LMFDB - Embedded newform 1824.2.k.b.607.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1824,2,Mod(607,1824)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1824, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 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("1824.607"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1824 = 2^{5} \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1824.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$14.5647133287$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 52 x^{18} + 986 x^{16} + 8824 x^{14} + 42757 x^{12} + 118964 x^{10} + 190576 x^{8} + 166880 x^{6} + \cdots + 784 $ x^20 + 52x^18 + 986x^16 + 8824x^14 + 42757x^12 + 118964x^10 + 190576x^8 + 166880x^6 + 70964x^4 + 12912*x^2 + 784
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{27} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			607.6
Root			$-1.45345i$ of defining polynomial
Character	$\chi$	$=$	1824.607
Dual form			1824.2.k.b.607.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.69641 q^{5} +3.75190i q^{7} +1.00000 q^{9} +1.14256i q^{11} -3.38630i q^{13} -1.69641 q^{15} +4.59679 q^{17} +(3.11731 - 3.04670i) q^{19} +3.75190i q^{21} +5.01159i q^{23} -2.12219 q^{25} +1.00000 q^{27} +1.50164i q^{29} -2.60304 q^{31} +1.14256i q^{33} -6.36477i q^{35} +1.50425i q^{37} -3.38630i q^{39} +9.19750i q^{41} +7.26752i q^{43} -1.69641 q^{45} +9.56310i q^{47} -7.07677 q^{49} +4.59679 q^{51} +9.19750i q^{53} -1.93826i q^{55} +(3.11731 - 3.04670i) q^{57} -4.71047 q^{59} -13.0242 q^{61} +3.75190i q^{63} +5.74456i q^{65} +13.5100 q^{67} +5.01159i q^{69} +5.90096 q^{71} -13.7286 q^{73} -2.12219 q^{75} -4.28679 q^{77} -11.3085 q^{79} +1.00000 q^{81} +5.07385i q^{83} -7.79805 q^{85} +1.50164i q^{87} +4.42244i q^{89} +12.7051 q^{91} -2.60304 q^{93} +(-5.28825 + 5.16845i) q^{95} -4.74665i q^{97} +1.14256i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} - 8 q^{31} - 44 q^{49} + 4 q^{57} - 8 q^{61} + 8 q^{67} + 64 q^{71} + 8 q^{73} + 20 q^{75} + 40 q^{77} - 16 q^{79} + 20 q^{81} + 40 q^{85} - 8 q^{93}+ \cdots - 8 q^{95}+O(q^{100}) $ 20 * q + 20 * q^3 + 20 * q^9 + 4 * q^19 + 20 * q^25 + 20 * q^27 - 8 * q^31 - 44 * q^49 + 4 * q^57 - 8 * q^61 + 8 * q^67 + 64 * q^71 + 8 * q^73 + 20 * q^75 + 40 * q^77 - 16 * q^79 + 20 * q^81 + 40 * q^85 - 8 * q^93 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1824\mathbb{Z}\right)^\times$.

$n$	$97$	$229$	$799$	$1217$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$	−1.69641			−0.758658			−0.379329	−	0.925262i	$-0.623845\pi$
$5$	−1.69641			−0.758658			−0.379329	+	0.925262i	$0.623845\pi$
$6$		0			0
$7$			3.75190i			1.41809i	0.705165	+	0.709043i	$0.250873\pi$
$7$			3.75190i			1.41809i	−0.705165	+	0.709043i	$0.749127\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			1.14256i			0.344496i	0.985054	+	0.172248i	$0.0551031\pi$
$11$			1.14256i			0.344496i	−0.985054	+	0.172248i	$0.944897\pi$
$12$		0			0
$13$		−	3.38630i		−	0.939190i	−0.882882	−	0.469595i	$-0.844400\pi$
$13$		−	3.38630i		−	0.939190i	0.882882	−	0.469595i	$-0.155600\pi$
$14$		0			0
$15$	−1.69641			−0.438012
$16$		0			0
$17$	4.59679			1.11489			0.557443	−	0.830215i	$-0.311783\pi$
$17$	4.59679			1.11489			0.557443	+	0.830215i	$0.311783\pi$
$18$		0			0
$19$	3.11731	−	3.04670i	0.715161	−	0.698960i
$19$	3.11731	−	3.04670i	0.715161	−	0.698960i
$20$		0			0
$21$			3.75190i			0.818732i
$22$		0			0
$23$			5.01159i			1.04499i	0.852643	+	0.522494i	$0.174998\pi$
$23$			5.01159i			1.04499i	−0.852643	+	0.522494i	$0.825002\pi$
$24$		0			0
$25$	−2.12219			−0.424438
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			1.50164i			0.278848i	0.990233	+	0.139424i	$0.0445251\pi$
$29$			1.50164i			0.278848i	−0.990233	+	0.139424i	$0.955475\pi$
$30$		0			0
$31$	−2.60304			−0.467520			−0.233760	−	0.972294i	$-0.575103\pi$
$31$	−2.60304			−0.467520			−0.233760	+	0.972294i	$0.575103\pi$
$32$		0			0
$33$			1.14256i			0.198895i
$34$		0			0
$35$		−	6.36477i		−	1.07584i
$36$		0			0
$37$			1.50425i			0.247297i	0.992326	+	0.123649i	$0.0394596\pi$
$37$			1.50425i			0.247297i	−0.992326	+	0.123649i	$0.960540\pi$
$38$		0			0
$39$		−	3.38630i		−	0.542242i
$40$		0			0
$41$			9.19750i			1.43641i	0.695833	+	0.718204i	$0.255035\pi$
$41$			9.19750i			1.43641i	−0.695833	+	0.718204i	$0.744965\pi$
$42$		0			0
$43$			7.26752i			1.10829i	0.832421	+	0.554144i	$0.186954\pi$
$43$			7.26752i			1.10829i	−0.832421	+	0.554144i	$0.813046\pi$
$44$		0			0
$45$	−1.69641			−0.252886
$46$		0			0
$47$			9.56310i			1.39492i	0.716623	+	0.697461i	$0.245687\pi$
$47$			9.56310i			1.39492i	−0.716623	+	0.697461i	$0.754313\pi$
$48$		0			0
$49$	−7.07677			−1.01097
$50$		0			0
$51$	4.59679			0.643679
$52$		0			0
$53$			9.19750i			1.26337i	0.775224	+	0.631687i	$0.217637\pi$
$53$			9.19750i			1.26337i	−0.775224	+	0.631687i	$0.782363\pi$
$54$		0			0
$55$		−	1.93826i		−	0.261355i
$56$		0			0
$57$	3.11731	−	3.04670i	0.412898	−	0.403545i
$58$		0			0
$59$	−4.71047			−0.613252			−0.306626	−	0.951830i	$-0.599200\pi$
$59$	−4.71047			−0.613252			−0.306626	+	0.951830i	$0.599200\pi$
$60$		0			0
$61$	−13.0242			−1.66757			−0.833787	−	0.552086i	$-0.813832\pi$
$61$	−13.0242			−1.66757			−0.833787	+	0.552086i	$0.813832\pi$
$62$		0			0
$63$			3.75190i			0.472695i
$64$		0			0
$65$			5.74456i			0.712525i
$66$		0			0
$67$	13.5100			1.65051			0.825253	−	0.564763i	$-0.191032\pi$
