LMFDB - Embedded newform 2400.2.o.c.2399.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(2399,2400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.2399"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.o (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,8,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1640964851$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 480)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2399.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2400.2399
Dual form			2400.2.o.c.2399.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 - 0.292893i) q^{3} +3.41421 q^{7} +(2.82843 + 1.00000i) q^{9} +2.82843 q^{11} -2.00000i q^{13} +7.65685 q^{17} -2.82843i q^{19} +(-5.82843 - 1.00000i) q^{21} -7.41421i q^{23} +(-4.53553 - 2.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(-4.82843 - 0.828427i) q^{33} +0.343146i q^{37} +(-0.585786 + 3.41421i) q^{39} +2.00000i q^{41} -7.89949 q^{43} -6.24264i q^{47} +4.65685 q^{49} +(-13.0711 - 2.24264i) q^{51} +3.65685 q^{53} +(-0.828427 + 4.82843i) q^{57} +1.65685 q^{59} +5.65685 q^{61} +(9.65685 + 3.41421i) q^{63} -1.07107 q^{67} +(-2.17157 + 12.6569i) q^{69} -1.17157 q^{71} +15.6569i q^{73} +9.65685 q^{77} -4.48528i q^{79} +(7.00000 + 5.65685i) q^{81} -5.07107i q^{83} +(2.34315 - 13.6569i) q^{87} +7.31371i q^{89} -6.82843i q^{91} +(1.65685 - 9.65685i) q^{93} -18.9706i q^{97} +(8.00000 + 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{7} + 8 q^{17} - 12 q^{21} - 4 q^{27} - 8 q^{33} - 8 q^{39} + 8 q^{43} - 4 q^{49} - 24 q^{51} - 8 q^{53} + 8 q^{57} - 16 q^{59} + 16 q^{63} + 24 q^{67} - 20 q^{69} - 16 q^{71} + 16 q^{77}+ \cdots + 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^7 + 8 * q^17 - 12 * q^21 - 4 * q^27 - 8 * q^33 - 8 * q^39 + 8 * q^43 - 4 * q^49 - 24 * q^51 - 8 * q^53 + 8 * q^57 - 16 * q^59 + 16 * q^63 + 24 * q^67 - 20 * q^69 - 16 * q^71 + 16 * q^77 + 28 * q^81 + 32 * q^87 - 16 * q^93 + 32 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1601$	$1951$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$3$	−1.70711	−	0.292893i	−0.985599	−	0.169102i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	3.41421			1.29045			0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421			1.29045			0.645226	+	0.763992i	$0.276763\pi$
$8$		0			0
$9$	2.82843	+	1.00000i	0.942809	+	0.333333i
$10$		0			0
$11$	2.82843			0.852803			0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843			0.852803			0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$		−	2.00000i		−	0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$		−	2.00000i		−	0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	7.65685			1.85706			0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685			1.85706			0.928530	+	0.371257i	$0.121073\pi$
$18$		0			0
$19$		−	2.82843i		−	0.648886i	−0.945905	−	0.324443i	$-0.894823\pi$
$19$		−	2.82843i		−	0.648886i	0.945905	−	0.324443i	$-0.105177\pi$
$20$		0			0
$21$	−5.82843	−	1.00000i	−1.27187	−	0.218218i
$22$		0			0
$23$		−	7.41421i		−	1.54597i	−0.634424	−	0.772985i	$-0.718763\pi$
$23$		−	7.41421i		−	1.54597i	0.634424	−	0.772985i	$-0.281237\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−4.53553	−	2.53553i	−0.872864	−	0.487964i
$28$		0			0
$29$			8.00000i			1.48556i	0.669534	+	0.742781i	$0.266494\pi$
$29$			8.00000i			1.48556i	−0.669534	+	0.742781i	$0.733506\pi$
$30$		0			0
$31$			5.65685i			1.01600i	0.861357	+	0.508001i	$0.169615\pi$
$31$			5.65685i			1.01600i	−0.861357	+	0.508001i	$0.830385\pi$
$32$		0			0
$33$	−4.82843	−	0.828427i	−0.840521	−	0.144211i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			0.343146i			0.0564128i	0.999602	+	0.0282064i	$0.00897957\pi$
$37$			0.343146i			0.0564128i	−0.999602	+	0.0282064i	$0.991020\pi$
$38$		0			0
$39$	−0.585786	+	3.41421i	−0.0938009	+	0.546712i
$40$		0			0
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	−7.89949			−1.20466			−0.602331	−	0.798247i	$-0.705761\pi$
$43$	−7.89949			−1.20466			−0.602331	+	0.798247i	$0.705761\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.24264i		−	0.910583i	−0.890343	−	0.455291i	$-0.849535\pi$
$47$		−	6.24264i		−	0.910583i	0.890343	−	0.455291i	$-0.150465\pi$
$48$		0			0
$49$	4.65685			0.665265
$50$		0			0
$51$	−13.0711	−	2.24264i	−1.83032	−	0.314033i
$52$		0			0
$53$	3.65685			0.502308			0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685			0.502308			0.251154	+	0.967947i	$0.419190\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.828427	+	4.82843i	−0.109728	+	0.639541i
$58$		0			0
$59$	1.65685			0.215704			0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685			0.215704			0.107852	+	0.994167i	$0.465603\pi$
$60$		0			0
$61$	5.65685			0.724286			0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685			0.724286			0.362143	+	0.932123i	$0.382045\pi$
$62$		0			0
$63$	9.65685	+	3.41421i	1.21665	+	0.430150i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.07107			−0.130852			−0.0654259	−	0.997857i	$-0.520841\pi$
$67$	−1.07107			−0.130852			−0.0654259	+	0.997857i	$0.520841\pi$
$68$		0			0
$69$	−2.17157	+	12.6569i	−0.261427	+	1.52371i
$70$		0			0
$71$	−1.17157			−0.139040			−0.0695201	−	0.997581i	$-0.522147\pi$
$71$	−1.17157			−0.139040			−0.0695201	+	0.997581i	$0.522147\pi$
$72$		0			0
$73$			15.6569i			1.83250i	0.400611	+	0.916248i	$0.368798\pi$
$73$			15.6569i			1.83250i	−0.400611	+	0.916248i	$0.631202\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.65685			1.10050
