LMFDB - Embedded newform 351.2.b.d.298.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [351,2,Mod(298,351)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(351, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("351.298"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 351 = 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	351.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-2,0,0,0,0,0,8,0,0,6,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	no
Analytic conductor:	$2.80274911095$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.8112.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 5x^{2} + 3 $ x^4 + 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			298.1
Root			$-2.07431i$ of defining polynomial
Character	$\chi$	$=$	351.298
Dual form			351.2.b.d.298.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.07431i q^{2} -2.30278 q^{4} +2.70236i q^{5} -4.77668i q^{7} +0.628052i q^{8} +5.60555 q^{10} -2.07431i q^{11} +(3.30278 - 1.44626i) q^{13} -9.90833 q^{14} -3.30278 q^{16} -6.90833 q^{17} -3.33042i q^{19} -6.22294i q^{20} -4.30278 q^{22} +3.90833 q^{23} -2.30278 q^{25} +(-3.00000 - 6.85099i) q^{26} +10.9996i q^{28} +3.00000 q^{29} -2.89252i q^{31} +8.10709i q^{32} +14.3300i q^{34} +12.9083 q^{35} +7.66920i q^{37} -6.90833 q^{38} -1.69722 q^{40} -0.628052i q^{41} +5.90833 q^{43} +4.77668i q^{44} -8.10709i q^{46} +5.59489i q^{47} -15.8167 q^{49} +4.77668i q^{50} +(-7.60555 + 3.33042i) q^{52} +6.90833 q^{53} +5.60555 q^{55} +3.00000 q^{56} -6.22294i q^{58} -7.28888i q^{59} +9.60555 q^{61} -6.00000 q^{62} +10.2111 q^{64} +(3.90833 + 8.92530i) q^{65} +8.10709i q^{67} +15.9083 q^{68} -26.7759i q^{70} -2.51221i q^{71} +4.33879i q^{73} +15.9083 q^{74} +7.66920i q^{76} -9.90833 q^{77} +2.69722 q^{79} -8.92530i q^{80} -1.30278 q^{82} +2.26447i q^{83} -18.6688i q^{85} -12.2557i q^{86} +1.30278 q^{88} -2.51221i q^{89} +(-6.90833 - 15.7763i) q^{91} -9.00000 q^{92} +11.6056 q^{94} +9.00000 q^{95} +9.55336i q^{97} +32.8087i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 8 q^{10} + 6 q^{13} - 18 q^{14} - 6 q^{16} - 6 q^{17} - 10 q^{22} - 6 q^{23} - 2 q^{25} - 12 q^{26} + 12 q^{29} + 30 q^{35} - 6 q^{38} - 14 q^{40} + 2 q^{43} - 20 q^{49} - 16 q^{52} + 6 q^{53}+ \cdots + 36 q^{95}+O(q^{100}) $ 4 * q - 2 * q^4 + 8 * q^10 + 6 * q^13 - 18 * q^14 - 6 * q^16 - 6 * q^17 - 10 * q^22 - 6 * q^23 - 2 * q^25 - 12 * q^26 + 12 * q^29 + 30 * q^35 - 6 * q^38 - 14 * q^40 + 2 * q^43 - 20 * q^49 - 16 * q^52 + 6 * q^53 + 8 * q^55 + 12 * q^56 + 24 * q^61 - 24 * q^62 + 12 * q^64 - 6 * q^65 + 42 * q^68 + 42 * q^74 - 18 * q^77 + 18 * q^79 + 2 * q^82 - 2 * q^88 - 6 * q^91 - 36 * q^92 + 32 * q^94 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/351\mathbb{Z}\right)^\times$.

$n$	$28$	$326$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.07431i		−	1.46676i	−0.679818	−	0.733380i	$-0.737941\pi$
$2$		−	2.07431i		−	1.46676i	0.679818	−	0.733380i	$-0.262059\pi$
$3$		0			0
$3$		0			0
$4$	−2.30278			−1.15139
$5$			2.70236i			1.20853i	0.796782	+	0.604267i	$0.206534\pi$
$5$			2.70236i			1.20853i	−0.796782	+	0.604267i	$0.793466\pi$
$6$		0			0
$7$		−	4.77668i		−	1.80541i	−0.430255	−	0.902707i	$-0.641576\pi$
$7$		−	4.77668i		−	1.80541i	0.430255	−	0.902707i	$-0.358424\pi$
$8$			0.628052i			0.222050i
$9$		0			0
$10$	5.60555			1.77263
$11$		−	2.07431i		−	0.625429i	−0.949847	−	0.312714i	$-0.898762\pi$
$11$		−	2.07431i		−	0.625429i	0.949847	−	0.312714i	$-0.101238\pi$
$12$		0			0
$13$	3.30278	−	1.44626i	0.916025	−	0.401121i
$13$	3.30278	−	1.44626i	0.916025	−	0.401121i
$14$	−9.90833			−2.64811
$15$		0			0
$16$	−3.30278			−0.825694
$17$	−6.90833			−1.67552			−0.837758	−	0.546042i	$-0.816134\pi$
$17$	−6.90833			−1.67552			−0.837758	+	0.546042i	$0.816134\pi$
$18$		0			0
$19$		−	3.33042i		−	0.764050i	−0.924152	−	0.382025i	$-0.875227\pi$
$19$		−	3.33042i		−	0.764050i	0.924152	−	0.382025i	$-0.124773\pi$
$20$		−	6.22294i		−	1.39149i
$21$		0			0
$22$	−4.30278			−0.917355
$23$	3.90833			0.814942			0.407471	−	0.913218i	$-0.366411\pi$
$23$	3.90833			0.814942			0.407471	+	0.913218i	$0.366411\pi$
$24$		0			0
$25$	−2.30278			−0.460555
$26$	−3.00000	−	6.85099i	−0.588348	−	1.34359i
$27$		0			0
$28$			10.9996i			2.07873i
$29$	3.00000			0.557086			0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000			0.557086			0.278543	+	0.960424i	$0.410149\pi$
$30$		0			0
$31$		−	2.89252i		−	0.519513i	−0.965674	−	0.259756i	$-0.916358\pi$
$31$		−	2.89252i		−	0.519513i	0.965674	−	0.259756i	$-0.0836422\pi$
$32$			8.10709i			1.43315i
$33$		0			0
$34$			14.3300i			2.45758i
$35$	12.9083			2.18191
$36$		0			0
$37$			7.66920i			1.26081i	0.776267	+	0.630404i	$0.217111\pi$
$37$			7.66920i			1.26081i	−0.776267	+	0.630404i	$0.782889\pi$
$38$	−6.90833			−1.12068
$39$		0			0
$40$	−1.69722			−0.268355
$41$		−	0.628052i		−	0.0980852i	−0.998797	−	0.0490426i	$-0.984383\pi$
$41$		−	0.628052i		−	0.0980852i	0.998797	−	0.0490426i	$-0.0156170\pi$
$42$		0			0
$43$	5.90833			0.901011			0.450506	−	0.892774i	$-0.351244\pi$
$43$	5.90833			0.901011			0.450506	+	0.892774i	$0.351244\pi$
$44$			4.77668i			0.720111i
$45$		0			0
$46$		−	8.10709i		−	1.19533i
$47$			5.59489i			0.816098i	0.912960	+	0.408049i	$0.133791\pi$
$47$			5.59489i			0.816098i	−0.912960	+	0.408049i	$0.866209\pi$
$48$		0			0
$49$	−15.8167			−2.25952
$50$			4.77668i			0.675524i
$51$		0			0
$52$	−7.60555	+	3.33042i	−1.05470	+	0.461846i
$53$	6.90833			0.948932			0.474466	−	0.880274i	$-0.342641\pi$
$53$	6.90833			0.948932			0.474466	+	0.880274i	$0.342641\pi$
$54$		0			0
$55$	5.60555			0.755852
$56$	3.00000			0.400892
$57$		0			0
$58$		−	6.22294i		−	0.817112i
$59$		−	7.28888i		−	0.948932i	−0.880274	−	0.474466i	$-0.842641\pi$
$59$		−	7.28888i		−	0.948932i	0.880274	−	0.474466i	$-0.157359\pi$
$60$		0			0
