LMFDB - Embedded newform 532.2.g.b.265.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [532,2,Mod(265,532)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("532.265"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(532, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 532 = 2^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	532.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.24804138753$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.113164960000.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 17x^{6} - 30x^{5} + 174x^{4} - 208x^{3} + 962x^{2} - 382x + 2449 $ x^8 - 2x^7 + 17x^6 - 30x^5 + 174x^4 - 208x^3 + 962x^2 - 382*x + 2449
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			265.7
Root			$-0.477260 - 2.09331i$ of defining polynomial
Character	$\chi$	$=$	532.265
Dual form			532.2.g.b.265.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57255 q^{3} -3.38705i q^{5} +(-1.61803 - 2.09331i) q^{7} +3.61803 q^{9} -0.618034 q^{11} -1.58993 q^{13} -8.71338i q^{15} +2.09331i q^{17} +(4.16248 + 1.29374i) q^{19} +(-4.16248 - 5.38516i) q^{21} -1.47214 q^{23} -6.47214 q^{25} +1.58993 q^{27} -3.32821i q^{29} +7.34233 q^{31} -1.58993 q^{33} +(-7.09017 + 5.48037i) q^{35} +5.38516i q^{37} -4.09017 q^{39} +7.34233 q^{41} +7.70820 q^{43} -12.2545i q^{45} +10.1612i q^{47} +(-1.76393 + 6.77411i) q^{49} +5.38516i q^{51} -2.05695i q^{53} +2.09331i q^{55} +(10.7082 + 3.32821i) q^{57} +6.73503 q^{59} -0.799575i q^{61} +(-5.85410 - 7.57368i) q^{63} +5.38516i q^{65} +14.0985i q^{67} -3.78715 q^{69} -12.0416i q^{71} +2.89289i q^{73} -16.6499 q^{75} +(1.00000 + 1.29374i) q^{77} -10.7703i q^{79} -6.76393 q^{81} -16.4411i q^{83} +7.09017 q^{85} -8.56201i q^{87} -10.2902 q^{89} +(2.57255 + 3.32821i) q^{91} +18.8885 q^{93} +(4.38197 - 14.0985i) q^{95} +4.76978 q^{97} -2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{7} + 20 q^{9} + 4 q^{11} + 24 q^{23} - 16 q^{25} - 12 q^{35} + 12 q^{39} + 8 q^{43} - 32 q^{49} + 32 q^{57} - 20 q^{63} + 8 q^{77} - 72 q^{81} + 12 q^{85} + 8 q^{93} + 44 q^{95}+O(q^{100}) $ 8 * q - 4 * q^7 + 20 * q^9 + 4 * q^11 + 24 * q^23 - 16 * q^25 - 12 * q^35 + 12 * q^39 + 8 * q^43 - 32 * q^49 + 32 * q^57 - 20 * q^63 + 8 * q^77 - 72 * q^81 + 12 * q^85 + 8 * q^93 + 44 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/532\mathbb{Z}\right)^\times$.

$n$	$267$	$381$	$477$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	2.57255			1.48526			0.742632	−	0.669699i	$-0.233577\pi$
$3$	2.57255			1.48526			0.742632	+	0.669699i	$0.233577\pi$
$4$		0			0
$5$		−	3.38705i		−	1.51474i	−0.652988	−	0.757368i	$-0.726485\pi$
$5$		−	3.38705i		−	1.51474i	0.652988	−	0.757368i	$-0.273515\pi$
$6$		0			0
$7$	−1.61803	−	2.09331i	−0.611559	−	0.791199i
$7$	−1.61803	−	2.09331i	−0.611559	−	0.791199i
$8$		0			0
$9$	3.61803			1.20601
$10$		0			0
$11$	−0.618034			−0.186344			−0.0931721	−	0.995650i	$-0.529701\pi$
$11$	−0.618034			−0.186344			−0.0931721	+	0.995650i	$0.529701\pi$
$12$		0			0
$13$	−1.58993			−0.440966			−0.220483	−	0.975391i	$-0.570763\pi$
$13$	−1.58993			−0.440966			−0.220483	+	0.975391i	$0.570763\pi$
$14$		0			0
$15$		−	8.71338i		−	2.24978i
$16$		0			0
$17$			2.09331i			0.507703i	0.967243	+	0.253852i	$0.0816975\pi$
$17$			2.09331i			0.507703i	−0.967243	+	0.253852i	$0.918302\pi$
$18$		0			0
$19$	4.16248	+	1.29374i	0.954938	+	0.296804i
$19$	4.16248	+	1.29374i	0.954938	+	0.296804i
$20$		0			0
$21$	−4.16248	−	5.38516i	−0.908328	−	1.17514i
$22$		0			0
$23$	−1.47214			−0.306962			−0.153481	−	0.988152i	$-0.549048\pi$
$23$	−1.47214			−0.306962			−0.153481	+	0.988152i	$0.549048\pi$
$24$		0			0
$25$	−6.47214			−1.29443
$26$		0			0
$27$	1.58993			0.305981
$28$		0			0
$29$		−	3.32821i		−	0.618034i	−0.951057	−	0.309017i	$-0.900000\pi$
$29$		−	3.32821i		−	0.618034i	0.951057	−	0.309017i	$-0.100000\pi$
$30$		0			0
$31$	7.34233			1.31872			0.659361	−	0.751827i	$-0.270827\pi$
$31$	7.34233			1.31872			0.659361	+	0.751827i	$0.270827\pi$
$32$		0			0
$33$	−1.58993			−0.276771
$34$		0			0
$35$	−7.09017	+	5.48037i	−1.19846	+	0.926351i
$36$		0			0
$37$			5.38516i			0.885316i	0.896691	+	0.442658i	$0.145964\pi$
$37$			5.38516i			0.885316i	−0.896691	+	0.442658i	$0.854036\pi$
$38$		0			0
$39$	−4.09017			−0.654951
$40$		0			0
$41$	7.34233			1.14668			0.573340	−	0.819318i	$-0.305648\pi$
$41$	7.34233			1.14668			0.573340	+	0.819318i	$0.305648\pi$
$42$		0			0
$43$	7.70820			1.17549			0.587745	−	0.809046i	$-0.300016\pi$
$43$	7.70820			1.17549			0.587745	+	0.809046i	$0.300016\pi$
$44$		0			0
$45$		−	12.2545i		−	1.82679i
$46$		0			0
$47$			10.1612i			1.48216i	0.671418	+	0.741079i	$0.265686\pi$
$47$			10.1612i			1.48216i	−0.671418	+	0.741079i	$0.734314\pi$
$48$		0			0
$49$	−1.76393	+	6.77411i	−0.251990	+	0.967730i
$50$		0			0
$51$			5.38516i			0.754074i
$52$		0			0
$53$		−	2.05695i		−	0.282544i	−0.989971	−	0.141272i	$-0.954881\pi$
$53$		−	2.05695i		−	0.282544i	0.989971	−	0.141272i	$-0.0451192\pi$
$54$		0			0
$55$			2.09331i			0.282262i
$56$		0			0
$57$	10.7082	+	3.32821i	1.41834	+	0.440833i
$58$		0			0
$59$	6.73503			0.876827			0.438413	−	0.898773i	$-0.355541\pi$
$59$	6.73503			0.876827			0.438413	+	0.898773i	$0.355541\pi$
$60$		0			0
$61$		−	0.799575i		−	0.102375i	−0.998689	−	0.0511875i	$-0.983699\pi$
$61$		−	0.799575i		−	0.102375i	0.998689	−	0.0511875i	$-0.0163006\pi$
$62$		0			0
$63$	−5.85410	−	7.57368i	−0.737548	−	0.954194i
$64$		0			0
$65$			5.38516i			0.667947i
$66$		0			0
$67$			14.0985i			1.72241i	0.508257	+	0.861206i	$0.330290\pi$
