LMFDB - Embedded newform 540.2.d.c.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [540,2,Mod(109,540)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(540, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("540.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 540 = 2^{2} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	540.d (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:	$4.31192170915$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			109.4
Root			$1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	540.109
Dual form			540.2.d.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)

$f(q)$	$=$	$q+(1.58114 + 1.58114i) q^{5} +1.00000i q^{7} +3.16228 q^{11} -3.00000i q^{13} +6.32456i q^{17} -3.00000 q^{19} +3.16228i q^{23} +5.00000i q^{25} +9.48683 q^{29} -2.00000 q^{31} +(-1.58114 + 1.58114i) q^{35} +1.00000i q^{37} +3.16228 q^{41} -10.0000i q^{43} -6.32456i q^{47} +6.00000 q^{49} +9.48683i q^{53} +(5.00000 + 5.00000i) q^{55} -6.32456 q^{59} -1.00000 q^{61} +(4.74342 - 4.74342i) q^{65} +11.0000i q^{67} -9.48683 q^{71} -13.0000i q^{73} +3.16228i q^{77} +3.00000 q^{79} -15.8114i q^{83} +(-10.0000 + 10.0000i) q^{85} -12.6491 q^{89} +3.00000 q^{91} +(-4.74342 - 4.74342i) q^{95} -1.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{19} - 8 q^{31} + 24 q^{49} + 20 q^{55} - 4 q^{61} + 12 q^{79} - 40 q^{85} + 12 q^{91}+O(q^{100}) $ 4 * q - 12 * q^19 - 8 * q^31 + 24 * q^49 + 20 * q^55 - 4 * q^61 + 12 * q^79 - 40 * q^85 + 12 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$461$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$5$	1.58114	+	1.58114i	0.707107	+	0.707107i
$6$		0			0
$7$			1.00000i			0.377964i	0.981981	+	0.188982i	$0.0605189\pi$
$7$			1.00000i			0.377964i	−0.981981	+	0.188982i	$0.939481\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.16228			0.953463			0.476731	−	0.879049i	$-0.341821\pi$
$11$	3.16228			0.953463			0.476731	+	0.879049i	$0.341821\pi$
$12$		0			0
$13$		−	3.00000i		−	0.832050i	−0.909353	−	0.416025i	$-0.863423\pi$
$13$		−	3.00000i		−	0.832050i	0.909353	−	0.416025i	$-0.136577\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.32456i			1.53393i	0.641689	+	0.766965i	$0.278234\pi$
$17$			6.32456i			1.53393i	−0.641689	+	0.766965i	$0.721766\pi$
$18$		0			0
$19$	−3.00000			−0.688247			−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000			−0.688247			−0.344124	+	0.938924i	$0.611824\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			3.16228i			0.659380i	0.944089	+	0.329690i	$0.106944\pi$
$23$			3.16228i			0.659380i	−0.944089	+	0.329690i	$0.893056\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	9.48683			1.76166			0.880830	−	0.473432i	$-0.156985\pi$
$29$	9.48683			1.76166			0.880830	+	0.473432i	$0.156985\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−1.58114	+	1.58114i	−0.267261	+	0.267261i
$36$		0			0
$37$			1.00000i			0.164399i	0.996616	+	0.0821995i	$0.0261945\pi$
$37$			1.00000i			0.164399i	−0.996616	+	0.0821995i	$0.973806\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.16228			0.493865			0.246932	−	0.969033i	$-0.420577\pi$
$41$	3.16228			0.493865			0.246932	+	0.969033i	$0.420577\pi$
$42$		0			0
$43$		−	10.0000i		−	1.52499i	−0.646997	−	0.762493i	$-0.723975\pi$
$43$		−	10.0000i		−	1.52499i	0.646997	−	0.762493i	$-0.276025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	6.32456i		−	0.922531i	−0.887262	−	0.461266i	$-0.847396\pi$
$47$		−	6.32456i		−	0.922531i	0.887262	−	0.461266i	$-0.152604\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$			9.48683i			1.30312i	0.758599	+	0.651558i	$0.225884\pi$
$53$			9.48683i			1.30312i	−0.758599	+	0.651558i	$0.774116\pi$
$54$		0			0
$55$	5.00000	+	5.00000i	0.674200	+	0.674200i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.32456			−0.823387			−0.411693	−	0.911322i	$-0.635063\pi$
$59$	−6.32456			−0.823387			−0.411693	+	0.911322i	$0.635063\pi$
$60$		0			0
$61$	−1.00000			−0.128037			−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000			−0.128037			−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.74342	−	4.74342i	0.588348	−	0.588348i
$66$		0			0
$67$			11.0000i			1.34386i	0.740613	+	0.671932i	$0.234535\pi$
$67$			11.0000i			1.34386i	−0.740613	+	0.671932i	$0.765465\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−9.48683			−1.12588			−0.562940	−	0.826498i	$-0.690330\pi$
$71$	−9.48683			−1.12588			−0.562940	+	0.826498i	$0.690330\pi$
$72$		0			0
$73$		−	13.0000i		−	1.52153i	−0.649025	−	0.760767i	$-0.724823\pi$
$73$		−	13.0000i		−	1.52153i	0.649025	−	0.760767i	$-0.275177\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			3.16228i			0.360375i
$78$		0			0
$79$	3.00000			0.337526			0.168763	−	0.985657i	$-0.446023\pi$
$79$	3.00000			0.337526			0.168763	+	0.985657i	$0.446023\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	15.8114i		−	1.73553i	−0.496979	−	0.867763i	$-0.665557\pi$
