LMFDB - Embedded newform 540.2.d.a.109.2

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:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
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			109.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	540.109
Dual form			540.2.d.a.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +O(q^{10})$	$q+(-2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +2.00000 q^{11} -6.00000i q^{13} +1.00000i q^{17} +3.00000 q^{19} -7.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +1.00000 q^{31} +(2.00000 + 4.00000i) q^{35} -8.00000i q^{37} -10.0000 q^{41} +2.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -9.00000i q^{53} +(-4.00000 + 2.00000i) q^{55} -10.0000 q^{59} +5.00000 q^{61} +(6.00000 + 12.0000i) q^{65} +8.00000i q^{67} +6.00000 q^{71} +2.00000i q^{73} -4.00000i q^{77} -15.0000 q^{79} -13.0000i q^{83} +(-1.00000 - 2.00000i) q^{85} -2.00000 q^{89} -12.0000 q^{91} +(-6.00000 + 3.00000i) q^{95} +8.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} + 4 q^{11} + 6 q^{19} + 6 q^{25} + 12 q^{29} + 2 q^{31} + 4 q^{35} - 20 q^{41} + 6 q^{49} - 8 q^{55} - 20 q^{59} + 10 q^{61} + 12 q^{65} + 12 q^{71} - 30 q^{79} - 2 q^{85} - 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100}) $ 2 * q - 4 * q^5 + 4 * q^11 + 6 * q^19 + 6 * q^25 + 12 * q^29 + 2 * q^31 + 4 * q^35 - 20 * q^41 + 6 * q^49 - 8 * q^55 - 20 * q^59 + 10 * q^61 + 12 * q^65 + 12 * q^71 - 30 * q^79 - 2 * q^85 - 4 * q^89 - 24 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

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
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$		0			0
$7$		−	2.00000i		−	0.755929i	−0.925820	−	0.377964i	$-0.876624\pi$
$7$		−	2.00000i		−	0.755929i	0.925820	−	0.377964i	$-0.123376\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.00000			0.603023			0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000			0.603023			0.301511	+	0.953463i	$0.402509\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.00000i			0.242536i	0.992620	+	0.121268i	$0.0386960\pi$
$17$			1.00000i			0.242536i	−0.992620	+	0.121268i	$0.961304\pi$
$18$		0			0
$19$	3.00000			0.688247			0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000			0.688247			0.344124	+	0.938924i	$0.388176\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	7.00000i		−	1.45960i	−0.683660	−	0.729800i	$-0.739613\pi$
$23$		−	7.00000i		−	1.45960i	0.683660	−	0.729800i	$-0.260387\pi$
$24$		0			0
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	1.00000			0.179605			0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000			0.179605			0.0898027	+	0.995960i	$0.471376\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	+	4.00000i	0.338062	+	0.676123i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−10.0000			−1.56174			−0.780869	−	0.624695i	$-0.785223\pi$
$41$	−10.0000			−1.56174			−0.780869	+	0.624695i	$0.785223\pi$
$42$		0			0
$43$			2.00000i			0.304997i	0.988304	+	0.152499i	$0.0487319\pi$
$43$			2.00000i			0.304997i	−0.988304	+	0.152499i	$0.951268\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			8.00000i			1.16692i	0.812142	+	0.583460i	$0.198301\pi$
$47$			8.00000i			1.16692i	−0.812142	+	0.583460i	$0.801699\pi$
$48$		0			0
$49$	3.00000			0.428571
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.00000i		−	1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$		−	9.00000i		−	1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$		0			0
$55$	−4.00000	+	2.00000i	−0.539360	+	0.269680i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.0000			−1.30189			−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000			−1.30189			−0.650945	+	0.759125i	$0.725627\pi$
$60$		0			0
$61$	5.00000			0.640184			0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000			0.640184			0.320092	+	0.947386i	$0.396286\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000	+	12.0000i	0.744208	+	1.48842i
$66$		0			0
$67$			8.00000i			0.977356i	0.872464	+	0.488678i	$0.162521\pi$
$67$			8.00000i			0.977356i	−0.872464	+	0.488678i	$0.837479\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$			2.00000i			0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$			2.00000i			0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	4.00000i		−	0.455842i
$78$		0			0
$79$	−15.0000			−1.68763			−0.843816	−	0.536633i	$-0.819696\pi$
$79$	−15.0000			−1.68763			−0.843816	+	0.536633i	$0.819696\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	13.0000i		−	1.42694i	−0.700688	−	0.713468i	$-0.747124\pi$
$83$		−	13.0000i		−	1.42694i	0.700688	−	0.713468i	$-0.252876\pi$
$84$		0			0
