LMFDB - Embedded newform 1071.2.f.a.883.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1071,2,Mod(883,1071)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1071, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1071.883"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1071 = 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1071.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	no
Analytic conductor:	$8.55197805648$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 357)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			883.3
Root			$1.45161 + 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1071.883
Dual form			1071.2.f.a.883.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.688892 q^{2} -1.52543 q^{4} -3.21432i q^{5} +1.00000i q^{7} +2.42864 q^{8} +2.21432i q^{10} -3.90321i q^{11} -2.59210 q^{13} -0.688892i q^{14} +1.37778 q^{16} +(-0.214320 - 4.11753i) q^{17} +2.31111 q^{19} +4.90321i q^{20} +2.68889i q^{22} +3.62222i q^{23} -5.33185 q^{25} +1.78568 q^{26} -1.52543i q^{28} -2.90321i q^{29} +1.59210i q^{31} -5.80642 q^{32} +(0.147643 + 2.83654i) q^{34} +3.21432 q^{35} -1.73975i q^{37} -1.59210 q^{38} -7.80642i q^{40} +2.31111i q^{41} -4.18421 q^{43} +5.95407i q^{44} -2.49532i q^{46} -11.2652 q^{47} -1.00000 q^{49} +3.67307 q^{50} +3.95407 q^{52} +6.49532 q^{53} -12.5462 q^{55} +2.42864i q^{56} +2.00000i q^{58} -6.96989 q^{59} +0.0809666i q^{61} -1.09679i q^{62} +1.24443 q^{64} +8.33185i q^{65} -15.0049 q^{67} +(0.326929 + 6.28100i) q^{68} -2.21432 q^{70} +5.03657i q^{71} -11.2859i q^{73} +1.19850i q^{74} -3.52543 q^{76} +3.90321 q^{77} +17.1590i q^{79} -4.42864i q^{80} -1.59210i q^{82} -14.2351 q^{83} +(-13.2351 + 0.688892i) q^{85} +2.88247 q^{86} -9.47949i q^{88} -1.67307 q^{89} -2.59210i q^{91} -5.52543i q^{92} +7.76049 q^{94} -7.42864i q^{95} +5.57136i q^{97} +0.688892 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8} - 2 q^{13} + 8 q^{16} + 12 q^{17} + 14 q^{19} + 8 q^{25} + 24 q^{26} - 8 q^{32} - 12 q^{34} + 6 q^{35} + 4 q^{38} + 2 q^{43} - 28 q^{47} - 6 q^{49} - 4 q^{50} - 16 q^{52}+ \cdots + 4 q^{98}+O(q^{100}) $ 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 - 2 * q^13 + 8 * q^16 + 12 * q^17 + 14 * q^19 + 8 * q^25 + 24 * q^26 - 8 * q^32 - 12 * q^34 + 6 * q^35 + 4 * q^38 + 2 * q^43 - 28 * q^47 - 6 * q^49 - 4 * q^50 - 16 * q^52 + 12 * q^53 - 22 * q^55 - 28 * q^59 + 8 * q^64 - 24 * q^67 + 28 * q^68 - 8 * q^76 + 10 * q^77 - 32 * q^83 - 26 * q^85 + 44 * q^86 + 16 * q^89 - 20 * q^94 + 4 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1071\mathbb{Z}\right)^\times$.

$n$	$190$	$596$	$766$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	−0.688892			−0.487120			−0.243560	−	0.969886i	$-0.578315\pi$
$2$	−0.688892			−0.487120			−0.243560	+	0.969886i	$0.578315\pi$
$3$		0			0
$3$		0			0
$4$	−1.52543			−0.762714
$5$		−	3.21432i		−	1.43749i	−0.695275	−	0.718744i	$-0.744717\pi$
$5$		−	3.21432i		−	1.43749i	0.695275	−	0.718744i	$-0.255283\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$	2.42864			0.858654
$9$		0			0
$10$			2.21432i			0.700229i
$11$		−	3.90321i		−	1.17686i	−0.808547	−	0.588431i	$-0.799746\pi$
$11$		−	3.90321i		−	1.17686i	0.808547	−	0.588431i	$-0.200254\pi$
$12$		0			0
$13$	−2.59210			−0.718920			−0.359460	−	0.933160i	$-0.617039\pi$
$13$	−2.59210			−0.718920			−0.359460	+	0.933160i	$0.617039\pi$
$14$		−	0.688892i		−	0.184114i
$15$		0			0
$16$	1.37778			0.344446
$17$	−0.214320	−	4.11753i	−0.0519802	−	0.998648i
$17$	−0.214320	−	4.11753i	−0.0519802	−	0.998648i
$18$		0			0
$19$	2.31111			0.530204			0.265102	−	0.964220i	$-0.414594\pi$
$19$	2.31111			0.530204			0.265102	+	0.964220i	$0.414594\pi$
$20$			4.90321i			1.09639i
$21$		0			0
$22$			2.68889i			0.573274i
$23$			3.62222i			0.755284i	0.925952	+	0.377642i	$0.123265\pi$
$23$			3.62222i			0.755284i	−0.925952	+	0.377642i	$0.876735\pi$
$24$		0			0
$25$	−5.33185			−1.06637
$26$	1.78568			0.350201
$27$		0			0
$28$		−	1.52543i		−	0.288279i
$29$		−	2.90321i		−	0.539113i	−0.962985	−	0.269556i	$-0.913123\pi$
$29$		−	2.90321i		−	0.539113i	0.962985	−	0.269556i	$-0.0868771\pi$
$30$		0			0
$31$			1.59210i			0.285950i	0.989726	+	0.142975i	$0.0456669\pi$
$31$			1.59210i			0.285950i	−0.989726	+	0.142975i	$0.954333\pi$
$32$	−5.80642			−1.02644
$33$		0			0
$34$	0.147643	+	2.83654i	0.0253206	+	0.486462i
$35$	3.21432			0.543319
$36$		0			0
$37$		−	1.73975i		−	0.286013i	−0.989722	−	0.143006i	$-0.954323\pi$
$37$		−	1.73975i		−	0.286013i	0.989722	−	0.143006i	$-0.0456769\pi$
$38$	−1.59210			−0.258273
$39$		0			0
$40$		−	7.80642i		−	1.23430i
$41$			2.31111i			0.360934i	0.983581	+	0.180467i	$0.0577610\pi$
$41$			2.31111i			0.360934i	−0.983581	+	0.180467i	$0.942239\pi$
$42$		0			0
$43$	−4.18421			−0.638086			−0.319043	−	0.947740i	$-0.603361\pi$
$43$	−4.18421			−0.638086			−0.319043	+	0.947740i	$0.603361\pi$
$44$			5.95407i			0.897609i
$45$		0			0
$46$		−	2.49532i		−	0.367914i
$47$	−11.2652			−1.64319			−0.821597	−	0.570068i	$-0.806917\pi$
$47$	−11.2652			−1.64319			−0.821597	+	0.570068i	$0.806917\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$	3.67307			0.519451
$51$		0			0
$52$	3.95407			0.548330
$53$	6.49532			0.892200			0.446100	−	0.894983i	$-0.352813\pi$
$53$	6.49532			0.892200			0.446100	+	0.894983i	$0.352813\pi$
$54$		0			0
$55$	−12.5462			−1.69173
$56$			2.42864i			0.324541i
$57$		0			0
$58$			2.00000i			0.262613i
$59$	−6.96989			−0.907402			−0.453701	−	0.891154i	$-0.649897\pi$
$59$	−6.96989			−0.907402			−0.453701	+	0.891154i	$0.649897\pi$
$60$		0			0
$61$			0.0809666i			0.0103667i	0.999987	+	0.00518336i	$0.00164992\pi$
$61$			0.0809666i			0.0103667i	−0.999987	+	0.00518336i	$0.998350\pi$
$62$		−	1.09679i		−	0.139292i
$63$		0			0
$64$	1.24443			0.155554
$65$			8.33185i			1.03344i
$66$		0			0
