LMFDB - Embedded newform 1620.4.d.c.649.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(649,1620)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.649");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$95.5830942093$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 21 x^{14} + 18613 x^{12} + 454908 x^{10} + 675247800 x^{8} + 10468697712 x^{6} + \cdots + 46\!\cdots\!16 $ x^16 - 21x^14 + 18613x^12 + 454908x^10 + 675247800x^8 + 10468697712x^6 + 7045978548688x^4 + 30690005743296*x^2 + 46709023184271616
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.15
Root			$-8.13083 - 5.26533i$ of defining polynomial
Character	$\chi$	$=$	1620.649
Dual form			1620.4.d.c.649.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.86288 - 5.26533i) q^{5} +21.0065i q^{7} +O(q^{10})$	$q+(9.86288 - 5.26533i) q^{5} +21.0065i q^{7} -65.1343 q^{11} +58.5280i q^{13} +112.747i q^{17} +26.1803 q^{19} +90.8428i q^{23} +(69.5527 - 103.863i) q^{25} -155.254 q^{29} +5.87661 q^{31} +(110.606 + 207.184i) q^{35} -156.720i q^{37} -64.4994 q^{41} -479.359i q^{43} -420.578i q^{47} -98.2715 q^{49} -182.499i q^{53} +(-642.412 + 342.953i) q^{55} -573.173 q^{59} -7.77692 q^{61} +(308.169 + 577.254i) q^{65} -691.747i q^{67} +169.387 q^{71} -758.429i q^{73} -1368.24i q^{77} -1228.82 q^{79} -583.924i q^{83} +(593.650 + 1112.01i) q^{85} -1119.84 q^{89} -1229.47 q^{91} +(258.213 - 137.848i) q^{95} +874.892i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 84 q^{19} + 54 q^{25} + 88 q^{31} + 84 q^{49} - 1120 q^{55} - 4 q^{61} - 1344 q^{79} + 482 q^{85} - 2100 q^{91}+O(q^{100}) $ 16 * q + 84 * q^19 + 54 * q^25 + 88 * q^31 + 84 * q^49 - 1120 * q^55 - 4 * q^61 - 1344 * q^79 + 482 * q^85 - 2100 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	9.86288	−	5.26533i	0.882163	−	0.470945i
$5$	9.86288	−	5.26533i	0.882163	−	0.470945i
$6$		0			0
$7$			21.0065i			1.13424i	0.823634	+	0.567121i	$0.191943\pi$
$7$			21.0065i			1.13424i	−0.823634	+	0.567121i	$0.808057\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−65.1343			−1.78534			−0.892670	−	0.450711i	$-0.851170\pi$
$11$	−65.1343			−1.78534			−0.892670	+	0.450711i	$0.851170\pi$
$12$		0			0
$13$			58.5280i			1.24867i	0.781156	+	0.624336i	$0.214631\pi$
$13$			58.5280i			1.24867i	−0.781156	+	0.624336i	$0.785369\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			112.747i			1.60854i	0.594264	+	0.804270i	$0.297443\pi$
$17$			112.747i			1.60854i	−0.594264	+	0.804270i	$0.702557\pi$
$18$		0			0
$19$	26.1803			0.316114			0.158057	−	0.987430i	$-0.449477\pi$
$19$	26.1803			0.316114			0.158057	+	0.987430i	$0.449477\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			90.8428i			0.823567i	0.911282	+	0.411783i	$0.135094\pi$
$23$			90.8428i			0.823567i	−0.911282	+	0.411783i	$0.864906\pi$
$24$		0			0
$25$	69.5527	−	103.863i	0.556421	−	0.830900i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−155.254			−0.994133			−0.497067	−	0.867712i	$-0.665590\pi$
$29$	−155.254			−0.994133			−0.497067	+	0.867712i	$0.665590\pi$
$30$		0			0
$31$	5.87661			0.0340474			0.0170237	−	0.999855i	$-0.494581\pi$
$31$	5.87661			0.0340474			0.0170237	+	0.999855i	$0.494581\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	110.606	+	207.184i	0.534166	+	1.00059i
$36$		0			0
$37$		−	156.720i		−	0.696340i	−0.937431	−	0.348170i	$-0.886803\pi$
$37$		−	156.720i		−	0.696340i	0.937431	−	0.348170i	$-0.113197\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−64.4994			−0.245686			−0.122843	−	0.992426i	$-0.539201\pi$
$41$	−64.4994			−0.245686			−0.122843	+	0.992426i	$0.539201\pi$
$42$		0			0
$43$		−	479.359i		−	1.70003i	−0.526754	−	0.850017i	$-0.676591\pi$
$43$		−	479.359i		−	1.70003i	0.526754	−	0.850017i	$-0.323409\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	420.578i		−	1.30527i	−0.757673	−	0.652634i	$-0.773664\pi$
$47$		−	420.578i		−	1.30527i	0.757673	−	0.652634i	$-0.226336\pi$
$48$		0			0
$49$	−98.2715			−0.286506
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	182.499i		−	0.472984i	−0.971633	−	0.236492i	$-0.924002\pi$
$53$		−	182.499i		−	0.472984i	0.971633	−	0.236492i	$-0.0759977\pi$
$54$		0			0
$55$	−642.412	+	342.953i	−1.57496	+	0.840797i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−573.173			−1.26476			−0.632379	−	0.774659i	$-0.717922\pi$
$59$	−573.173			−1.26476			−0.632379	+	0.774659i	$0.717922\pi$
$60$		0			0
$61$	−7.77692			−0.0163235			−0.00816175	−	0.999967i	$-0.502598\pi$
$61$	−7.77692			−0.0163235			−0.00816175	+	0.999967i	$0.502598\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	308.169	+	577.254i	0.588056	+	1.10153i
$66$		0			0
$67$		−	691.747i		−	1.26135i	−0.776048	−	0.630674i	$-0.782778\pi$
$67$		−	691.747i		−	1.26135i	0.776048	−	0.630674i	$-0.217222\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	169.387			0.283135			0.141568	−	0.989929i	$-0.454786\pi$
$71$	169.387			0.283135			0.141568	+	0.989929i	$0.454786\pi$
$72$		0			0
$73$		−	758.429i		−	1.21599i	−0.793940	−	0.607996i	$-0.791974\pi$
$73$		−	758.429i		−	1.21599i	0.793940	−	0.607996i	$-0.208026\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	1368.24i		−	2.02501i