$83$		−	15.8114i		−	1.73553i	0.496979	−	0.867763i	$-0.334443\pi$
$84$		0			0
$85$	−10.0000	+	10.0000i	−1.08465	+	1.08465i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.6491			−1.34080			−0.670402	−	0.741999i	$-0.733878\pi$
$89$	−12.6491			−1.34080			−0.670402	+	0.741999i	$0.733878\pi$
$90$		0			0
$91$	3.00000			0.314485
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.74342	−	4.74342i	−0.486664	−	0.486664i
$96$		0			0
$97$		−	1.00000i		−	0.101535i	−0.998711	−	0.0507673i	$-0.983833\pi$
$97$		−	1.00000i		−	0.101535i	0.998711	−	0.0507673i	$-0.0161667\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.32456			0.629317			0.314658	−	0.949205i	$-0.398110\pi$
$101$	6.32456			0.629317			0.314658	+	0.949205i	$0.398110\pi$
$102$		0			0
$103$		−	17.0000i		−	1.67506i	−0.546392	−	0.837530i	$-0.683999\pi$
$103$		−	17.0000i		−	1.67506i	0.546392	−	0.837530i	$-0.316001\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	9.48683i		−	0.917127i	−0.888662	−	0.458563i	$-0.848364\pi$
$107$		−	9.48683i		−	0.917127i	0.888662	−	0.458563i	$-0.151636\pi$
$108$		0			0
$109$	−16.0000			−1.53252			−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000			−1.53252			−0.766261	+	0.642529i	$0.777885\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	9.48683i		−	0.892446i	−0.894922	−	0.446223i	$-0.852769\pi$
$113$		−	9.48683i		−	0.892446i	0.894922	−	0.446223i	$-0.147231\pi$
$114$		0			0
$115$	−5.00000	+	5.00000i	−0.466252	+	0.466252i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−6.32456			−0.579771
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−7.90569	+	7.90569i	−0.707107	+	0.707107i
$126$		0			0
$127$			10.0000i			0.887357i	0.896186	+	0.443678i	$0.146327\pi$
$127$			10.0000i			0.887357i	−0.896186	+	0.443678i	$0.853673\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−18.9737			−1.65774			−0.828868	−	0.559444i	$-0.811015\pi$
$131$	−18.9737			−1.65774			−0.828868	+	0.559444i	$0.811015\pi$
$132$		0			0
$133$		−	3.00000i		−	0.260133i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	9.48683i		−	0.810515i	−0.914203	−	0.405257i	$-0.867182\pi$
$137$		−	9.48683i		−	0.810515i	0.914203	−	0.405257i	$-0.132818\pi$
$138$		0			0
$139$	17.0000			1.44192			0.720961	−	0.692976i	$-0.243701\pi$
$139$	17.0000			1.44192			0.720961	+	0.692976i	$0.243701\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	9.48683i		−	0.793329i
$144$		0			0
$145$	15.0000	+	15.0000i	1.24568	+	1.24568i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	12.6491			1.03626			0.518128	−	0.855303i	$-0.326629\pi$
$149$	12.6491			1.03626			0.518128	+	0.855303i	$0.326629\pi$
$150$		0			0
$151$	1.00000			0.0813788			0.0406894	−	0.999172i	$-0.487045\pi$
$151$	1.00000			0.0813788			0.0406894	+	0.999172i	$0.487045\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3.16228	−	3.16228i	−0.254000	−	0.254000i
$156$		0			0
$157$		−	10.0000i		−	0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$		−	10.0000i		−	0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.16228			−0.249222
$162$		0			0
$163$		−	17.0000i		−	1.33154i	−0.746156	−	0.665771i	$-0.768103\pi$
$163$		−	17.0000i		−	1.33154i	0.746156	−	0.665771i	$-0.231897\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	4.00000			0.307692
$170$		0			0
$171$		0			0
$172$		0			0
$173$			6.32456i			0.480847i	0.970668	+	0.240424i	$0.0772863\pi$
$173$			6.32456i			0.480847i	−0.970668	+	0.240424i	$0.922714\pi$
$174$		0			0
$175$	−5.00000			−0.377964
$176$		0			0
$177$		0			0
$178$		0			0
$179$	3.16228			0.236360			0.118180	−	0.992992i	$-0.462294\pi$
$179$	3.16228			0.236360			0.118180	+	0.992992i	$0.462294\pi$
$180$		0			0
$181$	11.0000			0.817624			0.408812	−	0.912619i	$-0.365943\pi$
$181$	11.0000			0.817624			0.408812	+	0.912619i	$0.365943\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.58114	+	1.58114i	−0.116248	+	0.116248i
$186$		0			0
$187$			20.0000i			1.46254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		−	7.00000i		−	0.503871i	−0.967744	−	0.251936i	$-0.918933\pi$
$193$		−	7.00000i		−	0.503871i	0.967744	−	0.251936i	$-0.0810671\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			12.6491i			0.901212i	0.892723	+	0.450606i	$0.148792\pi$
$197$			12.6491i			0.901212i	−0.892723	+	0.450606i	$0.851208\pi$
$198$		0			0
$199$	−27.0000			−1.91398			−0.956990	−	0.290122i	$-0.906304\pi$
$199$	−27.0000			−1.91398			−0.956990	+	0.290122i	$0.906304\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			9.48683i			0.665845i
$204$		0			0
$205$	5.00000	+	5.00000i	0.349215	+	0.349215i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.48683			−0.656218
$210$		0			0
$211$	−21.0000			−1.44570			−0.722850	−	0.691005i	$-0.757168\pi$
$211$	−21.0000			−1.44570			−0.722850	+	0.691005i	$0.757168\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	15.8114	−	15.8114i	1.07833	−	1.07833i
