LMFDB - Embedded newform 1120.2.h.b.111.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1120,2,Mod(111,1120)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1120, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1120.111");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1120 = 2^{5} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1120.h (of order $2$, degree $1$, not 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:	$8.94324502638$
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} - x^{15} - 2x^{12} + 6x^{11} - 12x^{9} + 8x^{8} - 24x^{7} + 48x^{5} - 32x^{4} - 128x + 256 $ x^16 - x^15 - 2x^12 + 6x^11 - 12x^9 + 8x^8 - 24x^7 + 48x^5 - 32x^4 - 128x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 280)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			111.14
Root			$1.41214 + 0.0765298i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.b.111.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.21915i q^{3} +1.00000 q^{5} +(-1.20923 - 2.35325i) q^{7} -1.92464 q^{9} +O(q^{10})$	$q+2.21915i q^{3} +1.00000 q^{5} +(-1.20923 - 2.35325i) q^{7} -1.92464 q^{9} +3.88394 q^{11} -5.67503 q^{13} +2.21915i q^{15} +5.63892i q^{17} +1.31134i q^{19} +(5.22221 - 2.68346i) q^{21} +7.37559i q^{23} +1.00000 q^{25} +2.38639i q^{27} +9.07201i q^{29} +2.23073 q^{31} +8.61906i q^{33} +(-1.20923 - 2.35325i) q^{35} -6.98438i q^{37} -12.5938i q^{39} +7.47757i q^{41} +1.46735 q^{43} -1.92464 q^{45} -0.567314 q^{47} +(-4.07554 + 5.69122i) q^{49} -12.5136 q^{51} -0.100950i q^{53} +3.88394 q^{55} -2.91006 q^{57} -2.93497i q^{59} +13.8324 q^{61} +(2.32733 + 4.52915i) q^{63} -5.67503 q^{65} -5.54736 q^{67} -16.3676 q^{69} +2.42368i q^{71} -6.08011i q^{73} +2.21915i q^{75} +(-4.69657 - 9.13987i) q^{77} -2.83370i q^{79} -11.0697 q^{81} +2.52687i q^{83} +5.63892i q^{85} -20.1322 q^{87} -10.9114i q^{89} +(6.86240 + 13.3547i) q^{91} +4.95033i q^{93} +1.31134i q^{95} -9.93165i q^{97} -7.47519 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) $ 16 * q + 16 * q^5 - 16 * q^9	$ 16 q + 16 q^{5} - 16 q^{9} + 4 q^{11} - 4 q^{21} + 16 q^{25} + 16 q^{31} + 4 q^{43} - 16 q^{45} - 8 q^{49} + 40 q^{51} + 4 q^{55} - 16 q^{57} - 8 q^{61} - 28 q^{63} - 20 q^{67} - 40 q^{69} - 4 q^{77} + 24 q^{81} - 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) $ 16 * q + 16 * q^5 - 16 * q^9 + 4 * q^11 - 4 * q^21 + 16 * q^25 + 16 * q^31 + 4 * q^43 - 16 * q^45 - 8 * q^49 + 40 * q^51 + 4 * q^55 - 16 * q^57 - 8 * q^61 - 28 * q^63 - 20 * q^67 - 40 * q^69 - 4 * q^77 + 24 * q^81 - 72 * q^87 + 32 * q^91 - 20 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times$.

$n$	$351$	$421$	$801$	$897$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$			2.21915i			1.28123i	0.767863	+	0.640614i	$0.221320\pi$
$3$			2.21915i			1.28123i	−0.767863	+	0.640614i	$0.778680\pi$
$4$		0			0
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−1.20923	−	2.35325i	−0.457045	−	0.889444i
$7$	−1.20923	−	2.35325i	−0.457045	−	0.889444i
$8$		0			0
$9$	−1.92464			−0.641546
$10$		0			0
$11$	3.88394			1.17105			0.585526	−	0.810653i	$-0.300888\pi$
$11$	3.88394			1.17105			0.585526	+	0.810653i	$0.300888\pi$
$12$		0			0
$13$	−5.67503			−1.57397			−0.786985	−	0.616972i	$-0.788359\pi$
$13$	−5.67503			−1.57397			−0.786985	+	0.616972i	$0.788359\pi$
$14$		0			0
$15$			2.21915i			0.572983i
$16$		0			0
$17$			5.63892i			1.36764i	0.729651	+	0.683820i	$0.239683\pi$
$17$			5.63892i			1.36764i	−0.729651	+	0.683820i	$0.760317\pi$
$18$		0			0
$19$			1.31134i			0.300841i	0.988622	+	0.150421i	$0.0480628\pi$
$19$			1.31134i			0.300841i	−0.988622	+	0.150421i	$0.951937\pi$
$20$		0			0
$21$	5.22221	−	2.68346i	1.13958	−	0.585579i
$22$		0			0
$23$			7.37559i			1.53792i	0.639299	+	0.768958i	$0.279225\pi$
$23$			7.37559i			1.53792i	−0.639299	+	0.768958i	$0.720775\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$			2.38639i			0.459261i
$28$		0			0
$29$			9.07201i			1.68463i	0.538986	+	0.842315i	$0.318808\pi$
$29$			9.07201i			1.68463i	−0.538986	+	0.842315i	$0.681192\pi$
$30$		0			0
$31$	2.23073			0.400651			0.200326	−	0.979729i	$-0.435800\pi$
$31$	2.23073			0.400651			0.200326	+	0.979729i	$0.435800\pi$
$32$		0			0
$33$			8.61906i			1.50039i
$34$		0			0
$35$	−1.20923	−	2.35325i	−0.204397	−	0.397771i
$36$		0			0
$37$		−	6.98438i		−	1.14822i	−0.818777	−	0.574112i	$-0.805347\pi$
$37$		−	6.98438i		−	1.14822i	0.818777	−	0.574112i	$-0.194653\pi$
$38$		0			0
$39$		−	12.5938i		−	2.01662i
$40$		0			0
$41$			7.47757i			1.16780i	0.811826	+	0.583900i	$0.198474\pi$
$41$			7.47757i			1.16780i	−0.811826	+	0.583900i	$0.801526\pi$
$42$		0			0
$43$	1.46735			0.223769			0.111884	−	0.993721i	$-0.464311\pi$
$43$	1.46735			0.223769			0.111884	+	0.993721i	$0.464311\pi$
$44$		0			0
$45$	−1.92464			−0.286908
$46$		0			0
$47$	−0.567314			−0.0827512			−0.0413756	−	0.999144i	$-0.513174\pi$
$47$	−0.567314			−0.0827512			−0.0413756	+	0.999144i	$0.513174\pi$
$48$		0			0
$49$	−4.07554	+	5.69122i	−0.582220	+	0.813032i
$50$		0			0
$51$	−12.5136			−1.75226
$52$		0			0
$53$		−	0.100950i		−	0.0138665i	−0.999976	−	0.00693326i	$-0.997793\pi$
$53$		−	0.100950i		−	0.0138665i	0.999976	−	0.00693326i	$-0.00220694\pi$
$54$		0			0
$55$	3.88394			0.523711
$56$		0			0
$57$	−2.91006			−0.385446
$58$		0			0
$59$		−	2.93497i		−	0.382101i	−0.981580	−	0.191050i	$-0.938811\pi$
$59$		−	2.93497i		−	0.382101i	0.981580	−	0.191050i	$-0.0611894\pi$
$60$		0			0
$61$	13.8324			1.77106			0.885531	−	0.464580i	$-0.153795\pi$
$61$	13.8324			1.77106			0.885531	+	0.464580i	$0.153795\pi$
