LMFDB - Embedded newform 684.3.y.a.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,3,Mod(145,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("684.145");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	684.y (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.6376500822$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	684.145
Dual form			684.3.y.a.217.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} +O(q^{10})$	$q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} -16.0000 q^{11} +(13.5000 + 7.79423i) q^{13} +(11.0000 + 19.0526i) q^{17} +19.0000 q^{19} +(20.0000 - 34.6410i) q^{23} +(10.5000 - 18.1865i) q^{25} +(-15.0000 - 8.66025i) q^{29} -50.2295i q^{31} +(1.00000 + 1.73205i) q^{35} +15.5885i q^{37} +(12.0000 - 6.92820i) q^{41} +(24.5000 + 42.4352i) q^{43} +(23.0000 - 39.8372i) q^{47} -48.0000 q^{49} +(42.0000 + 24.2487i) q^{53} +(16.0000 + 27.7128i) q^{55} +(57.0000 - 32.9090i) q^{59} +(48.5000 - 84.0045i) q^{61} -31.1769i q^{65} +(-22.5000 - 12.9904i) q^{67} +(42.0000 - 24.2487i) q^{71} +(-17.5000 - 30.3109i) q^{73} +16.0000 q^{77} +(-76.5000 + 44.1673i) q^{79} +146.000 q^{83} +(22.0000 - 38.1051i) q^{85} +(33.0000 + 19.0526i) q^{89} +(-13.5000 - 7.79423i) q^{91} +(-19.0000 - 32.9090i) q^{95} +(54.0000 - 31.1769i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 - 2 * q^7	$ 2 q - 2 q^{5} - 2 q^{7} - 32 q^{11} + 27 q^{13} + 22 q^{17} + 38 q^{19} + 40 q^{23} + 21 q^{25} - 30 q^{29} + 2 q^{35} + 24 q^{41} + 49 q^{43} + 46 q^{47} - 96 q^{49} + 84 q^{53} + 32 q^{55} + 114 q^{59} + 97 q^{61} - 45 q^{67} + 84 q^{71} - 35 q^{73} + 32 q^{77} - 153 q^{79} + 292 q^{83} + 44 q^{85} + 66 q^{89} - 27 q^{91} - 38 q^{95} + 108 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 - 32 * q^11 + 27 * q^13 + 22 * q^17 + 38 * q^19 + 40 * q^23 + 21 * q^25 - 30 * q^29 + 2 * q^35 + 24 * q^41 + 49 * q^43 + 46 * q^47 - 96 * q^49 + 84 * q^53 + 32 * q^55 + 114 * q^59 + 97 * q^61 - 45 * q^67 + 84 * q^71 - 35 * q^73 + 32 * q^77 - 153 * q^79 + 292 * q^83 + 44 * q^85 + 66 * q^89 - 27 * q^91 - 38 * q^95 + 108 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$

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			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.200000	−	0.346410i	0.748528	−	0.663103i	$-0.230761\pi$
$5$	−1.00000	−	1.73205i	−0.200000	−	0.346410i	−0.948528	+	0.316693i	$0.897428\pi$
$6$		0			0
$7$	−1.00000			−0.142857			−0.0714286	−	0.997446i	$-0.522756\pi$
$7$	−1.00000			−0.142857			−0.0714286	+	0.997446i	$0.522756\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−16.0000			−1.45455			−0.727273	−	0.686349i	$-0.759212\pi$
$11$	−16.0000			−1.45455			−0.727273	+	0.686349i	$0.759212\pi$
$12$		0			0
$13$	13.5000	+	7.79423i	1.03846	+	0.599556i	0.919397	−	0.393330i	$-0.128677\pi$
$13$	13.5000	+	7.79423i	1.03846	+	0.599556i	0.119064	+	0.992887i	$0.462011\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.0000	+	19.0526i	0.647059	+	1.12074i	0.983822	+	0.179149i	$0.0573345\pi$
$17$	11.0000	+	19.0526i	0.647059	+	1.12074i	−0.336763	+	0.941589i	$0.609332\pi$
$18$		0			0
$19$	19.0000			1.00000
$19$	19.0000			1.00000
$20$		0			0
$21$		0			0
$22$		0			0
$23$	20.0000	−	34.6410i	0.869565	−	1.50613i	0.00712357	−	0.999975i	$-0.497732\pi$
$23$	20.0000	−	34.6410i	0.869565	−	1.50613i	0.862442	−	0.506157i	$-0.168934\pi$
$24$		0			0
$25$	10.5000	−	18.1865i	0.420000	−	0.727461i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−15.0000	−	8.66025i	−0.517241	−	0.298629i	0.218564	−	0.975823i	$-0.429863\pi$
$29$	−15.0000	−	8.66025i	−0.517241	−	0.298629i	−0.735805	+	0.677193i	$0.763196\pi$
$30$		0			0
$31$		−	50.2295i		−	1.62031i	−0.586219	−	0.810153i	$-0.699384\pi$
$31$		−	50.2295i		−	1.62031i	0.586219	−	0.810153i	$-0.300616\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000	+	1.73205i	0.0285714	+	0.0494872i
$36$		0			0
$37$			15.5885i			0.421310i	0.977561	+	0.210655i	$0.0675596\pi$
$37$			15.5885i			0.421310i	−0.977561	+	0.210655i	$0.932440\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	12.0000	−	6.92820i	0.292683	−	0.168981i	−0.346468	−	0.938062i	$-0.612619\pi$
$41$	12.0000	−	6.92820i	0.292683	−	0.168981i	0.639151	+	0.769081i	$0.279286\pi$
$42$		0			0
$43$	24.5000	+	42.4352i	0.569767	+	0.986866i	0.996589	+	0.0825301i	$0.0263001\pi$
$43$	24.5000	+	42.4352i	0.569767	+	0.986866i	−0.426821	+	0.904336i	$0.640367\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	23.0000	−	39.8372i	0.489362	−	0.847599i	−0.510563	−	0.859840i	$-0.670563\pi$
$47$	23.0000	−	39.8372i	0.489362	−	0.847599i	0.999925	+	0.0122408i	$0.00389646\pi$
$48$		0			0
$49$	−48.0000			−0.979592
$50$		0			0
$51$		0			0
$52$		0			0
$53$	42.0000	+	24.2487i	0.792453	+	0.457523i	0.840825	−	0.541307i	$-0.182070\pi$
$53$	42.0000	+	24.2487i	0.792453	+	0.457523i	−0.0483725	+	0.998829i	$0.515403\pi$
$54$		0			0
$55$	16.0000	+	27.7128i	0.290909	+	0.503869i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	57.0000	−	32.9090i	0.966102	−	0.557779i	0.0680561	−	0.997681i	$-0.478320\pi$
$59$	57.0000	−	32.9090i	0.966102	−	0.557779i	0.898046	+	0.439902i	$0.144987\pi$
$60$		0			0
$61$	48.5000	−	84.0045i	0.795082	−	1.37712i	−0.127705	−	0.991812i	$-0.540761\pi$
$61$	48.5000	−	84.0045i	0.795082	−	1.37712i	0.922787	−	0.385310i	$-0.125906\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	31.1769i		−	0.479645i
$66$		0			0
$67$	−22.5000	−	12.9904i	−0.335821	−	0.193886i	0.322602	−	0.946535i	$-0.395443\pi$
$67$	−22.5000	−	12.9904i	−0.335821	−	0.193886i	−0.658422	+	0.752649i	$0.728776\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	42.0000	−	24.2487i	0.591549	−	0.341531i	−0.174161	−	0.984717i	$-0.555721\pi$
$71$	42.0000	−	24.2487i	0.591549	−	0.341531i	0.765710	+	0.643186i	$0.222388\pi$
$72$		0			0
$73$	−17.5000	−	30.3109i	−0.239726	−	0.415218i	0.720910	−	0.693029i	$-0.243724\pi$
