LMFDB - Embedded newform 3360.2.ba.c.2591.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3360,2,Mod(2591,3360)] 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("3360.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3360, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,-4,0,0,0,0,0,8,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} + 8 x^{18} - 20 x^{17} + 50 x^{16} - 92 x^{15} + 144 x^{14} - 244 x^{13} + \cdots + 59049 $ x^20 - 4x^19 + 8x^18 - 20x^17 + 50x^16 - 92x^15 + 144x^14 - 244x^13 + 481x^12 - 696x^11 + 848x^10 - 2088x^9 + 4329x^8 - 6588x^7 + 11664x^6 - 22356x^5 + 36450x^4 - 43740x^3 + 52488x^2 - 78732*x + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.9
Root			$1.64300 - 0.548239i$ of defining polynomial
Character	$\chi$	$=$	3360.2591
Dual form			3360.2.ba.c.2591.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.774110 - 1.54944i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-1.80151 + 2.39887i) q^{9} -0.559419 q^{11} -3.35848 q^{13} +(-1.54944 + 0.774110i) q^{15} -3.44380i q^{17} -1.52129i q^{19} +(1.54944 - 0.774110i) q^{21} -4.13079 q^{23} -1.00000 q^{25} +(5.11146 + 0.934336i) q^{27} -10.0534i q^{29} -0.148209i q^{31} +(0.433052 + 0.866784i) q^{33} +1.00000 q^{35} +10.5011 q^{37} +(2.59983 + 5.20375i) q^{39} +6.79091i q^{41} +11.3827i q^{43} +(2.39887 + 1.80151i) q^{45} -9.99341 q^{47} -1.00000 q^{49} +(-5.33594 + 2.66588i) q^{51} +8.05096i q^{53} +0.559419i q^{55} +(-2.35714 + 1.17764i) q^{57} -4.78044 q^{59} +2.04177 q^{61} +(-2.39887 - 1.80151i) q^{63} +3.35848i q^{65} +2.20738i q^{67} +(3.19768 + 6.40039i) q^{69} -1.64755 q^{71} -9.19346 q^{73} +(0.774110 + 1.54944i) q^{75} -0.559419i q^{77} +15.5490i q^{79} +(-2.50914 - 8.64316i) q^{81} -5.78506 q^{83} -3.44380 q^{85} +(-15.5771 + 7.78241i) q^{87} +1.31868i q^{89} -3.35848i q^{91} +(-0.229640 + 0.114730i) q^{93} -1.52129 q^{95} +17.9337 q^{97} +(1.00780 - 1.34197i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{3} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 8 q^{23} - 20 q^{25} + 20 q^{27} - 40 q^{33} + 20 q^{35} + 16 q^{37} + 4 q^{39} - 20 q^{49} + 4 q^{51} - 16 q^{57} - 64 q^{59} - 64 q^{61}+ \cdots + 28 q^{99}+O(q^{100}) $ 20 * q - 4 * q^3 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 8 * q^23 - 20 * q^25 + 20 * q^27 - 40 * q^33 + 20 * q^35 + 16 * q^37 + 4 * q^39 - 20 * q^49 + 4 * q^51 - 16 * q^57 - 64 * q^59 - 64 * q^61 + 8 * q^69 - 40 * q^71 + 8 * q^73 + 4 * q^75 + 8 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 + 72 * q^93 + 16 * q^95 + 88 * q^97 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−0.774110	−	1.54944i	−0.446932	−	0.894568i
$3$	−0.774110	−	1.54944i	−0.446932	−	0.894568i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.80151	+	2.39887i	−0.600503	+	0.799623i
$10$		0			0
$11$	−0.559419			−0.168671			−0.0843356	−	0.996437i	$-0.526877\pi$
$11$	−0.559419			−0.168671			−0.0843356	+	0.996437i	$0.526877\pi$
$12$		0			0
$13$	−3.35848			−0.931474			−0.465737	−	0.884923i	$-0.654211\pi$
$13$	−3.35848			−0.931474			−0.465737	+	0.884923i	$0.654211\pi$
$14$		0			0
$15$	−1.54944	+	0.774110i	−0.400063	+	0.199874i
$16$		0			0
$17$		−	3.44380i		−	0.835243i	−0.908621	−	0.417622i	$-0.862864\pi$
$17$		−	3.44380i		−	0.835243i	0.908621	−	0.417622i	$-0.137136\pi$
$18$		0			0
$19$		−	1.52129i		−	0.349007i	−0.984657	−	0.174503i	$-0.944168\pi$
$19$		−	1.52129i		−	0.349007i	0.984657	−	0.174503i	$-0.0558320\pi$
$20$		0			0
$21$	1.54944	−	0.774110i	0.338115	−	0.168925i
$22$		0			0
$23$	−4.13079			−0.861328			−0.430664	−	0.902512i	$-0.641721\pi$
$23$	−4.13079			−0.861328			−0.430664	+	0.902512i	$0.641721\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.11146	+	0.934336i	0.983701	+	0.179813i
$28$		0			0
$29$		−	10.0534i		−	1.86686i	−0.358754	−	0.933432i	$-0.616798\pi$
$29$		−	10.0534i		−	1.86686i	0.358754	−	0.933432i	$-0.383202\pi$
$30$		0			0
$31$		−	0.148209i		−	0.0266191i	−0.999911	−	0.0133095i	$-0.995763\pi$
$31$		−	0.148209i		−	0.0266191i	0.999911	−	0.0133095i	$-0.00423668\pi$
$32$		0			0
$33$	0.433052	+	0.866784i	0.0753846	+	0.150888i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	10.5011			1.72638			0.863188	−	0.504882i	$-0.168464\pi$
$37$	10.5011			1.72638			0.863188	+	0.504882i	$0.168464\pi$
$38$		0			0
$39$	2.59983	+	5.20375i	0.416306	+	0.833267i
$40$		0			0
$41$			6.79091i			1.06056i	0.847822	+	0.530281i	$0.177914\pi$
$41$			6.79091i			1.06056i	−0.847822	+	0.530281i	$0.822086\pi$
$42$		0			0
$43$			11.3827i			1.73584i	0.496704	+	0.867920i	$0.334543\pi$
$43$			11.3827i			1.73584i	−0.496704	+	0.867920i	$0.665457\pi$
$44$		0			0
$45$	2.39887	+	1.80151i	0.357602	+	0.268553i
$46$		0			0
$47$	−9.99341			−1.45769			−0.728844	−	0.684680i	$-0.759942\pi$
$47$	−9.99341			−1.45769			−0.728844	+	0.684680i	$0.759942\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.33594	+	2.66588i	−0.747182	+	0.373297i
$52$		0			0
$53$			8.05096i			1.10588i	0.833220	+	0.552942i	$0.186495\pi$
$53$			8.05096i			1.10588i	−0.833220	+	0.552942i	$0.813505\pi$
$54$		0			0
$55$			0.559419i			0.0754320i
$56$		0			0
$57$	−2.35714	+	1.17764i	−0.312210	+	0.155983i
$58$		0			0
$59$	−4.78044			−0.622360			−0.311180	−	0.950351i	$-0.600724\pi$
$59$	−4.78044			−0.622360			−0.311180	+	0.950351i	$0.600724\pi$
$60$		0			0
$61$	2.04177			0.261421			0.130711	−	0.991421i	$-0.458274\pi$
$61$	2.04177			0.261421			0.130711	+	0.991421i	$0.458274\pi$
$62$		0			0
$63$	−2.39887	−	1.80151i	−0.302229	−	0.226969i
$64$		0			0
$65$			3.35848i			0.416568i
$66$		0			0
$67$			2.20738i			0.269675i	0.990868	+	0.134837i	$0.0430512\pi$
$67$			2.20738i			0.269675i	−0.990868	+	0.134837i	$0.956949\pi$
$68$		0			0
$69$	3.19768	+	6.40039i	0.384956	+	0.770516i
