LMFDB - Embedded newform 495.3.b.c.406.7

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)

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")

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);

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			406.7
Root			$-0.664903i$ of defining polynomial
Character	$\chi$	$=$	495.406
Dual form			495.3.b.c.406.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.664903i q^{2} +3.55790 q^{4} +2.23607 q^{5} +8.36247i q^{7} -5.02527i q^{8} -1.48677i q^{10} +(-7.04830 + 8.44521i) q^{11} +16.0778i q^{13} +5.56023 q^{14} +10.8903 q^{16} +22.0359i q^{17} -6.10810i q^{19} +7.95572 q^{20} +(5.61524 + 4.68643i) q^{22} -6.43313 q^{23} +5.00000 q^{25} +10.6902 q^{26} +29.7529i q^{28} -20.7699i q^{29} -52.4060 q^{31} -27.3421i q^{32} +14.6517 q^{34} +18.6990i q^{35} +23.9112 q^{37} -4.06129 q^{38} -11.2368i q^{40} +66.7913i q^{41} -56.5789i q^{43} +(-25.0772 + 30.0472i) q^{44} +4.27741i q^{46} +35.9964 q^{47} -20.9309 q^{49} -3.32451i q^{50} +57.2034i q^{52} +83.7803 q^{53} +(-15.7605 + 18.8841i) q^{55} +42.0237 q^{56} -13.8099 q^{58} +13.5741 q^{59} -120.042i q^{61} +34.8449i q^{62} +25.3814 q^{64} +35.9511i q^{65} +69.5457 q^{67} +78.4015i q^{68} +12.4330 q^{70} +69.1708 q^{71} +78.0255i q^{73} -15.8986i q^{74} -21.7320i q^{76} +(-70.6228 - 58.9412i) q^{77} -7.65867i q^{79} +24.3515 q^{80} +44.4097 q^{82} +5.69286i q^{83} +49.2737i q^{85} -37.6195 q^{86} +(42.4394 + 35.4196i) q^{88} -73.0710 q^{89} -134.450 q^{91} -22.8885 q^{92} -23.9341i q^{94} -13.6581i q^{95} -37.9109 q^{97} +13.9170i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	0.664903i		−	0.332451i	−0.986088	−	0.166226i	$-0.946842\pi$
$2$		−	0.664903i		−	0.332451i	0.986088	−	0.166226i	$-0.0531580\pi$
$3$		0			0
$3$		0			0
$4$	3.55790			0.889476
$5$	2.23607			0.447214
$5$	2.23607			0.447214
$6$		0			0
$7$			8.36247i			1.19464i	0.802004	+	0.597319i	$0.203767\pi$
$7$			8.36247i			1.19464i	−0.802004	+	0.597319i	$0.796233\pi$
$8$		−	5.02527i		−	0.628159i
$9$		0			0
$10$		−	1.48677i		−	0.148677i
$11$	−7.04830	+	8.44521i	−0.640754	+	0.767746i
$11$	−7.04830	+	8.44521i	−0.640754	+	0.767746i
$12$		0			0
$13$			16.0778i			1.23676i	0.785881	+	0.618378i	$0.212210\pi$
$13$			16.0778i			1.23676i	−0.785881	+	0.618378i	$0.787790\pi$
$14$	5.56023			0.397159
$15$		0			0
$16$	10.8903			0.680644
$17$			22.0359i			1.29623i	0.761544	+	0.648114i	$0.224442\pi$
$17$			22.0359i			1.29623i	−0.761544	+	0.648114i	$0.775558\pi$
$18$		0			0
$19$		−	6.10810i		−	0.321479i	−0.986997	−	0.160739i	$-0.948612\pi$
$19$		−	6.10810i		−	0.321479i	0.986997	−	0.160739i	$-0.0513879\pi$
$20$	7.95572			0.397786
$21$		0			0
$22$	5.61524	+	4.68643i	0.255238	+	0.213020i
$23$	−6.43313			−0.279701			−0.139851	−	0.990173i	$-0.544662\pi$
$23$	−6.43313			−0.279701			−0.139851	+	0.990173i	$0.544662\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$	10.6902			0.411161
$27$		0			0
$28$			29.7529i			1.06260i
$29$		−	20.7699i		−	0.716202i	−0.933683	−	0.358101i	$-0.883424\pi$
$29$		−	20.7699i		−	0.716202i	0.933683	−	0.358101i	$-0.116576\pi$
$30$		0			0
$31$	−52.4060			−1.69052			−0.845258	−	0.534358i	$-0.820554\pi$
$31$	−52.4060			−1.69052			−0.845258	+	0.534358i	$0.820554\pi$
$32$		−	27.3421i		−	0.854440i
$33$		0			0
$34$	14.6517			0.430932
$35$			18.6990i			0.534258i
$36$		0			0
$37$	23.9112			0.646249			0.323124	−	0.946356i	$-0.395267\pi$
$37$	23.9112			0.646249			0.323124	+	0.946356i	$0.395267\pi$
$38$	−4.06129			−0.106876
$39$		0			0
$40$		−	11.2368i		−	0.280921i
$41$			66.7913i			1.62906i	0.580124	+	0.814528i	$0.303004\pi$
$41$			66.7913i			1.62906i	−0.580124	+	0.814528i	$0.696996\pi$
$42$		0			0
$43$		−	56.5789i		−	1.31579i	−0.753110	−	0.657894i	$-0.771447\pi$
$43$		−	56.5789i		−	1.31579i	0.753110	−	0.657894i	$-0.228553\pi$
$44$	−25.0772	+	30.0472i	−0.569936	+	0.682892i
$45$		0			0
$46$			4.27741i			0.0929871i
$47$	35.9964			0.765881			0.382940	−	0.923773i	$-0.374911\pi$
$47$	35.9964			0.765881			0.382940	+	0.923773i	$0.374911\pi$
$48$		0			0
$49$	−20.9309			−0.427160
$50$		−	3.32451i		−	0.0664903i
$51$		0			0
$52$			57.2034i			1.10006i
$53$	83.7803			1.58076			0.790381	−	0.612616i	$-0.209883\pi$
$53$	83.7803			1.58076			0.790381	+	0.612616i	$0.209883\pi$
$54$		0			0
$55$	−15.7605	+	18.8841i	−0.286554	+	0.343346i
$56$	42.0237			0.750422
$57$		0			0
$58$	−13.8099			−0.238102
$59$	13.5741			0.230070			0.115035	−	0.993361i	$-0.463302\pi$
$59$	13.5741			0.230070			0.115035	+	0.993361i	$0.463302\pi$
$60$		0			0
$61$		−	120.042i		−	1.96790i	−0.178455	−	0.983948i	$-0.557110\pi$
$61$		−	120.042i		−	1.96790i	0.178455	−	0.983948i	$-0.442890\pi$
$62$			34.8449i			0.562015i
$63$		0			0
$64$	25.3814			0.396584
$65$			35.9511i			0.553094i
$66$		0			0
$67$	69.5457			1.03799			0.518997	−	0.854776i	$-0.326305\pi$
$67$	69.5457			1.03799			0.518997	+	0.854776i	$0.326305\pi$
$68$			78.4015i			1.15296i
$69$		0			0
$70$	12.4330			0.177615
$71$	69.1708			0.974237			0.487119	−	0.873336i	$-0.338048\pi$
$71$	69.1708			0.974237			0.487119	+	0.873336i	$0.338048\pi$
$72$		0			0
$73$			78.0255i			1.06884i	0.845218	+	0.534421i	$0.179470\pi$
$73$			78.0255i			1.06884i	−0.845218	+	0.534421i	$0.820530\pi$
$74$		−	15.8986i		−	0.214846i
$75$		0			0
$76$		−	21.7320i		−	0.285948i
$77$	−70.6228	−	58.9412i	−0.917179	−	0.765470i
$78$		0			0
$79$		−	7.65867i		−	0.0969451i	−0.998825	−	0.0484726i	$-0.984565\pi$
$79$		−	7.65867i		−	0.0969451i	0.998825	−	0.0484726i	$-0.0154353\pi$
$80$	24.3515			0.304393
$81$		0			0
$82$	44.4097			0.541582
$83$			5.69286i			0.0685887i	0.999412	+	0.0342943i	$0.0109184\pi$
$83$			5.69286i			0.0685887i	−0.999412	+	0.0342943i	$0.989082\pi$
$84$		0			0
