LMFDB - Embedded newform 1452.3.e.i.485.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,3,Mod(485,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.485"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1452.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,-16,0,-6,0,0,0,6,0,-12,0,0,0,-12,0,-4,0,0,0,-44, 0,10,0,0,0,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)

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

Self dual:	no
Analytic conductor:	$39.5641343851$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.46805967296.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 6x^{4} + 2x^{3} + 62x^{2} + 174x + 486 $ x^6 - 2x^5 + 6x^4 + 2x^3 + 62x^2 + 174*x + 486
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.6
Root			$-1.69245 - 1.32314i$ of defining polynomial
Character	$\chi$	$=$	1452.485
Dual form			1452.3.e.i.485.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.69245 + 1.32314i) q^{3} +8.37865i q^{5} -7.27992 q^{7} +(5.49862 + 7.12497i) q^{9} +17.4871 q^{13} +(-11.0861 + 22.5591i) q^{15} +19.9823i q^{17} +20.5046 q^{19} +(-19.6009 - 9.63233i) q^{21} +31.1461i q^{23} -45.2017 q^{25} +(5.37749 + 26.4591i) q^{27} -45.5581i q^{29} -8.48714 q^{31} -60.9959i q^{35} +26.5073 q^{37} +(47.0833 + 23.1379i) q^{39} +26.6247i q^{41} -6.57729 q^{43} +(-59.6976 + 46.0710i) q^{45} +9.84434i q^{47} +3.99724 q^{49} +(-26.4393 + 53.8015i) q^{51} +80.9782i q^{53} +(55.2077 + 27.1304i) q^{57} -73.6716i q^{59} -81.2541 q^{61} +(-40.0295 - 51.8692i) q^{63} +146.519i q^{65} -86.4357 q^{67} +(-41.2106 + 83.8596i) q^{69} -105.088i q^{71} -106.036 q^{73} +(-121.704 - 59.8080i) q^{75} +74.6567 q^{79} +(-20.5303 + 78.3550i) q^{81} -56.4514i q^{83} -167.425 q^{85} +(60.2795 - 122.663i) q^{87} -45.8286i q^{89} -127.305 q^{91} +(-22.8512 - 11.2296i) q^{93} +171.801i q^{95} +5.58831 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 16 q^{7} - 6 q^{9} + 6 q^{13} - 12 q^{15} - 12 q^{19} - 4 q^{21} - 44 q^{25} + 10 q^{27} + 48 q^{31} + 102 q^{37} + 150 q^{39} + 52 q^{43} - 14 q^{45} - 54 q^{49} + 22 q^{51} + 8 q^{57}+ \cdots + 254 q^{97}+O(q^{100}) $ 6 * q + 4 * q^3 - 16 * q^7 - 6 * q^9 + 6 * q^13 - 12 * q^15 - 12 * q^19 - 4 * q^21 - 44 * q^25 + 10 * q^27 + 48 * q^31 + 102 * q^37 + 150 * q^39 + 52 * q^43 - 14 * q^45 - 54 * q^49 + 22 * q^51 + 8 * q^57 - 152 * q^61 - 122 * q^63 - 24 * q^67 - 112 * q^69 - 280 * q^73 - 290 * q^75 + 272 * q^79 - 186 * q^81 - 198 * q^85 - 154 * q^87 - 68 * q^91 - 114 * q^93 + 254 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	2.69245	+	1.32314i	0.897485	+	0.441045i
$3$	2.69245	+	1.32314i	0.897485	+	0.441045i
$4$		0			0
$5$			8.37865i			1.67573i	0.545878	+	0.837865i	$0.316196\pi$
$5$			8.37865i			1.67573i	−0.545878	+	0.837865i	$0.683804\pi$
$6$		0			0
$7$	−7.27992			−1.03999			−0.519994	−	0.854170i	$-0.674066\pi$
$7$	−7.27992			−1.03999			−0.519994	+	0.854170i	$0.674066\pi$
$8$		0			0
$9$	5.49862	+	7.12497i	0.610958	+	0.791663i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$	17.4871			1.34516			0.672582	−	0.740022i	$-0.265185\pi$
$13$	17.4871			1.34516			0.672582	+	0.740022i	$0.265185\pi$
$14$		0			0
$15$	−11.0861	+	22.5591i	−0.739073	+	1.50394i
$16$		0			0
$17$			19.9823i			1.17543i	0.809068	+	0.587715i	$0.199972\pi$
$17$			19.9823i			1.17543i	−0.809068	+	0.587715i	$0.800028\pi$
$18$		0			0
$19$	20.5046			1.07919			0.539595	−	0.841925i	$-0.318578\pi$
$19$	20.5046			1.07919			0.539595	+	0.841925i	$0.318578\pi$
$20$		0			0
$21$	−19.6009	−	9.63233i	−0.933374	−	0.458682i
$22$		0			0
$23$			31.1461i			1.35418i	0.735900	+	0.677090i	$0.236759\pi$
$23$			31.1461i			1.35418i	−0.735900	+	0.677090i	$0.763241\pi$
$24$		0			0
$25$	−45.2017			−1.80807
$26$		0			0
$27$	5.37749	+	26.4591i	0.199166	+	0.979966i
$28$		0			0
$29$		−	45.5581i		−	1.57097i	−0.618882	−	0.785484i	$-0.712414\pi$
$29$		−	45.5581i		−	1.57097i	0.618882	−	0.785484i	$-0.287586\pi$
$30$		0			0
$31$	−8.48714			−0.273779			−0.136889	−	0.990586i	$-0.543710\pi$
$31$	−8.48714			−0.273779			−0.136889	+	0.990586i	$0.543710\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	60.9959i		−	1.74274i
$36$		0			0
$37$	26.5073			0.716415			0.358207	−	0.933642i	$-0.383388\pi$
$37$	26.5073			0.716415			0.358207	+	0.933642i	$0.383388\pi$
$38$		0			0
$39$	47.0833	+	23.1379i	1.20726	+	0.593279i
$40$		0			0
$41$			26.6247i			0.649383i	0.945820	+	0.324692i	$0.105261\pi$
$41$			26.6247i			0.649383i	−0.945820	+	0.324692i	$0.894739\pi$
$42$		0			0
$43$	−6.57729			−0.152960			−0.0764801	−	0.997071i	$-0.524368\pi$
$43$	−6.57729			−0.152960			−0.0764801	+	0.997071i	$0.524368\pi$
$44$		0			0
$45$	−59.6976	+	46.0710i	−1.32661	+	1.02380i
$46$		0			0
$47$			9.84434i			0.209454i	0.994501	+	0.104727i	$0.0333969\pi$
$47$			9.84434i			0.209454i	−0.994501	+	0.104727i	$0.966603\pi$
$48$		0			0
$49$	3.99724			0.0815764
$50$		0			0
$51$	−26.4393	+	53.8015i	−0.518418	+	1.05493i
$52$		0			0
$53$			80.9782i			1.52789i	0.645281	+	0.763945i	$0.276740\pi$
$53$			80.9782i			1.52789i	−0.645281	+	0.763945i	$0.723260\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	55.2077	+	27.1304i	0.968556	+	0.475971i
$58$		0			0
$59$		−	73.6716i		−	1.24867i	−0.781156	−	0.624335i	$-0.785370\pi$
$59$		−	73.6716i		−	1.24867i	0.781156	−	0.624335i	$-0.214630\pi$
$60$		0			0
$61$	−81.2541			−1.33203			−0.666017	−	0.745937i	$-0.732002\pi$
$61$	−81.2541			−1.33203			−0.666017	+	0.745937i	$0.732002\pi$
$62$		0			0
$63$	−40.0295	−	51.8692i	−0.635389	−	0.823321i
$64$		0			0
$65$			146.519i			2.25413i
$66$		0			0
$67$	−86.4357			−1.29009			−0.645043	−	0.764147i	$-0.723160\pi$
