LMFDB - Embedded newform 8379.2.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] 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("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,0,6,0,10,2,0,0,0,0,6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.9066518536$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 399)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +3.82843 q^{5} +4.41421 q^{8} +9.24264 q^{10} +3.82843 q^{11} +2.82843 q^{13} +3.00000 q^{16} +3.65685 q^{17} -1.00000 q^{19} +14.6569 q^{20} +9.24264 q^{22} +1.82843 q^{23} +9.65685 q^{25} +6.82843 q^{26} +4.82843 q^{29} -3.17157 q^{31} -1.58579 q^{32} +8.82843 q^{34} +1.17157 q^{37} -2.41421 q^{38} +16.8995 q^{40} -7.65685 q^{41} -12.6569 q^{43} +14.6569 q^{44} +4.41421 q^{46} -6.17157 q^{47} +23.3137 q^{50} +10.8284 q^{52} +2.00000 q^{53} +14.6569 q^{55} +11.6569 q^{58} -10.8284 q^{59} -14.6569 q^{61} -7.65685 q^{62} -9.82843 q^{64} +10.8284 q^{65} -12.0000 q^{67} +14.0000 q^{68} -4.34315 q^{71} -10.3137 q^{73} +2.82843 q^{74} -3.82843 q^{76} +2.34315 q^{79} +11.4853 q^{80} -18.4853 q^{82} -4.17157 q^{83} +14.0000 q^{85} -30.5563 q^{86} +16.8995 q^{88} -10.8284 q^{89} +7.00000 q^{92} -14.8995 q^{94} -3.82843 q^{95} +16.8284 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 10 q^{10} + 2 q^{11} + 6 q^{16} - 4 q^{17} - 2 q^{19} + 18 q^{20} + 10 q^{22} - 2 q^{23} + 8 q^{25} + 8 q^{26} + 4 q^{29} - 12 q^{31} - 6 q^{32} + 12 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 10 * q^10 + 2 * q^11 + 6 * q^16 - 4 * q^17 - 2 * q^19 + 18 * q^20 + 10 * q^22 - 2 * q^23 + 8 * q^25 + 8 * q^26 + 4 * q^29 - 12 * q^31 - 6 * q^32 + 12 * q^34 + 8 * q^37 - 2 * q^38 + 14 * q^40 - 4 * q^41 - 14 * q^43 + 18 * q^44 + 6 * q^46 - 18 * q^47 + 24 * q^50 + 16 * q^52 + 4 * q^53 + 18 * q^55 + 12 * q^58 - 16 * q^59 - 18 * q^61 - 4 * q^62 - 14 * q^64 + 16 * q^65 - 24 * q^67 + 28 * q^68 - 20 * q^71 + 2 * q^73 - 2 * q^76 + 16 * q^79 + 6 * q^80 - 20 * q^82 - 14 * q^83 + 28 * q^85 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 14 * q^92 - 10 * q^94 - 2 * q^95 + 28 * q^97

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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	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
$7$	0	0
$7$	0	0
$8$	4.41421	1.56066
$9$	0	0
$10$	9.24264	2.92278
$11$	3.82843	1.15431	0.577157	−	0.816633i	$-0.304162\pi$
$11$	3.82843	1.15431	0.577157	+	0.816633i	$0.304162\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	14.6569	3.27737
$21$	0	0
$22$	9.24264	1.97054
$23$	1.82843	0.381253	0.190627	−	0.981663i	$-0.438948\pi$
$23$	1.82843	0.381253	0.190627	+	0.981663i	$0.438948\pi$
$24$	0	0
$25$	9.65685	1.93137
$26$	6.82843	1.33916
$27$	0	0
$28$	0	0
$29$	4.82843	0.896616	0.448308	−	0.893879i	$-0.352027\pi$
$29$	4.82843	0.896616	0.448308	+	0.893879i	$0.352027\pi$
$30$	0	0
$31$	−3.17157	−0.569631	−0.284816	−	0.958582i	$-0.591932\pi$
$31$	−3.17157	−0.569631	−0.284816	+	0.958582i	$0.591932\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	8.82843	1.51406
$35$	0	0
$36$	0	0
$37$	1.17157	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$37$	1.17157	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$38$	−2.41421	−0.391637
$39$	0	0
$40$	16.8995	2.67204
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	−12.6569	−1.93015	−0.965076	−	0.261970i	$-0.915628\pi$
$43$	−12.6569	−1.93015	−0.965076	+	0.261970i	$0.915628\pi$
$44$	14.6569	2.20960
$45$	0	0
$46$	4.41421	0.650840
$47$	−6.17157	−0.900216	−0.450108	−	0.892974i	$-0.648615\pi$
$47$	−6.17157	−0.900216	−0.450108	+	0.892974i	$0.648615\pi$
$48$	0	0
$49$	0	0
$50$	23.3137	3.29706
$51$	0	0
$52$	10.8284	1.50163
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	14.6569	1.97633
$56$	0	0
$57$	0	0
$58$	11.6569	1.53062
$59$	−10.8284	−1.40974	−0.704871	−	0.709336i	$-0.748995\pi$
$59$	−10.8284	−1.40974	−0.704871	+	0.709336i	$0.748995\pi$
$60$	0	0
$61$	−14.6569	−1.87662	−0.938309	−	0.345798i	$-0.887608\pi$
$61$	−14.6569	−1.87662	−0.938309	+	0.345798i	$0.887608\pi$
$62$	−7.65685	−0.972421
$63$	0	0
$64$	−9.82843	−1.22855
$65$	10.8284	1.34310
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	14.0000	1.69775
$69$	0	0
$70$	0	0
$71$	−4.34315	−0.515437	−0.257718	−	0.966220i	$-0.582971\pi$
$71$	−4.34315	−0.515437	−0.257718	+	0.966220i	$0.582971\pi$
$72$	0	0
$73$	−10.3137	−1.20713	−0.603564	−	0.797314i	$-0.706253\pi$
$73$	−10.3137	−1.20713	−0.603564	+	0.797314i	$0.706253\pi$
$74$	2.82843	0.328798
$75$	0	0
$76$	−3.82843	−0.439151
$77$	0	0
$78$	0	0
$79$	2.34315	0.263624	0.131812	−	0.991275i	$-0.457920\pi$
$79$	2.34315	0.263624	0.131812	+	0.991275i	$0.457920\pi$
