LMFDB - Embedded newform 600.3.u.c.193.2

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,-8,0,0,0,-8,0,-48,0,0,0,-8,0,0,0,-12] 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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			193.2
Root			$1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	600.193
Dual form			600.3.u.c.457.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 1.22474i) q^{3} +(-3.22474 + 3.22474i) q^{7} +3.00000i q^{9} -6.89898 q^{11} +(-18.1237 - 18.1237i) q^{13} +(0.449490 - 0.449490i) q^{17} -9.89898i q^{19} -7.89898 q^{21} +(-10.6515 - 10.6515i) q^{23} +(-3.67423 + 3.67423i) q^{27} -36.2929i q^{29} +25.6969 q^{31} +(-8.44949 - 8.44949i) q^{33} +(-13.3031 + 13.3031i) q^{37} -44.3939i q^{39} -3.10102 q^{41} +(-2.72985 - 2.72985i) q^{43} +(-37.1464 + 37.1464i) q^{47} +28.2020i q^{49} +1.10102 q^{51} +(-65.1918 - 65.1918i) q^{53} +(12.1237 - 12.1237i) q^{57} -80.3837i q^{59} +13.7878 q^{61} +(-9.67423 - 9.67423i) q^{63} +(-84.3712 + 84.3712i) q^{67} -26.0908i q^{69} +98.2929 q^{71} +(-52.4949 - 52.4949i) q^{73} +(22.2474 - 22.2474i) q^{77} -68.2020i q^{79} -9.00000 q^{81} +(89.7321 + 89.7321i) q^{83} +(44.4495 - 44.4495i) q^{87} -40.5857i q^{89} +116.889 q^{91} +(31.4722 + 31.4722i) q^{93} +(-105.720 + 105.720i) q^{97} -20.6969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{7} - 8 q^{11} - 48 q^{13} - 8 q^{17} - 12 q^{21} - 72 q^{23} + 44 q^{31} - 24 q^{33} - 112 q^{37} - 32 q^{41} - 104 q^{43} - 80 q^{47} + 24 q^{51} - 104 q^{53} + 24 q^{57} - 180 q^{61} - 24 q^{63}+ \cdots - 320 q^{97}+O(q^{100}) $ 4 * q - 8 * q^7 - 8 * q^11 - 48 * q^13 - 8 * q^17 - 12 * q^21 - 72 * q^23 + 44 * q^31 - 24 * q^33 - 112 * q^37 - 32 * q^41 - 104 * q^43 - 80 * q^47 + 24 * q^51 - 104 * q^53 + 24 * q^57 - 180 * q^61 - 24 * q^63 - 264 * q^67 + 256 * q^71 - 112 * q^73 + 40 * q^77 - 36 * q^81 + 16 * q^83 + 168 * q^87 + 252 * q^91 + 72 * q^93 - 320 * q^97

Character values

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

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$7$	−3.22474	+	3.22474i	−0.460678	+	0.460678i	−0.898878	−	0.438200i	$-0.855616\pi$
$7$	−3.22474	+	3.22474i	−0.460678	+	0.460678i	0.438200	+	0.898878i	$0.355616\pi$
$8$		0			0
$9$			3.00000i			0.333333i
$10$		0			0
$11$	−6.89898			−0.627180			−0.313590	−	0.949558i	$-0.601532\pi$
$11$	−6.89898			−0.627180			−0.313590	+	0.949558i	$0.601532\pi$
$12$		0			0
$13$	−18.1237	−	18.1237i	−1.39413	−	1.39413i	−0.815804	−	0.578329i	$-0.803705\pi$
$13$	−18.1237	−	18.1237i	−1.39413	−	1.39413i	−0.578329	−	0.815804i	$-0.696295\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.449490	−	0.449490i	0.0264406	−	0.0264406i	−0.693763	−	0.720203i	$-0.744048\pi$
$17$	0.449490	−	0.449490i	0.0264406	−	0.0264406i	0.720203	+	0.693763i	$0.244048\pi$
$18$		0			0
$19$		−	9.89898i		−	0.520999i	−0.965474	−	0.260499i	$-0.916113\pi$
$19$		−	9.89898i		−	0.520999i	0.965474	−	0.260499i	$-0.0838872\pi$
$20$		0			0
$21$	−7.89898			−0.376142
$22$		0			0
$23$	−10.6515	−	10.6515i	−0.463110	−	0.463110i	0.436563	−	0.899673i	$-0.356195\pi$
$23$	−10.6515	−	10.6515i	−0.463110	−	0.463110i	−0.899673	+	0.436563i	$0.856195\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−3.67423	+	3.67423i	−0.136083	+	0.136083i
$28$		0			0
$29$		−	36.2929i		−	1.25148i	−0.780033	−	0.625739i	$-0.784798\pi$
$29$		−	36.2929i		−	1.25148i	0.780033	−	0.625739i	$-0.215202\pi$
$30$		0			0
$31$	25.6969			0.828933			0.414467	−	0.910064i	$-0.363968\pi$
$31$	25.6969			0.828933			0.414467	+	0.910064i	$0.363968\pi$
$32$		0			0
$33$	−8.44949	−	8.44949i	−0.256045	−	0.256045i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−13.3031	+	13.3031i	−0.359542	+	0.359542i	−0.863644	−	0.504102i	$-0.831824\pi$
$37$	−13.3031	+	13.3031i	−0.359542	+	0.359542i	0.504102	+	0.863644i	$0.331824\pi$
$38$		0			0
$39$		−	44.3939i		−	1.13830i
$40$		0			0
$41$	−3.10102			−0.0756346			−0.0378173	−	0.999285i	$-0.512040\pi$
$41$	−3.10102			−0.0756346			−0.0378173	+	0.999285i	$0.512040\pi$
$42$		0			0
$43$	−2.72985	−	2.72985i	−0.0634848	−	0.0634848i	0.674652	−	0.738136i	$-0.264294\pi$
$43$	−2.72985	−	2.72985i	−0.0634848	−	0.0634848i	−0.738136	+	0.674652i	$0.764294\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−37.1464	+	37.1464i	−0.790350	+	0.790350i	−0.981551	−	0.191201i	$-0.938762\pi$
$47$	−37.1464	+	37.1464i	−0.790350	+	0.790350i	0.191201	+	0.981551i	$0.438762\pi$
$48$		0			0
$49$			28.2020i			0.575552i
$50$		0			0
$51$	1.10102			0.0215886
$52$		0			0
$53$	−65.1918	−	65.1918i	−1.23003	−	1.23003i	−0.963950	−	0.266085i	$-0.914270\pi$
$53$	−65.1918	−	65.1918i	−1.23003	−	1.23003i	−0.266085	−	0.963950i	$-0.585730\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	12.1237	−	12.1237i	0.212697	−	0.212697i
$58$		0			0
$59$		−	80.3837i		−	1.36244i	−0.732081	−	0.681218i	$-0.761451\pi$
$59$		−	80.3837i		−	1.36244i	0.732081	−	0.681218i	$-0.238549\pi$
$60$		0			0
$61$	13.7878			0.226029			0.113014	−	0.993593i	$-0.463949\pi$
$61$	13.7878			0.226029			0.113014	+	0.993593i	$0.463949\pi$
$62$		0			0
$63$	−9.67423	−	9.67423i	−0.153559	−	0.153559i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−84.3712	+	84.3712i	−1.25927	+	1.25927i	−0.307830	+	0.951441i	$0.599603\pi$
$67$	−84.3712	+	84.3712i	−1.25927	+	1.25927i	−0.951441	+	0.307830i	$0.900397\pi$
$68$		0			0
$69$		−	26.0908i		−	0.378128i
$70$		0			0
$71$	98.2929			1.38441			0.692203	−	0.721703i	$-0.256640\pi$
$71$	98.2929			1.38441			0.692203	+	0.721703i	$0.256640\pi$
$72$		0			0
$73$	−52.4949	−	52.4949i	−0.719108	−	0.719108i	0.249314	−	0.968423i	$-0.419795\pi$
$73$	−52.4949	−	52.4949i	−0.719108	−	0.719108i	−0.968423	+	0.249314i	$0.919795\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22.2474	−	22.2474i	0.288928	−	0.288928i
$78$		0			0
$79$		−	68.2020i		−	0.863317i	−0.902037	−	0.431658i	$-0.857929\pi$
