LMFDB - Embedded newform 960.2.o.d.959.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,2,Mod(959,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("960.959");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 960 = 2^{6} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	960.o (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:	$7.66563859404$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10}\cdot 3 $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			959.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	960.959
Dual form			960.2.o.d.959.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 - 1.00000i) q^{3} +(-1.73205 - 1.41421i) q^{5} +(1.00000 + 2.82843i) q^{9} +O(q^{10})$	$q+(-1.41421 - 1.00000i) q^{3} +(-1.73205 - 1.41421i) q^{5} +(1.00000 + 2.82843i) q^{9} +4.89898 q^{11} +4.89898i q^{13} +(1.03528 + 3.73205i) q^{15} +3.46410 q^{17} -3.46410i q^{19} -6.00000i q^{23} +(1.00000 + 4.89898i) q^{25} +(1.41421 - 5.00000i) q^{27} +2.82843i q^{29} -3.46410i q^{31} +(-6.92820 - 4.89898i) q^{33} -4.89898i q^{37} +(4.89898 - 6.92820i) q^{39} -5.65685i q^{41} -8.48528 q^{43} +(2.26795 - 6.31319i) q^{45} -6.00000i q^{47} -7.00000 q^{49} +(-4.89898 - 3.46410i) q^{51} +10.3923 q^{53} +(-8.48528 - 6.92820i) q^{55} +(-3.46410 + 4.89898i) q^{57} +4.89898 q^{59} +2.00000 q^{61} +(6.92820 - 8.48528i) q^{65} +8.48528 q^{67} +(-6.00000 + 8.48528i) q^{69} +9.79796 q^{71} -9.79796i q^{73} +(3.48477 - 7.92820i) q^{75} +10.3923i q^{79} +(-7.00000 + 5.65685i) q^{81} +6.00000i q^{83} +(-6.00000 - 4.89898i) q^{85} +(2.82843 - 4.00000i) q^{87} +5.65685i q^{89} +(-3.46410 + 4.89898i) q^{93} +(-4.89898 + 6.00000i) q^{95} +(4.89898 + 13.8564i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 8 q^{25} + 32 q^{45} - 56 q^{49} + 16 q^{61} - 48 q^{69} - 56 q^{81} - 48 q^{85}+O(q^{100}) $ 8 * q + 8 * q^9 + 8 * q^25 + 32 * q^45 - 56 * q^49 + 16 * q^61 - 48 * q^69 - 56 * q^81 - 48 * q^85

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/960\mathbb{Z}\right)^\times$.

$n$	$511$	$577$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−1.41421	−	1.00000i	−0.816497	−	0.577350i
$3$	−1.41421	−	1.00000i	−0.816497	−	0.577350i
$4$		0			0
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$5$	−1.73205	−	1.41421i	−0.774597	−	0.632456i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		0			0
$9$	1.00000	+	2.82843i	0.333333	+	0.942809i
$10$		0			0
$11$	4.89898			1.47710			0.738549	−	0.674200i	$-0.235511\pi$
$11$	4.89898			1.47710			0.738549	+	0.674200i	$0.235511\pi$
$12$		0			0
$13$			4.89898i			1.35873i	0.733799	+	0.679366i	$0.237745\pi$
$13$			4.89898i			1.35873i	−0.733799	+	0.679366i	$0.762255\pi$
$14$		0			0
$15$	1.03528	+	3.73205i	0.267307	+	0.963611i
$16$		0			0
$17$	3.46410			0.840168			0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410			0.840168			0.420084	+	0.907485i	$0.362001\pi$
$18$		0			0
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	6.00000i		−	1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$		−	6.00000i		−	1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$		0			0
$25$	1.00000	+	4.89898i	0.200000	+	0.979796i
$26$		0			0
$27$	1.41421	−	5.00000i	0.272166	−	0.962250i
$28$		0			0
$29$			2.82843i			0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$			2.82843i			0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$	−6.92820	−	4.89898i	−1.20605	−	0.852803i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	4.89898i		−	0.805387i	−0.915335	−	0.402694i	$-0.868074\pi$
$37$		−	4.89898i		−	0.805387i	0.915335	−	0.402694i	$-0.131926\pi$
$38$		0			0
$39$	4.89898	−	6.92820i	0.784465	−	1.10940i
$40$		0			0
$41$		−	5.65685i		−	0.883452i	−0.897150	−	0.441726i	$-0.854366\pi$
$41$		−	5.65685i		−	0.883452i	0.897150	−	0.441726i	$-0.145634\pi$
$42$		0			0
$43$	−8.48528			−1.29399			−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528			−1.29399			−0.646997	+	0.762493i	$0.723975\pi$
$44$		0			0
$45$	2.26795	−	6.31319i	0.338086	−	0.941115i
$46$		0			0
$47$		−	6.00000i		−	0.875190i	−0.899172	−	0.437595i	$-0.855830\pi$
$47$		−	6.00000i		−	0.875190i	0.899172	−	0.437595i	$-0.144170\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−4.89898	−	3.46410i	−0.685994	−	0.485071i
$52$		0			0
$53$	10.3923			1.42749			0.713746	−	0.700404i	$-0.246997\pi$