$80$	11.4853	1.28409
$81$	0	0
$82$	−18.4853	−2.04136
$83$	−4.17157	−0.457890	−0.228945	−	0.973439i	$-0.573528\pi$
$83$	−4.17157	−0.457890	−0.228945	+	0.973439i	$0.573528\pi$
$84$	0	0
$85$	14.0000	1.51851
$86$	−30.5563	−3.29498
$87$	0	0
$88$	16.8995	1.80149
$89$	−10.8284	−1.14781	−0.573905	−	0.818922i	$-0.694572\pi$
$89$	−10.8284	−1.14781	−0.573905	+	0.818922i	$0.694572\pi$
$90$	0	0
$91$	0	0
$92$	7.00000	0.729800
$93$	0	0
$94$	−14.8995	−1.53677
$95$	−3.82843	−0.392788
$96$	0	0
$97$	16.8284	1.70867	0.854334	−	0.519724i	$-0.173965\pi$
$97$	16.8284	1.70867	0.854334	+	0.519724i	$0.173965\pi$
$98$	0	0
$99$	0	0
$100$	36.9706	3.69706
$101$	3.48528	0.346798	0.173399	−	0.984852i	$-0.444525\pi$
$101$	3.48528	0.346798	0.173399	+	0.984852i	$0.444525\pi$
$102$	0	0
$103$	3.65685	0.360321	0.180160	−	0.983637i	$-0.442338\pi$
$103$	3.65685	0.360321	0.180160	+	0.983637i	$0.442338\pi$
$104$	12.4853	1.22428
$105$	0	0
$106$	4.82843	0.468978
$107$	2.82843	0.273434	0.136717	−	0.990610i	$-0.456345\pi$
$107$	2.82843	0.273434	0.136717	+	0.990610i	$0.456345\pi$
$108$	0	0
$109$	18.1421	1.73770	0.868851	−	0.495074i	$-0.164859\pi$
$109$	18.1421	1.73770	0.868851	+	0.495074i	$0.164859\pi$
$110$	35.3848	3.37381
$111$	0	0
$112$	0	0
$113$	12.8284	1.20680	0.603398	−	0.797440i	$-0.293813\pi$
$113$	12.8284	1.20680	0.603398	+	0.797440i	$0.293813\pi$
$114$	0	0
$115$	7.00000	0.652753
$116$	18.4853	1.71632
$117$	0	0
$118$	−26.1421	−2.40658
$119$	0	0
$120$	0	0
$121$	3.65685	0.332441
$122$	−35.3848	−3.20359
$123$	0	0
$124$	−12.1421	−1.09040
$125$	17.8284	1.59462
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	26.1421	2.29282
$131$	−10.3431	−0.903685	−0.451842	−	0.892098i	$-0.649233\pi$
$131$	−10.3431	−0.903685	−0.451842	+	0.892098i	$0.649233\pi$
$132$	0	0
$133$	0	0
$134$	−28.9706	−2.50268
$135$	0	0
$136$	16.1421	1.38418
$137$	13.8284	1.18144	0.590721	−	0.806876i	$-0.298843\pi$
$137$	13.8284	1.18144	0.590721	+	0.806876i	$0.298843\pi$
$138$	0	0
$139$	−6.65685	−0.564627	−0.282314	−	0.959322i	$-0.591102\pi$
$139$	−6.65685	−0.564627	−0.282314	+	0.959322i	$0.591102\pi$
$140$	0	0
$141$	0	0
$142$	−10.4853	−0.879905
$143$	10.8284	0.905519
$144$	0	0
$145$	18.4853	1.53512
$146$	−24.8995	−2.06070
$147$	0	0
$148$	4.48528	0.368688
$149$	−3.82843	−0.313637	−0.156818	−	0.987627i	$-0.550124\pi$
$149$	−3.82843	−0.313637	−0.156818	+	0.987627i	$0.550124\pi$
$150$	0	0
$151$	−9.31371	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$151$	−9.31371	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$152$	−4.41421	−0.358040
$153$	0	0
$154$	0	0
$155$	−12.1421	−0.975280
$156$	0	0
$157$	10.6569	0.850510	0.425255	−	0.905074i	$-0.360184\pi$
$157$	10.6569	0.850510	0.425255	+	0.905074i	$0.360184\pi$
$158$	5.65685	0.450035
$159$	0	0
$160$	−6.07107	−0.479960
$161$	0	0
$162$	0	0
$163$	21.9706	1.72087	0.860434	−	0.509563i	$-0.170193\pi$
$163$	21.9706	1.72087	0.860434	+	0.509563i	$0.170193\pi$
$164$	−29.3137	−2.28902
$165$	0	0
$166$	−10.0711	−0.781666
$167$	−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
$169$	−5.00000	−0.384615
$170$	33.7990	2.59226
$171$	0	0
$172$	−48.4558	−3.69472
$173$	−2.82843	−0.215041	−0.107521	−	0.994203i	$-0.534291\pi$
$173$	−2.82843	−0.215041	−0.107521	+	0.994203i	$0.534291\pi$
$174$	0	0
$175$	0	0
$176$	11.4853	0.865736
$177$	0	0
$178$	−26.1421	−1.95944
$179$	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
$181$	−11.6569	−0.866447	−0.433224	−	0.901286i	$-0.642624\pi$
$181$	−11.6569	−0.866447	−0.433224	+	0.901286i	$0.642624\pi$
$182$	0	0
$183$	0	0
$184$	8.07107	0.595007
$185$	4.48528	0.329764
$186$	0	0
$187$	14.0000	1.02378
$188$	−23.6274	−1.72321
$189$	0	0
$190$	−9.24264	−0.670532
$191$	−11.8284	−0.855875	−0.427937	−	0.903808i	$-0.640760\pi$
$191$	−11.8284	−0.855875	−0.427937	+	0.903808i	$0.640760\pi$
$192$	0	0
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	40.6274	2.91688
$195$	0	0
$196$	0	0
$197$	17.4853	1.24577	0.622887	−	0.782312i	$-0.285959\pi$
$197$	17.4853	1.24577	0.622887	+	0.782312i	$0.285959\pi$
$198$	0	0
$199$	3.34315	0.236989	0.118495	−	0.992955i	$-0.462193\pi$
$199$	3.34315	0.236989	0.118495	+	0.992955i	$0.462193\pi$
$200$	42.6274	3.01421
$201$	0	0
$202$	8.41421	0.592022
$203$	0	0
$204$	0	0
$205$	−29.3137	−2.04736
$206$	8.82843	0.615106
$207$	0	0
$208$	8.48528	0.588348
$209$	−3.82843	−0.264818
$210$	0	0
$211$	16.4853	1.13489	0.567447	−	0.823410i	$-0.307931\pi$
$211$	16.4853	1.13489	0.567447	+	0.823410i	$0.307931\pi$
$212$	7.65685	0.525875
$213$	0	0
$214$	6.82843	0.466782
$215$	−48.4558	−3.30466
$216$	0	0
$217$	0	0
$218$	43.7990	2.96644
$219$	0	0
$220$	56.1127	3.78312
