LMFDB - Embedded newform 60.5.g.b.41.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [60,5,Mod(41,60)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(60, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("60.41");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	60.g (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.20219778503$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-5}, \sqrt{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 58x^{2} + 1521 $ x^4 - 58*x^2 + 1521
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.2
Root			$5.83095 - 2.23607i$ of defining polynomial
Character	$\chi$	$=$	60.41
Dual form			60.5.g.b.41.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.91548 + 4.28313i) q^{3} -11.1803i q^{5} +7.49286 q^{7} +(44.3095 - 67.8061i) q^{9} +O(q^{10})$	\(q+(-7.91548 + 4.28313i) q^{3} -11.1803i q^{5} +7.49286 q^{7} +(44.3095 - 67.8061i) q^{9} -185.562i q^{11} +269.886 q^{13} +(47.8869 + 88.4977i) q^{15} -438.205i q^{17} -310.929 q^{19} +(-59.3095 + 32.0929i) q^{21} +436.880i q^{23} -125.000 q^{25} +(-60.3083 + 726.501i) q^{27} -813.673i q^{29} -115.714 q^{31} +(794.786 + 1468.81i) q^{33} -83.7727i q^{35} -279.914 q^{37} +(-2136.27 + 1155.96i) q^{39} -121.131i q^{41} +1342.71 q^{43} +(-758.095 - 495.395i) q^{45} +950.856i q^{47} -2344.86 q^{49} +(1876.89 + 3468.60i) q^{51} -2299.13i q^{53} -2074.64 q^{55} +(2461.15 - 1331.75i) q^{57} -6352.54i q^{59} +5231.86 q^{61} +(332.005 - 508.061i) q^{63} -3017.41i q^{65} +761.278 q^{67} +(-1871.21 - 3458.11i) q^{69} +2084.62i q^{71} +8598.37 q^{73} +(989.434 - 535.392i) q^{75} -1390.39i q^{77} -8650.21 q^{79} +(-2634.33 - 6008.91i) q^{81} +7854.04i q^{83} -4899.29 q^{85} +(3485.07 + 6440.61i) q^{87} +9177.21i q^{89} +2022.21 q^{91} +(915.933 - 495.620i) q^{93} +3476.29i q^{95} -10936.5 q^{97} +(-12582.2 - 8222.15i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9}+O(q^{10}) $ 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9	$ 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9} - 40 q^{13} - 100 q^{15} - 544 q^{19} - 4 q^{21} - 500 q^{25} + 820 q^{27} + 2336 q^{31} + 1080 q^{33} - 280 q^{37} - 3064 q^{39} + 7400 q^{43} - 700 q^{45} - 7980 q^{49} + 1560 q^{51} - 4800 q^{55} + 4760 q^{57} - 64 q^{61} + 4640 q^{63} - 8920 q^{67} - 19380 q^{69} + 18440 q^{73} + 2500 q^{75} - 7312 q^{79} + 2524 q^{81} - 12600 q^{85} + 14640 q^{87} + 19984 q^{91} - 3520 q^{93} - 9880 q^{97} - 23040 q^{99}+O(q^{100}) $ 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9 - 40 * q^13 - 100 * q^15 - 544 * q^19 - 4 * q^21 - 500 * q^25 + 820 * q^27 + 2336 * q^31 + 1080 * q^33 - 280 * q^37 - 3064 * q^39 + 7400 * q^43 - 700 * q^45 - 7980 * q^49 + 1560 * q^51 - 4800 * q^55 + 4760 * q^57 - 64 * q^61 + 4640 * q^63 - 8920 * q^67 - 19380 * q^69 + 18440 * q^73 + 2500 * q^75 - 7312 * q^79 + 2524 * q^81 - 12600 * q^85 + 14640 * q^87 + 19984 * q^91 - 3520 * q^93 - 9880 * q^97 - 23040 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−7.91548	+	4.28313i	−0.879497	+	0.475904i
$3$	−7.91548	+	4.28313i	−0.879497	+	0.475904i
$4$		0			0
$5$		−	11.1803i		−	0.447214i
$5$		−	11.1803i		−	0.447214i
$6$		0			0
$7$	7.49286			0.152915			0.0764577	−	0.997073i	$-0.475639\pi$
$7$	7.49286			0.152915			0.0764577	+	0.997073i	$0.475639\pi$
$8$		0			0
$9$	44.3095	−	67.8061i	0.547031	−	0.837112i
$10$		0			0
$11$		−	185.562i		−	1.53357i	−0.641905	−	0.766784i	$-0.721856\pi$
$11$		−	185.562i		−	1.53357i	0.641905	−	0.766784i	$-0.278144\pi$
$12$		0			0
$13$	269.886			1.59696			0.798478	−	0.602023i	$-0.205639\pi$
$13$	269.886			1.59696			0.798478	+	0.602023i	$0.205639\pi$
$14$		0			0
$15$	47.8869	+	88.4977i	0.212831	+	0.393323i
$16$		0			0
$17$		−	438.205i		−	1.51628i	−0.652091	−	0.758141i	$-0.726108\pi$
$17$		−	438.205i		−	1.51628i	0.652091	−	0.758141i	$-0.273892\pi$
$18$		0			0
$19$	−310.929			−0.861298			−0.430649	−	0.902520i	$-0.641715\pi$
$19$	−310.929			−0.861298			−0.430649	+	0.902520i	$0.641715\pi$
$20$		0			0
$21$	−59.3095	+	32.0929i	−0.134489	+	0.0727730i
$22$		0			0
$23$			436.880i			0.825860i	0.910763	+	0.412930i	$0.135495\pi$
$23$			436.880i			0.825860i	−0.910763	+	0.412930i	$0.864505\pi$
$24$		0			0
$25$	−125.000			−0.200000
$26$		0			0
$27$	−60.3083	+	726.501i	−0.0827275	+	0.996572i
$28$		0			0
$29$		−	813.673i		−	0.967507i	−0.875204	−	0.483753i	$-0.839273\pi$
$29$		−	813.673i		−	0.967507i	0.875204	−	0.483753i	$-0.160727\pi$
$30$		0			0
$31$	−115.714			−0.120410			−0.0602051	−	0.998186i	$-0.519175\pi$
$31$	−115.714			−0.120410			−0.0602051	+	0.998186i	$0.519175\pi$
$32$		0			0
$33$	794.786	+	1468.81i	0.729831	+	1.34877i
$34$		0			0
$35$		−	83.7727i		−	0.0683859i
$36$		0			0
$37$	−279.914			−0.204466			−0.102233	−	0.994760i	$-0.532599\pi$
$37$	−279.914			−0.204466			−0.102233	+	0.994760i	$0.532599\pi$
$38$		0			0
$39$	−2136.27	+	1155.96i	−1.40452	+	0.759998i
$40$		0			0
$41$		−	121.131i		−	0.0720589i	−0.999351	−	0.0360295i	$-0.988529\pi$
$41$		−	121.131i		−	0.0720589i	0.999351	−	0.0360295i	$-0.0114710\pi$
$42$		0			0
$43$	1342.71			0.726180			0.363090	−	0.931754i	$-0.381722\pi$
$43$	1342.71			0.726180			0.363090	+	0.931754i	$0.381722\pi$
$44$		0			0
$45$	−758.095	−	495.395i	−0.374368	−	0.244640i
$46$		0			0
$47$			950.856i			0.430446i	0.976565	+	0.215223i	$0.0690479\pi$
$47$			950.856i			0.430446i	−0.976565	+	0.215223i	$0.930952\pi$
$48$		0			0
$49$	−2344.86			−0.976617
$50$		0			0
$51$	1876.89	+	3468.60i	0.721604	+	1.33357i
$52$		0			0
$53$		−	2299.13i		−	0.818485i	−0.912426	−	0.409243i	$-0.865793\pi$
$53$		−	2299.13i		−	0.818485i	0.912426	−	0.409243i	$-0.134207\pi$
$54$		0			0
$55$	−2074.64			−0.685832
$56$		0			0
$57$	2461.15	−	1331.75i	0.757509	−	0.409895i
$58$		0			0
$59$		−	6352.54i		−	1.82492i	−0.409168	−	0.912459i	$-0.634181\pi$
$59$		−	6352.54i		−	1.82492i	0.409168	−	0.912459i	$-0.365819\pi$
$60$		0			0