$53$	10.3923			1.42749			0.713746	+	0.700404i	$0.246997\pi$
$54$		0			0
$55$	−8.48528	−	6.92820i	−1.14416	−	0.934199i
$56$		0			0
$57$	−3.46410	+	4.89898i	−0.458831	+	0.648886i
$58$		0			0
$59$	4.89898			0.637793			0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898			0.637793			0.318896	+	0.947790i	$0.396688\pi$
$60$		0			0
$61$	2.00000			0.256074			0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000			0.256074			0.128037	+	0.991769i	$0.459132\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.92820	−	8.48528i	0.859338	−	1.05247i
$66$		0			0
$67$	8.48528			1.03664			0.518321	−	0.855186i	$-0.326557\pi$
$67$	8.48528			1.03664			0.518321	+	0.855186i	$0.326557\pi$
$68$		0			0
$69$	−6.00000	+	8.48528i	−0.722315	+	1.02151i
$70$		0			0
$71$	9.79796			1.16280			0.581402	−	0.813617i	$-0.302504\pi$
$71$	9.79796			1.16280			0.581402	+	0.813617i	$0.302504\pi$
$72$		0			0
$73$		−	9.79796i		−	1.14676i	−0.819288	−	0.573382i	$-0.805631\pi$
$73$		−	9.79796i		−	1.14676i	0.819288	−	0.573382i	$-0.194369\pi$
$74$		0			0
$75$	3.48477	−	7.92820i	0.402386	−	0.915470i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			10.3923i			1.16923i	0.811312	+	0.584613i	$0.198754\pi$
$79$			10.3923i			1.16923i	−0.811312	+	0.584613i	$0.801246\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$			6.00000i			0.658586i	0.944228	+	0.329293i	$0.106810\pi$
$83$			6.00000i			0.658586i	−0.944228	+	0.329293i	$0.893190\pi$
$84$		0			0
$85$	−6.00000	−	4.89898i	−0.650791	−	0.531369i
$86$		0			0
$87$	2.82843	−	4.00000i	0.303239	−	0.428845i
$88$		0			0
$89$			5.65685i			0.599625i	0.953998	+	0.299813i	$0.0969242\pi$
$89$			5.65685i			0.599625i	−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−3.46410	+	4.89898i	−0.359211	+	0.508001i
$94$		0			0
$95$	−4.89898	+	6.00000i	−0.502625	+	0.615587i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$	4.89898	+	13.8564i	0.492366	+	1.39262i
$100$		0			0
$101$		−	14.1421i		−	1.40720i	−0.710599	−	0.703598i	$-0.751576\pi$
$101$		−	14.1421i		−	1.40720i	0.710599	−	0.703598i	$-0.248424\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	18.0000i		−	1.74013i	−0.492941	−	0.870063i	$-0.664078\pi$
$107$		−	18.0000i		−	1.74013i	0.492941	−	0.870063i	$-0.335922\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	−4.89898	+	6.92820i	−0.464991	+	0.657596i
$112$		0			0
$113$	3.46410			0.325875			0.162938	−	0.986636i	$-0.447903\pi$
$113$	3.46410			0.325875			0.162938	+	0.986636i	$0.447903\pi$
$114$		0			0
$115$	−8.48528	+	10.3923i	−0.791257	+	0.969087i
$116$		0			0
$117$	−13.8564	+	4.89898i	−1.28103	+	0.452911i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	13.0000			1.18182
$122$		0			0
$123$	−5.65685	+	8.00000i	−0.510061	+	0.721336i
$124$		0			0
$125$	5.19615	−	9.89949i	0.464758	−	0.885438i
$126$		0			0
$127$	16.9706			1.50589			0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706			1.50589			0.752947	+	0.658081i	$0.228632\pi$
$128$		0			0
$129$	12.0000	+	8.48528i	1.05654	+	0.747087i
$130$		0			0
$131$	−4.89898			−0.428026			−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898			−0.428026			−0.214013	+	0.976831i	$0.568653\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−9.52056	+	6.66025i	−0.819399	+	0.573223i
$136$		0			0
$137$	3.46410			0.295958			0.147979	−	0.988990i	$-0.452723\pi$
$137$	3.46410			0.295958			0.147979	+	0.988990i	$0.452723\pi$
$138$		0			0
$139$		−	17.3205i		−	1.46911i	−0.678551	−	0.734553i	$-0.737392\pi$
$139$		−	17.3205i		−	1.46911i	0.678551	−	0.734553i	$-0.262608\pi$
$140$		0			0
$141$	−6.00000	+	8.48528i	−0.505291	+	0.714590i
$142$		0			0
$143$			24.0000i			2.00698i
$144$		0			0
$145$	4.00000	−	4.89898i	0.332182	−	0.406838i
$146$		0			0
$147$	9.89949	+	7.00000i	0.816497	+	0.577350i
$148$		0			0
$149$		−	2.82843i		−	0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$		−	2.82843i		−	0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$		0			0
$151$		−	3.46410i		−	0.281905i	−0.990016	−	0.140952i	$-0.954984\pi$
$151$		−	3.46410i		−	0.281905i	0.990016	−	0.140952i	$-0.0450164\pi$
$152$		0			0
$153$	3.46410	+	9.79796i	0.280056	+	0.792118i
$154$		0			0
$155$	−4.89898	+	6.00000i	−0.393496	+	0.481932i
$156$		0			0
$157$			4.89898i			0.390981i	0.980706	+	0.195491i	$0.0626299\pi$