$221$	10.3431	0.695755
$222$	0	0
$223$	21.7990	1.45977	0.729884	−	0.683571i	$-0.239574\pi$
$223$	21.7990	1.45977	0.729884	+	0.683571i	$0.239574\pi$
$224$	0	0
$225$	0	0
$226$	30.9706	2.06013
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	−15.6569	−1.03463	−0.517317	−	0.855794i	$-0.673069\pi$
$229$	−15.6569	−1.03463	−0.517317	+	0.855794i	$0.673069\pi$
$230$	16.8995	1.11432
$231$	0	0
$232$	21.3137	1.39931
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−23.6274	−1.54128
$236$	−41.4558	−2.69855
$237$	0	0
$238$	0	0
$239$	−2.34315	−0.151565	−0.0757827	−	0.997124i	$-0.524146\pi$
$239$	−2.34315	−0.151565	−0.0757827	+	0.997124i	$0.524146\pi$
$240$	0	0
$241$	19.7990	1.27537	0.637683	−	0.770299i	$-0.279893\pi$
$241$	19.7990	1.27537	0.637683	+	0.770299i	$0.279893\pi$
$242$	8.82843	0.567513
$243$	0	0
$244$	−56.1127	−3.59225
$245$	0	0
$246$	0	0
$247$	−2.82843	−0.179969
$248$	−14.0000	−0.889001
$249$	0	0
$250$	43.0416	2.72219
$251$	13.4853	0.851183	0.425592	−	0.904915i	$-0.360066\pi$
$251$	13.4853	0.851183	0.425592	+	0.904915i	$0.360066\pi$
$252$	0	0
$253$	7.00000	0.440086
$254$	40.9706	2.57072
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−26.0000	−1.62184	−0.810918	−	0.585160i	$-0.801032\pi$
$257$	−26.0000	−1.62184	−0.810918	+	0.585160i	$0.801032\pi$
$258$	0	0
$259$	0	0
$260$	41.4558	2.57098
$261$	0	0
$262$	−24.9706	−1.54269
$263$	−11.3137	−0.697633	−0.348817	−	0.937191i	$-0.613416\pi$
$263$	−11.3137	−0.697633	−0.348817	+	0.937191i	$0.613416\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	0	0
$267$	0	0
$268$	−45.9411	−2.80630
$269$	8.00000	0.487769	0.243884	−	0.969804i	$-0.421578\pi$
$269$	8.00000	0.487769	0.243884	+	0.969804i	$0.421578\pi$
$270$	0	0
$271$	−19.6274	−1.19228	−0.596140	−	0.802880i	$-0.703300\pi$
$271$	−19.6274	−1.19228	−0.596140	+	0.802880i	$0.703300\pi$
$272$	10.9706	0.665188
$273$	0	0
$274$	33.3848	2.01685
$275$	36.9706	2.22941
$276$	0	0
$277$	13.0000	0.781094	0.390547	−	0.920583i	$-0.372286\pi$
$277$	13.0000	0.781094	0.390547	+	0.920583i	$0.372286\pi$
$278$	−16.0711	−0.963879
$279$	0	0
$280$	0	0
$281$	−6.97056	−0.415829	−0.207914	−	0.978147i	$-0.566668\pi$
$281$	−6.97056	−0.415829	−0.207914	+	0.978147i	$0.566668\pi$
$282$	0	0
$283$	31.6274	1.88005	0.940027	−	0.341099i	$-0.110799\pi$
$283$	31.6274	1.88005	0.940027	+	0.341099i	$0.110799\pi$
$284$	−16.6274	−0.986656
$285$	0	0
$286$	26.1421	1.54582
$287$	0	0
$288$	0	0
$289$	−3.62742	−0.213377
$290$	44.6274	2.62061
$291$	0	0
$292$	−39.4853	−2.31070
$293$	0.828427	0.0483972	0.0241986	−	0.999707i	$-0.492297\pi$
$293$	0.828427	0.0483972	0.0241986	+	0.999707i	$0.492297\pi$
$294$	0	0
$295$	−41.4558	−2.41365
$296$	5.17157	0.300592
$297$	0	0
$298$	−9.24264	−0.535412
$299$	5.17157	0.299080
$300$	0	0
$301$	0	0
$302$	−22.4853	−1.29388
$303$	0	0
$304$	−3.00000	−0.172062
$305$	−56.1127	−3.21300
$306$	0	0
$307$	2.48528	0.141843	0.0709213	−	0.997482i	$-0.477406\pi$
$307$	2.48528	0.141843	0.0709213	+	0.997482i	$0.477406\pi$
$308$	0	0
$309$	0	0
$310$	−29.3137	−1.66491
$311$	−22.6274	−1.28308	−0.641542	−	0.767088i	$-0.721705\pi$
$311$	−22.6274	−1.28308	−0.641542	+	0.767088i	$0.721705\pi$
$312$	0	0
$313$	31.2843	1.76829	0.884146	−	0.467211i	$-0.154741\pi$
$313$	31.2843	1.76829	0.884146	+	0.467211i	$0.154741\pi$
$314$	25.7279	1.45191
$315$	0	0
$316$	8.97056	0.504634
$317$	−29.7990	−1.67368	−0.836839	−	0.547449i	$-0.815599\pi$
$317$	−29.7990	−1.67368	−0.836839	+	0.547449i	$0.815599\pi$
$318$	0	0
$319$	18.4853	1.03498
$320$	−37.6274	−2.10344
$321$	0	0
$322$	0	0
$323$	−3.65685	−0.203473
$324$	0	0
$325$	27.3137	1.51509
$326$	53.0416	2.93770
$327$	0	0
$328$	−33.7990	−1.86624
$329$	0	0
$330$	0	0
$331$	−5.51472	−0.303116	−0.151558	−	0.988448i	$-0.548429\pi$
$331$	−5.51472	−0.303116	−0.151558	+	0.988448i	$0.548429\pi$
$332$	−15.9706	−0.876499
$333$	0	0
$334$	−21.3137	−1.16623
$335$	−45.9411	−2.51003
$336$	0	0
$337$	−6.48528	−0.353276	−0.176638	−	0.984276i	$-0.556522\pi$
$337$	−6.48528	−0.353276	−0.176638	+	0.984276i	$0.556522\pi$
$338$	−12.0711	−0.656580
$339$	0	0
$340$	53.5980	2.90676
$341$	−12.1421	−0.657534
$342$	0	0
$343$	0	0
$344$	−55.8701	−3.01231
$345$	0	0
$346$	−6.82843	−0.367099
$347$	−20.4558	−1.09813	−0.549064	−	0.835781i	$-0.685016\pi$
$347$	−20.4558	−1.09813	−0.549064	+	0.835781i	$0.685016\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	0	0
$351$	0	0
$352$	−6.07107	−0.323589
$353$	1.31371	0.0699216	0.0349608	−	0.999389i	$-0.488869\pi$
$353$	1.31371	0.0699216	0.0349608	+	0.999389i	$0.488869\pi$
$354$	0	0
$355$	−16.6274	−0.882492
$356$	−41.4558	−2.19716
$357$	0	0
$358$	25.3137	1.33787
