LMFDB - Embedded newform 2400.2.h.b.1151.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2400,2,Mod(1151,2400)] 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, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2400.1151"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2400 = 2^{5} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2400.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == 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			1151.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2400.1151
Dual form			2400.2.h.b.1151.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.292893 - 1.70711i) q^{3} -3.41421i q^{7} +(-2.82843 + 1.00000i) q^{9} -2.82843 q^{11} +2.00000 q^{13} +7.65685i q^{17} +2.82843i q^{19} +(-5.82843 + 1.00000i) q^{21} -7.41421 q^{23} +(2.53553 + 4.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(0.828427 + 4.82843i) q^{33} +0.343146 q^{37} +(-0.585786 - 3.41421i) q^{39} -2.00000i q^{41} -7.89949i q^{43} +6.24264 q^{47} -4.65685 q^{49} +(13.0711 - 2.24264i) q^{51} -3.65685i q^{53} +(4.82843 - 0.828427i) q^{57} +1.65685 q^{59} +5.65685 q^{61} +(3.41421 + 9.65685i) q^{63} +1.07107i q^{67} +(2.17157 + 12.6569i) q^{69} +1.17157 q^{71} -15.6569 q^{73} +9.65685i q^{77} +4.48528i q^{79} +(7.00000 - 5.65685i) q^{81} -5.07107 q^{83} +(13.6569 - 2.34315i) q^{87} +7.31371i q^{89} -6.82843i q^{91} +(9.65685 - 1.65685i) q^{93} -18.9706 q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 8 q^{13} - 12 q^{21} - 24 q^{23} - 4 q^{27} - 8 q^{33} + 24 q^{37} - 8 q^{39} + 8 q^{47} + 4 q^{49} + 24 q^{51} + 8 q^{57} - 16 q^{59} + 8 q^{63} + 20 q^{69} + 16 q^{71} - 40 q^{73} + 28 q^{81}+ \cdots + 32 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^13 - 12 * q^21 - 24 * q^23 - 4 * q^27 - 8 * q^33 + 24 * q^37 - 8 * q^39 + 8 * q^47 + 4 * q^49 + 24 * q^51 + 8 * q^57 - 16 * q^59 + 8 * q^63 + 20 * q^69 + 16 * q^71 - 40 * q^73 + 28 * q^81 + 8 * q^83 + 32 * q^87 + 16 * q^93 - 8 * q^97 + 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
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$3$	−0.292893	−	1.70711i	−0.169102	−	0.985599i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	3.41421i		−	1.29045i	−0.763992	−	0.645226i	$-0.776763\pi$
$7$		−	3.41421i		−	1.29045i	0.763992	−	0.645226i	$-0.223237\pi$
$8$		0			0
$9$	−2.82843	+	1.00000i	−0.942809	+	0.333333i
$10$		0			0
$11$	−2.82843			−0.852803			−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843			−0.852803			−0.426401	+	0.904534i	$0.640219\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			7.65685i			1.85706i	0.371257	+	0.928530i	$0.378927\pi$
$17$			7.65685i			1.85706i	−0.371257	+	0.928530i	$0.621073\pi$
$18$		0			0
$19$			2.82843i			0.648886i	0.945905	+	0.324443i	$0.105177\pi$
$19$			2.82843i			0.648886i	−0.945905	+	0.324443i	$0.894823\pi$
$20$		0			0
$21$	−5.82843	+	1.00000i	−1.27187	+	0.218218i
$22$		0			0
$23$	−7.41421			−1.54597			−0.772985	−	0.634424i	$-0.781237\pi$
$23$	−7.41421			−1.54597			−0.772985	+	0.634424i	$0.781237\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	2.53553	+	4.53553i	0.487964	+	0.872864i
$28$		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
$33$	0.828427	+	4.82843i	0.144211	+	0.840521i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.343146			0.0564128			0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146			0.0564128			0.0282064	+	0.999602i	$0.491020\pi$
$38$		0			0
$39$	−0.585786	−	3.41421i	−0.0938009	−	0.546712i
$40$		0			0
$41$		−	2.00000i		−	0.312348i	−0.987730	−	0.156174i	$-0.950084\pi$
$41$		−	2.00000i		−	0.312348i	0.987730	−	0.156174i	$-0.0499160\pi$
$42$		0			0
$43$		−	7.89949i		−	1.20466i	−0.798247	−	0.602331i	$-0.794239\pi$
$43$		−	7.89949i		−	1.20466i	0.798247	−	0.602331i	$-0.205761\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.24264			0.910583			0.455291	−	0.890343i	$-0.349535\pi$
$47$	6.24264			0.910583			0.455291	+	0.890343i	$0.349535\pi$
$48$		0			0
$49$	−4.65685			−0.665265
$50$		0			0
$51$	13.0711	−	2.24264i	1.83032	−	0.314033i
$52$		0			0
$53$		−	3.65685i		−	0.502308i	−0.967947	−	0.251154i	$-0.919190\pi$
$53$		−	3.65685i		−	0.502308i	0.967947	−	0.251154i	$-0.0808100\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.82843	−	0.828427i	0.639541	−	0.109728i
$58$		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
$63$	3.41421	+	9.65685i	0.430150	+	1.21665i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			1.07107i			0.130852i	0.997857	+	0.0654259i	$0.0208406\pi$
$67$			1.07107i			0.130852i	−0.997857	+	0.0654259i	$0.979159\pi$
$68$		0			0
$69$	2.17157	+	12.6569i	0.261427	+	1.52371i
$70$		0			0
$71$	1.17157			0.139040			0.0695201	−	0.997581i	$-0.477853\pi$
$71$	1.17157			0.139040			0.0695201	+	0.997581i	$0.477853\pi$
$72$		0			0
$73$	−15.6569			−1.83250			−0.916248	−	0.400611i	$-0.868798\pi$