$157$			4.89898i			0.390981i	−0.980706	+	0.195491i	$0.937370\pi$
$158$		0			0
$159$	−14.6969	−	10.3923i	−1.16554	−	0.824163i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	8.48528			0.664619			0.332309	−	0.943170i	$-0.392172\pi$
$163$	8.48528			0.664619			0.332309	+	0.943170i	$0.392172\pi$
$164$		0			0
$165$	5.07180	+	18.2832i	0.394839	+	1.42335i
$166$		0			0
$167$		−	6.00000i		−	0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$		−	6.00000i		−	0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$		0			0
$169$	−11.0000			−0.846154
$170$		0			0
$171$	9.79796	−	3.46410i	0.749269	−	0.264906i
$172$		0			0
$173$	−3.46410			−0.263371			−0.131685	−	0.991292i	$-0.542039\pi$
$173$	−3.46410			−0.263371			−0.131685	+	0.991292i	$0.542039\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−6.92820	−	4.89898i	−0.520756	−	0.368230i
$178$		0			0
$179$	−24.4949			−1.83083			−0.915417	−	0.402506i	$-0.868139\pi$
$179$	−24.4949			−1.83083			−0.915417	+	0.402506i	$0.868139\pi$
$180$		0			0
$181$	10.0000			0.743294			0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000			0.743294			0.371647	+	0.928374i	$0.378793\pi$
$182$		0			0
$183$	−2.82843	−	2.00000i	−0.209083	−	0.147844i
$184$		0			0
$185$	−6.92820	+	8.48528i	−0.509372	+	0.623850i
$186$		0			0
$187$	16.9706			1.24101
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−19.5959			−1.41791			−0.708955	−	0.705253i	$-0.750833\pi$
$191$	−19.5959			−1.41791			−0.708955	+	0.705253i	$0.750833\pi$
$192$		0			0
$193$			19.5959i			1.41055i	0.708936	+	0.705273i	$0.249175\pi$
$193$			19.5959i			1.41055i	−0.708936	+	0.705273i	$0.750825\pi$
$194$		0			0
$195$	−18.2832	+	5.07180i	−1.30929	+	0.363199i
$196$		0			0
$197$	24.2487			1.72765			0.863825	−	0.503793i	$-0.168062\pi$
$197$	24.2487			1.72765			0.863825	+	0.503793i	$0.168062\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$	−12.0000	−	8.48528i	−0.846415	−	0.598506i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−8.00000	+	9.79796i	−0.558744	+	0.684319i
$206$		0			0
$207$	16.9706	−	6.00000i	1.17954	−	0.417029i
$208$		0			0
$209$		−	16.9706i		−	1.17388i
$210$		0			0
$211$			24.2487i			1.66935i	0.550743	+	0.834675i	$0.314345\pi$
$211$			24.2487i			1.66935i	−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$	−13.8564	−	9.79796i	−0.949425	−	0.671345i
$214$		0			0
$215$	14.6969	+	12.0000i	1.00232	+	0.818393i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−9.79796	+	13.8564i	−0.662085	+	0.936329i
$220$		0			0
$221$			16.9706i			1.14156i
$222$		0			0
$223$	−16.9706			−1.13643			−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706			−1.13643			−0.568216	+	0.822879i	$0.692366\pi$
$224$		0			0
$225$	−12.8564	+	7.72741i	−0.857094	+	0.515160i
$226$		0			0
$227$			6.00000i			0.398234i	0.979976	+	0.199117i	$0.0638074\pi$
$227$			6.00000i			0.398234i	−0.979976	+	0.199117i	$0.936193\pi$
$228$		0			0
$229$	10.0000			0.660819			0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000			0.660819			0.330409	+	0.943838i	$0.392813\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.46410			0.226941			0.113470	−	0.993541i	$-0.463803\pi$
$233$	3.46410			0.226941			0.113470	+	0.993541i	$0.463803\pi$
$234$		0			0
$235$	−8.48528	+	10.3923i	−0.553519	+	0.677919i
$236$		0			0
$237$	10.3923	−	14.6969i	0.675053	−	0.954669i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	10.0000			0.644157			0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000			0.644157			0.322078	+	0.946713i	$0.395619\pi$
$242$		0			0
$243$	15.5563	−	1.00000i	0.997940	−	0.0641500i
$244$		0			0
$245$	12.1244	+	9.89949i	0.774597	+	0.632456i
$246$		0			0
$247$	16.9706			1.07981
$248$		0			0
$249$	6.00000	−	8.48528i	0.380235	−	0.537733i
$250$		0			0
$251$	−14.6969			−0.927663			−0.463831	−	0.885924i	$-0.653526\pi$
$251$	−14.6969			−0.927663			−0.463831	+	0.885924i	$0.653526\pi$
$252$		0			0
$253$		−	29.3939i		−	1.84798i
$254$		0			0
$255$	3.58630	+	12.9282i	0.224583	+	0.809595i
$256$		0			0
$257$	3.46410			0.216085			0.108042	−	0.994146i	$-0.465542\pi$
$257$	3.46410			0.216085			0.108042	+	0.994146i	$0.465542\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−8.00000	+	2.82843i	−0.495188	+	0.175075i
$262$		0			0
$263$		−	6.00000i		−	0.369976i	−0.982741	−	0.184988i	$-0.940775\pi$
