LMFDB - Embedded newform 1287.2.b.b.298.1

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			298.1
Root			$-2.66546i$ of defining polynomial
Character	$\chi$	$=$	1287.298
Dual form			1287.2.b.b.298.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66546i q^{2} -5.10468 q^{4} +0.458686i q^{5} +1.17072i q^{7} +8.27540i q^{8} +O(q^{10})$	\(q-2.66546i q^{2} -5.10468 q^{4} +0.458686i q^{5} +1.17072i q^{7} +8.27540i q^{8} +1.22261 q^{10} -1.00000i q^{11} +(-3.50921 - 0.827915i) q^{13} +3.12051 q^{14} +11.8484 q^{16} +6.55353 q^{17} +5.85202i q^{19} -2.34145i q^{20} -2.66546 q^{22} +3.16452 q^{23} +4.78961 q^{25} +(-2.20677 + 9.35366i) q^{26} -5.97616i q^{28} +2.13998 q^{29} +8.66805i q^{31} -15.0306i q^{32} -17.4682i q^{34} -0.536994 q^{35} -8.23483i q^{37} +15.5983 q^{38} -3.79581 q^{40} -5.38281i q^{41} -3.12946 q^{43} +5.10468i q^{44} -8.43490i q^{46} -10.5508i q^{47} +5.62941 q^{49} -12.7665i q^{50} +(17.9134 + 4.22624i) q^{52} +1.26682 q^{53} +0.458686 q^{55} -9.68820 q^{56} -5.70404i q^{58} -0.884643i q^{59} +5.03768 q^{61} +23.1043 q^{62} -16.3668 q^{64} +(0.379753 - 1.60963i) q^{65} -3.84383i q^{67} -33.4537 q^{68} +1.43134i q^{70} +5.11327i q^{71} +5.92140i q^{73} -21.9496 q^{74} -29.8727i q^{76} +1.17072 q^{77} +2.48143 q^{79} +5.43469i q^{80} -14.3477 q^{82} +7.40248i q^{83} +3.00601i q^{85} +8.34145i q^{86} +8.27540 q^{88} +7.39861i q^{89} +(0.969259 - 4.10831i) q^{91} -16.1539 q^{92} -28.1228 q^{94} -2.68424 q^{95} +14.6680i q^{97} -15.0050i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.66546i		−	1.88477i	−0.334537	−	0.942383i	$-0.608580\pi$
$2$		−	2.66546i		−	1.88477i	0.334537	−	0.942383i	$-0.391420\pi$
$3$		0			0
$3$		0			0
$4$	−5.10468			−2.55234
$5$			0.458686i			0.205131i	0.994726	+	0.102565i	$0.0327051\pi$
$5$			0.458686i			0.205131i	−0.994726	+	0.102565i	$0.967295\pi$
$6$		0			0
$7$			1.17072i			0.442492i	0.975218	+	0.221246i	$0.0710123\pi$
$7$			1.17072i			0.442492i	−0.975218	+	0.221246i	$0.928988\pi$
$8$			8.27540i			2.92580i
$9$		0			0
$10$	1.22261			0.386623
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.50921	−	0.827915i	−0.973280	−	0.229622i
$13$	−3.50921	−	0.827915i	−0.973280	−	0.229622i
$14$	3.12051			0.833993
$15$		0			0
$16$	11.8484			2.96210
$17$	6.55353			1.58946			0.794732	−	0.606960i	$-0.207611\pi$
$17$	6.55353			1.58946			0.794732	+	0.606960i	$0.207611\pi$
$18$		0			0
$19$			5.85202i			1.34254i	0.741211	+	0.671272i	$0.234252\pi$
$19$			5.85202i			1.34254i	−0.741211	+	0.671272i	$0.765748\pi$
$20$		−	2.34145i		−	0.523563i
$21$		0			0
$22$	−2.66546			−0.568278
$23$	3.16452			0.659848			0.329924	−	0.944008i	$-0.392977\pi$
$23$	3.16452			0.659848			0.329924	+	0.944008i	$0.392977\pi$
$24$		0			0
$25$	4.78961			0.957921
$26$	−2.20677	+	9.35366i	−0.432784	+	1.83440i
$27$		0			0
$28$		−	5.97616i		−	1.12939i
$29$	2.13998			0.397385			0.198692	−	0.980062i	$-0.436331\pi$
$29$	2.13998			0.397385			0.198692	+	0.980062i	$0.436331\pi$
$30$		0			0
$31$			8.66805i			1.55683i	0.627752	+	0.778414i	$0.283975\pi$
$31$			8.66805i			1.55683i	−0.627752	+	0.778414i	$0.716025\pi$
$32$		−	15.0306i		−	2.65707i
$33$		0			0
$34$		−	17.4682i		−	2.99577i
$35$	−0.536994			−0.0907686
$36$		0			0
$37$		−	8.23483i		−	1.35380i	−0.736076	−	0.676899i	$-0.763324\pi$
$37$		−	8.23483i		−	1.35380i	0.736076	−	0.676899i	$-0.236676\pi$
$38$	15.5983			2.53038
$39$		0			0
$40$	−3.79581			−0.600170
$41$		−	5.38281i		−	0.840653i	−0.907373	−	0.420327i	$-0.861915\pi$
$41$		−	5.38281i		−	0.840653i	0.907373	−	0.420327i	$-0.138085\pi$
$42$		0			0
$43$	−3.12946			−0.477238			−0.238619	−	0.971113i	$-0.576695\pi$
$43$	−3.12946			−0.477238			−0.238619	+	0.971113i	$0.576695\pi$
$44$			5.10468i			0.769559i
$45$		0			0
$46$		−	8.43490i		−	1.24366i
$47$		−	10.5508i		−	1.53899i	−0.638651	−	0.769497i	$-0.720507\pi$
$47$		−	10.5508i		−	1.53899i	0.638651	−	0.769497i	$-0.279493\pi$
$48$		0			0
$49$	5.62941			0.804201
$50$		−	12.7665i		−	1.80546i
$51$		0			0
$52$	17.9134	+	4.22624i	2.48414	+	0.586074i
$53$	1.26682			0.174011			0.0870054	−	0.996208i	$-0.472270\pi$
$53$	1.26682			0.174011			0.0870054	+	0.996208i	$0.472270\pi$
$54$		0			0
$55$	0.458686			0.0618492
$56$	−9.68820			−1.29464
$57$		0			0
$58$		−	5.70404i		−	0.748977i
$59$		−	0.884643i		−	0.115171i	−0.998341	−	0.0575854i	$-0.981660\pi$
$59$		−	0.884643i		−	0.115171i	0.998341	−	0.0575854i	$-0.0183401\pi$
$60$		0			0
$61$	5.03768			0.645009			0.322505	−	0.946568i	$-0.395475\pi$
$61$	5.03768			0.645009			0.322505	+	0.946568i	$0.395475\pi$
$62$	23.1043			2.93425
$63$		0			0
$64$	−16.3668			−2.04585
$65$	0.379753	−	1.60963i	0.0471026	−	0.199649i
$66$		0			0
$67$		−	3.84383i		−	0.469598i	−0.972044	−	0.234799i	$-0.924557\pi$
$67$		−	3.84383i		−	0.469598i	0.972044	−	0.234799i	$-0.0754432\pi$
$68$	−33.4537			−4.05685
$69$		0			0
$70$			1.43134i			0.171077i
$71$			5.11327i			0.606833i	0.952858	+	0.303416i	$0.0981273\pi$
$71$			5.11327i			0.606833i	−0.952858	+	0.303416i	$0.901873\pi$
$72$		0			0
$73$			5.92140i			0.693047i	0.938041	+	0.346524i	$0.112638\pi$
$73$			5.92140i			0.693047i	−0.938041	+	0.346524i	$0.887362\pi$
$74$	−21.9496			−2.55159
$75$		0			0
$76$		−	29.8727i		−	3.42663i
$77$	1.17072			0.133416
$78$		0			0
$79$	2.48143			0.279182			0.139591	−	0.990209i	$-0.455421\pi$
$79$	2.48143			0.279182			0.139591	+	0.990209i	$0.455421\pi$
$80$			5.43469i			0.607617i
$81$		0			0
$82$	−14.3477			−1.58443
$83$			7.40248i			0.812528i	0.913756	+	0.406264i	$0.133169\pi$
$83$			7.40248i			0.812528i	−0.913756	+	0.406264i	$0.866831\pi$
$84$		0			0
$85$			3.00601i			0.326048i
$86$			8.34145i			0.899481i
