LMFDB - Embedded newform 1120.2.h.a.111.15

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			111.15
Root			$1.07046 - 0.924187i$ of defining polynomial
Character	$\chi$	$=$	1120.111
Dual form			1120.2.h.a.111.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.99734i q^{3} -1.00000 q^{5} +(0.183359 - 2.63939i) q^{7} -5.98405 q^{9} +O(q^{10})$	$q+2.99734i q^{3} -1.00000 q^{5} +(0.183359 - 2.63939i) q^{7} -5.98405 q^{9} -4.87706 q^{11} +2.42800 q^{13} -2.99734i q^{15} -3.92955i q^{17} -5.24043i q^{19} +(7.91115 + 0.549588i) q^{21} +0.114122i q^{23} +1.00000 q^{25} -8.94421i q^{27} -3.60881i q^{29} +4.62694 q^{31} -14.6182i q^{33} +(-0.183359 + 2.63939i) q^{35} -7.83800i q^{37} +7.27754i q^{39} +10.4815i q^{41} -2.76350 q^{43} +5.98405 q^{45} -12.0817 q^{47} +(-6.93276 - 0.967910i) q^{49} +11.7782 q^{51} +0.668088i q^{53} +4.87706 q^{55} +15.7073 q^{57} -1.37541i q^{59} +1.17808 q^{61} +(-1.09723 + 15.7942i) q^{63} -2.42800 q^{65} -2.57766 q^{67} -0.342063 q^{69} -9.43699i q^{71} +5.62623i q^{73} +2.99734i q^{75} +(-0.894251 + 12.8725i) q^{77} -11.8804i q^{79} +8.85669 q^{81} -5.52348i q^{83} +3.92955i q^{85} +10.8168 q^{87} -6.21826i q^{89} +(0.445195 - 6.40844i) q^{91} +13.8685i q^{93} +5.24043i q^{95} -7.85094i q^{97} +29.1845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			2.99734i			1.73052i	0.501328	+	0.865258i	$0.332845\pi$
$3$			2.99734i			1.73052i	−0.501328	+	0.865258i	$0.667155\pi$
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$	0.183359	−	2.63939i	0.0693031	−	0.997596i
$7$	0.183359	−	2.63939i	0.0693031	−	0.997596i
$8$		0			0
$9$	−5.98405			−1.99468
$10$		0			0
$11$	−4.87706			−1.47049			−0.735244	−	0.677803i	$-0.762932\pi$
$11$	−4.87706			−1.47049			−0.735244	+	0.677803i	$0.762932\pi$
$12$		0			0
$13$	2.42800			0.673406			0.336703	−	0.941611i	$-0.390688\pi$
$13$	2.42800			0.673406			0.336703	+	0.941611i	$0.390688\pi$
$14$		0			0
$15$		−	2.99734i		−	0.773910i
$16$		0			0
$17$		−	3.92955i		−	0.953057i	−0.879159	−	0.476528i	$-0.841895\pi$
$17$		−	3.92955i		−	0.953057i	0.879159	−	0.476528i	$-0.158105\pi$
$18$		0			0
$19$		−	5.24043i		−	1.20224i	−0.799160	−	0.601118i	$-0.794722\pi$
$19$		−	5.24043i		−	1.20224i	0.799160	−	0.601118i	$-0.205278\pi$
$20$		0			0
$21$	7.91115	+	0.549588i	1.72635	+	0.119930i
$22$		0			0
$23$			0.114122i			0.0237961i	0.999929	+	0.0118981i	$0.00378736\pi$
$23$			0.114122i			0.0237961i	−0.999929	+	0.0118981i	$0.996213\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		−	8.94421i		−	1.72131i
$28$		0			0
$29$		−	3.60881i		−	0.670139i	−0.942193	−	0.335069i	$-0.891240\pi$
$29$		−	3.60881i		−	0.670139i	0.942193	−	0.335069i	$-0.108760\pi$
$30$		0			0
$31$	4.62694			0.831023			0.415511	−	0.909588i	$-0.363603\pi$
$31$	4.62694			0.831023			0.415511	+	0.909588i	$0.363603\pi$
$32$		0			0
$33$		−	14.6182i		−	2.54470i
$34$		0			0
$35$	−0.183359	+	2.63939i	−0.0309933	+	0.446138i
$36$		0			0
$37$		−	7.83800i		−	1.28856i	−0.764790	−	0.644279i	$-0.777157\pi$
$37$		−	7.83800i		−	1.28856i	0.764790	−	0.644279i	$-0.222843\pi$
$38$		0			0
$39$			7.27754i			1.16534i
$40$		0			0
$41$			10.4815i			1.63694i	0.574552	+	0.818468i	$0.305176\pi$
$41$			10.4815i			1.63694i	−0.574552	+	0.818468i	$0.694824\pi$
$42$		0			0
$43$	−2.76350			−0.421430			−0.210715	−	0.977548i	$-0.567579\pi$
$43$	−2.76350			−0.421430			−0.210715	+	0.977548i	$0.567579\pi$
$44$		0			0
$45$	5.98405			0.892049
$46$		0			0
$47$	−12.0817			−1.76229			−0.881145	−	0.472846i	$-0.843227\pi$
$47$	−12.0817			−1.76229			−0.881145	+	0.472846i	$0.843227\pi$
$48$		0			0
$49$	−6.93276	−	0.967910i	−0.990394	−	0.138273i
$50$		0			0
$51$	11.7782			1.64928
$52$		0			0
$53$			0.668088i			0.0917689i	0.998947	+	0.0458845i	$0.0146106\pi$
$53$			0.668088i			0.0917689i	−0.998947	+	0.0458845i	$0.985389\pi$
$54$		0			0
$55$	4.87706			0.657622
$56$		0			0
$57$	15.7073			2.08049
$58$		0			0
$59$		−	1.37541i		−	0.179063i	−0.995984	−	0.0895314i	$-0.971463\pi$
$59$		−	1.37541i		−	0.179063i	0.995984	−	0.0895314i	$-0.0285369\pi$
$60$		0			0
$61$	1.17808			0.150837			0.0754186	−	0.997152i	$-0.475971\pi$
$61$	1.17808			0.150837			0.0754186	+	0.997152i	$0.475971\pi$
$62$		0			0
$63$	−1.09723	+	15.7942i	−0.138238	+	1.98989i
$64$		0			0
$65$	−2.42800			−0.301156
$66$		0			0
$67$	−2.57766			−0.314911			−0.157456	−	0.987526i	$-0.550329\pi$
$67$	−2.57766			−0.314911			−0.157456	+	0.987526i	$0.550329\pi$
$68$		0			0
$69$	−0.342063			−0.0411795
$70$		0			0
$71$		−	9.43699i		−	1.11996i	−0.828505	−	0.559982i	$-0.810808\pi$
$71$		−	9.43699i		−	1.11996i	0.828505	−	0.559982i	$-0.189192\pi$
$72$		0			0
$73$			5.62623i			0.658500i	0.944243	+	0.329250i	$0.106796\pi$
$73$			5.62623i			0.658500i	−0.944243	+	0.329250i	$0.893204\pi$
$74$		0			0
$75$			2.99734i			0.346103i
$76$		0			0
$77$	−0.894251	+	12.8725i	−0.101909	+	1.46695i
$78$		0			0
$79$		−	11.8804i		−	1.33665i	−0.743872	−	0.668323i	$-0.767013\pi$
$79$		−	11.8804i		−	1.33665i	0.743872	−	0.668323i	$-0.232987\pi$
$80$		0			0
$81$	8.85669			0.984077
$82$		0			0
$83$		−	5.52348i		−	0.606280i	−0.952946	−	0.303140i	$-0.901965\pi$
$83$		−	5.52348i		−	0.606280i	0.952946	−	0.303140i	$-0.0980350\pi$
$84$		0			0
$85$			3.92955i			0.426220i
$86$		0			0
$87$	10.8168			1.15969
$88$		0			0
$89$		−	6.21826i		−	0.659135i	−0.944132	−	0.329567i	$-0.893097\pi$