$263$		−	6.00000i		−	0.369976i	0.982741	−	0.184988i	$-0.0592246\pi$
$264$		0			0
$265$	−18.0000	−	14.6969i	−1.10573	−	0.902826i
$266$		0			0
$267$	5.65685	−	8.00000i	0.346194	−	0.489592i
$268$		0			0
$269$		−	19.7990i		−	1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$		−	19.7990i		−	1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$		0			0
$271$			10.3923i			0.631288i	0.948878	+	0.315644i	$0.102220\pi$
$271$			10.3923i			0.631288i	−0.948878	+	0.315644i	$0.897780\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.89898	+	24.0000i	0.295420	+	1.44725i
$276$		0			0
$277$			14.6969i			0.883053i	0.897248	+	0.441527i	$0.145563\pi$
$277$			14.6969i			0.883053i	−0.897248	+	0.441527i	$0.854437\pi$
$278$		0			0
$279$	9.79796	−	3.46410i	0.586588	−	0.207390i
$280$		0			0
$281$		−	5.65685i		−	0.337460i	−0.985662	−	0.168730i	$-0.946033\pi$
$281$		−	5.65685i		−	0.337460i	0.985662	−	0.168730i	$-0.0539665\pi$
$282$		0			0
$283$	−8.48528			−0.504398			−0.252199	−	0.967675i	$-0.581154\pi$
$283$	−8.48528			−0.504398			−0.252199	+	0.967675i	$0.581154\pi$
$284$		0			0
$285$	12.9282	−	3.58630i	0.765801	−	0.212434i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.00000			−0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−17.3205			−1.01187			−0.505937	−	0.862570i	$-0.668853\pi$
$293$	−17.3205			−1.01187			−0.505937	+	0.862570i	$0.668853\pi$
$294$		0			0
$295$	−8.48528	−	6.92820i	−0.494032	−	0.403376i
$296$		0			0
$297$	6.92820	−	24.4949i	0.402015	−	1.42134i
$298$		0			0
$299$	29.3939			1.69989
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−14.1421	+	20.0000i	−0.812444	+	1.14897i
$304$		0			0
$305$	−3.46410	−	2.82843i	−0.198354	−	0.161955i
$306$		0			0
$307$	8.48528			0.484281			0.242140	−	0.970241i	$-0.422151\pi$
$307$	8.48528			0.484281			0.242140	+	0.970241i	$0.422151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.79796			−0.555591			−0.277796	−	0.960640i	$-0.589604\pi$
$311$	−9.79796			−0.555591			−0.277796	+	0.960640i	$0.589604\pi$
$312$		0			0
$313$			9.79796i			0.553813i	0.960897	+	0.276907i	$0.0893093\pi$
$313$			9.79796i			0.553813i	−0.960897	+	0.276907i	$0.910691\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−17.3205			−0.972817			−0.486408	−	0.873732i	$-0.661693\pi$
$317$	−17.3205			−0.972817			−0.486408	+	0.873732i	$0.661693\pi$
$318$		0			0
$319$			13.8564i			0.775810i
$320$		0			0
$321$	−18.0000	+	25.4558i	−1.00466	+	1.42081i
$322$		0			0
$323$		−	12.0000i		−	0.667698i
$324$		0			0
$325$	−24.0000	+	4.89898i	−1.33128	+	0.271746i
$326$		0			0
$327$	−2.82843	−	2.00000i	−0.156412	−	0.110600i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	17.3205i		−	0.952021i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$		−	17.3205i		−	0.952021i	0.879440	−	0.476011i	$-0.157918\pi$
$332$		0			0
$333$	13.8564	−	4.89898i	0.759326	−	0.268462i
$334$		0			0
$335$	−14.6969	−	12.0000i	−0.802980	−	0.655630i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−4.89898	−	3.46410i	−0.266076	−	0.188144i
$340$		0			0
$341$		−	16.9706i		−	0.919007i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	22.3923	−	6.21166i	1.20556	−	0.334424i
$346$		0			0
$347$			6.00000i			0.322097i	0.986947	+	0.161048i	$0.0514875\pi$
$347$			6.00000i			0.322097i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$	−14.0000			−0.749403			−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000			−0.749403			−0.374701	+	0.927146i	$0.622255\pi$
$350$		0			0
$351$	24.4949	+	6.92820i	1.30744	+	0.369800i
$352$		0			0
$353$	31.1769			1.65938			0.829690	−	0.558225i	$-0.188517\pi$
$353$	31.1769			1.65938			0.829690	+	0.558225i	$0.188517\pi$
$354$		0			0
$355$	−16.9706	−	13.8564i	−0.900704	−	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−9.79796			−0.517116			−0.258558	−	0.965996i	$-0.583247\pi$
$359$	−9.79796			−0.517116			−0.258558	+	0.965996i	$0.583247\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−18.3848	−	13.0000i	−0.964951	−	0.682323i
$364$		0			0
$365$	−13.8564	+	16.9706i	−0.725277	+	0.888280i
$366$		0			0
$367$	16.9706			0.885856			0.442928	−	0.896557i	$-0.353940\pi$
$367$	16.9706			0.885856			0.442928	+	0.896557i	$0.353940\pi$
$368$		0			0
$369$	16.0000	−	5.65685i	0.832927	−	0.294484i
$370$		0			0
$371$		0			0
$372$		0			0
