LMFDB - Embedded newform 4080.2.m.r.2449.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4080,2,Mod(2449,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4080.2449"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.m (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,0,0,0,0,-10,0,-12,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	no
Analytic conductor:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} + 2x^{8} + 14x^{7} + 42x^{6} + 2x^{5} + 10x^{4} + 54x^{3} + 121x^{2} + 44x + 8 $ x^10 - 2x^9 + 2x^8 + 14x^7 + 42x^6 + 2x^5 + 10x^4 + 54x^3 + 121x^2 + 44*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1020)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2449.6
Root			$2.37468 - 2.37468i$ of defining polynomial
Character	$\chi$	$=$	4080.2449
Dual form			4080.2.m.r.2449.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-2.16715 + 0.550869i) q^{5} -2.94396i q^{7} -1.00000 q^{9} -2.22048 q^{11} -2.92695i q^{13} +(-0.550869 - 2.16715i) q^{15} -1.00000i q^{17} +7.45305 q^{19} +2.94396 q^{21} +8.15951i q^{23} +(4.39309 - 2.38763i) q^{25} -1.00000i q^{27} -6.55478 q^{29} +7.49874 q^{31} -2.22048i q^{33} +(1.62173 + 6.38000i) q^{35} +2.74048i q^{37} +2.92695 q^{39} -6.88909 q^{41} -9.78125i q^{43} +(2.16715 - 0.550869i) q^{45} +8.16561i q^{47} -1.66687 q^{49} +1.00000 q^{51} -8.94396i q^{53} +(4.81212 - 1.22320i) q^{55} +7.45305i q^{57} +2.18569 q^{59} -14.7082 q^{61} +2.94396i q^{63} +(1.61237 + 6.34314i) q^{65} +2.18569i q^{67} -8.15951 q^{69} -5.85390 q^{71} +7.81604i q^{73} +(2.38763 + 4.39309i) q^{75} +6.53700i q^{77} -12.9975 q^{79} +1.00000 q^{81} -11.5383i q^{83} +(0.550869 + 2.16715i) q^{85} -6.55478i q^{87} -8.82029 q^{89} -8.61681 q^{91} +7.49874i q^{93} +(-16.1519 + 4.10565i) q^{95} +3.01818i q^{97} +2.22048 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{9} - 12 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} - 12 q^{31} - 4 q^{35} + 28 q^{41} - 30 q^{49} + 10 q^{51} - 26 q^{55} + 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots + 12 q^{99}+O(q^{100}) $ 10 * q - 10 * q^9 - 12 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 - 12 * q^31 - 4 * q^35 + 28 * q^41 - 30 * q^49 + 10 * q^51 - 26 * q^55 + 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 - 8 * q^75 + 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 - 40 * q^91 - 44 * q^95 + 12 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1361$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−2.16715	+	0.550869i	−0.969179	+	0.246356i
$5$	−2.16715	+	0.550869i	−0.969179	+	0.246356i
$6$		0			0
$7$		−	2.94396i		−	1.11271i	−0.830945	−	0.556355i	$-0.812199\pi$
$7$		−	2.94396i		−	1.11271i	0.830945	−	0.556355i	$-0.187801\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−2.22048			−0.669501			−0.334750	−	0.942307i	$-0.608652\pi$
$11$	−2.22048			−0.669501			−0.334750	+	0.942307i	$0.608652\pi$
$12$		0			0
$13$		−	2.92695i		−	0.811790i	−0.913920	−	0.405895i	$-0.866960\pi$
$13$		−	2.92695i		−	0.811790i	0.913920	−	0.405895i	$-0.133040\pi$
$14$		0			0
$15$	−0.550869	−	2.16715i	−0.142234	−	0.559556i
$16$		0			0
$17$		−	1.00000i		−	0.242536i
$17$		−	1.00000i		−	0.242536i
$18$		0			0
$19$	7.45305			1.70985			0.854923	−	0.518755i	$-0.173604\pi$
$19$	7.45305			1.70985			0.854923	+	0.518755i	$0.173604\pi$
$20$		0			0
$21$	2.94396			0.642424
$22$		0			0
$23$			8.15951i			1.70138i	0.525671	+	0.850688i	$0.323814\pi$
$23$			8.15951i			1.70138i	−0.525671	+	0.850688i	$0.676186\pi$
$24$		0			0
$25$	4.39309	−	2.38763i	0.878617	−	0.477527i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−6.55478			−1.21719			−0.608596	−	0.793480i	$-0.708267\pi$
$29$	−6.55478			−1.21719			−0.608596	+	0.793480i	$0.708267\pi$
$30$		0			0
$31$	7.49874			1.34681			0.673407	−	0.739272i	$-0.264830\pi$
$31$	7.49874			1.34681			0.673407	+	0.739272i	$0.264830\pi$
$32$		0			0
$33$		−	2.22048i		−	0.386536i
$34$		0			0
$35$	1.62173	+	6.38000i	0.274123	+	1.07842i
$36$		0			0
$37$			2.74048i			0.450532i	0.974297	+	0.225266i	$0.0723251\pi$
$37$			2.74048i			0.450532i	−0.974297	+	0.225266i	$0.927675\pi$
$38$		0			0
$39$	2.92695			0.468687
$40$		0			0
$41$	−6.88909			−1.07589			−0.537947	−	0.842978i	$-0.680800\pi$
$41$	−6.88909			−1.07589			−0.537947	+	0.842978i	$0.680800\pi$
$42$		0			0
$43$		−	9.78125i		−	1.49163i	−0.666155	−	0.745813i	$-0.732061\pi$
$43$		−	9.78125i		−	1.49163i	0.666155	−	0.745813i	$-0.267939\pi$
$44$		0			0
$45$	2.16715	−	0.550869i	0.323060	−	0.0821187i
$46$		0			0
$47$			8.16561i			1.19108i	0.803327	+	0.595539i	$0.203061\pi$
$47$			8.16561i			1.19108i	−0.803327	+	0.595539i	$0.796939\pi$
$48$		0			0
$49$	−1.66687			−0.238125
$50$		0			0
$51$	1.00000			0.140028
$52$		0			0
$53$		−	8.94396i		−	1.22855i	−0.789093	−	0.614273i	$-0.789449\pi$
$53$		−	8.94396i		−	1.22855i	0.789093	−	0.614273i	$-0.210551\pi$
$54$		0			0
$55$	4.81212	−	1.22320i	0.648866	−	0.164936i
$56$		0			0
$57$			7.45305i			0.987180i
$58$		0			0
$59$	2.18569			0.284553			0.142277	−	0.989827i	$-0.454558\pi$
$59$	2.18569			0.284553			0.142277	+	0.989827i	$0.454558\pi$
$60$		0			0
$61$	−14.7082			−1.88319			−0.941596	−	0.336745i	$-0.890674\pi$
$61$	−14.7082			−1.88319			−0.941596	+	0.336745i	$0.890674\pi$
$62$		0			0
$63$			2.94396i			0.370904i
$64$		0			0
$65$	1.61237	+	6.34314i	0.199989	+	0.786770i
$66$		0			0
$67$			2.18569i			0.267025i	0.991047	+	0.133513i	$0.0426256\pi$
$67$			2.18569i			0.267025i	−0.991047	+	0.133513i	$0.957374\pi$
$68$		0			0
$69$	−8.15951			−0.982290
$70$		0			0
$71$	−5.85390			−0.694730			−0.347365	−	0.937730i	$-0.612924\pi$
