LMFDB - Embedded newform 1340.3.g.a.669.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1340,3,Mod(669,1340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1340, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1340.669");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1340 = 2^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1340.g (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:	$36.5123554243$
Analytic rank:	$0$
Dimension:	$68$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			669.15
Character	$\chi$	$=$	1340.669
Dual form			1340.3.g.a.669.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.94848 q^{3} +(4.78326 + 1.45617i) q^{5} +0.855885 q^{7} +6.59049 q^{9} +O(q^{10})$	$q-3.94848 q^{3} +(4.78326 + 1.45617i) q^{5} +0.855885 q^{7} +6.59049 q^{9} +4.32833i q^{11} -4.61600 q^{13} +(-18.8866 - 5.74968i) q^{15} -10.0593i q^{17} +24.3780 q^{19} -3.37944 q^{21} +9.78769i q^{23} +(20.7591 + 13.9305i) q^{25} +9.51391 q^{27} -22.5771 q^{29} +47.6840i q^{31} -17.0903i q^{33} +(4.09392 + 1.24632i) q^{35} -62.4066i q^{37} +18.2262 q^{39} -21.4289i q^{41} +30.3265 q^{43} +(31.5240 + 9.59690i) q^{45} -25.3886i q^{47} -48.2675 q^{49} +39.7189i q^{51} -41.4423 q^{53} +(-6.30281 + 20.7035i) q^{55} -96.2562 q^{57} -28.8520 q^{59} +98.4653i q^{61} +5.64070 q^{63} +(-22.0795 - 6.72170i) q^{65} +(18.9961 - 64.2507i) q^{67} -38.6465i q^{69} +118.560 q^{71} -90.6692i q^{73} +(-81.9669 - 55.0044i) q^{75} +3.70455i q^{77} +137.981i q^{79} -96.8799 q^{81} +17.9932i q^{83} +(14.6481 - 48.1162i) q^{85} +89.1451 q^{87} +107.490 q^{89} -3.95076 q^{91} -188.279i q^{93} +(116.606 + 35.4987i) q^{95} +121.609 q^{97} +28.5258i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 68 q + 208 q^{9}+O(q^{10}) $ 68 * q + 208 * q^9	$ 68 q + 208 q^{9} - 18 q^{15} + 48 q^{19} - 48 q^{21} + 34 q^{25} + 20 q^{29} + 22 q^{35} - 24 q^{39} + 656 q^{49} + 44 q^{55} + 124 q^{59} - 106 q^{65} - 28 q^{71} + 660 q^{81} + 12 q^{89} - 684 q^{91}+O(q^{100}) $ 68 * q + 208 * q^9 - 18 * q^15 + 48 * q^19 - 48 * q^21 + 34 * q^25 + 20 * q^29 + 22 * q^35 - 24 * q^39 + 656 * q^49 + 44 * q^55 + 124 * q^59 - 106 * q^65 - 28 * q^71 + 660 * q^81 + 12 * q^89 - 684 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times$.

$n$	$537$	$671$	$1141$
$\chi(n)$	$-1$	$1$	$-1$

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			0
$2$		0			0
$3$	−3.94848			−1.31616			−0.658080	−	0.752948i	$-0.728631\pi$
$3$	−3.94848			−1.31616			−0.658080	+	0.752948i	$0.728631\pi$
$4$		0			0
$5$	4.78326	+	1.45617i	0.956652	+	0.291235i
$5$	4.78326	+	1.45617i	0.956652	+	0.291235i
$6$		0			0
$7$	0.855885			0.122269			0.0611346	−	0.998130i	$-0.480528\pi$
$7$	0.855885			0.122269			0.0611346	+	0.998130i	$0.480528\pi$
$8$		0			0
$9$	6.59049			0.732276
$10$		0			0
$11$			4.32833i			0.393485i	0.980455	+	0.196742i	$0.0630362\pi$
$11$			4.32833i			0.393485i	−0.980455	+	0.196742i	$0.936964\pi$
$12$		0			0
$13$	−4.61600			−0.355077			−0.177538	−	0.984114i	$-0.556813\pi$
$13$	−4.61600			−0.355077			−0.177538	+	0.984114i	$0.556813\pi$
$14$		0			0
$15$	−18.8866	−	5.74968i	−1.25911	−	0.383312i
$16$		0			0
$17$		−	10.0593i		−	0.591723i	−0.955231	−	0.295862i	$-0.904393\pi$
$17$		−	10.0593i		−	0.591723i	0.955231	−	0.295862i	$-0.0956067\pi$
$18$		0			0
$19$	24.3780			1.28305			0.641527	−	0.767100i	$-0.278301\pi$
$19$	24.3780			1.28305			0.641527	+	0.767100i	$0.278301\pi$
$20$		0			0
$21$	−3.37944			−0.160926
$22$		0			0
$23$			9.78769i			0.425552i	0.977101	+	0.212776i	$0.0682504\pi$
$23$			9.78769i			0.425552i	−0.977101	+	0.212776i	$0.931750\pi$
$24$		0			0
$25$	20.7591	+	13.9305i	0.830364	+	0.557221i
$26$		0			0
$27$	9.51391			0.352367
$28$		0			0
$29$	−22.5771			−0.778519			−0.389260	−	0.921128i	$-0.627269\pi$
$29$	−22.5771			−0.778519			−0.389260	+	0.921128i	$0.627269\pi$
$30$		0			0
$31$			47.6840i			1.53819i	0.639132	+	0.769097i	$0.279294\pi$
$31$			47.6840i			1.53819i	−0.639132	+	0.769097i	$0.720706\pi$
$32$		0			0
$33$		−	17.0903i		−	0.517889i
$34$		0			0
$35$	4.09392	+	1.24632i	0.116969	+	0.0356091i
$36$		0			0
$37$		−	62.4066i		−	1.68666i	−0.537392	−	0.843332i	$-0.680590\pi$
$37$		−	62.4066i		−	1.68666i	0.537392	−	0.843332i	$-0.319410\pi$
$38$		0			0
$39$	18.2262			0.467337
$40$		0			0
$41$		−	21.4289i		−	0.522656i	−0.965250	−	0.261328i	$-0.915840\pi$
$41$		−	21.4289i		−	0.522656i	0.965250	−	0.261328i	$-0.0841604\pi$
$42$		0			0
$43$	30.3265			0.705268			0.352634	−	0.935761i	$-0.385286\pi$
$43$	30.3265			0.705268			0.352634	+	0.935761i	$0.385286\pi$
$44$		0			0
$45$	31.5240	+	9.59690i	0.700533	+	0.213265i
$46$		0			0
$47$		−	25.3886i		−	0.540184i	−0.962835	−	0.270092i	$-0.912946\pi$
$47$		−	25.3886i		−	0.540184i	0.962835	−	0.270092i	$-0.0870541\pi$
$48$		0			0
$49$	−48.2675			−0.985050
$50$		0			0
$51$			39.7189i			0.778802i
$52$		0			0
$53$	−41.4423			−0.781930			−0.390965	−	0.920406i	$-0.627859\pi$
$53$	−41.4423			−0.781930			−0.390965	+	0.920406i	$0.627859\pi$
$54$		0			0
$55$	−6.30281	+	20.7035i	−0.114596	+	0.376428i
$56$		0			0
$57$	−96.2562			−1.68870
$58$		0			0
$59$	−28.8520			−0.489018			−0.244509	−	0.969647i	$-0.578627\pi$
$59$	−28.8520			−0.489018			−0.244509	+	0.969647i	$0.578627\pi$
$60$		0			0
$61$			98.4653i			1.61418i	0.590425	+	0.807092i	$0.298960\pi$
$61$			98.4653i			1.61418i	−0.590425	+	0.807092i	$0.701040\pi$
$62$		0			0
$63$	5.64070			0.0895349
$64$		0			0
$65$	−22.0795	−	6.72170i	−0.339685	−	0.103411i
$66$		0			0
$67$	18.9961	−	64.2507i	0.283524	−	0.958965i
$67$	18.9961	−	64.2507i	0.283524	−	0.958965i
$68$		0			0