$373$			14.6969i			0.760979i	0.924785	+	0.380489i	$0.124244\pi$
$373$			14.6969i			0.760979i	−0.924785	+	0.380489i	$0.875756\pi$
$374$		0			0
$375$	−17.2480	+	8.80385i	−0.890681	+	0.454629i
$376$		0			0
$377$	−13.8564			−0.713641
$378$		0			0
$379$			10.3923i			0.533817i	0.963722	+	0.266908i	$0.0860021\pi$
$379$			10.3923i			0.533817i	−0.963722	+	0.266908i	$0.913998\pi$
$380$		0			0
$381$	−24.0000	−	16.9706i	−1.22956	−	0.869428i
$382$		0			0
$383$		−	6.00000i		−	0.306586i	−0.988181	−	0.153293i	$-0.951012\pi$
$383$		−	6.00000i		−	0.306586i	0.988181	−	0.153293i	$-0.0489878\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8.48528	−	24.0000i	−0.431331	−	1.21999i
$388$		0			0
$389$		−	2.82843i		−	0.143407i	−0.997426	−	0.0717035i	$-0.977156\pi$
$389$		−	2.82843i		−	0.143407i	0.997426	−	0.0717035i	$-0.0228435\pi$
$390$		0			0
$391$		−	20.7846i		−	1.05112i
$392$		0			0
$393$	6.92820	+	4.89898i	0.349482	+	0.247121i
$394$		0			0
$395$	14.6969	−	18.0000i	0.739483	−	0.905678i
$396$		0			0
$397$			4.89898i			0.245873i	0.992415	+	0.122936i	$0.0392311\pi$
$397$			4.89898i			0.245873i	−0.992415	+	0.122936i	$0.960769\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.6274i			1.12996i	0.825105	+	0.564980i	$0.191116\pi$
$401$			22.6274i			1.12996i	−0.825105	+	0.564980i	$0.808884\pi$
$402$		0			0
$403$	16.9706			0.845364
$404$		0			0
$405$	20.1244	+	0.101536i	0.999987	+	0.00504536i
$406$		0			0
$407$		−	24.0000i		−	1.18964i
$408$		0			0
$409$	10.0000			0.494468			0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000			0.494468			0.247234	+	0.968956i	$0.420478\pi$
$410$		0			0
$411$	−4.89898	−	3.46410i	−0.241649	−	0.170872i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	8.48528	−	10.3923i	0.416526	−	0.510138i
$416$		0			0
$417$	−17.3205	+	24.4949i	−0.848189	+	1.19952i
$418$		0			0
$419$	14.6969			0.717992			0.358996	−	0.933339i	$-0.383119\pi$
$419$	14.6969			0.717992			0.358996	+	0.933339i	$0.383119\pi$
$420$		0			0
$421$	26.0000			1.26716			0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000			1.26716			0.633581	+	0.773676i	$0.281584\pi$
$422$		0			0
$423$	16.9706	−	6.00000i	0.825137	−	0.291730i
$424$		0			0
$425$	3.46410	+	16.9706i	0.168034	+	0.823193i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000	−	33.9411i	1.15873	−	1.63869i
$430$		0			0
$431$	19.5959			0.943902			0.471951	−	0.881625i	$-0.343550\pi$
$431$	19.5959			0.943902			0.471951	+	0.881625i	$0.343550\pi$
$432$		0			0
$433$		−	19.5959i		−	0.941720i	−0.882208	−	0.470860i	$-0.843944\pi$
$433$		−	19.5959i		−	0.941720i	0.882208	−	0.470860i	$-0.156056\pi$
$434$		0			0
$435$	−10.5558	+	2.92820i	−0.506113	+	0.140397i
$436$		0			0
$437$	−20.7846			−0.994263
$438$		0			0
$439$		−	17.3205i		−	0.826663i	−0.910581	−	0.413331i	$-0.864365\pi$
$439$		−	17.3205i		−	0.826663i	0.910581	−	0.413331i	$-0.135635\pi$
$440$		0			0
$441$	−7.00000	−	19.7990i	−0.333333	−	0.942809i
$442$		0			0
$443$			30.0000i			1.42534i	0.701498	+	0.712672i	$0.252515\pi$
$443$			30.0000i			1.42534i	−0.701498	+	0.712672i	$0.747485\pi$
$444$		0			0
$445$	8.00000	−	9.79796i	0.379236	−	0.464468i
$446$		0			0
$447$	−2.82843	+	4.00000i	−0.133780	+	0.189194i
$448$		0			0
$449$		−	22.6274i		−	1.06785i	−0.845531	−	0.533927i	$-0.820716\pi$
$449$		−	22.6274i		−	1.06785i	0.845531	−	0.533927i	$-0.179284\pi$
$450$		0			0
$451$		−	27.7128i		−	1.30495i
$452$		0			0
$453$	−3.46410	+	4.89898i	−0.162758	+	0.230174i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			9.79796i			0.458329i	0.973388	+	0.229165i	$0.0735994\pi$
$457$			9.79796i			0.458329i	−0.973388	+	0.229165i	$0.926401\pi$
$458$		0			0
$459$	4.89898	−	17.3205i	0.228665	−	0.808452i
$460$		0			0
$461$			36.7696i			1.71253i	0.516538	+	0.856264i	$0.327221\pi$
$461$			36.7696i			1.71253i	−0.516538	+	0.856264i	$0.672779\pi$
$462$		0			0
$463$	−16.9706			−0.788689			−0.394344	−	0.918963i	$-0.629028\pi$
$463$	−16.9706			−0.788689			−0.394344	+	0.918963i	$0.629028\pi$
$464$		0			0
$465$	12.9282	−	3.58630i	0.599531	−	0.166311i
$466$		0			0
$467$		−	18.0000i		−	0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$		−	18.0000i		−	0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	4.89898	−	6.92820i	0.225733	−	0.319235i
$472$		0			0
$473$	−41.5692			−1.91135
$474$		0			0
$475$	16.9706	−	3.46410i	0.778663	−	0.158944i
