LMFDB - Embedded newform 1320.2.d.c.529.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1320,2,Mod(529,1320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1320, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1320.529");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1320.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.5402530668$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 23x^{8} + 187x^{6} + 657x^{4} + 928x^{2} + 324 $ x^10 + 23x^8 + 187x^6 + 657x^4 + 928x^2 + 324
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.7
Root			$-3.13869i$ of defining polynomial
Character	$\chi$	$=$	1320.529
Dual form			1320.2.d.c.529.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +(-1.76211 - 1.37658i) q^{5} -2.71268i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.34872i q^{13} +(1.37658 - 1.76211i) q^{15} -4.23689i q^{17} -0.771041 q^{19} +2.71268 q^{21} +7.39451i q^{23} +(1.21003 + 4.85137i) q^{25} -1.00000i q^{27} -4.64134 q^{29} -3.41239 q^{31} -1.00000i q^{33} +(-3.73424 + 4.78003i) q^{35} +4.27738i q^{37} -4.34872 q^{39} -5.06140 q^{41} +11.8729i q^{43} +(1.76211 + 1.37658i) q^{45} +1.96915i q^{47} -0.358655 q^{49} +4.23689 q^{51} -4.29525i q^{53} +(1.76211 + 1.37658i) q^{55} -0.771041i q^{57} -1.98702 q^{59} -3.16025 q^{61} +2.71268i q^{63} +(5.98638 - 7.66290i) q^{65} +12.7512i q^{67} -7.39451 q^{69} -6.94958 q^{71} -2.46054i q^{73} +(-4.85137 + 1.21003i) q^{75} +2.71268i q^{77} -2.88287 q^{79} +1.00000 q^{81} +2.18316i q^{83} +(-5.83244 + 7.46585i) q^{85} -4.64134i q^{87} -2.58761 q^{89} +11.7967 q^{91} -3.41239i q^{93} +(1.35866 + 1.06140i) q^{95} -11.2448i q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} - 4 q^{19} + 4 q^{21} - 2 q^{25} - 8 q^{29} + 8 q^{31} + 8 q^{35} + 16 q^{41} + 2 q^{45} - 42 q^{49} - 12 q^{51} + 2 q^{55} - 24 q^{59} + 20 q^{61} + 24 q^{65} - 8 q^{69}+ \cdots + 10 q^{99}+O(q^{100}) $ 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 - 4 * q^19 + 4 * q^21 - 2 * q^25 - 8 * q^29 + 8 * q^31 + 8 * q^35 + 16 * q^41 + 2 * q^45 - 42 * q^49 - 12 * q^51 + 2 * q^55 - 24 * q^59 + 20 * q^61 + 24 * q^65 - 8 * q^69 + 8 * q^71 + 4 * q^75 - 36 * q^79 + 10 * q^81 + 12 * q^85 - 68 * q^89 + 52 * q^95 + 10 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1320\mathbb{Z}\right)^\times$.

$n$	$661$	$881$	$991$	$1057$	$1201$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

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
$5$	−1.76211	−	1.37658i	−0.788037	−	0.615627i
$5$	−1.76211	−	1.37658i	−0.788037	−	0.615627i
$6$		0			0
$7$		−	2.71268i		−	1.02530i	−0.858598	−	0.512649i	$-0.828664\pi$
$7$		−	2.71268i		−	1.02530i	0.858598	−	0.512649i	$-0.171336\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−1.00000			−0.301511
$11$	−1.00000			−0.301511
$12$		0			0
$13$			4.34872i			1.20612i	0.797697	+	0.603059i	$0.206052\pi$
$13$			4.34872i			1.20612i	−0.797697	+	0.603059i	$0.793948\pi$
$14$		0			0
$15$	1.37658	−	1.76211i	0.355433	−	0.454974i
$16$		0			0
$17$		−	4.23689i		−	1.02760i	−0.857911	−	0.513799i	$-0.828238\pi$
$17$		−	4.23689i		−	1.02760i	0.857911	−	0.513799i	$-0.171762\pi$
$18$		0			0
$19$	−0.771041			−0.176889			−0.0884445	−	0.996081i	$-0.528190\pi$
$19$	−0.771041			−0.176889			−0.0884445	+	0.996081i	$0.528190\pi$
$20$		0			0
$21$	2.71268			0.591956
$22$		0			0
$23$			7.39451i			1.54186i	0.636918	+	0.770931i	$0.280209\pi$
$23$			7.39451i			1.54186i	−0.636918	+	0.770931i	$0.719791\pi$
$24$		0			0
$25$	1.21003	+	4.85137i	0.242006	+	0.970275i
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.64134			−0.861876			−0.430938	−	0.902382i	$-0.641817\pi$
$29$	−4.64134			−0.861876			−0.430938	+	0.902382i	$0.641817\pi$
$30$		0			0
$31$	−3.41239			−0.612883			−0.306441	−	0.951890i	$-0.599138\pi$
$31$	−3.41239			−0.612883			−0.306441	+	0.951890i	$0.599138\pi$
$32$		0			0
$33$		−	1.00000i		−	0.174078i
$34$		0			0
$35$	−3.73424	+	4.78003i	−0.631202	+	0.807973i
$36$		0			0
$37$			4.27738i			0.703197i	0.936151	+	0.351598i	$0.114362\pi$
$37$			4.27738i			0.703197i	−0.936151	+	0.351598i	$0.885638\pi$
$38$		0			0
$39$	−4.34872			−0.696352
$40$		0			0
$41$	−5.06140			−0.790458			−0.395229	−	0.918583i	$-0.629335\pi$
$41$	−5.06140			−0.790458			−0.395229	+	0.918583i	$0.629335\pi$
$42$		0			0
$43$			11.8729i			1.81060i	0.424768	+	0.905302i	$0.360356\pi$
$43$			11.8729i			1.81060i	−0.424768	+	0.905302i	$0.639644\pi$
$44$		0			0
$45$	1.76211	+	1.37658i	0.262679	+	0.205209i
$46$		0			0
$47$			1.96915i			0.287229i	0.989634	+	0.143615i	$0.0458726\pi$
$47$			1.96915i			0.287229i	−0.989634	+	0.143615i	$0.954127\pi$
$48$		0			0
$49$	−0.358655			−0.0512365
$50$		0			0
$51$	4.23689			0.593284
$52$		0			0
$53$		−	4.29525i		−	0.589998i	−0.955498	−	0.294999i	$-0.904681\pi$
$53$		−	4.29525i		−	0.589998i	0.955498	−	0.294999i	$-0.0953193\pi$
$54$		0			0
$55$	1.76211	+	1.37658i	0.237602	+	0.185619i
$56$		0			0
$57$		−	0.771041i		−	0.102127i
$58$		0			0
$59$	−1.98702			−0.258688			−0.129344	−	0.991600i	$-0.541287\pi$
$59$	−1.98702			−0.258688			−0.129344	+	0.991600i	$0.541287\pi$
$60$		0			0
$61$	−3.16025			−0.404628			−0.202314	−	0.979321i	$-0.564846\pi$
$61$	−3.16025			−0.404628			−0.202314	+	0.979321i	$0.564846\pi$
$62$		0			0
$63$			2.71268i			0.341766i
$64$		0			0
$65$	5.98638	−	7.66290i	0.742519	−	0.950466i
$66$		0			0
$67$			12.7512i			1.55780i	0.627146	+	0.778902i	$0.284223\pi$
$67$			12.7512i			1.55780i	−0.627146	+	0.778902i	$0.715777\pi$
$68$		0			0
$69$	−7.39451			−0.890195
$70$		0			0
$71$	−6.94958			−0.824763			−0.412382	−	0.911011i	$-0.635303\pi$
