LMFDB - Embedded newform 1287.2.b.b.298.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(298,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1287.298");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.b (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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 19x^{10} + 133x^{8} + 423x^{6} + 601x^{4} + 312x^{2} + 36 $ x^12 + 19x^10 + 133x^8 + 423x^6 + 601x^4 + 312*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			298.9
Root			$1.50594i$ of defining polynomial
Character	$\chi$	$=$	1287.298
Dual form			1287.2.b.b.298.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.50594i q^{2} -0.267864 q^{4} +2.54333i q^{5} +0.340634i q^{7} +2.60850i q^{8} +O(q^{10})$	\(q+1.50594i q^{2} -0.267864 q^{4} +2.54333i q^{5} +0.340634i q^{7} +2.60850i q^{8} -3.83012 q^{10} -1.00000i q^{11} +(-2.40208 - 2.68887i) q^{13} -0.512975 q^{14} -4.46398 q^{16} -6.84200 q^{17} +2.05385i q^{19} -0.681268i q^{20} +1.50594 q^{22} -6.76220 q^{23} -1.46855 q^{25} +(4.04928 - 3.61740i) q^{26} -0.0912436i q^{28} +1.25655 q^{29} +1.07906i q^{31} -1.50550i q^{32} -10.3037i q^{34} -0.866346 q^{35} +8.12878i q^{37} -3.09298 q^{38} -6.63428 q^{40} +7.18264i q^{41} +4.43660 q^{43} +0.267864i q^{44} -10.1835i q^{46} +0.783004i q^{47} +6.88397 q^{49} -2.21155i q^{50} +(0.643432 + 0.720250i) q^{52} +8.45753 q^{53} +2.54333 q^{55} -0.888543 q^{56} +1.89230i q^{58} -8.65045i q^{59} -12.9632 q^{61} -1.62501 q^{62} -6.66076 q^{64} +(6.83868 - 6.10930i) q^{65} +5.40112i q^{67} +1.83273 q^{68} -1.30467i q^{70} -12.5812i q^{71} -6.66697i q^{73} -12.2415 q^{74} -0.550153i q^{76} +0.340634 q^{77} -0.0621796 q^{79} -11.3534i q^{80} -10.8166 q^{82} +16.1238i q^{83} -17.4015i q^{85} +6.68127i q^{86} +2.60850 q^{88} +8.25911i q^{89} +(0.915919 - 0.818231i) q^{91} +1.81135 q^{92} -1.17916 q^{94} -5.22363 q^{95} +7.07906i q^{97} +10.3669i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 14 q^{4}+O(q^{10}) $ 12 * q - 14 * q^4	$ 12 q - 14 q^{4} + 8 q^{10} - 8 q^{13} + 10 q^{16} + 12 q^{17} - 2 q^{22} + 4 q^{23} - 16 q^{25} - 10 q^{26} - 8 q^{29} - 16 q^{35} - 18 q^{38} + 16 q^{40} + 12 q^{43} + 32 q^{49} + 36 q^{52} + 20 q^{53} - 8 q^{55} - 22 q^{56} - 12 q^{61} + 72 q^{62} - 10 q^{64} + 20 q^{65} - 68 q^{68} - 20 q^{74} - 8 q^{77} - 48 q^{79} - 44 q^{82} + 30 q^{88} + 6 q^{92} - 64 q^{94} + 4 q^{95}+O(q^{100}) $ 12 * q - 14 * q^4 + 8 * q^10 - 8 * q^13 + 10 * q^16 + 12 * q^17 - 2 * q^22 + 4 * q^23 - 16 * q^25 - 10 * q^26 - 8 * q^29 - 16 * q^35 - 18 * q^38 + 16 * q^40 + 12 * q^43 + 32 * q^49 + 36 * q^52 + 20 * q^53 - 8 * q^55 - 22 * q^56 - 12 * q^61 + 72 * q^62 - 10 * q^64 + 20 * q^65 - 68 * q^68 - 20 * q^74 - 8 * q^77 - 48 * q^79 - 44 * q^82 + 30 * q^88 + 6 * q^92 - 64 * q^94 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times$.

$n$	$496$	$937$	$1145$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$			1.50594i			1.06486i	0.846473	+	0.532431i	$0.178721\pi$
$2$			1.50594i			1.06486i	−0.846473	+	0.532431i	$0.821279\pi$
$3$		0			0
$3$		0			0
$4$	−0.267864			−0.133932
$5$			2.54333i			1.13741i	0.822540	+	0.568707i	$0.192556\pi$
$5$			2.54333i			1.13741i	−0.822540	+	0.568707i	$0.807444\pi$
$6$		0			0
$7$			0.340634i			0.128748i	0.997926	+	0.0643738i	$0.0205050\pi$
$7$			0.340634i			0.128748i	−0.997926	+	0.0643738i	$0.979495\pi$
$8$			2.60850i			0.922243i
$9$		0			0
$10$	−3.83012			−1.21119
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−2.40208	−	2.68887i	−0.666218	−	0.745757i
$13$	−2.40208	−	2.68887i	−0.666218	−	0.745757i
$14$	−0.512975			−0.137098
$15$		0			0
$16$	−4.46398			−1.11599
$17$	−6.84200			−1.65943			−0.829715	−	0.558188i	$-0.811497\pi$
$17$	−6.84200			−1.65943			−0.829715	+	0.558188i	$0.811497\pi$
$18$		0			0
$19$			2.05385i			0.471186i	0.971852	+	0.235593i	$0.0757032\pi$
$19$			2.05385i			0.471186i	−0.971852	+	0.235593i	$0.924297\pi$
$20$		−	0.681268i		−	0.152336i
$21$		0			0
$22$	1.50594			0.321068
$23$	−6.76220			−1.41002			−0.705008	−	0.709199i	$-0.749057\pi$
$23$	−6.76220			−1.41002			−0.705008	+	0.709199i	$0.749057\pi$
$24$		0			0
$25$	−1.46855			−0.293710
$26$	4.04928	−	3.61740i	0.794129	−	0.709431i
$27$		0			0
$28$		−	0.0912436i		−	0.0172434i
$29$	1.25655			0.233336			0.116668	−	0.993171i	$-0.462779\pi$
$29$	1.25655			0.233336			0.116668	+	0.993171i	$0.462779\pi$
$30$		0			0
$31$			1.07906i			0.193805i	0.995294	+	0.0969027i	$0.0308936\pi$
$31$			1.07906i			0.193805i	−0.995294	+	0.0969027i	$0.969106\pi$
$32$		−	1.50550i		−	0.266137i
$33$		0			0
$34$		−	10.3037i		−	1.76706i
$35$	−0.866346			−0.146439
$36$		0			0
$37$			8.12878i			1.33636i	0.743998	+	0.668182i	$0.232927\pi$
$37$			8.12878i			1.33636i	−0.743998	+	0.668182i	$0.767073\pi$
$38$	−3.09298			−0.501748
$39$		0			0
$40$	−6.63428			−1.04897
$41$			7.18264i			1.12174i	0.827904	+	0.560870i	$0.189533\pi$
$41$			7.18264i			1.12174i	−0.827904	+	0.560870i	$0.810467\pi$
$42$		0			0
$43$	4.43660			0.676575			0.338288	−	0.941043i	$-0.390152\pi$
$43$	4.43660			0.676575			0.338288	+	0.941043i	$0.390152\pi$
$44$			0.267864i			0.0403820i
$45$		0			0
$46$		−	10.1835i		−	1.50147i
$47$			0.783004i			0.114213i	0.998368	+	0.0571065i	$0.0181874\pi$
$47$			0.783004i			0.114213i	−0.998368	+	0.0571065i	$0.981813\pi$
$48$		0			0
$49$	6.88397			0.983424
$50$		−	2.21155i		−	0.312761i
$51$		0			0
$52$	0.643432	+	0.720250i	0.0892279	+	0.0998807i
$53$	8.45753			1.16173			0.580866	−	0.813999i	$-0.302714\pi$
$53$	8.45753			1.16173			0.580866	+	0.813999i	$0.302714\pi$
$54$		0			0
$55$	2.54333			0.342943
$56$	−0.888543			−0.118737
$57$		0			0
$58$			1.89230i			0.248471i
$59$		−	8.65045i		−	1.12619i	−0.826391	−	0.563096i	$-0.809610\pi$
$59$		−	8.65045i		−	1.12619i	0.826391	−	0.563096i	$-0.190390\pi$
$60$		0			0
$61$	−12.9632			−1.65977			−0.829883	−	0.557938i	$-0.811593\pi$
$61$	−12.9632			−1.65977			−0.829883	+	0.557938i	$0.811593\pi$