$61$	9.60555			1.22986			0.614932	−	0.788580i	$-0.289183\pi$
$61$	9.60555			1.22986			0.614932	+	0.788580i	$0.289183\pi$
$62$	−6.00000			−0.762001
$63$		0			0
$64$	10.2111			1.27639
$65$	3.90833	+	8.92530i	0.484768	+	1.10705i
$66$		0			0
$67$			8.10709i			0.990439i	0.868768	+	0.495220i	$0.164912\pi$
$67$			8.10709i			0.990439i	−0.868768	+	0.495220i	$0.835088\pi$
$68$	15.9083			1.92917
$69$		0			0
$70$		−	26.7759i		−	3.20033i
$71$		−	2.51221i		−	0.298144i	−0.988826	−	0.149072i	$-0.952371\pi$
$71$		−	2.51221i		−	0.298144i	0.988826	−	0.149072i	$-0.0476286\pi$
$72$		0			0
$73$			4.33879i			0.507816i	0.967228	+	0.253908i	$0.0817161\pi$
$73$			4.33879i			0.507816i	−0.967228	+	0.253908i	$0.918284\pi$
$74$	15.9083			1.84931
$75$		0			0
$76$			7.66920i			0.879718i
$77$	−9.90833			−1.12916
$78$		0			0
$79$	2.69722			0.303461			0.151731	−	0.988422i	$-0.451515\pi$
$79$	2.69722			0.303461			0.151731	+	0.988422i	$0.451515\pi$
$80$		−	8.92530i		−	0.997879i
$81$		0			0
$82$	−1.30278			−0.143868
$83$			2.26447i			0.248558i	0.992247	+	0.124279i	$0.0396618\pi$
$83$			2.26447i			0.248558i	−0.992247	+	0.124279i	$0.960338\pi$
$84$		0			0
$85$		−	18.6688i		−	2.02492i
$86$		−	12.2557i		−	1.32157i
$87$		0			0
$88$	1.30278			0.138876
$89$		−	2.51221i		−	0.266293i	−0.991096	−	0.133147i	$-0.957492\pi$
$89$		−	2.51221i		−	0.266293i	0.991096	−	0.133147i	$-0.0425081\pi$
$90$		0			0
$91$	−6.90833	−	15.7763i	−0.724189	−	1.65381i
$92$	−9.00000			−0.938315
$93$		0			0
$94$	11.6056			1.19702
$95$	9.00000			0.923381
$96$		0			0
$97$			9.55336i			0.969996i	0.874515	+	0.484998i	$0.161180\pi$
$97$			9.55336i			0.969996i	−0.874515	+	0.484998i	$0.838820\pi$
$98$			32.8087i			3.31418i
$99$		0			0
$100$	5.30278			0.530278
$101$	−15.0000			−1.49256			−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000			−1.49256			−0.746278	+	0.665635i	$0.768161\pi$
$102$		0			0
$103$	−10.9083			−1.07483			−0.537415	−	0.843318i	$-0.680599\pi$
$103$	−10.9083			−1.07483			−0.537415	+	0.843318i	$0.680599\pi$
$104$	0.908327	+	2.07431i	0.0890688	+	0.203403i
$105$		0			0
$106$		−	14.3300i		−	1.39186i
$107$	2.09167			0.202210			0.101105	−	0.994876i	$-0.467762\pi$
$107$	2.09167			0.202210			0.101105	+	0.994876i	$0.467762\pi$
$108$		0			0
$109$			3.33042i			0.318996i	0.987198	+	0.159498i	$0.0509876\pi$
$109$			3.33042i			0.318996i	−0.987198	+	0.159498i	$0.949012\pi$
$110$		−	11.6277i		−	1.10865i
$111$		0			0
$112$			15.7763i			1.49072i
$113$	15.0000			1.41108			0.705541	−	0.708669i	$-0.250704\pi$
$113$	15.0000			1.41108			0.705541	+	0.708669i	$0.250704\pi$
$114$		0			0
$115$			10.5617i			0.984886i
$116$	−6.90833			−0.641422
$117$		0			0
$118$	−15.1194			−1.39186
$119$			32.9989i			3.02500i
$120$		0			0
$121$	6.69722			0.608839
$122$		−	19.9249i		−	1.80392i
$123$		0			0
$124$			6.66083i			0.598160i
$125$			7.28888i			0.651938i
$126$		0			0
$127$	17.4222			1.54597			0.772985	−	0.634424i	$-0.218763\pi$
$127$	17.4222			1.54597			0.772985	+	0.634424i	$0.218763\pi$
$128$		−	4.96684i		−	0.439010i
$129$		0			0
$130$	18.5139	−	8.10709i	1.62377	−	0.711039i
$131$	10.8167			0.945055			0.472528	−	0.881316i	$-0.343342\pi$
$131$	10.8167			0.945055			0.472528	+	0.881316i	$0.343342\pi$
$132$		0			0
$133$	−15.9083			−1.37943
$134$	16.8167			1.45274
$135$		0			0
$136$		−	4.33879i		−	0.372048i
$137$		−	14.0823i		−	1.20313i	−0.798823	−	0.601566i	$-0.794544\pi$
$137$		−	14.0823i		−	1.20313i	0.798823	−	0.601566i	$-0.205456\pi$
$138$		0			0
$139$	−13.0000			−1.10265			−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000			−1.10265			−0.551323	+	0.834292i	$0.685877\pi$
$140$	−29.7250			−2.51222
$141$		0			0
$142$	−5.21110			−0.437306
$143$	−3.00000	−	6.85099i	−0.250873	−	0.572909i
$144$		0			0
$145$			8.10709i			0.673258i
$146$	9.00000			0.744845
$147$		0			0
$148$		−	17.6605i		−	1.45168i
$149$		−	4.96684i		−	0.406899i	−0.979085	−	0.203450i	$-0.934785\pi$
$149$		−	4.96684i		−	0.406899i	0.979085	−	0.203450i	$-0.0652153\pi$
$150$		0			0
$151$			10.5617i			0.859501i	0.902948	+	0.429751i	$0.141398\pi$
$151$			10.5617i			0.859501i	−0.902948	+	0.429751i	$0.858602\pi$
$152$	2.09167			0.169657
$153$		0			0
$154$			20.5530i			1.65621i
$155$	7.81665			0.627849
$156$		0			0
$157$	7.72498			0.616521			0.308260	−	0.951302i	$-0.400253\pi$
$157$	7.72498			0.616521			0.308260	+	0.951302i	$0.400253\pi$
$158$		−	5.59489i		−	0.445105i
$159$		0			0
$160$	−21.9083			−1.73201
$161$		−	18.6688i		−	1.47131i
$162$		0			0
$163$			2.32205i			0.181877i	0.995857	+	0.0909384i	$0.0289866\pi$
$163$			2.32205i			0.181877i	−0.995857	+	0.0909384i	$0.971013\pi$
$164$			1.44626i			0.112934i
$165$		0			0
$166$	4.69722			0.364575
$167$			6.03278i			0.466831i	0.972377	+	0.233415i	$0.0749902\pi$
$167$			6.03278i			0.466831i	−0.972377	+	0.233415i	$0.925010\pi$
$168$		0			0
$169$	8.81665	−	9.55336i	0.678204	−	0.734874i
$170$	−38.7250			−2.97007
$171$		0			0
$172$	−13.6056			−1.03741
$173$	−3.90833			−0.297145			−0.148572	−	0.988902i	$-0.547468\pi$
$173$	−3.90833			−0.297145			−0.148572	+	0.988902i	$0.547468\pi$
$174$		0			0
$175$			10.9996i			0.831493i
$176$			6.85099i			0.516413i
$177$		0			0
$178$	−5.21110			−0.390589
$179$	−11.0917			−0.829031			−0.414515	−	0.910042i	$-0.636049\pi$
$179$	−11.0917			−0.829031			−0.414515	+	0.910042i	$0.636049\pi$
$180$		0			0
$181$	8.90833			0.662151			0.331075	−	0.943604i	$-0.392589\pi$
$181$	8.90833			0.662151			0.331075	+	0.943604i	$0.392589\pi$
$182$	−32.7250	+	14.3300i	−2.42574	+	1.06221i
$183$		0			0
$184$			2.45463i			0.180958i
$185$	−20.7250			−1.52373
$186$		0			0
$187$			14.3300i			1.04792i
$188$		−	12.8838i		−	0.939646i
$189$		0			0
$190$		−	18.6688i		−	1.35438i
$191$	−8.72498			−0.631317			−0.315659	−	0.948873i	$-0.602225\pi$
$191$	−8.72498			−0.631317			−0.315659	+	0.948873i	$0.602225\pi$
$192$		0			0