$73$	−17.5000	−	30.3109i	−0.239726	−	0.415218i	−0.960636	+	0.277811i	$0.910391\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	16.0000			0.207792
$78$		0			0
$79$	−76.5000	+	44.1673i	−0.968354	+	0.559080i	−0.898734	−	0.438494i	$-0.855512\pi$
$79$	−76.5000	+	44.1673i	−0.968354	+	0.559080i	−0.0696203	+	0.997574i	$0.522179\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	146.000			1.75904			0.879518	−	0.475865i	$-0.157865\pi$
$83$	146.000			1.75904			0.879518	+	0.475865i	$0.157865\pi$
$84$		0			0
$85$	22.0000	−	38.1051i	0.258824	−	0.448296i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	33.0000	+	19.0526i	0.370787	+	0.214074i	0.673802	−	0.738912i	$-0.264660\pi$
$89$	33.0000	+	19.0526i	0.370787	+	0.214074i	−0.303015	+	0.952986i	$0.597993\pi$
$90$		0			0
$91$	−13.5000	−	7.79423i	−0.148352	−	0.0856509i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−19.0000	−	32.9090i	−0.200000	−	0.346410i
$96$		0			0
$97$	54.0000	−	31.1769i	0.556701	−	0.321411i	−0.195119	−	0.980780i	$-0.562509\pi$
$97$	54.0000	−	31.1769i	0.556701	−	0.321411i	0.751820	+	0.659368i	$0.229176\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.0000	−	24.2487i	0.138614	−	0.240086i	−0.788358	−	0.615216i	$-0.789069\pi$
$101$	14.0000	−	24.2487i	0.138614	−	0.240086i	0.926972	+	0.375130i	$0.122402\pi$
$102$		0			0
$103$			126.440i			1.22757i	0.789473	+	0.613785i	$0.210354\pi$
$103$			126.440i			1.22757i	−0.789473	+	0.613785i	$0.789646\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	83.1384i		−	0.776995i	−0.921450	−	0.388497i	$-0.872994\pi$
$107$		−	83.1384i		−	0.776995i	0.921450	−	0.388497i	$-0.127006\pi$
$108$		0			0
$109$	−66.0000	+	38.1051i	−0.605505	+	0.349588i	−0.771204	−	0.636588i	$-0.780345\pi$
$109$	−66.0000	+	38.1051i	−0.605505	+	0.349588i	0.165699	+	0.986176i	$0.447012\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			183.597i			1.62476i	0.583131	+	0.812378i	$0.301827\pi$
$113$			183.597i			1.62476i	−0.583131	+	0.812378i	$0.698173\pi$
$114$		0			0
$115$	−80.0000			−0.695652
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−11.0000	−	19.0526i	−0.0924370	−	0.160106i
$120$		0			0
$121$	135.000			1.11570
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−92.0000			−0.736000
$126$		0			0
$127$	24.0000	+	13.8564i	0.188976	+	0.109106i	0.591503	−	0.806303i	$-0.298535\pi$
$127$	24.0000	+	13.8564i	0.188976	+	0.109106i	−0.402527	+	0.915408i	$0.631868\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−64.0000	−	110.851i	−0.488550	−	0.846193i	0.511364	−	0.859364i	$-0.329141\pi$
$131$	−64.0000	−	110.851i	−0.488550	−	0.846193i	−0.999913	+	0.0131717i	$0.995807\pi$
$132$		0			0
$133$	−19.0000			−0.142857
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−58.0000	+	100.459i	−0.423358	+	0.733277i	−0.996265	−	0.0863428i	$-0.972482\pi$
$137$	−58.0000	+	100.459i	−0.423358	+	0.733277i	0.572908	+	0.819620i	$0.305815\pi$
$138$		0			0
$139$	87.5000	−	151.554i	0.629496	−	1.09032i	−0.358156	−	0.933662i	$-0.616595\pi$
$139$	87.5000	−	151.554i	0.629496	−	1.09032i	0.987653	−	0.156658i	$-0.0500721\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−216.000	−	124.708i	−1.51049	−	0.872082i
$144$		0			0
$145$			34.6410i			0.238904i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	74.0000	+	128.172i	0.496644	+	0.860213i	0.999993	−	0.00387051i	$-0.00123202\pi$
$149$	74.0000	+	128.172i	0.496644	+	0.860213i	−0.503348	+	0.864084i	$0.667899\pi$
$150$		0			0
$151$		−	193.990i		−	1.28470i	−0.766411	−	0.642350i	$-0.777960\pi$
$151$		−	193.990i		−	1.28470i	0.766411	−	0.642350i	$-0.222040\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−87.0000	+	50.2295i	−0.561290	+	0.324061i
$156$		0			0
$157$	96.5000	+	167.143i	0.614650	+	1.06460i	0.990446	+	0.137902i	$0.0440359\pi$
$157$	96.5000	+	167.143i	0.614650	+	1.06460i	−0.375796	+	0.926702i	$0.622631\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−20.0000	+	34.6410i	−0.124224	+	0.215162i
$162$		0			0
$163$	233.000			1.42945			0.714724	−	0.699407i	$-0.246552\pi$
$163$	233.000			1.42945			0.714724	+	0.699407i	$0.246552\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−63.0000	−	36.3731i	−0.377246	−	0.217803i	0.299374	−	0.954136i	$-0.403222\pi$
$167$	−63.0000	−	36.3731i	−0.377246	−	0.217803i	−0.676619	+	0.736333i	$0.736556\pi$
$168$		0			0
$169$	37.0000	+	64.0859i	0.218935	+	0.379206i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−99.0000	+	57.1577i	−0.572254	+	0.330391i	−0.758049	−	0.652197i	$-0.773847\pi$
$173$	−99.0000	+	57.1577i	−0.572254	+	0.330391i	0.185795	+	0.982589i	$0.440514\pi$
$174$		0			0
$175$	−10.5000	+	18.1865i	−0.0600000	+	0.103923i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			83.1384i			0.464461i	0.972661	+	0.232230i	$0.0746023\pi$
$179$			83.1384i			0.464461i	−0.972661	+	0.232230i	$0.925398\pi$
$180$		0			0
$181$	42.0000	+	24.2487i	0.232044	+	0.133971i	0.611515	−	0.791233i	$-0.290561\pi$
$181$	42.0000	+	24.2487i	0.232044	+	0.133971i	−0.379471	+	0.925204i	$0.623894\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	27.0000	−	15.5885i	0.145946	−	0.0842619i
$186$		0			0
$187$	−176.000	−	304.841i	−0.941176	−	1.63017i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−70.0000			−0.366492			−0.183246	−	0.983067i	$-0.558661\pi$
$191$	−70.0000			−0.366492			−0.183246	+	0.983067i	$0.558661\pi$
$192$		0			0
$193$	−262.500	+	151.554i	−1.36010	+	0.785256i	−0.989637	−	0.143590i	$-0.954135\pi$
$193$	−262.500	+	151.554i	−1.36010	+	0.785256i	−0.370466	+	0.928846i	$0.620802\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.0101523			0.00507614	−	0.999987i	$-0.498384\pi$
$197$	2.00000			0.0101523			0.00507614	+	0.999987i	$0.498384\pi$
$198$		0			0
$199$	−98.5000	+	170.607i	−0.494975	+	0.857322i	−0.999983	−	0.00579283i	$-0.998156\pi$