$71$	−5.85390			−0.694730			−0.347365	+	0.937730i	$0.612924\pi$
$72$		0			0
$73$			7.81604i			0.914798i	0.889262	+	0.457399i	$0.151219\pi$
$73$			7.81604i			0.914798i	−0.889262	+	0.457399i	$0.848781\pi$
$74$		0			0
$75$	2.38763	+	4.39309i	0.275700	+	0.507270i
$76$		0			0
$77$			6.53700i			0.744960i
$78$		0			0
$79$	−12.9975			−1.46233			−0.731165	−	0.682200i	$-0.761023\pi$
$79$	−12.9975			−1.46233			−0.731165	+	0.682200i	$0.761023\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	11.5383i		−	1.26650i	−0.773949	−	0.633248i	$-0.781721\pi$
$83$		−	11.5383i		−	1.26650i	0.773949	−	0.633248i	$-0.218279\pi$
$84$		0			0
$85$	0.550869	+	2.16715i	0.0597502	+	0.235061i
$86$		0			0
$87$		−	6.55478i		−	0.702747i
$88$		0			0
$89$	−8.82029			−0.934948			−0.467474	−	0.884007i	$-0.654836\pi$
$89$	−8.82029			−0.934948			−0.467474	+	0.884007i	$0.654836\pi$
$90$		0			0
$91$	−8.61681			−0.903287
$92$		0			0
$93$			7.49874i			0.777583i
$94$		0			0
$95$	−16.1519	+	4.10565i	−1.65715	+	0.421231i
$96$		0			0
$97$			3.01818i			0.306450i	0.988191	+	0.153225i	$0.0489659\pi$
$97$			3.01818i			0.306450i	−0.988191	+	0.153225i	$0.951034\pi$
$98$		0			0
$99$	2.22048			0.223167
$100$		0			0
$101$	6.09139			0.606116			0.303058	−	0.952972i	$-0.401992\pi$
$101$	6.09139			0.606116			0.303058	+	0.952972i	$0.401992\pi$
$102$		0			0
$103$			9.78125i			0.963775i	0.876233	+	0.481887i	$0.160049\pi$
$103$			9.78125i			0.963775i	−0.876233	+	0.481887i	$0.839951\pi$
$104$		0			0
$105$	−6.38000	+	1.62173i	−0.622624	+	0.158265i
$106$		0			0
$107$			2.78617i			0.269349i	0.990890	+	0.134675i	$0.0429990\pi$
$107$			2.78617i			0.269349i	−0.990890	+	0.134675i	$0.957001\pi$
$108$		0			0
$109$	−12.9061			−1.23618			−0.618090	−	0.786108i	$-0.712093\pi$
$109$	−12.9061			−1.23618			−0.618090	+	0.786108i	$0.712093\pi$
$110$		0			0
$111$	−2.74048			−0.260115
$112$		0			0
$113$		−	19.2691i		−	1.81268i	−0.422546	−	0.906341i	$-0.638864\pi$
$113$		−	19.2691i		−	1.81268i	0.422546	−	0.906341i	$-0.361136\pi$
$114$		0			0
$115$	−4.49482	−	17.6829i	−0.419145	−	1.64894i
$116$		0			0
$117$			2.92695i			0.270597i
$118$		0			0
$119$	−2.94396			−0.269872
$120$		0			0
$121$	−6.06946			−0.551769
$122$		0			0
$123$		−	6.88909i		−	0.621168i
$124$		0			0
$125$	−8.20521	+	7.59438i	−0.733896	+	0.679262i
$126$		0			0
$127$			16.0705i			1.42603i	0.701149	+	0.713014i	$0.252671\pi$
$127$			16.0705i			1.42603i	−0.701149	+	0.713014i	$0.747329\pi$
$128$		0			0
$129$	9.78125			0.861191
$130$		0			0
$131$	−15.5900			−1.36210			−0.681051	−	0.732236i	$-0.738477\pi$
$131$	−15.5900			−1.36210			−0.681051	+	0.732236i	$0.738477\pi$
$132$		0			0
$133$		−	21.9414i		−	1.90256i
$134$		0			0
$135$	0.550869	+	2.16715i	0.0474113	+	0.186519i
$136$		0			0
$137$		−	11.4091i		−	0.974744i	−0.873195	−	0.487372i	$-0.837956\pi$
$137$		−	11.4091i		−	0.974744i	0.873195	−	0.487372i	$-0.162044\pi$
$138$		0			0
$139$	12.2949			1.04284			0.521418	−	0.853301i	$-0.325403\pi$
$139$	12.2949			1.04284			0.521418	+	0.853301i	$0.325403\pi$
$140$		0			0
$141$	−8.16561			−0.687669
$142$		0			0
$143$			6.49924i			0.543494i
$144$		0			0
$145$	14.2052	−	3.61083i	1.17968	−	0.299863i
$146$		0			0
$147$		−	1.66687i		−	0.137481i
$148$		0			0
$149$	−11.0918			−0.908675			−0.454337	−	0.890830i	$-0.650124\pi$
$149$	−11.0918			−0.908675			−0.454337	+	0.890830i	$0.650124\pi$
$150$		0			0
$151$	−14.7824			−1.20298			−0.601488	−	0.798882i	$-0.705425\pi$
$151$	−14.7824			−1.20298			−0.601488	+	0.798882i	$0.705425\pi$
$152$		0			0
$153$			1.00000i			0.0808452i
$154$		0			0
$155$	−16.2509	+	4.13082i	−1.30530	+	0.331796i
$156$		0			0
$157$		−	5.55361i		−	0.443226i	−0.975135	−	0.221613i	$-0.928868\pi$
$157$		−	5.55361i		−	0.443226i	0.975135	−	0.221613i	$-0.0711322\pi$
$158$		0			0
$159$	8.94396			0.709302
$160$		0			0
$161$	24.0212			1.89314
$162$		0			0
$163$		−	16.4945i		−	1.29195i	−0.763359	−	0.645974i	$-0.776451\pi$
$163$		−	16.4945i		−	1.29195i	0.763359	−	0.645974i	$-0.223549\pi$
$164$		0			0
$165$	1.22320	+	4.81212i	0.0952256	+	0.374623i
$166$		0			0
$167$			5.12590i			0.396654i	0.980136	+	0.198327i	$0.0635508\pi$
$167$			5.12590i			0.396654i	−0.980136	+	0.198327i	$0.936449\pi$
$168$		0			0
$169$	4.43297			0.340997
$170$		0			0
$171$	−7.45305			−0.569949
$172$		0			0
$173$			9.75256i			0.741473i	0.928738	+	0.370737i	$0.120895\pi$
$173$			9.75256i			0.741473i	−0.928738	+	0.370737i	$0.879105\pi$
$174$		0			0
$175$	−7.02909	−	12.9330i	−0.531349	−	0.977647i
$176$		0			0
$177$			2.18569i			0.164287i
$178$		0			0
$179$	−4.31903			−0.322819			−0.161410	−	0.986888i	$-0.551604\pi$
$179$	−4.31903			−0.322819			−0.161410	+	0.986888i	$0.551604\pi$
$180$		0			0
$181$	−0.219705			−0.0163306			−0.00816529	−	0.999967i	$-0.502599\pi$
$181$	−0.219705			−0.0163306			−0.00816529	+	0.999967i	$0.502599\pi$
$182$		0			0
$183$		−	14.7082i		−	1.08726i
$184$		0			0
$185$	−1.50965	−	5.93903i	−0.110991	−	0.436646i
$186$		0			0
$187$			2.22048i			0.162378i
$188$		0			0
$189$	−2.94396			−0.214141
$190$		0			0
$191$	22.2927			1.61305			0.806523	−	0.591203i	$-0.201347\pi$
$191$	22.2927			1.61305			0.806523	+	0.591203i	$0.201347\pi$
$192$		0			0
$193$			24.5225i			1.76517i	0.470154	+	0.882584i	$0.344198\pi$
$193$			24.5225i			1.76517i	−0.470154	+	0.882584i	$0.655802\pi$
$194$		0			0
$195$	−6.34314	+	1.61237i	−0.454242	+	0.115464i
$196$		0			0
$197$		−	11.6923i		−	0.833039i	−0.909127	−	0.416520i	$-0.863250\pi$
$197$		−	11.6923i		−	0.833039i	0.909127	−	0.416520i	$-0.136750\pi$
$198$		0			0