$62$		0			0
$63$	2.32733	+	4.52915i	0.293216	+	0.570619i
$64$		0			0
$65$	−5.67503			−0.703901
$66$		0			0
$67$	−5.54736			−0.677718			−0.338859	−	0.940837i	$-0.610041\pi$
$67$	−5.54736			−0.677718			−0.338859	+	0.940837i	$0.610041\pi$
$68$		0			0
$69$	−16.3676			−1.97042
$70$		0			0
$71$			2.42368i			0.287638i	0.989604	+	0.143819i	$0.0459384\pi$
$71$			2.42368i			0.287638i	−0.989604	+	0.143819i	$0.954062\pi$
$72$		0			0
$73$		−	6.08011i		−	0.711623i	−0.934558	−	0.355811i	$-0.884205\pi$
$73$		−	6.08011i		−	0.711623i	0.934558	−	0.355811i	$-0.115795\pi$
$74$		0			0
$75$			2.21915i			0.256246i
$76$		0			0
$77$	−4.69657	−	9.13987i	−0.535224	−	1.04159i
$78$		0			0
$79$		−	2.83370i		−	0.318816i	−0.987213	−	0.159408i	$-0.949041\pi$
$79$		−	2.83370i		−	0.318816i	0.987213	−	0.159408i	$-0.0509586\pi$
$80$		0			0
$81$	−11.0697			−1.22996
$82$		0			0
$83$			2.52687i			0.277360i	0.990337	+	0.138680i	$0.0442860\pi$
$83$			2.52687i			0.277360i	−0.990337	+	0.138680i	$0.955714\pi$
$84$		0			0
$85$			5.63892i			0.611627i
$86$		0			0
$87$	−20.1322			−2.15840
$88$		0			0
$89$		−	10.9114i		−	1.15661i	−0.815822	−	0.578303i	$-0.803715\pi$
$89$		−	10.9114i		−	1.15661i	0.815822	−	0.578303i	$-0.196285\pi$
$90$		0			0
$91$	6.86240	+	13.3547i	0.719375	+	1.39996i
$92$		0			0
$93$			4.95033i			0.513326i
$94$		0			0
$95$			1.31134i			0.134540i
$96$		0			0
$97$		−	9.93165i		−	1.00841i	−0.863585	−	0.504203i	$-0.831786\pi$
$97$		−	9.93165i		−	1.00841i	0.863585	−	0.504203i	$-0.168214\pi$
$98$		0			0
$99$	−7.47519			−0.751285
$100$		0			0
$101$	3.07042			0.305519			0.152759	−	0.988263i	$-0.451184\pi$
$101$	3.07042			0.305519			0.152759	+	0.988263i	$0.451184\pi$
$102$		0			0
$103$	−5.89112			−0.580470			−0.290235	−	0.956955i	$-0.593733\pi$
$103$	−5.89112			−0.580470			−0.290235	+	0.956955i	$0.593733\pi$
$104$		0			0
$105$	5.22221	−	2.68346i	0.509636	−	0.261879i
$106$		0			0
$107$	11.0481			1.06806			0.534032	−	0.845464i	$-0.320676\pi$
$107$	11.0481			1.06806			0.534032	+	0.845464i	$0.320676\pi$
$108$		0			0
$109$		−	4.12606i		−	0.395205i	−0.980282	−	0.197603i	$-0.936684\pi$
$109$		−	4.12606i		−	0.395205i	0.980282	−	0.197603i	$-0.0633156\pi$
$110$		0			0
$111$	15.4994			1.47114
$112$		0			0
$113$	−5.28550			−0.497218			−0.248609	−	0.968604i	$-0.579973\pi$
$113$	−5.28550			−0.497218			−0.248609	+	0.968604i	$0.579973\pi$
$114$		0			0
$115$			7.37559i			0.687777i
$116$		0			0
$117$	10.9224			1.00977
$118$		0			0
$119$	13.2698	−	6.81874i	1.21644	−	0.625073i
$120$		0			0
$121$	4.08501			0.371364
$122$		0			0
$123$	−16.5939			−1.49622
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$			7.13436i			0.633072i	0.948580	+	0.316536i	$0.102520\pi$
$127$			7.13436i			0.633072i	−0.948580	+	0.316536i	$0.897480\pi$
$128$		0			0
$129$			3.25627i			0.286699i
$130$		0			0
$131$		−	6.86654i		−	0.599932i	−0.953950	−	0.299966i	$-0.903025\pi$
$131$		−	6.86654i		−	0.599932i	0.953950	−	0.299966i	$-0.0969754\pi$
$132$		0			0
$133$	3.08590	−	1.58570i	0.267581	−	0.137498i
$134$		0			0
$135$			2.38639i			0.205388i
$136$		0			0
$137$	5.74697			0.490996			0.245498	−	0.969397i	$-0.421048\pi$
$137$	5.74697			0.490996			0.245498	+	0.969397i	$0.421048\pi$
$138$		0			0
$139$			12.6356i			1.07174i	0.844301	+	0.535869i	$0.180016\pi$
$139$			12.6356i			1.07174i	−0.844301	+	0.535869i	$0.819984\pi$
$140$		0			0
$141$		−	1.25896i		−	0.106023i
$142$		0			0
$143$	−22.0415			−1.84320
$144$		0			0
$145$			9.07201i			0.753389i
$146$		0			0
$147$	−12.6297	−	9.04424i	−1.04168	−	0.745956i
$148$		0			0
$149$			11.9402i			0.978181i	0.872233	+	0.489091i	$0.162671\pi$
$149$			11.9402i			0.978181i	−0.872233	+	0.489091i	$0.837329\pi$
$150$		0			0
$151$		−	7.62474i		−	0.620493i	−0.950656	−	0.310246i	$-0.899588\pi$
$151$		−	7.62474i		−	0.620493i	0.950656	−	0.310246i	$-0.100412\pi$
$152$		0			0
$153$		−	10.8529i		−	0.877404i
$154$		0			0
$155$	2.23073			0.179177
$156$		0			0
$157$	−15.2157			−1.21435			−0.607174	−	0.794569i	$-0.707697\pi$
$157$	−15.2157			−1.21435			−0.607174	+	0.794569i	$0.707697\pi$
$158$		0			0
$159$	0.224023			0.0177662
$160$		0			0
$161$	17.3566	−	8.91876i	1.36789	−	0.702897i
$162$		0			0
$163$	0.486426			0.0380998			0.0190499	−	0.999819i	$-0.493936\pi$
$163$	0.486426			0.0380998			0.0190499	+	0.999819i	$0.493936\pi$
$164$		0			0
$165$			8.61906i			0.670993i
$166$		0			0
$167$	19.6159			1.51793			0.758963	−	0.651134i	$-0.225706\pi$
$167$	19.6159			1.51793			0.758963	+	0.651134i	$0.225706\pi$
$168$		0			0
$169$	19.2060			1.47738
$170$		0			0
$171$		−	2.52385i		−	0.193004i
$172$		0			0
$173$	8.14454			0.619218			0.309609	−	0.950864i	$-0.399802\pi$
$173$	8.14454			0.619218			0.309609	+	0.950864i	$0.399802\pi$
$174$		0			0
$175$	−1.20923	−	2.35325i	−0.0914090	−	0.177889i
$176$		0			0
$177$	6.51315			0.489558
$178$		0			0
$179$	20.3812			1.52336			0.761680	−	0.647953i	$-0.224375\pi$
$179$	20.3812			1.52336			0.761680	+	0.647953i	$0.224375\pi$
$180$		0			0
$181$	−6.68992			−0.497258			−0.248629	−	0.968599i	$-0.579980\pi$
$181$	−6.68992			−0.497258			−0.248629	+	0.968599i	$0.579980\pi$
$182$		0			0
$183$			30.6963i			2.26914i
$184$		0			0
$185$		−	6.98438i		−	0.513502i
$186$		0			0
$187$			21.9012i			1.60158i
$188$		0			0
$189$	5.61576	−	2.88569i	0.408487	−	0.209903i
$190$		0			0
$191$		−	5.95641i		−	0.430991i	−0.976505	−	0.215495i	$-0.930863\pi$
$191$		−	5.95641i		−	0.430991i	0.976505	−	0.215495i	$-0.0691366\pi$
$192$		0			0
$193$	10.9141			0.785617			0.392808	−	0.919620i	$-0.371504\pi$