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	18.9737			1.27631
$222$		0			0
$223$		−	10.0000i		−	0.669650i	−0.942280	−	0.334825i	$-0.891323\pi$
$223$		−	10.0000i		−	0.669650i	0.942280	−	0.334825i	$-0.108677\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			15.8114i			1.04944i	0.851275	+	0.524719i	$0.175830\pi$
$227$			15.8114i			1.04944i	−0.851275	+	0.524719i	$0.824170\pi$
$228$		0			0
$229$	16.0000			1.05731			0.528655	−	0.848837i	$-0.322697\pi$
$229$	16.0000			1.05731			0.528655	+	0.848837i	$0.322697\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			6.32456i			0.414335i	0.978305	+	0.207168i	$0.0664246\pi$
$233$			6.32456i			0.414335i	−0.978305	+	0.207168i	$0.933575\pi$
$234$		0			0
$235$	10.0000	−	10.0000i	0.652328	−	0.652328i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3.16228			−0.204551			−0.102275	−	0.994756i	$-0.532612\pi$
$239$	−3.16228			−0.204551			−0.102275	+	0.994756i	$0.532612\pi$
$240$		0			0
$241$	−11.0000			−0.708572			−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000			−0.708572			−0.354286	+	0.935137i	$0.615276\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.48683	+	9.48683i	0.606092	+	0.606092i
$246$		0			0
$247$			9.00000i			0.572656i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	28.4605			1.79641			0.898205	−	0.439576i	$-0.144871\pi$
$251$	28.4605			1.79641			0.898205	+	0.439576i	$0.144871\pi$
$252$		0			0
$253$			10.0000i			0.628695i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.9737i		−	1.18354i	−0.806105	−	0.591772i	$-0.798428\pi$
$257$		−	18.9737i		−	1.18354i	0.806105	−	0.591772i	$-0.201572\pi$
$258$		0			0
$259$	−1.00000			−0.0621370
$260$		0			0
$261$		0			0
$262$		0			0
$263$			31.6228i			1.94994i	0.222328	+	0.974972i	$0.428634\pi$
$263$			31.6228i			1.94994i	−0.222328	+	0.974972i	$0.571366\pi$
$264$		0			0
$265$	−15.0000	+	15.0000i	−0.921443	+	0.921443i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	18.9737			1.15684			0.578422	−	0.815737i	$-0.303669\pi$
$269$	18.9737			1.15684			0.578422	+	0.815737i	$0.303669\pi$
$270$		0			0
$271$	19.0000			1.15417			0.577084	−	0.816685i	$-0.304191\pi$
$271$	19.0000			1.15417			0.577084	+	0.816685i	$0.304191\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			15.8114i			0.953463i
$276$		0			0
$277$		−	20.0000i		−	1.20168i	−0.799368	−	0.600842i	$-0.794832\pi$
$277$		−	20.0000i		−	1.20168i	0.799368	−	0.600842i	$-0.205168\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−18.9737			−1.13187			−0.565937	−	0.824448i	$-0.691485\pi$
$281$	−18.9737			−1.13187			−0.565937	+	0.824448i	$0.691485\pi$
$282$		0			0
$283$			10.0000i			0.594438i	0.954809	+	0.297219i	$0.0960592\pi$
$283$			10.0000i			0.594438i	−0.954809	+	0.297219i	$0.903941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			3.16228i			0.186663i
$288$		0			0
$289$	−23.0000			−1.35294
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.48683i			0.554227i	0.960837	+	0.277113i	$0.0893778\pi$
$293$			9.48683i			0.554227i	−0.960837	+	0.277113i	$0.910622\pi$
$294$		0			0
$295$	−10.0000	−	10.0000i	−0.582223	−	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.48683			0.548638
$300$		0			0
$301$	10.0000			0.576390
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.58114	−	1.58114i	−0.0905357	−	0.0905357i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.9737			1.07590			0.537949	−	0.842977i	$-0.319199\pi$
$311$	18.9737			1.07590			0.537949	+	0.842977i	$0.319199\pi$
$312$		0			0
$313$		−	7.00000i		−	0.395663i	−0.980236	−	0.197832i	$-0.936610\pi$
$313$		−	7.00000i		−	0.395663i	0.980236	−	0.197832i	$-0.0633900\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.2982i			1.42089i	0.703753	+	0.710445i	$0.251506\pi$
$317$			25.2982i			1.42089i	−0.703753	+	0.710445i	$0.748494\pi$
$318$		0			0
$319$	30.0000			1.67968
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	18.9737i		−	1.05572i
$324$		0			0
$325$	15.0000			0.832050
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6.32456			0.348684
$330$		0			0
$331$	19.0000			1.04433			0.522167	−	0.852843i	$-0.325124\pi$
$331$	19.0000			1.04433			0.522167	+	0.852843i	$0.325124\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−17.3925	+	17.3925i	−0.950255	+	0.950255i
$336$		0			0
$337$		−	11.0000i		−	0.599208i	−0.954064	−	0.299604i	$-0.903145\pi$
$337$		−	11.0000i		−	0.599208i	0.954064	−	0.299604i	$-0.0968546\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.32456			−0.342494
$342$		0			0
$343$			13.0000i			0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			28.4605i			1.52784i	0.645311	+	0.763920i	$0.276728\pi$
$347$			28.4605i			1.52784i	−0.645311	+	0.763920i	$0.723272\pi$
$348$		0			0
$349$	17.0000			0.909989			0.454995	−	0.890494i	$-0.349641\pi$
$349$	17.0000			0.909989			0.454995	+	0.890494i	$0.349641\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	12.6491i		−	0.673244i	−0.941640	−	0.336622i	$-0.890716\pi$