$79$		−	68.2020i		−	0.863317i	0.902037	−	0.431658i	$-0.142071\pi$
$80$		0			0
$81$	−9.00000			−0.111111
$82$		0			0
$83$	89.7321	+	89.7321i	1.08111	+	1.08111i	0.996406	+	0.0847040i	$0.0269945\pi$
$83$	89.7321	+	89.7321i	1.08111	+	1.08111i	0.0847040	+	0.996406i	$0.473006\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	44.4495	−	44.4495i	0.510914	−	0.510914i
$88$		0			0
$89$		−	40.5857i		−	0.456019i	−0.973659	−	0.228010i	$-0.926778\pi$
$89$		−	40.5857i		−	0.456019i	0.973659	−	0.228010i	$-0.0732218\pi$
$90$		0			0
$91$	116.889			1.28449
$92$		0			0
$93$	31.4722	+	31.4722i	0.338411	+	0.338411i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−105.720	+	105.720i	−1.08989	+	1.08989i	−0.0943546	+	0.995539i	$0.530079\pi$
$97$	−105.720	+	105.720i	−1.08989	+	1.08989i	−0.995539	+	0.0943546i	$0.969921\pi$
$98$		0			0
$99$		−	20.6969i		−	0.209060i
$100$		0			0
$101$	−197.485			−1.95529			−0.977647	−	0.210253i	$-0.932571\pi$
$101$	−197.485			−1.95529			−0.977647	+	0.210253i	$0.932571\pi$
$102$		0			0
$103$	−45.7980	−	45.7980i	−0.444640	−	0.444640i	0.448928	−	0.893568i	$-0.351806\pi$
$103$	−45.7980	−	45.7980i	−0.444640	−	0.444640i	−0.893568	+	0.448928i	$0.851806\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	139.485	−	139.485i	1.30360	−	1.30360i	0.377645	−	0.925951i	$-0.376734\pi$
$107$	139.485	−	139.485i	1.30360	−	1.30360i	0.925951	−	0.377645i	$-0.123266\pi$
$108$		0			0
$109$			140.576i			1.28968i	0.764316	+	0.644842i	$0.223077\pi$
$109$			140.576i			1.28968i	−0.764316	+	0.644842i	$0.776923\pi$
$110$		0			0
$111$	−32.5857			−0.293565
$112$		0			0
$113$	−105.980	−	105.980i	−0.937872	−	0.937872i	0.0603074	−	0.998180i	$-0.480792\pi$
$113$	−105.980	−	105.980i	−0.937872	−	0.937872i	−0.998180	+	0.0603074i	$0.980792\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	54.3712	−	54.3712i	0.464711	−	0.464711i
$118$		0			0
$119$			2.89898i			0.0243612i
$120$		0			0
$121$	−73.4041			−0.606645
$122$		0			0
$123$	−3.79796	−	3.79796i	−0.0308777	−	0.0308777i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−66.6969	+	66.6969i	−0.525173	+	0.525173i	−0.919129	−	0.393956i	$-0.871106\pi$
$127$	−66.6969	+	66.6969i	−0.525173	+	0.525173i	0.393956	+	0.919129i	$0.371106\pi$
$128$		0			0
$129$		−	6.68673i		−	0.0518351i
$130$		0			0
$131$	−32.6765			−0.249439			−0.124720	−	0.992192i	$-0.539803\pi$
$131$	−32.6765			−0.249439			−0.124720	+	0.992192i	$0.539803\pi$
$132$		0			0
$133$	31.9217	+	31.9217i	0.240013	+	0.240013i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−45.3281	+	45.3281i	−0.330862	+	0.330862i	−0.852914	−	0.522052i	$-0.825167\pi$
$137$	−45.3281	+	45.3281i	−0.330862	+	0.330862i	0.522052	+	0.852914i	$0.325167\pi$
$138$		0			0
$139$			252.747i			1.81832i	0.416443	+	0.909162i	$0.363276\pi$
$139$			252.747i			1.81832i	−0.416443	+	0.909162i	$0.636724\pi$
$140$		0			0
$141$	−90.9898			−0.645318
$142$		0			0
$143$	125.035	+	125.035i	0.874372	+	0.874372i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−34.5403	+	34.5403i	−0.234968	+	0.234968i
$148$		0			0
$149$		−	75.2122i		−	0.504780i	−0.967626	−	0.252390i	$-0.918783\pi$
$149$		−	75.2122i		−	0.504780i	0.967626	−	0.252390i	$-0.0812166\pi$
$150$		0			0
$151$	187.091			1.23901			0.619506	−	0.784992i	$-0.287333\pi$
$151$	187.091			1.23901			0.619506	+	0.784992i	$0.287333\pi$
$152$		0			0
$153$	1.34847	+	1.34847i	0.00881352	+	0.00881352i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	119.452	−	119.452i	0.760839	−	0.760839i	−0.215635	−	0.976474i	$-0.569182\pi$
$157$	119.452	−	119.452i	0.760839	−	0.760839i	0.976474	+	0.215635i	$0.0691820\pi$
$158$		0			0
$159$		−	159.687i		−	1.00432i
$160$		0			0
$161$	68.6969			0.426689
$162$		0			0
$163$	104.866	+	104.866i	0.643350	+	0.643350i	0.951377	−	0.308027i	$-0.0996688\pi$
$163$	104.866	+	104.866i	0.643350	+	0.643350i	−0.308027	+	0.951377i	$0.599669\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	67.3031	−	67.3031i	0.403012	−	0.403012i	−0.476281	−	0.879293i	$-0.658015\pi$
$167$	67.3031	−	67.3031i	0.403012	−	0.403012i	0.879293	+	0.476281i	$0.158015\pi$
$168$		0			0
$169$			487.939i			2.88721i
$170$		0			0
$171$	29.6969			0.173666
$172$		0			0
$173$	47.8638	+	47.8638i	0.276669	+	0.276669i	0.831778	−	0.555109i	$-0.187323\pi$
$173$	47.8638	+	47.8638i	0.276669	+	0.276669i	−0.555109	+	0.831778i	$0.687323\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	98.4495	−	98.4495i	0.556212	−	0.556212i
$178$		0			0
$179$			134.000i			0.748603i	0.927307	+	0.374302i	$0.122118\pi$
$179$			134.000i			0.748603i	−0.927307	+	0.374302i	$0.877882\pi$
$180$		0			0
$181$	−34.4143			−0.190134			−0.0950671	−	0.995471i	$-0.530307\pi$
$181$	−34.4143			−0.190134			−0.0950671	+	0.995471i	$0.530307\pi$
$182$		0			0
$183$	16.8865	+	16.8865i	0.0922759	+	0.0922759i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−3.10102	+	3.10102i	−0.0165830	+	0.0165830i
$188$		0			0
$189$		−	23.6969i		−	0.125381i
$190$		0			0
$191$	−238.272			−1.24750			−0.623750	−	0.781624i	$-0.714392\pi$
$191$	−238.272			−1.24750			−0.623750	+	0.781624i	$0.714392\pi$
$192$		0			0
$193$	61.8763	+	61.8763i	0.320602	+	0.320602i	0.848998	−	0.528396i	$-0.177206\pi$
$193$	61.8763	+	61.8763i	0.320602	+	0.320602i	−0.528396	+	0.848998i	$0.677206\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	72.2474	−	72.2474i	0.366738	−	0.366738i	−0.499548	−	0.866286i	$-0.666501\pi$
$197$	72.2474	−	72.2474i	0.366738	−	0.366738i	0.866286	+	0.499548i	$0.166501\pi$
$198$		0			0
$199$			85.4541i			0.429417i	0.976678	+	0.214709i	$0.0688802\pi$
$199$			85.4541i			0.429417i	−0.976678	+	0.214709i	$0.931120\pi$
$200$		0			0
$201$	−206.666			−1.02819
$202$		0			0
$203$	117.035	+	117.035i	0.576528	+	0.576528i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	31.9546	−	31.9546i	0.154370	−	0.154370i
$208$		0			0
$209$			68.2929i			0.326760i
$210$		0			0