$67$	13.5100			1.65051			0.825253	+	0.564763i	$0.191032\pi$
$68$		0			0
$69$			5.01159i			0.603324i
$70$		0			0
$71$	5.90096			0.700315			0.350158	−	0.936691i	$-0.386128\pi$
$71$	5.90096			0.700315			0.350158	+	0.936691i	$0.386128\pi$
$72$		0			0
$73$	−13.7286			−1.60681			−0.803407	−	0.595430i	$-0.796982\pi$
$73$	−13.7286			−1.60681			−0.803407	+	0.595430i	$0.796982\pi$
$74$		0			0
$75$	−2.12219			−0.245049
$76$		0			0
$77$	−4.28679			−0.488525
$78$		0			0
$79$	−11.3085			−1.27231			−0.636154	−	0.771562i	$-0.719476\pi$
$79$	−11.3085			−1.27231			−0.636154	+	0.771562i	$0.719476\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			5.07385i			0.556928i	0.960447	+	0.278464i	$0.0898252\pi$
$83$			5.07385i			0.556928i	−0.960447	+	0.278464i	$0.910175\pi$
$84$		0			0
$85$	−7.79805			−0.845817
$86$		0			0
$87$			1.50164i			0.160993i
$88$		0			0
$89$			4.42244i			0.468778i	0.972143	+	0.234389i	$0.0753090\pi$
$89$			4.42244i			0.468778i	−0.972143	+	0.234389i	$0.924691\pi$
$90$		0			0
$91$	12.7051			1.33185
$92$		0			0
$93$	−2.60304			−0.269923
$94$		0			0
$95$	−5.28825	+	5.16845i	−0.542563	+	0.530272i
$96$		0			0
$97$		−	4.74665i		−	0.481949i	−0.970531	−	0.240975i	$-0.922533\pi$
$97$		−	4.74665i		−	0.481949i	0.970531	−	0.240975i	$-0.0774670\pi$
$98$		0			0
$99$			1.14256i			0.114832i
$100$		0			0
$101$	6.74733			0.671384			0.335692	−	0.941972i	$-0.391030\pi$
$101$	6.74733			0.671384			0.335692	+	0.941972i	$0.391030\pi$
$102$		0			0
$103$	12.0610			1.18841			0.594205	−	0.804314i	$-0.297467\pi$
$103$	12.0610			1.18841			0.594205	+	0.804314i	$0.297467\pi$
$104$		0			0
$105$		−	6.36477i		−	0.621138i
$106$		0			0
$107$	15.9929			1.54609			0.773047	−	0.634349i	$-0.218732\pi$
$107$	15.9929			1.54609			0.773047	+	0.634349i	$0.218732\pi$
$108$		0			0
$109$			2.41446i			0.231263i	0.993292	+	0.115632i	$0.0368892\pi$
$109$			2.41446i			0.231263i	−0.993292	+	0.115632i	$0.963111\pi$
$110$		0			0
$111$			1.50425i			0.142777i
$112$		0			0
$113$		−	10.9278i		−	1.02800i	−0.857790	−	0.514001i	$-0.828163\pi$
$113$		−	10.9278i		−	1.02800i	0.857790	−	0.514001i	$-0.171837\pi$
$114$		0			0
$115$		−	8.50171i		−	0.792789i
$116$		0			0
$117$		−	3.38630i		−	0.313063i
$118$		0			0
$119$			17.2467i			1.58100i
$120$		0			0
$121$	9.69455			0.881322
$122$		0			0
$123$			9.19750i			0.829310i
$124$		0			0
$125$	12.0822			1.08066
$126$		0			0
$127$	9.92662			0.880845			0.440422	−	0.897791i	$-0.354829\pi$
$127$	9.92662			0.880845			0.440422	+	0.897791i	$0.354829\pi$
$128$		0			0
$129$			7.26752i			0.639870i
$130$		0			0
$131$		−	4.67124i		−	0.408128i	−0.978958	−	0.204064i	$-0.934585\pi$
$131$		−	4.67124i		−	0.408128i	0.978958	−	0.204064i	$-0.0654151\pi$
$132$		0			0
$133$	11.4309	+	11.6959i	0.991185	+	1.01416i
$134$		0			0
$135$	−1.69641			−0.146004
$136$		0			0
$137$	−16.0194			−1.36863			−0.684317	−	0.729185i	$-0.739900\pi$
$137$	−16.0194			−1.36863			−0.684317	+	0.729185i	$0.739900\pi$
$138$		0			0
$139$		−	11.8297i		−	1.00339i	−0.865046	−	0.501693i	$-0.832711\pi$
$139$		−	11.8297i		−	1.00339i	0.865046	−	0.501693i	$-0.167289\pi$
$140$		0			0
$141$			9.56310i			0.805358i
$142$		0			0
$143$	3.86906			0.323547
$144$		0			0
$145$		−	2.54740i		−	0.211550i
$146$		0			0
$147$	−7.07677			−0.583682
$148$		0			0
$149$	−8.91978			−0.730737			−0.365369	−	0.930863i	$-0.619057\pi$
$149$	−8.91978			−0.730737			−0.365369	+	0.930863i	$0.619057\pi$
$150$		0			0
$151$	13.2770			1.08047			0.540233	−	0.841516i	$-0.318336\pi$
$151$	13.2770			1.08047			0.540233	+	0.841516i	$0.318336\pi$
$152$		0			0
$153$	4.59679			0.371628
$154$		0			0
$155$	4.41583			0.354688
$156$		0			0
$157$	−4.39158			−0.350486			−0.175243	−	0.984525i	$-0.556071\pi$
$157$	−4.39158			−0.350486			−0.175243	+	0.984525i	$0.556071\pi$
$158$		0			0
$159$			9.19750i			0.729409i
$160$		0			0
$161$	−18.8030			−1.48188
$162$		0			0
$163$		−	12.9407i		−	1.01360i	−0.862065	−	0.506798i	$-0.830829\pi$
$163$		−	12.9407i		−	1.01360i	0.862065	−	0.506798i	$-0.169171\pi$
$164$		0			0
$165$		−	1.93826i		−	0.150893i
$166$		0			0
$167$	−16.6261			−1.28657			−0.643283	−	0.765628i	$-0.722428\pi$
$167$	−16.6261			−1.28657			−0.643283	+	0.765628i	$0.722428\pi$
$168$		0			0
$169$	1.53298			0.117922
$170$		0			0
$171$	3.11731	−	3.04670i	0.238387	−	0.232987i
$172$		0			0
$173$		−	7.63053i		−	0.580138i	−0.957006	−	0.290069i	$-0.906322\pi$
$173$		−	7.63053i		−	0.580138i	0.957006	−	0.290069i	$-0.0936783\pi$
$174$		0			0
$175$		−	7.96224i		−	0.601889i
$176$		0			0
$177$	−4.71047			−0.354061
$178$		0			0
$179$	−1.31105			−0.0979925			−0.0489962	−	0.998799i	$-0.515602\pi$
$179$	−1.31105			−0.0979925			−0.0489962	+	0.998799i	$0.515602\pi$
$180$		0			0
$181$		−	12.8711i		−	0.956700i	−0.878169	−	0.478350i	$-0.841235\pi$
$181$		−	12.8711i		−	0.956700i	0.878169	−	0.478350i	$-0.158765\pi$
$182$		0			0
$183$	−13.0242			−0.962774
$184$		0			0
$185$		−	2.55183i		−	0.187614i
$186$		0			0
$187$			5.25213i			0.384074i
$188$		0			0
$189$			3.75190i			0.272911i
$190$		0			0
$191$		−	2.99551i		−	0.216747i	−0.994110	−	0.108374i	$-0.965436\pi$
$191$		−	2.99551i		−	0.216747i	0.994110	−	0.108374i	$-0.0345643\pi$
$192$		0			0
$193$		−	3.76959i		−	0.271341i	−0.990754	−	0.135671i	$-0.956681\pi$
$193$		−	3.76959i		−	0.271341i	0.990754	−	0.135671i	$-0.0433189\pi$
$194$		0			0
$195$			5.74456i			0.411376i
$196$		0			0
$197$	13.6614			0.973336			0.486668	−	0.873587i	$-0.338212\pi$
