LMFDB - Embedded newform 270.2.c.c.109.1

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			109.1
Root			$-2.17945 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	270.109
Dual form			270.2.c.c.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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	$-0.728229\pi$
$11$	−4.35890			−1.31426			−0.657129	+	0.753778i	$0.728229\pi$
$12$		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	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		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	$-0.933137\pi$
$23$		−	2.00000i		−	0.417029i	0.978019	−	0.208514i	$-0.0668628\pi$
$24$		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	$-0.261656\pi$
$41$	8.71780			1.36149			0.680746	+	0.732520i	$0.261656\pi$
$42$		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	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$		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	$-0.933942\pi$
$53$		−	3.00000i		−	0.412082i	0.978543	−	0.206041i	$-0.0660580\pi$
$54$		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	$-0.307918\pi$
$59$	8.71780			1.13496			0.567480	+	0.823387i	$0.307918\pi$
$60$		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	$-0.911517\pi$
$83$		−	5.00000i		−	0.548821i	0.961613	−	0.274411i	$-0.0884828\pi$
$84$		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	$-0.652883\pi$
$89$	−8.71780			−0.924085			−0.462042	+	0.886858i	$0.652883\pi$
$90$		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	$-0.430417\pi$
$101$	4.35890			0.433727			0.216863	+	0.976202i	$0.430417\pi$
$102$		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	$-0.741816\pi$
$107$		−	15.0000i		−	1.45010i	0.688694	−	0.725052i	$-0.258184\pi$
$108$		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	$-0.678617\pi$
$113$		−	18.0000i		−	1.69330i	0.532152	−	0.846649i	$-0.321383\pi$
$114$		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	$-0.693544\pi$
$131$	−13.0767			−1.14252			−0.571258	+	0.820770i	$0.693544\pi$
$132$		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	$0.279207\pi$
$137$			18.0000i			1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$		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	$-0.557140\pi$
$149$	−4.35890			−0.357095			−0.178547	+	0.983931i	$0.557140\pi$
$150$		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	$0.245234\pi$
$167$			18.0000i			1.39288i	−0.717614	+	0.696441i	$0.754766\pi$
$168$		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	$-0.743098\pi$
$173$		−	19.0000i		−	1.44454i	0.691609	−	0.722272i	$-0.256902\pi$
$174$		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	$-0.197012\pi$
$179$	21.7945			1.62900			0.814499	+	0.580166i	$0.197012\pi$
$180$		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	$-0.942999\pi$
$197$		−	5.00000i		−	0.356235i	0.984009	−	0.178118i	$-0.0570008\pi$
$198$		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	$-0.957621\pi$
$227$		−	4.00000i		−	0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$		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	$0.151644\pi$
$233$			14.0000i			0.917170i	−0.888650	+	0.458585i	$0.848356\pi$
$234$		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	$-0.590982\pi$
$239$	−8.71780			−0.563907			−0.281954	+	0.959428i	$0.590982\pi$
$240$		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	$-0.190953\pi$
$251$	26.1534			1.65079			0.825394	+	0.564557i	$0.190953\pi$
$252$		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	$0.122107\pi$
$257$			12.0000i			0.748539i	−0.927320	+	0.374270i	$0.877893\pi$
$258$		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	$-0.980360\pi$
$263$		−	2.00000i		−	0.123325i	0.998097	−	0.0616626i	$-0.0196403\pi$