$193$	10.9141			0.785617			0.392808	+	0.919620i	$0.371504\pi$
$194$		0			0
$195$		−	12.5938i		−	0.901858i
$196$		0			0
$197$		−	3.76694i		−	0.268384i	−0.990955	−	0.134192i	$-0.957156\pi$
$197$		−	3.76694i		−	0.268384i	0.990955	−	0.134192i	$-0.0428438\pi$
$198$		0			0
$199$	−7.52925			−0.533734			−0.266867	−	0.963733i	$-0.585988\pi$
$199$	−7.52925			−0.533734			−0.266867	+	0.963733i	$0.585988\pi$
$200$		0			0
$201$		−	12.3104i		−	0.868311i
$202$		0			0
$203$	21.3487	−	10.9701i	1.49838	−	0.769952i
$204$		0			0
$205$			7.47757i			0.522256i
$206$		0			0
$207$		−	14.1953i		−	0.986644i
$208$		0			0
$209$			5.09315i			0.352301i
$210$		0			0
$211$	3.93687			0.271026			0.135513	−	0.990776i	$-0.456732\pi$
$211$	3.93687			0.271026			0.135513	+	0.990776i	$0.456732\pi$
$212$		0			0
$213$	−5.37852			−0.368530
$214$		0			0
$215$	1.46735			0.100072
$216$		0			0
$217$	−2.69746	−	5.24946i	−0.183116	−	0.356357i
$218$		0			0
$219$	13.4927			0.911752
$220$		0			0
$221$		−	32.0011i		−	2.15262i
$222$		0			0
$223$	−17.3868			−1.16431			−0.582154	−	0.813078i	$-0.697790\pi$
$223$	−17.3868			−1.16431			−0.582154	+	0.813078i	$0.697790\pi$
$224$		0			0
$225$	−1.92464			−0.128309
$226$		0			0
$227$		−	22.9676i		−	1.52441i	−0.647334	−	0.762206i	$-0.724116\pi$
$227$		−	22.9676i		−	1.52441i	0.647334	−	0.762206i	$-0.275884\pi$
$228$		0			0
$229$	14.1945			0.938000			0.469000	−	0.883198i	$-0.344614\pi$
$229$	14.1945			0.938000			0.469000	+	0.883198i	$0.344614\pi$
$230$		0			0
$231$	20.2828	−	10.4224i	1.33451	−	0.685744i
$232$		0			0
$233$	14.0393			0.919743			0.459872	−	0.887985i	$-0.347895\pi$
$233$	14.0393			0.919743			0.459872	+	0.887985i	$0.347895\pi$
$234$		0			0
$235$	−0.567314			−0.0370075
$236$		0			0
$237$	6.28841			0.408476
$238$		0			0
$239$		−	12.7467i		−	0.824517i	−0.911067	−	0.412259i	$-0.864740\pi$
$239$		−	12.7467i		−	0.824517i	0.911067	−	0.412259i	$-0.135260\pi$
$240$		0			0
$241$		−	5.80924i		−	0.374206i	−0.982340	−	0.187103i	$-0.940090\pi$
$241$		−	5.80924i		−	0.374206i	0.982340	−	0.187103i	$-0.0599098\pi$
$242$		0			0
$243$		−	17.4061i		−	1.11660i
$244$		0			0
$245$	−4.07554	+	5.69122i	−0.260377	+	0.363599i
$246$		0			0
$247$		−	7.44187i		−	0.473515i
$248$		0			0
$249$	−5.60752			−0.355362
$250$		0			0
$251$		−	26.0680i		−	1.64540i	−0.568478	−	0.822698i	$-0.692468\pi$
$251$		−	26.0680i		−	1.64540i	0.568478	−	0.822698i	$-0.307532\pi$
$252$		0			0
$253$			28.6464i			1.80098i
$254$		0			0
$255$	−12.5136			−0.783634
$256$		0			0
$257$			18.1281i			1.13080i	0.824816	+	0.565401i	$0.191278\pi$
$257$			18.1281i			1.13080i	−0.824816	+	0.565401i	$0.808722\pi$
$258$		0			0
$259$	−16.4360	+	8.44570i	−1.02128	+	0.524790i
$260$		0			0
$261$		−	17.4603i		−	1.08077i
$262$		0			0
$263$			22.7053i			1.40007i	0.714110	+	0.700034i	$0.246832\pi$
$263$			22.7053i			1.40007i	−0.714110	+	0.700034i	$0.753168\pi$
$264$		0			0
$265$		−	0.100950i		−	0.00620130i
$266$		0			0
$267$	24.2140			1.48188
$268$		0			0
$269$	−9.24420			−0.563629			−0.281814	−	0.959469i	$-0.590936\pi$
$269$	−9.24420			−0.563629			−0.281814	+	0.959469i	$0.590936\pi$
$270$		0			0
$271$	14.5888			0.886206			0.443103	−	0.896471i	$-0.353878\pi$
$271$	14.5888			0.886206			0.443103	+	0.896471i	$0.353878\pi$
$272$		0			0
$273$	−29.6362	+	15.2287i	−1.79367	+	0.921684i
$274$		0			0
$275$	3.88394			0.234211
$276$		0			0
$277$		−	17.6637i		−	1.06131i	−0.847588	−	0.530654i	$-0.821946\pi$
$277$		−	17.6637i		−	1.06131i	0.847588	−	0.530654i	$-0.178054\pi$
$278$		0			0
$279$	−4.29335			−0.257036
$280$		0			0
$281$	−20.0874			−1.19832			−0.599158	−	0.800631i	$-0.704498\pi$
$281$	−20.0874			−1.19832			−0.599158	+	0.800631i	$0.704498\pi$
$282$		0			0
$283$			1.40295i			0.0833969i	0.999130	+	0.0416985i	$0.0132769\pi$
$283$			1.40295i			0.0833969i	−0.999130	+	0.0416985i	$0.986723\pi$
$284$		0			0
$285$	−2.91006			−0.172377
$286$		0			0
$287$	17.5966	−	9.04208i	1.03869	−	0.533737i
$288$		0			0
$289$	−14.7974			−0.870438
$290$		0			0
$291$	22.0399			1.29200
$292$		0			0
$293$	−14.9167			−0.871446			−0.435723	−	0.900081i	$-0.643507\pi$
$293$	−14.9167			−0.871446			−0.435723	+	0.900081i	$0.643507\pi$
$294$		0			0
$295$		−	2.93497i		−	0.170881i
$296$		0			0
$297$			9.26860i			0.537819i
$298$		0			0
$299$		−	41.8567i		−	2.42063i
$300$		0			0
$301$	−1.77436	−	3.45304i	−0.102272	−	0.199030i
$302$		0			0
$303$			6.81374i			0.391439i
$304$		0			0
$305$	13.8324			0.792043
$306$		0			0
$307$		−	3.67891i		−	0.209966i	−0.994474	−	0.104983i	$-0.966521\pi$
$307$		−	3.67891i		−	0.209966i	0.994474	−	0.104983i	$-0.0334789\pi$
$308$		0			0
$309$		−	13.0733i		−	0.743714i
$310$		0			0
$311$	−31.1614			−1.76700			−0.883501	−	0.468429i	$-0.844820\pi$
$311$	−31.1614			−1.76700			−0.883501	+	0.468429i	$0.844820\pi$
$312$		0			0
$313$		−	24.0566i		−	1.35976i	−0.733323	−	0.679880i	$-0.762032\pi$
$313$		−	24.0566i		−	1.35976i	0.733323	−	0.679880i	$-0.237968\pi$
$314$		0			0
$315$	2.32733	+	4.52915i	0.131130	+	0.255189i
$316$		0			0
$317$			19.3274i			1.08554i	0.839883	+	0.542768i	$0.182624\pi$
$317$			19.3274i			1.08554i	−0.839883	+	0.542768i	$0.817376\pi$
$318$		0			0
$319$			35.2352i			1.97279i
$320$		0			0
$321$			24.5175i			1.36843i
$322$		0			0
$323$	−7.39452			−0.411442
$324$		0			0
$325$	−5.67503			−0.314794
$326$		0			0
$327$	9.15636			0.506348
$328$		0			0
$329$	0.686011	+	1.33503i	0.0378210	+	0.0736025i
$330$		0			0
$331$	−19.5607			−1.07515			−0.537576	−	0.843216i	$-0.680660\pi$
$331$	−19.5607			−1.07515			−0.537576	+	0.843216i	$0.680660\pi$
$332$		0			0
$333$			13.4424i			0.736639i