$89$		−	6.21826i		−	0.659135i	0.944132	−	0.329567i	$-0.106903\pi$
$90$		0			0
$91$	0.445195	−	6.40844i	0.0466691	−	0.671787i
$92$		0			0
$93$			13.8685i			1.43810i
$94$		0			0
$95$			5.24043i			0.537657i
$96$		0			0
$97$		−	7.85094i		−	0.797142i	−0.917137	−	0.398571i	$-0.869506\pi$
$97$		−	7.85094i		−	0.797142i	0.917137	−	0.398571i	$-0.130494\pi$
$98$		0			0
$99$	29.1845			2.93316
$100$		0			0
$101$	10.9057			1.08516			0.542580	−	0.840004i	$-0.317448\pi$
$101$	10.9057			1.08516			0.542580	+	0.840004i	$0.317448\pi$
$102$		0			0
$103$	15.8202			1.55881			0.779403	−	0.626523i	$-0.215523\pi$
$103$	15.8202			1.55881			0.779403	+	0.626523i	$0.215523\pi$
$104$		0			0
$105$	−7.91115	−	0.549588i	−0.772049	−	0.0536343i
$106$		0			0
$107$	−6.53444			−0.631708			−0.315854	−	0.948808i	$-0.602291\pi$
$107$	−6.53444			−0.631708			−0.315854	+	0.948808i	$0.602291\pi$
$108$		0			0
$109$			10.9309i			1.04699i	0.852028	+	0.523496i	$0.175372\pi$
$109$			10.9309i			1.04699i	−0.852028	+	0.523496i	$0.824628\pi$
$110$		0			0
$111$	23.4931			2.22987
$112$		0			0
$113$	3.72004			0.349951			0.174976	−	0.984573i	$-0.444015\pi$
$113$	3.72004			0.349951			0.174976	+	0.984573i	$0.444015\pi$
$114$		0			0
$115$		−	0.114122i		−	0.0106419i
$116$		0			0
$117$	−14.5293			−1.34323
$118$		0			0
$119$	−10.3716	−	0.720518i	−0.950766	−	0.0660498i
$120$		0			0
$121$	12.7857			1.16233
$122$		0			0
$123$	−31.4166			−2.83274
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		−	3.80326i		−	0.337485i	−0.985660	−	0.168742i	$-0.946029\pi$
$127$		−	3.80326i		−	0.337485i	0.985660	−	0.168742i	$-0.0539706\pi$
$128$		0			0
$129$		−	8.28315i		−	0.729291i
$130$		0			0
$131$			7.68109i			0.671100i	0.942022	+	0.335550i	$0.108922\pi$
$131$			7.68109i			0.671100i	−0.942022	+	0.335550i	$0.891078\pi$
$132$		0			0
$133$	−13.8315	−	0.960878i	−1.19935	−	0.0833187i
$134$		0			0
$135$			8.94421i			0.769795i
$136$		0			0
$137$	−16.9739			−1.45018			−0.725090	−	0.688655i	$-0.758202\pi$
$137$	−16.9739			−1.45018			−0.725090	+	0.688655i	$0.758202\pi$
$138$		0			0
$139$			2.54598i			0.215947i	0.994154	+	0.107973i	$0.0344362\pi$
$139$			2.54598i			0.215947i	−0.994154	+	0.107973i	$0.965564\pi$
$140$		0			0
$141$		−	36.2128i		−	3.04967i
$142$		0			0
$143$	−11.8415			−0.990235
$144$		0			0
$145$			3.60881i			0.299695i
$146$		0			0
$147$	2.90116	−	20.7798i	0.239283	−	1.71389i
$148$		0			0
$149$		−	13.7925i		−	1.12992i	−0.825118	−	0.564961i	$-0.808891\pi$
$149$		−	13.7925i		−	1.12992i	0.825118	−	0.564961i	$-0.191109\pi$
$150$		0			0
$151$		−	0.128374i		−	0.0104469i	−0.999986	−	0.00522347i	$-0.998337\pi$
$151$		−	0.128374i		−	0.0104469i	0.999986	−	0.00522347i	$-0.00166269\pi$
$152$		0			0
$153$			23.5146i			1.90105i
$154$		0			0
$155$	−4.62694			−0.371645
$156$		0			0
$157$	−13.1585			−1.05017			−0.525083	−	0.851051i	$-0.675966\pi$
$157$	−13.1585			−1.05017			−0.525083	+	0.851051i	$0.675966\pi$
$158$		0			0
$159$	−2.00249			−0.158808
$160$		0			0
$161$	0.301213	+	0.0209253i	0.0237389	+	0.00164914i
$162$		0			0
$163$	−0.158264			−0.0123962			−0.00619809	−	0.999981i	$-0.501973\pi$
$163$	−0.158264			−0.0123962			−0.00619809	+	0.999981i	$0.501973\pi$
$164$		0			0
$165$			14.6182i			1.13802i
$166$		0			0
$167$	−16.7105			−1.29310			−0.646550	−	0.762871i	$-0.723789\pi$
$167$	−16.7105			−1.29310			−0.646550	+	0.762871i	$0.723789\pi$
$168$		0			0
$169$	−7.10482			−0.546524
$170$		0			0
$171$			31.3590i			2.39808i
$172$		0			0
$173$	−5.19433			−0.394918			−0.197459	−	0.980311i	$-0.563269\pi$
$173$	−5.19433			−0.394918			−0.197459	+	0.980311i	$0.563269\pi$
$174$		0			0
$175$	0.183359	−	2.63939i	0.0138606	−	0.199519i
$176$		0			0
$177$	4.12256			0.309871
$178$		0			0
$179$	17.3274			1.29511			0.647555	−	0.762019i	$-0.275792\pi$
$179$	17.3274			1.29511			0.647555	+	0.762019i	$0.275792\pi$
$180$		0			0
$181$	−21.0585			−1.56526			−0.782631	−	0.622485i	$-0.786123\pi$
$181$	−21.0585			−1.56526			−0.782631	+	0.622485i	$0.786123\pi$
$182$		0			0
$183$			3.53110i			0.261026i
$184$		0			0
$185$			7.83800i			0.576261i
$186$		0			0
$187$			19.1647i			1.40146i
$188$		0			0
$189$	−23.6073	−	1.64000i	−1.71718	−	0.119292i
$190$		0			0
$191$		−	8.43465i		−	0.610310i	−0.952303	−	0.305155i	$-0.901292\pi$
$191$		−	8.43465i		−	0.610310i	0.952303	−	0.305155i	$-0.0987083\pi$
$192$		0			0
$193$	−25.6337			−1.84516			−0.922578	−	0.385811i	$-0.873922\pi$
$193$	−25.6337			−1.84516			−0.922578	+	0.385811i	$0.873922\pi$
$194$		0			0
$195$		−	7.27754i		−	0.521156i
$196$		0			0
$197$		−	1.78729i		−	0.127339i	−0.997971	−	0.0636697i	$-0.979720\pi$
$197$		−	1.78729i		−	0.127339i	0.997971	−	0.0636697i	$-0.0202804\pi$
$198$		0			0
$199$	22.3178			1.58206			0.791032	−	0.611774i	$-0.209544\pi$
$199$	22.3178			1.58206			0.791032	+	0.611774i	$0.209544\pi$
$200$		0			0
$201$		−	7.72612i		−	0.544958i
$202$		0			0
$203$	−9.52505	−	0.661706i	−0.668527	−	0.0464427i
$204$		0			0
$205$		−	10.4815i		−	0.732060i
$206$		0			0
$207$		−	0.682913i		−	0.0474657i
$208$		0			0
$209$			25.5579i			1.76787i
$210$		0			0
$211$	−8.58471			−0.590996			−0.295498	−	0.955343i	$-0.595486\pi$
$211$	−8.58471			−0.590996			−0.295498	+	0.955343i	$0.595486\pi$
$212$		0			0
$213$	28.2859			1.93812
$214$		0			0
$215$	2.76350			0.188469
$216$		0			0
$217$	0.848389	−	12.2123i	0.0575924	−	0.829024i
