LMFDB - Embedded newform 2320.2.d.g.929.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2320,2,Mod(929,2320)] 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("2320.929"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2320 = 2^{4} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2320.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,3,0,0,0,-12,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$18.5252932689$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.84345856.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 13x^{4} + 41x^{2} + 1 $ x^6 + 13x^4 + 41x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			929.5
Root			$-0.156785i$ of defining polynomial
Character	$\chi$	$=$	2320.929
Dual form			2320.2.d.g.929.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56387i q^{3} +(1.28672 + 1.82876i) q^{5} -1.09364i q^{7} -3.57344 q^{9} +4.40198 q^{11} -3.97108i q^{13} +(-4.68870 + 3.29899i) q^{15} +6.22138i q^{17} +6.97542 q^{19} +2.80395 q^{21} -0.780070i q^{23} +(-1.68870 + 4.70620i) q^{25} -1.47023i q^{27} +1.00000 q^{29} -6.40198 q^{31} +11.2861i q^{33} +(2.00000 - 1.40721i) q^{35} -1.09364i q^{37} +10.1813 q^{39} -0.376593i q^{43} +(-4.59802 - 6.53495i) q^{45} +4.75115i q^{47} +5.80395 q^{49} -15.9508 q^{51} +11.2861i q^{53} +(5.66412 + 8.05014i) q^{55} +17.8841i q^{57} +10.0000 q^{59} -1.14688 q^{61} +3.90806i q^{63} +(7.26214 - 5.10967i) q^{65} +5.90782i q^{67} +2.00000 q^{69} -2.00000 q^{71} -8.72223i q^{73} +(-12.0661 - 4.32960i) q^{75} -4.81418i q^{77} -5.54886 q^{79} -6.95084 q^{81} +6.22138i q^{83} +(-11.3774 + 8.00519i) q^{85} +2.56387i q^{87} +10.8040 q^{89} -4.34293 q^{91} -16.4139i q^{93} +(8.97542 + 12.7563i) q^{95} -17.4176i q^{97} -15.7302 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q + 3 * q^5 - 12 * q^9 + 10 * q^11 - 7 * q^15 + 16 * q^19 - 16 * q^21 + 11 * q^25 + 6 * q^29 - 22 * q^31 + 12 * q^35 - 14 * q^39 - 44 * q^45 + 2 * q^49 - 44 * q^51 - 13 * q^55 + 60 * q^59 + 12 * q^61 + 13 * q^65 + 12 * q^69 - 12 * q^71 - 9 * q^75 + 2 * q^79 + 10 * q^81 - 26 * q^85 + 32 * q^89 - 40 * q^91 + 28 * q^95 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2320\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$1857$	$2031$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$			2.56387i			1.48025i	0.672468	+	0.740126i	$0.265234\pi$
$3$			2.56387i			1.48025i	−0.672468	+	0.740126i	$0.734766\pi$
$4$		0			0
$5$	1.28672	+	1.82876i	0.575439	+	0.817845i
$5$	1.28672	+	1.82876i	0.575439	+	0.817845i
$6$		0			0
$7$		−	1.09364i		−	0.413357i	−0.978409	−	0.206678i	$-0.933735\pi$
$7$		−	1.09364i		−	0.413357i	0.978409	−	0.206678i	$-0.0662654\pi$
$8$		0			0
$9$	−3.57344			−1.19115
$10$		0			0
$11$	4.40198			1.32725			0.663623	−	0.748067i	$-0.269018\pi$
$11$	4.40198			1.32725			0.663623	+	0.748067i	$0.269018\pi$
$12$		0			0
$13$		−	3.97108i		−	1.10138i	−0.834710	−	0.550690i	$-0.814365\pi$
$13$		−	3.97108i		−	1.10138i	0.834710	−	0.550690i	$-0.185635\pi$
$14$		0			0
$15$	−4.68870	+	3.29899i	−1.21062	+	0.851795i
$16$		0			0
$17$			6.22138i			1.50891i	0.656353	+	0.754454i	$0.272098\pi$
$17$			6.22138i			1.50891i	−0.656353	+	0.754454i	$0.727902\pi$
$18$		0			0
$19$	6.97542			1.60027			0.800135	−	0.599819i	$-0.204761\pi$
$19$	6.97542			1.60027			0.800135	+	0.599819i	$0.204761\pi$
$20$		0			0
$21$	2.80395			0.611873
$22$		0			0
$23$		−	0.780070i		−	0.162656i	−0.996687	−	0.0813279i	$-0.974084\pi$
$23$		−	0.780070i		−	0.162656i	0.996687	−	0.0813279i	$-0.0259161\pi$
$24$		0			0
$25$	−1.68870	+	4.70620i	−0.337739	+	0.941240i
$26$		0			0
$27$		−	1.47023i		−	0.282946i
$28$		0			0
$29$	1.00000			0.185695
$29$	1.00000			0.185695
$30$		0			0
$31$	−6.40198			−1.14983			−0.574914	−	0.818214i	$-0.694965\pi$
$31$	−6.40198			−1.14983			−0.574914	+	0.818214i	$0.694965\pi$
$32$		0			0
$33$			11.2861i			1.96466i
$34$		0			0
$35$	2.00000	−	1.40721i	0.338062	−	0.237862i
$36$		0			0
$37$		−	1.09364i		−	0.179793i	−0.995951	−	0.0898966i	$-0.971346\pi$
$37$		−	1.09364i		−	0.179793i	0.995951	−	0.0898966i	$-0.0286537\pi$
$38$		0			0
$39$	10.1813			1.63032
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		−	0.376593i		−	0.0574299i	−0.999588	−	0.0287150i	$-0.990858\pi$
$43$		−	0.376593i		−	0.0574299i	0.999588	−	0.0287150i	$-0.00914152\pi$
$44$		0			0
$45$	−4.59802	−	6.53495i	−0.685433	−	0.974173i
$46$		0			0
$47$			4.75115i			0.693027i	0.938045	+	0.346513i	$0.112634\pi$
$47$			4.75115i			0.693027i	−0.938045	+	0.346513i	$0.887366\pi$
$48$		0			0
$49$	5.80395			0.829136
$50$		0			0
$51$	−15.9508			−2.23356
$52$		0			0
$53$			11.2861i			1.55027i	0.631798	+	0.775133i	$0.282317\pi$
$53$			11.2861i			1.55027i	−0.631798	+	0.775133i	$0.717683\pi$
$54$		0			0
$55$	5.66412	+	8.05014i	0.763749	+	1.08548i
$56$		0			0
$57$			17.8841i			2.36880i
$58$		0			0
$59$	10.0000			1.30189			0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000			1.30189			0.650945	+	0.759125i	$0.274373\pi$
$60$		0			0
$61$	−1.14688			−0.146844			−0.0734218	−	0.997301i	$-0.523392\pi$
$61$	−1.14688			−0.146844			−0.0734218	+	0.997301i	$0.523392\pi$
$62$		0			0
$63$			3.90806i			0.492369i
$64$		0			0
$65$	7.26214	−	5.10967i	0.900758	−	0.633777i
$66$		0			0
$67$			5.90782i			0.721754i	0.932613	+	0.360877i	$0.117523\pi$
$67$			5.90782i			0.721754i	−0.932613	+	0.360877i	$0.882477\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$		−	8.72223i		−	1.02086i	−0.859919	−	0.510430i	$-0.829486\pi$
$73$		−	8.72223i		−	1.02086i	0.859919	−	0.510430i	$-0.170514\pi$
