LMFDB - Embedded newform 1408.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1408,2,Mod(1,1408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1408, 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("1408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1408 = 2^{7} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1408.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:	$11.2429366046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} +3.82843 q^{5} +1.41421 q^{7} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} +3.82843 q^{5} +1.41421 q^{7} +2.82843 q^{9} -1.00000 q^{11} -4.24264 q^{13} +9.24264 q^{15} -4.58579 q^{17} +0.585786 q^{19} +3.41421 q^{21} +5.58579 q^{23} +9.65685 q^{25} -0.414214 q^{27} +8.00000 q^{29} +8.07107 q^{31} -2.41421 q^{33} +5.41421 q^{35} -5.82843 q^{37} -10.2426 q^{39} -7.07107 q^{41} -0.828427 q^{43} +10.8284 q^{45} -4.48528 q^{47} -5.00000 q^{49} -11.0711 q^{51} +2.82843 q^{53} -3.82843 q^{55} +1.41421 q^{57} -11.7279 q^{59} +5.65685 q^{61} +4.00000 q^{63} -16.2426 q^{65} +15.7279 q^{67} +13.4853 q^{69} -6.07107 q^{71} -15.0711 q^{73} +23.3137 q^{75} -1.41421 q^{77} -0.828427 q^{79} -9.48528 q^{81} +11.1716 q^{83} -17.5563 q^{85} +19.3137 q^{87} -3.82843 q^{89} -6.00000 q^{91} +19.4853 q^{93} +2.24264 q^{95} -9.00000 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{11} + 10 q^{15} - 12 q^{17} + 4 q^{19} + 4 q^{21} + 14 q^{23} + 8 q^{25} + 2 q^{27} + 16 q^{29} + 2 q^{31} - 2 q^{33} + 8 q^{35} - 6 q^{37} - 12 q^{39} + 4 q^{43} + 16 q^{45} + 8 q^{47} - 10 q^{49} - 8 q^{51} - 2 q^{55} + 2 q^{59} + 8 q^{63} - 24 q^{65} + 6 q^{67} + 10 q^{69} + 2 q^{71} - 16 q^{73} + 24 q^{75} + 4 q^{79} - 2 q^{81} + 28 q^{83} - 4 q^{85} + 16 q^{87} - 2 q^{89} - 12 q^{91} + 22 q^{93} - 4 q^{95} - 18 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 + 10 * q^15 - 12 * q^17 + 4 * q^19 + 4 * q^21 + 14 * q^23 + 8 * q^25 + 2 * q^27 + 16 * q^29 + 2 * q^31 - 2 * q^33 + 8 * q^35 - 6 * q^37 - 12 * q^39 + 4 * q^43 + 16 * q^45 + 8 * q^47 - 10 * q^49 - 8 * q^51 - 2 * q^55 + 2 * q^59 + 8 * q^63 - 24 * q^65 + 6 * q^67 + 10 * q^69 + 2 * q^71 - 16 * q^73 + 24 * q^75 + 4 * q^79 - 2 * q^81 + 28 * q^83 - 4 * q^85 + 16 * q^87 - 2 * q^89 - 12 * q^91 + 22 * q^93 - 4 * q^95 - 18 * q^97

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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	3.82843	1.71212	0.856062	−	0.516873i	$-0.172904\pi$
$5$	3.82843	1.71212	0.856062	+	0.516873i	$0.172904\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	0	0
$15$	9.24264	2.38644
$16$	0	0
$17$	−4.58579	−1.11222	−0.556108	−	0.831110i	$-0.687706\pi$
$17$	−4.58579	−1.11222	−0.556108	+	0.831110i	$0.687706\pi$
$18$	0	0
$19$	0.585786	0.134389	0.0671943	−	0.997740i	$-0.478595\pi$
$19$	0.585786	0.134389	0.0671943	+	0.997740i	$0.478595\pi$
$20$	0	0
$21$	3.41421	0.745042
$22$	0	0
$23$	5.58579	1.16472	0.582358	−	0.812932i	$-0.302130\pi$
$23$	5.58579	1.16472	0.582358	+	0.812932i	$0.302130\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	8.00000	1.48556	0.742781	−	0.669534i	$-0.233506\pi$
$29$	8.00000	1.48556	0.742781	+	0.669534i	$0.233506\pi$
$30$	0	0
$31$	8.07107	1.44961	0.724803	−	0.688956i	$-0.241931\pi$
$31$	8.07107	1.44961	0.724803	+	0.688956i	$0.241931\pi$
$32$	0	0
$33$	−2.41421	−0.420261
$34$	0	0
$35$	5.41421	0.915169
$36$	0	0
$37$	−5.82843	−0.958188	−0.479094	−	0.877764i	$-0.659035\pi$
$37$	−5.82843	−0.958188	−0.479094	+	0.877764i	$0.659035\pi$
$38$	0	0
$39$	−10.2426	−1.64014
$40$	0	0
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	−0.828427	−0.126334	−0.0631670	−	0.998003i	$-0.520120\pi$
$43$	−0.828427	−0.126334	−0.0631670	+	0.998003i	$0.520120\pi$
$44$	0	0
$45$	10.8284	1.61421
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−11.0711	−1.55026
$52$	0	0
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	−3.82843	−0.516225
$56$	0	0
$57$	1.41421	0.187317
$58$	0	0
$59$	−11.7279	−1.52685	−0.763423	−	0.645899i	$-0.776483\pi$
$59$	−11.7279	−1.52685	−0.763423	+	0.645899i	$0.776483\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	0	0
$65$	−16.2426	−2.01465
$66$	0	0
$67$	15.7279	1.92147	0.960736	−	0.277465i	$-0.0894943\pi$
$67$	15.7279	1.92147	0.960736	+	0.277465i	$0.0894943\pi$
$68$	0	0
$69$	13.4853	1.62344
$70$	0	0
$71$	−6.07107	−0.720503	−0.360252	−	0.932855i	$-0.617309\pi$
$71$	−6.07107	−0.720503	−0.360252	+	0.932855i	$0.617309\pi$
$72$	0	0
$73$	−15.0711	−1.76394	−0.881968	−	0.471310i	$-0.843781\pi$
$73$	−15.0711	−1.76394	−0.881968	+	0.471310i	$0.843781\pi$
$74$	0	0
$75$	23.3137	2.69204
$76$	0	0
$77$	−1.41421	−0.161165
$78$	0	0
