LMFDB - Embedded newform 1287.2.b.c.298.10

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 23x^{12} + 201x^{10} + 835x^{8} + 1695x^{6} + 1565x^{4} + 511x^{2} + 49 $ x^14 + 23x^12 + 201x^10 + 835x^8 + 1695x^6 + 1565x^4 + 511x^2 + 49
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 429)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			298.10
Root			$1.36814i$ of defining polynomial
Character	$\chi$	$=$	1287.298
Dual form			1287.2.b.c.298.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.36814i q^{2} +0.128197 q^{4} -2.18365i q^{5} +4.27070i q^{7} +2.91167i q^{8} +O(q^{10})$	\(q+1.36814i q^{2} +0.128197 q^{4} -2.18365i q^{5} +4.27070i q^{7} +2.91167i q^{8} +2.98753 q^{10} +1.00000i q^{11} +(-0.922450 + 3.48555i) q^{13} -5.84291 q^{14} -3.72717 q^{16} -3.79173 q^{17} -3.17118i q^{19} -0.279937i q^{20} -1.36814 q^{22} -4.94243 q^{23} +0.231676 q^{25} +(-4.76872 - 1.26204i) q^{26} +0.547491i q^{28} +7.35483 q^{29} +0.0727448i q^{31} +0.724050i q^{32} -5.18761i q^{34} +9.32571 q^{35} -3.66658i q^{37} +4.33862 q^{38} +6.35806 q^{40} +4.16722i q^{41} -7.11697 q^{43} +0.128197i q^{44} -6.76193i q^{46} +11.6337i q^{47} -11.2389 q^{49} +0.316965i q^{50} +(-0.118255 + 0.446838i) q^{52} -5.11268 q^{53} +2.18365 q^{55} -12.4349 q^{56} +10.0624i q^{58} +5.62765i q^{59} -5.40800 q^{61} -0.0995250 q^{62} -8.44494 q^{64} +(7.61123 + 2.01431i) q^{65} +10.1145i q^{67} -0.486088 q^{68} +12.7589i q^{70} -9.62239i q^{71} -6.44736i q^{73} +5.01639 q^{74} -0.406536i q^{76} -4.27070 q^{77} +10.1209 q^{79} +8.13884i q^{80} -5.70134 q^{82} +7.80918i q^{83} +8.27981i q^{85} -9.73700i q^{86} -2.91167 q^{88} +2.87912i q^{89} +(-14.8858 - 3.93951i) q^{91} -0.633605 q^{92} -15.9166 q^{94} -6.92475 q^{95} +3.79281i q^{97} -15.3764i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 18 q^{4}+O(q^{10}) $ 14 * q - 18 * q^4	$ 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) $ 14 * q - 18 * q^4 - 16 * q^14 + 34 * q^16 - 4 * q^17 + 6 * q^22 + 8 * q^23 - 26 * q^25 + 6 * q^26 + 24 * q^29 + 8 * q^35 + 32 * q^38 - 20 * q^40 + 32 * q^43 - 46 * q^49 + 4 * q^52 - 20 * q^53 + 12 * q^55 + 32 * q^56 - 20 * q^61 - 72 * q^62 - 58 * q^64 - 12 * q^65 + 20 * q^68 + 12 * q^77 + 12 * q^79 + 20 * q^82 - 30 * q^88 + 16 * q^91 + 24 * q^92 + 64 * q^94 + 36 * 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 $

$2$			1.36814i			0.967420i	0.875228	+	0.483710i	$0.160711\pi$
$2$			1.36814i			0.967420i	−0.875228	+	0.483710i	$0.839289\pi$
$3$		0			0
$3$		0			0
$4$	0.128197			0.0640985
$5$		−	2.18365i		−	0.976558i	−0.872688	−	0.488279i	$-0.837625\pi$
$5$		−	2.18365i		−	0.976558i	0.872688	−	0.488279i	$-0.162375\pi$
$6$		0			0
$7$			4.27070i			1.61417i	0.590433	+	0.807087i	$0.298957\pi$
$7$			4.27070i			1.61417i	−0.590433	+	0.807087i	$0.701043\pi$
$8$			2.91167i			1.02943i
$9$		0			0
$10$	2.98753			0.944741
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−0.922450	+	3.48555i	−0.255842	+	0.966719i
$13$	−0.922450	+	3.48555i	−0.255842	+	0.966719i
$14$	−5.84291			−1.56158
$15$		0			0
$16$	−3.72717			−0.931793
$17$	−3.79173			−0.919629			−0.459815	−	0.888015i	$-0.652084\pi$
$17$	−3.79173			−0.919629			−0.459815	+	0.888015i	$0.652084\pi$
$18$		0			0
$19$		−	3.17118i		−	0.727519i	−0.931493	−	0.363760i	$-0.881493\pi$
$19$		−	3.17118i		−	0.727519i	0.931493	−	0.363760i	$-0.118507\pi$
$20$		−	0.279937i		−	0.0625959i
$21$		0			0
$22$	−1.36814			−0.291688
$23$	−4.94243			−1.03057			−0.515284	−	0.857020i	$-0.672313\pi$
$23$	−4.94243			−1.03057			−0.515284	+	0.857020i	$0.672313\pi$
$24$		0			0
$25$	0.231676			0.0463352
$26$	−4.76872	−	1.26204i	−0.935223	−	0.247506i
$27$		0			0
$28$			0.547491i			0.103466i
$29$	7.35483			1.36576			0.682879	−	0.730531i	$-0.260728\pi$
$29$	7.35483			1.36576			0.682879	+	0.730531i	$0.260728\pi$
$30$		0			0
$31$			0.0727448i			0.0130654i	0.999979	+	0.00653268i	$0.00207943\pi$
$31$			0.0727448i			0.0130654i	−0.999979	+	0.00653268i	$0.997921\pi$
$32$			0.724050i			0.127995i
$33$		0			0
$34$		−	5.18761i		−	0.889668i
$35$	9.32571			1.57633
$36$		0			0
$37$		−	3.66658i		−	0.602782i	−0.953501	−	0.301391i	$-0.902549\pi$
$37$		−	3.66658i		−	0.602782i	0.953501	−	0.301391i	$-0.0974510\pi$
$38$	4.33862			0.703817
$39$		0			0
$40$	6.35806			1.00530
$41$			4.16722i			0.650811i	0.945575	+	0.325405i	$0.105501\pi$
$41$			4.16722i			0.650811i	−0.945575	+	0.325405i	$0.894499\pi$
$42$		0			0
$43$	−7.11697			−1.08533			−0.542664	−	0.839950i	$-0.682584\pi$
$43$	−7.11697			−1.08533			−0.542664	+	0.839950i	$0.682584\pi$
$44$			0.128197i			0.0193264i
$45$		0			0
$46$		−	6.76193i		−	0.996992i
$47$			11.6337i			1.69695i	0.529232	+	0.848477i	$0.322480\pi$
$47$			11.6337i			1.69695i	−0.529232	+	0.848477i	$0.677520\pi$
$48$		0			0
$49$	−11.2389			−1.60556
$50$			0.316965i			0.0448256i
$51$		0			0
$52$	−0.118255	+	0.446838i	−0.0163991	+	0.0619652i
$53$	−5.11268			−0.702281			−0.351140	−	0.936323i	$-0.614206\pi$
$53$	−5.11268			−0.702281			−0.351140	+	0.936323i	$0.614206\pi$
$54$		0			0
$55$	2.18365			0.294443
$56$	−12.4349			−1.66168
$57$		0			0
$58$			10.0624i			1.32126i
$59$			5.62765i			0.732658i	0.930485	+	0.366329i	$0.119386\pi$
$59$			5.62765i			0.732658i	−0.930485	+	0.366329i	$0.880614\pi$
$60$		0			0
$61$	−5.40800			−0.692424			−0.346212	−	0.938156i	$-0.612532\pi$
$61$	−5.40800			−0.692424			−0.346212	+	0.938156i	$0.612532\pi$
$62$	−0.0995250			−0.0126397
$63$		0			0
$64$	−8.44494			−1.05562
$65$	7.61123	+	2.01431i	0.944057	+	0.249844i
$66$		0			0
$67$			10.1145i			1.23569i	0.786302	+	0.617843i	$0.211993\pi$
$67$			10.1145i			1.23569i	−0.786302	+	0.617843i	$0.788007\pi$
$68$	−0.486088			−0.0589469
$69$		0			0
$70$			12.7589i			1.52498i
$71$		−	9.62239i		−	1.14197i	−0.820961	−	0.570984i	$-0.806562\pi$
$71$		−	9.62239i		−	1.14197i	0.820961	−	0.570984i	$-0.193438\pi$