$87$		0			0
$88$	8.27540			0.882161
$89$			7.39861i			0.784251i	0.919912	+	0.392125i	$0.128260\pi$
$89$			7.39861i			0.784251i	−0.919912	+	0.392125i	$0.871740\pi$
$90$		0			0
$91$	0.969259	−	4.10831i	0.101606	−	0.430668i
$92$	−16.1539			−1.68416
$93$		0			0
$94$	−28.1228			−2.90064
$95$	−2.68424			−0.275397
$96$		0			0
$97$			14.6680i			1.48931i	0.667447	+	0.744657i	$0.267387\pi$
$97$			14.6680i			1.48931i	−0.667447	+	0.744657i	$0.732613\pi$
$98$		−	15.0050i		−	1.51573i
$99$		0			0
$100$	−24.4494			−2.44494
$101$	11.6018			1.15442			0.577209	−	0.816596i	$-0.304142\pi$
$101$	11.6018			1.15442			0.577209	+	0.816596i	$0.304142\pi$
$102$		0			0
$103$	−3.98768			−0.392918			−0.196459	−	0.980512i	$-0.562944\pi$
$103$	−3.98768			−0.392918			−0.196459	+	0.980512i	$0.562944\pi$
$104$	6.85133	−	29.0401i	0.671828	−	2.84762i
$105$		0			0
$106$		−	3.37665i		−	0.327970i
$107$	13.7192			1.32628			0.663142	−	0.748493i	$-0.269222\pi$
$107$	13.7192			1.32628			0.663142	+	0.748493i	$0.269222\pi$
$108$		0			0
$109$		−	7.12146i		−	0.682112i	−0.940043	−	0.341056i	$-0.889215\pi$
$109$		−	7.12146i		−	0.682112i	0.940043	−	0.341056i	$-0.110785\pi$
$110$		−	1.22261i		−	0.116571i
$111$		0			0
$112$			13.8712i			1.31070i
$113$	−8.48050			−0.797778			−0.398889	−	0.916999i	$-0.630604\pi$
$113$	−8.48050			−0.797778			−0.398889	+	0.916999i	$0.630604\pi$
$114$		0			0
$115$			1.45152i			0.135355i
$116$	−10.9239			−1.01426
$117$		0			0
$118$	−2.35798			−0.217070
$119$			7.67237i			0.703325i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		−	13.4277i		−	1.21569i
$123$		0			0
$124$		−	44.2476i		−	3.97355i
$125$			4.49036i			0.401630i
$126$		0			0
$127$	14.9557			1.32711			0.663553	−	0.748130i	$-0.269048\pi$
$127$	14.9557			1.32711			0.663553	+	0.748130i	$0.269048\pi$
$128$			13.5637i			1.19887i
$129$		0			0
$130$	−4.29039	−	1.01222i	−0.376292	−	0.0887773i
$131$	12.1089			1.05796			0.528982	−	0.848633i	$-0.322574\pi$
$131$	12.1089			1.05796			0.528982	+	0.848633i	$0.322574\pi$
$132$		0			0
$133$	−6.85109			−0.594065
$134$	−10.2456			−0.885082
$135$		0			0
$136$			54.2331i			4.65045i
$137$			7.58680i			0.648184i	0.946026	+	0.324092i	$0.105059\pi$
$137$			7.58680i			0.648184i	−0.946026	+	0.324092i	$0.894941\pi$
$138$		0			0
$139$	10.7464			0.911494			0.455747	−	0.890109i	$-0.349372\pi$
$139$	10.7464			0.911494			0.455747	+	0.890109i	$0.349372\pi$
$140$	2.74118			0.231672
$141$		0			0
$142$	13.6292			1.14374
$143$	−0.827915	+	3.50921i	−0.0692337	+	0.293455i
$144$		0			0
$145$			0.981579i			0.0815157i
$146$	15.7833			1.30623
$147$		0			0
$148$			42.0362i			3.45535i
$149$		−	11.4579i		−	0.938672i	−0.883020	−	0.469336i	$-0.844493\pi$
$149$		−	11.4579i		−	0.938672i	0.883020	−	0.469336i	$-0.155507\pi$
$150$		0			0
$151$			16.6187i			1.35241i	0.736714	+	0.676205i	$0.236376\pi$
$151$			16.6187i			1.35241i	−0.736714	+	0.676205i	$0.763624\pi$
$152$	−48.4278			−3.92801
$153$		0			0
$154$		−	3.12051i		−	0.251458i
$155$	−3.97591			−0.319353
$156$		0			0
$157$	−14.1605			−1.13013			−0.565067	−	0.825045i	$-0.691150\pi$
$157$	−14.1605			−1.13013			−0.565067	+	0.825045i	$0.691150\pi$
$158$		−	6.61414i		−	0.526193i
$159$		0			0
$160$	6.89434			0.545046
$161$			3.70477i			0.291977i
$162$		0			0
$163$		−	5.44196i		−	0.426247i	−0.977025	−	0.213124i	$-0.931636\pi$
$163$		−	5.44196i		−	0.426247i	0.977025	−	0.213124i	$-0.0683637\pi$
$164$			27.4775i			2.14563i
$165$		0			0
$166$	19.7310			1.53142
$167$		−	0.886440i		−	0.0685948i	−0.999412	−	0.0342974i	$-0.989081\pi$
$167$		−	0.886440i		−	0.0685948i	0.999412	−	0.0342974i	$-0.0109193\pi$
$168$		0			0
$169$	11.6291	+	5.81066i	0.894547	+	0.446973i
$170$	8.01241			0.614524
$171$		0			0
$172$	15.9749			1.21807
$173$	−6.77560			−0.515140			−0.257570	−	0.966260i	$-0.582922\pi$
$173$	−6.77560			−0.515140			−0.257570	+	0.966260i	$0.582922\pi$
$174$		0			0
$175$			5.60730i			0.423872i
$176$		−	11.8484i		−	0.893107i
$177$		0			0
$178$	19.7207			1.47813
$179$	−1.31745			−0.0984711			−0.0492356	−	0.998787i	$-0.515679\pi$
$179$	−1.31745			−0.0984711			−0.0492356	+	0.998787i	$0.515679\pi$
$180$		0			0
$181$	11.0987			0.824961			0.412481	−	0.910966i	$-0.364662\pi$
$181$	11.0987			0.824961			0.412481	+	0.910966i	$0.364662\pi$
$182$	−10.9505	−	2.58352i	−0.811708	−	0.191503i
$183$		0			0
$184$			26.1877i			1.93058i
$185$	3.77720			0.277705
$186$		0			0
$187$		−	6.55353i		−	0.479242i
$188$			53.8585i			3.92803i
$189$		0			0
$190$			7.15473i			0.519059i
$191$	19.2393			1.39210			0.696052	−	0.717991i	$-0.254938\pi$
$191$	19.2393			1.39210			0.696052	+	0.717991i	$0.254938\pi$
$192$		0			0
$193$			19.7460i			1.42135i	0.703523	+	0.710673i	$0.251609\pi$
$193$			19.7460i			1.42135i	−0.703523	+	0.710673i	$0.748391\pi$
$194$	39.0971			2.80701
$195$		0			0
$196$	−28.7363			−2.05260
$197$		−	21.0198i		−	1.49760i	−0.662796	−	0.748800i	$-0.730630\pi$
$197$		−	21.0198i		−	1.49760i	0.662796	−	0.748800i	$-0.269370\pi$
$198$		0			0
$199$	−3.72266			−0.263892			−0.131946	−	0.991257i	$-0.542123\pi$
$199$	−3.72266			−0.263892			−0.131946	+	0.991257i	$0.542123\pi$
$200$			39.6359i			2.80268i
$201$		0			0
$202$		−	30.9240i		−	2.17581i
$203$			2.50532i			0.175839i
$204$		0			0
$205$	2.46902			0.172444
$206$			10.6290i			0.740558i
$207$		0			0
$208$	−41.5785	−	9.80947i	−2.88295	−	0.680164i
$209$	5.85202			0.404793
$210$		0			0
$211$	−11.9327			−0.821478			−0.410739	−	0.911753i	$-0.634729\pi$
$211$	−11.9327			−0.821478			−0.410739	+	0.911753i	$0.634729\pi$
$212$	−6.46670			−0.444135
$213$		0			0
$214$		−	36.5680i		−	2.49974i
$215$		−	1.43544i		−	0.0978961i
