LMFDB - Embedded newform 9680.2.a.bu.1.2

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 605)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.73205 q^{7} -2.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.73205 q^{7} -2.00000 q^{9} +3.46410 q^{13} -1.00000 q^{15} -6.92820 q^{17} +3.46410 q^{19} +1.73205 q^{21} +1.00000 q^{25} -5.00000 q^{27} +8.00000 q^{31} -1.73205 q^{35} -8.00000 q^{37} +3.46410 q^{39} +12.1244 q^{41} -8.66025 q^{43} +2.00000 q^{45} -9.00000 q^{47} -4.00000 q^{49} -6.92820 q^{51} +6.00000 q^{53} +3.46410 q^{57} +12.0000 q^{59} +8.66025 q^{61} -3.46410 q^{63} -3.46410 q^{65} +5.00000 q^{67} +12.0000 q^{71} +1.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -3.46410 q^{83} +6.92820 q^{85} +3.00000 q^{89} +6.00000 q^{91} +8.00000 q^{93} -3.46410 q^{95} -10.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9} - 2 q^{15} + 2 q^{25} - 10 q^{27} + 16 q^{31} - 16 q^{37} + 4 q^{45} - 18 q^{47} - 8 q^{49} + 12 q^{53} + 24 q^{59} + 10 q^{67} + 24 q^{71} + 2 q^{75} + 2 q^{81} + 6 q^{89} + 12 q^{91} + 16 q^{93} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^9 - 2 * q^15 + 2 * q^25 - 10 * q^27 + 16 * q^31 - 16 * q^37 + 4 * q^45 - 18 * q^47 - 8 * q^49 + 12 * q^53 + 24 * q^59 + 10 * q^67 + 24 * q^71 + 2 * q^75 + 2 * q^81 + 6 * q^89 + 12 * q^91 + 16 * q^93 - 20 * 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
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.73205	0.654654	0.327327	−	0.944911i	$-0.393852\pi$
$7$	1.73205	0.654654	0.327327	+	0.944911i	$0.393852\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−6.92820	−1.68034	−0.840168	−	0.542326i	$-0.817544\pi$
$17$	−6.92820	−1.68034	−0.840168	+	0.542326i	$0.817544\pi$
$18$	0	0
$19$	3.46410	0.794719	0.397360	−	0.917663i	$-0.369927\pi$
$19$	3.46410	0.794719	0.397360	+	0.917663i	$0.369927\pi$
$20$	0	0
$21$	1.73205	0.377964
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	3.46410	0.554700
$40$	0	0
$41$	12.1244	1.89351	0.946753	−	0.321960i	$-0.104342\pi$
$41$	12.1244	1.89351	0.946753	+	0.321960i	$0.104342\pi$
$42$	0	0
$43$	−8.66025	−1.32068	−0.660338	−	0.750968i	$-0.729587\pi$
$43$	−8.66025	−1.32068	−0.660338	+	0.750968i	$0.729587\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−4.00000	−0.571429
$50$	0	0
$51$	−6.92820	−0.970143
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.46410	0.458831
$58$	0	0
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	8.66025	1.10883	0.554416	−	0.832240i	$-0.312942\pi$
$61$	8.66025	1.10883	0.554416	+	0.832240i	$0.312942\pi$
$62$	0	0
$63$	−3.46410	−0.436436
$64$	0	0
$65$	−3.46410	−0.429669
$66$	0	0
$67$	5.00000	0.610847	0.305424	−	0.952217i	$-0.401202\pi$
$67$	5.00000	0.610847	0.305424	+	0.952217i	$0.401202\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	10.3923	1.16923	0.584613	−	0.811312i	$-0.301246\pi$
$79$	10.3923	1.16923	0.584613	+	0.811312i	$0.301246\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.46410	−0.380235	−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410	−0.380235	−0.190117	+	0.981761i	$0.560887\pi$
$84$	0	0
$85$	6.92820	0.751469
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.00000	0.317999	0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−3.46410	−0.355409
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.73205	−0.172345	−0.0861727	−	0.996280i	$-0.527464\pi$
$101$	−1.73205	−0.172345	−0.0861727	+	0.996280i	$0.527464\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	−1.73205	−0.169031
$106$	0	0
$107$	−1.73205	−0.167444	−0.0837218	−	0.996489i	$-0.526681\pi$
$107$	−1.73205	−0.167444	−0.0837218	+	0.996489i	$0.526681\pi$
$108$	0	0
$109$	1.73205	0.165900	0.0829502	−	0.996554i	$-0.473566\pi$