$67$			14.0985i			1.72241i	−0.508257	+	0.861206i	$0.669710\pi$
$68$		0			0
$69$	−3.78715			−0.455919
$70$		0			0
$71$		−	12.0416i		−	1.42907i	−0.699597	−	0.714537i	$-0.746637\pi$
$71$		−	12.0416i		−	1.42907i	0.699597	−	0.714537i	$-0.253363\pi$
$72$		0			0
$73$			2.89289i			0.338587i	0.985566	+	0.169294i	$0.0541486\pi$
$73$			2.89289i			0.338587i	−0.985566	+	0.169294i	$0.945851\pi$
$74$		0			0
$75$	−16.6499			−1.92257
$76$		0			0
$77$	1.00000	+	1.29374i	0.113961	+	0.147435i
$78$		0			0
$79$		−	10.7703i		−	1.21176i	−0.795557	−	0.605878i	$-0.792822\pi$
$79$		−	10.7703i		−	1.21176i	0.795557	−	0.605878i	$-0.207178\pi$
$80$		0			0
$81$	−6.76393			−0.751548
$82$		0			0
$83$		−	16.4411i		−	1.80465i	−0.431061	−	0.902323i	$-0.641861\pi$
$83$		−	16.4411i		−	1.80465i	0.431061	−	0.902323i	$-0.358139\pi$
$84$		0			0
$85$	7.09017			0.769037
$86$		0			0
$87$		−	8.56201i		−	0.917944i
$88$		0			0
$89$	−10.2902			−1.09076			−0.545380	−	0.838189i	$-0.683615\pi$
$89$	−10.2902			−1.09076			−0.545380	+	0.838189i	$0.683615\pi$
$90$		0			0
$91$	2.57255	+	3.32821i	0.269677	+	0.348892i
$92$		0			0
$93$	18.8885			1.95865
$94$		0			0
$95$	4.38197	−	14.0985i	0.449580	−	1.44648i
$96$		0			0
$97$	4.76978			0.484298			0.242149	−	0.970239i	$-0.422148\pi$
$97$	4.76978			0.484298			0.242149	+	0.970239i	$0.422148\pi$
$98$		0			0
$99$	−2.23607			−0.224733
$100$		0			0
$101$			3.38705i			0.337024i	0.985700	+	0.168512i	$0.0538963\pi$
$101$			3.38705i			0.337024i	−0.985700	+	0.168512i	$0.946104\pi$
$102$		0			0
$103$	−13.0947			−1.29026			−0.645131	−	0.764072i	$-0.723197\pi$
$103$	−13.0947			−1.29026			−0.645131	+	0.764072i	$0.723197\pi$
$104$		0			0
$105$	−18.2398	+	14.0985i	−1.78003	+	1.37588i
$106$		0			0
$107$		−	5.38516i		−	0.520604i	−0.965527	−	0.260302i	$-0.916178\pi$
$107$		−	5.38516i		−	0.520604i	0.965527	−	0.260302i	$-0.0838220\pi$
$108$		0			0
$109$			12.0416i			1.15338i	0.816965	+	0.576688i	$0.195655\pi$
$109$			12.0416i			1.15338i	−0.816965	+	0.576688i	$0.804345\pi$
$110$		0			0
$111$			13.8536i			1.31493i
$112$		0			0
$113$			19.4837i			1.83287i	0.400180	+	0.916437i	$0.368948\pi$
$113$			19.4837i			1.83287i	−0.400180	+	0.916437i	$0.631052\pi$
$114$		0			0
$115$			4.98620i			0.464966i
$116$		0			0
$117$	−5.75241			−0.531810
$118$		0			0
$119$	4.38197	−	3.38705i	0.401694	−	0.310491i
$120$		0			0
$121$	−10.6180			−0.965276
$122$		0			0
$123$	18.8885			1.70312
$124$		0			0
$125$			4.98620i			0.445980i
$126$		0			0
$127$			10.7703i			0.955712i	0.878438	+	0.477856i	$0.158586\pi$
$127$			10.7703i			0.955712i	−0.878438	+	0.477856i	$0.841414\pi$
$128$		0			0
$129$	19.8298			1.74591
$130$		0			0
$131$			0.494165i			0.0431754i	0.999767	+	0.0215877i	$0.00687211\pi$
$131$			0.494165i			0.0431754i	−0.999767	+	0.0215877i	$0.993128\pi$
$132$		0			0
$133$	−4.02683	−	10.8067i	−0.349170	−	0.937059i
$134$		0			0
$135$		−	5.38516i		−	0.463481i
$136$		0			0
$137$	−19.7082			−1.68379			−0.841893	−	0.539645i	$-0.818559\pi$
$137$	−19.7082			−1.68379			−0.841893	+	0.539645i	$0.818559\pi$
$138$		0			0
$139$		−	11.7603i		−	0.997497i	−0.866747	−	0.498748i	$-0.833793\pi$
$139$		−	11.7603i		−	0.997497i	0.866747	−	0.498748i	$-0.166207\pi$
$140$		0			0
$141$			26.1401i			2.20140i
$142$		0			0
$143$	0.982628			0.0821715
$144$		0			0
$145$	−11.2728			−0.936159
$146$		0			0
$147$	−4.53781	+	17.4268i	−0.374272	+	1.43733i
$148$		0			0
$149$	−7.47214			−0.612141			−0.306071	−	0.952009i	$-0.599014\pi$
$149$	−7.47214			−0.612141			−0.306071	+	0.952009i	$0.599014\pi$
$150$		0			0
$151$			7.44211i			0.605631i	0.953049	+	0.302815i	$0.0979265\pi$
$151$			7.44211i			0.605631i	−0.953049	+	0.302815i	$0.902073\pi$
$152$		0			0
$153$			7.57368i			0.612296i
$154$		0			0
$155$		−	24.8689i		−	1.99752i
$156$		0			0
$157$			17.2407i			1.37596i	0.725731	+	0.687978i	$0.241501\pi$
$157$			17.2407i			1.37596i	−0.725731	+	0.687978i	$0.758499\pi$
$158$		0			0
$159$		−	5.29161i		−	0.419652i
$160$		0			0
$161$	2.38197	+	3.08164i	0.187725	+	0.242868i
$162$		0			0
$163$	−11.6180			−0.909995			−0.454997	−	0.890493i	$-0.650360\pi$
$163$	−11.6180			−0.909995			−0.454997	+	0.890493i	$0.650360\pi$
$164$		0			0
$165$			5.38516i			0.419235i
$166$		0			0
$167$	20.2051			1.56352			0.781759	−	0.623581i	$-0.214323\pi$
$167$	20.2051			1.56352			0.781759	+	0.623581i	$0.214323\pi$
$168$		0			0
$169$	−10.4721			−0.805549
$170$		0			0
$171$	15.0600	+	4.68079i	1.15167	+	0.357949i
$172$		0			0
$173$	18.6152			1.41529			0.707643	−	0.706570i	$-0.249759\pi$
$173$	18.6152			1.41529			0.707643	+	0.706570i	$0.249759\pi$
$174$		0			0
$175$	10.4721	+	13.5482i	0.791619	+	1.02415i
$176$		0			0
$177$	17.3262			1.30232
$178$		0			0
$179$		−	20.7550i		−	1.55130i	−0.631164	−	0.775650i	$-0.717422\pi$
$179$		−	20.7550i		−	1.55130i	0.631164	−	0.775650i	$-0.282578\pi$
$180$		0			0
$181$	−16.4180			−1.22034			−0.610168	−	0.792272i	$-0.708898\pi$
$181$	−16.4180			−1.22034			−0.610168	+	0.792272i	$0.708898\pi$
$182$		0			0
$183$		−	2.05695i		−	0.152054i
$184$		0			0
$185$	18.2398			1.34102
$186$		0			0
$187$		−	1.29374i		−	0.0946076i
$188$		0			0
$189$	−2.57255	−	3.32821i	−0.187126	−	0.242092i
$190$		0			0
$191$	−23.0344			−1.66671			−0.833357	−	0.552735i	$-0.813584\pi$
$191$	−23.0344			−1.66671			−0.833357	+	0.552735i	$0.813584\pi$
$192$		0			0
$193$			3.32821i			0.239570i	0.992800	+	0.119785i	$0.0382205\pi$
$193$			3.32821i			0.239570i	−0.992800	+	0.119785i	$0.961779\pi$
$194$		0			0
$195$			13.8536i			0.992079i
$196$		0			0
$197$	0.381966			0.0272140			0.0136070	−	0.999907i	$-0.495669\pi$
