LMFDB - Embedded newform 2550.2.f.p.1699.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2550,2,Mod(1699,2550)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2550, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2550.1699"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,-4,0,0,4,0,4,0,0,4,0,0,0,4,-8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.3618525154$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 7x^{2} + 9 $ x^4 + 7*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 510)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1699.4
Root			$1.30278i$ of defining polynomial
Character	$\chi$	$=$	2550.1699
Dual form			2550.2.f.p.1699.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{12} -0.605551i q^{13} +4.60555i q^{14} +1.00000 q^{16} +(-2.00000 + 3.60555i) q^{17} +1.00000i q^{18} -4.60555 q^{21} -2.00000 q^{23} +1.00000i q^{24} +0.605551 q^{26} -1.00000 q^{27} -4.60555 q^{28} +7.21110i q^{29} +2.00000i q^{31} +1.00000i q^{32} +(-3.60555 - 2.00000i) q^{34} -1.00000 q^{36} +11.2111 q^{37} +0.605551i q^{39} +1.39445i q^{41} -4.60555i q^{42} -2.60555i q^{43} -2.00000i q^{46} -4.00000i q^{47} -1.00000 q^{48} +14.2111 q^{49} +(2.00000 - 3.60555i) q^{51} +0.605551i q^{52} -7.21110i q^{53} -1.00000i q^{54} -4.60555i q^{56} -7.21110 q^{58} +10.6056 q^{59} +3.21110i q^{61} -2.00000 q^{62} +4.60555 q^{63} -1.00000 q^{64} -14.6056i q^{67} +(2.00000 - 3.60555i) q^{68} +2.00000 q^{69} +13.8167i q^{71} -1.00000i q^{72} -6.60555 q^{73} +11.2111i q^{74} -0.605551 q^{78} +7.21110i q^{79} +1.00000 q^{81} -1.39445 q^{82} +5.21110i q^{83} +4.60555 q^{84} +2.60555 q^{86} -7.21110i q^{87} +0.788897 q^{89} -2.78890i q^{91} +2.00000 q^{92} -2.00000i q^{93} +4.00000 q^{94} -1.00000i q^{96} -2.60555 q^{97} +14.2111i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{7} + 4 q^{9} + 4 q^{12} + 4 q^{16} - 8 q^{17} - 4 q^{21} - 8 q^{23} - 12 q^{26} - 4 q^{27} - 4 q^{28} - 4 q^{36} + 16 q^{37} - 4 q^{48} + 28 q^{49} + 8 q^{51} + 28 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^7 + 4 * q^9 + 4 * q^12 + 4 * q^16 - 8 * q^17 - 4 * q^21 - 8 * q^23 - 12 * q^26 - 4 * q^27 - 4 * q^28 - 4 * q^36 + 16 * q^37 - 4 * q^48 + 28 * q^49 + 8 * q^51 + 28 * q^59 - 8 * q^62 + 4 * q^63 - 4 * q^64 + 8 * q^68 + 8 * q^69 - 12 * q^73 + 12 * q^78 + 4 * q^81 - 20 * q^82 + 4 * q^84 - 4 * q^86 + 32 * q^89 + 8 * q^92 + 16 * q^94 + 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2550\mathbb{Z}\right)^\times$.

$n$	$751$	$851$	$1327$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	1.00000i		−	0.408248i
$7$	4.60555			1.74073			0.870367	−	0.492403i	$-0.163881\pi$
$7$	4.60555			1.74073			0.870367	+	0.492403i	$0.163881\pi$
$8$		−	1.00000i		−	0.353553i
$9$	1.00000			0.333333
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	1.00000			0.288675
$13$		−	0.605551i		−	0.167950i	−0.996468	−	0.0839749i	$-0.973238\pi$
$13$		−	0.605551i		−	0.167950i	0.996468	−	0.0839749i	$-0.0267615\pi$
$14$			4.60555i			1.23089i
$15$		0			0
$16$	1.00000			0.250000
$17$	−2.00000	+	3.60555i	−0.485071	+	0.874475i
$17$	−2.00000	+	3.60555i	−0.485071	+	0.874475i
$18$			1.00000i			0.235702i
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$	−4.60555			−1.00501
$22$		0			0
$23$	−2.00000			−0.417029			−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000			−0.417029			−0.208514	+	0.978019i	$0.566863\pi$
$24$			1.00000i			0.204124i
$25$		0			0
$26$	0.605551			0.118758
$27$	−1.00000			−0.192450
$28$	−4.60555			−0.870367
$29$			7.21110i			1.33907i	0.742781	+	0.669534i	$0.233506\pi$
$29$			7.21110i			1.33907i	−0.742781	+	0.669534i	$0.766494\pi$
$30$		0			0
$31$			2.00000i			0.359211i	0.983739	+	0.179605i	$0.0574821\pi$
$31$			2.00000i			0.359211i	−0.983739	+	0.179605i	$0.942518\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$	−3.60555	−	2.00000i	−0.618347	−	0.342997i
$35$		0			0
$36$	−1.00000			−0.166667
$37$	11.2111			1.84309			0.921547	−	0.388267i	$-0.126926\pi$
$37$	11.2111			1.84309			0.921547	+	0.388267i	$0.126926\pi$
$38$		0			0
$39$			0.605551i			0.0969658i
$40$		0			0
$41$			1.39445i			0.217776i	0.994054	+	0.108888i	$0.0347290\pi$
$41$			1.39445i			0.217776i	−0.994054	+	0.108888i	$0.965271\pi$
$42$		−	4.60555i		−	0.710652i
$43$		−	2.60555i		−	0.397343i	−0.980066	−	0.198671i	$-0.936337\pi$
$43$		−	2.60555i		−	0.397343i	0.980066	−	0.198671i	$-0.0636627\pi$
$44$		0			0
$45$		0			0
$46$		−	2.00000i		−	0.294884i
$47$		−	4.00000i		−	0.583460i	−0.956501	−	0.291730i	$-0.905769\pi$
$47$		−	4.00000i		−	0.583460i	0.956501	−	0.291730i	$-0.0942309\pi$
$48$	−1.00000			−0.144338
$49$	14.2111			2.03016
$50$		0			0
$51$	2.00000	−	3.60555i	0.280056	−	0.504878i
$52$			0.605551i			0.0839749i
$53$		−	7.21110i		−	0.990521i	−0.868744	−	0.495261i	$-0.835073\pi$
$53$		−	7.21110i		−	0.990521i	0.868744	−	0.495261i	$-0.164927\pi$
$54$		−	1.00000i		−	0.136083i
$55$		0			0
$56$		−	4.60555i		−	0.615443i
$57$		0			0
$58$	−7.21110			−0.946864
$59$	10.6056			1.38073			0.690363	−	0.723464i	$-0.257451\pi$
$59$	10.6056			1.38073			0.690363	+	0.723464i	$0.257451\pi$
$60$		0			0
$61$			3.21110i			0.411140i	0.978642	+	0.205570i	$0.0659048\pi$
$61$			3.21110i			0.411140i	−0.978642	+	0.205570i	$0.934095\pi$
$62$	−2.00000			−0.254000
$63$	4.60555			0.580245
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$		−	14.6056i		−	1.78435i	−0.451688	−	0.892176i	$-0.649178\pi$
$67$		−	14.6056i		−	1.78435i	0.451688	−	0.892176i	$-0.350822\pi$
$68$	2.00000	−	3.60555i	0.242536	−	0.437237i
$69$	2.00000			0.240772
$70$		0			0
$71$			13.8167i			1.63974i	0.572553	+	0.819868i	$0.305953\pi$
$71$			13.8167i			1.63974i	−0.572553	+	0.819868i	$0.694047\pi$
$72$		−	1.00000i		−	0.117851i
$73$	−6.60555			−0.773121			−0.386561	−	0.922264i	$-0.626337\pi$
$73$	−6.60555			−0.773121			−0.386561	+	0.922264i	$0.626337\pi$