$85$			49.2737i			0.579690i
$86$	−37.6195			−0.437436
$87$		0			0
$88$	42.4394	+	35.4196i	0.482266	+	0.402495i
$89$	−73.0710			−0.821023			−0.410512	−	0.911855i	$-0.634650\pi$
$89$	−73.0710			−0.821023			−0.410512	+	0.911855i	$0.634650\pi$
$90$		0			0
$91$	−134.450			−1.47748
$92$	−22.8885			−0.248788
$93$		0			0
$94$		−	23.9341i		−	0.254618i
$95$		−	13.6581i		−	0.143770i
$96$		0			0
$97$	−37.9109			−0.390834			−0.195417	−	0.980720i	$-0.562606\pi$
$97$	−37.9109			−0.390834			−0.195417	+	0.980720i	$0.562606\pi$
$98$			13.9170i			0.142010i
$99$		0			0
$100$	17.7895			0.177895
$101$			21.2764i			0.210658i	0.994437	+	0.105329i	$0.0335895\pi$
$101$			21.2764i			0.210658i	−0.994437	+	0.105329i	$0.966411\pi$
$102$		0			0
$103$	147.861			1.43555			0.717773	−	0.696277i	$-0.245162\pi$
$103$	147.861			1.43555			0.717773	+	0.696277i	$0.245162\pi$
$104$	80.7954			0.776879
$105$		0			0
$106$		−	55.7058i		−	0.525526i
$107$		−	115.599i		−	1.08037i	−0.841548	−	0.540183i	$-0.818355\pi$
$107$		−	115.599i		−	1.08037i	0.841548	−	0.540183i	$-0.181645\pi$
$108$		0			0
$109$			138.985i			1.27510i	0.770411	+	0.637548i	$0.220051\pi$
$109$			138.985i			1.27510i	−0.770411	+	0.637548i	$0.779949\pi$
$110$	12.5561	+	10.4792i	0.114146	+	0.0952653i
$111$		0			0
$112$			91.0698i			0.813123i
$113$	37.9798			0.336104			0.168052	−	0.985778i	$-0.446252\pi$
$113$	37.9798			0.336104			0.168052	+	0.985778i	$0.446252\pi$
$114$		0			0
$115$	−14.3849			−0.125086
$116$		−	73.8972i		−	0.637045i
$117$		0			0
$118$		−	9.02546i		−	0.0764869i
$119$	−184.274			−1.54852
$120$		0			0
$121$	−21.6430	−	119.049i	−0.178868	−	0.983873i
$122$	−79.8160			−0.654230
$123$		0			0
$124$	−186.456			−1.50367
$125$	11.1803			0.0894427
$126$		0			0
$127$		−	8.43502i		−	0.0664175i	−0.999448	−	0.0332087i	$-0.989427\pi$
$127$		−	8.43502i		−	0.0664175i	0.999448	−	0.0332087i	$-0.0105726\pi$
$128$		−	126.244i		−	0.986285i
$129$		0			0
$130$	23.9040			0.183877
$131$		−	186.122i		−	1.42078i	−0.703811	−	0.710388i	$-0.748520\pi$
$131$		−	186.122i		−	1.42078i	0.703811	−	0.710388i	$-0.251480\pi$
$132$		0			0
$133$	51.0788			0.384051
$134$		−	46.2411i		−	0.345083i
$135$		0			0
$136$	110.736			0.814237
$137$	−40.5816			−0.296216			−0.148108	−	0.988971i	$-0.547318\pi$
$137$	−40.5816			−0.296216			−0.148108	+	0.988971i	$0.547318\pi$
$138$		0			0
$139$		−	88.1363i		−	0.634074i	−0.948413	−	0.317037i	$-0.897312\pi$
$139$		−	88.1363i		−	0.634074i	0.948413	−	0.317037i	$-0.102688\pi$
$140$			66.5294i			0.475210i
$141$		0			0
$142$		−	45.9919i		−	0.323886i
$143$	−135.781	−	113.321i	−0.949514	−	0.792457i
$144$		0			0
$145$		−	46.4428i		−	0.320295i
$146$	51.8793			0.355338
$147$		0			0
$148$	85.0738			0.574823
$149$		−	90.6013i		−	0.608062i	−0.952662	−	0.304031i	$-0.901667\pi$
$149$		−	90.6013i		−	0.608062i	0.952662	−	0.304031i	$-0.0983327\pi$
$150$		0			0
$151$		−	38.1059i		−	0.252357i	−0.992008	−	0.126178i	$-0.959729\pi$
$151$		−	38.1059i		−	0.252357i	0.992008	−	0.126178i	$-0.0402712\pi$
$152$	−30.6948			−0.201940
$153$		0			0
$154$	−39.1901	+	46.9573i	−0.254481	+	0.304917i
$155$	−117.183			−0.756022
$156$		0			0
$157$	−271.722			−1.73071			−0.865356	−	0.501157i	$-0.832908\pi$
$157$	−271.722			−1.73071			−0.865356	+	0.501157i	$0.832908\pi$
$158$	−5.09227			−0.0322295
$159$		0			0
$160$		−	61.1387i		−	0.382117i
$161$		−	53.7968i		−	0.334142i
$162$		0			0
$163$	128.405			0.787759			0.393879	−	0.919162i	$-0.371133\pi$
$163$	128.405			0.787759			0.393879	+	0.919162i	$0.371133\pi$
$164$			237.637i			1.44901i
$165$		0			0
$166$	3.78520			0.0228024
$167$		−	172.263i		−	1.03152i	−0.856734	−	0.515758i	$-0.827510\pi$
$167$		−	172.263i		−	1.03152i	0.856734	−	0.515758i	$-0.172490\pi$
$168$		0			0
$169$	−89.4964			−0.529565
$170$	32.7622			0.192719
$171$		0			0
$172$		−	201.302i		−	1.17036i
$173$			125.237i			0.723914i	0.932195	+	0.361957i	$0.117891\pi$
$173$			125.237i			0.723914i	−0.932195	+	0.361957i	$0.882109\pi$
$174$		0			0
$175$			41.8123i			0.238928i
$176$	−76.7581	+	91.9709i	−0.436126	+	0.522562i
$177$		0			0
$178$			48.5851i			0.272950i
$179$	−75.0674			−0.419371			−0.209685	−	0.977769i	$-0.567244\pi$
$179$	−75.0674			−0.419371			−0.209685	+	0.977769i	$0.567244\pi$
$180$		0			0
$181$	−285.524			−1.57748			−0.788740	−	0.614727i	$-0.789266\pi$
$181$	−285.524			−1.57748			−0.788740	+	0.614727i	$0.789266\pi$
$182$			89.3963i			0.491189i
$183$		0			0
$184$			32.3282i			0.175697i
$185$	53.4671			0.289011
$186$		0			0
$187$	−186.097	−	155.315i	−0.995173	−	0.830563i
$188$	128.072			0.681233
$189$		0			0
$190$	−9.08132			−0.0477964
$191$	−256.484			−1.34285			−0.671423	−	0.741074i	$-0.734317\pi$
$191$	−256.484			−1.34285			−0.671423	+	0.741074i	$0.734317\pi$
$192$		0			0
$193$			157.978i			0.818541i	0.912413	+	0.409271i	$0.134217\pi$
$193$			157.978i			0.818541i	−0.912413	+	0.409271i	$0.865783\pi$
$194$			25.2070i			0.129933i
$195$		0			0
$196$	−74.4700			−0.379949
$197$			203.131i			1.03112i	0.856852	+	0.515562i	$0.172417\pi$
$197$			203.131i			1.03112i	−0.856852	+	0.515562i	$0.827583\pi$
$198$		0			0
$199$	286.490			1.43965			0.719823	−	0.694158i	$-0.244223\pi$
$199$	286.490			1.43965			0.719823	+	0.694158i	$0.244223\pi$
$200$		−	25.1264i		−	0.125632i
$201$		0			0
$202$	14.1467			0.0700334
$203$	173.687			0.855603
$204$		0			0
$205$			149.350i			0.728536i
$206$		−	98.3133i		−	0.477249i
$207$		0			0
$208$			175.092i			0.841790i
$209$	51.5841	+	43.0517i	0.246814	+	0.205989i
$210$		0			0
$211$			138.840i			0.658012i	0.944328	+	0.329006i	$0.106714\pi$
$211$			138.840i			0.658012i	−0.944328	+	0.329006i	$0.893286\pi$
$212$	298.082			1.40605
$213$		0			0
$214$	−76.8621			−0.359169
$215$		−	126.514i		−	0.588438i
$216$		0			0