$197$	13.6614			0.973336			0.486668	+	0.873587i	$0.338212\pi$
$198$		0			0
$199$			3.75712i			0.266335i	0.991094	+	0.133168i	$0.0425148\pi$
$199$			3.75712i			0.266335i	−0.991094	+	0.133168i	$0.957485\pi$
$200$		0			0
$201$	13.5100			0.952920
$202$		0			0
$203$	−5.63402			−0.395431
$204$		0			0
$205$		−	15.6027i		−	1.08974i
$206$		0			0
$207$			5.01159i			0.348329i
$208$		0			0
$209$	3.48105	+	3.56173i	0.240789	+	0.246370i
$210$		0			0
$211$	17.0469			1.17356			0.586779	−	0.809747i	$-0.300396\pi$
$211$	17.0469			1.17356			0.586779	+	0.809747i	$0.300396\pi$
$212$		0			0
$213$	5.90096			0.404327
$214$		0			0
$215$		−	12.3287i		−	0.840811i
$216$		0			0
$217$		−	9.76635i		−	0.662983i
$218$		0			0
$219$	−13.7286			−0.927695
$220$		0			0
$221$		−	15.5661i		−	1.04709i
$222$		0			0
$223$	−10.5808			−0.708542			−0.354271	−	0.935143i	$-0.615271\pi$
$223$	−10.5808			−0.708542			−0.354271	+	0.935143i	$0.615271\pi$
$224$		0			0
$225$	−2.12219			−0.141479
$226$		0			0
$227$	−9.97902			−0.662331			−0.331165	−	0.943573i	$-0.607442\pi$
$227$	−9.97902			−0.662331			−0.331165	+	0.943573i	$0.607442\pi$
$228$		0			0
$229$	−11.9053			−0.786724			−0.393362	−	0.919384i	$-0.628688\pi$
$229$	−11.9053			−0.786724			−0.393362	+	0.919384i	$0.628688\pi$
$230$		0			0
$231$	−4.28679			−0.282050
$232$		0			0
$233$	5.86245			0.384062			0.192031	−	0.981389i	$-0.438493\pi$
$233$	5.86245			0.384062			0.192031	+	0.981389i	$0.438493\pi$
$234$		0			0
$235$		−	16.2230i		−	1.05827i
$236$		0			0
$237$	−11.3085			−0.734567
$238$		0			0
$239$		−	15.6175i		−	1.01021i	−0.863057	−	0.505107i	$-0.831453\pi$
$239$		−	15.6175i		−	1.01021i	0.863057	−	0.505107i	$-0.168547\pi$
$240$		0			0
$241$			19.2781i			1.24181i	0.783886	+	0.620905i	$0.213235\pi$
$241$			19.2781i			1.24181i	−0.783886	+	0.620905i	$0.786765\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	12.0051			0.766979
$246$		0			0
$247$	−10.3170	−	10.5562i	−0.656457	−	0.671672i
$248$		0			0
$249$			5.07385i			0.321542i
$250$		0			0
$251$		−	0.633320i		−	0.0399748i	−0.999800	−	0.0199874i	$-0.993637\pi$
$251$		−	0.633320i		−	0.0399748i	0.999800	−	0.0199874i	$-0.00636261\pi$
$252$		0			0
$253$	−5.72606			−0.359994
$254$		0			0
$255$	−7.79805			−0.488333
$256$		0			0
$257$			27.6282i			1.72340i	0.507417	+	0.861701i	$0.330600\pi$
$257$			27.6282i			1.72340i	−0.507417	+	0.861701i	$0.669400\pi$
$258$		0			0
$259$	−5.64381			−0.350689
$260$		0			0
$261$			1.50164i			0.0929494i
$262$		0			0
$263$			5.27755i			0.325428i	0.986673	+	0.162714i	$0.0520247\pi$
$263$			5.27755i			0.325428i	−0.986673	+	0.162714i	$0.947975\pi$
$264$		0			0
$265$		−	15.6027i		−	0.958469i
$266$		0			0
$267$			4.42244i			0.270649i
$268$		0			0
$269$		−	2.43012i		−	0.148167i	−0.997252	−	0.0740834i	$-0.976397\pi$
$269$		−	2.43012i		−	0.148167i	0.997252	−	0.0740834i	$-0.0236031\pi$
$270$		0			0
$271$			18.1691i			1.10369i	0.833945	+	0.551847i	$0.186077\pi$
$271$			18.1691i			1.10369i	−0.833945	+	0.551847i	$0.813923\pi$
$272$		0			0
$273$	12.7051			0.768945
$274$		0			0
$275$		−	2.42474i		−	0.146217i
$276$		0			0
$277$	22.6189			1.35904			0.679519	−	0.733658i	$-0.262188\pi$
$277$	22.6189			1.35904			0.679519	+	0.733658i	$0.262188\pi$
$278$		0			0
$279$	−2.60304			−0.155840
$280$		0			0
$281$			20.2656i			1.20894i	0.796626	+	0.604472i	$0.206616\pi$
$281$			20.2656i			1.20894i	−0.796626	+	0.604472i	$0.793384\pi$
$282$		0			0
$283$		−	13.6187i		−	0.809546i	−0.914417	−	0.404773i	$-0.867351\pi$
$283$		−	13.6187i		−	0.809546i	0.914417	−	0.404773i	$-0.132649\pi$
$284$		0			0
$285$	−5.28825	+	5.16845i	−0.313249	+	0.306153i
$286$		0			0
$287$	−34.5081			−2.03695
$288$		0			0
$289$	4.13048			0.242969
$290$		0			0
$291$		−	4.74665i		−	0.278253i
$292$		0			0
$293$			4.61940i			0.269868i	0.990855	+	0.134934i	$0.0430822\pi$
$293$			4.61940i			0.269868i	−0.990855	+	0.134934i	$0.956918\pi$
$294$		0			0
$295$	7.99090			0.465248
$296$		0			0
$297$			1.14256i			0.0662983i
$298$		0			0
$299$	16.9707			0.981443
$300$		0			0
$301$	−27.2670			−1.57165
$302$		0			0
$303$	6.74733			0.387624
$304$		0			0
$305$	22.0944			1.26512
$306$		0			0
$307$	−3.37396			−0.192562			−0.0962811	−	0.995354i	$-0.530695\pi$
$307$	−3.37396			−0.192562			−0.0962811	+	0.995354i	$0.530695\pi$
$308$		0			0
$309$	12.0610			0.686128
$310$		0			0
$311$			27.0077i			1.53147i	0.643158	+	0.765734i	$0.277624\pi$
$311$			27.0077i			1.53147i	−0.643158	+	0.765734i	$0.722376\pi$
$312$		0			0
$313$	−4.86622			−0.275055			−0.137527	−	0.990498i	$-0.543916\pi$
$313$	−4.86622			−0.275055			−0.137527	+	0.990498i	$0.543916\pi$
$314$		0			0
$315$		−	6.36477i		−	0.358614i
$316$		0			0
$317$		−	27.1413i		−	1.52441i	−0.647336	−	0.762205i	$-0.724117\pi$
$317$		−	27.1413i		−	1.52441i	0.647336	−	0.762205i	$-0.275883\pi$
$318$		0			0
$319$	−1.71572			−0.0960621
$320$		0			0
$321$	15.9929			0.892638
$322$		0			0
$323$	14.3296	−	14.0050i	0.797322	−	0.779260i
$324$		0			0
$325$			7.18636i			0.398628i
$326$		0			0
$327$			2.41446i			0.133520i
$328$		0			0
$329$	−35.8798			−1.97812
$330$		0			0
$331$	24.3832			1.34022			0.670111	−	0.742261i	$-0.266246\pi$
$331$	24.3832			1.34022			0.670111	+	0.742261i	$0.266246\pi$
$332$		0			0
$333$			1.50425i			0.0824325i
$334$		0			0
$335$	−22.9185			−1.25217
$336$		0			0
$337$			6.27799i			0.341984i	0.985272	+	0.170992i	$0.0546972\pi$
$337$			6.27799i			0.341984i	−0.985272	+	0.170992i	$0.945303\pi$