$67$	−86.4357			−1.29009			−0.645043	+	0.764147i	$0.723160\pi$
$68$		0			0
$69$	−41.2106	+	83.8596i	−0.597255	+	1.21536i
$70$		0			0
$71$		−	105.088i		−	1.48012i	−0.672543	−	0.740058i	$-0.734798\pi$
$71$		−	105.088i		−	1.48012i	0.672543	−	0.740058i	$-0.265202\pi$
$72$		0			0
$73$	−106.036			−1.45255			−0.726275	−	0.687405i	$-0.758750\pi$
$73$	−106.036			−1.45255			−0.726275	+	0.687405i	$0.758750\pi$
$74$		0			0
$75$	−121.704	−	59.8080i	−1.62271	−	0.797440i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	74.6567			0.945022			0.472511	−	0.881325i	$-0.343348\pi$
$79$	74.6567			0.945022			0.472511	+	0.881325i	$0.343348\pi$
$80$		0			0
$81$	−20.5303	+	78.3550i	−0.253461	+	0.967346i
$82$		0			0
$83$		−	56.4514i		−	0.680137i	−0.940401	−	0.340069i	$-0.889550\pi$
$83$		−	56.4514i		−	0.680137i	0.940401	−	0.340069i	$-0.110450\pi$
$84$		0			0
$85$	−167.425			−1.96970
$86$		0			0
$87$	60.2795	−	122.663i	0.692868	−	1.40992i
$88$		0			0
$89$		−	45.8286i		−	0.514928i	−0.966288	−	0.257464i	$-0.917113\pi$
$89$		−	45.8286i		−	0.514928i	0.966288	−	0.257464i	$-0.0828869\pi$
$90$		0			0
$91$	−127.305			−1.39896
$92$		0			0
$93$	−22.8512	−	11.2296i	−0.245712	−	0.120749i
$94$		0			0
$95$			171.801i			1.80843i
$96$		0			0
$97$	5.58831			0.0576115			0.0288057	−	0.999585i	$-0.490830\pi$
$97$	5.58831			0.0576115			0.0288057	+	0.999585i	$0.490830\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	112.903i		−	1.11785i	−0.829219	−	0.558924i	$-0.811214\pi$
$101$		−	112.903i		−	1.11785i	0.829219	−	0.558924i	$-0.188786\pi$
$102$		0			0
$103$	84.9164			0.824431			0.412216	−	0.911086i	$-0.364755\pi$
$103$	84.9164			0.824431			0.412216	+	0.911086i	$0.364755\pi$
$104$		0			0
$105$	80.7058	−	164.229i	0.768627	−	1.56408i
$106$		0			0
$107$			40.7199i			0.380560i	0.981730	+	0.190280i	$0.0609396\pi$
$107$			40.7199i			0.380560i	−0.981730	+	0.190280i	$0.939060\pi$
$108$		0			0
$109$	38.3607			0.351933			0.175967	−	0.984396i	$-0.443695\pi$
$109$	38.3607			0.351933			0.175967	+	0.984396i	$0.443695\pi$
$110$		0			0
$111$	71.3698	+	35.0728i	0.642971	+	0.315971i
$112$		0			0
$113$			89.2556i			0.789872i	0.918709	+	0.394936i	$0.129233\pi$
$113$			89.2556i			0.789872i	−0.918709	+	0.394936i	$0.870767\pi$
$114$		0			0
$115$	−260.962			−2.26924
$116$		0			0
$117$	96.1552	+	124.595i	0.821839	+	1.06492i
$118$		0			0
$119$		−	145.470i		−	1.22243i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	−35.2281	+	71.6859i	−0.286408	+	0.582812i
$124$		0			0
$125$		−	169.263i		−	1.35410i
$126$		0			0
$127$	213.975			1.68484			0.842420	−	0.538822i	$-0.181131\pi$
$127$	213.975			1.68484			0.842420	+	0.538822i	$0.181131\pi$
$128$		0			0
$129$	−17.7091	−	8.70265i	−0.137280	−	0.0674624i
$130$		0			0
$131$		−	132.940i		−	1.01481i	−0.861708	−	0.507405i	$-0.830605\pi$
$131$		−	132.940i		−	1.01481i	0.861708	−	0.507405i	$-0.169395\pi$
$132$		0			0
$133$	−149.272			−1.12234
$134$		0			0
$135$	−221.691	+	45.0561i	−1.64216	+	0.333749i
$136$		0			0
$137$			79.2420i			0.578408i	0.957267	+	0.289204i	$0.0933907\pi$
$137$			79.2420i			0.578408i	−0.957267	+	0.289204i	$0.906609\pi$
$138$		0			0
$139$	−183.265			−1.31845			−0.659225	−	0.751946i	$-0.729116\pi$
$139$	−183.265			−1.31845			−0.659225	+	0.751946i	$0.729116\pi$
$140$		0			0
$141$	−13.0254	+	26.5054i	−0.0923787	+	0.187982i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	381.715			2.63252
$146$		0			0
$147$	10.7624	+	5.28890i	0.0732136	+	0.0359789i
$148$		0			0
$149$		−	109.756i		−	0.736617i	−0.929704	−	0.368308i	$-0.879937\pi$
$149$		−	109.756i		−	0.736617i	0.929704	−	0.368308i	$-0.120063\pi$
$150$		0			0
$151$	−160.632			−1.06379			−0.531894	−	0.846811i	$-0.678520\pi$
$151$	−160.632			−1.06379			−0.531894	+	0.846811i	$0.678520\pi$
$152$		0			0
$153$	−142.373	+	109.875i	−0.930544	+	0.718138i
$154$		0			0
$155$		−	71.1108i		−	0.458779i
$156$		0			0
$157$	27.0429			0.172248			0.0861238	−	0.996284i	$-0.472552\pi$
$157$	27.0429			0.172248			0.0861238	+	0.996284i	$0.472552\pi$
$158$		0			0
$159$	−107.145	+	218.030i	−0.673869	+	1.37126i
$160$		0			0
$161$		−	226.741i		−	1.40833i
$162$		0			0
$163$	239.660			1.47031			0.735155	−	0.677900i	$-0.237110\pi$
$163$	239.660			1.47031			0.735155	+	0.677900i	$0.237110\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	19.9272i		−	0.119325i	−0.998219	−	0.0596624i	$-0.980998\pi$
$167$		−	19.9272i		−	0.119325i	0.998219	−	0.0596624i	$-0.0190024\pi$
$168$		0			0
$169$	136.800			0.809468
$170$		0			0
$171$	112.747	+	146.095i	0.659339	+	0.854354i
$172$		0			0
$173$			310.628i			1.79554i	0.440468	+	0.897769i	$0.354813\pi$
$173$			310.628i			1.79554i	−0.440468	+	0.897769i	$0.645187\pi$
$174$		0			0
$175$	329.065			1.88037
$176$		0			0
$177$	97.4775	−	198.357i	0.550720	−	1.12066i
$178$		0			0
$179$		−	9.96867i		−	0.0556909i	−0.999612	−	0.0278455i	$-0.991135\pi$
$179$		−	9.96867i		−	0.0556909i	0.999612	−	0.0278455i	$-0.00886463\pi$
$180$		0			0
$181$	309.372			1.70924			0.854618	−	0.519257i	$-0.173791\pi$
$181$	309.372			1.70924			0.854618	+	0.519257i	$0.173791\pi$
$182$		0			0
$183$	−218.773	−	107.510i	−1.19548	−	0.587487i
$184$		0			0
$185$			222.096i			1.20052i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−39.1477	−	192.620i	−0.207131	−	1.01915i
$190$		0			0
$191$			116.083i			0.607763i	0.952710	+	0.303882i	$0.0982827\pi$
$191$			116.083i			0.607763i	−0.952710	+	0.303882i	$0.901717\pi$
$192$		0			0
$193$	−14.9939			−0.0776884			−0.0388442	−	0.999245i	$-0.512368\pi$
$193$	−14.9939			−0.0776884			−0.0388442	+	0.999245i	$0.512368\pi$
$194$		0			0
$195$	−193.864	+	394.495i	−0.994174	+	2.02305i
$196$		0			0
