LMFDB - Embedded newform 9680.2.a.cm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.477260$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91300 q^{3} -1.00000 q^{5} -3.06719 q^{7} +0.659557 q^{9} +O(q^{10})$	$q-1.91300 q^{3} -1.00000 q^{5} -3.06719 q^{7} +0.659557 q^{9} -3.04981 q^{13} +1.91300 q^{15} +0.463845 q^{17} -7.89563 q^{19} +5.86752 q^{21} +1.39026 q^{23} +1.00000 q^{25} +4.47726 q^{27} -3.72162 q^{29} +10.4765 q^{31} +3.06719 q^{35} +1.84453 q^{37} +5.83428 q^{39} +4.40763 q^{41} -1.31478 q^{43} -0.659557 q^{45} +2.98018 q^{47} +2.40763 q^{49} -0.887334 q^{51} +4.18814 q^{53} +15.1043 q^{57} -2.81502 q^{59} +2.01737 q^{61} -2.02298 q^{63} +3.04981 q^{65} +6.75753 q^{67} -2.65956 q^{69} +6.52195 q^{71} +9.87581 q^{73} -1.91300 q^{75} -11.5579 q^{79} -10.5437 q^{81} +8.91861 q^{83} -0.463845 q^{85} +7.11945 q^{87} -6.76978 q^{89} +9.35435 q^{91} -20.0415 q^{93} +7.89563 q^{95} +15.3543 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} - 3 q^{7}+O(q^{10}) $ 4 * q - 4 * q^5 - 3 * q^7	$ 4 q - 4 q^{5} - 3 q^{7} + q^{13} - q^{17} - 20 q^{19} + 10 q^{21} - 5 q^{23} + 4 q^{25} + 15 q^{27} + 12 q^{29} + 5 q^{31} + 3 q^{35} + 7 q^{37} - 7 q^{39} + 11 q^{41} - 19 q^{43} - 5 q^{47} + 3 q^{49} - 7 q^{51} - 11 q^{53} - 5 q^{57} - 9 q^{59} + 12 q^{61} + 5 q^{63} - q^{65} + 19 q^{67} - 8 q^{69} - 5 q^{71} + 11 q^{73} - 34 q^{79} + 4 q^{81} + 11 q^{83} + q^{85} + 19 q^{87} - 8 q^{89} + 8 q^{91} - 5 q^{93} + 20 q^{95} + 32 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 - 3 * q^7 + q^13 - q^17 - 20 * q^19 + 10 * q^21 - 5 * q^23 + 4 * q^25 + 15 * q^27 + 12 * q^29 + 5 * q^31 + 3 * q^35 + 7 * q^37 - 7 * q^39 + 11 * q^41 - 19 * q^43 - 5 * q^47 + 3 * q^49 - 7 * q^51 - 11 * q^53 - 5 * q^57 - 9 * q^59 + 12 * q^61 + 5 * q^63 - q^65 + 19 * q^67 - 8 * q^69 - 5 * q^71 + 11 * q^73 - 34 * q^79 + 4 * q^81 + 11 * q^83 + q^85 + 19 * q^87 - 8 * q^89 + 8 * q^91 - 5 * q^93 + 20 * q^95 + 32 * 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
$3$	−1.91300	−1.10447	−0.552235	−	0.833689i	$-0.686225\pi$
$3$	−1.91300	−1.10447	−0.552235	+	0.833689i	$0.686225\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.06719	−1.15929	−0.579644	−	0.814870i	$-0.696808\pi$
$7$	−3.06719	−1.15929	−0.579644	+	0.814870i	$0.696808\pi$
$8$	0	0
$9$	0.659557	0.219852
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.04981	−0.845866	−0.422933	−	0.906161i	$-0.638999\pi$
$13$	−3.04981	−0.845866	−0.422933	+	0.906161i	$0.638999\pi$
$14$	0	0
$15$	1.91300	0.493934
$16$	0	0
$17$	0.463845	0.112499	0.0562495	−	0.998417i	$-0.482086\pi$
$17$	0.463845	0.112499	0.0562495	+	0.998417i	$0.482086\pi$
$18$	0	0
$19$	−7.89563	−1.81138	−0.905690	−	0.423940i	$-0.860647\pi$
$19$	−7.89563	−1.81138	−0.905690	+	0.423940i	$0.860647\pi$
$20$	0	0
$21$	5.86752	1.28040
$22$	0	0
$23$	1.39026	0.289889	0.144944	−	0.989440i	$-0.453700\pi$
$23$	1.39026	0.289889	0.144944	+	0.989440i	$0.453700\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.47726	0.861649
$28$	0	0
$29$	−3.72162	−0.691087	−0.345544	−	0.938403i	$-0.612305\pi$
$29$	−3.72162	−0.691087	−0.345544	+	0.938403i	$0.612305\pi$
$30$	0	0
$31$	10.4765	1.88163	0.940815	−	0.338921i	$-0.110062\pi$
$31$	10.4765	1.88163	0.940815	+	0.338921i	$0.110062\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.06719	0.518449
$36$	0	0
$37$	1.84453	0.303239	0.151620	−	0.988439i	$-0.451551\pi$
$37$	1.84453	0.303239	0.151620	+	0.988439i	$0.451551\pi$
$38$	0	0
$39$	5.83428	0.934233
$40$	0	0
$41$	4.40763	0.688356	0.344178	−	0.938904i	$-0.388158\pi$
$41$	4.40763	0.688356	0.344178	+	0.938904i	$0.388158\pi$
$42$	0	0
$43$	−1.31478	−0.200502	−0.100251	−	0.994962i	$-0.531965\pi$
$43$	−1.31478	−0.200502	−0.100251	+	0.994962i	$0.531965\pi$
$44$	0	0
$45$	−0.659557	−0.0983210
$46$	0	0
$47$	2.98018	0.434704	0.217352	−	0.976093i	$-0.430258\pi$
$47$	2.98018	0.434704	0.217352	+	0.976093i	$0.430258\pi$
$48$	0	0
$49$	2.40763	0.343947
$50$	0	0
$51$	−0.887334	−0.124252
$52$	0	0
$53$	4.18814	0.575286	0.287643	−	0.957738i	$-0.407128\pi$
$53$	4.18814	0.575286	0.287643	+	0.957738i	$0.407128\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.1043	2.00061
$58$	0	0
$59$	−2.81502	−0.366485	−0.183242	−	0.983068i	$-0.558659\pi$
$59$	−2.81502	−0.366485	−0.183242	+	0.983068i	$0.558659\pi$
$60$	0	0
$61$	2.01737	0.258298	0.129149	−	0.991625i	$-0.458775\pi$
$61$	2.01737	0.258298	0.129149	+	0.991625i	$0.458775\pi$
$62$	0	0
$63$	−2.02298	−0.254872
$64$	0	0
$65$	3.04981	0.378283
$66$	0	0
$67$	6.75753	0.825564	0.412782	−	0.910830i	$-0.364557\pi$
$67$	6.75753	0.825564	0.412782	+	0.910830i	$0.364557\pi$
$68$	0	0
$69$	−2.65956	−0.320173
$70$	0	0
$71$	6.52195	0.774013	0.387007	−	0.922077i	$-0.373509\pi$
$71$	6.52195	0.774013	0.387007	+	0.922077i	$0.373509\pi$
$72$	0	0
$73$	9.87581	1.15588	0.577938	−	0.816081i	$-0.303858\pi$
