LMFDB - Embedded newform 980.2.q.e.949.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(569,980)] 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("980.569"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			949.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	980.949
Dual form			980.2.q.e.569.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.23205 + 1.86603i) q^{5} +(-1.50000 - 2.59808i) q^{9} +4.00000i q^{13} +(3.46410 + 2.00000i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-6.92820 + 4.00000i) q^{23} +(-1.96410 - 4.59808i) q^{25} -2.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-6.92820 + 4.00000i) q^{37} -6.00000 q^{41} -8.00000i q^{43} +(6.69615 + 0.401924i) q^{45} +(-6.92820 + 4.00000i) q^{47} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-7.46410 - 4.92820i) q^{65} +(-6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(3.46410 + 2.00000i) q^{73} +(-2.00000 - 3.46410i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(5.00000 + 8.66025i) q^{89} +(8.92820 + 0.535898i) q^{95} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) $ 4 * q + 2 * q^5 - 6 * q^9 - 8 * q^19 + 6 * q^25 - 8 * q^29 - 16 * q^31 - 24 * q^41 + 6 * q^45 + 8 * q^59 - 12 * q^61 - 16 * q^65 + 48 * q^71 - 8 * q^79 - 18 * q^81 - 32 * q^85 + 20 * q^89 + 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$		0			0
$5$	−1.23205	+	1.86603i	−0.550990	+	0.834512i
$5$	−1.23205	+	1.86603i	−0.550990	+	0.834512i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−1.50000	−	2.59808i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.46410	+	2.00000i	0.840168	+	0.485071i	0.857321	−	0.514782i	$-0.172127\pi$
$17$	3.46410	+	2.00000i	0.840168	+	0.485071i	−0.0171533	+	0.999853i	$0.505460\pi$
$18$		0			0
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	0.540068	−	0.841621i	$-0.318398\pi$
$19$	−2.00000	−	3.46410i	−0.458831	−	0.794719i	−0.998899	+	0.0469020i	$0.985065\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.92820	+	4.00000i	−1.44463	+	0.834058i	−0.998154	−	0.0607377i	$-0.980655\pi$
$23$	−6.92820	+	4.00000i	−1.44463	+	0.834058i	−0.446476	+	0.894795i	$0.647321\pi$
$24$		0			0
$25$	−1.96410	−	4.59808i	−0.392820	−	0.919615i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	0.243204	+	0.969975i	$0.421802\pi$
$31$	−4.00000	+	6.92820i	−0.718421	+	1.24434i	−0.961625	+	0.274367i	$0.911532\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6.92820	+	4.00000i	−1.13899	+	0.657596i	−0.946180	−	0.323640i	$-0.895093\pi$
$37$	−6.92820	+	4.00000i	−1.13899	+	0.657596i	−0.192809	+	0.981236i	$0.561760\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$		−	8.00000i		−	1.21999i	−0.792406	−	0.609994i	$-0.791172\pi$
$43$		−	8.00000i		−	1.21999i	0.792406	−	0.609994i	$-0.208828\pi$
$44$		0			0
$45$	6.69615	+	0.401924i	0.998203	+	0.0599153i
$46$		0			0
$47$	−6.92820	+	4.00000i	−1.01058	+	0.583460i	−0.911362	−	0.411606i	$-0.864968\pi$
$47$	−6.92820	+	4.00000i	−1.01058	+	0.583460i	−0.0992202	+	0.995066i	$0.531635\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	−0.705965	−	0.708247i	$-0.749486\pi$
$59$	2.00000	−	3.46410i	0.260378	−	0.450988i	0.966342	+	0.257260i	$0.0828195\pi$
$60$		0			0
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	0.607535	−	0.794293i	$-0.292159\pi$
$61$	−3.00000	−	5.19615i	−0.384111	−	0.665299i	−0.991645	+	0.128994i	$0.958825\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.46410	−	4.92820i	−0.925808	−	0.611268i
$66$		0			0
$67$	−6.92820	−	4.00000i	−0.846415	−	0.488678i	0.0130248	−	0.999915i	$-0.495854\pi$
$67$	−6.92820	−	4.00000i	−0.846415	−	0.488678i	−0.859440	+	0.511237i	$0.829187\pi$
$68$		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
$73$	3.46410	+	2.00000i	0.405442	+	0.234082i	0.688830	−	0.724923i	$-0.258125\pi$
$73$	3.46410	+	2.00000i	0.405442	+	0.234082i	−0.283387	+	0.959006i	$0.591458\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	0.731307	−	0.682048i	$-0.238911\pi$
$79$	−2.00000	−	3.46410i	−0.225018	−	0.389742i	−0.956325	+	0.292306i	$0.905577\pi$
$80$		0			0
$81$	−4.50000	+	7.79423i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	−8.00000	+	4.00000i	−0.867722	+	0.433861i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	0.999388	+	0.0349934i	$0.0111410\pi$
$89$	5.00000	+	8.66025i	0.529999	+	0.917985i	−0.469389	+	0.882992i	$0.655526\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	8.92820	+	0.535898i	0.916014	+	0.0549820i
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.0623905	+	0.998052i	$0.519872\pi$
$101$	−9.00000	+	15.5885i	−0.895533	+	1.55111i	−0.833143	+	0.553058i	$0.813461\pi$
$102$		0			0
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	−0.800855	−	0.598858i	$-0.795621\pi$
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	0.118199	+	0.992990i	$0.462288\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.92820	−	4.00000i	0.669775	−	0.386695i	−0.126217	−	0.992003i	$-0.540283\pi$
