LMFDB - Embedded newform 495.3.b.c.406.12

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			406.12
Root			$1.15335i$ of defining polynomial
Character	$\chi$	$=$	495.406
Dual form			495.3.b.c.406.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.15335i q^{2} +2.66978 q^{4} -2.23607 q^{5} +12.5112i q^{7} +7.69260i q^{8} -2.57897i q^{10} +(-10.8148 - 2.00994i) q^{11} +3.74983i q^{13} -14.4298 q^{14} +1.80684 q^{16} -0.100823i q^{17} -21.0495i q^{19} -5.96981 q^{20} +(2.31817 - 12.4733i) q^{22} -30.1161 q^{23} +5.00000 q^{25} -4.32487 q^{26} +33.4021i q^{28} -10.4508i q^{29} +6.86373 q^{31} +32.8543i q^{32} +0.116284 q^{34} -27.9758i q^{35} -58.1617 q^{37} +24.2775 q^{38} -17.2012i q^{40} +49.2050i q^{41} +76.4814i q^{43} +(-28.8732 - 5.36611i) q^{44} -34.7344i q^{46} +63.7154 q^{47} -107.530 q^{49} +5.76676i q^{50} +10.0112i q^{52} +8.45006 q^{53} +(24.1827 + 4.49437i) q^{55} -96.2435 q^{56} +12.0535 q^{58} +20.5266 q^{59} +81.4502i q^{61} +7.91630i q^{62} -30.6652 q^{64} -8.38487i q^{65} +44.0966 q^{67} -0.269175i q^{68} +32.2660 q^{70} -44.1124 q^{71} -134.904i q^{73} -67.0809i q^{74} -56.1975i q^{76} +(25.1468 - 135.306i) q^{77} -66.2504i q^{79} -4.04022 q^{80} -56.7507 q^{82} +101.508i q^{83} +0.225447i q^{85} -88.2100 q^{86} +(15.4617 - 83.1940i) q^{88} +57.8175 q^{89} -46.9147 q^{91} -80.4032 q^{92} +73.4862i q^{94} +47.0681i q^{95} -44.4855 q^{97} -124.019i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			1.15335i			0.576676i	0.957529	+	0.288338i	$0.0931027\pi$
$2$			1.15335i			0.576676i	−0.957529	+	0.288338i	$0.906897\pi$
$3$		0			0
$3$		0			0
$4$	2.66978			0.667445
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$			12.5112i			1.78731i	0.448754	+	0.893655i	$0.351868\pi$
$7$			12.5112i			1.78731i	−0.448754	+	0.893655i	$0.648132\pi$
$8$			7.69260i			0.961575i
$9$		0			0
$10$		−	2.57897i		−	0.257897i
$11$	−10.8148	−	2.00994i	−0.983165	−	0.182722i
$11$	−10.8148	−	2.00994i	−0.983165	−	0.182722i
$12$		0			0
$13$			3.74983i			0.288448i	0.989545	+	0.144224i	$0.0460686\pi$
$13$			3.74983i			0.288448i	−0.989545	+	0.144224i	$0.953931\pi$
$14$	−14.4298			−1.03070
$15$		0			0
$16$	1.80684			0.112928
$17$		−	0.100823i		−	0.00593076i	−0.999996	−	0.00296538i	$-0.999056\pi$
$17$		−	0.100823i		−	0.00593076i	0.999996	−	0.00296538i	$-0.000943911\pi$
$18$		0			0
$19$		−	21.0495i		−	1.10787i	−0.832561	−	0.553934i	$-0.813126\pi$
$19$		−	21.0495i		−	1.10787i	0.832561	−	0.553934i	$-0.186874\pi$
$20$	−5.96981			−0.298490
$21$		0			0
$22$	2.31817	−	12.4733i	0.105371	−	0.566967i
$23$	−30.1161			−1.30939			−0.654697	−	0.755891i	$-0.727204\pi$
$23$	−30.1161			−1.30939			−0.654697	+	0.755891i	$0.727204\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$	−4.32487			−0.166341
$27$		0			0
$28$			33.4021i			1.19293i
$29$		−	10.4508i		−	0.360373i	−0.983632	−	0.180186i	$-0.942330\pi$
$29$		−	10.4508i		−	0.360373i	0.983632	−	0.180186i	$-0.0576701\pi$
$30$		0			0
$31$	6.86373			0.221411			0.110705	−	0.993853i	$-0.464689\pi$
$31$	6.86373			0.221411			0.110705	+	0.993853i	$0.464689\pi$
$32$			32.8543i			1.02670i
$33$		0			0
$34$	0.116284			0.00342013
$35$		−	27.9758i		−	0.799310i
$36$		0			0
$37$	−58.1617			−1.57194			−0.785968	−	0.618267i	$-0.787835\pi$
$37$	−58.1617			−1.57194			−0.785968	+	0.618267i	$0.787835\pi$
$38$	24.2775			0.638880
$39$		0			0
$40$		−	17.2012i		−	0.430030i
$41$			49.2050i			1.20012i	0.799954	+	0.600061i	$0.204857\pi$
$41$			49.2050i			1.20012i	−0.799954	+	0.600061i	$0.795143\pi$
$42$		0			0
$43$			76.4814i			1.77864i	0.457287	+	0.889319i	$0.348821\pi$
$43$			76.4814i			1.77864i	−0.457287	+	0.889319i	$0.651179\pi$
$44$	−28.8732	−	5.36611i	−0.656208	−	0.121957i
$45$		0			0
$46$		−	34.7344i		−	0.755096i
$47$	63.7154			1.35565			0.677823	−	0.735225i	$-0.262924\pi$
$47$	63.7154			1.35565			0.677823	+	0.735225i	$0.262924\pi$
$48$		0			0
$49$	−107.530			−2.19448
$50$			5.76676i			0.115335i
$51$		0			0
$52$			10.0112i			0.192523i
$53$	8.45006			0.159435			0.0797175	−	0.996817i	$-0.474598\pi$
$53$	8.45006			0.159435			0.0797175	+	0.996817i	$0.474598\pi$
$54$		0			0
$55$	24.1827	+	4.49437i	0.439685	+	0.0817159i
$56$	−96.2435			−1.71863
$57$		0			0
$58$	12.0535			0.207818
$59$	20.5266			0.347908			0.173954	−	0.984754i	$-0.444346\pi$
$59$	20.5266			0.347908			0.173954	+	0.984754i	$0.444346\pi$
$60$		0			0
$61$			81.4502i			1.33525i	0.744498	+	0.667624i	$0.232689\pi$
$61$			81.4502i			1.33525i	−0.744498	+	0.667624i	$0.767311\pi$
$62$			7.91630i			0.127682i
$63$		0			0
$64$	−30.6652			−0.479144
$65$		−	8.38487i		−	0.128998i
$66$		0			0
$67$	44.0966			0.658158			0.329079	−	0.944302i	$-0.393262\pi$
$67$	44.0966			0.658158			0.329079	+	0.944302i	$0.393262\pi$
$68$		−	0.269175i		−	0.00395845i
$69$		0			0
$70$	32.2660			0.460943
$71$	−44.1124			−0.621301			−0.310650	−	0.950524i	$-0.600547\pi$
$71$	−44.1124			−0.621301			−0.310650	+	0.950524i	$0.600547\pi$
$72$		0			0
$73$		−	134.904i		−	1.84799i	−0.382399	−	0.923997i	$-0.624902\pi$
$73$		−	134.904i		−	1.84799i	0.382399	−	0.923997i	$-0.375098\pi$
$74$		−	67.0809i		−	0.906498i
$75$		0			0
$76$		−	56.1975i		−	0.739440i
$77$	25.1468	−	135.306i	0.326581	−	1.75722i
$78$		0			0
$79$		−	66.2504i		−	0.838613i	−0.907845	−	0.419307i	$-0.862273\pi$
$79$		−	66.2504i		−	0.838613i	0.907845	−	0.419307i	$-0.137727\pi$
$80$	−4.04022			−0.0505028
$81$		0			0
$82$	−56.7507			−0.692081
$83$			101.508i			1.22299i	0.791247	+	0.611497i	$0.209432\pi$
$83$			101.508i			1.22299i	−0.791247	+	0.611497i	$0.790568\pi$
$84$		0			0
$85$			0.225447i			0.00265232i
$86$	−88.2100			−1.02570
$87$		0			0
$88$	15.4617	−	83.1940i	0.175701	−	0.945387i
$89$	57.8175			0.649634			0.324817	−	0.945777i	$-0.394697\pi$