$199$	−2.79400			−0.198062			−0.0990308	−	0.995084i	$-0.531574\pi$
$199$	−2.79400			−0.198062			−0.0990308	+	0.995084i	$0.531574\pi$
$200$		0			0
$201$	−2.18569			−0.154167
$202$		0			0
$203$			19.2970i			1.35438i
$204$		0			0
$205$	14.9297	−	3.79499i	1.04273	−	0.265053i
$206$		0			0
$207$		−	8.15951i		−	0.567125i
$208$		0			0
$209$	−16.5494			−1.14474
$210$		0			0
$211$	−2.75653			−0.189767			−0.0948837	−	0.995488i	$-0.530248\pi$
$211$	−2.75653			−0.189767			−0.0948837	+	0.995488i	$0.530248\pi$
$212$		0			0
$213$		−	5.85390i		−	0.401103i
$214$		0			0
$215$	5.38819	+	21.1974i	0.367471	+	1.44565i
$216$		0			0
$217$		−	22.0760i		−	1.49861i
$218$		0			0
$219$	−7.81604			−0.528159
$220$		0			0
$221$	−2.92695			−0.196888
$222$		0			0
$223$			3.76929i			0.252410i	0.992004	+	0.126205i	$0.0402797\pi$
$223$			3.76929i			0.252410i	−0.992004	+	0.126205i	$0.959720\pi$
$224$		0			0
$225$	−4.39309	+	2.38763i	−0.292872	+	0.159176i
$226$		0			0
$227$		−	19.4726i		−	1.29244i	−0.763152	−	0.646220i	$-0.776349\pi$
$227$		−	19.4726i		−	1.29244i	0.763152	−	0.646220i	$-0.223651\pi$
$228$		0			0
$229$	−7.62799			−0.504072			−0.252036	−	0.967718i	$-0.581100\pi$
$229$	−7.62799			−0.504072			−0.252036	+	0.967718i	$0.581100\pi$
$230$		0			0
$231$	−6.53700			−0.430103
$232$		0			0
$233$			9.78812i			0.641241i	0.947208	+	0.320621i	$0.103891\pi$
$233$			9.78812i			0.641241i	−0.947208	+	0.320621i	$0.896109\pi$
$234$		0			0
$235$	−4.49819	−	17.6961i	−0.293429	−	1.15437i
$236$		0			0
$237$		−	12.9975i		−	0.844277i
$238$		0			0
$239$	−1.78029			−0.115158			−0.0575788	−	0.998341i	$-0.518338\pi$
$239$	−1.78029			−0.115158			−0.0575788	+	0.998341i	$0.518338\pi$
$240$		0			0
$241$	12.5961			0.811387			0.405693	−	0.914009i	$-0.367030\pi$
$241$	12.5961			0.811387			0.405693	+	0.914009i	$0.367030\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	3.61237	−	0.918229i	0.230786	−	0.0586635i
$246$		0			0
$247$		−	21.8147i		−	1.38804i
$248$		0			0
$249$	11.5383			0.731212
$250$		0			0
$251$	−21.4987			−1.35699			−0.678494	−	0.734606i	$-0.737367\pi$
$251$	−21.4987			−1.35699			−0.678494	+	0.734606i	$0.737367\pi$
$252$		0			0
$253$		−	18.1181i		−	1.13907i
$254$		0			0
$255$	−2.16715	+	0.550869i	−0.135712	+	0.0344968i
$256$		0			0
$257$			27.5505i			1.71856i	0.511510	+	0.859278i	$0.329086\pi$
$257$			27.5505i			1.71856i	−0.511510	+	0.859278i	$0.670914\pi$
$258$		0			0
$259$	8.06785			0.501312
$260$		0			0
$261$	6.55478			0.405731
$262$		0			0
$263$		−	10.1415i		−	0.625349i	−0.949860	−	0.312674i	$-0.898775\pi$
$263$		−	10.1415i		−	0.625349i	0.949860	−	0.312674i	$-0.101225\pi$
$264$		0			0
$265$	4.92695	+	19.3829i	0.302660	+	1.19068i
$266$		0			0
$267$		−	8.82029i		−	0.539793i
$268$		0			0
$269$	10.0303			0.611559			0.305780	−	0.952102i	$-0.401083\pi$
$269$	10.0303			0.611559			0.305780	+	0.952102i	$0.401083\pi$
$270$		0			0
$271$	−22.4148			−1.36160			−0.680801	−	0.732469i	$-0.738368\pi$
$271$	−22.4148			−1.36160			−0.680801	+	0.732469i	$0.738368\pi$
$272$		0			0
$273$		−	8.61681i		−	0.521513i
$274$		0			0
$275$	−9.75477	+	5.30170i	−0.588235	+	0.319704i
$276$		0			0
$277$			12.9975i			0.780943i	0.920615	+	0.390471i	$0.127688\pi$
$277$			12.9975i			0.780943i	−0.920615	+	0.390471i	$0.872312\pi$
$278$		0			0
$279$	−7.49874			−0.448938
$280$		0			0
$281$	−22.8514			−1.36320			−0.681599	−	0.731725i	$-0.738715\pi$
$281$	−22.8514			−1.36320			−0.681599	+	0.731725i	$0.738715\pi$
$282$		0			0
$283$		−	13.7820i		−	0.819256i	−0.912253	−	0.409628i	$-0.865658\pi$
$283$		−	13.7820i		−	0.819256i	0.912253	−	0.409628i	$-0.134342\pi$
$284$		0			0
$285$	−4.10565	−	16.1519i	−0.243198	−	0.956755i
$286$		0			0
$287$			20.2812i			1.19716i
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$	−3.01818			−0.176929
$292$		0			0
$293$		−	1.03188i		−	0.0602832i	−0.999546	−	0.0301416i	$-0.990404\pi$
$293$		−	1.03188i		−	0.0602832i	0.999546	−	0.0301416i	$-0.00959583\pi$
$294$		0			0
$295$	−4.73673	+	1.20403i	−0.275783	+	0.0701014i
$296$		0			0
$297$			2.22048i			0.128845i
$298$		0			0
$299$	23.8825			1.38116
$300$		0			0
$301$	−28.7956			−1.65975
$302$		0			0
$303$			6.09139i			0.349941i
$304$		0			0
$305$	31.8749	−	8.10229i	1.82515	−	0.463936i
$306$		0			0
$307$			21.0756i			1.20285i	0.798931	+	0.601423i	$0.205399\pi$
$307$			21.0756i			1.20285i	−0.798931	+	0.601423i	$0.794601\pi$
$308$		0			0
$309$	−9.78125			−0.556436
$310$		0			0
$311$	−25.1113			−1.42393			−0.711966	−	0.702214i	$-0.752195\pi$
$311$	−25.1113			−1.42393			−0.711966	+	0.702214i	$0.752195\pi$
$312$		0			0
$313$			15.2728i			0.863271i	0.902048	+	0.431636i	$0.142063\pi$
$313$			15.2728i			0.863271i	−0.902048	+	0.431636i	$0.857937\pi$
$314$		0			0
$315$	−1.62173	−	6.38000i	−0.0913744	−	0.359472i
$316$		0			0
$317$			7.29291i			0.409611i	0.978803	+	0.204805i	$0.0656561\pi$
$317$			7.29291i			0.409611i	−0.978803	+	0.204805i	$0.934344\pi$
$318$		0			0
$319$	14.5548			0.814911
$320$		0			0
$321$	−2.78617			−0.155509
$322$		0			0
$323$		−	7.45305i		−	0.414699i
$324$		0			0
$325$	−6.98848	−	12.8583i	−0.387651	−	0.713253i
$326$		0			0
$327$		−	12.9061i		−	0.713709i
$328$		0			0
$329$	24.0392			1.32532
$330$		0			0
$331$	−24.5504			−1.34941			−0.674706	−	0.738086i	$-0.735730\pi$
$331$	−24.5504			−1.34941			−0.674706	+	0.738086i	$0.735730\pi$
$332$		0			0
$333$		−	2.74048i		−	0.150177i
$334$		0			0
$335$	−1.20403	−	4.73673i	−0.0657833	−	0.258795i
$336$		0			0
$337$			6.35689i			0.346282i	0.984897	+	0.173141i	$0.0553916\pi$
$337$			6.35689i			0.346282i	−0.984897	+	0.173141i	$0.944608\pi$
$338$		0			0
$339$	19.2691			1.04655
