LMFDB - Embedded newform 2112.2.k.k.287.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2112,2,Mod(287,2112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2112, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2112.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2112 = 2^{6} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2112.k (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$16.8644049069$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	16.0.968265199641600000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 9x^{14} + 44x^{12} + 261x^{10} + 1029x^{8} + 1044x^{6} + 704x^{4} + 576x^{2} + 256 $ x^16 + 9x^14 + 44x^12 + 261x^10 + 1029x^8 + 1044x^6 + 704x^4 + 576*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.4
Root			$-0.676408 - 0.553538i$ of defining polynomial
Character	$\chi$	$=$	2112.287
Dual form			2112.2.k.k.287.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.62968 + 0.586627i) q^{3} +2.40932 q^{5} -1.23607i q^{7} +(2.31174 - 1.91203i) q^{9} +O(q^{10})$	$q+(-1.62968 + 0.586627i) q^{3} +2.40932 q^{5} -1.23607i q^{7} +(2.31174 - 1.91203i) q^{9} +1.00000i q^{11} +4.37780i q^{13} +(-3.92643 + 1.41337i) q^{15} +2.89940i q^{17} -0.564698 q^{19} +(0.725111 + 2.01440i) q^{21} -5.96500 q^{23} +0.804832 q^{25} +(-2.64575 + 4.47214i) q^{27} +1.29888i q^{31} +(-0.586627 - 1.62968i) q^{33} -2.97809i q^{35} +5.65147i q^{37} +(-2.56814 - 7.13443i) q^{39} +0.662534i q^{41} +8.87536 q^{43} +(5.56972 - 4.60670i) q^{45} -5.76026 q^{47} +5.47214 q^{49} +(-1.70087 - 4.72511i) q^{51} +9.03280 q^{53} +2.40932i q^{55} +(0.920279 - 0.331267i) q^{57} +9.17325i q^{59} -4.34233i q^{61} +(-2.36340 - 2.85746i) q^{63} +10.5475i q^{65} -5.49624 q^{67} +(9.72106 - 3.49923i) q^{69} +11.7638 q^{71} -12.1432 q^{73} +(-1.31162 + 0.472136i) q^{75} +1.23607 q^{77} -2.56066i q^{79} +(1.68826 - 8.84024i) q^{81} +15.7191i q^{83} +6.98560i q^{85} +0.240205i q^{89} +5.41126 q^{91} +(-0.761959 - 2.11677i) q^{93} -1.36054 q^{95} -15.0956 q^{97} +(1.91203 + 2.31174i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{5} - 4 q^{9}+O(q^{10}) $ 16 * q - 16 * q^5 - 4 * q^9	$ 16 q - 16 q^{5} - 4 q^{9} - 20 q^{21} + 40 q^{25} + 4 q^{45} + 16 q^{49} + 8 q^{53} - 44 q^{57} + 60 q^{69} - 56 q^{73} - 16 q^{77} + 68 q^{81} + 92 q^{93} - 88 q^{97}+O(q^{100}) $ 16 * q - 16 * q^5 - 4 * q^9 - 20 * q^21 + 40 * q^25 + 4 * q^45 + 16 * q^49 + 8 * q^53 - 44 * q^57 + 60 * q^69 - 56 * q^73 - 16 * q^77 + 68 * q^81 + 92 * q^93 - 88 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2112\mathbb{Z}\right)^\times$.

$n$	$133$	$1409$	$1729$	$2047$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.62968	+	0.586627i	−0.940898	+	0.338689i
$3$	−1.62968	+	0.586627i	−0.940898	+	0.338689i
$4$		0			0
$5$	2.40932			1.07748			0.538741	−	0.842472i	$-0.318900\pi$
$5$	2.40932			1.07748			0.538741	+	0.842472i	$0.318900\pi$
$6$		0			0
$7$		−	1.23607i		−	0.467190i	−0.972334	−	0.233595i	$-0.924951\pi$
$7$		−	1.23607i		−	0.467190i	0.972334	−	0.233595i	$-0.0750489\pi$
$8$		0			0
$9$	2.31174	−	1.91203i	0.770579	−	0.637344i
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$			4.37780i			1.21418i	0.794632	+	0.607092i	$0.207664\pi$
$13$			4.37780i			1.21418i	−0.794632	+	0.607092i	$0.792336\pi$
$14$		0			0
$15$	−3.92643	+	1.41337i	−1.01380	+	0.364931i
$16$		0			0
$17$			2.89940i			0.703209i	0.936149	+	0.351604i	$0.114364\pi$
$17$			2.89940i			0.703209i	−0.936149	+	0.351604i	$0.885636\pi$
$18$		0			0
$19$	−0.564698			−0.129551			−0.0647753	−	0.997900i	$-0.520633\pi$
$19$	−0.564698			−0.129551			−0.0647753	+	0.997900i	$0.520633\pi$
$20$		0			0
$21$	0.725111	+	2.01440i	0.158232	+	0.439578i
$22$		0			0
$23$	−5.96500			−1.24379			−0.621894	−	0.783101i	$-0.713637\pi$
$23$	−5.96500			−1.24379			−0.621894	+	0.783101i	$0.713637\pi$
$24$		0			0
$25$	0.804832			0.160966
$26$		0			0
$27$	−2.64575	+	4.47214i	−0.509175	+	0.860663i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$			1.29888i			0.233286i	0.993174	+	0.116643i	$0.0372134\pi$
$31$			1.29888i			0.233286i	−0.993174	+	0.116643i	$0.962787\pi$
$32$		0			0
$33$	−0.586627	−	1.62968i	−0.102119	−	0.283692i
$34$		0			0
$35$		−	2.97809i		−	0.503388i
$36$		0			0
$37$			5.65147i			0.929095i	0.885548	+	0.464548i	$0.153783\pi$
$37$			5.65147i			0.929095i	−0.885548	+	0.464548i	$0.846217\pi$
$38$		0			0
$39$	−2.56814	−	7.13443i	−0.411231	−	1.14242i
$40$		0			0
$41$			0.662534i			0.103470i	0.998661	+	0.0517352i	$0.0164752\pi$
$41$			0.662534i			0.103470i	−0.998661	+	0.0517352i	$0.983525\pi$
$42$		0			0
$43$	8.87536			1.35348			0.676740	−	0.736222i	$-0.263392\pi$
$43$	8.87536			1.35348			0.676740	+	0.736222i	$0.263392\pi$
$44$		0			0
$45$	5.56972	−	4.60670i	0.830285	−	0.686727i
$46$		0			0
$47$	−5.76026			−0.840221			−0.420110	−	0.907473i	$-0.638009\pi$
$47$	−5.76026			−0.840221			−0.420110	+	0.907473i	$0.638009\pi$
$48$		0			0
$49$	5.47214			0.781734
$50$		0			0
$51$	−1.70087	−	4.72511i	−0.238169	−	0.661648i
$52$		0			0
$53$	9.03280			1.24075			0.620375	−	0.784305i	$-0.286980\pi$
$53$	9.03280			1.24075			0.620375	+	0.784305i	$0.286980\pi$
$54$		0			0
$55$			2.40932i			0.324873i
$56$		0			0
$57$	0.920279	−	0.331267i	0.121894	−	0.0438774i
$58$		0			0
$59$			9.17325i			1.19426i	0.802146	+	0.597128i	$0.203692\pi$
$59$			9.17325i			1.19426i	−0.802146	+	0.597128i	$0.796308\pi$
$60$		0			0
$61$		−	4.34233i		−	0.555979i	−0.960584	−	0.277989i	$-0.910332\pi$
$61$		−	4.34233i		−	0.555979i	0.960584	−	0.277989i	$-0.0896680\pi$
$62$		0			0
$63$	−2.36340	−	2.85746i	−0.297761	−	0.360007i
$64$		0			0
$65$			10.5475i			1.30826i
$66$		0			0
$67$	−5.49624			−0.671472			−0.335736	−	0.941956i	$-0.608985\pi$
$67$	−5.49624			−0.671472			−0.335736	+	0.941956i	$0.608985\pi$
