LMFDB - Embedded newform 572.2.f.a.441.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [572,2,Mod(441,572)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(572, 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("572.441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 572 = 2^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	572.f (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:	$4.56744299562$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	572.441
Dual form			572.2.f.a.441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.00000i q^{7} -3.00000 q^{9} +O(q^{10})$	$q+4.00000i q^{7} -3.00000 q^{9} -1.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000 q^{23} +5.00000 q^{25} -6.00000 q^{29} -4.00000i q^{31} +8.00000i q^{37} +12.0000i q^{41} +12.0000 q^{43} -4.00000i q^{47} -9.00000 q^{49} +6.00000 q^{53} +4.00000i q^{59} -10.0000 q^{61} -12.0000i q^{63} +4.00000i q^{67} -4.00000i q^{71} +4.00000i q^{73} +4.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} -16.0000i q^{89} +(-8.00000 - 12.0000i) q^{91} +8.00000i q^{97} +3.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} + 10 q^{25} - 12 q^{29} + 24 q^{43} - 18 q^{49} + 12 q^{53} - 20 q^{61} + 8 q^{77} - 16 q^{79} + 18 q^{81} - 16 q^{91}+O(q^{100}) $ 2 * q - 6 * q^9 - 6 * q^13 - 4 * q^17 - 8 * q^23 + 10 * q^25 - 12 * q^29 + 24 * q^43 - 18 * q^49 + 12 * q^53 - 20 * q^61 + 8 * q^77 - 16 * q^79 + 18 * q^81 - 16 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$.

$n$	$287$	$353$	$365$
$\chi(n)$	$1$	$-1$	$1$

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			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$			4.00000i			1.51186i	0.654654	+	0.755929i	$0.272814\pi$
$7$			4.00000i			1.51186i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		0			0
$9$	−3.00000			−1.00000
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$		0			0
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$13$	−3.00000	+	2.00000i	−0.832050	+	0.554700i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.00000			−1.11417			−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000			−1.11417			−0.557086	+	0.830455i	$0.688081\pi$
$30$		0			0
$31$		−	4.00000i		−	0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$		−	4.00000i		−	0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$			8.00000i			1.31519i	0.753371	+	0.657596i	$0.228427\pi$
$37$			8.00000i			1.31519i	−0.753371	+	0.657596i	$0.771573\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			12.0000i			1.87409i	0.349215	+	0.937043i	$0.386448\pi$
$41$			12.0000i			1.87409i	−0.349215	+	0.937043i	$0.613552\pi$
$42$		0			0
$43$	12.0000			1.82998			0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000			1.82998			0.914991	+	0.403473i	$0.132197\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$		0			0
$49$	−9.00000			−1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$	−10.0000			−1.28037			−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000			−1.28037			−0.640184	+	0.768221i	$0.721142\pi$
$62$		0			0
$63$		−	12.0000i		−	1.51186i
$64$		0			0
$65$		0			0
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	4.00000i		−	0.474713i	−0.971423	−	0.237356i	$-0.923719\pi$
$71$		−	4.00000i		−	0.474713i	0.971423	−	0.237356i	$-0.0762809\pi$
$72$		0			0
$73$			4.00000i			0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$			4.00000i			0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.00000			0.455842
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	9.00000			1.00000
$82$		0			0
$83$		−	12.0000i		−	1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$		−	12.0000i		−	1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	16.0000i		−	1.69600i	−0.529999	−	0.847998i	$-0.677808\pi$
$89$		−	16.0000i		−	1.69600i	0.529999	−	0.847998i	$-0.322192\pi$
$90$		0			0
$91$	−8.00000	−	12.0000i	−0.838628	−	1.25794i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		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
$99$			3.00000i			0.301511i
$100$		0			0