$340$		0			0
$341$	−16.6508			−0.901693
$342$		0			0
$343$		−	15.7005i		−	0.847747i
$344$		0			0
$345$	17.6829	−	4.49482i	0.952015	−	0.241993i
$346$		0			0
$347$			15.0371i			0.807232i	0.914928	+	0.403616i	$0.132247\pi$
$347$			15.0371i			0.807232i	−0.914928	+	0.403616i	$0.867753\pi$
$348$		0			0
$349$	−34.9730			−1.87206			−0.936032	−	0.351916i	$-0.885530\pi$
$349$	−34.9730			−1.87206			−0.936032	+	0.351916i	$0.885530\pi$
$350$		0			0
$351$	−2.92695			−0.156229
$352$		0			0
$353$		−	13.7039i		−	0.729387i	−0.931128	−	0.364694i	$-0.881174\pi$
$353$		−	13.7039i		−	0.729387i	0.931128	−	0.364694i	$-0.118826\pi$
$354$		0			0
$355$	12.6863	−	3.22473i	0.673318	−	0.171151i
$356$		0			0
$357$		−	2.94396i		−	0.155811i
$358$		0			0
$359$	−13.1274			−0.692835			−0.346418	−	0.938080i	$-0.612602\pi$
$359$	−13.1274			−0.692835			−0.346418	+	0.938080i	$0.612602\pi$
$360$		0			0
$361$	36.5479			1.92357
$362$		0			0
$363$		−	6.06946i		−	0.318564i
$364$		0			0
$365$	−4.30561	−	16.9385i	−0.225366	−	0.886603i
$366$		0			0
$367$		−	20.7635i		−	1.08384i	−0.840429	−	0.541922i	$-0.817697\pi$
$367$		−	20.7635i		−	1.08384i	0.840429	−	0.541922i	$-0.182303\pi$
$368$		0			0
$369$	6.88909			0.358632
$370$		0			0
$371$	−26.3306			−1.36702
$372$		0			0
$373$			17.8879i			0.926201i	0.886306	+	0.463100i	$0.153263\pi$
$373$			17.8879i			0.926201i	−0.886306	+	0.463100i	$0.846737\pi$
$374$		0			0
$375$	−7.59438	−	8.20521i	−0.392172	−	0.423715i
$376$		0			0
$377$			19.1855i			0.988105i
$378$		0			0
$379$	6.52015			0.334918			0.167459	−	0.985879i	$-0.446444\pi$
$379$	6.52015			0.334918			0.167459	+	0.985879i	$0.446444\pi$
$380$		0			0
$381$	−16.0705			−0.823318
$382$		0			0
$383$		−	8.43497i		−	0.431007i	−0.976503	−	0.215503i	$-0.930861\pi$
$383$		−	8.43497i		−	0.431007i	0.976503	−	0.215503i	$-0.0691392\pi$
$384$		0			0
$385$	−3.60103	−	14.1667i	−0.183526	−	0.722000i
$386$		0			0
$387$			9.78125i			0.497209i
$388$		0			0
$389$	20.9635			1.06289			0.531445	−	0.847093i	$-0.321649\pi$
$389$	20.9635			1.06289			0.531445	+	0.847093i	$0.321649\pi$
$390$		0			0
$391$	8.15951			0.412644
$392$		0			0
$393$		−	15.5900i		−	0.786410i
$394$		0			0
$395$	28.1675	−	7.15991i	1.41726	−	0.360254i
$396$		0			0
$397$		−	24.6066i		−	1.23497i	−0.786583	−	0.617484i	$-0.788152\pi$
$397$		−	24.6066i		−	1.23497i	0.786583	−	0.617484i	$-0.211848\pi$
$398$		0			0
$399$	21.9414			1.09845
$400$		0			0
$401$	−16.5153			−0.824737			−0.412369	−	0.911017i	$-0.635298\pi$
$401$	−16.5153			−0.824737			−0.412369	+	0.911017i	$0.635298\pi$
$402$		0			0
$403$		−	21.9484i		−	1.09333i
$404$		0			0
$405$	−2.16715	+	0.550869i	−0.107687	+	0.0273729i
$406$		0			0
$407$		−	6.08518i		−	0.301631i
$408$		0			0
$409$	−10.9340			−0.540652			−0.270326	−	0.962769i	$-0.587131\pi$
$409$	−10.9340			−0.540652			−0.270326	+	0.962769i	$0.587131\pi$
$410$		0			0
$411$	11.4091			0.562769
$412$		0			0
$413$		−	6.43459i		−	0.316625i
$414$		0			0
$415$	6.35611	+	25.0053i	0.312009	+	1.22746i
$416$		0			0
$417$			12.2949i			0.602082i
$418$		0			0
$419$	21.5631			1.05343			0.526714	−	0.850043i	$-0.323424\pi$
$419$	21.5631			1.05343			0.526714	+	0.850043i	$0.323424\pi$
$420$		0			0
$421$	−12.5868			−0.613442			−0.306721	−	0.951799i	$-0.599232\pi$
$421$	−12.5868			−0.613442			−0.306721	+	0.951799i	$0.599232\pi$
$422$		0			0
$423$		−	8.16561i		−	0.397026i
$424$		0			0
$425$	−2.38763	−	4.39309i	−0.115817	−	0.213096i
$426$		0			0
$427$			43.3003i			2.09545i
$428$		0			0
$429$	−6.49924			−0.313786
$430$		0			0
$431$	9.88964			0.476367			0.238184	−	0.971220i	$-0.423448\pi$
$431$	9.88964			0.476367			0.238184	+	0.971220i	$0.423448\pi$
$432$		0			0
$433$		−	11.1185i		−	0.534319i	−0.963652	−	0.267159i	$-0.913915\pi$
$433$		−	11.1185i		−	0.534319i	0.963652	−	0.267159i	$-0.0860850\pi$
$434$		0			0
$435$	3.61083	+	14.2052i	0.173126	+	0.681088i
$436$		0			0
$437$			60.8132i			2.90909i
$438$		0			0
$439$	−7.87168			−0.375695			−0.187847	−	0.982198i	$-0.560151\pi$
$439$	−7.87168			−0.375695			−0.187847	+	0.982198i	$0.560151\pi$
$440$		0			0
$441$	1.66687			0.0793749
$442$		0			0
$443$			24.7964i			1.17811i	0.808093	+	0.589055i	$0.200500\pi$
$443$			24.7964i			1.17811i	−0.808093	+	0.589055i	$0.799500\pi$
$444$		0			0
$445$	19.1149	−	4.85882i	0.906133	−	0.230330i
$446$		0			0
$447$		−	11.0918i		−	0.524623i
$448$		0			0
$449$	−0.451817			−0.0213226			−0.0106613	−	0.999943i	$-0.503394\pi$
$449$	−0.451817			−0.0213226			−0.0106613	+	0.999943i	$0.503394\pi$
$450$		0			0
$451$	15.2971			0.720312
$452$		0			0
$453$		−	14.7824i		−	0.694539i
$454$		0			0
$455$	18.6739	−	4.74674i	0.875447	−	0.222530i
$456$		0			0
$457$		−	32.2642i		−	1.50925i	−0.656154	−	0.754627i	$-0.727818\pi$
$457$		−	32.2642i		−	1.50925i	0.656154	−	0.754627i	$-0.272182\pi$
$458$		0			0
$459$	−1.00000			−0.0466760
$460$		0			0
$461$	−6.01778			−0.280276			−0.140138	−	0.990132i	$-0.544755\pi$
$461$	−6.01778			−0.280276			−0.140138	+	0.990132i	$0.544755\pi$
$462$		0			0
$463$		−	7.48331i		−	0.347779i	−0.984765	−	0.173889i	$-0.944366\pi$
$463$		−	7.48331i		−	0.347779i	0.984765	−	0.173889i	$-0.0556335\pi$
$464$		0			0
$465$	−4.13082	−	16.2509i	−0.191562	−	0.753617i
$466$		0			0
$467$		−	8.18742i		−	0.378869i	−0.981893	−	0.189434i	$-0.939335\pi$
$467$		−	8.18742i		−	0.378869i	0.981893	−	0.189434i	$-0.0606654\pi$
$468$		0			0
$469$	6.43459			0.297122
$470$		0			0
$471$	5.55361			0.255897
$472$		0			0
$473$			21.7191i			0.998645i
$474$		0			0
$475$	32.7419	−	17.7951i	1.50230	−	0.816497i
$476$		0			0
$477$			8.94396i			0.409516i