$61$	5231.86			1.40604			0.703018	−	0.711172i	$-0.251836\pi$
$61$	5231.86			1.40604			0.703018	+	0.711172i	$0.251836\pi$
$62$		0			0
$63$	332.005	−	508.061i	0.0836495	−	0.128007i
$64$		0			0
$65$		−	3017.41i		−	0.714181i
$66$		0			0
$67$	761.278			0.169588			0.0847938	−	0.996399i	$-0.472977\pi$
$67$	761.278			0.169588			0.0847938	+	0.996399i	$0.472977\pi$
$68$		0			0
$69$	−1871.21	−	3458.11i	−0.393030	−	0.726341i
$70$		0			0
$71$			2084.62i			0.413533i	0.978390	+	0.206767i	$0.0662941\pi$
$71$			2084.62i			0.413533i	−0.978390	+	0.206767i	$0.933706\pi$
$72$		0			0
$73$	8598.37			1.61351			0.806753	−	0.590889i	$-0.201223\pi$
$73$	8598.37			1.61351			0.806753	+	0.590889i	$0.201223\pi$
$74$		0			0
$75$	989.434	−	535.392i	0.175899	−	0.0951808i
$76$		0			0
$77$		−	1390.39i		−	0.234506i
$78$		0			0
$79$	−8650.21			−1.38603			−0.693015	−	0.720923i	$-0.743718\pi$
$79$	−8650.21			−1.38603			−0.693015	+	0.720923i	$0.743718\pi$
$80$		0			0
$81$	−2634.33	−	6008.91i	−0.401514	−	0.915853i
$82$		0			0
$83$			7854.04i			1.14008i	0.821615	+	0.570042i	$0.193073\pi$
$83$			7854.04i			1.14008i	−0.821615	+	0.570042i	$0.806927\pi$
$84$		0			0
$85$	−4899.29			−0.678102
$86$		0			0
$87$	3485.07	+	6440.61i	0.460440	+	0.850920i
$88$		0			0
$89$			9177.21i			1.15859i	0.815117	+	0.579296i	$0.196672\pi$
$89$			9177.21i			1.15859i	−0.815117	+	0.579296i	$0.803328\pi$
$90$		0			0
$91$	2022.21			0.244199
$92$		0			0
$93$	915.933	−	495.620i	0.105900	−	0.0573037i
$94$		0			0
$95$			3476.29i			0.385184i
$96$		0			0
$97$	−10936.5			−1.16235			−0.581175	−	0.813779i	$-0.697407\pi$
$97$	−10936.5			−1.16235			−0.581175	+	0.813779i	$0.697407\pi$
$98$		0			0
$99$	−12582.2	−	8222.15i	−1.28377	−	0.838909i
$100$		0			0
$101$			9133.76i			0.895379i	0.894189	+	0.447690i	$0.147753\pi$
$101$			9133.76i			0.895379i	−0.894189	+	0.447690i	$0.852247\pi$
$102$		0			0
$103$	4767.26			0.449360			0.224680	−	0.974433i	$-0.427866\pi$
$103$	4767.26			0.449360			0.224680	+	0.974433i	$0.427866\pi$
$104$		0			0
$105$	358.810	+	663.101i	0.0325451	+	0.0601452i
$106$		0			0
$107$			1424.04i			0.124381i	0.998064	+	0.0621906i	$0.0198087\pi$
$107$			1424.04i			0.124381i	−0.998064	+	0.0621906i	$0.980191\pi$
$108$		0			0
$109$	13612.4			1.14573			0.572865	−	0.819649i	$-0.305832\pi$
$109$	13612.4			1.14573			0.572865	+	0.819649i	$0.305832\pi$
$110$		0			0
$111$	2215.65	−	1198.91i	0.179828	−	0.0973063i
$112$		0			0
$113$			24150.0i			1.89130i	0.325182	+	0.945651i	$0.394574\pi$
$113$			24150.0i			1.89130i	−0.325182	+	0.945651i	$0.605426\pi$
$114$		0			0
$115$	4884.46			0.369336
$116$		0			0
$117$	11958.5	−	18299.9i	0.873585	−	1.33683i
$118$		0			0
$119$		−	3283.41i		−	0.231863i
$120$		0			0
$121$	−19792.1			−1.35183
$122$		0			0
$123$	518.821	+	958.810i	0.0342931	+	0.0633756i
$124$		0			0
$125$			1397.54i			0.0894427i
$126$		0			0
$127$	−17779.2			−1.10231			−0.551157	−	0.834401i	$-0.685814\pi$
$127$	−17779.2			−1.10231			−0.551157	+	0.834401i	$0.685814\pi$
$128$		0			0
$129$	−10628.2	+	5751.00i	−0.638674	+	0.345592i
$130$		0			0
$131$		−	19682.8i		−	1.14695i	−0.819223	−	0.573475i	$-0.805595\pi$
$131$		−	19682.8i		−	1.14695i	0.819223	−	0.573475i	$-0.194405\pi$
$132$		0			0
$133$	−2329.74			−0.131706
$134$		0			0
$135$	8122.53	+	674.267i	0.445681	+	0.0369968i
$136$		0			0
$137$			24052.6i			1.28150i	0.767748	+	0.640752i	$0.221377\pi$
$137$			24052.6i			1.28150i	−0.767748	+	0.640752i	$0.778623\pi$
$138$		0			0
$139$	32890.8			1.70233			0.851167	−	0.524895i	$-0.175896\pi$
$139$	32890.8			1.70233			0.851167	+	0.524895i	$0.175896\pi$
$140$		0			0
$141$	−4072.64	−	7526.48i	−0.204851	−	0.378576i
$142$		0			0
$143$		−	50080.4i		−	2.44904i
$144$		0			0
$145$	−9097.14			−0.432682
$146$		0			0
$147$	18560.7	−	10043.3i	0.858932	−	0.464776i
$148$		0			0
$149$			5263.83i			0.237099i	0.992948	+	0.118549i	$0.0378244\pi$
$149$			5263.83i			0.237099i	−0.992948	+	0.118549i	$0.962176\pi$
$150$		0			0
$151$	36541.6			1.60263			0.801315	−	0.598243i	$-0.204134\pi$
$151$	36541.6			1.60263			0.801315	+	0.598243i	$0.204134\pi$
$152$		0			0
$153$	−29713.0	−	19416.7i	−1.26930	−	0.829453i
$154$		0			0
$155$			1293.72i			0.0538491i
$156$		0			0
$157$	2630.88			0.106734			0.0533669	−	0.998575i	$-0.483005\pi$
$157$	2630.88			0.106734			0.0533669	+	0.998575i	$0.483005\pi$
$158$		0			0
$159$	9847.46	+	18198.7i	0.389520	+	0.719856i
$160$		0			0
$161$			3273.48i			0.126287i
$162$		0			0
$163$	−20407.6			−0.768100			−0.384050	−	0.923312i	$-0.625471\pi$
$163$	−20407.6			−0.768100			−0.384050	+	0.923312i	$0.625471\pi$
$164$		0			0
$165$	16421.8	−	8885.97i	0.603188	−	0.326390i
$166$		0			0
$167$		−	45547.7i		−	1.63318i	−0.577220	−	0.816588i	$-0.695863\pi$
$167$		−	45547.7i		−	1.63318i	0.577220	−	0.816588i	$-0.304137\pi$
$168$		0			0
$169$	44277.3			1.55027
$170$		0			0
$171$	−13777.1	+	21082.9i	−0.471157	+	0.721003i
$172$		0			0
$173$		−	17995.0i		−	0.601256i	−0.953742	−	0.300628i	$-0.902804\pi$
$173$		−	17995.0i		−	0.601256i	0.953742	−	0.300628i	$-0.0971962\pi$
$174$		0			0
$175$	−936.607			−0.0305831
$176$		0			0
$177$	27208.8	+	50283.4i	0.868486	+	1.60501i
$178$		0			0
$179$			5358.45i			0.167237i	0.996498	+	0.0836186i	$0.0266477\pi$
$179$			5358.45i			0.167237i	−0.996498	+	0.0836186i	$0.973352\pi$
$180$		0			0
$181$	−42560.6			−1.29912			−0.649562	−	0.760309i	$-0.725048\pi$
$181$	−42560.6			−1.29912			−0.649562	+	0.760309i	$0.725048\pi$
$182$		0			0
$183$	−41412.6	+	22408.7i	−1.23660	+	0.669137i
$184$		0			0
$185$			3129.54i			0.0914401i
$186$		0			0
$187$	−81314.1			−2.32532
$188$		0			0
$189$	−451.881	+	5443.57i	−0.0126503	+	0.152391i
$190$		0			0
$191$			40085.0i			1.09879i	0.835562	+	0.549396i	$0.185142\pi$
$191$			40085.0i			1.09879i	−0.835562	+	0.549396i	$0.814858\pi$
$192$		0			0
$193$	−16107.4			−0.432425			−0.216212	−	0.976346i	$-0.569370\pi$
$193$	−16107.4			−0.432425			−0.216212	+	0.976346i	$0.569370\pi$
$194$		0			0