$218$		0			0
$219$	−16.8637			−1.13954
$220$		0			0
$221$		−	9.54096i		−	0.641794i
$222$		0			0
$223$	−14.0359			−0.939916			−0.469958	−	0.882689i	$-0.655731\pi$
$223$	−14.0359			−0.939916			−0.469958	+	0.882689i	$0.655731\pi$
$224$		0			0
$225$	−5.98405			−0.398937
$226$		0			0
$227$			29.9063i			1.98495i	0.122442	+	0.992476i	$0.460927\pi$
$227$			29.9063i			1.98495i	−0.122442	+	0.992476i	$0.539073\pi$
$228$		0			0
$229$	8.92103			0.589518			0.294759	−	0.955572i	$-0.404761\pi$
$229$	8.92103			0.589518			0.294759	+	0.955572i	$0.404761\pi$
$230$		0			0
$231$	−38.5831	−	2.68037i	−2.53858	−	0.176356i
$232$		0			0
$233$	0.565155			0.0370245			0.0185123	−	0.999829i	$-0.494107\pi$
$233$	0.565155			0.0370245			0.0185123	+	0.999829i	$0.494107\pi$
$234$		0			0
$235$	12.0817			0.788120
$236$		0			0
$237$	35.6095			2.31309
$238$		0			0
$239$		−	8.51136i		−	0.550554i	−0.961365	−	0.275277i	$-0.911230\pi$
$239$		−	8.51136i		−	0.550554i	0.961365	−	0.275277i	$-0.0887695\pi$
$240$		0			0
$241$		−	2.17523i		−	0.140119i	−0.997543	−	0.0700595i	$-0.977681\pi$
$241$		−	2.17523i		−	0.140119i	0.997543	−	0.0700595i	$-0.0223189\pi$
$242$		0			0
$243$		−	0.286112i		−	0.0183541i
$244$		0			0
$245$	6.93276	+	0.967910i	0.442918	+	0.0618375i
$246$		0			0
$247$		−	12.7238i		−	0.809593i
$248$		0			0
$249$	16.5557			1.04918
$250$		0			0
$251$		−	9.41690i		−	0.594390i	−0.954817	−	0.297195i	$-0.903949\pi$
$251$		−	9.41690i		−	0.594390i	0.954817	−	0.297195i	$-0.0960511\pi$
$252$		0			0
$253$		−	0.556580i		−	0.0349919i
$254$		0			0
$255$	−11.7782			−0.737580
$256$		0			0
$257$			4.44138i			0.277045i	0.990359	+	0.138523i	$0.0442354\pi$
$257$			4.44138i			0.277045i	−0.990359	+	0.138523i	$0.955765\pi$
$258$		0			0
$259$	−20.6875	−	1.43717i	−1.28546	−	0.0893011i
$260$		0			0
$261$			21.5953i			1.33671i
$262$		0			0
$263$		−	13.6247i		−	0.840132i	−0.907493	−	0.420066i	$-0.862007\pi$
$263$		−	13.6247i		−	0.840132i	0.907493	−	0.420066i	$-0.137993\pi$
$264$		0			0
$265$		−	0.668088i		−	0.0410403i
$266$		0			0
$267$	18.6383			1.14064
$268$		0			0
$269$	10.2713			0.626253			0.313127	−	0.949711i	$-0.398624\pi$
$269$	10.2713			0.626253			0.313127	+	0.949711i	$0.398624\pi$
$270$		0			0
$271$	−2.67264			−0.162352			−0.0811758	−	0.996700i	$-0.525868\pi$
$271$	−2.67264			−0.162352			−0.0811758	+	0.996700i	$0.525868\pi$
$272$		0			0
$273$	19.2083	+	1.33440i	1.16254	+	0.0807616i
$274$		0			0
$275$	−4.87706			−0.294098
$276$		0			0
$277$		−	28.1383i		−	1.69067i	−0.534239	−	0.845333i	$-0.679402\pi$
$277$		−	28.1383i		−	1.69067i	0.534239	−	0.845333i	$-0.320598\pi$
$278$		0			0
$279$	−27.6878			−1.65763
$280$		0			0
$281$	0.466842			0.0278494			0.0139247	−	0.999903i	$-0.495567\pi$
$281$	0.466842			0.0278494			0.0139247	+	0.999903i	$0.495567\pi$
$282$		0			0
$283$		−	10.8004i		−	0.642019i	−0.947076	−	0.321009i	$-0.895978\pi$
$283$		−	10.8004i		−	0.642019i	0.947076	−	0.321009i	$-0.104022\pi$
$284$		0			0
$285$	−15.7073			−0.930423
$286$		0			0
$287$	27.6648	+	1.92188i	1.63300	+	0.113445i
$288$		0			0
$289$	1.55860			0.0916824
$290$		0			0
$291$	23.5319			1.37947
$292$		0			0
$293$	−7.34532			−0.429118			−0.214559	−	0.976711i	$-0.568831\pi$
$293$	−7.34532			−0.429118			−0.214559	+	0.976711i	$0.568831\pi$
$294$		0			0
$295$			1.37541i			0.0800793i
$296$		0			0
$297$			43.6214i			2.53117i
$298$		0			0
$299$			0.277089i			0.0160244i
$300$		0			0
$301$	−0.506712	+	7.29396i	−0.0292064	+	0.420417i
$302$		0			0
$303$			32.6882i			1.87789i
$304$		0			0
$305$	−1.17808			−0.0674564
$306$		0			0
$307$		−	5.08078i		−	0.289975i	−0.989433	−	0.144988i	$-0.953686\pi$
$307$		−	5.08078i		−	0.289975i	0.989433	−	0.144988i	$-0.0463142\pi$
$308$		0			0
$309$			47.4184i			2.69754i
$310$		0			0
$311$	−17.8319			−1.01116			−0.505578	−	0.862781i	$-0.668721\pi$
$311$	−17.8319			−1.01116			−0.505578	+	0.862781i	$0.668721\pi$
$312$		0			0
$313$		−	3.54008i		−	0.200097i	−0.994983	−	0.100049i	$-0.968100\pi$
$313$		−	3.54008i		−	0.200097i	0.994983	−	0.100049i	$-0.0318998\pi$
$314$		0			0
$315$	1.09723	−	15.7942i	0.0618218	−	0.889904i
$316$		0			0
$317$			4.86360i			0.273167i	0.990629	+	0.136583i	$0.0436122\pi$
$317$			4.86360i			0.273167i	−0.990629	+	0.136583i	$0.956388\pi$
$318$		0			0
$319$			17.6004i			0.985431i
$320$		0			0
$321$		−	19.5859i		−	1.09318i
$322$		0			0
$323$	−20.5925			−1.14580
$324$		0			0
$325$	2.42800			0.134681
$326$		0			0
$327$	−32.7637			−1.81183
$328$		0			0
$329$	−2.21528	+	31.8882i	−0.122132	+	1.75805i
$330$		0			0
$331$	9.38870			0.516050			0.258025	−	0.966138i	$-0.416928\pi$
$331$	9.38870			0.516050			0.258025	+	0.966138i	$0.416928\pi$
$332$		0			0
$333$			46.9030i			2.57027i
$334$		0			0
$335$	2.57766			0.140832
$336$		0			0
$337$	19.7551			1.07613			0.538064	−	0.842904i	$-0.319156\pi$
$337$	19.7551			1.07613			0.538064	+	0.842904i	$0.319156\pi$
$338$		0			0
$339$			11.1502i			0.605596i
$340$		0			0
$341$	−22.5658			−1.22201
$342$		0			0
$343$	−3.82587	+	18.1208i	−0.206578	+	0.978430i
$344$		0			0
$345$	0.342063			0.0184161
$346$		0			0
$347$	29.8093			1.60025			0.800124	−	0.599835i	$-0.204767\pi$
$347$	29.8093			1.60025			0.800124	+	0.599835i	$0.204767\pi$
$348$		0			0
$349$	15.6993			0.840364			0.420182	−	0.907440i	$-0.361966\pi$
$349$	15.6993			0.840364			0.420182	+	0.907440i	$0.361966\pi$
