LMFDB - Embedded newform 1824.2.k.b.607.1

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)

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);

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")

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			607.1
Root			$0.709059i$ of defining polynomial
Character	$\chi$	$=$	1824.607
Dual form			1824.2.k.b.607.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -4.06455 q^{5} -3.06179i q^{7} +1.00000 q^{9} +2.03439i q^{11} -4.86979i q^{13} -4.06455 q^{15} -7.51623 q^{17} +(4.01890 + 1.68773i) q^{19} -3.06179i q^{21} +3.92817i q^{23} +11.5206 q^{25} +1.00000 q^{27} +7.10170i q^{29} -3.75541 q^{31} +2.03439i q^{33} +12.4448i q^{35} +10.3386i q^{37} -4.86979i q^{39} -1.20334i q^{41} +9.25741i q^{43} -4.06455 q^{45} -9.13493i q^{47} -2.37457 q^{49} -7.51623 q^{51} -1.20334i q^{53} -8.26888i q^{55} +(4.01890 + 1.68773i) q^{57} -2.64038 q^{59} -0.164898 q^{61} -3.06179i q^{63} +19.7935i q^{65} -5.34581 q^{67} +3.92817i q^{69} +5.59793 q^{71} -3.78023 q^{73} +11.5206 q^{75} +6.22888 q^{77} -16.4583 q^{79} +1.00000 q^{81} +10.8400i q^{83} +30.5501 q^{85} +7.10170i q^{87} +13.6422i q^{89} -14.9103 q^{91} -3.75541 q^{93} +(-16.3350 - 6.85987i) q^{95} +2.29631i q^{97} +2.03439i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.06455			−1.81772			−0.908862	−	0.417098i	$-0.863047\pi$
$5$	−4.06455			−1.81772			−0.908862	+	0.417098i	$0.863047\pi$
$6$		0			0
$7$		−	3.06179i		−	1.15725i	−0.815594	−	0.578624i	$-0.803590\pi$
$7$		−	3.06179i		−	1.15725i	0.815594	−	0.578624i	$-0.196410\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.03439i			0.613391i	0.951808	+	0.306696i	$0.0992234\pi$
$11$			2.03439i			0.613391i	−0.951808	+	0.306696i	$0.900777\pi$
$12$		0			0
$13$		−	4.86979i		−	1.35064i	−0.737526	−	0.675319i	$-0.764006\pi$
$13$		−	4.86979i		−	1.35064i	0.737526	−	0.675319i	$-0.235994\pi$
$14$		0			0
$15$	−4.06455			−1.04946
$16$		0			0
$17$	−7.51623			−1.82295			−0.911476	−	0.411352i	$-0.865057\pi$
$17$	−7.51623			−1.82295			−0.911476	+	0.411352i	$0.865057\pi$
$18$		0			0
$19$	4.01890	+	1.68773i	0.921999	+	0.387192i
$19$	4.01890	+	1.68773i	0.921999	+	0.387192i
$20$		0			0
$21$		−	3.06179i		−	0.668138i
$22$		0			0
$23$			3.92817i			0.819080i	0.912292	+	0.409540i	$0.134311\pi$
$23$			3.92817i			0.819080i	−0.912292	+	0.409540i	$0.865689\pi$
$24$		0			0
$25$	11.5206			2.30412
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$			7.10170i			1.31875i	0.751813	+	0.659377i	$0.229180\pi$
$29$			7.10170i			1.31875i	−0.751813	+	0.659377i	$0.770820\pi$
$30$		0			0
$31$	−3.75541			−0.674492			−0.337246	−	0.941416i	$-0.609495\pi$
$31$	−3.75541			−0.674492			−0.337246	+	0.941416i	$0.609495\pi$
$32$		0			0
$33$			2.03439i			0.354142i
$34$		0			0
$35$			12.4448i			2.10356i
$36$		0			0
$37$			10.3386i			1.69966i	0.527061	+	0.849828i	$0.323294\pi$
$37$			10.3386i			1.69966i	−0.527061	+	0.849828i	$0.676706\pi$
$38$		0			0
$39$		−	4.86979i		−	0.779791i
$40$		0			0
$41$		−	1.20334i		−	0.187930i	−0.995575	−	0.0939651i	$-0.970046\pi$
$41$		−	1.20334i		−	0.187930i	0.995575	−	0.0939651i	$-0.0299542\pi$
$42$		0			0
$43$			9.25741i			1.41174i	0.708341	+	0.705871i	$0.249444\pi$
$43$			9.25741i			1.41174i	−0.708341	+	0.705871i	$0.750556\pi$
$44$		0			0
$45$	−4.06455			−0.605908
$46$		0			0
$47$		−	9.13493i		−	1.33247i	−0.745744	−	0.666233i	$-0.767906\pi$
$47$		−	9.13493i		−	1.33247i	0.745744	−	0.666233i	$-0.232094\pi$
$48$		0			0
$49$	−2.37457			−0.339225
$50$		0			0
$51$	−7.51623			−1.05248
$52$		0			0
$53$		−	1.20334i		−	0.165292i	−0.996579	−	0.0826458i	$-0.973663\pi$
$53$		−	1.20334i		−	0.165292i	0.996579	−	0.0826458i	$-0.0263370\pi$
$54$		0			0
$55$		−	8.26888i		−	1.11498i
$56$		0			0
$57$	4.01890	+	1.68773i	0.532317	+	0.223545i
$58$		0			0
$59$	−2.64038			−0.343748			−0.171874	−	0.985119i	$-0.554982\pi$
$59$	−2.64038			−0.343748			−0.171874	+	0.985119i	$0.554982\pi$
$60$		0			0
$61$	−0.164898			−0.0211130			−0.0105565	−	0.999944i	$-0.503360\pi$
$61$	−0.164898			−0.0211130			−0.0105565	+	0.999944i	$0.503360\pi$
$62$		0			0
$63$		−	3.06179i		−	0.385750i
$64$		0			0
$65$			19.7935i			2.45508i
$66$		0			0
$67$	−5.34581			−0.653095			−0.326547	−	0.945181i	$-0.605885\pi$
$67$	−5.34581			−0.653095			−0.326547	+	0.945181i	$0.605885\pi$
$68$		0			0
$69$			3.92817i			0.472896i
$70$		0			0
$71$	5.59793			0.664352			0.332176	−	0.943217i	$-0.392217\pi$
$71$	5.59793			0.664352			0.332176	+	0.943217i	$0.392217\pi$
$72$		0			0
$73$	−3.78023			−0.442443			−0.221221	−	0.975224i	$-0.571004\pi$
$73$	−3.78023			−0.442443			−0.221221	+	0.975224i	$0.571004\pi$
$74$		0			0
$75$	11.5206			1.33028
$76$		0			0
$77$	6.22888			0.709846
$78$		0			0
$79$	−16.4583			−1.85170			−0.925852	−	0.377886i	$-0.876651\pi$
$79$	−16.4583			−1.85170			−0.925852	+	0.377886i	$0.876651\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			10.8400i			1.18984i	0.803784	+	0.594921i	$0.202817\pi$
$83$			10.8400i			1.18984i	−0.803784	+	0.594921i	$0.797183\pi$
$84$		0			0
$85$	30.5501			3.31362
$86$		0			0
$87$			7.10170i			0.761382i
$88$		0			0
$89$			13.6422i			1.44607i	0.690810	+	0.723037i	$0.257254\pi$
$89$			13.6422i			1.44607i	−0.690810	+	0.723037i	$0.742746\pi$
