LMFDB - Embedded newform 4176.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4176,2,Mod(1,4176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4176 = 2^{4} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4176.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$33.3455278841$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.82843 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.82843 q^{7} +2.41421 q^{11} +1.82843 q^{13} +4.82843 q^{17} -6.00000 q^{19} -7.65685 q^{23} -4.00000 q^{25} -1.00000 q^{29} +4.07107 q^{31} -2.82843 q^{35} -4.00000 q^{37} -12.4853 q^{41} -6.41421 q^{43} +5.24264 q^{47} +1.00000 q^{49} +7.48528 q^{53} +2.41421 q^{55} +7.65685 q^{59} +0.828427 q^{61} +1.82843 q^{65} +5.65685 q^{67} -3.17157 q^{71} +4.00000 q^{73} -6.82843 q^{77} -0.414214 q^{79} -3.65685 q^{83} +4.82843 q^{85} -4.48528 q^{89} -5.17157 q^{91} -6.00000 q^{95} -12.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 12 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} - 6 q^{31} - 8 q^{37} - 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 2 q^{53} + 2 q^{55} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 12 q^{71} + 8 q^{73} - 8 q^{77} + 2 q^{79} + 4 q^{83} + 4 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{95} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 - 12 * q^19 - 4 * q^23 - 8 * q^25 - 2 * q^29 - 6 * q^31 - 8 * q^37 - 8 * q^41 - 10 * q^43 + 2 * q^47 + 2 * q^49 - 2 * q^53 + 2 * q^55 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 12 * q^71 + 8 * q^73 - 8 * q^77 + 2 * q^79 + 4 * q^83 + 4 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^95 - 8 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.41421	0.727913	0.363956	−	0.931416i	$-0.381426\pi$
$11$	2.41421	0.727913	0.363956	+	0.931416i	$0.381426\pi$
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.07107	0.731185	0.365593	−	0.930775i	$-0.380866\pi$
$31$	4.07107	0.731185	0.365593	+	0.930775i	$0.380866\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.4853	−1.94987	−0.974937	−	0.222483i	$-0.928584\pi$
$41$	−12.4853	−1.94987	−0.974937	+	0.222483i	$0.928584\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.24264	0.764718	0.382359	−	0.924014i	$-0.375112\pi$
$47$	5.24264	0.764718	0.382359	+	0.924014i	$0.375112\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.48528	1.02818	0.514091	−	0.857736i	$-0.328129\pi$
$53$	7.48528	1.02818	0.514091	+	0.857736i	$0.328129\pi$
$54$	0	0
$55$	2.41421	0.325532
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	0.828427	0.106069	0.0530346	−	0.998593i	$-0.483111\pi$
$61$	0.828427	0.106069	0.0530346	+	0.998593i	$0.483111\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.82843	0.226788
$66$	0	0
$67$	5.65685	0.691095	0.345547	−	0.938401i	$-0.387693\pi$
$67$	5.65685	0.691095	0.345547	+	0.938401i	$0.387693\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.17157	−0.376396	−0.188198	−	0.982131i	$-0.560265\pi$
$71$	−3.17157	−0.376396	−0.188198	+	0.982131i	$0.560265\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.82843	−0.778171
$78$	0	0
$79$	−0.414214	−0.0466027	−0.0233013	−	0.999728i	$-0.507418\pi$
$79$	−0.414214	−0.0466027	−0.0233013	+	0.999728i	$0.507418\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.65685	−0.401392	−0.200696	−	0.979654i	$-0.564320\pi$
$83$	−3.65685	−0.401392	−0.200696	+	0.979654i	$0.564320\pi$
$84$	0	0
$85$	4.82843	0.523716
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−4.48528	−0.475439	−0.237719	−	0.971334i	$-0.576400\pi$
$89$	−4.48528	−0.475439	−0.237719	+	0.971334i	$0.576400\pi$
$90$	0	0
$91$	−5.17157	−0.542128
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−12.4853	−1.26769	−0.633844	−	0.773461i	$-0.718524\pi$
$97$	−12.4853	−1.26769	−0.633844	+	0.773461i	$0.718524\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	13.6569	1.35891	0.679454	−	0.733718i	$-0.262217\pi$
$101$	13.6569	1.35891	0.679454	+	0.733718i	$0.262217\pi$
$102$	0	0
$103$	−0.828427	−0.0816274	−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427	−0.0816274	−0.0408137	+	0.999167i	$0.512995\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−9.17157	−0.886649	−0.443325	−	0.896361i	$-0.646201\pi$
$107$	−9.17157	−0.886649	−0.443325	+	0.896361i	$0.646201\pi$
$108$	0	0
$109$	1.34315	0.128650	0.0643250	−	0.997929i	$-0.479511\pi$