$89$	57.8175			0.649634			0.324817	+	0.945777i	$0.394697\pi$
$90$		0			0
$91$	−46.9147			−0.515547
$92$	−80.4032			−0.873948
$93$		0			0
$94$			73.4862i			0.781768i
$95$			47.0681i			0.495453i
$96$		0			0
$97$	−44.4855			−0.458614			−0.229307	−	0.973354i	$-0.573646\pi$
$97$	−44.4855			−0.458614			−0.229307	+	0.973354i	$0.573646\pi$
$98$		−	124.019i		−	1.26550i
$99$		0			0
$100$	13.3489			0.133489
$101$			25.4637i			0.252116i	0.992023	+	0.126058i	$0.0402325\pi$
$101$			25.4637i			0.252116i	−0.992023	+	0.126058i	$0.959767\pi$
$102$		0			0
$103$	53.0663			0.515207			0.257603	−	0.966251i	$-0.417067\pi$
$103$	53.0663			0.515207			0.257603	+	0.966251i	$0.417067\pi$
$104$	−28.8459			−0.277365
$105$		0			0
$106$			9.74589i			0.0919423i
$107$			97.0812i			0.907301i	0.891180	+	0.453650i	$0.149878\pi$
$107$			97.0812i			0.907301i	−0.891180	+	0.453650i	$0.850122\pi$
$108$		0			0
$109$			64.1908i			0.588906i	0.955666	+	0.294453i	$0.0951375\pi$
$109$			64.1908i			0.588906i	−0.955666	+	0.294453i	$0.904863\pi$
$110$	−5.18359	+	27.8911i	−0.0471236	+	0.253555i
$111$		0			0
$112$			22.6057i			0.201837i
$113$	161.921			1.43293			0.716465	−	0.697623i	$-0.245759\pi$
$113$	161.921			1.43293			0.716465	+	0.697623i	$0.245759\pi$
$114$		0			0
$115$	67.3416			0.585579
$116$		−	27.9014i		−	0.240529i
$117$		0			0
$118$			23.6744i			0.200630i
$119$	1.26141			0.0106001
$120$		0			0
$121$	112.920	+	43.4743i	0.933225	+	0.359292i
$122$	−93.9407			−0.770006
$123$		0			0
$124$	18.3247			0.147779
$125$	−11.1803			−0.0894427
$126$		0			0
$127$		−	47.3863i		−	0.373120i	−0.982444	−	0.186560i	$-0.940266\pi$
$127$		−	47.3863i		−	0.373120i	0.982444	−	0.186560i	$-0.0597339\pi$
$128$			96.0495i			0.750387i
$129$		0			0
$130$	9.67070			0.0743900
$131$			218.239i			1.66594i	0.553315	+	0.832972i	$0.313362\pi$
$131$			218.239i			1.66594i	−0.553315	+	0.832972i	$0.686638\pi$
$132$		0			0
$133$	263.354			1.98010
$134$			50.8589i			0.379544i
$135$		0			0
$136$	0.775590			0.00570287
$137$	78.5537			0.573385			0.286692	−	0.958023i	$-0.407444\pi$
$137$	78.5537			0.573385			0.286692	+	0.958023i	$0.407444\pi$
$138$		0			0
$139$			60.3621i			0.434260i	0.976143	+	0.217130i	$0.0696695\pi$
$139$			60.3621i			0.434260i	−0.976143	+	0.217130i	$0.930331\pi$
$140$		−	74.6893i		−	0.533495i
$141$		0			0
$142$		−	50.8771i		−	0.358289i
$143$	7.53694	−	40.5537i	0.0527059	−	0.283592i
$144$		0			0
$145$			23.3687i			0.161164i
$146$	155.591			1.06569
$147$		0			0
$148$	−155.279			−1.04918
$149$		−	219.162i		−	1.47088i	−0.677588	−	0.735442i	$-0.736974\pi$
$149$		−	219.162i		−	1.47088i	0.677588	−	0.735442i	$-0.263026\pi$
$150$		0			0
$151$		−	36.3914i		−	0.241002i	−0.992713	−	0.120501i	$-0.961550\pi$
$151$		−	36.3914i		−	0.241002i	0.992713	−	0.120501i	$-0.0384502\pi$
$152$	161.925			1.06530
$153$		0			0
$154$	156.055	+	29.0031i	1.01335	+	0.188332i
$155$	−15.3478			−0.0990179
$156$		0			0
$157$	−146.443			−0.932757			−0.466378	−	0.884585i	$-0.654441\pi$
$157$	−146.443			−0.932757			−0.466378	+	0.884585i	$0.654441\pi$
$158$	76.4101			0.483608
$159$		0			0
$160$		−	73.4645i		−	0.459153i
$161$		−	376.787i		−	2.34029i
$162$		0			0
$163$	218.594			1.34107			0.670534	−	0.741879i	$-0.266065\pi$
$163$	218.594			1.34107			0.670534	+	0.741879i	$0.266065\pi$
$164$			131.366i			0.801015i
$165$		0			0
$166$	−117.075			−0.705271
$167$		−	109.361i		−	0.654857i	−0.944876	−	0.327428i	$-0.893818\pi$
$167$		−	109.361i		−	0.654857i	0.944876	−	0.327428i	$-0.106182\pi$
$168$		0			0
$169$	154.939			0.916798
$170$	−0.260019			−0.00152953
$171$		0			0
$172$			204.189i			1.18714i
$173$			86.0961i			0.497665i	0.968547	+	0.248833i	$0.0800469\pi$
$173$			86.0961i			0.497665i	−0.968547	+	0.248833i	$0.919953\pi$
$174$		0			0
$175$			62.5559i			0.357462i
$176$	−19.5407	−	3.63165i	−0.111026	−	0.0206344i
$177$		0			0
$178$			66.6839i			0.374628i
$179$	253.754			1.41762			0.708809	−	0.705400i	$-0.249233\pi$
$179$	253.754			1.41762			0.708809	+	0.705400i	$0.249233\pi$
$180$		0			0
$181$	−191.878			−1.06010			−0.530050	−	0.847966i	$-0.677827\pi$
$181$	−191.878			−1.06010			−0.530050	+	0.847966i	$0.677827\pi$
$182$		−	54.1092i		−	0.297303i
$183$		0			0
$184$		−	231.671i		−	1.25908i
$185$	130.053			0.702992
$186$		0			0
$187$	−0.202648	+	1.09038i	−0.00108368	+	0.00583091i
$188$	170.106			0.904819
$189$		0			0
$190$	−54.2860			−0.285716
$191$	105.312			0.551374			0.275687	−	0.961247i	$-0.411095\pi$
$191$	105.312			0.551374			0.275687	+	0.961247i	$0.411095\pi$
$192$		0			0
$193$		−	139.442i		−	0.722499i	−0.932469	−	0.361249i	$-0.882350\pi$
$193$		−	139.442i		−	0.722499i	0.932469	−	0.361249i	$-0.117650\pi$
$194$		−	51.3075i		−	0.264472i
$195$		0			0
$196$	−287.080			−1.46469
$197$			132.635i			0.673275i	0.941634	+	0.336638i	$0.109290\pi$
$197$			132.635i			0.673275i	−0.941634	+	0.336638i	$0.890710\pi$
$198$		0			0
$199$	37.5929			0.188909			0.0944545	−	0.995529i	$-0.469889\pi$
$199$	37.5929			0.188909			0.0944545	+	0.995529i	$0.469889\pi$
$200$			38.4630i			0.192315i
$201$		0			0
$202$	−29.3686			−0.145389
$203$	130.752			0.644098
$204$		0			0
$205$		−	110.026i		−	0.536711i
$206$			61.2041i			0.297107i
$207$		0			0
$208$			6.77535i			0.0325738i
$209$	−42.3083	+	227.646i	−0.202432	+	1.08922i
$210$		0			0
$211$			299.313i			1.41854i	0.704935	+	0.709272i	$0.250976\pi$
$211$			299.313i			1.41854i	−0.704935	+	0.709272i	$0.749024\pi$
$212$	22.5598			0.106414
$213$		0			0
$214$	−111.969			−0.523218
$215$		−	171.018i		−	0.795431i
$216$		0			0
$217$			85.8734i			0.395730i
$218$	−74.0345			−0.339608
$219$		0			0
$220$	64.5624	+	11.9990i	0.293465	+	0.0545408i
$221$	0.378068			0.00171072
$222$		0			0