$359$	6.17157	0.325723	0.162862	−	0.986649i	$-0.447928\pi$
$359$	6.17157	0.325723	0.162862	+	0.986649i	$0.447928\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−28.1421	−1.47912
$363$	0	0
$364$	0	0
$365$	−39.4853	−2.06675
$366$	0	0
$367$	29.6569	1.54808	0.774038	−	0.633140i	$-0.218234\pi$
$367$	29.6569	1.54808	0.774038	+	0.633140i	$0.218234\pi$
$368$	5.48528	0.285940
$369$	0	0
$370$	10.8284	0.562943
$371$	0	0
$372$	0	0
$373$	20.2843	1.05028	0.525140	−	0.851016i	$-0.324013\pi$
$373$	20.2843	1.05028	0.525140	+	0.851016i	$0.324013\pi$
$374$	33.7990	1.74770
$375$	0	0
$376$	−27.2426	−1.40493
$377$	13.6569	0.703364
$378$	0	0
$379$	−16.4853	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$379$	−16.4853	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$380$	−14.6569	−0.751881
$381$	0	0
$382$	−28.5563	−1.46107
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	0	0
$385$	0	0
$386$	9.65685	0.491521
$387$	0	0
$388$	64.4264	3.27076
$389$	−4.34315	−0.220206	−0.110103	−	0.993920i	$-0.535118\pi$
$389$	−4.34315	−0.220206	−0.110103	+	0.993920i	$0.535118\pi$
$390$	0	0
$391$	6.68629	0.338140
$392$	0	0
$393$	0	0
$394$	42.2132	2.12667
$395$	8.97056	0.451358
$396$	0	0
$397$	5.31371	0.266687	0.133344	−	0.991070i	$-0.457429\pi$
$397$	5.31371	0.266687	0.133344	+	0.991070i	$0.457429\pi$
$398$	8.07107	0.404566
$399$	0	0
$400$	28.9706	1.44853
$401$	0.686292	0.0342718	0.0171359	−	0.999853i	$-0.494545\pi$
$401$	0.686292	0.0342718	0.0171359	+	0.999853i	$0.494545\pi$
$402$	0	0
$403$	−8.97056	−0.446856
$404$	13.3431	0.663846
$405$	0	0
$406$	0	0
$407$	4.48528	0.222327
$408$	0	0
$409$	5.31371	0.262746	0.131373	−	0.991333i	$-0.458061\pi$
$409$	5.31371	0.262746	0.131373	+	0.991333i	$0.458061\pi$
$410$	−70.7696	−3.49506
$411$	0	0
$412$	14.0000	0.689730
$413$	0	0
$414$	0	0
$415$	−15.9706	−0.783964
$416$	−4.48528	−0.219909
$417$	0	0
$418$	−9.24264	−0.452072
$419$	10.1716	0.496914	0.248457	−	0.968643i	$-0.420077\pi$
$419$	10.1716	0.496914	0.248457	+	0.968643i	$0.420077\pi$
$420$	0	0
$421$	19.1716	0.934365	0.467183	−	0.884161i	$-0.345269\pi$
$421$	19.1716	0.934365	0.467183	+	0.884161i	$0.345269\pi$
$422$	39.7990	1.93738
$423$	0	0
$424$	8.82843	0.428746
$425$	35.3137	1.71297
$426$	0	0
$427$	0	0
$428$	10.8284	0.523412
$429$	0	0
$430$	−116.983	−5.64141
$431$	23.1716	1.11614	0.558068	−	0.829795i	$-0.311543\pi$
$431$	23.1716	1.11614	0.558068	+	0.829795i	$0.311543\pi$
$432$	0	0
$433$	8.48528	0.407777	0.203888	−	0.978994i	$-0.434642\pi$
$433$	8.48528	0.407777	0.203888	+	0.978994i	$0.434642\pi$
$434$	0	0
$435$	0	0
$436$	69.4558	3.32633
$437$	−1.82843	−0.0874655
$438$	0	0
$439$	−19.7990	−0.944954	−0.472477	−	0.881343i	$-0.656640\pi$
$439$	−19.7990	−0.944954	−0.472477	+	0.881343i	$0.656640\pi$
$440$	64.6985	3.08438
$441$	0	0
$442$	24.9706	1.18773
$443$	33.6569	1.59909	0.799543	−	0.600609i	$-0.205075\pi$
$443$	33.6569	1.59909	0.799543	+	0.600609i	$0.205075\pi$
$444$	0	0
$445$	−41.4558	−1.96520
$446$	52.6274	2.49198
$447$	0	0
$448$	0	0
$449$	−26.1421	−1.23372	−0.616862	−	0.787071i	$-0.711596\pi$
$449$	−26.1421	−1.23372	−0.616862	+	0.787071i	$0.711596\pi$
$450$	0	0
$451$	−29.3137	−1.38033
$452$	49.1127	2.31007
$453$	0	0
$454$	15.3137	0.718708
$455$	0	0
$456$	0	0
$457$	−14.6569	−0.685619	−0.342809	−	0.939405i	$-0.611378\pi$
$457$	−14.6569	−0.685619	−0.342809	+	0.939405i	$0.611378\pi$
$458$	−37.7990	−1.76623
$459$	0	0
$460$	26.7990	1.24951
$461$	−22.4558	−1.04587	−0.522936	−	0.852372i	$-0.675164\pi$
$461$	−22.4558	−1.04587	−0.522936	+	0.852372i	$0.675164\pi$
$462$	0	0
$463$	−20.6569	−0.960005	−0.480003	−	0.877267i	$-0.659364\pi$
$463$	−20.6569	−0.960005	−0.480003	+	0.877267i	$0.659364\pi$
$464$	14.4853	0.672462
$465$	0	0
$466$	−14.4853	−0.671018
$467$	−25.4853	−1.17932	−0.589659	−	0.807652i	$-0.700738\pi$
$467$	−25.4853	−1.17932	−0.589659	+	0.807652i	$0.700738\pi$
$468$	0	0
$469$	0	0
$470$	−57.0416	−2.63113
$471$	0	0
$472$	−47.7990	−2.20013
$473$	−48.4558	−2.22800
$474$	0	0
$475$	−9.65685	−0.443087
$476$	0	0
$477$	0	0
$478$	−5.65685	−0.258738
$479$	−12.1716	−0.556133	−0.278067	−	0.960562i	$-0.589694\pi$
$479$	−12.1716	−0.556133	−0.278067	+	0.960562i	$0.589694\pi$
$480$	0	0
$481$	3.31371	0.151092
$482$	47.7990	2.17718
$483$	0	0
$484$	14.0000	0.636364
$485$	64.4264	2.92545
$486$	0	0
$487$	−0.343146	−0.0155494	−0.00777471	−	0.999970i	$-0.502475\pi$
$487$	−0.343146	−0.0155494	−0.00777471	+	0.999970i	$0.502475\pi$
$488$	−64.6985	−2.92876
$489$	0	0
$490$	0	0
$491$	8.51472	0.384264	0.192132	−	0.981369i	$-0.438460\pi$
$491$	8.51472	0.384264	0.192132	+	0.981369i	$0.438460\pi$
$492$	0	0
$493$	17.6569	0.795225
$494$	−6.82843	−0.307225
$495$	0	0
