LMFDB - Embedded newform 495.3.b.c.406.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(406,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.406"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 36x^{14} + 500x^{12} + 3364x^{10} + 11310x^{8} + 17932x^{6} + 12708x^{4} + 3244x^{2} + 121 $ x^16 + 36x^14 + 500x^12 + 3364x^10 + 11310x^8 + 17932x^6 + 12708x^4 + 3244*x^2 + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 165)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			406.4
Root			$-2.33103i$ of defining polynomial
Character	$\chi$	$=$	495.406
Dual form			495.3.b.c.406.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.33103i q^{2} -1.43371 q^{4} -2.23607 q^{5} +6.32430i q^{7} -5.98210i q^{8} +5.21235i q^{10} +(10.3943 + 3.59991i) q^{11} +17.3965i q^{13} +14.7421 q^{14} -19.6793 q^{16} +12.2603i q^{17} +10.6069i q^{19} +3.20588 q^{20} +(8.39151 - 24.2294i) q^{22} +9.16729 q^{23} +5.00000 q^{25} +40.5519 q^{26} -9.06722i q^{28} -0.592304i q^{29} +17.1631 q^{31} +21.9447i q^{32} +28.5791 q^{34} -14.1416i q^{35} +55.3012 q^{37} +24.7249 q^{38} +13.3764i q^{40} -13.5046i q^{41} +36.0017i q^{43} +(-14.9024 - 5.16124i) q^{44} -21.3693i q^{46} +19.2531 q^{47} +9.00325 q^{49} -11.6552i q^{50} -24.9416i q^{52} -70.4442 q^{53} +(-23.2423 - 8.04965i) q^{55} +37.8326 q^{56} -1.38068 q^{58} -14.9479 q^{59} -36.1401i q^{61} -40.0077i q^{62} -27.5634 q^{64} -38.8998i q^{65} +127.842 q^{67} -17.5777i q^{68} -32.9644 q^{70} +27.8440 q^{71} +61.3643i q^{73} -128.909i q^{74} -15.2072i q^{76} +(-22.7669 + 65.7364i) q^{77} -141.550i q^{79} +44.0043 q^{80} -31.4798 q^{82} +137.804i q^{83} -27.4148i q^{85} +83.9212 q^{86} +(21.5350 - 62.1795i) q^{88} -160.353 q^{89} -110.021 q^{91} -13.1433 q^{92} -44.8797i q^{94} -23.7177i q^{95} -57.7240 q^{97} -20.9869i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4} + 28 q^{11} - 16 q^{16} - 40 q^{20} - 20 q^{22} - 56 q^{23} + 80 q^{25} + 88 q^{26} - 96 q^{31} - 200 q^{34} + 184 q^{37} - 296 q^{38} - 300 q^{44} + 200 q^{47} - 496 q^{49} + 80 q^{53} + 20 q^{55}+ \cdots - 144 q^{97}+O(q^{100}) $ 16 * q - 8 * q^4 + 28 * q^11 - 16 * q^16 - 40 * q^20 - 20 * q^22 - 56 * q^23 + 80 * q^25 + 88 * q^26 - 96 * q^31 - 200 * q^34 + 184 * q^37 - 296 * q^38 - 300 * q^44 + 200 * q^47 - 496 * q^49 + 80 * q^53 + 20 * q^55 + 32 * q^56 - 16 * q^58 + 136 * q^59 + 456 * q^64 + 256 * q^67 - 216 * q^71 + 248 * q^77 + 160 * q^80 + 232 * q^82 - 192 * q^86 - 116 * q^88 + 336 * q^89 + 256 * q^91 + 440 * q^92 - 144 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		−	2.33103i		−	1.16552i	−0.812646	−	0.582758i	$-0.801974\pi$
$2$		−	2.33103i		−	1.16552i	0.812646	−	0.582758i	$-0.198026\pi$
$3$		0			0
$3$		0			0
$4$	−1.43371			−0.358428
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$			6.32430i			0.903471i	0.892152	+	0.451736i	$0.149195\pi$
$7$			6.32430i			0.903471i	−0.892152	+	0.451736i	$0.850805\pi$
$8$		−	5.98210i		−	0.747763i
$9$		0			0
$10$			5.21235i			0.521235i
$11$	10.3943	+	3.59991i	0.944933	+	0.327265i
$11$	10.3943	+	3.59991i	0.944933	+	0.327265i
$12$		0			0
$13$			17.3965i			1.33820i	0.743175	+	0.669098i	$0.233319\pi$
$13$			17.3965i			1.33820i	−0.743175	+	0.669098i	$0.766681\pi$
$14$	14.7421			1.05301
$15$		0			0
$16$	−19.6793			−1.22996
$17$			12.2603i			0.721192i	0.932722	+	0.360596i	$0.117427\pi$
$17$			12.2603i			0.721192i	−0.932722	+	0.360596i	$0.882573\pi$
$18$		0			0
$19$			10.6069i			0.558256i	0.960254	+	0.279128i	$0.0900454\pi$
$19$			10.6069i			0.558256i	−0.960254	+	0.279128i	$0.909955\pi$
$20$	3.20588			0.160294
$21$		0			0
$22$	8.39151	−	24.2294i	0.381432	−	1.10133i
$23$	9.16729			0.398578			0.199289	−	0.979941i	$-0.436137\pi$
$23$	9.16729			0.398578			0.199289	+	0.979941i	$0.436137\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$	40.5519			1.55969
$27$		0			0
$28$		−	9.06722i		−	0.323829i
$29$		−	0.592304i		−	0.0204243i	−0.999948	−	0.0102121i	$-0.996749\pi$
$29$		−	0.592304i		−	0.0204243i	0.999948	−	0.0102121i	$-0.00325068\pi$
$30$		0			0
$31$	17.1631			0.553648			0.276824	−	0.960921i	$-0.410718\pi$
$31$	17.1631			0.553648			0.276824	+	0.960921i	$0.410718\pi$
$32$			21.9447i			0.685773i
$33$		0			0
$34$	28.5791			0.840561
$35$		−	14.1416i		−	0.404045i
$36$		0			0
$37$	55.3012			1.49463			0.747314	−	0.664472i	$-0.231343\pi$
$37$	55.3012			1.49463			0.747314	+	0.664472i	$0.231343\pi$
$38$	24.7249			0.650656
$39$		0			0
$40$			13.3764i			0.334410i
$41$		−	13.5046i		−	0.329382i	−0.986345	−	0.164691i	$-0.947337\pi$
$41$		−	13.5046i		−	0.329382i	0.986345	−	0.164691i	$-0.0526626\pi$
$42$		0			0
$43$			36.0017i			0.837249i	0.908159	+	0.418625i	$0.137488\pi$
$43$			36.0017i			0.837249i	−0.908159	+	0.418625i	$0.862512\pi$
$44$	−14.9024	−	5.16124i	−0.338690	−	0.117301i
$45$		0			0
$46$		−	21.3693i		−	0.464549i
$47$	19.2531			0.409641			0.204821	−	0.978800i	$-0.434339\pi$
$47$	19.2531			0.409641			0.204821	+	0.978800i	$0.434339\pi$
$48$		0			0
$49$	9.00325			0.183740
$50$		−	11.6552i		−	0.233103i
$51$		0			0
$52$		−	24.9416i		−	0.479646i
$53$	−70.4442			−1.32914			−0.664568	−	0.747228i	$-0.731384\pi$
$53$	−70.4442			−1.32914			−0.664568	+	0.747228i	$0.731384\pi$
$54$		0			0
$55$	−23.2423	−	8.04965i	−0.422587	−	0.146357i
$56$	37.8326			0.675582
$57$		0			0
$58$	−1.38068			−0.0238048
$59$	−14.9479			−0.253354			−0.126677	−	0.991944i	$-0.540431\pi$
$59$	−14.9479			−0.253354			−0.126677	+	0.991944i	$0.540431\pi$
$60$		0			0
$61$		−	36.1401i		−	0.592461i	−0.955116	−	0.296231i	$-0.904270\pi$
$61$		−	36.1401i		−	0.592461i	0.955116	−	0.296231i	$-0.0957297\pi$
$62$		−	40.0077i		−	0.645285i
$63$		0			0
$64$	−27.5634			−0.430678
$65$		−	38.8998i		−	0.598459i
$66$		0			0
$67$	127.842			1.90809			0.954046	−	0.299660i	$-0.0968732\pi$
$67$	127.842			1.90809			0.954046	+	0.299660i	$0.0968732\pi$
$68$		−	17.5777i		−	0.258495i
$69$		0			0