$476$		0			0
$477$	10.3923	+	29.3939i	0.475831	+	1.34585i
$478$		0			0
$479$	19.5959			0.895360			0.447680	−	0.894194i	$-0.352250\pi$
$479$	19.5959			0.895360			0.447680	+	0.894194i	$0.352250\pi$
$480$		0			0
$481$	24.0000			1.09431
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−33.9411			−1.53802			−0.769010	−	0.639237i	$-0.779250\pi$
$487$	−33.9411			−1.53802			−0.769010	+	0.639237i	$0.779250\pi$
$488$		0			0
$489$	−12.0000	−	8.48528i	−0.542659	−	0.383718i
$490$		0			0
$491$	24.4949			1.10544			0.552720	−	0.833367i	$-0.313590\pi$
$491$	24.4949			1.10544			0.552720	+	0.833367i	$0.313590\pi$
$492$		0			0
$493$			9.79796i			0.441278i
$494$		0			0
$495$	11.1106	−	30.9282i	0.499386	−	1.39012i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	3.46410i		−	0.155074i	−0.996989	−	0.0775372i	$-0.975294\pi$
$499$		−	3.46410i		−	0.155074i	0.996989	−	0.0775372i	$-0.0247057\pi$
$500$		0			0
$501$	−6.00000	+	8.48528i	−0.268060	+	0.379094i
$502$		0			0
$503$		−	6.00000i		−	0.267527i	−0.991013	−	0.133763i	$-0.957294\pi$
$503$		−	6.00000i		−	0.267527i	0.991013	−	0.133763i	$-0.0427062\pi$
$504$		0			0
$505$	−20.0000	+	24.4949i	−0.889988	+	1.09001i
$506$		0			0
$507$	15.5563	+	11.0000i	0.690882	+	0.488527i
$508$		0			0
$509$			36.7696i			1.62978i	0.579614	+	0.814891i	$0.303203\pi$
$509$			36.7696i			1.62978i	−0.579614	+	0.814891i	$0.696797\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−17.3205	−	4.89898i	−0.764719	−	0.216295i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	29.3939i		−	1.29274i
$518$		0			0
$519$	4.89898	+	3.46410i	0.215041	+	0.152057i
$520$		0			0
$521$			5.65685i			0.247831i	0.992293	+	0.123916i	$0.0395452\pi$
$521$			5.65685i			0.247831i	−0.992293	+	0.123916i	$0.960455\pi$
$522$		0			0
$523$	−42.4264			−1.85518			−0.927589	−	0.373603i	$-0.878122\pi$
$523$	−42.4264			−1.85518			−0.927589	+	0.373603i	$0.878122\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	12.0000i		−	0.522728i
$528$		0			0
$529$	−13.0000			−0.565217
$530$		0			0
$531$	4.89898	+	13.8564i	0.212598	+	0.601317i
$532$		0			0
$533$	27.7128			1.20038
$534$		0			0
$535$	−25.4558	+	31.1769i	−1.10055	+	1.34790i
$536$		0			0
$537$	34.6410	+	24.4949i	1.49487	+	1.05703i
$538$		0			0
$539$	−34.2929			−1.47710
$540$		0			0
$541$	34.0000			1.46177			0.730887	−	0.682498i	$-0.239107\pi$
$541$	34.0000			1.46177			0.730887	+	0.682498i	$0.239107\pi$
$542$		0			0
$543$	−14.1421	−	10.0000i	−0.606897	−	0.429141i
$544$		0			0
$545$	−3.46410	−	2.82843i	−0.148386	−	0.121157i
$546$		0			0
$547$	−25.4558			−1.08841			−0.544207	−	0.838951i	$-0.683169\pi$
$547$	−25.4558			−1.08841			−0.544207	+	0.838951i	$0.683169\pi$
$548$		0			0
$549$	2.00000	+	5.65685i	0.0853579	+	0.241429i
$550$		0			0
$551$	9.79796			0.417407
$552$		0			0
$553$		0			0
$554$		0			0
$555$	18.2832	−	5.07180i	0.776080	−	0.215286i
$556$		0			0
$557$	−3.46410			−0.146779			−0.0733893	−	0.997303i	$-0.523382\pi$
$557$	−3.46410			−0.146779			−0.0733893	+	0.997303i	$0.523382\pi$
$558$		0			0
$559$		−	41.5692i		−	1.75819i
$560$		0			0
$561$	−24.0000	−	16.9706i	−1.01328	−	0.716498i
$562$		0			0
$563$		−	18.0000i		−	0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$		−	18.0000i		−	0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$		0			0
$565$	−6.00000	−	4.89898i	−0.252422	−	0.206102i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			28.2843i			1.18574i	0.805299	+	0.592869i	$0.202005\pi$
$569$			28.2843i			1.18574i	−0.805299	+	0.592869i	$0.797995\pi$
$570$		0			0
$571$			38.1051i			1.59465i	0.603550	+	0.797325i	$0.293752\pi$
$571$			38.1051i			1.59465i	−0.603550	+	0.797325i	$0.706248\pi$
$572$		0			0
$573$	27.7128	+	19.5959i	1.15772	+	0.818631i
$574$		0			0
$575$	29.3939	−	6.00000i	1.22581	−	0.250217i
$576$		0			0
$577$		−	19.5959i		−	0.815789i	−0.913029	−	0.407894i	$-0.866263\pi$
$577$		−	19.5959i		−	0.815789i	0.913029	−	0.407894i	$-0.133737\pi$
$578$		0			0
$579$	19.5959	−	27.7128i	0.814379	−	1.15171i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	50.9117			2.10855
$584$		0			0
$585$	30.9282	+	11.1106i	1.27872	+	0.459368i
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$	−12.0000			−0.494451
$590$		0			0
$591$	−34.2929	−	24.2487i	−1.41062	−	0.997459i
$592$		0			0