$85$	−1.00000	−	2.00000i	−0.108465	−	0.216930i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−2.00000			−0.212000			−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000			−0.212000			−0.106000	+	0.994366i	$0.533804\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.00000	+	3.00000i	−0.615587	+	0.307794i
$96$		0			0
$97$			8.00000i			0.812277i	0.913812	+	0.406138i	$0.133125\pi$
$97$			8.00000i			0.812277i	−0.913812	+	0.406138i	$0.866875\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	16.0000			1.59206			0.796030	−	0.605257i	$-0.206930\pi$
$101$	16.0000			1.59206			0.796030	+	0.605257i	$0.206930\pi$
$102$		0			0
$103$			16.0000i			1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$			16.0000i			1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			12.0000i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			12.0000i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	11.0000			1.05361			0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000			1.05361			0.526804	+	0.849987i	$0.323390\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	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
$115$	7.00000	+	14.0000i	0.652753	+	1.30551i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000			0.183340
$120$		0			0
$121$	−7.00000			−0.636364
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$		0			0
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		−	6.00000i		−	0.520266i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			3.00000i			0.256307i	0.991754	+	0.128154i	$0.0409051\pi$
$137$			3.00000i			0.256307i	−0.991754	+	0.128154i	$0.959095\pi$
$138$		0			0
$139$	−4.00000			−0.339276			−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000			−0.339276			−0.169638	+	0.985506i	$0.554260\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$	−12.0000	+	6.00000i	−0.996546	+	0.498273i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$	4.00000			0.325515			0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000			0.325515			0.162758	+	0.986666i	$0.447961\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.00000	+	1.00000i	−0.160644	+	0.0803219i
$156$		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
$161$	−14.0000			−1.10335
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.0000i			1.16073i	0.814355	+	0.580367i	$0.197091\pi$
$167$			15.0000i			1.16073i	−0.814355	+	0.580367i	$0.802909\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$		0			0
$171$		0			0
$172$		0			0
$173$			1.00000i			0.0760286i	0.999277	+	0.0380143i	$0.0121032\pi$
$173$			1.00000i			0.0760286i	−0.999277	+	0.0380143i	$0.987897\pi$
$174$		0			0
$175$	−8.00000	−	6.00000i	−0.604743	−	0.453557i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	5.00000			0.371647			0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000			0.371647			0.185824	+	0.982583i	$0.440505\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.00000	+	16.0000i	0.588172	+	1.17634i
$186$		0			0
$187$			2.00000i			0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$			2.00000i			0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$			2.00000i			0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	13.0000i		−	0.926212i	−0.886303	−	0.463106i	$-0.846735\pi$
$197$		−	13.0000i		−	0.926212i	0.886303	−	0.463106i	$-0.153265\pi$
$198$		0			0
$199$	24.0000			1.70131			0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000			1.70131			0.850657	+	0.525720i	$0.176204\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	12.0000i		−	0.842235i
$204$		0			0
$205$	20.0000	−	10.0000i	1.39686	−	0.698430i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	6.00000			0.415029
$210$		0			0
$211$	3.00000			0.206529			0.103264	−	0.994654i	$-0.467071\pi$
$211$	3.00000			0.206529			0.103264	+	0.994654i	$0.467071\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2.00000	−	4.00000i	−0.136399	−	0.272798i
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000			0.403604
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			1.00000i			0.0663723i	0.999449	+	0.0331862i	$0.0105654\pi$
$227$			1.00000i			0.0663723i	−0.999449	+	0.0331862i	$0.989435\pi$
$228$		0			0
$229$	13.0000			0.859064			0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000			0.859064			0.429532	+	0.903052i	$0.358679\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	2.00000i		−	0.131024i	−0.997852	−	0.0655122i	$-0.979132\pi$
$233$		−	2.00000i		−	0.131024i	0.997852	−	0.0655122i	$-0.0208681\pi$
$234$		0			0
$235$	−8.00000	−	16.0000i	−0.521862	−	1.04372i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	28.0000			1.81117			0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000			1.81117			0.905585	+	0.424165i	$0.139432\pi$
$240$		0			0
