LMFDB - Embedded newform 8512.2.a.cb.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8512,2,Mod(1,8512)] 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("8512.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8512, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8512 = 2^{6} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8512.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-1,0,-1,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.9686622005$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.60663248.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 9x^{3} + 20x^{2} - 23x + 5 $ x^6 - x^5 - 10x^4 + 9x^3 + 20x^2 - 23x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4256)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.826398$ of defining polynomial
Character	$\chi$	$=$	8512.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.826398 q^{3} +2.92671 q^{5} -1.00000 q^{7} -2.31707 q^{9} +5.09479 q^{11} +2.10876 q^{13} -2.41863 q^{15} -3.51528 q^{17} -1.00000 q^{19} +0.826398 q^{21} -4.51894 q^{23} +3.56563 q^{25} +4.39401 q^{27} -2.02952 q^{29} +2.48472 q^{31} -4.21032 q^{33} -2.92671 q^{35} -11.9405 q^{37} -1.74268 q^{39} -4.80954 q^{41} +2.25546 q^{43} -6.78138 q^{45} +0.594730 q^{47} +1.00000 q^{49} +2.90502 q^{51} +0.0438832 q^{53} +14.9110 q^{55} +0.826398 q^{57} -8.74780 q^{59} -6.13147 q^{61} +2.31707 q^{63} +6.17174 q^{65} -4.24857 q^{67} +3.73444 q^{69} -9.58912 q^{71} -15.6788 q^{73} -2.94663 q^{75} -5.09479 q^{77} +15.0226 q^{79} +3.31999 q^{81} -2.75311 q^{83} -10.2882 q^{85} +1.67719 q^{87} +8.43759 q^{89} -2.10876 q^{91} -2.05337 q^{93} -2.92671 q^{95} +0.776992 q^{97} -11.8050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} - q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} + 4 q^{15} - 14 q^{17} - 6 q^{19} + q^{21} + 6 q^{23} + q^{25} + 2 q^{27} - 5 q^{29} + 22 q^{31} - 15 q^{33} + q^{35} - 7 q^{37} + 18 q^{39}+ \cdots - 7 q^{99}+O(q^{100}) $ 6 * q - q^3 - q^5 - 6 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 + 4 * q^15 - 14 * q^17 - 6 * q^19 + q^21 + 6 * q^23 + q^25 + 2 * q^27 - 5 * q^29 + 22 * q^31 - 15 * q^33 + q^35 - 7 * q^37 + 18 * q^39 - 11 * q^41 - 3 * q^43 + 2 * q^45 + 15 * q^47 + 6 * q^49 - 20 * q^51 - 19 * q^53 + 32 * q^55 + q^57 + 3 * q^59 + 9 * q^61 - 3 * q^63 - 4 * q^65 - 22 * q^67 - 26 * q^69 + 15 * q^71 - 26 * q^73 - 4 * q^75 + 3 * q^77 + 13 * q^79 - 10 * q^81 + 6 * q^83 - 6 * q^85 + 22 * q^87 - 19 * q^89 - 4 * q^91 - 26 * q^93 + q^95 + q^97 - 7 * q^99

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
$3$	−0.826398	−0.477121	−0.238561	−	0.971128i	$-0.576676\pi$
$3$	−0.826398	−0.477121	−0.238561	+	0.971128i	$0.576676\pi$
$4$	0	0
$5$	2.92671	1.30886	0.654432	−	0.756121i	$-0.272908\pi$
$5$	2.92671	1.30886	0.654432	+	0.756121i	$0.272908\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.31707	−0.772355
$10$	0	0
$11$	5.09479	1.53614	0.768068	−	0.640368i	$-0.221218\pi$
$11$	5.09479	1.53614	0.768068	+	0.640368i	$0.221218\pi$
$12$	0	0
$13$	2.10876	0.584865	0.292433	−	0.956286i	$-0.405535\pi$
$13$	2.10876	0.584865	0.292433	+	0.956286i	$0.405535\pi$
$14$	0	0
$15$	−2.41863	−0.624487
$16$	0	0
$17$	−3.51528	−0.852581	−0.426291	−	0.904586i	$-0.640180\pi$
$17$	−3.51528	−0.852581	−0.426291	+	0.904586i	$0.640180\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.826398	0.180335
$22$	0	0
$23$	−4.51894	−0.942264	−0.471132	−	0.882063i	$-0.656155\pi$
$23$	−4.51894	−0.942264	−0.471132	+	0.882063i	$0.656155\pi$
$24$	0	0
$25$	3.56563	0.713127
$26$	0	0
$27$	4.39401	0.845628
$28$	0	0
$29$	−2.02952	−0.376873	−0.188436	−	0.982085i	$-0.560342\pi$
$29$	−2.02952	−0.376873	−0.188436	+	0.982085i	$0.560342\pi$
$30$	0	0
$31$	2.48472	0.446268	0.223134	−	0.974788i	$-0.428371\pi$
$31$	2.48472	0.446268	0.223134	+	0.974788i	$0.428371\pi$
$32$	0	0
$33$	−4.21032	−0.732923
$34$	0	0
$35$	−2.92671	−0.494704
$36$	0	0
$37$	−11.9405	−1.96301	−0.981503	−	0.191448i	$-0.938682\pi$
$37$	−11.9405	−1.96301	−0.981503	+	0.191448i	$0.938682\pi$
$38$	0	0
$39$	−1.74268	−0.279052
$40$	0	0
$41$	−4.80954	−0.751124	−0.375562	−	0.926797i	$-0.622550\pi$
$41$	−4.80954	−0.751124	−0.375562	+	0.926797i	$0.622550\pi$
$42$	0	0
$43$	2.25546	0.343954	0.171977	−	0.985101i	$-0.444985\pi$
$43$	2.25546	0.343954	0.171977	+	0.985101i	$0.444985\pi$
$44$	0	0
$45$	−6.78138	−1.01091
$46$	0	0
$47$	0.594730	0.0867503	0.0433751	−	0.999059i	$-0.486189\pi$
$47$	0.594730	0.0867503	0.0433751	+	0.999059i	$0.486189\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.90502	0.406785
$52$	0	0
$53$	0.0438832	0.00602782	0.00301391	−	0.999995i	$-0.499041\pi$
$53$	0.0438832	0.00602782	0.00301391	+	0.999995i	$0.499041\pi$
$54$	0	0
$55$	14.9110	2.01060
$56$	0	0
$57$	0.826398	0.109459