$223$	372.747			1.67151			0.835756	−	0.549101i	$-0.185030\pi$
$223$	372.747			1.67151			0.835756	+	0.549101i	$0.185030\pi$
$224$	−411.046			−1.83503
$225$		0			0
$226$			186.752i			0.826336i
$227$			142.172i			0.626309i	0.949702	+	0.313154i	$0.101386\pi$
$227$			142.172i			0.626309i	−0.949702	+	0.313154i	$0.898614\pi$
$228$		0			0
$229$	−238.078			−1.03964			−0.519820	−	0.854276i	$-0.674001\pi$
$229$	−238.078			−1.03964			−0.519820	+	0.854276i	$0.674001\pi$
$230$			77.6685i			0.337689i
$231$		0			0
$232$	80.3939			0.346525
$233$			236.432i			1.01473i	0.861732	+	0.507365i	$0.169380\pi$
$233$			236.432i			1.01473i	−0.861732	+	0.507365i	$0.830620\pi$
$234$		0			0
$235$	−142.472			−0.606263
$236$	54.8014			0.232209
$237$		0			0
$238$			1.45485i			0.00611283i
$239$		−	357.930i		−	1.49762i	−0.662787	−	0.748808i	$-0.730626\pi$
$239$		−	357.930i		−	1.49762i	0.662787	−	0.748808i	$-0.269374\pi$
$240$		0			0
$241$		−	126.973i		−	0.526861i	−0.964678	−	0.263430i	$-0.915146\pi$
$241$		−	126.973i		−	0.526861i	0.964678	−	0.263430i	$-0.0848539\pi$
$242$	−50.1412	+	130.237i	−0.207195	+	0.538168i
$243$		0			0
$244$			217.454i			0.891205i
$245$	240.443			0.981401
$246$		0			0
$247$	78.9319			0.319562
$248$			52.8000i			0.212903i
$249$		0			0
$250$		−	12.8949i		−	0.0515795i
$251$	−219.707			−0.875325			−0.437663	−	0.899139i	$-0.644194\pi$
$251$	−219.707			−0.875325			−0.437663	+	0.899139i	$0.644194\pi$
$252$		0			0
$253$	325.699	+	60.5316i	1.28735	+	0.239255i
$254$	54.6531			0.215170
$255$		0			0
$256$	−233.440			−0.911874
$257$	−428.677			−1.66800			−0.834001	−	0.551763i	$-0.813955\pi$
$257$	−428.677			−1.66800			−0.834001	+	0.551763i	$0.813955\pi$
$258$		0			0
$259$		−	727.671i		−	2.80954i
$260$		−	22.3858i		−	0.0860990i
$261$		0			0
$262$	−251.706			−0.960710
$263$		−	252.442i		−	0.959855i	−0.877308	−	0.479928i	$-0.840663\pi$
$263$		−	252.442i		−	0.959855i	0.877308	−	0.479928i	$-0.159337\pi$
$264$		0			0
$265$	−18.8949			−0.0713015
$266$			303.739i			1.14188i
$267$		0			0
$268$	117.728			0.439284
$269$	176.094			0.654624			0.327312	−	0.944916i	$-0.393857\pi$
$269$	176.094			0.654624			0.327312	+	0.944916i	$0.393857\pi$
$270$		0			0
$271$		−	301.498i		−	1.11254i	−0.831003	−	0.556269i	$-0.812233\pi$
$271$		−	301.498i		−	1.11254i	0.831003	−	0.556269i	$-0.187767\pi$
$272$		−	0.182171i		−	0.000669747i
$273$		0			0
$274$			90.6001i			0.330657i
$275$	−54.0741	−	10.0497i	−0.196633	−	0.0365444i
$276$		0			0
$277$		−	100.640i		−	0.363322i	−0.983361	−	0.181661i	$-0.941853\pi$
$277$		−	100.640i		−	0.363322i	0.983361	−	0.181661i	$-0.0581474\pi$
$278$	−69.6187			−0.250427
$279$		0			0
$280$	215.207			0.768596
$281$			163.042i			0.580222i	0.956993	+	0.290111i	$0.0936922\pi$
$281$			163.042i			0.580222i	−0.956993	+	0.290111i	$0.906308\pi$
$282$		0			0
$283$			555.179i			1.96176i	0.194603	+	0.980882i	$0.437658\pi$
$283$			555.179i			1.96176i	−0.194603	+	0.980882i	$0.562342\pi$
$284$	−117.770			−0.414684
$285$		0			0
$286$	46.7726	+	8.69275i	0.163541	+	0.0303942i
$287$	−615.612			−2.14499
$288$		0			0
$289$	288.990			0.999965
$290$	−26.9524			−0.0929391
$291$		0			0
$292$		−	360.163i		−	1.23343i
$293$		−	404.729i		−	1.38133i	−0.723177	−	0.690663i	$-0.757319\pi$
$293$		−	404.729i		−	1.38133i	0.723177	−	0.690663i	$-0.242681\pi$
$294$		0			0
$295$	−45.8988			−0.155589
$296$		−	447.415i		−	1.51154i
$297$		0			0
$298$	252.771			0.848224
$299$		−	112.930i		−	0.377692i
$300$		0			0
$301$	−956.873			−3.17898
$302$	41.9720			0.138980
$303$		0			0
$304$		−	38.0331i		−	0.125109i
$305$		−	182.128i		−	0.597141i
$306$		0			0
$307$			225.744i			0.735323i	0.929960	+	0.367662i	$0.119842\pi$
$307$			225.744i			0.735323i	−0.929960	+	0.367662i	$0.880158\pi$
$308$	67.1363	−	361.237i	0.217975	−	1.17285i
$309$		0			0
$310$		−	17.7014i		−	0.0571012i
$311$	95.4488			0.306909			0.153455	−	0.988156i	$-0.450960\pi$
$311$	95.4488			0.306909			0.153455	+	0.988156i	$0.450960\pi$
$312$		0			0
$313$	65.9274			0.210631			0.105315	−	0.994439i	$-0.466415\pi$
$313$	65.9274			0.210631			0.105315	+	0.994439i	$0.466415\pi$
$314$		−	168.900i		−	0.537898i
$315$		0			0
$316$		−	176.874i		−	0.559728i
$317$	−277.442			−0.875211			−0.437605	−	0.899167i	$-0.644173\pi$
$317$	−277.442			−0.875211			−0.437605	+	0.899167i	$0.644173\pi$
$318$		0			0
$319$	−21.0055	+	113.024i	−0.0658481	+	0.354306i
$320$	68.5695			0.214280
$321$		0			0
$322$	434.568			1.34959
$323$	−2.12227			−0.00657049
$324$		0			0
$325$			18.7491i			0.0576897i
$326$			252.116i			0.773361i
$327$		0			0
$328$	−378.514			−1.15401
$329$			797.154i			2.42296i
$330$		0			0
$331$	277.559			0.838546			0.419273	−	0.907860i	$-0.362285\pi$
$331$	277.559			0.838546			0.419273	+	0.907860i	$0.362285\pi$
$332$			271.005i			0.816281i
$333$		0			0
$334$	126.132			0.377640
$335$	−98.6030			−0.294337
$336$		0			0
$337$		−	7.58975i		−	0.0225215i	−0.999937	−	0.0112608i	$-0.996416\pi$
$337$		−	7.58975i		−	0.0225215i	0.999937	−	0.0112608i	$-0.00358449\pi$
$338$			178.699i			0.528695i
$339$		0			0
$340$			0.601893i			0.00177027i
$341$	−74.2300	−	13.7957i	−0.217683	−	0.0404567i
$342$		0			0
$343$		−	732.273i		−	2.13491i
$344$	−588.341			−1.71029
$345$		0			0
$346$	−99.2991			−0.286992
$347$		−	114.576i		−	0.330190i	−0.986278	−	0.165095i	$-0.947207\pi$
$347$		−	114.576i		−	0.330190i	0.986278	−	0.165095i	$-0.0527930\pi$
$348$		0			0
$349$			559.957i			1.60446i	0.597015	+	0.802230i	$0.296353\pi$
$349$			559.957i			1.60446i	−0.597015	+	0.802230i	$0.703647\pi$
$350$	−72.1489			−0.206140
$351$		0			0
$352$	66.0354	−	355.313i	0.187601	−	1.00941i
$353$	−99.8950			−0.282989			−0.141494	−	0.989939i	$-0.545191\pi$
$353$	−99.8950			−0.282989			−0.141494	+	0.989939i	$0.545191\pi$