$67$	−15.0049			−1.83314			−0.916572	−	0.399871i	$-0.869055\pi$
$67$	−15.0049			−1.83314			−0.916572	+	0.399871i	$0.869055\pi$
$68$	0.326929	+	6.28100i	0.0396460	+	0.761683i
$69$		0			0
$70$	−2.21432			−0.264662
$71$			5.03657i			0.597730i	0.954295	+	0.298865i	$0.0966081\pi$
$71$			5.03657i			0.597730i	−0.954295	+	0.298865i	$0.903392\pi$
$72$		0			0
$73$		−	11.2859i		−	1.32092i	−0.750863	−	0.660458i	$-0.770362\pi$
$73$		−	11.2859i		−	1.32092i	0.750863	−	0.660458i	$-0.229638\pi$
$74$			1.19850i			0.139323i
$75$		0			0
$76$	−3.52543			−0.404394
$77$	3.90321			0.444812
$78$		0			0
$79$			17.1590i			1.93054i	0.261254	+	0.965270i	$0.415864\pi$
$79$			17.1590i			1.93054i	−0.261254	+	0.965270i	$0.584136\pi$
$80$		−	4.42864i		−	0.495137i
$81$		0			0
$82$		−	1.59210i		−	0.175818i
$83$	−14.2351			−1.56250			−0.781251	−	0.624218i	$-0.785418\pi$
$83$	−14.2351			−1.56250			−0.781251	+	0.624218i	$0.785418\pi$
$84$		0			0
$85$	−13.2351	+	0.688892i	−1.43554	+	0.0747208i
$86$	2.88247			0.310825
$87$		0			0
$88$		−	9.47949i		−	1.01052i
$89$	−1.67307			−0.177345			−0.0886726	−	0.996061i	$-0.528262\pi$
$89$	−1.67307			−0.177345			−0.0886726	+	0.996061i	$0.528262\pi$
$90$		0			0
$91$		−	2.59210i		−	0.271726i
$92$		−	5.52543i		−	0.576066i
$93$		0			0
$94$	7.76049			0.800434
$95$		−	7.42864i		−	0.762162i
$96$		0			0
$97$			5.57136i			0.565686i	0.959166	+	0.282843i	$0.0912775\pi$
$97$			5.57136i			0.565686i	−0.959166	+	0.282843i	$0.908722\pi$
$98$	0.688892			0.0695886
$99$		0			0
$100$	8.13335			0.813335
$101$	−14.5827			−1.45104			−0.725518	−	0.688203i	$-0.758400\pi$
$101$	−14.5827			−1.45104			−0.725518	+	0.688203i	$0.758400\pi$
$102$		0			0
$103$	11.5827			1.14128			0.570640	−	0.821200i	$-0.306695\pi$
$103$	11.5827			1.14128			0.570640	+	0.821200i	$0.306695\pi$
$104$	−6.29529			−0.617304
$105$		0			0
$106$	−4.47457			−0.434609
$107$		−	17.3368i		−	1.67601i	−0.545663	−	0.838005i	$-0.683722\pi$
$107$		−	17.3368i		−	1.67601i	0.545663	−	0.838005i	$-0.316278\pi$
$108$		0			0
$109$		−	15.9748i		−	1.53011i	−0.643965	−	0.765055i	$-0.722712\pi$
$109$		−	15.9748i		−	1.53011i	0.643965	−	0.765055i	$-0.277288\pi$
$110$	8.64296			0.824074
$111$		0			0
$112$			1.37778i			0.130188i
$113$		−	3.01429i		−	0.283561i	−0.989898	−	0.141780i	$-0.954717\pi$
$113$		−	3.01429i		−	0.283561i	0.989898	−	0.141780i	$-0.0452826\pi$
$114$		0			0
$115$	11.6430			1.08571
$116$			4.42864i			0.411189i
$117$		0			0
$118$	4.80150			0.442014
$119$	4.11753	−	0.214320i	0.377454	−	0.0196467i
$120$		0			0
$121$	−4.23506			−0.385006
$122$		−	0.0557773i		−	0.00504984i
$123$		0			0
$124$		−	2.42864i		−	0.218098i
$125$			1.06668i			0.0954065i
$126$		0			0
$127$	14.3319			1.27175			0.635873	−	0.771794i	$-0.280640\pi$
$127$	14.3319			1.27175			0.635873	+	0.771794i	$0.280640\pi$
$128$	10.7556			0.950667
$129$		0			0
$130$		−	5.73975i		−	0.503409i
$131$		−	1.44938i		−	0.126633i	−0.997993	−	0.0633166i	$-0.979832\pi$
$131$		−	1.44938i		−	0.126633i	0.997993	−	0.0633166i	$-0.0201678\pi$
$132$		0			0
$133$			2.31111i			0.200398i
$134$	10.3368			0.892961
$135$		0			0
$136$	−0.520505	−	10.0000i	−0.0446330	−	0.857493i
$137$	6.36196			0.543539			0.271770	−	0.962362i	$-0.412391\pi$
$137$	6.36196			0.543539			0.271770	+	0.962362i	$0.412391\pi$
$138$		0			0
$139$			7.93978i			0.673443i	0.941604	+	0.336722i	$0.109318\pi$
$139$			7.93978i			0.673443i	−0.941604	+	0.336722i	$0.890682\pi$
$140$	−4.90321			−0.414397
$141$		0			0
$142$		−	3.46965i		−	0.291167i
$143$			10.1175i			0.846071i
$144$		0			0
$145$	−9.33185			−0.774968
$146$			7.77478i			0.643445i
$147$		0			0
$148$			2.65386i			0.218146i
$149$	−15.7748			−1.29232			−0.646160	−	0.763202i	$-0.723626\pi$
$149$	−15.7748			−1.29232			−0.646160	+	0.763202i	$0.723626\pi$
$150$		0			0
$151$	20.9032			1.70108			0.850540	−	0.525911i	$-0.176275\pi$
$151$	20.9032			1.70108			0.850540	+	0.525911i	$0.176275\pi$
$152$	5.61285			0.455262
$153$		0			0
$154$	−2.68889			−0.216677
$155$	5.11753			0.411050
$156$		0			0
$157$	−15.9842			−1.27568			−0.637838	−	0.770170i	$-0.720171\pi$
$157$	−15.9842			−1.27568			−0.637838	+	0.770170i	$0.720171\pi$
$158$		−	11.8207i		−	0.940406i
$159$		0			0
$160$			18.6637i			1.47550i
$161$	−3.62222			−0.285471
$162$		0			0
$163$			17.9081i			1.40267i	0.712830	+	0.701337i	$0.247413\pi$
$163$			17.9081i			1.40267i	−0.712830	+	0.701337i	$0.752587\pi$
$164$		−	3.52543i		−	0.275290i
$165$		0			0
$166$	9.80642			0.761126
$167$			3.79060i			0.293326i	0.989187	+	0.146663i	$0.0468532\pi$
$167$			3.79060i			0.293326i	−0.989187	+	0.146663i	$0.953147\pi$
$168$		0			0
$169$	−6.28100			−0.483154
$170$	9.11753	−	0.474572i	0.699283	−	0.0363980i
$171$		0			0
$172$	6.38271			0.486677
$173$		−	8.20495i		−	0.623811i	−0.950113	−	0.311905i	$-0.899033\pi$
$173$		−	8.20495i		−	0.623811i	0.950113	−	0.311905i	$-0.100967\pi$
$174$		0			0
$175$		−	5.33185i		−	0.403050i
$176$		−	5.37778i		−	0.405366i
$177$		0			0
$178$	1.15257			0.0863884
$179$	16.6035			1.24100			0.620501	−	0.784206i	$-0.286929\pi$
$179$	16.6035			1.24100			0.620501	+	0.784206i	$0.286929\pi$
$180$		0			0
$181$			6.36842i			0.473361i	0.971588	+	0.236680i	$0.0760594\pi$
$181$			6.36842i			0.473361i	−0.971588	+	0.236680i	$0.923941\pi$
$182$			1.78568i			0.132363i
$183$		0			0
$184$			8.79706i			0.648528i
$185$	−5.59210			−0.411140
$186$		0			0
$187$	−16.0716	+	0.836535i	−1.17527	+	0.0611735i
$188$	17.1842			1.25329
$189$		0			0
$190$			5.11753i			0.371265i
$191$	−7.61285			−0.550846			−0.275423	−	0.961323i	$-0.588818\pi$
$191$	−7.61285			−0.550846			−0.275423	+	0.961323i	$0.588818\pi$
$192$		0			0
$193$		−	3.21924i		−	0.231726i	−0.993265	−	0.115863i	$-0.963037\pi$
$193$		−	3.21924i		−	0.231726i	0.993265	−	0.115863i	$-0.0369634\pi$
$194$		−	3.83807i		−	0.275557i
$195$		0			0
$196$	1.52543			0.108959