$109$	1.34315	0.128650	0.0643250	+	0.997929i	$0.479511\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.31371	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$113$	−9.31371	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$114$	0	0
$115$	−7.65685	−0.714005
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−13.6569	−1.25192
$120$	0	0
$121$	−5.17157	−0.470143
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	15.6569	1.38932	0.694661	−	0.719338i	$-0.255555\pi$
$127$	15.6569	1.38932	0.694661	+	0.719338i	$0.255555\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1.31371	−0.114779	−0.0573896	−	0.998352i	$-0.518278\pi$
$131$	−1.31371	−0.114779	−0.0573896	+	0.998352i	$0.518278\pi$
$132$	0	0
$133$	16.9706	1.47153
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.41421	0.369135
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.17157	0.177902	0.0889511	−	0.996036i	$-0.471649\pi$
$149$	2.17157	0.177902	0.0889511	+	0.996036i	$0.471649\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.07107	0.326996
$156$	0	0
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	21.6569	1.70680
$162$	0	0
$163$	−18.0711	−1.41544	−0.707718	−	0.706495i	$-0.750275\pi$
$163$	−18.0711	−1.41544	−0.707718	+	0.706495i	$0.750275\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.82843	−0.683164	−0.341582	−	0.939852i	$-0.610963\pi$
$167$	−8.82843	−0.683164	−0.341582	+	0.939852i	$0.610963\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−23.6569	−1.79860	−0.899299	−	0.437335i	$-0.855922\pi$
$173$	−23.6569	−1.79860	−0.899299	+	0.437335i	$0.855922\pi$
$174$	0	0
$175$	11.3137	0.855236
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.4853	0.783707	0.391853	−	0.920028i	$-0.371834\pi$
$179$	10.4853	0.783707	0.391853	+	0.920028i	$0.371834\pi$
$180$	0	0
$181$	−14.3137	−1.06393	−0.531965	−	0.846766i	$-0.678546\pi$
$181$	−14.3137	−1.06393	−0.531965	+	0.846766i	$0.678546\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	11.6569	0.852434
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.68629	0.194373	0.0971866	−	0.995266i	$-0.469016\pi$
$191$	2.68629	0.194373	0.0971866	+	0.995266i	$0.469016\pi$
$192$	0	0
$193$	−10.8284	−0.779447	−0.389724	−	0.920932i	$-0.627429\pi$
$193$	−10.8284	−0.779447	−0.389724	+	0.920932i	$0.627429\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	−16.4853	−1.16861	−0.584305	−	0.811534i	$-0.698633\pi$
$199$	−16.4853	−1.16861	−0.584305	+	0.811534i	$0.698633\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.82843	0.198517
$204$	0	0
$205$	−12.4853	−0.872010
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.4853	−1.00197
$210$	0	0
$211$	−17.3848	−1.19682	−0.598409	−	0.801191i	$-0.704200\pi$
$211$	−17.3848	−1.19682	−0.598409	+	0.801191i	$0.704200\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.41421	−0.437446
$216$	0	0
$217$	−11.5147	−0.781670
$218$	0	0
$219$	0	0
$220$	0	0
$221$	8.82843	0.593864
$222$	0	0
$223$	8.82843	0.591195	0.295598	−	0.955313i	$-0.404481\pi$
$223$	8.82843	0.591195	0.295598	+	0.955313i	$0.404481\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	20.1421	1.33688	0.668440	−	0.743766i	$-0.266962\pi$
$227$	20.1421	1.33688	0.668440	+	0.743766i	$0.266962\pi$
$228$	0	0
$229$	−20.4853	−1.35371	−0.676853	−	0.736118i	$-0.736657\pi$
$229$	−20.4853	−1.35371	−0.676853	+	0.736118i	$0.736657\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.31371	0.282600	0.141300	−	0.989967i	$-0.454872\pi$
$233$	4.31371	0.282600	0.141300	+	0.989967i	$0.454872\pi$
$234$	0	0
$235$	5.24264	0.341992
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.34315	−0.539673	−0.269837	−	0.962906i	$-0.586970\pi$
$239$	−8.34315	−0.539673	−0.269837	+	0.962906i	$0.586970\pi$
$240$	0	0
$241$	4.31371	0.277870	0.138935	−	0.990301i	$-0.455632\pi$
$241$	4.31371	0.277870	0.138935	+	0.990301i	$0.455632\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−10.9706	−0.698040
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.92893	0.374231	0.187115	−	0.982338i	$-0.440086\pi$
$251$	5.92893	0.374231	0.187115	+	0.982338i	$0.440086\pi$
$252$	0	0
$253$	−18.4853	−1.16216
$254$	0	0
$255$	0	0
$256$	0	0
$257$	23.8284	1.48638	0.743188	−	0.669082i	$-0.233313\pi$
$257$	23.8284	1.48638	0.743188	+	0.669082i	$0.233313\pi$
$258$	0	0
$259$	11.3137	0.703000
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.2426	0.693251	0.346625	−	0.938004i	$-0.387327\pi$
$263$	11.2426	0.693251	0.346625	+	0.938004i	$0.387327\pi$
$264$	0	0
$265$	7.48528	0.459817
$266$	0	0
$267$	0	0
$268$	0	0
$269$	19.4558	1.18624	0.593122	−	0.805113i	$-0.297895\pi$