$74$		0			0
$75$	−12.0661	−	4.32960i	−1.39327	−	0.499940i
$76$		0			0
$77$		−	4.81418i		−	0.548626i
$78$		0			0
$79$	−5.54886			−0.624296			−0.312148	−	0.950034i	$-0.601048\pi$
$79$	−5.54886			−0.624296			−0.312148	+	0.950034i	$0.601048\pi$
$80$		0			0
$81$	−6.95084			−0.772315
$82$		0			0
$83$			6.22138i			0.682886i	0.939903	+	0.341443i	$0.110916\pi$
$83$			6.22138i			0.682886i	−0.939903	+	0.341443i	$0.889084\pi$
$84$		0			0
$85$	−11.3774	+	8.00519i	−1.23405	+	0.868284i
$86$		0			0
$87$			2.56387i			0.274876i
$88$		0			0
$89$	10.8040			1.14522			0.572608	−	0.819829i	$-0.305932\pi$
$89$	10.8040			1.14522			0.572608	+	0.819829i	$0.305932\pi$
$90$		0			0
$91$	−4.34293			−0.455263
$92$		0			0
$93$		−	16.4139i		−	1.70204i
$94$		0			0
$95$	8.97542	+	12.7563i	0.920859	+	1.30877i
$96$		0			0
$97$		−	17.4176i		−	1.76849i	−0.467026	−	0.884244i	$-0.654674\pi$
$97$		−	17.4176i		−	1.76849i	0.467026	−	0.884244i	$-0.345326\pi$
$98$		0			0
$99$	−15.7302			−1.58095
$100$		0			0
$101$	−12.7548			−1.26915			−0.634574	−	0.772862i	$-0.718825\pi$
$101$	−12.7548			−1.26915			−0.634574	+	0.772862i	$0.718825\pi$
$102$		0			0
$103$		−	5.15463i		−	0.507901i	−0.967217	−	0.253950i	$-0.918270\pi$
$103$		−	5.15463i		−	0.507901i	0.967217	−	0.253950i	$-0.0817300\pi$
$104$		0			0
$105$	3.60790	+	5.12775i	0.352095	+	0.500417i
$106$		0			0
$107$			1.09364i			0.105726i	0.998602	+	0.0528631i	$0.0168347\pi$
$107$			1.09364i			0.105726i	−0.998602	+	0.0528631i	$0.983165\pi$
$108$		0			0
$109$	−15.3282			−1.46818			−0.734089	−	0.679053i	$-0.762391\pi$
$109$	−15.3282			−1.46818			−0.734089	+	0.679053i	$0.762391\pi$
$110$		0			0
$111$	2.80395			0.266139
$112$		0			0
$113$		−	7.28814i		−	0.685611i	−0.939406	−	0.342805i	$-0.888623\pi$
$113$		−	7.28814i		−	0.685611i	0.939406	−	0.342805i	$-0.111377\pi$
$114$		0			0
$115$	1.42656	−	1.00373i	0.133027	−	0.0935985i
$116$		0			0
$117$			14.1904i			1.31191i
$118$		0			0
$119$	6.80395			0.623717
$120$		0			0
$121$	8.37739			0.761581
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.7794	+	2.96735i	−0.964136	+	0.265408i
$126$		0			0
$127$		−	17.7312i		−	1.57339i	−0.617345	−	0.786693i	$-0.711792\pi$
$127$		−	17.7312i		−	1.57339i	0.617345	−	0.786693i	$-0.288208\pi$
$128$		0			0
$129$	0.965537			0.0850108
$130$		0			0
$131$	−7.31835			−0.639407			−0.319704	−	0.947518i	$-0.603583\pi$
$131$	−7.31835			−0.639407			−0.319704	+	0.947518i	$0.603583\pi$
$132$		0			0
$133$		−	7.62859i		−	0.661483i
$134$		0			0
$135$	2.68870	−	1.89178i	0.231406	−	0.162818i
$136$		0			0
$137$			9.78899i			0.836330i	0.908371	+	0.418165i	$0.137327\pi$
$137$			9.78899i			0.836330i	−0.908371	+	0.418165i	$0.862673\pi$
$138$		0			0
$139$	−2.75479			−0.233658			−0.116829	−	0.993152i	$-0.537273\pi$
$139$	−2.75479			−0.233658			−0.116829	+	0.993152i	$0.537273\pi$
$140$		0			0
$141$	−12.1813			−1.02585
$142$		0			0
$143$		−	17.4806i		−	1.46180i
$144$		0			0
$145$	1.28672	+	1.82876i	0.106856	+	0.151870i
$146$		0			0
$147$			14.8806i			1.22733i
$148$		0			0
$149$	4.52428			0.370643			0.185322	−	0.982678i	$-0.440667\pi$
$149$	4.52428			0.370643			0.185322	+	0.982678i	$0.440667\pi$
$150$		0			0
$151$	−2.80395			−0.228182			−0.114091	−	0.993470i	$-0.536396\pi$
$151$	−2.80395			−0.228182			−0.114091	+	0.993470i	$0.536396\pi$
$152$		0			0
$153$		−	22.2318i		−	1.79733i
$154$		0			0
$155$	−8.23756	−	11.7077i	−0.661657	−	0.940381i
$156$		0			0
$157$		−	4.97481i		−	0.397033i	−0.980098	−	0.198517i	$-0.936388\pi$
$157$		−	4.97481i		−	0.397033i	0.980098	−	0.198517i	$-0.0636124\pi$
$158$		0			0
$159$	−28.9361			−2.29478
$160$		0			0
$161$	−0.853115			−0.0672349
$162$		0			0
$163$		−	20.7615i		−	1.62617i	−0.582146	−	0.813084i	$-0.697787\pi$
$163$		−	20.7615i		−	1.62617i	0.582146	−	0.813084i	$-0.302213\pi$
$164$		0			0
$165$	−20.6395	+	14.5221i	−1.60679	+	1.13054i
$166$		0			0
$167$			21.9182i			1.69608i	0.529932	+	0.848040i	$0.322217\pi$
$167$			21.9182i			1.69608i	−0.529932	+	0.848040i	$0.677783\pi$
$168$		0			0
$169$	−2.76949			−0.213038
$170$		0			0
$171$	−24.9263			−1.90616
$172$		0			0
$173$		−	3.25404i		−	0.247400i	−0.992320	−	0.123700i	$-0.960524\pi$
$173$		−	3.25404i		−	0.247400i	0.992320	−	0.123700i	$-0.0394760\pi$
$174$		0			0
$175$	5.14688	+	1.84683i	0.389068	+	0.139607i
$176$		0			0
$177$			25.6387i			1.92712i
$178$		0			0
$179$	−7.19605			−0.537858			−0.268929	−	0.963160i	$-0.586670\pi$
$179$	−7.19605			−0.537858			−0.268929	+	0.963160i	$0.586670\pi$
$180$		0			0
$181$	13.0345			0.968844			0.484422	−	0.874834i	$-0.339030\pi$
$181$	13.0345			0.968844			0.484422	+	0.874834i	$0.339030\pi$
$182$		0			0
$183$		−	2.94047i		−	0.217365i
$184$		0			0
$185$	2.00000	−	1.40721i	0.147043	−	0.103460i
$186$		0			0
$187$			27.3864i			2.00269i
$188$		0			0
$189$	−1.60790			−0.116958
$190$		0			0
$191$	10.1223			0.732424			0.366212	−	0.930531i	$-0.380654\pi$
$191$	10.1223			0.732424			0.366212	+	0.930531i	$0.380654\pi$
$192$		0			0
$193$			22.8589i			1.64542i	0.568462	+	0.822710i	$0.307539\pi$
$193$			22.8589i			1.64542i	−0.568462	+	0.822710i	$0.692461\pi$
$194$		0			0
$195$	13.1006	+	18.6192i	0.938150	+	1.33335i
$196$		0			0
$197$		−	22.0711i		−	1.57250i	−0.617907	−	0.786251i	$-0.712019\pi$
$197$		−	22.0711i		−	1.57250i	0.617907	−	0.786251i	$-0.287981\pi$
$198$		0			0
$199$	5.19605			0.368338			0.184169	−	0.982895i	$-0.441041\pi$
$199$	5.19605			0.368338			0.184169	+	0.982895i	$0.441041\pi$
$200$		0			0
$201$	−15.1469			−1.06838
$202$		0			0