$68$		0			0
$69$	9.72106	−	3.49923i	1.17028	−	0.421258i
$70$		0			0
$71$	11.7638			1.39611			0.698053	−	0.716046i	$-0.254050\pi$
$71$	11.7638			1.39611			0.698053	+	0.716046i	$0.254050\pi$
$72$		0			0
$73$	−12.1432			−1.42126			−0.710629	−	0.703567i	$-0.751590\pi$
$73$	−12.1432			−1.42126			−0.710629	+	0.703567i	$0.751590\pi$
$74$		0			0
$75$	−1.31162	+	0.472136i	−0.151453	+	0.0545176i
$76$		0			0
$77$	1.23607			0.140863
$78$		0			0
$79$		−	2.56066i		−	0.288097i	−0.989571	−	0.144048i	$-0.953988\pi$
$79$		−	2.56066i		−	0.288097i	0.989571	−	0.144048i	$-0.0460121\pi$
$80$		0			0
$81$	1.68826	−	8.84024i	0.187585	−	0.982248i
$82$		0			0
$83$			15.7191i			1.72539i	0.505721	+	0.862697i	$0.331226\pi$
$83$			15.7191i			1.72539i	−0.505721	+	0.862697i	$0.668774\pi$
$84$		0			0
$85$			6.98560i			0.757694i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			0.240205i			0.0254617i	0.999919	+	0.0127309i	$0.00405247\pi$
$89$			0.240205i			0.0254617i	−0.999919	+	0.0127309i	$0.995948\pi$
$90$		0			0
$91$	5.41126			0.567254
$92$		0			0
$93$	−0.761959	−	2.11677i	−0.0790115	−	0.219499i
$94$		0			0
$95$	−1.36054			−0.139588
$96$		0			0
$97$	−15.0956			−1.53273			−0.766364	−	0.642407i	$-0.777936\pi$
$97$	−15.0956			−1.53273			−0.766364	+	0.642407i	$0.777936\pi$
$98$		0			0
$99$	1.91203	+	2.31174i	0.192167	+	0.232338i
$100$		0			0
$101$	11.9224			1.18632			0.593159	−	0.805085i	$-0.297880\pi$
$101$	11.9224			1.18632			0.593159	+	0.805085i	$0.297880\pi$
$102$		0			0
$103$		−	4.00000i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$103$		−	4.00000i		−	0.394132i	0.980390	−	0.197066i	$-0.0631413\pi$
$104$		0			0
$105$	1.74703	+	4.85334i	0.170492	+	0.473637i
$106$		0			0
$107$			7.49405i			0.724477i	0.932085	+	0.362239i	$0.117987\pi$
$107$			7.49405i			0.724477i	−0.932085	+	0.362239i	$0.882013\pi$
$108$		0			0
$109$			6.14781i			0.588853i	0.955674	+	0.294427i	$0.0951287\pi$
$109$			6.14781i			0.588853i	−0.955674	+	0.294427i	$0.904871\pi$
$110$		0			0
$111$	−3.31530	−	9.21010i	−0.314675	−	0.874184i
$112$		0			0
$113$			11.4722i			1.07921i	0.841917	+	0.539607i	$0.181427\pi$
$113$			11.4722i			1.07921i	−0.841917	+	0.539607i	$0.818573\pi$
$114$		0			0
$115$	−14.3716			−1.34016
$116$		0			0
$117$	8.37050	+	10.1203i	0.773853	+	0.935625i
$118$		0			0
$119$	3.58386			0.328532
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	−0.388660	−	1.07972i	−0.0350443	−	0.0973552i
$124$		0			0
$125$	−10.1075			−0.904043
$126$		0			0
$127$		−	11.5826i		−	1.02779i	−0.857854	−	0.513894i	$-0.828203\pi$
$127$		−	11.5826i		−	1.02779i	0.857854	−	0.513894i	$-0.171797\pi$
$128$		0			0
$129$	−14.4640	+	5.20653i	−1.27349	+	0.458409i
$130$		0			0
$131$			18.6592i			1.63026i	0.579277	+	0.815131i	$0.303335\pi$
$131$			18.6592i			1.63026i	−0.579277	+	0.815131i	$0.696665\pi$
$132$		0			0
$133$			0.698005i			0.0605247i
$134$		0			0
$135$	−6.37447	+	10.7748i	−0.548627	+	0.927348i
$136$		0			0
$137$			7.40793i			0.632902i	0.948609	+	0.316451i	$0.102491\pi$
$137$			7.40793i			0.632902i	−0.948609	+	0.316451i	$0.897509\pi$
$138$		0			0
$139$	−7.39389			−0.627142			−0.313571	−	0.949565i	$-0.601525\pi$
$139$	−7.39389			−0.627142			−0.313571	+	0.949565i	$0.601525\pi$
$140$		0			0
$141$	9.38741	−	3.37913i	0.790562	−	0.284574i
$142$		0			0
$143$	−4.37780			−0.366090
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−8.91785	+	3.21010i	−0.735532	+	0.264765i
$148$		0			0
$149$	−17.4502			−1.42958			−0.714789	−	0.699340i	$-0.753477\pi$
$149$	−17.4502			−1.42958			−0.714789	+	0.699340i	$0.753477\pi$
$150$		0			0
$151$			12.6045i			1.02574i	0.858467	+	0.512869i	$0.171418\pi$
$151$			12.6045i			1.02574i	−0.858467	+	0.512869i	$0.828582\pi$
$152$		0			0
$153$	5.54375	+	6.70266i	0.448186	+	0.541878i
$154$		0			0
$155$			3.12942i			0.251361i
$156$		0			0
$157$		−	13.2996i		−	1.06142i	−0.847552	−	0.530712i	$-0.821925\pi$
$157$		−	13.2996i		−	1.06142i	0.847552	−	0.530712i	$-0.178075\pi$
$158$		0			0
$159$	−14.7206	+	5.29888i	−1.16742	+	0.420229i
$160$		0			0
$161$			7.37314i			0.581085i
$162$		0			0
$163$	7.74597			0.606711			0.303355	−	0.952877i	$-0.401893\pi$
$163$	7.74597			0.606711			0.303355	+	0.952877i	$0.401893\pi$
$164$		0			0
$165$	−1.41337	−	3.92643i	−0.110031	−	0.305672i
$166$		0			0
$167$	20.4325			1.58112			0.790558	−	0.612387i	$-0.209790\pi$
$167$	20.4325			1.58112			0.790558	+	0.612387i	$0.209790\pi$
$168$		0			0
$169$	−6.16515			−0.474242
$170$		0			0
$171$	−1.30543	+	1.07972i	−0.0998290	+	0.0825683i
$172$		0			0
$173$	−8.55394			−0.650344			−0.325172	−	0.945655i	$-0.605422\pi$
$173$	−8.55394			−0.650344			−0.325172	+	0.945655i	$0.605422\pi$
$174$		0			0
$175$		−	0.994827i		−	0.0752018i
$176$		0			0
$177$	−5.38128	−	14.9495i	−0.404482	−	1.12367i
$178$		0			0
$179$			12.0737i			0.902430i	0.892415	+	0.451215i	$0.149009\pi$
$179$			12.0737i			0.902430i	−0.892415	+	0.451215i	$0.850991\pi$
$180$		0			0
$181$			1.77907i			0.132237i	0.997812	+	0.0661186i	$0.0210616\pi$
$181$			1.77907i			0.132237i	−0.997812	+	0.0661186i	$0.978938\pi$
$182$		0			0
$183$	2.54733	+	7.07663i	0.188304	+	0.523119i
$184$		0			0
$185$			13.6162i			1.00108i
$186$		0			0
$187$	−2.89940			−0.212025
$188$		0			0
$189$	5.52786	+	3.27033i	0.402093	+	0.237881i
$190$		0			0
$191$	−5.96500			−0.431612			−0.215806	−	0.976436i	$-0.569238\pi$
$191$	−5.96500			−0.431612			−0.215806	+	0.976436i	$0.569238\pi$
$192$		0			0
$193$	9.29078			0.668765			0.334382	−	0.942437i	$-0.391472\pi$
$193$	9.29078			0.668765			0.334382	+	0.942437i	$0.391472\pi$
$194$		0			0
$195$	−6.18747	−	17.1891i	−0.443094	−	1.23094i
$196$		0			0
$197$	23.7191			1.68991			0.844957	−	0.534833i	$-0.179626\pi$
$197$	23.7191			1.68991			0.844957	+	0.534833i	$0.179626\pi$