$109$	1.73205	0.165900	0.0829502	+	0.996554i	$0.473566\pi$
$110$	0	0
$111$	−8.00000	−0.759326
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−6.92820	−0.640513
$118$	0	0
$119$	−12.0000	−1.10004
$120$	0	0
$121$	0	0
$122$	0	0
$123$	12.1244	1.09322
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.73205	−0.153695	−0.0768473	−	0.997043i	$-0.524485\pi$
$127$	−1.73205	−0.153695	−0.0768473	+	0.997043i	$0.524485\pi$
$128$	0	0
$129$	−8.66025	−0.762493
$130$	0	0
$131$	3.46410	0.302660	0.151330	−	0.988483i	$-0.451644\pi$
$131$	3.46410	0.302660	0.151330	+	0.988483i	$0.451644\pi$
$132$	0	0
$133$	6.00000	0.520266
$134$	0	0
$135$	5.00000	0.430331
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	−13.8564	−1.17529	−0.587643	−	0.809121i	$-0.699944\pi$
$139$	−13.8564	−1.17529	−0.587643	+	0.809121i	$0.699944\pi$
$140$	0	0
$141$	−9.00000	−0.757937
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.00000	−0.329914
$148$	0	0
$149$	−19.0526	−1.56085	−0.780423	−	0.625252i	$-0.784996\pi$
$149$	−19.0526	−1.56085	−0.780423	+	0.625252i	$0.784996\pi$
$150$	0	0
$151$	20.7846	1.69143	0.845714	−	0.533637i	$-0.179175\pi$
$151$	20.7846	1.69143	0.845714	+	0.533637i	$0.179175\pi$
$152$	0	0
$153$	13.8564	1.12022
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.0000	1.48819	0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000	1.48819	0.744097	+	0.668071i	$0.232880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.19615	0.402090	0.201045	−	0.979582i	$-0.435566\pi$
$167$	5.19615	0.402090	0.201045	+	0.979582i	$0.435566\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−6.92820	−0.529813
$172$	0	0
$173$	10.3923	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$173$	10.3923	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$174$	0	0
$175$	1.73205	0.130931
$176$	0	0
$177$	12.0000	0.901975
$178$	0	0
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	0	0
$181$	11.0000	0.817624	0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000	0.817624	0.408812	+	0.912619i	$0.365943\pi$
$182$	0	0
$183$	8.66025	0.640184
$184$	0	0
$185$	8.00000	0.588172
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−8.66025	−0.629941
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	−3.46410	−0.249351	−0.124676	−	0.992198i	$-0.539789\pi$
$193$	−3.46410	−0.249351	−0.124676	+	0.992198i	$0.539789\pi$
$194$	0	0
$195$	−3.46410	−0.248069
$196$	0	0
$197$	−10.3923	−0.740421	−0.370211	−	0.928948i	$-0.620714\pi$
$197$	−10.3923	−0.740421	−0.370211	+	0.928948i	$0.620714\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	0	0
$201$	5.00000	0.352673
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.1244	−0.846802
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	8.66025	0.590624
$216$	0	0
$217$	13.8564	0.940634
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−24.0000	−1.61441
$222$	0	0
$223$	19.0000	1.27233	0.636167	−	0.771551i	$-0.280519\pi$
$223$	19.0000	1.27233	0.636167	+	0.771551i	$0.280519\pi$
$224$	0	0
$225$	−2.00000	−0.133333
$226$	0	0
$227$	−19.0526	−1.26456	−0.632281	−	0.774739i	$-0.717881\pi$
$227$	−19.0526	−1.26456	−0.632281	+	0.774739i	$0.717881\pi$
$228$	0	0
$229$	7.00000	0.462573	0.231287	−	0.972886i	$-0.425707\pi$
$229$	7.00000	0.462573	0.231287	+	0.972886i	$0.425707\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.46410	−0.226941	−0.113470	−	0.993541i	$-0.536197\pi$
$233$	−3.46410	−0.226941	−0.113470	+	0.993541i	$0.536197\pi$
$234$	0	0
$235$	9.00000	0.587095
$236$	0	0
$237$	10.3923	0.675053
$238$	0	0
$239$	−3.46410	−0.224074	−0.112037	−	0.993704i	$-0.535738\pi$
$239$	−3.46410	−0.224074	−0.112037	+	0.993704i	$0.535738\pi$
$240$	0	0
$241$	−19.0526	−1.22728	−0.613642	−	0.789585i	$-0.710296\pi$
$241$	−19.0526	−1.22728	−0.613642	+	0.789585i	$0.710296\pi$