$101$	14.0000			1.39305			0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000			1.39305			0.696526	+	0.717532i	$0.254728\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			4.00000i			0.383131i	0.981480	+	0.191565i	$0.0613564\pi$
$109$			4.00000i			0.383131i	−0.981480	+	0.191565i	$0.938644\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.00000			−0.188144			−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000			−0.188144			−0.0940721	+	0.995565i	$0.529988\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	9.00000	−	6.00000i	0.832050	−	0.554700i
$118$		0			0
$119$		−	8.00000i		−	0.733359i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	12.0000			1.04844			0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000			1.04844			0.524222	+	0.851581i	$0.324356\pi$
$132$		0			0
$133$	−16.0000			−1.38738
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	8.00000i		−	0.683486i	−0.939793	−	0.341743i	$-0.888983\pi$
$137$		−	8.00000i		−	0.683486i	0.939793	−	0.341743i	$-0.111017\pi$
$138$		0			0
$139$	12.0000			1.01783			0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000			1.01783			0.508913	+	0.860818i	$0.330047\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	+	3.00000i	0.167248	+	0.250873i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$			20.0000i			1.63846i	0.573462	+	0.819232i	$0.305600\pi$
$149$			20.0000i			1.63846i	−0.573462	+	0.819232i	$0.694400\pi$
$150$		0			0
$151$		−	12.0000i		−	0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$		−	12.0000i		−	0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$		0			0
$153$	6.00000			0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−14.0000			−1.11732			−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000			−1.11732			−0.558661	+	0.829396i	$0.688685\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$		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
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	5.00000	−	12.0000i	0.384615	−	0.923077i
$170$		0			0
$171$		−	12.0000i		−	0.917663i
$172$		0			0
$173$	−10.0000			−0.760286			−0.380143	−	0.924928i	$-0.624125\pi$
$173$	−10.0000			−0.760286			−0.380143	+	0.924928i	$0.624125\pi$
$174$		0			0
$175$			20.0000i			1.51186i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−16.0000			−1.19590			−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000			−1.19590			−0.597948	+	0.801535i	$0.704017\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$			2.00000i			0.146254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	20.0000			1.44715			0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000			1.44715			0.723575	+	0.690246i	$0.242498\pi$
$192$		0			0
$193$		−	20.0000i		−	1.43963i	−0.694165	−	0.719816i	$-0.744226\pi$
$193$		−	20.0000i		−	1.43963i	0.694165	−	0.719816i	$-0.255774\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.0000i			1.42494i	0.701702	+	0.712470i	$0.252424\pi$
$197$			20.0000i			1.42494i	−0.701702	+	0.712470i	$0.747576\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	24.0000i		−	1.68447i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	12.0000			0.834058
$208$		0			0
$209$	4.00000			0.276686
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	16.0000			1.08615
$218$		0			0
$219$		0			0
$220$		0			0
$221$	6.00000	−	4.00000i	0.403604	−	0.269069i
$222$		0			0
$223$			28.0000i			1.87502i	0.347960	+	0.937509i	$0.386874\pi$
$223$			28.0000i			1.87502i	−0.347960	+	0.937509i	$0.613126\pi$
$224$		0			0
$225$	−15.0000			−1.00000
$226$		0			0
$227$			20.0000i			1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$			20.0000i			1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$		0			0
$229$			24.0000i			1.58596i	0.609245	+	0.792982i	$0.291473\pi$
$229$			24.0000i			1.58596i	−0.609245	+	0.792982i	$0.708527\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			20.0000i			1.29369i	0.762620	+	0.646846i	$0.223912\pi$
$239$			20.0000i			1.29369i	−0.762620	+	0.646846i	$0.776088\pi$
$240$		0			0
$241$			12.0000i			0.772988i	0.922292	+	0.386494i	$0.126314\pi$
