LMFDB - Embedded newform 2016.2.s.o.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2016,2,Mod(289,2016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2016, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2016.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2016 = 2^{5} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2016.s (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.0978410475$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 49 $ x^4 + 7*x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$-1.32288 - 2.29129i$ of defining polynomial
Character	$\chi$	$=$	2016.289
Dual form			2016.2.s.o.865.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 2.59808i) q^{5} +2.64575 q^{7} +O(q^{10})$	$q+(-1.50000 + 2.59808i) q^{5} +2.64575 q^{7} +(1.32288 + 2.29129i) q^{11} -4.00000 q^{13} +(0.500000 + 0.866025i) q^{17} +(-3.96863 + 6.87386i) q^{19} +(1.32288 - 2.29129i) q^{23} +(-2.00000 - 3.46410i) q^{25} +4.00000 q^{29} +(-1.32288 - 2.29129i) q^{31} +(-3.96863 + 6.87386i) q^{35} +(2.50000 - 4.33013i) q^{37} -8.00000 q^{41} -10.5830 q^{43} +(-1.32288 + 2.29129i) q^{47} +7.00000 q^{49} +(3.50000 + 6.06218i) q^{53} -7.93725 q^{55} +(-1.32288 - 2.29129i) q^{59} +(2.50000 - 4.33013i) q^{61} +(6.00000 - 10.3923i) q^{65} +(1.32288 + 2.29129i) q^{67} +(4.50000 + 7.79423i) q^{73} +(3.50000 + 6.06218i) q^{77} +(-1.32288 + 2.29129i) q^{79} -10.5830 q^{83} -3.00000 q^{85} +(-4.50000 + 7.79423i) q^{89} -10.5830 q^{91} +(-11.9059 - 20.6216i) q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5}+O(q^{10}) $ 4 * q - 6 * q^5	$ 4 q - 6 q^{5} - 16 q^{13} + 2 q^{17} - 8 q^{25} + 16 q^{29} + 10 q^{37} - 32 q^{41} + 28 q^{49} + 14 q^{53} + 10 q^{61} + 24 q^{65} + 18 q^{73} + 14 q^{77} - 12 q^{85} - 18 q^{89} - 32 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 16 * q^13 + 2 * q^17 - 8 * q^25 + 16 * q^29 + 10 * q^37 - 32 * q^41 + 28 * q^49 + 14 * q^53 + 10 * q^61 + 24 * q^65 + 18 * q^73 + 14 * q^77 - 12 * q^85 - 18 * q^89 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1765$	$1793$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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
$3$		0			0
$4$		0			0
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	0.306851	+	0.951757i	$0.400725\pi$
$5$	−1.50000	+	2.59808i	−0.670820	+	1.16190i	−0.977672	+	0.210138i	$0.932609\pi$
$6$		0			0
$7$	2.64575			1.00000
$7$	2.64575			1.00000
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.32288	+	2.29129i	0.398862	+	0.690849i	0.993586	−	0.113081i	$-0.0360719\pi$
$11$	1.32288	+	2.29129i	0.398862	+	0.690849i	−0.594724	+	0.803930i	$0.702739\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.500000	+	0.866025i	0.121268	+	0.210042i	0.920268	−	0.391289i	$-0.127971\pi$
$17$	0.500000	+	0.866025i	0.121268	+	0.210042i	−0.799000	+	0.601331i	$0.794637\pi$
$18$		0			0
$19$	−3.96863	+	6.87386i	−0.910465	+	1.57697i	−0.0970575	+	0.995279i	$0.530943\pi$
$19$	−3.96863	+	6.87386i	−0.910465	+	1.57697i	−0.813408	+	0.581694i	$0.802390\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.32288	−	2.29129i	0.275839	−	0.477767i	−0.694508	−	0.719485i	$-0.744378\pi$
$23$	1.32288	−	2.29129i	0.275839	−	0.477767i	0.970346	+	0.241719i	$0.0777111\pi$
$24$		0			0
$25$	−2.00000	−	3.46410i	−0.400000	−	0.692820i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$	−1.32288	−	2.29129i	−0.237595	−	0.411527i	0.722428	−	0.691446i	$-0.243026\pi$
$31$	−1.32288	−	2.29129i	−0.237595	−	0.411527i	−0.960024	+	0.279918i	$0.909693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.96863	+	6.87386i	−0.670820	+	1.16190i
$36$		0			0
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	−0.584002	−	0.811752i	$-0.698514\pi$
$37$	2.50000	−	4.33013i	0.410997	−	0.711868i	0.994999	+	0.0998840i	$0.0318472\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	−10.5830			−1.61389			−0.806947	−	0.590624i	$-0.798881\pi$
$43$	−10.5830			−1.61389			−0.806947	+	0.590624i	$0.798881\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.32288	+	2.29129i	−0.192961	+	0.334219i	−0.946230	−	0.323494i	$-0.895142\pi$
$47$	−1.32288	+	2.29129i	−0.192961	+	0.334219i	0.753269	+	0.657713i	$0.228476\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.50000	+	6.06218i	0.480762	+	0.832704i	0.999756	−	0.0220735i	$-0.00702678\pi$
$53$	3.50000	+	6.06218i	0.480762	+	0.832704i	−0.518994	+	0.854778i	$0.673693\pi$
$54$		0			0
$55$	−7.93725			−1.07026
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.32288	−	2.29129i	−0.172224	−	0.298300i	0.766973	−	0.641679i	$-0.221762\pi$
$59$	−1.32288	−	2.29129i	−0.172224	−	0.298300i	−0.939197	+	0.343379i	$0.888429\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	−	10.3923i	0.744208	−	1.28901i
$66$		0			0
$67$	1.32288	+	2.29129i	0.161615	+	0.279925i	0.935448	−	0.353464i	$-0.114996\pi$
$67$	1.32288	+	2.29129i	0.161615	+	0.279925i	−0.773833	+	0.633390i	$0.781663\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	4.50000	+	7.79423i	0.526685	+	0.912245i	0.999517	+	0.0310925i	$0.00989865\pi$
$73$	4.50000	+	7.79423i	0.526685	+	0.912245i	−0.472831	+	0.881153i	$0.656768\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.50000	+	6.06218i	0.398862	+	0.690849i
$78$		0			0
$79$	−1.32288	+	2.29129i	−0.148835	+	0.257790i	−0.930797	−	0.365536i	$-0.880886\pi$
$79$	−1.32288	+	2.29129i	−0.148835	+	0.257790i	0.781962	+	0.623326i	$0.214219\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−10.5830			−1.16164			−0.580818	−	0.814034i	$-0.697267\pi$
$83$	−10.5830			−1.16164			−0.580818	+	0.814034i	$0.697267\pi$
$84$		0			0
$85$	−3.00000			−0.325396
