LMFDB - Embedded newform 1632.2.l.a.1393.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1632,2,Mod(1393,1632)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1632, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1632.1393");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1632 = 2^{5} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1632.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$13.0315856099$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - x^{16} - 2x^{14} + 2x^{12} - 4x^{11} + 4x^{10} + 8x^{8} - 16x^{7} + 16x^{6} - 64x^{4} - 128x^{2} + 512 $ x^18 - x^16 - 2x^14 + 2x^12 - 4x^11 + 4x^10 + 8x^8 - 16x^7 + 16x^6 - 64x^4 - 128*x^2 + 512
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 408)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1393.18
Root			$0.937200 - 1.05908i$ of defining polynomial
Character	$\chi$	$=$	1632.1393
Dual form			1632.2.l.a.1393.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +4.06326 q^{5} +1.33474i q^{7} +1.00000 q^{9} +1.38281 q^{11} +2.52933i q^{13} -4.06326 q^{15} +(0.873177 + 4.02959i) q^{17} +4.60881i q^{19} -1.33474i q^{21} -3.08302i q^{23} +11.5101 q^{25} -1.00000 q^{27} -4.59613 q^{29} +2.64041i q^{31} -1.38281 q^{33} +5.42340i q^{35} -7.56937 q^{37} -2.52933i q^{39} -1.67814i q^{41} +10.9014i q^{43} +4.06326 q^{45} +2.19704 q^{47} +5.21846 q^{49} +(-0.873177 - 4.02959i) q^{51} -3.93905i q^{53} +5.61870 q^{55} -4.60881i q^{57} -12.8504i q^{59} -7.00079 q^{61} +1.33474i q^{63} +10.2773i q^{65} -2.01087i q^{67} +3.08302i q^{69} -3.38152i q^{71} -12.6981i q^{73} -11.5101 q^{75} +1.84569i q^{77} -10.0914i q^{79} +1.00000 q^{81} +13.3967i q^{83} +(3.54794 + 16.3732i) q^{85} +4.59613 q^{87} +10.8271 q^{89} -3.37601 q^{91} -2.64041i q^{93} +18.7268i q^{95} +11.8383i q^{97} +1.38281 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 18 q^{3} - 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} - 18 q^{27} + 12 q^{29} - 16 q^{37} - 4 q^{45} - 18 q^{49} + 2 q^{51} + 16 q^{55} - 16 q^{61} - 22 q^{75} + 18 q^{81} + 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) $ 18 * q - 18 * q^3 - 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 - 18 * q^27 + 12 * q^29 - 16 * q^37 - 4 * q^45 - 18 * q^49 + 2 * q^51 + 16 * q^55 - 16 * q^61 - 22 * q^75 + 18 * q^81 + 16 * q^85 - 12 * q^87 + 20 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1632\mathbb{Z}\right)^\times$.

$n$	$511$	$545$	$613$	$1057$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$		0			0
$5$	4.06326			1.81714			0.908572	−	0.417728i	$-0.137173\pi$
$5$	4.06326			1.81714			0.908572	+	0.417728i	$0.137173\pi$
$6$		0			0
$7$			1.33474i			0.504485i	0.967664	+	0.252243i	$0.0811681\pi$
$7$			1.33474i			0.504485i	−0.967664	+	0.252243i	$0.918832\pi$
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$	1.38281			0.416932			0.208466	−	0.978030i	$-0.433153\pi$
$11$	1.38281			0.416932			0.208466	+	0.978030i	$0.433153\pi$
$12$		0			0
$13$			2.52933i			0.701511i	0.936467	+	0.350756i	$0.114075\pi$
$13$			2.52933i			0.701511i	−0.936467	+	0.350756i	$0.885925\pi$
$14$		0			0
$15$	−4.06326			−1.04913
$16$		0			0
$17$	0.873177	+	4.02959i	0.211777	+	0.977318i
$17$	0.873177	+	4.02959i	0.211777	+	0.977318i
$18$		0			0
$19$			4.60881i			1.05733i	0.848830	+	0.528667i	$0.177308\pi$
$19$			4.60881i			1.05733i	−0.848830	+	0.528667i	$0.822692\pi$
$20$		0			0
$21$		−	1.33474i		−	0.291265i
$22$		0			0
$23$		−	3.08302i		−	0.642854i	−0.946934	−	0.321427i	$-0.895837\pi$
$23$		−	3.08302i		−	0.642854i	0.946934	−	0.321427i	$-0.104163\pi$
$24$		0			0
$25$	11.5101			2.30201
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−4.59613			−0.853479			−0.426740	−	0.904374i	$-0.640338\pi$
$29$	−4.59613			−0.853479			−0.426740	+	0.904374i	$0.640338\pi$
$30$		0			0
$31$			2.64041i			0.474231i	0.971481	+	0.237116i	$0.0762020\pi$
$31$			2.64041i			0.474231i	−0.971481	+	0.237116i	$0.923798\pi$
$32$		0			0
$33$	−1.38281			−0.240716
$34$		0			0
$35$			5.42340i			0.916722i
$36$		0			0
$37$	−7.56937			−1.24440			−0.622198	−	0.782860i	$-0.713760\pi$
$37$	−7.56937			−1.24440			−0.622198	+	0.782860i	$0.713760\pi$
$38$		0			0
$39$		−	2.52933i		−	0.405018i
$40$		0			0
$41$		−	1.67814i		−	0.262082i	−0.991377	−	0.131041i	$-0.958168\pi$
$41$		−	1.67814i		−	0.262082i	0.991377	−	0.131041i	$-0.0418320\pi$
$42$		0			0
$43$			10.9014i			1.66245i	0.555939	+	0.831223i	$0.312359\pi$
$43$			10.9014i			1.66245i	−0.555939	+	0.831223i	$0.687641\pi$
$44$		0			0
$45$	4.06326			0.605715
$46$		0			0
$47$	2.19704			0.320471			0.160235	−	0.987079i	$-0.448775\pi$
$47$	2.19704			0.320471			0.160235	+	0.987079i	$0.448775\pi$
$48$		0			0
$49$	5.21846			0.745495
$50$		0			0
$51$	−0.873177	−	4.02959i	−0.122269	−	0.564255i
$52$		0			0
$53$		−	3.93905i		−	0.541070i	−0.962710	−	0.270535i	$-0.912799\pi$
$53$		−	3.93905i		−	0.541070i	0.962710	−	0.270535i	$-0.0872006\pi$
$54$		0			0
$55$	5.61870			0.757625
$56$		0			0
$57$		−	4.60881i		−	0.610452i
$58$		0			0
$59$		−	12.8504i		−	1.67298i	−0.547979	−	0.836492i	$-0.684603\pi$
$59$		−	12.8504i		−	1.67298i	0.547979	−	0.836492i	$-0.315397\pi$
$60$		0			0
$61$	−7.00079			−0.896360			−0.448180	−	0.893943i	$-0.647928\pi$
$61$	−7.00079			−0.896360			−0.448180	+	0.893943i	$0.647928\pi$
$62$		0			0
$63$			1.33474i			0.168162i
$64$		0			0
$65$			10.2773i			1.27475i
$66$		0			0
$67$		−	2.01087i		−	0.245666i	−0.992427	−	0.122833i	$-0.960802\pi$
$67$		−	2.01087i		−	0.245666i	0.992427	−	0.122833i	$-0.0391980\pi$