$241$			12.0000i			0.772988i	−0.922292	+	0.386494i	$0.873686\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−8.00000	−	12.0000i	−0.509028	−	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−24.0000			−1.51487			−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000			−1.51487			−0.757433	+	0.652913i	$0.773547\pi$
$252$		0			0
$253$			4.00000i			0.251478i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	18.0000			1.12281			0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000			1.12281			0.561405	+	0.827541i	$0.310261\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$		0			0
$261$	18.0000			1.11417
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−14.0000			−0.853595			−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000			−0.853595			−0.426798	+	0.904347i	$0.640358\pi$
$270$		0			0
$271$		−	12.0000i		−	0.728948i	−0.931214	−	0.364474i	$-0.881249\pi$
$271$		−	12.0000i		−	0.728948i	0.931214	−	0.364474i	$-0.118751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	5.00000i		−	0.301511i
$276$		0			0
$277$	−2.00000			−0.120168			−0.0600842	−	0.998193i	$-0.519137\pi$
$277$	−2.00000			−0.120168			−0.0600842	+	0.998193i	$0.519137\pi$
$278$		0			0
$279$			12.0000i			0.718421i
$280$		0			0
$281$			20.0000i			1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$281$			20.0000i			1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−48.0000			−2.83335
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	4.00000i		−	0.233682i	−0.993151	−	0.116841i	$-0.962723\pi$
$293$		−	4.00000i		−	0.233682i	0.993151	−	0.116841i	$-0.0372769\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	8.00000i	0.693978	−	0.462652i
$300$		0			0
$301$			48.0000i			2.76667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			4.00000i			0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$			4.00000i			0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.00000			−0.226819			−0.113410	−	0.993548i	$-0.536177\pi$
$311$	−4.00000			−0.226819			−0.113410	+	0.993548i	$0.536177\pi$
$312$		0			0
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	32.0000i		−	1.79730i	−0.438667	−	0.898650i	$-0.644549\pi$
$317$		−	32.0000i		−	1.79730i	0.438667	−	0.898650i	$-0.355451\pi$
$318$		0			0
$319$			6.00000i			0.335936i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	8.00000i		−	0.445132i
$324$		0			0
$325$	−15.0000	+	10.0000i	−0.832050	+	0.554700i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	16.0000			0.882109
$330$		0			0
$331$			4.00000i			0.219860i	0.993939	+	0.109930i	$0.0350627\pi$
$331$			4.00000i			0.219860i	−0.993939	+	0.109930i	$0.964937\pi$
$332$		0			0
$333$		−	24.0000i		−	1.31519i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.00000			−0.216612
$342$		0			0
$343$		−	8.00000i		−	0.431959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−12.0000			−0.644194			−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000			−0.644194			−0.322097	+	0.946707i	$0.604388\pi$
$348$		0			0
$349$			20.0000i			1.07058i	0.844670	+	0.535288i	$0.179797\pi$
$349$			20.0000i			1.07058i	−0.844670	+	0.535288i	$0.820203\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	16.0000i		−	0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$		−	16.0000i		−	0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$			20.0000i			1.05556i	0.849381	+	0.527780i	$0.176975\pi$
$359$			20.0000i			1.05556i	−0.849381	+	0.527780i	$0.823025\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−12.0000			−0.626395			−0.313197	−	0.949688i	$-0.601400\pi$
$367$	−12.0000			−0.626395			−0.313197	+	0.949688i	$0.601400\pi$
$368$		0			0
$369$		−	36.0000i		−	1.87409i
$370$		0			0
$371$			24.0000i			1.24602i
$372$		0			0
$373$	2.00000			0.103556			0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000			0.103556			0.0517780	+	0.998659i	$0.483511\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	18.0000	−	12.0000i	0.927047	−	0.618031i
$378$		0			0
$379$			36.0000i			1.84920i	0.380945	+	0.924598i	$0.375599\pi$
$379$			36.0000i			1.84920i	−0.380945	+	0.924598i	$0.624401\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$			28.0000i			1.43073i	0.698749	+	0.715367i	$0.253740\pi$