$496$	−9.51472	−0.427223
$497$	0	0
$498$	0	0
$499$	19.9706	0.894005	0.447003	−	0.894533i	$-0.352491\pi$
$499$	19.9706	0.894005	0.447003	+	0.894533i	$0.352491\pi$
$500$	68.2548	3.05245
$501$	0	0
$502$	32.5563	1.45306
$503$	−20.1127	−0.896781	−0.448390	−	0.893838i	$-0.648003\pi$
$503$	−20.1127	−0.896781	−0.448390	+	0.893838i	$0.648003\pi$
$504$	0	0
$505$	13.3431	0.593762
$506$	16.8995	0.751274
$507$	0	0
$508$	64.9706	2.88260
$509$	10.4853	0.464752	0.232376	−	0.972626i	$-0.425350\pi$
$509$	10.4853	0.464752	0.232376	+	0.972626i	$0.425350\pi$
$510$	0	0
$511$	0	0
$512$	−31.2426	−1.38074
$513$	0	0
$514$	−62.7696	−2.76865
$515$	14.0000	0.616914
$516$	0	0
$517$	−23.6274	−1.03913
$518$	0	0
$519$	0	0
$520$	47.7990	2.09612
$521$	40.9706	1.79495	0.897476	−	0.441062i	$-0.145398\pi$
$521$	40.9706	1.79495	0.897476	+	0.441062i	$0.145398\pi$
$522$	0	0
$523$	−37.7990	−1.65283	−0.826417	−	0.563058i	$-0.809625\pi$
$523$	−37.7990	−1.65283	−0.826417	+	0.563058i	$0.809625\pi$
$524$	−39.5980	−1.72985
$525$	0	0
$526$	−27.3137	−1.19093
$527$	−11.5980	−0.505216
$528$	0	0
$529$	−19.6569	−0.854646
$530$	18.4853	0.802949
$531$	0	0
$532$	0	0
$533$	−21.6569	−0.938062
$534$	0	0
$535$	10.8284	0.468154
$536$	−52.9706	−2.28798
$537$	0	0
$538$	19.3137	0.832673
$539$	0	0
$540$	0	0
$541$	23.0000	0.988847	0.494424	−	0.869221i	$-0.335379\pi$
$541$	23.0000	0.988847	0.494424	+	0.869221i	$0.335379\pi$
$542$	−47.3848	−2.03535
$543$	0	0
$544$	−5.79899	−0.248630
$545$	69.4558	2.97516
$546$	0	0
$547$	−5.79899	−0.247947	−0.123973	−	0.992286i	$-0.539564\pi$
$547$	−5.79899	−0.247947	−0.123973	+	0.992286i	$0.539564\pi$
$548$	52.9411	2.26153
$549$	0	0
$550$	89.2548	3.80584
$551$	−4.82843	−0.205698
$552$	0	0
$553$	0	0
$554$	31.3848	1.33341
$555$	0	0
$556$	−25.4853	−1.08082
$557$	−14.4558	−0.612514	−0.306257	−	0.951949i	$-0.599077\pi$
$557$	−14.4558	−0.612514	−0.306257	+	0.951949i	$0.599077\pi$
$558$	0	0
$559$	−35.7990	−1.51414
$560$	0	0
$561$	0	0
$562$	−16.8284	−0.709864
$563$	−0.485281	−0.0204522	−0.0102261	−	0.999948i	$-0.503255\pi$
$563$	−0.485281	−0.0204522	−0.0102261	+	0.999948i	$0.503255\pi$
$564$	0	0
$565$	49.1127	2.06619
$566$	76.3553	3.20945
$567$	0	0
$568$	−19.1716	−0.804421
$569$	45.7990	1.91999	0.959997	−	0.280011i	$-0.0903381\pi$
$569$	45.7990	1.91999	0.959997	+	0.280011i	$0.0903381\pi$
$570$	0	0
$571$	27.2843	1.14181	0.570906	−	0.821016i	$-0.306592\pi$
$571$	27.2843	1.14181	0.570906	+	0.821016i	$0.306592\pi$
$572$	41.4558	1.73336
$573$	0	0
$574$	0	0
$575$	17.6569	0.736342
$576$	0	0
$577$	18.3137	0.762410	0.381205	−	0.924491i	$-0.375509\pi$
$577$	18.3137	0.762410	0.381205	+	0.924491i	$0.375509\pi$
$578$	−8.75736	−0.364258
$579$	0	0
$580$	70.7696	2.93855
$581$	0	0
$582$	0	0
$583$	7.65685	0.317115
$584$	−45.5269	−1.88392
$585$	0	0
$586$	2.00000	0.0826192
$587$	26.6274	1.09903	0.549516	−	0.835483i	$-0.314812\pi$
$587$	26.6274	1.09903	0.549516	+	0.835483i	$0.314812\pi$
$588$	0	0
$589$	3.17157	0.130682
$590$	−100.083	−4.12036
$591$	0	0
$592$	3.51472	0.144454
$593$	−27.4853	−1.12869	−0.564343	−	0.825541i	$-0.690870\pi$
$593$	−27.4853	−1.12869	−0.564343	+	0.825541i	$0.690870\pi$
$594$	0	0
$595$	0	0
$596$	−14.6569	−0.600368
$597$	0	0
$598$	12.4853	0.510561
$599$	29.1127	1.18951	0.594756	−	0.803906i	$-0.297249\pi$
$599$	29.1127	1.18951	0.594756	+	0.803906i	$0.297249\pi$
$600$	0	0
$601$	−8.68629	−0.354321	−0.177161	−	0.984182i	$-0.556691\pi$
$601$	−8.68629	−0.354321	−0.177161	+	0.984182i	$0.556691\pi$
$602$	0	0
$603$	0	0
$604$	−35.6569	−1.45086
$605$	14.0000	0.569181
$606$	0	0
$607$	13.8579	0.562473	0.281237	−	0.959638i	$-0.409255\pi$
$607$	13.8579	0.562473	0.281237	+	0.959638i	$0.409255\pi$
$608$	1.58579	0.0643121
$609$	0	0
$610$	−135.468	−5.48494
$611$	−17.4558	−0.706188
$612$	0	0
$613$	−36.6274	−1.47937	−0.739684	−	0.672955i	$-0.765025\pi$
$613$	−36.6274	−1.47937	−0.739684	+	0.672955i	$0.765025\pi$
$614$	6.00000	0.242140
$615$	0	0
$616$	0	0
$617$	32.1716	1.29518	0.647589	−	0.761989i	$-0.275777\pi$
$617$	32.1716	1.29518	0.647589	+	0.761989i	$0.275777\pi$
$618$	0	0
$619$	9.34315	0.375533	0.187766	−	0.982214i	$-0.439875\pi$
$619$	9.34315	0.375533	0.187766	+	0.982214i	$0.439875\pi$
$620$	−46.4853	−1.86689
$621$	0	0
$622$	−54.6274	−2.19036
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	75.5269	3.01866
$627$	0	0
$628$	40.7990	1.62806
$629$	4.28427	0.170825
$630$	0	0
$631$	−16.6569	−0.663099	−0.331549	−	0.943438i	$-0.607571\pi$
$631$	−16.6569	−0.663099	−0.331549	+	0.943438i	$0.607571\pi$
$632$	10.3431	0.411428
$633$	0	0
$634$	−71.9411	−2.85715
$635$	64.9706	2.57828
$636$	0	0
$637$	0	0
$638$	44.6274	1.76682
$639$	0	0
$640$	−78.6985	−3.11083