$90$		0			0
$91$	−14.9103			−1.56302
$92$		0			0
$93$	−3.75541			−0.389418
$94$		0			0
$95$	−16.3350	−	6.85987i	−1.67594	−	0.703807i
$96$		0			0
$97$			2.29631i			0.233155i	0.993182	+	0.116578i	$0.0371924\pi$
$97$			2.29631i			0.233155i	−0.993182	+	0.116578i	$0.962808\pi$
$98$		0			0
$99$			2.03439i			0.204464i
$100$		0			0
$101$	2.19966			0.218874			0.109437	−	0.993994i	$-0.465095\pi$
$101$	2.19966			0.218874			0.109437	+	0.993994i	$0.465095\pi$
$102$		0			0
$103$	−15.1140			−1.48923			−0.744616	−	0.667493i	$-0.767367\pi$
$103$	−15.1140			−1.48923			−0.744616	+	0.667493i	$0.767367\pi$
$104$		0			0
$105$			12.4448i			1.21449i
$106$		0			0
$107$	−12.5242			−1.21076			−0.605380	−	0.795936i	$-0.706979\pi$
$107$	−12.5242			−1.21076			−0.605380	+	0.795936i	$0.706979\pi$
$108$		0			0
$109$			18.2918i			1.75203i	0.482280	+	0.876017i	$0.339809\pi$
$109$			18.2918i			1.75203i	−0.482280	+	0.876017i	$0.660191\pi$
$110$		0			0
$111$			10.3386i			0.981297i
$112$		0			0
$113$		−	8.29474i		−	0.780303i	−0.920751	−	0.390152i	$-0.872423\pi$
$113$		−	8.29474i		−	0.780303i	0.920751	−	0.390152i	$-0.127577\pi$
$114$		0			0
$115$		−	15.9663i		−	1.48886i
$116$		0			0
$117$		−	4.86979i		−	0.450212i
$118$		0			0
$119$			23.0131i			2.10961i
$120$		0			0
$121$	6.86126			0.623751
$122$		0			0
$123$		−	1.20334i		−	0.108502i
$124$		0			0
$125$	−26.5033			−2.37053
$126$		0			0
$127$	9.60764			0.852540			0.426270	−	0.904596i	$-0.359827\pi$
$127$	9.60764			0.852540			0.426270	+	0.904596i	$0.359827\pi$
$128$		0			0
$129$			9.25741i			0.815069i
$130$		0			0
$131$			4.87062i			0.425548i	0.977101	+	0.212774i	$0.0682499\pi$
$131$			4.87062i			0.425548i	−0.977101	+	0.212774i	$0.931750\pi$
$132$		0			0
$133$	5.16748	−	12.3050i	0.448077	−	1.06698i
$134$		0			0
$135$	−4.06455			−0.349821
$136$		0			0
$137$	5.12929			0.438225			0.219113	−	0.975700i	$-0.429684\pi$
$137$	5.12929			0.438225			0.219113	+	0.975700i	$0.429684\pi$
$138$		0			0
$139$		−	7.88793i		−	0.669046i	−0.942388	−	0.334523i	$-0.891425\pi$
$139$		−	7.88793i		−	0.669046i	0.942388	−	0.334523i	$-0.108575\pi$
$140$		0			0
$141$		−	9.13493i		−	0.769299i
$142$		0			0
$143$	9.90705			0.828469
$144$		0			0
$145$		−	28.8652i		−	2.39713i
$146$		0			0
$147$	−2.37457			−0.195852
$148$		0			0
$149$	18.8427			1.54366			0.771828	−	0.635832i	$-0.219343\pi$
$149$	18.8427			1.54366			0.771828	+	0.635832i	$0.219343\pi$
$150$		0			0
$151$	9.56271			0.778203			0.389101	−	0.921195i	$-0.372786\pi$
$151$	9.56271			0.778203			0.389101	+	0.921195i	$0.372786\pi$
$152$		0			0
$153$	−7.51623			−0.607651
$154$		0			0
$155$	15.2641			1.22604
$156$		0			0
$157$	−14.5309			−1.15969			−0.579846	−	0.814726i	$-0.696887\pi$
$157$	−14.5309			−1.15969			−0.579846	+	0.814726i	$0.696887\pi$
$158$		0			0
$159$		−	1.20334i		−	0.0954311i
$160$		0			0
$161$	12.0272			0.947880
$162$		0			0
$163$			0.676154i			0.0529605i	0.999649	+	0.0264802i	$0.00842990\pi$
$163$			0.676154i			0.0529605i	−0.999649	+	0.0264802i	$0.991570\pi$
$164$		0			0
$165$		−	8.26888i		−	0.643732i
$166$		0			0
$167$	−4.52041			−0.349800			−0.174900	−	0.984586i	$-0.555960\pi$
$167$	−4.52041			−0.349800			−0.174900	+	0.984586i	$0.555960\pi$
$168$		0			0
$169$	−10.7149			−0.824221
$170$		0			0
$171$	4.01890	+	1.68773i	0.307333	+	0.129064i
$172$		0			0
$173$		−	4.86251i		−	0.369690i	−0.982768	−	0.184845i	$-0.940822\pi$
$173$		−	4.86251i		−	0.369690i	0.982768	−	0.184845i	$-0.0591783\pi$
$174$		0			0
$175$		−	35.2737i		−	2.66644i
$176$		0			0
$177$	−2.64038			−0.198463
$178$		0			0
$179$	−2.48026			−0.185384			−0.0926918	−	0.995695i	$-0.529547\pi$
$179$	−2.48026			−0.185384			−0.0926918	+	0.995695i	$0.529547\pi$
$180$		0			0
$181$		−	2.50994i		−	0.186562i	−0.995640	−	0.0932812i	$-0.970264\pi$
$181$		−	2.50994i		−	0.186562i	0.995640	−	0.0932812i	$-0.0297356\pi$
$182$		0			0
$183$	−0.164898			−0.0121896
$184$		0			0
$185$		−	42.0218i		−	3.08950i
$186$		0			0
$187$		−	15.2909i		−	1.11818i
$188$		0			0
$189$		−	3.06179i		−	0.222713i
$190$		0			0
$191$		−	0.120985i		−	0.00875419i	−0.999990	−	0.00437709i	$-0.998607\pi$
$191$		−	0.120985i		−	0.00875419i	0.999990	−	0.00437709i	$-0.00139328\pi$
$192$		0			0
$193$		−	4.65187i		−	0.334849i	−0.985885	−	0.167424i	$-0.946455\pi$
$193$		−	4.65187i		−	0.334849i	0.985885	−	0.167424i	$-0.0535450\pi$
$194$		0			0
$195$			19.7935i			1.41744i
$196$		0			0
$197$	0.540217			0.0384889			0.0192444	−	0.999815i	$-0.493874\pi$
$197$	0.540217			0.0384889			0.0192444	+	0.999815i	$0.493874\pi$
$198$		0			0
$199$			3.41200i			0.241870i	0.992660	+	0.120935i	$0.0385893\pi$
$199$			3.41200i			0.241870i	−0.992660	+	0.120935i	$0.961411\pi$
$200$		0			0
$201$	−5.34581			−0.377064
$202$		0			0
$203$	21.7439			1.52613
$204$		0			0
$205$			4.89104i			0.341605i
$206$		0			0
$207$			3.92817i			0.273027i
$208$		0			0
$209$	−3.43350	+	8.17601i	−0.237500	+	0.565546i
$210$		0			0
$211$	9.72198			0.669289			0.334644	−	0.942344i	$-0.391384\pi$
$211$	9.72198			0.669289			0.334644	+	0.942344i	$0.391384\pi$
$212$		0			0
$213$	5.59793			0.383564
$214$		0			0
$215$		−	37.6272i		−	2.56615i
$216$		0			0
$217$			11.4983i			0.780556i
$218$		0			0
$219$	−3.78023			−0.255445
$220$		0			0
$221$			36.6025i			2.46215i