$241$	7.00000			0.450910			0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000			0.450910			0.225455	+	0.974254i	$0.427613\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	+	3.00000i	−0.383326	+	0.191663i
$246$		0			0
$247$		−	18.0000i		−	1.14531i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		−	14.0000i		−	0.880172i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	9.00000i		−	0.561405i	−0.959795	−	0.280702i	$-0.909433\pi$
$257$		−	9.00000i		−	0.561405i	0.959795	−	0.280702i	$-0.0905674\pi$
$258$		0			0
$259$	−16.0000			−0.994192
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	16.0000i		−	0.986602i	−0.869859	−	0.493301i	$-0.835790\pi$
$263$		−	16.0000i		−	0.986602i	0.869859	−	0.493301i	$-0.164210\pi$
$264$		0			0
$265$	9.00000	+	18.0000i	0.552866	+	1.10573i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$	−11.0000			−0.668202			−0.334101	−	0.942537i	$-0.608433\pi$
$271$	−11.0000			−0.668202			−0.334101	+	0.942537i	$0.608433\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	6.00000	−	8.00000i	0.361814	−	0.482418i
$276$		0			0
$277$		−	26.0000i		−	1.56219i	−0.624413	−	0.781094i	$-0.714662\pi$
$277$		−	26.0000i		−	1.56219i	0.624413	−	0.781094i	$-0.285338\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	26.0000i		−	1.54554i	−0.634686	−	0.772770i	$-0.718871\pi$
$283$		−	26.0000i		−	1.54554i	0.634686	−	0.772770i	$-0.281129\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			20.0000i			1.18056i
$288$		0			0
$289$	16.0000			0.941176
$290$		0			0
$291$		0			0
$292$		0			0
$293$			33.0000i			1.92788i	0.266119	+	0.963940i	$0.414259\pi$
$293$			33.0000i			1.92788i	−0.266119	+	0.963940i	$0.585741\pi$
$294$		0			0
$295$	20.0000	−	10.0000i	1.16445	−	0.582223i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−42.0000			−2.42892
$300$		0			0
$301$	4.00000			0.230556
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10.0000	+	5.00000i	−0.572598	+	0.286299i
$306$		0			0
$307$		−	12.0000i		−	0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$		−	12.0000i		−	0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$			20.0000i			1.13047i	0.824931	+	0.565233i	$0.191214\pi$
$313$			20.0000i			1.13047i	−0.824931	+	0.565233i	$0.808786\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11.0000i		−	0.617822i	−0.951091	−	0.308911i	$-0.900036\pi$
$317$		−	11.0000i		−	0.617822i	0.951091	−	0.308911i	$-0.0999645\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		0			0
$322$		0			0
$323$			3.00000i			0.166924i
$324$		0			0
$325$	−24.0000	−	18.0000i	−1.33128	−	0.998460i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$	28.0000			1.53902			0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000			1.53902			0.769510	+	0.638635i	$0.220501\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−8.00000	−	16.0000i	−0.437087	−	0.874173i
$336$		0			0
$337$			22.0000i			1.19842i	0.800593	+	0.599208i	$0.204518\pi$
$337$			22.0000i			1.19842i	−0.800593	+	0.599208i	$0.795482\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.00000			0.108306
$342$		0			0
$343$		−	20.0000i		−	1.07990i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			24.0000i			1.28839i	0.764862	+	0.644194i	$0.222807\pi$
$347$			24.0000i			1.28839i	−0.764862	+	0.644194i	$0.777193\pi$
$348$		0			0
$349$	−25.0000			−1.33822			−0.669110	−	0.743164i	$-0.733324\pi$
$349$	−25.0000			−1.33822			−0.669110	+	0.743164i	$0.733324\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.00000i		−	0.106449i	−0.998583	−	0.0532246i	$-0.983050\pi$
$353$		−	2.00000i		−	0.106449i	0.998583	−	0.0532246i	$-0.0169499\pi$
$354$		0			0
$355$	−12.0000	+	6.00000i	−0.636894	+	0.318447i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.0000			−0.633336			−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000			−0.633336			−0.316668	+	0.948536i	$0.602564\pi$
$360$		0			0
$361$	−10.0000			−0.526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.00000	−	4.00000i	−0.104685	−	0.209370i
$366$		0			0
$367$		−	2.00000i		−	0.104399i	−0.998637	−	0.0521996i	$-0.983377\pi$
$367$		−	2.00000i		−	0.104399i	0.998637	−	0.0521996i	$-0.0166232\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$		−	12.0000i		−	0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$		−	12.0000i		−	0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	36.0000i		−	1.85409i
$378$		0			0
$379$	25.0000			1.28416			0.642082	−	0.766636i	$-0.278071\pi$
$379$	25.0000			1.28416			0.642082	+	0.766636i	$0.278071\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			25.0000i			1.27744i	0.769439	+	0.638720i	$0.220536\pi$
$383$			25.0000i			1.27744i	−0.769439	+	0.638720i	$0.779464\pi$
$384$		0			0