$58$	0	0
$59$	−8.74780	−1.13887	−0.569433	−	0.822037i	$-0.692837\pi$
$59$	−8.74780	−1.13887	−0.569433	+	0.822037i	$0.692837\pi$
$60$	0	0
$61$	−6.13147	−0.785055	−0.392527	−	0.919740i	$-0.628399\pi$
$61$	−6.13147	−0.785055	−0.392527	+	0.919740i	$0.628399\pi$
$62$	0	0
$63$	2.31707	0.291923
$64$	0	0
$65$	6.17174	0.765510
$66$	0	0
$67$	−4.24857	−0.519045	−0.259523	−	0.965737i	$-0.583565\pi$
$67$	−4.24857	−0.519045	−0.259523	+	0.965737i	$0.583565\pi$
$68$	0	0
$69$	3.73444	0.449574
$70$	0	0
$71$	−9.58912	−1.13802	−0.569009	−	0.822331i	$-0.692673\pi$
$71$	−9.58912	−1.13802	−0.569009	+	0.822331i	$0.692673\pi$
$72$	0	0
$73$	−15.6788	−1.83506	−0.917532	−	0.397663i	$-0.869821\pi$
$73$	−15.6788	−1.83506	−0.917532	+	0.397663i	$0.869821\pi$
$74$	0	0
$75$	−2.94663	−0.340248
$76$	0	0
$77$	−5.09479	−0.580605
$78$	0	0
$79$	15.0226	1.69018	0.845089	−	0.534626i	$-0.179548\pi$
$79$	15.0226	1.69018	0.845089	+	0.534626i	$0.179548\pi$
$80$	0	0
$81$	3.31999	0.368888
$82$	0	0
$83$	−2.75311	−0.302193	−0.151096	−	0.988519i	$-0.548280\pi$
$83$	−2.75311	−0.302193	−0.151096	+	0.988519i	$0.548280\pi$
$84$	0	0
$85$	−10.2882	−1.11591
$86$	0	0
$87$	1.67719	0.179814
$88$	0	0
$89$	8.43759	0.894383	0.447192	−	0.894438i	$-0.352424\pi$
$89$	8.43759	0.894383	0.447192	+	0.894438i	$0.352424\pi$
$90$	0	0
$91$	−2.10876	−0.221058
$92$	0	0
$93$	−2.05337	−0.212924
$94$	0	0
$95$	−2.92671	−0.300274
$96$	0	0
$97$	0.776992	0.0788916	0.0394458	−	0.999222i	$-0.487441\pi$
$97$	0.776992	0.0788916	0.0394458	+	0.999222i	$0.487441\pi$
$98$	0	0
$99$	−11.8050	−1.18644
$100$	0	0
$101$	−1.47283	−0.146552	−0.0732758	−	0.997312i	$-0.523345\pi$
$101$	−1.47283	−0.146552	−0.0732758	+	0.997312i	$0.523345\pi$
$102$	0	0
$103$	−11.2348	−1.10699	−0.553496	−	0.832852i	$-0.686707\pi$
$103$	−11.2348	−1.10699	−0.553496	+	0.832852i	$0.686707\pi$
$104$	0	0
$105$	2.41863	0.236034
$106$	0	0
$107$	7.57148	0.731963	0.365981	−	0.930622i	$-0.380733\pi$
$107$	7.57148	0.731963	0.365981	+	0.930622i	$0.380733\pi$
$108$	0	0
$109$	2.17101	0.207945	0.103972	−	0.994580i	$-0.466845\pi$
$109$	2.17101	0.207945	0.103972	+	0.994580i	$0.466845\pi$
$110$	0	0
$111$	9.86760	0.936592
$112$	0	0
$113$	−10.0649	−0.946823	−0.473412	−	0.880841i	$-0.656978\pi$
$113$	−10.0649	−0.946823	−0.473412	+	0.880841i	$0.656978\pi$
$114$	0	0
$115$	−13.2256	−1.23330
$116$	0	0
$117$	−4.88614	−0.451724
$118$	0	0
$119$	3.51528	0.322245
$120$	0	0
$121$	14.9569	1.35972
$122$	0	0
$123$	3.97459	0.358377
$124$	0	0
$125$	−4.19797	−0.375478
$126$	0	0
$127$	18.6973	1.65911	0.829557	−	0.558422i	$-0.188593\pi$
$127$	18.6973	1.65911	0.829557	+	0.558422i	$0.188593\pi$
$128$	0	0
$129$	−1.86391	−0.164108
$130$	0	0
$131$	−0.977150	−0.0853740	−0.0426870	−	0.999088i	$-0.513592\pi$
$131$	−0.977150	−0.0853740	−0.0426870	+	0.999088i	$0.513592\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	12.8600	1.10681
$136$	0	0
$137$	−22.7365	−1.94251	−0.971257	−	0.238032i	$-0.923498\pi$
$137$	−22.7365	−1.94251	−0.971257	+	0.238032i	$0.923498\pi$
$138$	0	0
$139$	6.59485	0.559368	0.279684	−	0.960092i	$-0.409770\pi$
$139$	6.59485	0.559368	0.279684	+	0.960092i	$0.409770\pi$
$140$	0	0
$141$	−0.491484	−0.0413904
$142$	0	0
$143$	10.7437	0.898433
$144$	0	0
$145$	−5.93983	−0.493276
$146$	0	0
$147$	−0.826398	−0.0681602
$148$	0	0
$149$	6.47737	0.530647	0.265323	−	0.964159i	$-0.414521\pi$
$149$	6.47737	0.530647	0.265323	+	0.964159i	$0.414521\pi$
$150$	0	0
$151$	−2.05526	−0.167255	−0.0836274	−	0.996497i	$-0.526651\pi$
$151$	−2.05526	−0.167255	−0.0836274	+	0.996497i	$0.526651\pi$
$152$	0	0
$153$	8.14514	0.658496
$154$	0	0
$155$	7.27205	0.584105
$156$	0	0
$157$	1.36462	0.108908	0.0544542	−	0.998516i	$-0.482658\pi$
$157$	1.36462	0.108908	0.0544542	+	0.998516i	$0.482658\pi$
$158$	0	0
$159$	−0.0362650	−0.00287600
$160$	0	0
$161$	4.51894	0.356142
$162$	0	0
$163$	−12.0821	−0.946340	−0.473170	−	0.880971i	$-0.656890\pi$
$163$	−12.0821	−0.946340	−0.473170	+	0.880971i	$0.656890\pi$
$164$	0	0
$165$	−12.3224	−0.959298
$166$	0	0
$167$	10.5107	0.813343	0.406671	−	0.913574i	$-0.366689\pi$
$167$	10.5107	0.813343	0.406671	+	0.913574i	$0.366689\pi$
$168$	0	0
$169$	−8.55312	−0.657932
$170$	0	0
$171$	2.31707	0.177190
$172$	0	0
$173$	−8.51550	−0.647421	−0.323711	−	0.946156i	$-0.604930\pi$
$173$	−8.51550	−0.647421	−0.323711	+	0.946156i	$0.604930\pi$
$174$	0	0
$175$	−3.56563	−0.269537
$176$	0	0
$177$	7.22917	0.543378
$178$	0	0
$179$	2.95967	0.221216	0.110608	−	0.993864i	$-0.464720\pi$
$179$	2.95967	0.221216	0.110608	+	0.993864i	$0.464720\pi$
$180$	0	0
$181$	7.01283	0.521260	0.260630	−	0.965439i	$-0.416070\pi$
$181$	7.01283	0.521260	0.260630	+	0.965439i	$0.416070\pi$
$182$	0	0
$183$	5.06704	0.374566
$184$	0	0
$185$	−34.9464	−2.56931
$186$	0	0
$187$	−17.9096	−1.30968
