LMFDB - Embedded newform 2420.2.b.e.969.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2420,2,Mod(969,2420)] 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("2420.969"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2420 = 2^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2420.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,1,0,0,0,-10,0,0,0,0,0,11,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$19.3237972891$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 220)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			969.3
Root			$2.13746 + 0.656712i$ of defining polynomial
Character	$\chi$	$=$	2420.969
Dual form			2420.2.b.e.969.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.31342i q^{3} +(2.13746 - 0.656712i) q^{5} +3.46410i q^{7} +1.27492 q^{9} -6.09095i q^{13} +(0.862541 + 2.80739i) q^{15} -3.46410i q^{17} +4.00000 q^{19} -4.54983 q^{21} -8.24163i q^{23} +(4.13746 - 2.80739i) q^{25} +5.61478i q^{27} +6.54983 q^{29} +1.72508 q^{31} +(2.27492 + 7.40437i) q^{35} -8.24163i q^{37} +8.00000 q^{39} +6.54983 q^{41} -3.46410i q^{43} +(2.72508 - 0.837253i) q^{45} +2.62685i q^{47} -5.00000 q^{49} +4.54983 q^{51} +5.25370i q^{57} -14.2749 q^{59} -6.54983 q^{61} +4.41644i q^{63} +(-4.00000 - 13.0192i) q^{65} +10.8685i q^{67} +10.8248 q^{69} -2.27492 q^{71} +11.3446i q^{73} +(3.68729 + 5.43424i) q^{75} +4.54983 q^{79} -3.54983 q^{81} +1.78959i q^{83} +(-2.27492 - 7.40437i) q^{85} +8.60271i q^{87} -8.27492 q^{89} +21.0997 q^{91} +2.26577i q^{93} +(8.54983 - 2.62685i) q^{95} +3.94027i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} - 10 q^{9} + 11 q^{15} + 16 q^{19} + 12 q^{21} + 9 q^{25} - 4 q^{29} + 22 q^{31} - 6 q^{35} + 32 q^{39} - 4 q^{41} + 26 q^{45} - 20 q^{49} - 12 q^{51} - 42 q^{59} + 4 q^{61} - 16 q^{65} - 2 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) $ 4 * q + q^5 - 10 * q^9 + 11 * q^15 + 16 * q^19 + 12 * q^21 + 9 * q^25 - 4 * q^29 + 22 * q^31 - 6 * q^35 + 32 * q^39 - 4 * q^41 + 26 * q^45 - 20 * q^49 - 12 * q^51 - 42 * q^59 + 4 * q^61 - 16 * q^65 - 2 * q^69 + 6 * q^71 - 23 * q^75 - 12 * q^79 + 16 * q^81 + 6 * q^85 - 18 * q^89 + 24 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2420\mathbb{Z}\right)^\times$.

$n$	$1211$	$1937$	$2301$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			1.31342i			0.758306i	0.925334	+	0.379153i	$0.123785\pi$
$3$			1.31342i			0.758306i	−0.925334	+	0.379153i	$0.876215\pi$
$4$		0			0
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$6$		0			0
$7$			3.46410i			1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$			3.46410i			1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$	1.27492			0.424972
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		−	6.09095i		−	1.68933i	−0.535299	−	0.844663i	$-0.679801\pi$
$13$		−	6.09095i		−	1.68933i	0.535299	−	0.844663i	$-0.320199\pi$
$14$		0			0
$15$	0.862541	+	2.80739i	0.222707	+	0.724865i
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$	−4.54983			−0.992855
$22$		0			0
$23$		−	8.24163i		−	1.71850i	−0.511558	−	0.859249i	$-0.670931\pi$
$23$		−	8.24163i		−	1.71850i	0.511558	−	0.859249i	$-0.329069\pi$
$24$		0			0
$25$	4.13746	−	2.80739i	0.827492	−	0.561478i
$26$		0			0
$27$			5.61478i			1.08056i
$28$		0			0
$29$	6.54983			1.21627			0.608137	−	0.793832i	$-0.291917\pi$
$29$	6.54983			1.21627			0.608137	+	0.793832i	$0.291917\pi$
$30$		0			0
$31$	1.72508			0.309834			0.154917	−	0.987927i	$-0.450489\pi$
$31$	1.72508			0.309834			0.154917	+	0.987927i	$0.450489\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.27492	+	7.40437i	0.384531	+	1.25157i
$36$		0			0
$37$		−	8.24163i		−	1.35492i	−0.735562	−	0.677458i	$-0.763082\pi$
$37$		−	8.24163i		−	1.35492i	0.735562	−	0.677458i	$-0.236918\pi$
$38$		0			0
$39$	8.00000			1.28103
$40$		0			0
$41$	6.54983			1.02291			0.511456	−	0.859309i	$-0.329106\pi$
$41$	6.54983			1.02291			0.511456	+	0.859309i	$0.329106\pi$
$42$		0			0
$43$		−	3.46410i		−	0.528271i	−0.964486	−	0.264135i	$-0.914913\pi$
$43$		−	3.46410i		−	0.528271i	0.964486	−	0.264135i	$-0.0850865\pi$
$44$		0			0
$45$	2.72508	−	0.837253i	0.406231	−	0.124810i
$46$		0			0
$47$			2.62685i			0.383165i	0.981476	+	0.191583i	$0.0613620\pi$
$47$			2.62685i			0.383165i	−0.981476	+	0.191583i	$0.938638\pi$
$48$		0			0
$49$	−5.00000			−0.714286
$50$		0			0
$51$	4.54983			0.637104
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$			5.25370i			0.695869i
$58$		0			0
$59$	−14.2749			−1.85844			−0.929218	−	0.369532i	$-0.879518\pi$
$59$	−14.2749			−1.85844			−0.929218	+	0.369532i	$0.879518\pi$
$60$		0			0
$61$	−6.54983			−0.838620			−0.419310	−	0.907843i	$-0.637728\pi$
$61$	−6.54983			−0.838620			−0.419310	+	0.907843i	$0.637728\pi$
$62$		0			0
$63$			4.41644i			0.556419i
$64$		0			0
$65$	−4.00000	−	13.0192i	−0.496139	−	1.61483i
$66$		0			0
$67$			10.8685i			1.32780i	0.747823	+	0.663898i	$0.231099\pi$
$67$			10.8685i			1.32780i	−0.747823	+	0.663898i	$0.768901\pi$
$68$		0			0
$69$	10.8248			1.30315
$70$		0			0
$71$	−2.27492			−0.269983			−0.134992	−	0.990847i	$-0.543101\pi$
$71$	−2.27492			−0.269983			−0.134992	+	0.990847i	$0.543101\pi$
$72$		0			0
$73$			11.3446i			1.32779i	0.747826	+	0.663895i	$0.231098\pi$
$73$			11.3446i			1.32779i	−0.747826	+	0.663895i	$0.768902\pi$
$74$		0			0
$75$	3.68729	+	5.43424i	0.425772	+	0.627492i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	4.54983			0.511896			0.255948	−	0.966691i	$-0.417612\pi$
