LMFDB - Embedded newform 270.2.c.c.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,2,Mod(109,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("270.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	270.c (of order $2$, degree $1$, 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:	$2.15596085457$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{19})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 25 $ x^4 - 9*x^2 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.4
Root			$2.17945 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	270.109
Dual form			270.2.c.c.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +(2.17945 - 0.500000i) q^{5} -4.35890i q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.17945 - 0.500000i) q^{5} -4.35890i q^{7} -1.00000i q^{8} +(0.500000 + 2.17945i) q^{10} +4.35890 q^{11} +4.35890 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000 q^{19} +(-2.17945 + 0.500000i) q^{20} +4.35890i q^{22} +2.00000i q^{23} +(4.50000 - 2.17945i) q^{25} +4.35890i q^{28} +7.00000 q^{31} +1.00000i q^{32} -4.00000 q^{34} +(-2.17945 - 9.50000i) q^{35} +8.71780i q^{37} -6.00000i q^{38} +(-0.500000 - 2.17945i) q^{40} -8.71780 q^{41} -8.71780i q^{43} -4.35890 q^{44} -2.00000 q^{46} +2.00000i q^{47} -12.0000 q^{49} +(2.17945 + 4.50000i) q^{50} +3.00000i q^{53} +(9.50000 - 2.17945i) q^{55} -4.35890 q^{56} -8.71780 q^{59} -4.00000 q^{61} +7.00000i q^{62} -1.00000 q^{64} -8.71780i q^{67} -4.00000i q^{68} +(9.50000 - 2.17945i) q^{70} +4.35890i q^{73} -8.71780 q^{74} +6.00000 q^{76} -19.0000i q^{77} +(2.17945 - 0.500000i) q^{80} -8.71780i q^{82} +5.00000i q^{83} +(2.00000 + 8.71780i) q^{85} +8.71780 q^{86} -4.35890i q^{88} +8.71780 q^{89} -2.00000i q^{92} -2.00000 q^{94} +(-13.0767 + 3.00000i) q^{95} +4.35890i q^{97} -12.0000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 2 q^{10} + 4 q^{16} - 24 q^{19} + 18 q^{25} + 28 q^{31} - 16 q^{34} - 2 q^{40} - 8 q^{46} - 48 q^{49} + 38 q^{55} - 16 q^{61} - 4 q^{64} + 38 q^{70} + 24 q^{76} + 8 q^{85} - 8 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^10 + 4 * q^16 - 24 * q^19 + 18 * q^25 + 28 * q^31 - 16 * q^34 - 2 * q^40 - 8 * q^46 - 48 * q^49 + 38 * q^55 - 16 * q^61 - 4 * q^64 + 38 * q^70 + 24 * q^76 + 8 * q^85 - 8 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$	2.17945	−	0.500000i	0.974679	−	0.223607i
$5$	2.17945	−	0.500000i	0.974679	−	0.223607i
$6$		0			0
$7$		−	4.35890i		−	1.64751i	−0.566947	−	0.823754i	$-0.691875\pi$
$7$		−	4.35890i		−	1.64751i	0.566947	−	0.823754i	$-0.308125\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	0.500000	+	2.17945i	0.158114	+	0.689202i
$11$	4.35890			1.31426			0.657129	−	0.753778i	$-0.271771\pi$
$11$	4.35890			1.31426			0.657129	+	0.753778i	$0.271771\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	4.35890			1.16496
$15$		0			0
$16$	1.00000			0.250000
$17$			4.00000i			0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$			4.00000i			0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	−2.17945	+	0.500000i	−0.487340	+	0.111803i
$21$		0			0
$22$			4.35890i			0.929320i
$23$			2.00000i			0.417029i	0.978019	+	0.208514i	$0.0668628\pi$
$23$			2.00000i			0.417029i	−0.978019	+	0.208514i	$0.933137\pi$
$24$		0			0
$25$	4.50000	−	2.17945i	0.900000	−	0.435890i
$26$		0			0
$27$		0			0
$28$			4.35890i			0.823754i
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$	7.00000			1.25724			0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000			1.25724			0.628619	+	0.777714i	$0.283621\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−4.00000			−0.685994
$35$	−2.17945	−	9.50000i	−0.368394	−	1.60579i
$36$		0			0
$37$			8.71780i			1.43320i	0.697486	+	0.716599i	$0.254302\pi$
$37$			8.71780i			1.43320i	−0.697486	+	0.716599i	$0.745698\pi$
$38$		−	6.00000i		−	0.973329i
$39$		0			0
$40$	−0.500000	−	2.17945i	−0.0790569	−	0.344601i
$41$	−8.71780			−1.36149			−0.680746	−	0.732520i	$-0.738344\pi$
$41$	−8.71780			−1.36149			−0.680746	+	0.732520i	$0.738344\pi$
$42$		0			0
$43$		−	8.71780i		−	1.32945i	−0.747087	−	0.664726i	$-0.768548\pi$
$43$		−	8.71780i		−	1.32945i	0.747087	−	0.664726i	$-0.231452\pi$
$44$	−4.35890			−0.657129
$45$		0			0
$46$	−2.00000			−0.294884
$47$			2.00000i			0.291730i	0.989305	+	0.145865i	$0.0465965\pi$
$47$			2.00000i			0.291730i	−0.989305	+	0.145865i	$0.953403\pi$
$48$		0			0
$49$	−12.0000			−1.71429
$50$	2.17945	+	4.50000i	0.308221	+	0.636396i
$51$		0			0
$52$		0			0
