LMFDB - Embedded newform 5929.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5929,2,Mod(1,5929)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5929.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5929, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5929 = 7^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5929.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,12,0,0,0,0,-12,0,0,0,0,0,0,-4,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$47.3433033584$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.28825$ of defining polynomial
Character	$\chi$	$=$	5929.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607 q^{2} +3.00000 q^{4} +1.41421 q^{5} -2.23607 q^{8} -3.00000 q^{9} -3.16228 q^{10} +3.16228 q^{13} -1.00000 q^{16} -3.16228 q^{17} +6.70820 q^{18} +6.32456 q^{19} +4.24264 q^{20} -4.00000 q^{23} -3.00000 q^{25} -7.07107 q^{26} +4.47214 q^{29} -8.48528 q^{31} +6.70820 q^{32} +7.07107 q^{34} -9.00000 q^{36} +8.00000 q^{37} -14.1421 q^{38} -3.16228 q^{40} -9.48683 q^{41} +8.94427 q^{43} -4.24264 q^{45} +8.94427 q^{46} -2.82843 q^{47} +6.70820 q^{50} +9.48683 q^{52} -6.00000 q^{53} -10.0000 q^{58} +11.3137 q^{59} -3.16228 q^{61} +18.9737 q^{62} -13.0000 q^{64} +4.47214 q^{65} -8.00000 q^{67} -9.48683 q^{68} +6.70820 q^{72} -15.8114 q^{73} -17.8885 q^{74} +18.9737 q^{76} +8.94427 q^{79} -1.41421 q^{80} +9.00000 q^{81} +21.2132 q^{82} +6.32456 q^{83} -4.47214 q^{85} -20.0000 q^{86} +4.24264 q^{89} +9.48683 q^{90} -12.0000 q^{92} +6.32456 q^{94} +8.94427 q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{4} - 12 q^{9} - 4 q^{16} - 16 q^{23} - 12 q^{25} - 36 q^{36} + 32 q^{37} - 24 q^{53} - 40 q^{58} - 52 q^{64} - 32 q^{67} + 36 q^{81} - 80 q^{86} - 48 q^{92}+O(q^{100}) $ 4 * q + 12 * q^4 - 12 * q^9 - 4 * q^16 - 16 * q^23 - 12 * q^25 - 36 * q^36 + 32 * q^37 - 24 * q^53 - 40 * q^58 - 52 * q^64 - 32 * q^67 + 36 * q^81 - 80 * q^86 - 48 * q^92

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 $

$2$	−2.23607	−1.58114	−0.790569	−	0.612372i	$-0.790215\pi$
$2$	−2.23607	−1.58114	−0.790569	+	0.612372i	$0.790215\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	3.00000	1.50000
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.23607	−0.790569
$9$	−3.00000	−1.00000
$10$	−3.16228	−1.00000
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.16228	0.877058	0.438529	−	0.898717i	$-0.355500\pi$
$13$	3.16228	0.877058	0.438529	+	0.898717i	$0.355500\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	−3.16228	−0.766965	−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228	−0.766965	−0.383482	+	0.923548i	$0.625275\pi$
$18$	6.70820	1.58114
$19$	6.32456	1.45095	0.725476	−	0.688247i	$-0.241620\pi$
$19$	6.32456	1.45095	0.725476	+	0.688247i	$0.241620\pi$
$20$	4.24264	0.948683
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	−7.07107	−1.38675
$27$	0	0
$28$	0	0
$29$	4.47214	0.830455	0.415227	−	0.909718i	$-0.363702\pi$
$29$	4.47214	0.830455	0.415227	+	0.909718i	$0.363702\pi$
$30$	0	0
$31$	−8.48528	−1.52400	−0.762001	−	0.647576i	$-0.775783\pi$
$31$	−8.48528	−1.52400	−0.762001	+	0.647576i	$0.775783\pi$
$32$	6.70820	1.18585
$33$	0	0
$34$	7.07107	1.21268
$35$	0	0
$36$	−9.00000	−1.50000
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	−14.1421	−2.29416
$39$	0	0
$40$	−3.16228	−0.500000
$41$	−9.48683	−1.48159	−0.740797	−	0.671729i	$-0.765552\pi$
$41$	−9.48683	−1.48159	−0.740797	+	0.671729i	$0.765552\pi$
$42$	0	0
$43$	8.94427	1.36399	0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427	1.36399	0.681994	+	0.731357i	$0.261113\pi$
$44$	0	0
$45$	−4.24264	−0.632456
$46$	8.94427	1.31876
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	0	0
$50$	6.70820	0.948683
$51$	0	0
$52$	9.48683	1.31559
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−10.0000	−1.31306
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−3.16228	−0.404888	−0.202444	−	0.979294i	$-0.564888\pi$
$61$	−3.16228	−0.404888	−0.202444	+	0.979294i	$0.564888\pi$
$62$	18.9737	2.40966
$63$	0	0
$64$	−13.0000	−1.62500
$65$	4.47214	0.554700
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	−9.48683	−1.15045
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	6.70820	0.790569
$73$	−15.8114	−1.85058	−0.925292	−	0.379257i	$-0.876180\pi$
$73$	−15.8114	−1.85058	−0.925292	+	0.379257i	$0.876180\pi$
$74$	−17.8885	−2.07950
$75$	0	0
