LMFDB - Embedded newform 1216.3.g.d.417.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(417,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.417");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.g (of order $2$, degree $1$, 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:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 34x^{14} + 509x^{12} - 4794x^{10} + 30356x^{8} - 106386x^{6} + 288389x^{4} - 166634x^{2} + 28561 $ x^16 - 34x^14 + 509x^12 - 4794x^10 + 30356x^8 - 106386x^6 + 288389x^4 - 166634*x^2 + 28561
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			417.1
Root			$-0.592744 - 0.0718888i$ of defining polynomial
Character	$\chi$	$=$	1216.417
Dual form			1216.3.g.d.417.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.24985 q^{3} -3.61513i q^{5} -8.24529 q^{7} +1.56155 q^{9} +O(q^{10})$	$q-3.24985 q^{3} -3.61513i q^{5} -8.24529 q^{7} +1.56155 q^{9} +15.8078i q^{11} +15.0474 q^{13} +11.7486i q^{15} +10.3693 q^{17} +(-18.4743 - 4.43845i) q^{19} +26.7960 q^{21} +39.6414 q^{23} +11.9309 q^{25} +24.1739 q^{27} +3.29874 q^{29} -41.8434i q^{31} -51.3729i q^{33} +29.8078i q^{35} -65.3406 q^{37} -48.9018 q^{39} -57.8726i q^{41} +60.5464i q^{43} -5.64521i q^{45} -38.6264 q^{47} +18.9848 q^{49} -33.6988 q^{51} +43.6958 q^{53} +57.1470 q^{55} +(60.0388 + 14.4243i) q^{57} -6.27505 q^{59} +75.6677i q^{61} -12.8755 q^{63} -54.3981i q^{65} -125.495 q^{67} -128.829 q^{69} +113.781i q^{71} -71.3542 q^{73} -38.7736 q^{75} -130.340i q^{77} -142.430i q^{79} -92.6155 q^{81} -19.9697i q^{83} -37.4864i q^{85} -10.7204 q^{87} -16.0246i q^{89} -124.070 q^{91} +135.985i q^{93} +(-16.0455 + 66.7869i) q^{95} -12.5501i q^{97} +24.6847i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} - 32 q^{17} - 40 q^{25} - 224 q^{49} + 136 q^{57} - 416 q^{73} - 1152 q^{81}+O(q^{100}) $ 16 * q - 8 * q^9 - 32 * q^17 - 40 * q^25 - 224 * q^49 + 136 * q^57 - 416 * q^73 - 1152 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	−3.24985			−1.08328			−0.541642	−	0.840609i	$-0.682197\pi$
$3$	−3.24985			−1.08328			−0.541642	+	0.840609i	$0.682197\pi$
$4$		0			0
$5$		−	3.61513i		−	0.723025i	−0.932367	−	0.361513i	$-0.882260\pi$
$5$		−	3.61513i		−	0.723025i	0.932367	−	0.361513i	$-0.117740\pi$
$6$		0			0
$7$	−8.24529			−1.17790			−0.588949	−	0.808170i	$-0.700458\pi$
$7$	−8.24529			−1.17790			−0.588949	+	0.808170i	$0.700458\pi$
$8$		0			0
$9$	1.56155			0.173506
$10$		0			0
$11$			15.8078i			1.43707i	0.695491	+	0.718535i	$0.255187\pi$
$11$			15.8078i			1.43707i	−0.695491	+	0.718535i	$0.744813\pi$
$12$		0			0
$13$	15.0474			1.15749			0.578745	−	0.815509i	$-0.303543\pi$
$13$	15.0474			1.15749			0.578745	+	0.815509i	$0.303543\pi$
$14$		0			0
$15$			11.7486i			0.783242i
$16$		0			0
$17$	10.3693			0.609960			0.304980	−	0.952359i	$-0.401350\pi$
$17$	10.3693			0.609960			0.304980	+	0.952359i	$0.401350\pi$
$18$		0			0
$19$	−18.4743	−	4.43845i	−0.972332	−	0.233602i
$19$	−18.4743	−	4.43845i	−0.972332	−	0.233602i
$20$		0			0
$21$	26.7960			1.27600
$22$		0			0
$23$	39.6414			1.72354			0.861770	−	0.507299i	$-0.169356\pi$
$23$	39.6414			1.72354			0.861770	+	0.507299i	$0.169356\pi$
$24$		0			0
$25$	11.9309			0.477235
$26$		0			0
$27$	24.1739			0.895328
$28$		0			0
$29$	3.29874			0.113750			0.0568748	−	0.998381i	$-0.481886\pi$
$29$	3.29874			0.113750			0.0568748	+	0.998381i	$0.481886\pi$
$30$		0			0
$31$		−	41.8434i		−	1.34979i	−0.737916	−	0.674893i	$-0.764190\pi$
$31$		−	41.8434i		−	1.34979i	0.737916	−	0.674893i	$-0.235810\pi$
$32$		0			0
$33$		−	51.3729i		−	1.55676i
$34$		0			0
$35$			29.8078i			0.851650i
$36$		0			0
$37$	−65.3406			−1.76596			−0.882981	−	0.469408i	$-0.844467\pi$
$37$	−65.3406			−1.76596			−0.882981	+	0.469408i	$0.844467\pi$
$38$		0			0
$39$	−48.9018			−1.25389
$40$		0			0
$41$		−	57.8726i		−	1.41153i	−0.708447	−	0.705764i	$-0.750604\pi$
$41$		−	57.8726i		−	1.41153i	0.708447	−	0.705764i	$-0.249396\pi$
$42$		0			0
$43$			60.5464i			1.40806i	0.710172	+	0.704028i	$0.248617\pi$
$43$			60.5464i			1.40806i	−0.710172	+	0.704028i	$0.751383\pi$
$44$		0			0
$45$		−	5.64521i		−	0.125449i
$46$		0			0
$47$	−38.6264			−0.821838			−0.410919	−	0.911672i	$-0.634792\pi$
$47$	−38.6264			−0.821838			−0.410919	+	0.911672i	$0.634792\pi$
$48$		0			0
$49$	18.9848			0.387446
$50$		0			0
$51$	−33.6988			−0.660760
$52$		0			0
$53$	43.6958			0.824449			0.412224	−	0.911082i	$-0.364752\pi$
$53$	43.6958			0.824449			0.412224	+	0.911082i	$0.364752\pi$
$54$		0			0
$55$	57.1470			1.03904
$56$		0			0
$57$	60.0388	+	14.4243i	1.05331	+	0.253058i
$58$		0			0
$59$	−6.27505			−0.106357			−0.0531783	−	0.998585i	$-0.516935\pi$
$59$	−6.27505			−0.106357			−0.0531783	+	0.998585i	$0.516935\pi$
$60$		0			0
$61$			75.6677i			1.24045i	0.784422	+	0.620227i	$0.212960\pi$
$61$			75.6677i			1.24045i	−0.784422	+	0.620227i	$0.787040\pi$
$62$		0			0
$63$	−12.8755			−0.204372
$64$		0			0
$65$		−	54.3981i		−	0.836894i
$66$		0			0
$67$	−125.495			−1.87306			−0.936529	−	0.350591i	$-0.885981\pi$
$67$	−125.495			−1.87306			−0.936529	+	0.350591i	$0.885981\pi$
$68$		0			0
$69$	−128.829			−1.86708
$70$		0			0
$71$			113.781i			1.60256i	0.598292	+	0.801278i	$0.295846\pi$
$71$			113.781i			1.60256i	−0.598292	+	0.801278i	$0.704154\pi$
$72$		0			0
$73$	−71.3542			−0.977454			−0.488727	−	0.872437i	$-0.662539\pi$
