{59} -32882.0 q^{61} -41310.0 q^{63} +2366.00 q^{65} +69566.0 q^{67} +50542.0 q^{71} -46750.0 q^{73} +42500.0 q^{77} +19348.0 q^{79} +59049.0 q^{81} +87438.0 q^{83} -14868.0 q^{85} +94170.0 q^{89} -28730.0 q^{91} -1092.00 q^{95} +182786. q^{97} -60750.0 q^{99} +O(q^{100})$

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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−14.0000	−0.250440	−0.125220	−	0.992129i	$-0.539964\pi$
$5$	−14.0000	−0.250440	−0.125220	+	0.992129i	$0.539964\pi$
$6$	0	0
$7$	170.000	1.31131	0.655653	−	0.755063i	$-0.272394\pi$
$7$	170.000	1.31131	0.655653	+	0.755063i	$0.272394\pi$
$8$	0	0
$9$	−243.000	−1.00000
$10$	0	0
$11$	250.000	0.622957	0.311479	−	0.950253i	$-0.399176\pi$
$11$	250.000	0.622957	0.311479	+	0.950253i	$0.399176\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1062.00	0.891255	0.445628	−	0.895218i	$-0.352981\pi$
$17$	1062.00	0.891255	0.445628	+	0.895218i	$0.352981\pi$
$18$	0	0
$19$	78.0000	0.0495691	0.0247845	−	0.999693i	$-0.492110\pi$
$19$	78.0000	0.0495691	0.0247845	+	0.999693i	$0.492110\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1576.00	−0.621207	−0.310604	−	0.950539i	$-0.600531\pi$
$23$	−1576.00	−0.621207	−0.310604	+	0.950539i	$0.600531\pi$
$24$	0	0
$25$	−2929.00	−0.937280
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2578.00	0.569230	0.284615	−	0.958642i	$-0.408134\pi$
$29$	2578.00	0.569230	0.284615	+	0.958642i	$0.408134\pi$
$30$	0	0
$31$	8654.00	1.61738	0.808691	−	0.588234i	$-0.200176\pi$
$31$	8654.00	1.61738	0.808691	+	0.588234i	$0.200176\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2380.00	−0.328403
$36$	0	0
$37$	10986.0	1.31927	0.659637	−	0.751584i	$-0.270710\pi$
$37$	10986.0	1.31927	0.659637	+	0.751584i	$0.270710\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1050.00	0.0975505	0.0487753	−	0.998810i	$-0.484468\pi$
$41$	1050.00	0.0975505	0.0487753	+	0.998810i	$0.484468\pi$
$42$	0	0
$43$	5900.00	0.486610	0.243305	−	0.969950i	$-0.421768\pi$
$43$	5900.00	0.486610	0.243305	+	0.969950i	$0.421768\pi$
$44$	0	0
$45$	3402.00	0.250440
$46$	0	0
$47$	5962.00	0.393684	0.196842	−	0.980435i	$-0.436931\pi$
$47$	5962.00	0.393684	0.196842	+	0.980435i	$0.436931\pi$
$48$	0	0
$49$	12093.0	0.719522
$50$	0	0
$51$	0	0
$52$	0	0
$53$	29046.0	1.42035	0.710177	−	0.704023i	$-0.248615\pi$
$53$	29046.0	1.42035	0.710177	+	0.704023i	$0.248615\pi$
$54$	0	0
$55$	−3500.00	−0.156013
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13922.0	0.520681	0.260340	−	0.965517i	$-0.416165\pi$
$59$	13922.0	0.520681	0.260340	+	0.965517i	$0.416165\pi$
$60$	0	0
$61$	−32882.0	−1.13145	−0.565723	−	0.824596i	$-0.691403\pi$
$61$	−32882.0	−1.13145	−0.565723	+	0.824596i	$0.691403\pi$
$62$	0	0
$63$	−41310.0	−1.31131
$64$	0	0
$65$	2366.00	0.0694595
$66$	0	0
$67$	69566.0	1.89326	0.946629	−	0.322324i	$-0.104464\pi$
$67$	69566.0	1.89326	0.946629	+	0.322324i	$0.104464\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	50542.0	1.18989	0.594945	−	0.803767i	$-0.297174\pi$
$71$	50542.0	1.18989	0.594945	+	0.803767i	$0.297174\pi$
$72$	0	0
$73$	−46750.0	−1.02677	−0.513387	−	0.858157i	$-0.671609\pi$
$73$	−46750.0	−1.02677	−0.513387	+	0.858157i	$0.671609\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	42500.0	0.816887
$78$	0	0
$79$	19348.0	0.348793	0.174397	−	0.984675i	$-0.444202\pi$
$79$	19348.0	0.348793	0.174397	+	0.984675i	$0.444202\pi$
$80$	0	0
$81$	59049.0	1.00000
$82$	0	0
$83$	87438.0	1.39317	0.696586	−	0.717473i	$-0.254701\pi$
$83$	87438.0	1.39317	0.696586	+	0.717473i	$0.254701\pi$
$84$	0	0
$85$	−14868.0	−0.223206
$86$	0	0
$87$	0	0
$88$	0	0
$89$	94170.0	1.26019	0.630097	−	0.776516i	$-0.283015\pi$
$89$	94170.0	1.26019	0.630097	+	0.776516i	$0.283015\pi$
$90$	0	0
$91$	−28730.0	−0.363691
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1092.00	−0.0124141
$96$	0	0
$97$	182786.	1.97248	0.986242	−	0.165307i	$-0.0528613\pi$
$97$	182786.	1.97248	0.986242	+	0.165307i	$0.0528613\pi$
$98$	0	0
$99$	−60750.0	−0.622957
$100$	0	0
$101$	−18514.0	−0.180591	−0.0902957	−	0.995915i	$-0.528781\pi$
$101$	−18514.0	−0.180591	−0.0902957	+	0.995915i	$0.528781\pi$
$102$	0	0
$103$	−116056.	−1.07789	−0.538945	−	0.842341i	$-0.681177\pi$
$103$	−116056.	−1.07789	−0.538945	+	0.842341i	$0.681177\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−153520.	−1.29630	−0.648150	−	0.761513i	$-0.724457\pi$