$217$		−	438.244i		−	2.01956i
$218$	92.4117			0.423907
$219$		0			0
$220$	−56.0743	+	67.1877i	−0.254883	+	0.305398i
$221$	−354.289			−1.60312
$222$		0			0
$223$	−250.067			−1.12138			−0.560688	−	0.828027i	$-0.689463\pi$
$223$	−250.067			−1.12138			−0.560688	+	0.828027i	$0.689463\pi$
$224$	228.647			1.02075
$225$		0			0
$226$		−	25.2528i		−	0.111738i
$227$		−	247.410i		−	1.08991i	−0.838465	−	0.544955i	$-0.816547\pi$
$227$		−	247.410i		−	1.08991i	0.838465	−	0.544955i	$-0.183453\pi$
$228$		0			0
$229$	363.652			1.58800			0.794000	−	0.607918i	$-0.207995\pi$
$229$	363.652			1.58800			0.794000	+	0.607918i	$0.207995\pi$
$230$			9.56457i			0.0415851i
$231$		0			0
$232$	−104.374			−0.449889
$233$			263.367i			1.13033i	0.824978	+	0.565165i	$0.191188\pi$
$233$			263.367i			1.13033i	−0.824978	+	0.565165i	$0.808812\pi$
$234$		0			0
$235$	80.4904			0.342512
$236$	48.2954			0.204641
$237$		0			0
$238$			122.524i			0.514808i
$239$		−	364.771i		−	1.52624i	−0.646257	−	0.763120i	$-0.723667\pi$
$239$		−	364.771i		−	1.52624i	0.646257	−	0.763120i	$-0.276333\pi$
$240$		0			0
$241$		−	149.640i		−	0.620912i	−0.950588	−	0.310456i	$-0.899518\pi$
$241$		−	149.640i		−	0.620912i	0.950588	−	0.310456i	$-0.100482\pi$
$242$	−79.1558	+	14.3905i	−0.327090	+	0.0594649i
$243$		0			0
$244$		−	427.097i		−	1.75040i
$245$	−46.8028			−0.191032
$246$		0			0
$247$	98.2049			0.397591
$248$			263.354i			1.06191i
$249$		0			0
$250$		−	7.43384i		−	0.0297353i
$251$	118.965			0.473962			0.236981	−	0.971514i	$-0.423842\pi$
$251$	118.965			0.473962			0.236981	+	0.971514i	$0.423842\pi$
$252$		0			0
$253$	45.3426	−	54.3291i	0.179220	−	0.214740i
$254$	−5.60847			−0.0220806
$255$		0			0
$256$	17.5853			0.0686927
$257$	182.143			0.708729			0.354364	−	0.935107i	$-0.384697\pi$
$257$	182.143			0.708729			0.354364	+	0.935107i	$0.384697\pi$
$258$		0			0
$259$			199.957i			0.772034i
$260$			127.911i			0.491964i
$261$		0			0
$262$	−123.753			−0.472339
$263$		−	55.7205i		−	0.211865i	−0.994373	−	0.105933i	$-0.966217\pi$
$263$		−	55.7205i		−	0.211865i	0.994373	−	0.105933i	$-0.0337828\pi$
$264$		0			0
$265$	187.339			0.706938
$266$		−	33.9624i		−	0.127678i
$267$		0			0
$268$	247.437			0.923272
$269$	46.7482			0.173785			0.0868925	−	0.996218i	$-0.472306\pi$
$269$	46.7482			0.173785			0.0868925	+	0.996218i	$0.472306\pi$
$270$		0			0
$271$			81.2380i			0.299771i	0.988703	+	0.149886i	$0.0478906\pi$
$271$			81.2380i			0.299771i	−0.988703	+	0.149886i	$0.952109\pi$
$272$			239.977i			0.882269i
$273$		0			0
$274$			26.9828i			0.0984774i
$275$	−35.2415	+	42.2260i	−0.128151	+	0.153549i
$276$		0			0
$277$		−	309.041i		−	1.11567i	−0.829952	−	0.557835i	$-0.811632\pi$
$277$		−	309.041i		−	1.11567i	0.829952	−	0.557835i	$-0.188368\pi$
$278$	−58.6020			−0.210799
$279$		0			0
$280$	93.9678			0.335599
$281$		−	306.448i		−	1.09056i	−0.838253	−	0.545282i	$-0.816423\pi$
$281$		−	306.448i		−	1.09056i	0.838253	−	0.545282i	$-0.183577\pi$
$282$		0			0
$283$			495.935i			1.75242i	0.481929	+	0.876210i	$0.339936\pi$
$283$			495.935i			1.75242i	−0.481929	+	0.876210i	$0.660064\pi$
$284$	246.103			0.866561
$285$		0			0
$286$	−75.3476	+	90.2808i	−0.263453	+	0.315667i
$287$	−558.540			−1.94613
$288$		0			0
$289$	−196.579			−0.680205
$290$	−30.8800			−0.106483
$291$		0			0
$292$			277.607i			0.950710i
$293$			391.532i			1.33629i	0.744032	+	0.668144i	$0.232911\pi$
$293$			391.532i			1.33629i	−0.744032	+	0.668144i	$0.767089\pi$
$294$		0			0
$295$	30.3526			0.102890
$296$		−	120.160i		−	0.405947i
$297$		0			0
$298$	−60.2410			−0.202151
$299$		−	103.431i		−	0.345922i
$300$		0			0
$301$	473.139			1.57189
$302$	−25.3367			−0.0838964
$303$		0			0
$304$		−	66.5190i		−	0.218813i
$305$		−	268.421i		−	0.880070i
$306$		0			0
$307$			92.7037i			0.301966i	0.988536	+	0.150983i	$0.0482439\pi$
$307$			92.7037i			0.301966i	−0.988536	+	0.150983i	$0.951756\pi$
$308$	−251.269	−	209.707i	−0.815809	−	0.680867i
$309$		0			0
$310$			77.9156i			0.251341i
$311$	−442.851			−1.42396			−0.711980	−	0.702200i	$-0.752201\pi$
$311$	−442.851			−1.42396			−0.711980	+	0.702200i	$0.752201\pi$
$312$		0			0
$313$	97.9824			0.313043			0.156521	−	0.987675i	$-0.449972\pi$
$313$	97.9824			0.313043			0.156521	+	0.987675i	$0.449972\pi$
$314$			180.669i			0.575378i
$315$		0			0
$316$		−	27.2488i		−	0.0862304i
$317$	433.796			1.36844			0.684221	−	0.729275i	$-0.260143\pi$
$317$	433.796			1.36844			0.684221	+	0.729275i	$0.260143\pi$
$318$		0			0
$319$	175.406	+	146.392i	0.549862	+	0.458910i
$320$	56.7545			0.177358
$321$		0			0
$322$	−35.7697			−0.111086
$323$	134.597			0.416710
$324$		0			0
$325$			80.3891i			0.247351i
$326$		−	85.3766i		−	0.261891i
$327$		0			0
$328$	335.644			1.02331
$329$			301.019i			0.914950i
$330$		0			0
$331$	−270.505			−0.817235			−0.408618	−	0.912706i	$-0.633989\pi$
$331$	−270.505			−0.817235			−0.408618	+	0.912706i	$0.633989\pi$
$332$			20.2547i			0.0610080i
$333$		0			0
$334$	−114.538			−0.342929
$335$	155.509			0.464205
$336$		0			0
$337$			80.7287i			0.239551i	0.992801	+	0.119775i	$0.0382175\pi$
$337$			80.7287i			0.239551i	−0.992801	+	0.119775i	$0.961783\pi$
$338$			59.5064i			0.176054i
$339$		0			0
$340$			175.311i			0.515621i
$341$	369.373	−	442.580i	1.08321	−	1.29789i
$342$		0			0
$343$			234.727i			0.684336i
$344$	−284.324			−0.826524
$345$		0			0
$346$	83.2705			0.240666
$347$			105.638i			0.304433i	0.988347	+	0.152217i	$0.0486411\pi$
$347$			105.638i			0.304433i	−0.988347	+	0.152217i	$0.951359\pi$
$348$		0			0
$349$		−	515.712i		−	1.47769i	−0.673878	−	0.738843i	$-0.735373\pi$
$349$		−	515.712i		−	1.47769i	0.673878	−	0.738843i	$-0.264627\pi$
$350$	27.8011			0.0794318
$351$		0			0
$352$	230.909	+	192.715i	0.655993	+	0.547486i