$478$		0			0
$479$	37.1891			1.69922			0.849608	−	0.527415i	$-0.176839\pi$
$479$	37.1891			1.69922			0.849608	+	0.527415i	$0.176839\pi$
$480$		0			0
$481$	8.02124			0.365737
$482$		0			0
$483$			24.0212i			1.09300i
$484$		0			0
$485$	−1.66262	−	6.54085i	−0.0754958	−	0.297005i
$486$		0			0
$487$			37.6780i			1.70736i	0.520802	+	0.853678i	$0.325633\pi$
$487$			37.6780i			1.70736i	−0.520802	+	0.853678i	$0.674367\pi$
$488$		0			0
$489$	16.4945			0.745907
$490$		0			0
$491$	37.9828			1.71414			0.857069	−	0.515202i	$-0.172283\pi$
$491$	37.9828			1.71414			0.857069	+	0.515202i	$0.172283\pi$
$492$		0			0
$493$			6.55478i			0.295213i
$494$		0			0
$495$	−4.81212	+	1.22320i	−0.216289	+	0.0549785i
$496$		0			0
$497$			17.2336i			0.773033i
$498$		0			0
$499$	−6.63997			−0.297246			−0.148623	−	0.988894i	$-0.547484\pi$
$499$	−6.63997			−0.297246			−0.148623	+	0.988894i	$0.547484\pi$
$500$		0			0
$501$	−5.12590			−0.229008
$502$		0			0
$503$		−	14.9224i		−	0.665358i	−0.943040	−	0.332679i	$-0.892047\pi$
$503$		−	14.9224i		−	0.665358i	0.943040	−	0.332679i	$-0.107953\pi$
$504$		0			0
$505$	−13.2010	+	3.35556i	−0.587435	+	0.149320i
$506$		0			0
$507$			4.43297i			0.196875i
$508$		0			0
$509$	−5.68041			−0.251780			−0.125890	−	0.992044i	$-0.540179\pi$
$509$	−5.68041			−0.251780			−0.125890	+	0.992044i	$0.540179\pi$
$510$		0			0
$511$	23.0101			1.01791
$512$		0			0
$513$		−	7.45305i		−	0.329060i
$514$		0			0
$515$	−5.38819	−	21.1974i	−0.237432	−	0.934071i
$516$		0			0
$517$		−	18.1316i		−	0.797427i
$518$		0			0
$519$	−9.75256			−0.428090
$520$		0			0
$521$	−10.0303			−0.439436			−0.219718	−	0.975563i	$-0.570514\pi$
$521$	−10.0303			−0.439436			−0.219718	+	0.975563i	$0.570514\pi$
$522$		0			0
$523$			35.1154i			1.53549i	0.640756	+	0.767744i	$0.278621\pi$
$523$			35.1154i			1.53549i	−0.640756	+	0.767744i	$0.721379\pi$
$524$		0			0
$525$	12.9330	−	7.02909i	0.564445	−	0.306774i
$526$		0			0
$527$		−	7.49874i		−	0.326650i
$528$		0			0
$529$	−43.5777			−1.89468
$530$		0			0
$531$	−2.18569			−0.0948510
$532$		0			0
$533$			20.1640i			0.873400i
$534$		0			0
$535$	−1.53482	−	6.03806i	−0.0663559	−	0.261048i
$536$		0			0
$537$		−	4.31903i		−	0.186380i
$538$		0			0
$539$	3.70126			0.159425
$540$		0			0
$541$	−6.77276			−0.291184			−0.145592	−	0.989345i	$-0.546509\pi$
$541$	−6.77276			−0.291184			−0.145592	+	0.989345i	$0.546509\pi$
$542$		0			0
$543$		−	0.219705i		−	0.00942847i
$544$		0			0
$545$	27.9694	−	7.10957i	1.19808	−	0.304540i
$546$		0			0
$547$			39.0154i			1.66818i	0.551629	+	0.834090i	$0.314007\pi$
$547$			39.0154i			1.66818i	−0.551629	+	0.834090i	$0.685993\pi$
$548$		0			0
$549$	14.7082			0.627731
$550$		0			0
$551$	−48.8531			−2.08121
$552$		0			0
$553$			38.2640i			1.62715i
$554$		0			0
$555$	5.93903	−	1.50965i	0.252098	−	0.0640809i
$556$		0			0
$557$		−	4.89389i		−	0.207361i	−0.994611	−	0.103680i	$-0.966938\pi$
$557$		−	4.89389i		−	0.207361i	0.994611	−	0.103680i	$-0.0330619\pi$
$558$		0			0
$559$	−28.6292			−1.21089
$560$		0			0
$561$	−2.22048			−0.0937488
$562$		0			0
$563$			29.2863i			1.23427i	0.786856	+	0.617136i	$0.211707\pi$
$563$			29.2863i			1.23427i	−0.786856	+	0.617136i	$0.788293\pi$
$564$		0			0
$565$	10.6147	+	41.7590i	0.446566	+	1.75681i
$566$		0			0
$567$		−	2.94396i		−	0.123635i
$568$		0			0
$569$	−27.7551			−1.16356			−0.581778	−	0.813348i	$-0.697643\pi$
$569$	−27.7551			−1.16356			−0.581778	+	0.813348i	$0.697643\pi$
$570$		0			0
$571$	−32.9491			−1.37888			−0.689439	−	0.724343i	$-0.742143\pi$
$571$	−32.9491			−1.37888			−0.689439	+	0.724343i	$0.742143\pi$
$572$		0			0
$573$			22.2927i			0.931293i
$574$		0			0
$575$	19.4819	+	35.8454i	0.812453	+	1.49486i
$576$		0			0
$577$		−	31.3658i		−	1.30578i	−0.757455	−	0.652888i	$-0.773557\pi$
$577$		−	31.3658i		−	1.30578i	0.757455	−	0.652888i	$-0.226443\pi$
$578$		0			0
$579$	−24.5225			−1.01912
$580$		0			0
$581$	−33.9683			−1.40924
$582$		0			0
$583$			19.8599i			0.822513i
$584$		0			0
$585$	−1.61237	−	6.34314i	−0.0666632	−	0.262257i
$586$		0			0
$587$			33.7149i			1.39156i	0.718253	+	0.695782i	$0.244942\pi$
$587$			33.7149i			1.39156i	−0.718253	+	0.695782i	$0.755058\pi$
$588$		0			0
$589$	55.8885			2.30284
$590$		0			0
$591$	11.6923			0.480955
$592$		0			0
$593$			5.28289i			0.216942i	0.994100	+	0.108471i	$0.0345955\pi$
$593$			5.28289i			0.216942i	−0.994100	+	0.108471i	$0.965405\pi$
$594$		0			0
$595$	6.38000	−	1.62173i	0.261554	−	0.0664846i
$596$		0			0
$597$		−	2.79400i		−	0.114351i
$598$		0			0
$599$	−11.8098			−0.482537			−0.241268	−	0.970458i	$-0.577563\pi$
$599$	−11.8098			−0.482537			−0.241268	+	0.970458i	$0.577563\pi$
$600$		0			0
$601$	−7.80211			−0.318255			−0.159127	−	0.987258i	$-0.550868\pi$
$601$	−7.80211			−0.318255			−0.159127	+	0.987258i	$0.550868\pi$
$602$		0			0
$603$		−	2.18569i		−	0.0890083i
$604$		0			0
$605$	13.1534	−	3.34348i	0.534763	−	0.135932i
$606$		0			0
$607$		−	5.19576i		−	0.210889i	−0.994425	−	0.105445i	$-0.966373\pi$
$607$		−	5.19576i		−	0.210889i	0.994425	−	0.105445i	$-0.0336266\pi$
$608$		0			0
$609$	−19.2970			−0.781954
$610$		0			0
$611$	23.9003			0.966904
$612$		0			0
$613$		−	9.52430i		−	0.384683i	−0.981328	−	0.192341i	$-0.938392\pi$
$613$		−	9.52430i		−	0.384683i	0.981328	−	0.192341i	$-0.0616081\pi$
$614$		0			0
$615$	3.79499	+	14.9297i	0.153029	+	0.602023i
$616$		0			0
$617$			34.5985i			1.39288i	0.717614	+	0.696441i	$0.245234\pi$
$617$			34.5985i			1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$618$		0			0
$619$	−12.5225			−0.503322			−0.251661	−	0.967815i	$-0.580977\pi$