$107$	−153520.	−1.29630	−0.648150	+	0.761513i	$0.724457\pi$
$108$	0	0
$109$	−178622.	−1.44002	−0.720010	−	0.693963i	$-0.755863\pi$
$109$	−178622.	−1.44002	−0.720010	+	0.693963i	$0.755863\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−244754.	−1.80316	−0.901579	−	0.432615i	$-0.857591\pi$
$113$	−244754.	−1.80316	−0.901579	+	0.432615i	$0.857591\pi$
$114$	0	0
$115$	22064.0	0.155575
$116$	0	0
$117$	41067.0	0.277350
$118$	0	0
$119$	180540.	1.16871
$120$	0	0
$121$	−98551.0	−0.611924
$122$	0	0
$123$	0	0
$124$	0	0
$125$	84756.0	0.485172
$126$	0	0
$127$	−256600.	−1.41172	−0.705858	−	0.708353i	$-0.749438\pi$
$127$	−256600.	−1.41172	−0.705858	+	0.708353i	$0.749438\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	262736.	1.33765	0.668823	−	0.743421i	$-0.266798\pi$
$131$	262736.	1.33765	0.668823	+	0.743421i	$0.266798\pi$
$132$	0	0
$133$	13260.0	0.0650002
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−38286.0	−0.174276	−0.0871382	−	0.996196i	$-0.527772\pi$
$137$	−38286.0	−0.174276	−0.0871382	+	0.996196i	$0.527772\pi$
$138$	0	0
$139$	57776.0	0.253636	0.126818	−	0.991926i	$-0.459524\pi$
$139$	57776.0	0.253636	0.126818	+	0.991926i	$0.459524\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−42250.0	−0.172777
$144$	0	0
$145$	−36092.0	−0.142558
$146$	0	0
$147$	0	0
$148$	0	0
$149$	28866.0	0.106517	0.0532587	−	0.998581i	$-0.483039\pi$
$149$	28866.0	0.106517	0.0532587	+	0.998581i	$0.483039\pi$
$150$	0	0
$151$	−39870.0	−0.142300	−0.0711498	−	0.997466i	$-0.522667\pi$
$151$	−39870.0	−0.142300	−0.0711498	+	0.997466i	$0.522667\pi$
$152$	0	0
$153$	−258066.	−0.891255
$154$	0	0
$155$	−121156.	−0.405057
$156$	0	0
$157$	161042.	0.521423	0.260711	−	0.965417i	$-0.416043\pi$
$157$	161042.	0.521423	0.260711	+	0.965417i	$0.416043\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−267920.	−0.814593
$162$	0	0
$163$	−312830.	−0.922230	−0.461115	−	0.887340i	$-0.652550\pi$
$163$	−312830.	−0.922230	−0.461115	+	0.887340i	$0.652550\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−532926.	−1.47869	−0.739343	−	0.673329i	$-0.764864\pi$
$167$	−532926.	−1.47869	−0.739343	+	0.673329i	$0.764864\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	−18954.0	−0.0495691
$172$	0	0
$173$	−630458.	−1.60155	−0.800776	−	0.598964i	$-0.795579\pi$
$173$	−630458.	−1.60155	−0.800776	+	0.598964i	$0.795579\pi$
$174$	0	0
$175$	−497930.	−1.22906
$176$	0	0
$177$	0	0
$178$	0	0
$179$	674916.	1.57441	0.787204	−	0.616693i	$-0.211528\pi$
$179$	674916.	1.57441	0.787204	+	0.616693i	$0.211528\pi$
$180$	0	0
$181$	186282.	0.422644	0.211322	−	0.977417i	$-0.432223\pi$
$181$	186282.	0.422644	0.211322	+	0.977417i	$0.432223\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−153804.	−0.330399
$186$	0	0
$187$	265500.	0.555214
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−812180.	−1.61090	−0.805451	−	0.592663i	$-0.798077\pi$
$191$	−812180.	−1.61090	−0.805451	+	0.592663i	$0.798077\pi$
$192$	0	0
$193$	−150142.	−0.290141	−0.145070	−	0.989421i	$-0.546341\pi$
$193$	−150142.	−0.290141	−0.145070	+	0.989421i	$0.546341\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	236394.	0.433981	0.216991	−	0.976174i	$-0.430376\pi$
$197$	236394.	0.433981	0.216991	+	0.976174i	$0.430376\pi$
$198$	0	0
$199$	39376.0	0.0704854	0.0352427	−	0.999379i	$-0.488780\pi$
$199$	39376.0	0.0704854	0.0352427	+	0.999379i	$0.488780\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	438260.	0.746435
$204$	0	0
$205$	−14700.0	−0.0244305
$206$	0	0
$207$	382968.	0.621207
$208$	0	0
$209$	19500.0	0.0308794
$210$	0	0
$211$	410776.	0.635183	0.317592	−	0.948228i	$-0.397126\pi$
$211$	410776.	0.635183	0.317592	+	0.948228i	$0.397126\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−82600.0	−0.121866
$216$	0	0
$217$	1.47118e6	2.12088
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−179478.	−0.247190
$222$	0	0
$223$	−1.08688e6	−1.46359	−0.731796	−	0.681523i	$-0.761318\pi$
$223$	−1.08688e6	−1.46359	−0.731796	+	0.681523i	$0.761318\pi$
$224$	0	0
$225$	711747.	0.937280
$226$	0	0
$227$	256470.	0.330348	0.165174	−	0.986264i	$-0.447181\pi$
$227$	256470.	0.330348	0.165174	+	0.986264i	$0.447181\pi$
$228$	0	0
$229$	−298110.	−0.375654	−0.187827	−	0.982202i	$-0.560144\pi$
$229$	−298110.	−0.375654	−0.187827	+	0.982202i	$0.560144\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−611926.	−0.738430	−0.369215	−	0.929344i	$-0.620373\pi$
