LMFDB - Embedded newform 2700.2.j.a.593.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2700,2,Mod(593,2700)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2700, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2700.593"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2700.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-24, 0,-20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(61)] == traces)

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

Self dual:	no
Analytic conductor:	$21.5596085457$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.1
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	2700.593
Dual form			2700.2.j.a.1457.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.44949 + 2.44949i) q^{7} +6.00000i q^{11} +(-2.44949 - 2.44949i) q^{13} +(-3.67423 - 3.67423i) q^{17} +1.00000i q^{19} +(3.67423 - 3.67423i) q^{23} -6.00000 q^{29} -5.00000 q^{31} +(4.89898 - 4.89898i) q^{37} -6.00000i q^{41} +(7.34847 + 7.34847i) q^{43} +(-7.34847 - 7.34847i) q^{47} -5.00000i q^{49} +(3.67423 - 3.67423i) q^{53} +6.00000 q^{59} +5.00000 q^{61} +(4.89898 - 4.89898i) q^{67} -6.00000i q^{71} +(-2.44949 - 2.44949i) q^{73} +(-14.6969 - 14.6969i) q^{77} +5.00000i q^{79} +(11.0227 - 11.0227i) q^{83} +6.00000 q^{89} +12.0000 q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{29} - 20 q^{31} + 24 q^{59} + 20 q^{61} + 24 q^{89} + 48 q^{91}+O(q^{100}) $ 4 * q - 24 * q^29 - 20 * q^31 + 24 * q^59 + 20 * q^61 + 24 * q^89 + 48 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times$.

$n$	$1001$	$1351$	$2377$
$\chi(n)$	$-1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	−0.997433	−	0.0716124i	$-0.977186\pi$
$7$	−2.44949	+	2.44949i	−0.925820	+	0.925820i	0.0716124	+	0.997433i	$0.477186\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			6.00000i			1.80907i	0.426401	+	0.904534i	$0.359781\pi$
$11$			6.00000i			1.80907i	−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	−2.44949	−	2.44949i	−0.679366	−	0.679366i	0.280491	−	0.959857i	$-0.409503\pi$
$13$	−2.44949	−	2.44949i	−0.679366	−	0.679366i	−0.959857	+	0.280491i	$0.909503\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−3.67423	−	3.67423i	−0.891133	−	0.891133i	0.103497	−	0.994630i	$-0.466997\pi$
$17$	−3.67423	−	3.67423i	−0.891133	−	0.891133i	−0.994630	+	0.103497i	$0.966997\pi$
$18$		0			0
$19$			1.00000i			0.229416i	0.993399	+	0.114708i	$0.0365932\pi$
$19$			1.00000i			0.229416i	−0.993399	+	0.114708i	$0.963407\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.67423	−	3.67423i	0.766131	−	0.766131i	−0.211292	−	0.977423i	$-0.567767\pi$
$23$	3.67423	−	3.67423i	0.766131	−	0.766131i	0.977423	+	0.211292i	$0.0677671\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	−0.178545	−	0.983932i	$-0.557139\pi$
$37$	4.89898	−	4.89898i	0.805387	−	0.805387i	0.983932	+	0.178545i	$0.0571389\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.00000i		−	0.937043i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$		−	6.00000i		−	0.937043i	0.883452	−	0.468521i	$-0.155213\pi$
$42$		0			0
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.991647	+	0.128984i	$0.0411717\pi$
$43$	7.34847	+	7.34847i	1.12063	+	1.12063i	0.128984	+	0.991647i	$0.458828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−7.34847	−	7.34847i	−1.07188	−	1.07188i	−0.997208	−	0.0746766i	$-0.976208\pi$
$47$	−7.34847	−	7.34847i	−1.07188	−	1.07188i	−0.0746766	−	0.997208i	$-0.523792\pi$
$48$		0			0
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.67423	−	3.67423i	0.504695	−	0.504695i	−0.408198	−	0.912893i	$-0.633843\pi$
$53$	3.67423	−	3.67423i	0.504695	−	0.504695i	0.912893	+	0.408198i	$0.133843\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	6.00000			0.781133			0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000			0.781133			0.390567	+	0.920575i	$0.372279\pi$
$60$		0			0
$61$	5.00000			0.640184			0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000			0.640184			0.320092	+	0.947386i	$0.396286\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.89898	−	4.89898i	0.598506	−	0.598506i	−0.341409	−	0.939915i	$-0.610904\pi$
$67$	4.89898	−	4.89898i	0.598506	−	0.598506i	0.939915	+	0.341409i	$0.110904\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	6.00000i		−	0.712069i	−0.934473	−	0.356034i	$-0.884129\pi$
$71$		−	6.00000i		−	0.712069i	0.934473	−	0.356034i	$-0.115871\pi$
$72$		0			0
$73$	−2.44949	−	2.44949i	−0.286691	−	0.286691i	0.549079	−	0.835770i	$-0.314979\pi$
$73$	−2.44949	−	2.44949i	−0.286691	−	0.286691i	−0.835770	+	0.549079i	$0.814979\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−14.6969	−	14.6969i	−1.67487	−	1.67487i
$78$		0			0
$79$			5.00000i			0.562544i	0.959628	+	0.281272i	$0.0907563\pi$