$334$		0			0
$335$	−5.54736			−0.303085
$336$		0			0
$337$	−10.2860			−0.560314			−0.280157	−	0.959954i	$-0.590387\pi$
$337$	−10.2860			−0.560314			−0.280157	+	0.959954i	$0.590387\pi$
$338$		0			0
$339$		−	11.7293i		−	0.637050i
$340$		0			0
$341$	8.66403			0.469183
$342$		0			0
$343$	18.3211	+	2.70876i	0.989246	+	0.146260i
$344$		0			0
$345$	−16.3676			−0.881199
$346$		0			0
$347$	−3.49360			−0.187546			−0.0937732	−	0.995594i	$-0.529893\pi$
$347$	−3.49360			−0.187546			−0.0937732	+	0.995594i	$0.529893\pi$
$348$		0			0
$349$	10.8420			0.580361			0.290180	−	0.956972i	$-0.406285\pi$
$349$	10.8420			0.580361			0.290180	+	0.956972i	$0.406285\pi$
$350$		0			0
$351$		−	13.5428i		−	0.722863i
$352$		0			0
$353$		−	0.494060i		−	0.0262961i	−0.999914	−	0.0131481i	$-0.995815\pi$
$353$		−	0.494060i		−	0.0262961i	0.999914	−	0.0131481i	$-0.00418528\pi$
$354$		0			0
$355$			2.42368i			0.128636i
$356$		0			0
$357$	15.1318	+	29.4477i	0.800861	+	1.55854i
$358$		0			0
$359$			4.33537i			0.228812i	0.993434	+	0.114406i	$0.0364965\pi$
$359$			4.33537i			0.228812i	−0.993434	+	0.114406i	$0.963503\pi$
$360$		0			0
$361$	17.2804			0.909495
$362$		0			0
$363$			9.06525i			0.475802i
$364$		0			0
$365$		−	6.08011i		−	0.318247i
$366$		0			0
$367$	32.9836			1.72173			0.860866	−	0.508832i	$-0.169923\pi$
$367$	32.9836			1.72173			0.860866	+	0.508832i	$0.169923\pi$
$368$		0			0
$369$		−	14.3916i		−	0.749198i
$370$		0			0
$371$	−0.237560	+	0.122071i	−0.0123335	+	0.00633762i
$372$		0			0
$373$			14.2525i			0.737967i	0.929436	+	0.368984i	$0.120294\pi$
$373$			14.2525i			0.737967i	−0.929436	+	0.368984i	$0.879706\pi$
$374$		0			0
$375$			2.21915i			0.114597i
$376$		0			0
$377$		−	51.4839i		−	2.65156i
$378$		0			0
$379$	−20.1528			−1.03518			−0.517590	−	0.855629i	$-0.673171\pi$
$379$	−20.1528			−1.03518			−0.517590	+	0.855629i	$0.673171\pi$
$380$		0			0
$381$	−15.8322			−0.811110
$382$		0			0
$383$	−6.39302			−0.326668			−0.163334	−	0.986571i	$-0.552225\pi$
$383$	−6.39302			−0.326668			−0.163334	+	0.986571i	$0.552225\pi$
$384$		0			0
$385$	−4.69657	−	9.13987i	−0.239359	−	0.465811i
$386$		0			0
$387$	−2.82412			−0.143558
$388$		0			0
$389$			15.2150i			0.771432i	0.922618	+	0.385716i	$0.126045\pi$
$389$			15.2150i			0.771432i	−0.922618	+	0.385716i	$0.873955\pi$
$390$		0			0
$391$	−41.5904			−2.10332
$392$		0			0
$393$	15.2379			0.768650
$394$		0			0
$395$		−	2.83370i		−	0.142579i
$396$		0			0
$397$	27.8429			1.39739			0.698696	−	0.715418i	$-0.253764\pi$
$397$	27.8429			1.39739			0.698696	+	0.715418i	$0.253764\pi$
$398$		0			0
$399$	3.51892	+	6.84808i	0.176166	+	0.342833i
$400$		0			0
$401$	4.49415			0.224427			0.112214	−	0.993684i	$-0.464206\pi$
$401$	4.49415			0.224427			0.112214	+	0.993684i	$0.464206\pi$
$402$		0			0
$403$	−12.6595			−0.630613
$404$		0			0
$405$	−11.0697			−0.550057
$406$		0			0
$407$		−	27.1269i		−	1.34463i
$408$		0			0
$409$		−	9.16042i		−	0.452954i	−0.974017	−	0.226477i	$-0.927279\pi$
$409$		−	9.16042i		−	0.452954i	0.974017	−	0.226477i	$-0.0727207\pi$
$410$		0			0
$411$			12.7534i			0.629079i
$412$		0			0
$413$	−6.90671	+	3.54905i	−0.339857	+	0.174637i
$414$		0			0
$415$			2.52687i			0.124039i
$416$		0			0
$417$	−28.0403			−1.37314
$418$		0			0
$419$			18.8059i			0.918726i	0.888249	+	0.459363i	$0.151922\pi$
$419$			18.8059i			0.918726i	−0.888249	+	0.459363i	$0.848078\pi$
$420$		0			0
$421$		−	34.3816i		−	1.67566i	−0.545934	−	0.837828i	$-0.683825\pi$
$421$		−	34.3816i		−	1.67566i	0.545934	−	0.837828i	$-0.316175\pi$
$422$		0			0
$423$	1.09187			0.0530887
$424$		0			0
$425$			5.63892i			0.273528i
$426$		0			0
$427$	−16.7266	−	32.5511i	−0.809455	−	1.57526i
$428$		0			0
$429$		−	48.9134i		−	2.36156i
$430$		0			0
$431$		−	3.54360i		−	0.170689i	−0.996351	−	0.0853447i	$-0.972801\pi$
$431$		−	3.54360i		−	0.170689i	0.996351	−	0.0853447i	$-0.0271991\pi$
$432$		0			0
$433$			35.6627i			1.71384i	0.515451	+	0.856919i	$0.327624\pi$
$433$			35.6627i			1.71384i	−0.515451	+	0.856919i	$0.672376\pi$
$434$		0			0
$435$	−20.1322			−0.965264
$436$		0			0
$437$	−9.67188			−0.462669
$438$		0			0
$439$	12.9572			0.618415			0.309207	−	0.950995i	$-0.399936\pi$
$439$	12.9572			0.618415			0.309207	+	0.950995i	$0.399936\pi$
$440$		0			0
$441$	7.84394	−	10.9535i	0.373521	−	0.521597i
$442$		0			0
$443$	−0.593530			−0.0281994			−0.0140997	−	0.999901i	$-0.504488\pi$
$443$	−0.593530			−0.0281994			−0.0140997	+	0.999901i	$0.504488\pi$
$444$		0			0
$445$		−	10.9114i		−	0.517250i
$446$		0			0
$447$	−26.4972			−1.25327
$448$		0			0
$449$	26.9305			1.27093			0.635464	−	0.772130i	$-0.280809\pi$
$449$	26.9305			1.27093			0.635464	+	0.772130i	$0.280809\pi$
$450$		0			0
$451$			29.0424i			1.36756i
$452$		0			0
$453$	16.9205			0.794993
$454$		0			0
$455$	6.86240	+	13.3547i	0.321714	+	0.626080i
$456$		0			0
$457$	30.8127			1.44136			0.720678	−	0.693270i	$-0.243831\pi$
$457$	30.8127			1.44136			0.720678	+	0.693270i	$0.243831\pi$
$458$		0			0
$459$	−13.4567			−0.628104
$460$		0			0
$461$	0.701982			0.0326946			0.0163473	−	0.999866i	$-0.494796\pi$
$461$	0.701982			0.0326946			0.0163473	+	0.999866i	$0.494796\pi$
$462$		0			0
$463$		−	26.4984i		−	1.23148i	−0.787948	−	0.615742i	$-0.788857\pi$
$463$		−	26.4984i		−	1.23148i	0.787948	−	0.615742i	$-0.211143\pi$
$464$		0			0
$465$			4.95033i			0.229566i
$466$		0			0
$467$			41.6876i			1.92907i	0.263947	+	0.964537i	$0.414976\pi$
$467$			41.6876i			1.92907i	−0.263947	+	0.964537i	$0.585024\pi$
$468$		0			0
$469$	6.70802	+	13.0543i	0.309748	+	0.602792i
$470$		0			0