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	−0.999653	−	0.0263586i	$-0.991609\pi$
$89$	−4.50000	+	7.79423i	−0.476999	+	0.826187i	0.522654	+	0.852545i	$0.324942\pi$
$90$		0			0
$91$	−10.5830			−1.10940
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−11.9059	−	20.6216i	−1.22152	−	2.11573i
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	−0.949595	−	0.313478i	$-0.898506\pi$
$101$	−7.50000	−	12.9904i	−0.746278	−	1.29259i	0.203317	−	0.979113i	$-0.434828\pi$
$102$		0			0
$103$	−1.32288	+	2.29129i	−0.130347	+	0.225767i	−0.923810	−	0.382851i	$-0.874942\pi$
$103$	−1.32288	+	2.29129i	−0.130347	+	0.225767i	0.793463	+	0.608618i	$0.208276\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.61438	+	11.4564i	−0.639436	+	1.10754i	0.346121	+	0.938190i	$0.387499\pi$
$107$	−6.61438	+	11.4564i	−0.639436	+	1.10754i	−0.985557	+	0.169346i	$0.945835\pi$
$108$		0			0
$109$	−0.500000	−	0.866025i	−0.0478913	−	0.0829502i	0.841086	−	0.540901i	$-0.181917\pi$
$109$	−0.500000	−	0.866025i	−0.0478913	−	0.0829502i	−0.888977	+	0.457951i	$0.848583\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	3.96863	+	6.87386i	0.370076	+	0.640991i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.32288	+	2.29129i	0.121268	+	0.210042i
$120$		0			0
$121$	2.00000	−	3.46410i	0.181818	−	0.314918i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−3.00000			−0.268328
$126$		0			0
$127$	−21.1660			−1.87818			−0.939090	−	0.343672i	$-0.888329\pi$
$127$	−21.1660			−1.87818			−0.939090	+	0.343672i	$0.888329\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.96863	+	6.87386i	−0.346741	+	0.600572i	−0.985668	−	0.168694i	$-0.946045\pi$
$131$	−3.96863	+	6.87386i	−0.346741	+	0.600572i	0.638928	+	0.769267i	$0.279378\pi$
$132$		0			0
$133$	−10.5000	+	18.1865i	−0.910465	+	1.57697i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.500000	+	0.866025i	0.0427179	+	0.0739895i	0.886594	−	0.462549i	$-0.153065\pi$
$137$	0.500000	+	0.866025i	0.0427179	+	0.0739895i	−0.843876	+	0.536538i	$0.819732\pi$
$138$		0			0
$139$	10.5830			0.897639			0.448819	−	0.893622i	$-0.351845\pi$
$139$	10.5830			0.897639			0.448819	+	0.893622i	$0.351845\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−5.29150	−	9.16515i	−0.442498	−	0.766428i
$144$		0			0
$145$	−6.00000	+	10.3923i	−0.498273	+	0.863034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	−0.0115483	−	0.999933i	$-0.503676\pi$
$149$	10.5000	−	18.1865i	0.860194	−	1.48990i	0.871742	−	0.489966i	$-0.162991\pi$
$150$		0			0
$151$	−1.32288	−	2.29129i	−0.107654	−	0.186462i	0.807165	−	0.590325i	$-0.201001\pi$
$151$	−1.32288	−	2.29129i	−0.107654	−	0.186462i	−0.914819	+	0.403863i	$0.867667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.93725			0.637536
$156$		0			0
$157$	4.50000	+	7.79423i	0.359139	+	0.622047i	0.987817	−	0.155618i	$-0.0497370\pi$
$157$	4.50000	+	7.79423i	0.359139	+	0.622047i	−0.628678	+	0.777666i	$0.716404\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.50000	−	6.06218i	0.275839	−	0.477767i
$162$		0			0
$163$	−6.61438	+	11.4564i	−0.518078	+	0.897338i	0.481701	+	0.876335i	$0.340019\pi$
$163$	−6.61438	+	11.4564i	−0.518078	+	0.897338i	−0.999779	+	0.0210021i	$0.993314\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	5.29150			0.409469			0.204734	−	0.978818i	$-0.434367\pi$
$167$	5.29150			0.409469			0.204734	+	0.978818i	$0.434367\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.50000	−	2.59808i	0.114043	−	0.197528i	−0.803354	−	0.595502i	$-0.796953\pi$
$173$	1.50000	−	2.59808i	0.114043	−	0.197528i	0.917397	+	0.397974i	$0.130287\pi$
$174$		0			0
$175$	−5.29150	−	9.16515i	−0.400000	−	0.692820i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	11.9059	+	20.6216i	0.889887	+	1.54133i	0.840007	+	0.542575i	$0.182551\pi$
$179$	11.9059	+	20.6216i	0.889887	+	1.54133i	0.0498798	+	0.998755i	$0.484116\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	7.50000	+	12.9904i	0.551411	+	0.955072i
$186$		0			0
$187$	−1.32288	+	2.29129i	−0.0967382	+	0.167556i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−11.9059	+	20.6216i	−0.861479	+	1.49213i	0.00902170	+	0.999959i	$0.497128\pi$
$191$	−11.9059	+	20.6216i	−0.861479	+	1.49213i	−0.870501	+	0.492167i	$0.836205\pi$
$192$		0			0
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	0.964059	−	0.265689i	$-0.0855996\pi$
$193$	3.50000	+	6.06218i	0.251936	+	0.436365i	−0.712123	+	0.702055i	$0.752266\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	9.26013	+	16.0390i	0.656433	+	1.13698i	0.981532	+	0.191296i	$0.0612689\pi$
$199$	9.26013	+	16.0390i	0.656433	+	1.13698i	−0.325099	+	0.945680i	$0.605398\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.5830			0.742781
$204$		0			0
$205$	12.0000	−	20.7846i	0.838116	−	1.45166i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−21.0000			−1.45260
$210$		0			0
$211$	10.5830			0.728564			0.364282	−	0.931289i	$-0.381314\pi$
$211$	10.5830			0.728564			0.364282	+	0.931289i	$0.381314\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.8745	−	27.4955i	1.08263	−	1.87517i
$216$		0			0
$217$	−3.50000	−	6.06218i	−0.237595	−	0.411527i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.00000	−	3.46410i	−0.134535	−	0.233021i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.32288	+	2.29129i	0.0878023	+	0.152078i	0.906582	−	0.422030i	$-0.138682\pi$
$227$	1.32288	+	2.29129i	0.0878023	+	0.152078i	−0.818780	+	0.574108i	$0.805349\pi$
$228$		0			0
$229$	5.50000	−	9.52628i	0.363450	−	0.629514i	−0.625076	−	0.780564i	$-0.714932\pi$
$229$	5.50000	−	9.52628i	0.363450	−	0.629514i	0.988526	+	0.151050i	$0.0482653\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	−0.680135	−	0.733087i	$-0.738079\pi$
