LMFDB - Embedded newform 208.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [208,2,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, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("208.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
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:	$1.66088836204$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
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)

$f(q)$	$=$	$q+1.56155 q^{3} +3.56155 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} +3.56155 q^{5} -1.56155 q^{7} -0.561553 q^{9} -3.12311 q^{11} +1.00000 q^{13} +5.56155 q^{15} -6.68466 q^{17} +3.12311 q^{19} -2.43845 q^{21} +8.00000 q^{23} +7.68466 q^{25} -5.56155 q^{27} -2.00000 q^{29} -4.00000 q^{31} -4.87689 q^{33} -5.56155 q^{35} -2.68466 q^{37} +1.56155 q^{39} +5.12311 q^{41} -9.56155 q^{43} -2.00000 q^{45} +12.6847 q^{47} -4.56155 q^{49} -10.4384 q^{51} -5.12311 q^{53} -11.1231 q^{55} +4.87689 q^{57} +3.12311 q^{59} +2.87689 q^{61} +0.876894 q^{63} +3.56155 q^{65} -3.12311 q^{67} +12.4924 q^{69} -4.68466 q^{71} -6.00000 q^{73} +12.0000 q^{75} +4.87689 q^{77} -8.00000 q^{79} -7.00000 q^{81} +14.2462 q^{83} -23.8078 q^{85} -3.12311 q^{87} +10.0000 q^{89} -1.56155 q^{91} -6.24621 q^{93} +11.1231 q^{95} +8.24621 q^{97} +1.75379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 + 3 * q^5 + q^7 + 3 * q^9	$ 2 q - q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 7 q^{15} - q^{17} - 2 q^{19} - 9 q^{21} + 16 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 8 q^{31} - 18 q^{33} - 7 q^{35} + 7 q^{37} - q^{39} + 2 q^{41} - 15 q^{43} - 4 q^{45} + 13 q^{47} - 5 q^{49} - 25 q^{51} - 2 q^{53} - 14 q^{55} + 18 q^{57} - 2 q^{59} + 14 q^{61} + 10 q^{63} + 3 q^{65} + 2 q^{67} - 8 q^{69} + 3 q^{71} - 12 q^{73} + 24 q^{75} + 18 q^{77} - 16 q^{79} - 14 q^{81} + 12 q^{83} - 27 q^{85} + 2 q^{87} + 20 q^{89} + q^{91} + 4 q^{93} + 14 q^{95} + 20 q^{99}+O(q^{100}) $ 2 * q - q^3 + 3 * q^5 + q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 7 * q^15 - q^17 - 2 * q^19 - 9 * q^21 + 16 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 8 * q^31 - 18 * q^33 - 7 * q^35 + 7 * q^37 - q^39 + 2 * q^41 - 15 * q^43 - 4 * q^45 + 13 * q^47 - 5 * q^49 - 25 * q^51 - 2 * q^53 - 14 * q^55 + 18 * q^57 - 2 * q^59 + 14 * q^61 + 10 * q^63 + 3 * q^65 + 2 * q^67 - 8 * q^69 + 3 * q^71 - 12 * q^73 + 24 * q^75 + 18 * q^77 - 16 * q^79 - 14 * q^81 + 12 * q^83 - 27 * q^85 + 2 * q^87 + 20 * q^89 + q^91 + 4 * q^93 + 14 * q^95 + 20 * q^99

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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−3.12311	−0.941652	−0.470826	−	0.882226i	$-0.656044\pi$
$11$	−3.12311	−0.941652	−0.470826	+	0.882226i	$0.656044\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	5.56155	1.43599
$16$	0	0
$17$	−6.68466	−1.62127	−0.810634	−	0.585553i	$-0.800877\pi$
$17$	−6.68466	−1.62127	−0.810634	+	0.585553i	$0.800877\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	−2.43845	−0.532113
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−4.87689	−0.848958
$34$	0	0
$35$	−5.56155	−0.940074
$36$	0	0
$37$	−2.68466	−0.441355	−0.220678	−	0.975347i	$-0.570827\pi$
$37$	−2.68466	−0.441355	−0.220678	+	0.975347i	$0.570827\pi$
$38$	0	0
$39$	1.56155	0.250049
$40$	0	0
$41$	5.12311	0.800095	0.400047	−	0.916494i	$-0.368994\pi$
$41$	5.12311	0.800095	0.400047	+	0.916494i	$0.368994\pi$
$42$	0	0
$43$	−9.56155	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	12.6847	1.85025	0.925124	−	0.379666i	$-0.123961\pi$
$47$	12.6847	1.85025	0.925124	+	0.379666i	$0.123961\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−10.4384	−1.46167
$52$	0	0
$53$	−5.12311	−0.703713	−0.351856	−	0.936054i	$-0.614449\pi$
$53$	−5.12311	−0.703713	−0.351856	+	0.936054i	$0.614449\pi$
$54$	0	0
$55$	−11.1231	−1.49984
$56$	0	0
$57$	4.87689	0.645960
$58$	0	0
$59$	3.12311	0.406594	0.203297	−	0.979117i	$-0.434834\pi$
$59$	3.12311	0.406594	0.203297	+	0.979117i	$0.434834\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	0.876894	0.110478
$64$	0	0
$65$	3.56155	0.441756
$66$	0	0
$67$	−3.12311	−0.381548	−0.190774	−	0.981634i	$-0.561100\pi$
$67$	−3.12311	−0.381548	−0.190774	+	0.981634i	$0.561100\pi$
$68$	0	0
$69$	12.4924	1.50391
$70$	0	0
$71$	−4.68466	−0.555967	−0.277983	−	0.960586i	$-0.589666\pi$
$71$	−4.68466	−0.555967	−0.277983	+	0.960586i	$0.589666\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	4.87689	0.555774
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	14.2462	1.56372	0.781862	−	0.623451i	$-0.214270\pi$
$83$	14.2462	1.56372	0.781862	+	0.623451i	$0.214270\pi$
$84$	0	0
$85$	−23.8078	−2.58231
$86$	0	0
$87$	−3.12311	−0.334832
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−1.56155	−0.163695