$198$		0			0
$199$			9.12132i			0.646593i	0.946298	+	0.323297i	$0.104791\pi$
$199$			9.12132i			0.646593i	−0.946298	+	0.323297i	$0.895209\pi$
$200$		0			0
$201$	8.95713	−	3.22424i	0.631787	−	0.227420i
$202$		0			0
$203$		0			0
$204$		0			0
$205$			1.59626i			0.111487i
$206$		0			0
$207$	−13.7895	+	11.4053i	−0.958437	+	0.792721i
$208$		0			0
$209$		−	0.564698i		−	0.0390610i
$210$		0			0
$211$	−2.42757			−0.167121			−0.0835604	−	0.996503i	$-0.526629\pi$
$211$	−2.42757			−0.167121			−0.0835604	+	0.996503i	$0.526629\pi$
$212$		0			0
$213$	−19.1713	+	6.90096i	−1.31359	+	0.472846i
$214$		0			0
$215$	21.3836			1.45835
$216$		0			0
$217$	1.60551			0.108989
$218$		0			0
$219$	19.7896	−	7.12355i	1.33726	−	0.481365i
$220$		0			0
$221$	−12.6930			−0.853825
$222$		0			0
$223$			9.72719i			0.651381i	0.945476	+	0.325690i	$0.105597\pi$
$223$			9.72719i			0.651381i	−0.945476	+	0.325690i	$0.894403\pi$
$224$		0			0
$225$	1.86056	−	1.53886i	0.124037	−	0.102591i
$226$		0			0
$227$		−	21.0437i		−	1.39672i	−0.715748	−	0.698359i	$-0.753914\pi$
$227$		−	21.0437i		−	1.39672i	0.715748	−	0.698359i	$-0.246086\pi$
$228$		0			0
$229$		−	17.3639i		−	1.14744i	−0.819053	−	0.573718i	$-0.805500\pi$
$229$		−	17.3639i		−	1.14744i	0.819053	−	0.573718i	$-0.194500\pi$
$230$		0			0
$231$	−2.01440	+	0.725111i	−0.132538	+	0.0477088i
$232$		0			0
$233$			25.5688i			1.67507i	0.546385	+	0.837534i	$0.316004\pi$
$233$			25.5688i			1.67507i	−0.546385	+	0.837534i	$0.683996\pi$
$234$		0			0
$235$	−13.8783			−0.905322
$236$		0			0
$237$	1.50215	+	4.17307i	0.0975753	+	0.271070i
$238$		0			0
$239$	1.73454			0.112198			0.0560989	−	0.998425i	$-0.482134\pi$
$239$	1.73454			0.112198			0.0560989	+	0.998425i	$0.482134\pi$
$240$		0			0
$241$	29.8285			1.92142			0.960712	−	0.277549i	$-0.0895219\pi$
$241$	29.8285			1.92142			0.960712	+	0.277549i	$0.0895219\pi$
$242$		0			0
$243$	2.43459	+	15.3972i	0.156179	+	0.987729i
$244$		0			0
$245$	13.1841			0.842304
$246$		0			0
$247$		−	2.47214i		−	0.157298i
$248$		0			0
$249$	−9.22124	−	25.6171i	−0.584372	−	1.62342i
$250$		0			0
$251$			1.42451i			0.0899143i	0.998989	+	0.0449571i	$0.0143151\pi$
$251$			1.42451i			0.0899143i	−0.998989	+	0.0449571i	$0.985685\pi$
$252$		0			0
$253$		−	5.96500i		−	0.375016i
$254$		0			0
$255$	−4.09794	−	11.3843i	−0.256623	−	0.712913i
$256$		0			0
$257$		−	2.25879i		−	0.140900i	−0.997515	−	0.0704498i	$-0.977557\pi$
$257$		−	2.25879i		−	0.140900i	0.997515	−	0.0704498i	$-0.0224435\pi$
$258$		0			0
$259$	6.98560			0.434064
$260$		0			0
$261$		0			0
$262$		0			0
$263$	25.1948			1.55358			0.776789	−	0.629761i	$-0.216847\pi$
$263$	25.1948			1.55358			0.776789	+	0.629761i	$0.216847\pi$
$264$		0			0
$265$	21.7629			1.33689
$266$		0			0
$267$	−0.140911	−	0.391459i	−0.00862361	−	0.0239569i
$268$		0			0
$269$	−9.43934			−0.575527			−0.287763	−	0.957702i	$-0.592912\pi$
$269$	−9.43934			−0.575527			−0.287763	+	0.957702i	$0.592912\pi$
$270$		0			0
$271$		−	19.6043i		−	1.19088i	−0.803400	−	0.595439i	$-0.796978\pi$
$271$		−	19.6043i		−	1.19088i	0.803400	−	0.595439i	$-0.203022\pi$
$272$		0			0
$273$	−8.81864	+	3.17439i	−0.533729	+	0.192123i
$274$		0			0
$275$			0.804832i			0.0485332i
$276$		0			0
$277$			28.6118i			1.71912i	0.511038	+	0.859558i	$0.329261\pi$
$277$			28.6118i			1.71912i	−0.511038	+	0.859558i	$0.670739\pi$
$278$		0			0
$279$	2.48350	+	3.00267i	0.148684	+	0.179765i
$280$		0			0
$281$		−	26.0969i		−	1.55681i	−0.627764	−	0.778404i	$-0.716030\pi$
$281$		−	26.0969i		−	1.55681i	0.627764	−	0.778404i	$-0.283970\pi$
$282$		0			0
$283$	−33.2158			−1.97447			−0.987236	−	0.159263i	$-0.949088\pi$
$283$	−33.2158			−1.97447			−0.987236	+	0.159263i	$0.949088\pi$
$284$		0			0
$285$	2.21725	−	0.798129i	0.131338	−	0.0472771i
$286$		0			0
$287$	0.818937			0.0483403
$288$		0			0
$289$	8.59346			0.505498
$290$		0			0
$291$	24.6011	−	8.85549i	1.44214	−	0.519118i
$292$		0			0
$293$	−10.9619			−0.640400			−0.320200	−	0.947350i	$-0.603750\pi$
$293$	−10.9619			−0.640400			−0.320200	+	0.947350i	$0.603750\pi$
$294$		0			0
$295$			22.1013i			1.28679i
$296$		0			0
$297$	−4.47214	−	2.64575i	−0.259500	−	0.153522i
$298$		0			0
$299$		−	26.1136i		−	1.51019i
$300$		0			0
$301$		−	10.9706i		−	0.632332i
$302$		0			0
$303$	−19.4297	+	6.99398i	−1.11621	+	0.401793i
$304$		0			0
$305$		−	10.4621i		−	0.599057i
$306$		0			0
$307$	12.2771			0.700691			0.350346	−	0.936621i	$-0.386064\pi$
$307$	12.2771			0.700691			0.350346	+	0.936621i	$0.386064\pi$
$308$		0			0
$309$	2.34651	+	6.51873i	0.133488	+	0.370838i
$310$		0			0
$311$	1.36361			0.0773233			0.0386617	−	0.999252i	$-0.487691\pi$
$311$	1.36361			0.0773233			0.0386617	+	0.999252i	$0.487691\pi$
$312$		0			0
$313$	−34.5120			−1.95073			−0.975367	−	0.220587i	$-0.929203\pi$
$313$	−34.5120			−1.95073			−0.975367	+	0.220587i	$0.929203\pi$
$314$		0			0
$315$	−5.69420	−	6.88455i	−0.320832	−	0.387901i
$316$		0			0
$317$	0.239865			0.0134721			0.00673607	−	0.999977i	$-0.497856\pi$
$317$	0.239865			0.0134721			0.00673607	+	0.999977i	$0.497856\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.39621	−	12.2129i	−0.245373	−	0.681659i
$322$		0			0
$323$		−	1.63729i		−	0.0911011i
$324$		0			0
$325$			3.52339i			0.195443i
$326$		0			0
$327$	−3.60647	−	10.0190i	−0.199438	−	0.554051i
$328$		0			0
$329$			7.12008i			0.392543i
$330$		0			0
$331$	23.9450			1.31613			0.658067	−	0.752959i	$-0.271374\pi$
$331$	23.9450			1.31613			0.658067	+	0.752959i	$0.271374\pi$
$332$		0			0
$333$	10.8058	+	13.0647i	0.592154	+	0.715942i
$334$		0			0
$335$	−13.2422			−0.723499
$336$		0			0
$337$	8.53357			0.464853			0.232427	−	0.972614i	$-0.425333\pi$
$337$	8.53357			0.464853			0.232427	+	0.972614i	$0.425333\pi$
$338$		0			0
$339$	−6.72990	−	18.6961i	−0.365518	−	1.01543i