$73$	−71.3542			−0.977454			−0.488727	+	0.872437i	$0.662539\pi$
$74$		0			0
$75$	−38.7736			−0.516981
$76$		0			0
$77$		−	130.340i		−	1.69272i
$78$		0			0
$79$		−	142.430i		−	1.80291i	−0.432873	−	0.901455i	$-0.642500\pi$
$79$		−	142.430i		−	1.80291i	0.432873	−	0.901455i	$-0.357500\pi$
$80$		0			0
$81$	−92.6155			−1.14340
$82$		0			0
$83$		−	19.9697i		−	0.240599i	−0.992738	−	0.120299i	$-0.961615\pi$
$83$		−	19.9697i		−	0.240599i	0.992738	−	0.120299i	$-0.0383854\pi$
$84$		0			0
$85$		−	37.4864i		−	0.441016i
$86$		0			0
$87$	−10.7204			−0.123223
$88$		0			0
$89$		−	16.0246i		−	0.180052i	−0.995939	−	0.0900259i	$-0.971305\pi$
$89$		−	16.0246i		−	0.180052i	0.995939	−	0.0900259i	$-0.0286950\pi$
$90$		0			0
$91$	−124.070			−1.36341
$92$		0			0
$93$			135.985i			1.46220i
$94$		0			0
$95$	−16.0455	+	66.7869i	−0.168900	+	0.703021i
$96$		0			0
$97$		−	12.5501i		−	0.129382i	−0.997905	−	0.0646912i	$-0.979394\pi$
$97$		−	12.5501i		−	0.129382i	0.997905	−	0.0646912i	$-0.0206062\pi$
$98$		0			0
$99$			24.6847i			0.249340i
$100$		0			0
$101$			24.8608i			0.246147i	0.992398	+	0.123073i	$0.0392751\pi$
$101$			24.8608i			0.246147i	−0.992398	+	0.123073i	$0.960725\pi$
$102$		0			0
$103$			100.587i			0.976568i	0.872685	+	0.488284i	$0.162377\pi$
$103$			100.587i			0.976568i	−0.872685	+	0.488284i	$0.837623\pi$
$104$		0			0
$105$		−	96.8709i		−	0.922580i
$106$		0			0
$107$	−83.0221			−0.775907			−0.387954	−	0.921679i	$-0.626818\pi$
$107$	−83.0221			−0.775907			−0.387954	+	0.921679i	$0.626818\pi$
$108$		0			0
$109$	63.4882			0.582461			0.291230	−	0.956653i	$-0.405935\pi$
$109$	63.4882			0.582461			0.291230	+	0.956653i	$0.405935\pi$
$110$		0			0
$111$	212.348			1.91304
$112$		0			0
$113$		−	128.920i		−	1.14089i	−0.821337	−	0.570443i	$-0.806772\pi$
$113$		−	128.920i		−	1.14089i	0.821337	−	0.570443i	$-0.193228\pi$
$114$		0			0
$115$		−	143.309i		−	1.24616i
$116$		0			0
$117$	23.4973			0.200831
$118$		0			0
$119$	−85.4980			−0.718471
$120$		0			0
$121$	−128.885			−1.06517
$122$		0			0
$123$			188.078i			1.52909i
$124$		0			0
$125$		−	133.510i		−	1.06808i
$126$		0			0
$127$		−	8.04380i		−	0.0633370i	−0.999498	−	0.0316685i	$-0.989918\pi$
$127$		−	8.04380i		−	0.0633370i	0.999498	−	0.0316685i	$-0.0100821\pi$
$128$		0			0
$129$		−	196.767i		−	1.52533i
$130$		0			0
$131$		−	230.348i		−	1.75838i	−0.476473	−	0.879189i	$-0.658085\pi$
$131$		−	230.348i		−	1.75838i	0.476473	−	0.879189i	$-0.341915\pi$
$132$		0			0
$133$	152.326	+	36.5963i	1.14531	+	0.275160i
$134$		0			0
$135$		−	87.3916i		−	0.647345i
$136$		0			0
$137$	−134.477			−0.981586			−0.490793	−	0.871276i	$-0.663293\pi$
$137$	−134.477			−0.981586			−0.490793	+	0.871276i	$0.663293\pi$
$138$		0			0
$139$			4.68466i			0.0337026i	0.999858	+	0.0168513i	$0.00536419\pi$
$139$			4.68466i			0.0337026i	−0.999858	+	0.0168513i	$0.994636\pi$
$140$		0			0
$141$	125.530			0.890284
$142$		0			0
$143$			237.865i			1.66339i
$144$		0			0
$145$		−	11.9254i		−	0.0822438i
$146$		0			0
$147$	−61.6980			−0.419714
$148$		0			0
$149$			225.863i			1.51586i	0.652336	+	0.757930i	$0.273789\pi$
$149$			225.863i			1.51586i	−0.652336	+	0.757930i	$0.726211\pi$
$150$		0			0
$151$		−	26.3899i		−	0.174768i	−0.996175	−	0.0873838i	$-0.972149\pi$
$151$		−	26.3899i		−	0.174768i	0.996175	−	0.0873838i	$-0.0278506\pi$
$152$		0			0
$153$	16.1922			0.105832
$154$		0			0
$155$	−151.269			−0.975929
$156$		0			0
$157$		−	236.068i		−	1.50362i	−0.659380	−	0.751810i	$-0.729181\pi$
$157$		−	236.068i		−	1.50362i	0.659380	−	0.751810i	$-0.270819\pi$
$158$		0			0
$159$	−142.005			−0.893113
$160$		0			0
$161$	−326.855			−2.03016
$162$		0			0
$163$		−	208.047i		−	1.27636i	−0.769886	−	0.638182i	$-0.779687\pi$
$163$		−	208.047i		−	1.27636i	0.769886	−	0.638182i	$-0.220313\pi$
$164$		0			0
$165$	−185.720			−1.12557
$166$		0			0
$167$		−	273.111i		−	1.63540i	−0.575647	−	0.817698i	$-0.695250\pi$
$167$		−	273.111i		−	1.63540i	0.575647	−	0.817698i	$-0.304750\pi$
$168$		0			0
$169$	57.4233			0.339783
$170$		0			0
$171$	−28.8486	−	6.93087i	−0.168705	−	0.0405314i
$172$		0			0
$173$	13.1950			0.0762714			0.0381357	−	0.999273i	$-0.487858\pi$
$173$	13.1950			0.0762714			0.0381357	+	0.999273i	$0.487858\pi$
$174$		0			0
$175$	−98.3735			−0.562134
$176$		0			0
$177$	20.3930			0.115215
$178$		0			0
$179$	−113.071			−0.631681			−0.315841	−	0.948812i	$-0.602286\pi$
$179$	−113.071			−0.631681			−0.315841	+	0.948812i	$0.602286\pi$
$180$		0			0
$181$	110.077			0.608158			0.304079	−	0.952647i	$-0.401651\pi$
$181$	110.077			0.608158			0.304079	+	0.952647i	$0.401651\pi$
$182$		0			0
$183$		−	245.909i		−	1.34377i
$184$		0			0
$185$			236.215i			1.27684i
$186$		0			0
$187$			163.916i			0.876555i
$188$		0			0
$189$	−199.321			−1.05461
$190$		0			0
$191$	−93.6184			−0.490149			−0.245074	−	0.969504i	$-0.578812\pi$
$191$	−93.6184			−0.490149			−0.245074	+	0.969504i	$0.578812\pi$
$192$		0			0
$193$		−	32.6740i		−	0.169295i	−0.996411	−	0.0846476i	$-0.973024\pi$
$193$		−	32.6740i		−	0.169295i	0.996411	−	0.0846476i	$-0.0269764\pi$
$194$		0			0
$195$			176.786i			0.906595i
$196$		0			0
$197$			110.484i			0.560832i	0.959879	+	0.280416i	$0.0904724\pi$
$197$			110.484i			0.560832i	−0.959879	+	0.280416i	$0.909528\pi$
$198$		0			0
$199$	−22.9557			−0.115355			−0.0576777	−	0.998335i	$-0.518370\pi$