$193$			22.4371i			1.61506i	0.589827	+	0.807530i	$0.299196\pi$
$193$			22.4371i			1.61506i	−0.589827	+	0.807530i	$0.700804\pi$
$194$	19.8167			1.42275
$195$		0			0
$196$	36.4222			2.60159
$197$		−	26.5282i		−	1.89005i	−0.326991	−	0.945027i	$-0.606035\pi$
$197$		−	26.5282i		−	1.89005i	0.326991	−	0.945027i	$-0.393965\pi$
$198$		0			0
$199$	−26.1194			−1.85156			−0.925779	−	0.378066i	$-0.876589\pi$
$199$	−26.1194			−1.85156			−0.925779	+	0.378066i	$0.876589\pi$
$200$		−	1.44626i		−	0.102266i
$201$		0			0
$202$			31.1147i			2.18922i
$203$		−	14.3300i		−	1.00577i
$204$		0			0
$205$	1.69722			0.118539
$206$			22.6273i			1.57652i
$207$		0			0
$208$	−10.9083	+	4.77668i	−0.756356	+	0.331203i
$209$	−6.90833			−0.477859
$210$		0			0
$211$	−5.11943			−0.352436			−0.176218	−	0.984351i	$-0.556386\pi$
$211$	−5.11943			−0.352436			−0.176218	+	0.984351i	$0.556386\pi$
$212$	−15.9083			−1.09259
$213$		0			0
$214$		−	4.33879i		−	0.296593i
$215$			15.9665i			1.08890i
$216$		0			0
$217$	−13.8167			−0.937936
$218$	6.90833			0.467891
$219$		0			0
$220$	−12.9083			−0.870279
$221$	−22.8167	+	9.99125i	−1.53481	+	0.672084i
$222$		0			0
$223$		−	0.437893i		−	0.0293235i	−0.999893	−	0.0146617i	$-0.995333\pi$
$223$		−	0.437893i		−	0.0293235i	0.999893	−	0.0146617i	$-0.00466714\pi$
$224$	38.7250			2.58742
$225$		0			0
$226$		−	31.1147i		−	2.06972i
$227$		−	3.08268i		−	0.204605i	−0.994753	−	0.102302i	$-0.967379\pi$
$227$		−	3.08268i		−	0.204605i	0.994753	−	0.102302i	$-0.0326209\pi$
$228$		0			0
$229$		−	19.5446i		−	1.29154i	−0.763530	−	0.645772i	$-0.776536\pi$
$229$		−	19.5446i		−	1.29154i	0.763530	−	0.645772i	$-0.223464\pi$
$230$	21.9083			1.44459
$231$		0			0
$232$			1.88415i			0.123701i
$233$	4.81665			0.315549			0.157775	−	0.987475i	$-0.449568\pi$
$233$	4.81665			0.315549			0.157775	+	0.987475i	$0.449568\pi$
$234$		0			0
$235$	−15.1194			−0.986283
$236$			16.7847i			1.09259i
$237$		0			0
$238$	68.4500			4.43695
$239$			21.3712i			1.38239i	0.722670	+	0.691194i	$0.242915\pi$
$239$			21.3712i			1.38239i	−0.722670	+	0.691194i	$0.757085\pi$
$240$		0			0
$241$		−	15.7763i		−	1.01624i	−0.861286	−	0.508120i	$-0.830341\pi$
$241$		−	15.7763i		−	1.01624i	0.861286	−	0.508120i	$-0.169659\pi$
$242$		−	13.8921i		−	0.893021i
$243$		0			0
$244$	−22.1194			−1.41605
$245$		−	42.7424i		−	2.73071i
$246$		0			0
$247$	−4.81665	−	10.9996i	−0.306476	−	0.699889i
$248$	1.81665			0.115358
$249$		0			0
$250$	15.1194			0.956237
$251$	24.9083			1.57220			0.786100	−	0.618100i	$-0.212097\pi$
$251$	24.9083			1.57220			0.786100	+	0.618100i	$0.212097\pi$
$252$		0			0
$253$		−	8.10709i		−	0.509689i
$254$		−	36.1391i		−	2.26757i
$255$		0			0
$256$	10.1194			0.632464
$257$	−14.0917			−0.879014			−0.439507	−	0.898239i	$-0.644847\pi$
$257$	−14.0917			−0.879014			−0.439507	+	0.898239i	$0.644847\pi$
$258$		0			0
$259$	36.6333			2.27628
$260$	−9.00000	−	20.5530i	−0.558156	−	1.27464i
$261$		0			0
$262$		−	22.4371i		−	1.38617i
$263$	18.9083			1.16594			0.582969	−	0.812495i	$-0.301891\pi$
$263$	18.9083			1.16594			0.582969	+	0.812495i	$0.301891\pi$
$264$		0			0
$265$			18.6688i			1.14682i
$266$			32.9989i			2.02329i
$267$		0			0
$268$		−	18.6688i		−	1.14038i
$269$	−25.5416			−1.55730			−0.778650	−	0.627458i	$-0.784095\pi$
$269$	−25.5416			−1.55730			−0.778650	+	0.627458i	$0.784095\pi$
$270$		0			0
$271$		−	23.0076i		−	1.39761i	−0.715311	−	0.698806i	$-0.753715\pi$
$271$		−	23.0076i		−	1.39761i	0.715311	−	0.698806i	$-0.246285\pi$
$272$	22.8167			1.38346
$273$		0			0
$274$	−29.2111			−1.76471
$275$			4.77668i			0.288045i
$276$		0			0
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$			26.9661i			1.61732i
$279$		0			0
$280$			8.10709i			0.484492i
$281$			16.5945i			0.989945i	0.868909	+	0.494973i	$0.164822\pi$
$281$			16.5945i			0.989945i	−0.868909	+	0.494973i	$0.835178\pi$
$282$		0			0
$283$	13.5139			0.803317			0.401658	−	0.915790i	$-0.368434\pi$
$283$	13.5139			0.803317			0.401658	+	0.915790i	$0.368434\pi$
$284$			5.78505i			0.343279i
$285$		0			0
$286$	−14.2111	+	6.22294i	−0.840320	+	0.367970i
$287$	−3.00000			−0.177084
$288$		0			0
$289$	30.7250			1.80735
$290$	16.8167			0.987508
$291$		0			0
$292$		−	9.99125i		−	0.584694i
$293$			6.03278i			0.352439i	0.984351	+	0.176219i	$0.0563868\pi$
$293$			6.03278i			0.352439i	−0.984351	+	0.176219i	$0.943613\pi$
$294$		0			0
$295$	19.6972			1.14682
$296$	−4.81665			−0.279962
$297$		0			0
$298$	−10.3028			−0.596824
$299$	12.9083	−	5.65246i	0.746508	−	0.326890i
$300$		0			0
$301$		−	28.2222i		−	1.62670i
$302$	21.9083			1.26068
$303$		0			0
$304$			10.9996i			0.630871i
$305$			25.9577i			1.48633i
$306$		0			0
$307$			24.3213i			1.38809i	0.719932	+	0.694044i	$0.244173\pi$
$307$			24.3213i			1.38809i	−0.719932	+	0.694044i	$0.755827\pi$
$308$	22.8167			1.30010
$309$		0			0
$310$		−	16.2142i		−	0.920904i
$311$	−5.09167			−0.288722			−0.144361	−	0.989525i	$-0.546113\pi$
$311$	−5.09167			−0.288722			−0.144361	+	0.989525i	$0.546113\pi$
$312$		0			0
$313$	−9.02776			−0.510279			−0.255139	−	0.966904i	$-0.582121\pi$
$313$	−9.02776			−0.510279			−0.255139	+	0.966904i	$0.582121\pi$
$314$		−	16.0240i		−	0.904288i
$315$		0			0
$316$	−6.21110			−0.349402
$317$		−	3.08268i		−	0.173141i	−0.996246	−	0.0865703i	$-0.972409\pi$
$317$		−	3.08268i		−	0.173141i	0.996246	−	0.0865703i	$-0.0275907\pi$
$318$		0			0
$319$		−	6.22294i		−	0.348418i
$320$			27.5941i			1.54256i
$321$		0			0
$322$	−38.7250			−2.15806
$323$			23.0076i			1.28018i
$324$		0			0
$325$	−7.60555	+	3.33042i	−0.421880	+	0.184738i
$326$	4.81665			0.266770
$327$		0			0
$328$	0.394449			0.0217798
$329$	26.7250			1.47340
$330$		0			0
$331$		−	2.75994i		−	0.151700i	−0.997119	−	0.0758500i	$-0.975833\pi$
$331$		−	2.75994i		−	0.151700i	0.997119	−	0.0758500i	$-0.0241670\pi$
$332$		−	5.21457i		−	0.286187i
$333$		0			0