$222$		0			0
$223$	−14.2911			−0.957004			−0.478502	−	0.878086i	$-0.658820\pi$
$223$	−14.2911			−0.957004			−0.478502	+	0.878086i	$0.658820\pi$
$224$		0			0
$225$	11.5206			0.768039
$226$		0			0
$227$	−15.0916			−1.00166			−0.500832	−	0.865544i	$-0.666973\pi$
$227$	−15.0916			−1.00166			−0.500832	+	0.865544i	$0.666973\pi$
$228$		0			0
$229$	7.95563			0.525723			0.262862	−	0.964834i	$-0.415334\pi$
$229$	7.95563			0.525723			0.262862	+	0.964834i	$0.415334\pi$
$230$		0			0
$231$	6.22888			0.409830
$232$		0			0
$233$	−15.7550			−1.03215			−0.516073	−	0.856545i	$-0.672607\pi$
$233$	−15.7550			−1.03215			−0.516073	+	0.856545i	$0.672607\pi$
$234$		0			0
$235$			37.1294i			2.42205i
$236$		0			0
$237$	−16.4583			−1.06908
$238$		0			0
$239$			9.52997i			0.616442i	0.951315	+	0.308221i	$0.0997337\pi$
$239$			9.52997i			0.616442i	−0.951315	+	0.308221i	$0.900266\pi$
$240$		0			0
$241$			19.8680i			1.27981i	0.768454	+	0.639906i	$0.221027\pi$
$241$			19.8680i			1.27981i	−0.768454	+	0.639906i	$0.778973\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$	9.65158			0.616617
$246$		0			0
$247$	8.21889	−	19.5712i	0.522956	−	1.24529i
$248$		0			0
$249$			10.8400i			0.686956i
$250$		0			0
$251$		−	29.9086i		−	1.88781i	−0.330211	−	0.943907i	$-0.607120\pi$
$251$		−	29.9086i		−	1.88781i	0.330211	−	0.943907i	$-0.392880\pi$
$252$		0			0
$253$	−7.99143			−0.502417
$254$		0			0
$255$	30.5501			1.91312
$256$		0			0
$257$		−	24.3801i		−	1.52079i	−0.649461	−	0.760395i	$-0.725005\pi$
$257$		−	24.3801i		−	1.52079i	0.649461	−	0.760395i	$-0.274995\pi$
$258$		0			0
$259$	31.6547			1.96692
$260$		0			0
$261$			7.10170i			0.439584i
$262$		0			0
$263$		−	15.0267i		−	0.926583i	−0.886206	−	0.463292i	$-0.846668\pi$
$263$		−	15.0267i		−	0.926583i	0.886206	−	0.463292i	$-0.153332\pi$
$264$		0			0
$265$			4.89104i			0.300454i
$266$		0			0
$267$			13.6422i			0.834891i
$268$		0			0
$269$			4.46913i			0.272488i	0.990675	+	0.136244i	$0.0435030\pi$
$269$			4.46913i			0.272488i	−0.990675	+	0.136244i	$0.956497\pi$
$270$		0			0
$271$			23.0905i			1.40265i	0.712843	+	0.701323i	$0.247407\pi$
$271$			23.0905i			1.40265i	−0.712843	+	0.701323i	$0.752593\pi$
$272$		0			0
$273$	−14.9103			−0.902412
$274$		0			0
$275$			23.4374i			1.41333i
$276$		0			0
$277$	−20.9212			−1.25703			−0.628517	−	0.777796i	$-0.716338\pi$
$277$	−20.9212			−1.25703			−0.628517	+	0.777796i	$0.716338\pi$
$278$		0			0
$279$	−3.75541			−0.224831
$280$		0			0
$281$		−	30.7862i		−	1.83655i	−0.395940	−	0.918277i	$-0.629581\pi$
$281$		−	30.7862i		−	1.83655i	0.395940	−	0.918277i	$-0.370419\pi$
$282$		0			0
$283$		−	13.8331i		−	0.822294i	−0.911569	−	0.411147i	$-0.865128\pi$
$283$		−	13.8331i		−	0.822294i	0.911569	−	0.411147i	$-0.134872\pi$
$284$		0			0
$285$	−16.3350	−	6.85987i	−0.967604	−	0.406343i
$286$		0			0
$287$	−3.68438			−0.217482
$288$		0			0
$289$	39.4937			2.32316
$290$		0			0
$291$			2.29631i			0.134612i
$292$		0			0
$293$			9.58353i			0.559876i	0.960018	+	0.279938i	$0.0903138\pi$
$293$			9.58353i			0.559876i	−0.960018	+	0.279938i	$0.909686\pi$
$294$		0			0
$295$	10.7320			0.624839
$296$		0			0
$297$			2.03439i			0.118047i
$298$		0			0
$299$	19.1294			1.10628
$300$		0			0
$301$	28.3443			1.63374
$302$		0			0
$303$	2.19966			0.126367
$304$		0			0
$305$	0.670235			0.0383775
$306$		0			0
$307$	21.5591			1.23044			0.615221	−	0.788354i	$-0.289067\pi$
$307$	21.5591			1.23044			0.615221	+	0.788354i	$0.289067\pi$
$308$		0			0
$309$	−15.1140			−0.859808
$310$		0			0
$311$			20.8245i			1.18085i	0.807092	+	0.590426i	$0.201040\pi$
$311$			20.8245i			1.18085i	−0.807092	+	0.590426i	$0.798960\pi$
$312$		0			0
$313$	9.85255			0.556899			0.278450	−	0.960451i	$-0.410179\pi$
$313$	9.85255			0.556899			0.278450	+	0.960451i	$0.410179\pi$
$314$		0			0
$315$			12.4448i			0.701186i
$316$		0			0
$317$			7.93043i			0.445417i	0.974885	+	0.222709i	$0.0714899\pi$
$317$			7.93043i			0.445417i	−0.974885	+	0.222709i	$0.928510\pi$
$318$		0			0
$319$	−14.4476			−0.808912
$320$		0			0
$321$	−12.5242			−0.699033
$322$		0			0
$323$	−30.2070	−	12.6854i	−1.68076	−	0.705832i
$324$		0			0
$325$		−	56.1029i		−	3.11203i
$326$		0			0
$327$			18.2918i			1.01154i
$328$		0			0
$329$	−27.9692			−1.54199
$330$		0			0
$331$	−33.1163			−1.82024			−0.910118	−	0.414348i	$-0.864010\pi$
$331$	−33.1163			−1.82024			−0.910118	+	0.414348i	$0.864010\pi$
$332$		0			0
$333$			10.3386i			0.566552i
$334$		0			0
$335$	21.7283			1.18715
$336$		0			0
$337$			8.19214i			0.446254i	0.974789	+	0.223127i	$0.0716265\pi$
$337$			8.19214i			0.446254i	−0.974789	+	0.223127i	$0.928373\pi$
$338$		0			0
$339$		−	8.29474i		−	0.450508i
$340$		0			0
$341$		−	7.63997i		−	0.413728i
$342$		0			0
$343$		−	14.1621i		−	0.764681i
$344$		0			0
$345$		−	15.9663i		−	0.859594i
$346$		0			0
$347$			22.3320i			1.19884i	0.800434	+	0.599421i	$0.204602\pi$
$347$			22.3320i			1.19884i	−0.800434	+	0.599421i	$0.795398\pi$
$348$		0			0
$349$	−15.7367			−0.842365			−0.421182	−	0.906976i	$-0.638385\pi$
$349$	−15.7367			−0.842365			−0.421182	+	0.906976i	$0.638385\pi$
$350$		0			0
$351$		−	4.86979i		−	0.259930i
$352$		0			0
$353$	4.19003			0.223013			0.111506	−	0.993764i	$-0.464432\pi$
$353$	4.19003			0.223013			0.111506	+	0.993764i	$0.464432\pi$