$203$		−	1.09364i		−	0.0767585i
$204$		0			0
$205$		0			0
$206$		0			0
$207$			2.78754i			0.193747i
$208$		0			0
$209$	30.7056			2.12395
$210$		0			0
$211$	−1.64719			−0.113397			−0.0566985	−	0.998391i	$-0.518057\pi$
$211$	−1.64719			−0.113397			−0.0566985	+	0.998391i	$0.518057\pi$
$212$		0			0
$213$		−	5.12775i		−	0.351347i
$214$		0			0
$215$	0.688697	−	0.484571i	0.0469688	−	0.0330474i
$216$		0			0
$217$			7.00145i			0.475290i
$218$		0			0
$219$	22.3627			1.51113
$220$		0			0
$221$	24.7056			1.66188
$222$		0			0
$223$		−	9.47542i		−	0.634521i	−0.948338	−	0.317261i	$-0.897237\pi$
$223$		−	9.47542i		−	0.634521i	0.948338	−	0.317261i	$-0.102763\pi$
$224$		0			0
$225$	6.03446	−	16.8173i	0.402298	−	1.12116i
$226$		0			0
$227$			5.72800i			0.380181i	0.981767	+	0.190090i	$0.0608781\pi$
$227$			5.72800i			0.380181i	−0.981767	+	0.190090i	$0.939122\pi$
$228$		0			0
$229$	−15.1469			−1.00093			−0.500467	−	0.865756i	$-0.666838\pi$
$229$	−15.1469			−1.00093			−0.500467	+	0.865756i	$0.666838\pi$
$230$		0			0
$231$	12.3429			0.812105
$232$		0			0
$233$		−	0.717046i		−	0.0469753i	−0.999724	−	0.0234876i	$-0.992523\pi$
$233$		−	0.717046i		−	0.0469753i	0.999724	−	0.0234876i	$-0.00747703\pi$
$234$		0			0
$235$	−8.68870	+	6.11341i	−0.566788	+	0.398795i
$236$		0			0
$237$		−	14.2266i		−	0.924115i
$238$		0			0
$239$	−2.00000			−0.129369			−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000			−0.129369			−0.0646846	+	0.997906i	$0.520604\pi$
$240$		0			0
$241$	17.3282			1.11621			0.558105	−	0.829770i	$-0.311529\pi$
$241$	17.3282			1.11621			0.558105	+	0.829770i	$0.311529\pi$
$242$		0			0
$243$		−	22.2318i		−	1.42617i
$244$		0			0
$245$	7.46807	+	10.6140i	0.477117	+	0.678104i
$246$		0			0
$247$		−	27.7000i		−	1.76251i
$248$		0			0
$249$	−15.9508			−1.01084
$250$		0			0
$251$	−26.0099			−1.64173			−0.820865	−	0.571123i	$-0.806508\pi$
$251$	−26.0099			−1.64173			−0.820865	+	0.571123i	$0.806508\pi$
$252$		0			0
$253$		−	3.43385i		−	0.215884i
$254$		0			0
$255$	−20.5243	−	29.1702i	−1.28528	−	1.82671i
$256$		0			0
$257$		−	15.0335i		−	0.937766i	−0.883260	−	0.468883i	$-0.844657\pi$
$257$		−	15.0335i		−	0.937766i	0.883260	−	0.468883i	$-0.155343\pi$
$258$		0			0
$259$	−1.19605			−0.0743188
$260$		0			0
$261$	−3.57344			−0.221191
$262$		0			0
$263$			12.0662i			0.744032i	0.928226	+	0.372016i	$0.121333\pi$
$263$			12.0662i			0.744032i	−0.928226	+	0.372016i	$0.878667\pi$
$264$		0			0
$265$	−20.6395	+	14.5221i	−1.26788	+	0.892084i
$266$		0			0
$267$			27.7000i			1.69521i
$268$		0			0
$269$	10.8531			0.661726			0.330863	−	0.943679i	$-0.392660\pi$
$269$	10.8531			0.661726			0.330863	+	0.943679i	$0.392660\pi$
$270$		0			0
$271$	12.7449			0.774198			0.387099	−	0.922038i	$-0.373477\pi$
$271$	12.7449			0.774198			0.387099	+	0.922038i	$0.373477\pi$
$272$		0			0
$273$		−	11.1347i		−	0.673904i
$274$		0			0
$275$	−7.43361	+	20.7166i	−0.448263	+	1.24926i
$276$		0			0
$277$			20.0714i			1.20597i	0.797752	+	0.602986i	$0.206022\pi$
$277$			20.0714i			1.20597i	−0.797752	+	0.602986i	$0.793978\pi$
$278$		0			0
$279$	22.8771			1.36962
$280$		0			0
$281$	−20.9361			−1.24895			−0.624473	−	0.781047i	$-0.714686\pi$
$281$	−20.9361			−1.24895			−0.624473	+	0.781047i	$0.714686\pi$
$282$		0			0
$283$			16.9703i			1.00878i	0.863477	+	0.504389i	$0.168282\pi$
$283$			16.9703i			1.00878i	−0.863477	+	0.504389i	$0.831718\pi$
$284$		0			0
$285$	−32.7056	+	23.0118i	−1.93731	+	1.36310i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−21.7056			−1.27680
$290$		0			0
$291$	44.6565			2.61781
$292$		0			0
$293$			9.47542i			0.553560i	0.960933	+	0.276780i	$0.0892673\pi$
$293$			9.47542i			0.553560i	−0.960933	+	0.276780i	$0.910733\pi$
$294$		0			0
$295$	12.8672	+	18.2876i	0.749158	+	1.06474i
$296$		0			0
$297$		−	6.47193i		−	0.375539i
$298$		0			0
$299$	−3.09772			−0.179146
$300$		0			0
$301$	−0.411857			−0.0237391
$302$		0			0
$303$		−	32.7017i		−	1.87866i
$304$		0			0
$305$	−1.47572	−	2.09737i	−0.0844995	−	0.120095i
$306$		0			0
$307$		−	19.3812i		−	1.10614i	−0.833134	−	0.553072i	$-0.813456\pi$
$307$		−	19.3812i		−	1.10614i	0.833134	−	0.553072i	$-0.186544\pi$
$308$		0			0
$309$	13.2158			0.751821
$310$		0			0
$311$	9.26919			0.525607			0.262804	−	0.964849i	$-0.415353\pi$
$311$	9.26919			0.525607			0.262804	+	0.964849i	$0.415353\pi$
$312$		0			0
$313$			14.5324i			0.821422i	0.911766	+	0.410711i	$0.134719\pi$
$313$			14.5324i			0.821422i	−0.911766	+	0.410711i	$0.865281\pi$
$314$		0			0
$315$	−7.14688	+	5.02858i	−0.402681	+	0.283328i
$316$		0			0
$317$		−	22.4116i		−	1.25876i	−0.777098	−	0.629380i	$-0.783309\pi$
$317$		−	22.4116i		−	1.25876i	0.777098	−	0.629380i	$-0.216691\pi$
$318$		0			0
$319$	4.40198			0.246463
$320$		0			0
$321$	−2.80395			−0.156501
$322$		0			0
$323$			43.3968i			2.41466i
$324$		0			0
$325$	18.6887	+	6.70596i	1.03666	+	0.371979i
$326$		0			0
$327$		−	39.2996i		−	2.17327i
$328$		0			0
$329$	5.19605			0.286467
$330$		0			0
$331$	−7.89179			−0.433772			−0.216886	−	0.976197i	$-0.569590\pi$
$331$	−7.89179			−0.433772			−0.216886	+	0.976197i	$0.569590\pi$
$332$		0			0
$333$			3.90806i			0.214160i
$334$		0			0
$335$	−10.8040	+	7.60171i	−0.590283	+	0.415326i
$336$		0			0
$337$			33.4280i			1.82094i	0.413579	+	0.910468i	$0.364279\pi$
$337$			33.4280i			1.82094i	−0.413579	+	0.910468i	$0.635721\pi$
$338$		0			0
$339$	18.6859			1.01488
$340$		0			0
$341$	−28.1813			−1.52611
$342$		0			0
$343$		−	14.0029i		−	0.756086i
$344$		0			0
$345$	2.57344	+	3.65751i	0.138549	+	0.196914i