$641$	5.17157	0.204265	0.102132	−	0.994771i	$-0.467433\pi$
$641$	5.17157	0.204265	0.102132	+	0.994771i	$0.467433\pi$
$642$	0	0
$643$	1.37258	0.0541294	0.0270647	−	0.999634i	$-0.491384\pi$
$643$	1.37258	0.0541294	0.0270647	+	0.999634i	$0.491384\pi$
$644$	0	0
$645$	0	0
$646$	−8.82843	−0.347350
$647$	−6.85786	−0.269610	−0.134805	−	0.990872i	$-0.543041\pi$
$647$	−6.85786	−0.269610	−0.134805	+	0.990872i	$0.543041\pi$
$648$	0	0
$649$	−41.4558	−1.62728
$650$	65.9411	2.58642
$651$	0	0
$652$	84.1127	3.29411
$653$	−39.9411	−1.56302	−0.781509	−	0.623895i	$-0.785549\pi$
$653$	−39.9411	−1.56302	−0.781509	+	0.623895i	$0.785549\pi$
$654$	0	0
$655$	−39.5980	−1.54722
$656$	−22.9706	−0.896850
$657$	0	0
$658$	0	0
$659$	21.5147	0.838094	0.419047	−	0.907964i	$-0.362364\pi$
$659$	21.5147	0.838094	0.419047	+	0.907964i	$0.362364\pi$
$660$	0	0
$661$	12.2843	0.477803	0.238901	−	0.971044i	$-0.423213\pi$
$661$	12.2843	0.477803	0.238901	+	0.971044i	$0.423213\pi$
$662$	−13.3137	−0.517452
$663$	0	0
$664$	−18.4142	−0.714610
$665$	0	0
$666$	0	0
$667$	8.82843	0.341838
$668$	−33.7990	−1.30772
$669$	0	0
$670$	−110.912	−4.28489
$671$	−56.1127	−2.16621
$672$	0	0
$673$	2.82843	0.109028	0.0545139	−	0.998513i	$-0.482639\pi$
$673$	2.82843	0.109028	0.0545139	+	0.998513i	$0.482639\pi$
$674$	−15.6569	−0.603079
$675$	0	0
$676$	−19.1421	−0.736236
$677$	16.0000	0.614930	0.307465	−	0.951559i	$-0.400519\pi$
$677$	16.0000	0.614930	0.307465	+	0.951559i	$0.400519\pi$
$678$	0	0
$679$	0	0
$680$	61.7990	2.36988
$681$	0	0
$682$	−29.3137	−1.12248
$683$	−5.02944	−0.192446	−0.0962230	−	0.995360i	$-0.530676\pi$
$683$	−5.02944	−0.192446	−0.0962230	+	0.995360i	$0.530676\pi$
$684$	0	0
$685$	52.9411	2.02278
$686$	0	0
$687$	0	0
$688$	−37.9706	−1.44761
$689$	5.65685	0.215509
$690$	0	0
$691$	12.0000	0.456502	0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000	0.456502	0.228251	+	0.973602i	$0.426699\pi$
$692$	−10.8284	−0.411635
$693$	0	0
$694$	−49.3848	−1.87462
$695$	−25.4853	−0.966712
$696$	0	0
$697$	−28.0000	−1.06058
$698$	33.7990	1.27931
$699$	0	0
$700$	0	0
$701$	−2.45584	−0.0927560	−0.0463780	−	0.998924i	$-0.514768\pi$
$701$	−2.45584	−0.0927560	−0.0463780	+	0.998924i	$0.514768\pi$
$702$	0	0
$703$	−1.17157	−0.0441867
$704$	−37.6274	−1.41814
$705$	0	0
$706$	3.17157	0.119364
$707$	0	0
$708$	0	0
$709$	−15.9706	−0.599787	−0.299894	−	0.953973i	$-0.596951\pi$
$709$	−15.9706	−0.599787	−0.299894	+	0.953973i	$0.596951\pi$
$710$	−40.1421	−1.50651
$711$	0	0
$712$	−47.7990	−1.79134
$713$	−5.79899	−0.217174
$714$	0	0
$715$	41.4558	1.55036
$716$	40.1421	1.50018
$717$	0	0
$718$	14.8995	0.556044
$719$	0.970563	0.0361959	0.0180979	−	0.999836i	$-0.494239\pi$
$719$	0.970563	0.0361959	0.0180979	+	0.999836i	$0.494239\pi$
$720$	0	0
$721$	0	0
$722$	2.41421	0.0898477
$723$	0	0
$724$	−44.6274	−1.65856
$725$	46.6274	1.73170
$726$	0	0
$727$	−13.0000	−0.482143	−0.241072	−	0.970507i	$-0.577499\pi$
$727$	−13.0000	−0.482143	−0.241072	+	0.970507i	$0.577499\pi$
$728$	0	0
$729$	0	0
$730$	−95.3259	−3.52817
$731$	−46.2843	−1.71189
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	71.5980	2.64273
$735$	0	0
$736$	−2.89949	−0.106877
$737$	−45.9411	−1.69226
$738$	0	0
$739$	−8.00000	−0.294285	−0.147142	−	0.989115i	$-0.547008\pi$
$739$	−8.00000	−0.294285	−0.147142	+	0.989115i	$0.547008\pi$
$740$	17.1716	0.631240
$741$	0	0
$742$	0	0
$743$	−33.1716	−1.21695	−0.608473	−	0.793574i	$-0.708218\pi$
$743$	−33.1716	−1.21695	−0.608473	+	0.793574i	$0.708218\pi$
$744$	0	0
$745$	−14.6569	−0.536986
$746$	48.9706	1.79294
$747$	0	0
$748$	53.5980	1.95974
$749$	0	0
$750$	0	0
$751$	34.6274	1.26357	0.631786	−	0.775143i	$-0.282322\pi$
$751$	34.6274	1.26357	0.631786	+	0.775143i	$0.282322\pi$
$752$	−18.5147	−0.675162
$753$	0	0
$754$	32.9706	1.20072
$755$	−35.6569	−1.29769
$756$	0	0
$757$	9.34315	0.339582	0.169791	−	0.985480i	$-0.445691\pi$
$757$	9.34315	0.339582	0.169791	+	0.985480i	$0.445691\pi$
$758$	−39.7990	−1.44556
$759$	0	0
$760$	−16.8995	−0.613009
$761$	18.7990	0.681463	0.340731	−	0.940161i	$-0.389325\pi$
$761$	18.7990	0.681463	0.340731	+	0.940161i	$0.389325\pi$
$762$	0	0
$763$	0	0
$764$	−45.2843	−1.63833
$765$	0	0
$766$	−38.6274	−1.39567
$767$	−30.6274	−1.10589
$768$	0	0
$769$	14.3137	0.516166	0.258083	−	0.966123i	$-0.416909\pi$
$769$	14.3137	0.516166	0.258083	+	0.966123i	$0.416909\pi$
$770$	0	0
$771$	0	0
$772$	15.3137	0.551152
$773$	−2.82843	−0.101731	−0.0508657	−	0.998706i	$-0.516198\pi$
$773$	−2.82843	−0.101731	−0.0508657	+	0.998706i	$0.516198\pi$
$774$	0	0
$775$	−30.6274	−1.10017
$776$	74.2843	2.66665
$777$	0	0
$778$	−10.4853	−0.375916
$779$	7.65685	0.274335
$780$	0	0