$233$	−611926.	−0.738430	−0.369215	+	0.929344i	$0.620373\pi$
$234$	0	0
$235$	−83468.0	−0.0985940
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−36570.0	−0.0414124	−0.0207062	−	0.999786i	$-0.506591\pi$
$239$	−36570.0	−0.0414124	−0.0207062	+	0.999786i	$0.506591\pi$
$240$	0	0
$241$	380922.	0.422468	0.211234	−	0.977436i	$-0.432252\pi$
$241$	380922.	0.422468	0.211234	+	0.977436i	$0.432252\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−169302.	−0.180197
$246$	0	0
$247$	−13182.0	−0.0137480
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.22807e6	1.23038	0.615188	−	0.788380i	$-0.289080\pi$
$251$	1.22807e6	1.23038	0.615188	+	0.788380i	$0.289080\pi$
$252$	0	0
$253$	−394000.	−0.386986
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−439278.	−0.414865	−0.207432	−	0.978249i	$-0.566511\pi$
$257$	−439278.	−0.414865	−0.207432	+	0.978249i	$0.566511\pi$
$258$	0	0
$259$	1.86762e6	1.72997
$260$	0	0
$261$	−626454.	−0.569230
$262$	0	0
$263$	1.67987e6	1.49757	0.748783	−	0.662816i	$-0.230639\pi$
$263$	1.67987e6	1.49757	0.748783	+	0.662816i	$0.230639\pi$
$264$	0	0
$265$	−406644.	−0.355713
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.93840e6	1.63329	0.816645	−	0.577141i	$-0.195832\pi$
$269$	1.93840e6	1.63329	0.816645	+	0.577141i	$0.195832\pi$
$270$	0	0
$271$	695498.	0.575271	0.287636	−	0.957740i	$-0.407131\pi$
$271$	695498.	0.575271	0.287636	+	0.957740i	$0.407131\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−732250.	−0.583885
$276$	0	0
$277$	−1.13138e6	−0.885948	−0.442974	−	0.896534i	$-0.646077\pi$
$277$	−1.13138e6	−0.885948	−0.442974	+	0.896534i	$0.646077\pi$
$278$	0	0
$279$	−2.10292e6	−1.61738
$280$	0	0
$281$	1.73122e6	1.30793	0.653967	−	0.756523i	$-0.273103\pi$
$281$	1.73122e6	1.30793	0.653967	+	0.756523i	$0.273103\pi$
$282$	0	0
$283$	1.47124e6	1.09199	0.545995	−	0.837788i	$-0.316152\pi$
$283$	1.47124e6	1.09199	0.545995	+	0.837788i	$0.316152\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	178500.	0.127919
$288$	0	0
$289$	−292013.	−0.205664
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.88855e6	1.96567	0.982834	−	0.184491i	$-0.0590637\pi$
$293$	2.88855e6	1.96567	0.982834	+	0.184491i	$0.0590637\pi$
$294$	0	0
$295$	−194908.	−0.130399
$296$	0	0
$297$	0	0
$298$	0	0
$299$	266344.	0.172292
$300$	0	0
$301$	1.00300e6	0.638094
$302$	0	0
$303$	0	0
$304$	0	0
$305$	460348.	0.283359
$306$	0	0
$307$	−874118.	−0.529327	−0.264664	−	0.964341i	$-0.585261\pi$
$307$	−874118.	−0.529327	−0.264664	+	0.964341i	$0.585261\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.68224e6	−1.57252	−0.786261	−	0.617895i	$-0.787986\pi$
$311$	−2.68224e6	−1.57252	−0.786261	+	0.617895i	$0.787986\pi$
$312$	0	0
$313$	−1.34459e6	−0.775761	−0.387880	−	0.921710i	$-0.626793\pi$
$313$	−1.34459e6	−0.775761	−0.387880	+	0.921710i	$0.626793\pi$
$314$	0	0
$315$	578340.	0.328403
$316$	0	0
$317$	1.32074e6	0.738191	0.369095	−	0.929392i	$-0.379668\pi$
$317$	1.32074e6	0.738191	0.369095	+	0.929392i	$0.379668\pi$
$318$	0	0
$319$	644500.	0.354606
$320$	0	0
$321$	0	0
$322$	0	0
$323$	82836.0	0.0441787
$324$	0	0
$325$	495001.	0.259955
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.01354e6	0.516239
$330$	0	0
$331$	2.05728e6	1.03210	0.516051	−	0.856558i	$-0.327401\pi$
$331$	2.05728e6	1.03210	0.516051	+	0.856558i	$0.327401\pi$
$332$	0	0
$333$	−2.66960e6	−1.31927
$334$	0	0
$335$	−973924.	−0.474147
$336$	0	0
$337$	453398.	0.217473	0.108736	−	0.994071i	$-0.465320\pi$
$337$	453398.	0.217473	0.108736	+	0.994071i	$0.465320\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.16350e6	1.00756
$342$	0	0
$343$	−801380.	−0.367793
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.23065e6	0.548669	0.274334	−	0.961634i	$-0.411543\pi$
$347$	1.23065e6	0.548669	0.274334	+	0.961634i	$0.411543\pi$
$348$	0	0
$349$	−2.43825e6	−1.07155	−0.535777	−	0.844360i	$-0.679981\pi$
$349$	−2.43825e6	−1.07155	−0.535777	+	0.844360i	$0.679981\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.68315e6	−1.14606	−0.573031	−	0.819534i	$-0.694233\pi$
$353$	−2.68315e6	−1.14606	−0.573031	+	0.819534i	$0.694233\pi$
$354$	0	0
$355$	−707588.	−0.297995
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.58693e6	−0.649864	−0.324932	−	0.945737i	$-0.605341\pi$
$359$	−1.58693e6	−0.649864	−0.324932	+	0.945737i	$0.605341\pi$
$360$	0	0
