LMFDB - Embedded newform 1287.2.b.c.298.6

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			298.6
Root			$-0.584778i$ of defining polynomial
Character	$\chi$	$=$	1287.298
Dual form			1287.2.b.c.298.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.584778i q^{2} +1.65803 q^{4} -1.95350i q^{5} +1.51078i q^{7} -2.13914i q^{8} +O(q^{10})$	\(q-0.584778i q^{2} +1.65803 q^{4} -1.95350i q^{5} +1.51078i q^{7} -2.13914i q^{8} -1.14237 q^{10} -1.00000i q^{11} +(-3.49467 - 0.887280i) q^{13} +0.883474 q^{14} +2.06515 q^{16} +3.44198 q^{17} -5.09587i q^{19} -3.23897i q^{20} -0.584778 q^{22} -0.701611 q^{23} +1.18383 q^{25} +(-0.518862 + 2.04361i) q^{26} +2.50493i q^{28} -5.04937 q^{29} -7.26957i q^{31} -5.48593i q^{32} -2.01280i q^{34} +2.95132 q^{35} -2.08160i q^{37} -2.97995 q^{38} -4.17882 q^{40} +1.03658i q^{41} +5.48659 q^{43} -1.65803i q^{44} +0.410287i q^{46} -3.75225i q^{47} +4.71753 q^{49} -0.692276i q^{50} +(-5.79429 - 1.47114i) q^{52} +0.502747 q^{53} -1.95350 q^{55} +3.23178 q^{56} +2.95276i q^{58} +2.65023i q^{59} -5.38195 q^{61} -4.25109 q^{62} +0.922236 q^{64} +(-1.73330 + 6.82685i) q^{65} +8.47197i q^{67} +5.70692 q^{68} -1.72587i q^{70} -2.31533i q^{71} +2.21094i q^{73} -1.21727 q^{74} -8.44913i q^{76} +1.51078 q^{77} -5.54589 q^{79} -4.03427i q^{80} +0.606168 q^{82} +14.5137i q^{83} -6.72392i q^{85} -3.20844i q^{86} -2.13914 q^{88} +1.80966i q^{89} +(1.34049 - 5.27970i) q^{91} -1.16330 q^{92} -2.19424 q^{94} -9.95480 q^{95} -14.5085i q^{97} -2.75871i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.584778i		−	0.413501i	−0.978394	−	0.206750i	$-0.933711\pi$
$2$		−	0.584778i		−	0.413501i	0.978394	−	0.206750i	$-0.0662888\pi$
$3$		0			0
$3$		0			0
$4$	1.65803			0.829017
$5$		−	1.95350i		−	0.873633i	−0.899551	−	0.436817i	$-0.856106\pi$
$5$		−	1.95350i		−	0.873633i	0.899551	−	0.436817i	$-0.143894\pi$
$6$		0			0
$7$			1.51078i			0.571023i	0.958375	+	0.285511i	$0.0921634\pi$
$7$			1.51078i			0.571023i	−0.958375	+	0.285511i	$0.907837\pi$
$8$		−	2.13914i		−	0.756300i
$9$		0			0
$10$	−1.14237			−0.361248
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.49467	−	0.887280i	−0.969248	−	0.246087i
$13$	−3.49467	−	0.887280i	−0.969248	−	0.246087i
$14$	0.883474			0.236118
$15$		0			0
$16$	2.06515			0.516286
$17$	3.44198			0.834803			0.417401	−	0.908722i	$-0.362941\pi$
$17$	3.44198			0.834803			0.417401	+	0.908722i	$0.362941\pi$
$18$		0			0
$19$		−	5.09587i		−	1.16907i	−0.811368	−	0.584536i	$-0.801276\pi$
$19$		−	5.09587i		−	1.16907i	0.811368	−	0.584536i	$-0.198724\pi$
$20$		−	3.23897i		−	0.724257i
$21$		0			0
$22$	−0.584778			−0.124675
$23$	−0.701611			−0.146296			−0.0731480	−	0.997321i	$-0.523305\pi$
$23$	−0.701611			−0.146296			−0.0731480	+	0.997321i	$0.523305\pi$
$24$		0			0
$25$	1.18383			0.236765
$26$	−0.518862	+	2.04361i	−0.101757	+	0.400785i
$27$		0			0
$28$			2.50493i			0.473388i
$29$	−5.04937			−0.937645			−0.468822	−	0.883292i	$-0.655322\pi$
$29$	−5.04937			−0.937645			−0.468822	+	0.883292i	$0.655322\pi$
$30$		0			0
$31$		−	7.26957i		−	1.30565i	−0.757507	−	0.652827i	$-0.773583\pi$
$31$		−	7.26957i		−	1.30565i	0.757507	−	0.652827i	$-0.226417\pi$
$32$		−	5.48593i		−	0.969785i
$33$		0			0
$34$		−	2.01280i		−	0.345192i
$35$	2.95132			0.498864
$36$		0			0
$37$		−	2.08160i		−	0.342213i	−0.985253	−	0.171106i	$-0.945266\pi$
$37$		−	2.08160i		−	0.342213i	0.985253	−	0.171106i	$-0.0547342\pi$
$38$	−2.97995			−0.483413
$39$		0			0
$40$	−4.17882			−0.660729
$41$			1.03658i			0.161886i	0.996719	+	0.0809430i	$0.0257932\pi$
$41$			1.03658i			0.161886i	−0.996719	+	0.0809430i	$0.974207\pi$
$42$		0			0
$43$	5.48659			0.836698			0.418349	−	0.908286i	$-0.362609\pi$
$43$	5.48659			0.836698			0.418349	+	0.908286i	$0.362609\pi$
$44$		−	1.65803i		−	0.249958i
$45$		0			0
$46$			0.410287i			0.0604935i
$47$		−	3.75225i		−	0.547322i	−0.961826	−	0.273661i	$-0.911765\pi$
$47$		−	3.75225i		−	0.547322i	0.961826	−	0.273661i	$-0.0882347\pi$
$48$		0			0
$49$	4.71753			0.673933
$50$		−	0.692276i		−	0.0979026i
$51$		0			0
$52$	−5.79429	−	1.47114i	−0.803523	−	0.204010i
$53$	0.502747			0.0690575			0.0345288	−	0.999404i	$-0.489007\pi$
$53$	0.502747			0.0690575			0.0345288	+	0.999404i	$0.489007\pi$
$54$		0			0
$55$	−1.95350			−0.263410
$56$	3.23178			0.431865
$57$		0			0
$58$			2.95276i			0.387717i
$59$			2.65023i			0.345031i	0.985007	+	0.172515i	$0.0551894\pi$
$59$			2.65023i			0.345031i	−0.985007	+	0.172515i	$0.944811\pi$
$60$		0			0
$61$	−5.38195			−0.689088			−0.344544	−	0.938770i	$-0.611967\pi$
$61$	−5.38195			−0.689088			−0.344544	+	0.938770i	$0.611967\pi$
$62$	−4.25109			−0.539889
$63$		0			0
$64$	0.922236			0.115279
$65$	−1.73330	+	6.82685i	−0.214990	+	0.846767i
$66$		0			0
$67$			8.47197i			1.03502i	0.855679	+	0.517508i	$0.173140\pi$
$67$			8.47197i			1.03502i	−0.855679	+	0.517508i	$0.826860\pi$
$68$	5.70692			0.692066
$69$		0			0
$70$		−	1.72587i		−	0.206281i
$71$		−	2.31533i		−	0.274779i	−0.990517	−	0.137390i	$-0.956129\pi$
$71$		−	2.31533i		−	0.274779i	0.990517	−	0.137390i	$-0.0438712\pi$
$72$		0			0
$73$			2.21094i			0.258771i	0.991594	+	0.129385i	$0.0413004\pi$
$73$			2.21094i			0.258771i	−0.991594	+	0.129385i	$0.958700\pi$
$74$	−1.21727			−0.141505
$75$		0			0
$76$		−	8.44913i		−	0.969181i
$77$	1.51078			0.172170
$78$		0			0
$79$	−5.54589			−0.623961			−0.311981	−	0.950088i	$-0.600992\pi$
$79$	−5.54589			−0.623961			−0.311981	+	0.950088i	$0.600992\pi$
$80$		−	4.03427i		−	0.451045i
$81$		0			0
$82$	0.606168			0.0669400
$83$			14.5137i			1.59308i	0.604585	+	0.796541i	$0.293339\pi$
$83$			14.5137i			1.59308i	−0.604585	+	0.796541i	$0.706661\pi$