$73$	9.87581	1.15588	0.577938	+	0.816081i	$0.303858\pi$
$74$	0	0
$75$	−1.91300	−0.220894
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.5579	−1.30036	−0.650180	−	0.759780i	$-0.725307\pi$
$79$	−11.5579	−1.30036	−0.650180	+	0.759780i	$0.725307\pi$
$80$	0	0
$81$	−10.5437	−1.17152
$82$	0	0
$83$	8.91861	0.978945	0.489472	−	0.872019i	$-0.337189\pi$
$83$	8.91861	0.978945	0.489472	+	0.872019i	$0.337189\pi$
$84$	0	0
$85$	−0.463845	−0.0503111
$86$	0	0
$87$	7.11945	0.763285
$88$	0	0
$89$	−6.76978	−0.717595	−0.358797	−	0.933415i	$-0.616813\pi$
$89$	−6.76978	−0.717595	−0.358797	+	0.933415i	$0.616813\pi$
$90$	0	0
$91$	9.35435	0.980602
$92$	0	0
$93$	−20.0415	−2.07820
$94$	0	0
$95$	7.89563	0.810074
$96$	0	0
$97$	15.3543	1.55900	0.779499	−	0.626404i	$-0.215474\pi$
$97$	15.3543	1.55900	0.779499	+	0.626404i	$0.215474\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.7326	−1.16744	−0.583718	−	0.811956i	$-0.698403\pi$
$101$	−11.7326	−1.16744	−0.583718	+	0.811956i	$0.698403\pi$
$102$	0	0
$103$	−13.8881	−1.36843	−0.684215	−	0.729280i	$-0.739855\pi$
$103$	−13.8881	−1.36843	−0.684215	+	0.729280i	$0.739855\pi$
$104$	0	0
$105$	−5.86752	−0.572611
$106$	0	0
$107$	−7.32100	−0.707748	−0.353874	−	0.935293i	$-0.615136\pi$
$107$	−7.32100	−0.707748	−0.353874	+	0.935293i	$0.615136\pi$
$108$	0	0
$109$	7.43306	0.711958	0.355979	−	0.934494i	$-0.384147\pi$
$109$	7.43306	0.711958	0.355979	+	0.934494i	$0.384147\pi$
$110$	0	0
$111$	−3.52859	−0.334919
$112$	0	0
$113$	−3.03640	−0.285640	−0.142820	−	0.989749i	$-0.545617\pi$
$113$	−3.03640	−0.285640	−0.142820	+	0.989749i	$0.545617\pi$
$114$	0	0
$115$	−1.39026	−0.129642
$116$	0	0
$117$	−2.01153	−0.185966
$118$	0	0
$119$	−1.42270	−0.130419
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−8.43178	−0.760268
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−0.451597	−0.0400728	−0.0200364	−	0.999799i	$-0.506378\pi$
$127$	−0.451597	−0.0400728	−0.0200364	+	0.999799i	$0.506378\pi$
$128$	0	0
$129$	2.51517	0.221448
$130$	0	0
$131$	0.629003	0.0549563	0.0274781	−	0.999622i	$-0.491252\pi$
$131$	0.629003	0.0549563	0.0274781	+	0.999622i	$0.491252\pi$
$132$	0	0
$133$	24.2173	2.09991
$134$	0	0
$135$	−4.47726	−0.385341
$136$	0	0
$137$	11.1834	0.955462	0.477731	−	0.878506i	$-0.341459\pi$
$137$	11.1834	0.955462	0.477731	+	0.878506i	$0.341459\pi$
$138$	0	0
$139$	−2.37495	−0.201441	−0.100720	−	0.994915i	$-0.532115\pi$
$139$	−2.37495	−0.201441	−0.100720	+	0.994915i	$0.532115\pi$
$140$	0	0
$141$	−5.70108	−0.480118
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.72162	0.309064
$146$	0	0
$147$	−4.60579	−0.379879
$148$	0	0
$149$	8.72034	0.714398	0.357199	−	0.934028i	$-0.383732\pi$
$149$	8.72034	0.714398	0.357199	+	0.934028i	$0.383732\pi$
$150$	0	0
$151$	11.9354	0.971291	0.485646	−	0.874156i	$-0.338585\pi$
$151$	11.9354	0.971291	0.485646	+	0.874156i	$0.338585\pi$
$152$	0	0
$153$	0.305932	0.0247332
$154$	0	0
$155$	−10.4765	−0.841490
$156$	0	0
$157$	3.87532	0.309284	0.154642	−	0.987971i	$-0.450578\pi$
$157$	3.87532	0.309284	0.154642	+	0.987971i	$0.450578\pi$
$158$	0	0
$159$	−8.01190	−0.635385
$160$	0	0
$161$	−4.26418	−0.336064
$162$	0	0
$163$	9.93621	0.778264	0.389132	−	0.921182i	$-0.372775\pi$
$163$	9.93621	0.778264	0.389132	+	0.921182i	$0.372775\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.9200	1.07716	0.538581	−	0.842574i	$-0.318960\pi$
$167$	13.9200	1.07716	0.538581	+	0.842574i	$0.318960\pi$
$168$	0	0
$169$	−3.69863	−0.284510
$170$	0	0
$171$	−5.20762	−0.398236
$172$	0	0
$173$	−10.8311	−0.823475	−0.411737	−	0.911303i	$-0.635078\pi$
$173$	−10.8311	−0.823475	−0.411737	+	0.911303i	$0.635078\pi$
$174$	0	0
$175$	−3.06719	−0.231857
$176$	0	0
$177$	5.38513	0.404771
$178$	0	0
$179$	−22.7335	−1.69918	−0.849589	−	0.527445i	$-0.823150\pi$
$179$	−22.7335	−1.69918	−0.849589	+	0.527445i	$0.823150\pi$
$180$	0	0
$181$	2.39831	0.178265	0.0891327	−	0.996020i	$-0.471590\pi$
$181$	2.39831	0.178265	0.0891327	+	0.996020i	$0.471590\pi$
$182$	0	0
$183$	−3.85923	−0.285282
$184$	0	0
$185$	−1.84453	−0.135613
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−13.7326	−0.998899
$190$	0	0
$191$	−17.2462	−1.24789	−0.623947	−	0.781466i	$-0.714472\pi$
$191$	−17.2462	−1.24789	−0.623947	+	0.781466i	$0.714472\pi$
$192$	0	0
$193$	2.58574	0.186125	0.0930627	−	0.995660i	$-0.470334\pi$
$193$	2.58574	0.186125	0.0930627	+	0.995660i	$0.470334\pi$
$194$	0	0
$195$	−5.83428	−0.417802
$196$	0	0
$197$	−0.144731	−0.0103116	−0.00515582	−	0.999987i	$-0.501641\pi$
$197$	−0.144731	−0.0103116	−0.00515582	+	0.999987i	$0.501641\pi$
$198$	0	0
$199$	7.54177	0.534622	0.267311	−	0.963610i	$-0.413865\pi$
$199$	7.54177	0.534622	0.267311	+	0.963610i	$0.413865\pi$
$200$	0	0
$201$	−12.9271	−0.911810
$202$	0	0
$203$	11.4149	0.801169
$204$	0	0
$205$	−4.40763	−0.307842
$206$	0	0
$207$	0.916954	0.0637327
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−2.26881	−0.156191	−0.0780957	−	0.996946i	$-0.524884\pi$