$350$		0			0
$351$		−	21.7165i		−	1.15914i
$352$		0			0
$353$			10.2186i			0.543879i	0.962314	+	0.271939i	$0.0876651\pi$
$353$			10.2186i			0.543879i	−0.962314	+	0.271939i	$0.912335\pi$
$354$		0			0
$355$			9.43699i			0.500863i
$356$		0			0
$357$	2.15964	−	31.0873i	0.114300	−	1.64531i
$358$		0			0
$359$			23.4758i			1.23900i	0.784995	+	0.619502i	$0.212666\pi$
$359$			23.4758i			1.23900i	−0.784995	+	0.619502i	$0.787334\pi$
$360$		0			0
$361$	−8.46208			−0.445373
$362$		0			0
$363$			38.3230i			2.01144i
$364$		0			0
$365$		−	5.62623i		−	0.294490i
$366$		0			0
$367$	21.5002			1.12230			0.561151	−	0.827714i	$-0.310359\pi$
$367$	21.5002			1.12230			0.561151	+	0.827714i	$0.310359\pi$
$368$		0			0
$369$		−	62.7218i		−	3.26517i
$370$		0			0
$371$	1.76334	+	0.122500i	0.0915483	+	0.00635987i
$372$		0			0
$373$			0.781051i			0.0404413i	0.999796	+	0.0202206i	$0.00643687\pi$
$373$			0.781051i			0.0404413i	−0.999796	+	0.0202206i	$0.993563\pi$
$374$		0			0
$375$		−	2.99734i		−	0.154782i
$376$		0			0
$377$		−	8.76218i		−	0.451275i
$378$		0			0
$379$	−4.53381			−0.232886			−0.116443	−	0.993197i	$-0.537149\pi$
$379$	−4.53381			−0.232886			−0.116443	+	0.993197i	$0.537149\pi$
$380$		0			0
$381$	11.3997			0.584023
$382$		0			0
$383$	−36.0555			−1.84235			−0.921176	−	0.389147i	$-0.872770\pi$
$383$	−36.0555			−1.84235			−0.921176	+	0.389147i	$0.872770\pi$
$384$		0			0
$385$	0.894251	−	12.8725i	0.0455752	−	0.656041i
$386$		0			0
$387$	16.5369			0.840619
$388$		0			0
$389$		−	21.1691i		−	1.07332i	−0.843800	−	0.536658i	$-0.819687\pi$
$389$		−	21.1691i		−	1.07332i	0.843800	−	0.536658i	$-0.180313\pi$
$390$		0			0
$391$	0.448449			0.0226791
$392$		0			0
$393$	−23.0228			−1.16135
$394$		0			0
$395$			11.8804i			0.597766i
$396$		0			0
$397$	−14.1445			−0.709891			−0.354946	−	0.934887i	$-0.615501\pi$
$397$	−14.1445			−0.709891			−0.354946	+	0.934887i	$0.615501\pi$
$398$		0			0
$399$	2.88008	−	41.4578i	0.144184	−	2.07549i
$400$		0			0
$401$	17.8168			0.889729			0.444865	−	0.895598i	$-0.353252\pi$
$401$	17.8168			0.889729			0.444865	+	0.895598i	$0.353252\pi$
$402$		0			0
$403$	11.2342			0.559616
$404$		0			0
$405$	−8.85669			−0.440092
$406$		0			0
$407$			38.2264i			1.89481i
$408$		0			0
$409$			32.5292i			1.60846i	0.594315	+	0.804232i	$0.297423\pi$
$409$			32.5292i			1.60846i	−0.594315	+	0.804232i	$0.702577\pi$
$410$		0			0
$411$		−	50.8766i		−	2.50956i
$412$		0			0
$413$	−3.63024	−	0.252193i	−0.178632	−	0.0124096i
$414$		0			0
$415$			5.52348i			0.271137i
$416$		0			0
$417$	−7.63116			−0.373699
$418$		0			0
$419$			34.4793i			1.68443i	0.539146	+	0.842213i	$0.318747\pi$
$419$			34.4793i			1.68443i	−0.539146	+	0.842213i	$0.681253\pi$
$420$		0			0
$421$			12.2635i			0.597688i	0.954302	+	0.298844i	$0.0966010\pi$
$421$			12.2635i			0.597688i	−0.954302	+	0.298844i	$0.903399\pi$
$422$		0			0
$423$	72.2972			3.51521
$424$		0			0
$425$		−	3.92955i		−	0.190611i
$426$		0			0
$427$	0.216011	−	3.10940i	0.0104535	−	0.150475i
$428$		0			0
$429$		−	35.4930i		−	1.71362i
$430$		0			0
$431$			21.7718i			1.04871i	0.851499	+	0.524357i	$0.175694\pi$
$431$			21.7718i			1.04871i	−0.851499	+	0.524357i	$0.824306\pi$
$432$		0			0
$433$		−	26.1597i		−	1.25715i	−0.777747	−	0.628577i	$-0.783638\pi$
$433$		−	26.1597i		−	1.25715i	0.777747	−	0.628577i	$-0.216362\pi$
$434$		0			0
$435$	−10.8168			−0.518627
$436$		0			0
$437$	0.598049			0.0286086
$438$		0			0
$439$	29.5333			1.40955			0.704775	−	0.709431i	$-0.251048\pi$
$439$	29.5333			1.40955			0.704775	+	0.709431i	$0.251048\pi$
$440$		0			0
$441$	41.4860	+	5.79202i	1.97552	+	0.275811i
$442$		0			0
$443$	21.7915			1.03535			0.517673	−	0.855579i	$-0.326798\pi$
$443$	21.7915			1.03535			0.517673	+	0.855579i	$0.326798\pi$
$444$		0			0
$445$			6.21826i			0.294774i
$446$		0			0
$447$	41.3407			1.95535
$448$		0			0
$449$	−1.75711			−0.0829231			−0.0414615	−	0.999140i	$-0.513201\pi$
$449$	−1.75711			−0.0829231			−0.0414615	+	0.999140i	$0.513201\pi$
$450$		0			0
$451$		−	51.1189i		−	2.40709i
$452$		0			0
$453$	0.384781			0.0180786
$454$		0			0
$455$	−0.445195	+	6.40844i	−0.0208711	+	0.300432i
$456$		0			0
$457$	3.78856			0.177221			0.0886106	−	0.996066i	$-0.471757\pi$
$457$	3.78856			0.177221			0.0886106	+	0.996066i	$0.471757\pi$
$458$		0			0
$459$	−35.1468			−1.64051
$460$		0			0
$461$	−32.3541			−1.50688			−0.753439	−	0.657517i	$-0.771607\pi$
$461$	−32.3541			−1.50688			−0.753439	+	0.657517i	$0.771607\pi$
$462$		0			0
$463$			31.2530i			1.45245i	0.687456	+	0.726226i	$0.258727\pi$
$463$			31.2530i			1.45245i	−0.687456	+	0.726226i	$0.741273\pi$
$464$		0			0
$465$		−	13.8685i		−	0.643137i
$466$		0			0
$467$			22.6800i			1.04950i	0.851255	+	0.524752i	$0.175842\pi$
$467$			22.6800i			1.04950i	−0.851255	+	0.524752i	$0.824158\pi$
$468$		0			0
$469$	−0.472636	+	6.80344i	−0.0218243	+	0.314154i
$470$		0			0
$471$		−	39.4406i		−	1.81733i
$472$		0			0
$473$	13.4777			0.619708
$474$		0			0
$475$		−	5.24043i		−	0.240447i
$476$		0			0
$477$		−	3.99787i		−	0.183050i
$478$		0			0
$479$	12.8639			0.587765			0.293883	−	0.955842i	$-0.405053\pi$
$479$	12.8639			0.587765			0.293883	+	0.955842i	$0.405053\pi$
$480$		0			0
$481$		−	19.0307i		−	0.867723i
$482$		0			0
$483$	−0.0627202	+	0.902838i	−0.00285387	+	0.0410805i
$484$		0			0
$485$			7.85094i			0.356493i
$486$		0			0