$84$		0			0
$85$		−	6.72392i		−	0.729311i
$86$		−	3.20844i		−	0.345975i
$87$		0			0
$88$	−2.13914			−0.228033
$89$			1.80966i			0.191824i	0.995390	+	0.0959119i	$0.0305767\pi$
$89$			1.80966i			0.191824i	−0.995390	+	0.0959119i	$0.969423\pi$
$90$		0			0
$91$	1.34049	−	5.27970i	0.140521	−	0.553462i
$92$	−1.16330			−0.121282
$93$		0			0
$94$	−2.19424			−0.226318
$95$	−9.95480			−1.02134
$96$		0			0
$97$		−	14.5085i		−	1.47312i	−0.676372	−	0.736560i	$-0.736449\pi$
$97$		−	14.5085i		−	1.47312i	0.676372	−	0.736560i	$-0.263551\pi$
$98$		−	2.75871i		−	0.278672i
$99$		0			0
$100$	1.96282			0.196282
$101$	8.63799			0.859512			0.429756	−	0.902945i	$-0.358600\pi$
$101$	8.63799			0.859512			0.429756	+	0.902945i	$0.358600\pi$
$102$		0			0
$103$	2.90048			0.285792			0.142896	−	0.989738i	$-0.454358\pi$
$103$	2.90048			0.285792			0.142896	+	0.989738i	$0.454358\pi$
$104$	−1.89802	+	7.47559i	−0.186116	+	0.733042i
$105$		0			0
$106$		−	0.293995i		−	0.0285554i
$107$	2.29234			0.221609			0.110805	−	0.993842i	$-0.464657\pi$
$107$	2.29234			0.221609			0.110805	+	0.993842i	$0.464657\pi$
$108$		0			0
$109$		−	12.8818i		−	1.23386i	−0.787019	−	0.616929i	$-0.788377\pi$
$109$		−	12.8818i		−	1.23386i	0.787019	−	0.616929i	$-0.211623\pi$
$110$			1.14237i			0.108920i
$111$		0			0
$112$			3.11999i			0.294811i
$113$	12.6273			1.18788			0.593940	−	0.804509i	$-0.297572\pi$
$113$	12.6273			1.18788			0.593940	+	0.804509i	$0.297572\pi$
$114$		0			0
$115$			1.37060i			0.127809i
$116$	−8.37203			−0.777324
$117$		0			0
$118$	1.54980			0.142670
$119$			5.20009i			0.476691i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$			3.14725i			0.284939i
$123$		0			0
$124$		−	12.0532i		−	1.08241i
$125$		−	12.0801i		−	1.08048i
$126$		0			0
$127$	−5.05085			−0.448190			−0.224095	−	0.974567i	$-0.571943\pi$
$127$	−5.05085			−0.448190			−0.224095	+	0.974567i	$0.571943\pi$
$128$		−	11.5112i		−	1.01745i
$129$		0			0
$130$	3.99220	+	1.01360i	0.350139	+	0.0888985i
$131$	−15.9772			−1.39594			−0.697968	−	0.716129i	$-0.745912\pi$
$131$	−15.9772			−1.39594			−0.697968	+	0.716129i	$0.745912\pi$
$132$		0			0
$133$	7.69876			0.667567
$134$	4.95422			0.427980
$135$		0			0
$136$		−	7.36288i		−	0.631361i
$137$			17.2582i			1.47447i	0.675639	+	0.737233i	$0.263868\pi$
$137$			17.2582i			1.47447i	−0.675639	+	0.737233i	$0.736132\pi$
$138$		0			0
$139$	−5.70544			−0.483929			−0.241965	−	0.970285i	$-0.577792\pi$
$139$	−5.70544			−0.483929			−0.241965	+	0.970285i	$0.577792\pi$
$140$	4.89339			0.413567
$141$		0			0
$142$	−1.35395			−0.113621
$143$	−0.887280	+	3.49467i	−0.0741981	+	0.292239i
$144$		0			0
$145$			9.86396i			0.819158i
$146$	1.29291			0.107002
$147$		0			0
$148$		−	3.45136i		−	0.283700i
$149$		−	0.161015i		−	0.0131908i	−0.999978	−	0.00659542i	$-0.997901\pi$
$149$		−	0.161015i		−	0.0131908i	0.999978	−	0.00659542i	$-0.00209940\pi$
$150$		0			0
$151$			16.3107i			1.32734i	0.748025	+	0.663671i	$0.231003\pi$
$151$			16.3107i			1.32734i	−0.748025	+	0.663671i	$0.768997\pi$
$152$	−10.9008			−0.884170
$153$		0			0
$154$		−	0.883474i		−	0.0711924i
$155$	−14.2011			−1.14066
$156$		0			0
$157$	−14.2192			−1.13481			−0.567407	−	0.823437i	$-0.692053\pi$
$157$	−14.2192			−1.13481			−0.567407	+	0.823437i	$0.692053\pi$
$158$			3.24312i			0.258008i
$159$		0			0
$160$	−10.7168			−0.847236
$161$		−	1.05998i		−	0.0835384i
$162$		0			0
$163$			13.0186i			1.01970i	0.860264	+	0.509848i	$0.170298\pi$
$163$			13.0186i			1.01970i	−0.860264	+	0.509848i	$0.829702\pi$
$164$			1.71868i			0.134206i
$165$		0			0
$166$	8.48728			0.658740
$167$		−	14.0271i		−	1.08545i	−0.839911	−	0.542724i	$-0.817393\pi$
$167$		−	14.0271i		−	1.08545i	0.839911	−	0.542724i	$-0.182607\pi$
$168$		0			0
$169$	11.4255	+	6.20151i	0.878882	+	0.477039i
$170$	−3.93200			−0.301571
$171$		0			0
$172$	9.09696			0.693637
$173$	17.8674			1.35844			0.679218	−	0.733936i	$-0.262319\pi$
$173$	17.8674			1.35844			0.679218	+	0.733936i	$0.262319\pi$
$174$		0			0
$175$			1.78851i			0.135198i
$176$		−	2.06515i		−	0.155666i
$177$		0			0
$178$	1.05825			0.0793193
$179$	17.2723			1.29099			0.645495	−	0.763764i	$-0.276651\pi$
$179$	17.2723			1.29099			0.645495	+	0.763764i	$0.276651\pi$
$180$		0			0
$181$	−3.70318			−0.275255			−0.137628	−	0.990484i	$-0.543948\pi$
$181$	−3.70318			−0.275255			−0.137628	+	0.990484i	$0.543948\pi$
$182$	−3.08745	−	0.783889i	−0.228857	−	0.0581057i
$183$		0			0
$184$			1.50084i			0.110644i
$185$	−4.06641			−0.298969
$186$		0			0
$187$		−	3.44198i		−	0.251703i
$188$		−	6.22136i		−	0.453740i
$189$		0			0
$190$			5.82135i			0.422325i
$191$	20.6259			1.49243			0.746217	−	0.665702i	$-0.231868\pi$
$191$	20.6259			1.49243			0.746217	+	0.665702i	$0.231868\pi$
$192$		0			0
$193$			18.8475i			1.35667i	0.734753	+	0.678335i	$0.237298\pi$
$193$			18.8475i			1.35667i	−0.734753	+	0.678335i	$0.762702\pi$
$194$	−8.48429			−0.609136
$195$		0			0
$196$	7.82183			0.558702
$197$		−	7.18370i		−	0.511818i	−0.966701	−	0.255909i	$-0.917625\pi$
$197$		−	7.18370i		−	0.511818i	0.966701	−	0.255909i	$-0.0823747\pi$
$198$		0			0
$199$	18.9591			1.34398			0.671989	−	0.740562i	$-0.265440\pi$
$199$	18.9591			1.34398			0.671989	+	0.740562i	$0.265440\pi$
$200$		−	2.53237i		−	0.179066i
$201$		0			0
$202$		−	5.05131i		−	0.355409i
$203$		−	7.62851i		−	0.535417i
$204$		0			0
$205$	2.02496			0.141429
$206$		−	1.69614i		−	0.118175i
$207$		0			0
$208$	−7.21701	−	1.83236i	−0.500409	−	0.127051i
$209$	−5.09587			−0.352489
$210$		0			0
$211$	−19.9916			−1.37628			−0.688139	−	0.725578i	$-0.741572\pi$
$211$	−19.9916			−1.37628			−0.688139	+	0.725578i	$0.741572\pi$
$212$	0.833571			0.0572499