$197$	0.381966			0.0272140			0.0136070	+	0.999907i	$0.495669\pi$
$198$		0			0
$199$		−	23.7094i		−	1.68071i	−0.542034	−	0.840357i	$-0.682346\pi$
$199$		−	23.7094i		−	1.68071i	0.542034	−	0.840357i	$-0.317654\pi$
$200$		0			0
$201$			36.2693i			2.55824i
$202$		0			0
$203$	−6.96700	+	5.38516i	−0.488988	+	0.377964i
$204$		0			0
$205$		−	24.8689i		−	1.73692i
$206$		0			0
$207$	−5.32624			−0.370199
$208$		0			0
$209$	−2.57255	−	0.799575i	−0.177947	−	0.0553078i
$210$		0			0
$211$			26.1401i			1.79956i	0.436342	+	0.899781i	$0.356274\pi$
$211$			26.1401i			1.79956i	−0.436342	+	0.899781i	$0.643726\pi$
$212$		0			0
$213$		−	30.9777i		−	2.12255i
$214$		0			0
$215$		−	26.1081i		−	1.78056i
$216$		0			0
$217$	−11.8801	−	15.3698i	−0.806477	−	1.04337i
$218$		0			0
$219$			7.44211i			0.502892i
$220$		0			0
$221$		−	3.32821i		−	0.223880i
$222$		0			0
$223$	−13.0947			−0.876888			−0.438444	−	0.898758i	$-0.644470\pi$
$223$	−13.0947			−0.876888			−0.438444	+	0.898758i	$0.644470\pi$
$224$		0			0
$225$	−23.4164			−1.56109
$226$		0			0
$227$	−0.982628			−0.0652193			−0.0326097	−	0.999468i	$-0.510382\pi$
$227$	−0.982628			−0.0652193			−0.0326097	+	0.999468i	$0.510382\pi$
$228$		0			0
$229$			16.7465i			1.10664i	0.832968	+	0.553320i	$0.186640\pi$
$229$			16.7465i			1.10664i	−0.832968	+	0.553320i	$0.813360\pi$
$230$		0			0
$231$	2.57255	+	3.32821i	0.169262	+	0.218980i
$232$		0			0
$233$	−3.56231			−0.233374			−0.116687	−	0.993169i	$-0.537228\pi$
$233$	−3.56231			−0.233374			−0.116687	+	0.993169i	$0.537228\pi$
$234$		0			0
$235$	34.4164			2.24508
$236$		0			0
$237$		−	27.7073i		−	1.79978i
$238$		0			0
$239$	−1.65248			−0.106890			−0.0534449	−	0.998571i	$-0.517020\pi$
$239$	−1.65248			−0.106890			−0.0534449	+	0.998571i	$0.517020\pi$
$240$		0			0
$241$	−3.17985			−0.204832			−0.102416	−	0.994742i	$-0.532657\pi$
$241$	−3.17985			−0.204832			−0.102416	+	0.994742i	$0.532657\pi$
$242$		0			0
$243$	−22.1704			−1.42223
$244$		0			0
$245$	22.9443	+	5.97453i	1.46586	+	0.381699i
$246$		0			0
$247$	−6.61803	−	2.05695i	−0.421095	−	0.130881i
$248$		0			0
$249$		−	42.2956i		−	2.68038i
$250$		0			0
$251$		−	3.08164i		−	0.194512i	−0.995259	−	0.0972558i	$-0.968994\pi$
$251$		−	3.08164i		−	0.194512i	0.995259	−	0.0972558i	$-0.0310065\pi$
$252$		0			0
$253$	0.909830			0.0572005
$254$		0			0
$255$	18.2398			1.14222
$256$		0			0
$257$	14.8280			0.924947			0.462473	−	0.886633i	$-0.346962\pi$
$257$	14.8280			0.924947			0.462473	+	0.886633i	$0.346962\pi$
$258$		0			0
$259$	11.2728	−	8.71338i	0.700460	−	0.541423i
$260$		0			0
$261$		−	12.0416i		−	0.745356i
$262$		0			0
$263$	26.5623			1.63790			0.818951	−	0.573863i	$-0.194556\pi$
$263$	26.5623			1.63790			0.818951	+	0.573863i	$0.194556\pi$
$264$		0			0
$265$	−6.96700			−0.427979
$266$		0			0
$267$	−26.4721			−1.62007
$268$		0			0
$269$	10.8975			0.664433			0.332217	−	0.943203i	$-0.392204\pi$
$269$	10.8975			0.664433			0.332217	+	0.943203i	$0.392204\pi$
$270$		0			0
$271$			13.0541i			0.792977i	0.918040	+	0.396489i	$0.129771\pi$
$271$			13.0541i			0.792977i	−0.918040	+	0.396489i	$0.870229\pi$
$272$		0			0
$273$	6.61803	+	8.56201i	0.400542	+	0.518197i
$274$		0			0
$275$	4.00000			0.241209
$276$		0			0
$277$	−2.94427			−0.176904			−0.0884521	−	0.996080i	$-0.528192\pi$
$277$	−2.94427			−0.176904			−0.0884521	+	0.996080i	$0.528192\pi$
$278$		0			0
$279$	26.5648			1.59039
$280$		0			0
$281$		−	16.1555i		−	0.963756i	−0.876238	−	0.481878i	$-0.839955\pi$
$281$		−	16.1555i		−	0.963756i	0.876238	−	0.481878i	$-0.160045\pi$
$282$		0			0
$283$		−	23.2152i		−	1.38000i	−0.723809	−	0.690001i	$-0.757610\pi$
$283$		−	23.2152i		−	1.38000i	0.723809	−	0.690001i	$-0.242390\pi$
$284$		0			0
$285$	11.2728	−	36.2693i	0.667746	−	2.14841i
$286$		0			0
$287$	−11.8801	−	15.3698i	−0.701263	−	0.907251i
$288$		0			0
$289$	12.6180			0.742237
$290$		0			0
$291$	12.2705			0.719310
$292$		0			0
$293$	−9.91489			−0.579234			−0.289617	−	0.957143i	$-0.593528\pi$
$293$	−9.91489			−0.579234			−0.289617	+	0.957143i	$0.593528\pi$
$294$		0			0
$295$		−	22.8119i		−	1.32816i
$296$		0			0
$297$	−0.982628			−0.0570179
$298$		0			0
$299$	2.34059			0.135360
$300$		0			0
$301$	−12.4721	−	16.1357i	−0.718882	−	0.930046i
$302$		0			0
$303$			8.71338i			0.500571i
$304$		0			0
$305$	−2.70820			−0.155071
$306$		0			0
$307$	20.8124			1.18783			0.593913	−	0.804529i	$-0.297582\pi$
$307$	20.8124			1.18783			0.593913	+	0.804529i	$0.297582\pi$
$308$		0			0
$309$	−33.6869			−1.91638
$310$		0			0
$311$		−	14.6532i		−	0.830907i	−0.909614	−	0.415453i	$-0.863623\pi$
$311$		−	14.6532i		−	0.830907i	0.909614	−	0.415453i	$-0.136377\pi$
$312$		0			0
$313$		−	20.9331i		−	1.18321i	−0.806227	−	0.591606i	$-0.798494\pi$
$313$		−	20.9331i		−	1.18321i	0.806227	−	0.591606i	$-0.201506\pi$
$314$		0			0
$315$	−25.6525	+	19.8282i	−1.44535	+	1.11719i
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$			2.05695i			0.115167i
$320$		0			0
$321$		−	13.8536i		−	0.773234i
$322$		0			0
$323$	−2.70820	+	8.71338i	−0.150688	+	0.484825i
$324$		0			0
$325$	10.2902			0.570798
$326$		0			0
$327$			30.9777i			1.71307i
$328$		0			0
$329$	21.2705	−	16.4411i	1.17268	−	0.906428i
$330$		0			0
$331$			15.3698i			0.844801i	0.906409	+	0.422401i	$0.138812\pi$
$331$			15.3698i			0.844801i	−0.906409	+	0.422401i	$0.861188\pi$
$332$		0			0
$333$			19.4837i			1.06770i
$334$		0			0
$335$	47.7525			2.60900
$336$		0			0
$337$		−	24.8689i		−	1.35469i	−0.735664	−	0.677347i	$-0.763130\pi$
$337$		−	24.8689i		−	1.35469i	0.735664	−	0.677347i	$-0.236870\pi$
$338$		0			0
$339$			50.1229i			2.72230i