$619$	−12.5225			−0.503322			−0.251661	+	0.967815i	$0.580977\pi$
$620$		0			0
$621$	8.15951			0.327430
$622$		0			0
$623$			25.9665i			1.04033i
$624$		0			0
$625$	13.5984	−	20.9782i	0.543937	−	0.839126i
$626$		0			0
$627$		−	16.5494i		−	0.660918i
$628$		0			0
$629$	2.74048			0.109270
$630$		0			0
$631$	−14.2986			−0.569219			−0.284609	−	0.958644i	$-0.591864\pi$
$631$	−14.2986			−0.569219			−0.284609	+	0.958644i	$0.591864\pi$
$632$		0			0
$633$		−	2.75653i		−	0.109562i
$634$		0			0
$635$	−8.85276	−	34.8273i	−0.351311	−	1.38208i
$636$		0			0
$637$			4.87886i			0.193307i
$638$		0			0
$639$	5.85390			0.231577
$640$		0			0
$641$	6.33683			0.250290			0.125145	−	0.992138i	$-0.460060\pi$
$641$	6.33683			0.250290			0.125145	+	0.992138i	$0.460060\pi$
$642$		0			0
$643$			13.5652i			0.534961i	0.963563	+	0.267480i	$0.0861911\pi$
$643$			13.5652i			0.534961i	−0.963563	+	0.267480i	$0.913809\pi$
$644$		0			0
$645$	−21.1974	+	5.38819i	−0.834648	+	0.212160i
$646$		0			0
$647$		−	1.03764i		−	0.0407939i	−0.999792	−	0.0203969i	$-0.993507\pi$
$647$		−	1.03764i		−	0.0407939i	0.999792	−	0.0203969i	$-0.00649300\pi$
$648$		0			0
$649$	−4.85330			−0.190509
$650$		0			0
$651$	22.0760			0.865225
$652$		0			0
$653$		−	18.0163i		−	0.705034i	−0.935805	−	0.352517i	$-0.885326\pi$
$653$		−	18.0163i		−	0.705034i	0.935805	−	0.352517i	$-0.114674\pi$
$654$		0			0
$655$	33.7858	−	8.58804i	1.32012	−	0.335562i
$656$		0			0
$657$		−	7.81604i		−	0.304933i
$658$		0			0
$659$	12.8514			0.500619			0.250309	−	0.968166i	$-0.419468\pi$
$659$	12.8514			0.500619			0.250309	+	0.968166i	$0.419468\pi$
$660$		0			0
$661$	−33.5247			−1.30396			−0.651981	−	0.758236i	$-0.726062\pi$
$661$	−33.5247			−1.30396			−0.651981	+	0.758236i	$0.726062\pi$
$662$		0			0
$663$		−	2.92695i		−	0.113673i
$664$		0			0
$665$	12.0869	+	47.5504i	0.468708	+	1.84393i
$666$		0			0
$667$		−	53.4838i		−	2.07090i
$668$		0			0
$669$	−3.76929			−0.145729
$670$		0			0
$671$	32.6593			1.26080
$672$		0			0
$673$		−	43.1130i		−	1.66189i	−0.556358	−	0.830943i	$-0.687802\pi$
$673$		−	43.1130i		−	1.66189i	0.556358	−	0.830943i	$-0.312198\pi$
$674$		0			0
$675$	−2.38763	−	4.39309i	−0.0919001	−	0.169090i
$676$		0			0
$677$		−	0.254132i		−	0.00976710i	−0.999988	−	0.00488355i	$-0.998446\pi$
$677$		−	0.254132i		−	0.00976710i	0.999988	−	0.00488355i	$-0.00155449\pi$
$678$		0			0
$679$	8.88539			0.340990
$680$		0			0
$681$	19.4726			0.746190
$682$		0			0
$683$		−	31.2210i		−	1.19464i	−0.802004	−	0.597319i	$-0.796233\pi$
$683$		−	31.2210i		−	1.19464i	0.802004	−	0.597319i	$-0.203767\pi$
$684$		0			0
$685$	6.28491	+	24.7252i	0.240134	+	0.944701i
$686$		0			0
$687$		−	7.62799i		−	0.291026i
$688$		0			0
$689$	−26.1785			−0.997322
$690$		0			0
$691$	1.31305			0.0499506			0.0249753	−	0.999688i	$-0.492049\pi$
$691$	1.31305			0.0499506			0.0249753	+	0.999688i	$0.492049\pi$
$692$		0			0
$693$		−	6.53700i		−	0.248320i
$694$		0			0
$695$	−26.6448	+	6.77286i	−1.01070	+	0.256909i
$696$		0			0
$697$			6.88909i			0.260943i
$698$		0			0
$699$	−9.78812			−0.370221
$700$		0			0
$701$	−39.0578			−1.47519			−0.737596	−	0.675242i	$-0.764039\pi$
$701$	−39.0578			−1.47519			−0.737596	+	0.675242i	$0.764039\pi$
$702$		0			0
$703$			20.4249i			0.770340i
$704$		0			0
$705$	17.6961	−	4.49819i	0.666474	−	0.169411i
$706$		0			0
$707$		−	17.9328i		−	0.674431i
$708$		0			0
$709$	−3.72235			−0.139796			−0.0698979	−	0.997554i	$-0.522267\pi$
$709$	−3.72235			−0.139796			−0.0698979	+	0.997554i	$0.522267\pi$
$710$		0			0
$711$	12.9975			0.487444
$712$		0			0
$713$			61.1861i			2.29144i
$714$		0			0
$715$	−3.58023	−	14.0848i	−0.133893	−	0.526743i
$716$		0			0
$717$		−	1.78029i		−	0.0664863i
$718$		0			0
$719$	43.2995			1.61480			0.807399	−	0.590005i	$-0.200874\pi$
$719$	43.2995			1.61480			0.807399	+	0.590005i	$0.200874\pi$
$720$		0			0
$721$	28.7956			1.07240
$722$		0			0
$723$			12.5961i			0.468454i
$724$		0			0
$725$	−28.7957	+	15.6504i	−1.06945	+	0.581242i
$726$		0			0
$727$		−	3.62706i		−	0.134520i	−0.997735	−	0.0672601i	$-0.978574\pi$
$727$		−	3.62706i		−	0.134520i	0.997735	−	0.0672601i	$-0.0214257\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−9.78125			−0.361773
$732$		0			0
$733$			24.3743i			0.900285i	0.892957	+	0.450142i	$0.148627\pi$
$733$			24.3743i			0.900285i	−0.892957	+	0.450142i	$0.851373\pi$
$734$		0			0
$735$	0.918229	+	3.61237i	0.0338694	+	0.133244i
$736$		0			0
$737$		−	4.85330i		−	0.178773i
$738$		0			0
$739$	3.76879			0.138637			0.0693186	−	0.997595i	$-0.477918\pi$
$739$	3.76879			0.138637			0.0693186	+	0.997595i	$0.477918\pi$
$740$		0			0
$741$	21.8147			0.801383
$742$		0			0
$743$		−	45.5234i		−	1.67009i	−0.550179	−	0.835047i	$-0.685441\pi$
$743$		−	45.5234i		−	1.67009i	0.550179	−	0.835047i	$-0.314559\pi$
$744$		0			0
$745$	24.0376	−	6.11012i	0.880669	−	0.223858i
$746$		0			0
$747$			11.5383i			0.422166i
$748$		0			0
$749$	8.20237			0.299708
$750$		0			0
$751$	5.23474			0.191018			0.0955092	−	0.995429i	$-0.469552\pi$
$751$	5.23474			0.191018			0.0955092	+	0.995429i	$0.469552\pi$
$752$		0			0
$753$		−	21.4987i		−	0.783458i
$754$		0			0
$755$	32.0357	−	8.14318i	1.16590	−	0.296361i
$756$		0			0
$757$		−	39.7836i		−	1.44596i	−0.690869	−	0.722980i	$-0.742772\pi$
$757$		−	39.7836i		−	1.44596i	0.690869	−	0.722980i	$-0.257228\pi$
$758$		0			0
$759$	18.1181			0.657644
$760$		0			0
$761$	27.4012			0.993293			0.496646	−	0.867953i	$-0.334565\pi$
$761$	27.4012			0.993293			0.496646	+	0.867953i	$0.334565\pi$
$762$		0			0
$763$			37.9950i			1.37551i
$764$		0			0
$765$	−0.550869	−	2.16715i	−0.0199167	−	0.0783535i