$213$		0			0
$214$		−	1.34051i		−	0.0916356i
$215$		−	10.7181i		−	0.730967i
$216$		0			0
$217$	10.9828			0.745558
$218$	−7.53303			−0.510201
$219$		0			0
$220$	−3.23897			−0.218372
$221$	−12.0286	−	3.05400i	−0.809131	−	0.205434i
$222$		0			0
$223$			27.7784i			1.86018i	0.367330	+	0.930091i	$0.380272\pi$
$223$			27.7784i			1.86018i	−0.367330	+	0.930091i	$0.619728\pi$
$224$	8.28806			0.553769
$225$		0			0
$226$		−	7.38419i		−	0.491189i
$227$			28.6524i			1.90173i	0.309612	+	0.950863i	$0.399801\pi$
$227$			28.6524i			1.90173i	−0.309612	+	0.950863i	$0.600199\pi$
$228$		0			0
$229$			4.83459i			0.319479i	0.987159	+	0.159739i	$0.0510654\pi$
$229$			4.83459i			0.319479i	−0.987159	+	0.159739i	$0.948935\pi$
$230$	0.801497			0.0528491
$231$		0			0
$232$			10.8013i			0.709141i
$233$	13.6237			0.892518			0.446259	−	0.894904i	$-0.352756\pi$
$233$	13.6237			0.892518			0.446259	+	0.894904i	$0.352756\pi$
$234$		0			0
$235$	−7.33004			−0.478159
$236$			4.39417i			0.286036i
$237$		0			0
$238$	3.04090			0.197112
$239$		−	5.99903i		−	0.388045i	−0.980997	−	0.194023i	$-0.937846\pi$
$239$		−	5.99903i		−	0.388045i	0.980997	−	0.194023i	$-0.0621535\pi$
$240$		0			0
$241$		−	12.9290i		−	0.832831i	−0.909174	−	0.416416i	$-0.863286\pi$
$241$		−	12.9290i		−	0.832831i	0.909174	−	0.416416i	$-0.136714\pi$
$242$			0.584778i			0.0375910i
$243$		0			0
$244$	−8.92346			−0.571266
$245$		−	9.21571i		−	0.588770i
$246$		0			0
$247$	−4.52146	+	17.8084i	−0.287694	+	1.13312i
$248$	−15.5506			−0.987466
$249$		0			0
$250$	−7.06420			−0.446779
$251$	−23.9724			−1.51313			−0.756563	−	0.653921i	$-0.773123\pi$
$251$	−23.9724			−1.51313			−0.756563	+	0.653921i	$0.773123\pi$
$252$		0			0
$253$			0.701611i			0.0441099i
$254$			2.95363i			0.185327i
$255$		0			0
$256$	−4.88701			−0.305438
$257$	16.3441			1.01952			0.509758	−	0.860318i	$-0.329735\pi$
$257$	16.3441			1.01952			0.509758	+	0.860318i	$0.329735\pi$
$258$		0			0
$259$	3.14485			0.195411
$260$	−2.87388	+	11.3192i	−0.178230	+	0.701984i
$261$		0			0
$262$			9.34313i			0.577220i
$263$	−10.1510			−0.625936			−0.312968	−	0.949764i	$-0.601323\pi$
$263$	−10.1510			−0.625936			−0.312968	+	0.949764i	$0.601323\pi$
$264$		0			0
$265$		−	0.982117i		−	0.0603310i
$266$		−	4.50207i		−	0.276040i
$267$		0			0
$268$			14.0468i			0.858045i
$269$	−16.5294			−1.00781			−0.503907	−	0.863758i	$-0.668105\pi$
$269$	−16.5294			−1.00781			−0.503907	+	0.863758i	$0.668105\pi$
$270$		0			0
$271$			20.8927i			1.26914i	0.772865	+	0.634570i	$0.218823\pi$
$271$			20.8927i			1.26914i	−0.772865	+	0.634570i	$0.781177\pi$
$272$	7.10819			0.430997
$273$		0			0
$274$	10.0922			0.609693
$275$		−	1.18383i		−	0.0713874i
$276$		0			0
$277$	17.3070			1.03988			0.519938	−	0.854204i	$-0.325955\pi$
$277$	17.3070			1.03988			0.519938	+	0.854204i	$0.325955\pi$
$278$			3.33642i			0.200105i
$279$		0			0
$280$		−	6.31329i		−	0.377291i
$281$			16.0666i			0.958450i	0.877692	+	0.479225i	$0.159082\pi$
$281$			16.0666i			0.958450i	−0.877692	+	0.479225i	$0.840918\pi$
$282$		0			0
$283$	17.2180			1.02350			0.511751	−	0.859134i	$-0.328997\pi$
$283$	17.2180			1.02350			0.511751	+	0.859134i	$0.328997\pi$
$284$		−	3.83890i		−	0.227797i
$285$		0			0
$286$	2.04361	+	0.518862i	0.120841	+	0.0306810i
$287$	−1.56604			−0.0924406
$288$		0			0
$289$	−5.15277			−0.303104
$290$	5.76823			0.338722
$291$		0			0
$292$			3.66581i			0.214525i
$293$		−	5.29670i		−	0.309436i	−0.987959	−	0.154718i	$-0.950553\pi$
$293$		−	5.29670i		−	0.309436i	0.987959	−	0.154718i	$-0.0494470\pi$
$294$		0			0
$295$	5.17723			0.301430
$296$	−4.45283			−0.258816
$297$		0			0
$298$	−0.0941580			−0.00545443
$299$	2.45190	+	0.622525i	0.141797	+	0.0360016i
$300$		0			0
$301$			8.28906i			0.477773i
$302$	9.53812			0.548857
$303$		0			0
$304$		−	10.5237i		−	0.603576i
$305$			10.5137i			0.602010i
$306$		0			0
$307$			26.3020i			1.50113i	0.660794	+	0.750567i	$0.270220\pi$
$307$			26.3020i			1.50113i	−0.660794	+	0.750567i	$0.729780\pi$
$308$	2.50493			0.142732
$309$		0			0
$310$			8.30451i			0.471665i
$311$	32.8590			1.86326			0.931632	−	0.363403i	$-0.118385\pi$
$311$	32.8590			1.86326			0.931632	+	0.363403i	$0.118385\pi$
$312$		0			0
$313$	33.2130			1.87731			0.938655	−	0.344858i	$-0.112073\pi$
$313$	33.2130			1.87731			0.938655	+	0.344858i	$0.112073\pi$
$314$			8.31507i			0.469247i
$315$		0			0
$316$	−9.19527			−0.517274
$317$			30.8436i			1.73235i	0.499740	+	0.866175i	$0.333429\pi$
$317$			30.8436i			1.73235i	−0.499740	+	0.866175i	$0.666571\pi$
$318$		0			0
$319$			5.04937i			0.282711i
$320$		−	1.80159i		−	0.100712i
$321$		0			0
$322$	−0.619855			−0.0345432
$323$		−	17.5399i		−	0.975945i
$324$		0			0
$325$	−4.13709	−	1.05039i	−0.229484	−	0.0582649i
$326$	7.61300			0.421645
$327$		0			0
$328$	2.21738			0.122434
$329$	5.66884			0.312534
$330$		0			0
$331$		−	5.75414i		−	0.316276i	−0.987417	−	0.158138i	$-0.949451\pi$
$331$		−	5.75414i		−	0.316276i	0.987417	−	0.158138i	$-0.0505491\pi$
$332$			24.0641i			1.32069i
$333$		0			0
$334$	−8.20273			−0.448833
$335$	16.5500			0.904224
$336$		0			0
$337$	−10.9245			−0.595094			−0.297547	−	0.954707i	$-0.596169\pi$
$337$	−10.9245			−0.595094			−0.297547	+	0.954707i	$0.596169\pi$
$338$	3.62651	−	6.68137i	0.197256	−	0.363419i
$339$		0			0
$340$		−	11.1485i		−	0.604612i
$341$	−7.26957			−0.393669
$342$		0			0
$343$			17.7027i			0.955854i
$344$		−	11.7366i		−	0.632795i
$345$		0			0
$346$		−	10.4485i		−	0.561715i
$347$	−16.3474			−0.877572			−0.438786	−	0.898591i	$-0.644591\pi$
$347$	−16.3474			−0.877572			−0.438786	+	0.898591i	$0.644591\pi$
$348$		0			0
$349$			8.62758i			0.461824i	0.972975	+	0.230912i	$0.0741709\pi$