$354$		0			0
$355$	−22.7531			−1.20761
$356$		0			0
$357$			23.0131i			1.21798i
$358$		0			0
$359$		−	15.5576i		−	0.821098i	−0.911838	−	0.410549i	$-0.865337\pi$
$359$		−	15.5576i		−	0.821098i	0.911838	−	0.410549i	$-0.134663\pi$
$360$		0			0
$361$	13.3031	+	13.5656i	0.700165	+	0.713981i
$362$		0			0
$363$	6.86126			0.360123
$364$		0			0
$365$	15.3650			0.804239
$366$		0			0
$367$			30.1542i			1.57404i	0.616930	+	0.787018i	$0.288376\pi$
$367$			30.1542i			1.57404i	−0.616930	+	0.787018i	$0.711624\pi$
$368$		0			0
$369$		−	1.20334i		−	0.0626434i
$370$		0			0
$371$	−3.68438			−0.191283
$372$		0			0
$373$		−	24.9916i		−	1.29401i	−0.762484	−	0.647007i	$-0.776020\pi$
$373$		−	24.9916i		−	1.29401i	0.762484	−	0.647007i	$-0.223980\pi$
$374$		0			0
$375$	−26.5033			−1.36862
$376$		0			0
$377$	34.5838			1.78116
$378$		0			0
$379$	−9.34382			−0.479960			−0.239980	−	0.970778i	$-0.577141\pi$
$379$	−9.34382			−0.479960			−0.239980	+	0.970778i	$0.577141\pi$
$380$		0			0
$381$	9.60764			0.492214
$382$		0			0
$383$	−31.2055			−1.59453			−0.797263	−	0.603632i	$-0.793720\pi$
$383$	−31.2055			−1.59453			−0.797263	+	0.603632i	$0.793720\pi$
$384$		0			0
$385$	−25.3176			−1.29030
$386$		0			0
$387$			9.25741i			0.470580i
$388$		0			0
$389$	2.27513			0.115354			0.0576768	−	0.998335i	$-0.481631\pi$
$389$	2.27513			0.115354			0.0576768	+	0.998335i	$0.481631\pi$
$390$		0			0
$391$		−	29.5250i		−	1.49314i
$392$		0			0
$393$			4.87062i			0.245690i
$394$		0			0
$395$	66.8957			3.36589
$396$		0			0
$397$	−3.99809			−0.200659			−0.100329	−	0.994954i	$-0.531990\pi$
$397$	−3.99809			−0.200659			−0.100329	+	0.994954i	$0.531990\pi$
$398$		0			0
$399$	5.16748	−	12.3050i	0.258697	−	0.616023i
$400$		0			0
$401$			13.9342i			0.695843i	0.937524	+	0.347921i	$0.113112\pi$
$401$			13.9342i			0.695843i	−0.937524	+	0.347921i	$0.886888\pi$
$402$		0			0
$403$			18.2881i			0.910995i
$404$		0			0
$405$	−4.06455			−0.201969
$406$		0			0
$407$	−21.0327			−1.04255
$408$		0			0
$409$		−	13.3747i		−	0.661335i	−0.943747	−	0.330668i	$-0.892726\pi$
$409$		−	13.3747i		−	0.661335i	0.943747	−	0.330668i	$-0.107274\pi$
$410$		0			0
$411$	5.12929			0.253009
$412$		0			0
$413$			8.08430i			0.397802i
$414$		0			0
$415$		−	44.0597i		−	2.16280i
$416$		0			0
$417$		−	7.88793i		−	0.386274i
$418$		0			0
$419$		−	14.1805i		−	0.692764i	−0.938094	−	0.346382i	$-0.887410\pi$
$419$		−	14.1805i		−	0.692764i	0.938094	−	0.346382i	$-0.112590\pi$
$420$		0			0
$421$			7.59833i			0.370320i	0.982708	+	0.185160i	$0.0592803\pi$
$421$			7.59833i			0.370320i	−0.982708	+	0.185160i	$0.940720\pi$
$422$		0			0
$423$		−	9.13493i		−	0.444155i
$424$		0			0
$425$	−86.5914			−4.20030
$426$		0			0
$427$			0.504882i			0.0244330i
$428$		0			0
$429$	9.90705			0.478317
$430$		0			0
$431$	−30.0511			−1.44751			−0.723756	−	0.690056i	$-0.757586\pi$
$431$	−30.0511			−1.44751			−0.723756	+	0.690056i	$0.757586\pi$
$432$		0			0
$433$		−	17.8938i		−	0.859921i	−0.902848	−	0.429960i	$-0.858528\pi$
$433$		−	17.8938i		−	0.859921i	0.902848	−	0.429960i	$-0.141472\pi$
$434$		0			0
$435$		−	28.8652i		−	1.38398i
$436$		0			0
$437$	−6.62969	+	15.7869i	−0.317141	+	0.755191i
$438$		0			0
$439$	10.0092			0.477713			0.238857	−	0.971055i	$-0.423227\pi$
$439$	10.0092			0.477713			0.238857	+	0.971055i	$0.423227\pi$
$440$		0			0
$441$	−2.37457			−0.113075
$442$		0			0
$443$			29.3957i			1.39663i	0.715790	+	0.698315i	$0.246067\pi$
$443$			29.3957i			1.39663i	−0.715790	+	0.698315i	$0.753933\pi$
$444$		0			0
$445$		−	55.4496i		−	2.62856i
$446$		0			0
$447$	18.8427			0.891230
$448$		0			0
$449$		−	11.4726i		−	0.541426i	−0.962660	−	0.270713i	$-0.912741\pi$
$449$		−	11.4726i		−	0.541426i	0.962660	−	0.270713i	$-0.0872595\pi$
$450$		0			0
$451$	2.44806			0.115275
$452$		0			0
$453$	9.56271			0.449295
$454$		0			0
$455$	60.6037			2.84114
$456$		0			0
$457$	−13.6106			−0.636679			−0.318339	−	0.947977i	$-0.603125\pi$
$457$	−13.6106			−0.636679			−0.318339	+	0.947977i	$0.603125\pi$
$458$		0			0
$459$	−7.51623			−0.350827
$460$		0			0
$461$	−28.0238			−1.30520			−0.652599	−	0.757704i	$-0.726321\pi$
$461$	−28.0238			−1.30520			−0.652599	+	0.757704i	$0.726321\pi$
$462$		0			0
$463$			6.24370i			0.290169i	0.989419	+	0.145085i	$0.0463455\pi$
$463$			6.24370i			0.290169i	−0.989419	+	0.145085i	$0.953655\pi$
$464$		0			0
$465$	15.2641			0.707855
$466$		0			0
$467$			1.58161i			0.0731882i	0.999330	+	0.0365941i	$0.0116509\pi$
$467$			1.58161i			0.0731882i	−0.999330	+	0.0365941i	$0.988349\pi$
$468$		0			0
$469$			16.3678i			0.755793i
$470$		0			0
$471$	−14.5309			−0.669548
$472$		0			0
$473$	−18.8332			−0.865950
$474$		0			0
$475$	46.3001	+	19.4436i	2.12440	+	0.892135i
$476$		0			0
$477$		−	1.20334i		−	0.0550972i
$478$		0			0
$479$			21.4085i			0.978179i	0.872234	+	0.489089i	$0.162671\pi$
$479$			21.4085i			0.978179i	−0.872234	+	0.489089i	$0.837329\pi$
$480$		0			0
$481$	50.3468			2.29562
$482$		0			0
$483$	12.0272			0.547259
$484$		0			0
$485$		−	9.33349i		−	0.423812i
$486$		0			0
$487$	−1.63273			−0.0739862			−0.0369931	−	0.999316i	$-0.511778\pi$
$487$	−1.63273			−0.0739862			−0.0369931	+	0.999316i	$0.511778\pi$
$488$		0			0
$489$			0.676154i			0.0305767i
$490$		0			0
