LMFDB - Embedded newform 1140.2.f.a.229.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1140,2,Mod(229,1140)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1140, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1140.229"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1140.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$9.10294583043$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			229.6
Root			$1.45161 - 1.45161i$ of defining polynomial
Character	$\chi$	$=$	1140.229
Dual form			1140.2.f.a.229.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(2.21432 - 0.311108i) q^{5} +2.21432i q^{7} -1.00000 q^{9} -1.31111 q^{11} +0.836535i q^{13} +(0.311108 + 2.21432i) q^{15} +4.90321i q^{17} -1.00000 q^{19} -2.21432 q^{21} +5.33185i q^{23} +(4.80642 - 1.37778i) q^{25} -1.00000i q^{27} -1.31111 q^{29} -1.71900 q^{31} -1.31111i q^{33} +(0.688892 + 4.90321i) q^{35} +1.03011i q^{37} -0.836535 q^{39} +5.87310 q^{41} +0.214320i q^{43} +(-2.21432 + 0.311108i) q^{45} +5.33185i q^{47} +2.09679 q^{49} -4.90321 q^{51} -8.70964i q^{53} +(-2.90321 + 0.407896i) q^{55} -1.00000i q^{57} -8.10171 q^{59} +3.33185 q^{61} -2.21432i q^{63} +(0.260253 + 1.85236i) q^{65} +9.61285i q^{67} -5.33185 q^{69} -7.80642 q^{71} -0.193576i q^{73} +(1.37778 + 4.80642i) q^{75} -2.90321i q^{77} +12.9906 q^{79} +1.00000 q^{81} -0.0459330i q^{83} +(1.52543 + 10.8573i) q^{85} -1.31111i q^{87} +2.92396 q^{89} -1.85236 q^{91} -1.71900i q^{93} +(-2.21432 + 0.311108i) q^{95} +2.60147i q^{97} +1.31111 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} - 8 q^{11} + 2 q^{15} - 6 q^{19} + 2 q^{25} - 8 q^{29} - 24 q^{31} + 4 q^{35} + 8 q^{39} + 8 q^{41} + 26 q^{49} - 16 q^{51} - 4 q^{55} + 4 q^{59} - 20 q^{61} + 28 q^{65} + 8 q^{69} - 20 q^{71}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 - 8 * q^11 + 2 * q^15 - 6 * q^19 + 2 * q^25 - 8 * q^29 - 24 * q^31 + 4 * q^35 + 8 * q^39 + 8 * q^41 + 26 * q^49 - 16 * q^51 - 4 * q^55 + 4 * q^59 - 20 * q^61 + 28 * q^65 + 8 * q^69 - 20 * q^71 + 8 * q^75 + 24 * q^79 + 6 * q^81 - 4 * q^85 - 36 * q^89 - 24 * q^91 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times$.

$n$	$457$	$571$	$761$	$781$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	2.21432	−	0.311108i	0.990274	−	0.139132i
$5$	2.21432	−	0.311108i	0.990274	−	0.139132i
$6$		0			0
$7$			2.21432i			0.836934i	0.908232	+	0.418467i	$0.137432\pi$
$7$			2.21432i			0.836934i	−0.908232	+	0.418467i	$0.862568\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.31111			−0.395314			−0.197657	−	0.980271i	$-0.563333\pi$
$11$	−1.31111			−0.395314			−0.197657	+	0.980271i	$0.563333\pi$
$12$		0			0
$13$			0.836535i			0.232013i	0.993248	+	0.116007i	$0.0370094\pi$
$13$			0.836535i			0.232013i	−0.993248	+	0.116007i	$0.962991\pi$
$14$		0			0
$15$	0.311108	+	2.21432i	0.0803277	+	0.571735i
$16$		0			0
$17$			4.90321i			1.18920i	0.804020	+	0.594602i	$0.202690\pi$
$17$			4.90321i			1.18920i	−0.804020	+	0.594602i	$0.797310\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−2.21432			−0.483204
$22$		0			0
$23$			5.33185i			1.11177i	0.831260	+	0.555884i	$0.187620\pi$
$23$			5.33185i			1.11177i	−0.831260	+	0.555884i	$0.812380\pi$
$24$		0			0
$25$	4.80642	−	1.37778i	0.961285	−	0.275557i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−1.31111			−0.243467			−0.121733	−	0.992563i	$-0.538845\pi$
$29$	−1.31111			−0.243467			−0.121733	+	0.992563i	$0.538845\pi$
$30$		0			0
$31$	−1.71900			−0.308742			−0.154371	−	0.988013i	$-0.549335\pi$
$31$	−1.71900			−0.308742			−0.154371	+	0.988013i	$0.549335\pi$
$32$		0			0
$33$		−	1.31111i		−	0.228235i
$34$		0			0
$35$	0.688892	+	4.90321i	0.116444	+	0.828794i
$36$		0			0
$37$			1.03011i			0.169349i	0.996409	+	0.0846746i	$0.0269851\pi$
$37$			1.03011i			0.169349i	−0.996409	+	0.0846746i	$0.973015\pi$
$38$		0			0
$39$	−0.836535			−0.133953
$40$		0			0
$41$	5.87310			0.917224			0.458612	−	0.888637i	$-0.348347\pi$
$41$	5.87310			0.917224			0.458612	+	0.888637i	$0.348347\pi$
$42$		0			0
$43$			0.214320i			0.0326835i	0.999866	+	0.0163417i	$0.00520196\pi$
$43$			0.214320i			0.0326835i	−0.999866	+	0.0163417i	$0.994798\pi$
$44$		0			0
$45$	−2.21432	+	0.311108i	−0.330091	+	0.0463772i
$46$		0			0
$47$			5.33185i			0.777730i	0.921295	+	0.388865i	$0.127133\pi$
$47$			5.33185i			0.777730i	−0.921295	+	0.388865i	$0.872867\pi$
$48$		0			0
$49$	2.09679			0.299541
$50$		0			0
$51$	−4.90321			−0.686587
$52$		0			0
$53$		−	8.70964i		−	1.19636i	−0.801362	−	0.598180i	$-0.795891\pi$
$53$		−	8.70964i		−	1.19636i	0.801362	−	0.598180i	$-0.204109\pi$
$54$		0			0
$55$	−2.90321	+	0.407896i	−0.391469	+	0.0550007i
$56$		0			0
$57$		−	1.00000i		−	0.132453i
$58$		0			0
$59$	−8.10171			−1.05475			−0.527376	−	0.849632i	$-0.676824\pi$
$59$	−8.10171			−1.05475			−0.527376	+	0.849632i	$0.676824\pi$
$60$		0			0
$61$	3.33185			0.426600			0.213300	−	0.976987i	$-0.431579\pi$
$61$	3.33185			0.426600			0.213300	+	0.976987i	$0.431579\pi$
$62$		0			0
$63$		−	2.21432i		−	0.278978i
$64$		0			0
$65$	0.260253	+	1.85236i	0.0322804	+	0.229757i
$66$		0			0
$67$			9.61285i			1.17440i	0.809443	+	0.587198i	$0.199769\pi$
$67$			9.61285i			1.17440i	−0.809443	+	0.587198i	$0.800231\pi$
$68$		0			0
$69$	−5.33185			−0.641879
$70$		0			0
$71$	−7.80642			−0.926452			−0.463226	−	0.886240i	$-0.653308\pi$
$71$	−7.80642			−0.926452			−0.463226	+	0.886240i	$0.653308\pi$
$72$		0			0
$73$		−	0.193576i		−	0.0226564i	−0.999936	−	0.0113282i	$-0.996394\pi$
$73$		−	0.193576i		−	0.0226564i	0.999936	−	0.0113282i	$-0.00360595\pi$