$71$	−6.94958			−0.824763			−0.412382	+	0.911011i	$0.635303\pi$
$72$		0			0
$73$		−	2.46054i		−	0.287985i	−0.989579	−	0.143992i	$-0.954006\pi$
$73$		−	2.46054i		−	0.287985i	0.989579	−	0.143992i	$-0.0459941\pi$
$74$		0			0
$75$	−4.85137	+	1.21003i	−0.560188	+	0.139722i
$76$		0			0
$77$			2.71268i			0.309139i
$78$		0			0
$79$	−2.88287			−0.324348			−0.162174	−	0.986762i	$-0.551851\pi$
$79$	−2.88287			−0.324348			−0.162174	+	0.986762i	$0.551851\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			2.18316i			0.239633i	0.992796	+	0.119817i	$0.0382307\pi$
$83$			2.18316i			0.239633i	−0.992796	+	0.119817i	$0.961769\pi$
$84$		0			0
$85$	−5.83244	+	7.46585i	−0.632617	+	0.809786i
$86$		0			0
$87$		−	4.64134i		−	0.497604i
$88$		0			0
$89$	−2.58761			−0.274287			−0.137143	−	0.990551i	$-0.543792\pi$
$89$	−2.58761			−0.274287			−0.137143	+	0.990551i	$0.543792\pi$
$90$		0			0
$91$	11.7967			1.23663
$92$		0			0
$93$		−	3.41239i		−	0.353848i
$94$		0			0
$95$	1.35866	+	1.06140i	0.139395	+	0.108898i
$96$		0			0
$97$		−	11.2448i		−	1.14174i	−0.821041	−	0.570870i	$-0.806606\pi$
$97$		−	11.2448i		−	1.14174i	0.821041	−	0.570870i	$-0.193394\pi$
$98$		0			0
$99$	1.00000			0.100504
$100$		0			0
$101$	−13.1961			−1.31306			−0.656531	−	0.754299i	$-0.727977\pi$
$101$	−13.1961			−1.31306			−0.656531	+	0.754299i	$0.727977\pi$
$102$		0			0
$103$			2.57463i			0.253686i	0.991923	+	0.126843i	$0.0404844\pi$
$103$			2.57463i			0.253686i	−0.991923	+	0.126843i	$0.959516\pi$
$104$		0			0
$105$	−4.78003	−	3.73424i	−0.466484	−	0.364424i
$106$		0			0
$107$			2.46054i			0.237870i	0.992902	+	0.118935i	$0.0379480\pi$
$107$			2.46054i			0.237870i	−0.992902	+	0.118935i	$0.962052\pi$
$108$		0			0
$109$	−18.0110			−1.72514			−0.862570	−	0.505938i	$-0.831146\pi$
$109$	−18.0110			−1.72514			−0.862570	+	0.505938i	$0.831146\pi$
$110$		0			0
$111$	−4.27738			−0.405991
$112$		0			0
$113$		−	13.3388i		−	1.25481i	−0.778694	−	0.627404i	$-0.784118\pi$
$113$		−	13.3388i		−	1.25481i	0.778694	−	0.627404i	$-0.215882\pi$
$114$		0			0
$115$	10.1792	−	13.0299i	0.949213	−	1.21505i
$116$		0			0
$117$		−	4.34872i		−	0.402039i
$118$		0			0
$119$	−11.4934			−1.05359
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		−	5.06140i		−	0.456371i
$124$		0			0
$125$	4.54613	−	10.2143i	0.406618	−	0.913598i
$126$		0			0
$127$		−	6.79365i		−	0.602839i	−0.953492	−	0.301420i	$-0.902539\pi$
$127$		−	6.79365i		−	0.602839i	0.953492	−	0.301420i	$-0.0974605\pi$
$128$		0			0
$129$	−11.8729			−1.04535
$130$		0			0
$131$	1.08628			0.0949088			0.0474544	−	0.998873i	$-0.484889\pi$
$131$	1.08628			0.0949088			0.0474544	+	0.998873i	$0.484889\pi$
$132$		0			0
$133$			2.09159i			0.181364i
$134$		0			0
$135$	−1.37658	+	1.76211i	−0.118478	+	0.151658i
$136$		0			0
$137$			11.7515i			1.00400i	0.864869	+	0.501998i	$0.167402\pi$
$137$			11.7515i			1.00400i	−0.864869	+	0.501998i	$0.832598\pi$
$138$		0			0
$139$	19.4486			1.64961			0.824805	−	0.565418i	$-0.191285\pi$
$139$	19.4486			1.64961			0.824805	+	0.565418i	$0.191285\pi$
$140$		0			0
$141$	−1.96915			−0.165832
$142$		0			0
$143$		−	4.34872i		−	0.363658i
$144$		0			0
$145$	8.17854	+	6.38920i	0.679191	+	0.530595i
$146$		0			0
$147$		−	0.358655i		−	0.0295814i
$148$		0			0
$149$	−14.1477			−1.15902			−0.579512	−	0.814964i	$-0.696757\pi$
$149$	−14.1477			−1.15902			−0.579512	+	0.814964i	$0.696757\pi$
$150$		0			0
$151$	13.4482			1.09440			0.547201	−	0.837001i	$-0.315693\pi$
$151$	13.4482			1.09440			0.547201	+	0.837001i	$0.315693\pi$
$152$		0			0
$153$			4.23689i			0.342533i
$154$		0			0
$155$	6.01298	+	4.69744i	0.482974	+	0.377307i
$156$		0			0
$157$			2.04311i			0.163058i	0.996671	+	0.0815290i	$0.0259803\pi$
$157$			2.04311i			0.163058i	−0.996671	+	0.0815290i	$0.974020\pi$
$158$		0			0
$159$	4.29525			0.340636
$160$		0			0
$161$	20.0590			1.58087
$162$		0			0
$163$			14.9057i			1.16751i	0.811931	+	0.583754i	$0.198417\pi$
$163$			14.9057i			1.16751i	−0.811931	+	0.583754i	$0.801583\pi$
$164$		0			0
$165$	−1.37658	+	1.76211i	−0.107167	+	0.137180i
$166$		0			0
$167$			13.7001i			1.06015i	0.847952	+	0.530074i	$0.177836\pi$
$167$			13.7001i			1.06015i	−0.847952	+	0.530074i	$0.822164\pi$
$168$		0			0
$169$	−5.91136			−0.454720
$170$		0			0
$171$	0.771041			0.0589630
$172$		0			0
$173$			6.16329i			0.468586i	0.972166	+	0.234293i	$0.0752776\pi$
$173$			6.16329i			0.468586i	−0.972166	+	0.234293i	$0.924722\pi$
$174$		0			0
$175$	13.1602	−	3.28243i	0.994821	−	0.248128i
$176$		0			0
$177$		−	1.98702i		−	0.149353i
$178$		0			0
$179$	19.3438			1.44582			0.722911	−	0.690941i	$-0.242804\pi$
$179$	19.3438			1.44582			0.722911	+	0.690941i	$0.242804\pi$
$180$		0			0
$181$	22.1098			1.64341			0.821705	−	0.569913i	$-0.193023\pi$
$181$	22.1098			1.64341			0.821705	+	0.569913i	$0.193023\pi$
$182$		0			0
$183$		−	3.16025i		−	0.233612i
$184$		0			0
$185$	5.88818	−	7.53719i	0.432907	−	0.554145i
$186$		0			0
$187$			4.23689i			0.309832i
$188$		0			0
$189$	−2.71268			−0.197319
$190$		0			0
$191$	−22.1835			−1.60514			−0.802572	−	0.596555i	$-0.796536\pi$
$191$	−22.1835			−1.60514			−0.802572	+	0.596555i	$0.796536\pi$
$192$		0			0
$193$		−	1.39678i		−	0.100542i	−0.998736	−	0.0502711i	$-0.983991\pi$
$193$		−	1.39678i		−	0.100542i	0.998736	−	0.0502711i	$-0.0160085\pi$
$194$		0			0
$195$	7.66290	+	5.98638i	0.548752	+	0.428694i
$196$		0			0
$197$		−	20.2601i		−	1.44347i	−0.692167	−	0.721737i	$-0.743344\pi$
$197$		−	20.2601i		−	1.44347i	0.692167	−	0.721737i	$-0.256656\pi$
$198$		0			0
$199$	−25.8992			−1.83594			−0.917971	−	0.396647i	$-0.870174\pi$