$353$	467.858			1.32538			0.662688	−	0.748896i	$-0.269416\pi$
$353$	467.858			1.32538			0.662688	+	0.748896i	$0.269416\pi$
$354$		0			0
$355$	154.671			0.435692
$356$	−259.980			−0.730280
$357$		0			0
$358$			49.9125i			0.139420i
$359$			274.186i			0.763751i	0.924214	+	0.381875i	$0.124722\pi$
$359$			274.186i			0.763751i	−0.924214	+	0.381875i	$0.875278\pi$
$360$		0			0
$361$	323.691			0.896651
$362$			189.846i			0.524435i
$363$		0			0
$364$	−478.361			−1.31418
$365$			174.470i			0.478001i
$366$		0			0
$367$	−285.758			−0.778631			−0.389316	−	0.921104i	$-0.627288\pi$
$367$	−285.758			−0.778631			−0.389316	+	0.921104i	$0.627288\pi$
$368$	−70.0587			−0.190377
$369$		0			0
$370$		−	35.5504i		−	0.0960822i
$371$			700.610i			1.88844i
$372$		0			0
$373$		−	18.2467i		−	0.0489188i	−0.999701	−	0.0244594i	$-0.992214\pi$
$373$		−	18.2467i		−	0.0489188i	0.999701	−	0.0244594i	$-0.00778644\pi$
$374$	−103.270	+	123.737i	−0.276122	+	0.330847i
$375$		0			0
$376$		−	180.892i		−	0.481095i
$377$	333.934			0.885767
$378$		0			0
$379$	−336.559			−0.888018			−0.444009	−	0.896022i	$-0.646444\pi$
$379$	−336.559			−0.888018			−0.444009	+	0.896022i	$0.646444\pi$
$380$		−	48.5943i		−	0.127880i
$381$		0			0
$382$			170.537i			0.446431i
$383$	212.241			0.554154			0.277077	−	0.960848i	$-0.410634\pi$
$383$	212.241			0.554154			0.277077	+	0.960848i	$0.410634\pi$
$384$		0			0
$385$	−157.917	−	131.796i	−0.410175	−	0.342328i
$386$	105.040			0.272125
$387$		0			0
$388$	−134.883			−0.347637
$389$	530.891			1.36476			0.682379	−	0.730999i	$-0.260946\pi$
$389$	530.891			1.36476			0.682379	+	0.730999i	$0.260946\pi$
$390$		0			0
$391$		−	141.760i		−	0.362556i
$392$			105.183i			0.268324i
$393$		0			0
$394$	135.063			0.342798
$395$		−	17.1253i		−	0.0433552i
$396$		0			0
$397$	−616.868			−1.55382			−0.776911	−	0.629610i	$-0.783215\pi$
$397$	−616.868			−1.55382			−0.776911	+	0.629610i	$0.783215\pi$
$398$		−	190.488i		−	0.478612i
$399$		0			0
$400$	54.4515			0.136129
$401$	300.579			0.749574			0.374787	−	0.927111i	$-0.377716\pi$
$401$	300.579			0.749574			0.374787	+	0.927111i	$0.377716\pi$
$402$		0			0
$403$		−	842.575i		−	2.09076i
$404$			75.6994i			0.187375i
$405$		0			0
$406$		−	115.485i		−	0.284446i
$407$	−168.533	+	201.935i	−0.414087	+	0.496155i
$408$		0			0
$409$			696.288i			1.70242i	0.524828	+	0.851208i	$0.324130\pi$
$409$			696.288i			1.70242i	−0.524828	+	0.851208i	$0.675870\pi$
$410$	99.3032			0.242203
$411$		0			0
$412$	526.076			1.27688
$413$			113.513i			0.274850i
$414$		0			0
$415$			12.7296i			0.0306738i
$416$	439.601			1.05673
$417$		0			0
$418$	28.6252	−	34.2984i	0.0684813	−	0.0820537i
$419$	326.612			0.779504			0.389752	−	0.920920i	$-0.372561\pi$
$419$	326.612			0.779504			0.389752	+	0.920920i	$0.372561\pi$
$420$		0			0
$421$	23.5479			0.0559333			0.0279666	−	0.999609i	$-0.491097\pi$
$421$	23.5479			0.0559333			0.0279666	+	0.999609i	$0.491097\pi$
$422$	92.3154			0.218757
$423$		0			0
$424$		−	421.019i		−	0.992969i
$425$			110.179i			0.259245i
$426$		0			0
$427$	1003.84			2.35092
$428$		−	411.290i		−	0.960959i
$429$		0			0
$430$	−84.1197			−0.195627
$431$			223.614i			0.518825i	0.965767	+	0.259413i	$0.0835290\pi$
$431$			223.614i			0.518825i	−0.965767	+	0.259413i	$0.916471\pi$
$432$		0			0
$433$	−181.032			−0.418087			−0.209044	−	0.977906i	$-0.567035\pi$
$433$	−181.032			−0.418087			−0.209044	+	0.977906i	$0.567035\pi$
$434$	−291.389			−0.671404
$435$		0			0
$436$			494.497i			1.13417i
$437$			39.2942i			0.0899181i
$438$		0			0
$439$		−	66.4465i		−	0.151359i	−0.997132	−	0.0756794i	$-0.975887\pi$
$439$		−	66.4465i		−	0.151359i	0.997132	−	0.0756794i	$-0.0241125\pi$
$440$	94.8975	+	79.2006i	0.215676	+	0.180001i
$441$		0			0
$442$			235.567i			0.532958i
$443$	−430.438			−0.971642			−0.485821	−	0.874058i	$-0.661479\pi$
$443$	−430.438			−0.971642			−0.485821	+	0.874058i	$0.661479\pi$
$444$		0			0
$445$	−163.392			−0.367173
$446$			166.270i			0.372803i
$447$		0			0
$448$			212.251i			0.473775i
$449$	−261.662			−0.582766			−0.291383	−	0.956607i	$-0.594115\pi$
$449$	−261.662			−0.582766			−0.291383	+	0.956607i	$0.594115\pi$
$450$		0			0
$451$	−564.066	−	470.765i	−1.25070	−	1.04382i
$452$	135.128			0.298957
$453$		0			0
$454$	−164.503			−0.362342
$455$	−300.640			−0.660747
$456$		0			0
$457$		−	187.453i		−	0.410181i	−0.978743	−	0.205090i	$-0.934251\pi$
$457$		−	187.453i		−	0.410181i	0.978743	−	0.205090i	$-0.0657488\pi$
$458$		−	241.793i		−	0.527932i
$459$		0			0
$460$	−51.1802			−0.111261
$461$			683.006i			1.48157i	0.671740	+	0.740787i	$0.265547\pi$
$461$			683.006i			1.48157i	−0.671740	+	0.740787i	$0.734453\pi$
$462$		0			0
$463$	72.4460			0.156471			0.0782354	−	0.996935i	$-0.475071\pi$
$463$	72.4460			0.156471			0.0782354	+	0.996935i	$0.475071\pi$
$464$		−	226.190i		−	0.487479i
$465$		0			0
$466$	175.113			0.375780
$467$	−401.507			−0.859759			−0.429879	−	0.902886i	$-0.641444\pi$
$467$	−401.507			−0.859759			−0.429879	+	0.902886i	$0.641444\pi$
$468$		0			0
$469$			581.573i			1.24003i
$470$		−	53.5183i		−	0.113869i
$471$		0			0
$472$		−	68.2135i		−	0.144520i
$473$	477.820	+	398.785i	1.01019	+	0.843097i
$474$		0			0
$475$		−	30.5405i		−	0.0642958i
$476$	−655.630			−1.37737
$477$		0			0
$478$	−242.537			−0.507400
$479$		−	620.609i		−	1.29563i	−0.761796	−	0.647817i	$-0.775682\pi$
$479$		−	620.609i		−	1.29563i	0.761796	−	0.647817i	$-0.224318\pi$
$480$		0			0
$481$			384.440i			0.799252i
$482$	−99.4958			−0.206423
$483$		0			0
$484$	−77.0038	−	423.564i	−0.159099	−	0.875132i
$485$	−84.7713			−0.174786
$486$		0			0
$487$	−539.097			−1.10697			−0.553487	−	0.832857i	$-0.686703\pi$
$487$	−539.097			−1.10697			−0.553487	+	0.832857i	$0.686703\pi$
$488$	−603.242			−1.23615