$107$	6.92820	−	4.00000i	0.669775	−	0.386695i	0.795991	+	0.605308i	$0.206950\pi$
$108$		0			0
$109$	7.00000	−	12.1244i	0.670478	−	1.16130i	−0.307290	−	0.951616i	$-0.599422\pi$
$109$	7.00000	−	12.1244i	0.670478	−	1.16130i	0.977769	−	0.209687i	$-0.0672444\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			16.0000i			1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$			16.0000i			1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	1.07180	−	17.8564i	0.0999456	−	1.66512i
$116$		0			0
$117$	10.3923	−	6.00000i	0.960769	−	0.554700i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	5.50000	+	9.52628i	0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.0000	+	2.00000i	0.983870	+	0.178885i
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	0.765331	−	0.643637i	$-0.222575\pi$
$131$	−2.00000	−	3.46410i	−0.174741	−	0.302660i	−0.940072	+	0.340977i	$0.889242\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	0.765855	−	0.643013i	$-0.222316\pi$
$137$	6.92820	+	4.00000i	0.591916	+	0.341743i	−0.173939	+	0.984757i	$0.555649\pi$
$138$		0			0
$139$	4.00000			0.339276			0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000			0.339276			0.169638	+	0.985506i	$0.445740\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	2.46410	−	3.73205i	0.204633	−	0.309930i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	0.962348	−	0.271821i	$-0.0876260\pi$
$149$	3.00000	+	5.19615i	0.245770	+	0.425685i	−0.716578	+	0.697507i	$0.754293\pi$
$150$		0			0
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	−0.999909	−	0.0134886i	$-0.995706\pi$
$151$	−6.00000	+	10.3923i	−0.488273	+	0.845714i	0.511636	+	0.859202i	$0.329040\pi$
$152$		0			0
$153$		−	12.0000i		−	0.970143i
$154$		0			0
$155$	−8.00000	−	16.0000i	−0.642575	−	1.28515i
$156$		0			0
$157$	−10.3923	−	6.00000i	−0.829396	−	0.478852i	0.0242497	−	0.999706i	$-0.492280\pi$
$157$	−10.3923	−	6.00000i	−0.829396	−	0.478852i	−0.853646	+	0.520854i	$0.825614\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	20.7846	−	12.0000i	1.62798	−	0.939913i	0.643280	−	0.765631i	$-0.277573\pi$
$163$	20.7846	−	12.0000i	1.62798	−	0.939913i	0.984696	−	0.174282i	$-0.0557604\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	24.0000i		−	1.85718i	−0.371113	−	0.928588i	$-0.621024\pi$
$167$		−	24.0000i		−	1.85718i	0.371113	−	0.928588i	$-0.378976\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−6.00000	+	10.3923i	−0.458831	+	0.794719i
$172$		0			0
$173$	10.3923	−	6.00000i	0.790112	−	0.456172i	−0.0498898	−	0.998755i	$-0.515887\pi$
$173$	10.3923	−	6.00000i	0.790112	−	0.456172i	0.840002	+	0.542583i	$0.182554\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000	+	6.92820i	−0.298974	+	0.517838i	−0.975901	−	0.218212i	$-0.929978\pi$
$179$	−4.00000	+	6.92820i	−0.298974	+	0.517838i	0.676927	+	0.736050i	$0.263311\pi$
$180$		0			0
$181$	−14.0000			−1.04061			−0.520306	−	0.853980i	$-0.674182\pi$
$181$	−14.0000			−1.04061			−0.520306	+	0.853980i	$0.674182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.07180	−	17.8564i	0.0788001	−	1.31283i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0000	+	17.3205i	0.723575	+	1.25327i	0.959558	+	0.281511i	$0.0908356\pi$
$191$	10.0000	+	17.3205i	0.723575	+	1.25327i	−0.235983	+	0.971757i	$0.575831\pi$
$192$		0			0
$193$	−13.8564	−	8.00000i	−0.997406	−	0.575853i	−0.0899262	−	0.995948i	$-0.528663\pi$
$193$	−13.8564	−	8.00000i	−0.997406	−	0.575853i	−0.907480	+	0.420096i	$0.861996\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			16.0000i			1.13995i	0.821661	+	0.569976i	$0.193048\pi$
$197$			16.0000i			1.13995i	−0.821661	+	0.569976i	$0.806952\pi$
$198$		0			0
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	−0.0299585	−	0.999551i	$-0.509538\pi$
$199$	12.0000	−	20.7846i	0.850657	−	1.47338i	0.880616	−	0.473831i	$-0.157129\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	7.39230	−	11.1962i	0.516301	−	0.781973i
$206$		0			0
$207$	20.7846	+	12.0000i	1.44463	+	0.834058i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	24.0000			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	14.9282	+	9.85641i	1.01810	+	0.672201i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−8.00000	+	13.8564i	−0.538138	+	0.932083i
$222$		0			0
$223$		−	8.00000i		−	0.535720i	−0.963458	−	0.267860i	$-0.913684\pi$
$223$		−	8.00000i		−	0.535720i	0.963458	−	0.267860i	$-0.0863164\pi$
$224$		0			0
$225$	−9.00000	+	12.0000i	−0.600000	+	0.800000i
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	7.00000	+	12.1244i	0.462573	+	0.801200i	0.999088	−	0.0426906i	$-0.0135930\pi$
$229$	7.00000	+	12.1244i	0.462573	+	0.801200i	−0.536515	+	0.843891i	$0.680260\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.92820	−	4.00000i	0.453882	−	0.262049i	−0.255586	−	0.966786i	$-0.582269\pi$
$233$	6.92820	−	4.00000i	0.453882	−	0.262049i	0.709468	+	0.704737i	$0.248935\pi$
$234$		0			0
$235$	1.07180	−	17.8564i	0.0699163	−	1.16482i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.0000			−1.29369			−0.646846	−	0.762620i	$-0.723912\pi$