$76$	18.9737	2.17643
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	−1.41421	−0.158114
$81$	9.00000	1.00000
$82$	21.2132	2.34261
$83$	6.32456	0.694210	0.347105	−	0.937826i	$-0.387165\pi$
$83$	6.32456	0.694210	0.347105	+	0.937826i	$0.387165\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	−20.0000	−2.15666
$87$	0	0
$88$	0	0
$89$	4.24264	0.449719	0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264	0.449719	0.224860	+	0.974391i	$0.427808\pi$
$90$	9.48683	1.00000
$91$	0	0
$92$	−12.0000	−1.25109
$93$	0	0
$94$	6.32456	0.652328
$95$	8.94427	0.917663
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	0	0
$100$	−9.00000	−0.900000
$101$	−9.48683	−0.943975	−0.471988	−	0.881605i	$-0.656463\pi$
$101$	−9.48683	−0.943975	−0.471988	+	0.881605i	$0.656463\pi$
$102$	0	0
$103$	−8.48528	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$103$	−8.48528	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$104$	−7.07107	−0.693375
$105$	0	0
$106$	13.4164	1.30312
$107$	−8.94427	−0.864675	−0.432338	−	0.901712i	$-0.642311\pi$
$107$	−8.94427	−0.864675	−0.432338	+	0.901712i	$0.642311\pi$
$108$	0	0
$109$	13.4164	1.28506	0.642529	−	0.766261i	$-0.277885\pi$
$109$	13.4164	1.28506	0.642529	+	0.766261i	$0.277885\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	13.4164	1.24568
$117$	−9.48683	−0.877058
$118$	−25.2982	−2.32889
$119$	0	0
$120$	0	0
$121$	0	0
$122$	7.07107	0.640184
$123$	0	0
$124$	−25.4558	−2.28600
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−8.94427	−0.793676	−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427	−0.793676	−0.396838	+	0.917889i	$0.629892\pi$
$128$	15.6525	1.38350
$129$	0	0
$130$	−10.0000	−0.877058
$131$	−18.9737	−1.65774	−0.828868	−	0.559444i	$-0.811015\pi$
$131$	−18.9737	−1.65774	−0.828868	+	0.559444i	$0.811015\pi$
$132$	0	0
$133$	0	0
$134$	17.8885	1.54533
$135$	0	0
$136$	7.07107	0.606339
$137$	−8.00000	−0.683486	−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000	−0.683486	−0.341743	+	0.939793i	$0.611017\pi$
$138$	0	0
$139$	18.9737	1.60933	0.804663	−	0.593732i	$-0.202346\pi$
$139$	18.9737	1.60933	0.804663	+	0.593732i	$0.202346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	3.00000	0.250000
$145$	6.32456	0.525226
$146$	35.3553	2.92603
$147$	0	0
$148$	24.0000	1.97279
$149$	−17.8885	−1.46549	−0.732743	−	0.680505i	$-0.761760\pi$
$149$	−17.8885	−1.46549	−0.732743	+	0.680505i	$0.761760\pi$
$150$	0	0
$151$	−17.8885	−1.45575	−0.727875	−	0.685710i	$-0.759492\pi$
$151$	−17.8885	−1.45575	−0.727875	+	0.685710i	$0.759492\pi$
$152$	−14.1421	−1.14708
$153$	9.48683	0.766965
$154$	0	0
$155$	−12.0000	−0.963863
$156$	0	0
$157$	7.07107	0.564333	0.282166	−	0.959366i	$-0.408947\pi$
$157$	7.07107	0.564333	0.282166	+	0.959366i	$0.408947\pi$
$158$	−20.0000	−1.59111
$159$	0	0
$160$	9.48683	0.750000
$161$	0	0
$162$	−20.1246	−1.58114
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−28.4605	−2.22239
$165$	0	0
$166$	−14.1421	−1.09764
$167$	12.6491	0.978818	0.489409	−	0.872054i	$-0.337213\pi$
$167$	12.6491	0.978818	0.489409	+	0.872054i	$0.337213\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	10.0000	0.766965
$171$	−18.9737	−1.45095
$172$	26.8328	2.04598
$173$	−3.16228	−0.240424	−0.120212	−	0.992748i	$-0.538357\pi$
$173$	−3.16228	−0.240424	−0.120212	+	0.992748i	$0.538357\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	−9.48683	−0.711068
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−12.7279	−0.948683
$181$	−12.7279	−0.946059	−0.473029	−	0.881047i	$-0.656840\pi$
$181$	−12.7279	−0.946059	−0.473029	+	0.881047i	$0.656840\pi$
$182$	0	0
$183$	0	0
$184$	8.94427	0.659380
$185$	11.3137	0.831800
$186$	0	0
$187$	0	0
$188$	−8.48528	−0.618853
$189$	0	0
$190$	−20.0000	−1.45095
$191$	−20.0000	−1.44715	−0.723575	−	0.690246i	$-0.757502\pi$
$191$	−20.0000	−1.44715	−0.723575	+	0.690246i	$0.757502\pi$
$192$	0	0
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−22.1359	−1.58927
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	−2.82843	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$199$	−2.82843	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$200$	6.70820	0.474342
$201$	0	0
$202$	21.2132	1.49256
$203$	0	0
$204$	0	0
$205$	−13.4164	−0.937043
$206$	18.9737	1.32196
$207$	12.0000	0.834058
$208$	−3.16228	−0.219265
$209$	0	0
$210$	0	0
$211$	8.94427	0.615749	0.307875	−	0.951427i	$-0.400382\pi$