$338$		0			0
$339$		−	10.9278i		−	0.593517i
$340$		0			0
$341$		−	2.97414i		−	0.161059i
$342$		0			0
$343$		−	0.288034i		−	0.0155524i
$344$		0			0
$345$		−	8.50171i		−	0.457717i
$346$		0			0
$347$			33.0018i			1.77163i	0.464038	+	0.885816i	$0.346400\pi$
$347$			33.0018i			1.77163i	−0.464038	+	0.885816i	$0.653600\pi$
$348$		0			0
$349$	−7.40263			−0.396254			−0.198127	−	0.980176i	$-0.563486\pi$
$349$	−7.40263			−0.396254			−0.198127	+	0.980176i	$0.563486\pi$
$350$		0			0
$351$		−	3.38630i		−	0.180747i
$352$		0			0
$353$	23.6333			1.25787			0.628937	−	0.777457i	$-0.283491\pi$
$353$	23.6333			1.25787			0.628937	+	0.777457i	$0.283491\pi$
$354$		0			0
$355$	−10.0105			−0.531300
$356$		0			0
$357$			17.2467i			0.912793i
$358$		0			0
$359$		−	17.3533i		−	0.915875i	−0.888984	−	0.457937i	$-0.848588\pi$
$359$		−	17.3533i		−	0.915875i	0.888984	−	0.457937i	$-0.151412\pi$
$360$		0			0
$361$	0.435277	−	18.9950i	0.0229093	−	0.999738i
$362$		0			0
$363$	9.69455			0.508832
$364$		0			0
$365$	23.2894			1.21902
$366$		0			0
$367$		−	14.0084i		−	0.731233i	−0.930766	−	0.365617i	$-0.880858\pi$
$367$		−	14.0084i		−	0.731233i	0.930766	−	0.365617i	$-0.119142\pi$
$368$		0			0
$369$			9.19750i			0.478803i
$370$		0			0
$371$	−34.5081			−1.79157
$372$		0			0
$373$		−	18.1502i		−	0.939784i	−0.882724	−	0.469892i	$-0.844293\pi$
$373$		−	18.1502i		−	0.939784i	0.882724	−	0.469892i	$-0.155707\pi$
$374$		0			0
$375$	12.0822			0.623920
$376$		0			0
$377$	5.08501			0.261891
$378$		0			0
$379$	−31.5940			−1.62288			−0.811438	−	0.584439i	$-0.801315\pi$
$379$	−31.5940			−1.62288			−0.811438	+	0.584439i	$0.801315\pi$
$380$		0			0
$381$	9.92662			0.508556
$382$		0			0
$383$	10.9392			0.558969			0.279485	−	0.960150i	$-0.409836\pi$
$383$	10.9392			0.558969			0.279485	+	0.960150i	$0.409836\pi$
$384$		0			0
$385$	7.27216			0.370624
$386$		0			0
$387$			7.26752i			0.369429i
$388$		0			0
$389$	13.6666			0.692923			0.346462	−	0.938064i	$-0.387383\pi$
$389$	13.6666			0.692923			0.346462	+	0.938064i	$0.387383\pi$
$390$		0			0
$391$			23.0372i			1.16504i
$392$		0			0
$393$		−	4.67124i		−	0.235633i
$394$		0			0
$395$	19.1839			0.965247
$396$		0			0
$397$	4.75975			0.238885			0.119443	−	0.992841i	$-0.461889\pi$
$397$	4.75975			0.238885			0.119443	+	0.992841i	$0.461889\pi$
$398$		0			0
$399$	11.4309	+	11.6959i	0.572261	+	0.585525i
$400$		0			0
$401$		−	28.8543i		−	1.44092i	−0.693499	−	0.720458i	$-0.743932\pi$
$401$		−	28.8543i		−	1.44092i	0.693499	−	0.720458i	$-0.256068\pi$
$402$		0			0
$403$			8.81467i			0.439090i
$404$		0			0
$405$	−1.69641			−0.0842954
$406$		0			0
$407$	−1.71870			−0.0851930
$408$		0			0
$409$		−	31.6463i		−	1.56481i	−0.622771	−	0.782404i	$-0.713993\pi$
$409$		−	31.6463i		−	1.56481i	0.622771	−	0.782404i	$-0.286007\pi$
$410$		0			0
$411$	−16.0194			−0.790181
$412$		0			0
$413$		−	17.6732i		−	0.869643i
$414$		0			0
$415$		−	8.60734i		−	0.422518i
$416$		0			0
$417$		−	11.8297i		−	0.579305i
$418$		0			0
$419$		−	15.4272i		−	0.753667i	−0.926281	−	0.376833i	$-0.877013\pi$
$419$		−	15.4272i		−	0.753667i	0.926281	−	0.376833i	$-0.122987\pi$
$420$		0			0
$421$		−	33.1566i		−	1.61596i	−0.589213	−	0.807978i	$-0.700562\pi$
$421$		−	33.1566i		−	1.61596i	0.589213	−	0.807978i	$-0.299438\pi$
$422$		0			0
$423$			9.56310i			0.464974i
$424$		0			0
$425$	−9.75525			−0.473199
$426$		0			0
$427$		−	48.8654i		−	2.36476i
$428$		0			0
$429$	3.86906			0.186800
$430$		0			0
$431$	34.3209			1.65318			0.826591	−	0.562803i	$-0.190277\pi$
$431$	34.3209			1.65318			0.826591	+	0.562803i	$0.190277\pi$
$432$		0			0
$433$		−	28.2056i		−	1.35547i	−0.735305	−	0.677737i	$-0.762961\pi$
$433$		−	28.2056i		−	1.35547i	0.735305	−	0.677737i	$-0.237039\pi$
$434$		0			0
$435$		−	2.54740i		−	0.122139i
$436$		0			0
$437$	15.2688	+	15.6227i	0.730405	+	0.747334i
$438$		0			0
$439$	−25.8050			−1.23161			−0.615804	−	0.787899i	$-0.711169\pi$
$439$	−25.8050			−1.23161			−0.615804	+	0.787899i	$0.711169\pi$
$440$		0			0
$441$	−7.07677			−0.336989
$442$		0			0
$443$			0.824311i			0.0391642i	0.999808	+	0.0195821i	$0.00623357\pi$
$443$			0.824311i			0.0391642i	−0.999808	+	0.0195821i	$0.993766\pi$
$444$		0			0
$445$		−	7.50229i		−	0.355642i
$446$		0			0
$447$	−8.91978			−0.421891
$448$		0			0
$449$			36.3663i			1.71623i	0.513457	+	0.858115i	$0.328364\pi$
$449$			36.3663i			1.71623i	−0.513457	+	0.858115i	$0.671636\pi$
$450$		0			0
$451$	−10.5087			−0.494837
$452$		0			0
$453$	13.2770			0.623807
$454$		0			0
$455$	−21.5530			−1.01042
$456$		0			0
$457$	2.74881			0.128584			0.0642921	−	0.997931i	$-0.479521\pi$
$457$	2.74881			0.128584			0.0642921	+	0.997931i	$0.479521\pi$
$458$		0			0
$459$	4.59679			0.214560
$460$		0			0
$461$	30.2335			1.40812			0.704058	−	0.710142i	$-0.251369\pi$
$461$	30.2335			1.40812			0.704058	+	0.710142i	$0.251369\pi$
$462$		0			0
$463$		−	6.38444i		−	0.296710i	−0.988934	−	0.148355i	$-0.952602\pi$
$463$		−	6.38444i		−	0.296710i	0.988934	−	0.148355i	$-0.0473979\pi$
$464$		0			0
$465$	4.41583			0.204779
$466$		0			0
$467$			13.1338i			0.607762i	0.952710	+	0.303881i	$0.0982825\pi$
$467$			13.1338i			0.607762i	−0.952710	+	0.303881i	$0.901718\pi$
$468$		0			0
$469$			50.6881i			2.34056i
$470$		0			0
$471$	−4.39158			−0.202353
$472$		0			0
$473$	−8.30361			−0.381801
$474$		0			0
$475$	−6.61552	+	6.46566i	−0.303541	+	0.296665i
$476$		0			0
$477$			9.19750i			0.421124i