$78$		0			0
$79$		−	4.48528i		−	0.504634i	−0.967645	−	0.252317i	$-0.918807\pi$
$79$		−	4.48528i		−	0.504634i	0.967645	−	0.252317i	$-0.0811925\pi$
$80$		0			0
$81$	7.00000	+	5.65685i	0.777778	+	0.628539i
$82$		0			0
$83$		−	5.07107i		−	0.556622i	−0.960491	−	0.278311i	$-0.910225\pi$
$83$		−	5.07107i		−	0.556622i	0.960491	−	0.278311i	$-0.0897746\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	2.34315	−	13.6569i	0.251212	−	1.46417i
$88$		0			0
$89$			7.31371i			0.775252i	0.921817	+	0.387626i	$0.126705\pi$
$89$			7.31371i			0.775252i	−0.921817	+	0.387626i	$0.873295\pi$
$90$		0			0
$91$		−	6.82843i		−	0.715814i
$92$		0			0
$93$	1.65685	−	9.65685i	0.171808	−	1.00137i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	18.9706i		−	1.92617i	−0.269200	−	0.963084i	$-0.586759\pi$
$97$		−	18.9706i		−	1.92617i	0.269200	−	0.963084i	$-0.413241\pi$
$98$		0			0
$99$	8.00000	+	2.82843i	0.804030	+	0.284268i
$100$		0			0
$101$		−	7.31371i		−	0.727741i	−0.931450	−	0.363871i	$-0.881455\pi$
$101$		−	7.31371i		−	0.727741i	0.931450	−	0.363871i	$-0.118545\pi$
$102$		0			0
$103$	2.24264			0.220974			0.110487	−	0.993878i	$-0.464759\pi$
$103$	2.24264			0.220974			0.110487	+	0.993878i	$0.464759\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	15.4142i		−	1.49015i	−0.666982	−	0.745074i	$-0.732414\pi$
$107$		−	15.4142i		−	1.49015i	0.666982	−	0.745074i	$-0.267586\pi$
$108$		0			0
$109$	−17.6569			−1.69122			−0.845610	−	0.533801i	$-0.820763\pi$
$109$	−17.6569			−1.69122			−0.845610	+	0.533801i	$0.820763\pi$
$110$		0			0
$111$	0.100505	−	0.585786i	0.00953952	−	0.0556004i
$112$		0			0
$113$	10.0000			0.940721			0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000			0.940721			0.470360	+	0.882474i	$0.344124\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.00000	−	5.65685i	0.184900	−	0.522976i
$118$		0			0
$119$	26.1421			2.39645
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	0.585786	−	3.41421i	0.0528186	−	0.307849i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−17.0711			−1.51481			−0.757406	−	0.652944i	$-0.773534\pi$
$127$	−17.0711			−1.51481			−0.757406	+	0.652944i	$0.773534\pi$
$128$		0			0
$129$	13.4853	+	2.31371i	1.18731	+	0.203711i
$130$		0			0
$131$	10.8284			0.946084			0.473042	−	0.881040i	$-0.343156\pi$
$131$	10.8284			0.946084			0.473042	+	0.881040i	$0.343156\pi$
$132$		0			0
$133$		−	9.65685i		−	0.837355i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	17.3137			1.47921			0.739605	−	0.673041i	$-0.235012\pi$
$137$	17.3137			1.47921			0.739605	+	0.673041i	$0.235012\pi$
$138$		0			0
$139$			13.1716i			1.11720i	0.829438	+	0.558599i	$0.188661\pi$
$139$			13.1716i			1.11720i	−0.829438	+	0.558599i	$0.811339\pi$
$140$		0			0
$141$	−1.82843	+	10.6569i	−0.153981	+	0.897469i
$142$		0			0
$143$		−	5.65685i		−	0.473050i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−7.94975	−	1.36396i	−0.655684	−	0.112498i
$148$		0			0
$149$		−	13.3137i		−	1.09070i	−0.838208	−	0.545351i	$-0.816396\pi$
$149$		−	13.3137i		−	1.09070i	0.838208	−	0.545351i	$-0.183604\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	21.6569	+	7.65685i	1.75085	+	0.619020i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			7.65685i			0.611083i	0.952179	+	0.305542i	$0.0988376\pi$
$157$			7.65685i			0.611083i	−0.952179	+	0.305542i	$0.901162\pi$
$158$		0			0
$159$	−6.24264	−	1.07107i	−0.495074	−	0.0849412i
$160$		0			0
$161$		−	25.3137i		−	1.99500i
$162$		0			0
$163$	10.2426			0.802266			0.401133	−	0.916020i	$-0.368617\pi$
$163$	10.2426			0.802266			0.401133	+	0.916020i	$0.368617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			18.7279i			1.44921i	0.689164	+	0.724605i	$0.257978\pi$
$167$			18.7279i			1.44921i	−0.689164	+	0.724605i	$0.742022\pi$
$168$		0			0
$169$	9.00000			0.692308
$170$		0			0
$171$	2.82843	−	8.00000i	0.216295	−	0.611775i
$172$		0			0
$173$	−2.00000			−0.152057			−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000			−0.152057			−0.0760286	+	0.997106i	$0.524224\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.82843	−	0.485281i	−0.212598	−	0.0364760i
$178$		0			0
$179$	17.6569			1.31974			0.659868	−	0.751382i	$-0.270612\pi$
$179$	17.6569			1.31974			0.659868	+	0.751382i	$0.270612\pi$
$180$		0			0
$181$	−9.31371			−0.692283			−0.346141	−	0.938182i	$-0.612508\pi$
$181$	−9.31371			−0.692283			−0.346141	+	0.938182i	$0.612508\pi$
$182$		0			0
$183$	−9.65685	−	1.65685i	−0.713855	−	0.122478i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	21.6569			1.58371
$188$		0			0
$189$	−15.4853	−	8.65685i	−1.12639	−	0.629693i
$190$		0			0
$191$	18.1421			1.31272			0.656359	−	0.754448i	$-0.272096\pi$
$191$	18.1421			1.31272			0.656359	+	0.754448i	$0.272096\pi$
$192$		0			0
$193$		−	7.65685i		−	0.551152i	−0.961279	−	0.275576i	$-0.911131\pi$
$193$		−	7.65685i		−	0.551152i	0.961279	−	0.275576i	$-0.0888686\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.9706			1.63658			0.818292	−	0.574802i	$-0.194921\pi$
$197$	22.9706			1.63658			0.818292	+	0.574802i	$0.194921\pi$
$198$		0			0
$199$		−	22.8284i		−	1.61826i	−0.587627	−	0.809132i	$-0.699938\pi$
$199$		−	22.8284i		−	1.61826i	0.587627	−	0.809132i	$-0.300062\pi$
$200$		0			0
$201$	1.82843	+	0.313708i	0.128967	+	0.0221273i
$202$		0			0
$203$			27.3137i			1.91705i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	7.41421	−	20.9706i	0.515323	−	1.45755i
$208$		0			0
$209$		−	8.00000i		−	0.553372i
$210$		0			0
$211$			12.0000i			0.826114i	0.910705	+	0.413057i	$0.135539\pi$