$211$	8.94427	0.615749	0.307875	+	0.951427i	$0.400382\pi$
$212$	−18.0000	−1.23625
$213$	0	0
$214$	20.0000	1.36717
$215$	12.6491	0.862662
$216$	0	0
$217$	0	0
$218$	−30.0000	−2.03186
$219$	0	0
$220$	0	0
$221$	−10.0000	−0.672673
$222$	0	0
$223$	−14.1421	−0.947027	−0.473514	−	0.880786i	$-0.657015\pi$
$223$	−14.1421	−0.947027	−0.473514	+	0.880786i	$0.657015\pi$
$224$	0	0
$225$	9.00000	0.600000
$226$	13.4164	0.892446
$227$	−6.32456	−0.419775	−0.209888	−	0.977725i	$-0.567310\pi$
$227$	−6.32456	−0.419775	−0.209888	+	0.977725i	$0.567310\pi$
$228$	0	0
$229$	−24.0416	−1.58872	−0.794358	−	0.607450i	$-0.792192\pi$
$229$	−24.0416	−1.58872	−0.794358	+	0.607450i	$0.792192\pi$
$230$	12.6491	0.834058
$231$	0	0
$232$	−10.0000	−0.656532
$233$	−13.4164	−0.878938	−0.439469	−	0.898258i	$-0.644833\pi$
$233$	−13.4164	−0.878938	−0.439469	+	0.898258i	$0.644833\pi$
$234$	21.2132	1.38675
$235$	−4.00000	−0.260931
$236$	33.9411	2.20938
$237$	0	0
$238$	0	0
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	22.1359	1.42590	0.712951	−	0.701214i	$-0.247358\pi$
$241$	22.1359	1.42590	0.712951	+	0.701214i	$0.247358\pi$
$242$	0	0
$243$	0	0
$244$	−9.48683	−0.607332
$245$	0	0
$246$	0	0
$247$	20.0000	1.27257
$248$	18.9737	1.20483
$249$	0	0
$250$	25.2982	1.60000
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	0	0
$254$	20.0000	1.25491
$255$	0	0
$256$	−9.00000	−0.562500
$257$	24.0416	1.49968	0.749838	−	0.661622i	$-0.230131\pi$
$257$	24.0416	1.49968	0.749838	+	0.661622i	$0.230131\pi$
$258$	0	0
$259$	0	0
$260$	13.4164	0.832050
$261$	−13.4164	−0.830455
$262$	42.4264	2.62111
$263$	17.8885	1.10305	0.551527	−	0.834157i	$-0.314045\pi$
$263$	17.8885	1.10305	0.551527	+	0.834157i	$0.314045\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	0	0
$268$	−24.0000	−1.46603
$269$	18.3848	1.12094	0.560470	−	0.828175i	$-0.310621\pi$
$269$	18.3848	1.12094	0.560470	+	0.828175i	$0.310621\pi$
$270$	0	0
$271$	25.2982	1.53676	0.768379	−	0.639995i	$-0.221064\pi$
$271$	25.2982	1.53676	0.768379	+	0.639995i	$0.221064\pi$
$272$	3.16228	0.191741
$273$	0	0
$274$	17.8885	1.08069
$275$	0	0
$276$	0	0
$277$	17.8885	1.07482	0.537409	−	0.843322i	$-0.319403\pi$
$277$	17.8885	1.07482	0.537409	+	0.843322i	$0.319403\pi$
$278$	−42.4264	−2.54457
$279$	25.4558	1.52400
$280$	0	0
$281$	22.3607	1.33393	0.666963	−	0.745091i	$-0.267594\pi$
$281$	22.3607	1.33393	0.666963	+	0.745091i	$0.267594\pi$
$282$	0	0
$283$	−6.32456	−0.375956	−0.187978	−	0.982173i	$-0.560193\pi$
$283$	−6.32456	−0.375956	−0.187978	+	0.982173i	$0.560193\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	−20.1246	−1.18585
$289$	−7.00000	−0.411765
$290$	−14.1421	−0.830455
$291$	0	0
$292$	−47.4342	−2.77587
$293$	9.48683	0.554227	0.277113	−	0.960837i	$-0.410622\pi$
$293$	9.48683	0.554227	0.277113	+	0.960837i	$0.410622\pi$
$294$	0	0
$295$	16.0000	0.931556
$296$	−17.8885	−1.03975
$297$	0	0
$298$	40.0000	2.31714
$299$	−12.6491	−0.731517
$300$	0	0
$301$	0	0
$302$	40.0000	2.30174
$303$	0	0
$304$	−6.32456	−0.362738
$305$	−4.47214	−0.256074
$306$	−21.2132	−1.21268
$307$	−18.9737	−1.08288	−0.541442	−	0.840738i	$-0.682121\pi$
$307$	−18.9737	−1.08288	−0.541442	+	0.840738i	$0.682121\pi$
$308$	0	0
$309$	0	0
$310$	26.8328	1.52400
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−15.5563	−0.879297	−0.439648	−	0.898170i	$-0.644897\pi$
$313$	−15.5563	−0.879297	−0.439648	+	0.898170i	$0.644897\pi$
$314$	−15.8114	−0.892288
$315$	0	0
$316$	26.8328	1.50946
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	0	0
$320$	−18.3848	−1.02774
$321$	0	0
$322$	0	0
$323$	−20.0000	−1.11283
$324$	27.0000	1.50000
$325$	−9.48683	−0.526235
$326$	−8.94427	−0.495377
$327$	0	0
$328$	21.2132	1.17130
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	18.9737	1.04132
$333$	−24.0000	−1.31519
$334$	−28.2843	−1.54765
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−4.47214	−0.243613	−0.121806	−	0.992554i	$-0.538869\pi$
$337$	−4.47214	−0.243613	−0.121806	+	0.992554i	$0.538869\pi$
$338$	6.70820	0.364878
$339$	0	0
$340$	−13.4164	−0.727607
$341$	0	0
$342$	42.4264	2.29416
$343$	0	0
$344$	−20.0000	−1.07833
$345$	0	0
$346$	7.07107	0.380143
$347$	−26.8328	−1.44046	−0.720231	−	0.693735i	$-0.755964\pi$
$347$	−26.8328	−1.44046	−0.720231	+	0.693735i	$0.755964\pi$
$348$	0	0
$349$	3.16228	0.169273	0.0846364	−	0.996412i	$-0.473027\pi$