$269$	19.4558	1.18624	0.593122	+	0.805113i	$0.297895\pi$
$270$	0	0
$271$	14.5563	0.884235	0.442118	−	0.896957i	$-0.354227\pi$
$271$	14.5563	0.884235	0.442118	+	0.896957i	$0.354227\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−9.65685	−0.582330
$276$	0	0
$277$	5.31371	0.319270	0.159635	−	0.987176i	$-0.448968\pi$
$277$	5.31371	0.319270	0.159635	+	0.987176i	$0.448968\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.97056	0.117554	0.0587770	−	0.998271i	$-0.481280\pi$
$281$	1.97056	0.117554	0.0587770	+	0.998271i	$0.481280\pi$
$282$	0	0
$283$	−0.343146	−0.0203979	−0.0101989	−	0.999948i	$-0.503246\pi$
$283$	−0.343146	−0.0203979	−0.0101989	+	0.999948i	$0.503246\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	35.3137	2.08450
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.65685	0.213636	0.106818	−	0.994279i	$-0.465934\pi$
$293$	3.65685	0.213636	0.106818	+	0.994279i	$0.465934\pi$
$294$	0	0
$295$	7.65685	0.445799
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.0000	−0.809641
$300$	0	0
$301$	18.1421	1.04570
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.828427	0.0474356
$306$	0	0
$307$	16.8995	0.964505	0.482253	−	0.876032i	$-0.339819\pi$
$307$	16.8995	0.964505	0.482253	+	0.876032i	$0.339819\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	25.3137	1.43541	0.717704	−	0.696348i	$-0.245193\pi$
$311$	25.3137	1.43541	0.717704	+	0.696348i	$0.245193\pi$
$312$	0	0
$313$	4.17157	0.235791	0.117896	−	0.993026i	$-0.462385\pi$
$313$	4.17157	0.235791	0.117896	+	0.993026i	$0.462385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−19.4558	−1.09275	−0.546375	−	0.837541i	$-0.683992\pi$
$317$	−19.4558	−1.09275	−0.546375	+	0.837541i	$0.683992\pi$
$318$	0	0
$319$	−2.41421	−0.135170
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−28.9706	−1.61197
$324$	0	0
$325$	−7.31371	−0.405692
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−14.8284	−0.817518
$330$	0	0
$331$	−0.414214	−0.0227672	−0.0113836	−	0.999935i	$-0.503624\pi$
$331$	−0.414214	−0.0227672	−0.0113836	+	0.999935i	$0.503624\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.65685	0.309067
$336$	0	0
$337$	−17.7990	−0.969573	−0.484786	−	0.874633i	$-0.661103\pi$
$337$	−17.7990	−0.969573	−0.484786	+	0.874633i	$0.661103\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.82843	0.532239
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−14.4853	−0.777611	−0.388805	−	0.921320i	$-0.627112\pi$
$347$	−14.4853	−0.777611	−0.388805	+	0.921320i	$0.627112\pi$
$348$	0	0
$349$	23.1421	1.23877	0.619385	−	0.785087i	$-0.287382\pi$
$349$	23.1421	1.23877	0.619385	+	0.785087i	$0.287382\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.97056	0.371006	0.185503	−	0.982644i	$-0.440609\pi$
$353$	6.97056	0.371006	0.185503	+	0.982644i	$0.440609\pi$
$354$	0	0
$355$	−3.17157	−0.168330
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.0711	0.953754	0.476877	−	0.878970i	$-0.341769\pi$
$359$	18.0711	0.953754	0.476877	+	0.878970i	$0.341769\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.00000	0.209370
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−21.1716	−1.09917
$372$	0	0
$373$	−3.68629	−0.190869	−0.0954345	−	0.995436i	$-0.530424\pi$
$373$	−3.68629	−0.190869	−0.0954345	+	0.995436i	$0.530424\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.82843	−0.0941688
$378$	0	0
$379$	−26.9706	−1.38538	−0.692692	−	0.721233i	$-0.743576\pi$
$379$	−26.9706	−1.38538	−0.692692	+	0.721233i	$0.743576\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−20.4853	−1.04675	−0.523374	−	0.852103i	$-0.675327\pi$
$383$	−20.4853	−1.04675	−0.523374	+	0.852103i	$0.675327\pi$
$384$	0	0
$385$	−6.82843	−0.348009
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−36.9706	−1.87448	−0.937241	−	0.348682i	$-0.886629\pi$
$389$	−36.9706	−1.87448	−0.937241	+	0.348682i	$0.886629\pi$
$390$	0	0
$391$	−36.9706	−1.86968
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−0.414214	−0.0208413
$396$	0	0
$397$	30.6569	1.53862	0.769312	−	0.638874i	$-0.220599\pi$
$397$	30.6569	1.53862	0.769312	+	0.638874i	$0.220599\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7.34315	0.366699	0.183350	−	0.983048i	$-0.441306\pi$
$401$	7.34315	0.366699	0.183350	+	0.983048i	$0.441306\pi$
$402$	0	0
$403$	7.44365	0.370795
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−9.65685	−0.478672
$408$	0	0
$409$	14.9706	0.740247	0.370123	−	0.928983i	$-0.379315\pi$