$79$	4.54983			0.511896			0.255948	+	0.966691i	$0.417612\pi$
$80$		0			0
$81$	−3.54983			−0.394426
$82$		0			0
$83$			1.78959i			0.196434i	0.995165	+	0.0982168i	$0.0313139\pi$
$83$			1.78959i			0.196434i	−0.995165	+	0.0982168i	$0.968686\pi$
$84$		0			0
$85$	−2.27492	−	7.40437i	−0.246749	−	0.803117i
$86$		0			0
$87$			8.60271i			0.922307i
$88$		0			0
$89$	−8.27492			−0.877139			−0.438570	−	0.898697i	$-0.644515\pi$
$89$	−8.27492			−0.877139			−0.438570	+	0.898697i	$0.644515\pi$
$90$		0			0
$91$	21.0997			2.21185
$92$		0			0
$93$			2.26577i			0.234949i
$94$		0			0
$95$	8.54983	−	2.62685i	0.877195	−	0.269509i
$96$		0			0
$97$			3.94027i			0.400074i	0.979788	+	0.200037i	$0.0641062\pi$
$97$			3.94027i			0.400074i	−0.979788	+	0.200037i	$0.935894\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	10.0000			0.995037			0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000			0.995037			0.497519	+	0.867453i	$0.334245\pi$
$102$		0			0
$103$			9.55505i			0.941487i	0.882270	+	0.470744i	$0.156014\pi$
$103$			9.55505i			0.941487i	−0.882270	+	0.470744i	$0.843986\pi$
$104$		0			0
$105$	−9.72508	+	2.98793i	−0.949071	+	0.291592i
$106$		0			0
$107$			3.46410i			0.334887i	0.985882	+	0.167444i	$0.0535512\pi$
$107$			3.46410i			0.334887i	−0.985882	+	0.167444i	$0.946449\pi$
$108$		0			0
$109$	−19.0997			−1.82942			−0.914708	−	0.404115i	$-0.867580\pi$
$109$	−19.0997			−1.82942			−0.914708	+	0.404115i	$0.867580\pi$
$110$		0			0
$111$	10.8248			1.02744
$112$		0			0
$113$		−	3.94027i		−	0.370670i	−0.982675	−	0.185335i	$-0.940663\pi$
$113$		−	3.94027i		−	0.370670i	0.982675	−	0.185335i	$-0.0593370\pi$
$114$		0			0
$115$	−5.41238	−	17.6161i	−0.504707	−	1.64271i
$116$		0			0
$117$		−	7.76546i		−	0.717917i
$118$		0			0
$119$	12.0000			1.10004
$120$		0			0
$121$		0			0
$122$		0			0
$123$			8.60271i			0.775680i
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		−	6.09095i		−	0.540484i	−0.962792	−	0.270242i	$-0.912896\pi$
$127$		−	6.09095i		−	0.540484i	0.962792	−	0.270242i	$-0.0871038\pi$
$128$		0			0
$129$	4.54983			0.400591
$130$		0			0
$131$	−8.54983			−0.747003			−0.373501	−	0.927630i	$-0.621843\pi$
$131$	−8.54983			−0.747003			−0.373501	+	0.927630i	$0.621843\pi$
$132$		0			0
$133$			13.8564i			1.20150i
$134$		0			0
$135$	3.68729	+	12.0014i	0.317352	+	1.03291i
$136$		0			0
$137$		−	3.94027i		−	0.336640i	−0.985732	−	0.168320i	$-0.946166\pi$
$137$		−	3.94027i		−	0.336640i	0.985732	−	0.168320i	$-0.0538342\pi$
$138$		0			0
$139$	−8.54983			−0.725187			−0.362594	−	0.931947i	$-0.618109\pi$
$139$	−8.54983			−0.725187			−0.362594	+	0.931947i	$0.618109\pi$
$140$		0			0
$141$	−3.45017			−0.290556
$142$		0			0
$143$		0			0
$144$		0			0
$145$	14.0000	−	4.30136i	1.16264	−	0.357208i
$146$		0			0
$147$		−	6.56712i		−	0.541647i
$148$		0			0
$149$	−2.00000			−0.163846			−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000			−0.163846			−0.0819232	+	0.996639i	$0.526106\pi$
$150$		0			0
$151$	−3.45017			−0.280770			−0.140385	−	0.990097i	$-0.544834\pi$
$151$	−3.45017			−0.280770			−0.140385	+	0.990097i	$0.544834\pi$
$152$		0			0
$153$		−	4.41644i		−	0.357048i
$154$		0			0
$155$	3.68729	−	1.13288i	0.296171	−	0.0909953i
$156$		0			0
$157$			23.0504i			1.83962i	0.392364	+	0.919810i	$0.371657\pi$
$157$			23.0504i			1.83962i	−0.392364	+	0.919810i	$0.628343\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	28.5498			2.25004
$162$		0			0
$163$			0.952341i			0.0745931i	0.999304	+	0.0372966i	$0.0118746\pi$
$163$			0.952341i			0.0745931i	−0.999304	+	0.0372966i	$0.988125\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	6.09095i		−	0.471332i	−0.971834	−	0.235666i	$-0.924273\pi$
$167$		−	6.09095i		−	0.471332i	0.971834	−	0.235666i	$-0.0757271\pi$
$168$		0			0
$169$	−24.0997			−1.85382
$170$		0			0
$171$	5.09967			0.389981
$172$		0			0
$173$			13.9715i			1.06223i	0.847299	+	0.531117i	$0.178227\pi$
$173$			13.9715i			1.06223i	−0.847299	+	0.531117i	$0.821773\pi$
$174$		0			0
$175$	9.72508	+	14.3326i	0.735147	+	1.08344i
$176$		0			0
$177$		−	18.7490i		−	1.40926i
$178$		0			0
$179$	10.2749			0.767983			0.383992	−	0.923337i	$-0.374549\pi$
$179$	10.2749			0.767983			0.383992	+	0.923337i	$0.374549\pi$
$180$		0			0
$181$	3.72508			0.276883			0.138442	−	0.990371i	$-0.455791\pi$
$181$	3.72508			0.276883			0.138442	+	0.990371i	$0.455791\pi$
$182$		0			0
$183$		−	8.60271i		−	0.635931i
$184$		0			0
$185$	−5.41238	−	17.6161i	−0.397926	−	1.29516i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−19.4502			−1.41479
$190$		0			0
$191$	6.27492			0.454037			0.227019	−	0.973890i	$-0.427102\pi$
$191$	6.27492			0.454037			0.227019	+	0.973890i	$0.427102\pi$
$192$		0			0
$193$			19.9474i			1.43584i	0.696125	+	0.717921i	$0.254906\pi$
$193$			19.9474i			1.43584i	−0.696125	+	0.717921i	$0.745094\pi$
$194$		0			0
$195$	17.0997	−	5.25370i	1.22453	−	0.376225i
$196$		0			0
$197$			1.78959i			0.127503i	0.997966	+	0.0637517i	$0.0203066\pi$
$197$			1.78959i			0.127503i	−0.997966	+	0.0637517i	$0.979693\pi$
$198$		0			0
$199$	5.09967			0.361506			0.180753	−	0.983529i	$-0.442147\pi$
$199$	5.09967			0.361506			0.180753	+	0.983529i	$0.442147\pi$
$200$		0			0
$201$	−14.2749			−1.00688
$202$		0			0
$203$			22.6893i			1.59248i
$204$		0			0
$205$	14.0000	−	4.30136i	0.977802	−	0.300420i
$206$		0			0
$207$		−	10.5074i		−	0.730314i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	13.0997			0.901818			0.450909	−	0.892570i	$-0.351100\pi$