$334$	12.5139			0.684729
$335$	−21.9083			−1.19698
$336$		0			0
$337$	−16.2111			−0.883075			−0.441538	−	0.897243i	$-0.645567\pi$
$337$	−16.2111			−0.883075			−0.441538	+	0.897243i	$0.645567\pi$
$338$	−19.8167	−	18.2885i	−1.07788	−	0.994763i
$339$		0			0
$340$			42.9901i			2.33147i
$341$	−6.00000			−0.324918
$342$		0			0
$343$			42.1143i			2.27396i
$344$			3.71073i			0.200069i
$345$		0			0
$346$			8.10709i			0.435840i
$347$	−16.8167			−0.902765			−0.451383	−	0.892330i	$-0.649069\pi$
$347$	−16.8167			−0.902765			−0.451383	+	0.892330i	$0.649069\pi$
$348$		0			0
$349$		−	10.1238i		−	0.541916i	−0.962591	−	0.270958i	$-0.912659\pi$
$349$		−	10.1238i		−	0.541916i	0.962591	−	0.270958i	$-0.0873405\pi$
$350$	22.8167			1.21960
$351$		0			0
$352$	16.8167			0.896331
$353$		−	4.39636i		−	0.233995i	−0.993132	−	0.116997i	$-0.962673\pi$
$353$		−	4.39636i		−	0.233995i	0.993132	−	0.116997i	$-0.0373269\pi$
$354$		0			0
$355$	6.78890			0.360317
$356$			5.78505i			0.306607i
$357$		0			0
$358$			23.0076i			1.21599i
$359$		−	10.1814i		−	0.537354i	−0.963230	−	0.268677i	$-0.913414\pi$
$359$		−	10.1814i		−	0.537354i	0.963230	−	0.268677i	$-0.0865864\pi$
$360$		0			0
$361$	7.90833			0.416228
$362$		−	18.4787i		−	0.971217i
$363$		0			0
$364$	15.9083	+	36.3293i	0.833823	+	1.90417i
$365$	−11.7250			−0.613714
$366$		0			0
$367$	−5.18335			−0.270568			−0.135284	−	0.990807i	$-0.543195\pi$
$367$	−5.18335			−0.270568			−0.135284	+	0.990807i	$0.543195\pi$
$368$	−12.9083			−0.672893
$369$		0			0
$370$			42.9901i			2.23495i
$371$		−	32.9989i		−	1.71322i
$372$		0			0
$373$	−4.48612			−0.232283			−0.116141	−	0.993233i	$-0.537053\pi$
$373$	−4.48612			−0.232283			−0.116141	+	0.993233i	$0.537053\pi$
$374$	29.7250			1.53704
$375$		0			0
$376$	−3.51388			−0.181214
$377$	9.90833	−	4.33879i	0.510305	−	0.223459i
$378$		0			0
$379$		−	29.6684i		−	1.52397i	−0.647597	−	0.761983i	$-0.724226\pi$
$379$		−	29.6684i		−	1.52397i	0.647597	−	0.761983i	$-0.275774\pi$
$380$	−20.7250			−1.06317
$381$		0			0
$382$			18.0983i			0.925992i
$383$		−	34.9406i		−	1.78538i	−0.450671	−	0.892690i	$-0.648815\pi$
$383$		−	34.9406i		−	1.78538i	0.450671	−	0.892690i	$-0.351185\pi$
$384$		0			0
$385$		−	26.7759i		−	1.36463i
$386$	46.5416			2.36891
$387$		0			0
$388$		−	21.9992i		−	1.11684i
$389$	−33.6333			−1.70528			−0.852638	−	0.522502i	$-0.824999\pi$
$389$	−33.6333			−1.70528			−0.852638	+	0.522502i	$0.824999\pi$
$390$		0			0
$391$	−27.0000			−1.36545
$392$		−	9.93367i		−	0.501726i
$393$		0			0
$394$	−55.0278			−2.77226
$395$			7.28888i			0.366744i
$396$		0			0
$397$		−	12.3133i		−	0.617987i	−0.951064	−	0.308993i	$-0.900008\pi$
$397$		−	12.3133i		−	0.617987i	0.951064	−	0.308993i	$-0.0999921\pi$
$398$			54.1799i			2.71579i
$399$		0			0
$400$	7.60555			0.380278
$401$			20.3628i			1.01687i	0.861100	+	0.508435i	$0.169776\pi$
$401$			20.3628i			1.01687i	−0.861100	+	0.508435i	$0.830224\pi$
$402$		0			0
$403$	−4.18335	−	9.55336i	−0.208387	−	0.475887i
$404$	34.5416			1.71851
$405$		0			0
$406$	−29.7250			−1.47523
$407$	15.9083			0.788546
$408$		0			0
$409$			25.7675i			1.27412i	0.770813	+	0.637062i	$0.219850\pi$
$409$			25.7675i			1.27412i	−0.770813	+	0.637062i	$0.780150\pi$
$410$		−	3.52058i		−	0.173869i
$411$		0			0
$412$	25.1194			1.23755
$413$	−34.8167			−1.71322
$414$		0			0
$415$	−6.11943			−0.300391
$416$	11.7250	+	26.7759i	0.574864	+	1.31280i
$417$		0			0
$418$			14.3300i			0.700905i
$419$	19.8167			0.968107			0.484053	−	0.875038i	$-0.339164\pi$
$419$	19.8167			0.968107			0.484053	+	0.875038i	$0.339164\pi$
$420$		0			0
$421$			18.2309i			0.888521i	0.895898	+	0.444261i	$0.146534\pi$
$421$			18.2309i			0.888521i	−0.895898	+	0.444261i	$0.853466\pi$
$422$			10.6193i			0.516939i
$423$		0			0
$424$			4.33879i			0.210710i
$425$	15.9083			0.771667
$426$		0			0
$427$		−	45.8826i		−	2.22042i
$428$	−4.81665			−0.232822
$429$		0			0
$430$	33.1194			1.59716
$431$		−	18.8590i		−	0.908405i	−0.890898	−	0.454203i	$-0.849924\pi$
$431$		−	18.8590i		−	0.908405i	0.890898	−	0.454203i	$-0.150076\pi$
$432$		0			0
$433$	15.3305			0.736738			0.368369	−	0.929680i	$-0.379916\pi$
$433$	15.3305			0.736738			0.368369	+	0.929680i	$0.379916\pi$
$434$			28.6601i			1.37573i
$435$		0			0
$436$		−	7.66920i		−	0.367288i
$437$		−	13.0164i		−	0.622657i
$438$		0			0
$439$	8.00000			0.381819			0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000			0.381819			0.190910	+	0.981608i	$0.438856\pi$
$440$			3.52058i			0.167837i
$441$		0			0
$442$	20.7250	+	47.3289i	0.985787	+	2.25121i
$443$	16.8167			0.798983			0.399492	−	0.916737i	$-0.369187\pi$
$443$	16.8167			0.798983			0.399492	+	0.916737i	$0.369187\pi$
$444$		0			0
$445$	6.78890			0.321825
$446$	−0.908327			−0.0430105
$447$		0			0
$448$		−	48.7752i		−	2.30441i
$449$			27.1562i			1.28158i	0.767715	+	0.640791i	$0.221393\pi$
$449$			27.1562i			1.28158i	−0.767715	+	0.640791i	$0.778607\pi$
$450$		0			0
$451$	−1.30278			−0.0613453
$452$	−34.5416			−1.62470
$453$		0			0
$454$	−6.39445			−0.300106
$455$	42.6333	−	18.6688i	1.99868	−	0.875208i
$456$		0			0
$457$		−	0.132583i		−	0.00620198i	−0.999995	−	0.00310099i	$-0.999013\pi$
$457$		−	0.132583i		−	0.00620198i	0.999995	−	0.00310099i	$-0.000987078\pi$
$458$	−40.5416			−1.89439
$459$		0			0
$460$		−	24.3213i		−	1.13399i
$461$			17.9082i			0.834067i	0.908891	+	0.417034i	$0.136930\pi$
$461$			17.9082i			0.834067i	−0.908891	+	0.417034i	$0.863070\pi$
$462$		0			0
$463$		−	8.23968i		−	0.382930i	−0.981499	−	0.191465i	$-0.938676\pi$
$463$		−	8.23968i		−	0.382930i	0.981499	−	0.191465i	$-0.0613239\pi$
$464$	−9.90833			−0.459983
$465$		0			0
$466$		−	9.99125i		−	0.462836i
$467$	−36.6333			−1.69519			−0.847594	−	0.530646i	$-0.821949\pi$
$467$	−36.6333			−1.69519			−0.847594	+	0.530646i	$0.821949\pi$
$468$		0			0
$469$	38.7250			1.78815