$195$	12924.0	+	23884.3i	0.339881	+	0.628120i
$196$		0			0
$197$			6357.91i			0.163826i	0.996640	+	0.0819128i	$0.0261029\pi$
$197$			6357.91i			0.163826i	−0.996640	+	0.0819128i	$0.973897\pi$
$198$		0			0
$199$	42969.4			1.08506			0.542529	−	0.840037i	$-0.317467\pi$
$199$	42969.4			1.08506			0.542529	+	0.840037i	$0.317467\pi$
$200$		0			0
$201$	−6025.88	+	3260.66i	−0.149152	+	0.0807073i
$202$		0			0
$203$		−	6096.74i		−	0.147947i
$204$		0			0
$205$	−1354.29			−0.0322257
$206$		0			0
$207$	29623.1	+	19357.9i	0.691337	+	0.451771i
$208$		0			0
$209$			57696.4i			1.32086i
$210$		0			0
$211$	15301.0			0.343681			0.171840	−	0.985125i	$-0.445029\pi$
$211$	15301.0			0.343681			0.171840	+	0.985125i	$0.445029\pi$
$212$		0			0
$213$	−8928.72	−	16500.8i	−0.196802	−	0.363702i
$214$		0			0
$215$		−	15011.9i		−	0.324758i
$216$		0			0
$217$	−867.030			−0.0184126
$218$		0			0
$219$	−68060.2	+	36828.0i	−1.41907	+	0.767873i
$220$		0			0
$221$		−	118265.i		−	2.42144i
$222$		0			0
$223$	−6149.72			−0.123665			−0.0618324	−	0.998087i	$-0.519694\pi$
$223$	−6149.72			−0.123665			−0.0618324	+	0.998087i	$0.519694\pi$
$224$		0			0
$225$	−5538.69	+	8475.76i	−0.109406	+	0.167422i
$226$		0			0
$227$		−	18408.5i		−	0.357246i	−0.983918	−	0.178623i	$-0.942836\pi$
$227$		−	18408.5i		−	0.357246i	0.983918	−	0.178623i	$-0.0571642\pi$
$228$		0			0
$229$	78317.1			1.49343			0.746717	−	0.665142i	$-0.231629\pi$
$229$	78317.1			1.49343			0.746717	+	0.665142i	$0.231629\pi$
$230$		0			0
$231$	5955.21	+	11005.6i	0.111602	+	0.206248i
$232$		0			0
$233$			12038.2i			0.221742i	0.993835	+	0.110871i	$0.0353641\pi$
$233$			12038.2i			0.221742i	−0.993835	+	0.110871i	$0.964636\pi$
$234$		0			0
$235$	10630.9			0.192501
$236$		0			0
$237$	68470.6	−	37050.0i	1.21901	−	0.659617i
$238$		0			0
$239$		−	50014.8i		−	0.875593i	−0.899074	−	0.437797i	$-0.855759\pi$
$239$		−	50014.8i		−	0.875593i	0.899074	−	0.437797i	$-0.144241\pi$
$240$		0			0
$241$	−45019.3			−0.775112			−0.387556	−	0.921846i	$-0.626681\pi$
$241$	−45019.3			−0.775112			−0.387556	+	0.921846i	$0.626681\pi$
$242$		0			0
$243$	46589.0	+	36280.2i	0.788988	+	0.614408i
$244$		0			0
$245$			26216.3i			0.436756i
$246$		0			0
$247$	−83915.2			−1.37546
$248$		0			0
$249$	−33639.9	−	62168.5i	−0.542571	−	1.00270i
$250$		0			0
$251$			5129.44i			0.0814184i	0.999171	+	0.0407092i	$0.0129617\pi$
$251$			5129.44i			0.0814184i	−0.999171	+	0.0407092i	$0.987038\pi$
$252$		0			0
$253$	81068.1			1.26651
$254$		0			0
$255$	38780.2	−	20984.3i	0.596389	−	0.322711i
$256$		0			0
$257$			10584.1i			0.160247i	0.996785	+	0.0801234i	$0.0255314\pi$
$257$			10584.1i			0.160247i	−0.996785	+	0.0801234i	$0.974469\pi$
$258$		0			0
$259$	−2097.36			−0.0312660
$260$		0			0
$261$	−55172.0	−	36053.5i	−0.809912	−	0.529256i
$262$		0			0
$263$			60860.5i			0.879881i	0.898027	+	0.439940i	$0.145000\pi$
$263$			60860.5i			0.879881i	−0.898027	+	0.439940i	$0.855000\pi$
$264$		0			0
$265$	−25705.0			−0.366038
$266$		0			0
$267$	−39307.2	−	72642.0i	−0.551378	−	1.01898i
$268$		0			0
$269$		−	116001.i		−	1.60309i	−0.597932	−	0.801547i	$-0.704011\pi$
$269$		−	116001.i		−	1.60309i	0.597932	−	0.801547i	$-0.295989\pi$
$270$		0			0
$271$	89298.3			1.21592			0.607959	−	0.793968i	$-0.291988\pi$
$271$	89298.3			1.21592			0.607959	+	0.793968i	$0.291988\pi$
$272$		0			0
$273$	−16006.8	+	8661.42i	−0.214773	+	0.116215i
$274$		0			0
$275$			23195.2i			0.306714i
$276$		0			0
$277$	42073.5			0.548338			0.274169	−	0.961681i	$-0.411597\pi$
$277$	42073.5			0.548338			0.274169	+	0.961681i	$0.411597\pi$
$278$		0			0
$279$	−5127.24	+	7846.13i	−0.0658681	+	0.100797i
$280$		0			0
$281$			80101.2i			1.01444i	0.861817	+	0.507220i	$0.169327\pi$
$281$			80101.2i			1.01444i	−0.861817	+	0.507220i	$0.830673\pi$
$282$		0			0
$283$	60061.3			0.749933			0.374966	−	0.927038i	$-0.377654\pi$
$283$	60061.3			0.749933			0.374966	+	0.927038i	$0.377654\pi$
$284$		0			0
$285$	−14889.4	−	27516.5i	−0.183311	−	0.338768i
$286$		0			0
$287$		−	907.618i		−	0.0110189i
$288$		0			0
$289$	−108503.			−1.29911
$290$		0			0
$291$	86567.9	−	46842.7i	1.02228	−	0.553166i
$292$		0			0
$293$		−	50062.0i		−	0.583140i	−0.956549	−	0.291570i	$-0.905822\pi$
$293$		−	50062.0i		−	0.583140i	0.956549	−	0.291570i	$-0.0941776\pi$
$294$		0			0
$295$	−71023.6			−0.816128
$296$		0			0
$297$	134811.	+	11190.9i	1.52831	+	0.126868i
$298$		0			0
$299$			117908.i			1.31886i
$300$		0			0
$301$	10060.7			0.111044
$302$		0			0
$303$	−39121.1	−	72298.1i	−0.426114	−	0.787484i
$304$		0			0
$305$		−	58493.9i		−	0.628798i
$306$		0			0
$307$	−37829.6			−0.401380			−0.200690	−	0.979655i	$-0.564318\pi$
$307$	−37829.6			−0.401380			−0.200690	+	0.979655i	$0.564318\pi$
$308$		0			0
$309$	−37735.2	+	20418.8i	−0.395211	+	0.213852i
$310$		0			0
$311$			134422.i			1.38980i	0.719108	+	0.694898i	$0.244551\pi$
$311$			134422.i			1.38980i	−0.719108	+	0.694898i	$0.755449\pi$
$312$		0			0
$313$	93976.5			0.959247			0.479623	−	0.877474i	$-0.340773\pi$
$313$	93976.5			0.959247			0.479623	+	0.877474i	$0.340773\pi$
$314$		0			0
$315$	−5680.30	−	3711.93i	−0.0572466	−	0.0374092i
$316$		0			0
$317$			193466.i			1.92524i	0.270849	+	0.962622i	$0.412696\pi$
$317$			193466.i			1.92524i	−0.270849	+	0.962622i	$0.587304\pi$
$318$		0			0
$319$	−150987.			−1.48374
$320$		0			0
$321$	−6099.35	−	11272.0i	−0.0591935	−	0.109393i
$322$		0			0
$323$			136251.i			1.30597i
$324$		0			0
$325$	−33735.7			−0.319391
$326$		0			0
$327$	−107749.	+	58303.9i	−1.00767	+	0.545258i
$328$		0			0
$329$			7124.63i			0.0658219i
$330$		0			0
$331$	47066.4			0.429591			0.214796	−	0.976659i	$-0.431091\pi$
$331$	47066.4			0.429591			0.214796	+	0.976659i	$0.431091\pi$
$332$		0			0
$333$	−12402.9	+	18979.9i	−0.111849	+	0.171161i
$334$		0			0
$335$		−	8511.35i		−	0.0758418i
$336$		0			0
$337$	−18146.4			−0.159783			−0.0798915	−	0.996804i	$-0.525457\pi$