$199$	−25.8992			−1.83594			−0.917971	+	0.396647i	$0.870174\pi$
$200$		0			0
$201$	−12.7512			−0.899398
$202$		0			0
$203$			12.5905i			0.883680i
$204$		0			0
$205$	8.91872	+	6.96745i	0.622911	+	0.486628i
$206$		0			0
$207$		−	7.39451i		−	0.513954i
$208$		0			0
$209$	0.771041			0.0533340
$210$		0			0
$211$	14.8321			1.02109			0.510543	−	0.859852i	$-0.329444\pi$
$211$	14.8321			1.02109			0.510543	+	0.859852i	$0.329444\pi$
$212$		0			0
$213$		−	6.94958i		−	0.476177i
$214$		0			0
$215$	16.3441	−	20.9213i	1.11466	−	1.42682i
$216$		0			0
$217$			9.25672i			0.628387i
$218$		0			0
$219$	2.46054			0.166268
$220$		0			0
$221$	18.4251			1.23940
$222$		0			0
$223$			8.73318i			0.584817i	0.956293	+	0.292409i	$0.0944567\pi$
$223$			8.73318i			0.584817i	−0.956293	+	0.292409i	$0.905543\pi$
$224$		0			0
$225$	−1.21003	−	4.85137i	−0.0806686	−	0.323425i
$226$		0			0
$227$		−	17.0259i		−	1.13005i	−0.825074	−	0.565025i	$-0.808866\pi$
$227$		−	17.0259i		−	1.13005i	0.825074	−	0.565025i	$-0.191134\pi$
$228$		0			0
$229$	−16.3205			−1.07849			−0.539244	−	0.842149i	$-0.681290\pi$
$229$	−16.3205			−1.07849			−0.539244	+	0.842149i	$0.681290\pi$
$230$		0			0
$231$	−2.71268			−0.178482
$232$		0			0
$233$		−	10.2748i		−	0.673122i	−0.941662	−	0.336561i	$-0.890736\pi$
$233$		−	10.2748i		−	0.673122i	0.941662	−	0.336561i	$-0.109264\pi$
$234$		0			0
$235$	2.71070	−	3.46984i	0.176826	−	0.226348i
$236$		0			0
$237$		−	2.88287i		−	0.187262i
$238$		0			0
$239$	−0.223649			−0.0144667			−0.00723333	−	0.999974i	$-0.502302\pi$
$239$	−0.223649			−0.0144667			−0.00723333	+	0.999974i	$0.502302\pi$
$240$		0			0
$241$	−1.77635			−0.114425			−0.0572124	−	0.998362i	$-0.518221\pi$
$241$	−1.77635			−0.114425			−0.0572124	+	0.998362i	$0.518221\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	0.631988	+	0.493719i	0.0403762	+	0.0315426i
$246$		0			0
$247$		−	3.35304i		−	0.213349i
$248$		0			0
$249$	−2.18316			−0.138352
$250$		0			0
$251$	−2.82713			−0.178447			−0.0892236	−	0.996012i	$-0.528439\pi$
$251$	−2.82713			−0.178447			−0.0892236	+	0.996012i	$0.528439\pi$
$252$		0			0
$253$		−	7.39451i		−	0.464889i
$254$		0			0
$255$	−7.46585	−	5.83244i	−0.467530	−	0.365242i
$256$		0			0
$257$			25.7316i			1.60509i	0.596589	+	0.802547i	$0.296522\pi$
$257$			25.7316i			1.60509i	−0.596589	+	0.802547i	$0.703478\pi$
$258$		0			0
$259$	11.6032			0.720987
$260$		0			0
$261$	4.64134			0.287292
$262$		0			0
$263$		−	13.0889i		−	0.807097i	−0.914958	−	0.403548i	$-0.867777\pi$
$263$		−	13.0889i		−	0.807097i	0.914958	−	0.403548i	$-0.132223\pi$
$264$		0			0
$265$	−5.91278	+	7.56868i	−0.363219	+	0.464941i
$266$		0			0
$267$		−	2.58761i		−	0.158359i
$268$		0			0
$269$	23.8484			1.45406			0.727032	−	0.686603i	$-0.240899\pi$
$269$	23.8484			1.45406			0.727032	+	0.686603i	$0.240899\pi$
$270$		0			0
$271$	−5.69786			−0.346120			−0.173060	−	0.984911i	$-0.555366\pi$
$271$	−5.69786			−0.346120			−0.173060	+	0.984911i	$0.555366\pi$
$272$		0			0
$273$			11.7967i			0.713969i
$274$		0			0
$275$	−1.21003	−	4.85137i	−0.0729675	−	0.292549i
$276$		0			0
$277$			19.8969i			1.19549i	0.801687	+	0.597744i	$0.203936\pi$
$277$			19.8969i			1.19549i	−0.801687	+	0.597744i	$0.796064\pi$
$278$		0			0
$279$	3.41239			0.204294
$280$		0			0
$281$	1.79061			0.106819			0.0534094	−	0.998573i	$-0.482991\pi$
$281$	1.79061			0.106819			0.0534094	+	0.998573i	$0.482991\pi$
$282$		0			0
$283$			23.5378i			1.39918i	0.714545	+	0.699589i	$0.246634\pi$
$283$			23.5378i			1.39918i	−0.714545	+	0.699589i	$0.753366\pi$
$284$		0			0
$285$	−1.06140	+	1.35866i	−0.0628721	+	0.0804798i
$286$		0			0
$287$			13.7300i			0.810456i
$288$		0			0
$289$	−0.951274			−0.0559573
$290$		0			0
$291$	11.2448			0.659184
$292$		0			0
$293$		−	27.4102i		−	1.60132i	−0.599117	−	0.800662i	$-0.704482\pi$
$293$		−	27.4102i		−	1.60132i	0.599117	−	0.800662i	$-0.295518\pi$
$294$		0			0
$295$	3.50133	+	2.73530i	0.203856	+	0.159255i
$296$		0			0
$297$			1.00000i			0.0580259i
$298$		0			0
$299$	−32.1567			−1.85967
$300$		0			0
$301$	32.2075			1.85641
$302$		0			0
$303$		−	13.1961i		−	0.758096i
$304$		0			0
$305$	5.56868	+	4.35035i	0.318862	+	0.249100i
$306$		0			0
$307$		−	1.74035i		−	0.0993267i	−0.998766	−	0.0496634i	$-0.984185\pi$
$307$		−	1.74035i		−	0.0993267i	0.998766	−	0.0496634i	$-0.0158148\pi$
$308$		0			0
$309$	−2.57463			−0.146466
$310$		0			0
$311$	17.9623			1.01855			0.509273	−	0.860605i	$-0.329914\pi$
$311$	17.9623			1.01855			0.509273	+	0.860605i	$0.329914\pi$
$312$		0			0
$313$		−	4.83487i		−	0.273283i	−0.990621	−	0.136641i	$-0.956369\pi$
$313$		−	4.83487i		−	0.273283i	0.990621	−	0.136641i	$-0.0436308\pi$
$314$		0			0
$315$	3.73424	−	4.78003i	0.210401	−	0.269324i
$316$		0			0
$317$			14.6365i			0.822065i	0.911621	+	0.411033i	$0.134832\pi$
$317$			14.6365i			0.822065i	−0.911621	+	0.411033i	$0.865168\pi$
$318$		0			0
$319$	4.64134			0.259865
$320$		0			0
$321$	−2.46054			−0.137334
$322$		0			0
$323$			3.26682i			0.181771i
$324$		0			0
$325$	−21.0973	+	5.26208i	−1.17027	+	0.291887i
$326$		0			0
$327$		−	18.0110i		−	0.996010i
$328$		0			0
$329$	5.34167			0.294496
$330$		0			0
$331$	−15.4356			−0.848419			−0.424209	−	0.905564i	$-0.639448\pi$
$331$	−15.4356			−0.848419			−0.424209	+	0.905564i	$0.639448\pi$
$332$		0			0
$333$		−	4.27738i		−	0.234399i
$334$		0			0
$335$	17.5531	−	22.4689i	0.959026	−	1.22761i
$336$		0			0
$337$		−	31.9576i		−	1.74084i	−0.492307	−	0.870422i	$-0.663846\pi$
$337$		−	31.9576i		−	1.74084i	0.492307	−	0.870422i	$-0.336154\pi$
$338$		0			0
$339$	13.3388			0.724463