$69$		−	38.6465i		−	0.560094i
$70$		0			0
$71$	118.560			1.66986			0.834928	−	0.550359i	$-0.185509\pi$
$71$	118.560			1.66986			0.834928	+	0.550359i	$0.185509\pi$
$72$		0			0
$73$		−	90.6692i		−	1.24204i	−0.783793	−	0.621022i	$-0.786718\pi$
$73$		−	90.6692i		−	1.24204i	0.783793	−	0.621022i	$-0.213282\pi$
$74$		0			0
$75$	−81.9669	−	55.0044i	−1.09289	−	0.733392i
$76$		0			0
$77$			3.70455i			0.0481111i
$78$		0			0
$79$			137.981i			1.74659i	0.487191	+	0.873295i	$0.338021\pi$
$79$			137.981i			1.74659i	−0.487191	+	0.873295i	$0.661979\pi$
$80$		0			0
$81$	−96.8799			−1.19605
$82$		0			0
$83$			17.9932i			0.216785i	0.994108	+	0.108393i	$0.0345704\pi$
$83$			17.9932i			0.216785i	−0.994108	+	0.108393i	$0.965430\pi$
$84$		0			0
$85$	14.6481	−	48.1162i	0.172330	−	0.566073i
$86$		0			0
$87$	89.1451			1.02466
$88$		0			0
$89$	107.490			1.20775			0.603876	−	0.797079i	$-0.293622\pi$
$89$	107.490			1.20775			0.603876	+	0.797079i	$0.293622\pi$
$90$		0			0
$91$	−3.95076			−0.0434150
$92$		0			0
$93$		−	188.279i		−	2.02451i
$94$		0			0
$95$	116.606	+	35.4987i	1.22744	+	0.373670i
$96$		0			0
$97$	121.609			1.25370			0.626852	−	0.779139i	$-0.284343\pi$
$97$	121.609			1.25370			0.626852	+	0.779139i	$0.284343\pi$
$98$		0			0
$99$			28.5258i			0.288140i
$100$		0			0
$101$			133.352i			1.32032i	0.751125	+	0.660161i	$0.229512\pi$
$101$			133.352i			1.32032i	−0.751125	+	0.660161i	$0.770488\pi$
$102$		0			0
$103$			145.855i			1.41607i	0.706177	+	0.708035i	$0.250418\pi$
$103$			145.855i			1.41607i	−0.706177	+	0.708035i	$0.749582\pi$
$104$		0			0
$105$	−16.1648	−	4.92106i	−0.153950	−	0.0468673i
$106$		0			0
$107$			179.341i			1.67608i	0.545609	+	0.838040i	$0.316299\pi$
$107$			179.341i			1.67608i	−0.545609	+	0.838040i	$0.683701\pi$
$108$		0			0
$109$			117.813i			1.08085i	0.841391	+	0.540427i	$0.181737\pi$
$109$			117.813i			1.08085i	−0.841391	+	0.540427i	$0.818263\pi$
$110$		0			0
$111$			246.411i			2.21992i
$112$		0			0
$113$	81.1018			0.717715			0.358858	−	0.933392i	$-0.383166\pi$
$113$	81.1018			0.717715			0.358858	+	0.933392i	$0.383166\pi$
$114$		0			0
$115$	−14.2526	+	46.8171i	−0.123936	+	0.407105i
$116$		0			0
$117$	−30.4217			−0.260014
$118$		0			0
$119$		−	8.60960i		−	0.0723496i
$120$		0			0
$121$	102.266			0.845170
$122$		0			0
$123$			84.6116i			0.687899i
$124$		0			0
$125$	79.0109	+	96.8622i	0.632087	+	0.774897i
$126$		0			0
$127$		−	31.2818i		−	0.246314i	−0.992387	−	0.123157i	$-0.960698\pi$
$127$		−	31.2818i		−	0.246314i	0.992387	−	0.123157i	$-0.0393018\pi$
$128$		0			0
$129$	−119.744			−0.928245
$130$		0			0
$131$	181.724			1.38721			0.693603	−	0.720358i	$-0.256022\pi$
$131$	181.724			1.38721			0.693603	+	0.720358i	$0.256022\pi$
$132$		0			0
$133$	20.8648			0.156878
$134$		0			0
$135$	45.5075	+	13.8539i	0.337092	+	0.102622i
$136$		0			0
$137$	−194.446			−1.41931			−0.709657	−	0.704547i	$-0.751150\pi$
$137$	−194.446			−1.41931			−0.709657	+	0.704547i	$0.751150\pi$
$138$		0			0
$139$		−	194.020i		−	1.39583i	−0.716180	−	0.697915i	$-0.754111\pi$
$139$		−	194.020i		−	1.39583i	0.716180	−	0.697915i	$-0.245889\pi$
$140$		0			0
$141$			100.247i			0.710968i
$142$		0			0
$143$		−	19.9796i		−	0.139717i
$144$		0			0
$145$	−107.992	−	32.8761i	−0.744772	−	0.226732i
$146$		0			0
$147$	190.583			1.29648
$148$		0			0
$149$	100.073			0.671634			0.335817	−	0.941927i	$-0.390988\pi$
$149$	100.073			0.671634			0.335817	+	0.941927i	$0.390988\pi$
$150$		0			0
$151$	−32.3260			−0.214080			−0.107040	−	0.994255i	$-0.534137\pi$
$151$	−32.3260			−0.214080			−0.107040	+	0.994255i	$0.534137\pi$
$152$		0			0
$153$		−	66.2956i		−	0.433305i
$154$		0			0
$155$	−69.4363	+	228.085i	−0.447976	+	1.47152i
$156$		0			0
$157$			251.363i			1.60104i	0.599309	+	0.800518i	$0.295442\pi$
$157$			251.363i			1.60104i	−0.599309	+	0.800518i	$0.704558\pi$
$158$		0			0
$159$	163.634			1.02914
$160$		0			0
$161$			8.37714i			0.0520319i
$162$		0			0
$163$			232.856i			1.42856i	0.699858	+	0.714282i	$0.253247\pi$
$163$			232.856i			1.42856i	−0.699858	+	0.714282i	$0.746753\pi$
$164$		0			0
$165$	24.8865	−	81.7474i	0.150827	−	0.495439i
$166$		0			0
$167$			230.268i			1.37885i	0.724358	+	0.689424i	$0.242136\pi$
$167$			230.268i			1.37885i	−0.724358	+	0.689424i	$0.757864\pi$
$168$		0			0
$169$	−147.693			−0.873921
$170$		0			0
$171$	160.663			0.939551
$172$		0			0
$173$			11.0120i			0.0636532i	0.999493	+	0.0318266i	$0.0101324\pi$
$173$			11.0120i			0.0636532i	−0.999493	+	0.0318266i	$0.989868\pi$
$174$		0			0
$175$	17.7674	+	11.9229i	0.101528	+	0.0681310i
$176$		0			0
$177$	113.922			0.643625
$178$		0			0
$179$		−	43.1431i		−	0.241023i	−0.992712	−	0.120511i	$-0.961547\pi$
$179$		−	43.1431i		−	0.241023i	0.992712	−	0.120511i	$-0.0384534\pi$
$180$		0			0
$181$	−29.5436			−0.163225			−0.0816123	−	0.996664i	$-0.526007\pi$
$181$	−29.5436			−0.163225			−0.0816123	+	0.996664i	$0.526007\pi$
$182$		0			0
$183$		−	388.788i		−	2.12452i
$184$		0			0
$185$	90.8749	−	298.507i	0.491216	−	1.61355i
$186$		0			0
$187$	43.5399			0.232834
$188$		0			0
$189$	8.14281			0.0430837
$190$		0			0
$191$			148.942i			0.779803i	0.920857	+	0.389902i	$0.127491\pi$
$191$			148.942i			0.779803i	−0.920857	+	0.389902i	$0.872509\pi$
$192$		0			0
$193$			48.6412i			0.252027i	0.992029	+	0.126013i	$0.0402183\pi$
$193$			48.6412i			0.252027i	−0.992029	+	0.126013i	$0.959782\pi$
$194$		0			0
$195$	87.1804	+	26.5405i	0.447079	+	0.136105i
$196$		0			0
$197$	143.069			0.726237			0.363119	−	0.931743i	$-0.381712\pi$
$197$	143.069			0.726237			0.363119	+	0.931743i	$0.381712\pi$
$198$		0			0