$340$		0			0
$341$	−4.53781			−0.245736
$342$		0			0
$343$	17.0344	−	7.26827i	0.919773	−	0.392450i
$344$		0			0
$345$			12.8273i			0.690598i
$346$		0			0
$347$	−20.5623			−1.10384			−0.551921	−	0.833896i	$-0.686105\pi$
$347$	−20.5623			−1.10384			−0.551921	+	0.833896i	$0.686105\pi$
$348$		0			0
$349$			24.2035i			1.29559i	0.761816	+	0.647793i	$0.224308\pi$
$349$			24.2035i			1.29559i	−0.761816	+	0.647793i	$0.775692\pi$
$350$		0			0
$351$	−2.52786			−0.134927
$352$		0			0
$353$		−	1.48249i		−	0.0789052i	−0.999221	−	0.0394526i	$-0.987439\pi$
$353$		−	1.48249i		−	0.0789052i	0.999221	−	0.0394526i	$-0.0125614\pi$
$354$		0			0
$355$	−40.7855			−2.16467
$356$		0			0
$357$	11.2728	−	8.71338i	0.596622	−	0.461161i
$358$		0			0
$359$	−9.74265			−0.514197			−0.257099	−	0.966385i	$-0.582767\pi$
$359$	−9.74265			−0.514197			−0.257099	+	0.966385i	$0.582767\pi$
$360$		0			0
$361$	15.6525	+	10.7703i	0.823815	+	0.566859i
$362$		0			0
$363$	−27.3155			−1.43369
$364$		0			0
$365$	9.79837			0.512870
$366$		0			0
$367$			17.9236i			0.935604i	0.883833	+	0.467802i	$0.154954\pi$
$367$			17.9236i			0.935604i	−0.883833	+	0.467802i	$0.845046\pi$
$368$		0			0
$369$	26.5648			1.38291
$370$		0			0
$371$	−4.30584	+	3.32821i	−0.223548	+	0.172792i
$372$		0			0
$373$			19.4837i			1.00883i	0.863462	+	0.504414i	$0.168291\pi$
$373$			19.4837i			1.00883i	−0.863462	+	0.504414i	$0.831709\pi$
$374$		0			0
$375$			12.8273i			0.662398i
$376$		0			0
$377$			5.29161i			0.272532i
$378$		0			0
$379$			12.0416i			0.618535i	0.950975	+	0.309268i	$0.100084\pi$
$379$			12.0416i			0.618535i	−0.950975	+	0.309268i	$0.899916\pi$
$380$		0			0
$381$			27.7073i			1.41949i
$382$		0			0
$383$	11.1295			0.568690			0.284345	−	0.958722i	$-0.408224\pi$
$383$	11.1295			0.568690			0.284345	+	0.958722i	$0.408224\pi$
$384$		0			0
$385$	4.38197	−	3.38705i	0.223326	−	0.172620i
$386$		0			0
$387$	27.8885			1.41765
$388$		0			0
$389$	−10.5066			−0.532705			−0.266352	−	0.963876i	$-0.585818\pi$
$389$	−10.5066			−0.532705			−0.266352	+	0.963876i	$0.585818\pi$
$390$		0			0
$391$		−	3.08164i		−	0.155845i
$392$		0			0
$393$			1.27126i			0.0641268i
$394$		0			0
$395$	−36.4797			−1.83549
$396$		0			0
$397$		−	8.37326i		−	0.420242i	−0.977675	−	0.210121i	$-0.932614\pi$
$397$		−	8.37326i		−	0.420242i	0.977675	−	0.210121i	$-0.0673858\pi$
$398$		0			0
$399$	−10.3592	−	27.8008i	−0.518611	−	1.39178i
$400$		0			0
$401$			20.7550i			1.03645i	0.855243	+	0.518227i	$0.173408\pi$
$401$			20.7550i			1.03645i	−0.855243	+	0.518227i	$0.826592\pi$
$402$		0			0
$403$	−11.6738			−0.581512
$404$		0			0
$405$			22.9098i			1.13840i
$406$		0			0
$407$		−	3.32821i		−	0.164973i
$408$		0			0
$409$	21.1877			1.04767			0.523833	−	0.851821i	$-0.324502\pi$
$409$	21.1877			1.04767			0.523833	+	0.851821i	$0.324502\pi$
$410$		0			0
$411$	−50.7004			−2.50087
$412$		0			0
$413$	−10.8975	−	14.0985i	−0.536232	−	0.693744i
$414$		0			0
$415$	−55.6869			−2.73356
$416$		0			0
$417$		−	30.2540i		−	1.48155i
$418$		0			0
$419$		−	11.1495i		−	0.544688i	−0.962200	−	0.272344i	$-0.912201\pi$
$419$		−	11.1495i		−	0.544688i	0.962200	−	0.272344i	$-0.0877989\pi$
$420$		0			0
$421$		−	1.27126i		−	0.0619577i	−0.999520	−	0.0309788i	$-0.990138\pi$
$421$		−	1.27126i		−	0.0619577i	0.999520	−	0.0309788i	$-0.00986245\pi$
$422$		0			0
$423$			36.7634i			1.78750i
$424$		0			0
$425$		−	13.5482i		−	0.657185i
$426$		0			0
$427$	−1.67376	+	1.29374i	−0.0809990	+	0.0626084i
$428$		0			0
$429$	2.52786			0.122046
$430$		0			0
$431$		−	17.4268i		−	0.839417i	−0.907659	−	0.419709i	$-0.862132\pi$
$431$		−	17.4268i		−	0.839417i	0.907659	−	0.419709i	$-0.137868\pi$
$432$		0			0
$433$	5.52044			0.265295			0.132648	−	0.991163i	$-0.457652\pi$
$433$	5.52044			0.265295			0.132648	+	0.991163i	$0.457652\pi$
$434$		0			0
$435$	−29.0000			−1.39044
$436$		0			0
$437$	−6.12774	−	1.90456i	−0.293129	−	0.0911075i
$438$		0			0
$439$	−2.42919			−0.115939			−0.0579695	−	0.998318i	$-0.518463\pi$
$439$	−2.42919			−0.115939			−0.0579695	+	0.998318i	$0.518463\pi$
$440$		0			0
$441$	−6.38197	+	24.5090i	−0.303903	+	1.16709i
$442$		0			0
$443$	22.7426			1.08054			0.540268	−	0.841493i	$-0.318323\pi$
$443$	22.7426			1.08054			0.540268	+	0.841493i	$0.318323\pi$
$444$		0			0
$445$			34.8535i			1.65222i
$446$		0			0
$447$	−19.2225			−0.909192
$448$		0			0
$449$		−	4.59948i		−	0.217063i	−0.994093	−	0.108531i	$-0.965385\pi$
$449$		−	4.59948i		−	0.217063i	0.994093	−	0.108531i	$-0.0346148\pi$
$450$		0			0
$451$	−4.53781			−0.213677
$452$		0			0
$453$			19.1452i			0.899522i
$454$		0			0
$455$	11.2728	−	8.71338i	0.528479	−	0.408490i
$456$		0			0
$457$	14.0902			0.659110			0.329555	−	0.944136i	$-0.393101\pi$
$457$	14.0902			0.659110			0.329555	+	0.944136i	$0.393101\pi$
$458$		0			0
$459$			3.32821i			0.155348i
$460$		0			0
$461$		−	0.305410i		−	0.0142244i	−0.999975	−	0.00711219i	$-0.997736\pi$
$461$		−	0.305410i		−	0.0142244i	0.999975	−	0.00711219i	$-0.00226390\pi$
$462$		0			0
$463$	34.0132			1.58073			0.790363	−	0.612639i	$-0.209892\pi$
$463$	34.0132			1.58073			0.790363	+	0.612639i	$0.209892\pi$
$464$		0			0
$465$		−	63.9765i		−	2.96684i
$466$		0			0
$467$			38.3626i			1.77521i	0.460607	+	0.887604i	$0.347632\pi$
$467$			38.3626i			1.77521i	−0.460607	+	0.887604i	$0.652368\pi$
$468$		0			0
$469$	29.5127	−	22.8119i	1.36277	−	1.05336i
$470$		0			0
$471$			44.3526i			2.04366i
$472$		0			0
$473$	−4.76393			−0.219046
$474$		0			0
$475$	−26.9401	−	8.37326i	−1.23610	−	0.384191i
$476$		0			0
$477$		−	7.44211i		−	0.340751i
$478$		0			0