$354$		0			0
$355$	98.6382			0.277854
$356$	154.360			0.433595
$357$		0			0
$358$			292.667i			0.817506i
$359$			326.583i			0.909702i	0.890568	+	0.454851i	$0.150307\pi$
$359$			326.583i			0.909702i	−0.890568	+	0.454851i	$0.849693\pi$
$360$		0			0
$361$	−82.0805			−0.227370
$362$		−	221.303i		−	0.611335i
$363$		0			0
$364$	−125.252			−0.344099
$365$			301.654i			0.826448i
$366$		0			0
$367$	29.5143			0.0804203			0.0402102	−	0.999191i	$-0.487197\pi$
$367$	29.5143			0.0804203			0.0402102	+	0.999191i	$0.487197\pi$
$368$	−54.4150			−0.147867
$369$		0			0
$370$			149.997i			0.405398i
$371$			105.720i			0.284960i
$372$		0			0
$373$		−	589.241i		−	1.57973i	−0.613279	−	0.789867i	$-0.710150\pi$
$373$		−	589.241i		−	1.57973i	0.613279	−	0.789867i	$-0.289850\pi$
$374$	−1.25759	−	0.233725i	−0.00336255	−	0.000624933i
$375$		0			0
$376$			490.137i			1.30356i
$377$	39.1887			0.103949
$378$		0			0
$379$	−307.718			−0.811921			−0.405961	−	0.913891i	$-0.633063\pi$
$379$	−307.718			−0.811921			−0.405961	+	0.913891i	$0.633063\pi$
$380$			125.661i			0.330688i
$381$		0			0
$382$			121.462i			0.317964i
$383$	−8.26040			−0.0215676			−0.0107838	−	0.999942i	$-0.503433\pi$
$383$	−8.26040			−0.0215676			−0.0107838	+	0.999942i	$0.503433\pi$
$384$		0			0
$385$	−56.2299	+	302.553i	−0.146052	+	0.785853i
$386$	160.826			0.416648
$387$		0			0
$388$	−118.767			−0.306099
$389$	335.772			0.863166			0.431583	−	0.902073i	$-0.357955\pi$
$389$	335.772			0.863166			0.431583	+	0.902073i	$0.357955\pi$
$390$		0			0
$391$			3.03639i			0.00776570i
$392$		−	827.182i		−	2.11016i
$393$		0			0
$394$	−152.975			−0.388261
$395$			148.140i			0.375039i
$396$		0			0
$397$	43.7968			0.110319			0.0551596	−	0.998478i	$-0.482433\pi$
$397$	43.7968			0.110319			0.0551596	+	0.998478i	$0.482433\pi$
$398$			43.3578i			0.108939i
$399$		0			0
$400$	9.03421			0.0225855
$401$	−510.351			−1.27269			−0.636347	−	0.771403i	$-0.719555\pi$
$401$	−510.351			−1.27269			−0.636347	+	0.771403i	$0.719555\pi$
$402$		0			0
$403$			25.7378i			0.0638655i
$404$			67.9824i			0.168273i
$405$		0			0
$406$			150.803i			0.371436i
$407$	629.007	+	116.902i	1.54547	+	0.287228i
$408$		0			0
$409$			332.673i			0.813382i	0.913566	+	0.406691i	$0.133317\pi$
$409$			332.673i			0.813382i	−0.913566	+	0.406691i	$0.866683\pi$
$410$	126.898			0.309508
$411$		0			0
$412$	141.675			0.343872
$413$			256.812i			0.621820i
$414$		0			0
$415$		−	226.980i		−	0.546939i
$416$	−123.198			−0.296149
$417$		0			0
$418$	−262.556	−	48.7963i	−0.628125	−	0.116738i
$419$	303.619			0.724628			0.362314	−	0.932056i	$-0.381987\pi$
$419$	303.619			0.724628			0.362314	+	0.932056i	$0.381987\pi$
$420$		0			0
$421$	4.25383			0.0101041			0.00505206	−	0.999987i	$-0.498392\pi$
$421$	4.25383			0.0101041			0.00505206	+	0.999987i	$0.498392\pi$
$422$	−345.213			−0.818040
$423$		0			0
$424$			65.0029i			0.153309i
$425$		−	0.504114i		−	0.00118615i
$426$		0			0
$427$	−1019.04			−2.38650
$428$			259.185i			0.605573i
$429$		0			0
$430$	197.244			0.458706
$431$		−	783.951i		−	1.81891i	−0.415800	−	0.909456i	$-0.636498\pi$
$431$		−	783.951i		−	1.81891i	0.415800	−	0.909456i	$-0.363502\pi$
$432$		0			0
$433$	−767.681			−1.77294			−0.886468	−	0.462790i	$-0.846848\pi$
$433$	−767.681			−1.77294			−0.886468	+	0.462790i	$0.846848\pi$
$434$	−99.0422			−0.228208
$435$		0			0
$436$			171.375i			0.393062i
$437$			633.927i			1.45063i
$438$		0			0
$439$			221.039i			0.503506i	0.967792	+	0.251753i	$0.0810070\pi$
$439$			221.039i			0.503506i	−0.967792	+	0.251753i	$0.918993\pi$
$440$	−34.5734	+	186.028i	−0.0785759	+	0.422790i
$441$		0			0
$442$			0.436046i			0.000986529i
$443$	−334.048			−0.754058			−0.377029	−	0.926201i	$-0.623054\pi$
$443$	−334.048			−0.754058			−0.377029	+	0.926201i	$0.623054\pi$
$444$		0			0
$445$	−129.284			−0.290525
$446$			429.909i			0.963921i
$447$		0			0
$448$		−	383.658i		−	0.856380i
$449$	−382.507			−0.851909			−0.425954	−	0.904745i	$-0.640062\pi$
$449$	−382.507			−0.851909			−0.425954	+	0.904745i	$0.640062\pi$
$450$		0			0
$451$	98.8993	−	532.143i	0.219289	−	1.17992i
$452$	432.294			0.956402
$453$		0			0
$454$	−163.974			−0.361177
$455$	104.905			0.230559
$456$		0			0
$457$			683.546i			1.49573i	0.663854	+	0.747863i	$0.268920\pi$
$457$			683.546i			1.49573i	−0.663854	+	0.747863i	$0.731080\pi$
$458$		−	274.587i		−	0.599536i
$459$		0			0
$460$	179.787			0.390842
$461$		−	343.807i		−	0.745785i	−0.927874	−	0.372893i	$-0.878366\pi$
$461$		−	343.807i		−	0.745785i	0.927874	−	0.372893i	$-0.121634\pi$
$462$		0			0
$463$	90.9035			0.196336			0.0981679	−	0.995170i	$-0.468702\pi$
$463$	90.9035			0.196336			0.0981679	+	0.995170i	$0.468702\pi$
$464$		−	18.8830i		−	0.0406960i
$465$		0			0
$466$	−272.689			−0.585170
$467$	605.618			1.29683			0.648413	−	0.761289i	$-0.275433\pi$
$467$	605.618			1.29683			0.648413	+	0.761289i	$0.275433\pi$
$468$		0			0
$469$			551.700i			1.17633i
$470$		−	164.320i		−	0.349617i
$471$		0			0
$472$			157.903i			0.334540i
$473$	153.723	−	827.132i	0.324997	−	1.74869i
$474$		0			0
$475$		−	105.247i		−	0.221573i
$476$	3.36769			0.00707499
$477$		0			0
$478$	412.820			0.863639
$479$		−	218.798i		−	0.456781i	−0.973570	−	0.228390i	$-0.926654\pi$
$479$		−	218.798i		−	0.456781i	0.973570	−	0.228390i	$-0.0733463\pi$
$480$		0			0
$481$		−	218.096i		−	0.453422i
$482$	146.445			0.303828
$483$		0			0
$484$	301.472	+	116.067i	0.622876	+	0.239808i
$485$	99.4727			0.205098
$486$		0			0
$487$	944.923			1.94029			0.970147	−	0.242519i	$-0.0779736\pi$
$487$	944.923			1.94029			0.970147	+	0.242519i	$0.0779736\pi$
$488$	−626.564			−1.28394
$489$		0			0
$490$			277.316i			0.565950i
$491$		−	367.168i		−	0.747797i	−0.927470	−	0.373898i	$-0.878021\pi$
