LMFDB - Embedded newform 980.2.q.e.569.2

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			569.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	980.569
Dual form			980.2.q.e.949.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.23205 + 0.133975i) q^{5} +(-1.50000 + 2.59808i) q^{9} +4.00000i q^{13} +(-3.46410 + 2.00000i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(6.92820 + 4.00000i) q^{23} +(4.96410 + 0.598076i) q^{25} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(6.92820 + 4.00000i) q^{37} -6.00000 q^{41} -8.00000i q^{43} +(-3.69615 + 5.59808i) q^{45} +(6.92820 + 4.00000i) q^{47} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-0.535898 + 8.92820i) q^{65} +(6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(-3.46410 + 2.00000i) q^{73} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(5.00000 - 8.66025i) q^{89} +(-4.92820 + 7.46410i) q^{95} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

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

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	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$		0			0
$5$	2.23205	+	0.133975i	0.998203	+	0.0599153i
$5$	2.23205	+	0.133975i	0.998203	+	0.0599153i
$6$		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	$-0.166667\pi$
$11$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$12$		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	$-0.827873\pi$
$17$	−3.46410	+	2.00000i	−0.840168	+	0.485071i	0.0171533	+	0.999853i	$0.494540\pi$
$18$		0			0
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	−0.998899	−	0.0469020i	$-0.985065\pi$
$19$	−2.00000	+	3.46410i	−0.458831	+	0.794719i	0.540068	+	0.841621i	$0.318398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	6.92820	+	4.00000i	1.44463	+	0.834058i	0.998154	−	0.0607377i	$-0.0193453\pi$
$23$	6.92820	+	4.00000i	1.44463	+	0.834058i	0.446476	+	0.894795i	$0.352679\pi$
$24$		0			0
$25$	4.96410	+	0.598076i	0.992820	+	0.119615i
$26$		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	−0.961625	−	0.274367i	$-0.911532\pi$
$31$	−4.00000	−	6.92820i	−0.718421	−	1.24434i	0.243204	−	0.969975i	$-0.421802\pi$
$32$		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	$-0.104907\pi$
$37$	6.92820	+	4.00000i	1.13899	+	0.657596i	0.192809	+	0.981236i	$0.438240\pi$
$38$		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
$45$	−3.69615	+	5.59808i	−0.550990	+	0.834512i
$46$		0			0
$47$	6.92820	+	4.00000i	1.01058	+	0.583460i	0.911362	−	0.411606i	$-0.135032\pi$
$47$	6.92820	+	4.00000i	1.01058	+	0.583460i	0.0992202	+	0.995066i	$0.468365\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	−0.991645	−	0.128994i	$-0.958825\pi$
$61$	−3.00000	+	5.19615i	−0.384111	+	0.665299i	0.607535	+	0.794293i	$0.292159\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.535898	+	8.92820i	−0.0664700	+	1.10741i
$66$		0			0
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	−0.0130248	−	0.999915i	$-0.504146\pi$
$67$	6.92820	−	4.00000i	0.846415	−	0.488678i	0.859440	+	0.511237i	$0.170813\pi$
$68$		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	$-0.741875\pi$
$73$	−3.46410	+	2.00000i	−0.405442	+	0.234082i	0.283387	+	0.959006i	$0.408542\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	−0.956325	−	0.292306i	$-0.905577\pi$
$79$	−2.00000	+	3.46410i	−0.225018	+	0.389742i	0.731307	+	0.682048i	$0.238911\pi$
$80$		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	−0.469389	−	0.882992i	$-0.655526\pi$
$89$	5.00000	−	8.66025i	0.529999	−	0.917985i	0.999388	−	0.0349934i	$-0.0111410\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.92820	+	7.46410i	−0.505623	+	0.765801i