$479$		−	26.6023i		−	1.21549i	−0.794133	−	0.607744i	$-0.792075\pi$
$479$		−	26.6023i		−	1.21549i	0.794133	−	0.607744i	$-0.207925\pi$
$480$		0			0
$481$		−	8.56201i		−	0.390394i
$482$		0			0
$483$	6.12774	+	7.92769i	0.278822	+	0.360723i
$484$		0			0
$485$		−	16.1555i		−	0.733583i
$486$		0			0
$487$			7.92769i			0.359238i	0.983736	+	0.179619i	$0.0574865\pi$
$487$			7.92769i			0.359238i	−0.983736	+	0.179619i	$0.942513\pi$
$488$		0			0
$489$	−29.8880			−1.35158
$490$		0			0
$491$	13.4164			0.605474			0.302737	−	0.953074i	$-0.402100\pi$
$491$	13.4164			0.605474			0.302737	+	0.953074i	$0.402100\pi$
$492$		0			0
$493$	6.96700			0.313778
$494$		0			0
$495$			7.57368i			0.340412i
$496$		0			0
$497$	−25.2068	+	19.4837i	−1.13068	+	0.873964i
$498$		0			0
$499$	24.4508			1.09457			0.547285	−	0.836946i	$-0.315661\pi$
$499$	24.4508			1.09457			0.547285	+	0.836946i	$0.315661\pi$
$500$		0			0
$501$	51.9787			2.32224
$502$		0			0
$503$		−	2.58748i		−	0.115370i	−0.998335	−	0.0576850i	$-0.981628\pi$
$503$		−	2.58748i		−	0.115370i	0.998335	−	0.0576850i	$-0.0183719\pi$
$504$		0			0
$505$	11.4721			0.510503
$506$		0			0
$507$	−26.9401			−1.19645
$508$		0			0
$509$	−9.07562			−0.402270			−0.201135	−	0.979564i	$-0.564463\pi$
$509$	−9.07562			−0.402270			−0.201135	+	0.979564i	$0.564463\pi$
$510$		0			0
$511$	6.05573	−	4.68079i	0.267890	−	0.207066i
$512$		0			0
$513$	6.61803	+	2.05695i	0.292193	+	0.0908166i
$514$		0			0
$515$			44.3526i			1.95441i
$516$		0			0
$517$		−	6.27994i		−	0.276192i
$518$		0			0
$519$	47.8885			2.10207
$520$		0			0
$521$	−36.1044			−1.58176			−0.790880	−	0.611971i	$-0.790377\pi$
$521$	−36.1044			−1.58176			−0.790880	+	0.611971i	$0.790377\pi$
$522$		0			0
$523$	36.3911			1.59127			0.795636	−	0.605776i	$-0.207137\pi$
$523$	36.3911			1.59127			0.795636	+	0.605776i	$0.207137\pi$
$524$		0			0
$525$	26.9401	+	34.8535i	1.17576	+	1.52113i
$526$		0			0
$527$			15.3698i			0.669519i
$528$		0			0
$529$	−20.8328			−0.905775
$530$		0			0
$531$	24.3676			1.05746
$532$		0			0
$533$	−11.6738			−0.505647
$534$		0			0
$535$	−18.2398			−0.788577
$536$		0			0
$537$		−	53.3933i		−	2.30409i
$538$		0			0
$539$	1.09017	−	4.18663i	0.0469569	−	0.180331i
$540$		0			0
$541$	−19.2361			−0.827023			−0.413512	−	0.910499i	$-0.635698\pi$
$541$	−19.2361			−0.827023			−0.413512	+	0.910499i	$0.635698\pi$
$542$		0			0
$543$	−42.2361			−1.81252
$544$		0			0
$545$	40.7855			1.74706
$546$		0			0
$547$			30.2540i			1.29357i	0.762673	+	0.646785i	$0.223887\pi$
$547$			30.2540i			1.29357i	−0.762673	+	0.646785i	$0.776113\pi$
$548$		0			0
$549$		−	2.89289i		−	0.123466i
$550$		0			0
$551$	4.30584	−	13.8536i	0.183435	−	0.590184i
$552$		0			0
$553$	−22.5457	+	17.4268i	−0.958740	+	0.741061i
$554$		0			0
$555$	46.9230			1.99177
$556$		0			0
$557$	−25.3820			−1.07547			−0.537734	−	0.843114i	$-0.680720\pi$
$557$	−25.3820			−1.07547			−0.537734	+	0.843114i	$0.680720\pi$
$558$		0			0
$559$	−12.2555			−0.518351
$560$		0			0
$561$		−	3.32821i		−	0.140517i
$562$		0			0
$563$	−40.0349			−1.68727			−0.843634	−	0.536918i	$-0.819588\pi$
$563$	−40.0349			−1.68727			−0.843634	+	0.536918i	$0.819588\pi$
$564$		0			0
$565$	65.9924			2.77632
$566$		0			0
$567$	10.9443	+	14.1590i	0.459616	+	0.594624i
$568$		0			0
$569$		−	5.38516i		−	0.225758i	−0.993609	−	0.112879i	$-0.963993\pi$
$569$		−	5.38516i		−	0.225758i	0.993609	−	0.112879i	$-0.0360072\pi$
$570$		0			0
$571$	19.1803			0.802672			0.401336	−	0.915931i	$-0.368546\pi$
$571$	19.1803			0.802672			0.401336	+	0.915931i	$0.368546\pi$
$572$		0			0
$573$	−59.2573			−2.47551
$574$		0			0
$575$	9.52786			0.397339
$576$		0			0
$577$		−	27.4018i		−	1.14075i	−0.821383	−	0.570377i	$-0.806797\pi$
$577$		−	27.4018i		−	1.14075i	0.821383	−	0.570377i	$-0.193203\pi$
$578$		0			0
$579$			8.56201i			0.355825i
$580$		0			0
$581$	−34.4164	+	26.6023i	−1.42783	+	1.10365i
$582$		0			0
$583$			1.27126i			0.0526504i
$584$		0			0
$585$			19.4837i			0.805552i
$586$		0			0
$587$			12.5599i			0.518402i	0.965823	+	0.259201i	$0.0834592\pi$
$587$			12.5599i			0.518402i	−0.965823	+	0.259201i	$0.916541\pi$
$588$		0			0
$589$	30.5623	+	9.49906i	1.25930	+	0.391402i
$590$		0			0
$591$	0.982628			0.0404199
$592$		0			0
$593$			25.9194i			1.06438i	0.846625	+	0.532190i	$0.178631\pi$
$593$			25.9194i			1.06438i	−0.846625	+	0.532190i	$0.821369\pi$
$594$		0			0
$595$	−11.4721	−	14.8420i	−0.470312	−	0.608461i
$596$		0			0
$597$		−	60.9937i		−	2.49630i
$598$		0			0
$599$		−	34.3679i		−	1.40424i	−0.712061	−	0.702118i	$-0.752238\pi$
$599$		−	34.3679i		−	1.40424i	0.712061	−	0.702118i	$-0.247762\pi$
$600$		0			0
$601$	3.32322			0.135557			0.0677784	−	0.997700i	$-0.478409\pi$
$601$	3.32322			0.135557			0.0677784	+	0.997700i	$0.478409\pi$
$602$		0			0
$603$			51.0090i			2.07725i
$604$		0			0
$605$			35.9639i			1.46214i
$606$		0			0
$607$	0.839265			0.0340647			0.0170324	−	0.999855i	$-0.494578\pi$
$607$	0.839265			0.0340647			0.0170324	+	0.999855i	$0.494578\pi$
$608$		0			0
$609$	−17.9230	+	13.8536i	−0.726276	+	0.561377i
$610$		0			0
$611$		−	16.1555i		−	0.653581i
$612$		0			0
$613$	−17.0344			−0.688015			−0.344007	−	0.938967i	$-0.611785\pi$
$613$	−17.0344			−0.688015			−0.344007	+	0.938967i	$0.611785\pi$
$614$		0			0
$615$		−	63.9765i		−	2.57978i
$616$		0			0
$617$	30.7984			1.23990			0.619948	−	0.784643i	$-0.287154\pi$
$617$	30.7984			1.23990			0.619948	+	0.784643i	$0.287154\pi$
$618$		0			0
$619$		−	35.1643i		−	1.41337i	−0.707527	−	0.706686i	$-0.750189\pi$
$619$		−	35.1643i		−	1.41337i	0.707527	−	0.706686i	$-0.249811\pi$
$620$		0			0
$621$	−2.34059			−0.0939245