$242$	0	0
$243$	16.0000	1.02640
$244$	0	0
$245$	4.00000	0.255551
$246$	0	0
$247$	12.0000	0.763542
$248$	0	0
$249$	−3.46410	−0.219529
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	6.92820	0.433861
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−13.8564	−0.860995
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−24.2487	−1.49524	−0.747620	−	0.664127i	$-0.768803\pi$
$263$	−24.2487	−1.49524	−0.747620	+	0.664127i	$0.768803\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	3.00000	0.183597
$268$	0	0
$269$	−15.0000	−0.914566	−0.457283	−	0.889321i	$-0.651177\pi$
$269$	−15.0000	−0.914566	−0.457283	+	0.889321i	$0.651177\pi$
$270$	0	0
$271$	13.8564	0.841717	0.420858	−	0.907126i	$-0.361729\pi$
$271$	13.8564	0.841717	0.420858	+	0.907126i	$0.361729\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	0	0
$276$	0	0
$277$	10.3923	0.624413	0.312207	−	0.950014i	$-0.398932\pi$
$277$	10.3923	0.624413	0.312207	+	0.950014i	$0.398932\pi$
$278$	0	0
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−6.92820	−0.413302	−0.206651	−	0.978415i	$-0.566256\pi$
$281$	−6.92820	−0.413302	−0.206651	+	0.978415i	$0.566256\pi$
$282$	0	0
$283$	5.19615	0.308879	0.154440	−	0.988002i	$-0.450643\pi$
$283$	5.19615	0.308879	0.154440	+	0.988002i	$0.450643\pi$
$284$	0	0
$285$	−3.46410	−0.205196
$286$	0	0
$287$	21.0000	1.23959
$288$	0	0
$289$	31.0000	1.82353
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−6.92820	−0.404750	−0.202375	−	0.979308i	$-0.564866\pi$
$293$	−6.92820	−0.404750	−0.202375	+	0.979308i	$0.564866\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−15.0000	−0.864586
$302$	0	0
$303$	−1.73205	−0.0995037
$304$	0	0
$305$	−8.66025	−0.495885
$306$	0	0
$307$	10.3923	0.593120	0.296560	−	0.955014i	$-0.404160\pi$
$307$	10.3923	0.593120	0.296560	+	0.955014i	$0.404160\pi$
$308$	0	0
$309$	4.00000	0.227552
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	3.46410	0.195180
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−1.73205	−0.0966736
$322$	0	0
$323$	−24.0000	−1.33540
$324$	0	0
$325$	3.46410	0.192154
$326$	0	0
$327$	1.73205	0.0957826
$328$	0	0
$329$	−15.5885	−0.859419
$330$	0	0
$331$	34.0000	1.86881	0.934405	−	0.356214i	$-0.115932\pi$
$331$	34.0000	1.86881	0.934405	+	0.356214i	$0.115932\pi$
$332$	0	0
$333$	16.0000	0.876795
$334$	0	0
$335$	−5.00000	−0.273179
$336$	0	0
$337$	10.3923	0.566105	0.283052	−	0.959104i	$-0.408653\pi$
$337$	10.3923	0.566105	0.283052	+	0.959104i	$0.408653\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−19.0526	−1.02874
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.1244	0.650870	0.325435	−	0.945564i	$-0.394489\pi$
$347$	12.1244	0.650870	0.325435	+	0.945564i	$0.394489\pi$
$348$	0	0
$349$	34.6410	1.85429	0.927146	−	0.374701i	$-0.122255\pi$
$349$	34.6410	1.85429	0.927146	+	0.374701i	$0.122255\pi$
$350$	0	0
$351$	−17.3205	−0.924500
$352$	0	0
$353$	24.0000	1.27739	0.638696	−	0.769460i	$-0.279474\pi$
$353$	24.0000	1.27739	0.638696	+	0.769460i	$0.279474\pi$
$354$	0	0
$355$	−12.0000	−0.636894
$356$	0	0
$357$	−12.0000	−0.635107
$358$	0	0
$359$	3.46410	0.182828	0.0914141	−	0.995813i	$-0.470861\pi$
$359$	3.46410	0.182828	0.0914141	+	0.995813i	$0.470861\pi$
$360$	0	0
$361$	−7.00000	−0.368421
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−17.0000	−0.887393	−0.443696	−	0.896177i	$-0.646333\pi$
$367$	−17.0000	−0.887393	−0.443696	+	0.896177i	$0.646333\pi$
$368$	0	0
$369$	−24.2487	−1.26234
$370$	0	0
$371$	10.3923	0.539542
$372$	0	0
$373$	−13.8564	−0.717458	−0.358729	−	0.933442i	$-0.616790\pi$
$373$	−13.8564	−0.717458	−0.358729	+	0.933442i	$0.616790\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	−1.73205	−0.0887357