$491$		−	367.168i		−	0.747797i	0.927470	−	0.373898i	$-0.121979\pi$
$492$		0			0
$493$	−1.05368			−0.00213728
$494$			91.0363i			0.184284i
$495$		0			0
$496$	12.4017			0.0250034
$497$		−	551.897i		−	1.11046i
$498$		0			0
$499$	−345.580			−0.692545			−0.346272	−	0.938134i	$-0.612553\pi$
$499$	−345.580			−0.692545			−0.346272	+	0.938134i	$0.612553\pi$
$500$	−29.8490			−0.0596981
$501$		0			0
$502$		−	253.399i		−	0.504779i
$503$			118.948i			0.236477i	0.992985	+	0.118238i	$0.0377247\pi$
$503$			118.948i			0.236477i	−0.992985	+	0.118238i	$0.962275\pi$
$504$		0			0
$505$		−	56.9385i		−	0.112750i
$506$	−69.8142	+	375.646i	−0.137973	+	0.742383i
$507$		0			0
$508$		−	126.511i		−	0.249037i
$509$	−479.729			−0.942493			−0.471246	−	0.882002i	$-0.656196\pi$
$509$	−479.729			−0.942493			−0.471246	+	0.882002i	$0.656196\pi$
$510$		0			0
$511$	1687.80			3.30294
$512$			114.960i			0.224531i
$513$		0			0
$514$		−	494.415i		−	0.961897i
$515$	−118.660			−0.230408
$516$		0			0
$517$	−689.070	−	128.064i	−1.33282	−	0.247707i
$518$	839.260			1.62019
$519$		0			0
$520$	64.5015			0.124041
$521$	653.167			1.25368			0.626840	−	0.779148i	$-0.284348\pi$
$521$	653.167			1.25368			0.626840	+	0.779148i	$0.284348\pi$
$522$		0			0
$523$			587.477i			1.12328i	0.827381	+	0.561641i	$0.189830\pi$
$523$			587.477i			1.12328i	−0.827381	+	0.561641i	$0.810170\pi$
$524$			582.649i			1.11193i
$525$		0			0
$526$	291.154			0.553525
$527$		−	0.692021i		−	0.00131313i
$528$		0			0
$529$	377.977			0.714512
$530$		−	21.7925i		−	0.0411179i
$531$		0			0
$532$	703.096			1.32161
$533$	−184.510			−0.346173
$534$		0			0
$535$		−	217.080i		−	0.405757i
$536$			339.217i			0.632868i
$537$		0			0
$538$			203.098i			0.377506i
$539$	1162.91	+	216.128i	2.15753	+	0.400980i
$540$		0			0
$541$		−	633.500i		−	1.17098i	−0.810680	−	0.585490i	$-0.800902\pi$
$541$		−	633.500i		−	1.17098i	0.810680	−	0.585490i	$-0.199098\pi$
$542$	347.733			0.641573
$543$		0			0
$544$	3.31247			0.00608910
$545$		−	143.535i		−	0.263367i
$546$		0			0
$547$		−	402.570i		−	0.735960i	−0.929834	−	0.367980i	$-0.880049\pi$
$547$		−	402.570i		−	0.735960i	0.929834	−	0.367980i	$-0.119951\pi$
$548$	209.721			0.382703
$549$		0			0
$550$	11.5909	−	62.3664i	0.0210743	−	0.113393i
$551$	−219.984			−0.399245
$552$		0			0
$553$	828.871			1.49886
$554$	116.074			0.209519
$555$		0			0
$556$			161.153i			0.289844i
$557$			889.872i			1.59762i	0.601586	+	0.798808i	$0.294536\pi$
$557$			889.872i			1.59762i	−0.601586	+	0.798808i	$0.705464\pi$
$558$		0			0
$559$	−286.792			−0.513045
$560$		−	50.5479i		−	0.0902642i
$561$		0			0
$562$	−188.045			−0.334600
$563$			486.498i			0.864117i	0.901846	+	0.432058i	$0.142213\pi$
$563$			486.498i			0.864117i	−0.901846	+	0.432058i	$0.857787\pi$
$564$		0			0
$565$	−362.067			−0.640826
$566$	−640.317			−1.13130
$567$		0			0
$568$		−	339.339i		−	0.597427i
$569$		−	270.054i		−	0.474611i	−0.971435	−	0.237306i	$-0.923736\pi$
$569$		−	270.054i		−	0.474611i	0.971435	−	0.237306i	$-0.0762643\pi$
$570$		0			0
$571$			632.947i			1.10849i	0.832354	+	0.554244i	$0.186993\pi$
$571$			632.947i			1.10849i	−0.832354	+	0.554244i	$0.813007\pi$
$572$	20.1220	−	108.269i	0.0351783	−	0.189282i
$573$		0			0
$574$		−	710.018i		−	1.23696i
$575$	−150.580			−0.261879
$576$		0			0
$577$	−787.637			−1.36505			−0.682527	−	0.730860i	$-0.739119\pi$
$577$	−787.637			−1.36505			−0.682527	+	0.730860i	$0.739119\pi$
$578$			333.307i			0.576656i
$579$		0			0
$580$			62.3893i			0.107568i
$581$	−1269.99			−2.18587
$582$		0			0
$583$	−91.3858	−	16.9841i	−0.156751	−	0.0291323i
$584$	1037.76			1.77699
$585$		0			0
$586$	466.794			0.796577
$587$	279.624			0.476361			0.238180	−	0.971221i	$-0.423449\pi$
$587$	279.624			0.476361			0.238180	+	0.971221i	$0.423449\pi$
$588$		0			0
$589$		−	144.478i		−	0.245294i
$590$		−	52.9375i		−	0.0897245i
$591$		0			0
$592$	−105.089			−0.177515
$593$		−	221.094i		−	0.372840i	−0.982470	−	0.186420i	$-0.940311\pi$
$593$		−	221.094i		−	0.372840i	0.982470	−	0.186420i	$-0.0596886\pi$
$594$		0			0
$595$	−2.82060			−0.00474051
$596$		−	585.114i		−	0.981734i
$597$		0			0
$598$	130.248			0.217806
$599$	119.880			0.200133			0.100066	−	0.994981i	$-0.468094\pi$
$599$	119.880			0.200133			0.100066	+	0.994981i	$0.468094\pi$
$600$		0			0
$601$		−	435.414i		−	0.724483i	−0.932084	−	0.362242i	$-0.882012\pi$
$601$		−	435.414i		−	0.724483i	0.932084	−	0.362242i	$-0.117988\pi$
$602$		−	1103.61i		−	1.83324i
$603$		0			0
$604$		−	97.1569i		−	0.160856i
$605$	−252.497	−	97.2116i	−0.417351	−	0.160680i
$606$		0			0
$607$			470.310i			0.774811i	0.921909	+	0.387406i	$0.126629\pi$
$607$			470.310i			0.774811i	−0.921909	+	0.387406i	$0.873371\pi$
$608$	691.567			1.13745
$609$		0			0
$610$	210.058			0.344357
$611$			238.922i			0.391034i
$612$		0			0
$613$			376.345i			0.613940i	0.951719	+	0.306970i	$0.0993151\pi$
$613$			376.345i			0.613940i	−0.951719	+	0.306970i	$0.900685\pi$
$614$	−260.363			−0.424043
$615$		0			0
$616$	1040.86	+	193.444i	1.68970	+	0.314033i
$617$	−266.016			−0.431145			−0.215572	−	0.976488i	$-0.569162\pi$
$617$	−266.016			−0.431145			−0.215572	+	0.976488i	$0.569162\pi$
$618$		0			0
$619$	988.237			1.59651			0.798253	−	0.602322i	$-0.205758\pi$
$619$	988.237			1.59651			0.798253	+	0.602322i	$0.205758\pi$
$620$	−40.9752			−0.0660890
$621$		0			0
$622$			110.086i			0.176987i
$623$			723.364i			1.16110i
$624$		0			0
$625$	25.0000			0.0400000
$626$			76.0375i			0.121466i
$627$		0			0
$628$	−390.970			−0.622564
$629$			5.86403i			0.00932278i
$630$		0			0
$631$	102.951			0.163155			0.0815774	−	0.996667i	$-0.474004\pi$
$631$	102.951			0.163155			0.0815774	+	0.996667i	$0.474004\pi$
$632$	509.638			0.806390