$353$		−	12.6491i		−	0.673244i	0.941640	−	0.336622i	$-0.109284\pi$
$354$		0			0
$355$	−15.0000	−	15.0000i	−0.796117	−	0.796117i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	28.4605			1.50209			0.751044	−	0.660252i	$-0.229551\pi$
$359$	28.4605			1.50209			0.751044	+	0.660252i	$0.229551\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	20.5548	−	20.5548i	1.07589	−	1.07589i
$366$		0			0
$367$			19.0000i			0.991792i	0.868382	+	0.495896i	$0.165160\pi$
$367$			19.0000i			0.991792i	−0.868382	+	0.495896i	$0.834840\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−9.48683			−0.492532
$372$		0			0
$373$			27.0000i			1.39801i	0.715118	+	0.699004i	$0.246373\pi$
$373$			27.0000i			1.39801i	−0.715118	+	0.699004i	$0.753627\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	28.4605i		−	1.46579i
$378$		0			0
$379$	−23.0000			−1.18143			−0.590715	−	0.806880i	$-0.701154\pi$
$379$	−23.0000			−1.18143			−0.590715	+	0.806880i	$0.701154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	12.6491i		−	0.646339i	−0.946341	−	0.323170i	$-0.895252\pi$
$383$		−	12.6491i		−	0.646339i	0.946341	−	0.323170i	$-0.104748\pi$
$384$		0			0
$385$	−5.00000	+	5.00000i	−0.254824	+	0.254824i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.6491			−0.641335			−0.320668	−	0.947192i	$-0.603907\pi$
$389$	−12.6491			−0.641335			−0.320668	+	0.947192i	$0.603907\pi$
$390$		0			0
$391$	−20.0000			−1.01144
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.74342	+	4.74342i	0.238667	+	0.238667i
$396$		0			0
$397$		−	20.0000i		−	1.00377i	−0.864934	−	0.501886i	$-0.832640\pi$
$397$		−	20.0000i		−	1.00377i	0.864934	−	0.501886i	$-0.167360\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.1359			−1.10542			−0.552708	−	0.833375i	$-0.686406\pi$
$401$	−22.1359			−1.10542			−0.552708	+	0.833375i	$0.686406\pi$
$402$		0			0
$403$			6.00000i			0.298881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3.16228i			0.156748i
$408$		0			0
$409$	7.00000			0.346128			0.173064	−	0.984911i	$-0.444633\pi$
$409$	7.00000			0.346128			0.173064	+	0.984911i	$0.444633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	6.32456i		−	0.311211i
$414$		0			0
$415$	25.0000	−	25.0000i	1.22720	−	1.22720i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.6491			−0.617949			−0.308975	−	0.951070i	$-0.599986\pi$
$419$	−12.6491			−0.617949			−0.308975	+	0.951070i	$0.599986\pi$
$420$		0			0
$421$	−29.0000			−1.41337			−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000			−1.41337			−0.706687	+	0.707527i	$0.749811\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−31.6228			−1.53393
$426$		0			0
$427$		−	1.00000i		−	0.0483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	22.1359			1.06625			0.533125	−	0.846036i	$-0.321017\pi$
$431$	22.1359			1.06625			0.533125	+	0.846036i	$0.321017\pi$
$432$		0			0
$433$			20.0000i			0.961139i	0.876957	+	0.480569i	$0.159570\pi$
$433$			20.0000i			0.961139i	−0.876957	+	0.480569i	$0.840430\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	9.48683i		−	0.453817i
$438$		0			0
$439$	−6.00000			−0.286364			−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000			−0.286364			−0.143182	+	0.989696i	$0.545733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			18.9737i			0.901466i	0.892659	+	0.450733i	$0.148837\pi$
$443$			18.9737i			0.901466i	−0.892659	+	0.450733i	$0.851163\pi$
$444$		0			0
$445$	−20.0000	−	20.0000i	−0.948091	−	0.948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−37.9473			−1.79085			−0.895423	−	0.445217i	$-0.853127\pi$
$449$	−37.9473			−1.79085			−0.895423	+	0.445217i	$0.853127\pi$
$450$		0			0
$451$	10.0000			0.470882
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.74342	+	4.74342i	0.222375	+	0.222375i
$456$		0			0
$457$		−	20.0000i		−	0.935561i	−0.883845	−	0.467780i	$-0.845054\pi$
$457$		−	20.0000i		−	0.935561i	0.883845	−	0.467780i	$-0.154946\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		−	3.00000i		−	0.139422i	−0.997567	−	0.0697109i	$-0.977792\pi$
$463$		−	3.00000i		−	0.139422i	0.997567	−	0.0697109i	$-0.0222077\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	37.9473i		−	1.75599i	−0.478667	−	0.877997i	$-0.658880\pi$
$467$		−	37.9473i		−	1.75599i	0.478667	−	0.877997i	$-0.341120\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	31.6228i		−	1.45402i
$474$		0			0
$475$		−	15.0000i		−	0.688247i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−22.1359			−1.01142			−0.505709	−	0.862704i	$-0.668769\pi$
$479$	−22.1359			−1.01142			−0.505709	+	0.862704i	$0.668769\pi$
$480$		0			0
$481$	3.00000			0.136788
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.58114	−	1.58114i	0.0717958	−	0.0717958i
$486$		0			0
$487$			21.0000i			0.951601i	0.879553	+	0.475800i	$0.157842\pi$
$487$			21.0000i			0.951601i	−0.879553	+	0.475800i	$0.842158\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−9.48683			−0.428135			−0.214067	−	0.976819i	$-0.568671\pi$