$70$		0			0
$71$	−1.64755			−0.195528			−0.0977641	−	0.995210i	$-0.531169\pi$
$71$	−1.64755			−0.195528			−0.0977641	+	0.995210i	$0.531169\pi$
$72$		0			0
$73$	−9.19346			−1.07601			−0.538007	−	0.842940i	$-0.680823\pi$
$73$	−9.19346			−1.07601			−0.538007	+	0.842940i	$0.680823\pi$
$74$		0			0
$75$	0.774110	+	1.54944i	0.0893865	+	0.178914i
$76$		0			0
$77$		−	0.559419i		−	0.0637517i
$78$		0			0
$79$			15.5490i			1.74940i	0.484663	+	0.874701i	$0.338942\pi$
$79$			15.5490i			1.74940i	−0.484663	+	0.874701i	$0.661058\pi$
$80$		0			0
$81$	−2.50914	−	8.64316i	−0.278793	−	0.960351i
$82$		0			0
$83$	−5.78506			−0.634993			−0.317497	−	0.948259i	$-0.602842\pi$
$83$	−5.78506			−0.634993			−0.317497	+	0.948259i	$0.602842\pi$
$84$		0			0
$85$	−3.44380			−0.373532
$86$		0			0
$87$	−15.5771	+	7.78241i	−1.67004	+	0.834362i
$88$		0			0
$89$			1.31868i			0.139780i	0.997555	+	0.0698901i	$0.0222649\pi$
$89$			1.31868i			0.139780i	−0.997555	+	0.0698901i	$0.977735\pi$
$90$		0			0
$91$		−	3.35848i		−	0.352064i
$92$		0			0
$93$	−0.229640	+	0.114730i	−0.0238126	+	0.0118969i
$94$		0			0
$95$	−1.52129			−0.156081
$96$		0			0
$97$	17.9337			1.82089			0.910444	−	0.413631i	$-0.135740\pi$
$97$	17.9337			1.82089			0.910444	+	0.413631i	$0.135740\pi$
$98$		0			0
$99$	1.00780	−	1.34197i	0.101288	−	0.134873i
$100$		0			0
$101$		−	13.3381i		−	1.32720i	−0.748090	−	0.663598i	$-0.769029\pi$
$101$		−	13.3381i		−	1.32720i	0.748090	−	0.663598i	$-0.230971\pi$
$102$		0			0
$103$			9.59178i			0.945107i	0.881302	+	0.472553i	$0.156668\pi$
$103$			9.59178i			0.945107i	−0.881302	+	0.472553i	$0.843332\pi$
$104$		0			0
$105$	−0.774110	−	1.54944i	−0.0755454	−	0.151210i
$106$		0			0
$107$	1.28937			0.124648			0.0623241	−	0.998056i	$-0.480149\pi$
$107$	1.28937			0.124648			0.0623241	+	0.998056i	$0.480149\pi$
$108$		0			0
$109$	13.7624			1.31820			0.659101	−	0.752054i	$-0.270937\pi$
$109$	13.7624			1.31820			0.659101	+	0.752054i	$0.270937\pi$
$110$		0			0
$111$	−8.12903	−	16.2708i	−0.771574	−	1.54436i
$112$		0			0
$113$			17.8453i			1.67874i	0.543559	+	0.839371i	$0.317076\pi$
$113$			17.8453i			1.67874i	−0.543559	+	0.839371i	$0.682924\pi$
$114$		0			0
$115$			4.13079i			0.385198i
$116$		0			0
$117$	6.05033	−	8.05655i	0.559353	−	0.744828i
$118$		0			0
$119$	3.44380			0.315692
$120$		0			0
$121$	−10.6871			−0.971550
$122$		0			0
$123$	10.5221	−	5.25691i	0.948744	−	0.474000i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$			8.36158i			0.741970i	0.928639	+	0.370985i	$0.120980\pi$
$127$			8.36158i			0.741970i	−0.928639	+	0.370985i	$0.879020\pi$
$128$		0			0
$129$	17.6367	−	8.81143i	1.55283	−	0.775803i
$130$		0			0
$131$	5.67560			0.495879			0.247940	−	0.968775i	$-0.420246\pi$
$131$	5.67560			0.495879			0.247940	+	0.968775i	$0.420246\pi$
$132$		0			0
$133$	1.52129			0.131912
$134$		0			0
$135$	0.934336	−	5.11146i	0.0804148	−	0.439924i
$136$		0			0
$137$			9.53927i			0.814994i	0.913206	+	0.407497i	$0.133598\pi$
$137$			9.53927i			0.814994i	−0.913206	+	0.407497i	$0.866402\pi$
$138$		0			0
$139$			17.9076i			1.51890i	0.650565	+	0.759451i	$0.274532\pi$
$139$			17.9076i			1.51890i	−0.650565	+	0.759451i	$0.725468\pi$
$140$		0			0
$141$	7.73599	+	15.4842i	0.651488	+	1.30400i
$142$		0			0
$143$	1.87880			0.157113
$144$		0			0
$145$	−10.0534			−0.834887
$146$		0			0
$147$	0.774110	+	1.54944i	0.0638475	+	0.127795i
$148$		0			0
$149$		−	7.54380i		−	0.618012i	−0.951060	−	0.309006i	$-0.900004\pi$
$149$		−	7.54380i		−	0.618012i	0.951060	−	0.309006i	$-0.0999964\pi$
$150$		0			0
$151$			1.48926i			0.121194i	0.998162	+	0.0605971i	$0.0193005\pi$
$151$			1.48926i			0.121194i	−0.998162	+	0.0605971i	$0.980700\pi$
$152$		0			0
$153$	8.26121	+	6.20403i	0.667879	+	0.501566i
$154$		0			0
$155$	−0.148209			−0.0119044
$156$		0			0
$157$	−16.1710			−1.29059			−0.645293	−	0.763935i	$-0.723265\pi$
$157$	−16.1710			−1.29059			−0.645293	+	0.763935i	$0.723265\pi$
$158$		0			0
$159$	12.4745	−	6.23233i	0.989288	−	0.494256i
$160$		0			0
$161$		−	4.13079i		−	0.325551i
$162$		0			0
$163$			17.0063i			1.33204i	0.745935	+	0.666019i	$0.232003\pi$
$163$			17.0063i			1.33204i	−0.745935	+	0.666019i	$0.767997\pi$
$164$		0			0
$165$	0.866784	−	0.433052i	0.0674791	−	0.0337130i
$166$		0			0
$167$	−6.40209			−0.495408			−0.247704	−	0.968836i	$-0.579676\pi$
$167$	−6.40209			−0.495408			−0.247704	+	0.968836i	$0.579676\pi$
$168$		0			0
$169$	−1.72062			−0.132356
$170$		0			0
$171$	3.64936	+	2.74061i	0.279074	+	0.209580i
$172$		0			0
$173$			2.70411i			0.205590i	0.994703	+	0.102795i	$0.0327786\pi$
$173$			2.70411i			0.205590i	−0.994703	+	0.102795i	$0.967221\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$	3.70059	+	7.40699i	0.278153	+	0.556743i
$178$		0			0
$179$	6.71305			0.501757			0.250879	−	0.968019i	$-0.419281\pi$
$179$	6.71305			0.501757			0.250879	+	0.968019i	$0.419281\pi$
$180$		0			0
$181$	11.4983			0.854659			0.427329	−	0.904096i	$-0.359454\pi$
$181$	11.4983			0.854659			0.427329	+	0.904096i	$0.359454\pi$
$182$		0			0
$183$	−1.58055	−	3.16359i	−0.116838	−	0.233859i
$184$		0			0
$185$		−	10.5011i		−	0.772059i
$186$		0			0
$187$			1.92652i			0.140881i
$188$		0			0
$189$	−0.934336	+	5.11146i	−0.0679629	+	0.371804i
$190$		0			0
$191$	−22.3079			−1.61414			−0.807071	−	0.590454i	$-0.798949\pi$
$191$	−22.3079			−1.61414			−0.807071	+	0.590454i	$0.798949\pi$
$192$		0			0
$193$	8.19154			0.589640			0.294820	−	0.955553i	$-0.404740\pi$
$193$	8.19154			0.589640			0.294820	+	0.955553i	$0.404740\pi$
$194$		0			0
$195$	5.20375	−	2.59983i	0.372648	−	0.186178i
$196$		0			0
$197$		−	7.23102i		−	0.515189i	−0.966253	−	0.257594i	$-0.917070\pi$
$197$		−	7.23102i		−	0.515189i	0.966253	−	0.257594i	$-0.0829298\pi$