$197$		−	159.048i		−	0.807351i	−0.914902	−	0.403675i	$-0.867733\pi$
$197$		−	159.048i		−	0.807351i	0.914902	−	0.403675i	$-0.132267\pi$
$198$		0			0
$199$	249.578			1.25416			0.627079	−	0.778955i	$-0.284250\pi$
$199$	249.578			1.25416			0.627079	+	0.778955i	$0.284250\pi$
$200$		0			0
$201$	−232.724	−	114.366i	−1.15783	−	0.568986i
$202$		0			0
$203$			331.659i			1.63379i
$204$		0			0
$205$	−223.079			−1.08819
$206$		0			0
$207$	−221.915	+	171.261i	−1.07205	+	0.827347i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	22.2657			0.105525			0.0527624	−	0.998607i	$-0.483197\pi$
$211$	22.2657			0.105525			0.0527624	+	0.998607i	$0.483197\pi$
$212$		0			0
$213$	139.046	−	282.945i	0.652798	−	1.32838i
$214$		0			0
$215$		−	55.1088i		−	0.256320i
$216$		0			0
$217$	61.7857			0.284727
$218$		0			0
$219$	−285.497	−	140.300i	−1.30364	−	0.640640i
$220$		0			0
$221$			349.433i			1.58115i
$222$		0			0
$223$	59.9979			0.269049			0.134524	−	0.990910i	$-0.457049\pi$
$223$	59.9979			0.269049			0.134524	+	0.990910i	$0.457049\pi$
$224$		0			0
$225$	−248.547	−	322.061i	−1.10465	−	1.43138i
$226$		0			0
$227$			139.821i			0.615952i	0.951394	+	0.307976i	$0.0996516\pi$
$227$			139.821i			0.615952i	−0.951394	+	0.307976i	$0.900348\pi$
$228$		0			0
$229$	−104.494			−0.456306			−0.228153	−	0.973625i	$-0.573269\pi$
$229$	−104.494			−0.456306			−0.228153	+	0.973625i	$0.573269\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			162.489i			0.697376i	0.937239	+	0.348688i	$0.113373\pi$
$233$			162.489i			0.697376i	−0.937239	+	0.348688i	$0.886627\pi$
$234$		0			0
$235$	−82.4823			−0.350988
$236$		0			0
$237$	201.010	+	98.7810i	0.848143	+	0.416797i
$238$		0			0
$239$		−	96.7647i		−	0.404873i	−0.979295	−	0.202437i	$-0.935114\pi$
$239$		−	96.7647i		−	0.404873i	0.979295	−	0.202437i	$-0.0648860\pi$
$240$		0			0
$241$	28.3306			0.117554			0.0587771	−	0.998271i	$-0.481280\pi$
$241$	28.3306			0.117554			0.0587771	+	0.998271i	$0.481280\pi$
$242$		0			0
$243$	−158.951	+	183.803i	−0.654120	+	0.756390i
$244$		0			0
$245$			33.4915i			0.136700i
$246$		0			0
$247$	358.567			1.45169
$248$		0			0
$249$	74.6928	−	151.993i	0.299971	−	0.610413i
$250$		0			0
$251$		−	107.795i		−	0.429463i	−0.976673	−	0.214732i	$-0.931112\pi$
$251$		−	107.795i		−	0.429463i	0.976673	−	0.214732i	$-0.0688877\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−450.783	−	221.526i	−1.76778	−	0.868728i
$256$		0			0
$257$			317.215i			1.23430i	0.786845	+	0.617150i	$0.211713\pi$
$257$			317.215i			1.23430i	−0.786845	+	0.617150i	$0.788287\pi$
$258$		0			0
$259$	−192.971			−0.745063
$260$		0			0
$261$	324.600	−	250.507i	1.24368	−	0.959796i
$262$		0			0
$263$		−	116.343i		−	0.442370i	−0.975232	−	0.221185i	$-0.929008\pi$
$263$		−	116.343i		−	0.442370i	0.975232	−	0.221185i	$-0.0709924\pi$
$264$		0			0
$265$	−678.488			−2.56033
$266$		0			0
$267$	60.6375	−	123.391i	0.227107	−	0.462140i
$268$		0			0
$269$			90.3058i			0.335709i	0.985812	+	0.167855i	$0.0536839\pi$
$269$			90.3058i			0.335709i	−0.985812	+	0.167855i	$0.946316\pi$
$270$		0			0
$271$	157.530			0.581290			0.290645	−	0.956831i	$-0.406130\pi$
$271$	157.530			0.581290			0.290645	+	0.956831i	$0.406130\pi$
$272$		0			0
$273$	−342.763	−	168.442i	−1.25554	−	0.617003i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	185.978			0.671402			0.335701	−	0.941969i	$-0.391027\pi$
$277$	185.978			0.671402			0.335701	+	0.941969i	$0.391027\pi$
$278$		0			0
$279$	−46.6676	−	60.4706i	−0.167267	−	0.216741i
$280$		0			0
$281$			470.138i			1.67309i	0.547900	+	0.836544i	$0.315427\pi$
$281$			470.138i			1.67309i	−0.547900	+	0.836544i	$0.684573\pi$
$282$		0			0
$283$	−238.743			−0.843616			−0.421808	−	0.906685i	$-0.638604\pi$
$283$	−238.743			−0.843616			−0.421808	+	0.906685i	$0.638604\pi$
$284$		0			0
$285$	−227.316	+	462.566i	−0.797599	+	1.62304i
$286$		0			0
$287$		−	193.826i		−	0.675351i
$288$		0			0
$289$	−110.293			−0.381635
$290$		0			0
$291$	15.0463	+	7.39410i	0.0517054	+	0.0254093i
$292$		0			0
$293$			507.781i			1.73304i	0.499141	+	0.866521i	$0.333649\pi$
$293$			507.781i			1.73304i	−0.499141	+	0.866521i	$0.666351\pi$
$294$		0			0
$295$	617.268			2.09243
$296$		0			0
$297$		0			0
$298$		0			0
$299$			544.657i			1.82160i
$300$		0			0
$301$	47.8822			0.159077
$302$		0			0
$303$	149.386	−	303.986i	0.493022	−	1.00325i
$304$		0			0
$305$		−	680.799i		−	2.23213i
$306$		0			0
$307$	464.465			1.51291			0.756457	−	0.654043i	$-0.226928\pi$
$307$	464.465			1.51291			0.756457	+	0.654043i	$0.226928\pi$
$308$		0			0
$309$	228.634	+	112.356i	0.739915	+	0.363612i
$310$		0			0
$311$			540.293i			1.73728i	0.495447	+	0.868638i	$0.335004\pi$
$311$			540.293i			1.73728i	−0.495447	+	0.868638i	$0.664996\pi$
$312$		0			0
$313$	−236.564			−0.755796			−0.377898	−	0.925847i	$-0.623353\pi$
$313$	−236.564			−0.755796			−0.377898	+	0.925847i	$0.623353\pi$
$314$		0			0
$315$	434.594	−	335.393i	1.37966	−	1.06474i
$316$		0			0
$317$		−	175.782i		−	0.554518i	−0.960795	−	0.277259i	$-0.910574\pi$
$317$		−	175.782i		−	0.554518i	0.960795	−	0.277259i	$-0.0894260\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−53.8780	+	109.637i	−0.167844	+	0.341547i
$322$		0			0
$323$			409.729i			1.26851i
$324$		0			0
$325$	−790.449			−2.43215
$326$		0			0
$327$	103.285	+	50.7565i	0.315855	+	0.155219i
$328$		0			0
$329$		−	71.6660i		−	0.217830i
$330$		0			0
$331$	231.001			0.697888			0.348944	−	0.937144i	$-0.386540\pi$
$331$	231.001			0.697888			0.348944	+	0.937144i	$0.386540\pi$
$332$		0			0
$333$	145.754	+	188.864i	0.437699	+	0.567159i
$334$		0			0
$335$		−	724.214i		−	2.16183i
$336$		0			0