$766$		0			0
$767$		−	6.39742i		−	0.230997i
$768$		0			0
$769$	28.7265			1.03590			0.517952	−	0.855410i	$-0.326695\pi$
$769$	28.7265			1.03590			0.517952	+	0.855410i	$0.326695\pi$
$770$		0			0
$771$	−27.5505			−0.992208
$772$		0			0
$773$		−	9.08155i		−	0.326641i	−0.986573	−	0.163320i	$-0.947780\pi$
$773$		−	9.08155i		−	0.326641i	0.986573	−	0.163320i	$-0.0522204\pi$
$774$		0			0
$775$	32.9426	−	17.9042i	1.18333	−	0.643139i
$776$		0			0
$777$			8.06785i			0.289432i
$778$		0			0
$779$	−51.3447			−1.83961
$780$		0			0
$781$	12.9985			0.465122
$782$		0			0
$783$			6.55478i			0.234249i
$784$		0			0
$785$	3.05931	+	12.0355i	0.109192	+	0.429566i
$786$		0			0
$787$		−	26.6783i		−	0.950978i	−0.879722	−	0.475489i	$-0.842271\pi$
$787$		−	26.6783i		−	0.950978i	0.879722	−	0.475489i	$-0.157729\pi$
$788$		0			0
$789$	10.1415			0.361045
$790$		0			0
$791$	−56.7273			−2.01699
$792$		0			0
$793$			43.0502i			1.52876i
$794$		0			0
$795$	−19.3829	+	4.92695i	−0.687441	+	0.174741i
$796$		0			0
$797$		−	15.8270i		−	0.560621i	−0.959909	−	0.280310i	$-0.909563\pi$
$797$		−	15.8270i		−	0.560621i	0.959909	−	0.280310i	$-0.0904374\pi$
$798$		0			0
$799$	8.16561			0.288879
$800$		0			0
$801$	8.82029			0.311649
$802$		0			0
$803$		−	17.3554i		−	0.612458i
$804$		0			0
$805$	−52.0577	+	13.2326i	−1.83479	+	0.466387i
$806$		0			0
$807$			10.0303i			0.353084i
$808$		0			0
$809$	22.0429			0.774987			0.387494	−	0.921872i	$-0.373341\pi$
$809$	22.0429			0.774987			0.387494	+	0.921872i	$0.373341\pi$
$810$		0			0
$811$	41.3144			1.45074			0.725372	−	0.688357i	$-0.241668\pi$
$811$	41.3144			1.45074			0.725372	+	0.688357i	$0.241668\pi$
$812$		0			0
$813$		−	22.4148i		−	0.786121i
$814$		0			0
$815$	9.08631	+	35.7460i	0.318279	+	1.25213i
$816$		0			0
$817$		−	72.9001i		−	2.55045i
$818$		0			0
$819$	8.61681			0.301096
$820$		0			0
$821$	−0.00462923			−0.000161561			−8.07806e−5	−	1.00000i	$-0.500026\pi$
$821$	−0.00462923			−0.000161561			−8.07806e−5		1.00000i	$0.500026\pi$
$822$		0			0
$823$			35.0154i			1.22056i	0.792185	+	0.610281i	$0.208943\pi$
$823$			35.0154i			1.22056i	−0.792185	+	0.610281i	$0.791057\pi$
$824$		0			0
$825$	−5.30170	−	9.75477i	−0.184581	−	0.339618i
$826$		0			0
$827$		−	21.1206i		−	0.734436i	−0.930135	−	0.367218i	$-0.880310\pi$
$827$		−	21.1206i		−	0.734436i	0.930135	−	0.367218i	$-0.119690\pi$
$828$		0			0
$829$	22.3747			0.777104			0.388552	−	0.921427i	$-0.372975\pi$
$829$	22.3747			0.777104			0.388552	+	0.921427i	$0.372975\pi$
$830$		0			0
$831$	−12.9975			−0.450878
$832$		0			0
$833$			1.66687i			0.0577537i
$834$		0			0
$835$	−2.82370	−	11.1086i	−0.0977182	−	0.384429i
$836$		0			0
$837$		−	7.49874i		−	0.259194i
$838$		0			0
$839$	11.2265			0.387581			0.193790	−	0.981043i	$-0.437922\pi$
$839$	11.2265			0.387581			0.193790	+	0.981043i	$0.437922\pi$
$840$		0			0
$841$	13.9652			0.481559
$842$		0			0
$843$		−	22.8514i		−	0.787043i
$844$		0			0
$845$	−9.60690	+	2.44198i	−0.330488	+	0.0840068i
$846$		0			0
$847$			17.8682i			0.613959i
$848$		0			0
$849$	13.7820			0.472998
$850$		0			0
$851$	−22.3610			−0.766524
$852$		0			0
$853$			27.9221i			0.956035i	0.878350	+	0.478018i	$0.158645\pi$
$853$			27.9221i			0.956035i	−0.878350	+	0.478018i	$0.841355\pi$
$854$		0			0
$855$	16.1519	−	4.10565i	0.552382	−	0.140410i
$856$		0			0
$857$		−	38.5855i		−	1.31806i	−0.752119	−	0.659028i	$-0.770968\pi$
$857$		−	38.5855i		−	1.31806i	0.752119	−	0.659028i	$-0.229032\pi$
$858$		0			0
$859$	−30.1869			−1.02996			−0.514981	−	0.857202i	$-0.672201\pi$
$859$	−30.1869			−1.02996			−0.514981	+	0.857202i	$0.672201\pi$
$860$		0			0
$861$	−20.2812			−0.691180
$862$		0			0
$863$		−	2.10589i		−	0.0716852i	−0.999357	−	0.0358426i	$-0.988589\pi$
$863$		−	2.10589i		−	0.0716852i	0.999357	−	0.0358426i	$-0.0114115\pi$
$864$		0			0
$865$	−5.37238	−	21.1353i	−0.182667	−	0.718621i
$866$		0			0
$867$		−	1.00000i		−	0.0339618i
$868$		0			0
$869$	28.8607			0.979031
$870$		0			0
$871$	6.39742			0.216768
$872$		0			0
$873$		−	3.01818i		−	0.102150i
$874$		0			0
$875$	22.3575	+	24.1558i	0.755822	+	0.816614i
$876$		0			0
$877$		−	0.0171621i		−	0.000579523i	−1.00000		0.000289762i	$-0.999908\pi$
$877$		−	0.0171621i		−	0.000579523i	1.00000		0.000289762i	$-9.22340e-5\pi$
$878$		0			0
$879$	1.03188			0.0348045
$880$		0			0
$881$	−17.6280			−0.593902			−0.296951	−	0.954893i	$-0.595970\pi$
$881$	−17.6280			−0.593902			−0.296951	+	0.954893i	$0.595970\pi$
$882$		0			0
$883$			35.7475i			1.20300i	0.798874	+	0.601499i	$0.205430\pi$
$883$			35.7475i			1.20300i	−0.798874	+	0.601499i	$0.794570\pi$
$884$		0			0
$885$	−1.20403	−	4.73673i	−0.0404731	−	0.159223i
$886$		0			0
$887$			31.3002i			1.05096i	0.850807	+	0.525479i	$0.176114\pi$
$887$			31.3002i			1.05096i	−0.850807	+	0.525479i	$0.823886\pi$
$888$		0			0
$889$	47.3109			1.58676
$890$		0			0
$891$	−2.22048			−0.0743890
$892$		0			0
$893$			60.8587i			2.03656i
$894$		0			0
$895$	9.35998	−	2.37922i	0.312870	−	0.0795285i
$896$		0			0
$897$			23.8825i			0.797413i
$898$		0			0
$899$	−49.1526			−1.63933
$900$		0			0
$901$	−8.94396			−0.297966
$902$		0			0
$903$		−	28.7956i		−	0.958256i
$904$		0			0
$905$	0.476135	−	0.121029i	0.0158273	−	0.00402314i
$906$		0			0
$907$		−	38.6563i		−	1.28356i	−0.766889	−	0.641780i	$-0.778196\pi$
$907$		−	38.6563i		−	1.28356i	0.766889	−	0.641780i	$-0.221804\pi$
$908$		0			0
$909$	−6.09139			−0.202039
$910$		0			0
$911$	7.51866			0.249104			0.124552	−	0.992213i	$-0.460251\pi$
$911$	7.51866			0.249104			0.124552	+	0.992213i	$0.460251\pi$
$912$		0			0
$913$			25.6207i			0.847920i
$914$		0			0