$198$		0			0
$199$		−	22.5289i		−	1.59703i	−0.601973	−	0.798517i	$-0.705618\pi$
$199$		−	22.5289i		−	1.59703i	0.601973	−	0.798517i	$-0.294382\pi$
$200$		0			0
$201$	3.42020	−	1.70876i	0.241243	−	0.120526i
$202$		0			0
$203$	10.0534			0.705608
$204$		0			0
$205$	6.79091			0.474298
$206$		0			0
$207$	7.44164	−	9.90921i	0.517230	−	0.688738i
$208$		0			0
$209$			0.851036i			0.0588674i
$210$		0			0
$211$			6.88610i			0.474059i	0.971502	+	0.237029i	$0.0761738\pi$
$211$			6.88610i			0.474059i	−0.971502	+	0.237029i	$0.923826\pi$
$212$		0			0
$213$	1.27538	+	2.55277i	0.0873879	+	0.174913i
$214$		0			0
$215$	11.3827			0.776291
$216$		0			0
$217$	0.148209			0.0100611
$218$		0			0
$219$	7.11675	+	14.2447i	0.480905	+	0.962567i
$220$		0			0
$221$			11.5659i			0.778008i
$222$		0			0
$223$		−	14.4511i		−	0.967718i	−0.875146	−	0.483859i	$-0.839235\pi$
$223$		−	14.4511i		−	0.967718i	0.875146	−	0.483859i	$-0.160765\pi$
$224$		0			0
$225$	1.80151	−	2.39887i	0.120101	−	0.159925i
$226$		0			0
$227$	18.6351			1.23686			0.618428	−	0.785842i	$-0.287770\pi$
$227$	18.6351			1.23686			0.618428	+	0.785842i	$0.287770\pi$
$228$		0			0
$229$	0.601274			0.0397333			0.0198667	−	0.999803i	$-0.493676\pi$
$229$	0.601274			0.0397333			0.0198667	+	0.999803i	$0.493676\pi$
$230$		0			0
$231$	−0.866784	+	0.433052i	−0.0570302	+	0.0284927i
$232$		0			0
$233$			6.96923i			0.456569i	0.973594	+	0.228285i	$0.0733117\pi$
$233$			6.96923i			0.456569i	−0.973594	+	0.228285i	$0.926688\pi$
$234$		0			0
$235$			9.99341i			0.651898i
$236$		0			0
$237$	24.0922	−	12.0366i	1.56496	−	0.781864i
$238$		0			0
$239$	−15.2936			−0.989258			−0.494629	−	0.869104i	$-0.664696\pi$
$239$	−15.2936			−0.989258			−0.494629	+	0.869104i	$0.664696\pi$
$240$		0			0
$241$	18.4312			1.18726			0.593629	−	0.804739i	$-0.297695\pi$
$241$	18.4312			1.18726			0.593629	+	0.804739i	$0.297695\pi$
$242$		0			0
$243$	−11.4497	+	10.5785i	−0.734498	+	0.678611i
$244$		0			0
$245$			1.00000i			0.0638877i
$246$		0			0
$247$			5.10921i			0.325091i
$248$		0			0
$249$	4.47827	+	8.96359i	0.283799	+	0.568044i
$250$		0			0
$251$	2.46546			0.155618			0.0778091	−	0.996968i	$-0.475208\pi$
$251$	2.46546			0.155618			0.0778091	+	0.996968i	$0.475208\pi$
$252$		0			0
$253$	2.31084			0.145281
$254$		0			0
$255$	2.66588	+	5.33594i	0.166944	+	0.334150i
$256$		0			0
$257$			3.20727i			0.200064i	0.994984	+	0.100032i	$0.0318945\pi$
$257$			3.20727i			0.200064i	−0.994984	+	0.100032i	$0.968105\pi$
$258$		0			0
$259$			10.5011i			0.652509i
$260$		0			0
$261$	24.1167	+	18.1112i	1.49279	+	1.12106i
$262$		0			0
$263$	8.38162			0.516833			0.258416	−	0.966034i	$-0.416799\pi$
$263$	8.38162			0.516833			0.258416	+	0.966034i	$0.416799\pi$
$264$		0			0
$265$	8.05096			0.494567
$266$		0			0
$267$	2.04322	−	1.02081i	0.125043	−	0.0624723i
$268$		0			0
$269$			9.30871i			0.567562i	0.958889	+	0.283781i	$0.0915889\pi$
$269$			9.30871i			0.567562i	−0.958889	+	0.283781i	$0.908411\pi$
$270$		0			0
$271$			4.72185i			0.286832i	0.989662	+	0.143416i	$0.0458087\pi$
$271$			4.72185i			0.286832i	−0.989662	+	0.143416i	$0.954191\pi$
$272$		0			0
$273$	−5.20375	+	2.59983i	−0.314945	+	0.157349i
$274$		0			0
$275$	0.559419			0.0337342
$276$		0			0
$277$	−24.7697			−1.48827			−0.744135	−	0.668030i	$-0.767138\pi$
$277$	−24.7697			−1.48827			−0.744135	+	0.668030i	$0.767138\pi$
$278$		0			0
$279$	0.355533	+	0.266999i	0.0212852	+	0.0159848i
$280$		0			0
$281$			7.70344i			0.459549i	0.973244	+	0.229774i	$0.0737988\pi$
$281$			7.70344i			0.459549i	−0.973244	+	0.229774i	$0.926201\pi$
$282$		0			0
$283$			13.4931i			0.802084i	0.916060	+	0.401042i	$0.131352\pi$
$283$			13.4931i			0.802084i	−0.916060	+	0.401042i	$0.868648\pi$
$284$		0			0
$285$	1.17764	+	2.35714i	0.0697575	+	0.139625i
$286$		0			0
$287$	−6.79091			−0.400855
$288$		0			0
$289$	5.14027			0.302369
$290$		0			0
$291$	−13.8826	−	27.7871i	−0.813814	−	1.62891i
$292$		0			0
$293$			6.48006i			0.378569i	0.981922	+	0.189285i	$0.0606169\pi$
$293$			6.48006i			0.378569i	−0.981922	+	0.189285i	$0.939383\pi$
$294$		0			0
$295$			4.78044i			0.278328i
$296$		0			0
$297$	−2.85945	−	0.522685i	−0.165922	−	0.0303293i
$298$		0			0
$299$	13.8732			0.802305
$300$		0			0
$301$	−11.3827			−0.656086
$302$		0			0
$303$	−20.6666	+	10.3252i	−1.18727	+	0.593167i
$304$		0			0
$305$		−	2.04177i		−	0.116911i
$306$		0			0
$307$		−	27.0199i		−	1.54211i	−0.636771	−	0.771053i	$-0.719730\pi$
$307$		−	27.0199i		−	1.54211i	0.636771	−	0.771053i	$-0.280270\pi$
$308$		0			0
$309$	14.8619	−	7.42509i	0.845462	−	0.422399i
$310$		0			0
$311$	−17.3968			−0.986480			−0.493240	−	0.869893i	$-0.664188\pi$
$311$	−17.3968			−0.986480			−0.493240	+	0.869893i	$0.664188\pi$
$312$		0			0
$313$	−22.0197			−1.24463			−0.622315	−	0.782767i	$-0.713808\pi$
$313$	−22.0197			−1.24463			−0.622315	+	0.782767i	$0.713808\pi$
$314$		0			0
$315$	−1.80151	+	2.39887i	−0.101503	+	0.135161i
$316$		0			0
$317$			22.1799i			1.24575i	0.782322	+	0.622874i	$0.214035\pi$
$317$			22.1799i			1.24575i	−0.782322	+	0.622874i	$0.785965\pi$
$318$		0			0
$319$			5.62405i			0.314886i
$320$		0			0
$321$	−0.998115	−	1.99780i	−0.0557094	−	0.111506i
$322$		0			0
$323$	−5.23900			−0.291506
$324$		0			0
$325$	3.35848			0.186295
$326$		0			0
$327$	−10.6536	−	21.3240i	−0.589148	−	1.17922i
$328$		0			0
$329$		−	9.99341i		−	0.550954i
$330$		0			0
$331$			1.38708i			0.0762409i	0.999273	+	0.0381204i	$0.0121371\pi$
$331$			1.38708i			0.0762409i	−0.999273	+	0.0381204i	$0.987863\pi$
$332$		0			0
$333$	−18.9179	+	25.1908i	−1.03669	+	1.38045i
$334$		0			0
$335$	2.20738			0.120602
$336$		0			0
$337$	2.33963			0.127448			0.0637240	−	0.997968i	$-0.479702\pi$
$337$	2.33963			0.127448			0.0637240	+	0.997968i	$0.479702\pi$