$239$	−20.0000			−1.29369			−0.646846	+	0.762620i	$0.723912\pi$
$240$		0			0
$241$	−9.00000	+	15.5885i	−0.579741	+	1.00414i	0.415768	+	0.909471i	$0.363513\pi$
$241$	−9.00000	+	15.5885i	−0.579741	+	1.00414i	−0.995509	+	0.0946700i	$0.969820\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	13.8564	−	8.00000i	0.881662	−	0.509028i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	−0.139533	−	0.990217i	$-0.544560\pi$
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	0.787787	+	0.615948i	$0.211227\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.00000	+	5.19615i	0.185695	+	0.321634i
$262$		0			0
$263$	20.7846	+	12.0000i	1.28163	+	0.739952i	0.977147	−	0.212565i	$-0.0681817\pi$
$263$	20.7846	+	12.0000i	1.28163	+	0.739952i	0.304487	+	0.952517i	$0.401515\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	−0.672374	−	0.740212i	$-0.734725\pi$
$269$	5.00000	−	8.66025i	0.304855	−	0.528025i	0.977229	+	0.212187i	$0.0680585\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−13.8564	−	8.00000i	−0.832551	−	0.480673i	0.0221745	−	0.999754i	$-0.492941\pi$
$277$	−13.8564	−	8.00000i	−0.832551	−	0.480673i	−0.854725	+	0.519081i	$0.826274\pi$
$278$		0			0
$279$	24.0000			1.43684
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$	−27.7128	−	16.0000i	−1.64736	−	0.951101i	−0.978117	−	0.208053i	$-0.933287\pi$
$283$	−27.7128	−	16.0000i	−1.64736	−	0.951101i	−0.669238	−	0.743048i	$-0.733379\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.0294118	−	0.0509427i
$290$		0			0
$291$		0			0
$292$		0			0
$293$			12.0000i			0.701047i	0.936554	+	0.350524i	$0.113996\pi$
$293$			12.0000i			0.701047i	−0.936554	+	0.350524i	$0.886004\pi$
$294$		0			0
$295$	4.00000	+	8.00000i	0.232889	+	0.465778i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.0000	−	27.7128i	−0.925304	−	1.60267i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	13.3923	+	0.803848i	0.766841	+	0.0460282i
$306$		0			0
$307$			16.0000i			0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$			16.0000i			0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	0.294384	+	0.955687i	$0.404886\pi$
$311$	−12.0000	+	20.7846i	−0.680458	+	1.17859i	−0.974841	+	0.222900i	$0.928448\pi$
$312$		0			0
$313$	−3.46410	+	2.00000i	−0.195803	+	0.113047i	−0.594696	−	0.803951i	$-0.702728\pi$
$313$	−3.46410	+	2.00000i	−0.195803	+	0.113047i	0.398894	+	0.916997i	$0.369394\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.8564	−	8.00000i	0.778253	−	0.449325i	−0.0575576	−	0.998342i	$-0.518331\pi$
$317$	13.8564	−	8.00000i	0.778253	−	0.449325i	0.835811	+	0.549017i	$0.184998\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	18.3923	−	7.85641i	1.02022	−	0.435795i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.00000	+	13.8564i	0.439720	+	0.761617i	0.997668	−	0.0682590i	$-0.0217444\pi$
$331$	8.00000	+	13.8564i	0.439720	+	0.761617i	−0.557948	+	0.829876i	$0.688411\pi$
$332$		0			0
$333$	20.7846	+	12.0000i	1.13899	+	0.657596i
$334$		0			0
$335$	16.0000	−	8.00000i	0.874173	−	0.437087i
$336$		0			0
$337$		−	8.00000i		−	0.435788i	−0.975972	−	0.217894i	$-0.930081\pi$
$337$		−	8.00000i		−	0.435788i	0.975972	−	0.217894i	$-0.0699187\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−20.7846	−	12.0000i	−1.11578	−	0.644194i	−0.175457	−	0.984487i	$-0.556140\pi$
$347$	−20.7846	−	12.0000i	−1.11578	−	0.644194i	−0.940319	+	0.340293i	$0.889474\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	10.3923	+	6.00000i	0.553127	+	0.319348i	0.750382	−	0.661004i	$-0.229870\pi$
$353$	10.3923	+	6.00000i	0.553127	+	0.319348i	−0.197256	+	0.980352i	$0.563203\pi$
$354$		0			0
$355$	−14.7846	+	22.3923i	−0.784686	+	1.18846i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	−0.999476	−	0.0323801i	$-0.989691\pi$
$359$	−10.0000	−	17.3205i	−0.527780	−	0.914141i	0.471696	−	0.881761i	$-0.343642\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−8.00000	+	4.00000i	−0.418739	+	0.209370i
$366$		0			0
$367$	−6.92820	−	4.00000i	−0.361649	−	0.208798i	0.308155	−	0.951336i	$-0.400289\pi$
$367$	−6.92820	−	4.00000i	−0.361649	−	0.208798i	−0.669804	+	0.742538i	$0.733622\pi$
$368$		0			0
$369$	9.00000	+	15.5885i	0.468521	+	0.811503i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−27.7128	+	16.0000i	−1.43492	+	0.828449i	−0.997490	−	0.0708063i	$-0.977443\pi$
$373$	−27.7128	+	16.0000i	−1.43492	+	0.828449i	−0.437425	+	0.899255i	$0.644109\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.00000i		−	0.412021i
$378$		0			0
$379$	8.00000			0.410932			0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000			0.410932			0.205466	+	0.978664i	$0.434129\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	20.7846	−	12.0000i	1.06204	−	0.613171i	0.136047	−	0.990702i	$-0.456560\pi$
$383$	20.7846	−	12.0000i	1.06204	−	0.613171i	0.925997	+	0.377531i	$0.123227\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−20.7846	+	12.0000i	−1.05654	+	0.609994i