$53$			3.00000i			0.412082i	0.978543	+	0.206041i	$0.0660580\pi$
$53$			3.00000i			0.412082i	−0.978543	+	0.206041i	$0.933942\pi$
$54$		0			0
$55$	9.50000	−	2.17945i	1.28098	−	0.293877i
$56$	−4.35890			−0.582482
$57$		0			0
$58$		0			0
$59$	−8.71780			−1.13496			−0.567480	−	0.823387i	$-0.692082\pi$
$59$	−8.71780			−1.13496			−0.567480	+	0.823387i	$0.692082\pi$
$60$		0			0
$61$	−4.00000			−0.512148			−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000			−0.512148			−0.256074	+	0.966657i	$0.582429\pi$
$62$			7.00000i			0.889001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		−	8.71780i		−	1.06505i	−0.846415	−	0.532524i	$-0.821244\pi$
$67$		−	8.71780i		−	1.06505i	0.846415	−	0.532524i	$-0.178756\pi$
$68$		−	4.00000i		−	0.485071i
$69$		0			0
$70$	9.50000	−	2.17945i	1.13547	−	0.260494i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			4.35890i			0.510171i	0.966919	+	0.255085i	$0.0821035\pi$
$73$			4.35890i			0.510171i	−0.966919	+	0.255085i	$0.917896\pi$
$74$	−8.71780			−1.01342
$75$		0			0
$76$	6.00000			0.688247
$77$		−	19.0000i		−	2.16525i
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	2.17945	−	0.500000i	0.243670	−	0.0559017i
$81$		0			0
$82$		−	8.71780i		−	0.962720i
$83$			5.00000i			0.548821i	0.961613	+	0.274411i	$0.0884828\pi$
$83$			5.00000i			0.548821i	−0.961613	+	0.274411i	$0.911517\pi$
$84$		0			0
$85$	2.00000	+	8.71780i	0.216930	+	0.945578i
$86$	8.71780			0.940064
$87$		0			0
$88$		−	4.35890i		−	0.464660i
$89$	8.71780			0.924085			0.462042	−	0.886858i	$-0.347117\pi$
$89$	8.71780			0.924085			0.462042	+	0.886858i	$0.347117\pi$
$90$		0			0
$91$		0			0
$92$		−	2.00000i		−	0.208514i
$93$		0			0
$94$	−2.00000			−0.206284
$95$	−13.0767	+	3.00000i	−1.34164	+	0.307794i
$96$		0			0
$97$			4.35890i			0.442579i	0.975208	+	0.221290i	$0.0710266\pi$
$97$			4.35890i			0.442579i	−0.975208	+	0.221290i	$0.928973\pi$
$98$		−	12.0000i		−	1.21218i
$99$		0			0
$100$	−4.50000	+	2.17945i	−0.450000	+	0.217945i
$101$	−4.35890			−0.433727			−0.216863	−	0.976202i	$-0.569583\pi$
$101$	−4.35890			−0.433727			−0.216863	+	0.976202i	$0.569583\pi$
$102$		0			0
$103$			8.71780i			0.858990i	0.903069	+	0.429495i	$0.141308\pi$
$103$			8.71780i			0.858990i	−0.903069	+	0.429495i	$0.858692\pi$
$104$		0			0
$105$		0			0
$106$	−3.00000			−0.291386
$107$			15.0000i			1.45010i	0.688694	+	0.725052i	$0.258184\pi$
$107$			15.0000i			1.45010i	−0.688694	+	0.725052i	$0.741816\pi$
$108$		0			0
$109$	−10.0000			−0.957826			−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000			−0.957826			−0.478913	+	0.877862i	$0.658969\pi$
$110$	2.17945	+	9.50000i	0.207802	+	0.905789i
$111$		0			0
$112$		−	4.35890i		−	0.411877i
$113$			18.0000i			1.69330i	0.532152	+	0.846649i	$0.321383\pi$
$113$			18.0000i			1.69330i	−0.532152	+	0.846649i	$0.678617\pi$
$114$		0			0
$115$	1.00000	+	4.35890i	0.0932505	+	0.406469i
$116$		0			0
$117$		0			0
$118$		−	8.71780i		−	0.802538i
$119$	17.4356			1.59832
$120$		0			0
$121$	8.00000			0.727273
$122$		−	4.00000i		−	0.362143i
$123$		0			0
$124$	−7.00000			−0.628619
$125$	8.71780	−	7.00000i	0.779744	−	0.626099i
$126$		0			0
$127$		−	4.35890i		−	0.386790i	−0.981121	−	0.193395i	$-0.938050\pi$
$127$		−	4.35890i		−	0.386790i	0.981121	−	0.193395i	$-0.0619498\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		0			0
$130$		0			0
$131$	13.0767			1.14252			0.571258	−	0.820770i	$-0.306456\pi$
$131$	13.0767			1.14252			0.571258	+	0.820770i	$0.306456\pi$
$132$		0			0
$133$			26.1534i			2.26779i
$134$	8.71780			0.753103
$135$		0			0
$136$	4.00000			0.342997
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$	2.17945	+	9.50000i	0.184197	+	0.802897i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$	−4.35890			−0.360745
$147$		0			0
$148$		−	8.71780i		−	0.716599i
$149$	4.35890			0.357095			0.178547	−	0.983931i	$-0.442860\pi$
$149$	4.35890			0.357095			0.178547	+	0.983931i	$0.442860\pi$
$150$		0			0
$151$	−11.0000			−0.895167			−0.447584	−	0.894242i	$-0.647715\pi$
$151$	−11.0000			−0.895167			−0.447584	+	0.894242i	$0.647715\pi$
$152$			6.00000i			0.486664i
$153$		0			0
$154$	19.0000			1.53106
$155$	15.2561	−	3.50000i	1.22540	−	0.281127i
$156$		0			0
$157$			17.4356i			1.39151i	0.718278	+	0.695756i	$0.244931\pi$
$157$			17.4356i			1.39151i	−0.718278	+	0.695756i	$0.755069\pi$
$158$		0			0
$159$		0			0
$160$	0.500000	+	2.17945i	0.0395285	+	0.172301i
$161$	8.71780			0.687059
$162$		0			0