$338$		0			0
$339$	27.6501	−	13.8142i	1.50175	−	0.750284i
$340$		0			0
$341$			0.0829108i			0.00448987i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	6.40039	−	3.19768i	0.344585	−	0.172157i
$346$		0			0
$347$	12.8348			0.689008			0.344504	−	0.938785i	$-0.388047\pi$
$347$	12.8348			0.689008			0.344504	+	0.938785i	$0.388047\pi$
$348$		0			0
$349$	−4.19892			−0.224763			−0.112382	−	0.993665i	$-0.535848\pi$
$349$	−4.19892			−0.224763			−0.112382	+	0.993665i	$0.535848\pi$
$350$		0			0
$351$	−17.1667	−	3.13795i	−0.916292	−	0.167491i
$352$		0			0
$353$			23.0939i			1.22916i	0.788854	+	0.614581i	$0.210675\pi$
$353$			23.0939i			1.22916i	−0.788854	+	0.614581i	$0.789325\pi$
$354$		0			0
$355$			1.64755i			0.0874428i
$356$		0			0
$357$	−2.66588	−	5.33594i	−0.141093	−	0.282408i
$358$		0			0
$359$	−13.2600			−0.699838			−0.349919	−	0.936780i	$-0.613791\pi$
$359$	−13.2600			−0.699838			−0.349919	+	0.936780i	$0.613791\pi$
$360$		0			0
$361$	16.6857			0.878194
$362$		0			0
$363$	8.27295	+	16.5589i	0.434217	+	0.869117i
$364$		0			0
$365$			9.19346i			0.481208i
$366$		0			0
$367$		−	25.6105i		−	1.33686i	−0.743776	−	0.668429i	$-0.766967\pi$
$367$		−	25.6105i		−	1.33686i	0.743776	−	0.668429i	$-0.233033\pi$
$368$		0			0
$369$	−16.2905	−	12.2339i	−0.848049	−	0.636870i
$370$		0			0
$371$	−8.05096			−0.417985
$372$		0			0
$373$	−30.4020			−1.57416			−0.787079	−	0.616853i	$-0.788407\pi$
$373$	−30.4020			−1.57416			−0.787079	+	0.616853i	$0.788407\pi$
$374$		0			0
$375$	1.54944	−	0.774110i	0.0800126	−	0.0399749i
$376$		0			0
$377$			33.7640i			1.73894i
$378$		0			0
$379$			2.35256i			0.120843i	0.998173	+	0.0604213i	$0.0192444\pi$
$379$			2.35256i			0.120843i	−0.998173	+	0.0604213i	$0.980756\pi$
$380$		0			0
$381$	12.9557	−	6.47278i	0.663743	−	0.331611i
$382$		0			0
$383$	7.68362			0.392615			0.196307	−	0.980542i	$-0.437105\pi$
$383$	7.68362			0.392615			0.196307	+	0.980542i	$0.437105\pi$
$384$		0			0
$385$	−0.559419			−0.0285106
$386$		0			0
$387$	−27.3055	−	20.5060i	−1.38802	−	1.04238i
$388$		0			0
$389$		−	30.6797i		−	1.55552i	−0.628560	−	0.777761i	$-0.716355\pi$
$389$		−	30.6797i		−	1.55552i	0.628560	−	0.777761i	$-0.283645\pi$
$390$		0			0
$391$			14.2256i			0.719419i
$392$		0			0
$393$	−4.39354	−	8.79398i	−0.221625	−	0.443598i
$394$		0			0
$395$	15.5490			0.782356
$396$		0			0
$397$	−4.52854			−0.227281			−0.113640	−	0.993522i	$-0.536251\pi$
$397$	−4.52854			−0.227281			−0.113640	+	0.993522i	$0.536251\pi$
$398$		0			0
$399$	−1.17764	−	2.35714i	−0.0589559	−	0.118004i
$400$		0			0
$401$		−	23.9410i		−	1.19555i	−0.801662	−	0.597777i	$-0.796051\pi$
$401$		−	23.9410i		−	1.19555i	0.801662	−	0.597777i	$-0.203949\pi$
$402$		0			0
$403$			0.497756i			0.0247950i
$404$		0			0
$405$	−8.64316	+	2.50914i	−0.429482	+	0.124680i
$406$		0			0
$407$	−5.87454			−0.291190
$408$		0			0
$409$	10.9393			0.540915			0.270458	−	0.962732i	$-0.412825\pi$
$409$	10.9393			0.540915			0.270458	+	0.962732i	$0.412825\pi$
$410$		0			0
$411$	14.7805	−	7.38444i	0.729068	−	0.364247i
$412$		0			0
$413$		−	4.78044i		−	0.235230i
$414$		0			0
$415$			5.78506i			0.283978i
$416$		0			0
$417$	27.7467	−	13.8624i	1.35876	−	0.678846i
$418$		0			0
$419$	9.24700			0.451746			0.225873	−	0.974157i	$-0.427477\pi$
$419$	9.24700			0.451746			0.225873	+	0.974157i	$0.427477\pi$
$420$		0			0
$421$	−14.4097			−0.702284			−0.351142	−	0.936322i	$-0.614207\pi$
$421$	−14.4097			−0.702284			−0.351142	+	0.936322i	$0.614207\pi$
$422$		0			0
$423$	18.0032	−	23.9729i	0.875346	−	1.16560i
$424$		0			0
$425$			3.44380i			0.167049i
$426$		0			0
$427$			2.04177i			0.0988080i
$428$		0			0
$429$	−1.45439	−	2.91108i	−0.0702188	−	0.140548i
$430$		0			0
$431$	10.6564			0.513301			0.256651	−	0.966504i	$-0.417381\pi$
$431$	10.6564			0.513301			0.256651	+	0.966504i	$0.417381\pi$
$432$		0			0
$433$	1.74635			0.0839241			0.0419621	−	0.999119i	$-0.486639\pi$
$433$	1.74635			0.0839241			0.0419621	+	0.999119i	$0.486639\pi$
$434$		0			0
$435$	7.78241	+	15.5771i	0.373138	+	0.746863i
$436$		0			0
$437$			6.28411i			0.300610i
$438$		0			0
$439$			36.6430i			1.74887i	0.485140	+	0.874436i	$0.338769\pi$
$439$			36.6430i			1.74887i	−0.485140	+	0.874436i	$0.661231\pi$
$440$		0			0
$441$	1.80151	−	2.39887i	0.0857861	−	0.114232i
$442$		0			0
$443$	−23.9701			−1.13885			−0.569426	−	0.822042i	$-0.692835\pi$
$443$	−23.9701			−1.13885			−0.569426	+	0.822042i	$0.692835\pi$
$444$		0			0
$445$	1.31868			0.0625116
$446$		0			0
$447$	−11.6886	+	5.83973i	−0.552854	+	0.276210i
$448$		0			0
$449$		−	28.5921i		−	1.34934i	−0.738117	−	0.674672i	$-0.764285\pi$
$449$		−	28.5921i		−	1.34934i	0.738117	−	0.674672i	$-0.235715\pi$
$450$		0			0
$451$		−	3.79896i		−	0.178886i
$452$		0			0
$453$	2.30751	−	1.15285i	0.108416	−	0.0541656i
$454$		0			0
$455$	−3.35848			−0.157448
$456$		0			0
$457$	−10.4416			−0.488436			−0.244218	−	0.969720i	$-0.578531\pi$
$457$	−10.4416			−0.488436			−0.244218	+	0.969720i	$0.578531\pi$
$458$		0			0
$459$	3.21766	−	17.6028i	0.150188	−	0.821629i
$460$		0			0
$461$			28.9503i			1.34835i	0.738572	+	0.674175i	$0.235501\pi$
$461$			28.9503i			1.34835i	−0.738572	+	0.674175i	$0.764499\pi$
$462$		0			0
$463$		−	29.0619i		−	1.35062i	−0.737534	−	0.675310i	$-0.764010\pi$
$463$		−	29.0619i		−	1.35062i	0.737534	−	0.675310i	$-0.235990\pi$
$464$		0			0
$465$	0.114730	+	0.229640i	0.00532047	+	0.0106493i
$466$		0			0
$467$	−15.0751			−0.697592			−0.348796	−	0.937199i	$-0.613409\pi$
$467$	−15.0751			−0.697592			−0.348796	+	0.937199i	$0.613409\pi$
$468$		0			0
$469$	−2.20738			−0.101928
$470$		0			0
$471$	12.5181	+	25.0559i	0.576805	+	1.15452i
$472$		0			0
$473$		−	6.36768i		−	0.292786i
$474$		0			0
$475$			1.52129i			0.0698014i
$476$		0			0