$199$	−98.5000	+	170.607i	−0.494975	+	0.857322i	0.505008	+	0.863114i	$0.331489\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	15.0000	+	8.66025i	0.0738916	+	0.0426613i
$204$		0			0
$205$	−24.0000	−	13.8564i	−0.117073	−	0.0675922i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−304.000			−1.45455
$210$		0			0
$211$	−316.500	+	182.731i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$211$	−316.500	+	182.731i	−1.50000	+	0.866025i	−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	49.0000	−	84.8705i	0.227907	−	0.394746i
$216$		0			0
$217$			50.2295i			0.231472i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			342.946i			1.55179i
$222$		0			0
$223$	211.500	−	122.110i	0.948430	−	0.547577i	0.0558375	−	0.998440i	$-0.482217\pi$
$223$	211.500	−	122.110i	0.948430	−	0.547577i	0.892593	+	0.450863i	$0.148884\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			148.956i			0.656195i	0.944644	+	0.328098i	$0.106408\pi$
$227$			148.956i			0.656195i	−0.944644	+	0.328098i	$0.893592\pi$
$228$		0			0
$229$	83.0000			0.362445			0.181223	−	0.983442i	$-0.441995\pi$
$229$	83.0000			0.362445			0.181223	+	0.983442i	$0.441995\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−40.0000	−	69.2820i	−0.171674	−	0.297348i	0.767331	−	0.641251i	$-0.221584\pi$
$233$	−40.0000	−	69.2820i	−0.171674	−	0.297348i	−0.939005	+	0.343903i	$0.888251\pi$
$234$		0			0
$235$	−92.0000			−0.391489
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−58.0000			−0.242678			−0.121339	−	0.992611i	$-0.538719\pi$
$239$	−58.0000			−0.242678			−0.121339	+	0.992611i	$0.538719\pi$
$240$		0			0
$241$	−244.500	−	141.162i	−1.01452	−	0.585735i	−0.102010	−	0.994783i	$-0.532527\pi$
$241$	−244.500	−	141.162i	−1.01452	−	0.585735i	−0.912513	+	0.409048i	$0.865861\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	48.0000	+	83.1384i	0.195918	+	0.339341i
$246$		0			0
$247$	256.500	+	148.090i	1.03846	+	0.599556i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−133.000	+	230.363i	−0.529880	+	0.917780i	0.469512	+	0.882926i	$0.344430\pi$
$251$	−133.000	+	230.363i	−0.529880	+	0.917780i	−0.999392	+	0.0348538i	$0.988903\pi$
$252$		0			0
$253$	−320.000	+	554.256i	−1.26482	+	2.19074i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	162.000	+	93.5307i	0.630350	+	0.363933i	0.780888	−	0.624671i	$-0.214767\pi$
$257$	162.000	+	93.5307i	0.630350	+	0.363933i	−0.150537	+	0.988604i	$0.548100\pi$
$258$		0			0
$259$		−	15.5885i		−	0.0601871i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−82.0000	−	142.028i	−0.311787	−	0.540031i	0.666962	−	0.745092i	$-0.267594\pi$
$263$	−82.0000	−	142.028i	−0.311787	−	0.540031i	−0.978749	+	0.205060i	$0.934261\pi$
$264$		0			0
$265$		−	96.9948i		−	0.366018i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	447.000	−	258.076i	1.66171	−	0.959389i	0.689811	−	0.723989i	$-0.257693\pi$
$269$	447.000	−	258.076i	1.66171	−	0.959389i	0.971899	−	0.235400i	$-0.0756399\pi$
$270$		0			0
$271$	−7.00000	−	12.1244i	−0.0258303	−	0.0447393i	0.852821	−	0.522203i	$-0.174890\pi$
$271$	−7.00000	−	12.1244i	−0.0258303	−	0.0447393i	−0.878652	+	0.477464i	$0.841556\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−168.000	+	290.985i	−0.610909	+	1.05813i
$276$		0			0
$277$	206.000			0.743682			0.371841	−	0.928296i	$-0.378727\pi$
$277$	206.000			0.743682			0.371841	+	0.928296i	$0.378727\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−87.0000	−	50.2295i	−0.309609	−	0.178753i	0.337143	−	0.941454i	$-0.390539\pi$
$281$	−87.0000	−	50.2295i	−0.309609	−	0.178753i	−0.646751	+	0.762701i	$0.723873\pi$
$282$		0			0
$283$	−271.000	−	469.386i	−0.957597	−	1.65861i	−0.728310	−	0.685248i	$-0.759694\pi$
$283$	−271.000	−	469.386i	−0.957597	−	1.65861i	−0.229288	−	0.973359i	$-0.573640\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000	+	6.92820i	−0.0418118	+	0.0241401i
$288$		0			0
$289$	−97.5000	+	168.875i	−0.337370	+	0.584342i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	481.510i		−	1.64338i	−0.569935	−	0.821690i	$-0.693032\pi$
$293$		−	481.510i		−	1.64338i	0.569935	−	0.821690i	$-0.306968\pi$
$294$		0			0
$295$	−114.000	−	65.8179i	−0.386441	−	0.223112i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	540.000	−	311.769i	1.80602	−	1.04271i
$300$		0			0
$301$	−24.5000	−	42.4352i	−0.0813953	−	0.140981i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−194.000			−0.636066
$306$		0			0
$307$	−138.000	+	79.6743i	−0.449511	+	0.259526i	−0.707624	−	0.706589i	$-0.750233\pi$
$307$	−138.000	+	79.6743i	−0.449511	+	0.259526i	0.258112	+	0.966115i	$0.416900\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−322.000			−1.03537			−0.517685	−	0.855571i	$-0.673206\pi$
$311$	−322.000			−1.03537			−0.517685	+	0.855571i	$0.673206\pi$
$312$		0			0
$313$	−13.0000	+	22.5167i	−0.0415335	+	0.0719382i	−0.886045	−	0.463599i	$-0.846558\pi$
$313$	−13.0000	+	22.5167i	−0.0415335	+	0.0719382i	0.844511	+	0.535538i	$0.179891\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	291.000	+	168.009i	0.917981	+	0.529997i	0.882990	−	0.469391i	$-0.155527\pi$
$317$	291.000	+	168.009i	0.917981	+	0.529997i	0.0349907	+	0.999388i	$0.488860\pi$
$318$		0			0
$319$	240.000	+	138.564i	0.752351	+	0.434370i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	209.000	+	361.999i	0.647059	+	1.12074i
$324$		0			0
$325$	283.500	−	163.679i	0.872308	−	0.503627i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−23.0000	+	39.8372i	−0.0699088	+	0.121086i
$330$		0			0
$331$		−	219.970i		−	0.664563i	−0.943180	−	0.332282i	$-0.892182\pi$
$331$		−	219.970i		−	0.664563i	0.943180	−	0.332282i	$-0.107818\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			51.9615i			0.155109i
$336$		0			0
$337$	−505.500	+	291.851i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$337$	−505.500	+	291.851i	−1.50000	+	0.866025i	−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			803.672i			2.35681i