$163$		−	17.4356i		−	1.36566i	−0.730577	−	0.682831i	$-0.760749\pi$
$163$		−	17.4356i		−	1.36566i	0.730577	−	0.682831i	$-0.239251\pi$
$164$	8.71780			0.680746
$165$		0			0
$166$	−5.00000			−0.388075
$167$		−	18.0000i		−	1.39288i	−0.717614	−	0.696441i	$-0.754766\pi$
$167$		−	18.0000i		−	1.39288i	0.717614	−	0.696441i	$-0.245234\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$	−8.71780	+	2.00000i	−0.668625	+	0.153393i
$171$		0			0
$172$			8.71780i			0.664726i
$173$			19.0000i			1.44454i	0.691609	+	0.722272i	$0.256902\pi$
$173$			19.0000i			1.44454i	−0.691609	+	0.722272i	$0.743098\pi$
$174$		0			0
$175$	−9.50000	−	19.6150i	−0.718132	−	1.48276i
$176$	4.35890			0.328564
$177$		0			0
$178$			8.71780i			0.653427i
$179$	−21.7945			−1.62900			−0.814499	−	0.580166i	$-0.802988\pi$
$179$	−21.7945			−1.62900			−0.814499	+	0.580166i	$0.802988\pi$
$180$		0			0
$181$	8.00000			0.594635			0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000			0.594635			0.297318	+	0.954779i	$0.403908\pi$
$182$		0			0
$183$		0			0
$184$	2.00000			0.147442
$185$	4.35890	+	19.0000i	0.320473	+	1.39691i
$186$		0			0
$187$			17.4356i			1.27502i
$188$		−	2.00000i		−	0.145865i
$189$		0			0
$190$	−3.00000	−	13.0767i	−0.217643	−	0.948683i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	21.7945i		−	1.56880i	−0.620254	−	0.784401i	$-0.712970\pi$
$193$		−	21.7945i		−	1.56880i	0.620254	−	0.784401i	$-0.287030\pi$
$194$	−4.35890			−0.312951
$195$		0			0
$196$	12.0000			0.857143
$197$			5.00000i			0.356235i	0.984009	+	0.178118i	$0.0570008\pi$
$197$			5.00000i			0.356235i	−0.984009	+	0.178118i	$0.942999\pi$
$198$		0			0
$199$	3.00000			0.212664			0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000			0.212664			0.106332	+	0.994331i	$0.466089\pi$
$200$	−2.17945	−	4.50000i	−0.154110	−	0.318198i
$201$		0			0
$202$		−	4.35890i		−	0.306691i
$203$		0			0
$204$		0			0
$205$	−19.0000	+	4.35890i	−1.32702	+	0.304439i
$206$	−8.71780			−0.607398
$207$		0			0
$208$		0			0
$209$	−26.1534			−1.80907
$210$		0			0
$211$	6.00000			0.413057			0.206529	−	0.978441i	$-0.433783\pi$
$211$	6.00000			0.413057			0.206529	+	0.978441i	$0.433783\pi$
$212$		−	3.00000i		−	0.206041i
$213$		0			0
$214$	−15.0000			−1.02538
$215$	−4.35890	−	19.0000i	−0.297274	−	1.29579i
$216$		0			0
$217$		−	30.5123i		−	2.07131i
$218$		−	10.0000i		−	0.677285i
$219$		0			0
$220$	−9.50000	+	2.17945i	−0.640490	+	0.146938i
$221$		0			0
$222$		0			0
$223$		−	8.71780i		−	0.583787i	−0.956451	−	0.291893i	$-0.905715\pi$
$223$		−	8.71780i		−	0.583787i	0.956451	−	0.291893i	$-0.0942853\pi$
$224$	4.35890			0.291241
$225$		0			0
$226$	−18.0000			−1.19734
$227$			4.00000i			0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$			4.00000i			0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$		0			0
$229$	22.0000			1.45380			0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000			1.45380			0.726900	+	0.686743i	$0.240960\pi$
$230$	−4.35890	+	1.00000i	−0.287417	+	0.0659380i
$231$		0			0
$232$		0			0
$233$		−	14.0000i		−	0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$		−	14.0000i		−	0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$		0			0
$235$	1.00000	+	4.35890i	0.0652328	+	0.284343i
$236$	8.71780			0.567480
$237$		0			0
$238$			17.4356i			1.13018i
$239$	8.71780			0.563907			0.281954	−	0.959428i	$-0.409018\pi$
$239$	8.71780			0.563907			0.281954	+	0.959428i	$0.409018\pi$
$240$		0			0
$241$	−14.0000			−0.901819			−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000			−0.901819			−0.450910	+	0.892570i	$0.648900\pi$
$242$			8.00000i			0.514259i
$243$		0			0
$244$	4.00000			0.256074
$245$	−26.1534	+	6.00000i	−1.67088	+	0.383326i
$246$		0			0
$247$		0			0
$248$		−	7.00000i		−	0.444500i
$249$		0			0
$250$	7.00000	+	8.71780i	0.442719	+	0.551362i
$251$	−26.1534			−1.65079			−0.825394	−	0.564557i	$-0.809047\pi$
$251$	−26.1534			−1.65079			−0.825394	+	0.564557i	$0.809047\pi$
$252$		0			0
$253$			8.71780i			0.548083i
$254$	4.35890			0.273502
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	12.0000i		−	0.748539i	−0.927320	−	0.374270i	$-0.877893\pi$
$257$		−	12.0000i		−	0.748539i	0.927320	−	0.374270i	$-0.122107\pi$
$258$		0			0
$259$	38.0000			2.36121
$260$		0			0
$261$		0			0
$262$			13.0767i			0.807881i
$263$			2.00000i			0.123325i	0.998097	+	0.0616626i	$0.0196403\pi$
$263$			2.00000i			0.123325i	−0.998097	+	0.0616626i	$0.980360\pi$
$264$		0			0
$265$	1.50000	+	6.53835i	0.0921443	+	0.401648i
$266$	−26.1534			−1.60357