$487$		−	20.3133i		−	0.920483i	−0.887794	−	0.460241i	$-0.847763\pi$
$487$		−	20.3133i		−	0.920483i	0.887794	−	0.460241i	$-0.152237\pi$
$488$		0			0
$489$		−	0.474370i		−	0.0214518i
$490$		0			0
$491$	−16.4582			−0.742746			−0.371373	−	0.928484i	$-0.621113\pi$
$491$	−16.4582			−0.742746			−0.371373	+	0.928484i	$0.621113\pi$
$492$		0			0
$493$	−14.1810			−0.638680
$494$		0			0
$495$	−29.1845			−1.31175
$496$		0			0
$497$	−24.9079	−	1.73035i	−1.11727	−	0.0776170i
$498$		0			0
$499$	−18.7483			−0.839291			−0.419645	−	0.907688i	$-0.637846\pi$
$499$	−18.7483			−0.839291			−0.419645	+	0.907688i	$0.637846\pi$
$500$		0			0
$501$		−	50.0872i		−	2.23773i
$502$		0			0
$503$	19.5629			0.872269			0.436134	−	0.899882i	$-0.356347\pi$
$503$	19.5629			0.872269			0.436134	+	0.899882i	$0.356347\pi$
$504$		0			0
$505$	−10.9057			−0.485298
$506$		0			0
$507$		−	21.2956i		−	0.945769i
$508$		0			0
$509$	19.9845			0.885797			0.442898	−	0.896572i	$-0.353950\pi$
$509$	19.9845			0.885797			0.442898	+	0.896572i	$0.353950\pi$
$510$		0			0
$511$	14.8498	+	1.03162i	0.656917	+	0.0456361i
$512$		0			0
$513$	−46.8715			−2.06943
$514$		0			0
$515$	−15.8202			−0.697119
$516$		0			0
$517$	58.9229			2.59143
$518$		0			0
$519$		−	15.5692i		−	0.683411i
$520$		0			0
$521$			1.35111i			0.0591932i	0.999562	+	0.0295966i	$0.00942226\pi$
$521$			1.35111i			0.0591932i	−0.999562	+	0.0295966i	$0.990578\pi$
$522$		0			0
$523$		−	28.1452i		−	1.23071i	−0.788252	−	0.615353i	$-0.789014\pi$
$523$		−	28.1452i		−	1.23071i	0.788252	−	0.615353i	$-0.210986\pi$
$524$		0			0
$525$	7.91115	+	0.549588i	0.345271	+	0.0239860i
$526$		0			0
$527$		−	18.1818i		−	0.792012i
$528$		0			0
$529$	22.9870			0.999434
$530$		0			0
$531$			8.23051i			0.357174i
$532$		0			0
$533$			25.4491i			1.10232i
$534$		0			0
$535$	6.53444			0.282508
$536$		0			0
$537$			51.9360i			2.24121i
$538$		0			0
$539$	33.8115	+	4.72055i	1.45636	+	0.203329i
$540$		0			0
$541$		−	0.0636523i		−	0.00273663i	−0.999999	−	0.00136831i	$-0.999564\pi$
$541$		−	0.0636523i		−	0.00273663i	0.999999	−	0.00136831i	$-0.000435548\pi$
$542$		0			0
$543$		−	63.1194i		−	2.70871i
$544$		0			0
$545$		−	10.9309i		−	0.468229i
$546$		0			0
$547$	−26.6632			−1.14004			−0.570018	−	0.821632i	$-0.693064\pi$
$547$	−26.6632			−1.14004			−0.570018	+	0.821632i	$0.693064\pi$
$548$		0			0
$549$	−7.04966			−0.300872
$550$		0			0
$551$	−18.9117			−0.805665
$552$		0			0
$553$	−31.3569	−	2.17837i	−1.33343	−	0.0926336i
$554$		0			0
$555$	−23.4931			−0.997228
$556$		0			0
$557$			9.81678i			0.415950i	0.978134	+	0.207975i	$0.0666873\pi$
$557$			9.81678i			0.415950i	−0.978134	+	0.207975i	$0.933313\pi$
$558$		0			0
$559$	−6.70978			−0.283794
$560$		0			0
$561$	−57.4430			−2.42525
$562$		0			0
$563$			30.8680i			1.30093i	0.759535	+	0.650466i	$0.225426\pi$
$563$			30.8680i			1.30093i	−0.759535	+	0.650466i	$0.774574\pi$
$564$		0			0
$565$	−3.72004			−0.156503
$566$		0			0
$567$	1.62395	−	23.3763i	0.0681995	−	0.981711i
$568$		0			0
$569$	30.9000			1.29540			0.647698	−	0.761897i	$-0.275732\pi$
$569$	30.9000			1.29540			0.647698	+	0.761897i	$0.275732\pi$
$570$		0			0
$571$	37.5033			1.56946			0.784732	−	0.619835i	$-0.212801\pi$
$571$	37.5033			1.56946			0.784732	+	0.619835i	$0.212801\pi$
$572$		0			0
$573$	25.2815			1.05615
$574$		0			0
$575$			0.114122i			0.00475922i
$576$		0			0
$577$		−	17.4216i		−	0.725272i	−0.931931	−	0.362636i	$-0.881877\pi$
$577$		−	17.4216i		−	0.725272i	0.931931	−	0.362636i	$-0.118123\pi$
$578$		0			0
$579$		−	76.8330i		−	3.19307i
$580$		0			0
$581$	−14.5786	−	1.01278i	−0.604822	−	0.0420171i
$582$		0			0
$583$		−	3.25830i		−	0.134945i
$584$		0			0
$585$	14.5293			0.600711
$586$		0			0
$587$		−	12.4270i		−	0.512918i	−0.966555	−	0.256459i	$-0.917444\pi$
$587$		−	12.4270i		−	0.512918i	0.966555	−	0.256459i	$-0.0825558\pi$
$588$		0			0
$589$		−	24.2471i		−	0.999086i
$590$		0			0
$591$	5.35713			0.220363
$592$		0			0
$593$		−	13.7257i		−	0.563646i	−0.959466	−	0.281823i	$-0.909061\pi$
$593$		−	13.7257i		−	0.563646i	0.959466	−	0.281823i	$-0.0909391\pi$
$594$		0			0
$595$	10.3716	+	0.720518i	0.425195	+	0.0295384i
$596$		0			0
$597$			66.8940i			2.73779i
$598$		0			0
$599$		−	27.7947i		−	1.13566i	−0.823146	−	0.567829i	$-0.807783\pi$
$599$		−	27.7947i		−	1.13566i	0.823146	−	0.567829i	$-0.192217\pi$
$600$		0			0
$601$			38.9855i			1.59025i	0.606443	+	0.795127i	$0.292596\pi$
$601$			38.9855i			1.59025i	−0.606443	+	0.795127i	$0.707404\pi$
$602$		0			0
$603$	15.4248			0.628148
$604$		0			0
$605$	−12.7857			−0.519811
$606$		0			0
$607$	−11.7981			−0.478872			−0.239436	−	0.970912i	$-0.576962\pi$
$607$	−11.7981			−0.478872			−0.239436	+	0.970912i	$0.576962\pi$
$608$		0			0
$609$	1.98336	−	28.5498i	0.0803697	−	1.15690i
$610$		0			0
$611$	−29.3342			−1.18674
$612$		0			0
$613$		−	13.4628i		−	0.543756i	−0.962332	−	0.271878i	$-0.912355\pi$
$613$		−	13.4628i		−	0.543756i	0.962332	−	0.271878i	$-0.0876447\pi$
$614$		0			0
$615$	31.4166			1.26684
$616$		0			0
$617$	−15.2924			−0.615650			−0.307825	−	0.951443i	$-0.599601\pi$
$617$	−15.2924			−0.615650			−0.307825	+	0.951443i	$0.599601\pi$
$618$		0			0
$619$			3.76548i			0.151347i	0.997133	+	0.0756736i	$0.0241107\pi$
$619$			3.76548i			0.151347i	−0.997133	+	0.0756736i	$0.975889\pi$
$620$		0			0
$621$	1.02073			0.0409606
$622$		0			0
