LMFDB - Embedded newform 140.2.c.a.139.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,2,Mod(139,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("140.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 140 = 2^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	140.c (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:	$1.11790562830$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			139.1
Root			$0.707107 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	140.139
Dual form			140.2.c.a.139.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -3.16228i q^{3} +2.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +(2.12132 + 1.58114i) q^{7} -2.82843 q^{8} -7.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -3.16228i q^{3} +2.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +(2.12132 + 1.58114i) q^{7} -2.82843 q^{8} -7.00000 q^{9} +3.16228i q^{10} -6.32456i q^{12} +(-3.00000 - 2.23607i) q^{14} -7.07107 q^{15} +4.00000 q^{16} +9.89949 q^{18} -4.47214i q^{20} +(5.00000 - 6.70820i) q^{21} +1.41421 q^{23} +8.94427i q^{24} -5.00000 q^{25} +12.6491i q^{27} +(4.24264 + 3.16228i) q^{28} +6.00000 q^{29} +10.0000 q^{30} -5.65685 q^{32} +(3.53553 - 4.74342i) q^{35} -14.0000 q^{36} +6.32456i q^{40} +4.47214i q^{41} +(-7.07107 + 9.48683i) q^{42} +12.7279 q^{43} +15.6525i q^{45} -2.00000 q^{46} -9.48683i q^{47} -12.6491i q^{48} +(2.00000 + 6.70820i) q^{49} +7.07107 q^{50} -17.8885i q^{54} +(-6.00000 - 4.47214i) q^{56} -8.48528 q^{58} -14.1421 q^{60} -13.4164i q^{61} +(-14.8492 - 11.0680i) q^{63} +8.00000 q^{64} -4.24264 q^{67} -4.47214i q^{69} +(-5.00000 + 6.70820i) q^{70} +19.7990 q^{72} +15.8114i q^{75} -8.94427i q^{80} +19.0000 q^{81} -6.32456i q^{82} +9.48683i q^{83} +(10.0000 - 13.4164i) q^{84} -18.0000 q^{86} -18.9737i q^{87} +17.8885i q^{89} -22.1359i q^{90} +2.82843 q^{92} +13.4164i q^{94} +17.8885i q^{96} +(-2.82843 - 9.48683i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 28 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 28 * q^9	$ 4 q + 8 q^{4} - 28 q^{9} - 12 q^{14} + 16 q^{16} + 20 q^{21} - 20 q^{25} + 24 q^{29} + 40 q^{30} - 56 q^{36} - 8 q^{46} + 8 q^{49} - 24 q^{56} + 32 q^{64} - 20 q^{70} + 76 q^{81} + 40 q^{84} - 72 q^{86}+O(q^{100}) $ 4 * q + 8 * q^4 - 28 * q^9 - 12 * q^14 + 16 * q^16 + 20 * q^21 - 20 * q^25 + 24 * q^29 + 40 * q^30 - 56 * q^36 - 8 * q^46 + 8 * q^49 - 24 * q^56 + 32 * q^64 - 20 * q^70 + 76 * q^81 + 40 * q^84 - 72 * q^86

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$.

$n$	$57$	$71$	$101$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−1.41421			−1.00000
$2$	−1.41421			−1.00000
$3$		−	3.16228i		−	1.82574i	−0.408248	−	0.912871i	$-0.633860\pi$
$3$		−	3.16228i		−	1.82574i	0.408248	−	0.912871i	$-0.366140\pi$
$4$	2.00000			1.00000
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$			4.47214i			1.82574i
$7$	2.12132	+	1.58114i	0.801784	+	0.597614i
$7$	2.12132	+	1.58114i	0.801784	+	0.597614i
$8$	−2.82843			−1.00000
$9$	−7.00000			−2.33333
$10$			3.16228i			1.00000i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	6.32456i		−	1.82574i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	−3.00000	−	2.23607i	−0.801784	−	0.597614i
$15$	−7.07107			−1.82574
$16$	4.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	9.89949			2.33333
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		−	4.47214i		−	1.00000i
$21$	5.00000	−	6.70820i	1.09109	−	1.46385i
$22$		0			0
$23$	1.41421			0.294884			0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421			0.294884			0.147442	+	0.989071i	$0.452896\pi$
$24$			8.94427i			1.82574i
$25$	−5.00000			−1.00000
$26$		0			0
$27$			12.6491i			2.43432i
$28$	4.24264	+	3.16228i	0.801784	+	0.597614i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$	10.0000			1.82574
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−5.65685			−1.00000
$33$		0			0
$34$		0			0
$35$	3.53553	−	4.74342i	0.597614	−	0.801784i
$36$	−14.0000			−2.33333
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$			6.32456i			1.00000i
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$	−7.07107	+	9.48683i	−1.09109	+	1.46385i
$43$	12.7279			1.94099			0.970495	−	0.241121i	$-0.0775152\pi$
$43$	12.7279			1.94099			0.970495	+	0.241121i	$0.0775152\pi$
$44$		0			0
$45$			15.6525i			2.33333i
$46$	−2.00000			−0.294884
$47$		−	9.48683i		−	1.38380i	−0.721995	−	0.691898i	$-0.756775\pi$
$47$		−	9.48683i		−	1.38380i	0.721995	−	0.691898i	$-0.243225\pi$
$48$		−	12.6491i		−	1.82574i
$49$	2.00000	+	6.70820i	0.285714	+	0.958315i
$50$	7.07107			1.00000
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		−	17.8885i		−	2.43432i