$211$	13.0997			0.901818			0.450909	+	0.892570i	$0.351100\pi$
$212$		0			0
$213$		−	2.98793i		−	0.204730i
$214$		0			0
$215$	−2.27492	−	7.40437i	−0.155148	−	0.504974i
$216$		0			0
$217$			5.97586i			0.405668i
$218$		0			0
$219$	−14.9003			−1.00687
$220$		0			0
$221$	−21.0997			−1.41932
$222$		0			0
$223$		−	13.4953i		−	0.903714i	−0.892090	−	0.451857i	$-0.850762\pi$
$223$		−	13.4953i		−	0.903714i	0.892090	−	0.451857i	$-0.149238\pi$
$224$		0			0
$225$	5.27492	−	3.57919i	0.351661	−	0.238613i
$226$		0			0
$227$			10.3923i			0.689761i	0.938647	+	0.344881i	$0.112081\pi$
$227$			10.3923i			0.689761i	−0.938647	+	0.344881i	$0.887919\pi$
$228$		0			0
$229$	20.2749			1.33980			0.669902	−	0.742449i	$-0.266336\pi$
$229$	20.2749			1.33980			0.669902	+	0.742449i	$0.266336\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	19.9474i		−	1.30679i	−0.757015	−	0.653397i	$-0.773343\pi$
$233$		−	19.9474i		−	1.30679i	0.757015	−	0.653397i	$-0.226657\pi$
$234$		0			0
$235$	1.72508	+	5.61478i	0.112532	+	0.366268i
$236$		0			0
$237$			5.97586i			0.388174i
$238$		0			0
$239$	4.54983			0.294304			0.147152	−	0.989114i	$-0.452989\pi$
$239$	4.54983			0.294304			0.147152	+	0.989114i	$0.452989\pi$
$240$		0			0
$241$	6.54983			0.421912			0.210956	−	0.977496i	$-0.432342\pi$
$241$	6.54983			0.421912			0.210956	+	0.977496i	$0.432342\pi$
$242$		0			0
$243$			12.1819i			0.781469i
$244$		0			0
$245$	−10.6873	+	3.28356i	−0.682786	+	0.209779i
$246$		0			0
$247$		−	24.3638i		−	1.55023i
$248$		0			0
$249$	−2.35050			−0.148957
$250$		0			0
$251$	2.82475			0.178297			0.0891484	−	0.996018i	$-0.471585\pi$
$251$	2.82475			0.178297			0.0891484	+	0.996018i	$0.471585\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	9.72508	−	2.98793i	0.609008	−	0.187111i
$256$		0			0
$257$			19.1101i			1.19206i	0.802964	+	0.596028i	$0.203255\pi$
$257$			19.1101i			1.19206i	−0.802964	+	0.596028i	$0.796745\pi$
$258$		0			0
$259$	28.5498			1.77400
$260$		0			0
$261$	8.35050			0.516883
$262$		0			0
$263$		−	3.46410i		−	0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$263$		−	3.46410i		−	0.213606i	0.994280	−	0.106803i	$-0.0340614\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		−	10.8685i		−	0.665140i
$268$		0			0
$269$	18.0000			1.09748			0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000			1.09748			0.548740	+	0.835993i	$0.315108\pi$
$270$		0			0
$271$	−25.0997			−1.52470			−0.762348	−	0.647167i	$-0.775954\pi$
$271$	−25.0997			−1.52470			−0.762348	+	0.647167i	$0.775954\pi$
$272$		0			0
$273$			27.7128i			1.67726i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			13.9715i			0.839466i	0.907648	+	0.419733i	$0.137876\pi$
$277$			13.9715i			0.839466i	−0.907648	+	0.419733i	$0.862124\pi$
$278$		0			0
$279$	2.19934			0.131671
$280$		0			0
$281$	−6.00000			−0.357930			−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000			−0.357930			−0.178965	+	0.983855i	$0.557275\pi$
$282$		0			0
$283$			13.0192i			0.773908i	0.922099	+	0.386954i	$0.126473\pi$
$283$			13.0192i			0.773908i	−0.922099	+	0.386954i	$0.873527\pi$
$284$		0			0
$285$	3.45017	+	11.2296i	0.204370	+	0.665182i
$286$		0			0
$287$			22.6893i			1.33931i
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$	−5.17525			−0.303378
$292$		0			0
$293$			13.0192i			0.760587i	0.924866	+	0.380294i	$0.124177\pi$
$293$			13.0192i			0.760587i	−0.924866	+	0.380294i	$0.875823\pi$
$294$		0			0
$295$	−30.5120	+	9.37451i	−1.77648	+	0.545805i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−50.1993			−2.90310
$300$		0			0
$301$	12.0000			0.691669
$302$		0			0
$303$			13.1342i			0.754542i
$304$		0			0
$305$	−14.0000	+	4.30136i	−0.801638	+	0.246295i
$306$		0			0
$307$		−	26.8756i		−	1.53387i	−0.641725	−	0.766935i	$-0.721781\pi$
$307$		−	26.8756i		−	1.53387i	0.641725	−	0.766935i	$-0.278219\pi$
$308$		0			0
$309$	−12.5498			−0.713935
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$			8.24163i			0.465844i	0.972495	+	0.232922i	$0.0748287\pi$
$313$			8.24163i			0.465844i	−0.972495	+	0.232922i	$0.925171\pi$
$314$		0			0
$315$	2.90033	+	9.43996i	0.163415	+	0.531882i
$316$		0			0
$317$			14.4477i			0.811462i	0.913993	+	0.405731i	$0.132983\pi$
$317$			14.4477i			0.811462i	−0.913993	+	0.405731i	$0.867017\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−4.54983			−0.253947
$322$		0			0
$323$		−	13.8564i		−	0.770991i
$324$		0			0
$325$	−17.0997	−	25.2011i	−0.948519	−	1.39790i
$326$		0			0
$327$		−	25.0860i		−	1.38726i
$328$		0			0
$329$	−9.09967			−0.501681
$330$		0			0
$331$	21.7251			1.19412			0.597059	−	0.802197i	$-0.296336\pi$
$331$	21.7251			1.19412			0.597059	+	0.802197i	$0.296336\pi$
$332$		0			0
$333$		−	10.5074i		−	0.575802i
$334$		0			0
$335$	7.13746	+	23.2309i	0.389961	+	1.26924i
$336$		0			0
$337$		−	13.9715i		−	0.761076i	−0.924765	−	0.380538i	$-0.875739\pi$
$337$		−	13.9715i		−	0.761076i	0.924765	−	0.380538i	$-0.124261\pi$
$338$		0			0
$339$	5.17525			0.281081
$340$		0			0
$341$		0			0
$342$		0			0
$343$			6.92820i			0.374088i
$344$		0			0
$345$	23.1375	−	7.10874i	1.24568	−	0.382722i
$346$		0			0
$347$			11.3446i			0.609013i	0.952510	+	0.304506i	$0.0984915\pi$
$347$			11.3446i			0.609013i	−0.952510	+	0.304506i	$0.901509\pi$
$348$		0			0
$349$	−31.0997			−1.66473			−0.832364	−	0.554230i	$-0.813013\pi$
$349$	−31.0997			−1.66473			−0.832364	+	0.554230i	$0.813013\pi$