$211$			12.0000i			0.826114i	−0.910705	+	0.413057i	$0.864461\pi$
$212$		0			0
$213$	2.00000	+	0.343146i	0.137038	+	0.0235120i
$214$		0			0
$215$		0			0
$216$		0			0
$217$			19.3137i			1.31110i
$218$		0			0
$219$	4.58579	−	26.7279i	0.309879	−	1.80611i
$220$		0			0
$221$		−	15.3137i		−	1.03011i
$222$		0			0
$223$	−7.89949			−0.528989			−0.264495	−	0.964387i	$-0.585205\pi$
$223$	−7.89949			−0.528989			−0.264495	+	0.964387i	$0.585205\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			7.41421i			0.492099i	0.969257	+	0.246049i	$0.0791325\pi$
$227$			7.41421i			0.492099i	−0.969257	+	0.246049i	$0.920867\pi$
$228$		0			0
$229$	−2.68629			−0.177515			−0.0887576	−	0.996053i	$-0.528290\pi$
$229$	−2.68629			−0.177515			−0.0887576	+	0.996053i	$0.528290\pi$
$230$		0			0
$231$	−16.4853	−	2.82843i	−1.08465	−	0.186097i
$232$		0			0
$233$	−6.97056			−0.456657			−0.228328	−	0.973584i	$-0.573326\pi$
$233$	−6.97056			−0.456657			−0.228328	+	0.973584i	$0.573326\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.31371	+	7.65685i	−0.0853345	+	0.497366i
$238$		0			0
$239$	2.34315			0.151565			0.0757827	−	0.997124i	$-0.475854\pi$
$239$	2.34315			0.151565			0.0757827	+	0.997124i	$0.475854\pi$
$240$		0			0
$241$	9.65685			0.622053			0.311026	−	0.950401i	$-0.399327\pi$
$241$	9.65685			0.622053			0.311026	+	0.950401i	$0.399327\pi$
$242$		0			0
$243$	−10.2929	−	11.7071i	−0.660289	−	0.751011i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5.65685			−0.359937
$248$		0			0
$249$	−1.48528	+	8.65685i	−0.0941259	+	0.548606i
$250$		0			0
$251$	0.485281			0.0306307			0.0153153	−	0.999883i	$-0.495125\pi$
$251$	0.485281			0.0306307			0.0153153	+	0.999883i	$0.495125\pi$
$252$		0			0
$253$		−	20.9706i		−	1.31841i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.0000			−1.37232			−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000			−1.37232			−0.686161	+	0.727450i	$0.740706\pi$
$258$		0			0
$259$			1.17157i			0.0727980i
$260$		0			0
$261$	−8.00000	+	22.6274i	−0.495188	+	1.40060i
$262$		0			0
$263$			11.8995i			0.733754i	0.930269	+	0.366877i	$0.119573\pi$
$263$			11.8995i			0.733754i	−0.930269	+	0.366877i	$0.880427\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.14214	−	12.4853i	0.131097	−	0.764087i
$268$		0			0
$269$			18.0000i			1.09748i	0.835993	+	0.548740i	$0.184892\pi$
$269$			18.0000i			1.09748i	−0.835993	+	0.548740i	$0.815108\pi$
$270$		0			0
$271$			2.34315i			0.142336i	0.997464	+	0.0711680i	$0.0226726\pi$
$271$			2.34315i			0.142336i	−0.997464	+	0.0711680i	$0.977327\pi$
$272$		0			0
$273$	−2.00000	+	11.6569i	−0.121046	+	0.705505i
$274$		0			0
$275$		0			0
$276$		0			0
$277$		−	11.6569i		−	0.700392i	−0.936676	−	0.350196i	$-0.886115\pi$
$277$		−	11.6569i		−	0.700392i	0.936676	−	0.350196i	$-0.113885\pi$
$278$		0			0
$279$	−5.65685	+	16.0000i	−0.338667	+	0.957895i
$280$		0			0
$281$		−	13.3137i		−	0.794229i	−0.917769	−	0.397115i	$-0.870012\pi$
$281$		−	13.3137i		−	0.794229i	0.917769	−	0.397115i	$-0.129988\pi$
$282$		0			0
$283$	13.5563			0.805841			0.402921	−	0.915235i	$-0.367995\pi$
$283$	13.5563			0.805841			0.402921	+	0.915235i	$0.367995\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			6.82843i			0.403069i
$288$		0			0
$289$	41.6274			2.44867
$290$		0			0
$291$	−5.55635	+	32.3848i	−0.325719	+	1.89843i
$292$		0			0
$293$	−22.0000			−1.28525			−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000			−1.28525			−0.642627	+	0.766179i	$0.722155\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−12.8284	−	7.17157i	−0.744381	−	0.416137i
$298$		0			0
$299$	−14.8284			−0.857550
$300$		0			0
$301$	−26.9706			−1.55456
$302$		0			0
$303$	−2.14214	+	12.4853i	−0.123062	+	0.717261i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−7.89949			−0.450848			−0.225424	−	0.974261i	$-0.572377\pi$
$307$	−7.89949			−0.450848			−0.225424	+	0.974261i	$0.572377\pi$
$308$		0			0
$309$	−3.82843	−	0.656854i	−0.217792	−	0.0373671i
$310$		0			0
$311$	−11.5147			−0.652940			−0.326470	−	0.945208i	$-0.605859\pi$
$311$	−11.5147			−0.652940			−0.326470	+	0.945208i	$0.605859\pi$
$312$		0			0
$313$			17.3137i			0.978629i	0.872107	+	0.489314i	$0.162753\pi$
$313$			17.3137i			0.978629i	−0.872107	+	0.489314i	$0.837247\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.343146			−0.0192730			−0.00963649	−	0.999954i	$-0.503067\pi$
$317$	−0.343146			−0.0192730			−0.00963649	+	0.999954i	$0.503067\pi$
$318$		0			0
$319$			22.6274i			1.26689i
$320$		0			0
$321$	−4.51472	+	26.3137i	−0.251987	+	1.46869i
$322$		0			0
$323$		−	21.6569i		−	1.20502i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	30.1421	+	5.17157i	1.66686	+	0.285989i
$328$		0			0
$329$		−	21.3137i		−	1.17506i
$330$		0			0
$331$		−	8.68629i		−	0.477442i	−0.971088	−	0.238721i	$-0.923272\pi$
$331$		−	8.68629i		−	0.477442i	0.971088	−	0.238721i	$-0.0767281\pi$
$332$		0			0
$333$	−0.343146	+	0.970563i	−0.0188043	+	0.0531865i
$334$		0			0
$335$		0			0
$336$		0			0
$337$			9.31371i			0.507350i	0.967290	+	0.253675i	$0.0816394\pi$
$337$			9.31371i			0.507350i	−0.967290	+	0.253675i	$0.918361\pi$
$338$		0			0
$339$	−17.0711	−	2.92893i	−0.927173	−	0.159078i
$340$		0			0
$341$			16.0000i			0.866449i
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$			27.6985i			1.48693i	0.668773	+	0.743466i	$0.266820\pi$
$347$			27.6985i			1.48693i	−0.668773	+	0.743466i	$0.733180\pi$
$348$		0			0
$349$	−16.6274			−0.890045			−0.445023	−	0.895519i	$-0.646804\pi$