$623$	−16.4124	−	1.14017i	−0.657550	−	0.0456801i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−76.6056			−3.05933
$628$		0			0
$629$	−30.7998			−1.22807
$630$		0			0
$631$			21.1777i			0.843070i	0.906812	+	0.421535i	$0.138508\pi$
$631$			21.1777i			0.843070i	−0.906812	+	0.421535i	$0.861492\pi$
$632$		0			0
$633$		−	25.7313i		−	1.02273i
$634$		0			0
$635$			3.80326i			0.150928i
$636$		0			0
$637$	−16.8327	−	2.35009i	−0.666937	−	0.0931138i
$638$		0			0
$639$			56.4714i			2.23397i
$640$		0			0
$641$	29.5569			1.16743			0.583713	−	0.811960i	$-0.301599\pi$
$641$	29.5569			1.16743			0.583713	+	0.811960i	$0.301599\pi$
$642$		0			0
$643$		−	19.8063i		−	0.781085i	−0.920585	−	0.390543i	$-0.872287\pi$
$643$		−	19.8063i		−	0.781085i	0.920585	−	0.390543i	$-0.127713\pi$
$644$		0			0
$645$			8.28315i			0.326149i
$646$		0			0
$647$	12.4755			0.490461			0.245231	−	0.969465i	$-0.421136\pi$
$647$	12.4755			0.490461			0.245231	+	0.969465i	$0.421136\pi$
$648$		0			0
$649$			6.70794i			0.263310i
$650$		0			0
$651$	36.6044	+	2.54291i	1.43464	+	0.0996646i
$652$		0			0
$653$		−	9.99757i		−	0.391235i	−0.980680	−	0.195618i	$-0.937329\pi$
$653$		−	9.99757i		−	0.391235i	0.980680	−	0.195618i	$-0.0626712\pi$
$654$		0			0
$655$		−	7.68109i		−	0.300125i
$656$		0			0
$657$		−	33.6676i		−	1.31350i
$658$		0			0
$659$	−27.2870			−1.06295			−0.531475	−	0.847074i	$-0.678362\pi$
$659$	−27.2870			−1.06295			−0.531475	+	0.847074i	$0.678362\pi$
$660$		0			0
$661$	38.2384			1.48730			0.743650	−	0.668569i	$-0.233093\pi$
$661$	38.2384			1.48730			0.743650	+	0.668569i	$0.233093\pi$
$662$		0			0
$663$	28.5975			1.11063
$664$		0			0
$665$	13.8315	+	0.960878i	0.536364	+	0.0372613i
$666$		0			0
$667$	0.411845			0.0159467
$668$		0			0
$669$		−	42.0705i		−	1.62654i
$670$		0			0
$671$	−5.74554			−0.221804
$672$		0			0
$673$	8.47579			0.326718			0.163359	−	0.986567i	$-0.447767\pi$
$673$	8.47579			0.326718			0.163359	+	0.986567i	$0.447767\pi$
$674$		0			0
$675$		−	8.94421i		−	0.344263i
$676$		0			0
$677$	−23.2065			−0.891897			−0.445948	−	0.895059i	$-0.647134\pi$
$677$	−23.2065			−0.891897			−0.445948	+	0.895059i	$0.647134\pi$
$678$		0			0
$679$	−20.7217	−	1.43954i	−0.795225	−	0.0552444i
$680$		0			0
$681$	−89.6394			−3.43499
$682$		0			0
$683$	25.7680			0.985986			0.492993	−	0.870033i	$-0.335903\pi$
$683$	25.7680			0.985986			0.492993	+	0.870033i	$0.335903\pi$
$684$		0			0
$685$	16.9739			0.648540
$686$		0			0
$687$			26.7394i			1.02017i
$688$		0			0
$689$			1.62212i			0.0617978i
$690$		0			0
$691$			24.5498i			0.933917i	0.884279	+	0.466958i	$0.154650\pi$
$691$			24.5498i			0.933917i	−0.884279	+	0.466958i	$0.845350\pi$
$692$		0			0
$693$	5.35124	−	77.0294i	0.203277	−	2.92610i
$694$		0			0
$695$		−	2.54598i		−	0.0965744i
$696$		0			0
$697$	41.1877			1.56009
$698$		0			0
$699$			1.69396i			0.0640715i
$700$		0			0
$701$		−	2.33837i		−	0.0883188i	−0.999024	−	0.0441594i	$-0.985939\pi$
$701$		−	2.33837i		−	0.0883188i	0.999024	−	0.0441594i	$-0.0140610\pi$
$702$		0			0
$703$	−41.0745			−1.54915
$704$		0			0
$705$			36.2128i			1.36385i
$706$		0			0
$707$	1.99966	−	28.7845i	0.0752049	−	1.08255i
$708$		0			0
$709$			36.8318i			1.38325i	0.722257	+	0.691624i	$0.243105\pi$
$709$			36.8318i			1.38325i	−0.722257	+	0.691624i	$0.756895\pi$
$710$		0			0
$711$			71.0927i			2.66618i
$712$		0			0
$713$			0.528036i			0.0197751i
$714$		0			0
$715$	11.8415			0.442847
$716$		0			0
$717$	25.5114			0.952742
$718$		0			0
$719$	33.8925			1.26398			0.631989	−	0.774978i	$-0.282239\pi$
$719$	33.8925			1.26398			0.631989	+	0.774978i	$0.282239\pi$
$720$		0			0
$721$	2.90076	−	41.7556i	0.108030	−	1.55506i
$722$		0			0
$723$	6.51991			0.242478
$724$		0			0
$725$		−	3.60881i		−	0.134028i
$726$		0			0
$727$	−4.28102			−0.158774			−0.0793871	−	0.996844i	$-0.525296\pi$
$727$	−4.28102			−0.158774			−0.0793871	+	0.996844i	$0.525296\pi$
$728$		0			0
$729$	27.4276			1.01584
$730$		0			0
$731$			10.8593i			0.401647i
$732$		0			0
$733$	38.0426			1.40513			0.702567	−	0.711618i	$-0.252037\pi$
$733$	38.0426			1.40513			0.702567	+	0.711618i	$0.252037\pi$
$734$		0			0
$735$	−2.90116	+	20.7798i	−0.107011	+	0.766476i
$736$		0			0
$737$	12.5714			0.463073
$738$		0			0
$739$	−9.93398			−0.365427			−0.182714	−	0.983166i	$-0.558488\pi$
$739$	−9.93398			−0.365427			−0.182714	+	0.983166i	$0.558488\pi$
$740$		0			0
$741$	38.1374			1.40101
$742$		0			0
$743$			1.68905i			0.0619651i	0.999520	+	0.0309826i	$0.00986363\pi$
$743$			1.68905i			0.0619651i	−0.999520	+	0.0309826i	$0.990136\pi$
$744$		0			0
$745$			13.7925i			0.505316i
$746$		0			0
$747$			33.0527i			1.20934i
$748$		0			0
$749$	−1.19815	+	17.2469i	−0.0437793	+	0.630189i
$750$		0			0
$751$		−	36.9451i		−	1.34815i	−0.738664	−	0.674073i	$-0.764543\pi$
$751$		−	36.9451i		−	1.34815i	0.738664	−	0.674073i	$-0.235457\pi$
$752$		0			0
$753$	28.2257			1.02860
$754$		0			0
$755$			0.128374i			0.00467202i
$756$		0			0
$757$			45.8391i			1.66605i	0.553235	+	0.833025i	$0.313393\pi$
$757$			45.8391i			1.66605i	−0.553235	+	0.833025i	$0.686607\pi$
$758$		0			0
$759$	1.66826			0.0605540
$760$		0			0
$761$		−	19.9588i		−	0.723506i	−0.932274	−	0.361753i	$-0.882178\pi$
$761$		−	19.9588i		−	0.723506i	0.932274	−	0.361753i	$-0.117822\pi$
$762$		0			0
$763$	28.8509	+	2.00428i	1.04447	+	0.0725597i
$764$		0			0