$489$		0			0
$490$			31.1193i			0.0635088i
$491$		−	480.768i		−	0.979161i	−0.871958	−	0.489580i	$-0.837150\pi$
$491$		−	480.768i		−	0.979161i	0.871958	−	0.489580i	$-0.162850\pi$
$492$		0			0
$493$	457.682			0.928361
$494$		−	65.2967i		−	0.132180i
$495$		0			0
$496$	−570.717			−1.15064
$497$			578.439i			1.16386i
$498$		0			0
$499$	566.286			1.13484			0.567421	−	0.823428i	$-0.307941\pi$
$499$	566.286			1.13484			0.567421	+	0.823428i	$0.307941\pi$
$500$	39.7786			0.0795572
$501$		0			0
$502$		−	79.0998i		−	0.157569i
$503$			622.513i			1.23760i	0.785549	+	0.618800i	$0.212381\pi$
$503$			622.513i			1.23760i	−0.785549	+	0.618800i	$0.787619\pi$
$504$		0			0
$505$			47.5755i			0.0942089i
$506$	−36.1236	−	30.1484i	−0.0713905	−	0.0595819i
$507$		0			0
$508$		−	30.0110i		−	0.0590768i
$509$	−506.799			−0.995677			−0.497838	−	0.867270i	$-0.665873\pi$
$509$	−506.799			−0.995677			−0.497838	+	0.867270i	$0.665873\pi$
$510$		0			0
$511$	−652.485			−1.27688
$512$		−	516.670i		−	1.00912i
$513$		0			0
$514$		−	121.108i		−	0.235618i
$515$	330.628			0.641996
$516$		0			0
$517$	−253.713	+	303.997i	−0.490741	+	0.588002i
$518$	132.952			0.256664
$519$		0			0
$520$	180.664			0.347431
$521$	486.736			0.934233			0.467117	−	0.884196i	$-0.345293\pi$
$521$	486.736			0.934233			0.467117	+	0.884196i	$0.345293\pi$
$522$		0			0
$523$		−	390.347i		−	0.746362i	−0.927759	−	0.373181i	$-0.878267\pi$
$523$		−	390.347i		−	0.746362i	0.927759	−	0.373181i	$-0.121733\pi$
$524$		−	662.203i		−	1.26375i
$525$		0			0
$526$	−37.0487			−0.0704348
$527$		−	1154.81i		−	2.19129i
$528$		0			0
$529$	−487.615			−0.921767
$530$		−	124.562i		−	0.235022i
$531$		0			0
$532$	181.733			0.341604
$533$	−1073.86			−2.01474
$534$		0			0
$535$		−	258.487i		−	0.483154i
$536$		−	349.486i		−	0.652026i
$537$		0			0
$538$		−	31.0830i		−	0.0577751i
$539$	147.527	−	176.765i	0.273705	−	0.327951i
$540$		0			0
$541$		−	43.4279i		−	0.0802735i	−0.999194	−	0.0401367i	$-0.987221\pi$
$541$		−	43.4279i		−	0.0802735i	0.999194	−	0.0401367i	$-0.0127794\pi$
$542$	54.0154			0.0996594
$543$		0			0
$544$	602.506			1.10755
$545$			310.781i			0.570240i
$546$		0			0
$547$			298.284i			0.545309i	0.962112	+	0.272655i	$0.0879016\pi$
$547$			298.284i			0.545309i	−0.962112	+	0.272655i	$0.912098\pi$
$548$	−144.386			−0.263477
$549$		0			0
$550$	28.0762	+	23.4322i	0.0510476	+	0.0426039i
$551$	−126.864			−0.230244
$552$		0			0
$553$	64.0453			0.115814
$554$	−205.482			−0.370906
$555$		0			0
$556$		−	313.580i		−	0.563994i
$557$		−	401.329i		−	0.720520i	−0.932852	−	0.360260i	$-0.882688\pi$
$557$		−	401.329i		−	0.720520i	0.932852	−	0.360260i	$-0.117312\pi$
$558$		0			0
$559$	909.666			1.62731
$560$			203.638i			0.363640i
$561$		0			0
$562$	−203.758			−0.362559
$563$		−	190.526i		−	0.338412i	−0.985581	−	0.169206i	$-0.945880\pi$
$563$		−	190.526i		−	0.338412i	0.985581	−	0.169206i	$-0.0541203\pi$
$564$		0			0
$565$	84.9253			0.150310
$566$	329.749			0.582595
$567$		0			0
$568$		−	347.602i		−	0.611976i
$569$		−	718.614i		−	1.26294i	−0.775400	−	0.631471i	$-0.782452\pi$
$569$		−	718.614i		−	1.26294i	0.775400	−	0.631471i	$-0.217548\pi$
$570$		0			0
$571$		−	324.008i		−	0.567440i	−0.958907	−	0.283720i	$-0.908431\pi$
$571$		−	324.008i		−	0.567440i	0.958907	−	0.283720i	$-0.0915686\pi$
$572$	−483.094	−	403.186i	−0.844570	−	0.704871i
$573$		0			0
$574$			371.375i			0.646994i
$575$	−32.1657			−0.0559403
$576$		0			0
$577$	1131.01			1.96016			0.980079	−	0.198606i	$-0.0636415\pi$
$577$	1131.01			1.96016			0.980079	+	0.198606i	$0.0636415\pi$
$578$			130.706i			0.226135i
$579$		0			0
$580$		−	165.239i		−	0.284895i
$581$	−47.6064			−0.0819387
$582$		0			0
$583$	−590.509	+	707.542i	−1.01288	+	1.21362i
$584$	392.099			0.671403
$585$		0			0
$586$	260.331			0.444251
$587$	856.901			1.45980			0.729898	−	0.683556i	$-0.239567\pi$
$587$	856.901			1.45980			0.729898	+	0.683556i	$0.239567\pi$
$588$		0			0
$589$			320.101i			0.543465i
$590$		−	20.1815i		−	0.0342060i
$591$		0			0
$592$	260.400			0.439865
$593$		−	712.663i		−	1.20179i	−0.799327	−	0.600896i	$-0.794810\pi$
$593$		−	712.663i		−	1.20179i	0.799327	−	0.600896i	$-0.205190\pi$
$594$		0			0
$595$	−412.050			−0.692520
$596$		−	322.351i		−	0.540857i
$597$		0			0
$598$	−68.7714			−0.115002
$599$	−822.850			−1.37371			−0.686853	−	0.726796i	$-0.741008\pi$
$599$	−822.850			−1.37371			−0.686853	+	0.726796i	$0.741008\pi$
$600$		0			0
$601$		−	350.575i		−	0.583319i	−0.956522	−	0.291659i	$-0.905793\pi$
$601$		−	350.575i		−	0.583319i	0.956522	−	0.291659i	$-0.0942074\pi$
$602$		−	314.591i		−	0.522577i
$603$		0			0
$604$		−	135.577i		−	0.224465i
$605$	−48.3952	−	266.201i	−0.0799921	−	0.440001i
$606$		0			0
$607$			358.620i			0.590808i	0.955372	+	0.295404i	$0.0954542\pi$
$607$			358.620i			0.590808i	−0.955372	+	0.295404i	$0.904546\pi$
$608$	−167.008			−0.274684
$609$		0			0
$610$	−178.474			−0.292580
$611$			578.744i			0.947208i
$612$		0			0
$613$			28.2177i			0.0460322i	0.999735	+	0.0230161i	$0.00732690\pi$
$613$			28.2177i			0.0460322i	−0.999735	+	0.0230161i	$0.992673\pi$
$614$	61.6389			0.100389
$615$		0			0
$616$	−296.195	+	354.898i	−0.480836	+	0.576134i
$617$	−550.410			−0.892074			−0.446037	−	0.895014i	$-0.647165\pi$
$617$	−550.410			−0.892074			−0.446037	+	0.895014i	$0.647165\pi$
$618$		0			0
$619$	−373.784			−0.603852			−0.301926	−	0.953331i	$-0.597629\pi$
$619$	−373.784			−0.603852			−0.301926	+	0.953331i	$0.597629\pi$
$620$	−416.927			−0.672464
$621$		0			0
$622$			294.453i			0.473397i
$623$		−	611.054i		−	0.980825i
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	65.1488i		−	0.104072i
$627$		0			0
$628$	−966.761			−1.53943
$629$			526.904i			0.837685i
$630$		0			0