$471$		−	33.7660i		−	1.55586i
$472$		0			0
$473$	5.69910			0.262045
$474$		0			0
$475$			1.31134i			0.0601682i
$476$		0			0
$477$			0.194292i			0.00889602i
$478$		0			0
$479$	1.36128			0.0621983			0.0310991	−	0.999516i	$-0.490099\pi$
$479$	1.36128			0.0621983			0.0310991	+	0.999516i	$0.490099\pi$
$480$		0			0
$481$			39.6366i			1.80727i
$482$		0			0
$483$	19.7921	+	38.5169i	0.900572	+	1.75258i
$484$		0			0
$485$		−	9.93165i		−	0.450973i
$486$		0			0
$487$			10.7466i			0.486976i	0.969904	+	0.243488i	$0.0782916\pi$
$487$			10.7466i			0.486976i	−0.969904	+	0.243488i	$0.921708\pi$
$488$		0			0
$489$			1.07945i			0.0488146i
$490$		0			0
$491$	−23.1188			−1.04334			−0.521670	−	0.853148i	$-0.674691\pi$
$491$	−23.1188			−1.04334			−0.521670	+	0.853148i	$0.674691\pi$
$492$		0			0
$493$	−51.1564			−2.30397
$494$		0			0
$495$	−7.47519			−0.335985
$496$		0			0
$497$	5.70353	−	2.93079i	0.255838	−	0.131464i
$498$		0			0
$499$	23.5311			1.05340			0.526698	−	0.850053i	$-0.323430\pi$
$499$	23.5311			1.05340			0.526698	+	0.850053i	$0.323430\pi$
$500$		0			0
$501$			43.5307i			1.94481i
$502$		0			0
$503$	20.3215			0.906093			0.453046	−	0.891487i	$-0.350337\pi$
$503$	20.3215			0.906093			0.453046	+	0.891487i	$0.350337\pi$
$504$		0			0
$505$	3.07042			0.136632
$506$		0			0
$507$			42.6210i			1.89286i
$508$		0			0
$509$	32.6233			1.44600			0.723001	−	0.690847i	$-0.242762\pi$
$509$	32.6233			1.44600			0.723001	+	0.690847i	$0.242762\pi$
$510$		0			0
$511$	−14.3080	+	7.35224i	−0.632948	+	0.325244i
$512$		0			0
$513$	−3.12936			−0.138165
$514$		0			0
$515$	−5.89112			−0.259594
$516$		0			0
$517$	−2.20341			−0.0969060
$518$		0			0
$519$			18.0740i			0.793360i
$520$		0			0
$521$		−	30.2799i		−	1.32659i	−0.748360	−	0.663293i	$-0.769158\pi$
$521$		−	30.2799i		−	1.32659i	0.748360	−	0.663293i	$-0.230842\pi$
$522$		0			0
$523$			4.23380i			0.185131i	0.995707	+	0.0925655i	$0.0295067\pi$
$523$			4.23380i			0.185131i	−0.995707	+	0.0925655i	$0.970493\pi$
$524$		0			0
$525$	5.22221	−	2.68346i	0.227916	−	0.117116i
$526$		0			0
$527$			12.5789i			0.547946i
$528$		0			0
$529$	−31.3993			−1.36519
$530$		0			0
$531$			5.64876i			0.245135i
$532$		0			0
$533$		−	42.4354i		−	1.83808i
$534$		0			0
$535$	11.0481			0.477653
$536$		0			0
$537$			45.2289i			1.95177i
$538$		0			0
$539$	−15.8292	+	22.1044i	−0.681810	+	0.952103i
$540$		0			0
$541$			20.3843i			0.876390i	0.898880	+	0.438195i	$0.144382\pi$
$541$			20.3843i			0.876390i	−0.898880	+	0.438195i	$0.855618\pi$
$542$		0			0
$543$		−	14.8459i		−	0.637101i
$544$		0			0
$545$		−	4.12606i		−	0.176741i
$546$		0			0
$547$	38.6889			1.65422			0.827110	−	0.562041i	$-0.189984\pi$
$547$	38.6889			1.65422			0.827110	+	0.562041i	$0.189984\pi$
$548$		0			0
$549$	−26.6225			−1.13622
$550$		0			0
$551$	−11.8965			−0.506806
$552$		0			0
$553$	−6.66839	+	3.42659i	−0.283569	+	0.145713i
$554$		0			0
$555$	15.4994			0.657913
$556$		0			0
$557$			27.3750i			1.15991i	0.814647	+	0.579957i	$0.196931\pi$
$557$			27.3750i			1.15991i	−0.814647	+	0.579957i	$0.803069\pi$
$558$		0			0
$559$	−8.32726			−0.352206
$560$		0			0
$561$	−48.6022			−2.05199
$562$		0			0
$563$		−	13.1643i		−	0.554811i	−0.960753	−	0.277405i	$-0.910526\pi$
$563$		−	13.1643i		−	0.554811i	0.960753	−	0.277405i	$-0.0894745\pi$
$564$		0			0
$565$	−5.28550			−0.222363
$566$		0			0
$567$	13.3858	+	26.0497i	0.562149	+	1.09398i
$568$		0			0
$569$	−16.3200			−0.684168			−0.342084	−	0.939669i	$-0.611133\pi$
$569$	−16.3200			−0.684168			−0.342084	+	0.939669i	$0.611133\pi$
$570$		0			0
$571$	23.5552			0.985754			0.492877	−	0.870099i	$-0.335945\pi$
$571$	23.5552			0.985754			0.492877	+	0.870099i	$0.335945\pi$
$572$		0			0
$573$	13.2182			0.552198
$574$		0			0
$575$			7.37559i			0.307583i
$576$		0			0
$577$			29.3020i			1.21986i	0.792457	+	0.609928i	$0.208802\pi$
$577$			29.3020i			1.21986i	−0.792457	+	0.609928i	$0.791198\pi$
$578$		0			0
$579$			24.2201i			1.00655i
$580$		0			0
$581$	5.94635	−	3.05556i	0.246696	−	0.126766i
$582$		0			0
$583$		−	0.392083i		−	0.0162384i
$584$		0			0
$585$	10.9224			0.451585
$586$		0			0
$587$		−	18.3219i		−	0.756225i	−0.925760	−	0.378113i	$-0.876573\pi$
$587$		−	18.3219i		−	0.756225i	0.925760	−	0.378113i	$-0.123427\pi$
$588$		0			0
$589$			2.92524i			0.120532i
$590$		0			0
$591$	8.35942			0.343861
$592$		0			0
$593$		−	35.9919i		−	1.47801i	−0.673699	−	0.739006i	$-0.735296\pi$
$593$		−	35.9919i		−	1.47801i	0.673699	−	0.739006i	$-0.264704\pi$
$594$		0			0
$595$	13.2698	−	6.81874i	0.544008	−	0.279541i
$596$		0			0
$597$		−	16.7085i		−	0.683835i
$598$		0			0
$599$		−	8.44016i		−	0.344856i	−0.985022	−	0.172428i	$-0.944839\pi$
$599$		−	8.44016i		−	0.344856i	0.985022	−	0.172428i	$-0.0551611\pi$
$600$		0			0
$601$		−	1.25783i		−	0.0513079i	−0.999671	−	0.0256539i	$-0.991833\pi$
$601$		−	1.25783i		−	0.0513079i	0.999671	−	0.0256539i	$-0.00816680\pi$
$602$		0			0
$603$	10.6767			0.434787
$604$		0			0
$605$	4.08501			0.166079
$606$		0			0
$607$	9.60051			0.389673			0.194836	−	0.980836i	$-0.437582\pi$
$607$	9.60051			0.389673			0.194836	+	0.980836i	$0.437582\pi$
$608$		0			0
$609$	24.3444	+	47.3760i	0.986484	+	1.91977i
$610$		0			0
$611$	3.21952			0.130248
$612$		0			0
$613$			31.9083i			1.28877i	0.764703	+	0.644383i	$0.222886\pi$
$613$			31.9083i			1.28877i	−0.764703	+	0.644383i	$0.777114\pi$
$614$		0			0
$615$	−16.5939			−0.669129
$616$		0			0
$617$	28.7928			1.15916			0.579578	−	0.814917i	$-0.303217\pi$
$617$	28.7928			1.15916			0.579578	+	0.814917i	$0.303217\pi$
$618$		0			0