$385$	4.00000	+	8.00000i	0.203859	+	0.407718i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.0000			0.507020			0.253510	−	0.967333i	$-0.418415\pi$
$389$	10.0000			0.507020			0.253510	+	0.967333i	$0.418415\pi$
$390$		0			0
$391$	7.00000			0.354005
$392$		0			0
$393$		0			0
$394$		0			0
$395$	30.0000	−	15.0000i	1.50946	−	0.754732i
$396$		0			0
$397$			22.0000i			1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$			22.0000i			1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	28.0000			1.39825			0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000			1.39825			0.699127	+	0.714998i	$0.253572\pi$
$402$		0			0
$403$		−	6.00000i		−	0.298881i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16.0000i		−	0.793091i
$408$		0			0
$409$	−5.00000			−0.247234			−0.123617	−	0.992330i	$-0.539449\pi$
$409$	−5.00000			−0.247234			−0.123617	+	0.992330i	$0.539449\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			20.0000i			0.984136i
$414$		0			0
$415$	13.0000	+	26.0000i	0.638145	+	1.27629i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.00000			−0.0977064			−0.0488532	−	0.998806i	$-0.515557\pi$
$419$	−2.00000			−0.0977064			−0.0488532	+	0.998806i	$0.515557\pi$
$420$		0			0
$421$	−5.00000			−0.243685			−0.121843	−	0.992549i	$-0.538880\pi$
$421$	−5.00000			−0.243685			−0.121843	+	0.992549i	$0.538880\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	4.00000	+	3.00000i	0.194029	+	0.145521i
$426$		0			0
$427$		−	10.0000i		−	0.483934i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−22.0000			−1.05970			−0.529851	−	0.848091i	$-0.677752\pi$
$431$	−22.0000			−1.05970			−0.529851	+	0.848091i	$0.677752\pi$
$432$		0			0
$433$			38.0000i			1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$			38.0000i			1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	21.0000i		−	1.00457i
$438$		0			0
$439$	−33.0000			−1.57500			−0.787502	−	0.616312i	$-0.788626\pi$
$439$	−33.0000			−1.57500			−0.787502	+	0.616312i	$0.788626\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			15.0000i			0.712672i	0.934358	+	0.356336i	$0.115974\pi$
$443$			15.0000i			0.712672i	−0.934358	+	0.356336i	$0.884026\pi$
$444$		0			0
$445$	4.00000	−	2.00000i	0.189618	−	0.0948091i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6.00000			0.283158			0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000			0.283158			0.141579	+	0.989927i	$0.454782\pi$
$450$		0			0
$451$	−20.0000			−0.941763
$452$		0			0
$453$		0			0
$454$		0			0
$455$	24.0000	−	12.0000i	1.12514	−	0.562569i
$456$		0			0
$457$			28.0000i			1.30978i	0.755722	+	0.654892i	$0.227286\pi$
$457$			28.0000i			1.30978i	−0.755722	+	0.654892i	$0.772714\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−24.0000			−1.11779			−0.558896	−	0.829238i	$-0.688775\pi$
$461$	−24.0000			−1.11779			−0.558896	+	0.829238i	$0.688775\pi$
$462$		0			0
$463$			24.0000i			1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$			24.0000i			1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			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
$469$	16.0000			0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$			4.00000i			0.183920i
$474$		0			0
$475$	9.00000	−	12.0000i	0.412948	−	0.550598i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.00000			−0.0913823			−0.0456912	−	0.998956i	$-0.514549\pi$
$479$	−2.00000			−0.0913823			−0.0456912	+	0.998956i	$0.514549\pi$
$480$		0			0
$481$	−48.0000			−2.18861
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−8.00000	−	16.0000i	−0.363261	−	0.726523i
$486$		0			0
$487$		−	18.0000i		−	0.815658i	−0.913058	−	0.407829i	$-0.866286\pi$
$487$		−	18.0000i		−	0.815658i	0.913058	−	0.407829i	$-0.133714\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.0000			−0.541552			−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000			−0.541552			−0.270776	+	0.962642i	$0.587280\pi$
$492$		0			0
$493$			6.00000i			0.270226i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	12.0000i		−	0.538274i
$498$		0			0
$499$	−9.00000			−0.402895			−0.201448	−	0.979499i	$-0.564565\pi$
$499$	−9.00000			−0.402895			−0.201448	+	0.979499i	$0.564565\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			21.0000i			0.936344i	0.883637	+	0.468172i	$0.155087\pi$
$503$			21.0000i			0.936344i	−0.883637	+	0.468172i	$0.844913\pi$
$504$		0			0
$505$	−32.0000	+	16.0000i	−1.42398	+	0.711991i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	24.0000			1.06378			0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000			1.06378			0.531891	+	0.846813i	$0.321482\pi$
$510$		0			0
$511$	4.00000			0.176950
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−16.0000	−	32.0000i	−0.705044	−	1.41009i
$516$		0			0
$517$			16.0000i			0.703679i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−2.00000			−0.0876216			−0.0438108	−	0.999040i	$-0.513950\pi$