$346$		0			0
$347$		−	31.1069i		−	1.66991i	−0.550320	−	0.834954i	$-0.685494\pi$
$347$		−	31.1069i		−	1.66991i	0.550320	−	0.834954i	$-0.314506\pi$
$348$		0			0
$349$	5.76949			0.308834			0.154417	−	0.988006i	$-0.450650\pi$
$349$	5.76949			0.308834			0.154417	+	0.988006i	$0.450650\pi$
$350$		0			0
$351$	−5.83842			−0.311632
$352$		0			0
$353$			13.5095i			0.719039i	0.933137	+	0.359520i	$0.117059\pi$
$353$			13.5095i			0.719039i	−0.933137	+	0.359520i	$0.882941\pi$
$354$		0			0
$355$	−2.57344	−	3.65751i	−0.136584	−	0.194121i
$356$		0			0
$357$			17.4445i			0.923259i
$358$		0			0
$359$	−1.15677			−0.0610518			−0.0305259	−	0.999534i	$-0.509718\pi$
$359$	−1.15677			−0.0610518			−0.0305259	+	0.999534i	$0.509718\pi$
$360$		0			0
$361$	29.6565			1.56087
$362$		0			0
$363$			21.4786i			1.12733i
$364$		0			0
$365$	15.9508	−	11.2231i	0.834905	−	0.587443i
$366$		0			0
$367$			10.5959i			0.553104i	0.960999	+	0.276552i	$0.0891918\pi$
$367$			10.5959i			0.553104i	−0.960999	+	0.276552i	$0.910808\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.3429			0.640813
$372$		0			0
$373$			7.09907i			0.367576i	0.982966	+	0.183788i	$0.0588360\pi$
$373$			7.09907i			0.367576i	−0.982966	+	0.183788i	$0.941164\pi$
$374$		0			0
$375$	−7.60790	−	27.6369i	−0.392871	−	1.42717i
$376$		0			0
$377$		−	3.97108i		−	0.204521i
$378$		0			0
$379$	18.1223			0.930880			0.465440	−	0.885079i	$-0.345896\pi$
$379$	18.1223			0.930880			0.465440	+	0.885079i	$0.345896\pi$
$380$		0			0
$381$	45.4604			2.32901
$382$		0			0
$383$		−	3.28092i		−	0.167647i	−0.996481	−	0.0838236i	$-0.973287\pi$
$383$		−	3.28092i		−	0.167647i	0.996481	−	0.0838236i	$-0.0267132\pi$
$384$		0			0
$385$	8.80395	−	6.19450i	0.448691	−	0.315701i
$386$		0			0
$387$			1.34573i			0.0684075i
$388$		0			0
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	4.85312			0.245433
$392$		0			0
$393$		−	18.7633i		−	0.946484i
$394$		0			0
$395$	−7.13984	−	10.1475i	−0.359244	−	0.510577i
$396$		0			0
$397$		−	36.3054i		−	1.82212i	−0.412278	−	0.911058i	$-0.635267\pi$
$397$		−	36.3054i		−	1.82212i	0.412278	−	0.911058i	$-0.364733\pi$
$398$		0			0
$399$	19.5587			0.979162
$400$		0			0
$401$	26.0830			1.30252			0.651262	−	0.758853i	$-0.274240\pi$
$401$	26.0830			1.30252			0.651262	+	0.758853i	$0.274240\pi$
$402$		0			0
$403$			25.4228i			1.26640i
$404$		0			0
$405$	−8.94379	−	12.7114i	−0.444420	−	0.631634i
$406$		0			0
$407$		−	4.81418i		−	0.238630i
$408$		0			0
$409$	−13.0486			−0.645210			−0.322605	−	0.946534i	$-0.604558\pi$
$409$	−13.0486			−0.645210			−0.322605	+	0.946534i	$0.604558\pi$
$410$		0			0
$411$	−25.0977			−1.23798
$412$		0			0
$413$		−	10.9364i		−	0.538145i
$414$		0			0
$415$	−11.3774	+	8.00519i	−0.558494	+	0.392959i
$416$		0			0
$417$		−	7.06293i		−	0.345873i
$418$		0			0
$419$	35.8525			1.75151			0.875755	−	0.482756i	$-0.160364\pi$
$419$	35.8525			1.75151			0.875755	+	0.482756i	$0.160364\pi$
$420$		0			0
$421$	16.7056			0.814182			0.407091	−	0.913388i	$-0.366543\pi$
$421$	16.7056			0.814182			0.407091	+	0.913388i	$0.366543\pi$
$422$		0			0
$423$		−	16.9780i		−	0.825497i
$424$		0			0
$425$	−29.2791	−	10.5060i	−1.42024	−	0.509618i
$426$		0			0
$427$			1.25428i			0.0606988i
$428$		0			0
$429$	44.8180			2.16384
$430$		0			0
$431$	26.3135			1.26748			0.633739	−	0.773547i	$-0.281519\pi$
$431$	26.3135			1.26748			0.633739	+	0.773547i	$0.281519\pi$
$432$		0			0
$433$		−	10.7220i		−	0.515266i	−0.966243	−	0.257633i	$-0.917057\pi$
$433$		−	10.7220i		−	0.515266i	0.966243	−	0.257633i	$-0.0829425\pi$
$434$		0			0
$435$	−4.68870	+	3.29899i	−0.224806	+	0.158174i
$436$		0			0
$437$		−	5.44131i		−	0.260293i
$438$		0			0
$439$	29.5587			1.41076			0.705381	−	0.708828i	$-0.250776\pi$
$439$	29.5587			1.41076			0.705381	+	0.708828i	$0.250776\pi$
$440$		0			0
$441$	−20.7401			−0.987623
$442$		0			0
$443$		−	3.40697i		−	0.161870i	−0.996719	−	0.0809349i	$-0.974209\pi$
$443$		−	3.40697i		−	0.161870i	0.996719	−	0.0809349i	$-0.0257906\pi$
$444$		0			0
$445$	13.9017	+	19.7578i	0.659003	+	0.936609i
$446$		0			0
$447$			11.5997i			0.548646i
$448$		0			0
$449$	−0.461020			−0.0217569			−0.0108784	−	0.999941i	$-0.503463\pi$
$449$	−0.461020			−0.0217569			−0.0108784	+	0.999941i	$0.503463\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	7.18898i		−	0.337768i
$454$		0			0
$455$	−5.58814	−	7.94216i	−0.261976	−	0.372334i
$456$		0			0
$457$			1.37262i			0.0642084i	0.999485	+	0.0321042i	$0.0102208\pi$
$457$			1.37262i			0.0642084i	−0.999485	+	0.0321042i	$0.989779\pi$
$458$		0			0
$459$	9.14688			0.426940
$460$		0			0
$461$	14.5875			0.679409			0.339705	−	0.940532i	$-0.389673\pi$
$461$	14.5875			0.679409			0.339705	+	0.940532i	$0.389673\pi$
$462$		0			0
$463$		−	18.8517i		−	0.876112i	−0.898948	−	0.438056i	$-0.855667\pi$
$463$		−	18.8517i		−	0.876112i	0.898948	−	0.438056i	$-0.144333\pi$
$464$		0			0
$465$	30.0169	−	21.1200i	1.39200	−	0.979419i
$466$		0			0
$467$		−	29.9580i		−	1.38629i	−0.720798	−	0.693145i	$-0.756225\pi$
$467$		−	29.9580i		−	1.38629i	0.720798	−	0.693145i	$-0.243775\pi$
$468$		0			0
$469$	6.46102			0.298342
$470$		0			0
$471$	12.7548			0.587710
$472$		0			0
$473$		−	1.65776i		−	0.0762237i
$474$		0			0
$475$	−11.7794	+	32.8277i	−0.540475	+	1.50624i
$476$		0			0
$477$		−	40.3302i		−	1.84660i
$478$		0			0
$479$	−32.7647			−1.49706			−0.748528	−	0.663103i	$-0.769239\pi$
$479$	−32.7647			−1.49706			−0.748528	+	0.663103i	$0.769239\pi$
$480$		0			0
$481$	−4.34293			−0.198021
$482$		0			0