$62$	−1.62501			−0.206376
$63$		0			0
$64$	−6.66076			−0.832595
$65$	6.83868	−	6.10930i	0.848234	−	0.757766i
$66$		0			0
$67$			5.40112i			0.659852i	0.944007	+	0.329926i	$0.107024\pi$
$67$			5.40112i			0.659852i	−0.944007	+	0.329926i	$0.892976\pi$
$68$	1.83273			0.222251
$69$		0			0
$70$		−	1.30467i		−	0.155938i
$71$		−	12.5812i		−	1.49311i	−0.665324	−	0.746555i	$-0.731706\pi$
$71$		−	12.5812i		−	1.49311i	0.665324	−	0.746555i	$-0.268294\pi$
$72$		0			0
$73$		−	6.66697i		−	0.780310i	−0.920749	−	0.390155i	$-0.872421\pi$
$73$		−	6.66697i		−	0.780310i	0.920749	−	0.390155i	$-0.127579\pi$
$74$	−12.2415			−1.42304
$75$		0			0
$76$		−	0.550153i		−	0.0631069i
$77$	0.340634			0.0388188
$78$		0			0
$79$	−0.0621796			−0.00699576			−0.00349788	−	0.999994i	$-0.501113\pi$
$79$	−0.0621796			−0.00699576			−0.00349788	+	0.999994i	$0.501113\pi$
$80$		−	11.3534i		−	1.26935i
$81$		0			0
$82$	−10.8166			−1.19450
$83$			16.1238i			1.76981i	0.465768	+	0.884907i	$0.345778\pi$
$83$			16.1238i			1.76981i	−0.465768	+	0.884907i	$0.654222\pi$
$84$		0			0
$85$		−	17.4015i		−	1.88746i
$86$			6.68127i			0.720460i
$87$		0			0
$88$	2.60850			0.278067
$89$			8.25911i			0.875464i	0.899105	+	0.437732i	$0.144218\pi$
$89$			8.25911i			0.875464i	−0.899105	+	0.437732i	$0.855782\pi$
$90$		0			0
$91$	0.915919	−	0.818231i	0.0960144	−	0.0857739i
$92$	1.81135			0.188846
$93$		0			0
$94$	−1.17916			−0.121621
$95$	−5.22363			−0.535933
$96$		0			0
$97$			7.07906i			0.718770i	0.933189	+	0.359385i	$0.117013\pi$
$97$			7.07906i			0.718770i	−0.933189	+	0.359385i	$0.882987\pi$
$98$			10.3669i			1.04721i
$99$		0			0
$100$	0.393372			0.0393372
$101$	−6.30800			−0.627669			−0.313834	−	0.949478i	$-0.601614\pi$
$101$	−6.30800			−0.627669			−0.313834	+	0.949478i	$0.601614\pi$
$102$		0			0
$103$	−1.72009			−0.169485			−0.0847425	−	0.996403i	$-0.527007\pi$
$103$	−1.72009			−0.169485			−0.0847425	+	0.996403i	$0.527007\pi$
$104$	7.01390	−	6.26583i	0.687769	−	0.614415i
$105$		0			0
$106$			12.7366i			1.23708i
$107$	0.319450			0.0308824			0.0154412	−	0.999881i	$-0.495085\pi$
$107$	0.319450			0.0308824			0.0154412	+	0.999881i	$0.495085\pi$
$108$		0			0
$109$			5.12620i			0.491001i	0.969397	+	0.245500i	$0.0789522\pi$
$109$			5.12620i			0.491001i	−0.969397	+	0.245500i	$0.921048\pi$
$110$			3.83012i			0.365187i
$111$		0			0
$112$		−	1.52058i		−	0.143681i
$113$	−3.58358			−0.337115			−0.168557	−	0.985692i	$-0.553911\pi$
$113$	−3.58358			−0.337115			−0.168557	+	0.985692i	$0.553911\pi$
$114$		0			0
$115$		−	17.1985i		−	1.60377i
$116$	−0.336585			−0.0312512
$117$		0			0
$118$	13.0271			1.19924
$119$		−	2.33062i		−	0.213647i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		−	19.5218i		−	1.76742i
$123$		0			0
$124$		−	0.289042i		−	0.0259567i
$125$			8.98166i			0.803344i
$126$		0			0
$127$	−4.06220			−0.360462			−0.180231	−	0.983624i	$-0.557684\pi$
$127$	−4.06220			−0.360462			−0.180231	+	0.983624i	$0.557684\pi$
$128$		−	13.0417i		−	1.15274i
$129$		0			0
$130$	9.20026	+	10.2987i	0.806916	+	0.903253i
$131$	−3.17149			−0.277094			−0.138547	−	0.990356i	$-0.544243\pi$
$131$	−3.17149			−0.277094			−0.138547	+	0.990356i	$0.544243\pi$
$132$		0			0
$133$	−0.699612			−0.0606640
$134$	−8.13378			−0.702652
$135$		0			0
$136$		−	17.8473i		−	1.53040i
$137$		−	3.75436i		−	0.320757i	−0.987056	−	0.160378i	$-0.948729\pi$
$137$		−	3.75436i		−	0.320757i	0.987056	−	0.160378i	$-0.0512715\pi$
$138$		0			0
$139$	1.40207			0.118922			0.0594612	−	0.998231i	$-0.481062\pi$
$139$	1.40207			0.118922			0.0594612	+	0.998231i	$0.481062\pi$
$140$	0.232063			0.0196129
$141$		0			0
$142$	18.9465			1.58996
$143$	−2.68887	+	2.40208i	−0.224854	+	0.200872i
$144$		0			0
$145$			3.19583i			0.265400i
$146$	10.0401			0.830923
$147$		0			0
$148$		−	2.17741i		−	0.178982i
$149$		−	11.9545i		−	0.979349i	−0.871905	−	0.489674i	$-0.837116\pi$
$149$		−	11.9545i		−	0.979349i	0.871905	−	0.489674i	$-0.162884\pi$
$150$		0			0
$151$			15.2534i			1.24130i	0.784087	+	0.620651i	$0.213132\pi$
$151$			15.2534i			1.24130i	−0.784087	+	0.620651i	$0.786868\pi$
$152$	−5.35747			−0.434548
$153$		0			0
$154$			0.512975i			0.0413367i
$155$	−2.74442			−0.220437
$156$		0			0
$157$	17.2054			1.37314			0.686572	−	0.727062i	$-0.259115\pi$
$157$	17.2054			1.37314			0.686572	+	0.727062i	$0.259115\pi$
$158$		−	0.0936390i		−	0.00744952i
$159$		0			0
$160$	3.82899			0.302708
$161$		−	2.30343i		−	0.181536i
$162$		0			0
$163$			8.25265i			0.646398i	0.946331	+	0.323199i	$0.104758\pi$
$163$			8.25265i			0.646398i	−0.946331	+	0.323199i	$0.895242\pi$
$164$		−	1.92397i		−	0.150237i
$165$		0			0
$166$	−24.2815			−1.88461
$167$		−	4.23721i		−	0.327885i	−0.986470	−	0.163943i	$-0.947579\pi$
$167$		−	4.23721i		−	0.327885i	0.986470	−	0.163943i	$-0.0524212\pi$
$168$		0			0
$169$	−1.45999	+	12.9178i	−0.112307	+	0.993674i
$170$	26.2057			2.00988
$171$		0			0
$172$	−1.18841			−0.0906151
$173$	17.3235			1.31708			0.658542	−	0.752544i	$-0.271173\pi$
$173$	17.3235			1.31708			0.658542	+	0.752544i	$0.271173\pi$
$174$		0			0
$175$		−	0.500238i		−	0.0378145i
$176$			4.46398i			0.336485i
$177$		0			0
$178$	−12.4377			−0.932249
$179$	19.2155			1.43623			0.718115	−	0.695924i	$-0.245005\pi$
$179$	19.2155			1.43623			0.718115	+	0.695924i	$0.245005\pi$
$180$		0			0
$181$	3.47932			0.258616			0.129308	−	0.991605i	$-0.458724\pi$
$181$	3.47932			0.258616			0.129308	+	0.991605i	$0.458724\pi$
$182$	1.23221	+	1.37932i	0.0913374	+	0.102242i
$183$		0			0
$184$		−	17.6392i		−	1.30038i
$185$	−20.6742			−1.52000
$186$		0			0
$187$			6.84200i			0.500337i
$188$		−	0.209739i		−	0.0152968i
$189$		0			0
$190$		−	7.86649i		−	0.570695i
$191$	23.1913			1.67807			0.839033	−	0.544081i	$-0.183122\pi$
$191$	23.1913			1.67807			0.839033	+	0.544081i	$0.183122\pi$
$192$		0			0
$193$		−	15.0397i		−	1.08258i	−0.840835	−	0.541291i	$-0.817936\pi$
$193$		−	15.0397i		−	1.08258i	0.840835	−	0.541291i	$-0.182064\pi$