$199$	13.8535			0.0696156			0.0348078	−	0.999394i	$-0.488918\pi$
$199$	13.8535			0.0696156			0.0348078	+	0.999394i	$0.488918\pi$
$200$		0			0
$201$	−75.0057	+	253.692i	−0.373163	+	1.26215i
$202$		0			0
$203$	−19.3234			−0.0951890
$204$		0			0
$205$	31.2042	−	102.500i	0.152216	−	0.500000i
$206$		0			0
$207$			64.5057i			0.311622i
$208$		0			0
$209$			105.516i			0.504862i
$210$		0			0
$211$	5.59113			0.0264983			0.0132491	−	0.999912i	$-0.495783\pi$
$211$	5.59113			0.0264983			0.0132491	+	0.999912i	$0.495783\pi$
$212$		0			0
$213$	−468.131			−2.19780
$214$		0			0
$215$	145.060	+	44.1607i	0.674695	+	0.205399i
$216$		0			0
$217$			40.8120i			0.188074i
$218$		0			0
$219$			358.005i			1.63473i
$220$		0			0
$221$			46.4336i			0.210107i
$222$		0			0
$223$		−	101.379i		−	0.454613i	−0.973823	−	0.227306i	$-0.927008\pi$
$223$		−	101.379i		−	0.454613i	0.973823	−	0.227306i	$-0.0729919\pi$
$224$		0			0
$225$	136.813	+	91.8089i	0.608056	+	0.408040i
$226$		0			0
$227$			255.116i			1.12386i	0.827184	+	0.561930i	$0.189941\pi$
$227$			255.116i			1.12386i	−0.827184	+	0.561930i	$0.810059\pi$
$228$		0			0
$229$		−	26.7615i		−	0.116862i	−0.998291	−	0.0584311i	$-0.981390\pi$
$229$		−	26.7615i		−	0.116862i	0.998291	−	0.0584311i	$-0.0186098\pi$
$230$		0			0
$231$		−	14.6274i		−	0.0633219i
$232$		0			0
$233$	413.650			1.77532			0.887662	−	0.460496i	$-0.152328\pi$
$233$	413.650			1.77532			0.887662	+	0.460496i	$0.152328\pi$
$234$		0			0
$235$	36.9703	−	121.440i	0.157320	−	0.516768i
$236$		0			0
$237$		−	544.814i		−	2.29879i
$238$		0			0
$239$		−	422.347i		−	1.76714i	−0.468297	−	0.883571i	$-0.655132\pi$
$239$		−	422.347i		−	1.76714i	0.468297	−	0.883571i	$-0.344868\pi$
$240$		0			0
$241$	317.835			1.31882			0.659409	−	0.751785i	$-0.270807\pi$
$241$	317.835			1.31882			0.659409	+	0.751785i	$0.270807\pi$
$242$		0			0
$243$	296.903			1.22182
$244$		0			0
$245$	−230.876	−	70.2859i	−0.942350	−	0.286881i
$246$		0			0
$247$	−112.529			−0.455583
$248$		0			0
$249$		−	71.0457i		−	0.285324i
$250$		0			0
$251$			371.440i			1.47984i	0.672695	+	0.739920i	$0.265136\pi$
$251$			371.440i			1.47984i	−0.672695	+	0.739920i	$0.734864\pi$
$252$		0			0
$253$	−42.3644			−0.167448
$254$		0			0
$255$	−57.8377	+	189.986i	−0.226814	+	0.745042i
$256$		0			0
$257$		−	188.712i		−	0.734290i	−0.930164	−	0.367145i	$-0.880335\pi$
$257$		−	188.712i		−	0.734290i	0.930164	−	0.367145i	$-0.119665\pi$
$258$		0			0
$259$		−	53.4129i		−	0.206227i
$260$		0			0
$261$	−148.794			−0.570091
$262$		0			0
$263$		−	246.987i		−	0.939113i	−0.882902	−	0.469557i	$-0.844414\pi$
$263$		−	246.987i		−	0.939113i	0.882902	−	0.469557i	$-0.155586\pi$
$264$		0			0
$265$	−198.229	−	60.3472i	−0.748034	−	0.227725i
$266$		0			0
$267$	−424.421			−1.58959
$268$		0			0
$269$	−141.598			−0.526386			−0.263193	−	0.964743i	$-0.584776\pi$
$269$	−141.598			−0.526386			−0.263193	+	0.964743i	$0.584776\pi$
$270$		0			0
$271$			414.334i			1.52891i	0.644679	+	0.764453i	$0.276991\pi$
$271$			414.334i			1.52891i	−0.644679	+	0.764453i	$0.723009\pi$
$272$		0			0
$273$	15.5995			0.0571410
$274$		0			0
$275$	−60.2959	+	89.8523i	−0.219258	+	0.326736i
$276$		0			0
$277$			53.9517i			0.194771i	0.995247	+	0.0973857i	$0.0310480\pi$
$277$			53.9517i			0.194771i	−0.995247	+	0.0973857i	$0.968952\pi$
$278$		0			0
$279$			314.261i			1.12638i
$280$		0			0
$281$		−	326.327i		−	1.16130i	−0.814152	−	0.580652i	$-0.802798\pi$
$281$		−	326.327i		−	1.16130i	0.814152	−	0.580652i	$-0.197202\pi$
$282$		0			0
$283$		−	287.048i		−	1.01430i	−0.861857	−	0.507151i	$-0.830698\pi$
$283$		−	287.048i		−	1.01430i	0.861857	−	0.507151i	$-0.169302\pi$
$284$		0			0
$285$	−460.418	−	140.166i	−1.61550	−	0.491810i
$286$		0			0
$287$		−	18.3407i		−	0.0639048i
$288$		0			0
$289$	187.811			0.649864
$290$		0			0
$291$	−480.172			−1.65007
$292$		0			0
$293$		−	144.806i		−	0.494220i	−0.968987	−	0.247110i	$-0.920519\pi$
$293$		−	144.806i		−	0.494220i	0.968987	−	0.247110i	$-0.0794808\pi$
$294$		0			0
$295$	−138.007	−	42.0136i	−0.467820	−	0.142419i
$296$		0			0
$297$			41.1793i			0.138651i
$298$		0			0
$299$		−	45.1799i		−	0.151103i
$300$		0			0
$301$	25.9560			0.0862326
$302$		0			0
$303$		−	526.539i		−	1.73775i
$304$		0			0
$305$	−143.383	+	470.985i	−0.470107	+	1.54421i
$306$		0			0
$307$		−	119.545i		−	0.389399i	−0.980863	−	0.194699i	$-0.937627\pi$
$307$		−	119.545i		−	0.389399i	0.980863	−	0.194699i	$-0.0623731\pi$
$308$		0			0
$309$		−	575.907i		−	1.86378i
$310$		0			0
$311$		−	192.248i		−	0.618160i	−0.951036	−	0.309080i	$-0.899979\pi$
$311$		−	192.248i		−	0.618160i	0.951036	−	0.309080i	$-0.100021\pi$
$312$		0			0
$313$	−358.885			−1.14660			−0.573299	−	0.819346i	$-0.694337\pi$
$313$	−358.885			−1.14660			−0.573299	+	0.819346i	$0.694337\pi$
$314$		0			0
$315$	26.9809	+	8.21385i	0.0856537	+	0.0260757i
$316$		0			0
$317$			297.142i			0.937356i	0.883369	+	0.468678i	$0.155270\pi$
$317$			297.142i			0.937356i	−0.883369	+	0.468678i	$0.844730\pi$
$318$		0			0
$319$		−	97.7210i		−	0.306335i
$320$		0			0
$321$		−	708.123i		−	2.20599i
$322$		0			0
$323$		−	245.226i		−	0.759213i
$324$		0			0
$325$	−95.8240	−	64.3032i	−0.294843	−	0.197856i
$326$		0			0
$327$		−	465.183i		−	1.42258i
$328$		0			0
$329$		−	21.7298i		−	0.0660479i
$330$		0			0
$331$			488.872i			1.47696i	0.674278	+	0.738478i	$0.264455\pi$
$331$			488.872i			1.47696i	−0.674278	+	0.738478i	$0.735545\pi$
$332$		0			0
$333$		−	411.290i		−	1.23511i
$334$		0			0
$335$	184.423	−	279.666i	0.550518	−	0.834823i
$336$		0			0
$337$	−265.223			−0.787012			−0.393506	−	0.919322i	$-0.628738\pi$
