LMFDB - Embedded newform 600.3.u.e.457.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(193,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.193"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.u (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,12,0,0,0,-16,0,-48,0,0,0,24,0,0,0,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)

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

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			457.2
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	600.457
Dual form			600.3.u.e.193.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 - 1.22474i) q^{3} +(5.44949 + 5.44949i) q^{7} -3.00000i q^{9} -6.44949 q^{11} +(-14.4495 + 14.4495i) q^{13} +(23.1464 + 23.1464i) q^{17} +16.6969i q^{19} +13.3485 q^{21} +(6.65153 - 6.65153i) q^{23} +(-3.67423 - 3.67423i) q^{27} -0.0454077i q^{29} +4.49490 q^{31} +(-7.89898 + 7.89898i) q^{33} +(-35.3485 - 35.3485i) q^{37} +35.3939i q^{39} +20.2929 q^{41} +(-32.2929 + 32.2929i) q^{43} +(50.5403 + 50.5403i) q^{47} +10.3939i q^{49} +56.6969 q^{51} +(5.50510 - 5.50510i) q^{53} +(20.4495 + 20.4495i) q^{57} +55.4393i q^{59} +47.8888 q^{61} +(16.3485 - 16.3485i) q^{63} +(85.2827 + 85.2827i) q^{67} -16.2929i q^{69} +48.4041 q^{71} +(21.9898 - 21.9898i) q^{73} +(-35.1464 - 35.1464i) q^{77} -126.697i q^{79} -9.00000 q^{81} +(-94.9444 + 94.9444i) q^{83} +(-0.0556128 - 0.0556128i) q^{87} -71.7980i q^{89} -157.485 q^{91} +(5.50510 - 5.50510i) q^{93} +(37.0000 + 37.0000i) q^{97} +19.3485i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} - 16 q^{11} - 48 q^{13} + 24 q^{17} + 24 q^{21} + 56 q^{23} - 80 q^{31} - 12 q^{33} - 112 q^{37} - 56 q^{41} + 8 q^{43} + 16 q^{47} + 168 q^{51} + 120 q^{53} + 72 q^{57} - 24 q^{61} + 36 q^{63}+ \cdots + 148 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 - 16 * q^11 - 48 * q^13 + 24 * q^17 + 24 * q^21 + 56 * q^23 - 80 * q^31 - 12 * q^33 - 112 * q^37 - 56 * q^41 + 8 * q^43 + 16 * q^47 + 168 * q^51 + 120 * q^53 + 72 * q^57 - 24 * q^61 + 36 * q^63 + 8 * q^67 + 272 * q^71 - 108 * q^73 - 72 * q^77 - 36 * q^81 - 272 * q^83 - 108 * q^87 - 336 * q^91 + 120 * q^93 + 148 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$3$	1.22474	−	1.22474i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	5.44949	+	5.44949i	0.778499	+	0.778499i	0.979575	−	0.201077i	$-0.0644441\pi$
$7$	5.44949	+	5.44949i	0.778499	+	0.778499i	−0.201077	+	0.979575i	$0.564444\pi$
$8$		0			0
$9$		−	3.00000i		−	0.333333i
$10$		0			0
$11$	−6.44949			−0.586317			−0.293159	−	0.956064i	$-0.594706\pi$
$11$	−6.44949			−0.586317			−0.293159	+	0.956064i	$0.594706\pi$
$12$		0			0
$13$	−14.4495	+	14.4495i	−1.11150	+	1.11150i	−0.118551	+	0.992948i	$0.537825\pi$
$13$	−14.4495	+	14.4495i	−1.11150	+	1.11150i	−0.992948	+	0.118551i	$0.962175\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	23.1464	+	23.1464i	1.36155	+	1.36155i	0.871938	+	0.489617i	$0.162863\pi$
$17$	23.1464	+	23.1464i	1.36155	+	1.36155i	0.489617	+	0.871938i	$0.337137\pi$
$18$		0			0
$19$			16.6969i			0.878786i	0.898295	+	0.439393i	$0.144806\pi$
$19$			16.6969i			0.878786i	−0.898295	+	0.439393i	$0.855194\pi$
$20$		0			0
$21$	13.3485			0.635641
$22$		0			0
$23$	6.65153	−	6.65153i	0.289197	−	0.289197i	−0.547566	−	0.836763i	$-0.684445\pi$
$23$	6.65153	−	6.65153i	0.289197	−	0.289197i	0.836763	+	0.547566i	$0.184445\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	−	3.67423i	−0.136083	−	0.136083i
$28$		0			0
$29$		−	0.0454077i		−	0.00156578i	−1.00000		0.000782891i	$-0.999751\pi$
$29$		−	0.0454077i		−	0.00156578i	1.00000		0.000782891i	$-0.000249202\pi$
$30$		0			0
$31$	4.49490			0.144997			0.0724983	−	0.997369i	$-0.476903\pi$
$31$	4.49490			0.144997			0.0724983	+	0.997369i	$0.476903\pi$
$32$		0			0
$33$	−7.89898	+	7.89898i	−0.239363	+	0.239363i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−35.3485	−	35.3485i	−0.955364	−	0.955364i	0.0436815	−	0.999046i	$-0.486091\pi$
$37$	−35.3485	−	35.3485i	−0.955364	−	0.955364i	−0.999046	+	0.0436815i	$0.986091\pi$
$38$		0			0
$39$			35.3939i			0.907535i
$40$		0			0
$41$	20.2929			0.494948			0.247474	−	0.968895i	$-0.420400\pi$
$41$	20.2929			0.494948			0.247474	+	0.968895i	$0.420400\pi$
$42$		0			0
$43$	−32.2929	+	32.2929i	−0.750997	+	0.750997i	−0.974665	−	0.223669i	$-0.928197\pi$
$43$	−32.2929	+	32.2929i	−0.750997	+	0.750997i	0.223669	+	0.974665i	$0.428197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	50.5403	+	50.5403i	1.07533	+	1.07533i	0.996922	+	0.0784040i	$0.0249824\pi$
$47$	50.5403	+	50.5403i	1.07533	+	1.07533i	0.0784040	+	0.996922i	$0.475018\pi$
$48$		0			0
$49$			10.3939i			0.212120i
$50$		0			0
$51$	56.6969			1.11170
$52$		0			0
$53$	5.50510	−	5.50510i	0.103870	−	0.103870i	−0.653262	−	0.757132i	$-0.726600\pi$
$53$	5.50510	−	5.50510i	0.103870	−	0.103870i	0.757132	+	0.653262i	$0.226600\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	20.4495	+	20.4495i	0.358763	+	0.358763i
$58$		0			0
$59$			55.4393i			0.939649i	0.882760	+	0.469824i	$0.155683\pi$
$59$			55.4393i			0.939649i	−0.882760	+	0.469824i	$0.844317\pi$
$60$		0			0
$61$	47.8888			0.785062			0.392531	−	0.919739i	$-0.371600\pi$
$61$	47.8888			0.785062			0.392531	+	0.919739i	$0.371600\pi$
$62$		0			0
$63$	16.3485	−	16.3485i	0.259500	−	0.259500i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	85.2827	+	85.2827i	1.27288	+	1.27288i	0.944573	+	0.328303i	$0.106477\pi$
$67$	85.2827	+	85.2827i	1.27288	+	1.27288i	0.328303	+	0.944573i	$0.393523\pi$
$68$		0			0
$69$		−	16.2929i		−	0.236128i
$70$		0			0
$71$	48.4041			0.681748			0.340874	−	0.940109i	$-0.389277\pi$
$71$	48.4041			0.681748			0.340874	+	0.940109i	$0.389277\pi$
$72$		0			0
$73$	21.9898	−	21.9898i	0.301230	−	0.301230i	−0.540265	−	0.841495i	$-0.681676\pi$
$73$	21.9898	−	21.9898i	0.301230	−	0.301230i	0.841495	+	0.540265i	$0.181676\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−35.1464	−	35.1464i	−0.456447	−	0.456447i