$622$		0			0
$623$	16.6499	+	21.5407i	0.667065	+	0.863008i
$624$		0			0
$625$	−15.4721			−0.618885
$626$		0			0
$627$	−6.61803	−	2.05695i	−0.264299	−	0.0821467i
$628$		0			0
$629$	−11.2728			−0.449478
$630$		0			0
$631$	19.2918			0.767994			0.383997	−	0.923334i	$-0.374547\pi$
$631$	19.2918			0.767994			0.383997	+	0.923334i	$0.374547\pi$
$632$		0			0
$633$			67.2469i			2.67283i
$634$		0			0
$635$	36.4797			1.44765
$636$		0			0
$637$	2.80452	−	10.7703i	0.111119	−	0.426736i
$638$		0			0
$639$		−	43.5669i		−	1.72348i
$640$		0			0
$641$		−	24.8689i		−	0.982261i	−0.871086	−	0.491131i	$-0.836584\pi$
$641$		−	24.8689i		−	0.982261i	0.871086	−	0.491131i	$-0.163416\pi$
$642$		0			0
$643$			41.4442i			1.63440i	0.576354	+	0.817200i	$0.304475\pi$
$643$			41.4442i			1.63440i	−0.576354	+	0.817200i	$0.695525\pi$
$644$		0			0
$645$		−	67.1645i		−	2.64460i
$646$		0			0
$647$			13.7370i			0.540056i	0.962852	+	0.270028i	$0.0870330\pi$
$647$			13.7370i			0.540056i	−0.962852	+	0.270028i	$0.912967\pi$
$648$		0			0
$649$	−4.16248			−0.163392
$650$		0			0
$651$	−30.5623	−	39.5397i	−1.19783	−	1.54968i
$652$		0			0
$653$	−3.87539			−0.151656			−0.0758278	−	0.997121i	$-0.524160\pi$
$653$	−3.87539			−0.151656			−0.0758278	+	0.997121i	$0.524160\pi$
$654$		0			0
$655$	1.67376			0.0653993
$656$		0			0
$657$			10.4666i			0.408340i
$658$		0			0
$659$		−	14.0985i		−	0.549201i	−0.961558	−	0.274601i	$-0.911454\pi$
$659$		−	14.0985i		−	0.549201i	0.961558	−	0.274601i	$-0.0885457\pi$
$660$		0			0
$661$	−0.750661			−0.0291973			−0.0145987	−	0.999893i	$-0.504647\pi$
$661$	−0.750661			−0.0291973			−0.0145987	+	0.999893i	$0.504647\pi$
$662$		0			0
$663$		−	8.56201i		−	0.332521i
$664$		0			0
$665$	−36.6029	+	13.6391i	−1.41940	+	0.528901i
$666$		0			0
$667$			4.89958i			0.189713i
$668$		0			0
$669$	−33.6869			−1.30241
$670$		0			0
$671$			0.494165i			0.0190770i
$672$		0			0
$673$		−	32.7966i		−	1.26421i	−0.774881	−	0.632107i	$-0.782190\pi$
$673$		−	32.7966i		−	1.26421i	0.774881	−	0.632107i	$-0.217810\pi$
$674$		0			0
$675$	−10.2902			−0.396071
$676$		0			0
$677$	−38.5883			−1.48307			−0.741535	−	0.670915i	$-0.765902\pi$
$677$	−38.5883			−1.48307			−0.741535	+	0.670915i	$0.765902\pi$
$678$		0			0
$679$	−7.71766	−	9.98464i	−0.296177	−	0.383175i
$680$		0			0
$681$	−2.52786			−0.0968680
$682$		0			0
$683$			7.92769i			0.303345i	0.988431	+	0.151672i	$0.0484659\pi$
$683$			7.92769i			0.303345i	−0.988431	+	0.151672i	$0.951534\pi$
$684$		0			0
$685$			66.7528i			2.55049i
$686$		0			0
$687$			43.0813i			1.64365i
$688$		0			0
$689$			3.27040i			0.124592i
$690$		0			0
$691$			19.7115i			0.749861i	0.927053	+	0.374930i	$0.122333\pi$
$691$			19.7115i			0.749861i	−0.927053	+	0.374930i	$0.877667\pi$
$692$		0			0
$693$	3.61803	+	4.68079i	0.137438	+	0.177809i
$694$		0			0
$695$	−39.8328			−1.51094
$696$		0			0
$697$			15.3698i			0.582173i
$698$		0			0
$699$	−9.16422			−0.346623
$700$		0			0
$701$	11.1246			0.420171			0.210085	−	0.977683i	$-0.432626\pi$
$701$	11.1246			0.420171			0.210085	+	0.977683i	$0.432626\pi$
$702$		0			0
$703$	−6.96700	+	22.4156i	−0.262765	+	0.845422i
$704$		0			0
$705$	88.5381			3.33454
$706$		0			0
$707$	7.09017	−	5.48037i	0.266653	−	0.206110i
$708$		0			0
$709$	−38.4853			−1.44535			−0.722673	−	0.691190i	$-0.757087\pi$
$709$	−38.4853			−1.44535			−0.722673	+	0.691190i	$0.757087\pi$
$710$		0			0
$711$		−	38.9674i		−	1.46139i
$712$		0			0
$713$	−10.8089			−0.404797
$714$		0			0
$715$		−	3.32821i		−	0.124468i
$716$		0			0
$717$	−4.25108			−0.158760
$718$		0			0
$719$		−	17.1240i		−	0.638618i	−0.947651	−	0.319309i	$-0.896549\pi$
$719$		−	17.1240i		−	0.638618i	0.947651	−	0.319309i	$-0.103451\pi$
$720$		0			0
$721$	21.1877	+	27.4114i	0.789072	+	1.02085i
$722$		0			0
$723$	−8.18034			−0.304230
$724$		0			0
$725$			21.5407i			0.800000i
$726$		0			0
$727$			8.18450i			0.303546i	0.988415	+	0.151773i	$0.0484983\pi$
$727$			8.18450i			0.303546i	−0.988415	+	0.151773i	$0.951502\pi$
$728$		0			0
$729$	−36.7426			−1.36084
$730$		0			0
$731$			16.1357i			0.596800i
$732$		0			0
$733$			1.59915i			0.0590660i	0.999564	+	0.0295330i	$0.00940201\pi$
$733$			1.59915i			0.0590660i	−0.999564	+	0.0295330i	$0.990598\pi$
$734$		0			0
$735$	59.0254	+	15.3698i	2.17718	+	0.566924i
$736$		0			0
$737$		−	8.71338i		−	0.320961i
$738$		0			0
$739$	−13.1803			−0.484847			−0.242423	−	0.970171i	$-0.577942\pi$
$739$	−13.1803			−0.484847			−0.242423	+	0.970171i	$0.577942\pi$
$740$		0			0
$741$	−17.0252	−	5.29161i	−0.625438	−	0.194392i
$742$		0			0
$743$		−	7.92769i		−	0.290839i	−0.989370	−	0.145419i	$-0.953547\pi$
$743$		−	7.92769i		−	0.290839i	0.989370	−	0.145419i	$-0.0464532\pi$
$744$		0			0
$745$			25.3085i			0.927233i
$746$		0			0
$747$		−	59.4845i		−	2.17642i
$748$		0			0
$749$	−11.2728	+	8.71338i	−0.411901	+	0.318380i
$750$		0			0
$751$		−	34.3679i		−	1.25410i	−0.778977	−	0.627052i	$-0.784261\pi$
$751$		−	34.3679i		−	1.25410i	0.778977	−	0.627052i	$-0.215739\pi$
$752$		0			0
$753$		−	7.92769i		−	0.288901i
$754$		0			0
$755$	25.2068			0.917371
$756$		0			0
$757$	−25.8885			−0.940935			−0.470468	−	0.882417i	$-0.655915\pi$
$757$	−25.8885			−0.940935			−0.470468	+	0.882417i	$0.655915\pi$
$758$		0			0
$759$	2.34059			0.0849579
$760$		0			0
$761$		−	42.4326i		−	1.53818i	−0.639141	−	0.769089i	$-0.720710\pi$
$761$		−	42.4326i		−	1.53818i	0.639141	−	0.769089i	$-0.279290\pi$
$762$		0			0
$763$	25.2068	−	19.4837i	0.912549	−	0.705358i
$764$		0			0
$765$	25.6525			0.927467
$766$		0			0
$767$	−10.7082			−0.386651
$768$		0			0