$74$		0			0
$75$	1.37778	+	4.80642i	0.159093	+	0.554998i
$76$		0			0
$77$		−	2.90321i		−	0.330852i
$78$		0			0
$79$	12.9906			1.46156			0.730780	−	0.682613i	$-0.239156\pi$
$79$	12.9906			1.46156			0.730780	+	0.682613i	$0.239156\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	0.0459330i		−	0.00504180i	−0.999997	−	0.00252090i	$-0.999198\pi$
$83$		−	0.0459330i		−	0.00504180i	0.999997	−	0.00252090i	$-0.000802428\pi$
$84$		0			0
$85$	1.52543	+	10.8573i	0.165456	+	1.17764i
$86$		0			0
$87$		−	1.31111i		−	0.140566i
$88$		0			0
$89$	2.92396			0.309939			0.154969	−	0.987919i	$-0.450472\pi$
$89$	2.92396			0.309939			0.154969	+	0.987919i	$0.450472\pi$
$90$		0			0
$91$	−1.85236			−0.194180
$92$		0			0
$93$		−	1.71900i		−	0.178252i
$94$		0			0
$95$	−2.21432	+	0.311108i	−0.227184	+	0.0319190i
$96$		0			0
$97$			2.60147i			0.264139i	0.991240	+	0.132070i	$0.0421623\pi$
$97$			2.60147i			0.264139i	−0.991240	+	0.132070i	$0.957838\pi$
$98$		0			0
$99$	1.31111			0.131771
$100$		0			0
$101$	−7.61285			−0.757507			−0.378753	−	0.925498i	$-0.623647\pi$
$101$	−7.61285			−0.757507			−0.378753	+	0.925498i	$0.623647\pi$
$102$		0			0
$103$			3.05086i			0.300610i	0.988640	+	0.150305i	$0.0480255\pi$
$103$			3.05086i			0.300610i	−0.988640	+	0.150305i	$0.951974\pi$
$104$		0			0
$105$	−4.90321	+	0.688892i	−0.478504	+	0.0672290i
$106$		0			0
$107$			8.99063i			0.869157i	0.900634	+	0.434579i	$0.143103\pi$
$107$			8.99063i			0.869157i	−0.900634	+	0.434579i	$0.856897\pi$
$108$		0			0
$109$	13.9081			1.33216			0.666079	−	0.745881i	$-0.267971\pi$
$109$	13.9081			1.33216			0.666079	+	0.745881i	$0.267971\pi$
$110$		0			0
$111$	−1.03011			−0.0977739
$112$		0			0
$113$		−	10.7699i		−	1.01314i	−0.862198	−	0.506572i	$-0.830913\pi$
$113$		−	10.7699i		−	1.01314i	0.862198	−	0.506572i	$-0.169087\pi$
$114$		0			0
$115$	1.65878	+	11.8064i	0.154682	+	1.10095i
$116$		0			0
$117$		−	0.836535i		−	0.0773377i
$118$		0			0
$119$	−10.8573			−0.995285
$120$		0			0
$121$	−9.28100			−0.843727
$122$		0			0
$123$			5.87310i			0.529560i
$124$		0			0
$125$	10.2143	−	4.54617i	0.913597	−	0.406622i
$126$		0			0
$127$			2.29529i			0.203674i	0.994801	+	0.101837i	$0.0324720\pi$
$127$			2.29529i			0.203674i	−0.994801	+	0.101837i	$0.967528\pi$
$128$		0			0
$129$	−0.214320			−0.0188698
$130$		0			0
$131$	−3.87310			−0.338394			−0.169197	−	0.985582i	$-0.554117\pi$
$131$	−3.87310			−0.338394			−0.169197	+	0.985582i	$0.554117\pi$
$132$		0			0
$133$		−	2.21432i		−	0.192006i
$134$		0			0
$135$	−0.311108	−	2.21432i	−0.0267759	−	0.190578i
$136$		0			0
$137$			13.6128i			1.16302i	0.813538	+	0.581512i	$0.197539\pi$
$137$			13.6128i			1.16302i	−0.813538	+	0.581512i	$0.802461\pi$
$138$		0			0
$139$	6.19358			0.525332			0.262666	−	0.964887i	$-0.415398\pi$
$139$	6.19358			0.525332			0.262666	+	0.964887i	$0.415398\pi$
$140$		0			0
$141$	−5.33185			−0.449023
$142$		0			0
$143$		−	1.09679i		−	0.0917180i
$144$		0			0
$145$	−2.90321	+	0.407896i	−0.241099	+	0.0338739i
$146$		0			0
$147$			2.09679i			0.172940i
$148$		0			0
$149$	−22.1748			−1.81663			−0.908317	−	0.418283i	$-0.862632\pi$
$149$	−22.1748			−1.81663			−0.908317	+	0.418283i	$0.862632\pi$
$150$		0			0
$151$	15.2716			1.24279			0.621394	−	0.783498i	$-0.286567\pi$
$151$	15.2716			1.24279			0.621394	+	0.783498i	$0.286567\pi$
$152$		0			0
$153$		−	4.90321i		−	0.396401i
$154$		0			0
$155$	−3.80642	+	0.534795i	−0.305739	+	0.0429558i
$156$		0			0
$157$		−	8.23506i		−	0.657230i	−0.944464	−	0.328615i	$-0.893418\pi$
$157$		−	8.23506i		−	0.657230i	0.944464	−	0.328615i	$-0.106582\pi$
$158$		0			0
$159$	8.70964			0.690719
$160$		0			0
$161$	−11.8064			−0.930477
$162$		0			0
$163$		−	21.8272i		−	1.70964i	−0.518928	−	0.854818i	$-0.673669\pi$
$163$		−	21.8272i		−	1.70964i	0.518928	−	0.854818i	$-0.326331\pi$
$164$		0			0
$165$	−0.407896	−	2.90321i	−0.0317547	−	0.226015i
$166$		0			0
$167$		−	18.9447i		−	1.46598i	−0.680237	−	0.732992i	$-0.738123\pi$
$167$		−	18.9447i		−	1.46598i	0.680237	−	0.732992i	$-0.261877\pi$
$168$		0			0
$169$	12.3002			0.946170
$170$		0			0
$171$	1.00000			0.0764719
$172$		0			0
$173$		−	17.5254i		−	1.33243i	−0.745758	−	0.666217i	$-0.767913\pi$
$173$		−	17.5254i		−	1.33243i	0.745758	−	0.666217i	$-0.232087\pi$
$174$		0			0
$175$	3.05086	+	10.6430i	0.230623	+	0.804532i
$176$		0			0
$177$		−	8.10171i		−	0.608962i
$178$		0			0
$179$	0.101710			0.00760218			0.00380109	−	0.999993i	$-0.498790\pi$
$179$	0.101710			0.00760218			0.00380109	+	0.999993i	$0.498790\pi$
$180$		0			0
$181$	13.3461			0.992011			0.496005	−	0.868319i	$-0.334800\pi$
$181$	13.3461			0.992011			0.496005	+	0.868319i	$0.334800\pi$
$182$		0			0
$183$			3.33185i			0.246298i
$184$		0			0
$185$	0.320476	+	2.28100i	0.0235618	+	0.167702i
$186$		0			0
$187$		−	6.42864i		−	0.470109i
$188$		0			0
$189$	2.21432			0.161068
$190$		0			0
$191$	5.01582			0.362932			0.181466	−	0.983397i	$-0.441916\pi$
$191$	5.01582			0.362932			0.181466	+	0.983397i	$0.441916\pi$
$192$		0			0
$193$		−	19.7669i		−	1.42286i	−0.702759	−	0.711428i	$-0.748049\pi$
$193$		−	19.7669i		−	1.42286i	0.702759	−	0.711428i	$-0.251951\pi$
$194$		0			0
$195$	−1.85236	+	0.260253i	−0.132650	+	0.0186371i
$196$		0			0
$197$			9.19850i			0.655366i	0.944788	+	0.327683i	$0.106268\pi$
$197$			9.19850i			0.655366i	−0.944788	+	0.327683i	$0.893732\pi$
$198$		0			0
$199$	−4.29529			−0.304485			−0.152242	−	0.988343i	$-0.548649\pi$
$199$	−4.29529			−0.304485			−0.152242	+	0.988343i	$0.548649\pi$
$200$		0			0
$201$	−9.61285			−0.678038
$202$		0			0
