LMFDB - Embedded newform 528.2.o.b.175.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [528,2,Mod(175,528)] 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("528.175"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(528, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 528 = 2^{4} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	528.o (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:	$4.21610122672$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.454201344.7
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 $ x^8 - 2x^7 - 4x^6 + 24x^5 - 25x^4 - 12x^3 + 128x^2 - 182*x + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			175.3
Root			$1.02715 + 1.10132i$ of defining polynomial
Character	$\chi$	$=$	528.175
Dual form			528.2.o.b.175.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.73205 q^{5} -5.13734 q^{7} -1.00000 q^{9} +(1.88040 - 2.73205i) q^{11} -5.13734i q^{13} -2.73205i q^{15} -3.76080i q^{17} +3.76080 q^{19} +5.13734i q^{21} -1.26795i q^{23} +2.46410 q^{25} +1.00000i q^{27} -2.00000i q^{31} +(-2.73205 - 1.88040i) q^{33} -14.0355 q^{35} +0.535898 q^{37} -5.13734 q^{39} +10.2747i q^{41} +3.76080 q^{43} -2.73205 q^{45} -4.19615i q^{47} +19.3923 q^{49} -3.76080 q^{51} -10.7321 q^{53} +(5.13734 - 7.46410i) q^{55} -3.76080i q^{57} -9.46410i q^{59} +2.38425i q^{61} +5.13734 q^{63} -14.0355i q^{65} +10.3923i q^{67} -1.26795 q^{69} +10.7321i q^{71} -10.2747i q^{73} -2.46410i q^{75} +(-9.66025 + 14.0355i) q^{77} +5.13734 q^{79} +1.00000 q^{81} +10.2747 q^{83} -10.2747i q^{85} -10.3923 q^{89} +26.3923i q^{91} -2.00000 q^{93} +10.2747 q^{95} +5.46410 q^{97} +(-1.88040 + 2.73205i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} + 32 q^{37} - 8 q^{45} + 72 q^{49} - 72 q^{53} - 24 q^{69} - 8 q^{77} + 8 q^{81} - 16 q^{93} + 16 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 8 * q^9 - 8 * q^25 - 8 * q^33 + 32 * q^37 - 8 * q^45 + 72 * q^49 - 72 * q^53 - 24 * q^69 - 8 * q^77 + 8 * q^81 - 16 * q^93 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/528\mathbb{Z}\right)^\times$.

$n$	$133$	$145$	$353$	$463$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$		−	1.00000i		−	0.577350i
$3$		−	1.00000i		−	0.577350i
$4$		0			0
$5$	2.73205			1.22181			0.610905	−	0.791704i	$-0.290806\pi$
$5$	2.73205			1.22181			0.610905	+	0.791704i	$0.290806\pi$
$6$		0			0
$7$	−5.13734			−1.94173			−0.970867	−	0.239620i	$-0.922977\pi$
$7$	−5.13734			−1.94173			−0.970867	+	0.239620i	$0.922977\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	1.88040	−	2.73205i	0.566961	−	0.823744i
$11$	1.88040	−	2.73205i	0.566961	−	0.823744i
$12$		0			0
$13$		−	5.13734i		−	1.42484i	−0.701752	−	0.712421i	$-0.747598\pi$
$13$		−	5.13734i		−	1.42484i	0.701752	−	0.712421i	$-0.252402\pi$
$14$		0			0
$15$		−	2.73205i		−	0.705412i
$16$		0			0
$17$		−	3.76080i		−	0.912127i	−0.889947	−	0.456064i	$-0.849259\pi$
$17$		−	3.76080i		−	0.912127i	0.889947	−	0.456064i	$-0.150741\pi$
$18$		0			0
$19$	3.76080			0.862786			0.431393	−	0.902164i	$-0.358022\pi$
$19$	3.76080			0.862786			0.431393	+	0.902164i	$0.358022\pi$
$20$		0			0
$21$			5.13734i			1.12106i
$22$		0			0
$23$		−	1.26795i		−	0.264386i	−0.991224	−	0.132193i	$-0.957798\pi$
$23$		−	1.26795i		−	0.264386i	0.991224	−	0.132193i	$-0.0422018\pi$
$24$		0			0
$25$	2.46410			0.492820
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	2.00000i		−	0.359211i	−0.983739	−	0.179605i	$-0.942518\pi$
$31$		−	2.00000i		−	0.359211i	0.983739	−	0.179605i	$-0.0574821\pi$
$32$		0			0
$33$	−2.73205	−	1.88040i	−0.475589	−	0.327335i
$34$		0			0
$35$	−14.0355			−2.37243
$36$		0			0
$37$	0.535898			0.0881012			0.0440506	−	0.999029i	$-0.485974\pi$
$37$	0.535898			0.0881012			0.0440506	+	0.999029i	$0.485974\pi$
$38$		0			0
$39$	−5.13734			−0.822633
$40$		0			0
$41$			10.2747i			1.60464i	0.596896	+	0.802318i	$0.296400\pi$
$41$			10.2747i			1.60464i	−0.596896	+	0.802318i	$0.703600\pi$
$42$		0			0
$43$	3.76080			0.573516			0.286758	−	0.958003i	$-0.407422\pi$
$43$	3.76080			0.573516			0.286758	+	0.958003i	$0.407422\pi$
$44$		0			0
$45$	−2.73205			−0.407270
$46$		0			0
$47$		−	4.19615i		−	0.612072i	−0.952020	−	0.306036i	$-0.900997\pi$
$47$		−	4.19615i		−	0.612072i	0.952020	−	0.306036i	$-0.0990028\pi$
$48$		0			0
$49$	19.3923			2.77033
$50$		0			0
$51$	−3.76080			−0.526617
$52$		0			0
$53$	−10.7321			−1.47416			−0.737080	−	0.675805i	$-0.763796\pi$
$53$	−10.7321			−1.47416			−0.737080	+	0.675805i	$0.763796\pi$
$54$		0			0
$55$	5.13734	−	7.46410i	0.692719	−	1.00646i
$56$		0			0
$57$		−	3.76080i		−	0.498130i
$58$		0			0
$59$		−	9.46410i		−	1.23212i	−0.787699	−	0.616061i	$-0.788728\pi$
$59$		−	9.46410i		−	1.23212i	0.787699	−	0.616061i	$-0.211272\pi$
$60$		0			0
$61$			2.38425i			0.305272i	0.988282	+	0.152636i	$0.0487762\pi$
$61$			2.38425i			0.305272i	−0.988282	+	0.152636i	$0.951224\pi$
$62$		0			0
$63$	5.13734			0.647245
$64$		0			0
$65$		−	14.0355i		−	1.74089i
$66$		0			0
$67$			10.3923i			1.26962i	0.772667	+	0.634811i	$0.218922\pi$
$67$			10.3923i			1.26962i	−0.772667	+	0.634811i	$0.781078\pi$
$68$		0			0
$69$	−1.26795			−0.152643
$70$		0			0
$71$			10.7321i			1.27366i	0.771004	+	0.636830i	$0.219755\pi$
$71$			10.7321i			1.27366i	−0.771004	+	0.636830i	$0.780245\pi$
$72$		0			0
$73$		−	10.2747i		−	1.20256i	−0.799038	−	0.601281i	$-0.794657\pi$