$337$	11.6789			0.0346555			0.0173277	−	0.999850i	$-0.494484\pi$
$337$	11.6789			0.0346555			0.0173277	+	0.999850i	$0.494484\pi$
$338$		0			0
$339$	−118.097	+	240.317i	−0.348370	+	0.708899i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	327.616			0.955150
$344$		0			0
$345$	−702.630	−	345.289i	−2.03661	−	1.00084i
$346$		0			0
$347$		−	559.872i		−	1.61346i	−0.590918	−	0.806731i	$-0.701234\pi$
$347$		−	559.872i		−	1.61346i	0.590918	−	0.806731i	$-0.298766\pi$
$348$		0			0
$349$	528.668			1.51481			0.757404	−	0.652947i	$-0.226467\pi$
$349$	528.668			1.51481			0.757404	+	0.652947i	$0.226467\pi$
$350$		0			0
$351$	94.0369	+	462.694i	0.267911	+	1.31822i
$352$		0			0
$353$			218.146i			0.617978i	0.951066	+	0.308989i	$0.0999906\pi$
$353$			218.146i			0.617978i	−0.951066	+	0.308989i	$0.900009\pi$
$354$		0			0
$355$	880.497			2.48027
$356$		0			0
$357$	192.476	−	391.670i	0.539149	−	1.09712i
$358$		0			0
$359$			243.577i			0.678487i	0.940699	+	0.339244i	$0.110171\pi$
$359$			243.577i			0.678487i	−0.940699	+	0.339244i	$0.889829\pi$
$360$		0			0
$361$	59.4383			0.164649
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	888.439i		−	2.43408i
$366$		0			0
$367$	−21.2934			−0.0580200			−0.0290100	−	0.999579i	$-0.509235\pi$
$367$	−21.2934			−0.0580200			−0.0290100	+	0.999579i	$0.509235\pi$
$368$		0			0
$369$	−189.700	+	146.399i	−0.514093	+	0.396746i
$370$		0			0
$371$		−	589.515i		−	1.58899i
$372$		0			0
$373$	−557.973			−1.49591			−0.747953	−	0.663752i	$-0.768963\pi$
$373$	−557.973			−1.49591			−0.747953	+	0.663752i	$0.768963\pi$
$374$		0			0
$375$	223.958	−	455.733i	0.597221	−	1.21529i
$376$		0			0
$377$		−	796.681i		−	2.11321i
$378$		0			0
$379$	−304.928			−0.804559			−0.402280	−	0.915517i	$-0.631782\pi$
$379$	−304.928			−0.804559			−0.402280	+	0.915517i	$0.631782\pi$
$380$		0			0
$381$	576.117	+	283.117i	1.51212	+	0.743090i
$382$		0			0
$383$		−	104.006i		−	0.271556i	−0.990739	−	0.135778i	$-0.956647\pi$
$383$		−	104.006i		−	0.271556i	0.990739	−	0.135778i	$-0.0433535\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−36.1660	−	46.8630i	−0.0934523	−	0.121093i
$388$		0			0
$389$			242.201i			0.622625i	0.950308	+	0.311312i	$0.100769\pi$
$389$			242.201i			0.622625i	−0.950308	+	0.311312i	$0.899231\pi$
$390$		0			0
$391$	−622.372			−1.59174
$392$		0			0
$393$	175.898	−	357.935i	0.447577	−	0.910777i
$394$		0			0
$395$			625.522i			1.58360i
$396$		0			0
$397$	−7.60465			−0.0191553			−0.00957764	−	0.999954i	$-0.503049\pi$
$397$	−7.60465			−0.0191553			−0.00957764	+	0.999954i	$0.503049\pi$
$398$		0			0
$399$	−401.908	−	197.507i	−1.00729	−	0.495005i
$400$		0			0
$401$			91.6483i			0.228549i	0.993449	+	0.114275i	$0.0364544\pi$
$401$			91.6483i			0.228549i	−0.993449	+	0.114275i	$0.963546\pi$
$402$		0			0
$403$	−148.416			−0.368278
$404$		0			0
$405$	−656.509	−	172.016i	−1.62101	−	0.424731i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−424.666			−1.03830			−0.519152	−	0.854682i	$-0.673752\pi$
$409$	−424.666			−1.03830			−0.519152	+	0.854682i	$0.673752\pi$
$410$		0			0
$411$	−104.848	+	213.355i	−0.255104	+	0.519113i
$412$		0			0
$413$			536.323i			1.29860i
$414$		0			0
$415$	472.986			1.13973
$416$		0			0
$417$	−493.432	−	242.484i	−1.18329	−	0.581496i
$418$		0			0
$419$		−	166.320i		−	0.396946i	−0.980106	−	0.198473i	$-0.936402\pi$
$419$		−	166.320i		−	0.396946i	0.980106	−	0.198473i	$-0.0635982\pi$
$420$		0			0
$421$	648.061			1.53934			0.769668	−	0.638444i	$-0.220422\pi$
$421$	648.061			1.53934			0.769668	+	0.638444i	$0.220422\pi$
$422$		0			0
$423$	−70.1406	+	54.1303i	−0.165817	+	0.127968i
$424$		0			0
$425$		−	903.234i		−	2.12526i
$426$		0			0
$427$	591.523			1.38530
$428$		0			0
$429$		0			0
$430$		0			0
$431$			747.095i			1.73340i	0.498831	+	0.866699i	$0.333763\pi$
$431$			747.095i			1.73340i	−0.498831	+	0.866699i	$0.666237\pi$
$432$		0			0
$433$	15.6775			0.0362066			0.0181033	−	0.999836i	$-0.494237\pi$
$433$	15.6775			0.0362066			0.0181033	+	0.999836i	$0.494237\pi$
$434$		0			0
$435$	1027.75	+	505.061i	2.36264	+	1.16106i
$436$		0			0
$437$			638.639i			1.46142i
$438$		0			0
$439$	−0.168208			−0.000383162			−0.000191581	−	1.00000i	$-0.500061\pi$
$439$	−0.168208			−0.000383162			−0.000191581		1.00000i	$0.500061\pi$
$440$		0			0
$441$	21.9793	+	28.4802i	0.0498398	+	0.0645810i
$442$		0			0
$443$		−	266.640i		−	0.601895i	−0.953641	−	0.300948i	$-0.902697\pi$
$443$		−	266.640i		−	0.601895i	0.953641	−	0.300948i	$-0.0973030\pi$
$444$		0			0
$445$	383.982			0.862880
$446$		0			0
$447$	145.222	−	295.513i	0.324881	−	0.661102i
$448$		0			0
$449$		−	237.442i		−	0.528825i	−0.964410	−	0.264412i	$-0.914822\pi$
$449$		−	237.442i		−	0.528825i	0.964410	−	0.264412i	$-0.0851780\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−432.494	−	212.538i	−0.954734	−	0.469179i
$454$		0			0
$455$		−	1066.64i		−	2.34427i
$456$		0			0
$457$	−179.258			−0.392249			−0.196124	−	0.980579i	$-0.562836\pi$
$457$	−179.258			−0.392249			−0.196124	+	0.980579i	$0.562836\pi$
$458$		0			0
$459$	−528.713	+	107.455i	−1.15188	+	0.234106i
$460$		0			0
$461$		−	299.328i		−	0.649301i	−0.945834	−	0.324651i	$-0.894753\pi$
$461$		−	299.328i		−	0.649301i	0.945834	−	0.324651i	$-0.105247\pi$
$462$		0			0
$463$	669.950			1.44698			0.723488	−	0.690336i	$-0.242537\pi$
$463$	669.950			1.44698			0.723488	+	0.690336i	$0.242537\pi$
$464$		0			0
$465$	94.0892	−	191.462i	0.202342	−	0.411747i
$466$		0			0
$467$			233.594i			0.500201i	0.968220	+	0.250101i	$0.0804637\pi$
$467$			233.594i			0.500201i	−0.968220	+	0.250101i	$0.919536\pi$
$468$		0			0
$469$	629.245			1.34167
$470$		0			0