$211$	−2.26881	−0.156191	−0.0780957	+	0.996946i	$0.524884\pi$
$212$	0	0
$213$	−12.4765	−0.854874
$214$	0	0
$215$	1.31478	0.0896673
$216$	0	0
$217$	−32.1333	−2.18135
$218$	0	0
$219$	−18.8924	−1.27663
$220$	0	0
$221$	−1.41464	−0.0951591
$222$	0	0
$223$	−8.57968	−0.574538	−0.287269	−	0.957850i	$-0.592747\pi$
$223$	−8.57968	−0.574538	−0.287269	+	0.957850i	$0.592747\pi$
$224$	0	0
$225$	0.659557	0.0439705
$226$	0	0
$227$	6.20039	0.411534	0.205767	−	0.978601i	$-0.434031\pi$
$227$	6.20039	0.411534	0.205767	+	0.978601i	$0.434031\pi$
$228$	0	0
$229$	23.1659	1.53084	0.765422	−	0.643528i	$-0.222530\pi$
$229$	23.1659	1.53084	0.765422	+	0.643528i	$0.222530\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.8627	1.82535	0.912673	−	0.408690i	$-0.134014\pi$
$233$	27.8627	1.82535	0.912673	+	0.408690i	$0.134014\pi$
$234$	0	0
$235$	−2.98018	−0.194406
$236$	0	0
$237$	22.1102	1.43621
$238$	0	0
$239$	−16.2862	−1.05347	−0.526734	−	0.850030i	$-0.676584\pi$
$239$	−16.2862	−1.05347	−0.526734	+	0.850030i	$0.676584\pi$
$240$	0	0
$241$	−4.39063	−0.282826	−0.141413	−	0.989951i	$-0.545164\pi$
$241$	−4.39063	−0.282826	−0.141413	+	0.989951i	$0.545164\pi$
$242$	0	0
$243$	6.73820	0.432256
$244$	0	0
$245$	−2.40763	−0.153818
$246$	0	0
$247$	24.0802	1.53219
$248$	0	0
$249$	−17.0613	−1.08121
$250$	0	0
$251$	2.66668	0.168319	0.0841597	−	0.996452i	$-0.473179\pi$
$251$	2.66668	0.168319	0.0841597	+	0.996452i	$0.473179\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.887334	0.0555670
$256$	0	0
$257$	−21.6327	−1.34941	−0.674704	−	0.738088i	$-0.735729\pi$
$257$	−21.6327	−1.34941	−0.674704	+	0.738088i	$0.735729\pi$
$258$	0	0
$259$	−5.65752	−0.351541
$260$	0	0
$261$	−2.45462	−0.151937
$262$	0	0
$263$	22.1392	1.36516	0.682581	−	0.730810i	$-0.260858\pi$
$263$	22.1392	1.36516	0.682581	+	0.730810i	$0.260858\pi$
$264$	0	0
$265$	−4.18814	−0.257276
$266$	0	0
$267$	12.9506	0.792562
$268$	0	0
$269$	20.7184	1.26322	0.631611	−	0.775285i	$-0.282394\pi$
$269$	20.7184	1.26322	0.631611	+	0.775285i	$0.282394\pi$
$270$	0	0
$271$	−0.423112	−0.0257022	−0.0128511	−	0.999917i	$-0.504091\pi$
$271$	−0.423112	−0.0257022	−0.0128511	+	0.999917i	$0.504091\pi$
$272$	0	0
$273$	−17.8948	−1.08304
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.57255	0.515075	0.257537	−	0.966268i	$-0.417089\pi$
$277$	8.57255	0.515075	0.257537	+	0.966268i	$0.417089\pi$
$278$	0	0
$279$	6.90983	0.413681
$280$	0	0
$281$	7.01108	0.418246	0.209123	−	0.977889i	$-0.432939\pi$
$281$	7.01108	0.418246	0.209123	+	0.977889i	$0.432939\pi$
$282$	0	0
$283$	9.95317	0.591655	0.295827	−	0.955241i	$-0.404405\pi$
$283$	9.95317	0.591655	0.295827	+	0.955241i	$0.404405\pi$
$284$	0	0
$285$	−15.1043	−0.894702
$286$	0	0
$287$	−13.5190	−0.798002
$288$	0	0
$289$	−16.7848	−0.987344
$290$	0	0
$291$	−29.3728	−1.72186
$292$	0	0
$293$	−21.8209	−1.27479	−0.637394	−	0.770538i	$-0.719988\pi$
$293$	−21.8209	−1.27479	−0.637394	+	0.770538i	$0.719988\pi$
$294$	0	0
$295$	2.81502	0.163897
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.24002	−0.245207
$300$	0	0
$301$	4.03268	0.232440
$302$	0	0
$303$	22.4444	1.28940
$304$	0	0
$305$	−2.01737	−0.115514
$306$	0	0
$307$	−30.8674	−1.76170	−0.880849	−	0.473397i	$-0.843028\pi$
$307$	−30.8674	−1.76170	−0.880849	+	0.473397i	$0.843028\pi$
$308$	0	0
$309$	26.5678	1.51139
$310$	0	0
$311$	19.4150	1.10092	0.550462	−	0.834860i	$-0.314451\pi$
$311$	19.4150	1.10092	0.550462	+	0.834860i	$0.314451\pi$
$312$	0	0
$313$	1.05147	0.0594326	0.0297163	−	0.999558i	$-0.490540\pi$
$313$	1.05147	0.0594326	0.0297163	+	0.999558i	$0.490540\pi$
$314$	0	0
$315$	2.02298	0.113982
$316$	0	0
$317$	−23.8314	−1.33851	−0.669253	−	0.743034i	$-0.733386\pi$
$317$	−23.8314	−1.33851	−0.669253	+	0.743034i	$0.733386\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	14.0051	0.781686
$322$	0	0
$323$	−3.66235	−0.203778
$324$	0	0
$325$	−3.04981	−0.169173
$326$	0	0
$327$	−14.2194	−0.786336
$328$	0	0
$329$	−9.14077	−0.503947
$330$	0	0
$331$	−25.6693	−1.41091	−0.705457	−	0.708753i	$-0.749258\pi$
$331$	−25.6693	−1.41091	−0.705457	+	0.708753i	$0.749258\pi$
$332$	0	0
$333$	1.21657	0.0666679
$334$	0	0
$335$	−6.75753	−0.369203
$336$	0	0
$337$	23.8922	1.30149	0.650744	−	0.759297i	$-0.274457\pi$
$337$	23.8922	1.30149	0.650744	+	0.759297i	$0.274457\pi$
$338$	0	0
$339$	5.80862	0.315481
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.0857	0.760554
$344$	0	0
$345$	2.65956	0.143186
$346$	0	0
$347$	−0.332152	−0.0178309	−0.00891543	−	0.999960i	$-0.502838\pi$
$347$	−0.332152	−0.0178309	−0.00891543	+	0.999960i	$0.502838\pi$
$348$	0	0
$349$	−1.22149	−0.0653846	−0.0326923	−	0.999465i	$-0.510408\pi$
$349$	−1.22149	−0.0653846	−0.0326923	+	0.999465i	$0.510408\pi$
$350$	0	0
$351$	−13.6548	−0.728840
$352$	0	0
$353$	−25.7038	−1.36808	−0.684039	−	0.729446i	$-0.739778\pi$