$203$		−	2.90321i		−	0.203766i
$204$		0			0
$205$	13.0049	−	1.82717i	0.908303	−	0.127615i
$206$		0			0
$207$		−	5.33185i		−	0.370589i
$208$		0			0
$209$	1.31111			0.0906912
$210$		0			0
$211$	15.0923			1.03900			0.519500	−	0.854471i	$-0.326118\pi$
$211$	15.0923			1.03900			0.519500	+	0.854471i	$0.326118\pi$
$212$		0			0
$213$		−	7.80642i		−	0.534887i
$214$		0			0
$215$	0.0666765	+	0.474572i	0.00454730	+	0.0323656i
$216$		0			0
$217$		−	3.80642i		−	0.258397i
$218$		0			0
$219$	0.193576			0.0130807
$220$		0			0
$221$	−4.10171			−0.275911
$222$		0			0
$223$		−	7.05086i		−	0.472160i	−0.971734	−	0.236080i	$-0.924137\pi$
$223$		−	7.05086i		−	0.472160i	0.971734	−	0.236080i	$-0.0758628\pi$
$224$		0			0
$225$	−4.80642	+	1.37778i	−0.320428	+	0.0918523i
$226$		0			0
$227$		−	6.94470i		−	0.460936i	−0.973080	−	0.230468i	$-0.925974\pi$
$227$		−	6.94470i		−	0.460936i	0.973080	−	0.230468i	$-0.0740257\pi$
$228$		0			0
$229$	19.6686			1.29974			0.649870	−	0.760046i	$-0.274823\pi$
$229$	19.6686			1.29974			0.649870	+	0.760046i	$0.274823\pi$
$230$		0			0
$231$	2.90321			0.191017
$232$		0			0
$233$		−	16.5303i		−	1.08294i	−0.840720	−	0.541470i	$-0.817868\pi$
$233$		−	16.5303i		−	1.08294i	0.840720	−	0.541470i	$-0.182132\pi$
$234$		0			0
$235$	1.65878	+	11.8064i	0.108207	+	0.770166i
$236$		0			0
$237$			12.9906i			0.843832i
$238$		0			0
$239$	11.0573			0.715238			0.357619	−	0.933868i	$-0.383589\pi$
$239$	11.0573			0.715238			0.357619	+	0.933868i	$0.383589\pi$
$240$		0			0
$241$	8.75557			0.563996			0.281998	−	0.959415i	$-0.409003\pi$
$241$	8.75557			0.563996			0.281998	+	0.959415i	$0.409003\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	4.64296	−	0.652327i	0.296628	−	0.0416757i
$246$		0			0
$247$		−	0.836535i		−	0.0532275i
$248$		0			0
$249$	0.0459330			0.00291088
$250$		0			0
$251$	25.0257			1.57961			0.789803	−	0.613361i	$-0.210183\pi$
$251$	25.0257			1.57961			0.789803	+	0.613361i	$0.210183\pi$
$252$		0			0
$253$		−	6.99063i		−	0.439497i
$254$		0			0
$255$	−10.8573	+	1.52543i	−0.679909	+	0.0955260i
$256$		0			0
$257$		−	12.0558i		−	0.752019i	−0.926616	−	0.376009i	$-0.877296\pi$
$257$		−	12.0558i		−	0.752019i	0.926616	−	0.376009i	$-0.122704\pi$
$258$		0			0
$259$	−2.28100			−0.141734
$260$		0			0
$261$	1.31111			0.0811555
$262$		0			0
$263$		−	8.08742i		−	0.498692i	−0.968415	−	0.249346i	$-0.919784\pi$
$263$		−	8.08742i		−	0.498692i	0.968415	−	0.249346i	$-0.0802156\pi$
$264$		0			0
$265$	−2.70964	−	19.2859i	−0.166452	−	1.18472i
$266$		0			0
$267$			2.92396i			0.178943i
$268$		0			0
$269$	−6.49532			−0.396026			−0.198013	−	0.980199i	$-0.563449\pi$
$269$	−6.49532			−0.396026			−0.198013	+	0.980199i	$0.563449\pi$
$270$		0			0
$271$	−21.3274			−1.29555			−0.647774	−	0.761833i	$-0.724300\pi$
$271$	−21.3274			−1.29555			−0.647774	+	0.761833i	$0.724300\pi$
$272$		0			0
$273$		−	1.85236i		−	0.112110i
$274$		0			0
$275$	−6.30174	+	1.80642i	−0.380009	+	0.108931i
$276$		0			0
$277$		−	2.10171i		−	0.126280i	−0.998005	−	0.0631398i	$-0.979889\pi$
$277$		−	2.10171i		−	0.126280i	0.998005	−	0.0631398i	$-0.0201114\pi$
$278$		0			0
$279$	1.71900			0.102914
$280$		0			0
$281$	8.85083			0.527996			0.263998	−	0.964523i	$-0.414959\pi$
$281$	8.85083			0.527996			0.263998	+	0.964523i	$0.414959\pi$
$282$		0			0
$283$		−	19.0716i		−	1.13369i	−0.823825	−	0.566844i	$-0.808164\pi$
$283$		−	19.0716i		−	1.13369i	0.823825	−	0.566844i	$-0.191836\pi$
$284$		0			0
$285$	−0.311108	−	2.21432i	−0.0184284	−	0.131165i
$286$		0			0
$287$			13.0049i			0.767656i
$288$		0			0
$289$	−7.04149			−0.414205
$290$		0			0
$291$	−2.60147			−0.152501
$292$		0			0
$293$		−	17.3002i		−	1.01069i	−0.862918	−	0.505344i	$-0.831365\pi$
$293$		−	17.3002i		−	1.01069i	0.862918	−	0.505344i	$-0.168635\pi$
$294$		0			0
$295$	−17.9398	+	2.52051i	−1.04449	+	0.146749i
$296$		0			0
$297$			1.31111i			0.0760782i
$298$		0			0
$299$	−4.46028			−0.257945
$300$		0			0
$301$	−0.474572			−0.0273539
$302$		0			0
$303$		−	7.61285i		−	0.437347i
$304$		0			0
$305$	7.37778	−	1.03657i	0.422451	−	0.0593535i
$306$		0			0
$307$		−	16.6953i		−	0.952854i	−0.879214	−	0.476427i	$-0.841932\pi$
$307$		−	16.6953i		−	0.952854i	0.879214	−	0.476427i	$-0.158068\pi$
$308$		0			0
$309$	−3.05086			−0.173557
$310$		0			0
$311$	−8.85083			−0.501884			−0.250942	−	0.968002i	$-0.580740\pi$
$311$	−8.85083			−0.501884			−0.250942	+	0.968002i	$0.580740\pi$
$312$		0			0
$313$			21.5625i			1.21878i	0.792870	+	0.609391i	$0.208586\pi$
$313$			21.5625i			1.21878i	−0.792870	+	0.609391i	$0.791414\pi$
$314$		0			0
$315$	−0.688892	−	4.90321i	−0.0388147	−	0.276265i
$316$		0			0
$317$			20.2810i			1.13909i	0.821959	+	0.569547i	$0.192881\pi$
$317$			20.2810i			1.13909i	−0.821959	+	0.569547i	$0.807119\pi$
$318$		0			0
$319$	1.71900			0.0962457
$320$		0			0
$321$	−8.99063			−0.501808
$322$		0			0
$323$		−	4.90321i		−	0.272822i
$324$		0			0
$325$	1.15257	+	4.02074i	0.0639328	+	0.223031i
$326$		0			0
$327$			13.9081i			0.769122i
$328$		0			0
$329$	−11.8064			−0.650909
$330$		0			0
$331$	−2.11906			−0.116474			−0.0582371	−	0.998303i	$-0.518548\pi$
$331$	−2.11906			−0.116474			−0.0582371	+	0.998303i	$0.518548\pi$
$332$		0			0
$333$		−	1.03011i		−	0.0564498i
$334$		0			0
$335$	2.99063	+	21.2859i	0.163396	+	1.16297i
$336$		0			0
$337$		−	12.5511i		−	0.683702i	−0.939754	−	0.341851i	$-0.888946\pi$
$337$		−	12.5511i		−	0.683702i	0.939754	−	0.341851i	$-0.111054\pi$
$338$		0			0
$339$	10.7699			0.584938
$340$		0			0
$341$	2.25380			0.122050
$342$		0			0
$343$			20.1432i			1.08763i