$79$			5.00000i			0.562544i	−0.959628	+	0.281272i	$0.909244\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	11.0227	−	11.0227i	1.20990	−	1.20990i	0.238840	−	0.971059i	$-0.423233\pi$
$83$	11.0227	−	11.0227i	1.20990	−	1.20990i	0.971059	−	0.238840i	$-0.0767671\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	12.0000			1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	−0.910633	−	0.413217i	$-0.864405\pi$
$97$	−4.89898	+	4.89898i	−0.497416	+	0.497416i	0.413217	+	0.910633i	$0.364405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$		0			0
$103$	−4.89898	−	4.89898i	−0.482711	−	0.482711i	0.423286	−	0.905996i	$-0.360877\pi$
$103$	−4.89898	−	4.89898i	−0.482711	−	0.482711i	−0.905996	+	0.423286i	$0.860877\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.34847	−	7.34847i	−0.710403	−	0.710403i	0.256216	−	0.966620i	$-0.417524\pi$
$107$	−7.34847	−	7.34847i	−0.710403	−	0.710403i	−0.966620	+	0.256216i	$0.917524\pi$
$108$		0			0
$109$			11.0000i			1.05361i	0.849987	+	0.526804i	$0.176610\pi$
$109$			11.0000i			1.05361i	−0.849987	+	0.526804i	$0.823390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	18.0000			1.65006
$120$		0			0
$121$	−25.0000			−2.27273
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−14.6969	+	14.6969i	−1.30414	+	1.30414i	−0.378570	+	0.925573i	$0.623584\pi$
$127$	−14.6969	+	14.6969i	−1.30414	+	1.30414i	−0.925573	+	0.378570i	$0.876416\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			12.0000i			1.04844i	0.851581	+	0.524222i	$0.175644\pi$
$131$			12.0000i			1.04844i	−0.851581	+	0.524222i	$0.824356\pi$
$132$		0			0
$133$	−2.44949	−	2.44949i	−0.212398	−	0.212398i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.0227	−	11.0227i	−0.941733	−	0.941733i	0.0566604	−	0.998394i	$-0.481955\pi$
$137$	−11.0227	−	11.0227i	−0.941733	−	0.941733i	−0.998394	+	0.0566604i	$0.981955\pi$
$138$		0			0
$139$		−	20.0000i		−	1.69638i	−0.529694	−	0.848189i	$-0.677693\pi$
$139$		−	20.0000i		−	1.69638i	0.529694	−	0.848189i	$-0.322307\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.6969	−	14.6969i	1.22902	−	1.22902i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−7.34847	+	7.34847i	−0.586472	+	0.586472i	−0.936674	−	0.350202i	$-0.886113\pi$
$157$	−7.34847	+	7.34847i	−0.586472	+	0.586472i	0.350202	+	0.936674i	$0.386113\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			18.0000i			1.41860i
$162$		0			0
$163$	−4.89898	−	4.89898i	−0.383718	−	0.383718i	0.488722	−	0.872440i	$-0.337463\pi$
$163$	−4.89898	−	4.89898i	−0.383718	−	0.383718i	−0.872440	+	0.488722i	$0.837463\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3.67423	−	3.67423i	−0.284321	−	0.284321i	0.550509	−	0.834829i	$-0.314434\pi$
$167$	−3.67423	−	3.67423i	−0.284321	−	0.284321i	−0.834829	+	0.550509i	$0.814434\pi$
$168$		0			0
$169$		−	1.00000i		−	0.0769231i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−3.67423	+	3.67423i	−0.279347	+	0.279347i	−0.832848	−	0.553501i	$-0.813291\pi$
$173$	−3.67423	+	3.67423i	−0.279347	+	0.279347i	0.553501	+	0.832848i	$0.313291\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−24.0000			−1.79384			−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000			−1.79384			−0.896922	+	0.442189i	$0.854202\pi$
$180$		0			0
$181$	19.0000			1.41226			0.706129	−	0.708083i	$-0.250440\pi$
$181$	19.0000			1.41226			0.706129	+	0.708083i	$0.250440\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	22.0454	−	22.0454i	1.61212	−	1.61212i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	12.0000i		−	0.868290i	−0.900843	−	0.434145i	$-0.857051\pi$
$191$		−	12.0000i		−	0.868290i	0.900843	−	0.434145i	$-0.142949\pi$
$192$		0			0
$193$	17.1464	+	17.1464i	1.23423	+	1.23423i	0.962325	+	0.271902i	$0.0876526\pi$
$193$	17.1464	+	17.1464i	1.23423	+	1.23423i	0.271902	+	0.962325i	$0.412347\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−18.3712	−	18.3712i	−1.30889	−	1.30889i	−0.922215	−	0.386676i	$-0.873623\pi$
$197$	−18.3712	−	18.3712i	−1.30889	−	1.30889i	−0.386676	−	0.922215i	$-0.626377\pi$
$198$		0			0
$199$			8.00000i			0.567105i	0.958957	+	0.283552i	$0.0915130\pi$
$199$			8.00000i			0.567105i	−0.958957	+	0.283552i	$0.908487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	14.6969	−	14.6969i	1.03152	−	1.03152i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−6.00000			−0.415029
$210$		0			0
$211$	−25.0000			−1.72107			−0.860535	−	0.509390i	$-0.829871\pi$
$211$	−25.0000			−1.72107			−0.860535	+	0.509390i	$0.829871\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	12.2474	−	12.2474i	0.831411	−	0.831411i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			18.0000i			1.21081i
$222$		0			0