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.3205	0.880451
$388$	0	0
$389$	3.00000	0.152106	0.0760530	−	0.997104i	$-0.475768\pi$
$389$	3.00000	0.152106	0.0760530	+	0.997104i	$0.475768\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	3.46410	0.174741
$394$	0	0
$395$	−10.3923	−0.522894
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	0	0
$399$	6.00000	0.300376
$400$	0	0
$401$	27.0000	1.34832	0.674158	−	0.738587i	$-0.264507\pi$
$401$	27.0000	1.34832	0.674158	+	0.738587i	$0.264507\pi$
$402$	0	0
$403$	27.7128	1.38047
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	5.19615	0.256933	0.128467	−	0.991714i	$-0.458994\pi$
$409$	5.19615	0.256933	0.128467	+	0.991714i	$0.458994\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	20.7846	1.02274
$414$	0	0
$415$	3.46410	0.170046
$416$	0	0
$417$	−13.8564	−0.678551
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	0	0
$421$	−17.0000	−0.828529	−0.414265	−	0.910156i	$-0.635961\pi$
$421$	−17.0000	−0.828529	−0.414265	+	0.910156i	$0.635961\pi$
$422$	0	0
$423$	18.0000	0.875190
$424$	0	0
$425$	−6.92820	−0.336067
$426$	0	0
$427$	15.0000	0.725901
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−38.1051	−1.83546	−0.917729	−	0.397206i	$-0.869980\pi$
$431$	−38.1051	−1.83546	−0.917729	+	0.397206i	$0.869980\pi$
$432$	0	0
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	27.7128	1.32266	0.661330	−	0.750095i	$-0.269992\pi$
$439$	27.7128	1.32266	0.661330	+	0.750095i	$0.269992\pi$
$440$	0	0
$441$	8.00000	0.380952
$442$	0	0
$443$	15.0000	0.712672	0.356336	−	0.934358i	$-0.384026\pi$
$443$	15.0000	0.712672	0.356336	+	0.934358i	$0.384026\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	0	0
$447$	−19.0526	−0.901155
$448$	0	0
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	20.7846	0.976546
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	20.7846	0.972263	0.486132	−	0.873886i	$-0.338408\pi$
$457$	20.7846	0.972263	0.486132	+	0.873886i	$0.338408\pi$
$458$	0	0
$459$	34.6410	1.61690
$460$	0	0
$461$	−12.1244	−0.564688	−0.282344	−	0.959313i	$-0.591112\pi$
$461$	−12.1244	−0.564688	−0.282344	+	0.959313i	$0.591112\pi$
$462$	0	0
$463$	−41.0000	−1.90543	−0.952716	−	0.303863i	$-0.901724\pi$
$463$	−41.0000	−1.90543	−0.952716	+	0.303863i	$0.901724\pi$
$464$	0	0
$465$	−8.00000	−0.370991
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	8.66025	0.399893
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	0	0
$474$	0	0
$475$	3.46410	0.158944
$476$	0	0
$477$	−12.0000	−0.549442
$478$	0	0
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	−27.7128	−1.26360
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.0000	0.454077
$486$	0	0
$487$	20.0000	0.906287	0.453143	−	0.891438i	$-0.350303\pi$
$487$	20.0000	0.906287	0.453143	+	0.891438i	$0.350303\pi$
$488$	0	0
$489$	19.0000	0.859210
$490$	0	0
$491$	−24.2487	−1.09433	−0.547165	−	0.837025i	$-0.684293\pi$
$491$	−24.2487	−1.09433	−0.547165	+	0.837025i	$0.684293\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.7846	0.932317
$498$	0	0
$499$	38.0000	1.70111	0.850557	−	0.525883i	$-0.176265\pi$
$499$	38.0000	1.70111	0.850557	+	0.525883i	$0.176265\pi$
$500$	0	0
$501$	5.19615	0.232147
$502$	0	0
$503$	−8.66025	−0.386142	−0.193071	−	0.981185i	$-0.561845\pi$
$503$	−8.66025	−0.386142	−0.193071	+	0.981185i	$0.561845\pi$
$504$	0	0
$505$	1.73205	0.0770752
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−17.3205	−0.764719
$514$	0	0
$515$	−4.00000	−0.176261
$516$	0	0
$517$	0	0
$518$	0	0
$519$	10.3923	0.456172
$520$	0	0
$521$	15.0000	0.657162	0.328581	−	0.944476i	$-0.393430\pi$
$521$	15.0000	0.657162	0.328581	+	0.944476i	$0.393430\pi$
$522$	0	0
$523$	17.3205	0.757373	0.378686	−	0.925525i	$-0.376376\pi$
$523$	17.3205	0.757373	0.378686	+	0.925525i	$0.376376\pi$