$72$		0			0
$73$		−	6.44736i		−	0.754607i	−0.926090	−	0.377303i	$-0.876851\pi$
$73$		−	6.44736i		−	0.754607i	0.926090	−	0.377303i	$-0.123149\pi$
$74$	5.01639			0.583144
$75$		0			0
$76$		−	0.406536i		−	0.0466329i
$77$	−4.27070			−0.486692
$78$		0			0
$79$	10.1209			1.13869			0.569347	−	0.822097i	$-0.307196\pi$
$79$	10.1209			1.13869			0.569347	+	0.822097i	$0.307196\pi$
$80$			8.13884i			0.909949i
$81$		0			0
$82$	−5.70134			−0.629607
$83$			7.80918i			0.857169i	0.903502	+	0.428585i	$0.140988\pi$
$83$			7.80918i			0.857169i	−0.903502	+	0.428585i	$0.859012\pi$
$84$		0			0
$85$			8.27981i			0.898071i
$86$		−	9.73700i		−	1.04997i
$87$		0			0
$88$	−2.91167			−0.310385
$89$			2.87912i			0.305187i	0.988289	+	0.152593i	$0.0487624\pi$
$89$			2.87912i			0.305187i	−0.988289	+	0.152593i	$0.951238\pi$
$90$		0			0
$91$	−14.8858	−	3.93951i	−1.56045	−	0.412973i
$92$	−0.633605			−0.0660579
$93$		0			0
$94$	−15.9166			−1.64167
$95$	−6.92475			−0.710465
$96$		0			0
$97$			3.79281i			0.385101i	0.981287	+	0.192551i	$0.0616760\pi$
$97$			3.79281i			0.385101i	−0.981287	+	0.192551i	$0.938324\pi$
$98$		−	15.3764i		−	1.55325i
$99$		0			0
$100$	0.0297002			0.00297002
$101$	12.8641			1.28003			0.640014	−	0.768363i	$-0.278928\pi$
$101$	12.8641			1.28003			0.640014	+	0.768363i	$0.278928\pi$
$102$		0			0
$103$	16.9975			1.67482			0.837409	−	0.546577i	$-0.184070\pi$
$103$	16.9975			1.67482			0.837409	+	0.546577i	$0.184070\pi$
$104$	−10.1488	−	2.68587i	−0.995169	−	0.263371i
$105$		0			0
$106$		−	6.99486i		−	0.679400i
$107$	12.6819			1.22600			0.613001	−	0.790082i	$-0.289962\pi$
$107$	12.6819			1.22600			0.613001	+	0.790082i	$0.289962\pi$
$108$		0			0
$109$			8.89607i			0.852089i	0.904702	+	0.426045i	$0.140093\pi$
$109$			8.89607i			0.852089i	−0.904702	+	0.426045i	$0.859907\pi$
$110$			2.98753i			0.284850i
$111$		0			0
$112$		−	15.9176i		−	1.50408i
$113$	−1.36552			−0.128457			−0.0642287	−	0.997935i	$-0.520459\pi$
$113$	−1.36552			−0.128457			−0.0642287	+	0.997935i	$0.520459\pi$
$114$		0			0
$115$			10.7925i			1.00641i
$116$	0.942868			0.0875431
$117$		0			0
$118$	−7.69941			−0.708788
$119$		−	16.1933i		−	1.48444i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		−	7.39890i		−	0.669865i
$123$		0			0
$124$			0.00932567i			0.000837470i
$125$		−	11.4241i		−	1.02181i
$126$		0			0
$127$	−10.4027			−0.923093			−0.461546	−	0.887116i	$-0.652705\pi$
$127$	−10.4027			−0.923093			−0.461546	+	0.887116i	$0.652705\pi$
$128$		−	10.1058i		−	0.893231i
$129$		0			0
$130$	−2.75585	+	10.4132i	−0.241704	+	0.913299i
$131$	12.0678			1.05437			0.527183	−	0.849752i	$-0.323248\pi$
$131$	12.0678			1.05437			0.527183	+	0.849752i	$0.323248\pi$
$132$		0			0
$133$	13.5432			1.17434
$134$	−13.8381			−1.19543
$135$		0			0
$136$		−	11.0403i		−	0.946694i
$137$			3.96114i			0.338423i	0.985580	+	0.169211i	$0.0541221\pi$
$137$			3.96114i			0.338423i	−0.985580	+	0.169211i	$0.945878\pi$
$138$		0			0
$139$	18.2437			1.54741			0.773703	−	0.633548i	$-0.218402\pi$
$139$	18.2437			1.54741			0.773703	+	0.633548i	$0.218402\pi$
$140$	1.19553			0.101041
$141$		0			0
$142$	13.1648			1.10476
$143$	−3.48555	−	0.922450i	−0.291477	−	0.0771391i
$144$		0			0
$145$		−	16.0604i		−	1.33374i
$146$	8.82089			0.730022
$147$		0			0
$148$		−	0.470045i		−	0.0386375i
$149$		−	13.8984i		−	1.13860i	−0.822131	−	0.569299i	$-0.807215\pi$
$149$		−	13.8984i		−	1.13860i	0.822131	−	0.569299i	$-0.192785\pi$
$150$		0			0
$151$			2.29739i			0.186959i	0.995621	+	0.0934795i	$0.0297990\pi$
$151$			2.29739i			0.186959i	−0.995621	+	0.0934795i	$0.970201\pi$
$152$	9.23344			0.748931
$153$		0			0
$154$		−	5.84291i		−	0.470835i
$155$	0.158849			0.0127591
$156$		0			0
$157$	−15.6293			−1.24736			−0.623678	−	0.781682i	$-0.714362\pi$
$157$	−15.6293			−1.24736			−0.623678	+	0.781682i	$0.714362\pi$
$158$			13.8468i			1.10160i
$159$		0			0
$160$	1.58107			0.124995
$161$		−	21.1076i		−	1.66351i
$162$		0			0
$163$		−	1.25583i		−	0.0983643i	−0.998790	−	0.0491822i	$-0.984339\pi$
$163$		−	1.25583i		−	0.0983643i	0.998790	−	0.0491822i	$-0.0156615\pi$
$164$			0.534226i			0.0417160i
$165$		0			0
$166$	−10.6840			−0.829243
$167$			17.3823i			1.34509i	0.740058	+	0.672543i	$0.234798\pi$
$167$			17.3823i			1.34509i	−0.740058	+	0.672543i	$0.765202\pi$
$168$		0			0
$169$	−11.2982	−	6.43050i	−0.869090	−	0.494654i
$170$	−11.3279			−0.868812
$171$		0			0
$172$	−0.912375			−0.0695679
$173$	−12.0899			−0.919178			−0.459589	−	0.888132i	$-0.652003\pi$
$173$	−12.0899			−0.919178			−0.459589	+	0.888132i	$0.652003\pi$
$174$		0			0
$175$			0.989418i			0.0747930i
$176$		−	3.72717i		−	0.280946i
$177$		0			0
$178$	−3.93904			−0.295244
$179$	−5.13094			−0.383504			−0.191752	−	0.981443i	$-0.561417\pi$
$179$	−5.13094			−0.383504			−0.191752	+	0.981443i	$0.561417\pi$
$180$		0			0
$181$	23.3170			1.73314			0.866568	−	0.499059i	$-0.166321\pi$
$181$	23.3170			1.73314			0.866568	+	0.499059i	$0.166321\pi$
$182$	5.38979	−	20.3658i	0.399518	−	1.50961i
$183$		0			0
$184$		−	14.3907i		−	1.06090i
$185$	−8.00653			−0.588652
$186$		0			0
$187$		−	3.79173i		−	0.277279i
$188$			1.49141i			0.108772i
$189$		0			0
$190$		−	9.47402i		−	0.687318i
$191$	−11.1231			−0.804838			−0.402419	−	0.915456i	$-0.631831\pi$
$191$	−11.1231			−0.804838			−0.402419	+	0.915456i	$0.631831\pi$
$192$		0			0
$193$		−	23.1664i		−	1.66756i	−0.552099	−	0.833778i	$-0.686173\pi$
$193$		−	23.1664i		−	1.66756i	0.552099	−	0.833778i	$-0.313827\pi$
$194$	−5.18909			−0.372555
$195$		0			0
$196$	−1.44079			−0.102914
$197$		−	9.18435i		−	0.654358i	−0.944962	−	0.327179i	$-0.893902\pi$
$197$		−	9.18435i		−	0.654358i	0.944962	−	0.327179i	$-0.106098\pi$
$198$		0			0
$199$	−6.38937			−0.452930			−0.226465	−	0.974019i	$-0.572717\pi$
$199$	−6.38937			−0.452930			−0.226465	+	0.974019i	$0.572717\pi$