$223$	−14.6969	−	14.6969i	−0.984180	−	0.984180i	0.0156970	−	0.999877i	$-0.495003\pi$
$223$	−14.6969	−	14.6969i	−0.984180	−	0.984180i	−0.999877	+	0.0156970i	$0.995003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	3.67423	+	3.67423i	0.243868	+	0.243868i	0.818448	−	0.574581i	$-0.194835\pi$
$227$	3.67423	+	3.67423i	0.243868	+	0.243868i	−0.574581	+	0.818448i	$0.694835\pi$
$228$		0			0
$229$		−	5.00000i		−	0.330409i	−0.986259	−	0.165205i	$-0.947172\pi$
$229$		−	5.00000i		−	0.330409i	0.986259	−	0.165205i	$-0.0528285\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	−0.999333	−	0.0365050i	$-0.988378\pi$
$233$	−14.6969	+	14.6969i	−0.962828	+	0.962828i	0.0365050	+	0.999333i	$0.488378\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	23.0000			1.48156			0.740780	−	0.671748i	$-0.234456\pi$
$241$	23.0000			1.48156			0.740780	+	0.671748i	$0.234456\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.44949	−	2.44949i	0.155857	−	0.155857i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$	22.0454	+	22.0454i	1.38598	+	1.38598i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	3.67423	+	3.67423i	0.229192	+	0.229192i	0.812355	−	0.583163i	$-0.198185\pi$
$257$	3.67423	+	3.67423i	0.229192	+	0.229192i	−0.583163	+	0.812355i	$0.698185\pi$
$258$		0			0
$259$			24.0000i			1.49129i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	7.34847	−	7.34847i	0.453126	−	0.453126i	−0.443265	−	0.896391i	$-0.646180\pi$
$263$	7.34847	−	7.34847i	0.453126	−	0.453126i	0.896391	+	0.443265i	$0.146180\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	24.0000			1.46331			0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000			1.46331			0.731653	+	0.681677i	$0.238749\pi$
$270$		0			0
$271$	17.0000			1.03268			0.516338	−	0.856385i	$-0.327295\pi$
$271$	17.0000			1.03268			0.516338	+	0.856385i	$0.327295\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−7.34847	+	7.34847i	−0.441527	+	0.441527i	−0.892525	−	0.450998i	$-0.851068\pi$
$277$	−7.34847	+	7.34847i	−0.441527	+	0.441527i	0.450998	+	0.892525i	$0.351068\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			18.0000i			1.07379i	0.843649	+	0.536895i	$0.180403\pi$
$281$			18.0000i			1.07379i	−0.843649	+	0.536895i	$0.819597\pi$
$282$		0			0
$283$	7.34847	+	7.34847i	0.436821	+	0.436821i	0.890941	−	0.454120i	$-0.150046\pi$
$283$	7.34847	+	7.34847i	0.436821	+	0.436821i	−0.454120	+	0.890941i	$0.650046\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	14.6969	+	14.6969i	0.867533	+	0.867533i
$288$		0			0
$289$			10.0000i			0.588235i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.67423	+	3.67423i	−0.214651	+	0.214651i	−0.806240	−	0.591589i	$-0.798501\pi$
$293$	−3.67423	+	3.67423i	−0.214651	+	0.214651i	0.591589	+	0.806240i	$0.298501\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−18.0000			−1.04097
$300$		0			0
$301$	−36.0000			−2.07501
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	30.0000i		−	1.70114i	−0.525859	−	0.850572i	$-0.676256\pi$
$311$		−	30.0000i		−	1.70114i	0.525859	−	0.850572i	$-0.323744\pi$
$312$		0			0
$313$	−19.5959	−	19.5959i	−1.10763	−	1.10763i	−0.993462	−	0.114165i	$-0.963581\pi$
$313$	−19.5959	−	19.5959i	−1.10763	−	1.10763i	−0.114165	−	0.993462i	$-0.536419\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−11.0227	−	11.0227i	−0.619097	−	0.619097i	0.326203	−	0.945300i	$-0.394231\pi$
$317$	−11.0227	−	11.0227i	−0.619097	−	0.619097i	−0.945300	+	0.326203i	$0.894231\pi$
$318$		0			0
$319$		−	36.0000i		−	2.01561i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.67423	−	3.67423i	0.204440	−	0.204440i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	36.0000			1.98474
$330$		0			0
$331$	−4.00000			−0.219860			−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000			−0.219860			−0.109930	+	0.993939i	$0.535063\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−12.2474	+	12.2474i	−0.667161	+	0.667161i	−0.957058	−	0.289897i	$-0.906379\pi$
$337$	−12.2474	+	12.2474i	−0.667161	+	0.667161i	0.289897	+	0.957058i	$0.406379\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	30.0000i		−	1.62459i
$342$		0			0
$343$	−4.89898	−	4.89898i	−0.264520	−	0.264520i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	7.34847	+	7.34847i	0.394486	+	0.394486i	0.876283	−	0.481797i	$-0.160016\pi$
$347$	7.34847	+	7.34847i	0.394486	+	0.394486i	−0.481797	+	0.876283i	$0.660016\pi$
$348$		0			0
$349$		−	23.0000i		−	1.23116i	−0.788074	−	0.615581i	$-0.788921\pi$
$349$		−	23.0000i		−	1.23116i	0.788074	−	0.615581i	$-0.211079\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	−0.980208	−	0.197969i	$-0.936565\pi$
$353$	−14.6969	+	14.6969i	−0.782239	+	0.782239i	0.197969	+	0.980208i	$0.436565\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	12.0000			0.633336			0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000			0.633336			0.316668	+	0.948536i	$0.397436\pi$