$337$	−265.223			−0.787012			−0.393506	+	0.919322i	$0.628738\pi$
$338$		0			0
$339$	−320.229			−0.944628
$340$		0			0
$341$	−206.392			−0.605256
$342$		0			0
$343$	−83.2498			−0.242711
$344$		0			0
$345$	56.2761	−	184.856i	0.163119	−	0.535815i
$346$		0			0
$347$	4.22662			0.0121805			0.00609023	−	0.999981i	$-0.498061\pi$
$347$	4.22662			0.0121805			0.00609023	+	0.999981i	$0.498061\pi$
$348$		0			0
$349$	86.7812			0.248657			0.124328	−	0.992241i	$-0.460322\pi$
$349$	86.7812			0.248657			0.124328	+	0.992241i	$0.460322\pi$
$350$		0			0
$351$	−43.9162			−0.125117
$352$		0			0
$353$	−476.102			−1.34873			−0.674365	−	0.738398i	$-0.735583\pi$
$353$	−476.102			−1.34873			−0.674365	+	0.738398i	$0.735583\pi$
$354$		0			0
$355$	567.102	+	172.644i	1.59747	+	0.486320i
$356$		0			0
$357$			33.9948i			0.0952236i
$358$		0			0
$359$	−392.646			−1.09372			−0.546860	−	0.837224i	$-0.684177\pi$
$359$	−392.646			−1.09372			−0.546860	+	0.837224i	$0.684177\pi$
$360$		0			0
$361$	233.289			0.646229
$362$		0			0
$363$	−403.793			−1.11238
$364$		0			0
$365$	132.030	−	433.694i	0.361726	−	1.18820i
$366$		0			0
$367$	454.204			1.23761			0.618806	−	0.785544i	$-0.287617\pi$
$367$	454.204			1.23761			0.618806	+	0.785544i	$0.287617\pi$
$368$		0			0
$369$		−	141.227i		−	0.382729i
$370$		0			0
$371$	−35.4698			−0.0956060
$372$		0			0
$373$	596.464			1.59910			0.799549	−	0.600600i	$-0.205072\pi$
$373$	596.464			1.59910			0.799549	+	0.600600i	$0.205072\pi$
$374$		0			0
$375$	−311.973	−	382.458i	−0.831928	−	1.01989i
$376$		0			0
$377$	104.216			0.276434
$378$		0			0
$379$		−	232.359i		−	0.613085i	−0.951857	−	0.306542i	$-0.900828\pi$
$379$		−	232.359i		−	0.613085i	0.951857	−	0.306542i	$-0.0991721\pi$
$380$		0			0
$381$			123.516i			0.324188i
$382$		0			0
$383$	−669.728			−1.74864			−0.874318	−	0.485354i	$-0.838691\pi$
$383$	−669.728			−1.74864			−0.874318	+	0.485354i	$0.838691\pi$
$384$		0			0
$385$	−5.39448	+	17.7198i	−0.0140116	+	0.0460256i
$386$		0			0
$387$	199.867			0.516451
$388$		0			0
$389$	192.318			0.494391			0.247195	−	0.968966i	$-0.420491\pi$
$389$	192.318			0.494391			0.247195	+	0.968966i	$0.420491\pi$
$390$		0			0
$391$	98.4573			0.251809
$392$		0			0
$393$	−717.533			−1.82578
$394$		0			0
$395$	−200.924	+	659.997i	−0.508668	+	1.67088i
$396$		0			0
$397$			198.527i			0.500067i	0.968237	+	0.250034i	$0.0804417\pi$
$397$			198.527i			0.500067i	−0.968237	+	0.250034i	$0.919558\pi$
$398$		0			0
$399$	−82.3842			−0.206477
$400$		0			0
$401$		−	484.388i		−	1.20795i	−0.797004	−	0.603975i	$-0.793583\pi$
$401$		−	484.388i		−	1.20795i	0.797004	−	0.603975i	$-0.206417\pi$
$402$		0			0
$403$		−	220.109i		−	0.546177i
$404$		0			0
$405$	−463.401	−	141.074i	−1.14420	−	0.348331i
$406$		0			0
$407$	270.116			0.663677
$408$		0			0
$409$			37.3002i			0.0911984i	0.998960	+	0.0455992i	$0.0145197\pi$
$409$			37.3002i			0.0911984i	−0.998960	+	0.0455992i	$0.985480\pi$
$410$		0			0
$411$	767.766			1.86804
$412$		0			0
$413$	−24.6940			−0.0597919
$414$		0			0
$415$	−26.2012	+	86.0661i	−0.0631355	+	0.207388i
$416$		0			0
$417$			766.086i			1.83714i
$418$		0			0
$419$	572.901			1.36730			0.683652	−	0.729808i	$-0.260390\pi$
$419$	572.901			1.36730			0.683652	+	0.729808i	$0.260390\pi$
$420$		0			0
$421$	−308.037			−0.731679			−0.365839	−	0.930678i	$-0.619218\pi$
$421$	−308.037			−0.731679			−0.365839	+	0.930678i	$0.619218\pi$
$422$		0			0
$423$		−	167.324i		−	0.395564i
$424$		0			0
$425$	140.131	−	208.822i	0.329720	−	0.491346i
$426$		0			0
$427$			84.2749i			0.197365i
$428$		0			0
$429$			78.8889i			0.183890i
$430$		0			0
$431$	−127.773			−0.296458			−0.148229	−	0.988953i	$-0.547357\pi$
$431$	−127.773			−0.296458			−0.148229	+	0.988953i	$0.547357\pi$
$432$		0			0
$433$	305.662			0.705918			0.352959	−	0.935639i	$-0.385176\pi$
$433$	305.662			0.705918			0.352959	+	0.935639i	$0.385176\pi$
$434$		0			0
$435$	426.404	+	129.811i	0.980239	+	0.298416i
$436$		0			0
$437$			238.605i			0.546006i
$438$		0			0
$439$	−823.344			−1.87550			−0.937750	−	0.347311i	$-0.887095\pi$
$439$	−823.344			−1.87550			−0.937750	+	0.347311i	$0.887095\pi$
$440$		0			0
$441$	−318.106			−0.721329
$442$		0			0
$443$	49.6007			0.111966			0.0559828	−	0.998432i	$-0.482171\pi$
$443$	49.6007			0.111966			0.0559828	+	0.998432i	$0.482171\pi$
$444$		0			0
$445$	514.152	+	156.524i	1.15540	+	0.351739i
$446$		0			0
$447$	−395.138			−0.883978
$448$		0			0
$449$	−411.512			−0.916507			−0.458253	−	0.888822i	$-0.651525\pi$
$449$	−411.512			−0.916507			−0.458253	+	0.888822i	$0.651525\pi$
$450$		0			0
$451$	92.7514			0.205657
$452$		0			0
$453$	127.639			0.281763
$454$		0			0
$455$	−18.8975	−	5.75300i	−0.0415330	−	0.0126440i
$456$		0			0
$457$			874.960i			1.91457i	0.289141	+	0.957287i	$0.406630\pi$
$457$			874.960i			1.91457i	−0.289141	+	0.957287i	$0.593370\pi$
$458$		0			0
$459$		−	95.7032i		−	0.208504i
$460$		0			0
$461$	254.000			0.550976			0.275488	−	0.961305i	$-0.411161\pi$
$461$	254.000			0.550976			0.275488	+	0.961305i	$0.411161\pi$
$462$		0			0
$463$	286.494			0.618778			0.309389	−	0.950935i	$-0.399875\pi$
$463$	286.494			0.618778			0.309389	+	0.950935i	$0.399875\pi$
$464$		0			0
$465$	274.168	−	900.589i	0.589608	−	1.93675i
$466$		0			0
$467$			50.3602i			0.107838i	0.998545	+	0.0539188i	$0.0171712\pi$
$467$			50.3602i			0.107838i	−0.998545	+	0.0539188i	$0.982829\pi$
$468$		0			0
$469$	16.2585	−	54.9912i	0.0346663	−	0.117252i
$470$		0			0
$471$		−	992.500i		−	2.10722i
$472$		0			0
$473$			131.263i			0.277512i
$474$		0			0
$475$	506.066	+	339.599i	1.06540	+	0.714945i
$476$		0			0
$477$	−273.125			−0.572589