$55$		0			0
$56$	−6.00000	−	4.47214i	−0.801784	−	0.597614i
$57$		0			0
$58$	−8.48528			−1.11417
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−14.1421			−1.82574
$61$		−	13.4164i		−	1.71780i	−0.512148	−	0.858898i	$-0.671150\pi$
$61$		−	13.4164i		−	1.71780i	0.512148	−	0.858898i	$-0.328850\pi$
$62$		0			0
$63$	−14.8492	−	11.0680i	−1.87083	−	1.39443i
$64$	8.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−4.24264			−0.518321			−0.259161	−	0.965834i	$-0.583446\pi$
$67$	−4.24264			−0.518321			−0.259161	+	0.965834i	$0.583446\pi$
$68$		0			0
$69$		−	4.47214i		−	0.538382i
$70$	−5.00000	+	6.70820i	−0.597614	+	0.801784i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	19.7990			2.33333
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$			15.8114i			1.82574i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		−	8.94427i		−	1.00000i
$81$	19.0000			2.11111
$82$		−	6.32456i		−	0.698430i
$83$			9.48683i			1.04132i	0.853766	+	0.520658i	$0.174313\pi$
$83$			9.48683i			1.04132i	−0.853766	+	0.520658i	$0.825687\pi$
$84$	10.0000	−	13.4164i	1.09109	−	1.46385i
$85$		0			0
$86$	−18.0000			−1.94099
$87$		−	18.9737i		−	2.03419i
$88$		0			0
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$		−	22.1359i		−	2.33333i
$91$		0			0
$92$	2.82843			0.294884
$93$		0			0
$94$			13.4164i			1.38380i
$95$		0			0
$96$			17.8885i			1.82574i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−2.82843	−	9.48683i	−0.285714	−	0.958315i
$99$		0			0
$100$	−10.0000			−1.00000
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$			15.8114i			1.55794i	0.627060	+	0.778971i	$0.284258\pi$
$103$			15.8114i			1.55794i	−0.627060	+	0.778971i	$0.715742\pi$
$104$		0			0
$105$	−15.0000	−	11.1803i	−1.46385	−	1.09109i
$106$		0			0
$107$	−18.3848			−1.77732			−0.888662	−	0.458563i	$-0.848364\pi$
$107$	−18.3848			−1.77732			−0.888662	+	0.458563i	$0.848364\pi$
$108$			25.2982i			2.43432i
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$	8.48528	+	6.32456i	0.801784	+	0.597614i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	3.16228i		−	0.294884i
$116$	12.0000			1.11417
$117$		0			0
$118$		0			0
$119$		0			0
$120$	20.0000			1.82574
$121$	11.0000			1.00000
$122$			18.9737i			1.71780i
$123$	14.1421			1.27515
$124$		0			0
$125$			11.1803i			1.00000i
$126$	21.0000	+	15.6525i	1.87083	+	1.39443i
$127$	4.24264			0.376473			0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264			0.376473			0.188237	+	0.982124i	$0.439723\pi$
$128$	−11.3137			−1.00000
$129$		−	40.2492i		−	3.54375i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$	6.00000			0.518321
$135$	28.2843			2.43432
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$			6.32456i			0.538382i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	7.07107	−	9.48683i	0.597614	−	0.801784i
$141$	−30.0000			−2.52646
$142$		0			0
$143$		0			0
$144$	−28.0000			−2.33333
$145$		−	13.4164i		−	1.11417i
$146$		0			0
$147$	21.2132	−	6.32456i	1.74964	−	0.521641i
$148$		0			0
$149$	24.0000			1.96616			0.983078	−	0.183186i	$-0.0586410\pi$
$149$	24.0000			1.96616			0.983078	+	0.183186i	$0.0586410\pi$
$150$		−	22.3607i		−	1.82574i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$			12.6491i			1.00000i
$161$	3.00000	+	2.23607i	0.236433	+	0.176227i
$162$	−26.8701			−2.11111
$163$	−12.7279			−0.996928			−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279			−0.996928			−0.498464	+	0.866910i	$0.666102\pi$
$164$			8.94427i			0.698430i
$165$		0			0
$166$		−	13.4164i		−	1.04132i
$167$			9.48683i			0.734113i	0.930199	+	0.367057i	$0.119634\pi$
$167$			9.48683i			0.734113i	−0.930199	+	0.367057i	$0.880366\pi$
$168$	−14.1421	+	18.9737i	−1.09109	+	1.46385i
$169$	−13.0000			−1.00000
$170$		0			0
$171$		0			0
$172$	25.4558			1.94099
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$			26.8328i			2.03419i
$175$	−10.6066	−	7.90569i	−0.801784	−	0.597614i
$176$		0			0
$177$		0			0
$178$		−	25.2982i		−	1.89618i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$			31.3050i			2.33333i
$181$		−	26.8328i		−	1.99447i	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$		−	26.8328i		−	1.99447i	0.0743294	−	0.997234i	$-0.476318\pi$
$182$		0			0
$183$	−42.4264			−3.13625
$184$	−4.00000			−0.294884
$185$		0			0
$186$		0			0
$187$		0			0
$188$		−	18.9737i		−	1.38380i
$189$	−20.0000	+	26.8328i	−1.45479	+	1.95180i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	25.2982i		−	1.82574i
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	4.00000	+	13.4164i	0.285714	+	0.958315i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	14.1421			1.00000
$201$			13.4164i			0.946320i