$197$			6.13828i			0.437334i	0.975800	+	0.218667i	$0.0701708\pi$
$197$			6.13828i			0.437334i	−0.975800	+	0.218667i	$0.929829\pi$
$198$		0			0
$199$		−	3.65233i		−	0.258907i	−0.991586	−	0.129453i	$-0.958678\pi$
$199$		−	3.65233i		−	0.258907i	0.991586	−	0.129453i	$-0.0413222\pi$
$200$	−12.9491			−0.915643
$201$		0			0
$202$	10.0459			0.706829
$203$	2.90321			0.203766
$204$		0			0
$205$	7.42864			0.518839
$206$	−7.97926			−0.555941
$207$		0			0
$208$	−3.57136			−0.247629
$209$		−	9.02074i		−	0.623978i
$210$		0			0
$211$		−	13.4445i		−	0.925555i	−0.886475	−	0.462777i	$-0.846853\pi$
$211$		−	13.4445i		−	0.925555i	0.886475	−	0.462777i	$-0.153147\pi$
$212$	−9.90813			−0.680493
$213$		0			0
$214$			11.9432i			0.816418i
$215$			13.4494i			0.917240i
$216$		0			0
$217$	−1.59210			−0.108079
$218$			11.0049i			0.745347i
$219$		0			0
$220$	19.1383			1.29030
$221$	0.555539	+	10.6731i	0.0373696	+	0.717948i
$222$		0			0
$223$	10.0716			0.674444			0.337222	−	0.941425i	$-0.390513\pi$
$223$	10.0716			0.674444			0.337222	+	0.941425i	$0.390513\pi$
$224$		−	5.80642i		−	0.387958i
$225$		0			0
$226$			2.07652i			0.138128i
$227$		−	22.2192i		−	1.47474i	−0.675487	−	0.737371i	$-0.736067\pi$
$227$		−	22.2192i		−	1.47474i	0.675487	−	0.737371i	$-0.263933\pi$
$228$		0			0
$229$	22.3225			1.47511			0.737556	−	0.675286i	$-0.235980\pi$
$229$	22.3225			1.47511			0.737556	+	0.675286i	$0.235980\pi$
$230$	−8.02074			−0.528872
$231$		0			0
$232$		−	7.05086i		−	0.462911i
$233$			27.5116i			1.80235i	0.433460	+	0.901173i	$0.357293\pi$
$233$			27.5116i			1.80235i	−0.433460	+	0.901173i	$0.642707\pi$
$234$		0			0
$235$			36.2099i			2.36207i
$236$	10.6321			0.692088
$237$		0			0
$238$	−2.83654	+	0.147643i	−0.183865	+	0.00957029i
$239$	8.22861			0.532265			0.266132	−	0.963937i	$-0.414254\pi$
$239$	8.22861			0.532265			0.266132	+	0.963937i	$0.414254\pi$
$240$		0			0
$241$			16.2953i			1.04967i	0.851203	+	0.524836i	$0.175873\pi$
$241$			16.2953i			1.04967i	−0.851203	+	0.524836i	$0.824127\pi$
$242$	2.91750			0.187544
$243$		0			0
$244$		−	0.123509i		−	0.00790684i
$245$			3.21432i			0.205355i
$246$		0			0
$247$	−5.99063			−0.381175
$248$			3.86665i			0.245532i
$249$		0			0
$250$		−	0.734825i		−	0.0464744i
$251$	28.7862			1.81697			0.908483	−	0.417922i	$-0.137241\pi$
$251$	28.7862			1.81697			0.908483	+	0.417922i	$0.137241\pi$
$252$		0			0
$253$	14.1383			0.888866
$254$	−9.87310			−0.619493
$255$		0			0
$256$	−9.89829			−0.618643
$257$	−15.2050			−0.948459			−0.474229	−	0.880401i	$-0.657273\pi$
$257$	−15.2050			−0.948459			−0.474229	+	0.880401i	$0.657273\pi$
$258$		0			0
$259$	1.73975			0.108103
$260$		−	12.7096i		−	0.788218i
$261$		0			0
$262$			0.998469i			0.0616856i
$263$	16.9906			1.04769			0.523844	−	0.851814i	$-0.324498\pi$
$263$	16.9906			1.04769			0.523844	+	0.851814i	$0.324498\pi$
$264$		0			0
$265$		−	20.8780i		−	1.28253i
$266$		−	1.59210i		−	0.0976182i
$267$		0			0
$268$	22.8889			1.39816
$269$		−	9.49532i		−	0.578940i	−0.957187	−	0.289470i	$-0.906521\pi$
$269$		−	9.49532i		−	0.578940i	0.957187	−	0.289470i	$-0.0934790\pi$
$270$		0			0
$271$	−10.8316			−0.657974			−0.328987	−	0.944335i	$-0.606707\pi$
$271$	−10.8316			−0.657974			−0.328987	+	0.944335i	$0.606707\pi$
$272$	−0.295286	−	5.67307i	−0.0179044	−	0.343980i
$273$		0			0
$274$	−4.38271			−0.264769
$275$			20.8113i			1.25497i
$276$		0			0
$277$		−	29.3590i		−	1.76401i	−0.471236	−	0.882007i	$-0.656192\pi$
$277$		−	29.3590i		−	1.76401i	0.471236	−	0.882007i	$-0.343808\pi$
$278$		−	5.46965i		−	0.328048i
$279$		0			0
$280$	7.80642			0.466523
$281$	7.76494			0.463217			0.231609	−	0.972809i	$-0.425601\pi$
$281$	7.76494			0.463217			0.231609	+	0.972809i	$0.425601\pi$
$282$		0			0
$283$		−	32.3575i		−	1.92345i	−0.274008	−	0.961727i	$-0.588350\pi$
$283$		−	32.3575i		−	1.92345i	0.274008	−	0.961727i	$-0.411650\pi$
$284$		−	7.68292i		−	0.455897i
$285$		0			0
$286$		−	6.96989i		−	0.412138i
$287$	−2.31111			−0.136420
$288$		0			0
$289$	−16.9081	+	1.76494i	−0.994596	+	0.103820i
$290$	6.42864			0.377503
$291$		0			0
$292$			17.2159i			1.00748i
$293$	27.6731			1.61668			0.808339	−	0.588717i	$-0.200367\pi$
$293$	27.6731			1.61668			0.808339	+	0.588717i	$0.200367\pi$
$294$		0			0
$295$			22.4035i			1.30438i
$296$		−	4.22522i		−	0.245586i
$297$		0			0
$298$	10.8671			0.629516
$299$		−	9.38916i		−	0.542989i
$300$		0			0
$301$		−	4.18421i		−	0.241174i
$302$	−14.4001			−0.828630
$303$		0			0
$304$	3.18421			0.182627
$305$	0.260253			0.0149020
$306$		0			0
$307$	−23.3876			−1.33480			−0.667401	−	0.744698i	$-0.732593\pi$
$307$	−23.3876			−1.33480			−0.667401	+	0.744698i	$0.732593\pi$
$308$	−5.95407			−0.339264
$309$		0			0
$310$	−3.52543			−0.200231
$311$		−	2.71408i		−	0.153901i	−0.997035	−	0.0769507i	$-0.975482\pi$
$311$		−	2.71408i		−	0.153901i	0.997035	−	0.0769507i	$-0.0245184\pi$
$312$		0			0
$313$		−	31.6019i		−	1.78625i	−0.449811	−	0.893124i	$-0.648509\pi$
$313$		−	31.6019i		−	1.78625i	0.449811	−	0.893124i	$-0.351491\pi$
$314$	11.0114			0.621408
$315$		0			0
$316$		−	26.1748i		−	1.47245i
$317$		−	8.91750i		−	0.500857i	−0.968135	−	0.250428i	$-0.919429\pi$
$317$		−	8.91750i		−	0.500857i	0.968135	−	0.250428i	$-0.0805715\pi$
$318$		0			0
$319$	−11.3319			−0.634462
$320$		−	4.00000i		−	0.223607i
$321$		0			0
$322$	2.49532			0.139059
$323$	−0.495316	−	9.51606i	−0.0275601	−	0.529488i
$324$		0			0
$325$	13.8207			0.766635
$326$		−	12.3368i		−	0.683271i
$327$		0			0
$328$			5.61285i			0.309918i
$329$		−	11.2652i		−	0.621069i
$330$		0			0
$331$	11.9634			0.657570			0.328785	−	0.944405i	$-0.393361\pi$
$331$	11.9634			0.657570			0.328785	+	0.944405i	$0.393361\pi$
$332$	21.7146			1.19174
$333$		0			0
$334$		−	2.61132i		−	0.142885i
$335$			48.2306i			2.63512i
$336$		0			0
$337$			10.7620i			0.586245i	0.956075	+	0.293122i	$0.0946943\pi$