$73$	−15.6569			−1.83250			−0.916248	+	0.400611i	$0.868798\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			9.65685i			1.10050i
$78$		0			0
$79$			4.48528i			0.504634i	0.967645	+	0.252317i	$0.0811925\pi$
$79$			4.48528i			0.504634i	−0.967645	+	0.252317i	$0.918807\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	−5.07107			−0.556622			−0.278311	−	0.960491i	$-0.589775\pi$
$83$	−5.07107			−0.556622			−0.278311	+	0.960491i	$0.589775\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	13.6569	−	2.34315i	1.46417	−	0.251212i
$88$		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
$93$	9.65685	−	1.65685i	1.00137	−	0.171808i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−18.9706			−1.92617			−0.963084	−	0.269200i	$-0.913241\pi$
$97$	−18.9706			−1.92617			−0.963084	+	0.269200i	$0.913241\pi$
$98$		0			0
$99$	8.00000	−	2.82843i	0.804030	−	0.284268i
$100$		0			0
$101$			7.31371i			0.727741i	0.931450	+	0.363871i	$0.118545\pi$
$101$			7.31371i			0.727741i	−0.931450	+	0.363871i	$0.881455\pi$
$102$		0			0
$103$			2.24264i			0.220974i	0.993878	+	0.110487i	$0.0352410\pi$
$103$			2.24264i			0.220974i	−0.993878	+	0.110487i	$0.964759\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	15.4142			1.49015			0.745074	−	0.666982i	$-0.232414\pi$
$107$	15.4142			1.49015			0.745074	+	0.666982i	$0.232414\pi$
$108$		0			0
$109$	17.6569			1.69122			0.845610	−	0.533801i	$-0.179237\pi$
$109$	17.6569			1.69122			0.845610	+	0.533801i	$0.179237\pi$
$110$		0			0
$111$	−0.100505	−	0.585786i	−0.00953952	−	0.0556004i
$112$		0			0
$113$		−	10.0000i		−	0.940721i	−0.882474	−	0.470360i	$-0.844124\pi$
$113$		−	10.0000i		−	0.940721i	0.882474	−	0.470360i	$-0.155876\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−5.65685	+	2.00000i	−0.522976	+	0.184900i
$118$		0			0
$119$	26.1421			2.39645
$120$		0			0
$121$	−3.00000			−0.272727
$122$		0			0
$123$	−3.41421	+	0.585786i	−0.307849	+	0.0528186i
$124$		0			0
$125$		0			0
$126$		0			0
$127$			17.0711i			1.51481i	0.652944	+	0.757406i	$0.273534\pi$
$127$			17.0711i			1.51481i	−0.652944	+	0.757406i	$0.726466\pi$
$128$		0			0
$129$	−13.4853	+	2.31371i	−1.18731	+	0.203711i
$130$		0			0
$131$	−10.8284			−0.946084			−0.473042	−	0.881040i	$-0.656844\pi$
$131$	−10.8284			−0.946084			−0.473042	+	0.881040i	$0.656844\pi$
$132$		0			0
$133$	9.65685			0.837355
$134$		0			0
$135$		0			0
$136$		0			0
$137$			17.3137i			1.47921i	0.673041	+	0.739605i	$0.264988\pi$
$137$			17.3137i			1.47921i	−0.673041	+	0.739605i	$0.735012\pi$
$138$		0			0
$139$		−	13.1716i		−	1.11720i	−0.829438	−	0.558599i	$-0.811339\pi$
$139$		−	13.1716i		−	1.11720i	0.829438	−	0.558599i	$-0.188661\pi$
$140$		0			0
$141$	−1.82843	−	10.6569i	−0.153981	−	0.897469i
$142$		0			0
$143$	−5.65685			−0.473050
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.36396	+	7.94975i	0.112498	+	0.655684i
$148$		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
$153$	−7.65685	−	21.6569i	−0.619020	−	1.75085i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	7.65685			0.611083			0.305542	−	0.952179i	$-0.401162\pi$
$157$	7.65685			0.611083			0.305542	+	0.952179i	$0.401162\pi$
$158$		0			0
$159$	−6.24264	+	1.07107i	−0.495074	+	0.0849412i
$160$		0			0
$161$			25.3137i			1.99500i
$162$		0			0
$163$			10.2426i			0.802266i	0.916020	+	0.401133i	$0.131383\pi$
$163$			10.2426i			0.802266i	−0.916020	+	0.401133i	$0.868617\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−18.7279			−1.44921			−0.724605	−	0.689164i	$-0.757978\pi$
$167$	−18.7279			−1.44921			−0.724605	+	0.689164i	$0.757978\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	−2.82843	−	8.00000i	−0.216295	−	0.611775i
$172$		0			0
$173$			2.00000i			0.152057i	0.997106	+	0.0760286i	$0.0242240\pi$
$173$			2.00000i			0.152057i	−0.997106	+	0.0760286i	$0.975776\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.485281	−	2.82843i	−0.0364760	−	0.212598i
$178$		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
$183$	−1.65685	−	9.65685i	−0.122478	−	0.713855i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	21.6569i		−	1.58371i
$188$		0			0
$189$	15.4853	−	8.65685i	1.12639	−	0.629693i
$190$		0			0
$191$	−18.1421			−1.31272			−0.656359	−	0.754448i	$-0.727904\pi$
$191$	−18.1421			−1.31272			−0.656359	+	0.754448i	$0.727904\pi$
$192$		0			0
$193$	7.65685			0.551152			0.275576	−	0.961279i	$-0.411131\pi$
$193$	7.65685			0.551152			0.275576	+	0.961279i	$0.411131\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.9706i			1.63658i	0.574802	+	0.818292i	$0.305079\pi$
$197$			22.9706i			1.63658i	−0.574802	+	0.818292i	$0.694921\pi$
$198$		0			0
$199$			22.8284i			1.61826i	0.587627	+	0.809132i	$0.300062\pi$
$199$			22.8284i			1.61826i	−0.587627	+	0.809132i	$0.699938\pi$
$200$		0			0
$201$	1.82843	−	0.313708i	0.128967	−	0.0221273i
$202$		0			0
$203$	27.3137			1.91705
$204$		0			0
$205$		0			0
$206$		0			0