$96$		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	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	0.800855	−	0.598858i	$-0.204379\pi$
$103$	6.92820	+	4.00000i	0.682656	+	0.394132i	−0.118199	+	0.992990i	$0.537712\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.92820	−	4.00000i	−0.669775	−	0.386695i	0.126217	−	0.992003i	$-0.459717\pi$
$107$	−6.92820	−	4.00000i	−0.669775	−	0.386695i	−0.795991	+	0.605308i	$0.793050\pi$
$108$		0			0
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	0.977769	+	0.209687i	$0.0672444\pi$
$109$	7.00000	+	12.1244i	0.670478	+	1.16130i	−0.307290	+	0.951616i	$0.599422\pi$
$110$		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
$115$	14.9282	+	9.85641i	1.39206	+	0.919115i
$116$		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	−0.940072	−	0.340977i	$-0.889242\pi$
$131$	−2.00000	+	3.46410i	−0.174741	+	0.302660i	0.765331	+	0.643637i	$0.222575\pi$
$132$		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	$-0.777684\pi$
$137$	−6.92820	+	4.00000i	−0.591916	+	0.341743i	0.173939	+	0.984757i	$0.444351\pi$
$138$		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
$145$	−4.46410	−	0.267949i	−0.370723	−	0.0222520i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	−0.716578	−	0.697507i	$-0.754293\pi$
$149$	3.00000	−	5.19615i	0.245770	−	0.425685i	0.962348	+	0.271821i	$0.0876260\pi$
$150$		0			0
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	0.511636	−	0.859202i	$-0.329040\pi$
$151$	−6.00000	−	10.3923i	−0.488273	−	0.845714i	−0.999909	+	0.0134886i	$0.995706\pi$
$152$		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	$-0.507720\pi$
$157$	10.3923	−	6.00000i	0.829396	−	0.478852i	0.853646	+	0.520854i	$0.174386\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−20.7846	−	12.0000i	−1.62798	−	0.939913i	−0.984696	−	0.174282i	$-0.944240\pi$
$163$	−20.7846	−	12.0000i	−1.62798	−	0.939913i	−0.643280	−	0.765631i	$-0.722427\pi$
$164$		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	$-0.484113\pi$
$173$	−10.3923	−	6.00000i	−0.790112	−	0.456172i	−0.840002	+	0.542583i	$0.817446\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−4.00000	−	6.92820i	−0.298974	−	0.517838i	0.676927	−	0.736050i	$-0.263311\pi$
$179$	−4.00000	−	6.92820i	−0.298974	−	0.517838i	−0.975901	+	0.218212i	$0.929978\pi$
$180$		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
$185$	14.9282	+	9.85641i	1.09754	+	0.724657i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	−0.235983	−	0.971757i	$-0.575831\pi$
$191$	10.0000	−	17.3205i	0.723575	−	1.25327i	0.959558	−	0.281511i	$-0.0908356\pi$
$192$		0			0
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.0899262	−	0.995948i	$-0.471337\pi$
$193$	13.8564	−	8.00000i	0.997406	−	0.575853i	0.907480	+	0.420096i	$0.138004\pi$
$194$		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	0.880616	+	0.473831i	$0.157129\pi$
$199$	12.0000	+	20.7846i	0.850657	+	1.47338i	−0.0299585	+	0.999551i	$0.509538\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−13.3923	−	0.803848i	−0.935359	−	0.0561432i
$206$		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
$215$	1.07180	−	17.8564i	0.0730959	−	1.21780i
$216$		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	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	−0.536515	−	0.843891i	$-0.680260\pi$
$229$	7.00000	−	12.1244i	0.462573	−	0.801200i	0.999088	+	0.0426906i	$0.0135930\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.92820	−	4.00000i	−0.453882	−	0.262049i	0.255586	−	0.966786i	$-0.417731\pi$
$233$	−6.92820	−	4.00000i	−0.453882	−	0.262049i	−0.709468	+	0.704737i	$0.751065\pi$
$234$		0			0
$235$	14.9282	+	9.85641i	0.973809	+	0.642961i
$236$		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	−0.995509	−	0.0946700i	$-0.969820\pi$