$409$	14.9706	0.740247	0.370123	+	0.928983i	$0.379315\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−21.6569	−1.06566
$414$	0	0
$415$	−3.65685	−0.179508
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−26.4853	−1.29389	−0.646945	−	0.762536i	$-0.723954\pi$
$419$	−26.4853	−1.29389	−0.646945	+	0.762536i	$0.723954\pi$
$420$	0	0
$421$	−25.1127	−1.22392	−0.611959	−	0.790889i	$-0.709618\pi$
$421$	−25.1127	−1.22392	−0.611959	+	0.790889i	$0.709618\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−19.3137	−0.936852
$426$	0	0
$427$	−2.34315	−0.113393
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.34315	0.401875	0.200938	−	0.979604i	$-0.435601\pi$
$431$	8.34315	0.401875	0.200938	+	0.979604i	$0.435601\pi$
$432$	0	0
$433$	−14.6274	−0.702949	−0.351474	−	0.936197i	$-0.614320\pi$
$433$	−14.6274	−0.702949	−0.351474	+	0.936197i	$0.614320\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	45.9411	2.19766
$438$	0	0
$439$	11.6569	0.556351	0.278176	−	0.960530i	$-0.410270\pi$
$439$	11.6569	0.556351	0.278176	+	0.960530i	$0.410270\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.6569	−1.69411	−0.847054	−	0.531507i	$-0.821626\pi$
$443$	−35.6569	−1.69411	−0.847054	+	0.531507i	$0.821626\pi$
$444$	0	0
$445$	−4.48528	−0.212623
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.02944	0.0485821	0.0242911	−	0.999705i	$-0.492267\pi$
$449$	1.02944	0.0485821	0.0242911	+	0.999705i	$0.492267\pi$
$450$	0	0
$451$	−30.1421	−1.41934
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.17157	−0.242447
$456$	0	0
$457$	34.9706	1.63585	0.817927	−	0.575322i	$-0.195123\pi$
$457$	34.9706	1.63585	0.817927	+	0.575322i	$0.195123\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−14.0000	−0.652045	−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000	−0.652045	−0.326023	+	0.945362i	$0.605709\pi$
$462$	0	0
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	32.3553	1.49723	0.748613	−	0.663007i	$-0.230720\pi$
$467$	32.3553	1.49723	0.748613	+	0.663007i	$0.230720\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−15.4853	−0.712014
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−12.8995	−0.589393	−0.294696	−	0.955591i	$-0.595219\pi$
$479$	−12.8995	−0.589393	−0.294696	+	0.955591i	$0.595219\pi$
$480$	0	0
$481$	−7.31371	−0.333476
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.4853	−0.566927
$486$	0	0
$487$	28.4853	1.29079	0.645396	−	0.763848i	$-0.276693\pi$
$487$	28.4853	1.29079	0.645396	+	0.763848i	$0.276693\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−12.7574	−0.575732	−0.287866	−	0.957671i	$-0.592946\pi$
$491$	−12.7574	−0.575732	−0.287866	+	0.957671i	$0.592946\pi$
$492$	0	0
$493$	−4.82843	−0.217461
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.97056	0.402385
$498$	0	0
$499$	14.9706	0.670174	0.335087	−	0.942187i	$-0.391234\pi$
$499$	14.9706	0.670174	0.335087	+	0.942187i	$0.391234\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	25.7279	1.14715	0.573576	−	0.819153i	$-0.305556\pi$
$503$	25.7279	1.14715	0.573576	+	0.819153i	$0.305556\pi$
$504$	0	0
$505$	13.6569	0.607722
$506$	0	0
$507$	0	0
$508$	0	0
$509$	27.4853	1.21826	0.609132	−	0.793069i	$-0.291518\pi$
$509$	27.4853	1.21826	0.609132	+	0.793069i	$0.291518\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−0.828427	−0.0365049
$516$	0	0
$517$	12.6569	0.556648
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.857864	0.0375837	0.0187919	−	0.999823i	$-0.494018\pi$
$521$	0.857864	0.0375837	0.0187919	+	0.999823i	$0.494018\pi$
$522$	0	0
$523$	−27.3137	−1.19435	−0.597173	−	0.802113i	$-0.703709\pi$
$523$	−27.3137	−1.19435	−0.597173	+	0.802113i	$0.703709\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.6569	0.856266
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22.8284	−0.988809
$534$	0	0
$535$	−9.17157	−0.396522
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.41421	0.103988
$540$	0	0
$541$	−21.6569	−0.931101	−0.465550	−	0.885021i	$-0.654144\pi$
$541$	−21.6569	−0.931101	−0.465550	+	0.885021i	$0.654144\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.34315	0.0575340
$546$	0	0
$547$	3.79899	0.162433	0.0812165	−	0.996696i	$-0.474119\pi$
$547$	3.79899	0.162433	0.0812165	+	0.996696i	$0.474119\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	1.17157	0.0498203
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−5.31371	−0.225149	−0.112575	−	0.993643i	$-0.535910\pi$
$557$	−5.31371	−0.225149	−0.112575	+	0.993643i	$0.535910\pi$
$558$	0	0
$559$	−11.7279	−0.496038