$383$			28.0000i			1.43073i	−0.698749	+	0.715367i	$0.746260\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−36.0000			−1.82998
$388$		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
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			16.0000i			0.803017i	0.915855	+	0.401508i	$0.131514\pi$
$397$			16.0000i			0.803017i	−0.915855	+	0.401508i	$0.868486\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	24.0000i		−	1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$		−	24.0000i		−	1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$		0			0
$403$	8.00000	+	12.0000i	0.398508	+	0.597763i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.00000			0.396545
$408$		0			0
$409$			28.0000i			1.38451i	0.721653	+	0.692255i	$0.243383\pi$
$409$			28.0000i			1.38451i	−0.721653	+	0.692255i	$0.756617\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−16.0000			−0.787309
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−8.00000			−0.390826			−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000			−0.390826			−0.195413	+	0.980721i	$0.562605\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$			12.0000i			0.583460i
$424$		0			0
$425$	−10.0000			−0.485071
$426$		0			0
$427$		−	40.0000i		−	1.93574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			4.00000i			0.192673i	0.995349	+	0.0963366i	$0.0307125\pi$
$431$			4.00000i			0.192673i	−0.995349	+	0.0963366i	$0.969287\pi$
$432$		0			0
$433$	34.0000			1.63394			0.816968	−	0.576683i	$-0.195653\pi$
$433$	34.0000			1.63394			0.816968	+	0.576683i	$0.195653\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	16.0000i		−	0.765384i
$438$		0			0
$439$	32.0000			1.52728			0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000			1.52728			0.763638	+	0.645644i	$0.223411\pi$
$440$		0			0
$441$	27.0000			1.28571
$442$		0			0
$443$	24.0000			1.14027			0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000			1.14027			0.570137	+	0.821549i	$0.306890\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$	12.0000			0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	36.0000i		−	1.68401i	−0.539471	−	0.842004i	$-0.681376\pi$
$457$		−	36.0000i		−	1.68401i	0.539471	−	0.842004i	$-0.318624\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	20.0000i		−	0.931493i	−0.884918	−	0.465746i	$-0.845786\pi$
$461$		−	20.0000i		−	0.931493i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$		−	36.0000i		−	1.67306i	−0.547920	−	0.836531i	$-0.684580\pi$
$463$		−	36.0000i		−	1.67306i	0.547920	−	0.836531i	$-0.315420\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	8.00000			0.370196			0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000			0.370196			0.185098	+	0.982720i	$0.440740\pi$
$468$		0			0
$469$	−16.0000			−0.738811
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$			20.0000i			0.917663i
$476$		0			0
$477$	−18.0000			−0.824163
$478$		0			0
$479$		−	12.0000i		−	0.548294i	−0.961688	−	0.274147i	$-0.911605\pi$
$479$		−	12.0000i		−	0.548294i	0.961688	−	0.274147i	$-0.0883955\pi$
$480$		0			0
$481$	−16.0000	−	24.0000i	−0.729537	−	1.09431i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		−	36.0000i		−	1.63132i	−0.578535	−	0.815658i	$-0.696375\pi$
$487$		−	36.0000i		−	1.63132i	0.578535	−	0.815658i	$-0.303625\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−20.0000			−0.902587			−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000			−0.902587			−0.451294	+	0.892375i	$0.649037\pi$
$492$		0			0
$493$	12.0000			0.540453
$494$		0			0
$495$		0			0
$496$		0			0
$497$	16.0000			0.717698
$498$		0			0
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16.0000			−0.713405			−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000			−0.713405			−0.356702	+	0.934218i	$0.616099\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$			24.0000i			1.06378i	0.846813	+	0.531891i	$0.178518\pi$
$509$			24.0000i			1.06378i	−0.846813	+	0.531891i	$0.821482\pi$
$510$		0			0
$511$	−16.0000			−0.707798
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−4.00000			−0.175920
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.00000			0.262865			0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000			0.262865			0.131432	+	0.991325i	$0.458042\pi$