$521$	−2.00000			−0.0876216			−0.0438108	+	0.999040i	$0.513950\pi$
$522$		0			0
$523$			22.0000i			0.961993i	0.876723	+	0.480996i	$0.159725\pi$
$523$			22.0000i			0.961993i	−0.876723	+	0.480996i	$0.840275\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.00000i			0.0435607i
$528$		0			0
$529$	−26.0000			−1.13043
$530$		0			0
$531$		0			0
$532$		0			0
$533$			60.0000i			2.59889i
$534$		0			0
$535$	−12.0000	−	24.0000i	−0.518805	−	1.03761i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	6.00000			0.258438
$540$		0			0
$541$	22.0000			0.945854			0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000			0.945854			0.472927	+	0.881102i	$0.343197\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−22.0000	+	11.0000i	−0.942376	+	0.471188i
$546$		0			0
$547$			22.0000i			0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$			22.0000i			0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	18.0000			0.766826
$552$		0			0
$553$			30.0000i			1.27573i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	18.0000i		−	0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$		−	18.0000i		−	0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$		0			0
$559$	12.0000			0.507546
$560$		0			0
$561$		0			0
$562$		0			0
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$	18.0000	+	36.0000i	0.757266	+	1.51453i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−14.0000			−0.586911			−0.293455	−	0.955973i	$-0.594805\pi$
$569$	−14.0000			−0.586911			−0.293455	+	0.955973i	$0.594805\pi$
$570$		0			0
$571$	37.0000			1.54840			0.774201	−	0.632940i	$-0.218152\pi$
$571$	37.0000			1.54840			0.774201	+	0.632940i	$0.218152\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−28.0000	−	21.0000i	−1.16768	−	0.875761i
$576$		0			0
$577$			22.0000i			0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$			22.0000i			0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−26.0000			−1.07866
$582$		0			0
$583$		−	18.0000i		−	0.745484i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			15.0000i			0.619116i	0.950881	+	0.309558i	$0.100181\pi$
$587$			15.0000i			0.619116i	−0.950881	+	0.309558i	$0.899819\pi$
$588$		0			0
$589$	3.00000			0.123613
$590$		0			0
$591$		0			0
$592$		0			0
$593$			9.00000i			0.369586i	0.982777	+	0.184793i	$0.0591614\pi$
$593$			9.00000i			0.369586i	−0.982777	+	0.184793i	$0.940839\pi$
$594$		0			0
$595$	−4.00000	+	2.00000i	−0.163984	+	0.0819920i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−6.00000			−0.245153			−0.122577	−	0.992459i	$-0.539116\pi$
$599$	−6.00000			−0.245153			−0.122577	+	0.992459i	$0.539116\pi$
$600$		0			0
$601$	5.00000			0.203954			0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000			0.203954			0.101977	+	0.994787i	$0.467483\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	14.0000	−	7.00000i	0.569181	−	0.284590i
$606$		0			0
$607$			14.0000i			0.568242i	0.958788	+	0.284121i	$0.0917018\pi$
$607$			14.0000i			0.568242i	−0.958788	+	0.284121i	$0.908298\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	48.0000			1.94187
$612$		0			0
$613$		−	16.0000i		−	0.646234i	−0.946359	−	0.323117i	$-0.895269\pi$
$613$		−	16.0000i		−	0.646234i	0.946359	−	0.323117i	$-0.104731\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			1.00000i			0.0402585i	0.999797	+	0.0201292i	$0.00640777\pi$
$617$			1.00000i			0.0402585i	−0.999797	+	0.0201292i	$0.993592\pi$
$618$		0			0
$619$	4.00000			0.160774			0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000			0.160774			0.0803868	+	0.996764i	$0.474384\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			4.00000i			0.160257i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8.00000			0.318981
$630$		0			0
$631$	−37.0000			−1.47295			−0.736473	−	0.676467i	$-0.763510\pi$
$631$	−37.0000			−1.47295			−0.736473	+	0.676467i	$0.763510\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−16.0000	−	32.0000i	−0.634941	−	1.26988i
$636$		0			0
$637$		−	18.0000i		−	0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	32.0000			1.26392			0.631962	−	0.774999i	$-0.282250\pi$
$641$	32.0000			1.26392			0.631962	+	0.774999i	$0.282250\pi$
$642$		0			0
$643$			6.00000i			0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$			6.00000i			0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	21.0000i		−	0.825595i	−0.910823	−	0.412798i	$-0.864552\pi$
$647$		−	21.0000i		−	0.825595i	0.910823	−	0.412798i	$-0.135448\pi$
$648$		0			0
$649$	−20.0000			−0.785069
$650$		0			0
$651$		0			0
$652$		0			0
$653$			23.0000i			0.900060i	0.893014	+	0.450030i	$0.148587\pi$