$202$		−	12.6491i		−	0.889988i
$203$	12.7279	+	9.48683i	0.893325	+	0.665845i
$204$		0			0
$205$	10.0000			0.698430
$206$		−	22.3607i		−	1.55794i
$207$	−9.89949			−0.688062
$208$		0			0
$209$		0			0
$210$	21.2132	+	15.8114i	1.46385	+	1.09109i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	26.0000			1.77732
$215$		−	28.4605i		−	1.94099i
$216$		−	35.7771i		−	2.43432i
$217$		0			0
$218$	22.6274			1.53252
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			3.16228i			0.211762i	0.994379	+	0.105881i	$0.0337662\pi$
$223$			3.16228i			0.211762i	−0.994379	+	0.105881i	$0.966234\pi$
$224$	−12.0000	−	8.94427i	−0.801784	−	0.597614i
$225$	35.0000			2.33333
$226$		0			0
$227$			28.4605i			1.88899i	0.328526	+	0.944495i	$0.393448\pi$
$227$			28.4605i			1.88899i	−0.328526	+	0.944495i	$0.606552\pi$
$228$		0			0
$229$			26.8328i			1.77316i	0.462573	+	0.886581i	$0.346926\pi$
$229$			26.8328i			1.77316i	−0.462573	+	0.886581i	$0.653074\pi$
$230$			4.47214i			0.294884i
$231$		0			0
$232$	−16.9706			−1.11417
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−21.2132			−1.38380
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−28.2843			−1.82574
$241$			13.4164i			0.864227i	0.901819	+	0.432113i	$0.142232\pi$
$241$			13.4164i			0.864227i	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−15.5563			−1.00000
$243$		−	22.1359i		−	1.42002i
$244$		−	26.8328i		−	1.71780i
$245$	15.0000	−	4.47214i	0.958315	−	0.285714i
$246$	−20.0000			−1.27515
$247$		0			0
$248$		0			0
$249$	30.0000			1.90117
$250$		−	15.8114i		−	1.00000i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−29.6985	−	22.1359i	−1.87083	−	1.39443i
$253$		0			0
$254$	−6.00000			−0.376473
$255$		0			0
$256$	16.0000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$			56.9210i			3.54375i
$259$		0			0
$260$		0			0
$261$	−42.0000			−2.59973
$262$		0			0
$263$	15.5563			0.959246			0.479623	−	0.877475i	$-0.340774\pi$
$263$	15.5563			0.959246			0.479623	+	0.877475i	$0.340774\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	56.5685			3.46194
$268$	−8.48528			−0.518321
$269$		−	22.3607i		−	1.36335i	−0.731653	−	0.681677i	$-0.761251\pi$
$269$		−	22.3607i		−	1.36335i	0.731653	−	0.681677i	$-0.238749\pi$
$270$	−40.0000			−2.43432
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		−	8.94427i		−	0.538382i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	−10.0000	+	13.4164i	−0.597614	+	0.801784i
$281$	−12.0000			−0.715860			−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000			−0.715860			−0.357930	+	0.933748i	$0.616517\pi$
$282$	42.4264			2.52646
$283$		−	15.8114i		−	0.939889i	−0.882696	−	0.469945i	$-0.844274\pi$
$283$		−	15.8114i		−	0.939889i	0.882696	−	0.469945i	$-0.155726\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−7.07107	+	9.48683i	−0.417392	+	0.559990i
$288$	39.5980			2.33333
$289$	−17.0000			−1.00000
$290$			18.9737i			1.11417i
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−30.0000	+	8.94427i	−1.74964	+	0.521641i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−33.9411			−1.96616
$299$		0			0
$300$			31.6228i			1.82574i
$301$	27.0000	+	20.1246i	1.55625	+	1.15996i
$302$		0			0
$303$	28.2843			1.62489
$304$		0			0
$305$	−30.0000			−1.71780
$306$		0			0
$307$		−	34.7851i		−	1.98529i	−0.121070	−	0.992644i	$-0.538633\pi$
$307$		−	34.7851i		−	1.98529i	0.121070	−	0.992644i	$-0.461367\pi$
$308$		0			0
$309$	50.0000			2.84440
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0			−	1.00000i	$-0.5\pi$
$313$		0			0				1.00000i	$0.5\pi$
$314$		0			0
$315$	−24.7487	+	33.2039i	−1.39443	+	1.87083i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		−	17.8885i		−	1.00000i
$321$			58.1378i			3.24493i
$322$	−4.24264	−	3.16228i	−0.236433	−	0.176227i
$323$		0			0
$324$	38.0000			2.11111
$325$		0			0
$326$	18.0000			0.996928
$327$			50.5964i			2.79799i
$328$		−	12.6491i		−	0.698430i
$329$	15.0000	−	20.1246i	0.826977	−	1.10951i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			18.9737i			1.04132i
$333$		0			0
$334$		−	13.4164i		−	0.734113i
$335$			9.48683i			0.518321i
$336$	20.0000	−	26.8328i	1.09109	−	1.46385i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	18.3848			1.00000
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.36396	+	17.3925i	−0.343622	+	0.939108i
$344$	−36.0000			−1.94099
$345$	−10.0000			−0.538382
$346$		0			0
$347$	−24.0416			−1.29062			−0.645311	−	0.763920i	$-0.723272\pi$
$347$	−24.0416			−1.29062			−0.645311	+	0.763920i	$0.723272\pi$
$348$		−	37.9473i		−	2.03419i
$349$		−	26.8328i		−	1.43633i	−0.695874	−	0.718164i	$-0.744983\pi$
$349$		−	26.8328i		−	1.43633i	0.695874	−	0.718164i	$-0.255017\pi$