$915$	8.10229	+	31.8749i	0.267854	+	1.05375i
$916$		0			0
$917$			45.8962i			1.51563i
$918$		0			0
$919$	8.24705			0.272045			0.136023	−	0.990706i	$-0.456568\pi$
$919$	8.24705			0.272045			0.136023	+	0.990706i	$0.456568\pi$
$920$		0			0
$921$	−21.0756			−0.694463
$922$		0			0
$923$			17.1341i			0.563975i
$924$		0			0
$925$	6.54326	+	12.0392i	0.215141	+	0.395845i
$926$		0			0
$927$		−	9.78125i		−	0.321258i
$928$		0			0
$929$	38.5663			1.26532			0.632660	−	0.774430i	$-0.281963\pi$
$929$	38.5663			1.26532			0.632660	+	0.774430i	$0.281963\pi$
$930$		0			0
$931$	−12.4233			−0.407157
$932$		0			0
$933$		−	25.1113i		−	0.822107i
$934$		0			0
$935$	−1.22320	−	4.81212i	−0.0400028	−	0.157373i
$936$		0			0
$937$			21.8518i			0.713866i	0.934130	+	0.356933i	$0.116178\pi$
$937$			21.8518i			0.713866i	−0.934130	+	0.356933i	$0.883822\pi$
$938$		0			0
$939$	−15.2728			−0.498410
$940$		0			0
$941$	−8.65899			−0.282275			−0.141138	−	0.989990i	$-0.545076\pi$
$941$	−8.65899			−0.282275			−0.141138	+	0.989990i	$0.545076\pi$
$942$		0			0
$943$		−	56.2116i		−	1.83050i
$944$		0			0
$945$	6.38000	−	1.62173i	0.207541	−	0.0527550i
$946$		0			0
$947$		−	6.28370i		−	0.204193i	−0.994775	−	0.102096i	$-0.967445\pi$
$947$		−	6.28370i		−	0.204193i	0.994775	−	0.102096i	$-0.0325550\pi$
$948$		0			0
$949$	22.8771			0.742624
$950$		0			0
$951$	−7.29291			−0.236489
$952$		0			0
$953$		−	44.7341i		−	1.44908i	−0.689233	−	0.724540i	$-0.742052\pi$
$953$		−	44.7341i		−	1.44908i	0.689233	−	0.724540i	$-0.257948\pi$
$954$		0			0
$955$	−48.3117	+	12.2804i	−1.56333	+	0.397384i
$956$		0			0
$957$			14.5548i			0.470489i
$958$		0			0
$959$	−33.5878			−1.08461
$960$		0			0
$961$	25.2311			0.813906
$962$		0			0
$963$		−	2.78617i		−	0.0897832i
$964$		0			0
$965$	−13.5087	−	53.1440i	−0.434860	−	1.71076i
$966$		0			0
$967$		−	4.14972i		−	0.133446i	−0.997772	−	0.0667230i	$-0.978746\pi$
$967$		−	4.14972i		−	0.133446i	0.997772	−	0.0667230i	$-0.0212544\pi$
$968$		0			0
$969$	7.45305			0.239426
$970$		0			0
$971$	−12.6864			−0.407126			−0.203563	−	0.979062i	$-0.565252\pi$
$971$	−12.6864			−0.407126			−0.203563	+	0.979062i	$0.565252\pi$
$972$		0			0
$973$		−	36.1955i		−	1.16038i
$974$		0			0
$975$	12.8583	−	6.98848i	0.411797	−	0.223811i
$976$		0			0
$977$			20.2144i			0.646717i	0.946277	+	0.323359i	$0.104812\pi$
$977$			20.2144i			0.646717i	−0.946277	+	0.323359i	$0.895188\pi$
$978$		0			0
$979$	19.5853			0.625949
$980$		0			0
$981$	12.9061			0.412060
$982$		0			0
$983$			39.9959i			1.27567i	0.770173	+	0.637835i	$0.220170\pi$
$983$			39.9959i			1.27567i	−0.770173	+	0.637835i	$0.779830\pi$
$984$		0			0
$985$	6.44091	+	25.3389i	0.205224	+	0.807365i
$986$		0			0
$987$			24.0392i			0.765176i
$988$		0			0
$989$	79.8102			2.53782
$990$		0			0
$991$	−21.7724			−0.691622			−0.345811	−	0.938304i	$-0.612396\pi$
$991$	−21.7724			−0.691622			−0.345811	+	0.938304i	$0.612396\pi$
$992$		0			0
$993$		−	24.5504i		−	0.779084i
$994$		0			0
$995$	6.05503	−	1.53913i	0.191957	−	0.0487937i
$996$		0			0
$997$		−	39.4553i		−	1.24956i	−0.780800	−	0.624781i	$-0.785188\pi$
$997$		−	39.4553i		−	1.24956i	0.780800	−	0.624781i	$-0.214812\pi$
$998$		0			0
$999$	2.74048			0.0867049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.m.r.2449.6		10
4.3	odd	2		1020.2.g.d.409.1	✓	10
5.4	even	2	inner	4080.2.m.r.2449.1		10
12.11	even	2		3060.2.g.g.2449.9		10
20.3	even	4		5100.2.a.bd.1.5		5
20.7	even	4		5100.2.a.bc.1.1		5
20.19	odd	2		1020.2.g.d.409.6	yes	10
60.59	even	2		3060.2.g.g.2449.10		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1020.2.g.d.409.1	✓	10	4.3	odd	2
1020.2.g.d.409.6	yes	10	20.19	odd	2
3060.2.g.g.2449.9		10	12.11	even	2
3060.2.g.g.2449.10		10	60.59	even	2
4080.2.m.r.2449.1		10	5.4	even	2	inner
4080.2.m.r.2449.6		10	1.1	even	1	trivial
5100.2.a.bc.1.1		5	20.7	even	4
5100.2.a.bd.1.5		5	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.m (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-2.16715 + 0.550869i) q^{5} -2.94396i q^{7} -1.00000 q^{9} -2.22048 q^{11} -2.92695i q^{13} +(-0.550869 - 2.16715i) q^{15} -1.00000i q^{17} +7.45305 q^{19} +2.94396 q^{21} +8.15951i q^{23} +(4.39309 - 2.38763i) q^{25} -1.00000i q^{27} -6.55478 q^{29} +7.49874 q^{31} -2.22048i q^{33} +(1.62173 + 6.38000i) q^{35} +2.74048i q^{37} +2.92695 q^{39} -6.88909 q^{41} -9.78125i q^{43} +(2.16715 - 0.550869i) q^{45} +8.16561i q^{47} -1.66687 q^{49} +1.00000 q^{51} -8.94396i q^{53} +(4.81212 - 1.22320i) q^{55} +7.45305i q^{57} +2.18569 q^{59} -14.7082 q^{61} +2.94396i q^{63} +(1.61237 + 6.34314i) q^{65} +2.18569i q^{67} -8.15951 q^{69} -5.85390 q^{71} +7.81604i q^{73} +(2.38763 + 4.39309i) q^{75} +6.53700i q^{77} -12.9975 q^{79} +1.00000 q^{81} -11.5383i q^{83} +(0.550869 + 2.16715i) q^{85} -6.55478i q^{87} -8.82029 q^{89} -8.61681 q^{91} +7.49874i q^{93} +(-16.1519 + 4.10565i) q^{95} +3.01818i q^{97} +2.22048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{9} - 12 q^{11} - 4 q^{15} + 24 q^{19} - 4 q^{21} + 12 q^{25} - 12 q^{29} - 12 q^{31} - 4 q^{35} + 28 q^{41} - 30 q^{49} + 10 q^{51} - 26 q^{55} + 48 q^{59} - 28 q^{61} + 48 q^{65} - 12 q^{69}+ \cdots + 12 q^{99}+O(q^{100}) \) 10 * q - 10 * q^9 - 12 * q^11 - 4 * q^15 + 24 * q^19 - 4 * q^21 + 12 * q^25 - 12 * q^29 - 12 * q^31 - 4 * q^35 + 28 * q^41 - 30 * q^49 + 10 * q^51 - 26 * q^55 + 48 * q^59 - 28 * q^61 + 48 * q^65 - 12 * q^69 - 8 * q^75 + 44 * q^79 + 10 * q^81 + 4 * q^85 - 36 * q^89 - 40 * q^91 - 44 * q^95 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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