$342$		0			0
$343$	97.0000			0.282799
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−235.000	−	407.032i	−0.677233	−	1.17300i	−0.975811	−	0.218617i	$-0.929845\pi$
$347$	−235.000	−	407.032i	−0.677233	−	1.17300i	0.298577	−	0.954385i	$-0.403488\pi$
$348$		0			0
$349$	−295.000			−0.845272			−0.422636	−	0.906299i	$-0.638895\pi$
$349$	−295.000			−0.845272			−0.422636	+	0.906299i	$0.638895\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−262.000			−0.742210			−0.371105	−	0.928591i	$-0.621021\pi$
$353$	−262.000			−0.742210			−0.371105	+	0.928591i	$0.621021\pi$
$354$		0			0
$355$	−84.0000	−	48.4974i	−0.236620	−	0.136612i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−232.000	−	401.836i	−0.646240	−	1.11932i	−0.984014	−	0.178093i	$-0.943007\pi$
$359$	−232.000	−	401.836i	−0.646240	−	1.11932i	0.337774	−	0.941227i	$-0.390326\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−35.0000	+	60.6218i	−0.0958904	+	0.166087i
$366$		0			0
$367$	−116.500	+	201.784i	−0.317439	+	0.549820i	−0.979953	−	0.199229i	$-0.936156\pi$
$367$	−116.500	+	201.784i	−0.317439	+	0.549820i	0.662514	+	0.749049i	$0.269489\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−42.0000	−	24.2487i	−0.113208	−	0.0653604i
$372$		0			0
$373$		−	533.472i		−	1.43022i	−0.699013	−	0.715109i	$-0.746377\pi$
$373$		−	533.472i		−	1.43022i	0.699013	−	0.715109i	$-0.253623\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−135.000	−	233.827i	−0.358090	−	0.620230i
$378$		0			0
$379$			161.081i			0.425015i	0.977159	+	0.212508i	$0.0681630\pi$
$379$			161.081i			0.425015i	−0.977159	+	0.212508i	$0.931837\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	498.000	−	287.520i	1.30026	−	0.750706i	0.319812	−	0.947481i	$-0.396380\pi$
$383$	498.000	−	287.520i	1.30026	−	0.750706i	0.980449	+	0.196775i	$0.0630468\pi$
$384$		0			0
$385$	−16.0000	−	27.7128i	−0.0415584	−	0.0719813i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−76.0000	+	131.636i	−0.195373	+	0.338396i	−0.947023	−	0.321167i	$-0.895925\pi$
$389$	−76.0000	+	131.636i	−0.195373	+	0.338396i	0.751650	+	0.659562i	$0.229258\pi$
$390$		0			0
$391$	880.000			2.25064
$392$		0			0
$393$		0			0
$394$		0			0
$395$	153.000	+	88.3346i	0.387342	+	0.223632i
$396$		0			0
$397$	249.500	+	432.147i	0.628463	+	1.08853i	0.987860	+	0.155346i	$0.0496492\pi$
$397$	249.500	+	432.147i	0.628463	+	1.08853i	−0.359397	+	0.933185i	$0.617017\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−87.0000	+	50.2295i	−0.216958	+	0.125261i	−0.604541	−	0.796574i	$-0.706643\pi$
$401$	−87.0000	+	50.2295i	−0.216958	+	0.125261i	0.387583	+	0.921835i	$0.373310\pi$
$402$		0			0
$403$	391.500	−	678.098i	0.971464	−	1.68263i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	249.415i		−	0.612814i
$408$		0			0
$409$	342.000	+	197.454i	0.836186	+	0.482772i	0.855966	−	0.517032i	$-0.172963\pi$
$409$	342.000	+	197.454i	0.836186	+	0.482772i	−0.0197801	+	0.999804i	$0.506297\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−57.0000	+	32.9090i	−0.138015	+	0.0796827i
$414$		0			0
$415$	−146.000	−	252.879i	−0.351807	−	0.609348i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−628.000			−1.49881			−0.749403	−	0.662114i	$-0.769660\pi$
$419$	−628.000			−1.49881			−0.749403	+	0.662114i	$0.769660\pi$
$420$		0			0
$421$	558.000	−	322.161i	1.32542	−	0.765229i	0.340829	−	0.940125i	$-0.389292\pi$
$421$	558.000	−	322.161i	1.32542	−	0.765229i	0.984587	+	0.174896i	$0.0559590\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	462.000			1.08706
$426$		0			0
$427$	−48.5000	+	84.0045i	−0.113583	+	0.196732i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	393.000	+	226.899i	0.911833	+	0.526447i	0.881020	−	0.473078i	$-0.156857\pi$
$431$	393.000	+	226.899i	0.911833	+	0.526447i	0.0308125	+	0.999525i	$0.490191\pi$
$432$		0			0
$433$	433.500	+	250.281i	1.00115	+	0.578017i	0.908590	−	0.417690i	$-0.137160\pi$
$433$	433.500	+	250.281i	1.00115	+	0.578017i	0.0925650	+	0.995707i	$0.470493\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	380.000	−	658.179i	0.869565	−	1.50613i
$438$		0			0
$439$	292.500	−	168.875i	0.666287	−	0.384681i	−0.128381	−	0.991725i	$-0.540978\pi$
$439$	292.500	−	168.875i	0.666287	−	0.384681i	0.794668	+	0.607044i	$0.207645\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	365.000	−	632.199i	0.823928	−	1.42708i	−0.0788082	−	0.996890i	$-0.525111\pi$
$443$	365.000	−	632.199i	0.823928	−	1.42708i	0.902736	−	0.430195i	$-0.141555\pi$
$444$		0			0
$445$		−	76.2102i		−	0.171259i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	446.869i		−	0.995254i	−0.867391	−	0.497627i	$-0.834205\pi$
$449$		−	446.869i		−	0.995254i	0.867391	−	0.497627i	$-0.165795\pi$
$450$		0			0
$451$	−192.000	+	110.851i	−0.425721	+	0.245790i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			31.1769i			0.0685207i
$456$		0			0
$457$	−397.000			−0.868709			−0.434354	−	0.900742i	$-0.643023\pi$
$457$	−397.000			−0.868709			−0.434354	+	0.900742i	$0.643023\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	287.000	+	497.099i	0.622560	+	1.07830i	0.989007	+	0.147866i	$0.0472405\pi$
$461$	287.000	+	497.099i	0.622560	+	1.07830i	−0.366448	+	0.930439i	$0.619426\pi$
$462$		0			0
$463$	−367.000			−0.792657			−0.396328	−	0.918109i	$-0.629716\pi$
$463$	−367.000			−0.792657			−0.396328	+	0.918109i	$0.629716\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−424.000			−0.907923			−0.453961	−	0.891021i	$-0.649990\pi$
$467$	−424.000			−0.907923			−0.453961	+	0.891021i	$0.649990\pi$
$468$		0			0
$469$	22.5000	+	12.9904i	0.0479744	+	0.0276980i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−392.000	−	678.964i	−0.828753	−	1.43544i
$474$		0			0
$475$	199.500	−	345.544i	0.420000	−	0.727461i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−364.000	+	630.466i	−0.759916	+	1.31621i	0.182976	+	0.983117i	$0.441427\pi$