$361$	−2.47002e6	−0.997543
$362$	0	0
$363$	0	0
$364$	0	0
$365$	654500.	0.257145
$366$	0	0
$367$	60052.0	0.0232735	0.0116368	−	0.999932i	$-0.496296\pi$
$367$	60052.0	0.0232735	0.0116368	+	0.999932i	$0.496296\pi$
$368$	0	0
$369$	−255150.	−0.0975505
$370$	0	0
$371$	4.93782e6	1.86252
$372$	0	0
$373$	−4.01853e6	−1.49553	−0.747766	−	0.663963i	$-0.768873\pi$
$373$	−4.01853e6	−1.49553	−0.747766	+	0.663963i	$0.768873\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−435682.	−0.157876
$378$	0	0
$379$	−1.67581e6	−0.599276	−0.299638	−	0.954053i	$-0.596866\pi$
$379$	−1.67581e6	−0.599276	−0.299638	+	0.954053i	$0.596866\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−687258.	−0.239399	−0.119700	−	0.992810i	$-0.538193\pi$
$383$	−687258.	−0.239399	−0.119700	+	0.992810i	$0.538193\pi$
$384$	0	0
$385$	−595000.	−0.204581
$386$	0	0
$387$	−1.43370e6	−0.486610
$388$	0	0
$389$	1.37611e6	0.461082	0.230541	−	0.973063i	$-0.425950\pi$
$389$	1.37611e6	0.461082	0.230541	+	0.973063i	$0.425950\pi$
$390$	0	0
$391$	−1.67371e6	−0.553655
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−270872.	−0.0873517
$396$	0	0
$397$	−721198.	−0.229656	−0.114828	−	0.993385i	$-0.536632\pi$
$397$	−721198.	−0.229656	−0.114828	+	0.993385i	$0.536632\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.22681e6	0.691548	0.345774	−	0.938318i	$-0.387616\pi$
$401$	2.22681e6	0.691548	0.345774	+	0.938318i	$0.387616\pi$
$402$	0	0
$403$	−1.46253e6	−0.448581
$404$	0	0
$405$	−826686.	−0.250440
$406$	0	0
$407$	2.74650e6	0.821852
$408$	0	0
$409$	2.00783e6	0.593496	0.296748	−	0.954956i	$-0.404098\pi$
$409$	2.00783e6	0.593496	0.296748	+	0.954956i	$0.404098\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.36674e6	0.682772
$414$	0	0
$415$	−1.22413e6	−0.348906
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−5.99378e6	−1.66788	−0.833942	−	0.551852i	$-0.813921\pi$
$419$	−5.99378e6	−1.66788	−0.833942	+	0.551852i	$0.813921\pi$
$420$	0	0
$421$	−5.32737e6	−1.46490	−0.732449	−	0.680822i	$-0.761623\pi$
$421$	−5.32737e6	−1.46490	−0.732449	+	0.680822i	$0.761623\pi$
$422$	0	0
$423$	−1.44877e6	−0.393684
$424$	0	0
$425$	−3.11060e6	−0.835356
$426$	0	0
$427$	−5.58994e6	−1.48367
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.42972e6	1.40794	0.703970	−	0.710230i	$-0.251409\pi$
$431$	5.42972e6	1.40794	0.703970	+	0.710230i	$0.251409\pi$
$432$	0	0
$433$	7.43979e6	1.90696	0.953479	−	0.301459i	$-0.0974737\pi$
$433$	7.43979e6	1.90696	0.953479	+	0.301459i	$0.0974737\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−122928.	−0.0307927
$438$	0	0
$439$	−6.86418e6	−1.69991	−0.849957	−	0.526852i	$-0.823372\pi$
$439$	−6.86418e6	−1.69991	−0.849957	+	0.526852i	$0.823372\pi$
$440$	0	0
$441$	−2.93860e6	−0.719522
$442$	0	0
$443$	3.46630e6	0.839182	0.419591	−	0.907713i	$-0.362173\pi$
$443$	3.46630e6	0.839182	0.419591	+	0.907713i	$0.362173\pi$
$444$	0	0
$445$	−1.31838e6	−0.315603
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−1.40426e6	−0.328725	−0.164362	−	0.986400i	$-0.552557\pi$
$449$	−1.40426e6	−0.328725	−0.164362	+	0.986400i	$0.552557\pi$
$450$	0	0
$451$	262500.	0.0607698
$452$	0	0
$453$	0	0
$454$	0	0
$455$	402220.	0.0910825
$456$	0	0
$457$	−5.95072e6	−1.33284	−0.666421	−	0.745575i	$-0.732175\pi$
$457$	−5.95072e6	−1.33284	−0.666421	+	0.745575i	$0.732175\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.25465e6	−1.37073	−0.685363	−	0.728202i	$-0.740356\pi$
$461$	−6.25465e6	−1.37073	−0.685363	+	0.728202i	$0.740356\pi$
$462$	0	0
$463$	1.55055e6	0.336149	0.168075	−	0.985774i	$-0.446245\pi$
$463$	1.55055e6	0.336149	0.168075	+	0.985774i	$0.446245\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.80480e6	0.382945	0.191472	−	0.981498i	$-0.438674\pi$
$467$	1.80480e6	0.382945	0.191472	+	0.981498i	$0.438674\pi$
$468$	0	0
$469$	1.18262e7	2.48264
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.47500e6	0.303137
$474$	0	0
$475$	−228462.	−0.0464601
$476$	0	0
$477$	−7.05818e6	−1.42035
$478$	0	0
$479$	−2.21809e6	−0.441712	−0.220856	−	0.975306i	$-0.570885\pi$
$479$	−2.21809e6	−0.441712	−0.220856	+	0.975306i	$0.570885\pi$
$480$	0	0
$481$	−1.85663e6	−0.365901
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.55900e6	−0.493988
$486$	0	0
$487$	6.14268e6	1.17364	0.586821	−	0.809717i	$-0.300379\pi$
$487$	6.14268e6	1.17364	0.586821	+	0.809717i	$0.300379\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.44486e6	−1.20645	−0.603226	−	0.797571i	$-0.706118\pi$