$483$		−	2.18728i		−	0.0995246i
$484$		0			0
$485$	31.8525	−	22.4116i	1.44635	−	1.01766i
$486$		0			0
$487$			14.1098i			0.639375i	0.947523	+	0.319688i	$0.103578\pi$
$487$			14.1098i			0.639375i	−0.947523	+	0.319688i	$0.896422\pi$
$488$		0			0
$489$	53.2299			2.40714
$490$		0			0
$491$	−4.10821			−0.185401			−0.0927004	−	0.995694i	$-0.529550\pi$
$491$	−4.10821			−0.185401			−0.0927004	+	0.995694i	$0.529550\pi$
$492$		0			0
$493$			6.22138i			0.280197i
$494$		0			0
$495$	−20.2404	−	28.7667i	−0.909738	−	1.29297i
$496$		0			0
$497$			2.18728i			0.0981129i
$498$		0			0
$499$	30.0689			1.34607			0.673035	−	0.739611i	$-0.264990\pi$
$499$	30.0689			1.34607			0.673035	+	0.739611i	$0.264990\pi$
$500$		0			0
$501$	−56.1954			−2.51063
$502$		0			0
$503$		−	14.8806i		−	0.663493i	−0.943369	−	0.331746i	$-0.892362\pi$
$503$		−	14.8806i		−	0.663493i	0.943369	−	0.331746i	$-0.107638\pi$
$504$		0			0
$505$	−16.4119	−	23.3254i	−0.730318	−	1.03797i
$506$		0			0
$507$		−	7.10062i		−	0.315350i
$508$		0			0
$509$	−1.93674			−0.0858445			−0.0429223	−	0.999078i	$-0.513667\pi$
$509$	−1.93674			−0.0858445			−0.0429223	+	0.999078i	$0.513667\pi$
$510$		0			0
$511$	−9.53898			−0.421980
$512$		0			0
$513$		−	10.2555i		−	0.452791i
$514$		0			0
$515$	9.42656	−	6.63257i	0.415384	−	0.292266i
$516$		0			0
$517$			20.9145i			0.919817i
$518$		0			0
$519$	8.34293			0.366214
$520$		0			0
$521$	−4.23051			−0.185342			−0.0926710	−	0.995697i	$-0.529540\pi$
$521$	−4.23051			−0.185342			−0.0926710	+	0.995697i	$0.529540\pi$
$522$		0			0
$523$		−	4.40144i		−	0.192462i	−0.995359	−	0.0962308i	$-0.969321\pi$
$523$		−	4.40144i		−	0.192462i	0.995359	−	0.0962308i	$-0.0306787\pi$
$524$		0			0
$525$	−4.73503	+	13.1960i	−0.206654	+	0.575919i
$526$		0			0
$527$		−	39.8292i		−	1.73499i
$528$		0			0
$529$	22.3915			0.973543
$530$		0			0
$531$	−35.7344			−1.55074
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−2.00000	+	1.40721i	−0.0864675	+	0.0608390i
$536$		0			0
$537$		−	18.4497i		−	0.796165i
$538$		0			0
$539$	25.5489			1.10047
$540$		0			0
$541$	−32.4119			−1.39349			−0.696747	−	0.717317i	$-0.745370\pi$
$541$	−32.4119			−1.39349			−0.696747	+	0.717317i	$0.745370\pi$
$542$		0			0
$543$			33.4187i			1.43413i
$544$		0			0
$545$	−19.7232	−	28.0316i	−0.844847	−	1.20074i
$546$		0			0
$547$			12.9093i			0.551961i	0.961163	+	0.275980i	$0.0890024\pi$
$547$			12.9093i			0.551961i	−0.961163	+	0.275980i	$0.910998\pi$
$548$		0			0
$549$	4.09833			0.174912
$550$		0			0
$551$	6.97542			0.297163
$552$		0			0
$553$			6.06845i			0.258057i
$554$		0			0
$555$	3.60790	+	5.12775i	0.153147	+	0.217661i
$556$		0			0
$557$			23.0195i			0.975369i	0.873020	+	0.487685i	$0.162158\pi$
$557$			23.0195i			0.975369i	−0.873020	+	0.487685i	$0.837842\pi$
$558$		0			0
$559$	−1.49548			−0.0632522
$560$		0			0
$561$	−70.2152			−2.96449
$562$		0			0
$563$		−	27.3234i		−	1.15154i	−0.817611	−	0.575771i	$-0.804702\pi$
$563$		−	27.3234i		−	1.15154i	0.817611	−	0.575771i	$-0.195298\pi$
$564$		0			0
$565$	13.3282	−	9.37781i	0.560723	−	0.394527i
$566$		0			0
$567$			7.60171i			0.319242i
$568$		0			0
$569$	0.803952			0.0337034			0.0168517	−	0.999858i	$-0.494636\pi$
$569$	0.803952			0.0337034			0.0168517	+	0.999858i	$0.494636\pi$
$570$		0			0
$571$	−43.4604			−1.81876			−0.909381	−	0.415964i	$-0.863444\pi$
$571$	−43.4604			−1.81876			−0.909381	+	0.415964i	$0.863444\pi$
$572$		0			0
$573$			25.9523i			1.08417i
$574$		0			0
$575$	3.67116	+	1.31730i	0.153098	+	0.0549353i
$576$		0			0
$577$		−	1.27345i		−	0.0530146i	−0.999649	−	0.0265073i	$-0.991561\pi$
$577$		−	1.27345i		−	0.0530146i	0.999649	−	0.0265073i	$-0.00843852\pi$
$578$		0			0
$579$	−58.6073			−2.43564
$580$		0			0
$581$	6.80395			0.282276
$582$		0			0
$583$			49.6812i			2.05758i
$584$		0			0
$585$	−25.9508	+	18.2591i	−1.07294	+	0.754922i
$586$		0			0
$587$		−	31.8678i		−	1.31533i	−0.753312	−	0.657663i	$-0.771545\pi$
$587$		−	31.8678i		−	1.31533i	0.753312	−	0.657663i	$-0.228455\pi$
$588$		0			0
$589$	−44.6565			−1.84004
$590$		0			0
$591$	56.5875			2.32770
$592$		0			0
$593$		−	1.21043i		−	0.0497064i	−0.999691	−	0.0248532i	$-0.992088\pi$
$593$		−	1.21043i		−	0.0497064i	0.999691	−	0.0248532i	$-0.00791183\pi$
$594$		0			0
$595$	8.75479	+	12.4428i	0.358911	+	0.510104i
$596$		0			0
$597$			13.3220i			0.545233i
$598$		0			0
$599$	24.2053			0.989003			0.494501	−	0.869177i	$-0.335351\pi$
$599$	24.2053			0.989003			0.494501	+	0.869177i	$0.335351\pi$
$600$		0			0
$601$	4.68586			0.191140			0.0955702	−	0.995423i	$-0.469533\pi$
$601$	4.68586			0.191140			0.0955702	+	0.995423i	$0.469533\pi$
$602$		0			0
$603$		−	21.1112i		−	0.859716i
$604$		0			0
$605$	10.7794	+	15.3202i	0.438244	+	0.622855i
$606$		0			0
$607$		−	0.502641i		−	0.0204016i	−0.999948	−	0.0102008i	$-0.996753\pi$
$607$		−	0.502641i		−	0.0204016i	0.999948	−	0.0102008i	$-0.00324707\pi$
$608$		0			0
$609$	2.80395			0.113622
$610$		0			0
$611$	18.8672			0.763286
$612$		0			0
$613$			22.9142i			0.925496i	0.886490	+	0.462748i	$0.153137\pi$
$613$			22.9142i			0.925496i	−0.886490	+	0.462748i	$0.846863\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.8574i			0.638397i	0.947688	+	0.319198i	$0.103414\pi$
$617$			15.8574i			0.638397i	−0.947688	+	0.319198i	$0.896586\pi$
$618$		0			0
$619$	−35.4014			−1.42290			−0.711451	−	0.702736i	$-0.751961\pi$
$619$	−35.4014			−1.42290			−0.711451	+	0.702736i	$0.751961\pi$
$620$		0			0
$621$	−1.14688			−0.0460229
$622$		0			0
$623$		−	11.8156i		−	0.473383i
$624$		0			0