$353$	−25.7038	−1.36808	−0.684039	+	0.729446i	$0.739778\pi$
$354$	0	0
$355$	−6.52195	−0.346149
$356$	0	0
$357$	2.72162	0.144043
$358$	0	0
$359$	−17.9315	−0.946387	−0.473193	−	0.880959i	$-0.656899\pi$
$359$	−17.9315	−0.946387	−0.473193	+	0.880959i	$0.656899\pi$
$360$	0	0
$361$	43.3409	2.28110
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.87581	−0.516923
$366$	0	0
$367$	8.49091	0.443222	0.221611	−	0.975135i	$-0.428869\pi$
$367$	8.49091	0.443222	0.221611	+	0.975135i	$0.428869\pi$
$368$	0	0
$369$	2.90708	0.151337
$370$	0	0
$371$	−12.8458	−0.666921
$372$	0	0
$373$	−35.8450	−1.85598	−0.927991	−	0.372604i	$-0.878465\pi$
$373$	−35.8450	−1.85598	−0.927991	+	0.372604i	$0.878465\pi$
$374$	0	0
$375$	1.91300	0.0987867
$376$	0	0
$377$	11.3502	0.584567
$378$	0	0
$379$	−17.5516	−0.901564	−0.450782	−	0.892634i	$-0.648855\pi$
$379$	−17.5516	−0.901564	−0.450782	+	0.892634i	$0.648855\pi$
$380$	0	0
$381$	0.863904	0.0442592
$382$	0	0
$383$	21.6250	1.10499	0.552494	−	0.833517i	$-0.313676\pi$
$383$	21.6250	1.10499	0.552494	+	0.833517i	$0.313676\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.867173	−0.0440809
$388$	0	0
$389$	35.7416	1.81217	0.906086	−	0.423093i	$-0.139056\pi$
$389$	35.7416	1.81217	0.906086	+	0.423093i	$0.139056\pi$
$390$	0	0
$391$	0.644864	0.0326122
$392$	0	0
$393$	−1.20328	−0.0606975
$394$	0	0
$395$	11.5579	0.581539
$396$	0	0
$397$	−20.0447	−1.00601	−0.503007	−	0.864282i	$-0.667773\pi$
$397$	−20.0447	−1.00601	−0.503007	+	0.864282i	$0.667773\pi$
$398$	0	0
$399$	−46.3277	−2.31929
$400$	0	0
$401$	−20.8987	−1.04363	−0.521815	−	0.853059i	$-0.674745\pi$
$401$	−20.8987	−1.04363	−0.521815	+	0.853059i	$0.674745\pi$
$402$	0	0
$403$	−31.9513	−1.59161
$404$	0	0
$405$	10.5437	0.523918
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−23.4982	−1.16191	−0.580955	−	0.813936i	$-0.697321\pi$
$409$	−23.4982	−1.16191	−0.580955	+	0.813936i	$0.697321\pi$
$410$	0	0
$411$	−21.3938	−1.05528
$412$	0	0
$413$	8.63420	0.424861
$414$	0	0
$415$	−8.91861	−0.437797
$416$	0	0
$417$	4.54328	0.222485
$418$	0	0
$419$	10.1128	0.494043	0.247022	−	0.969010i	$-0.420548\pi$
$419$	10.1128	0.494043	0.247022	+	0.969010i	$0.420548\pi$
$420$	0	0
$421$	8.92283	0.434872	0.217436	−	0.976075i	$-0.430231\pi$
$421$	8.92283	0.434872	0.217436	+	0.976075i	$0.430231\pi$
$422$	0	0
$423$	1.96560	0.0955708
$424$	0	0
$425$	0.463845	0.0224998
$426$	0	0
$427$	−6.18765	−0.299442
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.6122	−0.848352	−0.424176	−	0.905580i	$-0.639436\pi$
$431$	−17.6122	−0.848352	−0.424176	+	0.905580i	$0.639436\pi$
$432$	0	0
$433$	9.14397	0.439431	0.219716	−	0.975564i	$-0.429487\pi$
$433$	9.14397	0.439431	0.219716	+	0.975564i	$0.429487\pi$
$434$	0	0
$435$	−7.11945	−0.341351
$436$	0	0
$437$	−10.9769	−0.525099
$438$	0	0
$439$	6.46946	0.308770	0.154385	−	0.988011i	$-0.450660\pi$
$439$	6.46946	0.308770	0.154385	+	0.988011i	$0.450660\pi$
$440$	0	0
$441$	1.58797	0.0756176
$442$	0	0
$443$	40.8842	1.94247	0.971233	−	0.238132i	$-0.0765350\pi$
$443$	40.8842	1.94247	0.971233	+	0.238132i	$0.0765350\pi$
$444$	0	0
$445$	6.76978	0.320918
$446$	0	0
$447$	−16.6820	−0.789031
$448$	0	0
$449$	−12.1608	−0.573902	−0.286951	−	0.957945i	$-0.592642\pi$
$449$	−12.1608	−0.573902	−0.286951	+	0.957945i	$0.592642\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−22.8324	−1.07276
$454$	0	0
$455$	−9.35435	−0.438539
$456$	0	0
$457$	9.90240	0.463215	0.231607	−	0.972809i	$-0.425602\pi$
$457$	9.90240	0.463215	0.231607	+	0.972809i	$0.425602\pi$
$458$	0	0
$459$	2.07676	0.0969346
$460$	0	0
$461$	3.12529	0.145559	0.0727796	−	0.997348i	$-0.476813\pi$
$461$	3.12529	0.145559	0.0727796	+	0.997348i	$0.476813\pi$
$462$	0	0
$463$	24.3518	1.13173	0.565863	−	0.824499i	$-0.308543\pi$
$463$	24.3518	1.13173	0.565863	+	0.824499i	$0.308543\pi$
$464$	0	0
$465$	20.0415	0.929400
$466$	0	0
$467$	−33.1737	−1.53510	−0.767548	−	0.640991i	$-0.778523\pi$
$467$	−33.1737	−1.53510	−0.767548	+	0.640991i	$0.778523\pi$
$468$	0	0
$469$	−20.7266	−0.957065
$470$	0	0
$471$	−7.41347	−0.341595
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−7.89563	−0.362276
$476$	0	0
$477$	2.76232	0.126478
$478$	0	0
$479$	−17.2977	−0.790353	−0.395176	−	0.918605i	$-0.629317\pi$
$479$	−17.2977	−0.790353	−0.395176	+	0.918605i	$0.629317\pi$
$480$	0	0
$481$	−5.62548	−0.256500
$482$	0	0
$483$	8.15736	0.371173
$484$	0	0
$485$	−15.3543	−0.697205
$486$	0	0
$487$	−7.64061	−0.346229	−0.173114	−	0.984902i	$-0.555383\pi$
$487$	−7.64061	−0.346229	−0.173114	+	0.984902i	$0.555383\pi$
$488$	0	0
$489$	−19.0079	−0.859569
$490$	0	0
$491$	−30.6563	−1.38350	−0.691750	−	0.722137i	$-0.743160\pi$
$491$	−30.6563	−1.38350	−0.691750	+	0.722137i	$0.743160\pi$
$492$	0	0
$493$	−1.72625	−0.0777466
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−20.0040	−0.897303
$498$	0	0
$499$	37.4783	1.67776	0.838879	−	0.544317i	$-0.183211\pi$