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.24264	−0.389531	−0.194765	−	0.980850i	$-0.562395\pi$
$563$	−9.24264	−0.389531	−0.194765	+	0.980850i	$0.562395\pi$
$564$	0	0
$565$	−9.31371	−0.391831
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.3431	1.18821	0.594103	−	0.804389i	$-0.297507\pi$
$569$	28.3431	1.18821	0.594103	+	0.804389i	$0.297507\pi$
$570$	0	0
$571$	30.6274	1.28172	0.640859	−	0.767659i	$-0.278578\pi$
$571$	30.6274	1.28172	0.640859	+	0.767659i	$0.278578\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.6274	1.27725
$576$	0	0
$577$	9.79899	0.407937	0.203969	−	0.978977i	$-0.434616\pi$
$577$	9.79899	0.407937	0.203969	+	0.978977i	$0.434616\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	10.3431	0.429106
$582$	0	0
$583$	18.0711	0.748427
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.65685	−0.150935	−0.0754673	−	0.997148i	$-0.524045\pi$
$587$	−3.65685	−0.150935	−0.0754673	+	0.997148i	$0.524045\pi$
$588$	0	0
$589$	−24.4264	−1.00647
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.51472	0.103267	0.0516336	−	0.998666i	$-0.483557\pi$
$593$	2.51472	0.103267	0.0516336	+	0.998666i	$0.483557\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−43.8701	−1.79248	−0.896241	−	0.443567i	$-0.853713\pi$
$599$	−43.8701	−1.79248	−0.896241	+	0.443567i	$0.853713\pi$
$600$	0	0
$601$	−22.8284	−0.931191	−0.465595	−	0.884998i	$-0.654160\pi$
$601$	−22.8284	−0.931191	−0.465595	+	0.884998i	$0.654160\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.17157	−0.210254
$606$	0	0
$607$	−17.7279	−0.719554	−0.359777	−	0.933038i	$-0.617147\pi$
$607$	−17.7279	−0.719554	−0.359777	+	0.933038i	$0.617147\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9.58579	0.387799
$612$	0	0
$613$	−9.00000	−0.363507	−0.181753	−	0.983344i	$-0.558177\pi$
$613$	−9.00000	−0.363507	−0.181753	+	0.983344i	$0.558177\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23.3137	−0.938575	−0.469287	−	0.883046i	$-0.655489\pi$
$617$	−23.3137	−0.938575	−0.469287	+	0.883046i	$0.655489\pi$
$618$	0	0
$619$	−36.4142	−1.46361	−0.731805	−	0.681514i	$-0.761322\pi$
$619$	−36.4142	−1.46361	−0.731805	+	0.681514i	$0.761322\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12.6863	0.508266
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19.3137	−0.770088
$630$	0	0
$631$	31.1716	1.24092	0.620460	−	0.784238i	$-0.286946\pi$
$631$	31.1716	1.24092	0.620460	+	0.784238i	$0.286946\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15.6569	0.621323
$636$	0	0
$637$	1.82843	0.0724449
$638$	0	0
$639$	0	0
$640$	0	0
$641$	21.7990	0.861008	0.430504	−	0.902589i	$-0.358336\pi$
$641$	21.7990	0.861008	0.430504	+	0.902589i	$0.358336\pi$
$642$	0	0
$643$	−15.5147	−0.611841	−0.305920	−	0.952057i	$-0.598964\pi$
$643$	−15.5147	−0.611841	−0.305920	+	0.952057i	$0.598964\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	28.3431	1.11428	0.557142	−	0.830417i	$-0.311898\pi$
$647$	28.3431	1.11428	0.557142	+	0.830417i	$0.311898\pi$
$648$	0	0
$649$	18.4853	0.725611
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.85786	0.0727039	0.0363519	−	0.999339i	$-0.488426\pi$
$653$	1.85786	0.0727039	0.0363519	+	0.999339i	$0.488426\pi$
$654$	0	0
$655$	−1.31371	−0.0513308
$656$	0	0
$657$	0	0
$658$	0	0
$659$	11.5858	0.451318	0.225659	−	0.974206i	$-0.427546\pi$
$659$	11.5858	0.451318	0.225659	+	0.974206i	$0.427546\pi$
$660$	0	0
$661$	10.6863	0.415649	0.207824	−	0.978166i	$-0.433362\pi$
$661$	10.6863	0.415649	0.207824	+	0.978166i	$0.433362\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.9706	0.658090
$666$	0	0
$667$	7.65685	0.296475
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	23.6274	0.910770	0.455385	−	0.890295i	$-0.349502\pi$
$673$	23.6274	0.910770	0.455385	+	0.890295i	$0.349502\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	0	0
$679$	35.3137	1.35522
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.9706	−0.496305	−0.248152	−	0.968721i	$-0.579823\pi$
$683$	−12.9706	−0.496305	−0.248152	+	0.968721i	$0.579823\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13.6863	0.521406
$690$	0	0
$691$	−48.0000	−1.82601	−0.913003	−	0.407953i	$-0.866243\pi$
$691$	−48.0000	−1.82601	−0.913003	+	0.407953i	$0.866243\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.0000	−0.531050
$696$	0	0
$697$	−60.2843	−2.28343
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−22.1127	−0.835185	−0.417593	−	0.908634i	$-0.637126\pi$
$701$	−22.1127	−0.835185	−0.417593	+	0.908634i	$0.637126\pi$