$653$			23.0000i			0.900060i	−0.893014	+	0.450030i	$0.851413\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	38.0000			1.48027			0.740135	−	0.672458i	$-0.234762\pi$
$659$	38.0000			1.48027			0.740135	+	0.672458i	$0.234762\pi$
$660$		0			0
$661$	−10.0000			−0.388955			−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000			−0.388955			−0.194477	+	0.980907i	$0.562301\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	6.00000	+	12.0000i	0.232670	+	0.465340i
$666$		0			0
$667$		−	42.0000i		−	1.62625i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.0000			0.386046
$672$		0			0
$673$		−	6.00000i		−	0.231283i	−0.993291	−	0.115642i	$-0.963108\pi$
$673$		−	6.00000i		−	0.231283i	0.993291	−	0.115642i	$-0.0368924\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	38.0000i		−	1.46046i	−0.683202	−	0.730229i	$-0.739413\pi$
$677$		−	38.0000i		−	1.46046i	0.683202	−	0.730229i	$-0.260587\pi$
$678$		0			0
$679$	16.0000			0.614024
$680$		0			0
$681$		0			0
$682$		0			0
$683$			23.0000i			0.880071i	0.897980	+	0.440035i	$0.145034\pi$
$683$			23.0000i			0.880071i	−0.897980	+	0.440035i	$0.854966\pi$
$684$		0			0
$685$	−3.00000	−	6.00000i	−0.114624	−	0.229248i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−54.0000			−2.05724
$690$		0			0
$691$	−17.0000			−0.646710			−0.323355	−	0.946278i	$-0.604811\pi$
$691$	−17.0000			−0.646710			−0.323355	+	0.946278i	$0.604811\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.00000	−	4.00000i	0.303457	−	0.151729i
$696$		0			0
$697$		−	10.0000i		−	0.378777i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28.0000			−1.05755			−0.528773	−	0.848763i	$-0.677348\pi$
$701$	−28.0000			−1.05755			−0.528773	+	0.848763i	$0.677348\pi$
$702$		0			0
$703$		−	24.0000i		−	0.905177i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	32.0000i		−	1.20348i
$708$		0			0
$709$	6.00000			0.225335			0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000			0.225335			0.112667	+	0.993633i	$0.464061\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	7.00000i		−	0.262152i
$714$		0			0
$715$	12.0000	+	24.0000i	0.448775	+	0.897549i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	16.0000			0.596699			0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000			0.596699			0.298350	+	0.954457i	$0.403564\pi$
$720$		0			0
$721$	32.0000			1.19174
$722$		0			0
$723$		0			0
$724$		0			0
$725$	18.0000	−	24.0000i	0.668503	−	0.891338i
$726$		0			0
$727$		−	12.0000i		−	0.445055i	−0.974926	−	0.222528i	$-0.928569\pi$
$727$		−	12.0000i		−	0.445055i	0.974926	−	0.222528i	$-0.0714308\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.00000			−0.0739727
$732$		0			0
$733$		−	8.00000i		−	0.295487i	−0.989026	−	0.147743i	$-0.952799\pi$
$733$		−	8.00000i		−	0.295487i	0.989026	−	0.147743i	$-0.0472010\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16.0000i			0.589368i
$738$		0			0
$739$	29.0000			1.06678			0.533391	−	0.845869i	$-0.320917\pi$
$739$	29.0000			1.06678			0.533391	+	0.845869i	$0.320917\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			12.0000i			0.440237i	0.975473	+	0.220119i	$0.0706445\pi$
$743$			12.0000i			0.440237i	−0.975473	+	0.220119i	$0.929356\pi$
$744$		0			0
$745$	20.0000	−	10.0000i	0.732743	−	0.366372i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	24.0000			0.876941
$750$		0			0
$751$	11.0000			0.401396			0.200698	−	0.979653i	$-0.435679\pi$
$751$	11.0000			0.401396			0.200698	+	0.979653i	$0.435679\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000	+	4.00000i	−0.291150	+	0.145575i
$756$		0			0
$757$		−	20.0000i		−	0.726912i	−0.931611	−	0.363456i	$-0.881597\pi$
$757$		−	20.0000i		−	0.726912i	0.931611	−	0.363456i	$-0.118403\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−18.0000			−0.652499			−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000			−0.652499			−0.326250	+	0.945284i	$0.605785\pi$
$762$		0			0
$763$		−	22.0000i		−	0.796453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			60.0000i			2.16647i
$768$		0			0
$769$	15.0000			0.540914			0.270457	−	0.962732i	$-0.412825\pi$
$769$	15.0000			0.540914			0.270457	+	0.962732i	$0.412825\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			39.0000i			1.40273i	0.712801	+	0.701366i	$0.247426\pi$
$773$			39.0000i			1.40273i	−0.712801	+	0.701366i	$0.752574\pi$
$774$		0			0
$775$	3.00000	−	4.00000i	0.107763	−	0.143684i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−30.0000			−1.07486
$780$		0			0
$781$	12.0000			0.429394
$782$		0			0
$783$		0			0
$784$		0			0
$785$	10.0000	+	20.0000i	0.356915	+	0.713831i
$786$		0			0
$787$			10.0000i			0.356462i	0.983989	+	0.178231i	$0.0570374\pi$
$787$			10.0000i			0.356462i	−0.983989	+	0.178231i	$0.942963\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−36.0000			−1.28001