$350$		0			0
$351$	34.1993			1.82543
$352$		0			0
$353$		−	27.3517i		−	1.45579i	−0.685691	−	0.727893i	$-0.740500\pi$
$353$		−	27.3517i		−	1.45579i	0.685691	−	0.727893i	$-0.259500\pi$
$354$		0			0
$355$	−4.86254	+	1.49397i	−0.258077	+	0.0792915i
$356$		0			0
$357$			15.7611i			0.834165i
$358$		0			0
$359$	−33.0997			−1.74693			−0.873467	−	0.486884i	$-0.838134\pi$
$359$	−33.0997			−1.74693			−0.873467	+	0.486884i	$0.838134\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.45017	+	24.2487i	0.389959	+	1.26924i
$366$		0			0
$367$			5.61478i			0.293089i	0.989204	+	0.146545i	$0.0468152\pi$
$367$			5.61478i			0.293089i	−0.989204	+	0.146545i	$0.953185\pi$
$368$		0			0
$369$	8.35050			0.434709
$370$		0			0
$371$		0			0
$372$		0			0
$373$			8.71780i			0.451390i	0.974198	+	0.225695i	$0.0724654\pi$
$373$			8.71780i			0.451390i	−0.974198	+	0.225695i	$0.927535\pi$
$374$		0			0
$375$	11.4502	+	9.19397i	0.591284	+	0.474774i
$376$		0			0
$377$		−	39.8947i		−	2.05468i
$378$		0			0
$379$	−35.9244			−1.84531			−0.922657	−	0.385622i	$-0.873987\pi$
$379$	−35.9244			−1.84531			−0.922657	+	0.385622i	$0.873987\pi$
$380$		0			0
$381$	8.00000			0.409852
$382$		0			0
$383$		−	31.6531i		−	1.61740i	−0.588223	−	0.808699i	$-0.700172\pi$
$383$		−	31.6531i		−	1.61740i	0.588223	−	0.808699i	$-0.299828\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	4.41644i		−	0.224500i
$388$		0			0
$389$	−20.2749			−1.02798			−0.513990	−	0.857796i	$-0.671833\pi$
$389$	−20.2749			−1.02798			−0.513990	+	0.857796i	$0.671833\pi$
$390$		0			0
$391$	−28.5498			−1.44383
$392$		0			0
$393$		−	11.2296i		−	0.566456i
$394$		0			0
$395$	9.72508	−	2.98793i	0.489322	−	0.150339i
$396$		0			0
$397$		−	26.0383i		−	1.30683i	−0.757002	−	0.653413i	$-0.773337\pi$
$397$		−	26.0383i		−	1.30683i	0.757002	−	0.653413i	$-0.226663\pi$
$398$		0			0
$399$	−18.1993			−0.911106
$400$		0			0
$401$	24.1993			1.20846			0.604229	−	0.796811i	$-0.293481\pi$
$401$	24.1993			1.20846			0.604229	+	0.796811i	$0.293481\pi$
$402$		0			0
$403$		−	10.5074i		−	0.523411i
$404$		0			0
$405$	−7.58762	+	2.33122i	−0.377032	+	0.115839i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.00000			0.0988936			0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000			0.0988936			0.0494468	+	0.998777i	$0.484254\pi$
$410$		0			0
$411$	5.17525			0.255276
$412$		0			0
$413$		−	49.4498i		−	2.43326i
$414$		0			0
$415$	1.17525	+	3.82518i	0.0576907	+	0.187771i
$416$		0			0
$417$		−	11.2296i		−	0.549914i
$418$		0			0
$419$	−4.00000			−0.195413			−0.0977064	−	0.995215i	$-0.531151\pi$
$419$	−4.00000			−0.195413			−0.0977064	+	0.995215i	$0.531151\pi$
$420$		0			0
$421$	−19.0997			−0.930861			−0.465430	−	0.885084i	$-0.654100\pi$
$421$	−19.0997			−0.930861			−0.465430	+	0.885084i	$0.654100\pi$
$422$		0			0
$423$			3.34901i			0.162835i
$424$		0			0
$425$	−9.72508	−	14.3326i	−0.471736	−	0.695232i
$426$		0			0
$427$		−	22.6893i		−	1.09801i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.4502			0.551535			0.275768	−	0.961224i	$-0.411068\pi$
$431$	11.4502			0.551535			0.275768	+	0.961224i	$0.411068\pi$
$432$		0			0
$433$			15.1698i			0.729016i	0.931200	+	0.364508i	$0.118763\pi$
$433$			15.1698i			0.729016i	−0.931200	+	0.364508i	$0.881237\pi$
$434$		0			0
$435$	5.64950	+	18.3879i	0.270873	+	0.881634i
$436$		0			0
$437$		−	32.9665i		−	1.57700i
$438$		0			0
$439$	25.0997			1.19794			0.598971	−	0.800771i	$-0.295576\pi$
$439$	25.0997			1.19794			0.598971	+	0.800771i	$0.295576\pi$
$440$		0			0
$441$	−6.37459			−0.303552
$442$		0			0
$443$			34.2799i			1.62869i	0.580382	+	0.814344i	$0.302903\pi$
$443$			34.2799i			1.62869i	−0.580382	+	0.814344i	$0.697097\pi$
$444$		0			0
$445$	−17.6873	+	5.43424i	−0.838458	+	0.257608i
$446$		0			0
$447$		−	2.62685i		−	0.124246i
$448$		0			0
$449$	3.72508			0.175798			0.0878988	−	0.996129i	$-0.471985\pi$
$449$	3.72508			0.175798			0.0878988	+	0.996129i	$0.471985\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	4.53153i		−	0.212910i
$454$		0			0
$455$	45.0997	−	13.8564i	2.11431	−	0.649598i
$456$		0			0
$457$		−	2.51176i		−	0.117495i	−0.998273	−	0.0587476i	$-0.981289\pi$
$457$		−	2.51176i		−	0.117495i	0.998273	−	0.0587476i	$-0.0187107\pi$
$458$		0			0
$459$	19.4502			0.907856
$460$		0			0
$461$	14.5498			0.677653			0.338827	−	0.940849i	$-0.389970\pi$
$461$	14.5498			0.677653			0.338827	+	0.940849i	$0.389970\pi$
$462$		0			0
$463$		−	13.4953i		−	0.627181i	−0.949558	−	0.313590i	$-0.898468\pi$
$463$		−	13.4953i		−	0.627181i	0.949558	−	0.313590i	$-0.101532\pi$
$464$		0			0
$465$	1.48796	+	4.84298i	0.0690023	+	0.224588i
$466$		0			0
$467$		−	32.6054i		−	1.50880i	−0.656415	−	0.754400i	$-0.727928\pi$
$467$		−	32.6054i		−	1.50880i	0.656415	−	0.754400i	$-0.272072\pi$
$468$		0			0
$469$	−37.6495			−1.73849
$470$		0			0
$471$	−30.2749			−1.39499
$472$		0			0
$473$		0			0
$474$		0			0
$475$	16.5498	−	11.2296i	0.759358	−	0.515247i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	41.0997			1.87789			0.938946	−	0.344065i	$-0.111804\pi$
$479$	41.0997			1.87789			0.938946	+	0.344065i	$0.111804\pi$
$480$		0			0
$481$	−50.1993			−2.28889
$482$		0			0
$483$			37.4980i			1.70622i
$484$		0			0
$485$	2.58762	+	8.42217i	0.117498	+	0.382431i
$486$		0			0
$487$		−	8.24163i		−	0.373464i	−0.982411	−	0.186732i	$-0.940210\pi$
$487$		−	8.24163i		−	0.373464i	0.982411	−	0.186732i	$-0.0597896\pi$
$488$		0			0
$489$	−1.25083			−0.0565644