$491$	−6.44486e6	−1.20645	−0.603226	+	0.797571i	$0.706118\pi$
$492$	0	0
$493$	2.73784e6	0.507330
$494$	0	0
$495$	850500.	0.156013
$496$	0	0
$497$	8.59214e6	1.56031
$498$	0	0
$499$	−4.25838e6	−0.765584	−0.382792	−	0.923835i	$-0.625037\pi$
$499$	−4.25838e6	−0.765584	−0.382792	+	0.923835i	$0.625037\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.56242e6	0.627806	0.313903	−	0.949455i	$-0.398363\pi$
$503$	3.56242e6	0.627806	0.313903	+	0.949455i	$0.398363\pi$
$504$	0	0
$505$	259196.	0.0452272
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.23936e6	0.725281	0.362640	−	0.931929i	$-0.381875\pi$
$509$	4.23936e6	0.725281	0.362640	+	0.931929i	$0.381875\pi$
$510$	0	0
$511$	−7.94750e6	−1.34641
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.62478e6	0.269946
$516$	0	0
$517$	1.49050e6	0.245248
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.38657e6	0.385194	0.192597	−	0.981278i	$-0.438309\pi$
$521$	2.38657e6	0.385194	0.192597	+	0.981278i	$0.438309\pi$
$522$	0	0
$523$	8.84129e6	1.41339	0.706694	−	0.707519i	$-0.250186\pi$
$523$	8.84129e6	1.41339	0.706694	+	0.707519i	$0.250186\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.19055e6	1.44150
$528$	0	0
$529$	−3.95257e6	−0.614101
$530$	0	0
$531$	−3.38305e6	−0.520681
$532$	0	0
$533$	−177450.	−0.0270557
$534$	0	0
$535$	2.14928e6	0.324645
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.02325e6	0.448231
$540$	0	0
$541$	70058.0	0.0102912	0.00514558	−	0.999987i	$-0.498362\pi$
$541$	70058.0	0.0102912	0.00514558	+	0.999987i	$0.498362\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.50071e6	0.360638
$546$	0	0
$547$	6.60752e6	0.944213	0.472107	−	0.881541i	$-0.343494\pi$
$547$	6.60752e6	0.944213	0.472107	+	0.881541i	$0.343494\pi$
$548$	0	0
$549$	7.99033e6	1.13145
$550$	0	0
$551$	201084.	0.0282162
$552$	0	0
$553$	3.28916e6	0.457375
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.10726e7	−1.51221	−0.756107	−	0.654448i	$-0.772901\pi$
$557$	−1.10726e7	−1.51221	−0.756107	+	0.654448i	$0.772901\pi$
$558$	0	0
$559$	−997100.	−0.134961
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.43532e6	0.190843	0.0954216	−	0.995437i	$-0.469580\pi$
$563$	1.43532e6	0.190843	0.0954216	+	0.995437i	$0.469580\pi$
$564$	0	0
$565$	3.42656e6	0.451582
$566$	0	0
$567$	1.00383e7	1.31131
$568$	0	0
$569$	−1.17051e7	−1.51564	−0.757818	−	0.652466i	$-0.773734\pi$
$569$	−1.17051e7	−1.51564	−0.757818	+	0.652466i	$0.773734\pi$
$570$	0	0
$571$	−4.81885e6	−0.618519	−0.309260	−	0.950978i	$-0.600081\pi$
$571$	−4.81885e6	−0.618519	−0.309260	+	0.950978i	$0.600081\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.61610e6	0.582245
$576$	0	0
$577$	−1.35572e6	−0.169523	−0.0847617	−	0.996401i	$-0.527013\pi$
$577$	−1.35572e6	−0.169523	−0.0847617	+	0.996401i	$0.527013\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.48645e7	1.82687
$582$	0	0
$583$	7.26150e6	0.884820
$584$	0	0
$585$	−574938.	−0.0694595
$586$	0	0
$587$	−5.03941e6	−0.603649	−0.301824	−	0.953364i	$-0.597596\pi$
$587$	−5.03941e6	−0.603649	−0.301824	+	0.953364i	$0.597596\pi$
$588$	0	0
$589$	675012.	0.0801721
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.16124e6	1.06984	0.534919	−	0.844904i	$-0.320342\pi$
$593$	9.16124e6	1.06984	0.534919	+	0.844904i	$0.320342\pi$
$594$	0	0
$595$	−2.52756e6	−0.292691
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.46635e6	0.736363	0.368182	−	0.929754i	$-0.379980\pi$
$599$	6.46635e6	0.736363	0.368182	+	0.929754i	$0.379980\pi$
$600$	0	0
$601$	−1.18021e7	−1.33282	−0.666411	−	0.745585i	$-0.732170\pi$
$601$	−1.18021e7	−1.33282	−0.666411	+	0.745585i	$0.732170\pi$
$602$	0	0
$603$	−1.69045e7	−1.89326
$604$	0	0
$605$	1.37971e6	0.153250
$606$	0	0
$607$	−2.25748e6	−0.248686	−0.124343	−	0.992239i	$-0.539682\pi$
$607$	−2.25748e6	−0.248686	−0.124343	+	0.992239i	$0.539682\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−1.00758e6	−0.109188
$612$	0	0
$613$	−2.75378e6	−0.295991	−0.147995	−	0.988988i	$-0.547282\pi$
$613$	−2.75378e6	−0.295991	−0.147995	+	0.988988i	$0.547282\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.41607e6	0.361255	0.180627	−	0.983552i	$-0.442187\pi$
$617$	3.41607e6	0.361255	0.180627	+	0.983552i	$0.442187\pi$
$618$	0	0
$619$	−9.43169e6	−0.989379	−0.494690	−	0.869070i	$-0.664718\pi$