$491$	−9.48683			−0.428135			−0.214067	+	0.976819i	$0.568671\pi$
$492$		0			0
$493$			60.0000i			2.70226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	9.48683i		−	0.425543i
$498$		0			0
$499$	24.0000			1.07439			0.537194	−	0.843459i	$-0.319484\pi$
$499$	24.0000			1.07439			0.537194	+	0.843459i	$0.319484\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			9.48683i			0.422997i	0.977378	+	0.211498i	$0.0678343\pi$
$503$			9.48683i			0.422997i	−0.977378	+	0.211498i	$0.932166\pi$
$504$		0			0
$505$	10.0000	+	10.0000i	0.444994	+	0.444994i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−28.4605			−1.26149			−0.630745	−	0.775990i	$-0.717250\pi$
$509$	−28.4605			−1.26149			−0.630745	+	0.775990i	$0.717250\pi$
$510$		0			0
$511$	13.0000			0.575086
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26.8794	−	26.8794i	1.18445	−	1.18445i
$516$		0			0
$517$		−	20.0000i		−	0.879599i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−22.1359			−0.969793			−0.484897	−	0.874571i	$-0.661143\pi$
$521$	−22.1359			−0.969793			−0.484897	+	0.874571i	$0.661143\pi$
$522$		0			0
$523$			7.00000i			0.306089i	0.988219	+	0.153044i	$0.0489077\pi$
$523$			7.00000i			0.306089i	−0.988219	+	0.153044i	$0.951092\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.6491i		−	0.551004i
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	9.48683i		−	0.410920i
$534$		0			0
$535$	15.0000	−	15.0000i	0.648507	−	0.648507i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	18.9737			0.817254
$540$		0			0
$541$	19.0000			0.816874			0.408437	−	0.912787i	$-0.366074\pi$
$541$	19.0000			0.816874			0.408437	+	0.912787i	$0.366074\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−25.2982	−	25.2982i	−1.08366	−	1.08366i
$546$		0			0
$547$		−	11.0000i		−	0.470326i	−0.971956	−	0.235163i	$-0.924438\pi$
$547$		−	11.0000i		−	0.470326i	0.971956	−	0.235163i	$-0.0755624\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−28.4605			−1.21246
$552$		0			0
$553$			3.00000i			0.127573i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			18.9737i			0.803940i	0.915653	+	0.401970i	$0.131674\pi$
$557$			18.9737i			0.803940i	−0.915653	+	0.401970i	$0.868326\pi$
$558$		0			0
$559$	−30.0000			−1.26886
$560$		0			0
$561$		0			0
$562$		0			0
$563$			25.2982i			1.06619i	0.846054	+	0.533096i	$0.178972\pi$
$563$			25.2982i			1.06619i	−0.846054	+	0.533096i	$0.821028\pi$
$564$		0			0
$565$	15.0000	−	15.0000i	0.631055	−	0.631055i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.32456			0.265139			0.132570	−	0.991174i	$-0.457677\pi$
$569$	6.32456			0.265139			0.132570	+	0.991174i	$0.457677\pi$
$570$		0			0
$571$	−11.0000			−0.460336			−0.230168	−	0.973151i	$-0.573928\pi$
$571$	−11.0000			−0.460336			−0.230168	+	0.973151i	$0.573928\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−15.8114			−0.659380
$576$		0			0
$577$		−	41.0000i		−	1.70685i	−0.521214	−	0.853426i	$-0.674521\pi$
$577$		−	41.0000i		−	1.70685i	0.521214	−	0.853426i	$-0.325479\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	15.8114			0.655967
$582$		0			0
$583$			30.0000i			1.24247i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	9.48683i		−	0.391564i	−0.980647	−	0.195782i	$-0.937276\pi$
$587$		−	9.48683i		−	0.391564i	0.980647	−	0.195782i	$-0.0627244\pi$
$588$		0			0
$589$	6.00000			0.247226
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	28.4605i		−	1.16873i	−0.811490	−	0.584366i	$-0.801343\pi$
$593$		−	28.4605i		−	1.16873i	0.811490	−	0.584366i	$-0.198657\pi$
$594$		0			0
$595$	−10.0000	−	10.0000i	−0.409960	−	0.409960i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	28.4605			1.16286			0.581432	−	0.813595i	$-0.302493\pi$
$599$	28.4605			1.16286			0.581432	+	0.813595i	$0.302493\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.58114	−	1.58114i	−0.0642824	−	0.0642824i
$606$		0			0
$607$			11.0000i			0.446476i	0.974764	+	0.223238i	$0.0716627\pi$
$607$			11.0000i			0.446476i	−0.974764	+	0.223238i	$0.928337\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−18.9737			−0.767592
$612$		0			0
$613$			23.0000i			0.928961i	0.885583	+	0.464481i	$0.153759\pi$
$613$			23.0000i			0.928961i	−0.885583	+	0.464481i	$0.846241\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	3.16228i		−	0.127309i	−0.997972	−	0.0636543i	$-0.979725\pi$
$617$		−	3.16228i		−	0.127309i	0.997972	−	0.0636543i	$-0.0202755\pi$
$618$		0			0
$619$	−17.0000			−0.683288			−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000			−0.683288			−0.341644	+	0.939829i	$0.610984\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	12.6491i		−	0.506776i
$624$		0			0
$625$	−25.0000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−6.32456			−0.252177
$630$		0			0
$631$	−1.00000			−0.0398094			−0.0199047	−	0.999802i	$-0.506336\pi$