$478$		0			0
$479$		−	20.3564i		−	0.930108i	−0.885282	−	0.465054i	$-0.846035\pi$
$479$		−	20.3564i		−	0.930108i	0.885282	−	0.465054i	$-0.153965\pi$
$480$		0			0
$481$	5.09385			0.232259
$482$		0			0
$483$	−18.8030			−0.855565
$484$		0			0
$485$			8.05227i			0.365635i
$486$		0			0
$487$	22.2061			1.00626			0.503128	−	0.864212i	$-0.332182\pi$
$487$	22.2061			1.00626			0.503128	+	0.864212i	$0.332182\pi$
$488$		0			0
$489$		−	12.9407i		−	0.585200i
$490$		0			0
$491$		−	14.9595i		−	0.675114i	−0.941305	−	0.337557i	$-0.890400\pi$
$491$		−	14.9595i		−	0.675114i	0.941305	−	0.337557i	$-0.109600\pi$
$492$		0			0
$493$			6.90274i			0.310884i
$494$		0			0
$495$		−	1.93826i		−	0.0871183i
$496$		0			0
$497$			22.1398i			0.993107i
$498$		0			0
$499$			38.5856i			1.72733i	0.504069	+	0.863663i	$0.331836\pi$
$499$			38.5856i			1.72733i	−0.504069	+	0.863663i	$0.668164\pi$
$500$		0			0
$501$	−16.6261			−0.742800
$502$		0			0
$503$		−	22.1268i		−	0.986586i	−0.869863	−	0.493293i	$-0.835793\pi$
$503$		−	22.1268i		−	0.986586i	0.869863	−	0.493293i	$-0.164207\pi$
$504$		0			0
$505$	−11.4462			−0.509351
$506$		0			0
$507$	1.53298			0.0680820
$508$		0			0
$509$		−	44.6469i		−	1.97894i	−0.144738	−	0.989470i	$-0.546234\pi$
$509$		−	44.6469i		−	1.97894i	0.144738	−	0.989470i	$-0.453766\pi$
$510$		0			0
$511$		−	51.5085i		−	2.27860i
$512$		0			0
$513$	3.11731	−	3.04670i	0.137633	−	0.134515i
$514$		0			0
$515$	−20.4605			−0.901597
$516$		0			0
$517$	−10.9265			−0.480545
$518$		0			0
$519$		−	7.63053i		−	0.334943i
$520$		0			0
$521$		−	11.4136i		−	0.500040i	−0.968241	−	0.250020i	$-0.919563\pi$
$521$		−	11.4136i		−	0.500040i	0.968241	−	0.250020i	$-0.0804372\pi$
$522$		0			0
$523$	−12.7208			−0.556241			−0.278120	−	0.960546i	$-0.589711\pi$
$523$	−12.7208			−0.556241			−0.278120	+	0.960546i	$0.589711\pi$
$524$		0			0
$525$		−	7.96224i		−	0.347501i
$526$		0			0
$527$	−11.9656			−0.521231
$528$		0			0
$529$	−2.11600			−0.0919999
$530$		0			0
$531$	−4.71047			−0.204417
$532$		0			0
$533$	31.1455			1.34906
$534$		0			0
$535$	−27.1306			−1.17296
$536$		0			0
$537$	−1.31105			−0.0565760
$538$		0			0
$539$		−	8.08566i		−	0.348274i
$540$		0			0
$541$	−35.1306			−1.51038			−0.755192	−	0.655504i	$-0.772456\pi$
$541$	−35.1306			−1.51038			−0.755192	+	0.655504i	$0.772456\pi$
$542$		0			0
$543$		−	12.8711i		−	0.552351i
$544$		0			0
$545$		−	4.09591i		−	0.175450i
$546$		0			0
$547$	29.4541			1.25937			0.629684	−	0.776852i	$-0.283184\pi$
$547$	29.4541			1.25937			0.629684	+	0.776852i	$0.283184\pi$
$548$		0			0
$549$	−13.0242			−0.555858
$550$		0			0
$551$	4.57505	+	4.68109i	0.194904	+	0.199421i
$552$		0			0
$553$		−	42.4285i		−	1.80424i
$554$		0			0
$555$		−	2.55183i		−	0.108319i
$556$		0			0
$557$	−10.6241			−0.450157			−0.225079	−	0.974341i	$-0.572264\pi$
$557$	−10.6241			−0.450157			−0.225079	+	0.974341i	$0.572264\pi$
$558$		0			0
$559$	24.6100			1.04089
$560$		0			0
$561$			5.25213i			0.221745i
$562$		0			0
$563$	−16.8954			−0.712056			−0.356028	−	0.934475i	$-0.615869\pi$
$563$	−16.8954			−0.712056			−0.356028	+	0.934475i	$0.615869\pi$
$564$		0			0
$565$			18.5381i			0.779902i
$566$		0			0
$567$			3.75190i			0.157565i
$568$		0			0
$569$			11.6671i			0.489109i	0.969636	+	0.244555i	$0.0786417\pi$
$569$			11.6671i			0.489109i	−0.969636	+	0.244555i	$0.921358\pi$
$570$		0			0
$571$			23.6655i			0.990370i	0.868787	+	0.495185i	$0.164900\pi$
$571$			23.6655i			0.990370i	−0.868787	+	0.495185i	$0.835100\pi$
$572$		0			0
$573$		−	2.99551i		−	0.125139i
$574$		0			0
$575$		−	10.6355i		−	0.443532i
$576$		0			0
$577$	43.8578			1.82583			0.912913	−	0.408155i	$-0.133828\pi$
$577$	43.8578			1.82583			0.912913	+	0.408155i	$0.133828\pi$
$578$		0			0
$579$		−	3.76959i		−	0.156659i
$580$		0			0
$581$	−19.0366			−0.789771
$582$		0			0
$583$	−10.5087			−0.435227
$584$		0			0
$585$			5.74456i			0.237508i
$586$		0			0
$587$			3.86984i			0.159725i	0.996806	+	0.0798627i	$0.0254482\pi$
$587$			3.86984i			0.159725i	−0.996806	+	0.0798627i	$0.974552\pi$
$588$		0			0
$589$	−8.11449	+	7.93067i	−0.334352	+	0.326778i
$590$		0			0
$591$	13.6614			0.561956
$592$		0			0
$593$	3.15728			0.129654			0.0648270	−	0.997897i	$-0.479350\pi$
$593$	3.15728			0.129654			0.0648270	+	0.997897i	$0.479350\pi$
$594$		0			0
$595$		−	29.2575i		−	1.19944i
$596$		0			0
$597$			3.75712i			0.153769i
$598$		0			0
$599$	17.6525			0.721261			0.360631	−	0.932709i	$-0.382561\pi$
$599$	17.6525			0.721261			0.360631	+	0.932709i	$0.382561\pi$
$600$		0			0
$601$		−	23.3358i		−	0.951887i	−0.879476	−	0.475944i	$-0.842107\pi$
$601$		−	23.3358i		−	0.951887i	0.879476	−	0.475944i	$-0.157893\pi$
$602$		0			0
$603$	13.5100			0.550169
$604$		0			0
$605$	−16.4459			−0.668623
$606$		0			0
$607$	23.1559			0.939870			0.469935	−	0.882701i	$-0.344277\pi$
$607$	23.1559			0.939870			0.469935	+	0.882701i	$0.344277\pi$
$608$		0			0
$609$	−5.63402			−0.228302
$610$		0			0
$611$	32.3835			1.31010
$612$		0			0
$613$	26.3611			1.06471			0.532357	−	0.846520i	$-0.321307\pi$
$613$	26.3611			1.06471			0.532357	+	0.846520i	$0.321307\pi$
$614$		0			0
$615$		−	15.6027i		−	0.629163i
$616$		0			0
$617$	−38.3002			−1.54191			−0.770954	−	0.636891i	$-0.780220\pi$
$617$	−38.3002			−1.54191			−0.770954	+	0.636891i	$0.780220\pi$
$618$		0			0
$619$		−	3.61343i		−	0.145236i	−0.997360	−	0.0726181i	$-0.976865\pi$
$619$		−	3.61343i		−	0.145236i	0.997360	−	0.0726181i	$-0.0231354\pi$