$349$	3.16228	0.169273	0.0846364	+	0.996412i	$0.473027\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	35.3553	1.88177	0.940887	−	0.338719i	$-0.109994\pi$
$353$	35.3553	1.88177	0.940887	+	0.338719i	$0.109994\pi$
$354$	0	0
$355$	0	0
$356$	12.7279	0.674579
$357$	0	0
$358$	−8.94427	−0.472719
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	9.48683	0.500000
$361$	21.0000	1.10526
$362$	28.4605	1.49585
$363$	0	0
$364$	0	0
$365$	−22.3607	−1.17041
$366$	0	0
$367$	−25.4558	−1.32878	−0.664392	−	0.747384i	$-0.731309\pi$
$367$	−25.4558	−1.32878	−0.664392	+	0.747384i	$0.731309\pi$
$368$	4.00000	0.208514
$369$	28.4605	1.48159
$370$	−25.2982	−1.31519
$371$	0	0
$372$	0	0
$373$	35.7771	1.85247	0.926234	−	0.376950i	$-0.123027\pi$
$373$	35.7771	1.85247	0.926234	+	0.376950i	$0.123027\pi$
$374$	0	0
$375$	0	0
$376$	6.32456	0.326164
$377$	14.1421	0.728357
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	26.8328	1.37649
$381$	0	0
$382$	44.7214	2.28814
$383$	−14.1421	−0.722629	−0.361315	−	0.932444i	$-0.617672\pi$
$383$	−14.1421	−0.722629	−0.361315	+	0.932444i	$0.617672\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−26.8328	−1.36399
$388$	29.6985	1.50771
$389$	−24.0000	−1.21685	−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000	−1.21685	−0.608424	+	0.793612i	$0.708198\pi$
$390$	0	0
$391$	12.6491	0.639693
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.6491	0.636446
$396$	0	0
$397$	−18.3848	−0.922705	−0.461353	−	0.887217i	$-0.652636\pi$
$397$	−18.3848	−0.922705	−0.461353	+	0.887217i	$0.652636\pi$
$398$	6.32456	0.317021
$399$	0	0
$400$	3.00000	0.150000
$401$	−8.00000	−0.399501	−0.199750	−	0.979847i	$-0.564013\pi$
$401$	−8.00000	−0.399501	−0.199750	+	0.979847i	$0.564013\pi$
$402$	0	0
$403$	−26.8328	−1.33664
$404$	−28.4605	−1.41596
$405$	12.7279	0.632456
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9.48683	0.469094	0.234547	−	0.972105i	$-0.424639\pi$
$409$	9.48683	0.469094	0.234547	+	0.972105i	$0.424639\pi$
$410$	30.0000	1.48159
$411$	0	0
$412$	−25.4558	−1.25412
$413$	0	0
$414$	−26.8328	−1.31876
$415$	8.94427	0.439057
$416$	21.2132	1.04006
$417$	0	0
$418$	0	0
$419$	−39.5980	−1.93449	−0.967244	−	0.253849i	$-0.918303\pi$
$419$	−39.5980	−1.93449	−0.967244	+	0.253849i	$0.918303\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−20.0000	−0.973585
$423$	8.48528	0.412568
$424$	13.4164	0.651558
$425$	9.48683	0.460179
$426$	0	0
$427$	0	0
$428$	−26.8328	−1.29701
$429$	0	0
$430$	−28.2843	−1.36399
$431$	−35.7771	−1.72332	−0.861661	−	0.507485i	$-0.830575\pi$
$431$	−35.7771	−1.72332	−0.861661	+	0.507485i	$0.830575\pi$
$432$	0	0
$433$	1.41421	0.0679628	0.0339814	−	0.999422i	$-0.489181\pi$
$433$	1.41421	0.0679628	0.0339814	+	0.999422i	$0.489181\pi$
$434$	0	0
$435$	0	0
$436$	40.2492	1.92759
$437$	−25.2982	−1.21018
$438$	0	0
$439$	12.6491	0.603709	0.301855	−	0.953354i	$-0.402394\pi$
$439$	12.6491	0.603709	0.301855	+	0.953354i	$0.402394\pi$
$440$	0	0
$441$	0	0
$442$	22.3607	1.06359
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	31.6228	1.49738
$447$	0	0
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	−20.1246	−0.948683
$451$	0	0
$452$	−18.0000	−0.846649
$453$	0	0
$454$	14.1421	0.663723
$455$	0	0
$456$	0	0
$457$	−35.7771	−1.67358	−0.836791	−	0.547523i	$-0.815571\pi$
$457$	−35.7771	−1.67358	−0.836791	+	0.547523i	$0.815571\pi$
$458$	53.7587	2.51198
$459$	0	0
$460$	−16.9706	−0.791257
$461$	−22.1359	−1.03097	−0.515487	−	0.856897i	$-0.672389\pi$
$461$	−22.1359	−1.03097	−0.515487	+	0.856897i	$0.672389\pi$
$462$	0	0
$463$	16.0000	0.743583	0.371792	−	0.928316i	$-0.378744\pi$
$463$	16.0000	0.743583	0.371792	+	0.928316i	$0.378744\pi$
$464$	−4.47214	−0.207614
$465$	0	0
$466$	30.0000	1.38972
$467$	39.5980	1.83238	0.916188	−	0.400749i	$-0.131250\pi$
$467$	39.5980	1.83238	0.916188	+	0.400749i	$0.131250\pi$
$468$	−28.4605	−1.31559
$469$	0	0
$470$	8.94427	0.412568
$471$	0	0
$472$	−25.2982	−1.16445
$473$	0	0
$474$	0	0
$475$	−18.9737	−0.870572
$476$	0	0
$477$	18.0000	0.824163
$478$	20.0000	0.914779
$479$	−25.2982	−1.15591	−0.577953	−	0.816070i	$-0.696148\pi$
$479$	−25.2982	−1.15591	−0.577953	+	0.816070i	$0.696148\pi$
$480$	0	0
$481$	25.2982	1.15350
$482$	−49.4975	−2.25455
$483$	0	0
$484$	0	0
$485$	14.0000	0.635707
$486$	0	0
$487$	−12.0000	−0.543772	−0.271886	−	0.962329i	$-0.587647\pi$
$487$	−12.0000	−0.543772	−0.271886	+	0.962329i	$0.587647\pi$
$488$	7.07107	0.320092
$489$	0	0
