LMFDB - Embedded newform 495.3.b.c.406.10

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

$q$-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	$0.0531580\pi$
$2$			0.664903i			0.332451i	−0.986088	+	0.166226i	$0.946842\pi$
$3$		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	$-0.796233\pi$
$7$		−	8.36247i		−	1.19464i	0.802004	−	0.597319i	$-0.203767\pi$
$8$			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	$-0.787790\pi$
$13$		−	16.0778i		−	1.23676i	0.785881	−	0.618378i	$-0.212210\pi$
$14$	5.56023			0.397159
$15$		0			0
$16$	10.8903			0.680644
$17$		−	22.0359i		−	1.29623i	−0.761544	−	0.648114i	$-0.775558\pi$
$17$		−	22.0359i		−	1.29623i	0.761544	−	0.648114i	$-0.224442\pi$
$18$		0			0
$19$			6.10810i			0.321479i	0.986997	+	0.160739i	$0.0513879\pi$
$19$			6.10810i			0.321479i	−0.986997	+	0.160739i	$0.948612\pi$
$20$	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	$0.116576\pi$
$29$			20.7699i			0.716202i	−0.933683	+	0.358101i	$0.883424\pi$
$30$		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	$-0.696996\pi$
$41$		−	66.7913i		−	1.62906i	0.580124	−	0.814528i	$-0.303004\pi$
$42$		0			0
$43$			56.5789i			1.31579i	0.753110	+	0.657894i	$0.228553\pi$
$43$			56.5789i			1.31579i	−0.753110	+	0.657894i	$0.771447\pi$
$44$	−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	$0.442890\pi$
$61$			120.042i			1.96790i	−0.178455	+	0.983948i	$0.557110\pi$
$62$		−	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	$-0.820530\pi$
$73$		−	78.0255i		−	1.06884i	0.845218	−	0.534421i	$-0.179470\pi$
$74$			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	$0.0154353\pi$
$79$			7.65867i			0.0969451i	−0.998825	+	0.0484726i	$0.984565\pi$
$80$	24.3515			0.304393
$81$		0			0
$82$	44.4097			0.541582
$83$		−	5.69286i		−	0.0685887i	−0.999412	−	0.0342943i	$-0.989082\pi$
$83$		−	5.69286i		−	0.0685887i	0.999412	−	0.0342943i	$-0.0109184\pi$
$84$		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	$-0.966411\pi$
$101$		−	21.2764i		−	0.210658i	0.994437	−	0.105329i	$-0.0335895\pi$
$102$		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	$0.181645\pi$
$107$			115.599i			1.08037i	−0.841548	+	0.540183i	$0.818355\pi$
$108$		0			0
$109$		−	138.985i		−	1.27510i	−0.770411	−	0.637548i	$-0.779949\pi$
$109$		−	138.985i		−	1.27510i	0.770411	−	0.637548i	$-0.220051\pi$
$110$	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	$0.0105726\pi$
$127$			8.43502i			0.0664175i	−0.999448	+	0.0332087i	$0.989427\pi$
$128$			126.244i			0.986285i
$129$		0			0
$130$	23.9040			0.183877
$131$			186.122i			1.42078i	0.703811	+	0.710388i	$0.251480\pi$
$131$			186.122i			1.42078i	−0.703811	+	0.710388i	$0.748520\pi$
$132$		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	$0.102688\pi$
$139$			88.1363i			0.634074i	−0.948413	+	0.317037i	$0.897312\pi$
$140$		−	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	$0.0983327\pi$
$149$			90.6013i			0.608062i	−0.952662	+	0.304031i	$0.901667\pi$
$150$		0			0
$151$			38.1059i			0.252357i	0.992008	+	0.126178i	$0.0402712\pi$
$151$			38.1059i			0.252357i	−0.992008	+	0.126178i	$0.959729\pi$
$152$	−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	$0.172490\pi$
$167$			172.263i			1.03152i	−0.856734	+	0.515758i	$0.827510\pi$
$168$		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	$-0.882109\pi$