$593$	−24.2487			−0.995775			−0.497888	−	0.867242i	$-0.665891\pi$
$593$	−24.2487			−0.995775			−0.497888	+	0.867242i	$0.665891\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	10.3923	−	14.6969i	0.425329	−	0.601506i
$598$		0			0
$599$	9.79796			0.400334			0.200167	−	0.979762i	$-0.435852\pi$
$599$	9.79796			0.400334			0.200167	+	0.979762i	$0.435852\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$	8.48528	+	24.0000i	0.345547	+	0.977356i
$604$		0			0
$605$	−22.5167	−	18.3848i	−0.915432	−	0.747447i
$606$		0			0
$607$	−16.9706			−0.688814			−0.344407	−	0.938820i	$-0.611920\pi$
$607$	−16.9706			−0.688814			−0.344407	+	0.938820i	$0.611920\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	29.3939			1.18915
$612$		0			0
$613$			14.6969i			0.593604i	0.954939	+	0.296802i	$0.0959201\pi$
$613$			14.6969i			0.593604i	−0.954939	+	0.296802i	$0.904080\pi$
$614$		0			0
$615$	21.1117	−	5.85641i	0.851305	−	0.236153i
$616$		0			0
$617$	−24.2487			−0.976216			−0.488108	−	0.872783i	$-0.662313\pi$
$617$	−24.2487			−0.976216			−0.488108	+	0.872783i	$0.662313\pi$
$618$		0			0
$619$			10.3923i			0.417702i	0.977947	+	0.208851i	$0.0669724\pi$
$619$			10.3923i			0.417702i	−0.977947	+	0.208851i	$0.933028\pi$
$620$		0			0
$621$	−30.0000	−	8.48528i	−1.20386	−	0.340503i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−23.0000	+	9.79796i	−0.920000	+	0.391918i
$626$		0			0
$627$	−16.9706	+	24.0000i	−0.677739	+	0.958468i
$628$		0			0
$629$		−	16.9706i		−	0.676661i
$630$		0			0
$631$			38.1051i			1.51694i	0.651707	+	0.758470i	$0.274053\pi$
$631$			38.1051i			1.51694i	−0.651707	+	0.758470i	$0.725947\pi$
$632$		0			0
$633$	24.2487	−	34.2929i	0.963800	−	1.36302i
$634$		0			0
$635$	−29.3939	−	24.0000i	−1.16646	−	0.952411i
$636$		0			0
$637$		−	34.2929i		−	1.35873i
$638$		0			0
$639$	9.79796	+	27.7128i	0.387601	+	1.09630i
$640$		0			0
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$		0			0
$643$	8.48528			0.334627			0.167313	−	0.985904i	$-0.446491\pi$
$643$	8.48528			0.334627			0.167313	+	0.985904i	$0.446491\pi$
$644$		0			0
$645$	−8.78461	−	31.6675i	−0.345894	−	1.24691i
$646$		0			0
$647$			42.0000i			1.65119i	0.564263	+	0.825595i	$0.309160\pi$
$647$			42.0000i			1.65119i	−0.564263	+	0.825595i	$0.690840\pi$
$648$		0			0
$649$	24.0000			0.942082
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−31.1769			−1.22005			−0.610023	−	0.792383i	$-0.708840\pi$
$653$	−31.1769			−1.22005			−0.610023	+	0.792383i	$0.708840\pi$
$654$		0			0
$655$	8.48528	+	6.92820i	0.331547	+	0.270707i
$656$		0			0
$657$	27.7128	−	9.79796i	1.08118	−	0.382255i
$658$		0			0
$659$	14.6969			0.572511			0.286256	−	0.958153i	$-0.407589\pi$
$659$	14.6969			0.572511			0.286256	+	0.958153i	$0.407589\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$	16.9706	−	24.0000i	0.659082	−	0.932083i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	16.9706			0.657103
$668$		0			0
$669$	24.0000	+	16.9706i	0.927894	+	0.656120i
$670$		0			0
$671$	9.79796			0.378246
$672$		0			0
$673$		−	39.1918i		−	1.51073i	−0.655302	−	0.755367i	$-0.727459\pi$
$673$		−	39.1918i		−	1.51073i	0.655302	−	0.755367i	$-0.272541\pi$
$674$		0			0
$675$	25.9091	+	1.92820i	0.997242	+	0.0742166i
$676$		0			0
$677$	10.3923			0.399409			0.199704	−	0.979856i	$-0.436002\pi$
$677$	10.3923			0.399409			0.199704	+	0.979856i	$0.436002\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	6.00000	−	8.48528i	0.229920	−	0.325157i
$682$		0			0
$683$			6.00000i			0.229584i	0.993390	+	0.114792i	$0.0366201\pi$
$683$			6.00000i			0.229584i	−0.993390	+	0.114792i	$0.963380\pi$
$684$		0			0
$685$	−6.00000	−	4.89898i	−0.229248	−	0.187180i
$686$		0			0
$687$	−14.1421	−	10.0000i	−0.539556	−	0.381524i
$688$		0			0
$689$			50.9117i			1.93958i
$690$		0			0
$691$		−	31.1769i		−	1.18603i	−0.805193	−	0.593013i	$-0.797938\pi$
$691$		−	31.1769i		−	1.18603i	0.805193	−	0.593013i	$-0.202062\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−24.4949	+	30.0000i	−0.929144	+	1.13796i
$696$		0			0
$697$		−	19.5959i		−	0.742248i
$698$		0			0
$699$	−4.89898	−	3.46410i	−0.185296	−	0.131024i
$700$		0			0
$701$		−	19.7990i		−	0.747798i	−0.927470	−	0.373899i	$-0.878021\pi$
$701$		−	19.7990i		−	0.747798i	0.927470	−	0.373899i	$-0.121979\pi$
$702$		0			0
$703$	−16.9706			−0.640057
$704$		0			0