$194$	−10.6607			−0.765391
$195$		0			0
$196$	−1.84397			−0.131712
$197$			13.5316i			0.964088i	0.876147	+	0.482044i	$0.160105\pi$
$197$			13.5316i			0.964088i	−0.876147	+	0.482044i	$0.839895\pi$
$198$		0			0
$199$	10.1507			0.719563			0.359782	−	0.933037i	$-0.382851\pi$
$199$	10.1507			0.719563			0.359782	+	0.933037i	$0.382851\pi$
$200$		−	3.83071i		−	0.270872i
$201$		0			0
$202$		−	9.49948i		−	0.668381i
$203$			0.428024i			0.0300414i
$204$		0			0
$205$	−18.2678			−1.27588
$206$		−	2.59035i		−	0.180478i
$207$		0			0
$208$	10.7228	+	12.0030i	0.743495	+	0.832261i
$209$	2.05385			0.142068
$210$		0			0
$211$	21.1239			1.45423			0.727115	−	0.686516i	$-0.240861\pi$
$211$	21.1239			1.45423			0.727115	+	0.686516i	$0.240861\pi$
$212$	−2.26547			−0.155593
$213$		0			0
$214$			0.481074i			0.0328855i
$215$			11.2838i			0.769546i
$216$		0			0
$217$	−0.367565			−0.0249520
$218$	−7.71976			−0.522848
$219$		0			0
$220$	−0.681268			−0.0459311
$221$	16.4351	+	18.3972i	1.10554	+	1.23753i
$222$		0			0
$223$			19.2046i			1.28603i	0.765852	+	0.643017i	$0.222317\pi$
$223$			19.2046i			1.28603i	−0.765852	+	0.643017i	$0.777683\pi$
$224$	0.512824			0.0342645
$225$		0			0
$226$		−	5.39667i		−	0.358981i
$227$			22.3252i			1.48177i	0.671630	+	0.740886i	$0.265594\pi$
$227$			22.3252i			1.48177i	−0.671630	+	0.740886i	$0.734406\pi$
$228$		0			0
$229$			2.07642i			0.137214i	0.997644	+	0.0686069i	$0.0218554\pi$
$229$			2.07642i			0.137214i	−0.997644	+	0.0686069i	$0.978145\pi$
$230$	25.9000			1.70780
$231$		0			0
$232$			3.27771i			0.215193i
$233$	−24.6326			−1.61373			−0.806867	−	0.590733i	$-0.798839\pi$
$233$	−24.6326			−1.61373			−0.806867	+	0.590733i	$0.798839\pi$
$234$		0			0
$235$	−1.99144			−0.129907
$236$			2.31714i			0.150833i
$237$		0			0
$238$	3.50978			0.227505
$239$		−	25.8292i		−	1.67075i	−0.549680	−	0.835376i	$-0.685250\pi$
$239$		−	25.8292i		−	1.67075i	0.549680	−	0.835376i	$-0.314750\pi$
$240$		0			0
$241$			19.7699i			1.27349i	0.771073	+	0.636747i	$0.219720\pi$
$241$			19.7699i			1.27349i	−0.771073	+	0.636747i	$0.780280\pi$
$242$		−	1.50594i		−	0.0968057i
$243$		0			0
$244$	3.47237			0.222296
$245$			17.5082i			1.11856i
$246$		0			0
$247$	5.52253	−	4.93352i	0.351390	−	0.313913i
$248$	−2.81473			−0.178736
$249$		0			0
$250$	−13.5259			−0.855451
$251$	15.2452			0.962269			0.481135	−	0.876647i	$-0.340225\pi$
$251$	15.2452			0.962269			0.481135	+	0.876647i	$0.340225\pi$
$252$		0			0
$253$			6.76220i			0.425136i
$254$		−	6.11744i		−	0.383842i
$255$		0			0
$256$	6.31857			0.394910
$257$	−28.6815			−1.78911			−0.894553	−	0.446962i	$-0.852506\pi$
$257$	−28.6815			−1.78911			−0.894553	+	0.446962i	$0.852506\pi$
$258$		0			0
$259$	−2.76894			−0.172054
$260$	−1.83184	+	1.63646i	−0.113606	+	0.101489i
$261$		0			0
$262$		−	4.77608i		−	0.295067i
$263$	21.0305			1.29680			0.648399	−	0.761301i	$-0.275439\pi$
$263$	21.0305			1.29680			0.648399	+	0.761301i	$0.275439\pi$
$264$		0			0
$265$			21.5103i			1.32137i
$266$		−	1.05358i		−	0.0645988i
$267$		0			0
$268$		−	1.44677i		−	0.0883753i
$269$	−16.7807			−1.02314			−0.511568	−	0.859243i	$-0.670935\pi$
$269$	−16.7807			−1.02314			−0.511568	+	0.859243i	$0.670935\pi$
$270$		0			0
$271$		−	16.4206i		−	0.997480i	−0.866752	−	0.498740i	$-0.833796\pi$
$271$		−	16.4206i		−	0.997480i	0.866752	−	0.498740i	$-0.166204\pi$
$272$	30.5425			1.85191
$273$		0			0
$274$	5.65386			0.341562
$275$			1.46855i			0.0885570i
$276$		0			0
$277$	−15.9823			−0.960283			−0.480141	−	0.877191i	$-0.659415\pi$
$277$	−15.9823			−0.960283			−0.480141	+	0.877191i	$0.659415\pi$
$278$			2.11144i			0.126636i
$279$		0			0
$280$		−	2.25986i		−	0.135053i
$281$			9.03706i			0.539106i	0.962986	+	0.269553i	$0.0868759\pi$
$281$			9.03706i			0.539106i	−0.962986	+	0.269553i	$0.913124\pi$
$282$		0			0
$283$	−10.7518			−0.639128			−0.319564	−	0.947565i	$-0.603537\pi$
$283$	−10.7518			−0.639128			−0.319564	+	0.947565i	$0.603537\pi$
$284$			3.37004i			0.199975i
$285$		0			0
$286$	−3.61740	−	4.04928i	−0.213901	−	0.239439i
$287$	−2.44665			−0.144421
$288$		0			0
$289$	29.8130			1.75371
$290$	−4.81274			−0.282614
$291$		0			0
$292$			1.78584i			0.104508i
$293$		−	26.3300i		−	1.53822i	−0.639118	−	0.769109i	$-0.720700\pi$
$293$		−	26.3300i		−	1.53822i	0.639118	−	0.769109i	$-0.279300\pi$
$294$		0			0
$295$	22.0010			1.28095
$296$	−21.2039			−1.23245
$297$		0			0
$298$	18.0028			1.04287
$299$	16.2434	+	18.1826i	0.939378	+	1.05153i
$300$		0			0
$301$			1.51126i			0.0871074i
$302$	−22.9707			−1.32182
$303$		0			0
$304$		−	9.16835i		−	0.525841i
$305$		−	32.9697i		−	1.88784i
$306$		0			0
$307$			19.2448i			1.09836i	0.835705	+	0.549179i	$0.185060\pi$
$307$			19.2448i			1.09836i	−0.835705	+	0.549179i	$0.814940\pi$
$308$	−0.0912436			−0.00519908
$309$		0			0
$310$		−	4.13294i		−	0.234735i
$311$	2.28996			0.129852			0.0649259	−	0.997890i	$-0.479319\pi$
$311$	2.28996			0.129852			0.0649259	+	0.997890i	$0.479319\pi$
$312$		0			0
$313$	−18.8754			−1.06690			−0.533451	−	0.845831i	$-0.679105\pi$
$313$	−18.8754			−1.06690			−0.533451	+	0.845831i	$0.679105\pi$
$314$			25.9104i			1.46221i
$315$		0			0
$316$	0.0166557			0.000936956
$317$			32.7533i			1.83961i	0.392376	+	0.919805i	$0.371653\pi$
$317$			32.7533i			1.83961i	−0.392376	+	0.919805i	$0.628347\pi$
$318$		0			0
$319$		−	1.25655i		−	0.0703534i
$320$		−	16.9405i		−	0.947005i
$321$		0			0
$322$	3.46884			0.193311
$323$		−	14.0525i		−	0.781900i
$324$		0			0
$325$	3.52758	+	3.94874i	0.195675	+	0.219036i
$326$	−12.4280			−0.688325
$327$		0			0
$328$	−18.7359			−1.03452
$329$	−0.266718			−0.0147046
$330$		0			0
$331$			22.4648i			1.23478i	0.786659	+	0.617388i	$0.211809\pi$
$331$			22.4648i			1.23478i	−0.786659	+	0.617388i	$0.788191\pi$
$332$		−	4.31898i		−	0.237035i
$333$		0			0
$334$	6.38100			0.349153
$335$	−13.7369			−0.750525
$336$		0			0
$337$	−11.9293			−0.649828			−0.324914	−	0.945744i	$-0.605335\pi$