$200$			0.674563i			0.0476988i
$201$		0			0
$202$			17.5999i			1.23833i
$203$			31.4103i			2.20457i
$204$		0			0
$205$	9.09975			0.635554
$206$			23.2550i			1.62025i
$207$		0			0
$208$	3.43813	−	12.9913i	0.238391	−	0.900782i
$209$	3.17118			0.219355
$210$		0			0
$211$	22.4436			1.54508			0.772541	−	0.634965i	$-0.218985\pi$
$211$	22.4436			1.54508			0.772541	+	0.634965i	$0.218985\pi$
$212$	−0.655431			−0.0450152
$213$		0			0
$214$			17.3505i			1.18606i
$215$			15.5410i			1.05989i
$216$		0			0
$217$	−0.310671			−0.0210897
$218$	−12.1711			−0.824328
$219$		0			0
$220$	0.279937			0.0188734
$221$	3.49768	−	13.2163i	0.235279	−	0.889023i
$222$		0			0
$223$		−	16.5344i		−	1.10722i	−0.832775	−	0.553611i	$-0.813249\pi$
$223$		−	16.5344i		−	1.10722i	0.832775	−	0.553611i	$-0.186751\pi$
$224$	−3.09220			−0.206606
$225$		0			0
$226$		−	1.86822i		−	0.124272i
$227$			19.9826i			1.32630i	0.748489	+	0.663148i	$0.230780\pi$
$227$			19.9826i			1.32630i	−0.748489	+	0.663148i	$0.769220\pi$
$228$		0			0
$229$		−	15.8459i		−	1.04713i	−0.851986	−	0.523564i	$-0.824602\pi$
$229$		−	15.8459i		−	1.04713i	0.851986	−	0.523564i	$-0.175398\pi$
$230$	−14.7657			−0.973620
$231$		0			0
$232$			21.4148i			1.40595i
$233$	15.9195			1.04292			0.521460	−	0.853276i	$-0.325388\pi$
$233$	15.9195			1.04292			0.521460	+	0.853276i	$0.325388\pi$
$234$		0			0
$235$	25.4040			1.65717
$236$			0.721449i			0.0469623i
$237$		0			0
$238$	22.1547			1.43608
$239$		−	16.9898i		−	1.09898i	−0.835500	−	0.549491i	$-0.814822\pi$
$239$		−	16.9898i		−	1.09898i	0.835500	−	0.549491i	$-0.185178\pi$
$240$		0			0
$241$			21.9883i			1.41639i	0.706015	+	0.708197i	$0.250491\pi$
$241$			21.9883i			1.41639i	−0.706015	+	0.708197i	$0.749509\pi$
$242$		−	1.36814i		−	0.0879473i
$243$		0			0
$244$	−0.693290			−0.0443834
$245$			24.5418i			1.56792i
$246$		0			0
$247$	11.0533	+	2.92526i	0.703307	+	0.186130i
$248$	−0.211809			−0.0134499
$249$		0			0
$250$	15.6298			0.988516
$251$	11.9461			0.754029			0.377015	−	0.926207i	$-0.376951\pi$
$251$	11.9461			0.754029			0.377015	+	0.926207i	$0.376951\pi$
$252$		0			0
$253$		−	4.94243i		−	0.310728i
$254$		−	14.2324i		−	0.893019i
$255$		0			0
$256$	−3.06382			−0.191489
$257$	28.3671			1.76949			0.884746	−	0.466074i	$-0.154332\pi$
$257$	28.3671			1.76949			0.884746	+	0.466074i	$0.154332\pi$
$258$		0			0
$259$	15.6589			0.972995
$260$	0.975737	+	0.258228i	0.0605126	+	0.0160146i
$261$		0			0
$262$			16.5104i			1.02001i
$263$	1.37798			0.0849699			0.0424849	−	0.999097i	$-0.486473\pi$
$263$	1.37798			0.0849699			0.0424849	+	0.999097i	$0.486473\pi$
$264$		0			0
$265$			11.1643i			0.685818i
$266$			18.5289i			1.13608i
$267$		0			0
$268$			1.29665i			0.0792056i
$269$	14.6274			0.891847			0.445923	−	0.895071i	$-0.352875\pi$
$269$	14.6274			0.891847			0.445923	+	0.895071i	$0.352875\pi$
$270$		0			0
$271$		−	13.3040i		−	0.808158i	−0.914724	−	0.404079i	$-0.867592\pi$
$271$		−	13.3040i		−	0.808158i	0.914724	−	0.404079i	$-0.132408\pi$
$272$	14.1324			0.856904
$273$		0			0
$274$	−5.41938			−0.327397
$275$			0.231676i			0.0139706i
$276$		0			0
$277$	−22.1581			−1.33135			−0.665676	−	0.746241i	$-0.731857\pi$
$277$	−22.1581			−1.33135			−0.665676	+	0.746241i	$0.731857\pi$
$278$			24.9598i			1.49699i
$279$		0			0
$280$			27.1534i			1.62273i
$281$		−	2.12589i		−	0.126820i	−0.997988	−	0.0634100i	$-0.979802\pi$
$281$		−	2.12589i		−	0.126820i	0.997988	−	0.0634100i	$-0.0201976\pi$
$282$		0			0
$283$	31.1036			1.84892			0.924458	−	0.381283i	$-0.124518\pi$
$283$	31.1036			1.84892			0.924458	+	0.381283i	$0.124518\pi$
$284$		−	1.23356i		−	0.0731984i
$285$		0			0
$286$	1.26204	−	4.76872i	0.0746259	−	0.281980i
$287$	−17.7970			−1.05052
$288$		0			0
$289$	−2.62279			−0.154282
$290$	21.9728			1.29029
$291$		0			0
$292$		−	0.826533i		−	0.0483692i
$293$			16.9009i			0.987363i	0.869643	+	0.493682i	$0.164349\pi$
$293$			16.9009i			0.987363i	−0.869643	+	0.493682i	$0.835651\pi$
$294$		0			0
$295$	12.2888			0.715483
$296$	10.6759			0.620523
$297$		0			0
$298$	19.0149			1.10150
$299$	4.55914	−	17.2271i	0.263662	−	0.996269i
$300$		0			0
$301$		−	30.3945i		−	1.75191i
$302$	−3.14315			−0.180868
$303$		0			0
$304$			11.8195i			0.677897i
$305$			11.8092i			0.676192i
$306$		0			0
$307$			24.6110i			1.40462i	0.711869	+	0.702312i	$0.247849\pi$
$307$			24.6110i			1.40462i	−0.711869	+	0.702312i	$0.752151\pi$
$308$	−0.547491			−0.0311962
$309$		0			0
$310$			0.217328i			0.0123434i
$311$	−1.00504			−0.0569904			−0.0284952	−	0.999594i	$-0.509072\pi$
$311$	−1.00504			−0.0569904			−0.0284952	+	0.999594i	$0.509072\pi$
$312$		0			0
$313$	−13.8744			−0.784227			−0.392114	−	0.919917i	$-0.628256\pi$
$313$	−13.8744			−0.784227			−0.392114	+	0.919917i	$0.628256\pi$
$314$		−	21.3831i		−	1.20672i
$315$		0			0
$316$	1.29747			0.0729886
$317$		−	15.6405i		−	0.878460i	−0.898375	−	0.439230i	$-0.855251\pi$
$317$		−	15.6405i		−	0.878460i	0.898375	−	0.439230i	$-0.144749\pi$
$318$		0			0
$319$			7.35483i			0.411792i
$320$			18.4408i			1.03087i
$321$		0			0
$322$	28.8782			1.60932
$323$			12.0243i			0.669048i
$324$		0			0
$325$	−0.213709	+	0.807519i	−0.0118545	+	0.0447931i
$326$	1.71815			0.0951596
$327$		0			0
$328$	−12.1336			−0.669964
$329$	−49.6842			−2.73918
$330$		0			0
$331$			23.1426i			1.27203i	0.771675	+	0.636017i	$0.219419\pi$
$331$			23.1426i			1.27203i	−0.771675	+	0.636017i	$0.780581\pi$
$332$			1.00111i			0.0549433i
$333$		0			0
$334$	−23.7814			−1.30126
$335$	22.0866			1.20672
$336$		0			0
$337$	−15.1946			−0.827705			−0.413853	−	0.910344i	$-0.635817\pi$
$337$	−15.1946			−0.827705			−0.413853	+	0.910344i	$0.635817\pi$
$338$	8.79781	−	15.4575i	0.478538	−	0.840775i
$339$		0			0
$340$			1.06145i			0.0575650i
$341$	−0.0727448			−0.00393935
$342$		0			0
$343$		−	18.1030i		−	0.977472i
$344$		−	20.7223i		−	1.11727i