$499$	37.4783	1.67776	0.838879	+	0.544317i	$0.183211\pi$
$500$	0	0
$501$	−26.6289	−1.18969
$502$	0	0
$503$	8.47695	0.377969	0.188984	−	0.981980i	$-0.439480\pi$
$503$	8.47695	0.377969	0.188984	+	0.981980i	$0.439480\pi$
$504$	0	0
$505$	11.7326	0.522093
$506$	0	0
$507$	7.07548	0.314233
$508$	0	0
$509$	−1.69723	−0.0752286	−0.0376143	−	0.999292i	$-0.511976\pi$
$509$	−1.69723	−0.0752286	−0.0376143	+	0.999292i	$0.511976\pi$
$510$	0	0
$511$	−30.2909	−1.33999
$512$	0	0
$513$	−35.3508	−1.56077
$514$	0	0
$515$	13.8881	0.611981
$516$	0	0
$517$	0	0
$518$	0	0
$519$	20.7199	0.909502
$520$	0	0
$521$	−37.0929	−1.62507	−0.812535	−	0.582912i	$-0.801913\pi$
$521$	−37.0929	−1.62507	−0.812535	+	0.582912i	$0.801913\pi$
$522$	0	0
$523$	−18.0818	−0.790662	−0.395331	−	0.918539i	$-0.629370\pi$
$523$	−18.0818	−0.790662	−0.395331	+	0.918539i	$0.629370\pi$
$524$	0	0
$525$	5.86752	0.256079
$526$	0	0
$527$	4.85946	0.211681
$528$	0	0
$529$	−21.0672	−0.915965
$530$	0	0
$531$	−1.85667	−0.0805726
$532$	0	0
$533$	−13.4424	−0.582257
$534$	0	0
$535$	7.32100	0.316515
$536$	0	0
$537$	43.4890	1.87669
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.7524	−0.505275	−0.252638	−	0.967561i	$-0.581298\pi$
$541$	−11.7524	−0.505275	−0.252638	+	0.967561i	$0.581298\pi$
$542$	0	0
$543$	−4.58797	−0.196889
$544$	0	0
$545$	−7.43306	−0.318397
$546$	0	0
$547$	−21.7569	−0.930256	−0.465128	−	0.885243i	$-0.653992\pi$
$547$	−21.7569	−0.930256	−0.465128	+	0.885243i	$0.653992\pi$
$548$	0	0
$549$	1.33057	0.0567874
$550$	0	0
$551$	29.3845	1.25182
$552$	0	0
$553$	35.4501	1.50749
$554$	0	0
$555$	3.52859	0.149780
$556$	0	0
$557$	−4.83432	−0.204837	−0.102418	−	0.994741i	$-0.532658\pi$
$557$	−4.83432	−0.204837	−0.102418	+	0.994741i	$0.532658\pi$
$558$	0	0
$559$	4.00984	0.169598
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.77199	0.201115	0.100558	−	0.994931i	$-0.467937\pi$
$563$	4.77199	0.201115	0.100558	+	0.994931i	$0.467937\pi$
$564$	0	0
$565$	3.03640	0.127742
$566$	0	0
$567$	32.3394	1.35813
$568$	0	0
$569$	−35.7187	−1.49741	−0.748703	−	0.662905i	$-0.769323\pi$
$569$	−35.7187	−1.49741	−0.748703	+	0.662905i	$0.769323\pi$
$570$	0	0
$571$	−33.9838	−1.42218	−0.711090	−	0.703101i	$-0.751798\pi$
$571$	−33.9838	−1.42218	−0.711090	+	0.703101i	$0.751798\pi$
$572$	0	0
$573$	32.9920	1.37826
$574$	0	0
$575$	1.39026	0.0579777
$576$	0	0
$577$	−20.6579	−0.860000	−0.430000	−	0.902829i	$-0.641486\pi$
$577$	−20.6579	−0.860000	−0.430000	+	0.902829i	$0.641486\pi$
$578$	0	0
$579$	−4.94650	−0.205570
$580$	0	0
$581$	−27.3550	−1.13488
$582$	0	0
$583$	0	0
$584$	0	0
$585$	2.01153	0.0831664
$586$	0	0
$587$	−13.3600	−0.551428	−0.275714	−	0.961240i	$-0.588914\pi$
$587$	−13.3600	−0.551428	−0.275714	+	0.961240i	$0.588914\pi$
$588$	0	0
$589$	−82.7183	−3.40835
$590$	0	0
$591$	0.276870	0.0113889
$592$	0	0
$593$	−20.8062	−0.854410	−0.427205	−	0.904155i	$-0.640502\pi$
$593$	−20.8062	−0.854410	−0.427205	+	0.904155i	$0.640502\pi$
$594$	0	0
$595$	1.42270	0.0583250
$596$	0	0
$597$	−14.4274	−0.590473
$598$	0	0
$599$	14.2456	0.582061	0.291030	−	0.956714i	$-0.406002\pi$
$599$	14.2456	0.582061	0.291030	+	0.956714i	$0.406002\pi$
$600$	0	0
$601$	20.7462	0.846255	0.423127	−	0.906070i	$-0.360932\pi$
$601$	20.7462	0.846255	0.423127	+	0.906070i	$0.360932\pi$
$602$	0	0
$603$	4.45698	0.181502
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.9219	0.727428	0.363714	−	0.931511i	$-0.381508\pi$
$607$	17.9219	0.727428	0.363714	+	0.931511i	$0.381508\pi$
$608$	0	0
$609$	−21.8367	−0.884866
$610$	0	0
$611$	−9.08900	−0.367702
$612$	0	0
$613$	28.2019	1.13906	0.569531	−	0.821970i	$-0.307125\pi$
$613$	28.2019	1.13906	0.569531	+	0.821970i	$0.307125\pi$
$614$	0	0
$615$	8.43178	0.340002
$616$	0	0
$617$	−4.72930	−0.190394	−0.0951972	−	0.995458i	$-0.530348\pi$
$617$	−4.72930	−0.190394	−0.0951972	+	0.995458i	$0.530348\pi$
$618$	0	0
$619$	30.0575	1.20811	0.604056	−	0.796942i	$-0.293550\pi$
$619$	30.0575	1.20811	0.604056	+	0.796942i	$0.293550\pi$
$620$	0	0
$621$	6.22454	0.249782
$622$	0	0
$623$	20.7642	0.831899
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.855577	0.0341141
$630$	0	0
$631$	31.7036	1.26210	0.631050	−	0.775742i	$-0.282624\pi$
$631$	31.7036	1.26210	0.631050	+	0.775742i	$0.282624\pi$
$632$	0	0
$633$	4.34023	0.172509
$634$	0	0
$635$	0.451597	0.0179211
$636$	0	0
$637$	−7.34282	−0.290933
$638$	0	0
$639$	4.30160	0.170169
$640$	0	0
$641$	−18.9573	−0.748770	−0.374385	−	0.927273i	$-0.622146\pi$
$641$	−18.9573	−0.748770	−0.374385	+	0.927273i	$0.622146\pi$
$642$	0	0
$643$	36.2489	1.42952	0.714758	−	0.699372i	$-0.246537\pi$
$643$	36.2489	1.42952	0.714758	+	0.699372i	$0.246537\pi$
$644$	0	0
$645$	−2.51517	−0.0990347
$646$	0	0
$647$	34.9519	1.37410	0.687050	−	0.726610i	$-0.258905\pi$
$647$	34.9519	1.37410	0.687050	+	0.726610i	$0.258905\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	61.4709	2.40923
$652$	0	0