$349$			8.62758i			0.461824i	−0.972975	+	0.230912i	$0.925829\pi$
$350$	1.04588			0.0559046
$351$		0			0
$352$	−5.48593			−0.292401
$353$			16.5793i			0.882425i	0.897403	+	0.441212i	$0.145451\pi$
$353$			16.5793i			0.882425i	−0.897403	+	0.441212i	$0.854549\pi$
$354$		0			0
$355$	−4.52300			−0.240056
$356$			3.00048i			0.159025i
$357$		0			0
$358$		−	10.1005i		−	0.533826i
$359$			15.5233i			0.819289i	0.912245	+	0.409645i	$0.134347\pi$
$359$			15.5233i			0.819289i	−0.912245	+	0.409645i	$0.865653\pi$
$360$		0			0
$361$	−6.96788			−0.366731
$362$			2.16554i			0.113818i
$363$		0			0
$364$	2.22258	−	8.75392i	0.116495	−	0.458830i
$365$	4.31907			0.226071
$366$		0			0
$367$	23.4682			1.22503			0.612514	−	0.790460i	$-0.290158\pi$
$367$	23.4682			1.22503			0.612514	+	0.790460i	$0.290158\pi$
$368$	−1.44893			−0.0755306
$369$		0			0
$370$			2.37795i			0.123624i
$371$			0.759541i			0.0394334i
$372$		0			0
$373$	12.3222			0.638018			0.319009	−	0.947752i	$-0.396650\pi$
$373$	12.3222			0.638018			0.319009	+	0.947752i	$0.396650\pi$
$374$	−2.01280			−0.104079
$375$		0			0
$376$	−8.02659			−0.413940
$377$	17.6459	+	4.48021i	0.908810	+	0.230742i
$378$		0			0
$379$			22.8695i			1.17473i	0.809322	+	0.587365i	$0.199835\pi$
$379$			22.8695i			1.17473i	−0.809322	+	0.587365i	$0.800165\pi$
$380$	−16.5054			−0.846709
$381$		0			0
$382$		−	12.0616i		−	0.617123i
$383$		−	35.5259i		−	1.81529i	−0.419737	−	0.907646i	$-0.637878\pi$
$383$		−	35.5259i		−	1.81529i	0.419737	−	0.907646i	$-0.362122\pi$
$384$		0			0
$385$		−	2.95132i		−	0.150413i
$386$	11.0216			0.560984
$387$		0			0
$388$		−	24.0557i		−	1.22124i
$389$	−2.67074			−0.135412			−0.0677060	−	0.997705i	$-0.521568\pi$
$389$	−2.67074			−0.135412			−0.0677060	+	0.997705i	$0.521568\pi$
$390$		0			0
$391$	−2.41493			−0.122128
$392$		−	10.0915i		−	0.509696i
$393$		0			0
$394$	−4.20087			−0.211637
$395$			10.8339i			0.545113i
$396$		0			0
$397$		−	36.4406i		−	1.82890i	−0.404698	−	0.914451i	$-0.632623\pi$
$397$		−	36.4406i		−	1.82890i	0.404698	−	0.914451i	$-0.367377\pi$
$398$		−	11.0869i		−	0.555736i
$399$		0			0
$400$	2.44477			0.122239
$401$		−	13.6597i		−	0.682132i	−0.940039	−	0.341066i	$-0.889212\pi$
$401$		−	13.6597i		−	0.682132i	0.940039	−	0.341066i	$-0.110788\pi$
$402$		0			0
$403$	−6.45015	+	25.4048i	−0.321305	+	1.26550i
$404$	14.3221			0.712550
$405$		0			0
$406$	−4.46099			−0.221395
$407$	−2.08160			−0.103181
$408$		0			0
$409$		−	25.8507i		−	1.27823i	−0.769110	−	0.639117i	$-0.779300\pi$
$409$		−	25.8507i		−	1.27823i	0.769110	−	0.639117i	$-0.220700\pi$
$410$		−	1.18415i		−	0.0584810i
$411$		0			0
$412$	4.80909			0.236927
$413$	−4.00393			−0.197020
$414$		0			0
$415$	28.3525			1.39177
$416$	−4.86756	+	19.1715i	−0.238652	+	0.939962i
$417$		0			0
$418$			2.97995i			0.145754i
$419$	−22.6067			−1.10441			−0.552205	−	0.833708i	$-0.686213\pi$
$419$	−22.6067			−1.10441			−0.552205	+	0.833708i	$0.686213\pi$
$420$		0			0
$421$		−	24.6918i		−	1.20341i	−0.798720	−	0.601703i	$-0.794489\pi$
$421$		−	24.6918i		−	1.20341i	0.798720	−	0.601703i	$-0.205511\pi$
$422$			11.6907i			0.569092i
$423$		0			0
$424$		−	1.07544i		−	0.0522282i
$425$	4.07471			0.197652
$426$		0			0
$427$		−	8.13097i		−	0.393485i
$428$	3.80079			0.183718
$429$		0			0
$430$	−6.26770			−0.302255
$431$			30.1126i			1.45047i	0.688499	+	0.725237i	$0.258270\pi$
$431$			30.1126i			1.45047i	−0.688499	+	0.725237i	$0.741730\pi$
$432$		0			0
$433$	−30.9402			−1.48689			−0.743445	−	0.668797i	$-0.766809\pi$
$433$	−30.9402			−1.48689			−0.743445	+	0.668797i	$0.766809\pi$
$434$		−	6.42248i		−	0.308289i
$435$		0			0
$436$		−	21.3585i		−	1.02289i
$437$			3.57532i			0.171031i
$438$		0			0
$439$	7.02158			0.335122			0.167561	−	0.985862i	$-0.446411\pi$
$439$	7.02158			0.335122			0.167561	+	0.985862i	$0.446411\pi$
$440$			4.17882i			0.199217i
$441$		0			0
$442$	−1.78591	+	7.03406i	−0.0849472	+	0.334576i
$443$	22.5411			1.07096			0.535480	−	0.844548i	$-0.320131\pi$
$443$	22.5411			1.07096			0.535480	+	0.844548i	$0.320131\pi$
$444$		0			0
$445$	3.53518			0.167584
$446$	16.2442			0.769187
$447$		0			0
$448$			1.39330i			0.0658272i
$449$			9.41670i			0.444401i	0.975001	+	0.222201i	$0.0713240\pi$
$449$			9.41670i			0.444401i	−0.975001	+	0.222201i	$0.928676\pi$
$450$		0			0
$451$	1.03658			0.0488105
$452$	20.9365			0.984772
$453$		0			0
$454$	16.7553			0.786365
$455$	−10.3139	−	2.61865i	−0.483523	−	0.122764i
$456$		0			0
$457$			3.48691i			0.163111i	0.996669	+	0.0815554i	$0.0259887\pi$
$457$			3.48691i			0.163111i	−0.996669	+	0.0815554i	$0.974011\pi$
$458$	2.82717			0.132105
$459$		0			0
$460$			2.27250i			0.105956i
$461$			15.1983i			0.707855i	0.935273	+	0.353928i	$0.115154\pi$
$461$			15.1983i			0.707855i	−0.935273	+	0.353928i	$0.884846\pi$
$462$		0			0
$463$			15.9882i			0.743035i	0.928426	+	0.371518i	$0.121162\pi$
$463$			15.9882i			0.743035i	−0.928426	+	0.371518i	$0.878838\pi$
$464$	−10.4277			−0.484093
$465$		0			0
$466$		−	7.96684i		−	0.369057i
$467$	32.9225			1.52347			0.761736	−	0.647888i	$-0.224348\pi$
$467$	32.9225			1.52347			0.761736	+	0.647888i	$0.224348\pi$
$468$		0			0
$469$	−12.7993			−0.591017
$470$			4.28645i			0.197719i
$471$		0			0
$472$	5.66921			0.260947
$473$		−	5.48659i		−	0.252274i
$474$		0			0
$475$		−	6.03262i		−	0.276796i
$476$			8.62193i			0.395185i
$477$		0			0
$478$	−3.50811			−0.160457
$479$		−	33.6023i		−	1.53533i	−0.640853	−	0.767664i	$-0.721419\pi$
$479$		−	33.6023i		−	1.53533i	0.640853	−	0.767664i	$-0.278581\pi$
$480$		0			0
$481$	−1.84696	+	7.27451i	−0.0842142	+	0.331689i
$482$	−7.56061			−0.344376
$483$		0			0