$79$	−0.828427	−0.0932053	−0.0466027	−	0.998914i	$-0.514839\pi$
$79$	−0.828427	−0.0932053	−0.0466027	+	0.998914i	$0.514839\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	11.1716	1.22624	0.613120	−	0.789990i	$-0.289914\pi$
$83$	11.1716	1.22624	0.613120	+	0.789990i	$0.289914\pi$
$84$	0	0
$85$	−17.5563	−1.90425
$86$	0	0
$87$	19.3137	2.07065
$88$	0	0
$89$	−3.82843	−0.405812	−0.202906	−	0.979198i	$-0.565039\pi$
$89$	−3.82843	−0.405812	−0.202906	+	0.979198i	$0.565039\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	0	0
$93$	19.4853	2.02053
$94$	0	0
$95$	2.24264	0.230090
$96$	0	0
$97$	−9.00000	−0.913812	−0.456906	−	0.889515i	$-0.651042\pi$
$97$	−9.00000	−0.913812	−0.456906	+	0.889515i	$0.651042\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	−15.8995	−1.58206	−0.791029	−	0.611778i	$-0.790455\pi$
$101$	−15.8995	−1.58206	−0.791029	+	0.611778i	$0.790455\pi$
$102$	0	0
$103$	7.65685	0.754452	0.377226	−	0.926121i	$-0.376878\pi$
$103$	7.65685	0.754452	0.377226	+	0.926121i	$0.376878\pi$
$104$	0	0
$105$	13.0711	1.27561
$106$	0	0
$107$	3.07107	0.296891	0.148446	−	0.988921i	$-0.452573\pi$
$107$	3.07107	0.296891	0.148446	+	0.988921i	$0.452573\pi$
$108$	0	0
$109$	−13.3137	−1.27522	−0.637611	−	0.770358i	$-0.720077\pi$
$109$	−13.3137	−1.27522	−0.637611	+	0.770358i	$0.720077\pi$
$110$	0	0
$111$	−14.0711	−1.33557
$112$	0	0
$113$	7.48528	0.704156	0.352078	−	0.935971i	$-0.385475\pi$
$113$	7.48528	0.704156	0.352078	+	0.935971i	$0.385475\pi$
$114$	0	0
$115$	21.3848	1.99414
$116$	0	0
$117$	−12.0000	−1.10940
$118$	0	0
$119$	−6.48528	−0.594505
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−17.0711	−1.53925
$124$	0	0
$125$	17.8284	1.59462
$126$	0	0
$127$	−11.8995	−1.05591	−0.527955	−	0.849273i	$-0.677041\pi$
$127$	−11.8995	−1.05591	−0.527955	+	0.849273i	$0.677041\pi$
$128$	0	0
$129$	−2.00000	−0.176090
$130$	0	0
$131$	7.07107	0.617802	0.308901	−	0.951094i	$-0.400039\pi$
$131$	7.07107	0.617802	0.308901	+	0.951094i	$0.400039\pi$
$132$	0	0
$133$	0.828427	0.0718337
$134$	0	0
$135$	−1.58579	−0.136483
$136$	0	0
$137$	11.8284	1.01057	0.505285	−	0.862952i	$-0.331387\pi$
$137$	11.8284	1.01057	0.505285	+	0.862952i	$0.331387\pi$
$138$	0	0
$139$	16.2426	1.37768	0.688841	−	0.724912i	$-0.258120\pi$
$139$	16.2426	1.37768	0.688841	+	0.724912i	$0.258120\pi$
$140$	0	0
$141$	−10.8284	−0.911918
$142$	0	0
$143$	4.24264	0.354787
$144$	0	0
$145$	30.6274	2.54347
$146$	0	0
$147$	−12.0711	−0.995605
$148$	0	0
$149$	−12.5858	−1.03107	−0.515534	−	0.856869i	$-0.672406\pi$
$149$	−12.5858	−1.03107	−0.515534	+	0.856869i	$0.672406\pi$
$150$	0	0
$151$	4.82843	0.392932	0.196466	−	0.980511i	$-0.437053\pi$
$151$	4.82843	0.392932	0.196466	+	0.980511i	$0.437053\pi$
$152$	0	0
$153$	−12.9706	−1.04861
$154$	0	0
$155$	30.8995	2.48191
$156$	0	0
$157$	−1.48528	−0.118538	−0.0592692	−	0.998242i	$-0.518877\pi$
$157$	−1.48528	−0.118538	−0.0592692	+	0.998242i	$0.518877\pi$
$158$	0	0
$159$	6.82843	0.541529
$160$	0	0
$161$	7.89949	0.622567
$162$	0	0
$163$	−12.9706	−1.01593	−0.507966	−	0.861377i	$-0.669603\pi$
$163$	−12.9706	−1.01593	−0.507966	+	0.861377i	$0.669603\pi$
$164$	0	0
$165$	−9.24264	−0.719539
$166$	0	0
$167$	−14.2426	−1.10213	−0.551064	−	0.834463i	$-0.685778\pi$
$167$	−14.2426	−1.10213	−0.551064	+	0.834463i	$0.685778\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	1.65685	0.126703
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	13.6569	1.03236
$176$	0	0
$177$	−28.3137	−2.12819
$178$	0	0
$179$	−24.8995	−1.86108	−0.930538	−	0.366196i	$-0.880660\pi$
$179$	−24.8995	−1.86108	−0.930538	+	0.366196i	$0.880660\pi$
$180$	0	0
$181$	12.6569	0.940777	0.470388	−	0.882460i	$-0.344114\pi$
$181$	12.6569	0.940777	0.470388	+	0.882460i	$0.344114\pi$
$182$	0	0
$183$	13.6569	1.00954
$184$	0	0
$185$	−22.3137	−1.64054
$186$	0	0
$187$	4.58579	0.335346
$188$	0	0
$189$	−0.585786	−0.0426097
$190$	0	0
$191$	−11.2426	−0.813489	−0.406744	−	0.913542i	$-0.633336\pi$
$191$	−11.2426	−0.813489	−0.406744	+	0.913542i	$0.633336\pi$
$192$	0	0
$193$	−5.65685	−0.407189	−0.203595	−	0.979055i	$-0.565262\pi$
$193$	−5.65685	−0.407189	−0.203595	+	0.979055i	$0.565262\pi$
$194$	0	0
$195$	−39.2132	−2.80812
$196$	0	0
$197$	−3.65685	−0.260540	−0.130270	−	0.991479i	$-0.541584\pi$
$197$	−3.65685	−0.260540	−0.130270	+	0.991479i	$0.541584\pi$
$198$	0	0
$199$	11.3137	0.802008	0.401004	−	0.916076i	$-0.368661\pi$
$199$	11.3137	0.802008	0.401004	+	0.916076i	$0.368661\pi$
$200$	0	0
$201$	37.9706	2.67824
$202$	0	0
$203$	11.3137	0.794067
$204$	0	0
$205$	−27.0711	−1.89073
$206$	0	0
$207$	15.7990	1.09811
$208$	0	0
$209$	−0.585786	−0.0405197
$210$	0	0