$792$		0			0
$793$		−	30.0000i		−	1.06533i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	41.0000i		−	1.45229i	−0.687539	−	0.726147i	$-0.741309\pi$
$797$		−	41.0000i		−	1.45229i	0.687539	−	0.726147i	$-0.258691\pi$
$798$		0			0
$799$	−8.00000			−0.283020
$800$		0			0
$801$		0			0
$802$		0			0
$803$			4.00000i			0.141157i
$804$		0			0
$805$	28.0000	−	14.0000i	0.986870	−	0.493435i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−8.00000			−0.280918			−0.140459	−	0.990086i	$-0.544858\pi$
$811$	−8.00000			−0.280918			−0.140459	+	0.990086i	$0.544858\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−4.00000	−	8.00000i	−0.140114	−	0.280228i
$816$		0			0
$817$			6.00000i			0.209913i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	4.00000			0.139601			0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000			0.139601			0.0698005	+	0.997561i	$0.477764\pi$
$822$		0			0
$823$		−	54.0000i		−	1.88232i	−0.337959	−	0.941161i	$-0.609737\pi$
$823$		−	54.0000i		−	1.88232i	0.337959	−	0.941161i	$-0.390263\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			7.00000i			0.243414i	0.992566	+	0.121707i	$0.0388368\pi$
$827$			7.00000i			0.243414i	−0.992566	+	0.121707i	$0.961163\pi$
$828$		0			0
$829$	−6.00000			−0.208389			−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000			−0.208389			−0.104194	+	0.994557i	$0.533226\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.00000i			0.103944i
$834$		0			0
$835$	−15.0000	−	30.0000i	−0.519096	−	1.03819i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4.00000			−0.138095			−0.0690477	−	0.997613i	$-0.521996\pi$
$839$	−4.00000			−0.138095			−0.0690477	+	0.997613i	$0.521996\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$	46.0000	−	23.0000i	1.58245	−	0.791224i
$846$		0			0
$847$			14.0000i			0.481046i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−56.0000			−1.91966
$852$		0			0
$853$			4.00000i			0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$			4.00000i			0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	3.00000i		−	0.102478i	−0.998686	−	0.0512390i	$-0.983683\pi$
$857$		−	3.00000i		−	0.102478i	0.998686	−	0.0512390i	$-0.0163170\pi$
$858$		0			0
$859$	−23.0000			−0.784750			−0.392375	−	0.919805i	$-0.628346\pi$
$859$	−23.0000			−0.784750			−0.392375	+	0.919805i	$0.628346\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	39.0000i		−	1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$		−	39.0000i		−	1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$		0			0
$865$	−1.00000	−	2.00000i	−0.0340010	−	0.0680020i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−30.0000			−1.01768
$870$		0			0
$871$	48.0000			1.62642
$872$		0			0
$873$		0			0
$874$		0			0
$875$	22.0000	+	4.00000i	0.743736	+	0.135225i
$876$		0			0
$877$		−	46.0000i		−	1.55331i	−0.629926	−	0.776655i	$-0.716915\pi$
$877$		−	46.0000i		−	1.55331i	0.629926	−	0.776655i	$-0.283085\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			42.0000i			1.41341i	0.707507	+	0.706706i	$0.249820\pi$
$883$			42.0000i			1.41341i	−0.707507	+	0.706706i	$0.750180\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			27.0000i			0.906571i	0.891365	+	0.453286i	$0.149748\pi$
$887$			27.0000i			0.906571i	−0.891365	+	0.453286i	$0.850252\pi$
$888$		0			0
$889$	32.0000			1.07325
$890$		0			0
$891$		0			0
$892$		0			0
$893$			24.0000i			0.803129i
$894$		0			0
$895$	−40.0000	+	20.0000i	−1.33705	+	0.668526i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.00000			0.200111
$900$		0			0
$901$	9.00000			0.299833
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.0000	+	5.00000i	−0.332411	+	0.166206i
$906$		0			0
$907$		−	24.0000i		−	0.796907i	−0.917189	−	0.398453i	$-0.869547\pi$
$907$		−	24.0000i		−	0.796907i	0.917189	−	0.398453i	$-0.130453\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	54.0000			1.78910			0.894550	−	0.446968i	$-0.147496\pi$
$911$	54.0000			1.78910			0.894550	+	0.446968i	$0.147496\pi$
$912$		0			0
$913$		−	26.0000i		−	0.860474i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	36.0000i		−	1.18495i
$924$		0			0
$925$	−32.0000	−	24.0000i	−1.05215	−	0.789115i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−6.00000			−0.196854			−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000			−0.196854			−0.0984268	+	0.995144i	$0.531381\pi$
$930$		0			0
$931$	9.00000			0.294963
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−2.00000	−	4.00000i	−0.0654070	−	0.130814i
$936$		0			0
$937$		−	28.0000i		−	0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$		−	28.0000i		−	0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	20.0000			0.651981			0.325991	−	0.945373i	$-0.394302\pi$