$199$	−22.9557			−0.115355			−0.0576777	+	0.998335i	$0.518370\pi$
$200$		0			0
$201$	407.840			2.02905
$202$		0			0
$203$	−27.1991			−0.133986
$204$		0			0
$205$	−209.217			−1.02057
$206$		0			0
$207$	61.9022			0.299044
$208$		0			0
$209$	70.1619	−	292.038i	0.335703	−	1.39731i
$210$		0			0
$211$	−31.4737			−0.149165			−0.0745823	−	0.997215i	$-0.523762\pi$
$211$	−31.4737			−0.149165			−0.0745823	+	0.997215i	$0.523762\pi$
$212$		0			0
$213$		−	369.773i		−	1.73602i
$214$		0			0
$215$	218.883			1.01806
$216$		0			0
$217$			345.011i			1.58991i
$218$		0			0
$219$	231.891			1.05886
$220$		0			0
$221$	156.031			0.706022
$222$		0			0
$223$		−	286.306i		−	1.28388i	−0.766753	−	0.641942i	$-0.778129\pi$
$223$		−	286.306i		−	1.28388i	0.766753	−	0.641942i	$-0.221871\pi$
$224$		0			0
$225$	18.6307			0.0828030
$226$		0			0
$227$	−284.962			−1.25534			−0.627670	−	0.778479i	$-0.715991\pi$
$227$	−284.962			−1.25534			−0.627670	+	0.778479i	$0.715991\pi$
$228$		0			0
$229$		−	226.613i		−	0.989576i	−0.869014	−	0.494788i	$-0.835246\pi$
$229$		−	226.613i		−	0.989576i	0.869014	−	0.494788i	$-0.164754\pi$
$230$		0			0
$231$			423.585i			1.83370i
$232$		0			0
$233$	−74.7926			−0.320998			−0.160499	−	0.987036i	$-0.551310\pi$
$233$	−74.7926			−0.320998			−0.160499	+	0.987036i	$0.551310\pi$
$234$		0			0
$235$			139.639i			0.594209i
$236$		0			0
$237$			462.876i			1.95306i
$238$		0			0
$239$	281.605			1.17826			0.589132	−	0.808037i	$-0.299470\pi$
$239$	281.605			1.17826			0.589132	+	0.808037i	$0.299470\pi$
$240$		0			0
$241$		−	176.468i		−	0.732231i	−0.930569	−	0.366116i	$-0.880687\pi$
$241$		−	176.468i		−	0.732231i	0.930569	−	0.366116i	$-0.119313\pi$
$242$		0			0
$243$	83.4221			0.343301
$244$		0			0
$245$		−	68.6326i		−	0.280133i
$246$		0			0
$247$	−277.990	−	66.7869i	−1.12546	−	0.270393i
$248$		0			0
$249$			64.8986i			0.260637i
$250$		0			0
$251$			109.747i			0.437240i	0.975810	+	0.218620i	$0.0701554\pi$
$251$			109.747i			0.437240i	−0.975810	+	0.218620i	$0.929845\pi$
$252$		0			0
$253$			626.642i			2.47685i
$254$		0			0
$255$			121.825i			0.477746i
$256$		0			0
$257$		−	164.444i		−	0.639859i	−0.947441	−	0.319930i	$-0.896341\pi$
$257$		−	164.444i		−	0.639859i	0.947441	−	0.319930i	$-0.103659\pi$
$258$		0			0
$259$	538.753			2.08013
$260$		0			0
$261$	5.15115			0.0197362
$262$		0			0
$263$	357.647			1.35988			0.679938	−	0.733269i	$-0.262007\pi$
$263$	357.647			1.35988			0.679938	+	0.733269i	$0.262007\pi$
$264$		0			0
$265$		−	157.966i		−	0.596097i
$266$		0			0
$267$			52.0776i			0.195047i
$268$		0			0
$269$	85.9452			0.319499			0.159750	−	0.987158i	$-0.448931\pi$
$269$	85.9452			0.319499			0.159750	+	0.987158i	$0.448931\pi$
$270$		0			0
$271$	−225.988			−0.833904			−0.416952	−	0.908928i	$-0.636902\pi$
$271$	−225.988			−0.833904			−0.416952	+	0.908928i	$0.636902\pi$
$272$		0			0
$273$	403.209			1.47696
$274$		0			0
$275$			188.600i			0.685820i
$276$		0			0
$277$			233.734i			0.843803i	0.906642	+	0.421902i	$0.138637\pi$
$277$			233.734i			0.843803i	−0.906642	+	0.421902i	$0.861363\pi$
$278$		0			0
$279$		−	65.3406i		−	0.234196i
$280$		0			0
$281$			149.395i			0.531654i	0.964021	+	0.265827i	$0.0856450\pi$
$281$			149.395i			0.531654i	−0.964021	+	0.265827i	$0.914355\pi$
$282$		0			0
$283$		−	240.438i		−	0.849606i	−0.905286	−	0.424803i	$-0.860343\pi$
$283$		−	240.438i		−	0.849606i	0.905286	−	0.424803i	$-0.139657\pi$
$284$		0			0
$285$	52.1457	−	217.048i	0.182967	−	0.761571i
$286$		0			0
$287$			477.177i			1.66264i
$288$		0			0
$289$	−181.477			−0.627949
$290$		0			0
$291$			40.7860i			0.140158i
$292$		0			0
$293$	−286.712			−0.978540			−0.489270	−	0.872132i	$-0.662737\pi$
$293$	−286.712			−0.978540			−0.489270	+	0.872132i	$0.662737\pi$
$294$		0			0
$295$			22.6851i			0.0768986i
$296$		0			0
$297$			382.135i			1.28665i
$298$		0			0
$299$	596.499			1.99498
$300$		0			0
$301$		−	499.223i		−	1.65855i
$302$		0			0
$303$		−	80.7941i		−	0.266647i
$304$		0			0
$305$	273.548			0.896880
$306$		0			0
$307$	152.069			0.495339			0.247670	−	0.968845i	$-0.420335\pi$
$307$	152.069			0.495339			0.247670	+	0.968845i	$0.420335\pi$
$308$		0			0
$309$		−	326.892i		−	1.05790i
$310$		0			0
$311$	−448.541			−1.44225			−0.721127	−	0.692803i	$-0.756375\pi$
$311$	−448.541			−1.44225			−0.721127	+	0.692803i	$0.756375\pi$
$312$		0			0
$313$	71.6373			0.228873			0.114437	−	0.993431i	$-0.463494\pi$
$313$	71.6373			0.228873			0.114437	+	0.993431i	$0.463494\pi$
$314$		0			0
$315$			46.5464i			0.147766i
$316$		0			0
$317$	−373.292			−1.17758			−0.588788	−	0.808288i	$-0.700395\pi$
$317$	−373.292			−1.17758			−0.588788	+	0.808288i	$0.700395\pi$
$318$		0			0
$319$			52.1457i			0.163466i
$320$		0			0
$321$	269.810			0.840528
$322$		0			0
$323$	−191.566	−	46.0237i	−0.593084	−	0.142488i
$324$		0			0
$325$	179.528			0.552394
$326$		0			0
$327$	−206.327			−0.630971
$328$		0			0
$329$	318.486			0.968042
$330$		0			0
$331$	404.084			1.22080			0.610398	−	0.792095i	$-0.291009\pi$
$331$	404.084			1.22080			0.610398	+	0.792095i	$0.291009\pi$
$332$		0			0
$333$	−102.033			−0.306405
$334$		0			0
$335$			453.680i			1.35427i
$336$		0			0
$337$			412.331i			1.22354i	0.791037	+	0.611768i	$0.209541\pi$
$337$			412.331i			1.22354i	−0.791037	+	0.611768i	$0.790459\pi$
$338$		0			0
$339$			418.972i			1.23590i