$340$		0			0
$341$	3.41239			0.184791
$342$		0			0
$343$		−	18.0159i		−	0.972766i
$344$		0			0
$345$	13.0299	+	10.1792i	0.701507	+	0.548028i
$346$		0			0
$347$		−	3.49182i		−	0.187451i	−0.995598	−	0.0937253i	$-0.970122\pi$
$347$		−	3.49182i		−	0.187451i	0.995598	−	0.0937253i	$-0.0298775\pi$
$348$		0			0
$349$	29.8378			1.59718			0.798591	−	0.601874i	$-0.205579\pi$
$349$	29.8378			1.59718			0.798591	+	0.601874i	$0.205579\pi$
$350$		0			0
$351$	4.34872			0.232117
$352$		0			0
$353$		−	29.4198i		−	1.56586i	−0.622113	−	0.782928i	$-0.713725\pi$
$353$		−	29.4198i		−	1.56586i	0.622113	−	0.782928i	$-0.286275\pi$
$354$		0			0
$355$	12.2459	+	9.56668i	0.649944	+	0.507747i
$356$		0			0
$357$		−	11.4934i		−	0.608293i
$358$		0			0
$359$	−34.6418			−1.82833			−0.914163	−	0.405347i	$-0.867151\pi$
$359$	−34.6418			−1.82833			−0.914163	+	0.405347i	$0.867151\pi$
$360$		0			0
$361$	−18.4055			−0.968710
$362$		0			0
$363$			1.00000i			0.0524864i
$364$		0			0
$365$	−3.38715	+	4.33574i	−0.177291	+	0.226943i
$366$		0			0
$367$			8.49975i			0.443684i	0.975083	+	0.221842i	$0.0712068\pi$
$367$			8.49975i			0.443684i	−0.975083	+	0.221842i	$0.928793\pi$
$368$		0			0
$369$	5.06140			0.263486
$370$		0			0
$371$	−11.6517			−0.604924
$372$		0			0
$373$		−	13.8908i		−	0.719238i	−0.933099	−	0.359619i	$-0.882907\pi$
$373$		−	13.8908i		−	0.719238i	0.933099	−	0.359619i	$-0.117093\pi$
$374$		0			0
$375$	10.2143	+	4.54613i	0.527466	+	0.234761i
$376$		0			0
$377$		−	20.1839i		−	1.03952i
$378$		0			0
$379$	−38.7878			−1.99239			−0.996197	−	0.0871348i	$-0.972229\pi$
$379$	−38.7878			−1.99239			−0.996197	+	0.0871348i	$0.972229\pi$
$380$		0			0
$381$	6.79365			0.348049
$382$		0			0
$383$			14.7735i			0.754892i	0.926032	+	0.377446i	$0.123198\pi$
$383$			14.7735i			0.754892i	−0.926032	+	0.377446i	$0.876802\pi$
$384$		0			0
$385$	3.73424	−	4.78003i	0.190314	−	0.243613i
$386$		0			0
$387$		−	11.8729i		−	0.603535i
$388$		0			0
$389$	−31.4059			−1.59234			−0.796170	−	0.605073i	$-0.793144\pi$
$389$	−31.4059			−1.59234			−0.796170	+	0.605073i	$0.793144\pi$
$390$		0			0
$391$	31.3298			1.58441
$392$		0			0
$393$			1.08628i			0.0547956i
$394$		0			0
$395$	5.07991	+	3.96851i	0.255598	+	0.199677i
$396$		0			0
$397$		−	7.27079i		−	0.364911i	−0.983214	−	0.182455i	$-0.941595\pi$
$397$		−	7.27079i		−	0.364911i	0.983214	−	0.182455i	$-0.0584045\pi$
$398$		0			0
$399$	−2.09159			−0.104710
$400$		0			0
$401$	38.7456			1.93486			0.967430	−	0.253138i	$-0.0814625\pi$
$401$	38.7456			1.93486			0.967430	+	0.253138i	$0.0814625\pi$
$402$		0			0
$403$		−	14.8395i		−	0.739208i
$404$		0			0
$405$	−1.76211	−	1.37658i	−0.0875597	−	0.0684030i
$406$		0			0
$407$		−	4.27738i		−	0.212022i
$408$		0			0
$409$	30.9378			1.52978			0.764888	−	0.644163i	$-0.222794\pi$
$409$	30.9378			1.52978			0.764888	+	0.644163i	$0.222794\pi$
$410$		0			0
$411$	−11.7515			−0.579657
$412$		0			0
$413$			5.39015i			0.265232i
$414$		0			0
$415$	3.00531	−	3.84696i	0.147525	−	0.188840i
$416$		0			0
$417$			19.4486i			0.952402i
$418$		0			0
$419$	−27.4153			−1.33932			−0.669662	−	0.742666i	$-0.733561\pi$
$419$	−27.4153			−1.33932			−0.669662	+	0.742666i	$0.733561\pi$
$420$		0			0
$421$	−5.51932			−0.268995			−0.134498	−	0.990914i	$-0.542942\pi$
$421$	−5.51932			−0.268995			−0.134498	+	0.990914i	$0.542942\pi$
$422$		0			0
$423$		−	1.96915i		−	0.0957432i
$424$		0			0
$425$	20.5548	−	5.12677i	0.997052	−	0.248685i
$426$		0			0
$427$			8.57275i			0.414864i
$428$		0			0
$429$	4.34872			0.209958
$430$		0			0
$431$	−9.70402			−0.467426			−0.233713	−	0.972306i	$-0.575088\pi$
$431$	−9.70402			−0.467426			−0.233713	+	0.972306i	$0.575088\pi$
$432$		0			0
$433$			4.55476i			0.218888i	0.993993	+	0.109444i	$0.0349070\pi$
$433$			4.55476i			0.218888i	−0.993993	+	0.109444i	$0.965093\pi$
$434$		0			0
$435$	−6.38920	+	8.17854i	−0.306339	+	0.392131i
$436$		0			0
$437$		−	5.70147i		−	0.272738i
$438$		0			0
$439$	14.0647			0.671272			0.335636	−	0.941992i	$-0.391049\pi$
$439$	14.0647			0.671272			0.335636	+	0.941992i	$0.391049\pi$
$440$		0			0
$441$	0.358655			0.0170788
$442$		0			0
$443$			25.1400i			1.19444i	0.802078	+	0.597219i	$0.203728\pi$
$443$			25.1400i			1.19444i	−0.802078	+	0.597219i	$0.796272\pi$
$444$		0			0
$445$	4.55965	+	3.56207i	0.216148	+	0.168858i
$446$		0			0
$447$		−	14.1477i		−	0.669162i
$448$		0			0
$449$	−36.2014			−1.70845			−0.854225	−	0.519903i	$-0.825968\pi$
$449$	−36.2014			−1.70845			−0.854225	+	0.519903i	$0.825968\pi$
$450$		0			0
$451$	5.06140			0.238332
$452$		0			0
$453$			13.4482i			0.631854i
$454$		0			0
$455$	−20.7870	−	16.2392i	−0.974511	−	0.761303i
$456$		0			0
$457$		−	14.0750i		−	0.658399i	−0.944260	−	0.329199i	$-0.893221\pi$
$457$		−	14.0750i		−	0.658399i	0.944260	−	0.329199i	$-0.106779\pi$
$458$		0			0
$459$	−4.23689			−0.197761
$460$		0			0
$461$	7.78722			0.362687			0.181343	−	0.983420i	$-0.441955\pi$
$461$	7.78722			0.362687			0.181343	+	0.983420i	$0.441955\pi$
$462$		0			0
$463$		−	40.4698i		−	1.88079i	−0.340083	−	0.940395i	$-0.610455\pi$
$463$		−	40.4698i		−	1.88079i	0.340083	−	0.940395i	$-0.389545\pi$
$464$		0			0
$465$	−4.69744	+	6.01298i	−0.217838	+	0.278845i
$466$		0			0
$467$			17.6755i			0.817925i	0.912551	+	0.408962i	$0.134109\pi$
$467$			17.6755i			0.817925i	−0.912551	+	0.408962i	$0.865891\pi$
$468$		0			0
$469$	34.5899			1.59721
$470$		0			0
$471$	−2.04311			−0.0941416
$472$		0			0
$473$		−	11.8729i		−	0.545918i
$474$		0			0
$475$	−0.932982	−	3.74061i	−0.0428081	−	0.171631i
$476$		0			0
$477$			4.29525i			0.196666i
$478$		0			0