$73$		−	10.2747i		−	1.20256i	0.799038	−	0.601281i	$-0.205343\pi$
$74$		0			0
$75$		−	2.46410i		−	0.284530i
$76$		0			0
$77$	−9.66025	+	14.0355i	−1.10089	+	1.59949i
$78$		0			0
$79$	5.13734			0.577996			0.288998	−	0.957330i	$-0.406678\pi$
$79$	5.13734			0.577996			0.288998	+	0.957330i	$0.406678\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	10.2747			1.12779			0.563897	−	0.825845i	$-0.309302\pi$
$83$	10.2747			1.12779			0.563897	+	0.825845i	$0.309302\pi$
$84$		0			0
$85$		−	10.2747i		−	1.11445i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−10.3923			−1.10158			−0.550791	−	0.834643i	$-0.685674\pi$
$89$	−10.3923			−1.10158			−0.550791	+	0.834643i	$0.685674\pi$
$90$		0			0
$91$			26.3923i			2.76667i
$92$		0			0
$93$	−2.00000			−0.207390
$94$		0			0
$95$	10.2747			1.05416
$96$		0			0
$97$	5.46410			0.554795			0.277398	−	0.960755i	$-0.410528\pi$
$97$	5.46410			0.554795			0.277398	+	0.960755i	$0.410528\pi$
$98$		0			0
$99$	−1.88040	+	2.73205i	−0.188987	+	0.274581i
$100$		0			0
$101$			17.7963i			1.77080i	0.464833	+	0.885398i	$0.346114\pi$
$101$			17.7963i			1.77080i	−0.464833	+	0.885398i	$0.653886\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$		0			0
$105$			14.0355i			1.36972i
$106$		0			0
$107$	−3.76080			−0.363570			−0.181785	−	0.983338i	$-0.558187\pi$
$107$	−3.76080			−0.363570			−0.181785	+	0.983338i	$0.558187\pi$
$108$		0			0
$109$			12.6589i			1.21251i	0.795272	+	0.606253i	$0.207328\pi$
$109$			12.6589i			1.21251i	−0.795272	+	0.606253i	$0.792672\pi$
$110$		0			0
$111$		−	0.535898i		−	0.0508652i
$112$		0			0
$113$	17.3205			1.62938			0.814688	−	0.579899i	$-0.196908\pi$
$113$	17.3205			1.62938			0.814688	+	0.579899i	$0.196908\pi$
$114$		0			0
$115$		−	3.46410i		−	0.323029i
$116$		0			0
$117$			5.13734i			0.474948i
$118$		0			0
$119$			19.3205i			1.77111i
$120$		0			0
$121$	−3.92820	−	10.2747i	−0.357109	−	0.934063i
$122$		0			0
$123$	10.2747			0.926437
$124$		0			0
$125$	−6.92820			−0.619677
$126$		0			0
$127$	−5.13734			−0.455866			−0.227933	−	0.973677i	$-0.573197\pi$
$127$	−5.13734			−0.455866			−0.227933	+	0.973677i	$0.573197\pi$
$128$		0			0
$129$		−	3.76080i		−	0.331120i
$130$		0			0
$131$	2.75309			0.240539			0.120269	−	0.992741i	$-0.461624\pi$
$131$	2.75309			0.240539			0.120269	+	0.992741i	$0.461624\pi$
$132$		0			0
$133$	−19.3205			−1.67530
$134$		0			0
$135$			2.73205i			0.235137i
$136$		0			0
$137$	4.92820			0.421045			0.210522	−	0.977589i	$-0.432484\pi$
$137$	4.92820			0.421045			0.210522	+	0.977589i	$0.432484\pi$
$138$		0			0
$139$	14.0355			1.19047			0.595237	−	0.803550i	$-0.297058\pi$
$139$	14.0355			1.19047			0.595237	+	0.803550i	$0.297058\pi$
$140$		0			0
$141$	−4.19615			−0.353380
$142$		0			0
$143$	−14.0355	−	9.66025i	−1.17371	−	0.807831i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	19.3923i		−	1.59945i
$148$		0			0
$149$		−	17.7963i		−	1.45793i	−0.684552	−	0.728964i	$-0.740002\pi$
$149$		−	17.7963i		−	1.45793i	0.684552	−	0.728964i	$-0.259998\pi$
$150$		0			0
$151$	−2.38425			−0.194027			−0.0970137	−	0.995283i	$-0.530929\pi$
$151$	−2.38425			−0.194027			−0.0970137	+	0.995283i	$0.530929\pi$
$152$		0			0
$153$			3.76080i			0.304042i
$154$		0			0
$155$		−	5.46410i		−	0.438887i
$156$		0			0
$157$	11.8564			0.946244			0.473122	−	0.880997i	$-0.343127\pi$
$157$	11.8564			0.946244			0.473122	+	0.880997i	$0.343127\pi$
$158$		0			0
$159$			10.7321i			0.851107i
$160$		0			0
$161$			6.51389i			0.513367i
$162$		0			0
$163$		−	2.92820i		−	0.229355i	−0.993403	−	0.114677i	$-0.963417\pi$
$163$		−	2.92820i		−	0.229355i	0.993403	−	0.114677i	$-0.0365834\pi$
$164$		0			0
$165$	−7.46410	−	5.13734i	−0.581080	−	0.399942i
$166$		0			0
$167$	17.7963			1.37712			0.688559	−	0.725180i	$-0.258244\pi$
$167$	17.7963			1.37712			0.688559	+	0.725180i	$0.258244\pi$
$168$		0			0
$169$	−13.3923			−1.03018
$170$		0			0
$171$	−3.76080			−0.287595
$172$		0			0
$173$		−	13.0278i		−	0.990484i	−0.868755	−	0.495242i	$-0.835079\pi$
$173$		−	13.0278i		−	0.990484i	0.868755	−	0.495242i	$-0.164921\pi$
$174$		0			0
$175$	−12.6589			−0.956926
$176$		0			0
$177$	−9.46410			−0.711365
$178$		0			0
$179$		−	0.392305i		−	0.0293222i	−0.999893	−	0.0146611i	$-0.995333\pi$
$179$		−	0.392305i		−	0.0293222i	0.999893	−	0.0146611i	$-0.00466695\pi$
$180$		0			0
$181$	2.39230			0.177819			0.0889093	−	0.996040i	$-0.471662\pi$
$181$	2.39230			0.177819			0.0889093	+	0.996040i	$0.471662\pi$
$182$		0			0
$183$	2.38425			0.176249
$184$		0			0
$185$	1.46410			0.107643
$186$		0			0
$187$	−10.2747	−	7.07180i	−0.751360	−	0.517141i
$188$		0			0
$189$		−	5.13734i		−	0.373687i
$190$		0			0
$191$			11.1244i			0.804930i	0.915435	+	0.402465i	$0.131847\pi$
$191$			11.1244i			0.804930i	−0.915435	+	0.402465i	$0.868153\pi$
$192$		0			0
$193$			13.0278i			0.937760i	0.883262	+	0.468880i	$0.155342\pi$
$193$			13.0278i			0.937760i	−0.883262	+	0.468880i	$0.844658\pi$
$194$		0			0
$195$	−14.0355			−1.00510
$196$		0			0
$197$			2.75309i			0.196150i	0.995179	+	0.0980749i	$0.0312685\pi$
$197$			2.75309i			0.196150i	−0.995179	+	0.0980749i	$0.968732\pi$
$198$		0			0
$199$		−	23.4641i		−	1.66333i	−0.555281	−	0.831663i	$-0.687389\pi$
$199$		−	23.4641i		−	1.66333i	0.555281	−	0.831663i	$-0.312611\pi$
$200$		0			0
$201$	10.3923			0.733017
$202$		0			0
$203$		0			0
$204$		0			0
$205$			28.0710i			1.96056i