$267$		0			0
$268$			8.71780i			0.532524i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	−5.00000			−0.303728			−0.151864	−	0.988401i	$-0.548528\pi$
$271$	−5.00000			−0.303728			−0.151864	+	0.988401i	$0.548528\pi$
$272$			4.00000i			0.242536i
$273$		0			0
$274$	18.0000			1.08742
$275$	19.6150	−	9.50000i	1.18283	−	0.572872i
$276$		0			0
$277$		−	17.4356i		−	1.04760i	−0.851840	−	0.523802i	$-0.824513\pi$
$277$		−	17.4356i		−	1.04760i	0.851840	−	0.523802i	$-0.175487\pi$
$278$		−	4.00000i		−	0.239904i
$279$		0			0
$280$	−9.50000	+	2.17945i	−0.567734	+	0.130247i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$			8.71780i			0.518219i	0.965848	+	0.259110i	$0.0834291\pi$
$283$			8.71780i			0.518219i	−0.965848	+	0.259110i	$0.916571\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			38.0000i			2.24307i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$		0			0
$292$		−	4.35890i		−	0.255085i
$293$			6.00000i			0.350524i	0.984522	+	0.175262i	$0.0560772\pi$
$293$			6.00000i			0.350524i	−0.984522	+	0.175262i	$0.943923\pi$
$294$		0			0
$295$	−19.0000	+	4.35890i	−1.10622	+	0.253785i
$296$	8.71780			0.506712
$297$		0			0
$298$			4.35890i			0.252504i
$299$		0			0
$300$		0			0
$301$	−38.0000			−2.19028
$302$		−	11.0000i		−	0.632979i
$303$		0			0
$304$	−6.00000			−0.344124
$305$	−8.71780	+	2.00000i	−0.499180	+	0.114520i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$			19.0000i			1.08263i
$309$		0			0
$310$	3.50000	+	15.2561i	0.198787	+	0.866491i
$311$	−26.1534			−1.48302			−0.741511	−	0.670940i	$-0.765891\pi$
$311$	−26.1534			−1.48302			−0.741511	+	0.670940i	$0.765891\pi$
$312$		0			0
$313$		−	21.7945i		−	1.23190i	−0.787786	−	0.615949i	$-0.788773\pi$
$313$		−	21.7945i		−	1.23190i	0.787786	−	0.615949i	$-0.211227\pi$
$314$	−17.4356			−0.983948
$315$		0			0
$316$		0			0
$317$		−	17.0000i		−	0.954815i	−0.878682	−	0.477408i	$-0.841577\pi$
$317$		−	17.0000i		−	0.954815i	0.878682	−	0.477408i	$-0.158423\pi$
$318$		0			0
$319$		0			0
$320$	−2.17945	+	0.500000i	−0.121835	+	0.0279508i
$321$		0			0
$322$			8.71780i			0.485824i
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$		0			0
$326$	17.4356			0.965668
$327$		0			0
$328$			8.71780i			0.481360i
$329$	8.71780			0.480628
$330$		0			0
$331$	10.0000			0.549650			0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000			0.549650			0.274825	+	0.961494i	$0.411380\pi$
$332$		−	5.00000i		−	0.274411i
$333$		0			0
$334$	18.0000			0.984916
$335$	−4.35890	−	19.0000i	−0.238152	−	1.03808i
$336$		0			0
$337$		−	17.4356i		−	0.949777i	−0.880046	−	0.474889i	$-0.842488\pi$
$337$		−	17.4356i		−	0.949777i	0.880046	−	0.474889i	$-0.157512\pi$
$338$			13.0000i			0.707107i
$339$		0			0
$340$	−2.00000	−	8.71780i	−0.108465	−	0.472789i
$341$	30.5123			1.65233
$342$		0			0
$343$			21.7945i			1.17679i
$344$	−8.71780			−0.470032
$345$		0			0
$346$	−19.0000			−1.02145
$347$			27.0000i			1.44944i	0.689046	+	0.724718i	$0.258030\pi$
$347$			27.0000i			1.44944i	−0.689046	+	0.724718i	$0.741970\pi$
$348$		0			0
$349$	14.0000			0.749403			0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000			0.749403			0.374701	+	0.927146i	$0.377745\pi$
$350$	19.6150	−	9.50000i	1.04847	−	0.507796i
$351$		0			0
$352$			4.35890i			0.232330i
$353$		−	26.0000i		−	1.38384i	−0.721974	−	0.691920i	$-0.756765\pi$
$353$		−	26.0000i		−	1.38384i	0.721974	−	0.691920i	$-0.243235\pi$
$354$		0			0
$355$		0			0
$356$	−8.71780			−0.462042
$357$		0			0
$358$		−	21.7945i		−	1.15187i
$359$	26.1534			1.38032			0.690162	−	0.723655i	$-0.257539\pi$
$359$	26.1534			1.38032			0.690162	+	0.723655i	$0.257539\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$			8.00000i			0.420471i
$363$		0			0
$364$		0			0
$365$	2.17945	+	9.50000i	0.114078	+	0.497253i
$366$		0			0
$367$		−	4.35890i		−	0.227533i	−0.993508	−	0.113766i	$-0.963708\pi$
$367$		−	4.35890i		−	0.227533i	0.993508	−	0.113766i	$-0.0362915\pi$
$368$			2.00000i			0.104257i
$369$		0			0
$370$	−19.0000	+	4.35890i	−0.987763	+	0.226608i
$371$	13.0767			0.678908
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−17.4356			−0.901573
$375$		0			0
$376$	2.00000			0.103142
$377$		0			0
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$	13.0767	−	3.00000i	0.670820	−	0.153897i
$381$		0			0
$382$		0			0
$383$			4.00000i			0.204390i	0.994764	+	0.102195i	$0.0325866\pi$
$383$			4.00000i			0.204390i	−0.994764	+	0.102195i	$0.967413\pi$
$384$		0			0
$385$	−9.50000	−	41.4095i	−0.484165	−	2.11043i