$68$		0			0
$69$			3.08302i			0.371152i
$70$		0			0
$71$		−	3.38152i		−	0.401312i	−0.979662	−	0.200656i	$-0.935693\pi$
$71$		−	3.38152i		−	0.401312i	0.979662	−	0.200656i	$-0.0643074\pi$
$72$		0			0
$73$		−	12.6981i		−	1.48620i	−0.669179	−	0.743101i	$-0.733354\pi$
$73$		−	12.6981i		−	1.48620i	0.669179	−	0.743101i	$-0.266646\pi$
$74$		0			0
$75$	−11.5101			−1.32907
$76$		0			0
$77$			1.84569i			0.210336i
$78$		0			0
$79$		−	10.0914i		−	1.13537i	−0.823244	−	0.567687i	$-0.807838\pi$
$79$		−	10.0914i		−	1.13537i	0.823244	−	0.567687i	$-0.192162\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			13.3967i			1.47048i	0.677808	+	0.735239i	$0.262930\pi$
$83$			13.3967i			1.47048i	−0.677808	+	0.735239i	$0.737070\pi$
$84$		0			0
$85$	3.54794	+	16.3732i	0.384829	+	1.77593i
$86$		0			0
$87$	4.59613			0.492757
$88$		0			0
$89$	10.8271			1.14767			0.573837	−	0.818970i	$-0.305454\pi$
$89$	10.8271			1.14767			0.573837	+	0.818970i	$0.305454\pi$
$90$		0			0
$91$	−3.37601			−0.353902
$92$		0			0
$93$		−	2.64041i		−	0.273797i
$94$		0			0
$95$			18.7268i			1.92133i
$96$		0			0
$97$			11.8383i			1.20200i	0.799248	+	0.601001i	$0.205231\pi$
$97$			11.8383i			1.20200i	−0.799248	+	0.601001i	$0.794769\pi$
$98$		0			0
$99$	1.38281			0.138977
$100$		0			0
$101$			19.2918i			1.91960i	0.280683	+	0.959801i	$0.409439\pi$
$101$			19.2918i			1.91960i	−0.280683	+	0.959801i	$0.590561\pi$
$102$		0			0
$103$	11.3576			1.11909			0.559547	−	0.828799i	$-0.310975\pi$
$103$	11.3576			1.11909			0.559547	+	0.828799i	$0.310975\pi$
$104$		0			0
$105$		−	5.42340i		−	0.529270i
$106$		0			0
$107$	0.539159			0.0521224			0.0260612	−	0.999660i	$-0.491704\pi$
$107$	0.539159			0.0521224			0.0260612	+	0.999660i	$0.491704\pi$
$108$		0			0
$109$	−9.10216			−0.871828			−0.435914	−	0.899988i	$-0.643575\pi$
$109$	−9.10216			−0.871828			−0.435914	+	0.899988i	$0.643575\pi$
$110$		0			0
$111$	7.56937			0.718453
$112$		0			0
$113$			6.53188i			0.614468i	0.951634	+	0.307234i	$0.0994034\pi$
$113$			6.53188i			0.614468i	−0.951634	+	0.307234i	$0.900597\pi$
$114$		0			0
$115$		−	12.5271i		−	1.16816i
$116$		0			0
$117$			2.52933i			0.233837i
$118$		0			0
$119$	−5.37846	+	1.16547i	−0.493042	+	0.106838i
$120$		0			0
$121$	−9.08785			−0.826168
$122$		0			0
$123$			1.67814i			0.151313i
$124$		0			0
$125$	26.4521			2.36595
$126$		0			0
$127$	12.5968			1.11779			0.558894	−	0.829239i	$-0.311226\pi$
$127$	12.5968			1.11779			0.558894	+	0.829239i	$0.311226\pi$
$128$		0			0
$129$		−	10.9014i		−	0.959814i
$130$		0			0
$131$	15.0923			1.31862			0.659311	−	0.751870i	$-0.270848\pi$
$131$	15.0923			1.31862			0.659311	+	0.751870i	$0.270848\pi$
$132$		0			0
$133$	−6.15157			−0.533409
$134$		0			0
$135$	−4.06326			−0.349710
$136$		0			0
$137$	1.12940			0.0964912			0.0482456	−	0.998836i	$-0.484637\pi$
$137$	1.12940			0.0964912			0.0482456	+	0.998836i	$0.484637\pi$
$138$		0			0
$139$	10.6515			0.903450			0.451725	−	0.892157i	$-0.350809\pi$
$139$	10.6515			0.903450			0.451725	+	0.892157i	$0.350809\pi$
$140$		0			0
$141$	−2.19704			−0.185024
$142$		0			0
$143$			3.49758i			0.292482i
$144$		0			0
$145$	−18.6753			−1.55090
$146$		0			0
$147$	−5.21846			−0.430412
$148$		0			0
$149$		−	0.280184i		−	0.0229536i	−0.999934	−	0.0114768i	$-0.996347\pi$
$149$		−	0.280184i		−	0.0229536i	0.999934	−	0.0114768i	$-0.00365325\pi$
$150$		0			0
$151$	−9.48355			−0.771760			−0.385880	−	0.922549i	$-0.626102\pi$
$151$	−9.48355			−0.771760			−0.385880	+	0.922549i	$0.626102\pi$
$152$		0			0
$153$	0.873177	+	4.02959i	0.0705922	+	0.325773i
$154$		0			0
$155$			10.7287i			0.861746i
$156$		0			0
$157$		−	23.6841i		−	1.89019i	−0.326789	−	0.945097i	$-0.605967\pi$
$157$		−	23.6841i		−	1.89019i	0.326789	−	0.945097i	$-0.394033\pi$
$158$		0			0
$159$			3.93905i			0.312387i
$160$		0			0
$161$	4.11504			0.324310
$162$		0			0
$163$	18.4216			1.44289			0.721446	−	0.692471i	$-0.243478\pi$
$163$	18.4216			1.44289			0.721446	+	0.692471i	$0.243478\pi$
$164$		0			0
$165$	−5.61870			−0.437415
$166$		0			0
$167$		−	9.89324i		−	0.765562i	−0.923839	−	0.382781i	$-0.874966\pi$
$167$		−	9.89324i		−	0.765562i	0.923839	−	0.382781i	$-0.125034\pi$
$168$		0			0
$169$	6.60246			0.507882
$170$		0			0
$171$			4.60881i			0.352444i
$172$		0			0
$173$	18.5486			1.41022			0.705111	−	0.709097i	$-0.250897\pi$
$173$	18.5486			1.41022			0.705111	+	0.709097i	$0.250897\pi$
$174$		0			0
$175$			15.3630i			1.16133i
$176$		0			0
$177$			12.8504i			0.965897i
$178$		0			0
$179$			9.68489i			0.723883i	0.932201	+	0.361941i	$0.117886\pi$
$179$			9.68489i			0.723883i	−0.932201	+	0.361941i	$0.882114\pi$
$180$		0			0
$181$	6.22823			0.462941			0.231470	−	0.972842i	$-0.425646\pi$
$181$	6.22823			0.462941			0.231470	+	0.972842i	$0.425646\pi$
$182$		0			0
$183$	7.00079			0.517514
$184$		0			0
$185$	−30.7563			−2.26125
$186$		0			0
$187$	1.20743	+	5.57214i	0.0882964	+	0.407475i
$188$		0			0
$189$		−	1.33474i		−	0.0970882i
$190$		0			0
$191$	18.5765			1.34415			0.672074	−	0.740484i	$-0.265404\pi$
$191$	18.5765			1.34415			0.672074	+	0.740484i	$0.265404\pi$
$192$		0			0
$193$			13.6072i			0.979471i	0.871871	+	0.489736i	$0.162907\pi$
$193$			13.6072i			0.979471i	−0.871871	+	0.489736i	$0.837093\pi$
$194$		0			0
$195$		−	10.2773i		−	0.735976i
$196$		0			0
$197$	−7.88624			−0.561871			−0.280936	−	0.959727i	$-0.590645\pi$
$197$	−7.88624			−0.561871			−0.280936	+	0.959727i	$0.590645\pi$
$198$		0			0