$702$	0	0
$703$	24.0000	0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−38.6274	−1.45273
$708$	0	0
$709$	0.857864	0.0322178	0.0161089	−	0.999870i	$-0.494872\pi$
$709$	0.857864	0.0322178	0.0161089	+	0.999870i	$0.494872\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−31.1716	−1.16738
$714$	0	0
$715$	4.41421	0.165082
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.14214	0.303650	0.151825	−	0.988407i	$-0.451485\pi$
$719$	8.14214	0.303650	0.151825	+	0.988407i	$0.451485\pi$
$720$	0	0
$721$	2.34315	0.0872633
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.00000	0.148556
$726$	0	0
$727$	21.3137	0.790482	0.395241	−	0.918578i	$-0.370661\pi$
$727$	21.3137	0.790482	0.395241	+	0.918578i	$0.370661\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−30.9706	−1.14549
$732$	0	0
$733$	49.2548	1.81927	0.909634	−	0.415410i	$-0.136362\pi$
$733$	49.2548	1.81927	0.909634	+	0.415410i	$0.136362\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.6569	0.503057
$738$	0	0
$739$	10.0711	0.370470	0.185235	−	0.982694i	$-0.440695\pi$
$739$	10.0711	0.370470	0.185235	+	0.982694i	$0.440695\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12.3431	0.452826	0.226413	−	0.974031i	$-0.427300\pi$
$743$	12.3431	0.452826	0.226413	+	0.974031i	$0.427300\pi$
$744$	0	0
$745$	2.17157	0.0795603
$746$	0	0
$747$	0	0
$748$	0	0
$749$	25.9411	0.947868
$750$	0	0
$751$	−2.68629	−0.0980242	−0.0490121	−	0.998798i	$-0.515607\pi$
$751$	−2.68629	−0.0980242	−0.0490121	+	0.998798i	$0.515607\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−14.1421	−0.514685
$756$	0	0
$757$	42.4853	1.54415	0.772077	−	0.635529i	$-0.219218\pi$
$757$	42.4853	1.54415	0.772077	+	0.635529i	$0.219218\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33.5980	1.21793	0.608963	−	0.793199i	$-0.291586\pi$
$761$	33.5980	1.21793	0.608963	+	0.793199i	$0.291586\pi$
$762$	0	0
$763$	−3.79899	−0.137533
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.0000	0.505511
$768$	0	0
$769$	13.1127	0.472856	0.236428	−	0.971649i	$-0.424023\pi$
$769$	13.1127	0.472856	0.236428	+	0.971649i	$0.424023\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	36.4853	1.31228	0.656142	−	0.754637i	$-0.272187\pi$
$773$	36.4853	1.31228	0.656142	+	0.754637i	$0.272187\pi$
$774$	0	0
$775$	−16.2843	−0.584948
$776$	0	0
$777$	0	0
$778$	0	0
$779$	74.9117	2.68399
$780$	0	0
$781$	−7.65685	−0.273984
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.48528	−0.302853
$786$	0	0
$787$	−42.0833	−1.50011	−0.750053	−	0.661378i	$-0.769972\pi$
$787$	−42.0833	−1.50011	−0.750053	+	0.661378i	$0.769972\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26.3431	0.936654
$792$	0	0
$793$	1.51472	0.0537892
$794$	0	0
$795$	0	0
$796$	0	0
$797$	55.7401	1.97442	0.987208	−	0.159437i	$-0.0509679\pi$
$797$	55.7401	1.97442	0.987208	+	0.159437i	$0.0509679\pi$
$798$	0	0
$799$	25.3137	0.895535
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.65685	0.340783
$804$	0	0
$805$	21.6569	0.763304
$806$	0	0
$807$	0	0
$808$	0	0
$809$	20.2843	0.713157	0.356578	−	0.934265i	$-0.383943\pi$
$809$	20.2843	0.713157	0.356578	+	0.934265i	$0.383943\pi$
$810$	0	0
$811$	−5.17157	−0.181598	−0.0907992	−	0.995869i	$-0.528942\pi$
$811$	−5.17157	−0.181598	−0.0907992	+	0.995869i	$0.528942\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−18.0711	−0.633002
$816$	0	0
$817$	38.4853	1.34643
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.4853	−0.540440	−0.270220	−	0.962799i	$-0.587096\pi$
$821$	−15.4853	−0.540440	−0.270220	+	0.962799i	$0.587096\pi$
$822$	0	0
$823$	−2.28427	−0.0796247	−0.0398123	−	0.999207i	$-0.512676\pi$
$823$	−2.28427	−0.0796247	−0.0398123	+	0.999207i	$0.512676\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13.1005	0.455549	0.227775	−	0.973714i	$-0.426855\pi$
$827$	13.1005	0.455549	0.227775	+	0.973714i	$0.426855\pi$
$828$	0	0
$829$	9.79899	0.340333	0.170166	−	0.985415i	$-0.445569\pi$
$829$	9.79899	0.340333	0.170166	+	0.985415i	$0.445569\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.82843	0.167295
$834$	0	0
$835$	−8.82843	−0.305520
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.0711	−0.761978	−0.380989	−	0.924580i	$-0.624416\pi$
$839$	−22.0711	−0.761978	−0.380989	+	0.924580i	$0.624416\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9.65685	−0.332206
$846$	0	0
$847$	14.6274	0.502604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	30.6274	1.04989
$852$	0	0