$70$	−32.9644			−0.470921
$71$	27.8440			0.392169			0.196084	−	0.980587i	$-0.437177\pi$
$71$	27.8440			0.392169			0.196084	+	0.980587i	$0.437177\pi$
$72$		0			0
$73$			61.3643i			0.840607i	0.907383	+	0.420304i	$0.138076\pi$
$73$			61.3643i			0.840607i	−0.907383	+	0.420304i	$0.861924\pi$
$74$		−	128.909i		−	1.74201i
$75$		0			0
$76$		−	15.2072i		−	0.200095i
$77$	−22.7669	+	65.7364i	−0.295674	+	0.853719i
$78$		0			0
$79$		−	141.550i		−	1.79177i	−0.444289	−	0.895884i	$-0.646544\pi$
$79$		−	141.550i		−	1.79177i	0.444289	−	0.895884i	$-0.353456\pi$
$80$	44.0043			0.550054
$81$		0			0
$82$	−31.4798			−0.383900
$83$			137.804i			1.66028i	0.557552	+	0.830142i	$0.311741\pi$
$83$			137.804i			1.66028i	−0.557552	+	0.830142i	$0.688259\pi$
$84$		0			0
$85$		−	27.4148i		−	0.322527i
$86$	83.9212			0.975828
$87$		0			0
$88$	21.5350	−	62.1795i	0.244716	−	0.706585i
$89$	−160.353			−1.80172			−0.900858	−	0.434113i	$-0.857062\pi$
$89$	−160.353			−1.80172			−0.900858	+	0.434113i	$0.857062\pi$
$90$		0			0
$91$	−110.021			−1.20902
$92$	−13.1433			−0.142861
$93$		0			0
$94$		−	44.8797i		−	0.477443i
$95$		−	23.7177i		−	0.249660i
$96$		0			0
$97$	−57.7240			−0.595092			−0.297546	−	0.954707i	$-0.596168\pi$
$97$	−57.7240			−0.595092			−0.297546	+	0.954707i	$0.596168\pi$
$98$		−	20.9869i		−	0.214152i
$99$		0			0
$100$	−7.16856			−0.0716856
$101$			147.055i			1.45599i	0.685585	+	0.727993i	$0.259547\pi$
$101$			147.055i			1.45599i	−0.685585	+	0.727993i	$0.740453\pi$
$102$		0			0
$103$	−98.7792			−0.959022			−0.479511	−	0.877536i	$-0.659186\pi$
$103$	−98.7792			−0.959022			−0.479511	+	0.877536i	$0.659186\pi$
$104$	104.068			1.00065
$105$		0			0
$106$			164.208i			1.54913i
$107$		−	107.953i		−	1.00891i	−0.863439	−	0.504453i	$-0.831694\pi$
$107$		−	107.953i		−	1.00891i	0.863439	−	0.504453i	$-0.168306\pi$
$108$		0			0
$109$		−	33.0995i		−	0.303666i	−0.988406	−	0.151833i	$-0.951482\pi$
$109$		−	33.0995i		−	0.303666i	0.988406	−	0.151833i	$-0.0485175\pi$
$110$	−18.7640	+	54.1785i	−0.170582	+	0.492532i
$111$		0			0
$112$		−	124.458i		−	1.11123i
$113$	163.297			1.44510			0.722552	−	0.691316i	$-0.242969\pi$
$113$	163.297			1.44510			0.722552	+	0.691316i	$0.242969\pi$
$114$		0			0
$115$	−20.4987			−0.178249
$116$			0.849193i			0.00732063i
$117$		0			0
$118$			34.8441i			0.295289i
$119$	−77.5376			−0.651576
$120$		0			0
$121$	95.0813	+	74.8369i	0.785795	+	0.618486i
$122$	−84.2438			−0.690523
$123$		0			0
$124$	−24.6069			−0.198443
$125$	−11.1803			−0.0894427
$126$		0			0
$127$			204.213i			1.60798i	0.594645	+	0.803988i	$0.297293\pi$
$127$			204.213i			1.60798i	−0.594645	+	0.803988i	$0.702707\pi$
$128$			152.030i			1.18774i
$129$		0			0
$130$	−90.6768			−0.697514
$131$		−	243.373i		−	1.85781i	−0.370316	−	0.928906i	$-0.620751\pi$
$131$		−	243.373i		−	1.85781i	0.370316	−	0.928906i	$-0.379249\pi$
$132$		0			0
$133$	−67.0810			−0.504368
$134$		−	298.004i		−	2.22391i
$135$		0			0
$136$	73.3421			0.539280
$137$	−68.3154			−0.498652			−0.249326	−	0.968420i	$-0.580209\pi$
$137$	−68.3154			−0.498652			−0.249326	+	0.968420i	$0.580209\pi$
$138$		0			0
$139$			21.5172i			0.154800i	0.997000	+	0.0774001i	$0.0246619\pi$
$139$			21.5172i			0.154800i	−0.997000	+	0.0774001i	$0.975338\pi$
$140$			20.2749i			0.144821i
$141$		0			0
$142$		−	64.9052i		−	0.457079i
$143$	−62.6260	+	180.824i	−0.437944	+	1.26450i
$144$		0			0
$145$			1.32443i			0.00913401i
$146$	143.042			0.979742
$147$		0			0
$148$	−79.2860			−0.535716
$149$			45.9449i			0.308355i	0.988043	+	0.154178i	$0.0492728\pi$
$149$			45.9449i			0.308355i	−0.988043	+	0.154178i	$0.950727\pi$
$150$		0			0
$151$			123.582i			0.818425i	0.912439	+	0.409212i	$0.134197\pi$
$151$			123.582i			0.818425i	−0.912439	+	0.409212i	$0.865803\pi$
$152$	63.4513			0.417443
$153$		0			0
$154$	153.234	+	53.0704i	0.995024	+	0.344613i
$155$	−38.3778			−0.247599
$156$		0			0
$157$	65.4047			0.416590			0.208295	−	0.978066i	$-0.433209\pi$
$157$	65.4047			0.416590			0.208295	+	0.978066i	$0.433209\pi$
$158$	−329.957			−2.08833
$159$		0			0
$160$		−	49.0699i		−	0.306687i
$161$			57.9767i			0.360104i
$162$		0			0
$163$	−318.348			−1.95305			−0.976527	−	0.215395i	$-0.930896\pi$
$163$	−318.348			−1.95305			−0.976527	+	0.215395i	$0.930896\pi$
$164$			19.3618i			0.118060i
$165$		0			0
$166$	321.225			1.93509
$167$		−	17.2478i		−	0.103280i	−0.998666	−	0.0516400i	$-0.983555\pi$
$167$		−	17.2478i		−	0.103280i	0.998666	−	0.0516400i	$-0.0164448\pi$
$168$		0			0
$169$	−133.639			−0.790766
$170$	−63.9048			−0.375910
$171$		0			0
$172$		−	51.6161i		−	0.300094i
$173$			322.103i			1.86187i	0.365190	+	0.930933i	$0.381004\pi$
$173$			322.103i			1.86187i	−0.365190	+	0.930933i	$0.618996\pi$
$174$		0			0
$175$			31.6215i			0.180694i
$176$	−204.552	−	70.8438i	−1.16223	−	0.402522i
$177$		0			0
$178$			373.788i			2.09993i
$179$	−45.5484			−0.254460			−0.127230	−	0.991873i	$-0.540609\pi$
$179$	−45.5484			−0.254460			−0.127230	+	0.991873i	$0.540609\pi$
$180$		0			0
$181$	134.271			0.741830			0.370915	−	0.928667i	$-0.379044\pi$
$181$	134.271			0.741830			0.370915	+	0.928667i	$0.379044\pi$
$182$			256.462i			1.40913i
$183$		0			0
$184$		−	54.8397i		−	0.298042i
$185$	−123.657			−0.668418
$186$		0			0
$187$	−44.1359	+	127.436i	−0.236021	+	0.681478i
$188$	−27.6034			−0.146827
$189$		0			0
$190$	−55.2867			−0.290982
$191$	68.0366			0.356212			0.178106	−	0.984011i	$-0.443003\pi$
$191$	68.0366			0.356212			0.178106	+	0.984011i	$0.443003\pi$
$192$		0			0
$193$			127.465i			0.660443i	0.943904	+	0.330221i	$0.107123\pi$
$193$			127.465i			0.660443i	−0.943904	+	0.330221i	$0.892877\pi$
$194$			134.556i			0.693590i
$195$		0			0
$196$	−12.9081			−0.0658575
$197$		−	148.516i		−	0.753888i	−0.926236	−	0.376944i	$-0.876975\pi$
$197$		−	148.516i		−	0.753888i	0.926236	−	0.376944i	$-0.123025\pi$
$198$		0			0
$199$	63.0543			0.316856			0.158428	−	0.987371i	$-0.449357\pi$