$78$		0			0
$79$	−1228.82			−1.75004			−0.875022	−	0.484083i	$-0.839153\pi$
$79$	−1228.82			−1.75004			−0.875022	+	0.484083i	$0.839153\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	583.924i		−	0.772217i	−0.922453	−	0.386109i	$-0.873819\pi$
$83$		−	583.924i		−	0.772217i	0.922453	−	0.386109i	$-0.126181\pi$
$84$		0			0
$85$	593.650	+	1112.01i	0.757534	+	1.41899i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1119.84			−1.33374			−0.666869	−	0.745175i	$-0.732366\pi$
$89$	−1119.84			−1.33374			−0.666869	+	0.745175i	$0.732366\pi$
$90$		0			0
$91$	−1229.47			−1.41630
$92$		0			0
$93$		0			0
$94$		0			0
$95$	258.213	−	137.848i	0.278864	−	0.148872i
$96$		0			0
$97$			874.892i			0.915792i	0.889006	+	0.457896i	$0.151397\pi$
$97$			874.892i			0.915792i	−0.889006	+	0.457896i	$0.848603\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−25.1395			−0.0247670			−0.0123835	−	0.999923i	$-0.503942\pi$
$101$	−25.1395			−0.0247670			−0.0123835	+	0.999923i	$0.503942\pi$
$102$		0			0
$103$			186.130i			0.178057i	0.996029	+	0.0890287i	$0.0283763\pi$
$103$			186.130i			0.178057i	−0.996029	+	0.0890287i	$0.971624\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			419.870i			0.379349i	0.981847	+	0.189675i	$0.0607433\pi$
$107$			419.870i			0.379349i	−0.981847	+	0.189675i	$0.939257\pi$
$108$		0			0
$109$	2186.39			1.92127			0.960635	−	0.277815i	$-0.0896101\pi$
$109$	2186.39			1.92127			0.960635	+	0.277815i	$0.0896101\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1610.54i			1.34077i	0.742012	+	0.670386i	$0.233872\pi$
$113$			1610.54i			1.34077i	−0.742012	+	0.670386i	$0.766128\pi$
$114$		0			0
$115$	478.317	+	895.972i	0.387855	+	0.726520i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−2368.42			−1.82447
$120$		0			0
$121$	2911.48			2.18744
$122$		0			0
$123$		0			0
$124$		0			0
$125$	139.119	−	1390.60i	0.0995457	−	0.995033i
$126$		0			0
$127$		−	1896.01i		−	1.32475i	−0.749172	−	0.662376i	$-0.769548\pi$
$127$		−	1896.01i		−	1.32475i	0.749172	−	0.662376i	$-0.230452\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	402.752			0.268615			0.134308	−	0.990940i	$-0.457119\pi$
$131$	402.752			0.268615			0.134308	+	0.990940i	$0.457119\pi$
$132$		0			0
$133$			549.955i			0.358550i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1817.91i		−	1.13368i	−0.823827	−	0.566841i	$-0.808165\pi$
$137$		−	1817.91i		−	1.13368i	0.823827	−	0.566841i	$-0.191835\pi$
$138$		0			0
$139$	2050.95			1.25150			0.625751	−	0.780022i	$-0.284792\pi$
$139$	2050.95			1.25150			0.625751	+	0.780022i	$0.284792\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	3812.18i		−	2.22931i
$144$		0			0
$145$	−1531.25	+	817.461i	−0.876987	+	0.468182i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1865.52			−1.02570			−0.512850	−	0.858478i	$-0.671410\pi$
$149$	−1865.52			−1.02570			−0.512850	+	0.858478i	$0.671410\pi$
$150$		0			0
$151$	685.335			0.369349			0.184675	−	0.982800i	$-0.440877\pi$
$151$	685.335			0.369349			0.184675	+	0.982800i	$0.440877\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	57.9603	−	30.9423i	0.0300354	−	0.0160345i
$156$		0			0
$157$			2498.01i			1.26983i	0.772583	+	0.634914i	$0.218965\pi$
$157$			2498.01i			1.26983i	−0.772583	+	0.634914i	$0.781035\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1908.29			−0.934125
$162$		0			0
$163$			2000.21i			0.961155i	0.876952	+	0.480577i	$0.159573\pi$
$163$			2000.21i			0.961155i	−0.876952	+	0.480577i	$0.840427\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			72.3913i			0.0335438i	0.999859	+	0.0167719i	$0.00533891\pi$
$167$			72.3913i			0.0335438i	−0.999859	+	0.0167719i	$0.994661\pi$
$168$		0			0
$169$	−1228.53			−0.559184
$170$		0			0
$171$		0			0
$172$		0			0
$173$			2571.41i			1.13006i	0.825070	+	0.565030i	$0.191135\pi$
$173$			2571.41i			1.13006i	−0.825070	+	0.565030i	$0.808865\pi$
$174$		0			0
$175$	2181.78	+	1461.06i	0.942442	+	0.631117i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3521.38			−1.47039			−0.735196	−	0.677855i	$-0.762910\pi$
$179$	−3521.38			−1.47039			−0.735196	+	0.677855i	$0.762910\pi$
$180$		0			0
$181$	−3469.30			−1.42470			−0.712350	−	0.701824i	$-0.752369\pi$
$181$	−3469.30			−1.42470			−0.712350	+	0.701824i	$0.752369\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−825.181	−	1545.71i	−0.327938	−	0.614285i
$186$		0			0
$187$		−	7343.70i		−	2.87179i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	114.358			0.0433227			0.0216614	−	0.999765i	$-0.493104\pi$
$191$	114.358			0.0433227			0.0216614	+	0.999765i	$0.493104\pi$
$192$		0			0
$193$			2999.34i			1.11864i	0.828952	+	0.559319i	$0.188937\pi$
$193$			2999.34i			1.11864i	−0.828952	+	0.559319i	$0.811063\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4529.70i			1.63821i	0.573641	+	0.819107i	$0.305530\pi$
$197$			4529.70i			1.63821i	−0.573641	+	0.819107i	$0.694470\pi$
$198$		0			0
$199$	−4099.57			−1.46036			−0.730178	−	0.683257i	$-0.760563\pi$
$199$	−4099.57			−1.46036			−0.730178	+	0.683257i	$0.760563\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	3261.33i		−	1.12759i