$337$	−11.9293			−0.649828			−0.324914	+	0.945744i	$0.605335\pi$
$338$	−19.4534	−	2.19867i	−1.05813	−	0.119592i
$339$		0			0
$340$			4.66124i			0.252791i
$341$	1.07906			0.0584345
$342$		0			0
$343$			4.72935i			0.255361i
$344$			11.5729i			0.623967i
$345$		0			0
$346$			26.0883i			1.40251i
$347$	−33.7422			−1.81138			−0.905688	−	0.423944i	$-0.860645\pi$
$347$	−33.7422			−1.81138			−0.905688	+	0.423944i	$0.860645\pi$
$348$		0			0
$349$		−	8.54150i		−	0.457216i	−0.973519	−	0.228608i	$-0.926583\pi$
$349$		−	8.54150i		−	0.457216i	0.973519	−	0.228608i	$-0.0734174\pi$
$350$	0.753330			0.0402672
$351$		0			0
$352$	−1.50550			−0.0802433
$353$			1.14982i			0.0611987i	0.999532	+	0.0305993i	$0.00974159\pi$
$353$			1.14982i			0.0611987i	−0.999532	+	0.0305993i	$0.990258\pi$
$354$		0			0
$355$	31.9981			1.69828
$356$		−	2.21232i		−	0.117253i
$357$		0			0
$358$			28.9374i			1.52939i
$359$			10.8893i			0.574713i	0.957824	+	0.287357i	$0.0927765\pi$
$359$			10.8893i			0.574713i	−0.957824	+	0.287357i	$0.907223\pi$
$360$		0			0
$361$	14.7817			0.777984
$362$			5.23965i			0.275390i
$363$		0			0
$364$	−0.245342	+	0.219175i	−0.0128594	+	0.0114879i
$365$	16.9563			0.887536
$366$		0			0
$367$	6.29101			0.328388			0.164194	−	0.986428i	$-0.447498\pi$
$367$	6.29101			0.328388			0.164194	+	0.986428i	$0.447498\pi$
$368$	30.1863			1.57357
$369$		0			0
$370$		−	31.1342i		−	1.61859i
$371$			2.88092i			0.149570i
$372$		0			0
$373$	−0.517636			−0.0268022			−0.0134011	−	0.999910i	$-0.504266\pi$
$373$	−0.517636			−0.0268022			−0.0134011	+	0.999910i	$0.504266\pi$
$374$	−10.3037			−0.532790
$375$		0			0
$376$	−2.04246			−0.105332
$377$	−3.01834	−	3.37870i	−0.155453	−	0.174012i
$378$		0			0
$379$		−	1.27591i		−	0.0655392i	−0.999463	−	0.0327696i	$-0.989567\pi$
$379$		−	1.27591i		−	0.0655392i	0.999463	−	0.0327696i	$-0.0104328\pi$
$380$	1.39922			0.0717786
$381$		0			0
$382$			34.9248i			1.78691i
$383$			36.7458i			1.87762i	0.344430	+	0.938812i	$0.388072\pi$
$383$			36.7458i			1.87762i	−0.344430	+	0.938812i	$0.611928\pi$
$384$		0			0
$385$			0.866346i			0.0441531i
$386$	22.6489			1.15280
$387$		0			0
$388$		−	1.89623i		−	0.0962663i
$389$	−18.3577			−0.930772			−0.465386	−	0.885108i	$-0.654084\pi$
$389$	−18.3577			−0.930772			−0.465386	+	0.885108i	$0.654084\pi$
$390$		0			0
$391$	46.2670			2.33982
$392$			17.9568i			0.906956i
$393$		0			0
$394$	−20.3778			−1.02662
$395$		−	0.158144i		−	0.00795707i
$396$		0			0
$397$		−	34.4036i		−	1.72667i	−0.504635	−	0.863333i	$-0.668373\pi$
$397$		−	34.4036i		−	1.72667i	0.504635	−	0.863333i	$-0.331627\pi$
$398$			15.2864i			0.766236i
$399$		0			0
$400$	6.55558			0.327779
$401$			2.55010i			0.127346i	0.997971	+	0.0636728i	$0.0202814\pi$
$401$			2.55010i			0.127346i	−0.997971	+	0.0636728i	$0.979719\pi$
$402$		0			0
$403$	2.90145	−	2.59200i	0.144532	−	0.129117i
$404$	1.68968			0.0840650
$405$		0			0
$406$	−0.644580			−0.0319900
$407$	8.12878			0.402929
$408$		0			0
$409$		−	12.0370i		−	0.595194i	−0.954692	−	0.297597i	$-0.903815\pi$
$409$		−	12.0370i		−	0.595194i	0.954692	−	0.297597i	$-0.0961851\pi$
$410$		−	27.5103i		−	1.35864i
$411$		0			0
$412$	0.460749			0.0226995
$413$	2.94664			0.144994
$414$		0			0
$415$	−41.0081			−2.01301
$416$	−4.04808	+	3.61633i	−0.198474	+	0.177305i
$417$		0			0
$418$			3.09298i			0.151283i
$419$	10.7176			0.523591			0.261796	−	0.965123i	$-0.415685\pi$
$419$	10.7176			0.523591			0.261796	+	0.965123i	$0.415685\pi$
$420$		0			0
$421$		−	5.82807i		−	0.284043i	−0.989864	−	0.142021i	$-0.954640\pi$
$421$		−	5.82807i		−	0.284043i	0.989864	−	0.142021i	$-0.0453601\pi$
$422$			31.8114i			1.54856i
$423$		0			0
$424$			22.0615i			1.07140i
$425$	10.0478			0.487391
$426$		0			0
$427$		−	4.41570i		−	0.213691i
$428$	−0.0855692			−0.00413614
$429$		0			0
$430$	−16.9927			−0.819461
$431$			10.8775i			0.523950i	0.965075	+	0.261975i	$0.0843739\pi$
$431$			10.8775i			0.523950i	−0.965075	+	0.261975i	$0.915626\pi$
$432$		0			0
$433$	7.35152			0.353291			0.176646	−	0.984275i	$-0.443475\pi$
$433$	7.35152			0.353291			0.176646	+	0.984275i	$0.443475\pi$
$434$		−	0.553532i		−	0.0265704i
$435$		0			0
$436$		−	1.37312i		−	0.0657607i
$437$		−	13.8886i		−	0.664380i
$438$		0			0
$439$	−19.7212			−0.941240			−0.470620	−	0.882336i	$-0.655970\pi$
$439$	−19.7212			−0.941240			−0.470620	+	0.882336i	$0.655970\pi$
$440$			6.63428i			0.316277i
$441$		0			0
$442$	−27.7052	+	24.7503i	−1.31780	+	1.17725i
$443$	29.6357			1.40803			0.704016	−	0.710184i	$-0.251388\pi$
$443$	29.6357			1.40803			0.704016	+	0.710184i	$0.251388\pi$
$444$		0			0
$445$	−21.0057			−0.995765
$446$	−28.9210			−1.36945
$447$		0			0
$448$		−	2.26888i		−	0.107195i
$449$		−	15.7638i		−	0.743938i	−0.928245	−	0.371969i	$-0.878683\pi$
$449$		−	15.7638i		−	0.743938i	0.928245	−	0.371969i	$-0.121317\pi$
$450$		0			0
$451$	7.18264			0.338217
$452$	0.959912			0.0451505
$453$		0			0
$454$	−33.6204			−1.57788
$455$	2.08104	+	2.32949i	0.0975604	+	0.109208i
$456$		0			0
$457$			31.9831i			1.49611i	0.663639	+	0.748053i	$0.269011\pi$
$457$			31.9831i			1.49611i	−0.663639	+	0.748053i	$0.730989\pi$
$458$	−3.12697			−0.146114
$459$		0			0
$460$			4.60687i			0.214796i
$461$		−	25.4915i		−	1.18726i	−0.804740	−	0.593628i	$-0.797695\pi$
$461$		−	25.4915i		−	1.18726i	0.804740	−	0.593628i	$-0.202305\pi$
$462$		0			0
$463$			33.7011i			1.56622i	0.621883	+	0.783111i	$0.286368\pi$
$463$			33.7011i			1.56622i	−0.621883	+	0.783111i	$0.713632\pi$
$464$	−5.60922			−0.260402
$465$		0			0
$466$		−	37.0953i		−	1.71840i
$467$	−7.55866			−0.349773			−0.174887	−	0.984589i	$-0.555956\pi$
$467$	−7.55866			−0.349773			−0.174887	+	0.984589i	$0.555956\pi$
$468$		0			0
$469$	−1.83981			−0.0849543
$470$		−	2.99900i		−	0.138333i
$471$		0			0
$472$	22.5647			1.03862
$473$		−	4.43660i		−	0.203995i
$474$		0			0
$475$		−	3.01619i		−	0.138392i
$476$			0.624289i			0.0286142i
$477$		0			0
$478$	38.8973			1.77912