$349$	−16.6274			−0.890045			−0.445023	+	0.895519i	$0.646804\pi$
$350$		0			0
$351$	−5.07107	+	9.07107i	−0.270674	+	0.484178i
$352$		0			0
$353$	6.00000			0.319348			0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000			0.319348			0.159674	+	0.987170i	$0.448956\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−44.6274	−	7.65685i	−2.36193	−	0.405244i
$358$		0			0
$359$	−29.6569			−1.56523			−0.782614	−	0.622507i	$-0.786114\pi$
$359$	−29.6569			−1.56523			−0.782614	+	0.622507i	$0.786114\pi$
$360$		0			0
$361$	11.0000			0.578947
$362$		0			0
$363$	5.12132	+	0.878680i	0.268800	+	0.0461187i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−14.9289			−0.779284			−0.389642	−	0.920966i	$-0.627401\pi$
$367$	−14.9289			−0.779284			−0.389642	+	0.920966i	$0.627401\pi$
$368$		0			0
$369$	−2.00000	+	5.65685i	−0.104116	+	0.294484i
$370$		0			0
$371$	12.4853			0.648204
$372$		0			0
$373$		−	19.6569i		−	1.01779i	−0.860828	−	0.508897i	$-0.830054\pi$
$373$		−	19.6569i		−	1.01779i	0.860828	−	0.508897i	$-0.169946\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	16.0000			0.824042
$378$		0			0
$379$			14.1421i			0.726433i	0.931705	+	0.363216i	$0.118321\pi$
$379$			14.1421i			0.726433i	−0.931705	+	0.363216i	$0.881679\pi$
$380$		0			0
$381$	29.1421	+	5.00000i	1.49300	+	0.256158i
$382$		0			0
$383$		−	0.585786i		−	0.0299323i	−0.999888	−	0.0149661i	$-0.995236\pi$
$383$		−	0.585786i		−	0.0299323i	0.999888	−	0.0149661i	$-0.00476405\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−22.3431	−	7.89949i	−1.13577	−	0.401554i
$388$		0			0
$389$		−	9.31371i		−	0.472224i	−0.971726	−	0.236112i	$-0.924127\pi$
$389$		−	9.31371i		−	0.472224i	0.971726	−	0.236112i	$-0.0758732\pi$
$390$		0			0
$391$		−	56.7696i		−	2.87096i
$392$		0			0
$393$	−18.4853	−	3.17157i	−0.932459	−	0.159985i
$394$		0			0
$395$		0			0
$396$		0			0
$397$			20.6274i			1.03526i	0.855604	+	0.517630i	$0.173186\pi$
$397$			20.6274i			1.03526i	−0.855604	+	0.517630i	$0.826814\pi$
$398$		0			0
$399$	−2.82843	+	16.4853i	−0.141598	+	0.825296i
$400$		0			0
$401$			8.00000i			0.399501i	0.979847	+	0.199750i	$0.0640132\pi$
$401$			8.00000i			0.399501i	−0.979847	+	0.199750i	$0.935987\pi$
$402$		0			0
$403$	11.3137			0.563576
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.970563i			0.0481090i
$408$		0			0
$409$	−5.65685			−0.279713			−0.139857	−	0.990172i	$-0.544664\pi$
$409$	−5.65685			−0.279713			−0.139857	+	0.990172i	$0.544664\pi$
$410$		0			0
$411$	−29.5563	−	5.07107i	−1.45791	−	0.250137i
$412$		0			0
$413$	5.65685			0.278356
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.85786	−	22.4853i	0.188920	−	1.10111i
$418$		0			0
$419$	−20.0000			−0.977064			−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000			−0.977064			−0.488532	+	0.872546i	$0.662467\pi$
$420$		0			0
$421$	16.9706			0.827095			0.413547	−	0.910483i	$-0.364290\pi$
$421$	16.9706			0.827095			0.413547	+	0.910483i	$0.364290\pi$
$422$		0			0
$423$	6.24264	−	17.6569i	0.303528	−	0.858506i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	19.3137			0.934656
$428$		0			0
$429$	−1.65685	+	9.65685i	−0.0799937	+	0.466237i
$430$		0			0
$431$	11.5147			0.554644			0.277322	−	0.960777i	$-0.410553\pi$
$431$	11.5147			0.554644			0.277322	+	0.960777i	$0.410553\pi$
$432$		0			0
$433$			15.6569i			0.752420i	0.926534	+	0.376210i	$0.122773\pi$
$433$			15.6569i			0.752420i	−0.926534	+	0.376210i	$0.877227\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−20.9706			−1.00316
$438$		0			0
$439$		−	17.1716i		−	0.819554i	−0.912186	−	0.409777i	$-0.865606\pi$
$439$		−	17.1716i		−	0.819554i	0.912186	−	0.409777i	$-0.134394\pi$
$440$		0			0
$441$	13.1716	+	4.65685i	0.627218	+	0.221755i
$442$		0			0
$443$		−	35.8995i		−	1.70564i	−0.522208	−	0.852818i	$-0.674891\pi$
$443$		−	35.8995i		−	1.70564i	0.522208	−	0.852818i	$-0.325109\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−3.89949	+	22.7279i	−0.184440	+	1.07499i
$448$		0			0
$449$			2.00000i			0.0943858i	0.998886	+	0.0471929i	$0.0150276\pi$
$449$			2.00000i			0.0943858i	−0.998886	+	0.0471929i	$0.984972\pi$
$450$		0			0
$451$			5.65685i			0.266371i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			5.31371i			0.248565i	0.992247	+	0.124282i	$0.0396629\pi$
$457$			5.31371i			0.248565i	−0.992247	+	0.124282i	$0.960337\pi$
$458$		0			0
$459$	−34.7279	−	19.4142i	−1.62096	−	0.906178i
$460$		0			0
$461$		−	26.6274i		−	1.24016i	−0.784538	−	0.620081i	$-0.787100\pi$
$461$		−	26.6274i		−	1.24016i	0.784538	−	0.620081i	$-0.212900\pi$
$462$		0			0
$463$	−20.5858			−0.956703			−0.478351	−	0.878169i	$-0.658765\pi$
$463$	−20.5858			−0.956703			−0.478351	+	0.878169i	$0.658765\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			35.8995i			1.66123i	0.556847	+	0.830615i	$0.312011\pi$
$467$			35.8995i			1.66123i	−0.556847	+	0.830615i	$0.687989\pi$
$468$		0			0
$469$	−3.65685			−0.168858
$470$		0			0
$471$	2.24264	−	13.0711i	0.103335	−	0.602283i
$472$		0			0
$473$	−22.3431			−1.02734
$474$		0			0
$475$		0			0
$476$		0			0
$477$	10.3431	+	3.65685i	0.473580	+	0.167436i
$478$		0			0
$479$	0.970563			0.0443461			0.0221731	−	0.999754i	$-0.492942\pi$
$479$	0.970563			0.0443461			0.0221731	+	0.999754i	$0.492942\pi$
$480$		0			0
$481$	0.686292			0.0312922
$482$		0			0
$483$	−7.41421	+	43.2132i	−0.337358	+	1.96627i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−8.10051			−0.367069			−0.183534	−	0.983013i	$-0.558754\pi$
$487$	−8.10051			−0.367069			−0.183534	+	0.983013i	$0.558754\pi$
$488$		0			0