$241$	−9.00000	−	15.5885i	−0.579741	−	1.00414i	0.415768	−	0.909471i	$-0.363513\pi$
$242$		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	$-0.455440\pi$
$257$	−10.3923	−	6.00000i	−0.648254	−	0.374270i	−0.787787	+	0.615948i	$0.788773\pi$
$258$		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	$-0.931818\pi$
$263$	−20.7846	+	12.0000i	−1.28163	+	0.739952i	−0.304487	+	0.952517i	$0.598485\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		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	$-0.507059\pi$
$277$	13.8564	−	8.00000i	0.832551	−	0.480673i	0.854725	+	0.519081i	$0.173726\pi$
$278$		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	0.669238	−	0.743048i	$-0.266621\pi$
$283$	27.7128	−	16.0000i	1.64736	−	0.951101i	0.978117	−	0.208053i	$-0.0667128\pi$
$284$		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
$305$	−7.39230	+	11.1962i	−0.423282	+	0.641090i
$306$		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	−0.974841	−	0.222900i	$-0.928448\pi$
$311$	−12.0000	−	20.7846i	−0.680458	−	1.17859i	0.294384	−	0.955687i	$-0.404886\pi$
$312$		0			0
$313$	3.46410	+	2.00000i	0.195803	+	0.113047i	0.594696	−	0.803951i	$-0.297272\pi$
$313$	3.46410	+	2.00000i	0.195803	+	0.113047i	−0.398894	+	0.916997i	$0.630606\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−13.8564	−	8.00000i	−0.778253	−	0.449325i	0.0575576	−	0.998342i	$-0.481669\pi$
$317$	−13.8564	−	8.00000i	−0.778253	−	0.449325i	−0.835811	+	0.549017i	$0.815002\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$	−2.39230	+	19.8564i	−0.132701	+	1.10144i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.00000	−	13.8564i	0.439720	−	0.761617i	−0.557948	−	0.829876i	$-0.688411\pi$
$331$	8.00000	−	13.8564i	0.439720	−	0.761617i	0.997668	+	0.0682590i	$0.0217444\pi$
$332$		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	$-0.443860\pi$
$347$	20.7846	−	12.0000i	1.11578	−	0.644194i	0.940319	+	0.340293i	$0.110526\pi$
$348$		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	$-0.770130\pi$
$353$	−10.3923	+	6.00000i	−0.553127	+	0.319348i	0.197256	+	0.980352i	$0.436797\pi$
$354$		0			0
$355$	26.7846	+	1.60770i	1.42158	+	0.0853276i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	0.471696	+	0.881761i	$0.343642\pi$
$359$	−10.0000	+	17.3205i	−0.527780	+	0.914141i	−0.999476	+	0.0323801i	$0.989691\pi$
$360$		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	$-0.599711\pi$
$367$	6.92820	−	4.00000i	0.361649	−	0.208798i	0.669804	+	0.742538i	$0.266378\pi$
$368$		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	$-0.0225572\pi$
$373$	27.7128	+	16.0000i	1.43492	+	0.828449i	0.437425	+	0.899255i	$0.355891\pi$
$374$		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	$-0.543440\pi$
$383$	−20.7846	−	12.0000i	−1.06204	−	0.613171i	−0.925997	+	0.377531i	$0.876773\pi$
$384$		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	0.932002	−	0.362454i	$-0.118061\pi$
$389$	3.00000	+	5.19615i	0.152106	+	0.263455i	−0.779895	+	0.625910i	$0.784728\pi$
$390$		0			0
$391$	−32.0000			−1.61831
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−4.92820	+	7.46410i	−0.247965	+	0.375560i
$396$		0			0
$397$	24.2487	+	14.0000i	1.21701	+	0.702640i	0.964277	−	0.264897i	$-0.0853379\pi$
$397$	24.2487	+	14.0000i	1.21701	+	0.702640i	0.252731	+	0.967537i	$0.418671\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		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	0.715523	−	0.698589i	$-0.246188\pi$
$409$	−5.00000	−	8.66025i	−0.247234	−	0.428222i	−0.962757	+	0.270367i	$0.912855\pi$
$410$		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