$211$	99.4745			0.471443			0.235722	−	0.971821i	$-0.424255\pi$
$211$	99.4745			0.471443			0.235722	+	0.971821i	$0.424255\pi$
$212$		0			0
$213$	120.384	+	120.384i	0.565182	+	0.565182i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−82.8661	+	82.8661i	−0.381871	+	0.381871i
$218$		0			0
$219$		−	128.586i		−	0.587149i
$220$		0			0
$221$	−16.2929			−0.0737233
$222$		0			0
$223$	−2.35076	−	2.35076i	−0.0105415	−	0.0105415i	0.701816	−	0.712358i	$-0.252373\pi$
$223$	−2.35076	−	2.35076i	−0.0105415	−	0.0105415i	−0.712358	+	0.701816i	$0.752373\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	204.474	−	204.474i	0.900769	−	0.900769i	−0.0947340	−	0.995503i	$-0.530200\pi$
$227$	204.474	−	204.474i	0.900769	−	0.900769i	0.995503	+	0.0947340i	$0.0302000\pi$
$228$		0			0
$229$		−	131.808i		−	0.575582i	−0.957693	−	0.287791i	$-0.907079\pi$
$229$		−	131.808i		−	0.575582i	0.957693	−	0.287791i	$-0.0929208\pi$
$230$		0			0
$231$	54.4949			0.235909
$232$		0			0
$233$	139.616	+	139.616i	0.599212	+	0.599212i	0.940103	−	0.340891i	$-0.110729\pi$
$233$	139.616	+	139.616i	0.599212	+	0.599212i	−0.340891	+	0.940103i	$0.610729\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	83.5301	−	83.5301i	0.352448	−	0.352448i
$238$		0			0
$239$			118.697i			0.496640i	0.968678	+	0.248320i	$0.0798784\pi$
$239$			118.697i			0.496640i	−0.968678	+	0.248320i	$0.920122\pi$
$240$		0			0
$241$	277.384			1.15097			0.575485	−	0.817812i	$-0.304813\pi$
$241$	277.384			1.15097			0.575485	+	0.817812i	$0.304813\pi$
$242$		0			0
$243$	−11.0227	−	11.0227i	−0.0453609	−	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−179.406	+	179.406i	−0.726342	+	0.726342i
$248$		0			0
$249$			219.798i			0.882723i
$250$		0			0
$251$	37.5755			0.149703			0.0748516	−	0.997195i	$-0.476152\pi$
$251$	37.5755			0.149703			0.0748516	+	0.997195i	$0.476152\pi$
$252$		0			0
$253$	73.4847	+	73.4847i	0.290453	+	0.290453i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−79.8684	+	79.8684i	−0.310772	+	0.310772i	−0.845209	−	0.534437i	$-0.820524\pi$
$257$	−79.8684	+	79.8684i	−0.310772	+	0.310772i	0.534437	+	0.845209i	$0.320524\pi$
$258$		0			0
$259$		−	85.7980i		−	0.331266i
$260$		0			0
$261$	108.879			0.417159
$262$		0			0
$263$	222.767	+	222.767i	0.847024	+	0.847024i	0.989761	−	0.142737i	$-0.0455902\pi$
$263$	222.767	+	222.767i	0.847024	+	0.847024i	−0.142737	+	0.989761i	$0.545590\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	49.7071	−	49.7071i	0.186169	−	0.186169i
$268$		0			0
$269$		−	450.091i		−	1.67320i	−0.547814	−	0.836600i	$-0.684540\pi$
$269$		−	450.091i		−	1.67320i	0.547814	−	0.836600i	$-0.315460\pi$
$270$		0			0
$271$	−99.2122			−0.366097			−0.183048	−	0.983104i	$-0.558597\pi$
$271$	−99.2122			−0.366097			−0.183048	+	0.983104i	$0.558597\pi$
$272$		0			0
$273$	143.159	+	143.159i	0.524392	+	0.524392i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−26.5936	+	26.5936i	−0.0960059	+	0.0960059i	−0.753478	−	0.657473i	$-0.771626\pi$
$277$	−26.5936	+	26.5936i	−0.0960059	+	0.0960059i	0.657473	+	0.753478i	$0.271626\pi$
$278$		0			0
$279$			77.0908i			0.276311i
$280$		0			0
$281$	325.101			1.15694			0.578472	−	0.815703i	$-0.303649\pi$
$281$	325.101			1.15694			0.578472	+	0.815703i	$0.303649\pi$
$282$		0			0
$283$	−158.351	−	158.351i	−0.559543	−	0.559543i	0.369634	−	0.929177i	$-0.379483\pi$
$283$	−158.351	−	158.351i	−0.559543	−	0.559543i	−0.929177	+	0.369634i	$0.879483\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	10.0000	−	10.0000i	0.0348432	−	0.0348432i
$288$		0			0
$289$			288.596i			0.998602i
$290$		0			0
$291$	−258.959			−0.889894
$292$		0			0
$293$	−347.126	−	347.126i	−1.18473	−	1.18473i	−0.978504	−	0.206226i	$-0.933882\pi$
$293$	−347.126	−	347.126i	−1.18473	−	1.18473i	−0.206226	−	0.978504i	$-0.566118\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	25.3485	−	25.3485i	0.0853484	−	0.0853484i
$298$		0			0
$299$			386.091i			1.29127i
$300$		0			0
$301$	17.6061			0.0584921
$302$		0			0
$303$	−241.868	−	241.868i	−0.798245	−	0.798245i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−107.250	+	107.250i	−0.349348	+	0.349348i	−0.859867	−	0.510519i	$-0.829453\pi$
$307$	−107.250	+	107.250i	−0.349348	+	0.349348i	0.510519	+	0.859867i	$0.329453\pi$
$308$		0			0
$309$		−	112.182i		−	0.363047i
$310$		0			0
$311$	−356.788			−1.14723			−0.573614	−	0.819126i	$-0.694459\pi$
$311$	−356.788			−1.14723			−0.573614	+	0.819126i	$0.694459\pi$
$312$		0			0
$313$	103.386	+	103.386i	0.330307	+	0.330307i	0.852703	−	0.522396i	$-0.174962\pi$
$313$	103.386	+	103.386i	0.330307	+	0.330307i	−0.522396	+	0.852703i	$0.674962\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	207.060	−	207.060i	0.653187	−	0.653187i	−0.300572	−	0.953759i	$-0.597178\pi$
$317$	207.060	−	207.060i	0.653187	−	0.653187i	0.953759	+	0.300572i	$0.0971777\pi$
$318$		0			0
$319$			250.384i			0.784902i
$320$		0			0
$321$	341.666			1.06438
$322$		0			0
$323$	−4.44949	−	4.44949i	−0.0137755	−	0.0137755i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−172.169	+	172.169i	−0.526511	+	0.526511i
$328$		0			0
$329$		−	239.576i		−	0.728193i
$330$		0			0
$331$	−565.555			−1.70863			−0.854313	−	0.519759i	$-0.826022\pi$
$331$	−565.555			−1.70863			−0.854313	+	0.519759i	$0.826022\pi$
$332$		0			0
$333$	−39.9092	−	39.9092i	−0.119847	−	0.119847i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−65.6538	+	65.6538i	−0.194818	+	0.194818i	−0.797774	−	0.602956i	$-0.793989\pi$
$337$	−65.6538	+	65.6538i	−0.194818	+	0.194818i	0.602956	+	0.797774i	$0.293989\pi$
$338$		0			0
$339$		−	259.596i		−	0.765770i
$340$		0			0
$341$	−177.283			−0.519890
$342$		0			0
$343$	−248.957	−	248.957i	−0.725822	−	0.725822i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−106.177	+	106.177i	−0.305986	+	0.305986i	−0.843350	−	0.537364i	$-0.819420\pi$
$347$	−106.177	+	106.177i	−0.305986	+	0.305986i	0.537364	+	0.843350i	$0.319420\pi$