$344$		0			0
$345$	−11.8064	+	1.65878i	−0.635636	+	0.0893057i
$346$		0			0
$347$		−	8.63651i		−	0.463632i	−0.972760	−	0.231816i	$-0.925533\pi$
$347$		−	8.63651i		−	0.463632i	0.972760	−	0.231816i	$-0.0744667\pi$
$348$		0			0
$349$	9.45091			0.505896			0.252948	−	0.967480i	$-0.418600\pi$
$349$	9.45091			0.505896			0.252948	+	0.967480i	$0.418600\pi$
$350$		0			0
$351$	0.836535			0.0446510
$352$		0			0
$353$		−	5.21279i		−	0.277449i	−0.990331	−	0.138724i	$-0.955700\pi$
$353$		−	5.21279i		−	0.277449i	0.990331	−	0.138724i	$-0.0443002\pi$
$354$		0			0
$355$	−17.2859	+	2.42864i	−0.917441	+	0.128899i
$356$		0			0
$357$		−	10.8573i		−	0.574628i
$358$		0			0
$359$	4.22861			0.223177			0.111589	−	0.993754i	$-0.464406\pi$
$359$	4.22861			0.223177			0.111589	+	0.993754i	$0.464406\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		−	9.28100i		−	0.487126i
$364$		0			0
$365$	−0.0602231	−	0.428639i	−0.00315222	−	0.0224360i
$366$		0			0
$367$			6.15410i			0.321241i	0.987016	+	0.160621i	$0.0513496\pi$
$367$			6.15410i			0.321241i	−0.987016	+	0.160621i	$0.948650\pi$
$368$		0			0
$369$	−5.87310			−0.305741
$370$		0			0
$371$	19.2859			1.00127
$372$		0			0
$373$		−	15.1635i		−	0.785134i	−0.919723	−	0.392567i	$-0.871587\pi$
$373$		−	15.1635i		−	0.785134i	0.919723	−	0.392567i	$-0.128413\pi$
$374$		0			0
$375$	4.54617	+	10.2143i	0.234763	+	0.527465i
$376$		0			0
$377$		−	1.09679i		−	0.0564875i
$378$		0			0
$379$	−12.5161			−0.642907			−0.321453	−	0.946925i	$-0.604171\pi$
$379$	−12.5161			−0.642907			−0.321453	+	0.946925i	$0.604171\pi$
$380$		0			0
$381$	−2.29529			−0.117591
$382$		0			0
$383$			19.5210i			0.997476i	0.866753	+	0.498738i	$0.166203\pi$
$383$			19.5210i			0.997476i	−0.866753	+	0.498738i	$0.833797\pi$
$384$		0			0
$385$	−0.903212	−	6.42864i	−0.0460319	−	0.327634i
$386$		0			0
$387$		−	0.214320i		−	0.0108945i
$388$		0			0
$389$	−29.5812			−1.49983			−0.749913	−	0.661536i	$-0.769905\pi$
$389$	−29.5812			−1.49983			−0.749913	+	0.661536i	$0.769905\pi$
$390$		0			0
$391$	−26.1432			−1.32212
$392$		0			0
$393$		−	3.87310i		−	0.195372i
$394$		0			0
$395$	28.7654	−	4.04149i	1.44735	−	0.203349i
$396$		0			0
$397$			29.1526i			1.46313i	0.681774	+	0.731563i	$0.261209\pi$
$397$			29.1526i			1.46313i	−0.681774	+	0.731563i	$0.738791\pi$
$398$		0			0
$399$	2.21432			0.110855
$400$		0			0
$401$	−14.7304			−0.735600			−0.367800	−	0.929905i	$-0.619889\pi$
$401$	−14.7304			−0.735600			−0.367800	+	0.929905i	$0.619889\pi$
$402$		0			0
$403$		−	1.43801i		−	0.0716323i
$404$		0			0
$405$	2.21432	−	0.311108i	0.110030	−	0.0154591i
$406$		0			0
$407$		−	1.35059i		−	0.0669461i
$408$		0			0
$409$	−10.3368			−0.511121			−0.255560	−	0.966793i	$-0.582260\pi$
$409$	−10.3368			−0.511121			−0.255560	+	0.966793i	$0.582260\pi$
$410$		0			0
$411$	−13.6128			−0.671472
$412$		0			0
$413$		−	17.9398i		−	0.882759i
$414$		0			0
$415$	−0.0142901	−	0.101710i	−0.000701473	−	0.00499276i
$416$		0			0
$417$			6.19358i			0.303301i
$418$		0			0
$419$	−30.9842			−1.51368			−0.756838	−	0.653602i	$-0.773257\pi$
$419$	−30.9842			−1.51368			−0.756838	+	0.653602i	$0.773257\pi$
$420$		0			0
$421$	32.1432			1.56656			0.783282	−	0.621667i	$-0.213544\pi$
$421$	32.1432			1.56656			0.783282	+	0.621667i	$0.213544\pi$
$422$		0			0
$423$		−	5.33185i		−	0.259243i
$424$		0			0
$425$	6.75557	+	23.5669i	0.327693	+	1.14316i
$426$		0			0
$427$			7.37778i			0.357036i
$428$		0			0
$429$	1.09679			0.0529534
$430$		0			0
$431$	30.9719			1.49186			0.745932	−	0.666022i	$-0.232004\pi$
$431$	30.9719			1.49186			0.745932	+	0.666022i	$0.232004\pi$
$432$		0			0
$433$		−	40.9195i		−	1.96647i	−0.182353	−	0.983233i	$-0.558371\pi$
$433$		−	40.9195i		−	1.96647i	0.182353	−	0.983233i	$-0.441629\pi$
$434$		0			0
$435$	−0.407896	−	2.90321i	−0.0195571	−	0.139198i
$436$		0			0
$437$		−	5.33185i		−	0.255057i
$438$		0			0
$439$	4.26671			0.203639			0.101819	−	0.994803i	$-0.467534\pi$
$439$	4.26671			0.203639			0.101819	+	0.994803i	$0.467534\pi$
$440$		0			0
$441$	−2.09679			−0.0998471
$442$		0			0
$443$		−	4.34122i		−	0.206258i	−0.994668	−	0.103129i	$-0.967115\pi$
$443$		−	4.34122i		−	0.206258i	0.994668	−	0.103129i	$-0.0328854\pi$
$444$		0			0
$445$	6.47457	−	0.909665i	0.306924	−	0.0431223i
$446$		0			0
$447$		−	22.1748i		−	1.04883i
$448$		0			0
$449$	2.56845			0.121212			0.0606062	−	0.998162i	$-0.480697\pi$
$449$	2.56845			0.121212			0.0606062	+	0.998162i	$0.480697\pi$
$450$		0			0
$451$	−7.70027			−0.362591
$452$		0			0
$453$			15.2716i			0.717524i
$454$		0			0
$455$	−4.10171	+	0.576283i	−0.192291	+	0.0270165i
$456$		0			0
$457$			27.2859i			1.27638i	0.769878	+	0.638191i	$0.220317\pi$
$457$			27.2859i			1.27638i	−0.769878	+	0.638191i	$0.779683\pi$
$458$		0			0
$459$	4.90321			0.228862
$460$		0			0
$461$	6.00000			0.279448			0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000			0.279448			0.139724	+	0.990190i	$0.455378\pi$
$462$		0			0
$463$			37.2464i			1.73099i	0.500919	+	0.865494i	$0.332996\pi$
$463$			37.2464i			1.73099i	−0.500919	+	0.865494i	$0.667004\pi$
$464$		0			0
$465$	−0.534795	−	3.80642i	−0.0248005	−	0.176519i
$466$		0			0
$467$			13.8809i			0.642333i	0.947023	+	0.321167i	$0.104075\pi$
$467$			13.8809i			0.642333i	−0.947023	+	0.321167i	$0.895925\pi$
$468$		0			0
$469$	−21.2859			−0.982892
$470$		0			0
$471$	8.23506			0.379452
$472$		0			0
$473$		−	0.280996i		−	0.0129202i
$474$		0			0
$475$	−4.80642	+	1.37778i	−0.220534	+	0.0632171i
$476$		0			0
$477$			8.70964i			0.398787i
$478$		0			0