$853$	10.9706	0.375625	0.187812	−	0.982205i	$-0.439860\pi$
$853$	10.9706	0.375625	0.187812	+	0.982205i	$0.439860\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.8284	0.404051	0.202026	−	0.979380i	$-0.435248\pi$
$857$	11.8284	0.404051	0.202026	+	0.979380i	$0.435248\pi$
$858$	0	0
$859$	5.72792	0.195434	0.0977171	−	0.995214i	$-0.468846\pi$
$859$	5.72792	0.195434	0.0977171	+	0.995214i	$0.468846\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−45.1127	−1.53565	−0.767827	−	0.640657i	$-0.778662\pi$
$863$	−45.1127	−1.53565	−0.767827	+	0.640657i	$0.778662\pi$
$864$	0	0
$865$	−23.6569	−0.804357
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	10.3431	0.350464
$872$	0	0
$873$	0	0
$874$	0	0
$875$	25.4558	0.860565
$876$	0	0
$877$	−8.85786	−0.299109	−0.149554	−	0.988753i	$-0.547784\pi$
$877$	−8.85786	−0.299109	−0.149554	+	0.988753i	$0.547784\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−14.0000	−0.471672	−0.235836	−	0.971793i	$-0.575783\pi$
$881$	−14.0000	−0.471672	−0.235836	+	0.971793i	$0.575783\pi$
$882$	0	0
$883$	46.4264	1.56237	0.781186	−	0.624298i	$-0.214615\pi$
$883$	46.4264	1.56237	0.781186	+	0.624298i	$0.214615\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.8995	1.23896	0.619482	−	0.785011i	$-0.287343\pi$
$887$	36.8995	1.23896	0.619482	+	0.785011i	$0.287343\pi$
$888$	0	0
$889$	−44.2843	−1.48525
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.4558	−1.05263
$894$	0	0
$895$	10.4853	0.350484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4.07107	−0.135778
$900$	0	0
$901$	36.1421	1.20407
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14.3137	−0.475804
$906$	0	0
$907$	34.2843	1.13839	0.569195	−	0.822202i	$-0.307255\pi$
$907$	34.2843	1.13839	0.569195	+	0.822202i	$0.307255\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−46.5563	−1.54248	−0.771240	−	0.636544i	$-0.780363\pi$
$911$	−46.5563	−1.54248	−0.771240	+	0.636544i	$0.780363\pi$
$912$	0	0
$913$	−8.82843	−0.292178
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.71573	0.122704
$918$	0	0
$919$	20.1421	0.664428	0.332214	−	0.943204i	$-0.392204\pi$
$919$	20.1421	0.664428	0.332214	+	0.943204i	$0.392204\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5.79899	−0.190876
$924$	0	0
$925$	16.0000	0.526077
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41.3137	−1.35546	−0.677729	−	0.735311i	$-0.737036\pi$
$929$	−41.3137	−1.35546	−0.677729	+	0.735311i	$0.737036\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	0	0
$934$	0	0
$935$	11.6569	0.381220
$936$	0	0
$937$	28.6274	0.935217	0.467608	−	0.883936i	$-0.345116\pi$
$937$	28.6274	0.935217	0.467608	+	0.883936i	$0.345116\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.5980	−0.736673	−0.368337	−	0.929693i	$-0.620073\pi$
$941$	−22.5980	−0.736673	−0.368337	+	0.929693i	$0.620073\pi$
$942$	0	0
$943$	95.5980	3.11310
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.3848	1.27983	0.639917	−	0.768444i	$-0.278969\pi$
$947$	39.3848	1.27983	0.639917	+	0.768444i	$0.278969\pi$
$948$	0	0
$949$	7.31371	0.237413
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.62742	−0.311863	−0.155931	−	0.987768i	$-0.549838\pi$
$953$	−9.62742	−0.311863	−0.155931	+	0.987768i	$0.549838\pi$
$954$	0	0
$955$	2.68629	0.0869264
$956$	0	0
$957$	0	0
$958$	0	0
$959$	33.9411	1.09602
$960$	0	0
$961$	−14.4264	−0.465368
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.8284	−0.348579
$966$	0	0
$967$	26.7574	0.860459	0.430229	−	0.902720i	$-0.358433\pi$
$967$	26.7574	0.860459	0.430229	+	0.902720i	$0.358433\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.34315	−0.139378	−0.0696891	−	0.997569i	$-0.522201\pi$
$971$	−4.34315	−0.139378	−0.0696891	+	0.997569i	$0.522201\pi$
$972$	0	0
$973$	39.5980	1.26945
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−41.8284	−1.33821	−0.669105	−	0.743168i	$-0.733322\pi$
$977$	−41.8284	−1.33821	−0.669105	+	0.743168i	$0.733322\pi$
$978$	0	0
$979$	−10.8284	−0.346078
$980$	0	0
$981$	0	0
$982$	0	0
$983$	31.8701	1.01650	0.508248	−	0.861210i	$-0.330293\pi$
$983$	31.8701	1.01650	0.508248	+	0.861210i	$0.330293\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	49.1127	1.56169
$990$	0	0
$991$	7.17157	0.227813	0.113906	−	0.993492i	$-0.463664\pi$
$991$	7.17157	0.227813	0.113906	+	0.993492i	$0.463664\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.4853	−0.522619
$996$	0	0
$997$	−28.2843	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$997$	−28.2843	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4176.2.a.bq.1.1		2