$484$	−1.65803			−0.0753652
$485$	−28.3425			−1.28697
$486$		0			0
$487$		−	3.70001i		−	0.167663i	−0.996480	−	0.0838316i	$-0.973284\pi$
$487$		−	3.70001i		−	0.167663i	0.996480	−	0.0838316i	$-0.0267158\pi$
$488$			11.5127i			0.521158i
$489$		0			0
$490$	−5.38915			−0.243457
$491$	−10.1397			−0.457599			−0.228799	−	0.973474i	$-0.573480\pi$
$491$	−10.1397			−0.457599			−0.228799	+	0.973474i	$0.573480\pi$
$492$		0			0
$493$	−17.3798			−0.782749
$494$	10.4140	+	2.64405i	0.468546	+	0.118962i
$495$		0			0
$496$		−	15.0127i		−	0.674091i
$497$	3.49796			0.156905
$498$		0			0
$499$			15.1124i			0.676525i	0.941052	+	0.338263i	$0.109839\pi$
$499$			15.1124i			0.676525i	−0.941052	+	0.338263i	$0.890161\pi$
$500$		−	20.0293i		−	0.895736i
$501$		0			0
$502$			14.0185i			0.625678i
$503$	−11.9190			−0.531441			−0.265721	−	0.964050i	$-0.585610\pi$
$503$	−11.9190			−0.531441			−0.265721	+	0.964050i	$0.585610\pi$
$504$		0			0
$505$		−	16.8743i		−	0.750898i
$506$	0.410287			0.0182395
$507$		0			0
$508$	−8.37448			−0.371558
$509$		−	35.2059i		−	1.56047i	−0.625485	−	0.780236i	$-0.715099\pi$
$509$		−	35.2059i		−	1.56047i	0.625485	−	0.780236i	$-0.284901\pi$
$510$		0			0
$511$	−3.34025			−0.147764
$512$		−	20.1645i		−	0.891154i
$513$		0			0
$514$		−	9.55767i		−	0.421571i
$515$		−	5.66609i		−	0.249678i
$516$		0			0
$517$	−3.75225			−0.165024
$518$		−	1.83904i		−	0.0808028i
$519$		0			0
$520$	14.6036	+	3.70778i	0.640410	+	0.162597i
$521$	2.84143			0.124485			0.0622426	−	0.998061i	$-0.480175\pi$
$521$	2.84143			0.124485			0.0622426	+	0.998061i	$0.480175\pi$
$522$		0			0
$523$	−17.8564			−0.780804			−0.390402	−	0.920644i	$-0.627664\pi$
$523$	−17.8564			−0.780804			−0.390402	+	0.920644i	$0.627664\pi$
$524$	−26.4908			−1.15725
$525$		0			0
$526$			5.93607i			0.258825i
$527$		−	25.0217i		−	1.08996i
$528$		0			0
$529$	−22.5077			−0.978597
$530$	−0.574321			−0.0249469
$531$		0			0
$532$	12.7648			0.553424
$533$	0.919734	−	3.62249i	0.0398381	−	0.156908i
$534$		0			0
$535$		−	4.47810i		−	0.193605i
$536$	18.1227			0.782782
$537$		0			0
$538$			9.66602i			0.416732i
$539$		−	4.71753i		−	0.203198i
$540$		0			0
$541$		−	3.56883i		−	0.153436i	−0.997053	−	0.0767179i	$-0.975556\pi$
$541$		−	3.56883i		−	0.153436i	0.997053	−	0.0767179i	$-0.0244441\pi$
$542$	12.2176			0.524791
$543$		0			0
$544$		−	18.8825i		−	0.809579i
$545$	−25.1647			−1.07794
$546$		0			0
$547$	3.50639			0.149923			0.0749613	−	0.997186i	$-0.476117\pi$
$547$	3.50639			0.149923			0.0749613	+	0.997186i	$0.476117\pi$
$548$			28.6146i			1.22236i
$549$		0			0
$550$	−0.692276			−0.0295188
$551$			25.7309i			1.09617i
$552$		0			0
$553$		−	8.37864i		−	0.356296i
$554$		−	10.1207i		−	0.429989i
$555$		0			0
$556$	−9.45982			−0.401186
$557$		−	7.38164i		−	0.312770i	−0.987696	−	0.156385i	$-0.950016\pi$
$557$		−	7.38164i		−	0.312770i	0.987696	−	0.156385i	$-0.0499841\pi$
$558$		0			0
$559$	−19.1738	−	4.86815i	−0.810967	−	0.205901i
$560$	6.09491			0.257557
$561$		0			0
$562$	9.39537			0.396320
$563$	−35.5383			−1.49776			−0.748879	−	0.662706i	$-0.769408\pi$
$563$	−35.5383			−1.49776			−0.748879	+	0.662706i	$0.769408\pi$
$564$		0			0
$565$		−	24.6675i		−	1.03777i
$566$		−	10.0687i		−	0.423219i
$567$		0			0
$568$	−4.95281			−0.207815
$569$	−19.6546			−0.823966			−0.411983	−	0.911192i	$-0.635164\pi$
$569$	−19.6546			−0.823966			−0.411983	+	0.911192i	$0.635164\pi$
$570$		0			0
$571$	18.6081			0.778726			0.389363	−	0.921084i	$-0.372695\pi$
$571$	18.6081			0.778726			0.389363	+	0.921084i	$0.372695\pi$
$572$	−1.47114	+	5.79429i	−0.0615115	+	0.242271i
$573$		0			0
$574$			0.915788i			0.0382243i
$575$	−0.830586			−0.0346378
$576$		0			0
$577$		−	25.5485i		−	1.06360i	−0.846871	−	0.531799i	$-0.821516\pi$
$577$		−	25.5485i		−	1.06360i	0.846871	−	0.531799i	$-0.178484\pi$
$578$			3.01323i			0.125334i
$579$		0			0
$580$			16.3548i			0.679096i
$581$	−21.9270			−0.909686
$582$		0			0
$583$		−	0.502747i		−	0.0208216i
$584$	4.72950			0.195708
$585$		0			0
$586$	−3.09739			−0.127952
$587$			2.35936i			0.0973813i	0.998814	+	0.0486907i	$0.0155049\pi$
$587$			2.35936i			0.0973813i	−0.998814	+	0.0486907i	$0.984495\pi$
$588$		0			0
$589$	−37.0448			−1.52640
$590$		−	3.02753i		−	0.124642i
$591$		0			0
$592$		−	4.29881i		−	0.176680i
$593$		−	11.6707i		−	0.479257i	−0.970865	−	0.239628i	$-0.922974\pi$
$593$		−	11.6707i		−	0.479257i	0.970865	−	0.239628i	$-0.0770256\pi$
$594$		0			0
$595$	10.1584			0.416453
$596$		−	0.266968i		−	0.0109354i
$597$		0			0
$598$	0.364040	−	1.43382i	0.0148867	−	0.0586332i
$599$	−9.12181			−0.372707			−0.186354	−	0.982483i	$-0.559667\pi$
$599$	−9.12181			−0.372707			−0.186354	+	0.982483i	$0.559667\pi$
$600$		0			0
$601$	6.90316			0.281586			0.140793	−	0.990039i	$-0.455035\pi$
$601$	6.90316			0.281586			0.140793	+	0.990039i	$0.455035\pi$
$602$	4.84726			0.197560
$603$		0			0
$604$			27.0436i			1.10039i
$605$			1.95350i			0.0794212i
$606$		0			0
$607$	11.5559			0.469038			0.234519	−	0.972112i	$-0.424648\pi$
$607$	11.5559			0.469038			0.234519	+	0.972112i	$0.424648\pi$
$608$	−27.9556			−1.13375
$609$		0			0
$610$	6.14816			0.248932
$611$	−3.32930	+	13.1129i	−0.134689	+	0.530491i
$612$		0			0
$613$			37.7362i			1.52415i	0.647488	+	0.762075i	$0.275819\pi$
$613$			37.7362i			1.52415i	−0.647488	+	0.762075i	$0.724181\pi$
$614$	15.3808			0.620720
$615$		0			0
$616$		−	3.23178i		−	0.130212i
$617$		−	28.2159i		−	1.13593i	−0.823053	−	0.567964i	$-0.807731\pi$
$617$		−	28.2159i		−	1.13593i	0.823053	−	0.567964i	$-0.192269\pi$
$618$		0			0
$619$		−	38.4407i		−	1.54506i	−0.634978	−	0.772530i	$-0.718991\pi$
$619$		−	38.4407i		−	1.54506i	0.634978	−	0.772530i	$-0.281009\pi$