$211$	−2.34315	−0.161309	−0.0806544	−	0.996742i	$-0.525701\pi$
$211$	−2.34315	−0.161309	−0.0806544	+	0.996742i	$0.525701\pi$
$212$	0	0
$213$	−14.6569	−1.00427
$214$	0	0
$215$	−3.17157	−0.216299
$216$	0	0
$217$	11.4142	0.774847
$218$	0	0
$219$	−36.3848	−2.45866
$220$	0	0
$221$	19.4558	1.30874
$222$	0	0
$223$	−13.2426	−0.886793	−0.443396	−	0.896326i	$-0.646227\pi$
$223$	−13.2426	−0.886793	−0.443396	+	0.896326i	$0.646227\pi$
$224$	0	0
$225$	27.3137	1.82091
$226$	0	0
$227$	28.7279	1.90674	0.953370	−	0.301805i	$-0.0975893\pi$
$227$	28.7279	1.90674	0.953370	+	0.301805i	$0.0975893\pi$
$228$	0	0
$229$	17.9706	1.18753	0.593764	−	0.804639i	$-0.297641\pi$
$229$	17.9706	1.18753	0.593764	+	0.804639i	$0.297641\pi$
$230$	0	0
$231$	−3.41421	−0.224639
$232$	0	0
$233$	−11.3137	−0.741186	−0.370593	−	0.928795i	$-0.620845\pi$
$233$	−11.3137	−0.741186	−0.370593	+	0.928795i	$0.620845\pi$
$234$	0	0
$235$	−17.1716	−1.12015
$236$	0	0
$237$	−2.00000	−0.129914
$238$	0	0
$239$	15.1716	0.981367	0.490684	−	0.871338i	$-0.336747\pi$
$239$	15.1716	0.981367	0.490684	+	0.871338i	$0.336747\pi$
$240$	0	0
$241$	4.68629	0.301871	0.150935	−	0.988544i	$-0.451771\pi$
$241$	4.68629	0.301871	0.150935	+	0.988544i	$0.451771\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	−19.1421	−1.22295
$246$	0	0
$247$	−2.48528	−0.158135
$248$	0	0
$249$	26.9706	1.70919
$250$	0	0
$251$	6.89949	0.435492	0.217746	−	0.976005i	$-0.430130\pi$
$251$	6.89949	0.435492	0.217746	+	0.976005i	$0.430130\pi$
$252$	0	0
$253$	−5.58579	−0.351175
$254$	0	0
$255$	−42.3848	−2.65424
$256$	0	0
$257$	−26.1421	−1.63070	−0.815351	−	0.578967i	$-0.803456\pi$
$257$	−26.1421	−1.63070	−0.815351	+	0.578967i	$0.803456\pi$
$258$	0	0
$259$	−8.24264	−0.512173
$260$	0	0
$261$	22.6274	1.40060
$262$	0	0
$263$	22.8701	1.41023	0.705114	−	0.709094i	$-0.250896\pi$
$263$	22.8701	1.41023	0.705114	+	0.709094i	$0.250896\pi$
$264$	0	0
$265$	10.8284	0.665185
$266$	0	0
$267$	−9.24264	−0.565640
$268$	0	0
$269$	13.6569	0.832673	0.416337	−	0.909211i	$-0.363314\pi$
$269$	13.6569	0.832673	0.416337	+	0.909211i	$0.363314\pi$
$270$	0	0
$271$	−19.4142	−1.17933	−0.589665	−	0.807648i	$-0.700740\pi$
$271$	−19.4142	−1.17933	−0.589665	+	0.807648i	$0.700740\pi$
$272$	0	0
$273$	−14.4853	−0.876689
$274$	0	0
$275$	−9.65685	−0.582330
$276$	0	0
$277$	18.9706	1.13983	0.569915	−	0.821703i	$-0.306976\pi$
$277$	18.9706	1.13983	0.569915	+	0.821703i	$0.306976\pi$
$278$	0	0
$279$	22.8284	1.36670
$280$	0	0
$281$	6.97056	0.415829	0.207914	−	0.978147i	$-0.433332\pi$
$281$	6.97056	0.415829	0.207914	+	0.978147i	$0.433332\pi$
$282$	0	0
$283$	−2.34315	−0.139286	−0.0696428	−	0.997572i	$-0.522186\pi$
$283$	−2.34315	−0.139286	−0.0696428	+	0.997572i	$0.522186\pi$
$284$	0	0
$285$	5.41421	0.320710
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	4.02944	0.237026
$290$	0	0
$291$	−21.7279	−1.27371
$292$	0	0
$293$	27.5563	1.60986	0.804930	−	0.593370i	$-0.202203\pi$
$293$	27.5563	1.60986	0.804930	+	0.593370i	$0.202203\pi$
$294$	0	0
$295$	−44.8995	−2.61415
$296$	0	0
$297$	0.414214	0.0240351
$298$	0	0
$299$	−23.6985	−1.37052
$300$	0	0
$301$	−1.17157	−0.0675283
$302$	0	0
$303$	−38.3848	−2.20515
$304$	0	0
$305$	21.6569	1.24007
$306$	0	0
$307$	25.0711	1.43088	0.715441	−	0.698673i	$-0.246226\pi$
$307$	25.0711	1.43088	0.715441	+	0.698673i	$0.246226\pi$
$308$	0	0
$309$	18.4853	1.05159
$310$	0	0
$311$	15.5147	0.879759	0.439879	−	0.898057i	$-0.355021\pi$
$311$	15.5147	0.879759	0.439879	+	0.898057i	$0.355021\pi$
$312$	0	0
$313$	11.8284	0.668582	0.334291	−	0.942470i	$-0.391503\pi$
$313$	11.8284	0.668582	0.334291	+	0.942470i	$0.391503\pi$
$314$	0	0
$315$	15.3137	0.862830
$316$	0	0
$317$	2.51472	0.141241	0.0706203	−	0.997503i	$-0.477502\pi$
$317$	2.51472	0.141241	0.0706203	+	0.997503i	$0.477502\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	7.41421	0.413821
$322$	0	0
$323$	−2.68629	−0.149469
$324$	0	0
$325$	−40.9706	−2.27264
$326$	0	0
$327$	−32.1421	−1.77746
$328$	0	0
$329$	−6.34315	−0.349709
$330$	0	0
$331$	−29.3848	−1.61513	−0.807567	−	0.589776i	$-0.799216\pi$
$331$	−29.3848	−1.61513	−0.807567	+	0.589776i	$0.799216\pi$
$332$	0	0
$333$	−16.4853	−0.903388
$334$	0	0
$335$	60.2132	3.28980
$336$	0	0
$337$	−18.7279	−1.02017	−0.510087	−	0.860123i	$-0.670387\pi$
$337$	−18.7279	−1.02017	−0.510087	+	0.860123i	$0.670387\pi$
$338$	0	0
$339$	18.0711	0.981486
$340$	0	0
$341$	−8.07107	−0.437073
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	51.6274	2.77953
$346$	0	0
$347$	2.92893	0.157233	0.0786167	−	0.996905i	$-0.474950\pi$
$347$	2.92893	0.157233	0.0786167	+	0.996905i	$0.474950\pi$
$348$	0	0
$349$	−20.5858	−1.10193	−0.550966	−	0.834528i	$-0.685741\pi$