$388$		0			0
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$	−32.0000			−1.61831
$392$		0			0
$393$		0			0
$394$		0			0
$395$	8.92820	+	0.535898i	0.449227	+	0.0269640i
$396$		0			0
$397$	−24.2487	+	14.0000i	−1.21701	+	0.702640i	−0.964277	−	0.264897i	$-0.914662\pi$
$397$	−24.2487	+	14.0000i	−1.21701	+	0.702640i	−0.252731	+	0.967537i	$0.581329\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−27.7128	−	16.0000i	−1.38047	−	0.797017i
$404$		0			0
$405$	−9.00000	−	18.0000i	−0.447214	−	0.894427i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	0.715523	+	0.698589i	$0.246188\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−28.0000			−1.36789			−0.683945	−	0.729534i	$-0.739737\pi$
$419$	−28.0000			−1.36789			−0.683945	+	0.729534i	$0.739737\pi$
$420$		0			0
$421$	10.0000			0.487370			0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000			0.487370			0.243685	+	0.969854i	$0.421644\pi$
$422$		0			0
$423$	20.7846	+	12.0000i	1.01058	+	0.583460i
$424$		0			0
$425$	2.39230	−	19.8564i	0.116044	−	0.963177i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.0000	+	17.3205i	−0.481683	+	0.834300i	−0.999779	−	0.0210230i	$-0.993308\pi$
$431$	−10.0000	+	17.3205i	−0.481683	+	0.834300i	0.518096	+	0.855323i	$0.326641\pi$
$432$		0			0
$433$			28.0000i			1.34559i	0.739827	+	0.672797i	$0.234907\pi$
$433$			28.0000i			1.34559i	−0.739827	+	0.672797i	$0.765093\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	27.7128	+	16.0000i	1.32568	+	0.765384i
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.7846	+	12.0000i	−0.987507	+	0.570137i	−0.904528	−	0.426414i	$-0.859777\pi$
$443$	−20.7846	+	12.0000i	−0.987507	+	0.570137i	−0.0829786	+	0.996551i	$0.526443\pi$
$444$		0			0
$445$	−22.3205	−	1.33975i	−1.05809	−	0.0635100i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−2.00000			−0.0943858			−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000			−0.0943858			−0.0471929	+	0.998886i	$0.515028\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−13.8564	+	8.00000i	−0.648175	+	0.374224i	−0.787757	−	0.615986i	$-0.788758\pi$
$457$	−13.8564	+	8.00000i	−0.648175	+	0.374224i	0.139581	+	0.990211i	$0.455424\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	26.0000			1.21094			0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000			1.21094			0.605470	+	0.795868i	$0.292985\pi$
$462$		0			0
$463$			8.00000i			0.371792i	0.982569	+	0.185896i	$0.0595187\pi$
$463$			8.00000i			0.371792i	−0.982569	+	0.185896i	$0.940481\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−13.8564	+	8.00000i	−0.641198	+	0.370196i	−0.785076	−	0.619400i	$-0.787376\pi$
$467$	−13.8564	+	8.00000i	−0.641198	+	0.370196i	0.143878	+	0.989595i	$0.454043\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−12.0000	+	16.0000i	−0.550598	+	0.734130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−16.0000	−	27.7128i	−0.729537	−	1.26360i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.3923	−	14.7846i	−1.01678	−	0.671335i
$486$		0			0
$487$	6.92820	+	4.00000i	0.313947	+	0.181257i	0.648691	−	0.761052i	$-0.275317\pi$
$487$	6.92820	+	4.00000i	0.313947	+	0.181257i	−0.334744	+	0.942309i	$0.608650\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$	−6.92820	−	4.00000i	−0.312031	−	0.180151i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−4.00000	−	6.92820i	−0.179065	−	0.310149i	0.762496	−	0.646993i	$-0.223974\pi$
$499$	−4.00000	−	6.92820i	−0.179065	−	0.310149i	−0.941560	+	0.336844i	$0.890640\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	8.00000i		−	0.356702i	−0.983967	−	0.178351i	$-0.942924\pi$
$503$		−	8.00000i		−	0.356702i	0.983967	−	0.178351i	$-0.0570763\pi$
$504$		0			0
$505$	−18.0000	−	36.0000i	−0.800989	−	1.60198i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.0000	+	19.0526i	0.487566	+	0.844490i	0.999898	−	0.0142980i	$-0.00455136\pi$
$509$	11.0000	+	19.0526i	0.487566	+	0.844490i	−0.512331	+	0.858788i	$0.671218\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.07180	−	17.8564i	0.0472290	−	0.786847i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	−0.924229	−	0.381839i	$-0.875291\pi$
$521$	−3.00000	+	5.19615i	−0.131432	+	0.227648i	0.792797	+	0.609486i	$0.208624\pi$
$522$		0			0
$523$	−27.7128	+	16.0000i	−1.21180	+	0.699631i	−0.963150	−	0.268963i	$-0.913319\pi$
$523$	−27.7128	+	16.0000i	−1.21180	+	0.699631i	−0.248646	+	0.968594i	$0.579986\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−27.7128	+	16.0000i	−1.20719	+	0.696971i
$528$		0			0
$529$	20.5000	−	35.5070i	0.891304	−	1.54378i
$530$		0			0
$531$	−12.0000			−0.520756
$532$		0			0
$533$		−	24.0000i		−	1.03956i
$534$		0			0
$535$	−1.07180	+	17.8564i	−0.0463378	+	0.772000i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.00000	−	12.1244i	−0.300954	−	0.521267i	0.675399	−	0.737453i	$-0.263972\pi$
$541$	−7.00000	−	12.1244i	−0.300954	−	0.521267i	−0.976352	+	0.216186i	$0.930638\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.0000	+	28.0000i	0.599694	+	1.19939i