$522$		0			0
$523$	20.0000			0.874539			0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000			0.874539			0.437269	+	0.899331i	$0.355946\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			8.00000i			0.348485i
$528$		0			0
$529$	−7.00000			−0.304348
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$	−24.0000	−	36.0000i	−1.03956	−	1.55933i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			9.00000i			0.387657i
$540$		0			0
$541$		−	12.0000i		−	0.515920i	−0.966156	−	0.257960i	$-0.916950\pi$
$541$		−	12.0000i		−	0.515920i	0.966156	−	0.257960i	$-0.0830503\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	28.0000			1.19719			0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000			1.19719			0.598597	+	0.801050i	$0.295725\pi$
$548$		0			0
$549$	30.0000			1.28037
$550$		0			0
$551$		−	24.0000i		−	1.02243i
$552$		0			0
$553$		−	32.0000i		−	1.36078i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	4.00000i		−	0.169485i	−0.996403	−	0.0847427i	$-0.972993\pi$
$557$		−	4.00000i		−	0.169485i	0.996403	−	0.0847427i	$-0.0270068\pi$
$558$		0			0
$559$	−36.0000	+	24.0000i	−1.52264	+	1.01509i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	28.0000			1.18006			0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000			1.18006			0.590030	+	0.807382i	$0.299116\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			36.0000i			1.51186i
$568$		0			0
$569$	10.0000			0.419222			0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000			0.419222			0.209611	+	0.977785i	$0.432780\pi$
$570$		0			0
$571$	4.00000			0.167395			0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000			0.167395			0.0836974	+	0.996491i	$0.473327\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−20.0000			−0.834058
$576$		0			0
$577$		−	8.00000i		−	0.333044i	−0.986038	−	0.166522i	$-0.946746\pi$
$577$		−	8.00000i		−	0.333044i	0.986038	−	0.166522i	$-0.0532537\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	48.0000			1.99138
$582$		0			0
$583$		−	6.00000i		−	0.248495i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			4.00000i			0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$			4.00000i			0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$		0			0
$589$	16.0000			0.659269
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	4.00000i		−	0.164260i	−0.996622	−	0.0821302i	$-0.973828\pi$
$593$		−	4.00000i		−	0.164260i	0.996622	−	0.0821302i	$-0.0261723\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000			0.490307			0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000			0.490307			0.245153	+	0.969484i	$0.421162\pi$
$600$		0			0
$601$	−26.0000			−1.06056			−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000			−1.06056			−0.530281	+	0.847822i	$0.677914\pi$
$602$		0			0
$603$		−	12.0000i		−	0.488678i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−32.0000			−1.29884			−0.649420	−	0.760430i	$-0.724988\pi$
$607$	−32.0000			−1.29884			−0.649420	+	0.760430i	$0.724988\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.00000	+	12.0000i	0.323645	+	0.485468i
$612$		0			0
$613$			28.0000i			1.13091i	0.824779	+	0.565455i	$0.191299\pi$
$613$			28.0000i			1.13091i	−0.824779	+	0.565455i	$0.808701\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			24.0000i			0.966204i	0.875564	+	0.483102i	$0.160490\pi$
$617$			24.0000i			0.966204i	−0.875564	+	0.483102i	$0.839510\pi$
$618$		0			0
$619$		−	28.0000i		−	1.12542i	−0.826656	−	0.562708i	$-0.809760\pi$
$619$		−	28.0000i		−	1.12542i	0.826656	−	0.562708i	$-0.190240\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	64.0000			2.56411
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	16.0000i		−	0.637962i
$630$		0			0
$631$		−	20.0000i		−	0.796187i	−0.917345	−	0.398094i	$-0.869672\pi$
$631$		−	20.0000i		−	0.796187i	0.917345	−	0.398094i	$-0.130328\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	27.0000	−	18.0000i	1.06978	−	0.713186i
$638$		0			0
$639$			12.0000i			0.474713i
$640$		0			0