$653$	−29.7893	−1.16574	−0.582872	−	0.812564i	$-0.698071\pi$
$653$	−29.7893	−1.16574	−0.582872	+	0.812564i	$0.698071\pi$
$654$	0	0
$655$	−0.629003	−0.0245772
$656$	0	0
$657$	6.51366	0.254122
$658$	0	0
$659$	−7.30532	−0.284575	−0.142287	−	0.989825i	$-0.545446\pi$
$659$	−7.30532	−0.284575	−0.142287	+	0.989825i	$0.545446\pi$
$660$	0	0
$661$	−22.7352	−0.884296	−0.442148	−	0.896942i	$-0.645783\pi$
$661$	−22.7352	−0.884296	−0.442148	+	0.896942i	$0.645783\pi$
$662$	0	0
$663$	2.70620	0.105100
$664$	0	0
$665$	−24.2173	−0.939109
$666$	0	0
$667$	−5.17401	−0.200338
$668$	0	0
$669$	16.4129	0.634559
$670$	0	0
$671$	0	0
$672$	0	0
$673$	17.7451	0.684024	0.342012	−	0.939696i	$-0.388892\pi$
$673$	17.7451	0.684024	0.342012	+	0.939696i	$0.388892\pi$
$674$	0	0
$675$	4.47726	0.172330
$676$	0	0
$677$	−8.16216	−0.313697	−0.156849	−	0.987623i	$-0.550133\pi$
$677$	−8.16216	−0.313697	−0.156849	+	0.987623i	$0.550133\pi$
$678$	0	0
$679$	−47.0946	−1.80733
$680$	0	0
$681$	−11.8613	−0.454527
$682$	0	0
$683$	6.19100	0.236892	0.118446	−	0.992960i	$-0.462209\pi$
$683$	6.19100	0.236892	0.118446	+	0.992960i	$0.462209\pi$
$684$	0	0
$685$	−11.1834	−0.427296
$686$	0	0
$687$	−44.3163	−1.69077
$688$	0	0
$689$	−12.7731	−0.486615
$690$	0	0
$691$	−23.6051	−0.897981	−0.448990	−	0.893537i	$-0.648216\pi$
$691$	−23.6051	−0.897981	−0.448990	+	0.893537i	$0.648216\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.37495	0.0900871
$696$	0	0
$697$	2.04446	0.0774393
$698$	0	0
$699$	−53.3013	−2.01604
$700$	0	0
$701$	−37.2284	−1.40610	−0.703049	−	0.711142i	$-0.748178\pi$
$701$	−37.2284	−1.40610	−0.703049	+	0.711142i	$0.748178\pi$
$702$	0	0
$703$	−14.5637	−0.549282
$704$	0	0
$705$	5.70108	0.214715
$706$	0	0
$707$	35.9860	1.35339
$708$	0	0
$709$	18.2537	0.685534	0.342767	−	0.939420i	$-0.388636\pi$
$709$	18.2537	0.685534	0.342767	+	0.939420i	$0.388636\pi$
$710$	0	0
$711$	−7.62307	−0.285887
$712$	0	0
$713$	14.5650	0.545463
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.1555	1.16352
$718$	0	0
$719$	−9.57389	−0.357046	−0.178523	−	0.983936i	$-0.557132\pi$
$719$	−9.57389	−0.357046	−0.178523	+	0.983936i	$0.557132\pi$
$720$	0	0
$721$	42.5972	1.58640
$722$	0	0
$723$	8.39927	0.312372
$724$	0	0
$725$	−3.72162	−0.138217
$726$	0	0
$727$	14.0175	0.519882	0.259941	−	0.965625i	$-0.416297\pi$
$727$	14.0175	0.519882	0.259941	+	0.965625i	$0.416297\pi$
$728$	0	0
$729$	18.7408	0.694104
$730$	0	0
$731$	−0.609854	−0.0225563
$732$	0	0
$733$	−9.28772	−0.343050	−0.171525	−	0.985180i	$-0.554869\pi$
$733$	−9.28772	−0.343050	−0.171525	+	0.985180i	$0.554869\pi$
$734$	0	0
$735$	4.60579	0.169887
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−38.9586	−1.43312	−0.716558	−	0.697527i	$-0.754284\pi$
$739$	−38.9586	−1.43312	−0.716558	+	0.697527i	$0.754284\pi$
$740$	0	0
$741$	−46.0653	−1.69225
$742$	0	0
$743$	45.9296	1.68499	0.842497	−	0.538701i	$-0.181085\pi$
$743$	45.9296	1.68499	0.842497	+	0.538701i	$0.181085\pi$
$744$	0	0
$745$	−8.72034	−0.319489
$746$	0	0
$747$	5.88233	0.215223
$748$	0	0
$749$	22.4549	0.820483
$750$	0	0
$751$	−23.1928	−0.846318	−0.423159	−	0.906055i	$-0.639079\pi$
$751$	−23.1928	−0.846318	−0.423159	+	0.906055i	$0.639079\pi$
$752$	0	0
$753$	−5.10135	−0.185904
$754$	0	0
$755$	−11.9354	−0.434375
$756$	0	0
$757$	−6.52202	−0.237047	−0.118523	−	0.992951i	$-0.537816\pi$
$757$	−6.52202	−0.237047	−0.118523	+	0.992951i	$0.537816\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	6.56682	0.238047	0.119024	−	0.992891i	$-0.462024\pi$
$761$	6.56682	0.238047	0.119024	+	0.992891i	$0.462024\pi$
$762$	0	0
$763$	−22.7986	−0.825364
$764$	0	0
$765$	−0.305932	−0.0110610
$766$	0	0
$767$	8.58530	0.309997
$768$	0	0
$769$	−12.5950	−0.454188	−0.227094	−	0.973873i	$-0.572922\pi$
$769$	−12.5950	−0.454188	−0.227094	+	0.973873i	$0.572922\pi$
$770$	0	0
$771$	41.3832	1.49038
$772$	0	0
$773$	21.9448	0.789299	0.394650	−	0.918832i	$-0.370866\pi$
$773$	21.9448	0.789299	0.394650	+	0.918832i	$0.370866\pi$
$774$	0	0
$775$	10.4765	0.376326
$776$	0	0
$777$	10.8228	0.388267
$778$	0	0
$779$	−34.8010	−1.24687
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−16.6627	−0.595475
$784$	0	0
$785$	−3.87532	−0.138316
$786$	0	0
$787$	16.7298	0.596354	0.298177	−	0.954511i	$-0.403621\pi$
$787$	16.7298	0.596354	0.298177	+	0.954511i	$0.403621\pi$
$788$	0	0
$789$	−42.3522	−1.50778
$790$	0	0
$791$	9.31320	0.331139
$792$	0	0
$793$	−6.15261	−0.218486
$794$	0	0
$795$	8.01190	0.284153
$796$	0	0
$797$	−11.7707	−0.416940	−0.208470	−	0.978029i	$-0.566848\pi$
$797$	−11.7707	−0.416940	−0.208470	+	0.978029i	$0.566848\pi$
$798$	0	0
$799$	1.38234	0.0489038
$800$	0	0
$801$	−4.46505	−0.157765
$802$	0	0
$803$	0	0
$804$	0	0
$805$	4.26418	0.150292
$806$	0	0
$807$	−39.6342	−1.39519
$808$	0	0
$809$	−1.19595	−0.0420472	−0.0210236	−	0.999779i	$-0.506693\pi$
$809$	−1.19595	−0.0420472	−0.0210236	+	0.999779i	$0.506693\pi$
$810$	0	0