$546$		0			0
$547$		−	8.00000i		−	0.342055i	−0.985266	−	0.171028i	$-0.945291\pi$
$547$		−	8.00000i		−	0.342055i	0.985266	−	0.171028i	$-0.0547087\pi$
$548$		0			0
$549$	−9.00000	+	15.5885i	−0.384111	+	0.665299i
$550$		0			0
$551$	4.00000	+	6.92820i	0.170406	+	0.295151i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	13.8564	+	8.00000i	0.587115	+	0.338971i	0.763956	−	0.645269i	$-0.223255\pi$
$557$	13.8564	+	8.00000i	0.587115	+	0.338971i	−0.176841	+	0.984239i	$0.556588\pi$
$558$		0			0
$559$	32.0000			1.35346
$560$		0			0
$561$		0			0
$562$		0			0
$563$	13.8564	+	8.00000i	0.583978	+	0.337160i	0.762713	−	0.646737i	$-0.223867\pi$
$563$	13.8564	+	8.00000i	0.583978	+	0.337160i	−0.178735	+	0.983897i	$0.557200\pi$
$564$		0			0
$565$	−29.8564	−	19.7128i	−1.25607	−	0.829324i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.0000	+	36.3731i	0.880366	+	1.52484i	0.850935	+	0.525271i	$0.176036\pi$
$569$	21.0000	+	36.3731i	0.880366	+	1.52484i	0.0294311	+	0.999567i	$0.490630\pi$
$570$		0			0
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	−0.0554391	−	0.998462i	$-0.517656\pi$
$571$	20.0000	−	34.6410i	0.836974	−	1.44968i	0.892413	−	0.451219i	$-0.149011\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	32.0000	+	24.0000i	1.33449	+	1.00087i
$576$		0			0
$577$	−3.46410	−	2.00000i	−0.144212	−	0.0832611i	0.426158	−	0.904649i	$-0.359867\pi$
$577$	−3.46410	−	2.00000i	−0.144212	−	0.0832611i	−0.570370	+	0.821388i	$0.693200\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.60770	+	26.7846i	−0.0664700	+	1.10741i
$586$		0			0
$587$			32.0000i			1.32078i	0.750922	+	0.660391i	$0.229609\pi$
$587$			32.0000i			1.32078i	−0.750922	+	0.660391i	$0.770391\pi$
$588$		0			0
$589$	32.0000			1.31854
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−31.1769	+	18.0000i	−1.28028	+	0.739171i	−0.976900	−	0.213697i	$-0.931449\pi$
$593$	−31.1769	+	18.0000i	−1.28028	+	0.739171i	−0.303383	+	0.952869i	$0.598116\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	0.219061	+	0.975711i	$0.429701\pi$
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	−0.954521	+	0.298143i	$0.903633\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$			24.0000i			0.977356i
$604$		0			0
$605$	−24.5526	−	1.47372i	−0.998203	−	0.0599153i
$606$		0			0
$607$	34.6410	−	20.0000i	1.40604	−	0.811775i	0.411033	−	0.911621i	$-0.365168\pi$
$607$	34.6410	−	20.0000i	1.40604	−	0.811775i	0.995003	+	0.0998457i	$0.0318349\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.0000	−	27.7128i	−0.647291	−	1.12114i
$612$		0			0
$613$	−20.7846	−	12.0000i	−0.839482	−	0.484675i	0.0176058	−	0.999845i	$-0.494396\pi$
$613$	−20.7846	−	12.0000i	−0.839482	−	0.484675i	−0.857088	+	0.515170i	$0.827729\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			8.00000i			0.322068i	0.986949	+	0.161034i	$0.0514829\pi$
$617$			8.00000i			0.322068i	−0.986949	+	0.161034i	$0.948517\pi$
$618$		0			0
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	−0.903416	−	0.428765i	$-0.858949\pi$
$619$	−2.00000	+	3.46410i	−0.0803868	+	0.139234i	0.823029	+	0.567999i	$0.192282\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−17.2846	+	18.0622i	−0.691384	+	0.722487i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−32.0000			−1.27592
$630$		0			0
$631$	−36.0000			−1.43314			−0.716569	−	0.697517i	$-0.754288\pi$
$631$	−36.0000			−1.43314			−0.716569	+	0.697517i	$0.754288\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	14.9282	+	9.85641i	0.592408	+	0.391140i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−18.0000	−	31.1769i	−0.712069	−	1.23334i
$640$		0			0
$641$	1.00000	−	1.73205i	0.0394976	−	0.0684119i	−0.845601	−	0.533816i	$-0.820758\pi$
$641$	1.00000	−	1.73205i	0.0394976	−	0.0684119i	0.885098	+	0.465404i	$0.154091\pi$
$642$		0			0
$643$			16.0000i			0.630978i	0.948929	+	0.315489i	$0.102169\pi$
$643$			16.0000i			0.630978i	−0.948929	+	0.315489i	$0.897831\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	20.7846	+	12.0000i	0.817127	+	0.471769i	0.849425	−	0.527710i	$-0.176949\pi$
$647$	20.7846	+	12.0000i	0.817127	+	0.471769i	−0.0322975	+	0.999478i	$0.510282\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	20.7846	−	12.0000i	0.813365	−	0.469596i	−0.0347583	−	0.999396i	$-0.511066\pi$
$653$	20.7846	−	12.0000i	0.813365	−	0.469596i	0.848123	+	0.529799i	$0.177733\pi$
$654$		0			0
$655$	8.92820	+	0.535898i	0.348854	+	0.0209393i
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	−24.0000			−0.934907			−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000			−0.934907			−0.467454	+	0.884018i	$0.654829\pi$
$660$		0			0
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	−0.319071	−	0.947731i	$-0.603371\pi$
$661$	17.0000	−	29.4449i	0.661223	−	1.14527i	0.980294	−	0.197542i	$-0.0632958\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	13.8564	−	8.00000i	0.536522	−	0.309761i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	8.00000i		−	0.308377i	−0.988041	−	0.154189i	$-0.950724\pi$