$625$	−19.2966	−	15.8947i	−0.771864	−	0.635788i
$626$		0			0
$627$			78.7253i			3.14399i
$628$		0			0
$629$	6.80395			0.271291
$630$		0			0
$631$	−44.7056			−1.77970			−0.889851	−	0.456250i	$-0.849192\pi$
$631$	−44.7056			−1.77970			−0.889851	+	0.456250i	$0.849192\pi$
$632$		0			0
$633$		−	4.22318i		−	0.167856i
$634$		0			0
$635$	32.4260	−	22.8150i	1.28678	−	0.905388i
$636$		0			0
$637$		−	23.0480i		−	0.913194i
$638$		0			0
$639$	7.14688			0.282726
$640$		0			0
$641$	10.0689			0.397699			0.198849	−	0.980030i	$-0.436280\pi$
$641$	10.0689			0.397699			0.198849	+	0.980030i	$0.436280\pi$
$642$		0			0
$643$		−	13.9760i		−	0.551161i	−0.961278	−	0.275580i	$-0.911130\pi$
$643$		−	13.9760i		−	0.551161i	0.961278	−	0.275580i	$-0.0888700\pi$
$644$		0			0
$645$	1.24238	+	1.76573i	0.0489186	+	0.0695256i
$646$		0			0
$647$			31.1607i			1.22505i	0.790450	+	0.612527i	$0.209847\pi$
$647$			31.1607i			1.22505i	−0.790450	+	0.612527i	$0.790153\pi$
$648$		0			0
$649$	44.0198			1.72793
$650$		0			0
$651$	−17.9508			−0.703549
$652$		0			0
$653$		−	26.1129i		−	1.02188i	−0.859617	−	0.510939i	$-0.829298\pi$
$653$		−	26.1129i		−	1.02188i	0.859617	−	0.510939i	$-0.170702\pi$
$654$		0			0
$655$	−9.41668	−	13.3835i	−0.367940	−	0.522936i
$656$		0			0
$657$			31.1684i			1.21600i
$658$		0			0
$659$	−28.5974			−1.11400			−0.556999	−	0.830513i	$-0.688047\pi$
$659$	−28.5974			−1.11400			−0.556999	+	0.830513i	$0.688047\pi$
$660$		0			0
$661$	−38.9994			−1.51690			−0.758450	−	0.651731i	$-0.774043\pi$
$661$	−38.9994			−1.51690			−0.758450	+	0.651731i	$0.774043\pi$
$662$		0			0
$663$			63.3421i			2.46000i
$664$		0			0
$665$	13.9508	−	9.81587i	0.540990	−	0.380643i
$666$		0			0
$667$		−	0.780070i		−	0.0302044i
$668$		0			0
$669$	24.2938			0.939251
$670$		0			0
$671$	−5.04856			−0.194897
$672$		0			0
$673$		−	38.6725i		−	1.49072i	−0.666665	−	0.745358i	$-0.732279\pi$
$673$		−	38.6725i		−	1.49072i	0.666665	−	0.745358i	$-0.267721\pi$
$674$		0			0
$675$	6.91921	+	2.48278i	0.266320	+	0.0955622i
$676$		0			0
$677$		−	28.4800i		−	1.09458i	−0.836944	−	0.547288i	$-0.815660\pi$
$677$		−	28.4800i		−	1.09458i	0.836944	−	0.547288i	$-0.184340\pi$
$678$		0			0
$679$	−19.0486			−0.731017
$680$		0			0
$681$	−14.6859			−0.562764
$682$		0			0
$683$		−	12.4159i		−	0.475081i	−0.971378	−	0.237540i	$-0.923659\pi$
$683$		−	12.4159i		−	0.475081i	0.971378	−	0.237540i	$-0.0763412\pi$
$684$		0			0
$685$	−17.9017	+	12.5957i	−0.683988	+	0.481257i
$686$		0			0
$687$		−	38.8347i		−	1.48164i
$688$		0			0
$689$	44.8180			1.70743
$690$		0			0
$691$	−10.8040			−0.411002			−0.205501	−	0.978657i	$-0.565882\pi$
$691$	−10.8040			−0.411002			−0.205501	+	0.978657i	$0.565882\pi$
$692$		0			0
$693$			17.2032i			0.653495i
$694$		0			0
$695$	−3.54465	−	5.03784i	−0.134456	−	0.191096i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	1.83842			0.0695352
$700$		0			0
$701$	8.47512			0.320101			0.160050	−	0.987109i	$-0.448834\pi$
$701$	8.47512			0.320101			0.160050	+	0.987109i	$0.448834\pi$
$702$		0			0
$703$		−	7.62859i		−	0.287718i
$704$		0			0
$705$	−15.6740	−	22.2767i	−0.590317	−	0.838990i
$706$		0			0
$707$			13.9491i			0.524612i
$708$		0			0
$709$	32.8180			1.23251			0.616254	−	0.787548i	$-0.288650\pi$
$709$	32.8180			1.23251			0.616254	+	0.787548i	$0.288650\pi$
$710$		0			0
$711$	19.8285			0.743628
$712$		0			0
$713$			4.99399i			0.187026i
$714$		0			0
$715$	31.9678	−	22.4927i	1.19553	−	0.841178i
$716$		0			0
$717$		−	5.12775i		−	0.191499i
$718$		0			0
$719$	39.7542			1.48258			0.741290	−	0.671184i	$-0.234214\pi$
$719$	39.7542			1.48258			0.741290	+	0.671184i	$0.234214\pi$
$720$		0			0
$721$	−5.63731			−0.209944
$722$		0			0
$723$			44.4274i			1.65227i
$724$		0			0
$725$	−1.68870	+	4.70620i	−0.0627166	+	0.174784i
$726$		0			0
$727$		−	49.4844i		−	1.83527i	−0.397419	−	0.917637i	$-0.630094\pi$
$727$		−	49.4844i		−	1.83527i	0.397419	−	0.917637i	$-0.369906\pi$
$728$		0			0
$729$	36.1469			1.33877
$730$		0			0
$731$	2.34293			0.0866565
$732$		0			0
$733$			46.2381i			1.70784i	0.520403	+	0.853921i	$0.325782\pi$
$733$			46.2381i			1.70784i	−0.520403	+	0.853921i	$0.674218\pi$
$734$		0			0
$735$	−27.2130	+	19.1472i	−1.00377	+	0.706254i
$736$		0			0
$737$			26.0061i			0.957946i
$738$		0			0
$739$	28.7155			1.05632			0.528159	−	0.849146i	$-0.322883\pi$
$739$	28.7155			1.05632			0.528159	+	0.849146i	$0.322883\pi$
$740$		0			0
$741$	71.0192			2.60895
$742$		0			0
$743$			27.0536i			0.992502i	0.868179	+	0.496251i	$0.165290\pi$
$743$			27.0536i			0.992502i	−0.868179	+	0.496251i	$0.834710\pi$
$744$		0			0
$745$	5.82149	+	8.27380i	0.213283	+	0.303129i
$746$		0			0
$747$		−	22.2318i		−	0.813418i
$748$		0			0
$749$	1.19605			0.0437026
$750$		0			0
$751$	26.8771			0.980759			0.490380	−	0.871509i	$-0.336858\pi$
$751$	26.8771			0.980759			0.490380	+	0.871509i	$0.336858\pi$
$752$		0			0
$753$		−	66.6860i		−	2.43017i
$754$		0			0
$755$	−3.60790	−	5.12775i	−0.131305	−	0.186618i
$756$		0			0
$757$		−	7.28814i		−	0.264892i	−0.991190	−	0.132446i	$-0.957717\pi$
$757$		−	7.28814i		−	0.264892i	0.991190	−	0.132446i	$-0.0422831\pi$
$758$		0			0
$759$	8.80395			0.319563
$760$		0			0
$761$	−31.6079			−1.14579			−0.572893	−	0.819630i	$-0.694179\pi$
$761$	−31.6079			−1.14579			−0.572893	+	0.819630i	$0.694179\pi$
$762$		0			0
$763$			16.7636i			0.606882i
$764$		0			0
$765$	40.6565	−	28.6061i	1.46994	−	1.03425i
$766$		0			0
$767$		−	39.7108i		−	1.43387i
$768$		0			0
$769$	0.293769			0.0105936			0.00529679	−	0.999986i	$-0.498314\pi$