$478$		0			0
$479$	425.646			0.888614			0.444307	−	0.895875i	$-0.353450\pi$
$479$	425.646			0.888614			0.444307	+	0.895875i	$0.353450\pi$
$480$		0			0
$481$			288.069i			0.598895i
$482$		0			0
$483$		−	33.0770i		−	0.0684823i
$484$		0			0
$485$	581.688	+	177.084i	1.19936	+	0.365122i
$486$		0			0
$487$	−538.993			−1.10676			−0.553381	−	0.832928i	$-0.686663\pi$
$487$	−538.993			−1.10676			−0.553381	+	0.832928i	$0.686663\pi$
$488$		0			0
$489$		−	919.427i		−	1.88022i
$490$		0			0
$491$	−245.190			−0.499369			−0.249684	−	0.968327i	$-0.580327\pi$
$491$	−245.190			−0.499369			−0.249684	+	0.968327i	$0.580327\pi$
$492$		0			0
$493$			227.109i			0.460668i
$494$		0			0
$495$	−41.5386	+	136.446i	−0.0839163	+	0.275649i
$496$		0			0
$497$	101.474			0.204172
$498$		0			0
$499$		−	75.1075i		−	0.150516i	−0.997164	−	0.0752580i	$-0.976022\pi$
$499$		−	75.1075i		−	0.150516i	0.997164	−	0.0752580i	$-0.0239780\pi$
$500$		0			0
$501$		−	909.207i		−	1.81478i
$502$		0			0
$503$	−792.671			−1.57589			−0.787943	−	0.615748i	$-0.788854\pi$
$503$	−792.671			−1.57589			−0.787943	+	0.615748i	$0.788854\pi$
$504$		0			0
$505$	−194.184	+	637.859i	−0.384524	+	1.26309i
$506$		0			0
$507$	583.161			1.15022
$508$		0			0
$509$	−340.537			−0.669031			−0.334516	−	0.942390i	$-0.608573\pi$
$509$	−340.537			−0.669031			−0.334516	+	0.942390i	$0.608573\pi$
$510$		0			0
$511$		−	77.6024i		−	0.151864i
$512$		0			0
$513$	231.930			0.452106
$514$		0			0
$515$	−212.391	+	697.663i	−0.412409	+	1.35469i
$516$		0			0
$517$	109.890			0.212554
$518$		0			0
$519$		−	43.4807i		−	0.0837778i
$520$		0			0
$521$		−	286.334i		−	0.549585i	−0.961503	−	0.274793i	$-0.911391\pi$
$521$		−	286.334i		−	0.549585i	0.961503	−	0.274793i	$-0.0886092\pi$
$522$		0			0
$523$		−	433.254i		−	0.828402i	−0.910185	−	0.414201i	$-0.864061\pi$
$523$		−	433.254i		−	0.828402i	0.910185	−	0.414201i	$-0.135939\pi$
$524$		0			0
$525$	−70.1543	−	47.0774i	−0.133627	−	0.0896713i
$526$		0			0
$527$	479.667			0.910185
$528$		0			0
$529$	433.201			0.818906
$530$		0			0
$531$	−190.149			−0.358096
$532$		0			0
$533$			98.9157i			0.185583i
$534$		0			0
$535$	−261.151	+	857.832i	−0.488133	+	1.60342i
$536$		0			0
$537$			170.349i			0.317224i
$538$		0			0
$539$		−	208.918i		−	0.387602i
$540$		0			0
$541$		−	444.429i		−	0.821495i	−0.911749	−	0.410747i	$-0.865268\pi$
$541$		−	444.429i		−	0.821495i	0.911749	−	0.410747i	$-0.134732\pi$
$542$		0			0
$543$	116.652			0.214830
$544$		0			0
$545$	−171.557	+	563.531i	−0.314783	+	1.03400i
$546$		0			0
$547$	−597.311			−1.09198			−0.545988	−	0.837793i	$-0.683846\pi$
$547$	−597.311			−1.09198			−0.545988	+	0.837793i	$0.683846\pi$
$548$		0			0
$549$			648.934i			1.18203i
$550$		0			0
$551$	−550.384			−0.998883
$552$		0			0
$553$			118.096i			0.213554i
$554$		0			0
$555$	−358.818	+	1178.65i	−0.646518	+	2.12369i
$556$		0			0
$557$			70.7209i			0.126968i	0.997983	+	0.0634838i	$0.0202211\pi$
$557$			70.7209i			0.126968i	−0.997983	+	0.0634838i	$0.979779\pi$
$558$		0			0
$559$	−139.987			−0.250424
$560$		0			0
$561$	−171.917			−0.306447
$562$		0			0
$563$	587.393			1.04333			0.521663	−	0.853152i	$-0.325312\pi$
$563$	587.393			1.04333			0.521663	+	0.853152i	$0.325312\pi$
$564$		0			0
$565$	387.931	+	118.098i	0.686603	+	0.209024i
$566$		0			0
$567$	−82.9180			−0.146240
$568$		0			0
$569$	676.561			1.18903			0.594517	−	0.804083i	$-0.297343\pi$
$569$	676.561			1.18903			0.594517	+	0.804083i	$0.297343\pi$
$570$		0			0
$571$	954.240			1.67117			0.835587	−	0.549359i	$-0.185128\pi$
$571$	954.240			1.67117			0.835587	+	0.549359i	$0.185128\pi$
$572$		0			0
$573$		−	588.096i		−	1.02635i
$574$		0			0
$575$	−136.348	+	203.184i	−0.237126	+	0.353363i
$576$		0			0
$577$	−74.9435			−0.129885			−0.0649423	−	0.997889i	$-0.520686\pi$
$577$	−74.9435			−0.129885			−0.0649423	+	0.997889i	$0.520686\pi$
$578$		0			0
$579$		−	192.059i		−	0.331708i
$580$		0			0
$581$			15.4001i			0.0265062i
$582$		0			0
$583$		−	179.376i		−	0.307677i
$584$		0			0
$585$	−145.515	−	44.2993i	−0.248743	−	0.0757252i
$586$		0			0
$587$	477.974			0.814266			0.407133	−	0.913369i	$-0.366528\pi$
$587$	477.974			0.814266			0.407133	+	0.913369i	$0.366528\pi$
$588$		0			0
$589$			1162.44i			1.97359i
$590$		0			0
$591$	−564.904			−0.955844
$592$		0			0
$593$	526.649			0.888109			0.444055	−	0.896000i	$-0.353540\pi$
$593$	526.649			0.888109			0.444055	+	0.896000i	$0.353540\pi$
$594$		0			0
$595$	12.5371	−	41.1819i	0.0210707	−	0.0692133i
$596$		0			0
$597$	−54.7003			−0.0916252
$598$		0			0
$599$		−	279.733i		−	0.467000i	−0.972357	−	0.233500i	$-0.924982\pi$
$599$		−	279.733i		−	0.467000i	0.972357	−	0.233500i	$-0.0750178\pi$
$600$		0			0
$601$	−417.113			−0.694032			−0.347016	−	0.937859i	$-0.612805\pi$
$601$	−417.113			−0.694032			−0.347016	+	0.937859i	$0.612805\pi$
$602$		0			0
$603$	125.194	−	423.443i	0.207618	−	0.702228i
$604$		0			0
$605$	489.162	+	148.917i	0.808533	+	0.246143i
$606$		0			0
$607$		−	481.778i		−	0.793703i	−0.917883	−	0.396852i	$-0.870103\pi$
$607$		−	481.778i		−	0.793703i	0.917883	−	0.396852i	$-0.129897\pi$
$608$		0			0
$609$	76.2979			0.125284
$610$		0			0
$611$			117.194i			0.191807i
$612$		0			0
$613$			120.957i			0.197320i	0.995121	+	0.0986600i	$0.0314556\pi$
$613$			120.957i			0.197320i	−0.995121	+	0.0986600i	$0.968544\pi$
$614$		0			0
$615$	−123.209	+	404.719i	−0.200340	+	0.658080i
$616$		0			0
$617$		−	665.767i		−	1.07904i	−0.841973	−	0.539520i	$-0.818606\pi$
$617$		−	665.767i		−	1.07904i	0.841973	−	0.539520i	$-0.181394\pi$
$618$		0			0
$619$	659.991			1.06622			0.533110	−	0.846046i	$-0.321023\pi$