$490$		0			0
$491$	−21.0997			−0.952215			−0.476107	−	0.879387i	$-0.657953\pi$
$491$	−21.0997			−0.952215			−0.476107	+	0.879387i	$0.657953\pi$
$492$		0			0
$493$		−	22.6893i		−	1.02187i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	7.88054i		−	0.353491i
$498$		0			0
$499$	−41.0997			−1.83987			−0.919937	−	0.392066i	$-0.871760\pi$
$499$	−41.0997			−1.83987			−0.919937	+	0.392066i	$0.871760\pi$
$500$		0			0
$501$	8.00000			0.357414
$502$		0			0
$503$		−	12.2970i		−	0.548296i	−0.961688	−	0.274148i	$-0.911604\pi$
$503$		−	12.2970i		−	0.548296i	0.961688	−	0.274148i	$-0.0883957\pi$
$504$		0			0
$505$	21.3746	−	6.56712i	0.951157	−	0.292233i
$506$		0			0
$507$		−	31.6531i		−	1.40576i
$508$		0			0
$509$	−37.9244			−1.68097			−0.840485	−	0.541835i	$-0.817730\pi$
$509$	−37.9244			−1.68097			−0.840485	+	0.541835i	$0.817730\pi$
$510$		0			0
$511$	−39.2990			−1.73849
$512$		0			0
$513$			22.4591i			0.991594i
$514$		0			0
$515$	6.27492	+	20.4235i	0.276506	+	0.899968i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−18.3505			−0.805497
$520$		0			0
$521$	30.4743			1.33510			0.667551	−	0.744564i	$-0.267343\pi$
$521$	30.4743			1.33510			0.667551	+	0.744564i	$0.267343\pi$
$522$		0			0
$523$		−	23.5265i		−	1.02874i	−0.857567	−	0.514372i	$-0.828025\pi$
$523$		−	23.5265i		−	1.02874i	0.857567	−	0.514372i	$-0.171975\pi$
$524$		0			0
$525$	−18.8248	+	12.7732i	−0.821580	+	0.557466i
$526$		0			0
$527$		−	5.97586i		−	0.260313i
$528$		0			0
$529$	−44.9244			−1.95324
$530$		0			0
$531$	−18.1993			−0.789784
$532$		0			0
$533$		−	39.8947i		−	1.72803i
$534$		0			0
$535$	2.27492	+	7.40437i	0.0983532	+	0.320119i
$536$		0			0
$537$			13.4953i			0.582366i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−9.45017			−0.406294			−0.203147	−	0.979148i	$-0.565117\pi$
$541$	−9.45017			−0.406294			−0.203147	+	0.979148i	$0.565117\pi$
$542$		0			0
$543$			4.89261i			0.209962i
$544$		0			0
$545$	−40.8248	+	12.5430i	−1.74874	+	0.537282i
$546$		0			0
$547$			23.5265i			1.00592i	0.864309	+	0.502961i	$0.167756\pi$
$547$			23.5265i			1.00592i	−0.864309	+	0.502961i	$0.832244\pi$
$548$		0			0
$549$	−8.35050			−0.356391
$550$		0			0
$551$	26.1993			1.11613
$552$		0			0
$553$			15.7611i			0.670230i
$554$		0			0
$555$	23.1375	−	7.10874i	0.982130	−	0.301749i
$556$		0			0
$557$			20.8997i			0.885549i	0.896633	+	0.442774i	$0.146006\pi$
$557$			20.8997i			0.885549i	−0.896633	+	0.442774i	$0.853994\pi$
$558$		0			0
$559$	−21.0997			−0.892421
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	40.7320i		−	1.71665i	−0.513108	−	0.858324i	$-0.671506\pi$
$563$		−	40.7320i		−	1.71665i	0.513108	−	0.858324i	$-0.328494\pi$
$564$		0			0
$565$	−2.58762	−	8.42217i	−0.108862	−	0.354323i
$566$		0			0
$567$		−	12.2970i		−	0.516425i
$568$		0			0
$569$	−2.54983			−0.106895			−0.0534473	−	0.998571i	$-0.517021\pi$
$569$	−2.54983			−0.106895			−0.0534473	+	0.998571i	$0.517021\pi$
$570$		0			0
$571$	2.90033			0.121375			0.0606875	−	0.998157i	$-0.480671\pi$
$571$	2.90033			0.121375			0.0606875	+	0.998157i	$0.480671\pi$
$572$		0			0
$573$			8.24163i			0.344299i
$574$		0			0
$575$	−23.1375	−	34.0994i	−0.964899	−	1.42204i
$576$		0			0
$577$			20.4235i			0.850243i	0.905136	+	0.425121i	$0.139769\pi$
$577$			20.4235i			0.850243i	−0.905136	+	0.425121i	$0.860231\pi$
$578$		0			0
$579$	−26.1993			−1.08881
$580$		0			0
$581$	−6.19934			−0.257192
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−5.09967	−	16.5983i	−0.210845	−	0.686257i
$586$		0			0
$587$			18.3879i			0.758951i	0.925202	+	0.379476i	$0.123896\pi$
$587$			18.3879i			0.758951i	−0.925202	+	0.379476i	$0.876104\pi$
$588$		0			0
$589$	6.90033			0.284323
$590$		0			0
$591$	−2.35050			−0.0966865
$592$		0			0
$593$			9.43996i			0.387653i	0.981036	+	0.193826i	$0.0620899\pi$
$593$			9.43996i			0.387653i	−0.981036	+	0.193826i	$0.937910\pi$
$594$		0			0
$595$	25.6495	−	7.88054i	1.05153	−	0.323071i
$596$		0			0
$597$			6.69803i			0.274132i
$598$		0			0
$599$	26.1993			1.07048			0.535238	−	0.844701i	$-0.320222\pi$
$599$	26.1993			1.07048			0.535238	+	0.844701i	$0.320222\pi$
$600$		0			0
$601$	−14.0000			−0.571072			−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000			−0.571072			−0.285536	+	0.958368i	$0.592172\pi$
$602$		0			0
$603$			13.8564i			0.564276i
$604$		0			0
$605$		0			0
$606$		0			0
$607$			1.78959i			0.0726374i	0.999340	+	0.0363187i	$0.0115631\pi$
$607$			1.78959i			0.0726374i	−0.999340	+	0.0363187i	$0.988437\pi$
$608$		0			0
$609$	−29.8007			−1.20758
$610$		0			0
$611$	16.0000			0.647291
$612$		0			0
$613$		−	34.0339i		−	1.37462i	−0.726365	−	0.687309i	$-0.758792\pi$
$613$		−	34.0339i		−	1.37462i	0.726365	−	0.687309i	$-0.241208\pi$
$614$		0			0
$615$	5.64950	+	18.3879i	0.227810	+	0.741473i
$616$		0			0
$617$			34.6410i			1.39459i	0.716782	+	0.697297i	$0.245614\pi$
$617$			34.6410i			1.39459i	−0.716782	+	0.697297i	$0.754386\pi$
$618$		0			0
$619$	7.92442			0.318509			0.159255	−	0.987238i	$-0.449091\pi$
$619$	7.92442			0.318509			0.159255	+	0.987238i	$0.449091\pi$
$620$		0			0
$621$	46.2749			1.85695
$622$		0			0
$623$		−	28.6652i		−	1.14845i
$624$		0			0
$625$	9.23713	−	23.2309i	0.369485	−	0.929237i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−28.5498			−1.13836
$630$		0			0
$631$	43.3746			1.72672			0.863358	−	0.504593i	$-0.168357\pi$
$631$	43.3746			1.72672			0.863358	+	0.504593i	$0.168357\pi$
$632$		0			0
$633$			17.2054i			0.683854i