$92$	0	0
$93$	−6.24621	−0.647702
$94$	0	0
$95$	11.1231	1.14121
$96$	0	0
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	0	0
$99$	1.75379	0.176262
$100$	0	0
$101$	1.12311	0.111753	0.0558766	−	0.998438i	$-0.482205\pi$
$101$	1.12311	0.111753	0.0558766	+	0.998438i	$0.482205\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	−8.68466	−0.847536
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	17.8078	1.70567	0.852837	−	0.522177i	$-0.174880\pi$
$109$	17.8078	1.70567	0.852837	+	0.522177i	$0.174880\pi$
$110$	0	0
$111$	−4.19224	−0.397909
$112$	0	0
$113$	14.4924	1.36333	0.681666	−	0.731663i	$-0.261256\pi$
$113$	14.4924	1.36333	0.681666	+	0.731663i	$0.261256\pi$
$114$	0	0
$115$	28.4924	2.65693
$116$	0	0
$117$	−0.561553	−0.0519156
$118$	0	0
$119$	10.4384	0.956891
$120$	0	0
$121$	−1.24621	−0.113292
$122$	0	0
$123$	8.00000	0.721336
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−6.24621	−0.554262	−0.277131	−	0.960832i	$-0.589384\pi$
$127$	−6.24621	−0.554262	−0.277131	+	0.960832i	$0.589384\pi$
$128$	0	0
$129$	−14.9309	−1.31459
$130$	0	0
$131$	−15.8078	−1.38113	−0.690565	−	0.723270i	$-0.742638\pi$
$131$	−15.8078	−1.38113	−0.690565	+	0.723270i	$0.742638\pi$
$132$	0	0
$133$	−4.87689	−0.422880
$134$	0	0
$135$	−19.8078	−1.70478
$136$	0	0
$137$	5.12311	0.437696	0.218848	−	0.975759i	$-0.429770\pi$
$137$	5.12311	0.437696	0.218848	+	0.975759i	$0.429770\pi$
$138$	0	0
$139$	4.68466	0.397348	0.198674	−	0.980066i	$-0.436337\pi$
$139$	4.68466	0.397348	0.198674	+	0.980066i	$0.436337\pi$
$140$	0	0
$141$	19.8078	1.66811
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	−7.12311	−0.591542
$146$	0	0
$147$	−7.12311	−0.587504
$148$	0	0
$149$	−0.246211	−0.0201704	−0.0100852	−	0.999949i	$-0.503210\pi$
$149$	−0.246211	−0.0201704	−0.0100852	+	0.999949i	$0.503210\pi$
$150$	0	0
$151$	−6.43845	−0.523953	−0.261977	−	0.965074i	$-0.584374\pi$
$151$	−6.43845	−0.523953	−0.261977	+	0.965074i	$0.584374\pi$
$152$	0	0
$153$	3.75379	0.303476
$154$	0	0
$155$	−14.2462	−1.14428
$156$	0	0
$157$	10.4924	0.837386	0.418693	−	0.908128i	$-0.362488\pi$
$157$	10.4924	0.837386	0.418693	+	0.908128i	$0.362488\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	−12.4924	−0.984541
$162$	0	0
$163$	−12.4924	−0.978482	−0.489241	−	0.872149i	$-0.662726\pi$
$163$	−12.4924	−0.978482	−0.489241	+	0.872149i	$0.662726\pi$
$164$	0	0
$165$	−17.3693	−1.35220
$166$	0	0
$167$	2.24621	0.173817	0.0869085	−	0.996216i	$-0.472301\pi$
$167$	2.24621	0.173817	0.0869085	+	0.996216i	$0.472301\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.75379	−0.134116
$172$	0	0
$173$	−13.1231	−0.997731	−0.498866	−	0.866679i	$-0.666250\pi$
$173$	−13.1231	−0.997731	−0.498866	+	0.866679i	$0.666250\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	4.87689	0.366570
$178$	0	0
$179$	4.68466	0.350148	0.175074	−	0.984555i	$-0.443984\pi$
$179$	4.68466	0.350148	0.175074	+	0.984555i	$0.443984\pi$
$180$	0	0
$181$	−14.8769	−1.10579	−0.552895	−	0.833251i	$-0.686477\pi$
$181$	−14.8769	−1.10579	−0.552895	+	0.833251i	$0.686477\pi$
$182$	0	0
$183$	4.49242	0.332089
$184$	0	0
$185$	−9.56155	−0.702979
$186$	0	0
$187$	20.8769	1.52667
$188$	0	0
$189$	8.68466	0.631716
$190$	0	0
$191$	9.36932	0.677940	0.338970	−	0.940797i	$-0.389921\pi$
$191$	9.36932	0.677940	0.338970	+	0.940797i	$0.389921\pi$
$192$	0	0
$193$	3.36932	0.242529	0.121264	−	0.992620i	$-0.461305\pi$
$193$	3.36932	0.242529	0.121264	+	0.992620i	$0.461305\pi$
$194$	0	0
$195$	5.56155	0.398271
$196$	0	0
$197$	14.6847	1.04624	0.523119	−	0.852259i	$-0.324768\pi$
$197$	14.6847	1.04624	0.523119	+	0.852259i	$0.324768\pi$
$198$	0	0
$199$	3.12311	0.221391	0.110696	−	0.993854i	$-0.464692\pi$
$199$	3.12311	0.221391	0.110696	+	0.993854i	$0.464692\pi$
$200$	0	0
$201$	−4.87689	−0.343990
$202$	0	0
$203$	3.12311	0.219199
$204$	0	0
$205$	18.2462	1.27437
$206$	0	0
$207$	−4.49242	−0.312245
$208$	0	0
$209$	−9.75379	−0.674684
$210$	0	0
$211$	−3.31534	−0.228238	−0.114119	−	0.993467i	$-0.536404\pi$
$211$	−3.31534	−0.228238	−0.114119	+	0.993467i	$0.536404\pi$
$212$	0	0
$213$	−7.31534	−0.501239
$214$	0	0
$215$	−34.0540	−2.32246
$216$	0	0
$217$	6.24621	0.424020
$218$	0	0
$219$	−9.36932	−0.633120
$220$	0	0
$221$	−6.68466	−0.449659
$222$	0	0
$223$	3.31534	0.222012	0.111006	−	0.993820i	$-0.464593\pi$
$223$	3.31534	0.222012	0.111006	+	0.993820i	$0.464593\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	0	0
$229$	−13.8078	−0.912443	−0.456221	−	0.889866i	$-0.650797\pi$
$229$	−13.8078	−0.912443	−0.456221	+	0.889866i	$0.650797\pi$
$230$	0	0
$231$	7.61553	0.501065
$232$	0	0
$233$	13.8078	0.904577	0.452288	−	0.891872i	$-0.350608\pi$
$233$	13.8078	0.904577	0.452288	+	0.891872i	$0.350608\pi$
$234$	0	0
$235$	45.1771	2.94703