$345$		0			0
$346$		−	16.5407i		−	0.889231i
$347$	−16.6545			−0.894061			−0.447030	−	0.894519i	$-0.647518\pi$
$347$	−16.6545			−0.894061			−0.447030	+	0.894519i	$0.647518\pi$
$348$		0			0
$349$			29.1080i			1.55812i	0.626952	+	0.779058i	$0.284302\pi$
$349$			29.1080i			1.55812i	−0.626952	+	0.779058i	$0.715698\pi$
$350$	−1.35366			−0.0723562
$351$		0			0
$352$	−0.724050			−0.0385920
$353$		−	29.1723i		−	1.55268i	−0.630312	−	0.776342i	$-0.717073\pi$
$353$		−	29.1723i		−	1.55268i	0.630312	−	0.776342i	$-0.282927\pi$
$354$		0			0
$355$	−21.0119			−1.11520
$356$			0.369095i			0.0195620i
$357$		0			0
$358$		−	7.01983i		−	0.371010i
$359$		−	18.0809i		−	0.954271i	−0.878830	−	0.477136i	$-0.841675\pi$
$359$		−	18.0809i		−	0.954271i	0.878830	−	0.477136i	$-0.158325\pi$
$360$		0			0
$361$	8.94359			0.470715
$362$			31.9008i			1.67667i
$363$		0			0
$364$	−1.90831	−	0.505033i	−0.100023	−	0.0264709i
$365$	−14.0788			−0.736917
$366$		0			0
$367$	18.1338			0.946578			0.473289	−	0.880907i	$-0.343067\pi$
$367$	18.1338			0.946578			0.473289	+	0.880907i	$0.343067\pi$
$368$	18.4213			0.960275
$369$		0			0
$370$		−	10.9540i		−	0.569474i
$371$		−	21.8347i		−	1.13360i
$372$		0			0
$373$	−15.8564			−0.821012			−0.410506	−	0.911858i	$-0.634648\pi$
$373$	−15.8564			−0.821012			−0.410506	+	0.911858i	$0.634648\pi$
$374$	5.18761			0.268245
$375$		0			0
$376$	−33.8736			−1.74690
$377$	−6.78447	+	25.6357i	−0.349418	+	1.32030i
$378$		0			0
$379$		−	18.3431i		−	0.942224i	−0.882073	−	0.471112i	$-0.843853\pi$
$379$		−	18.3431i		−	0.942224i	0.882073	−	0.471112i	$-0.156147\pi$
$380$	−0.887733			−0.0455397
$381$		0			0
$382$		−	15.2179i		−	0.778617i
$383$		−	16.1019i		−	0.822768i	−0.911462	−	0.411384i	$-0.865046\pi$
$383$		−	16.1019i		−	0.822768i	0.911462	−	0.411384i	$-0.134954\pi$
$384$		0			0
$385$			9.32571i			0.475282i
$386$	31.6949			1.61323
$387$		0			0
$388$			0.486227i			0.0246844i
$389$	32.7166			1.65880			0.829400	−	0.558655i	$-0.188683\pi$
$389$	32.7166			1.65880			0.829400	+	0.558655i	$0.188683\pi$
$390$		0			0
$391$	18.7403			0.947740
$392$		−	32.7239i		−	1.65281i
$393$		0			0
$394$	12.5655			0.633039
$395$		−	22.1006i		−	1.11200i
$396$		0			0
$397$		−	3.51685i		−	0.176506i	−0.996098	−	0.0882529i	$-0.971872\pi$
$397$		−	3.51685i		−	0.176506i	0.996098	−	0.0882529i	$-0.0281284\pi$
$398$		−	8.74154i		−	0.438174i
$399$		0			0
$400$	−0.863496			−0.0431748
$401$		−	18.6231i		−	0.929992i	−0.885313	−	0.464996i	$-0.846056\pi$
$401$		−	18.6231i		−	0.929992i	0.885313	−	0.464996i	$-0.153944\pi$
$402$		0			0
$403$	−0.253556	−	0.0671034i	−0.0126305	−	0.00334266i
$404$	1.64914			0.0820480
$405$		0			0
$406$	−42.9736			−2.13275
$407$	3.66658			0.181746
$408$		0			0
$409$			17.6809i			0.874262i	0.899398	+	0.437131i	$0.144005\pi$
$409$			17.6809i			0.874262i	−0.899398	+	0.437131i	$0.855995\pi$
$410$			12.4497i			0.614848i
$411$		0			0
$412$	2.17903			0.107353
$413$	−24.0340			−1.18264
$414$		0			0
$415$	17.0525			0.837075
$416$	−2.52371	−	0.667900i	−0.123735	−	0.0327465i
$417$		0			0
$418$			4.33862i			0.212209i
$419$	1.93036			0.0943044			0.0471522	−	0.998888i	$-0.484985\pi$
$419$	1.93036			0.0943044			0.0471522	+	0.998888i	$0.484985\pi$
$420$		0			0
$421$		−	7.21742i		−	0.351756i	−0.984412	−	0.175878i	$-0.943724\pi$
$421$		−	7.21742i		−	0.351756i	0.984412	−	0.175878i	$-0.0562764\pi$
$422$			30.7060i			1.49474i
$423$		0			0
$424$		−	14.8864i		−	0.722949i
$425$	−0.878452			−0.0426112
$426$		0			0
$427$		−	23.0960i		−	1.11769i
$428$	1.62578			0.0785849
$429$		0			0
$430$	−21.2622			−1.02535
$431$			29.7876i			1.43482i	0.696653	+	0.717408i	$0.254672\pi$
$431$			29.7876i			1.43482i	−0.696653	+	0.717408i	$0.745328\pi$
$432$		0			0
$433$	25.2468			1.21329			0.606643	−	0.794975i	$-0.292516\pi$
$433$	25.2468			1.21329			0.606643	+	0.794975i	$0.292516\pi$
$434$		−	0.425041i		−	0.0204026i
$435$		0			0
$436$			1.14045i			0.0546177i
$437$			15.6733i			0.749758i
$438$		0			0
$439$	9.07110			0.432940			0.216470	−	0.976289i	$-0.430546\pi$
$439$	9.07110			0.432940			0.216470	+	0.976289i	$0.430546\pi$
$440$			6.35806i			0.303109i
$441$		0			0
$442$	18.0817	+	4.78531i	0.860059	+	0.227614i
$443$	12.0753			0.573715			0.286858	−	0.957973i	$-0.407389\pi$
$443$	12.0753			0.573715			0.286858	+	0.957973i	$0.407389\pi$
$444$		0			0
$445$	6.28700			0.298032
$446$	22.6213			1.07115
$447$		0			0
$448$		−	36.0658i		−	1.70395i
$449$			8.94438i			0.422111i	0.977474	+	0.211056i	$0.0676901\pi$
$449$			8.94438i			0.422111i	−0.977474	+	0.211056i	$0.932310\pi$
$450$		0			0
$451$	−4.16722			−0.196227
$452$	−0.175056			−0.00823394
$453$		0			0
$454$	−27.3390			−1.28308
$455$	−8.60250	+	32.5053i	−0.403292	+	1.52387i
$456$		0			0
$457$		−	28.4975i		−	1.33305i	−0.745481	−	0.666527i	$-0.767780\pi$
$457$		−	28.4975i		−	1.33305i	0.745481	−	0.666527i	$-0.232220\pi$
$458$	21.6794			1.01301
$459$		0			0
$460$			1.38357i			0.0645093i
$461$			19.2580i			0.896932i	0.893800	+	0.448466i	$0.148030\pi$
$461$			19.2580i			0.896932i	−0.893800	+	0.448466i	$0.851970\pi$
$462$		0			0
$463$		−	36.3639i		−	1.68997i	−0.534787	−	0.844987i	$-0.679608\pi$
$463$		−	36.3639i		−	1.68997i	0.534787	−	0.844987i	$-0.320392\pi$
$464$	−27.4127			−1.27260
$465$		0			0
$466$			21.7801i			1.00894i
$467$	1.70345			0.0788264			0.0394132	−	0.999223i	$-0.487451\pi$
$467$	1.70345			0.0788264			0.0394132	+	0.999223i	$0.487451\pi$
$468$		0			0
$469$	−43.1961			−1.99461
$470$			34.7562i			1.60318i
$471$		0			0
$472$	−16.3859			−0.754220
$473$		−	7.11697i		−	0.327239i
$474$		0			0
$475$		−	0.734687i		−	0.0337097i
$476$		−	2.07594i		−	0.0951505i
$477$		0			0
$478$	23.2444			1.06318
$479$		−	24.5812i		−	1.12314i	−0.827429	−	0.561571i	$-0.810197\pi$
$479$		−	24.5812i		−	1.12314i	0.827429	−	0.561571i	$-0.189803\pi$
$480$		0			0