$349$	−20.5858	−1.10193	−0.550966	+	0.834528i	$0.685741\pi$
$350$	0	0
$351$	1.75736	0.0938009
$352$	0	0
$353$	16.7990	0.894120	0.447060	−	0.894504i	$-0.352471\pi$
$353$	16.7990	0.894120	0.447060	+	0.894504i	$0.352471\pi$
$354$	0	0
$355$	−23.2426	−1.23359
$356$	0	0
$357$	−15.6569	−0.828649
$358$	0	0
$359$	−19.8995	−1.05026	−0.525128	−	0.851024i	$-0.675982\pi$
$359$	−19.8995	−1.05026	−0.525128	+	0.851024i	$0.675982\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	2.41421	0.126713
$364$	0	0
$365$	−57.6985	−3.02008
$366$	0	0
$367$	−8.89949	−0.464550	−0.232275	−	0.972650i	$-0.574617\pi$
$367$	−8.89949	−0.464550	−0.232275	+	0.972650i	$0.574617\pi$
$368$	0	0
$369$	−20.0000	−1.04116
$370$	0	0
$371$	4.00000	0.207670
$372$	0	0
$373$	9.55635	0.494809	0.247405	−	0.968912i	$-0.420422\pi$
$373$	9.55635	0.494809	0.247405	+	0.968912i	$0.420422\pi$
$374$	0	0
$375$	43.0416	2.22266
$376$	0	0
$377$	−33.9411	−1.74806
$378$	0	0
$379$	−5.44365	−0.279622	−0.139811	−	0.990178i	$-0.544649\pi$
$379$	−5.44365	−0.279622	−0.139811	+	0.990178i	$0.544649\pi$
$380$	0	0
$381$	−28.7279	−1.47178
$382$	0	0
$383$	−4.41421	−0.225556	−0.112778	−	0.993620i	$-0.535975\pi$
$383$	−4.41421	−0.225556	−0.112778	+	0.993620i	$0.535975\pi$
$384$	0	0
$385$	−5.41421	−0.275934
$386$	0	0
$387$	−2.34315	−0.119109
$388$	0	0
$389$	−7.97056	−0.404124	−0.202062	−	0.979373i	$-0.564764\pi$
$389$	−7.97056	−0.404124	−0.202062	+	0.979373i	$0.564764\pi$
$390$	0	0
$391$	−25.6152	−1.29542
$392$	0	0
$393$	17.0711	0.861121
$394$	0	0
$395$	−3.17157	−0.159579
$396$	0	0
$397$	13.1716	0.661062	0.330531	−	0.943795i	$-0.392772\pi$
$397$	13.1716	0.661062	0.330531	+	0.943795i	$0.392772\pi$
$398$	0	0
$399$	2.00000	0.100125
$400$	0	0
$401$	−21.4558	−1.07145	−0.535727	−	0.844391i	$-0.679962\pi$
$401$	−21.4558	−1.07145	−0.535727	+	0.844391i	$0.679962\pi$
$402$	0	0
$403$	−34.2426	−1.70575
$404$	0	0
$405$	−36.3137	−1.80444
$406$	0	0
$407$	5.82843	0.288904
$408$	0	0
$409$	24.6274	1.21775	0.608874	−	0.793267i	$-0.291622\pi$
$409$	24.6274	1.21775	0.608874	+	0.793267i	$0.291622\pi$
$410$	0	0
$411$	28.5563	1.40858
$412$	0	0
$413$	−16.5858	−0.816133
$414$	0	0
$415$	42.7696	2.09947
$416$	0	0
$417$	39.2132	1.92028
$418$	0	0
$419$	35.7990	1.74890	0.874448	−	0.485120i	$-0.161224\pi$
$419$	35.7990	1.74890	0.874448	+	0.485120i	$0.161224\pi$
$420$	0	0
$421$	7.51472	0.366245	0.183122	−	0.983090i	$-0.441380\pi$
$421$	7.51472	0.366245	0.183122	+	0.983090i	$0.441380\pi$
$422$	0	0
$423$	−12.6863	−0.616829
$424$	0	0
$425$	−44.2843	−2.14810
$426$	0	0
$427$	8.00000	0.387147
$428$	0	0
$429$	10.2426	0.494519
$430$	0	0
$431$	15.5563	0.749323	0.374661	−	0.927162i	$-0.377759\pi$
$431$	15.5563	0.749323	0.374661	+	0.927162i	$0.377759\pi$
$432$	0	0
$433$	1.68629	0.0810380	0.0405190	−	0.999179i	$-0.487099\pi$
$433$	1.68629	0.0810380	0.0405190	+	0.999179i	$0.487099\pi$
$434$	0	0
$435$	73.9411	3.54521
$436$	0	0
$437$	3.27208	0.156525
$438$	0	0
$439$	−15.3137	−0.730883	−0.365442	−	0.930834i	$-0.619082\pi$
$439$	−15.3137	−0.730883	−0.365442	+	0.930834i	$0.619082\pi$
$440$	0	0
$441$	−14.1421	−0.673435
$442$	0	0
$443$	10.8995	0.517851	0.258925	−	0.965897i	$-0.416632\pi$
$443$	10.8995	0.517851	0.258925	+	0.965897i	$0.416632\pi$
$444$	0	0
$445$	−14.6569	−0.694802
$446$	0	0
$447$	−30.3848	−1.43715
$448$	0	0
$449$	17.1421	0.808987	0.404494	−	0.914541i	$-0.367448\pi$
$449$	17.1421	0.808987	0.404494	+	0.914541i	$0.367448\pi$
$450$	0	0
$451$	7.07107	0.332964
$452$	0	0
$453$	11.6569	0.547687
$454$	0	0
$455$	−22.9706	−1.07688
$456$	0	0
$457$	−20.0000	−0.935561	−0.467780	−	0.883845i	$-0.654946\pi$
$457$	−20.0000	−0.935561	−0.467780	+	0.883845i	$0.654946\pi$
$458$	0	0
$459$	1.89949	0.0886608
$460$	0	0
$461$	20.9706	0.976696	0.488348	−	0.872649i	$-0.337600\pi$
$461$	20.9706	0.976696	0.488348	+	0.872649i	$0.337600\pi$
$462$	0	0
$463$	19.7279	0.916834	0.458417	−	0.888737i	$-0.348417\pi$
$463$	19.7279	0.916834	0.458417	+	0.888737i	$0.348417\pi$
$464$	0	0
$465$	74.5980	3.45940
$466$	0	0
$467$	−1.38478	−0.0640798	−0.0320399	−	0.999487i	$-0.510200\pi$
$467$	−1.38478	−0.0640798	−0.0320399	+	0.999487i	$0.510200\pi$
$468$	0	0
$469$	22.2426	1.02707
$470$	0	0
$471$	−3.58579	−0.165224
$472$	0	0
$473$	0.828427	0.0380911
$474$	0	0
$475$	5.65685	0.259554
$476$	0	0
$477$	8.00000	0.366295
$478$	0	0
$479$	13.5563	0.619405	0.309703	−	0.950833i	$-0.399770\pi$
$479$	13.5563	0.619405	0.309703	+	0.950833i	$0.399770\pi$
$480$	0	0
$481$	24.7279	1.12750
$482$	0	0
$483$	19.0711	0.867764
$484$	0	0
$485$	−34.4558	−1.56456
$486$	0	0
$487$	−16.6985	−0.756681	−0.378340	−	0.925667i	$-0.623505\pi$