$477$	−19.3132	−	14.5039i	−0.884290	−	0.664087i
$478$		0			0
$479$	19.6925			0.899773			0.449887	−	0.893086i	$-0.351464\pi$
$479$	19.6925			0.899773			0.449887	+	0.893086i	$0.351464\pi$
$480$		0			0
$481$	−35.2678			−1.60808
$482$		0			0
$483$	−6.40039	+	3.19768i	−0.291228	+	0.145500i
$484$		0			0
$485$		−	17.9337i		−	0.814326i
$486$		0			0
$487$			33.5499i			1.52029i	0.649754	+	0.760145i	$0.274872\pi$
$487$			33.5499i			1.52029i	−0.649754	+	0.760145i	$0.725128\pi$
$488$		0			0
$489$	26.3502	−	13.1648i	1.19160	−	0.595331i
$490$		0			0
$491$	−6.95363			−0.313813			−0.156906	−	0.987613i	$-0.550152\pi$
$491$	−6.95363			−0.313813			−0.156906	+	0.987613i	$0.550152\pi$
$492$		0			0
$493$	−34.6218			−1.55929
$494$		0			0
$495$	−1.34197	−	1.00780i	−0.0603172	−	0.0452972i
$496$		0			0
$497$		−	1.64755i		−	0.0739027i
$498$		0			0
$499$			13.3865i			0.599261i	0.954055	+	0.299631i	$0.0968634\pi$
$499$			13.3865i			0.599261i	−0.954055	+	0.299631i	$0.903137\pi$
$500$		0			0
$501$	4.95592	+	9.91963i	0.221414	+	0.443176i
$502$		0			0
$503$	−25.7962			−1.15019			−0.575097	−	0.818085i	$-0.695036\pi$
$503$	−25.7962			−1.15019			−0.575097	+	0.818085i	$0.695036\pi$
$504$		0			0
$505$	−13.3381			−0.593540
$506$		0			0
$507$	1.33195	+	2.66600i	0.0591540	+	0.118401i
$508$		0			0
$509$		−	42.6623i		−	1.89097i	−0.325661	−	0.945487i	$-0.605587\pi$
$509$		−	42.6623i		−	1.89097i	0.325661	−	0.945487i	$-0.394413\pi$
$510$		0			0
$511$		−	9.19346i		−	0.406695i
$512$		0			0
$513$	1.42139	−	7.77599i	0.0627560	−	0.343318i
$514$		0			0
$515$	9.59178			0.422665
$516$		0			0
$517$	5.59050			0.245870
$518$		0			0
$519$	4.18985	−	2.09328i	0.183914	−	0.0918849i
$520$		0			0
$521$		−	13.0770i		−	0.572912i	−0.958093	−	0.286456i	$-0.907523\pi$
$521$		−	13.0770i		−	0.572912i	0.958093	−	0.286456i	$-0.0924772\pi$
$522$		0			0
$523$			0.170026i			0.00743473i	0.999993	+	0.00371737i	$0.00118328\pi$
$523$			0.170026i			0.00743473i	−0.999993	+	0.00371737i	$0.998817\pi$
$524$		0			0
$525$	−1.54944	+	0.774110i	−0.0676230	+	0.0337849i
$526$		0			0
$527$	−0.510401			−0.0222334
$528$		0			0
$529$	−5.93661			−0.258114
$530$		0			0
$531$	8.61200	−	11.4676i	0.373729	−	0.497653i
$532$		0			0
$533$		−	22.8071i		−	0.987886i
$534$		0			0
$535$		−	1.28937i		−	0.0557444i
$536$		0			0
$537$	−5.19664	−	10.4015i	−0.224252	−	0.448856i
$538$		0			0
$539$	0.559419			0.0240959
$540$		0			0
$541$	−16.1798			−0.695622			−0.347811	−	0.937565i	$-0.613075\pi$
$541$	−16.1798			−0.695622			−0.347811	+	0.937565i	$0.613075\pi$
$542$		0			0
$543$	−8.90091	−	17.8158i	−0.381975	−	0.764550i
$544$		0			0
$545$		−	13.7624i		−	0.589518i
$546$		0			0
$547$			41.6207i			1.77957i	0.456377	+	0.889787i	$0.349147\pi$
$547$			41.6207i			1.77957i	−0.456377	+	0.889787i	$0.650853\pi$
$548$		0			0
$549$	−3.67826	+	4.89793i	−0.156984	+	0.209038i
$550$		0			0
$551$	−15.2941			−0.651549
$552$		0			0
$553$	−15.5490			−0.661212
$554$		0			0
$555$	−16.2708	+	8.12903i	−0.690659	+	0.345058i
$556$		0			0
$557$		−	15.8996i		−	0.673687i	−0.941561	−	0.336843i	$-0.890641\pi$
$557$		−	15.8996i		−	0.673687i	0.941561	−	0.336843i	$-0.109359\pi$
$558$		0			0
$559$		−	38.2284i		−	1.61689i
$560$		0			0
$561$	2.98503	−	1.49134i	0.126028	−	0.0629645i
$562$		0			0
$563$	−18.7579			−0.790549			−0.395275	−	0.918563i	$-0.629351\pi$
$563$	−18.7579			−0.790549			−0.395275	+	0.918563i	$0.629351\pi$
$564$		0			0
$565$	17.8453			0.750756
$566$		0			0
$567$	8.64316	−	2.50914i	0.362979	−	0.105374i
$568$		0			0
$569$			29.5520i			1.23889i	0.785042	+	0.619443i	$0.212641\pi$
$569$			29.5520i			1.23889i	−0.785042	+	0.619443i	$0.787359\pi$
$570$		0			0
$571$		−	34.9401i		−	1.46220i	−0.682272	−	0.731098i	$-0.739008\pi$
$571$		−	34.9401i		−	1.46220i	0.682272	−	0.731098i	$-0.260992\pi$
$572$		0			0
$573$	17.2688	+	34.5647i	0.721413	+	1.44396i
$574$		0			0
$575$	4.13079			0.172266
$576$		0			0
$577$	−39.0082			−1.62393			−0.811966	−	0.583705i	$-0.801602\pi$
$577$	−39.0082			−1.62393			−0.811966	+	0.583705i	$0.801602\pi$
$578$		0			0
$579$	−6.34115	−	12.6923i	−0.263529	−	0.527473i
$580$		0			0
$581$		−	5.78506i		−	0.240005i
$582$		0			0
$583$		−	4.50386i		−	0.186531i
$584$		0			0
$585$	−8.05655	−	6.05033i	−0.333097	−	0.250150i
$586$		0			0
$587$	28.4743			1.17526			0.587630	−	0.809130i	$-0.300061\pi$
$587$	28.4743			1.17526			0.587630	+	0.809130i	$0.300061\pi$
$588$		0			0
$589$	−0.225468			−0.00929024
$590$		0			0
$591$	−11.2040	+	5.59760i	−0.460871	+	0.230255i
$592$		0			0
$593$			30.1937i			1.23991i	0.784638	+	0.619954i	$0.212849\pi$
$593$			30.1937i			1.23991i	−0.784638	+	0.619954i	$0.787151\pi$
$594$		0			0
$595$		−	3.44380i		−	0.141182i
$596$		0			0
$597$	−34.9071	+	17.4399i	−1.42865	+	0.713766i
$598$		0			0
$599$	41.0384			1.67678			0.838392	−	0.545068i	$-0.183496\pi$
$599$	41.0384			1.67678			0.838392	+	0.545068i	$0.183496\pi$
$600$		0			0
$601$	−3.92391			−0.160060			−0.0800298	−	0.996792i	$-0.525502\pi$
$601$	−3.92391			−0.160060			−0.0800298	+	0.996792i	$0.525502\pi$
$602$		0			0
$603$	−5.29522	−	3.97662i	−0.215638	−	0.161941i
$604$		0			0
$605$			10.6871i			0.434490i
$606$		0			0
$607$			0.710992i			0.0288583i	0.999896	+	0.0144291i	$0.00459310\pi$
$607$			0.710992i			0.0288583i	−0.999896	+	0.0144291i	$0.995407\pi$
$608$		0			0
$609$	−7.78241	−	15.5771i	−0.315359	−	0.631214i
$610$		0			0
$611$	33.5626			1.35780
$612$		0			0
$613$	11.1958			0.452192			0.226096	−	0.974105i	$-0.427404\pi$
$613$	11.1958			0.452192			0.226096	+	0.974105i	$0.427404\pi$
$614$		0			0
$615$	−5.25691	−	10.5221i	−0.211979	−	0.424291i
$616$		0			0
$617$			24.8681i			1.00115i	0.865693	+	0.500576i	$0.166878\pi$
$617$			24.8681i			1.00115i	−0.865693	+	0.500576i	$0.833122\pi$
$618$		0			0