$74$			11.2111i			1.30326i
$75$		0			0
$76$		0			0
$77$		0			0
$78$	−0.605551			−0.0685652
$79$			7.21110i			0.811312i	0.914026	+	0.405656i	$0.132957\pi$
$79$			7.21110i			0.811312i	−0.914026	+	0.405656i	$0.867043\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$	−1.39445			−0.153991
$83$			5.21110i			0.571993i	0.958231	+	0.285996i	$0.0923245\pi$
$83$			5.21110i			0.571993i	−0.958231	+	0.285996i	$0.907675\pi$
$84$	4.60555			0.502507
$85$		0			0
$86$	2.60555			0.280964
$87$		−	7.21110i		−	0.773111i
$88$		0			0
$89$	0.788897			0.0836230			0.0418115	−	0.999126i	$-0.486687\pi$
$89$	0.788897			0.0836230			0.0418115	+	0.999126i	$0.486687\pi$
$90$		0			0
$91$		−	2.78890i		−	0.292356i
$92$	2.00000			0.208514
$93$		−	2.00000i		−	0.207390i
$94$	4.00000			0.412568
$95$		0			0
$96$		−	1.00000i		−	0.102062i
$97$	−2.60555			−0.264554			−0.132277	−	0.991213i	$-0.542229\pi$
$97$	−2.60555			−0.264554			−0.132277	+	0.991213i	$0.542229\pi$
$98$			14.2111i			1.43554i
$99$		0			0
$100$		0			0
$101$	16.6056			1.65231			0.826157	−	0.563440i	$-0.190522\pi$
$101$	16.6056			1.65231			0.826157	+	0.563440i	$0.190522\pi$
$102$	3.60555	+	2.00000i	0.357003	+	0.198030i
$103$			14.4222i			1.42106i	0.703666	+	0.710531i	$0.251545\pi$
$103$			14.4222i			1.42106i	−0.703666	+	0.710531i	$0.748455\pi$
$104$	−0.605551			−0.0593792
$105$		0			0
$106$	7.21110			0.700404
$107$	1.21110			0.117082			0.0585409	−	0.998285i	$-0.481355\pi$
$107$	1.21110			0.117082			0.0585409	+	0.998285i	$0.481355\pi$
$108$	1.00000			0.0962250
$109$		−	0.788897i		−	0.0755627i	−0.999286	−	0.0377813i	$-0.987971\pi$
$109$		−	0.788897i		−	0.0755627i	0.999286	−	0.0377813i	$-0.0120290\pi$
$110$		0			0
$111$	−11.2111			−1.06411
$112$	4.60555			0.435184
$113$	−5.21110			−0.490219			−0.245110	−	0.969495i	$-0.578824\pi$
$113$	−5.21110			−0.490219			−0.245110	+	0.969495i	$0.578824\pi$
$114$		0			0
$115$		0			0
$116$		−	7.21110i		−	0.669534i
$117$		−	0.605551i		−	0.0559832i
$118$			10.6056i			0.976320i
$119$	−9.21110	+	16.6056i	−0.844380	+	1.52223i
$120$		0			0
$121$	11.0000			1.00000
$122$	−3.21110			−0.290720
$123$		−	1.39445i		−	0.125733i
$124$		−	2.00000i		−	0.179605i
$125$		0			0
$126$			4.60555i			0.410295i
$127$		−	9.21110i		−	0.817353i	−0.912679	−	0.408677i	$-0.865990\pi$
$127$		−	9.21110i		−	0.817353i	0.912679	−	0.408677i	$-0.134010\pi$
$128$		−	1.00000i		−	0.0883883i
$129$			2.60555i			0.229406i
$130$		0			0
$131$			10.4222i			0.910592i	0.890340	+	0.455296i	$0.150467\pi$
$131$			10.4222i			0.910592i	−0.890340	+	0.455296i	$0.849533\pi$
$132$		0			0
$133$		0			0
$134$	14.6056			1.26173
$135$		0			0
$136$	3.60555	+	2.00000i	0.309173	+	0.171499i
$137$			7.21110i			0.616086i	0.951372	+	0.308043i	$0.0996741\pi$
$137$			7.21110i			0.616086i	−0.951372	+	0.308043i	$0.900326\pi$
$138$			2.00000i			0.170251i
$139$			17.2111i			1.45983i	0.683540	+	0.729913i	$0.260440\pi$
$139$			17.2111i			1.45983i	−0.683540	+	0.729913i	$0.739560\pi$
$140$		0			0
$141$			4.00000i			0.336861i
$142$	−13.8167			−1.15947
$143$		0			0
$144$	1.00000			0.0833333
$145$		0			0
$146$		−	6.60555i		−	0.546679i
$147$	−14.2111			−1.17211
$148$	−11.2111			−0.921547
$149$	−23.0278			−1.88651			−0.943254	−	0.332073i	$-0.892252\pi$
$149$	−23.0278			−1.88651			−0.943254	+	0.332073i	$0.892252\pi$
$150$		0			0
$151$	6.42221			0.522632			0.261316	−	0.965253i	$-0.415844\pi$
$151$	6.42221			0.522632			0.261316	+	0.965253i	$0.415844\pi$
$152$		0			0
$153$	−2.00000	+	3.60555i	−0.161690	+	0.291492i
$154$		0			0
$155$		0			0
$156$		−	0.605551i		−	0.0484829i
$157$			20.6056i			1.64450i	0.569125	+	0.822251i	$0.307282\pi$
$157$			20.6056i			1.64450i	−0.569125	+	0.822251i	$0.692718\pi$
$158$	−7.21110			−0.573685
$159$			7.21110i			0.571878i
$160$		0			0
$161$	−9.21110			−0.725937
$162$			1.00000i			0.0785674i
$163$	20.0000			1.56652			0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000			1.56652			0.783260	+	0.621694i	$0.213555\pi$
$164$		−	1.39445i		−	0.108888i
$165$		0			0
$166$	−5.21110			−0.404460
$167$	3.21110			0.248483			0.124241	−	0.992252i	$-0.460350\pi$
$167$	3.21110			0.248483			0.124241	+	0.992252i	$0.460350\pi$
$168$			4.60555i			0.355326i
$169$	12.6333			0.971793
$170$		0			0
$171$		0			0
$172$			2.60555i			0.198671i
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$	7.21110			0.546672
$175$		0			0
$176$		0			0
$177$	−10.6056			−0.797162
$178$			0.788897i			0.0591304i
$179$	−25.0278			−1.87066			−0.935331	−	0.353773i	$-0.884898\pi$
$179$	−25.0278			−1.87066			−0.935331	+	0.353773i	$0.884898\pi$
$180$		0			0
$181$			7.21110i			0.535997i	0.963419	+	0.267999i	$0.0863622\pi$
$181$			7.21110i			0.535997i	−0.963419	+	0.267999i	$0.913638\pi$
$182$	2.78890			0.206727
$183$		−	3.21110i		−	0.237372i
$184$			2.00000i			0.147442i
$185$		0			0
$186$	2.00000			0.146647
$187$		0			0
$188$			4.00000i			0.291730i
$189$	−4.60555			−0.335005
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000			0.0721688
$193$	−10.6056			−0.763404			−0.381702	−	0.924285i	$-0.624662\pi$
$193$	−10.6056			−0.763404			−0.381702	+	0.924285i	$0.624662\pi$
$194$		−	2.60555i		−	0.187068i
$195$		0			0
$196$	−14.2111			−1.01508
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$		−	14.0000i		−	0.992434i	−0.868199	−	0.496217i	$-0.834722\pi$
$199$		−	14.0000i		−	0.992434i	0.868199	−	0.496217i	$-0.165278\pi$
$200$		0			0
$201$			14.6056i			1.03020i
$202$			16.6056i			1.16836i
$203$			33.2111i			2.33096i
$204$	−2.00000	+	3.60555i	−0.140028	+	0.252439i
$205$		0			0
$206$	−14.4222			−1.00484
$207$	−2.00000			−0.139010
$208$		−	0.605551i		−	0.0419874i
$209$		0			0
$210$		0			0
$211$			16.0000i			1.10149i	0.834675	+	0.550743i	$0.185655\pi$
$211$			16.0000i			1.10149i	−0.834675	+	0.550743i	$0.814345\pi$
$212$			7.21110i			0.495261i