$206$		0			0
$207$			1.26795i			0.0881286i
$208$		0			0
$209$	7.07180	−	10.2747i	0.489166	−	0.710715i
$210$		0			0
$211$	21.5571			1.48405			0.742025	−	0.670372i	$-0.233865\pi$
$211$	21.5571			1.48405			0.742025	+	0.670372i	$0.233865\pi$
$212$		0			0
$213$	10.7321			0.735348
$214$		0			0
$215$	10.2747			0.700728
$216$		0			0
$217$			10.2747i			0.697491i
$218$		0			0
$219$	−10.2747			−0.694299
$220$		0			0
$221$	−19.3205			−1.29964
$222$		0			0
$223$			11.4641i			0.767693i	0.923397	+	0.383847i	$0.125401\pi$
$223$			11.4641i			0.767693i	−0.923397	+	0.383847i	$0.874599\pi$
$224$		0			0
$225$	−2.46410			−0.164273
$226$		0			0
$227$	−16.7886			−1.11430			−0.557149	−	0.830413i	$-0.688105\pi$
$227$	−16.7886			−1.11430			−0.557149	+	0.830413i	$0.688105\pi$
$228$		0			0
$229$	23.8564			1.57648			0.788238	−	0.615371i	$-0.210994\pi$
$229$	23.8564			1.57648			0.788238	+	0.615371i	$0.210994\pi$
$230$		0			0
$231$	14.0355	+	9.66025i	0.923467	+	0.635598i
$232$		0			0
$233$			10.2747i			0.673117i	0.941663	+	0.336559i	$0.109263\pi$
$233$			10.2747i			0.673117i	−0.941663	+	0.336559i	$0.890737\pi$
$234$		0			0
$235$		−	11.4641i		−	0.747836i
$236$		0			0
$237$		−	5.13734i		−	0.333706i
$238$		0			0
$239$	−23.3025			−1.50731			−0.753656	−	0.657269i	$-0.771711\pi$
$239$	−23.3025			−1.50731			−0.753656	+	0.657269i	$0.771711\pi$
$240$		0			0
$241$		−	28.0710i		−	1.80821i	−0.427310	−	0.904105i	$-0.640539\pi$
$241$		−	28.0710i		−	1.80821i	0.427310	−	0.904105i	$-0.359461\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$	52.9808			3.38482
$246$		0			0
$247$		−	19.3205i		−	1.22933i
$248$		0			0
$249$		−	10.2747i		−	0.651132i
$250$		0			0
$251$			21.8564i			1.37956i	0.724017	+	0.689782i	$0.242294\pi$
$251$			21.8564i			1.37956i	−0.724017	+	0.689782i	$0.757706\pi$
$252$		0			0
$253$	−3.46410	−	2.38425i	−0.217786	−	0.149896i
$254$		0			0
$255$	−10.2747			−0.643426
$256$		0			0
$257$	−15.4641			−0.964624			−0.482312	−	0.875999i	$-0.660203\pi$
$257$	−15.4641			−0.964624			−0.482312	+	0.875999i	$0.660203\pi$
$258$		0			0
$259$	−2.75309			−0.171069
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.76850			0.294038			0.147019	−	0.989134i	$-0.453032\pi$
$263$	4.76850			0.294038			0.147019	+	0.989134i	$0.453032\pi$
$264$		0			0
$265$	−29.3205			−1.80114
$266$		0			0
$267$			10.3923i			0.635999i
$268$		0			0
$269$	−0.875644			−0.0533890			−0.0266945	−	0.999644i	$-0.508498\pi$
$269$	−0.875644			−0.0533890			−0.0266945	+	0.999644i	$0.508498\pi$
$270$		0			0
$271$	−2.38425			−0.144833			−0.0724164	−	0.997374i	$-0.523071\pi$
$271$	−2.38425			−0.144833			−0.0724164	+	0.997374i	$0.523071\pi$
$272$		0			0
$273$	26.3923			1.59733
$274$		0			0
$275$	4.63349	−	6.73205i	0.279410	−	0.405958i
$276$		0			0
$277$		−	9.90584i		−	0.595184i	−0.954693	−	0.297592i	$-0.903816\pi$
$277$		−	9.90584i		−	0.595184i	0.954693	−	0.297592i	$-0.0961836\pi$
$278$		0			0
$279$			2.00000i			0.119737i
$280$		0			0
$281$		−	9.26699i		−	0.552822i	−0.961040	−	0.276411i	$-0.910855\pi$
$281$		−	9.26699i		−	0.552822i	0.961040	−	0.276411i	$-0.0891451\pi$
$282$		0			0
$283$	−14.0355			−0.834323			−0.417161	−	0.908832i	$-0.636975\pi$
$283$	−14.0355			−0.834323			−0.417161	+	0.908832i	$0.636975\pi$
$284$		0			0
$285$		−	10.2747i		−	0.608620i
$286$		0			0
$287$		−	52.7846i		−	3.11578i
$288$		0			0
$289$	2.85641			0.168024
$290$		0			0
$291$		−	5.46410i		−	0.320311i
$292$		0			0
$293$		−	20.5494i		−	1.20051i	−0.799810	−	0.600254i	$-0.795066\pi$
$293$		−	20.5494i		−	1.20051i	0.799810	−	0.600254i	$-0.204934\pi$
$294$		0			0
$295$		−	25.8564i		−	1.50542i
$296$		0			0
$297$	2.73205	+	1.88040i	0.158530	+	0.109112i
$298$		0			0
$299$	−6.51389			−0.376708
$300$		0			0
$301$	−19.3205			−1.11362
$302$		0			0
$303$	17.7963			1.02237
$304$		0			0
$305$			6.51389i			0.372984i
$306$		0			0
$307$	−6.51389			−0.371767			−0.185884	−	0.982572i	$-0.559515\pi$
$307$	−6.51389			−0.371767			−0.185884	+	0.982572i	$0.559515\pi$
$308$		0			0
$309$	10.0000			0.568880
$310$		0			0
$311$		−	20.9808i		−	1.18971i	−0.803833	−	0.594855i	$-0.797209\pi$
$311$		−	20.9808i		−	1.18971i	0.803833	−	0.594855i	$-0.202791\pi$
$312$		0			0
$313$	11.8564			0.670164			0.335082	−	0.942189i	$-0.391236\pi$
$313$	11.8564			0.670164			0.335082	+	0.942189i	$0.391236\pi$
$314$		0			0
$315$	14.0355			0.790810
$316$		0			0
$317$	9.66025			0.542574			0.271287	−	0.962499i	$-0.412551\pi$
$317$	9.66025			0.542574			0.271287	+	0.962499i	$0.412551\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$			3.76080i			0.209907i
$322$		0			0
$323$		−	14.1436i		−	0.786971i
$324$		0			0
$325$		−	12.6589i		−	0.702192i
$326$		0			0
$327$	12.6589			0.700041
$328$		0			0
$329$			21.5571i			1.18848i
$330$		0			0
$331$			1.07180i			0.0589113i	0.999566	+	0.0294556i	$0.00937738\pi$
$331$			1.07180i			0.0589113i	−0.999566	+	0.0294556i	$0.990623\pi$
$332$		0			0
$333$	−0.535898			−0.0293671
$334$		0			0
$335$			28.3923i			1.55124i
$336$		0			0
$337$			2.75309i			0.149971i	0.997185	+	0.0749853i	$0.0238910\pi$
$337$			2.75309i			0.149971i	−0.997185	+	0.0749853i	$0.976109\pi$
$338$		0			0
$339$		−	17.3205i		−	0.940721i
$340$		0			0
$341$	−5.46410	−	3.76080i	−0.295898	−	0.203659i
$342$		0			0
$343$	−63.6635			−3.43751
$344$		0			0
$345$	−3.46410			−0.186501
$346$		0			0