$479$	−7.74061			−0.353677			−0.176839	−	0.984240i	$-0.556587\pi$
$479$	−7.74061			−0.353677			−0.176839	+	0.984240i	$0.556587\pi$
$480$		0			0
$481$	−18.6011			−0.848138
$482$		0			0
$483$			20.0590i			0.912715i
$484$		0			0
$485$	−15.4795	+	19.8146i	−0.702886	+	0.899733i
$486$		0			0
$487$		−	11.8838i		−	0.538507i	−0.963069	−	0.269253i	$-0.913223\pi$
$487$		−	11.8838i		−	0.538507i	0.963069	−	0.269253i	$-0.0867769\pi$
$488$		0			0
$489$	−14.9057			−0.674061
$490$		0			0
$491$	20.1334			0.908609			0.454304	−	0.890847i	$-0.349888\pi$
$491$	20.1334			0.908609			0.454304	+	0.890847i	$0.349888\pi$
$492$		0			0
$493$			19.6649i			0.885662i
$494$		0			0
$495$	−1.76211	−	1.37658i	−0.0792007	−	0.0618729i
$496$		0			0
$497$			18.8520i			0.845628i
$498$		0			0
$499$	−1.79297			−0.0802644			−0.0401322	−	0.999194i	$-0.512778\pi$
$499$	−1.79297			−0.0802644			−0.0401322	+	0.999194i	$0.512778\pi$
$500$		0			0
$501$	−13.7001			−0.612076
$502$		0			0
$503$		−	27.3284i		−	1.21851i	−0.792973	−	0.609257i	$-0.791468\pi$
$503$		−	27.3284i		−	1.21851i	0.792973	−	0.609257i	$-0.208532\pi$
$504$		0			0
$505$	23.2529	+	18.1656i	1.03474	+	0.808357i
$506$		0			0
$507$		−	5.91136i		−	0.262533i
$508$		0			0
$509$	21.3588			0.946711			0.473355	−	0.880872i	$-0.343043\pi$
$509$	21.3588			0.946711			0.473355	+	0.880872i	$0.343043\pi$
$510$		0			0
$511$	−6.67468			−0.295270
$512$		0			0
$513$			0.771041i			0.0340423i
$514$		0			0
$515$	3.54420	−	4.53677i	0.156176	−	0.199914i
$516$		0			0
$517$		−	1.96915i		−	0.0866029i
$518$		0			0
$519$	−6.16329			−0.270538
$520$		0			0
$521$	−9.30092			−0.407481			−0.203740	−	0.979025i	$-0.565310\pi$
$521$	−9.30092			−0.407481			−0.203740	+	0.979025i	$0.565310\pi$
$522$		0			0
$523$			31.2732i			1.36748i	0.729726	+	0.683740i	$0.239648\pi$
$523$			31.2732i			1.36748i	−0.729726	+	0.683740i	$0.760352\pi$
$524$		0			0
$525$	3.28243	+	13.1602i	0.143257	+	0.574360i
$526$		0			0
$527$			14.4579i			0.629797i
$528$		0			0
$529$	−31.6788			−1.37734
$530$		0			0
$531$	1.98702			0.0862292
$532$		0			0
$533$		−	22.0106i		−	0.953386i
$534$		0			0
$535$	3.38715	−	4.33574i	0.146439	−	0.187450i
$536$		0			0
$537$			19.3438i			0.834746i
$538$		0			0
$539$	0.358655			0.0154484
$540$		0			0
$541$	−36.9479			−1.58852			−0.794258	−	0.607581i	$-0.792140\pi$
$541$	−36.9479			−1.58852			−0.794258	+	0.607581i	$0.792140\pi$
$542$		0			0
$543$			22.1098i			0.948823i
$544$		0			0
$545$	31.7372	+	24.7936i	1.35947	+	1.06204i
$546$		0			0
$547$		−	38.9738i		−	1.66640i	−0.552972	−	0.833200i	$-0.686506\pi$
$547$		−	38.9738i		−	1.66640i	0.552972	−	0.833200i	$-0.313494\pi$
$548$		0			0
$549$	3.16025			0.134876
$550$		0			0
$551$	3.57867			0.152456
$552$		0			0
$553$			7.82030i			0.332553i
$554$		0			0
$555$	7.53719	+	5.88818i	0.319936	+	0.249939i
$556$		0			0
$557$			30.6469i			1.29855i	0.760554	+	0.649275i	$0.224927\pi$
$557$			30.6469i			1.29855i	−0.760554	+	0.649275i	$0.775073\pi$
$558$		0			0
$559$	−51.6320			−2.18380
$560$		0			0
$561$	−4.23689			−0.178882
$562$		0			0
$563$			3.90517i			0.164583i	0.996608	+	0.0822917i	$0.0262239\pi$
$563$			3.90517i			0.164583i	−0.996608	+	0.0822917i	$0.973776\pi$
$564$		0			0
$565$	−18.3620	+	23.5043i	−0.772494	+	0.988835i
$566$		0			0
$567$		−	2.71268i		−	0.113922i
$568$		0			0
$569$	−24.3062			−1.01897			−0.509485	−	0.860479i	$-0.670164\pi$
$569$	−24.3062			−1.01897			−0.509485	+	0.860479i	$0.670164\pi$
$570$		0			0
$571$	23.0298			0.963768			0.481884	−	0.876235i	$-0.339953\pi$
$571$	23.0298			0.963768			0.481884	+	0.876235i	$0.339953\pi$
$572$		0			0
$573$		−	22.1835i		−	0.926731i
$574$		0			0
$575$	−35.8736	+	8.94758i	−1.49603	+	0.373140i
$576$		0			0
$577$			14.3497i			0.597386i	0.954349	+	0.298693i	$0.0965507\pi$
$577$			14.3497i			0.597386i	−0.954349	+	0.298693i	$0.903449\pi$
$578$		0			0
$579$	1.39678			0.0580481
$580$		0			0
$581$	5.92223			0.245696
$582$		0			0
$583$			4.29525i			0.177891i
$584$		0			0
$585$	−5.98638	+	7.66290i	−0.247506	+	0.316822i
$586$		0			0
$587$		−	22.5254i		−	0.929723i	−0.885383	−	0.464861i	$-0.846104\pi$
$587$		−	22.5254i		−	0.929723i	0.885383	−	0.464861i	$-0.153896\pi$
$588$		0			0
$589$	2.63109			0.108412
$590$		0			0
$591$	20.2601			0.833391
$592$		0			0
$593$		−	26.8832i		−	1.10396i	−0.833857	−	0.551981i	$-0.813872\pi$
$593$		−	26.8832i		−	1.10396i	0.833857	−	0.551981i	$-0.186128\pi$
$594$		0			0
$595$	20.2525	+	15.8216i	0.830272	+	0.648621i
$596$		0			0
$597$		−	25.8992i		−	1.05998i
$598$		0			0
$599$	17.8769			0.730432			0.365216	−	0.930923i	$-0.380995\pi$
$599$	17.8769			0.730432			0.365216	+	0.930923i	$0.380995\pi$
$600$		0			0
$601$	−10.6782			−0.435572			−0.217786	−	0.975997i	$-0.569883\pi$
$601$	−10.6782			−0.435572			−0.217786	+	0.975997i	$0.569883\pi$
$602$		0			0
$603$		−	12.7512i		−	0.519268i
$604$		0			0
$605$	−1.76211	−	1.37658i	−0.0716398	−	0.0559661i
$606$		0			0
$607$		−	1.87782i		−	0.0762183i	−0.999274	−	0.0381091i	$-0.987867\pi$
$607$		−	1.87782i		−	0.0762183i	0.999274	−	0.0381091i	$-0.0121335\pi$
$608$		0			0
$609$	−12.5905			−0.510193
$610$		0			0
$611$	−8.56326			−0.346433
$612$		0			0
$613$		−	30.8066i		−	1.24427i	−0.782911	−	0.622134i	$-0.786266\pi$
$613$		−	30.8066i		−	1.24427i	0.782911	−	0.622134i	$-0.213734\pi$
$614$		0			0
$615$	−6.96745	+	8.91872i	−0.280955	+	0.359638i
$616$		0			0
$617$			34.1529i			1.37495i	0.726210	+	0.687473i	$0.241280\pi$
$617$			34.1529i			1.37495i	−0.726210	+	0.687473i	$0.758720\pi$
$618$		0			0
$619$	−27.1895			−1.09284			−0.546418	−	0.837512i	$-0.684009\pi$