$641$	−18.0000			−0.710957			−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000			−0.710957			−0.355479	+	0.934684i	$0.615682\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.00000			−0.157256			−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000			−0.157256			−0.0786281	+	0.996904i	$0.525054\pi$
$648$		0			0
$649$	4.00000			0.157014
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.0000			0.704394			0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000			0.704394			0.352197	+	0.935926i	$0.385435\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	12.0000i		−	0.468165i
$658$		0			0
$659$	−36.0000			−1.40236			−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000			−1.40236			−0.701180	+	0.712984i	$0.747343\pi$
$660$		0			0
$661$			16.0000i			0.622328i	0.950356	+	0.311164i	$0.100719\pi$
$661$			16.0000i			0.622328i	−0.950356	+	0.311164i	$0.899281\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	24.0000			0.929284
$668$		0			0
$669$		0			0
$670$		0			0
$671$			10.0000i			0.386046i
$672$		0			0
$673$	14.0000			0.539660			0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000			0.539660			0.269830	+	0.962908i	$0.413032\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	28.0000i		−	1.07139i	−0.844411	−	0.535695i	$-0.820050\pi$
$683$		−	28.0000i		−	1.07139i	0.844411	−	0.535695i	$-0.179950\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	+	12.0000i	−0.685745	+	0.457164i
$690$		0			0
$691$		−	28.0000i		−	1.06517i	−0.846376	−	0.532585i	$-0.821221\pi$
$691$		−	28.0000i		−	1.06517i	0.846376	−	0.532585i	$-0.178779\pi$
$692$		0			0
$693$	−12.0000			−0.455842
$694$		0			0
$695$		0			0
$696$		0			0
$697$		−	24.0000i		−	0.909065i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.0000			0.377695			0.188847	−	0.982006i	$-0.439525\pi$
$701$	10.0000			0.377695			0.188847	+	0.982006i	$0.439525\pi$
$702$		0			0
$703$	−32.0000			−1.20690
$704$		0			0
$705$		0			0
$706$		0			0
$707$			56.0000i			2.10610i
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$	24.0000			0.900070
$712$		0			0
$713$			16.0000i			0.599205i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−36.0000			−1.34257			−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000			−1.34257			−0.671287	+	0.741198i	$0.734258\pi$
$720$		0			0
$721$			16.0000i			0.595871i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−30.0000			−1.11417
$726$		0			0
$727$	−12.0000			−0.445055			−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000			−0.445055			−0.222528	+	0.974926i	$0.571431\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−24.0000			−0.887672
$732$		0			0
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.00000			0.147342
$738$		0			0
$739$		−	28.0000i		−	1.03000i	−0.857191	−	0.514998i	$-0.827793\pi$
$739$		−	28.0000i		−	1.03000i	0.857191	−	0.514998i	$-0.172207\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	12.0000i		−	0.440237i	−0.975473	−	0.220119i	$-0.929356\pi$
$743$		−	12.0000i		−	0.440237i	0.975473	−	0.220119i	$-0.0706445\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			36.0000i			1.31717i
$748$		0			0
$749$			48.0000i			1.75388i
$750$		0			0
$751$	28.0000			1.02173			0.510867	−	0.859660i	$-0.329324\pi$
$751$	28.0000			1.02173			0.510867	+	0.859660i	$0.329324\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−10.0000			−0.363456			−0.181728	−	0.983349i	$-0.558169\pi$
$757$	−10.0000			−0.363456			−0.181728	+	0.983349i	$0.558169\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	20.0000i		−	0.724999i	−0.931984	−	0.362500i	$-0.881923\pi$
$761$		−	20.0000i		−	0.724999i	0.931984	−	0.362500i	$-0.118077\pi$
$762$		0			0
$763$	−16.0000			−0.579239
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−8.00000	−	12.0000i	−0.288863	−	0.433295i
$768$		0			0
$769$			4.00000i			0.144244i	0.997396	+	0.0721218i	$0.0229770\pi$
$769$			4.00000i			0.144244i	−0.997396	+	0.0721218i	$0.977023\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			24.0000i			0.863220i	0.902060	+	0.431610i	$0.142054\pi$
$773$			24.0000i			0.863220i	−0.902060	+	0.431610i	$0.857946\pi$
$774$		0			0
$775$		−	20.0000i		−	0.718421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.0000			−1.71978