$213$		−	13.8167i		−	0.946702i
$214$			1.21110i			0.0827893i
$215$		0			0
$216$			1.00000i			0.0680414i
$217$			9.21110i			0.625290i
$218$	0.788897			0.0534309
$219$	6.60555			0.446362
$220$		0			0
$221$	2.18335	+	1.21110i	0.146868	+	0.0814676i
$222$		−	11.2111i		−	0.752440i
$223$		−	20.0000i		−	1.33930i	−0.742677	−	0.669650i	$-0.766444\pi$
$223$		−	20.0000i		−	1.33930i	0.742677	−	0.669650i	$-0.233556\pi$
$224$			4.60555i			0.307721i
$225$		0			0
$226$		−	5.21110i		−	0.346637i
$227$	29.2111			1.93881			0.969404	−	0.245469i	$-0.0789419\pi$
$227$	29.2111			1.93881			0.969404	+	0.245469i	$0.0789419\pi$
$228$		0			0
$229$	−2.00000			−0.132164			−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000			−0.132164			−0.0660819	+	0.997814i	$0.521050\pi$
$230$		0			0
$231$		0			0
$232$	7.21110			0.473432
$233$	−8.00000			−0.524097			−0.262049	−	0.965055i	$-0.584398\pi$
$233$	−8.00000			−0.524097			−0.262049	+	0.965055i	$0.584398\pi$
$234$	0.605551			0.0395861
$235$		0			0
$236$	−10.6056			−0.690363
$237$		−	7.21110i		−	0.468411i
$238$	−16.6056	−	9.21110i	−1.07638	−	0.597067i
$239$	−10.7889			−0.697876			−0.348938	−	0.937146i	$-0.613458\pi$
$239$	−10.7889			−0.697876			−0.348938	+	0.937146i	$0.613458\pi$
$240$		0			0
$241$		−	14.4222i		−	0.929016i	−0.885569	−	0.464508i	$-0.846231\pi$
$241$		−	14.4222i		−	0.929016i	0.885569	−	0.464508i	$-0.153769\pi$
$242$			11.0000i			0.707107i
$243$	−1.00000			−0.0641500
$244$		−	3.21110i		−	0.205570i
$245$		0			0
$246$	1.39445			0.0889068
$247$		0			0
$248$	2.00000			0.127000
$249$		−	5.21110i		−	0.330240i
$250$		0			0
$251$	14.6056			0.921894			0.460947	−	0.887428i	$-0.347510\pi$
$251$	14.6056			0.921894			0.460947	+	0.887428i	$0.347510\pi$
$252$	−4.60555			−0.290122
$253$		0			0
$254$	9.21110			0.577956
$255$		0			0
$256$	1.00000			0.0625000
$257$			16.4222i			1.02439i	0.858869	+	0.512195i	$0.171167\pi$
$257$			16.4222i			1.02439i	−0.858869	+	0.512195i	$0.828833\pi$
$258$	−2.60555			−0.162215
$259$	51.6333			3.20834
$260$		0			0
$261$			7.21110i			0.446356i
$262$	−10.4222			−0.643886
$263$		−	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
$265$		0			0
$266$		0			0
$267$	−0.788897			−0.0482797
$268$			14.6056i			0.892176i
$269$		−	0.422205i		−	0.0257423i	−0.999917	−	0.0128711i	$-0.995903\pi$
$269$		−	0.422205i		−	0.0257423i	0.999917	−	0.0128711i	$-0.00409713\pi$
$270$		0			0
$271$	20.0000			1.21491			0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000			1.21491			0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	+	3.60555i	−0.121268	+	0.218619i
$273$			2.78890i			0.168792i
$274$	−7.21110			−0.435639
$275$		0			0
$276$	−2.00000			−0.120386
$277$	−19.2111			−1.15428			−0.577142	−	0.816644i	$-0.695832\pi$
$277$	−19.2111			−1.15428			−0.577142	+	0.816644i	$0.695832\pi$
$278$	−17.2111			−1.03225
$279$			2.00000i			0.119737i
$280$		0			0
$281$	−23.2111			−1.38466			−0.692329	−	0.721582i	$-0.743415\pi$
$281$	−23.2111			−1.38466			−0.692329	+	0.721582i	$0.743415\pi$
$282$	−4.00000			−0.238197
$283$	23.6333			1.40485			0.702427	−	0.711756i	$-0.252100\pi$
$283$	23.6333			1.40485			0.702427	+	0.711756i	$0.252100\pi$
$284$		−	13.8167i		−	0.819868i
$285$		0			0
$286$		0			0
$287$			6.42221i			0.379091i
$288$			1.00000i			0.0589256i
$289$	−9.00000	−	14.4222i	−0.529412	−	0.848365i
$290$		0			0
$291$	2.60555			0.152740
$292$	6.60555			0.386561
$293$			0.422205i			0.0246655i	0.999924	+	0.0123327i	$0.00392573\pi$
$293$			0.422205i			0.0246655i	−0.999924	+	0.0123327i	$0.996074\pi$
$294$		−	14.2111i		−	0.828808i
$295$		0			0
$296$		−	11.2111i		−	0.651632i
$297$		0			0
$298$		−	23.0278i		−	1.33396i
$299$			1.21110i			0.0700399i
$300$		0			0
$301$		−	12.0000i		−	0.691669i
$302$			6.42221i			0.369556i
$303$	−16.6056			−0.953964
$304$		0			0
$305$		0			0
$306$	−3.60555	−	2.00000i	−0.206116	−	0.114332i
$307$		−	18.6056i		−	1.06187i	−0.847411	−	0.530937i	$-0.821840\pi$
$307$		−	18.6056i		−	1.06187i	0.847411	−	0.530937i	$-0.178160\pi$
$308$		0			0
$309$		−	14.4222i		−	0.820451i
$310$		0			0
$311$		−	3.39445i		−	0.192482i	−0.995358	−	0.0962408i	$-0.969318\pi$
$311$		−	3.39445i		−	0.192482i	0.995358	−	0.0962408i	$-0.0306819\pi$
$312$	0.605551			0.0342826
$313$	−8.18335			−0.462550			−0.231275	−	0.972888i	$-0.574290\pi$
$313$	−8.18335			−0.462550			−0.231275	+	0.972888i	$0.574290\pi$
$314$	−20.6056			−1.16284
$315$		0			0
$316$		−	7.21110i		−	0.405656i
$317$	−16.4222			−0.922363			−0.461181	−	0.887306i	$-0.652574\pi$
$317$	−16.4222			−0.922363			−0.461181	+	0.887306i	$0.652574\pi$
$318$	−7.21110			−0.404379
$319$		0			0
$320$		0			0
$321$	−1.21110			−0.0675972
$322$		−	9.21110i		−	0.513315i
$323$		0			0
$324$	−1.00000			−0.0555556
$325$		0			0
$326$			20.0000i			1.10770i
$327$			0.788897i			0.0436261i
$328$	1.39445			0.0769956
$329$		−	18.4222i		−	1.01565i
$330$		0			0
$331$	11.6333			0.639424			0.319712	−	0.947515i	$-0.396414\pi$
$331$	11.6333			0.639424			0.319712	+	0.947515i	$0.396414\pi$
$332$		−	5.21110i		−	0.285996i
$333$	11.2111			0.614365
$334$			3.21110i			0.175704i
$335$		0			0
$336$	−4.60555			−0.251253
$337$	6.60555			0.359827			0.179914	−	0.983682i	$-0.442418\pi$
$337$	6.60555			0.359827			0.179914	+	0.983682i	$0.442418\pi$
$338$			12.6333i			0.687161i
$339$	5.21110			0.283028
$340$		0			0
$341$		0			0
$342$		0			0
$343$	33.2111			1.79323
$344$	−2.60555			−0.140482
$345$		0			0
$346$			6.00000i			0.322562i
$347$	15.6333			0.839240			0.419620	−	0.907700i	$-0.362163\pi$
$347$	15.6333			0.839240			0.419620	+	0.907700i	$0.362163\pi$
$348$			7.21110i			0.386556i
$349$	−11.2111			−0.600117			−0.300058	−	0.953921i	$-0.597006\pi$
$349$	−11.2111			−0.600117			−0.300058	+	0.953921i	$0.597006\pi$
$350$		0			0
$351$			0.605551i			0.0323219i
$352$		0			0