$490$	0	0
$491$	−17.8885	−0.807299	−0.403649	−	0.914914i	$-0.632258\pi$
$491$	−17.8885	−0.807299	−0.403649	+	0.914914i	$0.632258\pi$
$492$	0	0
$493$	−14.1421	−0.636930
$494$	−44.7214	−2.01211
$495$	0	0
$496$	8.48528	0.381000
$497$	0	0
$498$	0	0
$499$	−16.0000	−0.716258	−0.358129	−	0.933672i	$-0.616585\pi$
$499$	−16.0000	−0.716258	−0.358129	+	0.933672i	$0.616585\pi$
$500$	−33.9411	−1.51789
$501$	0	0
$502$	12.6491	0.564557
$503$	37.9473	1.69199	0.845994	−	0.533192i	$-0.179008\pi$
$503$	37.9473	1.69199	0.845994	+	0.533192i	$0.179008\pi$
$504$	0	0
$505$	−13.4164	−0.597022
$506$	0	0
$507$	0	0
$508$	−26.8328	−1.19051
$509$	−4.24264	−0.188052	−0.0940259	−	0.995570i	$-0.529974\pi$
$509$	−4.24264	−0.188052	−0.0940259	+	0.995570i	$0.529974\pi$
$510$	0	0
$511$	0	0
$512$	−11.1803	−0.494106
$513$	0	0
$514$	−53.7587	−2.37120
$515$	−12.0000	−0.528783
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−10.0000	−0.438529
$521$	15.5563	0.681536	0.340768	−	0.940147i	$-0.389313\pi$
$521$	15.5563	0.681536	0.340768	+	0.940147i	$0.389313\pi$
$522$	30.0000	1.31306
$523$	−6.32456	−0.276553	−0.138277	−	0.990394i	$-0.544156\pi$
$523$	−6.32456	−0.276553	−0.138277	+	0.990394i	$0.544156\pi$
$524$	−56.9210	−2.48661
$525$	0	0
$526$	−40.0000	−1.74408
$527$	26.8328	1.16886
$528$	0	0
$529$	−7.00000	−0.304348
$530$	18.9737	0.824163
$531$	−33.9411	−1.47292
$532$	0	0
$533$	−30.0000	−1.29944
$534$	0	0
$535$	−12.6491	−0.546869
$536$	17.8885	0.772667
$537$	0	0
$538$	−41.1096	−1.77236
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	−56.5685	−2.42983
$543$	0	0
$544$	−21.2132	−0.909509
$545$	18.9737	0.812743
$546$	0	0
$547$	35.7771	1.52972	0.764859	−	0.644198i	$-0.222809\pi$
$547$	35.7771	1.52972	0.764859	+	0.644198i	$0.222809\pi$
$548$	−24.0000	−1.02523
$549$	9.48683	0.404888
$550$	0	0
$551$	28.2843	1.20495
$552$	0	0
$553$	0	0
$554$	−40.0000	−1.69944
$555$	0	0
$556$	56.9210	2.41399
$557$	−35.7771	−1.51592	−0.757962	−	0.652299i	$-0.773805\pi$
$557$	−35.7771	−1.51592	−0.757962	+	0.652299i	$0.773805\pi$
$558$	−56.9210	−2.40966
$559$	28.2843	1.19630
$560$	0	0
$561$	0	0
$562$	−50.0000	−2.10912
$563$	−18.9737	−0.799645	−0.399822	−	0.916593i	$-0.630928\pi$
$563$	−18.9737	−0.799645	−0.399822	+	0.916593i	$0.630928\pi$
$564$	0	0
$565$	−8.48528	−0.356978
$566$	14.1421	0.594438
$567$	0	0
$568$	0	0
$569$	−13.4164	−0.562445	−0.281223	−	0.959643i	$-0.590740\pi$
$569$	−13.4164	−0.562445	−0.281223	+	0.959643i	$0.590740\pi$
$570$	0	0
$571$	−17.8885	−0.748612	−0.374306	−	0.927305i	$-0.622119\pi$
$571$	−17.8885	−0.748612	−0.374306	+	0.927305i	$0.622119\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	12.0000	0.500435
$576$	39.0000	1.62500
$577$	−21.2132	−0.883117	−0.441559	−	0.897232i	$-0.645574\pi$
$577$	−21.2132	−0.883117	−0.441559	+	0.897232i	$0.645574\pi$
$578$	15.6525	0.651057
$579$	0	0
$580$	18.9737	0.787839
$581$	0	0
$582$	0	0
$583$	0	0
$584$	35.3553	1.46301
$585$	−13.4164	−0.554700
$586$	−21.2132	−0.876309
$587$	−45.2548	−1.86787	−0.933933	−	0.357447i	$-0.883647\pi$
$587$	−45.2548	−1.86787	−0.933933	+	0.357447i	$0.883647\pi$
$588$	0	0
$589$	−53.6656	−2.21125
$590$	−35.7771	−1.47292
$591$	0	0
$592$	−8.00000	−0.328798
$593$	−22.1359	−0.909014	−0.454507	−	0.890743i	$-0.650185\pi$
$593$	−22.1359	−0.909014	−0.454507	+	0.890743i	$0.650185\pi$
$594$	0	0
$595$	0	0
$596$	−53.6656	−2.19823
$597$	0	0
$598$	28.2843	1.15663
$599$	−40.0000	−1.63436	−0.817178	−	0.576386i	$-0.804463\pi$
$599$	−40.0000	−1.63436	−0.817178	+	0.576386i	$0.804463\pi$
$600$	0	0
$601$	−34.7851	−1.41891	−0.709456	−	0.704750i	$-0.751059\pi$
$601$	−34.7851	−1.41891	−0.709456	+	0.704750i	$0.751059\pi$
$602$	0	0
$603$	24.0000	0.977356
$604$	−53.6656	−2.18362
$605$	0	0
$606$	0	0
$607$	25.2982	1.02682	0.513412	−	0.858143i	$-0.328381\pi$
$607$	25.2982	1.02682	0.513412	+	0.858143i	$0.328381\pi$
$608$	42.4264	1.72062
$609$	0	0
$610$	10.0000	0.404888
$611$	−8.94427	−0.361847
$612$	28.4605	1.15045
$613$	13.4164	0.541884	0.270942	−	0.962596i	$-0.412665\pi$
$613$	13.4164	0.541884	0.270942	+	0.962596i	$0.412665\pi$
$614$	42.4264	1.71219
$615$	0	0
$616$	0	0
$617$	−48.0000	−1.93241	−0.966204	−	0.257780i	$-0.917009\pi$
$617$	−48.0000	−1.93241	−0.966204	+	0.257780i	$0.917009\pi$
$618$	0	0
$619$	16.9706	0.682105	0.341052	−	0.940044i	$-0.389217\pi$
$619$	16.9706	0.682105	0.341052	+	0.940044i	$0.389217\pi$
$620$	−36.0000	−1.44579