$470$			31.3624i			1.44664i
$471$		0			0
$472$	4.57779			0.210710
$473$		−	12.2557i		−	0.563519i
$474$		0			0
$475$			7.66920i			0.351887i
$476$		−	75.9890i		−	3.48295i
$477$		0			0
$478$	44.3305			2.02763
$479$			4.58652i			0.209563i	0.994495	+	0.104782i	$0.0334144\pi$
$479$			4.58652i			0.209563i	−0.994495	+	0.104782i	$0.966586\pi$
$480$		0			0
$481$	11.0917	+	25.3297i	0.505737	+	1.15493i
$482$	−32.7250			−1.49058
$483$		0			0
$484$	−15.4222			−0.701009
$485$	−25.8167			−1.17227
$486$		0			0
$487$		−	6.66083i		−	0.301831i	−0.988547	−	0.150916i	$-0.951778\pi$
$487$		−	6.66083i		−	0.301831i	0.988547	−	0.150916i	$-0.0482222\pi$
$488$			6.03278i			0.273091i
$489$		0			0
$490$	−88.6611			−4.00530
$491$	−9.90833			−0.447157			−0.223578	−	0.974686i	$-0.571774\pi$
$491$	−9.90833			−0.447157			−0.223578	+	0.974686i	$0.571774\pi$
$492$		0			0
$493$	−20.7250			−0.933406
$494$	−22.8167	+	9.99125i	−1.02657	+	0.449528i
$495$		0			0
$496$			9.55336i			0.428958i
$497$	−12.0000			−0.538274
$498$		0			0
$499$			2.45463i			0.109884i	0.998490	+	0.0549422i	$0.0174974\pi$
$499$			2.45463i			0.109884i	−0.998490	+	0.0549422i	$0.982503\pi$
$500$		−	16.7847i		−	0.750633i
$501$		0			0
$502$		−	51.6677i		−	2.30604i
$503$	18.6333			0.830818			0.415409	−	0.909635i	$-0.363638\pi$
$503$	18.6333			0.830818			0.415409	+	0.909635i	$0.363638\pi$
$504$		0			0
$505$		−	40.5355i		−	1.80380i
$506$	−16.8167			−0.747591
$507$		0			0
$508$	−40.1194			−1.78001
$509$		−	21.6189i		−	0.958242i	−0.877749	−	0.479121i	$-0.840956\pi$
$509$		−	21.6189i		−	0.958242i	0.877749	−	0.479121i	$-0.159044\pi$
$510$		0			0
$511$	20.7250			0.916819
$512$		−	30.9245i		−	1.36668i
$513$		0			0
$514$			29.2305i			1.28930i
$515$		−	29.4783i		−	1.29897i
$516$		0			0
$517$	11.6056			0.510412
$518$		−	75.9890i		−	3.33876i
$519$		0			0
$520$	−5.60555	+	2.45463i	−0.245820	+	0.107643i
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$		0			0
$523$	−13.9083			−0.608169			−0.304084	−	0.952645i	$-0.598350\pi$
$523$	−13.9083			−0.608169			−0.304084	+	0.952645i	$0.598350\pi$
$524$	−24.9083			−1.08813
$525$		0			0
$526$		−	39.2218i		−	1.71015i
$527$			19.9825i			0.870451i
$528$		0			0
$529$	−7.72498			−0.335869
$530$	38.7250			1.68211
$531$		0			0
$532$	36.6333			1.58826
$533$	−0.908327	−	2.07431i	−0.0393440	−	0.0898485i
$534$		0			0
$535$			5.65246i			0.244377i
$536$	−5.09167			−0.219927
$537$		0			0
$538$			52.9814i			2.28419i
$539$			32.8087i			1.41317i
$540$		0			0
$541$			17.6605i			0.759282i	0.925134	+	0.379641i	$0.123952\pi$
$541$			17.6605i			0.759282i	−0.925134	+	0.379641i	$0.876048\pi$
$542$	−47.7250			−2.04996
$543$		0			0
$544$		−	56.0065i		−	2.40126i
$545$	−9.00000			−0.385518
$546$		0			0
$547$	19.0917			0.816301			0.408150	−	0.912915i	$-0.366174\pi$
$547$	19.0917			0.816301			0.408150	+	0.912915i	$0.366174\pi$
$548$			32.4284i			1.38527i
$549$		0			0
$550$	9.90833			0.422492
$551$		−	9.99125i		−	0.425642i
$552$		0			0
$553$		−	12.8838i		−	0.547874i
$554$			20.7431i			0.881291i
$555$		0			0
$556$	29.9361			1.26957
$557$			23.1228i			0.979743i	0.871795	+	0.489871i	$0.162956\pi$
$557$			23.1228i			0.979743i	−0.871795	+	0.489871i	$0.837044\pi$
$558$		0			0
$559$	19.5139	−	8.54499i	0.825349	−	0.361414i
$560$	−42.6333			−1.80159
$561$		0			0
$562$	34.4222			1.45201
$563$	−12.6333			−0.532430			−0.266215	−	0.963914i	$-0.585773\pi$
$563$	−12.6333			−0.532430			−0.266215	+	0.963914i	$0.585773\pi$
$564$		0			0
$565$			40.5355i			1.70534i
$566$		−	28.0320i		−	1.17827i
$567$		0			0
$568$	1.57779			0.0662028
$569$	30.6333			1.28422			0.642108	−	0.766615i	$-0.278060\pi$
$569$	30.6333			1.28422			0.642108	+	0.766615i	$0.278060\pi$
$570$		0			0
$571$	16.2389			0.679575			0.339788	−	0.940502i	$-0.389645\pi$
$571$	16.2389			0.679575			0.339788	+	0.940502i	$0.389645\pi$
$572$	6.90833	+	15.7763i	0.288852	+	0.659640i
$573$		0			0
$574$			6.22294i			0.259740i
$575$	−9.00000			−0.375326
$576$		0			0
$577$			13.0164i			0.541878i	0.962597	+	0.270939i	$0.0873342\pi$
$577$			13.0164i			0.541878i	−0.962597	+	0.270939i	$0.912666\pi$
$578$		−	63.7332i		−	2.65095i
$579$		0			0
$580$		−	18.6688i		−	0.775181i
$581$	10.8167			0.448750
$582$		0			0
$583$		−	14.3300i		−	0.593489i
$584$	−2.72498			−0.112761
$585$		0			0
$586$	12.5139			0.516944
$587$			27.5941i			1.13893i	0.822015	+	0.569466i	$0.192850\pi$
$587$			27.5941i			1.13893i	−0.822015	+	0.569466i	$0.807150\pi$
$588$		0			0
$589$	−9.63331			−0.396934
$590$		−	40.8582i		−	1.68211i
$591$		0			0
$592$		−	25.3297i		−	1.04104i
$593$			45.8251i			1.88181i	0.338672	+	0.940905i	$0.390022\pi$
$593$			45.8251i			1.88181i	−0.338672	+	0.940905i	$0.609978\pi$
$594$		0			0
$595$	−89.1749			−3.65582
$596$			11.4375i			0.468499i
$597$		0			0
$598$	−11.7250	−	26.7759i	−0.479470	−	1.09495i
$599$	−21.6333			−0.883913			−0.441956	−	0.897036i	$-0.645715\pi$
$599$	−21.6333			−0.883913			−0.441956	+	0.897036i	$0.645715\pi$
$600$		0			0
$601$	−33.3028			−1.35845			−0.679224	−	0.733931i	$-0.737684\pi$
$601$	−33.3028			−1.35845			−0.679224	+	0.733931i	$0.737684\pi$
$602$	−58.5416			−2.38598
$603$		0			0
$604$		−	24.3213i		−	0.989619i
$605$			18.0983i			0.735802i
$606$		0			0
$607$	7.72498			0.313547			0.156774	−	0.987635i	$-0.449891\pi$
$607$	7.72498			0.313547			0.156774	+	0.987635i	$0.449891\pi$
$608$	27.0000			1.09499
$609$		0			0
$610$	53.8444			2.18010
$611$	8.09167	+	18.4787i	0.327354	+	0.747567i
$612$		0			0
$613$		−	5.65246i		−	0.228301i	−0.993463	−	0.114150i	$-0.963585\pi$
$613$		−	5.65246i		−	0.228301i	0.993463	−	0.114150i	$-0.0364146\pi$
$614$	50.4500			2.03599
$615$		0			0
$616$		−	6.22294i		−	0.250729i
$617$		−	24.9493i		−	1.00442i	−0.864745	−	0.502211i	$-0.832520\pi$
$617$		−	24.9493i		−	1.00442i	0.864745	−	0.502211i	$-0.167480\pi$
$618$		0			0