$337$			10.7620i			0.586245i	−0.956075	+	0.293122i	$0.905306\pi$
$338$	4.32693			0.235354
$339$		0			0
$340$	20.1891	−	1.05086i	1.09491	−	0.0569906i
$341$	6.21432			0.336524
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$	−10.1619			−0.547895
$345$		0			0
$346$			5.65233i			0.303871i
$347$			14.4844i			0.777564i	0.921330	+	0.388782i	$0.127104\pi$
$347$			14.4844i			0.777564i	−0.921330	+	0.388782i	$0.872896\pi$
$348$		0			0
$349$	31.7402			1.69902			0.849508	−	0.527576i	$-0.176899\pi$
$349$	31.7402			1.69902			0.849508	+	0.527576i	$0.176899\pi$
$350$			3.67307i			0.196334i
$351$		0			0
$352$			22.6637i			1.20798i
$353$	−3.32540			−0.176993			−0.0884965	−	0.996076i	$-0.528206\pi$
$353$	−3.32540			−0.176993			−0.0884965	+	0.996076i	$0.528206\pi$
$354$		0			0
$355$	16.1891			0.859230
$356$	2.55215			0.135264
$357$		0			0
$358$	−11.4380			−0.604517
$359$	−17.5462			−0.926051			−0.463026	−	0.886345i	$-0.653236\pi$
$359$	−17.5462			−0.926051			−0.463026	+	0.886345i	$0.653236\pi$
$360$		0			0
$361$	−13.6588			−0.718883
$362$		−	4.38715i		−	0.230584i
$363$		0			0
$364$			3.95407i			0.207249i
$365$	−36.2766			−1.89880
$366$		0			0
$367$		−	29.5941i		−	1.54480i	−0.635136	−	0.772400i	$-0.719056\pi$
$367$		−	29.5941i		−	1.54480i	0.635136	−	0.772400i	$-0.280944\pi$
$368$			4.99063i			0.260155i
$369$		0			0
$370$	3.85236			0.200274
$371$			6.49532i			0.337220i
$372$		0			0
$373$	−7.85236			−0.406580			−0.203290	−	0.979119i	$-0.565163\pi$
$373$	−7.85236			−0.406580			−0.203290	+	0.979119i	$0.565163\pi$
$374$	11.0716	−	0.576283i	0.572499	−	0.0297989i
$375$		0			0
$376$	−27.3590			−1.41094
$377$			7.52543i			0.387579i
$378$		0			0
$379$			5.22570i			0.268426i	0.990953	+	0.134213i	$0.0428506\pi$
$379$			5.22570i			0.268426i	−0.990953	+	0.134213i	$0.957149\pi$
$380$			11.3319i			0.581312i
$381$		0			0
$382$	5.24443			0.268328
$383$	−22.3575			−1.14242			−0.571208	−	0.820805i	$-0.693525\pi$
$383$	−22.3575			−1.14242			−0.571208	+	0.820805i	$0.693525\pi$
$384$		0			0
$385$		−	12.5462i		−	0.639412i
$386$			2.21771i			0.112878i
$387$		0			0
$388$		−	8.49871i		−	0.431456i
$389$	−38.4385			−1.94891			−0.974454	−	0.224586i	$-0.927897\pi$
$389$	−38.4385			−1.94891			−0.974454	+	0.224586i	$0.927897\pi$
$390$		0			0
$391$	14.9146	−	0.776312i	0.754263	−	0.0392598i
$392$	−2.42864			−0.122665
$393$		0			0
$394$		−	4.22861i		−	0.213034i
$395$	55.1546			2.77513
$396$		0			0
$397$		−	24.2973i		−	1.21945i	−0.792614	−	0.609723i	$-0.791281\pi$
$397$		−	24.2973i		−	1.21945i	0.792614	−	0.609723i	$-0.208719\pi$
$398$			2.51606i			0.126119i
$399$		0			0
$400$	−7.34614			−0.367307
$401$			14.0651i			0.702380i	0.936304	+	0.351190i	$0.114223\pi$
$401$			14.0651i			0.702380i	−0.936304	+	0.351190i	$0.885777\pi$
$402$		0			0
$403$		−	4.12690i		−	0.205576i
$404$	22.2449			1.10673
$405$		0			0
$406$	−2.00000			−0.0992583
$407$	−6.79060			−0.336598
$408$		0			0
$409$	−18.7353			−0.926401			−0.463201	−	0.886254i	$-0.653299\pi$
$409$	−18.7353			−0.926401			−0.463201	+	0.886254i	$0.653299\pi$
$410$	−5.11753			−0.252737
$411$		0			0
$412$	−17.6686			−0.870471
$413$		−	6.96989i		−	0.342966i
$414$		0			0
$415$			45.7560i			2.24608i
$416$	15.0509			0.737929
$417$		0			0
$418$			6.21432i			0.303952i
$419$		−	34.2908i		−	1.67522i	−0.546271	−	0.837609i	$-0.683953\pi$
$419$		−	34.2908i		−	1.67522i	0.546271	−	0.837609i	$-0.316047\pi$
$420$		0			0
$421$	−6.07805			−0.296226			−0.148113	−	0.988970i	$-0.547320\pi$
$421$	−6.07805			−0.296226			−0.148113	+	0.988970i	$0.547320\pi$
$422$			9.26178i			0.450857i
$423$		0			0
$424$	15.7748			0.766091
$425$	1.14272	+	21.9541i	0.0554301	+	1.06493i
$426$		0			0
$427$	−0.0809666			−0.00391825
$428$			26.4460i			1.27832i
$429$		0			0
$430$		−	9.26517i		−	0.446806i
$431$		−	1.04101i		−	0.0501437i	−0.999686	−	0.0250719i	$-0.992019\pi$
$431$		−	1.04101i		−	0.0501437i	0.999686	−	0.0250719i	$-0.00798146\pi$
$432$		0			0
$433$	0.771390			0.0370706			0.0185353	−	0.999828i	$-0.494100\pi$
$433$	0.771390			0.0370706			0.0185353	+	0.999828i	$0.494100\pi$
$434$	1.09679			0.0526475
$435$		0			0
$436$			24.3684i			1.16704i
$437$			8.37133i			0.400455i
$438$		0			0
$439$		−	16.9097i		−	0.807054i	−0.914968	−	0.403527i	$-0.867784\pi$
$439$		−	16.9097i		−	0.807054i	0.914968	−	0.403527i	$-0.132216\pi$
$440$	−30.4701			−1.45261
$441$		0			0
$442$	−0.382707	−	7.35260i	−0.0182035	−	0.349727i
$443$	8.40345			0.399260			0.199630	−	0.979871i	$-0.436026\pi$
$443$	8.40345			0.399260			0.199630	+	0.979871i	$0.436026\pi$
$444$		0			0
$445$			5.37778i			0.254931i
$446$	−6.93825			−0.328535
$447$		0			0
$448$			1.24443i			0.0587939i
$449$		−	8.44293i		−	0.398446i	−0.979954	−	0.199223i	$-0.936158\pi$
$449$		−	8.44293i		−	0.398446i	0.979954	−	0.199223i	$-0.0638419\pi$
$450$		0			0
$451$	9.02074			0.424770
$452$			4.59808i			0.216276i
$453$		0			0
$454$			15.3067i			0.718377i
$455$	−8.33185			−0.390603
$456$		0			0
$457$	−35.4558			−1.65855			−0.829277	−	0.558838i	$-0.811247\pi$
$457$	−35.4558			−1.65855			−0.829277	+	0.558838i	$0.811247\pi$
$458$	−15.3778			−0.718557
$459$		0			0
$460$	−17.7605			−0.828087
$461$	14.3763			0.669569			0.334784	−	0.942295i	$-0.391337\pi$
$461$	14.3763			0.669569			0.334784	+	0.942295i	$0.391337\pi$
$462$		0			0
$463$	35.0464			1.62874			0.814372	−	0.580343i	$-0.197081\pi$
$463$	35.0464			1.62874			0.814372	+	0.580343i	$0.197081\pi$
$464$		−	4.00000i		−	0.185695i
$465$		0			0
$466$		−	18.9525i		−	0.877959i
$467$	−28.4701			−1.31744			−0.658720	−	0.752388i	$-0.728902\pi$
$467$	−28.4701			−1.31744			−0.658720	+	0.752388i	$0.728902\pi$
$468$		0			0
$469$		−	15.0049i		−	0.692863i
$470$		−	24.9447i		−	1.15061i
$471$		0			0
$472$	−16.9273			−0.779144
$473$			16.3319i			0.750939i
$474$		0			0
$475$	−12.3225			−0.565394