$360$		0			0
$361$	18.0000			0.947368
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.2474	+	12.2474i	−0.639312	+	0.639312i	−0.950386	−	0.311074i	$-0.899311\pi$
$367$	−12.2474	+	12.2474i	−0.639312	+	0.639312i	0.311074	+	0.950386i	$0.399311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			18.0000i			0.934513i
$372$		0			0
$373$	−24.4949	−	24.4949i	−1.26830	−	1.26830i	−0.946967	−	0.321331i	$-0.895870\pi$
$373$	−24.4949	−	24.4949i	−1.26830	−	1.26830i	−0.321331	−	0.946967i	$-0.604130\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	14.6969	+	14.6969i	0.756931	+	0.756931i
$378$		0			0
$379$			19.0000i			0.975964i	0.872854	+	0.487982i	$0.162267\pi$
$379$			19.0000i			0.975964i	−0.872854	+	0.487982i	$0.837733\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.67423	+	3.67423i	−0.187745	+	0.187745i	−0.794720	−	0.606976i	$-0.792383\pi$
$383$	−3.67423	+	3.67423i	−0.187745	+	0.187745i	0.606976	+	0.794720i	$0.292383\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	18.0000			0.912636			0.456318	−	0.889817i	$-0.349168\pi$
$389$	18.0000			0.912636			0.456318	+	0.889817i	$0.349168\pi$
$390$		0			0
$391$	−27.0000			−1.36545
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−7.34847	+	7.34847i	−0.368809	+	0.368809i	−0.867043	−	0.498234i	$-0.833982\pi$
$397$	−7.34847	+	7.34847i	−0.368809	+	0.368809i	0.498234	+	0.867043i	$0.333982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	36.0000i		−	1.79775i	−0.438201	−	0.898877i	$-0.644384\pi$
$401$		−	36.0000i		−	1.79775i	0.438201	−	0.898877i	$-0.355616\pi$
$402$		0			0
$403$	12.2474	+	12.2474i	0.610089	+	0.610089i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	29.3939	+	29.3939i	1.45700	+	1.45700i
$408$		0			0
$409$		−	11.0000i		−	0.543915i	−0.962309	−	0.271957i	$-0.912329\pi$
$409$		−	11.0000i		−	0.543915i	0.962309	−	0.271957i	$-0.0876710\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−14.6969	+	14.6969i	−0.723189	+	0.723189i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0000			−0.879358			−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000			−0.879358			−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$	13.0000			0.633581			0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000			0.633581			0.316791	+	0.948495i	$0.397395\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−12.2474	+	12.2474i	−0.592696	+	0.592696i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	6.00000i		−	0.289010i	−0.989504	−	0.144505i	$-0.953841\pi$
$431$		−	6.00000i		−	0.289010i	0.989504	−	0.144505i	$-0.0461589\pi$
$432$		0			0
$433$	−7.34847	−	7.34847i	−0.353145	−	0.353145i	0.508133	−	0.861278i	$-0.330336\pi$
$433$	−7.34847	−	7.34847i	−0.353145	−	0.353145i	−0.861278	+	0.508133i	$0.830336\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.67423	+	3.67423i	0.175762	+	0.175762i
$438$		0			0
$439$			35.0000i			1.67046i	0.549902	+	0.835229i	$0.314665\pi$
$439$			35.0000i			1.67046i	−0.549902	+	0.835229i	$0.685335\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.67423	−	3.67423i	0.174568	−	0.174568i	−0.614415	−	0.788983i	$-0.710608\pi$
$443$	3.67423	−	3.67423i	0.174568	−	0.174568i	0.788983	+	0.614415i	$0.210608\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−30.0000			−1.41579			−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000			−1.41579			−0.707894	+	0.706319i	$0.750354\pi$
$450$		0			0
$451$	36.0000			1.69517
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	14.6969	−	14.6969i	0.687494	−	0.687494i	−0.274184	−	0.961677i	$-0.588408\pi$
$457$	14.6969	−	14.6969i	0.687494	−	0.687494i	0.961677	+	0.274184i	$0.0884076\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$		0			0
$463$	19.5959	+	19.5959i	0.910700	+	0.910700i	0.996327	−	0.0856277i	$-0.0272896\pi$
$463$	19.5959	+	19.5959i	0.910700	+	0.910700i	−0.0856277	+	0.996327i	$0.527290\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.3712	+	18.3712i	0.850117	+	0.850117i	0.990147	−	0.140031i	$-0.0447201\pi$
$467$	18.3712	+	18.3712i	0.850117	+	0.850117i	−0.140031	+	0.990147i	$0.544720\pi$
$468$		0			0
$469$			24.0000i			1.10822i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−44.0908	+	44.0908i	−2.02730	+	2.02730i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−30.0000			−1.37073			−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000			−1.37073			−0.685367	+	0.728197i	$0.740358\pi$
$480$		0			0
$481$	−24.0000			−1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	12.2474	−	12.2474i	0.554985	−	0.554985i	−0.372890	−	0.927875i	$-0.621633\pi$