$619$	−9.43169e6	−0.989379	−0.494690	+	0.869070i	$0.664718\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.60089e7	1.65250
$624$	0	0
$625$	7.96654e6	0.815774
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.16671e7	1.17581
$630$	0	0
$631$	4.87474e6	0.487391	0.243696	−	0.969852i	$-0.421640\pi$
$631$	4.87474e6	0.487391	0.243696	+	0.969852i	$0.421640\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.59240e6	0.353550
$636$	0	0
$637$	−2.04372e6	−0.199559
$638$	0	0
$639$	−1.22817e7	−1.18989
$640$	0	0
$641$	9.74279e6	0.936566	0.468283	−	0.883579i	$-0.344873\pi$
$641$	9.74279e6	0.936566	0.468283	+	0.883579i	$0.344873\pi$
$642$	0	0
$643$	−1.63894e6	−0.156327	−0.0781637	−	0.996941i	$-0.524906\pi$
$643$	−1.63894e6	−0.156327	−0.0781637	+	0.996941i	$0.524906\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.59069e6	0.149391	0.0746955	−	0.997206i	$-0.476202\pi$
$647$	1.59069e6	0.149391	0.0746955	+	0.997206i	$0.476202\pi$
$648$	0	0
$649$	3.48050e6	0.324362
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.59778e7	1.46634	0.733170	−	0.680045i	$-0.238040\pi$
$653$	1.59778e7	1.46634	0.733170	+	0.680045i	$0.238040\pi$
$654$	0	0
$655$	−3.67830e6	−0.335000
$656$	0	0
$657$	1.13602e7	1.02677
$658$	0	0
$659$	6.02458e6	0.540397	0.270199	−	0.962805i	$-0.412911\pi$
$659$	6.02458e6	0.540397	0.270199	+	0.962805i	$0.412911\pi$
$660$	0	0
$661$	−2.00705e7	−1.78671	−0.893355	−	0.449352i	$-0.851655\pi$
$661$	−2.00705e7	−1.78671	−0.893355	+	0.449352i	$0.851655\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−185640.	−0.0162786
$666$	0	0
$667$	−4.06293e6	−0.353610
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.22050e6	−0.704842
$672$	0	0
$673$	−5.48575e6	−0.466873	−0.233436	−	0.972372i	$-0.574997\pi$
$673$	−5.48575e6	−0.466873	−0.233436	+	0.972372i	$0.574997\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.74926e6	−0.398248	−0.199124	−	0.979974i	$-0.563810\pi$
$677$	−4.74926e6	−0.398248	−0.199124	+	0.979974i	$0.563810\pi$
$678$	0	0
$679$	3.10736e7	2.58653
$680$	0	0
$681$	0	0
$682$	0	0
$683$	6.13964e6	0.503606	0.251803	−	0.967778i	$-0.418976\pi$
$683$	6.13964e6	0.503606	0.251803	+	0.967778i	$0.418976\pi$
$684$	0	0
$685$	536004.	0.0436457
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.90877e6	−0.393935
$690$	0	0
$691$	−1.57617e7	−1.25577	−0.627883	−	0.778308i	$-0.716078\pi$
$691$	−1.57617e7	−1.25577	−0.627883	+	0.778308i	$0.716078\pi$
$692$	0	0
$693$	−1.03275e7	−0.816887
$694$	0	0
$695$	−808864.	−0.0635204
$696$	0	0
$697$	1.11510e6	0.0869424
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.42036e7	−1.09170	−0.545851	−	0.837882i	$-0.683793\pi$
$701$	−1.42036e7	−1.09170	−0.545851	+	0.837882i	$0.683793\pi$
$702$	0	0
$703$	856908.	0.0653952
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.14738e6	−0.236810
$708$	0	0
$709$	1.60718e7	1.20074	0.600369	−	0.799723i	$-0.295020\pi$
$709$	1.60718e7	1.20074	0.600369	+	0.799723i	$0.295020\pi$
$710$	0	0
$711$	−4.70156e6	−0.348793
$712$	0	0
$713$	−1.36387e7	−1.00473
$714$	0	0
$715$	591500.	0.0432703
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.07078e7	1.49387	0.746933	−	0.664900i	$-0.231526\pi$
$719$	2.07078e7	1.49387	0.746933	+	0.664900i	$0.231526\pi$
$720$	0	0
$721$	−1.97295e7	−1.41344
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.55096e6	−0.533528
$726$	0	0
$727$	−5.04803e6	−0.354231	−0.177115	−	0.984190i	$-0.556677\pi$
$727$	−5.04803e6	−0.354231	−0.177115	+	0.984190i	$0.556677\pi$
$728$	0	0
$729$	−1.43489e7	−1.00000
$730$	0	0
$731$	6.26580e6	0.433694
$732$	0	0
$733$	−2.10377e7	−1.44623	−0.723115	−	0.690728i	$-0.757290\pi$
$733$	−2.10377e7	−1.44623	−0.723115	+	0.690728i	$0.757290\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.73915e7	1.17942
$738$	0	0
$739$	−1.38992e7	−0.936218	−0.468109	−	0.883671i	$-0.655065\pi$
$739$	−1.38992e7	−0.936218	−0.468109	+	0.883671i	$0.655065\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.23267e6	−0.0819169	−0.0409584	−	0.999161i	$-0.513041\pi$
$743$	−1.23267e6	−0.0819169	−0.0409584	+	0.999161i	$0.513041\pi$
$744$	0	0
$745$	−404124.	−0.0266762
$746$	0	0
$747$	−2.12474e7	−1.39317
$748$	0	0
$749$	−2.60984e7	−1.69985
$750$	0	0
$751$	1.62624e6	0.105217	0.0526084	−	0.998615i	$-0.483247\pi$
$751$	1.62624e6	0.105217	0.0526084	+	0.998615i	$0.483247\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	558180.	0.0356375