$199$		−	19.7275i		−	1.39844i	−0.714906	−	0.699221i	$-0.753530\pi$
$199$		−	19.7275i		−	1.39844i	0.714906	−	0.699221i	$-0.246470\pi$
$200$		0			0
$201$			2.01087i			0.141836i
$202$		0			0
$203$		−	6.13465i		−	0.430568i
$204$		0			0
$205$		−	6.81873i		−	0.476241i
$206$		0			0
$207$		−	3.08302i		−	0.214285i
$208$		0			0
$209$			6.37309i			0.440836i
$210$		0			0
$211$	−23.9727			−1.65035			−0.825175	−	0.564877i	$-0.808924\pi$
$211$	−23.9727			−1.65035			−0.825175	+	0.564877i	$0.808924\pi$
$212$		0			0
$213$			3.38152i			0.231698i
$214$		0			0
$215$			44.2952i			3.02090i
$216$		0			0
$217$	−3.52426			−0.239243
$218$		0			0
$219$			12.6981i			0.858060i
$220$		0			0
$221$	−10.1922	+	2.20856i	−0.685600	+	0.148564i
$222$		0			0
$223$	−17.6208			−1.17997			−0.589987	−	0.807413i	$-0.700867\pi$
$223$	−17.6208			−1.17997			−0.589987	+	0.807413i	$0.700867\pi$
$224$		0			0
$225$	11.5101			0.767338
$226$		0			0
$227$	−14.7789			−0.980911			−0.490456	−	0.871466i	$-0.663170\pi$
$227$	−14.7789			−0.980911			−0.490456	+	0.871466i	$0.663170\pi$
$228$		0			0
$229$		−	21.7445i		−	1.43692i	−0.695569	−	0.718459i	$-0.744848\pi$
$229$		−	21.7445i		−	1.43692i	0.695569	−	0.718459i	$-0.255152\pi$
$230$		0			0
$231$		−	1.84569i		−	0.121438i
$232$		0			0
$233$		−	10.4914i		−	0.687314i	−0.939095	−	0.343657i	$-0.888334\pi$
$233$		−	10.4914i		−	0.687314i	0.939095	−	0.343657i	$-0.111666\pi$
$234$		0			0
$235$	8.92713			0.582342
$236$		0			0
$237$			10.0914i			0.655509i
$238$		0			0
$239$	−14.7604			−0.954770			−0.477385	−	0.878694i	$-0.658415\pi$
$239$	−14.7604			−0.954770			−0.477385	+	0.878694i	$0.658415\pi$
$240$		0			0
$241$		−	21.7730i		−	1.40252i	−0.712903	−	0.701262i	$-0.752620\pi$
$241$		−	21.7730i		−	1.40252i	0.712903	−	0.701262i	$-0.247380\pi$
$242$		0			0
$243$	−1.00000			−0.0641500
$244$		0			0
$245$	21.2040			1.35467
$246$		0			0
$247$	−11.6572			−0.741731
$248$		0			0
$249$		−	13.3967i		−	0.848981i
$250$		0			0
$251$		−	18.0029i		−	1.13634i	−0.822913	−	0.568168i	$-0.807653\pi$
$251$		−	18.0029i		−	1.13634i	0.822913	−	0.568168i	$-0.192347\pi$
$252$		0			0
$253$		−	4.26322i		−	0.268026i
$254$		0			0
$255$	−3.54794	−	16.3732i	−0.222181	−	1.02533i
$256$		0			0
$257$	4.44516			0.277281			0.138641	−	0.990343i	$-0.455727\pi$
$257$	4.44516			0.277281			0.138641	+	0.990343i	$0.455727\pi$
$258$		0			0
$259$		−	10.1032i		−	0.627780i
$260$		0			0
$261$	−4.59613			−0.284493
$262$		0			0
$263$	18.7200			1.15433			0.577163	−	0.816629i	$-0.304160\pi$
$263$	18.7200			1.15433			0.577163	+	0.816629i	$0.304160\pi$
$264$		0			0
$265$		−	16.0054i		−	0.983203i
$266$		0			0
$267$	−10.8271			−0.662610
$268$		0			0
$269$	9.33536			0.569187			0.284594	−	0.958648i	$-0.408141\pi$
$269$	9.33536			0.569187			0.284594	+	0.958648i	$0.408141\pi$
$270$		0			0
$271$	−12.0895			−0.734388			−0.367194	−	0.930144i	$-0.619681\pi$
$271$	−12.0895			−0.734388			−0.367194	+	0.930144i	$0.619681\pi$
$272$		0			0
$273$	3.37601			0.204325
$274$		0			0
$275$	15.9162			0.959783
$276$		0			0
$277$	−21.2973			−1.27963			−0.639816	−	0.768528i	$-0.720989\pi$
$277$	−21.2973			−1.27963			−0.639816	+	0.768528i	$0.720989\pi$
$278$		0			0
$279$			2.64041i			0.158077i
$280$		0			0
$281$	7.37238			0.439799			0.219900	−	0.975522i	$-0.429427\pi$
$281$	7.37238			0.439799			0.219900	+	0.975522i	$0.429427\pi$
$282$		0			0
$283$	−22.3443			−1.32823			−0.664114	−	0.747631i	$-0.731191\pi$
$283$	−22.3443			−1.32823			−0.664114	+	0.747631i	$0.731191\pi$
$284$		0			0
$285$		−	18.7268i		−	1.10928i
$286$		0			0
$287$	2.23989			0.132216
$288$		0			0
$289$	−15.4751	+	7.03708i	−0.910301	+	0.413946i
$290$		0			0
$291$		−	11.8383i		−	0.693976i
$292$		0			0
$293$		−	0.823580i		−	0.0481141i	−0.999711	−	0.0240570i	$-0.992342\pi$
$293$		−	0.823580i		−	0.0481141i	0.999711	−	0.0240570i	$-0.00765833\pi$
$294$		0			0
$295$		−	52.2146i		−	3.04005i
$296$		0			0
$297$	−1.38281			−0.0802386
$298$		0			0
$299$	7.79799			0.450969
$300$		0			0
$301$	−14.5505			−0.838679
$302$		0			0
$303$		−	19.2918i		−	1.10828i
$304$		0			0
$305$	−28.4460			−1.62882
$306$		0			0
$307$		−	1.85701i		−	0.105985i	−0.998595	−	0.0529925i	$-0.983124\pi$
$307$		−	1.85701i		−	0.105985i	0.998595	−	0.0529925i	$-0.0168759\pi$
$308$		0			0
$309$	−11.3576			−0.646109
$310$		0			0
$311$		−	4.96908i		−	0.281771i	−0.990026	−	0.140885i	$-0.955005\pi$
$311$		−	4.96908i		−	0.281771i	0.990026	−	0.140885i	$-0.0449949\pi$
$312$		0			0
$313$			4.54467i			0.256880i	0.991717	+	0.128440i	$0.0409969\pi$
$313$			4.54467i			0.256880i	−0.991717	+	0.128440i	$0.959003\pi$
$314$		0			0
$315$			5.42340i			0.305574i
$316$		0			0
$317$	−24.5554			−1.37917			−0.689585	−	0.724205i	$-0.742207\pi$
$317$	−24.5554			−1.37917			−0.689585	+	0.724205i	$0.742207\pi$
$318$		0			0
$319$	−6.35556			−0.355843
$320$		0			0
$321$	−0.539159			−0.0300929
$322$		0			0
$323$	−18.5716	+	4.02431i	−1.03335	+	0.223918i
$324$		0			0
$325$			29.1128i			1.61489i
$326$		0			0
$327$	9.10216			0.503350
$328$		0			0
$329$			2.93248i			0.161673i
$330$		0			0
$331$		−	23.9681i		−	1.31740i	−0.752404	−	0.658702i	$-0.771106\pi$
$331$		−	23.9681i		−	1.31740i	0.752404	−	0.658702i	$-0.228894\pi$
$332$		0			0
$333$	−7.56937			−0.414799
$334$		0			0
$335$		−	8.17067i		−	0.446411i
$336$		0			0
$337$		−	31.1833i		−	1.69866i	−0.527861	−	0.849331i	$-0.677006\pi$
$337$		−	31.1833i		−	1.69866i	0.527861	−	0.849331i	$-0.322994\pi$
$338$		0			0
$339$		−	6.53188i		−	0.354763i
$340$		0			0
$341$			3.65117i			0.197722i