$765$		−	23.5146i		−	0.850174i
$766$		0			0
$767$		−	3.33949i		−	0.120582i
$768$		0			0
$769$			13.9529i			0.503153i	0.967837	+	0.251576i	$0.0809490\pi$
$769$			13.9529i			0.503153i	−0.967837	+	0.251576i	$0.919051\pi$
$770$		0			0
$771$	−13.3123			−0.479431
$772$		0			0
$773$	−33.8775			−1.21849			−0.609245	−	0.792982i	$-0.708527\pi$
$773$	−33.8775			−1.21849			−0.609245	+	0.792982i	$0.708527\pi$
$774$		0			0
$775$	4.62694			0.166205
$776$		0			0
$777$	4.30767	−	62.0076i	0.154537	−	2.22451i
$778$		0			0
$779$	54.9276			1.96798
$780$		0			0
$781$			46.0247i			1.64689i
$782$		0			0
$783$	−32.2779			−1.15352
$784$		0			0
$785$	13.1585			0.469649
$786$		0			0
$787$		−	8.07748i		−	0.287931i	−0.989583	−	0.143965i	$-0.954015\pi$
$787$		−	8.07748i		−	0.287931i	0.989583	−	0.143965i	$-0.0459854\pi$
$788$		0			0
$789$	40.8377			1.45386
$790$		0			0
$791$	0.682101	−	9.81862i	0.0242527	−	0.349110i
$792$		0			0
$793$	2.86037			0.101575
$794$		0			0
$795$	2.00249			0.0710209
$796$		0			0
$797$	−30.9873			−1.09763			−0.548813	−	0.835945i	$-0.684920\pi$
$797$	−30.9873			−1.09763			−0.548813	+	0.835945i	$0.684920\pi$
$798$		0			0
$799$			47.4755i			1.67956i
$800$		0			0
$801$			37.2104i			1.31476i
$802$		0			0
$803$		−	27.4394i		−	0.968316i
$804$		0			0
$805$	−0.301213	−	0.0209253i	−0.0106164	−	0.000737520i
$806$		0			0
$807$			30.7866i			1.08374i
$808$		0			0
$809$	−52.5895			−1.84895			−0.924474	−	0.381245i	$-0.875495\pi$
$809$	−52.5895			−1.84895			−0.924474	+	0.381245i	$0.875495\pi$
$810$		0			0
$811$			37.3872i			1.31284i	0.754395	+	0.656421i	$0.227931\pi$
$811$			37.3872i			1.31284i	−0.754395	+	0.656421i	$0.772069\pi$
$812$		0			0
$813$		−	8.01082i		−	0.280952i
$814$		0			0
$815$	0.158264			0.00554374
$816$		0			0
$817$			14.4819i			0.506659i
$818$		0			0
$819$	−2.66407	+	38.3484i	−0.0930901	+	1.34000i
$820$		0			0
$821$		−	23.1500i		−	0.807941i	−0.914772	−	0.403970i	$-0.867630\pi$
$821$		−	23.1500i		−	0.807941i	0.914772	−	0.403970i	$-0.132370\pi$
$822$		0			0
$823$			10.1543i			0.353957i	0.984215	+	0.176979i	$0.0566323\pi$
$823$			10.1543i			0.353957i	−0.984215	+	0.176979i	$0.943368\pi$
$824$		0			0
$825$		−	14.6182i		−	0.508940i
$826$		0			0
$827$	25.7346			0.894881			0.447440	−	0.894314i	$-0.352336\pi$
$827$	25.7346			0.894881			0.447440	+	0.894314i	$0.352336\pi$
$828$		0			0
$829$	29.2920			1.01735			0.508677	−	0.860958i	$-0.330135\pi$
$829$	29.2920			1.01735			0.508677	+	0.860958i	$0.330135\pi$
$830$		0			0
$831$	84.3400			2.92572
$832$		0			0
$833$	−3.80346	+	27.2427i	−0.131782	+	0.943902i
$834$		0			0
$835$	16.7105			0.578292
$836$		0			0
$837$		−	41.3843i		−	1.43045i
$838$		0			0
$839$	−11.5479			−0.398676			−0.199338	−	0.979931i	$-0.563879\pi$
$839$	−11.5479			−0.398676			−0.199338	+	0.979931i	$0.563879\pi$
$840$		0			0
$841$	15.9765			0.550914
$842$		0			0
$843$			1.39928i			0.0481939i
$844$		0			0
$845$	7.10482			0.244413
$846$		0			0
$847$	2.34436	−	33.7464i	0.0805533	−	1.15954i
$848$		0			0
$849$	32.3726			1.11102
$850$		0			0
$851$	0.894489			0.0306627
$852$		0			0
$853$	−35.5827			−1.21833			−0.609164	−	0.793045i	$-0.708495\pi$
$853$	−35.5827			−1.21833			−0.609164	+	0.793045i	$0.708495\pi$
$854$		0			0
$855$		−	31.3590i		−	1.07245i
$856$		0			0
$857$			1.71368i			0.0585383i	0.999572	+	0.0292691i	$0.00931799\pi$
$857$			1.71368i			0.0585383i	−0.999572	+	0.0292691i	$0.990682\pi$
$858$		0			0
$859$		−	14.7176i		−	0.502158i	−0.967967	−	0.251079i	$-0.919215\pi$
$859$		−	14.7176i		−	0.502158i	0.967967	−	0.251079i	$-0.0807855\pi$
$860$		0			0
$861$	−5.76052	+	82.9208i	−0.196318	+	2.82593i
$862$		0			0
$863$		−	37.4241i		−	1.27393i	−0.770892	−	0.636966i	$-0.780189\pi$
$863$		−	37.4241i		−	1.27393i	0.770892	−	0.636966i	$-0.219811\pi$
$864$		0			0
$865$	5.19433			0.176612
$866$		0			0
$867$			4.67166i			0.158658i
$868$		0			0
$869$			57.9412i			1.96552i
$870$		0			0
$871$	−6.25855			−0.212063
$872$		0			0
$873$			46.9804i			1.59005i
$874$		0			0
$875$	−0.183359	+	2.63939i	−0.00619866	+	0.0892277i
$876$		0			0
$877$		−	24.9933i		−	0.843965i	−0.906604	−	0.421982i	$-0.861334\pi$
$877$		−	24.9933i		−	0.843965i	0.906604	−	0.421982i	$-0.138666\pi$
$878$		0			0
$879$		−	22.0164i		−	0.742596i
$880$		0			0
$881$		−	27.7020i		−	0.933303i	−0.884441	−	0.466651i	$-0.845460\pi$
$881$		−	27.7020i		−	0.933303i	0.884441	−	0.466651i	$-0.154540\pi$
$882$		0			0
$883$	−3.66299			−0.123269			−0.0616347	−	0.998099i	$-0.519631\pi$
$883$	−3.66299			−0.123269			−0.0616347	+	0.998099i	$0.519631\pi$
$884$		0			0
$885$	−4.12256			−0.138579
$886$		0			0
$887$	−40.0849			−1.34592			−0.672960	−	0.739679i	$-0.734978\pi$
$887$	−40.0849			−1.34592			−0.672960	+	0.739679i	$0.734978\pi$
$888$		0			0
$889$	−10.0383	−	0.697361i	−0.336674	−	0.0233887i
$890$		0			0
$891$	−43.1946			−1.44707
$892$		0			0
$893$			63.3130i			2.11869i
$894$		0			0
$895$	−17.3274			−0.579190
$896$		0			0
$897$	−0.830529			−0.0277306
$898$		0			0
$899$		−	16.6977i		−	0.556900i
$900$		0			0
$901$	2.62529			0.0874610
$902$		0			0
$903$	−21.8625	−	1.51879i	−0.727538	−	0.0505421i
$904$		0			0
$905$	21.0585			0.700007
$906$		0			0
$907$	33.0878			1.09866			0.549331	−	0.835605i	$-0.314883\pi$
$907$	33.0878			1.09866			0.549331	+	0.835605i	$0.314883\pi$
$908$		0			0
$909$	−65.2604			−2.16455
$910$		0			0