$621$	0	0
$622$	−18.9737	−0.760775
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	34.7851	1.39029
$627$	0	0
$628$	21.2132	0.846499
$629$	−25.2982	−1.00871
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	−20.0000	−0.795557
$633$	0	0
$634$	−40.2492	−1.59850
$635$	−12.6491	−0.501965
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	22.1359	0.875000
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	0	0
$643$	5.65685	0.223085	0.111542	−	0.993760i	$-0.464421\pi$
$643$	5.65685	0.223085	0.111542	+	0.993760i	$0.464421\pi$
$644$	0	0
$645$	0	0
$646$	44.7214	1.75954
$647$	−31.1127	−1.22317	−0.611583	−	0.791180i	$-0.709467\pi$
$647$	−31.1127	−1.22317	−0.611583	+	0.791180i	$0.709467\pi$
$648$	−20.1246	−0.790569
$649$	0	0
$650$	21.2132	0.832050
$651$	0	0
$652$	12.0000	0.469956
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	0	0
$655$	−26.8328	−1.04844
$656$	9.48683	0.370399
$657$	47.4342	1.85058
$658$	0	0
$659$	−8.94427	−0.348419	−0.174210	−	0.984709i	$-0.555737\pi$
$659$	−8.94427	−0.348419	−0.174210	+	0.984709i	$0.555737\pi$
$660$	0	0
$661$	12.7279	0.495059	0.247529	−	0.968880i	$-0.420381\pi$
$661$	12.7279	0.495059	0.247529	+	0.968880i	$0.420381\pi$
$662$	0	0
$663$	0	0
$664$	−14.1421	−0.548821
$665$	0	0
$666$	53.6656	2.07950
$667$	−17.8885	−0.692647
$668$	37.9473	1.46823
$669$	0	0
$670$	25.2982	0.977356
$671$	0	0
$672$	0	0
$673$	40.2492	1.55149	0.775747	−	0.631044i	$-0.217373\pi$
$673$	40.2492	1.55149	0.775747	+	0.631044i	$0.217373\pi$
$674$	10.0000	0.385186
$675$	0	0
$676$	−9.00000	−0.346154
$677$	9.48683	0.364609	0.182304	−	0.983242i	$-0.441644\pi$
$677$	9.48683	0.364609	0.182304	+	0.983242i	$0.441644\pi$
$678$	0	0
$679$	0	0
$680$	10.0000	0.383482
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	−56.9210	−2.17643
$685$	−11.3137	−0.432275
$686$	0	0
$687$	0	0
$688$	−8.94427	−0.340997
$689$	−18.9737	−0.722839
$690$	0	0
$691$	33.9411	1.29118	0.645591	−	0.763684i	$-0.276611\pi$
$691$	33.9411	1.29118	0.645591	+	0.763684i	$0.276611\pi$
$692$	−9.48683	−0.360635
$693$	0	0
$694$	60.0000	2.27757
$695$	26.8328	1.01783
$696$	0	0
$697$	30.0000	1.13633
$698$	−7.07107	−0.267644
$699$	0	0
$700$	0	0
$701$	−13.4164	−0.506731	−0.253365	−	0.967371i	$-0.581537\pi$
$701$	−13.4164	−0.506731	−0.253365	+	0.967371i	$0.581537\pi$
$702$	0	0
$703$	50.5964	1.90828
$704$	0	0
$705$	0	0
$706$	−79.0569	−2.97535
$707$	0	0
$708$	0	0
$709$	−40.0000	−1.50223	−0.751116	−	0.660171i	$-0.770484\pi$
$709$	−40.0000	−1.50223	−0.751116	+	0.660171i	$0.770484\pi$
$710$	0	0
$711$	−26.8328	−1.00631
$712$	−9.48683	−0.355534
$713$	33.9411	1.27111
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	40.0000	1.49279
$719$	−2.82843	−0.105483	−0.0527413	−	0.998608i	$-0.516796\pi$
$719$	−2.82843	−0.105483	−0.0527413	+	0.998608i	$0.516796\pi$
$720$	4.24264	0.158114
$721$	0	0
$722$	−46.9574	−1.74757
$723$	0	0
$724$	−38.1838	−1.41909
$725$	−13.4164	−0.498273
$726$	0	0
$727$	2.82843	0.104901	0.0524503	−	0.998624i	$-0.483297\pi$
$727$	2.82843	0.104901	0.0524503	+	0.998624i	$0.483297\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	50.0000	1.85058
$731$	−28.2843	−1.04613
$732$	0	0
$733$	28.4605	1.05121	0.525606	−	0.850728i	$-0.323839\pi$
$733$	28.4605	1.05121	0.525606	+	0.850728i	$0.323839\pi$
$734$	56.9210	2.10099
$735$	0	0
$736$	−26.8328	−0.989071
$737$	0	0
$738$	−63.6396	−2.34261
$739$	17.8885	0.658041	0.329020	−	0.944323i	$-0.393282\pi$
$739$	17.8885	0.658041	0.329020	+	0.944323i	$0.393282\pi$
$740$	33.9411	1.24770
$741$	0	0
$742$	0	0
$743$	8.94427	0.328134	0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427	0.328134	0.164067	+	0.986449i	$0.447539\pi$
$744$	0	0
$745$	−25.2982	−0.926855
$746$	−80.0000	−2.92901
$747$	−18.9737	−0.694210
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	2.82843	0.103142
$753$	0	0
$754$	−31.6228	−1.15163
$755$	−25.2982	−0.920697
$756$	0	0
$757$	−8.00000	−0.290765	−0.145382	−	0.989376i	$-0.546441\pi$
$757$	−8.00000	−0.290765	−0.145382	+	0.989376i	$0.546441\pi$
$758$	−8.94427	−0.324871
$759$	0	0
$760$	−20.0000	−0.725476
$761$	15.8114	0.573162	0.286581	−	0.958056i	$-0.407481\pi$
$761$	15.8114	0.573162	0.286581	+	0.958056i	$0.407481\pi$
$762$	0	0
$763$	0	0
$764$	−60.0000	−2.17072
$765$	13.4164	0.485071
$766$	31.6228	1.14258
$767$	35.7771	1.29184
$768$	0	0
$769$	−22.1359	−0.798243	−0.399121	−	0.916898i	$-0.630685\pi$