$204$		0			0
$205$	−636.150	+	339.610i	−0.216735	+	0.115705i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1705.23			−0.564371
$210$		0			0
$211$	14.6626			0.00478394			0.00239197	−	0.999997i	$-0.499239\pi$
$211$	14.6626			0.00478394			0.00239197	+	0.999997i	$0.499239\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2523.98	−	4727.85i	−0.800623	−	1.49971i
$216$		0			0
$217$			123.447i			0.0386180i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−6598.86			−2.00854
$222$		0			0
$223$			1062.05i			0.318923i	0.987204	+	0.159461i	$0.0509758\pi$
$223$			1062.05i			0.318923i	−0.987204	+	0.159461i	$0.949024\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	76.6458i		−	0.0224104i	−0.999937	−	0.0112052i	$-0.996433\pi$
$227$		−	76.6458i		−	0.0224104i	0.999937	−	0.0112052i	$-0.00356680\pi$
$228$		0			0
$229$	5262.83			1.51868			0.759340	−	0.650694i	$-0.225522\pi$
$229$	5262.83			1.51868			0.759340	+	0.650694i	$0.225522\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			2938.11i			0.826104i	0.910708	+	0.413052i	$0.135537\pi$
$233$			2938.11i			0.826104i	−0.910708	+	0.413052i	$0.864463\pi$
$234$		0			0
$235$	−2214.48	−	4148.11i	−0.614710	−	1.15146i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−355.001			−0.0960798			−0.0480399	−	0.998845i	$-0.515297\pi$
$239$	−355.001			−0.0960798			−0.0480399	+	0.998845i	$0.515297\pi$
$240$		0			0
$241$	−6455.52			−1.72546			−0.862731	−	0.505663i	$-0.831248\pi$
$241$	−6455.52			−1.72546			−0.862731	+	0.505663i	$0.831248\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−969.240	+	517.432i	−0.252745	+	0.134929i
$246$		0			0
$247$			1532.28i			0.394723i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3776.19			−0.949605			−0.474803	−	0.880092i	$-0.657481\pi$
$251$	−3776.19			−0.949605			−0.474803	+	0.880092i	$0.657481\pi$
$252$		0			0
$253$		−	5916.99i		−	1.47035i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			259.121i			0.0628932i	0.999505	+	0.0314466i	$0.0100114\pi$
$257$			259.121i			0.0628932i	−0.999505	+	0.0314466i	$0.989989\pi$
$258$		0			0
$259$	3292.13			0.789818
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	5994.14i		−	1.40538i	−0.711496	−	0.702690i	$-0.751982\pi$
$263$		−	5994.14i		−	1.40538i	0.711496	−	0.702690i	$-0.248018\pi$
$264$		0			0
$265$	−960.916	−	1799.96i	−0.222750	−	0.417249i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6135.04			1.39056			0.695279	−	0.718740i	$-0.255281\pi$
$269$	6135.04			1.39056			0.695279	+	0.718740i	$0.255281\pi$
$270$		0			0
$271$	2065.77			0.463051			0.231525	−	0.972829i	$-0.425628\pi$
$271$	2065.77			0.463051			0.231525	+	0.972829i	$0.425628\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−4530.27	+	6765.02i	−0.993401	+	1.48344i
$276$		0			0
$277$		−	3370.60i		−	0.731119i	−0.930788	−	0.365559i	$-0.880878\pi$
$277$		−	3370.60i		−	0.731119i	0.930788	−	0.365559i	$-0.119122\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8312.91			−1.76479			−0.882396	−	0.470508i	$-0.844071\pi$
$281$	−8312.91			−1.76479			−0.882396	+	0.470508i	$0.844071\pi$
$282$		0			0
$283$			3565.91i			0.749015i	0.927224	+	0.374507i	$0.122188\pi$
$283$			3565.91i			0.749015i	−0.927224	+	0.374507i	$0.877812\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	1354.90i		−	0.278667i
$288$		0			0
$289$	−7798.90			−1.58740
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	7615.21i		−	1.51838i	−0.650869	−	0.759190i	$-0.725595\pi$
$293$		−	7615.21i		−	1.51838i	0.650869	−	0.759190i	$-0.274405\pi$
$294$		0			0
$295$	−5653.13	+	3017.94i	−1.11572	+	0.595632i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−5316.85			−1.02837
$300$		0			0
$301$	10069.6			1.92825
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−76.7028	+	40.9480i	−0.0144000	+	0.00768747i
$306$		0			0
$307$		−	8480.90i		−	1.57665i	−0.615261	−	0.788323i	$-0.710949\pi$
$307$		−	8480.90i		−	1.57665i	0.615261	−	0.788323i	$-0.289051\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−628.242			−0.114548			−0.0572739	−	0.998359i	$-0.518241\pi$
$311$	−628.242			−0.114548			−0.0572739	+	0.998359i	$0.518241\pi$
$312$		0			0
$313$		−	314.646i		−	0.0568206i	−0.999596	−	0.0284103i	$-0.990956\pi$
$313$		−	314.646i		−	0.0568206i	0.999596	−	0.0284103i	$-0.00904450\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	8403.84i		−	1.48898i	−0.667634	−	0.744490i	$-0.732693\pi$
$317$		−	8403.84i		−	1.48898i	0.667634	−	0.744490i	$-0.267307\pi$
$318$		0			0
$319$	10112.3			1.77487
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2951.75i			0.508482i
$324$		0			0
$325$	6078.87	+	4070.78i	1.03752	+	0.694788i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8834.86			1.48049
$330$		0			0
$331$	−909.509			−0.151031			−0.0755153	−	0.997145i	$-0.524060\pi$
$331$	−909.509			−0.151031			−0.0755153	+	0.997145i	$0.524060\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3642.27	−	6822.62i	−0.594026	−	1.11271i
$336$		0			0
$337$		−	3998.91i		−	0.646393i	−0.946332	−	0.323196i	$-0.895243\pi$