$340$		0			0
$341$	−1.29888			−0.0703384
$342$		0			0
$343$		−	15.4164i		−	0.832408i
$344$		0			0
$345$	23.4212	−	8.43077i	1.26095	−	0.453897i
$346$		0			0
$347$			14.0176i			0.752505i	0.926517	+	0.376252i	$0.122787\pi$
$347$			14.0176i			0.752505i	−0.926517	+	0.376252i	$0.877213\pi$
$348$		0			0
$349$		−	13.5429i		−	0.724933i	−0.931997	−	0.362467i	$-0.881935\pi$
$349$		−	13.5429i		−	0.724933i	0.931997	−	0.362467i	$-0.118065\pi$
$350$		0			0
$351$	−19.5781	−	11.5826i	−1.04500	−	0.618232i
$352$		0			0
$353$			28.1645i			1.49904i	0.661980	+	0.749522i	$0.269716\pi$
$353$			28.1645i			1.49904i	−0.661980	+	0.749522i	$0.730284\pi$
$354$		0			0
$355$	28.3428			1.50428
$356$		0			0
$357$	−5.84056	+	2.10239i	−0.309115	+	0.111270i
$358$		0			0
$359$	11.5621			0.610227			0.305113	−	0.952316i	$-0.401306\pi$
$359$	11.5621			0.610227			0.305113	+	0.952316i	$0.401306\pi$
$360$		0			0
$361$	−18.6811			−0.983217
$362$		0			0
$363$	1.62968	−	0.586627i	0.0855362	−	0.0307899i
$364$		0			0
$365$	−29.2570			−1.53138
$366$		0			0
$367$		−	32.0575i		−	1.67339i	−0.547671	−	0.836694i	$-0.684485\pi$
$367$		−	32.0575i		−	1.67339i	0.547671	−	0.836694i	$-0.315515\pi$
$368$		0			0
$369$	1.26679	+	1.53160i	0.0659463	+	0.0797322i
$370$		0			0
$371$		−	11.1652i		−	0.579666i
$372$		0			0
$373$		−	26.5033i		−	1.37229i	−0.727466	−	0.686143i	$-0.759302\pi$
$373$		−	26.5033i		−	1.37229i	0.727466	−	0.686143i	$-0.240698\pi$
$374$		0			0
$375$	16.4720	−	5.92934i	0.850613	−	0.306190i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.520165			−0.0267191			−0.0133595	−	0.999911i	$-0.504253\pi$
$379$	−0.520165			−0.0267191			−0.0133595	+	0.999911i	$0.504253\pi$
$380$		0			0
$381$	6.79465	+	18.8759i	0.348101	+	0.967043i
$382$		0			0
$383$	−0.237133			−0.0121169			−0.00605845	−	0.999982i	$-0.501928\pi$
$383$	−0.237133			−0.0121169			−0.00605845	+	0.999982i	$0.501928\pi$
$384$		0			0
$385$	2.97809			0.151777
$386$		0			0
$387$	20.5175	−	16.9700i	1.04296	−	0.862633i
$388$		0			0
$389$	−27.8913			−1.41415			−0.707073	−	0.707140i	$-0.749985\pi$
$389$	−27.8913			−1.41415			−0.707073	+	0.707140i	$0.749985\pi$
$390$		0			0
$391$		−	17.2949i		−	0.874643i
$392$		0			0
$393$	−10.9460	−	30.4086i	−0.552152	−	1.53391i
$394$		0			0
$395$		−	6.16946i		−	0.310419i
$396$		0			0
$397$			15.4919i			0.777518i	0.921340	+	0.388759i	$0.127096\pi$
$397$			15.4919i			0.777518i	−0.921340	+	0.388759i	$0.872904\pi$
$398$		0			0
$399$	−0.409469	−	1.13753i	−0.0204991	−	0.0569476i
$400$		0			0
$401$		−	34.2473i		−	1.71023i	−0.518439	−	0.855115i	$-0.673487\pi$
$401$		−	34.2473i		−	1.71023i	0.518439	−	0.855115i	$-0.326513\pi$
$402$		0			0
$403$	−5.68625			−0.283252
$404$		0			0
$405$	4.06757	−	21.2990i	0.202119	−	1.05835i
$406$		0			0
$407$	−5.65147			−0.280133
$408$		0			0
$409$	8.33030			0.411907			0.205953	−	0.978562i	$-0.433970\pi$
$409$	8.33030			0.411907			0.205953	+	0.978562i	$0.433970\pi$
$410$		0			0
$411$	−4.34569	−	12.0726i	−0.214357	−	0.595496i
$412$		0			0
$413$	11.3388			0.557944
$414$		0			0
$415$			37.8723i			1.85908i
$416$		0			0
$417$	12.0497	−	4.33746i	0.590077	−	0.212406i
$418$		0			0
$419$		−	0.537730i		−	0.0262698i	−0.999914	−	0.0131349i	$-0.995819\pi$
$419$		−	0.537730i		−	0.0262698i	0.999914	−	0.0131349i	$-0.00418110\pi$
$420$		0			0
$421$		−	33.7450i		−	1.64463i	−0.569032	−	0.822315i	$-0.692682\pi$
$421$		−	33.7450i		−	1.64463i	0.569032	−	0.822315i	$-0.307318\pi$
$422$		0			0
$423$	−13.3162	+	11.0138i	−0.647457	+	0.535510i
$424$		0			0
$425$			2.33353i			0.113193i
$426$		0			0
$427$	−5.36742			−0.259747
$428$		0			0
$429$	7.13443	−	2.56814i	0.344454	−	0.123991i
$430$		0			0
$431$	8.50254			0.409553			0.204776	−	0.978809i	$-0.434353\pi$
$431$	8.50254			0.409553			0.204776	+	0.978809i	$0.434353\pi$
$432$		0			0
$433$	−13.8704			−0.666570			−0.333285	−	0.942826i	$-0.608157\pi$
$433$	−13.8704			−0.666570			−0.333285	+	0.942826i	$0.608157\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.36842			0.161133
$438$		0			0
$439$		−	29.8380i		−	1.42409i	−0.702134	−	0.712045i	$-0.747769\pi$
$439$		−	29.8380i		−	1.42409i	0.702134	−	0.712045i	$-0.252231\pi$
$440$		0			0
$441$	12.6501	−	10.4629i	0.602388	−	0.498233i
$442$		0			0
$443$			1.88247i			0.0894390i	0.999000	+	0.0447195i	$0.0142394\pi$
$443$			1.88247i			0.0894390i	−0.999000	+	0.0447195i	$0.985761\pi$
$444$		0			0
$445$			0.578732i			0.0274345i
$446$		0			0
$447$	28.4383	−	10.2368i	1.34509	−	0.484183i
$448$		0			0
$449$		−	29.0543i		−	1.37116i	−0.727998	−	0.685580i	$-0.759549\pi$
$449$		−	29.0543i		−	1.37116i	0.727998	−	0.685580i	$-0.240451\pi$
$450$		0			0
$451$	−0.662534			−0.0311975
$452$		0			0
$453$	−7.39413	−	20.5413i	−0.347407	−	0.965116i
$454$		0			0
$455$	13.0375			0.611206
$456$		0			0
$457$	20.5978			0.963523			0.481761	−	0.876302i	$-0.339997\pi$
$457$	20.5978			0.963523			0.481761	+	0.876302i	$0.339997\pi$
$458$		0			0
$459$	−12.9665	−	7.67110i	−0.605226	−	0.358056i
$460$		0			0
$461$	−10.2209			−0.476034			−0.238017	−	0.971261i	$-0.576497\pi$
$461$	−10.2209			−0.476034			−0.238017	+	0.971261i	$0.576497\pi$
$462$		0			0
$463$		−	29.3044i		−	1.36189i	−0.732333	−	0.680946i	$-0.761569\pi$
$463$		−	29.3044i		−	1.36189i	0.732333	−	0.680946i	$-0.238431\pi$
$464$		0			0
$465$	−1.83580	−	5.09997i	−0.0851334	−	0.236506i
$466$		0			0
$467$		−	6.11753i		−	0.283085i	−0.989932	−	0.141543i	$-0.954794\pi$
$467$		−	6.11753i		−	0.283085i	0.989932	−	0.141543i	$-0.0452062\pi$
$468$		0			0
$469$			6.79372i			0.313705i
$470$		0			0
$471$	7.80190	+	21.6741i	0.359493	+	0.998691i
$472$		0			0
$473$			8.87536i			0.408090i
$474$		0			0
$475$	−0.454487			−0.0208533
$476$		0			0
$477$	20.8815	−	17.2710i	0.956096	−	0.790785i
$478$		0			0