$479$	−1.07604			−0.0491657			−0.0245829	−	0.999698i	$-0.507826\pi$
$479$	−1.07604			−0.0491657			−0.0245829	+	0.999698i	$0.507826\pi$
$480$		0			0
$481$	−0.861725			−0.0392913
$482$		0			0
$483$		−	11.8064i		−	0.537211i
$484$		0			0
$485$	0.809338	+	5.76049i	0.0367502	+	0.261570i
$486$		0			0
$487$			26.8069i			1.21474i	0.794420	+	0.607368i	$0.207775\pi$
$487$			26.8069i			1.21474i	−0.794420	+	0.607368i	$0.792225\pi$
$488$		0			0
$489$	21.8272			0.987059
$490$		0			0
$491$	18.6474			0.841546			0.420773	−	0.907166i	$-0.361759\pi$
$491$	18.6474			0.841546			0.420773	+	0.907166i	$0.361759\pi$
$492$		0			0
$493$		−	6.42864i		−	0.289531i
$494$		0			0
$495$	2.90321	−	0.407896i	0.130490	−	0.0183336i
$496$		0			0
$497$		−	17.2859i		−	0.775379i
$498$		0			0
$499$	8.94914			0.400619			0.200309	−	0.979733i	$-0.435805\pi$
$499$	8.94914			0.400619			0.200309	+	0.979733i	$0.435805\pi$
$500$		0			0
$501$	18.9447			0.846387
$502$		0			0
$503$			21.3604i			0.952415i	0.879333	+	0.476207i	$0.157989\pi$
$503$			21.3604i			0.952415i	−0.879333	+	0.476207i	$0.842011\pi$
$504$		0			0
$505$	−16.8573	+	2.36842i	−0.750139	+	0.105393i
$506$		0			0
$507$			12.3002i			0.546271i
$508$		0			0
$509$	−12.4953			−0.553845			−0.276923	−	0.960892i	$-0.589315\pi$
$509$	−12.4953			−0.553845			−0.276923	+	0.960892i	$0.589315\pi$
$510$		0			0
$511$	0.428639			0.0189619
$512$		0			0
$513$			1.00000i			0.0441511i
$514$		0			0
$515$	0.949145	+	6.75557i	0.0418243	+	0.297686i
$516$		0			0
$517$		−	6.99063i		−	0.307448i
$518$		0			0
$519$	17.5254			0.769281
$520$		0			0
$521$	34.7116			1.52074			0.760372	−	0.649487i	$-0.225016\pi$
$521$	34.7116			1.52074			0.760372	+	0.649487i	$0.225016\pi$
$522$		0			0
$523$		−	41.2958i		−	1.80574i	−0.429916	−	0.902869i	$-0.641457\pi$
$523$		−	41.2958i		−	1.80574i	0.429916	−	0.902869i	$-0.358543\pi$
$524$		0			0
$525$	−10.6430	+	3.05086i	−0.464497	+	0.133150i
$526$		0			0
$527$		−	8.42864i		−	0.367157i
$528$		0			0
$529$	−5.42864			−0.236028
$530$		0			0
$531$	8.10171			0.351584
$532$		0			0
$533$			4.91306i			0.212808i
$534$		0			0
$535$	2.79706	+	19.9081i	0.120927	+	0.860704i
$536$		0			0
$537$			0.101710i			0.00438912i
$538$		0			0
$539$	−2.74912			−0.118413
$540$		0			0
$541$	21.4924			0.924031			0.462015	−	0.886872i	$-0.347127\pi$
$541$	21.4924			0.924031			0.462015	+	0.886872i	$0.347127\pi$
$542$		0			0
$543$			13.3461i			0.572738i
$544$		0			0
$545$	30.7971	−	4.32693i	1.31920	−	0.185345i
$546$		0			0
$547$		−	4.45722i		−	0.190577i	−0.995450	−	0.0952885i	$-0.969623\pi$
$547$		−	4.45722i		−	0.190577i	0.995450	−	0.0952885i	$-0.0303774\pi$
$548$		0			0
$549$	−3.33185			−0.142200
$550$		0			0
$551$	1.31111			0.0558551
$552$		0			0
$553$			28.7654i			1.22323i
$554$		0			0
$555$	−2.28100	+	0.320476i	−0.0968229	+	0.0136034i
$556$		0			0
$557$		−	23.6860i		−	1.00361i	−0.864982	−	0.501804i	$-0.832670\pi$
$557$		−	23.6860i		−	1.00361i	0.864982	−	0.501804i	$-0.167330\pi$
$558$		0			0
$559$	−0.179286			−0.00758299
$560$		0			0
$561$	6.42864			0.271417
$562$		0			0
$563$			22.5575i			0.950687i	0.879800	+	0.475344i	$0.157676\pi$
$563$			22.5575i			0.950687i	−0.879800	+	0.475344i	$0.842324\pi$
$564$		0			0
$565$	−3.35059	−	23.8479i	−0.140960	−	1.00329i
$566$		0			0
$567$			2.21432i			0.0929927i
$568$		0			0
$569$	−34.8222			−1.45982			−0.729912	−	0.683541i	$-0.760439\pi$
$569$	−34.8222			−1.45982			−0.729912	+	0.683541i	$0.760439\pi$
$570$		0			0
$571$	−25.0192			−1.04702			−0.523511	−	0.852019i	$-0.675378\pi$
$571$	−25.0192			−1.04702			−0.523511	+	0.852019i	$0.675378\pi$
$572$		0			0
$573$			5.01582i			0.209539i
$574$		0			0
$575$	7.34614	+	25.6271i	0.306355	+	1.06873i
$576$		0			0
$577$			26.1245i			1.08758i	0.839223	+	0.543788i	$0.183010\pi$
$577$			26.1245i			1.08758i	−0.839223	+	0.543788i	$0.816990\pi$
$578$		0			0
$579$	19.7669			0.821486
$580$		0			0
$581$	0.101710			0.00421965
$582$		0			0
$583$			11.4193i			0.472938i
$584$		0			0
$585$	−0.260253	−	1.85236i	−0.0107601	−	0.0765855i
$586$		0			0
$587$		−	40.8943i		−	1.68789i	−0.536430	−	0.843945i	$-0.680228\pi$
$587$		−	40.8943i		−	1.68789i	0.536430	−	0.843945i	$-0.319772\pi$
$588$		0			0
$589$	1.71900			0.0708303
$590$		0			0
$591$	−9.19850			−0.378376
$592$		0			0
$593$			8.78415i			0.360722i	0.983601	+	0.180361i	$0.0577266\pi$
$593$			8.78415i			0.360722i	−0.983601	+	0.180361i	$0.942273\pi$
$594$		0			0
$595$	−24.0415	+	3.37778i	−0.985605	+	0.138476i
$596$		0			0
$597$		−	4.29529i		−	0.175794i
$598$		0			0
$599$	−14.0415			−0.573720			−0.286860	−	0.957973i	$-0.592611\pi$
$599$	−14.0415			−0.573720			−0.286860	+	0.957973i	$0.592611\pi$
$600$		0			0
$601$	−24.3051			−0.991427			−0.495713	−	0.868486i	$-0.665093\pi$
$601$	−24.3051			−0.991427			−0.495713	+	0.868486i	$0.665093\pi$
$602$		0			0
$603$		−	9.61285i		−	0.391465i
$604$		0			0
$605$	−20.5511	+	2.88739i	−0.835521	+	0.117389i
$606$		0			0
$607$			23.1655i			0.940258i	0.882598	+	0.470129i	$0.155793\pi$
$607$			23.1655i			0.940258i	−0.882598	+	0.470129i	$0.844207\pi$
$608$		0			0
$609$	2.90321			0.117644
$610$		0			0
$611$	−4.46028			−0.180444
$612$		0			0
$613$			33.7462i			1.36300i	0.731820	+	0.681498i	$0.238671\pi$
$613$			33.7462i			1.36300i	−0.731820	+	0.681498i	$0.761329\pi$
$614$		0			0
$615$	1.82717	+	13.0049i	0.0736785	+	0.524409i
$616$		0			0
$617$			41.5768i			1.67382i	0.547343	+	0.836909i	$0.315639\pi$
$617$			41.5768i			1.67382i	−0.547343	+	0.836909i	$0.684361\pi$
$618$		0			0
$619$	−16.8988			−0.679219			−0.339609	−	0.940567i	$-0.610295\pi$