$476$	−6.28100	+	0.326929i	−0.287889	+	0.0149848i
$477$		0			0
$478$	−5.66862			−0.259277
$479$		−	3.12690i		−	0.142872i	−0.997445	−	0.0714358i	$-0.977242\pi$
$479$		−	3.12690i		−	0.142872i	0.997445	−	0.0714358i	$-0.0227581\pi$
$480$		0			0
$481$			4.50961i			0.205620i
$482$		−	11.2257i		−	0.511316i
$483$		0			0
$484$	6.46028			0.293649
$485$	17.9081			0.813166
$486$		0			0
$487$			10.2667i			0.465229i	0.972569	+	0.232614i	$0.0747280\pi$
$487$			10.2667i			0.465229i	−0.972569	+	0.232614i	$0.925272\pi$
$488$			0.196639i			0.00890142i
$489$		0			0
$490$		−	2.21432i		−	0.100033i
$491$	0.663703			0.0299525			0.0149762	−	0.999888i	$-0.495233\pi$
$491$	0.663703			0.0299525			0.0149762	+	0.999888i	$0.495233\pi$
$492$		0			0
$493$	−11.9541	+	0.622216i	−0.538384	+	0.0280232i
$494$	4.12690			0.185678
$495$		0			0
$496$			2.19358i			0.0984945i
$497$	−5.03657			−0.225921
$498$		0			0
$499$		−	3.99355i		−	0.178776i	−0.995997	−	0.0893878i	$-0.971509\pi$
$499$		−	3.99355i		−	0.178776i	0.995997	−	0.0893878i	$-0.0284911\pi$
$500$		−	1.62714i		−	0.0727678i
$501$		0			0
$502$	−19.8306			−0.885081
$503$		−	14.8588i		−	0.662522i	−0.943539	−	0.331261i	$-0.892526\pi$
$503$		−	14.8588i		−	0.662522i	0.943539	−	0.331261i	$-0.107474\pi$
$504$		0			0
$505$			46.8736i			2.08585i
$506$	−9.73975			−0.432985
$507$		0			0
$508$	−21.8622			−0.969978
$509$	12.5018			0.554131			0.277066	−	0.960851i	$-0.410638\pi$
$509$	12.5018			0.554131			0.277066	+	0.960851i	$0.410638\pi$
$510$		0			0
$511$	11.2859			0.499260
$512$	−14.6923			−0.649313
$513$		0			0
$514$	10.4746			0.462014
$515$		−	37.2306i		−	1.64058i
$516$		0			0
$517$			43.9704i			1.93381i
$518$	−1.19850			−0.0526590
$519$		0			0
$520$			20.2351i			0.887366i
$521$		−	29.5970i		−	1.29667i	−0.761356	−	0.648335i	$-0.775466\pi$
$521$		−	29.5970i		−	1.29667i	0.761356	−	0.648335i	$-0.224534\pi$
$522$		0			0
$523$	27.8163			1.21632			0.608160	−	0.793814i	$-0.291908\pi$
$523$	27.8163			1.21632			0.608160	+	0.793814i	$0.291908\pi$
$524$			2.21093i			0.0965849i
$525$		0			0
$526$	−11.7047			−0.510350
$527$	6.55554	−	0.341219i	0.285564	−	0.0148637i
$528$		0			0
$529$	9.87955			0.429546
$530$			14.3827i			0.624745i
$531$		0			0
$532$		−	3.52543i		−	0.152847i
$533$		−	5.99063i		−	0.259483i
$534$		0			0
$535$	−55.7259			−2.40924
$536$	−36.4415			−1.57404
$537$		0			0
$538$			6.54125i			0.282013i
$539$			3.90321i			0.168123i
$540$		0			0
$541$		−	3.18421i		−	0.136900i	−0.997655	−	0.0684499i	$-0.978195\pi$
$541$		−	3.18421i		−	0.136900i	0.997655	−	0.0684499i	$-0.0218053\pi$
$542$	7.46181			0.320512
$543$		0			0
$544$	1.24443	+	23.9081i	0.0533546	+	1.02505i
$545$	−51.3481			−2.19951
$546$		0			0
$547$			11.8415i			0.506304i	0.967426	+	0.253152i	$0.0814673\pi$
$547$			11.8415i			0.506304i	−0.967426	+	0.253152i	$0.918533\pi$
$548$	−9.70471			−0.414565
$549$		0			0
$550$		−	14.3368i		−	0.611322i
$551$		−	6.70964i		−	0.285840i
$552$		0			0
$553$	−17.1590			−0.729676
$554$			20.2252i			0.859287i
$555$		0			0
$556$		−	12.1116i		−	0.513644i
$557$	−10.4035			−0.440808			−0.220404	−	0.975409i	$-0.570738\pi$
$557$	−10.4035			−0.440808			−0.220404	+	0.975409i	$0.570738\pi$
$558$		0			0
$559$	10.8459			0.458733
$560$	4.42864			0.187144
$561$		0			0
$562$	−5.34920			−0.225643
$563$	−23.3176			−0.982718			−0.491359	−	0.870957i	$-0.663500\pi$
$563$	−23.3176			−0.982718			−0.491359	+	0.870957i	$0.663500\pi$
$564$		0			0
$565$	−9.68889			−0.407615
$566$			22.2908i			0.936954i
$567$		0			0
$568$			12.2320i			0.513243i
$569$	18.7239			0.784948			0.392474	−	0.919763i	$-0.371619\pi$
$569$	18.7239			0.784948			0.392474	+	0.919763i	$0.371619\pi$
$570$		0			0
$571$			33.7431i			1.41211i	0.708159	+	0.706053i	$0.249526\pi$
$571$			33.7431i			1.41211i	−0.708159	+	0.706053i	$0.750474\pi$
$572$		−	15.4336i		−	0.645310i
$573$		0			0
$574$	1.59210			0.0664531
$575$		−	19.3131i		−	0.805413i
$576$		0			0
$577$	−34.9101			−1.45333			−0.726664	−	0.686993i	$-0.758930\pi$
$577$	−34.9101			−1.45333			−0.726664	+	0.686993i	$0.758930\pi$
$578$	11.6479	−	1.21585i	0.484488	−	0.0505727i
$579$		0			0
$580$	14.2351			0.591079
$581$		−	14.2351i		−	0.590570i
$582$		0			0
$583$		−	25.3526i		−	1.05000i
$584$		−	27.4094i		−	1.13421i
$585$		0			0
$586$	−19.0638			−0.787517
$587$	−27.1635			−1.12116			−0.560578	−	0.828102i	$-0.689421\pi$
$587$	−27.1635			−1.12116			−0.560578	+	0.828102i	$0.689421\pi$
$588$		0			0
$589$			3.67952i			0.151612i
$590$		−	15.4336i		−	0.635390i
$591$		0			0
$592$		−	2.39700i		−	0.0985160i
$593$	23.6019			0.969216			0.484608	−	0.874731i	$-0.338962\pi$
$593$	23.6019			0.969216			0.484608	+	0.874731i	$0.338962\pi$
$594$		0			0
$595$	−0.688892	−	13.2351i	−0.0282418	−	0.542585i
$596$	24.0633			0.985671
$597$		0			0
$598$			6.46812i			0.264501i
$599$	31.2257			1.27585			0.637924	−	0.770100i	$-0.279794\pi$
$599$	31.2257			1.27585			0.637924	+	0.770100i	$0.279794\pi$
$600$		0			0
$601$		−	20.1541i		−	0.822103i	−0.911612	−	0.411051i	$-0.865162\pi$
$601$		−	20.1541i		−	0.822103i	0.911612	−	0.411051i	$-0.134838\pi$
$602$			2.88247i			0.117481i
$603$		0			0
$604$	−31.8863			−1.29744
$605$			13.6128i			0.553441i
$606$		0			0
$607$		−	34.1354i		−	1.38551i	−0.721172	−	0.692756i	$-0.756396\pi$
$607$		−	34.1354i		−	1.38551i	0.721172	−	0.692756i	$-0.243604\pi$
$608$	−13.4193			−0.544223
$609$		0			0
$610$	−0.179286			−0.00725908
$611$	29.2005			1.18133
$612$		0			0
$613$	−16.0923			−0.649964			−0.324982	−	0.945720i	$-0.605358\pi$
$613$	−16.0923			−0.649964			−0.324982	+	0.945720i	$0.605358\pi$
$614$	16.1116			0.650209
$615$		0			0
$616$	9.47949			0.381940
$617$			23.8020i			0.958232i	0.877752	+	0.479116i	$0.159043\pi$
$617$			23.8020i			0.958232i	−0.877752	+	0.479116i	$0.840957\pi$
$618$		0			0
$619$			6.58274i			0.264583i	0.991211	+	0.132291i	$0.0422334\pi$