$425$	−18.3923	+	7.85641i	−0.892158	+	0.381092i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−10.0000	−	17.3205i	−0.481683	−	0.834300i	0.518096	−	0.855323i	$-0.326641\pi$
$431$	−10.0000	−	17.3205i	−0.481683	−	0.834300i	−0.999779	+	0.0210230i	$0.993308\pi$
$432$		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	−0.945552	−	0.325471i	$-0.894477\pi$
$439$	−4.00000	+	6.92820i	−0.190910	+	0.330665i	0.754642	+	0.656136i	$0.227810\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	20.7846	+	12.0000i	0.987507	+	0.570137i	0.904528	−	0.426414i	$-0.140223\pi$
$443$	20.7846	+	12.0000i	0.987507	+	0.570137i	0.0829786	+	0.996551i	$0.473557\pi$
$444$		0			0
$445$	12.3205	−	18.6603i	0.584048	−	0.884581i
$446$		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	$-0.211242\pi$
$457$	13.8564	+	8.00000i	0.648175	+	0.374224i	−0.139581	+	0.990211i	$0.544576\pi$
$458$		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	$-0.212624\pi$
$467$	13.8564	+	8.00000i	0.641198	+	0.370196i	−0.143878	+	0.989595i	$0.545957\pi$
$468$		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	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−16.0000	+	27.7128i	−0.729537	+	1.26360i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.60770	+	26.7846i	−0.0730017	+	1.21623i
$486$		0			0
$487$	−6.92820	+	4.00000i	−0.313947	+	0.181257i	−0.648691	−	0.761052i	$-0.724683\pi$
$487$	−6.92820	+	4.00000i	−0.313947	+	0.181257i	0.334744	+	0.942309i	$0.391350\pi$
$488$		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	−0.941560	−	0.336844i	$-0.890640\pi$
$499$	−4.00000	+	6.92820i	−0.179065	+	0.310149i	0.762496	+	0.646993i	$0.223974\pi$
$500$		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	−0.512331	−	0.858788i	$-0.671218\pi$
$509$	11.0000	−	19.0526i	0.487566	−	0.844490i	0.999898	+	0.0142980i	$0.00455136\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	14.9282	+	9.85641i	0.657815	+	0.434325i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	0.792797	−	0.609486i	$-0.208624\pi$
$521$	−3.00000	−	5.19615i	−0.131432	−	0.227648i	−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$	27.7128	+	16.0000i	1.21180	+	0.699631i	0.963150	−	0.268963i	$-0.0866810\pi$
$523$	27.7128	+	16.0000i	1.21180	+	0.699631i	0.248646	+	0.968594i	$0.420014\pi$
$524$		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
$535$	−14.9282	−	9.85641i	−0.645403	−	0.426130i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−7.00000	+	12.1244i	−0.300954	+	0.521267i	−0.976352	−	0.216186i	$-0.930638\pi$
$541$	−7.00000	+	12.1244i	−0.300954	+	0.521267i	0.675399	+	0.737453i	$0.263972\pi$
$542$		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	$-0.776745\pi$
$557$	−13.8564	+	8.00000i	−0.587115	+	0.338971i	0.176841	+	0.984239i	$0.443412\pi$
$558$		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	$-0.776133\pi$
$563$	−13.8564	+	8.00000i	−0.583978	+	0.337160i	0.178735	+	0.983897i	$0.442800\pi$
$564$		0			0
$565$	−2.14359	+	35.7128i	−0.0901817	+	1.50245i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.0294311	−	0.999567i	$-0.490630\pi$
$569$	21.0000	−	36.3731i	0.880366	−	1.52484i	0.850935	−	0.525271i	$-0.176036\pi$
$570$		0			0
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	0.892413	+	0.451219i	$0.149011\pi$
$571$	20.0000	+	34.6410i	0.836974	+	1.44968i	−0.0554391	+	0.998462i	$0.517656\pi$
$572$		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	$-0.640133\pi$
$577$	3.46410	−	2.00000i	0.144212	−	0.0832611i	0.570370	+	0.821388i	$0.306800\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−22.3923	−	14.7846i	−0.925808	−	0.611268i
$586$		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	$-0.0685507\pi$
$593$	31.1769	+	18.0000i	1.28028	+	0.739171i	0.303383	+	0.952869i	$0.401884\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	−0.954521	−	0.298143i	$-0.903633\pi$