$911$			41.4259i			1.37250i	0.727365	+	0.686250i	$0.240745\pi$
$911$			41.4259i			1.37250i	−0.727365	+	0.686250i	$0.759255\pi$
$912$		0			0
$913$			26.9383i			0.891527i
$914$		0			0
$915$		−	3.53110i		−	0.116734i
$916$		0			0
$917$	20.2734	+	1.40840i	0.669487	+	0.0465093i
$918$		0			0
$919$		−	2.61219i		−	0.0861682i	−0.999071	−	0.0430841i	$-0.986282\pi$
$919$		−	2.61219i		−	0.0861682i	0.999071	−	0.0430841i	$-0.0137183\pi$
$920$		0			0
$921$	15.2288			0.501807
$922$		0			0
$923$		−	22.9130i		−	0.754191i
$924$		0			0
$925$		−	7.83800i		−	0.257712i
$926$		0			0
$927$	−94.6686			−3.10932
$928$		0			0
$929$			29.7850i			0.977213i	0.872504	+	0.488606i	$0.162495\pi$
$929$			29.7850i			0.977213i	−0.872504	+	0.488606i	$0.837505\pi$
$930$		0			0
$931$	−5.07226	+	36.3306i	−0.166237	+	1.19069i
$932$		0			0
$933$		−	53.4484i		−	1.74982i
$934$		0			0
$935$		−	19.1647i		−	0.626751i
$936$		0			0
$937$		−	20.9620i		−	0.684800i	−0.939554	−	0.342400i	$-0.888760\pi$
$937$		−	20.9620i		−	0.684800i	0.939554	−	0.342400i	$-0.111240\pi$
$938$		0			0
$939$	10.6108			0.346271
$940$		0			0
$941$	21.3589			0.696280			0.348140	−	0.937443i	$-0.386813\pi$
$941$	21.3589			0.696280			0.348140	+	0.937443i	$0.386813\pi$
$942$		0			0
$943$	−1.19617			−0.0389527
$944$		0			0
$945$	23.6073	+	1.64000i	0.767944	+	0.0533492i
$946$		0			0
$947$	54.2008			1.76129			0.880645	−	0.473777i	$-0.157110\pi$
$947$	54.2008			1.76129			0.880645	+	0.473777i	$0.157110\pi$
$948$		0			0
$949$			13.6605i			0.443438i
$950$		0			0
$951$	−14.5779			−0.472719
$952$		0			0
$953$	26.3943			0.854994			0.427497	−	0.904017i	$-0.359395\pi$
$953$	26.3943			0.854994			0.427497	+	0.904017i	$0.359395\pi$
$954$		0			0
$955$			8.43465i			0.272939i
$956$		0			0
$957$	−52.7542			−1.70530
$958$		0			0
$959$	−3.11231	+	44.8008i	−0.100502	+	1.44669i
$960$		0			0
$961$	−9.59145			−0.309402
$962$		0			0
$963$	39.1024			1.26006
$964$		0			0
$965$	25.6337			0.825179
$966$		0			0
$967$		−	32.9216i		−	1.05869i	−0.848408	−	0.529343i	$-0.822438\pi$
$967$		−	32.9216i		−	1.05869i	0.848408	−	0.529343i	$-0.177562\pi$
$968$		0			0
$969$		−	61.7229i		−	1.98282i
$970$		0			0
$971$			32.2598i			1.03527i	0.855603	+	0.517633i	$0.173187\pi$
$971$			32.2598i			1.03527i	−0.855603	+	0.517633i	$0.826813\pi$
$972$		0			0
$973$	6.71982	+	0.466827i	0.215428	+	0.0149658i
$974$		0			0
$975$			7.27754i			0.233068i
$976$		0			0
$977$	−21.3881			−0.684265			−0.342132	−	0.939652i	$-0.611149\pi$
$977$	−21.3881			−0.684265			−0.342132	+	0.939652i	$0.611149\pi$
$978$		0			0
$979$			30.3268i			0.969249i
$980$		0			0
$981$		−	65.4111i		−	2.08842i
$982$		0			0
$983$	18.6503			0.594852			0.297426	−	0.954745i	$-0.403872\pi$
$983$	18.6503			0.594852			0.297426	+	0.954745i	$0.403872\pi$
$984$		0			0
$985$			1.78729i			0.0569479i
$986$		0			0
$987$	−95.5798	−	6.63994i	−3.04234	−	0.211351i
$988$		0			0
$989$		−	0.315377i		−	0.0100284i
$990$		0			0
$991$		−	13.5821i		−	0.431448i	−0.976454	−	0.215724i	$-0.930789\pi$
$991$		−	13.5821i		−	0.431448i	0.976454	−	0.215724i	$-0.0692112\pi$
$992$		0			0
$993$			28.1411i			0.893032i
$994$		0			0
$995$	−22.3178			−0.707521
$996$		0			0
$997$	−25.3616			−0.803210			−0.401605	−	0.915813i	$-0.631548\pi$
$997$	−25.3616			−0.803210			−0.401605	+	0.915813i	$0.631548\pi$
$998$		0			0
$999$	−70.1047			−2.21801

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1120.2.h.a.111.15		16
4.3	odd	2		280.2.h.a.251.11	✓	16
7.6	odd	2		1120.2.h.b.111.2		16
8.3	odd	2		1120.2.h.b.111.15		16
8.5	even	2		280.2.h.b.251.12	yes	16
28.27	even	2		280.2.h.b.251.11	yes	16
56.13	odd	2		280.2.h.a.251.12	yes	16
56.27	even	2	inner	1120.2.h.a.111.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.h.a.251.11	✓	16	4.3	odd	2
280.2.h.a.251.12	yes	16	56.13	odd	2
280.2.h.b.251.11	yes	16	28.27	even	2
280.2.h.b.251.12	yes	16	8.5	even	2
1120.2.h.a.111.2		16	56.27	even	2	inner
1120.2.h.a.111.15		16	1.1	even	1	trivial
1120.2.h.b.111.2		16	7.6	odd	2
1120.2.h.b.111.15		16	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.99734i q^{3} -1.00000 q^{5} +(0.183359 - 2.63939i) q^{7} -5.98405 q^{9} +O(q^{10})\)	\(q+2.99734i q^{3} -1.00000 q^{5} +(0.183359 - 2.63939i) q^{7} -5.98405 q^{9} -4.87706 q^{11} +2.42800 q^{13} -2.99734i q^{15} -3.92955i q^{17} -5.24043i q^{19} +(7.91115 + 0.549588i) q^{21} +0.114122i q^{23} +1.00000 q^{25} -8.94421i q^{27} -3.60881i q^{29} +4.62694 q^{31} -14.6182i q^{33} +(-0.183359 + 2.63939i) q^{35} -7.83800i q^{37} +7.27754i q^{39} +10.4815i q^{41} -2.76350 q^{43} +5.98405 q^{45} -12.0817 q^{47} +(-6.93276 - 0.967910i) q^{49} +11.7782 q^{51} +0.668088i q^{53} +4.87706 q^{55} +15.7073 q^{57} -1.37541i q^{59} +1.17808 q^{61} +(-1.09723 + 15.7942i) q^{63} -2.42800 q^{65} -2.57766 q^{67} -0.342063 q^{69} -9.43699i q^{71} +5.62623i q^{73} +2.99734i q^{75} +(-0.894251 + 12.8725i) q^{77} -11.8804i q^{79} +8.85669 q^{81} -5.52348i q^{83} +3.92955i q^{85} +10.8168 q^{87} -6.21826i q^{89} +(0.445195 - 6.40844i) q^{91} +13.8685i q^{93} +5.24043i q^{95} -7.85094i q^{97} +29.1845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 16 q^{9}+O(q^{10}) \) 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

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Representations

Groups

Database

Properties

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