$173$		−	125.237i		−	0.723914i	0.932195	−	0.361957i	$-0.117891\pi$
$174$		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	$-0.865783\pi$
$193$		−	157.978i		−	0.818541i	0.912413	−	0.409271i	$-0.134217\pi$
$194$		−	25.2070i		−	0.129933i
$195$		0			0
$196$	−74.4700			−0.379949
$197$		−	203.131i		−	1.03112i	−0.856852	−	0.515562i	$-0.827583\pi$
$197$		−	203.131i		−	1.03112i	0.856852	−	0.515562i	$-0.172417\pi$
$198$		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	$-0.893286\pi$
$211$		−	138.840i		−	0.658012i	0.944328	−	0.329006i	$-0.106714\pi$
$212$	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	$0.183453\pi$
$227$			247.410i			1.08991i	−0.838465	+	0.544955i	$0.816547\pi$
$228$		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	$-0.808812\pi$
$233$		−	263.367i		−	1.13033i	0.824978	−	0.565165i	$-0.191188\pi$
$234$		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	$0.276333\pi$
$239$			364.771i			1.52624i	−0.646257	+	0.763120i	$0.723667\pi$
$240$		0			0
$241$			149.640i			0.620912i	0.950588	+	0.310456i	$0.100482\pi$
$241$			149.640i			0.620912i	−0.950588	+	0.310456i	$0.899518\pi$
$242$	−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	$0.0337828\pi$
$263$			55.7205i			0.211865i	−0.994373	+	0.105933i	$0.966217\pi$
$264$		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	$-0.952109\pi$
$271$		−	81.2380i		−	0.299771i	0.988703	−	0.149886i	$-0.0478906\pi$
$272$		−	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	$0.188368\pi$
$277$			309.041i			1.11567i	−0.829952	+	0.557835i	$0.811632\pi$
$278$	−58.6020			−0.210799
$279$		0			0
$280$	93.9678			0.335599
$281$			306.448i			1.09056i	0.838253	+	0.545282i	$0.183577\pi$
$281$			306.448i			1.09056i	−0.838253	+	0.545282i	$0.816423\pi$
$282$		0			0
$283$		−	495.935i		−	1.75242i	−0.481929	−	0.876210i	$-0.660064\pi$
$283$		−	495.935i		−	1.75242i	0.481929	−	0.876210i	$-0.339936\pi$
$284$	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	$-0.767089\pi$
$293$		−	391.532i		−	1.33629i	0.744032	−	0.668144i	$-0.232911\pi$
$294$		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	$-0.951756\pi$
$307$		−	92.7037i		−	0.301966i	0.988536	−	0.150983i	$-0.0482439\pi$
$308$	−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	$-0.961783\pi$
$337$		−	80.7287i		−	0.239551i	0.992801	−	0.119775i	$-0.0382175\pi$
$338$		−	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	$-0.951359\pi$
$347$		−	105.638i		−	0.304433i	0.988347	−	0.152217i	$-0.0486411\pi$
$348$		0			0
$349$			515.712i			1.47769i	0.673878	+	0.738843i	$0.264627\pi$
$349$			515.712i			1.47769i	−0.673878	+	0.738843i	$0.735373\pi$
$350$	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	$-0.875278\pi$
$359$		−	274.186i		−	0.763751i	0.924214	−	0.381875i	$-0.124722\pi$
$360$		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	$0.00778644\pi$
$373$			18.2467i			0.0489188i	−0.999701	+	0.0244594i	$0.992214\pi$
$374$	−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	$-0.675870\pi$
$409$		−	696.288i		−	1.70242i	0.524828	−	0.851208i	$-0.324130\pi$
$410$	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	$-0.916471\pi$
$431$		−	223.614i		−	0.518825i	0.965767	−	0.259413i	$-0.0835290\pi$
$432$		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	$0.0241125\pi$
$439$			66.4465i			0.151359i	−0.997132	+	0.0756794i	$0.975887\pi$