$347$	−30.8241			−1.65472			−0.827361	−	0.561670i	$-0.810159\pi$
$347$	−30.8241			−1.65472			−0.827361	+	0.561670i	$0.810159\pi$
$348$		0			0
$349$			7.89044i			0.422365i	0.977447	+	0.211183i	$0.0677315\pi$
$349$			7.89044i			0.422365i	−0.977447	+	0.211183i	$0.932268\pi$
$350$		0			0
$351$	5.13734			0.274211
$352$		0			0
$353$	8.14359			0.433440			0.216720	−	0.976234i	$-0.430464\pi$
$353$	8.14359			0.433440			0.216720	+	0.976234i	$0.430464\pi$
$354$		0			0
$355$			29.3205i			1.55617i
$356$		0			0
$357$	19.3205			1.02255
$358$		0			0
$359$	−7.52159			−0.396975			−0.198487	−	0.980103i	$-0.563603\pi$
$359$	−7.52159			−0.396975			−0.198487	+	0.980103i	$0.563603\pi$
$360$		0			0
$361$	−4.85641			−0.255600
$362$		0			0
$363$	−10.2747	+	3.92820i	−0.539281	+	0.206177i
$364$		0			0
$365$		−	28.0710i		−	1.46930i
$366$		0			0
$367$		−	10.3923i		−	0.542474i	−0.962513	−	0.271237i	$-0.912567\pi$
$367$		−	10.3923i		−	0.542474i	0.962513	−	0.271237i	$-0.0874327\pi$
$368$		0			0
$369$		−	10.2747i		−	0.534879i
$370$		0			0
$371$	55.1342			2.86243
$372$		0			0
$373$			18.1651i			0.940555i	0.882519	+	0.470277i	$0.155846\pi$
$373$			18.1651i			0.940555i	−0.882519	+	0.470277i	$0.844154\pi$
$374$		0			0
$375$			6.92820i			0.357771i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			2.92820i			0.150412i	0.997168	+	0.0752058i	$0.0239614\pi$
$379$			2.92820i			0.150412i	−0.997168	+	0.0752058i	$0.976039\pi$
$380$		0			0
$381$			5.13734i			0.263194i
$382$		0			0
$383$			20.9808i			1.07207i	0.844197	+	0.536033i	$0.180078\pi$
$383$			20.9808i			1.07207i	−0.844197	+	0.536033i	$0.819922\pi$
$384$		0			0
$385$	−26.3923	+	38.3457i	−1.34508	+	1.95428i
$386$		0			0
$387$	−3.76080			−0.191172
$388$		0			0
$389$	−1.26795			−0.0642876			−0.0321438	−	0.999483i	$-0.510233\pi$
$389$	−1.26795			−0.0642876			−0.0321438	+	0.999483i	$0.510233\pi$
$390$		0			0
$391$	−4.76850			−0.241153
$392$		0			0
$393$		−	2.75309i		−	0.138875i
$394$		0			0
$395$	14.0355			0.706202
$396$		0			0
$397$	−21.3205			−1.07005			−0.535023	−	0.844838i	$-0.679697\pi$
$397$	−21.3205			−1.07005			−0.535023	+	0.844838i	$0.679697\pi$
$398$		0			0
$399$			19.3205i			0.967235i
$400$		0			0
$401$	3.46410			0.172989			0.0864945	−	0.996252i	$-0.472434\pi$
$401$	3.46410			0.172989			0.0864945	+	0.996252i	$0.472434\pi$
$402$		0			0
$403$	−10.2747			−0.511819
$404$		0			0
$405$	2.73205			0.135757
$406$		0			0
$407$	1.00770	−	1.46410i	0.0499500	−	0.0725728i
$408$		0			0
$409$			25.3179i			1.25189i	0.779868	+	0.625944i	$0.215286\pi$
$409$			25.3179i			1.25189i	−0.779868	+	0.625944i	$0.784714\pi$
$410$		0			0
$411$		−	4.92820i		−	0.243090i
$412$		0			0
$413$			48.6203i			2.39245i
$414$		0			0
$415$	28.0710			1.37795
$416$		0			0
$417$		−	14.0355i		−	0.687321i
$418$		0			0
$419$			6.14359i			0.300134i	0.988676	+	0.150067i	$0.0479490\pi$
$419$			6.14359i			0.300134i	−0.988676	+	0.150067i	$0.952051\pi$
$420$		0			0
$421$	9.32051			0.454254			0.227127	−	0.973865i	$-0.427067\pi$
$421$	9.32051			0.454254			0.227127	+	0.973865i	$0.427067\pi$
$422$		0			0
$423$			4.19615i			0.204024i
$424$		0			0
$425$		−	9.26699i		−	0.449515i
$426$		0			0
$427$		−	12.2487i		−	0.592757i
$428$		0			0
$429$	−9.66025	+	14.0355i	−0.466401	+	0.677640i
$430$		0			0
$431$	23.3025			1.12244			0.561220	−	0.827666i	$-0.310332\pi$
$431$	23.3025			1.12244			0.561220	+	0.827666i	$0.310332\pi$
$432$		0			0
$433$	−16.9282			−0.813518			−0.406759	−	0.913536i	$-0.633341\pi$
$433$	−16.9282			−0.813518			−0.406759	+	0.913536i	$0.633341\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	4.76850i		−	0.228108i
$438$		0			0
$439$	28.4398			1.35736			0.678679	−	0.734435i	$-0.262553\pi$
$439$	28.4398			1.35736			0.678679	+	0.734435i	$0.262553\pi$
$440$		0			0
$441$	−19.3923			−0.923443
$442$		0			0
$443$			2.14359i			0.101845i	0.998703	+	0.0509226i	$0.0162162\pi$
$443$			2.14359i			0.101845i	−0.998703	+	0.0509226i	$0.983784\pi$
$444$		0			0
$445$	−28.3923			−1.34592
$446$		0			0
$447$	−17.7963			−0.841735
$448$		0			0
$449$	−15.0718			−0.711282			−0.355641	−	0.934623i	$-0.615737\pi$
$449$	−15.0718			−0.711282			−0.355641	+	0.934623i	$0.615737\pi$
$450$		0			0
$451$	28.0710	+	19.3205i	1.32181	+	0.909767i
$452$		0			0
$453$			2.38425i			0.112022i
$454$		0			0
$455$			72.1051i			3.38034i
$456$		0			0
$457$		−	2.75309i		−	0.128784i	−0.997925	−	0.0643922i	$-0.979489\pi$
$457$		−	2.75309i		−	0.128784i	0.997925	−	0.0643922i	$-0.0205109\pi$
$458$		0			0
$459$	3.76080			0.175539
$460$		0			0
$461$		−	5.50619i		−	0.256449i	−0.991745	−	0.128224i	$-0.959072\pi$
$461$		−	5.50619i		−	0.256449i	0.991745	−	0.128224i	$-0.0409278\pi$
$462$		0			0
$463$		−	32.9282i		−	1.53030i	−0.643850	−	0.765152i	$-0.722664\pi$
$463$		−	32.9282i		−	1.53030i	0.643850	−	0.765152i	$-0.277336\pi$
$464$		0			0
$465$	−5.46410			−0.253392
$466$		0			0
$467$			9.46410i			0.437946i	0.975731	+	0.218973i	$0.0702707\pi$
$467$			9.46410i			0.437946i	−0.975731	+	0.218973i	$0.929729\pi$
$468$		0			0
$469$		−	53.3888i		−	2.46527i
$470$		0			0
$471$		−	11.8564i		−	0.546314i
$472$		0			0
$473$	7.07180	−	10.2747i	0.325162	−	0.472431i
$474$		0			0
$475$	9.26699			0.425198
$476$		0			0
$477$	10.7321			0.491387
$478$		0			0
$479$	−10.2747			−0.469462			−0.234731	−	0.972060i	$-0.575421\pi$
$479$	−10.2747			−0.469462			−0.234731	+	0.972060i	$0.575421\pi$