$353$		−	16.4222i		−	0.874066i	−0.899446	−	0.437033i	$-0.856029\pi$
$353$		−	16.4222i		−	0.874066i	0.899446	−	0.437033i	$-0.143971\pi$
$354$		−	10.6056i		−	0.563679i
$355$		0			0
$356$	−0.788897			−0.0418115
$357$	9.21110	−	16.6056i	0.487503	−	0.878859i
$358$		−	25.0278i		−	1.32276i
$359$	−7.63331			−0.402871			−0.201435	−	0.979502i	$-0.564561\pi$
$359$	−7.63331			−0.402871			−0.201435	+	0.979502i	$0.564561\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$	−7.21110			−0.379007
$363$	−11.0000			−0.577350
$364$			2.78890i			0.146178i
$365$		0			0
$366$	3.21110			0.167847
$367$	−33.4500			−1.74607			−0.873037	−	0.487654i	$-0.837853\pi$
$367$	−33.4500			−1.74607			−0.873037	+	0.487654i	$0.837853\pi$
$368$	−2.00000			−0.104257
$369$			1.39445i			0.0725921i
$370$		0			0
$371$		−	33.2111i		−	1.72423i
$372$			2.00000i			0.103695i
$373$			23.0278i			1.19233i	0.802861	+	0.596166i	$0.203310\pi$
$373$			23.0278i			1.19233i	−0.802861	+	0.596166i	$0.796690\pi$
$374$		0			0
$375$		0			0
$376$	−4.00000			−0.206284
$377$	4.36669			0.224896
$378$		−	4.60555i		−	0.236884i
$379$		−	37.2111i		−	1.91141i	−0.294333	−	0.955703i	$-0.595097\pi$
$379$		−	37.2111i		−	1.91141i	0.294333	−	0.955703i	$-0.404903\pi$
$380$		0			0
$381$			9.21110i			0.471899i
$382$		−	8.00000i		−	0.409316i
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$			1.00000i			0.0510310i
$385$		0			0
$386$		−	10.6056i		−	0.539808i
$387$		−	2.60555i		−	0.132448i
$388$	2.60555			0.132277
$389$	12.6056			0.639127			0.319563	−	0.947565i	$-0.396464\pi$
$389$	12.6056			0.639127			0.319563	+	0.947565i	$0.396464\pi$
$390$		0			0
$391$	4.00000	−	7.21110i	0.202289	−	0.364681i
$392$		−	14.2111i		−	0.717769i
$393$		−	10.4222i		−	0.525731i
$394$			18.0000i			0.906827i
$395$		0			0
$396$		0			0
$397$	−32.4222			−1.62722			−0.813612	−	0.581408i	$-0.802502\pi$
$397$	−32.4222			−1.62722			−0.813612	+	0.581408i	$0.802502\pi$
$398$	14.0000			0.701757
$399$		0			0
$400$		0			0
$401$			14.6056i			0.729366i	0.931132	+	0.364683i	$0.118823\pi$
$401$			14.6056i			0.729366i	−0.931132	+	0.364683i	$0.881177\pi$
$402$	−14.6056			−0.728459
$403$	1.21110			0.0603293
$404$	−16.6056			−0.826157
$405$		0			0
$406$	−33.2111			−1.64824
$407$		0			0
$408$	−3.60555	−	2.00000i	−0.178501	−	0.0990148i
$409$	13.6333			0.674124			0.337062	−	0.941483i	$-0.390567\pi$
$409$	13.6333			0.674124			0.337062	+	0.941483i	$0.390567\pi$
$410$		0			0
$411$		−	7.21110i		−	0.355697i
$412$		−	14.4222i		−	0.710531i
$413$	48.8444			2.40348
$414$		−	2.00000i		−	0.0982946i
$415$		0			0
$416$	0.605551			0.0296896
$417$		−	17.2111i		−	0.842831i
$418$		0			0
$419$		−	27.6333i		−	1.34998i	−0.737829	−	0.674988i	$-0.764149\pi$
$419$		−	27.6333i		−	1.34998i	0.737829	−	0.674988i	$-0.235851\pi$
$420$		0			0
$421$	28.4222			1.38521			0.692607	−	0.721315i	$-0.256462\pi$
$421$	28.4222			1.38521			0.692607	+	0.721315i	$0.256462\pi$
$422$	−16.0000			−0.778868
$423$		−	4.00000i		−	0.194487i
$424$	−7.21110			−0.350202
$425$		0			0
$426$	13.8167			0.669419
$427$			14.7889i			0.715685i
$428$	−1.21110			−0.0585409
$429$		0			0
$430$		0			0
$431$		−	7.02776i		−	0.338515i	−0.985572	−	0.169258i	$-0.945863\pi$
$431$		−	7.02776i		−	0.338515i	0.985572	−	0.169258i	$-0.0541370\pi$
$432$	−1.00000			−0.0481125
$433$		−	16.4222i		−	0.789201i	−0.918853	−	0.394600i	$-0.870883\pi$
$433$		−	16.4222i		−	0.789201i	0.918853	−	0.394600i	$-0.129117\pi$
$434$	−9.21110			−0.442147
$435$		0			0
$436$			0.788897i			0.0377813i
$437$		0			0
$438$			6.60555i			0.315625i
$439$			17.6333i			0.841592i	0.907155	+	0.420796i	$0.138249\pi$
$439$			17.6333i			0.841592i	−0.907155	+	0.420796i	$0.861751\pi$
$440$		0			0
$441$	14.2111			0.676719
$442$	−1.21110	+	2.18335i	−0.0576063	+	0.103851i
$443$			27.6333i			1.31290i	0.754370	+	0.656449i	$0.227942\pi$
$443$			27.6333i			1.31290i	−0.754370	+	0.656449i	$0.772058\pi$
$444$	11.2111			0.532055
$445$		0			0
$446$	20.0000			0.947027
$447$	23.0278			1.08918
$448$	−4.60555			−0.217592
$449$			25.3944i			1.19844i	0.800585	+	0.599219i	$0.204522\pi$
$449$			25.3944i			1.19844i	−0.800585	+	0.599219i	$0.795478\pi$
$450$		0			0
$451$		0			0
$452$	5.21110			0.245110
$453$	−6.42221			−0.301742
$454$			29.2111i			1.37094i
$455$		0			0
$456$		0			0
$457$		−	30.8444i		−	1.44284i	−0.692497	−	0.721420i	$-0.743490\pi$
$457$		−	30.8444i		−	1.44284i	0.692497	−	0.721420i	$-0.256510\pi$
$458$		−	2.00000i		−	0.0934539i
$459$	2.00000	−	3.60555i	0.0933520	−	0.168293i
$460$		0			0
$461$	−0.972244			−0.0452819			−0.0226409	−	0.999744i	$-0.507207\pi$
$461$	−0.972244			−0.0452819			−0.0226409	+	0.999744i	$0.507207\pi$
$462$		0			0
$463$		−	20.0000i		−	0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$		−	20.0000i		−	0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$			7.21110i			0.334767i
$465$		0			0
$466$		−	8.00000i		−	0.370593i
$467$		−	1.57779i		−	0.0730116i	−0.999333	−	0.0365058i	$-0.988377\pi$
$467$		−	1.57779i		−	0.0730116i	0.999333	−	0.0365058i	$-0.0116227\pi$
$468$			0.605551i			0.0279916i
$469$		−	67.2666i		−	3.10608i
$470$		0			0
$471$		−	20.6056i		−	0.949454i
$472$		−	10.6056i		−	0.488160i
$473$		0			0
$474$	7.21110			0.331217
$475$		0			0
$476$	9.21110	−	16.6056i	0.422190	−	0.761114i
$477$		−	7.21110i		−	0.330174i
$478$		−	10.7889i		−	0.493473i
$479$			20.6056i			0.941492i	0.882269	+	0.470746i	$0.156015\pi$
$479$			20.6056i			0.941492i	−0.882269	+	0.470746i	$0.843985\pi$
$480$		0			0
$481$		−	6.78890i		−	0.309547i
$482$	14.4222			0.656913
$483$	9.21110			0.419120
$484$	−11.0000			−0.500000
$485$		0			0
$486$		−	1.00000i		−	0.0453609i
$487$	21.4500			0.971991			0.485995	−	0.873961i	$-0.338457\pi$
$487$	21.4500			0.971991			0.485995	+	0.873961i	$0.338457\pi$
$488$	3.21110			0.145360
$489$	−20.0000			−0.904431
$490$		0			0
$491$	33.0278			1.49052			0.745261	−	0.666773i	$-0.232325\pi$