$619$		−	15.2907i		−	0.614586i	−0.951615	−	0.307293i	$-0.900577\pi$
$619$		−	15.2907i		−	0.614586i	0.951615	−	0.307293i	$-0.0994232\pi$
$620$		0			0
$621$	−17.6010			−0.706305
$622$		0			0
$623$	−25.6772	+	13.1944i	−1.02874	+	0.528621i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−11.3025			−0.451378
$628$		0			0
$629$	39.3844			1.57036
$630$		0			0
$631$		−	38.9523i		−	1.55067i	−0.631551	−	0.775334i	$-0.717581\pi$
$631$		−	38.9523i		−	1.55067i	0.631551	−	0.775334i	$-0.282419\pi$
$632$		0			0
$633$			8.73652i			0.347246i
$634$		0			0
$635$			7.13436i			0.283119i
$636$		0			0
$637$	23.1288	−	32.2979i	0.916397	−	1.27969i
$638$		0			0
$639$		−	4.66472i		−	0.184533i
$640$		0			0
$641$	−8.27890			−0.326997			−0.163498	−	0.986544i	$-0.552278\pi$
$641$	−8.27890			−0.326997			−0.163498	+	0.986544i	$0.552278\pi$
$642$		0			0
$643$			18.0050i			0.710048i	0.934857	+	0.355024i	$0.115527\pi$
$643$			18.0050i			0.710048i	−0.934857	+	0.355024i	$0.884473\pi$
$644$		0			0
$645$			3.25627i			0.128216i
$646$		0			0
$647$	−36.7978			−1.44667			−0.723336	−	0.690496i	$-0.757392\pi$
$647$	−36.7978			−1.44667			−0.723336	+	0.690496i	$0.757392\pi$
$648$		0			0
$649$		−	11.3993i		−	0.447460i
$650$		0			0
$651$	11.6494	−	5.98608i	0.456574	−	0.234613i
$652$		0			0
$653$			13.5053i			0.528504i	0.964454	+	0.264252i	$0.0851250\pi$
$653$			13.5053i			0.528504i	−0.964454	+	0.264252i	$0.914875\pi$
$654$		0			0
$655$		−	6.86654i		−	0.268298i
$656$		0			0
$657$			11.7020i			0.456539i
$658$		0			0
$659$	21.7852			0.848629			0.424315	−	0.905515i	$-0.360515\pi$
$659$	21.7852			0.848629			0.424315	+	0.905515i	$0.360515\pi$
$660$		0			0
$661$	−27.4541			−1.06784			−0.533921	−	0.845534i	$-0.679282\pi$
$661$	−27.4541			−1.06784			−0.533921	+	0.845534i	$0.679282\pi$
$662$		0			0
$663$	71.0152			2.75800
$664$		0			0
$665$	3.08590	−	1.58570i	0.119666	−	0.0614910i
$666$		0			0
$667$	−66.9114			−2.59082
$668$		0			0
$669$		−	38.5841i		−	1.49175i
$670$		0			0
$671$	53.7244			2.07401
$672$		0			0
$673$	−15.2602			−0.588238			−0.294119	−	0.955769i	$-0.595026\pi$
$673$	−15.2602			−0.588238			−0.294119	+	0.955769i	$0.595026\pi$
$674$		0			0
$675$			2.38639i			0.0918522i
$676$		0			0
$677$	−29.8069			−1.14557			−0.572787	−	0.819704i	$-0.694138\pi$
$677$	−29.8069			−1.14557			−0.572787	+	0.819704i	$0.694138\pi$
$678$		0			0
$679$	−23.3716	+	12.0096i	−0.896921	+	0.460887i
$680$		0			0
$681$	50.9686			1.95312
$682$		0			0
$683$	−45.4219			−1.73802			−0.869010	−	0.494795i	$-0.835243\pi$
$683$	−45.4219			−1.73802			−0.869010	+	0.494795i	$0.835243\pi$
$684$		0			0
$685$	5.74697			0.219580
$686$		0			0
$687$			31.4998i			1.20179i
$688$		0			0
$689$			0.572893i			0.0218255i
$690$		0			0
$691$		−	25.4854i		−	0.969510i	−0.874650	−	0.484755i	$-0.838909\pi$
$691$		−	25.4854i		−	0.969510i	0.874650	−	0.484755i	$-0.161091\pi$
$692$		0			0
$693$	9.03920	+	17.5910i	0.343371	+	0.668225i
$694$		0			0
$695$			12.6356i			0.479296i
$696$		0			0
$697$	−42.1654			−1.59713
$698$		0			0
$699$			31.1553i			1.17840i
$700$		0			0
$701$			24.4503i			0.923473i	0.887017	+	0.461737i	$0.152773\pi$
$701$			24.4503i			0.923473i	−0.887017	+	0.461737i	$0.847227\pi$
$702$		0			0
$703$	9.15887			0.345433
$704$		0			0
$705$		−	1.25896i		−	0.0474150i
$706$		0			0
$707$	−3.71284	−	7.22546i	−0.139636	−	0.271742i
$708$		0			0
$709$			32.1784i			1.20848i	0.796801	+	0.604242i	$0.206524\pi$
$709$			32.1784i			1.20848i	−0.796801	+	0.604242i	$0.793476\pi$
$710$		0			0
$711$			5.45385i			0.204535i
$712$		0			0
$713$			16.4529i			0.616168i
$714$		0			0
$715$	−22.0415			−0.824305
$716$		0			0
$717$	28.2869			1.05639
$718$		0			0
$719$	14.5688			0.543324			0.271662	−	0.962393i	$-0.412427\pi$
$719$	14.5688			0.543324			0.271662	+	0.962393i	$0.412427\pi$
$720$		0			0
$721$	7.12371	+	13.8633i	0.265301	+	0.516295i
$722$		0			0
$723$	12.8916			0.479443
$724$		0			0
$725$			9.07201i			0.336926i
$726$		0			0
$727$	−36.7720			−1.36380			−0.681899	−	0.731447i	$-0.738846\pi$
$727$	−36.7720			−1.36380			−0.681899	+	0.731447i	$0.738846\pi$
$728$		0			0
$729$	5.41785			0.200661
$730$		0			0
$731$			8.27428i			0.306035i
$732$		0			0
$733$	−44.4405			−1.64145			−0.820725	−	0.571324i	$-0.806430\pi$
$733$	−44.4405			−1.64145			−0.820725	+	0.571324i	$0.806430\pi$
$734$		0			0
$735$	−12.6297	−	9.04424i	−0.465853	−	0.333602i
$736$		0			0
$737$	−21.5456			−0.793643
$738$		0			0
$739$	−4.41504			−0.162410			−0.0812049	−	0.996697i	$-0.525877\pi$
$739$	−4.41504			−0.162410			−0.0812049	+	0.996697i	$0.525877\pi$
$740$		0			0
$741$	16.5147			0.606681
$742$		0			0
$743$		−	47.0540i		−	1.72625i	−0.504994	−	0.863123i	$-0.668505\pi$
$743$		−	47.0540i		−	1.72625i	0.504994	−	0.863123i	$-0.331495\pi$
$744$		0			0
$745$			11.9402i			0.437456i
$746$		0			0
$747$		−	4.86332i		−	0.177939i
$748$		0			0
$749$	−13.3597	−	25.9990i	−0.488153	−	0.949983i
$750$		0			0
$751$			32.7742i			1.19595i	0.801516	+	0.597974i	$0.204027\pi$
$751$			32.7742i			1.19595i	−0.801516	+	0.597974i	$0.795973\pi$
$752$		0			0
$753$	57.8488			2.10813
$754$		0			0
$755$		−	7.62474i		−	0.277493i
$756$		0			0
$757$		−	40.7277i		−	1.48027i	−0.672457	−	0.740136i	$-0.734761\pi$
$757$		−	40.7277i		−	1.48027i	0.672457	−	0.740136i	$-0.265239\pi$
$758$		0			0
$759$	−63.5706			−2.30747
$760$		0			0
$761$		−	17.5757i		−	0.637118i	−0.947903	−	0.318559i	$-0.896801\pi$
$761$		−	17.5757i		−	0.637118i	0.947903	−	0.318559i	$-0.103199\pi$
$762$		0			0
$763$	−9.70964	+	4.98935i	−0.351513	+	0.180627i
$764$		0			0
$765$		−	10.8529i		−	0.392387i