$78$		0			0
$79$		−	126.697i		−	1.60376i	−0.597486	−	0.801879i	$-0.703834\pi$
$79$		−	126.697i		−	1.60376i	0.597486	−	0.801879i	$-0.296166\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	−94.9444	+	94.9444i	−1.14391	+	1.14391i	−0.156180	+	0.987729i	$0.549918\pi$
$83$	−94.9444	+	94.9444i	−1.14391	+	1.14391i	−0.987729	+	0.156180i	$0.950082\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.0556128	−	0.0556128i	−0.000639228	−	0.000639228i
$88$		0			0
$89$		−	71.7980i		−	0.806719i	−0.915042	−	0.403359i	$-0.867842\pi$
$89$		−	71.7980i		−	0.806719i	0.915042	−	0.403359i	$-0.132158\pi$
$90$		0			0
$91$	−157.485			−1.73060
$92$		0			0
$93$	5.50510	−	5.50510i	0.0591947	−	0.0591947i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	0.871622	−	0.490179i	$-0.163068\pi$
$97$	37.0000	+	37.0000i	0.381443	+	0.381443i	−0.490179	+	0.871622i	$0.663068\pi$
$98$		0			0
$99$			19.3485i			0.195439i
$100$		0			0
$101$	94.3383			0.934042			0.467021	−	0.884246i	$-0.345327\pi$
$101$	94.3383			0.934042			0.467021	+	0.884246i	$0.345327\pi$
$102$		0			0
$103$	−72.6413	+	72.6413i	−0.705256	+	0.705256i	−0.965534	−	0.260278i	$-0.916186\pi$
$103$	−72.6413	+	72.6413i	−0.705256	+	0.705256i	0.260278	+	0.965534i	$0.416186\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.0556	+	13.0556i	0.122015	+	0.122015i	0.765478	−	0.643463i	$-0.222503\pi$
$107$	13.0556	+	13.0556i	0.122015	+	0.122015i	−0.643463	+	0.765478i	$0.722503\pi$
$108$		0			0
$109$		−	69.2827i		−	0.635621i	−0.948154	−	0.317810i	$-0.897052\pi$
$109$		−	69.2827i		−	0.635621i	0.948154	−	0.317810i	$-0.102948\pi$
$110$		0			0
$111$	−86.5857			−0.780051
$112$		0			0
$113$	102.136	−	102.136i	0.903860	−	0.903860i	−0.0919072	−	0.995768i	$-0.529296\pi$
$113$	102.136	−	102.136i	0.903860	−	0.903860i	0.995768	+	0.0919072i	$0.0292963\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	43.3485	+	43.3485i	0.370500	+	0.370500i
$118$		0			0
$119$			252.272i			2.11994i
$120$		0			0
$121$	−79.4041			−0.656232
$122$		0			0
$123$	24.8536	−	24.8536i	0.202062	−	0.202062i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−119.944	−	119.944i	−0.944444	−	0.944444i	0.0540920	−	0.998536i	$-0.482774\pi$
$127$	−119.944	−	119.944i	−0.944444	−	0.944444i	−0.998536	+	0.0540920i	$0.982774\pi$
$128$		0			0
$129$			79.1010i			0.613186i
$130$		0			0
$131$	121.146			0.924782			0.462391	−	0.886676i	$-0.346992\pi$
$131$	121.146			0.924782			0.462391	+	0.886676i	$0.346992\pi$
$132$		0			0
$133$	−90.9898	+	90.9898i	−0.684134	+	0.684134i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−119.530	−	119.530i	−0.872482	−	0.872482i	0.120260	−	0.992742i	$-0.461627\pi$
$137$	−119.530	−	119.530i	−0.872482	−	0.872482i	−0.992742	+	0.120260i	$0.961627\pi$
$138$		0			0
$139$			140.788i			1.01286i	0.862281	+	0.506431i	$0.169035\pi$
$139$			140.788i			1.01286i	−0.862281	+	0.506431i	$0.830965\pi$
$140$		0			0
$141$	123.798			0.878000
$142$		0			0
$143$	93.1918	−	93.1918i	0.651691	−	0.651691i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	12.7298	+	12.7298i	0.0865976	+	0.0865976i
$148$		0			0
$149$		−	53.5301i		−	0.359262i	−0.983734	−	0.179631i	$-0.942510\pi$
$149$		−	53.5301i		−	0.359262i	0.983734	−	0.179631i	$-0.0574904\pi$
$150$		0			0
$151$	−232.606			−1.54044			−0.770219	−	0.637780i	$-0.779853\pi$
$151$	−232.606			−1.54044			−0.770219	+	0.637780i	$0.779853\pi$
$152$		0			0
$153$	69.4393	−	69.4393i	0.453852	−	0.453852i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−128.631	−	128.631i	−0.819306	−	0.819306i	0.166701	−	0.986007i	$-0.446689\pi$
$157$	−128.631	−	128.631i	−0.819306	−	0.819306i	−0.986007	+	0.166701i	$0.946689\pi$
$158$		0			0
$159$		−	13.4847i		−	0.0848094i
$160$		0			0
$161$	72.4949			0.450279
$162$		0			0
$163$	117.576	−	117.576i	0.721322	−	0.721322i	−0.247552	−	0.968875i	$-0.579626\pi$
$163$	117.576	−	117.576i	0.721322	−	0.721322i	0.968875	+	0.247552i	$0.0796262\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	33.6617	+	33.6617i	0.201567	+	0.201567i	0.800671	−	0.599104i	$-0.204476\pi$
$167$	33.6617	+	33.6617i	0.201567	+	0.201567i	−0.599104	+	0.800671i	$0.704476\pi$
$168$		0			0
$169$		−	248.576i		−	1.47086i
$170$		0			0
$171$	50.0908			0.292929
$172$		0			0
$173$	−28.2474	+	28.2474i	−0.163280	+	0.163280i	−0.784018	−	0.620738i	$-0.786833\pi$
$173$	−28.2474	+	28.2474i	−0.163280	+	0.163280i	0.620738	+	0.784018i	$0.286833\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	67.8990	+	67.8990i	0.383610	+	0.383610i
$178$		0			0
$179$		−	45.2372i		−	0.252722i	−0.991984	−	0.126361i	$-0.959670\pi$
$179$		−	45.2372i		−	0.252722i	0.991984	−	0.126361i	$-0.0403298\pi$
$180$		0			0
$181$	−260.656			−1.44009			−0.720045	−	0.693928i	$-0.755879\pi$
$181$	−260.656			−1.44009			−0.720045	+	0.693928i	$0.755879\pi$
$182$		0			0
$183$	58.6515	−	58.6515i	0.320500	−	0.320500i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−149.283	−	149.283i	−0.798303	−	0.798303i
$188$		0			0
$189$		−	40.0454i		−	0.211880i
$190$		0			0
$191$	−51.8684			−0.271562			−0.135781	−	0.990739i	$-0.543354\pi$
$191$	−51.8684			−0.271562			−0.135781	+	0.990739i	$0.543354\pi$
$192$		0			0
$193$	−16.6163	+	16.6163i	−0.0860950	+	0.0860950i	−0.748843	−	0.662748i	$-0.769390\pi$
$193$	−16.6163	+	16.6163i	−0.0860950	+	0.0860950i	0.662748	+	0.748843i	$0.269390\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−86.0908	−	86.0908i	−0.437009	−	0.437009i	0.453995	−	0.891004i	$-0.349998\pi$
$197$	−86.0908	−	86.0908i	−0.437009	−	0.437009i	−0.891004	+	0.453995i	$0.849998\pi$
$198$		0			0
$199$			28.5653i			0.143544i	0.997421	+	0.0717721i	$0.0228654\pi$
$199$			28.5653i			0.143544i	−0.997421	+	0.0717721i	$0.977135\pi$
$200$		0			0
$201$	208.899			1.03930
$202$		0			0
$203$	0.247449	−	0.247449i	0.00121896	−	0.00121896i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−19.9546	−	19.9546i	−0.0963990	−	0.0963990i
$208$		0			0
$209$		−	107.687i		−	0.515248i