$479$	−7.13225			−0.325881			−0.162940	−	0.986636i	$-0.552098\pi$
$479$	−7.13225			−0.325881			−0.162940	+	0.986636i	$0.552098\pi$
$480$		0			0
$481$	−24.7410			−1.12809
$482$		0			0
$483$	−4.32528	−	12.0159i	−0.196807	−	0.546742i
$484$		0			0
$485$	−36.3702			−1.65149
$486$		0			0
$487$			31.2226i			1.41483i	0.706797	+	0.707417i	$0.250139\pi$
$487$			31.2226i			1.41483i	−0.706797	+	0.707417i	$0.749861\pi$
$488$		0			0
$489$	−12.6235	+	4.54399i	−0.570853	+	0.205486i
$490$		0			0
$491$		−	21.2307i		−	0.958130i	−0.877779	−	0.479065i	$-0.840976\pi$
$491$		−	21.2307i		−	0.958130i	0.877779	−	0.479065i	$-0.159024\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	4.60670	+	5.56972i	0.207056	+	0.250340i
$496$		0			0
$497$		−	14.5409i		−	0.652247i
$498$		0			0
$499$	3.03271			0.135763			0.0678814	−	0.997693i	$-0.478376\pi$
$499$	3.03271			0.135763			0.0678814	+	0.997693i	$0.478376\pi$
$500$		0			0
$501$	−33.2986	+	11.9863i	−1.48767	+	0.535507i
$502$		0			0
$503$	39.5657			1.76415			0.882074	−	0.471111i	$-0.156147\pi$
$503$	39.5657			1.76415			0.882074	+	0.471111i	$0.156147\pi$
$504$		0			0
$505$	28.7248			1.27824
$506$		0			0
$507$	10.0472	−	3.61664i	0.446214	−	0.160621i
$508$		0			0
$509$	−19.1900			−0.850582			−0.425291	−	0.905057i	$-0.639828\pi$
$509$	−19.1900			−0.850582			−0.425291	+	0.905057i	$0.639828\pi$
$510$		0			0
$511$			15.0099i			0.663997i
$512$		0			0
$513$	1.49405	−	2.52541i	0.0659639	−	0.111499i
$514$		0			0
$515$		−	9.63729i		−	0.424670i
$516$		0			0
$517$		−	5.76026i		−	0.253336i
$518$		0			0
$519$	13.9402	−	5.01797i	0.611907	−	0.220264i
$520$		0			0
$521$			14.0966i			0.617584i	0.951130	+	0.308792i	$0.0999247\pi$
$521$			14.0966i			0.617584i	−0.951130	+	0.308792i	$0.900075\pi$
$522$		0			0
$523$	28.8546			1.26172			0.630861	−	0.775896i	$-0.282702\pi$
$523$	28.8546			1.26172			0.630861	+	0.775896i	$0.282702\pi$
$524$		0			0
$525$	0.583592	+	1.62125i	0.0254700	+	0.0707573i
$526$		0			0
$527$	−3.76598			−0.164049
$528$		0			0
$529$	12.5812			0.547009
$530$		0			0
$531$	17.5396	+	21.2062i	0.761152	+	0.920269i
$532$		0			0
$533$	−2.90044			−0.125632
$534$		0			0
$535$			18.0556i			0.780611i
$536$		0			0
$537$	−7.08276	−	19.6763i	−0.305643	−	0.849095i
$538$		0			0
$539$			5.47214i			0.235702i
$540$		0			0
$541$			26.1709i			1.12517i	0.826738	+	0.562587i	$0.190194\pi$
$541$			26.1709i			1.12517i	−0.826738	+	0.562587i	$0.809806\pi$
$542$		0			0
$543$	−1.04365	−	2.89932i	−0.0447873	−	0.124422i
$544$		0			0
$545$			14.8121i			0.634479i
$546$		0			0
$547$	12.6292			0.539985			0.269992	−	0.962862i	$-0.412979\pi$
$547$	12.6292			0.539985			0.269992	+	0.962862i	$0.412979\pi$
$548$		0			0
$549$	−8.30268	−	10.0383i	−0.354350	−	0.428426i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.16515			−0.134596
$554$		0			0
$555$	−7.98763	−	22.1901i	−0.339056	−	0.941917i
$556$		0			0
$557$	−42.7552			−1.81159			−0.905797	−	0.423711i	$-0.860727\pi$
$557$	−42.7552			−1.81159			−0.905797	+	0.423711i	$0.860727\pi$
$558$		0			0
$559$			38.8546i			1.64337i
$560$		0			0
$561$	4.72511	−	1.70087i	0.199494	−	0.0718107i
$562$		0			0
$563$			24.1312i			1.01701i	0.861060	+	0.508504i	$0.169801\pi$
$563$			24.1312i			1.01701i	−0.861060	+	0.508504i	$0.830199\pi$
$564$		0			0
$565$			27.6402i			1.16283i
$566$		0			0
$567$	−10.9271	−	2.08681i	−0.458896	−	0.0876377i
$568$		0			0
$569$		−	39.1086i		−	1.63952i	−0.572708	−	0.819759i	$-0.694107\pi$
$569$		−	39.1086i		−	1.63952i	0.572708	−	0.819759i	$-0.305893\pi$
$570$		0			0
$571$	6.01143			0.251571			0.125785	−	0.992057i	$-0.459855\pi$
$571$	6.01143			0.251571			0.125785	+	0.992057i	$0.459855\pi$
$572$		0			0
$573$	9.72106	−	3.49923i	0.406103	−	0.146182i
$574$		0			0
$575$	−4.80082			−0.200208
$576$		0			0
$577$	1.70958			0.0711708			0.0355854	−	0.999367i	$-0.488670\pi$
$577$	1.70958			0.0711708			0.0355854	+	0.999367i	$0.488670\pi$
$578$		0			0
$579$	−15.1410	+	5.45022i	−0.629240	+	0.226503i
$580$		0			0
$581$	19.4299			0.806086
$582$		0			0
$583$			9.03280i			0.374100i
$584$		0			0
$585$	20.1672	+	24.3831i	0.833812	+	1.00812i
$586$		0			0
$587$		−	10.3627i		−	0.427715i	−0.976865	−	0.213857i	$-0.931397\pi$
$587$		−	10.3627i		−	0.427715i	0.976865	−	0.213857i	$-0.0686028\pi$
$588$		0			0
$589$		−	0.733476i		−	0.0302224i
$590$		0			0
$591$	−38.6546	+	13.9143i	−1.59004	+	0.572356i
$592$		0			0
$593$			12.2147i			0.501599i	0.968039	+	0.250800i	$0.0806935\pi$
$593$			12.2147i			0.501599i	−0.968039	+	0.250800i	$0.919307\pi$
$594$		0			0
$595$	8.63467			0.353987
$596$		0			0
$597$	−5.35081	−	14.8649i	−0.218994	−	0.608379i
$598$		0			0
$599$	26.8553			1.09728			0.548640	−	0.836059i	$-0.315146\pi$
$599$	26.8553			1.09728			0.548640	+	0.836059i	$0.315146\pi$
$600$		0			0
$601$	11.1475			0.454718			0.227359	−	0.973811i	$-0.426991\pi$
$601$	11.1475			0.454718			0.227359	+	0.973811i	$0.426991\pi$
$602$		0			0
$603$	−12.7059	+	10.5090i	−0.517423	+	0.427959i
$604$		0			0
$605$	−2.40932			−0.0979529
$606$		0			0
$607$		−	21.0328i		−	0.853695i	−0.904324	−	0.426847i	$-0.859624\pi$
$607$		−	21.0328i		−	0.853695i	0.904324	−	0.426847i	$-0.140376\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	25.2173i		−	1.02018i
$612$		0			0
$613$		−	23.3705i		−	0.943925i	−0.881619	−	0.471962i	$-0.843546\pi$
$613$		−	23.3705i		−	0.943925i	0.881619	−	0.471962i	$-0.156454\pi$
$614$		0			0
$615$	−0.936408	−	2.60140i	−0.0377596	−	0.104898i
$616$		0			0
$617$			40.0023i			1.61043i	0.592982	+	0.805216i	$0.297951\pi$
$617$			40.0023i			1.61043i	−0.592982	+	0.805216i	$0.702049\pi$
$618$		0			0
$619$	14.0342			0.564084			0.282042	−	0.959402i	$-0.408988\pi$
$619$	14.0342			0.564084			0.282042	+	0.959402i	$0.408988\pi$
$620$		0			0
$621$	15.7819	−	26.6763i	0.633306	−	1.07048i