$491$	33.0278			1.49052			0.745261	+	0.666773i	$0.232325\pi$
$492$			1.39445i			0.0628666i
$493$	−26.0000	−	14.4222i	−1.17098	−	0.649543i
$494$		0			0
$495$		0			0
$496$			2.00000i			0.0898027i
$497$			63.6333i			2.85434i
$498$	5.21110			0.233515
$499$			16.0000i			0.716258i	0.933672	+	0.358129i	$0.116585\pi$
$499$			16.0000i			0.716258i	−0.933672	+	0.358129i	$0.883415\pi$
$500$		0			0
$501$	−3.21110			−0.143461
$502$			14.6056i			0.651878i
$503$	22.0000			0.980932			0.490466	−	0.871460i	$-0.336827\pi$
$503$	22.0000			0.980932			0.490466	+	0.871460i	$0.336827\pi$
$504$		−	4.60555i		−	0.205148i
$505$		0			0
$506$		0			0
$507$	−12.6333			−0.561065
$508$			9.21110i			0.408677i
$509$	−31.0278			−1.37528			−0.687641	−	0.726051i	$-0.741353\pi$
$509$	−31.0278			−1.37528			−0.687641	+	0.726051i	$0.741353\pi$
$510$		0			0
$511$	−30.4222			−1.34580
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−16.4222			−0.724352
$515$		0			0
$516$		−	2.60555i		−	0.114703i
$517$		0			0
$518$			51.6333i			2.26864i
$519$	−6.00000			−0.263371
$520$		0			0
$521$			38.6056i			1.69134i	0.533706	+	0.845670i	$0.320799\pi$
$521$			38.6056i			1.69134i	−0.533706	+	0.845670i	$0.679201\pi$
$522$	−7.21110			−0.315621
$523$			1.39445i			0.0609750i	0.999535	+	0.0304875i	$0.00970597\pi$
$523$			1.39445i			0.0609750i	−0.999535	+	0.0304875i	$0.990294\pi$
$524$		−	10.4222i		−	0.455296i
$525$		0			0
$526$	16.0000			0.697633
$527$	−7.21110	−	4.00000i	−0.314121	−	0.174243i
$528$		0			0
$529$	−19.0000			−0.826087
$530$		0			0
$531$	10.6056			0.460242
$532$		0			0
$533$	0.844410			0.0365755
$534$		−	0.788897i		−	0.0341389i
$535$		0			0
$536$	−14.6056			−0.630864
$537$	25.0278			1.08003
$538$	0.422205			0.0182026
$539$		0			0
$540$		0			0
$541$		−	16.7889i		−	0.721811i	−0.932602	−	0.360906i	$-0.882468\pi$
$541$		−	16.7889i		−	0.721811i	0.932602	−	0.360906i	$-0.117532\pi$
$542$			20.0000i			0.859074i
$543$		−	7.21110i		−	0.309458i
$544$	−3.60555	−	2.00000i	−0.154587	−	0.0857493i
$545$		0			0
$546$	−2.78890			−0.119354
$547$	23.6333			1.01049			0.505244	−	0.862977i	$-0.331403\pi$
$547$	23.6333			1.01049			0.505244	+	0.862977i	$0.331403\pi$
$548$		−	7.21110i		−	0.308043i
$549$			3.21110i			0.137047i
$550$		0			0
$551$		0			0
$552$		−	2.00000i		−	0.0851257i
$553$			33.2111i			1.41228i
$554$		−	19.2111i		−	0.816202i
$555$		0			0
$556$		−	17.2111i		−	0.729913i
$557$		−	15.2111i		−	0.644515i	−0.946652	−	0.322258i	$-0.895558\pi$
$557$		−	15.2111i		−	0.644515i	0.946652	−	0.322258i	$-0.104442\pi$
$558$	−2.00000			−0.0846668
$559$	−1.57779			−0.0667336
$560$		0			0
$561$		0			0
$562$		−	23.2111i		−	0.979101i
$563$		−	39.6333i		−	1.67034i	−0.549988	−	0.835172i	$-0.685368\pi$
$563$		−	39.6333i		−	1.67034i	0.549988	−	0.835172i	$-0.314632\pi$
$564$		−	4.00000i		−	0.168430i
$565$		0			0
$566$			23.6333i			0.993382i
$567$	4.60555			0.193415
$568$	13.8167			0.579734
$569$	−25.6333			−1.07460			−0.537302	−	0.843390i	$-0.680556\pi$
$569$	−25.6333			−1.07460			−0.537302	+	0.843390i	$0.680556\pi$
$570$		0			0
$571$			10.7889i			0.451501i	0.974185	+	0.225751i	$0.0724835\pi$
$571$			10.7889i			0.451501i	−0.974185	+	0.225751i	$0.927517\pi$
$572$		0			0
$573$	8.00000			0.334205
$574$	−6.42221			−0.268058
$575$		0			0
$576$	−1.00000			−0.0416667
$577$			30.8444i			1.28407i	0.766675	+	0.642035i	$0.221910\pi$
$577$			30.8444i			1.28407i	−0.766675	+	0.642035i	$0.778090\pi$
$578$	14.4222	−	9.00000i	0.599885	−	0.374351i
$579$	10.6056			0.440752
$580$		0			0
$581$			24.0000i			0.995688i
$582$			2.60555i			0.108004i
$583$		0			0
$584$			6.60555i			0.273340i
$585$		0			0
$586$	−0.422205			−0.0174411
$587$			30.0555i			1.24052i	0.784395	+	0.620262i	$0.212974\pi$
$587$			30.0555i			1.24052i	−0.784395	+	0.620262i	$0.787026\pi$
$588$	14.2111			0.586056
$589$		0			0
$590$		0			0
$591$	−18.0000			−0.740421
$592$	11.2111			0.460773
$593$		−	36.4222i		−	1.49568i	−0.663879	−	0.747840i	$-0.731091\pi$
$593$		−	36.4222i		−	1.49568i	0.663879	−	0.747840i	$-0.268909\pi$
$594$		0			0
$595$		0			0
$596$	23.0278			0.943254
$597$			14.0000i			0.572982i
$598$	−1.21110			−0.0495257
$599$	13.2111			0.539791			0.269896	−	0.962890i	$-0.413011\pi$
$599$	13.2111			0.539791			0.269896	+	0.962890i	$0.413011\pi$
$600$		0			0
$601$		−	37.2111i		−	1.51787i	−0.651165	−	0.758936i	$-0.725719\pi$
$601$		−	37.2111i		−	1.51787i	0.651165	−	0.758936i	$-0.274281\pi$
$602$	12.0000			0.489083
$603$		−	14.6056i		−	0.594784i
$604$	−6.42221			−0.261316
$605$		0			0
$606$		−	16.6056i		−	0.674554i
$607$	12.2389			0.496760			0.248380	−	0.968663i	$-0.420102\pi$
$607$	12.2389			0.496760			0.248380	+	0.968663i	$0.420102\pi$
$608$		0			0
$609$		−	33.2111i		−	1.34578i
$610$		0			0
$611$	−2.42221			−0.0979919
$612$	2.00000	−	3.60555i	0.0808452	−	0.145746i
$613$			21.4500i			0.866356i	0.901308	+	0.433178i	$0.142608\pi$
$613$			21.4500i			0.866356i	−0.901308	+	0.433178i	$0.857392\pi$
$614$	18.6056			0.750859
$615$		0			0
$616$		0			0
$617$	−34.4222			−1.38579			−0.692893	−	0.721041i	$-0.743664\pi$
$617$	−34.4222			−1.38579			−0.692893	+	0.721041i	$0.743664\pi$
$618$	14.4222			0.580146
$619$		−	30.4222i		−	1.22277i	−0.791333	−	0.611386i	$-0.790612\pi$
$619$		−	30.4222i		−	1.22277i	0.791333	−	0.611386i	$-0.209388\pi$
$620$		0			0
$621$	2.00000			0.0802572
$622$	3.39445			0.136105
$623$	3.63331			0.145565
$624$			0.605551i			0.0242415i
$625$		0			0
$626$		−	8.18335i		−	0.327072i
$627$		0			0
$628$		−	20.6056i		−	0.822251i
$629$	−22.4222	+	40.4222i	−0.894032	+	1.61174i
$630$		0			0
$631$	−39.2666			−1.56318			−0.781590	−	0.623793i	$-0.785591\pi$
$631$	−39.2666			−1.56318			−0.781590	+	0.623793i	$0.785591\pi$
$632$	7.21110			0.286842
$633$		−	16.0000i		−	0.635943i
$634$		−	16.4222i		−	0.652209i
$635$		0			0