$337$		−	3998.91i		−	0.646393i	0.946332	−	0.323196i	$-0.104757\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−382.769			−0.0607862
$342$		0			0
$343$			5140.88i			0.809275i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	3974.88i		−	0.614935i	−0.951559	−	0.307468i	$-0.900518\pi$
$347$		−	3974.88i		−	0.614935i	0.951559	−	0.307468i	$-0.0994816\pi$
$348$		0			0
$349$	3534.51			0.542115			0.271058	−	0.962563i	$-0.412627\pi$
$349$	3534.51			0.542115			0.271058	+	0.962563i	$0.412627\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			69.8587i			0.0105332i	0.999986	+	0.00526658i	$0.00167641\pi$
$353$			69.8587i			0.0105332i	−0.999986	+	0.00526658i	$0.998324\pi$
$354$		0			0
$355$	1670.65	−	891.880i	0.249771	−	0.133341i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2034.82			0.299146			0.149573	−	0.988751i	$-0.452210\pi$
$359$	2034.82			0.299146			0.149573	+	0.988751i	$0.452210\pi$
$360$		0			0
$361$	−6173.59			−0.900072
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3993.38	−	7480.29i	−0.572665	−	1.07270i
$366$		0			0
$367$			13899.6i			1.97699i	0.151259	+	0.988494i	$0.451667\pi$
$367$			13899.6i			1.97699i	−0.151259	+	0.988494i	$0.548333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3833.66			0.536479
$372$		0			0
$373$			5908.54i			0.820194i	0.912042	+	0.410097i	$0.134505\pi$
$373$			5908.54i			0.820194i	−0.912042	+	0.410097i	$0.865495\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	9086.68i		−	1.24135i
$378$		0			0
$379$	−7485.37			−1.01451			−0.507253	−	0.861797i	$-0.669339\pi$
$379$	−7485.37			−1.01451			−0.507253	+	0.861797i	$0.669339\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	4219.52i		−	0.562944i	−0.959569	−	0.281472i	$-0.909177\pi$
$383$		−	4219.52i		−	0.562944i	0.959569	−	0.281472i	$-0.0908226\pi$
$384$		0			0
$385$	−7204.24	−	13494.8i	−0.953668	−	1.78639i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3281.73			0.427738			0.213869	−	0.976862i	$-0.431393\pi$
$389$	3281.73			0.427738			0.213869	+	0.976862i	$0.431393\pi$
$390$		0			0
$391$	−10242.3			−1.32474
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−12119.7	+	6470.16i	−1.54382	+	0.824175i
$396$		0			0
$397$			5336.57i			0.674647i	0.941389	+	0.337324i	$0.109522\pi$
$397$			5336.57i			0.674647i	−0.941389	+	0.337324i	$0.890478\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1062.66			−0.132336			−0.0661680	−	0.997808i	$-0.521077\pi$
$401$	−1062.66			−0.132336			−0.0661680	+	0.997808i	$0.521077\pi$
$402$		0			0
$403$			343.946i			0.0425141i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			10207.8i			1.24320i
$408$		0			0
$409$	6610.27			0.799161			0.399581	−	0.916698i	$-0.369156\pi$
$409$	6610.27			0.799161			0.399581	+	0.916698i	$0.369156\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	12040.3i		−	1.43454i
$414$		0			0
$415$	−3074.55	−	5759.17i	−0.363672	−	0.681221i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2326.80			−0.271293			−0.135646	−	0.990757i	$-0.543311\pi$
$419$	−2326.80			−0.271293			−0.135646	+	0.990757i	$0.543311\pi$
$420$		0			0
$421$	−4000.58			−0.463127			−0.231563	−	0.972820i	$-0.574384\pi$
$421$	−4000.58			−0.463127			−0.231563	+	0.972820i	$0.574384\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11710.2	+	7841.86i	1.33654	+	0.895026i
$426$		0			0
$427$		−	163.366i		−	0.0185148i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1669.23			−0.186552			−0.0932759	−	0.995640i	$-0.529734\pi$
$431$	−1669.23			−0.186552			−0.0932759	+	0.995640i	$0.529734\pi$
$432$		0			0
$433$		−	2609.86i		−	0.289658i	−0.989457	−	0.144829i	$-0.953737\pi$
$433$		−	2609.86i		−	0.289658i	0.989457	−	0.144829i	$-0.0462632\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			2378.29i			0.260341i
$438$		0			0
$439$	7526.08			0.818223			0.409112	−	0.912484i	$-0.365839\pi$
$439$	7526.08			0.818223			0.409112	+	0.912484i	$0.365839\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			9263.17i			0.993469i	0.867903	+	0.496734i	$0.165468\pi$
$443$			9263.17i			0.993469i	−0.867903	+	0.496734i	$0.834532\pi$
$444$		0			0
$445$	−11044.8	+	5896.32i	−1.17657	+	0.628118i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−3801.00			−0.399510			−0.199755	−	0.979846i	$-0.564015\pi$
$449$	−3801.00			−0.399510			−0.199755	+	0.979846i	$0.564015\pi$
$450$		0			0
$451$	4201.13			0.438633
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−12126.1	+	6473.54i	−1.24940	+	0.666999i
$456$		0			0
$457$			6616.42i			0.677249i	0.940922	+	0.338625i	$0.109962\pi$
$457$			6616.42i			0.677249i	−0.940922	+	0.338625i	$0.890038\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	12390.3			1.25179			0.625893	−	0.779909i	$-0.284735\pi$
$461$	12390.3			1.25179			0.625893	+	0.779909i	$0.284735\pi$
$462$		0			0
$463$		−	5424.74i		−	0.544512i	−0.962225	−	0.272256i	$-0.912230\pi$
$463$		−	5424.74i		−	0.544512i	0.962225	−	0.272256i	$-0.0877697\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	14946.0i		−	1.48099i	−0.672064	−	0.740493i	$-0.734592\pi$
$467$		−	14946.0i		−	1.48099i	0.672064	−	0.740493i	$-0.265408\pi$