$386$	21.7945			1.10931
$387$		0			0
$388$		−	4.35890i		−	0.221290i
$389$	−4.35890			−0.221005			−0.110502	−	0.993876i	$-0.535246\pi$
$389$	−4.35890			−0.221005			−0.110502	+	0.993876i	$0.535246\pi$
$390$		0			0
$391$	−8.00000			−0.404577
$392$			12.0000i			0.606092i
$393$		0			0
$394$	−5.00000			−0.251896
$395$		0			0
$396$		0			0
$397$			34.8712i			1.75013i	0.484001	+	0.875067i	$0.339183\pi$
$397$			34.8712i			1.75013i	−0.484001	+	0.875067i	$0.660817\pi$
$398$			3.00000i			0.150376i
$399$		0			0
$400$	4.50000	−	2.17945i	0.225000	−	0.108972i
$401$	34.8712			1.74138			0.870692	−	0.491828i	$-0.163671\pi$
$401$	34.8712			1.74138			0.870692	+	0.491828i	$0.163671\pi$
$402$		0			0
$403$		0			0
$404$	4.35890			0.216863
$405$		0			0
$406$		0			0
$407$			38.0000i			1.88359i
$408$		0			0
$409$	31.0000			1.53285			0.766426	−	0.642333i	$-0.222033\pi$
$409$	31.0000			1.53285			0.766426	+	0.642333i	$0.222033\pi$
$410$	−4.35890	−	19.0000i	−0.215271	−	0.938343i
$411$		0			0
$412$		−	8.71780i		−	0.429495i
$413$			38.0000i			1.86986i
$414$		0			0
$415$	2.50000	+	10.8972i	0.122720	+	0.534925i
$416$		0			0
$417$		0			0
$418$		−	26.1534i		−	1.27920i
$419$	8.71780			0.425892			0.212946	−	0.977064i	$-0.431694\pi$
$419$	8.71780			0.425892			0.212946	+	0.977064i	$0.431694\pi$
$420$		0			0
$421$	16.0000			0.779792			0.389896	−	0.920859i	$-0.372511\pi$
$421$	16.0000			0.779792			0.389896	+	0.920859i	$0.372511\pi$
$422$			6.00000i			0.292075i
$423$		0			0
$424$	3.00000			0.145693
$425$	8.71780	+	18.0000i	0.422875	+	0.873128i
$426$		0			0
$427$			17.4356i			0.843768i
$428$		−	15.0000i		−	0.725052i
$429$		0			0
$430$	19.0000	−	4.35890i	0.916261	−	0.210205i
$431$	−8.71780			−0.419922			−0.209961	−	0.977710i	$-0.567334\pi$
$431$	−8.71780			−0.419922			−0.209961	+	0.977710i	$0.567334\pi$
$432$		0			0
$433$		−	21.7945i		−	1.04738i	−0.851910	−	0.523688i	$-0.824556\pi$
$433$		−	21.7945i		−	1.04738i	0.851910	−	0.523688i	$-0.175444\pi$
$434$	30.5123			1.46464
$435$		0			0
$436$	10.0000			0.478913
$437$		−	12.0000i		−	0.574038i
$438$		0			0
$439$	−9.00000			−0.429547			−0.214773	−	0.976664i	$-0.568901\pi$
$439$	−9.00000			−0.429547			−0.214773	+	0.976664i	$0.568901\pi$
$440$	−2.17945	−	9.50000i	−0.103901	−	0.452895i
$441$		0			0
$442$		0			0
$443$		−	12.0000i		−	0.570137i	−0.958507	−	0.285069i	$-0.907984\pi$
$443$		−	12.0000i		−	0.570137i	0.958507	−	0.285069i	$-0.0920164\pi$
$444$		0			0
$445$	19.0000	−	4.35890i	0.900686	−	0.206632i
$446$	8.71780			0.412800
$447$		0			0
$448$			4.35890i			0.205939i
$449$	26.1534			1.23425			0.617127	−	0.786863i	$-0.288296\pi$
$449$	26.1534			1.23425			0.617127	+	0.786863i	$0.288296\pi$
$450$		0			0
$451$	−38.0000			−1.78935
$452$		−	18.0000i		−	0.846649i
$453$		0			0
$454$	−4.00000			−0.187729
$455$		0			0
$456$		0			0
$457$		−	4.35890i		−	0.203901i	−0.994789	−	0.101950i	$-0.967492\pi$
$457$		−	4.35890i		−	0.203901i	0.994789	−	0.101950i	$-0.0325083\pi$
$458$			22.0000i			1.02799i
$459$		0			0
$460$	−1.00000	−	4.35890i	−0.0466252	−	0.203235i
$461$	−39.2301			−1.82713			−0.913564	−	0.406696i	$-0.866681\pi$
$461$	−39.2301			−1.82713			−0.913564	+	0.406696i	$0.866681\pi$
$462$		0			0
$463$			13.0767i			0.607726i	0.952716	+	0.303863i	$0.0982765\pi$
$463$			13.0767i			0.607726i	−0.952716	+	0.303863i	$0.901724\pi$
$464$		0			0
$465$		0			0
$466$	14.0000			0.648537
$467$		−	21.0000i		−	0.971764i	−0.874024	−	0.485882i	$-0.838498\pi$
$467$		−	21.0000i		−	0.971764i	0.874024	−	0.485882i	$-0.161502\pi$
$468$		0			0
$469$	−38.0000			−1.75468
$470$	−4.35890	+	1.00000i	−0.201061	+	0.0461266i
$471$		0			0
$472$			8.71780i			0.401269i
$473$		−	38.0000i		−	1.74724i
$474$		0			0
$475$	−27.0000	+	13.0767i	−1.23884	+	0.600000i
$476$	−17.4356			−0.799159
$477$		0			0
$478$			8.71780i			0.398743i
$479$	8.71780			0.398326			0.199163	−	0.979966i	$-0.436178\pi$
$479$	8.71780			0.398326			0.199163	+	0.979966i	$0.436178\pi$
$480$		0			0
$481$		0			0
$482$		−	14.0000i		−	0.637683i
$483$		0			0
$484$	−8.00000			−0.363636
$485$	2.17945	+	9.50000i	0.0989637	+	0.431373i
$486$		0			0
$487$			26.1534i			1.18512i	0.805525	+	0.592562i	$0.201883\pi$
$487$			26.1534i			1.18512i	−0.805525	+	0.592562i	$0.798117\pi$
$488$			4.00000i			0.181071i
$489$		0			0
$490$	−6.00000	−	26.1534i	−0.271052	−	1.18149i
$491$	13.0767			0.590143			0.295072	−	0.955475i	$-0.404657\pi$
$491$	13.0767			0.590143			0.295072	+	0.955475i	$0.404657\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	7.00000			0.314309