$636$		−	7.21110i		−	0.285939i
$637$		−	8.60555i		−	0.340964i
$638$		0			0
$639$			13.8167i			0.546578i
$640$		0			0
$641$		−	35.4500i		−	1.40019i	−0.714050	−	0.700095i	$-0.753141\pi$
$641$		−	35.4500i		−	1.40019i	0.714050	−	0.700095i	$-0.246859\pi$
$642$		−	1.21110i		−	0.0477984i
$643$	8.84441			0.348789			0.174395	−	0.984676i	$-0.444203\pi$
$643$	8.84441			0.348789			0.174395	+	0.984676i	$0.444203\pi$
$644$	9.21110			0.362968
$645$		0			0
$646$		0			0
$647$			18.4222i			0.724252i	0.932129	+	0.362126i	$0.117949\pi$
$647$			18.4222i			0.724252i	−0.932129	+	0.362126i	$0.882051\pi$
$648$		−	1.00000i		−	0.0392837i
$649$		0			0
$650$		0			0
$651$		−	9.21110i		−	0.361012i
$652$	−20.0000			−0.783260
$653$	22.0000			0.860927			0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000			0.860927			0.430463	+	0.902608i	$0.358350\pi$
$654$	−0.788897			−0.0308483
$655$		0			0
$656$			1.39445i			0.0544441i
$657$	−6.60555			−0.257707
$658$	18.4222			0.718172
$659$	29.0278			1.13076			0.565380	−	0.824830i	$-0.308729\pi$
$659$	29.0278			1.13076			0.565380	+	0.824830i	$0.308729\pi$
$660$		0			0
$661$	−4.78890			−0.186267			−0.0931333	−	0.995654i	$-0.529688\pi$
$661$	−4.78890			−0.186267			−0.0931333	+	0.995654i	$0.529688\pi$
$662$			11.6333i			0.452141i
$663$	−2.18335	−	1.21110i	−0.0847941	−	0.0470353i
$664$	5.21110			0.202230
$665$		0			0
$666$			11.2111i			0.434421i
$667$		−	14.4222i		−	0.558430i
$668$	−3.21110			−0.124241
$669$			20.0000i			0.773245i
$670$		0			0
$671$		0			0
$672$		−	4.60555i		−	0.177663i
$673$	2.97224			0.114572			0.0572858	−	0.998358i	$-0.481755\pi$
$673$	2.97224			0.114572			0.0572858	+	0.998358i	$0.481755\pi$
$674$			6.60555i			0.254436i
$675$		0			0
$676$	−12.6333			−0.485896
$677$	−30.8444			−1.18545			−0.592723	−	0.805406i	$-0.701947\pi$
$677$	−30.8444			−1.18545			−0.592723	+	0.805406i	$0.701947\pi$
$678$			5.21110i			0.200131i
$679$	−12.0000			−0.460518
$680$		0			0
$681$	−29.2111			−1.11937
$682$		0			0
$683$	4.00000			0.153056			0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000			0.153056			0.0765279	+	0.997067i	$0.475617\pi$
$684$		0			0
$685$		0			0
$686$			33.2111i			1.26801i
$687$	2.00000			0.0763048
$688$		−	2.60555i		−	0.0993357i
$689$	−4.36669			−0.166358
$690$		0			0
$691$			5.57779i			0.212189i	0.994356	+	0.106095i	$0.0338347\pi$
$691$			5.57779i			0.212189i	−0.994356	+	0.106095i	$0.966165\pi$
$692$	−6.00000			−0.228086
$693$		0			0
$694$			15.6333i			0.593432i
$695$		0			0
$696$	−7.21110			−0.273336
$697$	−5.02776	−	2.78890i	−0.190440	−	0.105637i
$698$		−	11.2111i		−	0.424346i
$699$	8.00000			0.302588
$700$		0			0
$701$	−20.6056			−0.778261			−0.389130	−	0.921183i	$-0.627224\pi$
$701$	−20.6056			−0.778261			−0.389130	+	0.921183i	$0.627224\pi$
$702$	−0.605551			−0.0228551
$703$		0			0
$704$		0			0
$705$		0			0
$706$	16.4222			0.618058
$707$	76.4777			2.87624
$708$	10.6056			0.398581
$709$		−	3.21110i		−	0.120595i	−0.998180	−	0.0602977i	$-0.980795\pi$
$709$		−	3.21110i		−	0.120595i	0.998180	−	0.0602977i	$-0.0192050\pi$
$710$		0			0
$711$			7.21110i			0.270437i
$712$		−	0.788897i		−	0.0295652i
$713$		−	4.00000i		−	0.149801i
$714$	16.6056	+	9.21110i	0.621447	+	0.344717i
$715$		0			0
$716$	25.0278			0.935331
$717$	10.7889			0.402919
$718$		−	7.63331i		−	0.284873i
$719$		−	3.39445i		−	0.126592i	−0.997995	−	0.0632958i	$-0.979839\pi$
$719$		−	3.39445i		−	0.126592i	0.997995	−	0.0632958i	$-0.0201612\pi$
$720$		0			0
$721$			66.4222i			2.47369i
$722$		−	19.0000i		−	0.707107i
$723$			14.4222i			0.536368i
$724$		−	7.21110i		−	0.267999i
$725$		0			0
$726$		−	11.0000i		−	0.408248i
$727$		−	30.0555i		−	1.11470i	−0.830279	−	0.557349i	$-0.811819\pi$
$727$		−	30.0555i		−	1.11470i	0.830279	−	0.557349i	$-0.188181\pi$
$728$	−2.78890			−0.103363
$729$	1.00000			0.0370370
$730$		0			0
$731$	9.39445	+	5.21110i	0.347466	+	0.192740i
$732$			3.21110i			0.118686i
$733$		−	18.1833i		−	0.671617i	−0.941930	−	0.335809i	$-0.890990\pi$
$733$		−	18.1833i		−	0.671617i	0.941930	−	0.335809i	$-0.109010\pi$
$734$		−	33.4500i		−	1.23466i
$735$		0			0
$736$		−	2.00000i		−	0.0737210i
$737$		0			0
$738$	−1.39445			−0.0513304
$739$	−2.78890			−0.102591			−0.0512956	−	0.998684i	$-0.516335\pi$
$739$	−2.78890			−0.102591			−0.0512956	+	0.998684i	$0.516335\pi$
$740$		0			0
$741$		0			0
$742$	33.2111			1.21922
$743$	−27.2111			−0.998279			−0.499139	−	0.866522i	$-0.666350\pi$
$743$	−27.2111			−0.998279			−0.499139	+	0.866522i	$0.666350\pi$
$744$	−2.00000			−0.0733236
$745$		0			0
$746$	−23.0278			−0.843106
$747$			5.21110i			0.190664i
$748$		0			0
$749$	5.57779			0.203808
$750$		0			0
$751$			37.6333i			1.37326i	0.727008	+	0.686629i	$0.240910\pi$
$751$			37.6333i			1.37326i	−0.727008	+	0.686629i	$0.759090\pi$
$752$		−	4.00000i		−	0.145865i
$753$	−14.6056			−0.532256
$754$			4.36669i			0.159026i
$755$		0			0
$756$	4.60555			0.167502
$757$		−	47.0278i		−	1.70925i	−0.519243	−	0.854626i	$-0.673786\pi$
$757$		−	47.0278i		−	1.70925i	0.519243	−	0.854626i	$-0.326214\pi$
$758$	37.2111			1.35157
$759$		0			0
$760$		0			0
$761$	−23.2111			−0.841402			−0.420701	−	0.907199i	$-0.638216\pi$
$761$	−23.2111			−0.841402			−0.420701	+	0.907199i	$0.638216\pi$
$762$	−9.21110			−0.333683
$763$		−	3.63331i		−	0.131535i
$764$	8.00000			0.289430
$765$		0			0
$766$		0			0
$767$		−	6.42221i		−	0.231892i
$768$	−1.00000			−0.0360844
$769$	−1.63331			−0.0588986			−0.0294493	−	0.999566i	$-0.509375\pi$
$769$	−1.63331			−0.0588986			−0.0294493	+	0.999566i	$0.509375\pi$
$770$		0			0
$771$		−	16.4222i		−	0.591431i
$772$	10.6056			0.381702
$773$			15.2111i			0.547105i	0.961857	+	0.273553i	$0.0881988\pi$
$773$			15.2111i			0.547105i	−0.961857	+	0.273553i	$0.911801\pi$
$774$	2.60555			0.0936546
$775$		0			0
$776$			2.60555i			0.0935338i
$777$	−51.6333			−1.85233