$769$	0.293769			0.0105936			0.00529679	+	0.999986i	$0.498314\pi$
$770$		0			0
$771$	38.5440			1.38813
$772$		0			0
$773$		−	28.4877i		−	1.02463i	−0.858797	−	0.512316i	$-0.828788\pi$
$773$		−	28.4877i		−	1.02463i	0.858797	−	0.512316i	$-0.171212\pi$
$774$		0			0
$775$	10.8110	−	30.1290i	0.388343	−	1.08226i
$776$		0			0
$777$		−	3.06651i		−	0.110011i
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−8.80395			−0.315030
$782$		0			0
$783$		−	1.47023i		−	0.0525418i
$784$		0			0
$785$	9.09772	−	6.40120i	0.324712	−	0.228469i
$786$		0			0
$787$			32.9883i			1.17591i	0.808895	+	0.587954i	$0.200066\pi$
$787$			32.9883i			1.17591i	−0.808895	+	0.587954i	$0.799934\pi$
$788$		0			0
$789$	−30.9361			−1.10136
$790$		0			0
$791$	−7.97060			−0.283402
$792$		0			0
$793$			4.55437i			0.161731i
$794$		0			0
$795$	−37.2327	−	52.9171i	−1.32051	−	1.87678i
$796$		0			0
$797$		−	30.9194i		−	1.09522i	−0.836733	−	0.547611i	$-0.815537\pi$
$797$		−	30.9194i		−	1.09522i	0.836733	−	0.547611i	$-0.184463\pi$
$798$		0			0
$799$	−29.5587			−1.04571
$800$		0			0
$801$	−38.6073			−1.36412
$802$		0			0
$803$		−	38.3951i		−	1.35493i
$804$		0			0
$805$	−1.09772	−	1.56014i	−0.0386896	−	0.0549877i
$806$		0			0
$807$			27.8260i			0.979522i
$808$		0			0
$809$	−21.0977			−0.741756			−0.370878	−	0.928682i	$-0.620943\pi$
$809$	−21.0977			−0.741756			−0.370878	+	0.928682i	$0.620943\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$			32.6763i			1.14601i
$814$		0			0
$815$	37.9678	−	26.7143i	1.32995	−	0.935761i
$816$		0			0
$817$		−	2.62690i		−	0.0919035i
$818$		0			0
$819$	15.5192			0.542285
$820$		0			0
$821$	−15.2791			−0.533243			−0.266622	−	0.963801i	$-0.585907\pi$
$821$	−15.2791			−0.533243			−0.266622	+	0.963801i	$0.585907\pi$
$822$		0			0
$823$			7.78152i			0.271247i	0.990760	+	0.135623i	$0.0433037\pi$
$823$			7.78152i			0.271247i	−0.990760	+	0.135623i	$0.956696\pi$
$824$		0			0
$825$	−53.1147	−	19.0588i	−1.84921	−	0.663543i
$826$		0			0
$827$		−	0.376593i		−	0.0130954i	−0.999979	−	0.00654772i	$-0.997916\pi$
$827$		−	0.376593i		−	0.0130954i	0.999979	−	0.00654772i	$-0.00208422\pi$
$828$		0			0
$829$	49.8723			1.73214			0.866068	−	0.499927i	$-0.166640\pi$
$829$	49.8723			1.73214			0.866068	+	0.499927i	$0.166640\pi$
$830$		0			0
$831$	−51.4604			−1.78514
$832$		0			0
$833$			36.1086i			1.25109i
$834$		0			0
$835$	−40.0830	+	28.2026i	−1.38713	+	0.975991i
$836$		0			0
$837$			9.41240i			0.325340i
$838$		0			0
$839$	−0.0590441			−0.00203843			−0.00101921	−	0.999999i	$-0.500324\pi$
$839$	−0.0590441			−0.00203843			−0.00101921	+	0.999999i	$0.500324\pi$
$840$		0			0
$841$	1.00000			0.0344828
$842$		0			0
$843$		−	53.6776i		−	1.84875i
$844$		0			0
$845$	−3.56356	−	5.06472i	−0.122590	−	0.174232i
$846$		0			0
$847$		−	9.16185i		−	0.314805i
$848$		0			0
$849$	−43.5096			−1.49324
$850$		0			0
$851$	−0.853115			−0.0292444
$852$		0			0
$853$			2.77983i			0.0951795i	0.998867	+	0.0475897i	$0.0151540\pi$
$853$			2.77983i			0.0951795i	−0.998867	+	0.0475897i	$0.984846\pi$
$854$		0			0
$855$	−32.0731	−	45.5840i	−1.09688	−	1.55894i
$856$		0			0
$857$		−	0.850802i		−	0.0290628i	−0.999894	−	0.0145314i	$-0.995374\pi$
$857$		−	0.850802i		−	0.0290628i	0.999894	−	0.0145314i	$-0.00462566\pi$
$858$		0			0
$859$	7.35342			0.250895			0.125448	−	0.992100i	$-0.459963\pi$
$859$	7.35342			0.250895			0.125448	+	0.992100i	$0.459963\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	42.6167i		−	1.45069i	−0.688386	−	0.725345i	$-0.741680\pi$
$863$		−	42.6167i		−	1.45069i	0.688386	−	0.725345i	$-0.258320\pi$
$864$		0			0
$865$	5.95084	−	4.18704i	0.202335	−	0.142364i
$866$		0			0
$867$		−	55.6505i		−	1.88999i
$868$		0			0
$869$	−24.4260			−0.828594
$870$		0			0
$871$	23.4604			0.794926
$872$		0			0
$873$			62.2407i			2.10653i
$874$		0			0
$875$	3.24521	+	11.7887i	0.109708	+	0.398532i
$876$		0			0
$877$		−	13.4196i		−	0.453148i	−0.973994	−	0.226574i	$-0.927247\pi$
$877$		−	13.4196i		−	0.453148i	0.973994	−	0.226574i	$-0.0727526\pi$
$878$		0			0
$879$	−24.2938			−0.819408
$880$		0			0
$881$	−38.3135			−1.29082			−0.645408	−	0.763838i	$-0.723313\pi$
$881$	−38.3135			−1.29082			−0.645408	+	0.763838i	$0.723313\pi$
$882$		0			0
$883$			31.5542i			1.06188i	0.847408	+	0.530942i	$0.178162\pi$
$883$			31.5542i			1.06188i	−0.847408	+	0.530942i	$0.821838\pi$
$884$		0			0
$885$	−46.8870	+	32.9899i	−1.57609	+	1.10894i
$886$		0			0
$887$			23.3961i			0.785565i	0.919631	+	0.392783i	$0.128488\pi$
$887$			23.3961i			0.785565i	−0.919631	+	0.392783i	$0.871512\pi$
$888$		0			0
$889$	−19.3915			−0.650370
$890$		0			0
$891$	−30.5974			−1.02505
$892$		0			0
$893$			33.1413i			1.10903i
$894$		0			0
$895$	−9.25931	−	13.1598i	−0.309504	−	0.439884i
$896$		0			0
$897$		−	7.94216i		−	0.265181i
$898$		0			0
$899$	−6.40198			−0.213518
$900$		0			0
$901$	−70.2152			−2.33921
$902$		0			0
$903$		−	1.05595i		−	0.0351398i
$904$		0			0
$905$	16.7717	+	23.8369i	0.557511	+	0.792364i
$906$		0			0
$907$			32.4150i			1.07632i	0.842842	+	0.538161i	$0.180881\pi$
$907$			32.4150i			1.07632i	−0.842842	+	0.538161i	$0.819119\pi$
$908$		0			0
$909$	45.5785			1.51174
$910$		0			0
$911$	45.2749			1.50002			0.750011	−	0.661425i	$-0.230048\pi$
$911$	45.2749			1.50002			0.750011	+	0.661425i	$0.230048\pi$
$912$		0			0
$913$			27.3864i			0.906357i
$914$		0			0
$915$	5.37739	−	3.78356i	0.177771	−	0.125081i
$916$		0			0
$917$			8.00364i			0.264303i
$918$		0			0
$919$	−5.90167			−0.194678			−0.0973391	−	0.995251i	$-0.531033\pi$