$479$	−364.000	+	630.466i	−0.759916	+	1.31621i	−0.942893	+	0.333097i	$0.891906\pi$
$480$		0			0
$481$	−121.500	+	210.444i	−0.252599	+	0.437514i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−108.000	−	62.3538i	−0.222680	−	0.128565i
$486$		0			0
$487$			270.200i			0.554825i	0.960751	+	0.277413i	$0.0894769\pi$
$487$			270.200i			0.554825i	−0.960751	+	0.277413i	$0.910523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	380.000	+	658.179i	0.773931	+	1.34049i	0.935393	+	0.353609i	$0.115046\pi$
$491$	380.000	+	658.179i	0.773931	+	1.34049i	−0.161463	+	0.986879i	$0.551621\pi$
$492$		0			0
$493$		−	381.051i		−	0.772923i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−42.0000	+	24.2487i	−0.0845070	+	0.0487902i
$498$		0			0
$499$	231.500	+	400.970i	0.463928	+	0.803547i	0.999152	−	0.0411632i	$-0.0131064\pi$
$499$	231.500	+	400.970i	0.463928	+	0.803547i	−0.535225	+	0.844710i	$0.679773\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	65.0000	−	112.583i	0.129225	−	0.223824i	−0.794152	−	0.607720i	$-0.792084\pi$
$503$	65.0000	−	112.583i	0.129225	−	0.223824i	0.923376	+	0.383896i	$0.125418\pi$
$504$		0			0
$505$	−56.0000			−0.110891
$506$		0			0
$507$		0			0
$508$		0			0
$509$	459.000	+	265.004i	0.901768	+	0.520636i	0.877773	−	0.479076i	$-0.159028\pi$
$509$	459.000	+	265.004i	0.901768	+	0.520636i	0.0239947	+	0.999712i	$0.492362\pi$
$510$		0			0
$511$	17.5000	+	30.3109i	0.0342466	+	0.0593168i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	219.000	−	126.440i	0.425243	−	0.245514i
$516$		0			0
$517$	−368.000	+	637.395i	−0.711799	+	1.23287i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			602.754i			1.15692i	0.815712	+	0.578458i	$0.196345\pi$
$521$			602.754i			1.15692i	−0.815712	+	0.578458i	$0.803655\pi$
$522$		0			0
$523$	−277.500	−	160.215i	−0.530593	−	0.306338i	0.210665	−	0.977558i	$-0.432437\pi$
$523$	−277.500	−	160.215i	−0.530593	−	0.306338i	−0.741258	+	0.671220i	$0.765770\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	957.000	−	552.524i	1.81594	−	1.04843i
$528$		0			0
$529$	−535.500	−	927.513i	−1.01229	−	1.75333i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	216.000			0.405253
$534$		0			0
$535$	−144.000	+	83.1384i	−0.269159	+	0.155399i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	768.000			1.42486
$540$		0			0
$541$	−182.500	+	316.099i	−0.337338	+	0.584287i	−0.983931	−	0.178548i	$-0.942860\pi$
$541$	−182.500	+	316.099i	−0.337338	+	0.584287i	0.646593	+	0.762835i	$0.276193\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	132.000	+	76.2102i	0.242202	+	0.139835i
$546$		0			0
$547$	−199.500	−	115.181i	−0.364717	−	0.210569i	0.306431	−	0.951893i	$-0.400865\pi$
$547$	−199.500	−	115.181i	−0.364717	−	0.210569i	−0.671148	+	0.741324i	$0.734198\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−285.000	−	164.545i	−0.517241	−	0.298629i
$552$		0			0
$553$	76.5000	−	44.1673i	0.138336	−	0.0798685i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	320.000	−	554.256i	0.574506	−	0.995074i	−0.421589	−	0.906787i	$-0.638527\pi$
$557$	320.000	−	554.256i	0.574506	−	0.995074i	0.996095	−	0.0882870i	$-0.0281392\pi$
$558$		0			0
$559$			763.834i			1.36643i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	96.9948i		−	0.172282i	−0.996283	−	0.0861411i	$-0.972546\pi$
$563$		−	96.9948i		−	0.172282i	0.996283	−	0.0861411i	$-0.0274536\pi$
$564$		0			0
$565$	318.000	−	183.597i	0.562832	−	0.324951i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	152.420i		−	0.267874i	−0.990990	−	0.133937i	$-0.957238\pi$
$569$		−	152.420i		−	0.267874i	0.990990	−	0.133937i	$-0.0427620\pi$
$570$		0			0
$571$	335.000			0.586690			0.293345	−	0.956007i	$-0.405232\pi$
$571$	335.000			0.586690			0.293345	+	0.956007i	$0.405232\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−420.000	−	727.461i	−0.730435	−	1.26515i
$576$		0			0
$577$	746.000			1.29289			0.646447	−	0.762959i	$-0.276254\pi$
$577$	746.000			1.29289			0.646447	+	0.762959i	$0.276254\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−146.000			−0.251291
$582$		0			0
$583$	−672.000	−	387.979i	−1.15266	−	0.665488i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	89.0000	+	154.153i	0.151618	+	0.262611i	0.931823	−	0.362914i	$-0.118218\pi$
$587$	89.0000	+	154.153i	0.151618	+	0.262611i	−0.780204	+	0.625525i	$0.784885\pi$
$588$		0			0
$589$		−	954.360i		−	1.62031i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	137.000	−	237.291i	0.231029	−	0.400153i	−0.727082	−	0.686550i	$-0.759124\pi$
$593$	137.000	−	237.291i	0.231029	−	0.400153i	0.958111	+	0.286397i	$0.0924575\pi$
$594$		0			0
$595$	−22.0000	+	38.1051i	−0.0369748	+	0.0640422i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	252.000	+	145.492i	0.420701	+	0.242892i	0.695377	−	0.718645i	$-0.255237\pi$
$599$	252.000	+	145.492i	0.420701	+	0.242892i	−0.274676	+	0.961537i	$0.588571\pi$
$600$		0			0
$601$		−	1020.18i		−	1.69747i	−0.528820	−	0.848734i	$-0.677366\pi$
$601$		−	1020.18i		−	1.69747i	0.528820	−	0.848734i	$-0.322634\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−135.000	−	233.827i	−0.223140	−	0.386491i
$606$		0			0
$607$			157.617i			0.259665i	0.991536	+	0.129832i	$0.0414440\pi$
$607$			157.617i			0.259665i	−0.991536	+	0.129832i	$0.958556\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	621.000	−	358.535i	1.01637	−	0.586800i
$612$		0			0
$613$	401.000	+	694.552i	0.654160	+	1.13304i	0.982104	+	0.188341i	$0.0603110\pi$
$613$	401.000	+	694.552i	0.654160	+	1.13304i	−0.327944	+	0.944697i	$0.606356\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−382.000	+	661.643i	−0.619125	+	1.07236i	0.370521	+	0.928824i	$0.379179\pi$
$617$	−382.000	+	661.643i	−0.619125	+	1.07236i	−0.989646	+	0.143531i	$0.954154\pi$
$618$		0			0