$233$	4.50000	−	7.79423i	0.294805	−	0.510617i	0.974939	+	0.222470i	$0.0714120\pi$
$234$		0			0
$235$	−3.96863	−	6.87386i	−0.258885	−	0.448401i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	15.8745			1.02684			0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745			1.02684			0.513418	+	0.858138i	$0.328379\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	0.881680	−	0.471848i	$-0.156413\pi$
$241$	0.500000	+	0.866025i	0.0322078	+	0.0557856i	−0.849472	+	0.527633i	$0.823079\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.5000	+	18.1865i	−0.670820	+	1.16190i
$246$		0			0
$247$	15.8745	−	27.4955i	1.01007	−	1.74949i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	5.29150			0.333997			0.166998	−	0.985957i	$-0.446593\pi$
$251$	5.29150			0.333997			0.166998	+	0.985957i	$0.446593\pi$
$252$		0			0
$253$	7.00000			0.440086
$254$		0			0
$255$		0			0
$256$		0			0
$257$	8.50000	−	14.7224i	0.530215	−	0.918360i	−0.469163	−	0.883112i	$-0.655444\pi$
$257$	8.50000	−	14.7224i	0.530215	−	0.918360i	0.999379	−	0.0352486i	$-0.0112223\pi$
$258$		0			0
$259$	6.61438	−	11.4564i	0.410997	−	0.711868i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.26013	+	16.0390i	0.571004	+	0.989008i	0.996463	+	0.0840304i	$0.0267793\pi$
$263$	9.26013	+	16.0390i	0.571004	+	0.989008i	−0.425459	+	0.904978i	$0.639887\pi$
$264$		0			0
$265$	−21.0000			−1.29002
$266$		0			0
$267$		0			0
$268$		0			0
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	0.998816	−	0.0486418i	$-0.0154893\pi$
$269$	7.50000	+	12.9904i	0.457283	+	0.792038i	−0.541533	+	0.840679i	$0.682156\pi$
$270$		0			0
$271$	−1.32288	+	2.29129i	−0.0803590	+	0.139186i	−0.903404	−	0.428790i	$-0.858940\pi$
$271$	−1.32288	+	2.29129i	−0.0803590	+	0.139186i	0.823045	+	0.567976i	$0.192273\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	5.29150	−	9.16515i	0.319090	−	0.552679i
$276$		0			0
$277$	12.5000	+	21.6506i	0.751052	+	1.30086i	0.947313	+	0.320309i	$0.103787\pi$
$277$	12.5000	+	21.6506i	0.751052	+	1.30086i	−0.196261	+	0.980552i	$0.562880\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	24.0000			1.43172			0.715860	−	0.698244i	$-0.246035\pi$
$281$	24.0000			1.43172			0.715860	+	0.698244i	$0.246035\pi$
$282$		0			0
$283$	9.26013	+	16.0390i	0.550458	+	0.953420i	0.998241	+	0.0592787i	$0.0188801\pi$
$283$	9.26013	+	16.0390i	0.550458	+	0.953420i	−0.447784	+	0.894142i	$0.647787\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−21.1660			−1.24939
$288$		0			0
$289$	8.00000	−	13.8564i	0.470588	−	0.815083i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.0000			0.584206			0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000			0.584206			0.292103	+	0.956387i	$0.405645\pi$
$294$		0			0
$295$	7.93725			0.462125
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5.29150	+	9.16515i	−0.306015	+	0.530034i
$300$		0			0
$301$	−28.0000			−1.61389
$302$		0			0
$303$		0			0
$304$		0			0
$305$	7.50000	+	12.9904i	0.429449	+	0.743827i
$306$		0			0
$307$	−10.5830			−0.604004			−0.302002	−	0.953307i	$-0.597655\pi$
$307$	−10.5830			−0.604004			−0.302002	+	0.953307i	$0.597655\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.9059	−	20.6216i	−0.675121	−	1.16934i	−0.976434	−	0.215818i	$-0.930758\pi$
$311$	−11.9059	−	20.6216i	−0.675121	−	1.16934i	0.301313	−	0.953525i	$-0.402575\pi$
$312$		0			0
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	−0.519300	−	0.854592i	$-0.673807\pi$
$313$	8.50000	−	14.7224i	0.480448	−	0.832161i	0.999748	+	0.0224310i	$0.00714060\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.50000	+	11.2583i	−0.365076	+	0.632331i	−0.988788	−	0.149323i	$-0.952290\pi$
$317$	−6.50000	+	11.2583i	−0.365076	+	0.632331i	0.623712	+	0.781654i	$0.285624\pi$
$318$		0			0
$319$	5.29150	+	9.16515i	0.296267	+	0.513150i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−7.93725			−0.441641
$324$		0			0
$325$	8.00000	+	13.8564i	0.443760	+	0.768615i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.50000	+	6.06218i	−0.192961	+	0.334219i
$330$		0			0
$331$	3.96863	−	6.87386i	0.218135	−	0.377822i	−0.736102	−	0.676870i	$-0.763336\pi$
$331$	3.96863	−	6.87386i	0.218135	−	0.377822i	0.954238	+	0.299048i	$0.0966692\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.93725			−0.433659
$336$		0			0
$337$	24.0000			1.30736			0.653682	−	0.756770i	$-0.273224\pi$
$337$	24.0000			1.30736			0.653682	+	0.756770i	$0.273224\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3.50000	−	6.06218i	0.189536	−	0.328285i
$342$		0			0
$343$	18.5203			1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.9059	−	20.6216i	−0.639141	−	1.10702i	−0.985622	−	0.168968i	$-0.945957\pi$
$347$	−11.9059	−	20.6216i	−0.639141	−	1.10702i	0.346480	−	0.938057i	$-0.387377\pi$
$348$		0			0
$349$	20.0000			1.07058			0.535288	−	0.844670i	$-0.320203\pi$
$349$	20.0000			1.07058			0.535288	+	0.844670i	$0.320203\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.50000	+	7.79423i	0.239511	+	0.414845i	0.960574	−	0.278024i	$-0.0896796\pi$
$353$	4.50000	+	7.79423i	0.239511	+	0.414845i	−0.721063	+	0.692869i	$0.756346\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	9.26013	−	16.0390i	0.488731	−	0.846507i	−0.511185	−	0.859471i	$-0.670793\pi$
$359$	9.26013	−	16.0390i	0.488731	−	0.846507i	0.999916	+	0.0129639i	$0.00412664\pi$
$360$		0			0
$361$	−22.0000	−	38.1051i	−1.15789	−	2.00553i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−27.0000			−1.41324
$366$		0			0
$367$	−9.26013	−	16.0390i	−0.483375	−	0.837230i	0.516443	−	0.856322i	$-0.327256\pi$