$487$	−16.6985	−0.756681	−0.378340	+	0.925667i	$0.623505\pi$
$488$	0	0
$489$	−31.3137	−1.41605
$490$	0	0
$491$	−18.6274	−0.840644	−0.420322	−	0.907375i	$-0.638083\pi$
$491$	−18.6274	−0.840644	−0.420322	+	0.907375i	$0.638083\pi$
$492$	0	0
$493$	−36.6863	−1.65227
$494$	0	0
$495$	−10.8284	−0.486702
$496$	0	0
$497$	−8.58579	−0.385125
$498$	0	0
$499$	15.3137	0.685536	0.342768	−	0.939420i	$-0.388636\pi$
$499$	15.3137	0.685536	0.342768	+	0.939420i	$0.388636\pi$
$500$	0	0
$501$	−34.3848	−1.53620
$502$	0	0
$503$	9.51472	0.424240	0.212120	−	0.977244i	$-0.431963\pi$
$503$	9.51472	0.424240	0.212120	+	0.977244i	$0.431963\pi$
$504$	0	0
$505$	−60.8701	−2.70868
$506$	0	0
$507$	12.0711	0.536095
$508$	0	0
$509$	38.6569	1.71343	0.856717	−	0.515786i	$-0.172500\pi$
$509$	38.6569	1.71343	0.856717	+	0.515786i	$0.172500\pi$
$510$	0	0
$511$	−21.3137	−0.942863
$512$	0	0
$513$	−0.242641	−0.0107128
$514$	0	0
$515$	29.3137	1.29172
$516$	0	0
$517$	4.48528	0.197262
$518$	0	0
$519$	43.4558	1.90750
$520$	0	0
$521$	16.6569	0.729750	0.364875	−	0.931057i	$-0.381112\pi$
$521$	16.6569	0.729750	0.364875	+	0.931057i	$0.381112\pi$
$522$	0	0
$523$	−33.3553	−1.45853	−0.729264	−	0.684233i	$-0.760137\pi$
$523$	−33.3553	−1.45853	−0.729264	+	0.684233i	$0.760137\pi$
$524$	0	0
$525$	32.9706	1.43895
$526$	0	0
$527$	−37.0122	−1.61228
$528$	0	0
$529$	8.20101	0.356566
$530$	0	0
$531$	−33.1716	−1.43952
$532$	0	0
$533$	30.0000	1.29944
$534$	0	0
$535$	11.7574	0.508315
$536$	0	0
$537$	−60.1127	−2.59405
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	−23.0711	−0.991903	−0.495951	−	0.868350i	$-0.665181\pi$
$541$	−23.0711	−0.991903	−0.495951	+	0.868350i	$0.665181\pi$
$542$	0	0
$543$	30.5563	1.31130
$544$	0	0
$545$	−50.9706	−2.18334
$546$	0	0
$547$	6.34315	0.271213	0.135607	−	0.990763i	$-0.456702\pi$
$547$	6.34315	0.271213	0.135607	+	0.990763i	$0.456702\pi$
$548$	0	0
$549$	16.0000	0.682863
$550$	0	0
$551$	4.68629	0.199643
$552$	0	0
$553$	−1.17157	−0.0498203
$554$	0	0
$555$	−53.8701	−2.28666
$556$	0	0
$557$	6.72792	0.285071	0.142536	−	0.989790i	$-0.454474\pi$
$557$	6.72792	0.285071	0.142536	+	0.989790i	$0.454474\pi$
$558$	0	0
$559$	3.51472	0.148657
$560$	0	0
$561$	11.0711	0.467421
$562$	0	0
$563$	−18.6274	−0.785052	−0.392526	−	0.919741i	$-0.628399\pi$
$563$	−18.6274	−0.785052	−0.392526	+	0.919741i	$0.628399\pi$
$564$	0	0
$565$	28.6569	1.20560
$566$	0	0
$567$	−13.4142	−0.563344
$568$	0	0
$569$	26.6274	1.11628	0.558140	−	0.829747i	$-0.311515\pi$
$569$	26.6274	1.11628	0.558140	+	0.829747i	$0.311515\pi$
$570$	0	0
$571$	−6.34315	−0.265452	−0.132726	−	0.991153i	$-0.542373\pi$
$571$	−6.34315	−0.265452	−0.132726	+	0.991153i	$0.542373\pi$
$572$	0	0
$573$	−27.1421	−1.13388
$574$	0	0
$575$	53.9411	2.24950
$576$	0	0
$577$	−21.1421	−0.880159	−0.440079	−	0.897959i	$-0.645050\pi$
$577$	−21.1421	−0.880159	−0.440079	+	0.897959i	$0.645050\pi$
$578$	0	0
$579$	−13.6569	−0.567559
$580$	0	0
$581$	15.7990	0.655453
$582$	0	0
$583$	−2.82843	−0.117141
$584$	0	0
$585$	−45.9411	−1.89943
$586$	0	0
$587$	13.3137	0.549516	0.274758	−	0.961513i	$-0.411402\pi$
$587$	13.3137	0.549516	0.274758	+	0.961513i	$0.411402\pi$
$588$	0	0
$589$	4.72792	0.194811
$590$	0	0
$591$	−8.82843	−0.363153
$592$	0	0
$593$	25.9411	1.06527	0.532637	−	0.846344i	$-0.321201\pi$
$593$	25.9411	1.06527	0.532637	+	0.846344i	$0.321201\pi$
$594$	0	0
$595$	−24.8284	−1.01787
$596$	0	0
$597$	27.3137	1.11788
$598$	0	0
$599$	−5.65685	−0.231133	−0.115566	−	0.993300i	$-0.536868\pi$
$599$	−5.65685	−0.231133	−0.115566	+	0.993300i	$0.536868\pi$
$600$	0	0
$601$	−23.4142	−0.955086	−0.477543	−	0.878608i	$-0.658472\pi$
$601$	−23.4142	−0.955086	−0.477543	+	0.878608i	$0.658472\pi$
$602$	0	0
$603$	44.4853	1.81158
$604$	0	0
$605$	3.82843	0.155648
$606$	0	0
$607$	−17.7990	−0.722439	−0.361219	−	0.932481i	$-0.617639\pi$
$607$	−17.7990	−0.722439	−0.361219	+	0.932481i	$0.617639\pi$
$608$	0	0
$609$	27.3137	1.10681
$610$	0	0
$611$	19.0294	0.769849
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	−65.3553	−2.63538
$616$	0	0
$617$	16.2843	0.655580	0.327790	−	0.944751i	$-0.393696\pi$
$617$	16.2843	0.655580	0.327790	+	0.944751i	$0.393696\pi$
$618$	0	0
$619$	−29.7279	−1.19487	−0.597433	−	0.801919i	$-0.703813\pi$
$619$	−29.7279	−1.19487	−0.597433	+	0.801919i	$0.703813\pi$
$620$	0	0
$621$	−2.31371	−0.0928459
$622$	0	0
$623$	−5.41421	−0.216916
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−1.41421	−0.0564782
$628$	0	0
$629$	26.7279	1.06571
$630$	0	0
$631$	−11.2426	−0.447562	−0.223781	−	0.974639i	$-0.571840\pi$
$631$	−11.2426	−0.447562	−0.223781	+	0.974639i	$0.571840\pi$
$632$	0	0
$633$	−5.65685	−0.224840