$524$	0	0
$525$	1.73205	0.0755929
$526$	0	0
$527$	−55.4256	−2.41438
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−24.0000	−1.04151
$532$	0	0
$533$	42.0000	1.81922
$534$	0	0
$535$	1.73205	0.0748831
$536$	0	0
$537$	18.0000	0.776757
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−5.19615	−0.223400	−0.111700	−	0.993742i	$-0.535630\pi$
$541$	−5.19615	−0.223400	−0.111700	+	0.993742i	$0.535630\pi$
$542$	0	0
$543$	11.0000	0.472055
$544$	0	0
$545$	−1.73205	−0.0741929
$546$	0	0
$547$	45.0333	1.92549	0.962743	−	0.270418i	$-0.0871621\pi$
$547$	45.0333	1.92549	0.962743	+	0.270418i	$0.0871621\pi$
$548$	0	0
$549$	−17.3205	−0.739221
$550$	0	0
$551$	0	0
$552$	0	0
$553$	18.0000	0.765438
$554$	0	0
$555$	8.00000	0.339581
$556$	0	0
$557$	−6.92820	−0.293557	−0.146779	−	0.989169i	$-0.546891\pi$
$557$	−6.92820	−0.293557	−0.146779	+	0.989169i	$0.546891\pi$
$558$	0	0
$559$	−30.0000	−1.26886
$560$	0	0
$561$	0	0
$562$	0	0
$563$	15.5885	0.656975	0.328488	−	0.944508i	$-0.393461\pi$
$563$	15.5885	0.656975	0.328488	+	0.944508i	$0.393461\pi$
$564$	0	0
$565$	6.00000	0.252422
$566$	0	0
$567$	1.73205	0.0727393
$568$	0	0
$569$	−1.73205	−0.0726113	−0.0363057	−	0.999341i	$-0.511559\pi$
$569$	−1.73205	−0.0726113	−0.0363057	+	0.999341i	$0.511559\pi$
$570$	0	0
$571$	3.46410	0.144968	0.0724841	−	0.997370i	$-0.476907\pi$
$571$	3.46410	0.144968	0.0724841	+	0.997370i	$0.476907\pi$
$572$	0	0
$573$	6.00000	0.250654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−20.0000	−0.832611	−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000	−0.832611	−0.416305	+	0.909225i	$0.636675\pi$
$578$	0	0
$579$	−3.46410	−0.143963
$580$	0	0
$581$	−6.00000	−0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	6.92820	0.286446
$586$	0	0
$587$	−21.0000	−0.866763	−0.433381	−	0.901211i	$-0.642680\pi$
$587$	−21.0000	−0.866763	−0.433381	+	0.901211i	$0.642680\pi$
$588$	0	0
$589$	27.7128	1.14189
$590$	0	0
$591$	−10.3923	−0.427482
$592$	0	0
$593$	24.2487	0.995775	0.497888	−	0.867242i	$-0.334109\pi$
$593$	24.2487	0.995775	0.497888	+	0.867242i	$0.334109\pi$
$594$	0	0
$595$	12.0000	0.491952
$596$	0	0
$597$	14.0000	0.572982
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	0	0
$601$	20.7846	0.847822	0.423911	−	0.905704i	$-0.360657\pi$
$601$	20.7846	0.847822	0.423911	+	0.905704i	$0.360657\pi$
$602$	0	0
$603$	−10.0000	−0.407231
$604$	0	0
$605$	0	0
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−31.1769	−1.26128
$612$	0	0
$613$	−10.3923	−0.419741	−0.209871	−	0.977729i	$-0.567304\pi$
$613$	−10.3923	−0.419741	−0.209871	+	0.977729i	$0.567304\pi$
$614$	0	0
$615$	−12.1244	−0.488901
$616$	0	0
$617$	−36.0000	−1.44931	−0.724653	−	0.689114i	$-0.758000\pi$
$617$	−36.0000	−1.44931	−0.724653	+	0.689114i	$0.758000\pi$
$618$	0	0
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	5.19615	0.208179
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	55.4256	2.20996
$630$	0	0
$631$	−28.0000	−1.11466	−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000	−1.11466	−0.557331	+	0.830290i	$0.688175\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.73205	0.0687343
$636$	0	0
$637$	−13.8564	−0.549011
$638$	0	0
$639$	−24.0000	−0.949425
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	31.0000	1.22252	0.611260	−	0.791430i	$-0.290663\pi$
$643$	31.0000	1.22252	0.611260	+	0.791430i	$0.290663\pi$
$644$	0	0
$645$	8.66025	0.340997
$646$	0	0
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	13.8564	0.543075
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	−3.46410	−0.135354
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−3.46410	−0.134942	−0.0674711	−	0.997721i	$-0.521493\pi$
$659$	−3.46410	−0.134942	−0.0674711	+	0.997721i	$0.521493\pi$