$491$		−	13.6535i		−	0.616173i	−0.951358	−	0.308086i	$-0.900311\pi$
$491$		−	13.6535i		−	0.616173i	0.951358	−	0.308086i	$-0.0996886\pi$
$492$		0			0
$493$		−	53.3780i		−	2.40402i
$494$		0			0
$495$		−	8.26888i		−	0.371659i
$496$		0			0
$497$		−	17.1397i		−	0.768821i
$498$		0			0
$499$		−	13.8795i		−	0.621330i	−0.950519	−	0.310665i	$-0.899448\pi$
$499$		−	13.8795i		−	0.621330i	0.950519	−	0.310665i	$-0.100552\pi$
$500$		0			0
$501$	−4.52041			−0.201957
$502$		0			0
$503$		−	14.5055i		−	0.646770i	−0.946267	−	0.323385i	$-0.895179\pi$
$503$		−	14.5055i		−	0.646770i	0.946267	−	0.323385i	$-0.104821\pi$
$504$		0			0
$505$	−8.94063			−0.397853
$506$		0			0
$507$	−10.7149			−0.475864
$508$		0			0
$509$			29.5849i			1.31133i	0.755053	+	0.655664i	$0.227611\pi$
$509$			29.5849i			1.31133i	−0.755053	+	0.655664i	$0.772389\pi$
$510$		0			0
$511$			11.5743i			0.512017i
$512$		0			0
$513$	4.01890	+	1.68773i	0.177439	+	0.0745151i
$514$		0			0
$515$	61.4319			2.70701
$516$		0			0
$517$	18.5840			0.817323
$518$		0			0
$519$		−	4.86251i		−	0.213440i
$520$		0			0
$521$			6.48642i			0.284175i	0.989854	+	0.142088i	$0.0453815\pi$
$521$			6.48642i			0.284175i	−0.989854	+	0.142088i	$0.954619\pi$
$522$		0			0
$523$	−29.1143			−1.27308			−0.636540	−	0.771244i	$-0.719635\pi$
$523$	−29.1143			−1.27308			−0.636540	+	0.771244i	$0.719635\pi$
$524$		0			0
$525$		−	35.2737i		−	1.53947i
$526$		0			0
$527$	28.2266			1.22957
$528$		0			0
$529$	7.56948			0.329108
$530$		0			0
$531$	−2.64038			−0.114583
$532$		0			0
$533$	−5.86002			−0.253826
$534$		0			0
$535$	50.9053			2.20083
$536$		0			0
$537$	−2.48026			−0.107031
$538$		0			0
$539$		−	4.83081i		−	0.208078i
$540$		0			0
$541$	−3.32843			−0.143100			−0.0715501	−	0.997437i	$-0.522795\pi$
$541$	−3.32843			−0.143100			−0.0715501	+	0.997437i	$0.522795\pi$
$542$		0			0
$543$		−	2.50994i		−	0.107712i
$544$		0			0
$545$		−	74.3479i		−	3.18471i
$546$		0			0
$547$	−3.13876			−0.134203			−0.0671017	−	0.997746i	$-0.521375\pi$
$547$	−3.13876			−0.134203			−0.0671017	+	0.997746i	$0.521375\pi$
$548$		0			0
$549$	−0.164898			−0.00703766
$550$		0			0
$551$	−11.9858	+	28.5410i	−0.510610	+	1.21589i
$552$		0			0
$553$			50.3919i			2.14288i
$554$		0			0
$555$		−	42.0218i		−	1.78373i
$556$		0			0
$557$	33.1616			1.40510			0.702552	−	0.711633i	$-0.252044\pi$
$557$	33.1616			1.40510			0.702552	+	0.711633i	$0.252044\pi$
$558$		0			0
$559$	45.0816			1.90675
$560$		0			0
$561$		−	15.2909i		−	0.645584i
$562$		0			0
$563$	5.77422			0.243354			0.121677	−	0.992570i	$-0.461173\pi$
$563$	5.77422			0.243354			0.121677	+	0.992570i	$0.461173\pi$
$564$		0			0
$565$			33.7144i			1.41838i
$566$		0			0
$567$		−	3.06179i		−	0.128583i
$568$		0			0
$569$			15.4858i			0.649197i	0.945852	+	0.324599i	$0.105229\pi$
$569$			15.4858i			0.649197i	−0.945852	+	0.324599i	$0.894771\pi$
$570$		0			0
$571$		−	46.1672i		−	1.93204i	−0.258476	−	0.966018i	$-0.583220\pi$
$571$		−	46.1672i		−	1.93204i	0.258476	−	0.966018i	$-0.416780\pi$
$572$		0			0
$573$		−	0.120985i		−	0.00505423i
$574$		0			0
$575$			45.2548i			1.88726i
$576$		0			0
$577$	0.208981			0.00870000			0.00435000	−	0.999991i	$-0.498615\pi$
$577$	0.208981			0.00870000			0.00435000	+	0.999991i	$0.498615\pi$
$578$		0			0
$579$		−	4.65187i		−	0.193325i
$580$		0			0
$581$	33.1898			1.37694
$582$		0			0
$583$	2.44806			0.101388
$584$		0			0
$585$			19.7935i			0.818362i
$586$		0			0
$587$		−	1.63072i		−	0.0673069i	−0.999434	−	0.0336534i	$-0.989286\pi$
$587$		−	1.63072i		−	0.0673069i	0.999434	−	0.0336534i	$-0.0107142\pi$
$588$		0			0
$589$	−15.0926	−	6.33812i	−0.621881	−	0.261158i
$590$		0			0
$591$	0.540217			0.0222216
$592$		0			0
$593$	0.819037			0.0336338			0.0168169	−	0.999859i	$-0.494647\pi$
$593$	0.819037			0.0336338			0.0168169	+	0.999859i	$0.494647\pi$
$594$		0			0
$595$		−	93.5381i		−	3.83469i
$596$		0			0
$597$			3.41200i			0.139644i
$598$		0			0
$599$	7.49935			0.306415			0.153208	−	0.988194i	$-0.451040\pi$
$599$	7.49935			0.306415			0.153208	+	0.988194i	$0.451040\pi$
$600$		0			0
$601$			8.60924i			0.351178i	0.984464	+	0.175589i	$0.0561830\pi$
$601$			8.60924i			0.351178i	−0.984464	+	0.175589i	$0.943817\pi$
$602$		0			0
$603$	−5.34581			−0.217698
$604$		0			0
$605$	−27.8880			−1.13381
$606$		0			0
$607$	−41.4380			−1.68192			−0.840959	−	0.541099i	$-0.818008\pi$
$607$	−41.4380			−1.68192			−0.840959	+	0.541099i	$0.818008\pi$
$608$		0			0
$609$	21.7439			0.881109
$610$		0			0
$611$	−44.4852			−1.79968
$612$		0			0
$613$	−0.281050			−0.0113515			−0.00567575	−	0.999984i	$-0.501807\pi$
$613$	−0.281050			−0.0113515			−0.00567575	+	0.999984i	$0.501807\pi$
$614$		0			0
$615$			4.89104i			0.197226i
$616$		0			0
$617$	−3.55763			−0.143225			−0.0716123	−	0.997433i	$-0.522814\pi$
$617$	−3.55763			−0.143225			−0.0716123	+	0.997433i	$0.522814\pi$
$618$		0			0
$619$			2.43426i			0.0978411i	0.998803	+	0.0489205i	$0.0155781\pi$
$619$			2.43426i			0.0978411i	−0.998803	+	0.0489205i	$0.984422\pi$
$620$		0			0
$621$			3.92817i			0.157632i
$622$		0			0
$623$	41.7697			1.67347
$624$		0			0
$625$	50.1211			2.00484
$626$		0			0
$627$	−3.43350	+	8.17601i	−0.137121	+	0.326518i
$628$		0			0
$629$		−	77.7073i		−	3.09839i
$630$		0			0