$780$		0			0
$781$	−4.00000			−0.143131
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			36.0000i			1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$			36.0000i			1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	8.00000i		−	0.284447i
$792$		0			0
$793$	30.0000	−	20.0000i	1.06533	−	0.710221i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0000			0.637593			0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000			0.637593			0.318796	+	0.947823i	$0.396721\pi$
$798$		0			0
$799$			8.00000i			0.283020i
$800$		0			0
$801$			48.0000i			1.69600i
$802$		0			0
$803$	4.00000			0.141157
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	54.0000			1.89854			0.949269	−	0.314464i	$-0.101825\pi$
$809$	54.0000			1.89854			0.949269	+	0.314464i	$0.101825\pi$
$810$		0			0
$811$			4.00000i			0.140459i	0.997531	+	0.0702295i	$0.0223732\pi$
$811$			4.00000i			0.140459i	−0.997531	+	0.0702295i	$0.977627\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$			48.0000i			1.67931i
$818$		0			0
$819$	24.0000	+	36.0000i	0.838628	+	1.25794i
$820$		0			0
$821$		−	36.0000i		−	1.25641i	−0.778048	−	0.628204i	$-0.783790\pi$
$821$		−	36.0000i		−	1.25641i	0.778048	−	0.628204i	$-0.216210\pi$
$822$		0			0
$823$	4.00000			0.139431			0.0697156	−	0.997567i	$-0.477791\pi$
$823$	4.00000			0.139431			0.0697156	+	0.997567i	$0.477791\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			36.0000i			1.25184i	0.779886	+	0.625921i	$0.215277\pi$
$827$			36.0000i			1.25184i	−0.779886	+	0.625921i	$0.784723\pi$
$828$		0			0
$829$	−34.0000			−1.18087			−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000			−1.18087			−0.590434	+	0.807086i	$0.701044\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	18.0000			0.623663
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$			44.0000i			1.51905i	0.650479	+	0.759524i	$0.274568\pi$
$839$			44.0000i			1.51905i	−0.650479	+	0.759524i	$0.725432\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		−	4.00000i		−	0.137442i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	32.0000i		−	1.09695i
$852$		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
$857$	−54.0000			−1.84460			−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000			−1.84460			−0.922302	+	0.386469i	$0.873695\pi$
$858$		0			0
$859$	8.00000			0.272956			0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000			0.272956			0.136478	+	0.990643i	$0.456422\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			44.0000i			1.49778i	0.662696	+	0.748889i	$0.269412\pi$
$863$			44.0000i			1.49778i	−0.662696	+	0.748889i	$0.730588\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	−8.00000	−	12.0000i	−0.271070	−	0.406604i
$872$		0			0
$873$		−	24.0000i		−	0.812277i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	28.0000i		−	0.945493i	−0.881199	−	0.472746i	$-0.843263\pi$
$877$		−	28.0000i		−	0.945493i	0.881199	−	0.472746i	$-0.156737\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	−16.0000			−0.538443			−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000			−0.538443			−0.269221	+	0.963078i	$0.586766\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.00000			0.268614			0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000			0.268614			0.134307	+	0.990940i	$0.457119\pi$
$888$		0			0
$889$			32.0000i			1.07325i
$890$		0			0
$891$		−	9.00000i		−	0.301511i
$892$		0			0
$893$	16.0000			0.535420
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$			24.0000i			0.800445i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	40.0000			1.32818			0.664089	−	0.747653i	$-0.268820\pi$
$907$	40.0000			1.32818			0.664089	+	0.747653i	$0.268820\pi$
$908$		0			0
$909$	−42.0000			−1.39305
$910$		0			0
$911$	36.0000			1.19273			0.596367	−	0.802712i	$-0.296610\pi$
$911$	36.0000			1.19273			0.596367	+	0.802712i	$0.296610\pi$
$912$		0			0
$913$	−12.0000			−0.397142
$914$		0			0
$915$		0			0
$916$		0			0
$917$			48.0000i			1.58510i
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8.00000	+	12.0000i	0.263323	+	0.394985i
$924$		0			0
$925$			40.0000i			1.31519i