$348$		0			0
$349$		−	335.980i		−	0.962692i	−0.876531	−	0.481346i	$-0.840148\pi$
$349$		−	335.980i		−	0.962692i	0.876531	−	0.481346i	$-0.159852\pi$
$350$		0			0
$351$	133.182			0.379435
$352$		0			0
$353$	−419.328	−	419.328i	−1.18790	−	1.18790i	−0.977648	−	0.210251i	$-0.932572\pi$
$353$	−419.328	−	419.328i	−1.18790	−	1.18790i	−0.210251	−	0.977648i	$-0.567428\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−3.55051	+	3.55051i	−0.00994541	+	0.00994541i
$358$		0			0
$359$		−	191.019i		−	0.532087i	−0.963961	−	0.266044i	$-0.914283\pi$
$359$		−	191.019i		−	0.532087i	0.963961	−	0.266044i	$-0.0857166\pi$
$360$		0			0
$361$	263.010			0.728560
$362$		0			0
$363$	−89.9013	−	89.9013i	−0.247662	−	0.247662i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	207.341	−	207.341i	0.564961	−	0.564961i	−0.365752	−	0.930712i	$-0.619188\pi$
$367$	207.341	−	207.341i	0.564961	−	0.564961i	0.930712	+	0.365752i	$0.119188\pi$
$368$		0			0
$369$		−	9.30306i		−	0.0252115i
$370$		0			0
$371$	420.454			1.13330
$372$		0			0
$373$	353.052	+	353.052i	0.946521	+	0.946521i	0.998641	−	0.0521200i	$-0.0165978\pi$
$373$	353.052	+	353.052i	0.946521	+	0.946521i	−0.0521200	+	0.998641i	$0.516598\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−657.762	+	657.762i	−1.74473	+	1.74473i
$378$		0			0
$379$		−	607.393i		−	1.60262i	−0.598250	−	0.801310i	$-0.704137\pi$
$379$		−	607.393i		−	1.60262i	0.598250	−	0.801310i	$-0.295863\pi$
$380$		0			0
$381$	−163.373			−0.428802
$382$		0			0
$383$	−207.414	−	207.414i	−0.541552	−	0.541552i	0.382432	−	0.923984i	$-0.375087\pi$
$383$	−207.414	−	207.414i	−0.541552	−	0.541552i	−0.923984	+	0.382432i	$0.875087\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.18954	−	8.18954i	0.0211616	−	0.0211616i
$388$		0			0
$389$		−	593.889i		−	1.52671i	−0.645981	−	0.763353i	$-0.723552\pi$
$389$		−	593.889i		−	1.52671i	0.645981	−	0.763353i	$-0.276448\pi$
$390$		0			0
$391$	−9.57551			−0.0244898
$392$		0			0
$393$	−40.0204	−	40.0204i	−0.101833	−	0.101833i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	288.062	−	288.062i	0.725598	−	0.725598i	−0.244141	−	0.969740i	$-0.578506\pi$
$397$	288.062	−	288.062i	0.725598	−	0.725598i	0.969740	+	0.244141i	$0.0785061\pi$
$398$		0			0
$399$			78.1918i			0.195970i
$400$		0			0
$401$	129.748			0.323561			0.161781	−	0.986827i	$-0.448276\pi$
$401$	129.748			0.323561			0.161781	+	0.986827i	$0.448276\pi$
$402$		0			0
$403$	−465.724	−	465.724i	−1.15564	−	1.15564i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	91.7775	−	91.7775i	0.225498	−	0.225498i
$408$		0			0
$409$		−	262.696i		−	0.642288i	−0.947030	−	0.321144i	$-0.895933\pi$
$409$		−	262.696i		−	0.642288i	0.947030	−	0.321144i	$-0.104067\pi$
$410$		0			0
$411$	−111.031			−0.270147
$412$		0			0
$413$	259.217	+	259.217i	0.627644	+	0.627644i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−309.551	+	309.551i	−0.742327	+	0.742327i
$418$		0			0
$419$			638.030i			1.52274i	0.648315	+	0.761372i	$0.275474\pi$
$419$			638.030i			1.52274i	−0.648315	+	0.761372i	$0.724526\pi$
$420$		0			0
$421$	−272.606			−0.647520			−0.323760	−	0.946139i	$-0.604947\pi$
$421$	−272.606			−0.647520			−0.323760	+	0.946139i	$0.604947\pi$
$422$		0			0
$423$	−111.439	−	111.439i	−0.263450	−	0.263450i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−44.4620	+	44.4620i	−0.104126	+	0.104126i
$428$		0			0
$429$			306.272i			0.713922i
$430$		0			0
$431$	−264.960			−0.614757			−0.307378	−	0.951587i	$-0.599452\pi$
$431$	−264.960			−0.614757			−0.307378	+	0.951587i	$0.599452\pi$
$432$		0			0
$433$	−210.619	−	210.619i	−0.486417	−	0.486417i	0.420756	−	0.907174i	$-0.361765\pi$
$433$	−210.619	−	210.619i	−0.486417	−	0.486417i	−0.907174	+	0.420756i	$0.861765\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−105.439	+	105.439i	−0.241280	+	0.241280i
$438$		0			0
$439$		−	423.999i		−	0.965829i	−0.875667	−	0.482915i	$-0.839578\pi$
$439$		−	423.999i		−	0.965829i	0.875667	−	0.482915i	$-0.160422\pi$
$440$		0			0
$441$	−84.6061			−0.191851
$442$		0			0
$443$	−533.040	−	533.040i	−1.20325	−	1.20325i	−0.973172	−	0.230078i	$-0.926102\pi$
$443$	−533.040	−	533.040i	−1.20325	−	1.20325i	−0.230078	−	0.973172i	$-0.573898\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	92.1158	−	92.1158i	0.206076	−	0.206076i
$448$		0			0
$449$			7.23266i			0.0161084i	0.999968	+	0.00805418i	$0.00256375\pi$
$449$			7.23266i			0.0161084i	−0.999968	+	0.00805418i	$0.997436\pi$
$450$		0			0
$451$	21.3939			0.0474365
$452$		0			0
$453$	229.139	+	229.139i	0.505825	+	0.505825i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−183.918	+	183.918i	−0.402447	+	0.402447i	−0.879095	−	0.476647i	$-0.841852\pi$
$457$	−183.918	+	183.918i	−0.402447	+	0.402447i	0.476647	+	0.879095i	$0.341852\pi$
$458$		0			0
$459$			3.30306i			0.00719621i
$460$		0			0
$461$	632.595			1.37222			0.686112	−	0.727496i	$-0.259316\pi$
$461$	632.595			1.37222			0.686112	+	0.727496i	$0.259316\pi$
$462$		0			0
$463$	−382.838	−	382.838i	−0.826863	−	0.826863i	0.160218	−	0.987082i	$-0.448780\pi$
$463$	−382.838	−	382.838i	−0.826863	−	0.826863i	−0.987082	+	0.160218i	$0.948780\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−164.581	+	164.581i	−0.352422	+	0.352422i	−0.861010	−	0.508588i	$-0.830168\pi$
$467$	−164.581	+	164.581i	−0.352422	+	0.352422i	0.508588	+	0.861010i	$0.330168\pi$
$468$		0			0
$469$		−	544.151i		−	1.16024i
$470$		0			0
$471$	292.596			0.621223
$472$		0			0
$473$	18.8332	+	18.8332i	0.0398164	+	0.0398164i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	195.576	−	195.576i	0.410012	−	0.410012i
$478$		0			0
$479$			356.606i			0.744480i	0.928136	+	0.372240i	$0.121410\pi$
$479$			356.606i			0.744480i	−0.928136	+	0.372240i	$0.878590\pi$
$480$		0			0
$481$	482.202			1.00250
$482$		0			0