$619$	−127.000			−0.205170			−0.102585	−	0.994724i	$-0.532711\pi$
$619$	−127.000			−0.205170			−0.102585	+	0.994724i	$0.532711\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−33.0000	−	19.0526i	−0.0529695	−	0.0305820i
$624$		0			0
$625$	−170.500	−	295.315i	−0.272800	−	0.472503i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−297.000	+	171.473i	−0.472178	+	0.272612i
$630$		0			0
$631$	−161.500	+	279.726i	−0.255943	+	0.443306i	−0.965151	−	0.261693i	$-0.915719\pi$
$631$	−161.500	+	279.726i	−0.255943	+	0.443306i	0.709208	+	0.704999i	$0.249053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	55.4256i		−	0.0872845i
$636$		0			0
$637$	−648.000	−	374.123i	−1.01727	−	0.587320i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	873.000	−	504.027i	1.36193	−	0.786313i	0.372053	−	0.928211i	$-0.378654\pi$
$641$	873.000	−	504.027i	1.36193	−	0.786313i	0.989881	+	0.141898i	$0.0453206\pi$
$642$		0			0
$643$	222.500	+	385.381i	0.346034	+	0.599349i	0.985541	−	0.169437i	$-0.0541948\pi$
$643$	222.500	+	385.381i	0.346034	+	0.599349i	−0.639507	+	0.768785i	$0.720861\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−106.000			−0.163833			−0.0819165	−	0.996639i	$-0.526104\pi$
$647$	−106.000			−0.163833			−0.0819165	+	0.996639i	$0.526104\pi$
$648$		0			0
$649$	−912.000	+	526.543i	−1.40524	+	0.811315i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−328.000			−0.502297			−0.251149	−	0.967949i	$-0.580808\pi$
$653$	−328.000			−0.502297			−0.251149	+	0.967949i	$0.580808\pi$
$654$		0			0
$655$	−128.000	+	221.703i	−0.195420	+	0.338477i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−348.000	−	200.918i	−0.528073	−	0.304883i	0.212158	−	0.977235i	$-0.431951\pi$
$659$	−348.000	−	200.918i	−0.528073	−	0.304883i	−0.740231	+	0.672352i	$0.765284\pi$
$660$		0			0
$661$	828.000	+	478.046i	1.25265	+	0.723216i	0.971635	−	0.236488i	$-0.0759963\pi$
$661$	828.000	+	478.046i	1.25265	+	0.723216i	0.281013	+	0.959704i	$0.409330\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19.0000	+	32.9090i	0.0285714	+	0.0494872i
$666$		0			0
$667$	−600.000	+	346.410i	−0.899550	+	0.519356i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−776.000	+	1344.07i	−1.15648	+	2.00309i
$672$		0			0
$673$		−	521.347i		−	0.774662i	−0.921941	−	0.387331i	$-0.873397\pi$
$673$		−	521.347i		−	0.774662i	0.921941	−	0.387331i	$-0.126603\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			439.941i			0.649839i	0.945742	+	0.324919i	$0.105337\pi$
$677$			439.941i			0.649839i	−0.945742	+	0.324919i	$0.894663\pi$
$678$		0			0
$679$	−54.0000	+	31.1769i	−0.0795287	+	0.0459159i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			717.069i			1.04988i	0.851139	+	0.524941i	$0.175913\pi$
$683$			717.069i			1.04988i	−0.851139	+	0.524941i	$0.824087\pi$
$684$		0			0
$685$	232.000			0.338686
$686$		0			0
$687$		0			0
$688$		0			0
$689$	378.000	+	654.715i	0.548621	+	0.950240i
$690$		0			0
$691$	422.000			0.610709			0.305355	−	0.952239i	$-0.401225\pi$
$691$	422.000			0.610709			0.305355	+	0.952239i	$0.401225\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−350.000			−0.503597
$696$		0			0
$697$	264.000	+	152.420i	0.378766	+	0.218681i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	185.000	+	320.429i	0.263909	+	0.457103i	0.967277	−	0.253722i	$-0.0816549\pi$
$701$	185.000	+	320.429i	0.263909	+	0.457103i	−0.703368	+	0.710825i	$0.748322\pi$
$702$		0			0
$703$			296.181i			0.421310i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−14.0000	+	24.2487i	−0.0198020	+	0.0342980i
$708$		0			0
$709$	45.5000	−	78.8083i	0.0641749	−	0.111154i	−0.832153	−	0.554546i	$-0.812892\pi$
$709$	45.5000	−	78.8083i	0.0641749	−	0.111154i	0.896328	+	0.443392i	$0.146225\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1740.00	−	1004.59i	−2.44039	−	1.40896i
$714$		0			0
$715$			498.831i			0.697665i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−502.000	−	869.490i	−0.698192	−	1.20930i	−0.969093	−	0.246697i	$-0.920655\pi$
$719$	−502.000	−	869.490i	−0.698192	−	1.20930i	0.270901	−	0.962607i	$-0.412678\pi$
$720$		0			0
$721$		−	126.440i		−	0.175367i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−315.000	+	181.865i	−0.434483	+	0.250849i
$726$		0			0
$727$	147.500	+	255.477i	0.202889	+	0.351413i	0.949458	−	0.313894i	$-0.101634\pi$
$727$	147.500	+	255.477i	0.202889	+	0.351413i	−0.746569	+	0.665308i	$0.768300\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−539.000	+	933.575i	−0.737346	+	1.27712i
$732$		0			0
$733$	−250.000			−0.341064			−0.170532	−	0.985352i	$-0.554549\pi$
$733$	−250.000			−0.341064			−0.170532	+	0.985352i	$0.554549\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	360.000	+	207.846i	0.488467	+	0.282016i
$738$		0			0
$739$	285.500	+	494.501i	0.386333	+	0.669148i	0.991953	−	0.126606i	$-0.0404083\pi$
$739$	285.500	+	494.501i	0.386333	+	0.669148i	−0.605620	+	0.795754i	$0.707075\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−6.00000	+	3.46410i	−0.00807537	+	0.00466232i	−0.504032	−	0.863685i	$-0.668151\pi$
$743$	−6.00000	+	3.46410i	−0.00807537	+	0.00466232i	0.495957	+	0.868347i	$0.334817\pi$
$744$		0			0
$745$	148.000	−	256.344i	0.198658	−	0.344085i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			83.1384i			0.110999i
$750$		0			0
$751$	−841.500	−	485.840i	−1.12051	−	0.646924i	−0.178976	−	0.983853i	$-0.557278\pi$
$751$	−841.500	−	485.840i	−1.12051	−	0.646924i	−0.941530	+	0.336929i	$0.890612\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−336.000	+	193.990i	−0.445033	+	0.256940i
$756$		0			0
$757$	−119.500	−	206.980i	−0.157860	−	0.273421i	0.776237	−	0.630442i	$-0.217126\pi$
$757$	−119.500	−	206.980i	−0.157860	−	0.273421i	−0.934097	+	0.357020i	$0.883793\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1402.00			−1.84231			−0.921156	−	0.389193i	$-0.872754\pi$
$761$	−1402.00			−1.84231			−0.921156	+	0.389193i	$0.872754\pi$