$199$	63.0543			0.316856			0.158428	+	0.987371i	$0.449357\pi$
$200$		−	29.9105i		−	0.149553i
$201$		0			0
$202$	342.789			1.69697
$203$	3.74591			0.0184527
$204$		0			0
$205$			30.1973i			0.147304i
$206$			230.258i			1.11776i
$207$		0			0
$208$		−	342.352i		−	1.64592i
$209$	−38.1838	+	110.250i	−0.182698	+	0.527514i
$210$		0			0
$211$		−	228.972i		−	1.08517i	−0.840000	−	0.542587i	$-0.817445\pi$
$211$		−	228.972i		−	1.08517i	0.840000	−	0.542587i	$-0.182555\pi$
$212$	100.997			0.476400
$213$		0			0
$214$	−251.642			−1.17590
$215$		−	80.5023i		−	0.374429i
$216$		0			0
$217$			108.544i			0.500205i
$218$	−77.1561			−0.353927
$219$		0			0
$220$	33.3227	+	11.5409i	0.151467	+	0.0524585i
$221$	−213.286			−0.965096
$222$		0			0
$223$	−101.764			−0.456339			−0.228169	−	0.973621i	$-0.573274\pi$
$223$	−101.764			−0.456339			−0.228169	+	0.973621i	$0.573274\pi$
$224$	−138.785			−0.619576
$225$		0			0
$226$		−	380.650i		−	1.68429i
$227$			351.517i			1.54853i	0.632861	+	0.774266i	$0.281881\pi$
$227$			351.517i			1.54853i	−0.632861	+	0.774266i	$0.718119\pi$
$228$		0			0
$229$	141.322			0.617125			0.308563	−	0.951204i	$-0.400152\pi$
$229$	141.322			0.617125			0.308563	+	0.951204i	$0.400152\pi$
$230$			47.7831i			0.207753i
$231$		0			0
$232$	−3.54322			−0.0152725
$233$		−	370.243i		−	1.58903i	−0.607246	−	0.794514i	$-0.707726\pi$
$233$		−	370.243i		−	1.58903i	0.607246	−	0.794514i	$-0.292274\pi$
$234$		0			0
$235$	−43.0513			−0.183197
$236$	21.4310			0.0908093
$237$		0			0
$238$			180.743i			0.759423i
$239$		−	68.3043i		−	0.285792i	−0.989738	−	0.142896i	$-0.954359\pi$
$239$		−	68.3043i		−	0.285792i	0.989738	−	0.142896i	$-0.0456414\pi$
$240$		0			0
$241$		−	200.283i		−	0.831052i	−0.909581	−	0.415526i	$-0.863598\pi$
$241$		−	200.283i		−	0.831052i	0.909581	−	0.415526i	$-0.136402\pi$
$242$	174.447	−	221.637i	0.720856	−	0.915857i
$243$		0			0
$244$			51.8145i			0.212355i
$245$	−20.1319			−0.0821709
$246$		0			0
$247$	−184.523			−0.747055
$248$		−	102.671i		−	0.413997i
$249$		0			0
$250$			26.0617i			0.104247i
$251$	−213.104			−0.849019			−0.424510	−	0.905423i	$-0.639553\pi$
$251$	−213.104			−0.849019			−0.424510	+	0.905423i	$0.639553\pi$
$252$		0			0
$253$	95.2872	+	33.0015i	0.376629	+	0.130441i
$254$	476.027			1.87412
$255$		0			0
$256$	244.133			0.953646
$257$	30.3594			0.118130			0.0590650	−	0.998254i	$-0.481188\pi$
$257$	30.3594			0.118130			0.0590650	+	0.998254i	$0.481188\pi$
$258$		0			0
$259$			349.741i			1.35035i
$260$			55.7712i			0.214504i
$261$		0			0
$262$	−567.311			−2.16531
$263$		−	359.742i		−	1.36784i	−0.729557	−	0.683920i	$-0.760274\pi$
$263$		−	359.742i		−	1.36784i	0.729557	−	0.683920i	$-0.239726\pi$
$264$		0			0
$265$	157.518			0.594408
$266$			156.368i			0.587849i
$267$		0			0
$268$	−183.289			−0.683914
$269$	320.990			1.19327			0.596637	−	0.802512i	$-0.296503\pi$
$269$	320.990			1.19327			0.596637	+	0.802512i	$0.296503\pi$
$270$		0			0
$271$		−	419.497i		−	1.54796i	−0.633211	−	0.773979i	$-0.718264\pi$
$271$		−	419.497i		−	1.54796i	0.633211	−	0.773979i	$-0.281736\pi$
$272$		−	241.274i		−	0.887036i
$273$		0			0
$274$			159.245i			0.581187i
$275$	51.9713	+	17.9996i	0.188987	+	0.0654530i
$276$		0			0
$277$		−	86.8213i		−	0.313434i	−0.987644	−	0.156717i	$-0.949909\pi$
$277$		−	86.8213i		−	0.313434i	0.987644	−	0.156717i	$-0.0500911\pi$
$278$	50.1574			0.180422
$279$		0			0
$280$	−84.5962			−0.302129
$281$		−	300.755i		−	1.07030i	−0.844756	−	0.535152i	$-0.820255\pi$
$281$		−	300.755i		−	1.07030i	0.844756	−	0.535152i	$-0.179745\pi$
$282$		0			0
$283$		−	319.781i		−	1.12997i	−0.825102	−	0.564984i	$-0.808882\pi$
$283$		−	319.781i		−	1.12997i	0.825102	−	0.564984i	$-0.191118\pi$
$284$	−39.9202			−0.140564
$285$		0			0
$286$	421.507	+	145.983i	1.47380	+	0.510431i
$287$	85.4074			0.297587
$288$		0			0
$289$	138.686			0.479882
$290$	3.08729			0.0106458
$291$		0			0
$292$		−	87.9788i		−	0.301297i
$293$		−	421.749i		−	1.43942i	−0.694276	−	0.719709i	$-0.744275\pi$
$293$		−	421.749i		−	1.43942i	0.694276	−	0.719709i	$-0.255725\pi$
$294$		0			0
$295$	33.4245			0.113304
$296$		−	330.817i		−	1.11763i
$297$		0			0
$298$	107.099			0.359393
$299$			159.479i			0.533375i
$300$		0			0
$301$	−227.686			−0.756431
$302$	288.074			0.953887
$303$		0			0
$304$		−	208.736i		−	0.686631i
$305$			80.8118i			0.264957i
$306$		0			0
$307$		−	65.1772i		−	0.212303i	−0.994350	−	0.106152i	$-0.966147\pi$
$307$		−	65.1772i		−	0.212303i	0.994350	−	0.106152i	$-0.0338529\pi$
$308$	32.6412	−	94.2470i	0.105978	−	0.305997i
$309$		0			0
$310$			89.4599i			0.288580i
$311$	296.311			0.952768			0.476384	−	0.879237i	$-0.341947\pi$
$311$	296.311			0.952768			0.476384	+	0.879237i	$0.341947\pi$
$312$		0			0
$313$	−430.760			−1.37623			−0.688115	−	0.725601i	$-0.741562\pi$
$313$	−430.760			−1.37623			−0.688115	+	0.725601i	$0.741562\pi$
$314$		−	152.460i		−	0.485543i
$315$		0			0
$316$			202.941i			0.642219i
$317$	248.606			0.784246			0.392123	−	0.919913i	$-0.371741\pi$
$317$	248.606			0.784246			0.392123	+	0.919913i	$0.371741\pi$
$318$		0			0
$319$	2.13224	−	6.15656i	0.00668414	−	0.0192996i
$320$	61.6337			0.192605
$321$		0			0
$322$	135.146			0.419707
$323$	−130.043			−0.402610
$324$		0			0
$325$			86.9827i			0.267639i
$326$			742.079i			2.27632i
$327$		0			0
$328$	−80.7862			−0.246299
$329$			121.763i			0.370099i
$330$		0			0
$331$	−379.595			−1.14681			−0.573406	−	0.819271i	$-0.694378\pi$
$331$	−379.595			−1.14681			−0.573406	+	0.819271i	$0.694378\pi$
$332$		−	197.571i		−	0.595092i
$333$		0			0
$334$	−40.2051			−0.120375
$335$	−285.864			−0.853325
$336$		0			0
$337$		−	528.010i		−	1.56680i	−0.621521	−	0.783398i	$-0.713485\pi$
$337$		−	528.010i		−	1.56680i	0.621521	−	0.783398i	$-0.286515\pi$
$338$			311.518i			0.921651i
$339$		0			0
$340$			39.3049i			0.115603i
$341$	178.398	+	61.7856i	0.523160	+	0.181189i
$342$		0			0