$619$			5.77627i			0.232168i	0.993239	+	0.116084i	$0.0370342\pi$
$619$			5.77627i			0.232168i	−0.993239	+	0.116084i	$0.962966\pi$
$620$		0			0
$621$	−21.1143	−	3.85954i	−0.847289	−	0.154878i
$622$		0			0
$623$	−1.31868			−0.0528320
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	1.31863	−	0.658796i	0.0526609	−	0.0263098i
$628$		0			0
$629$		−	36.1638i		−	1.44194i
$630$		0			0
$631$			4.45353i			0.177292i	0.996063	+	0.0886462i	$0.0282541\pi$
$631$			4.45353i			0.177292i	−0.996063	+	0.0886462i	$0.971746\pi$
$632$		0			0
$633$	10.6696	−	5.33060i	0.424078	−	0.211872i
$634$		0			0
$635$	8.36158			0.331819
$636$		0			0
$637$	3.35848			0.133068
$638$		0			0
$639$	2.96807	−	3.95225i	0.117415	−	0.156349i
$640$		0			0
$641$		−	22.6239i		−	0.893592i	−0.894636	−	0.446796i	$-0.852565\pi$
$641$		−	22.6239i		−	0.893592i	0.894636	−	0.446796i	$-0.147435\pi$
$642$		0			0
$643$		−	7.80291i		−	0.307717i	−0.988093	−	0.153858i	$-0.950830\pi$
$643$		−	7.80291i		−	0.307717i	0.988093	−	0.153858i	$-0.0491700\pi$
$644$		0			0
$645$	−8.81143	−	17.6367i	−0.346950	−	0.694445i
$646$		0			0
$647$	5.78288			0.227349			0.113674	−	0.993518i	$-0.463738\pi$
$647$	5.78288			0.227349			0.113674	+	0.993518i	$0.463738\pi$
$648$		0			0
$649$	2.67427			0.104974
$650$		0			0
$651$	−0.114730	−	0.229640i	−0.00449662	−	0.00900030i
$652$		0			0
$653$			17.7631i			0.695123i	0.937657	+	0.347561i	$0.112990\pi$
$653$			17.7631i			0.695123i	−0.937657	+	0.347561i	$0.887010\pi$
$654$		0			0
$655$		−	5.67560i		−	0.221764i
$656$		0			0
$657$	16.5621	−	22.0539i	0.646149	−	0.860405i
$658$		0			0
$659$	38.1448			1.48591			0.742956	−	0.669340i	$-0.233423\pi$
$659$	38.1448			1.48591			0.742956	+	0.669340i	$0.233423\pi$
$660$		0			0
$661$	6.85707			0.266709			0.133354	−	0.991068i	$-0.457425\pi$
$661$	6.85707			0.266709			0.133354	+	0.991068i	$0.457425\pi$
$662$		0			0
$663$	17.9207	−	8.95329i	0.695980	−	0.347717i
$664$		0			0
$665$		−	1.52129i		−	0.0589929i
$666$		0			0
$667$			41.5283i			1.60798i
$668$		0			0
$669$	−22.3911	+	11.1868i	−0.865690	+	0.432505i
$670$		0			0
$671$	−1.14220			−0.0440942
$672$		0			0
$673$	−10.2067			−0.393441			−0.196721	−	0.980460i	$-0.563029\pi$
$673$	−10.2067			−0.393441			−0.196721	+	0.980460i	$0.563029\pi$
$674$		0			0
$675$	−5.11146	−	0.934336i	−0.196740	−	0.0359626i
$676$		0			0
$677$		−	50.6791i		−	1.94776i	−0.227070	−	0.973879i	$-0.572914\pi$
$677$		−	50.6791i		−	1.94776i	0.227070	−	0.973879i	$-0.427086\pi$
$678$		0			0
$679$			17.9337i			0.688231i
$680$		0			0
$681$	−14.4256	−	28.8739i	−0.552791	−	1.10645i
$682$		0			0
$683$	−35.8473			−1.37166			−0.685830	−	0.727762i	$-0.740561\pi$
$683$	−35.8473			−1.37166			−0.685830	+	0.727762i	$0.740561\pi$
$684$		0			0
$685$	9.53927			0.364477
$686$		0			0
$687$	−0.465452	−	0.931636i	−0.0177581	−	0.0355441i
$688$		0			0
$689$		−	27.0390i		−	1.03010i
$690$		0			0
$691$		−	0.581892i		−	0.0221362i	−0.999939	−	0.0110681i	$-0.996477\pi$
$691$		−	0.581892i		−	0.0221362i	0.999939	−	0.0110681i	$-0.00352316\pi$
$692$		0			0
$693$	1.34197	+	1.00780i	0.0509773	+	0.0382831i
$694$		0			0
$695$	17.9076			0.679273
$696$		0			0
$697$	23.3865			0.885827
$698$		0			0
$699$	10.7984	−	5.39495i	0.408432	−	0.204056i
$700$		0			0
$701$		−	17.7229i		−	0.669384i	−0.942328	−	0.334692i	$-0.891368\pi$
$701$		−	17.7229i		−	0.669384i	0.942328	−	0.334692i	$-0.108632\pi$
$702$		0			0
$703$		−	15.9752i		−	0.602517i
$704$		0			0
$705$	15.4842	−	7.73599i	0.583167	−	0.291354i
$706$		0			0
$707$	13.3381			0.501633
$708$		0			0
$709$	−20.7870			−0.780672			−0.390336	−	0.920672i	$-0.627641\pi$
$709$	−20.7870			−0.780672			−0.390336	+	0.920672i	$0.627641\pi$
$710$		0			0
$711$	−37.3001	−	28.0117i	−1.39886	−	1.05052i
$712$		0			0
$713$			0.612219i			0.0229278i
$714$		0			0
$715$		−	1.87880i		−	0.0702630i
$716$		0			0
$717$	11.8389	+	23.6964i	0.442132	+	0.884959i
$718$		0			0
$719$	−50.0506			−1.86657			−0.933286	−	0.359134i	$-0.883072\pi$
$719$	−50.0506			−1.86657			−0.933286	+	0.359134i	$0.883072\pi$
$720$		0			0
$721$	−9.59178			−0.357217
$722$		0			0
$723$	−14.2678	−	28.5580i	−0.530624	−	1.06208i
$724$		0			0
$725$			10.0534i			0.373373i
$726$		0			0
$727$		−	26.8190i		−	0.994661i	−0.867561	−	0.497331i	$-0.834314\pi$
$727$		−	26.8190i		−	0.994661i	0.867561	−	0.497331i	$-0.165686\pi$
$728$		0			0
$729$	25.2540	+	9.55164i	0.935335	+	0.353764i
$730$		0			0
$731$	39.1996			1.44985
$732$		0			0
$733$	−33.7399			−1.24621			−0.623106	−	0.782137i	$-0.714130\pi$
$733$	−33.7399			−1.24621			−0.623106	+	0.782137i	$0.714130\pi$
$734$		0			0
$735$	1.54944	−	0.774110i	0.0571518	−	0.0285535i
$736$		0			0
$737$		−	1.23485i		−	0.0454864i
$738$		0			0
$739$		−	0.473982i		−	0.0174357i	−0.999962	−	0.00871785i	$-0.997225\pi$
$739$		−	0.473982i		−	0.0174357i	0.999962	−	0.00871785i	$-0.00277501\pi$
$740$		0			0
$741$	7.91639	−	3.95509i	0.290816	−	0.145294i
$742$		0			0
$743$	−45.8108			−1.68063			−0.840317	−	0.542095i	$-0.817631\pi$
$743$	−45.8108			−1.68063			−0.840317	+	0.542095i	$0.817631\pi$
$744$		0			0
$745$	−7.54380			−0.276384
$746$		0			0
$747$	10.4218	−	13.8776i	0.381315	−	0.507755i
$748$		0			0
$749$			1.28937i			0.0471126i
$750$		0			0
$751$			30.2865i			1.10517i	0.833456	+	0.552586i	$0.186359\pi$
$751$			30.2865i			1.10517i	−0.833456	+	0.552586i	$0.813641\pi$
$752$		0			0
$753$	−1.90853	−	3.82007i	−0.0695508	−	0.139211i
$754$		0			0
$755$	1.48926			0.0541997
$756$		0			0
$757$	−44.6109			−1.62141			−0.810705	−	0.585455i	$-0.800916\pi$
$757$	−44.6109			−1.62141			−0.810705	+	0.585455i	$0.800916\pi$
$758$		0			0
$759$	−1.78884	−	3.58050i	−0.0649309	−	0.129964i
$760$		0			0
$761$			17.4006i			0.630770i	0.948964	+	0.315385i	$0.102134\pi$