$619$	659.991			1.06622			0.533110	+	0.846046i	$0.321023\pi$
$620$		0			0
$621$			93.1192i			0.149950i
$622$		0			0
$623$	91.9990			0.147671
$624$		0			0
$625$	236.881	+	578.370i	0.379010	+	0.925393i
$626$		0			0
$627$		−	416.629i		−	0.664479i
$628$		0			0
$629$	−627.766			−0.998039
$630$		0			0
$631$			343.720i			0.544723i	0.962195	+	0.272361i	$0.0878046\pi$
$631$			343.720i			0.544723i	−0.962195	+	0.272361i	$0.912195\pi$
$632$		0			0
$633$	−22.0765			−0.0348759
$634$		0			0
$635$	45.5518	−	149.629i	0.0717351	−	0.235636i
$636$		0			0
$637$	222.802			0.349768
$638$		0			0
$639$	781.367			1.22280
$640$		0			0
$641$		−	1020.34i		−	1.59179i	−0.605432	−	0.795897i	$-0.706999\pi$
$641$		−	1020.34i		−	1.59179i	0.605432	−	0.795897i	$-0.293001\pi$
$642$		0			0
$643$			450.851i			0.701168i	0.936531	+	0.350584i	$0.114017\pi$
$643$			450.851i			0.701168i	−0.936531	+	0.350584i	$0.885983\pi$
$644$		0			0
$645$	−572.765	−	174.368i	−0.888007	−	0.270337i
$646$		0			0
$647$	−782.802			−1.20989			−0.604947	−	0.796265i	$-0.706806\pi$
$647$	−782.802			−1.20989			−0.604947	+	0.796265i	$0.706806\pi$
$648$		0			0
$649$		−	124.881i		−	0.192421i
$650$		0			0
$651$		−	161.145i		−	0.247535i
$652$		0			0
$653$	−1083.86			−1.65981			−0.829906	−	0.557904i	$-0.811606\pi$
$653$	−1083.86			−1.65981			−0.829906	+	0.557904i	$0.811606\pi$
$654$		0			0
$655$	869.232	+	264.622i	1.32707	+	0.404003i
$656$		0			0
$657$		−	597.554i		−	0.909519i
$658$		0			0
$659$	−153.081			−0.232293			−0.116147	−	0.993232i	$-0.537054\pi$
$659$	−153.081			−0.232293			−0.116147	+	0.993232i	$0.537054\pi$
$660$		0			0
$661$		−	1000.18i		−	1.51313i	−0.653917	−	0.756566i	$-0.726875\pi$
$661$		−	1000.18i		−	1.51313i	0.653917	−	0.756566i	$-0.273125\pi$
$662$		0			0
$663$		−	183.342i		−	0.276534i
$664$		0			0
$665$	99.8017	+	30.3828i	0.150078	+	0.0456884i
$666$		0			0
$667$		−	220.977i		−	0.331300i
$668$		0			0
$669$			400.291i			0.598343i
$670$		0			0
$671$	−426.190			−0.635157
$672$		0			0
$673$	−1121.64			−1.66662			−0.833310	−	0.552806i	$-0.813557\pi$
$673$	−1121.64			−1.66662			−0.833310	+	0.552806i	$0.813557\pi$
$674$		0			0
$675$	197.500	+	132.534i	0.292593	+	0.196346i
$676$		0			0
$677$	−892.185			−1.31785			−0.658925	−	0.752209i	$-0.728989\pi$
$677$	−892.185			−1.31785			−0.658925	+	0.752209i	$0.728989\pi$
$678$		0			0
$679$	104.084			0.153289
$680$		0			0
$681$		−	1007.32i		−	1.47918i
$682$		0			0
$683$	843.556			1.23507			0.617537	−	0.786542i	$-0.288130\pi$
$683$	843.556			1.23507			0.617537	+	0.786542i	$0.288130\pi$
$684$		0			0
$685$	−930.086	−	283.148i	−1.35779	−	0.413354i
$686$		0			0
$687$			105.667i			0.153809i
$688$		0			0
$689$	191.297			0.277645
$690$		0			0
$691$	−76.3031			−0.110424			−0.0552121	−	0.998475i	$-0.517584\pi$
$691$	−76.3031			−0.110424			−0.0552121	+	0.998475i	$0.517584\pi$
$692$		0			0
$693$			24.4148i			0.0352306i
$694$		0			0
$695$	282.528	−	928.050i	0.406515	−	1.33532i
$696$		0			0
$697$	−215.560			−0.309268
$698$		0			0
$699$	−1633.29			−2.33661
$700$		0			0
$701$		−	124.798i		−	0.178028i	−0.996030	−	0.0890142i	$-0.971628\pi$
$701$		−	124.798i		−	0.178028i	0.996030	−	0.0890142i	$-0.0283717\pi$
$702$		0			0
$703$		−	1521.35i		−	2.16408i
$704$		0			0
$705$	−145.976	+	479.505i	−0.207059	+	0.680149i
$706$		0			0
$707$			114.134i			0.161435i
$708$		0			0
$709$	−12.0337			−0.0169728			−0.00848639	−	0.999964i	$-0.502701\pi$
$709$	−12.0337			−0.0169728			−0.00848639	+	0.999964i	$0.502701\pi$
$710$		0			0
$711$			909.360i			1.27899i
$712$		0			0
$713$	−466.716			−0.654581
$714$		0			0
$715$	29.0937	−	95.5674i	0.0406905	−	0.133661i
$716$		0			0
$717$			1667.63i			2.32584i
$718$		0			0
$719$	494.946			0.688381			0.344191	−	0.938900i	$-0.388153\pi$
$719$	494.946			0.688381			0.344191	+	0.938900i	$0.388153\pi$
$720$		0			0
$721$			124.835i			0.173142i
$722$		0			0
$723$	−1254.97			−1.73577
$724$		0			0
$725$	−468.680	−	314.510i	−0.646455	−	0.433807i
$726$		0			0
$727$	543.442			0.747514			0.373757	−	0.927527i	$-0.378069\pi$
$727$	543.442			0.747514			0.373757	+	0.927527i	$0.378069\pi$
$728$		0			0
$729$	−300.396			−0.412066
$730$		0			0
$731$		−	305.063i		−	0.417323i
$732$		0			0
$733$	588.900			0.803411			0.401706	−	0.915769i	$-0.368418\pi$
$733$	588.900			0.803411			0.401706	+	0.915769i	$0.368418\pi$
$734$		0			0
$735$	911.608	+	277.522i	1.24028	+	0.377581i
$736$		0			0
$737$	278.098	+	82.2214i	0.377338	+	0.111562i
$738$		0			0
$739$			799.430i			1.08177i	0.841096	+	0.540886i	$0.181911\pi$
$739$			799.430i			1.08177i	−0.841096	+	0.540886i	$0.818089\pi$
$740$		0			0
$741$	444.318			0.599619
$742$		0			0
$743$		−	726.353i		−	0.977595i	−0.872397	−	0.488797i	$-0.837436\pi$
$743$		−	726.353i		−	0.977595i	0.872397	−	0.488797i	$-0.162564\pi$
$744$		0			0
$745$	478.677	+	145.724i	0.642520	+	0.195603i
$746$		0			0
$747$			118.584i			0.158747i
$748$		0			0
$749$			153.495i			0.204933i
$750$		0			0
$751$	596.715			0.794561			0.397280	−	0.917697i	$-0.369954\pi$
$751$	596.715			0.794561			0.397280	+	0.917697i	$0.369954\pi$
$752$		0			0
$753$		−	1466.62i		−	1.94770i
$754$		0			0
$755$	−154.624	−	47.0724i	−0.204800	−	0.0623475i
$756$		0			0
$757$	1419.77			1.87553			0.937763	−	0.347275i	$-0.112893\pi$
$757$	1419.77			1.87553			0.937763	+	0.347275i	$0.112893\pi$
$758$		0			0
$759$	167.275			0.220388
$760$		0			0
$761$	−1309.80			−1.72116			−0.860578	−	0.509319i	$-0.829897\pi$
$761$	−1309.80			−1.72116			−0.860578	+	0.509319i	$0.829897\pi$
$762$		0			0
$763$			100.835i			0.132155i
$764$		0			0