$483$	84.1362	+	84.1362i	0.174195	+	0.174195i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	170.103	−	170.103i	0.349288	−	0.349288i	−0.510556	−	0.859844i	$-0.670560\pi$
$487$	170.103	−	170.103i	0.349288	−	0.349288i	0.859844	+	0.510556i	$0.170560\pi$
$488$		0			0
$489$			256.868i			0.525293i
$490$		0			0
$491$	−439.514			−0.895141			−0.447571	−	0.894249i	$-0.647711\pi$
$491$	−439.514			−0.895141			−0.447571	+	0.894249i	$0.647711\pi$
$492$		0			0
$493$	−16.3133	−	16.3133i	−0.0330898	−	0.0330898i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−316.969	+	316.969i	−0.637765	+	0.637765i
$498$		0			0
$499$		−	751.413i		−	1.50584i	−0.658113	−	0.752919i	$-0.728645\pi$
$499$		−	751.413i		−	1.50584i	0.658113	−	0.752919i	$-0.271355\pi$
$500$		0			0
$501$	164.858			0.329058
$502$		0			0
$503$	543.626	+	543.626i	1.08077	+	1.08077i	0.996438	+	0.0843284i	$0.0268745\pi$
$503$	543.626	+	543.626i	1.08077	+	1.08077i	0.0843284	+	0.996438i	$0.473126\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−597.601	+	597.601i	−1.17870	+	1.17870i
$508$		0			0
$509$		−	471.798i		−	0.926912i	−0.886120	−	0.463456i	$-0.846609\pi$
$509$		−	471.798i		−	0.926912i	0.886120	−	0.463456i	$-0.153391\pi$
$510$		0			0
$511$	338.565			0.662554
$512$		0			0
$513$	36.3712	+	36.3712i	0.0708990	+	0.0708990i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	256.272	−	256.272i	0.495691	−	0.495691i
$518$		0			0
$519$			117.242i			0.225899i
$520$		0			0
$521$	881.928			1.69276			0.846380	−	0.532580i	$-0.178777\pi$
$521$	881.928			1.69276			0.846380	+	0.532580i	$0.178777\pi$
$522$		0			0
$523$	305.493	+	305.493i	0.584116	+	0.584116i	0.936032	−	0.351916i	$-0.114470\pi$
$523$	305.493	+	305.493i	0.584116	+	0.584116i	−0.351916	+	0.936032i	$0.614470\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	11.5505	−	11.5505i	0.0219175	−	0.0219175i
$528$		0			0
$529$		−	302.090i		−	0.571058i
$530$		0			0
$531$	241.151			0.454145
$532$		0			0
$533$	56.2020	+	56.2020i	0.105445	+	0.105445i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−164.116	+	164.116i	−0.305616	+	0.305616i
$538$		0			0
$539$		−	194.565i		−	0.360975i
$540$		0			0
$541$	−761.867			−1.40826			−0.704129	−	0.710072i	$-0.748662\pi$
$541$	−761.867			−1.40826			−0.704129	+	0.710072i	$0.748662\pi$
$542$		0			0
$543$	−42.1487	−	42.1487i	−0.0776220	−	0.0776220i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	178.020	−	178.020i	0.325449	−	0.325449i	−0.525404	−	0.850853i	$-0.676086\pi$
$547$	178.020	−	178.020i	0.325449	−	0.325449i	0.850853	+	0.525404i	$0.176086\pi$
$548$		0			0
$549$			41.3633i			0.0753429i
$550$		0			0
$551$	−359.262			−0.652019
$552$		0			0
$553$	219.934	+	219.934i	0.397711	+	0.397711i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−82.9852	+	82.9852i	−0.148986	+	0.148986i	−0.777665	−	0.628679i	$-0.783596\pi$
$557$	−82.9852	+	82.9852i	−0.148986	+	0.148986i	0.628679	+	0.777665i	$0.283596\pi$
$558$		0			0
$559$			98.9500i			0.177013i
$560$		0			0
$561$	−7.59592			−0.0135400
$562$		0			0
$563$	−228.227	−	228.227i	−0.405377	−	0.405377i	0.474746	−	0.880123i	$-0.342540\pi$
$563$	−228.227	−	228.227i	−0.405377	−	0.405377i	−0.880123	+	0.474746i	$0.842540\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	29.0227	−	29.0227i	0.0511864	−	0.0511864i
$568$		0			0
$569$			613.787i			1.07871i	0.842078	+	0.539356i	$0.181332\pi$
$569$			613.787i			1.07871i	−0.842078	+	0.539356i	$0.818668\pi$
$570$		0			0
$571$	−541.656			−0.948610			−0.474305	−	0.880361i	$-0.657301\pi$
$571$	−541.656			−0.948610			−0.474305	+	0.880361i	$0.657301\pi$
$572$		0			0
$573$	−291.823	−	291.823i	−0.509290	−	0.509290i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	341.967	−	341.967i	0.592664	−	0.592664i	−0.345686	−	0.938350i	$-0.612354\pi$
$577$	341.967	−	341.967i	0.592664	−	0.592664i	0.938350	+	0.345686i	$0.112354\pi$
$578$		0			0
$579$			151.565i			0.261771i
$580$		0			0
$581$	−578.727			−0.996087
$582$		0			0
$583$	449.757	+	449.757i	0.771453	+	0.771453i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	454.141	−	454.141i	0.773664	−	0.773664i	−0.205081	−	0.978745i	$-0.565746\pi$
$587$	454.141	−	454.141i	0.773664	−	0.773664i	0.978745	+	0.205081i	$0.0657458\pi$
$588$		0			0
$589$		−	254.373i		−	0.431873i
$590$		0			0
$591$	176.969			0.299441
$592$		0			0
$593$	−50.6357	−	50.6357i	−0.0853891	−	0.0853891i	0.663122	−	0.748511i	$-0.269231\pi$
$593$	−50.6357	−	50.6357i	−0.0853891	−	0.0853891i	−0.748511	+	0.663122i	$0.769231\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−104.659	+	104.659i	−0.175309	+	0.175309i
$598$		0			0
$599$			506.443i			0.845481i	0.906251	+	0.422740i	$0.138932\pi$
$599$			506.443i			0.845481i	−0.906251	+	0.422740i	$0.861068\pi$
$600$		0			0
$601$	−785.120			−1.30636			−0.653178	−	0.757204i	$-0.726565\pi$
$601$	−785.120			−1.30636			−0.653178	+	0.757204i	$0.726565\pi$
$602$		0			0
$603$	−253.114	−	253.114i	−0.419757	−	0.419757i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	766.252	−	766.252i	1.26236	−	1.26236i	0.312413	−	0.949946i	$-0.398863\pi$
$607$	766.252	−	766.252i	1.26236	−	1.26236i	0.949946	−	0.312413i	$-0.101137\pi$
$608$		0			0
$609$			286.677i			0.470733i
$610$		0			0
$611$	1346.46			2.20370
$612$		0			0
$613$	182.697	+	182.697i	0.298037	+	0.298037i	0.840245	−	0.542207i	$-0.182411\pi$
$613$	182.697	+	182.697i	0.298037	+	0.298037i	−0.542207	+	0.840245i	$0.682411\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	71.1214	−	71.1214i	0.115270	−	0.115270i	−0.647119	−	0.762389i	$-0.724026\pi$
$617$	71.1214	−	71.1214i	0.115270	−	0.115270i	0.762389	+	0.647119i	$0.224026\pi$
$618$		0			0
$619$			1031.35i			1.66616i	0.553154	+	0.833079i	$0.313424\pi$
$619$			1031.35i			1.66616i	−0.553154	+	0.833079i	$0.686576\pi$
$620$		0			0
$621$	78.2724			0.126043
$622$		0			0
$623$	130.879	+	130.879i	0.210078	+	0.210078i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−83.6413	+	83.6413i	−0.133399	+	0.133399i