$367$	−9.26013	−	16.0390i	−0.483375	−	0.837230i	−0.999818	+	0.0190919i	$0.993923\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	9.26013	+	16.0390i	0.480762	+	0.832704i
$372$		0			0
$373$	10.5000	−	18.1865i	0.543669	−	0.941663i	−0.455020	−	0.890481i	$-0.650368\pi$
$373$	10.5000	−	18.1865i	0.543669	−	0.941663i	0.998689	−	0.0511818i	$-0.0162988\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−16.0000			−0.824042
$378$		0			0
$379$	5.29150			0.271806			0.135903	−	0.990722i	$-0.456606\pi$
$379$	5.29150			0.271806			0.135903	+	0.990722i	$0.456606\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	9.26013	−	16.0390i	0.473171	−	0.819555i	−0.526358	−	0.850263i	$-0.676443\pi$
$383$	9.26013	−	16.0390i	0.473171	−	0.819555i	0.999528	+	0.0307077i	$0.00977611\pi$
$384$		0			0
$385$	−21.0000			−1.07026
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.50000	−	12.9904i	−0.380265	−	0.658638i	0.610835	−	0.791758i	$-0.290834\pi$
$389$	−7.50000	−	12.9904i	−0.380265	−	0.658638i	−0.991100	+	0.133120i	$0.957501\pi$
$390$		0			0
$391$	2.64575			0.133801
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.96863	−	6.87386i	−0.199683	−	0.345862i
$396$		0			0
$397$	−2.50000	+	4.33013i	−0.125471	+	0.217323i	−0.921917	−	0.387387i	$-0.873378\pi$
$397$	−2.50000	+	4.33013i	−0.125471	+	0.217323i	0.796446	+	0.604710i	$0.206711\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−11.5000	+	19.9186i	−0.574283	+	0.994687i	0.421837	+	0.906672i	$0.361386\pi$
$401$	−11.5000	+	19.9186i	−0.574283	+	0.994687i	−0.996119	+	0.0880147i	$0.971948\pi$
$402$		0			0
$403$	5.29150	+	9.16515i	0.263589	+	0.456549i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	13.2288			0.655725
$408$		0			0
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	0.996701	+	0.0811615i	$0.0258630\pi$
$409$	11.5000	+	19.9186i	0.568638	+	0.984911i	−0.428063	+	0.903749i	$0.640804\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−3.50000	−	6.06218i	−0.172224	−	0.298300i
$414$		0			0
$415$	15.8745	−	27.4955i	0.779249	−	1.34970i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.7490			−1.55104			−0.775520	−	0.631322i	$-0.782512\pi$
$419$	−31.7490			−1.55104			−0.775520	+	0.631322i	$0.782512\pi$
$420$		0			0
$421$	−12.0000			−0.584844			−0.292422	−	0.956289i	$-0.594461\pi$
$421$	−12.0000			−0.584844			−0.292422	+	0.956289i	$0.594461\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.00000	−	3.46410i	0.0970143	−	0.168034i
$426$		0			0
$427$	6.61438	−	11.4564i	0.320092	−	0.554416i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.32288	+	2.29129i	0.0637207	+	0.110367i	0.896126	−	0.443800i	$-0.146370\pi$
$431$	1.32288	+	2.29129i	0.0637207	+	0.110367i	−0.832405	+	0.554168i	$0.813037\pi$
$432$		0			0
$433$	−32.0000			−1.53782			−0.768911	−	0.639356i	$-0.779201\pi$
$433$	−32.0000			−1.53782			−0.768911	+	0.639356i	$0.779201\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	10.5000	+	18.1865i	0.502283	+	0.869980i
$438$		0			0
$439$	11.9059	−	20.6216i	0.568237	−	0.984215i	−0.428504	−	0.903540i	$-0.640959\pi$
$439$	11.9059	−	20.6216i	0.568237	−	0.984215i	0.996740	−	0.0806748i	$-0.0257075\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.96863	−	6.87386i	0.188555	−	0.326587i	−0.756214	−	0.654325i	$-0.772953\pi$
$443$	3.96863	−	6.87386i	0.188555	−	0.326587i	0.944769	+	0.327738i	$0.106286\pi$
$444$		0			0
$445$	−13.5000	−	23.3827i	−0.639961	−	1.10845i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$	−10.5830	−	18.3303i	−0.498334	−	0.863140i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15.8745	−	27.4955i	0.744208	−	1.28901i
$456$		0			0
$457$	−3.50000	+	6.06218i	−0.163723	+	0.283577i	−0.936201	−	0.351465i	$-0.885684\pi$
$457$	−3.50000	+	6.06218i	−0.163723	+	0.283577i	0.772478	+	0.635042i	$0.219017\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	20.0000			0.931493			0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000			0.931493			0.465746	+	0.884918i	$0.345786\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.96863	+	6.87386i	−0.183646	+	0.318084i	−0.943119	−	0.332454i	$-0.892123\pi$
$467$	−3.96863	+	6.87386i	−0.183646	+	0.318084i	0.759473	+	0.650538i	$0.225457\pi$
$468$		0			0
$469$	3.50000	+	6.06218i	0.161615	+	0.279925i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−14.0000	−	24.2487i	−0.643721	−	1.11496i
$474$		0			0
$475$	31.7490			1.45674
$476$		0			0
$477$		0			0
$478$		0			0
$479$	19.8431	+	34.3693i	0.906656	+	1.57037i	0.818679	+	0.574252i	$0.194707\pi$
$479$	19.8431	+	34.3693i	0.906656	+	1.57037i	0.0879772	+	0.996122i	$0.471960\pi$
$480$		0			0
$481$	−10.0000	+	17.3205i	−0.455961	+	0.789747i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	12.0000	−	20.7846i	0.544892	−	0.943781i
$486$		0			0
$487$	11.9059	+	20.6216i	0.539507	+	0.934453i	0.998931	+	0.0462362i	$0.0147227\pi$
$487$	11.9059	+	20.6216i	0.539507	+	0.934453i	−0.459424	+	0.888217i	$0.651944\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	37.0405			1.67162			0.835808	−	0.549022i	$-0.185000\pi$
$491$	37.0405			1.67162			0.835808	+	0.549022i	$0.185000\pi$
$492$		0			0
$493$	2.00000	+	3.46410i	0.0900755	+	0.156015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−6.61438	+	11.4564i	−0.296100	+	0.512861i	−0.975240	−	0.221148i	$-0.929020\pi$
$499$	−6.61438	+	11.4564i	−0.296100	+	0.512861i	0.679140	+	0.734009i	$0.262353\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	21.1660			0.943746			0.471873	−	0.881667i	$-0.343578\pi$
$503$	21.1660			0.943746			0.471873	+	0.881667i	$0.343578\pi$
$504$		0			0
$505$	45.0000			2.00247