$619$	−27.1895			−1.09284			−0.546418	+	0.837512i	$0.684009\pi$
$620$		0			0
$621$	7.39451			0.296732
$622$		0			0
$623$			7.01938i			0.281226i
$624$		0			0
$625$	−22.0717	+	11.7406i	−0.882866	+	0.469624i
$626$		0			0
$627$			0.771041i			0.0307924i
$628$		0			0
$629$	18.1228			0.722604
$630$		0			0
$631$	−25.9237			−1.03201			−0.516003	−	0.856587i	$-0.672580\pi$
$631$	−25.9237			−1.03201			−0.516003	+	0.856587i	$0.672580\pi$
$632$		0			0
$633$			14.8321i			0.589525i
$634$		0			0
$635$	−9.35204	+	11.9711i	−0.371124	+	0.475060i
$636$		0			0
$637$		−	1.55969i		−	0.0617972i
$638$		0			0
$639$	6.94958			0.274921
$640$		0			0
$641$	−31.9467			−1.26182			−0.630910	−	0.775856i	$-0.717318\pi$
$641$	−31.9467			−1.26182			−0.630910	+	0.775856i	$0.717318\pi$
$642$		0			0
$643$		−	31.5097i		−	1.24262i	−0.783564	−	0.621311i	$-0.786600\pi$
$643$		−	31.5097i		−	1.24262i	0.783564	−	0.621311i	$-0.213400\pi$
$644$		0			0
$645$	20.9213	+	16.3441i	0.823777	+	0.643548i
$646$		0			0
$647$			40.6467i			1.59799i	0.601340	+	0.798994i	$0.294634\pi$
$647$			40.6467i			1.59799i	−0.601340	+	0.798994i	$0.705366\pi$
$648$		0			0
$649$	1.98702			0.0779973
$650$		0			0
$651$	−9.25672			−0.362800
$652$		0			0
$653$			35.4109i			1.38573i	0.721065	+	0.692867i	$0.243653\pi$
$653$			35.4109i			1.38573i	−0.721065	+	0.692867i	$0.756347\pi$
$654$		0			0
$655$	−1.91414	−	1.49536i	−0.0747917	−	0.0584284i
$656$		0			0
$657$			2.46054i			0.0959949i
$658$		0			0
$659$	23.7983			0.927051			0.463525	−	0.886084i	$-0.346584\pi$
$659$	23.7983			0.927051			0.463525	+	0.886084i	$0.346584\pi$
$660$		0			0
$661$	−20.8965			−0.812779			−0.406390	−	0.913700i	$-0.633212\pi$
$661$	−20.8965			−0.812779			−0.406390	+	0.913700i	$0.633212\pi$
$662$		0			0
$663$			18.4251i			0.715570i
$664$		0			0
$665$	2.87925	−	3.68560i	0.111653	−	0.142922i
$666$		0			0
$667$		−	34.3205i		−	1.32889i
$668$		0			0
$669$	−8.73318			−0.337644
$670$		0			0
$671$	3.16025			0.122000
$672$		0			0
$673$			32.3432i			1.24674i	0.781927	+	0.623370i	$0.214237\pi$
$673$			32.3432i			1.24674i	−0.781927	+	0.623370i	$0.785763\pi$
$674$		0			0
$675$	4.85137	−	1.21003i	0.186729	−	0.0465740i
$676$		0			0
$677$			43.4460i			1.66976i	0.550428	+	0.834882i	$0.314464\pi$
$677$			43.4460i			1.66976i	−0.550428	+	0.834882i	$0.685536\pi$
$678$		0			0
$679$	−30.5037			−1.17062
$680$		0			0
$681$	17.0259			0.652435
$682$		0			0
$683$			24.4983i			0.937401i	0.883357	+	0.468701i	$0.155278\pi$
$683$			24.4983i			0.937401i	−0.883357	+	0.468701i	$0.844722\pi$
$684$		0			0
$685$	16.1769	−	20.7073i	0.618088	−	0.791186i
$686$		0			0
$687$		−	16.3205i		−	0.622666i
$688$		0			0
$689$	18.6788			0.711607
$690$		0			0
$691$	−28.9476			−1.10122			−0.550609	−	0.834763i	$-0.685604\pi$
$691$	−28.9476			−1.10122			−0.550609	+	0.834763i	$0.685604\pi$
$692$		0			0
$693$		−	2.71268i		−	0.103046i
$694$		0			0
$695$	−34.2705	−	26.7727i	−1.29995	−	1.01554i
$696$		0			0
$697$			21.4446i			0.812273i
$698$		0			0
$699$	10.2748			0.388627
$700$		0			0
$701$	−14.3811			−0.543167			−0.271584	−	0.962415i	$-0.587547\pi$
$701$	−14.3811			−0.543167			−0.271584	+	0.962415i	$0.587547\pi$
$702$		0			0
$703$		−	3.29803i		−	0.124388i
$704$		0			0
$705$	3.46984	+	2.71070i	0.130682	+	0.102091i
$706$		0			0
$707$			35.7969i			1.34628i
$708$		0			0
$709$	4.92459			0.184947			0.0924733	−	0.995715i	$-0.470523\pi$
$709$	4.92459			0.184947			0.0924733	+	0.995715i	$0.470523\pi$
$710$		0			0
$711$	2.88287			0.108116
$712$		0			0
$713$		−	25.2329i		−	0.944981i
$714$		0			0
$715$	−5.98638	+	7.66290i	−0.223878	+	0.286576i
$716$		0			0
$717$		−	0.223649i		−	0.00835233i
$718$		0			0
$719$	11.2943			0.421206			0.210603	−	0.977572i	$-0.432457\pi$
$719$	11.2943			0.421206			0.210603	+	0.977572i	$0.432457\pi$
$720$		0			0
$721$	6.98416			0.260104
$722$		0			0
$723$		−	1.77635i		−	0.0660632i
$724$		0			0
$725$	−5.61616	−	22.5169i	−0.208579	−	0.836257i
$726$		0			0
$727$		−	15.1606i		−	0.562276i	−0.959667	−	0.281138i	$-0.909288\pi$
$727$		−	15.1606i		−	0.562276i	0.959667	−	0.281138i	$-0.0907118\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$	50.3043			1.86057
$732$		0			0
$733$			36.6851i			1.35499i	0.735525	+	0.677497i	$0.236935\pi$
$733$			36.6851i			1.35499i	−0.735525	+	0.677497i	$0.763065\pi$
$734$		0			0
$735$	−0.493719	+	0.631988i	−0.0182111	+	0.0233112i
$736$		0			0
$737$		−	12.7512i		−	0.469695i
$738$		0			0
$739$	23.2149			0.853976			0.426988	−	0.904257i	$-0.359575\pi$
$739$	23.2149			0.853976			0.426988	+	0.904257i	$0.359575\pi$
$740$		0			0
$741$	3.35304			0.123177
$742$		0			0
$743$			42.9933i			1.57727i	0.614861	+	0.788636i	$0.289212\pi$
$743$			42.9933i			1.57727i	−0.614861	+	0.788636i	$0.710788\pi$
$744$		0			0
$745$	24.9297	+	19.4755i	0.913354	+	0.713527i
$746$		0			0
$747$		−	2.18316i		−	0.0798778i
$748$		0			0
$749$	6.67468			0.243887
$750$		0			0
$751$	−6.56538			−0.239574			−0.119787	−	0.992800i	$-0.538221\pi$
$751$	−6.56538			−0.239574			−0.119787	+	0.992800i	$0.538221\pi$
$752$		0			0
$753$		−	2.82713i		−	0.103026i
$754$		0			0
$755$	−23.6972	−	18.5126i	−0.862430	−	0.673744i
$756$		0			0
$757$		−	3.69666i		−	0.134357i	−0.997741	−	0.0671786i	$-0.978600\pi$
$757$		−	3.69666i		−	0.134357i	0.997741	−	0.0671786i	$-0.0213997\pi$
$758$		0			0
$759$	7.39451			0.268404
$760$		0			0
$761$	43.2474			1.56772			0.783859	−	0.620939i	$-0.213249\pi$
$761$	43.2474			1.56772			0.783859	+	0.620939i	$0.213249\pi$
$762$		0			0
$763$			48.8581i			1.76878i
$764$		0			0
$765$	5.83244	−	7.46585i	0.210872	−	0.269929i
$766$		0			0