$216$		0			0
$217$	−10.1479			−0.688883
$218$	−18.9820			−1.28562
$219$		0			0
$220$	−2.34145			−0.157860
$221$	−22.9977	−	5.42577i	−1.54699	−	0.364977i
$222$		0			0
$223$		−	2.88605i		−	0.193264i	−0.995320	−	0.0966320i	$-0.969193\pi$
$223$		−	2.88605i		−	0.193264i	0.995320	−	0.0966320i	$-0.0308070\pi$
$224$	17.5967			1.17573
$225$		0			0
$226$			22.6044i			1.50362i
$227$		−	8.00487i		−	0.531302i	−0.964069	−	0.265651i	$-0.914413\pi$
$227$		−	8.00487i		−	0.531302i	0.964069	−	0.265651i	$-0.0855868\pi$
$228$		0			0
$229$		−	26.0987i		−	1.72465i	−0.506356	−	0.862324i	$-0.669008\pi$
$229$		−	26.0987i		−	1.72465i	0.506356	−	0.862324i	$-0.330992\pi$
$230$	3.86897			0.255112
$231$		0			0
$232$			17.7092i			1.16267i
$233$	−0.160396			−0.0105079			−0.00525394	−	0.999986i	$-0.501672\pi$
$233$	−0.160396			−0.0105079			−0.00525394	+	0.999986i	$0.501672\pi$
$234$		0			0
$235$	4.83951			0.315695
$236$			4.51582i			0.293955i
$237$		0			0
$238$	20.4504			1.32560
$239$			9.08010i			0.587343i	0.955906	+	0.293672i	$0.0948772\pi$
$239$			9.08010i			0.587343i	−0.955906	+	0.293672i	$0.905123\pi$
$240$		0			0
$241$		−	0.132631i		−	0.00854350i	−0.999991	−	0.00427175i	$-0.998640\pi$
$241$		−	0.132631i		−	0.00854350i	0.999991	−	0.00427175i	$-0.00135974\pi$
$242$			2.66546i			0.171342i
$243$		0			0
$244$	−25.7158			−1.64628
$245$			2.58213i			0.164966i
$246$		0			0
$247$	4.84497	−	20.5360i	0.308278	−	1.30667i
$248$	−71.7316			−4.55496
$249$		0			0
$250$	11.9689			0.756978
$251$	−27.5007			−1.73583			−0.867915	−	0.496713i	$-0.834540\pi$
$251$	−27.5007			−1.73583			−0.867915	+	0.496713i	$0.834540\pi$
$252$		0			0
$253$		−	3.16452i		−	0.198952i
$254$		−	39.8639i		−	2.50128i
$255$		0			0
$256$	3.41999			0.213750
$257$	−7.78887			−0.485856			−0.242928	−	0.970044i	$-0.578108\pi$
$257$	−7.78887			−0.485856			−0.242928	+	0.970044i	$0.578108\pi$
$258$		0			0
$259$	9.64070			0.599044
$260$	−1.93852	+	8.21662i	−0.120222	+	0.509573i
$261$		0			0
$262$		−	32.2759i		−	1.99401i
$263$	−7.74585			−0.477629			−0.238815	−	0.971065i	$-0.576759\pi$
$263$	−7.74585			−0.477629			−0.238815	+	0.971065i	$0.576759\pi$
$264$		0			0
$265$			0.581072i			0.0356949i
$266$			18.2613i			1.11967i
$267$		0			0
$268$			19.6215i			1.19857i
$269$	3.34660			0.204046			0.102023	−	0.994782i	$-0.467468\pi$
$269$	3.34660			0.204046			0.102023	+	0.994782i	$0.467468\pi$
$270$		0			0
$271$			10.5534i			0.641075i	0.947236	+	0.320538i	$0.103864\pi$
$271$			10.5534i			0.641075i	−0.947236	+	0.320538i	$0.896136\pi$
$272$	77.6489			4.70815
$273$		0			0
$274$	20.2223			1.22167
$275$		−	4.78961i		−	0.288824i
$276$		0			0
$277$	10.9571			0.658348			0.329174	−	0.944269i	$-0.393230\pi$
$277$	10.9571			0.658348			0.329174	+	0.944269i	$0.393230\pi$
$278$		−	28.6440i		−	1.71795i
$279$		0			0
$280$		−	4.44384i		−	0.265570i
$281$			16.5005i			0.984338i	0.870500	+	0.492169i	$0.163796\pi$
$281$			16.5005i			0.984338i	−0.870500	+	0.492169i	$0.836204\pi$
$282$		0			0
$283$	−1.15458			−0.0686326			−0.0343163	−	0.999411i	$-0.510925\pi$
$283$	−1.15458			−0.0686326			−0.0343163	+	0.999411i	$0.510925\pi$
$284$		−	26.1016i		−	1.54884i
$285$		0			0
$286$	9.35366	+	2.20677i	0.553094	+	0.130489i
$287$	6.30177			0.371982
$288$		0			0
$289$	25.9488			1.52640
$290$	2.61636			0.153638
$291$		0			0
$292$		−	30.2268i		−	1.76889i
$293$			13.4247i			0.784281i	0.919905	+	0.392141i	$0.128265\pi$
$293$			13.4247i			0.784281i	−0.919905	+	0.392141i	$0.871735\pi$
$294$		0			0
$295$	0.405773			0.0236250
$296$	68.1465			3.96093
$297$		0			0
$298$	−30.5407			−1.76918
$299$	−11.1050	−	2.61995i	−0.642216	−	0.151516i
$300$		0			0
$301$		−	3.66373i		−	0.211174i
$302$	44.2964			2.54897
$303$		0			0
$304$			69.3370i			3.97675i
$305$			2.31071i			0.132311i
$306$		0			0
$307$			30.2535i			1.72666i	0.504643	+	0.863328i	$0.331624\pi$
$307$			30.2535i			1.72666i	−0.504643	+	0.863328i	$0.668376\pi$
$308$	−5.97616			−0.340524
$309$		0			0
$310$			10.5976i			0.601905i
$311$	8.48341			0.481050			0.240525	−	0.970643i	$-0.422680\pi$
$311$	8.48341			0.481050			0.240525	+	0.970643i	$0.422680\pi$
$312$		0			0
$313$	−10.7899			−0.609881			−0.304941	−	0.952371i	$-0.598637\pi$
$313$	−10.7899			−0.609881			−0.304941	+	0.952371i	$0.598637\pi$
$314$			37.7444i			2.13004i
$315$		0			0
$316$	−12.6669			−0.712568
$317$		−	19.8347i		−	1.11403i	−0.830504	−	0.557013i	$-0.811947\pi$
$317$		−	19.8347i		−	1.11403i	0.830504	−	0.557013i	$-0.188053\pi$
$318$		0			0
$319$		−	2.13998i		−	0.119816i
$320$		−	7.50721i		−	0.419666i
$321$		0			0
$322$	9.87493			0.550308
$323$			38.3514i			2.13393i
$324$		0			0
$325$	−16.8077	−	3.96539i	−0.932326	−	0.219960i
$326$	−14.5053			−0.803376
$327$		0			0
$328$	44.5449			2.45958
$329$	12.3521			0.680991
$330$		0			0
$331$			3.41102i			0.187486i	0.995596	+	0.0937432i	$0.0298833\pi$
$331$			3.41102i			0.187486i	−0.995596	+	0.0937432i	$0.970117\pi$
$332$		−	37.7873i		−	2.07385i
$333$		0			0
$334$	−2.36277			−0.129285
$335$	1.76311			0.0963290
$336$		0			0
$337$	−7.64192			−0.416282			−0.208141	−	0.978099i	$-0.566741\pi$
$337$	−7.64192			−0.416282			−0.208141	+	0.978099i	$0.566741\pi$
$338$	15.4881	−	30.9969i	0.842440	−	1.68601i
$339$		0			0
$340$		−	15.3447i		−	0.832185i
$341$	8.66805			0.469401
$342$		0			0
$343$			14.7855i			0.798344i
$344$		−	25.8975i		−	1.39630i
$345$		0			0
$346$			18.0601i			0.970917i
$347$	23.9328			1.28478			0.642390	−	0.766378i	$-0.277943\pi$
$347$	23.9328			1.28478			0.642390	+	0.766378i	$0.277943\pi$
$348$		0			0
$349$		−	13.6173i		−	0.728917i	−0.931220	−	0.364458i	$-0.881254\pi$
$349$		−	13.6173i		−	0.728917i	0.931220	−	0.364458i	$-0.118746\pi$