$660$	0	0
$661$	37.0000	1.43913	0.719567	−	0.694423i	$-0.244340\pi$
$661$	37.0000	1.43913	0.719567	+	0.694423i	$0.244340\pi$
$662$	0	0
$663$	−24.0000	−0.932083
$664$	0	0
$665$	−6.00000	−0.232670
$666$	0	0
$667$	0	0
$668$	0	0
$669$	19.0000	0.734582
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−20.7846	−0.801188	−0.400594	−	0.916256i	$-0.631196\pi$
$673$	−20.7846	−0.801188	−0.400594	+	0.916256i	$0.631196\pi$
$674$	0	0
$675$	−5.00000	−0.192450
$676$	0	0
$677$	27.7128	1.06509	0.532545	−	0.846402i	$-0.321236\pi$
$677$	27.7128	1.06509	0.532545	+	0.846402i	$0.321236\pi$
$678$	0	0
$679$	−17.3205	−0.664700
$680$	0	0
$681$	−19.0526	−0.730096
$682$	0	0
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	0	0
$685$	−18.0000	−0.687745
$686$	0	0
$687$	7.00000	0.267067
$688$	0	0
$689$	20.7846	0.791831
$690$	0	0
$691$	10.0000	0.380418	0.190209	−	0.981744i	$-0.439083\pi$
$691$	10.0000	0.380418	0.190209	+	0.981744i	$0.439083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.8564	0.525603
$696$	0	0
$697$	−84.0000	−3.18173
$698$	0	0
$699$	−3.46410	−0.131024
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	−27.7128	−1.04521
$704$	0	0
$705$	9.00000	0.338960
$706$	0	0
$707$	−3.00000	−0.112827
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	−20.7846	−0.779484
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.46410	−0.129369
$718$	0	0
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	0	0
$721$	6.92820	0.258020
$722$	0	0
$723$	−19.0526	−0.708572
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.00000	−0.0370879	−0.0185440	−	0.999828i	$-0.505903\pi$
$727$	−1.00000	−0.0370879	−0.0185440	+	0.999828i	$0.505903\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	60.0000	2.21918
$732$	0	0
$733$	−20.7846	−0.767697	−0.383849	−	0.923396i	$-0.625402\pi$
$733$	−20.7846	−0.767697	−0.383849	+	0.923396i	$0.625402\pi$
$734$	0	0
$735$	4.00000	0.147542
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−3.46410	−0.127429	−0.0637145	−	0.997968i	$-0.520295\pi$
$739$	−3.46410	−0.127429	−0.0637145	+	0.997968i	$0.520295\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	−5.19615	−0.190628	−0.0953142	−	0.995447i	$-0.530386\pi$
$743$	−5.19615	−0.190628	−0.0953142	+	0.995447i	$0.530386\pi$
$744$	0	0
$745$	19.0526	0.698032
$746$	0	0
$747$	6.92820	0.253490
$748$	0	0
$749$	−3.00000	−0.109618
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−20.7846	−0.756429
$756$	0	0
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	34.6410	1.25574	0.627868	−	0.778320i	$-0.283928\pi$
$761$	34.6410	1.25574	0.627868	+	0.778320i	$0.283928\pi$
$762$	0	0
$763$	3.00000	0.108607
$764$	0	0
$765$	−13.8564	−0.500979
$766$	0	0
$767$	41.5692	1.50098
$768$	0	0
$769$	27.7128	0.999350	0.499675	−	0.866213i	$-0.333453\pi$
$769$	27.7128	0.999350	0.499675	+	0.866213i	$0.333453\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	8.00000	0.287368
$776$	0	0
$777$	−13.8564	−0.497096
$778$	0	0
$779$	42.0000	1.50481
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.00000	−0.142766
$786$	0	0
$787$	25.9808	0.926114	0.463057	−	0.886328i	$-0.346752\pi$
$787$	25.9808	0.926114	0.463057	+	0.886328i	$0.346752\pi$
$788$	0	0
$789$	−24.2487	−0.863277
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	30.0000	1.06533
$794$	0	0
$795$	−6.00000	−0.212798
$796$	0	0
$797$	−12.0000	−0.425062	−0.212531	−	0.977154i	$-0.568171\pi$
$797$	−12.0000	−0.425062	−0.212531	+	0.977154i	$0.568171\pi$
$798$	0	0
$799$	62.3538	2.20592
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−15.0000	−0.528025
$808$	0	0
$809$	−48.4974	−1.70508	−0.852539	−	0.522663i	$-0.824939\pi$
$809$	−48.4974	−1.70508	−0.852539	+	0.522663i	$0.824939\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	0	0