$619$			6.58274i			0.264583i	−0.991211	+	0.132291i	$0.957767\pi$
$620$	−7.80642			−0.313514
$621$		0			0
$622$			1.86971i			0.0749685i
$623$		−	1.67307i		−	0.0670302i
$624$		0			0
$625$	−23.2306			−0.929225
$626$			21.7703i			0.870118i
$627$		0			0
$628$	24.3827			0.972976
$629$	−7.16346	+	0.372862i	−0.285626	+	0.0148670i
$630$		0			0
$631$	3.20648			0.127648			0.0638240	−	0.997961i	$-0.479670\pi$
$631$	3.20648			0.127648			0.0638240	+	0.997961i	$0.479670\pi$
$632$			41.6731i			1.65767i
$633$		0			0
$634$			6.14320i			0.243978i
$635$		−	46.0672i		−	1.82812i
$636$		0			0
$637$	2.59210			0.102703
$638$	7.80642			0.309059
$639$		0			0
$640$		−	34.5718i		−	1.36657i
$641$		−	28.7275i		−	1.13467i	−0.823488	−	0.567333i	$-0.807975\pi$
$641$		−	28.7275i		−	1.13467i	0.823488	−	0.567333i	$-0.192025\pi$
$642$		0			0
$643$		−	39.2543i		−	1.54804i	−0.633163	−	0.774019i	$-0.718244\pi$
$643$		−	39.2543i		−	1.54804i	0.633163	−	0.774019i	$-0.281756\pi$
$644$	5.52543			0.217732
$645$		0			0
$646$	0.341219	+	6.55554i	0.0134251	+	0.257924i
$647$	−6.25380			−0.245862			−0.122931	−	0.992415i	$-0.539229\pi$
$647$	−6.25380			−0.245862			−0.122931	+	0.992415i	$0.539229\pi$
$648$		0			0
$649$			27.2050i			1.06789i
$650$	−9.52098			−0.373444
$651$		0			0
$652$		−	27.3176i		−	1.06984i
$653$		−	47.5303i		−	1.86001i	−0.367552	−	0.930003i	$-0.619804\pi$
$653$		−	47.5303i		−	1.86001i	0.367552	−	0.930003i	$-0.380196\pi$
$654$		0			0
$655$	−4.65878			−0.182034
$656$			3.18421i			0.124322i
$657$		0			0
$658$			7.76049i			0.302535i
$659$	−25.8227			−1.00591			−0.502955	−	0.864312i	$-0.667754\pi$
$659$	−25.8227			−1.00591			−0.502955	+	0.864312i	$0.667754\pi$
$660$		0			0
$661$	−28.0672			−1.09169			−0.545843	−	0.837888i	$-0.683790\pi$
$661$	−28.0672			−1.09169			−0.545843	+	0.837888i	$0.683790\pi$
$662$	−8.24152			−0.320316
$663$		0			0
$664$	−34.5718			−1.34165
$665$	7.42864			0.288070
$666$		0			0
$667$	10.5161			0.407183
$668$		−	5.78229i		−	0.223723i
$669$		0			0
$670$		−	33.2257i		−	1.28362i
$671$	0.316030			0.0122002
$672$		0			0
$673$		−	23.3907i		−	0.901645i	−0.892614	−	0.450822i	$-0.851131\pi$
$673$		−	23.3907i		−	0.901645i	0.892614	−	0.450822i	$-0.148869\pi$
$674$		−	7.41387i		−	0.285572i
$675$		0			0
$676$	9.58120			0.368508
$677$			3.46674i			0.133237i	0.997779	+	0.0666187i	$0.0212211\pi$
$677$			3.46674i			0.133237i	−0.997779	+	0.0666187i	$0.978779\pi$
$678$		0			0
$679$	−5.57136			−0.213809
$680$	−32.1432	+	1.67307i	−1.23264	+	0.0641593i
$681$		0			0
$682$	−4.28100			−0.163928
$683$			42.7101i			1.63426i	0.576456	+	0.817129i	$0.304435\pi$
$683$			42.7101i			1.63426i	−0.576456	+	0.817129i	$0.695565\pi$
$684$		0			0
$685$		−	20.4494i		−	0.781331i
$686$			0.688892i			0.0263020i
$687$		0			0
$688$	−5.76494			−0.219786
$689$	−16.8365			−0.641421
$690$		0			0
$691$			30.1037i			1.14520i	0.819835	+	0.572600i	$0.194065\pi$
$691$			30.1037i			1.14520i	−0.819835	+	0.572600i	$0.805935\pi$
$692$			12.5161i			0.475789i
$693$		0			0
$694$		−	9.97820i		−	0.378767i
$695$	25.5210			0.968066
$696$		0			0
$697$	9.51606	−	0.495316i	0.360446	−	0.0187614i
$698$	−21.8656			−0.827625
$699$		0			0
$700$			8.13335i			0.307412i
$701$	6.32693			0.238965			0.119482	−	0.992836i	$-0.461876\pi$
$701$	6.32693			0.238965			0.119482	+	0.992836i	$0.461876\pi$
$702$		0			0
$703$		−	4.02074i		−	0.151645i
$704$		−	4.85728i		−	0.183066i
$705$		0			0
$706$	2.29084			0.0862169
$707$		−	14.5827i		−	0.548440i
$708$		0			0
$709$			2.48886i			0.0934712i	0.998907	+	0.0467356i	$0.0148818\pi$
$709$			2.48886i			0.0934712i	−0.998907	+	0.0467356i	$0.985118\pi$
$710$	−11.1526			−0.418548
$711$		0			0
$712$	−4.06329			−0.152278
$713$	−5.76694			−0.215974
$714$		0			0
$715$	32.5210			1.21622
$716$	−25.3274			−0.946530
$717$		0			0
$718$	12.0874			0.451099
$719$		−	25.6113i		−	0.955141i	−0.878593	−	0.477570i	$-0.841518\pi$
$719$		−	25.6113i		−	0.955141i	0.878593	−	0.477570i	$-0.158482\pi$
$720$		0			0
$721$			11.5827i			0.431364i
$722$	9.40943			0.350183
$723$		0			0
$724$		−	9.71456i		−	0.361039i
$725$			15.4795i			0.574894i
$726$		0			0
$727$	−0.612371			−0.0227116			−0.0113558	−	0.999936i	$-0.503615\pi$
$727$	−0.612371			−0.0227116			−0.0113558	+	0.999936i	$0.503615\pi$
$728$		−	6.29529i		−	0.233319i
$729$		0			0
$730$	24.9906			0.924945
$731$	0.896758	+	17.2286i	0.0331678	+	0.637223i
$732$		0			0
$733$	33.9496			1.25396			0.626979	−	0.779036i	$-0.284291\pi$
$733$	33.9496			1.25396			0.626979	+	0.779036i	$0.284291\pi$
$734$			20.3872i			0.752504i
$735$		0			0
$736$		−	21.0321i		−	0.775254i
$737$			58.5674i			2.15736i
$738$		0			0
$739$	14.3907			0.529370			0.264685	−	0.964335i	$-0.414732\pi$
$739$	14.3907			0.529370			0.264685	+	0.964335i	$0.414732\pi$
$740$	8.53035			0.313582
$741$		0			0
$742$		−	4.47457i		−	0.164267i
$743$			45.4938i			1.66901i	0.551004	+	0.834503i	$0.314245\pi$
$743$			45.4938i			1.66901i	−0.551004	+	0.834503i	$0.685755\pi$
$744$		0			0
$745$			50.7052i			1.85769i
$746$	5.40943			0.198053
$747$		0			0
$748$	24.5161	−	1.27607i	0.896396	−	0.0466579i
$749$	17.3368			0.633472
$750$		0			0
$751$		−	39.9017i		−	1.45603i	−0.685560	−	0.728017i	$-0.740442\pi$
$751$		−	39.9017i		−	1.45603i	0.685560	−	0.728017i	$-0.259558\pi$
$752$	−15.5210			−0.565992
$753$		0			0
$754$		−	5.18421i		−	0.188798i
$755$		−	67.1896i		−	2.44528i
$756$		0			0
$757$	−23.1388			−0.840992			−0.420496	−	0.907294i	$-0.638144\pi$
$757$	−23.1388			−0.840992			−0.420496	+	0.907294i	$0.638144\pi$
$758$		−	3.59994i		−	0.130756i
$759$		0			0
$760$		−	18.0415i		−	0.654434i
$761$	31.2958			1.13447			0.567235	−	0.823556i	$-0.308013\pi$
$761$	31.2958			1.13447			0.567235	+	0.823556i	$0.308013\pi$
$762$		0			0
$763$	15.9748			0.578327
$764$	11.6128			0.420138
$765$		0			0
$766$	15.4019			0.556494