$236$	0	0
$237$	−12.4924	−0.811470
$238$	0	0
$239$	−1.56155	−0.101008	−0.0505042	−	0.998724i	$-0.516083\pi$
$239$	−1.56155	−0.101008	−0.0505042	+	0.998724i	$0.516083\pi$
$240$	0	0
$241$	−20.2462	−1.30417	−0.652087	−	0.758145i	$-0.726106\pi$
$241$	−20.2462	−1.30417	−0.652087	+	0.758145i	$0.726106\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−16.2462	−1.03793
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	22.2462	1.40980
$250$	0	0
$251$	−5.75379	−0.363176	−0.181588	−	0.983375i	$-0.558124\pi$
$251$	−5.75379	−0.363176	−0.181588	+	0.983375i	$0.558124\pi$
$252$	0	0
$253$	−24.9848	−1.57078
$254$	0	0
$255$	−37.1771	−2.32812
$256$	0	0
$257$	15.5616	0.970703	0.485351	−	0.874319i	$-0.338692\pi$
$257$	15.5616	0.970703	0.485351	+	0.874319i	$0.338692\pi$
$258$	0	0
$259$	4.19224	0.260493
$260$	0	0
$261$	1.12311	0.0695185
$262$	0	0
$263$	9.75379	0.601444	0.300722	−	0.953712i	$-0.402772\pi$
$263$	9.75379	0.601444	0.300722	+	0.953712i	$0.402772\pi$
$264$	0	0
$265$	−18.2462	−1.12086
$266$	0	0
$267$	15.6155	0.955655
$268$	0	0
$269$	−13.1231	−0.800130	−0.400065	−	0.916487i	$-0.631012\pi$
$269$	−13.1231	−0.800130	−0.400065	+	0.916487i	$0.631012\pi$
$270$	0	0
$271$	0.192236	0.0116775	0.00583875	−	0.999983i	$-0.498141\pi$
$271$	0.192236	0.0116775	0.00583875	+	0.999983i	$0.498141\pi$
$272$	0	0
$273$	−2.43845	−0.147582
$274$	0	0
$275$	−24.0000	−1.44725
$276$	0	0
$277$	4.63068	0.278231	0.139115	−	0.990276i	$-0.455574\pi$
$277$	4.63068	0.278231	0.139115	+	0.990276i	$0.455574\pi$
$278$	0	0
$279$	2.24621	0.134477
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−24.4924	−1.45592	−0.727962	−	0.685618i	$-0.759532\pi$
$283$	−24.4924	−1.45592	−0.727962	+	0.685618i	$0.759532\pi$
$284$	0	0
$285$	17.3693	1.02887
$286$	0	0
$287$	−8.00000	−0.472225
$288$	0	0
$289$	27.6847	1.62851
$290$	0	0
$291$	12.8769	0.754857
$292$	0	0
$293$	19.5616	1.14280	0.571399	−	0.820672i	$-0.306401\pi$
$293$	19.5616	1.14280	0.571399	+	0.820672i	$0.306401\pi$
$294$	0	0
$295$	11.1231	0.647612
$296$	0	0
$297$	17.3693	1.00787
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	14.9309	0.860601
$302$	0	0
$303$	1.75379	0.100753
$304$	0	0
$305$	10.2462	0.586696
$306$	0	0
$307$	−4.87689	−0.278339	−0.139170	−	0.990269i	$-0.544443\pi$
$307$	−4.87689	−0.278339	−0.139170	+	0.990269i	$0.544443\pi$
$308$	0	0
$309$	22.2462	1.26554
$310$	0	0
$311$	19.1231	1.08437	0.542186	−	0.840259i	$-0.317597\pi$
$311$	19.1231	1.08437	0.542186	+	0.840259i	$0.317597\pi$
$312$	0	0
$313$	6.19224	0.350006	0.175003	−	0.984568i	$-0.444007\pi$
$313$	6.19224	0.350006	0.175003	+	0.984568i	$0.444007\pi$
$314$	0	0
$315$	3.12311	0.175967
$316$	0	0
$317$	4.24621	0.238491	0.119245	−	0.992865i	$-0.461952\pi$
$317$	4.24621	0.238491	0.119245	+	0.992865i	$0.461952\pi$
$318$	0	0
$319$	6.24621	0.349721
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	−20.8769	−1.16162
$324$	0	0
$325$	7.68466	0.426268
$326$	0	0
$327$	27.8078	1.53777
$328$	0	0
$329$	−19.8078	−1.09204
$330$	0	0
$331$	−28.4924	−1.56609	−0.783043	−	0.621968i	$-0.786333\pi$
$331$	−28.4924	−1.56609	−0.783043	+	0.621968i	$0.786333\pi$
$332$	0	0
$333$	1.50758	0.0826147
$334$	0	0
$335$	−11.1231	−0.607720
$336$	0	0
$337$	−33.8078	−1.84163	−0.920813	−	0.390004i	$-0.872474\pi$
$337$	−33.8078	−1.84163	−0.920813	+	0.390004i	$0.872474\pi$
$338$	0	0
$339$	22.6307	1.22913
$340$	0	0
$341$	12.4924	0.676503
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	44.4924	2.39539
$346$	0	0
$347$	4.68466	0.251486	0.125743	−	0.992063i	$-0.459869\pi$
$347$	4.68466	0.251486	0.125743	+	0.992063i	$0.459869\pi$
$348$	0	0
$349$	−16.9309	−0.906289	−0.453144	−	0.891437i	$-0.649698\pi$
$349$	−16.9309	−0.906289	−0.453144	+	0.891437i	$0.649698\pi$
$350$	0	0
$351$	−5.56155	−0.296854
$352$	0	0
$353$	−15.3693	−0.818026	−0.409013	−	0.912529i	$-0.634127\pi$
$353$	−15.3693	−0.818026	−0.409013	+	0.912529i	$0.634127\pi$
$354$	0	0
$355$	−16.6847	−0.885530
$356$	0	0
$357$	16.3002	0.862697
$358$	0	0
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	−1.94602	−0.102140
$364$	0	0
$365$	−21.3693	−1.11852
$366$	0	0
$367$	31.6155	1.65032	0.825159	−	0.564901i	$-0.191086\pi$
$367$	31.6155	1.65032	0.825159	+	0.564901i	$0.191086\pi$
$368$	0	0
$369$	−2.87689	−0.149765
$370$	0	0
$371$	8.00000	0.415339
$372$	0	0
$373$	29.6155	1.53343	0.766717	−	0.641985i	$-0.221889\pi$
$373$	29.6155	1.53343	0.766717	+	0.641985i	$0.221889\pi$
$374$	0	0
$375$	14.9309	0.771027
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	−22.2462	−1.14271	−0.571356	−	0.820703i	$-0.693582\pi$
$379$	−22.2462	−1.14271	−0.571356	+	0.820703i	$0.693582\pi$
$380$	0	0
$381$	−9.75379	−0.499702
$382$	0	0
$383$	31.8078	1.62530	0.812650	−	0.582752i	$-0.198024\pi$