$762$		0			0
$763$	66.0000	−	38.1051i	0.0865007	−	0.0499412i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1026.00			1.33768
$768$		0			0
$769$	51.5000	−	89.2006i	0.0669701	−	0.115996i	−0.830596	−	0.556875i	$-0.812000\pi$
$769$	51.5000	−	89.2006i	0.0669701	−	0.115996i	0.897566	+	0.440880i	$0.145333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	69.0000	+	39.8372i	0.0892626	+	0.0515358i	0.543967	−	0.839107i	$-0.316922\pi$
$773$	69.0000	+	39.8372i	0.0892626	+	0.0515358i	−0.454704	+	0.890642i	$0.650255\pi$
$774$		0			0
$775$	−913.500	−	527.409i	−1.17871	−	0.680528i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	228.000	−	131.636i	0.292683	−	0.168981i
$780$		0			0
$781$	−672.000	+	387.979i	−0.860435	+	0.496773i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	193.000	−	334.286i	0.245860	−	0.425842i
$786$		0			0
$787$			188.794i			0.239890i	0.992781	+	0.119945i	$0.0382718\pi$
$787$			188.794i			0.239890i	−0.992781	+	0.119945i	$0.961728\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	183.597i		−	0.232108i
$792$		0			0
$793$	1309.50	−	756.040i	1.65132	−	0.953392i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			630.466i			0.791050i	0.918455	+	0.395525i	$0.129437\pi$
$797$			630.466i			0.791050i	−0.918455	+	0.395525i	$0.870563\pi$
$798$		0			0
$799$	1012.00			1.26658
$800$		0			0
$801$		0			0
$802$		0			0
$803$	280.000	+	484.974i	0.348692	+	0.603953i
$804$		0			0
$805$	80.0000			0.0993789
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1054.00			−1.30284			−0.651422	−	0.758716i	$-0.725827\pi$
$809$	−1054.00			−1.30284			−0.651422	+	0.758716i	$0.725827\pi$
$810$		0			0
$811$	960.000	+	554.256i	1.18372	+	0.683423i	0.956873	−	0.290506i	$-0.0938237\pi$
$811$	960.000	+	554.256i	1.18372	+	0.683423i	0.226851	+	0.973930i	$0.427157\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−233.000	−	403.568i	−0.285890	−	0.495175i
$816$		0			0
$817$	465.500	+	806.270i	0.569767	+	0.986866i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−487.000	+	843.509i	−0.593179	+	1.02742i	0.400622	+	0.916243i	$0.368794\pi$
$821$	−487.000	+	843.509i	−0.593179	+	1.02742i	−0.993801	+	0.111173i	$0.964539\pi$
$822$		0			0
$823$	317.000	−	549.060i	0.385176	−	0.667145i	−0.606617	−	0.794994i	$-0.707474\pi$
$823$	317.000	−	549.060i	0.385176	−	0.667145i	0.991794	+	0.127849i	$0.0408073\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1155.00	−	666.840i	−1.39661	−	0.806336i	−0.402578	−	0.915386i	$-0.631886\pi$
$827$	−1155.00	−	666.840i	−1.39661	−	0.806336i	−0.994036	+	0.109050i	$0.965219\pi$
$828$		0			0
$829$			296.181i			0.357275i	0.983915	+	0.178637i	$0.0571689\pi$
$829$			296.181i			0.357275i	−0.983915	+	0.178637i	$0.942831\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−528.000	−	914.523i	−0.633854	−	1.09787i
$834$		0			0
$835$			145.492i			0.174242i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−1395.00	+	805.404i	−1.66269	+	0.959957i	−0.691273	+	0.722594i	$0.742950\pi$
$839$	−1395.00	+	805.404i	−1.66269	+	0.959957i	−0.971421	+	0.237363i	$0.923717\pi$
$840$		0			0
$841$	−270.500	−	468.520i	−0.321641	−	0.557098i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	74.0000	−	128.172i	0.0875740	−	0.151683i
$846$		0			0
$847$	−135.000			−0.159386
$848$		0			0
$849$		0			0
$850$		0			0
$851$	540.000	+	311.769i	0.634548	+	0.366356i
$852$		0			0
$853$	219.500	+	380.185i	0.257327	+	0.445704i	0.965525	−	0.260310i	$-0.0838249\pi$
$853$	219.500	+	380.185i	0.257327	+	0.445704i	−0.708198	+	0.706014i	$0.750492\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−531.000	+	306.573i	−0.619603	+	0.357728i	−0.776715	−	0.629853i	$-0.783115\pi$
$857$	−531.000	+	306.573i	−0.619603	+	0.357728i	0.157111	+	0.987581i	$0.449782\pi$
$858$		0			0
$859$	−278.500	+	482.376i	−0.324214	+	0.561555i	−0.981353	−	0.192214i	$-0.938433\pi$
$859$	−278.500	+	482.376i	−0.324214	+	0.561555i	0.657139	+	0.753770i	$0.271767\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1340.61i			1.55343i	0.629854	+	0.776713i	$0.283115\pi$
$863$			1340.61i			1.55343i	−0.629854	+	0.776713i	$0.716885\pi$
$864$		0			0
$865$	198.000	+	114.315i	0.228902	+	0.132156i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1224.00	−	706.677i	1.40852	−	0.813207i
$870$		0			0
$871$	−202.500	−	350.740i	−0.232491	−	0.402687i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	92.0000			0.105143
$876$		0			0
$877$	−397.500	+	229.497i	−0.453250	+	0.261684i	−0.709202	−	0.705006i	$-0.750944\pi$
$877$	−397.500	+	229.497i	−0.453250	+	0.261684i	0.255952	+	0.966690i	$0.417611\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−910.000			−1.03292			−0.516459	−	0.856312i	$-0.672750\pi$
$881$	−910.000			−1.03292			−0.516459	+	0.856312i	$0.672750\pi$
$882$		0			0
$883$	−2.50000	+	4.33013i	−0.00283126	+	0.00490388i	−0.867438	−	0.497546i	$-0.834235\pi$
$883$	−2.50000	+	4.33013i	−0.00283126	+	0.00490388i	0.864606	+	0.502450i	$0.167568\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	927.000	+	535.204i	1.04510	+	0.603386i	0.921272	−	0.388918i	$-0.127151\pi$
$887$	927.000	+	535.204i	1.04510	+	0.603386i	0.123823	+	0.992304i	$0.460484\pi$
$888$		0			0
$889$	−24.0000	−	13.8564i	−0.0269966	−	0.0155865i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	437.000	−	756.906i	0.489362	−	0.847599i
$894$		0			0
$895$	144.000	−	83.1384i	0.160894	−	0.0928921i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−435.000	+	753.442i	−0.483871	+	0.838089i
$900$		0			0
$901$			1066.94i			1.18418i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	96.9948i		−	0.107177i
$906$		0			0
$907$	180.000	−	103.923i	0.198456	−	0.114579i	−0.397479	−	0.917611i	$-0.630115\pi$
$907$	180.000	−	103.923i	0.198456	−	0.114579i	0.595935	+	0.803032i	$0.296781\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			782.887i			0.859371i	0.902979	+	0.429685i	$0.141376\pi$