$634$	0	0
$635$	−45.5563	−1.80785
$636$	0	0
$637$	21.2132	0.840498
$638$	0	0
$639$	−17.1716	−0.679297
$640$	0	0
$641$	−15.9706	−0.630799	−0.315400	−	0.948959i	$-0.602139\pi$
$641$	−15.9706	−0.630799	−0.315400	+	0.948959i	$0.602139\pi$
$642$	0	0
$643$	−34.5563	−1.36277	−0.681385	−	0.731925i	$-0.738622\pi$
$643$	−34.5563	−1.36277	−0.681385	+	0.731925i	$0.738622\pi$
$644$	0	0
$645$	−7.65685	−0.301488
$646$	0	0
$647$	13.7279	0.539700	0.269850	−	0.962902i	$-0.413026\pi$
$647$	13.7279	0.539700	0.269850	+	0.962902i	$0.413026\pi$
$648$	0	0
$649$	11.7279	0.460361
$650$	0	0
$651$	27.5563	1.08002
$652$	0	0
$653$	38.6569	1.51276	0.756380	−	0.654133i	$-0.226966\pi$
$653$	38.6569	1.51276	0.756380	+	0.654133i	$0.226966\pi$
$654$	0	0
$655$	27.0711	1.05775
$656$	0	0
$657$	−42.6274	−1.66305
$658$	0	0
$659$	43.4558	1.69280	0.846400	−	0.532548i	$-0.178765\pi$
$659$	43.4558	1.69280	0.846400	+	0.532548i	$0.178765\pi$
$660$	0	0
$661$	−22.3137	−0.867903	−0.433951	−	0.900936i	$-0.642881\pi$
$661$	−22.3137	−0.867903	−0.433951	+	0.900936i	$0.642881\pi$
$662$	0	0
$663$	46.9706	1.82419
$664$	0	0
$665$	3.17157	0.122988
$666$	0	0
$667$	44.6863	1.73026
$668$	0	0
$669$	−31.9706	−1.23605
$670$	0	0
$671$	−5.65685	−0.218380
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−14.2426	−0.547389	−0.273695	−	0.961817i	$-0.588246\pi$
$677$	−14.2426	−0.547389	−0.273695	+	0.961817i	$0.588246\pi$
$678$	0	0
$679$	−12.7279	−0.488453
$680$	0	0
$681$	69.3553	2.65770
$682$	0	0
$683$	33.1716	1.26927	0.634637	−	0.772810i	$-0.281149\pi$
$683$	33.1716	1.26927	0.634637	+	0.772810i	$0.281149\pi$
$684$	0	0
$685$	45.2843	1.73022
$686$	0	0
$687$	43.3848	1.65523
$688$	0	0
$689$	−12.0000	−0.457164
$690$	0	0
$691$	−4.21320	−0.160278	−0.0801389	−	0.996784i	$-0.525536\pi$
$691$	−4.21320	−0.160278	−0.0801389	+	0.996784i	$0.525536\pi$
$692$	0	0
$693$	−4.00000	−0.151947
$694$	0	0
$695$	62.1838	2.35876
$696$	0	0
$697$	32.4264	1.22824
$698$	0	0
$699$	−27.3137	−1.03310
$700$	0	0
$701$	−2.72792	−0.103032	−0.0515161	−	0.998672i	$-0.516405\pi$
$701$	−2.72792	−0.103032	−0.0515161	+	0.998672i	$0.516405\pi$
$702$	0	0
$703$	−3.41421	−0.128770
$704$	0	0
$705$	−41.4558	−1.56132
$706$	0	0
$707$	−22.4853	−0.845646
$708$	0	0
$709$	41.0000	1.53979	0.769894	−	0.638172i	$-0.220309\pi$
$709$	41.0000	1.53979	0.769894	+	0.638172i	$0.220309\pi$
$710$	0	0
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	45.0833	1.68838
$714$	0	0
$715$	16.2426	0.607440
$716$	0	0
$717$	36.6274	1.36788
$718$	0	0
$719$	52.5563	1.96002	0.980011	−	0.198946i	$-0.0637517\pi$
$719$	52.5563	1.96002	0.980011	+	0.198946i	$0.0637517\pi$
$720$	0	0
$721$	10.8284	0.403272
$722$	0	0
$723$	11.3137	0.420761
$724$	0	0
$725$	77.2548	2.86917
$726$	0	0
$727$	26.7574	0.992376	0.496188	−	0.868215i	$-0.334733\pi$
$727$	26.7574	0.992376	0.496188	+	0.868215i	$0.334733\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	3.79899	0.140511
$732$	0	0
$733$	0.686292	0.0253488	0.0126744	−	0.999920i	$-0.495966\pi$
$733$	0.686292	0.0253488	0.0126744	+	0.999920i	$0.495966\pi$
$734$	0	0
$735$	−46.2132	−1.70460
$736$	0	0
$737$	−15.7279	−0.579345
$738$	0	0
$739$	7.85786	0.289056	0.144528	−	0.989501i	$-0.453834\pi$
$739$	7.85786	0.289056	0.144528	+	0.989501i	$0.453834\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−48.1838	−1.76532
$746$	0	0
$747$	31.5980	1.15611
$748$	0	0
$749$	4.34315	0.158695
$750$	0	0
$751$	6.75736	0.246580	0.123290	−	0.992371i	$-0.460656\pi$
$751$	6.75736	0.246580	0.123290	+	0.992371i	$0.460656\pi$
$752$	0	0
$753$	16.6569	0.607010
$754$	0	0
$755$	18.4853	0.672748
$756$	0	0
$757$	−8.28427	−0.301097	−0.150548	−	0.988603i	$-0.548104\pi$
$757$	−8.28427	−0.301097	−0.150548	+	0.988603i	$0.548104\pi$
$758$	0	0
$759$	−13.4853	−0.489485
$760$	0	0
$761$	−36.7279	−1.33139	−0.665693	−	0.746226i	$-0.731864\pi$
$761$	−36.7279	−1.33139	−0.665693	+	0.746226i	$0.731864\pi$
$762$	0	0
$763$	−18.8284	−0.681635
$764$	0	0
$765$	−49.6569	−1.79535
$766$	0	0
$767$	49.7574	1.79663
$768$	0	0
$769$	10.5858	0.381733	0.190867	−	0.981616i	$-0.438870\pi$
$769$	10.5858	0.381733	0.190867	+	0.981616i	$0.438870\pi$
$770$	0	0
$771$	−63.1127	−2.27295
$772$	0	0
$773$	−12.6274	−0.454177	−0.227088	−	0.973874i	$-0.572921\pi$
$773$	−12.6274	−0.454177	−0.227088	+	0.973874i	$0.572921\pi$
$774$	0	0
$775$	77.9411	2.79973
$776$	0	0
$777$	−19.8995	−0.713890
$778$	0	0
$779$	−4.14214	−0.148407
$780$	0	0
$781$	6.07107	0.217240
$782$	0	0
$783$	−3.31371	−0.118422
$784$	0	0
$785$	−5.68629	−0.202952
$786$	0	0
$787$	50.6274	1.80467	0.902336	−	0.431033i	$-0.141851\pi$
$787$	50.6274	1.80467	0.902336	+	0.431033i	$0.141851\pi$