$188$	0	0
$189$	−4.39401	−0.319617
$190$	0	0
$191$	−4.62228	−0.334456	−0.167228	−	0.985918i	$-0.553482\pi$
$191$	−4.62228	−0.334456	−0.167228	+	0.985918i	$0.553482\pi$
$192$	0	0
$193$	−14.2472	−1.02554	−0.512769	−	0.858527i	$-0.671380\pi$
$193$	−14.2472	−1.02554	−0.512769	+	0.858527i	$0.671380\pi$
$194$	0	0
$195$	−5.10031	−0.365241
$196$	0	0
$197$	−3.44932	−0.245754	−0.122877	−	0.992422i	$-0.539212\pi$
$197$	−3.44932	−0.245754	−0.122877	+	0.992422i	$0.539212\pi$
$198$	0	0
$199$	13.0655	0.926189	0.463095	−	0.886309i	$-0.346739\pi$
$199$	13.0655	0.926189	0.463095	+	0.886309i	$0.346739\pi$
$200$	0	0
$201$	3.51101	0.247647
$202$	0	0
$203$	2.02952	0.142445
$204$	0	0
$205$	−14.0761	−0.983119
$206$	0	0
$207$	10.4707	0.727763
$208$	0	0
$209$	−5.09479	−0.352414
$210$	0	0
$211$	−9.50182	−0.654133	−0.327066	−	0.945001i	$-0.606060\pi$
$211$	−9.50182	−0.654133	−0.327066	+	0.945001i	$0.606060\pi$
$212$	0	0
$213$	7.92443	0.542973
$214$	0	0
$215$	6.60107	0.450189
$216$	0	0
$217$	−2.48472	−0.168674
$218$	0	0
$219$	12.9569	0.875548
$220$	0	0
$221$	−7.41290	−0.498645
$222$	0	0
$223$	−4.68839	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$223$	−4.68839	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$224$	0	0
$225$	−8.26181	−0.550787
$226$	0	0
$227$	0.579568	0.0384673	0.0192336	−	0.999815i	$-0.493877\pi$
$227$	0.579568	0.0384673	0.0192336	+	0.999815i	$0.493877\pi$
$228$	0	0
$229$	16.4029	1.08393	0.541967	−	0.840400i	$-0.317680\pi$
$229$	16.4029	1.08393	0.541967	+	0.840400i	$0.317680\pi$
$230$	0	0
$231$	4.21032	0.277019
$232$	0	0
$233$	7.76992	0.509024	0.254512	−	0.967070i	$-0.418085\pi$
$233$	7.76992	0.509024	0.254512	+	0.967070i	$0.418085\pi$
$234$	0	0
$235$	1.74060	0.113544
$236$	0	0
$237$	−12.4147	−0.806419
$238$	0	0
$239$	26.7229	1.72856	0.864281	−	0.503009i	$-0.167774\pi$
$239$	26.7229	1.72856	0.864281	+	0.503009i	$0.167774\pi$
$240$	0	0
$241$	−10.6085	−0.683355	−0.341677	−	0.939817i	$-0.610995\pi$
$241$	−10.6085	−0.683355	−0.341677	+	0.939817i	$0.610995\pi$
$242$	0	0
$243$	−15.9257	−1.02163
$244$	0	0
$245$	2.92671	0.186981
$246$	0	0
$247$	−2.10876	−0.134177
$248$	0	0
$249$	2.27516	0.144183
$250$	0	0
$251$	−15.0827	−0.952015	−0.476007	−	0.879441i	$-0.657916\pi$
$251$	−15.0827	−0.952015	−0.476007	+	0.879441i	$0.657916\pi$
$252$	0	0
$253$	−23.0230	−1.44745
$254$	0	0
$255$	8.50216	0.532426
$256$	0	0
$257$	−16.3992	−1.02295	−0.511477	−	0.859297i	$-0.670901\pi$
$257$	−16.3992	−1.02295	−0.511477	+	0.859297i	$0.670901\pi$
$258$	0	0
$259$	11.9405	0.741946
$260$	0	0
$261$	4.70254	0.291080
$262$	0	0
$263$	8.18627	0.504787	0.252394	−	0.967625i	$-0.418782\pi$
$263$	8.18627	0.504787	0.252394	+	0.967625i	$0.418782\pi$
$264$	0	0
$265$	0.128433	0.00788960
$266$	0	0
$267$	−6.97281	−0.426729
$268$	0	0
$269$	−6.98253	−0.425733	−0.212866	−	0.977081i	$-0.568280\pi$
$269$	−6.98253	−0.425733	−0.212866	+	0.977081i	$0.568280\pi$
$270$	0	0
$271$	−6.74676	−0.409837	−0.204918	−	0.978779i	$-0.565693\pi$
$271$	−6.74676	−0.409837	−0.204918	+	0.978779i	$0.565693\pi$
$272$	0	0
$273$	1.74268	0.105472
$274$	0	0
$275$	18.1662	1.09546
$276$	0	0
$277$	28.6164	1.71940	0.859698	−	0.510803i	$-0.170652\pi$
$277$	28.6164	1.71940	0.859698	+	0.510803i	$0.170652\pi$
$278$	0	0
$279$	−5.75725	−0.344678
$280$	0	0
$281$	−25.8281	−1.54078	−0.770388	−	0.637575i	$-0.779938\pi$
$281$	−25.8281	−1.54078	−0.770388	+	0.637575i	$0.779938\pi$
$282$	0	0
$283$	−27.7197	−1.64776	−0.823882	−	0.566762i	$-0.808196\pi$
$283$	−27.7197	−1.64776	−0.823882	+	0.566762i	$0.808196\pi$
$284$	0	0
$285$	2.41863	0.143267
$286$	0	0
$287$	4.80954	0.283898
$288$	0	0
$289$	−4.64279	−0.273105
$290$	0	0
$291$	−0.642105	−0.0376408
$292$	0	0
$293$	26.8564	1.56897	0.784484	−	0.620149i	$-0.212928\pi$
$293$	26.8564	1.56897	0.784484	+	0.620149i	$0.212928\pi$
$294$	0	0
$295$	−25.6023	−1.49062
$296$	0	0
$297$	22.3866	1.29900
$298$	0	0
$299$	−9.52937	−0.551098
$300$	0	0
$301$	−2.25546	−0.130002
$302$	0	0
$303$	1.21714	0.0699229
$304$	0	0
$305$	−17.9450	−1.02753
$306$	0	0
$307$	15.2751	0.871795	0.435897	−	0.899996i	$-0.356431\pi$
$307$	15.2751	0.871795	0.435897	+	0.899996i	$0.356431\pi$
$308$	0	0
$309$	9.28438	0.528170
$310$	0	0
$311$	−0.0804288	−0.00456070	−0.00228035	−	0.999997i	$-0.500726\pi$
$311$	−0.0804288	−0.00456070	−0.00228035	+	0.999997i	$0.500726\pi$
$312$	0	0
$313$	34.2305	1.93482	0.967411	−	0.253210i	$-0.0814862\pi$
$313$	34.2305	1.93482	0.967411	+	0.253210i	$0.0814862\pi$
$314$	0	0
$315$	6.78138	0.382088
$316$	0	0
$317$	−28.0016	−1.57273	−0.786363	−	0.617765i	$-0.788038\pi$
$317$	−28.0016	−1.57273	−0.786363	+	0.617765i	$0.788038\pi$
$318$	0	0
$319$	−10.3400	−0.578928
$320$	0	0
$321$	−6.25706	−0.349235
$322$	0	0
$323$	3.51528	0.195596
$324$	0	0
$325$	7.51908	0.417083