$343$			366.830i			1.06947i
$344$	215.366			0.626064
$345$		0			0
$346$	750.832			2.17004
$347$		−	35.7638i		−	0.103066i	−0.998671	−	0.0515328i	$-0.983589\pi$
$347$		−	35.7638i		−	0.103066i	0.998671	−	0.0515328i	$-0.0164107\pi$
$348$		0			0
$349$		−	453.183i		−	1.29852i	−0.760568	−	0.649259i	$-0.775079\pi$
$349$		−	453.183i		−	1.29852i	0.760568	−	0.649259i	$-0.224921\pi$
$350$	73.7107			0.210602
$351$		0			0
$352$	−78.9991	+	228.099i	−0.224429	+	0.648009i
$353$	76.8798			0.217790			0.108895	−	0.994053i	$-0.465269\pi$
$353$	76.8798			0.217790			0.108895	+	0.994053i	$0.465269\pi$
$354$		0			0
$355$	−62.2610			−0.175383
$356$	229.900			0.645786
$357$		0			0
$358$			106.175i			0.296577i
$359$			406.327i			1.13183i	0.824463	+	0.565915i	$0.191477\pi$
$359$			406.327i			1.13183i	−0.824463	+	0.565915i	$0.808523\pi$
$360$		0			0
$361$	248.494			0.688350
$362$		−	312.991i		−	0.864615i
$363$		0			0
$364$	157.738			0.433347
$365$		−	137.215i		−	0.375931i
$366$		0			0
$367$	−287.030			−0.782098			−0.391049	−	0.920370i	$-0.627888\pi$
$367$	−287.030			−0.782098			−0.391049	+	0.920370i	$0.627888\pi$
$368$	−180.406			−0.490234
$369$		0			0
$370$			288.249i			0.779051i
$371$		−	445.510i		−	1.20084i
$372$		0			0
$373$		−	359.705i		−	0.964356i	−0.876073	−	0.482178i	$-0.839846\pi$
$373$		−	359.705i		−	0.964356i	0.876073	−	0.482178i	$-0.160154\pi$
$374$	297.058	+	102.882i	0.794274	+	0.275086i
$375$		0			0
$376$		−	115.174i		−	0.306314i
$377$	10.3040			0.0273316
$378$		0			0
$379$	94.3980			0.249071			0.124536	−	0.992215i	$-0.460256\pi$
$379$	94.3980			0.249071			0.124536	+	0.992215i	$0.460256\pi$
$380$			34.0043i			0.0894850i
$381$		0			0
$382$		−	158.595i		−	0.415171i
$383$	92.3643			0.241160			0.120580	−	0.992704i	$-0.461525\pi$
$383$	92.3643			0.241160			0.120580	+	0.992704i	$0.461525\pi$
$384$		0			0
$385$	50.9084	−	146.991i	0.132230	−	0.381795i
$386$	297.126			0.769757
$387$		0			0
$388$	82.7595			0.213298
$389$	−126.237			−0.324517			−0.162259	−	0.986748i	$-0.551878\pi$
$389$	−126.237			−0.324517			−0.162259	+	0.986748i	$0.551878\pi$
$390$		0			0
$391$			112.393i			0.287451i
$392$		−	53.8583i		−	0.137394i
$393$		0			0
$394$	−346.195			−0.878668
$395$			316.515i			0.801303i
$396$		0			0
$397$	732.111			1.84411			0.922054	−	0.387060i	$-0.126509\pi$
$397$	732.111			1.84411			0.922054	+	0.387060i	$0.126509\pi$
$398$		−	146.982i		−	0.369300i
$399$		0			0
$400$	−98.3966			−0.245991
$401$	−288.306			−0.718966			−0.359483	−	0.933152i	$-0.617047\pi$
$401$	−288.306			−0.718966			−0.359483	+	0.933152i	$0.617047\pi$
$402$		0			0
$403$			298.578i			0.740889i
$404$		−	210.834i		−	0.521866i
$405$		0			0
$406$		−	8.73183i		−	0.0215070i
$407$	574.815	+	199.080i	1.41232	+	0.489139i
$408$		0			0
$409$		−	536.385i		−	1.31146i	−0.754997	−	0.655728i	$-0.772362\pi$
$409$		−	536.385i		−	1.31146i	0.754997	−	0.655728i	$-0.227638\pi$
$410$	70.3909			0.171685
$411$		0			0
$412$	141.621			0.343740
$413$		−	94.5350i		−	0.228898i
$414$		0			0
$415$		−	308.138i		−	0.742501i
$416$	−381.762			−0.917697
$417$		0			0
$418$	256.997	+	89.0076i	0.614826	+	0.212937i
$419$	−772.235			−1.84304			−0.921521	−	0.388328i	$-0.873053\pi$
$419$	−772.235			−1.84304			−0.921521	+	0.388328i	$0.873053\pi$
$420$		0			0
$421$	530.329			1.25969			0.629844	−	0.776721i	$-0.283119\pi$
$421$	530.329			1.25969			0.629844	+	0.776721i	$0.283119\pi$
$422$	−533.740			−1.26479
$423$		0			0
$424$			421.405i			0.993879i
$425$			61.3013i			0.144238i
$426$		0			0
$427$	228.561			0.535272
$428$			154.773i			0.361620i
$429$		0			0
$430$	−187.653			−0.436403
$431$		−	98.0885i		−	0.227584i	−0.993505	−	0.113792i	$-0.963700\pi$
$431$		−	98.0885i		−	0.227584i	0.993505	−	0.113792i	$-0.0362997\pi$
$432$		0			0
$433$	−467.819			−1.08041			−0.540206	−	0.841533i	$-0.681654\pi$
$433$	−467.819			−1.08041			−0.540206	+	0.841533i	$0.681654\pi$
$434$	253.021			0.582997
$435$		0			0
$436$			47.4552i			0.108842i
$437$			97.2362i			0.222509i
$438$		0			0
$439$			732.331i			1.66818i	0.551628	+	0.834090i	$0.314007\pi$
$439$			732.331i			1.66818i	−0.551628	+	0.834090i	$0.685993\pi$
$440$	−48.1538	+	139.038i	−0.109440	+	0.315995i
$441$		0			0
$442$			497.177i			1.12483i
$443$	−522.885			−1.18033			−0.590164	−	0.807284i	$-0.700937\pi$
$443$	−522.885			−1.18033			−0.590164	+	0.807284i	$0.700937\pi$
$444$		0			0
$445$	358.560			0.805752
$446$			237.214i			0.531870i
$447$		0			0
$448$		−	174.319i		−	0.389105i
$449$	−119.788			−0.266788			−0.133394	−	0.991063i	$-0.542588\pi$
$449$	−119.788			−0.266788			−0.133394	+	0.991063i	$0.542588\pi$
$450$		0			0
$451$	48.6156	−	140.371i	0.107795	−	0.311243i
$452$	−234.121			−0.517966
$453$		0			0
$454$	819.397			1.80484
$455$	246.014			0.540691
$456$		0			0
$457$		−	4.81549i		−	0.0105372i	−0.999986	−	0.00526859i	$-0.998323\pi$
$457$		−	4.81549i		−	0.0105372i	0.999986	−	0.00526859i	$-0.00167705\pi$
$458$		−	329.425i		−	0.719269i
$459$		0			0
$460$	29.3892			0.0638896
$461$		−	570.680i		−	1.23792i	−0.785424	−	0.618958i	$-0.787555\pi$
$461$		−	570.680i		−	1.23792i	0.785424	−	0.618958i	$-0.212445\pi$
$462$		0			0
$463$	131.935			0.284956			0.142478	−	0.989798i	$-0.454493\pi$
$463$	131.935			0.284956			0.142478	+	0.989798i	$0.454493\pi$
$464$			11.6561i			0.0251210i
$465$		0			0
$466$	−863.050			−1.85204
$467$	89.8728			0.192447			0.0962235	−	0.995360i	$-0.469324\pi$
$467$	89.8728			0.192447			0.0962235	+	0.995360i	$0.469324\pi$
$468$		0			0
$469$			808.512i			1.72391i
$470$			100.354i			0.213519i
$471$		0			0
$472$			89.4199i			0.189449i
$473$	−129.603	+	374.211i	−0.274002	+	0.791144i
$474$		0			0
$475$			53.0343i			0.111651i
$476$	111.167			0.233543
$477$		0			0
$478$	−159.219			−0.333095
$479$			717.047i			1.49697i	0.663153	+	0.748484i	$0.269218\pi$
$479$			717.047i			1.49697i	−0.663153	+	0.748484i	$0.730782\pi$
$480$		0			0
$481$			962.049i			2.00010i