$619$			16.2142i			0.651703i	0.945421	+	0.325852i	$0.105651\pi$
$619$			16.2142i			0.651703i	−0.945421	+	0.325852i	$0.894349\pi$
$620$	−18.0000			−0.722897
$621$		0			0
$622$			10.5617i			0.423487i
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−31.2111			−1.24844
$626$			18.7264i			0.748457i
$627$		0			0
$628$	−17.7889			−0.709854
$629$		−	52.9814i		−	2.11250i
$630$		0			0
$631$			27.6517i			1.10080i	0.834902	+	0.550398i	$0.185524\pi$
$631$			27.6517i			1.10080i	−0.834902	+	0.550398i	$0.814476\pi$
$632$			1.69400i			0.0673835i
$633$		0			0
$634$	−6.39445			−0.253956
$635$			47.0812i			1.86836i
$636$		0			0
$637$	−52.2389	+	22.8750i	−2.06978	+	0.906341i
$638$	−12.9083			−0.511046
$639$		0			0
$640$	13.4222			0.530559
$641$	−1.81665			−0.0717535			−0.0358768	−	0.999356i	$-0.511422\pi$
$641$	−1.81665			−0.0717535			−0.0358768	+	0.999356i	$0.511422\pi$
$642$		0			0
$643$			26.7759i			1.05594i	0.849263	+	0.527969i	$0.177046\pi$
$643$			26.7759i			1.05594i	−0.849263	+	0.527969i	$0.822954\pi$
$644$			42.9901i			1.69405i
$645$		0			0
$646$	47.7250			1.87771
$647$	34.5416			1.35797			0.678986	−	0.734151i	$-0.262420\pi$
$647$	34.5416			1.35797			0.678986	+	0.734151i	$0.262420\pi$
$648$		0			0
$649$	−15.1194			−0.593490
$650$	6.90833	+	15.7763i	0.270967	+	0.618797i
$651$		0			0
$652$		−	5.34715i		−	0.209411i
$653$	1.81665			0.0710912			0.0355456	−	0.999368i	$-0.488683\pi$
$653$	1.81665			0.0710912			0.0355456	+	0.999368i	$0.488683\pi$
$654$		0			0
$655$			29.2305i			1.14213i
$656$			2.07431i			0.0809883i
$657$		0			0
$658$		−	55.4360i		−	2.16112i
$659$	12.6333			0.492124			0.246062	−	0.969254i	$-0.420863\pi$
$659$	12.6333			0.492124			0.246062	+	0.969254i	$0.420863\pi$
$660$		0			0
$661$			46.7584i			1.81869i	0.416041	+	0.909346i	$0.363417\pi$
$661$			46.7584i			1.81869i	−0.416041	+	0.909346i	$0.636583\pi$
$662$	−5.72498			−0.222508
$663$		0			0
$664$	−1.42221			−0.0551923
$665$		−	42.9901i		−	1.66708i
$666$		0			0
$667$	11.7250			0.453993
$668$		−	13.8921i		−	0.537503i
$669$		0			0
$670$			45.4447i			1.75568i
$671$		−	19.9249i		−	0.769193i
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$			33.6269i			1.29526i
$675$		0			0
$676$	−20.3028	+	21.9992i	−0.780876	+	0.846124i
$677$	39.0000			1.49889			0.749446	−	0.662066i	$-0.230320\pi$
$677$	39.0000			1.49889			0.749446	+	0.662066i	$0.230320\pi$
$678$		0			0
$679$	45.6333			1.75125
$680$	11.7250			0.449632
$681$		0			0
$682$			12.4459i			0.476577i
$683$			19.4870i			0.745650i	0.927902	+	0.372825i	$0.121611\pi$
$683$			19.4870i			0.745650i	−0.927902	+	0.372825i	$0.878389\pi$
$684$		0			0
$685$	38.0555			1.45403
$686$	87.3583			3.33535
$687$		0			0
$688$	−19.5139			−0.743960
$689$	22.8167	−	9.99125i	0.869245	−	0.380636i
$690$		0			0
$691$		−	25.7675i		−	0.980244i	−0.871654	−	0.490122i	$-0.836952\pi$
$691$		−	25.7675i		−	0.980244i	0.871654	−	0.490122i	$-0.163048\pi$
$692$	9.00000			0.342129
$693$		0			0
$694$			34.8830i			1.32414i
$695$		−	35.1307i		−	1.33259i
$696$		0			0
$697$			4.33879i			0.164343i
$698$	−21.0000			−0.794862
$699$		0			0
$700$		−	25.3297i		−	0.957371i
$701$	4.81665			0.181923			0.0909613	−	0.995854i	$-0.471006\pi$
$701$	4.81665			0.181923			0.0909613	+	0.995854i	$0.471006\pi$
$702$		0			0
$703$	25.5416			0.963321
$704$		−	21.1810i		−	0.798290i
$705$		0			0
$706$	−9.11943			−0.343214
$707$			71.6502i			2.69468i
$708$		0			0
$709$		−	10.1238i		−	0.380209i	−0.981764	−	0.190104i	$-0.939117\pi$
$709$		−	10.1238i		−	0.380209i	0.981764	−	0.190104i	$-0.0608826\pi$
$710$		−	14.0823i		−	0.528499i
$711$		0			0
$712$	1.57779			0.0591304
$713$		−	11.3049i		−	0.423373i
$714$		0			0
$715$	18.5139	−	8.10709i	0.692380	−	0.303188i
$716$	25.5416			0.954536
$717$		0			0
$718$	−21.1194			−0.788170
$719$	45.6333			1.70184			0.850918	−	0.525299i	$-0.176047\pi$
$719$	45.6333			1.70184			0.850918	+	0.525299i	$0.176047\pi$
$720$		0			0
$721$			52.1056i			1.94051i
$722$		−	16.4043i		−	0.610507i
$723$		0			0
$724$	−20.5139			−0.762392
$725$	−6.90833			−0.256569
$726$		0			0
$727$	−49.8444			−1.84863			−0.924313	−	0.381634i	$-0.875361\pi$
$727$	−49.8444			−1.84863			−0.924313	+	0.381634i	$0.875361\pi$
$728$	9.90833	−	4.33879i	0.367227	−	0.160806i
$729$		0			0
$730$			24.3213i			0.900171i
$731$	−40.8167			−1.50966
$732$		0			0
$733$			27.7843i			1.02624i	0.858318	+	0.513118i	$0.171510\pi$
$733$			27.7843i			1.02624i	−0.858318	+	0.513118i	$0.828490\pi$
$734$			10.7519i			0.396859i
$735$		0			0
$736$			31.6852i			1.16793i
$737$	16.8167			0.619449
$738$		0			0
$739$			16.6521i			0.612557i	0.951942	+	0.306278i	$0.0990838\pi$
$739$			16.6521i			0.612557i	−0.951942	+	0.306278i	$0.900916\pi$
$740$	47.7250			1.75441
$741$		0			0
$742$	−68.4500			−2.51288
$743$			26.7183i			0.980201i	0.871666	+	0.490100i	$0.163040\pi$
$743$			26.7183i			0.980201i	−0.871666	+	0.490100i	$0.836960\pi$
$744$		0			0
$745$	13.4222			0.491752
$746$			9.30562i			0.340703i
$747$		0			0
$748$		−	32.9989i		−	1.20656i
$749$		−	9.99125i		−	0.365072i
$750$		0			0
$751$	−24.7250			−0.902227			−0.451114	−	0.892466i	$-0.648973\pi$
$751$	−24.7250			−0.902227			−0.451114	+	0.892466i	$0.648973\pi$
$752$		−	18.4787i		−	0.673847i
$753$		0			0
$754$	−9.00000	−	20.5530i	−0.327761	−	0.748495i
$755$	−28.5416			−1.03874
$756$		0			0
$757$	−3.93608			−0.143059			−0.0715297	−	0.997438i	$-0.522788\pi$
$757$	−3.93608			−0.143059			−0.0715297	+	0.997438i	$0.522788\pi$
$758$	−61.5416			−2.23529
$759$		0			0
$760$			5.65246i			0.205036i
$761$			2.39706i			0.0868932i	0.999056	+	0.0434466i	$0.0138338\pi$
$761$			2.39706i			0.0868932i	−0.999056	+	0.0434466i	$0.986166\pi$
$762$		0			0
$763$	15.9083			0.575920
$764$	20.0917			0.726891
$765$		0			0
$766$	−72.4777			−2.61873
$767$	−10.5416	−	24.0735i	−0.380636	−	0.869245i
$768$		0			0
$769$		−	28.5275i		−	1.02873i	−0.857572	−	0.514364i	$-0.828028\pi$