$471$	72.8117	+	35.7814i	0.154590	+	0.0759690i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−926.843			−1.95125
$476$		0			0
$477$	−576.967	+	445.268i	−1.20957	+	0.933477i
$478$		0			0
$479$		−	105.783i		−	0.220842i	−0.993885	−	0.110421i	$-0.964780\pi$
$479$		−	105.783i		−	0.220842i	0.993885	−	0.110421i	$-0.0352199\pi$
$480$		0			0
$481$	463.538			0.963696
$482$		0			0
$483$	300.010	−	610.491i	0.621138	−	1.26396i
$484$		0			0
$485$			46.8225i			0.0965412i
$486$		0			0
$487$	682.210			1.40084			0.700421	−	0.713730i	$-0.252996\pi$
$487$	682.210			1.40084			0.700421	+	0.713730i	$0.252996\pi$
$488$		0			0
$489$	645.275	+	317.103i	1.31958	+	0.648473i
$490$		0			0
$491$		−	597.301i		−	1.21650i	−0.793746	−	0.608249i	$-0.791872\pi$
$491$		−	597.301i		−	1.21650i	0.793746	−	0.608249i	$-0.208128\pi$
$492$		0			0
$493$	910.356			1.84656
$494$		0			0
$495$		0			0
$496$		0			0
$497$			765.034i			1.53930i
$498$		0			0
$499$	985.228			1.97440			0.987202	−	0.159472i	$-0.0509793\pi$
$499$	985.228			1.97440			0.987202	+	0.159472i	$0.0509793\pi$
$500$		0			0
$501$	26.3665	−	53.6532i	0.0526277	−	0.107092i
$502$		0			0
$503$			329.323i			0.654717i	0.944900	+	0.327359i	$0.106158\pi$
$503$			329.323i			0.654717i	−0.944900	+	0.327359i	$0.893842\pi$
$504$		0			0
$505$	945.972			1.87321
$506$		0			0
$507$	368.328	+	181.005i	0.726485	+	0.357012i
$508$		0			0
$509$			584.161i			1.14766i	0.818973	+	0.573832i	$0.194544\pi$
$509$			584.161i			1.14766i	−0.818973	+	0.573832i	$0.805456\pi$
$510$		0			0
$511$	771.934			1.51063
$512$		0			0
$513$	110.263	+	542.533i	0.214938	+	1.05757i
$514$		0			0
$515$			711.485i			1.38152i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−411.003	+	836.352i	−0.791913	+	1.61147i
$520$		0			0
$521$			433.397i			0.831856i	0.909398	+	0.415928i	$0.136543\pi$
$521$			433.397i			0.831856i	−0.909398	+	0.415928i	$0.863457\pi$
$522$		0			0
$523$	−641.835			−1.22722			−0.613609	−	0.789610i	$-0.710283\pi$
$523$	−641.835			−1.22722			−0.613609	+	0.789610i	$0.710283\pi$
$524$		0			0
$525$	885.992	+	435.398i	1.68760	+	0.829329i
$526$		0			0
$527$		−	169.593i		−	0.321808i
$528$		0			0
$529$	−441.082			−0.833803
$530$		0			0
$531$	524.908	−	405.092i	0.988527	−	0.762886i
$532$		0			0
$533$			465.590i			0.873528i
$534$		0			0
$535$	−341.178			−0.637716
$536$		0			0
$537$	13.1899	−	26.8402i	0.0245622	−	0.0499817i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	210.690			0.389445			0.194722	−	0.980858i	$-0.437619\pi$
$541$	210.690			0.389445			0.194722	+	0.980858i	$0.437619\pi$
$542$		0			0
$543$	832.969	+	409.341i	1.53401	+	0.753851i
$544$		0			0
$545$			321.411i			0.589745i
$546$		0			0
$547$	−372.667			−0.681292			−0.340646	−	0.940192i	$-0.610646\pi$
$547$	−372.667			−0.681292			−0.340646	+	0.940192i	$0.610646\pi$
$548$		0			0
$549$	−446.785	−	578.932i	−0.813817	−	1.05452i
$550$		0			0
$551$		−	934.150i		−	1.69537i
$552$		0			0
$553$	−543.495			−0.982812
$554$		0			0
$555$	−293.863	+	597.983i	−0.529483	+	1.07745i
$556$		0			0
$557$		−	776.588i		−	1.39423i	−0.716958	−	0.697117i	$-0.754466\pi$
$557$		−	776.588i		−	1.39423i	0.716958	−	0.697117i	$-0.245534\pi$
$558$		0			0
$559$	−115.018			−0.205757
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1029.71i		−	1.82897i	−0.404614	−	0.914487i	$-0.632594\pi$
$563$		−	1029.71i		−	1.82897i	0.404614	−	0.914487i	$-0.367406\pi$
$564$		0			0
$565$	−747.841			−1.32361
$566$		0			0
$567$	149.459	−	570.418i	0.263596	−	1.00603i
$568$		0			0
$569$		−	169.522i		−	0.297930i	−0.988842	−	0.148965i	$-0.952406\pi$
$569$		−	169.522i		−	0.297930i	0.988842	−	0.148965i	$-0.0475942\pi$
$570$		0			0
$571$	235.262			0.412017			0.206009	−	0.978550i	$-0.433953\pi$
$571$	235.262			0.412017			0.206009	+	0.978550i	$0.433953\pi$
$572$		0			0
$573$	−153.593	+	312.548i	−0.268051	+	0.545458i
$574$		0			0
$575$		−	1407.86i		−	2.44845i
$576$		0			0
$577$	548.736			0.951015			0.475508	−	0.879712i	$-0.342264\pi$
$577$	548.736			0.951015			0.475508	+	0.879712i	$0.342264\pi$
$578$		0			0
$579$	−40.3703	−	19.8389i	−0.0697242	−	0.0342641i
$580$		0			0
$581$			410.962i			0.707335i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1043.94	+	805.650i	−1.78451	+	1.37718i
$586$		0			0
$587$		−	437.920i		−	0.746030i	−0.927825	−	0.373015i	$-0.878324\pi$
$587$		−	437.920i		−	0.746030i	0.927825	−	0.373015i	$-0.121676\pi$
$588$		0			0
$589$	−174.025			−0.295459
$590$		0			0
$591$	210.442	−	428.230i	0.356078	−	0.724585i
$592$		0			0
$593$		−	174.581i		−	0.294402i	−0.989107	−	0.147201i	$-0.952974\pi$
$593$		−	174.581i		−	0.294402i	0.989107	−	0.147201i	$-0.0470265\pi$
$594$		0			0
$595$	1218.84			2.04847
$596$		0			0
$597$	671.976	+	330.225i	1.12559	+	0.553141i
$598$		0			0
$599$		−	349.360i		−	0.583239i	−0.956534	−	0.291619i	$-0.905806\pi$
$599$		−	349.360i		−	0.583239i	0.956534	−	0.291619i	$-0.0941941\pi$
$600$		0			0
$601$	−773.473			−1.28698			−0.643488	−	0.765456i	$-0.722513\pi$
$601$	−773.473			−1.28698			−0.643488	+	0.765456i	$0.722513\pi$
$602$		0			0
$603$	−475.277	−	615.852i	−0.788188	−	1.02131i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−129.695			−0.213666			−0.106833	−	0.994277i	$-0.534071\pi$
$607$	−129.695			−0.213666			−0.106833	+	0.994277i	$0.534071\pi$
$608$		0			0
$609$	−438.830	+	892.977i	−0.720575	+	1.46630i
$610$		0			0
$611$			172.149i			0.281750i
$612$		0			0
$613$	−497.059			−0.810863			−0.405431	−	0.914125i	$-0.632879\pi$
$613$	−497.059			−0.810863			−0.405431	+	0.914125i	$0.632879\pi$
$614$		0			0
$615$	−600.630	−	295.164i	−0.976635	−	0.479942i
$616$		0			0
$617$			54.9436i			0.0890496i	0.999008	+	0.0445248i	$0.0141774\pi$