$599$	−18.0000	−	31.1769i	−0.735460	−	1.27385i	0.219061	−	0.975711i	$-0.429701\pi$
$600$		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
$605$	13.5526	−	20.5263i	0.550990	−	0.834512i
$606$		0			0
$607$	−34.6410	−	20.0000i	−1.40604	−	0.811775i	−0.411033	−	0.911621i	$-0.634832\pi$
$607$	−34.6410	−	20.0000i	−1.40604	−	0.811775i	−0.995003	+	0.0998457i	$0.968165\pi$
$608$		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	$-0.505604\pi$
$613$	20.7846	−	12.0000i	0.839482	−	0.484675i	0.857088	+	0.515170i	$0.172271\pi$
$614$		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	0.823029	−	0.567999i	$-0.192282\pi$
$619$	−2.00000	−	3.46410i	−0.0803868	−	0.139234i	−0.903416	+	0.428765i	$0.858949\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	24.2846	+	5.93782i	0.971384	+	0.237513i
$626$		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
$635$	1.07180	−	17.8564i	0.0425330	−	0.708610i
$636$		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	0.885098	−	0.465404i	$-0.154091\pi$
$641$	1.00000	+	1.73205i	0.0394976	+	0.0684119i	−0.845601	+	0.533816i	$0.820758\pi$
$642$		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	$-0.823051\pi$
$647$	−20.7846	+	12.0000i	−0.817127	+	0.471769i	0.0322975	+	0.999478i	$0.489718\pi$
$648$		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	$-0.488934\pi$
$653$	−20.7846	−	12.0000i	−0.813365	−	0.469596i	−0.848123	+	0.529799i	$0.822267\pi$
$654$		0			0
$655$	−4.92820	+	7.46410i	−0.192561	+	0.291647i
$656$		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	0.980294	+	0.197542i	$0.0632958\pi$
$661$	17.0000	+	29.4449i	0.661223	+	1.14527i	−0.319071	+	0.947731i	$0.603371\pi$
$662$		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	$-0.308842\pi$
$677$	3.46410	+	2.00000i	0.133136	+	0.0768662i	−0.431953	+	0.901896i	$0.642175\pi$
$678$		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	$-0.617755\pi$
$683$	6.92820	−	4.00000i	0.265100	−	0.153056i	0.626659	+	0.779294i	$0.284422\pi$
$684$		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	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.92820	+	0.535898i	0.338666	+	0.0203278i
$696$		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	−0.916845	−	0.399244i	$-0.869273\pi$
$709$	−3.00000	+	5.19615i	−0.112667	+	0.195146i	0.804178	+	0.594389i	$0.202606\pi$
$710$		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	0.396605	+	0.917989i	$0.370188\pi$
$719$	−16.0000	+	27.7128i	−0.596699	+	1.03351i	−0.993305	+	0.115524i	$0.963145\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−9.92820	−	1.19615i	−0.368724	−	0.0444240i
$726$		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	$-0.453756\pi$
$733$	−17.3205	−	10.0000i	−0.639748	−	0.369358i	−0.784517	+	0.620107i	$0.787089\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−4.00000	−	6.92820i	−0.147142	−	0.254858i	0.783028	−	0.621987i	$-0.213674\pi$
$739$	−4.00000	−	6.92820i	−0.147142	−	0.254858i	−0.930170	+	0.367129i	$0.880341\pi$
$740$		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
$745$	7.39230	−	11.1962i	0.270833	−	0.410195i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	−0.623856	−	0.781540i	$-0.714435\pi$
$751$	10.0000	−	17.3205i	0.364905	−	0.632034i	0.988761	+	0.149505i	$0.0477681\pi$
$752$		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	−0.761057	−	0.648686i	$-0.775319\pi$
$761$	5.00000	−	8.66025i	0.181250	−	0.313934i	0.942306	+	0.334752i	$0.108652\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.60770	−	26.7846i	0.0581263	−	0.968400i
$766$		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	$-0.442517\pi$