$619$	−16.8988			−0.679219			−0.339609	+	0.940567i	$0.610295\pi$
$620$		0			0
$621$	5.33185			0.213960
$622$		0			0
$623$			6.47457i			0.259398i
$624$		0			0
$625$	21.2034	−	13.2444i	0.848137	−	0.529777i
$626$		0			0
$627$			1.31111i			0.0523606i
$628$		0			0
$629$	−5.05086			−0.201391
$630$		0			0
$631$	−23.7748			−0.946459			−0.473230	−	0.880939i	$-0.656912\pi$
$631$	−23.7748			−0.946459			−0.473230	+	0.880939i	$0.656912\pi$
$632$		0			0
$633$			15.0923i			0.599867i
$634$		0			0
$635$	0.714082	+	5.08250i	0.0283375	+	0.201693i
$636$		0			0
$637$			1.75404i			0.0694975i
$638$		0			0
$639$	7.80642			0.308817
$640$		0			0
$641$	−21.5877			−0.852661			−0.426331	−	0.904567i	$-0.640194\pi$
$641$	−21.5877			−0.852661			−0.426331	+	0.904567i	$0.640194\pi$
$642$		0			0
$643$		−	26.2242i		−	1.03418i	−0.855931	−	0.517090i	$-0.827015\pi$
$643$		−	26.2242i		−	1.03418i	0.855931	−	0.517090i	$-0.172985\pi$
$644$		0			0
$645$	−0.474572	+	0.0666765i	−0.0186863	+	0.00262539i
$646$		0			0
$647$			23.9037i			0.939751i	0.882733	+	0.469875i	$0.155701\pi$
$647$			23.9037i			0.939751i	−0.882733	+	0.469875i	$0.844299\pi$
$648$		0			0
$649$	10.6222			0.416958
$650$		0			0
$651$	3.80642			0.149186
$652$		0			0
$653$			29.6686i			1.16102i	0.814252	+	0.580512i	$0.197147\pi$
$653$			29.6686i			1.16102i	−0.814252	+	0.580512i	$0.802853\pi$
$654$		0			0
$655$	−8.57628	+	1.20495i	−0.335103	+	0.0470814i
$656$		0			0
$657$			0.193576i			0.00755212i
$658$		0			0
$659$	28.3180			1.10311			0.551557	−	0.834137i	$-0.314034\pi$
$659$	28.3180			1.10311			0.551557	+	0.834137i	$0.314034\pi$
$660$		0			0
$661$	4.06022			0.157924			0.0789622	−	0.996878i	$-0.474839\pi$
$661$	4.06022			0.157924			0.0789622	+	0.996878i	$0.474839\pi$
$662$		0			0
$663$		−	4.10171i		−	0.159297i
$664$		0			0
$665$	−0.688892	−	4.90321i	−0.0267141	−	0.190138i
$666$		0			0
$667$		−	6.99063i		−	0.270678i
$668$		0			0
$669$	7.05086			0.272602
$670$		0			0
$671$	−4.36842			−0.168641
$672$		0			0
$673$			2.27454i			0.0876772i	0.999039	+	0.0438386i	$0.0139587\pi$
$673$			2.27454i			0.0876772i	−0.999039	+	0.0438386i	$0.986041\pi$
$674$		0			0
$675$	−1.37778	−	4.80642i	−0.0530309	−	0.184999i
$676$		0			0
$677$			23.9857i			0.921846i	0.887440	+	0.460923i	$0.152482\pi$
$677$			23.9857i			0.921846i	−0.887440	+	0.460923i	$0.847518\pi$
$678$		0			0
$679$	−5.76049			−0.221067
$680$		0			0
$681$	6.94470			0.266121
$682$		0			0
$683$		−	12.4001i		−	0.474475i	−0.971452	−	0.237238i	$-0.923758\pi$
$683$		−	12.4001i		−	0.474475i	0.971452	−	0.237238i	$-0.0762420\pi$
$684$		0			0
$685$	4.23506	+	30.1432i	0.161813	+	1.15171i
$686$		0			0
$687$			19.6686i			0.750405i
$688$		0			0
$689$	7.28592			0.277571
$690$		0			0
$691$	−35.9782			−1.36868			−0.684338	−	0.729165i	$-0.739909\pi$
$691$	−35.9782			−1.36868			−0.684338	+	0.729165i	$0.739909\pi$
$692$		0			0
$693$			2.90321i			0.110284i
$694$		0			0
$695$	13.7146	−	1.92687i	0.520223	−	0.0730903i
$696$		0			0
$697$			28.7971i			1.09077i
$698$		0			0
$699$	16.5303			0.625235
$700$		0			0
$701$	2.65386			0.100235			0.0501174	−	0.998743i	$-0.484040\pi$
$701$	2.65386			0.100235			0.0501174	+	0.998743i	$0.484040\pi$
$702$		0			0
$703$		−	1.03011i		−	0.0388514i
$704$		0			0
$705$	−11.8064	+	1.65878i	−0.444656	+	0.0624733i
$706$		0			0
$707$		−	16.8573i		−	0.633983i
$708$		0			0
$709$	28.4844			1.06975			0.534877	−	0.844930i	$-0.320358\pi$
$709$	28.4844			1.06975			0.534877	+	0.844930i	$0.320358\pi$
$710$		0			0
$711$	−12.9906			−0.487187
$712$		0			0
$713$		−	9.16547i		−	0.343250i
$714$		0			0
$715$	−0.341219	−	2.42864i	−0.0127609	−	0.0908260i
$716$		0			0
$717$			11.0573i			0.412943i
$718$		0			0
$719$	12.6800			0.472884			0.236442	−	0.971646i	$-0.424019\pi$
$719$	12.6800			0.472884			0.236442	+	0.971646i	$0.424019\pi$
$720$		0			0
$721$	−6.75557			−0.251591
$722$		0			0
$723$			8.75557i			0.325623i
$724$		0			0
$725$	−6.30174	+	1.80642i	−0.234041	+	0.0670889i
$726$		0			0
$727$		−	50.5096i		−	1.87330i	−0.350270	−	0.936649i	$-0.613910\pi$
$727$		−	50.5096i		−	1.87330i	0.350270	−	0.936649i	$-0.386090\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	−1.05086			−0.0388673
$732$		0			0
$733$		−	17.4380i		−	0.644087i	−0.946725	−	0.322044i	$-0.895630\pi$
$733$		−	17.4380i		−	0.644087i	0.946725	−	0.322044i	$-0.104370\pi$
$734$		0			0
$735$	0.652327	+	4.64296i	0.0240614	+	0.171258i
$736$		0			0
$737$		−	12.6035i		−	0.464255i
$738$		0			0
$739$	41.6731			1.53297			0.766484	−	0.642263i	$-0.222004\pi$
$739$	41.6731			1.53297			0.766484	+	0.642263i	$0.222004\pi$
$740$		0			0
$741$	0.836535			0.0307309
$742$		0			0
$743$			6.07313i			0.222802i	0.993776	+	0.111401i	$0.0355337\pi$
$743$			6.07313i			0.222802i	−0.993776	+	0.111401i	$0.964466\pi$
$744$		0			0
$745$	−49.1022	+	6.89877i	−1.79896	+	0.252751i
$746$		0			0
$747$			0.0459330i			0.00168060i
$748$		0			0
$749$	−19.9081			−0.727427
$750$		0			0
$751$	−14.0272			−0.511860			−0.255930	−	0.966695i	$-0.582382\pi$
$751$	−14.0272			−0.511860			−0.255930	+	0.966695i	$0.582382\pi$
$752$		0			0
$753$			25.0257i			0.911986i
$754$		0			0
$755$	33.8163	−	4.75112i	1.23070	−	0.172911i
$756$		0			0
$757$		−	5.76494i		−	0.209530i	−0.994497	−	0.104765i	$-0.966591\pi$
$757$		−	5.76494i		−	0.209530i	0.994497	−	0.104765i	$-0.0334091\pi$
$758$		0			0
$759$	6.99063			0.253744
$760$		0			0
$761$	−47.5910			−1.72517			−0.862587	−	0.505909i	$-0.831157\pi$
$761$	−47.5910			−1.72517			−0.862587	+	0.505909i	$0.831157\pi$
$762$		0			0
$763$			30.7971i			1.11493i
$764$		0			0