$482$	−466.867			−0.968604
$483$		0			0
$484$	−136.319	−	107.294i	−0.281651	−	0.221683i
$485$	129.075			0.266133
$486$		0			0
$487$	916.798			1.88254			0.941271	−	0.337653i	$-0.109633\pi$
$487$	916.798			1.88254			0.941271	+	0.337653i	$0.109633\pi$
$488$	−216.194			−0.443020
$489$		0			0
$490$			46.9280i			0.0957715i
$491$		−	302.142i		−	0.615361i	−0.951490	−	0.307680i	$-0.900447\pi$
$491$		−	302.142i		−	0.615361i	0.951490	−	0.307680i	$-0.0995527\pi$
$492$		0			0
$493$	7.26180			0.0147298
$494$			430.128i			0.870705i
$495$		0			0
$496$	−337.758			−0.680963
$497$			176.094i			0.354313i
$498$		0			0
$499$	−102.798			−0.206008			−0.103004	−	0.994681i	$-0.532846\pi$
$499$	−102.798			−0.206008			−0.103004	+	0.994681i	$0.532846\pi$
$500$	16.0294			0.0320588
$501$		0			0
$502$			496.752i			0.989545i
$503$			567.728i			1.12868i	0.825541	+	0.564342i	$0.190870\pi$
$503$			567.728i			1.12868i	−0.825541	+	0.564342i	$0.809130\pi$
$504$		0			0
$505$		−	328.824i		−	0.651137i
$506$	76.9275	−	222.118i	0.152031	−	0.438968i
$507$		0			0
$508$		−	292.783i		−	0.576344i
$509$	423.197			0.831428			0.415714	−	0.909495i	$-0.363532\pi$
$509$	423.197			0.831428			0.415714	+	0.909495i	$0.363532\pi$
$510$		0			0
$511$	−388.086			−0.759465
$512$			39.0375i			0.0762451i
$513$		0			0
$514$		−	70.7687i		−	0.137682i
$515$	220.877			0.428887
$516$		0			0
$517$	200.122	+	69.3096i	0.387083	+	0.134061i
$518$	815.258			1.57386
$519$		0			0
$520$	−232.703			−0.447505
$521$	658.340			1.26361			0.631805	−	0.775128i	$-0.282315\pi$
$521$	658.340			1.26361			0.631805	+	0.775128i	$0.282315\pi$
$522$		0			0
$523$		−	568.723i		−	1.08743i	−0.839272	−	0.543713i	$-0.817018\pi$
$523$		−	568.723i		−	1.08743i	0.839272	−	0.543713i	$-0.182982\pi$
$524$			348.927i			0.665892i
$525$		0			0
$526$	−838.570			−1.59424
$527$			210.424i			0.399286i
$528$		0			0
$529$	−444.961			−0.841136
$530$		−	367.180i		−	0.692792i
$531$		0			0
$532$	96.1748			0.180780
$533$	234.934			0.440777
$534$		0			0
$535$			241.390i			0.451197i
$536$		−	764.765i		−	1.42680i
$537$		0			0
$538$		−	748.239i		−	1.39078i
$539$	93.5821	+	32.4109i	0.173622	+	0.0601316i
$540$		0			0
$541$		−	451.918i		−	0.835339i	−0.908599	−	0.417669i	$-0.862847\pi$
$541$		−	451.918i		−	0.835339i	0.908599	−	0.417669i	$-0.137153\pi$
$542$	−977.860			−1.80417
$543$		0			0
$544$	−269.048			−0.494574
$545$			74.0128i			0.135803i
$546$		0			0
$547$		−	149.931i		−	0.274097i	−0.990564	−	0.137048i	$-0.956238\pi$
$547$		−	149.931i		−	0.274097i	0.990564	−	0.137048i	$-0.0437615\pi$
$548$	97.9446			0.178731
$549$		0			0
$550$	41.9576	−	121.147i	0.0762865	−	0.220267i
$551$	6.28248			0.0114020
$552$		0			0
$553$	895.202			1.61881
$554$	−202.383			−0.365313
$555$		0			0
$556$		−	30.8495i		−	0.0554847i
$557$		−	14.5869i		−	0.0261883i	−0.999914	−	0.0130941i	$-0.995832\pi$
$557$		−	14.5869i		−	0.0261883i	0.999914	−	0.0130941i	$-0.00416812\pi$
$558$		0			0
$559$	−626.305			−1.12040
$560$			278.296i			0.496958i
$561$		0			0
$562$	−701.070			−1.24746
$563$		−	123.461i		−	0.219291i	−0.993971	−	0.109645i	$-0.965028\pi$
$563$		−	123.461i		−	0.219291i	0.993971	−	0.109645i	$-0.0349715\pi$
$564$		0			0
$565$	−365.143			−0.646270
$566$	−745.420			−1.31700
$567$		0			0
$568$		−	166.565i		−	0.293249i
$569$		−	849.647i		−	1.49323i	−0.665257	−	0.746614i	$-0.731678\pi$
$569$		−	849.647i		−	1.49323i	0.665257	−	0.746614i	$-0.268322\pi$
$570$		0			0
$571$			607.460i			1.06385i	0.846791	+	0.531926i	$0.178532\pi$
$571$			607.460i			1.06385i	−0.846791	+	0.531926i	$0.821468\pi$
$572$	89.7876	−	259.250i	0.156971	−	0.453234i
$573$		0			0
$574$		−	199.087i		−	0.346842i
$575$	45.8365			0.0797156
$576$		0			0
$577$	−345.393			−0.598602			−0.299301	−	0.954159i	$-0.596754\pi$
$577$	−345.393			−0.598602			−0.299301	+	0.954159i	$0.596754\pi$
$578$		−	323.281i		−	0.559310i
$579$		0			0
$580$		−	1.89885i		−	0.00327388i
$581$	−871.511			−1.50002
$582$		0			0
$583$	−732.216	−	253.593i	−1.25594	−	0.434980i
$584$	367.088			0.628575
$585$		0			0
$586$	−983.111			−1.67766
$587$	−800.135			−1.36309			−0.681546	−	0.731775i	$-0.738692\pi$
$587$	−800.135			−1.36309			−0.681546	+	0.731775i	$0.738692\pi$
$588$		0			0
$589$			182.046i			0.309077i
$590$		−	77.9137i		−	0.132057i
$591$		0			0
$592$	−1088.29			−1.83833
$593$			157.850i			0.266189i	0.991103	+	0.133094i	$0.0424913\pi$
$593$			157.850i			0.266189i	−0.991103	+	0.133094i	$0.957509\pi$
$594$		0			0
$595$	173.379			0.291394
$596$		−	65.8718i		−	0.110523i
$597$		0			0
$598$	371.751			0.621657
$599$	−623.613			−1.04109			−0.520545	−	0.853834i	$-0.674271\pi$
$599$	−623.613			−1.04109			−0.520545	+	0.853834i	$0.674271\pi$
$600$		0			0
$601$		−	140.001i		−	0.232947i	−0.993194	−	0.116474i	$-0.962841\pi$
$601$		−	140.001i		−	0.232947i	0.993194	−	0.116474i	$-0.0371591\pi$
$602$			530.743i			0.881632i
$603$		0			0
$604$		−	177.181i		−	0.293346i
$605$	−212.608	−	167.340i	−0.351418	−	0.276596i
$606$		0			0
$607$		−	758.571i		−	1.24971i	−0.780743	−	0.624853i	$-0.785159\pi$
$607$		−	758.571i		−	1.24971i	0.780743	−	0.624853i	$-0.214841\pi$
$608$	−232.765			−0.382837
$609$		0			0
$610$	188.375			0.308811
$611$			334.938i			0.548180i
$612$		0			0
$613$			487.490i			0.795253i	0.917547	+	0.397627i	$0.130166\pi$
$613$			487.490i			0.795253i	−0.917547	+	0.397627i	$0.869834\pi$
$614$	−151.930			−0.247443
$615$		0			0
$616$	393.242	+	136.194i	0.638380	+	0.221094i
$617$	773.785			1.25411			0.627054	−	0.778975i	$-0.284260\pi$
$617$	773.785			1.25411			0.627054	+	0.778975i	$0.284260\pi$
$618$		0			0
$619$	1182.12			1.90972			0.954859	−	0.297059i	$-0.0960058\pi$
$619$	1182.12			1.90972			0.954859	+	0.297059i	$0.0960058\pi$
$620$	55.0227			0.0887463
$621$		0			0
$622$		−	690.710i		−	1.11047i
$623$		−	1014.12i		−	1.62780i
$624$		0			0
$625$	25.0000			0.0400000
$626$			1004.12i			1.60402i
$627$		0			0