$342$		0			0
$343$			16.3085i			0.880576i
$344$		0			0
$345$			12.5271i			0.674437i
$346$		0			0
$347$	29.7572			1.59745			0.798726	−	0.601695i	$-0.205508\pi$
$347$	29.7572			1.59745			0.798726	+	0.601695i	$0.205508\pi$
$348$		0			0
$349$		−	17.3027i		−	0.926191i	−0.886308	−	0.463095i	$-0.846739\pi$
$349$		−	17.3027i		−	0.926191i	0.886308	−	0.463095i	$-0.153261\pi$
$350$		0			0
$351$		−	2.52933i		−	0.135006i
$352$		0			0
$353$	16.4532			0.875713			0.437857	−	0.899045i	$-0.355738\pi$
$353$	16.4532			0.875713			0.437857	+	0.899045i	$0.355738\pi$
$354$		0			0
$355$		−	13.7400i		−	0.729242i
$356$		0			0
$357$	5.37846	−	1.16547i	0.284658	−	0.0616830i
$358$		0			0
$359$	−23.2851			−1.22894			−0.614472	−	0.788939i	$-0.710631\pi$
$359$	−23.2851			−1.22894			−0.614472	+	0.788939i	$0.710631\pi$
$360$		0			0
$361$	−2.24111			−0.117953
$362$		0			0
$363$	9.08785			0.476988
$364$		0			0
$365$		−	51.5958i		−	2.70065i
$366$		0			0
$367$			4.59048i			0.239621i	0.992797	+	0.119811i	$0.0382287\pi$
$367$			4.59048i			0.239621i	−0.992797	+	0.119811i	$0.961771\pi$
$368$		0			0
$369$		−	1.67814i		−	0.0873607i
$370$		0			0
$371$	5.25762			0.272962
$372$		0			0
$373$			15.3977i			0.797263i	0.917111	+	0.398631i	$0.130515\pi$
$373$			15.3977i			0.797263i	−0.917111	+	0.398631i	$0.869485\pi$
$374$		0			0
$375$	−26.4521			−1.36598
$376$		0			0
$377$		−	11.6251i		−	0.598726i
$378$		0			0
$379$	−1.84157			−0.0945951			−0.0472975	−	0.998881i	$-0.515061\pi$
$379$	−1.84157			−0.0945951			−0.0472975	+	0.998881i	$0.515061\pi$
$380$		0			0
$381$	−12.5968			−0.645356
$382$		0			0
$383$	−34.1856			−1.74680			−0.873401	−	0.487002i	$-0.838090\pi$
$383$	−34.1856			−1.74680			−0.873401	+	0.487002i	$0.838090\pi$
$384$		0			0
$385$			7.49952i			0.382211i
$386$		0			0
$387$			10.9014i			0.554149i
$388$		0			0
$389$			10.1711i			0.515693i	0.966186	+	0.257846i	$0.0830129\pi$
$389$			10.1711i			0.515693i	−0.966186	+	0.257846i	$0.916987\pi$
$390$		0			0
$391$	12.4233	−	2.69202i	0.628273	−	0.136141i
$392$		0			0
$393$	−15.0923			−0.761307
$394$		0			0
$395$		−	41.0041i		−	2.06314i
$396$		0			0
$397$	24.1191			1.21050			0.605252	−	0.796034i	$-0.293072\pi$
$397$	24.1191			1.21050			0.605252	+	0.796034i	$0.293072\pi$
$398$		0			0
$399$	6.15157			0.307964
$400$		0			0
$401$		−	12.1154i		−	0.605013i	−0.953147	−	0.302506i	$-0.902177\pi$
$401$		−	12.1154i		−	0.605013i	0.953147	−	0.302506i	$-0.0978234\pi$
$402$		0			0
$403$	−6.67847			−0.332678
$404$		0			0
$405$	4.06326			0.201905
$406$		0			0
$407$	−10.4670			−0.518829
$408$		0			0
$409$	5.35379			0.264728			0.132364	−	0.991201i	$-0.457743\pi$
$409$	5.35379			0.264728			0.132364	+	0.991201i	$0.457743\pi$
$410$		0			0
$411$	−1.12940			−0.0557092
$412$		0			0
$413$	17.1520			0.843995
$414$		0			0
$415$			54.4342i			2.67207i
$416$		0			0
$417$	−10.6515			−0.521607
$418$		0			0
$419$	−18.3063			−0.894322			−0.447161	−	0.894454i	$-0.647565\pi$
$419$	−18.3063			−0.894322			−0.447161	+	0.894454i	$0.647565\pi$
$420$		0			0
$421$			2.57989i			0.125736i	0.998022	+	0.0628682i	$0.0200248\pi$
$421$			2.57989i			0.125736i	−0.998022	+	0.0628682i	$0.979975\pi$
$422$		0			0
$423$	2.19704			0.106824
$424$		0			0
$425$	10.0503	+	46.3808i	0.487512	+	2.24980i
$426$		0			0
$427$		−	9.34425i		−	0.452200i
$428$		0			0
$429$		−	3.49758i		−	0.168865i
$430$		0			0
$431$		−	23.8390i		−	1.14829i	−0.818755	−	0.574143i	$-0.805335\pi$
$431$		−	23.8390i		−	1.14829i	0.818755	−	0.574143i	$-0.194665\pi$
$432$		0			0
$433$	19.7075			0.947084			0.473542	−	0.880771i	$-0.342975\pi$
$433$	19.7075			0.947084			0.473542	+	0.880771i	$0.342975\pi$
$434$		0			0
$435$	18.6753			0.895410
$436$		0			0
$437$	14.2090			0.679711
$438$		0			0
$439$		−	11.3880i		−	0.543518i	−0.962365	−	0.271759i	$-0.912395\pi$
$439$		−	11.3880i		−	0.543518i	0.962365	−	0.271759i	$-0.0876054\pi$
$440$		0			0
$441$	5.21846			0.248498
$442$		0			0
$443$			3.77327i			0.179273i	0.995975	+	0.0896367i	$0.0285706\pi$
$443$			3.77327i			0.179273i	−0.995975	+	0.0896367i	$0.971429\pi$
$444$		0			0
$445$	43.9934			2.08549
$446$		0			0
$447$			0.280184i			0.0132522i
$448$		0			0
$449$		−	16.5349i		−	0.780331i	−0.920745	−	0.390165i	$-0.872418\pi$
$449$		−	16.5349i		−	0.780331i	0.920745	−	0.390165i	$-0.127582\pi$
$450$		0			0
$451$		−	2.32055i		−	0.109270i
$452$		0			0
$453$	9.48355			0.445576
$454$		0			0
$455$	−13.7176			−0.643091
$456$		0			0
$457$	−7.78148			−0.364002			−0.182001	−	0.983298i	$-0.558257\pi$
$457$	−7.78148			−0.364002			−0.182001	+	0.983298i	$0.558257\pi$
$458$		0			0
$459$	−0.873177	−	4.02959i	−0.0407564	−	0.188085i
$460$		0			0
$461$		−	9.02457i		−	0.420316i	−0.977667	−	0.210158i	$-0.932602\pi$
$461$		−	9.02457i		−	0.420316i	0.977667	−	0.210158i	$-0.0673978\pi$
$462$		0			0
$463$	−4.13029			−0.191951			−0.0959754	−	0.995384i	$-0.530597\pi$
$463$	−4.13029			−0.191951			−0.0959754	+	0.995384i	$0.530597\pi$
$464$		0			0
$465$		−	10.7287i		−	0.497530i
$466$		0			0
$467$			26.5850i			1.23021i	0.788447	+	0.615103i	$0.210885\pi$
$467$			26.5850i			1.23021i	−0.788447	+	0.615103i	$0.789115\pi$
$468$		0			0
$469$	2.68399			0.123935
$470$		0			0
$471$			23.6841i			1.09130i
$472$		0			0
$473$			15.0745i			0.693127i
$474$		0			0
$475$			53.0477i			2.43399i
$476$		0			0
$477$		−	3.93905i		−	0.180357i
$478$		0			0
$479$			9.58908i			0.438136i	0.975710	+	0.219068i	$0.0703017\pi$
$479$			9.58908i			0.438136i	−0.975710	+	0.219068i	$0.929698\pi$
$480$		0			0