$440$	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	$0.0657488\pi$
$457$			187.453i			0.410181i	−0.978743	+	0.205090i	$0.934251\pi$
$458$			241.793i			0.527932i
$459$		0			0
$460$	−51.1802			−0.111261
$461$		−	683.006i		−	1.48157i	−0.671740	−	0.740787i	$-0.734453\pi$
$461$		−	683.006i		−	1.48157i	0.671740	−	0.740787i	$-0.265547\pi$
$462$		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	$0.224318\pi$
$479$			620.609i			1.29563i	−0.761796	+	0.647817i	$0.775682\pi$
$480$		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	$0.162850\pi$
$491$			480.768i			0.979161i	−0.871958	+	0.489580i	$0.837150\pi$
$492$		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	$-0.787619\pi$
$503$		−	622.513i		−	1.23760i	0.785549	−	0.618800i	$-0.212381\pi$
$504$		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	$0.121733\pi$
$523$			390.347i			0.746362i	−0.927759	+	0.373181i	$0.878267\pi$
$524$			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	$0.0127794\pi$
$541$			43.4279i			0.0802735i	−0.999194	+	0.0401367i	$0.987221\pi$
$542$	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	$-0.912098\pi$
$547$		−	298.284i		−	0.545309i	0.962112	−	0.272655i	$-0.0879016\pi$
$548$	−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	$0.117312\pi$
$557$			401.329i			0.720520i	−0.932852	+	0.360260i	$0.882688\pi$
$558$		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	$0.0541203\pi$
$563$			190.526i			0.338412i	−0.985581	+	0.169206i	$0.945880\pi$
$564$		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	$0.217548\pi$
$569$			718.614i			1.26294i	−0.775400	+	0.631471i	$0.782452\pi$
$570$		0			0
$571$			324.008i			0.567440i	0.958907	+	0.283720i	$0.0915686\pi$
$571$			324.008i			0.567440i	−0.958907	+	0.283720i	$0.908431\pi$
$572$	−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	$0.205190\pi$
$593$			712.663i			1.20179i	−0.799327	+	0.600896i	$0.794810\pi$
$594$		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	$0.0942074\pi$
$601$			350.575i			0.583319i	−0.956522	+	0.291659i	$0.905793\pi$
$602$			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	$-0.904546\pi$
$607$		−	358.620i		−	0.590808i	0.955372	−	0.295404i	$-0.0954542\pi$
$608$	−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	$-0.992673\pi$
$613$		−	28.2177i		−	0.0460322i	0.999735	−	0.0230161i	$-0.00732690\pi$
$614$	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	$-0.969120\pi$
$659$		−	127.662i		−	0.193721i	0.995298	−	0.0968607i	$-0.0308801\pi$
$660$		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	$0.250179\pi$
$673$			952.301i			1.41501i	−0.706709	+	0.707505i	$0.749821\pi$
$674$	53.6767			0.0796390
$675$		0			0
$676$	−318.420			−0.471035
$677$		−	258.470i		−	0.381788i	−0.981611	−	0.190894i	$-0.938861\pi$
$677$		−	258.470i		−	0.381788i	0.981611	−	0.190894i	$-0.0611387\pi$
$678$		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	$-0.958634\pi$
$701$		−	181.684i		−	0.259179i	0.991568	−	0.129589i	$-0.0413659\pi$
$702$		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	$-0.726218\pi$
$733$		−	1111.10i		−	1.51583i	0.652353	−	0.757915i	$-0.273782\pi$
$734$		−	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	$0.225851\pi$
$739$			962.884i			1.30296i	−0.758668	+	0.651478i	$0.774149\pi$
$740$	190.231			0.257069