$628$	−93.7714			−0.149318
$629$			678.007i			1.07791i
$630$		0			0
$631$	−1089.91			−1.72727			−0.863634	−	0.504120i	$-0.831817\pi$
$631$	−1089.91			−1.72727			−0.863634	+	0.504120i	$0.831817\pi$
$632$	−846.764			−1.33982
$633$		0			0
$634$		−	579.509i		−	0.914051i
$635$		−	456.634i		−	0.719109i
$636$		0			0
$637$			156.625i			0.245880i
$638$	−14.3511	−	4.97032i	−0.0224939	−	0.00779048i
$639$		0			0
$640$		−	339.950i		−	0.531171i
$641$	661.586			1.03212			0.516058	−	0.856554i	$-0.327399\pi$
$641$	661.586			1.03212			0.516058	+	0.856554i	$0.327399\pi$
$642$		0			0
$643$	700.467			1.08937			0.544687	−	0.838639i	$-0.316648\pi$
$643$	700.467			1.08937			0.544687	+	0.838639i	$0.316648\pi$
$644$		−	83.1219i		−	0.129071i
$645$		0			0
$646$			303.134i			0.469248i
$647$	−128.968			−0.199332			−0.0996658	−	0.995021i	$-0.531777\pi$
$647$	−128.968			−0.199332			−0.0996658	+	0.995021i	$0.531777\pi$
$648$		0			0
$649$	−155.372	−	53.8112i	−0.239403	−	0.0829140i
$650$	202.759			0.311938
$651$		0			0
$652$	456.419			0.700029
$653$	828.659			1.26900			0.634502	−	0.772921i	$-0.281205\pi$
$653$	828.659			1.26900			0.634502	+	0.772921i	$0.281205\pi$
$654$		0			0
$655$			544.199i			0.830839i
$656$			265.762i			0.405125i
$657$		0			0
$658$	283.832			0.431356
$659$		−	216.327i		−	0.328265i	−0.986438	−	0.164133i	$-0.947517\pi$
$659$		−	216.327i		−	0.328265i	0.986438	−	0.164133i	$-0.0524825\pi$
$660$		0			0
$661$	114.149			0.172691			0.0863457	−	0.996265i	$-0.472481\pi$
$661$	114.149			0.172691			0.0863457	+	0.996265i	$0.472481\pi$
$662$			884.848i			1.33663i
$663$		0			0
$664$	824.355			1.24150
$665$	149.998			0.225560
$666$		0			0
$667$		−	5.42982i		−	0.00814066i
$668$			24.7283i			0.0370184i
$669$		0			0
$670$			666.358i			0.994564i
$671$	130.101	−	375.650i	0.193892	−	0.559836i
$672$		0			0
$673$			1173.35i			1.74346i	0.489989	+	0.871729i	$0.337001\pi$
$673$			1173.35i			1.74346i	−0.489989	+	0.871729i	$0.662999\pi$
$674$	−1230.81			−1.82613
$675$		0			0
$676$	191.600			0.283433
$677$		−	145.079i		−	0.214297i	−0.994243	−	0.107148i	$-0.965828\pi$
$677$		−	145.079i		−	0.214297i	0.994243	−	0.107148i	$-0.0341720\pi$
$678$		0			0
$679$		−	365.064i		−	0.537649i
$680$	−163.998			−0.241174
$681$		0			0
$682$	144.024	−	415.850i	0.211179	−	0.609751i
$683$	−542.580			−0.794407			−0.397204	−	0.917731i	$-0.630019\pi$
$683$	−542.580			−0.794407			−0.397204	+	0.917731i	$0.630019\pi$
$684$		0			0
$685$	152.758			0.223004
$686$	855.092			1.24649
$687$		0			0
$688$		−	708.489i		−	1.02978i
$689$		−	1225.49i		−	1.77864i
$690$		0			0
$691$	−190.564			−0.275780			−0.137890	−	0.990448i	$-0.544032\pi$
$691$	−190.564			−0.275780			−0.137890	+	0.990448i	$0.544032\pi$
$692$		−	461.803i		−	0.667345i
$693$		0			0
$694$	−83.3665			−0.120125
$695$		−	48.1140i		−	0.0692288i
$696$		0			0
$697$	165.571			0.237547
$698$	−1056.38			−1.51344
$699$		0			0
$700$		−	45.3361i		−	0.0647659i
$701$			149.736i			0.213603i	0.994280	+	0.106801i	$0.0340609\pi$
$701$			149.736i			0.213603i	−0.994280	+	0.106801i	$0.965939\pi$
$702$		0			0
$703$			586.572i			0.834385i
$704$	−286.501	−	99.2259i	−0.406962	−	0.140946i
$705$		0			0
$706$		−	179.209i		−	0.253838i
$707$	−930.017			−1.31544
$708$		0			0
$709$	−225.902			−0.318621			−0.159310	−	0.987229i	$-0.550927\pi$
$709$	−225.902			−0.318621			−0.159310	+	0.987229i	$0.550927\pi$
$710$			145.132i			0.204412i
$711$		0			0
$712$			959.247i			1.34726i
$713$	157.339			0.220672
$714$		0			0
$715$	140.036	−	404.335i	0.195855	−	0.565503i
$716$	65.3032			0.0912056
$717$		0			0
$718$	947.162			1.31917
$719$	−508.963			−0.707877			−0.353938	−	0.935269i	$-0.615158\pi$
$719$	−508.963			−0.707877			−0.353938	+	0.935269i	$0.615158\pi$
$720$		0			0
$721$		−	624.709i		−	0.866448i
$722$		−	579.249i		−	0.802283i
$723$		0			0
$724$	−192.506			−0.265893
$725$		−	2.96152i		−	0.00408485i
$726$		0			0
$727$	−155.888			−0.214426			−0.107213	−	0.994236i	$-0.534193\pi$
$727$	−155.888			−0.214426			−0.107213	+	0.994236i	$0.534193\pi$
$728$			658.156i			0.904061i
$729$		0			0
$730$	−319.852			−0.438154
$731$	−441.391			−0.603818
$732$		0			0
$733$		−	393.332i		−	0.536605i	−0.963335	−	0.268303i	$-0.913537\pi$
$733$		−	393.332i		−	0.536605i	0.963335	−	0.268303i	$-0.0864627\pi$
$734$			669.076i			0.911548i
$735$		0			0
$736$			201.174i			0.273334i
$737$	1328.82	+	460.221i	1.80302	+	0.624451i
$738$		0			0
$739$		−	224.197i		−	0.303379i	−0.988428	−	0.151689i	$-0.951529\pi$
$739$		−	224.197i		−	0.303379i	0.988428	−	0.151689i	$-0.0484713\pi$
$740$	177.289			0.239580
$741$		0			0
$742$	−1038.50			−1.39959
$743$		−	115.127i		−	0.154948i	−0.996994	−	0.0774741i	$-0.975314\pi$
$743$		−	115.127i		−	0.154948i	0.996994	−	0.0774741i	$-0.0246855\pi$
$744$		0			0
$745$		−	102.736i		−	0.137901i
$746$	−838.484			−1.12397
$747$		0			0
$748$	63.2781	−	182.707i	0.0845964	−	0.244261i
$749$	682.727			0.911518
$750$		0			0
$751$	−494.453			−0.658392			−0.329196	−	0.944262i	$-0.606778\pi$
$751$	−494.453			−0.658392			−0.329196	+	0.944262i	$0.606778\pi$
$752$	−378.888			−0.503841
$753$		0			0
$754$		−	24.0190i		−	0.0318555i
$755$		−	276.338i		−	0.366011i
$756$		0			0
$757$	531.890			0.702629			0.351314	−	0.936258i	$-0.385735\pi$
$757$	531.890			0.702629			0.351314	+	0.936258i	$0.385735\pi$
$758$		−	220.045i		−	0.290297i
$759$		0			0
$760$	−141.881			−0.186686
$761$			1343.41i			1.76532i	0.470013	+	0.882660i	$0.344249\pi$
$761$			1343.41i			1.76532i	−0.470013	+	0.882660i	$0.655751\pi$
$762$		0			0
$763$	209.331			0.274353
$764$	−97.5448			−0.127676
$765$		0			0
$766$		−	215.304i		−	0.281076i
$767$		−	260.042i		−	0.339038i
$768$		0			0
$769$			735.802i			0.956830i	0.878134	+	0.478415i	$0.158788\pi$
$769$			735.802i			0.956830i	−0.878134	+	0.478415i	$0.841212\pi$
$770$	−342.641	−	118.669i	−0.444988	−	0.154116i
$771$		0			0
$772$		−	182.749i		−	0.236721i