$617$			54.9436i			0.0890496i	−0.999008	+	0.0445248i	$0.985823\pi$
$618$		0			0
$619$	872.824			1.41005			0.705027	−	0.709180i	$-0.250935\pi$
$619$	872.824			1.41005			0.705027	+	0.709180i	$0.250935\pi$
$620$		0			0
$621$	−824.098	+	167.488i	−1.32705	+	0.269707i
$622$		0			0
$623$			333.629i			0.535520i
$624$		0			0
$625$	288.152			0.461043
$626$		0			0
$627$		0			0
$628$		0			0
$629$			529.678i			0.842095i
$630$		0			0
$631$	130.706			0.207141			0.103570	−	0.994622i	$-0.466973\pi$
$631$	130.706			0.207141			0.103570	+	0.994622i	$0.466973\pi$
$632$		0			0
$633$	59.9494	+	29.4606i	0.0947068	+	0.0465412i
$634$		0			0
$635$			1792.82i			2.82333i
$636$		0			0
$637$	69.9004			0.109734
$638$		0			0
$639$	748.750	−	577.841i	1.17175	−	0.904289i
$640$		0			0
$641$		−	883.070i		−	1.37764i	−0.724930	−	0.688822i	$-0.758128\pi$
$641$		−	883.070i		−	1.37764i	0.724930	−	0.688822i	$-0.241872\pi$
$642$		0			0
$643$	−322.647			−0.501784			−0.250892	−	0.968015i	$-0.580724\pi$
$643$	−322.647			−0.501784			−0.250892	+	0.968015i	$0.580724\pi$
$644$		0			0
$645$	72.9164	−	148.378i	0.113049	−	0.230043i
$646$		0			0
$647$		−	1091.68i		−	1.68729i	−0.536899	−	0.843646i	$-0.680405\pi$
$647$		−	1091.68i		−	1.68729i	0.536899	−	0.843646i	$-0.319595\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	166.355	+	81.7509i	0.255538	+	0.125577i
$652$		0			0
$653$		−	769.561i		−	1.17850i	−0.807951	−	0.589250i	$-0.799423\pi$
$653$		−	769.561i		−	1.17850i	0.807951	−	0.589250i	$-0.200577\pi$
$654$		0			0
$655$	1113.86			1.70055
$656$		0			0
$657$	−583.052	−	755.504i	−0.887447	−	1.14993i
$658$		0			0
$659$			399.019i			0.605492i	0.953071	+	0.302746i	$0.0979034\pi$
$659$			399.019i			0.605492i	−0.953071	+	0.302746i	$0.902097\pi$
$660$		0			0
$661$	122.307			0.185033			0.0925163	−	0.995711i	$-0.470509\pi$
$661$	122.307			0.185033			0.0925163	+	0.995711i	$0.470509\pi$
$662$		0			0
$663$	−462.348	+	940.834i	−0.697357	+	1.41906i
$664$		0			0
$665$		−	1250.70i		−	1.88075i
$666$		0			0
$667$	1418.96			2.12737
$668$		0			0
$669$	161.542	+	79.3854i	0.241467	+	0.118663i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1065.47			1.58316			0.791580	−	0.611066i	$-0.209259\pi$
$673$	1065.47			1.58316			0.791580	+	0.611066i	$0.209259\pi$
$674$		0			0
$675$	−243.072	−	1196.00i	−0.360106	−	1.77185i
$676$		0			0
$677$		−	557.073i		−	0.822856i	−0.911442	−	0.411428i	$-0.865030\pi$
$677$		−	557.073i		−	0.822856i	0.911442	−	0.411428i	$-0.134970\pi$
$678$		0			0
$679$	−40.6825			−0.0599153
$680$		0			0
$681$	−185.002	+	376.462i	−0.271663	+	0.552808i
$682$		0			0
$683$		−	756.144i		−	1.10709i	−0.832818	−	0.553546i	$-0.813274\pi$
$683$		−	756.144i		−	1.10709i	0.832818	−	0.553546i	$-0.186726\pi$
$684$		0			0
$685$	−663.940			−0.969256
$686$		0			0
$687$	−281.346	−	138.260i	−0.409528	−	0.201252i
$688$		0			0
$689$			1416.08i			2.05526i
$690$		0			0
$691$	1.91179			0.00276670			0.00138335	−	0.999999i	$-0.499560\pi$
$691$	1.91179			0.00276670			0.00138335	+	0.999999i	$0.499560\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	1535.51i		−	2.20937i
$696$		0			0
$697$	−532.023			−0.763305
$698$		0			0
$699$	−214.994	+	437.493i	−0.307574	+	0.625884i
$700$		0			0
$701$		−	1217.71i		−	1.73710i	−0.495602	−	0.868550i	$-0.665053\pi$
$701$		−	1217.71i		−	1.73710i	0.495602	−	0.868550i	$-0.334947\pi$
$702$		0			0
$703$	543.522			0.773147
$704$		0			0
$705$	−222.080	−	109.135i	−0.315007	−	0.154802i
$706$		0			0
$707$			821.923i			1.16255i
$708$		0			0
$709$	−573.617			−0.809051			−0.404525	−	0.914527i	$-0.632563\pi$
$709$	−573.617			−0.809051			−0.404525	+	0.914527i	$0.632563\pi$
$710$		0			0
$711$	410.509	+	531.927i	0.577369	+	0.748139i
$712$		0			0
$713$		−	264.342i		−	0.370746i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	128.033	−	260.534i	0.178567	−	0.363367i
$718$		0			0
$719$		−	396.242i		−	0.551101i	−0.961287	−	0.275551i	$-0.911140\pi$
$719$		−	396.242i		−	0.551101i	0.961287	−	0.275551i	$-0.0888602\pi$
$720$		0			0
$721$	−618.185			−0.857399
$722$		0			0
$723$	76.2788	+	37.4852i	0.105503	+	0.0518468i
$724$		0			0
$725$			2059.30i			2.84042i
$726$		0			0
$727$	−956.423			−1.31557			−0.657787	−	0.753204i	$-0.728507\pi$
$727$	−956.423			−1.31557			−0.657787	+	0.753204i	$0.728507\pi$
$728$		0			0
$729$	−671.165	+	284.567i	−0.920666	+	0.390352i
$730$		0			0
$731$		−	131.429i		−	0.179794i
$732$		0			0
$733$	468.899			0.639699			0.319849	−	0.947468i	$-0.396368\pi$
$733$	468.899			0.639699			0.319849	+	0.947468i	$0.396368\pi$
$734$		0			0
$735$	−44.3138	+	90.1743i	−0.0602909	+	0.122686i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−285.619			−0.386494			−0.193247	−	0.981150i	$-0.561902\pi$
$739$	−285.619			−0.386494			−0.193247	+	0.981150i	$0.561902\pi$
$740$		0			0
$741$	965.425	+	474.433i	1.30287	+	0.640260i
$742$		0			0
$743$		−	489.071i		−	0.658238i	−0.944288	−	0.329119i	$-0.893248\pi$
$743$		−	489.071i		−	0.658238i	0.944288	−	0.329119i	$-0.106752\pi$
$744$		0			0
$745$	919.606			1.23437
$746$		0			0
$747$	402.214	−	310.405i	0.538439	−	0.415535i
$748$		0			0
$749$		−	296.438i		−	0.395778i
$750$		0			0
$751$	1257.57			1.67453			0.837263	−	0.546800i	$-0.184154\pi$
$751$	1257.57			1.67453			0.837263	+	0.546800i	$0.184154\pi$
$752$		0			0
$753$	142.628	−	290.234i	0.189413	−	0.385437i
$754$		0			0
$755$		−	1345.88i		−	1.78262i
$756$		0			0
$757$	−389.306			−0.514274			−0.257137	−	0.966375i	$-0.582779\pi$
$757$	−389.306			−0.514274			−0.257137	+	0.966375i	$0.582779\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	106.334i		−	0.139729i	−0.997556	−	0.0698645i	$-0.977743\pi$