$620$	−23.5460			−0.945628
$621$		0			0
$622$		−	19.2152i		−	0.770461i
$623$	−2.73401			−0.109536
$624$		0			0
$625$	−17.6794			−0.707177
$626$		−	19.4222i		−	0.776269i
$627$		0			0
$628$	−23.5759			−0.940781
$629$		−	7.16483i		−	0.285680i
$630$		0			0
$631$			16.9688i			0.675519i	0.941232	+	0.337759i	$0.109669\pi$
$631$			16.9688i			0.675519i	−0.941232	+	0.337759i	$0.890331\pi$
$632$			11.8634i			0.471902i
$633$		0			0
$634$	18.0367			0.716328
$635$			9.86685i			0.391554i
$636$		0			0
$637$	−16.4862	−	4.18577i	−0.653208	−	0.165846i
$638$	2.95276			0.116901
$639$		0			0
$640$	−22.4871			−0.888881
$641$	6.00876			0.237332			0.118666	−	0.992934i	$-0.462138\pi$
$641$	6.00876			0.237332			0.118666	+	0.992934i	$0.462138\pi$
$642$		0			0
$643$			9.64269i			0.380270i	0.981758	+	0.190135i	$0.0608927\pi$
$643$			9.64269i			0.380270i	−0.981758	+	0.190135i	$0.939107\pi$
$644$		−	1.75749i		−	0.0692547i
$645$		0			0
$646$	−10.2569			−0.403554
$647$	−34.1912			−1.34420			−0.672098	−	0.740463i	$-0.734606\pi$
$647$	−34.1912			−1.34420			−0.672098	+	0.740463i	$0.734606\pi$
$648$		0			0
$649$	2.65023			0.104031
$650$	−0.614243	+	2.41928i	−0.0240926	+	0.0948919i
$651$		0			0
$652$			21.5853i			0.845345i
$653$	2.32307			0.0909086			0.0454543	−	0.998966i	$-0.485526\pi$
$653$	2.32307			0.0909086			0.0454543	+	0.998966i	$0.485526\pi$
$654$		0			0
$655$			31.2115i			1.21954i
$656$			2.14068i			0.0835796i
$657$		0			0
$658$		−	3.31502i		−	0.129233i
$659$	−4.07928			−0.158906			−0.0794532	−	0.996839i	$-0.525317\pi$
$659$	−4.07928			−0.158906			−0.0794532	+	0.996839i	$0.525317\pi$
$660$		0			0
$661$		−	42.2740i		−	1.64427i	−0.569293	−	0.822135i	$-0.692783\pi$
$661$		−	42.2740i		−	1.64427i	0.569293	−	0.822135i	$-0.307217\pi$
$662$	−3.36489			−0.130780
$663$		0			0
$664$	31.0467			1.20485
$665$		−	15.0395i		−	0.583209i
$666$		0			0
$667$	3.54270			0.137174
$668$		−	23.2574i		−	0.899854i
$669$		0			0
$670$		−	9.67809i		−	0.373897i
$671$			5.38195i			0.207768i
$672$		0			0
$673$	10.6688			0.411252			0.205626	−	0.978631i	$-0.434077\pi$
$673$	10.6688			0.411252			0.205626	+	0.978631i	$0.434077\pi$
$674$			6.38840i			0.246072i
$675$		0			0
$676$	18.9438	+	10.2823i	0.728608	+	0.395473i
$677$	−24.2174			−0.930752			−0.465376	−	0.885113i	$-0.654081\pi$
$677$	−24.2174			−0.930752			−0.465376	+	0.885113i	$0.654081\pi$
$678$		0			0
$679$	21.9193			0.841185
$680$	−14.3834			−0.551578
$681$		0			0
$682$			4.25109i			0.162783i
$683$		−	37.1698i		−	1.42226i	−0.703059	−	0.711132i	$-0.748183\pi$
$683$		−	37.1698i		−	1.42226i	0.703059	−	0.711132i	$-0.251817\pi$
$684$		0			0
$685$	33.7139			1.28814
$686$	10.3521			0.395246
$687$		0			0
$688$	11.3306			0.431976
$689$	−1.75693	−	0.446077i	−0.0669339	−	0.0169942i
$690$		0			0
$691$			15.9632i			0.607268i	0.952789	+	0.303634i	$0.0981999\pi$
$691$			15.9632i			0.607268i	−0.952789	+	0.303634i	$0.901800\pi$
$692$	29.6248			1.12617
$693$		0			0
$694$			9.55959i			0.362877i
$695$			11.1456i			0.422777i
$696$		0			0
$697$			3.56788i			0.135143i
$698$	5.04522			0.190965
$699$		0			0
$700$			2.96540i			0.112082i
$701$	−2.01736			−0.0761947			−0.0380974	−	0.999274i	$-0.512130\pi$
$701$	−2.01736			−0.0761947			−0.0380974	+	0.999274i	$0.512130\pi$
$702$		0			0
$703$	−10.6076			−0.400072
$704$		−	0.922236i		−	0.0347581i
$705$		0			0
$706$	9.69519			0.364883
$707$			13.0501i			0.490801i
$708$		0			0
$709$		−	29.6428i		−	1.11326i	−0.830761	−	0.556629i	$-0.812095\pi$
$709$		−	29.6428i		−	1.11326i	0.830761	−	0.556629i	$-0.187905\pi$
$710$			2.64495i			0.0992634i
$711$		0			0
$712$	3.87112			0.145076
$713$			5.10041i			0.191012i
$714$		0			0
$715$	6.82685	+	1.73330i	0.255310	+	0.0648219i
$716$	28.6380			1.07025
$717$		0			0
$718$	9.07770			0.338777
$719$	32.3389			1.20604			0.603018	−	0.797728i	$-0.293965\pi$
$719$	32.3389			1.20604			0.603018	+	0.797728i	$0.293965\pi$
$720$		0			0
$721$			4.38199i			0.163194i
$722$			4.07467i			0.151643i
$723$		0			0
$724$	−6.13999			−0.228191
$725$	−5.97758			−0.222002
$726$		0			0
$727$	−37.9325			−1.40684			−0.703419	−	0.710775i	$-0.748344\pi$
$727$	−37.9325			−1.40684			−0.703419	+	0.710775i	$0.748344\pi$
$728$	−11.2940	−	2.86749i	−0.418584	−	0.106276i
$729$		0			0
$730$		−	2.52570i		−	0.0934803i
$731$	18.8847			0.698478
$732$		0			0
$733$			16.7917i			0.620217i	0.950701	+	0.310109i	$0.100365\pi$
$733$			16.7917i			0.620217i	−0.950701	+	0.310109i	$0.899635\pi$
$734$		−	13.7237i		−	0.506550i
$735$		0			0
$736$			3.84899i			0.141876i
$737$	8.47197			0.312069
$738$		0			0
$739$		−	25.4629i		−	0.936668i	−0.883551	−	0.468334i	$-0.844854\pi$
$739$		−	25.4629i		−	0.936668i	0.883551	−	0.468334i	$-0.155146\pi$
$740$	−6.74225			−0.247850
$741$		0			0
$742$	0.444164			0.0163058
$743$		−	4.81269i		−	0.176560i	−0.996096	−	0.0882802i	$-0.971863\pi$
$743$		−	4.81269i		−	0.176560i	0.996096	−	0.0882802i	$-0.0281371\pi$
$744$		0			0
$745$	−0.314543			−0.0115240
$746$		−	7.20574i		−	0.263821i
$747$		0			0
$748$		−	5.70692i		−	0.208666i
$749$			3.46324i			0.126544i
$750$		0			0
$751$	24.9537			0.910574			0.455287	−	0.890345i	$-0.349537\pi$
$751$	24.9537			0.910574			0.455287	+	0.890345i	$0.349537\pi$
$752$		−	7.74895i		−	0.282575i
$753$		0			0
$754$	2.61993	−	10.3189i	0.0954122	−	0.375794i
$755$	31.8629			1.15961
$756$		0			0
$757$	−33.8373			−1.22984			−0.614919	−	0.788591i	$-0.710811\pi$
$757$	−33.8373			−1.22984			−0.614919	+	0.788591i	$0.710811\pi$
$758$	13.3736			0.485752
$759$		0			0
$760$			21.2947i			0.772440i
$761$		−	50.5981i		−	1.83418i	−0.398679	−	0.917091i	$-0.630531\pi$