$350$	15.0000	+	11.1803i	0.801784	+	0.597614i
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$			35.7771i			1.89618i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		−	44.2719i		−	2.33333i
$361$	−19.0000			−1.00000
$362$			37.9473i			1.99447i
$363$		−	34.7851i		−	1.82574i
$364$		0			0
$365$		0			0
$366$	60.0000			3.13625
$367$		−	3.16228i		−	0.165070i	−0.996588	−	0.0825348i	$-0.973698\pi$
$367$		−	3.16228i		−	0.165070i	0.996588	−	0.0825348i	$-0.0263016\pi$
$368$	5.65685			0.294884
$369$		−	31.3050i		−	1.62967i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	35.3553			1.82574
$376$			26.8328i			1.38380i
$377$		0			0
$378$	28.2843	−	37.9473i	1.45479	−	1.95180i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		−	13.4164i		−	0.687343i
$382$		0			0
$383$		−	28.4605i		−	1.45426i	−0.686498	−	0.727132i	$-0.740853\pi$
$383$		−	28.4605i		−	1.45426i	0.686498	−	0.727132i	$-0.259147\pi$
$384$			35.7771i			1.82574i
$385$		0			0
$386$		0			0
$387$	−89.0955			−4.52898
$388$		0			0
$389$	−24.0000			−1.21685			−0.608424	−	0.793612i	$-0.708198\pi$
$389$	−24.0000			−1.21685			−0.608424	+	0.793612i	$0.708198\pi$
$390$		0			0
$391$		0			0
$392$	−5.65685	−	18.9737i	−0.285714	−	0.958315i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$	−18.0000			−0.898877			−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000			−0.898877			−0.449439	+	0.893311i	$0.648376\pi$
$402$		−	18.9737i		−	0.946320i
$403$		0			0
$404$			17.8885i			0.889988i
$405$		−	42.4853i		−	2.11111i
$406$	−18.0000	−	13.4164i	−0.893325	−	0.665845i
$407$		0			0
$408$		0			0
$409$			40.2492i			1.99020i	0.0988936	+	0.995098i	$0.468470\pi$
$409$			40.2492i			1.99020i	−0.0988936	+	0.995098i	$0.531530\pi$
$410$	−14.1421			−0.698430
$411$		0			0
$412$			31.6228i			1.55794i
$413$		0			0
$414$	14.0000			0.688062
$415$	21.2132			1.04132
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−30.0000	−	22.3607i	−1.46385	−	1.09109i
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$			66.4078i			3.22886i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	21.2132	−	28.4605i	1.02658	−	1.37730i
$428$	−36.7696			−1.77732
$429$		0			0
$430$			40.2492i			1.94099i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			50.5964i			2.43432i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	−42.4264			−2.03419
$436$	−32.0000			−1.53252
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−14.0000	−	46.9574i	−0.666667	−	2.23607i
$442$		0			0
$443$	41.0122			1.94855			0.974274	−	0.225367i	$-0.0723580\pi$
$443$	41.0122			1.94855			0.974274	+	0.225367i	$0.0723580\pi$
$444$		0			0
$445$	40.0000			1.89618
$446$		−	4.47214i		−	0.211762i
$447$		−	75.8947i		−	3.58969i
$448$	16.9706	+	12.6491i	0.801784	+	0.597614i
$449$	−36.0000			−1.69895			−0.849473	−	0.527633i	$-0.823080\pi$
$449$	−36.0000			−1.69895			−0.849473	+	0.527633i	$0.823080\pi$
$450$	−49.4975			−2.33333
$451$		0			0
$452$		0			0
$453$		0			0
$454$		−	40.2492i		−	1.88899i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		−	37.9473i		−	1.77316i
$459$		0			0
$460$		−	6.32456i		−	0.294884i
$461$		−	8.94427i		−	0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$		−	8.94427i		−	0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$		0			0
$463$	12.7279			0.591517			0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279			0.591517			0.295758	+	0.955263i	$0.404428\pi$
$464$	24.0000			1.11417
$465$		0			0
$466$		0			0
$467$		−	28.4605i		−	1.31699i	−0.752583	−	0.658497i	$-0.771192\pi$
$467$		−	28.4605i		−	1.31699i	0.752583	−	0.658497i	$-0.228808\pi$
$468$		0			0
$469$	−9.00000	−	6.70820i	−0.415581	−	0.309756i
$470$	30.0000			1.38380
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	40.0000			1.82574
$481$		0			0
$482$		−	18.9737i		−	0.864227i
$483$	7.07107	−	9.48683i	0.321745	−	0.431666i
$484$	22.0000			1.00000
$485$		0			0
$486$			31.3050i			1.42002i
$487$	38.1838			1.73027			0.865136	−	0.501538i	$-0.167232\pi$
$487$	38.1838			1.73027			0.865136	+	0.501538i	$0.167232\pi$
$488$			37.9473i			1.71780i
$489$			40.2492i			1.82013i
$490$	−21.2132	+	6.32456i	−0.958315	+	0.285714i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	28.2843			1.27515
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−42.4264			−1.90117
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$			22.3607i			1.00000i
$501$	30.0000			1.34030
$502$		0			0
$503$			9.48683i			0.422997i	0.977378	+	0.211498i	$0.0678343\pi$
$503$			9.48683i			0.422997i	−0.977378	+	0.211498i	$0.932166\pi$
$504$	42.0000	+	31.3050i	1.87083	+	1.39443i
$505$	20.0000			0.889988
$506$		0			0