$761$		−	106.334i		−	0.139729i	0.997556	−	0.0698645i	$-0.0222567\pi$
$762$		0			0
$763$	−279.263			−0.366007
$764$		0			0
$765$	−920.605	−	1192.90i	−1.20341	−	1.55934i
$766$		0			0
$767$		−	1288.31i		−	1.67967i
$768$		0			0
$769$	−410.462			−0.533761			−0.266881	−	0.963730i	$-0.585993\pi$
$769$	−410.462			−0.533761			−0.266881	+	0.963730i	$0.585993\pi$
$770$		0			0
$771$	−419.719	+	854.088i	−0.544383	+	1.10777i
$772$		0			0
$773$			1373.12i			1.77635i	0.459509	+	0.888173i	$0.348025\pi$
$773$			1373.12i			1.77635i	−0.459509	+	0.888173i	$0.651975\pi$
$774$		0			0
$775$	383.633			0.495011
$776$		0			0
$777$	−519.567	−	255.327i	−0.668683	−	0.328607i
$778$		0			0
$779$			545.929i			0.700808i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1205.42	−	244.988i	1.53949	−	0.312884i
$784$		0			0
$785$			226.583i			0.288640i
$786$		0			0
$787$	263.656			0.335014			0.167507	−	0.985871i	$-0.446428\pi$
$787$	263.656			0.335014			0.167507	+	0.985871i	$0.446428\pi$
$788$		0			0
$789$	153.938	−	313.249i	0.195105	−	0.397020i
$790$		0			0
$791$		−	649.774i		−	0.821458i
$792$		0			0
$793$	−1420.90			−1.79180
$794$		0			0
$795$	−1826.80	−	897.731i	−2.29786	−	1.12922i
$796$		0			0
$797$			33.2222i			0.0416840i	0.999783	+	0.0208420i	$0.00663470\pi$
$797$			33.2222i			0.0416840i	−0.999783	+	0.0208420i	$0.993365\pi$
$798$		0			0
$799$	−196.713			−0.246199
$800$		0			0
$801$	326.527	−	251.994i	0.407650	−	0.314600i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	1899.79			2.35998
$806$		0			0
$807$	−119.487	+	243.144i	−0.148063	+	0.301294i
$808$		0			0
$809$		−	294.199i		−	0.363658i	−0.983330	−	0.181829i	$-0.941798\pi$
$809$		−	294.199i		−	0.363658i	0.983330	−	0.181829i	$-0.0582017\pi$
$810$		0			0
$811$	−788.003			−0.971644			−0.485822	−	0.874058i	$-0.661480\pi$
$811$	−788.003			−0.971644			−0.485822	+	0.874058i	$0.661480\pi$
$812$		0			0
$813$	424.141	+	208.433i	0.521699	+	0.256375i
$814$		0			0
$815$			2008.03i			2.46384i
$816$		0			0
$817$	−134.865			−0.165073
$818$		0			0
$819$	−700.002	−	907.044i	−0.854703	−	1.10750i
$820$		0			0
$821$			377.639i			0.459975i	0.973194	+	0.229987i	$0.0738685\pi$
$821$			377.639i			0.459975i	−0.973194	+	0.229987i	$0.926132\pi$
$822$		0			0
$823$	190.681			0.231690			0.115845	−	0.993267i	$-0.463042\pi$
$823$	190.681			0.231690			0.115845	+	0.993267i	$0.463042\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1170.25i			1.41505i	0.706686	+	0.707527i	$0.250189\pi$
$827$			1170.25i			1.41505i	−0.706686	+	0.707527i	$0.749811\pi$
$828$		0			0
$829$	573.179			0.691411			0.345705	−	0.938343i	$-0.387640\pi$
$829$	573.179			0.691411			0.345705	+	0.938343i	$0.387640\pi$
$830$		0			0
$831$	500.738	+	246.075i	0.602573	+	0.296119i
$832$		0			0
$833$			79.8742i			0.0958874i
$834$		0			0
$835$	166.963			0.199956
$836$		0			0
$837$	−45.6395	−	224.562i	−0.0545275	−	0.268294i
$838$		0			0
$839$		−	479.253i		−	0.571220i	−0.958346	−	0.285610i	$-0.907804\pi$
$839$		−	479.253i		−	0.571220i	0.958346	−	0.285610i	$-0.0921962\pi$
$840$		0			0
$841$	−1234.54			−1.46794
$842$		0			0
$843$	−622.056	+	1265.82i	−0.737908	+	1.50157i
$844$		0			0
$845$			1146.20i			1.35645i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−642.806	−	315.890i	−0.757133	−	0.372073i
$850$		0			0
$851$			825.602i			0.970155i
$852$		0			0
$853$	41.7560			0.0489519			0.0244759	−	0.999700i	$-0.492208\pi$
$853$	41.7560			0.0489519			0.0244759	+	0.999700i	$0.492208\pi$
$854$		0			0
$855$	−1224.07	+	944.667i	−1.43167	+	1.10487i
$856$		0			0
$857$		−	394.533i		−	0.460365i	−0.973147	−	0.230182i	$-0.926068\pi$
$857$		−	394.533i		−	0.460365i	0.973147	−	0.230182i	$-0.0739323\pi$
$858$		0			0
$859$	−142.959			−0.166425			−0.0832124	−	0.996532i	$-0.526518\pi$
$859$	−142.959			−0.166425			−0.0832124	+	0.996532i	$0.526518\pi$
$860$		0			0
$861$	256.458	−	521.867i	0.297861	−	0.606118i
$862$		0			0
$863$			89.9020i			0.104174i	0.998643	+	0.0520869i	$0.0165873\pi$
$863$			89.9020i			0.104174i	−0.998643	+	0.0520869i	$0.983413\pi$
$864$		0			0
$865$	−2602.64			−3.00883
$866$		0			0
$867$	−296.958	−	145.932i	−0.342512	−	0.168319i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1511.51			−1.73538
$872$		0			0
$873$	30.7280	+	39.8166i	0.0351982	+	0.0456089i
$874$		0			0
$875$			1232.22i			1.40825i
$876$		0			0
$877$	−643.269			−0.733487			−0.366744	−	0.930322i	$-0.619527\pi$
$877$	−643.269			−0.733487			−0.366744	+	0.930322i	$0.619527\pi$
$878$		0			0
$879$	−671.864	+	1367.18i	−0.764350	+	1.55538i
$880$		0			0
$881$			156.970i			0.178172i	0.996024	+	0.0890860i	$0.0283946\pi$
$881$			156.970i			0.178172i	−0.996024	+	0.0890860i	$0.971605\pi$
$882$		0			0
$883$	−1504.02			−1.70330			−0.851651	−	0.524109i	$-0.824398\pi$
$883$	−1504.02			−1.70330			−0.851651	+	0.524109i	$0.824398\pi$
$884$		0			0
$885$	1661.97	+	816.730i	1.87793	+	0.922858i
$886$		0			0
$887$			1243.76i			1.40221i	0.713057	+	0.701106i	$0.247310\pi$
$887$			1243.76i			1.40221i	−0.713057	+	0.701106i	$0.752690\pi$
$888$		0			0
$889$	−1557.72			−1.75221
$890$		0			0
$891$		0			0
$892$		0			0
$893$			201.854i			0.226041i
$894$		0			0
$895$	83.5240			0.0933229
$896$		0			0
$897$	−720.655	+	1466.46i	−0.803406	+	1.63485i
$898$		0			0
$899$			386.658i			0.430098i
$900$		0			0
$901$	−1618.13			−1.79593
$902$		0			0
$903$	128.921	+	63.3546i	0.142769	+	0.0701601i
$904$		0			0
$905$			2592.12i			2.86422i
$906$		0			0
$907$	198.465			0.218814			0.109407	−	0.993997i	$-0.465105\pi$
$907$	198.465			0.218814			0.109407	+	0.993997i	$0.465105\pi$
$908$		0			0
$909$	804.428	−	620.810i	0.884960	−	0.682959i
$910$		0			0
$911$		−	923.510i		−	1.01373i	−0.862025	−	0.506866i	$-0.830804\pi$