$628$		0			0
$629$			11.9592i			0.0190130i
$630$		0			0
$631$	−374.201			−0.593029			−0.296514	−	0.955028i	$-0.595824\pi$
$631$	−374.201			−0.593029			−0.296514	+	0.955028i	$0.595824\pi$
$632$		0			0
$633$	121.831	+	121.831i	0.192466	+	0.192466i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	511.126	−	511.126i	0.802396	−	0.802396i
$638$		0			0
$639$			294.879i			0.461469i
$640$		0			0
$641$	−884.827			−1.38038			−0.690192	−	0.723626i	$-0.742474\pi$
$641$	−884.827			−1.38038			−0.690192	+	0.723626i	$0.742474\pi$
$642$		0			0
$643$	683.787	+	683.787i	1.06343	+	1.06343i	0.997847	+	0.0655849i	$0.0208913\pi$
$643$	683.787	+	683.787i	1.06343	+	1.06343i	0.0655849	+	0.997847i	$0.479109\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−597.898	+	597.898i	−0.924108	+	0.924108i	−0.997317	−	0.0732085i	$-0.976676\pi$
$647$	−597.898	+	597.898i	−0.924108	+	0.924108i	0.0732085	+	0.997317i	$0.476676\pi$
$648$		0			0
$649$			554.565i			0.854492i
$650$		0			0
$651$	−202.980			−0.311797
$652$		0			0
$653$	−282.136	−	282.136i	−0.432062	−	0.432062i	0.457268	−	0.889329i	$-0.348828\pi$
$653$	−282.136	−	282.136i	−0.432062	−	0.432062i	−0.889329	+	0.457268i	$0.848828\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	157.485	−	157.485i	0.239703	−	0.239703i
$658$		0			0
$659$		−	503.505i		−	0.764044i	−0.924153	−	0.382022i	$-0.875228\pi$
$659$		−	503.505i		−	0.764044i	0.924153	−	0.382022i	$-0.124772\pi$
$660$		0			0
$661$	38.7673			0.0586495			0.0293248	−	0.999570i	$-0.490664\pi$
$661$	38.7673			0.0586495			0.0293248	+	0.999570i	$0.490664\pi$
$662$		0			0
$663$	−19.9546	−	19.9546i	−0.0300974	−	0.0300974i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−386.574	+	386.574i	−0.579572	+	0.579572i
$668$		0			0
$669$		−	5.75817i		−	0.00860713i
$670$		0			0
$671$	−95.1214			−0.141761
$672$		0			0
$673$	359.728	+	359.728i	0.534513	+	0.534513i	0.921912	−	0.387399i	$-0.126626\pi$
$673$	359.728	+	359.728i	0.534513	+	0.534513i	−0.387399	+	0.921912i	$0.626626\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−40.0454	+	40.0454i	−0.0591513	+	0.0591513i	−0.736064	−	0.676912i	$-0.763318\pi$
$677$	−40.0454	+	40.0454i	−0.0591513	+	0.0591513i	0.676912	+	0.736064i	$0.263318\pi$
$678$		0			0
$679$		−	681.838i		−	1.00418i
$680$		0			0
$681$	500.858			0.735475
$682$		0			0
$683$	502.243	+	502.243i	0.735348	+	0.735348i	0.971674	−	0.236326i	$-0.0759432\pi$
$683$	502.243	+	502.243i	0.735348	+	0.735348i	−0.236326	+	0.971674i	$0.575943\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	161.431	−	161.431i	0.234980	−	0.234980i
$688$		0			0
$689$			2363.04i			3.42966i
$690$		0			0
$691$	−242.241			−0.350566			−0.175283	−	0.984518i	$-0.556084\pi$
$691$	−242.241			−0.350566			−0.175283	+	0.984518i	$0.556084\pi$
$692$		0			0
$693$	66.7423	+	66.7423i	0.0963093	+	0.0963093i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.39388	+	1.39388i	−0.00199982	+	0.00199982i
$698$		0			0
$699$			341.989i			0.489254i
$700$		0			0
$701$	619.181			0.883282			0.441641	−	0.897192i	$-0.354397\pi$
$701$	619.181			0.883282			0.441641	+	0.897192i	$0.354397\pi$
$702$		0			0
$703$	131.687	+	131.687i	0.187321	+	0.187321i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	636.838	−	636.838i	0.900761	−	0.900761i
$708$		0			0
$709$		−	618.514i		−	0.872376i	−0.899856	−	0.436188i	$-0.856328\pi$
$709$		−	618.514i		−	0.872376i	0.899856	−	0.436188i	$-0.143672\pi$
$710$		0			0
$711$	204.606			0.287772
$712$		0			0
$713$	−273.712	−	273.712i	−0.383887	−	0.383887i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−145.373	+	145.373i	−0.202752	+	0.202752i
$718$		0			0
$719$			555.989i			0.773281i	0.922231	+	0.386640i	$0.126365\pi$
$719$			555.989i			0.773281i	−0.922231	+	0.386640i	$0.873635\pi$
$720$		0			0
$721$	295.373			0.409672
$722$		0			0
$723$	339.724	+	339.724i	0.469881	+	0.469881i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	103.992	−	103.992i	0.143043	−	0.143043i	−0.631959	−	0.775002i	$-0.717749\pi$
$727$	103.992	−	103.992i	0.143043	−	0.143043i	0.775002	+	0.631959i	$0.217749\pi$
$728$		0			0
$729$		−	27.0000i		−	0.0370370i
$730$		0			0
$731$	−2.45408			−0.00335715
$732$		0			0
$733$	−462.979	−	462.979i	−0.631621	−	0.631621i	0.316853	−	0.948475i	$-0.397374\pi$
$733$	−462.979	−	462.979i	−0.631621	−	0.631621i	−0.948475	+	0.316853i	$0.897374\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	582.075	−	582.075i	0.789790	−	0.789790i
$738$		0			0
$739$		−	701.414i		−	0.949140i	−0.880218	−	0.474570i	$-0.842604\pi$
$739$		−	701.414i		−	0.949140i	0.880218	−	0.474570i	$-0.157396\pi$
$740$		0			0
$741$	−439.454			−0.593055
$742$		0			0
$743$	634.534	+	634.534i	0.854016	+	0.854016i	0.990625	−	0.136609i	$-0.0436205\pi$
$743$	634.534	+	634.534i	0.854016	+	0.854016i	−0.136609	+	0.990625i	$0.543620\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−269.196	+	269.196i	−0.360370	+	0.360370i
$748$		0			0
$749$			899.605i			1.20107i
$750$		0			0
$751$	−934.241			−1.24400			−0.621998	−	0.783019i	$-0.713679\pi$
$751$	−934.241			−1.24400			−0.621998	+	0.783019i	$0.713679\pi$
$752$		0			0
$753$	46.0204	+	46.0204i	0.0611161	+	0.0611161i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	656.598	−	656.598i	0.867369	−	0.867369i	−0.124812	−	0.992180i	$-0.539833\pi$
$757$	656.598	−	656.598i	0.867369	−	0.867369i	0.992180	+	0.124812i	$0.0398327\pi$
$758$		0			0
$759$			180.000i			0.237154i
$760$		0			0
$761$	−911.292			−1.19749			−0.598746	−	0.800939i	$-0.704334\pi$
$761$	−911.292			−1.19749			−0.598746	+	0.800939i	$0.704334\pi$
$762$		0			0
$763$	−453.320	−	453.320i	−0.594129	−	0.594129i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1456.85	+	1456.85i	−1.89942	+	1.89942i
$768$		0			0
$769$			769.847i			1.00110i	0.865707	+	0.500551i	$0.166869\pi$
$769$			769.847i			1.00110i	−0.865707	+	0.500551i	$0.833131\pi$
$770$		0			0