$210$		0			0
$211$	−197.151			−0.934365			−0.467183	−	0.884161i	$-0.654731\pi$
$211$	−197.151			−0.934365			−0.467183	+	0.884161i	$0.654731\pi$
$212$		0			0
$213$	59.2827	−	59.2827i	0.278322	−	0.278322i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	24.4949	+	24.4949i	0.112880	+	0.112880i
$218$		0			0
$219$		−	53.8638i		−	0.245953i
$220$		0			0
$221$	−668.908			−3.02673
$222$		0			0
$223$	−287.338	+	287.338i	−1.28851	+	1.28851i	−0.352822	+	0.935690i	$0.614778\pi$
$223$	−287.338	+	287.338i	−1.28851	+	1.28851i	−0.935690	+	0.352822i	$0.885222\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	234.384	+	234.384i	1.03253	+	1.03253i	0.999453	+	0.0330743i	$0.0105298\pi$
$227$	234.384	+	234.384i	1.03253	+	1.03253i	0.0330743	+	0.999453i	$0.489470\pi$
$228$		0			0
$229$		−	284.969i		−	1.24441i	−0.782855	−	0.622204i	$-0.786237\pi$
$229$		−	284.969i		−	1.24441i	0.782855	−	0.622204i	$-0.213763\pi$
$230$		0			0
$231$	−86.0908			−0.372688
$232$		0			0
$233$	241.530	−	241.530i	1.03661	−	1.03661i	0.0373060	−	0.999304i	$-0.488122\pi$
$233$	241.530	−	241.530i	1.03661	−	1.03661i	0.999304	−	0.0373060i	$-0.0118776\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−155.171	−	155.171i	−0.654732	−	0.654732i
$238$		0			0
$239$		−	327.737i		−	1.37128i	−0.727939	−	0.685642i	$-0.759522\pi$
$239$		−	327.737i		−	1.37128i	0.727939	−	0.685642i	$-0.240478\pi$
$240$		0			0
$241$	269.131			1.11672			0.558362	−	0.829597i	$-0.311430\pi$
$241$	269.131			1.11672			0.558362	+	0.829597i	$0.311430\pi$
$242$		0			0
$243$	−11.0227	+	11.0227i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−241.262	−	241.262i	−0.976770	−	0.976770i
$248$		0			0
$249$			232.565i			0.933997i
$250$		0			0
$251$	186.136			0.741579			0.370789	−	0.928717i	$-0.379087\pi$
$251$	186.136			0.741579			0.370789	+	0.928717i	$0.379087\pi$
$252$		0			0
$253$	−42.8990	+	42.8990i	−0.169561	+	0.169561i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−138.268	−	138.268i	−0.538007	−	0.538007i	0.384936	−	0.922943i	$-0.374224\pi$
$257$	−138.268	−	138.268i	−0.538007	−	0.538007i	−0.922943	+	0.384936i	$0.874224\pi$
$258$		0			0
$259$		−	385.262i		−	1.48750i
$260$		0			0
$261$	−0.136223			−0.000521927
$262$		0			0
$263$	163.146	−	163.146i	0.620329	−	0.620329i	−0.325287	−	0.945615i	$-0.605461\pi$
$263$	163.146	−	163.146i	0.620329	−	0.620329i	0.945615	+	0.325287i	$0.105461\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−87.9342	−	87.9342i	−0.329342	−	0.329342i
$268$		0			0
$269$			108.227i			0.402331i	0.979557	+	0.201165i	$0.0644729\pi$
$269$			108.227i			0.402331i	−0.979557	+	0.201165i	$0.935527\pi$
$270$		0			0
$271$	324.384			1.19699			0.598494	−	0.801127i	$-0.295766\pi$
$271$	324.384			1.19699			0.598494	+	0.801127i	$0.295766\pi$
$272$		0			0
$273$	−192.879	+	192.879i	−0.706515	+	0.706515i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	165.864	+	165.864i	0.598786	+	0.598786i	0.939990	−	0.341203i	$-0.110834\pi$
$277$	165.864	+	165.864i	0.598786	+	0.598786i	−0.341203	+	0.939990i	$0.610834\pi$
$278$		0			0
$279$		−	13.4847i		−	0.0483322i
$280$		0			0
$281$	300.434			1.06916			0.534579	−	0.845118i	$-0.320470\pi$
$281$	300.434			1.06916			0.534579	+	0.845118i	$0.320470\pi$
$282$		0			0
$283$	352.161	−	352.161i	1.24439	−	1.24439i	0.286223	−	0.958163i	$-0.407600\pi$
$283$	352.161	−	352.161i	1.24439	−	1.24439i	0.958163	−	0.286223i	$-0.0923998\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	110.586	+	110.586i	0.385316	+	0.385316i
$288$		0			0
$289$			782.514i			2.70766i
$290$		0			0
$291$	90.6311			0.311447
$292$		0			0
$293$	109.414	−	109.414i	0.373428	−	0.373428i	−0.495296	−	0.868724i	$-0.664941\pi$
$293$	109.414	−	109.414i	0.373428	−	0.373428i	0.868724	+	0.495296i	$0.164941\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	23.6969	+	23.6969i	0.0797877	+	0.0797877i
$298$		0			0
$299$			192.222i			0.642884i
$300$		0			0
$301$	−351.959			−1.16930
$302$		0			0
$303$	115.540	−	115.540i	0.381321	−	0.381321i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−110.627	−	110.627i	−0.360347	−	0.360347i	0.503594	−	0.863941i	$-0.332011\pi$
$307$	−110.627	−	110.627i	−0.360347	−	0.360347i	−0.863941	+	0.503594i	$0.832011\pi$
$308$		0			0
$309$			177.934i			0.575839i
$310$		0			0
$311$	2.38367			0.00766454			0.00383227	−	0.999993i	$-0.498780\pi$
$311$	2.38367			0.00766454			0.00383227	+	0.999993i	$0.498780\pi$
$312$		0			0
$313$	−132.959	+	132.959i	−0.424790	+	0.424790i	−0.886849	−	0.462059i	$-0.847111\pi$
$313$	−132.959	+	132.959i	−0.424790	+	0.424790i	0.462059	+	0.886849i	$0.347111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	37.1464	+	37.1464i	0.117181	+	0.117181i	0.763266	−	0.646085i	$-0.223595\pi$
$317$	37.1464	+	37.1464i	0.117181	+	0.117181i	−0.646085	+	0.763266i	$0.723595\pi$
$318$		0			0
$319$			0.292856i			0.000918045i
$320$		0			0
$321$	31.9796			0.0996249
$322$		0			0
$323$	−386.474	+	386.474i	−1.19652	+	1.19652i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−84.8536	−	84.8536i	−0.259491	−	0.259491i
$328$		0			0
$329$			550.838i			1.67428i
$330$		0			0
$331$	21.6459			0.0653955			0.0326978	−	0.999465i	$-0.489590\pi$
$331$	21.6459			0.0653955			0.0326978	+	0.999465i	$0.489590\pi$
$332$		0			0
$333$	−106.045	+	106.045i	−0.318455	+	0.318455i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	136.757	+	136.757i	0.405808	+	0.405808i	0.880274	−	0.474466i	$-0.157359\pi$
$337$	136.757	+	136.757i	0.405808	+	0.405808i	−0.474466	+	0.880274i	$0.657359\pi$
$338$		0			0
$339$		−	250.182i		−	0.737999i
$340$		0			0
$341$	−28.9898			−0.0850141
$342$		0			0
$343$	210.384	−	210.384i	0.613363	−	0.613363i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−33.8480	−	33.8480i	−0.0975445	−	0.0975445i	0.656650	−	0.754195i	$-0.271973\pi$
$347$	−33.8480	−	33.8480i	−0.0975445	−	0.0975445i	−0.754195	+	0.656650i	$0.771973\pi$
$348$		0			0
$349$		−	241.283i		−	0.691354i	−0.938354	−	0.345677i	$-0.887649\pi$