$765$	−1.52543	−	10.8573i	−0.0551519	−	0.392546i
$766$		0			0
$767$		−	6.77737i		−	0.244717i
$768$		0			0
$769$	7.39069			0.266515			0.133258	−	0.991081i	$-0.457456\pi$
$769$	7.39069			0.266515			0.133258	+	0.991081i	$0.457456\pi$
$770$		0			0
$771$	12.0558			0.434178
$772$		0			0
$773$			53.8533i			1.93697i	0.249074	+	0.968484i	$0.419874\pi$
$773$			53.8533i			1.93697i	−0.249074	+	0.968484i	$0.580126\pi$
$774$		0			0
$775$	−8.26226	+	2.36842i	−0.296789	+	0.0850760i
$776$		0			0
$777$		−	2.28100i		−	0.0818303i
$778$		0			0
$779$	−5.87310			−0.210426
$780$		0			0
$781$	10.2351			0.366239
$782$		0			0
$783$			1.31111i			0.0468552i
$784$		0			0
$785$	−2.56199	−	18.2351i	−0.0914414	−	0.650837i
$786$		0			0
$787$		−	42.8069i		−	1.52590i	−0.646457	−	0.762951i	$-0.723750\pi$
$787$		−	42.8069i		−	1.52590i	0.646457	−	0.762951i	$-0.276250\pi$
$788$		0			0
$789$	8.08742			0.287920
$790$		0			0
$791$	23.8479			0.847934
$792$		0			0
$793$			2.78721i			0.0989768i
$794$		0			0
$795$	19.2859	−	2.70964i	0.684001	−	0.0961009i
$796$		0			0
$797$			38.7828i			1.37376i	0.726773	+	0.686878i	$0.241019\pi$
$797$			38.7828i			1.37376i	−0.726773	+	0.686878i	$0.758981\pi$
$798$		0			0
$799$	−26.1432			−0.924880
$800$		0			0
$801$	−2.92396			−0.103313
$802$		0			0
$803$			0.253799i			0.00895638i
$804$		0			0
$805$	−26.1432	+	3.67307i	−0.921427	+	0.129459i
$806$		0			0
$807$		−	6.49532i		−	0.228646i
$808$		0			0
$809$	2.88586			0.101461			0.0507307	−	0.998712i	$-0.483845\pi$
$809$	2.88586			0.101461			0.0507307	+	0.998712i	$0.483845\pi$
$810$		0			0
$811$	−40.9403			−1.43761			−0.718803	−	0.695213i	$-0.755310\pi$
$811$	−40.9403			−1.43761			−0.718803	+	0.695213i	$0.755310\pi$
$812$		0			0
$813$		−	21.3274i		−	0.747985i
$814$		0			0
$815$	−6.79060	−	48.3323i	−0.237864	−	1.69301i
$816$		0			0
$817$		−	0.214320i		−	0.00749810i
$818$		0			0
$819$	1.85236			0.0647266
$820$		0			0
$821$	18.6637			0.651368			0.325684	−	0.945479i	$-0.394405\pi$
$821$	18.6637			0.651368			0.325684	+	0.945479i	$0.394405\pi$
$822$		0			0
$823$			27.5319i			0.959701i	0.877350	+	0.479851i	$0.159309\pi$
$823$			27.5319i			0.959701i	−0.877350	+	0.479851i	$0.840691\pi$
$824$		0			0
$825$	−1.80642	−	6.30174i	−0.0628916	−	0.219398i
$826$		0			0
$827$		−	44.3007i		−	1.54049i	−0.637751	−	0.770243i	$-0.720135\pi$
$827$		−	44.3007i		−	1.54049i	0.637751	−	0.770243i	$-0.279865\pi$
$828$		0			0
$829$	14.9621			0.519654			0.259827	−	0.965655i	$-0.416335\pi$
$829$	14.9621			0.519654			0.259827	+	0.965655i	$0.416335\pi$
$830$		0			0
$831$	2.10171			0.0729075
$832$		0			0
$833$			10.2810i			0.356215i
$834$		0			0
$835$	−5.89384	−	41.9496i	−0.203965	−	1.45173i
$836$		0			0
$837$			1.71900i			0.0594175i
$838$		0			0
$839$	−15.1526			−0.523125			−0.261562	−	0.965187i	$-0.584238\pi$
$839$	−15.1526			−0.523125			−0.261562	+	0.965187i	$0.584238\pi$
$840$		0			0
$841$	−27.2810			−0.940724
$842$		0			0
$843$			8.85083i			0.304839i
$844$		0			0
$845$	27.2366	−	3.82669i	0.936967	−	0.131642i
$846$		0			0
$847$		−	20.5511i		−	0.706144i
$848$		0			0
$849$	19.0716			0.654536
$850$		0			0
$851$	−5.49240			−0.188277
$852$		0			0
$853$		−	13.4608i		−	0.460888i	−0.973086	−	0.230444i	$-0.925982\pi$
$853$		−	13.4608i		−	0.460888i	0.973086	−	0.230444i	$-0.0740178\pi$
$854$		0			0
$855$	2.21432	−	0.311108i	0.0757281	−	0.0106397i
$856$		0			0
$857$			32.8845i			1.12331i	0.827371	+	0.561656i	$0.189836\pi$
$857$			32.8845i			1.12331i	−0.827371	+	0.561656i	$0.810164\pi$
$858$		0			0
$859$	9.76494			0.333175			0.166588	−	0.986027i	$-0.446725\pi$
$859$	9.76494			0.333175			0.166588	+	0.986027i	$0.446725\pi$
$860$		0			0
$861$	−13.0049			−0.443207
$862$		0			0
$863$			8.05038i			0.274038i	0.990568	+	0.137019i	$0.0437522\pi$
$863$			8.05038i			0.274038i	−0.990568	+	0.137019i	$0.956248\pi$
$864$		0			0
$865$	−5.45230	−	38.8069i	−0.185384	−	1.31947i
$866$		0			0
$867$		−	7.04149i		−	0.239141i
$868$		0			0
$869$	−17.0321			−0.577775
$870$		0			0
$871$	−8.04149			−0.272475
$872$		0			0
$873$		−	2.60147i		−	0.0880465i
$874$		0			0
$875$	10.0667	+	22.6178i	0.340316	+	0.764620i
$876$		0			0
$877$			11.4400i			0.386302i	0.981169	+	0.193151i	$0.0618707\pi$
$877$			11.4400i			0.386302i	−0.981169	+	0.193151i	$0.938129\pi$
$878$		0			0
$879$	17.3002			0.583522
$880$		0			0
$881$	42.3279			1.42606			0.713031	−	0.701132i	$-0.247322\pi$
$881$	42.3279			1.42606			0.713031	+	0.701132i	$0.247322\pi$
$882$		0			0
$883$		−	30.5926i		−	1.02952i	−0.857334	−	0.514761i	$-0.827881\pi$
$883$		−	30.5926i		−	1.02952i	0.857334	−	0.514761i	$-0.172119\pi$
$884$		0			0
$885$	−2.52051	−	17.9398i	−0.0847259	−	0.603039i
$886$		0			0
$887$			13.1240i			0.440660i	0.975425	+	0.220330i	$0.0707135\pi$
$887$			13.1240i			0.440660i	−0.975425	+	0.220330i	$0.929287\pi$
$888$		0			0
$889$	−5.08250			−0.170462
$890$		0			0
$891$	−1.31111			−0.0439238
$892$		0			0
$893$		−	5.33185i		−	0.178424i
$894$		0			0
$895$	0.225219	−	0.0316429i	0.00752824	−	0.00105770i
$896$		0			0
$897$		−	4.46028i		−	0.148924i
$898$		0			0
$899$	2.25380			0.0751684
$900$		0			0
$901$	42.7052			1.42272
$902$		0			0
$903$		−	0.474572i		−	0.0157928i
$904$		0			0
$905$	29.5526	−	4.15209i	0.982362	−	0.138020i
$906$		0			0
$907$		−	58.2449i		−	1.93399i	−0.254795	−	0.966995i	$-0.582008\pi$
$907$		−	58.2449i		−	1.93399i	0.254795	−	0.966995i	$-0.417992\pi$
$908$		0			0
$909$	7.61285			0.252502
$910$		0			0
$911$	40.3783			1.33779			0.668896	−	0.743356i	$-0.266767\pi$
$911$	40.3783			1.33779			0.668896	+	0.743356i	$0.266767\pi$