$771$	−195.637			−0.253744
$772$		0			0
$773$	480.515	+	480.515i	0.621624	+	0.621624i	0.945947	−	0.324323i	$-0.105136\pi$
$773$	480.515	+	480.515i	0.621624	+	0.621624i	−0.324323	+	0.945947i	$0.605136\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	105.081	−	105.081i	0.135239	−	0.135239i
$778$		0			0
$779$			30.6969i			0.0394056i
$780$		0			0
$781$	−678.120			−0.868272
$782$		0			0
$783$	133.348	+	133.348i	0.170305	+	0.170305i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	48.0079	−	48.0079i	0.0610012	−	0.0610012i	−0.675948	−	0.736949i	$-0.736266\pi$
$787$	48.0079	−	48.0079i	0.0610012	−	0.0610012i	0.736949	+	0.675948i	$0.236266\pi$
$788$		0			0
$789$			545.666i			0.691592i
$790$		0			0
$791$	683.514			0.864114
$792$		0			0
$793$	−249.885	−	249.885i	−0.315114	−	0.315114i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	650.080	−	650.080i	0.815658	−	0.815658i	−0.169817	−	0.985476i	$-0.554318\pi$
$797$	650.080	−	650.080i	0.815658	−	0.815658i	0.985476	+	0.169817i	$0.0543178\pi$
$798$		0			0
$799$			33.3939i			0.0417946i
$800$		0			0
$801$	121.757			0.152006
$802$		0			0
$803$	362.161	+	362.161i	0.451010	+	0.451010i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	551.246	−	551.246i	0.683081	−	0.683081i
$808$		0			0
$809$		−	941.281i		−	1.16351i	−0.813364	−	0.581756i	$-0.802366\pi$
$809$		−	941.281i		−	1.16351i	0.813364	−	0.581756i	$-0.197634\pi$
$810$		0			0
$811$	657.674			0.810943			0.405471	−	0.914108i	$-0.367107\pi$
$811$	657.674			0.810943			0.405471	+	0.914108i	$0.367107\pi$
$812$		0			0
$813$	−121.510	−	121.510i	−0.149458	−	0.149458i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−27.0227	+	27.0227i	−0.0330755	+	0.0330755i
$818$		0			0
$819$			350.666i			0.428164i
$820$		0			0
$821$	−574.041			−0.699197			−0.349599	−	0.936900i	$-0.613682\pi$
$821$	−574.041			−0.699197			−0.349599	+	0.936900i	$0.613682\pi$
$822$		0			0
$823$	547.027	+	547.027i	0.664675	+	0.664675i	0.956478	−	0.291804i	$-0.0942554\pi$
$823$	547.027	+	547.027i	0.664675	+	0.664675i	−0.291804	+	0.956478i	$0.594255\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	959.271	−	959.271i	1.15994	−	1.15994i	0.175454	−	0.984488i	$-0.443861\pi$
$827$	959.271	−	959.271i	1.15994	−	1.15994i	0.984488	−	0.175454i	$-0.0561392\pi$
$828$		0			0
$829$			156.020i			0.188203i	0.995563	+	0.0941016i	$0.0299978\pi$
$829$			156.020i			0.188203i	−0.995563	+	0.0941016i	$0.970002\pi$
$830$		0			0
$831$	−65.1408			−0.0783885
$832$		0			0
$833$	12.6765	+	12.6765i	0.0152179	+	0.0152179i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−94.4166	+	94.4166i	−0.112804	+	0.112804i
$838$		0			0
$839$		−	771.566i		−	0.919626i	−0.888016	−	0.459813i	$-0.847916\pi$
$839$		−	771.566i		−	0.919626i	0.888016	−	0.459813i	$-0.152084\pi$
$840$		0			0
$841$	−476.171			−0.566197
$842$		0			0
$843$	398.166	+	398.166i	0.472320	+	0.472320i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	236.709	−	236.709i	0.279468	−	0.279468i
$848$		0			0
$849$		−	387.879i		−	0.456865i
$850$		0			0
$851$	283.396			0.333015
$852$		0			0
$853$	162.553	+	162.553i	0.190566	+	0.190566i	0.795941	−	0.605375i	$-0.206977\pi$
$853$	162.553	+	162.553i	0.190566	+	0.190566i	−0.605375	+	0.795941i	$0.706977\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−426.434	+	426.434i	−0.497589	+	0.497589i	−0.910687	−	0.413098i	$-0.864447\pi$
$857$	−426.434	+	426.434i	−0.497589	+	0.497589i	0.413098	+	0.910687i	$0.364447\pi$
$858$		0			0
$859$		−	155.453i		−	0.180970i	−0.995898	−	0.0904849i	$-0.971158\pi$
$859$		−	155.453i		−	0.180970i	0.995898	−	0.0904849i	$-0.0288417\pi$
$860$		0			0
$861$	24.4949			0.0284494
$862$		0			0
$863$	8.30409	+	8.30409i	0.00962236	+	0.00962236i	0.711902	−	0.702279i	$-0.247834\pi$
$863$	8.30409	+	8.30409i	0.00962236	+	0.00962236i	−0.702279	+	0.711902i	$0.747834\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−353.456	+	353.456i	−0.407677	+	0.407677i
$868$		0			0
$869$			470.524i			0.541455i
$870$		0			0
$871$	3058.24			3.51118
$872$		0			0
$873$	−317.159	−	317.159i	−0.363298	−	0.363298i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−485.654	+	485.654i	−0.553767	+	0.553767i	−0.927526	−	0.373759i	$-0.878069\pi$
$877$	−485.654	+	485.654i	−0.553767	+	0.553767i	0.373759	+	0.927526i	$0.378069\pi$
$878$		0			0
$879$		−	850.282i		−	0.967328i
$880$		0			0
$881$	1068.34			1.21265			0.606323	−	0.795219i	$-0.292644\pi$
$881$	1068.34			1.21265			0.606323	+	0.795219i	$0.292644\pi$
$882$		0			0
$883$	1022.15	+	1022.15i	1.15758	+	1.15758i	0.984994	+	0.172590i	$0.0552137\pi$
$883$	1022.15	+	1022.15i	1.15758	+	1.15758i	0.172590	+	0.984994i	$0.444786\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−580.711	+	580.711i	−0.654691	+	0.654691i	−0.954119	−	0.299428i	$-0.903204\pi$
$887$	−580.711	+	580.711i	−0.654691	+	0.654691i	0.299428	+	0.954119i	$0.403204\pi$
$888$		0			0
$889$		−	430.161i		−	0.483871i
$890$		0			0
$891$	62.0908			0.0696867
$892$		0			0
$893$	367.712	+	367.712i	0.411771	+	0.411771i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−472.863	+	472.863i	−0.527160	+	0.527160i
$898$		0			0
$899$		−	932.615i		−	1.03739i
$900$		0			0
$901$	−58.6061			−0.0650456
$902$		0			0
$903$	21.5630	+	21.5630i	0.0238793	+	0.0238793i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	222.979	−	222.979i	0.245842	−	0.245842i	−0.573420	−	0.819262i	$-0.694384\pi$
$907$	222.979	−	222.979i	0.245842	−	0.245842i	0.819262	+	0.573420i	$0.194384\pi$
$908$		0			0
$909$		−	592.454i		−	0.651765i
$910$		0			0
$911$	−710.849			−0.780295			−0.390148	−	0.920752i	$-0.627576\pi$
$911$	−710.849			−0.780295			−0.390148	+	0.920752i	$0.627576\pi$
$912$		0			0
$913$	−619.060	−	619.060i	−0.678051	−	0.678051i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	105.373	−	105.373i	0.114911	−	0.114911i
$918$		0			0
$919$			535.595i			0.582802i	0.956601	+	0.291401i	$0.0941214\pi$