$337$	−18146.4			−0.159783			−0.0798915	+	0.996804i	$0.525457\pi$
$338$		0			0
$339$	−103438.	−	191159.i	−0.900078	−	1.66340i
$340$		0			0
$341$			21472.1i			0.184657i
$342$		0			0
$343$	−35560.0			−0.302255
$344$		0			0
$345$	−38662.9	+	20920.8i	−0.324830	+	0.175768i
$346$		0			0
$347$		−	38797.0i		−	0.322210i	−0.986937	−	0.161105i	$-0.948494\pi$
$347$		−	38797.0i		−	0.322210i	0.986937	−	0.161105i	$-0.0515059\pi$
$348$		0			0
$349$	23918.0			0.196369			0.0981847	−	0.995168i	$-0.468696\pi$
$349$	23918.0			0.196369			0.0981847	+	0.995168i	$0.468696\pi$
$350$		0			0
$351$	−16276.4	+	196072.i	−0.132112	+	1.59148i
$352$		0			0
$353$			121919.i			0.978413i	0.872168	+	0.489207i	$0.162714\pi$
$353$			121919.i			0.978413i	−0.872168	+	0.489207i	$0.837286\pi$
$354$		0			0
$355$	23306.8			0.184938
$356$		0			0
$357$	14063.3	+	25989.8i	0.110344	+	0.203923i
$358$		0			0
$359$		−	54589.0i		−	0.423561i	−0.977317	−	0.211781i	$-0.932074\pi$
$359$		−	54589.0i		−	0.423561i	0.977317	−	0.211781i	$-0.0679262\pi$
$360$		0			0
$361$	−33644.4			−0.258166
$362$		0			0
$363$	156664.	−	84772.4i	1.18893	−	0.643341i
$364$		0			0
$365$		−	96132.7i		−	0.721582i
$366$		0			0
$367$	173280.			1.28652			0.643259	−	0.765648i	$-0.277582\pi$
$367$	173280.			1.28652			0.643259	+	0.765648i	$0.277582\pi$
$368$		0			0
$369$	−8213.43	−	5367.26i	−0.0603214	−	0.0394185i
$370$		0			0
$371$		−	17227.0i		−	0.125159i
$372$		0			0
$373$	−33117.7			−0.238036			−0.119018	−	0.992892i	$-0.537975\pi$
$373$	−33117.7			−0.238036			−0.119018	+	0.992892i	$0.537975\pi$
$374$		0			0
$375$	−5985.86	−	11062.2i	−0.0425661	−	0.0786646i
$376$		0			0
$377$		−	219599.i		−	1.54507i
$378$		0			0
$379$	−5342.79			−0.0371954			−0.0185977	−	0.999827i	$-0.505920\pi$
$379$	−5342.79			−0.0371954			−0.0185977	+	0.999827i	$0.505920\pi$
$380$		0			0
$381$	140731.	−	76150.9i	0.969483	−	0.524596i
$382$		0			0
$383$		−	258063.i		−	1.75925i	−0.475669	−	0.879625i	$-0.657794\pi$
$383$		−	258063.i		−	1.75925i	0.475669	−	0.879625i	$-0.342206\pi$
$384$		0			0
$385$	−15545.0			−0.104874
$386$		0			0
$387$	59494.7	−	91043.7i	0.397243	−	0.607894i
$388$		0			0
$389$			39109.0i			0.258450i	0.991615	+	0.129225i	$0.0412490\pi$
$389$			39109.0i			0.258450i	−0.991615	+	0.129225i	$0.958751\pi$
$390$		0			0
$391$	191443.			1.25224
$392$		0			0
$393$	84304.1	+	155799.i	0.545838	+	1.00874i
$394$		0			0
$395$			96712.3i			0.619851i
$396$		0			0
$397$	−39193.5			−0.248676			−0.124338	−	0.992240i	$-0.539681\pi$
$397$	−39193.5			−0.248676			−0.124338	+	0.992240i	$0.539681\pi$
$398$		0			0
$399$	18441.0	−	9978.60i	0.115835	−	0.0626793i
$400$		0			0
$401$		−	109754.i		−	0.682545i	−0.939964	−	0.341272i	$-0.889142\pi$
$401$		−	109754.i		−	0.682545i	0.939964	−	0.341272i	$-0.110858\pi$
$402$		0			0
$403$	−31229.6			−0.192290
$404$		0			0
$405$	−67181.7	+	29452.7i	−0.409582	+	0.179562i
$406$		0			0
$407$			51941.4i			0.313563i
$408$		0			0
$409$	−47748.4			−0.285438			−0.142719	−	0.989763i	$-0.545585\pi$
$409$	−47748.4			−0.285438			−0.142719	+	0.989763i	$0.545585\pi$
$410$		0			0
$411$	−103020.	−	190387.i	−0.609873	−	1.12708i
$412$		0			0
$413$		−	47598.7i		−	0.279058i
$414$		0			0
$415$	87810.9			0.509861
$416$		0			0
$417$	−260346.	+	140876.i	−1.49720	+	0.810147i
$418$		0			0
$419$		−	305048.i		−	1.73756i	−0.495197	−	0.868781i	$-0.664904\pi$
$419$		−	305048.i		−	1.73756i	0.495197	−	0.868781i	$-0.335096\pi$
$420$		0			0
$421$	−307155.			−1.73298			−0.866489	−	0.499196i	$-0.833629\pi$
$421$	−307155.			−1.73298			−0.866489	+	0.499196i	$0.833629\pi$
$422$		0			0
$423$	64473.8	+	42132.0i	0.360332	+	0.235468i
$424$		0			0
$425$			54775.7i			0.303256i
$426$		0			0
$427$	39201.5			0.215004
$428$		0			0
$429$	214501.	+	396411.i	1.16551	+	2.15393i
$430$		0			0
$431$		−	291873.i		−	1.57123i	−0.618716	−	0.785615i	$-0.712347\pi$
$431$		−	291873.i		−	1.57123i	0.618716	−	0.785615i	$-0.287653\pi$
$432$		0			0
$433$	227573.			1.21380			0.606898	−	0.794780i	$-0.292414\pi$
$433$	227573.			1.21380			0.606898	+	0.794780i	$0.292414\pi$
$434$		0			0
$435$	72008.2	−	38964.3i	0.380543	−	0.205915i
$436$		0			0
$437$		−	135838.i		−	0.711311i
$438$		0			0
$439$	−85820.5			−0.445309			−0.222655	−	0.974897i	$-0.571472\pi$
$439$	−85820.5			−0.445309			−0.222655	+	0.974897i	$0.571472\pi$
$440$		0			0
$441$	−103899.	+	158996.i	−0.534240	+	0.817538i
$442$		0			0
$443$			294251.i			1.49938i	0.661791	+	0.749689i	$0.269797\pi$
$443$			294251.i			1.49938i	−0.661791	+	0.749689i	$0.730203\pi$
$444$		0			0
$445$	102604.			0.518138
$446$		0			0
$447$	−22545.7	−	41665.7i	−0.112836	−	0.208528i
$448$		0			0
$449$		−	10134.9i		−	0.0502720i	−0.999684	−	0.0251360i	$-0.991998\pi$
$449$		−	10134.9i		−	0.0502720i	0.999684	−	0.0251360i	$-0.00800188\pi$
$450$		0			0
$451$	−22477.3			−0.110507
$452$		0			0
$453$	−289244.	+	156512.i	−1.40951	+	0.762698i
$454$		0			0
$455$		−	22609.0i		−	0.109209i
$456$		0			0
$457$	171657.			0.821921			0.410960	−	0.911653i	$-0.365193\pi$
$457$	171657.			0.821921			0.410960	+	0.911653i	$0.365193\pi$
$458$		0			0
$459$	318357.	+	26427.4i	1.51108	+	0.125438i
$460$		0			0
$461$			35926.8i			0.169051i	0.996421	+	0.0845254i	$0.0269374\pi$
$461$			35926.8i			0.169051i	−0.996421	+	0.0845254i	$0.973063\pi$
$462$		0			0
$463$	−284140.			−1.32547			−0.662735	−	0.748854i	$-0.730605\pi$
$463$	−284140.			−1.32547			−0.662735	+	0.748854i	$0.730605\pi$
$464$		0			0
$465$	−5541.20	−	10240.4i	−0.0256270	−	0.0473601i
$466$		0			0
$467$			163891.i			0.751485i	0.926724	+	0.375743i	$0.122612\pi$
$467$			163891.i			0.751485i	−0.926724	+	0.375743i	$0.877388\pi$
$468$		0			0
$469$	5704.15			0.0259325
$470$		0			0
$471$	−20824.7	+	11268.4i	−0.0938722	+	0.0507951i
$472$		0			0
$473$		−	249155.i		−	1.11365i
$474$		0			0
$475$	38866.1			0.172260
$476$		0			0
$477$	−155895.	−	101873.i	−0.685164	−	0.447737i
$478$		0			0
$479$			192581.i			0.839347i	0.907675	+	0.419673i	$0.137855\pi$