$479$			0.166684i			0.00761597i	0.999993	+	0.00380798i	$0.00121212\pi$
$479$			0.166684i			0.00761597i	−0.999993	+	0.00380798i	$0.998788\pi$
$480$		0			0
$481$	21.8572	−	19.5260i	0.996603	−	0.890310i
$482$	−29.7724			−1.35610
$483$		0			0
$484$	0.267864			0.0121756
$485$	−18.0044			−0.817539
$486$		0			0
$487$		−	35.9972i		−	1.63119i	−0.578623	−	0.815595i	$-0.696410\pi$
$487$		−	35.9972i		−	1.63119i	0.578623	−	0.815595i	$-0.303590\pi$
$488$		−	33.8144i		−	1.53071i
$489$		0			0
$490$	−26.3664			−1.19111
$491$	−7.88958			−0.356052			−0.178026	−	0.984026i	$-0.556971\pi$
$491$	−7.88958			−0.356052			−0.178026	+	0.984026i	$0.556971\pi$
$492$		0			0
$493$	−8.59734			−0.387205
$494$	7.42960	+	8.31662i	0.334274	+	0.374182i
$495$		0			0
$496$		−	4.81691i		−	0.216286i
$497$	4.28557			0.192234
$498$		0			0
$499$		−	35.6986i		−	1.59809i	−0.601271	−	0.799045i	$-0.705339\pi$
$499$		−	35.6986i		−	1.59809i	0.601271	−	0.799045i	$-0.294661\pi$
$500$		−	2.40586i		−	0.107593i
$501$		0			0
$502$			22.9584i			1.02468i
$503$	20.3779			0.908604			0.454302	−	0.890848i	$-0.349889\pi$
$503$	20.3779			0.908604			0.454302	+	0.890848i	$0.349889\pi$
$504$		0			0
$505$		−	16.0433i		−	0.713919i
$506$	−10.1835			−0.452711
$507$		0			0
$508$	1.08812			0.0482774
$509$		−	19.3463i		−	0.857511i	−0.903420	−	0.428756i	$-0.858952\pi$
$509$		−	19.3463i		−	0.857511i	0.903420	−	0.428756i	$-0.141048\pi$
$510$		0			0
$511$	2.27100			0.100463
$512$		−	16.5680i		−	0.732211i
$513$		0			0
$514$		−	43.1928i		−	1.90515i
$515$		−	4.37475i		−	0.192775i
$516$		0			0
$517$	0.783004			0.0344365
$518$		−	4.16986i		−	0.183213i
$519$		0			0
$520$	15.9361	+	17.8387i	0.698844	+	0.782278i
$521$	12.0683			0.528724			0.264362	−	0.964424i	$-0.414839\pi$
$521$	12.0683			0.528724			0.264362	+	0.964424i	$0.414839\pi$
$522$		0			0
$523$	−32.1258			−1.40476			−0.702381	−	0.711801i	$-0.747880\pi$
$523$	−32.1258			−1.40476			−0.702381	+	0.711801i	$0.747880\pi$
$524$	0.849528			0.0371118
$525$		0			0
$526$			31.6708i			1.38091i
$527$		−	7.38295i		−	0.321606i
$528$		0			0
$529$	22.7274			0.988146
$530$	−32.3933			−1.40708
$531$		0			0
$532$	0.187401			0.00812485
$533$	19.3131	−	17.2533i	0.836545	−	0.747323i
$534$		0			0
$535$			0.812469i			0.0351261i
$536$	−14.0888			−0.608544
$537$		0			0
$538$		−	25.2708i		−	1.08950i
$539$		−	6.88397i		−	0.296514i
$540$		0			0
$541$		−	12.8562i		−	0.552733i	−0.961052	−	0.276367i	$-0.910870\pi$
$541$		−	12.8562i		−	0.552733i	0.961052	−	0.276367i	$-0.0891304\pi$
$542$	24.7285			1.06218
$543$		0			0
$544$			10.3006i			0.441636i
$545$	−13.0376			−0.558471
$546$		0			0
$547$	1.74358			0.0745501			0.0372750	−	0.999305i	$-0.488132\pi$
$547$	1.74358			0.0745501			0.0372750	+	0.999305i	$0.488132\pi$
$548$			1.00566i			0.0429596i
$549$		0			0
$550$	−2.21155			−0.0943010
$551$			2.58077i			0.109945i
$552$		0			0
$553$		−	0.0211805i		−	0.000900686i
$554$		−	24.0684i		−	1.02257i
$555$		0			0
$556$	−0.375565			−0.0159275
$557$			11.0140i			0.466679i	0.972395	+	0.233339i	$0.0749653\pi$
$557$			11.0140i			0.466679i	−0.972395	+	0.233339i	$0.925035\pi$
$558$		0			0
$559$	−10.6571	−	11.9294i	−0.450747	−	0.504561i
$560$	3.86735			0.163425
$561$		0			0
$562$	−13.6093			−0.574074
$563$	31.6588			1.33426			0.667129	−	0.744942i	$-0.267523\pi$
$563$	31.6588			1.33426			0.667129	+	0.744942i	$0.267523\pi$
$564$		0			0
$565$		−	9.11424i		−	0.383439i
$566$		−	16.1916i		−	0.680584i
$567$		0			0
$568$	32.8180			1.37701
$569$	14.0121			0.587418			0.293709	−	0.955895i	$-0.405110\pi$
$569$	14.0121			0.587418			0.293709	+	0.955895i	$0.405110\pi$
$570$		0			0
$571$	17.4648			0.730880			0.365440	−	0.930835i	$-0.380918\pi$
$571$	17.4648			0.730880			0.365440	+	0.930835i	$0.380918\pi$
$572$	0.720250	−	0.643432i	0.0301152	−	0.0269032i
$573$		0			0
$574$		−	3.68451i		−	0.153789i
$575$	9.93064			0.414136
$576$		0			0
$577$			33.7269i			1.40407i	0.712144	+	0.702034i	$0.247724\pi$
$577$			33.7269i			1.40407i	−0.712144	+	0.702034i	$0.752276\pi$
$578$			44.8967i			1.86746i
$579$		0			0
$580$		−	0.856049i		−	0.0355455i
$581$	−5.49230			−0.227859
$582$		0			0
$583$		−	8.45753i		−	0.350275i
$584$	17.3908			0.719636
$585$		0			0
$586$	39.6515			1.63799
$587$		−	35.0331i		−	1.44597i	−0.690864	−	0.722985i	$-0.742770\pi$
$587$		−	35.0331i		−	1.44597i	0.690864	−	0.722985i	$-0.257230\pi$
$588$		0			0
$589$	−2.21623			−0.0913184
$590$			33.1322i			1.36403i
$591$		0			0
$592$		−	36.2867i		−	1.49137i
$593$			7.37947i			0.303039i	0.988454	+	0.151519i	$0.0484166\pi$
$593$			7.37947i			0.303039i	−0.988454	+	0.151519i	$0.951583\pi$
$594$		0			0
$595$	5.92754			0.243006
$596$			3.20217i			0.131166i
$597$		0			0
$598$	−27.3820	+	24.4616i	−1.11973	+	1.00031i
$599$	39.3483			1.60773			0.803865	−	0.594812i	$-0.202774\pi$
$599$	39.3483			1.60773			0.803865	+	0.594812i	$0.202774\pi$
$600$		0			0
$601$	−16.6958			−0.681036			−0.340518	−	0.940238i	$-0.610602\pi$
$601$	−16.6958			−0.681036			−0.340518	+	0.940238i	$0.610602\pi$
$602$	−2.27587			−0.0927574
$603$		0			0
$604$		−	4.08583i		−	0.166250i
$605$		−	2.54333i		−	0.103401i
$606$		0			0
$607$	−31.4742			−1.27750			−0.638750	−	0.769414i	$-0.720548\pi$
$607$	−31.4742			−1.27750			−0.638750	+	0.769414i	$0.720548\pi$
$608$	3.09207			0.125400
$609$		0			0
$610$	49.6505			2.01029
$611$	2.10539	−	1.88084i	0.0851751	−	0.0760907i
$612$		0			0
$613$		−	9.52576i		−	0.384742i	−0.981322	−	0.192371i	$-0.938382\pi$
$613$		−	9.52576i		−	0.384742i	0.981322	−	0.192371i	$-0.0616177\pi$
$614$	−28.9816			−1.16960
$615$		0			0
$616$			0.888543i			0.0358004i
$617$		−	11.9917i		−	0.482768i	−0.970430	−	0.241384i	$-0.922399\pi$
$617$		−	11.9917i		−	0.482768i	0.970430	−	0.241384i	$-0.0776013\pi$
$618$		0			0
$619$			14.6900i			0.590442i	0.955429	+	0.295221i	$0.0953933\pi$
$619$			14.6900i			0.590442i	−0.955429	+	0.295221i	$0.904607\pi$
$620$	0.735131			0.0295236
$621$		0			0
$622$			3.44855i			0.138274i
$623$	−2.81333			−0.112714