$507$			41.1096i			1.82574i
$508$	8.48528			0.376473
$509$		−	44.7214i		−	1.98224i	−0.132973	−	0.991120i	$-0.542452\pi$
$509$		−	44.7214i		−	1.98224i	0.132973	−	0.991120i	$-0.457548\pi$
$510$		0			0
$511$		0			0
$512$	−22.6274			−1.00000
$513$		0			0
$514$		0			0
$515$	35.3553			1.55794
$516$		−	80.4984i		−	3.54375i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	17.8885i		−	0.783711i	−0.920027	−	0.391856i	$-0.871833\pi$
$521$		−	17.8885i		−	0.783711i	0.920027	−	0.391856i	$-0.128167\pi$
$522$	59.3970			2.59973
$523$			34.7851i			1.52104i	0.649312	+	0.760522i	$0.275057\pi$
$523$			34.7851i			1.52104i	−0.649312	+	0.760522i	$0.724943\pi$
$524$		0			0
$525$	−25.0000	+	33.5410i	−1.09109	+	1.46385i
$526$	−22.0000			−0.959246
$527$		0			0
$528$		0			0
$529$	−21.0000			−0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−80.0000			−3.46194
$535$			41.1096i			1.77732i
$536$	12.0000			0.518321
$537$		0			0
$538$			31.6228i			1.36335i
$539$		0			0
$540$	56.5685			2.43432
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	−84.8528			−3.64138
$544$		0			0
$545$			35.7771i			1.53252i
$546$		0			0
$547$	−46.6690			−1.99542			−0.997712	−	0.0676046i	$-0.978464\pi$
$547$	−46.6690			−1.99542			−0.997712	+	0.0676046i	$0.978464\pi$
$548$		0			0
$549$			93.9149i			4.00819i
$550$		0			0
$551$		0			0
$552$			12.6491i			0.538382i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	14.1421	−	18.9737i	0.597614	−	0.801784i
$561$		0			0
$562$	16.9706			0.715860
$563$		−	47.4342i		−	1.99911i	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$		−	47.4342i		−	1.99911i	0.0298010	−	0.999556i	$-0.490513\pi$
$564$	−60.0000			−2.52646
$565$		0			0
$566$			22.3607i			0.939889i
$567$	40.3051	+	30.0416i	1.69265	+	1.26163i
$568$		0			0
$569$	36.0000			1.50920			0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000			1.50920			0.754599	+	0.656186i	$0.227831\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	10.0000	−	13.4164i	0.417392	−	0.559990i
$575$	−7.07107			−0.294884
$576$	−56.0000			−2.33333
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	24.0416			1.00000
$579$		0			0
$580$		−	26.8328i		−	1.11417i
$581$	−15.0000	+	20.1246i	−0.622305	+	0.834910i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			47.4342i			1.95782i	0.204298	+	0.978909i	$0.434509\pi$
$587$			47.4342i			1.95782i	−0.204298	+	0.978909i	$0.565491\pi$
$588$	42.4264	−	12.6491i	1.74964	−	0.521641i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	48.0000			1.96616
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		−	44.7214i		−	1.82574i
$601$		−	40.2492i		−	1.64180i	−0.571072	−	0.820900i	$-0.693472\pi$
$601$		−	40.2492i		−	1.64180i	0.571072	−	0.820900i	$-0.306528\pi$
$602$	−38.1838	−	28.4605i	−1.55625	−	1.15996i
$603$	29.6985			1.20942
$604$		0			0
$605$		−	24.5967i		−	1.00000i
$606$	−40.0000			−1.62489
$607$			15.8114i			0.641764i	0.947119	+	0.320882i	$0.103979\pi$
$607$			15.8114i			0.641764i	−0.947119	+	0.320882i	$0.896021\pi$
$608$		0			0
$609$	30.0000	−	40.2492i	1.21566	−	1.63098i
$610$	42.4264			1.71780
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			49.1935i			1.98529i
$615$		−	31.6228i		−	1.27515i
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	−70.7107			−2.84440
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$			17.8885i			0.717843i
$622$		0			0
$623$	−28.2843	+	37.9473i	−1.13319	+	1.52033i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	35.0000	−	46.9574i	1.39443	−	1.87083i
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	9.48683i		−	0.376473i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$			25.2982i			1.00000i
$641$	12.0000			0.473972			0.236986	−	0.971513i	$-0.423841\pi$
$641$	12.0000			0.473972			0.236986	+	0.971513i	$0.423841\pi$
$642$		−	82.2192i		−	3.24493i
$643$		−	41.1096i		−	1.62120i	−0.585597	−	0.810602i	$-0.699140\pi$
$643$		−	41.1096i		−	1.62120i	0.585597	−	0.810602i	$-0.300860\pi$
$644$	6.00000	+	4.47214i	0.236433	+	0.176227i
$645$	−90.0000			−3.54375
$646$		0			0
$647$		−	47.4342i		−	1.86483i	−0.361390	−	0.932415i	$-0.617698\pi$
$647$		−	47.4342i		−	1.86483i	0.361390	−	0.932415i	$-0.382302\pi$
$648$	−53.7401			−2.11111
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−25.4558			−0.996928
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		−	71.5542i		−	2.79799i
$655$		0			0
$656$			17.8885i			0.698430i
$657$		0			0
$658$	−21.2132	+	28.4605i	−0.826977	+	1.10951i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			40.2492i			1.56551i	0.622328	+	0.782757i	$0.286187\pi$
$661$			40.2492i			1.56551i	−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$		0			0