$340$		0			0
$341$	661.450			1.93974
$342$		0			0
$343$	247.484			0.721527
$344$		0			0
$345$			465.732i			1.34995i
$346$		0			0
$347$		−	251.363i		−	0.724388i	−0.932103	−	0.362194i	$-0.882028\pi$
$347$		−	251.363i		−	0.724388i	0.932103	−	0.362194i	$-0.117972\pi$
$348$		0			0
$349$		−	393.439i		−	1.12733i	−0.826002	−	0.563667i	$-0.809390\pi$
$349$		−	393.439i		−	1.12733i	0.826002	−	0.563667i	$-0.190610\pi$
$350$		0			0
$351$	363.753			1.03633
$352$		0			0
$353$	−440.380			−1.24753			−0.623767	−	0.781610i	$-0.714399\pi$
$353$	−440.380			−1.24753			−0.623767	+	0.781610i	$0.714399\pi$
$354$		0			0
$355$	411.334			1.15869
$356$		0			0
$357$	277.856			0.778309
$358$		0			0
$359$	−87.7780			−0.244507			−0.122254	−	0.992499i	$-0.539012\pi$
$359$	−87.7780			−0.244507			−0.122254	+	0.992499i	$0.539012\pi$
$360$		0			0
$361$	321.600	+	163.995i	0.890860	+	0.454278i
$362$		0			0
$363$	418.859			1.15388
$364$		0			0
$365$			257.954i			0.706724i
$366$		0			0
$367$	335.152			0.913221			0.456611	−	0.889667i	$-0.349063\pi$
$367$	335.152			0.913221			0.456611	+	0.889667i	$0.349063\pi$
$368$		0			0
$369$		−	90.3712i		−	0.244908i
$370$		0			0
$371$	−360.284			−0.971117
$372$		0			0
$373$	−106.144			−0.284568			−0.142284	−	0.989826i	$-0.545445\pi$
$373$	−106.144			−0.284568			−0.142284	+	0.989826i	$0.545445\pi$
$374$		0			0
$375$			433.887i			1.15703i
$376$		0			0
$377$	49.6373			0.131664
$378$		0			0
$379$	−203.864			−0.537899			−0.268950	−	0.963154i	$-0.586676\pi$
$379$	−203.864			−0.537899			−0.268950	+	0.963154i	$0.586676\pi$
$380$		0			0
$381$			26.1412i			0.0686120i
$382$		0			0
$383$		−	553.454i		−	1.44505i	−0.691345	−	0.722525i	$-0.742982\pi$
$383$		−	553.454i		−	1.44505i	0.691345	−	0.722525i	$-0.257018\pi$
$384$		0			0
$385$	−471.194			−1.22388
$386$		0			0
$387$			94.5464i			0.244306i
$388$		0			0
$389$			451.141i			1.15975i	0.814707	+	0.579873i	$0.196898\pi$
$389$			451.141i			1.15975i	−0.814707	+	0.579873i	$0.803102\pi$
$390$		0			0
$391$	411.054			1.05129
$392$		0			0
$393$			748.596i			1.90482i
$394$		0			0
$395$	−514.902			−1.30355
$396$		0			0
$397$			564.045i			1.42077i	0.703814	+	0.710384i	$0.251479\pi$
$397$			564.045i			1.42077i	−0.703814	+	0.710384i	$0.748521\pi$
$398$		0			0
$399$	−495.038	−	118.933i	−1.24070	−	0.298077i
$400$		0			0
$401$			130.268i			0.324858i	0.986720	+	0.162429i	$0.0519329\pi$
$401$			130.268i			0.324858i	−0.986720	+	0.162429i	$0.948067\pi$
$402$		0			0
$403$		−	629.633i		−	1.56236i
$404$		0			0
$405$			334.817i			0.826708i
$406$		0			0
$407$		−	1032.89i		−	2.53781i
$408$		0			0
$409$			436.161i			1.06641i	0.845987	+	0.533204i	$0.179012\pi$
$409$			436.161i			1.06641i	−0.845987	+	0.533204i	$0.820988\pi$
$410$		0			0
$411$	437.032			1.06334
$412$		0			0
$413$	51.7396			0.125277
$414$		0			0
$415$	−72.1929			−0.173959
$416$		0			0
$417$		−	15.2245i		−	0.0365095i
$418$		0			0
$419$			149.110i			0.355871i	0.984042	+	0.177935i	$0.0569418\pi$
$419$			149.110i			0.355871i	−0.984042	+	0.177935i	$0.943058\pi$
$420$		0			0
$421$	−376.362			−0.893972			−0.446986	−	0.894541i	$-0.647503\pi$
$421$	−376.362			−0.893972			−0.446986	+	0.894541i	$0.647503\pi$
$422$		0			0
$423$	−60.3171			−0.142594
$424$		0			0
$425$	123.715			0.291094
$426$		0			0
$427$		−	623.902i		−	1.46113i
$428$		0			0
$429$		−	773.027i		−	1.80193i
$430$		0			0
$431$		−	342.157i		−	0.793867i	−0.917847	−	0.396933i	$-0.870074\pi$
$431$		−	342.157i		−	0.793867i	0.917847	−	0.396933i	$-0.129926\pi$
$432$		0			0
$433$		−	497.902i		−	1.14989i	−0.818193	−	0.574944i	$-0.805024\pi$
$433$		−	497.902i		−	1.14989i	0.818193	−	0.574944i	$-0.194976\pi$
$434$		0			0
$435$			38.7557i			0.0890935i
$436$		0			0
$437$	−732.348	−	175.946i	−1.67585	−	0.402623i
$438$		0			0
$439$		−	181.203i		−	0.412762i	−0.978472	−	0.206381i	$-0.933831\pi$
$439$		−	181.203i		−	0.412762i	0.978472	−	0.206381i	$-0.0661686\pi$
$440$		0			0
$441$	29.6458			0.0672241
$442$		0			0
$443$		−	634.965i		−	1.43333i	−0.697418	−	0.716665i	$-0.745668\pi$
$443$		−	634.965i		−	1.43333i	0.697418	−	0.716665i	$-0.254332\pi$
$444$		0			0
$445$	−57.9310			−0.130182
$446$		0			0
$447$		−	734.022i		−	1.64211i
$448$		0			0
$449$			560.224i			1.24772i	0.781538	+	0.623858i	$0.214436\pi$
$449$			560.224i			1.24772i	−0.781538	+	0.623858i	$0.785564\pi$
$450$		0			0
$451$	914.837			2.02846
$452$		0			0
$453$			85.7633i			0.189323i
$454$		0			0
$455$			448.528i			0.985777i
$456$		0			0
$457$	−216.110			−0.472888			−0.236444	−	0.971645i	$-0.575982\pi$
$457$	−216.110			−0.472888			−0.236444	+	0.971645i	$0.575982\pi$
$458$		0			0
$459$	250.667			0.546114
$460$		0			0
$461$		−	140.350i		−	0.304446i	−0.988346	−	0.152223i	$-0.951357\pi$
$461$		−	140.350i		−	0.304446i	0.988346	−	0.152223i	$-0.0486432\pi$
$462$		0			0
$463$	−631.897			−1.36479			−0.682394	−	0.730984i	$-0.739061\pi$
$463$	−631.897			−1.36479			−0.682394	+	0.730984i	$0.739061\pi$
$464$		0			0
$465$	491.602			1.05721
$466$		0			0
$467$			110.806i			0.237272i	0.992938	+	0.118636i	$0.0378521\pi$
$467$			110.806i			0.237272i	−0.992938	+	0.118636i	$0.962148\pi$
$468$		0			0
$469$	1034.74			2.20627
$470$		0			0
$471$			767.188i			1.62885i
$472$		0			0
$473$	−957.103			−2.02347
$474$		0			0
$475$	−220.415	−	52.9545i	−0.464031	−	0.111483i
$476$		0			0
$477$	68.2333			0.143047