$481$		−	19.1455i		−	0.872958i
$482$		0			0
$483$	−4.11504			−0.187241
$484$		0			0
$485$			48.1023i			2.18421i
$486$		0			0
$487$		−	11.3135i		−	0.512663i	−0.966589	−	0.256331i	$-0.917486\pi$
$487$		−	11.3135i		−	0.512663i	0.966589	−	0.256331i	$-0.0825138\pi$
$488$		0			0
$489$	−18.4216			−0.833054
$490$		0			0
$491$		−	12.2851i		−	0.554421i	−0.960809	−	0.277210i	$-0.910590\pi$
$491$		−	12.2851i		−	0.554421i	0.960809	−	0.277210i	$-0.0894099\pi$
$492$		0			0
$493$	−4.01323	−	18.5205i	−0.180747	−	0.834121i
$494$		0			0
$495$	5.61870			0.252542
$496$		0			0
$497$	4.51345			0.202456
$498$		0			0
$499$	7.65944			0.342884			0.171442	−	0.985194i	$-0.445157\pi$
$499$	7.65944			0.342884			0.171442	+	0.985194i	$0.445157\pi$
$500$		0			0
$501$			9.89324i			0.441997i
$502$		0			0
$503$			39.8142i			1.77523i	0.460589	+	0.887614i	$0.347638\pi$
$503$			39.8142i			1.77523i	−0.460589	+	0.887614i	$0.652362\pi$
$504$		0			0
$505$			78.3874i			3.48819i
$506$		0			0
$507$	−6.60246			−0.293226
$508$		0			0
$509$		−	25.2357i		−	1.11855i	−0.828982	−	0.559275i	$-0.811079\pi$
$509$		−	25.2357i		−	1.11855i	0.828982	−	0.559275i	$-0.188921\pi$
$510$		0			0
$511$	16.9487			0.749767
$512$		0			0
$513$		−	4.60881i		−	0.203484i
$514$		0			0
$515$	46.1487			2.03355
$516$		0			0
$517$	3.03808			0.133615
$518$		0			0
$519$	−18.5486			−0.814192
$520$		0			0
$521$			25.5294i			1.11846i	0.829012	+	0.559231i	$0.188904\pi$
$521$			25.5294i			1.11846i	−0.829012	+	0.559231i	$0.811096\pi$
$522$		0			0
$523$			4.68453i			0.204840i	0.994741	+	0.102420i	$0.0326586\pi$
$523$			4.68453i			0.204840i	−0.994741	+	0.102420i	$0.967341\pi$
$524$		0			0
$525$		−	15.3630i		−	0.670495i
$526$		0			0
$527$	−10.6397	+	2.30554i	−0.463475	+	0.100431i
$528$		0			0
$529$	13.4950			0.586739
$530$		0			0
$531$		−	12.8504i		−	0.557661i
$532$		0			0
$533$	4.24459			0.183854
$534$		0			0
$535$	2.19074			0.0947140
$536$		0			0
$537$		−	9.68489i		−	0.417934i
$538$		0			0
$539$	7.21613			0.310821
$540$		0			0
$541$	30.5595			1.31386			0.656929	−	0.753953i	$-0.271855\pi$
$541$	30.5595			1.31386			0.656929	+	0.753953i	$0.271855\pi$
$542$		0			0
$543$	−6.22823			−0.267279
$544$		0			0
$545$	−36.9844			−1.58424
$546$		0			0
$547$	−19.1292			−0.817904			−0.408952	−	0.912556i	$-0.634106\pi$
$547$	−19.1292			−0.817904			−0.408952	+	0.912556i	$0.634106\pi$
$548$		0			0
$549$	−7.00079			−0.298787
$550$		0			0
$551$		−	21.1827i		−	0.902412i
$552$		0			0
$553$	13.4695			0.572780
$554$		0			0
$555$	30.7563			1.30553
$556$		0			0
$557$		−	14.9766i		−	0.634581i	−0.948328	−	0.317290i	$-0.897227\pi$
$557$		−	14.9766i		−	0.634581i	0.948328	−	0.317290i	$-0.102773\pi$
$558$		0			0
$559$	−27.5733			−1.16622
$560$		0			0
$561$	−1.20743	−	5.57214i	−0.0509779	−	0.235256i
$562$		0			0
$563$			16.9215i			0.713157i	0.934265	+	0.356578i	$0.116057\pi$
$563$			16.9215i			0.713157i	−0.934265	+	0.356578i	$0.883943\pi$
$564$		0			0
$565$			26.5407i			1.11658i
$566$		0			0
$567$			1.33474i			0.0560539i
$568$		0			0
$569$	2.45534			0.102933			0.0514667	−	0.998675i	$-0.483610\pi$
$569$	2.45534			0.102933			0.0514667	+	0.998675i	$0.483610\pi$
$570$		0			0
$571$	−16.2429			−0.679744			−0.339872	−	0.940472i	$-0.610384\pi$
$571$	−16.2429			−0.679744			−0.339872	+	0.940472i	$0.610384\pi$
$572$		0			0
$573$	−18.5765			−0.776044
$574$		0			0
$575$		−	35.4858i		−	1.47986i
$576$		0			0
$577$	−19.8321			−0.825619			−0.412810	−	0.910817i	$-0.635453\pi$
$577$	−19.8321			−0.825619			−0.412810	+	0.910817i	$0.635453\pi$
$578$		0			0
$579$		−	13.6072i		−	0.565498i
$580$		0			0
$581$	−17.8811			−0.741834
$582$		0			0
$583$		−	5.44695i		−	0.225589i
$584$		0			0
$585$			10.2773i			0.424916i
$586$		0			0
$587$		−	16.1112i		−	0.664982i	−0.943106	−	0.332491i	$-0.892111\pi$
$587$		−	16.1112i		−	0.664982i	0.943106	−	0.332491i	$-0.107889\pi$
$588$		0			0
$589$	−12.1691			−0.501420
$590$		0			0
$591$	7.88624			0.324397
$592$		0			0
$593$	27.2516			1.11909			0.559544	−	0.828801i	$-0.310976\pi$
$593$	27.2516			1.11909			0.559544	+	0.828801i	$0.310976\pi$
$594$		0			0
$595$	−21.8541	+	4.73559i	−0.895929	+	0.194140i
$596$		0			0
$597$			19.7275i			0.807391i
$598$		0			0
$599$	−20.5532			−0.839780			−0.419890	−	0.907575i	$-0.637931\pi$
$599$	−20.5532			−0.839780			−0.419890	+	0.907575i	$0.637931\pi$
$600$		0			0
$601$		−	9.92222i		−	0.404736i	−0.979310	−	0.202368i	$-0.935136\pi$
$601$		−	9.92222i		−	0.404736i	0.979310	−	0.202368i	$-0.0648637\pi$
$602$		0			0
$603$		−	2.01087i		−	0.0818888i
$604$		0			0
$605$	−36.9263			−1.50127
$606$		0			0
$607$			21.2113i			0.860938i	0.902605	+	0.430469i	$0.141652\pi$
$607$			21.2113i			0.860938i	−0.902605	+	0.430469i	$0.858348\pi$
$608$		0			0
$609$			6.13465i			0.248588i
$610$		0			0
$611$			5.55704i			0.224814i
$612$		0			0
$613$		−	5.72825i		−	0.231362i	−0.993286	−	0.115681i	$-0.963095\pi$
$613$		−	5.72825i		−	0.231362i	0.993286	−	0.115681i	$-0.0369050\pi$
$614$		0			0
$615$			6.81873i			0.274958i
$616$		0			0
$617$		−	28.9519i		−	1.16556i	−0.812631	−	0.582779i	$-0.801965\pi$
$617$		−	28.9519i		−	1.16556i	0.812631	−	0.582779i	$-0.198035\pi$
$618$		0			0
$619$	−33.2069			−1.33470			−0.667349	−	0.744746i	$-0.732571\pi$
$619$	−33.2069			−1.33470			−0.667349	+	0.744746i	$0.732571\pi$
$620$		0			0
$621$			3.08302i			0.123717i
$622$		0			0
$623$			14.4514i			0.578984i
$624$		0			0
$625$	49.9313			1.99725
$626$		0			0
$627$		−	6.37309i		−	0.254517i
$628$		0			0