$773$	31.1769	−	18.0000i	1.12136	−	0.647415i	0.941747	+	0.336323i	$0.109183\pi$
$774$		0			0
$775$	−15.7128	−	36.7846i	−0.564421	−	1.32134i
$776$		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	$-0.339912\pi$
$787$	41.5692	−	24.0000i	1.48178	−	0.855508i	0.999786	+	0.0206657i	$0.00657856\pi$
$788$		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	0.541748	−	0.840541i	$-0.317763\pi$
$809$	−13.0000	−	22.5167i	−0.457056	−	0.791644i	−0.998804	+	0.0488972i	$0.984429\pi$
$810$		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
$815$	−44.7846	−	29.5692i	−1.56874	−	1.03576i
$816$		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	−0.808915	−	0.587925i	$-0.799945\pi$
$821$	3.00000	−	5.19615i	0.104701	−	0.181347i	0.913616	+	0.406578i	$0.133278\pi$
$822$		0			0
$823$	−20.7846	+	12.0000i	−0.724506	+	0.418294i	−0.816409	−	0.577474i	$-0.804038\pi$
$823$	−20.7846	+	12.0000i	−0.724506	+	0.418294i	0.0919029	+	0.995768i	$0.470705\pi$
$824$		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	0.609309	−	0.792933i	$-0.291447\pi$
$829$	−11.0000	−	19.0526i	−0.382046	−	0.661723i	−0.991355	+	0.131210i	$0.958114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	3.21539	−	53.5692i	0.111273	−	1.85384i
$836$		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
$845$	−6.69615	−	0.401924i	−0.230355	−	0.0138266i
$846$		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	0.539371	−	0.842068i	$-0.318662\pi$
$857$	45.0333	−	26.0000i	1.53831	−	0.888143i	0.998938	−	0.0460748i	$-0.0146713\pi$
$858$		0			0
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	0.376381	+	0.926465i	$0.377169\pi$
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	−0.990533	+	0.137277i	$0.956165\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−6.92820	−	4.00000i	−0.235839	−	0.136162i	0.377424	−	0.926041i	$-0.376810\pi$
$863$	−6.92820	−	4.00000i	−0.235839	−	0.136162i	−0.613263	+	0.789879i	$0.710143\pi$
$864$		0			0
$865$	−22.3923	−	14.7846i	−0.761361	−	0.502692i
$866$		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	$-0.569342\pi$
$877$	−34.6410	−	20.0000i	−1.16974	−	0.675352i	−0.953620	+	0.301014i	$0.902675\pi$
$878$		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	$-0.465338\pi$
$887$	−20.7846	−	12.0000i	−0.697879	−	0.402921i	−0.806557	+	0.591156i	$0.798672\pi$
$888$		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
$905$	−31.2487	−	1.87564i	−1.03874	−	0.0623485i
$906$		0			0
$907$	6.92820	−	4.00000i	0.230047	−	0.132818i	−0.380547	−	0.924762i	$-0.624264\pi$
$907$	6.92820	−	4.00000i	0.230047	−	0.132818i	0.610594	+	0.791944i	$0.290931\pi$
$908$		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	−0.897126	−	0.441776i	$-0.854349\pi$
$919$	−2.00000	+	3.46410i	−0.0659739	+	0.114270i	0.831152	+	0.556046i	$0.187682\pi$
$920$		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	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$		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	0.989599	+	0.143856i	$0.0459502\pi$
$941$	19.0000	+	32.9090i	0.619382	+	1.07280i	−0.370216	+	0.928946i	$0.620716\pi$
$942$		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	$-0.460840\pi$
$947$	−20.7846	−	12.0000i	−0.675409	−	0.389948i	−0.798123	+	0.602494i	$0.794174\pi$
$948$		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
$955$	24.6410	−	37.3205i	0.797365	−	1.20766i
$956$		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	0.247193	−	0.968966i	$-0.420492\pi$
$971$	30.0000	−	51.9615i	0.962746	−	1.66752i	0.715553	−	0.698558i	$-0.246175\pi$
$972$		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	$-0.792091\pi$
$977$	−20.7846	+	12.0000i	−0.664959	+	0.383914i	0.129205	+	0.991618i	$0.458757\pi$