$487$	12.2474	−	12.2474i	0.554985	−	0.554985i	0.927875	+	0.372890i	$0.121633\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			12.0000i			0.541552i	0.962642	+	0.270776i	$0.0872803\pi$
$491$			12.0000i			0.541552i	−0.962642	+	0.270776i	$0.912720\pi$
$492$		0			0
$493$	22.0454	+	22.0454i	0.992875	+	0.992875i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	14.6969	+	14.6969i	0.659248	+	0.659248i
$498$		0			0
$499$		−	13.0000i		−	0.581960i	−0.956729	−	0.290980i	$-0.906019\pi$
$499$		−	13.0000i		−	0.581960i	0.956729	−	0.290980i	$-0.0939813\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	18.3712	−	18.3712i	0.819130	−	0.819130i	−0.166852	−	0.985982i	$-0.553360\pi$
$503$	18.3712	−	18.3712i	0.819130	−	0.819130i	0.985982	+	0.166852i	$0.0533602\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	12.0000			0.530849
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	44.0908	−	44.0908i	1.93911	−	1.93911i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			18.0000i			0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$521$			18.0000i			0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$522$		0			0
$523$	−2.44949	−	2.44949i	−0.107109	−	0.107109i	0.651521	−	0.758630i	$-0.274131\pi$
$523$	−2.44949	−	2.44949i	−0.107109	−	0.107109i	−0.758630	+	0.651521i	$0.774131\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.3712	+	18.3712i	0.800261	+	0.800261i
$528$		0			0
$529$		−	4.00000i		−	0.173913i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−14.6969	+	14.6969i	−0.636595	+	0.636595i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	30.0000			1.29219
$540$		0			0
$541$	−10.0000			−0.429934			−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000			−0.429934			−0.214967	+	0.976621i	$0.568964\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−12.2474	+	12.2474i	−0.523663	+	0.523663i	−0.918676	−	0.395013i	$-0.870740\pi$
$547$	−12.2474	+	12.2474i	−0.523663	+	0.523663i	0.395013	+	0.918676i	$0.370740\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	6.00000i		−	0.255609i
$552$		0			0
$553$	−12.2474	−	12.2474i	−0.520814	−	0.520814i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		−	36.0000i		−	1.52264i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.34847	−	7.34847i	0.309701	−	0.309701i	−0.535092	−	0.844794i	$-0.679723\pi$
$563$	7.34847	−	7.34847i	0.309701	−	0.309701i	0.844794	+	0.535092i	$0.179723\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.00000			0.251533			0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000			0.251533			0.125767	+	0.992060i	$0.459861\pi$
$570$		0			0
$571$	−7.00000			−0.292941			−0.146470	−	0.989215i	$-0.546791\pi$
$571$	−7.00000			−0.292941			−0.146470	+	0.989215i	$0.546791\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.44949	−	2.44949i	0.101974	−	0.101974i	−0.654279	−	0.756253i	$-0.727028\pi$
$577$	2.44949	−	2.44949i	0.101974	−	0.101974i	0.756253	+	0.654279i	$0.227028\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			54.0000i			2.24030i
$582$		0			0
$583$	22.0454	+	22.0454i	0.913027	+	0.913027i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	33.0681	+	33.0681i	1.36487	+	1.36487i	0.867594	+	0.497273i	$0.165665\pi$
$587$	33.0681	+	33.0681i	1.36487	+	1.36487i	0.497273	+	0.867594i	$0.334335\pi$
$588$		0			0
$589$		−	5.00000i		−	0.206021i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−25.7196	+	25.7196i	−1.05618	+	1.05618i	−0.0578541	+	0.998325i	$0.518426\pi$
$593$	−25.7196	+	25.7196i	−1.05618	+	1.05618i	−0.998325	+	0.0578541i	$0.981574\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000			−0.245153			−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000			−0.245153			−0.122577	+	0.992459i	$0.539116\pi$
$600$		0			0
$601$	19.0000			0.775026			0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000			0.775026			0.387513	+	0.921864i	$0.373334\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	12.2474	−	12.2474i	0.497109	−	0.497109i	−0.413428	−	0.910537i	$-0.635669\pi$
$607$	12.2474	−	12.2474i	0.497109	−	0.497109i	0.910537	+	0.413428i	$0.135669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			36.0000i			1.45640i
$612$		0			0
$613$	−19.5959	−	19.5959i	−0.791472	−	0.791472i	0.190262	−	0.981733i	$-0.439066\pi$
$613$	−19.5959	−	19.5959i	−0.791472	−	0.791472i	−0.981733	+	0.190262i	$0.939066\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	25.7196	+	25.7196i	1.03543	+	1.03543i	0.999349	+	0.0360851i	$0.0114887\pi$
$617$	25.7196	+	25.7196i	1.03543	+	1.03543i	0.0360851	+	0.999349i	$0.488511\pi$
$618$		0			0
$619$		−	4.00000i		−	0.160774i	−0.996764	−	0.0803868i	$-0.974384\pi$
$619$		−	4.00000i		−	0.160774i	0.996764	−	0.0803868i	$-0.0256155\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−14.6969	+	14.6969i	−0.588820	+	0.588820i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−36.0000			−1.43541