$207$	20.9706	−	7.41421i	1.45755	−	0.515323i
$208$		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
$213$	−0.343146	−	2.00000i	−0.0235120	−	0.137038i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	19.3137			1.31110
$218$		0			0
$219$	4.58579	+	26.7279i	0.309879	+	1.80611i
$220$		0			0
$221$			15.3137i			1.03011i
$222$		0			0
$223$		−	7.89949i		−	0.528989i	−0.964387	−	0.264495i	$-0.914795\pi$
$223$		−	7.89949i		−	0.528989i	0.964387	−	0.264495i	$-0.0852052\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.41421			−0.492099			−0.246049	−	0.969257i	$-0.579133\pi$
$227$	−7.41421			−0.492099			−0.246049	+	0.969257i	$0.579133\pi$
$228$		0			0
$229$	2.68629			0.177515			0.0887576	−	0.996053i	$-0.471710\pi$
$229$	2.68629			0.177515			0.0887576	+	0.996053i	$0.471710\pi$
$230$		0			0
$231$	16.4853	−	2.82843i	1.08465	−	0.186097i
$232$		0			0
$233$			6.97056i			0.456657i	0.973584	+	0.228328i	$0.0733260\pi$
$233$			6.97056i			0.456657i	−0.973584	+	0.228328i	$0.926674\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	7.65685	−	1.31371i	0.497366	−	0.0853345i
$238$		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
$243$	−11.7071	−	10.2929i	−0.751011	−	0.660289i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			5.65685i			0.359937i
$248$		0			0
$249$	1.48528	+	8.65685i	0.0941259	+	0.548606i
$250$		0			0
$251$	−0.485281			−0.0306307			−0.0153153	−	0.999883i	$-0.504875\pi$
$251$	−0.485281			−0.0306307			−0.0153153	+	0.999883i	$0.504875\pi$
$252$		0			0
$253$	20.9706			1.31841
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	22.0000i		−	1.37232i	−0.727450	−	0.686161i	$-0.759294\pi$
$257$		−	22.0000i		−	1.37232i	0.727450	−	0.686161i	$-0.240706\pi$
$258$		0			0
$259$		−	1.17157i		−	0.0727980i
$260$		0			0
$261$	−8.00000	−	22.6274i	−0.495188	−	1.40060i
$262$		0			0
$263$	11.8995			0.733754			0.366877	−	0.930269i	$-0.380427\pi$
$263$	11.8995			0.733754			0.366877	+	0.930269i	$0.380427\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	12.4853	−	2.14214i	0.764087	−	0.131097i
$268$		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
$273$	−11.6569	+	2.00000i	−0.705505	+	0.121046i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−11.6569			−0.700392			−0.350196	−	0.936676i	$-0.613885\pi$
$277$	−11.6569			−0.700392			−0.350196	+	0.936676i	$0.613885\pi$
$278$		0			0
$279$	−5.65685	−	16.0000i	−0.338667	−	0.957895i
$280$		0			0
$281$			13.3137i			0.794229i	0.917769	+	0.397115i	$0.129988\pi$
$281$			13.3137i			0.794229i	−0.917769	+	0.397115i	$0.870012\pi$
$282$		0			0
$283$			13.5563i			0.805841i	0.915235	+	0.402921i	$0.132005\pi$
$283$			13.5563i			0.805841i	−0.915235	+	0.402921i	$0.867995\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.82843			−0.403069
$288$		0			0
$289$	−41.6274			−2.44867
$290$		0			0
$291$	5.55635	+	32.3848i	0.325719	+	1.89843i
$292$		0			0
$293$			22.0000i			1.28525i	0.766179	+	0.642627i	$0.222155\pi$
$293$			22.0000i			1.28525i	−0.766179	+	0.642627i	$0.777845\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−7.17157	−	12.8284i	−0.416137	−	0.744381i
$298$		0			0
$299$	−14.8284			−0.857550
$300$		0			0
$301$	−26.9706			−1.55456
$302$		0			0
$303$	12.4853	−	2.14214i	0.717261	−	0.123062i
$304$		0			0
$305$		0			0
$306$		0			0
$307$			7.89949i			0.450848i	0.974261	+	0.225424i	$0.0723767\pi$
$307$			7.89949i			0.450848i	−0.974261	+	0.225424i	$0.927623\pi$
$308$		0			0
$309$	3.82843	−	0.656854i	0.217792	−	0.0373671i
$310$		0			0
$311$	11.5147			0.652940			0.326470	−	0.945208i	$-0.394141\pi$
$311$	11.5147			0.652940			0.326470	+	0.945208i	$0.394141\pi$
$312$		0			0
$313$	−17.3137			−0.978629			−0.489314	−	0.872107i	$-0.662753\pi$
$313$	−17.3137			−0.978629			−0.489314	+	0.872107i	$0.662753\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	0.343146i		−	0.0192730i	−0.999954	−	0.00963649i	$-0.996933\pi$
$317$		−	0.343146i		−	0.0192730i	0.999954	−	0.00963649i	$-0.00306744\pi$
$318$		0			0
$319$		−	22.6274i		−	1.26689i
$320$		0			0
$321$	−4.51472	−	26.3137i	−0.251987	−	1.46869i
$322$		0			0
$323$	−21.6569			−1.20502
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−5.17157	−	30.1421i	−0.285989	−	1.66686i
$328$		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
$333$	−0.970563	+	0.343146i	−0.0531865	+	0.0188043i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	9.31371			0.507350			0.253675	−	0.967290i	$-0.418361\pi$
$337$	9.31371			0.507350			0.253675	+	0.967290i	$0.418361\pi$
$338$		0			0
$339$	−17.0711	+	2.92893i	−0.927173	+	0.159078i
$340$		0			0
$341$		−	16.0000i		−	0.866449i
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−27.6985			−1.48693			−0.743466	−	0.668773i	$-0.766820\pi$
$347$	−27.6985			−1.48693			−0.743466	+	0.668773i	$0.766820\pi$
$348$		0			0
$349$	16.6274			0.890045			0.445023	−	0.895519i	$-0.353196\pi$