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.50000	+	16.4545i	−0.421080	+	0.729332i	−0.996045	−	0.0888457i	$-0.971682\pi$
$509$	−9.50000	+	16.4545i	−0.421080	+	0.729332i	0.574965	+	0.818178i	$0.305016\pi$
$510$		0			0
$511$	11.9059	+	20.6216i	0.526685	+	0.912245i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3.96863	−	6.87386i	−0.174879	−	0.302899i
$516$		0			0
$517$	−7.00000			−0.307860
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11.5000	+	19.9186i	0.503824	+	0.872649i	0.999990	+	0.00442139i	$0.00140738\pi$
$521$	11.5000	+	19.9186i	0.503824	+	0.872649i	−0.496166	+	0.868228i	$0.665259\pi$
$522$		0			0
$523$	17.1974	−	29.7867i	0.751989	−	1.30248i	−0.194868	−	0.980829i	$-0.562428\pi$
$523$	17.1974	−	29.7867i	0.751989	−	1.30248i	0.946857	−	0.321654i	$-0.104239\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.32288	−	2.29129i	0.0576254	−	0.0998101i
$528$		0			0
$529$	8.00000	+	13.8564i	0.347826	+	0.602452i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	32.0000			1.38607
$534$		0			0
$535$	−19.8431	−	34.3693i	−0.857894	−	1.48592i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.26013	+	16.0390i	0.398862	+	0.690849i
$540$		0			0
$541$	−10.5000	+	18.1865i	−0.451430	+	0.781900i	−0.998475	−	0.0552031i	$-0.982419\pi$
$541$	−10.5000	+	18.1865i	−0.451430	+	0.781900i	0.547045	+	0.837103i	$0.315753\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3.00000			0.128506
$546$		0			0
$547$	−26.4575			−1.13124			−0.565621	−	0.824665i	$-0.691363\pi$
$547$	−26.4575			−1.13124			−0.565621	+	0.824665i	$0.691363\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−15.8745	+	27.4955i	−0.676277	+	1.17135i
$552$		0			0
$553$	−3.50000	+	6.06218i	−0.148835	+	0.257790i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−19.5000	−	33.7750i	−0.826242	−	1.43109i	−0.900967	−	0.433888i	$-0.857141\pi$
$557$	−19.5000	−	33.7750i	−0.826242	−	1.43109i	0.0747252	−	0.997204i	$-0.476192\pi$
$558$		0			0
$559$	42.3320			1.79045
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.32288	−	2.29129i	−0.0557526	−	0.0965663i	0.836802	−	0.547505i	$-0.184422\pi$
$563$	−1.32288	−	2.29129i	−0.0557526	−	0.0965663i	−0.892555	+	0.450939i	$0.851089\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−16.5000	+	28.5788i	−0.691716	+	1.19809i	0.279559	+	0.960128i	$0.409812\pi$
$569$	−16.5000	+	28.5788i	−0.691716	+	1.19809i	−0.971275	+	0.237959i	$0.923522\pi$
$570$		0			0
$571$	−1.32288	−	2.29129i	−0.0553606	−	0.0958874i	0.837017	−	0.547177i	$-0.184298\pi$
$571$	−1.32288	−	2.29129i	−0.0553606	−	0.0958874i	−0.892378	+	0.451290i	$0.850964\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−10.5830			−0.441342
$576$		0			0
$577$	20.5000	+	35.5070i	0.853426	+	1.47818i	0.878097	+	0.478482i	$0.158813\pi$
$577$	20.5000	+	35.5070i	0.853426	+	1.47818i	−0.0246713	+	0.999696i	$0.507854\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−28.0000			−1.16164
$582$		0			0
$583$	−9.26013	+	16.0390i	−0.383515	+	0.664268i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	31.7490			1.31042			0.655211	−	0.755446i	$-0.272580\pi$
$587$	31.7490			1.31042			0.655211	+	0.755446i	$0.272580\pi$
$588$		0			0
$589$	21.0000			0.865290
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.5000	−	28.5788i	0.677574	−	1.17359i	−0.298136	−	0.954524i	$-0.596365\pi$
$593$	16.5000	−	28.5788i	0.677574	−	1.17359i	0.975709	−	0.219069i	$-0.0703019\pi$
$594$		0			0
$595$	−7.93725			−0.325396
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−11.9059	−	20.6216i	−0.486461	−	0.842575i	0.513418	−	0.858139i	$-0.328379\pi$
$599$	−11.9059	−	20.6216i	−0.486461	−	0.842575i	−0.999879	+	0.0155634i	$0.995046\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	6.00000	+	10.3923i	0.243935	+	0.422507i
$606$		0			0
$607$	−19.8431	+	34.3693i	−0.805408	+	1.39501i	0.110607	+	0.993864i	$0.464720\pi$
$607$	−19.8431	+	34.3693i	−0.805408	+	1.39501i	−0.916015	+	0.401143i	$0.868613\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.29150	−	9.16515i	0.214071	−	0.370782i
$612$		0			0
$613$	−20.5000	−	35.5070i	−0.827987	−	1.43412i	−0.899615	−	0.436684i	$-0.856153\pi$
$613$	−20.5000	−	35.5070i	−0.827987	−	1.43412i	0.0716275	−	0.997431i	$-0.477181\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	32.0000			1.28827			0.644136	−	0.764911i	$-0.277217\pi$
$617$	32.0000			1.28827			0.644136	+	0.764911i	$0.277217\pi$
$618$		0			0
$619$	22.4889	+	38.9519i	0.903905	+	1.56561i	0.822381	+	0.568937i	$0.192645\pi$
$619$	22.4889	+	38.9519i	0.903905	+	1.56561i	0.0815238	+	0.996671i	$0.474021\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.9059	+	20.6216i	−0.476999	+	0.826187i
$624$		0			0
$625$	14.5000	−	25.1147i	0.580000	−	1.00459i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	5.00000			0.199363
$630$		0			0
$631$	42.3320			1.68521			0.842606	−	0.538531i	$-0.181021\pi$
$631$	42.3320			1.68521			0.842606	+	0.538531i	$0.181021\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	31.7490	−	54.9909i	1.25992	−	2.18225i
$636$		0			0
$637$	−28.0000			−1.10940
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	0.855982	−	0.517005i	$-0.172953\pi$
$641$	−0.500000	−	0.866025i	−0.0197488	−	0.0342059i	−0.875731	+	0.482800i	$0.839620\pi$
$642$		0			0
$643$	10.5830			0.417353			0.208676	−	0.977985i	$-0.433084\pi$
$643$	10.5830			0.417353			0.208676	+	0.977985i	$0.433084\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	19.8431	+	34.3693i	0.780114	+	1.35120i	0.931875	+	0.362780i	$0.118172\pi$
$647$	19.8431	+	34.3693i	0.780114	+	1.35120i	−0.151761	+	0.988417i	$0.548494\pi$
$648$		0			0