$326$	0	0
$327$	−1.79412	−0.0992148
$328$	0	0
$329$	−0.594730	−0.0327885
$330$	0	0
$331$	−28.0802	−1.54342	−0.771712	−	0.635972i	$-0.780600\pi$
$331$	−28.0802	−1.54342	−0.771712	+	0.635972i	$0.780600\pi$
$332$	0	0
$333$	27.6669	1.51614
$334$	0	0
$335$	−12.4343	−0.679360
$336$	0	0
$337$	16.5178	0.899780	0.449890	−	0.893084i	$-0.351463\pi$
$337$	16.5178	0.899780	0.449890	+	0.893084i	$0.351463\pi$
$338$	0	0
$339$	8.31759	0.451749
$340$	0	0
$341$	12.6591	0.685529
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	10.9296	0.588432
$346$	0	0
$347$	23.8339	1.27947	0.639735	−	0.768596i	$-0.279044\pi$
$347$	23.8339	1.27947	0.639735	+	0.768596i	$0.279044\pi$
$348$	0	0
$349$	8.73703	0.467682	0.233841	−	0.972275i	$-0.424870\pi$
$349$	8.73703	0.467682	0.233841	+	0.972275i	$0.424870\pi$
$350$	0	0
$351$	9.26593	0.494579
$352$	0	0
$353$	35.7089	1.90059	0.950296	−	0.311348i	$-0.100780\pi$
$353$	35.7089	1.90059	0.950296	+	0.311348i	$0.100780\pi$
$354$	0	0
$355$	−28.0646	−1.48951
$356$	0	0
$357$	−2.90502	−0.153750
$358$	0	0
$359$	28.7227	1.51593	0.757963	−	0.652298i	$-0.226195\pi$
$359$	28.7227	1.51593	0.757963	+	0.652298i	$0.226195\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−12.3603	−0.648749
$364$	0	0
$365$	−45.8873	−2.40185
$366$	0	0
$367$	−8.21735	−0.428942	−0.214471	−	0.976730i	$-0.568803\pi$
$367$	−8.21735	−0.428942	−0.214471	+	0.976730i	$0.568803\pi$
$368$	0	0
$369$	11.1440	0.580134
$370$	0	0
$371$	−0.0438832	−0.00227830
$372$	0	0
$373$	−20.9720	−1.08589	−0.542943	−	0.839769i	$-0.682690\pi$
$373$	−20.9720	−1.08589	−0.542943	+	0.839769i	$0.682690\pi$
$374$	0	0
$375$	3.46920	0.179149
$376$	0	0
$377$	−4.27978	−0.220420
$378$	0	0
$379$	−19.5705	−1.00527	−0.502635	−	0.864499i	$-0.667636\pi$
$379$	−19.5705	−1.00527	−0.502635	+	0.864499i	$0.667636\pi$
$380$	0	0
$381$	−15.4514	−0.791598
$382$	0	0
$383$	−15.8307	−0.808911	−0.404456	−	0.914558i	$-0.632539\pi$
$383$	−15.8307	−0.808911	−0.404456	+	0.914558i	$0.632539\pi$
$384$	0	0
$385$	−14.9110	−0.759934
$386$	0	0
$387$	−5.22604	−0.265655
$388$	0	0
$389$	−6.16562	−0.312609	−0.156305	−	0.987709i	$-0.549958\pi$
$389$	−6.16562	−0.312609	−0.156305	+	0.987709i	$0.549958\pi$
$390$	0	0
$391$	15.8854	0.803357
$392$	0	0
$393$	0.807515	0.0407337
$394$	0	0
$395$	43.9669	2.21221
$396$	0	0
$397$	2.91575	0.146337	0.0731685	−	0.997320i	$-0.476689\pi$
$397$	2.91575	0.146337	0.0731685	+	0.997320i	$0.476689\pi$
$398$	0	0
$399$	−0.826398	−0.0413717
$400$	0	0
$401$	25.7516	1.28597	0.642987	−	0.765877i	$-0.277695\pi$
$401$	25.7516	1.28597	0.642987	+	0.765877i	$0.277695\pi$
$402$	0	0
$403$	5.23968	0.261007
$404$	0	0
$405$	9.71666	0.482825
$406$	0	0
$407$	−60.8343	−3.01544
$408$	0	0
$409$	−16.2005	−0.801062	−0.400531	−	0.916283i	$-0.631174\pi$
$409$	−16.2005	−0.801062	−0.400531	+	0.916283i	$0.631174\pi$
$410$	0	0
$411$	18.7894	0.926815
$412$	0	0
$413$	8.74780	0.430451
$414$	0	0
$415$	−8.05755	−0.395530
$416$	0	0
$417$	−5.44997	−0.266886
$418$	0	0
$419$	19.0784	0.932042	0.466021	−	0.884774i	$-0.345687\pi$
$419$	19.0784	0.932042	0.466021	+	0.884774i	$0.345687\pi$
$420$	0	0
$421$	−17.3485	−0.845514	−0.422757	−	0.906243i	$-0.638938\pi$
$421$	−17.3485	−0.845514	−0.422757	+	0.906243i	$0.638938\pi$
$422$	0	0
$423$	−1.37803	−0.0670020
$424$	0	0
$425$	−12.5342	−0.607999
$426$	0	0
$427$	6.13147	0.296723
$428$	0	0
$429$	−8.87857	−0.428662
$430$	0	0
$431$	−9.08484	−0.437601	−0.218801	−	0.975770i	$-0.570214\pi$
$431$	−9.08484	−0.437601	−0.218801	+	0.975770i	$0.570214\pi$
$432$	0	0
$433$	−34.7200	−1.66854	−0.834269	−	0.551358i	$-0.814110\pi$
$433$	−34.7200	−1.66854	−0.834269	+	0.551358i	$0.814110\pi$
$434$	0	0
$435$	4.90866	0.235352
$436$	0	0
$437$	4.51894	0.216170
$438$	0	0
$439$	−33.0559	−1.57767	−0.788835	−	0.614605i	$-0.789316\pi$
$439$	−33.0559	−1.57767	−0.788835	+	0.614605i	$0.789316\pi$
$440$	0	0
$441$	−2.31707	−0.110336
$442$	0	0
$443$	−19.5932	−0.930899	−0.465449	−	0.885074i	$-0.654107\pi$
$443$	−19.5932	−0.930899	−0.465449	+	0.885074i	$0.654107\pi$
$444$	0	0
$445$	24.6944	1.17063
$446$	0	0
$447$	−5.35289	−0.253183
$448$	0	0
$449$	−5.44859	−0.257135	−0.128568	−	0.991701i	$-0.541038\pi$
$449$	−5.44859	−0.257135	−0.128568	+	0.991701i	$0.541038\pi$
$450$	0	0
$451$	−24.5036	−1.15383
$452$	0	0
$453$	1.69846	0.0798008
$454$	0	0
$455$	−6.17174	−0.289335
$456$	0	0
$457$	−16.4564	−0.769797	−0.384899	−	0.922959i	$-0.625764\pi$
$457$	−16.4564	−0.769797	−0.384899	+	0.922959i	$0.625764\pi$
$458$	0	0
$459$	−15.4462	−0.720967
$460$	0	0
$461$	−8.25925	−0.384672	−0.192336	−	0.981329i	$-0.561606\pi$
$461$	−8.25925	−0.384672	−0.192336	+	0.981329i	$0.561606\pi$
$462$	0	0
$463$	24.2729	1.12806	0.564028	−	0.825756i	$-0.309251\pi$
$463$	24.2729	1.12806	0.564028	+	0.825756i	$0.309251\pi$