$478$		0			0
$479$	−8.62016			−0.0179962			−0.00899808	−	0.999960i	$-0.502864\pi$
$479$	−8.62016			−0.0179962			−0.00899808	+	0.999960i	$0.502864\pi$
$480$		0			0
$481$	−983.204			−2.04408
$482$		0			0
$483$	1062.23			2.19924
$484$		0			0
$485$	−45.3701			−0.0935467
$486$		0			0
$487$			69.6796i			0.143079i	0.997438	+	0.0715396i	$0.0227912\pi$
$487$			69.6796i			0.143079i	−0.997438	+	0.0715396i	$0.977209\pi$
$488$		0			0
$489$			676.124i			1.38267i
$490$		0			0
$491$		−	268.634i		−	0.547117i	−0.961855	−	0.273558i	$-0.911799\pi$
$491$		−	268.634i		−	0.547117i	0.961855	−	0.273558i	$-0.0882007\pi$
$492$		0			0
$493$	34.2057			0.0693827
$494$		0			0
$495$	89.2381			0.180279
$496$		0			0
$497$		−	938.161i		−	1.88765i
$498$		0			0
$499$		−	259.730i		−	0.520501i	−0.965541	−	0.260251i	$-0.916195\pi$
$499$		−	259.730i		−	0.520501i	0.965541	−	0.260251i	$-0.0838052\pi$
$500$		0			0
$501$			887.571i			1.77160i
$502$		0			0
$503$	−481.577			−0.957409			−0.478705	−	0.877976i	$-0.658893\pi$
$503$	−481.577			−0.957409			−0.478705	+	0.877976i	$0.658893\pi$
$504$		0			0
$505$	89.8750			0.177970
$506$		0			0
$507$	−186.617			−0.368082
$508$		0			0
$509$	−396.561			−0.779098			−0.389549	−	0.921006i	$-0.627369\pi$
$509$	−396.561			−0.779098			−0.389549	+	0.921006i	$0.627369\pi$
$510$		0			0
$511$	588.336			1.15134
$512$		0			0
$513$	−446.596	−	107.294i	−0.870557	−	0.209151i
$514$		0			0
$515$	363.633			0.706083
$516$		0			0
$517$		−	610.597i		−	1.18104i
$518$		0			0
$519$	−42.8817			−0.0826236
$520$		0			0
$521$			691.994i			1.32820i	0.747642	+	0.664102i	$0.231186\pi$
$521$			691.994i			1.32820i	−0.747642	+	0.664102i	$0.768814\pi$
$522$		0			0
$523$	−601.497			−1.15009			−0.575045	−	0.818122i	$-0.695015\pi$
$523$	−601.497			−1.15009			−0.575045	+	0.818122i	$0.695015\pi$
$524$		0			0
$525$	319.700			0.608952
$526$		0			0
$527$		−	433.887i		−	0.823315i
$528$		0			0
$529$	1042.44			1.97059
$530$		0			0
$531$	−9.79881			−0.0184535
$532$		0			0
$533$		−	870.831i		−	1.63383i
$534$		0			0
$535$			300.135i			0.561000i
$536$		0			0
$537$	367.464			0.684290
$538$		0			0
$539$			300.108i			0.556787i
$540$		0			0
$541$		−	280.925i		−	0.519270i	−0.965707	−	0.259635i	$-0.916398\pi$
$541$		−	280.925i		−	0.519270i	0.965707	−	0.259635i	$-0.0836023\pi$
$542$		0			0
$543$	−357.733			−0.658809
$544$		0			0
$545$		−	229.518i		−	0.421134i
$546$		0			0
$547$	−366.932			−0.670808			−0.335404	−	0.942074i	$-0.608873\pi$
$547$	−366.932			−0.670808			−0.335404	+	0.942074i	$0.608873\pi$
$548$		0			0
$549$			118.159i			0.215226i
$550$		0			0
$551$	−60.9419	−	14.6413i	−0.110602	−	0.0265722i
$552$		0			0
$553$			1174.38i			2.12365i
$554$		0			0
$555$		−	767.663i		−	1.38318i
$556$		0			0
$557$		−	118.799i		−	0.213284i	−0.994297	−	0.106642i	$-0.965990\pi$
$557$		−	118.799i		−	0.213284i	0.994297	−	0.106642i	$-0.0340099\pi$
$558$		0			0
$559$			911.064i			1.62981i
$560$		0			0
$561$		−	532.702i		−	0.949558i
$562$		0			0
$563$	−95.4244			−0.169493			−0.0847464	−	0.996403i	$-0.527008\pi$
$563$	−95.4244			−0.169493			−0.0847464	+	0.996403i	$0.527008\pi$
$564$		0			0
$565$	−466.062			−0.824889
$566$		0			0
$567$	763.642			1.34681
$568$		0			0
$569$		−	1072.80i		−	1.88542i	−0.333616	−	0.942709i	$-0.608269\pi$
$569$		−	1072.80i		−	1.88542i	0.333616	−	0.942709i	$-0.391731\pi$
$570$		0			0
$571$		−	262.371i		−	0.459494i	−0.973250	−	0.229747i	$-0.926210\pi$
$571$		−	262.371i		−	0.459494i	0.973250	−	0.229747i	$-0.0737899\pi$
$572$		0			0
$573$	304.246			0.530971
$574$		0			0
$575$	472.957			0.822533
$576$		0			0
$577$	84.7841			0.146940			0.0734698	−	0.997297i	$-0.476593\pi$
$577$	84.7841			0.146940			0.0734698	+	0.997297i	$0.476593\pi$
$578$		0			0
$579$			106.186i			0.183395i
$580$		0			0
$581$			164.656i			0.283401i
$582$		0			0
$583$			690.733i			1.18479i
$584$		0			0
$585$		−	84.9455i		−	0.145206i
$586$		0			0
$587$			248.287i			0.422976i	0.977381	+	0.211488i	$0.0678309\pi$
$587$			248.287i			0.422976i	−0.977381	+	0.211488i	$0.932169\pi$
$588$		0			0
$589$	−185.720	+	773.027i	−0.315313	+	1.31244i
$590$		0			0
$591$		−	359.056i		−	0.607540i
$592$		0			0
$593$	−915.909			−1.54453			−0.772267	−	0.635298i	$-0.780877\pi$
$593$	−915.909			−1.54453			−0.772267	+	0.635298i	$0.780877\pi$
$594$		0			0
$595$			309.086i			0.519473i
$596$		0			0
$597$	74.6027			0.124963
$598$		0			0
$599$		−	172.525i		−	0.288021i	−0.989576	−	0.144011i	$-0.954000\pi$
$599$		−	172.525i		−	0.288021i	0.989576	−	0.144011i	$-0.0459999\pi$
$600$		0			0
$601$			999.158i			1.66249i	0.555905	+	0.831246i	$0.312372\pi$
$601$			999.158i			1.66249i	−0.555905	+	0.831246i	$0.687628\pi$
$602$		0			0
$603$	−195.967			−0.324986
$604$		0			0
$605$			465.937i			0.770144i
$606$		0			0
$607$			166.561i			0.274401i	0.990543	+	0.137200i	$0.0438104\pi$
$607$			166.561i			0.274401i	−0.990543	+	0.137200i	$0.956190\pi$
$608$		0			0
$609$	88.3930			0.145144
$610$		0			0
$611$	−581.225			−0.951269
$612$		0			0
$613$		−	89.3785i		−	0.145805i	−0.997339	−	0.0729025i	$-0.976774\pi$
$613$		−	89.3785i		−	0.145805i	0.997339	−	0.0729025i	$-0.0232262\pi$
$614$		0			0
$615$	679.924			1.10557
$616$		0			0
$617$	357.626			0.579621			0.289810	−	0.957084i	$-0.406408\pi$
$617$	357.626			0.579621			0.289810	+	0.957084i	$0.406408\pi$
$618$		0			0