$479$			192581.i			0.839347i	−0.907675	+	0.419673i	$0.862145\pi$
$480$		0			0
$481$	−75544.9			−0.326524
$482$		0			0
$483$	−14020.7	−	25911.1i	−0.0601003	−	0.111069i
$484$		0			0
$485$			122274.i			0.519818i
$486$		0			0
$487$	58765.7			0.247780			0.123890	−	0.992296i	$-0.460463\pi$
$487$	58765.7			0.247780			0.123890	+	0.992296i	$0.460463\pi$
$488$		0			0
$489$	161536.	−	87408.7i	0.675542	−	0.365542i
$490$		0			0
$491$			250734.i			1.04004i	0.854154	+	0.520020i	$0.174075\pi$
$491$			250734.i			1.04004i	−0.854154	+	0.520020i	$0.825925\pi$
$492$		0			0
$493$	−356556.			−1.46701
$494$		0			0
$495$	−91926.4	+	140673.i	−0.375172	+	0.574119i
$496$		0			0
$497$			15619.8i			0.0632356i
$498$		0			0
$499$	21438.1			0.0860963			0.0430482	−	0.999073i	$-0.486293\pi$
$499$	21438.1			0.0860963			0.0430482	+	0.999073i	$0.486293\pi$
$500$		0			0
$501$	195087.	+	360531.i	0.777235	+	1.43637i
$502$		0			0
$503$		−	312093.i		−	1.23353i	−0.787149	−	0.616763i	$-0.788444\pi$
$503$		−	312093.i		−	1.23353i	0.787149	−	0.616763i	$-0.211556\pi$
$504$		0			0
$505$	102119.			0.400426
$506$		0			0
$507$	−350476.	+	189646.i	−1.36346	+	0.737780i
$508$		0			0
$509$			365909.i			1.41233i	0.708045	+	0.706167i	$0.249577\pi$
$509$			365909.i			1.41233i	−0.708045	+	0.706167i	$0.750423\pi$
$510$		0			0
$511$	64426.4			0.246730
$512$		0			0
$513$	18751.6	−	225890.i	0.0712530	−	0.858346i
$514$		0			0
$515$		−	53299.6i		−	0.200960i
$516$		0			0
$517$	176442.			0.660119
$518$		0			0
$519$	77074.9	+	142439.i	0.286140	+	0.528803i
$520$		0			0
$521$		−	285210.i		−	1.05072i	−0.850879	−	0.525362i	$-0.823930\pi$
$521$		−	285210.i		−	1.05072i	0.850879	−	0.525362i	$-0.176070\pi$
$522$		0			0
$523$	−423227.			−1.54729			−0.773643	−	0.633622i	$-0.781567\pi$
$523$	−423227.			−1.54729			−0.773643	+	0.633622i	$0.781567\pi$
$524$		0			0
$525$	7413.69	−	4011.61i	0.0268977	−	0.0145546i
$526$		0			0
$527$			50706.6i			0.182576i
$528$		0			0
$529$	88977.1			0.317956
$530$		0			0
$531$	−430741.	−	281478.i	−1.52766	−	0.998287i
$532$		0			0
$533$		−	32691.5i		−	0.115075i
$534$		0			0
$535$	15921.3			0.0556250
$536$		0			0
$537$	−22950.9	−	42414.7i	−0.0795888	−	0.147085i
$538$		0			0
$539$			435116.i			1.49771i
$540$		0			0
$541$	−171697.			−0.586636			−0.293318	−	0.956015i	$-0.594759\pi$
$541$	−171697.			−0.586636			−0.293318	+	0.956015i	$0.594759\pi$
$542$		0			0
$543$	336887.	−	182293.i	1.14258	−	0.618258i
$544$		0			0
$545$		−	152192.i		−	0.512386i
$546$		0			0
$547$	302744.			1.01181			0.505907	−	0.862588i	$-0.331158\pi$
$547$	302744.			1.01181			0.505907	+	0.862588i	$0.331158\pi$
$548$		0			0
$549$	231821.	−	354752.i	0.769145	−	1.17701i
$550$		0			0
$551$			252994.i			0.833312i
$552$		0			0
$553$	−64814.8			−0.211945
$554$		0			0
$555$	−13404.2	−	24771.8i	−0.0435167	−	0.0804213i
$556$		0			0
$557$		−	258715.i		−	0.833896i	−0.908930	−	0.416948i	$-0.863100\pi$
$557$		−	258715.i		−	0.833896i	0.908930	−	0.416948i	$-0.136900\pi$
$558$		0			0
$559$	362377.			1.15968
$560$		0			0
$561$	643640.	−	348279.i	2.04511	−	1.10663i
$562$		0			0
$563$			7185.47i			0.0226693i	0.999936	+	0.0113346i	$0.00360801\pi$
$563$			7185.47i			0.0226693i	−0.999936	+	0.0113346i	$0.996392\pi$
$564$		0			0
$565$	270006.			0.845816
$566$		0			0
$567$	−19738.7	−	45023.9i	−0.0613977	−	0.140048i
$568$		0			0
$569$		−	465206.i		−	1.43688i	−0.695589	−	0.718440i	$-0.744857\pi$
$569$		−	465206.i		−	1.43688i	0.695589	−	0.718440i	$-0.255143\pi$
$570$		0			0
$571$	−176094.			−0.540099			−0.270049	−	0.962847i	$-0.587040\pi$
$571$	−176094.			−0.540099			−0.270049	+	0.962847i	$0.587040\pi$
$572$		0			0
$573$	−171689.	−	317292.i	−0.522919	−	0.966384i
$574$		0			0
$575$		−	54610.0i		−	0.165172i
$576$		0			0
$577$	31818.4			0.0955712			0.0477856	−	0.998858i	$-0.484784\pi$
$577$	31818.4			0.0955712			0.0477856	+	0.998858i	$0.484784\pi$
$578$		0			0
$579$	127498.	−	68990.1i	0.380317	−	0.205793i
$580$		0			0
$581$			58849.2i			0.174337i
$582$		0			0
$583$	−426630.			−1.25520
$584$		0			0
$585$	−204599.	−	133700.i	−0.597849	−	0.390679i
$586$		0			0
$587$			7750.43i			0.0224931i	0.999937	+	0.0112466i	$0.00357997\pi$
$587$			7750.43i			0.0224931i	−0.999937	+	0.0112466i	$0.996420\pi$
$588$		0			0
$589$	35978.9			0.103709
$590$		0			0
$591$	−27231.8	−	50325.9i	−0.0779652	−	0.144084i
$592$		0			0
$593$			91847.4i			0.261191i	0.991436	+	0.130595i	$0.0416889\pi$
$593$			91847.4i			0.261191i	−0.991436	+	0.130595i	$0.958311\pi$
$594$		0			0
$595$	−36709.6			−0.103692
$596$		0			0
$597$	−340123.	+	184044.i	−0.954305	+	0.516383i
$598$		0			0
$599$		−	410.933i		−	0.00114529i	−1.00000		0.000572647i	$-0.999818\pi$
$599$		−	410.933i		−	0.00114529i	1.00000		0.000572647i	$-0.000182279\pi$
$600$		0			0
$601$	597721.			1.65482			0.827408	−	0.561602i	$-0.189815\pi$
$601$	597721.			1.65482			0.827408	+	0.561602i	$0.189815\pi$
$602$		0			0
$603$	33731.9	−	51619.3i	0.0927696	−	0.141964i
$604$		0			0
$605$			221283.i			0.604557i
$606$		0			0
$607$	609937.			1.65542			0.827709	−	0.561158i	$-0.189644\pi$
$607$	609937.			1.65542			0.827709	+	0.561158i	$0.189644\pi$
$608$		0			0
$609$	26113.1	+	48258.6i	0.0704084	+	0.130119i
$610$		0			0
$611$			256622.i			0.687404i
$612$		0			0
$613$	462712.			1.23137			0.615687	−	0.787991i	$-0.288879\pi$
$613$	462712.			1.23137			0.615687	+	0.787991i	$0.288879\pi$
$614$		0			0
$615$	10719.8	−	5800.59i	0.0283424	−	0.0153364i
$616$		0			0
$617$			205753.i			0.540474i	0.962794	+	0.270237i	$0.0871021\pi$
$617$			205753.i			0.540474i	−0.962794	+	0.270237i	$0.912898\pi$
$618$		0			0
$619$	−467560.			−1.22027			−0.610135	−	0.792298i	$-0.708885\pi$
$619$	−467560.			−1.22027			−0.610135	+	0.792298i	$0.708885\pi$
$620$		0			0
$621$	−317394.	−	26347.5i	−0.823029	−	0.0683213i
$622$		0			0
$623$			68763.5i			0.177167i
$624$		0			0
$625$	15625.0			0.0400000
$626$		0			0
$627$	−247122.	−	456695.i	−0.628602	−	1.16169i