$349$		−	241.283i		−	0.691354i	0.938354	−	0.345677i	$-0.112351\pi$
$350$		0			0
$351$	106.182			0.302512
$352$		0			0
$353$	−126.833	+	126.833i	−0.359301	+	0.359301i	−0.863555	−	0.504254i	$-0.831767\pi$
$353$	−126.833	+	126.833i	−0.359301	+	0.359301i	0.504254	+	0.863555i	$0.331767\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	308.969	+	308.969i	0.865460	+	0.865460i
$358$		0			0
$359$		−	22.1112i		−	0.0615912i	−0.999526	−	0.0307956i	$-0.990196\pi$
$359$		−	22.1112i		−	0.0615912i	0.999526	−	0.0307956i	$-0.00980409\pi$
$360$		0			0
$361$	82.2122			0.227735
$362$		0			0
$363$	−97.2497	+	97.2497i	−0.267906	+	0.267906i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	213.944	+	213.944i	0.582955	+	0.582955i	0.935714	−	0.352759i	$-0.114757\pi$
$367$	213.944	+	213.944i	0.582955	+	0.582955i	−0.352759	+	0.935714i	$0.614757\pi$
$368$		0			0
$369$		−	60.8786i		−	0.164983i
$370$		0			0
$371$	60.0000			0.161725
$372$		0			0
$373$	−210.025	+	210.025i	−0.563070	+	0.563070i	−0.930178	−	0.367108i	$-0.880348\pi$
$373$	−210.025	+	210.025i	−0.563070	+	0.563070i	0.367108	+	0.930178i	$0.380348\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.656118	+	0.656118i	0.00174037	+	0.00174037i
$378$		0			0
$379$		−	124.343i		−	0.328081i	−0.986454	−	0.164041i	$-0.947547\pi$
$379$		−	124.343i		−	0.328081i	0.986454	−	0.164041i	$-0.0524528\pi$
$380$		0			0
$381$	−293.803			−0.771135
$382$		0			0
$383$	−418.540	+	418.540i	−1.09279	+	1.09279i	−0.0975654	+	0.995229i	$0.531106\pi$
$383$	−418.540	+	418.540i	−1.09279	+	1.09279i	−0.995229	+	0.0975654i	$0.968894\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	96.8786	+	96.8786i	0.250332	+	0.250332i
$388$		0			0
$389$		−	369.884i		−	0.950859i	−0.879754	−	0.475430i	$-0.842293\pi$
$389$		−	369.884i		−	0.950859i	0.879754	−	0.475430i	$-0.157707\pi$
$390$		0			0
$391$	307.918			0.787515
$392$		0			0
$393$	148.373	−	148.373i	0.377541	−	0.377541i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	373.984	+	373.984i	0.942026	+	0.942026i	0.998409	−	0.0563835i	$-0.0179570\pi$
$397$	373.984	+	373.984i	0.942026	+	0.942026i	−0.0563835	+	0.998409i	$0.517957\pi$
$398$		0			0
$399$			222.879i			0.558593i
$400$		0			0
$401$	113.151			0.282172			0.141086	−	0.989997i	$-0.454941\pi$
$401$	113.151			0.282172			0.141086	+	0.989997i	$0.454941\pi$
$402$		0			0
$403$	−64.9490	+	64.9490i	−0.161164	+	0.161164i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	227.980	+	227.980i	0.560146	+	0.560146i
$408$		0			0
$409$			90.3837i			0.220987i	0.993877	+	0.110493i	$0.0352431\pi$
$409$			90.3837i			0.220987i	−0.993877	+	0.110493i	$0.964757\pi$
$410$		0			0
$411$	−292.788			−0.712379
$412$		0			0
$413$	−302.116	+	302.116i	−0.731515	+	0.731515i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	172.429	+	172.429i	0.413499	+	0.413499i
$418$		0			0
$419$			334.772i			0.798978i	0.916738	+	0.399489i	$0.130812\pi$
$419$			334.772i			0.798978i	−0.916738	+	0.399489i	$0.869188\pi$
$420$		0			0
$421$	57.2735			0.136042			0.0680208	−	0.997684i	$-0.478332\pi$
$421$	57.2735			0.136042			0.0680208	+	0.997684i	$0.478332\pi$
$422$		0			0
$423$	151.621	−	151.621i	0.358442	−	0.358442i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	260.969	+	260.969i	0.611170	+	0.611170i
$428$		0			0
$429$		−	228.272i		−	0.532104i
$430$		0			0
$431$	442.656			1.02704			0.513522	−	0.858076i	$-0.328340\pi$
$431$	442.656			1.02704			0.513522	+	0.858076i	$0.328340\pi$
$432$		0			0
$433$	14.8684	−	14.8684i	0.0343380	−	0.0343380i	−0.689729	−	0.724067i	$-0.742270\pi$
$433$	14.8684	−	14.8684i	0.0343380	−	0.0343380i	0.724067	+	0.689729i	$0.242270\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	111.060	+	111.060i	0.254142	+	0.254142i
$438$		0			0
$439$			233.818i			0.532616i	0.963888	+	0.266308i	$0.0858038\pi$
$439$			233.818i			0.532616i	−0.963888	+	0.266308i	$0.914196\pi$
$440$		0			0
$441$	31.1816			0.0707066
$442$		0			0
$443$	246.747	−	246.747i	0.556991	−	0.556991i	−0.371459	−	0.928449i	$-0.621142\pi$
$443$	246.747	−	246.747i	0.556991	−	0.556991i	0.928449	+	0.371459i	$0.121142\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−65.5607	−	65.5607i	−0.146668	−	0.146668i
$448$		0			0
$449$		−	282.758i		−	0.629751i	−0.949133	−	0.314875i	$-0.898037\pi$
$449$		−	282.758i		−	0.629751i	0.949133	−	0.314875i	$-0.101963\pi$
$450$		0			0
$451$	−130.879			−0.290196
$452$		0			0
$453$	−284.883	+	284.883i	−0.628881	+	0.628881i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	298.171	+	298.171i	0.652454	+	0.652454i	0.953583	−	0.301129i	$-0.0973636\pi$
$457$	298.171	+	298.171i	0.652454	+	0.652454i	−0.301129	+	0.953583i	$0.597364\pi$
$458$		0			0
$459$		−	170.091i		−	0.370568i
$460$		0			0
$461$	−264.318			−0.573358			−0.286679	−	0.958027i	$-0.592551\pi$
$461$	−264.318			−0.573358			−0.286679	+	0.958027i	$0.592551\pi$
$462$		0			0
$463$	−135.116	+	135.116i	−0.291827	+	0.291827i	−0.837802	−	0.545975i	$-0.816159\pi$
$463$	−135.116	+	135.116i	−0.291827	+	0.291827i	0.545975	+	0.837802i	$0.316159\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−558.631	−	558.631i	−1.19621	−	1.19621i	−0.975292	−	0.220920i	$-0.929094\pi$
$467$	−558.631	−	558.631i	−1.19621	−	1.19621i	−0.220920	−	0.975292i	$-0.570906\pi$
$468$		0			0
$469$			929.494i			1.98186i
$470$		0			0
$471$	−315.081			−0.668961
$472$		0			0
$473$	208.272	−	208.272i	0.440322	−	0.440322i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−16.5153	−	16.5153i	−0.0346233	−	0.0346233i
$478$		0			0
$479$		−	456.141i		−	0.952277i	−0.879370	−	0.476139i	$-0.842036\pi$
$479$		−	456.141i		−	0.952277i	0.879370	−	0.476139i	$-0.157964\pi$
$480$		0			0
$481$	1021.53			2.12377
$482$		0			0
$483$	88.7878	−	88.7878i	0.183826	−	0.183826i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−11.9444	−	11.9444i	−0.0245265	−	0.0245265i	0.694737	−	0.719264i	$-0.255521\pi$
$487$	−11.9444	−	11.9444i	−0.0245265	−	0.0245265i	−0.719264	+	0.694737i	$0.755521\pi$