$629$	−6.60940	−	30.5014i	−0.263534	−	1.21617i
$630$		0			0
$631$	9.18742			0.365745			0.182873	−	0.983137i	$-0.441460\pi$
$631$	9.18742			0.365745			0.182873	+	0.983137i	$0.441460\pi$
$632$		0			0
$633$	23.9727			0.952830
$634$		0			0
$635$	51.1842			2.03118
$636$		0			0
$637$			13.1992i			0.522973i
$638$		0			0
$639$		−	3.38152i		−	0.133771i
$640$		0			0
$641$			1.98751i			0.0785019i	0.999229	+	0.0392509i	$0.0124972\pi$
$641$			1.98751i			0.0785019i	−0.999229	+	0.0392509i	$0.987503\pi$
$642$		0			0
$643$	15.6525			0.617274			0.308637	−	0.951180i	$-0.400127\pi$
$643$	15.6525			0.617274			0.308637	+	0.951180i	$0.400127\pi$
$644$		0			0
$645$		−	44.2952i		−	1.74412i
$646$		0			0
$647$	50.3006			1.97752			0.988760	−	0.149512i	$-0.0477703\pi$
$647$	50.3006			1.97752			0.988760	+	0.149512i	$0.0477703\pi$
$648$		0			0
$649$		−	17.7697i		−	0.697520i
$650$		0			0
$651$	3.52426			0.138127
$652$		0			0
$653$	−9.17554			−0.359067			−0.179533	−	0.983752i	$-0.557459\pi$
$653$	−9.17554			−0.359067			−0.179533	+	0.983752i	$0.557459\pi$
$654$		0			0
$655$	61.3240			2.39613
$656$		0			0
$657$		−	12.6981i		−	0.495401i
$658$		0			0
$659$			20.4499i			0.796616i	0.917252	+	0.398308i	$0.130403\pi$
$659$			20.4499i			0.796616i	−0.917252	+	0.398308i	$0.869597\pi$
$660$		0			0
$661$			41.2625i			1.60492i	0.596703	+	0.802462i	$0.296477\pi$
$661$			41.2625i			1.60492i	−0.596703	+	0.802462i	$0.703523\pi$
$662$		0			0
$663$	10.1922	−	2.20856i	0.395831	−	0.0857733i
$664$		0			0
$665$	−24.9954			−0.969281
$666$		0			0
$667$			14.1700i			0.548663i
$668$		0			0
$669$	17.6208			0.681258
$670$		0			0
$671$	−9.68074			−0.373721
$672$		0			0
$673$		−	23.7133i		−	0.914079i	−0.889446	−	0.457040i	$-0.848910\pi$
$673$		−	23.7133i		−	0.914079i	0.889446	−	0.457040i	$-0.151090\pi$
$674$		0			0
$675$	−11.5101			−0.443023
$676$		0			0
$677$	−14.9173			−0.573317			−0.286659	−	0.958033i	$-0.592545\pi$
$677$	−14.9173			−0.573317			−0.286659	+	0.958033i	$0.592545\pi$
$678$		0			0
$679$	−15.8011			−0.606392
$680$		0			0
$681$	14.7789			0.566329
$682$		0			0
$683$	−7.24088			−0.277065			−0.138532	−	0.990358i	$-0.544238\pi$
$683$	−7.24088			−0.277065			−0.138532	+	0.990358i	$0.544238\pi$
$684$		0			0
$685$	4.58905			0.175338
$686$		0			0
$687$			21.7445i			0.829605i
$688$		0			0
$689$	9.96318			0.379567
$690$		0			0
$691$	17.5509			0.667666			0.333833	−	0.942632i	$-0.391658\pi$
$691$	17.5509			0.667666			0.333833	+	0.942632i	$0.391658\pi$
$692$		0			0
$693$			1.84569i			0.0701120i
$694$		0			0
$695$	43.2799			1.64170
$696$		0			0
$697$	6.76223	−	1.46532i	0.256138	−	0.0555028i
$698$		0			0
$699$			10.4914i			0.396821i
$700$		0			0
$701$			22.1815i			0.837783i	0.908036	+	0.418891i	$0.137581\pi$
$701$			22.1815i			0.837783i	−0.908036	+	0.418891i	$0.862419\pi$
$702$		0			0
$703$		−	34.8858i		−	1.31574i
$704$		0			0
$705$	−8.92713			−0.336215
$706$		0			0
$707$	−25.7495			−0.968410
$708$		0			0
$709$	−4.78270			−0.179618			−0.0898091	−	0.995959i	$-0.528626\pi$
$709$	−4.78270			−0.179618			−0.0898091	+	0.995959i	$0.528626\pi$
$710$		0			0
$711$		−	10.0914i		−	0.378458i
$712$		0			0
$713$	8.14043			0.304861
$714$		0			0
$715$			14.2116i			0.531483i
$716$		0			0
$717$	14.7604			0.551237
$718$		0			0
$719$		−	40.8961i		−	1.52517i	−0.646889	−	0.762584i	$-0.723930\pi$
$719$		−	40.8961i		−	1.52517i	0.646889	−	0.762584i	$-0.276070\pi$
$720$		0			0
$721$			15.1594i			0.564566i
$722$		0			0
$723$			21.7730i			0.809748i
$724$		0			0
$725$	−52.9017			−1.96472
$726$		0			0
$727$	10.7907			0.400204			0.200102	−	0.979775i	$-0.435873\pi$
$727$	10.7907			0.400204			0.200102	+	0.979775i	$0.435873\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	−43.9281	+	9.51884i	−1.62474	+	0.352067i
$732$		0			0
$733$			8.86600i			0.327473i	0.986504	+	0.163737i	$0.0523547\pi$
$733$			8.86600i			0.327473i	−0.986504	+	0.163737i	$0.947645\pi$
$734$		0			0
$735$	−21.2040			−0.782120
$736$		0			0
$737$		−	2.78064i		−	0.102426i
$738$		0			0
$739$		−	14.2082i		−	0.522656i	−0.965250	−	0.261328i	$-0.915840\pi$
$739$		−	14.2082i		−	0.522656i	0.965250	−	0.261328i	$-0.0841604\pi$
$740$		0			0
$741$	11.6572			0.428239
$742$		0			0
$743$		−	33.9021i		−	1.24375i	−0.783118	−	0.621874i	$-0.786372\pi$
$743$		−	33.9021i		−	1.24375i	0.783118	−	0.621874i	$-0.213628\pi$
$744$		0			0
$745$		−	1.13846i		−	0.0417099i
$746$		0			0
$747$			13.3967i			0.490159i
$748$		0			0
$749$			0.719638i			0.0262950i
$750$		0			0
$751$			0.832425i			0.0303756i	0.999885	+	0.0151878i	$0.00483462\pi$
$751$			0.832425i			0.0303756i	−0.999885	+	0.0151878i	$0.995165\pi$
$752$		0			0
$753$			18.0029i			0.656063i
$754$		0			0
$755$	−38.5341			−1.40240
$756$		0			0
$757$		−	26.1220i		−	0.949420i	−0.880142	−	0.474710i	$-0.842553\pi$
$757$		−	26.1220i		−	0.949420i	0.880142	−	0.474710i	$-0.157447\pi$
$758$		0			0
$759$			4.26322i			0.154745i
$760$		0			0
$761$	−36.9537			−1.33957			−0.669784	−	0.742556i	$-0.733614\pi$
$761$	−36.9537			−1.33957			−0.669784	+	0.742556i	$0.733614\pi$
$762$		0			0
$763$		−	12.1490i		−	0.439824i
$764$		0			0
$765$	3.54794	+	16.3732i	0.128276	+	0.591976i
$766$		0			0
$767$	32.5030			1.17362
$768$		0			0
$769$	−51.3629			−1.85219			−0.926097	−	0.377285i	$-0.876858\pi$
$769$	−51.3629			−1.85219			−0.926097	+	0.377285i	$0.876858\pi$
$770$		0			0
$771$	−4.44516			−0.160088
$772$		0			0
$773$			47.0804i			1.69336i	0.532100	+	0.846682i	$0.321403\pi$
$773$			47.0804i			1.69336i	−0.532100	+	0.846682i	$0.678597\pi$