$489$	−17.4853	−	3.00000i	−0.790712	−	0.135665i
$490$		0			0
$491$	39.1127			1.76513			0.882566	−	0.470189i	$-0.155814\pi$
$491$	39.1127			1.76513			0.882566	+	0.470189i	$0.155814\pi$
$492$		0			0
$493$			61.2548i			2.75878i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.00000			−0.179425
$498$		0			0
$499$			33.4558i			1.49769i	0.662746	+	0.748845i	$0.269391\pi$
$499$			33.4558i			1.49769i	−0.662746	+	0.748845i	$0.730609\pi$
$500$		0			0
$501$	5.48528	−	31.9706i	0.245064	−	1.42834i
$502$		0			0
$503$			0.786797i			0.0350815i	0.999846	+	0.0175408i	$0.00558368\pi$
$503$			0.786797i			0.0350815i	−0.999846	+	0.0175408i	$0.994416\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−15.3640	−	2.63604i	−0.682337	−	0.117071i
$508$		0			0
$509$		−	22.6274i		−	1.00294i	−0.865174	−	0.501471i	$-0.832792\pi$
$509$		−	22.6274i		−	1.00294i	0.865174	−	0.501471i	$-0.167208\pi$
$510$		0			0
$511$			53.4558i			2.36475i
$512$		0			0
$513$	−7.17157	+	12.8284i	−0.316633	+	0.566389i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	17.6569i		−	0.776548i
$518$		0			0
$519$	3.41421	+	0.585786i	0.149867	+	0.0257132i
$520$		0			0
$521$			16.0000i			0.700973i	0.936568	+	0.350486i	$0.113984\pi$
$521$			16.0000i			0.700973i	−0.936568	+	0.350486i	$0.886016\pi$
$522$		0			0
$523$	11.4142			0.499109			0.249554	−	0.968361i	$-0.419716\pi$
$523$	11.4142			0.499109			0.249554	+	0.968361i	$0.419716\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			43.3137i			1.88677i
$528$		0			0
$529$	−31.9706			−1.39002
$530$		0			0
$531$	4.68629	+	1.65685i	0.203368	+	0.0719014i
$532$		0			0
$533$	4.00000			0.173259
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−30.1421	−	5.17157i	−1.30073	−	0.223170i
$538$		0			0
$539$	13.1716			0.567340
$540$		0			0
$541$	−14.6863			−0.631413			−0.315706	−	0.948857i	$-0.602241\pi$
$541$	−14.6863			−0.631413			−0.315706	+	0.948857i	$0.602241\pi$
$542$		0			0
$543$	15.8995	+	2.72792i	0.682313	+	0.117066i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	18.0416			0.771404			0.385702	−	0.922623i	$-0.373959\pi$
$547$	18.0416			0.771404			0.385702	+	0.922623i	$0.373959\pi$
$548$		0			0
$549$	16.0000	+	5.65685i	0.682863	+	0.241429i
$550$		0			0
$551$	22.6274			0.963960
$552$		0			0
$553$		−	15.3137i		−	0.651205i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$			15.7990i			0.668226i
$560$		0			0
$561$	−36.9706	−	6.34315i	−1.56090	−	0.267808i
$562$		0			0
$563$			25.5563i			1.07707i	0.842603	+	0.538536i	$0.181022\pi$
$563$			25.5563i			1.07707i	−0.842603	+	0.538536i	$0.818978\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	23.8995	+	19.3137i	1.00368	+	0.811100i
$568$		0			0
$569$		−	28.6274i		−	1.20012i	−0.799954	−	0.600062i	$-0.795143\pi$
$569$		−	28.6274i		−	1.20012i	0.799954	−	0.600062i	$-0.204857\pi$
$570$		0			0
$571$		−	16.2843i		−	0.681476i	−0.940158	−	0.340738i	$-0.889323\pi$
$571$		−	16.2843i		−	0.681476i	0.940158	−	0.340738i	$-0.110677\pi$
$572$		0			0
$573$	−30.9706	−	5.31371i	−1.29381	−	0.221983i
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	22.0000i		−	0.915872i	−0.888985	−	0.457936i	$-0.848589\pi$
$577$		−	22.0000i		−	0.915872i	0.888985	−	0.457936i	$-0.151411\pi$
$578$		0			0
$579$	−2.24264	+	13.0711i	−0.0932010	+	0.543215i
$580$		0			0
$581$		−	17.3137i		−	0.718294i
$582$		0			0
$583$	10.3431			0.428369
$584$		0			0
$585$		0			0
$586$		0			0
$587$			18.7279i			0.772984i	0.922293	+	0.386492i	$0.126313\pi$
$587$			18.7279i			0.772984i	−0.922293	+	0.386492i	$0.873687\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$	−39.2132	−	6.72792i	−1.61302	−	0.276750i
$592$		0			0
$593$	−32.6274			−1.33985			−0.669924	−	0.742430i	$-0.733673\pi$
$593$	−32.6274			−1.33985			−0.669924	+	0.742430i	$0.733673\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−6.68629	+	38.9706i	−0.273652	+	1.59496i
$598$		0			0
$599$	−30.6274			−1.25140			−0.625701	−	0.780063i	$-0.715187\pi$
$599$	−30.6274			−1.25140			−0.625701	+	0.780063i	$0.715187\pi$
$600$		0			0
$601$	4.97056			0.202753			0.101377	−	0.994848i	$-0.467675\pi$
$601$	4.97056			0.202753			0.101377	+	0.994848i	$0.467675\pi$
$602$		0			0
$603$	−3.02944	−	1.07107i	−0.123368	−	0.0436173i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−41.0711			−1.66702			−0.833512	−	0.552502i	$-0.813673\pi$
$607$	−41.0711			−1.66702			−0.833512	+	0.552502i	$0.813673\pi$
$608$		0			0
$609$	8.00000	−	46.6274i	0.324176	−	1.88944i
$610$		0			0
$611$	−12.4853			−0.505100
$612$		0			0
$613$		−	37.3137i		−	1.50709i	−0.657398	−	0.753543i	$-0.728343\pi$
$613$		−	37.3137i		−	1.50709i	0.657398	−	0.753543i	$-0.271657\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.65685			−0.147219			−0.0736097	−	0.997287i	$-0.523452\pi$
$617$	−3.65685			−0.147219			−0.0736097	+	0.997287i	$0.523452\pi$
$618$		0			0
$619$		−	17.4558i		−	0.701610i	−0.936449	−	0.350805i	$-0.885908\pi$
$619$		−	17.4558i		−	0.701610i	0.936449	−	0.350805i	$-0.114092\pi$
$620$		0			0
$621$	−18.7990	+	33.6274i	−0.754377	+	1.34942i
$622$		0			0
$623$			24.9706i			1.00042i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−2.34315	+	13.6569i	−0.0935762	+	0.545402i
$628$		0			0
$629$			2.62742i			0.104762i
$630$		0			0
$631$			3.31371i			0.131917i	0.997822	+	0.0659583i	$0.0210104\pi$
$631$			3.31371i			0.131917i	−0.997822	+	0.0659583i	$0.978990\pi$
$632$		0			0
$633$	3.51472	−	20.4853i	0.139698	−	0.814217i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	9.31371i		−	0.369023i