$788$	0	0
$789$	55.2132	1.96564
$790$	0	0
$791$	10.5858	0.376387
$792$	0	0
$793$	−24.0000	−0.852265
$794$	0	0
$795$	26.1421	0.927166
$796$	0	0
$797$	−17.6274	−0.624395	−0.312198	−	0.950017i	$-0.601065\pi$
$797$	−17.6274	−0.624395	−0.312198	+	0.950017i	$0.601065\pi$
$798$	0	0
$799$	20.5685	0.727663
$800$	0	0
$801$	−10.8284	−0.382604
$802$	0	0
$803$	15.0711	0.531846
$804$	0	0
$805$	30.2426	1.06591
$806$	0	0
$807$	32.9706	1.16062
$808$	0	0
$809$	−35.0711	−1.23303	−0.616517	−	0.787342i	$-0.711457\pi$
$809$	−35.0711	−1.23303	−0.616517	+	0.787342i	$0.711457\pi$
$810$	0	0
$811$	2.48528	0.0872700	0.0436350	−	0.999048i	$-0.486106\pi$
$811$	2.48528	0.0872700	0.0436350	+	0.999048i	$0.486106\pi$
$812$	0	0
$813$	−46.8701	−1.64380
$814$	0	0
$815$	−49.6569	−1.73940
$816$	0	0
$817$	−0.485281	−0.0169778
$818$	0	0
$819$	−16.9706	−0.592999
$820$	0	0
$821$	−3.37258	−0.117704	−0.0588520	−	0.998267i	$-0.518744\pi$
$821$	−3.37258	−0.117704	−0.0588520	+	0.998267i	$0.518744\pi$
$822$	0	0
$823$	−5.92893	−0.206670	−0.103335	−	0.994647i	$-0.532951\pi$
$823$	−5.92893	−0.206670	−0.103335	+	0.994647i	$0.532951\pi$
$824$	0	0
$825$	−23.3137	−0.811679
$826$	0	0
$827$	21.0711	0.732713	0.366356	−	0.930475i	$-0.380605\pi$
$827$	21.0711	0.732713	0.366356	+	0.930475i	$0.380605\pi$
$828$	0	0
$829$	6.11270	0.212303	0.106151	−	0.994350i	$-0.466147\pi$
$829$	6.11270	0.212303	0.106151	+	0.994350i	$0.466147\pi$
$830$	0	0
$831$	45.7990	1.58875
$832$	0	0
$833$	22.9289	0.794440
$834$	0	0
$835$	−54.5269	−1.88698
$836$	0	0
$837$	−3.34315	−0.115556
$838$	0	0
$839$	35.8701	1.23837	0.619186	−	0.785244i	$-0.287463\pi$
$839$	35.8701	1.23837	0.619186	+	0.785244i	$0.287463\pi$
$840$	0	0
$841$	35.0000	1.20690
$842$	0	0
$843$	16.8284	0.579602
$844$	0	0
$845$	19.1421	0.658509
$846$	0	0
$847$	1.41421	0.0485930
$848$	0	0
$849$	−5.65685	−0.194143
$850$	0	0
$851$	−32.5563	−1.11602
$852$	0	0
$853$	33.3137	1.14064	0.570320	−	0.821423i	$-0.306819\pi$
$853$	33.3137	1.14064	0.570320	+	0.821423i	$0.306819\pi$
$854$	0	0
$855$	6.34315	0.216931
$856$	0	0
$857$	43.3137	1.47957	0.739784	−	0.672844i	$-0.234928\pi$
$857$	43.3137	1.47957	0.739784	+	0.672844i	$0.234928\pi$
$858$	0	0
$859$	−14.6985	−0.501506	−0.250753	−	0.968051i	$-0.580678\pi$
$859$	−14.6985	−0.501506	−0.250753	+	0.968051i	$0.580678\pi$
$860$	0	0
$861$	−24.1421	−0.822762
$862$	0	0
$863$	−17.9411	−0.610723	−0.305362	−	0.952236i	$-0.598777\pi$
$863$	−17.9411	−0.610723	−0.305362	+	0.952236i	$0.598777\pi$
$864$	0	0
$865$	68.9117	2.34307
$866$	0	0
$867$	9.72792	0.330378
$868$	0	0
$869$	0.828427	0.0281025
$870$	0	0
$871$	−66.7279	−2.26099
$872$	0	0
$873$	−25.4558	−0.861550
$874$	0	0
$875$	25.2132	0.852362
$876$	0	0
$877$	−32.5269	−1.09836	−0.549178	−	0.835705i	$-0.685059\pi$
$877$	−32.5269	−1.09836	−0.549178	+	0.835705i	$0.685059\pi$
$878$	0	0
$879$	66.5269	2.24390
$880$	0	0
$881$	−2.17157	−0.0731621	−0.0365811	−	0.999331i	$-0.511647\pi$
$881$	−2.17157	−0.0731621	−0.0365811	+	0.999331i	$0.511647\pi$
$882$	0	0
$883$	29.3137	0.986485	0.493242	−	0.869892i	$-0.335812\pi$
$883$	29.3137	0.986485	0.493242	+	0.869892i	$0.335812\pi$
$884$	0	0
$885$	−108.397	−3.64372
$886$	0	0
$887$	46.4853	1.56082	0.780411	−	0.625266i	$-0.215010\pi$
$887$	46.4853	1.56082	0.780411	+	0.625266i	$0.215010\pi$
$888$	0	0
$889$	−16.8284	−0.564407
$890$	0	0
$891$	9.48528	0.317769
$892$	0	0
$893$	−2.62742	−0.0879232
$894$	0	0
$895$	−95.3259	−3.18639
$896$	0	0
$897$	−57.2132	−1.91029
$898$	0	0
$899$	64.5685	2.15348
$900$	0	0
$901$	−12.9706	−0.432112
$902$	0	0
$903$	−2.82843	−0.0941242
$904$	0	0
$905$	48.4558	1.61073
$906$	0	0
$907$	−20.9706	−0.696316	−0.348158	−	0.937436i	$-0.613193\pi$
$907$	−20.9706	−0.696316	−0.348158	+	0.937436i	$0.613193\pi$
$908$	0	0
$909$	−44.9706	−1.49158
$910$	0	0
$911$	−51.7990	−1.71618	−0.858089	−	0.513502i	$-0.828348\pi$
$911$	−51.7990	−1.71618	−0.858089	+	0.513502i	$0.828348\pi$
$912$	0	0
$913$	−11.1716	−0.369725
$914$	0	0
$915$	52.2843	1.72846
$916$	0	0
$917$	10.0000	0.330229
$918$	0	0
$919$	15.4558	0.509841	0.254921	−	0.966962i	$-0.417951\pi$
$919$	15.4558	0.509841	0.254921	+	0.966962i	$0.417951\pi$
$920$	0	0
$921$	60.5269	1.99443
$922$	0	0
$923$	25.7574	0.847814
$924$	0	0
$925$	−56.2843	−1.85062
$926$	0	0
$927$	21.6569	0.711304
$928$	0	0
$929$	7.79899	0.255877	0.127938	−	0.991782i	$-0.459164\pi$
$929$	7.79899	0.255877	0.127938	+	0.991782i	$0.459164\pi$
$930$	0	0
$931$	−2.92893	−0.0959919
$932$	0	0
$933$	37.4558	1.22625
$934$	0	0
$935$	17.5563	0.574154
$936$	0	0
$937$	−51.5563	−1.68427	−0.842136	−	0.539265i	$-0.818702\pi$
$937$	−51.5563	−1.68427	−0.842136	+	0.539265i	$0.818702\pi$