$628$		0			0
$629$			122660.i			0.310028i
$630$		0			0
$631$	−639276.			−1.60557			−0.802786	−	0.596267i	$-0.796650\pi$
$631$	−639276.			−1.60557			−0.802786	+	0.596267i	$0.796650\pi$
$632$		0			0
$633$	−121115.	+	65536.2i	−0.302266	+	0.163559i
$634$		0			0
$635$			198778.i			0.492970i
$636$		0			0
$637$	−632843.			−1.55961
$638$		0			0
$639$	141350.	+	92368.6i	0.346174	+	0.226216i
$640$		0			0
$641$			207410.i			0.504794i	0.967624	+	0.252397i	$0.0812189\pi$
$641$			207410.i			0.504794i	−0.967624	+	0.252397i	$0.918781\pi$
$642$		0			0
$643$	396677.			0.959434			0.479717	−	0.877423i	$-0.340739\pi$
$643$	396677.			0.959434			0.479717	+	0.877423i	$0.340739\pi$
$644$		0			0
$645$	64298.1	+	118827.i	0.154553	+	0.285623i
$646$		0			0
$647$			287606.i			0.687052i	0.939143	+	0.343526i	$0.111621\pi$
$647$			287606.i			0.687052i	−0.939143	+	0.343526i	$0.888379\pi$
$648$		0			0
$649$	−1.17879e6			−2.79864
$650$		0			0
$651$	6862.96	−	3713.61i	0.0161938	−	0.00876262i
$652$		0			0
$653$			481395.i			1.12895i	0.825450	+	0.564476i	$0.190922\pi$
$653$			481395.i			1.12895i	−0.825450	+	0.564476i	$0.809078\pi$
$654$		0			0
$655$	−220060.			−0.512931
$656$		0			0
$657$	380990.	−	583022.i	0.882638	−	1.35069i
$658$		0			0
$659$			300201.i			0.691260i	0.938371	+	0.345630i	$0.112335\pi$
$659$			300201.i			0.691260i	−0.938371	+	0.345630i	$0.887665\pi$
$660$		0			0
$661$	−84703.8			−0.193865			−0.0969327	−	0.995291i	$-0.530903\pi$
$661$	−84703.8			−0.193865			−0.0969327	+	0.995291i	$0.530903\pi$
$662$		0			0
$663$	506546.	+	936127.i	1.15237	+	2.12965i
$664$		0			0
$665$			26047.3i			0.0589006i
$666$		0			0
$667$	355477.			0.799025
$668$		0			0
$669$	48678.0	−	26340.1i	0.108763	−	0.0588525i
$670$		0			0
$671$		−	970832.i		−	2.15625i
$672$		0			0
$673$	440538.			0.972643			0.486322	−	0.873780i	$-0.338338\pi$
$673$	440538.			0.972643			0.486322	+	0.873780i	$0.338338\pi$
$674$		0			0
$675$	7538.54	−	90812.6i	0.0165455	−	0.199314i
$676$		0			0
$677$			298268.i			0.650772i	0.945581	+	0.325386i	$0.105494\pi$
$677$			298268.i			0.650772i	−0.945581	+	0.325386i	$0.894506\pi$
$678$		0			0
$679$	−81945.9			−0.177741
$680$		0			0
$681$	78846.1	+	145712.i	0.170015	+	0.314196i
$682$		0			0
$683$			450549.i			0.965829i	0.875668	+	0.482915i	$0.160422\pi$
$683$			450549.i			0.965829i	−0.875668	+	0.482915i	$0.839578\pi$
$684$		0			0
$685$	268916.			0.573106
$686$		0			0
$687$	−619917.	+	335443.i	−1.31347	+	0.710731i
$688$		0			0
$689$		−	620501.i		−	1.30709i
$690$		0			0
$691$	587313.			1.23002			0.615012	−	0.788518i	$-0.289151\pi$
$691$	587313.			1.23002			0.615012	+	0.788518i	$0.289151\pi$
$692$		0			0
$693$	−94276.7	−	61607.4i	−0.196308	−	0.128282i
$694$		0			0
$695$		−	367730.i		−	0.761307i
$696$		0			0
$697$	−53080.3			−0.109262
$698$		0			0
$699$	−51561.0	−	95287.7i	−0.105528	−	0.195022i
$700$		0			0
$701$			521524.i			1.06130i	0.847591	+	0.530650i	$0.178052\pi$
$701$			521524.i			1.06130i	−0.847591	+	0.530650i	$0.821948\pi$
$702$		0			0
$703$	87033.3			0.176106
$704$		0			0
$705$	−84148.6	+	45533.5i	−0.169304	+	0.0916122i
$706$		0			0
$707$			68438.0i			0.136917i
$708$		0			0
$709$	−129165.			−0.256951			−0.128476	−	0.991713i	$-0.541008\pi$
$709$	−129165.			−0.256951			−0.128476	+	0.991713i	$0.541008\pi$
$710$		0			0
$711$	−383287.	+	586537.i	−0.758202	+	1.16026i
$712$		0			0
$713$		−	50553.2i		−	0.0994419i
$714$		0			0
$715$	−559916.			−1.09524
$716$		0			0
$717$	214220.	+	395891.i	0.416698	+	0.770082i
$718$		0			0
$719$		−	747554.i		−	1.44606i	−0.690819	−	0.723028i	$-0.742750\pi$
$719$		−	747554.i		−	1.44606i	0.690819	−	0.723028i	$-0.257250\pi$
$720$		0			0
$721$	35720.4			0.0687141
$722$		0			0
$723$	356349.	−	192824.i	0.681709	−	0.368879i
$724$		0			0
$725$			101709.i			0.193501i
$726$		0			0
$727$	−19031.0			−0.0360075			−0.0180037	−	0.999838i	$-0.505731\pi$
$727$	−19031.0			−0.0360075			−0.0180037	+	0.999838i	$0.505731\pi$
$728$		0			0
$729$	−524167.	−	87628.1i	−0.986312	−	0.164888i
$730$		0			0
$731$		−	588382.i		−	1.10109i
$732$		0			0
$733$	−209303.			−0.389553			−0.194777	−	0.980848i	$-0.562398\pi$
$733$	−209303.			−0.389553			−0.194777	+	0.980848i	$0.562398\pi$
$734$		0			0
$735$	−112288.	−	207514.i	−0.207854	−	0.384126i
$736$		0			0
$737$		−	141264.i		−	0.260074i
$738$		0			0
$739$	247993.			0.454099			0.227049	−	0.973883i	$-0.427092\pi$
$739$	247993.			0.454099			0.227049	+	0.973883i	$0.427092\pi$
$740$		0			0
$741$	664228.	−	359420.i	1.20971	−	0.654585i
$742$		0			0
$743$			247338.i			0.448037i	0.974585	+	0.224018i	$0.0719176\pi$
$743$			247338.i			0.448037i	−0.974585	+	0.224018i	$0.928082\pi$
$744$		0			0
$745$	58851.4			0.106034
$746$		0			0
$747$	532552.	+	348009.i	0.954379	+	0.623662i
$748$		0			0
$749$			10670.1i			0.0190198i
$750$		0			0
$751$	146465.			0.259689			0.129845	−	0.991534i	$-0.458552\pi$
$751$	146465.			0.259689			0.129845	+	0.991534i	$0.458552\pi$
$752$		0			0
$753$	−21970.1	−	40602.0i	−0.0387473	−	0.0716073i
$754$		0			0
$755$		−	408547.i		−	0.716718i
$756$		0			0
$757$	38708.6			0.0675485			0.0337742	−	0.999429i	$-0.489247\pi$
$757$	38708.6			0.0675485			0.0337742	+	0.999429i	$0.489247\pi$
$758$		0			0
$759$	−641693.	+	347226.i	−1.11389	+	0.602738i
$760$		0			0
$761$			712289.i			1.22995i	0.788547	+	0.614974i	$0.210833\pi$
$761$			712289.i			1.22995i	−0.788547	+	0.614974i	$0.789167\pi$
$762$		0			0
$763$	101996.			0.175200
$764$		0			0
$765$	−217085.	+	332201.i	−0.370943	+	0.567647i
$766$		0			0
$767$		−	1.71446e6i		−	2.91432i
$768$		0			0
$769$	−591771.			−1.00069			−0.500347	−	0.865825i	$-0.666794\pi$
$769$	−591771.			−1.00069			−0.500347	+	0.865825i	$0.666794\pi$
$770$		0			0
$771$	−45333.3	−	83778.5i	−0.0762620	−	0.140937i
$772$		0			0
$773$		−	791169.i		−	1.32407i	−0.749474	−	0.662034i	$-0.769694\pi$
$773$		−	791169.i		−	1.32407i	0.749474	−	0.662034i	$-0.230306\pi$