$619$			887.373i			1.43356i	0.697300	+	0.716780i	$0.254385\pi$
$619$			887.373i			1.43356i	−0.697300	+	0.716780i	$0.745615\pi$
$620$		0			0
$621$	958.287			1.54313
$622$		0			0
$623$			132.128i			0.212083i
$624$		0			0
$625$	−184.383			−0.295012
$626$		0			0
$627$	−228.016	+	949.080i	−0.363662	+	1.51368i
$628$		0			0
$629$	−677.538			−1.07717
$630$		0			0
$631$	537.904			0.852463			0.426231	−	0.904614i	$-0.359841\pi$
$631$	537.904			0.852463			0.426231	+	0.904614i	$0.359841\pi$
$632$		0			0
$633$	102.285			0.161588
$634$		0			0
$635$	−29.0793			−0.0457942
$636$		0			0
$637$	285.672			0.448465
$638$		0			0
$639$			177.676i			0.278053i
$640$		0			0
$641$		−	140.199i		−	0.218719i	−0.994002	−	0.109360i	$-0.965120\pi$
$641$		−	140.199i		−	0.218719i	0.994002	−	0.109360i	$-0.0348800\pi$
$642$		0			0
$643$			78.7926i			0.122539i	0.998121	+	0.0612695i	$0.0195149\pi$
$643$			78.7926i			0.122539i	−0.998121	+	0.0612695i	$0.980485\pi$
$644$		0			0
$645$	−711.337			−1.10285
$646$		0			0
$647$	−396.430			−0.612720			−0.306360	−	0.951916i	$-0.599111\pi$
$647$	−396.430			−0.612720			−0.306360	+	0.951916i	$0.599111\pi$
$648$		0			0
$649$		−	99.1944i		−	0.152842i
$650$		0			0
$651$		−	1121.23i		−	1.72233i
$652$		0			0
$653$		−	451.531i		−	0.691472i	−0.938332	−	0.345736i	$-0.887629\pi$
$653$		−	451.531i		−	0.691472i	0.938332	−	0.345736i	$-0.112371\pi$
$654$		0			0
$655$	−832.735			−1.27135
$656$		0			0
$657$	−111.423			−0.169594
$658$		0			0
$659$	−417.511			−0.633552			−0.316776	−	0.948500i	$-0.602600\pi$
$659$	−417.511			−0.633552			−0.316776	+	0.948500i	$0.602600\pi$
$660$		0			0
$661$	230.228			0.348302			0.174151	−	0.984719i	$-0.444282\pi$
$661$	230.228			0.348302			0.174151	+	0.984719i	$0.444282\pi$
$662$		0			0
$663$	−507.078			−0.764823
$664$		0			0
$665$	132.300	−	550.678i	0.198948	−	0.828087i
$666$		0			0
$667$	130.767			0.196052
$668$		0			0
$669$			930.453i			1.39081i
$670$		0			0
$671$	−1196.14			−1.78262
$672$		0			0
$673$			1016.00i			1.50966i	0.655918	+	0.754832i	$0.272282\pi$
$673$			1016.00i			1.50966i	−0.655918	+	0.754832i	$0.727718\pi$
$674$		0			0
$675$	288.415			0.427282
$676$		0			0
$677$	103.073			0.152250			0.0761249	−	0.997098i	$-0.475745\pi$
$677$	103.073			0.152250			0.0761249	+	0.997098i	$0.475745\pi$
$678$		0			0
$679$			103.479i			0.152399i
$680$		0			0
$681$	926.086			1.35989
$682$		0			0
$683$	−1104.30			−1.61684			−0.808422	−	0.588604i	$-0.799678\pi$
$683$	−1104.30			−1.61684			−0.808422	+	0.588604i	$0.799678\pi$
$684$		0			0
$685$			486.152i			0.709711i
$686$		0			0
$687$			736.459i			1.07199i
$688$		0			0
$689$	657.507			0.954291
$690$		0			0
$691$			973.285i			1.40852i	0.709944	+	0.704258i	$0.248720\pi$
$691$			973.285i			1.40852i	−0.709944	+	0.704258i	$0.751280\pi$
$692$		0			0
$693$		−	203.532i		−	0.293697i
$694$		0			0
$695$	16.9356			0.0243678
$696$		0			0
$697$		−	600.100i		−	0.860975i
$698$		0			0
$699$	243.065			0.347733
$700$		0			0
$701$		−	830.370i		−	1.18455i	−0.805736	−	0.592275i	$-0.798230\pi$
$701$		−	830.370i		−	1.18455i	0.805736	−	0.592275i	$-0.201770\pi$
$702$		0			0
$703$	1207.12	+	290.011i	1.71710	+	0.412533i
$704$		0			0
$705$		−	453.807i		−	0.643698i
$706$		0			0
$707$		−	204.985i		−	0.289936i
$708$		0			0
$709$		−	160.346i		−	0.226158i	−0.993586	−	0.113079i	$-0.963929\pi$
$709$		−	160.346i		−	0.226158i	0.993586	−	0.113079i	$-0.0360713\pi$
$710$		0			0
$711$		−	222.412i		−	0.312815i
$712$		0			0
$713$		−	1658.73i		−	2.32641i
$714$		0			0
$715$	859.913			1.20268
$716$		0			0
$717$	−915.175			−1.27639
$718$		0			0
$719$	1030.80			1.43366			0.716830	−	0.697248i	$-0.245592\pi$
$719$	1030.80			1.43366			0.716830	+	0.697248i	$0.245592\pi$
$720$		0			0
$721$		−	829.365i		−	1.15030i
$722$		0			0
$723$			573.494i			0.793215i
$724$		0			0
$725$	39.3568			0.0542853
$726$		0			0
$727$	−592.646			−0.815194			−0.407597	−	0.913162i	$-0.633633\pi$
$727$	−592.646			−0.815194			−0.407597	+	0.913162i	$0.633633\pi$
$728$		0			0
$729$	562.430			0.771509
$730$		0			0
$731$			627.825i			0.858857i
$732$		0			0
$733$		−	218.547i		−	0.298155i	−0.988826	−	0.149077i	$-0.952370\pi$
$733$		−	218.547i		−	0.298155i	0.988826	−	0.149077i	$-0.0476303\pi$
$734$		0			0
$735$			223.046i			0.303464i
$736$		0			0
$737$		−	1983.79i		−	2.69171i
$738$		0			0
$739$			425.198i			0.575369i	0.957725	+	0.287685i	$0.0928855\pi$
$739$			425.198i			0.575369i	−0.957725	+	0.287685i	$0.907115\pi$
$740$		0			0
$741$	903.426	+	217.048i	1.21920	+	0.292912i
$742$		0			0
$743$		−	1195.75i		−	1.60935i	−0.593717	−	0.804674i	$-0.702340\pi$
$743$		−	1195.75i		−	1.60935i	0.593717	−	0.804674i	$-0.297660\pi$
$744$		0			0
$745$	816.524			1.09600
$746$		0			0
$747$		−	31.1837i		−	0.0417453i
$748$		0			0
$749$	684.541			0.913940
$750$		0			0
$751$		−	566.015i		−	0.753681i	−0.926278	−	0.376841i	$-0.877010\pi$
$751$		−	566.015i		−	0.753681i	0.926278	−	0.376841i	$-0.122990\pi$
$752$		0			0
$753$		−	356.662i		−	0.473655i
$754$		0			0
$755$	−95.4028			−0.126361
$756$		0			0
$757$		−	720.831i		−	0.952220i	−0.879386	−	0.476110i	$-0.842046\pi$
$757$		−	720.831i		−	0.952220i	0.879386	−	0.476110i	$-0.157954\pi$
$758$		0			0
$759$		−	2036.50i		−	2.68313i
$760$		0			0
$761$	−1338.79			−1.75925			−0.879624	−	0.475670i	$-0.842206\pi$
$761$	−1338.79			−1.75925			−0.879624	+	0.475670i	$0.842206\pi$