$978$		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	$-0.446470\pi$
$983$	34.6410	−	20.0000i	1.10488	−	0.637901i	0.937498	+	0.347992i	$0.113136\pi$
$984$		0			0
$985$	−2.14359	+	35.7128i	−0.0683006	+	1.13790i
$986$		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	0.832510	−	0.554010i	$-0.186903\pi$
$991$	−2.00000	−	3.46410i	−0.0635321	−	0.110041i	−0.896042	+	0.443969i	$0.853570\pi$
$992$		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	$-0.812889\pi$
$997$	−24.2487	+	14.0000i	−0.767964	+	0.443384i	0.0641836	+	0.997938i	$0.479556\pi$
$998$		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.569.2		4
5.4	even	2	inner	980.2.q.e.569.1		4
7.2	even	3		980.2.e.a.589.1		2
7.3	odd	6		980.2.q.d.949.2		4
7.4	even	3	inner	980.2.q.e.949.1		4
7.5	odd	6		140.2.e.b.29.2	yes	2
7.6	odd	2		980.2.q.d.569.1		4
21.5	even	6		1260.2.k.b.1009.1		2
28.19	even	6		560.2.g.c.449.2		2
35.2	odd	12		4900.2.a.m.1.1		1
35.4	even	6	inner	980.2.q.e.949.2		4
35.9	even	6		980.2.e.a.589.2		2
35.12	even	12		700.2.a.f.1.1		1
35.19	odd	6		140.2.e.b.29.1	✓	2
35.23	odd	12		4900.2.a.l.1.1		1
35.24	odd	6		980.2.q.d.949.1		4
35.33	even	12		700.2.a.h.1.1		1
35.34	odd	2		980.2.q.d.569.2		4
56.5	odd	6		2240.2.g.d.449.1		2
56.19	even	6		2240.2.g.c.449.1		2
84.47	odd	6		5040.2.t.g.1009.1		2
105.47	odd	12		6300.2.a.g.1.1		1
105.68	odd	12		6300.2.a.y.1.1		1
105.89	even	6		1260.2.k.b.1009.2		2
140.19	even	6		560.2.g.c.449.1		2
140.47	odd	12		2800.2.a.s.1.1		1
140.103	odd	12		2800.2.a.o.1.1		1
280.19	even	6		2240.2.g.c.449.2		2
280.229	odd	6		2240.2.g.d.449.2		2
420.299	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	35.19	odd	6
140.2.e.b.29.2	yes	2	7.5	odd	6
560.2.g.c.449.1		2	140.19	even	6
560.2.g.c.449.2		2	28.19	even	6
700.2.a.f.1.1		1	35.12	even	12
700.2.a.h.1.1		1	35.33	even	12
980.2.e.a.589.1		2	7.2	even	3
980.2.e.a.589.2		2	35.9	even	6
980.2.q.d.569.1		4	7.6	odd	2
980.2.q.d.569.2		4	35.34	odd	2
980.2.q.d.949.1		4	35.24	odd	6
980.2.q.d.949.2		4	7.3	odd	6
980.2.q.e.569.1		4	5.4	even	2	inner
980.2.q.e.569.2		4	1.1	even	1	trivial
980.2.q.e.949.1		4	7.4	even	3	inner
980.2.q.e.949.2		4	35.4	even	6	inner
1260.2.k.b.1009.1		2	21.5	even	6
1260.2.k.b.1009.2		2	105.89	even	6
2240.2.g.c.449.1		2	56.19	even	6
2240.2.g.c.449.2		2	280.19	even	6
2240.2.g.d.449.1		2	56.5	odd	6
2240.2.g.d.449.2		2	280.229	odd	6
2800.2.a.o.1.1		1	140.103	odd	12
2800.2.a.s.1.1		1	140.47	odd	12
4900.2.a.l.1.1		1	35.23	odd	12
4900.2.a.m.1.1		1	35.2	odd	12
5040.2.t.g.1009.1		2	84.47	odd	6
5040.2.t.g.1009.2		2	420.299	odd	6
6300.2.a.g.1.1		1	105.47	odd	12
6300.2.a.y.1.1		1	105.68	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)

\(f(q)\)	\(=\)	\(q+(2.23205 + 0.133975i) q^{5} +(-1.50000 + 2.59808i) q^{9} +4.00000i q^{13} +(-3.46410 + 2.00000i) q^{17} +(-2.00000 + 3.46410i) q^{19} +(6.92820 + 4.00000i) q^{23} +(4.96410 + 0.598076i) q^{25} -2.00000 q^{29} +(-4.00000 - 6.92820i) q^{31} +(6.92820 + 4.00000i) q^{37} -6.00000 q^{41} -8.00000i q^{43} +(-3.69615 + 5.59808i) q^{45} +(6.92820 + 4.00000i) q^{47} +(2.00000 + 3.46410i) q^{59} +(-3.00000 + 5.19615i) q^{61} +(-0.535898 + 8.92820i) q^{65} +(6.92820 - 4.00000i) q^{67} +12.0000 q^{71} +(-3.46410 + 2.00000i) q^{73} +(-2.00000 + 3.46410i) q^{79} +(-4.50000 - 7.79423i) q^{81} +(-8.00000 + 4.00000i) q^{85} +(5.00000 - 8.66025i) q^{89} +(-4.92820 + 7.46410i) q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} - 6 q^{9} - 8 q^{19} + 6 q^{25} - 8 q^{29} - 16 q^{31} - 24 q^{41} + 6 q^{45} + 8 q^{59} - 12 q^{61} - 16 q^{65} + 48 q^{71} - 8 q^{79} - 18 q^{81} - 32 q^{85} + 20 q^{89} + 8 q^{95}+O(q^{100}) \) 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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