$624$		0			0
$625$	−30.1861			−1.20744
$626$		−	28.4253i		−	1.13610i
$627$		0			0
$628$	−4.60872			−0.183908
$629$		−	55.6172i		−	2.21760i
$630$		0			0
$631$			7.18673i			0.286099i	0.989716	+	0.143050i	$0.0456909\pi$
$631$			7.18673i			0.286099i	−0.989716	+	0.143050i	$0.954309\pi$
$632$		−	0.162195i		−	0.00645179i
$633$		0			0
$634$	−49.3246			−1.95893
$635$		−	10.3315i		−	0.409994i
$636$		0			0
$637$	−16.5359	−	18.5101i	−0.655175	−	0.733395i
$638$	1.89230			0.0749167
$639$		0			0
$640$	33.1695			1.31114
$641$	14.1749			0.559874			0.279937	−	0.960018i	$-0.409686\pi$
$641$	14.1749			0.559874			0.279937	+	0.960018i	$0.409686\pi$
$642$		0			0
$643$		−	42.2453i		−	1.66599i	−0.553278	−	0.832997i	$-0.686623\pi$
$643$		−	42.2453i		−	1.66599i	0.553278	−	0.832997i	$-0.313377\pi$
$644$			0.617007i			0.0243135i
$645$		0			0
$646$	21.1622			0.832616
$647$	−5.44640			−0.214120			−0.107060	−	0.994253i	$-0.534144\pi$
$647$	−5.44640			−0.214120			−0.107060	+	0.994253i	$0.534144\pi$
$648$		0			0
$649$	−8.65045			−0.339560
$650$	−5.94657	+	5.31234i	−0.233244	+	0.208367i
$651$		0			0
$652$		−	2.21059i		−	0.0865733i
$653$	24.8928			0.974132			0.487066	−	0.873365i	$-0.338067\pi$
$653$	24.8928			0.974132			0.487066	+	0.873365i	$0.338067\pi$
$654$		0			0
$655$		−	8.06616i		−	0.315171i
$656$		−	32.0631i		−	1.25185i
$657$		0			0
$658$		−	0.401662i		−	0.0156584i
$659$	−23.2444			−0.905474			−0.452737	−	0.891644i	$-0.649552\pi$
$659$	−23.2444			−0.905474			−0.452737	+	0.891644i	$0.649552\pi$
$660$		0			0
$661$		−	10.7246i		−	0.417140i	−0.978007	−	0.208570i	$-0.933119\pi$
$661$		−	10.7246i		−	0.417140i	0.978007	−	0.208570i	$-0.0668810\pi$
$662$	−33.8307			−1.31487
$663$		0			0
$664$	−42.0588			−1.63220
$665$		−	1.77935i		−	0.0690001i
$666$		0			0
$667$	−8.49706			−0.329007
$668$			1.13500i			0.0439143i
$669$		0			0
$670$		−	20.6869i		−	0.799206i
$671$			12.9632i			0.500438i
$672$		0			0
$673$	28.4409			1.09632			0.548158	−	0.836375i	$-0.315329\pi$
$673$	28.4409			1.09632			0.548158	+	0.836375i	$0.315329\pi$
$674$		−	17.9648i		−	0.691978i
$675$		0			0
$676$	0.391080	−	3.46020i	0.0150415	−	0.133085i
$677$	3.60489			0.138547			0.0692736	−	0.997598i	$-0.477932\pi$
$677$	3.60489			0.138547			0.0692736	+	0.997598i	$0.477932\pi$
$678$		0			0
$679$	−2.41137			−0.0925398
$680$	45.3918			1.74070
$681$		0			0
$682$			1.62501i			0.0622247i
$683$			33.4027i			1.27812i	0.769158	+	0.639059i	$0.220676\pi$
$683$			33.4027i			1.27812i	−0.769158	+	0.639059i	$0.779324\pi$
$684$		0			0
$685$	9.54860			0.364833
$686$	−7.12213			−0.271924
$687$		0			0
$688$	−19.8049			−0.755054
$689$	−20.3157	−	22.7412i	−0.773966	−	0.866369i
$690$		0			0
$691$			49.0437i			1.86571i	0.360251	+	0.932856i	$0.382691\pi$
$691$			49.0437i			1.86571i	−0.360251	+	0.932856i	$0.617309\pi$
$692$	−4.64035			−0.176400
$693$		0			0
$694$		−	50.8138i		−	1.92887i
$695$			3.56594i			0.135264i
$696$		0			0
$697$		−	49.1436i		−	1.86145i
$698$	12.8630			0.486872
$699$		0			0
$700$			0.133996i			0.00506457i
$701$	−14.0083			−0.529085			−0.264542	−	0.964374i	$-0.585221\pi$
$701$	−14.0083			−0.529085			−0.264542	+	0.964374i	$0.585221\pi$
$702$		0			0
$703$	−16.6953			−0.629676
$704$			6.66076i			0.251037i
$705$		0			0
$706$	−1.73156			−0.0651682
$707$		−	2.14872i		−	0.0808108i
$708$		0			0
$709$		−	38.0403i		−	1.42863i	−0.699823	−	0.714317i	$-0.746738\pi$
$709$		−	38.0403i		−	1.42863i	0.699823	−	0.714317i	$-0.253262\pi$
$710$			48.1874i			1.80844i
$711$		0			0
$712$	−21.5439			−0.807391
$713$		−	7.29684i		−	0.273269i
$714$		0			0
$715$	−6.10930	−	6.83868i	−0.228475	−	0.255752i
$716$	−5.14713			−0.192357
$717$		0			0
$718$	−16.3986			−0.611991
$719$	41.2344			1.53778			0.768891	−	0.639380i	$-0.220809\pi$
$719$	41.2344			1.53778			0.768891	+	0.639380i	$0.220809\pi$
$720$		0			0
$721$		−	0.585919i		−	0.0218208i
$722$			22.2604i			0.828446i
$723$		0			0
$724$	−0.931984			−0.0346369
$725$	−1.84531			−0.0685332
$726$		0			0
$727$	−5.96467			−0.221217			−0.110609	−	0.993864i	$-0.535280\pi$
$727$	−5.96467			−0.221217			−0.110609	+	0.993864i	$0.535280\pi$
$728$	2.13435	+	2.38917i	0.0791044	+	0.0885486i
$729$		0			0
$730$			25.5353i			0.945103i
$731$	−30.3552			−1.12273
$732$		0			0
$733$			10.8681i			0.401422i	0.979651	+	0.200711i	$0.0643252\pi$
$733$			10.8681i			0.401422i	−0.979651	+	0.200711i	$0.935675\pi$
$734$			9.47390i			0.349688i
$735$		0			0
$736$			10.1805i			0.375258i
$737$	5.40112			0.198953
$738$		0			0
$739$		−	9.82461i		−	0.361404i	−0.983538	−	0.180702i	$-0.942163\pi$
$739$		−	9.82461i		−	0.361404i	0.983538	−	0.180702i	$-0.0578370\pi$
$740$	5.53788			0.203576
$741$		0			0
$742$	−4.33850			−0.159272
$743$			3.77796i			0.138600i	0.997596	+	0.0692999i	$0.0220765\pi$
$743$			3.77796i			0.138600i	−0.997596	+	0.0692999i	$0.977923\pi$
$744$		0			0
$745$	30.4042			1.11392
$746$		−	0.779531i		−	0.0285406i
$747$		0			0
$748$		−	1.83273i		−	0.0670111i
$749$			0.108816i			0.00397604i
$750$		0			0
$751$	−6.22875			−0.227290			−0.113645	−	0.993521i	$-0.536253\pi$
$751$	−6.22875			−0.227290			−0.113645	+	0.993521i	$0.536253\pi$
$752$		−	3.49531i		−	0.127461i
$753$		0			0
$754$	5.08813	−	4.54545i	0.185299	−	0.165536i
$755$	−38.7944			−1.41187
$756$		0			0
$757$	7.19065			0.261349			0.130674	−	0.991425i	$-0.458286\pi$
$757$	7.19065			0.261349			0.130674	+	0.991425i	$0.458286\pi$
$758$	1.92145			0.0697903
$759$		0			0
$760$		−	13.6258i		−	0.494261i
$761$			25.6946i			0.931427i	0.884936	+	0.465714i	$0.154202\pi$
$761$			25.6946i			0.931427i	−0.884936	+	0.465714i	$0.845798\pi$
$762$		0			0
$763$	−1.74616			−0.0632151
$764$	−6.21212			−0.224747
$765$		0			0
$766$	−55.3371			−1.99941
$767$	−23.2599	+	20.7791i	−0.839866	+	0.750290i
$768$		0			0
$769$			27.5371i			0.993013i	0.868033	+	0.496507i	$0.165384\pi$
$769$			27.5371i			0.993013i	−0.868033	+	0.496507i	$0.834616\pi$
$770$	−1.30467			−0.0470170