$481$	12.7801	+	3.38224i	0.582721	+	0.154217i
$482$	−30.0831			−1.37025
$483$		0			0
$484$	−0.128197			−0.00582714
$485$	8.28216			0.376074
$486$		0			0
$487$			40.5270i			1.83646i	0.396053	+	0.918228i	$0.370380\pi$
$487$			40.5270i			1.83646i	−0.396053	+	0.918228i	$0.629620\pi$
$488$		−	15.7463i		−	0.712802i
$489$		0			0
$490$	−33.5766			−1.51683
$491$	16.8245			0.759277			0.379639	−	0.925135i	$-0.376048\pi$
$491$	16.8245			0.759277			0.379639	+	0.925135i	$0.376048\pi$
$492$		0			0
$493$	−27.8875			−1.25599
$494$	−4.00216	+	15.1225i	−0.180066	+	0.680393i
$495$		0			0
$496$		−	0.271132i		−	0.0121742i
$497$	41.0943			1.84333
$498$		0			0
$499$			28.5150i			1.27651i	0.769826	+	0.638254i	$0.220343\pi$
$499$			28.5150i			1.27651i	−0.769826	+	0.638254i	$0.779657\pi$
$500$		−	1.46454i		−	0.0654963i
$501$		0			0
$502$			16.3439i			0.729463i
$503$	−27.4200			−1.22260			−0.611299	−	0.791400i	$-0.709353\pi$
$503$	−27.4200			−1.22260			−0.611299	+	0.791400i	$0.709353\pi$
$504$		0			0
$505$		−	28.0908i		−	1.25002i
$506$	6.76193			0.300604
$507$		0			0
$508$	−1.33360			−0.0591689
$509$			25.4659i			1.12876i	0.825516	+	0.564378i	$0.190884\pi$
$509$			25.4659i			1.12876i	−0.825516	+	0.564378i	$0.809116\pi$
$510$		0			0
$511$	27.5348			1.21807
$512$		−	24.4032i		−	1.07848i
$513$		0			0
$514$			38.8101i			1.71184i
$515$		−	37.1167i		−	1.63556i
$516$		0			0
$517$	−11.6337			−0.511651
$518$			21.4235i			0.941295i
$519$		0			0
$520$	−5.86499	+	22.1614i	−0.257197	+	0.971840i
$521$	−2.38806			−0.104623			−0.0523113	−	0.998631i	$-0.516659\pi$
$521$	−2.38806			−0.104623			−0.0523113	+	0.998631i	$0.516659\pi$
$522$		0			0
$523$	45.6041			1.99413			0.997064	−	0.0765769i	$-0.0243991\pi$
$523$	45.6041			1.99413			0.997064	+	0.0765769i	$0.0243991\pi$
$524$	1.54705			0.0675833
$525$		0			0
$526$			1.88527i			0.0822015i
$527$		−	0.275829i		−	0.0120153i
$528$		0			0
$529$	1.42760			0.0620694
$530$	−15.2743			−0.663474
$531$		0			0
$532$	1.73620			0.0752736
$533$	−14.5251	−	3.84405i	−0.629151	−	0.166504i
$534$		0			0
$535$		−	27.6927i		−	1.19726i
$536$	−29.4501			−1.27205
$537$		0			0
$538$			20.0123i			0.862791i
$539$		−	11.2389i		−	0.484093i
$540$		0			0
$541$			29.4997i			1.26829i	0.773214	+	0.634145i	$0.218648\pi$
$541$			29.4997i			1.26829i	−0.773214	+	0.634145i	$0.781352\pi$
$542$	18.2017			0.781828
$543$		0			0
$544$		−	2.74540i		−	0.117708i
$545$	19.4259			0.832114
$546$		0			0
$547$	7.75142			0.331427			0.165713	−	0.986174i	$-0.447007\pi$
$547$	7.75142			0.331427			0.165713	+	0.986174i	$0.447007\pi$
$548$			0.507806i			0.0216924i
$549$		0			0
$550$	−0.316965			−0.0135154
$551$		−	23.3235i		−	0.993616i
$552$		0			0
$553$			43.2235i			1.83805i
$554$		−	30.3153i		−	1.28798i
$555$		0			0
$556$	2.33878			0.0991865
$557$		−	20.2857i		−	0.859534i	−0.902940	−	0.429767i	$-0.858596\pi$
$557$		−	20.2857i		−	0.859534i	0.902940	−	0.429767i	$-0.141404\pi$
$558$		0			0
$559$	6.56505	−	24.8066i	0.277672	−	1.04921i
$560$	−34.7585			−1.46882
$561$		0			0
$562$	2.90851			0.122688
$563$	−32.7633			−1.38081			−0.690403	−	0.723425i	$-0.742567\pi$
$563$	−32.7633			−1.38081			−0.690403	+	0.723425i	$0.742567\pi$
$564$		0			0
$565$			2.98182i			0.125446i
$566$			42.5540i			1.78868i
$567$		0			0
$568$	28.0172			1.17558
$569$	27.8002			1.16545			0.582723	−	0.812671i	$-0.301987\pi$
$569$	27.8002			1.16545			0.582723	+	0.812671i	$0.301987\pi$
$570$		0			0
$571$	−43.4425			−1.81801			−0.909007	−	0.416782i	$-0.863158\pi$
$571$	−43.4425			−1.81801			−0.909007	+	0.416782i	$0.863158\pi$
$572$	−0.446838	−	0.118255i	−0.0186832	−	0.00494450i
$573$		0			0
$574$		−	24.3487i		−	1.01630i
$575$	−1.14504			−0.0477515
$576$		0			0
$577$			25.7084i			1.07025i	0.844772	+	0.535127i	$0.179736\pi$
$577$			25.7084i			1.07025i	−0.844772	+	0.535127i	$0.820264\pi$
$578$		−	3.58834i		−	0.149255i
$579$		0			0
$580$		−	2.05889i		−	0.0854909i
$581$	−33.3507			−1.38362
$582$		0			0
$583$		−	5.11268i		−	0.211746i
$584$	18.7726			0.776815
$585$		0			0
$586$	−23.1228			−0.955195
$587$		−	1.44805i		−	0.0597673i	−0.999553	−	0.0298837i	$-0.990486\pi$
$587$		−	1.44805i		−	0.0597673i	0.999553	−	0.0298837i	$-0.00951368\pi$
$588$		0			0
$589$	0.230687			0.00950530
$590$			16.8128i			0.692173i
$591$		0			0
$592$			13.6660i			0.561668i
$593$			4.27539i			0.175569i	0.996139	+	0.0877845i	$0.0279787\pi$
$593$			4.27539i			0.175569i	−0.996139	+	0.0877845i	$0.972021\pi$
$594$		0			0
$595$	−35.3606			−1.44964
$596$		−	1.78173i		−	0.0729824i
$597$		0			0
$598$	23.5691	+	6.23754i	0.963810	+	0.255072i
$599$	−15.9807			−0.652953			−0.326476	−	0.945205i	$-0.605861\pi$
$599$	−15.9807			−0.652953			−0.326476	+	0.945205i	$0.605861\pi$
$600$		0			0
$601$	−25.2647			−1.03057			−0.515283	−	0.857020i	$-0.672313\pi$
$601$	−25.2647			−1.03057			−0.515283	+	0.857020i	$0.672313\pi$
$602$	41.5838			1.69483
$603$		0			0
$604$			0.294519i			0.0119838i
$605$			2.18365i			0.0887780i
$606$		0			0
$607$	41.6506			1.69055			0.845273	−	0.534334i	$-0.179437\pi$
$607$	41.6506			1.69055			0.845273	+	0.534334i	$0.179437\pi$
$608$	2.29610			0.0931190
$609$		0			0
$610$	−16.1566			−0.654162
$611$	−40.5500	−	10.7315i	−1.64048	−	0.434151i
$612$		0			0
$613$			16.1620i			0.652777i	0.945236	+	0.326388i	$0.105832\pi$
$613$			16.1620i			0.652777i	−0.945236	+	0.326388i	$0.894168\pi$
$614$	−33.6713			−1.35886
$615$		0			0
$616$		−	12.4349i		−	0.501015i
$617$		−	8.12561i		−	0.327125i	−0.986533	−	0.163562i	$-0.947701\pi$
$617$		−	8.12561i		−	0.327125i	0.986533	−	0.163562i	$-0.0522985\pi$
$618$		0			0
$619$		−	7.09593i		−	0.285209i	−0.989780	−	0.142605i	$-0.954452\pi$
$619$		−	7.09593i		−	0.285209i	0.989780	−	0.142605i	$-0.0455478\pi$
$620$	0.0203640			0.000817838
$621$		0			0
$622$		−	1.37503i		−	0.0551337i
$623$	−12.2959			−0.492624
$624$		0			0
$625$	−23.7879			−0.951518