$649$	3.50000	−	6.06218i	0.137387	−	0.237961i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	9.50000	−	16.4545i	0.371764	−	0.643914i	−0.618073	−	0.786121i	$-0.712086\pi$
$653$	9.50000	−	16.4545i	0.371764	−	0.643914i	0.989837	+	0.142207i	$0.0454198\pi$
$654$		0			0
$655$	−11.9059	−	20.6216i	−0.465201	−	0.805752i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−31.7490			−1.23677			−0.618383	−	0.785877i	$-0.712212\pi$
$659$	−31.7490			−1.23677			−0.618383	+	0.785877i	$0.712212\pi$
$660$		0			0
$661$	16.5000	+	28.5788i	0.641776	+	1.11159i	0.985036	+	0.172348i	$0.0551353\pi$
$661$	16.5000	+	28.5788i	0.641776	+	1.11159i	−0.343261	+	0.939240i	$0.611531\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−31.5000	−	54.5596i	−1.22152	−	2.11573i
$666$		0			0
$667$	5.29150	−	9.16515i	0.204888	−	0.354876i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	13.2288			0.510690
$672$		0			0
$673$	−24.0000			−0.925132			−0.462566	−	0.886585i	$-0.653071\pi$
$673$	−24.0000			−0.925132			−0.462566	+	0.886585i	$0.653071\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.50000	+	2.59808i	−0.0576497	+	0.0998522i	−0.893410	−	0.449242i	$-0.851694\pi$
$677$	−1.50000	+	2.59808i	−0.0576497	+	0.0998522i	0.835760	+	0.549095i	$0.185027\pi$
$678$		0			0
$679$	−21.1660			−0.812277
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.9059	+	20.6216i	0.455566	+	0.789063i	0.998721	−	0.0505694i	$-0.0161036\pi$
$683$	11.9059	+	20.6216i	0.455566	+	0.789063i	−0.543155	+	0.839633i	$0.682770\pi$
$684$		0			0
$685$	−3.00000			−0.114624
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.0000	−	24.2487i	−0.533358	−	0.923802i
$690$		0			0
$691$	14.5516	−	25.2042i	0.553570	−	0.958812i	−0.444443	−	0.895807i	$-0.646598\pi$
$691$	14.5516	−	25.2042i	0.553570	−	0.958812i	0.998013	−	0.0630046i	$-0.0200683\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−15.8745	+	27.4955i	−0.602154	+	1.04296i
$696$		0			0
$697$	−4.00000	−	6.92820i	−0.151511	−	0.262424i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	19.8431	+	34.3693i	0.748398	+	1.29626i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−19.8431	−	34.3693i	−0.746278	−	1.29259i
$708$		0			0
$709$	18.5000	−	32.0429i	0.694782	−	1.20340i	−0.275472	−	0.961309i	$-0.588834\pi$
$709$	18.5000	−	32.0429i	0.694782	−	1.20340i	0.970254	−	0.242089i	$-0.0778325\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−7.00000			−0.262152
$714$		0			0
$715$	31.7490			1.18735
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.8431	+	34.3693i	−0.740024	+	1.28176i	0.212460	+	0.977170i	$0.431853\pi$
$719$	−19.8431	+	34.3693i	−0.740024	+	1.28176i	−0.952484	+	0.304589i	$0.901481\pi$
$720$		0			0
$721$	−3.50000	+	6.06218i	−0.130347	+	0.225767i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.00000	−	13.8564i	−0.297113	−	0.514614i
$726$		0			0
$727$	−42.3320			−1.57001			−0.785004	−	0.619491i	$-0.787339\pi$
$727$	−42.3320			−1.57001			−0.785004	+	0.619491i	$0.787339\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−5.29150	−	9.16515i	−0.195713	−	0.338985i
$732$		0			0
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	−0.949541	−	0.313644i	$-0.898450\pi$
$733$	−5.50000	+	9.52628i	−0.203147	+	0.351861i	0.746394	+	0.665505i	$0.231784\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.50000	+	6.06218i	−0.128924	+	0.223303i
$738$		0			0
$739$	−22.4889	−	38.9519i	−0.827267	−	1.43287i	−0.900174	−	0.435530i	$-0.856561\pi$
$739$	−22.4889	−	38.9519i	−0.827267	−	1.43287i	0.0729072	−	0.997339i	$-0.476772\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−21.1660			−0.776506			−0.388253	−	0.921553i	$-0.626921\pi$
$743$	−21.1660			−0.776506			−0.388253	+	0.921553i	$0.626921\pi$
$744$		0			0
$745$	31.5000	+	54.5596i	1.15407	+	1.99891i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−17.5000	+	30.3109i	−0.639436	+	1.10754i
$750$		0			0
$751$	−19.8431	+	34.3693i	−0.724086	+	1.25415i	0.235263	+	0.971932i	$0.424405\pi$
$751$	−19.8431	+	34.3693i	−0.724086	+	1.25415i	−0.959349	+	0.282222i	$0.908928\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.93725			0.288866
$756$		0			0
$757$	4.00000			0.145382			0.0726912	−	0.997354i	$-0.476841\pi$
$757$	4.00000			0.145382			0.0726912	+	0.997354i	$0.476841\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	4.50000	−	7.79423i	0.163125	−	0.282541i	−0.772863	−	0.634573i	$-0.781176\pi$
$761$	4.50000	−	7.79423i	0.163125	−	0.282541i	0.935988	+	0.352032i	$0.114509\pi$
$762$		0			0
$763$	−1.32288	−	2.29129i	−0.0478913	−	0.0829502i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.29150	+	9.16515i	0.191065	+	0.330934i
$768$		0			0
$769$	−24.0000			−0.865462			−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000			−0.865462			−0.432731	+	0.901523i	$0.642450\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	16.5000	+	28.5788i	0.593464	+	1.02791i	0.993762	+	0.111524i	$0.0355733\pi$
$773$	16.5000	+	28.5788i	0.593464	+	1.02791i	−0.400298	+	0.916385i	$0.631093\pi$
$774$		0			0
$775$	−5.29150	+	9.16515i	−0.190076	+	0.329222i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	31.7490	−	54.9909i	1.13753	−	1.97025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−27.0000			−0.963671
$786$		0			0
$787$	−9.26013	−	16.0390i	−0.330088	−	0.571729i	0.652441	−	0.757840i	$-0.273745\pi$
$787$	−9.26013	−	16.0390i	−0.330088	−	0.571729i	−0.982529	+	0.186111i	$0.940412\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−10.0000	+	17.3205i	−0.355110	+	0.615069i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	52.0000			1.84193			0.920967	−	0.389640i	$-0.127401\pi$