$480$		0			0
$481$		−	2.75309i		−	0.125530i
$482$		0			0
$483$	6.51389			0.296392
$484$		0			0
$485$	14.9282			0.677855
$486$		0			0
$487$		−	18.3923i		−	0.833435i	−0.909036	−	0.416717i	$-0.863180\pi$
$487$		−	18.3923i		−	0.833435i	0.909036	−	0.416717i	$-0.136820\pi$
$488$		0			0
$489$	−2.92820			−0.132418
$490$		0			0
$491$	−11.2824			−0.509167			−0.254584	−	0.967051i	$-0.581938\pi$
$491$	−11.2824			−0.509167			−0.254584	+	0.967051i	$0.581938\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−5.13734	+	7.46410i	−0.230906	+	0.335486i
$496$		0			0
$497$		−	55.1342i		−	2.47311i
$498$		0			0
$499$		−	17.0718i		−	0.764239i	−0.924113	−	0.382119i	$-0.875194\pi$
$499$		−	17.0718i		−	0.764239i	0.924113	−	0.382119i	$-0.124806\pi$
$500$		0			0
$501$		−	17.7963i		−	0.795079i
$502$		0			0
$503$	2.75309			0.122754			0.0613772	−	0.998115i	$-0.480451\pi$
$503$	2.75309			0.122754			0.0613772	+	0.998115i	$0.480451\pi$
$504$		0			0
$505$			48.6203i			2.16358i
$506$		0			0
$507$			13.3923i			0.594773i
$508$		0			0
$509$	26.7321			1.18488			0.592439	−	0.805616i	$-0.298165\pi$
$509$	26.7321			1.18488			0.592439	+	0.805616i	$0.298165\pi$
$510$		0			0
$511$			52.7846i			2.33505i
$512$		0			0
$513$			3.76080i			0.166043i
$514$		0			0
$515$			27.3205i			1.20389i
$516$		0			0
$517$	−11.4641	−	7.89044i	−0.504191	−	0.347021i
$518$		0			0
$519$	−13.0278			−0.571856
$520$		0			0
$521$	−41.3205			−1.81028			−0.905142	−	0.425109i	$-0.860236\pi$
$521$	−41.3205			−1.81028			−0.905142	+	0.425109i	$0.860236\pi$
$522$		0			0
$523$	37.3380			1.63267			0.816337	−	0.577575i	$-0.196001\pi$
$523$	37.3380			1.63267			0.816337	+	0.577575i	$0.196001\pi$
$524$		0			0
$525$			12.6589i			0.552481i
$526$		0			0
$527$	−7.52159			−0.327646
$528$		0			0
$529$	21.3923			0.930100
$530$		0			0
$531$			9.46410i			0.410707i
$532$		0			0
$533$	52.7846			2.28636
$534$		0			0
$535$	−10.2747			−0.444214
$536$		0			0
$537$	−0.392305			−0.0169292
$538$		0			0
$539$	36.4653	−	52.9808i	1.57067	−	2.28204i
$540$		0			0
$541$			27.7021i			1.19101i	0.803353	+	0.595504i	$0.203047\pi$
$541$			27.7021i			1.19101i	−0.803353	+	0.595504i	$0.796953\pi$
$542$		0			0
$543$		−	2.39230i		−	0.102664i
$544$		0			0
$545$			34.5849i			1.48145i
$546$		0			0
$547$	−9.26699			−0.396228			−0.198114	−	0.980179i	$-0.563482\pi$
$547$	−9.26699			−0.396228			−0.198114	+	0.980179i	$0.563482\pi$
$548$		0			0
$549$		−	2.38425i		−	0.101757i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−26.3923			−1.12231
$554$		0			0
$555$		−	1.46410i		−	0.0621477i
$556$		0			0
$557$			20.5494i			0.870705i	0.900260	+	0.435353i	$0.143376\pi$
$557$			20.5494i			0.870705i	−0.900260	+	0.435353i	$0.856624\pi$
$558$		0			0
$559$		−	19.3205i		−	0.817170i
$560$		0			0
$561$	−7.07180	+	10.2747i	−0.298571	+	0.433798i
$562$		0			0
$563$	−31.8318			−1.34155			−0.670775	−	0.741661i	$-0.734038\pi$
$563$	−31.8318			−1.34155			−0.670775	+	0.741661i	$0.734038\pi$
$564$		0			0
$565$	47.3205			1.99079
$566$		0			0
$567$	−5.13734			−0.215748
$568$		0			0
$569$		−	9.26699i		−	0.388492i	−0.980953	−	0.194246i	$-0.937774\pi$
$569$		−	9.26699i		−	0.388492i	0.980953	−	0.194246i	$-0.0622260\pi$
$570$		0			0
$571$	−14.0355			−0.587367			−0.293683	−	0.955903i	$-0.594881\pi$
$571$	−14.0355			−0.587367			−0.293683	+	0.955903i	$0.594881\pi$
$572$		0			0
$573$	11.1244			0.464727
$574$		0			0
$575$		−	3.12436i		−	0.130295i
$576$		0			0
$577$	40.3923			1.68155			0.840777	−	0.541382i	$-0.182099\pi$
$577$	40.3923			1.68155			0.840777	+	0.541382i	$0.182099\pi$
$578$		0			0
$579$	13.0278			0.541416
$580$		0			0
$581$	−52.7846			−2.18987
$582$		0			0
$583$	−20.1805	+	29.3205i	−0.835792	+	1.21433i
$584$		0			0
$585$			14.0355i			0.580296i
$586$		0			0
$587$			33.1769i			1.36936i	0.728845	+	0.684679i	$0.240058\pi$
$587$			33.1769i			1.36936i	−0.728845	+	0.684679i	$0.759942\pi$
$588$		0			0
$589$		−	7.52159i		−	0.309922i
$590$		0			0
$591$	2.75309			0.113247
$592$		0			0
$593$			24.3102i			0.998299i	0.866516	+	0.499150i	$0.166354\pi$
$593$			24.3102i			0.998299i	−0.866516	+	0.499150i	$0.833646\pi$
$594$		0			0
$595$			52.7846i			2.16396i
$596$		0			0
$597$	−23.4641			−0.960322
$598$		0			0
$599$		−	15.8038i		−	0.645728i	−0.946445	−	0.322864i	$-0.895354\pi$
$599$		−	15.8038i		−	0.645728i	0.946445	−	0.322864i	$-0.104646\pi$
$600$		0			0
$601$			32.8395i			1.33955i	0.742564	+	0.669775i	$0.233609\pi$
$601$			32.8395i			1.33955i	−0.742564	+	0.669775i	$0.766391\pi$
$602$		0			0
$603$		−	10.3923i		−	0.423207i
$604$		0			0
$605$	−10.7321	−	28.0710i	−0.436320	−	1.14125i
$606$		0			0
$607$	27.7021			1.12439			0.562197	−	0.827003i	$-0.309956\pi$
$607$	27.7021			1.12439			0.562197	+	0.827003i	$0.309956\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−21.5571			−0.872106
$612$		0			0
$613$			33.2083i			1.34127i	0.741787	+	0.670636i	$0.233979\pi$
$613$			33.2083i			1.34127i	−0.741787	+	0.670636i	$0.766021\pi$
$614$		0			0
$615$	28.0710			1.13193
$616$		0			0
$617$	8.53590			0.343642			0.171821	−	0.985128i	$-0.445035\pi$
$617$	8.53590			0.343642			0.171821	+	0.985128i	$0.445035\pi$
$618$		0			0
$619$		−	8.53590i		−	0.343087i	−0.985177	−	0.171543i	$-0.945125\pi$
$619$		−	8.53590i		−	0.343087i	0.985177	−	0.171543i	$-0.0548754\pi$
$620$		0			0
$621$	1.26795			0.0508810
$622$		0			0
$623$	53.3888			2.13898