$626$		−	18.9821i		−	0.758677i
$627$		0			0
$628$	−2.00363			−0.0799536
$629$			13.9027i			0.554337i
$630$		0			0
$631$		−	3.62153i		−	0.144171i	−0.997398	−	0.0720854i	$-0.977035\pi$
$631$		−	3.62153i		−	0.144171i	0.997398	−	0.0720854i	$-0.0229654\pi$
$632$			29.4688i			1.17221i
$633$		0			0
$634$	21.3984			0.849840
$635$			22.7159i			0.901453i
$636$		0			0
$637$	10.3673	−	39.1738i	0.410768	−	1.55212i
$638$	−10.0624			−0.398375
$639$		0			0
$640$	−22.0674			−0.872291
$641$	−8.81244			−0.348070			−0.174035	−	0.984739i	$-0.555681\pi$
$641$	−8.81244			−0.348070			−0.174035	+	0.984739i	$0.555681\pi$
$642$		0			0
$643$		−	47.2262i		−	1.86242i	−0.364484	−	0.931210i	$-0.618755\pi$
$643$		−	47.2262i		−	1.86242i	0.364484	−	0.931210i	$-0.381245\pi$
$644$		−	2.70594i		−	0.106629i
$645$		0			0
$646$	−16.4509			−0.647251
$647$	41.0552			1.61405			0.807023	−	0.590520i	$-0.201077\pi$
$647$	41.0552			1.61405			0.807023	+	0.590520i	$0.201077\pi$
$648$		0			0
$649$	−5.62765			−0.220905
$650$	−1.10480	−	0.292384i	−0.0433337	−	0.0114682i
$651$		0			0
$652$		−	0.160994i		−	0.00630501i
$653$	−36.9362			−1.44542			−0.722712	−	0.691149i	$-0.757105\pi$
$653$	−36.9362			−1.44542			−0.722712	+	0.691149i	$0.757105\pi$
$654$		0			0
$655$		−	26.3518i		−	1.02965i
$656$		−	15.5320i		−	0.606421i
$657$		0			0
$658$		−	67.9748i		−	2.64994i
$659$	−39.7980			−1.55031			−0.775154	−	0.631772i	$-0.782328\pi$
$659$	−39.7980			−1.55031			−0.775154	+	0.631772i	$0.782328\pi$
$660$		0			0
$661$			21.8622i			0.850342i	0.905113	+	0.425171i	$0.139786\pi$
$661$			21.8622i			0.850342i	−0.905113	+	0.425171i	$0.860214\pi$
$662$	−31.6623			−1.23059
$663$		0			0
$664$	−22.7378			−0.882396
$665$		−	29.5736i		−	1.14681i
$666$		0			0
$667$	−36.3507			−1.40751
$668$			2.22836i			0.0862180i
$669$		0			0
$670$			30.2175i			1.16740i
$671$		−	5.40800i		−	0.208774i
$672$		0			0
$673$	32.3152			1.24566			0.622830	−	0.782357i	$-0.285983\pi$
$673$	32.3152			1.24566			0.622830	+	0.782357i	$0.285983\pi$
$674$		−	20.7884i		−	0.800739i
$675$		0			0
$676$	−1.44839	−	0.824371i	−0.0557074	−	0.0317066i
$677$	21.1472			0.812753			0.406376	−	0.913706i	$-0.366792\pi$
$677$	21.1472			0.812753			0.406376	+	0.913706i	$0.366792\pi$
$678$		0			0
$679$	−16.1979			−0.621620
$680$	−24.1081			−0.924502
$681$		0			0
$682$		−	0.0995250i		−	0.00381101i
$683$		−	24.7401i		−	0.946654i	−0.880887	−	0.473327i	$-0.843053\pi$
$683$		−	24.7401i		−	0.946654i	0.880887	−	0.473327i	$-0.156947\pi$
$684$		0			0
$685$	8.64973			0.330489
$686$	24.7675			0.945626
$687$		0			0
$688$	26.5262			1.01130
$689$	4.71619	−	17.8205i	0.179673	−	0.678908i
$690$		0			0
$691$			22.7394i			0.865047i	0.901623	+	0.432523i	$0.142377\pi$
$691$			22.7394i			0.865047i	−0.901623	+	0.432523i	$0.857623\pi$
$692$	−1.54989			−0.0589180
$693$		0			0
$694$		−	22.7857i		−	0.864932i
$695$		−	39.8377i		−	1.51113i
$696$		0			0
$697$		−	15.8010i		−	0.598505i
$698$	−39.8238			−1.50735
$699$		0			0
$700$			0.126840i			0.00479412i
$701$	−23.8173			−0.899567			−0.449783	−	0.893138i	$-0.648499\pi$
$701$	−23.8173			−0.899567			−0.449783	+	0.893138i	$0.648499\pi$
$702$		0			0
$703$	−11.6274			−0.438536
$704$		−	8.44494i		−	0.318281i
$705$		0			0
$706$	39.9117			1.50210
$707$			54.9389i			2.06619i
$708$		0			0
$709$		−	15.8902i		−	0.596770i	−0.954446	−	0.298385i	$-0.903552\pi$
$709$		−	15.8902i		−	0.596770i	0.954446	−	0.298385i	$-0.0964480\pi$
$710$		−	28.7472i		−	1.07886i
$711$		0			0
$712$	−8.38305			−0.314168
$713$		−	0.359536i		−	0.0134647i
$714$		0			0
$715$	−2.01431	+	7.61123i	−0.0753308	+	0.284644i
$716$	−0.657771			−0.0245821
$717$		0			0
$718$	24.7371			0.923181
$719$	4.64051			0.173062			0.0865309	−	0.996249i	$-0.472422\pi$
$719$	4.64051			0.173062			0.0865309	+	0.996249i	$0.472422\pi$
$720$		0			0
$721$			72.5914i			2.70345i
$722$			12.2361i			0.455379i
$723$		0			0
$724$	2.98917			0.111091
$725$	1.70394			0.0632826
$726$		0			0
$727$	14.5583			0.539938			0.269969	−	0.962869i	$-0.412987\pi$
$727$	14.5583			0.539938			0.269969	+	0.962869i	$0.412987\pi$
$728$	11.4705	−	43.3424i	0.425127	−	1.60638i
$729$		0			0
$730$		−	19.2617i		−	0.712908i
$731$	26.9856			0.998100
$732$		0			0
$733$			34.2475i			1.26496i	0.774576	+	0.632480i	$0.217963\pi$
$733$			34.2475i			1.26496i	−0.774576	+	0.632480i	$0.782037\pi$
$734$			24.8096i			0.915738i
$735$		0			0
$736$		−	3.57856i		−	0.131908i
$737$	−10.1145			−0.372573
$738$		0			0
$739$			30.5325i			1.12316i	0.827424	+	0.561578i	$0.189806\pi$
$739$			30.5325i			1.12316i	−0.827424	+	0.561578i	$0.810194\pi$
$740$	−1.02641			−0.0377317
$741$		0			0
$742$	29.8729			1.09667
$743$		−	2.86476i		−	0.105098i	−0.998618	−	0.0525490i	$-0.983265\pi$
$743$		−	2.86476i		−	0.105098i	0.998618	−	0.0525490i	$-0.0167346\pi$
$744$		0			0
$745$	−30.3491			−1.11191
$746$		−	21.6937i		−	0.794263i
$747$		0			0
$748$		−	0.486088i		−	0.0177732i
$749$			54.1604i			1.97898i
$750$		0			0
$751$	−38.1810			−1.39324			−0.696621	−	0.717439i	$-0.745314\pi$
$751$	−38.1810			−1.39324			−0.696621	+	0.717439i	$0.745314\pi$
$752$		−	43.3609i		−	1.58121i
$753$		0			0
$754$	−35.0731	−	9.28209i	−1.27729	−	0.338034i
$755$	5.01670			0.182576
$756$		0			0
$757$	−38.9456			−1.41550			−0.707750	−	0.706463i	$-0.750290\pi$
$757$	−38.9456			−1.41550			−0.707750	+	0.706463i	$0.750290\pi$
$758$	25.0960			0.911526
$759$		0			0
$760$		−	20.1626i		−	0.731374i
$761$		−	26.9359i		−	0.976427i	−0.872724	−	0.488214i	$-0.837649\pi$
$761$		−	26.9359i		−	0.976427i	0.872724	−	0.488214i	$-0.162351\pi$
$762$		0			0
$763$	−37.9925			−1.37542
$764$	−1.42595			−0.0515889
$765$		0			0
$766$	22.0296			0.795962
$767$	−19.6155	−	5.19123i	−0.708274	−	0.187444i
$768$		0			0
$769$			3.83575i			0.138321i	0.997606	+	0.0691603i	$0.0220320\pi$
$769$			3.83575i			0.138321i	−0.997606	+	0.0691603i	$0.977968\pi$