$774$		0			0
$775$	14464.3			0.0240820
$776$		0			0
$777$	16601.6	−	8983.26i	0.0274984	−	0.0148796i
$778$		0			0
$779$			37663.1i			0.0620642i
$780$		0			0
$781$	386826.			0.634181
$782$		0			0
$783$	591134.	+	49071.3i	0.964190	+	0.0800394i
$784$		0			0
$785$		−	29414.2i		−	0.0477329i
$786$		0			0
$787$	90648.7			0.146357			0.0731783	−	0.997319i	$-0.476686\pi$
$787$	90648.7			0.146357			0.0731783	+	0.997319i	$0.476686\pi$
$788$		0			0
$789$	−260674.	−	481740.i	−0.418739	−	0.773853i
$790$		0			0
$791$			180953.i			0.289209i
$792$		0			0
$793$	1.41200e6			2.24538
$794$		0			0
$795$	203467.	−	110098.i	0.321929	−	0.174199i
$796$		0			0
$797$		−	374938.i		−	0.590259i	−0.955457	−	0.295129i	$-0.904637\pi$
$797$		−	374938.i		−	0.590259i	0.955457	−	0.295129i	$-0.0953627\pi$
$798$		0			0
$799$	416670.			0.652678
$800$		0			0
$801$	622271.	+	406638.i	0.969872	+	0.633786i
$802$		0			0
$803$		−	1.59553e6i		−	2.47442i
$804$		0			0
$805$	36598.6			0.0564771
$806$		0			0
$807$	496850.	+	918207.i	0.762918	+	1.40992i
$808$		0			0
$809$		−	302871.i		−	0.462765i	−0.972863	−	0.231383i	$-0.925675\pi$
$809$		−	302871.i		−	0.462765i	0.972863	−	0.231383i	$-0.0743249\pi$
$810$		0			0
$811$	−78720.5			−0.119687			−0.0598434	−	0.998208i	$-0.519060\pi$
$811$	−78720.5			−0.119687			−0.0598434	+	0.998208i	$0.519060\pi$
$812$		0			0
$813$	−706838.	+	382477.i	−1.06940	+	0.578660i
$814$		0			0
$815$			228164.i			0.343505i
$816$		0			0
$817$	−417486.			−0.625458
$818$		0			0
$819$	89603.4	−	137118.i	0.133585	−	0.204422i
$820$		0			0
$821$		−	126623.i		−	0.187856i	−0.995579	−	0.0939280i	$-0.970058\pi$
$821$		−	126623.i		−	0.187856i	0.995579	−	0.0939280i	$-0.0299423\pi$
$822$		0			0
$823$	−574953.			−0.848854			−0.424427	−	0.905462i	$-0.639524\pi$
$823$	−574953.			−0.848854			−0.424427	+	0.905462i	$0.639524\pi$
$824$		0			0
$825$	−99348.2	−	183601.i	−0.145966	−	0.269754i
$826$		0			0
$827$			724955.i			1.05999i	0.848002	+	0.529993i	$0.177805\pi$
$827$			724955.i			1.05999i	−0.848002	+	0.529993i	$0.822195\pi$
$828$		0			0
$829$	643760.			0.936732			0.468366	−	0.883535i	$-0.344843\pi$
$829$	643760.			0.936732			0.468366	+	0.883535i	$0.344843\pi$
$830$		0			0
$831$	−333031.	+	180206.i	−0.482262	+	0.260956i
$832$		0			0
$833$			1.02753e6i			1.48083i
$834$		0			0
$835$	−509238.			−0.730379
$836$		0			0
$837$	6978.53	−	84066.5i	0.00996123	−	0.119997i
$838$		0			0
$839$		−	639908.i		−	0.909063i	−0.890731	−	0.454531i	$-0.849807\pi$
$839$		−	639908.i		−	0.909063i	0.890731	−	0.454531i	$-0.150193\pi$
$840$		0			0
$841$	45217.0			0.0639308
$842$		0			0
$843$	−343084.	−	634039.i	−0.482776	−	0.892197i
$844$		0			0
$845$		−	495035.i		−	0.693302i
$846$		0			0
$847$	−148300.			−0.206716
$848$		0			0
$849$	−475414.	+	257251.i	−0.659564	+	0.356896i
$850$		0			0
$851$		−	122289.i		−	0.168860i
$852$		0			0
$853$	−1.23118e6			−1.69210			−0.846048	−	0.533107i	$-0.821024\pi$
$853$	−1.23118e6			−1.69210			−0.846048	+	0.533107i	$0.821024\pi$
$854$		0			0
$855$	235713.	+	154033.i	0.322442	+	0.210708i
$856$		0			0
$857$			1.33870e6i			1.82273i	0.411597	+	0.911366i	$0.364971\pi$
$857$			1.33870e6i			1.82273i	−0.411597	+	0.911366i	$0.635029\pi$
$858$		0			0
$859$	94388.3			0.127918			0.0639590	−	0.997953i	$-0.479627\pi$
$859$	94388.3			0.127918			0.0639590	+	0.997953i	$0.479627\pi$
$860$		0			0
$861$	3887.45	+	7184.23i	0.00524395	+	0.00969111i
$862$		0			0
$863$			774167.i			1.03947i	0.854327	+	0.519736i	$0.173970\pi$
$863$			774167.i			1.03947i	−0.854327	+	0.519736i	$0.826030\pi$
$864$		0			0
$865$	−201190.			−0.268890
$866$		0			0
$867$	858853.	−	464733.i	1.14256	−	0.618252i
$868$		0			0
$869$			1.60515e6i			2.12557i
$870$		0			0
$871$	205458.			0.270824
$872$		0			0
$873$	−484593.	+	741564.i	−0.635841	+	0.973017i
$874$		0			0
$875$			10471.6i			0.0136772i
$876$		0			0
$877$	−1.12546e6			−1.46329			−0.731647	−	0.681684i	$-0.761248\pi$
$877$	−1.12546e6			−1.46329			−0.731647	+	0.681684i	$0.761248\pi$
$878$		0			0
$879$	214422.	+	396264.i	0.277519	+	0.512870i
$880$		0			0
$881$			474036.i			0.610745i	0.952233	+	0.305372i	$0.0987809\pi$
$881$			474036.i			0.610745i	−0.952233	+	0.305372i	$0.901219\pi$
$882$		0			0
$883$	79072.8			0.101416			0.0507079	−	0.998714i	$-0.483852\pi$
$883$	79072.8			0.101416			0.0507079	+	0.998714i	$0.483852\pi$
$884$		0			0
$885$	562185.	−	304203.i	0.717783	−	0.388399i
$886$		0			0
$887$			366939.i			0.466387i	0.972430	+	0.233194i	$0.0749176\pi$
$887$			366939.i			0.466387i	−0.972430	+	0.233194i	$0.925082\pi$
$888$		0			0
$889$	−133217.			−0.168561
$890$		0			0
$891$	−1.11502e6	+	488831.i	−1.40452	+	0.615749i
$892$		0			0
$893$		−	295648.i		−	0.370742i
$894$		0			0
$895$	59909.3			0.0747907
$896$		0			0
$897$	−505014.	−	933295.i	−0.627651	−	1.15994i
$898$		0			0
$899$			94153.6i			0.116498i
$900$		0			0
$901$	−1.00749e6			−1.24105
$902$		0			0
$903$	−79635.3	+	43091.4i	−0.0976630	+	0.0528463i
$904$		0			0
$905$			475842.i			0.580985i
$906$		0			0
$907$	418936.			0.509252			0.254626	−	0.967040i	$-0.418048\pi$
$907$	418936.			0.509252			0.254626	+	0.967040i	$0.418048\pi$
$908$		0			0
$909$	619325.	+	404713.i	0.749533	+	0.489800i
$910$		0			0
$911$		−	1.16696e6i		−	1.40611i	−0.711135	−	0.703056i	$-0.751818\pi$
$911$		−	1.16696e6i		−	1.40611i	0.711135	−	0.703056i	$-0.248182\pi$
$912$		0			0
$913$	1.45741e6			1.74840
$914$		0			0
$915$	250537.	+	463007.i	0.299247	+	0.553026i
$916$		0			0
$917$		−	147480.i		−	0.175386i
$918$		0			0
$919$	108728.			0.128739			0.0643697	−	0.997926i	$-0.479496\pi$
$919$	108728.			0.128739			0.0643697	+	0.997926i	$0.479496\pi$
$920$		0			0
$921$	299440.	−	162029.i	0.353013	−	0.191018i
$922$		0			0
$923$			562610.i			0.660395i
$924$		0			0
$925$	34989.3			0.0408932
$926$		0			0
$927$	211235.	−	323250.i	0.245814	−	0.376165i
$928$		0			0
$929$		−	139198.i		−	0.161288i	−0.996743	−	0.0806440i	$-0.974302\pi$