$468$		0			0
$469$	14531.2			1.43067
$470$		0			0
$471$		0			0
$472$		0			0
$473$			31222.7i			3.03514i
$474$		0			0
$475$	1820.91	−	2719.15i	0.175893	−	0.262659i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−13838.0			−1.31999			−0.659994	−	0.751271i	$-0.729441\pi$
$479$	−13838.0			−1.31999			−0.659994	+	0.751271i	$0.729441\pi$
$480$		0			0
$481$	9172.50			0.869501
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4606.59	+	8628.95i	0.431288	+	0.807877i
$486$		0			0
$487$			6488.31i			0.603723i	0.953352	+	0.301862i	$0.0976081\pi$
$487$			6488.31i			0.603723i	−0.953352	+	0.301862i	$0.902392\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−6211.89			−0.570955			−0.285477	−	0.958385i	$-0.592152\pi$
$491$	−6211.89			−0.570955			−0.285477	+	0.958385i	$0.592152\pi$
$492$		0			0
$493$		−	17504.4i		−	1.59910i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			3558.23i			0.321144i
$498$		0			0
$499$	−8593.36			−0.770925			−0.385462	−	0.922724i	$-0.625958\pi$
$499$	−8593.36			−0.770925			−0.385462	+	0.922724i	$0.625958\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			17902.7i			1.58696i	0.608595	+	0.793481i	$0.291733\pi$
$503$			17902.7i			1.58696i	−0.608595	+	0.793481i	$0.708267\pi$
$504$		0			0
$505$	−247.947	+	132.368i	−0.0218486	+	0.0116639i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	13894.1			1.20992			0.604958	−	0.796257i	$-0.293190\pi$
$509$	13894.1			1.20992			0.604958	+	0.796257i	$0.293190\pi$
$510$		0			0
$511$	15931.9			1.37923
$512$		0			0
$513$		0			0
$514$		0			0
$515$	980.034	+	1835.78i	0.0838553	+	0.157076i
$516$		0			0
$517$			27394.1i			2.33035i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10819.5			−0.909810			−0.454905	−	0.890540i	$-0.650327\pi$
$521$	−10819.5			−0.909810			−0.454905	+	0.890540i	$0.650327\pi$
$522$		0			0
$523$		−	2723.37i		−	0.227696i	−0.993498	−	0.113848i	$-0.963682\pi$
$523$		−	2723.37i		−	0.227696i	0.993498	−	0.113848i	$-0.0363176\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			662.571i			0.0547667i
$528$		0			0
$529$	3914.58			0.321738
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	3775.02i		−	0.306781i
$534$		0			0
$535$	2210.75	+	4141.13i	0.178653	+	0.334648i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6400.85			0.511510
$540$		0			0
$541$	−1451.93			−0.115385			−0.0576924	−	0.998334i	$-0.518374\pi$
$541$	−1451.93			−0.115385			−0.0576924	+	0.998334i	$0.518374\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	21564.1	−	11512.1i	1.69487	−	0.904812i
$546$		0			0
$547$			5460.52i			0.426828i	0.976962	+	0.213414i	$0.0684583\pi$
$547$			5460.52i			0.426828i	−0.976962	+	0.213414i	$0.931542\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4064.58			−0.314259
$552$		0			0
$553$		−	25813.2i		−	1.98497i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	3235.36i		−	0.246116i	−0.992399	−	0.123058i	$-0.960730\pi$
$557$		−	3235.36i		−	0.246116i	0.992399	−	0.123058i	$-0.0392701\pi$
$558$		0			0
$559$	28055.9			2.12279
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	13621.4i		−	1.01967i	−0.860273	−	0.509834i	$-0.829707\pi$
$563$		−	13621.4i		−	1.01967i	0.860273	−	0.509834i	$-0.170293\pi$
$564$		0			0
$565$	8480.04	+	15884.6i	0.631430	+	1.18278i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19780.3			−1.45735			−0.728677	−	0.684857i	$-0.759865\pi$
$569$	−19780.3			−1.45735			−0.728677	+	0.684857i	$0.759865\pi$
$570$		0			0
$571$	−26045.7			−1.90890			−0.954449	−	0.298375i	$-0.903555\pi$
$571$	−26045.7			−1.90890			−0.954449	+	0.298375i	$0.903555\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	9435.17	+	6318.36i	0.684302	+	0.458250i
$576$		0			0
$577$			26765.3i			1.93111i	0.260194	+	0.965556i	$0.416213\pi$
$577$			26765.3i			1.93111i	−0.260194	+	0.965556i	$0.583787\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12266.2			0.875881
$582$		0			0
$583$			11886.9i			0.844437i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			10391.1i			0.730639i	0.930882	+	0.365320i	$0.119040\pi$
$587$			10391.1i			0.730639i	−0.930882	+	0.365320i	$0.880960\pi$
$588$		0			0
$589$	153.851			0.0107629
$590$		0			0
$591$		0			0
$592$		0			0
$593$			2616.21i			0.181172i	0.995889	+	0.0905860i	$0.0288740\pi$
$593$			2616.21i			0.181172i	−0.995889	+	0.0905860i	$0.971126\pi$
$594$		0			0
$595$	−23359.4	+	12470.5i	−1.60948	+	0.859227i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	11974.6			0.816812			0.408406	−	0.912800i	$-0.366085\pi$
$599$	11974.6			0.816812			0.408406	+	0.912800i	$0.366085\pi$
$600$		0			0
$601$	−20624.2			−1.39979			−0.699897	−	0.714243i	$-0.746771\pi$
$601$	−20624.2			−1.39979			−0.699897	+	0.714243i	$0.746771\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	28715.6	−	15329.9i	1.92968	−	1.03016i
$606$		0			0
$607$		−	23380.8i		−	1.56342i	−0.623639	−	0.781712i	$-0.714347\pi$
$607$		−	23380.8i		−	1.56342i	0.623639	−	0.781712i	$-0.285653\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	24615.6			1.62985
$612$		0			0
$613$		−	287.368i		−	0.0189342i	−0.999955	−	0.00946711i	$-0.996986\pi$