$633$		0			0
$634$		−	319.988i		−	0.504713i
$635$			105.959i			0.166864i
$636$		0			0
$637$		−	403.217i		−	0.632994i
$638$	−130.356	−	24.2268i	−0.204320	−	0.0379730i
$639$		0			0
$640$		−	214.773i		−	0.335583i
$641$	704.065			1.09839			0.549193	−	0.835696i	$-0.314935\pi$
$641$	704.065			1.09839			0.549193	+	0.835696i	$0.314935\pi$
$642$		0			0
$643$	−899.353			−1.39868			−0.699341	−	0.714788i	$-0.746523\pi$
$643$	−899.353			−1.39868			−0.699341	+	0.714788i	$0.746523\pi$
$644$		−	1005.94i		−	1.56202i
$645$		0			0
$646$		−	2.44772i		−	0.00378904i
$647$	166.435			0.257241			0.128621	−	0.991694i	$-0.458945\pi$
$647$	166.435			0.257241			0.128621	+	0.991694i	$0.458945\pi$
$648$		0			0
$649$	−221.991	−	41.2573i	−0.342051	−	0.0635705i
$650$	−21.6243			−0.0332682
$651$		0			0
$652$	583.598			0.895089
$653$	−709.328			−1.08626			−0.543130	−	0.839648i	$-0.682761\pi$
$653$	−709.328			−1.08626			−0.543130	+	0.839648i	$0.682761\pi$
$654$		0			0
$655$		−	487.996i		−	0.745033i
$656$			88.9057i			0.135527i
$657$		0			0
$658$	−919.399			−1.39726
$659$		−	1058.83i		−	1.60672i	−0.595494	−	0.803360i	$-0.703044\pi$
$659$		−	1058.83i		−	1.60672i	0.595494	−	0.803360i	$-0.296956\pi$
$660$		0			0
$661$	−679.552			−1.02807			−0.514033	−	0.857770i	$-0.671849\pi$
$661$	−679.552			−1.02807			−0.514033	+	0.857770i	$0.671849\pi$
$662$			320.123i			0.483569i
$663$		0			0
$664$	−780.864			−1.17600
$665$	−588.877			−0.885529
$666$		0			0
$667$			314.737i			0.471870i
$668$		−	291.970i		−	0.437081i
$669$		0			0
$670$		−	113.724i		−	0.169737i
$671$	163.710	−	880.868i	0.243980	−	1.31277i
$672$		0			0
$673$		−	183.626i		−	0.272846i	−0.990651	−	0.136423i	$-0.956439\pi$
$673$		−	183.626i		−	0.272846i	0.990651	−	0.136423i	$-0.0435607\pi$
$674$	8.75365			0.0129876
$675$		0			0
$676$	413.652			0.611912
$677$			369.554i			0.545870i	0.962032	+	0.272935i	$0.0879945\pi$
$677$			369.554i			0.545870i	−0.962032	+	0.272935i	$0.912006\pi$
$678$		0			0
$679$		−	556.566i		−	0.819685i
$680$	−1.73427			−0.00255040
$681$		0			0
$682$	15.9113	−	85.6133i	0.0233304	−	0.125533i
$683$	1234.02			1.80677			0.903385	−	0.428831i	$-0.141074\pi$
$683$	1234.02			1.80677			0.903385	+	0.428831i	$0.141074\pi$
$684$		0			0
$685$	−175.651			−0.256426
$686$	844.568			1.23115
$687$		0			0
$688$			138.190i			0.200857i
$689$			31.6863i			0.0459888i
$690$		0			0
$691$	176.079			0.254818			0.127409	−	0.991850i	$-0.459334\pi$
$691$	176.079			0.254818			0.127409	+	0.991850i	$0.459334\pi$
$692$			229.858i			0.332164i
$693$		0			0
$694$	132.146			0.190412
$695$		−	134.974i		−	0.194207i
$696$		0			0
$697$	4.96099			0.00711763
$698$	−645.827			−0.925254
$699$		0			0
$700$			167.010i			0.238586i
$701$			622.951i			0.888660i	0.895863	+	0.444330i	$0.146558\pi$
$701$			622.951i			0.888660i	−0.895863	+	0.444330i	$0.853442\pi$
$702$		0			0
$703$			1224.27i			1.74150i
$704$	331.639	+	61.6354i	0.471078	+	0.0875503i
$705$		0			0
$706$		−	115.214i		−	0.163193i
$707$	−318.581			−0.450609
$708$		0			0
$709$	639.807			0.902408			0.451204	−	0.892421i	$-0.350995\pi$
$709$	639.807			0.902408			0.451204	+	0.892421i	$0.350995\pi$
$710$			113.765i			0.160232i
$711$		0			0
$712$			444.767i			0.624672i
$713$	−206.709			−0.289914
$714$		0			0
$715$	−16.8531	+	90.6808i	−0.0235708	+	0.126826i
$716$	677.466			0.946182
$717$		0			0
$718$	−376.665			−0.524603
$719$	−659.212			−0.916846			−0.458423	−	0.888734i	$-0.651586\pi$
$719$	−659.212			−0.916846			−0.458423	+	0.888734i	$0.651586\pi$
$720$		0			0
$721$			663.922i			0.920835i
$722$		−	94.6677i		−	0.131119i
$723$		0			0
$724$	−512.273			−0.707559
$725$		−	52.2540i		−	0.0720745i
$726$		0			0
$727$	−778.488			−1.07082			−0.535411	−	0.844592i	$-0.679843\pi$
$727$	−778.488			−1.07082			−0.535411	+	0.844592i	$0.679843\pi$
$728$		−	360.896i		−	0.495737i
$729$		0			0
$730$	−347.913			−0.476593
$731$	7.71108			0.0105487
$732$		0			0
$733$			449.200i			0.612825i	0.951899	+	0.306412i	$0.0991286\pi$
$733$			449.200i			0.612825i	−0.951899	+	0.306412i	$0.900871\pi$
$734$			34.0403i			0.0463764i
$735$		0			0
$736$		−	989.443i		−	1.34435i
$737$	−476.896	−	88.6317i	−0.647078	−	0.120260i
$738$		0			0
$739$			577.184i			0.781034i	0.920596	+	0.390517i	$0.127704\pi$
$739$			577.184i			0.781034i	−0.920596	+	0.390517i	$0.872296\pi$
$740$	347.214			0.469208
$741$		0			0
$742$	−121.933			−0.164330
$743$		−	435.120i		−	0.585626i	−0.956170	−	0.292813i	$-0.905409\pi$
$743$		−	435.120i		−	0.585626i	0.956170	−	0.292813i	$-0.0945913\pi$
$744$		0			0
$745$			490.061i			0.657799i
$746$	679.602			0.910994
$747$		0			0
$748$	−0.541027	+	2.91108i	−0.000723298	+	0.00389181i
$749$	−1214.60			−1.62163
$750$		0			0
$751$	335.814			0.447155			0.223578	−	0.974686i	$-0.428226\pi$
$751$	335.814			0.447155			0.223578	+	0.974686i	$0.428226\pi$
$752$	115.124			0.153090
$753$		0			0
$754$			45.1984i			0.0599448i
$755$			81.3736i			0.107780i
$756$		0			0
$757$	105.328			0.139139			0.0695696	−	0.997577i	$-0.477837\pi$
$757$	105.328			0.139139			0.0695696	+	0.997577i	$0.477837\pi$
$758$		−	354.907i		−	0.468215i
$759$		0			0
$760$	−362.076			−0.476416
$761$			702.618i			0.923283i	0.887067	+	0.461641i	$0.152739\pi$
$761$			702.618i			0.923283i	−0.887067	+	0.461641i	$0.847261\pi$
$762$		0			0
$763$	−803.102			−1.05256
$764$	281.161			0.368012
$765$		0			0
$766$		−	9.52715i		−	0.0124375i
$767$			76.9711i			0.100353i
$768$		0			0
$769$		−	1401.93i		−	1.82305i	−0.411241	−	0.911527i	$-0.634905\pi$
$769$		−	1401.93i		−	1.82305i	0.411241	−	0.911527i	$-0.365095\pi$
$770$	−348.950	−	64.8528i	−0.453182	−	0.0842245i
$771$		0			0
$772$		−	372.280i		−	0.482228i
$773$	−1172.25			−1.51649			−0.758247	−	0.651967i	$-0.773944\pi$