$769$	−22.1359	−0.798243	−0.399121	+	0.916898i	$0.630685\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−35.3553	−1.27164	−0.635822	−	0.771836i	$-0.719339\pi$
$773$	−35.3553	−1.27164	−0.635822	+	0.771836i	$0.719339\pi$
$774$	60.0000	2.15666
$775$	25.4558	0.914401
$776$	−22.1359	−0.794634
$777$	0	0
$778$	53.6656	1.92401
$779$	−60.0000	−2.14972
$780$	0	0
$781$	0	0
$782$	−28.2843	−1.01144
$783$	0	0
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	6.32456	0.225446	0.112723	−	0.993626i	$-0.464043\pi$
$787$	6.32456	0.225446	0.112723	+	0.993626i	$0.464043\pi$
$788$	0	0
$789$	0	0
$790$	−28.2843	−1.00631
$791$	0	0
$792$	0	0
$793$	−10.0000	−0.355110
$794$	41.1096	1.45893
$795$	0	0
$796$	−8.48528	−0.300753
$797$	−18.3848	−0.651222	−0.325611	−	0.945504i	$-0.605570\pi$
$797$	−18.3848	−0.651222	−0.325611	+	0.945504i	$0.605570\pi$
$798$	0	0
$799$	8.94427	0.316426
$800$	−20.1246	−0.711512
$801$	−12.7279	−0.449719
$802$	17.8885	0.631666
$803$	0	0
$804$	0	0
$805$	0	0
$806$	60.0000	2.11341
$807$	0	0
$808$	21.2132	0.746278
$809$	−35.7771	−1.25786	−0.628928	−	0.777464i	$-0.716506\pi$
$809$	−35.7771	−1.25786	−0.628928	+	0.777464i	$0.716506\pi$
$810$	−28.4605	−1.00000
$811$	−31.6228	−1.11043	−0.555213	−	0.831708i	$-0.687363\pi$
$811$	−31.6228	−1.11043	−0.555213	+	0.831708i	$0.687363\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5.65685	0.198151
$816$	0	0
$817$	56.5685	1.97908
$818$	−21.2132	−0.741702
$819$	0	0
$820$	−40.2492	−1.40556
$821$	−17.8885	−0.624314	−0.312157	−	0.950030i	$-0.601052\pi$
$821$	−17.8885	−0.624314	−0.312157	+	0.950030i	$0.601052\pi$
$822$	0	0
$823$	44.0000	1.53374	0.766872	−	0.641800i	$-0.221812\pi$
$823$	44.0000	1.53374	0.766872	+	0.641800i	$0.221812\pi$
$824$	18.9737	0.660979
$825$	0	0
$826$	0	0
$827$	−17.8885	−0.622046	−0.311023	−	0.950402i	$-0.600672\pi$
$827$	−17.8885	−0.622046	−0.311023	+	0.950402i	$0.600672\pi$
$828$	36.0000	1.25109
$829$	38.1838	1.32618	0.663089	−	0.748541i	$-0.269245\pi$
$829$	38.1838	1.32618	0.663089	+	0.748541i	$0.269245\pi$
$830$	−20.0000	−0.694210
$831$	0	0
$832$	−41.1096	−1.42522
$833$	0	0
$834$	0	0
$835$	17.8885	0.619059
$836$	0	0
$837$	0	0
$838$	88.5438	3.05869
$839$	2.82843	0.0976481	0.0488241	−	0.998807i	$-0.484453\pi$
$839$	2.82843	0.0976481	0.0488241	+	0.998807i	$0.484453\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	49.1935	1.69532
$843$	0	0
$844$	26.8328	0.923624
$845$	−4.24264	−0.145951
$846$	−18.9737	−0.652328
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	−21.2132	−0.727607
$851$	−32.0000	−1.09695
$852$	0	0
$853$	9.48683	0.324823	0.162411	−	0.986723i	$-0.448073\pi$
$853$	9.48683	0.324823	0.162411	+	0.986723i	$0.448073\pi$
$854$	0	0
$855$	−26.8328	−0.917663
$856$	20.0000	0.683586
$857$	15.8114	0.540107	0.270053	−	0.962845i	$-0.412959\pi$
$857$	15.8114	0.540107	0.270053	+	0.962845i	$0.412959\pi$
$858$	0	0
$859$	39.5980	1.35107	0.675533	−	0.737330i	$-0.263914\pi$
$859$	39.5980	1.35107	0.675533	+	0.737330i	$0.263914\pi$
$860$	37.9473	1.29399
$861$	0	0
$862$	80.0000	2.72481
$863$	−36.0000	−1.22545	−0.612727	−	0.790295i	$-0.709928\pi$
$863$	−36.0000	−1.22545	−0.612727	+	0.790295i	$0.709928\pi$
$864$	0	0
$865$	−4.47214	−0.152057
$866$	−3.16228	−0.107459
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−25.2982	−0.857198
$872$	−30.0000	−1.01593
$873$	−29.6985	−1.00514
$874$	56.5685	1.91346
$875$	0	0
$876$	0	0
$877$	13.4164	0.453040	0.226520	−	0.974007i	$-0.427265\pi$
$877$	13.4164	0.453040	0.226520	+	0.974007i	$0.427265\pi$
$878$	−28.2843	−0.954548
$879$	0	0
$880$	0	0
$881$	−43.8406	−1.47703	−0.738514	−	0.674238i	$-0.764472\pi$
$881$	−43.8406	−1.47703	−0.738514	+	0.674238i	$0.764472\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	−30.0000	−1.00901
$885$	0	0
$886$	−8.94427	−0.300489
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	−13.4164	−0.449719
$891$	0	0
$892$	−42.4264	−1.42054
$893$	−17.8885	−0.598617
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	0	0
$898$	22.3607	0.746186
$899$	−37.9473	−1.26561
$900$	27.0000	0.900000
$901$	18.9737	0.632104
$902$	0	0
$903$	0	0
$904$	13.4164	0.446223
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−8.00000	−0.265636	−0.132818	−	0.991140i	$-0.542403\pi$
$907$	−8.00000	−0.265636	−0.132818	+	0.991140i	$0.542403\pi$
$908$	−18.9737	−0.629663
$909$	28.4605	0.943975
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	80.0000	2.64616