$911$			782.887i			0.859371i	−0.902979	+	0.429685i	$0.858624\pi$
$912$		0			0
$913$	−2336.00			−2.55860
$914$		0			0
$915$		0			0
$916$		0			0
$917$	64.0000	+	110.851i	0.0697928	+	0.120885i
$918$		0			0
$919$	−1225.00			−1.33297			−0.666485	−	0.745518i	$-0.732202\pi$
$919$	−1225.00			−1.33297			−0.666485	+	0.745518i	$0.732202\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	756.000			0.819068
$924$		0			0
$925$	283.500	+	163.679i	0.306486	+	0.176950i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−145.000	−	251.147i	−0.156082	−	0.270342i	0.777371	−	0.629043i	$-0.216553\pi$
$929$	−145.000	−	251.147i	−0.156082	−	0.270342i	−0.933452	+	0.358701i	$0.883220\pi$
$930$		0			0
$931$	−912.000			−0.979592
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−352.000	+	609.682i	−0.376471	+	0.652066i
$936$		0			0
$937$	276.500	−	478.912i	0.295091	−	0.511112i	−0.679915	−	0.733291i	$-0.737983\pi$
$937$	276.500	−	478.912i	0.295091	−	0.511112i	0.975006	+	0.222179i	$0.0713168\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−972.000	−	561.184i	−1.03294	−	0.596370i	−0.115117	−	0.993352i	$-0.536724\pi$
$941$	−972.000	−	561.184i	−1.03294	−	0.596370i	−0.917827	+	0.396982i	$0.870058\pi$
$942$		0			0
$943$		−	554.256i		−	0.587758i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−370.000	−	640.859i	−0.390707	−	0.676725i	0.601836	−	0.798620i	$-0.294436\pi$
$947$	−370.000	−	640.859i	−0.390707	−	0.676725i	−0.992543	+	0.121895i	$0.961103\pi$
$948$		0			0
$949$		−	545.596i		−	0.574917i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1086.00	−	627.002i	1.13956	−	0.657925i	0.193238	−	0.981152i	$-0.438101\pi$
$953$	1086.00	−	627.002i	1.13956	−	0.657925i	0.946321	+	0.323227i	$0.104768\pi$
$954$		0			0
$955$	70.0000	+	121.244i	0.0732984	+	0.126957i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	58.0000	−	100.459i	0.0604797	−	0.104754i
$960$		0			0
$961$	−1562.00			−1.62539
$962$		0			0
$963$		0			0
$964$		0			0
$965$	525.000	+	303.109i	0.544041	+	0.314102i
$966$		0			0
$967$	−167.500	−	290.119i	−0.173216	−	0.300019i	0.766326	−	0.642451i	$-0.222083\pi$
$967$	−167.500	−	290.119i	−0.173216	−	0.300019i	−0.939542	+	0.342432i	$0.888749\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1083.00	+	625.270i	−1.11535	+	0.643945i	−0.940209	−	0.340599i	$-0.889370\pi$
$971$	−1083.00	+	625.270i	−1.11535	+	0.643945i	−0.175136	+	0.984544i	$0.556037\pi$
$972$		0			0
$973$	−87.5000	+	151.554i	−0.0899281	+	0.155760i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1292.11i		−	1.32253i	−0.750153	−	0.661264i	$-0.770020\pi$
$977$		−	1292.11i		−	1.32253i	0.750153	−	0.661264i	$-0.229980\pi$
$978$		0			0
$979$	−528.000	−	304.841i	−0.539326	−	0.311380i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	414.000	−	239.023i	0.421160	−	0.243157i	−0.274414	−	0.961612i	$-0.588484\pi$
$983$	414.000	−	239.023i	0.421160	−	0.243157i	0.695573	+	0.718455i	$0.255150\pi$
$984$		0			0
$985$	−2.00000	−	3.46410i	−0.00203046	−	0.00351685i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1960.00			1.98180
$990$		0			0
$991$	955.500	−	551.658i	0.964178	−	0.556668i	0.0667213	−	0.997772i	$-0.478746\pi$
$991$	955.500	−	551.658i	0.964178	−	0.556668i	0.897456	+	0.441103i	$0.145413\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	394.000			0.395980
$996$		0			0
$997$	270.500	−	468.520i	0.271314	−	0.469930i	−0.697885	−	0.716210i	$-0.745875\pi$
$997$	270.500	−	468.520i	0.271314	−	0.469930i	0.969199	+	0.246281i	$0.0792085\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.3.y.a.145.1		2
3.2	odd	2		228.3.l.b.145.1	✓	2
12.11	even	2		912.3.be.c.145.1		2
19.8	odd	6	inner	684.3.y.a.217.1		2
57.8	even	6		228.3.l.b.217.1	yes	2
228.179	odd	6		912.3.be.c.673.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.3.l.b.145.1	✓	2	3.2	odd	2
228.3.l.b.217.1	yes	2	57.8	even	6
684.3.y.a.145.1		2	1.1	even	1	trivial
684.3.y.a.217.1		2	19.8	odd	6	inner
912.3.be.c.145.1		2	12.11	even	2
912.3.be.c.673.1		2	228.179	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	684.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{5} -1.00000 q^{7} -16.0000 q^{11} +(13.5000 + 7.79423i) q^{13} +(11.0000 + 19.0526i) q^{17} +19.0000 q^{19} +(20.0000 - 34.6410i) q^{23} +(10.5000 - 18.1865i) q^{25} +(-15.0000 - 8.66025i) q^{29} -50.2295i q^{31} +(1.00000 + 1.73205i) q^{35} +15.5885i q^{37} +(12.0000 - 6.92820i) q^{41} +(24.5000 + 42.4352i) q^{43} +(23.0000 - 39.8372i) q^{47} -48.0000 q^{49} +(42.0000 + 24.2487i) q^{53} +(16.0000 + 27.7128i) q^{55} +(57.0000 - 32.9090i) q^{59} +(48.5000 - 84.0045i) q^{61} -31.1769i q^{65} +(-22.5000 - 12.9904i) q^{67} +(42.0000 - 24.2487i) q^{71} +(-17.5000 - 30.3109i) q^{73} +16.0000 q^{77} +(-76.5000 + 44.1673i) q^{79} +146.000 q^{83} +(22.0000 - 38.1051i) q^{85} +(33.0000 + 19.0526i) q^{89} +(-13.5000 - 7.79423i) q^{91} +(-19.0000 - 32.9090i) q^{95} +(54.0000 - 31.1769i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 - 2 * q^7	\( 2 q - 2 q^{5} - 2 q^{7} - 32 q^{11} + 27 q^{13} + 22 q^{17} + 38 q^{19} + 40 q^{23} + 21 q^{25} - 30 q^{29} + 2 q^{35} + 24 q^{41} + 49 q^{43} + 46 q^{47} - 96 q^{49} + 84 q^{53} + 32 q^{55} + 114 q^{59} + 97 q^{61} - 45 q^{67} + 84 q^{71} - 35 q^{73} + 32 q^{77} - 153 q^{79} + 292 q^{83} + 44 q^{85} + 66 q^{89} - 27 q^{91} - 38 q^{95} + 108 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 - 32 * q^11 + 27 * q^13 + 22 * q^17 + 38 * q^19 + 40 * q^23 + 21 * q^25 - 30 * q^29 + 2 * q^35 + 24 * q^41 + 49 * q^43 + 46 * q^47 - 96 * q^49 + 84 * q^53 + 32 * q^55 + 114 * q^59 + 97 * q^61 - 45 * q^67 + 84 * q^71 - 35 * q^73 + 32 * q^77 - 153 * q^79 + 292 * q^83 + 44 * q^85 + 66 * q^89 - 27 * q^91 - 38 * q^95 + 108 * q^97

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