$781$	−16.6274	−0.594976
$782$	16.1421	0.577242
$783$	0	0
$784$	0	0
$785$	40.7990	1.45618
$786$	0	0
$787$	16.2843	0.580472	0.290236	−	0.956955i	$-0.406266\pi$
$787$	16.2843	0.580472	0.290236	+	0.956955i	$0.406266\pi$
$788$	66.9411	2.38468
$789$	0	0
$790$	21.6569	0.770516
$791$	0	0
$792$	0	0
$793$	−41.4558	−1.47214
$794$	12.8284	0.455264
$795$	0	0
$796$	12.7990	0.453648
$797$	−7.51472	−0.266185	−0.133092	−	0.991104i	$-0.542491\pi$
$797$	−7.51472	−0.266185	−0.133092	+	0.991104i	$0.542491\pi$
$798$	0	0
$799$	−22.5685	−0.798418
$800$	−15.3137	−0.541421
$801$	0	0
$802$	1.65685	0.0585056
$803$	−39.4853	−1.39341
$804$	0	0
$805$	0	0
$806$	−21.6569	−0.762830
$807$	0	0
$808$	15.3848	0.541235
$809$	24.7990	0.871886	0.435943	−	0.899974i	$-0.356415\pi$
$809$	24.7990	0.871886	0.435943	+	0.899974i	$0.356415\pi$
$810$	0	0
$811$	−12.1421	−0.426368	−0.213184	−	0.977012i	$-0.568383\pi$
$811$	−12.1421	−0.426368	−0.213184	+	0.977012i	$0.568383\pi$
$812$	0	0
$813$	0	0
$814$	10.8284	0.379536
$815$	84.1127	2.94634
$816$	0	0
$817$	12.6569	0.442807
$818$	12.8284	0.448535
$819$	0	0
$820$	−112.225	−3.91908
$821$	−6.85786	−0.239341	−0.119671	−	0.992814i	$-0.538184\pi$
$821$	−6.85786	−0.239341	−0.119671	+	0.992814i	$0.538184\pi$
$822$	0	0
$823$	31.9706	1.11442	0.557212	−	0.830370i	$-0.311871\pi$
$823$	31.9706	1.11442	0.557212	+	0.830370i	$0.311871\pi$
$824$	16.1421	0.562338
$825$	0	0
$826$	0	0
$827$	15.1127	0.525520	0.262760	−	0.964861i	$-0.415367\pi$
$827$	15.1127	0.525520	0.262760	+	0.964861i	$0.415367\pi$
$828$	0	0
$829$	−38.1421	−1.32473	−0.662366	−	0.749181i	$-0.730447\pi$
$829$	−38.1421	−1.32473	−0.662366	+	0.749181i	$0.730447\pi$
$830$	−38.5563	−1.33831
$831$	0	0
$832$	−27.7990	−0.963757
$833$	0	0
$834$	0	0
$835$	−33.7990	−1.16966
$836$	−14.6569	−0.506918
$837$	0	0
$838$	24.5563	0.848285
$839$	45.2548	1.56237	0.781185	−	0.624299i	$-0.214615\pi$
$839$	45.2548	1.56237	0.781185	+	0.624299i	$0.214615\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	46.2843	1.59506
$843$	0	0
$844$	63.1127	2.17243
$845$	−19.1421	−0.658509
$846$	0	0
$847$	0	0
$848$	6.00000	0.206041
$849$	0	0
$850$	85.2548	2.92422
$851$	2.14214	0.0734315
$852$	0	0
$853$	−6.37258	−0.218193	−0.109097	−	0.994031i	$-0.534796\pi$
$853$	−6.37258	−0.218193	−0.109097	+	0.994031i	$0.534796\pi$
$854$	0	0
$855$	0	0
$856$	12.4853	0.426738
$857$	26.8284	0.916442	0.458221	−	0.888838i	$-0.348487\pi$
$857$	26.8284	0.916442	0.458221	+	0.888838i	$0.348487\pi$
$858$	0	0
$859$	30.9411	1.05570	0.527849	−	0.849338i	$-0.322999\pi$
$859$	30.9411	1.05570	0.527849	+	0.849338i	$0.322999\pi$
$860$	−185.510	−6.32583
$861$	0	0
$862$	55.9411	1.90536
$863$	−7.17157	−0.244123	−0.122062	−	0.992523i	$-0.538951\pi$
$863$	−7.17157	−0.244123	−0.122062	+	0.992523i	$0.538951\pi$
$864$	0	0
$865$	−10.8284	−0.368178
$866$	20.4853	0.696118
$867$	0	0
$868$	0	0
$869$	8.97056	0.304305
$870$	0	0
$871$	−33.9411	−1.15005
$872$	80.0833	2.71196
$873$	0	0
$874$	−4.41421	−0.149313
$875$	0	0
$876$	0	0
$877$	21.6569	0.731300	0.365650	−	0.930752i	$-0.380847\pi$
$877$	21.6569	0.731300	0.365650	+	0.930752i	$0.380847\pi$
$878$	−47.7990	−1.61314
$879$	0	0
$880$	43.9706	1.48225
$881$	4.62742	0.155902	0.0779508	−	0.996957i	$-0.475162\pi$
$881$	4.62742	0.155902	0.0779508	+	0.996957i	$0.475162\pi$
$882$	0	0
$883$	−44.9706	−1.51338	−0.756690	−	0.653774i	$-0.773185\pi$
$883$	−44.9706	−1.51338	−0.756690	+	0.653774i	$0.773185\pi$
$884$	39.5980	1.33182
$885$	0	0
$886$	81.2548	2.72981
$887$	15.4558	0.518956	0.259478	−	0.965749i	$-0.416449\pi$
$887$	15.4558	0.518956	0.259478	+	0.965749i	$0.416449\pi$
$888$	0	0
$889$	0	0
$890$	−100.083	−3.35480
$891$	0	0
$892$	83.4558	2.79431
$893$	6.17157	0.206524
$894$	0	0
$895$	40.1421	1.34180
$896$	0	0
$897$	0	0
$898$	−63.1127	−2.10610
$899$	−15.3137	−0.510741
$900$	0	0
$901$	7.31371	0.243655
$902$	−70.7696	−2.35637
$903$	0	0
$904$	56.6274	1.88340
$905$	−44.6274	−1.48347
$906$	0	0
$907$	−19.6569	−0.652695	−0.326348	−	0.945250i	$-0.605818\pi$
$907$	−19.6569	−0.652695	−0.326348	+	0.945250i	$0.605818\pi$
$908$	24.2843	0.805902
$909$	0	0
$910$	0	0
$911$	−18.0000	−0.596367	−0.298183	−	0.954509i	$-0.596381\pi$
$911$	−18.0000	−0.596367	−0.298183	+	0.954509i	$0.596381\pi$
$912$	0	0
$913$	−15.9706	−0.528548
$914$	−35.3848	−1.17042
$915$	0	0
$916$	−59.9411	−1.98051
$917$	0	0
$918$	0	0
$919$	−43.3431	−1.42976	−0.714879	−	0.699248i	$-0.753518\pi$
$919$	−43.3431	−1.42976	−0.714879	+	0.699248i	$0.753518\pi$
$920$	30.8995	1.01873
$921$	0	0
$922$	−54.2132	−1.78542
$923$	−12.2843	−0.404342
$924$	0	0
$925$	11.3137	0.371992
$926$	−49.8701	−1.63883
$927$	0	0
$928$	−7.65685	−0.251349