$350$	14.9460			0.798899
$351$		0			0
$352$	−15.0306			−0.801136
$353$		−	3.44816i		−	0.183527i	−0.995781	−	0.0917636i	$-0.970750\pi$
$353$		−	3.44816i		−	0.183527i	0.995781	−	0.0917636i	$-0.0292504\pi$
$354$		0			0
$355$	−2.34538			−0.124480
$356$		−	37.7675i		−	2.00167i
$357$		0			0
$358$			3.51162i			0.185595i
$359$		−	20.5605i		−	1.08514i	−0.840011	−	0.542570i	$-0.817451\pi$
$359$		−	20.5605i		−	1.08514i	0.840011	−	0.542570i	$-0.182549\pi$
$360$		0			0
$361$	−15.2461			−0.802427
$362$		−	29.5832i		−	1.55486i
$363$		0			0
$364$	−4.94776	+	20.9716i	−0.259333	+	1.09921i
$365$	−2.71606			−0.142165
$366$		0			0
$367$	−33.1416			−1.72998			−0.864989	−	0.501790i	$-0.832675\pi$
$367$	−33.1416			−1.72998			−0.864989	+	0.501790i	$0.832675\pi$
$368$	37.4945			1.95453
$369$		0			0
$370$		−	10.0680i		−	0.523409i
$371$			1.48309i			0.0769983i
$372$		0			0
$373$	7.24503			0.375134			0.187567	−	0.982252i	$-0.439940\pi$
$373$	7.24503			0.375134			0.187567	+	0.982252i	$0.439940\pi$
$374$	−17.4682			−0.903258
$375$		0			0
$376$	87.3122			4.50278
$377$	−7.50964	−	1.77172i	−0.386766	−	0.0912484i
$378$		0			0
$379$			26.9600i			1.38484i	0.721493	+	0.692421i	$0.243456\pi$
$379$			26.9600i			1.38484i	−0.721493	+	0.692421i	$0.756544\pi$
$380$	13.7022			0.702907
$381$		0			0
$382$		−	51.2815i		−	2.62379i
$383$			16.9498i			0.866094i	0.901371	+	0.433047i	$0.142562\pi$
$383$			16.9498i			0.866094i	−0.901371	+	0.433047i	$0.857438\pi$
$384$		0			0
$385$			0.536994i			0.0273678i
$386$	52.6321			2.67890
$387$		0			0
$388$		−	74.8757i		−	3.80124i
$389$	−2.61306			−0.132488			−0.0662438	−	0.997803i	$-0.521101\pi$
$389$	−2.61306			−0.132488			−0.0662438	+	0.997803i	$0.521101\pi$
$390$		0			0
$391$	20.7388			1.04880
$392$			46.5856i			2.35293i
$393$		0			0
$394$	−56.0275			−2.82262
$395$			1.13820i			0.0572688i
$396$		0			0
$397$		−	20.0083i		−	1.00419i	−0.864813	−	0.502095i	$-0.832563\pi$
$397$		−	20.0083i		−	1.00419i	0.864813	−	0.502095i	$-0.167437\pi$
$398$			9.92260i			0.497375i
$399$		0			0
$400$	56.7492			2.83746
$401$		−	28.1352i		−	1.40500i	−0.711682	−	0.702502i	$-0.752066\pi$
$401$		−	28.1352i		−	1.40500i	0.711682	−	0.702502i	$-0.247934\pi$
$402$		0			0
$403$	7.17641	−	30.4180i	0.357482	−	1.51523i
$404$	−59.2233			−2.94647
$405$		0			0
$406$	6.67784			0.331416
$407$	−8.23483			−0.408185
$408$		0			0
$409$			32.3950i			1.60183i	0.598779	+	0.800914i	$0.295653\pi$
$409$			32.3950i			1.60183i	−0.598779	+	0.800914i	$0.704347\pi$
$410$		−	6.58107i		−	0.325016i
$411$		0			0
$412$	20.3558			1.00286
$413$	1.03567			0.0509621
$414$		0			0
$415$	−3.39541			−0.166674
$416$	−12.4441	+	52.7457i	−0.610122	+	2.58607i
$417$		0			0
$418$		−	15.5983i		−	0.762939i
$419$	−7.23409			−0.353408			−0.176704	−	0.984264i	$-0.556544\pi$
$419$	−7.23409			−0.353408			−0.176704	+	0.984264i	$0.556544\pi$
$420$		0			0
$421$			6.86086i			0.334378i	0.985925	+	0.167189i	$0.0534690\pi$
$421$			6.86086i			0.334378i	−0.985925	+	0.167189i	$0.946531\pi$
$422$			31.8060i			1.54829i
$423$		0			0
$424$			10.4834i			0.509120i
$425$	31.3888			1.52258
$426$		0			0
$427$			5.89773i			0.285411i
$428$	−70.0321			−3.38513
$429$		0			0
$430$	−3.82610			−0.184511
$431$			13.2631i			0.638863i	0.947609	+	0.319431i	$0.103492\pi$
$431$			13.2631i			0.638863i	−0.947609	+	0.319431i	$0.896508\pi$
$432$		0			0
$433$	9.73932			0.468042			0.234021	−	0.972232i	$-0.424812\pi$
$433$	9.73932			0.468042			0.234021	+	0.972232i	$0.424812\pi$
$434$			27.0488i			1.29838i
$435$		0			0
$436$			36.3528i			1.74098i
$437$			18.5188i			0.885875i
$438$		0			0
$439$	−11.7838			−0.562412			−0.281206	−	0.959647i	$-0.590734\pi$
$439$	−11.7838			−0.562412			−0.281206	+	0.959647i	$0.590734\pi$
$440$			3.79581i			0.180958i
$441$		0			0
$442$	−14.4622	+	61.2995i	−0.687895	+	2.91572i
$443$	8.14169			0.386823			0.193412	−	0.981118i	$-0.438045\pi$
$443$	8.14169			0.386823			0.193412	+	0.981118i	$0.438045\pi$
$444$		0			0
$445$	−3.39364			−0.160874
$446$	−7.69264			−0.364257
$447$		0			0
$448$		−	19.1609i		−	0.905270i
$449$		−	35.3317i		−	1.66741i	−0.552211	−	0.833704i	$-0.686216\pi$
$449$		−	35.3317i		−	1.66741i	0.552211	−	0.833704i	$-0.313784\pi$
$450$		0			0
$451$	−5.38281			−0.253467
$452$	43.2902			2.03620
$453$		0			0
$454$	−21.3367			−1.00138
$455$	1.88443	+	0.444585i	0.0883432	+	0.0208425i
$456$		0			0
$457$		−	19.6535i		−	0.919350i	−0.888087	−	0.459675i	$-0.847966\pi$
$457$		−	19.6535i		−	0.919350i	0.888087	−	0.459675i	$-0.152034\pi$
$458$	−69.5650			−3.25056
$459$		0			0
$460$		−	7.40955i		−	0.345472i
$461$			11.8871i			0.553636i	0.960922	+	0.276818i	$0.0892799\pi$
$461$			11.8871i			0.553636i	−0.960922	+	0.276818i	$0.910720\pi$
$462$		0			0
$463$		−	33.4104i		−	1.55271i	−0.630293	−	0.776357i	$-0.717065\pi$
$463$		−	33.4104i		−	1.55271i	0.630293	−	0.776357i	$-0.282935\pi$
$464$	25.3554			1.17709
$465$		0			0
$466$			0.427529i			0.0198049i
$467$	−7.45002			−0.344746			−0.172373	−	0.985032i	$-0.555143\pi$
$467$	−7.45002			−0.344746			−0.172373	+	0.985032i	$0.555143\pi$
$468$		0			0
$469$	4.50005			0.207793
$470$		−	12.8995i		−	0.595010i
$471$		0			0
$472$	7.32078			0.336966
$473$			3.12946i			0.143893i
$474$		0			0
$475$			28.0289i			1.28605i
$476$		−	39.1650i		−	1.79512i
$477$		0			0
$478$	24.2027			1.10700
$479$			22.1756i			1.01323i	0.862173	+	0.506614i	$0.169103\pi$
$479$			22.1756i			1.01323i	−0.862173	+	0.506614i	$0.830897\pi$
$480$		0			0
$481$	−6.81774	+	28.8977i	−0.310862	+	1.31762i
$482$	−0.353522			−0.0161025
$483$		0			0
$484$	5.10468			0.232031
$485$	−6.72803			−0.305504