$766$		0			0
$767$			16.6561i			0.601415i
$768$		0			0
$769$			31.9583i			1.15245i	0.817292	+	0.576223i	$0.195474\pi$
$769$			31.9583i			1.15245i	−0.817292	+	0.576223i	$0.804526\pi$
$770$		0			0
$771$	−40.2291			−1.44882
$772$		0			0
$773$	15.3325			0.551470			0.275735	−	0.961234i	$-0.411079\pi$
$773$	15.3325			0.551470			0.275735	+	0.961234i	$0.411079\pi$
$774$		0			0
$775$	2.23073			0.0801302
$776$		0			0
$777$	−18.7423	−	36.4739i	−0.672376	−	1.30849i
$778$		0			0
$779$	−9.80561			−0.351322
$780$		0			0
$781$			9.41345i			0.336840i
$782$		0			0
$783$	−21.6494			−0.773685
$784$		0			0
$785$	−15.2157			−0.543073
$786$		0			0
$787$			7.87957i			0.280876i	0.990089	+	0.140438i	$0.0448511\pi$
$787$			7.87957i			0.280876i	−0.990089	+	0.140438i	$0.955149\pi$
$788$		0			0
$789$	−50.3865			−1.79381
$790$		0			0
$791$	6.39138	+	12.4381i	0.227251	+	0.442248i
$792$		0			0
$793$	−78.4995			−2.78760
$794$		0			0
$795$	0.224023			0.00794528
$796$		0			0
$797$	−25.0570			−0.887563			−0.443781	−	0.896135i	$-0.646363\pi$
$797$	−25.0570			−0.887563			−0.443781	+	0.896135i	$0.646363\pi$
$798$		0			0
$799$		−	3.19904i		−	0.113174i
$800$		0			0
$801$			21.0005i			0.742016i
$802$		0			0
$803$		−	23.6148i		−	0.833348i
$804$		0			0
$805$	17.3566	−	8.91876i	0.611739	−	0.314345i
$806$		0			0
$807$		−	20.5143i		−	0.722137i
$808$		0			0
$809$	−4.94114			−0.173721			−0.0868606	−	0.996220i	$-0.527683\pi$
$809$	−4.94114			−0.173721			−0.0868606	+	0.996220i	$0.527683\pi$
$810$		0			0
$811$		−	0.565648i		−	0.0198626i	−0.999951	−	0.00993130i	$-0.996839\pi$
$811$		−	0.565648i		−	0.0198626i	0.999951	−	0.00993130i	$-0.00316128\pi$
$812$		0			0
$813$			32.3748i			1.13543i
$814$		0			0
$815$	0.486426			0.0170388
$816$		0			0
$817$			1.92419i			0.0673189i
$818$		0			0
$819$	−13.2076	−	25.7031i	−0.461513	−	0.898138i
$820$		0			0
$821$		−	23.2920i		−	0.812897i	−0.913674	−	0.406448i	$-0.866767\pi$
$821$		−	23.2920i		−	0.812897i	0.913674	−	0.406448i	$-0.133233\pi$
$822$		0			0
$823$		−	30.2227i		−	1.05350i	−0.850021	−	0.526748i	$-0.823411\pi$
$823$		−	30.2227i		−	1.05350i	0.850021	−	0.526748i	$-0.176589\pi$
$824$		0			0
$825$			8.61906i			0.300077i
$826$		0			0
$827$	18.8086			0.654040			0.327020	−	0.945017i	$-0.393956\pi$
$827$	18.8086			0.654040			0.327020	+	0.945017i	$0.393956\pi$
$828$		0			0
$829$	12.6773			0.440301			0.220150	−	0.975466i	$-0.429345\pi$
$829$	12.6773			0.440301			0.220150	+	0.975466i	$0.429345\pi$
$830$		0			0
$831$	39.1984			1.35978
$832$		0			0
$833$	−32.0924	−	22.9816i	−1.11193	−	0.796267i
$834$		0			0
$835$	19.6159			0.678837
$836$		0			0
$837$			5.32339i			0.184003i
$838$		0			0
$839$	28.4650			0.982722			0.491361	−	0.870956i	$-0.336500\pi$
$839$	28.4650			0.982722			0.491361	+	0.870956i	$0.336500\pi$
$840$		0			0
$841$	−53.3014			−1.83798
$842$		0			0
$843$		−	44.5771i		−	1.53532i
$844$		0			0
$845$	19.2060			0.660706
$846$		0			0
$847$	−4.93970	−	9.61303i	−0.169730	−	0.330308i
$848$		0			0
$849$	−3.11337			−0.106851
$850$		0			0
$851$	51.5139			1.76587
$852$		0			0
$853$	30.6148			1.04823			0.524115	−	0.851647i	$-0.324396\pi$
$853$	30.6148			1.04823			0.524115	+	0.851647i	$0.324396\pi$
$854$		0			0
$855$		−	2.52385i		−	0.0863138i
$856$		0			0
$857$		−	29.2591i		−	0.999473i	−0.866177	−	0.499737i	$-0.833430\pi$
$857$		−	29.2591i		−	0.999473i	0.866177	−	0.499737i	$-0.166570\pi$
$858$		0			0
$859$		−	37.5302i		−	1.28052i	−0.768160	−	0.640258i	$-0.778828\pi$
$859$		−	37.5302i		−	1.28052i	0.768160	−	0.640258i	$-0.221172\pi$
$860$		0			0
$861$	20.0658	+	39.0495i	0.683839	+	1.33080i
$862$		0			0
$863$		−	42.5541i		−	1.44856i	−0.689507	−	0.724279i	$-0.742173\pi$
$863$		−	42.5541i		−	1.44856i	0.689507	−	0.724279i	$-0.257827\pi$
$864$		0			0
$865$	8.14454			0.276923
$866$		0			0
$867$		−	32.8378i		−	1.11523i
$868$		0			0
$869$		−	11.0059i		−	0.373351i
$870$		0			0
$871$	31.4814			1.06671
$872$		0			0
$873$			19.1148i			0.646939i
$874$		0			0
$875$	−1.20923	−	2.35325i	−0.0408794	−	0.0795543i
$876$		0			0
$877$		−	29.0477i		−	0.980870i	−0.871478	−	0.490435i	$-0.836838\pi$
$877$		−	29.0477i		−	0.980870i	0.871478	−	0.490435i	$-0.163162\pi$
$878$		0			0
$879$		−	33.1025i		−	1.11652i
$880$		0			0
$881$		−	45.7311i		−	1.54072i	−0.637610	−	0.770360i	$-0.720077\pi$
$881$		−	45.7311i		−	1.54072i	0.637610	−	0.770360i	$-0.279923\pi$
$882$		0			0
$883$	27.4055			0.922269			0.461135	−	0.887330i	$-0.347442\pi$
$883$	27.4055			0.922269			0.461135	+	0.887330i	$0.347442\pi$
$884$		0			0
$885$	6.51315			0.218937
$886$		0			0
$887$	7.53368			0.252956			0.126478	−	0.991969i	$-0.459633\pi$
$887$	7.53368			0.252956			0.126478	+	0.991969i	$0.459633\pi$
$888$		0			0
$889$	16.7889	−	8.62707i	0.563082	−	0.289343i
$890$		0			0
$891$	−42.9940			−1.44035
$892$		0			0
$893$		−	0.743939i		−	0.0248950i
$894$		0			0
$895$	20.3812			0.681268
$896$		0			0
$897$	92.8864			3.10139
$898$		0			0
$899$			20.2372i			0.674949i
$900$		0			0
$901$	0.569248			0.0189644
$902$		0			0
$903$	7.66282	−	3.93758i	0.255003	−	0.131034i
$904$		0			0
$905$	−6.68992			−0.222380
$906$		0			0
$907$	40.1412			1.33287			0.666433	−	0.745565i	$-0.267820\pi$
$907$	40.1412			1.33287			0.666433	+	0.745565i	$0.267820\pi$
$908$		0			0
$909$	−5.90946			−0.196004
$910$		0			0
$911$			47.5516i			1.57545i	0.616024	+	0.787727i	$0.288742\pi$
$911$			47.5516i			1.57545i	−0.616024	+	0.787727i	$0.711258\pi$
$912$		0			0
$913$			9.81423i			0.324804i
$914$		0			0
$915$			30.6963i			1.01479i
$916$		0			0
$917$	−16.1587	+	8.30321i	−0.533606	+	0.274196i