$631$	1100.42			1.74393			0.871965	−	0.489568i	$-0.162846\pi$
$631$	1100.42			1.74393			0.871965	+	0.489568i	$0.162846\pi$
$632$	−38.4869			−0.0608969
$633$		0			0
$634$		−	288.432i		−	0.454940i
$635$		−	18.8613i		−	0.0297028i
$636$		0			0
$637$		−	336.523i		−	0.528293i
$638$	97.3366	−	116.628i	0.152565	−	0.182802i
$639$		0			0
$640$		−	282.291i		−	0.441080i
$641$	−653.649			−1.01973			−0.509867	−	0.860253i	$-0.670305\pi$
$641$	−653.649			−1.01973			−0.509867	+	0.860253i	$0.670305\pi$
$642$		0			0
$643$	−502.794			−0.781950			−0.390975	−	0.920401i	$-0.627862\pi$
$643$	−502.794			−0.781950			−0.390975	+	0.920401i	$0.627862\pi$
$644$		−	191.404i		−	0.297211i
$645$		0			0
$646$		−	89.4940i		−	0.138536i
$647$	−897.406			−1.38703			−0.693513	−	0.720444i	$-0.743938\pi$
$647$	−897.406			−1.38703			−0.693513	+	0.720444i	$0.743938\pi$
$648$		0			0
$649$	−95.6743	+	114.636i	−0.147418	+	0.176635i
$650$	53.4509			0.0822322
$651$		0			0
$652$	456.852			0.700693
$653$	789.461			1.20898			0.604488	−	0.796615i	$-0.293378\pi$
$653$	789.461			1.20898			0.604488	+	0.796615i	$0.293378\pi$
$654$		0			0
$655$		−	416.181i		−	0.635390i
$656$			727.378i			1.10881i
$657$		0			0
$658$	200.148			0.304176
$659$			127.662i			0.193721i	0.995298	+	0.0968607i	$0.0308801\pi$
$659$			127.662i			0.193721i	−0.995298	+	0.0968607i	$0.969120\pi$
$660$		0			0
$661$	525.245			0.794621			0.397311	−	0.917684i	$-0.369944\pi$
$661$	525.245			0.794621			0.397311	+	0.917684i	$0.369944\pi$
$662$			179.859i			0.271691i
$663$		0			0
$664$	28.6082			0.0430846
$665$	114.216			0.171753
$666$		0			0
$667$			133.615i			0.200323i
$668$		−	612.896i		−	0.917510i
$669$		0			0
$670$		−	103.398i		−	0.154326i
$671$	1013.78	+	846.089i	1.51084	+	1.26094i
$672$		0			0
$673$		−	952.301i		−	1.41501i	−0.706709	−	0.707505i	$-0.749821\pi$
$673$		−	952.301i		−	1.41501i	0.706709	−	0.707505i	$-0.250179\pi$
$674$	53.6767			0.0796390
$675$		0			0
$676$	−318.420			−0.471035
$677$			258.470i			0.381788i	0.981611	+	0.190894i	$0.0611387\pi$
$677$			258.470i			0.381788i	−0.981611	+	0.190894i	$0.938861\pi$
$678$		0			0
$679$		−	317.028i		−	0.466905i
$680$	247.614			0.364138
$681$		0			0
$682$	−294.272	−	245.597i	−0.431484	−	0.360113i
$683$	−391.010			−0.572489			−0.286245	−	0.958157i	$-0.592407\pi$
$683$	−391.010			−0.572489			−0.286245	+	0.958157i	$0.592407\pi$
$684$		0			0
$685$	−90.7432			−0.132472
$686$	156.071			0.227508
$687$		0			0
$688$		−	616.161i		−	0.895583i
$689$			1347.01i			1.95502i
$690$		0			0
$691$	199.456			0.288648			0.144324	−	0.989531i	$-0.453899\pi$
$691$	199.456			0.288648			0.144324	+	0.989531i	$0.453899\pi$
$692$			445.582i			0.643904i
$693$		0			0
$694$	70.2392			0.101209
$695$		−	197.079i		−	0.283566i
$696$		0			0
$697$	−1471.80			−2.11163
$698$	−342.898			−0.491258
$699$		0			0
$700$			148.764i			0.212520i
$701$			181.684i			0.259179i	0.991568	+	0.129589i	$0.0413659\pi$
$701$			181.684i			0.259179i	−0.991568	+	0.129589i	$0.958634\pi$
$702$		0			0
$703$		−	146.052i		−	0.207755i
$704$	−178.896	+	214.351i	−0.254113	+	0.304476i
$705$		0			0
$706$		−	311.080i		−	0.440623i
$707$	−177.923			−0.251660
$708$		0			0
$709$	975.319			1.37563			0.687813	−	0.725888i	$-0.258571\pi$
$709$	975.319			1.37563			0.687813	+	0.725888i	$0.258571\pi$
$710$		−	102.841i		−	0.144846i
$711$		0			0
$712$			367.202i			0.515733i
$713$	337.135			0.472840
$714$		0			0
$715$	−303.615	−	253.394i	−0.424636	−	0.354397i
$716$	−267.083			−0.373020
$717$		0			0
$718$	182.307			0.253910
$719$	1352.64			1.88128			0.940641	−	0.339402i	$-0.110225\pi$
$719$	1352.64			1.88128			0.940641	+	0.339402i	$0.110225\pi$
$720$		0			0
$721$			1236.48i			1.71496i
$722$		−	215.223i		−	0.298093i
$723$		0			0
$724$	−1015.87			−1.40313
$725$		−	103.849i		−	0.143240i
$726$		0			0
$727$	−268.503			−0.369330			−0.184665	−	0.982801i	$-0.559120\pi$
$727$	−268.503			−0.369330			−0.184665	+	0.982801i	$0.559120\pi$
$728$			675.649i			0.928089i
$729$		0			0
$730$	116.006			0.158912
$731$	1246.76			1.70556
$732$		0			0
$733$			1111.10i			1.51583i	0.652353	+	0.757915i	$0.273782\pi$
$733$			1111.10i			1.51583i	−0.652353	+	0.757915i	$0.726218\pi$
$734$			190.001i			0.258857i
$735$		0			0
$736$			175.895i			0.238988i
$737$	−490.178	+	587.327i	−0.665100	+	0.796916i
$738$		0			0
$739$		−	962.884i		−	1.30296i	−0.758668	−	0.651478i	$-0.774149\pi$
$739$		−	962.884i		−	1.30296i	0.758668	−	0.651478i	$-0.225851\pi$
$740$	190.231			0.257069
$741$		0			0
$742$	465.838			0.627814
$743$		−	674.012i		−	0.907149i	−0.891218	−	0.453574i	$-0.850149\pi$
$743$		−	674.012i		−	0.907149i	0.891218	−	0.453574i	$-0.149851\pi$
$744$		0			0
$745$		−	202.591i		−	0.271934i
$746$	−12.1323			−0.0162631
$747$		0			0
$748$	−662.117	−	552.597i	−0.885183	−	0.738766i
$749$	966.693			1.29065
$750$		0			0
$751$	360.050			0.479428			0.239714	−	0.970844i	$-0.422946\pi$
$751$	360.050			0.479428			0.239714	+	0.970844i	$0.422946\pi$
$752$	392.012			0.521292
$753$		0			0
$754$		−	222.034i		−	0.294475i
$755$		−	85.2074i		−	0.112857i
$756$		0			0
$757$	573.607			0.757737			0.378869	−	0.925450i	$-0.376313\pi$
$757$	573.607			0.757737			0.378869	+	0.925450i	$0.376313\pi$
$758$			223.779i			0.295223i
$759$		0			0
$760$	−68.6358			−0.0903102
$761$		−	1229.64i		−	1.61582i	−0.589305	−	0.807910i	$-0.700599\pi$
$761$		−	1229.64i		−	1.61582i	0.589305	−	0.807910i	$-0.299401\pi$
$762$		0			0
$763$	−1162.26			−1.52328
$764$	−912.545			−1.19443
$765$		0			0
$766$		−	141.120i		−	0.184229i
$767$			218.242i			0.284540i
$768$		0			0
$769$			1225.78i			1.59399i	0.603985	+	0.796996i	$0.293579\pi$
$769$			1225.78i			1.59399i	−0.603985	+	0.796996i	$0.706421\pi$
$770$	−87.6318	+	105.000i	−0.113808	+	0.136363i
$771$		0			0