$349$	16.6274			0.890045			0.445023	+	0.895519i	$0.353196\pi$
$350$		0			0
$351$	5.07107	+	9.07107i	0.270674	+	0.484178i
$352$		0			0
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−7.65685	−	44.6274i	−0.405244	−	2.36193i
$358$		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
$363$	0.878680	+	5.12132i	0.0461187	+	0.268800i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			14.9289i			0.779284i	0.920966	+	0.389642i	$0.127401\pi$
$367$			14.9289i			0.779284i	−0.920966	+	0.389642i	$0.872599\pi$
$368$		0			0
$369$	2.00000	+	5.65685i	0.104116	+	0.294484i
$370$		0			0
$371$	−12.4853			−0.648204
$372$		0			0
$373$	19.6569			1.01779			0.508897	−	0.860828i	$-0.330054\pi$
$373$	19.6569			1.01779			0.508897	+	0.860828i	$0.330054\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			16.0000i			0.824042i
$378$		0			0
$379$		−	14.1421i		−	0.726433i	−0.931705	−	0.363216i	$-0.881679\pi$
$379$		−	14.1421i		−	0.726433i	0.931705	−	0.363216i	$-0.118321\pi$
$380$		0			0
$381$	29.1421	−	5.00000i	1.49300	−	0.256158i
$382$		0			0
$383$	−0.585786			−0.0299323			−0.0149661	−	0.999888i	$-0.504764\pi$
$383$	−0.585786			−0.0299323			−0.0149661	+	0.999888i	$0.504764\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.89949	+	22.3431i	0.401554	+	1.13577i
$388$		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
$393$	3.17157	+	18.4853i	0.159985	+	0.932459i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	20.6274			1.03526			0.517630	−	0.855604i	$-0.326814\pi$
$397$	20.6274			1.03526			0.517630	+	0.855604i	$0.326814\pi$
$398$		0			0
$399$	−2.82843	−	16.4853i	−0.141598	−	0.825296i
$400$		0			0
$401$		−	8.00000i		−	0.399501i	−0.979847	−	0.199750i	$-0.935987\pi$
$401$		−	8.00000i		−	0.399501i	0.979847	−	0.199750i	$-0.0640132\pi$
$402$		0			0
$403$			11.3137i			0.563576i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.970563			−0.0481090
$408$		0			0
$409$	5.65685			0.279713			0.139857	−	0.990172i	$-0.455336\pi$
$409$	5.65685			0.279713			0.139857	+	0.990172i	$0.455336\pi$
$410$		0			0
$411$	29.5563	−	5.07107i	1.45791	−	0.250137i
$412$		0			0
$413$		−	5.65685i		−	0.278356i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−22.4853	+	3.85786i	−1.10111	+	0.188920i
$418$		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
$423$	−17.6569	+	6.24264i	−0.858506	+	0.303528i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		−	19.3137i		−	0.934656i
$428$		0			0
$429$	1.65685	+	9.65685i	0.0799937	+	0.466237i
$430$		0			0
$431$	−11.5147			−0.554644			−0.277322	−	0.960777i	$-0.589447\pi$
$431$	−11.5147			−0.554644			−0.277322	+	0.960777i	$0.589447\pi$
$432$		0			0
$433$	−15.6569			−0.752420			−0.376210	−	0.926534i	$-0.622773\pi$
$433$	−15.6569			−0.752420			−0.376210	+	0.926534i	$0.622773\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	20.9706i		−	1.00316i
$438$		0			0
$439$			17.1716i			0.819554i	0.912186	+	0.409777i	$0.134394\pi$
$439$			17.1716i			0.819554i	−0.912186	+	0.409777i	$0.865606\pi$
$440$		0			0
$441$	13.1716	−	4.65685i	0.627218	−	0.221755i
$442$		0			0
$443$	−35.8995			−1.70564			−0.852818	−	0.522208i	$-0.825109\pi$
$443$	−35.8995			−1.70564			−0.852818	+	0.522208i	$0.825109\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−22.7279	+	3.89949i	−1.07499	+	0.184440i
$448$		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
$457$	5.31371			0.248565			0.124282	−	0.992247i	$-0.460337\pi$
$457$	5.31371			0.248565			0.124282	+	0.992247i	$0.460337\pi$
$458$		0			0
$459$	−34.7279	+	19.4142i	−1.62096	+	0.906178i
$460$		0			0
$461$			26.6274i			1.24016i	0.784538	+	0.620081i	$0.212900\pi$
$461$			26.6274i			1.24016i	−0.784538	+	0.620081i	$0.787100\pi$
$462$		0			0
$463$		−	20.5858i		−	0.956703i	−0.878169	−	0.478351i	$-0.841235\pi$
$463$		−	20.5858i		−	0.956703i	0.878169	−	0.478351i	$-0.158765\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−35.8995			−1.66123			−0.830615	−	0.556847i	$-0.812011\pi$
$467$	−35.8995			−1.66123			−0.830615	+	0.556847i	$0.812011\pi$
$468$		0			0
$469$	3.65685			0.168858
$470$		0			0
$471$	−2.24264	−	13.0711i	−0.103335	−	0.602283i
$472$		0			0
$473$			22.3431i			1.02734i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	3.65685	+	10.3431i	0.167436	+	0.473580i
$478$		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
$483$	43.2132	−	7.41421i	1.96627	−	0.337358i
$484$		0			0
$485$		0			0
$486$		0			0
$487$			8.10051i			0.367069i	0.983013	+	0.183534i	$0.0587539\pi$
$487$			8.10051i			0.367069i	−0.983013	+	0.183534i	$0.941246\pi$
$488$		0			0
$489$	17.4853	−	3.00000i	0.790712	−	0.135665i
$490$		0			0
$491$	−39.1127			−1.76513			−0.882566	−	0.470189i	$-0.844186\pi$
$491$	−39.1127			−1.76513			−0.882566	+	0.470189i	$0.844186\pi$
$492$		0			0
$493$	−61.2548			−2.75878
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	4.00000i		−	0.179425i
$498$		0			0