$761$			17.4006i			0.630770i	−0.948964	+	0.315385i	$0.897866\pi$
$762$		0			0
$763$			13.7624i			0.498234i
$764$		0			0
$765$	6.20403	−	8.26121i	0.224307	−	0.298685i
$766$		0			0
$767$	16.0550			0.579713
$768$		0			0
$769$	−12.6539			−0.456311			−0.228156	−	0.973625i	$-0.573269\pi$
$769$	−12.6539			−0.456311			−0.228156	+	0.973625i	$0.573269\pi$
$770$		0			0
$771$	4.96946	−	2.48278i	0.178971	−	0.0894151i
$772$		0			0
$773$		−	13.0718i		−	0.470161i	−0.971976	−	0.235081i	$-0.924465\pi$
$773$		−	13.0718i		−	0.470161i	0.971976	−	0.235081i	$-0.0755354\pi$
$774$		0			0
$775$			0.148209i			0.00532382i
$776$		0			0
$777$	16.2708	−	8.12903i	0.583713	−	0.291627i
$778$		0			0
$779$	10.3309			0.370144
$780$		0			0
$781$	0.921671			0.0329800
$782$		0			0
$783$	9.39323	−	51.3874i	0.335687	−	1.83644i
$784$		0			0
$785$			16.1710i			0.577168i
$786$		0			0
$787$		−	39.9700i		−	1.42478i	−0.701786	−	0.712388i	$-0.747614\pi$
$787$		−	39.9700i		−	1.42478i	0.701786	−	0.712388i	$-0.252386\pi$
$788$		0			0
$789$	−6.48829	−	12.9868i	−0.230989	−	0.462342i
$790$		0			0
$791$	−17.8453			−0.634505
$792$		0			0
$793$	−6.85722			−0.243507
$794$		0			0
$795$	−6.23233	−	12.4745i	−0.221038	−	0.442423i
$796$		0			0
$797$			5.21911i			0.184870i	0.995719	+	0.0924351i	$0.0294651\pi$
$797$			5.21911i			0.184870i	−0.995719	+	0.0924351i	$0.970535\pi$
$798$		0			0
$799$			34.4153i			1.21752i
$800$		0			0
$801$	−3.16335	−	2.37562i	−0.111771	−	0.0839384i
$802$		0			0
$803$	5.14300			0.181492
$804$		0			0
$805$	−4.13079			−0.145591
$806$		0			0
$807$	14.4233	−	7.20596i	0.507723	−	0.253662i
$808$		0			0
$809$		−	39.7500i		−	1.39753i	−0.715349	−	0.698767i	$-0.753732\pi$
$809$		−	39.7500i		−	1.39753i	0.715349	−	0.698767i	$-0.246268\pi$
$810$		0			0
$811$		−	38.4127i		−	1.34885i	−0.738343	−	0.674426i	$-0.764391\pi$
$811$		−	38.4127i		−	1.34885i	0.738343	−	0.674426i	$-0.235609\pi$
$812$		0			0
$813$	7.31621	−	3.65523i	0.256591	−	0.128195i
$814$		0			0
$815$	17.0063			0.595705
$816$		0			0
$817$	17.3163			0.605820
$818$		0			0
$819$	8.05655	+	6.05033i	0.281519	+	0.211416i
$820$		0			0
$821$			43.8482i			1.53031i	0.643845	+	0.765156i	$0.277338\pi$
$821$			43.8482i			1.53031i	−0.643845	+	0.765156i	$0.722662\pi$
$822$		0			0
$823$			1.83409i			0.0639323i	0.999489	+	0.0319661i	$0.0101769\pi$
$823$			1.83409i			0.0639323i	−0.999489	+	0.0319661i	$0.989823\pi$
$824$		0			0
$825$	−0.433052	−	0.866784i	−0.0150769	−	0.0301776i
$826$		0			0
$827$	−25.8626			−0.899329			−0.449665	−	0.893197i	$-0.648456\pi$
$827$	−25.8626			−0.899329			−0.449665	+	0.893197i	$0.648456\pi$
$828$		0			0
$829$	−13.5228			−0.469667			−0.234833	−	0.972036i	$-0.575454\pi$
$829$	−13.5228			−0.469667			−0.234833	+	0.972036i	$0.575454\pi$
$830$		0			0
$831$	19.1745	+	38.3791i	0.665156	+	1.33136i
$832$		0			0
$833$			3.44380i			0.119320i
$834$		0			0
$835$			6.40209i			0.221553i
$836$		0			0
$837$	0.138477	−	0.757563i	0.00478646	−	0.0261852i
$838$		0			0
$839$	37.8234			1.30581			0.652904	−	0.757441i	$-0.273550\pi$
$839$	37.8234			1.30581			0.652904	+	0.757441i	$0.273550\pi$
$840$		0			0
$841$	−72.0703			−2.48518
$842$		0			0
$843$	11.9360	−	5.96331i	0.411097	−	0.205387i
$844$		0			0
$845$			1.72062i			0.0591912i
$846$		0			0
$847$		−	10.6871i		−	0.367211i
$848$		0			0
$849$	20.9068	−	10.4452i	0.717518	−	0.358477i
$850$		0			0
$851$	−43.3779			−1.48698
$852$		0			0
$853$	14.2364			0.487445			0.243723	−	0.969845i	$-0.421631\pi$
$853$	14.2364			0.487445			0.243723	+	0.969845i	$0.421631\pi$
$854$		0			0
$855$	2.74061	−	3.64936i	0.0937269	−	0.124806i
$856$		0			0
$857$			25.0750i			0.856547i	0.903649	+	0.428274i	$0.140878\pi$
$857$			25.0750i			0.856547i	−0.903649	+	0.428274i	$0.859122\pi$
$858$		0			0
$859$			15.6236i			0.533071i	0.963825	+	0.266535i	$0.0858789\pi$
$859$			15.6236i			0.533071i	−0.963825	+	0.266535i	$0.914121\pi$
$860$		0			0
$861$	5.25691	+	10.5221i	0.179155	+	0.358592i
$862$		0			0
$863$	−4.70737			−0.160241			−0.0801204	−	0.996785i	$-0.525530\pi$
$863$	−4.70737			−0.160241			−0.0801204	+	0.996785i	$0.525530\pi$
$864$		0			0
$865$	2.70411			0.0919427
$866$		0			0
$867$	−3.97913	−	7.96452i	−0.135138	−	0.270489i
$868$		0			0
$869$		−	8.69842i		−	0.295074i
$870$		0			0
$871$		−	7.41345i		−	0.251195i
$872$		0			0
$873$	−32.3077	+	43.0205i	−1.09345	+	1.45602i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$	−20.6365			−0.696844			−0.348422	−	0.937338i	$-0.613282\pi$
$877$	−20.6365			−0.696844			−0.348422	+	0.937338i	$0.613282\pi$
$878$		0			0
$879$	10.0404	−	5.01628i	0.338656	−	0.169195i
$880$		0			0
$881$			45.3459i			1.52774i	0.645370	+	0.763870i	$0.276703\pi$
$881$			45.3459i			1.52774i	−0.645370	+	0.763870i	$0.723297\pi$
$882$		0			0
$883$		−	5.49234i		−	0.184832i	−0.995720	−	0.0924160i	$-0.970541\pi$
$883$		−	5.49234i		−	0.184832i	0.995720	−	0.0924160i	$-0.0294590\pi$
$884$		0			0
$885$	7.40699	−	3.70059i	0.248983	−	0.124394i
$886$		0			0
$887$	43.5975			1.46386			0.731931	−	0.681379i	$-0.238619\pi$
$887$	43.5975			1.46386			0.731931	+	0.681379i	$0.238619\pi$
$888$		0			0
$889$	−8.36158			−0.280438
$890$		0			0
$891$	1.40366	+	4.83515i	0.0470243	+	0.161984i
$892$		0			0
$893$			15.2028i			0.508743i
$894$		0			0
$895$		−	6.71305i		−	0.224393i
$896$		0			0
$897$	−10.7393	−	21.4956i	−0.358576	−	0.717716i
$898$		0			0
$899$	−1.49000			−0.0496942
$900$		0			0
$901$	27.7259			0.923682
$902$		0			0
$903$	8.81143	+	17.6367i	0.293226	+	0.586913i
$904$		0			0
$905$		−	11.4983i		−	0.382215i
$906$		0			0
$907$		−	53.5801i		−	1.77910i	−0.456839	−	0.889549i	$-0.651019\pi$
$907$		−	53.5801i		−	1.77910i	0.456839	−	0.889549i	$-0.348981\pi$
$908$		0			0
$909$	31.9965	+	24.0288i	1.06126	+	0.796984i
$910$		0			0