$771$		0			0
$772$			4.02860i			0.144992i
$773$		−	37.5945i		−	1.35218i	−0.736819	−	0.676091i	$-0.763673\pi$
$773$		−	37.5945i		−	1.35218i	0.736819	−	0.676091i	$-0.236327\pi$
$774$		0			0
$775$		−	1.58466i		−	0.0569226i
$776$	−18.4657			−0.662881
$777$		0			0
$778$		−	27.6456i		−	0.991144i
$779$	−14.7521			−0.528548
$780$		0			0
$781$	−12.5812			−0.450190
$782$			69.6754i			2.49159i
$783$		0			0
$784$	−30.7299			−1.09750
$785$			43.7592i			1.56183i
$786$		0			0
$787$		−	10.0158i		−	0.357026i	−0.983938	−	0.178513i	$-0.942871\pi$
$787$		−	10.0158i		−	0.357026i	0.983938	−	0.178513i	$-0.0571286\pi$
$788$		−	3.62463i		−	0.129122i
$789$		0			0
$790$	0.238155			0.00847319
$791$		−	1.22069i		−	0.0434027i
$792$		0			0
$793$	31.1386	+	34.8562i	1.10577	+	1.23778i
$794$	51.8098			1.83866
$795$		0			0
$796$	−2.71900			−0.0963725
$797$	29.8615			1.05775			0.528875	−	0.848700i	$-0.322614\pi$
$797$	29.8615			1.05775			0.528875	+	0.848700i	$0.322614\pi$
$798$		0			0
$799$		−	5.35732i		−	0.189528i
$800$			2.21090i			0.0781672i
$801$		0			0
$802$	−3.84030			−0.135606
$803$	−6.66697			−0.235272
$804$		0			0
$805$	5.85841			0.206482
$806$	3.90340	+	4.36942i	0.137491	+	0.153906i
$807$		0			0
$808$		−	16.4544i		−	0.578863i
$809$	−15.9859			−0.562035			−0.281018	−	0.959703i	$-0.590672\pi$
$809$	−15.9859			−0.562035			−0.281018	+	0.959703i	$0.590672\pi$
$810$		0			0
$811$			37.8164i			1.32791i	0.747771	+	0.663956i	$0.231124\pi$
$811$			37.8164i			1.32791i	−0.747771	+	0.663956i	$0.768876\pi$
$812$		−	0.114652i		−	0.00402351i
$813$		0			0
$814$			12.2415i			0.429064i
$815$	−20.9893			−0.735222
$816$		0			0
$817$			9.11212i			0.318793i
$818$	18.1271			0.633799
$819$		0			0
$820$	4.89330			0.170881
$821$			17.5207i			0.611478i	0.952115	+	0.305739i	$0.0989035\pi$
$821$			17.5207i			0.611478i	−0.952115	+	0.305739i	$0.901096\pi$
$822$		0			0
$823$	30.4981			1.06310			0.531549	−	0.847028i	$-0.321610\pi$
$823$	30.4981			1.06310			0.531549	+	0.847028i	$0.321610\pi$
$824$		−	4.48684i		−	0.156306i
$825$		0			0
$826$			4.43747i			0.154399i
$827$		−	33.3227i		−	1.15874i	−0.815064	−	0.579371i	$-0.803298\pi$
$827$		−	33.3227i		−	1.15874i	0.815064	−	0.579371i	$-0.196702\pi$
$828$		0			0
$829$	−24.9889			−0.867899			−0.433949	−	0.900937i	$-0.642880\pi$
$829$	−24.9889			−0.867899			−0.433949	+	0.900937i	$0.642880\pi$
$830$		−	61.7559i		−	2.14358i
$831$		0			0
$832$	15.9997	+	17.9099i	0.554690	+	0.620913i
$833$	−47.1001			−1.63192
$834$		0			0
$835$	10.7766			0.372941
$836$	−0.550153			−0.0190274
$837$		0			0
$838$			16.1402i			0.557553i
$839$		−	14.1528i		−	0.488609i	−0.969699	−	0.244305i	$-0.921440\pi$
$839$		−	14.1528i		−	0.488609i	0.969699	−	0.244305i	$-0.0785597\pi$
$840$		0			0
$841$	−27.4211			−0.945554
$842$	8.77673			0.302466
$843$		0			0
$844$	−5.65834			−0.194768
$845$	−32.8542	−	3.71325i	−1.13022	−	0.127740i
$846$		0			0
$847$		−	0.340634i		−	0.0117043i
$848$	−37.7542			−1.29649
$849$		0			0
$850$			15.1315i			0.519005i
$851$		−	54.9685i		−	1.88429i
$852$		0			0
$853$		−	44.5158i		−	1.52419i	−0.647465	−	0.762095i	$-0.724171\pi$
$853$		−	44.5158i		−	1.52419i	0.647465	−	0.762095i	$-0.275829\pi$
$854$	6.64979			0.227551
$855$		0			0
$856$			0.833285i			0.0284811i
$857$	−12.5580			−0.428974			−0.214487	−	0.976727i	$-0.568808\pi$
$857$	−12.5580			−0.428974			−0.214487	+	0.976727i	$0.568808\pi$
$858$		0			0
$859$	−1.35257			−0.0461490			−0.0230745	−	0.999734i	$-0.507345\pi$
$859$	−1.35257			−0.0461490			−0.0230745	+	0.999734i	$0.507345\pi$
$860$		−	3.02251i		−	0.103067i
$861$		0			0
$862$	−16.3809			−0.557935
$863$			21.7630i			0.740822i	0.928868	+	0.370411i	$0.120783\pi$
$863$			21.7630i			0.740822i	−0.928868	+	0.370411i	$0.879217\pi$
$864$		0			0
$865$			44.0596i			1.49807i
$866$			11.0710i			0.376207i
$867$		0			0
$868$	0.0984575			0.00334187
$869$			0.0621796i			0.00210930i
$870$		0			0
$871$	14.5229	−	12.9739i	0.492089	−	0.439605i
$872$	−13.3717			−0.452822
$873$		0			0
$874$	20.9154			0.707473
$875$	−3.05946			−0.103429
$876$		0			0
$877$			14.9309i			0.504180i	0.967704	+	0.252090i	$0.0811179\pi$
$877$			14.9309i			0.504180i	−0.967704	+	0.252090i	$0.918882\pi$
$878$		−	29.6990i		−	1.00229i
$879$		0			0
$880$	−11.3534			−0.382723
$881$	39.4409			1.32880			0.664399	−	0.747378i	$-0.268687\pi$
$881$	39.4409			1.32880			0.664399	+	0.747378i	$0.268687\pi$
$882$		0			0
$883$	−9.00648			−0.303092			−0.151546	−	0.988450i	$-0.548425\pi$
$883$	−9.00648			−0.303092			−0.151546	+	0.988450i	$0.548425\pi$
$884$	−4.40236	−	4.92795i	−0.148067	−	0.165745i
$885$		0			0
$886$			44.6296i			1.49936i
$887$	−8.41991			−0.282713			−0.141356	−	0.989959i	$-0.545146\pi$
$887$	−8.41991			−0.282713			−0.141356	+	0.989959i	$0.545146\pi$
$888$		0			0
$889$		−	1.38372i		−	0.0464086i
$890$		−	31.6334i		−	1.06035i
$891$		0			0
$892$		−	5.14421i		−	0.172241i
$893$	−1.60817			−0.0538155
$894$		0			0
$895$			48.8713i			1.63359i
$896$	4.44245			0.148412
$897$		0			0
$898$	23.7393			0.792192
$899$			1.35590i			0.0452218i
$900$		0			0
$901$	−57.8665			−1.92781
$902$			10.8166i			0.360155i
$903$		0			0
$904$		−	9.34776i		−	0.310902i
$905$			8.84907i			0.294153i
$906$		0			0
$907$	30.8200			1.02336			0.511681	−	0.859175i	$-0.329023\pi$
$907$	30.8200			1.02336			0.511681	+	0.859175i	$0.329023\pi$
$908$		−	5.98011i		−	0.198457i
$909$		0			0
$910$	−3.50808	+	3.13392i	−0.116292	+	0.103888i
$911$	17.3328			0.574260			0.287130	−	0.957892i	$-0.407299\pi$
$911$	17.3328			0.574260			0.287130	+	0.957892i	$0.407299\pi$
$912$		0			0
$913$	16.1238			0.533619
$914$	−48.1647			−1.59315
$915$		0			0
$916$		−	0.556199i		−	0.0183773i
$917$		−	1.08032i		−	0.0356752i
$918$		0			0
$919$	15.4659			0.510172			0.255086	−	0.966918i	$-0.417896\pi$
$919$	15.4659			0.510172			0.255086	+	0.966918i	$0.417896\pi$
$920$	44.8624			1.47907
$921$		0			0
$922$	38.3887			1.26426
$923$	−33.8291	+	30.2210i	−1.11350	+	0.994737i