$773$	−1172.25			−1.51649			−0.758247	+	0.651967i	$0.773944\pi$
$774$		0			0
$775$	34.3187			0.0442822
$776$		−	342.210i		−	0.440992i
$777$		0			0
$778$			387.263i			0.497767i
$779$	1035.74			1.32958
$780$		0			0
$781$	477.067	+	88.6634i	0.610841	+	0.113525i
$782$	−3.50202			−0.00447829
$783$		0			0
$784$	−194.289			−0.247817
$785$	327.456			0.417142
$786$		0			0
$787$		−	684.107i		−	0.869259i	−0.900609	−	0.434629i	$-0.856879\pi$
$787$		−	684.107i		−	0.869259i	0.900609	−	0.434629i	$-0.143121\pi$
$788$			354.107i			0.449374i
$789$		0			0
$790$	−170.858			−0.216276
$791$			2025.82i			2.56109i
$792$		0			0
$793$	−305.424			−0.385150
$794$			50.5131i			0.0636185i
$795$		0			0
$796$	100.365			0.126086
$797$	307.869			0.386285			0.193142	−	0.981171i	$-0.438132\pi$
$797$	307.869			0.386285			0.193142	+	0.981171i	$0.438132\pi$
$798$		0			0
$799$		−	6.42397i		−	0.00804001i
$800$			164.272i			0.205340i
$801$		0			0
$802$		−	588.614i		−	0.733932i
$803$	−271.149	+	1458.96i	−0.337670	+	1.81688i
$804$		0			0
$805$			842.522i			1.04661i
$806$	−29.6848			−0.0368297
$807$		0			0
$808$	−195.882			−0.242428
$809$			447.140i			0.552707i	0.961056	+	0.276353i	$0.0891260\pi$
$809$			447.140i			0.552707i	−0.961056	+	0.276353i	$0.910874\pi$
$810$		0			0
$811$		−	502.360i		−	0.619433i	−0.950829	−	0.309716i	$-0.899766\pi$
$811$		−	502.360i		−	0.619433i	0.950829	−	0.309716i	$-0.100234\pi$
$812$	349.079			0.429900
$813$		0			0
$814$	−134.829	+	725.467i	−0.165637	+	0.891237i
$815$	−488.791			−0.599744
$816$		0			0
$817$	1609.89			1.97049
$818$	−383.689			−0.469058
$819$		0			0
$820$		−	293.744i		−	0.358225i
$821$		−	65.1337i		−	0.0793346i	−0.999213	−	0.0396673i	$-0.987370\pi$
$821$		−	65.1337i		−	0.0793346i	0.999213	−	0.0396673i	$-0.0126298\pi$
$822$		0			0
$823$	−198.188			−0.240812			−0.120406	−	0.992725i	$-0.538420\pi$
$823$	−198.188			−0.240812			−0.120406	+	0.992725i	$0.538420\pi$
$824$			408.218i			0.495410i
$825$		0			0
$826$	−296.194			−0.358588
$827$		−	294.941i		−	0.356640i	−0.983973	−	0.178320i	$-0.942934\pi$
$827$		−	294.941i		−	0.356640i	0.983973	−	0.178320i	$-0.0570662\pi$
$828$		0			0
$829$	332.089			0.400589			0.200295	−	0.979736i	$-0.435810\pi$
$829$	332.089			0.400589			0.200295	+	0.979736i	$0.435810\pi$
$830$	261.788			0.315407
$831$		0			0
$832$		−	114.989i		−	0.138208i
$833$			10.8414i			0.0130149i
$834$		0			0
$835$			244.539i			0.292861i
$836$	−112.954	+	607.765i	−0.135112	+	0.726992i
$837$		0			0
$838$			350.179i			0.417875i
$839$	−316.736			−0.377516			−0.188758	−	0.982024i	$-0.560446\pi$
$839$	−316.736			−0.377516			−0.188758	+	0.982024i	$0.560446\pi$
$840$		0			0
$841$	731.781			0.870132
$842$			4.90617i			0.00582680i
$843$		0			0
$844$			799.100i			0.946800i
$845$	−346.454			−0.410004
$846$		0			0
$847$	−543.915	+	1412.77i	−0.642166	+	1.66796i
$848$	15.2679			0.0180046
$849$		0			0
$850$	0.581421			0.000684025
$851$	1751.60			2.05828
$852$		0			0
$853$		−	190.690i		−	0.223553i	−0.993733	−	0.111776i	$-0.964346\pi$
$853$		−	190.690i		−	0.223553i	0.993733	−	0.111776i	$-0.0356540\pi$
$854$		−	1175.31i		−	1.37624i
$855$		0			0
$856$	−746.807			−0.872438
$857$		−	1037.09i		−	1.21014i	−0.796173	−	0.605070i	$-0.793145\pi$
$857$		−	1037.09i		−	1.21014i	0.796173	−	0.605070i	$-0.206855\pi$
$858$		0			0
$859$	1537.20			1.78953			0.894764	−	0.446539i	$-0.147344\pi$
$859$	1537.20			1.78953			0.894764	+	0.446539i	$0.147344\pi$
$860$		−	456.580i		−	0.530906i
$861$		0			0
$862$	904.172			1.04892
$863$	−915.241			−1.06053			−0.530267	−	0.847831i	$-0.677908\pi$
$863$	−915.241			−1.06053			−0.530267	+	0.847831i	$0.677908\pi$
$864$		0			0
$865$		−	192.517i		−	0.222563i
$866$		−	885.406i		−	1.02241i
$867$		0			0
$868$			229.263i			0.264128i
$869$	−133.160	+	716.486i	−0.153233	+	0.824495i
$870$		0			0
$871$			165.355i			0.189845i
$872$	−493.794			−0.566278
$873$		0			0
$874$	−731.141			−0.836546
$875$		−	139.879i		−	0.159862i
$876$		0			0
$877$		−	979.084i		−	1.11640i	−0.829706	−	0.558201i	$-0.811492\pi$
$877$		−	979.084i		−	1.11640i	0.829706	−	0.558201i	$-0.188508\pi$
$878$	−254.936			−0.290360
$879$		0			0
$880$	43.6942	+	8.12062i	0.0496525	+	0.00922798i
$881$	−461.922			−0.524315			−0.262158	−	0.965025i	$-0.584434\pi$
$881$	−461.922			−0.524315			−0.262158	+	0.965025i	$0.584434\pi$
$882$		0			0
$883$	−388.559			−0.440045			−0.220022	−	0.975495i	$-0.570613\pi$
$883$	−388.559			−0.440045			−0.220022	+	0.975495i	$0.570613\pi$
$884$	1.00936			0.00114181
$885$		0			0
$886$		−	385.274i		−	0.434847i
$887$			353.901i			0.398987i	0.979899	+	0.199493i	$0.0639297\pi$
$887$			353.901i			0.398987i	−0.979899	+	0.199493i	$0.936070\pi$
$888$		0			0
$889$	592.858			0.666882
$890$		−	149.110i		−	0.167539i
$891$		0			0
$892$	995.153			1.11564
$893$		−	1341.18i		−	1.50188i
$894$		0			0
$895$	−567.410			−0.633978
$896$	−1201.69			−1.34117
$897$		0			0
$898$		−	441.165i		−	0.491275i
$899$		−	71.7316i		−	0.0797904i
$900$		0			0
$901$		−	0.851959i		−	0.000945571i
$902$	613.748	+	114.066i	0.680430	+	0.126459i
$903$		0			0
$904$			1245.59i			1.37787i
$905$	429.053			0.474092
$906$		0			0
$907$	−1653.76			−1.82333			−0.911663	−	0.410939i	$-0.865201\pi$
$907$	−1653.76			−1.82333			−0.911663	+	0.410939i	$0.865201\pi$
$908$			379.568i			0.418027i
$909$		0			0
$910$			120.992i			0.132958i
$911$	−1437.18			−1.57759			−0.788793	−	0.614659i	$-0.789294\pi$
$911$	−1437.18			−1.57759			−0.788793	+	0.614659i	$0.789294\pi$
$912$		0			0
$913$	204.026	−	1097.79i	0.223468	−	1.20240i
$914$	−788.370			−0.862549
$915$		0			0
$916$	−635.615			−0.693903
$917$	−2730.42			−2.97756
$918$		0			0
$919$		−	528.097i		−	0.574643i	−0.957834	−	0.287321i	$-0.907235\pi$