$813$	13.8564	0.485965
$814$	0	0
$815$	−19.0000	−0.665541
$816$	0	0
$817$	−30.0000	−1.04957
$818$	0	0
$819$	−12.0000	−0.419314
$820$	0	0
$821$	29.4449	1.02763	0.513816	−	0.857900i	$-0.328231\pi$
$821$	29.4449	1.02763	0.513816	+	0.857900i	$0.328231\pi$
$822$	0	0
$823$	−17.0000	−0.592583	−0.296291	−	0.955098i	$-0.595750\pi$
$823$	−17.0000	−0.592583	−0.296291	+	0.955098i	$0.595750\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−29.4449	−1.02390	−0.511949	−	0.859016i	$-0.671076\pi$
$827$	−29.4449	−1.02390	−0.511949	+	0.859016i	$0.671076\pi$
$828$	0	0
$829$	29.0000	1.00721	0.503606	−	0.863934i	$-0.332006\pi$
$829$	29.0000	1.00721	0.503606	+	0.863934i	$0.332006\pi$
$830$	0	0
$831$	10.3923	0.360505
$832$	0	0
$833$	27.7128	0.960192
$834$	0	0
$835$	−5.19615	−0.179820
$836$	0	0
$837$	−40.0000	−1.38260
$838$	0	0
$839$	42.0000	1.45000	0.725001	−	0.688748i	$-0.241839\pi$
$839$	42.0000	1.45000	0.725001	+	0.688748i	$0.241839\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	−6.92820	−0.238620
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	0	0
$848$	0	0
$849$	5.19615	0.178331
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−34.6410	−1.18609	−0.593043	−	0.805171i	$-0.702074\pi$
$853$	−34.6410	−1.18609	−0.593043	+	0.805171i	$0.702074\pi$
$854$	0	0
$855$	6.92820	0.236940
$856$	0	0
$857$	−10.3923	−0.354994	−0.177497	−	0.984121i	$-0.556800\pi$
$857$	−10.3923	−0.354994	−0.177497	+	0.984121i	$0.556800\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	21.0000	0.715678
$862$	0	0
$863$	−21.0000	−0.714848	−0.357424	−	0.933942i	$-0.616345\pi$
$863$	−21.0000	−0.714848	−0.357424	+	0.933942i	$0.616345\pi$
$864$	0	0
$865$	−10.3923	−0.353349
$866$	0	0
$867$	31.0000	1.05282
$868$	0	0
$869$	0	0
$870$	0	0
$871$	17.3205	0.586883
$872$	0	0
$873$	20.0000	0.676897
$874$	0	0
$875$	−1.73205	−0.0585540
$876$	0	0
$877$	−27.7128	−0.935795	−0.467898	−	0.883783i	$-0.654988\pi$
$877$	−27.7128	−0.935795	−0.467898	+	0.883783i	$0.654988\pi$
$878$	0	0
$879$	−6.92820	−0.233682
$880$	0	0
$881$	−21.0000	−0.707508	−0.353754	−	0.935339i	$-0.615095\pi$
$881$	−21.0000	−0.707508	−0.353754	+	0.935339i	$0.615095\pi$
$882$	0	0
$883$	56.0000	1.88455	0.942275	−	0.334840i	$-0.108682\pi$
$883$	56.0000	1.88455	0.942275	+	0.334840i	$0.108682\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	32.9090	1.10497	0.552487	−	0.833521i	$-0.313679\pi$
$887$	32.9090	1.10497	0.552487	+	0.833521i	$0.313679\pi$
$888$	0	0
$889$	−3.00000	−0.100617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−31.1769	−1.04330
$894$	0	0
$895$	−18.0000	−0.601674
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−41.5692	−1.38487
$902$	0	0
$903$	−15.0000	−0.499169
$904$	0	0
$905$	−11.0000	−0.365652
$906$	0	0
$907$	−17.0000	−0.564476	−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000	−0.564476	−0.282238	+	0.959344i	$0.591077\pi$
$908$	0	0
$909$	3.46410	0.114897
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−8.66025	−0.286299
$916$	0	0
$917$	6.00000	0.198137
$918$	0	0
$919$	−38.1051	−1.25697	−0.628486	−	0.777821i	$-0.716325\pi$
$919$	−38.1051	−1.25697	−0.628486	+	0.777821i	$0.716325\pi$
$920$	0	0
$921$	10.3923	0.342438
$922$	0	0
$923$	41.5692	1.36827
$924$	0	0
$925$	−8.00000	−0.263038
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−13.8564	−0.454125
$932$	0	0
$933$	6.00000	0.196431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−24.2487	−0.792171	−0.396085	−	0.918214i	$-0.629632\pi$
$937$	−24.2487	−0.792171	−0.396085	+	0.918214i	$0.629632\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	−50.2295	−1.63743	−0.818717	−	0.574197i	$-0.805314\pi$
$941$	−50.2295	−1.63743	−0.818717	+	0.574197i	$0.805314\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	8.66025	0.281718