$488$		0			0
$489$		−	288.000i		−	0.588957i
$490$		0			0
$491$	822.468			1.67509			0.837544	−	0.546370i	$-0.183991\pi$
$491$	822.468			1.67509			0.837544	+	0.546370i	$0.183991\pi$
$492$		0			0
$493$	1.05103	−	1.05103i	0.00213190	−	0.00213190i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	263.778	+	263.778i	0.530740	+	0.530740i
$498$		0			0
$499$			312.474i			0.626201i	0.949720	+	0.313101i	$0.101368\pi$
$499$			312.474i			0.626201i	−0.949720	+	0.313101i	$0.898632\pi$
$500$		0			0
$501$	82.4541			0.164579
$502$		0			0
$503$	371.530	−	371.530i	0.738628	−	0.738628i	−0.233684	−	0.972313i	$-0.575078\pi$
$503$	371.530	−	371.530i	0.738628	−	0.738628i	0.972313	+	0.233684i	$0.0750782\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−304.442	−	304.442i	−0.600476	−	0.600476i
$508$		0			0
$509$			171.228i			0.336401i	0.985753	+	0.168200i	$0.0537956\pi$
$509$			171.228i			0.336401i	−0.985753	+	0.168200i	$0.946204\pi$
$510$		0			0
$511$	239.666			0.469014
$512$		0			0
$513$	61.3485	−	61.3485i	0.119588	−	0.119588i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−325.959	−	325.959i	−0.630482	−	0.630482i
$518$		0			0
$519$			69.1918i			0.133318i
$520$		0			0
$521$	−206.313			−0.395995			−0.197997	−	0.980203i	$-0.563444\pi$
$521$	−206.313			−0.395995			−0.197997	+	0.980203i	$0.563444\pi$
$522$		0			0
$523$	135.526	−	135.526i	0.259131	−	0.259131i	−0.565570	−	0.824701i	$-0.691344\pi$
$523$	135.526	−	135.526i	0.259131	−	0.259131i	0.824701	+	0.565570i	$0.191344\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	104.041	+	104.041i	0.197421	+	0.197421i
$528$		0			0
$529$			440.514i			0.832730i
$530$		0			0
$531$	166.318			0.313216
$532$		0			0
$533$	−293.221	+	293.221i	−0.550134	+	0.550134i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−55.4041	−	55.4041i	−0.103173	−	0.103173i
$538$		0			0
$539$		−	67.0352i		−	0.124370i
$540$		0			0
$541$	303.485			0.560970			0.280485	−	0.959858i	$-0.409505\pi$
$541$	303.485			0.560970			0.280485	+	0.959858i	$0.409505\pi$
$542$		0			0
$543$	−319.237	+	319.237i	−0.587914	+	0.587914i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	71.3939	+	71.3939i	0.130519	+	0.130519i	0.769348	−	0.638829i	$-0.220581\pi$
$547$	71.3939	+	71.3939i	0.130519	+	0.130519i	−0.638829	+	0.769348i	$0.720581\pi$
$548$		0			0
$549$		−	143.666i		−	0.261687i
$550$		0			0
$551$	0.758169			0.00137599
$552$		0			0
$553$	690.434	−	690.434i	1.24852	−	1.24852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−753.019	−	753.019i	−1.35192	−	1.35192i	−0.883513	−	0.468407i	$-0.844828\pi$
$557$	−753.019	−	753.019i	−1.35192	−	1.35192i	−0.468407	−	0.883513i	$-0.655172\pi$
$558$		0			0
$559$		−	933.231i		−	1.66946i
$560$		0			0
$561$	−365.666			−0.651812
$562$		0			0
$563$	−703.464	+	703.464i	−1.24949	+	1.24949i	−0.293548	+	0.955944i	$0.594836\pi$
$563$	−703.464	+	703.464i	−1.24949	+	1.24949i	−0.955944	+	0.293548i	$0.905164\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−49.0454	−	49.0454i	−0.0864998	−	0.0864998i
$568$		0			0
$569$			434.504i			0.763628i	0.924239	+	0.381814i	$0.124700\pi$
$569$			434.504i			0.763628i	−0.924239	+	0.381814i	$0.875300\pi$
$570$		0			0
$571$	−131.040			−0.229492			−0.114746	−	0.993395i	$-0.536605\pi$
$571$	−131.040			−0.229492			−0.114746	+	0.993395i	$0.536605\pi$
$572$		0			0
$573$	−63.5255	+	63.5255i	−0.110865	+	0.110865i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−290.444	−	290.444i	−0.503369	−	0.503369i	0.409114	−	0.912483i	$-0.365838\pi$
$577$	−290.444	−	290.444i	−0.503369	−	0.503369i	−0.912483	+	0.409114i	$0.865838\pi$
$578$		0			0
$579$			40.7015i			0.0702962i
$580$		0			0
$581$	−1034.80			−1.78106
$582$		0			0
$583$	−35.5051	+	35.5051i	−0.0609007	+	0.0609007i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−180.288	−	180.288i	−0.307135	−	0.307135i	0.536662	−	0.843797i	$-0.319685\pi$
$587$	−180.288	−	180.288i	−0.307135	−	0.307135i	−0.843797	+	0.536662i	$0.819685\pi$
$588$		0			0
$589$			75.0510i			0.127421i
$590$		0			0
$591$	−210.879			−0.356817
$592$		0			0
$593$	25.0556	−	25.0556i	0.0422523	−	0.0422523i	−0.685665	−	0.727917i	$-0.740488\pi$
$593$	25.0556	−	25.0556i	0.0422523	−	0.0422523i	0.727917	+	0.685665i	$0.240488\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	34.9852	+	34.9852i	0.0586017	+	0.0586017i
$598$		0			0
$599$			509.807i			0.851097i	0.904936	+	0.425549i	$0.139919\pi$
$599$			509.807i			0.851097i	−0.904936	+	0.425549i	$0.860081\pi$
$600$		0			0
$601$	179.757			0.299097			0.149548	−	0.988754i	$-0.452218\pi$
$601$	179.757			0.299097			0.149548	+	0.988754i	$0.452218\pi$
$602$		0			0
$603$	255.848	−	255.848i	0.424292	−	0.424292i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−33.1566	−	33.1566i	−0.0546238	−	0.0546238i	0.679267	−	0.733891i	$-0.262298\pi$
$607$	−33.1566	−	33.1566i	−0.0546238	−	0.0546238i	−0.733891	+	0.679267i	$0.762298\pi$
$608$		0			0
$609$		−	0.606123i		−	0.000995276i
$610$		0			0
$611$	−1460.56			−2.39045
$612$		0			0
$613$	261.712	−	261.712i	0.426936	−	0.426936i	−0.460647	−	0.887583i	$-0.652383\pi$
$613$	261.712	−	261.712i	0.426936	−	0.426936i	0.887583	+	0.460647i	$0.152383\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	762.075	+	762.075i	1.23513	+	1.23513i	0.961968	+	0.273161i	$0.0880693\pi$
$617$	762.075	+	762.075i	1.23513	+	1.23513i	0.273161	+	0.961968i	$0.411931\pi$
$618$		0			0
$619$		−	81.6367i		−	0.131885i	−0.997823	−	0.0659424i	$-0.978995\pi$
$619$		−	81.6367i		−	0.131885i	0.997823	−	0.0659424i	$-0.0210054\pi$
$620$		0			0
$621$	−48.8786			−0.0787095
$622$		0			0
$623$	391.262	−	391.262i	0.628029	−	0.628029i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−131.889	−	131.889i	−0.210349	−	0.210349i
$628$		0			0
$629$		−	1636.38i		−	2.60156i
$630$		0			0
$631$	−16.4133			−0.0260115			−0.0130057	−	0.999915i	$-0.504140\pi$
$631$	−16.4133			−0.0260115			−0.0130057	+	0.999915i	$0.504140\pi$