$634$		0			0
$635$	−4.00000	−	13.0192i	−0.158735	−	0.516649i
$636$		0			0
$637$			30.4547i			1.20666i
$638$		0			0
$639$	−2.90033			−0.114735
$640$		0			0
$641$	−5.37459			−0.212283			−0.106142	−	0.994351i	$-0.533850\pi$
$641$	−5.37459			−0.212283			−0.106142	+	0.994351i	$0.533850\pi$
$642$		0			0
$643$		−	0.591258i		−	0.0233170i	−0.999932	−	0.0116585i	$-0.996289\pi$
$643$		−	0.591258i		−	0.0233170i	0.999932	−	0.0116585i	$-0.00371109\pi$
$644$		0			0
$645$	9.72508	−	2.98793i	0.382925	−	0.117650i
$646$		0			0
$647$			23.7725i			0.934595i	0.884100	+	0.467298i	$0.154772\pi$
$647$			23.7725i			0.934595i	−0.884100	+	0.467298i	$0.845228\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−7.84884			−0.307620
$652$		0			0
$653$			16.8443i			0.659170i	0.944126	+	0.329585i	$0.106909\pi$
$653$			16.8443i			0.659170i	−0.944126	+	0.329585i	$0.893091\pi$
$654$		0			0
$655$	−18.2749	+	5.61478i	−0.714060	+	0.219388i
$656$		0			0
$657$			14.4635i			0.564274i
$658$		0			0
$659$	8.54983			0.333054			0.166527	−	0.986037i	$-0.446745\pi$
$659$	8.54983			0.333054			0.166527	+	0.986037i	$0.446745\pi$
$660$		0			0
$661$	−11.1752			−0.434667			−0.217333	−	0.976097i	$-0.569736\pi$
$661$	−11.1752			−0.434667			−0.217333	+	0.976097i	$0.569736\pi$
$662$		0			0
$663$		−	27.7128i		−	1.07628i
$664$		0			0
$665$	9.09967	+	29.6175i	0.352870	+	1.14852i
$666$		0			0
$667$		−	53.9813i		−	2.09016i
$668$		0			0
$669$	17.7251			0.685291
$670$		0			0
$671$		0			0
$672$		0			0
$673$			43.5890i			1.68023i	0.542407	+	0.840116i	$0.317513\pi$
$673$			43.5890i			1.68023i	−0.542407	+	0.840116i	$0.682487\pi$
$674$		0			0
$675$	15.7629	+	23.2309i	0.606713	+	0.894158i
$676$		0			0
$677$			38.3353i			1.47335i	0.676250	+	0.736673i	$0.263604\pi$
$677$			38.3353i			1.47335i	−0.676250	+	0.736673i	$0.736396\pi$
$678$		0			0
$679$	−13.6495			−0.523820
$680$		0			0
$681$	−13.6495			−0.523050
$682$		0			0
$683$			2.62685i			0.100514i	0.998736	+	0.0502568i	$0.0160040\pi$
$683$			2.62685i			0.100514i	−0.998736	+	0.0502568i	$0.983996\pi$
$684$		0			0
$685$	−2.58762	−	8.42217i	−0.0988680	−	0.321795i
$686$		0			0
$687$			26.6296i			1.01598i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−7.37459			−0.280542			−0.140271	−	0.990113i	$-0.544797\pi$
$691$	−7.37459			−0.280542			−0.140271	+	0.990113i	$0.544797\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−18.2749	+	5.61478i	−0.693207	+	0.212981i
$696$		0			0
$697$		−	22.6893i		−	0.859418i
$698$		0			0
$699$	26.1993			0.990950
$700$		0			0
$701$	48.1993			1.82046			0.910232	−	0.414099i	$-0.135903\pi$
$701$	48.1993			1.82046			0.910232	+	0.414099i	$0.135903\pi$
$702$		0			0
$703$		−	32.9665i		−	1.24336i
$704$		0			0
$705$	−7.37459	+	2.26577i	−0.277743	+	0.0853337i
$706$		0			0
$707$			34.6410i			1.30281i
$708$		0			0
$709$	−30.4743			−1.14448			−0.572242	−	0.820085i	$-0.693926\pi$
$709$	−30.4743			−1.14448			−0.572242	+	0.820085i	$0.693926\pi$
$710$		0			0
$711$	5.80066			0.217542
$712$		0			0
$713$		−	14.2175i		−	0.532449i
$714$		0			0
$715$		0			0
$716$		0			0
$717$			5.97586i			0.223173i
$718$		0			0
$719$	9.72508			0.362684			0.181342	−	0.983420i	$-0.441956\pi$
$719$	9.72508			0.362684			0.181342	+	0.983420i	$0.441956\pi$
$720$		0			0
$721$	−33.0997			−1.23270
$722$		0			0
$723$			8.60271i			0.319938i
$724$		0			0
$725$	27.0997	−	18.3879i	1.00646	−	0.682911i
$726$		0			0
$727$			25.6772i			0.952315i	0.879360	+	0.476158i	$0.157971\pi$
$727$			25.6772i			0.952315i	−0.879360	+	0.476158i	$0.842029\pi$
$728$		0			0
$729$	−26.6495			−0.987019
$730$		0			0
$731$	−12.0000			−0.443836
$732$		0			0
$733$		−	37.3830i		−	1.38077i	−0.723442	−	0.690385i	$-0.757441\pi$
$733$		−	37.3830i		−	1.38077i	0.723442	−	0.690385i	$-0.242559\pi$
$734$		0			0
$735$	−4.31271	−	14.0369i	−0.159077	−	0.517761i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−33.6495			−1.23782			−0.618908	−	0.785463i	$-0.712425\pi$
$739$	−33.6495			−1.23782			−0.618908	+	0.785463i	$0.712425\pi$
$740$		0			0
$741$	32.0000			1.17555
$742$		0			0
$743$			40.7320i			1.49431i	0.664649	+	0.747155i	$0.268581\pi$
$743$			40.7320i			1.49431i	−0.664649	+	0.747155i	$0.731419\pi$
$744$		0			0
$745$	−4.27492	+	1.31342i	−0.156621	+	0.0481201i
$746$		0			0
$747$			2.28159i			0.0834788i
$748$		0			0
$749$	−12.0000			−0.438470
$750$		0			0
$751$	6.82475			0.249039			0.124519	−	0.992217i	$-0.460261\pi$
$751$	6.82475			0.249039			0.124519	+	0.992217i	$0.460261\pi$
$752$		0			0
$753$			3.71010i			0.135203i
$754$		0			0
$755$	−7.37459	+	2.26577i	−0.268389	+	0.0824596i
$756$		0			0
$757$		−	34.6410i		−	1.25905i	−0.776981	−	0.629525i	$-0.783250\pi$
$757$		−	34.6410i		−	1.25905i	0.776981	−	0.629525i	$-0.216750\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−22.5498			−0.817431			−0.408715	−	0.912662i	$-0.634023\pi$
$761$	−22.5498			−0.817431			−0.408715	+	0.912662i	$0.634023\pi$
$762$		0			0
$763$		−	66.1632i		−	2.39527i
$764$		0			0
$765$	−2.90033	−	9.43996i	−0.104862	−	0.341303i
$766$		0			0
$767$			86.9478i			3.13950i
$768$		0			0
$769$	−4.35050			−0.156883			−0.0784415	−	0.996919i	$-0.524994\pi$
$769$	−4.35050			−0.156883			−0.0784415	+	0.996919i	$0.524994\pi$
$770$		0			0
$771$	−25.0997			−0.903942
$772$		0			0
$773$		−	21.0148i		−	0.755849i	−0.925836	−	0.377925i	$-0.876638\pi$
$773$		−	21.0148i		−	0.755849i	0.925836	−	0.377925i	$-0.123362\pi$
$774$		0			0
$775$	7.13746	−	4.84298i	0.256385	−	0.173965i