$797$	52.0000			1.84193			0.920967	+	0.389640i	$0.127401\pi$
$798$		0			0
$799$	−2.64575			−0.0936000
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11.9059	+	20.6216i	−0.420149	+	0.727720i
$804$		0			0
$805$	10.5000	+	18.1865i	0.370076	+	0.640991i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.5000	+	26.8468i	0.544951	+	0.943883i	0.998610	+	0.0527074i	$0.0167851\pi$
$809$	15.5000	+	26.8468i	0.544951	+	0.943883i	−0.453659	+	0.891175i	$0.649882\pi$
$810$		0			0
$811$	−10.5830			−0.371620			−0.185810	−	0.982586i	$-0.559491\pi$
$811$	−10.5830			−0.371620			−0.185810	+	0.982586i	$0.559491\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−19.8431	−	34.3693i	−0.695075	−	1.20390i
$816$		0			0
$817$	42.0000	−	72.7461i	1.46939	−	2.54507i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.50000	−	11.2583i	0.226852	−	0.392918i	−0.730022	−	0.683424i	$-0.760490\pi$
$821$	6.50000	−	11.2583i	0.226852	−	0.392918i	0.956873	+	0.290505i	$0.0938234\pi$
$822$		0			0
$823$	−9.26013	−	16.0390i	−0.322788	−	0.559085i	0.658274	−	0.752778i	$-0.271287\pi$
$823$	−9.26013	−	16.0390i	−0.322788	−	0.559085i	−0.981062	+	0.193693i	$0.937953\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−31.7490			−1.10402			−0.552011	−	0.833837i	$-0.686139\pi$
$827$	−31.7490			−1.10402			−0.552011	+	0.833837i	$0.686139\pi$
$828$		0			0
$829$	7.50000	+	12.9904i	0.260486	+	0.451175i	0.966371	−	0.257152i	$-0.0827840\pi$
$829$	7.50000	+	12.9904i	0.260486	+	0.451175i	−0.705885	+	0.708326i	$0.749451\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.50000	+	6.06218i	0.121268	+	0.210042i
$834$		0			0
$835$	−7.93725	+	13.7477i	−0.274680	+	0.475760i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	21.1660			0.730732			0.365366	−	0.930864i	$-0.380944\pi$
$839$	21.1660			0.730732			0.365366	+	0.930864i	$0.380944\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−4.50000	+	7.79423i	−0.154805	+	0.268130i
$846$		0			0
$847$	5.29150	−	9.16515i	0.181818	−	0.314918i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−6.61438	−	11.4564i	−0.226738	−	0.392722i
$852$		0			0
$853$	−36.0000			−1.23262			−0.616308	−	0.787505i	$-0.711372\pi$
$853$	−36.0000			−1.23262			−0.616308	+	0.787505i	$0.711372\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	7.50000	+	12.9904i	0.256195	+	0.443743i	0.965219	−	0.261441i	$-0.0841977\pi$
$857$	7.50000	+	12.9904i	0.256195	+	0.443743i	−0.709024	+	0.705184i	$0.750864\pi$
$858$		0			0
$859$	−14.5516	+	25.2042i	−0.496495	+	0.859955i	−0.999992	−	0.00404218i	$-0.998713\pi$
$859$	−14.5516	+	25.2042i	−0.496495	+	0.859955i	0.503497	+	0.863997i	$0.332047\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−19.8431	+	34.3693i	−0.675468	+	1.16995i	0.300864	+	0.953667i	$0.402725\pi$
$863$	−19.8431	+	34.3693i	−0.675468	+	1.16995i	−0.976332	+	0.216278i	$0.930608\pi$
$864$		0			0
$865$	4.50000	+	7.79423i	0.153005	+	0.265012i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7.00000			−0.237459
$870$		0			0
$871$	−5.29150	−	9.16515i	−0.179296	−	0.310549i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.93725			−0.268328
$876$		0			0
$877$	−18.5000	+	32.0429i	−0.624701	+	1.08201i	0.363898	+	0.931439i	$0.381446\pi$
$877$	−18.5000	+	32.0429i	−0.624701	+	1.08201i	−0.988599	+	0.150574i	$0.951888\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−46.0000			−1.54978			−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000			−1.54978			−0.774890	+	0.632096i	$0.782195\pi$
$882$		0			0
$883$	10.5830			0.356146			0.178073	−	0.984017i	$-0.443014\pi$
$883$	10.5830			0.356146			0.178073	+	0.984017i	$0.443014\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.32288	+	2.29129i	−0.0444178	+	0.0769339i	−0.887380	−	0.461040i	$-0.847477\pi$
$887$	−1.32288	+	2.29129i	−0.0444178	+	0.0769339i	0.842962	+	0.537973i	$0.180810\pi$
$888$		0			0
$889$	−56.0000			−1.87818
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.5000	−	18.1865i	−0.351369	−	0.608589i
$894$		0			0
$895$	−71.4353			−2.38782
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−5.29150	−	9.16515i	−0.176481	−	0.305675i
$900$		0			0
$901$	−3.50000	+	6.06218i	−0.116602	+	0.201960i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	9.00000	−	15.5885i	0.299170	−	0.518178i
$906$		0			0
$907$	−9.26013	−	16.0390i	−0.307478	−	0.532567i	0.670332	−	0.742061i	$-0.266152\pi$
$907$	−9.26013	−	16.0390i	−0.307478	−	0.532567i	−0.977810	+	0.209494i	$0.932818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−21.1660			−0.701261			−0.350631	−	0.936514i	$-0.614033\pi$
$911$	−21.1660			−0.701261			−0.350631	+	0.936514i	$0.614033\pi$
$912$		0			0
$913$	−14.0000	−	24.2487i	−0.463332	−	0.802515i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−10.5000	+	18.1865i	−0.346741	+	0.600572i
$918$		0			0
$919$	−9.26013	+	16.0390i	−0.305463	+	0.529078i	−0.977364	−	0.211563i	$-0.932145\pi$
$919$	−9.26013	+	16.0390i	−0.305463	+	0.529078i	0.671901	+	0.740641i	$0.265478\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−20.0000			−0.657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−4.50000	+	7.79423i	−0.147640	+	0.255720i	−0.930355	−	0.366660i	$-0.880501\pi$
$929$	−4.50000	+	7.79423i	−0.147640	+	0.255720i	0.782715	+	0.622381i	$0.213834\pi$
$930$		0			0
$931$	−27.7804	+	48.1170i	−0.910465	+	1.57697i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.96863	−	6.87386i	−0.129788	−	0.224799i
$936$		0			0
$937$	−22.0000			−0.718709			−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000			−0.718709			−0.359354	+	0.933201i	$0.617003\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.500000	−	0.866025i	−0.0162995	−	0.0282316i	0.857761	−	0.514049i	$-0.171855\pi$