$774$		0			0
$775$			30.3913i			1.09169i
$776$		0			0
$777$			10.1032i			0.362449i
$778$		0			0
$779$	7.73424			0.277108
$780$		0			0
$781$		−	4.67599i		−	0.167320i
$782$		0			0
$783$	4.59613			0.164252
$784$		0			0
$785$		−	96.2345i		−	3.43476i
$786$		0			0
$787$	21.2273			0.756671			0.378336	−	0.925669i	$-0.376497\pi$
$787$	21.2273			0.756671			0.378336	+	0.925669i	$0.376497\pi$
$788$		0			0
$789$	−18.7200			−0.666450
$790$		0			0
$791$	−8.71838			−0.309990
$792$		0			0
$793$		−	17.7074i		−	0.628807i
$794$		0			0
$795$			16.0054i			0.567652i
$796$		0			0
$797$		−	1.24847i		−	0.0442232i	−0.999756	−	0.0221116i	$-0.992961\pi$
$797$		−	1.24847i		−	0.0442232i	0.999756	−	0.0221116i	$-0.00703891\pi$
$798$		0			0
$799$	1.91840	+	8.85315i	0.0678682	+	0.313202i
$800$		0			0
$801$	10.8271			0.382558
$802$		0			0
$803$		−	17.5591i		−	0.619645i
$804$		0			0
$805$	16.7205			0.589319
$806$		0			0
$807$	−9.33536			−0.328620
$808$		0			0
$809$			3.89688i			0.137007i	0.997651	+	0.0685034i	$0.0218224\pi$
$809$			3.89688i			0.137007i	−0.997651	+	0.0685034i	$0.978178\pi$
$810$		0			0
$811$	−39.7545			−1.39597			−0.697985	−	0.716112i	$-0.745920\pi$
$811$	−39.7545			−1.39597			−0.697985	+	0.716112i	$0.745920\pi$
$812$		0			0
$813$	12.0895			0.423999
$814$		0			0
$815$	74.8517			2.62194
$816$		0			0
$817$	−50.2424			−1.75776
$818$		0			0
$819$	−3.37601			−0.117967
$820$		0			0
$821$	−7.68119			−0.268076			−0.134038	−	0.990976i	$-0.542794\pi$
$821$	−7.68119			−0.268076			−0.134038	+	0.990976i	$0.542794\pi$
$822$		0			0
$823$			23.9730i			0.835647i	0.908528	+	0.417824i	$0.137207\pi$
$823$			23.9730i			0.835647i	−0.908528	+	0.417824i	$0.862793\pi$
$824$		0			0
$825$	−15.9162			−0.554131
$826$		0			0
$827$	−53.2001			−1.84995			−0.924974	−	0.380030i	$-0.875914\pi$
$827$	−53.2001			−1.84995			−0.924974	+	0.380030i	$0.875914\pi$
$828$		0			0
$829$		−	5.80159i		−	0.201498i	−0.994912	−	0.100749i	$-0.967876\pi$
$829$		−	5.80159i		−	0.201498i	0.994912	−	0.100749i	$-0.0321238\pi$
$830$		0			0
$831$	21.2973			0.738796
$832$		0			0
$833$	4.55664	+	21.0282i	0.157878	+	0.728586i
$834$		0			0
$835$		−	40.1988i		−	1.39114i
$836$		0			0
$837$		−	2.64041i		−	0.0912658i
$838$		0			0
$839$			21.8766i			0.755264i	0.925956	+	0.377632i	$0.123262\pi$
$839$			21.8766i			0.755264i	−0.925956	+	0.377632i	$0.876738\pi$
$840$		0			0
$841$	−7.87561			−0.271573
$842$		0			0
$843$	−7.37238			−0.253918
$844$		0			0
$845$	26.8275			0.922895
$846$		0			0
$847$		−	12.1299i		−	0.416789i
$848$		0			0
$849$	22.3443			0.766853
$850$		0			0
$851$			23.3365i			0.799966i
$852$		0			0
$853$	13.3416			0.456809			0.228405	−	0.973566i	$-0.426649\pi$
$853$	13.3416			0.456809			0.228405	+	0.973566i	$0.426649\pi$
$854$		0			0
$855$			18.7268i			0.640442i
$856$		0			0
$857$		−	0.322155i		−	0.0110046i	−0.999985	−	0.00550230i	$-0.998249\pi$
$857$		−	0.322155i		−	0.0110046i	0.999985	−	0.00550230i	$-0.00175145\pi$
$858$		0			0
$859$		−	28.0802i		−	0.958085i	−0.877792	−	0.479043i	$-0.840984\pi$
$859$		−	28.0802i		−	0.958085i	0.877792	−	0.479043i	$-0.159016\pi$
$860$		0			0
$861$	−2.23989			−0.0763352
$862$		0			0
$863$	27.1517			0.924255			0.462127	−	0.886814i	$-0.347086\pi$
$863$	27.1517			0.924255			0.462127	+	0.886814i	$0.347086\pi$
$864$		0			0
$865$	75.3676			2.56258
$866$		0			0
$867$	15.4751	−	7.03708i	0.525563	−	0.238992i
$868$		0			0
$869$		−	13.9545i		−	0.473374i
$870$		0			0
$871$	5.08616			0.172338
$872$		0			0
$873$			11.8383i			0.400667i
$874$		0			0
$875$			35.3067i			1.19358i
$876$		0			0
$877$	25.4455			0.859233			0.429617	−	0.903011i	$-0.358649\pi$
$877$	25.4455			0.859233			0.429617	+	0.903011i	$0.358649\pi$
$878$		0			0
$879$			0.823580i			0.0277787i
$880$		0			0
$881$			15.2859i			0.514995i	0.966279	+	0.257498i	$0.0828979\pi$
$881$			15.2859i			0.514995i	−0.966279	+	0.257498i	$0.917102\pi$
$882$		0			0
$883$			13.6008i			0.457703i	0.973461	+	0.228852i	$0.0734970\pi$
$883$			13.6008i			0.457703i	−0.973461	+	0.228852i	$0.926503\pi$
$884$		0			0
$885$			52.2146i			1.75518i
$886$		0			0
$887$		−	22.2405i		−	0.746764i	−0.927678	−	0.373382i	$-0.878198\pi$
$887$		−	22.2405i		−	0.746764i	0.927678	−	0.373382i	$-0.121802\pi$
$888$		0			0
$889$			16.8135i			0.563908i
$890$		0			0
$891$	1.38281			0.0463258
$892$		0			0
$893$			10.1257i			0.338844i
$894$		0			0
$895$			39.3522i			1.31540i
$896$		0			0
$897$	−7.79799			−0.260367
$898$		0			0
$899$		−	12.1356i		−	0.404747i
$900$		0			0
$901$	15.8727	−	3.43949i	0.528798	−	0.114586i
$902$		0			0
$903$	14.5505			0.484212
$904$		0			0
$905$	25.3069			0.841230
$906$		0			0
$907$	−1.22779			−0.0407680			−0.0203840	−	0.999792i	$-0.506489\pi$
$907$	−1.22779			−0.0407680			−0.0203840	+	0.999792i	$0.506489\pi$
$908$		0			0
$909$			19.2918i			0.639867i
$910$		0			0
$911$		−	4.50510i		−	0.149261i	−0.997211	−	0.0746304i	$-0.976222\pi$
$911$		−	4.50510i		−	0.149261i	0.997211	−	0.0746304i	$-0.0237777\pi$
$912$		0			0
$913$			18.5250i			0.613089i
$914$		0			0
$915$	28.4460			0.940397
$916$		0			0
$917$			20.1444i			0.665225i
$918$		0			0
$919$	−24.2454			−0.799781			−0.399890	−	0.916563i	$-0.630952\pi$
$919$	−24.2454			−0.799781			−0.399890	+	0.916563i	$0.630952\pi$
$920$		0			0
$921$			1.85701i			0.0611905i
$922$		0			0
$923$	8.55299			0.281525
$924$		0			0
$925$	−87.1239			−2.86462
$926$		0			0
$927$	11.3576			0.373031
$928$		0			0
$929$			55.5044i			1.82104i	0.413463	+	0.910521i	$0.364319\pi$