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−12.2474	+	12.2474i	−0.485262	+	0.485262i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	24.0000i		−	0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$		−	24.0000i		−	0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$		0			0
$643$	−22.0454	−	22.0454i	−0.869386	−	0.869386i	0.123018	−	0.992404i	$-0.460743\pi$
$643$	−22.0454	−	22.0454i	−0.869386	−	0.869386i	−0.992404	+	0.123018i	$0.960743\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	11.0227	+	11.0227i	0.433347	+	0.433347i	0.889765	−	0.456418i	$-0.150868\pi$
$647$	11.0227	+	11.0227i	0.433347	+	0.433347i	−0.456418	+	0.889765i	$0.650868\pi$
$648$		0			0
$649$			36.0000i			1.41312i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.67423	−	3.67423i	0.143784	−	0.143784i	−0.631551	−	0.775335i	$-0.717581\pi$
$653$	3.67423	−	3.67423i	0.143784	−	0.143784i	0.775335	+	0.631551i	$0.217581\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	18.0000			0.701180			0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000			0.701180			0.350590	+	0.936529i	$0.385981\pi$
$660$		0			0
$661$	−38.0000			−1.47803			−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−38.0000			−1.47803			−0.739014	+	0.673690i	$0.764708\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−22.0454	+	22.0454i	−0.853602	+	0.853602i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			30.0000i			1.15814i
$672$		0			0
$673$	26.9444	+	26.9444i	1.03863	+	1.03863i	0.999223	+	0.0394065i	$0.0125467\pi$
$673$	26.9444	+	26.9444i	1.03863	+	1.03863i	0.0394065	+	0.999223i	$0.487453\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.6969	−	14.6969i	−0.564849	−	0.564849i	0.365832	−	0.930681i	$-0.380785\pi$
$677$	−14.6969	−	14.6969i	−0.564849	−	0.564849i	−0.930681	+	0.365832i	$0.880785\pi$
$678$		0			0
$679$		−	24.0000i		−	0.921035i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−25.7196	+	25.7196i	−0.984135	+	0.984135i	−0.999876	−	0.0157413i	$-0.994989\pi$
$683$	−25.7196	+	25.7196i	−0.984135	+	0.984135i	0.0157413	+	0.999876i	$0.494989\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000			−0.685745
$690$		0			0
$691$	−35.0000			−1.33146			−0.665731	−	0.746191i	$-0.731880\pi$
$691$	−35.0000			−1.33146			−0.665731	+	0.746191i	$0.731880\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−22.0454	+	22.0454i	−0.835029	+	0.835029i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12.0000i		−	0.453234i	−0.973984	−	0.226617i	$-0.927233\pi$
$701$		−	12.0000i		−	0.453234i	0.973984	−	0.226617i	$-0.0727665\pi$
$702$		0			0
$703$	4.89898	+	4.89898i	0.184769	+	0.184769i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	29.3939	+	29.3939i	1.10547	+	1.10547i
$708$		0			0
$709$		−	14.0000i		−	0.525781i	−0.964826	−	0.262891i	$-0.915324\pi$
$709$		−	14.0000i		−	0.525781i	0.964826	−	0.262891i	$-0.0846758\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−18.3712	+	18.3712i	−0.688006	+	0.688006i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	24.0000			0.895049			0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000			0.895049			0.447524	+	0.894272i	$0.352306\pi$
$720$		0			0
$721$	24.0000			0.893807
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$727$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	54.0000i		−	1.99726i
$732$		0			0
$733$	−14.6969	−	14.6969i	−0.542844	−	0.542844i	0.381518	−	0.924362i	$-0.375402\pi$
$733$	−14.6969	−	14.6969i	−0.542844	−	0.542844i	−0.924362	+	0.381518i	$0.875402\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	29.3939	+	29.3939i	1.08274	+	1.08274i
$738$		0			0
$739$			7.00000i			0.257499i	0.991677	+	0.128750i	$0.0410963\pi$
$739$			7.00000i			0.257499i	−0.991677	+	0.128750i	$0.958904\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.0454	−	22.0454i	0.808768	−	0.808768i	−0.175680	−	0.984447i	$-0.556212\pi$
$743$	22.0454	−	22.0454i	0.808768	−	0.808768i	0.984447	+	0.175680i	$0.0562123\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	36.0000			1.31541
$750$		0			0
$751$	25.0000			0.912263			0.456131	−	0.889912i	$-0.349235\pi$
$751$	25.0000			0.912263			0.456131	+	0.889912i	$0.349235\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−24.4949	+	24.4949i	−0.890282	+	0.890282i	−0.994549	−	0.104267i	$-0.966750\pi$
$757$	−24.4949	+	24.4949i	−0.890282	+	0.890282i	0.104267	+	0.994549i	$0.466750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	−26.9444	−	26.9444i	−0.975452	−	0.975452i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−14.6969	−	14.6969i	−0.530676	−	0.530676i
$768$		0			0
$769$			47.0000i			1.69486i	0.530904	+	0.847432i	$0.321852\pi$