$772$			562.072i			0.728073i
$773$	−86.0374			−0.111303			−0.0556516	−	0.998450i	$-0.517724\pi$
$773$	−86.0374			−0.111303			−0.0556516	+	0.998450i	$0.517724\pi$
$774$		0			0
$775$	−262.030			−0.338103
$776$			190.512i			0.245506i
$777$		0			0
$778$		−	352.991i		−	0.453715i
$779$	407.968			0.523707
$780$		0			0
$781$	−487.537	+	584.162i	−0.624247	+	0.747967i
$782$	−94.2563			−0.120532
$783$		0			0
$784$	−227.943			−0.290744
$785$	−607.589			−0.773998
$786$		0			0
$787$		−	1109.97i		−	1.41038i	−0.709021	−	0.705188i	$-0.750863\pi$
$787$		−	1109.97i		−	1.41038i	0.709021	−	0.705188i	$-0.249137\pi$
$788$			722.722i			0.917160i
$789$		0			0
$790$	−11.3867			−0.0144135
$791$			317.604i			0.401523i
$792$		0			0
$793$	1930.01			2.43381
$794$			410.157i			0.516570i
$795$		0			0
$796$	1019.30			1.28053
$797$	532.778			0.668480			0.334240	−	0.942488i	$-0.391520\pi$
$797$	532.778			0.668480			0.334240	+	0.942488i	$0.391520\pi$
$798$		0			0
$799$			793.212i			0.992756i
$800$		−	136.710i		−	0.170888i
$801$		0			0
$802$		−	199.856i		−	0.249197i
$803$	−658.941	−	549.947i	−0.820599	−	0.684865i
$804$		0			0
$805$		−	120.293i		−	0.149433i
$806$	−560.230			−0.695075
$807$		0			0
$808$	106.920			0.132326
$809$			803.028i			0.992619i	0.868146	+	0.496309i	$0.165312\pi$
$809$			803.028i			0.992619i	−0.868146	+	0.496309i	$0.834688\pi$
$810$		0			0
$811$			323.073i			0.398364i	0.979962	+	0.199182i	$0.0638285\pi$
$811$			323.073i			0.398364i	−0.979962	+	0.199182i	$0.936171\pi$
$812$	617.963			0.761038
$813$		0			0
$814$	134.267	+	112.058i	0.164947	+	0.137664i
$815$	287.122			0.352296
$816$		0			0
$817$	−345.589			−0.422998
$818$	462.964			0.565971
$819$		0			0
$820$			531.373i			0.648016i
$821$			345.071i			0.420306i	0.977668	+	0.210153i	$0.0673962\pi$
$821$			345.071i			0.420306i	−0.977668	+	0.210153i	$0.932604\pi$
$822$		0			0
$823$	−61.4018			−0.0746073			−0.0373036	−	0.999304i	$-0.511877\pi$
$823$	−61.4018			−0.0746073			−0.0373036	+	0.999304i	$0.511877\pi$
$824$		−	743.043i		−	0.901751i
$825$		0			0
$826$	75.4751			0.0913742
$827$		−	1331.04i		−	1.60948i	−0.593626	−	0.804741i	$-0.702304\pi$
$827$		−	1331.04i		−	1.60948i	0.593626	−	0.804741i	$-0.297696\pi$
$828$		0			0
$829$	177.243			0.213803			0.106902	−	0.994270i	$-0.465907\pi$
$829$	177.243			0.213803			0.106902	+	0.994270i	$0.465907\pi$
$830$	8.46396			0.0101975
$831$		0			0
$832$			408.078i			0.490478i
$833$		−	461.229i		−	0.553697i
$834$		0			0
$835$		−	385.193i		−	0.461308i
$836$	183.531	+	153.174i	0.219535	+	0.183222i
$837$		0			0
$838$		−	217.165i		−	0.259147i
$839$	−477.684			−0.569349			−0.284675	−	0.958624i	$-0.591886\pi$
$839$	−477.684			−0.569349			−0.284675	+	0.958624i	$0.591886\pi$
$840$		0			0
$841$	409.612			0.487054
$842$		−	15.6571i		−	0.0185951i
$843$		0			0
$844$			493.981i			0.585286i
$845$	−200.120			−0.236828
$846$		0			0
$847$	995.540	−	180.989i	1.17537	−	0.213682i
$848$	912.393			1.07594
$849$		0			0
$850$	73.2585			0.0861865
$851$	−153.824			−0.180757
$852$		0			0
$853$		−	129.866i		−	0.152247i	−0.997098	−	0.0761234i	$-0.975746\pi$
$853$		−	129.866i		−	0.152247i	0.997098	−	0.0761234i	$-0.0242543\pi$
$854$		−	667.459i		−	0.781568i
$855$		0			0
$856$	−580.917			−0.678641
$857$			419.114i			0.489048i	0.969643	+	0.244524i	$0.0786317\pi$
$857$			419.114i			0.489048i	−0.969643	+	0.244524i	$0.921368\pi$
$858$		0			0
$859$	254.977			0.296830			0.148415	−	0.988925i	$-0.452583\pi$
$859$	254.977			0.296830			0.148415	+	0.988925i	$0.452583\pi$
$860$		−	450.126i		−	0.523402i
$861$		0			0
$862$	148.681			0.172484
$863$	59.8124			0.0693075			0.0346538	−	0.999399i	$-0.488967\pi$
$863$	59.8124			0.0693075			0.0346538	+	0.999399i	$0.488967\pi$
$864$		0			0
$865$			280.039i			0.323744i
$866$			120.368i			0.138994i
$867$		0			0
$868$		−	1559.23i		−	1.79635i
$869$	64.6790	+	53.9806i	0.0744293	+	0.0621180i
$870$		0			0
$871$			1118.14i			1.28375i
$872$	698.439			0.800962
$873$		0			0
$874$	26.1268			0.0298934
$875$			93.4952i			0.106852i
$876$		0			0
$877$		−	563.009i		−	0.641971i	−0.947084	−	0.320986i	$-0.895986\pi$
$877$		−	563.009i		−	0.641971i	0.947084	−	0.320986i	$-0.104014\pi$
$878$	−44.1804			−0.0503194
$879$		0			0
$880$	−171.636	+	205.653i	−0.195041	+	0.233697i
$881$	−1178.14			−1.33728			−0.668641	−	0.743586i	$-0.733124\pi$
$881$	−1178.14			−1.33728			−0.668641	+	0.743586i	$0.733124\pi$
$882$		0			0
$883$	−704.399			−0.797733			−0.398867	−	0.917009i	$-0.630596\pi$
$883$	−704.399			−0.797733			−0.398867	+	0.917009i	$0.630596\pi$
$884$	−1260.53			−1.42593
$885$		0			0
$886$			286.199i			0.323024i
$887$			1582.21i			1.78378i	0.452257	+	0.891888i	$0.350619\pi$
$887$			1582.21i			1.78378i	−0.452257	+	0.891888i	$0.649381\pi$
$888$		0			0
$889$	70.5376			0.0793448
$890$			108.640i			0.122067i
$891$		0			0
$892$	−889.714			−0.997438
$893$		−	219.870i		−	0.246214i
$894$		0			0
$895$	−167.856			−0.187548
$896$	1055.71			1.17825
$897$		0			0
$898$			173.980i			0.193741i
$899$			1088.47i			1.21075i
$900$		0			0
$901$			1846.17i			2.04903i
$902$	−313.013	+	375.049i	−0.347021	+	0.415797i
$903$		0			0
$904$		−	190.859i		−	0.211127i
$905$	−638.451			−0.705471
$906$		0			0
$907$	−436.018			−0.480725			−0.240363	−	0.970683i	$-0.577266\pi$
$907$	−436.018			−0.480725			−0.240363	+	0.970683i	$0.577266\pi$
$908$		−	880.260i		−	0.969449i
$909$		0			0
$910$			199.896i			0.219666i
$911$	82.7040			0.0907838			0.0453919	−	0.998969i	$-0.485546\pi$
$911$	82.7040			0.0907838			0.0453919	+	0.998969i	$0.485546\pi$
$912$		0			0
$913$	−48.0774	−	40.1250i	−0.0526587	−	0.0439485i
$914$	−124.638			−0.136365
$915$		0			0
$916$	1293.84			1.41249
$917$	1556.44			1.69731
$918$		0			0