$464$	0	0
$465$	−6.00961	−0.278689
$466$	0	0
$467$	−25.6069	−1.18495	−0.592473	−	0.805590i	$-0.701848\pi$
$467$	−25.6069	−1.18495	−0.592473	+	0.805590i	$0.701848\pi$
$468$	0	0
$469$	4.24857	0.196181
$470$	0	0
$471$	−1.12772	−0.0519625
$472$	0	0
$473$	11.4911	0.528361
$474$	0	0
$475$	−3.56563	−0.163603
$476$	0	0
$477$	−0.101680	−0.00465562
$478$	0	0
$479$	−5.67956	−0.259506	−0.129753	−	0.991546i	$-0.541418\pi$
$479$	−5.67956	−0.259506	−0.129753	+	0.991546i	$0.541418\pi$
$480$	0	0
$481$	−25.1797	−1.14809
$482$	0	0
$483$	−3.73444	−0.169923
$484$	0	0
$485$	2.27403	0.103258
$486$	0	0
$487$	11.0444	0.500467	0.250234	−	0.968185i	$-0.419493\pi$
$487$	11.0444	0.500467	0.250234	+	0.968185i	$0.419493\pi$
$488$	0	0
$489$	9.98460	0.451519
$490$	0	0
$491$	24.5154	1.10636	0.553182	−	0.833061i	$-0.313414\pi$
$491$	24.5154	1.10636	0.553182	+	0.833061i	$0.313414\pi$
$492$	0	0
$493$	7.13435	0.321315
$494$	0	0
$495$	−34.5497	−1.55289
$496$	0	0
$497$	9.58912	0.430131
$498$	0	0
$499$	−22.9496	−1.02737	−0.513684	−	0.857980i	$-0.671719\pi$
$499$	−22.9496	−1.02737	−0.513684	+	0.857980i	$0.671719\pi$
$500$	0	0
$501$	−8.68603	−0.388063
$502$	0	0
$503$	−34.0578	−1.51856	−0.759281	−	0.650762i	$-0.774449\pi$
$503$	−34.0578	−1.51856	−0.759281	+	0.650762i	$0.774449\pi$
$504$	0	0
$505$	−4.31054	−0.191816
$506$	0	0
$507$	7.06828	0.313914
$508$	0	0
$509$	15.9218	0.705721	0.352860	−	0.935676i	$-0.385209\pi$
$509$	15.9218	0.705721	0.352860	+	0.935676i	$0.385209\pi$
$510$	0	0
$511$	15.6788	0.693589
$512$	0	0
$513$	−4.39401	−0.194000
$514$	0	0
$515$	−32.8809	−1.44890
$516$	0	0
$517$	3.03002	0.133260
$518$	0	0
$519$	7.03719	0.308899
$520$	0	0
$521$	−34.1831	−1.49759	−0.748794	−	0.662803i	$-0.769367\pi$
$521$	−34.1831	−1.49759	−0.748794	+	0.662803i	$0.769367\pi$
$522$	0	0
$523$	−36.7002	−1.60479	−0.802393	−	0.596796i	$-0.796440\pi$
$523$	−36.7002	−1.60479	−0.802393	+	0.596796i	$0.796440\pi$
$524$	0	0
$525$	2.94663	0.128602
$526$	0	0
$527$	−8.73448	−0.380480
$528$	0	0
$529$	−2.57918	−0.112138
$530$	0	0
$531$	20.2692	0.879610
$532$	0	0
$533$	−10.1422	−0.439306
$534$	0	0
$535$	22.1595	0.958041
$536$	0	0
$537$	−2.44587	−0.105547
$538$	0	0
$539$	5.09479	0.219448
$540$	0	0
$541$	−39.6142	−1.70315	−0.851575	−	0.524234i	$-0.824352\pi$
$541$	−39.6142	−1.70315	−0.851575	+	0.524234i	$0.824352\pi$
$542$	0	0
$543$	−5.79539	−0.248704
$544$	0	0
$545$	6.35391	0.272171
$546$	0	0
$547$	16.2404	0.694389	0.347194	−	0.937793i	$-0.387134\pi$
$547$	16.2404	0.694389	0.347194	+	0.937793i	$0.387134\pi$
$548$	0	0
$549$	14.2070	0.606341
$550$	0	0
$551$	2.02952	0.0864606
$552$	0	0
$553$	−15.0226	−0.638827
$554$	0	0
$555$	28.8796	1.22587
$556$	0	0
$557$	−9.88843	−0.418986	−0.209493	−	0.977810i	$-0.567181\pi$
$557$	−9.88843	−0.418986	−0.209493	+	0.977810i	$0.567181\pi$
$558$	0	0
$559$	4.75622	0.201167
$560$	0	0
$561$	14.8005	0.624877
$562$	0	0
$563$	8.79674	0.370738	0.185369	−	0.982669i	$-0.440652\pi$
$563$	8.79674	0.370738	0.185369	+	0.982669i	$0.440652\pi$
$564$	0	0
$565$	−29.4570	−1.23926
$566$	0	0
$567$	−3.31999	−0.139427
$568$	0	0
$569$	36.9080	1.54727	0.773633	−	0.633634i	$-0.218438\pi$
$569$	36.9080	1.54727	0.773633	+	0.633634i	$0.218438\pi$
$570$	0	0
$571$	−19.1368	−0.800848	−0.400424	−	0.916330i	$-0.631137\pi$
$571$	−19.1368	−0.800848	−0.400424	+	0.916330i	$0.631137\pi$
$572$	0	0
$573$	3.81984	0.159576
$574$	0	0
$575$	−16.1129	−0.671954
$576$	0	0
$577$	−18.5531	−0.772377	−0.386189	−	0.922420i	$-0.626209\pi$
$577$	−18.5531	−0.772377	−0.386189	+	0.922420i	$0.626209\pi$
$578$	0	0
$579$	11.7739	0.489306
$580$	0	0
$581$	2.75311	0.114218
$582$	0	0
$583$	0.223576	0.00925955
$584$	0	0
$585$	−14.3003	−0.591246
$586$	0	0
$587$	19.4916	0.804503	0.402251	−	0.915529i	$-0.368228\pi$
$587$	19.4916	0.804503	0.402251	+	0.915529i	$0.368228\pi$
$588$	0	0
$589$	−2.48472	−0.102381
$590$	0	0
$591$	2.85051	0.117254
$592$	0	0
$593$	0.164813	0.00676807	0.00338403	−	0.999994i	$-0.498923\pi$
$593$	0.164813	0.00676807	0.00338403	+	0.999994i	$0.498923\pi$
$594$	0	0
$595$	10.2882	0.421776
$596$	0	0
$597$	−10.7973	−0.441905
$598$	0	0
$599$	25.4998	1.04189	0.520946	−	0.853590i	$-0.325579\pi$
$599$	25.4998	1.04189	0.520946	+	0.853590i	$0.325579\pi$
$600$	0	0
$601$	30.8136	1.25691	0.628457	−	0.777844i	$-0.283687\pi$
$601$	30.8136	1.25691	0.628457	+	0.777844i	$0.283687\pi$
$602$	0	0
$603$	9.84421	0.400887
$604$	0	0
$605$	43.7744	1.77968
$606$	0	0
$607$	−20.6884	−0.839716	−0.419858	−	0.907590i	$-0.637920\pi$
$607$	−20.6884	−0.839716	−0.419858	+	0.907590i	$0.637920\pi$
$608$	0	0
$609$	−1.67719	−0.0679633
$610$	0	0
$611$	1.25414	0.0507372
$612$	0	0
$613$	7.11608	0.287416	0.143708	−	0.989620i	$-0.454097\pi$
$613$	7.11608	0.287416	0.143708	+	0.989620i	$0.454097\pi$
$614$	0	0