$486$		0			0
$487$		−	8.84223i		−	0.400680i	−0.979726	−	0.200340i	$-0.935795\pi$
$487$		−	8.84223i		−	0.400680i	0.979726	−	0.200340i	$-0.0642046\pi$
$488$			41.6888i			1.88717i
$489$		0			0
$490$	6.88257			0.310923
$491$	−27.1206			−1.22393			−0.611967	−	0.790883i	$-0.709621\pi$
$491$	−27.1206			−1.22393			−0.611967	+	0.790883i	$0.709621\pi$
$492$		0			0
$493$	14.0244			0.631629
$494$	−54.7378	−	12.9141i	−2.46277	−	0.581032i
$495$		0			0
$496$			102.702i			4.61148i
$497$	−5.98622			−0.268518
$498$		0			0
$499$			27.8865i			1.24837i	0.781276	+	0.624185i	$0.214569\pi$
$499$			27.8865i			1.24837i	−0.781276	+	0.624185i	$0.785431\pi$
$500$		−	22.9218i		−	1.02510i
$501$		0			0
$502$			73.3021i			3.27163i
$503$	18.4384			0.822127			0.411064	−	0.911607i	$-0.365157\pi$
$503$	18.4384			0.822127			0.411064	+	0.911607i	$0.365157\pi$
$504$		0			0
$505$			5.32157i			0.236807i
$506$	−8.43490			−0.374977
$507$		0			0
$508$	−76.3441			−3.38722
$509$			23.5533i			1.04398i	0.852951	+	0.521992i	$0.174811\pi$
$509$			23.5533i			1.04398i	−0.852951	+	0.521992i	$0.825189\pi$
$510$		0			0
$511$	−6.93231			−0.306667
$512$			18.0116i			0.796006i
$513$		0			0
$514$			20.7609i			0.915725i
$515$		−	1.82909i		−	0.0805994i
$516$		0			0
$517$	−10.5508			−0.464024
$518$		−	25.6969i		−	1.12906i
$519$		0			0
$520$	13.3203	+	3.14261i	0.584134	+	0.137813i
$521$	−22.7100			−0.994942			−0.497471	−	0.867481i	$-0.665738\pi$
$521$	−22.7100			−0.994942			−0.497471	+	0.867481i	$0.665738\pi$
$522$		0			0
$523$	10.4633			0.457526			0.228763	−	0.973482i	$-0.426532\pi$
$523$	10.4633			0.457526			0.228763	+	0.973482i	$0.426532\pi$
$524$	−61.8123			−2.70028
$525$		0			0
$526$			20.6463i			0.900219i
$527$			56.8063i			2.47452i
$528$		0			0
$529$	−12.9858			−0.564601
$530$	1.54882			0.0672766
$531$		0			0
$532$	34.9726			1.51626
$533$	−4.45651	+	18.8894i	−0.193033	+	0.818191i
$534$		0			0
$535$			6.29280i			0.272062i
$536$	31.8092			1.37395
$537$		0			0
$538$		−	8.92023i		−	0.384578i
$539$		−	5.62941i		−	0.242476i
$540$		0			0
$541$		−	42.1990i		−	1.81428i	−0.420832	−	0.907139i	$-0.638262\pi$
$541$		−	42.1990i		−	1.81428i	0.420832	−	0.907139i	$-0.361738\pi$
$542$	28.1297			1.20828
$543$		0			0
$544$		−	98.5037i		−	4.22331i
$545$	3.26651			0.139922
$546$		0			0
$547$	9.87841			0.422370			0.211185	−	0.977446i	$-0.432268\pi$
$547$	9.87841			0.422370			0.211185	+	0.977446i	$0.432268\pi$
$548$		−	38.7282i		−	1.65439i
$549$		0			0
$550$	−12.7665			−0.544366
$551$			12.5232i			0.533507i
$552$		0			0
$553$			2.90506i			0.123536i
$554$		−	29.2057i		−	1.24083i
$555$		0			0
$556$	−54.8567			−2.32644
$557$		−	27.7902i		−	1.17751i	−0.808312	−	0.588754i	$-0.799619\pi$
$557$		−	27.7902i		−	1.17751i	0.808312	−	0.588754i	$-0.200381\pi$
$558$		0			0
$559$	10.9819	+	2.59092i	0.464486	+	0.109584i
$560$	−6.36252			−0.268866
$561$		0			0
$562$	43.9815			1.85525
$563$	27.1946			1.14612			0.573058	−	0.819515i	$-0.305757\pi$
$563$	27.1946			1.14612			0.573058	+	0.819515i	$0.305757\pi$
$564$		0			0
$565$		−	3.88989i		−	0.163649i
$566$			3.07748i			0.129356i
$567$		0			0
$568$	−42.3143			−1.77547
$569$	−33.0535			−1.38567			−0.692837	−	0.721095i	$-0.743639\pi$
$569$	−33.0535			−1.38567			−0.692837	+	0.721095i	$0.743639\pi$
$570$		0			0
$571$	−30.0787			−1.25875			−0.629377	−	0.777100i	$-0.716690\pi$
$571$	−30.0787			−1.25875			−0.629377	+	0.777100i	$0.716690\pi$
$572$	4.22624	−	17.9134i	0.176708	−	0.748997i
$573$		0			0
$574$		−	16.7971i		−	0.701099i
$575$	15.1568			0.632082
$576$		0			0
$577$		−	6.67291i		−	0.277797i	−0.990307	−	0.138898i	$-0.955644\pi$
$577$		−	6.67291i		−	0.277797i	0.990307	−	0.138898i	$-0.0443561\pi$
$578$		−	69.1654i		−	2.87690i
$579$		0			0
$580$		−	5.01065i		−	0.208056i
$581$	−8.66625			−0.359537
$582$		0			0
$583$		−	1.26682i		−	0.0524662i
$584$	−49.0019			−2.02771
$585$		0			0
$586$	35.7831			1.47819
$587$		−	16.5282i		−	0.682191i	−0.940029	−	0.341096i	$-0.889202\pi$
$587$		−	16.5282i		−	0.682191i	0.940029	−	0.341096i	$-0.110798\pi$
$588$		0			0
$589$	−50.7256			−2.09011
$590$		−	1.08157i		−	0.0445277i
$591$		0			0
$592$		−	97.5695i		−	4.01008i
$593$		−	17.3007i		−	0.710456i	−0.934780	−	0.355228i	$-0.884403\pi$
$593$		−	17.3007i		−	0.710456i	0.934780	−	0.355228i	$-0.115597\pi$
$594$		0			0
$595$	−3.51921			−0.144273
$596$			58.4892i			2.39581i
$597$		0			0
$598$	−6.98338	+	29.5998i	−0.285572	+	1.21043i
$599$	−35.7952			−1.46255			−0.731276	−	0.682082i	$-0.761075\pi$
$599$	−35.7952			−1.46255			−0.731276	+	0.682082i	$0.761075\pi$
$600$		0			0
$601$	−45.7772			−1.86729			−0.933646	−	0.358196i	$-0.883392\pi$
$601$	−45.7772			−1.86729			−0.933646	+	0.358196i	$0.883392\pi$
$602$	−9.76552			−0.398013
$603$		0			0
$604$		−	84.8331i		−	3.45181i
$605$		−	0.458686i		−	0.0186482i
$606$		0			0
$607$	−7.60403			−0.308638			−0.154319	−	0.988021i	$-0.549318\pi$
$607$	−7.60403			−0.308638			−0.154319	+	0.988021i	$0.549318\pi$
$608$	87.9596			3.56723
$609$		0			0
$610$	6.15912			0.249375
$611$	−8.73517	+	37.0250i	−0.353387	+	1.49787i
$612$		0			0
$613$			5.27304i			0.212976i	0.994314	+	0.106488i	$0.0339606\pi$
$613$			5.27304i			0.212976i	−0.994314	+	0.106488i	$0.966039\pi$
$614$	80.6394			3.25434
$615$		0			0
$616$			9.68820i			0.390349i
$617$		−	15.9614i		−	0.642584i	−0.946980	−	0.321292i	$-0.895883\pi$
$617$		−	15.9614i		−	0.642584i	0.946980	−	0.321292i	$-0.104117\pi$
$618$		0			0
$619$		−	20.0005i		−	0.803887i	−0.915665	−	0.401943i	$-0.868335\pi$
$619$		−	20.0005i		−	0.803887i	0.915665	−	0.401943i	$-0.131665\pi$
$620$	20.2958			0.815097
$621$		0			0
$622$		−	22.6122i		−	0.906667i