$613$		−	287.368i		−	0.0189342i	0.999955	−	0.00946711i	$-0.00301352\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			13679.0i			0.892539i	0.894899	+	0.446269i	$0.147248\pi$
$617$			13679.0i			0.892539i	−0.894899	+	0.446269i	$0.852752\pi$
$618$		0			0
$619$	12361.3			0.802654			0.401327	−	0.915935i	$-0.368549\pi$
$619$	12361.3			0.802654			0.401327	+	0.915935i	$0.368549\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	23523.9i		−	1.51278i
$624$		0			0
$625$	−5949.85	−	14447.8i	−0.380790	−	0.924661i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17669.7			1.12009
$630$		0			0
$631$	2362.42			0.149043			0.0745216	−	0.997219i	$-0.476257\pi$
$631$	2362.42			0.149043			0.0745216	+	0.997219i	$0.476257\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−9983.10	−	18700.1i	−0.623885	−	1.16865i
$636$		0			0
$637$		−	5751.64i		−	0.357752i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	21711.2			1.33782			0.668910	−	0.743343i	$-0.266761\pi$
$641$	21711.2			1.33782			0.668910	+	0.743343i	$0.266761\pi$
$642$		0			0
$643$			752.470i			0.0461501i	0.999734	+	0.0230751i	$0.00734567\pi$
$643$			752.470i			0.0461501i	−0.999734	+	0.0230751i	$0.992654\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9304.86i			0.565397i	0.959209	+	0.282699i	$0.0912296\pi$
$647$			9304.86i			0.565397i	−0.959209	+	0.282699i	$0.908770\pi$
$648$		0			0
$649$	37333.2			2.25802
$650$		0			0
$651$		0			0
$652$		0			0
$653$			13939.6i			0.835372i	0.908591	+	0.417686i	$0.137159\pi$
$653$			13939.6i			0.835372i	−0.908591	+	0.417686i	$0.862841\pi$
$654$		0			0
$655$	3972.29	−	2120.62i	0.236962	−	0.126503i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−23203.9			−1.37162			−0.685809	−	0.727782i	$-0.740551\pi$
$659$	−23203.9			−1.37162			−0.685809	+	0.727782i	$0.740551\pi$
$660$		0			0
$661$	−15629.0			−0.919665			−0.459833	−	0.888006i	$-0.652091\pi$
$661$	−15629.0			−0.919665			−0.459833	+	0.888006i	$0.652091\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2895.69	+	5424.14i	0.168857	+	0.316299i
$666$		0			0
$667$		−	14103.7i		−	0.818735i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	506.545			0.0291430
$672$		0			0
$673$		−	8510.83i		−	0.487472i	−0.969842	−	0.243736i	$-0.921627\pi$
$673$		−	8510.83i		−	0.487472i	0.969842	−	0.243736i	$-0.0783730\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			32374.7i			1.83791i	0.394367	+	0.918953i	$0.370964\pi$
$677$			32374.7i			1.83791i	−0.394367	+	0.918953i	$0.629036\pi$
$678$		0			0
$679$	−18378.4			−1.03873
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	13427.7i		−	0.752265i	−0.926566	−	0.376132i	$-0.877254\pi$
$683$		−	13427.7i		−	0.752265i	0.926566	−	0.376132i	$-0.122746\pi$
$684$		0			0
$685$	−9571.89	−	17929.8i	−0.533902	−	1.00009i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	10681.3			0.590602
$690$		0			0
$691$	16206.7			0.892233			0.446117	−	0.894975i	$-0.352807\pi$
$691$	16206.7			0.892233			0.446117	+	0.894975i	$0.352807\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20228.2	−	10798.9i	1.10403	−	0.589389i
$696$		0			0
$697$		−	7272.12i		−	0.395195i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	21408.7			1.15349			0.576745	−	0.816924i	$-0.304323\pi$
$701$	21408.7			1.15349			0.576745	+	0.816924i	$0.304323\pi$
$702$		0			0
$703$		−	4102.97i		−	0.220123i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	528.091i		−	0.0280918i
$708$		0			0
$709$	−31484.5			−1.66774			−0.833868	−	0.551964i	$-0.813879\pi$
$709$	−31484.5			−1.66774			−0.833868	+	0.551964i	$0.813879\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			533.848i			0.0280403i
$714$		0			0
$715$	−20072.4	−	37599.1i	−1.04988	−	1.96661i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−2360.48			−0.122435			−0.0612177	−	0.998124i	$-0.519498\pi$
$719$	−2360.48			−0.122435			−0.0612177	+	0.998124i	$0.519498\pi$
$720$		0			0
$721$	−3909.93			−0.201960
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−10798.3	+	16125.0i	−0.553157	+	0.826025i
$726$		0			0
$727$		−	11437.9i		−	0.583507i	−0.956494	−	0.291754i	$-0.905761\pi$
$727$		−	11437.9i		−	0.583507i	0.956494	−	0.291754i	$-0.0942387\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	54046.3			2.73457
$732$		0			0
$733$		−	30907.0i		−	1.55740i	−0.627394	−	0.778702i	$-0.715879\pi$
$733$		−	30907.0i		−	1.55740i	0.627394	−	0.778702i	$-0.284121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			45056.5i			2.25194i
$738$		0			0
$739$	25620.9			1.27534			0.637672	−	0.770308i	$-0.279897\pi$
$739$	25620.9			1.27534			0.637672	+	0.770308i	$0.279897\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	20017.5i		−	0.988388i	−0.869352	−	0.494194i	$-0.835463\pi$
$743$		−	20017.5i		−	0.988388i	0.869352	−	0.494194i	$-0.164537\pi$
$744$		0			0
$745$	−18399.4	+	9822.56i	−0.904834	+	0.483048i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−8819.99			−0.430274
$750$		0			0
$751$	−29068.2			−1.41240			−0.706200	−	0.708013i	$-0.749592\pi$