$664$		−	26.8328i		−	1.04132i
$665$		0			0
$666$		0			0
$667$	8.48528			0.328551
$668$			18.9737i			0.734113i
$669$	10.0000			0.386622
$670$		−	13.4164i		−	0.518321i
$671$		0			0
$672$	−28.2843	+	37.9473i	−1.09109	+	1.46385i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		−	63.2456i		−	2.43432i
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	90.0000			3.44881
$682$		0			0
$683$	−43.8406			−1.67751			−0.838757	−	0.544505i	$-0.816717\pi$
$683$	−43.8406			−1.67751			−0.838757	+	0.544505i	$0.816717\pi$
$684$		0			0
$685$		0			0
$686$	9.00000	−	24.5967i	0.343622	−	0.939108i
$687$	84.8528			3.23734
$688$	50.9117			1.94099
$689$		0			0
$690$	14.1421			0.538382
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	34.0000			1.29062
$695$		0			0
$696$			53.6656i			2.03419i
$697$		0			0
$698$			37.9473i			1.43633i
$699$		0			0
$700$	−21.2132	−	15.8114i	−0.801784	−	0.597614i
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$			67.0820i			2.52646i
$706$		0			0
$707$	−14.1421	+	18.9737i	−0.531870	+	0.713578i
$708$		0			0
$709$	−46.0000			−1.72757			−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000			−1.72757			−0.863783	+	0.503864i	$0.831911\pi$
$710$		0			0
$711$		0			0
$712$		−	50.5964i		−	1.89618i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$			62.6099i			2.33333i
$721$	−25.0000	+	33.5410i	−0.931049	+	1.24913i
$722$	26.8701			1.00000
$723$	42.4264			1.57786
$724$		−	53.6656i		−	1.99447i
$725$	−30.0000			−1.11417
$726$			49.1935i			1.82574i
$727$			53.7587i			1.99380i	0.0786754	+	0.996900i	$0.474931\pi$
$727$			53.7587i			1.99380i	−0.0786754	+	0.996900i	$0.525069\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$		0			0
$732$	−84.8528			−3.13625
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$			4.47214i			0.165070i
$735$	−14.1421	−	47.4342i	−0.521641	−	1.74964i
$736$	−8.00000			−0.294884
$737$		0			0
$738$			44.2719i			1.62967i
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−26.8701			−0.985767			−0.492883	−	0.870095i	$-0.664057\pi$
$743$	−26.8701			−0.985767			−0.492883	+	0.870095i	$0.664057\pi$
$744$		0			0
$745$		−	53.6656i		−	1.96616i
$746$		0			0
$747$		−	66.4078i		−	2.42974i
$748$		0			0
$749$	−39.0000	−	29.0689i	−1.42503	−	1.06215i
$750$	−50.0000			−1.82574
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	37.9473i		−	1.38380i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−40.0000	+	53.6656i	−1.45479	+	1.95180i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	35.7771i		−	1.29692i	−0.761249	−	0.648459i	$-0.775414\pi$
$761$		−	35.7771i		−	1.29692i	0.761249	−	0.648459i	$-0.224586\pi$
$762$			18.9737i			0.687343i
$763$	−33.9411	−	25.2982i	−1.22875	−	0.915857i
$764$		0			0
$765$		0			0
$766$			40.2492i			1.45426i
$767$		0			0
$768$		−	50.5964i		−	1.82574i
$769$			53.6656i			1.93523i	0.252426	+	0.967616i	$0.418771\pi$
$769$			53.6656i			1.93523i	−0.252426	+	0.967616i	$0.581229\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	126.000			4.52898
$775$		0			0
$776$		0			0
$777$		0			0
$778$	33.9411			1.21685
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$			75.8947i			2.71225i
$784$	8.00000	+	26.8328i	0.285714	+	0.958315i
$785$		0			0
$786$		0			0
$787$			41.1096i			1.46540i	0.680552	+	0.732700i	$0.261740\pi$
$787$			41.1096i			1.46540i	−0.680552	+	0.732700i	$0.738260\pi$
$788$		0			0
$789$		−	49.1935i		−	1.75133i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	28.2843			1.00000
$801$		−	125.220i		−	4.42442i
$802$	25.4558			0.898877
$803$		0			0
$804$			26.8328i			0.946320i
$805$	5.00000	−	6.70820i	0.176227	−	0.236433i
$806$		0			0
$807$	−70.7107			−2.48913
$808$		−	25.2982i		−	0.889988i
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$			60.0833i			2.11111i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	25.4558	+	18.9737i	0.893325	+	0.665845i
$813$		0			0
$814$		0			0
$815$			28.4605i			0.996928i
$816$		0			0
$817$		0			0
$818$		−	56.9210i		−	1.99020i
$819$		0			0
$820$	20.0000			0.698430
$821$	−48.0000			−1.67521			−0.837606	−	0.546275i	$-0.816045\pi$
$821$	−48.0000			−1.67521			−0.837606	+	0.546275i	$0.816045\pi$
$822$		0			0
$823$	55.1543			1.92256			0.961280	−	0.275575i	$-0.0888683\pi$
$823$	55.1543			1.92256			0.961280	+	0.275575i	$0.0888683\pi$
$824$		−	44.7214i		−	1.55794i
$825$		0			0
$826$		0			0
$827$	32.5269			1.13107			0.565536	−	0.824724i	$-0.308669\pi$
$827$	32.5269			1.13107			0.565536	+	0.824724i	$0.308669\pi$
$828$	−19.7990			−0.688062
$829$			13.4164i			0.465971i	0.972480	+	0.232986i	$0.0748495\pi$
$829$			13.4164i			0.465971i	−0.972480	+	0.232986i	$0.925151\pi$