$615$	11.6325	0.469067
$616$	0	0
$617$	44.9620	1.81010	0.905051	−	0.425304i	$-0.139833\pi$
$617$	44.9620	1.81010	0.905051	+	0.425304i	$0.139833\pi$
$618$	0	0
$619$	23.9564	0.962887	0.481444	−	0.876477i	$-0.340113\pi$
$619$	23.9564	0.962887	0.481444	+	0.876477i	$0.340113\pi$
$620$	0	0
$621$	−19.8563	−0.796805
$622$	0	0
$623$	−8.43759	−0.338045
$624$	0	0
$625$	−30.1144	−1.20458
$626$	0	0
$627$	4.21032	0.168144
$628$	0	0
$629$	41.9742	1.67362
$630$	0	0
$631$	23.6538	0.941644	0.470822	−	0.882228i	$-0.343957\pi$
$631$	23.6538	0.941644	0.470822	+	0.882228i	$0.343957\pi$
$632$	0	0
$633$	7.85229	0.312100
$634$	0	0
$635$	54.7215	2.17156
$636$	0	0
$637$	2.10876	0.0835522
$638$	0	0
$639$	22.2186	0.878955
$640$	0	0
$641$	−1.53783	−0.0607406	−0.0303703	−	0.999539i	$-0.509669\pi$
$641$	−1.53783	−0.0607406	−0.0303703	+	0.999539i	$0.509669\pi$
$642$	0	0
$643$	10.4209	0.410962	0.205481	−	0.978661i	$-0.434124\pi$
$643$	10.4209	0.410962	0.205481	+	0.978661i	$0.434124\pi$
$644$	0	0
$645$	−5.45511	−0.214795
$646$	0	0
$647$	11.8538	0.466021	0.233011	−	0.972474i	$-0.425142\pi$
$647$	11.8538	0.466021	0.233011	+	0.972474i	$0.425142\pi$
$648$	0	0
$649$	−44.5682	−1.74946
$650$	0	0
$651$	2.05337	0.0804778
$652$	0	0
$653$	−32.8461	−1.28537	−0.642683	−	0.766132i	$-0.722179\pi$
$653$	−32.8461	−1.28537	−0.642683	+	0.766132i	$0.722179\pi$
$654$	0	0
$655$	−2.85983	−0.111743
$656$	0	0
$657$	36.3288	1.41732
$658$	0	0
$659$	27.2197	1.06033	0.530165	−	0.847895i	$-0.322130\pi$
$659$	27.2197	1.06033	0.530165	+	0.847895i	$0.322130\pi$
$660$	0	0
$661$	27.0256	1.05118	0.525588	−	0.850739i	$-0.323845\pi$
$661$	27.0256	1.05118	0.525588	+	0.850739i	$0.323845\pi$
$662$	0	0
$663$	6.12600	0.237914
$664$	0	0
$665$	2.92671	0.113493
$666$	0	0
$667$	9.17129	0.355114
$668$	0	0
$669$	3.87447	0.149796
$670$	0	0
$671$	−31.2386	−1.20595
$672$	0	0
$673$	−7.76019	−0.299133	−0.149567	−	0.988752i	$-0.547788\pi$
$673$	−7.76019	−0.299133	−0.149567	+	0.988752i	$0.547788\pi$
$674$	0	0
$675$	15.6674	0.603040
$676$	0	0
$677$	−22.3909	−0.860552	−0.430276	−	0.902697i	$-0.641584\pi$
$677$	−22.3909	−0.860552	−0.430276	+	0.902697i	$0.641584\pi$
$678$	0	0
$679$	−0.776992	−0.0298182
$680$	0	0
$681$	−0.478954	−0.0183535
$682$	0	0
$683$	−23.5055	−0.899413	−0.449707	−	0.893176i	$-0.648471\pi$
$683$	−23.5055	−0.899413	−0.449707	+	0.893176i	$0.648471\pi$
$684$	0	0
$685$	−66.5433	−2.54249
$686$	0	0
$687$	−13.5553	−0.517168
$688$	0	0
$689$	0.0925392	0.00352546
$690$	0	0
$691$	−37.3456	−1.42069	−0.710346	−	0.703853i	$-0.751461\pi$
$691$	−37.3456	−1.42069	−0.710346	+	0.703853i	$0.751461\pi$
$692$	0	0
$693$	11.8050	0.448433
$694$	0	0
$695$	19.3012	0.732137
$696$	0	0
$697$	16.9069	0.640394
$698$	0	0
$699$	−6.42105	−0.242866
$700$	0	0
$701$	−12.6610	−0.478198	−0.239099	−	0.970995i	$-0.576852\pi$
$701$	−12.6610	−0.478198	−0.239099	+	0.970995i	$0.576852\pi$
$702$	0	0
$703$	11.9405	0.450344
$704$	0	0
$705$	−1.43843	−0.0541744
$706$	0	0
$707$	1.47283	0.0553913
$708$	0	0
$709$	4.36478	0.163923	0.0819614	−	0.996636i	$-0.473882\pi$
$709$	4.36478	0.163923	0.0819614	+	0.996636i	$0.473882\pi$
$710$	0	0
$711$	−34.8084	−1.30542
$712$	0	0
$713$	−11.2283	−0.420503
$714$	0	0
$715$	31.4437	1.17593
$716$	0	0
$717$	−22.0838	−0.824734
$718$	0	0
$719$	45.9954	1.71534	0.857669	−	0.514202i	$-0.171912\pi$
$719$	45.9954	1.71534	0.857669	+	0.514202i	$0.171912\pi$
$720$	0	0
$721$	11.2348	0.418404
$722$	0	0
$723$	8.76686	0.326043
$724$	0	0
$725$	−7.23654	−0.268758
$726$	0	0
$727$	−0.317551	−0.0117773	−0.00588867	−	0.999983i	$-0.501874\pi$
$727$	−0.317551	−0.0117773	−0.00588867	+	0.999983i	$0.501874\pi$
$728$	0	0
$729$	3.20097	0.118555
$730$	0	0
$731$	−7.92857	−0.293249
$732$	0	0
$733$	50.1283	1.85153	0.925766	−	0.378096i	$-0.123421\pi$
$733$	50.1283	1.85153	0.925766	+	0.378096i	$0.123421\pi$
$734$	0	0
$735$	−2.41863	−0.0892124
$736$	0	0
$737$	−21.6456	−0.797324
$738$	0	0
$739$	−12.3099	−0.452828	−0.226414	−	0.974031i	$-0.572700\pi$
$739$	−12.3099	−0.452828	−0.226414	+	0.974031i	$0.572700\pi$
$740$	0	0
$741$	1.74268	0.0640189
$742$	0	0
$743$	−49.3177	−1.80929	−0.904646	−	0.426164i	$-0.859865\pi$
$743$	−49.3177	−1.80929	−0.904646	+	0.426164i	$0.859865\pi$
$744$	0	0
$745$	18.9574	0.694545
$746$	0	0
$747$	6.37913	0.233400
$748$	0	0
$749$	−7.57148	−0.276656
$750$	0	0
$751$	0.533001	0.0194495	0.00972475	−	0.999953i	$-0.496904\pi$
$751$	0.533001	0.0194495	0.00972475	+	0.999953i	$0.496904\pi$
$752$	0	0
$753$	12.4644	0.454226
$754$	0	0
$755$	−6.01515	−0.218914
$756$	0	0
$757$	−10.6559	−0.387296	−0.193648	−	0.981071i	$-0.562032\pi$
$757$	−10.6559	−0.387296	−0.193648	+	0.981071i	$0.562032\pi$
$758$	0	0
$759$	19.0262	0.690608
$760$	0	0
$761$	−15.0451	−0.545384	−0.272692	−	0.962101i	$-0.587914\pi$