$631$	−1.00000			−0.0398094			−0.0199047	+	0.999802i	$0.506336\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−15.8114	+	15.8114i	−0.627456	+	0.627456i
$636$		0			0
$637$		−	18.0000i		−	0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	12.6491			0.499610			0.249805	−	0.968296i	$-0.419634\pi$
$641$	12.6491			0.499610			0.249805	+	0.968296i	$0.419634\pi$
$642$		0			0
$643$			30.0000i			1.18308i	0.806274	+	0.591542i	$0.201481\pi$
$643$			30.0000i			1.18308i	−0.806274	+	0.591542i	$0.798519\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			28.4605i			1.11890i	0.828865	+	0.559449i	$0.188987\pi$
$647$			28.4605i			1.11890i	−0.828865	+	0.559449i	$0.811013\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$			12.6491i			0.494998i	0.968888	+	0.247499i	$0.0796087\pi$
$653$			12.6491i			0.494998i	−0.968888	+	0.247499i	$0.920391\pi$
$654$		0			0
$655$	−30.0000	−	30.0000i	−1.17220	−	1.17220i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.6491			0.492739			0.246370	−	0.969176i	$-0.420762\pi$
$659$	12.6491			0.492739			0.246370	+	0.969176i	$0.420762\pi$
$660$		0			0
$661$	11.0000			0.427850			0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000			0.427850			0.213925	+	0.976850i	$0.431375\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.74342	−	4.74342i	0.183942	−	0.183942i
$666$		0			0
$667$			30.0000i			1.16160i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−3.16228			−0.122078
$672$		0			0
$673$		−	3.00000i		−	0.115642i	−0.998327	−	0.0578208i	$-0.981585\pi$
$673$		−	3.00000i		−	0.115642i	0.998327	−	0.0578208i	$-0.0184152\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	31.6228i		−	1.21536i	−0.794181	−	0.607681i	$-0.792100\pi$
$677$		−	31.6228i		−	1.21536i	0.794181	−	0.607681i	$-0.207900\pi$
$678$		0			0
$679$	1.00000			0.0383765
$680$		0			0
$681$		0			0
$682$		0			0
$683$			22.1359i			0.847008i	0.905894	+	0.423504i	$0.139200\pi$
$683$			22.1359i			0.847008i	−0.905894	+	0.423504i	$0.860800\pi$
$684$		0			0
$685$	15.0000	−	15.0000i	0.573121	−	0.573121i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	28.4605			1.08426
$690$		0			0
$691$	−2.00000			−0.0760836			−0.0380418	−	0.999276i	$-0.512112\pi$
$691$	−2.00000			−0.0760836			−0.0380418	+	0.999276i	$0.512112\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	26.8794	+	26.8794i	1.01959	+	1.01959i
$696$		0			0
$697$			20.0000i			0.757554i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.16228			0.119438			0.0597188	−	0.998215i	$-0.480980\pi$
$701$	3.16228			0.119438			0.0597188	+	0.998215i	$0.480980\pi$
$702$		0			0
$703$		−	3.00000i		−	0.113147i
$704$		0			0
$705$		0			0
$706$		0			0
$707$			6.32456i			0.237859i
$708$		0			0
$709$	−3.00000			−0.112667			−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000			−0.112667			−0.0563337	+	0.998412i	$0.517941\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	6.32456i		−	0.236856i
$714$		0			0
$715$	15.0000	−	15.0000i	0.560968	−	0.560968i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−41.1096			−1.53313			−0.766565	−	0.642167i	$-0.778036\pi$
$719$	−41.1096			−1.53313			−0.766565	+	0.642167i	$0.778036\pi$
$720$		0			0
$721$	17.0000			0.633113
$722$		0			0
$723$		0			0
$724$		0			0
$725$			47.4342i			1.76166i
$726$		0			0
$727$		−	30.0000i		−	1.11264i	−0.830969	−	0.556319i	$-0.812213\pi$
$727$		−	30.0000i		−	1.11264i	0.830969	−	0.556319i	$-0.187787\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	63.2456			2.33922
$732$		0			0
$733$			40.0000i			1.47743i	0.674016	+	0.738717i	$0.264568\pi$
$733$			40.0000i			1.47743i	−0.674016	+	0.738717i	$0.735432\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			34.7851i			1.28132i
$738$		0			0
$739$	26.0000			0.956425			0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000			0.956425			0.478213	+	0.878244i	$0.341285\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	18.9737i		−	0.696076i	−0.937480	−	0.348038i	$-0.886848\pi$
$743$		−	18.9737i		−	0.696076i	0.937480	−	0.348038i	$-0.113152\pi$
$744$		0			0
$745$	20.0000	+	20.0000i	0.732743	+	0.732743i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.48683			0.346641
$750$		0			0
$751$	−49.0000			−1.78804			−0.894018	−	0.448032i	$-0.852125\pi$
$751$	−49.0000			−1.78804			−0.894018	+	0.448032i	$0.852125\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.58114	+	1.58114i	0.0575435	+	0.0575435i
$756$		0			0
$757$		−	29.0000i		−	1.05402i	−0.849858	−	0.527011i	$-0.823312\pi$
$757$		−	29.0000i		−	1.05402i	0.849858	−	0.527011i	$-0.176688\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.48683			−0.343897			−0.171949	−	0.985106i	$-0.555006\pi$
$761$	−9.48683			−0.343897			−0.171949	+	0.985106i	$0.555006\pi$
$762$		0			0
$763$		−	16.0000i		−	0.579239i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			18.9737i			0.685099i
$768$		0			0
$769$	3.00000			0.108183			0.0540914	−	0.998536i	$-0.482774\pi$