$911$	−20.9339			−0.693572			−0.346786	−	0.937944i	$-0.612727\pi$
$911$	−20.9339			−0.693572			−0.346786	+	0.937944i	$0.612727\pi$
$912$		0			0
$913$	3.23627			0.107105
$914$		0			0
$915$	−3.16359	+	1.58055i	−0.104585	+	0.0522514i
$916$		0			0
$917$			5.67560i			0.187425i
$918$		0			0
$919$		−	12.5797i		−	0.414966i	−0.978239	−	0.207483i	$-0.933473\pi$
$919$		−	12.5797i		−	0.414966i	0.978239	−	0.207483i	$-0.0665272\pi$
$920$		0			0
$921$	−41.8656	+	20.9163i	−1.37952	+	0.689217i
$922$		0			0
$923$	5.53326			0.182129
$924$		0			0
$925$	−10.5011			−0.345275
$926$		0			0
$927$	−23.0094	−	17.2797i	−0.755729	−	0.567539i
$928$		0			0
$929$			15.9869i			0.524513i	0.964998	+	0.262257i	$0.0844667\pi$
$929$			15.9869i			0.524513i	−0.964998	+	0.262257i	$0.915533\pi$
$930$		0			0
$931$			1.52129i			0.0498581i
$932$		0			0
$933$	13.4670	+	26.9552i	0.440890	+	0.882473i
$934$		0			0
$935$	1.92652			0.0630041
$936$		0			0
$937$	−24.2675			−0.792786			−0.396393	−	0.918081i	$-0.629738\pi$
$937$	−24.2675			−0.792786			−0.396393	+	0.918081i	$0.629738\pi$
$938$		0			0
$939$	17.0457	+	34.1182i	0.556265	+	1.11341i
$940$		0			0
$941$		−	12.5947i		−	0.410575i	−0.978702	−	0.205288i	$-0.934187\pi$
$941$		−	12.5947i		−	0.410575i	0.978702	−	0.205288i	$-0.0658130\pi$
$942$		0			0
$943$		−	28.0518i		−	0.913492i
$944$		0			0
$945$	5.11146	+	0.934336i	0.166276	+	0.0303939i
$946$		0			0
$947$	−6.75087			−0.219374			−0.109687	−	0.993966i	$-0.534985\pi$
$947$	−6.75087			−0.219374			−0.109687	+	0.993966i	$0.534985\pi$
$948$		0			0
$949$	30.8761			1.00228
$950$		0			0
$951$	34.3664	−	17.1697i	1.11441	−	0.556765i
$952$		0			0
$953$			5.76167i			0.186639i	0.995636	+	0.0933194i	$0.0297478\pi$
$953$			5.76167i			0.186639i	−0.995636	+	0.0933194i	$0.970252\pi$
$954$		0			0
$955$			22.3079i			0.721867i
$956$		0			0
$957$	8.71411	−	4.35363i	0.281687	−	0.140733i
$958$		0			0
$959$	−9.53927			−0.308039
$960$		0			0
$961$	30.9780			0.999291
$962$		0			0
$963$	−2.32281	+	3.09303i	−0.0748516	+	0.0996716i
$964$		0			0
$965$		−	8.19154i		−	0.263695i
$966$		0			0
$967$			59.3405i			1.90826i	0.299391	+	0.954131i	$0.403217\pi$
$967$			59.3405i			1.90826i	−0.299391	+	0.954131i	$0.596783\pi$
$968$		0			0
$969$	4.05556	+	8.11750i	0.130283	+	0.260772i
$970$		0			0
$971$	−32.8405			−1.05390			−0.526950	−	0.849896i	$-0.676665\pi$
$971$	−32.8405			−1.05390			−0.526950	+	0.849896i	$0.676665\pi$
$972$		0			0
$973$	−17.9076			−0.574091
$974$		0			0
$975$	−2.59983	−	5.20375i	−0.0832612	−	0.166653i
$976$		0			0
$977$			35.5664i			1.13787i	0.822382	+	0.568935i	$0.192644\pi$
$977$			35.5664i			1.13787i	−0.822382	+	0.568935i	$0.807356\pi$
$978$		0			0
$979$		−	0.737697i		−	0.0235769i
$980$		0			0
$981$	−24.7931	+	33.0143i	−0.791584	+	1.05406i
$982$		0			0
$983$	−18.8755			−0.602036			−0.301018	−	0.953618i	$-0.597326\pi$
$983$	−18.8755			−0.602036			−0.301018	+	0.953618i	$0.597326\pi$
$984$		0			0
$985$	−7.23102			−0.230399
$986$		0			0
$987$	−15.4842	+	7.73599i	−0.492866	+	0.246239i
$988$		0			0
$989$		−	47.0193i		−	1.49513i
$990$		0			0
$991$			23.3269i			0.741004i	0.928832	+	0.370502i	$0.120814\pi$
$991$			23.3269i			0.741004i	−0.928832	+	0.370502i	$0.879186\pi$
$992$		0			0
$993$	2.14919	−	1.07375i	0.0682026	−	0.0340745i
$994$		0			0
$995$	−22.5289			−0.714215
$996$		0			0
$997$	22.5693			0.714777			0.357389	−	0.933956i	$-0.383667\pi$
$997$	22.5693			0.714777			0.357389	+	0.933956i	$0.383667\pi$
$998$		0			0
$999$	53.6761	+	9.81159i	1.69824	+	0.310425i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.ba.c.2591.9	✓	20
3.2	odd	2		3360.2.ba.d.2591.11	yes	20
4.3	odd	2		3360.2.ba.d.2591.12	yes	20
12.11	even	2	inner	3360.2.ba.c.2591.10	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.ba.c.2591.9	✓	20	1.1	even	1	trivial
3360.2.ba.c.2591.10	yes	20	12.11	even	2	inner
3360.2.ba.d.2591.11	yes	20	3.2	odd	2
3360.2.ba.d.2591.12	yes	20	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.774110 - 1.54944i) q^{3} -1.00000i q^{5} +1.00000i q^{7} +(-1.80151 + 2.39887i) q^{9} -0.559419 q^{11} -3.35848 q^{13} +(-1.54944 + 0.774110i) q^{15} -3.44380i q^{17} -1.52129i q^{19} +(1.54944 - 0.774110i) q^{21} -4.13079 q^{23} -1.00000 q^{25} +(5.11146 + 0.934336i) q^{27} -10.0534i q^{29} -0.148209i q^{31} +(0.433052 + 0.866784i) q^{33} +1.00000 q^{35} +10.5011 q^{37} +(2.59983 + 5.20375i) q^{39} +6.79091i q^{41} +11.3827i q^{43} +(2.39887 + 1.80151i) q^{45} -9.99341 q^{47} -1.00000 q^{49} +(-5.33594 + 2.66588i) q^{51} +8.05096i q^{53} +0.559419i q^{55} +(-2.35714 + 1.17764i) q^{57} -4.78044 q^{59} +2.04177 q^{61} +(-2.39887 - 1.80151i) q^{63} +3.35848i q^{65} +2.20738i q^{67} +(3.19768 + 6.40039i) q^{69} -1.64755 q^{71} -9.19346 q^{73} +(0.774110 + 1.54944i) q^{75} -0.559419i q^{77} +15.5490i q^{79} +(-2.50914 - 8.64316i) q^{81} -5.78506 q^{83} -3.44380 q^{85} +(-15.5771 + 7.78241i) q^{87} +1.31868i q^{89} -3.35848i q^{91} +(-0.229640 + 0.114730i) q^{93} -1.52129 q^{95} +17.9337 q^{97} +(1.00780 - 1.34197i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{3} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 4 q^{15} + 4 q^{21} + 8 q^{23} - 20 q^{25} + 20 q^{27} - 40 q^{33} + 20 q^{35} + 16 q^{37} + 4 q^{39} - 20 q^{49} + 4 q^{51} - 16 q^{57} - 64 q^{59} - 64 q^{61}+ \cdots + 28 q^{99}+O(q^{100}) \) 20 * q - 4 * q^3 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 4 * q^15 + 4 * q^21 + 8 * q^23 - 20 * q^25 + 20 * q^27 - 40 * q^33 + 20 * q^35 + 16 * q^37 + 4 * q^39 - 20 * q^49 + 4 * q^51 - 16 * q^57 - 64 * q^59 - 64 * q^61 + 8 * q^69 - 40 * q^71 + 8 * q^73 + 4 * q^75 + 8 * q^81 + 24 * q^83 + 8 * q^85 - 48 * q^87 + 72 * q^93 + 16 * q^95 + 88 * q^97 + 28 * q^99

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