$741$		0			0
$742$	465.838			0.627814
$743$			674.012i			0.907149i	0.891218	+	0.453574i	$0.149851\pi$
$743$			674.012i			0.907149i	−0.891218	+	0.453574i	$0.850149\pi$
$744$		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	$0.299401\pi$
$761$			1229.64i			1.61582i	−0.589305	+	0.807910i	$0.700599\pi$
$762$		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	$-0.706421\pi$
$769$		−	1225.78i		−	1.59399i	0.603985	−	0.796996i	$-0.293579\pi$
$770$	−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	$0.249137\pi$
$787$			1109.97i			1.41038i	−0.709021	+	0.705188i	$0.750863\pi$
$788$		−	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	$-0.834688\pi$
$809$		−	803.028i		−	0.992619i	0.868146	−	0.496309i	$-0.165312\pi$
$810$		0			0
$811$		−	323.073i		−	0.398364i	−0.979962	−	0.199182i	$-0.936171\pi$
$811$		−	323.073i		−	0.398364i	0.979962	−	0.199182i	$-0.0638285\pi$
$812$	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	$-0.932604\pi$
$821$		−	345.071i		−	0.420306i	0.977668	−	0.210153i	$-0.0673962\pi$
$822$		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	$0.297696\pi$
$827$			1331.04i			1.60948i	−0.593626	+	0.804741i	$0.702304\pi$
$828$		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	$0.0242543\pi$
$853$			129.866i			0.152247i	−0.997098	+	0.0761234i	$0.975746\pi$
$854$			667.459i			0.781568i
$855$		0			0
$856$	−580.917			−0.678641
$857$		−	419.114i		−	0.489048i	−0.969643	−	0.244524i	$-0.921368\pi$
$857$		−	419.114i		−	0.489048i	0.969643	−	0.244524i	$-0.0786317\pi$
$858$		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	$0.104014\pi$
$877$			563.009i			0.641971i	−0.947084	+	0.320986i	$0.895986\pi$
$878$	−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	$-0.649381\pi$
$887$		−	1582.21i		−	1.78378i	0.452257	−	0.891888i	$-0.350619\pi$
$888$		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	$-0.617972\pi$
$919$		−	1713.21i		−	1.86421i	0.362193	−	0.932103i	$-0.382028\pi$
$920$		−	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	$0.0548958\pi$
$937$			321.591i			0.343213i	−0.985166	+	0.171607i	$0.945104\pi$
$938$	386.690			0.412249
$939$		0			0
$940$	286.377			0.304657
$941$		−	880.847i		−	0.936076i	−0.883708	−	0.468038i	$-0.844961\pi$
$941$		−	880.847i		−	0.936076i	0.883708	−	0.468038i	$-0.155039\pi$
$942$		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	$-0.992255\pi$
$953$		−	46.3722i		−	0.0486592i	0.999704	−	0.0243296i	$-0.00774512\pi$
$954$		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	$-0.956389\pi$
$967$		−	264.144i		−	0.273158i	0.990629	−	0.136579i	$-0.0436108\pi$
$968$	−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	$0.00321793\pi$
$997$			20.1579i			0.0202185i	−0.999949	+	0.0101093i	$0.996782\pi$
$998$			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.406.10		16
3.2	odd	2		165.3.b.a.76.7	✓	16
11.10	odd	2	inner	495.3.b.c.406.7		16
12.11	even	2		2640.3.c.c.241.3		16
15.2	even	4		825.3.h.b.274.20		32
15.8	even	4		825.3.h.b.274.14		32
15.14	odd	2		825.3.b.d.76.10		16
33.32	even	2		165.3.b.a.76.10	yes	16
132.131	odd	2		2640.3.c.c.241.2		16
165.32	odd	4		825.3.h.b.274.13		32
165.98	odd	4		825.3.h.b.274.19		32
165.164	even	2		825.3.b.d.76.7		16

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

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