$756$	0	0
$757$	3.49882e6	0.221913	0.110956	−	0.993825i	$-0.464609\pi$
$757$	3.49882e6	0.221913	0.110956	+	0.993825i	$0.464609\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.21713e7	1.38781	0.693905	−	0.720067i	$-0.255889\pi$
$761$	2.21713e7	1.38781	0.693905	+	0.720067i	$0.255889\pi$
$762$	0	0
$763$	−3.03657e7	−1.88831
$764$	0	0
$765$	3.61292e6	0.223206
$766$	0	0
$767$	−2.35282e6	−0.144411
$768$	0	0
$769$	1.08955e6	0.0664400	0.0332200	−	0.999448i	$-0.489424\pi$
$769$	1.08955e6	0.0664400	0.0332200	+	0.999448i	$0.489424\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.95219e6	0.117510	0.0587549	−	0.998272i	$-0.481287\pi$
$773$	1.95219e6	0.117510	0.0587549	+	0.998272i	$0.481287\pi$
$774$	0	0
$775$	−2.53476e7	−1.51594
$776$	0	0
$777$	0	0
$778$	0	0
$779$	81900.0	0.00483549
$780$	0	0
$781$	1.26355e7	0.741250
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.25459e6	−0.130585
$786$	0	0
$787$	1.44531e7	0.831809	0.415904	−	0.909408i	$-0.363465\pi$
$787$	1.44531e7	0.831809	0.415904	+	0.909408i	$0.363465\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−4.16082e7	−2.36449
$792$	0	0
$793$	5.55706e6	0.313807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.23500e7	0.688685	0.344343	−	0.938844i	$-0.388102\pi$
$797$	1.23500e7	0.688685	0.344343	+	0.938844i	$0.388102\pi$
$798$	0	0
$799$	6.33164e6	0.350873
$800$	0	0
$801$	−2.28833e7	−1.26019
$802$	0	0
$803$	−1.16875e7	−0.639636
$804$	0	0
$805$	3.75088e6	0.204006
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.15968e7	0.622970	0.311485	−	0.950251i	$-0.399174\pi$
$809$	1.15968e7	0.622970	0.311485	+	0.950251i	$0.399174\pi$
$810$	0	0
$811$	2.47534e7	1.32155	0.660774	−	0.750585i	$-0.270228\pi$
$811$	2.47534e7	1.32155	0.660774	+	0.750585i	$0.270228\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.37962e6	0.230963
$816$	0	0
$817$	460200.	0.0241208
$818$	0	0
$819$	6.98139e6	0.363691
$820$	0	0
$821$	−2.47470e6	−0.128134	−0.0640671	−	0.997946i	$-0.520407\pi$
$821$	−2.47470e6	−0.128134	−0.0640671	+	0.997946i	$0.520407\pi$
$822$	0	0
$823$	7.84754e6	0.403863	0.201932	−	0.979400i	$-0.435278\pi$
$823$	7.84754e6	0.403863	0.201932	+	0.979400i	$0.435278\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.26192e7	1.15004	0.575020	−	0.818140i	$-0.304994\pi$
$827$	2.26192e7	1.15004	0.575020	+	0.818140i	$0.304994\pi$
$828$	0	0
$829$	−1.73912e7	−0.878907	−0.439454	−	0.898265i	$-0.644828\pi$
$829$	−1.73912e7	−0.878907	−0.439454	+	0.898265i	$0.644828\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.28428e7	0.641278
$834$	0	0
$835$	7.46096e6	0.370321
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3.43825e7	1.68629	0.843147	−	0.537684i	$-0.180701\pi$
$839$	3.43825e7	1.68629	0.843147	+	0.537684i	$0.180701\pi$
$840$	0	0
$841$	−1.38651e7	−0.675977
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−399854.	−0.0192646
$846$	0	0
$847$	−1.67537e7	−0.802419
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.73139e7	−0.819543
$852$	0	0
$853$	2.31007e7	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$853$	2.31007e7	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$854$	0	0
$855$	265356.	0.0124141
$856$	0	0
$857$	−7.02305e6	−0.326643	−0.163322	−	0.986573i	$-0.552221\pi$
$857$	−7.02305e6	−0.326643	−0.163322	+	0.986573i	$0.552221\pi$
$858$	0	0
$859$	−8.82135e6	−0.407899	−0.203949	−	0.978981i	$-0.565378\pi$
$859$	−8.82135e6	−0.407899	−0.203949	+	0.978981i	$0.565378\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.39560e7	−1.09493	−0.547466	−	0.836828i	$-0.684408\pi$
$863$	−2.39560e7	−1.09493	−0.547466	+	0.836828i	$0.684408\pi$
$864$	0	0
$865$	8.82641e6	0.401092
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.83700e6	0.217283
$870$	0	0
$871$	−1.17567e7	−0.525096
$872$	0	0
$873$	−4.44170e7	−1.97248
$874$	0	0
$875$	1.44085e7	0.636208
$876$	0	0
$877$	−5.79805e6	−0.254556	−0.127278	−	0.991867i	$-0.540624\pi$
$877$	−5.79805e6	−0.254556	−0.127278	+	0.991867i	$0.540624\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.30527e7	−0.566580	−0.283290	−	0.959034i	$-0.591426\pi$
$881$	−1.30527e7	−0.566580	−0.283290	+	0.959034i	$0.591426\pi$
$882$	0	0
$883$	−4.73009e6	−0.204159	−0.102079	−	0.994776i	$-0.532550\pi$
$883$	−4.73009e6	−0.204159	−0.102079	+	0.994776i	$0.532550\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.80737e7	−1.19809	−0.599046	−	0.800714i	$-0.704453\pi$