$499$		−	33.4558i		−	1.49769i	−0.662746	−	0.748845i	$-0.730609\pi$
$499$		−	33.4558i		−	1.49769i	0.662746	−	0.748845i	$-0.269391\pi$
$500$		0			0
$501$	5.48528	+	31.9706i	0.245064	+	1.42834i
$502$		0			0
$503$	0.786797			0.0350815			0.0175408	−	0.999846i	$-0.494416\pi$
$503$	0.786797			0.0350815			0.0175408	+	0.999846i	$0.494416\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.63604	+	15.3640i	0.117071	+	0.682337i
$508$		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
$513$	−12.8284	+	7.17157i	−0.566389	+	0.316633i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−17.6569			−0.776548
$518$		0			0
$519$	3.41421	−	0.585786i	0.149867	−	0.0257132i
$520$		0			0
$521$		−	16.0000i		−	0.700973i	−0.936568	−	0.350486i	$-0.886016\pi$
$521$		−	16.0000i		−	0.700973i	0.936568	−	0.350486i	$-0.113984\pi$
$522$		0			0
$523$			11.4142i			0.499109i	0.968361	+	0.249554i	$0.0802841\pi$
$523$			11.4142i			0.499109i	−0.968361	+	0.249554i	$0.919716\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−43.3137			−1.88677
$528$		0			0
$529$	31.9706			1.39002
$530$		0			0
$531$	−4.68629	+	1.65685i	−0.203368	+	0.0719014i
$532$		0			0
$533$		−	4.00000i		−	0.173259i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−5.17157	−	30.1421i	−0.223170	−	1.30073i
$538$		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
$543$	2.72792	+	15.8995i	0.117066	+	0.682313i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	18.0416i		−	0.771404i	−0.922623	−	0.385702i	$-0.873959\pi$
$547$		−	18.0416i		−	0.771404i	0.922623	−	0.385702i	$-0.126041\pi$
$548$		0			0
$549$	−16.0000	+	5.65685i	−0.682863	+	0.241429i
$550$		0			0
$551$	−22.6274			−0.963960
$552$		0			0
$553$	15.3137			0.651205
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$		−	15.7990i		−	0.668226i
$560$		0			0
$561$	−36.9706	+	6.34315i	−1.56090	+	0.267808i
$562$		0			0
$563$	25.5563			1.07707			0.538536	−	0.842603i	$-0.318978\pi$
$563$	25.5563			1.07707			0.538536	+	0.842603i	$0.318978\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−19.3137	−	23.8995i	−0.811100	−	1.00368i
$568$		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
$573$	5.31371	+	30.9706i	0.221983	+	1.29381i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−22.0000			−0.915872			−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000			−0.915872			−0.457936	+	0.888985i	$0.651411\pi$
$578$		0			0
$579$	−2.24264	−	13.0711i	−0.0932010	−	0.543215i
$580$		0			0
$581$			17.3137i			0.718294i
$582$		0			0
$583$			10.3431i			0.428369i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.7279			−0.772984			−0.386492	−	0.922293i	$-0.626313\pi$
$587$	−18.7279			−0.772984			−0.386492	+	0.922293i	$0.626313\pi$
$588$		0			0
$589$	−16.0000			−0.659269
$590$		0			0
$591$	39.2132	−	6.72792i	1.61302	−	0.276750i
$592$		0			0
$593$			32.6274i			1.33985i	0.742430	+	0.669924i	$0.233673\pi$
$593$			32.6274i			1.33985i	−0.742430	+	0.669924i	$0.766327\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	38.9706	−	6.68629i	1.59496	−	0.273652i
$598$		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
$603$	−1.07107	−	3.02944i	−0.0436173	−	0.123368i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			41.0711i			1.66702i	0.552502	+	0.833512i	$0.313673\pi$
$607$			41.0711i			1.66702i	−0.552502	+	0.833512i	$0.686327\pi$
$608$		0			0
$609$	−8.00000	−	46.6274i	−0.324176	−	1.88944i
$610$		0			0
$611$	12.4853			0.505100
$612$		0			0
$613$	37.3137			1.50709			0.753543	−	0.657398i	$-0.228343\pi$
$613$	37.3137			1.50709			0.753543	+	0.657398i	$0.228343\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	3.65685i		−	0.147219i	−0.997287	−	0.0736097i	$-0.976548\pi$
$617$		−	3.65685i		−	0.147219i	0.997287	−	0.0736097i	$-0.0234519\pi$
$618$		0			0
$619$			17.4558i			0.701610i	0.936449	+	0.350805i	$0.114092\pi$
$619$			17.4558i			0.701610i	−0.936449	+	0.350805i	$0.885908\pi$
$620$		0			0
$621$	−18.7990	−	33.6274i	−0.754377	−	1.34942i
$622$		0			0
$623$	24.9706			1.00042
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−13.6569	+	2.34315i	−0.545402	+	0.0935762i
$628$		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
$633$	20.4853	−	3.51472i	0.814217	−	0.139698i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−9.31371			−0.369023
$638$		0			0
$639$	−3.31371	+	1.17157i	−0.131088	+	0.0463467i
$640$		0			0
$641$		−	5.31371i		−	0.209879i	−0.994479	−	0.104939i	$-0.966535\pi$
$641$		−	5.31371i		−	0.209879i	0.994479	−	0.104939i	$-0.0334649\pi$
$642$		0			0
$643$			16.1005i			0.634942i	0.948268	+	0.317471i	$0.102834\pi$
$643$			16.1005i			0.634942i	−0.948268	+	0.317471i	$0.897166\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.7574			1.32714			0.663569	−	0.748115i	$-0.269041\pi$
$647$	33.7574			1.32714			0.663569	+	0.748115i	$0.269041\pi$
$648$		0			0
$649$	−4.68629			−0.183953
$650$		0			0
$651$	−5.65685	−	32.9706i	−0.221710	−	1.29222i