$622$		0			0
$623$	0.296910			0.0118955
$624$		0			0
$625$	−28.3764			−1.13506
$626$		0			0
$627$	0.331267	+	0.920279i	0.0132295	+	0.0367524i
$628$		0			0
$629$	−16.3859			−0.653348
$630$		0			0
$631$			3.64539i			0.145121i	0.997364	+	0.0725603i	$0.0231170\pi$
$631$			3.64539i			0.145121i	−0.997364	+	0.0725603i	$0.976883\pi$
$632$		0			0
$633$	3.95617	−	1.42408i	0.157244	−	0.0566020i
$634$		0			0
$635$		−	27.9062i		−	1.10742i
$636$		0			0
$637$			23.9559i			0.949168i
$638$		0			0
$639$	27.1948	−	22.4928i	1.07581	−	0.889801i
$640$		0			0
$641$			30.3085i			1.19711i	0.801081	+	0.598556i	$0.204259\pi$
$641$			30.3085i			1.19711i	−0.801081	+	0.598556i	$0.795741\pi$
$642$		0			0
$643$	−42.9731			−1.69469			−0.847347	−	0.531039i	$-0.821802\pi$
$643$	−42.9731			−1.69469			−0.847347	+	0.531039i	$0.821802\pi$
$644$		0			0
$645$	−34.8485	+	12.5442i	−1.37216	+	0.493927i
$646$		0			0
$647$	−17.1713			−0.675072			−0.337536	−	0.941313i	$-0.609594\pi$
$647$	−17.1713			−0.675072			−0.337536	+	0.941313i	$0.609594\pi$
$648$		0			0
$649$	−9.17325			−0.360082
$650$		0			0
$651$	−2.61647	+	0.941833i	−0.102547	+	0.0369134i
$652$		0			0
$653$	8.08458			0.316374			0.158187	−	0.987409i	$-0.449435\pi$
$653$	8.08458			0.316374			0.158187	+	0.987409i	$0.449435\pi$
$654$		0			0
$655$			44.9560i			1.75658i
$656$		0			0
$657$	−28.0720	+	23.2183i	−1.09519	+	0.905831i
$658$		0			0
$659$			10.6330i			0.414202i	0.978320	+	0.207101i	$0.0664029\pi$
$659$			10.6330i			0.414202i	−0.978320	+	0.207101i	$0.933597\pi$
$660$		0			0
$661$			11.9036i			0.462996i	0.972835	+	0.231498i	$0.0743627\pi$
$661$			11.9036i			0.462996i	−0.972835	+	0.231498i	$0.925637\pi$
$662$		0			0
$663$	20.6856	−	7.44606i	0.803362	−	0.289181i
$664$		0			0
$665$			1.68172i			0.0652143i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−5.70623	−	15.8522i	−0.220616	−	0.612883i
$670$		0			0
$671$	4.34233			0.167634
$672$		0			0
$673$	10.0877			0.388851			0.194425	−	0.980917i	$-0.437716\pi$
$673$	10.0877			0.388851			0.194425	+	0.980917i	$0.437716\pi$
$674$		0			0
$675$	−2.12938	+	3.59932i	−0.0819600	+	0.138538i
$676$		0			0
$677$	20.0776			0.771646			0.385823	−	0.922573i	$-0.373917\pi$
$677$	20.0776			0.771646			0.385823	+	0.922573i	$0.373917\pi$
$678$		0			0
$679$			18.6592i			0.716074i
$680$		0			0
$681$	12.3448	+	34.2945i	0.473053	+	1.31417i
$682$		0			0
$683$			3.84471i			0.147114i	0.997291	+	0.0735570i	$0.0234351\pi$
$683$			3.84471i			0.147114i	−0.997291	+	0.0735570i	$0.976565\pi$
$684$		0			0
$685$			17.8481i			0.681940i
$686$		0			0
$687$	10.1861	+	28.2976i	0.388624	+	1.07962i
$688$		0			0
$689$			39.5438i			1.50650i
$690$		0			0
$691$	−34.8446			−1.32555			−0.662775	−	0.748819i	$-0.730621\pi$
$691$	−34.8446			−1.32555			−0.662775	+	0.748819i	$0.730621\pi$
$692$		0			0
$693$	2.85746	−	2.36340i	0.108546	−	0.0897782i
$694$		0			0
$695$	−17.8143			−0.675734
$696$		0			0
$697$	−1.92095			−0.0727613
$698$		0			0
$699$	−14.9994	−	41.6691i	−0.567328	−	1.57607i
$700$		0			0
$701$	22.7904			0.860781			0.430391	−	0.902643i	$-0.358376\pi$
$701$	22.7904			0.860781			0.430391	+	0.902643i	$0.358376\pi$
$702$		0			0
$703$		−	3.19137i		−	0.120365i
$704$		0			0
$705$	22.6173	−	8.14140i	0.851816	−	0.306623i
$706$		0			0
$707$		−	14.7368i		−	0.554236i
$708$		0			0
$709$			0.147341i			0.00553351i	0.999996	+	0.00276675i	$0.000880686\pi$
$709$			0.147341i			0.00553351i	−0.999996	+	0.00276675i	$0.999119\pi$
$710$		0			0
$711$	−4.89607	−	5.91958i	−0.183617	−	0.222001i
$712$		0			0
$713$		−	7.74783i		−	0.290158i
$714$		0			0
$715$	−10.5475			−0.394455
$716$		0			0
$717$	−2.82675	+	1.01753i	−0.105567	+	0.0380002i
$718$		0			0
$719$	23.1877			0.864754			0.432377	−	0.901693i	$-0.357675\pi$
$719$	23.1877			0.864754			0.432377	+	0.901693i	$0.357675\pi$
$720$		0			0
$721$	−4.94427			−0.184134
$722$		0			0
$723$	−48.6110	+	17.4982i	−1.80786	+	0.650765i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	5.84450i		−	0.216761i	−0.994110	−	0.108380i	$-0.965434\pi$
$727$		−	5.84450i		−	0.216761i	0.994110	−	0.108380i	$-0.0345664\pi$
$728$		0			0
$729$	−13.0000	−	23.6643i	−0.481481	−	0.876456i
$730$		0			0
$731$			25.7333i			0.951779i
$732$		0			0
$733$		−	22.4806i		−	0.830340i	−0.909744	−	0.415170i	$-0.863722\pi$
$733$		−	22.4806i		−	0.830340i	0.909744	−	0.415170i	$-0.136278\pi$
$734$		0			0
$735$	−21.4860	+	7.73417i	−0.792522	+	0.285279i
$736$		0			0
$737$		−	5.49624i		−	0.202456i
$738$		0			0
$739$	−0.436363			−0.0160519			−0.00802593	−	0.999968i	$-0.502555\pi$
$739$	−0.436363			−0.0160519			−0.00802593	+	0.999968i	$0.502555\pi$
$740$		0			0
$741$	1.45022	+	4.02880i	0.0532752	+	0.148002i
$742$		0			0
$743$	44.7117			1.64031			0.820157	−	0.572138i	$-0.193886\pi$
$743$	44.7117			1.64031			0.820157	+	0.572138i	$0.193886\pi$
$744$		0			0
$745$	−42.0432			−1.54034
$746$		0			0
$747$	30.0554	+	36.3384i	1.09967	+	1.32955i
$748$		0			0
$749$	9.26316			0.338468
$750$		0			0
$751$		−	36.1669i		−	1.31975i	−0.751376	−	0.659875i	$-0.770609\pi$
$751$		−	36.1669i		−	1.31975i	0.751376	−	0.659875i	$-0.229391\pi$
$752$		0			0
$753$	−0.835656	−	2.32150i	−0.0304530	−	0.0846002i
$754$		0			0
$755$			30.3683i			1.10521i
$756$		0			0
$757$		−	35.8890i		−	1.30441i	−0.758044	−	0.652204i	$-0.773845\pi$
$757$		−	35.8890i		−	1.30441i	0.758044	−	0.652204i	$-0.226155\pi$
$758$		0			0
$759$	3.49923	+	9.72106i	0.127014	+	0.352852i
$760$		0			0
$761$		−	20.4363i		−	0.740816i	−0.928869	−	0.370408i	$-0.879218\pi$
$761$		−	20.4363i		−	0.740816i	0.928869	−	0.370408i	$-0.120782\pi$
$762$		0			0
$763$	7.59911			0.275106
$764$		0			0
$765$	13.3567	+	16.1489i	0.482912	+	0.583864i
$766$		0			0
$767$	−40.1587			−1.45005
$768$		0			0
$769$	7.78914			0.280883			0.140442	−	0.990089i	$-0.455148\pi$