$767$	18.0667			0.652350
$768$		0			0
$769$	19.3446			0.697584			0.348792	−	0.937200i	$-0.386592\pi$
$769$	19.3446			0.697584			0.348792	+	0.937200i	$0.386592\pi$
$770$			8.64296i			0.311471i
$771$		0			0
$772$			4.91072i			0.176741i
$773$	5.56046			0.199996			0.0999979	−	0.994988i	$-0.468116\pi$
$773$	5.56046			0.199996			0.0999979	+	0.994988i	$0.468116\pi$
$774$		0			0
$775$		−	8.48886i		−	0.304929i
$776$			13.5308i			0.485728i
$777$		0			0
$778$	26.4800			0.949353
$779$			5.34122i			0.191369i
$780$		0			0
$781$	19.6588			0.703446
$782$	−10.2745	+	0.534795i	−0.367417	+	0.0191242i
$783$		0			0
$784$	−1.37778			−0.0492066
$785$			51.3783i			1.83377i
$786$		0			0
$787$			22.4177i			0.799106i	0.916710	+	0.399553i	$0.130835\pi$
$787$			22.4177i			0.799106i	−0.916710	+	0.399553i	$0.869165\pi$
$788$		−	9.36349i		−	0.333561i
$789$		0			0
$790$	−37.9956			−1.35182
$791$	3.01429			0.107176
$792$		0			0
$793$		−	0.209874i		−	0.00745284i
$794$			16.7382i			0.594017i
$795$		0			0
$796$			5.57136i			0.197472i
$797$	−11.5812			−0.410227			−0.205114	−	0.978738i	$-0.565756\pi$
$797$	−11.5812			−0.410227			−0.205114	+	0.978738i	$0.565756\pi$
$798$		0			0
$799$	2.41435	+	46.3847i	0.0854135	+	1.64097i
$800$	30.9590			1.09457
$801$		0			0
$802$		−	9.68937i		−	0.342143i
$803$	−44.0513			−1.55454
$804$		0			0
$805$			11.6430i			0.410360i
$806$			2.84299i			0.100140i
$807$		0			0
$808$	−35.4162			−1.24594
$809$			40.6084i			1.42772i	0.700291	+	0.713858i	$0.253054\pi$
$809$			40.6084i			1.42772i	−0.700291	+	0.713858i	$0.746946\pi$
$810$		0			0
$811$			43.7431i			1.53603i	0.640432	+	0.768015i	$0.278755\pi$
$811$			43.7431i			1.53603i	−0.640432	+	0.768015i	$0.721245\pi$
$812$	−4.42864			−0.155415
$813$		0			0
$814$	4.67799			0.163964
$815$	57.5625			2.01633
$816$		0			0
$817$	−9.67016			−0.338316
$818$	12.9066			0.451269
$819$		0			0
$820$	−11.3319			−0.395725
$821$		−	21.8889i		−	0.763929i	−0.924177	−	0.381964i	$-0.875248\pi$
$821$		−	21.8889i		−	0.763929i	0.924177	−	0.381964i	$-0.124752\pi$
$822$		0			0
$823$		−	54.4449i		−	1.89783i	−0.315530	−	0.948916i	$-0.602182\pi$
$823$		−	54.4449i		−	1.89783i	0.315530	−	0.948916i	$-0.397818\pi$
$824$	28.1303			0.979965
$825$		0			0
$826$			4.80150i			0.167066i
$827$			13.6593i			0.474979i	0.971390	+	0.237489i	$0.0763245\pi$
$827$			13.6593i			0.474979i	−0.971390	+	0.237489i	$0.923675\pi$
$828$		0			0
$829$	43.2070			1.50064			0.750320	−	0.661075i	$-0.229899\pi$
$829$	43.2070			1.50064			0.750320	+	0.661075i	$0.229899\pi$
$830$		−	31.5210i		−	1.09411i
$831$		0			0
$832$	−3.22570			−0.111831
$833$	0.214320	+	4.11753i	0.00742574	+	0.142664i
$834$		0			0
$835$	12.1842			0.421652
$836$			13.7605i			0.475917i
$837$		0			0
$838$			23.6227i			0.816032i
$839$			13.8352i			0.477643i	0.971064	+	0.238821i	$0.0767610\pi$
$839$			13.8352i			0.477643i	−0.971064	+	0.238821i	$0.923239\pi$
$840$		0			0
$841$	20.5714			0.709357
$842$	4.18712			0.144298
$843$		0			0
$844$			20.5086i			0.705933i
$845$			20.1891i			0.694527i
$846$		0			0
$847$		−	4.23506i		−	0.145518i
$848$	8.94914			0.307315
$849$		0			0
$850$	−0.787212	−	15.1240i	−0.0270011	−	0.518748i
$851$	6.30174			0.216021
$852$		0			0
$853$			30.7991i			1.05454i	0.849698	+	0.527270i	$0.176784\pi$
$853$			30.7991i			1.05454i	−0.849698	+	0.527270i	$0.823216\pi$
$854$	0.0557773			0.00190866
$855$		0			0
$856$		−	42.1048i		−	1.43911i
$857$			41.7877i			1.42744i	0.700431	+	0.713720i	$0.252991\pi$
$857$			41.7877i			1.42744i	−0.700431	+	0.713720i	$0.747009\pi$
$858$		0			0
$859$	32.7368			1.11697			0.558483	−	0.829516i	$-0.311384\pi$
$859$	32.7368			1.11697			0.558483	+	0.829516i	$0.311384\pi$
$860$		−	20.5161i		−	0.699592i
$861$		0			0
$862$			0.717144i			0.0244260i
$863$	20.7205			0.705335			0.352668	−	0.935749i	$-0.385275\pi$
$863$	20.7205			0.705335			0.352668	+	0.935749i	$0.385275\pi$
$864$		0			0
$865$	−26.3733			−0.896720
$866$	−0.531405			−0.0180579
$867$		0			0
$868$	2.42864			0.0824334
$869$	66.9753			2.27198
$870$		0			0
$871$	38.8943			1.31788
$872$		−	38.7971i		−	1.31383i
$873$		0			0
$874$		−	5.76694i		−	0.195070i
$875$	−1.06668			−0.0360602
$876$		0			0
$877$			14.1082i			0.476399i	0.971216	+	0.238199i	$0.0765572\pi$
$877$			14.1082i			0.476399i	−0.971216	+	0.238199i	$0.923443\pi$
$878$			11.6489i			0.393133i
$879$		0			0
$880$	−17.2859			−0.582708
$881$			0.805947i			0.0271531i	0.999908	+	0.0135765i	$0.00432168\pi$
$881$			0.805947i			0.0271531i	−0.999908	+	0.0135765i	$0.995678\pi$
$882$		0			0
$883$	6.17976			0.207966			0.103983	−	0.994579i	$-0.466841\pi$
$883$	6.17976			0.207966			0.103983	+	0.994579i	$0.466841\pi$
$884$	−0.847435	−	16.2810i	−0.0285023	−	0.547589i
$885$		0			0
$886$	−5.78907			−0.194488
$887$		−	1.35212i		−	0.0453997i	−0.999742	−	0.0226998i	$-0.992774\pi$
$887$		−	1.35212i		−	0.0453997i	0.999742	−	0.0226998i	$-0.00722621\pi$
$888$		0			0
$889$			14.3319i			0.480675i
$890$		−	3.70471i		−	0.124182i
$891$		0			0
$892$	−15.3635			−0.514408
$893$	−26.0350			−0.871229
$894$		0			0
$895$		−	53.3689i		−	1.78393i
$896$			10.7556i			0.359318i
$897$		0			0
$898$			5.81627i			0.194091i
$899$	4.62222			0.154160
$900$		0			0
$901$	−1.39207	−	26.7447i	−0.0463767	−	0.890994i
$902$	−6.21432			−0.206914
$903$		0			0
$904$		−	7.32062i		−	0.243480i
$905$	20.4701			0.680450
$906$		0			0
$907$			9.74620i			0.323617i	0.986822	+	0.161809i	$0.0517327\pi$
$907$			9.74620i			0.323617i	−0.986822	+	0.161809i	$0.948267\pi$
$908$			33.8938i			1.12481i
$909$		0			0
$910$	5.73975			0.190271
$911$		−	12.1570i		−	0.402780i	−0.979511	−	0.201390i	$-0.935454\pi$
$911$		−	12.1570i		−	0.402780i	0.979511	−	0.201390i	$-0.0645458\pi$
$912$		0			0
$913$			55.5625i			1.83885i
$914$	24.4252			0.807915
$915$		0			0
$916$	−34.0513			−1.12509
$917$	1.44938			0.0478628