$652$		0			0
$653$		−	32.3431i		−	1.26569i	−0.774281	−	0.632843i	$-0.781888\pi$
$653$		−	32.3431i		−	1.26569i	0.774281	−	0.632843i	$-0.218112\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	44.2843	−	15.6569i	1.72769	−	0.610832i
$658$		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
$663$	26.1421	−	4.48528i	1.01528	−	0.174194i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	59.3137i		−	2.29664i
$668$		0			0
$669$	−13.4853	+	2.31371i	−0.521371	+	0.0894531i
$670$		0			0
$671$	−16.0000			−0.617673
$672$		0			0
$673$	24.3431			0.938359			0.469180	−	0.883103i	$-0.344550\pi$
$673$	24.3431			0.938359			0.469180	+	0.883103i	$0.344550\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			0.343146i			0.0131882i	0.999978	+	0.00659408i	$0.00209898\pi$
$677$			0.343146i			0.0131882i	−0.999978	+	0.00659408i	$0.997901\pi$
$678$		0			0
$679$			64.7696i			2.48563i
$680$		0			0
$681$	2.17157	+	12.6569i	0.0832149	+	0.485012i
$682$		0			0
$683$	−9.55635			−0.365664			−0.182832	−	0.983144i	$-0.558526\pi$
$683$	−9.55635			−0.365664			−0.182832	+	0.983144i	$0.558526\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−0.786797	−	4.58579i	−0.0300182	−	0.174959i
$688$		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
$693$	−9.65685	−	27.3137i	−0.366834	−	1.03756i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	15.3137			0.580048
$698$		0			0
$699$	11.8995	−	2.04163i	0.450080	−	0.0772216i
$700$		0			0
$701$		−	15.9411i		−	0.602088i	−0.953610	−	0.301044i	$-0.902665\pi$
$701$		−	15.9411i		−	0.602088i	0.953610	−	0.301044i	$-0.0973351\pi$
$702$		0			0
$703$			0.970563i			0.0366055i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	24.9706			0.939115
$708$		0			0
$709$	−2.68629			−0.100886			−0.0504429	−	0.998727i	$-0.516063\pi$
$709$	−2.68629			−0.100886			−0.0504429	+	0.998727i	$0.516063\pi$
$710$		0			0
$711$	−4.48528	−	12.6863i	−0.168211	−	0.475773i
$712$		0			0
$713$		−	41.9411i		−	1.57071i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.686292	−	4.00000i	−0.0256300	−	0.149383i
$718$		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
$723$	−2.82843	−	16.4853i	−0.105190	−	0.613094i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	28.1838i		−	1.04528i	−0.852554	−	0.522639i	$-0.824948\pi$
$727$		−	28.1838i		−	1.04528i	0.852554	−	0.522639i	$-0.175052\pi$
$728$		0			0
$729$	−14.1421	+	23.0000i	−0.523783	+	0.851852i
$730$		0			0
$731$	60.4853			2.23713
$732$		0			0
$733$	−9.02944			−0.333510			−0.166755	−	0.985998i	$-0.553329\pi$
$733$	−9.02944			−0.333510			−0.166755	+	0.985998i	$0.553329\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	3.02944i		−	0.111591i
$738$		0			0
$739$		−	15.5147i		−	0.570718i	−0.958421	−	0.285359i	$-0.907887\pi$
$739$		−	15.5147i		−	0.570718i	0.958421	−	0.285359i	$-0.0921129\pi$
$740$		0			0
$741$	9.65685	−	1.65685i	0.354753	−	0.0608661i
$742$		0			0
$743$	−5.27208			−0.193414			−0.0967069	−	0.995313i	$-0.530831\pi$
$743$	−5.27208			−0.193414			−0.0967069	+	0.995313i	$0.530831\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	14.3431	−	5.07107i	0.524788	−	0.185541i
$748$		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
$753$	0.142136	+	0.828427i	0.00517971	+	0.0301896i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−16.3431			−0.594002			−0.297001	−	0.954877i	$-0.595986\pi$
$757$	−16.3431			−0.594002			−0.297001	+	0.954877i	$0.595986\pi$
$758$		0			0
$759$	−6.14214	−	35.7990i	−0.222945	−	1.29942i
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		−	60.2843i		−	2.18244i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.31371			0.119651
$768$		0			0
$769$	25.3137			0.912836			0.456418	−	0.889766i	$-0.349132\pi$
$769$	25.3137			0.912836			0.456418	+	0.889766i	$0.349132\pi$
$770$		0			0
$771$	−37.5563	+	6.44365i	−1.35256	+	0.232062i
$772$		0			0
$773$			26.2843i			0.945380i	0.881229	+	0.472690i	$0.156717\pi$
$773$			26.2843i			0.945380i	−0.881229	+	0.472690i	$0.843283\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−2.00000	+	0.343146i	−0.0717496	+	0.0123103i
$778$		0			0
$779$	5.65685			0.202678
$780$		0			0
$781$	−3.31371			−0.118574
$782$		0			0
$783$	−36.2843	+	20.2843i	−1.29669	+	0.724901i
$784$		0			0
$785$		0			0
$786$		0			0
$787$			21.5563i			0.768401i	0.923250	+	0.384200i	$0.125523\pi$
$787$			21.5563i			0.768401i	−0.923250	+	0.384200i	$0.874477\pi$
$788$		0			0
$789$	−3.48528	−	20.3137i	−0.124079	−	0.723187i
$790$		0			0
$791$	−34.1421			−1.21395
$792$		0			0
$793$	11.3137			0.401762
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	30.9706i		−	1.09703i	−0.836140	−	0.548517i	$-0.815193\pi$
$797$		−	30.9706i		−	1.09703i	0.836140	−	0.548517i	$-0.184807\pi$
$798$		0			0
$799$			47.7990i			1.69101i
$800$		0			0
$801$	−7.31371	−	20.6863i	−0.258417	−	0.730914i
$802$		0			0
$803$	44.2843			1.56276
$804$		0			0
$805$		0			0