$623$	−8.66172			−0.347024
$624$		0			0
$625$	21.8884			0.875535
$626$			28.7601i			1.14948i
$627$		0			0
$628$	72.2850			2.88449
$629$		−	53.9672i		−	2.15181i
$630$		0			0
$631$		−	28.7753i		−	1.14553i	−0.819721	−	0.572764i	$-0.805871\pi$
$631$		−	28.7753i		−	1.14553i	0.819721	−	0.572764i	$-0.194129\pi$
$632$			20.5348i			0.816831i
$633$		0			0
$634$	−52.8686			−2.09968
$635$			6.85998i			0.272230i
$636$		0			0
$637$	−19.7548	−	4.66067i	−0.782713	−	0.184663i
$638$	−5.70404			−0.225825
$639$		0			0
$640$	−6.22148			−0.245926
$641$	−39.5621			−1.56261			−0.781304	−	0.624150i	$-0.785445\pi$
$641$	−39.5621			−1.56261			−0.781304	+	0.624150i	$0.785445\pi$
$642$		0			0
$643$			8.36156i			0.329748i	0.986315	+	0.164874i	$0.0527217\pi$
$643$			8.36156i			0.329748i	−0.986315	+	0.164874i	$0.947278\pi$
$644$		−	18.9117i		−	0.745225i
$645$		0			0
$646$	102.224			4.02195
$647$	−9.64297			−0.379104			−0.189552	−	0.981871i	$-0.560704\pi$
$647$	−9.64297			−0.379104			−0.189552	+	0.981871i	$0.560704\pi$
$648$		0			0
$649$	−0.884643			−0.0347253
$650$	−10.5696	+	44.8004i	−0.414573	+	1.75721i
$651$		0			0
$652$			27.7795i			1.08793i
$653$	−25.0322			−0.979587			−0.489794	−	0.871838i	$-0.662928\pi$
$653$	−25.0322			−0.979587			−0.489794	+	0.871838i	$0.662928\pi$
$654$		0			0
$655$			5.55420i			0.217021i
$656$		−	63.7777i		−	2.49010i
$657$		0			0
$658$		−	32.9239i		−	1.28351i
$659$	−27.2819			−1.06275			−0.531375	−	0.847136i	$-0.678325\pi$
$659$	−27.2819			−1.06275			−0.531375	+	0.847136i	$0.678325\pi$
$660$		0			0
$661$		−	1.09069i		−	0.0424229i	−0.999775	−	0.0212114i	$-0.993248\pi$
$661$		−	1.09069i		−	0.0424229i	0.999775	−	0.0212114i	$-0.00675232\pi$
$662$	9.09193			0.353368
$663$		0			0
$664$	−61.2585			−2.37729
$665$		−	3.14250i		−	0.121861i
$666$		0			0
$667$	6.77201			0.262213
$668$			4.52499i			0.175077i
$669$		0			0
$670$		−	4.69950i		−	0.181557i
$671$		−	5.03768i		−	0.194478i
$672$		0			0
$673$	48.0360			1.85165			0.925826	−	0.377950i	$-0.123371\pi$
$673$	48.0360			1.85165			0.925826	+	0.377950i	$0.123371\pi$
$674$			20.3692i			0.784594i
$675$		0			0
$676$	−59.3629	−	29.6615i	−2.28319	−	1.14083i
$677$	−2.28541			−0.0878355			−0.0439178	−	0.999035i	$-0.513984\pi$
$677$	−2.28541			−0.0878355			−0.0439178	+	0.999035i	$0.513984\pi$
$678$		0			0
$679$	−17.1722			−0.659009
$680$	−24.8760			−0.953950
$681$		0			0
$682$		−	23.1043i		−	0.884711i
$683$		−	35.3074i		−	1.35100i	−0.737360	−	0.675500i	$-0.763928\pi$
$683$		−	35.3074i		−	1.35100i	0.737360	−	0.675500i	$-0.236072\pi$
$684$		0			0
$685$	−3.47996			−0.132962
$686$	39.4103			1.50469
$687$		0			0
$688$	−37.0791			−1.41363
$689$	−4.44553	−	1.04882i	−0.169361	−	0.0399568i
$690$		0			0
$691$			31.3666i			1.19324i	0.802523	+	0.596621i	$0.203491\pi$
$691$			31.3666i			1.19324i	−0.802523	+	0.596621i	$0.796509\pi$
$692$	34.5873			1.31481
$693$		0			0
$694$		−	63.7920i		−	2.42151i
$695$			4.92920i			0.186975i
$696$		0			0
$697$		−	35.2764i		−	1.33619i
$698$	−36.2963			−1.37384
$699$		0			0
$700$		−	28.6235i		−	1.08187i
$701$	21.8220			0.824206			0.412103	−	0.911137i	$-0.364794\pi$
$701$	21.8220			0.824206			0.412103	+	0.911137i	$0.364794\pi$
$702$		0			0
$703$	48.1903			1.81753
$704$			16.3668i			0.616846i
$705$		0			0
$706$	−9.19094			−0.345906
$707$			13.5824i			0.510821i
$708$		0			0
$709$			36.9873i			1.38909i	0.719451	+	0.694543i	$0.244393\pi$
$709$			36.9873i			1.38909i	−0.719451	+	0.694543i	$0.755607\pi$
$710$			6.25153i			0.234616i
$711$		0			0
$712$	−61.2265			−2.29456
$713$			27.4302i			1.02727i
$714$		0			0
$715$	−1.60963	−	0.379753i	−0.0601966	−	0.0142020i
$716$	6.72518			0.251332
$717$		0			0
$718$	−54.8031			−2.04523
$719$	11.0686			0.412788			0.206394	−	0.978469i	$-0.433827\pi$
$719$	11.0686			0.412788			0.206394	+	0.978469i	$0.433827\pi$
$720$		0			0
$721$		−	4.66847i		−	0.173863i
$722$			40.6379i			1.51239i
$723$		0			0
$724$	−56.6554			−2.10558
$725$	10.2497			0.380663
$726$		0			0
$727$	20.0495			0.743595			0.371798	−	0.928314i	$-0.378742\pi$
$727$	20.0495			0.743595			0.371798	+	0.928314i	$0.378742\pi$
$728$	33.9979	+	8.02101i	1.26005	+	0.297278i
$729$		0			0
$730$			7.23956i			0.267948i
$731$	−20.5090			−0.758553
$732$		0			0
$733$		−	17.6554i		−	0.652117i	−0.945350	−	0.326059i	$-0.894279\pi$
$733$		−	17.6554i		−	0.652117i	0.945350	−	0.326059i	$-0.105721\pi$
$734$			88.3377i			3.26060i
$735$		0			0
$736$		−	47.5647i		−	1.75326i
$737$	−3.84383			−0.141589
$738$		0			0
$739$		−	20.1704i		−	0.741981i	−0.928637	−	0.370990i	$-0.879018\pi$
$739$		−	20.1704i		−	0.741981i	0.928637	−	0.370990i	$-0.120982\pi$
$740$	−19.2814			−0.708798
$741$		0			0
$742$	3.95312			0.145124
$743$			2.45850i			0.0901934i	0.998983	+	0.0450967i	$0.0143596\pi$
$743$			2.45850i			0.0901934i	−0.998983	+	0.0450967i	$0.985640\pi$
$744$		0			0
$745$	5.25560			0.192550
$746$		−	19.3113i		−	0.707039i
$747$		0			0
$748$			33.4537i			1.22319i
$749$			16.0614i			0.586870i
$750$		0			0
$751$	−17.0807			−0.623285			−0.311642	−	0.950199i	$-0.600879\pi$
$751$	−17.0807			−0.623285			−0.311642	+	0.950199i	$0.600879\pi$
$752$		−	125.010i		−	4.55865i
$753$		0			0
$754$	−4.72246	+	20.0167i	−0.171982	+	0.728964i
$755$	−7.62276			−0.277421
$756$		0			0
$757$	22.8517			0.830560			0.415280	−	0.909694i	$-0.363684\pi$
$757$	22.8517			0.830560			0.415280	+	0.909694i	$0.363684\pi$
$758$	71.8608			2.61010
$759$		0			0
$760$		−	22.2132i		−	0.805756i
$761$		−	11.6590i		−	0.422639i	−0.977417	−	0.211319i	$-0.932224\pi$
$761$		−	11.6590i		−	0.422639i	0.977417	−	0.211319i	$-0.0677760\pi$
$762$		0			0
$763$	8.33725			0.301829