$383$	31.8078	1.62530	0.812650	+	0.582752i	$0.198024\pi$
$384$	0	0
$385$	17.3693	0.885222
$386$	0	0
$387$	5.36932	0.272938
$388$	0	0
$389$	−28.7386	−1.45711	−0.728553	−	0.684989i	$-0.759807\pi$
$389$	−28.7386	−1.45711	−0.728553	+	0.684989i	$0.759807\pi$
$390$	0	0
$391$	−53.4773	−2.70446
$392$	0	0
$393$	−24.6847	−1.24518
$394$	0	0
$395$	−28.4924	−1.43361
$396$	0	0
$397$	−2.00000	−0.100377	−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000	−0.100377	−0.0501886	+	0.998740i	$0.515982\pi$
$398$	0	0
$399$	−7.61553	−0.381253
$400$	0	0
$401$	−37.6155	−1.87843	−0.939215	−	0.343330i	$-0.888445\pi$
$401$	−37.6155	−1.87843	−0.939215	+	0.343330i	$0.888445\pi$
$402$	0	0
$403$	−4.00000	−0.199254
$404$	0	0
$405$	−24.9309	−1.23882
$406$	0	0
$407$	8.38447	0.415603
$408$	0	0
$409$	21.1231	1.04447	0.522235	−	0.852802i	$-0.325098\pi$
$409$	21.1231	1.04447	0.522235	+	0.852802i	$0.325098\pi$
$410$	0	0
$411$	8.00000	0.394611
$412$	0	0
$413$	−4.87689	−0.239976
$414$	0	0
$415$	50.7386	2.49066
$416$	0	0
$417$	7.31534	0.358234
$418$	0	0
$419$	−26.9309	−1.31566	−0.657830	−	0.753167i	$-0.728525\pi$
$419$	−26.9309	−1.31566	−0.657830	+	0.753167i	$0.728525\pi$
$420$	0	0
$421$	−18.6847	−0.910635	−0.455317	−	0.890329i	$-0.650474\pi$
$421$	−18.6847	−0.910635	−0.455317	+	0.890329i	$0.650474\pi$
$422$	0	0
$423$	−7.12311	−0.346337
$424$	0	0
$425$	−51.3693	−2.49178
$426$	0	0
$427$	−4.49242	−0.217404
$428$	0	0
$429$	−4.87689	−0.235459
$430$	0	0
$431$	14.4384	0.695476	0.347738	−	0.937592i	$-0.386950\pi$
$431$	14.4384	0.695476	0.347738	+	0.937592i	$0.386950\pi$
$432$	0	0
$433$	39.1771	1.88273	0.941365	−	0.337389i	$-0.109544\pi$
$433$	39.1771	1.88273	0.941365	+	0.337389i	$0.109544\pi$
$434$	0	0
$435$	−11.1231	−0.533312
$436$	0	0
$437$	24.9848	1.19519
$438$	0	0
$439$	−12.8769	−0.614581	−0.307290	−	0.951616i	$-0.599422\pi$
$439$	−12.8769	−0.614581	−0.307290	+	0.951616i	$0.599422\pi$
$440$	0	0
$441$	2.56155	0.121979
$442$	0	0
$443$	−0.192236	−0.00913340	−0.00456670	−	0.999990i	$-0.501454\pi$
$443$	−0.192236	−0.00913340	−0.00456670	+	0.999990i	$0.501454\pi$
$444$	0	0
$445$	35.6155	1.68834
$446$	0	0
$447$	−0.384472	−0.0181849
$448$	0	0
$449$	−16.7386	−0.789945	−0.394972	−	0.918693i	$-0.629246\pi$
$449$	−16.7386	−0.789945	−0.394972	+	0.918693i	$0.629246\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	−10.0540	−0.472377
$454$	0	0
$455$	−5.56155	−0.260730
$456$	0	0
$457$	−24.7386	−1.15722	−0.578612	−	0.815603i	$-0.696406\pi$
$457$	−24.7386	−1.15722	−0.578612	+	0.815603i	$0.696406\pi$
$458$	0	0
$459$	37.1771	1.73528
$460$	0	0
$461$	−14.1922	−0.660998	−0.330499	−	0.943806i	$-0.607217\pi$
$461$	−14.1922	−0.660998	−0.330499	+	0.943806i	$0.607217\pi$
$462$	0	0
$463$	−21.7538	−1.01098	−0.505492	−	0.862831i	$-0.668689\pi$
$463$	−21.7538	−1.01098	−0.505492	+	0.862831i	$0.668689\pi$
$464$	0	0
$465$	−22.2462	−1.03164
$466$	0	0
$467$	−7.50758	−0.347409	−0.173705	−	0.984798i	$-0.555574\pi$
$467$	−7.50758	−0.347409	−0.173705	+	0.984798i	$0.555574\pi$
$468$	0	0
$469$	4.87689	0.225194
$470$	0	0
$471$	16.3845	0.754957
$472$	0	0
$473$	29.8617	1.37304
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	2.87689	0.131724
$478$	0	0
$479$	−31.8078	−1.45333	−0.726667	−	0.686990i	$-0.758932\pi$
$479$	−31.8078	−1.45333	−0.726667	+	0.686990i	$0.758932\pi$
$480$	0	0
$481$	−2.68466	−0.122410
$482$	0	0
$483$	−19.5076	−0.887626
$484$	0	0
$485$	29.3693	1.33359
$486$	0	0
$487$	−18.2462	−0.826815	−0.413407	−	0.910546i	$-0.635661\pi$
$487$	−18.2462	−0.826815	−0.413407	+	0.910546i	$0.635661\pi$
$488$	0	0
$489$	−19.5076	−0.882163
$490$	0	0
$491$	−38.0540	−1.71735	−0.858676	−	0.512519i	$-0.828712\pi$
$491$	−38.0540	−1.71735	−0.858676	+	0.512519i	$0.828712\pi$
$492$	0	0
$493$	13.3693	0.602124
$494$	0	0
$495$	6.24621	0.280746
$496$	0	0
$497$	7.31534	0.328138
$498$	0	0
$499$	4.49242	0.201108	0.100554	−	0.994932i	$-0.467938\pi$
$499$	4.49242	0.201108	0.100554	+	0.994932i	$0.467938\pi$
$500$	0	0
$501$	3.50758	0.156707
$502$	0	0
$503$	−17.3693	−0.774460	−0.387230	−	0.921983i	$-0.626568\pi$
$503$	−17.3693	−0.774460	−0.387230	+	0.921983i	$0.626568\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	1.56155	0.0693510
$508$	0	0
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	9.36932	0.414474
$512$	0	0
$513$	−17.3693	−0.766874
$514$	0	0
$515$	50.7386	2.23581
$516$	0	0
$517$	−39.6155	−1.74229
$518$	0	0
$519$	−20.4924	−0.899518
$520$	0	0
$521$	2.68466	0.117617	0.0588085	−	0.998269i	$-0.481270\pi$
$521$	2.68466	0.117617	0.0588085	+	0.998269i	$0.481270\pi$
$522$	0	0
$523$	−32.4924	−1.42079	−0.710397	−	0.703801i	$-0.751485\pi$
$523$	−32.4924	−1.42079	−0.710397	+	0.703801i	$0.751485\pi$
$524$	0	0
$525$	−18.7386	−0.817821
$526$	0	0
$527$	26.7386	1.16475