$769$	3.00000			0.108183			0.0540914	+	0.998536i	$0.482774\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			47.4342i			1.70609i	0.521839	+	0.853044i	$0.325246\pi$
$773$			47.4342i			1.70609i	−0.521839	+	0.853044i	$0.674754\pi$
$774$		0			0
$775$		−	10.0000i		−	0.359211i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.48683			−0.339901
$780$		0			0
$781$	−30.0000			−1.07348
$782$		0			0
$783$		0			0
$784$		0			0
$785$	15.8114	−	15.8114i	0.564333	−	0.564333i
$786$		0			0
$787$			1.00000i			0.0356462i	0.999841	+	0.0178231i	$0.00567356\pi$
$787$			1.00000i			0.0356462i	−0.999841	+	0.0178231i	$0.994326\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	9.48683			0.337313
$792$		0			0
$793$			3.00000i			0.106533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			6.32456i			0.224027i	0.993707	+	0.112014i	$0.0357300\pi$
$797$			6.32456i			0.224027i	−0.993707	+	0.112014i	$0.964270\pi$
$798$		0			0
$799$	40.0000			1.41510
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	41.1096i		−	1.45073i
$804$		0			0
$805$	−5.00000	−	5.00000i	−0.176227	−	0.176227i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	18.9737			0.667079			0.333539	−	0.942736i	$-0.391757\pi$
$809$	18.9737			0.667079			0.333539	+	0.942736i	$0.391757\pi$
$810$		0			0
$811$	−2.00000			−0.0702295			−0.0351147	−	0.999383i	$-0.511180\pi$
$811$	−2.00000			−0.0702295			−0.0351147	+	0.999383i	$0.511180\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	26.8794	−	26.8794i	0.941543	−	0.941543i
$816$		0			0
$817$			30.0000i			1.04957i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	44.2719			1.54510			0.772550	−	0.634954i	$-0.218981\pi$
$821$	44.2719			1.54510			0.772550	+	0.634954i	$0.218981\pi$
$822$		0			0
$823$			3.00000i			0.104573i	0.998632	+	0.0522867i	$0.0166510\pi$
$823$			3.00000i			0.104573i	−0.998632	+	0.0522867i	$0.983349\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			44.2719i			1.53948i	0.638355	+	0.769742i	$0.279615\pi$
$827$			44.2719i			1.53948i	−0.638355	+	0.769742i	$0.720385\pi$
$828$		0			0
$829$	−3.00000			−0.104194			−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000			−0.104194			−0.0520972	+	0.998642i	$0.516591\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			37.9473i			1.31480i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	31.6228			1.09174			0.545870	−	0.837870i	$-0.316199\pi$
$839$	31.6228			1.09174			0.545870	+	0.837870i	$0.316199\pi$
$840$		0			0
$841$	61.0000			2.10345
$842$		0			0
$843$		0			0
$844$		0			0
$845$	6.32456	+	6.32456i	0.217571	+	0.217571i
$846$		0			0
$847$		−	1.00000i		−	0.0343604i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.16228			−0.108401
$852$		0			0
$853$			7.00000i			0.239675i	0.992793	+	0.119838i	$0.0382374\pi$
$853$			7.00000i			0.239675i	−0.992793	+	0.119838i	$0.961763\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			9.48683i			0.324064i	0.986786	+	0.162032i	$0.0518048\pi$
$857$			9.48683i			0.324064i	−0.986786	+	0.162032i	$0.948195\pi$
$858$		0			0
$859$	13.0000			0.443554			0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000			0.443554			0.221777	+	0.975097i	$0.428814\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−10.0000	+	10.0000i	−0.340010	+	0.340010i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	9.48683			0.321819
$870$		0			0
$871$	33.0000			1.11816
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−7.90569	−	7.90569i	−0.267261	−	0.267261i
$876$		0			0
$877$		−	49.0000i		−	1.65461i	−0.561751	−	0.827306i	$-0.689872\pi$
$877$		−	49.0000i		−	1.65461i	0.561751	−	0.827306i	$-0.310128\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.48683			−0.319620			−0.159810	−	0.987148i	$-0.551088\pi$
$881$	−9.48683			−0.319620			−0.159810	+	0.987148i	$0.551088\pi$
$882$		0			0
$883$			27.0000i			0.908622i	0.890843	+	0.454311i	$0.150115\pi$
$883$			27.0000i			0.908622i	−0.890843	+	0.454311i	$0.849885\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$	−10.0000			−0.335389
$890$		0			0
$891$		0			0
$892$		0			0
$893$			18.9737i			0.634930i
$894$		0			0
$895$	5.00000	+	5.00000i	0.167132	+	0.167132i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−18.9737			−0.632807
$900$		0			0
$901$	−60.0000			−1.99889
$902$		0			0
$903$		0			0
$904$		0			0
$905$	17.3925	+	17.3925i	0.578147	+	0.578147i
$906$		0			0
$907$			21.0000i			0.697294i	0.937254	+	0.348647i	$0.113359\pi$
$907$			21.0000i			0.697294i	−0.937254	+	0.348647i	$0.886641\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−18.9737			−0.628626			−0.314313	−	0.949319i	$-0.601774\pi$
$911$	−18.9737			−0.628626			−0.314313	+	0.949319i	$0.601774\pi$
$912$		0			0
$913$		−	50.0000i		−	1.65476i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	18.9737i		−	0.626566i
$918$		0			0
$919$	6.00000			0.197922			0.0989609	−	0.995091i	$-0.468448\pi$
$919$	6.00000			0.197922			0.0989609	+	0.995091i	$0.468448\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			28.4605i			0.936789i