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	55.4256	1.79541	0.897706	−	0.440595i	$-0.145232\pi$
$953$	55.4256	1.79541	0.897706	+	0.440595i	$0.145232\pi$
$954$	0	0
$955$	−6.00000	−0.194155
$956$	0	0
$957$	0	0
$958$	0	0
$959$	31.1769	1.00676
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	3.46410	0.111629
$964$	0	0
$965$	3.46410	0.111513
$966$	0	0
$967$	−38.1051	−1.22538	−0.612689	−	0.790324i	$-0.709912\pi$
$967$	−38.1051	−1.22538	−0.612689	+	0.790324i	$0.709912\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	0	0
$975$	3.46410	0.110940
$976$	0	0
$977$	−24.0000	−0.767828	−0.383914	−	0.923369i	$-0.625424\pi$
$977$	−24.0000	−0.767828	−0.383914	+	0.923369i	$0.625424\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−3.46410	−0.110600
$982$	0	0
$983$	3.00000	0.0956851	0.0478426	−	0.998855i	$-0.484765\pi$
$983$	3.00000	0.0956851	0.0478426	+	0.998855i	$0.484765\pi$
$984$	0	0
$985$	10.3923	0.331126
$986$	0	0
$987$	−15.5885	−0.496186
$988$	0	0
$989$	0	0
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	0	0
$993$	34.0000	1.07896
$994$	0	0
$995$	−14.0000	−0.443830
$996$	0	0
$997$	−45.0333	−1.42622	−0.713110	−	0.701052i	$-0.752714\pi$
$997$	−45.0333	−1.42622	−0.713110	+	0.701052i	$0.752714\pi$
$998$	0	0
$999$	40.0000	1.26554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9680.2.a.bu.1.2		2
4.3	odd	2		605.2.a.e.1.2	yes	2
11.10	odd	2	inner	9680.2.a.bu.1.1		2
12.11	even	2		5445.2.a.u.1.1		2
20.19	odd	2		3025.2.a.l.1.1		2
44.3	odd	10		605.2.g.i.251.2		8
44.7	even	10		605.2.g.i.511.1		8
44.15	odd	10		605.2.g.i.511.2		8
44.19	even	10		605.2.g.i.251.1		8
44.27	odd	10		605.2.g.i.366.1		8
44.31	odd	10		605.2.g.i.81.1		8
44.35	even	10		605.2.g.i.81.2		8
44.39	even	10		605.2.g.i.366.2		8
44.43	even	2		605.2.a.e.1.1	✓	2
132.131	odd	2		5445.2.a.u.1.2		2
220.219	even	2		3025.2.a.l.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
605.2.a.e.1.1	✓	2	44.43	even	2
605.2.a.e.1.2	yes	2	4.3	odd	2
605.2.g.i.81.1		8	44.31	odd	10
605.2.g.i.81.2		8	44.35	even	10
605.2.g.i.251.1		8	44.19	even	10
605.2.g.i.251.2		8	44.3	odd	10
605.2.g.i.366.1		8	44.27	odd	10
605.2.g.i.366.2		8	44.39	even	10
605.2.g.i.511.1		8	44.7	even	10
605.2.g.i.511.2		8	44.15	odd	10
3025.2.a.l.1.1		2	20.19	odd	2
3025.2.a.l.1.2		2	220.219	even	2
5445.2.a.u.1.1		2	12.11	even	2
5445.2.a.u.1.2		2	132.131	odd	2
9680.2.a.bu.1.1		2	11.10	odd	2	inner
9680.2.a.bu.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.73205 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.73205 q^{7} -2.00000 q^{9} +3.46410 q^{13} -1.00000 q^{15} -6.92820 q^{17} +3.46410 q^{19} +1.73205 q^{21} +1.00000 q^{25} -5.00000 q^{27} +8.00000 q^{31} -1.73205 q^{35} -8.00000 q^{37} +3.46410 q^{39} +12.1244 q^{41} -8.66025 q^{43} +2.00000 q^{45} -9.00000 q^{47} -4.00000 q^{49} -6.92820 q^{51} +6.00000 q^{53} +3.46410 q^{57} +12.0000 q^{59} +8.66025 q^{61} -3.46410 q^{63} -3.46410 q^{65} +5.00000 q^{67} +12.0000 q^{71} +1.00000 q^{75} +10.3923 q^{79} +1.00000 q^{81} -3.46410 q^{83} +6.92820 q^{85} +3.00000 q^{89} +6.00000 q^{91} +8.00000 q^{93} -3.46410 q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 4 q^{9} - 2 q^{15} + 2 q^{25} - 10 q^{27} + 16 q^{31} - 16 q^{37} + 4 q^{45} - 18 q^{47} - 8 q^{49} + 12 q^{53} + 24 q^{59} + 10 q^{67} + 24 q^{71} + 2 q^{75} + 2 q^{81} + 6 q^{89} + 12 q^{91} + 16 q^{93} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 4 * q^9 - 2 * q^15 + 2 * q^25 - 10 * q^27 + 16 * q^31 - 16 * q^37 + 4 * q^45 - 18 * q^47 - 8 * q^49 + 12 * q^53 + 24 * q^59 + 10 * q^67 + 24 * q^71 + 2 * q^75 + 2 * q^81 + 6 * q^89 + 12 * q^91 + 16 * q^93 - 20 * q^97

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