$778$			12.6056i			0.451931i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	7.21110	+	4.00000i	0.257869	+	0.143040i
$783$		−	7.21110i		−	0.257704i
$784$	14.2111			0.507539
$785$		0			0
$786$	10.4222			0.371748
$787$	2.78890			0.0994135			0.0497067	−	0.998764i	$-0.484171\pi$
$787$	2.78890			0.0994135			0.0497067	+	0.998764i	$0.484171\pi$
$788$	−18.0000			−0.641223
$789$			16.0000i			0.569615i
$790$		0			0
$791$	−24.0000			−0.853342
$792$		0			0
$793$	1.94449			0.0690508
$794$		−	32.4222i		−	1.15062i
$795$		0			0
$796$			14.0000i			0.496217i
$797$		−	48.4222i		−	1.71520i	−0.514315	−	0.857601i	$-0.671954\pi$
$797$		−	48.4222i		−	1.71520i	0.514315	−	0.857601i	$-0.328046\pi$
$798$		0			0
$799$	14.4222	+	8.00000i	0.510221	+	0.283020i
$800$		0			0
$801$	0.788897			0.0278743
$802$	−14.6056			−0.515740
$803$		0			0
$804$		−	14.6056i		−	0.515098i
$805$		0			0
$806$			1.21110i			0.0426593i
$807$			0.422205i			0.0148623i
$808$		−	16.6056i		−	0.584181i
$809$		−	6.97224i		−	0.245131i	−0.992460	−	0.122566i	$-0.960888\pi$
$809$		−	6.97224i		−	0.245131i	0.992460	−	0.122566i	$-0.0391122\pi$
$810$		0			0
$811$		−	31.6333i		−	1.11080i	−0.831585	−	0.555398i	$-0.812566\pi$
$811$		−	31.6333i		−	1.11080i	0.831585	−	0.555398i	$-0.187434\pi$
$812$		−	33.2111i		−	1.16548i
$813$	−20.0000			−0.701431
$814$		0			0
$815$		0			0
$816$	2.00000	−	3.60555i	0.0700140	−	0.126220i
$817$		0			0
$818$			13.6333i			0.476677i
$819$		−	2.78890i		−	0.0974520i
$820$		0			0
$821$		−	46.0000i		−	1.60541i	−0.596376	−	0.802706i	$-0.703393\pi$
$821$		−	46.0000i		−	1.60541i	0.596376	−	0.802706i	$-0.296607\pi$
$822$	7.21110			0.251516
$823$	−16.2389			−0.566051			−0.283026	−	0.959112i	$-0.591338\pi$
$823$	−16.2389			−0.566051			−0.283026	+	0.959112i	$0.591338\pi$
$824$	14.4222			0.502421
$825$		0			0
$826$			48.8444i			1.69951i
$827$	−2.42221			−0.0842283			−0.0421142	−	0.999113i	$-0.513409\pi$
$827$	−2.42221			−0.0842283			−0.0421142	+	0.999113i	$0.513409\pi$
$828$	2.00000			0.0695048
$829$	36.0555			1.25226			0.626130	−	0.779719i	$-0.284638\pi$
$829$	36.0555			1.25226			0.626130	+	0.779719i	$0.284638\pi$
$830$		0			0
$831$	19.2111			0.666426
$832$			0.605551i			0.0209937i
$833$	−28.4222	+	51.2389i	−0.984771	+	1.77532i
$834$	17.2111			0.595972
$835$		0			0
$836$		0			0
$837$		−	2.00000i		−	0.0691301i
$838$	27.6333			0.954577
$839$			3.02776i			0.104530i	0.998633	+	0.0522649i	$0.0166440\pi$
$839$			3.02776i			0.104530i	−0.998633	+	0.0522649i	$0.983356\pi$
$840$		0			0
$841$	−23.0000			−0.793103
$842$			28.4222i			0.979494i
$843$	23.2111			0.799433
$844$		−	16.0000i		−	0.550743i
$845$		0			0
$846$	4.00000			0.137523
$847$	50.6611			1.74073
$848$		−	7.21110i		−	0.247630i
$849$	−23.6333			−0.811093
$850$		0			0
$851$	−22.4222			−0.768623
$852$			13.8167i			0.473351i
$853$	35.2111			1.20561			0.602803	−	0.797890i	$-0.294051\pi$
$853$	35.2111			1.20561			0.602803	+	0.797890i	$0.294051\pi$
$854$	−14.7889			−0.506066
$855$		0			0
$856$		−	1.21110i		−	0.0413946i
$857$	−40.8444			−1.39522			−0.697609	−	0.716478i	$-0.745753\pi$
$857$	−40.8444			−1.39522			−0.697609	+	0.716478i	$0.745753\pi$
$858$		0			0
$859$	41.2111			1.40610			0.703052	−	0.711138i	$-0.251820\pi$
$859$	41.2111			1.40610			0.703052	+	0.711138i	$0.251820\pi$
$860$		0			0
$861$		−	6.42221i		−	0.218868i
$862$	7.02776			0.239366
$863$		−	54.4222i		−	1.85255i	−0.376844	−	0.926277i	$-0.622991\pi$
$863$		−	54.4222i		−	1.85255i	0.376844	−	0.926277i	$-0.377009\pi$
$864$		−	1.00000i		−	0.0340207i
$865$		0			0
$866$	16.4222			0.558049
$867$	9.00000	+	14.4222i	0.305656	+	0.489804i
$868$		−	9.21110i		−	0.312645i
$869$		0			0
$870$		0			0
$871$	−8.84441			−0.299681
$872$	−0.788897			−0.0267154
$873$	−2.60555			−0.0881845
$874$		0			0
$875$		0			0
$876$	−6.60555			−0.223181
$877$	−54.8444			−1.85196			−0.925982	−	0.377567i	$-0.876761\pi$
$877$	−54.8444			−1.85196			−0.925982	+	0.377567i	$0.876761\pi$
$878$	−17.6333			−0.595095
$879$		−	0.422205i		−	0.0142406i
$880$		0			0
$881$		−	53.0278i		−	1.78655i	−0.449510	−	0.893275i	$-0.648401\pi$
$881$		−	53.0278i		−	1.78655i	0.449510	−	0.893275i	$-0.351599\pi$
$882$			14.2111i			0.478513i
$883$			19.8167i			0.666883i	0.942771	+	0.333442i	$0.108210\pi$
$883$			19.8167i			0.666883i	−0.942771	+	0.333442i	$0.891790\pi$
$884$	−2.18335	−	1.21110i	−0.0734339	−	0.0407338i
$885$		0			0
$886$	−27.6333			−0.928359
$887$	28.4222			0.954324			0.477162	−	0.878815i	$-0.341665\pi$
$887$	28.4222			0.954324			0.477162	+	0.878815i	$0.341665\pi$
$888$			11.2111i			0.376220i
$889$		−	42.4222i		−	1.42280i
$890$		0			0
$891$		0			0
$892$			20.0000i			0.669650i
$893$		0			0
$894$			23.0278i			0.770163i
$895$		0			0
$896$		−	4.60555i		−	0.153861i
$897$		−	1.21110i		−	0.0404375i
$898$	−25.3944			−0.847424
$899$	−14.4222			−0.481007
$900$		0			0
$901$	26.0000	+	14.4222i	0.866186	+	0.480473i
$902$		0			0
$903$			12.0000i			0.399335i
$904$			5.21110i			0.173319i
$905$		0			0
$906$		−	6.42221i		−	0.213363i
$907$	−34.7889			−1.15515			−0.577573	−	0.816339i	$-0.696000\pi$
$907$	−34.7889			−1.15515			−0.577573	+	0.816339i	$0.696000\pi$
$908$	−29.2111			−0.969404
$909$	16.6056			0.550771
$910$		0			0
$911$		−	26.6611i		−	0.883320i	−0.897182	−	0.441660i	$-0.854390\pi$
$911$		−	26.6611i		−	0.883320i	0.897182	−	0.441660i	$-0.145610\pi$
$912$		0			0
$913$		0			0
$914$	30.8444			1.02024
$915$		0			0
$916$	2.00000			0.0660819
$917$			48.0000i			1.58510i
$918$	3.60555	+	2.00000i	0.119001	+	0.0660098i
$919$	−40.8444			−1.34733			−0.673666	−	0.739036i	$-0.735282\pi$
$919$	−40.8444			−1.34733			−0.673666	+	0.739036i	$0.735282\pi$
$920$		0			0
$921$			18.6056i			0.613074i
$922$		−	0.972244i		−	0.0320191i
$923$	8.36669			0.275393
$924$		0			0
$925$		0			0
$926$	20.0000			0.657241
$927$			14.4222i			0.473687i