$770$	−12.7589			−0.459798
$771$		0			0
$772$		−	2.96987i		−	0.106888i
$773$		−	21.7215i		−	0.781267i	−0.920546	−	0.390633i	$-0.872256\pi$
$773$		−	21.7215i		−	0.781267i	0.920546	−	0.390633i	$-0.127744\pi$
$774$		0			0
$775$			0.0168532i			0.000605385i
$776$	−11.0434			−0.396435
$777$		0			0
$778$			44.7609i			1.60476i
$779$	13.2150			0.473478
$780$		0			0
$781$	9.62239			0.344316
$782$			25.6394i			0.916863i
$783$		0			0
$784$	41.8893			1.49605
$785$			34.1290i			1.21811i
$786$		0			0
$787$		−	32.8705i		−	1.17171i	−0.810417	−	0.585853i	$-0.800760\pi$
$787$		−	32.8705i		−	1.17171i	0.810417	−	0.585853i	$-0.199240\pi$
$788$		−	1.17741i		−	0.0419434i
$789$		0			0
$790$	30.2366			1.07577
$791$		−	5.83174i		−	0.207353i
$792$		0			0
$793$	4.98861	−	18.8499i	0.177151	−	0.669379i
$794$	4.81154			0.170755
$795$		0			0
$796$	−0.819098			−0.0290322
$797$	−2.27867			−0.0807147			−0.0403573	−	0.999185i	$-0.512850\pi$
$797$	−2.27867			−0.0807147			−0.0403573	+	0.999185i	$0.512850\pi$
$798$		0			0
$799$		−	44.1120i		−	1.56057i
$800$			0.167745i			0.00593068i
$801$		0			0
$802$	25.4789			0.899693
$803$	6.44736			0.227523
$804$		0			0
$805$	−46.0917			−1.62452
$806$	0.0918068	−	0.346900i	0.00323376	−	0.0122190i
$807$		0			0
$808$			37.4561i			1.31770i
$809$	−25.3935			−0.892786			−0.446393	−	0.894837i	$-0.647292\pi$
$809$	−25.3935			−0.892786			−0.446393	+	0.894837i	$0.647292\pi$
$810$		0			0
$811$		−	32.7178i		−	1.14888i	−0.818548	−	0.574439i	$-0.805220\pi$
$811$		−	32.7178i		−	1.14888i	0.818548	−	0.574439i	$-0.194780\pi$
$812$			4.02671i			0.141310i
$813$		0			0
$814$			5.01639i			0.175824i
$815$	−2.74230			−0.0960584
$816$		0			0
$817$			22.5692i			0.789597i
$818$	−24.1899			−0.845779
$819$		0			0
$820$	1.16656			0.0407381
$821$			31.2361i			1.09015i	0.838388	+	0.545073i	$0.183498\pi$
$821$			31.2361i			1.09015i	−0.838388	+	0.545073i	$0.816502\pi$
$822$		0			0
$823$	−8.94424			−0.311777			−0.155888	−	0.987775i	$-0.549824\pi$
$823$	−8.94424			−0.311777			−0.155888	+	0.987775i	$0.549824\pi$
$824$			49.4912i			1.72411i
$825$		0			0
$826$		−	32.8819i		−	1.14411i
$827$		−	28.6667i		−	0.996838i	−0.866936	−	0.498419i	$-0.833914\pi$
$827$		−	28.6667i		−	0.996838i	0.866936	−	0.498419i	$-0.166086\pi$
$828$		0			0
$829$	−39.5515			−1.37368			−0.686839	−	0.726809i	$-0.741002\pi$
$829$	−39.5515			−1.37368			−0.686839	+	0.726809i	$0.741002\pi$
$830$			23.3302i			0.809803i
$831$		0			0
$832$	7.79004	−	29.4353i	0.270071	−	1.02049i
$833$	42.6148			1.47652
$834$		0			0
$835$	37.9569			1.31355
$836$	0.406536			0.0140604
$837$		0			0
$838$			2.64100i			0.0912320i
$839$			45.1833i			1.55990i	0.625840	+	0.779951i	$0.284756\pi$
$839$			45.1833i			1.55990i	−0.625840	+	0.779951i	$0.715244\pi$
$840$		0			0
$841$	25.0936			0.865296
$842$	9.87443			0.340295
$843$		0			0
$844$	2.87720			0.0990375
$845$	−14.0420	+	24.6712i	−0.483058	+	0.848717i
$846$		0			0
$847$		−	4.27070i		−	0.146743i
$848$	19.0558			0.654380
$849$		0			0
$850$		−	1.20184i		−	0.0412229i
$851$			18.1218i			0.621208i
$852$		0			0
$853$			13.2579i			0.453943i	0.973901	+	0.226972i	$0.0728825\pi$
$853$			13.2579i			0.453943i	−0.973901	+	0.226972i	$0.927118\pi$
$854$	31.5985			1.08128
$855$		0			0
$856$			36.9254i			1.26208i
$857$	38.9268			1.32971			0.664857	−	0.746971i	$-0.268493\pi$
$857$	38.9268			1.32971			0.664857	+	0.746971i	$0.268493\pi$
$858$		0			0
$859$	−21.2750			−0.725893			−0.362947	−	0.931810i	$-0.618229\pi$
$859$	−21.2750			−0.725893			−0.362947	+	0.931810i	$0.618229\pi$
$860$			1.99231i			0.0679371i
$861$		0			0
$862$	−40.7535			−1.38807
$863$			8.94497i			0.304490i	0.988343	+	0.152245i	$0.0486504\pi$
$863$			8.94497i			0.304490i	−0.988343	+	0.152245i	$0.951350\pi$
$864$		0			0
$865$			26.4001i			0.897630i
$866$			34.5412i			1.17376i
$867$		0			0
$868$	−0.0398271			−0.00135182
$869$			10.1209i			0.343329i
$870$		0			0
$871$	−35.2547	−	9.33014i	−1.19456	−	0.316140i
$872$	−25.9024			−0.877166
$873$		0			0
$874$	−21.4433			−0.725331
$875$	48.7891			1.64937
$876$		0			0
$877$			11.5696i			0.390679i	0.980736	+	0.195340i	$0.0625809\pi$
$877$			11.5696i			0.390679i	−0.980736	+	0.195340i	$0.937419\pi$
$878$			12.4105i			0.418835i
$879$		0			0
$880$	−8.13884			−0.274360
$881$	−13.2958			−0.447947			−0.223974	−	0.974595i	$-0.571903\pi$
$881$	−13.2958			−0.447947			−0.223974	+	0.974595i	$0.571903\pi$
$882$		0			0
$883$	18.7422			0.630726			0.315363	−	0.948971i	$-0.397874\pi$
$883$	18.7422			0.630726			0.315363	+	0.948971i	$0.397874\pi$
$884$	0.448392	−	1.69429i	0.0150811	−	0.0569851i
$885$		0			0
$886$			16.5207i			0.555024i
$887$	−41.6306			−1.39782			−0.698910	−	0.715210i	$-0.746331\pi$
$887$	−41.6306			−1.39782			−0.698910	+	0.715210i	$0.746331\pi$
$888$		0			0
$889$		−	44.4269i		−	1.49003i
$890$			8.60148i			0.288322i
$891$		0			0
$892$		−	2.11966i		−	0.0709713i
$893$	36.8927			1.23457
$894$		0			0
$895$			11.2042i			0.374514i
$896$	43.1586			1.44183
$897$		0			0
$898$	−12.2372			−0.408359
$899$			0.535026i			0.0178441i
$900$		0			0
$901$	19.3859			0.645838
$902$		−	5.70134i		−	0.189834i
$903$		0			0
$904$		−	3.97595i		−	0.132238i
$905$		−	50.9161i		−	1.69251i
$906$		0			0
$907$	−37.0540			−1.23036			−0.615179	−	0.788387i	$-0.710916\pi$
$907$	−37.0540			−1.23036			−0.615179	+	0.788387i	$0.710916\pi$
$908$			2.56172i			0.0850136i
$909$		0			0
$910$	−44.4717	−	11.7694i	−1.47422	−	0.390152i
$911$	43.8739			1.45361			0.726804	−	0.686845i	$-0.241005\pi$
$911$	43.8739			1.45361			0.726804	+	0.686845i	$0.241005\pi$
$912$		0			0
$913$	−7.80918			−0.258446
$914$	38.9885			1.28962
$915$		0			0
$916$		−	2.03140i		−	0.0671194i
$917$			51.5378i			1.70193i
$918$		0			0
$919$	−30.1561			−0.994759			−0.497380	−	0.867533i	$-0.665704\pi$
$919$	−30.1561			−0.994759			−0.497380	+	0.867533i	$0.665704\pi$