$624$		0			0
$625$	−31.2487			−1.24995
$626$		0			0
$627$	−10.2747	−	7.07180i	−0.410332	−	0.282420i
$628$		0			0
$629$		−	2.01540i		−	0.0803595i
$630$		0			0
$631$		−	6.78461i		−	0.270091i	−0.990839	−	0.135046i	$-0.956882\pi$
$631$		−	6.78461i		−	0.270091i	0.990839	−	0.135046i	$-0.0431181\pi$
$632$		0			0
$633$		−	21.5571i		−	0.856817i
$634$		0			0
$635$	−14.0355			−0.556981
$636$		0			0
$637$		−	99.6249i		−	3.94728i
$638$		0			0
$639$		−	10.7321i		−	0.424553i
$640$		0			0
$641$	−4.53590			−0.179157			−0.0895786	−	0.995980i	$-0.528552\pi$
$641$	−4.53590			−0.179157			−0.0895786	+	0.995980i	$0.528552\pi$
$642$		0			0
$643$			15.4641i			0.609845i	0.952377	+	0.304922i	$0.0986305\pi$
$643$			15.4641i			0.609845i	−0.952377	+	0.304922i	$0.901369\pi$
$644$		0			0
$645$		−	10.2747i		−	0.404565i
$646$		0			0
$647$		−	12.8756i		−	0.506194i	−0.967441	−	0.253097i	$-0.918551\pi$
$647$		−	12.8756i		−	0.506194i	0.967441	−	0.253097i	$-0.0814492\pi$
$648$		0			0
$649$	−25.8564	−	17.7963i	−1.01495	−	0.698565i
$650$		0			0
$651$	10.2747			0.402697
$652$		0			0
$653$	−21.2679			−0.832279			−0.416140	−	0.909301i	$-0.636617\pi$
$653$	−21.2679			−0.832279			−0.416140	+	0.909301i	$0.636617\pi$
$654$		0			0
$655$	7.52159			0.293893
$656$		0			0
$657$			10.2747i			0.400854i
$658$		0			0
$659$	−37.3380			−1.45448			−0.727240	−	0.686383i	$-0.759197\pi$
$659$	−37.3380			−1.45448			−0.727240	+	0.686383i	$0.759197\pi$
$660$		0			0
$661$	31.8564			1.23907			0.619535	−	0.784969i	$-0.287321\pi$
$661$	31.8564			1.23907			0.619535	+	0.784969i	$0.287321\pi$
$662$		0			0
$663$			19.3205i			0.750346i
$664$		0			0
$665$	−52.7846			−2.04690
$666$		0			0
$667$		0			0
$668$		0			0
$669$	11.4641			0.443228
$670$		0			0
$671$	6.51389	+	4.48334i	0.251466	+	0.173077i
$672$		0			0
$673$		−	32.8395i		−	1.26587i	−0.774206	−	0.632934i	$-0.781850\pi$
$673$		−	32.8395i		−	1.26587i	0.774206	−	0.632934i	$-0.218150\pi$
$674$		0			0
$675$			2.46410i			0.0948433i
$676$		0			0
$677$			4.76850i			0.183268i	0.995793	+	0.0916342i	$0.0292090\pi$
$677$			4.76850i			0.183268i	−0.995793	+	0.0916342i	$0.970791\pi$
$678$		0			0
$679$	−28.0710			−1.07726
$680$		0			0
$681$			16.7886i			0.643340i
$682$		0			0
$683$		−	19.7128i		−	0.754290i	−0.926154	−	0.377145i	$-0.876906\pi$
$683$		−	19.7128i		−	0.754290i	0.926154	−	0.377145i	$-0.123094\pi$
$684$		0			0
$685$	13.4641			0.514437
$686$		0			0
$687$		−	23.8564i		−	0.910179i
$688$		0			0
$689$			55.1342i			2.10045i
$690$		0			0
$691$			30.3923i			1.15618i	0.815974	+	0.578089i	$0.196201\pi$
$691$			30.3923i			1.15618i	−0.815974	+	0.578089i	$0.803799\pi$
$692$		0			0
$693$	9.66025	−	14.0355i	0.366963	−	0.533164i
$694$		0			0
$695$	38.3457			1.45453
$696$		0			0
$697$	38.6410			1.46363
$698$		0			0
$699$	10.2747			0.388624
$700$		0			0
$701$			13.0278i			0.492053i	0.969263	+	0.246026i	$0.0791250\pi$
$701$			13.0278i			0.492053i	−0.969263	+	0.246026i	$0.920875\pi$
$702$		0			0
$703$	2.01540			0.0760124
$704$		0			0
$705$	−11.4641			−0.431763
$706$		0			0
$707$		−	91.4256i		−	3.43841i
$708$		0			0
$709$	−22.7846			−0.855694			−0.427847	−	0.903851i	$-0.640728\pi$
$709$	−22.7846			−0.855694			−0.427847	+	0.903851i	$0.640728\pi$
$710$		0			0
$711$	−5.13734			−0.192665
$712$		0			0
$713$	−2.53590			−0.0949701
$714$		0			0
$715$	−38.3457	−	26.3923i	−1.43405	−	0.987016i
$716$		0			0
$717$			23.3025i			0.870247i
$718$		0			0
$719$		−	5.66025i		−	0.211092i	−0.994414	−	0.105546i	$-0.966341\pi$
$719$		−	5.66025i		−	0.211092i	0.994414	−	0.105546i	$-0.0336590\pi$
$720$		0			0
$721$		−	51.3734i		−	1.91325i
$722$		0			0
$723$	−28.0710			−1.04397
$724$		0			0
$725$		0			0
$726$		0			0
$727$			0.248711i			0.00922419i	0.999989	+	0.00461210i	$0.00146808\pi$
$727$			0.248711i			0.00922419i	−0.999989	+	0.00461210i	$0.998532\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	14.1436i		−	0.523120i
$732$		0			0
$733$			12.6589i			0.467569i	0.972288	+	0.233784i	$0.0751109\pi$
$733$			12.6589i			0.467569i	−0.972288	+	0.233784i	$0.924889\pi$
$734$		0			0
$735$		−	52.9808i		−	1.95422i
$736$		0			0
$737$	28.3923	+	19.5417i	1.04584	+	0.719827i
$738$		0			0
$739$	1.00770			0.0370689			0.0185345	−	0.999828i	$-0.494100\pi$
$739$	1.00770			0.0370689			0.0185345	+	0.999828i	$0.494100\pi$
$740$		0			0
$741$	−19.3205			−0.709757
$742$		0			0
$743$	22.5648			0.827822			0.413911	−	0.910317i	$-0.364163\pi$
$743$	22.5648			0.827822			0.413911	+	0.910317i	$0.364163\pi$
$744$		0			0
$745$		−	48.6203i		−	1.78131i
$746$		0			0
$747$	−10.2747			−0.375931
$748$		0			0
$749$	19.3205			0.705956
$750$		0			0
$751$			31.5692i			1.15198i	0.817458	+	0.575989i	$0.195383\pi$
$751$			31.5692i			1.15198i	−0.817458	+	0.575989i	$0.804617\pi$
$752$		0			0
$753$	21.8564			0.796492
$754$		0			0
$755$	−6.51389			−0.237065
$756$		0			0
$757$	43.1769			1.56929			0.784646	−	0.619944i	$-0.212845\pi$
$757$	43.1769			1.56929			0.784646	+	0.619944i	$0.212845\pi$
$758$		0			0
$759$	−2.38425	+	3.46410i	−0.0865428	+	0.125739i
$760$		0			0
$761$			37.3380i			1.35350i	0.736213	+	0.676750i	$0.236612\pi$
$761$			37.3380i			1.35350i	−0.736213	+	0.676750i	$0.763388\pi$
$762$		0			0
$763$		−	65.0333i		−	2.35436i
$764$		0			0
$765$			10.2747i			0.371482i
$766$		0			0
$767$	−48.6203			−1.75558
$768$		0			0
$769$		−	5.50619i		−	0.198558i	−0.995060	−	0.0992791i	$-0.968346\pi$