$632$		0			0
$633$	−241.460	+	241.460i	−0.381453	+	0.381453i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−150.186	−	150.186i	−0.235771	−	0.235771i
$638$		0			0
$639$		−	145.212i		−	0.227249i
$640$		0			0
$641$	−546.041			−0.851858			−0.425929	−	0.904757i	$-0.640053\pi$
$641$	−546.041			−0.851858			−0.425929	+	0.904757i	$0.640053\pi$
$642$		0			0
$643$	142.879	−	142.879i	0.222206	−	0.222206i	−0.587221	−	0.809427i	$-0.699778\pi$
$643$	142.879	−	142.879i	0.222206	−	0.222206i	0.809427	+	0.587221i	$0.199778\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	649.691	+	649.691i	1.00416	+	1.00416i	0.999991	+	0.00416838i	$0.00132684\pi$
$647$	649.691	+	649.691i	1.00416	+	1.00416i	0.00416838	+	0.999991i	$0.498673\pi$
$648$		0			0
$649$		−	357.555i		−	0.550932i
$650$		0			0
$651$	60.0000			0.0921659
$652$		0			0
$653$	150.904	−	150.904i	0.231093	−	0.231093i	−0.582056	−	0.813149i	$-0.697752\pi$
$653$	150.904	−	150.904i	0.231093	−	0.231093i	0.813149	+	0.582056i	$0.197752\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−65.9694	−	65.9694i	−0.100410	−	0.100410i
$658$		0			0
$659$		−	268.802i		−	0.407893i	−0.978982	−	0.203947i	$-0.934623\pi$
$659$		−	268.802i		−	0.407893i	0.978982	−	0.203947i	$-0.0653769\pi$
$660$		0			0
$661$	311.162			0.470745			0.235372	−	0.971905i	$-0.424369\pi$
$661$	311.162			0.470745			0.235372	+	0.971905i	$0.424369\pi$
$662$		0			0
$663$	−819.242	+	819.242i	−1.23566	+	1.23566i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.302031	−	0.302031i	−0.000452820	−	0.000452820i
$668$		0			0
$669$			703.832i			1.05207i
$670$		0			0
$671$	−308.858			−0.460295
$672$		0			0
$673$	547.756	−	547.756i	0.813902	−	0.813902i	−0.171314	−	0.985216i	$-0.554801\pi$
$673$	547.756	−	547.756i	0.813902	−	0.813902i	0.985216	+	0.171314i	$0.0548014\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−696.257	−	696.257i	−1.02844	−	1.02844i	−0.999583	−	0.0288606i	$-0.990812\pi$
$677$	−696.257	−	696.257i	−1.02844	−	1.02844i	−0.0288606	−	0.999583i	$-0.509188\pi$
$678$		0			0
$679$			403.262i			0.593906i
$680$		0			0
$681$	574.120			0.843055
$682$		0			0
$683$	84.0250	−	84.0250i	0.123023	−	0.123023i	−0.642915	−	0.765938i	$-0.722275\pi$
$683$	84.0250	−	84.0250i	0.123023	−	0.123023i	0.765938	+	0.642915i	$0.222275\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−349.015	−	349.015i	−0.508027	−	0.508027i
$688$		0			0
$689$			159.092i			0.230903i
$690$		0			0
$691$	1157.06			1.67447			0.837236	−	0.546842i	$-0.184170\pi$
$691$	1157.06			1.67447			0.837236	+	0.546842i	$0.184170\pi$
$692$		0			0
$693$	−105.439	+	105.439i	−0.152149	+	0.152149i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	469.707	+	469.707i	0.673898	+	0.673898i
$698$		0			0
$699$		−	591.626i		−	0.846388i
$700$		0			0
$701$	843.691			1.20355			0.601777	−	0.798664i	$-0.294460\pi$
$701$	843.691			1.20355			0.601777	+	0.798664i	$0.294460\pi$
$702$		0			0
$703$	590.211	−	590.211i	0.839561	−	0.839561i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	514.095	+	514.095i	0.727150	+	0.727150i
$708$		0			0
$709$		−	170.686i		−	0.240741i	−0.992729	−	0.120371i	$-0.961592\pi$
$709$		−	170.686i		−	0.240741i	0.992729	−	0.120371i	$-0.0384083\pi$
$710$		0			0
$711$	−380.091			−0.534586
$712$		0			0
$713$	29.8979	−	29.8979i	0.0419326	−	0.0419326i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−401.394	−	401.394i	−0.559824	−	0.559824i
$718$		0			0
$719$		−	605.435i		−	0.842051i	−0.907049	−	0.421026i	$-0.861670\pi$
$719$		−	605.435i		−	0.842051i	0.907049	−	0.421026i	$-0.138330\pi$
$720$		0			0
$721$	−791.716			−1.09808
$722$		0			0
$723$	329.616	−	329.616i	0.455901	−	0.455901i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	50.8025	+	50.8025i	0.0698797	+	0.0698797i	0.741183	−	0.671303i	$-0.234265\pi$
$727$	50.8025	+	50.8025i	0.0698797	+	0.0698797i	−0.671303	+	0.741183i	$0.734265\pi$
$728$		0			0
$729$			27.0000i			0.0370370i
$730$		0			0
$731$	−1494.93			−2.04505
$732$		0			0
$733$	−516.529	+	516.529i	−0.704678	+	0.704678i	−0.965411	−	0.260733i	$-0.916036\pi$
$733$	−516.529	+	516.529i	−0.704678	+	0.704678i	0.260733	+	0.965411i	$0.416036\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−550.030	−	550.030i	−0.746309	−	0.746309i
$738$		0			0
$739$			650.109i			0.879715i	0.898068	+	0.439857i	$0.144971\pi$
$739$			650.109i			0.879715i	−0.898068	+	0.439857i	$0.855029\pi$
$740$		0			0
$741$	−590.969			−0.797530
$742$		0			0
$743$	−90.6811	+	90.6811i	−0.122047	+	0.122047i	−0.765492	−	0.643445i	$-0.777504\pi$
$743$	−90.6811	+	90.6811i	−0.122047	+	0.122047i	0.643445	+	0.765492i	$0.277504\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	284.833	+	284.833i	0.381303	+	0.381303i
$748$		0			0
$749$			142.293i			0.189977i
$750$		0			0
$751$	−300.050			−0.399534			−0.199767	−	0.979843i	$-0.564019\pi$
$751$	−300.050			−0.399534			−0.199767	+	0.979843i	$0.564019\pi$
$752$		0			0
$753$	227.969	−	227.969i	0.302748	−	0.302748i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−523.176	−	523.176i	−0.691118	−	0.691118i	0.271360	−	0.962478i	$-0.412527\pi$
$757$	−523.176	−	523.176i	−0.691118	−	0.691118i	−0.962478	+	0.271360i	$0.912527\pi$
$758$		0			0
$759$			105.081i			0.138446i
$760$		0			0
$761$	−724.130			−0.951550			−0.475775	−	0.879567i	$-0.657832\pi$
$761$	−724.130			−0.951550			−0.475775	+	0.879567i	$0.657832\pi$
$762$		0			0
$763$	377.555	−	377.555i	0.494830	−	0.494830i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−801.069	−	801.069i	−1.04442	−	1.04442i
$768$		0			0
$769$		−	13.7775i		−	0.0179162i	−0.999960	−	0.00895809i	$-0.997149\pi$
$769$		−	13.7775i		−	0.0179162i	0.999960	−	0.00895809i	$-0.00285149\pi$
$770$		0			0
$771$	−338.686			−0.439281
$772$		0			0
$773$	564.207	−	564.207i	0.729892	−	0.729892i	−0.240706	−	0.970598i	$-0.577379\pi$
$773$	564.207	−	564.207i	0.729892	−	0.729892i	0.970598	+	0.240706i	$0.0773789\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−471.848	−	471.848i	−0.607269	−	0.607269i