$761$		−	50.5981i		−	1.83418i	0.398679	−	0.917091i	$-0.369469\pi$
$762$		0			0
$763$	19.4617			0.704561
$764$	34.1984			1.23725
$765$		0			0
$766$	−20.7748			−0.750624
$767$	2.35150	−	9.26169i	0.0849076	−	0.334420i
$768$		0			0
$769$			26.1333i			0.942389i	0.882029	+	0.471195i	$0.156177\pi$
$769$			26.1333i			0.942389i	−0.882029	+	0.471195i	$0.843823\pi$
$770$	−1.72587			−0.0621960
$771$		0			0
$772$			31.2497i			1.12470i
$773$			0.778578i			0.0280035i	0.999902	+	0.0140018i	$0.00445704\pi$
$773$			0.778578i			0.0280035i	−0.999902	+	0.0140018i	$0.995543\pi$
$774$		0			0
$775$		−	8.60591i		−	0.309133i
$776$	−31.0358			−1.11412
$777$		0			0
$778$			1.56179i			0.0559930i
$779$	5.28226			0.189257
$780$		0			0
$781$	−2.31533			−0.0828490
$782$			1.41220i			0.0505002i
$783$		0			0
$784$	9.74239			0.347942
$785$			27.7772i			0.991411i
$786$		0			0
$787$		−	33.9128i		−	1.20886i	−0.796658	−	0.604431i	$-0.793401\pi$
$787$		−	33.9128i		−	1.20886i	0.796658	−	0.604431i	$-0.206599\pi$
$788$		−	11.9108i		−	0.424305i
$789$		0			0
$790$	6.33544			0.225405
$791$			19.0772i			0.678306i
$792$		0			0
$793$	18.8082	+	4.77530i	0.667897	+	0.169576i
$794$	−21.3097			−0.756252
$795$		0			0
$796$	31.4349			1.11418
$797$	26.9305			0.953928			0.476964	−	0.878923i	$-0.341737\pi$
$797$	26.9305			0.953928			0.476964	+	0.878923i	$0.341737\pi$
$798$		0			0
$799$		−	12.9152i		−	0.456906i
$800$		−	6.49439i		−	0.229611i
$801$		0			0
$802$	−7.98789			−0.282062
$803$	2.21094			0.0780223
$804$		0			0
$805$	−2.07068			−0.0729819
$806$	14.8562	+	3.77191i	0.523286	+	0.132860i
$807$		0			0
$808$		−	18.4779i		−	0.650049i
$809$	−52.4814			−1.84515			−0.922575	−	0.385819i	$-0.873919\pi$
$809$	−52.4814			−1.84515			−0.922575	+	0.385819i	$0.873919\pi$
$810$		0			0
$811$		−	19.2236i		−	0.675033i	−0.941320	−	0.337516i	$-0.890413\pi$
$811$		−	19.2236i		−	0.675033i	0.941320	−	0.337516i	$-0.109587\pi$
$812$		−	12.6483i		−	0.443869i
$813$		0			0
$814$			1.21727i			0.0426655i
$815$	25.4319			0.890840
$816$		0			0
$817$		−	27.9590i		−	0.978160i
$818$	−15.1169			−0.528551
$819$		0			0
$820$	3.35744			0.117247
$821$			30.4099i			1.06131i	0.847587	+	0.530657i	$0.178055\pi$
$821$			30.4099i			1.06131i	−0.847587	+	0.530657i	$0.821945\pi$
$822$		0			0
$823$	−16.1575			−0.563216			−0.281608	−	0.959530i	$-0.590868\pi$
$823$	−16.1575			−0.563216			−0.281608	+	0.959530i	$0.590868\pi$
$824$		−	6.20452i		−	0.216145i
$825$		0			0
$826$			2.34141i			0.0814681i
$827$			47.5066i			1.65197i	0.563695	+	0.825983i	$0.309379\pi$
$827$			47.5066i			1.65197i	−0.563695	+	0.825983i	$0.690621\pi$
$828$		0			0
$829$	−56.6932			−1.96904			−0.984519	−	0.175278i	$-0.943917\pi$
$829$	−56.6932			−1.96904			−0.984519	+	0.175278i	$0.943917\pi$
$830$		−	16.5799i		−	0.575497i
$831$		0			0
$832$	−3.22291	−	0.818281i	−0.111734	−	0.0283688i
$833$	16.2376			0.562601
$834$		0			0
$835$	−27.4019			−0.948283
$836$	−8.44913			−0.292219
$837$		0			0
$838$			13.2199i			0.456674i
$839$		−	35.2293i		−	1.21625i	−0.793841	−	0.608125i	$-0.791922\pi$
$839$		−	35.2293i		−	1.21625i	0.793841	−	0.608125i	$-0.208078\pi$
$840$		0			0
$841$	−3.50384			−0.120822
$842$	−14.4393			−0.497610
$843$		0			0
$844$	−33.1468			−1.14096
$845$	12.1147	−	22.3197i	0.416757	−	0.767821i
$846$		0			0
$847$		−	1.51078i		−	0.0519112i
$848$	1.03824			0.0356535
$849$		0			0
$850$		−	2.38280i		−	0.0817294i
$851$			1.46047i			0.0500644i
$852$		0			0
$853$			18.7185i			0.640909i	0.947264	+	0.320454i	$0.103836\pi$
$853$			18.7185i			0.640909i	−0.947264	+	0.320454i	$0.896164\pi$
$854$	−4.75482			−0.162706
$855$		0			0
$856$		−	4.90365i		−	0.167603i
$857$	47.3032			1.61585			0.807924	−	0.589287i	$-0.200591\pi$
$857$	47.3032			1.61585			0.807924	+	0.589287i	$0.200591\pi$
$858$		0			0
$859$	−40.6908			−1.38835			−0.694176	−	0.719805i	$-0.744231\pi$
$859$	−40.6908			−1.38835			−0.694176	+	0.719805i	$0.744231\pi$
$860$		−	17.7709i		−	0.605984i
$861$		0			0
$862$	17.6092			0.599772
$863$		−	1.42374i		−	0.0484647i	−0.999706	−	0.0242324i	$-0.992286\pi$
$863$		−	1.42374i		−	0.0484647i	0.999706	−	0.0242324i	$-0.00771415\pi$
$864$		0			0
$865$		−	34.9041i		−	1.18678i
$866$			18.0931i			0.614830i
$867$		0			0
$868$	18.2098			0.618080
$869$			5.54589i			0.188131i
$870$		0			0
$871$	7.51701	−	29.6067i	0.254704	−	1.00319i
$872$	−27.5561			−0.933166
$873$		0			0
$874$	2.09077			0.0707213
$875$	18.2505			0.616978
$876$		0			0
$877$		−	56.1060i		−	1.89457i	−0.320397	−	0.947283i	$-0.603816\pi$
$877$		−	56.1060i		−	1.89457i	0.320397	−	0.947283i	$-0.396184\pi$
$878$		−	4.10607i		−	0.138573i
$879$		0			0
$880$	−4.03427			−0.135995
$881$	16.2234			0.546581			0.273290	−	0.961932i	$-0.411888\pi$
$881$	16.2234			0.546581			0.273290	+	0.961932i	$0.411888\pi$
$882$		0			0
$883$	33.0296			1.11153			0.555767	−	0.831338i	$-0.312425\pi$
$883$	33.0296			1.11153			0.555767	+	0.831338i	$0.312425\pi$
$884$	−19.9438	−	5.06364i	−0.670783	−	0.170309i
$885$		0			0
$886$		−	13.1815i		−	0.442843i
$887$	54.6149			1.83379			0.916894	−	0.399130i	$-0.130688\pi$
$887$	54.6149			1.83379			0.916894	+	0.399130i	$0.130688\pi$
$888$		0			0
$889$		−	7.63074i		−	0.255927i
$890$		−	2.06730i		−	0.0692960i
$891$		0			0
$892$			46.0576i			1.54212i
$893$	−19.1210			−0.639860
$894$		0			0
$895$		−	33.7414i		−	1.12785i
$896$	17.3909			0.580989
$897$		0			0
$898$	5.50668			0.183760
$899$			36.7068i			1.22424i
$900$		0			0
$901$	1.73044			0.0576494
$902$		−	0.606168i		−	0.0201832i
$903$		0			0
$904$		−	27.0116i		−	0.898393i
$905$			7.23417i			0.240472i
$906$		0			0
$907$	−31.1937			−1.03577			−0.517884	−	0.855451i	$-0.673280\pi$