$919$			1713.21i			1.86421i	0.362193	+	0.932103i	$0.382028\pi$
$919$			1713.21i			1.86421i	−0.362193	+	0.932103i	$0.617972\pi$
$920$			72.2881i			0.0785740i
$921$		0			0
$922$	454.132			0.492551
$923$			1112.12i			1.20489i
$924$		0			0
$925$	119.556			0.129250
$926$		−	48.1695i		−	0.0520189i
$927$		0			0
$928$	−567.891			−0.611952
$929$	−537.309			−0.578374			−0.289187	−	0.957273i	$-0.593385\pi$
$929$	−537.309			−0.578374			−0.289187	+	0.957273i	$0.593385\pi$
$930$		0			0
$931$			127.848i			0.137323i
$932$			937.034i			1.00540i
$933$		0			0
$934$			266.963i			0.285828i
$935$	−416.126	−	347.296i	−0.445055	−	0.371439i
$936$		0			0
$937$		−	321.591i		−	0.343213i	−0.985166	−	0.171607i	$-0.945104\pi$
$937$		−	321.591i		−	0.343213i	0.985166	−	0.171607i	$-0.0548958\pi$
$938$	386.690			0.412249
$939$		0			0
$940$	286.377			0.304657
$941$			880.847i			0.936076i	0.883708	+	0.468038i	$0.155039\pi$
$941$			880.847i			0.936076i	−0.883708	+	0.468038i	$0.844961\pi$
$942$		0			0
$943$		−	429.677i		−	0.455649i
$944$	147.826			0.156595
$945$		0			0
$946$	265.153	−	317.704i	0.280289	−	0.335839i
$947$	227.350			0.240074			0.120037	−	0.992769i	$-0.461699\pi$
$947$	227.350			0.240074			0.120037	+	0.992769i	$0.461699\pi$
$948$		0			0
$949$	−1254.48			−1.32190
$950$	−20.3064			−0.0213752
$951$		0			0
$952$			926.028i			0.972718i
$953$			46.3722i			0.0486592i	0.999704	+	0.0243296i	$0.00774512\pi$
$953$			46.3722i			0.0486592i	−0.999704	+	0.0243296i	$0.992255\pi$
$954$		0			0
$955$	−573.515			−0.600539
$956$		−	1297.82i		−	1.35755i
$957$		0			0
$958$	−412.644			−0.430735
$959$		−	339.362i		−	0.353871i
$960$		0			0
$961$	1785.39			1.85785
$962$	255.615			0.265712
$963$		0			0
$964$		−	532.404i		−	0.552286i
$965$			353.251i			0.366063i
$966$		0			0
$967$			264.144i			0.273158i	0.990629	+	0.136579i	$0.0436108\pi$
$967$			264.144i			0.273158i	−0.990629	+	0.136579i	$0.956389\pi$
$968$	−598.252	+	108.762i	−0.618029	+	0.112357i
$969$		0			0
$970$			56.3647i			0.0581079i
$971$	−666.134			−0.686029			−0.343015	−	0.939330i	$-0.611448\pi$
$971$	−666.134			−0.686029			−0.343015	+	0.939330i	$0.611448\pi$
$972$		0			0
$973$	737.037			0.757489
$974$			358.447i			0.368015i
$975$		0			0
$976$		−	1307.29i		−	1.33944i
$977$	−1339.01			−1.37053			−0.685266	−	0.728293i	$-0.740314\pi$
$977$	−1339.01			−1.37053			−0.685266	+	0.728293i	$0.740314\pi$
$978$		0			0
$979$	515.026	−	617.100i	0.526074	−	0.630337i
$980$	−166.520			−0.169918
$981$		0			0
$982$	−319.664			−0.325523
$983$	−1241.99			−1.26347			−0.631735	−	0.775185i	$-0.717657\pi$
$983$	−1241.99			−1.26347			−0.631735	+	0.775185i	$0.717657\pi$
$984$		0			0
$985$			454.216i			0.461133i
$986$		−	304.314i		−	0.308635i
$987$		0			0
$988$	349.404			0.353647
$989$			363.979i			0.368028i
$990$		0			0
$991$	886.745			0.894798			0.447399	−	0.894334i	$-0.352350\pi$
$991$	886.745			0.894798			0.447399	+	0.894334i	$0.352350\pi$
$992$			1432.89i			1.44444i
$993$		0			0
$994$	384.605			0.386927
$995$	640.610			0.643829
$996$		0			0
$997$		−	20.1579i		−	0.0202185i	−0.999949	−	0.0101093i	$-0.996782\pi$
$997$		−	20.1579i		−	0.0202185i	0.999949	−	0.0101093i	$-0.00321793\pi$
$998$		−	376.525i		−	0.377280i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.b.c.406.7		16
3.2	odd	2		165.3.b.a.76.10	yes	16
11.10	odd	2	inner	495.3.b.c.406.10		16
12.11	even	2		2640.3.c.c.241.2		16
15.2	even	4		825.3.h.b.274.13		32
15.8	even	4		825.3.h.b.274.19		32
15.14	odd	2		825.3.b.d.76.7		16
33.32	even	2		165.3.b.a.76.7	✓	16
132.131	odd	2		2640.3.c.c.241.3		16
165.32	odd	4		825.3.h.b.274.20		32
165.98	odd	4		825.3.h.b.274.14		32
165.164	even	2		825.3.b.d.76.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.3.b.a.76.7	✓	16	33.32	even	2
165.3.b.a.76.10	yes	16	3.2	odd	2
495.3.b.c.406.7		16	1.1	even	1	trivial
495.3.b.c.406.10		16	11.10	odd	2	inner
825.3.b.d.76.7		16	15.14	odd	2
825.3.b.d.76.10		16	165.164	even	2
825.3.h.b.274.13		32	15.2	even	4
825.3.h.b.274.14		32	165.98	odd	4
825.3.h.b.274.19		32	15.8	even	4
825.3.h.b.274.20		32	165.32	odd	4
2640.3.c.c.241.2		16	12.11	even	2
2640.3.c.c.241.3		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)

\(f(q)\)	\(=\)	\(q-0.664903i q^{2} +3.55790 q^{4} +2.23607 q^{5} +8.36247i q^{7} -5.02527i q^{8} -1.48677i q^{10} +(-7.04830 + 8.44521i) q^{11} +16.0778i q^{13} +5.56023 q^{14} +10.8903 q^{16} +22.0359i q^{17} -6.10810i q^{19} +7.95572 q^{20} +(5.61524 + 4.68643i) q^{22} -6.43313 q^{23} +5.00000 q^{25} +10.6902 q^{26} +29.7529i q^{28} -20.7699i q^{29} -52.4060 q^{31} -27.3421i q^{32} +14.6517 q^{34} +18.6990i q^{35} +23.9112 q^{37} -4.06129 q^{38} -11.2368i q^{40} +66.7913i q^{41} -56.5789i q^{43} +(-25.0772 + 30.0472i) q^{44} +4.27741i q^{46} +35.9964 q^{47} -20.9309 q^{49} -3.32451i q^{50} +57.2034i q^{52} +83.7803 q^{53} +(-15.7605 + 18.8841i) q^{55} +42.0237 q^{56} -13.8099 q^{58} +13.5741 q^{59} -120.042i q^{61} +34.8449i q^{62} +25.3814 q^{64} +35.9511i q^{65} +69.5457 q^{67} +78.4015i q^{68} +12.4330 q^{70} +69.1708 q^{71} +78.0255i q^{73} -15.8986i q^{74} -21.7320i q^{76} +(-70.6228 - 58.9412i) q^{77} -7.65867i q^{79} +24.3515 q^{80} +44.4097 q^{82} +5.69286i q^{83} +49.2737i q^{85} -37.6195 q^{86} +(42.4394 + 35.4196i) q^{88} -73.0710 q^{89} -134.450 q^{91} -22.8885 q^{92} -23.9341i q^{94} -13.6581i q^{95} -37.9109 q^{97} +13.9170i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} + 28 q^{11} - 16 q^{16} - 40 q^{20} - 20 q^{22} - 56 q^{23} + 80 q^{25} + 88 q^{26} - 96 q^{31} - 200 q^{34} + 184 q^{37} - 296 q^{38} - 300 q^{44} + 200 q^{47} - 496 q^{49} + 80 q^{53} + 20 q^{55}+ \cdots - 144 q^{97}+O(q^{100}) \) 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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