$751$	−29068.2			−1.41240			−0.706200	+	0.708013i	$0.749592\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6759.37	−	3608.51i	0.325826	−	0.173943i
$756$		0			0
$757$		−	14907.2i		−	0.715737i	−0.933772	−	0.357869i	$-0.883504\pi$
$757$		−	14907.2i		−	0.715737i	0.933772	−	0.357869i	$-0.116496\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−4146.06			−0.197496			−0.0987481	−	0.995112i	$-0.531484\pi$
$761$	−4146.06			−0.197496			−0.0987481	+	0.995112i	$0.531484\pi$
$762$		0			0
$763$			45928.4i			2.17919i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	33546.7i		−	1.57927i
$768$		0			0
$769$	−10010.0			−0.469402			−0.234701	−	0.972068i	$-0.575411\pi$
$769$	−10010.0			−0.469402			−0.234701	+	0.972068i	$0.575411\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	806.334i		−	0.0375185i	−0.999824	−	0.0187593i	$-0.994028\pi$
$773$		−	806.334i		−	0.0375185i	0.999824	−	0.0187593i	$-0.00597161\pi$
$774$		0			0
$775$	408.734	−	610.360i	0.0189447	−	0.0282900i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1688.61			−0.0776647
$780$		0			0
$781$	−11032.9			−0.505492
$782$		0			0
$783$		0			0
$784$		0			0
$785$	13152.8	+	24637.6i	0.598020	+	1.12020i
$786$		0			0
$787$			4647.33i			0.210495i	0.994446	+	0.105247i	$0.0335634\pi$
$787$			4647.33i			0.210495i	−0.994446	+	0.105247i	$0.966437\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−33831.8			−1.52076
$792$		0			0
$793$		−	455.168i		−	0.0203827i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			7442.62i			0.330779i	0.986228	+	0.165390i	$0.0528882\pi$
$797$			7442.62i			0.330779i	−0.986228	+	0.165390i	$0.947112\pi$
$798$		0			0
$799$	47418.9			2.09958
$800$		0			0
$801$		0			0
$802$		0			0
$803$			49399.8i			2.17096i
$804$		0			0
$805$	−18821.2	+	10047.8i	−0.824050	+	0.439921i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	31920.9			1.38724			0.693620	−	0.720341i	$-0.256015\pi$
$809$	31920.9			1.38724			0.693620	+	0.720341i	$0.256015\pi$
$810$		0			0
$811$	−9558.63			−0.413870			−0.206935	−	0.978355i	$-0.566349\pi$
$811$	−9558.63			−0.413870			−0.206935	+	0.978355i	$0.566349\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10531.7	+	19727.8i	0.452651	+	0.847895i
$816$		0			0
$817$		−	12549.7i		−	0.537405i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−19645.3			−0.835109			−0.417554	−	0.908652i	$-0.637113\pi$
$821$	−19645.3			−0.835109			−0.417554	+	0.908652i	$0.637113\pi$
$822$		0			0
$823$			38233.3i			1.61935i	0.586876	+	0.809677i	$0.300358\pi$
$823$			38233.3i			1.61935i	−0.586876	+	0.809677i	$0.699642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	24908.0i		−	1.04732i	−0.851926	−	0.523662i	$-0.824565\pi$
$827$		−	24908.0i		−	1.04732i	0.851926	−	0.523662i	$-0.175435\pi$
$828$		0			0
$829$	32695.8			1.36981			0.684904	−	0.728633i	$-0.259844\pi$
$829$	32695.8			1.36981			0.684904	+	0.728633i	$0.259844\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	11079.8i		−	0.460856i
$834$		0			0
$835$	381.164	+	713.987i	0.0157973	+	0.0295911i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−36302.1			−1.49379			−0.746895	−	0.664942i	$-0.768456\pi$
$839$	−36302.1			−1.49379			−0.746895	+	0.664942i	$0.768456\pi$
$840$		0			0
$841$	−285.335			−0.0116993
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12116.8	+	6468.60i	−0.493291	+	0.263345i
$846$		0			0
$847$			61159.9i			2.48108i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	14236.9			0.573483
$852$		0			0
$853$		−	25802.7i		−	1.03572i	−0.855466	−	0.517859i	$-0.826729\pi$
$853$		−	25802.7i		−	1.03572i	0.855466	−	0.517859i	$-0.173271\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			10276.9i			0.409629i	0.978801	+	0.204814i	$0.0656591\pi$
$857$			10276.9i			0.409629i	−0.978801	+	0.204814i	$0.934341\pi$
$858$		0			0
$859$	7368.38			0.292673			0.146336	−	0.989235i	$-0.453252\pi$
$859$	7368.38			0.292673			0.146336	+	0.989235i	$0.453252\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			46952.8i			1.85202i	0.377503	+	0.926009i	$0.376783\pi$
$863$			46952.8i			1.85202i	−0.377503	+	0.926009i	$0.623217\pi$
$864$		0			0
$865$	13539.3	+	25361.5i	0.532196	+	0.996897i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	80038.6			3.12442
$870$		0			0
$871$	40486.6			1.57501
$872$		0			0
$873$		0			0
$874$		0			0
$875$	29211.6	+	2922.41i	1.12861	+	0.112909i
$876$		0			0
$877$		−	23840.8i		−	0.917955i	−0.888448	−	0.458977i	$-0.848216\pi$
$877$		−	23840.8i		−	0.917955i	0.888448	−	0.458977i	$-0.151784\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	48305.5			1.84728			0.923640	−	0.383262i	$-0.125199\pi$
$881$	48305.5			1.84728			0.923640	+	0.383262i	$0.125199\pi$
$882$		0			0
$883$			44319.1i			1.68908i	0.535493	+	0.844540i	$0.320126\pi$
$883$			44319.1i			1.68908i	−0.535493	+	0.844540i	$0.679874\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			24194.8i			0.915875i	0.888984	+	0.457937i	$0.151412\pi$
$887$			24194.8i			0.915875i	−0.888984	+	0.457937i	$0.848588\pi$
$888$		0			0
$889$	39828.4			1.50259
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	11010.8i		−	0.412614i
$894$		0			0