$264$		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	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		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	$-0.234109\pi$
$311$	26.1534			1.48302			0.741511	+	0.670940i	$0.234109\pi$
$312$		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	$0.158423\pi$
$317$			17.0000i			0.954815i	−0.878682	+	0.477408i	$0.841577\pi$
$318$		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	$-0.741970\pi$
$347$		−	27.0000i		−	1.44944i	0.689046	−	0.724718i	$-0.258030\pi$
$348$		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	$0.243235\pi$
$353$			26.0000i			1.38384i	−0.721974	+	0.691920i	$0.756765\pi$
$354$		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	$-0.742461\pi$
$359$	−26.1534			−1.38032			−0.690162	+	0.723655i	$0.742461\pi$
$360$		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	$-0.967413\pi$
$383$		−	4.00000i		−	0.204390i	0.994764	−	0.102195i	$-0.0325866\pi$
$384$		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	$-0.464754\pi$
$389$	4.35890			0.221005			0.110502	+	0.993876i	$0.464754\pi$
$390$		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	$-0.836329\pi$
$401$	−34.8712			−1.74138			−0.870692	+	0.491828i	$0.836329\pi$
$402$		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	$-0.568306\pi$
$419$	−8.71780			−0.425892			−0.212946	+	0.977064i	$0.568306\pi$
$420$		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	$-0.432666\pi$
$431$	8.71780			0.419922			0.209961	+	0.977710i	$0.432666\pi$
$432$		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	$0.0920164\pi$
$443$			12.0000i			0.570137i	−0.958507	+	0.285069i	$0.907984\pi$
$444$		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	$-0.711704\pi$
$449$	−26.1534			−1.23425			−0.617127	+	0.786863i	$0.711704\pi$
$450$		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	$-0.133319\pi$
$461$	39.2301			1.82713			0.913564	+	0.406696i	$0.133319\pi$
$462$		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	$0.161502\pi$
$467$			21.0000i			0.971764i	−0.874024	+	0.485882i	$0.838498\pi$
$468$		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	$-0.563822\pi$
$479$	−8.71780			−0.398326			−0.199163	+	0.979966i	$0.563822\pi$
$480$		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	$-0.595343\pi$
$491$	−13.0767			−0.590143			−0.295072	+	0.955475i	$0.595343\pi$
$492$		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	$0.233199\pi$
$503$			30.0000i			1.33763i	−0.743427	+	0.668817i	$0.766801\pi$
$504$		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	$-0.406409\pi$
$509$	13.0767			0.579614			0.289807	+	0.957085i	$0.406409\pi$
$510$		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	$-0.375258\pi$
$521$	17.4356			0.763867			0.381934	+	0.924190i	$0.375258\pi$
$522$		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	$0.246428\pi$
$557$			33.0000i			1.39825i	−0.714997	+	0.699127i	$0.753572\pi$
$558$		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	$0.0744606\pi$
$563$			11.0000i			0.463595i	−0.972764	+	0.231797i	$0.925539\pi$
$564$		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	$-0.380909\pi$
$569$	17.4356			0.730938			0.365469	+	0.930823i	$0.380909\pi$
$570$		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	$-0.980280\pi$
$587$		−	3.00000i		−	0.123823i	0.998082	−	0.0619116i	$-0.0197197\pi$
$588$		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	$-0.960686\pi$
$593$		−	6.00000i		−	0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$		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	$-0.320576\pi$
$599$	26.1534			1.06860			0.534299	+	0.845295i	$0.320576\pi$