$811$	35.8253	1.25800	0.628998	−	0.777407i	$-0.283465\pi$
$811$	35.8253	1.25800	0.628998	+	0.777407i	$0.283465\pi$
$812$	0	0
$813$	0.809412	0.0283873
$814$	0	0
$815$	−9.93621	−0.348050
$816$	0	0
$817$	10.3810	0.363186
$818$	0	0
$819$	6.16973	0.215588
$820$	0	0
$821$	−5.49200	−0.191672	−0.0958360	−	0.995397i	$-0.530552\pi$
$821$	−5.49200	−0.191672	−0.0958360	+	0.995397i	$0.530552\pi$
$822$	0	0
$823$	−42.7252	−1.48931	−0.744654	−	0.667451i	$-0.767386\pi$
$823$	−42.7252	−1.48931	−0.744654	+	0.667451i	$0.767386\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.4208	1.12738	0.563691	−	0.825986i	$-0.309381\pi$
$827$	32.4208	1.12738	0.563691	+	0.825986i	$0.309381\pi$
$828$	0	0
$829$	−2.91149	−0.101120	−0.0505600	−	0.998721i	$-0.516101\pi$
$829$	−2.91149	−0.101120	−0.0505600	+	0.998721i	$0.516101\pi$
$830$	0	0
$831$	−16.3993	−0.568884
$832$	0	0
$833$	1.11677	0.0386937
$834$	0	0
$835$	−13.9200	−0.481722
$836$	0	0
$837$	46.9059	1.62130
$838$	0	0
$839$	20.8465	0.719701	0.359850	−	0.933010i	$-0.382828\pi$
$839$	20.8465	0.719701	0.359850	+	0.933010i	$0.382828\pi$
$840$	0	0
$841$	−15.1496	−0.522398
$842$	0	0
$843$	−13.4122	−0.461940
$844$	0	0
$845$	3.69863	0.127237
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−19.0404	−0.653464
$850$	0	0
$851$	2.56437	0.0879056
$852$	0	0
$853$	−34.5509	−1.18300	−0.591500	−	0.806305i	$-0.701464\pi$
$853$	−34.5509	−1.18300	−0.591500	+	0.806305i	$0.701464\pi$
$854$	0	0
$855$	5.20762	0.178097
$856$	0	0
$857$	−33.2969	−1.13740	−0.568699	−	0.822545i	$-0.692553\pi$
$857$	−33.2969	−1.13740	−0.568699	+	0.822545i	$0.692553\pi$
$858$	0	0
$859$	16.7665	0.572067	0.286034	−	0.958220i	$-0.407663\pi$
$859$	16.7665	0.572067	0.286034	+	0.958220i	$0.407663\pi$
$860$	0	0
$861$	25.8618	0.881369
$862$	0	0
$863$	1.48415	0.0505211	0.0252605	−	0.999681i	$-0.491958\pi$
$863$	1.48415	0.0505211	0.0252605	+	0.999681i	$0.491958\pi$
$864$	0	0
$865$	10.8311	0.368269
$866$	0	0
$867$	32.1094	1.09049
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−20.6092	−0.698316
$872$	0	0
$873$	10.1271	0.342749
$874$	0	0
$875$	3.06719	0.103690
$876$	0	0
$877$	24.8615	0.839513	0.419756	−	0.907637i	$-0.362115\pi$
$877$	24.8615	0.839513	0.419756	+	0.907637i	$0.362115\pi$
$878$	0	0
$879$	41.7433	1.40797
$880$	0	0
$881$	32.6968	1.10158	0.550792	−	0.834643i	$-0.314326\pi$
$881$	32.6968	1.10158	0.550792	+	0.834643i	$0.314326\pi$
$882$	0	0
$883$	−47.6218	−1.60260	−0.801300	−	0.598263i	$-0.795858\pi$
$883$	−47.6218	−1.60260	−0.801300	+	0.598263i	$0.795858\pi$
$884$	0	0
$885$	−5.38513	−0.181019
$886$	0	0
$887$	59.0960	1.98425	0.992125	−	0.125251i	$-0.0399736\pi$
$887$	59.0960	1.98425	0.992125	+	0.125251i	$0.0399736\pi$
$888$	0	0
$889$	1.38513	0.0464559
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−23.5304	−0.787415
$894$	0	0
$895$	22.7335	0.759896
$896$	0	0
$897$	8.11115	0.270824
$898$	0	0
$899$	−38.9894	−1.30037
$900$	0	0
$901$	1.94265	0.0647190
$902$	0	0
$903$	−7.71450	−0.256722
$904$	0	0
$905$	−2.39831	−0.0797227
$906$	0	0
$907$	−34.6576	−1.15079	−0.575393	−	0.817877i	$-0.695151\pi$
$907$	−34.6576	−1.15079	−0.575393	+	0.817877i	$0.695151\pi$
$908$	0	0
$909$	−7.73831	−0.256664
$910$	0	0
$911$	10.1883	0.337552	0.168776	−	0.985654i	$-0.446019\pi$
$911$	10.1883	0.337552	0.168776	+	0.985654i	$0.446019\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	3.85923	0.127582
$916$	0	0
$917$	−1.92927	−0.0637101
$918$	0	0
$919$	−36.1289	−1.19178	−0.595891	−	0.803065i	$-0.703201\pi$
$919$	−36.1289	−1.19178	−0.595891	+	0.803065i	$0.703201\pi$
$920$	0	0
$921$	59.0493	1.94574
$922$	0	0
$923$	−19.8907	−0.654711
$924$	0	0
$925$	1.84453	0.0606479
$926$	0	0
$927$	−9.15997	−0.300853
$928$	0	0
$929$	28.1240	0.922717	0.461359	−	0.887214i	$-0.347362\pi$
$929$	28.1240	0.922717	0.461359	+	0.887214i	$0.347362\pi$
$930$	0	0
$931$	−19.0097	−0.623019
$932$	0	0
$933$	−37.1409	−1.21594
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−0.0851677	−0.00278231	−0.00139115	−	0.999999i	$-0.500443\pi$
$937$	−0.0851677	−0.00278231	−0.00139115	+	0.999999i	$0.500443\pi$
$938$	0	0
$939$	−2.01146	−0.0656414
$940$	0	0
$941$	41.8154	1.36314	0.681572	−	0.731751i	$-0.261297\pi$
$941$	41.8154	1.36314	0.681572	+	0.731751i	$0.261297\pi$
$942$	0	0
$943$	6.12774	0.199547
$944$	0	0
$945$	13.7326	0.446721
$946$	0	0
$947$	−8.92463	−0.290012	−0.145006	−	0.989431i	$-0.546320\pi$
$947$	−8.92463	−0.290012	−0.145006	+	0.989431i	$0.546320\pi$
$948$	0	0
$949$	−30.1194	−0.977716
$950$	0	0
$951$	45.5894	1.47834
$952$	0	0
$953$	−5.26383	−0.170512	−0.0852561	−	0.996359i	$-0.527171\pi$
$953$	−5.26383	−0.170512	−0.0852561	+	0.996359i	$0.527171\pi$
$954$	0	0
$955$	17.2462	0.558075
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34.3016	−1.10765
$960$	0	0
$961$	78.7564	2.54053
$962$	0	0
$963$	−4.82862	−0.155600
$964$	0	0
$965$	−2.58574	−0.0832378
$966$	0	0