$631$			15.6441i			0.622781i	0.950282	+	0.311390i	$0.100795\pi$
$631$			15.6441i			0.622781i	−0.950282	+	0.311390i	$0.899205\pi$
$632$		0			0
$633$	9.72198			0.386414
$634$		0			0
$635$	−39.0508			−1.54968
$636$		0			0
$637$			11.5637i			0.458170i
$638$		0			0
$639$	5.59793			0.221451
$640$		0			0
$641$		−	11.3500i		−	0.448299i	−0.974555	−	0.224150i	$-0.928040\pi$
$641$		−	11.3500i		−	0.448299i	0.974555	−	0.224150i	$-0.0719605\pi$
$642$		0			0
$643$		−	25.8299i		−	1.01863i	−0.860579	−	0.509316i	$-0.829898\pi$
$643$		−	25.8299i		−	1.01863i	0.860579	−	0.509316i	$-0.170102\pi$
$644$		0			0
$645$		−	37.6272i		−	1.48157i
$646$		0			0
$647$		−	4.77393i		−	0.187683i	−0.995587	−	0.0938414i	$-0.970085\pi$
$647$		−	4.77393i		−	0.187683i	0.995587	−	0.0938414i	$-0.0299147\pi$
$648$		0			0
$649$		−	5.37156i		−	0.210852i
$650$		0			0
$651$			11.4983i			0.450654i
$652$		0			0
$653$	−3.24096			−0.126829			−0.0634143	−	0.997987i	$-0.520199\pi$
$653$	−3.24096			−0.126829			−0.0634143	+	0.997987i	$0.520199\pi$
$654$		0			0
$655$		−	19.7969i		−	0.773529i
$656$		0			0
$657$	−3.78023			−0.147481
$658$		0			0
$659$	−25.4491			−0.991357			−0.495679	−	0.868506i	$-0.665081\pi$
$659$	−25.4491			−0.991357			−0.495679	+	0.868506i	$0.665081\pi$
$660$		0			0
$661$		−	11.3163i		−	0.440153i	−0.975483	−	0.220076i	$-0.929369\pi$
$661$		−	11.3163i		−	0.440153i	0.975483	−	0.220076i	$-0.0706307\pi$
$662$		0			0
$663$			36.6025i			1.42152i
$664$		0			0
$665$	−21.0035	+	50.0145i	−0.814480	+	1.93948i
$666$		0			0
$667$	−27.8967			−1.08016
$668$		0			0
$669$	−14.2911			−0.552526
$670$		0			0
$671$		−	0.335466i		−	0.0129505i
$672$		0			0
$673$		−	14.4001i		−	0.555082i	−0.960714	−	0.277541i	$-0.910481\pi$
$673$		−	14.4001i		−	0.555082i	0.960714	−	0.277541i	$-0.0895195\pi$
$674$		0			0
$675$	11.5206			0.443428
$676$		0			0
$677$			48.6759i			1.87077i	0.353636	+	0.935383i	$0.384945\pi$
$677$			48.6759i			1.87077i	−0.353636	+	0.935383i	$0.615055\pi$
$678$		0			0
$679$	7.03084			0.269819
$680$		0			0
$681$	−15.0916			−0.578311
$682$		0			0
$683$	−20.4458			−0.782338			−0.391169	−	0.920319i	$-0.627929\pi$
$683$	−20.4458			−0.782338			−0.391169	+	0.920319i	$0.627929\pi$
$684$		0			0
$685$	−20.8483			−0.796572
$686$		0			0
$687$	7.95563			0.303526
$688$		0			0
$689$	−5.86002			−0.223249
$690$		0			0
$691$			40.2948i			1.53289i	0.642311	+	0.766444i	$0.277976\pi$
$691$			40.2948i			1.53289i	−0.642311	+	0.766444i	$0.722024\pi$
$692$		0			0
$693$	6.22888			0.236615
$694$		0			0
$695$			32.0609i			1.21614i
$696$		0			0
$697$			9.04458i			0.342588i
$698$		0			0
$699$	−15.7550			−0.595910
$700$		0			0
$701$	−2.34913			−0.0887255			−0.0443628	−	0.999015i	$-0.514126\pi$
$701$	−2.34913			−0.0887255			−0.0443628	+	0.999015i	$0.514126\pi$
$702$		0			0
$703$	−17.4488	+	41.5498i	−0.658092	+	1.56708i
$704$		0			0
$705$			37.1294i			1.39837i
$706$		0			0
$707$		−	6.73490i		−	0.253292i
$708$		0			0
$709$	8.24257			0.309556			0.154778	−	0.987949i	$-0.450534\pi$
$709$	8.24257			0.309556			0.154778	+	0.987949i	$0.450534\pi$
$710$		0			0
$711$	−16.4583			−0.617235
$712$		0			0
$713$		−	14.7519i		−	0.552463i
$714$		0			0
$715$	−40.2677			−1.50593
$716$		0			0
$717$			9.52997i			0.355903i
$718$		0			0
$719$		−	23.9958i		−	0.894892i	−0.894311	−	0.447446i	$-0.852334\pi$
$719$		−	23.9958i		−	0.894892i	0.894311	−	0.447446i	$-0.147666\pi$
$720$		0			0
$721$			46.2761i			1.72341i
$722$		0			0
$723$			19.8680i			0.738899i
$724$		0			0
$725$			81.8158i			3.03856i
$726$		0			0
$727$			42.3423i			1.57039i	0.619249	+	0.785195i	$0.287437\pi$
$727$			42.3423i			1.57039i	−0.619249	+	0.785195i	$0.712563\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$		−	69.5808i		−	2.57354i
$732$		0			0
$733$	−1.52984			−0.0565059			−0.0282529	−	0.999601i	$-0.508994\pi$
$733$	−1.52984			−0.0565059			−0.0282529	+	0.999601i	$0.508994\pi$
$734$		0			0
$735$	9.65158			0.356004
$736$		0			0
$737$		−	10.8755i		−	0.400603i
$738$		0			0
$739$			13.3504i			0.491104i	0.969383	+	0.245552i	$0.0789692\pi$
$739$			13.3504i			0.491104i	−0.969383	+	0.245552i	$0.921031\pi$
$740$		0			0
$741$	8.21889	−	19.5712i	0.301929	−	0.718967i
$742$		0			0
$743$	15.5603			0.570851			0.285426	−	0.958401i	$-0.407865\pi$
$743$	15.5603			0.570851			0.285426	+	0.958401i	$0.407865\pi$
$744$		0			0
$745$	−76.5872			−2.80594
$746$		0			0
$747$			10.8400i			0.396614i
$748$		0			0
$749$			38.3465i			1.40115i
$750$		0			0
$751$	−30.4073			−1.10958			−0.554788	−	0.831991i	$-0.687201\pi$
$751$	−30.4073			−1.10958			−0.554788	+	0.831991i	$0.687201\pi$
$752$		0			0
$753$		−	29.9086i		−	1.08993i
$754$		0			0
$755$	−38.8682			−1.41456
$756$		0			0
$757$	−3.49290			−0.126952			−0.0634758	−	0.997983i	$-0.520219\pi$
$757$	−3.49290			−0.126952			−0.0634758	+	0.997983i	$0.520219\pi$
$758$		0			0
$759$	−7.99143			−0.290070
$760$		0			0
$761$	17.3773			0.629928			0.314964	−	0.949104i	$-0.398008\pi$
$761$	17.3773			0.629928			0.314964	+	0.949104i	$0.398008\pi$
$762$		0			0
$763$	56.0056			2.02754
$764$		0			0
$765$	30.5501			1.10454
$766$		0			0
$767$			12.8581i			0.464279i
$768$		0			0
$769$	18.7505			0.676159			0.338079	−	0.941118i	$-0.390223\pi$
$769$	18.7505			0.676159			0.338079	+	0.941118i	$0.390223\pi$
$770$		0			0