$769$	7.78914			0.280883			0.140442	+	0.990089i	$0.455148\pi$
$770$		0			0
$771$	1.32507	+	3.68112i	0.0477212	+	0.132572i
$772$		0			0
$773$	20.8692			0.750613			0.375306	−	0.926901i	$-0.377537\pi$
$773$	20.8692			0.750613			0.375306	+	0.926901i	$0.377537\pi$
$774$		0			0
$775$			1.04538i			0.0375512i
$776$		0			0
$777$	−11.3843	+	4.09794i	−0.408410	+	0.147013i
$778$		0			0
$779$		−	0.374132i		−	0.0134047i
$780$		0			0
$781$			11.7638i			0.420942i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	32.0430i		−	1.14366i
$786$		0			0
$787$	−5.81576			−0.207309			−0.103655	−	0.994613i	$-0.533054\pi$
$787$	−5.81576			−0.207309			−0.103655	+	0.994613i	$0.533054\pi$
$788$		0			0
$789$	−41.0596	+	14.7800i	−1.46176	+	0.526180i
$790$		0			0
$791$	14.1804			0.504197
$792$		0			0
$793$	19.0099			0.675060
$794$		0			0
$795$	−35.4667	+	12.7667i	−1.25787	+	0.452789i
$796$		0			0
$797$	21.9648			0.778033			0.389017	−	0.921231i	$-0.372815\pi$
$797$	21.9648			0.778033			0.389017	+	0.921231i	$0.372815\pi$
$798$		0			0
$799$		−	16.7013i		−	0.590851i
$800$		0			0
$801$	0.459280	+	0.555292i	0.0162279	+	0.0196203i
$802$		0			0
$803$		−	12.1432i		−	0.428526i
$804$		0			0
$805$			17.7643i			0.626108i
$806$		0			0
$807$	15.3831	−	5.53737i	0.541512	−	0.194925i
$808$		0			0
$809$			4.31734i			0.151789i	0.997116	+	0.0758947i	$0.0241813\pi$
$809$			4.31734i			0.151789i	−0.997116	+	0.0758947i	$0.975819\pi$
$810$		0			0
$811$	26.3156			0.924067			0.462033	−	0.886863i	$-0.347120\pi$
$811$	26.3156			0.924067			0.462033	+	0.886863i	$0.347120\pi$
$812$		0			0
$813$	11.5004	+	31.9489i	0.403338	+	1.12050i
$814$		0			0
$815$	18.6625			0.653720
$816$		0			0
$817$	−5.01190			−0.175344
$818$		0			0
$819$	12.5094	−	10.3465i	0.437114	−	0.361536i
$820$		0			0
$821$	21.3212			0.744113			0.372057	−	0.928210i	$-0.378653\pi$
$821$	21.3212			0.744113			0.372057	+	0.928210i	$0.378653\pi$
$822$		0			0
$823$			52.3964i			1.82642i	0.407485	+	0.913212i	$0.366406\pi$
$823$			52.3964i			1.82642i	−0.407485	+	0.913212i	$0.633594\pi$
$824$		0			0
$825$	−0.472136	−	1.31162i	−0.0164377	−	0.0456648i
$826$		0			0
$827$			35.6197i			1.23862i	0.785148	+	0.619309i	$0.212587\pi$
$827$			35.6197i			1.23862i	−0.785148	+	0.619309i	$0.787413\pi$
$828$		0			0
$829$			32.6663i			1.13455i	0.823530	+	0.567273i	$0.192002\pi$
$829$			32.6663i			1.13455i	−0.823530	+	0.567273i	$0.807998\pi$
$830$		0			0
$831$	−16.7844	−	46.6282i	−0.582246	−	1.61751i
$832$		0			0
$833$			15.8659i			0.549722i
$834$		0			0
$835$	49.2286			1.70362
$836$		0			0
$837$	−5.80878	−	3.43652i	−0.200781	−	0.118783i
$838$		0			0
$839$	−23.1438			−0.799014			−0.399507	−	0.916730i	$-0.630819\pi$
$839$	−23.1438			−0.799014			−0.399507	+	0.916730i	$0.630819\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	15.3091	+	42.5296i	0.527274	+	1.46480i
$844$		0			0
$845$	−14.8538			−0.510987
$846$		0			0
$847$			1.23607i			0.0424718i
$848$		0			0
$849$	54.1312	−	19.4853i	1.85778	−	0.668732i
$850$		0			0
$851$		−	33.7110i		−	1.15560i
$852$		0			0
$853$			1.93689i			0.0663177i	0.999450	+	0.0331589i	$0.0105567\pi$
$853$			1.93689i			0.0663177i	−0.999450	+	0.0331589i	$0.989443\pi$
$854$		0			0
$855$	−3.14521	+	2.60140i	−0.107564	+	0.0889658i
$856$		0			0
$857$		−	52.8979i		−	1.80696i	−0.428634	−	0.903478i	$-0.641005\pi$
$857$		−	52.8979i		−	1.80696i	0.428634	−	0.903478i	$-0.358995\pi$
$858$		0			0
$859$	28.5177			0.973012			0.486506	−	0.873677i	$-0.338271\pi$
$859$	28.5177			0.973012			0.486506	+	0.873677i	$0.338271\pi$
$860$		0			0
$861$	−1.33461	+	0.480411i	−0.0454833	+	0.0163723i
$862$		0			0
$863$	−7.46908			−0.254250			−0.127125	−	0.991887i	$-0.540575\pi$
$863$	−7.46908			−0.254250			−0.127125	+	0.991887i	$0.540575\pi$
$864$		0			0
$865$	−20.6092			−0.700733
$866$		0			0
$867$	−14.0046	+	5.04115i	−0.475622	+	0.171207i
$868$		0			0
$869$	2.56066			0.0868645
$870$		0			0
$871$		−	24.0614i		−	0.815291i
$872$		0			0
$873$	−34.8971	+	28.8633i	−1.18109	+	0.976875i
$874$		0			0
$875$			12.4936i			0.422360i
$876$		0			0
$877$		−	56.4567i		−	1.90641i	−0.302327	−	0.953204i	$-0.597763\pi$
$877$		−	56.4567i		−	1.90641i	0.302327	−	0.953204i	$-0.402237\pi$
$878$		0			0
$879$	17.8644	−	6.43053i	0.602551	−	0.216897i
$880$		0			0
$881$			20.9967i			0.707398i	0.935359	+	0.353699i	$0.115076\pi$
$881$			20.9967i			0.707398i	−0.935359	+	0.353699i	$0.884924\pi$
$882$		0			0
$883$	−13.9224			−0.468525			−0.234263	−	0.972173i	$-0.575268\pi$
$883$	−13.9224			−0.468525			−0.234263	+	0.972173i	$0.575268\pi$
$884$		0			0
$885$	−12.9652	−	36.0182i	−0.435821	−	1.21074i
$886$		0			0
$887$	−25.0823			−0.842180			−0.421090	−	0.907019i	$-0.638352\pi$
$887$	−25.0823			−0.842180			−0.421090	+	0.907019i	$0.638352\pi$
$888$		0			0
$889$	−14.3169			−0.480172
$890$		0			0
$891$	8.84024	+	1.68826i	0.296159	+	0.0565589i
$892$		0			0
$893$	3.25281			0.108851
$894$		0			0
$895$			29.0894i			0.972352i
$896$		0			0
$897$	15.3189	+	42.5569i	0.511484	+	1.42093i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			26.1897i			0.872506i
$902$		0			0
$903$	6.43562	+	17.8785i	0.214164	+	0.594960i
$904$		0			0
$905$			4.28635i			0.142483i
$906$		0			0
$907$	−19.8459			−0.658973			−0.329486	−	0.944160i	$-0.606876\pi$
$907$	−19.8459			−0.658973			−0.329486	+	0.944160i	$0.606876\pi$
$908$		0			0
$909$	27.5614	−	22.7959i	0.914153	−	0.756093i
$910$		0			0
$911$	−53.4288			−1.77018			−0.885088	−	0.465424i	$-0.845902\pi$
$911$	−53.4288			−1.77018			−0.885088	+	0.465424i	$0.845902\pi$
$912$		0			0
$913$	−15.7191			−0.520226
$914$		0			0
$915$	6.13733	+	17.0499i	0.202894	+	0.563651i
$916$		0			0
$917$	23.0640			0.761642
$918$		0			0
$919$		−	45.4287i		−	1.49855i	−0.662257	−	0.749277i	$-0.730401\pi$