$919$		−	528.097i		−	0.574643i	0.957834	−	0.287321i	$-0.0927647\pi$
$920$			518.032i			0.563078i
$921$		0			0
$922$	396.531			0.430076
$923$		−	165.414i		−	0.179213i
$924$		0			0
$925$	−290.808			−0.314387
$926$			104.844i			0.113222i
$927$		0			0
$928$	343.354			0.369994
$929$	−1229.53			−1.32350			−0.661751	−	0.749724i	$-0.730186\pi$
$929$	−1229.53			−1.32350			−0.661751	+	0.749724i	$0.730186\pi$
$930$		0			0
$931$			2263.44i			2.43119i
$932$			631.221i			0.677276i
$933$		0			0
$934$			698.491i			0.747849i
$935$	0.453136	−	2.43816i	0.000484637	−	0.00260766i
$936$		0			0
$937$			403.452i			0.430579i	0.976550	+	0.215289i	$0.0690695\pi$
$937$			403.452i			0.430579i	−0.976550	+	0.215289i	$0.930931\pi$
$938$	−636.304			−0.678363
$939$		0			0
$940$	−380.369			−0.404647
$941$		−	1599.00i		−	1.69925i	−0.527386	−	0.849626i	$-0.676828\pi$
$941$		−	1599.00i		−	1.69925i	0.527386	−	0.849626i	$-0.323172\pi$
$942$		0			0
$943$		−	1481.86i		−	1.57143i
$944$	37.0883			0.0392884
$945$		0			0
$946$	953.974	+	177.297i	1.00843	+	0.187418i
$947$	−518.387			−0.547399			−0.273700	−	0.961815i	$-0.588247\pi$
$947$	−518.387			−0.547399			−0.273700	+	0.961815i	$0.588247\pi$
$948$		0			0
$949$	505.865			0.533051
$950$	121.387			0.127776
$951$		0			0
$952$			9.70355i			0.0101928i
$953$			1463.22i			1.53539i	0.640817	+	0.767693i	$0.278596\pi$
$953$			1463.22i			1.53539i	−0.640817	+	0.767693i	$0.721404\pi$
$954$		0			0
$955$	−235.486			−0.246582
$956$		−	955.595i		−	0.999576i
$957$		0			0
$958$	252.351			0.263414
$959$			982.800i			1.02482i
$960$		0			0
$961$	−913.889			−0.950977
$962$	251.542			0.261478
$963$		0			0
$964$		−	338.991i		−	0.351650i
$965$			311.802i			0.323111i
$966$		0			0
$967$		−	804.879i		−	0.832346i	−0.909285	−	0.416173i	$-0.863371\pi$
$967$		−	804.879i		−	0.832346i	0.909285	−	0.416173i	$-0.136629\pi$
$968$	−334.431	+	868.651i	−0.345486	+	0.897366i
$969$		0			0
$970$			114.727i			0.118275i
$971$	1459.59			1.50318			0.751590	−	0.659631i	$-0.229287\pi$
$971$	1459.59			1.50318			0.751590	+	0.659631i	$0.229287\pi$
$972$		0			0
$973$	−755.201			−0.776157
$974$			1089.83i			1.11892i
$975$		0			0
$976$			147.168i			0.150786i
$977$	−1227.01			−1.25589			−0.627947	−	0.778256i	$-0.716105\pi$
$977$	−1227.01			−1.25589			−0.627947	+	0.778256i	$0.716105\pi$
$978$		0			0
$979$	−625.285	−	116.210i	−0.638697	−	0.118703i
$980$	641.931			0.655031
$981$		0			0
$982$	423.474			0.431236
$983$	−566.797			−0.576599			−0.288300	−	0.957540i	$-0.593090\pi$
$983$	−566.797			−0.576599			−0.288300	+	0.957540i	$0.593090\pi$
$984$		0			0
$985$		−	296.581i		−	0.301098i
$986$		−	1.21526i		−	0.00123252i
$987$		0			0
$988$	210.731			0.213290
$989$		−	2303.32i		−	2.32894i
$990$		0			0
$991$	−289.458			−0.292087			−0.146043	−	0.989278i	$-0.546654\pi$
$991$	−289.458			−0.292087			−0.146043	+	0.989278i	$0.546654\pi$
$992$			225.503i			0.227322i
$993$		0			0
$994$	636.532			0.640374
$995$	−84.0603			−0.0844827
$996$		0			0
$997$			488.824i			0.490294i	0.969486	+	0.245147i	$0.0788363\pi$
$997$			488.824i			0.490294i	−0.969486	+	0.245147i	$0.921164\pi$
$998$		−	398.575i		−	0.399374i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.b.c.406.12		16
3.2	odd	2		165.3.b.a.76.5	✓	16
11.10	odd	2	inner	495.3.b.c.406.5		16
12.11	even	2		2640.3.c.c.241.13		16
15.2	even	4		825.3.h.b.274.23		32
15.8	even	4		825.3.h.b.274.9		32
15.14	odd	2		825.3.b.d.76.12		16
33.32	even	2		165.3.b.a.76.12	yes	16
132.131	odd	2		2640.3.c.c.241.16		16
165.32	odd	4		825.3.h.b.274.10		32
165.98	odd	4		825.3.h.b.274.24		32
165.164	even	2		825.3.b.d.76.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.3.b.a.76.5	✓	16	3.2	odd	2
165.3.b.a.76.12	yes	16	33.32	even	2
495.3.b.c.406.5		16	11.10	odd	2	inner
495.3.b.c.406.12		16	1.1	even	1	trivial
825.3.b.d.76.5		16	165.164	even	2
825.3.b.d.76.12		16	15.14	odd	2
825.3.h.b.274.9		32	15.8	even	4
825.3.h.b.274.10		32	165.32	odd	4
825.3.h.b.274.23		32	15.2	even	4
825.3.h.b.274.24		32	165.98	odd	4
2640.3.c.c.241.13		16	12.11	even	2
2640.3.c.c.241.16		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)

\(f(q)\)	\(=\)	\(q+1.15335i q^{2} +2.66978 q^{4} -2.23607 q^{5} +12.5112i q^{7} +7.69260i q^{8} -2.57897i q^{10} +(-10.8148 - 2.00994i) q^{11} +3.74983i q^{13} -14.4298 q^{14} +1.80684 q^{16} -0.100823i q^{17} -21.0495i q^{19} -5.96981 q^{20} +(2.31817 - 12.4733i) q^{22} -30.1161 q^{23} +5.00000 q^{25} -4.32487 q^{26} +33.4021i q^{28} -10.4508i q^{29} +6.86373 q^{31} +32.8543i q^{32} +0.116284 q^{34} -27.9758i q^{35} -58.1617 q^{37} +24.2775 q^{38} -17.2012i q^{40} +49.2050i q^{41} +76.4814i q^{43} +(-28.8732 - 5.36611i) q^{44} -34.7344i q^{46} +63.7154 q^{47} -107.530 q^{49} +5.76676i q^{50} +10.0112i q^{52} +8.45006 q^{53} +(24.1827 + 4.49437i) q^{55} -96.2435 q^{56} +12.0535 q^{58} +20.5266 q^{59} +81.4502i q^{61} +7.91630i q^{62} -30.6652 q^{64} -8.38487i q^{65} +44.0966 q^{67} -0.269175i q^{68} +32.2660 q^{70} -44.1124 q^{71} -134.904i q^{73} -67.0809i q^{74} -56.1975i q^{76} +(25.1468 - 135.306i) q^{77} -66.2504i q^{79} -4.04022 q^{80} -56.7507 q^{82} +101.508i q^{83} +0.225447i q^{85} -88.2100 q^{86} +(15.4617 - 83.1940i) q^{88} +57.8175 q^{89} -46.9147 q^{91} -80.4032 q^{92} +73.4862i q^{94} +47.0681i q^{95} -44.4855 q^{97} -124.019i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4} + 28 q^{11} - 16 q^{16} - 40 q^{20} - 20 q^{22} - 56 q^{23} + 80 q^{25} + 88 q^{26} - 96 q^{31} - 200 q^{34} + 184 q^{37} - 296 q^{38} - 300 q^{44} + 200 q^{47} - 496 q^{49} + 80 q^{53} + 20 q^{55}+ \cdots - 144 q^{97}+O(q^{100}) \) 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

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more