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−1.75379	−0.0761079
$532$	0	0
$533$	5.12311	0.221906
$534$	0	0
$535$	−14.2462	−0.615917
$536$	0	0
$537$	7.31534	0.315680
$538$	0	0
$539$	14.2462	0.613628
$540$	0	0
$541$	6.68466	0.287396	0.143698	−	0.989622i	$-0.454101\pi$
$541$	6.68466	0.287396	0.143698	+	0.989622i	$0.454101\pi$
$542$	0	0
$543$	−23.2311	−0.996940
$544$	0	0
$545$	63.4233	2.71676
$546$	0	0
$547$	33.5616	1.43499	0.717494	−	0.696564i	$-0.245289\pi$
$547$	33.5616	1.43499	0.717494	+	0.696564i	$0.245289\pi$
$548$	0	0
$549$	−1.61553	−0.0689491
$550$	0	0
$551$	−6.24621	−0.266098
$552$	0	0
$553$	12.4924	0.531232
$554$	0	0
$555$	−14.9309	−0.633780
$556$	0	0
$557$	3.17708	0.134617	0.0673086	−	0.997732i	$-0.478559\pi$
$557$	3.17708	0.134617	0.0673086	+	0.997732i	$0.478559\pi$
$558$	0	0
$559$	−9.56155	−0.404411
$560$	0	0
$561$	32.6004	1.37639
$562$	0	0
$563$	−30.0540	−1.26662	−0.633312	−	0.773897i	$-0.718305\pi$
$563$	−30.0540	−1.26662	−0.633312	+	0.773897i	$0.718305\pi$
$564$	0	0
$565$	51.6155	2.17148
$566$	0	0
$567$	10.9309	0.459053
$568$	0	0
$569$	−29.3153	−1.22896	−0.614482	−	0.788931i	$-0.710635\pi$
$569$	−29.3153	−1.22896	−0.614482	+	0.788931i	$0.710635\pi$
$570$	0	0
$571$	−8.19224	−0.342834	−0.171417	−	0.985199i	$-0.554835\pi$
$571$	−8.19224	−0.342834	−0.171417	+	0.985199i	$0.554835\pi$
$572$	0	0
$573$	14.6307	0.611206
$574$	0	0
$575$	61.4773	2.56378
$576$	0	0
$577$	−7.75379	−0.322794	−0.161397	−	0.986890i	$-0.551600\pi$
$577$	−7.75379	−0.322794	−0.161397	+	0.986890i	$0.551600\pi$
$578$	0	0
$579$	5.26137	0.218655
$580$	0	0
$581$	−22.2462	−0.922928
$582$	0	0
$583$	16.0000	0.662652
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	−40.9848	−1.69163	−0.845813	−	0.533480i	$-0.820884\pi$
$587$	−40.9848	−1.69163	−0.845813	+	0.533480i	$0.820884\pi$
$588$	0	0
$589$	−12.4924	−0.514741
$590$	0	0
$591$	22.9309	0.943250
$592$	0	0
$593$	5.50758	0.226169	0.113085	−	0.993585i	$-0.463927\pi$
$593$	5.50758	0.226169	0.113085	+	0.993585i	$0.463927\pi$
$594$	0	0
$595$	37.1771	1.52411
$596$	0	0
$597$	4.87689	0.199598
$598$	0	0
$599$	1.36932	0.0559488	0.0279744	−	0.999609i	$-0.491094\pi$
$599$	1.36932	0.0559488	0.0279744	+	0.999609i	$0.491094\pi$
$600$	0	0
$601$	−11.1771	−0.455923	−0.227961	−	0.973670i	$-0.573206\pi$
$601$	−11.1771	−0.455923	−0.227961	+	0.973670i	$0.573206\pi$
$602$	0	0
$603$	1.75379	0.0714198
$604$	0	0
$605$	−4.43845	−0.180449
$606$	0	0
$607$	9.36932	0.380289	0.190144	−	0.981756i	$-0.439104\pi$
$607$	9.36932	0.380289	0.190144	+	0.981756i	$0.439104\pi$
$608$	0	0
$609$	4.87689	0.197622
$610$	0	0
$611$	12.6847	0.513166
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	28.4924	1.14893
$616$	0	0
$617$	−35.8617	−1.44374	−0.721870	−	0.692029i	$-0.756717\pi$
$617$	−35.8617	−1.44374	−0.721870	+	0.692029i	$0.756717\pi$
$618$	0	0
$619$	46.2462	1.85879	0.929396	−	0.369084i	$-0.120328\pi$
$619$	46.2462	1.85879	0.929396	+	0.369084i	$0.120328\pi$
$620$	0	0
$621$	−44.4924	−1.78542
$622$	0	0
$623$	−15.6155	−0.625623
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−15.2311	−0.608270
$628$	0	0
$629$	17.9460	0.715555
$630$	0	0
$631$	35.3153	1.40588	0.702941	−	0.711248i	$-0.251870\pi$
$631$	35.3153	1.40588	0.702941	+	0.711248i	$0.251870\pi$
$632$	0	0
$633$	−5.17708	−0.205770
$634$	0	0
$635$	−22.2462	−0.882814
$636$	0	0
$637$	−4.56155	−0.180735
$638$	0	0
$639$	2.63068	0.104068
$640$	0	0
$641$	8.24621	0.325706	0.162853	−	0.986650i	$-0.447930\pi$
$641$	8.24621	0.325706	0.162853	+	0.986650i	$0.447930\pi$
$642$	0	0
$643$	21.8617	0.862143	0.431071	−	0.902318i	$-0.358136\pi$
$643$	21.8617	0.862143	0.431071	+	0.902318i	$0.358136\pi$
$644$	0	0
$645$	−53.1771	−2.09385
$646$	0	0
$647$	3.12311	0.122782	0.0613910	−	0.998114i	$-0.480446\pi$
$647$	3.12311	0.122782	0.0613910	+	0.998114i	$0.480446\pi$
$648$	0	0
$649$	−9.75379	−0.382870
$650$	0	0
$651$	9.75379	0.382281
$652$	0	0
$653$	−13.1231	−0.513547	−0.256773	−	0.966472i	$-0.582659\pi$
$653$	−13.1231	−0.513547	−0.256773	+	0.966472i	$0.582659\pi$
$654$	0	0
$655$	−56.3002	−2.19983
$656$	0	0
$657$	3.36932	0.131450
$658$	0	0
$659$	16.4924	0.642454	0.321227	−	0.947002i	$-0.395905\pi$
$659$	16.4924	0.642454	0.321227	+	0.947002i	$0.395905\pi$
$660$	0	0
$661$	8.73863	0.339893	0.169947	−	0.985453i	$-0.445640\pi$
$661$	8.73863	0.339893	0.169947	+	0.985453i	$0.445640\pi$
$662$	0	0
$663$	−10.4384	−0.405396
$664$	0	0
$665$	−17.3693	−0.673553
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	5.17708	0.200158
$670$	0	0
$671$	−8.98485	−0.346856
$672$	0	0
$673$	34.3002	1.32218	0.661088	−	0.750309i	$-0.270095\pi$
$673$	34.3002	1.32218	0.661088	+	0.750309i	$0.270095\pi$
$674$	0	0
$675$	−42.7386	−1.64501
$676$	0	0
$677$	23.3693	0.898156	0.449078	−	0.893493i	$-0.351753\pi$