$767$		−	8.64098i		−	0.312008i
$768$		0			0
$769$	−37.8488			−1.36486			−0.682431	−	0.730950i	$-0.739077\pi$
$769$	−37.8488			−1.36486			−0.682431	+	0.730950i	$0.739077\pi$
$770$		0			0
$771$	−25.7316			−0.926701
$772$		0			0
$773$			14.5250i			0.522428i	0.965281	+	0.261214i	$0.0841228\pi$
$773$			14.5250i			0.522428i	−0.965281	+	0.261214i	$0.915877\pi$
$774$		0			0
$775$	−4.12909	−	16.5548i	−0.148321	−	0.594665i
$776$		0			0
$777$			11.6032i			0.416262i
$778$		0			0
$779$	3.90255			0.139823
$780$		0			0
$781$	6.94958			0.248675
$782$		0			0
$783$			4.64134i			0.165868i
$784$		0			0
$785$	2.81251	−	3.60018i	0.100383	−	0.128496i
$786$		0			0
$787$		−	40.3415i		−	1.43802i	−0.695001	−	0.719009i	$-0.744596\pi$
$787$		−	40.3415i		−	1.43802i	0.695001	−	0.719009i	$-0.255404\pi$
$788$		0			0
$789$	13.0889			0.465977
$790$		0			0
$791$	−36.1839			−1.28655
$792$		0			0
$793$		−	13.7430i		−	0.488029i
$794$		0			0
$795$	−7.56868	−	5.91278i	−0.268434	−	0.209705i
$796$		0			0
$797$			30.5348i			1.08160i	0.841152	+	0.540799i	$0.181878\pi$
$797$			30.5348i			1.08160i	−0.841152	+	0.540799i	$0.818122\pi$
$798$		0			0
$799$	8.34306			0.295156
$800$		0			0
$801$	2.58761			0.0914289
$802$		0			0
$803$			2.46054i			0.0868307i
$804$		0			0
$805$	−35.3460	−	27.6129i	−1.24578	−	0.973226i
$806$		0			0
$807$			23.8484i			0.839505i
$808$		0			0
$809$	41.9375			1.47444			0.737222	−	0.675651i	$-0.236137\pi$
$809$	41.9375			1.47444			0.737222	+	0.675651i	$0.236137\pi$
$810$		0			0
$811$	−15.3869			−0.540307			−0.270154	−	0.962817i	$-0.587074\pi$
$811$	−15.3869			−0.540307			−0.270154	+	0.962817i	$0.587074\pi$
$812$		0			0
$813$		−	5.69786i		−	0.199833i
$814$		0			0
$815$	20.5190	−	26.2655i	0.718750	−	0.920040i
$816$		0			0
$817$		−	9.15451i		−	0.320276i
$818$		0			0
$819$	−11.7967			−0.412210
$820$		0			0
$821$	34.7244			1.21189			0.605945	−	0.795506i	$-0.292795\pi$
$821$	34.7244			1.21189			0.605945	+	0.795506i	$0.292795\pi$
$822$		0			0
$823$			11.8216i			0.412074i	0.978544	+	0.206037i	$0.0660568\pi$
$823$			11.8216i			0.412074i	−0.978544	+	0.206037i	$0.933943\pi$
$824$		0			0
$825$	4.85137	−	1.21003i	0.168903	−	0.0421278i
$826$		0			0
$827$			49.7533i			1.73009i	0.501693	+	0.865046i	$0.332711\pi$
$827$			49.7533i			1.73009i	−0.501693	+	0.865046i	$0.667289\pi$
$828$		0			0
$829$	−9.46484			−0.328728			−0.164364	−	0.986400i	$-0.552557\pi$
$829$	−9.46484			−0.328728			−0.164364	+	0.986400i	$0.552557\pi$
$830$		0			0
$831$	−19.8969			−0.690216
$832$		0			0
$833$			1.51958i			0.0526505i
$834$		0			0
$835$	18.8594	−	24.1411i	0.652656	−	0.835436i
$836$		0			0
$837$			3.41239i			0.117949i
$838$		0			0
$839$	17.4977			0.604088			0.302044	−	0.953294i	$-0.402331\pi$
$839$	17.4977			0.604088			0.302044	+	0.953294i	$0.402331\pi$
$840$		0			0
$841$	−7.45792			−0.257170
$842$		0			0
$843$			1.79061i			0.0616718i
$844$		0			0
$845$	10.4164	+	8.13748i	0.358336	+	0.279938i
$846$		0			0
$847$		−	2.71268i		−	0.0932089i
$848$		0			0
$849$	−23.5378			−0.807816
$850$		0			0
$851$	−31.6291			−1.08423
$852$		0			0
$853$		−	49.9585i		−	1.71055i	−0.518177	−	0.855274i	$-0.673389\pi$
$853$		−	49.9585i		−	1.71055i	0.518177	−	0.855274i	$-0.326611\pi$
$854$		0			0
$855$	−1.35866	−	1.06140i	−0.0464650	−	0.0362992i
$856$		0			0
$857$		−	31.3583i		−	1.07118i	−0.844478	−	0.535590i	$-0.820089\pi$
$857$		−	31.3583i		−	1.07118i	0.844478	−	0.535590i	$-0.179911\pi$
$858$		0			0
$859$	−37.0794			−1.26513			−0.632566	−	0.774506i	$-0.717998\pi$
$859$	−37.0794			−1.26513			−0.632566	+	0.774506i	$0.717998\pi$
$860$		0			0
$861$	−13.7300			−0.467917
$862$		0			0
$863$		−	35.4974i		−	1.20835i	−0.796853	−	0.604174i	$-0.793503\pi$
$863$		−	35.4974i		−	1.20835i	0.796853	−	0.604174i	$-0.206497\pi$
$864$		0			0
$865$	8.48429	−	10.8604i	0.288475	−	0.369264i
$866$		0			0
$867$		−	0.951274i		−	0.0323070i
$868$		0			0
$869$	2.88287			0.0977945
$870$		0			0
$871$	−55.4512			−1.87889
$872$		0			0
$873$			11.2448i			0.380580i
$874$		0			0
$875$	−27.7083	−	12.3322i	−0.936711	−	0.416905i
$876$		0			0
$877$			30.0290i			1.01401i	0.861944	+	0.507003i	$0.169247\pi$
$877$			30.0290i			1.01401i	−0.861944	+	0.507003i	$0.830753\pi$
$878$		0			0
$879$	27.4102			0.924524
$880$		0			0
$881$	35.0598			1.18119			0.590597	−	0.806967i	$-0.298892\pi$
$881$	35.0598			1.18119			0.590597	+	0.806967i	$0.298892\pi$
$882$		0			0
$883$			11.1016i			0.373599i	0.982398	+	0.186800i	$0.0598115\pi$
$883$			11.1016i			0.373599i	−0.982398	+	0.186800i	$0.940188\pi$
$884$		0			0
$885$	−2.73530	+	3.50133i	−0.0919460	+	0.117696i
$886$		0			0
$887$		−	3.43536i		−	0.115348i	−0.998335	−	0.0576741i	$-0.981632\pi$
$887$		−	3.43536i		−	0.115348i	0.998335	−	0.0576741i	$-0.0183684\pi$
$888$		0			0
$889$	−18.4290			−0.618090
$890$		0			0
$891$	−1.00000			−0.0335013
$892$		0			0
$893$		−	1.51829i		−	0.0508077i
$894$		0			0
$895$	−34.0858	−	26.6284i	−1.13936	−	0.890088i
$896$		0			0
$897$		−	32.1567i		−	1.07368i
$898$		0			0
$899$	15.8381			0.528229
$900$		0			0
$901$	−18.1985			−0.606281
$902$		0			0
$903$			32.2075i			1.07180i
$904$		0			0
$905$	−38.9598	−	30.4360i	−1.29507	−	1.01173i
$906$		0			0
$907$		−	38.8168i		−	1.28889i	−0.764650	−	0.644445i	$-0.777088\pi$
$907$		−	38.8168i		−	1.28889i	0.764650	−	0.644445i	$-0.222912\pi$
$908$		0			0
$909$	13.1961			0.437687
$910$		0			0
$911$	5.10977			0.169294			0.0846471	−	0.996411i	$-0.473024\pi$
$911$	5.10977			0.169294			0.0846471	+	0.996411i	$0.473024\pi$
$912$		0			0
$913$		−	2.18316i		−	0.0722522i
$914$		0			0
$915$	−4.35035	+	5.56868i	−0.143818	+	0.184095i