$762$		0			0
$763$	−523.479			−0.686080
$764$		0			0
$765$		−	58.5370i		−	0.0765189i
$766$		0			0
$767$	−94.4229			−0.123107
$768$		0			0
$769$	1257.05			1.63466			0.817328	−	0.576172i	$-0.195454\pi$
$769$	1257.05			1.63466			0.817328	+	0.576172i	$0.195454\pi$
$770$		0			0
$771$			534.419i			0.693150i
$772$		0			0
$773$	−1149.97			−1.48767			−0.743835	−	0.668363i	$-0.766995\pi$
$773$	−1149.97			−1.48767			−0.743835	+	0.668363i	$0.766995\pi$
$774$		0			0
$775$		−	499.228i		−	0.644165i
$776$		0			0
$777$	−1750.87			−2.25337
$778$		0			0
$779$	−256.865	+	1069.16i	−0.329736	+	1.37247i
$780$		0			0
$781$	−1798.63			−2.30298
$782$		0			0
$783$	79.7433			0.101843
$784$		0			0
$785$	−853.417			−1.08715
$786$		0			0
$787$	−521.408			−0.662526			−0.331263	−	0.943539i	$-0.607475\pi$
$787$	−521.408			−0.662526			−0.331263	+	0.943539i	$0.607475\pi$
$788$		0			0
$789$	−1162.30			−1.47313
$790$		0			0
$791$			1062.98i			1.34385i
$792$		0			0
$793$			1138.60i			1.43581i
$794$		0			0
$795$			513.366i			0.645743i
$796$		0			0
$797$	1042.61			1.30816			0.654082	−	0.756423i	$-0.273055\pi$
$797$	1042.61			1.30816			0.654082	+	0.756423i	$0.273055\pi$
$798$		0			0
$799$	−400.529			−0.501288
$800$		0			0
$801$		−	25.0233i		−	0.0312400i
$802$		0			0
$803$		−	1127.95i		−	1.40467i
$804$		0			0
$805$			1181.62i			1.46785i
$806$		0			0
$807$	−279.310			−0.346108
$808$		0			0
$809$	1156.58			1.42964			0.714818	−	0.699310i	$-0.246509\pi$
$809$	1156.58			1.42964			0.714818	+	0.699310i	$0.246509\pi$
$810$		0			0
$811$	751.144			0.926195			0.463097	−	0.886307i	$-0.346738\pi$
$811$	751.144			0.926195			0.463097	+	0.886307i	$0.346738\pi$
$812$		0			0
$813$	734.428			0.903356
$814$		0			0
$815$	−752.117			−0.922843
$816$		0			0
$817$	268.732	−	1118.55i	0.328925	−	1.36910i
$818$		0			0
$819$	−193.742			−0.236559
$820$		0			0
$821$			268.105i			0.326559i	0.986580	+	0.163279i	$0.0522072\pi$
$821$			268.105i			0.326559i	−0.986580	+	0.163279i	$0.947793\pi$
$822$		0			0
$823$	−190.282			−0.231205			−0.115603	−	0.993296i	$-0.536880\pi$
$823$	−190.282			−0.231205			−0.115603	+	0.993296i	$0.536880\pi$
$824$		0			0
$825$		−	612.924i		−	0.742938i
$826$		0			0
$827$	−1399.65			−1.69244			−0.846219	−	0.532835i	$-0.821127\pi$
$827$	−1399.65			−1.69244			−0.846219	+	0.532835i	$0.821127\pi$
$828$		0			0
$829$	172.931			0.208602			0.104301	−	0.994546i	$-0.466740\pi$
$829$	172.931			0.208602			0.104301	+	0.994546i	$0.466740\pi$
$830$		0			0
$831$		−	759.600i		−	0.914079i
$832$		0			0
$833$	196.860			0.236326
$834$		0			0
$835$	−987.331			−1.18243
$836$		0			0
$837$		−	1011.52i		−	1.20850i
$838$		0			0
$839$		−	916.849i		−	1.09279i	−0.837528	−	0.546394i	$-0.816000\pi$
$839$		−	916.849i		−	1.09279i	0.837528	−	0.546394i	$-0.184000\pi$
$840$		0			0
$841$	−830.118			−0.987061
$842$		0			0
$843$		−	485.511i		−	0.575933i
$844$		0			0
$845$		−	207.592i		−	0.245671i
$846$		0			0
$847$	1062.70			1.25466
$848$		0			0
$849$			781.390i			0.920365i
$850$		0			0
$851$	−2590.20			−3.04371
$852$		0			0
$853$			812.489i			0.952508i	0.879308	+	0.476254i	$0.158006\pi$
$853$			812.489i			0.952508i	−0.879308	+	0.476254i	$0.841994\pi$
$854$		0			0
$855$	−25.0560	+	104.291i	−0.0293052	+	0.121978i
$856$		0			0
$857$		−	834.572i		−	0.973830i	−0.873449	−	0.486915i	$-0.838122\pi$
$857$		−	834.572i		−	0.973830i	0.873449	−	0.486915i	$-0.161878\pi$
$858$		0			0
$859$		−	216.957i		−	0.252570i	−0.991994	−	0.126285i	$-0.959695\pi$
$859$		−	216.957i		−	0.252570i	0.991994	−	0.126285i	$-0.0403053\pi$
$860$		0			0
$861$		−	1550.76i		−	1.80111i
$862$		0			0
$863$			818.521i			0.948460i	0.880401	+	0.474230i	$0.157274\pi$
$863$			818.521i			0.948460i	−0.880401	+	0.474230i	$0.842726\pi$
$864$		0			0
$865$		−	47.7014i		−	0.0551461i
$866$		0			0
$867$	589.775			0.680248
$868$		0			0
$869$	2251.50			2.59091
$870$		0			0
$871$	−1888.37			−2.16804
$872$		0			0
$873$		−	19.5976i		−	0.0224486i
$874$		0			0
$875$			1100.83i			1.25809i
$876$		0			0
$877$	16.6717			0.0190100			0.00950498	−	0.999955i	$-0.496974\pi$
$877$	16.6717			0.0190100			0.00950498	+	0.999955i	$0.496974\pi$
$878$		0			0
$879$	931.773			1.06004
$880$		0			0
$881$	1185.57			1.34570			0.672852	−	0.739777i	$-0.265069\pi$
$881$	1185.57			1.34570			0.672852	+	0.739777i	$0.265069\pi$
$882$		0			0
$883$		−	1099.09i		−	1.24473i	−0.782728	−	0.622363i	$-0.786173\pi$
$883$		−	1099.09i		−	1.24473i	0.782728	−	0.622363i	$-0.213827\pi$
$884$		0			0
$885$		−	73.7232i		−	0.0833030i
$886$		0			0
$887$			160.320i			0.180744i	0.995908	+	0.0903720i	$0.0288056\pi$
$887$			160.320i			0.180744i	−0.995908	+	0.0903720i	$0.971194\pi$
$888$		0			0
$889$			66.3235i			0.0746046i
$890$		0			0
$891$		−	1464.04i		−	1.64315i
$892$		0			0
$893$	713.596	+	171.441i	0.799099	+	0.191983i
$894$		0			0
$895$			408.765i			0.456721i
$896$		0			0
$897$	−1938.53			−2.16113
$898$		0			0
$899$		−	138.030i		−	0.153538i
$900$		0			0
$901$	453.095			0.502881
$902$		0			0
$903$			1622.40i			1.79668i
$904$		0			0
$905$		−	397.941i		−	0.439714i
$906$		0			0
$907$	1028.35			1.13379			0.566897	−	0.823789i	$-0.308144\pi$
$907$	1028.35			1.13379			0.566897	+	0.823789i	$0.308144\pi$
$908$		0			0
$909$			38.8215i			0.0427079i
$910$		0			0
$911$			1404.51i			1.54172i	0.637005	+	0.770860i	$0.280173\pi$