$830$	−30.0000			−1.04132
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.2132			0.734113
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$	42.4264	+	31.6228i	1.46385	+	1.09109i
$841$	7.00000			0.241379
$842$	11.3137			0.389896
$843$			37.9473i			1.30698i
$844$		0			0
$845$			29.0689i			1.00000i
$846$		−	93.9149i		−	3.22886i
$847$	23.3345	+	17.3925i	0.801784	+	0.597614i
$848$		0			0
$849$	−50.0000			−1.71600
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$	−30.0000	+	40.2492i	−1.02658	+	1.37730i
$855$		0			0
$856$	52.0000			1.77732
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		−	56.9210i		−	1.94099i
$861$	30.0000	+	22.3607i	1.02240	+	0.762050i
$862$		0			0
$863$	−57.9828			−1.97376			−0.986878	−	0.161468i	$-0.948377\pi$
$863$	−57.9828			−1.97376			−0.986878	+	0.161468i	$0.948377\pi$
$864$		−	71.5542i		−	2.43432i
$865$		0			0
$866$		0			0
$867$			53.7587i			1.82574i
$868$		0			0
$869$		0			0
$870$	60.0000			2.03419
$871$		0			0
$872$	45.2548			1.53252
$873$		0			0
$874$		0			0
$875$	−17.6777	+	23.7171i	−0.597614	+	0.801784i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	58.1378i		−	1.95871i	−0.202145	−	0.979356i	$-0.564791\pi$
$881$		−	58.1378i		−	1.95871i	0.202145	−	0.979356i	$-0.435209\pi$
$882$	19.7990	+	66.4078i	0.666667	+	2.23607i
$883$	−55.1543			−1.85609			−0.928045	−	0.372467i	$-0.878512\pi$
$883$	−55.1543			−1.85609			−0.928045	+	0.372467i	$0.878512\pi$
$884$		0			0
$885$		0			0
$886$	−58.0000			−1.94855
$887$		−	28.4605i		−	0.955610i	−0.878466	−	0.477805i	$-0.841433\pi$
$887$		−	28.4605i		−	0.955610i	0.878466	−	0.477805i	$-0.158567\pi$
$888$		0			0
$889$	9.00000	+	6.70820i	0.301850	+	0.224986i
$890$	−56.5685			−1.89618
$891$		0			0
$892$			6.32456i			0.211762i
$893$		0			0
$894$			107.331i			3.58969i
$895$		0			0
$896$	−24.0000	−	17.8885i	−0.801784	−	0.597614i
$897$		0			0
$898$	50.9117			1.69895
$899$		0			0
$900$	70.0000			2.33333
$901$		0			0
$902$		0			0
$903$	63.6396	−	85.3815i	2.11779	−	2.84132i
$904$		0			0
$905$	−60.0000			−1.99447
$906$		0			0
$907$	−4.24264			−0.140875			−0.0704373	−	0.997516i	$-0.522439\pi$
$907$	−4.24264			−0.140875			−0.0704373	+	0.997516i	$0.522439\pi$
$908$			56.9210i			1.88899i
$909$		−	62.6099i		−	2.07664i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			94.8683i			3.13625i
$916$			53.6656i			1.77316i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$			8.94427i			0.294884i
$921$	−110.000			−3.62462
$922$			12.6491i			0.416576i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−18.0000			−0.591517
$927$		−	110.680i		−	3.63520i
$928$	−33.9411			−1.11417
$929$			49.1935i			1.61399i	0.590561	+	0.806993i	$0.298907\pi$
$929$			49.1935i			1.61399i	−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$			40.2492i			1.31699i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	12.7279	+	9.48683i	0.415581	+	0.309756i
$939$		0			0
$940$	−42.4264			−1.38380
$941$			44.7214i			1.45787i	0.684580	+	0.728937i	$0.259985\pi$
$941$			44.7214i			1.45787i	−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$			6.32456i			0.205956i
$944$		0			0
$945$	60.0000	+	44.7214i	1.95180	+	1.45479i
$946$		0			0
$947$	60.8112			1.97610			0.988049	−	0.154140i	$-0.0492608\pi$
$947$	60.8112			1.97610			0.988049	+	0.154140i	$0.0492608\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−56.5685			−1.82574
$961$	−31.0000			−1.00000
$962$		0			0
$963$	128.693			4.14709
$964$			26.8328i			0.864227i
$965$		0			0
$966$	−10.0000	+	13.4164i	−0.321745	+	0.431666i
$967$	−46.6690			−1.50078			−0.750388	−	0.660998i	$-0.770133\pi$
$967$	−46.6690			−1.50078			−0.750388	+	0.660998i	$0.770133\pi$
$968$	−31.1127			−1.00000
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	44.2719i		−	1.42002i
$973$		0			0
$974$	−54.0000			−1.73027
$975$		0			0
$976$		−	53.6656i		−	1.71780i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		−	56.9210i		−	1.82013i
$979$		0			0
$980$	30.0000	−	8.94427i	0.958315	−	0.285714i
$981$	112.000			3.57588
$982$		0			0
$983$		−	47.4342i		−	1.51291i	−0.654043	−	0.756457i	$-0.726928\pi$
$983$		−	47.4342i		−	1.51291i	0.654043	−	0.756457i	$-0.273072\pi$
$984$	−40.0000			−1.27515
$985$		0			0
$986$		0			0
$987$	−63.6396	−	47.4342i	−2.02567	−	1.50985i
$988$		0			0
$989$	18.0000			0.572367
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	60.0000			1.90117
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.2.c.a.139.1	✓	4
4.3	odd	2	inner	140.2.c.a.139.4	yes	4
5.2	odd	4		700.2.g.d.251.1		4
5.3	odd	4		700.2.g.d.251.4		4
5.4	even	2	inner	140.2.c.a.139.4	yes	4
7.2	even	3		980.2.s.b.619.4		8
7.3	odd	6		980.2.s.b.19.4		8