$806$		0			0
$807$	30.7279	−	5.27208i	1.08167	−	0.185586i
$808$		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
$813$	4.00000	−	0.686292i	0.140286	−	0.0240693i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	22.3431			0.781688
$818$		0			0
$819$	6.82843	+	19.3137i	0.238605	+	0.674876i
$820$		0			0
$821$		−	28.6274i		−	0.999104i	−0.866284	−	0.499552i	$-0.833498\pi$
$821$		−	28.6274i		−	0.999104i	0.866284	−	0.499552i	$-0.166502\pi$
$822$		0			0
$823$		−	16.1005i		−	0.561228i	−0.959821	−	0.280614i	$-0.909462\pi$
$823$		−	16.1005i		−	0.561228i	0.959821	−	0.280614i	$-0.0905381\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.5858			−0.854932			−0.427466	−	0.904032i	$-0.640594\pi$
$827$	−24.5858			−0.854932			−0.427466	+	0.904032i	$0.640594\pi$
$828$		0			0
$829$	−12.9706			−0.450486			−0.225243	−	0.974303i	$-0.572318\pi$
$829$	−12.9706			−0.450486			−0.225243	+	0.974303i	$0.572318\pi$
$830$		0			0
$831$	3.41421	+	19.8995i	0.118438	+	0.690306i
$832$		0			0
$833$		−	35.6569i		−	1.23544i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−25.6569	+	14.3431i	−0.886831	+	0.495772i
$838$		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
$843$	22.7279	−	3.89949i	0.782791	−	0.134306i
$844$		0			0
$845$		0			0
$846$		0			0
$847$			10.2426i			0.351941i
$848$		0			0
$849$	23.1421	−	3.97056i	0.794236	−	0.136269i
$850$		0			0
$851$	−2.54416			−0.0872125
$852$		0			0
$853$	−47.9411			−1.64147			−0.820736	−	0.571307i	$-0.806437\pi$
$853$	−47.9411			−1.64147			−0.820736	+	0.571307i	$0.806437\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	10.0000i		−	0.341593i	−0.985306	−	0.170797i	$-0.945366\pi$
$857$		−	10.0000i		−	0.341593i	0.985306	−	0.170797i	$-0.0546341\pi$
$858$		0			0
$859$			7.51472i			0.256399i	0.991748	+	0.128199i	$0.0409198\pi$
$859$			7.51472i			0.256399i	−0.991748	+	0.128199i	$0.959080\pi$
$860$		0			0
$861$	2.00000	+	11.6569i	0.0681598	+	0.397265i
$862$		0			0
$863$	−17.5563			−0.597625			−0.298813	−	0.954312i	$-0.596591\pi$
$863$	−17.5563			−0.597625			−0.298813	+	0.954312i	$0.596591\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.1924	+	71.0624i	0.414075	+	2.41341i
$868$		0			0
$869$		−	12.6863i		−	0.430353i
$870$		0			0
$871$			2.14214i			0.0725835i
$872$		0			0
$873$	53.6569	−	18.9706i	1.81601	−	0.642056i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	23.6569			0.798835			0.399418	−	0.916769i	$-0.369212\pi$
$877$	23.6569			0.798835			0.399418	+	0.916769i	$0.369212\pi$
$878$		0			0
$879$	37.5563	−	6.44365i	1.26674	−	0.217339i
$880$		0			0
$881$			28.6274i			0.964482i	0.876039	+	0.482241i	$0.160177\pi$
$881$			28.6274i			0.964482i	−0.876039	+	0.482241i	$0.839823\pi$
$882$		0			0
$883$			1.27208i			0.0428088i	0.999771	+	0.0214044i	$0.00681376\pi$
$883$			1.27208i			0.0428088i	−0.999771	+	0.0214044i	$0.993186\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−11.8995			−0.399546			−0.199773	−	0.979842i	$-0.564020\pi$
$887$	−11.8995			−0.399546			−0.199773	+	0.979842i	$0.564020\pi$
$888$		0			0
$889$	58.2843			1.95479
$890$		0			0
$891$	−19.7990	+	16.0000i	−0.663291	+	0.536020i
$892$		0			0
$893$			17.6569i			0.590864i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	4.34315	+	25.3137i	0.145013	+	0.845200i
$898$		0			0
$899$	−45.2548			−1.50933
$900$		0			0
$901$	28.0000			0.932815
$902$		0			0
$903$	7.89949	+	46.0416i	0.262879	+	1.53217i
$904$		0			0
$905$		0			0
$906$		0			0
$907$			1.07107i			0.0355642i	0.999842	+	0.0177821i	$0.00566052\pi$
$907$			1.07107i			0.0355642i	−0.999842	+	0.0177821i	$0.994339\pi$
$908$		0			0
$909$	−7.31371	−	20.6863i	−0.242580	−	0.686121i
$910$		0			0
$911$	11.1127			0.368180			0.184090	−	0.982909i	$-0.441066\pi$
$911$	11.1127			0.368180			0.184090	+	0.982909i	$0.441066\pi$
$912$		0			0
$913$	14.3431			0.474689
$914$		0			0
$915$		0			0
$916$		0			0
$917$			36.9706i			1.22088i
$918$		0			0
$919$		−	44.4853i		−	1.46743i	−0.679455	−	0.733717i	$-0.737784\pi$
$919$		−	44.4853i		−	1.46743i	0.679455	−	0.733717i	$-0.262216\pi$
$920$		0			0
$921$	13.4853	−	2.31371i	0.444355	−	0.0762393i
$922$		0			0
$923$	2.34315			0.0771256
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−2.24264	−	6.34315i	−0.0736580	−	0.208336i
$928$		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
$933$	−3.37258	−	19.6569i	−0.110413	−	0.643537i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−37.5980			−1.22827			−0.614136	−	0.789200i	$-0.710495\pi$
$937$	−37.5980			−1.22827			−0.614136	+	0.789200i	$0.710495\pi$
$938$		0			0
$939$	5.07107	+	29.5563i	0.165488	+	0.964535i
$940$		0			0
$941$		−	12.0000i		−	0.391189i	−0.980685	−	0.195594i	$-0.937336\pi$
$941$		−	12.0000i		−	0.391189i	0.980685	−	0.195594i	$-0.0626636\pi$
$942$		0			0
$943$			14.8284i			0.482880i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23.2132			0.754328			0.377164	−	0.926147i	$-0.376899\pi$