$769$			47.0000i			1.69486i	−0.530904	+	0.847432i	$0.678148\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−3.67423	+	3.67423i	−0.132153	+	0.132153i	−0.770089	−	0.637936i	$-0.779788\pi$
$773$	−3.67423	+	3.67423i	−0.132153	+	0.132153i	0.637936	+	0.770089i	$0.279788\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	6.00000			0.214972
$780$		0			0
$781$	36.0000			1.28818
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−17.1464	+	17.1464i	−0.611204	+	0.611204i	−0.943260	−	0.332056i	$-0.892258\pi$
$787$	−17.1464	+	17.1464i	−0.611204	+	0.611204i	0.332056	+	0.943260i	$0.392258\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−12.2474	−	12.2474i	−0.434920	−	0.434920i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−33.0681	−	33.0681i	−1.17133	−	1.17133i	−0.981892	−	0.189440i	$-0.939333\pi$
$797$	−33.0681	−	33.0681i	−1.17133	−	1.17133i	−0.189440	−	0.981892i	$-0.560667\pi$
$798$		0			0
$799$			54.0000i			1.91038i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	14.6969	−	14.6969i	0.518644	−	0.518644i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−42.0000			−1.47664			−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000			−1.47664			−0.738321	+	0.674450i	$0.764381\pi$
$810$		0			0
$811$	16.0000			0.561836			0.280918	−	0.959732i	$-0.409361\pi$
$811$	16.0000			0.561836			0.280918	+	0.959732i	$0.409361\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−7.34847	+	7.34847i	−0.257090	+	0.257090i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	24.0000i		−	0.837606i	−0.908077	−	0.418803i	$-0.862450\pi$
$821$		−	24.0000i		−	0.837606i	0.908077	−	0.418803i	$-0.137550\pi$
$822$		0			0
$823$	2.44949	+	2.44949i	0.0853838	+	0.0853838i	0.748509	−	0.663125i	$-0.230770\pi$
$823$	2.44949	+	2.44949i	0.0853838	+	0.0853838i	−0.663125	+	0.748509i	$0.730770\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.67423	+	3.67423i	0.127766	+	0.127766i	0.768098	−	0.640332i	$-0.221203\pi$
$827$	3.67423	+	3.67423i	0.127766	+	0.127766i	−0.640332	+	0.768098i	$0.721203\pi$
$828$		0			0
$829$		−	26.0000i		−	0.903017i	−0.892267	−	0.451509i	$-0.850886\pi$
$829$		−	26.0000i		−	0.903017i	0.892267	−	0.451509i	$-0.149114\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18.3712	+	18.3712i	−0.636523	+	0.636523i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	24.0000			0.828572			0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000			0.828572			0.414286	+	0.910147i	$0.364031\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	61.2372	−	61.2372i	2.10414	−	2.10414i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	36.0000i		−	1.23406i
$852$		0			0
$853$	−9.79796	−	9.79796i	−0.335476	−	0.335476i	0.519186	−	0.854661i	$-0.326235\pi$
$853$	−9.79796	−	9.79796i	−0.335476	−	0.335476i	−0.854661	+	0.519186i	$0.826235\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	40.4166	+	40.4166i	1.38060	+	1.38060i	0.843544	+	0.537060i	$0.180465\pi$
$857$	40.4166	+	40.4166i	1.38060	+	1.38060i	0.537060	+	0.843544i	$0.319535\pi$
$858$		0			0
$859$		−	43.0000i		−	1.46714i	−0.679613	−	0.733571i	$-0.737852\pi$
$859$		−	43.0000i		−	1.46714i	0.679613	−	0.733571i	$-0.262148\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.67423	+	3.67423i	−0.125072	+	0.125072i	−0.766872	−	0.641800i	$-0.778188\pi$
$863$	−3.67423	+	3.67423i	−0.125072	+	0.125072i	0.641800	+	0.766872i	$0.278188\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.0000			−1.01768
$870$		0			0
$871$	−24.0000			−0.813209
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	12.2474	−	12.2474i	0.413567	−	0.413567i	−0.469412	−	0.882979i	$-0.655534\pi$
$877$	12.2474	−	12.2474i	0.413567	−	0.413567i	0.882979	+	0.469412i	$0.155534\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	30.0000i		−	1.01073i	−0.862907	−	0.505363i	$-0.831359\pi$
$881$		−	30.0000i		−	1.01073i	0.862907	−	0.505363i	$-0.168641\pi$
$882$		0			0
$883$	12.2474	+	12.2474i	0.412159	+	0.412159i	0.882490	−	0.470331i	$-0.155865\pi$
$883$	12.2474	+	12.2474i	0.412159	+	0.412159i	−0.470331	+	0.882490i	$0.655865\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	18.3712	+	18.3712i	0.616844	+	0.616844i	0.944720	−	0.327877i	$-0.106333\pi$
$887$	18.3712	+	18.3712i	0.616844	+	0.616844i	−0.327877	+	0.944720i	$0.606333\pi$
$888$		0			0
$889$		−	72.0000i		−	2.41480i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	7.34847	−	7.34847i	0.245907	−	0.245907i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	30.0000			1.00056
$900$		0			0
$901$	−27.0000			−0.899500
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.89898	−	4.89898i	0.162668	−	0.162668i	−0.621080	−	0.783748i	$-0.713306\pi$