$765$	96.5380	−	317.109i	0.126194	−	0.414522i
$766$		0			0
$767$	133.181			0.173639
$768$		0			0
$769$			64.2236i			0.0835157i	0.999128	+	0.0417578i	$0.0132958\pi$
$769$			64.2236i			0.0835157i	−0.999128	+	0.0417578i	$0.986704\pi$
$770$		0			0
$771$			745.127i			0.966443i
$772$		0			0
$773$		−	1244.26i		−	1.60965i	−0.593510	−	0.804826i	$-0.702258\pi$
$773$		−	1244.26i		−	1.60965i	0.593510	−	0.804826i	$-0.297742\pi$
$774$		0			0
$775$	−664.263	+	989.878i	−0.857114	+	1.27726i
$776$		0			0
$777$			210.900i			0.271428i
$778$		0			0
$779$		−	522.394i		−	0.670596i
$780$		0			0
$781$			513.166i			0.657063i
$782$		0			0
$783$	−214.796			−0.274324
$784$		0			0
$785$	−366.028	+	1202.33i	−0.466278	+	1.53163i
$786$		0			0
$787$	−502.425			−0.638405			−0.319203	−	0.947686i	$-0.603415\pi$
$787$	−502.425			−0.638405			−0.319203	+	0.947686i	$0.603415\pi$
$788$		0			0
$789$			975.222i			1.23602i
$790$		0			0
$791$	69.4138			0.0877545
$792$		0			0
$793$		−	454.515i		−	0.573159i
$794$		0			0
$795$	782.703	+	238.280i	0.984533	+	0.299723i
$796$		0			0
$797$		−	161.557i		−	0.202707i	−0.994850	−	0.101353i	$-0.967683\pi$
$797$		−	161.557i		−	0.202707i	0.994850	−	0.101353i	$-0.0323173\pi$
$798$		0			0
$799$	−255.392			−0.319639
$800$		0			0
$801$	708.411			0.884408
$802$		0			0
$803$	392.446			0.488725
$804$		0			0
$805$	−12.1986	+	40.0700i	−0.0151535	+	0.0497764i
$806$		0			0
$807$	559.096			0.692808
$808$		0			0
$809$			415.623i			0.513749i	0.966445	+	0.256875i	$0.0826927\pi$
$809$			415.623i			0.513749i	−0.966445	+	0.256875i	$0.917307\pi$
$810$		0			0
$811$		−	37.3139i		−	0.0460098i	−0.999735	−	0.0230049i	$-0.992677\pi$
$811$		−	37.3139i		−	0.0460098i	0.999735	−	0.0230049i	$-0.00732333\pi$
$812$		0			0
$813$		−	1635.99i		−	2.01228i
$814$		0			0
$815$	−339.079	+	1113.81i	−0.416048	+	1.36664i
$816$		0			0
$817$	739.301			0.904897
$818$		0			0
$819$	−26.0374			−0.0317918
$820$		0			0
$821$	−113.419			−0.138147			−0.0690736	−	0.997612i	$-0.522004\pi$
$821$	−113.419			−0.138147			−0.0690736	+	0.997612i	$0.522004\pi$
$822$		0			0
$823$		−	927.557i		−	1.12704i	−0.826101	−	0.563522i	$-0.809446\pi$
$823$		−	927.557i		−	1.12704i	0.826101	−	0.563522i	$-0.190554\pi$
$824$		0			0
$825$	238.077	−	354.780i	0.288578	−	0.430036i
$826$		0			0
$827$		−	555.504i		−	0.671709i	−0.941914	−	0.335855i	$-0.890975\pi$
$827$		−	555.504i		−	0.671709i	0.941914	−	0.335855i	$-0.109025\pi$
$828$		0			0
$829$	1271.89			1.53425			0.767125	−	0.641498i	$-0.221687\pi$
$829$	1271.89			1.53425			0.767125	+	0.641498i	$0.221687\pi$
$830$		0			0
$831$		−	213.027i		−	0.256350i
$832$		0			0
$833$			485.536i			0.582877i
$834$		0			0
$835$	−335.310	+	1101.43i	−0.401569	+	1.31908i
$836$		0			0
$837$			453.661i			0.542009i
$838$		0			0
$839$	−296.616			−0.353536			−0.176768	−	0.984253i	$-0.556564\pi$
$839$	−296.616			−0.353536			−0.176768	+	0.984253i	$0.556564\pi$
$840$		0			0
$841$	−331.276			−0.393908
$842$		0			0
$843$			1288.49i			1.52846i
$844$		0			0
$845$	−706.452	−	215.066i	−0.836038	−	0.254516i
$846$		0			0
$847$	87.5276			0.103338
$848$		0			0
$849$			1133.40i			1.33498i
$850$		0			0
$851$	610.817			0.717763
$852$		0			0
$853$		−	522.324i		−	0.612337i	−0.951977	−	0.306169i	$-0.900953\pi$
$853$		−	522.324i		−	0.612337i	0.951977	−	0.306169i	$-0.0990471\pi$
$854$		0			0
$855$	768.493	+	233.954i	0.898822	+	0.273630i
$856$		0			0
$857$	1.49590			0.00174550			0.000872752	−	1.00000i	$-0.499722\pi$
$857$	1.49590			0.00174550			0.000872752		1.00000i	$0.499722\pi$
$858$		0			0
$859$	99.3975			0.115713			0.0578565	−	0.998325i	$-0.481573\pi$
$859$	99.3975			0.115713			0.0578565	+	0.998325i	$0.481573\pi$
$860$		0			0
$861$			72.4178i			0.0841089i
$862$		0			0
$863$			745.165i			0.863458i	0.902003	+	0.431729i	$0.142096\pi$
$863$			745.165i			0.863458i	−0.902003	+	0.431729i	$0.857904\pi$
$864$		0			0
$865$	−16.0354	+	52.6733i	−0.0185380	+	0.0608940i
$866$		0			0
$867$	−741.566			−0.855325
$868$		0			0
$869$	−597.226			−0.687257
$870$		0			0
$871$	−87.6859	+	296.581i	−0.100673	+	0.340506i
$872$		0			0
$873$	801.464			0.918058
$874$		0			0
$875$	67.6243	+	82.9029i	0.0772849	+	0.0947461i
$876$		0			0
$877$			711.369i			0.811139i	0.914064	+	0.405569i	$0.132927\pi$
$877$			711.369i			0.811139i	−0.914064	+	0.405569i	$0.867073\pi$
$878$		0			0
$879$			571.765i			0.650472i
$880$		0			0
$881$	846.618			0.960974			0.480487	−	0.877002i	$-0.340460\pi$
$881$	846.618			0.960974			0.480487	+	0.877002i	$0.340460\pi$
$882$		0			0
$883$	175.673			0.198950			0.0994751	−	0.995040i	$-0.468284\pi$
$883$	175.673			0.198950			0.0994751	+	0.995040i	$0.468284\pi$
$884$		0			0
$885$	544.917	+	165.890i	0.615725	+	0.187446i
$886$		0			0
$887$		−	947.905i		−	1.06866i	−0.845275	−	0.534332i	$-0.820563\pi$
$887$		−	947.905i		−	1.06866i	0.845275	−	0.534332i	$-0.179437\pi$
$888$		0			0
$889$		−	26.7736i		−	0.0301166i
$890$		0			0
$891$		−	419.328i		−	0.470626i
$892$		0			0
$893$		−	618.925i		−	0.693085i
$894$		0			0
$895$	62.8238	−	206.364i	0.0701942	−	0.230575i
$896$		0			0
$897$			178.392i			0.198876i
$898$		0			0
$899$		−	1076.56i		−	1.19751i
$900$		0			0
$901$			416.880i			0.462686i
$902$		0			0
$903$	−102.487			−0.113496
$904$		0			0
$905$	−141.315	−	43.0207i	−0.156149	−	0.0475367i
$906$		0			0
$907$			1490.99i			1.64387i	0.569584	+	0.821933i	$0.307105\pi$
$907$			1490.99i			1.64387i	−0.569584	+	0.821933i	$0.692895\pi$
$908$		0			0
$909$			878.858i			0.966840i
$910$		0			0
$911$	−944.741			−1.03704			−0.518519	−	0.855066i	$-0.673516\pi$
$911$	−944.741			−1.03704			−0.518519	+	0.855066i	$0.673516\pi$