$778$		0			0
$779$			338.829i			0.434953i
$780$		0			0
$781$	−312.182			−0.399720
$782$		0			0
$783$	−0.166838	+	0.166838i	−0.000213076	+	0.000213076i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	212.424	+	212.424i	0.269917	+	0.269917i	0.829067	−	0.559150i	$-0.188872\pi$
$787$	212.424	+	212.424i	0.269917	+	0.269917i	−0.559150	+	0.829067i	$0.688872\pi$
$788$		0			0
$789$		−	399.626i		−	0.506496i
$790$		0			0
$791$	1113.18			1.40731
$792$		0			0
$793$	−691.968	+	691.968i	−0.872596	+	0.872596i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	15.6617	+	15.6617i	0.0196509	+	0.0196509i	0.716864	−	0.697213i	$-0.245577\pi$
$797$	15.6617	+	15.6617i	0.0196509	+	0.0196509i	−0.697213	+	0.716864i	$0.745577\pi$
$798$		0			0
$799$			2339.66i			2.92823i
$800$		0			0
$801$	−215.394			−0.268906
$802$		0			0
$803$	−141.823	+	141.823i	−0.176616	+	0.176616i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	132.551	+	132.551i	0.164251	+	0.164251i
$808$		0			0
$809$		−	816.788i		−	1.00963i	−0.863229	−	0.504813i	$-0.831561\pi$
$809$		−	816.788i		−	1.00963i	0.863229	−	0.504813i	$-0.168439\pi$
$810$		0			0
$811$	830.504			1.02405			0.512025	−	0.858971i	$-0.328896\pi$
$811$	830.504			1.02405			0.512025	+	0.858971i	$0.328896\pi$
$812$		0			0
$813$	397.287	−	397.287i	0.488668	−	0.488668i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−539.192	−	539.192i	−0.659966	−	0.659966i
$818$		0			0
$819$			472.454i			0.576867i
$820$		0			0
$821$	569.217			0.693321			0.346661	−	0.937991i	$-0.387316\pi$
$821$	569.217			0.693321			0.346661	+	0.937991i	$0.387316\pi$
$822$		0			0
$823$	−497.095	+	497.095i	−0.604004	+	0.604004i	−0.941373	−	0.337369i	$-0.890463\pi$
$823$	−497.095	+	497.095i	−0.604004	+	0.604004i	0.337369	+	0.941373i	$0.390463\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−874.070	−	874.070i	−1.05692	−	1.05692i	−0.998279	−	0.0586377i	$-0.981324\pi$
$827$	−874.070	−	874.070i	−1.05692	−	1.05692i	−0.0586377	−	0.998279i	$-0.518676\pi$
$828$		0			0
$829$		−	1548.77i		−	1.86824i	−0.356954	−	0.934122i	$-0.616185\pi$
$829$		−	1548.77i		−	1.86824i	0.356954	−	0.934122i	$-0.383815\pi$
$830$		0			0
$831$	406.282			0.488907
$832$		0			0
$833$	−240.581	+	240.581i	−0.288813	+	0.288813i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−16.5153	−	16.5153i	−0.0197316	−	0.0197316i
$838$		0			0
$839$			731.523i			0.871899i	0.899971	+	0.435950i	$0.143587\pi$
$839$			731.523i			0.871899i	−0.899971	+	0.435950i	$0.856413\pi$
$840$		0			0
$841$	840.998			0.999998
$842$		0			0
$843$	367.955	−	367.955i	0.436482	−	0.436482i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−432.712	−	432.712i	−0.510876	−	0.510876i
$848$		0			0
$849$		−	862.615i		−	1.01604i
$850$		0			0
$851$	−470.243			−0.552577
$852$		0			0
$853$	554.166	−	554.166i	0.649667	−	0.649667i	−0.303246	−	0.952912i	$-0.598070\pi$
$853$	554.166	−	554.166i	0.649667	−	0.649667i	0.952912	+	0.303246i	$0.0980703\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−687.206	−	687.206i	−0.801874	−	0.801874i	0.181515	−	0.983388i	$-0.441900\pi$
$857$	−687.206	−	687.206i	−0.801874	−	0.801874i	−0.983388	+	0.181515i	$0.941900\pi$
$858$		0			0
$859$		−	1285.29i		−	1.49627i	−0.663549	−	0.748133i	$-0.730951\pi$
$859$		−	1285.29i		−	1.49627i	0.663549	−	0.748133i	$-0.269049\pi$
$860$		0			0
$861$	270.879			0.314609
$862$		0			0
$863$	712.985	−	712.985i	0.826171	−	0.826171i	−0.160814	−	0.986985i	$-0.551412\pi$
$863$	712.985	−	712.985i	0.826171	−	0.826171i	0.986985	+	0.160814i	$0.0514120\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	958.380	+	958.380i	1.10540	+	1.10540i
$868$		0			0
$869$			817.131i			0.940311i
$870$		0			0
$871$	−2464.58			−2.82960
$872$		0			0
$873$	111.000	−	111.000i	0.127148	−	0.127148i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	474.297	+	474.297i	0.540818	+	0.540818i	0.923769	−	0.382951i	$-0.125092\pi$
$877$	474.297	+	474.297i	0.540818	+	0.540818i	−0.382951	+	0.923769i	$0.625092\pi$
$878$		0			0
$879$		−	268.009i		−	0.304902i
$880$		0			0
$881$	1652.71			1.87594			0.937972	−	0.346712i	$-0.112702\pi$
$881$	1652.71			1.87594			0.937972	+	0.346712i	$0.112702\pi$
$882$		0			0
$883$	−18.9286	+	18.9286i	−0.0214367	+	0.0214367i	−0.717744	−	0.696307i	$-0.754825\pi$
$883$	−18.9286	+	18.9286i	−0.0214367	+	0.0214367i	0.696307	+	0.717744i	$0.254825\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	382.338	+	382.338i	0.431047	+	0.431047i	0.888984	−	0.457938i	$-0.151412\pi$
$887$	382.338	+	382.338i	0.431047	+	0.431047i	−0.457938	+	0.888984i	$0.651412\pi$
$888$		0			0
$889$		−	1307.27i		−	1.47050i
$890$		0			0
$891$	58.0454			0.0651464
$892$		0			0
$893$	−843.868	+	843.868i	−0.944981	+	0.944981i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	235.423	+	235.423i	0.262456	+	0.262456i
$898$		0			0
$899$		−	0.204103i		−	0.000227033i
$900$		0			0
$901$	254.847			0.282849
$902$		0			0
$903$	−431.060	+	431.060i	−0.477365	+	0.477365i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−725.485	−	725.485i	−0.799873	−	0.799873i	0.183202	−	0.983075i	$-0.441354\pi$
$907$	−725.485	−	725.485i	−0.799873	−	0.799873i	−0.983075	+	0.183202i	$0.941354\pi$
$908$		0			0
$909$		−	283.015i		−	0.311347i
$910$		0			0
$911$	−154.474			−0.169566			−0.0847829	−	0.996399i	$-0.527020\pi$
$911$	−154.474			−0.169566			−0.0847829	+	0.996399i	$0.527020\pi$
$912$		0			0
$913$	612.343	−	612.343i	0.670693	−	0.670693i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	660.186	+	660.186i	0.719941	+	0.719941i
$918$		0			0
$919$			659.079i			0.717169i	0.933497	+	0.358585i	$0.116741\pi$
$919$			659.079i			0.717169i	−0.933497	+	0.358585i	$0.883259\pi$
$920$		0			0
$921$	−270.979			−0.294222
$922$		0			0
$923$	−699.414	+	699.414i	−0.757762	+	0.757762i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	217.924	+	217.924i	0.235085	+	0.235085i
$928$		0			0