$769$		−	25.1198i		−	0.905842i	−0.891551	−	0.452921i	$-0.850382\pi$
$769$		−	25.1198i		−	0.905842i	0.891551	−	0.452921i	$-0.149618\pi$
$770$		0			0
$771$	38.1459			1.37379
$772$		0			0
$773$	10.7542			0.386800			0.193400	−	0.981120i	$-0.438049\pi$
$773$	10.7542			0.386800			0.193400	+	0.981120i	$0.438049\pi$
$774$		0			0
$775$	−47.5206			−1.70699
$776$		0			0
$777$	29.0000	−	22.4156i	1.04037	−	0.804157i
$778$		0			0
$779$	30.5623	+	9.49906i	1.09501	+	0.340339i
$780$		0			0
$781$			7.44211i			0.266300i
$782$		0			0
$783$		−	5.29161i		−	0.189107i
$784$		0			0
$785$	58.3951			2.08421
$786$		0			0
$787$	24.9749			0.890258			0.445129	−	0.895466i	$-0.353158\pi$
$787$	24.9749			0.890258			0.445129	+	0.895466i	$0.353158\pi$
$788$		0			0
$789$	68.3330			2.43272
$790$		0			0
$791$	40.7855	−	31.5253i	1.45017	−	1.12091i
$792$		0			0
$793$			1.27126i			0.0451439i
$794$		0			0
$795$	−17.9230			−0.635663
$796$		0			0
$797$	29.8880			1.05869			0.529344	−	0.848407i	$-0.322438\pi$
$797$	29.8880			1.05869			0.529344	+	0.848407i	$0.322438\pi$
$798$		0			0
$799$	−21.2705			−0.752497
$800$		0			0
$801$	−37.2304			−1.31547
$802$		0			0
$803$		−	1.78790i		−	0.0630938i
$804$		0			0
$805$	10.4377	−	8.06785i	0.367880	−	0.284354i
$806$		0			0
$807$	28.0344			0.986859
$808$		0			0
$809$	19.0689			0.670426			0.335213	−	0.942142i	$-0.391192\pi$
$809$	19.0689			0.670426			0.335213	+	0.942142i	$0.391192\pi$
$810$		0			0
$811$	16.1860			0.568367			0.284183	−	0.958770i	$-0.408278\pi$
$811$	16.1860			0.568367			0.284183	+	0.958770i	$0.408278\pi$
$812$		0			0
$813$			33.5823i			1.17778i
$814$		0			0
$815$			39.3509i			1.37840i
$816$		0			0
$817$	32.0852	+	9.97241i	1.12252	+	0.348890i
$818$		0			0
$819$	9.30759	+	12.0416i	0.325233	+	0.420767i
$820$		0			0
$821$	−40.5410			−1.41489			−0.707446	−	0.706768i	$-0.750153\pi$
$821$	−40.5410			−1.41489			−0.707446	+	0.706768i	$0.750153\pi$
$822$		0			0
$823$	−24.6525			−0.859331			−0.429666	−	0.902988i	$-0.641369\pi$
$823$	−24.6525			−0.859331			−0.429666	+	0.902988i	$0.641369\pi$
$824$		0			0
$825$	10.2902			0.358259
$826$		0			0
$827$			36.1248i			1.25618i	0.778140	+	0.628091i	$0.216163\pi$
$827$			36.1248i			1.25618i	−0.778140	+	0.628091i	$0.783837\pi$
$828$		0			0
$829$	−26.9401			−0.935670			−0.467835	−	0.883816i	$-0.654966\pi$
$829$	−26.9401			−0.935670			−0.467835	+	0.883816i	$0.654966\pi$
$830$		0			0
$831$	−7.57430			−0.262750
$832$		0			0
$833$	−14.1803	−	3.69246i	−0.491320	−	0.127936i
$834$		0			0
$835$		−	68.4358i		−	2.36832i
$836$		0			0
$837$	11.6738			0.403504
$838$		0			0
$839$	24.5109			0.846212			0.423106	−	0.906080i	$-0.360940\pi$
$839$	24.5109			0.846212			0.423106	+	0.906080i	$0.360940\pi$
$840$		0			0
$841$	17.9230			0.618034
$842$		0			0
$843$		−	41.5609i		−	1.43143i
$844$		0			0
$845$			35.4697i			1.22019i
$846$		0			0
$847$	17.1803	+	22.2269i	0.590323	+	0.763725i
$848$		0			0
$849$		−	59.7224i		−	2.04967i
$850$		0			0
$851$		−	7.92769i		−	0.271758i
$852$		0			0
$853$		−	43.1600i		−	1.47777i	−0.673831	−	0.738886i	$-0.735352\pi$
$853$		−	43.1600i		−	1.47777i	0.673831	−	0.738886i	$-0.264648\pi$
$854$		0			0
$855$	15.8541	−	51.0090i	0.542199	−	1.74447i
$856$		0			0
$857$	−57.2921			−1.95706			−0.978530	−	0.206103i	$-0.933922\pi$
$857$	−57.2921			−1.95706			−0.978530	+	0.206103i	$0.933922\pi$
$858$		0			0
$859$		−	3.69246i		−	0.125985i	−0.998014	−	0.0629926i	$-0.979936\pi$
$859$		−	3.69246i		−	0.125985i	0.998014	−	0.0629926i	$-0.0200645\pi$
$860$		0			0
$861$	−30.5623	−	39.5397i	−1.04156	−	1.34751i
$862$		0			0
$863$			36.9105i			1.25645i	0.778033	+	0.628223i	$0.216218\pi$
$863$			36.9105i			1.25645i	−0.778033	+	0.628223i	$0.783782\pi$
$864$		0			0
$865$		−	63.0506i		−	2.14378i
$866$		0			0
$867$	32.4606			1.10242
$868$		0			0
$869$			6.65643i			0.225804i
$870$		0			0
$871$		−	22.4156i		−	0.759525i
$872$		0			0
$873$	17.2572			0.584068
$874$		0			0
$875$	10.4377	−	8.06785i	0.352858	−	0.272743i
$876$		0			0
$877$			38.9674i			1.31584i	0.753089	+	0.657918i	$0.228563\pi$
$877$			38.9674i			1.31584i	−0.753089	+	0.657918i	$0.771437\pi$
$878$		0			0
$879$	−25.5066			−0.860316
$880$		0			0
$881$		−	29.3785i		−	0.989787i	−0.868954	−	0.494893i	$-0.835207\pi$
$881$		−	29.3785i		−	0.989787i	0.868954	−	0.494893i	$-0.164793\pi$
$882$		0			0
$883$	−24.5967			−0.827746			−0.413873	−	0.910335i	$-0.635824\pi$
$883$	−24.5967			−0.827746			−0.413873	+	0.910335i	$0.635824\pi$
$884$		0			0
$885$		−	58.6849i		−	1.97267i
$886$		0			0
$887$	−47.8959			−1.60819			−0.804093	−	0.594503i	$-0.797349\pi$
$887$	−47.8959			−1.60819			−0.804093	+	0.594503i	$0.797349\pi$
$888$		0			0
$889$	22.5457	−	17.4268i	0.756158	−	0.584475i
$890$		0			0
$891$	4.18034			0.140047
$892$		0			0
$893$	−13.1459	+	42.2956i	−0.439911	+	1.41537i
$894$		0			0
$895$	−70.2982			−2.34981
$896$		0			0
$897$	6.02129			0.201045
$898$		0			0
$899$		−	24.4369i		−	0.815015i
$900$		0			0
$901$	4.30584			0.143448
$902$		0			0
$903$	−32.0852	−	41.5099i	−1.06773	−	1.38136i
$904$		0			0
$905$			55.6085i			1.84849i
$906$		0			0
$907$		−	44.3526i		−	1.47270i	−0.676599	−	0.736352i	$-0.736547\pi$
$907$		−	44.3526i		−	1.47270i	0.676599	−	0.736352i	$-0.263453\pi$
$908$		0			0
$909$			12.2545i			0.406455i
$910$		0			0
$911$		−	6.65643i		−	0.220537i	−0.993902	−	0.110269i	$-0.964829\pi$
$911$		−	6.65643i		−	0.220537i	0.993902	−	0.110269i	$-0.0351711\pi$
$912$		0			0
$913$			10.1612i			0.336285i
$914$		0			0
$915$	−6.96700			−0.230322
$916$		0			0
$917$	1.03444	−	0.799575i	0.0341603	−	0.0264043i
$918$		0			0