$941$	−0.500000	−	0.866025i	−0.0162995	−	0.0282316i	−0.874060	+	0.485818i	$0.838522\pi$
$942$		0			0
$943$	−10.5830	+	18.3303i	−0.344630	+	0.596917i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.5516	+	25.2042i	−0.472865	+	0.819025i	−0.999518	−	0.0310549i	$-0.990113\pi$
$947$	−14.5516	+	25.2042i	−0.472865	+	0.819025i	0.526653	+	0.850080i	$0.323447\pi$
$948$		0			0
$949$	−18.0000	−	31.1769i	−0.584305	−	1.01205i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.0000			0.518291			0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000			0.518291			0.259145	+	0.965838i	$0.416559\pi$
$954$		0			0
$955$	−35.7176	−	61.8648i	−1.15580	−	2.00190i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.32288	+	2.29129i	0.0427179	+	0.0739895i
$960$		0			0
$961$	12.0000	−	20.7846i	0.387097	−	0.670471i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−21.0000			−0.676014
$966$		0			0
$967$	−42.3320			−1.36131			−0.680653	−	0.732606i	$-0.738304\pi$
$967$	−42.3320			−1.36131			−0.680653	+	0.732606i	$0.738304\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.96863	+	6.87386i	−0.127359	+	0.220593i	−0.922653	−	0.385632i	$-0.873983\pi$
$971$	−3.96863	+	6.87386i	−0.127359	+	0.220593i	0.795293	+	0.606225i	$0.207317\pi$
$972$		0			0
$973$	28.0000			0.897639
$974$		0			0
$975$		0			0
$976$		0			0
$977$	20.5000	+	35.5070i	0.655853	+	1.13597i	0.981679	+	0.190541i	$0.0610243\pi$
$977$	20.5000	+	35.5070i	0.655853	+	1.13597i	−0.325826	+	0.945430i	$0.605642\pi$
$978$		0			0
$979$	−23.8118			−0.761027
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.9059	+	20.6216i	0.379739	+	0.657727i	0.991024	−	0.133684i	$-0.0426806\pi$
$983$	11.9059	+	20.6216i	0.379739	+	0.657727i	−0.611285	+	0.791410i	$0.709347\pi$
$984$		0			0
$985$	18.0000	−	31.1769i	0.573528	−	0.993379i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.0000	+	24.2487i	−0.445174	+	0.771064i
$990$		0			0
$991$	22.4889	+	38.9519i	0.714383	+	1.23735i	0.963197	+	0.268796i	$0.0866259\pi$
$991$	22.4889	+	38.9519i	0.714383	+	1.23735i	−0.248814	+	0.968551i	$0.580041\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−55.5608			−1.76140
$996$		0			0
$997$	−11.5000	−	19.9186i	−0.364209	−	0.630828i	0.624440	−	0.781073i	$-0.285327\pi$
$997$	−11.5000	−	19.9186i	−0.364209	−	0.630828i	−0.988649	+	0.150245i	$0.951994\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2016.2.s.o.289.2		4
3.2	odd	2		224.2.i.c.65.2	yes	4
4.3	odd	2	inner	2016.2.s.o.289.1		4
7.4	even	3	inner	2016.2.s.o.865.2		4
12.11	even	2		224.2.i.c.65.1	✓	4
21.2	odd	6		1568.2.a.m.1.1		2
21.5	even	6		1568.2.a.t.1.2		2
21.11	odd	6		224.2.i.c.193.2	yes	4
21.17	even	6		1568.2.i.p.1537.1		4
21.20	even	2		1568.2.i.p.961.1		4
24.5	odd	2		448.2.i.h.65.1		4
24.11	even	2		448.2.i.h.65.2		4
28.11	odd	6	inner	2016.2.s.o.865.1		4
84.11	even	6		224.2.i.c.193.1	yes	4
84.23	even	6		1568.2.a.m.1.2		2
84.47	odd	6		1568.2.a.t.1.1		2
84.59	odd	6		1568.2.i.p.1537.2		4
84.83	odd	2		1568.2.i.p.961.2		4
168.5	even	6		3136.2.a.bg.1.1		2
168.11	even	6		448.2.i.h.193.2		4
168.53	odd	6		448.2.i.h.193.1		4
168.107	even	6		3136.2.a.bv.1.1		2
168.131	odd	6		3136.2.a.bg.1.2		2
168.149	odd	6		3136.2.a.bv.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.2.i.c.65.1	✓	4	12.11	even	2
224.2.i.c.65.2	yes	4	3.2	odd	2
224.2.i.c.193.1	yes	4	84.11	even	6
224.2.i.c.193.2	yes	4	21.11	odd	6
448.2.i.h.65.1		4	24.5	odd	2
448.2.i.h.65.2		4	24.11	even	2
448.2.i.h.193.1		4	168.53	odd	6
448.2.i.h.193.2		4	168.11	even	6
1568.2.a.m.1.1		2	21.2	odd	6
1568.2.a.m.1.2		2	84.23	even	6
1568.2.a.t.1.1		2	84.47	odd	6
1568.2.a.t.1.2		2	21.5	even	6
1568.2.i.p.961.1		4	21.20	even	2
1568.2.i.p.961.2		4	84.83	odd	2
1568.2.i.p.1537.1		4	21.17	even	6
1568.2.i.p.1537.2		4	84.59	odd	6
2016.2.s.o.289.1		4	4.3	odd	2	inner
2016.2.s.o.289.2		4	1.1	even	1	trivial
2016.2.s.o.865.1		4	28.11	odd	6	inner
2016.2.s.o.865.2		4	7.4	even	3	inner
3136.2.a.bg.1.1		2	168.5	even	6
3136.2.a.bg.1.2		2	168.131	odd	6
3136.2.a.bv.1.1		2	168.107	even	6
3136.2.a.bv.1.2		2	168.149	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2016.s (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 2.59808i) q^{5} +2.64575 q^{7} +O(q^{10})\)	\(q+(-1.50000 + 2.59808i) q^{5} +2.64575 q^{7} +(1.32288 + 2.29129i) q^{11} -4.00000 q^{13} +(0.500000 + 0.866025i) q^{17} +(-3.96863 + 6.87386i) q^{19} +(1.32288 - 2.29129i) q^{23} +(-2.00000 - 3.46410i) q^{25} +4.00000 q^{29} +(-1.32288 - 2.29129i) q^{31} +(-3.96863 + 6.87386i) q^{35} +(2.50000 - 4.33013i) q^{37} -8.00000 q^{41} -10.5830 q^{43} +(-1.32288 + 2.29129i) q^{47} +7.00000 q^{49} +(3.50000 + 6.06218i) q^{53} -7.93725 q^{55} +(-1.32288 - 2.29129i) q^{59} +(2.50000 - 4.33013i) q^{61} +(6.00000 - 10.3923i) q^{65} +(1.32288 + 2.29129i) q^{67} +(4.50000 + 7.79423i) q^{73} +(3.50000 + 6.06218i) q^{77} +(-1.32288 + 2.29129i) q^{79} -10.5830 q^{83} -3.00000 q^{85} +(-4.50000 + 7.79423i) q^{89} -10.5830 q^{91} +(-11.9059 - 20.6216i) q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5}+O(q^{10}) \) 4 * q - 6 * q^5	\( 4 q - 6 q^{5} - 16 q^{13} + 2 q^{17} - 8 q^{25} + 16 q^{29} + 10 q^{37} - 32 q^{41} + 28 q^{49} + 14 q^{53} + 10 q^{61} + 24 q^{65} + 18 q^{73} + 14 q^{77} - 12 q^{85} - 18 q^{89} - 32 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 16 * q^13 + 2 * q^17 - 8 * q^25 + 16 * q^29 + 10 * q^37 - 32 * q^41 + 28 * q^49 + 14 * q^53 + 10 * q^61 + 24 * q^65 + 18 * q^73 + 14 * q^77 - 12 * q^85 - 18 * q^89 - 32 * q^97

Introduction

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