$929$			368.556i			0.396723i	0.980129	+	0.198362i	$0.0635621\pi$
$929$			368.556i			0.396723i	−0.980129	+	0.198362i	$0.936438\pi$
$930$		0			0
$931$	−173.546			−0.186408
$932$		0			0
$933$	2.91939	−	2.91939i	0.00312904	−	0.00312904i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	820.898	+	820.898i	0.876092	+	0.876092i	0.993128	−	0.117036i	$-0.0373393\pi$
$937$	820.898	+	820.898i	0.876092	+	0.876092i	−0.117036	+	0.993128i	$0.537339\pi$
$938$		0			0
$939$			325.682i			0.346839i
$940$		0			0
$941$	1758.50			1.86875			0.934377	−	0.356286i	$-0.115957\pi$
$941$	1758.50			1.86875			0.934377	+	0.356286i	$0.115957\pi$
$942$		0			0
$943$	134.979	−	134.979i	0.143137	−	0.143137i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	118.080	+	118.080i	0.124688	+	0.124688i	0.766697	−	0.642009i	$-0.221899\pi$
$947$	118.080	+	118.080i	0.124688	+	0.124688i	−0.642009	+	0.766697i	$0.721899\pi$
$948$		0			0
$949$			635.483i			0.669634i
$950$		0			0
$951$	90.9898			0.0956780
$952$		0			0
$953$	217.955	−	217.955i	0.228704	−	0.228704i	−0.583447	−	0.812151i	$-0.698297\pi$
$953$	217.955	−	217.955i	0.228704	−	0.228704i	0.812151	+	0.583447i	$0.198297\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	0.358674	+	0.358674i	0.000374790	+	0.000374790i
$958$		0			0
$959$		−	1302.76i		−	1.35845i
$960$		0			0
$961$	−940.796			−0.978976
$962$		0			0
$963$	39.1668	−	39.1668i	0.0406717	−	0.0406717i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	323.983	+	323.983i	0.335039	+	0.335039i	0.854497	−	0.519457i	$-0.173866\pi$
$967$	323.983	+	323.983i	0.335039	+	0.335039i	−0.519457	+	0.854497i	$0.673866\pi$
$968$		0			0
$969$			946.665i			0.976951i
$970$		0			0
$971$	743.743			0.765956			0.382978	−	0.923757i	$-0.374898\pi$
$971$	743.743			0.765956			0.382978	+	0.923757i	$0.374898\pi$
$972$		0			0
$973$	−767.221	+	767.221i	−0.788511	+	0.788511i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1050.52	+	1050.52i	1.07525	+	1.07525i	0.996928	+	0.0783226i	$0.0249564\pi$
$977$	1050.52	+	1050.52i	1.07525	+	1.07525i	0.0783226	+	0.996928i	$0.475044\pi$
$978$		0			0
$979$			463.060i			0.472993i
$980$		0			0
$981$	−207.848			−0.211874
$982$		0			0
$983$	−72.9648	+	72.9648i	−0.0742267	+	0.0742267i	−0.743245	−	0.669019i	$-0.766715\pi$
$983$	−72.9648	+	72.9648i	−0.0742267	+	0.0742267i	0.669019	+	0.743245i	$0.266715\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	674.636	+	674.636i	0.683521	+	0.683521i
$988$		0			0
$989$			429.594i			0.434372i
$990$		0			0
$991$	444.624			0.448662			0.224331	−	0.974513i	$-0.427980\pi$
$991$	444.624			0.448662			0.224331	+	0.974513i	$0.427980\pi$
$992$		0			0
$993$	26.5107	−	26.5107i	0.0266976	−	0.0266976i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−12.6424	−	12.6424i	−0.0126804	−	0.0126804i	0.700738	−	0.713419i	$-0.252854\pi$
$997$	−12.6424	−	12.6424i	−0.0126804	−	0.0126804i	−0.713419	+	0.700738i	$0.752854\pi$
$998$		0			0
$999$			259.757i			0.260017i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.u.e.457.2		4
3.2	odd	2		1800.3.v.n.1657.2		4
4.3	odd	2		1200.3.bg.e.1057.1		4
5.2	odd	4		120.3.u.a.73.1	✓	4
5.3	odd	4	inner	600.3.u.e.193.2		4
5.4	even	2		120.3.u.a.97.1	yes	4
15.2	even	4		360.3.v.b.73.2		4
15.8	even	4		1800.3.v.n.793.2		4
15.14	odd	2		360.3.v.b.217.2		4
20.3	even	4		1200.3.bg.e.193.1		4
20.7	even	4		240.3.bg.c.193.2		4
20.19	odd	2		240.3.bg.c.97.2		4
40.19	odd	2		960.3.bg.d.577.1		4
40.27	even	4		960.3.bg.d.193.1		4
40.29	even	2		960.3.bg.c.577.2		4
40.37	odd	4		960.3.bg.c.193.2		4
60.47	odd	4		720.3.bh.g.433.2		4
60.59	even	2		720.3.bh.g.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.a.73.1	✓	4	5.2	odd	4
120.3.u.a.97.1	yes	4	5.4	even	2
240.3.bg.c.97.2		4	20.19	odd	2
240.3.bg.c.193.2		4	20.7	even	4
360.3.v.b.73.2		4	15.2	even	4
360.3.v.b.217.2		4	15.14	odd	2
600.3.u.e.193.2		4	5.3	odd	4	inner
600.3.u.e.457.2		4	1.1	even	1	trivial
720.3.bh.g.433.2		4	60.47	odd	4
720.3.bh.g.577.2		4	60.59	even	2
960.3.bg.c.193.2		4	40.37	odd	4
960.3.bg.c.577.2		4	40.29	even	2
960.3.bg.d.193.1		4	40.27	even	4
960.3.bg.d.577.1		4	40.19	odd	2
1200.3.bg.e.193.1		4	20.3	even	4
1200.3.bg.e.1057.1		4	4.3	odd	2
1800.3.v.n.793.2		4	15.8	even	4
1800.3.v.n.1657.2		4	3.2	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:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 - 1.22474i) q^{3} +(5.44949 + 5.44949i) q^{7} -3.00000i q^{9} -6.44949 q^{11} +(-14.4495 + 14.4495i) q^{13} +(23.1464 + 23.1464i) q^{17} +16.6969i q^{19} +13.3485 q^{21} +(6.65153 - 6.65153i) q^{23} +(-3.67423 - 3.67423i) q^{27} -0.0454077i q^{29} +4.49490 q^{31} +(-7.89898 + 7.89898i) q^{33} +(-35.3485 - 35.3485i) q^{37} +35.3939i q^{39} +20.2929 q^{41} +(-32.2929 + 32.2929i) q^{43} +(50.5403 + 50.5403i) q^{47} +10.3939i q^{49} +56.6969 q^{51} +(5.50510 - 5.50510i) q^{53} +(20.4495 + 20.4495i) q^{57} +55.4393i q^{59} +47.8888 q^{61} +(16.3485 - 16.3485i) q^{63} +(85.2827 + 85.2827i) q^{67} -16.2929i q^{69} +48.4041 q^{71} +(21.9898 - 21.9898i) q^{73} +(-35.1464 - 35.1464i) q^{77} -126.697i q^{79} -9.00000 q^{81} +(-94.9444 + 94.9444i) q^{83} +(-0.0556128 - 0.0556128i) q^{87} -71.7980i q^{89} -157.485 q^{91} +(5.50510 - 5.50510i) q^{93} +(37.0000 + 37.0000i) q^{97} +19.3485i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} - 16 q^{11} - 48 q^{13} + 24 q^{17} + 24 q^{21} + 56 q^{23} - 80 q^{31} - 12 q^{33} - 112 q^{37} - 56 q^{41} + 8 q^{43} + 16 q^{47} + 168 q^{51} + 120 q^{53} + 72 q^{57} - 24 q^{61} + 36 q^{63}+ \cdots + 148 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 - 16 * q^11 - 48 * q^13 + 24 * q^17 + 24 * q^21 + 56 * q^23 - 80 * q^31 - 12 * q^33 - 112 * q^37 - 56 * q^41 + 8 * q^43 + 16 * q^47 + 168 * q^51 + 120 * q^53 + 72 * q^57 - 24 * q^61 + 36 * q^63 + 8 * q^67 + 272 * q^71 - 108 * q^73 - 72 * q^77 - 36 * q^81 - 272 * q^83 - 108 * q^87 - 336 * q^91 + 120 * q^93 + 148 * q^97

Introduction

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