$769$		−	28.5275i		−	1.02873i	0.857572	−	0.514364i	$-0.171972\pi$
$770$	−55.5416			−2.00158
$771$		0			0
$772$		−	51.6677i		−	1.85956i
$773$			22.8174i			0.820686i	0.911931	+	0.410343i	$0.134591\pi$
$773$			22.8174i			0.820686i	−0.911931	+	0.410343i	$0.865409\pi$
$774$		0			0
$775$			6.66083i			0.239264i
$776$	−6.00000			−0.215387
$777$		0			0
$778$			69.7660i			2.50123i
$779$	−2.09167			−0.0749420
$780$		0			0
$781$	−5.21110			−0.186468
$782$			56.0065i			2.00279i
$783$		0			0
$784$	52.2389			1.86567
$785$			20.8757i			0.745086i
$786$		0			0
$787$		−	1.88415i		−	0.0671629i	−0.999436	−	0.0335814i	$-0.989309\pi$
$787$		−	1.88415i		−	0.0671629i	0.999436	−	0.0335814i	$-0.0106913\pi$
$788$			61.0884i			2.17619i
$789$		0			0
$790$	15.1194			0.537925
$791$		−	71.6502i		−	2.54759i
$792$		0			0
$793$	31.7250	−	13.8921i	1.12659	−	0.493324i
$794$	−25.5416			−0.906439
$795$		0			0
$796$	60.1472			2.13186
$797$	−39.0000			−1.38145			−0.690725	−	0.723117i	$-0.742709\pi$
$797$	−39.0000			−1.38145			−0.690725	+	0.723117i	$0.742709\pi$
$798$		0			0
$799$		−	38.6513i		−	1.36739i
$800$		−	18.6688i		−	0.660042i
$801$		0			0
$802$	42.2389			1.49151
$803$	9.00000			0.317603
$804$		0			0
$805$	50.4500			1.77813
$806$	−19.8167	+	8.67757i	−0.698012	+	0.305654i
$807$		0			0
$808$		−	9.42077i		−	0.331422i
$809$	21.3583			0.750917			0.375459	−	0.926839i	$-0.377485\pi$
$809$	21.3583			0.750917			0.375459	+	0.926839i	$0.377485\pi$
$810$		0			0
$811$		−	25.0243i		−	0.878724i	−0.898310	−	0.439362i	$-0.855205\pi$
$811$		−	25.0243i		−	0.878724i	0.898310	−	0.439362i	$-0.144795\pi$
$812$			32.9989i			1.15803i
$813$		0			0
$814$		−	32.9989i		−	1.15661i
$815$	−6.27502			−0.219804
$816$		0			0
$817$		−	19.6772i		−	0.688418i
$818$	53.4500			1.86883
$819$		0			0
$820$	−3.90833			−0.136485
$821$		−	7.42147i		−	0.259011i	−0.991579	−	0.129506i	$-0.958661\pi$
$821$		−	7.42147i		−	0.259011i	0.991579	−	0.129506i	$-0.0413390\pi$
$822$		0			0
$823$	−18.4500			−0.643125			−0.321563	−	0.946888i	$-0.604208\pi$
$823$	−18.4500			−0.643125			−0.321563	+	0.946888i	$0.604208\pi$
$824$		−	6.85099i		−	0.238666i
$825$		0			0
$826$			72.2206i			2.51288i
$827$		−	36.8247i		−	1.28052i	−0.768158	−	0.640261i	$-0.778826\pi$
$827$		−	36.8247i		−	1.28052i	0.768158	−	0.640261i	$-0.221174\pi$
$828$		0			0
$829$	−5.39445			−0.187357			−0.0936785	−	0.995603i	$-0.529863\pi$
$829$	−5.39445			−0.187357			−0.0936785	+	0.995603i	$0.529863\pi$
$830$			12.6936i			0.440602i
$831$		0			0
$832$	33.7250	−	14.7679i	1.16920	−	0.511986i
$833$	109.267			3.78586
$834$		0			0
$835$	−16.3028			−0.564181
$836$	15.9083			0.550201
$837$		0			0
$838$		−	41.1059i		−	1.41998i
$839$			48.5850i			1.67734i	0.544640	+	0.838670i	$0.316666\pi$
$839$			48.5850i			1.67734i	−0.544640	+	0.838670i	$0.683334\pi$
$840$		0			0
$841$	−20.0000			−0.689655
$842$	37.8167			1.30325
$843$		0			0
$844$	11.7889			0.405791
$845$	25.8167	+	23.8258i	0.888120	+	0.819633i
$846$		0			0
$847$		−	31.9905i		−	1.09921i
$848$	−22.8167			−0.783527
$849$		0			0
$850$		−	32.9989i		−	1.13185i
$851$			29.9737i			1.02749i
$852$		0			0
$853$		−	16.7847i		−	0.574696i	−0.957826	−	0.287348i	$-0.907226\pi$
$853$		−	16.7847i		−	0.574696i	0.957826	−	0.287348i	$-0.0927736\pi$
$854$	−95.1749			−3.25682
$855$		0			0
$856$			1.31368i			0.0449006i
$857$	32.7250			1.11786			0.558932	−	0.829213i	$-0.311211\pi$
$857$	32.7250			1.11786			0.558932	+	0.829213i	$0.311211\pi$
$858$		0			0
$859$	25.2389			0.861139			0.430569	−	0.902557i	$-0.358313\pi$
$859$	25.2389			0.861139			0.430569	+	0.902557i	$0.358313\pi$
$860$		−	36.7672i		−	1.25375i
$861$		0			0
$862$	−39.1194			−1.33241
$863$			2.13189i			0.0725703i	0.999341	+	0.0362852i	$0.0115525\pi$
$863$			2.13189i			0.0725703i	−0.999341	+	0.0362852i	$0.988448\pi$
$864$		0			0
$865$		−	10.5617i		−	0.359109i
$866$		−	31.8003i		−	1.08062i
$867$		0			0
$868$	31.8167			1.07993
$869$		−	5.59489i		−	0.189794i
$870$		0			0
$871$	11.7250	+	26.7759i	0.397286	+	0.907267i
$872$	−2.09167			−0.0708330
$873$		0			0
$874$	−27.0000			−0.913289
$875$	34.8167			1.17702
$876$		0			0
$877$		−	28.5275i		−	0.963305i	−0.876362	−	0.481652i	$-0.840037\pi$
$877$		−	28.5275i		−	0.963305i	0.876362	−	0.481652i	$-0.159963\pi$
$878$		−	16.5945i		−	0.560037i
$879$		0			0
$880$	−18.5139			−0.624103
$881$	−14.4500			−0.486832			−0.243416	−	0.969922i	$-0.578268\pi$
$881$	−14.4500			−0.486832			−0.243416	+	0.969922i	$0.578268\pi$
$882$		0			0
$883$	−8.11943			−0.273241			−0.136620	−	0.990623i	$-0.543624\pi$
$883$	−8.11943			−0.273241			−0.136620	+	0.990623i	$0.543624\pi$
$884$	52.5416	−	23.0076i	1.76717	−	0.773829i
$885$		0			0
$886$		−	34.8830i		−	1.17192i
$887$	−3.00000			−0.100730			−0.0503651	−	0.998731i	$-0.516038\pi$
$887$	−3.00000			−0.100730			−0.0503651	+	0.998731i	$0.516038\pi$
$888$		0			0
$889$		−	83.2203i		−	2.79112i
$890$		−	14.0823i		−	0.472040i
$891$		0			0
$892$			1.00837i			0.0337627i
$893$	18.6333			0.623540
$894$		0			0
$895$		−	29.9737i		−	1.00191i
$896$	−23.7250			−0.792596
$897$		0			0
$898$	56.3305			1.87977
$899$		−	8.67757i		−	0.289413i
$900$		0			0
$901$	−47.7250			−1.58995
$902$			2.70236i			0.0899789i
$903$		0			0
$904$			9.42077i			0.313330i
$905$			24.0735i			0.800232i
$906$		0			0
$907$	50.4222			1.67424			0.837121	−	0.547018i	$-0.184237\pi$
$907$	50.4222			1.67424			0.837121	+	0.547018i	$0.184237\pi$
$908$			7.09873i			0.235579i
$909$		0			0
$910$	−38.7250	−	88.4348i	−1.28372	−	2.93159i
$911$	23.7250			0.786044			0.393022	−	0.919529i	$-0.371430\pi$
$911$	23.7250			0.786044			0.393022	+	0.919529i	$0.371430\pi$
$912$		0			0
$913$	4.69722			0.155455
$914$	−0.275019			−0.00909683
$915$		0			0
$916$			45.0068i			1.48707i
$917$		−	51.6677i		−	1.70622i
$918$		0			0
$919$	−42.6611			−1.40726			−0.703629	−	0.710567i	$-0.748438\pi$