$918$		0			0
$919$	49.8800			1.64539			0.822695	−	0.568483i	$-0.192469\pi$
$919$	49.8800			1.64539			0.822695	+	0.568483i	$0.192469\pi$
$920$	28.2766			0.932250
$921$		0			0
$922$	−9.90369			−0.326161
$923$		−	13.0553i		−	0.429720i
$924$		0			0
$925$			9.27607i			0.304995i
$926$	−24.1432			−0.793395
$927$		0			0
$928$			16.8573i			0.553367i
$929$		−	6.61930i		−	0.217172i	−0.994087	−	0.108586i	$-0.965368\pi$
$929$		−	6.61930i		−	0.217172i	0.994087	−	0.108586i	$-0.0346323\pi$
$930$		0			0
$931$	−2.31111			−0.0757435
$932$		−	41.9670i		−	1.37467i
$933$		0			0
$934$	19.6128			0.641752
$935$	2.68889	+	51.6593i	0.0879362	+	1.68944i
$936$		0			0
$937$	9.28147			0.303212			0.151606	−	0.988441i	$-0.451555\pi$
$937$	9.28147			0.303212			0.151606	+	0.988441i	$0.451555\pi$
$938$			10.3368i			0.337508i
$939$		0			0
$940$		−	55.2355i		−	1.80158i
$941$		−	24.7338i		−	0.806298i	−0.915134	−	0.403149i	$-0.867916\pi$
$941$		−	24.7338i		−	0.806298i	0.915134	−	0.403149i	$-0.132084\pi$
$942$		0			0
$943$	−8.37133			−0.272608
$944$	−9.60300			−0.312551
$945$		0			0
$946$		−	11.2509i		−	0.365798i
$947$		−	27.0879i		−	0.880238i	−0.897939	−	0.440119i	$-0.854936\pi$
$947$		−	27.0879i		−	0.880238i	0.897939	−	0.440119i	$-0.145064\pi$
$948$		0			0
$949$			29.2543i			0.949634i
$950$	8.48886			0.275415
$951$		0			0
$952$	10.0000	−	0.520505i	0.324102	−	0.0168697i
$953$	−54.1367			−1.75366			−0.876831	−	0.480799i	$-0.840346\pi$
$953$	−54.1367			−1.75366			−0.876831	+	0.480799i	$0.840346\pi$
$954$		0			0
$955$			24.4701i			0.791835i
$956$	−12.5521			−0.405965
$957$		0			0
$958$			2.15410i			0.0695957i
$959$			6.36196i			0.205438i
$960$		0			0
$961$	28.4652			0.918232
$962$		−	3.10663i		−	0.100162i
$963$		0			0
$964$		−	24.8573i		−	0.800599i
$965$	−10.3477			−0.333103
$966$		0			0
$967$	26.4750			0.851380			0.425690	−	0.904869i	$-0.360031\pi$
$967$	26.4750			0.851380			0.425690	+	0.904869i	$0.360031\pi$
$968$	−10.2854			−0.330587
$969$		0			0
$970$	−12.3368			−0.396110
$971$	15.4479			0.495745			0.247873	−	0.968793i	$-0.420269\pi$
$971$	15.4479			0.495745			0.247873	+	0.968793i	$0.420269\pi$
$972$		0			0
$973$	−7.93978			−0.254538
$974$		−	7.07265i		−	0.226622i
$975$		0			0
$976$			0.111555i			0.00357078i
$977$	−58.6321			−1.87581			−0.937903	−	0.346898i	$-0.887235\pi$
$977$	−58.6321			−1.87581			−0.937903	+	0.346898i	$0.887235\pi$
$978$		0			0
$979$			6.53035i			0.208711i
$980$		−	4.90321i		−	0.156627i
$981$		0			0
$982$	−0.457220			−0.0145905
$983$			1.92396i			0.0613647i	0.999529	+	0.0306823i	$0.00976802\pi$
$983$			1.92396i			0.0613647i	−0.999529	+	0.0306823i	$0.990232\pi$
$984$		0			0
$985$	19.7304			0.628662
$986$	8.23506	−	0.428639i	0.262258	−	0.0136507i
$987$		0			0
$988$	9.13828			0.290727
$989$		−	15.1561i		−	0.481936i
$990$		0			0
$991$		−	19.6030i		−	0.622710i	−0.950294	−	0.311355i	$-0.899217\pi$
$991$		−	19.6030i		−	0.622710i	0.950294	−	0.311355i	$-0.100783\pi$
$992$		−	9.24443i		−	0.293511i
$993$		0			0
$994$	3.46965			0.110051
$995$	−11.7397			−0.372175
$996$		0			0
$997$		−	16.6726i		−	0.528026i	−0.964519	−	0.264013i	$-0.914954\pi$
$997$		−	16.6726i		−	0.528026i	0.964519	−	0.264013i	$-0.0850462\pi$
$998$			2.75112i			0.0870853i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1071.2.f.a.883.3		6
3.2	odd	2		357.2.f.a.169.3	✓	6
17.16	even	2	inner	1071.2.f.a.883.4		6
51.38	odd	4		6069.2.a.k.1.2		3
51.47	odd	4		6069.2.a.m.1.2		3
51.50	odd	2		357.2.f.a.169.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.f.a.169.3	✓	6	3.2	odd	2
357.2.f.a.169.4	yes	6	51.50	odd	2
1071.2.f.a.883.3		6	1.1	even	1	trivial
1071.2.f.a.883.4		6	17.16	even	2	inner
6069.2.a.k.1.2		3	51.38	odd	4
6069.2.a.m.1.2		3	51.47	odd	4

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:	\( N \)	\(=\)	\( 1071 = 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1071.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.688892 q^{2} -1.52543 q^{4} -3.21432i q^{5} +1.00000i q^{7} +2.42864 q^{8} +2.21432i q^{10} -3.90321i q^{11} -2.59210 q^{13} -0.688892i q^{14} +1.37778 q^{16} +(-0.214320 - 4.11753i) q^{17} +2.31111 q^{19} +4.90321i q^{20} +2.68889i q^{22} +3.62222i q^{23} -5.33185 q^{25} +1.78568 q^{26} -1.52543i q^{28} -2.90321i q^{29} +1.59210i q^{31} -5.80642 q^{32} +(0.147643 + 2.83654i) q^{34} +3.21432 q^{35} -1.73975i q^{37} -1.59210 q^{38} -7.80642i q^{40} +2.31111i q^{41} -4.18421 q^{43} +5.95407i q^{44} -2.49532i q^{46} -11.2652 q^{47} -1.00000 q^{49} +3.67307 q^{50} +3.95407 q^{52} +6.49532 q^{53} -12.5462 q^{55} +2.42864i q^{56} +2.00000i q^{58} -6.96989 q^{59} +0.0809666i q^{61} -1.09679i q^{62} +1.24443 q^{64} +8.33185i q^{65} -15.0049 q^{67} +(0.326929 + 6.28100i) q^{68} -2.21432 q^{70} +5.03657i q^{71} -11.2859i q^{73} +1.19850i q^{74} -3.52543 q^{76} +3.90321 q^{77} +17.1590i q^{79} -4.42864i q^{80} -1.59210i q^{82} -14.2351 q^{83} +(-13.2351 + 0.688892i) q^{85} +2.88247 q^{86} -9.47949i q^{88} -1.67307 q^{89} -2.59210i q^{91} -5.52543i q^{92} +7.76049 q^{94} -7.42864i q^{95} +5.57136i q^{97} +0.688892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{2} + 4 q^{4} - 12 q^{8} - 2 q^{13} + 8 q^{16} + 12 q^{17} + 14 q^{19} + 8 q^{25} + 24 q^{26} - 8 q^{32} - 12 q^{34} + 6 q^{35} + 4 q^{38} + 2 q^{43} - 28 q^{47} - 6 q^{49} - 4 q^{50} - 16 q^{52}+ \cdots + 4 q^{98}+O(q^{100}) \) 6 * q - 4 * q^2 + 4 * q^4 - 12 * q^8 - 2 * q^13 + 8 * q^16 + 12 * q^17 + 14 * q^19 + 8 * q^25 + 24 * q^26 - 8 * q^32 - 12 * q^34 + 6 * q^35 + 4 * q^38 + 2 * q^43 - 28 * q^47 - 6 * q^49 - 4 * q^50 - 16 * q^52 + 12 * q^53 - 22 * q^55 - 28 * q^59 + 8 * q^64 - 24 * q^67 + 28 * q^68 - 8 * q^76 + 10 * q^77 - 32 * q^83 - 26 * q^85 + 44 * q^86 + 16 * q^89 - 20 * q^94 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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