$907$	4.89898	−	4.89898i	0.162668	−	0.162668i	0.783748	+	0.621080i	$0.213306\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			30.0000i			0.993944i	0.867766	+	0.496972i	$0.165555\pi$
$911$			30.0000i			0.993944i	−0.867766	+	0.496972i	$0.834445\pi$
$912$		0			0
$913$	66.1362	+	66.1362i	2.18879	+	2.18879i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.3939	−	29.3939i	−0.970671	−	0.970671i
$918$		0			0
$919$			16.0000i			0.527791i	0.964551	+	0.263896i	$0.0850075\pi$
$919$			16.0000i			0.527791i	−0.964551	+	0.263896i	$0.914993\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−14.6969	+	14.6969i	−0.483756	+	0.483756i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−42.0000			−1.37798			−0.688988	−	0.724773i	$-0.741945\pi$
$929$	−42.0000			−1.37798			−0.688988	+	0.724773i	$0.741945\pi$
$930$		0			0
$931$	5.00000			0.163868
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.89898	−	4.89898i	0.160043	−	0.160043i	−0.622543	−	0.782586i	$-0.713900\pi$
$937$	4.89898	−	4.89898i	0.160043	−	0.160043i	0.782586	+	0.622543i	$0.213900\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$	−22.0454	−	22.0454i	−0.717897	−	0.717897i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−18.3712	−	18.3712i	−0.596983	−	0.596983i	0.342526	−	0.939508i	$-0.388718\pi$
$947$	−18.3712	−	18.3712i	−0.596983	−	0.596983i	−0.939508	+	0.342526i	$0.888718\pi$
$948$		0			0
$949$			12.0000i			0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	14.6969	−	14.6969i	0.476081	−	0.476081i	−0.427795	−	0.903876i	$-0.640710\pi$
$953$	14.6969	−	14.6969i	0.476081	−	0.476081i	0.903876	+	0.427795i	$0.140710\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	54.0000			1.74375
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	4.89898	−	4.89898i	0.157541	−	0.157541i	−0.623935	−	0.781476i	$-0.714467\pi$
$967$	4.89898	−	4.89898i	0.157541	−	0.157541i	0.781476	+	0.623935i	$0.214467\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	12.0000i		−	0.385098i	−0.981287	−	0.192549i	$-0.938325\pi$
$971$		−	12.0000i		−	0.385098i	0.981287	−	0.192549i	$-0.0616755\pi$
$972$		0			0
$973$	48.9898	+	48.9898i	1.57054	+	1.57054i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−14.6969	−	14.6969i	−0.470197	−	0.470197i	0.431782	−	0.901978i	$-0.357885\pi$
$977$	−14.6969	−	14.6969i	−0.470197	−	0.470197i	−0.901978	+	0.431782i	$0.857885\pi$
$978$		0			0
$979$			36.0000i			1.15056i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18.3712	−	18.3712i	0.585949	−	0.585949i	−0.350582	−	0.936532i	$-0.614016\pi$
$983$	18.3712	−	18.3712i	0.585949	−	0.585949i	0.936532	+	0.350582i	$0.114016\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	54.0000			1.71710
$990$		0			0
$991$	−37.0000			−1.17534			−0.587672	−	0.809099i	$-0.699955\pi$
$991$	−37.0000			−1.17534			−0.587672	+	0.809099i	$0.699955\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	22.0454	−	22.0454i	0.698185	−	0.698185i	−0.265834	−	0.964019i	$-0.585647\pi$
$997$	22.0454	−	22.0454i	0.698185	−	0.698185i	0.964019	+	0.265834i	$0.0856472\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2700.2.j.a.593.1	✓	4
3.2	odd	2		2700.2.j.g.593.1	yes	4
5.2	odd	4		2700.2.j.g.1457.1	yes	4
5.3	odd	4		2700.2.j.g.1457.2	yes	4
5.4	even	2	inner	2700.2.j.a.593.2	yes	4
15.2	even	4	inner	2700.2.j.a.1457.1	yes	4
15.8	even	4	inner	2700.2.j.a.1457.2	yes	4
15.14	odd	2		2700.2.j.g.593.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2700.2.j.a.593.1	✓	4	1.1	even	1	trivial
2700.2.j.a.593.2	yes	4	5.4	even	2	inner
2700.2.j.a.1457.1	yes	4	15.2	even	4	inner
2700.2.j.a.1457.2	yes	4	15.8	even	4	inner
2700.2.j.g.593.1	yes	4	3.2	odd	2
2700.2.j.g.593.2	yes	4	15.14	odd	2
2700.2.j.g.1457.1	yes	4	5.2	odd	4
2700.2.j.g.1457.2	yes	4	5.3	odd	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:	\( N \)	\(=\)	\( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2700.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.44949 + 2.44949i) q^{7} +6.00000i q^{11} +(-2.44949 - 2.44949i) q^{13} +(-3.67423 - 3.67423i) q^{17} +1.00000i q^{19} +(3.67423 - 3.67423i) q^{23} -6.00000 q^{29} -5.00000 q^{31} +(4.89898 - 4.89898i) q^{37} -6.00000i q^{41} +(7.34847 + 7.34847i) q^{43} +(-7.34847 - 7.34847i) q^{47} -5.00000i q^{49} +(3.67423 - 3.67423i) q^{53} +6.00000 q^{59} +5.00000 q^{61} +(4.89898 - 4.89898i) q^{67} -6.00000i q^{71} +(-2.44949 - 2.44949i) q^{73} +(-14.6969 - 14.6969i) q^{77} +5.00000i q^{79} +(11.0227 - 11.0227i) q^{83} +6.00000 q^{89} +12.0000 q^{91} +(-4.89898 + 4.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{29} - 20 q^{31} + 24 q^{59} + 20 q^{61} + 24 q^{89} + 48 q^{91}+O(q^{100}) \) 4 * q - 24 * q^29 - 20 * q^31 + 24 * q^59 + 20 * q^61 + 24 * q^89 + 48 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more