$907$	−31.1937			−1.03577			−0.517884	+	0.855451i	$0.673280\pi$
$908$			47.5066i			1.57656i
$909$		0			0
$910$	−1.53133	+	6.03135i	−0.0507631	+	0.199937i
$911$	−53.7300			−1.78015			−0.890077	−	0.455810i	$-0.849350\pi$
$911$	−53.7300			−1.78015			−0.890077	+	0.455810i	$0.849350\pi$
$912$		0			0
$913$	14.5137			0.480332
$914$	2.03907			0.0674464
$915$		0			0
$916$			8.01592i			0.264853i
$917$		−	24.1381i		−	0.797111i
$918$		0			0
$919$	16.3202			0.538353			0.269177	−	0.963091i	$-0.413248\pi$
$919$	16.3202			0.538353			0.269177	+	0.963091i	$0.413248\pi$
$920$	2.93190			0.0966620
$921$		0			0
$922$	8.88764			0.292699
$923$	−2.05435	+	8.09132i	−0.0676196	+	0.266329i
$924$		0			0
$925$		−	2.46425i		−	0.0810241i
$926$	9.34956			0.307246
$927$		0			0
$928$			27.7005i			0.909314i
$929$			56.4975i			1.85362i	0.375527	+	0.926811i	$0.377462\pi$
$929$			56.4975i			1.85362i	−0.375527	+	0.926811i	$0.622538\pi$
$930$		0			0
$931$		−	24.0399i		−	0.787877i
$932$	22.5885			0.739912
$933$		0			0
$934$		−	19.2524i		−	0.629957i
$935$	−6.72392			−0.219896
$936$		0			0
$937$	52.5328			1.71617			0.858086	−	0.513506i	$-0.171654\pi$
$937$	52.5328			1.71617			0.858086	+	0.513506i	$0.171654\pi$
$938$			7.48476i			0.244386i
$939$		0			0
$940$	−12.1535			−0.396402
$941$			24.3963i			0.795296i	0.917538	+	0.397648i	$0.130174\pi$
$941$			24.3963i			0.795296i	−0.917538	+	0.397648i	$0.869826\pi$
$942$		0			0
$943$		−	0.727273i		−	0.0236833i
$944$			5.47311i			0.178135i
$945$		0			0
$946$	−3.20844			−0.104315
$947$			38.9960i			1.26720i	0.773661	+	0.633600i	$0.218423\pi$
$947$			38.9960i			1.26720i	−0.773661	+	0.633600i	$0.781577\pi$
$948$		0			0
$949$	1.96172	−	7.72650i	0.0636801	−	0.250813i
$950$	−3.52775			−0.114455
$951$		0			0
$952$	11.1237			0.360522
$953$	−30.9120			−1.00134			−0.500670	−	0.865638i	$-0.666913\pi$
$953$	−30.9120			−1.00134			−0.500670	+	0.865638i	$0.666913\pi$
$954$		0			0
$955$		−	40.2927i		−	1.30384i
$956$		−	9.94660i		−	0.321696i
$957$		0			0
$958$	−19.6499			−0.634859
$959$	−26.0734			−0.841953
$960$		0			0
$961$	−21.8467			−0.704731
$962$	4.25398	+	1.08006i	0.137154	+	0.0348227i
$963$		0			0
$964$		−	21.4367i		−	0.690431i
$965$	36.8186			1.18523
$966$		0			0
$967$			31.5859i			1.01573i	0.861435	+	0.507867i	$0.169566\pi$
$967$			31.5859i			1.01573i	−0.861435	+	0.507867i	$0.830434\pi$
$968$			2.13914i			0.0687546i
$969$		0			0
$970$			16.5741i			0.532162i
$971$	32.5555			1.04476			0.522378	−	0.852714i	$-0.325045\pi$
$971$	32.5555			1.04476			0.522378	+	0.852714i	$0.325045\pi$
$972$		0			0
$973$		−	8.61969i		−	0.276335i
$974$	−2.16368			−0.0693289
$975$		0			0
$976$	−11.1145			−0.355767
$977$		−	3.54984i		−	0.113569i	−0.998386	−	0.0567847i	$-0.981915\pi$
$977$		−	3.54984i		−	0.113569i	0.998386	−	0.0567847i	$-0.0180848\pi$
$978$		0			0
$979$	1.80966			0.0578371
$980$		−	15.2800i		−	0.488101i
$981$		0			0
$982$			5.92949i			0.189218i
$983$			16.6279i			0.530349i	0.964200	+	0.265174i	$0.0854295\pi$
$983$			16.6279i			0.530349i	−0.964200	+	0.265174i	$0.914570\pi$
$984$		0			0
$985$	−14.0334			−0.447141
$986$			10.1634i			0.323667i
$987$		0			0
$988$	−7.49674	+	29.5269i	−0.238503	+	0.939377i
$989$	−3.84946			−0.122406
$990$		0			0
$991$	36.4447			1.15770			0.578852	−	0.815433i	$-0.303501\pi$
$991$	36.4447			1.15770			0.578852	+	0.815433i	$0.303501\pi$
$992$	−39.8804			−1.26620
$993$		0			0
$994$		−	2.04553i		−	0.0648804i
$995$		−	37.0367i		−	1.17414i
$996$		0			0
$997$	−2.23772			−0.0708692			−0.0354346	−	0.999372i	$-0.511282\pi$
$997$	−2.23772			−0.0708692			−0.0354346	+	0.999372i	$0.511282\pi$
$998$	8.83742			0.279744
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.b.c.298.6		14
3.2	odd	2		429.2.b.b.298.9	yes	14
13.12	even	2	inner	1287.2.b.c.298.9		14
39.5	even	4		5577.2.a.x.1.5		7
39.8	even	4		5577.2.a.y.1.3		7
39.38	odd	2		429.2.b.b.298.6	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.b.b.298.6	✓	14	39.38	odd	2
429.2.b.b.298.9	yes	14	3.2	odd	2
1287.2.b.c.298.6		14	1.1	even	1	trivial
1287.2.b.c.298.9		14	13.12	even	2	inner
5577.2.a.x.1.5		7	39.5	even	4
5577.2.a.y.1.3		7	39.8	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)

\(f(q)\)	\(=\)	\(q-0.584778i q^{2} +1.65803 q^{4} -1.95350i q^{5} +1.51078i q^{7} -2.13914i q^{8} +O(q^{10})\)	\(q-0.584778i q^{2} +1.65803 q^{4} -1.95350i q^{5} +1.51078i q^{7} -2.13914i q^{8} -1.14237 q^{10} -1.00000i q^{11} +(-3.49467 - 0.887280i) q^{13} +0.883474 q^{14} +2.06515 q^{16} +3.44198 q^{17} -5.09587i q^{19} -3.23897i q^{20} -0.584778 q^{22} -0.701611 q^{23} +1.18383 q^{25} +(-0.518862 + 2.04361i) q^{26} +2.50493i q^{28} -5.04937 q^{29} -7.26957i q^{31} -5.48593i q^{32} -2.01280i q^{34} +2.95132 q^{35} -2.08160i q^{37} -2.97995 q^{38} -4.17882 q^{40} +1.03658i q^{41} +5.48659 q^{43} -1.65803i q^{44} +0.410287i q^{46} -3.75225i q^{47} +4.71753 q^{49} -0.692276i q^{50} +(-5.79429 - 1.47114i) q^{52} +0.502747 q^{53} -1.95350 q^{55} +3.23178 q^{56} +2.95276i q^{58} +2.65023i q^{59} -5.38195 q^{61} -4.25109 q^{62} +0.922236 q^{64} +(-1.73330 + 6.82685i) q^{65} +8.47197i q^{67} +5.70692 q^{68} -1.72587i q^{70} -2.31533i q^{71} +2.21094i q^{73} -1.21727 q^{74} -8.44913i q^{76} +1.51078 q^{77} -5.54589 q^{79} -4.03427i q^{80} +0.606168 q^{82} +14.5137i q^{83} -6.72392i q^{85} -3.20844i q^{86} -2.13914 q^{88} +1.80966i q^{89} +(1.34049 - 5.27970i) q^{91} -1.16330 q^{92} -2.19424 q^{94} -9.95480 q^{95} -14.5085i q^{97} -2.75871i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 18 q^{4}+O(q^{10}) \) 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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