3.2	odd	2		464.2.a.h.1.2		2
4.3	odd	2		261.2.a.d.1.1		2
12.11	even	2		29.2.a.a.1.2	✓	2
20.19	odd	2		6525.2.a.o.1.2		2
24.5	odd	2		1856.2.a.w.1.1		2
24.11	even	2		1856.2.a.r.1.2		2
60.23	odd	4		725.2.b.b.349.2		4
60.47	odd	4		725.2.b.b.349.3		4
60.59	even	2		725.2.a.b.1.1		2
84.83	odd	2		1421.2.a.j.1.2		2
116.115	odd	2		7569.2.a.c.1.2		2
132.131	odd	2		3509.2.a.j.1.1		2
156.155	even	2		4901.2.a.g.1.1		2
204.203	even	2		8381.2.a.e.1.2		2
348.11	odd	28		841.2.e.k.63.3		24
348.23	even	14		841.2.d.j.645.1		12
348.35	even	14		841.2.d.f.645.2		12
348.47	odd	28		841.2.e.k.63.2		24
348.71	even	14		841.2.d.f.778.1		12
348.83	even	14		841.2.d.j.190.1		12
348.95	odd	28		841.2.e.k.267.3		24
348.107	even	14		841.2.d.j.574.2		12
348.119	odd	28		841.2.e.k.270.3		24
348.131	odd	28		841.2.e.k.196.3		24
348.143	odd	28		841.2.e.k.236.2		24
348.155	odd	28		841.2.e.k.651.3		24
348.167	even	14		841.2.d.f.571.2		12
348.179	even	14		841.2.d.f.605.2		12
348.191	odd	4		841.2.b.a.840.2		4
348.215	odd	4		841.2.b.a.840.3		4
348.227	even	14		841.2.d.j.605.1		12
348.239	even	14		841.2.d.j.571.1		12
348.251	odd	28		841.2.e.k.651.2		24
348.263	odd	28		841.2.e.k.236.3		24
348.275	odd	28		841.2.e.k.196.2		24
348.287	odd	28		841.2.e.k.270.2		24
348.299	even	14		841.2.d.f.574.1		12
348.311	odd	28		841.2.e.k.267.2		24
348.323	even	14		841.2.d.f.190.2		12
348.335	even	14		841.2.d.j.778.2		12
348.347	even	2		841.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.2	✓	2	12.11	even	2
261.2.a.d.1.1		2	4.3	odd	2
464.2.a.h.1.2		2	3.2	odd	2
725.2.a.b.1.1		2	60.59	even	2
725.2.b.b.349.2		4	60.23	odd	4
725.2.b.b.349.3		4	60.47	odd	4
841.2.a.d.1.1		2	348.347	even	2
841.2.b.a.840.2		4	348.191	odd	4
841.2.b.a.840.3		4	348.215	odd	4
841.2.d.f.190.2		12	348.323	even	14
841.2.d.f.571.2		12	348.167	even	14
841.2.d.f.574.1		12	348.299	even	14
841.2.d.f.605.2		12	348.179	even	14
841.2.d.f.645.2		12	348.35	even	14
841.2.d.f.778.1		12	348.71	even	14
841.2.d.j.190.1		12	348.83	even	14
841.2.d.j.571.1		12	348.239	even	14
841.2.d.j.574.2		12	348.107	even	14
841.2.d.j.605.1		12	348.227	even	14
841.2.d.j.645.1		12	348.23	even	14
841.2.d.j.778.2		12	348.335	even	14
841.2.e.k.63.2		24	348.47	odd	28
841.2.e.k.63.3		24	348.11	odd	28
841.2.e.k.196.2		24	348.275	odd	28
841.2.e.k.196.3		24	348.131	odd	28
841.2.e.k.236.2		24	348.143	odd	28
841.2.e.k.236.3		24	348.263	odd	28
841.2.e.k.267.2		24	348.311	odd	28
841.2.e.k.267.3		24	348.95	odd	28
841.2.e.k.270.2		24	348.287	odd	28
841.2.e.k.270.3		24	348.119	odd	28
841.2.e.k.651.2		24	348.251	odd	28
841.2.e.k.651.3		24	348.155	odd	28
1421.2.a.j.1.2		2	84.83	odd	2
1856.2.a.r.1.2		2	24.11	even	2
1856.2.a.w.1.1		2	24.5	odd	2
3509.2.a.j.1.1		2	132.131	odd	2
4176.2.a.bq.1.1		2	1.1	even	1	trivial
4901.2.a.g.1.1		2	156.155	even	2
6525.2.a.o.1.2		2	20.19	odd	2
7569.2.a.c.1.2		2	116.115	odd	2
8381.2.a.e.1.2		2	204.203	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4176 = 2^{4} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4176.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.82843 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.82843 q^{7} +2.41421 q^{11} +1.82843 q^{13} +4.82843 q^{17} -6.00000 q^{19} -7.65685 q^{23} -4.00000 q^{25} -1.00000 q^{29} +4.07107 q^{31} -2.82843 q^{35} -4.00000 q^{37} -12.4853 q^{41} -6.41421 q^{43} +5.24264 q^{47} +1.00000 q^{49} +7.48528 q^{53} +2.41421 q^{55} +7.65685 q^{59} +0.828427 q^{61} +1.82843 q^{65} +5.65685 q^{67} -3.17157 q^{71} +4.00000 q^{73} -6.82843 q^{77} -0.414214 q^{79} -3.65685 q^{83} +4.82843 q^{85} -4.48528 q^{89} -5.17157 q^{91} -6.00000 q^{95} -12.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} + 2 q^{11} - 2 q^{13} + 4 q^{17} - 12 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} - 6 q^{31} - 8 q^{37} - 8 q^{41} - 10 q^{43} + 2 q^{47} + 2 q^{49} - 2 q^{53} + 2 q^{55} + 4 q^{59} - 4 q^{61} - 2 q^{65} - 12 q^{71} + 8 q^{73} - 8 q^{77} + 2 q^{79} + 4 q^{83} + 4 q^{85} + 8 q^{89} - 16 q^{91} - 12 q^{95} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^11 - 2 * q^13 + 4 * q^17 - 12 * q^19 - 4 * q^23 - 8 * q^25 - 2 * q^29 - 6 * q^31 - 8 * q^37 - 8 * q^41 - 10 * q^43 + 2 * q^47 + 2 * q^49 - 2 * q^53 + 2 * q^55 + 4 * q^59 - 4 * q^61 - 2 * q^65 - 12 * q^71 + 8 * q^73 - 8 * q^77 + 2 * q^79 + 4 * q^83 + 4 * q^85 + 8 * q^89 - 16 * q^91 - 12 * q^95 - 8 * q^97

Introduction

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