$938$	0	0
$939$	28.5563	0.931901
$940$	0	0
$941$	10.6863	0.348363	0.174182	−	0.984714i	$-0.444272\pi$
$941$	10.6863	0.348363	0.174182	+	0.984714i	$0.444272\pi$
$942$	0	0
$943$	−39.4975	−1.28621
$944$	0	0
$945$	−2.24264	−0.0729531
$946$	0	0
$947$	−46.4142	−1.50826	−0.754130	−	0.656726i	$-0.771941\pi$
$947$	−46.4142	−1.50826	−0.754130	+	0.656726i	$0.771941\pi$
$948$	0	0
$949$	63.9411	2.07562
$950$	0	0
$951$	6.07107	0.196868
$952$	0	0
$953$	21.3553	0.691767	0.345884	−	0.938277i	$-0.387579\pi$
$953$	21.3553	0.691767	0.345884	+	0.938277i	$0.387579\pi$
$954$	0	0
$955$	−43.0416	−1.39279
$956$	0	0
$957$	−19.3137	−0.624324
$958$	0	0
$959$	16.7279	0.540173
$960$	0	0
$961$	34.1421	1.10136
$962$	0	0
$963$	8.68629	0.279912
$964$	0	0
$965$	−21.6569	−0.697159
$966$	0	0
$967$	−24.9706	−0.802999	−0.401500	−	0.915859i	$-0.631511\pi$
$967$	−24.9706	−0.802999	−0.401500	+	0.915859i	$0.631511\pi$
$968$	0	0
$969$	−6.48528	−0.208337
$970$	0	0
$971$	−7.72792	−0.248001	−0.124000	−	0.992282i	$-0.539572\pi$
$971$	−7.72792	−0.248001	−0.124000	+	0.992282i	$0.539572\pi$
$972$	0	0
$973$	22.9706	0.736402
$974$	0	0
$975$	−98.9117	−3.16771
$976$	0	0
$977$	−24.1716	−0.773317	−0.386659	−	0.922223i	$-0.626371\pi$
$977$	−24.1716	−0.773317	−0.386659	+	0.922223i	$0.626371\pi$
$978$	0	0
$979$	3.82843	0.122357
$980$	0	0
$981$	−37.6569	−1.20229
$982$	0	0
$983$	24.2721	0.774159	0.387080	−	0.922046i	$-0.373484\pi$
$983$	24.2721	0.774159	0.387080	+	0.922046i	$0.373484\pi$
$984$	0	0
$985$	−14.0000	−0.446077
$986$	0	0
$987$	−15.3137	−0.487441
$988$	0	0
$989$	−4.62742	−0.147143
$990$	0	0
$991$	−51.3137	−1.63003	−0.815017	−	0.579437i	$-0.803272\pi$
$991$	−51.3137	−1.63003	−0.815017	+	0.579437i	$0.803272\pi$
$992$	0	0
$993$	−70.9411	−2.25125
$994$	0	0
$995$	43.3137	1.37314
$996$	0	0
$997$	−35.8995	−1.13695	−0.568474	−	0.822701i	$-0.692466\pi$
$997$	−35.8995	−1.13695	−0.568474	+	0.822701i	$0.692466\pi$
$998$	0	0
$999$	2.41421	0.0763823

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1408.2.a.n.1.2	yes	2
4.3	odd	2		1408.2.a.i.1.1	yes	2
8.3	odd	2		1408.2.a.m.1.2	yes	2
8.5	even	2		1408.2.a.f.1.1	✓	2
16.3	odd	4		2816.2.c.u.1409.1		4
16.5	even	4		2816.2.c.v.1409.1		4
16.11	odd	4		2816.2.c.u.1409.4		4
16.13	even	4		2816.2.c.v.1409.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1408.2.a.f.1.1	✓	2	8.5	even	2
1408.2.a.i.1.1	yes	2	4.3	odd	2
1408.2.a.m.1.2	yes	2	8.3	odd	2
1408.2.a.n.1.2	yes	2	1.1	even	1	trivial
2816.2.c.u.1409.1		4	16.3	odd	4
2816.2.c.u.1409.4		4	16.11	odd	4
2816.2.c.v.1409.1		4	16.5	even	4
2816.2.c.v.1409.4		4	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:	\( N \)	\(=\)	\( 1408 = 2^{7} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1408.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} +3.82843 q^{5} +1.41421 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} +3.82843 q^{5} +1.41421 q^{7} +2.82843 q^{9} -1.00000 q^{11} -4.24264 q^{13} +9.24264 q^{15} -4.58579 q^{17} +0.585786 q^{19} +3.41421 q^{21} +5.58579 q^{23} +9.65685 q^{25} -0.414214 q^{27} +8.00000 q^{29} +8.07107 q^{31} -2.41421 q^{33} +5.41421 q^{35} -5.82843 q^{37} -10.2426 q^{39} -7.07107 q^{41} -0.828427 q^{43} +10.8284 q^{45} -4.48528 q^{47} -5.00000 q^{49} -11.0711 q^{51} +2.82843 q^{53} -3.82843 q^{55} +1.41421 q^{57} -11.7279 q^{59} +5.65685 q^{61} +4.00000 q^{63} -16.2426 q^{65} +15.7279 q^{67} +13.4853 q^{69} -6.07107 q^{71} -15.0711 q^{73} +23.3137 q^{75} -1.41421 q^{77} -0.828427 q^{79} -9.48528 q^{81} +11.1716 q^{83} -17.5563 q^{85} +19.3137 q^{87} -3.82843 q^{89} -6.00000 q^{91} +19.4853 q^{93} +2.24264 q^{95} -9.00000 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{11} + 10 q^{15} - 12 q^{17} + 4 q^{19} + 4 q^{21} + 14 q^{23} + 8 q^{25} + 2 q^{27} + 16 q^{29} + 2 q^{31} - 2 q^{33} + 8 q^{35} - 6 q^{37} - 12 q^{39} + 4 q^{43} + 16 q^{45} + 8 q^{47} - 10 q^{49} - 8 q^{51} - 2 q^{55} + 2 q^{59} + 8 q^{63} - 24 q^{65} + 6 q^{67} + 10 q^{69} + 2 q^{71} - 16 q^{73} + 24 q^{75} + 4 q^{79} - 2 q^{81} + 28 q^{83} - 4 q^{85} + 16 q^{87} - 2 q^{89} - 12 q^{91} + 22 q^{93} - 4 q^{95} - 18 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^11 + 10 * q^15 - 12 * q^17 + 4 * q^19 + 4 * q^21 + 14 * q^23 + 8 * q^25 + 2 * q^27 + 16 * q^29 + 2 * q^31 - 2 * q^33 + 8 * q^35 - 6 * q^37 - 12 * q^39 + 4 * q^43 + 16 * q^45 + 8 * q^47 - 10 * q^49 - 8 * q^51 - 2 * q^55 + 2 * q^59 + 8 * q^63 - 24 * q^65 + 6 * q^67 + 10 * q^69 + 2 * q^71 - 16 * q^73 + 24 * q^75 + 4 * q^79 - 2 * q^81 + 28 * q^83 - 4 * q^85 + 16 * q^87 - 2 * q^89 - 12 * q^91 + 22 * q^93 - 4 * q^95 - 18 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more