$673$		−	8.00000i		−	0.308377i	0.988041	−	0.154189i	$-0.0492764\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−3.46410	+	2.00000i	−0.133136	+	0.0768662i	−0.565089	−	0.825030i	$-0.691158\pi$
$677$	−3.46410	+	2.00000i	−0.133136	+	0.0768662i	0.431953	+	0.901896i	$0.357825\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.92820	−	4.00000i	−0.265100	−	0.153056i	0.361559	−	0.932349i	$-0.382245\pi$
$683$	−6.92820	−	4.00000i	−0.265100	−	0.153056i	−0.626659	+	0.779294i	$0.715578\pi$
$684$		0			0
$685$	−16.0000	+	8.00000i	−0.611329	+	0.305664i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	0.999276	+	0.0380440i	$0.0121127\pi$
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	−0.466691	+	0.884420i	$0.654554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−4.92820	+	7.46410i	−0.186937	+	0.283130i
$696$		0			0
$697$	−20.7846	−	12.0000i	−0.787273	−	0.454532i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	2.00000			0.0755390			0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000			0.0755390			0.0377695	+	0.999286i	$0.487975\pi$
$702$		0			0
$703$	27.7128	+	16.0000i	1.04521	+	0.603451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	0.804178	−	0.594389i	$-0.202606\pi$
$709$	−3.00000	−	5.19615i	−0.112667	−	0.195146i	−0.916845	+	0.399244i	$0.869273\pi$
$710$		0			0
$711$	−6.00000	+	10.3923i	−0.225018	+	0.389742i
$712$		0			0
$713$		−	64.0000i		−	2.39682i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16.0000	−	27.7128i	−0.596699	−	1.03351i	−0.993305	−	0.115524i	$-0.963145\pi$
$719$	−16.0000	−	27.7128i	−0.596699	−	1.03351i	0.396605	−	0.917989i	$-0.370188\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.92820	+	9.19615i	0.145890	+	0.341537i
$726$		0			0
$727$			8.00000i			0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$			8.00000i			0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$		0			0
$729$	27.0000			1.00000
$730$		0			0
$731$	16.0000	−	27.7128i	0.591781	−	1.02500i
$732$		0			0
$733$	17.3205	−	10.0000i	0.639748	−	0.369358i	−0.144770	−	0.989465i	$-0.546244\pi$
$733$	17.3205	−	10.0000i	0.639748	−	0.369358i	0.784517	+	0.620107i	$0.212911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−4.00000	+	6.92820i	−0.147142	+	0.254858i	−0.930170	−	0.367129i	$-0.880341\pi$
$739$	−4.00000	+	6.92820i	−0.147142	+	0.254858i	0.783028	+	0.621987i	$0.213674\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			24.0000i			0.880475i	0.897881	+	0.440237i	$0.145106\pi$
$743$			24.0000i			0.880475i	−0.897881	+	0.440237i	$0.854894\pi$
$744$		0			0
$745$	−13.3923	−	0.803848i	−0.490656	−	0.0294507i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	0.988761	−	0.149505i	$-0.0477681\pi$
$751$	10.0000	+	17.3205i	0.364905	+	0.632034i	−0.623856	+	0.781540i	$0.714435\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−12.0000	−	24.0000i	−0.436725	−	0.873449i
$756$		0			0
$757$		−	24.0000i		−	0.872295i	−0.899875	−	0.436147i	$-0.856343\pi$
$757$		−	24.0000i		−	0.872295i	0.899875	−	0.436147i	$-0.143657\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	5.00000	+	8.66025i	0.181250	+	0.313934i	0.942306	−	0.334752i	$-0.108652\pi$
$761$	5.00000	+	8.66025i	0.181250	+	0.313934i	−0.761057	+	0.648686i	$0.775319\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	22.3923	+	14.7846i	0.809595	+	0.534539i
$766$		0			0
$767$	13.8564	+	8.00000i	0.500326	+	0.288863i
$768$		0			0
$769$	−2.00000			−0.0721218			−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000			−0.0721218			−0.0360609	+	0.999350i	$0.511481\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−31.1769	−	18.0000i	−1.12136	−	0.647415i	−0.179609	−	0.983738i	$-0.557483\pi$
$773$	−31.1769	−	18.0000i	−1.12136	−	0.647415i	−0.941747	+	0.336323i	$0.890817\pi$
$774$		0			0
$775$	39.7128	+	4.78461i	1.42653	+	0.171868i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.0000	+	20.7846i	0.429945	+	0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	24.0000	−	12.0000i	0.856597	−	0.428298i
$786$		0			0
$787$	−41.5692	−	24.0000i	−1.48178	−	0.855508i	−0.481996	−	0.876173i	$-0.660088\pi$
$787$	−41.5692	−	24.0000i	−1.48178	−	0.855508i	−0.999786	+	0.0206657i	$0.993421\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	20.7846	−	12.0000i	0.738083	−	0.426132i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	28.0000i		−	0.991811i	−0.868377	−	0.495905i	$-0.834836\pi$
$797$		−	28.0000i		−	0.991811i	0.868377	−	0.495905i	$-0.165164\pi$
$798$		0			0
$799$	−32.0000			−1.13208
$800$		0			0
$801$	15.0000	−	25.9808i	0.529999	−	0.917985i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.0000	+	22.5167i	−0.457056	+	0.791644i	−0.998804	−	0.0488972i	$-0.984429\pi$
$809$	−13.0000	+	22.5167i	−0.457056	+	0.791644i	0.541748	+	0.840541i	$0.317763\pi$
$810$		0			0