$761$	−15.0451	−0.545384	−0.272692	+	0.962101i	$0.587914\pi$
$762$	0	0
$763$	−2.17101	−0.0785957
$764$	0	0
$765$	23.8385	0.861882
$766$	0	0
$767$	−18.4470	−0.666084
$768$	0	0
$769$	45.8351	1.65286	0.826428	−	0.563042i	$-0.190369\pi$
$769$	45.8351	1.65286	0.826428	+	0.563042i	$0.190369\pi$
$770$	0	0
$771$	13.5523	0.488073
$772$	0	0
$773$	−28.0716	−1.00967	−0.504833	−	0.863217i	$-0.668446\pi$
$773$	−28.0716	−1.00967	−0.504833	+	0.863217i	$0.668446\pi$
$774$	0	0
$775$	8.85959	0.318246
$776$	0	0
$777$	−9.86760	−0.353998
$778$	0	0
$779$	4.80954	0.172320
$780$	0	0
$781$	−48.8545	−1.74815
$782$	0	0
$783$	−8.91775	−0.318694
$784$	0	0
$785$	3.99384	0.142546
$786$	0	0
$787$	41.4821	1.47868	0.739339	−	0.673333i	$-0.235138\pi$
$787$	41.4821	1.47868	0.739339	+	0.673333i	$0.235138\pi$
$788$	0	0
$789$	−6.76512	−0.240845
$790$	0	0
$791$	10.0649	0.357866
$792$	0	0
$793$	−12.9298	−0.459151
$794$	0	0
$795$	−0.106137	−0.00376429
$796$	0	0
$797$	−9.42170	−0.333734	−0.166867	−	0.985979i	$-0.553365\pi$
$797$	−9.42170	−0.333734	−0.166867	+	0.985979i	$0.553365\pi$
$798$	0	0
$799$	−2.09064	−0.0739617
$800$	0	0
$801$	−19.5505	−0.690782
$802$	0	0
$803$	−79.8801	−2.81891
$804$	0	0
$805$	13.2256	0.466142
$806$	0	0
$807$	5.77035	0.203126
$808$	0	0
$809$	−14.5482	−0.511487	−0.255744	−	0.966745i	$-0.582320\pi$
$809$	−14.5482	−0.511487	−0.255744	+	0.966745i	$0.582320\pi$
$810$	0	0
$811$	−20.7348	−0.728097	−0.364048	−	0.931380i	$-0.618606\pi$
$811$	−20.7348	−0.728097	−0.364048	+	0.931380i	$0.618606\pi$
$812$	0	0
$813$	5.57551	0.195542
$814$	0	0
$815$	−35.3607	−1.23863
$816$	0	0
$817$	−2.25546	−0.0789085
$818$	0	0
$819$	4.88614	0.170736
$820$	0	0
$821$	−32.3341	−1.12847	−0.564234	−	0.825615i	$-0.690828\pi$
$821$	−32.3341	−1.12847	−0.564234	+	0.825615i	$0.690828\pi$
$822$	0	0
$823$	−3.45021	−0.120267	−0.0601334	−	0.998190i	$-0.519153\pi$
$823$	−3.45021	−0.120267	−0.0601334	+	0.998190i	$0.519153\pi$
$824$	0	0
$825$	−15.0125	−0.522667
$826$	0	0
$827$	−18.7714	−0.652745	−0.326372	−	0.945241i	$-0.605826\pi$
$827$	−18.7714	−0.652745	−0.326372	+	0.945241i	$0.605826\pi$
$828$	0	0
$829$	44.9733	1.56199	0.780993	−	0.624540i	$-0.214713\pi$
$829$	44.9733	1.56199	0.780993	+	0.624540i	$0.214713\pi$
$830$	0	0
$831$	−23.6486	−0.820360
$832$	0	0
$833$	−3.51528	−0.121797
$834$	0	0
$835$	30.7618	1.06456
$836$	0	0
$837$	10.9179	0.377377
$838$	0	0
$839$	11.7421	0.405382	0.202691	−	0.979243i	$-0.435031\pi$
$839$	11.7421	0.405382	0.202691	+	0.979243i	$0.435031\pi$
$840$	0	0
$841$	−24.8810	−0.857967
$842$	0	0
$843$	21.3443	0.735137
$844$	0	0
$845$	−25.0325	−0.861145
$846$	0	0
$847$	−14.9569	−0.513924
$848$	0	0
$849$	22.9075	0.786183
$850$	0	0
$851$	53.9584	1.84967
$852$	0	0
$853$	−47.2869	−1.61907	−0.809536	−	0.587070i	$-0.800281\pi$
$853$	−47.2869	−1.61907	−0.809536	+	0.587070i	$0.800281\pi$
$854$	0	0
$855$	6.78138	0.231918
$856$	0	0
$857$	16.6275	0.567984	0.283992	−	0.958827i	$-0.408341\pi$
$857$	16.6275	0.567984	0.283992	+	0.958827i	$0.408341\pi$
$858$	0	0
$859$	−32.6873	−1.11527	−0.557637	−	0.830085i	$-0.688292\pi$
$859$	−32.6873	−1.11527	−0.557637	+	0.830085i	$0.688292\pi$
$860$	0	0
$861$	−3.97459	−0.135454
$862$	0	0
$863$	42.0481	1.43134	0.715668	−	0.698441i	$-0.246122\pi$
$863$	42.0481	1.43134	0.715668	+	0.698441i	$0.246122\pi$
$864$	0	0
$865$	−24.9224	−0.847387
$866$	0	0
$867$	3.83679	0.130304
$868$	0	0
$869$	76.5371	2.59634
$870$	0	0
$871$	−8.95922	−0.303572
$872$	0	0
$873$	−1.80034	−0.0609323
$874$	0	0
$875$	4.19797	0.141917
$876$	0	0
$877$	−33.4922	−1.13095	−0.565475	−	0.824765i	$-0.691307\pi$
$877$	−33.4922	−1.13095	−0.565475	+	0.824765i	$0.691307\pi$
$878$	0	0
$879$	−22.1941	−0.748588
$880$	0	0
$881$	−9.34844	−0.314957	−0.157479	−	0.987522i	$-0.550337\pi$
$881$	−9.34844	−0.314957	−0.157479	+	0.987522i	$0.550337\pi$
$882$	0	0
$883$	−24.8006	−0.834606	−0.417303	−	0.908767i	$-0.637025\pi$
$883$	−24.8006	−0.834606	−0.417303	+	0.908767i	$0.637025\pi$
$884$	0	0
$885$	21.1577	0.711208
$886$	0	0
$887$	−9.25572	−0.310777	−0.155388	−	0.987853i	$-0.549663\pi$
$887$	−9.25572	−0.310777	−0.155388	+	0.987853i	$0.549663\pi$
$888$	0	0
$889$	−18.6973	−0.627086
$890$	0	0
$891$	16.9147	0.566663
$892$	0	0
$893$	−0.594730	−0.0199019
$894$	0	0
$895$	8.66211	0.289542
$896$	0	0
$897$	7.87506	0.262940
$898$	0	0
$899$	−5.04279	−0.168186
$900$	0	0
$901$	−0.154262	−0.00513920
$902$	0	0
$903$	1.86391	0.0620269
$904$	0	0
$905$	20.5245	0.682258
$906$	0	0
$907$	−52.9867	−1.75939	−0.879697	−	0.475535i	$-0.842254\pi$
$907$	−52.9867	−1.75939	−0.879697	+	0.475535i	$0.842254\pi$
$908$	0	0
$909$	3.41264	0.113190
$910$	0	0
$911$	7.68008	0.254452	0.127226	−	0.991874i	$-0.459393\pi$
$911$	7.68008	0.254452	0.127226	+	0.991874i	$0.459393\pi$
$912$	0	0
$913$	−14.0265	−0.464210
$914$	0	0