$926$		0			0
$927$	−12.0000			−0.394132
$928$		0			0
$929$		−	24.0000i		−	0.787414i	−0.919236	−	0.393707i	$-0.871192\pi$
$929$		−	24.0000i		−	0.787414i	0.919236	−	0.393707i	$-0.128808\pi$
$930$		0			0
$931$		−	36.0000i		−	1.17985i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−38.0000			−1.24141			−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000			−1.24141			−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	4.00000i		−	0.130396i	−0.997872	−	0.0651981i	$-0.979232\pi$
$941$		−	4.00000i		−	0.130396i	0.997872	−	0.0651981i	$-0.0207679\pi$
$942$		0			0
$943$		−	48.0000i		−	1.56310i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	12.0000i		−	0.389948i	−0.980808	−	0.194974i	$-0.937538\pi$
$947$		−	12.0000i		−	0.389948i	0.980808	−	0.194974i	$-0.0624622\pi$
$948$		0			0
$949$	−8.00000	−	12.0000i	−0.259691	−	0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−22.0000			−0.712650			−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000			−0.712650			−0.356325	+	0.934362i	$0.615970\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	32.0000			1.03333
$960$		0			0
$961$	15.0000			0.483871
$962$		0			0
$963$	−36.0000			−1.16008
$964$		0			0
$965$		0			0
$966$		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
$971$	40.0000			1.28366			0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000			1.28366			0.641831	+	0.766846i	$0.278175\pi$
$972$		0			0
$973$			48.0000i			1.53881i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	40.0000i		−	1.27971i	−0.768494	−	0.639857i	$-0.778994\pi$
$977$		−	40.0000i		−	1.27971i	0.768494	−	0.639857i	$-0.221006\pi$
$978$		0			0
$979$	−16.0000			−0.511362
$980$		0			0
$981$		−	12.0000i		−	0.383131i
$982$		0			0
$983$		−	36.0000i		−	1.14822i	−0.818778	−	0.574111i	$-0.805348\pi$
$983$		−	36.0000i		−	1.14822i	0.818778	−	0.574111i	$-0.194652\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−48.0000			−1.52631
$990$		0			0
$991$	36.0000			1.14358			0.571789	−	0.820401i	$-0.306250\pi$
$991$	36.0000			1.14358			0.571789	+	0.820401i	$0.306250\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.00000			−0.0633406			−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000			−0.0633406			−0.0316703	+	0.999498i	$0.510083\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	572.2.f.a.441.2	yes	2
3.2	odd	2		5148.2.e.a.1585.2		2
4.3	odd	2		2288.2.j.b.1585.1		2
13.5	odd	4		7436.2.a.a.1.1		1
13.8	odd	4		7436.2.a.b.1.1		1
13.12	even	2	inner	572.2.f.a.441.1	✓	2
39.38	odd	2		5148.2.e.a.1585.1		2
52.51	odd	2		2288.2.j.b.1585.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
572.2.f.a.441.1	✓	2	13.12	even	2	inner
572.2.f.a.441.2	yes	2	1.1	even	1	trivial
2288.2.j.b.1585.1		2	4.3	odd	2
2288.2.j.b.1585.2		2	52.51	odd	2
5148.2.e.a.1585.1		2	39.38	odd	2
5148.2.e.a.1585.2		2	3.2	odd	2
7436.2.a.a.1.1		1	13.5	odd	4
7436.2.a.b.1.1		1	13.8	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:	\( N \)	\(=\)	\( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	572.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.00000i q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q+4.00000i q^{7} -3.00000 q^{9} -1.00000i q^{11} +(-3.00000 + 2.00000i) q^{13} -2.00000 q^{17} +4.00000i q^{19} -4.00000 q^{23} +5.00000 q^{25} -6.00000 q^{29} -4.00000i q^{31} +8.00000i q^{37} +12.0000i q^{41} +12.0000 q^{43} -4.00000i q^{47} -9.00000 q^{49} +6.00000 q^{53} +4.00000i q^{59} -10.0000 q^{61} -12.0000i q^{63} +4.00000i q^{67} -4.00000i q^{71} +4.00000i q^{73} +4.00000 q^{77} -8.00000 q^{79} +9.00000 q^{81} -12.0000i q^{83} -16.0000i q^{89} +(-8.00000 - 12.0000i) q^{91} +8.00000i q^{97} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} - 6 q^{13} - 4 q^{17} - 8 q^{23} + 10 q^{25} - 12 q^{29} + 24 q^{43} - 18 q^{49} + 12 q^{53} - 20 q^{61} + 8 q^{77} - 16 q^{79} + 18 q^{81} - 16 q^{91}+O(q^{100}) \) 2 * q - 6 * q^9 - 6 * q^13 - 4 * q^17 - 8 * q^23 + 10 * q^25 - 12 * q^29 + 24 * q^43 - 18 * q^49 + 12 * q^53 - 20 * q^61 + 8 * q^77 - 16 * q^79 + 18 * q^81 - 16 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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