$620$		0			0
$621$			5.01159i			0.201108i
$622$		0			0
$623$	−16.5926			−0.664768
$624$		0			0
$625$	−9.88538			−0.395415
$626$		0			0
$627$	3.48105	+	3.56173i	0.139020	+	0.142242i
$628$		0			0
$629$			6.91473i			0.275708i
$630$		0			0
$631$		−	13.5482i		−	0.539347i	−0.962952	−	0.269673i	$-0.913084\pi$
$631$		−	13.5482i		−	0.539347i	0.962952	−	0.269673i	$-0.0869157\pi$
$632$		0			0
$633$	17.0469			0.677555
$634$		0			0
$635$	−16.8396			−0.668260
$636$		0			0
$637$			23.9641i			0.949491i
$638$		0			0
$639$	5.90096			0.233438
$640$		0			0
$641$			25.0663i			0.990058i	0.868877	+	0.495029i	$0.164842\pi$
$641$			25.0663i			0.990058i	−0.868877	+	0.495029i	$0.835158\pi$
$642$		0			0
$643$		−	10.6706i		−	0.420809i	−0.977614	−	0.210405i	$-0.932522\pi$
$643$		−	10.6706i		−	0.420809i	0.977614	−	0.210405i	$-0.0674781\pi$
$644$		0			0
$645$		−	12.3287i		−	0.485443i
$646$		0			0
$647$			17.5261i			0.689022i	0.938782	+	0.344511i	$0.111955\pi$
$647$			17.5261i			0.689022i	−0.938782	+	0.344511i	$0.888045\pi$
$648$		0			0
$649$		−	5.38202i		−	0.211263i
$650$		0			0
$651$		−	9.76635i		−	0.382774i
$652$		0			0
$653$	32.3872			1.26741			0.633704	−	0.773576i	$-0.281534\pi$
$653$	32.3872			1.26741			0.633704	+	0.773576i	$0.281534\pi$
$654$		0			0
$655$			7.92435i			0.309630i
$656$		0			0
$657$	−13.7286			−0.535605
$658$		0			0
$659$	−26.9073			−1.04816			−0.524080	−	0.851669i	$-0.675591\pi$
$659$	−26.9073			−1.04816			−0.524080	+	0.851669i	$0.675591\pi$
$660$		0			0
$661$			0.0716909i			0.00278845i	0.999999	+	0.00139423i	$0.000443796\pi$
$661$			0.0716909i			0.00278845i	−0.999999	+	0.00139423i	$0.999556\pi$
$662$		0			0
$663$		−	15.5661i		−	0.604537i
$664$		0			0
$665$	−19.3915	−	19.8410i	−0.751971	−	0.769400i
$666$		0			0
$667$	−7.52561			−0.291393
$668$		0			0
$669$	−10.5808			−0.409077
$670$		0			0
$671$		−	14.8809i		−	0.574473i
$672$		0			0
$673$		−	21.6090i		−	0.832967i	−0.909143	−	0.416483i	$-0.863262\pi$
$673$		−	21.6090i		−	0.832967i	0.909143	−	0.416483i	$-0.136738\pi$
$674$		0			0
$675$	−2.12219			−0.0816830
$676$		0			0
$677$		−	9.37382i		−	0.360265i	−0.983642	−	0.180133i	$-0.942347\pi$
$677$		−	9.37382i		−	0.360265i	0.983642	−	0.180133i	$-0.0576527\pi$
$678$		0			0
$679$	17.8090			0.683445
$680$		0			0
$681$	−9.97902			−0.382397
$682$		0			0
$683$	−34.3840			−1.31567			−0.657834	−	0.753163i	$-0.728527\pi$
$683$	−34.3840			−1.31567			−0.657834	+	0.753163i	$0.728527\pi$
$684$		0			0
$685$	27.1756			1.03833
$686$		0			0
$687$	−11.9053			−0.454215
$688$		0			0
$689$	31.1455			1.18655
$690$		0			0
$691$			46.8495i			1.78224i	0.453767	+	0.891120i	$0.350080\pi$
$691$			46.8495i			1.78224i	−0.453767	+	0.891120i	$0.649920\pi$
$692$		0			0
$693$	−4.28679			−0.162842
$694$		0			0
$695$			20.0681i			0.761227i
$696$		0			0
$697$			42.2790i			1.60143i
$698$		0			0
$699$	5.86245			0.221738
$700$		0			0
$701$	−45.8273			−1.73087			−0.865436	−	0.501019i	$-0.832959\pi$
$701$	−45.8273			−1.73087			−0.865436	+	0.501019i	$0.832959\pi$
$702$		0			0
$703$	4.58300	+	4.68922i	0.172851	+	0.176857i
$704$		0			0
$705$		−	16.2230i		−	0.610992i
$706$		0			0
$707$			25.3153i			0.952080i
$708$		0			0
$709$	−37.3018			−1.40090			−0.700449	−	0.713702i	$-0.747017\pi$
$709$	−37.3018			−1.40090			−0.700449	+	0.713702i	$0.747017\pi$
$710$		0			0
$711$	−11.3085			−0.424103
$712$		0			0
$713$		−	13.0454i		−	0.488553i
$714$		0			0
$715$	−6.56352			−0.245462
$716$		0			0
$717$		−	15.6175i		−	0.583248i
$718$		0			0
$719$		−	5.85053i		−	0.218188i	−0.994031	−	0.109094i	$-0.965205\pi$
$719$		−	5.85053i		−	0.218188i	0.994031	−	0.109094i	$-0.0347949\pi$
$720$		0			0
$721$			45.2518i			1.68527i
$722$		0			0
$723$			19.2781i			0.716960i
$724$		0			0
$725$		−	3.18677i		−	0.118354i
$726$		0			0
$727$		−	25.6941i		−	0.952941i	−0.879190	−	0.476471i	$-0.841916\pi$
$727$		−	25.6941i		−	0.952941i	0.879190	−	0.476471i	$-0.158084\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$			33.4073i			1.23561i
$732$		0			0
$733$	−20.2806			−0.749081			−0.374541	−	0.927210i	$-0.622200\pi$
$733$	−20.2806			−0.749081			−0.374541	+	0.927210i	$0.622200\pi$
$734$		0			0
$735$	12.0051			0.442815
$736$		0			0
$737$			15.4360i			0.568593i
$738$		0			0
$739$			35.8855i			1.32007i	0.751235	+	0.660035i	$0.229458\pi$
$739$			35.8855i			1.32007i	−0.751235	+	0.660035i	$0.770542\pi$
$740$		0			0
$741$	−10.3170	−	10.5562i	−0.379005	−	0.387790i
$742$		0			0
$743$	43.0655			1.57992			0.789960	−	0.613159i	$-0.210101\pi$
$743$	43.0655			1.57992			0.789960	+	0.613159i	$0.210101\pi$
$744$		0			0
$745$	15.1316			0.554380
$746$		0			0
$747$			5.07385i			0.185643i
$748$		0			0
$749$			60.0039i			2.19249i
$750$		0			0
$751$	54.5587			1.99088			0.995439	−	0.0954054i	$-0.0304147\pi$
$751$	54.5587			1.99088			0.995439	+	0.0954054i	$0.0304147\pi$
$752$		0			0
$753$		−	0.633320i		−	0.0230795i
$754$		0			0
$755$	−22.5232			−0.819704
$756$		0			0
$757$	−7.92358			−0.287987			−0.143994	−	0.989579i	$-0.545995\pi$
$757$	−7.92358			−0.287987			−0.143994	+	0.989579i	$0.545995\pi$
$758$		0			0
$759$	−5.72606			−0.207843
$760$		0			0
$761$	−1.46075			−0.0529520			−0.0264760	−	0.999649i	$-0.508429\pi$
$761$	−1.46075			−0.0529520			−0.0264760	+	0.999649i	$0.508429\pi$
$762$		0			0
$763$	−9.05881			−0.327951
$764$		0			0
$765$	−7.79805			−0.281939
$766$		0			0
$767$			15.9511i			0.575960i
$768$		0			0