$947$	23.2132			0.754328			0.377164	+	0.926147i	$0.376899\pi$
$948$		0			0
$949$	−31.3137			−1.01649
$950$		0			0
$951$	−0.585786	+	0.100505i	−0.0189954	+	0.00325910i
$952$		0			0
$953$			25.3137i			0.819991i	0.912087	+	0.409996i	$0.134470\pi$
$953$			25.3137i			0.819991i	−0.912087	+	0.409996i	$0.865530\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−38.6274	+	6.62742i	−1.24865	+	0.214234i
$958$		0			0
$959$	59.1127			1.90885
$960$		0			0
$961$	−1.00000			−0.0322581
$962$		0			0
$963$	−43.5980	+	15.4142i	−1.40493	+	0.496716i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	11.4142i		−	0.367056i	−0.983014	−	0.183528i	$-0.941248\pi$
$967$		−	11.4142i		−	0.367056i	0.983014	−	0.183528i	$-0.0587519\pi$
$968$		0			0
$969$	6.34315	+	36.9706i	0.203771	+	1.18767i
$970$		0			0
$971$	32.4853			1.04250			0.521251	−	0.853403i	$-0.325465\pi$
$971$	32.4853			1.04250			0.521251	+	0.853403i	$0.325465\pi$
$972$		0			0
$973$	−44.9706			−1.44169
$974$		0			0
$975$		0			0
$976$		0			0
$977$			21.0294i			0.672791i	0.941721	+	0.336396i	$0.109208\pi$
$977$			21.0294i			0.672791i	−0.941721	+	0.336396i	$0.890792\pi$
$978$		0			0
$979$		−	20.6863i		−	0.661137i
$980$		0			0
$981$	−49.9411	+	17.6569i	−1.59450	+	0.563740i
$982$		0			0
$983$	45.0711			1.43754			0.718772	−	0.695246i	$-0.244705\pi$
$983$	45.0711			1.43754			0.718772	+	0.695246i	$0.244705\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−36.3848	+	6.24264i	−1.15814	+	0.198705i
$988$		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
$993$	−14.8284	+	2.54416i	−0.470566	+	0.0807363i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	18.0000			0.570066			0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000			0.570066			0.285033	+	0.958518i	$0.407995\pi$
$998$		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.h.b.1151.3		4
3.2	odd	2		2400.2.h.e.1151.1		4
4.3	odd	2		2400.2.h.e.1151.2		4
5.2	odd	4		2400.2.o.b.2399.2		4
5.3	odd	4		2400.2.o.j.2399.3		4
5.4	even	2		480.2.h.d.191.2	yes	4
12.11	even	2	inner	2400.2.h.b.1151.4		4
15.2	even	4		2400.2.o.c.2399.1		4
15.8	even	4		2400.2.o.i.2399.4		4
15.14	odd	2		480.2.h.b.191.4	yes	4
20.3	even	4		2400.2.o.c.2399.2		4
20.7	even	4		2400.2.o.i.2399.3		4
20.19	odd	2		480.2.h.b.191.3	✓	4
40.19	odd	2		960.2.h.f.191.2		4
40.29	even	2		960.2.h.b.191.3		4
60.23	odd	4		2400.2.o.b.2399.1		4
60.47	odd	4		2400.2.o.j.2399.4		4
60.59	even	2		480.2.h.d.191.1	yes	4
120.29	odd	2		960.2.h.f.191.1		4
120.59	even	2		960.2.h.b.191.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.2.h.b.191.3	✓	4	20.19	odd	2
480.2.h.b.191.4	yes	4	15.14	odd	2
480.2.h.d.191.1	yes	4	60.59	even	2
480.2.h.d.191.2	yes	4	5.4	even	2
960.2.h.b.191.3		4	40.29	even	2
960.2.h.b.191.4		4	120.59	even	2
960.2.h.f.191.1		4	120.29	odd	2
960.2.h.f.191.2		4	40.19	odd	2
2400.2.h.b.1151.3		4	1.1	even	1	trivial
2400.2.h.b.1151.4		4	12.11	even	2	inner
2400.2.h.e.1151.1		4	3.2	odd	2
2400.2.h.e.1151.2		4	4.3	odd	2
2400.2.o.b.2399.1		4	60.23	odd	4
2400.2.o.b.2399.2		4	5.2	odd	4
2400.2.o.c.2399.1		4	15.2	even	4
2400.2.o.c.2399.2		4	20.3	even	4
2400.2.o.i.2399.3		4	20.7	even	4
2400.2.o.i.2399.4		4	15.8	even	4
2400.2.o.j.2399.3		4	5.3	odd	4
2400.2.o.j.2399.4		4	60.47	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:	\( N \)	\(=\)	\( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2400.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.292893 - 1.70711i) q^{3} -3.41421i q^{7} +(-2.82843 + 1.00000i) q^{9} -2.82843 q^{11} +2.00000 q^{13} +7.65685i q^{17} +2.82843i q^{19} +(-5.82843 + 1.00000i) q^{21} -7.41421 q^{23} +(2.53553 + 4.53553i) q^{27} +8.00000i q^{29} +5.65685i q^{31} +(0.828427 + 4.82843i) q^{33} +0.343146 q^{37} +(-0.585786 - 3.41421i) q^{39} -2.00000i q^{41} -7.89949i q^{43} +6.24264 q^{47} -4.65685 q^{49} +(13.0711 - 2.24264i) q^{51} -3.65685i q^{53} +(4.82843 - 0.828427i) q^{57} +1.65685 q^{59} +5.65685 q^{61} +(3.41421 + 9.65685i) q^{63} +1.07107i q^{67} +(2.17157 + 12.6569i) q^{69} +1.17157 q^{71} -15.6569 q^{73} +9.65685i q^{77} +4.48528i q^{79} +(7.00000 - 5.65685i) q^{81} -5.07107 q^{83} +(13.6569 - 2.34315i) q^{87} +7.31371i q^{89} -6.82843i q^{91} +(9.65685 - 1.65685i) q^{93} -18.9706 q^{97} +(8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 8 q^{13} - 12 q^{21} - 24 q^{23} - 4 q^{27} - 8 q^{33} + 24 q^{37} - 8 q^{39} + 8 q^{47} + 4 q^{49} + 24 q^{51} + 8 q^{57} - 16 q^{59} + 8 q^{63} + 20 q^{69} + 16 q^{71} - 40 q^{73} + 28 q^{81}+ \cdots + 32 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^13 - 12 * q^21 - 24 * q^23 - 4 * q^27 - 8 * q^33 + 24 * q^37 - 8 * q^39 + 8 * q^47 + 4 * q^49 + 24 * q^51 + 8 * q^57 - 16 * q^59 + 8 * q^63 + 20 * q^69 + 16 * q^71 - 40 * q^73 + 28 * q^81 + 8 * q^83 + 32 * q^87 + 16 * q^93 - 8 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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