$967$	−18.5421	−0.596275	−0.298138	−	0.954523i	$-0.596365\pi$
$967$	−18.5421	−0.596275	−0.298138	+	0.954523i	$0.596365\pi$
$968$	0	0
$969$	7.00606	0.225067
$970$	0	0
$971$	24.2230	0.777354	0.388677	−	0.921374i	$-0.372932\pi$
$971$	24.2230	0.777354	0.388677	+	0.921374i	$0.372932\pi$
$972$	0	0
$973$	7.28442	0.233528
$974$	0	0
$975$	5.83428	0.186847
$976$	0	0
$977$	49.0618	1.56963	0.784813	−	0.619732i	$-0.212759\pi$
$977$	49.0618	1.56963	0.784813	+	0.619732i	$0.212759\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.90253	0.156526
$982$	0	0
$983$	−49.1394	−1.56731	−0.783653	−	0.621199i	$-0.786646\pi$
$983$	−49.1394	−1.56731	−0.783653	+	0.621199i	$0.786646\pi$
$984$	0	0
$985$	0.144731	0.00461151
$986$	0	0
$987$	17.4863	0.556594
$988$	0	0
$989$	−1.82788	−0.0581233
$990$	0	0
$991$	−30.9620	−0.983541	−0.491771	−	0.870725i	$-0.663650\pi$
$991$	−30.9620	−0.983541	−0.491771	+	0.870725i	$0.663650\pi$
$992$	0	0
$993$	49.1053	1.55831
$994$	0	0
$995$	−7.54177	−0.239090
$996$	0	0
$997$	−22.8939	−0.725058	−0.362529	−	0.931972i	$-0.618087\pi$
$997$	−22.8939	−0.725058	−0.362529	+	0.931972i	$0.618087\pi$
$998$	0	0
$999$	8.25845	0.261286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.cm.1.1		4
4.3	odd	2		605.2.a.k.1.4		4
11.7	odd	10		880.2.bo.h.401.2		8
11.8	odd	10		880.2.bo.h.801.2		8
11.10	odd	2		9680.2.a.cn.1.1		4
12.11	even	2		5445.2.a.bi.1.1		4
20.19	odd	2		3025.2.a.w.1.1		4
44.3	odd	10		605.2.g.k.251.2		8
44.7	even	10		55.2.g.b.16.1	✓	8
44.15	odd	10		605.2.g.k.511.2		8
44.19	even	10		55.2.g.b.31.1	yes	8
44.27	odd	10		605.2.g.e.366.1		8
44.31	odd	10		605.2.g.e.81.1		8
44.35	even	10		605.2.g.m.81.2		8
44.39	even	10		605.2.g.m.366.2		8
44.43	even	2		605.2.a.j.1.1		4
132.95	odd	10		495.2.n.e.181.2		8
132.107	odd	10		495.2.n.e.361.2		8
132.131	odd	2		5445.2.a.bp.1.4		4
220.7	odd	20		275.2.z.a.49.4		16
220.19	even	10		275.2.h.a.251.2		8
220.63	odd	20		275.2.z.a.174.4		16
220.107	odd	20		275.2.z.a.174.1		16
220.139	even	10		275.2.h.a.126.2		8
220.183	odd	20		275.2.z.a.49.1		16
220.219	even	2		3025.2.a.bd.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.g.b.16.1	✓	8	44.7	even	10
55.2.g.b.31.1	yes	8	44.19	even	10
275.2.h.a.126.2		8	220.139	even	10
275.2.h.a.251.2		8	220.19	even	10
275.2.z.a.49.1		16	220.183	odd	20
275.2.z.a.49.4		16	220.7	odd	20
275.2.z.a.174.1		16	220.107	odd	20
275.2.z.a.174.4		16	220.63	odd	20
495.2.n.e.181.2		8	132.95	odd	10
495.2.n.e.361.2		8	132.107	odd	10
605.2.a.j.1.1		4	44.43	even	2
605.2.a.k.1.4		4	4.3	odd	2
605.2.g.e.81.1		8	44.31	odd	10
605.2.g.e.366.1		8	44.27	odd	10
605.2.g.k.251.2		8	44.3	odd	10
605.2.g.k.511.2		8	44.15	odd	10
605.2.g.m.81.2		8	44.35	even	10
605.2.g.m.366.2		8	44.39	even	10
880.2.bo.h.401.2		8	11.7	odd	10
880.2.bo.h.801.2		8	11.8	odd	10
3025.2.a.w.1.1		4	20.19	odd	2
3025.2.a.bd.1.4		4	220.219	even	2
5445.2.a.bi.1.1		4	12.11	even	2
5445.2.a.bp.1.4		4	132.131	odd	2
9680.2.a.cm.1.1		4	1.1	even	1	trivial
9680.2.a.cn.1.1		4	11.10	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91300 q^{3} -1.00000 q^{5} -3.06719 q^{7} +0.659557 q^{9} +O(q^{10})\)	\(q-1.91300 q^{3} -1.00000 q^{5} -3.06719 q^{7} +0.659557 q^{9} -3.04981 q^{13} +1.91300 q^{15} +0.463845 q^{17} -7.89563 q^{19} +5.86752 q^{21} +1.39026 q^{23} +1.00000 q^{25} +4.47726 q^{27} -3.72162 q^{29} +10.4765 q^{31} +3.06719 q^{35} +1.84453 q^{37} +5.83428 q^{39} +4.40763 q^{41} -1.31478 q^{43} -0.659557 q^{45} +2.98018 q^{47} +2.40763 q^{49} -0.887334 q^{51} +4.18814 q^{53} +15.1043 q^{57} -2.81502 q^{59} +2.01737 q^{61} -2.02298 q^{63} +3.04981 q^{65} +6.75753 q^{67} -2.65956 q^{69} +6.52195 q^{71} +9.87581 q^{73} -1.91300 q^{75} -11.5579 q^{79} -10.5437 q^{81} +8.91861 q^{83} -0.463845 q^{85} +7.11945 q^{87} -6.76978 q^{89} +9.35435 q^{91} -20.0415 q^{93} +7.89563 q^{95} +15.3543 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} - 3 q^{7}+O(q^{10}) \) 4 * q - 4 * q^5 - 3 * q^7	\( 4 q - 4 q^{5} - 3 q^{7} + q^{13} - q^{17} - 20 q^{19} + 10 q^{21} - 5 q^{23} + 4 q^{25} + 15 q^{27} + 12 q^{29} + 5 q^{31} + 3 q^{35} + 7 q^{37} - 7 q^{39} + 11 q^{41} - 19 q^{43} - 5 q^{47} + 3 q^{49} - 7 q^{51} - 11 q^{53} - 5 q^{57} - 9 q^{59} + 12 q^{61} + 5 q^{63} - q^{65} + 19 q^{67} - 8 q^{69} - 5 q^{71} + 11 q^{73} - 34 q^{79} + 4 q^{81} + 11 q^{83} + q^{85} + 19 q^{87} - 8 q^{89} + 8 q^{91} - 5 q^{93} + 20 q^{95} + 32 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 - 3 * q^7 + q^13 - q^17 - 20 * q^19 + 10 * q^21 - 5 * q^23 + 4 * q^25 + 15 * q^27 + 12 * q^29 + 5 * q^31 + 3 * q^35 + 7 * q^37 - 7 * q^39 + 11 * q^41 - 19 * q^43 - 5 * q^47 + 3 * q^49 - 7 * q^51 - 11 * q^53 - 5 * q^57 - 9 * q^59 + 12 * q^61 + 5 * q^63 - q^65 + 19 * q^67 - 8 * q^69 - 5 * q^71 + 11 * q^73 - 34 * q^79 + 4 * q^81 + 11 * q^83 + q^85 + 19 * q^87 - 8 * q^89 + 8 * q^91 - 5 * q^93 + 20 * q^95 + 32 * q^97

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