$929$		−	139198.i		−	0.161288i	0.996743	−	0.0806440i	$-0.0256977\pi$
$930$		0			0
$931$	729083.			0.841158
$932$		0			0
$933$	−575749.	−	1.06402e6i	−0.661409	−	1.22232i
$934$		0			0
$935$			909120.i			1.03992i
$936$		0			0
$937$	−1.66932e6			−1.90135			−0.950673	−	0.310196i	$-0.899605\pi$
$937$	−1.66932e6			−1.90135			−0.950673	+	0.310196i	$0.899605\pi$
$938$		0			0
$939$	−743868.	+	402514.i	−0.843655	+	0.456509i
$940$		0			0
$941$		−	309587.i		−	0.349626i	−0.984602	−	0.174813i	$-0.944068\pi$
$941$		−	309587.i		−	0.349626i	0.984602	−	0.174813i	$-0.0559321\pi$
$942$		0			0
$943$	52919.7			0.0595106
$944$		0			0
$945$	60860.9	+	5052.19i	0.0681514	+	0.00565739i
$946$		0			0
$947$			200056.i			0.223076i	0.993760	+	0.111538i	$0.0355776\pi$
$947$			200056.i			0.223076i	−0.993760	+	0.111538i	$0.964422\pi$
$948$		0			0
$949$	2.32058e6			2.57670
$950$		0			0
$951$	−828640.	−	1.53137e6i	−0.916231	−	1.69325i
$952$		0			0
$953$		−	33737.1i		−	0.0371469i	−0.999827	−	0.0185734i	$-0.994088\pi$
$953$		−	33737.1i		−	0.0371469i	0.999827	−	0.0185734i	$-0.00591245\pi$
$954$		0			0
$955$	448164.			0.491394
$956$		0			0
$957$	1.19513e6	−	646696.i	1.30494	−	0.706116i
$958$		0			0
$959$			180222.i			0.195962i
$960$		0			0
$961$	−910131.			−0.985501
$962$		0			0
$963$	96558.6	+	63098.5i	0.104121	+	0.0680404i
$964$		0			0
$965$			180086.i			0.193386i
$966$		0			0
$967$	−423000.			−0.452363			−0.226182	−	0.974085i	$-0.572624\pi$
$967$	−423000.			−0.452363			−0.226182	+	0.974085i	$0.572624\pi$
$968$		0			0
$969$	−583580.	−	1.07849e6i	−0.621516	−	1.14860i
$970$		0			0
$971$			1.28807e6i			1.36616i	0.730344	+	0.683080i	$0.239360\pi$
$971$			1.28807e6i			1.36616i	−0.730344	+	0.683080i	$0.760640\pi$
$972$		0			0
$973$	246446.			0.260313
$974$		0			0
$975$	267034.	−	144495.i	0.280904	−	0.152000i
$976$		0			0
$977$		−	1.20117e6i		−	1.25839i	−0.777247	−	0.629196i	$-0.783384\pi$
$977$		−	1.20117e6i		−	1.25839i	0.777247	−	0.629196i	$-0.216616\pi$
$978$		0			0
$979$	1.70294e6			1.77678
$980$		0			0
$981$	603160.	−	923006.i	0.626750	−	0.959105i
$982$		0			0
$983$		−	491587.i		−	0.508737i	−0.967107	−	0.254369i	$-0.918132\pi$
$983$		−	491587.i		−	0.508737i	0.967107	−	0.254369i	$-0.0818676\pi$
$984$		0			0
$985$	71083.5			0.0732650
$986$		0			0
$987$	−30515.7	−	56394.8i	−0.0313249	−	0.0578902i
$988$		0			0
$989$			586602.i			0.599723i
$990$		0			0
$991$	−543192.			−0.553103			−0.276551	−	0.960999i	$-0.589192\pi$
$991$	−543192.			−0.553103			−0.276551	+	0.960999i	$0.589192\pi$
$992$		0			0
$993$	−372553.	+	201592.i	−0.377824	+	0.204444i
$994$		0			0
$995$		−	480412.i		−	0.485252i
$996$		0			0
$997$	853733.			0.858879			0.429439	−	0.903096i	$-0.358711\pi$
$997$	853733.			0.858879			0.429439	+	0.903096i	$0.358711\pi$
$998$		0			0
$999$	16881.2	−	203358.i	0.0169150	−	0.203765i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.5.g.b.41.2	yes	4
3.2	odd	2	inner	60.5.g.b.41.1	✓	4
4.3	odd	2		240.5.l.c.161.3		4
5.2	odd	4		300.5.b.d.149.6		8
5.3	odd	4		300.5.b.d.149.3		8
5.4	even	2		300.5.g.g.101.3		4
12.11	even	2		240.5.l.c.161.4		4
15.2	even	4		300.5.b.d.149.4		8
15.8	even	4		300.5.b.d.149.5		8
15.14	odd	2		300.5.g.g.101.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.5.g.b.41.1	✓	4	3.2	odd	2	inner
60.5.g.b.41.2	yes	4	1.1	even	1	trivial
240.5.l.c.161.3		4	4.3	odd	2
240.5.l.c.161.4		4	12.11	even	2
300.5.b.d.149.3		8	5.3	odd	4
300.5.b.d.149.4		8	15.2	even	4
300.5.b.d.149.5		8	15.8	even	4
300.5.b.d.149.6		8	5.2	odd	4
300.5.g.g.101.3		4	5.4	even	2
300.5.g.g.101.4		4	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	60.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.91548 + 4.28313i) q^{3} -11.1803i q^{5} +7.49286 q^{7} +(44.3095 - 67.8061i) q^{9} +O(q^{10})\)	\(q+(-7.91548 + 4.28313i) q^{3} -11.1803i q^{5} +7.49286 q^{7} +(44.3095 - 67.8061i) q^{9} -185.562i q^{11} +269.886 q^{13} +(47.8869 + 88.4977i) q^{15} -438.205i q^{17} -310.929 q^{19} +(-59.3095 + 32.0929i) q^{21} +436.880i q^{23} -125.000 q^{25} +(-60.3083 + 726.501i) q^{27} -813.673i q^{29} -115.714 q^{31} +(794.786 + 1468.81i) q^{33} -83.7727i q^{35} -279.914 q^{37} +(-2136.27 + 1155.96i) q^{39} -121.131i q^{41} +1342.71 q^{43} +(-758.095 - 495.395i) q^{45} +950.856i q^{47} -2344.86 q^{49} +(1876.89 + 3468.60i) q^{51} -2299.13i q^{53} -2074.64 q^{55} +(2461.15 - 1331.75i) q^{57} -6352.54i q^{59} +5231.86 q^{61} +(332.005 - 508.061i) q^{63} -3017.41i q^{65} +761.278 q^{67} +(-1871.21 - 3458.11i) q^{69} +2084.62i q^{71} +8598.37 q^{73} +(989.434 - 535.392i) q^{75} -1390.39i q^{77} -8650.21 q^{79} +(-2634.33 - 6008.91i) q^{81} +7854.04i q^{83} -4899.29 q^{85} +(3485.07 + 6440.61i) q^{87} +9177.21i q^{89} +2022.21 q^{91} +(915.933 - 495.620i) q^{93} +3476.29i q^{95} -10936.5 q^{97} +(-12582.2 - 8222.15i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9}+O(q^{10}) \) 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9	\( 4 q - 20 q^{3} - 40 q^{7} - 56 q^{9} - 40 q^{13} - 100 q^{15} - 544 q^{19} - 4 q^{21} - 500 q^{25} + 820 q^{27} + 2336 q^{31} + 1080 q^{33} - 280 q^{37} - 3064 q^{39} + 7400 q^{43} - 700 q^{45} - 7980 q^{49} + 1560 q^{51} - 4800 q^{55} + 4760 q^{57} - 64 q^{61} + 4640 q^{63} - 8920 q^{67} - 19380 q^{69} + 18440 q^{73} + 2500 q^{75} - 7312 q^{79} + 2524 q^{81} - 12600 q^{85} + 14640 q^{87} + 19984 q^{91} - 3520 q^{93} - 9880 q^{97} - 23040 q^{99}+O(q^{100}) \) 4 * q - 20 * q^3 - 40 * q^7 - 56 * q^9 - 40 * q^13 - 100 * q^15 - 544 * q^19 - 4 * q^21 - 500 * q^25 + 820 * q^27 + 2336 * q^31 + 1080 * q^33 - 280 * q^37 - 3064 * q^39 + 7400 * q^43 - 700 * q^45 - 7980 * q^49 + 1560 * q^51 - 4800 * q^55 + 4760 * q^57 - 64 * q^61 + 4640 * q^63 - 8920 * q^67 - 19380 * q^69 + 18440 * q^73 + 2500 * q^75 - 7312 * q^79 + 2524 * q^81 - 12600 * q^85 + 14640 * q^87 + 19984 * q^91 - 3520 * q^93 - 9880 * q^97 - 23040 * q^99

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