$919$		−	45.4287i		−	1.49855i	0.662257	−	0.749277i	$-0.269599\pi$
$920$		0			0
$921$	−20.0078	+	7.20208i	−0.659279	+	0.237317i
$922$		0			0
$923$			51.4996i			1.69513i
$924$		0			0
$925$			4.54848i			0.149553i
$926$		0			0
$927$	−7.64813	−	9.24695i	−0.251198	−	0.303710i
$928$		0			0
$929$		−	44.4489i		−	1.45832i	−0.684342	−	0.729161i	$-0.739911\pi$
$929$		−	44.4489i		−	1.45832i	0.684342	−	0.729161i	$-0.260089\pi$
$930$		0			0
$931$	−3.09010			−0.101274
$932$		0			0
$933$	−2.22226	+	0.799931i	−0.0727534	+	0.0261886i
$934$		0			0
$935$	−6.98560			−0.228453
$936$		0			0
$937$	−1.31042			−0.0428096			−0.0214048	−	0.999771i	$-0.506814\pi$
$937$	−1.31042			−0.0428096			−0.0214048	+	0.999771i	$0.506814\pi$
$938$		0			0
$939$	56.2437	−	20.2457i	1.83544	−	0.660693i
$940$		0			0
$941$	9.76292			0.318262			0.159131	−	0.987257i	$-0.449131\pi$
$941$	9.76292			0.318262			0.159131	+	0.987257i	$0.449131\pi$
$942$		0			0
$943$		−	3.95201i		−	0.128695i
$944$		0			0
$945$	13.3184	+	7.87927i	0.433248	+	0.256313i
$946$		0			0
$947$		−	39.2254i		−	1.27465i	−0.770593	−	0.637327i	$-0.780040\pi$
$947$		−	39.2254i		−	1.27465i	0.770593	−	0.637327i	$-0.219960\pi$
$948$		0			0
$949$		−	53.1607i		−	1.72567i
$950$		0			0
$951$	−0.390904	+	0.140711i	−0.0126759	+	0.00456287i
$952$		0			0
$953$		−	19.9521i		−	0.646313i	−0.946346	−	0.323156i	$-0.895256\pi$
$953$		−	19.9521i		−	0.646313i	0.946346	−	0.323156i	$-0.104744\pi$
$954$		0			0
$955$	−14.3716			−0.465054
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.15670			0.295685
$960$		0			0
$961$	29.3129			0.945578
$962$		0			0
$963$	14.3289	+	17.3243i	0.461741	+	0.558267i
$964$		0			0
$965$	22.3845			0.720582
$966$		0			0
$967$		−	57.4125i		−	1.84626i	−0.384487	−	0.923131i	$-0.625621\pi$
$967$		−	57.4125i		−	1.84626i	0.384487	−	0.923131i	$-0.374379\pi$
$968$		0			0
$969$	0.960477	+	2.66826i	0.0308550	+	0.0857169i
$970$		0			0
$971$			11.4683i			0.368037i	0.982923	+	0.184018i	$0.0589106\pi$
$971$			11.4683i			0.368037i	−0.982923	+	0.184018i	$0.941089\pi$
$972$		0			0
$973$			9.13935i			0.292994i
$974$		0			0
$975$	−2.06692	−	5.74202i	−0.0661943	−	0.183892i
$976$		0			0
$977$		−	11.5431i		−	0.369298i	−0.982805	−	0.184649i	$-0.940885\pi$
$977$		−	11.5431i		−	0.369298i	0.982805	−	0.184649i	$-0.0591148\pi$
$978$		0			0
$979$	−0.240205			−0.00767699
$980$		0			0
$981$	11.7548	+	14.2121i	0.375302	+	0.453758i
$982$		0			0
$983$	−18.6701			−0.595483			−0.297742	−	0.954646i	$-0.596233\pi$
$983$	−18.6701			−0.595483			−0.297742	+	0.954646i	$0.596233\pi$
$984$		0			0
$985$	57.1469			1.82085
$986$		0			0
$987$	−4.17683	−	11.6035i	−0.132950	−	0.369343i
$988$		0			0
$989$	−52.9415			−1.68344
$990$		0			0
$991$		−	55.1411i		−	1.75161i	−0.482662	−	0.875807i	$-0.660330\pi$
$991$		−	55.1411i		−	1.75161i	0.482662	−	0.875807i	$-0.339670\pi$
$992$		0			0
$993$	−39.0227	+	14.0468i	−1.23835	+	0.445761i
$994$		0			0
$995$			21.9762i			0.696692i
$996$		0			0
$997$			8.63395i			0.273440i	0.990610	+	0.136720i	$0.0436560\pi$
$997$			8.63395i			0.273440i	−0.990610	+	0.136720i	$0.956344\pi$
$998$		0			0
$999$	−25.2741	−	14.9524i	−0.799638	−	0.473072i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2112.2.k.k.287.4	yes	16
3.2	odd	2		2112.2.k.l.287.1	yes	16
4.3	odd	2	inner	2112.2.k.k.287.14	yes	16
8.3	odd	2		2112.2.k.l.287.3	yes	16
8.5	even	2		2112.2.k.l.287.13	yes	16
12.11	even	2		2112.2.k.l.287.15	yes	16
24.5	odd	2	inner	2112.2.k.k.287.16	yes	16
24.11	even	2	inner	2112.2.k.k.287.2	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2112.2.k.k.287.2	✓	16	24.11	even	2	inner
2112.2.k.k.287.4	yes	16	1.1	even	1	trivial
2112.2.k.k.287.14	yes	16	4.3	odd	2	inner
2112.2.k.k.287.16	yes	16	24.5	odd	2	inner
2112.2.k.l.287.1	yes	16	3.2	odd	2
2112.2.k.l.287.3	yes	16	8.3	odd	2
2112.2.k.l.287.13	yes	16	8.5	even	2
2112.2.k.l.287.15	yes	16	12.11	even	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:	\( N \)	\(=\)	\( 2112 = 2^{6} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2112.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.62968 + 0.586627i) q^{3} +2.40932 q^{5} -1.23607i q^{7} +(2.31174 - 1.91203i) q^{9} +O(q^{10})\)	\(q+(-1.62968 + 0.586627i) q^{3} +2.40932 q^{5} -1.23607i q^{7} +(2.31174 - 1.91203i) q^{9} +1.00000i q^{11} +4.37780i q^{13} +(-3.92643 + 1.41337i) q^{15} +2.89940i q^{17} -0.564698 q^{19} +(0.725111 + 2.01440i) q^{21} -5.96500 q^{23} +0.804832 q^{25} +(-2.64575 + 4.47214i) q^{27} +1.29888i q^{31} +(-0.586627 - 1.62968i) q^{33} -2.97809i q^{35} +5.65147i q^{37} +(-2.56814 - 7.13443i) q^{39} +0.662534i q^{41} +8.87536 q^{43} +(5.56972 - 4.60670i) q^{45} -5.76026 q^{47} +5.47214 q^{49} +(-1.70087 - 4.72511i) q^{51} +9.03280 q^{53} +2.40932i q^{55} +(0.920279 - 0.331267i) q^{57} +9.17325i q^{59} -4.34233i q^{61} +(-2.36340 - 2.85746i) q^{63} +10.5475i q^{65} -5.49624 q^{67} +(9.72106 - 3.49923i) q^{69} +11.7638 q^{71} -12.1432 q^{73} +(-1.31162 + 0.472136i) q^{75} +1.23607 q^{77} -2.56066i q^{79} +(1.68826 - 8.84024i) q^{81} +15.7191i q^{83} +6.98560i q^{85} +0.240205i q^{89} +5.41126 q^{91} +(-0.761959 - 2.11677i) q^{93} -1.36054 q^{95} -15.0956 q^{97} +(1.91203 + 2.31174i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{5} - 4 q^{9}+O(q^{10}) \) 16 * q - 16 * q^5 - 4 * q^9	\( 16 q - 16 q^{5} - 4 q^{9} - 20 q^{21} + 40 q^{25} + 4 q^{45} + 16 q^{49} + 8 q^{53} - 44 q^{57} + 60 q^{69} - 56 q^{73} - 16 q^{77} + 68 q^{81} + 92 q^{93} - 88 q^{97}+O(q^{100}) \) 16 * q - 16 * q^5 - 4 * q^9 - 20 * q^21 + 40 * q^25 + 4 * q^45 + 16 * q^49 + 8 * q^53 - 44 * q^57 + 60 * q^69 - 56 * q^73 - 16 * q^77 + 68 * q^81 + 92 * q^93 - 88 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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