$600$		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	$0.0128181\pi$
$617$			2.00000i			0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$		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	$-0.444923\pi$
$641$	8.71780			0.344332			0.172166	+	0.985068i	$0.444923\pi$
$642$		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	$0.0757987\pi$
$647$			12.0000i			0.471769i	−0.971781	+	0.235884i	$0.924201\pi$
$648$		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	$-0.759877\pi$
$653$		−	35.0000i		−	1.36966i	0.728705	−	0.684828i	$-0.240123\pi$
$654$		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	$-0.527057\pi$
$659$	−4.35890			−0.169799			−0.0848993	+	0.996390i	$0.527057\pi$
$660$		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	$0.166531\pi$
$677$			26.0000i			0.999261i	−0.866239	+	0.499631i	$0.833469\pi$
$678$		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	$-0.875015\pi$
$683$		−	20.0000i		−	0.765279i	0.923898	−	0.382639i	$-0.124985\pi$
$684$		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	$-0.364976\pi$
$701$	21.7945			0.823167			0.411583	+	0.911372i	$0.364976\pi$
$702$		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	$-0.725331\pi$
$719$	−34.8712			−1.30048			−0.650238	+	0.759731i	$0.725331\pi$
$720$		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	$0.342776\pi$
$743$			48.0000i			1.76095i	−0.474093	+	0.880475i	$0.657224\pi$
$744$		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	$-0.103042\pi$
$761$	52.3068			1.89612			0.948060	+	0.318092i	$0.103042\pi$
$762$		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	$0.104929\pi$
$773$			18.0000i			0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$		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			0
$796$	−3.00000			−0.106332
$797$		−	37.0000i		−	1.31061i	−0.755366	−	0.655304i	$-0.772541\pi$
$797$		−	37.0000i		−	1.31061i	0.755366	−	0.655304i	$-0.227459\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$	−2.17945	−	4.50000i	−0.0770552	−	0.159099i
$801$		0			0
$802$			34.8712i			1.23134i
$803$		−	19.0000i		−	0.670495i
$804$		0			0
$805$	19.0000	−	4.35890i	0.669662	−	0.153631i
$806$		0			0
$807$		0			0
$808$			4.35890i			0.153346i
$809$	−26.1534			−0.919504			−0.459752	−	0.888047i	$-0.652062\pi$
$809$	−26.1534			−0.919504			−0.459752	+	0.888047i	$0.652062\pi$
$810$		0			0
$811$	−14.0000			−0.491606			−0.245803	−	0.969320i	$-0.579052\pi$
$811$	−14.0000			−0.491606			−0.245803	+	0.969320i	$0.579052\pi$
$812$		0			0
$813$		0			0
$814$	−38.0000			−1.33190
$815$	8.71780	+	38.0000i	0.305371	+	1.33108i
$816$		0			0
$817$			52.3068i			1.82998i
$818$		−	31.0000i		−	1.08389i
$819$		0			0
$820$	19.0000	−	4.35890i	0.663509	−	0.152219i
$821$	−34.8712			−1.21701			−0.608506	−	0.793549i	$-0.708231\pi$
$821$	−34.8712			−1.21701			−0.608506	+	0.793549i	$0.708231\pi$
$822$		0			0
$823$		−	39.2301i		−	1.36747i	−0.729728	−	0.683737i	$-0.760353\pi$
$823$		−	39.2301i		−	1.36747i	0.729728	−	0.683737i	$-0.239647\pi$
$824$	−8.71780			−0.303699
$825$		0			0
$826$	−38.0000			−1.32219
$827$		−	16.0000i		−	0.556375i	−0.960527	−	0.278187i	$-0.910266\pi$
$827$		−	16.0000i		−	0.556375i	0.960527	−	0.278187i	$-0.0897336\pi$
$828$		0			0
$829$	42.0000			1.45872			0.729360	−	0.684130i	$-0.239818\pi$
$829$	42.0000			1.45872			0.729360	+	0.684130i	$0.239818\pi$
$830$	10.8972	−	2.50000i	0.378249	−	0.0867763i
$831$		0			0
$832$		0			0
$833$			48.0000i			1.66310i
$834$		0			0
$835$	−9.00000	−	39.2301i	−0.311458	−	1.35761i
$836$	−26.1534			−0.904534
$837$		0			0
$838$			8.71780i			0.301151i
$839$	34.8712			1.20389			0.601944	−	0.798539i	$-0.294393\pi$
$839$	34.8712			1.20389			0.601944	+	0.798539i	$0.294393\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		−	16.0000i		−	0.551396i