$773$	1135.40			1.46882			0.734410	−	0.678706i	$-0.237459\pi$
$773$	1135.40			1.46882			0.734410	+	0.678706i	$0.237459\pi$
$774$		0			0
$775$	85.8154			0.110730
$776$			345.311i			0.444988i
$777$		0			0
$778$			294.263i			0.378230i
$779$	143.242			0.183879
$780$		0			0
$781$	289.417	+	100.236i	0.370573	+	0.128343i
$782$	261.993			0.335029
$783$		0			0
$784$	−177.178			−0.225992
$785$	−146.249			−0.186305
$786$		0			0
$787$		−	115.155i		−	0.146321i	−0.997320	−	0.0731605i	$-0.976691\pi$
$787$		−	115.155i		−	0.146321i	0.997320	−	0.0731605i	$-0.0233085\pi$
$788$			212.929i			0.270214i
$789$		0			0
$790$	737.806			0.933931
$791$			1032.74i			1.30561i
$792$		0			0
$793$	628.713			0.792828
$794$		−	1706.57i		−	2.14934i
$795$		0			0
$796$	−90.4017			−0.113570
$797$	1486.27			1.86483			0.932416	−	0.361387i	$-0.117697\pi$
$797$	1486.27			1.86483			0.932416	+	0.361387i	$0.117697\pi$
$798$		0			0
$799$			236.049i			0.295430i
$800$			109.724i			0.137155i
$801$		0			0
$802$			672.049i			0.837967i
$803$	−220.906	+	637.837i	−0.275101	+	0.794317i
$804$		0			0
$805$		−	129.640i		−	0.161043i
$806$	695.995			0.863518
$807$		0			0
$808$	879.695			1.08873
$809$			546.514i			0.675543i	0.941228	+	0.337771i	$0.109673\pi$
$809$			546.514i			0.675543i	−0.941228	+	0.337771i	$0.890327\pi$
$810$		0			0
$811$		−	1066.45i		−	1.31499i	−0.753460	−	0.657494i	$-0.771617\pi$
$811$		−	1066.45i		−	1.31499i	0.753460	−	0.657494i	$-0.228383\pi$
$812$	−5.37055			−0.00661398
$813$		0			0
$814$	464.061	−	1339.91i	0.570099	−	1.64608i
$815$	711.847			0.873432
$816$		0			0
$817$	−381.865			−0.467399
$818$	−1250.33			−1.52852
$819$		0			0
$820$		−	43.2942i		−	0.0527978i
$821$		−	1206.66i		−	1.46974i	−0.678207	−	0.734871i	$-0.737243\pi$
$821$		−	1206.66i		−	1.46974i	0.678207	−	0.734871i	$-0.262757\pi$
$822$		0			0
$823$	312.518			0.379730			0.189865	−	0.981810i	$-0.439195\pi$
$823$	312.518			0.379730			0.189865	+	0.981810i	$0.439195\pi$
$824$			590.907i			0.717120i
$825$		0			0
$826$	−220.364			−0.266785
$827$			1166.37i			1.41036i	0.709029	+	0.705180i	$0.249134\pi$
$827$			1166.37i			1.41036i	−0.709029	+	0.705180i	$0.750866\pi$
$828$		0			0
$829$	−205.935			−0.248413			−0.124207	−	0.992256i	$-0.539639\pi$
$829$	−205.935			−0.248413			−0.124207	+	0.992256i	$0.539639\pi$
$830$	−718.280			−0.865397
$831$		0			0
$832$		−	479.508i		−	0.576332i
$833$			110.382i			0.132512i
$834$		0			0
$835$			38.5672i			0.0461882i
$836$	54.7445	−	158.067i	0.0654839	−	0.189076i
$837$		0			0
$838$			1800.10i			2.14810i
$839$	619.889			0.738843			0.369422	−	0.929262i	$-0.379556\pi$
$839$	619.889			0.738843			0.369422	+	0.929262i	$0.379556\pi$
$840$		0			0
$841$	840.649			0.999583
$842$		−	1236.21i		−	1.46819i
$843$		0			0
$844$			328.279i			0.388956i
$845$	298.827			0.353641
$846$		0			0
$847$	−473.291	+	601.322i	−0.558785	+	0.709944i
$848$	1386.29			1.63478
$849$		0			0
$850$	142.895			0.168112
$851$	506.962			0.595725
$852$		0			0
$853$		−	1067.19i		−	1.25110i	−0.780185	−	0.625549i	$-0.784875\pi$
$853$		−	1067.19i		−	1.25110i	0.780185	−	0.625549i	$-0.215125\pi$
$854$		−	532.783i		−	0.623868i
$855$		0			0
$856$	−645.786			−0.754422
$857$			135.288i			0.157862i	0.996880	+	0.0789310i	$0.0251507\pi$
$857$			135.288i			0.157862i	−0.996880	+	0.0789310i	$0.974849\pi$
$858$		0			0
$859$	−88.1149			−0.102578			−0.0512892	−	0.998684i	$-0.516333\pi$
$859$	−88.1149			−0.102578			−0.0512892	+	0.998684i	$0.516333\pi$
$860$			115.417i			0.134206i
$861$		0			0
$862$	−228.648			−0.265252
$863$	1506.50			1.74566			0.872830	−	0.488024i	$-0.162282\pi$
$863$	1506.50			1.74566			0.872830	+	0.488024i	$0.162282\pi$
$864$		0			0
$865$		−	720.244i		−	0.832652i
$866$			1090.50i			1.25924i
$867$		0			0
$868$		−	155.621i		−	0.179287i
$869$	509.566	−	1471.30i	0.586382	−	1.69310i
$870$		0			0
$871$			2224.01i			2.55340i
$872$	−198.005			−0.227070
$873$		0			0
$874$	226.661			0.259337
$875$		−	70.7078i		−	0.0808089i
$876$		0			0
$877$			874.405i			0.997041i	0.866878	+	0.498521i	$0.166123\pi$
$877$			874.405i			0.997041i	−0.866878	+	0.498521i	$0.833877\pi$
$878$	1707.09			1.94429
$879$		0			0
$880$	457.392	+	158.412i	0.519764	+	0.180013i
$881$	−628.003			−0.712830			−0.356415	−	0.934328i	$-0.616001\pi$
$881$	−628.003			−0.712830			−0.356415	+	0.934328i	$0.616001\pi$
$882$		0			0
$883$	1234.85			1.39848			0.699238	−	0.714889i	$-0.253523\pi$
$883$	1234.85			1.39848			0.699238	+	0.714889i	$0.253523\pi$
$884$	305.791			0.345917
$885$		0			0
$886$			1218.86i			1.37569i
$887$		−	282.840i		−	0.318872i	−0.987208	−	0.159436i	$-0.949032\pi$
$887$		−	282.840i		−	0.318872i	0.987208	−	0.159436i	$-0.0509676\pi$
$888$		0			0
$889$	−1291.50			−1.45276
$890$		−	835.814i		−	0.939117i
$891$		0			0
$892$	145.900			0.163565
$893$			204.215i			0.228685i
$894$		0			0
$895$	101.849			0.113798
$896$	−961.484			−1.07308
$897$		0			0
$898$			279.229i			0.310946i
$899$		−	10.1658i		−	0.0113078i
$900$		0			0
$901$		−	863.665i		−	0.958563i
$902$	−327.209	−	113.324i	−0.362759	−	0.125637i
$903$		0			0
$904$		−	976.858i		−	1.08060i
$905$	−300.240			−0.331757
$906$		0			0
$907$	604.942			0.666970			0.333485	−	0.942755i	$-0.391775\pi$
$907$	604.942			0.666970			0.333485	+	0.942755i	$0.391775\pi$
$908$		−	503.974i		−	0.555037i
$909$		0			0
$910$		−	573.467i		−	0.630184i
$911$	−339.992			−0.373208			−0.186604	−	0.982435i	$-0.559748\pi$
$911$	−339.992			−0.373208			−0.186604	+	0.982435i	$0.559748\pi$
$912$		0			0
$913$	−496.081	+	1432.37i	−0.543352	+	1.56886i
$914$	−11.2251			−0.0122813
$915$		0			0
$916$	−202.615			−0.221195
$917$	1539.17			1.67848
$918$		0			0
$919$			885.348i			0.963382i	0.876341	+	0.481691i	$0.159977\pi$
$919$			885.348i			0.963382i	−0.876341	+	0.481691i	$0.840023\pi$
$920$			122.625i			0.133288i
$921$		0			0
$922$	−1330.27			−1.44281
$923$			484.389i			0.524798i
$924$		0			0