$771$		−	24.3801i		−	0.878029i
$772$		0			0
$773$			16.7315i			0.601790i	0.953657	+	0.300895i	$0.0972854\pi$
$773$			16.7315i			0.601790i	−0.953657	+	0.300895i	$0.902715\pi$
$774$		0			0
$775$	−43.2646			−1.55411
$776$		0			0
$777$	31.6547			1.13560
$778$		0			0
$779$	2.03091	−	4.83611i	0.0727650	−	0.173272i
$780$		0			0
$781$			11.3884i			0.407508i
$782$		0			0
$783$			7.10170i			0.253794i
$784$		0			0
$785$	59.0616			2.10800
$786$		0			0
$787$	13.9538			0.497398			0.248699	−	0.968581i	$-0.419997\pi$
$787$	13.9538			0.497398			0.248699	+	0.968581i	$0.419997\pi$
$788$		0			0
$789$		−	15.0267i		−	0.534963i
$790$		0			0
$791$	−25.3968			−0.903005
$792$		0			0
$793$			0.803017i			0.0285160i
$794$		0			0
$795$			4.89104i			0.173467i
$796$		0			0
$797$		−	21.2604i		−	0.753083i	−0.926400	−	0.376541i	$-0.877113\pi$
$797$		−	21.2604i		−	0.753083i	0.926400	−	0.376541i	$-0.122887\pi$
$798$		0			0
$799$			68.6602i			2.42902i
$800$		0			0
$801$			13.6422i			0.482024i
$802$		0			0
$803$		−	7.69047i		−	0.271391i
$804$		0			0
$805$	−48.8854			−1.72298
$806$		0			0
$807$			4.46913i			0.157321i
$808$		0			0
$809$	5.95675			0.209428			0.104714	−	0.994502i	$-0.466607\pi$
$809$	5.95675			0.209428			0.104714	+	0.994502i	$0.466607\pi$
$810$		0			0
$811$	−22.8854			−0.803617			−0.401808	−	0.915724i	$-0.631618\pi$
$811$	−22.8854			−0.803617			−0.401808	+	0.915724i	$0.631618\pi$
$812$		0			0
$813$			23.0905i			0.809818i
$814$		0			0
$815$		−	2.74826i		−	0.0962675i
$816$		0			0
$817$	−15.6240	+	37.2046i	−0.546614	+	1.30162i
$818$		0			0
$819$	−14.9103			−0.521008
$820$		0			0
$821$	−48.7037			−1.69977			−0.849885	−	0.526969i	$-0.823328\pi$
$821$	−48.7037			−1.69977			−0.849885	+	0.526969i	$0.823328\pi$
$822$		0			0
$823$			21.0140i			0.732501i	0.930516	+	0.366251i	$0.119359\pi$
$823$			21.0140i			0.732501i	−0.930516	+	0.366251i	$0.880641\pi$
$824$		0			0
$825$			23.4374i			0.815984i
$826$		0			0
$827$	−1.42632			−0.0495978			−0.0247989	−	0.999692i	$-0.507895\pi$
$827$	−1.42632			−0.0495978			−0.0247989	+	0.999692i	$0.507895\pi$
$828$		0			0
$829$		−	10.5557i		−	0.366616i	−0.983056	−	0.183308i	$-0.941319\pi$
$829$		−	10.5557i		−	0.366616i	0.983056	−	0.183308i	$-0.0586806\pi$
$830$		0			0
$831$	−20.9212			−0.725749
$832$		0			0
$833$	17.8478			0.618391
$834$		0			0
$835$	18.3735			0.635840
$836$		0			0
$837$	−3.75541			−0.129806
$838$		0			0
$839$	−45.4094			−1.56771			−0.783853	−	0.620947i	$-0.786748\pi$
$839$	−45.4094			−1.56771			−0.783853	+	0.620947i	$0.786748\pi$
$840$		0			0
$841$	−21.4342			−0.739110
$842$		0			0
$843$		−	30.7862i		−	1.06033i
$844$		0			0
$845$	43.5512			1.49821
$846$		0			0
$847$		−	21.0078i		−	0.721835i
$848$		0			0
$849$		−	13.8331i		−	0.474752i
$850$		0			0
$851$	−40.6118			−1.39215
$852$		0			0
$853$	40.2287			1.37741			0.688703	−	0.725044i	$-0.258181\pi$
$853$	40.2287			1.37741			0.688703	+	0.725044i	$0.258181\pi$
$854$		0			0
$855$	−16.3350	−	6.85987i	−0.558647	−	0.234602i
$856$		0			0
$857$		−	25.6521i		−	0.876260i	−0.898912	−	0.438130i	$-0.855641\pi$
$857$		−	25.6521i		−	0.876260i	0.898912	−	0.438130i	$-0.144359\pi$
$858$		0			0
$859$			45.1176i			1.53939i	0.638410	+	0.769697i	$0.279593\pi$
$859$			45.1176i			1.53939i	−0.638410	+	0.769697i	$0.720407\pi$
$860$		0			0
$861$	−3.68438			−0.125563
$862$		0			0
$863$	19.7164			0.671155			0.335577	−	0.942013i	$-0.391069\pi$
$863$	19.7164			0.671155			0.335577	+	0.942013i	$0.391069\pi$
$864$		0			0
$865$			19.7639i			0.671994i
$866$		0			0
$867$	39.4937			1.34128
$868$		0			0
$869$		−	33.4826i		−	1.13582i
$870$		0			0
$871$			26.0330i			0.882094i
$872$		0			0
$873$			2.29631i			0.0777185i
$874$		0			0
$875$			81.1476i			2.74329i
$876$		0			0
$877$		−	7.09803i		−	0.239683i	−0.992793	−	0.119842i	$-0.961761\pi$
$877$		−	7.09803i		−	0.239683i	0.992793	−	0.119842i	$-0.0382387\pi$
$878$		0			0
$879$			9.58353i			0.323244i
$880$		0			0
$881$	45.5558			1.53481			0.767406	−	0.641161i	$-0.221547\pi$
$881$	45.5558			1.53481			0.767406	+	0.641161i	$0.221547\pi$
$882$		0			0
$883$		−	46.1330i		−	1.55250i	−0.630425	−	0.776250i	$-0.717120\pi$
$883$		−	46.1330i		−	1.55250i	0.630425	−	0.776250i	$-0.282880\pi$
$884$		0			0
$885$	10.7320			0.360751
$886$		0			0
$887$	5.04155			0.169279			0.0846394	−	0.996412i	$-0.473026\pi$
$887$	5.04155			0.169279			0.0846394	+	0.996412i	$0.473026\pi$
$888$		0			0
$889$		−	29.4166i		−	0.986601i
$890$		0			0
$891$			2.03439i			0.0681546i
$892$		0			0
$893$	15.4173	−	36.7124i	0.515920	−	1.22853i
$894$		0			0
$895$	10.0812			0.336976
$896$		0			0
$897$	19.1294			0.638711
$898$		0			0
$899$		−	26.6698i		−	0.889489i
$900$		0			0
$901$			9.04458i			0.301319i
$902$		0			0
$903$	28.3443			0.943238
$904$		0			0
$905$			10.2018i			0.339119i
$906$		0			0
$907$	−26.0588			−0.865269			−0.432635	−	0.901569i	$-0.642416\pi$
$907$	−26.0588			−0.865269			−0.432635	+	0.901569i	$0.642416\pi$
$908$		0			0
$909$	2.19966			0.0729581
$910$		0			0
$911$	−6.15929			−0.204066			−0.102033	−	0.994781i	$-0.532535\pi$
$911$	−6.15929			−0.204066			−0.102033	+	0.994781i	$0.532535\pi$
$912$		0			0
$913$	−22.0527			−0.729839
$914$		0			0
$915$	0.670235			0.0221573
$916$		0			0
$917$	14.9128			0.492465
$918$		0			0