$677$	23.3693	0.898156	0.449078	+	0.893493i	$0.351753\pi$
$678$	0	0
$679$	−12.8769	−0.494170
$680$	0	0
$681$	12.4924	0.478711
$682$	0	0
$683$	24.0000	0.918334	0.459167	−	0.888350i	$-0.348148\pi$
$683$	24.0000	0.918334	0.459167	+	0.888350i	$0.348148\pi$
$684$	0	0
$685$	18.2462	0.697152
$686$	0	0
$687$	−21.5616	−0.822625
$688$	0	0
$689$	−5.12311	−0.195175
$690$	0	0
$691$	48.9848	1.86347	0.931736	−	0.363137i	$-0.118294\pi$
$691$	48.9848	1.86347	0.931736	+	0.363137i	$0.118294\pi$
$692$	0	0
$693$	−2.73863	−0.104032
$694$	0	0
$695$	16.6847	0.632885
$696$	0	0
$697$	−34.2462	−1.29717
$698$	0	0
$699$	21.5616	0.815533
$700$	0	0
$701$	47.3693	1.78911	0.894557	−	0.446953i	$-0.147491\pi$
$701$	47.3693	1.78911	0.894557	+	0.446953i	$0.147491\pi$
$702$	0	0
$703$	−8.38447	−0.316226
$704$	0	0
$705$	70.5464	2.65693
$706$	0	0
$707$	−1.75379	−0.0659580
$708$	0	0
$709$	24.7386	0.929079	0.464539	−	0.885552i	$-0.346220\pi$
$709$	24.7386	0.929079	0.464539	+	0.885552i	$0.346220\pi$
$710$	0	0
$711$	4.49242	0.168479
$712$	0	0
$713$	−32.0000	−1.19841
$714$	0	0
$715$	−11.1231	−0.415981
$716$	0	0
$717$	−2.43845	−0.0910655
$718$	0	0
$719$	4.87689	0.181877	0.0909387	−	0.995856i	$-0.471013\pi$
$719$	4.87689	0.181877	0.0909387	+	0.995856i	$0.471013\pi$
$720$	0	0
$721$	−22.2462	−0.828492
$722$	0	0
$723$	−31.6155	−1.17579
$724$	0	0
$725$	−15.3693	−0.570802
$726$	0	0
$727$	−28.1080	−1.04247	−0.521233	−	0.853414i	$-0.674528\pi$
$727$	−28.1080	−1.04247	−0.521233	+	0.853414i	$0.674528\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	63.9157	2.36401
$732$	0	0
$733$	−31.5616	−1.16575	−0.582876	−	0.812561i	$-0.698073\pi$
$733$	−31.5616	−1.16575	−0.582876	+	0.812561i	$0.698073\pi$
$734$	0	0
$735$	−25.3693	−0.935761
$736$	0	0
$737$	9.75379	0.359285
$738$	0	0
$739$	−6.24621	−0.229771	−0.114885	−	0.993379i	$-0.536650\pi$
$739$	−6.24621	−0.229771	−0.114885	+	0.993379i	$0.536650\pi$
$740$	0	0
$741$	4.87689	0.179157
$742$	0	0
$743$	43.9157	1.61111	0.805556	−	0.592520i	$-0.201867\pi$
$743$	43.9157	1.61111	0.805556	+	0.592520i	$0.201867\pi$
$744$	0	0
$745$	−0.876894	−0.0321269
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	6.24621	0.228232
$750$	0	0
$751$	40.9848	1.49556	0.747779	−	0.663948i	$-0.231120\pi$
$751$	40.9848	1.49556	0.747779	+	0.663948i	$0.231120\pi$
$752$	0	0
$753$	−8.98485	−0.327426
$754$	0	0
$755$	−22.9309	−0.834540
$756$	0	0
$757$	−5.12311	−0.186202	−0.0931012	−	0.995657i	$-0.529678\pi$
$757$	−5.12311	−0.186202	−0.0931012	+	0.995657i	$0.529678\pi$
$758$	0	0
$759$	−39.0152	−1.41616
$760$	0	0
$761$	13.5076	0.489649	0.244825	−	0.969567i	$-0.421270\pi$
$761$	13.5076	0.489649	0.244825	+	0.969567i	$0.421270\pi$
$762$	0	0
$763$	−27.8078	−1.00671
$764$	0	0
$765$	13.3693	0.483369
$766$	0	0
$767$	3.12311	0.112769
$768$	0	0
$769$	−43.8617	−1.58169	−0.790847	−	0.612013i	$-0.790360\pi$
$769$	−43.8617	−1.58169	−0.790847	+	0.612013i	$0.790360\pi$
$770$	0	0
$771$	24.3002	0.875150
$772$	0	0
$773$	0.822919	0.0295983	0.0147992	−	0.999890i	$-0.495289\pi$
$773$	0.822919	0.0295983	0.0147992	+	0.999890i	$0.495289\pi$
$774$	0	0
$775$	−30.7386	−1.10416
$776$	0	0
$777$	6.54640	0.234851
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	14.6307	0.523527
$782$	0	0
$783$	11.1231	0.397507
$784$	0	0
$785$	37.3693	1.33377
$786$	0	0
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	0	0
$789$	15.2311	0.542240
$790$	0	0
$791$	−22.6307	−0.804654
$792$	0	0
$793$	2.87689	0.102162
$794$	0	0
$795$	−28.4924	−1.01052
$796$	0	0
$797$	−46.4924	−1.64685	−0.823423	−	0.567428i	$-0.807939\pi$
$797$	−46.4924	−1.64685	−0.823423	+	0.567428i	$0.807939\pi$
$798$	0	0
$799$	−84.7926	−2.99975
$800$	0	0
$801$	−5.61553	−0.198415
$802$	0	0
$803$	18.7386	0.661272
$804$	0	0
$805$	−44.4924	−1.56815
$806$	0	0
$807$	−20.4924	−0.721367
$808$	0	0
$809$	15.1771	0.533598	0.266799	−	0.963752i	$-0.414034\pi$
$809$	15.1771	0.533598	0.266799	+	0.963752i	$0.414034\pi$
$810$	0	0
$811$	−2.73863	−0.0961664	−0.0480832	−	0.998843i	$-0.515311\pi$
$811$	−2.73863	−0.0961664	−0.0480832	+	0.998843i	$0.515311\pi$
$812$	0	0
$813$	0.300187	0.0105280
$814$	0	0
$815$	−44.4924	−1.55850
$816$	0	0
$817$	−29.8617	−1.04473
$818$	0	0
$819$	0.876894	0.0306412
$820$	0	0
$821$	19.5616	0.682703	0.341351	−	0.939936i	$-0.389115\pi$
$821$	19.5616	0.682703	0.341351	+	0.939936i	$0.389115\pi$
$822$	0	0
$823$	−31.6155	−1.10205	−0.551024	−	0.834489i	$-0.685763\pi$
$823$	−31.6155	−1.10205	−0.551024	+	0.834489i	$0.685763\pi$
$824$	0	0
$825$	−37.4773	−1.30479
$826$	0	0
$827$	−21.8617	−0.760207	−0.380104	−	0.924944i	$-0.624112\pi$
$827$	−21.8617	−0.760207	−0.380104	+	0.924944i	$0.624112\pi$
$828$	0	0
$829$	−14.4924	−0.503343	−0.251671	−	0.967813i	$-0.580980\pi$
$829$	−14.4924	−0.503343	−0.251671	+	0.967813i	$0.580980\pi$