$638$		0			0
$639$	−3.31371	−	1.17157i	−0.131088	−	0.0463467i
$640$		0			0
$641$			5.31371i			0.209879i	0.994479	+	0.104939i	$0.0334649\pi$
$641$			5.31371i			0.209879i	−0.994479	+	0.104939i	$0.966535\pi$
$642$		0			0
$643$	16.1005			0.634942			0.317471	−	0.948268i	$-0.397166\pi$
$643$	16.1005			0.634942			0.317471	+	0.948268i	$0.397166\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	33.7574i		−	1.32714i	−0.748115	−	0.663569i	$-0.769041\pi$
$647$		−	33.7574i		−	1.32714i	0.748115	−	0.663569i	$-0.230959\pi$
$648$		0			0
$649$	4.68629			0.183953
$650$		0			0
$651$	5.65685	−	32.9706i	0.221710	−	1.29222i
$652$		0			0
$653$	32.3431			1.26569			0.632843	−	0.774281i	$-0.281888\pi$
$653$	32.3431			1.26569			0.632843	+	0.774281i	$0.281888\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−15.6569	+	44.2843i	−0.610832	+	1.72769i
$658$		0			0
$659$	22.3431			0.870365			0.435183	−	0.900342i	$-0.356684\pi$
$659$	22.3431			0.870365			0.435183	+	0.900342i	$0.356684\pi$
$660$		0			0
$661$	10.3431			0.402302			0.201151	−	0.979560i	$-0.435532\pi$
$661$	10.3431			0.402302			0.201151	+	0.979560i	$0.435532\pi$
$662$		0			0
$663$	−4.48528	+	26.1421i	−0.174194	+	1.01528i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	59.3137			2.29664
$668$		0			0
$669$	13.4853	+	2.31371i	0.521371	+	0.0894531i
$670$		0			0
$671$	16.0000			0.617673
$672$		0			0
$673$		−	24.3431i		−	0.938359i	−0.883103	−	0.469180i	$-0.844550\pi$
$673$		−	24.3431i		−	0.938359i	0.883103	−	0.469180i	$-0.155450\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.343146			0.0131882			0.00659408	−	0.999978i	$-0.497901\pi$
$677$	0.343146			0.0131882			0.00659408	+	0.999978i	$0.497901\pi$
$678$		0			0
$679$		−	64.7696i		−	2.48563i
$680$		0			0
$681$	2.17157	−	12.6569i	0.0832149	−	0.485012i
$682$		0			0
$683$		−	9.55635i		−	0.365664i	−0.983144	−	0.182832i	$-0.941474\pi$
$683$		−	9.55635i		−	0.365664i	0.983144	−	0.182832i	$-0.0585264\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	4.58579	+	0.786797i	0.174959	+	0.0300182i
$688$		0			0
$689$		−	7.31371i		−	0.278630i
$690$		0			0
$691$			40.2843i			1.53249i	0.642551	+	0.766243i	$0.277876\pi$
$691$			40.2843i			1.53249i	−0.642551	+	0.766243i	$0.722124\pi$
$692$		0			0
$693$	27.3137	+	9.65685i	1.03756	+	0.366834i
$694$		0			0
$695$		0			0
$696$		0			0
$697$			15.3137i			0.580048i
$698$		0			0
$699$	11.8995	+	2.04163i	0.450080	+	0.0772216i
$700$		0			0
$701$			15.9411i			0.602088i	0.953610	+	0.301044i	$0.0973351\pi$
$701$			15.9411i			0.602088i	−0.953610	+	0.301044i	$0.902665\pi$
$702$		0			0
$703$	0.970563			0.0366055
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	24.9706i		−	0.939115i
$708$		0			0
$709$	2.68629			0.100886			0.0504429	−	0.998727i	$-0.483937\pi$
$709$	2.68629			0.100886			0.0504429	+	0.998727i	$0.483937\pi$
$710$		0			0
$711$	4.48528	−	12.6863i	0.168211	−	0.475773i
$712$		0			0
$713$	41.9411			1.57071
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−4.00000	−	0.686292i	−0.149383	−	0.0256300i
$718$		0			0
$719$	−12.2843			−0.458126			−0.229063	−	0.973412i	$-0.573566\pi$
$719$	−12.2843			−0.458126			−0.229063	+	0.973412i	$0.573566\pi$
$720$		0			0
$721$	7.65685			0.285156
$722$		0			0
$723$	−16.4853	−	2.82843i	−0.613094	−	0.105190i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	28.1838			1.04528			0.522639	−	0.852554i	$-0.324948\pi$
$727$	28.1838			1.04528			0.522639	+	0.852554i	$0.324948\pi$
$728$		0			0
$729$	14.1421	+	23.0000i	0.523783	+	0.851852i
$730$		0			0
$731$	−60.4853			−2.23713
$732$		0			0
$733$			9.02944i			0.333510i	0.985998	+	0.166755i	$0.0533289\pi$
$733$			9.02944i			0.333510i	−0.985998	+	0.166755i	$0.946671\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.02944			−0.111591
$738$		0			0
$739$			15.5147i			0.570718i	0.958421	+	0.285359i	$0.0921129\pi$
$739$			15.5147i			0.570718i	−0.958421	+	0.285359i	$0.907887\pi$
$740$		0			0
$741$	9.65685	+	1.65685i	0.354753	+	0.0608661i
$742$		0			0
$743$		−	5.27208i		−	0.193414i	−0.995313	−	0.0967069i	$-0.969169\pi$
$743$		−	5.27208i		−	0.193414i	0.995313	−	0.0967069i	$-0.0308309\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.07107	−	14.3431i	0.185541	−	0.524788i
$748$		0			0
$749$		−	52.6274i		−	1.92296i
$750$		0			0
$751$			38.6274i			1.40953i	0.709439	+	0.704767i	$0.248949\pi$
$751$			38.6274i			1.40953i	−0.709439	+	0.704767i	$0.751051\pi$
$752$		0			0
$753$	−0.828427	−	0.142136i	−0.0301896	−	0.00517971i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	16.3431i		−	0.594002i	−0.954877	−	0.297001i	$-0.904014\pi$
$757$		−	16.3431i		−	0.594002i	0.954877	−	0.297001i	$-0.0959864\pi$
$758$		0			0
$759$	−6.14214	+	35.7990i	−0.222945	+	1.29942i
$760$		0			0
$761$		−	11.3137i		−	0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$	−60.2843			−2.18244
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	3.31371i		−	0.119651i
$768$		0			0
$769$	−25.3137			−0.912836			−0.456418	−	0.889766i	$-0.650868\pi$
$769$	−25.3137			−0.912836			−0.456418	+	0.889766i	$0.650868\pi$
$770$		0			0
$771$	37.5563	+	6.44365i	1.35256	+	0.232062i
$772$		0			0
$773$	−26.2843			−0.945380			−0.472690	−	0.881229i	$-0.656717\pi$
$773$	−26.2843			−0.945380			−0.472690	+	0.881229i	$0.656717\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	0.343146	−	2.00000i	0.0123103	−	0.0717496i
$778$		0			0
$779$	5.65685			0.202678
$780$		0			0
$781$	−3.31371			−0.118574
$782$		0			0