$497$		0			0
$498$		0			0
$499$	−30.0000			−1.34298			−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000			−1.34298			−0.671492	+	0.741012i	$0.734346\pi$
$500$	−8.71780	+	7.00000i	−0.389872	+	0.313050i
$501$		0			0
$502$		−	26.1534i		−	1.16728i
$503$		−	30.0000i		−	1.33763i	−0.743427	−	0.668817i	$-0.766801\pi$
$503$		−	30.0000i		−	1.33763i	0.743427	−	0.668817i	$-0.233199\pi$
$504$		0			0
$505$	−9.50000	+	2.17945i	−0.422744	+	0.0969842i
$506$	−8.71780			−0.387553
$507$		0			0
$508$			4.35890i			0.193395i
$509$	−13.0767			−0.579614			−0.289807	−	0.957085i	$-0.593591\pi$
$509$	−13.0767			−0.579614			−0.289807	+	0.957085i	$0.593591\pi$
$510$		0			0
$511$	19.0000			0.840511
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	12.0000			0.529297
$515$	4.35890	+	19.0000i	0.192076	+	0.837240i
$516$		0			0
$517$			8.71780i			0.383408i
$518$			38.0000i			1.66962i
$519$		0			0
$520$		0			0
$521$	−17.4356			−0.763867			−0.381934	−	0.924190i	$-0.624742\pi$
$521$	−17.4356			−0.763867			−0.381934	+	0.924190i	$0.624742\pi$
$522$		0			0
$523$			34.8712i			1.52481i	0.647100	+	0.762405i	$0.275982\pi$
$523$			34.8712i			1.52481i	−0.647100	+	0.762405i	$0.724018\pi$
$524$	−13.0767			−0.571258
$525$		0			0
$526$	−2.00000			−0.0872041
$527$			28.0000i			1.21970i
$528$		0			0
$529$	19.0000			0.826087
$530$	−6.53835	+	1.50000i	−0.284008	+	0.0651558i
$531$		0			0
$532$		−	26.1534i		−	1.13389i
$533$		0			0
$534$		0			0
$535$	7.50000	+	32.6917i	0.324253	+	1.41339i
$536$	−8.71780			−0.376552
$537$		0			0
$538$		0			0
$539$	−52.3068			−2.25301
$540$		0			0
$541$	40.0000			1.71973			0.859867	−	0.510518i	$-0.170546\pi$
$541$	40.0000			1.71973			0.859867	+	0.510518i	$0.170546\pi$
$542$		−	5.00000i		−	0.214768i
$543$		0			0
$544$	−4.00000			−0.171499
$545$	−21.7945	+	5.00000i	−0.933574	+	0.214176i
$546$		0			0
$547$		−	17.4356i		−	0.745492i	−0.927933	−	0.372746i	$-0.878416\pi$
$547$		−	17.4356i		−	0.745492i	0.927933	−	0.372746i	$-0.121584\pi$
$548$			18.0000i			0.768922i
$549$		0			0
$550$	9.50000	+	19.6150i	0.405081	+	0.836388i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	17.4356			0.740767
$555$		0			0
$556$	4.00000			0.169638
$557$		−	33.0000i		−	1.39825i	−0.714997	−	0.699127i	$-0.753572\pi$
$557$		−	33.0000i		−	1.39825i	0.714997	−	0.699127i	$-0.246428\pi$
$558$		0			0
$559$		0			0
$560$	−2.17945	−	9.50000i	−0.0920985	−	0.401448i
$561$		0			0
$562$		0			0
$563$		−	11.0000i		−	0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$		−	11.0000i		−	0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$		0			0
$565$	9.00000	+	39.2301i	0.378633	+	1.65042i
$566$	−8.71780			−0.366436
$567$		0			0
$568$		0			0
$569$	−17.4356			−0.730938			−0.365469	−	0.930823i	$-0.619091\pi$
$569$	−17.4356			−0.730938			−0.365469	+	0.930823i	$0.619091\pi$
$570$		0			0
$571$	16.0000			0.669579			0.334790	−	0.942293i	$-0.391335\pi$
$571$	16.0000			0.669579			0.334790	+	0.942293i	$0.391335\pi$
$572$		0			0
$573$		0			0
$574$	−38.0000			−1.58609
$575$	4.35890	+	9.00000i	0.181779	+	0.375326i
$576$		0			0
$577$		−	17.4356i		−	0.725853i	−0.931818	−	0.362927i	$-0.881778\pi$
$577$		−	17.4356i		−	0.725853i	0.931818	−	0.362927i	$-0.118222\pi$
$578$			1.00000i			0.0415945i
$579$		0			0
$580$		0			0
$581$	21.7945			0.904188
$582$		0			0
$583$			13.0767i			0.541581i
$584$	4.35890			0.180373
$585$		0			0
$586$	−6.00000			−0.247858
$587$			3.00000i			0.123823i	0.998082	+	0.0619116i	$0.0197197\pi$
$587$			3.00000i			0.123823i	−0.998082	+	0.0619116i	$0.980280\pi$
$588$		0			0
$589$	−42.0000			−1.73058
$590$	−4.35890	−	19.0000i	−0.179453	−	0.782218i
$591$		0			0
$592$			8.71780i			0.358299i
$593$			6.00000i			0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$			6.00000i			0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$		0			0
$595$	38.0000	−	8.71780i	1.55785	−	0.357395i
$596$	−4.35890			−0.178547
$597$		0			0
$598$		0			0
$599$	−26.1534			−1.06860			−0.534299	−	0.845295i	$-0.679424\pi$
$599$	−26.1534			−1.06860			−0.534299	+	0.845295i	$0.679424\pi$
$600$		0			0
$601$	−13.0000			−0.530281			−0.265141	−	0.964210i	$-0.585418\pi$
$601$	−13.0000			−0.530281			−0.265141	+	0.964210i	$0.585418\pi$
$602$		−	38.0000i		−	1.54876i
$603$		0			0
$604$	11.0000			0.447584
$605$	17.4356	−	4.00000i	0.708858	−	0.162623i
$606$		0			0
$607$		−	8.71780i		−	0.353845i	−0.984225	−	0.176922i	$-0.943386\pi$