$920$	−31.4243			−1.03603
$921$		0			0
$922$	−26.3475			−0.867710
$923$	33.5393	+	8.87617i	1.10396	+	0.292163i
$924$		0			0
$925$		−	0.849459i		−	0.0279300i
$926$	49.7508			1.63491
$927$		0			0
$928$			5.32527i			0.174810i
$929$			29.2684i			0.960264i	0.877196	+	0.480132i	$0.159411\pi$
$929$			29.2684i			0.960264i	−0.877196	+	0.480132i	$0.840589\pi$
$930$		0			0
$931$			35.6406i			1.16807i
$932$	2.04083			0.0668496
$933$		0			0
$934$			2.33056i			0.0762583i
$935$	−8.27981			−0.270779
$936$		0			0
$937$	−14.7979			−0.483427			−0.241713	−	0.970348i	$-0.577709\pi$
$937$	−14.7979			−0.483427			−0.241713	+	0.970348i	$0.577709\pi$
$938$		−	59.0983i		−	1.92963i
$939$		0			0
$940$	3.25672			0.106222
$941$		−	17.5021i		−	0.570553i	−0.958445	−	0.285276i	$-0.907915\pi$
$941$		−	17.5021i		−	0.570553i	0.958445	−	0.285276i	$-0.0920854\pi$
$942$		0			0
$943$		−	20.5962i		−	0.670704i
$944$		−	20.9752i		−	0.682686i
$945$		0			0
$946$	9.73700			0.316577
$947$			32.5998i			1.05935i	0.848201	+	0.529675i	$0.177686\pi$
$947$			32.5998i			1.05935i	−0.848201	+	0.529675i	$0.822314\pi$
$948$		0			0
$949$	22.4726	+	5.94737i	0.729493	+	0.193060i
$950$	1.00515			0.0326115
$951$		0			0
$952$	47.1496			1.52813
$953$	−44.8368			−1.45241			−0.726204	−	0.687479i	$-0.758717\pi$
$953$	−44.8368			−1.45241			−0.726204	+	0.687479i	$0.758717\pi$
$954$		0			0
$955$			24.2889i			0.785971i
$956$		−	2.17805i		−	0.0704431i
$957$		0			0
$958$	33.6304			1.08655
$959$	−16.9168			−0.546273
$960$		0			0
$961$	30.9947			0.999829
$962$	−4.62737	+	17.4849i	−0.149192	+	0.563736i
$963$		0			0
$964$			2.81884i			0.0907887i
$965$	−50.5874			−1.62847
$966$		0			0
$967$			21.7225i			0.698550i	0.937020	+	0.349275i	$0.113572\pi$
$967$			21.7225i			0.698550i	−0.937020	+	0.349275i	$0.886428\pi$
$968$		−	2.91167i		−	0.0935846i
$969$		0			0
$970$			11.3311i			0.363821i
$971$	28.7521			0.922699			0.461349	−	0.887219i	$-0.347365\pi$
$971$	28.7521			0.922699			0.461349	+	0.887219i	$0.347365\pi$
$972$		0			0
$973$			77.9132i			2.49778i
$974$	−55.4466			−1.77662
$975$		0			0
$976$	20.1566			0.645196
$977$		−	13.7868i		−	0.441077i	−0.975378	−	0.220539i	$-0.929218\pi$
$977$		−	13.7868i		−	0.441077i	0.975378	−	0.220539i	$-0.0707815\pi$
$978$		0			0
$979$	−2.87912			−0.0920172
$980$			3.14619i			0.100501i
$981$		0			0
$982$			23.0182i			0.734540i
$983$		−	32.5478i		−	1.03811i	−0.854740	−	0.519056i	$-0.826284\pi$
$983$		−	32.5478i		−	1.03811i	0.854740	−	0.519056i	$-0.173716\pi$
$984$		0			0
$985$	−20.0554			−0.639018
$986$		−	38.1540i		−	1.21507i
$987$		0			0
$988$	1.41700	+	0.375009i	0.0450809	+	0.0119306i
$989$	35.1751			1.11850
$990$		0			0
$991$	−3.97937			−0.126409			−0.0632044	−	0.998001i	$-0.520132\pi$
$991$	−3.97937			−0.126409			−0.0632044	+	0.998001i	$0.520132\pi$
$992$	−0.0526709			−0.00167230
$993$		0			0
$994$			56.2227i			1.78328i
$995$			13.9521i			0.442312i
$996$		0			0
$997$	−24.9767			−0.791021			−0.395511	−	0.918461i	$-0.629432\pi$
$997$	−24.9767			−0.791021			−0.395511	+	0.918461i	$0.629432\pi$
$998$	−39.0125			−1.23492
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.b.c.298.10		14
3.2	odd	2		429.2.b.b.298.5	✓	14
13.12	even	2	inner	1287.2.b.c.298.5		14
39.5	even	4		5577.2.a.y.1.2		7
39.8	even	4		5577.2.a.x.1.6		7
39.38	odd	2		429.2.b.b.298.10	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.b.b.298.5	✓	14	3.2	odd	2
429.2.b.b.298.10	yes	14	39.38	odd	2
1287.2.b.c.298.5		14	13.12	even	2	inner
1287.2.b.c.298.10		14	1.1	even	1	trivial
5577.2.a.x.1.6		7	39.8	even	4
5577.2.a.y.1.2		7	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.36814i q^{2} +0.128197 q^{4} -2.18365i q^{5} +4.27070i q^{7} +2.91167i q^{8} +O(q^{10})\)	\(q+1.36814i q^{2} +0.128197 q^{4} -2.18365i q^{5} +4.27070i q^{7} +2.91167i q^{8} +2.98753 q^{10} +1.00000i q^{11} +(-0.922450 + 3.48555i) q^{13} -5.84291 q^{14} -3.72717 q^{16} -3.79173 q^{17} -3.17118i q^{19} -0.279937i q^{20} -1.36814 q^{22} -4.94243 q^{23} +0.231676 q^{25} +(-4.76872 - 1.26204i) q^{26} +0.547491i q^{28} +7.35483 q^{29} +0.0727448i q^{31} +0.724050i q^{32} -5.18761i q^{34} +9.32571 q^{35} -3.66658i q^{37} +4.33862 q^{38} +6.35806 q^{40} +4.16722i q^{41} -7.11697 q^{43} +0.128197i q^{44} -6.76193i q^{46} +11.6337i q^{47} -11.2389 q^{49} +0.316965i q^{50} +(-0.118255 + 0.446838i) q^{52} -5.11268 q^{53} +2.18365 q^{55} -12.4349 q^{56} +10.0624i q^{58} +5.62765i q^{59} -5.40800 q^{61} -0.0995250 q^{62} -8.44494 q^{64} +(7.61123 + 2.01431i) q^{65} +10.1145i q^{67} -0.486088 q^{68} +12.7589i q^{70} -9.62239i q^{71} -6.44736i q^{73} +5.01639 q^{74} -0.406536i q^{76} -4.27070 q^{77} +10.1209 q^{79} +8.13884i q^{80} -5.70134 q^{82} +7.80918i q^{83} +8.27981i q^{85} -9.73700i q^{86} -2.91167 q^{88} +2.87912i q^{89} +(-14.8858 - 3.93951i) q^{91} -0.633605 q^{92} -15.9166 q^{94} -6.92475 q^{95} +3.79281i q^{97} -15.3764i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{4}+O(q^{10}) \) 14 * q - 18 * q^4	\( 14 q - 18 q^{4} - 16 q^{14} + 34 q^{16} - 4 q^{17} + 6 q^{22} + 8 q^{23} - 26 q^{25} + 6 q^{26} + 24 q^{29} + 8 q^{35} + 32 q^{38} - 20 q^{40} + 32 q^{43} - 46 q^{49} + 4 q^{52} - 20 q^{53} + 12 q^{55} + 32 q^{56} - 20 q^{61} - 72 q^{62} - 58 q^{64} - 12 q^{65} + 20 q^{68} + 12 q^{77} + 12 q^{79} + 20 q^{82} - 30 q^{88} + 16 q^{91} + 24 q^{92} + 64 q^{94} + 36 q^{95}+O(q^{100}) \) 14 * q - 18 * q^4 - 16 * q^14 + 34 * q^16 - 4 * q^17 + 6 * q^22 + 8 * q^23 - 26 * q^25 + 6 * q^26 + 24 * q^29 + 8 * q^35 + 32 * q^38 - 20 * q^40 + 32 * q^43 - 46 * q^49 + 4 * q^52 - 20 * q^53 + 12 * q^55 + 32 * q^56 - 20 * q^61 - 72 * q^62 - 58 * q^64 - 12 * q^65 + 20 * q^68 + 12 * q^77 + 12 * q^79 + 20 * q^82 - 30 * q^88 + 16 * q^91 + 24 * q^92 + 64 * q^94 + 36 * q^95

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