$769$		−	5.50619i		−	0.198558i	0.995060	−	0.0992791i	$-0.0316537\pi$
$770$		0			0
$771$			15.4641i			0.556926i
$772$		0			0
$773$	−34.8372			−1.25301			−0.626503	−	0.779419i	$-0.715514\pi$
$773$	−34.8372			−1.25301			−0.626503	+	0.779419i	$0.715514\pi$
$774$		0			0
$775$		−	4.92820i		−	0.177026i
$776$		0			0
$777$			2.75309i			0.0987667i
$778$		0			0
$779$			38.6410i			1.38446i
$780$		0			0
$781$	29.3205	+	20.1805i	1.04917	+	0.722116i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	32.3923			1.15613
$786$		0			0
$787$	−9.26699			−0.330332			−0.165166	−	0.986266i	$-0.552816\pi$
$787$	−9.26699			−0.330332			−0.165166	+	0.986266i	$0.552816\pi$
$788$		0			0
$789$		−	4.76850i		−	0.169763i
$790$		0			0
$791$	−88.9814			−3.16381
$792$		0			0
$793$	12.2487			0.434964
$794$		0			0
$795$			29.3205i			1.03989i
$796$		0			0
$797$	17.6603			0.625558			0.312779	−	0.949826i	$-0.398740\pi$
$797$	17.6603			0.625558			0.312779	+	0.949826i	$0.398740\pi$
$798$		0			0
$799$	−15.7809			−0.558287
$800$		0			0
$801$	10.3923			0.367194
$802$		0			0
$803$	−28.0710	−	19.3205i	−0.990603	−	0.681806i
$804$		0			0
$805$			17.7963i			0.627237i
$806$		0			0
$807$			0.875644i			0.0308241i
$808$		0			0
$809$		−	25.3179i		−	0.890129i	−0.895498	−	0.445065i	$-0.853181\pi$
$809$		−	25.3179i		−	0.890129i	0.895498	−	0.445065i	$-0.146819\pi$
$810$		0			0
$811$	−3.76080			−0.132059			−0.0660297	−	0.997818i	$-0.521033\pi$
$811$	−3.76080			−0.132059			−0.0660297	+	0.997818i	$0.521033\pi$
$812$		0			0
$813$			2.38425i			0.0836193i
$814$		0			0
$815$		−	8.00000i		−	0.280228i
$816$		0			0
$817$	14.1436			0.494822
$818$		0			0
$819$		−	26.3923i		−	0.922222i
$820$		0			0
$821$		−	17.7963i		−	0.621095i	−0.950558	−	0.310547i	$-0.899488\pi$
$821$		−	17.7963i		−	0.621095i	0.950558	−	0.310547i	$-0.100512\pi$
$822$		0			0
$823$		−	44.2487i		−	1.54241i	−0.636585	−	0.771207i	$-0.719654\pi$
$823$		−	44.2487i		−	1.54241i	0.636585	−	0.771207i	$-0.280346\pi$
$824$		0			0
$825$	−6.73205	−	4.63349i	−0.234380	−	0.161318i
$826$		0			0
$827$	4.76850			0.165817			0.0829085	−	0.996557i	$-0.473579\pi$
$827$	4.76850			0.165817			0.0829085	+	0.996557i	$0.473579\pi$
$828$		0			0
$829$	39.8564			1.38427			0.692135	−	0.721768i	$-0.256670\pi$
$829$	39.8564			1.38427			0.692135	+	0.721768i	$0.256670\pi$
$830$		0			0
$831$	−9.90584			−0.343630
$832$		0			0
$833$		−	72.9305i		−	2.52689i
$834$		0			0
$835$	48.6203			1.68258
$836$		0			0
$837$	2.00000			0.0691301
$838$		0			0
$839$		−	23.1244i		−	0.798341i	−0.916877	−	0.399171i	$-0.869298\pi$
$839$		−	23.1244i		−	0.798341i	0.916877	−	0.399171i	$-0.130702\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	−9.26699			−0.319172
$844$		0			0
$845$	−36.5885			−1.25868
$846$		0			0
$847$	20.1805	+	52.7846i	0.693411	+	1.81370i
$848$		0			0
$849$			14.0355i			0.481697i
$850$		0			0
$851$		−	0.679492i		−	0.0232927i
$852$		0			0
$853$			35.9614i			1.23130i	0.788022	+	0.615648i	$0.211106\pi$
$853$			35.9614i			1.23130i	−0.788022	+	0.615648i	$0.788894\pi$
$854$		0			0
$855$	−10.2747			−0.351387
$856$		0			0
$857$		−	4.76850i		−	0.162889i	−0.996678	−	0.0814444i	$-0.974047\pi$
$857$		−	4.76850i		−	0.162889i	0.996678	−	0.0814444i	$-0.0259533\pi$
$858$		0			0
$859$			21.0718i			0.718960i	0.933153	+	0.359480i	$0.117046\pi$
$859$			21.0718i			0.718960i	−0.933153	+	0.359480i	$0.882954\pi$
$860$		0			0
$861$	−52.7846			−1.79889
$862$		0			0
$863$			4.87564i			0.165969i	0.996551	+	0.0829844i	$0.0264452\pi$
$863$			4.87564i			0.165969i	−0.996551	+	0.0829844i	$0.973555\pi$
$864$		0			0
$865$		−	35.5926i		−	1.21018i
$866$		0			0
$867$		−	2.85641i		−	0.0970086i
$868$		0			0
$869$	9.66025	−	14.0355i	0.327702	−	0.476121i
$870$		0			0
$871$	53.3888			1.80901
$872$		0			0
$873$	−5.46410			−0.184932
$874$		0			0
$875$	35.5926			1.20325
$876$		0			0
$877$		−	18.1651i		−	0.613393i	−0.951807	−	0.306696i	$-0.900776\pi$
$877$		−	18.1651i		−	0.613393i	0.951807	−	0.306696i	$-0.0992236\pi$
$878$		0			0
$879$	−20.5494			−0.693113
$880$		0			0
$881$	18.7846			0.632870			0.316435	−	0.948614i	$-0.397514\pi$
$881$	18.7846			0.632870			0.316435	+	0.948614i	$0.397514\pi$
$882$		0			0
$883$		−	43.4641i		−	1.46268i	−0.682011	−	0.731342i	$-0.738894\pi$
$883$		−	43.4641i		−	1.46268i	0.682011	−	0.731342i	$-0.261106\pi$
$884$		0			0
$885$	−25.8564			−0.869154
$886$		0			0
$887$	7.52159			0.252550			0.126275	−	0.991995i	$-0.459698\pi$
$887$	7.52159			0.252550			0.126275	+	0.991995i	$0.459698\pi$
$888$		0			0
$889$	26.3923			0.885169
$890$		0			0
$891$	1.88040	−	2.73205i	0.0629957	−	0.0915271i
$892$		0			0
$893$		−	15.7809i		−	0.528087i
$894$		0			0
$895$		−	1.07180i		−	0.0358262i
$896$		0			0
$897$			6.51389i			0.217493i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			40.3611i			1.34462i
$902$		0			0
$903$			19.3205i			0.642946i
$904$		0			0
$905$	6.53590			0.217261
$906$		0			0
$907$		−	17.6077i		−	0.584654i	−0.956318	−	0.292327i	$-0.905570\pi$
$907$		−	17.6077i		−	0.584654i	0.956318	−	0.292327i	$-0.0944296\pi$
$908$		0			0
$909$		−	17.7963i		−	0.590265i
$910$		0			0
$911$			0.980762i			0.0324941i	0.999868	+	0.0162470i	$0.00517182\pi$
$911$			0.980762i			0.0324941i	−0.999868	+	0.0162470i	$0.994828\pi$
$912$		0			0
$913$	19.3205	−	28.0710i	0.639415	−	0.929014i
$914$		0			0