$843$		0			0
$844$	−6.00000			−0.206529
$845$	−28.3328	+	6.50000i	−0.974679	+	0.223607i
$846$		0			0
$847$		−	34.8712i		−	1.19819i
$848$		−	3.00000i		−	0.103020i
$849$		0			0
$850$	−18.0000	+	8.71780i	−0.617395	+	0.299018i
$851$	17.4356			0.597685
$852$		0			0
$853$			8.71780i			0.298492i	0.988800	+	0.149246i	$0.0476846\pi$
$853$			8.71780i			0.298492i	−0.988800	+	0.149246i	$0.952315\pi$
$854$	17.4356			0.596634
$855$		0			0
$856$	15.0000			0.512689
$857$		−	48.0000i		−	1.63965i	−0.572615	−	0.819824i	$-0.694071\pi$
$857$		−	48.0000i		−	1.63965i	0.572615	−	0.819824i	$-0.305929\pi$
$858$		0			0
$859$	−20.0000			−0.682391			−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000			−0.682391			−0.341196	+	0.939992i	$0.610832\pi$
$860$	−4.35890	−	19.0000i	−0.148637	−	0.647895i
$861$		0			0
$862$		−	8.71780i		−	0.296929i
$863$		−	48.0000i		−	1.63394i	−0.576681	−	0.816970i	$-0.695652\pi$
$863$		−	48.0000i		−	1.63394i	0.576681	−	0.816970i	$-0.304348\pi$
$864$		0			0
$865$	9.50000	+	41.4095i	0.323010	+	1.40797i
$866$	−21.7945			−0.740607
$867$		0			0
$868$			30.5123i			1.03565i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		−	10.0000i		−	0.338643i
$873$		0			0
$874$	12.0000			0.405906
$875$	30.5123	+	38.0000i	1.03150	+	1.28463i
$876$		0			0
$877$		−	8.71780i		−	0.294379i	−0.989108	−	0.147190i	$-0.952977\pi$
$877$		−	8.71780i		−	0.294379i	0.989108	−	0.147190i	$-0.0470227\pi$
$878$			9.00000i			0.303735i
$879$		0			0
$880$	9.50000	−	2.17945i	0.320245	−	0.0734692i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		−	26.1534i		−	0.880132i	−0.897965	−	0.440066i	$-0.854955\pi$
$883$		−	26.1534i		−	0.880132i	0.897965	−	0.440066i	$-0.145045\pi$
$884$		0			0
$885$		0			0
$886$	12.0000			0.403148
$887$		−	24.0000i		−	0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$		−	24.0000i		−	0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$		0			0
$889$	−19.0000			−0.637240
$890$	−4.35890	−	19.0000i	−0.146111	−	0.636881i
$891$		0			0
$892$			8.71780i			0.291893i
$893$			12.0000i			0.401565i
$894$		0			0
$895$	−47.5000	+	10.8972i	−1.58775	+	0.364255i
$896$	4.35890			0.145621
$897$		0			0
$898$			26.1534i			0.872750i
$899$		0			0
$900$		0			0
$901$	−12.0000			−0.399778
$902$			38.0000i			1.26526i
$903$		0			0
$904$	18.0000			0.598671
$905$	−17.4356	+	4.00000i	−0.579579	+	0.132964i
$906$		0			0
$907$		−	26.1534i		−	0.868409i	−0.900814	−	0.434205i	$-0.857029\pi$
$907$		−	26.1534i		−	0.868409i	0.900814	−	0.434205i	$-0.142971\pi$
$908$			4.00000i			0.132745i
$909$		0			0
$910$		0			0
$911$	−26.1534			−0.866501			−0.433250	−	0.901274i	$-0.642633\pi$
$911$	−26.1534			−0.866501			−0.433250	+	0.901274i	$0.642633\pi$
$912$		0			0
$913$			21.7945i			0.721292i
$914$	−4.35890			−0.144180
$915$		0			0
$916$	−22.0000			−0.726900
$917$			57.0000i			1.88231i
$918$		0			0
$919$	−39.0000			−1.28649			−0.643246	−	0.765660i	$-0.722413\pi$
$919$	−39.0000			−1.28649			−0.643246	+	0.765660i	$0.722413\pi$
$920$	−4.35890	+	1.00000i	−0.143709	+	0.0329690i
$921$		0			0
$922$		−	39.2301i		−	1.29197i
$923$		0			0
$924$		0			0
$925$	19.0000	+	39.2301i	0.624716	+	1.28988i
$926$	13.0767			0.429727
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	72.0000			2.35970
$932$		−	14.0000i		−	0.458585i
$933$		0			0
$934$	21.0000			0.687141
$935$	−8.71780	−	38.0000i	−0.285102	−	1.24273i
$936$		0			0
$937$		−	47.9479i		−	1.56639i	−0.621777	−	0.783195i	$-0.713589\pi$
$937$		−	47.9479i		−	1.56639i	0.621777	−	0.783195i	$-0.286411\pi$
$938$			38.0000i			1.24074i
$939$		0			0
$940$	−1.00000	−	4.35890i	−0.0326164	−	0.142172i
$941$	47.9479			1.56306			0.781528	−	0.623870i	$-0.214440\pi$