$887$	−2.80737e7	−1.19809	−0.599046	+	0.800714i	$0.704453\pi$
$888$	0	0
$889$	−4.36220e7	−1.85119
$890$	0	0
$891$	1.47622e7	0.622957
$892$	0	0
$893$	465036.	0.0195145
$894$	0	0
$895$	−9.44882e6	−0.394294
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.23100e7	0.920663
$900$	0	0
$901$	3.08469e7	1.26590
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.60795e6	−0.105847
$906$	0	0
$907$	−2.28552e7	−0.922500	−0.461250	−	0.887270i	$-0.652599\pi$
$907$	−2.28552e7	−0.922500	−0.461250	+	0.887270i	$0.652599\pi$
$908$	0	0
$909$	4.49890e6	0.180591
$910$	0	0
$911$	3.27335e7	1.30676	0.653381	−	0.757029i	$-0.273350\pi$
$911$	3.27335e7	1.30676	0.653381	+	0.757029i	$0.273350\pi$
$912$	0	0
$913$	2.18595e7	0.867887
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.46651e7	1.75406
$918$	0	0
$919$	1.27717e7	0.498839	0.249419	−	0.968396i	$-0.419760\pi$
$919$	1.27717e7	0.498839	0.249419	+	0.968396i	$0.419760\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−8.54160e6	−0.330016
$924$	0	0
$925$	−3.21780e7	−1.23653
$926$	0	0
$927$	2.82016e7	1.07789
$928$	0	0
$929$	3.48297e7	1.32407	0.662034	−	0.749473i	$-0.269693\pi$
$929$	3.48297e7	1.32407	0.662034	+	0.749473i	$0.269693\pi$
$930$	0	0
$931$	943254.	0.0356660
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−3.71700e6	−0.139048
$936$	0	0
$937$	3.00172e7	1.11692	0.558459	−	0.829532i	$-0.311393\pi$
$937$	3.00172e7	1.11692	0.558459	+	0.829532i	$0.311393\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−4.50649e7	−1.65907	−0.829534	−	0.558457i	$-0.811394\pi$
$941$	−4.50649e7	−1.65907	−0.829534	+	0.558457i	$0.811394\pi$
$942$	0	0
$943$	−1.65480e6	−0.0605991
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.99276e7	−1.08442	−0.542210	−	0.840243i	$-0.682412\pi$
$947$	−2.99276e7	−1.08442	−0.542210	+	0.840243i	$0.682412\pi$
$948$	0	0
$949$	7.90075e6	0.284776
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.25147e7	−1.51638	−0.758188	−	0.652036i	$-0.773915\pi$
$953$	−4.25147e7	−1.51638	−0.758188	+	0.652036i	$0.773915\pi$
$954$	0	0
$955$	1.13705e7	0.403433
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.50862e6	−0.228530
$960$	0	0
$961$	4.62626e7	1.61593
$962$	0	0
$963$	3.73054e7	1.29630
$964$	0	0
$965$	2.10199e6	0.0726628
$966$	0	0
$967$	3.00251e7	1.03257	0.516284	−	0.856417i	$-0.327315\pi$
$967$	3.00251e7	1.03257	0.516284	+	0.856417i	$0.327315\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.00864e7	1.36442	0.682211	−	0.731155i	$-0.261018\pi$
$971$	4.00864e7	1.36442	0.682211	+	0.731155i	$0.261018\pi$
$972$	0	0
$973$	9.82192e6	0.332594
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.12151e7	1.71657	0.858284	−	0.513174i	$-0.171531\pi$
$977$	5.12151e7	1.71657	0.858284	+	0.513174i	$0.171531\pi$
$978$	0	0
$979$	2.35425e7	0.785047
$980$	0	0
$981$	4.34051e7	1.44002
$982$	0	0
$983$	−1.82382e7	−0.602004	−0.301002	−	0.953624i	$-0.597321\pi$
$983$	−1.82382e7	−0.602004	−0.301002	+	0.953624i	$0.597321\pi$
$984$	0	0
$985$	−3.30952e6	−0.108686
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−9.29840e6	−0.302286
$990$	0	0
$991$	3.24103e7	1.04833	0.524166	−	0.851616i	$-0.324377\pi$
$991$	3.24103e7	1.04833	0.524166	+	0.851616i	$0.324377\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−551264.	−0.0176523
$996$	0	0
$997$	−2.07867e7	−0.662289	−0.331145	−	0.943580i	$-0.607435\pi$
$997$	−2.07867e7	−0.662289	−0.331145	+	0.943580i	$0.607435\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.6.a.b.1.1		1
4.3	odd	2		26.6.a.a.1.1	✓	1
8.3	odd	2		832.6.a.d.1.1		1
8.5	even	2		832.6.a.e.1.1		1
12.11	even	2		234.6.a.g.1.1		1
20.3	even	4		650.6.b.a.599.2		2
20.7	even	4		650.6.b.a.599.1		2
20.19	odd	2		650.6.a.a.1.1		1
52.31	even	4		338.6.b.a.337.2		2
52.47	even	4		338.6.b.a.337.1		2
52.51	odd	2		338.6.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.6.a.a.1.1	✓	1	4.3	odd	2
208.6.a.b.1.1		1	1.1	even	1	trivial
234.6.a.g.1.1		1	12.11	even	2
338.6.a.d.1.1		1	52.51	odd	2
338.6.b.a.337.1		2	52.47	even	4
338.6.b.a.337.2		2	52.31	even	4
650.6.a.a.1.1		1	20.19	odd	2
650.6.b.a.599.1		2	20.7	even	4
650.6.b.a.599.2		2	20.3	even	4
832.6.a.d.1.1		1	8.3	odd	2
832.6.a.e.1.1		1	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	208.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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