$811$	36.0000			1.26413			0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000			1.26413			0.632065	+	0.774915i	$0.282207\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3.21539	+	53.5692i	−0.112630	+	1.87645i
$816$		0			0
$817$	−27.7128	+	16.0000i	−0.969549	+	0.559769i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	0.913616	−	0.406578i	$-0.133278\pi$
$821$	3.00000	+	5.19615i	0.104701	+	0.181347i	−0.808915	+	0.587925i	$0.799945\pi$
$822$		0			0
$823$	20.7846	+	12.0000i	0.724506	+	0.418294i	0.816409	−	0.577474i	$-0.195962\pi$
$823$	20.7846	+	12.0000i	0.724506	+	0.418294i	−0.0919029	+	0.995768i	$0.529295\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			8.00000i			0.278187i	0.990279	+	0.139094i	$0.0444189\pi$
$827$			8.00000i			0.278187i	−0.990279	+	0.139094i	$0.955581\pi$
$828$		0			0
$829$	−11.0000	+	19.0526i	−0.382046	+	0.661723i	−0.991355	−	0.131210i	$-0.958114\pi$
$829$	−11.0000	+	19.0526i	−0.382046	+	0.661723i	0.609309	+	0.792933i	$0.291447\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	44.7846	+	29.5692i	1.54984	+	1.02329i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−16.0000			−0.552381			−0.276191	−	0.961103i	$-0.589072\pi$
$839$	−16.0000			−0.552381			−0.276191	+	0.961103i	$0.589072\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.69615	−	5.59808i	0.127152	−	0.192580i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	32.0000	−	55.4256i	1.09695	−	1.89997i
$852$		0			0
$853$		−	4.00000i		−	0.136957i	−0.997653	−	0.0684787i	$-0.978185\pi$
$853$		−	4.00000i		−	0.136957i	0.997653	−	0.0684787i	$-0.0218145\pi$
$854$		0			0
$855$	−12.0000	−	24.0000i	−0.410391	−	0.820783i
$856$		0			0
$857$	−45.0333	−	26.0000i	−1.53831	−	0.888143i	−0.998938	−	0.0460748i	$-0.985329\pi$
$857$	−45.0333	−	26.0000i	−1.53831	−	0.888143i	−0.539371	−	0.842068i	$-0.681338\pi$
$858$		0			0
$859$	−18.0000	−	31.1769i	−0.614152	−	1.06374i	−0.990533	−	0.137277i	$-0.956165\pi$
$859$	−18.0000	−	31.1769i	−0.614152	−	1.06374i	0.376381	−	0.926465i	$-0.377169\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	6.92820	−	4.00000i	0.235839	−	0.136162i	−0.377424	−	0.926041i	$-0.623190\pi$
$863$	6.92820	−	4.00000i	0.235839	−	0.136162i	0.613263	+	0.789879i	$0.289857\pi$
$864$		0			0
$865$	−1.60770	+	26.7846i	−0.0546633	+	0.910704i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	16.0000	−	27.7128i	0.542139	−	0.939013i
$872$		0			0
$873$	31.1769	−	18.0000i	1.05518	−	0.609208i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	34.6410	−	20.0000i	1.16974	−	0.675352i	0.216124	−	0.976366i	$-0.430658\pi$
$877$	34.6410	−	20.0000i	1.16974	−	0.675352i	0.953620	+	0.301014i	$0.0973250\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		0			0
$883$			8.00000i			0.269221i	0.990899	+	0.134611i	$0.0429784\pi$
$883$			8.00000i			0.269221i	−0.990899	+	0.134611i	$0.957022\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.7846	−	12.0000i	0.697879	−	0.402921i	−0.108678	−	0.994077i	$-0.534662\pi$
$887$	20.7846	−	12.0000i	0.697879	−	0.402921i	0.806557	+	0.591156i	$0.201328\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	27.7128	+	16.0000i	0.927374	+	0.535420i
$894$		0			0
$895$	−8.00000	−	16.0000i	−0.267411	−	0.534821i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	8.00000	−	13.8564i	0.266815	−	0.462137i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.2487	−	26.1244i	0.573367	−	0.868403i
$906$		0			0
$907$	−6.92820	−	4.00000i	−0.230047	−	0.132818i	0.380547	−	0.924762i	$-0.375736\pi$
$907$	−6.92820	−	4.00000i	−0.230047	−	0.132818i	−0.610594	+	0.791944i	$0.709069\pi$
$908$		0			0
$909$	54.0000			1.79107
$910$		0			0
$911$	−28.0000			−0.927681			−0.463841	−	0.885919i	$-0.653529\pi$
$911$	−28.0000			−0.927681			−0.463841	+	0.885919i	$0.653529\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−2.00000	−	3.46410i	−0.0659739	−	0.114270i	0.831152	−	0.556046i	$-0.187682\pi$
$919$	−2.00000	−	3.46410i	−0.0659739	−	0.114270i	−0.897126	+	0.441776i	$0.854349\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			48.0000i			1.57994i
$924$		0			0
$925$	32.0000	+	24.0000i	1.05215	+	0.789115i
$926$		0			0
$927$	20.7846	+	12.0000i	0.682656	+	0.394132i
$928$		0			0
$929$	15.0000	+	25.9808i	0.492134	+	0.852401i	0.999959	−	0.00905914i	$-0.00288365\pi$
$929$	15.0000	+	25.9808i	0.492134	+	0.852401i	−0.507825	+	0.861460i	$0.669550\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	4.00000i		−	0.130674i	−0.997863	−	0.0653372i	$-0.979188\pi$
$937$		−	4.00000i		−	0.130674i	0.997863	−	0.0653372i	$-0.0208123\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	19.0000	−	32.9090i	0.619382	−	1.07280i	−0.370216	−	0.928946i	$-0.620716\pi$
$941$	19.0000	−	32.9090i	0.619382	−	1.07280i	0.989599	−	0.143856i	$-0.0459502\pi$
$942$		0			0
$943$	41.5692	−	24.0000i	1.35368	−	0.781548i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	20.7846	−	12.0000i	0.675409	−	0.389948i	−0.122714	−	0.992442i	$-0.539160\pi$
$947$	20.7846	−	12.0000i	0.675409	−	0.389948i	0.798123	+	0.602494i	$0.205826\pi$
$948$		0			0