$915$	0	0
$916$	−72.1249	−2.38307
$917$	0	0
$918$	0	0
$919$	−53.6656	−1.77027	−0.885133	−	0.465338i	$-0.845933\pi$
$919$	−53.6656	−1.77027	−0.885133	+	0.465338i	$0.845933\pi$
$920$	12.6491	0.417029
$921$	0	0
$922$	49.4975	1.63011
$923$	0	0
$924$	0	0
$925$	−24.0000	−0.789115
$926$	−35.7771	−1.17571
$927$	25.4558	0.836080
$928$	30.0000	0.984798
$929$	−32.5269	−1.06717	−0.533587	−	0.845745i	$-0.679156\pi$
$929$	−32.5269	−1.06717	−0.533587	+	0.845745i	$0.679156\pi$
$930$	0	0
$931$	0	0
$932$	−40.2492	−1.31841
$933$	0	0
$934$	−88.5438	−2.89724
$935$	0	0
$936$	21.2132	0.693375
$937$	34.7851	1.13638	0.568189	−	0.822898i	$-0.307644\pi$
$937$	34.7851	1.13638	0.568189	+	0.822898i	$0.307644\pi$
$938$	0	0
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−53.7587	−1.75248	−0.876242	−	0.481871i	$-0.839957\pi$
$941$	−53.7587	−1.75248	−0.876242	+	0.481871i	$0.839957\pi$
$942$	0	0
$943$	37.9473	1.23574
$944$	−11.3137	−0.368230
$945$	0	0
$946$	0	0
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	0	0
$949$	−50.0000	−1.62307
$950$	42.4264	1.37649
$951$	0	0
$952$	0	0
$953$	−35.7771	−1.15893	−0.579467	−	0.814996i	$-0.696739\pi$
$953$	−35.7771	−1.15893	−0.579467	+	0.814996i	$0.696739\pi$
$954$	−40.2492	−1.30312
$955$	−28.2843	−0.915258
$956$	−26.8328	−0.867835
$957$	0	0
$958$	56.5685	1.82765
$959$	0	0
$960$	0	0
$961$	41.0000	1.32258
$962$	−56.5685	−1.82384
$963$	26.8328	0.864675
$964$	66.4078	2.13885
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	−31.3050	−1.00514
$971$	5.65685	0.181537	0.0907685	−	0.995872i	$-0.471068\pi$
$971$	5.65685	0.181537	0.0907685	+	0.995872i	$0.471068\pi$
$972$	0	0
$973$	0	0
$974$	26.8328	0.859779
$975$	0	0
$976$	3.16228	0.101222
$977$	32.0000	1.02377	0.511885	−	0.859054i	$-0.328947\pi$
$977$	32.0000	1.02377	0.511885	+	0.859054i	$0.328947\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−40.2492	−1.28506
$982$	40.0000	1.27645
$983$	−8.48528	−0.270638	−0.135319	−	0.990802i	$-0.543206\pi$
$983$	−8.48528	−0.270638	−0.135319	+	0.990802i	$0.543206\pi$
$984$	0	0
$985$	0	0
$986$	31.6228	1.00707
$987$	0	0
$988$	60.0000	1.90885
$989$	−35.7771	−1.13765
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	−56.9210	−1.80724
$993$	0	0
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	9.48683	0.300451	0.150226	−	0.988652i	$-0.452000\pi$
$997$	9.48683	0.300451	0.150226	+	0.988652i	$0.452000\pi$
$998$	35.7771	1.13250
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5929.2.a.bf.1.2	yes	4
7.6	odd	2	inner	5929.2.a.bf.1.1	✓	4
11.10	odd	2	inner	5929.2.a.bf.1.4	yes	4
77.76	even	2	inner	5929.2.a.bf.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5929.2.a.bf.1.1	✓	4	7.6	odd	2	inner
5929.2.a.bf.1.2	yes	4	1.1	even	1	trivial
5929.2.a.bf.1.3	yes	4	77.76	even	2	inner
5929.2.a.bf.1.4	yes	4	11.10	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5929 = 7^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5929.a (trivial)

\(f(q)\)	\(=\)	\(q-2.23607 q^{2} +3.00000 q^{4} +1.41421 q^{5} -2.23607 q^{8} -3.00000 q^{9} -3.16228 q^{10} +3.16228 q^{13} -1.00000 q^{16} -3.16228 q^{17} +6.70820 q^{18} +6.32456 q^{19} +4.24264 q^{20} -4.00000 q^{23} -3.00000 q^{25} -7.07107 q^{26} +4.47214 q^{29} -8.48528 q^{31} +6.70820 q^{32} +7.07107 q^{34} -9.00000 q^{36} +8.00000 q^{37} -14.1421 q^{38} -3.16228 q^{40} -9.48683 q^{41} +8.94427 q^{43} -4.24264 q^{45} +8.94427 q^{46} -2.82843 q^{47} +6.70820 q^{50} +9.48683 q^{52} -6.00000 q^{53} -10.0000 q^{58} +11.3137 q^{59} -3.16228 q^{61} +18.9737 q^{62} -13.0000 q^{64} +4.47214 q^{65} -8.00000 q^{67} -9.48683 q^{68} +6.70820 q^{72} -15.8114 q^{73} -17.8885 q^{74} +18.9737 q^{76} +8.94427 q^{79} -1.41421 q^{80} +9.00000 q^{81} +21.2132 q^{82} +6.32456 q^{83} -4.47214 q^{85} -20.0000 q^{86} +4.24264 q^{89} +9.48683 q^{90} -12.0000 q^{92} +6.32456 q^{94} +8.94427 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{4} - 12 q^{9} - 4 q^{16} - 16 q^{23} - 12 q^{25} - 36 q^{36} + 32 q^{37} - 24 q^{53} - 40 q^{58} - 52 q^{64} - 32 q^{67} + 36 q^{81} - 80 q^{86} - 48 q^{92}+O(q^{100}) \) 4 * q + 12 * q^4 - 12 * q^9 - 4 * q^16 - 16 * q^23 - 12 * q^25 - 36 * q^36 + 32 * q^37 - 24 * q^53 - 40 * q^58 - 52 * q^64 - 32 * q^67 + 36 * q^81 - 80 * q^86 - 48 * q^92

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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