$919$	−42.6611			−1.40726			−0.703629	+	0.710567i	$0.748438\pi$
$920$	−6.63331			−0.218694
$921$		0			0
$922$	37.1472			1.22338
$923$	−3.63331	−	8.29725i	−0.119592	−	0.273107i
$924$		0			0
$925$		−	17.6605i		−	0.580672i
$926$	−17.0917			−0.561667
$927$		0			0
$928$			24.3213i			0.798385i
$929$			46.2629i			1.51784i	0.651185	+	0.758919i	$0.274272\pi$
$929$			46.2629i			1.51784i	−0.651185	+	0.758919i	$0.725728\pi$
$930$		0			0
$931$			52.6760i			1.72639i
$932$	−11.0917			−0.363320
$933$		0			0
$934$			75.9890i			2.48643i
$935$	−38.7250			−1.26644
$936$		0			0
$937$	0.880571			0.0287670			0.0143835	−	0.999897i	$-0.495421\pi$
$937$	0.880571			0.0287670			0.0143835	+	0.999897i	$0.495421\pi$
$938$		−	80.3277i		−	2.62279i
$939$		0			0
$940$	34.8167			1.13559
$941$		−	1.63642i		−	0.0533458i	−0.999644	−	0.0266729i	$-0.991509\pi$
$941$		−	1.63642i		−	0.0533458i	0.999644	−	0.0266729i	$-0.00849125\pi$
$942$		0			0
$943$		−	2.45463i		−	0.0799338i
$944$			24.0735i			0.783527i
$945$		0			0
$946$	−25.4222			−0.826547
$947$			28.9078i			0.939377i	0.882832	+	0.469689i	$0.155634\pi$
$947$			28.9078i			0.939377i	−0.882832	+	0.469689i	$0.844366\pi$
$948$		0			0
$949$	6.27502	+	14.3300i	0.203696	+	0.465173i
$950$	15.9083			0.516134
$951$		0			0
$952$	−20.7250			−0.671700
$953$	27.6333			0.895131			0.447565	−	0.894251i	$-0.352291\pi$
$953$	27.6333			0.895131			0.447565	+	0.894251i	$0.352291\pi$
$954$		0			0
$955$		−	23.5781i		−	0.762969i
$956$		−	49.2130i		−	1.59166i
$957$		0			0
$958$	9.51388			0.307379
$959$	−67.2666			−2.17215
$960$		0			0
$961$	22.6333			0.730107
$962$	52.5416	−	23.0076i	1.69401	−	0.741795i
$963$		0			0
$964$			36.3293i			1.17009i
$965$	−60.6333			−1.95186
$966$		0			0
$967$		−	24.8918i		−	0.800465i	−0.916414	−	0.400233i	$-0.868929\pi$
$967$		−	24.8918i		−	0.800465i	0.916414	−	0.400233i	$-0.131071\pi$
$968$			4.20620i			0.135192i
$969$		0			0
$970$			53.5518i			1.71945i
$971$	−49.8167			−1.59869			−0.799346	−	0.600871i	$-0.794821\pi$
$971$	−49.8167			−1.59869			−0.799346	+	0.600871i	$0.794821\pi$
$972$		0			0
$973$			62.0968i			1.99073i
$974$	−13.8167			−0.442714
$975$		0			0
$976$	−31.7250			−1.01549
$977$			29.0404i			0.929084i	0.885551	+	0.464542i	$0.153781\pi$
$977$			29.0404i			0.929084i	−0.885551	+	0.464542i	$0.846219\pi$
$978$		0			0
$979$	−5.21110			−0.166548
$980$			98.4261i			3.14411i
$981$		0			0
$982$			20.5530i			0.655872i
$983$			43.2378i			1.37907i	0.724251	+	0.689536i	$0.242186\pi$
$983$			43.2378i			1.37907i	−0.724251	+	0.689536i	$0.757814\pi$
$984$		0			0
$985$	71.6888			2.28420
$986$			42.9901i			1.36908i
$987$		0			0
$988$	11.0917	+	25.3297i	0.352873	+	0.805844i
$989$	23.0917			0.734272
$990$		0			0
$991$	−34.6333			−1.10016			−0.550082	−	0.835111i	$-0.685403\pi$
$991$	−34.6333			−1.10016			−0.550082	+	0.835111i	$0.685403\pi$
$992$	23.4500			0.744537
$993$		0			0
$994$			24.8918i			0.789519i
$995$		−	70.5842i		−	2.23767i
$996$		0			0
$997$	−22.3583			−0.708094			−0.354047	−	0.935228i	$-0.615195\pi$
$997$	−22.3583			−0.708094			−0.354047	+	0.935228i	$0.615195\pi$
$998$	5.09167			0.161174
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	351.2.b.d.298.1	✓	4
3.2	odd	2		351.2.b.e.298.4	yes	4
13.5	odd	4		4563.2.a.w.1.1		4
13.8	odd	4		4563.2.a.w.1.4		4
13.12	even	2	inner	351.2.b.d.298.4	yes	4
39.5	even	4		4563.2.a.x.1.4		4
39.8	even	4		4563.2.a.x.1.1		4
39.38	odd	2		351.2.b.e.298.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
351.2.b.d.298.1	✓	4	1.1	even	1	trivial
351.2.b.d.298.4	yes	4	13.12	even	2	inner
351.2.b.e.298.1	yes	4	39.38	odd	2
351.2.b.e.298.4	yes	4	3.2	odd	2
4563.2.a.w.1.1		4	13.5	odd	4
4563.2.a.w.1.4		4	13.8	odd	4
4563.2.a.x.1.1		4	39.8	even	4
4563.2.a.x.1.4		4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 351 = 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	351.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.07431i q^{2} -2.30278 q^{4} +2.70236i q^{5} -4.77668i q^{7} +0.628052i q^{8} +5.60555 q^{10} -2.07431i q^{11} +(3.30278 - 1.44626i) q^{13} -9.90833 q^{14} -3.30278 q^{16} -6.90833 q^{17} -3.33042i q^{19} -6.22294i q^{20} -4.30278 q^{22} +3.90833 q^{23} -2.30278 q^{25} +(-3.00000 - 6.85099i) q^{26} +10.9996i q^{28} +3.00000 q^{29} -2.89252i q^{31} +8.10709i q^{32} +14.3300i q^{34} +12.9083 q^{35} +7.66920i q^{37} -6.90833 q^{38} -1.69722 q^{40} -0.628052i q^{41} +5.90833 q^{43} +4.77668i q^{44} -8.10709i q^{46} +5.59489i q^{47} -15.8167 q^{49} +4.77668i q^{50} +(-7.60555 + 3.33042i) q^{52} +6.90833 q^{53} +5.60555 q^{55} +3.00000 q^{56} -6.22294i q^{58} -7.28888i q^{59} +9.60555 q^{61} -6.00000 q^{62} +10.2111 q^{64} +(3.90833 + 8.92530i) q^{65} +8.10709i q^{67} +15.9083 q^{68} -26.7759i q^{70} -2.51221i q^{71} +4.33879i q^{73} +15.9083 q^{74} +7.66920i q^{76} -9.90833 q^{77} +2.69722 q^{79} -8.92530i q^{80} -1.30278 q^{82} +2.26447i q^{83} -18.6688i q^{85} -12.2557i q^{86} +1.30278 q^{88} -2.51221i q^{89} +(-6.90833 - 15.7763i) q^{91} -9.00000 q^{92} +11.6056 q^{94} +9.00000 q^{95} +9.55336i q^{97} +32.8087i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 8 q^{10} + 6 q^{13} - 18 q^{14} - 6 q^{16} - 6 q^{17} - 10 q^{22} - 6 q^{23} - 2 q^{25} - 12 q^{26} + 12 q^{29} + 30 q^{35} - 6 q^{38} - 14 q^{40} + 2 q^{43} - 20 q^{49} - 16 q^{52} + 6 q^{53}+ \cdots + 36 q^{95}+O(q^{100}) \) 4 * q - 2 * q^4 + 8 * q^10 + 6 * q^13 - 18 * q^14 - 6 * q^16 - 6 * q^17 - 10 * q^22 - 6 * q^23 - 2 * q^25 - 12 * q^26 + 12 * q^29 + 30 * q^35 - 6 * q^38 - 14 * q^40 + 2 * q^43 - 20 * q^49 - 16 * q^52 + 6 * q^53 + 8 * q^55 + 12 * q^56 + 24 * q^61 - 24 * q^62 + 12 * q^64 - 6 * q^65 + 42 * q^68 + 42 * q^74 - 18 * q^77 + 18 * q^79 + 2 * q^82 - 2 * q^88 - 6 * q^91 - 36 * q^92 + 32 * q^94 + 36 * q^95

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