$607$		−	8.71780i		−	0.353845i	0.984225	−	0.176922i	$-0.0566141\pi$
$608$		−	6.00000i		−	0.243332i
$609$		0			0
$610$	−2.00000	−	8.71780i	−0.0809776	−	0.352973i
$611$		0			0
$612$		0			0
$613$		−	8.71780i		−	0.352109i	−0.984380	−	0.176054i	$-0.943667\pi$
$613$		−	8.71780i		−	0.352109i	0.984380	−	0.176054i	$-0.0563334\pi$
$614$		0			0
$615$		0			0
$616$	−19.0000			−0.765532
$617$		−	2.00000i		−	0.0805170i	−0.999189	−	0.0402585i	$-0.987182\pi$
$617$		−	2.00000i		−	0.0805170i	0.999189	−	0.0402585i	$-0.0128181\pi$
$618$		0			0
$619$	−32.0000			−1.28619			−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000			−1.28619			−0.643094	+	0.765787i	$0.722350\pi$
$620$	−15.2561	+	3.50000i	−0.612702	+	0.140563i
$621$		0			0
$622$		−	26.1534i		−	1.04866i
$623$		−	38.0000i		−	1.52244i
$624$		0			0
$625$	15.5000	−	19.6150i	0.620000	−	0.784602i
$626$	21.7945			0.871083
$627$		0			0
$628$		−	17.4356i		−	0.695756i
$629$	−34.8712			−1.39041
$630$		0			0
$631$	5.00000			0.199047			0.0995234	−	0.995035i	$-0.468268\pi$
$631$	5.00000			0.199047			0.0995234	+	0.995035i	$0.468268\pi$
$632$		0			0
$633$		0			0
$634$	17.0000			0.675156
$635$	−2.17945	−	9.50000i	−0.0864888	−	0.376996i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.500000	−	2.17945i	−0.0197642	−	0.0861503i
$641$	−8.71780			−0.344332			−0.172166	−	0.985068i	$-0.555077\pi$
$641$	−8.71780			−0.344332			−0.172166	+	0.985068i	$0.555077\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−8.71780			−0.343529
$645$		0			0
$646$	24.0000			0.944267
$647$		−	12.0000i		−	0.471769i	−0.971781	−	0.235884i	$-0.924201\pi$
$647$		−	12.0000i		−	0.471769i	0.971781	−	0.235884i	$-0.0757987\pi$
$648$		0			0
$649$	−38.0000			−1.49163
$650$		0			0
$651$		0			0
$652$			17.4356i			0.682831i
$653$			35.0000i			1.36966i	0.728705	+	0.684828i	$0.240123\pi$
$653$			35.0000i			1.36966i	−0.728705	+	0.684828i	$0.759877\pi$
$654$		0			0
$655$	28.5000	−	6.53835i	1.11359	−	0.255474i
$656$	−8.71780			−0.340373
$657$		0			0
$658$			8.71780i			0.339855i
$659$	4.35890			0.169799			0.0848993	−	0.996390i	$-0.472943\pi$
$659$	4.35890			0.169799			0.0848993	+	0.996390i	$0.472943\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$			10.0000i			0.388661i
$663$		0			0
$664$	5.00000			0.194038
$665$	13.0767	+	57.0000i	0.507093	+	2.21037i
$666$		0			0
$667$		0			0
$668$			18.0000i			0.696441i
$669$		0			0
$670$	19.0000	−	4.35890i	0.734034	−	0.168399i
$671$	−17.4356			−0.673094
$672$		0			0
$673$		−	13.0767i		−	0.504070i	−0.967718	−	0.252035i	$-0.918900\pi$
$673$		−	13.0767i		−	0.504070i	0.967718	−	0.252035i	$-0.0810997\pi$
$674$	17.4356			0.671594
$675$		0			0
$676$	−13.0000			−0.500000
$677$		−	26.0000i		−	0.999261i	−0.866239	−	0.499631i	$-0.833469\pi$
$677$		−	26.0000i		−	0.999261i	0.866239	−	0.499631i	$-0.166531\pi$
$678$		0			0
$679$	19.0000			0.729153
$680$	8.71780	−	2.00000i	0.334312	−	0.0766965i
$681$		0			0
$682$			30.5123i			1.16838i
$683$			20.0000i			0.765279i	0.923898	+	0.382639i	$0.124985\pi$
$683$			20.0000i			0.765279i	−0.923898	+	0.382639i	$0.875015\pi$
$684$		0			0
$685$	−9.00000	−	39.2301i	−0.343872	−	1.49890i
$686$	−21.7945			−0.832118
$687$		0			0
$688$		−	8.71780i		−	0.332363i
$689$		0			0
$690$		0			0
$691$	−20.0000			−0.760836			−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000			−0.760836			−0.380418	+	0.924815i	$0.624220\pi$
$692$		−	19.0000i		−	0.722272i
$693$		0			0
$694$	−27.0000			−1.02491
$695$	−8.71780	+	2.00000i	−0.330685	+	0.0758643i
$696$		0			0
$697$		−	34.8712i		−	1.32084i
$698$			14.0000i			0.529908i
$699$		0			0
$700$	9.50000	+	19.6150i	0.359066	+	0.741379i
$701$	−21.7945			−0.823167			−0.411583	−	0.911372i	$-0.635024\pi$
$701$	−21.7945			−0.823167			−0.411583	+	0.911372i	$0.635024\pi$
$702$		0			0
$703$		−	52.3068i		−	1.97279i
$704$	−4.35890			−0.164282
$705$		0			0
$706$	26.0000			0.978523
$707$			19.0000i			0.714569i
$708$		0			0
$709$	−12.0000			−0.450669			−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000			−0.450669			−0.225335	+	0.974281i	$0.572348\pi$
$710$		0			0
$711$		0			0
$712$		−	8.71780i		−	0.326713i
$713$			14.0000i			0.524304i
$714$		0			0
$715$		0			0
$716$	21.7945			0.814499
$717$		0			0
$718$			26.1534i			0.976036i
$719$	34.8712			1.30048			0.650238	−	0.759731i	$-0.274669\pi$
$719$	34.8712			1.30048			0.650238	+	0.759731i	$0.274669\pi$
$720$		0			0
$721$	38.0000			1.41519
$722$			17.0000i			0.632674i
$723$		0			0
$724$	−8.00000			−0.297318
$725$		0			0
$726$		0			0