$911$			1404.51i			1.54172i	−0.637005	+	0.770860i	$0.719827\pi$
$912$		0			0
$913$	315.676			0.345757
$914$		0			0
$915$	−888.992			−0.971576
$916$		0			0
$917$			1899.28i			2.07119i
$918$		0			0
$919$	877.710			0.955071			0.477536	−	0.878612i	$-0.341530\pi$
$919$	877.710			0.955071			0.477536	+	0.878612i	$0.341530\pi$
$920$		0			0
$921$	−494.203			−0.536593
$922$		0			0
$923$			1712.11i			1.85494i
$924$		0			0
$925$	−779.571			−0.842779
$926$		0			0
$927$			157.071i			0.169440i
$928$		0			0
$929$	−602.514			−0.648562			−0.324281	−	0.945961i	$-0.605122\pi$
$929$	−602.514			−0.648562			−0.324281	+	0.945961i	$0.605122\pi$
$930$		0			0
$931$	−350.732	−	84.2632i	−0.376726	−	0.0905083i
$932$		0			0
$933$	1457.69			1.56237
$934$		0			0
$935$	592.576			0.633771
$936$		0			0
$937$	374.006			0.399152			0.199576	−	0.979882i	$-0.436043\pi$
$937$	374.006			0.399152			0.199576	+	0.979882i	$0.436043\pi$
$938$		0			0
$939$	−232.811			−0.247935
$940$		0			0
$941$	−112.741			−0.119810			−0.0599050	−	0.998204i	$-0.519080\pi$
$941$	−112.741			−0.119810			−0.0599050	+	0.998204i	$0.519080\pi$
$942$		0			0
$943$		−	2294.15i		−	2.43282i
$944$		0			0
$945$			720.569i			0.762507i
$946$		0			0
$947$			777.474i			0.820986i	0.911864	+	0.410493i	$0.134643\pi$
$947$			777.474i			0.820986i	−0.911864	+	0.410493i	$0.865357\pi$
$948$		0			0
$949$	−1073.69			−1.13139
$950$		0			0
$951$	1213.14			1.27565
$952$		0			0
$953$			435.601i			0.457083i	0.973534	+	0.228542i	$0.0733957\pi$
$953$			435.601i			0.457083i	−0.973534	+	0.228542i	$0.926604\pi$
$954$		0			0
$955$			338.442i			0.354390i
$956$		0			0
$957$		−	169.466i		−	0.177080i
$958$		0			0
$959$	1108.80			1.15621
$960$		0			0
$961$	−789.867			−0.821922
$962$		0			0
$963$	−129.643			−0.134624
$964$		0			0
$965$	−118.120			−0.122405
$966$		0			0
$967$	330.092			0.341357			0.170679	−	0.985327i	$-0.445404\pi$
$967$	330.092			0.341357			0.170679	+	0.985327i	$0.445404\pi$
$968$		0			0
$969$	622.562	+	149.570i	0.642478	+	0.154355i
$970$		0			0
$971$	−720.723			−0.742248			−0.371124	−	0.928583i	$-0.621028\pi$
$971$	−720.723			−0.742248			−0.371124	+	0.928583i	$0.621028\pi$
$972$		0			0
$973$		−	38.6264i		−	0.0396982i
$974$		0			0
$975$	−583.440			−0.598400
$976$		0			0
$977$		−	147.674i		−	0.151151i	−0.997140	−	0.0755754i	$-0.975921\pi$
$977$		−	147.674i		−	0.151151i	0.997140	−	0.0755754i	$-0.0240794\pi$
$978$		0			0
$979$	253.313			0.258747
$980$		0			0
$981$	99.1402			0.101060
$982$		0			0
$983$			375.600i			0.382096i	0.981581	+	0.191048i	$0.0611886\pi$
$983$			375.600i			0.382096i	−0.981581	+	0.191048i	$0.938811\pi$
$984$		0			0
$985$	399.413			0.405495
$986$		0			0
$987$	−1035.03			−1.04867
$988$		0			0
$989$			2400.15i			2.42684i
$990$		0			0
$991$			89.2940i			0.0901049i	0.998985	+	0.0450525i	$0.0143455\pi$
$991$			89.2940i			0.0901049i	−0.998985	+	0.0450525i	$0.985655\pi$
$992$		0			0
$993$	−1313.21			−1.32247
$994$		0			0
$995$			82.9878i			0.0834048i
$996$		0			0
$997$		−	1194.36i		−	1.19795i	−0.800767	−	0.598976i	$-0.795575\pi$
$997$		−	1194.36i		−	1.19795i	0.800767	−	0.598976i	$-0.204425\pi$
$998$		0			0
$999$	−1579.54			−1.58112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.g.d.417.1	✓	16
4.3	odd	2	inner	1216.3.g.d.417.14	yes	16
8.3	odd	2	inner	1216.3.g.d.417.4	yes	16
8.5	even	2	inner	1216.3.g.d.417.15	yes	16
19.18	odd	2	inner	1216.3.g.d.417.13	yes	16
76.75	even	2	inner	1216.3.g.d.417.2	yes	16
152.37	odd	2	inner	1216.3.g.d.417.3	yes	16
152.75	even	2	inner	1216.3.g.d.417.16	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1216.3.g.d.417.1	✓	16	1.1	even	1	trivial
1216.3.g.d.417.2	yes	16	76.75	even	2	inner
1216.3.g.d.417.3	yes	16	152.37	odd	2	inner
1216.3.g.d.417.4	yes	16	8.3	odd	2	inner
1216.3.g.d.417.13	yes	16	19.18	odd	2	inner
1216.3.g.d.417.14	yes	16	4.3	odd	2	inner
1216.3.g.d.417.15	yes	16	8.5	even	2	inner
1216.3.g.d.417.16	yes	16	152.75	even	2	inner

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:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.24985 q^{3} -3.61513i q^{5} -8.24529 q^{7} +1.56155 q^{9} +O(q^{10})\)	\(q-3.24985 q^{3} -3.61513i q^{5} -8.24529 q^{7} +1.56155 q^{9} +15.8078i q^{11} +15.0474 q^{13} +11.7486i q^{15} +10.3693 q^{17} +(-18.4743 - 4.43845i) q^{19} +26.7960 q^{21} +39.6414 q^{23} +11.9309 q^{25} +24.1739 q^{27} +3.29874 q^{29} -41.8434i q^{31} -51.3729i q^{33} +29.8078i q^{35} -65.3406 q^{37} -48.9018 q^{39} -57.8726i q^{41} +60.5464i q^{43} -5.64521i q^{45} -38.6264 q^{47} +18.9848 q^{49} -33.6988 q^{51} +43.6958 q^{53} +57.1470 q^{55} +(60.0388 + 14.4243i) q^{57} -6.27505 q^{59} +75.6677i q^{61} -12.8755 q^{63} -54.3981i q^{65} -125.495 q^{67} -128.829 q^{69} +113.781i q^{71} -71.3542 q^{73} -38.7736 q^{75} -130.340i q^{77} -142.430i q^{79} -92.6155 q^{81} -19.9697i q^{83} -37.4864i q^{85} -10.7204 q^{87} -16.0246i q^{89} -124.070 q^{91} +135.985i q^{93} +(-16.0455 + 66.7869i) q^{95} -12.5501i q^{97} +24.6847i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} - 32 q^{17} - 40 q^{25} - 224 q^{49} + 136 q^{57} - 416 q^{73} - 1152 q^{81}+O(q^{100}) \) 16 * q - 8 * q^9 - 32 * q^17 - 40 * q^25 - 224 * q^49 + 136 * q^57 - 416 * q^73 - 1152 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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