$830$	0	0
$831$	7.23106	0.250843
$832$	0	0
$833$	30.4924	1.05650
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	22.2462	0.768942
$838$	0	0
$839$	−38.7386	−1.33741	−0.668703	−	0.743530i	$-0.733150\pi$
$839$	−38.7386	−1.33741	−0.668703	+	0.743530i	$0.733150\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	15.6155	0.537827
$844$	0	0
$845$	3.56155	0.122521
$846$	0	0
$847$	1.94602	0.0668662
$848$	0	0
$849$	−38.2462	−1.31261
$850$	0	0
$851$	−21.4773	−0.736231
$852$	0	0
$853$	−17.3153	−0.592866	−0.296433	−	0.955054i	$-0.595797\pi$
$853$	−17.3153	−0.592866	−0.296433	+	0.955054i	$0.595797\pi$
$854$	0	0
$855$	−6.24621	−0.213616
$856$	0	0
$857$	−8.73863	−0.298506	−0.149253	−	0.988799i	$-0.547687\pi$
$857$	−8.73863	−0.298506	−0.149253	+	0.988799i	$0.547687\pi$
$858$	0	0
$859$	52.9848	1.80782	0.903910	−	0.427723i	$-0.140684\pi$
$859$	52.9848	1.80782	0.903910	+	0.427723i	$0.140684\pi$
$860$	0	0
$861$	−12.4924	−0.425741
$862$	0	0
$863$	−4.30019	−0.146380	−0.0731900	−	0.997318i	$-0.523318\pi$
$863$	−4.30019	−0.146380	−0.0731900	+	0.997318i	$0.523318\pi$
$864$	0	0
$865$	−46.7386	−1.58916
$866$	0	0
$867$	43.2311	1.46820
$868$	0	0
$869$	24.9848	0.847553
$870$	0	0
$871$	−3.12311	−0.105822
$872$	0	0
$873$	−4.63068	−0.156725
$874$	0	0
$875$	−14.9309	−0.504756
$876$	0	0
$877$	−25.3153	−0.854838	−0.427419	−	0.904054i	$-0.640577\pi$
$877$	−25.3153	−0.854838	−0.427419	+	0.904054i	$0.640577\pi$
$878$	0	0
$879$	30.5464	1.03030
$880$	0	0
$881$	−10.1922	−0.343385	−0.171693	−	0.985151i	$-0.554924\pi$
$881$	−10.1922	−0.343385	−0.171693	+	0.985151i	$0.554924\pi$
$882$	0	0
$883$	−12.6847	−0.426873	−0.213436	−	0.976957i	$-0.568466\pi$
$883$	−12.6847	−0.426873	−0.213436	+	0.976957i	$0.568466\pi$
$884$	0	0
$885$	17.3693	0.583863
$886$	0	0
$887$	−53.4773	−1.79559	−0.897795	−	0.440413i	$-0.854832\pi$
$887$	−53.4773	−1.79559	−0.897795	+	0.440413i	$0.854832\pi$
$888$	0	0
$889$	9.75379	0.327132
$890$	0	0
$891$	21.8617	0.732396
$892$	0	0
$893$	39.6155	1.32568
$894$	0	0
$895$	16.6847	0.557707
$896$	0	0
$897$	12.4924	0.417110
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	34.2462	1.14091
$902$	0	0
$903$	23.3153	0.775886
$904$	0	0
$905$	−52.9848	−1.76128
$906$	0	0
$907$	0.192236	0.00638309	0.00319154	−	0.999995i	$-0.498984\pi$
$907$	0.192236	0.00638309	0.00319154	+	0.999995i	$0.498984\pi$
$908$	0	0
$909$	−0.630683	−0.0209184
$910$	0	0
$911$	9.36932	0.310419	0.155210	−	0.987882i	$-0.450395\pi$
$911$	9.36932	0.310419	0.155210	+	0.987882i	$0.450395\pi$
$912$	0	0
$913$	−44.4924	−1.47248
$914$	0	0
$915$	16.0000	0.528944
$916$	0	0
$917$	24.6847	0.815159
$918$	0	0
$919$	40.9848	1.35197	0.675983	−	0.736918i	$-0.263719\pi$
$919$	40.9848	1.35197	0.675983	+	0.736918i	$0.263719\pi$
$920$	0	0
$921$	−7.61553	−0.250940
$922$	0	0
$923$	−4.68466	−0.154197
$924$	0	0
$925$	−20.6307	−0.678333
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	47.8617	1.57029	0.785146	−	0.619310i	$-0.212588\pi$
$929$	47.8617	1.57029	0.785146	+	0.619310i	$0.212588\pi$
$930$	0	0
$931$	−14.2462	−0.466901
$932$	0	0
$933$	29.8617	0.977629
$934$	0	0
$935$	74.3542	2.43164
$936$	0	0
$937$	12.7386	0.416153	0.208077	−	0.978113i	$-0.433280\pi$
$937$	12.7386	0.416153	0.208077	+	0.978113i	$0.433280\pi$
$938$	0	0
$939$	9.66950	0.315552
$940$	0	0
$941$	−30.7926	−1.00381	−0.501905	−	0.864923i	$-0.667367\pi$
$941$	−30.7926	−1.00381	−0.501905	+	0.864923i	$0.667367\pi$
$942$	0	0
$943$	40.9848	1.33465
$944$	0	0
$945$	30.9309	1.00618
$946$	0	0
$947$	17.3693	0.564427	0.282213	−	0.959352i	$-0.408931\pi$
$947$	17.3693	0.564427	0.282213	+	0.959352i	$0.408931\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	6.63068	0.215015
$952$	0	0
$953$	59.6695	1.93288	0.966442	−	0.256883i	$-0.0826956\pi$
$953$	59.6695	1.93288	0.966442	+	0.256883i	$0.0826956\pi$
$954$	0	0
$955$	33.3693	1.07981
$956$	0	0
$957$	9.75379	0.315295
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	2.24621	0.0723831
$964$	0	0
$965$	12.0000	0.386294
$966$	0	0
$967$	1.56155	0.0502162	0.0251081	−	0.999685i	$-0.492007\pi$
$967$	1.56155	0.0502162	0.0251081	+	0.999685i	$0.492007\pi$
$968$	0	0
$969$	−32.6004	−1.04727
$970$	0	0
$971$	19.3153	0.619859	0.309929	−	0.950760i	$-0.399695\pi$
$971$	19.3153	0.619859	0.309929	+	0.950760i	$0.399695\pi$
$972$	0	0
$973$	−7.31534	−0.234519
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	21.5076	0.688088	0.344044	−	0.938953i	$-0.388203\pi$
$977$	21.5076	0.688088	0.344044	+	0.938953i	$0.388203\pi$
$978$	0	0
$979$	−31.2311	−0.998149
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	10.9309	0.348641	0.174320	−	0.984689i	$-0.444227\pi$
$983$	10.9309	0.348641	0.174320	+	0.984689i	$0.444227\pi$
$984$	0	0
$985$	52.3002	1.66642
$986$	0	0