$727$			39.2301i			1.45496i	0.686127	+	0.727482i	$0.259309\pi$
$727$			39.2301i			1.45496i	−0.686127	+	0.727482i	$0.740691\pi$
$728$		0			0
$729$		0			0
$730$	−9.50000	+	2.17945i	−0.351611	+	0.0806650i
$731$	34.8712			1.28976
$732$		0			0
$733$			8.71780i			0.321999i	0.986954	+	0.161000i	$0.0514718\pi$
$733$			8.71780i			0.321999i	−0.986954	+	0.161000i	$0.948528\pi$
$734$	4.35890			0.160890
$735$		0			0
$736$	−2.00000			−0.0737210
$737$		−	38.0000i		−	1.39975i
$738$		0			0
$739$	−16.0000			−0.588570			−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000			−0.588570			−0.294285	+	0.955718i	$0.595081\pi$
$740$	−4.35890	−	19.0000i	−0.160236	−	0.698454i
$741$		0			0
$742$			13.0767i			0.480061i
$743$		−	48.0000i		−	1.76095i	−0.474093	−	0.880475i	$-0.657224\pi$
$743$		−	48.0000i		−	1.76095i	0.474093	−	0.880475i	$-0.342776\pi$
$744$		0			0
$745$	9.50000	−	2.17945i	0.348053	−	0.0798489i
$746$		0			0
$747$		0			0
$748$		−	17.4356i		−	0.637509i
$749$	65.3835			2.38906
$750$		0			0
$751$	35.0000			1.27717			0.638584	−	0.769552i	$-0.279520\pi$
$751$	35.0000			1.27717			0.638584	+	0.769552i	$0.279520\pi$
$752$			2.00000i			0.0729325i
$753$		0			0
$754$		0			0
$755$	−23.9739	+	5.50000i	−0.872501	+	0.200165i
$756$		0			0
$757$			8.71780i			0.316854i	0.987371	+	0.158427i	$0.0506422\pi$
$757$			8.71780i			0.316854i	−0.987371	+	0.158427i	$0.949358\pi$
$758$			4.00000i			0.145287i
$759$		0			0
$760$	3.00000	+	13.0767i	0.108821	+	0.474342i
$761$	−52.3068			−1.89612			−0.948060	−	0.318092i	$-0.896958\pi$
$761$	−52.3068			−1.89612			−0.948060	+	0.318092i	$0.896958\pi$
$762$		0			0
$763$			43.5890i			1.57803i
$764$		0			0
$765$		0			0
$766$	−4.00000			−0.144526
$767$		0			0
$768$		0			0
$769$	−9.00000			−0.324548			−0.162274	−	0.986746i	$-0.551883\pi$
$769$	−9.00000			−0.324548			−0.162274	+	0.986746i	$0.551883\pi$
$770$	41.4095	−	9.50000i	1.49230	−	0.342356i
$771$		0			0
$772$			21.7945i			0.784401i
$773$		−	18.0000i		−	0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$		−	18.0000i		−	0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$		0			0
$775$	31.5000	−	15.2561i	1.13151	−	0.548017i
$776$	4.35890			0.156475
$777$		0			0
$778$		−	4.35890i		−	0.156274i
$779$	52.3068			1.87409
$780$		0			0
$781$		0			0
$782$		−	8.00000i		−	0.286079i
$783$		0			0
$784$	−12.0000			−0.428571
$785$	8.71780	+	38.0000i	0.311152	+	1.35628i
$786$		0			0
$787$		−	17.4356i		−	0.621512i	−0.950490	−	0.310756i	$-0.899418\pi$
$787$		−	17.4356i		−	0.621512i	0.950490	−	0.310756i	$-0.100582\pi$
$788$		−	5.00000i		−	0.178118i
$789$		0			0
$790$		0			0
$791$	78.4602			2.78972
$792$		0			0
$793$		0			0
$794$	−34.8712			−1.23753
$795$		0

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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.17945 - 0.500000i) q^{5} -4.35890i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +(2.17945 - 0.500000i) q^{5} -4.35890i q^{7} -1.00000i q^{8} +(0.500000 + 2.17945i) q^{10} +4.35890 q^{11} +4.35890 q^{14} +1.00000 q^{16} +4.00000i q^{17} -6.00000 q^{19} +(-2.17945 + 0.500000i) q^{20} +4.35890i q^{22} +2.00000i q^{23} +(4.50000 - 2.17945i) q^{25} +4.35890i q^{28} +7.00000 q^{31} +1.00000i q^{32} -4.00000 q^{34} +(-2.17945 - 9.50000i) q^{35} +8.71780i q^{37} -6.00000i q^{38} +(-0.500000 - 2.17945i) q^{40} -8.71780 q^{41} -8.71780i q^{43} -4.35890 q^{44} -2.00000 q^{46} +2.00000i q^{47} -12.0000 q^{49} +(2.17945 + 4.50000i) q^{50} +3.00000i q^{53} +(9.50000 - 2.17945i) q^{55} -4.35890 q^{56} -8.71780 q^{59} -4.00000 q^{61} +7.00000i q^{62} -1.00000 q^{64} -8.71780i q^{67} -4.00000i q^{68} +(9.50000 - 2.17945i) q^{70} +4.35890i q^{73} -8.71780 q^{74} +6.00000 q^{76} -19.0000i q^{77} +(2.17945 - 0.500000i) q^{80} -8.71780i q^{82} +5.00000i q^{83} +(2.00000 + 8.71780i) q^{85} +8.71780 q^{86} -4.35890i q^{88} +8.71780 q^{89} -2.00000i q^{92} -2.00000 q^{94} +(-13.0767 + 3.00000i) q^{95} +4.35890i q^{97} -12.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 2 q^{10} + 4 q^{16} - 24 q^{19} + 18 q^{25} + 28 q^{31} - 16 q^{34} - 2 q^{40} - 8 q^{46} - 48 q^{49} + 38 q^{55} - 16 q^{61} - 4 q^{64} + 38 q^{70} + 24 q^{76} + 8 q^{85} - 8 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^10 + 4 * q^16 - 24 * q^19 + 18 * q^25 + 28 * q^31 - 16 * q^34 - 2 * q^40 - 8 * q^46 - 48 * q^49 + 38 * q^55 - 16 * q^61 - 4 * q^64 + 38 * q^70 + 24 * q^76 + 8 * q^85 - 8 * q^94

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