$924$		0			0
$925$		−	11.9375i		−	0.392504i
$926$	−50.7519			−1.66781
$927$		0			0
$928$		−	1.89174i		−	0.0620993i
$929$		−	5.31893i		−	0.174509i	−0.996186	−	0.0872543i	$-0.972191\pi$
$929$		−	5.31893i		−	0.174509i	0.996186	−	0.0872543i	$-0.0278093\pi$
$930$		0			0
$931$			14.1387i			0.463376i
$932$	6.59818			0.216131
$933$		0			0
$934$		−	11.3829i		−	0.372460i
$935$	−17.4015			−0.569090
$936$		0			0
$937$	−5.47637			−0.178905			−0.0894526	−	0.995991i	$-0.528512\pi$
$937$	−5.47637			−0.178905			−0.0894526	+	0.995991i	$0.528512\pi$
$938$		−	2.77064i		−	0.0904647i
$939$		0			0
$940$	0.533436			0.0173987
$941$		−	34.7059i		−	1.13138i	−0.824618	−	0.565691i	$-0.808610\pi$
$941$		−	34.7059i		−	1.13138i	0.824618	−	0.565691i	$-0.191390\pi$
$942$		0			0
$943$		−	48.5704i		−	1.58167i
$944$			38.6154i			1.25682i
$945$		0			0
$946$	6.68127			0.217227
$947$			1.03745i			0.0337128i	0.999858	+	0.0168564i	$0.00536581\pi$
$947$			1.03745i			0.0337128i	−0.999858	+	0.0168564i	$0.994634\pi$
$948$		0			0
$949$	−17.9266	+	16.0146i	−0.581922	+	0.519857i
$950$	4.54220			0.147369
$951$		0			0
$952$	6.07941			0.197035
$953$	−7.13230			−0.231038			−0.115519	−	0.993305i	$-0.536853\pi$
$953$	−7.13230			−0.231038			−0.115519	+	0.993305i	$0.536853\pi$
$954$		0			0
$955$			58.9833i			1.90865i
$956$			6.91871i			0.223767i
$957$		0			0
$958$	−0.251016			−0.00810996
$959$	1.27886			0.0412967
$960$		0			0
$961$	29.8356			0.962439
$962$	29.4051	+	32.9157i	0.948057	+	1.06124i
$963$		0			0
$964$		−	5.29565i		−	0.170562i
$965$	38.2510			1.23134
$966$		0			0
$967$		−	31.7852i		−	1.02214i	−0.859538	−	0.511071i	$-0.829249\pi$
$967$		−	31.7852i		−	1.02214i	0.859538	−	0.511071i	$-0.170751\pi$
$968$		−	2.60850i		−	0.0838403i
$969$		0			0
$970$		−	27.1136i		−	0.870566i
$971$	−41.2369			−1.32335			−0.661677	−	0.749789i	$-0.730155\pi$
$971$	−41.2369			−1.32335			−0.661677	+	0.749789i	$0.730155\pi$
$972$		0			0
$973$			0.477594i			0.0153110i
$974$	54.2098			1.73699
$975$		0			0
$976$	57.8673			1.85229
$977$			52.5954i			1.68267i	0.540510	+	0.841337i	$0.318231\pi$
$977$			52.5954i			1.68267i	−0.540510	+	0.841337i	$0.681769\pi$
$978$		0			0
$979$	8.25911			0.263962
$980$		−	4.68983i		−	0.149811i
$981$		0			0
$982$		−	11.8813i		−	0.379146i
$983$			5.44329i			0.173614i	0.996225	+	0.0868070i	$0.0276663\pi$
$983$			5.44329i			0.173614i	−0.996225	+	0.0868070i	$0.972334\pi$
$984$		0			0
$985$	−34.4154			−1.09657
$986$		−	12.9471i		−	0.412320i
$987$		0			0
$988$	−1.47929	+	1.32151i	−0.0470624	+	0.0420429i
$989$	−30.0012			−0.953982
$990$		0			0
$991$	−14.9835			−0.475967			−0.237984	−	0.971269i	$-0.576486\pi$
$991$	−14.9835			−0.475967			−0.237984	+	0.971269i	$0.576486\pi$
$992$	1.62453			0.0515788
$993$		0			0
$994$			6.45383i			0.204703i
$995$			25.8166i			0.818441i
$996$		0			0
$997$	−32.4089			−1.02640			−0.513201	−	0.858269i	$-0.671540\pi$
$997$	−32.4089			−1.02640			−0.513201	+	0.858269i	$0.671540\pi$
$998$	53.7601			1.70175
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.b.b.298.9		12
3.2	odd	2		143.2.b.a.12.4	✓	12
12.11	even	2		2288.2.j.k.1585.7		12
13.12	even	2	inner	1287.2.b.b.298.4		12
39.5	even	4		1859.2.a.n.1.2		6
39.8	even	4		1859.2.a.j.1.5		6
39.38	odd	2		143.2.b.a.12.9	yes	12
156.155	even	2		2288.2.j.k.1585.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.b.a.12.4	✓	12	3.2	odd	2
143.2.b.a.12.9	yes	12	39.38	odd	2
1287.2.b.b.298.4		12	13.12	even	2	inner
1287.2.b.b.298.9		12	1.1	even	1	trivial
1859.2.a.j.1.5		6	39.8	even	4
1859.2.a.n.1.2		6	39.5	even	4
2288.2.j.k.1585.7		12	12.11	even	2
2288.2.j.k.1585.8		12	156.155	even	2

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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.50594i q^{2} -0.267864 q^{4} +2.54333i q^{5} +0.340634i q^{7} +2.60850i q^{8} +O(q^{10})\)	\(q+1.50594i q^{2} -0.267864 q^{4} +2.54333i q^{5} +0.340634i q^{7} +2.60850i q^{8} -3.83012 q^{10} -1.00000i q^{11} +(-2.40208 - 2.68887i) q^{13} -0.512975 q^{14} -4.46398 q^{16} -6.84200 q^{17} +2.05385i q^{19} -0.681268i q^{20} +1.50594 q^{22} -6.76220 q^{23} -1.46855 q^{25} +(4.04928 - 3.61740i) q^{26} -0.0912436i q^{28} +1.25655 q^{29} +1.07906i q^{31} -1.50550i q^{32} -10.3037i q^{34} -0.866346 q^{35} +8.12878i q^{37} -3.09298 q^{38} -6.63428 q^{40} +7.18264i q^{41} +4.43660 q^{43} +0.267864i q^{44} -10.1835i q^{46} +0.783004i q^{47} +6.88397 q^{49} -2.21155i q^{50} +(0.643432 + 0.720250i) q^{52} +8.45753 q^{53} +2.54333 q^{55} -0.888543 q^{56} +1.89230i q^{58} -8.65045i q^{59} -12.9632 q^{61} -1.62501 q^{62} -6.66076 q^{64} +(6.83868 - 6.10930i) q^{65} +5.40112i q^{67} +1.83273 q^{68} -1.30467i q^{70} -12.5812i q^{71} -6.66697i q^{73} -12.2415 q^{74} -0.550153i q^{76} +0.340634 q^{77} -0.0621796 q^{79} -11.3534i q^{80} -10.8166 q^{82} +16.1238i q^{83} -17.4015i q^{85} +6.68127i q^{86} +2.60850 q^{88} +8.25911i q^{89} +(0.915919 - 0.818231i) q^{91} +1.81135 q^{92} -1.17916 q^{94} -5.22363 q^{95} +7.07906i q^{97} +10.3669i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{4}+O(q^{10}) \) 12 * q - 14 * q^4	\( 12 q - 14 q^{4} + 8 q^{10} - 8 q^{13} + 10 q^{16} + 12 q^{17} - 2 q^{22} + 4 q^{23} - 16 q^{25} - 10 q^{26} - 8 q^{29} - 16 q^{35} - 18 q^{38} + 16 q^{40} + 12 q^{43} + 32 q^{49} + 36 q^{52} + 20 q^{53} - 8 q^{55} - 22 q^{56} - 12 q^{61} + 72 q^{62} - 10 q^{64} + 20 q^{65} - 68 q^{68} - 20 q^{74} - 8 q^{77} - 48 q^{79} - 44 q^{82} + 30 q^{88} + 6 q^{92} - 64 q^{94} + 4 q^{95}+O(q^{100}) \) 12 * q - 14 * q^4 + 8 * q^10 - 8 * q^13 + 10 * q^16 + 12 * q^17 - 2 * q^22 + 4 * q^23 - 16 * q^25 - 10 * q^26 - 8 * q^29 - 16 * q^35 - 18 * q^38 + 16 * q^40 + 12 * q^43 + 32 * q^49 + 36 * q^52 + 20 * q^53 - 8 * q^55 - 22 * q^56 - 12 * q^61 + 72 * q^62 - 10 * q^64 + 20 * q^65 - 68 * q^68 - 20 * q^74 - 8 * q^77 - 48 * q^79 - 44 * q^82 + 30 * q^88 + 6 * q^92 - 64 * q^94 + 4 * q^95

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