$919$	13.7639			0.454030			0.227015	−	0.973891i	$-0.427103\pi$
$919$	13.7639			0.454030			0.227015	+	0.973891i	$0.427103\pi$
$920$		0			0
$921$	53.5410			1.76424
$922$		0			0
$923$			19.1452i			0.630173i
$924$		0			0
$925$		−	34.8535i		−	1.14598i
$926$		0			0
$927$	−47.3772			−1.55607
$928$		0			0
$929$			20.6277i			0.676774i	0.941007	+	0.338387i	$0.109881\pi$
$929$			20.6277i			0.676774i	−0.941007	+	0.338387i	$0.890119\pi$
$930$		0			0
$931$	−16.1063	+	25.9150i	−0.527861	+	0.849330i
$932$		0			0
$933$		−	37.6962i		−	1.23412i
$934$		0			0
$935$	−4.38197			−0.143306
$936$		0			0
$937$		−	38.5513i		−	1.25942i	−0.776831	−	0.629709i	$-0.783174\pi$
$937$		−	38.5513i		−	1.25942i	0.776831	−	0.629709i	$-0.216826\pi$
$938$		0			0
$939$		−	53.8516i		−	1.75738i
$940$		0			0
$941$	4.68117			0.152602			0.0763010	−	0.997085i	$-0.475689\pi$
$941$	4.68117			0.152602			0.0763010	+	0.997085i	$0.475689\pi$
$942$		0			0
$943$	−10.8089			−0.351987
$944$		0			0
$945$	−11.2728	+	8.71338i	−0.366706	+	0.283446i
$946$		0			0
$947$	7.03444			0.228589			0.114294	−	0.993447i	$-0.463539\pi$
$947$	7.03444			0.228589			0.114294	+	0.993447i	$0.463539\pi$
$948$		0			0
$949$		−	4.59948i		−	0.149305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			38.1817i			1.23683i	0.785853	+	0.618414i	$0.212224\pi$
$953$			38.1817i			1.23683i	−0.785853	+	0.618414i	$0.787776\pi$
$954$		0			0
$955$			78.0189i			2.52463i
$956$		0			0
$957$			5.29161i			0.171054i
$958$		0			0
$959$	31.8885	+	41.2555i	1.02973	+	1.33221i
$960$		0			0
$961$	22.9098			0.739027
$962$		0			0
$963$		−	19.4837i		−	0.627854i
$964$		0			0
$965$	11.2728			0.362886
$966$		0			0
$967$	−36.9098			−1.18694			−0.593470	−	0.804856i	$-0.702242\pi$
$967$	−36.9098			−1.18694			−0.593470	+	0.804856i	$0.702242\pi$
$968$		0			0
$969$	−6.96700	+	22.4156i	−0.223812	+	0.720094i
$970$		0			0
$971$	7.48569			0.240227			0.120114	−	0.992760i	$-0.461674\pi$
$971$	7.48569			0.240227			0.120114	+	0.992760i	$0.461674\pi$
$972$		0			0
$973$	−24.6180	+	19.0286i	−0.789218	+	0.610028i
$974$		0			0
$975$	26.4721			0.847787
$976$		0			0
$977$			52.2803i			1.67259i	0.548276	+	0.836297i	$0.315284\pi$
$977$			52.2803i			1.67259i	−0.548276	+	0.836297i	$0.684716\pi$
$978$		0			0
$979$	6.35970			0.203257
$980$		0			0
$981$			43.5669i			1.39098i
$982$		0			0
$983$	−32.4606			−1.03533			−0.517666	−	0.855583i	$-0.673199\pi$
$983$	−32.4606			−1.03533			−0.517666	+	0.855583i	$0.673199\pi$
$984$		0			0
$985$		−	1.29374i		−	0.0412220i
$986$		0			0
$987$	54.7195	−	42.2956i	1.74174	−	1.34628i
$988$		0			0
$989$	−11.3475			−0.360830
$990$		0			0
$991$		−	24.0832i		−	0.765028i	−0.923950	−	0.382514i	$-0.875058\pi$
$991$		−	24.0832i		−	0.765028i	0.923950	−	0.382514i	$-0.124942\pi$
$992$		0			0
$993$			39.5397i			1.25475i
$994$		0			0
$995$	−80.3050			−2.54584
$996$		0			0
$997$		−	12.2545i		−	0.388103i	−0.980991	−	0.194052i	$-0.937837\pi$
$997$		−	12.2545i		−	0.388103i	0.980991	−	0.194052i	$-0.0621629\pi$
$998$		0			0
$999$			8.56201i			0.270890i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	532.2.g.b.265.7	yes	8
3.2	odd	2		4788.2.i.e.3457.7		8
4.3	odd	2		2128.2.m.e.1329.1		8
7.6	odd	2	inner	532.2.g.b.265.2	yes	8
19.18	odd	2	inner	532.2.g.b.265.1	✓	8
21.20	even	2		4788.2.i.e.3457.2		8
28.27	even	2		2128.2.m.e.1329.8		8
57.56	even	2		4788.2.i.e.3457.8		8
76.75	even	2		2128.2.m.e.1329.7		8
133.132	even	2	inner	532.2.g.b.265.8	yes	8
399.398	odd	2		4788.2.i.e.3457.1		8
532.531	odd	2		2128.2.m.e.1329.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
532.2.g.b.265.1	✓	8	19.18	odd	2	inner
532.2.g.b.265.2	yes	8	7.6	odd	2	inner
532.2.g.b.265.7	yes	8	1.1	even	1	trivial
532.2.g.b.265.8	yes	8	133.132	even	2	inner
2128.2.m.e.1329.1		8	4.3	odd	2
2128.2.m.e.1329.2		8	532.531	odd	2
2128.2.m.e.1329.7		8	76.75	even	2
2128.2.m.e.1329.8		8	28.27	even	2
4788.2.i.e.3457.1		8	399.398	odd	2
4788.2.i.e.3457.2		8	21.20	even	2
4788.2.i.e.3457.7		8	3.2	odd	2
4788.2.i.e.3457.8		8	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	532.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.57255 q^{3} -3.38705i q^{5} +(-1.61803 - 2.09331i) q^{7} +3.61803 q^{9} -0.618034 q^{11} -1.58993 q^{13} -8.71338i q^{15} +2.09331i q^{17} +(4.16248 + 1.29374i) q^{19} +(-4.16248 - 5.38516i) q^{21} -1.47214 q^{23} -6.47214 q^{25} +1.58993 q^{27} -3.32821i q^{29} +7.34233 q^{31} -1.58993 q^{33} +(-7.09017 + 5.48037i) q^{35} +5.38516i q^{37} -4.09017 q^{39} +7.34233 q^{41} +7.70820 q^{43} -12.2545i q^{45} +10.1612i q^{47} +(-1.76393 + 6.77411i) q^{49} +5.38516i q^{51} -2.05695i q^{53} +2.09331i q^{55} +(10.7082 + 3.32821i) q^{57} +6.73503 q^{59} -0.799575i q^{61} +(-5.85410 - 7.57368i) q^{63} +5.38516i q^{65} +14.0985i q^{67} -3.78715 q^{69} -12.0416i q^{71} +2.89289i q^{73} -16.6499 q^{75} +(1.00000 + 1.29374i) q^{77} -10.7703i q^{79} -6.76393 q^{81} -16.4411i q^{83} +7.09017 q^{85} -8.56201i q^{87} -10.2902 q^{89} +(2.57255 + 3.32821i) q^{91} +18.8885 q^{93} +(4.38197 - 14.0985i) q^{95} +4.76978 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{7} + 20 q^{9} + 4 q^{11} + 24 q^{23} - 16 q^{25} - 12 q^{35} + 12 q^{39} + 8 q^{43} - 32 q^{49} + 32 q^{57} - 20 q^{63} + 8 q^{77} - 72 q^{81} + 12 q^{85} + 8 q^{93} + 44 q^{95}+O(q^{100}) \) 8 * q - 4 * q^7 + 20 * q^9 + 4 * q^11 + 24 * q^23 - 16 * q^25 - 12 * q^35 + 12 * q^39 + 8 * q^43 - 32 * q^49 + 32 * q^57 - 20 * q^63 + 8 * q^77 - 72 * q^81 + 12 * q^85 + 8 * q^93 + 44 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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