$764$	−98.2103			−3.55312
$765$		0			0
$766$	45.1790			1.63238
$767$	−0.732409	+	3.10440i	−0.0264458	+	0.112093i
$768$		0			0
$769$			43.9248i			1.58397i	0.610540	+	0.791985i	$0.290952\pi$
$769$			43.9248i			1.58397i	−0.610540	+	0.791985i	$0.709048\pi$
$770$	1.43134			0.0515818
$771$		0			0
$772$		−	100.797i		−	3.62776i
$773$		−	44.5121i		−	1.60099i	−0.599341	−	0.800494i	$-0.704571\pi$
$773$		−	44.5121i		−	1.60099i	0.599341	−	0.800494i	$-0.295429\pi$
$774$		0			0
$775$			41.5165i			1.49132i
$776$	−121.384			−4.35743
$777$		0			0
$778$			6.96502i			0.249708i
$779$	31.5003			1.12861
$780$		0			0
$781$	5.11327			0.182967
$782$		−	55.2784i		−	1.97675i
$783$		0			0
$784$	66.6995			2.38212
$785$		−	6.49524i		−	0.231825i
$786$		0			0
$787$			26.3410i			0.938954i	0.882945	+	0.469477i	$0.155558\pi$
$787$			26.3410i			0.938954i	−0.882945	+	0.469477i	$0.844442\pi$
$788$			107.299i			3.82238i
$789$		0			0
$790$	3.03382			0.107938
$791$		−	9.92831i		−	0.353010i
$792$		0			0
$793$	−17.6783	−	4.17077i	−0.627774	−	0.148108i
$794$	−53.3314			−1.89266
$795$		0			0
$796$	19.0030			0.673543
$797$	−20.1094			−0.712312			−0.356156	−	0.934427i	$-0.615913\pi$
$797$	−20.1094			−0.712312			−0.356156	+	0.934427i	$0.615913\pi$
$798$		0			0
$799$		−	69.1450i		−	2.44618i
$800$		−	71.9908i		−	2.54526i
$801$		0			0
$802$	−74.9932			−2.64810
$803$	5.92140			0.208962
$804$		0			0
$805$	−1.69933			−0.0598934
$806$	−81.0780	−	19.1284i	−2.85585	−	0.673770i
$807$		0			0
$808$			96.0093i			3.37759i
$809$	−7.67428			−0.269813			−0.134907	−	0.990858i	$-0.543073\pi$
$809$	−7.67428			−0.269813			−0.134907	+	0.990858i	$0.543073\pi$
$810$		0			0
$811$			4.39093i			0.154187i	0.997024	+	0.0770933i	$0.0245639\pi$
$811$			4.39093i			0.154187i	−0.997024	+	0.0770933i	$0.975436\pi$
$812$		−	12.7889i		−	0.448802i
$813$		0			0
$814$			21.9496i			0.769333i
$815$	2.49615			0.0874363
$816$		0			0
$817$		−	18.3136i		−	0.640713i
$818$	86.3475			3.01907
$819$		0			0
$820$	−12.6035			−0.440135
$821$		−	4.48612i		−	0.156567i	−0.996931	−	0.0782833i	$-0.975056\pi$
$821$		−	4.48612i		−	0.156567i	0.996931	−	0.0782833i	$-0.0249439\pi$
$822$		0			0
$823$	−20.7536			−0.723426			−0.361713	−	0.932289i	$-0.617808\pi$
$823$	−20.7536			−0.723426			−0.361713	+	0.932289i	$0.617808\pi$
$824$		−	32.9997i		−	1.14960i
$825$		0			0
$826$		−	2.76054i		−	0.0960516i
$827$		−	35.0088i		−	1.21737i	−0.793410	−	0.608687i	$-0.791697\pi$
$827$		−	35.0088i		−	1.21737i	0.793410	−	0.608687i	$-0.208303\pi$
$828$		0			0
$829$	48.1248			1.67144			0.835721	−	0.549154i	$-0.185050\pi$
$829$	48.1248			1.67144			0.835721	+	0.549154i	$0.185050\pi$
$830$			9.05034i			0.314142i
$831$		0			0
$832$	57.4344	+	13.5503i	1.99118	+	0.469772i
$833$	36.8925			1.27825
$834$		0			0
$835$	0.406598			0.0140709
$836$	−29.8727			−1.03317
$837$		0			0
$838$			19.2822i			0.666092i
$839$			15.3315i			0.529304i	0.964344	+	0.264652i	$0.0852570\pi$
$839$			15.3315i			0.529304i	−0.964344	+	0.264652i	$0.914743\pi$
$840$		0			0
$841$	−24.4205			−0.842086
$842$	18.2874			0.630224
$843$		0			0
$844$	60.9124			2.09669
$845$	−2.66527	+	5.33411i	−0.0916879	+	0.183499i
$846$		0			0
$847$		−	1.17072i		−	0.0402265i
$848$	15.0098			0.515437
$849$		0			0
$850$		−	83.6657i		−	2.86971i
$851$		−	26.0593i		−	0.893300i
$852$		0			0
$853$			10.0332i			0.343531i	0.985138	+	0.171765i	$0.0549471\pi$
$853$			10.0332i			0.343531i	−0.985138	+	0.171765i	$0.945053\pi$
$854$	15.7202			0.537933
$855$		0			0
$856$			113.532i			3.88044i
$857$	−27.6616			−0.944903			−0.472452	−	0.881357i	$-0.656631\pi$
$857$	−27.6616			−0.944903			−0.472452	+	0.881357i	$0.656631\pi$
$858$		0			0
$859$	−43.3633			−1.47954			−0.739768	−	0.672862i	$-0.765065\pi$
$859$	−43.3633			−1.47954			−0.739768	+	0.672862i	$0.765065\pi$
$860$			7.32745i			0.249864i
$861$		0			0
$862$	35.3524			1.20411
$863$			38.2865i			1.30329i	0.758525	+	0.651644i	$0.225920\pi$
$863$			38.2865i			1.30329i	−0.758525	+	0.651644i	$0.774080\pi$
$864$		0			0
$865$		−	3.10787i		−	0.105671i
$866$		−	25.9598i		−	0.882149i
$867$		0			0
$868$	51.8017			1.75826
$869$		−	2.48143i		−	0.0841766i
$870$		0			0
$871$	−3.18236	+	13.4888i	−0.107830	+	0.457050i
$872$	58.9329			1.99572
$873$		0			0
$874$	49.3612			1.66967
$875$	−5.25696			−0.177718
$876$		0			0
$877$			4.41266i			0.149005i	0.997221	+	0.0745024i	$0.0237369\pi$
$877$			4.41266i			0.149005i	−0.997221	+	0.0745024i	$0.976263\pi$
$878$			31.4094i			1.06001i
$879$		0			0
$880$	5.43469			0.183204
$881$	9.20818			0.310232			0.155116	−	0.987896i	$-0.450425\pi$
$881$	9.20818			0.310232			0.155116	+	0.987896i	$0.450425\pi$
$882$		0			0
$883$	4.27713			0.143937			0.0719685	−	0.997407i	$-0.477072\pi$
$883$	4.27713			0.143937			0.0719685	+	0.997407i	$0.477072\pi$
$884$	117.396	+	27.6968i	3.94845	+	0.931544i
$885$		0			0
$886$		−	21.7014i		−	0.729071i
$887$	−46.4860			−1.56085			−0.780424	−	0.625251i	$-0.784997\pi$
$887$	−46.4860			−1.56085			−0.780424	+	0.625251i	$0.784997\pi$
$888$		0			0
$889$			17.5090i			0.587233i
$890$			9.04561i			0.303209i
$891$		0			0
$892$			14.7323i			0.493275i
$893$	61.7435			2.06617
$894$		0			0
$895$		−	0.604298i		−	0.0201994i
$896$	−15.8793			−0.530492
$897$		0			0
$898$	−94.1754			−3.14267
$899$			18.5495i			0.618659i
$900$		0			0
$901$	8.30213			0.276584
$902$			14.3477i			0.477725i
$903$		0			0
$904$		−	70.1795i		−	2.33414i
$905$			5.09083i			0.169225i
$906$		0			0
$907$	1.17192			0.0389129			0.0194564	−	0.999811i	$-0.493806\pi$
$907$	1.17192			0.0389129			0.0194564	+	0.999811i	$0.493806\pi$
$908$			40.8623i			1.35606i
$909$		0			0
$910$	1.18502	−	5.02286i	0.0392832	−	0.166506i