$929$	−16.4558	−0.539899	−0.269949	−	0.962875i	$-0.587007\pi$
$929$	−16.4558	−0.539899	−0.269949	+	0.962875i	$0.587007\pi$
$930$	0	0
$931$	0	0
$932$	−22.9706	−0.752426
$933$	0	0
$934$	−61.5269	−2.01322
$935$	53.5980	1.75284
$936$	0	0
$937$	−27.0000	−0.882052	−0.441026	−	0.897494i	$-0.645385\pi$
$937$	−27.0000	−0.882052	−0.441026	+	0.897494i	$0.645385\pi$
$938$	0	0
$939$	0	0
$940$	−90.4558	−2.95034
$941$	−59.9411	−1.95402	−0.977012	−	0.213182i	$-0.931617\pi$
$941$	−59.9411	−1.95402	−0.977012	+	0.213182i	$0.931617\pi$
$942$	0	0
$943$	−14.0000	−0.455903
$944$	−32.4853	−1.05731
$945$	0	0
$946$	−116.983	−3.80344
$947$	16.6863	0.542232	0.271116	−	0.962547i	$-0.412607\pi$
$947$	16.6863	0.542232	0.271116	+	0.962547i	$0.412607\pi$
$948$	0	0
$949$	−29.1716	−0.946949
$950$	−23.3137	−0.756397
$951$	0	0
$952$	0	0
$953$	26.6274	0.862547	0.431273	−	0.902221i	$-0.358064\pi$
$953$	26.6274	0.862547	0.431273	+	0.902221i	$0.358064\pi$
$954$	0	0
$955$	−45.2843	−1.46536
$956$	−8.97056	−0.290129
$957$	0	0
$958$	−29.3848	−0.949379
$959$	0	0
$960$	0	0
$961$	−20.9411	−0.675520
$962$	8.00000	0.257930
$963$	0	0
$964$	75.7990	2.44132
$965$	15.3137	0.492966
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	16.1421	0.518828
$969$	0	0
$970$	155.539	4.99406
$971$	39.7990	1.27721	0.638605	−	0.769535i	$-0.279512\pi$
$971$	39.7990	1.27721	0.638605	+	0.769535i	$0.279512\pi$
$972$	0	0
$973$	0	0
$974$	−0.828427	−0.0265445
$975$	0	0
$976$	−43.9706	−1.40746
$977$	−23.1127	−0.739441	−0.369720	−	0.929143i	$-0.620547\pi$
$977$	−23.1127	−0.739441	−0.369720	+	0.929143i	$0.620547\pi$
$978$	0	0
$979$	−41.4558	−1.32493
$980$	0	0
$981$	0	0
$982$	20.5563	0.655979
$983$	−3.45584	−0.110224	−0.0551122	−	0.998480i	$-0.517552\pi$
$983$	−3.45584	−0.110224	−0.0551122	+	0.998480i	$0.517552\pi$
$984$	0	0
$985$	66.9411	2.13292
$986$	42.6274	1.35753
$987$	0	0
$988$	−10.8284	−0.344498
$989$	−23.1421	−0.735877
$990$	0	0
$991$	53.2548	1.69170	0.845848	−	0.533424i	$-0.179095\pi$
$991$	53.2548	1.69170	0.845848	+	0.533424i	$0.179095\pi$
$992$	5.02944	0.159685
$993$	0	0
$994$	0	0
$995$	12.7990	0.405755
$996$	0	0
$997$	−30.0000	−0.950110	−0.475055	−	0.879956i	$-0.657572\pi$
$997$	−30.0000	−0.950110	−0.475055	+	0.879956i	$0.657572\pi$
$998$	48.2132	1.52616
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bm.1.2		2
3.2	odd	2		2793.2.a.o.1.1		2
7.2	even	3		1197.2.j.d.172.1		4
7.4	even	3		1197.2.j.d.856.1		4
7.6	odd	2		8379.2.a.bh.1.2		2
21.2	odd	6		399.2.j.c.172.2	yes	4
21.11	odd	6		399.2.j.c.58.2	✓	4
21.20	even	2		2793.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
399.2.j.c.58.2	✓	4	21.11	odd	6
399.2.j.c.172.2	yes	4	21.2	odd	6
1197.2.j.d.172.1		4	7.2	even	3
1197.2.j.d.856.1		4	7.4	even	3
2793.2.a.n.1.1		2	21.20	even	2
2793.2.a.o.1.1		2	3.2	odd	2
8379.2.a.bh.1.2		2	7.6	odd	2
8379.2.a.bm.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +3.82843 q^{5} +4.41421 q^{8} +9.24264 q^{10} +3.82843 q^{11} +2.82843 q^{13} +3.00000 q^{16} +3.65685 q^{17} -1.00000 q^{19} +14.6569 q^{20} +9.24264 q^{22} +1.82843 q^{23} +9.65685 q^{25} +6.82843 q^{26} +4.82843 q^{29} -3.17157 q^{31} -1.58579 q^{32} +8.82843 q^{34} +1.17157 q^{37} -2.41421 q^{38} +16.8995 q^{40} -7.65685 q^{41} -12.6569 q^{43} +14.6569 q^{44} +4.41421 q^{46} -6.17157 q^{47} +23.3137 q^{50} +10.8284 q^{52} +2.00000 q^{53} +14.6569 q^{55} +11.6569 q^{58} -10.8284 q^{59} -14.6569 q^{61} -7.65685 q^{62} -9.82843 q^{64} +10.8284 q^{65} -12.0000 q^{67} +14.0000 q^{68} -4.34315 q^{71} -10.3137 q^{73} +2.82843 q^{74} -3.82843 q^{76} +2.34315 q^{79} +11.4853 q^{80} -18.4853 q^{82} -4.17157 q^{83} +14.0000 q^{85} -30.5563 q^{86} +16.8995 q^{88} -10.8284 q^{89} +7.00000 q^{92} -14.8995 q^{94} -3.82843 q^{95} +16.8284 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 10 q^{10} + 2 q^{11} + 6 q^{16} - 4 q^{17} - 2 q^{19} + 18 q^{20} + 10 q^{22} - 2 q^{23} + 8 q^{25} + 8 q^{26} + 4 q^{29} - 12 q^{31} - 6 q^{32} + 12 q^{34}+ \cdots + 28 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 10 * q^10 + 2 * q^11 + 6 * q^16 - 4 * q^17 - 2 * q^19 + 18 * q^20 + 10 * q^22 - 2 * q^23 + 8 * q^25 + 8 * q^26 + 4 * q^29 - 12 * q^31 - 6 * q^32 + 12 * q^34 + 8 * q^37 - 2 * q^38 + 14 * q^40 - 4 * q^41 - 14 * q^43 + 18 * q^44 + 6 * q^46 - 18 * q^47 + 24 * q^50 + 16 * q^52 + 4 * q^53 + 18 * q^55 + 12 * q^58 - 16 * q^59 - 18 * q^61 - 4 * q^62 - 14 * q^64 + 16 * q^65 - 24 * q^67 + 28 * q^68 - 20 * q^71 + 2 * q^73 - 2 * q^76 + 16 * q^79 + 6 * q^80 - 20 * q^82 - 14 * q^83 + 28 * q^85 - 30 * q^86 + 14 * q^88 - 16 * q^89 + 14 * q^92 - 10 * q^94 - 2 * q^95 + 28 * q^97

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