$918$		0			0
$919$			51.6665i			1.70432i	0.523281	+	0.852160i	$0.324708\pi$
$919$			51.6665i			1.70432i	−0.523281	+	0.852160i	$0.675292\pi$
$920$		0			0
$921$	8.16405			0.269015
$922$		0			0
$923$		−	13.7545i		−	0.452734i
$924$		0			0
$925$		−	6.98438i		−	0.229645i
$926$		0			0
$927$	11.3383			0.372398
$928$		0			0
$929$		−	41.5413i		−	1.36293i	−0.731852	−	0.681464i	$-0.761344\pi$
$929$		−	41.5413i		−	1.36293i	0.731852	−	0.681464i	$-0.238656\pi$
$930$		0			0
$931$	−7.46310	−	5.34440i	−0.244593	−	0.175156i
$932$		0			0
$933$		−	69.1520i		−	2.26393i
$934$		0			0
$935$			21.9012i			0.716247i
$936$		0			0
$937$		−	15.1513i		−	0.494970i	−0.968892	−	0.247485i	$-0.920396\pi$
$937$		−	15.1513i		−	0.494970i	0.968892	−	0.247485i	$-0.0796041\pi$
$938$		0			0
$939$	53.3853			1.74216
$940$		0			0
$941$	12.3761			0.403451			0.201725	−	0.979442i	$-0.435345\pi$
$941$	12.3761			0.403451			0.201725	+	0.979442i	$0.435345\pi$
$942$		0			0
$943$	−55.1514			−1.79598
$944$		0			0
$945$	5.61576	−	2.88569i	0.182681	−	0.0938715i
$946$		0			0
$947$	37.6021			1.22190			0.610952	−	0.791668i	$-0.290787\pi$
$947$	37.6021			1.22190			0.610952	+	0.791668i	$0.290787\pi$
$948$		0			0
$949$			34.5048i			1.12007i
$950$		0			0
$951$	−42.8905			−1.39082
$952$		0			0
$953$	11.4843			0.372014			0.186007	−	0.982548i	$-0.440445\pi$
$953$	11.4843			0.372014			0.186007	+	0.982548i	$0.440445\pi$
$954$		0			0
$955$		−	5.95641i		−	0.192745i
$956$		0			0
$957$	−78.1922			−2.52760
$958$		0			0
$959$	−6.94939	−	13.5240i	−0.224407	−	0.436714i
$960$		0			0
$961$	−26.0238			−0.839479
$962$		0			0
$963$	−21.2637			−0.685213
$964$		0			0
$965$	10.9141			0.351339
$966$		0			0
$967$		−	23.0158i		−	0.740139i	−0.929004	−	0.370069i	$-0.879334\pi$
$967$		−	23.0158i		−	0.740139i	0.929004	−	0.370069i	$-0.120666\pi$
$968$		0			0
$969$		−	16.4096i		−	0.527152i
$970$		0			0
$971$		−	0.820067i		−	0.0263172i	−0.999913	−	0.0131586i	$-0.995811\pi$
$971$		−	0.820067i		−	0.0263172i	0.999913	−	0.0131586i	$-0.00418863\pi$
$972$		0			0
$973$	29.7347	−	15.2793i	0.953250	−	0.489832i
$974$		0			0
$975$		−	12.5938i		−	0.403323i
$976$		0			0
$977$	−37.3536			−1.19505			−0.597524	−	0.801851i	$-0.703849\pi$
$977$	−37.3536			−1.19505			−0.597524	+	0.801851i	$0.703849\pi$
$978$		0			0
$979$		−	42.3792i		−	1.35445i
$980$		0			0
$981$			7.94118i			0.253542i
$982$		0			0
$983$	−24.6621			−0.786598			−0.393299	−	0.919411i	$-0.628666\pi$
$983$	−24.6621			−0.786598			−0.393299	+	0.919411i	$0.628666\pi$
$984$		0			0
$985$		−	3.76694i		−	0.120025i
$986$		0			0
$987$	−2.96263	+	1.52236i	−0.0943016	+	0.0484574i
$988$		0			0
$989$			10.8226i			0.344138i
$990$		0			0
$991$			39.5414i			1.25608i	0.778183	+	0.628038i	$0.216142\pi$
$991$			39.5414i			1.25608i	−0.778183	+	0.628038i	$0.783858\pi$
$992$		0			0
$993$		−	43.4081i		−	1.37751i
$994$		0			0
$995$	−7.52925			−0.238693
$996$		0			0
$997$	11.9761			0.379286			0.189643	−	0.981853i	$-0.439267\pi$
$997$	11.9761			0.379286			0.189643	+	0.981853i	$0.439267\pi$
$998$		0			0
$999$	16.6674			0.527335

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.b.111.14		16
4.3	odd	2		280.2.h.b.251.15	yes	16
7.6	odd	2		1120.2.h.a.111.3		16
8.3	odd	2		1120.2.h.a.111.14		16
8.5	even	2		280.2.h.a.251.16	yes	16
28.27	even	2		280.2.h.a.251.15	✓	16
56.13	odd	2		280.2.h.b.251.16	yes	16
56.27	even	2	inner	1120.2.h.b.111.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.15	✓	16	28.27	even	2
280.2.h.a.251.16	yes	16	8.5	even	2
280.2.h.b.251.15	yes	16	4.3	odd	2
280.2.h.b.251.16	yes	16	56.13	odd	2
1120.2.h.a.111.3		16	7.6	odd	2
1120.2.h.a.111.14		16	8.3	odd	2
1120.2.h.b.111.3		16	56.27	even	2	inner
1120.2.h.b.111.14		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1120.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.21915i q^{3} +1.00000 q^{5} +(-1.20923 - 2.35325i) q^{7} -1.92464 q^{9} +O(q^{10})\)	\(q+2.21915i q^{3} +1.00000 q^{5} +(-1.20923 - 2.35325i) q^{7} -1.92464 q^{9} +3.88394 q^{11} -5.67503 q^{13} +2.21915i q^{15} +5.63892i q^{17} +1.31134i q^{19} +(5.22221 - 2.68346i) q^{21} +7.37559i q^{23} +1.00000 q^{25} +2.38639i q^{27} +9.07201i q^{29} +2.23073 q^{31} +8.61906i q^{33} +(-1.20923 - 2.35325i) q^{35} -6.98438i q^{37} -12.5938i q^{39} +7.47757i q^{41} +1.46735 q^{43} -1.92464 q^{45} -0.567314 q^{47} +(-4.07554 + 5.69122i) q^{49} -12.5136 q^{51} -0.100950i q^{53} +3.88394 q^{55} -2.91006 q^{57} -2.93497i q^{59} +13.8324 q^{61} +(2.32733 + 4.52915i) q^{63} -5.67503 q^{65} -5.54736 q^{67} -16.3676 q^{69} +2.42368i q^{71} -6.08011i q^{73} +2.21915i q^{75} +(-4.69657 - 9.13987i) q^{77} -2.83370i q^{79} -11.0697 q^{81} +2.52687i q^{83} +5.63892i q^{85} -20.1322 q^{87} -10.9114i q^{89} +(6.86240 + 13.3547i) q^{91} +4.95033i q^{93} +1.31134i q^{95} -9.93165i q^{97} -7.47519 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{5} - 16 q^{9}+O(q^{10}) \) 16 * q + 16 * q^5 - 16 * q^9	\( 16 q + 16 q^{5} - 16 q^{9} + 4 q^{11} - 4 q^{21} + 16 q^{25} + 16 q^{31} + 4 q^{43} - 16 q^{45} - 8 q^{49} + 40 q^{51} + 4 q^{55} - 16 q^{57} - 8 q^{61} - 28 q^{63} - 20 q^{67} - 40 q^{69} - 4 q^{77} + 24 q^{81} - 72 q^{87} + 32 q^{91} - 20 q^{99}+O(q^{100}) \) 16 * q + 16 * q^5 - 16 * q^9 + 4 * q^11 - 4 * q^21 + 16 * q^25 + 16 * q^31 + 4 * q^43 - 16 * q^45 - 8 * q^49 + 40 * q^51 + 4 * q^55 - 16 * q^57 - 8 * q^61 - 28 * q^63 - 20 * q^67 - 40 * q^69 - 4 * q^77 + 24 * q^81 - 72 * q^87 + 32 * q^91 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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