$911$	42.8661			1.42022			0.710109	−	0.704091i	$-0.248645\pi$
$911$	42.8661			1.42022			0.710109	+	0.704091i	$0.248645\pi$
$912$		0			0
$913$	7.40248			0.244986
$914$	−52.3855			−1.73276
$915$		0			0
$916$			133.225i			4.40189i
$917$			14.1762i			0.468140i
$918$		0			0
$919$	24.0196			0.792333			0.396167	−	0.918179i	$-0.370340\pi$
$919$	24.0196			0.792333			0.396167	+	0.918179i	$0.370340\pi$
$920$	−12.0119			−0.396021
$921$		0			0
$922$	31.6845			1.04347
$923$	4.23335	−	17.9435i	0.139342	−	0.590618i
$924$		0			0
$925$		−	39.4416i		−	1.29683i
$926$	−89.0542			−2.92650
$927$		0			0
$928$		−	32.1653i		−	1.05588i
$929$		−	19.2035i		−	0.630046i	−0.949084	−	0.315023i	$-0.897988\pi$
$929$		−	19.2035i		−	0.630046i	0.949084	−	0.315023i	$-0.102012\pi$
$930$		0			0
$931$			32.9434i			1.07968i
$932$	0.818770			0.0268197
$933$		0			0
$934$			19.8577i			0.649765i
$935$	3.00601			0.0983071
$936$		0			0
$937$	51.2628			1.67468			0.837342	−	0.546680i	$-0.184109\pi$
$937$	51.2628			1.67468			0.837342	+	0.546680i	$0.184109\pi$
$938$		−	11.9947i		−	0.391641i
$939$		0			0
$940$	−24.7041			−0.805760
$941$			1.61030i			0.0524943i	0.999655	+	0.0262471i	$0.00835568\pi$
$941$			1.61030i			0.0524943i	−0.999655	+	0.0262471i	$0.991644\pi$
$942$		0			0
$943$		−	17.0340i		−	0.554703i
$944$		−	10.4816i		−	0.341147i
$945$		0			0
$946$	8.34145			0.271204
$947$			11.4406i			0.371770i	0.982571	+	0.185885i	$0.0595152\pi$
$947$			11.4406i			0.371770i	−0.982571	+	0.185885i	$0.940485\pi$
$948$		0			0
$949$	4.90241	−	20.7794i	0.159139	−	0.674529i
$950$	74.7098			2.42391
$951$		0			0
$952$	−63.4919			−2.05779
$953$	−16.5093			−0.534789			−0.267394	−	0.963587i	$-0.586163\pi$
$953$	−16.5093			−0.534789			−0.267394	+	0.963587i	$0.586163\pi$
$954$		0			0
$955$			8.82478i			0.285563i
$956$		−	46.3510i		−	1.49910i
$957$		0			0
$958$	59.1082			1.90970
$959$	−8.88203			−0.286816
$960$		0			0
$961$	−44.1350			−1.42371
$962$	77.0258	+	18.1724i	2.48341	+	0.585902i
$963$		0			0
$964$			0.677038i			0.0218059i
$965$	−9.05720			−0.291561
$966$		0			0
$967$			30.8852i			0.993202i	0.867979	+	0.496601i	$0.165419\pi$
$967$			30.8852i			0.993202i	−0.867979	+	0.496601i	$0.834581\pi$
$968$		−	8.27540i		−	0.265982i
$969$		0			0
$970$			17.9333i			0.575803i
$971$	−23.7146			−0.761038			−0.380519	−	0.924773i	$-0.624255\pi$
$971$	−23.7146			−0.761038			−0.380519	+	0.924773i	$0.624255\pi$
$972$		0			0
$973$			12.5810i			0.403328i
$974$	−23.5686			−0.755187
$975$		0			0
$976$	59.6885			1.91058
$977$			52.6096i			1.68313i	0.540156	+	0.841565i	$0.318365\pi$
$977$			52.6096i			1.68313i	−0.540156	+	0.841565i	$0.681635\pi$
$978$		0			0
$979$	7.39861			0.236461
$980$		−	13.1810i		−	0.421050i
$981$		0			0
$982$			72.2888i			2.30683i
$983$		−	44.4857i		−	1.41887i	−0.704769	−	0.709437i	$-0.748949\pi$
$983$		−	44.4857i		−	1.41887i	0.704769	−	0.709437i	$-0.251051\pi$
$984$		0			0
$985$	9.64150			0.307204
$986$		−	37.3816i		−	1.19047i
$987$		0			0
$988$	−24.7320	+	104.830i	−0.786831	+	3.33507i
$989$	−9.90323			−0.314904
$990$		0			0
$991$	8.68365			0.275845			0.137923	−	0.990443i	$-0.455957\pi$
$991$	8.68365			0.275845			0.137923	+	0.990443i	$0.455957\pi$
$992$	130.286			4.13659
$993$		0			0
$994$			15.9560i			0.506094i
$995$		−	1.70753i		−	0.0541324i
$996$		0			0
$997$	−25.3265			−0.802100			−0.401050	−	0.916056i	$-0.631355\pi$
$997$	−25.3265			−0.802100			−0.401050	+	0.916056i	$0.631355\pi$
$998$	74.3303			2.35289
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.b.b.298.1		12
3.2	odd	2		143.2.b.a.12.12	yes	12
12.11	even	2		2288.2.j.k.1585.9		12
13.12	even	2	inner	1287.2.b.b.298.12		12
39.5	even	4		1859.2.a.n.1.6		6
39.8	even	4		1859.2.a.j.1.1		6
39.38	odd	2		143.2.b.a.12.1	✓	12
156.155	even	2		2288.2.j.k.1585.10		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.b.a.12.1	✓	12	39.38	odd	2
143.2.b.a.12.12	yes	12	3.2	odd	2
1287.2.b.b.298.1		12	1.1	even	1	trivial
1287.2.b.b.298.12		12	13.12	even	2	inner
1859.2.a.j.1.1		6	39.8	even	4
1859.2.a.n.1.6		6	39.5	even	4
2288.2.j.k.1585.9		12	12.11	even	2
2288.2.j.k.1585.10		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)

\(f(q)\)	\(=\)	\(q-2.66546i q^{2} -5.10468 q^{4} +0.458686i q^{5} +1.17072i q^{7} +8.27540i q^{8} +O(q^{10})\)	\(q-2.66546i q^{2} -5.10468 q^{4} +0.458686i q^{5} +1.17072i q^{7} +8.27540i q^{8} +1.22261 q^{10} -1.00000i q^{11} +(-3.50921 - 0.827915i) q^{13} +3.12051 q^{14} +11.8484 q^{16} +6.55353 q^{17} +5.85202i q^{19} -2.34145i q^{20} -2.66546 q^{22} +3.16452 q^{23} +4.78961 q^{25} +(-2.20677 + 9.35366i) q^{26} -5.97616i q^{28} +2.13998 q^{29} +8.66805i q^{31} -15.0306i q^{32} -17.4682i q^{34} -0.536994 q^{35} -8.23483i q^{37} +15.5983 q^{38} -3.79581 q^{40} -5.38281i q^{41} -3.12946 q^{43} +5.10468i q^{44} -8.43490i q^{46} -10.5508i q^{47} +5.62941 q^{49} -12.7665i q^{50} +(17.9134 + 4.22624i) q^{52} +1.26682 q^{53} +0.458686 q^{55} -9.68820 q^{56} -5.70404i q^{58} -0.884643i q^{59} +5.03768 q^{61} +23.1043 q^{62} -16.3668 q^{64} +(0.379753 - 1.60963i) q^{65} -3.84383i q^{67} -33.4537 q^{68} +1.43134i q^{70} +5.11327i q^{71} +5.92140i q^{73} -21.9496 q^{74} -29.8727i q^{76} +1.17072 q^{77} +2.48143 q^{79} +5.43469i q^{80} -14.3477 q^{82} +7.40248i q^{83} +3.00601i q^{85} +8.34145i q^{86} +8.27540 q^{88} +7.39861i q^{89} +(0.969259 - 4.10831i) q^{91} -16.1539 q^{92} -28.1228 q^{94} -2.68424 q^{95} +14.6680i q^{97} -15.0050i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 14 q^{4}+O(q^{10}) \) 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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