$783$	20.2843	−	36.2843i	0.724901	−	1.29669i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−21.5563			−0.768401			−0.384200	−	0.923250i	$-0.625523\pi$
$787$	−21.5563			−0.768401			−0.384200	+	0.923250i	$0.625523\pi$
$788$		0			0
$789$	3.48528	−	20.3137i	0.124079	−	0.723187i
$790$		0			0
$791$	34.1421			1.21395
$792$		0			0
$793$		−	11.3137i		−	0.401762i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−30.9706			−1.09703			−0.548517	−	0.836140i	$-0.684807\pi$
$797$	−30.9706			−1.09703			−0.548517	+	0.836140i	$0.684807\pi$
$798$		0			0
$799$		−	47.7990i		−	1.69101i
$800$		0			0
$801$	−7.31371	+	20.6863i	−0.258417	+	0.730914i
$802$		0			0
$803$			44.2843i			1.56276i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	5.27208	−	30.7279i	0.185586	−	1.08167i
$808$		0			0
$809$			26.6274i			0.936170i	0.883683	+	0.468085i	$0.155056\pi$
$809$			26.6274i			0.936170i	−0.883683	+	0.468085i	$0.844944\pi$
$810$		0			0
$811$		−	42.6274i		−	1.49685i	−0.663219	−	0.748426i	$-0.730810\pi$
$811$		−	42.6274i		−	1.49685i	0.663219	−	0.748426i	$-0.269190\pi$
$812$		0			0
$813$	0.686292	−	4.00000i	0.0240693	−	0.140286i
$814$		0			0
$815$		0			0
$816$		0			0
$817$			22.3431i			0.781688i
$818$		0			0
$819$	6.82843	−	19.3137i	0.238605	−	0.674876i
$820$		0			0
$821$			28.6274i			0.999104i	0.866284	+	0.499552i	$0.166502\pi$
$821$			28.6274i			0.999104i	−0.866284	+	0.499552i	$0.833498\pi$
$822$		0			0
$823$	−16.1005			−0.561228			−0.280614	−	0.959821i	$-0.590538\pi$
$823$	−16.1005			−0.561228			−0.280614	+	0.959821i	$0.590538\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			24.5858i			0.854932i	0.904032	+	0.427466i	$0.140594\pi$
$827$			24.5858i			0.854932i	−0.904032	+	0.427466i	$0.859406\pi$
$828$		0			0
$829$	12.9706			0.450486			0.225243	−	0.974303i	$-0.427682\pi$
$829$	12.9706			0.450486			0.225243	+	0.974303i	$0.427682\pi$
$830$		0			0
$831$	−3.41421	+	19.8995i	−0.118438	+	0.690306i
$832$		0			0
$833$	35.6569			1.23544
$834$		0			0
$835$		0			0
$836$		0			0
$837$	14.3431	−	25.6569i	0.495772	−	0.886831i
$838$		0			0
$839$	17.9411			0.619396			0.309698	−	0.950835i	$-0.399772\pi$
$839$	17.9411			0.619396			0.309698	+	0.950835i	$0.399772\pi$
$840$		0			0
$841$	−35.0000			−1.20690
$842$		0			0
$843$	−3.89949	+	22.7279i	−0.134306	+	0.782791i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−10.2426			−0.351941
$848$		0			0
$849$	−23.1421	−	3.97056i	−0.794236	−	0.136269i
$850$		0			0
$851$	2.54416			0.0872125
$852$		0			0
$853$			47.9411i			1.64147i	0.571307	+	0.820736i	$0.306437\pi$
$853$			47.9411i			1.64147i	−0.571307	+	0.820736i	$0.693563\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.0000			−0.341593			−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000			−0.341593			−0.170797	+	0.985306i	$0.554634\pi$
$858$		0			0
$859$		−	7.51472i		−	0.256399i	−0.991748	−	0.128199i	$-0.959080\pi$
$859$		−	7.51472i		−	0.256399i	0.991748	−	0.128199i	$-0.0409198\pi$
$860$		0			0
$861$	2.00000	−	11.6569i	0.0681598	−	0.397265i
$862$		0			0
$863$		−	17.5563i		−	0.597625i	−0.954312	−	0.298813i	$-0.903409\pi$
$863$		−	17.5563i		−	0.597625i	0.954312	−	0.298813i	$-0.0965905\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−71.0624	−	12.1924i	−2.41341	−	0.414075i
$868$		0			0
$869$		−	12.6863i		−	0.430353i
$870$		0			0
$871$			2.14214i			0.0725835i
$872$		0			0
$873$	18.9706	−	53.6569i	0.642056	−	1.81601i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			23.6569i			0.798835i	0.916769	+	0.399418i	$0.130788\pi$
$877$			23.6569i			0.798835i	−0.916769	+	0.399418i	$0.869212\pi$
$878$		0			0
$879$	37.5563	+	6.44365i	1.26674	+	0.217339i
$880$		0			0
$881$		−	28.6274i		−	0.964482i	−0.876039	−	0.482241i	$-0.839823\pi$
$881$		−	28.6274i		−	0.964482i	0.876039	−	0.482241i	$-0.160177\pi$
$882$		0			0
$883$	1.27208			0.0428088			0.0214044	−	0.999771i	$-0.493186\pi$
$883$	1.27208			0.0428088			0.0214044	+	0.999771i	$0.493186\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			11.8995i			0.399546i	0.979842	+	0.199773i	$0.0640205\pi$
$887$			11.8995i			0.399546i	−0.979842	+	0.199773i	$0.935980\pi$
$888$		0			0
$889$	−58.2843			−1.95479
$890$		0			0
$891$	19.7990	+	16.0000i	0.663291	+	0.536020i
$892$		0			0
$893$	−17.6569			−0.590864
$894$		0			0
$895$		0			0
$896$		0			0
$897$	25.3137	+	4.34315i	0.845200	+	0.145013i
$898$		0			0
$899$	−45.2548			−1.50933
$900$		0			0
$901$	28.0000			0.932815
$902$		0			0
$903$	46.0416	+	7.89949i	1.53217	+	0.262879i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.07107			−0.0355642			−0.0177821	−	0.999842i	$-0.505661\pi$
$907$	−1.07107			−0.0355642			−0.0177821	+	0.999842i	$0.505661\pi$
$908$		0			0
$909$	7.31371	−	20.6863i	0.242580	−	0.686121i
$910$		0			0
$911$	−11.1127			−0.368180			−0.184090	−	0.982909i	$-0.558934\pi$
$911$	−11.1127			−0.368180			−0.184090	+	0.982909i	$0.558934\pi$
$912$		0			0
$913$		−	14.3431i		−	0.474689i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	36.9706			1.22088
$918$		0			0
$919$			44.4853i			1.46743i	0.679455	+	0.733717i	$0.262216\pi$
$919$			44.4853i			1.46743i	−0.679455	+	0.733717i	$0.737784\pi$
$920$		0			0
$921$	13.4853	+	2.31371i	0.444355	+	0.0762393i
$922$		0			0
$923$			2.34315i			0.0771256i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	6.34315	+	2.24264i	0.208336	+	0.0736580i
$928$		0			0
$929$			46.0000i			1.50921i	0.656179	+	0.754606i	$0.272172\pi$
$929$			46.0000i			1.50921i	−0.656179	+	0.754606i	$0.727828\pi$
$930$		0			0
$931$		−	13.1716i		−	0.431681i
$932$		0			0