7.4	even	3		980.2.s.b.19.3		8
7.5	odd	6		980.2.s.b.619.3		8
7.6	odd	2	inner	140.2.c.a.139.2	yes	4
8.3	odd	2		2240.2.e.a.2239.2		4
8.5	even	2		2240.2.e.a.2239.4		4
20.3	even	4		700.2.g.d.251.1		4
20.7	even	4		700.2.g.d.251.4		4
20.19	odd	2	CM	140.2.c.a.139.1	✓	4
28.3	even	6		980.2.s.b.19.1		8
28.11	odd	6		980.2.s.b.19.2		8
28.19	even	6		980.2.s.b.619.2		8
28.23	odd	6		980.2.s.b.619.1		8
28.27	even	2	inner	140.2.c.a.139.3	yes	4
35.4	even	6		980.2.s.b.19.2		8
35.9	even	6		980.2.s.b.619.1		8
35.13	even	4		700.2.g.d.251.3		4
35.19	odd	6		980.2.s.b.619.2		8
35.24	odd	6		980.2.s.b.19.1		8
35.27	even	4		700.2.g.d.251.2		4
35.34	odd	2	inner	140.2.c.a.139.3	yes	4
40.19	odd	2		2240.2.e.a.2239.4		4
40.29	even	2		2240.2.e.a.2239.2		4
56.13	odd	2		2240.2.e.a.2239.1		4
56.27	even	2		2240.2.e.a.2239.3		4
140.19	even	6		980.2.s.b.619.3		8
140.27	odd	4		700.2.g.d.251.3		4
140.39	odd	6		980.2.s.b.19.3		8
140.59	even	6		980.2.s.b.19.4		8
140.79	odd	6		980.2.s.b.619.4		8
140.83	odd	4		700.2.g.d.251.2		4
140.139	even	2	inner	140.2.c.a.139.2	yes	4
280.69	odd	2		2240.2.e.a.2239.3		4
280.139	even	2		2240.2.e.a.2239.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.c.a.139.1	✓	4	1.1	even	1	trivial
140.2.c.a.139.1	✓	4	20.19	odd	2	CM
140.2.c.a.139.2	yes	4	7.6	odd	2	inner
140.2.c.a.139.2	yes	4	140.139	even	2	inner
140.2.c.a.139.3	yes	4	28.27	even	2	inner
140.2.c.a.139.3	yes	4	35.34	odd	2	inner
140.2.c.a.139.4	yes	4	4.3	odd	2	inner
140.2.c.a.139.4	yes	4	5.4	even	2	inner
700.2.g.d.251.1		4	5.2	odd	4
700.2.g.d.251.1		4	20.3	even	4
700.2.g.d.251.2		4	35.27	even	4
700.2.g.d.251.2		4	140.83	odd	4
700.2.g.d.251.3		4	35.13	even	4
700.2.g.d.251.3		4	140.27	odd	4
700.2.g.d.251.4		4	5.3	odd	4
700.2.g.d.251.4		4	20.7	even	4
980.2.s.b.19.1		8	28.3	even	6
980.2.s.b.19.1		8	35.24	odd	6
980.2.s.b.19.2		8	28.11	odd	6
980.2.s.b.19.2		8	35.4	even	6
980.2.s.b.19.3		8	7.4	even	3
980.2.s.b.19.3		8	140.39	odd	6
980.2.s.b.19.4		8	7.3	odd	6
980.2.s.b.19.4		8	140.59	even	6
980.2.s.b.619.1		8	28.23	odd	6
980.2.s.b.619.1		8	35.9	even	6
980.2.s.b.619.2		8	28.19	even	6
980.2.s.b.619.2		8	35.19	odd	6
980.2.s.b.619.3		8	7.5	odd	6
980.2.s.b.619.3		8	140.19	even	6
980.2.s.b.619.4		8	7.2	even	3
980.2.s.b.619.4		8	140.79	odd	6
2240.2.e.a.2239.1		4	56.13	odd	2
2240.2.e.a.2239.1		4	280.139	even	2
2240.2.e.a.2239.2		4	8.3	odd	2
2240.2.e.a.2239.2		4	40.29	even	2
2240.2.e.a.2239.3		4	56.27	even	2
2240.2.e.a.2239.3		4	280.69	odd	2
2240.2.e.a.2239.4		4	8.5	even	2
2240.2.e.a.2239.4		4	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 140 = 2^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	140.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -3.16228i q^{3} +2.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +(2.12132 + 1.58114i) q^{7} -2.82843 q^{8} -7.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -3.16228i q^{3} +2.00000 q^{4} -2.23607i q^{5} +4.47214i q^{6} +(2.12132 + 1.58114i) q^{7} -2.82843 q^{8} -7.00000 q^{9} +3.16228i q^{10} -6.32456i q^{12} +(-3.00000 - 2.23607i) q^{14} -7.07107 q^{15} +4.00000 q^{16} +9.89949 q^{18} -4.47214i q^{20} +(5.00000 - 6.70820i) q^{21} +1.41421 q^{23} +8.94427i q^{24} -5.00000 q^{25} +12.6491i q^{27} +(4.24264 + 3.16228i) q^{28} +6.00000 q^{29} +10.0000 q^{30} -5.65685 q^{32} +(3.53553 - 4.74342i) q^{35} -14.0000 q^{36} +6.32456i q^{40} +4.47214i q^{41} +(-7.07107 + 9.48683i) q^{42} +12.7279 q^{43} +15.6525i q^{45} -2.00000 q^{46} -9.48683i q^{47} -12.6491i q^{48} +(2.00000 + 6.70820i) q^{49} +7.07107 q^{50} -17.8885i q^{54} +(-6.00000 - 4.47214i) q^{56} -8.48528 q^{58} -14.1421 q^{60} -13.4164i q^{61} +(-14.8492 - 11.0680i) q^{63} +8.00000 q^{64} -4.24264 q^{67} -4.47214i q^{69} +(-5.00000 + 6.70820i) q^{70} +19.7990 q^{72} +15.8114i q^{75} -8.94427i q^{80} +19.0000 q^{81} -6.32456i q^{82} +9.48683i q^{83} +(10.0000 - 13.4164i) q^{84} -18.0000 q^{86} -18.9737i q^{87} +17.8885i q^{89} -22.1359i q^{90} +2.82843 q^{92} +13.4164i q^{94} +17.8885i q^{96} +(-2.82843 - 9.48683i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 28 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 28 * q^9	\( 4 q + 8 q^{4} - 28 q^{9} - 12 q^{14} + 16 q^{16} + 20 q^{21} - 20 q^{25} + 24 q^{29} + 40 q^{30} - 56 q^{36} - 8 q^{46} + 8 q^{49} - 24 q^{56} + 32 q^{64} - 20 q^{70} + 76 q^{81} + 40 q^{84} - 72 q^{86}+O(q^{100}) \) 4 * q + 8 * q^4 - 28 * q^9 - 12 * q^14 + 16 * q^16 + 20 * q^21 - 20 * q^25 + 24 * q^29 + 40 * q^30 - 56 * q^36 - 8 * q^46 + 8 * q^49 - 24 * q^56 + 32 * q^64 - 20 * q^70 + 76 * q^81 + 40 * q^84 - 72 * q^86

Introduction

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