$919$			8.25239i			0.272221i	0.990694	+	0.136111i	$0.0434603\pi$
$919$			8.25239i			0.272221i	−0.990694	+	0.136111i	$0.956540\pi$
$920$		0			0
$921$	21.5591			0.710396
$922$		0			0
$923$		−	27.2608i		−	0.897299i
$924$		0			0
$925$			119.107i			3.91621i
$926$		0			0
$927$	−15.1140			−0.496411
$928$		0			0
$929$	40.7458			1.33683			0.668414	−	0.743790i	$-0.266974\pi$
$929$	40.7458			1.33683			0.668414	+	0.743790i	$0.266974\pi$
$930$		0			0
$931$	−9.54318	−	4.00764i	−0.312765	−	0.131345i
$932$		0			0
$933$			20.8245i			0.681765i
$934$		0			0
$935$			62.1508i			2.03255i
$936$		0			0
$937$	−43.2705			−1.41359			−0.706793	−	0.707421i	$-0.749859\pi$
$937$	−43.2705			−1.41359			−0.706793	+	0.707421i	$0.749859\pi$
$938$		0			0
$939$	9.85255			0.321526
$940$		0			0
$941$			46.4011i			1.51263i	0.654207	+	0.756316i	$0.273003\pi$
$941$			46.4011i			1.51263i	−0.654207	+	0.756316i	$0.726997\pi$
$942$		0			0
$943$	4.72693			0.153930
$944$		0			0
$945$			12.4448i			0.404830i
$946$		0			0
$947$			13.5395i			0.439975i	0.975503	+	0.219988i	$0.0706017\pi$
$947$			13.5395i			0.439975i	−0.975503	+	0.219988i	$0.929398\pi$
$948$		0			0
$949$			18.4090i			0.597580i
$950$		0			0
$951$			7.93043i			0.257162i
$952$		0			0
$953$			33.4928i			1.08494i	0.840075	+	0.542470i	$0.182511\pi$
$953$			33.4928i			1.08494i	−0.840075	+	0.542470i	$0.817489\pi$
$954$		0			0
$955$			0.491751i			0.0159127i
$956$		0			0
$957$	−14.4476			−0.467025
$958$		0			0
$959$		−	15.7048i		−	0.507135i
$960$		0			0
$961$	−16.8969			−0.545060
$962$		0			0
$963$	−12.5242			−0.403587
$964$		0			0
$965$			18.9078i			0.608662i
$966$		0			0
$967$			13.2882i			0.427321i	0.976908	+	0.213660i	$0.0685386\pi$
$967$			13.2882i			0.427321i	−0.976908	+	0.213660i	$0.931461\pi$
$968$		0			0
$969$	−30.2070	−	12.6854i	−0.970388	−	0.407512i
$970$		0			0
$971$	−4.34514			−0.139442			−0.0697211	−	0.997567i	$-0.522211\pi$
$971$	−4.34514			−0.139442			−0.0697211	+	0.997567i	$0.522211\pi$
$972$		0			0
$973$	−24.1512			−0.774252
$974$		0			0
$975$		−	56.1029i		−	1.79673i
$976$		0			0
$977$			56.0805i			1.79418i	0.441853	+	0.897088i	$0.354321\pi$
$977$			56.0805i			1.79418i	−0.441853	+	0.897088i	$0.645679\pi$
$978$		0			0
$979$	−27.7536			−0.887009
$980$		0			0
$981$			18.2918i			0.584011i
$982$		0			0
$983$	20.8251			0.664217			0.332109	−	0.943241i	$-0.392240\pi$
$983$	20.8251			0.664217			0.332109	+	0.943241i	$0.392240\pi$
$984$		0			0
$985$	−2.19574			−0.0699621
$986$		0			0
$987$	−27.9692			−0.890271
$988$		0			0
$989$	−36.3647			−1.15633
$990$		0			0
$991$	56.0593			1.78078			0.890392	−	0.455195i	$-0.150431\pi$
$991$	56.0593			1.78078			0.890392	+	0.455195i	$0.150431\pi$
$992$		0			0
$993$	−33.1163			−1.05091
$994$		0			0
$995$		−	13.8683i		−	0.439654i
$996$		0			0
$997$	−31.7496			−1.00552			−0.502760	−	0.864426i	$-0.667682\pi$
$997$	−31.7496			−1.00552			−0.502760	+	0.864426i	$0.667682\pi$
$998$		0			0
$999$			10.3386i			0.327099i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1824.2.k.b.607.1	yes	20
3.2	odd	2		5472.2.k.b.2431.19		20
4.3	odd	2		1824.2.k.a.607.2	yes	20
8.3	odd	2		3648.2.k.l.2431.20		20
8.5	even	2		3648.2.k.k.2431.19		20
12.11	even	2		5472.2.k.a.2431.20		20
19.18	odd	2		1824.2.k.a.607.1	✓	20
57.56	even	2		5472.2.k.a.2431.19		20
76.75	even	2	inner	1824.2.k.b.607.2	yes	20
152.37	odd	2		3648.2.k.l.2431.19		20
152.75	even	2		3648.2.k.k.2431.20		20
228.227	odd	2		5472.2.k.b.2431.20		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1824.2.k.a.607.1	✓	20	19.18	odd	2
1824.2.k.a.607.2	yes	20	4.3	odd	2
1824.2.k.b.607.1	yes	20	1.1	even	1	trivial
1824.2.k.b.607.2	yes	20	76.75	even	2	inner
3648.2.k.k.2431.19		20	8.5	even	2
3648.2.k.k.2431.20		20	152.75	even	2
3648.2.k.l.2431.19		20	152.37	odd	2
3648.2.k.l.2431.20		20	8.3	odd	2
5472.2.k.a.2431.19		20	57.56	even	2
5472.2.k.a.2431.20		20	12.11	even	2
5472.2.k.b.2431.19		20	3.2	odd	2
5472.2.k.b.2431.20		20	228.227	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -4.06455 q^{5} -3.06179i q^{7} +1.00000 q^{9} +2.03439i q^{11} -4.86979i q^{13} -4.06455 q^{15} -7.51623 q^{17} +(4.01890 + 1.68773i) q^{19} -3.06179i q^{21} +3.92817i q^{23} +11.5206 q^{25} +1.00000 q^{27} +7.10170i q^{29} -3.75541 q^{31} +2.03439i q^{33} +12.4448i q^{35} +10.3386i q^{37} -4.86979i q^{39} -1.20334i q^{41} +9.25741i q^{43} -4.06455 q^{45} -9.13493i q^{47} -2.37457 q^{49} -7.51623 q^{51} -1.20334i q^{53} -8.26888i q^{55} +(4.01890 + 1.68773i) q^{57} -2.64038 q^{59} -0.164898 q^{61} -3.06179i q^{63} +19.7935i q^{65} -5.34581 q^{67} +3.92817i q^{69} +5.59793 q^{71} -3.78023 q^{73} +11.5206 q^{75} +6.22888 q^{77} -16.4583 q^{79} +1.00000 q^{81} +10.8400i q^{83} +30.5501 q^{85} +7.10170i q^{87} +13.6422i q^{89} -14.9103 q^{91} -3.75541 q^{93} +(-16.3350 - 6.85987i) q^{95} +2.29631i q^{97} +2.03439i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 20 q^{3} + 20 q^{9} + 4 q^{19} + 20 q^{25} + 20 q^{27} - 8 q^{31} - 44 q^{49} + 4 q^{57} - 8 q^{61} + 8 q^{67} + 64 q^{71} + 8 q^{73} + 20 q^{75} + 40 q^{77} - 16 q^{79} + 20 q^{81} + 40 q^{85} - 8 q^{93}+ \cdots - 8 q^{95}+O(q^{100}) \) 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

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