$928$	−7.21110			−0.236716
$929$		−	27.8167i		−	0.912635i	−0.889817	−	0.456317i	$-0.849168\pi$
$929$		−	27.8167i		−	0.912635i	0.889817	−	0.456317i	$-0.150832\pi$
$930$		0			0
$931$		0			0
$932$	8.00000			0.262049
$933$			3.39445i			0.111129i
$934$	1.57779			0.0516270
$935$		0			0
$936$	−0.605551			−0.0197931
$937$			39.2111i			1.28097i	0.767970	+	0.640485i	$0.221267\pi$
$937$			39.2111i			1.28097i	−0.767970	+	0.640485i	$0.778733\pi$
$938$	67.2666			2.19633
$939$	8.18335			0.267053
$940$		0			0
$941$			8.42221i			0.274556i	0.990533	+	0.137278i	$0.0438354\pi$
$941$			8.42221i			0.274556i	−0.990533	+	0.137278i	$0.956165\pi$
$942$	20.6056			0.671365
$943$		−	2.78890i		−	0.0908190i
$944$	10.6056			0.345181
$945$		0			0
$946$		0			0
$947$	9.21110			0.299321			0.149660	−	0.988737i	$-0.452182\pi$
$947$	9.21110			0.299321			0.149660	+	0.988737i	$0.452182\pi$
$948$			7.21110i			0.234206i
$949$			4.00000i			0.129845i
$950$		0			0
$951$	16.4222			0.532526
$952$	16.6056	+	9.21110i	0.538189	+	0.298534i
$953$		−	32.0555i		−	1.03838i	−0.854659	−	0.519190i	$-0.826234\pi$
$953$		−	32.0555i		−	1.03838i	0.854659	−	0.519190i	$-0.173766\pi$
$954$	7.21110			0.233468
$955$		0			0
$956$	10.7889			0.348938
$957$		0			0
$958$	−20.6056			−0.665735
$959$			33.2111i			1.07244i
$960$		0			0
$961$	27.0000			0.870968
$962$	6.78890			0.218883
$963$	1.21110			0.0390272
$964$			14.4222i			0.464508i
$965$		0			0
$966$			9.21110i			0.296362i
$967$		−	22.7889i		−	0.732842i	−0.930449	−	0.366421i	$-0.880583\pi$
$967$		−	22.7889i		−	0.732842i	0.930449	−	0.366421i	$-0.119417\pi$
$968$		−	11.0000i		−	0.353553i
$969$		0			0
$970$		0			0
$971$	23.4500			0.752545			0.376273	−	0.926509i	$-0.377206\pi$
$971$	23.4500			0.752545			0.376273	+	0.926509i	$0.377206\pi$
$972$	1.00000			0.0320750
$973$			79.2666i			2.54117i
$974$			21.4500i			0.687301i
$975$		0			0
$976$			3.21110i			0.102785i
$977$		−	26.8444i		−	0.858829i	−0.903108	−	0.429414i	$-0.858720\pi$
$977$		−	26.8444i		−	0.858829i	0.903108	−	0.429414i	$-0.141280\pi$
$978$		−	20.0000i		−	0.639529i
$979$		0			0
$980$		0			0
$981$		−	0.788897i		−	0.0251876i
$982$			33.0278i			1.05396i
$983$	−39.2111			−1.25064			−0.625320	−	0.780368i	$-0.715032\pi$
$983$	−39.2111			−1.25064			−0.625320	+	0.780368i	$0.715032\pi$
$984$	−1.39445			−0.0444534
$985$		0			0
$986$	14.4222	−	26.0000i	0.459297	−	0.828009i
$987$			18.4222i			0.586385i
$988$		0			0
$989$			5.21110i			0.165703i
$990$		0			0
$991$			11.5778i			0.367781i	0.982947	+	0.183890i	$0.0588691\pi$
$991$			11.5778i			0.367781i	−0.982947	+	0.183890i	$0.941131\pi$
$992$	−2.00000			−0.0635001
$993$	−11.6333			−0.369172
$994$	−63.6333			−2.01833
$995$		0			0
$996$			5.21110i			0.165120i
$997$	9.63331			0.305090			0.152545	−	0.988297i	$-0.451253\pi$
$997$	9.63331			0.305090			0.152545	+	0.988297i	$0.451253\pi$
$998$	−16.0000			−0.506471
$999$	−11.2111			−0.354704

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.f.p.1699.4		4
5.2	odd	4		510.2.c.c.271.4	yes	4
5.3	odd	4		2550.2.c.o.1801.1		4
5.4	even	2		2550.2.f.s.1699.1		4
15.2	even	4		1530.2.c.g.271.2		4
17.16	even	2		2550.2.f.s.1699.3		4
20.7	even	4		4080.2.h.q.3841.1		4
85.33	odd	4		2550.2.c.o.1801.4		4
85.47	odd	4		8670.2.a.bh.1.2		2
85.67	odd	4		510.2.c.c.271.1	✓	4
85.72	odd	4		8670.2.a.bk.1.1		2
85.84	even	2	inner	2550.2.f.p.1699.2		4
255.152	even	4		1530.2.c.g.271.3		4
340.67	even	4		4080.2.h.q.3841.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.c.c.271.1	✓	4	85.67	odd	4
510.2.c.c.271.4	yes	4	5.2	odd	4
1530.2.c.g.271.2		4	15.2	even	4
1530.2.c.g.271.3		4	255.152	even	4
2550.2.c.o.1801.1		4	5.3	odd	4
2550.2.c.o.1801.4		4	85.33	odd	4
2550.2.f.p.1699.2		4	85.84	even	2	inner
2550.2.f.p.1699.4		4	1.1	even	1	trivial
2550.2.f.s.1699.1		4	5.4	even	2
2550.2.f.s.1699.3		4	17.16	even	2
4080.2.h.q.3841.1		4	20.7	even	4
4080.2.h.q.3841.4		4	340.67	even	4
8670.2.a.bh.1.2		2	85.47	odd	4
8670.2.a.bk.1.1		2	85.72	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} -1.00000i q^{6} +4.60555 q^{7} -1.00000i q^{8} +1.00000 q^{9} +1.00000 q^{12} -0.605551i q^{13} +4.60555i q^{14} +1.00000 q^{16} +(-2.00000 + 3.60555i) q^{17} +1.00000i q^{18} -4.60555 q^{21} -2.00000 q^{23} +1.00000i q^{24} +0.605551 q^{26} -1.00000 q^{27} -4.60555 q^{28} +7.21110i q^{29} +2.00000i q^{31} +1.00000i q^{32} +(-3.60555 - 2.00000i) q^{34} -1.00000 q^{36} +11.2111 q^{37} +0.605551i q^{39} +1.39445i q^{41} -4.60555i q^{42} -2.60555i q^{43} -2.00000i q^{46} -4.00000i q^{47} -1.00000 q^{48} +14.2111 q^{49} +(2.00000 - 3.60555i) q^{51} +0.605551i q^{52} -7.21110i q^{53} -1.00000i q^{54} -4.60555i q^{56} -7.21110 q^{58} +10.6056 q^{59} +3.21110i q^{61} -2.00000 q^{62} +4.60555 q^{63} -1.00000 q^{64} -14.6056i q^{67} +(2.00000 - 3.60555i) q^{68} +2.00000 q^{69} +13.8167i q^{71} -1.00000i q^{72} -6.60555 q^{73} +11.2111i q^{74} -0.605551 q^{78} +7.21110i q^{79} +1.00000 q^{81} -1.39445 q^{82} +5.21110i q^{83} +4.60555 q^{84} +2.60555 q^{86} -7.21110i q^{87} +0.788897 q^{89} -2.78890i q^{91} +2.00000 q^{92} -2.00000i q^{93} +4.00000 q^{94} -1.00000i q^{96} -2.60555 q^{97} +14.2111i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{7} + 4 q^{9} + 4 q^{12} + 4 q^{16} - 8 q^{17} - 4 q^{21} - 8 q^{23} - 12 q^{26} - 4 q^{27} - 4 q^{28} - 4 q^{36} + 16 q^{37} - 4 q^{48} + 28 q^{49} + 8 q^{51} + 28 q^{59}+ \cdots + 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^7 + 4 * q^9 + 4 * q^12 + 4 * q^16 - 8 * q^17 - 4 * q^21 - 8 * q^23 - 12 * q^26 - 4 * q^27 - 4 * q^28 - 4 * q^36 + 16 * q^37 - 4 * q^48 + 28 * q^49 + 8 * q^51 + 28 * q^59 - 8 * q^62 + 4 * q^63 - 4 * q^64 + 8 * q^68 + 8 * q^69 - 12 * q^73 + 12 * q^78 + 4 * q^81 - 20 * q^82 + 4 * q^84 - 4 * q^86 + 32 * q^89 + 8 * q^92 + 16 * q^94 + 4 * q^97

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