$919$	−5.90167			−0.194678			−0.0973391	+	0.995251i	$0.531033\pi$
$920$		0			0
$921$	49.6909			1.63737
$922$		0			0
$923$			7.94216i			0.261419i
$924$		0			0
$925$	5.14688	+	1.84683i	0.169229	+	0.0607233i
$926$		0			0
$927$			18.4198i			0.604985i
$928$		0			0
$929$	6.00000			0.196854			0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000			0.196854			0.0984268	+	0.995144i	$0.468619\pi$
$930$		0			0
$931$	40.4850			1.32684
$932$		0			0
$933$			23.7650i			0.778032i
$934$		0			0
$935$	−50.0830	+	35.2386i	−1.63789	+	1.15243i
$936$		0			0
$937$		−	43.0755i		−	1.40721i	−0.710589	−	0.703607i	$-0.751571\pi$
$937$		−	43.0755i		−	1.40721i	0.710589	−	0.703607i	$-0.248429\pi$
$938$		0			0
$939$	−37.2593			−1.21591
$940$		0			0
$941$	2.91637			0.0950711			0.0475355	−	0.998870i	$-0.484863\pi$
$941$	2.91637			0.0950711			0.0475355	+	0.998870i	$0.484863\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−2.06893	−	2.94047i	−0.0673021	−	0.0956534i
$946$		0			0
$947$			0.556407i			0.0180808i	0.999959	+	0.00904041i	$0.00287769\pi$
$947$			0.556407i			0.0180808i	−0.999959	+	0.00904041i	$0.997122\pi$
$948$		0			0
$949$	−34.6367			−1.12435
$950$		0			0
$951$	57.4604			1.86328
$952$		0			0
$953$		−	37.1661i		−	1.20393i	−0.798523	−	0.601964i	$-0.794385\pi$
$953$		−	37.1661i		−	1.20393i	0.798523	−	0.601964i	$-0.205615\pi$
$954$		0			0
$955$	13.0246	+	18.5112i	0.421466	+	0.599009i
$956$		0			0
$957$			11.2861i			0.364828i
$958$		0			0
$959$	10.7056			0.345703
$960$		0			0
$961$	9.98530			0.322106
$962$		0			0
$963$		−	3.90806i		−	0.125935i
$964$		0			0
$965$	−41.8033	+	29.4130i	−1.34570	+	0.946839i
$966$		0			0
$967$		−	24.2754i		−	0.780643i	−0.920679	−	0.390322i	$-0.872364\pi$
$967$		−	24.2754i		−	0.780643i	0.920679	−	0.390322i	$-0.127636\pi$
$968$		0			0
$969$	−111.264			−3.57431
$970$		0			0
$971$	17.1709			0.551039			0.275520	−	0.961295i	$-0.411150\pi$
$971$	17.1709			0.551039			0.275520	+	0.961295i	$0.411150\pi$
$972$		0			0
$973$			3.01275i			0.0965842i
$974$		0			0
$975$	−17.1932	+	47.9154i	−0.550624	+	1.53452i
$976$		0			0
$977$			4.95785i			0.158616i	0.996850	+	0.0793078i	$0.0252710\pi$
$977$			4.95785i			0.158616i	−0.996850	+	0.0793078i	$0.974729\pi$
$978$		0			0
$979$	47.5587			1.51998
$980$		0			0
$981$	54.7746			1.74882
$982$		0			0
$983$			11.5651i			0.368869i	0.982845	+	0.184434i	$0.0590453\pi$
$983$			11.5651i			0.368869i	−0.982845	+	0.184434i	$0.940955\pi$
$984$		0			0
$985$	40.3627	−	28.3994i	1.28606	−	0.904879i
$986$		0			0
$987$			13.3220i			0.424044i
$988$		0			0
$989$	−0.293769			−0.00934132
$990$		0			0
$991$	18.3429			0.582682			0.291341	−	0.956619i	$-0.405898\pi$
$991$	18.3429			0.582682			0.291341	+	0.956619i	$0.405898\pi$
$992$		0			0
$993$		−	20.2336i		−	0.642092i
$994$		0			0
$995$	6.68586	+	9.50230i	0.211956	+	0.301243i
$996$		0			0
$997$		−	32.2997i		−	1.02294i	−0.859300	−	0.511471i	$-0.829101\pi$
$997$		−	32.2997i		−	1.02294i	0.859300	−	0.511471i	$-0.170899\pi$
$998$		0			0
$999$	−1.60790			−0.0508719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2320.2.d.g.929.5		6
4.3	odd	2		145.2.b.c.59.3	✓	6
5.4	even	2	inner	2320.2.d.g.929.2		6
12.11	even	2		1305.2.c.h.784.4		6
20.3	even	4		725.2.a.l.1.3		6
20.7	even	4		725.2.a.l.1.4		6
20.19	odd	2		145.2.b.c.59.4	yes	6
60.23	odd	4		6525.2.a.bt.1.4		6
60.47	odd	4		6525.2.a.bt.1.3		6
60.59	even	2		1305.2.c.h.784.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.3	✓	6	4.3	odd	2
145.2.b.c.59.4	yes	6	20.19	odd	2
725.2.a.l.1.3		6	20.3	even	4
725.2.a.l.1.4		6	20.7	even	4
1305.2.c.h.784.3		6	60.59	even	2
1305.2.c.h.784.4		6	12.11	even	2
2320.2.d.g.929.2		6	5.4	even	2	inner
2320.2.d.g.929.5		6	1.1	even	1	trivial
6525.2.a.bt.1.3		6	60.47	odd	4
6525.2.a.bt.1.4		6	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2320.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56387i q^{3} +(1.28672 + 1.82876i) q^{5} -1.09364i q^{7} -3.57344 q^{9} +4.40198 q^{11} -3.97108i q^{13} +(-4.68870 + 3.29899i) q^{15} +6.22138i q^{17} +6.97542 q^{19} +2.80395 q^{21} -0.780070i q^{23} +(-1.68870 + 4.70620i) q^{25} -1.47023i q^{27} +1.00000 q^{29} -6.40198 q^{31} +11.2861i q^{33} +(2.00000 - 1.40721i) q^{35} -1.09364i q^{37} +10.1813 q^{39} -0.376593i q^{43} +(-4.59802 - 6.53495i) q^{45} +4.75115i q^{47} +5.80395 q^{49} -15.9508 q^{51} +11.2861i q^{53} +(5.66412 + 8.05014i) q^{55} +17.8841i q^{57} +10.0000 q^{59} -1.14688 q^{61} +3.90806i q^{63} +(7.26214 - 5.10967i) q^{65} +5.90782i q^{67} +2.00000 q^{69} -2.00000 q^{71} -8.72223i q^{73} +(-12.0661 - 4.32960i) q^{75} -4.81418i q^{77} -5.54886 q^{79} -6.95084 q^{81} +6.22138i q^{83} +(-11.3774 + 8.00519i) q^{85} +2.56387i q^{87} +10.8040 q^{89} -4.34293 q^{91} -16.4139i q^{93} +(8.97542 + 12.7563i) q^{95} -17.4176i q^{97} -15.7302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} - 12 q^{9} + 10 q^{11} - 7 q^{15} + 16 q^{19} - 16 q^{21} + 11 q^{25} + 6 q^{29} - 22 q^{31} + 12 q^{35} - 14 q^{39} - 44 q^{45} + 2 q^{49} - 44 q^{51} - 13 q^{55} + 60 q^{59} + 12 q^{61}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q + 3 * q^5 - 12 * q^9 + 10 * q^11 - 7 * q^15 + 16 * q^19 - 16 * q^21 + 11 * q^25 + 6 * q^29 - 22 * q^31 + 12 * q^35 - 14 * q^39 - 44 * q^45 + 2 * q^49 - 44 * q^51 - 13 * q^55 + 60 * q^59 + 12 * q^61 + 13 * q^65 + 12 * q^69 - 12 * q^71 - 9 * q^75 + 2 * q^79 + 10 * q^81 - 26 * q^85 + 32 * q^89 - 40 * q^91 + 28 * q^95 + 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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