$769$	44.8348			1.61678			0.808392	−	0.588644i	$-0.200338\pi$
$769$	44.8348			1.61678			0.808392	+	0.588644i	$0.200338\pi$
$770$		0			0
$771$			27.6282i			0.995006i
$772$		0			0
$773$		−	24.5407i		−	0.882668i	−0.897343	−	0.441334i	$-0.854505\pi$
$773$		−	24.5407i		−	0.882668i	0.897343	−	0.441334i	$-0.145495\pi$
$774$		0			0
$775$	5.52414			0.198433
$776$		0			0
$777$	−5.64381			−0.202470
$778$		0			0
$779$	28.0220	+	28.6715i	1.00399	+	1.02726i
$780$		0			0
$781$			6.74223i			0.241256i
$782$		0			0
$783$			1.50164i			0.0536644i
$784$		0			0
$785$	7.44992			0.265899
$786$		0			0
$787$	8.16670			0.291111			0.145556	−	0.989350i	$-0.453503\pi$
$787$	8.16670			0.291111			0.145556	+	0.989350i	$0.453503\pi$
$788$		0			0
$789$			5.27755i			0.187886i
$790$		0			0
$791$	41.0001			1.45779
$792$		0			0
$793$			44.1037i			1.56617i
$794$		0			0
$795$		−	15.6027i		−	0.553372i
$796$		0			0
$797$		−	49.7979i		−	1.76393i	−0.471313	−	0.881966i	$-0.656220\pi$
$797$		−	49.7979i		−	1.76393i	0.471313	−	0.881966i	$-0.343780\pi$
$798$		0			0
$799$			43.9596i			1.55518i
$800$		0			0
$801$			4.42244i			0.156259i
$802$		0			0
$803$		−	15.6858i		−	0.553541i
$804$		0			0
$805$	31.8976			1.12424
$806$		0			0
$807$		−	2.43012i		−	0.0855441i
$808$		0			0
$809$	−5.26349			−0.185055			−0.0925273	−	0.995710i	$-0.529495\pi$
$809$	−5.26349			−0.185055			−0.0925273	+	0.995710i	$0.529495\pi$
$810$		0			0
$811$	−13.0982			−0.459941			−0.229970	−	0.973198i	$-0.573863\pi$
$811$	−13.0982			−0.459941			−0.229970	+	0.973198i	$0.573863\pi$
$812$		0			0
$813$			18.1691i			0.637218i
$814$		0			0
$815$			21.9528i			0.768974i
$816$		0			0
$817$	22.1419	+	22.6551i	0.774649	+	0.792603i
$818$		0			0
$819$	12.7051			0.443951
$820$		0			0
$821$	−11.1354			−0.388629			−0.194315	−	0.980939i	$-0.562248\pi$
$821$	−11.1354			−0.388629			−0.194315	+	0.980939i	$0.562248\pi$
$822$		0			0
$823$		−	44.3375i		−	1.54551i	−0.634706	−	0.772754i	$-0.718879\pi$
$823$		−	44.3375i		−	1.54551i	0.634706	−	0.772754i	$-0.281121\pi$
$824$		0			0
$825$		−	2.42474i		−	0.0844185i
$826$		0			0
$827$	0.746039			0.0259423			0.0129712	−	0.999916i	$-0.495871\pi$
$827$	0.746039			0.0259423			0.0129712	+	0.999916i	$0.495871\pi$
$828$		0			0
$829$		−	49.3157i		−	1.71281i	−0.516307	−	0.856403i	$-0.672694\pi$
$829$		−	49.3157i		−	1.71281i	0.516307	−	0.856403i	$-0.327306\pi$
$830$		0			0
$831$	22.6189			0.784641
$832$		0			0
$833$	−32.5304			−1.12711
$834$		0			0
$835$	28.2047			0.976065
$836$		0			0
$837$	−2.60304			−0.0899742
$838$		0			0
$839$	27.3775			0.945175			0.472587	−	0.881284i	$-0.343320\pi$
$839$	27.3775			0.945175			0.472587	+	0.881284i	$0.343320\pi$
$840$		0			0
$841$	26.7451			0.922244
$842$		0			0
$843$			20.2656i			0.697985i
$844$		0			0
$845$	−2.60057			−0.0894622
$846$		0			0
$847$			36.3730i			1.24979i
$848$		0			0
$849$		−	13.6187i		−	0.467392i
$850$		0			0
$851$	−7.53869			−0.258423
$852$		0			0
$853$	−23.1550			−0.792813			−0.396406	−	0.918075i	$-0.629743\pi$
$853$	−23.1550			−0.792813			−0.396406	+	0.918075i	$0.629743\pi$
$854$		0			0
$855$	−5.28825	+	5.16845i	−0.180854	+	0.176757i
$856$		0			0
$857$		−	9.01757i		−	0.308034i	−0.988068	−	0.154017i	$-0.950779\pi$
$857$		−	9.01757i		−	0.308034i	0.988068	−	0.154017i	$-0.0492211\pi$
$858$		0			0
$859$			17.1798i			0.586169i	0.956087	+	0.293084i	$0.0946817\pi$
$859$			17.1798i			0.586169i	−0.956087	+	0.293084i	$0.905318\pi$
$860$		0			0
$861$	−34.5081			−1.17603
$862$		0			0
$863$	21.7095			0.739000			0.369500	−	0.929231i	$-0.379529\pi$
$863$	21.7095			0.739000			0.369500	+	0.929231i	$0.379529\pi$
$864$		0			0
$865$			12.9445i			0.440127i
$866$		0			0
$867$	4.13048			0.140278
$868$		0			0
$869$		−	12.9207i		−	0.438305i
$870$		0			0
$871$		−	45.7488i		−	1.55014i
$872$		0			0
$873$		−	4.74665i		−	0.160650i
$874$		0			0
$875$			45.3311i			1.53247i
$876$		0			0
$877$			38.2240i			1.29073i	0.763873	+	0.645367i	$0.223296\pi$
$877$			38.2240i			1.29073i	−0.763873	+	0.645367i	$0.776704\pi$
$878$		0			0
$879$			4.61940i			0.155808i
$880$		0			0
$881$	−33.3148			−1.12241			−0.561203	−	0.827678i	$-0.689661\pi$
$881$	−33.3148			−1.12241			−0.561203	+	0.827678i	$0.689661\pi$
$882$		0			0
$883$			13.9552i			0.469630i	0.972040	+	0.234815i	$0.0754484\pi$
$883$			13.9552i			0.469630i	−0.972040	+	0.234815i	$0.924552\pi$
$884$		0			0
$885$	7.99090			0.268611
$886$		0			0
$887$	39.5218			1.32701			0.663507	−	0.748170i	$-0.269067\pi$
$887$	39.5218			1.32701			0.663507	+	0.748170i	$0.269067\pi$
$888$		0			0
$889$			37.2437i			1.24911i
$890$		0			0
$891$			1.14256i			0.0382773i
$892$		0			0
$893$	29.1359	+	29.8112i	0.974994	+	0.997593i
$894$		0			0
$895$	2.22408			0.0743428
$896$		0			0
$897$	16.9707			0.566636
$898$		0			0
$899$		−	3.90884i		−	0.130367i
$900$		0			0
$901$			42.2790i			1.40852i
$902$		0			0
$903$	−27.2670			−0.907390
$904$		0			0
$905$			21.8347i			0.725809i
$906$		0			0
$907$	−22.8201			−0.757730			−0.378865	−	0.925452i	$-0.623686\pi$
$907$	−22.8201			−0.757730			−0.378865	+	0.925452i	$0.623686\pi$
$908$		0			0
$909$	6.74733			0.223795
$910$		0			0
$911$	15.6901			0.519837			0.259918	−	0.965631i	$-0.416304\pi$
$911$	15.6901			0.519837			0.259918	+	0.965631i	$0.416304\pi$
$912$		0			0
$913$	−5.79720			−0.191859
$914$		0			0
$915$	22.0944			0.730417
$916$		0			0
$917$	17.5260			0.578761
$918$		0			0
$919$			39.6439i			1.30773i	0.756611	+	0.653865i	$0.226854\pi$