$776$		0			0
$777$			37.4980i			1.34523i
$778$		0			0
$779$	26.1993			0.938689
$780$		0			0
$781$		0			0
$782$		0			0
$783$			36.7759i			1.31426i
$784$		0			0
$785$	15.1375	+	49.2692i	0.540279	+	1.75849i
$786$		0			0
$787$			0.837253i			0.0298449i	0.999889	+	0.0149224i	$0.00475013\pi$
$787$			0.837253i			0.0298449i	−0.999889	+	0.0149224i	$0.995250\pi$
$788$		0			0
$789$	4.54983			0.161978
$790$		0			0
$791$	13.6495			0.485320
$792$		0			0
$793$			39.8947i			1.41670i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			17.7967i			0.630391i	0.949027	+	0.315195i	$0.102070\pi$
$797$			17.7967i			0.630391i	−0.949027	+	0.315195i	$0.897930\pi$
$798$		0			0
$799$	9.09967			0.321923
$800$		0			0
$801$	−10.5498			−0.372760
$802$		0			0
$803$		0			0
$804$		0			0
$805$	61.0241	−	18.7490i	2.15082	−	0.660816i
$806$		0			0
$807$			23.6416i			0.832225i
$808$		0			0
$809$	35.6495			1.25337			0.626685	−	0.779273i	$-0.284411\pi$
$809$	35.6495			1.25337			0.626685	+	0.779273i	$0.284411\pi$
$810$		0			0
$811$	−41.6495			−1.46251			−0.731256	−	0.682103i	$-0.761065\pi$
$811$	−41.6495			−1.46251			−0.731256	+	0.682103i	$0.761065\pi$
$812$		0			0
$813$		−	32.9665i		−	1.15619i
$814$		0			0
$815$	0.625414	+	2.03559i	0.0219073	+	0.0713036i
$816$		0			0
$817$		−	13.8564i		−	0.484774i
$818$		0			0
$819$	26.9003			0.939974
$820$		0			0
$821$	47.0997			1.64379			0.821895	−	0.569639i	$-0.192917\pi$
$821$	47.0997			1.64379			0.821895	+	0.569639i	$0.192917\pi$
$822$		0			0
$823$			48.1363i			1.67793i	0.544187	+	0.838964i	$0.316838\pi$
$823$			48.1363i			1.67793i	−0.544187	+	0.838964i	$0.683162\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	14.9238i		−	0.518953i	−0.965749	−	0.259476i	$-0.916450\pi$
$827$		−	14.9238i		−	0.518953i	0.965749	−	0.259476i	$-0.0835499\pi$
$828$		0			0
$829$	−19.7251			−0.685080			−0.342540	−	0.939503i	$-0.611287\pi$
$829$	−19.7251			−0.685080			−0.342540	+	0.939503i	$0.611287\pi$
$830$		0			0
$831$	−18.3505			−0.636572
$832$		0			0
$833$			17.3205i			0.600120i
$834$		0			0
$835$	−4.00000	−	13.0192i	−0.138426	−	0.450546i
$836$		0			0
$837$			9.68596i			0.334796i
$838$		0			0
$839$	2.27492			0.0785389			0.0392694	−	0.999229i	$-0.487497\pi$
$839$	2.27492			0.0785389			0.0392694	+	0.999229i	$0.487497\pi$
$840$		0			0
$841$	13.9003			0.479322
$842$		0			0
$843$		−	7.88054i		−	0.271420i
$844$		0			0
$845$	−51.5120	+	15.8265i	−1.77207	+	0.544450i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−17.0997			−0.586859
$850$		0			0
$851$	−67.9244			−2.32842
$852$		0			0
$853$			12.0668i			0.413160i	0.978430	+	0.206580i	$0.0662333\pi$
$853$			12.0668i			0.413160i	−0.978430	+	0.206580i	$0.933767\pi$
$854$		0			0
$855$	10.9003	−	3.34901i	0.372783	−	0.114534i
$856$		0			0
$857$			56.4931i			1.92977i	0.262680	+	0.964883i	$0.415394\pi$
$857$			56.4931i			1.92977i	−0.262680	+	0.964883i	$0.584606\pi$
$858$		0			0
$859$	−5.72508			−0.195337			−0.0976687	−	0.995219i	$-0.531139\pi$
$859$	−5.72508			−0.195337			−0.0976687	+	0.995219i	$0.531139\pi$
$860$		0			0
$861$	−29.8007			−1.01560
$862$		0			0
$863$		−	42.5216i		−	1.44745i	−0.690088	−	0.723725i	$-0.742428\pi$
$863$		−	42.5216i		−	1.44745i	0.690088	−	0.723725i	$-0.257572\pi$
$864$		0			0
$865$	9.17525	+	29.8635i	0.311968	+	1.01539i
$866$		0			0
$867$			6.56712i			0.223031i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	66.1993			2.24308
$872$		0			0
$873$			5.02352i			0.170020i
$874$		0			0
$875$	30.1993	+	24.2487i	1.02092	+	0.819756i
$876$		0			0
$877$		−	58.1676i		−	1.96418i	−0.188415	−	0.982089i	$-0.560335\pi$
$877$		−	58.1676i		−	1.96418i	0.188415	−	0.982089i	$-0.439665\pi$
$878$		0			0
$879$	−17.0997			−0.576758
$880$		0			0
$881$	24.8248			0.836367			0.418184	−	0.908363i	$-0.362667\pi$
$881$	24.8248			0.836367			0.418184	+	0.908363i	$0.362667\pi$
$882$		0			0
$883$			0.952341i			0.0320488i	0.999872	+	0.0160244i	$0.00510095\pi$
$883$			0.952341i			0.0320488i	−0.999872	+	0.0160244i	$0.994899\pi$
$884$		0			0
$885$	−12.3127	−	40.0753i	−0.413887	−	1.34711i
$886$		0			0
$887$			7.04329i			0.236491i	0.992984	+	0.118245i	$0.0377269\pi$
$887$			7.04329i			0.236491i	−0.992984	+	0.118245i	$0.962273\pi$
$888$		0			0
$889$	21.0997			0.707660
$890$		0			0
$891$		0			0
$892$		0			0
$893$			10.5074i			0.351616i
$894$		0			0
$895$	21.9622	−	6.74766i	0.734116	−	0.225549i
$896$		0			0
$897$		−	65.9330i		−	2.20144i
$898$		0			0
$899$	11.2990			0.376843
$900$		0			0
$901$		0			0
$902$		0			0
$903$			15.7611i			0.524496i
$904$		0			0
$905$	7.96221	−	2.44631i	0.264673	−	0.0813180i
$906$		0			0
$907$			14.5786i			0.484074i	0.970267	+	0.242037i	$0.0778155\pi$
$907$			14.5786i			0.484074i	−0.970267	+	0.242037i	$0.922185\pi$
$908$		0			0
$909$	12.7492			0.422863
$910$		0			0
$911$	−17.0997			−0.566537			−0.283269	−	0.959041i	$-0.591419\pi$
$911$	−17.0997			−0.566537			−0.283269	+	0.959041i	$0.591419\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−5.64950	−	18.3879i	−0.186767	−	0.607886i
$916$		0			0
$917$		−	29.6175i		−	0.978056i
$918$		0			0
$919$	−38.7492			−1.27822			−0.639109	−	0.769116i	$-0.720697\pi$
$919$	−38.7492			−1.27822			−0.639109	+	0.769116i	$0.720697\pi$
$920$		0			0
$921$	35.2990			1.16314
$922$		0			0
$923$			13.8564i			0.456089i
$924$		0			0
$925$	−23.1375	−	34.0994i	−0.760755	−	1.12118i
$926$		0			0
$927$			12.1819i			0.400106i
$928$		0			0