$916$		0			0
$917$		−	2.94674i		−	0.0973098i
$918$		0			0
$919$	−27.5970			−0.910342			−0.455171	−	0.890404i	$-0.650422\pi$
$919$	−27.5970			−0.910342			−0.455171	+	0.890404i	$0.650422\pi$
$920$		0			0
$921$	1.74035			0.0573463
$922$		0			0
$923$		−	30.2218i		−	0.994761i
$924$		0			0
$925$	−20.7512	+	5.17575i	−0.682294	+	0.170178i
$926$		0			0
$927$		−	2.57463i		−	0.0845620i
$928$		0			0
$929$	2.07172			0.0679709			0.0339854	−	0.999422i	$-0.489180\pi$
$929$	2.07172			0.0679709			0.0339854	+	0.999422i	$0.489180\pi$
$930$		0			0
$931$	0.276538			0.00906316
$932$		0			0
$933$			17.9623i			0.588058i
$934$		0			0
$935$	5.83244	−	7.46585i	0.190741	−	0.244160i
$936$		0			0
$937$			21.8925i			0.715198i	0.933875	+	0.357599i	$0.116404\pi$
$937$			21.8925i			0.715198i	−0.933875	+	0.357599i	$0.883596\pi$
$938$		0			0
$939$	4.83487			0.157780
$940$		0			0
$941$	−36.9844			−1.20566			−0.602829	−	0.797871i	$-0.705960\pi$
$941$	−36.9844			−1.20566			−0.602829	+	0.797871i	$0.705960\pi$
$942$		0			0
$943$		−	37.4266i		−	1.21878i
$944$		0			0
$945$	4.78003	+	3.73424i	0.155495	+	0.121475i
$946$		0			0
$947$			10.0916i			0.327933i	0.986466	+	0.163966i	$0.0524288\pi$
$947$			10.0916i			0.327933i	−0.986466	+	0.163966i	$0.947571\pi$
$948$		0			0
$949$	10.7002			0.347344
$950$		0			0
$951$	−14.6365			−0.474620
$952$		0			0
$953$			9.27795i			0.300542i	0.988645	+	0.150271i	$0.0480147\pi$
$953$			9.27795i			0.300542i	−0.988645	+	0.150271i	$0.951985\pi$
$954$		0			0
$955$	39.0897	+	30.5375i	1.26491	+	0.988171i
$956$		0			0
$957$			4.64134i			0.150033i
$958$		0			0
$959$	31.8780			1.02940
$960$		0			0
$961$	−19.3556			−0.624375
$962$		0			0
$963$		−	2.46054i		−	0.0792899i
$964$		0			0
$965$	−1.92278	+	2.46127i	−0.0618966	+	0.0792311i
$966$		0			0
$967$			8.51416i			0.273797i	0.990585	+	0.136898i	$0.0437134\pi$
$967$			8.51416i			0.273797i	−0.990585	+	0.136898i	$0.956287\pi$
$968$		0			0
$969$	−3.26682			−0.104945
$970$		0			0
$971$	26.9012			0.863301			0.431651	−	0.902041i	$-0.357931\pi$
$971$	26.9012			0.863301			0.431651	+	0.902041i	$0.357931\pi$
$972$		0			0
$973$		−	52.7579i		−	1.69134i
$974$		0			0
$975$	−5.26208	−	21.0973i	−0.168521	−	0.675653i
$976$		0			0
$977$			45.7509i			1.46370i	0.681466	+	0.731850i	$0.261343\pi$
$977$			45.7509i			1.46370i	−0.681466	+	0.731850i	$0.738657\pi$
$978$		0			0
$979$	2.58761			0.0827005
$980$		0			0
$981$	18.0110			0.575046
$982$		0			0
$983$		−	35.9148i		−	1.14550i	−0.819729	−	0.572752i	$-0.805876\pi$
$983$		−	35.9148i		−	1.14550i	0.819729	−	0.572752i	$-0.194124\pi$
$984$		0			0
$985$	−27.8898	+	35.7005i	−0.888643	+	1.13751i
$986$		0			0
$987$			5.34167i			0.170027i
$988$		0			0
$989$	−87.7945			−2.79170
$990$		0			0
$991$	49.3833			1.56871			0.784355	−	0.620312i	$-0.212994\pi$
$991$	49.3833			1.56871			0.784355	+	0.620312i	$0.212994\pi$
$992$		0			0
$993$		−	15.4356i		−	0.489835i
$994$		0			0
$995$	45.6370	+	35.6524i	1.44679	+	1.13026i
$996$		0			0
$997$		−	1.01442i		−	0.0321269i	−0.999871	−	0.0160634i	$-0.994887\pi$
$997$		−	1.01442i		−	0.0321269i	0.999871	−	0.0160634i	$-0.00511337\pi$
$998$		0			0
$999$	4.27738			0.135330

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1320.2.d.c.529.7	yes	10
3.2	odd	2		3960.2.d.h.3169.8		10
4.3	odd	2		2640.2.d.k.529.2		10
5.2	odd	4		6600.2.a.bz.1.4		5
5.3	odd	4		6600.2.a.bx.1.2		5
5.4	even	2	inner	1320.2.d.c.529.2	✓	10
15.14	odd	2		3960.2.d.h.3169.7		10
20.19	odd	2		2640.2.d.k.529.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1320.2.d.c.529.2	✓	10	5.4	even	2	inner
1320.2.d.c.529.7	yes	10	1.1	even	1	trivial
2640.2.d.k.529.2		10	4.3	odd	2
2640.2.d.k.529.7		10	20.19	odd	2
3960.2.d.h.3169.7		10	15.14	odd	2
3960.2.d.h.3169.8		10	3.2	odd	2
6600.2.a.bx.1.2		5	5.3	odd	4
6600.2.a.bz.1.4		5	5.2	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1320 = 2^{3} \cdot 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1320.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +(-1.76211 - 1.37658i) q^{5} -2.71268i q^{7} -1.00000 q^{9} -1.00000 q^{11} +4.34872i q^{13} +(1.37658 - 1.76211i) q^{15} -4.23689i q^{17} -0.771041 q^{19} +2.71268 q^{21} +7.39451i q^{23} +(1.21003 + 4.85137i) q^{25} -1.00000i q^{27} -4.64134 q^{29} -3.41239 q^{31} -1.00000i q^{33} +(-3.73424 + 4.78003i) q^{35} +4.27738i q^{37} -4.34872 q^{39} -5.06140 q^{41} +11.8729i q^{43} +(1.76211 + 1.37658i) q^{45} +1.96915i q^{47} -0.358655 q^{49} +4.23689 q^{51} -4.29525i q^{53} +(1.76211 + 1.37658i) q^{55} -0.771041i q^{57} -1.98702 q^{59} -3.16025 q^{61} +2.71268i q^{63} +(5.98638 - 7.66290i) q^{65} +12.7512i q^{67} -7.39451 q^{69} -6.94958 q^{71} -2.46054i q^{73} +(-4.85137 + 1.21003i) q^{75} +2.71268i q^{77} -2.88287 q^{79} +1.00000 q^{81} +2.18316i q^{83} +(-5.83244 + 7.46585i) q^{85} -4.64134i q^{87} -2.58761 q^{89} +11.7967 q^{91} -3.41239i q^{93} +(1.35866 + 1.06140i) q^{95} -11.2448i q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} - 4 q^{19} + 4 q^{21} - 2 q^{25} - 8 q^{29} + 8 q^{31} + 8 q^{35} + 16 q^{41} + 2 q^{45} - 42 q^{49} - 12 q^{51} + 2 q^{55} - 24 q^{59} + 20 q^{61} + 24 q^{65} - 8 q^{69}+ \cdots + 10 q^{99}+O(q^{100}) \) 10 * q - 2 * q^5 - 10 * q^9 - 10 * q^11 - 4 * q^19 + 4 * q^21 - 2 * q^25 - 8 * q^29 + 8 * q^31 + 8 * q^35 + 16 * q^41 + 2 * q^45 - 42 * q^49 - 12 * q^51 + 2 * q^55 - 24 * q^59 + 20 * q^61 + 24 * q^65 - 8 * q^69 + 8 * q^71 + 4 * q^75 - 36 * q^79 + 10 * q^81 + 12 * q^85 - 68 * q^89 + 52 * q^95 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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