$912$		0			0
$913$	−77.8805			−0.0853017
$914$		0			0
$915$	566.143	−	1859.67i	0.618736	−	2.03243i
$916$		0			0
$917$	155.535			0.169613
$918$		0			0
$919$			635.953i			0.692006i	0.938233	+	0.346003i	$0.112461\pi$
$919$			635.953i			0.692006i	−0.938233	+	0.346003i	$0.887539\pi$
$920$		0			0
$921$			472.022i			0.512511i
$922$		0			0
$923$	−547.271			−0.592927
$924$		0			0
$925$	869.356	−	1295.51i	0.939845	−	1.40055i
$926$		0			0
$927$			961.258i			1.03696i
$928$		0			0
$929$		−	197.204i		−	0.212276i	−0.994351	−	0.106138i	$-0.966152\pi$
$929$		−	197.204i		−	0.212276i	0.994351	−	0.106138i	$-0.0338485\pi$
$930$		0			0
$931$	−1176.67			−1.26387
$932$		0			0
$933$			759.087i			0.813598i
$934$		0			0
$935$	208.263	+	63.4018i	0.222741	+	0.0678094i
$936$		0			0
$937$	−135.186			−0.144275			−0.0721376	−	0.997395i	$-0.522982\pi$
$937$	−135.186			−0.144275			−0.0721376	+	0.997395i	$0.522982\pi$
$938$		0			0
$939$	1417.05			1.50911
$940$		0			0
$941$		−	682.975i		−	0.725796i	−0.931829	−	0.362898i	$-0.881787\pi$
$941$		−	682.975i		−	0.725796i	0.931829	−	0.362898i	$-0.118213\pi$
$942$		0			0
$943$	209.739			0.222417
$944$		0			0
$945$	38.9492	+	11.8574i	0.0412160	+	0.0125475i
$946$		0			0
$947$		−	852.671i		−	0.900392i	−0.892930	−	0.450196i	$-0.851354\pi$
$947$		−	852.671i		−	0.900392i	0.892930	−	0.450196i	$-0.148646\pi$
$948$		0			0
$949$			418.528i			0.441020i
$950$		0			0
$951$		−	1173.26i		−	1.23371i
$952$		0			0
$953$		−	613.623i		−	0.643885i	−0.946759	−	0.321943i	$-0.895664\pi$
$953$		−	613.623i		−	0.643885i	0.946759	−	0.321943i	$-0.104336\pi$
$954$		0			0
$955$	−216.886	+	712.430i	−0.227106	+	0.746000i
$956$		0			0
$957$			385.849i			0.403186i
$958$		0			0
$959$	−166.424			−0.173539
$960$		0			0
$961$	−1312.77			−1.36604
$962$		0			0
$963$			1181.94i			1.22735i
$964$		0			0
$965$	−70.8301	+	232.663i	−0.0733991	+	0.241102i
$966$		0			0
$967$			381.193i			0.394201i	0.980383	+	0.197101i	$0.0631526\pi$
$967$			381.193i			0.394201i	−0.980383	+	0.197101i	$0.936847\pi$
$968$		0			0
$969$			968.269i			0.999245i
$970$		0			0
$971$	−7.35190			−0.00757147			−0.00378574	−	0.999993i	$-0.501205\pi$
$971$	−7.35190			−0.00757147			−0.00378574	+	0.999993i	$0.501205\pi$
$972$		0			0
$973$		−	166.059i		−	0.170667i
$974$		0			0
$975$	378.359	+	253.900i	0.388060	+	0.260410i
$976$		0			0
$977$			216.621i			0.221721i	0.993836	+	0.110861i	$0.0353607\pi$
$977$			216.621i			0.221721i	−0.993836	+	0.110861i	$0.964639\pi$
$978$		0			0
$979$			465.252i			0.475232i
$980$		0			0
$981$			776.446i			0.791484i
$982$		0			0
$983$	1734.51			1.76451			0.882256	−	0.470770i	$-0.156024\pi$
$983$	1734.51			1.76451			0.882256	+	0.470770i	$0.156024\pi$
$984$		0			0
$985$	684.335	+	208.333i	0.694756	+	0.211506i
$986$		0			0
$987$			85.7995i			0.0869296i
$988$		0			0
$989$			296.827i			0.300128i
$990$		0			0
$991$			1044.62i			1.05411i	0.849831	+	0.527055i	$0.176704\pi$
$991$			1044.62i			1.05411i	−0.849831	+	0.527055i	$0.823296\pi$
$992$		0			0
$993$		−	1930.30i		−	1.94391i
$994$		0			0
$995$	66.2649	+	20.1731i	0.0665979	+	0.0202745i
$996$		0			0
$997$			1105.59i			1.10892i	0.832211	+	0.554460i	$0.187075\pi$
$997$			1105.59i			1.10892i	−0.832211	+	0.554460i	$0.812925\pi$
$998$		0			0
$999$		−	593.731i		−	0.594325i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1340.3.g.a.669.15	✓	68
5.4	even	2	inner	1340.3.g.a.669.53	yes	68
67.66	odd	2	inner	1340.3.g.a.669.54	yes	68
335.334	odd	2	inner	1340.3.g.a.669.16	yes	68

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1340.3.g.a.669.15	✓	68	1.1	even	1	trivial
1340.3.g.a.669.16	yes	68	335.334	odd	2	inner
1340.3.g.a.669.53	yes	68	5.4	even	2	inner
1340.3.g.a.669.54	yes	68	67.66	odd	2	inner

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:	\( N \)	\(=\)	\( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1340.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.94848 q^{3} +(4.78326 + 1.45617i) q^{5} +0.855885 q^{7} +6.59049 q^{9} +O(q^{10})\)	\(q-3.94848 q^{3} +(4.78326 + 1.45617i) q^{5} +0.855885 q^{7} +6.59049 q^{9} +4.32833i q^{11} -4.61600 q^{13} +(-18.8866 - 5.74968i) q^{15} -10.0593i q^{17} +24.3780 q^{19} -3.37944 q^{21} +9.78769i q^{23} +(20.7591 + 13.9305i) q^{25} +9.51391 q^{27} -22.5771 q^{29} +47.6840i q^{31} -17.0903i q^{33} +(4.09392 + 1.24632i) q^{35} -62.4066i q^{37} +18.2262 q^{39} -21.4289i q^{41} +30.3265 q^{43} +(31.5240 + 9.59690i) q^{45} -25.3886i q^{47} -48.2675 q^{49} +39.7189i q^{51} -41.4423 q^{53} +(-6.30281 + 20.7035i) q^{55} -96.2562 q^{57} -28.8520 q^{59} +98.4653i q^{61} +5.64070 q^{63} +(-22.0795 - 6.72170i) q^{65} +(18.9961 - 64.2507i) q^{67} -38.6465i q^{69} +118.560 q^{71} -90.6692i q^{73} +(-81.9669 - 55.0044i) q^{75} +3.70455i q^{77} +137.981i q^{79} -96.8799 q^{81} +17.9932i q^{83} +(14.6481 - 48.1162i) q^{85} +89.1451 q^{87} +107.490 q^{89} -3.95076 q^{91} -188.279i q^{93} +(116.606 + 35.4987i) q^{95} +121.609 q^{97} +28.5258i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 208 q^{9}+O(q^{10}) \) 68 * q + 208 * q^9	\( 68 q + 208 q^{9} - 18 q^{15} + 48 q^{19} - 48 q^{21} + 34 q^{25} + 20 q^{29} + 22 q^{35} - 24 q^{39} + 656 q^{49} + 44 q^{55} + 124 q^{59} - 106 q^{65} - 28 q^{71} + 660 q^{81} + 12 q^{89} - 684 q^{91}+O(q^{100}) \) 68 * q + 208 * q^9 - 18 * q^15 + 48 * q^19 - 48 * q^21 + 34 * q^25 + 20 * q^29 + 22 * q^35 - 24 * q^39 + 656 * q^49 + 44 * q^55 + 124 * q^59 - 106 * q^65 - 28 * q^71 + 660 * q^81 + 12 * q^89 - 684 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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