$919$			535.595i			0.582802i	−0.956601	+	0.291401i	$0.905879\pi$
$920$		0			0
$921$	−262.707			−0.285241
$922$		0			0
$923$	−1781.43	−	1781.43i	−1.93005	−	1.93005i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	137.394	−	137.394i	0.148213	−	0.148213i
$928$		0			0
$929$		−	1482.43i		−	1.59573i	−0.602837	−	0.797864i	$-0.705963\pi$
$929$		−	1482.43i		−	1.59573i	0.602837	−	0.797864i	$-0.294037\pi$
$930$		0			0
$931$	279.171			0.299862
$932$		0			0
$933$	−436.974	−	436.974i	−0.468354	−	0.468354i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−329.109	+	329.109i	−0.351237	+	0.351237i	−0.860570	−	0.509333i	$-0.829892\pi$
$937$	−329.109	+	329.109i	−0.351237	+	0.351237i	0.509333	+	0.860570i	$0.329892\pi$
$938$		0			0
$939$			253.243i			0.269694i
$940$		0			0
$941$	−1330.38			−1.41380			−0.706898	−	0.707316i	$-0.749906\pi$
$941$	−1330.38			−1.41380			−0.706898	+	0.707316i	$0.749906\pi$
$942$		0			0
$943$	33.0306	+	33.0306i	0.0350272	+	0.0350272i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−310.652	+	310.652i	−0.328038	+	0.328038i	−0.851840	−	0.523802i	$-0.824513\pi$
$947$	−310.652	+	310.652i	−0.328038	+	0.328038i	0.523802	+	0.851840i	$0.324513\pi$
$948$		0			0
$949$			1902.81i			2.00506i
$950$		0			0
$951$	507.192			0.533325
$952$		0			0
$953$	40.8990	+	40.8990i	0.0429160	+	0.0429160i	0.728239	−	0.685323i	$-0.240339\pi$
$953$	40.8990	+	40.8990i	0.0429160	+	0.0429160i	−0.685323	+	0.728239i	$0.740339\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−306.656	+	306.656i	−0.320435	+	0.320435i
$958$		0			0
$959$		−	292.343i		−	0.304841i
$960$		0			0
$961$	−300.667			−0.312869
$962$		0			0
$963$	418.454	+	418.454i	0.434532	+	0.434532i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	106.656	−	106.656i	0.110296	−	0.110296i	−0.649805	−	0.760101i	$-0.725149\pi$
$967$	106.656	−	106.656i	0.110296	−	0.110296i	0.760101	+	0.649805i	$0.225149\pi$
$968$		0			0
$969$		−	10.8990i		−	0.0112477i
$970$		0			0
$971$	1529.61			1.57530			0.787649	−	0.616124i	$-0.211298\pi$
$971$	1529.61			1.57530			0.787649	+	0.616124i	$0.211298\pi$
$972$		0			0
$973$	−815.044	−	815.044i	−0.837661	−	0.837661i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	148.974	−	148.974i	0.152481	−	0.152481i	−0.626744	−	0.779225i	$-0.715613\pi$
$977$	148.974	−	148.974i	0.152481	−	0.152481i	0.779225	+	0.626744i	$0.215613\pi$
$978$		0			0
$979$			280.000i			0.286006i
$980$		0			0
$981$	−421.727			−0.429895
$982$		0			0
$983$	−449.930	−	449.930i	−0.457711	−	0.457711i	0.440193	−	0.897903i	$-0.354910\pi$
$983$	−449.930	−	449.930i	−0.457711	−	0.457711i	−0.897903	+	0.440193i	$0.854910\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	293.419	−	293.419i	0.297284	−	0.297284i
$988$		0			0
$989$			58.1541i			0.0588009i
$990$		0			0
$991$	−1430.89			−1.44388			−0.721941	−	0.691955i	$-0.756750\pi$
$991$	−1430.89			−1.44388			−0.721941	+	0.691955i	$0.756750\pi$
$992$		0			0
$993$	−692.661	−	692.661i	−0.697544	−	0.697544i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−588.191	+	588.191i	−0.589961	+	0.589961i	−0.937621	−	0.347660i	$-0.886976\pi$
$997$	−588.191	+	588.191i	−0.589961	+	0.589961i	0.347660	+	0.937621i	$0.386976\pi$
$998$		0			0
$999$		−	97.7571i		−	0.0978550i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	600.3.u.c.193.2	✓	4
3.2	odd	2		1800.3.v.l.793.1		4
4.3	odd	2		1200.3.bg.l.193.1		4
5.2	odd	4	inner	600.3.u.c.457.2	yes	4
5.3	odd	4		600.3.u.d.457.1	yes	4
5.4	even	2		600.3.u.d.193.1	yes	4
15.2	even	4		1800.3.v.l.1657.1		4
15.8	even	4		1800.3.v.m.1657.2		4
15.14	odd	2		1800.3.v.m.793.2		4
20.3	even	4		1200.3.bg.f.1057.2		4
20.7	even	4		1200.3.bg.l.1057.1		4
20.19	odd	2		1200.3.bg.f.193.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
600.3.u.c.193.2	✓	4	1.1	even	1	trivial
600.3.u.c.457.2	yes	4	5.2	odd	4	inner
600.3.u.d.193.1	yes	4	5.4	even	2
600.3.u.d.457.1	yes	4	5.3	odd	4
1200.3.bg.f.193.2		4	20.19	odd	2
1200.3.bg.f.1057.2		4	20.3	even	4
1200.3.bg.l.193.1		4	4.3	odd	2
1200.3.bg.l.1057.1		4	20.7	even	4
1800.3.v.l.793.1		4	3.2	odd	2
1800.3.v.l.1657.1		4	15.2	even	4
1800.3.v.m.793.2		4	15.14	odd	2
1800.3.v.m.1657.2		4	15.8	even	4

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)

\(f(q)\)	\(=\)	\(q+(1.22474 + 1.22474i) q^{3} +(-3.22474 + 3.22474i) q^{7} +3.00000i q^{9} -6.89898 q^{11} +(-18.1237 - 18.1237i) q^{13} +(0.449490 - 0.449490i) q^{17} -9.89898i q^{19} -7.89898 q^{21} +(-10.6515 - 10.6515i) q^{23} +(-3.67423 + 3.67423i) q^{27} -36.2929i q^{29} +25.6969 q^{31} +(-8.44949 - 8.44949i) q^{33} +(-13.3031 + 13.3031i) q^{37} -44.3939i q^{39} -3.10102 q^{41} +(-2.72985 - 2.72985i) q^{43} +(-37.1464 + 37.1464i) q^{47} +28.2020i q^{49} +1.10102 q^{51} +(-65.1918 - 65.1918i) q^{53} +(12.1237 - 12.1237i) q^{57} -80.3837i q^{59} +13.7878 q^{61} +(-9.67423 - 9.67423i) q^{63} +(-84.3712 + 84.3712i) q^{67} -26.0908i q^{69} +98.2929 q^{71} +(-52.4949 - 52.4949i) q^{73} +(22.2474 - 22.2474i) q^{77} -68.2020i q^{79} -9.00000 q^{81} +(89.7321 + 89.7321i) q^{83} +(44.4495 - 44.4495i) q^{87} -40.5857i q^{89} +116.889 q^{91} +(31.4722 + 31.4722i) q^{93} +(-105.720 + 105.720i) q^{97} -20.6969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{7} - 8 q^{11} - 48 q^{13} - 8 q^{17} - 12 q^{21} - 72 q^{23} + 44 q^{31} - 24 q^{33} - 112 q^{37} - 32 q^{41} - 104 q^{43} - 80 q^{47} + 24 q^{51} - 104 q^{53} + 24 q^{57} - 180 q^{61} - 24 q^{63}+ \cdots - 320 q^{97}+O(q^{100}) \) 4 * q - 8 * q^7 - 8 * q^11 - 48 * q^13 - 8 * q^17 - 12 * q^21 - 72 * q^23 + 44 * q^31 - 24 * q^33 - 112 * q^37 - 32 * q^41 - 104 * q^43 - 80 * q^47 + 24 * q^51 - 104 * q^53 + 24 * q^57 - 180 * q^61 - 24 * q^63 - 264 * q^67 + 256 * q^71 - 112 * q^73 + 40 * q^77 - 36 * q^81 + 16 * q^83 + 168 * q^87 + 252 * q^91 + 72 * q^93 - 320 * q^97

Introduction

L-functions

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