$895$	−34730.9	+	18541.2i	−1.29712	+	0.692474i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−912.365			−0.0338477
$900$		0			0
$901$	20576.2			0.760814
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−34217.2	+	18267.0i	−1.25682	+	0.670956i
$906$		0			0
$907$			14163.1i			0.518498i	0.965810	+	0.259249i	$0.0834751\pi$
$907$			14163.1i			0.518498i	−0.965810	+	0.259249i	$0.916525\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−11021.9			−0.400848			−0.200424	−	0.979709i	$-0.564232\pi$
$911$	−11021.9			−0.400848			−0.200424	+	0.979709i	$0.564232\pi$
$912$		0			0
$913$			38033.5i			1.37867i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8460.39i			0.304675i
$918$		0			0
$919$	−6410.89			−0.230115			−0.115057	−	0.993359i	$-0.536705\pi$
$919$	−6410.89			−0.230115			−0.115057	+	0.993359i	$0.536705\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			9913.91i			0.353543i
$924$		0			0
$925$	−16277.3	−	10900.3i	−0.578589	−	0.387458i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−25036.0			−0.884181			−0.442090	−	0.896971i	$-0.645763\pi$
$929$	−25036.0			−0.884181			−0.442090	+	0.896971i	$0.645763\pi$
$930$		0			0
$931$	−2572.78			−0.0905686
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−38667.0	−	72430.0i	−1.35246	−	2.53339i
$936$		0			0
$937$		−	33214.7i		−	1.15803i	−0.815316	−	0.579016i	$-0.803437\pi$
$937$		−	33214.7i		−	1.15803i	0.815316	−	0.579016i	$-0.196563\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4586.93			0.158905			0.0794526	−	0.996839i	$-0.474683\pi$
$941$	4586.93			0.158905			0.0794526	+	0.996839i	$0.474683\pi$
$942$		0			0
$943$		−	5859.31i		−	0.202339i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	40309.2i		−	1.38318i	−0.722289	−	0.691591i	$-0.756910\pi$
$947$		−	40309.2i		−	1.38318i	0.722289	−	0.691591i	$-0.243090\pi$
$948$		0			0
$949$	44389.3			1.51838
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	2056.58i		−	0.0699048i	−0.999389	−	0.0349524i	$-0.988872\pi$
$953$		−	2056.58i		−	0.0699048i	0.999389	−	0.0349524i	$-0.0111280\pi$
$954$		0			0
$955$	1127.90	−	602.131i	0.0382177	−	0.0204026i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	38187.9			1.28587
$960$		0			0
$961$	−29756.5			−0.998841
$962$		0			0
$963$		0			0
$964$		0			0
$965$	15792.5	+	29582.1i	0.526817	+	0.986821i
$966$		0			0
$967$		−	45914.4i		−	1.52690i	−0.645869	−	0.763448i	$-0.723505\pi$
$967$		−	45914.4i		−	1.52690i	0.645869	−	0.763448i	$-0.276495\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−41761.0			−1.38020			−0.690100	−	0.723714i	$-0.742434\pi$
$971$	−41761.0			−1.38020			−0.690100	+	0.723714i	$0.742434\pi$
$972$		0			0
$973$			43083.1i			1.41951i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	13686.7i		−	0.448184i	−0.974568	−	0.224092i	$-0.928058\pi$
$977$		−	13686.7i		−	0.448184i	0.974568	−	0.224092i	$-0.0719416\pi$
$978$		0			0
$979$	72940.0			2.38118
$980$		0			0
$981$		0			0
$982$		0			0
$983$			20075.1i			0.651368i	0.945479	+	0.325684i	$0.105595\pi$
$983$			20075.1i			0.651368i	−0.945479	+	0.325684i	$0.894405\pi$
$984$		0			0
$985$	23850.4	+	44675.9i	0.771509	+	1.44517i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	43546.3			1.40009
$990$		0			0
$991$	9774.54			0.313318			0.156659	−	0.987653i	$-0.449928\pi$
$991$	9774.54			0.313318			0.156659	+	0.987653i	$0.449928\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−40433.5	+	21585.6i	−1.28827	+	0.687747i
$996$		0			0
$997$			3096.79i			0.0983715i	0.998790	+	0.0491858i	$0.0156626\pi$
$997$			3096.79i			0.0983715i	−0.998790	+	0.0491858i	$0.984337\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.d.c.649.15	yes	16
3.2	odd	2	inner	1620.4.d.c.649.2	yes	16
5.4	even	2	inner	1620.4.d.c.649.16	yes	16
15.14	odd	2	inner	1620.4.d.c.649.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.d.c.649.1	✓	16	15.14	odd	2	inner
1620.4.d.c.649.2	yes	16	3.2	odd	2	inner
1620.4.d.c.649.15	yes	16	1.1	even	1	trivial
1620.4.d.c.649.16	yes	16	5.4	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(9.86288 - 5.26533i) q^{5} +21.0065i q^{7} +O(q^{10})\)	\(q+(9.86288 - 5.26533i) q^{5} +21.0065i q^{7} -65.1343 q^{11} +58.5280i q^{13} +112.747i q^{17} +26.1803 q^{19} +90.8428i q^{23} +(69.5527 - 103.863i) q^{25} -155.254 q^{29} +5.87661 q^{31} +(110.606 + 207.184i) q^{35} -156.720i q^{37} -64.4994 q^{41} -479.359i q^{43} -420.578i q^{47} -98.2715 q^{49} -182.499i q^{53} +(-642.412 + 342.953i) q^{55} -573.173 q^{59} -7.77692 q^{61} +(308.169 + 577.254i) q^{65} -691.747i q^{67} +169.387 q^{71} -758.429i q^{73} -1368.24i q^{77} -1228.82 q^{79} -583.924i q^{83} +(593.650 + 1112.01i) q^{85} -1119.84 q^{89} -1229.47 q^{91} +(258.213 - 137.848i) q^{95} +874.892i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 84 q^{19} + 54 q^{25} + 88 q^{31} + 84 q^{49} - 1120 q^{55} - 4 q^{61} - 1344 q^{79} + 482 q^{85} - 2100 q^{91}+O(q^{100}) \) 16 * q + 84 * q^19 + 54 * q^25 + 88 * q^31 + 84 * q^49 - 1120 * q^55 - 4 * q^61 - 1344 * q^79 + 482 * q^85 - 2100 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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