$949$	−8.00000	+	13.8564i	−0.259691	+	0.449798i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−44.6410	−	2.67949i	−1.44455	−	0.0867063i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−16.5000	−	28.5788i	−0.532258	−	0.921898i
$962$		0			0
$963$	−20.7846	−	12.0000i	−0.669775	−	0.386695i
$964$		0			0
$965$	32.0000	−	16.0000i	1.03012	−	0.515058i
$966$		0			0
$967$			56.0000i			1.80084i	0.435023	+	0.900419i	$0.356740\pi$
$967$			56.0000i			1.80084i	−0.435023	+	0.900419i	$0.643260\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.0000	+	51.9615i	0.962746	+	1.66752i	0.715553	+	0.698558i	$0.246175\pi$
$971$	30.0000	+	51.9615i	0.962746	+	1.66752i	0.247193	+	0.968966i	$0.420492\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.7846	+	12.0000i	0.664959	+	0.383914i	0.794164	−	0.607704i	$-0.207909\pi$
$977$	20.7846	+	12.0000i	0.664959	+	0.383914i	−0.129205	+	0.991618i	$0.541243\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−42.0000			−1.34096
$982$		0			0
$983$	−34.6410	−	20.0000i	−1.10488	−	0.637901i	−0.167379	−	0.985893i	$-0.553530\pi$
$983$	−34.6410	−	20.0000i	−1.10488	−	0.637901i	−0.937498	+	0.347992i	$0.886864\pi$
$984$		0			0
$985$	−29.8564	−	19.7128i	−0.951304	−	0.628102i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	32.0000	+	55.4256i	1.01754	+	1.76243i
$990$		0			0
$991$	−2.00000	+	3.46410i	−0.0635321	+	0.110041i	−0.896042	−	0.443969i	$-0.853570\pi$
$991$	−2.00000	+	3.46410i	−0.0635321	+	0.110041i	0.832510	+	0.554010i	$0.186903\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	24.0000	+	48.0000i	0.760851	+	1.52170i
$996$		0			0
$997$	24.2487	+	14.0000i	0.767964	+	0.443384i	0.832148	−	0.554554i	$-0.187111\pi$
$997$	24.2487	+	14.0000i	0.767964	+	0.443384i	−0.0641836	+	0.997938i	$0.520444\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.q.e.949.1		4
5.4	even	2	inner	980.2.q.e.949.2		4
7.2	even	3	inner	980.2.q.e.569.2		4
7.3	odd	6		140.2.e.b.29.2	yes	2
7.4	even	3		980.2.e.a.589.1		2
7.5	odd	6		980.2.q.d.569.1		4
7.6	odd	2		980.2.q.d.949.2		4
21.17	even	6		1260.2.k.b.1009.1		2
28.3	even	6		560.2.g.c.449.2		2
35.3	even	12		700.2.a.h.1.1		1
35.4	even	6		980.2.e.a.589.2		2
35.9	even	6	inner	980.2.q.e.569.1		4
35.17	even	12		700.2.a.f.1.1		1
35.18	odd	12		4900.2.a.l.1.1		1
35.19	odd	6		980.2.q.d.569.2		4
35.24	odd	6		140.2.e.b.29.1	✓	2
35.32	odd	12		4900.2.a.m.1.1		1
35.34	odd	2		980.2.q.d.949.1		4
56.3	even	6		2240.2.g.c.449.1		2
56.45	odd	6		2240.2.g.d.449.1		2
84.59	odd	6		5040.2.t.g.1009.1		2
105.17	odd	12		6300.2.a.g.1.1		1
105.38	odd	12		6300.2.a.y.1.1		1
105.59	even	6		1260.2.k.b.1009.2		2
140.3	odd	12		2800.2.a.o.1.1		1
140.59	even	6		560.2.g.c.449.1		2
140.87	odd	12		2800.2.a.s.1.1		1
280.59	even	6		2240.2.g.c.449.2		2
280.269	odd	6		2240.2.g.d.449.2		2
420.59	odd	6		5040.2.t.g.1009.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.e.b.29.1	✓	2	35.24	odd	6
140.2.e.b.29.2	yes	2	7.3	odd	6
560.2.g.c.449.1		2	140.59	even	6
560.2.g.c.449.2		2	28.3	even	6
700.2.a.f.1.1		1	35.17	even	12
700.2.a.h.1.1		1	35.3	even	12
980.2.e.a.589.1		2	7.4	even	3
980.2.e.a.589.2		2	35.4	even	6
980.2.q.d.569.1		4	7.5	odd	6
980.2.q.d.569.2		4	35.19	odd	6
980.2.q.d.949.1		4	35.34	odd	2
980.2.q.d.949.2		4	7.6	odd	2
980.2.q.e.569.1		4	35.9	even	6	inner
980.2.q.e.569.2		4	7.2	even	3	inner
980.2.q.e.949.1		4	1.1	even	1	trivial
980.2.q.e.949.2		4	5.4	even	2	inner
1260.2.k.b.1009.1		2	21.17	even	6
1260.2.k.b.1009.2		2	105.59	even	6
2240.2.g.c.449.1		2	56.3	even	6
2240.2.g.c.449.2		2	280.59	even	6
2240.2.g.d.449.1		2	56.45	odd	6
2240.2.g.d.449.2		2	280.269	odd	6
2800.2.a.o.1.1		1	140.3	odd	12
2800.2.a.s.1.1		1	140.87	odd	12
4900.2.a.l.1.1		1	35.18	odd	12
4900.2.a.m.1.1		1	35.32	odd	12
5040.2.t.g.1009.1		2	84.59	odd	6
5040.2.t.g.1009.2		2	420.59	odd	6
6300.2.a.g.1.1		1	105.17	odd	12
6300.2.a.y.1.1		1	105.38	odd	12

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:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.23205 + 1.86603i) q^{5} +(-1.50000 - 2.59808i) q^{9} +4.00000i q^{13} +(3.46410 + 2.00000i) q^{17} +(-2.00000 - 3.46410i) q^{19} +(-6.92820 + 4.00000i) q^{23} +(-1.96410 - 4.59808i) q^{25} -2.00000 q^{29} +(-4.00000 + 6.92820i) q^{31} +(-6.92820 + 4.00000i) q^{37} -6.00000 q^{41} -8.00000i q^{43} +(6.69615 + 0.401924i) q^{45} +(-6.92820 + 4.00000i) q^{47} +(2.00000 - 3.46410i) q^{59} +(-3.00000 - 5.19615i) q^{61} +(-7.46410 - 4.92820i) q^{65} +(-6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(3.46410 + 2.00000i) q^{73} +(-2.00000 - 3.46410i) q^{79} +(-4.50000 + 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(5.00000 + 8.66025i) q^{89} +(8.92820 + 0.535898i) q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) \) 4 * q + 2 * q^5 - 6 * q^9 - 8 * q^19 + 6 * q^25 - 8 * q^29 - 16 * q^31 - 24 * q^41 + 6 * q^45 + 8 * q^59 - 12 * q^61 - 16 * q^65 + 48 * q^71 - 8 * q^79 - 18 * q^81 - 32 * q^85 + 20 * q^89 + 8 * q^95

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