$933$	19.6569	+	3.37258i	0.643537	+	0.110413i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	37.5980i		−	1.22827i	−0.789200	−	0.614136i	$-0.789505\pi$
$937$		−	37.5980i		−	1.22827i	0.789200	−	0.614136i	$-0.210495\pi$
$938$		0			0
$939$	5.07107	−	29.5563i	0.165488	−	0.964535i
$940$		0			0
$941$			12.0000i			0.391189i	0.980685	+	0.195594i	$0.0626636\pi$
$941$			12.0000i			0.391189i	−0.980685	+	0.195594i	$0.937336\pi$
$942$		0			0
$943$	14.8284			0.482880
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	23.2132i		−	0.754328i	−0.926147	−	0.377164i	$-0.876899\pi$
$947$		−	23.2132i		−	0.754328i	0.926147	−	0.377164i	$-0.123101\pi$
$948$		0			0
$949$	31.3137			1.01649
$950$		0			0
$951$	0.585786	+	0.100505i	0.0189954	+	0.00325910i
$952$		0			0
$953$	−25.3137			−0.819991			−0.409996	−	0.912087i	$-0.634470\pi$
$953$	−25.3137			−0.819991			−0.409996	+	0.912087i	$0.634470\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	6.62742	−	38.6274i	0.214234	−	1.24865i
$958$		0			0
$959$	59.1127			1.90885
$960$		0			0
$961$	−1.00000			−0.0322581
$962$		0			0
$963$	15.4142	−	43.5980i	0.496716	−	1.40493i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	11.4142			0.367056			0.183528	−	0.983014i	$-0.441248\pi$
$967$	11.4142			0.367056			0.183528	+	0.983014i	$0.441248\pi$
$968$		0			0
$969$	−6.34315	+	36.9706i	−0.203771	+	1.18767i
$970$		0			0
$971$	−32.4853			−1.04250			−0.521251	−	0.853403i	$-0.674535\pi$
$971$	−32.4853			−1.04250			−0.521251	+	0.853403i	$0.674535\pi$
$972$		0			0
$973$			44.9706i			1.44169i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	21.0294			0.672791			0.336396	−	0.941721i	$-0.390792\pi$
$977$	21.0294			0.672791			0.336396	+	0.941721i	$0.390792\pi$
$978$		0			0
$979$			20.6863i			0.661137i
$980$		0			0
$981$	−49.9411	−	17.6569i	−1.59450	−	0.563740i
$982$		0			0
$983$			45.0711i			1.43754i	0.695246	+	0.718772i	$0.255295\pi$
$983$			45.0711i			1.43754i	−0.695246	+	0.718772i	$0.744705\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−6.24264	+	36.3848i	−0.198705	+	1.15814i
$988$		0			0
$989$			58.5685i			1.86237i
$990$		0			0
$991$		−	28.2843i		−	0.898479i	−0.893411	−	0.449240i	$-0.851695\pi$
$991$		−	28.2843i		−	0.898479i	0.893411	−	0.449240i	$-0.148305\pi$
$992$		0			0
$993$	−2.54416	+	14.8284i	−0.0807363	+	0.470566i
$994$		0			0
$995$		0			0
$996$		0			0
$997$			18.0000i			0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$			18.0000i			0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$		0			0
$999$	0.870058	−	1.55635i	0.0275274	−	0.0492407i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2400.2.o.c.2399.1		4
3.2	odd	2		2400.2.o.b.2399.2		4
4.3	odd	2		2400.2.o.j.2399.4		4
5.2	odd	4		480.2.h.b.191.4	yes	4
5.3	odd	4		2400.2.h.e.1151.1		4
5.4	even	2		2400.2.o.i.2399.4		4
12.11	even	2		2400.2.o.i.2399.3		4
15.2	even	4		480.2.h.d.191.2	yes	4
15.8	even	4		2400.2.h.b.1151.3		4
15.14	odd	2		2400.2.o.j.2399.3		4
20.3	even	4		2400.2.h.b.1151.4		4
20.7	even	4		480.2.h.d.191.1	yes	4
20.19	odd	2		2400.2.o.b.2399.1		4
40.27	even	4		960.2.h.b.191.4		4
40.37	odd	4		960.2.h.f.191.1		4
60.23	odd	4		2400.2.h.e.1151.2		4
60.47	odd	4		480.2.h.b.191.3	✓	4
60.59	even	2	inner	2400.2.o.c.2399.2		4
120.77	even	4		960.2.h.b.191.3		4
120.107	odd	4		960.2.h.f.191.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.h.b.191.3	✓	4	60.47	odd	4
480.2.h.b.191.4	yes	4	5.2	odd	4
480.2.h.d.191.1	yes	4	20.7	even	4
480.2.h.d.191.2	yes	4	15.2	even	4
960.2.h.b.191.3		4	120.77	even	4
960.2.h.b.191.4		4	40.27	even	4
960.2.h.f.191.1		4	40.37	odd	4
960.2.h.f.191.2		4	120.107	odd	4
2400.2.h.b.1151.3		4	15.8	even	4
2400.2.h.b.1151.4		4	20.3	even	4
2400.2.h.e.1151.1		4	5.3	odd	4
2400.2.h.e.1151.2		4	60.23	odd	4
2400.2.o.b.2399.1		4	20.19	odd	2
2400.2.o.b.2399.2		4	3.2	odd	2
2400.2.o.c.2399.1		4	1.1	even	1	trivial
2400.2.o.c.2399.2		4	60.59	even	2	inner
2400.2.o.i.2399.3		4	12.11	even	2
2400.2.o.i.2399.4		4	5.4	even	2
2400.2.o.j.2399.3		4	15.14	odd	2
2400.2.o.j.2399.4		4	4.3	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 - 0.292893i) q^{3} +3.41421 q^{7} +(2.82843 + 1.00000i) q^{9} +2.82843 q^{11} -2.00000i q^{13} +7.65685 q^{17} -2.82843i q^{19} +(-5.82843 - 1.00000i) q^{21} -7.41421i q^{23} +(-4.53553 - 2.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(-4.82843 - 0.828427i) q^{33} +0.343146i q^{37} +(-0.585786 + 3.41421i) q^{39} +2.00000i q^{41} -7.89949 q^{43} -6.24264i q^{47} +4.65685 q^{49} +(-13.0711 - 2.24264i) q^{51} +3.65685 q^{53} +(-0.828427 + 4.82843i) q^{57} +1.65685 q^{59} +5.65685 q^{61} +(9.65685 + 3.41421i) q^{63} -1.07107 q^{67} +(-2.17157 + 12.6569i) q^{69} -1.17157 q^{71} +15.6569i q^{73} +9.65685 q^{77} -4.48528i q^{79} +(7.00000 + 5.65685i) q^{81} -5.07107i q^{83} +(2.34315 - 13.6569i) q^{87} +7.31371i q^{89} -6.82843i q^{91} +(1.65685 - 9.65685i) q^{93} -18.9706i q^{97} +(8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{7} + 8 q^{17} - 12 q^{21} - 4 q^{27} - 8 q^{33} - 8 q^{39} + 8 q^{43} - 4 q^{49} - 24 q^{51} - 8 q^{53} + 8 q^{57} - 16 q^{59} + 16 q^{63} + 24 q^{67} - 20 q^{69} - 16 q^{71} + 16 q^{77}+ \cdots + 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^7 + 8 * q^17 - 12 * q^21 - 4 * q^27 - 8 * q^33 - 8 * q^39 + 8 * q^43 - 4 * q^49 - 24 * q^51 - 8 * q^53 + 8 * q^57 - 16 * q^59 + 16 * q^63 + 24 * q^67 - 20 * q^69 - 16 * q^71 + 16 * q^77 + 28 * q^81 + 32 * q^87 - 16 * q^93 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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