$987$	−30.9309	−0.984540
$988$	0	0
$989$	−76.4924	−2.43232
$990$	0	0
$991$	−21.8617	−0.694461	−0.347231	−	0.937780i	$-0.612878\pi$
$991$	−21.8617	−0.694461	−0.347231	+	0.937780i	$0.612878\pi$
$992$	0	0
$993$	−44.4924	−1.41192
$994$	0	0
$995$	11.1231	0.352626
$996$	0	0
$997$	−16.2462	−0.514523	−0.257261	−	0.966342i	$-0.582820\pi$
$997$	−16.2462	−0.514523	−0.257261	+	0.966342i	$0.582820\pi$
$998$	0	0
$999$	14.9309	0.472392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.2.a.e.1.2		2
3.2	odd	2		1872.2.a.u.1.1		2
4.3	odd	2		104.2.a.b.1.1	✓	2
5.4	even	2		5200.2.a.bw.1.1		2
8.3	odd	2		832.2.a.k.1.2		2
8.5	even	2		832.2.a.n.1.1		2
12.11	even	2		936.2.a.j.1.1		2
13.5	odd	4		2704.2.f.k.337.3		4
13.8	odd	4		2704.2.f.k.337.4		4
13.12	even	2		2704.2.a.p.1.2		2
16.3	odd	4		3328.2.b.y.1665.2		4
16.5	even	4		3328.2.b.w.1665.2		4
16.11	odd	4		3328.2.b.y.1665.3		4
16.13	even	4		3328.2.b.w.1665.3		4
20.3	even	4		2600.2.d.k.1249.2		4
20.7	even	4		2600.2.d.k.1249.3		4
20.19	odd	2		2600.2.a.p.1.2		2
24.5	odd	2		7488.2.a.cv.1.2		2
24.11	even	2		7488.2.a.cu.1.2		2
28.27	even	2		5096.2.a.m.1.2		2
52.3	odd	6		1352.2.i.f.529.2		4
52.7	even	12		1352.2.o.d.361.4		8
52.11	even	12		1352.2.o.d.1161.4		8
52.15	even	12		1352.2.o.d.1161.3		8
52.19	even	12		1352.2.o.d.361.3		8
52.23	odd	6		1352.2.i.d.529.2		4
52.31	even	4		1352.2.f.c.337.1		4
52.35	odd	6		1352.2.i.f.1329.2		4
52.43	odd	6		1352.2.i.d.1329.2		4
52.47	even	4		1352.2.f.c.337.2		4
52.51	odd	2		1352.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.b.1.1	✓	2	4.3	odd	2
208.2.a.e.1.2		2	1.1	even	1	trivial
832.2.a.k.1.2		2	8.3	odd	2
832.2.a.n.1.1		2	8.5	even	2
936.2.a.j.1.1		2	12.11	even	2
1352.2.a.g.1.1		2	52.51	odd	2
1352.2.f.c.337.1		4	52.31	even	4
1352.2.f.c.337.2		4	52.47	even	4
1352.2.i.d.529.2		4	52.23	odd	6
1352.2.i.d.1329.2		4	52.43	odd	6
1352.2.i.f.529.2		4	52.3	odd	6
1352.2.i.f.1329.2		4	52.35	odd	6
1352.2.o.d.361.3		8	52.19	even	12
1352.2.o.d.361.4		8	52.7	even	12
1352.2.o.d.1161.3		8	52.15	even	12
1352.2.o.d.1161.4		8	52.11	even	12
1872.2.a.u.1.1		2	3.2	odd	2
2600.2.a.p.1.2		2	20.19	odd	2
2600.2.d.k.1249.2		4	20.3	even	4
2600.2.d.k.1249.3		4	20.7	even	4
2704.2.a.p.1.2		2	13.12	even	2
2704.2.f.k.337.3		4	13.5	odd	4
2704.2.f.k.337.4		4	13.8	odd	4
3328.2.b.w.1665.2		4	16.5	even	4
3328.2.b.w.1665.3		4	16.13	even	4
3328.2.b.y.1665.2		4	16.3	odd	4
3328.2.b.y.1665.3		4	16.11	odd	4
5096.2.a.m.1.2		2	28.27	even	2
5200.2.a.bw.1.1		2	5.4	even	2
7488.2.a.cu.1.2		2	24.11	even	2
7488.2.a.cv.1.2		2	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +3.56155 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} +3.56155 q^{5} -1.56155 q^{7} -0.561553 q^{9} -3.12311 q^{11} +1.00000 q^{13} +5.56155 q^{15} -6.68466 q^{17} +3.12311 q^{19} -2.43845 q^{21} +8.00000 q^{23} +7.68466 q^{25} -5.56155 q^{27} -2.00000 q^{29} -4.00000 q^{31} -4.87689 q^{33} -5.56155 q^{35} -2.68466 q^{37} +1.56155 q^{39} +5.12311 q^{41} -9.56155 q^{43} -2.00000 q^{45} +12.6847 q^{47} -4.56155 q^{49} -10.4384 q^{51} -5.12311 q^{53} -11.1231 q^{55} +4.87689 q^{57} +3.12311 q^{59} +2.87689 q^{61} +0.876894 q^{63} +3.56155 q^{65} -3.12311 q^{67} +12.4924 q^{69} -4.68466 q^{71} -6.00000 q^{73} +12.0000 q^{75} +4.87689 q^{77} -8.00000 q^{79} -7.00000 q^{81} +14.2462 q^{83} -23.8078 q^{85} -3.12311 q^{87} +10.0000 q^{89} -1.56155 q^{91} -6.24621 q^{93} +11.1231 q^{95} +8.24621 q^{97} +1.75379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 3 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 + 3 * q^5 + q^7 + 3 * q^9	\( 2 q - q^{3} + 3 q^{5} + q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} + 7 q^{15} - q^{17} - 2 q^{19} - 9 q^{21} + 16 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 8 q^{31} - 18 q^{33} - 7 q^{35} + 7 q^{37} - q^{39} + 2 q^{41} - 15 q^{43} - 4 q^{45} + 13 q^{47} - 5 q^{49} - 25 q^{51} - 2 q^{53} - 14 q^{55} + 18 q^{57} - 2 q^{59} + 14 q^{61} + 10 q^{63} + 3 q^{65} + 2 q^{67} - 8 q^{69} + 3 q^{71} - 12 q^{73} + 24 q^{75} + 18 q^{77} - 16 q^{79} - 14 q^{81} + 12 q^{83} - 27 q^{85} + 2 q^{87} + 20 q^{89} + q^{91} + 4 q^{93} + 14 q^{95} + 20 q^{99}+O(q^{100}) \) 2 * q - q^3 + 3 * q^5 + q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 + 7 * q^15 - q^17 - 2 * q^19 - 9 * q^21 + 16 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 8 * q^31 - 18 * q^33 - 7 * q^35 + 7 * q^37 - q^39 + 2 * q^41 - 15 * q^43 - 4 * q^45 + 13 * q^47 - 5 * q^49 - 25 * q^51 - 2 * q^53 - 14 * q^55 + 18 * q^57 - 2 * q^59 + 14 * q^61 + 10 * q^63 + 3 * q^65 + 2 * q^67 - 8 * q^69 + 3 * q^71 - 12 * q^73 + 24 * q^75 + 18 * q^77 - 16 * q^79 - 14 * q^81 + 12 * q^83 - 27 * q^85 + 2 * q^87 + 20 * q^89 + q^91 + 4 * q^93 + 14 * q^95 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more