$915$	14.8298	0.490256
$916$	0	0
$917$	0.977150	0.0322683
$918$	0	0
$919$	9.90937	0.326880	0.163440	−	0.986553i	$-0.447741\pi$
$919$	9.90937	0.326880	0.163440	+	0.986553i	$0.447741\pi$
$920$	0	0
$921$	−12.6233	−0.415952
$922$	0	0
$923$	−20.2212	−0.665588
$924$	0	0
$925$	−42.5754	−1.39987
$926$	0	0
$927$	26.0317	0.854992
$928$	0	0
$929$	−40.4529	−1.32722	−0.663609	−	0.748080i	$-0.730976\pi$
$929$	−40.4529	−1.32722	−0.663609	+	0.748080i	$0.730976\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0.0664662	0.00217601
$934$	0	0
$935$	−52.4163	−1.71420
$936$	0	0
$937$	14.9439	0.488196	0.244098	−	0.969751i	$-0.421508\pi$
$937$	14.9439	0.488196	0.244098	+	0.969751i	$0.421508\pi$
$938$	0	0
$939$	−28.2880	−0.923145
$940$	0	0
$941$	−24.2991	−0.792129	−0.396065	−	0.918223i	$-0.629624\pi$
$941$	−24.2991	−0.792129	−0.396065	+	0.918223i	$0.629624\pi$
$942$	0	0
$943$	21.7340	0.707757
$944$	0	0
$945$	−12.8600	−0.418336
$946$	0	0
$947$	41.7604	1.35703	0.678515	−	0.734587i	$-0.262624\pi$
$947$	41.7604	1.35703	0.678515	+	0.734587i	$0.262624\pi$
$948$	0	0
$949$	−33.0628	−1.07327
$950$	0	0
$951$	23.1405	0.750381
$952$	0	0
$953$	−22.4672	−0.727785	−0.363893	−	0.931441i	$-0.618553\pi$
$953$	−22.4672	−0.727785	−0.363893	+	0.931441i	$0.618553\pi$
$954$	0	0
$955$	−13.5281	−0.437758
$956$	0	0
$957$	8.54495	0.276219
$958$	0	0
$959$	22.7365	0.734201
$960$	0	0
$961$	−24.8262	−0.800844
$962$	0	0
$963$	−17.5436	−0.565336
$964$	0	0
$965$	−41.6975	−1.34229
$966$	0	0
$967$	13.1685	0.423471	0.211736	−	0.977327i	$-0.432088\pi$
$967$	13.1685	0.423471	0.211736	+	0.977327i	$0.432088\pi$
$968$	0	0
$969$	−2.90502	−0.0933228
$970$	0	0
$971$	11.2933	0.362418	0.181209	−	0.983445i	$-0.441999\pi$
$971$	11.2933	0.362418	0.181209	+	0.983445i	$0.441999\pi$
$972$	0	0
$973$	−6.59485	−0.211421
$974$	0	0
$975$	−6.21375	−0.198999
$976$	0	0
$977$	−16.4496	−0.526268	−0.263134	−	0.964759i	$-0.584756\pi$
$977$	−16.4496	−0.526268	−0.263134	+	0.964759i	$0.584756\pi$
$978$	0	0
$979$	42.9878	1.37389
$980$	0	0
$981$	−5.03036	−0.160607
$982$	0	0
$983$	−17.1057	−0.545586	−0.272793	−	0.962073i	$-0.587947\pi$
$983$	−17.1057	−0.545586	−0.272793	+	0.962073i	$0.587947\pi$
$984$	0	0
$985$	−10.0952	−0.321659
$986$	0	0
$987$	0.491484	0.0156441
$988$	0	0
$989$	−10.1923	−0.324096
$990$	0	0
$991$	19.2955	0.612942	0.306471	−	0.951880i	$-0.400852\pi$
$991$	19.2955	0.612942	0.306471	+	0.951880i	$0.400852\pi$
$992$	0	0
$993$	23.2054	0.736401
$994$	0	0
$995$	38.2390	1.21226
$996$	0	0
$997$	−38.8032	−1.22891	−0.614455	−	0.788952i	$-0.710624\pi$
$997$	−38.8032	−1.22891	−0.614455	+	0.788952i	$0.710624\pi$
$998$	0	0
$999$	−52.4667	−1.65997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8512.2.a.cb.1.3		6
4.3	odd	2		8512.2.a.cd.1.4		6
8.3	odd	2		4256.2.a.k.1.3	✓	6
8.5	even	2		4256.2.a.m.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4256.2.a.k.1.3	✓	6	8.3	odd	2
4256.2.a.m.1.4	yes	6	8.5	even	2
8512.2.a.cb.1.3		6	1.1	even	1	trivial
8512.2.a.cd.1.4		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

\(f(q)\)	\(=\)	\(q-0.826398 q^{3} +2.92671 q^{5} -1.00000 q^{7} -2.31707 q^{9} +5.09479 q^{11} +2.10876 q^{13} -2.41863 q^{15} -3.51528 q^{17} -1.00000 q^{19} +0.826398 q^{21} -4.51894 q^{23} +3.56563 q^{25} +4.39401 q^{27} -2.02952 q^{29} +2.48472 q^{31} -4.21032 q^{33} -2.92671 q^{35} -11.9405 q^{37} -1.74268 q^{39} -4.80954 q^{41} +2.25546 q^{43} -6.78138 q^{45} +0.594730 q^{47} +1.00000 q^{49} +2.90502 q^{51} +0.0438832 q^{53} +14.9110 q^{55} +0.826398 q^{57} -8.74780 q^{59} -6.13147 q^{61} +2.31707 q^{63} +6.17174 q^{65} -4.24857 q^{67} +3.73444 q^{69} -9.58912 q^{71} -15.6788 q^{73} -2.94663 q^{75} -5.09479 q^{77} +15.0226 q^{79} +3.31999 q^{81} -2.75311 q^{83} -10.2882 q^{85} +1.67719 q^{87} +8.43759 q^{89} -2.10876 q^{91} -2.05337 q^{93} -2.92671 q^{95} +0.776992 q^{97} -11.8050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} - q^{5} - 6 q^{7} + 3 q^{9} - 3 q^{11} + 4 q^{13} + 4 q^{15} - 14 q^{17} - 6 q^{19} + q^{21} + 6 q^{23} + q^{25} + 2 q^{27} - 5 q^{29} + 22 q^{31} - 15 q^{33} + q^{35} - 7 q^{37} + 18 q^{39}+ \cdots - 7 q^{99}+O(q^{100}) \) 6 * q - q^3 - q^5 - 6 * q^7 + 3 * q^9 - 3 * q^11 + 4 * q^13 + 4 * q^15 - 14 * q^17 - 6 * q^19 + q^21 + 6 * q^23 + q^25 + 2 * q^27 - 5 * q^29 + 22 * q^31 - 15 * q^33 + q^35 - 7 * q^37 + 18 * q^39 - 11 * q^41 - 3 * q^43 + 2 * q^45 + 15 * q^47 + 6 * q^49 - 20 * q^51 - 19 * q^53 + 32 * q^55 + q^57 + 3 * q^59 + 9 * q^61 - 3 * q^63 - 4 * q^65 - 22 * q^67 - 26 * q^69 + 15 * q^71 - 26 * q^73 - 4 * q^75 + 3 * q^77 + 13 * q^79 - 10 * q^81 + 6 * q^83 - 6 * q^85 + 22 * q^87 - 19 * q^89 - 4 * q^91 - 26 * q^93 + q^95 + q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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