$911$		−	923.510i		−	1.01373i	0.862025	−	0.506866i	$-0.169196\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	900.790	−	1833.02i	0.984470	−	2.00330i
$916$		0			0
$917$			967.793i			1.05539i
$918$		0			0
$919$	1340.69			1.45886			0.729429	−	0.684057i	$-0.239786\pi$
$919$	1340.69			1.45886			0.729429	+	0.684057i	$0.239786\pi$
$920$		0			0
$921$	1250.55	+	614.550i	1.35782	+	0.667264i
$922$		0			0
$923$		−	1837.69i		−	1.99100i
$924$		0			0
$925$	−1198.18			−1.29533
$926$		0			0
$927$	466.923	+	605.027i	0.503693	+	0.652672i
$928$		0			0
$929$			1528.15i			1.64495i	0.568805	+	0.822473i	$0.307406\pi$
$929$			1528.15i			1.64495i	−0.568805	+	0.822473i	$0.692594\pi$
$930$		0			0
$931$	81.9619			0.0880364
$932$		0			0
$933$	−714.881	+	1454.71i	−0.766218	+	1.55918i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	300.687			0.320904			0.160452	−	0.987044i	$-0.448705\pi$
$937$	300.687			0.320904			0.160452	+	0.987044i	$0.448705\pi$
$938$		0			0
$939$	−636.938	−	313.007i	−0.678315	−	0.333340i
$940$		0			0
$941$		−	581.086i		−	0.617520i	−0.951140	−	0.308760i	$-0.900086\pi$
$941$		−	581.086i		−	0.617520i	0.951140	−	0.308760i	$-0.0999140\pi$
$942$		0			0
$943$	−829.257			−0.879382
$944$		0			0
$945$	1613.89	−	328.005i	1.70782	−	0.347095i
$946$		0			0
$947$			1422.46i			1.50207i	0.660264	+	0.751034i	$0.270444\pi$
$947$			1422.46i			1.50207i	−0.660264	+	0.751034i	$0.729556\pi$
$948$		0			0
$949$	−1854.27			−1.95392
$950$		0			0
$951$	232.584	−	473.286i	0.244568	−	0.497672i
$952$		0			0
$953$			447.348i			0.469410i	0.972067	+	0.234705i	$0.0754124\pi$
$953$			447.348i			0.469410i	−0.972067	+	0.234705i	$0.924588\pi$
$954$		0			0
$955$	−972.617			−1.01845
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	576.875i		−	0.601538i
$960$		0			0
$961$	−888.968			−0.925045
$962$		0			0
$963$	−290.128	+	223.904i	−0.301275	+	0.232506i
$964$		0			0
$965$		−	125.628i		−	0.130185i
$966$		0			0
$967$	−1379.18			−1.42625			−0.713125	−	0.701037i	$-0.752721\pi$
$967$	−1379.18			−1.42625			−0.713125	+	0.701037i	$0.752721\pi$
$968$		0			0
$969$	−542.127	+	1103.18i	−0.559471	+	1.13847i
$970$		0			0
$971$		−	748.375i		−	0.770726i	−0.922765	−	0.385363i	$-0.874076\pi$
$971$		−	748.375i		−	0.770726i	0.922765	−	0.385363i	$-0.125924\pi$
$972$		0			0
$973$	1334.15			1.37117
$974$		0			0
$975$	−2128.25	−	1045.87i	−2.18282	−	1.07269i
$976$		0			0
$977$		−	37.2883i		−	0.0381662i	−0.999818	−	0.0190831i	$-0.993925\pi$
$977$		−	37.2883i		−	0.0381662i	0.999818	−	0.0190831i	$-0.00607470\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	210.931	+	273.319i	0.215017	+	0.278613i
$982$		0			0
$983$			394.691i			0.401517i	0.979641	+	0.200759i	$0.0643407\pi$
$983$			394.691i			0.401517i	−0.979641	+	0.200759i	$0.935659\pi$
$984$		0			0
$985$	1332.61			1.35290
$986$		0			0
$987$	94.8239	−	192.958i	0.0960728	−	0.195499i
$988$		0			0
$989$		−	204.857i		−	0.207136i
$990$		0			0
$991$	8.78560			0.00886538			0.00443269	−	0.999990i	$-0.498589\pi$
$991$	8.78560			0.00886538			0.00443269	+	0.999990i	$0.498589\pi$
$992$		0			0
$993$	621.959	+	305.646i	0.626344	+	0.307800i
$994$		0			0
$995$			2091.12i			2.10163i
$996$		0			0
$997$	−786.155			−0.788521			−0.394260	−	0.918999i	$-0.628999\pi$
$997$	−786.155			−0.788521			−0.394260	+	0.918999i	$0.628999\pi$
$998$		0			0
$999$	142.543	+	701.360i	0.142686	+	0.702062i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.3.e.i.485.6	yes	6
3.2	odd	2	inner	1452.3.e.i.485.5	✓	6
11.10	odd	2		1452.3.e.j.485.6	yes	6
33.32	even	2		1452.3.e.j.485.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1452.3.e.i.485.5	✓	6	3.2	odd	2	inner
1452.3.e.i.485.6	yes	6	1.1	even	1	trivial
1452.3.e.j.485.5	yes	6	33.32	even	2
1452.3.e.j.485.6	yes	6	11.10	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:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1452.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(2.69245 + 1.32314i) q^{3} +8.37865i q^{5} -7.27992 q^{7} +(5.49862 + 7.12497i) q^{9} +17.4871 q^{13} +(-11.0861 + 22.5591i) q^{15} +19.9823i q^{17} +20.5046 q^{19} +(-19.6009 - 9.63233i) q^{21} +31.1461i q^{23} -45.2017 q^{25} +(5.37749 + 26.4591i) q^{27} -45.5581i q^{29} -8.48714 q^{31} -60.9959i q^{35} +26.5073 q^{37} +(47.0833 + 23.1379i) q^{39} +26.6247i q^{41} -6.57729 q^{43} +(-59.6976 + 46.0710i) q^{45} +9.84434i q^{47} +3.99724 q^{49} +(-26.4393 + 53.8015i) q^{51} +80.9782i q^{53} +(55.2077 + 27.1304i) q^{57} -73.6716i q^{59} -81.2541 q^{61} +(-40.0295 - 51.8692i) q^{63} +146.519i q^{65} -86.4357 q^{67} +(-41.2106 + 83.8596i) q^{69} -105.088i q^{71} -106.036 q^{73} +(-121.704 - 59.8080i) q^{75} +74.6567 q^{79} +(-20.5303 + 78.3550i) q^{81} -56.4514i q^{83} -167.425 q^{85} +(60.2795 - 122.663i) q^{87} -45.8286i q^{89} -127.305 q^{91} +(-22.8512 - 11.2296i) q^{93} +171.801i q^{95} +5.58831 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 16 q^{7} - 6 q^{9} + 6 q^{13} - 12 q^{15} - 12 q^{19} - 4 q^{21} - 44 q^{25} + 10 q^{27} + 48 q^{31} + 102 q^{37} + 150 q^{39} + 52 q^{43} - 14 q^{45} - 54 q^{49} + 22 q^{51} + 8 q^{57}+ \cdots + 254 q^{97}+O(q^{100}) \) 6 * q + 4 * q^3 - 16 * q^7 - 6 * q^9 + 6 * q^13 - 12 * q^15 - 12 * q^19 - 4 * q^21 - 44 * q^25 + 10 * q^27 + 48 * q^31 + 102 * q^37 + 150 * q^39 + 52 * q^43 - 14 * q^45 - 54 * q^49 + 22 * q^51 + 8 * q^57 - 152 * q^61 - 122 * q^63 - 24 * q^67 - 112 * q^69 - 280 * q^73 - 290 * q^75 + 272 * q^79 - 186 * q^81 - 198 * q^85 - 154 * q^87 - 68 * q^91 - 114 * q^93 + 254 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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