$929$	−27.0997			−0.889111			−0.444556	−	0.895751i	$-0.646638\pi$
$929$	−27.0997			−0.889111			−0.444556	+	0.895751i	$0.646638\pi$
$930$		0			0
$931$	−20.0000			−0.655474
$932$		0			0
$933$		−	5.25370i		−	0.171998i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	30.2246i		−	0.987394i	−0.869634	−	0.493697i	$-0.835645\pi$
$937$		−	30.2246i		−	0.987394i	0.869634	−	0.493697i	$-0.164355\pi$
$938$		0			0
$939$	−10.8248			−0.353252
$940$		0			0
$941$	0.900331			0.0293500			0.0146750	−	0.999892i	$-0.495329\pi$
$941$	0.900331			0.0293500			0.0146750	+	0.999892i	$0.495329\pi$
$942$		0			0
$943$		−	53.9813i		−	1.75787i
$944$		0			0
$945$	−41.5739	+	12.7732i	−1.35240	+	0.415511i
$946$		0			0
$947$		−	8.24163i		−	0.267817i	−0.990994	−	0.133908i	$-0.957247\pi$
$947$		−	8.24163i		−	0.267817i	0.990994	−	0.133908i	$-0.0427528\pi$
$948$		0			0
$949$	69.0997			2.24307
$950$		0			0
$951$	−18.9759			−0.615336
$952$		0			0
$953$		−	6.81312i		−	0.220698i	−0.993893	−	0.110349i	$-0.964803\pi$
$953$		−	6.81312i		−	0.220698i	0.993893	−	0.110349i	$-0.0351969\pi$
$954$		0			0
$955$	13.4124	−	4.12081i	0.434014	−	0.133346i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	13.6495			0.440765
$960$		0			0
$961$	−28.0241			−0.904003
$962$		0			0
$963$			4.41644i			0.142318i
$964$		0			0
$965$	13.0997	+	42.6366i	0.421693	+	1.37252i
$966$		0			0
$967$			40.7320i			1.30985i	0.755693	+	0.654926i	$0.227300\pi$
$967$			40.7320i			1.30985i	−0.755693	+	0.654926i	$0.772700\pi$
$968$		0			0
$969$	18.1993			0.584647
$970$		0			0
$971$	29.7251			0.953923			0.476962	−	0.878924i	$-0.341738\pi$
$971$	29.7251			0.953923			0.476962	+	0.878924i	$0.341738\pi$
$972$		0			0
$973$		−	29.6175i		−	0.949493i
$974$		0			0
$975$	33.0997	−	22.4591i	1.06004	−	0.719267i
$976$		0			0
$977$			1.31342i			0.0420202i	0.999779	+	0.0210101i	$0.00668821\pi$
$977$			1.31342i			0.0420202i	−0.999779	+	0.0210101i	$0.993312\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−24.3505			−0.777452
$982$		0			0
$983$			30.9309i			0.986543i	0.869875	+	0.493272i	$0.164199\pi$
$983$			30.9309i			0.986543i	−0.869875	+	0.493272i	$0.835801\pi$
$984$		0			0
$985$	1.17525	+	3.82518i	0.0374465	+	0.121881i
$986$		0			0
$987$		−	11.9517i		−	0.380428i
$988$		0			0
$989$	−28.5498			−0.907832
$990$		0			0
$991$	34.1993			1.08638			0.543189	−	0.839611i	$-0.317217\pi$
$991$	34.1993			1.08638			0.543189	+	0.839611i	$0.317217\pi$
$992$		0			0
$993$			28.5342i			0.905507i
$994$		0			0
$995$	10.9003	−	3.34901i	0.345564	−	0.106171i
$996$		0			0
$997$			5.13861i			0.162741i	0.996684	+	0.0813707i	$0.0259298\pi$
$997$			5.13861i			0.162741i	−0.996684	+	0.0813707i	$0.974070\pi$
$998$		0			0
$999$	46.2749			1.46407

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2420.2.b.e.969.3		4
5.4	even	2	inner	2420.2.b.e.969.2		4
11.10	odd	2		220.2.b.b.89.3	yes	4
33.32	even	2		1980.2.c.g.1189.2		4
44.43	even	2		880.2.b.i.529.2		4
55.32	even	4		1100.2.a.j.1.3		4
55.43	even	4		1100.2.a.j.1.2		4
55.54	odd	2		220.2.b.b.89.2	✓	4
165.32	odd	4		9900.2.a.cb.1.4		4
165.98	odd	4		9900.2.a.cb.1.1		4
165.164	even	2		1980.2.c.g.1189.1		4
220.43	odd	4		4400.2.a.cd.1.3		4
220.87	odd	4		4400.2.a.cd.1.2		4
220.219	even	2		880.2.b.i.529.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.2.b.b.89.2	✓	4	55.54	odd	2
220.2.b.b.89.3	yes	4	11.10	odd	2
880.2.b.i.529.2		4	44.43	even	2
880.2.b.i.529.3		4	220.219	even	2
1100.2.a.j.1.2		4	55.43	even	4
1100.2.a.j.1.3		4	55.32	even	4
1980.2.c.g.1189.1		4	165.164	even	2
1980.2.c.g.1189.2		4	33.32	even	2
2420.2.b.e.969.2		4	5.4	even	2	inner
2420.2.b.e.969.3		4	1.1	even	1	trivial
4400.2.a.cd.1.2		4	220.87	odd	4
4400.2.a.cd.1.3		4	220.43	odd	4
9900.2.a.cb.1.1		4	165.98	odd	4
9900.2.a.cb.1.4		4	165.32	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2420 = 2^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2420.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.31342i q^{3} +(2.13746 - 0.656712i) q^{5} +3.46410i q^{7} +1.27492 q^{9} -6.09095i q^{13} +(0.862541 + 2.80739i) q^{15} -3.46410i q^{17} +4.00000 q^{19} -4.54983 q^{21} -8.24163i q^{23} +(4.13746 - 2.80739i) q^{25} +5.61478i q^{27} +6.54983 q^{29} +1.72508 q^{31} +(2.27492 + 7.40437i) q^{35} -8.24163i q^{37} +8.00000 q^{39} +6.54983 q^{41} -3.46410i q^{43} +(2.72508 - 0.837253i) q^{45} +2.62685i q^{47} -5.00000 q^{49} +4.54983 q^{51} +5.25370i q^{57} -14.2749 q^{59} -6.54983 q^{61} +4.41644i q^{63} +(-4.00000 - 13.0192i) q^{65} +10.8685i q^{67} +10.8248 q^{69} -2.27492 q^{71} +11.3446i q^{73} +(3.68729 + 5.43424i) q^{75} +4.54983 q^{79} -3.54983 q^{81} +1.78959i q^{83} +(-2.27492 - 7.40437i) q^{85} +8.60271i q^{87} -8.27492 q^{89} +21.0997 q^{91} +2.26577i q^{93} +(8.54983 - 2.62685i) q^{95} +3.94027i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} - 10 q^{9} + 11 q^{15} + 16 q^{19} + 12 q^{21} + 9 q^{25} - 4 q^{29} + 22 q^{31} - 6 q^{35} + 32 q^{39} - 4 q^{41} + 26 q^{45} - 20 q^{49} - 12 q^{51} - 42 q^{59} + 4 q^{61} - 16 q^{65} - 2 q^{69}+ \cdots + 4 q^{95}+O(q^{100}) \) 4 * q + q^5 - 10 * q^9 + 11 * q^15 + 16 * q^19 + 12 * q^21 + 9 * q^25 - 4 * q^29 + 22 * q^31 - 6 * q^35 + 32 * q^39 - 4 * q^41 + 26 * q^45 - 20 * q^49 - 12 * q^51 - 42 * q^59 + 4 * q^61 - 16 * q^65 - 2 * q^69 + 6 * q^71 - 23 * q^75 - 12 * q^79 + 16 * q^81 + 6 * q^85 - 18 * q^89 + 24 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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