$941$	47.9479			1.56306			0.781528	+	0.623870i	$0.214440\pi$
$942$		0			0
$943$		−	17.4356i		−	0.567781i
$944$	8.71780			0.283740
$945$		0			0
$946$	38.0000			1.23549
$947$		−	27.0000i		−	0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$		−	27.0000i		−	0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$		0			0
$949$		0			0
$950$	13.0767	+	27.0000i	0.424264	+	0.875996i
$951$		0			0
$952$		−	17.4356i		−	0.565091i
$953$			32.0000i			1.03658i	0.855204	+	0.518291i	$0.173432\pi$
$953$			32.0000i			1.03658i	−0.855204	+	0.518291i	$0.826568\pi$
$954$		0			0
$955$		0			0
$956$	8.71780			0.281954
$957$		0			0
$958$			8.71780i			0.281659i
$959$	78.4602			2.53361
$960$		0			0
$961$	18.0000			0.580645
$962$		0			0
$963$		0			0
$964$	14.0000			0.450910
$965$	10.8972	+	47.5000i	0.350795	+	1.52908i
$966$		0			0
$967$			30.5123i			0.981209i	0.871382	+	0.490605i	$0.163224\pi$
$967$			30.5123i			0.981209i	−0.871382	+	0.490605i	$0.836776\pi$
$968$			8.00000i			0.257130i
$969$		0			0
$970$	−9.50000	+	2.17945i	−0.305027	+	0.0699779i
$971$	39.2301			1.25895			0.629477	−	0.777019i	$-0.283269\pi$
$971$	39.2301			1.25895			0.629477	+	0.777019i	$0.283269\pi$
$972$		0			0
$973$			17.4356i			0.558960i
$974$	26.1534			0.838009
$975$		0			0
$976$	−4.00000			−0.128037
$977$		−	18.0000i		−	0.575871i	−0.957650	−	0.287936i	$-0.907031\pi$
$977$		−	18.0000i		−	0.575871i	0.957650	−	0.287936i	$-0.0929689\pi$
$978$		0			0
$979$	38.0000			1.21449
$980$	−26.1534	+	6.00000i	−0.835440	+	0.191663i
$981$		0			0
$982$			13.0767i			0.417294i
$983$		−	34.0000i		−	1.08443i	−0.840239	−	0.542216i	$-0.817586\pi$
$983$		−	34.0000i		−	1.08443i	0.840239	−	0.542216i	$-0.182414\pi$
$984$		0			0
$985$	2.50000	+	10.8972i	0.0796566	+	0.347215i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−17.4356			−0.554420
$990$		0			0
$991$	−53.0000			−1.68360			−0.841800	−	0.539789i	$-0.818504\pi$
$991$	−53.0000			−1.68360			−0.841800	+	0.539789i	$0.818504\pi$
$992$		−	7.00000i		−	0.222250i
$993$		0			0
$994$		0			0
$995$	−6.53835	+	1.50000i	−0.207280	+	0.0475532i
$996$		0			0
$997$		−	52.3068i		−	1.65657i	−0.560305	−	0.828286i	$-0.689316\pi$
$997$		−	52.3068i		−	1.65657i	0.560305	−	0.828286i	$-0.310684\pi$
$998$			30.0000i			0.949633i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.c.c.109.1	✓	4
3.2	odd	2	inner	270.2.c.c.109.4	yes	4
4.3	odd	2		2160.2.f.m.1729.2		4
5.2	odd	4		1350.2.a.x.1.2		2
5.3	odd	4		1350.2.a.w.1.1		2
5.4	even	2	inner	270.2.c.c.109.3	yes	4
9.2	odd	6		810.2.i.h.109.3		8
9.4	even	3		810.2.i.h.379.4		8
9.5	odd	6		810.2.i.h.379.1		8
9.7	even	3		810.2.i.h.109.2		8
12.11	even	2		2160.2.f.m.1729.3		4
15.2	even	4		1350.2.a.w.1.2		2
15.8	even	4		1350.2.a.x.1.1		2
15.14	odd	2	inner	270.2.c.c.109.2	yes	4
20.19	odd	2		2160.2.f.m.1729.1		4
45.4	even	6		810.2.i.h.379.2		8
45.14	odd	6		810.2.i.h.379.3		8
45.29	odd	6		810.2.i.h.109.1		8
45.34	even	6		810.2.i.h.109.4		8
60.59	even	2		2160.2.f.m.1729.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.2.c.c.109.1	✓	4	1.1	even	1	trivial
270.2.c.c.109.2	yes	4	15.14	odd	2	inner
270.2.c.c.109.3	yes	4	5.4	even	2	inner
270.2.c.c.109.4	yes	4	3.2	odd	2	inner
810.2.i.h.109.1		8	45.29	odd	6
810.2.i.h.109.2		8	9.7	even	3
810.2.i.h.109.3		8	9.2	odd	6
810.2.i.h.109.4		8	45.34	even	6
810.2.i.h.379.1		8	9.5	odd	6
810.2.i.h.379.2		8	45.4	even	6
810.2.i.h.379.3		8	45.14	odd	6
810.2.i.h.379.4		8	9.4	even	3
1350.2.a.w.1.1		2	5.3	odd	4
1350.2.a.w.1.2		2	15.2	even	4
1350.2.a.x.1.1		2	15.8	even	4
1350.2.a.x.1.2		2	5.2	odd	4
2160.2.f.m.1729.1		4	20.19	odd	2
2160.2.f.m.1729.2		4	4.3	odd	2
2160.2.f.m.1729.3		4	12.11	even	2
2160.2.f.m.1729.4		4	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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