$919$			39.6439i			1.30773i	−0.756611	+	0.653865i	$0.773146\pi$
$920$		0			0
$921$	−3.37396			−0.111176
$922$		0			0
$923$		−	19.9824i		−	0.657729i
$924$		0			0
$925$		−	3.19230i		−	0.104962i
$926$		0			0
$927$	12.0610			0.396136
$928$		0			0
$929$	−48.5092			−1.59154			−0.795768	−	0.605602i	$-0.792933\pi$
$929$	−48.5092			−1.59154			−0.795768	+	0.605602i	$0.792933\pi$
$930$		0			0
$931$	−22.0605	+	21.5608i	−0.723004	+	0.706626i
$932$		0			0
$933$			27.0077i			0.884193i
$934$		0			0
$935$		−	8.90977i		−	0.291381i
$936$		0			0
$937$	−25.3446			−0.827971			−0.413986	−	0.910283i	$-0.635864\pi$
$937$	−25.3446			−0.827971			−0.413986	+	0.910283i	$0.635864\pi$
$938$		0			0
$939$	−4.86622			−0.158803
$940$		0			0
$941$		−	5.02120i		−	0.163686i	−0.996645	−	0.0818432i	$-0.973919\pi$
$941$		−	5.02120i		−	0.163686i	0.996645	−	0.0818432i	$-0.0260807\pi$
$942$		0			0
$943$	−46.0940			−1.50103
$944$		0			0
$945$		−	6.36477i		−	0.207046i
$946$		0			0
$947$		−	50.5868i		−	1.64385i	−0.569595	−	0.821926i	$-0.692900\pi$
$947$		−	50.5868i		−	1.64385i	0.569595	−	0.821926i	$-0.307100\pi$
$948$		0			0
$949$			46.4893i			1.50910i
$950$		0			0
$951$		−	27.1413i		−	0.880118i
$952$		0			0
$953$		−	36.2995i		−	1.17586i	−0.808913	−	0.587928i	$-0.799944\pi$
$953$		−	36.2995i		−	1.17586i	0.808913	−	0.587928i	$-0.200056\pi$
$954$		0			0
$955$			5.08161i			0.164437i
$956$		0			0
$957$	−1.71572			−0.0554615
$958$		0			0
$959$		−	60.1034i		−	1.94084i
$960$		0			0
$961$	−24.2242			−0.781425
$962$		0			0
$963$	15.9929			0.515365
$964$		0			0
$965$			6.39478i			0.205855i
$966$		0			0
$967$			16.5835i			0.533291i	0.963795	+	0.266645i	$0.0859153\pi$
$967$			16.5835i			0.533291i	−0.963795	+	0.266645i	$0.914085\pi$
$968$		0			0
$969$	14.3296	−	14.0050i	0.460334	−	0.449906i
$970$		0			0
$971$	−20.4937			−0.657675			−0.328838	−	0.944386i	$-0.606657\pi$
$971$	−20.4937			−0.657675			−0.328838	+	0.944386i	$0.606657\pi$
$972$		0			0
$973$	44.3840			1.42289
$974$		0			0
$975$			7.18636i			0.230148i
$976$		0			0
$977$			56.4051i			1.80456i	0.431152	+	0.902279i	$0.358107\pi$
$977$			56.4051i			1.80456i	−0.431152	+	0.902279i	$0.641893\pi$
$978$		0			0
$979$	−5.05293			−0.161492
$980$		0			0
$981$			2.41446i			0.0770877i
$982$		0			0
$983$	−24.2710			−0.774124			−0.387062	−	0.922054i	$-0.626510\pi$
$983$	−24.2710			−0.774124			−0.387062	+	0.922054i	$0.626510\pi$
$984$		0			0
$985$	−23.1754			−0.738429
$986$		0			0
$987$	−35.8798			−1.14207
$988$		0			0
$989$	−36.4218			−1.15815
$990$		0			0
$991$	−53.9141			−1.71264			−0.856319	−	0.516447i	$-0.827254\pi$
$991$	−53.9141			−1.71264			−0.856319	+	0.516447i	$0.827254\pi$
$992$		0			0
$993$	24.3832			0.773778
$994$		0			0
$995$		−	6.37362i		−	0.202057i
$996$		0			0
$997$	19.3244			0.612010			0.306005	−	0.952030i	$-0.401008\pi$
$997$	19.3244			0.612010			0.306005	+	0.952030i	$0.401008\pi$
$998$		0			0
$999$			1.50425i			0.0475924i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.k.b.607.6	yes	20
3.2	odd	2		5472.2.k.b.2431.16		20
4.3	odd	2		1824.2.k.a.607.5	✓	20
8.3	odd	2		3648.2.k.l.2431.15		20
8.5	even	2		3648.2.k.k.2431.16		20
12.11	even	2		5472.2.k.a.2431.15		20
19.18	odd	2		1824.2.k.a.607.6	yes	20
57.56	even	2		5472.2.k.a.2431.16		20
76.75	even	2	inner	1824.2.k.b.607.5	yes	20
152.37	odd	2		3648.2.k.l.2431.16		20
152.75	even	2		3648.2.k.k.2431.15		20
228.227	odd	2		5472.2.k.b.2431.15		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.k.a.607.5	✓	20	4.3	odd	2
1824.2.k.a.607.6	yes	20	19.18	odd	2
1824.2.k.b.607.5	yes	20	76.75	even	2	inner
1824.2.k.b.607.6	yes	20	1.1	even	1	trivial
3648.2.k.k.2431.15		20	152.75	even	2
3648.2.k.k.2431.16		20	8.5	even	2
3648.2.k.l.2431.15		20	8.3	odd	2
3648.2.k.l.2431.16		20	152.37	odd	2
5472.2.k.a.2431.15		20	12.11	even	2
5472.2.k.a.2431.16		20	57.56	even	2
5472.2.k.b.2431.15		20	228.227	odd	2
5472.2.k.b.2431.16		20	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1824 = 2^{5} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1824.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.69641 q^{5} +3.75190i q^{7} +1.00000 q^{9} +1.14256i q^{11} -3.38630i q^{13} -1.69641 q^{15} +4.59679 q^{17} +(3.11731 - 3.04670i) q^{19} +3.75190i q^{21} +5.01159i q^{23} -2.12219 q^{25} +1.00000 q^{27} +1.50164i q^{29} -2.60304 q^{31} +1.14256i q^{33} -6.36477i q^{35} +1.50425i q^{37} -3.38630i q^{39} +9.19750i q^{41} +7.26752i q^{43} -1.69641 q^{45} +9.56310i q^{47} -7.07677 q^{49} +4.59679 q^{51} +9.19750i q^{53} -1.93826i q^{55} +(3.11731 - 3.04670i) q^{57} -4.71047 q^{59} -13.0242 q^{61} +3.75190i q^{63} +5.74456i q^{65} +13.5100 q^{67} +5.01159i q^{69} +5.90096 q^{71} -13.7286 q^{73} -2.12219 q^{75} -4.28679 q^{77} -11.3085 q^{79} +1.00000 q^{81} +5.07385i q^{83} -7.79805 q^{85} +1.50164i q^{87} +4.42244i q^{89} +12.7051 q^{91} -2.60304 q^{93} +(-5.28825 + 5.16845i) q^{95} -4.74665i q^{97} +1.14256i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} - 8 q^{31} - 44 q^{49} + 4 q^{57} - 8 q^{61} + 8 q^{67} + 64 q^{71} + 8 q^{73} + 20 q^{75} + 40 q^{77} - 16 q^{79} + 20 q^{81} + 40 q^{85} - 8 q^{93}+ \cdots - 8 q^{95}+O(q^{100}) \) 20 * q + 20 * q^3 + 20 * q^9 + 4 * q^19 + 20 * q^25 + 20 * q^27 - 8 * q^31 - 44 * q^49 + 4 * q^57 - 8 * q^61 + 8 * q^67 + 64 * q^71 + 8 * q^73 + 20 * q^75 + 40 * q^77 - 16 * q^79 + 20 * q^81 + 40 * q^85 - 8 * q^93 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more