$912$		0			0
$913$			0.0602231i			0.00199309i
$914$		0			0
$915$	1.03657	+	7.37778i	0.0342678	+	0.243902i
$916$		0			0
$917$		−	8.57628i		−	0.283214i
$918$		0			0
$919$	−53.2770			−1.75745			−0.878723	−	0.477331i	$-0.841604\pi$
$919$	−53.2770			−1.75745			−0.878723	+	0.477331i	$0.841604\pi$
$920$		0			0
$921$	16.6953			0.550130
$922$		0			0
$923$		−	6.53035i		−	0.214949i
$924$		0			0
$925$	1.41927	+	4.95115i	0.0466654	+	0.162793i
$926$		0			0
$927$		−	3.05086i		−	0.100203i
$928$		0			0
$929$	7.61285			0.249769			0.124885	−	0.992171i	$-0.460144\pi$
$929$	7.61285			0.249769			0.124885	+	0.992171i	$0.460144\pi$
$930$		0			0
$931$	−2.09679			−0.0687195
$932$		0			0
$933$		−	8.85083i		−	0.289763i
$934$		0			0
$935$	−2.00000	−	14.2351i	−0.0654070	−	0.465536i
$936$		0			0
$937$			17.0825i			0.558061i	0.960282	+	0.279030i	$0.0900130\pi$
$937$			17.0825i			0.558061i	−0.960282	+	0.279030i	$0.909987\pi$
$938$		0			0
$939$	−21.5625			−0.703665
$940$		0			0
$941$	−17.5965			−0.573631			−0.286816	−	0.957986i	$-0.592597\pi$
$941$	−17.5965			−0.573631			−0.286816	+	0.957986i	$0.592597\pi$
$942$		0			0
$943$			31.3145i			1.01974i
$944$		0			0
$945$	4.90321	−	0.688892i	0.159501	−	0.0224097i
$946$		0			0
$947$		−	49.2529i		−	1.60050i	−0.599664	−	0.800252i	$-0.704699\pi$
$947$		−	49.2529i		−	1.60050i	0.599664	−	0.800252i	$-0.295301\pi$
$948$		0			0
$949$	0.161933			0.00525658
$950$		0			0
$951$	−20.2810			−0.657656
$952$		0			0
$953$			28.8430i			0.934316i	0.884174	+	0.467158i	$0.154722\pi$
$953$			28.8430i			0.934316i	−0.884174	+	0.467158i	$0.845278\pi$
$954$		0			0
$955$	11.1066	−	1.56046i	0.359402	−	0.0504953i
$956$		0			0
$957$			1.71900i			0.0555675i
$958$		0			0
$959$	−30.1432			−0.973375
$960$		0			0
$961$	−28.0450			−0.904678
$962$		0			0
$963$		−	8.99063i		−	0.289719i
$964$		0			0
$965$	−6.14965	−	43.7703i	−0.197964	−	1.40902i
$966$		0			0
$967$		−	56.1867i		−	1.80684i	−0.428754	−	0.903421i	$-0.641047\pi$
$967$		−	56.1867i		−	1.80684i	0.428754	−	0.903421i	$-0.358953\pi$
$968$		0			0
$969$	4.90321			0.157514
$970$		0			0
$971$	−1.00937			−0.0323922			−0.0161961	−	0.999869i	$-0.505156\pi$
$971$	−1.00937			−0.0323922			−0.0161961	+	0.999869i	$0.505156\pi$
$972$		0			0
$973$			13.7146i			0.439669i
$974$		0			0
$975$	−4.02074	+	1.15257i	−0.128767	+	0.0369116i
$976$		0			0
$977$		−	27.8938i		−	0.892403i	−0.894932	−	0.446202i	$-0.852776\pi$
$977$		−	27.8938i		−	0.892403i	0.894932	−	0.446202i	$-0.147224\pi$
$978$		0			0
$979$	−3.83362			−0.122523
$980$		0			0
$981$	−13.9081			−0.444053
$982$		0			0
$983$		−	45.1294i		−	1.43940i	−0.694283	−	0.719702i	$-0.744278\pi$
$983$		−	45.1294i		−	1.43940i	0.694283	−	0.719702i	$-0.255722\pi$
$984$		0			0
$985$	2.86172	+	20.3684i	0.0911821	+	0.648992i
$986$		0			0
$987$		−	11.8064i		−	0.375803i
$988$		0			0
$989$	−1.14272			−0.0363364
$990$		0			0
$991$	−22.2854			−0.707920			−0.353960	−	0.935260i	$-0.615165\pi$
$991$	−22.2854			−0.707920			−0.353960	+	0.935260i	$0.615165\pi$
$992$		0			0
$993$		−	2.11906i		−	0.0672464i
$994$		0			0
$995$	−9.51114	+	1.33630i	−0.301523	+	0.0423635i
$996$		0			0
$997$			49.7748i			1.57638i	0.615430	+	0.788192i	$0.288982\pi$
$997$			49.7748i			1.57638i	−0.615430	+	0.788192i	$0.711018\pi$
$998$		0			0
$999$	1.03011			0.0325913

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1140.2.f.a.229.6	yes	6
3.2	odd	2		3420.2.f.b.1369.2		6
5.2	odd	4		5700.2.a.y.1.1		3
5.3	odd	4		5700.2.a.x.1.3		3
5.4	even	2	inner	1140.2.f.a.229.3	✓	6
15.14	odd	2		3420.2.f.b.1369.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.f.a.229.3	✓	6	5.4	even	2	inner
1140.2.f.a.229.6	yes	6	1.1	even	1	trivial
3420.2.f.b.1369.1		6	15.14	odd	2
3420.2.f.b.1369.2		6	3.2	odd	2
5700.2.a.x.1.3		3	5.3	odd	4
5700.2.a.y.1.1		3	5.2	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:	\( N \)	\(=\)	\( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(2.21432 - 0.311108i) q^{5} +2.21432i q^{7} -1.00000 q^{9} -1.31111 q^{11} +0.836535i q^{13} +(0.311108 + 2.21432i) q^{15} +4.90321i q^{17} -1.00000 q^{19} -2.21432 q^{21} +5.33185i q^{23} +(4.80642 - 1.37778i) q^{25} -1.00000i q^{27} -1.31111 q^{29} -1.71900 q^{31} -1.31111i q^{33} +(0.688892 + 4.90321i) q^{35} +1.03011i q^{37} -0.836535 q^{39} +5.87310 q^{41} +0.214320i q^{43} +(-2.21432 + 0.311108i) q^{45} +5.33185i q^{47} +2.09679 q^{49} -4.90321 q^{51} -8.70964i q^{53} +(-2.90321 + 0.407896i) q^{55} -1.00000i q^{57} -8.10171 q^{59} +3.33185 q^{61} -2.21432i q^{63} +(0.260253 + 1.85236i) q^{65} +9.61285i q^{67} -5.33185 q^{69} -7.80642 q^{71} -0.193576i q^{73} +(1.37778 + 4.80642i) q^{75} -2.90321i q^{77} +12.9906 q^{79} +1.00000 q^{81} -0.0459330i q^{83} +(1.52543 + 10.8573i) q^{85} -1.31111i q^{87} +2.92396 q^{89} -1.85236 q^{91} -1.71900i q^{93} +(-2.21432 + 0.311108i) q^{95} +2.60147i q^{97} +1.31111 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} - 8 q^{11} + 2 q^{15} - 6 q^{19} + 2 q^{25} - 8 q^{29} - 24 q^{31} + 4 q^{35} + 8 q^{39} + 8 q^{41} + 26 q^{49} - 16 q^{51} - 4 q^{55} + 4 q^{59} - 20 q^{61} + 28 q^{65} + 8 q^{69} - 20 q^{71}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 - 8 * q^11 + 2 * q^15 - 6 * q^19 + 2 * q^25 - 8 * q^29 - 24 * q^31 + 4 * q^35 + 8 * q^39 + 8 * q^41 + 26 * q^49 - 16 * q^51 - 4 * q^55 + 4 * q^59 - 20 * q^61 + 28 * q^65 + 8 * q^69 - 20 * q^71 + 8 * q^75 + 24 * q^79 + 6 * q^81 - 4 * q^85 - 36 * q^89 - 24 * q^91 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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