$925$	276.506			0.298925
$926$		−	307.544i		−	0.332121i
$927$		0			0
$928$	12.9979			0.0140064
$929$	821.063			0.883814			0.441907	−	0.897061i	$-0.354302\pi$
$929$	821.063			0.883814			0.441907	+	0.897061i	$0.354302\pi$
$930$		0			0
$931$			95.4962i			0.102574i
$932$			530.822i			0.569552i
$933$		0			0
$934$		−	209.496i		−	0.224300i
$935$	98.6908	−	284.956i	0.105552	−	0.304766i
$936$		0			0
$937$			1507.89i			1.60927i	0.593768	+	0.804636i	$0.297639\pi$
$937$			1507.89i			1.60927i	−0.593768	+	0.804636i	$0.702361\pi$
$938$	1884.67			2.00924
$939$		0			0
$940$	61.7232			0.0656629
$941$			138.310i			0.146982i	0.997296	+	0.0734909i	$0.0234140\pi$
$941$			138.310i			0.146982i	−0.997296	+	0.0734909i	$0.976586\pi$
$942$		0			0
$943$		−	123.801i		−	0.131284i
$944$	294.165			0.311615
$945$		0			0
$946$	872.298	+	302.109i	0.922091	+	0.319354i
$947$	−1559.87			−1.64717			−0.823586	−	0.567192i	$-0.808030\pi$
$947$	−1559.87			−1.64717			−0.823586	+	0.567192i	$0.808030\pi$
$948$		0			0
$949$	−1067.53			−1.12490
$950$	123.625			0.130131
$951$		0			0
$952$			463.838i			0.487224i
$953$			1073.64i			1.12659i	0.826255	+	0.563296i	$0.190467\pi$
$953$			1073.64i			1.12659i	−0.826255	+	0.563296i	$0.809533\pi$
$954$		0			0
$955$	−152.134			−0.159303
$956$			97.9286i			0.102436i
$957$		0			0
$958$	1671.46			1.74474
$959$		−	432.047i		−	0.450518i
$960$		0			0
$961$	−666.429			−0.693474
$962$	2242.57			2.33115
$963$		0			0
$964$			287.149i			0.297872i
$965$		−	285.021i		−	0.295359i
$966$		0			0
$967$		−	1403.40i		−	1.45130i	−0.688066	−	0.725648i	$-0.741540\pi$
$967$		−	1403.40i		−	1.45130i	0.688066	−	0.725648i	$-0.258460\pi$
$968$	447.682	−	568.786i	0.462481	−	0.587588i
$969$		0			0
$970$		−	300.877i		−	0.310183i
$971$	−724.260			−0.745891			−0.372945	−	0.927853i	$-0.621652\pi$
$971$	−724.260			−0.745891			−0.372945	+	0.927853i	$0.621652\pi$
$972$		0			0
$973$	−136.081			−0.139858
$974$		−	2137.09i		−	2.19413i
$975$		0			0
$976$			711.213i			0.728702i
$977$	−943.252			−0.965458			−0.482729	−	0.875770i	$-0.660354\pi$
$977$	−943.252			−0.965458			−0.482729	+	0.875770i	$0.660354\pi$
$978$		0			0
$979$	−1666.75	−	577.256i	−1.70250	−	0.589639i
$980$	28.8633			0.0294523
$981$		0			0
$982$	−704.303			−0.717213
$983$	−1101.28			−1.12033			−0.560163	−	0.828383i	$-0.689261\pi$
$983$	−1101.28			−1.12033			−0.560163	+	0.828383i	$0.689261\pi$
$984$		0			0
$985$			332.092i			0.337149i
$986$		−	16.9275i		−	0.0171678i
$987$		0			0
$988$	264.552			0.267766
$989$			330.038i			0.333709i
$990$		0			0
$991$	−639.463			−0.645271			−0.322635	−	0.946523i	$-0.604569\pi$
$991$	−639.463			−0.645271			−0.322635	+	0.946523i	$0.604569\pi$
$992$			376.639i			0.379676i
$993$		0			0
$994$	410.480			0.412958
$995$	−140.994			−0.141702
$996$		0			0
$997$		−	351.666i		−	0.352724i	−0.984325	−	0.176362i	$-0.943567\pi$
$997$		−	351.666i		−	0.352724i	0.984325	−	0.176362i	$-0.0564329\pi$
$998$			239.626i			0.240106i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.b.c.406.4		16
3.2	odd	2		165.3.b.a.76.13	yes	16
11.10	odd	2	inner	495.3.b.c.406.13		16
12.11	even	2		2640.3.c.c.241.14		16
15.2	even	4		825.3.h.b.274.7		32
15.8	even	4		825.3.h.b.274.25		32
15.14	odd	2		825.3.b.d.76.4		16
33.32	even	2		165.3.b.a.76.4	✓	16
132.131	odd	2		2640.3.c.c.241.15		16
165.32	odd	4		825.3.h.b.274.26		32
165.98	odd	4		825.3.h.b.274.8		32
165.164	even	2		825.3.b.d.76.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.3.b.a.76.4	✓	16	33.32	even	2
165.3.b.a.76.13	yes	16	3.2	odd	2
495.3.b.c.406.4		16	1.1	even	1	trivial
495.3.b.c.406.13		16	11.10	odd	2	inner
825.3.b.d.76.4		16	15.14	odd	2
825.3.b.d.76.13		16	165.164	even	2
825.3.h.b.274.7		32	15.2	even	4
825.3.h.b.274.8		32	165.98	odd	4
825.3.h.b.274.25		32	15.8	even	4
825.3.h.b.274.26		32	165.32	odd	4
2640.3.c.c.241.14		16	12.11	even	2
2640.3.c.c.241.15		16	132.131	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.33103i q^{2} -1.43371 q^{4} -2.23607 q^{5} +6.32430i q^{7} -5.98210i q^{8} +5.21235i q^{10} +(10.3943 + 3.59991i) q^{11} +17.3965i q^{13} +14.7421 q^{14} -19.6793 q^{16} +12.2603i q^{17} +10.6069i q^{19} +3.20588 q^{20} +(8.39151 - 24.2294i) q^{22} +9.16729 q^{23} +5.00000 q^{25} +40.5519 q^{26} -9.06722i q^{28} -0.592304i q^{29} +17.1631 q^{31} +21.9447i q^{32} +28.5791 q^{34} -14.1416i q^{35} +55.3012 q^{37} +24.7249 q^{38} +13.3764i q^{40} -13.5046i q^{41} +36.0017i q^{43} +(-14.9024 - 5.16124i) q^{44} -21.3693i q^{46} +19.2531 q^{47} +9.00325 q^{49} -11.6552i q^{50} -24.9416i q^{52} -70.4442 q^{53} +(-23.2423 - 8.04965i) q^{55} +37.8326 q^{56} -1.38068 q^{58} -14.9479 q^{59} -36.1401i q^{61} -40.0077i q^{62} -27.5634 q^{64} -38.8998i q^{65} +127.842 q^{67} -17.5777i q^{68} -32.9644 q^{70} +27.8440 q^{71} +61.3643i q^{73} -128.909i q^{74} -15.2072i q^{76} +(-22.7669 + 65.7364i) q^{77} -141.550i q^{79} +44.0043 q^{80} -31.4798 q^{82} +137.804i q^{83} -27.4148i q^{85} +83.9212 q^{86} +(21.5350 - 62.1795i) q^{88} -160.353 q^{89} -110.021 q^{91} -13.1433 q^{92} -44.8797i q^{94} -23.7177i q^{95} -57.7240 q^{97} -20.9869i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} + 28 q^{11} - 16 q^{16} - 40 q^{20} - 20 q^{22} - 56 q^{23} + 80 q^{25} + 88 q^{26} - 96 q^{31} - 200 q^{34} + 184 q^{37} - 296 q^{38} - 300 q^{44} + 200 q^{47} - 496 q^{49} + 80 q^{53} + 20 q^{55}+ \cdots - 144 q^{97}+O(q^{100}) \) 16 * q - 8 * q^4 + 28 * q^11 - 16 * q^16 - 40 * q^20 - 20 * q^22 - 56 * q^23 + 80 * q^25 + 88 * q^26 - 96 * q^31 - 200 * q^34 + 184 * q^37 - 296 * q^38 - 300 * q^44 + 200 * q^47 - 496 * q^49 + 80 * q^53 + 20 * q^55 + 32 * q^56 - 16 * q^58 + 136 * q^59 + 456 * q^64 + 256 * q^67 - 216 * q^71 + 248 * q^77 + 160 * q^80 + 232 * q^82 - 192 * q^86 - 116 * q^88 + 336 * q^89 + 256 * q^91 + 440 * q^92 - 144 * q^97

Introduction

L-functions

Modular forms

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