$924$		0			0
$925$	−5.00000			−0.164399
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.48683			0.311253			0.155626	−	0.987816i	$-0.450260\pi$
$929$	9.48683			0.311253			0.155626	+	0.987816i	$0.450260\pi$
$930$		0			0
$931$	−18.0000			−0.589926
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−31.6228	+	31.6228i	−1.03418	+	1.03418i
$936$		0			0
$937$			11.0000i			0.359354i	0.983726	+	0.179677i	$0.0575053\pi$
$937$			11.0000i			0.359354i	−0.983726	+	0.179677i	$0.942495\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−15.8114			−0.515437			−0.257718	−	0.966220i	$-0.582971\pi$
$941$	−15.8114			−0.515437			−0.257718	+	0.966220i	$0.582971\pi$
$942$		0			0
$943$			10.0000i			0.325645i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	56.9210i		−	1.84968i	−0.380352	−	0.924842i	$-0.624197\pi$
$947$		−	56.9210i		−	1.84968i	0.380352	−	0.924842i	$-0.375803\pi$
$948$		0			0
$949$	−39.0000			−1.26599
$950$		0			0
$951$		0			0
$952$		0			0
$953$			25.2982i			0.819490i	0.912200	+	0.409745i	$0.134382\pi$
$953$			25.2982i			0.819490i	−0.912200	+	0.409745i	$0.865618\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	9.48683			0.306346
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0680	−	11.0680i	0.356291	−	0.356291i
$966$		0			0
$967$			11.0000i			0.353736i	0.984235	+	0.176868i	$0.0565966\pi$
$967$			11.0000i			0.353736i	−0.984235	+	0.176868i	$0.943403\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−18.9737			−0.608894			−0.304447	−	0.952529i	$-0.598472\pi$
$971$	−18.9737			−0.608894			−0.304447	+	0.952529i	$0.598472\pi$
$972$		0			0
$973$			17.0000i			0.544995i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			28.4605i			0.910532i	0.890355	+	0.455266i	$0.150456\pi$
$977$			28.4605i			0.910532i	−0.890355	+	0.455266i	$0.849544\pi$
$978$		0			0
$979$	−40.0000			−1.27841
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	3.16228i		−	0.100861i	−0.998728	−	0.0504305i	$-0.983941\pi$
$983$		−	3.16228i		−	0.100861i	0.998728	−	0.0504305i	$-0.0160593\pi$
$984$		0			0
$985$	−20.0000	+	20.0000i	−0.637253	+	0.637253i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.6228			1.00555
$990$		0			0
$991$	19.0000			0.603555			0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000			0.603555			0.301777	+	0.953378i	$0.402420\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−42.6907	−	42.6907i	−1.35339	−	1.35339i
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.2.d.c.109.4	yes	4
3.2	odd	2	inner	540.2.d.c.109.1	✓	4
4.3	odd	2		2160.2.f.l.1729.4		4
5.2	odd	4		2700.2.a.v.1.2		2
5.3	odd	4		2700.2.a.w.1.2		2
5.4	even	2	inner	540.2.d.c.109.3	yes	4
9.2	odd	6		1620.2.r.g.109.4		8
9.4	even	3		1620.2.r.g.1189.3		8
9.5	odd	6		1620.2.r.g.1189.2		8
9.7	even	3		1620.2.r.g.109.1		8
12.11	even	2		2160.2.f.l.1729.1		4
15.2	even	4		2700.2.a.v.1.1		2
15.8	even	4		2700.2.a.w.1.1		2
15.14	odd	2	inner	540.2.d.c.109.2	yes	4
20.19	odd	2		2160.2.f.l.1729.3		4
45.4	even	6		1620.2.r.g.1189.1		8
45.14	odd	6		1620.2.r.g.1189.4		8
45.29	odd	6		1620.2.r.g.109.2		8
45.34	even	6		1620.2.r.g.109.3		8
60.59	even	2		2160.2.f.l.1729.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.d.c.109.1	✓	4	3.2	odd	2	inner
540.2.d.c.109.2	yes	4	15.14	odd	2	inner
540.2.d.c.109.3	yes	4	5.4	even	2	inner
540.2.d.c.109.4	yes	4	1.1	even	1	trivial
1620.2.r.g.109.1		8	9.7	even	3
1620.2.r.g.109.2		8	45.29	odd	6
1620.2.r.g.109.3		8	45.34	even	6
1620.2.r.g.109.4		8	9.2	odd	6
1620.2.r.g.1189.1		8	45.4	even	6
1620.2.r.g.1189.2		8	9.5	odd	6
1620.2.r.g.1189.3		8	9.4	even	3
1620.2.r.g.1189.4		8	45.14	odd	6
2160.2.f.l.1729.1		4	12.11	even	2
2160.2.f.l.1729.2		4	60.59	even	2
2160.2.f.l.1729.3		4	20.19	odd	2
2160.2.f.l.1729.4		4	4.3	odd	2
2700.2.a.v.1.1		2	15.2	even	4
2700.2.a.v.1.2		2	5.2	odd	4
2700.2.a.w.1.1		2	15.8	even	4
2700.2.a.w.1.2		2	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.58114 + 1.58114i) q^{5} +1.00000i q^{7} +3.16228 q^{11} -3.00000i q^{13} +6.32456i q^{17} -3.00000 q^{19} +3.16228i q^{23} +5.00000i q^{25} +9.48683 q^{29} -2.00000 q^{31} +(-1.58114 + 1.58114i) q^{35} +1.00000i q^{37} +3.16228 q^{41} -10.0000i q^{43} -6.32456i q^{47} +6.00000 q^{49} +9.48683i q^{53} +(5.00000 + 5.00000i) q^{55} -6.32456 q^{59} -1.00000 q^{61} +(4.74342 - 4.74342i) q^{65} +11.0000i q^{67} -9.48683 q^{71} -13.0000i q^{73} +3.16228i q^{77} +3.00000 q^{79} -15.8114i q^{83} +(-10.0000 + 10.0000i) q^{85} -12.6491 q^{89} +3.00000 q^{91} +(-4.74342 - 4.74342i) q^{95} -1.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{19} - 8 q^{31} + 24 q^{49} + 20 q^{55} - 4 q^{61} + 12 q^{79} - 40 q^{85} + 12 q^{91}+O(q^{100}) \) 4 * q - 12 * q^19 - 8 * q^31 + 24 * q^49 + 20 * q^55 - 4 * q^61 + 12 * q^79 - 40 * q^85 + 12 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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