$929$			55.5044i			1.82104i	−0.413463	+	0.910521i	$0.635681\pi$
$930$		0			0
$931$			24.0509i			0.788236i
$932$		0			0
$933$			4.96908i			0.162680i
$934$		0			0
$935$	4.90612	+	22.6410i	0.160447	+	0.740441i
$936$		0			0
$937$	−18.5511			−0.606038			−0.303019	−	0.952985i	$-0.597995\pi$
$937$	−18.5511			−0.606038			−0.303019	+	0.952985i	$0.597995\pi$
$938$		0			0
$939$		−	4.54467i		−	0.148310i
$940$		0			0
$941$	−35.9106			−1.17065			−0.585327	−	0.810798i	$-0.699034\pi$
$941$	−35.9106			−1.17065			−0.585327	+	0.810798i	$0.699034\pi$
$942$		0			0
$943$	−5.17375			−0.168481
$944$		0			0
$945$		−	5.42340i		−	0.176423i
$946$		0			0
$947$	−13.4238			−0.436216			−0.218108	−	0.975925i	$-0.569988\pi$
$947$	−13.4238			−0.436216			−0.218108	+	0.975925i	$0.569988\pi$
$948$		0			0
$949$	32.1178			1.04259
$950$		0			0
$951$	24.5554			0.796264
$952$		0			0
$953$	52.8115			1.71073			0.855366	−	0.518024i	$-0.173332\pi$
$953$	52.8115			1.71073			0.855366	+	0.518024i	$0.173332\pi$
$954$		0			0
$955$	75.4811			2.44251
$956$		0			0
$957$	6.35556			0.205446
$958$		0			0
$959$			1.50746i			0.0486784i
$960$		0			0
$961$	24.0282			0.775105
$962$		0			0
$963$	0.539159			0.0173741
$964$		0			0
$965$			55.2898i			1.77984i
$966$		0			0
$967$	7.20225			0.231609			0.115804	−	0.993272i	$-0.463055\pi$
$967$	7.20225			0.231609			0.115804	+	0.993272i	$0.463055\pi$
$968$		0			0
$969$	18.5716	−	4.02431i	0.596605	−	0.129279i
$970$		0			0
$971$			53.5069i			1.71712i	0.512715	+	0.858559i	$0.328640\pi$
$971$			53.5069i			1.71712i	−0.512715	+	0.858559i	$0.671360\pi$
$972$		0			0
$973$			14.2170i			0.455777i
$974$		0			0
$975$		−	29.1128i		−	0.932356i
$976$		0			0
$977$	−0.640518			−0.0204920			−0.0102460	−	0.999948i	$-0.503261\pi$
$977$	−0.640518			−0.0204920			−0.0102460	+	0.999948i	$0.503261\pi$
$978$		0			0
$979$	14.9718			0.478502
$980$		0			0
$981$	−9.10216			−0.290609
$982$		0			0
$983$			16.1739i			0.515866i	0.966163	+	0.257933i	$0.0830414\pi$
$983$			16.1739i			0.515866i	−0.966163	+	0.257933i	$0.916959\pi$
$984$		0			0
$985$	−32.0438			−1.02100
$986$		0			0
$987$		−	2.93248i		−	0.0933418i
$988$		0			0
$989$	33.6092			1.06871
$990$		0			0
$991$			4.18546i			0.132955i	0.997788	+	0.0664777i	$0.0211761\pi$
$991$			4.18546i			0.132955i	−0.997788	+	0.0664777i	$0.978824\pi$
$992$		0			0
$993$			23.9681i			0.760603i
$994$		0			0
$995$		−	80.1577i		−	2.54117i
$996$		0			0
$997$	−36.1353			−1.14442			−0.572209	−	0.820108i	$-0.693913\pi$
$997$	−36.1353			−1.14442			−0.572209	+	0.820108i	$0.693913\pi$
$998$		0			0
$999$	7.56937			0.239484

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1632.2.l.a.1393.18		18
3.2	odd	2		4896.2.l.d.3025.2		18
4.3	odd	2		408.2.l.b.373.5	yes	18
8.3	odd	2		408.2.l.a.373.6	yes	18
8.5	even	2		1632.2.l.b.1393.2		18
12.11	even	2		1224.2.l.d.1189.14		18
17.16	even	2		1632.2.l.b.1393.1		18
24.5	odd	2		4896.2.l.c.3025.18		18
24.11	even	2		1224.2.l.c.1189.13		18
51.50	odd	2		4896.2.l.c.3025.17		18
68.67	odd	2		408.2.l.a.373.5	✓	18
136.67	odd	2		408.2.l.b.373.6	yes	18
136.101	even	2	inner	1632.2.l.a.1393.17		18
204.203	even	2		1224.2.l.c.1189.14		18
408.101	odd	2		4896.2.l.d.3025.1		18
408.203	even	2		1224.2.l.d.1189.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.l.a.373.5	✓	18	68.67	odd	2
408.2.l.a.373.6	yes	18	8.3	odd	2
408.2.l.b.373.5	yes	18	4.3	odd	2
408.2.l.b.373.6	yes	18	136.67	odd	2
1224.2.l.c.1189.13		18	24.11	even	2
1224.2.l.c.1189.14		18	204.203	even	2
1224.2.l.d.1189.13		18	408.203	even	2
1224.2.l.d.1189.14		18	12.11	even	2
1632.2.l.a.1393.17		18	136.101	even	2	inner
1632.2.l.a.1393.18		18	1.1	even	1	trivial
1632.2.l.b.1393.1		18	17.16	even	2
1632.2.l.b.1393.2		18	8.5	even	2
4896.2.l.c.3025.17		18	51.50	odd	2
4896.2.l.c.3025.18		18	24.5	odd	2
4896.2.l.d.3025.1		18	408.101	odd	2
4896.2.l.d.3025.2		18	3.2	odd	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1632 = 2^{5} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1632.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.06326 q^{5} +1.33474i q^{7} +1.00000 q^{9} +1.38281 q^{11} +2.52933i q^{13} -4.06326 q^{15} +(0.873177 + 4.02959i) q^{17} +4.60881i q^{19} -1.33474i q^{21} -3.08302i q^{23} +11.5101 q^{25} -1.00000 q^{27} -4.59613 q^{29} +2.64041i q^{31} -1.38281 q^{33} +5.42340i q^{35} -7.56937 q^{37} -2.52933i q^{39} -1.67814i q^{41} +10.9014i q^{43} +4.06326 q^{45} +2.19704 q^{47} +5.21846 q^{49} +(-0.873177 - 4.02959i) q^{51} -3.93905i q^{53} +5.61870 q^{55} -4.60881i q^{57} -12.8504i q^{59} -7.00079 q^{61} +1.33474i q^{63} +10.2773i q^{65} -2.01087i q^{67} +3.08302i q^{69} -3.38152i q^{71} -12.6981i q^{73} -11.5101 q^{75} +1.84569i q^{77} -10.0914i q^{79} +1.00000 q^{81} +13.3967i q^{83} +(3.54794 + 16.3732i) q^{85} +4.59613 q^{87} +10.8271 q^{89} -3.37601 q^{91} -2.64041i q^{93} +18.7268i q^{95} +11.8383i q^{97} +1.38281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 18 q^{3} - 4 q^{5} + 18 q^{9} + 4 q^{15} - 2 q^{17} + 22 q^{25} - 18 q^{27} + 12 q^{29} - 16 q^{37} - 4 q^{45} - 18 q^{49} + 2 q^{51} + 16 q^{55} - 16 q^{61} - 22 q^{75} + 18 q^{81} + 16 q^{85} - 12 q^{87}+ \cdots + 20 q^{89}+O(q^{100}) \) 18 * q - 18 * q^3 - 4 * q^5 + 18 * q^9 + 4 * q^15 - 2 * q^17 + 22 * q^25 - 18 * q^27 + 12 * q^29 - 16 * q^37 - 4 * q^45 - 18 * q^49 + 2 * q^51 + 16 * q^55 - 16 * q^61 - 22 * q^75 + 18 * q^81 + 16 * q^85 - 12 * q^87 + 20 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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