$941$	20.0000			0.651981			0.325991	+	0.945373i	$0.394302\pi$
$942$		0			0
$943$			70.0000i			2.27951i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.00000i		−	0.0974869i	−0.998811	−	0.0487435i	$-0.984478\pi$
$947$		−	3.00000i		−	0.0974869i	0.998811	−	0.0487435i	$-0.0155217\pi$
$948$		0			0
$949$	12.0000			0.389536
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	38.0000i		−	1.23094i	−0.788160	−	0.615470i	$-0.788966\pi$
$953$		−	38.0000i		−	1.23094i	0.788160	−	0.615470i	$-0.211034\pi$
$954$		0			0
$955$	48.0000	−	24.0000i	1.55324	−	0.776622i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.00000			0.193750
$960$		0			0
$961$	−30.0000			−0.967742
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.00000	−	4.00000i	−0.0643823	−	0.128765i
$966$		0			0
$967$			20.0000i			0.643157i	0.946883	+	0.321578i	$0.104213\pi$
$967$			20.0000i			0.643157i	−0.946883	+	0.321578i	$0.895787\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	24.0000			0.770197			0.385098	−	0.922876i	$-0.374168\pi$
$971$	24.0000			0.770197			0.385098	+	0.922876i	$0.374168\pi$
$972$		0			0
$973$			8.00000i			0.256468i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	6.00000i		−	0.191957i	−0.995383	−	0.0959785i	$-0.969402\pi$
$977$		−	6.00000i		−	0.191957i	0.995383	−	0.0959785i	$-0.0305980\pi$
$978$		0			0
$979$	−4.00000			−0.127841
$980$		0			0
$981$		0			0
$982$		0			0
$983$			37.0000i			1.18012i	0.807361	+	0.590058i	$0.200895\pi$
$983$			37.0000i			1.18012i	−0.807361	+	0.590058i	$0.799105\pi$
$984$		0			0
$985$	13.0000	+	26.0000i	0.414214	+	0.828429i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.0000			0.445174
$990$		0			0
$991$	−17.0000			−0.540023			−0.270011	−	0.962857i	$-0.587027\pi$
$991$	−17.0000			−0.540023			−0.270011	+	0.962857i	$0.587027\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−48.0000	+	24.0000i	−1.52170	+	0.760851i
$996$		0			0
$997$		−	30.0000i		−	0.950110i	−0.879956	−	0.475055i	$-0.842428\pi$
$997$		−	30.0000i		−	0.950110i	0.879956	−	0.475055i	$-0.157572\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.2.d.a.109.2	yes	2
3.2	odd	2		540.2.d.b.109.1	yes	2
4.3	odd	2		2160.2.f.a.1729.2		2
5.2	odd	4		2700.2.a.p.1.1		1
5.3	odd	4		2700.2.a.h.1.1		1
5.4	even	2	inner	540.2.d.a.109.1	✓	2
9.2	odd	6		1620.2.r.a.109.2		4
9.4	even	3		1620.2.r.f.1189.2		4
9.5	odd	6		1620.2.r.a.1189.1		4
9.7	even	3		1620.2.r.f.109.1		4
12.11	even	2		2160.2.f.h.1729.1		2
15.2	even	4		2700.2.a.o.1.1		1
15.8	even	4		2700.2.a.e.1.1		1
15.14	odd	2		540.2.d.b.109.2	yes	2
20.19	odd	2		2160.2.f.a.1729.1		2
45.4	even	6		1620.2.r.f.1189.1		4
45.14	odd	6		1620.2.r.a.1189.2		4
45.29	odd	6		1620.2.r.a.109.1		4
45.34	even	6		1620.2.r.f.109.2		4
60.59	even	2		2160.2.f.h.1729.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.d.a.109.1	✓	2	5.4	even	2	inner
540.2.d.a.109.2	yes	2	1.1	even	1	trivial
540.2.d.b.109.1	yes	2	3.2	odd	2
540.2.d.b.109.2	yes	2	15.14	odd	2
1620.2.r.a.109.1		4	45.29	odd	6
1620.2.r.a.109.2		4	9.2	odd	6
1620.2.r.a.1189.1		4	9.5	odd	6
1620.2.r.a.1189.2		4	45.14	odd	6
1620.2.r.f.109.1		4	9.7	even	3
1620.2.r.f.109.2		4	45.34	even	6
1620.2.r.f.1189.1		4	45.4	even	6
1620.2.r.f.1189.2		4	9.4	even	3
2160.2.f.a.1729.1		2	20.19	odd	2
2160.2.f.a.1729.2		2	4.3	odd	2
2160.2.f.h.1729.1		2	12.11	even	2
2160.2.f.h.1729.2		2	60.59	even	2
2700.2.a.e.1.1		1	15.8	even	4
2700.2.a.h.1.1		1	5.3	odd	4
2700.2.a.o.1.1		1	15.2	even	4
2700.2.a.p.1.1		1	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +O(q^{10})\)	\(q+(-2.00000 + 1.00000i) q^{5} -2.00000i q^{7} +2.00000 q^{11} -6.00000i q^{13} +1.00000i q^{17} +3.00000 q^{19} -7.00000i q^{23} +(3.00000 - 4.00000i) q^{25} +6.00000 q^{29} +1.00000 q^{31} +(2.00000 + 4.00000i) q^{35} -8.00000i q^{37} -10.0000 q^{41} +2.00000i q^{43} +8.00000i q^{47} +3.00000 q^{49} -9.00000i q^{53} +(-4.00000 + 2.00000i) q^{55} -10.0000 q^{59} +5.00000 q^{61} +(6.00000 + 12.0000i) q^{65} +8.00000i q^{67} +6.00000 q^{71} +2.00000i q^{73} -4.00000i q^{77} -15.0000 q^{79} -13.0000i q^{83} +(-1.00000 - 2.00000i) q^{85} -2.00000 q^{89} -12.0000 q^{91} +(-6.00000 + 3.00000i) q^{95} +8.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} + 4 q^{11} + 6 q^{19} + 6 q^{25} + 12 q^{29} + 2 q^{31} + 4 q^{35} - 20 q^{41} + 6 q^{49} - 8 q^{55} - 20 q^{59} + 10 q^{61} + 12 q^{65} + 12 q^{71} - 30 q^{79} - 2 q^{85} - 4 q^{89} - 24 q^{91} - 12 q^{95}+O(q^{100}) \) 2 * q - 4 * q^5 + 4 * q^11 + 6 * q^19 + 6 * q^25 + 12 * q^29 + 2 * q^31 + 4 * q^35 - 20 * q^41 + 6 * q^49 - 8 * q^55 - 20 * q^59 + 10 * q^61 + 12 * q^65 + 12 * q^71 - 30 * q^79 - 2 * q^85 - 4 * q^89 - 24 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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