$915$	6.51389			0.215343
$916$		0			0
$917$	−14.1436			−0.467063
$918$		0			0
$919$	−28.4398			−0.938143			−0.469071	−	0.883160i	$-0.655411\pi$
$919$	−28.4398			−0.938143			−0.469071	+	0.883160i	$0.655411\pi$
$920$		0			0
$921$			6.51389i			0.214640i
$922$		0			0
$923$	55.1342			1.81477
$924$		0			0
$925$	1.32051			0.0434180
$926$		0			0
$927$		−	10.0000i		−	0.328443i
$928$		0			0
$929$	−15.8564			−0.520232			−0.260116	−	0.965577i	$-0.583761\pi$
$929$	−15.8564			−0.520232			−0.260116	+	0.965577i	$0.583761\pi$
$930$		0			0
$931$	72.9305			2.39020
$932$		0			0
$933$	−20.9808			−0.686879
$934$		0			0
$935$	−28.0710	−	19.3205i	−0.918019	−	0.631848i
$936$		0			0
$937$			7.52159i			0.245720i	0.992424	+	0.122860i	$0.0392066\pi$
$937$			7.52159i			0.245720i	−0.992424	+	0.122860i	$0.960793\pi$
$938$		0			0
$939$		−	11.8564i		−	0.386919i
$940$		0			0
$941$			10.2747i			0.334945i	0.985877	+	0.167473i	$0.0535606\pi$
$941$			10.2747i			0.334945i	−0.985877	+	0.167473i	$0.946439\pi$
$942$		0			0
$943$	13.0278			0.424243
$944$		0			0
$945$		−	14.0355i		−	0.456574i
$946$		0			0
$947$			49.8564i			1.62012i	0.586350	+	0.810058i	$0.300564\pi$
$947$			49.8564i			1.62012i	−0.586350	+	0.810058i	$0.699436\pi$
$948$		0			0
$949$	−52.7846			−1.71346
$950$		0			0
$951$		−	9.66025i		−	0.313255i
$952$		0			0
$953$		−	40.3611i		−	1.30742i	−0.756744	−	0.653712i	$-0.773211\pi$
$953$		−	40.3611i		−	1.30742i	0.756744	−	0.653712i	$-0.226789\pi$
$954$		0			0
$955$			30.3923i			0.983472i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−25.3179			−0.817557
$960$		0			0
$961$	27.0000			0.870968
$962$		0			0
$963$	3.76080			0.121190
$964$		0			0
$965$			35.5926i			1.14577i
$966$		0			0
$967$	−15.4120			−0.495618			−0.247809	−	0.968809i	$-0.579710\pi$
$967$	−15.4120			−0.495618			−0.247809	+	0.968809i	$0.579710\pi$
$968$		0			0
$969$	−14.1436			−0.454358
$970$		0			0
$971$			40.0000i			1.28366i	0.766846	+	0.641831i	$0.221825\pi$
$971$			40.0000i			1.28366i	−0.766846	+	0.641831i	$0.778175\pi$
$972$		0			0
$973$	−72.1051			−2.31158
$974$		0			0
$975$	−12.6589			−0.405410
$976$		0			0
$977$	8.24871			0.263900			0.131950	−	0.991256i	$-0.457876\pi$
$977$	8.24871			0.263900			0.131950	+	0.991256i	$0.457876\pi$
$978$		0			0
$979$	−19.5417	+	28.3923i	−0.624555	+	0.907422i
$980$		0			0
$981$		−	12.6589i		−	0.404169i
$982$		0			0
$983$			49.6603i			1.58392i	0.610575	+	0.791958i	$0.290938\pi$
$983$			49.6603i			1.58392i	−0.610575	+	0.791958i	$0.709062\pi$
$984$		0			0
$985$			7.52159i			0.239658i
$986$		0			0
$987$	21.5571			0.686169
$988$		0			0
$989$		−	4.76850i		−	0.151629i
$990$		0			0
$991$			22.7846i			0.723777i	0.932221	+	0.361888i	$0.117868\pi$
$991$			22.7846i			0.723777i	−0.932221	+	0.361888i	$0.882132\pi$
$992$		0			0
$993$	1.07180			0.0340124
$994$		0			0
$995$		−	64.1051i		−	2.03227i
$996$		0			0
$997$			24.9490i			0.790144i	0.918650	+	0.395072i	$0.129280\pi$
$997$			24.9490i			0.790144i	−0.918650	+	0.395072i	$0.870720\pi$
$998$		0			0
$999$			0.535898i			0.0169551i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	528.2.o.b.175.3	✓	8
3.2	odd	2		1584.2.o.g.703.2		8
4.3	odd	2	inner	528.2.o.b.175.8	yes	8
8.3	odd	2		2112.2.o.e.703.2		8
8.5	even	2		2112.2.o.e.703.5		8
11.10	odd	2	inner	528.2.o.b.175.4	yes	8
12.11	even	2		1584.2.o.g.703.3		8
33.32	even	2		1584.2.o.g.703.4		8
44.43	even	2	inner	528.2.o.b.175.7	yes	8
88.21	odd	2		2112.2.o.e.703.6		8
88.43	even	2		2112.2.o.e.703.1		8
132.131	odd	2		1584.2.o.g.703.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
528.2.o.b.175.3	✓	8	1.1	even	1	trivial
528.2.o.b.175.4	yes	8	11.10	odd	2	inner
528.2.o.b.175.7	yes	8	44.43	even	2	inner
528.2.o.b.175.8	yes	8	4.3	odd	2	inner
1584.2.o.g.703.1		8	132.131	odd	2
1584.2.o.g.703.2		8	3.2	odd	2
1584.2.o.g.703.3		8	12.11	even	2
1584.2.o.g.703.4		8	33.32	even	2
2112.2.o.e.703.1		8	88.43	even	2
2112.2.o.e.703.2		8	8.3	odd	2
2112.2.o.e.703.5		8	8.5	even	2
2112.2.o.e.703.6		8	88.21	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	528.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.73205 q^{5} -5.13734 q^{7} -1.00000 q^{9} +(1.88040 - 2.73205i) q^{11} -5.13734i q^{13} -2.73205i q^{15} -3.76080i q^{17} +3.76080 q^{19} +5.13734i q^{21} -1.26795i q^{23} +2.46410 q^{25} +1.00000i q^{27} -2.00000i q^{31} +(-2.73205 - 1.88040i) q^{33} -14.0355 q^{35} +0.535898 q^{37} -5.13734 q^{39} +10.2747i q^{41} +3.76080 q^{43} -2.73205 q^{45} -4.19615i q^{47} +19.3923 q^{49} -3.76080 q^{51} -10.7321 q^{53} +(5.13734 - 7.46410i) q^{55} -3.76080i q^{57} -9.46410i q^{59} +2.38425i q^{61} +5.13734 q^{63} -14.0355i q^{65} +10.3923i q^{67} -1.26795 q^{69} +10.7321i q^{71} -10.2747i q^{73} -2.46410i q^{75} +(-9.66025 + 14.0355i) q^{77} +5.13734 q^{79} +1.00000 q^{81} +10.2747 q^{83} -10.2747i q^{85} -10.3923 q^{89} +26.3923i q^{91} -2.00000 q^{93} +10.2747 q^{95} +5.46410 q^{97} +(-1.88040 + 2.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} + 32 q^{37} - 8 q^{45} + 72 q^{49} - 72 q^{53} - 24 q^{69} - 8 q^{77} + 8 q^{81} - 16 q^{93} + 16 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 8 * q^9 - 8 * q^25 - 8 * q^33 + 32 * q^37 - 8 * q^45 + 72 * q^49 - 72 * q^53 - 24 * q^69 - 8 * q^77 + 8 * q^81 - 16 * q^93 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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