$705$	22.3923	−	6.21166i	0.843343	−	0.233945i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−38.0000			−1.42712			−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000			−1.42712			−0.713560	+	0.700594i	$0.752918\pi$
$710$		0			0
$711$	−29.3939	+	10.3923i	−1.10236	+	0.389742i
$712$		0			0
$713$	−20.7846			−0.778390
$714$		0			0
$715$	33.9411	−	41.5692i	1.26933	−	1.55460i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−19.5959			−0.730804			−0.365402	−	0.930850i	$-0.619069\pi$
$719$	−19.5959			−0.730804			−0.365402	+	0.930850i	$0.619069\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−14.1421	−	10.0000i	−0.525952	−	0.371904i
$724$		0			0
$725$	−13.8564	+	2.82843i	−0.514614	+	0.105045i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	−23.0000	−	14.1421i	−0.851852	−	0.523783i
$730$		0			0
$731$	−29.3939			−1.08717
$732$		0			0
$733$			44.0908i			1.62853i	0.580492	+	0.814266i	$0.302860\pi$
$733$			44.0908i			1.62853i	−0.580492	+	0.814266i	$0.697140\pi$
$734$		0			0
$735$	−7.24693	−	26.1244i	−0.267307	−	0.963611i
$736$		0			0
$737$	41.5692			1.53122
$738$		0			0
$739$			24.2487i			0.892003i	0.895032	+	0.446002i	$0.147152\pi$
$739$			24.2487i			0.892003i	−0.895032	+	0.446002i	$0.852848\pi$
$740$		0			0
$741$	−24.0000	−	16.9706i	−0.881662	−	0.623429i
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$		0			0
$745$	−4.00000	+	4.89898i	−0.146549	+	0.179485i
$746$		0			0
$747$	−16.9706	+	6.00000i	−0.620920	+	0.219529i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	31.1769i		−	1.13766i	−0.822455	−	0.568831i	$-0.807396\pi$
$751$		−	31.1769i		−	1.13766i	0.822455	−	0.568831i	$-0.192604\pi$
$752$		0			0
$753$	20.7846	+	14.6969i	0.757433	+	0.535586i
$754$		0			0
$755$	−4.89898	+	6.00000i	−0.178292	+	0.218362i
$756$		0			0
$757$			14.6969i			0.534169i	0.963673	+	0.267085i	$0.0860603\pi$
$757$			14.6969i			0.534169i	−0.963673	+	0.267085i	$0.913940\pi$
$758$		0			0
$759$	−29.3939	+	41.5692i	−1.06693	+	1.50887i
$760$		0			0
$761$			39.5980i			1.43543i	0.696339	+	0.717713i	$0.254811\pi$
$761$			39.5980i			1.43543i	−0.696339	+	0.717713i	$0.745189\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	7.85641	−	21.8695i	0.284049	−	0.790695i
$766$		0			0
$767$			24.0000i			0.866590i
$768$		0			0
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	−4.89898	−	3.46410i	−0.176432	−	0.124757i
$772$		0			0
$773$	24.2487			0.872166			0.436083	−	0.899907i	$-0.356365\pi$
$773$	24.2487			0.872166			0.436083	+	0.899907i	$0.356365\pi$
$774$		0			0
$775$	16.9706	−	3.46410i	0.609601	−	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−19.5959			−0.702097
$780$		0			0
$781$	48.0000			1.71758
$782$		0			0
$783$	14.1421	+	4.00000i	0.505399	+	0.142948i
$784$		0			0
$785$	6.92820	−	8.48528i	0.247278	−	0.302853i
$786$		0			0
$787$	−25.4558			−0.907403			−0.453701	−	0.891154i	$-0.649897\pi$
$787$	−25.4558			−0.907403			−0.453701	+	0.891154i	$0.649897\pi$
$788$		0			0
$789$	−6.00000	+	8.48528i	−0.213606	+	0.302084i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			9.79796i			0.347936i
$794$		0			0
$795$	10.7589	+	38.7846i	0.381579	+	1.37555i
$796$		0			0
$797$	−17.3205			−0.613524			−0.306762	−	0.951786i	$-0.599246\pi$
$797$	−17.3205			−0.613524			−0.306762	+	0.951786i	$0.599246\pi$
$798$		0			0
$799$		−	20.7846i		−	0.735307i
$800$		0			0
$801$	−16.0000	+	5.65685i	−0.565332	+	0.199875i
$802$		0			0
$803$		−	48.0000i		−	1.69388i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−19.7990	+	28.0000i	−0.696957	+	0.985647i
$808$		0			0
$809$			5.65685i			0.198884i	0.995043	+	0.0994422i	$0.0317058\pi$
$809$			5.65685i			0.198884i	−0.995043	+	0.0994422i	$0.968294\pi$
$810$		0			0
$811$		−	17.3205i		−	0.608205i	−0.952639	−	0.304103i	$-0.901643\pi$
$811$		−	17.3205i		−	0.608205i	0.952639	−	0.304103i	$-0.0983566\pi$
$812$		0			0
$813$	10.3923	−	14.6969i	0.364474	−	0.515444i
$814$		0			0
$815$	−14.6969	−	12.0000i	−0.514811	−	0.420342i
$816$		0			0
$817$			29.3939i			1.02836i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	2.82843i		−	0.0987128i	−0.998781	−	0.0493564i	$-0.984283\pi$
$821$		−	2.82843i		−	0.0987128i	0.998781	−	0.0493564i	$-0.0157170\pi$
$822$		0			0
$823$	33.9411			1.18311			0.591557	−	0.806263i	$-0.298514\pi$
$823$	33.9411			1.18311			0.591557	+	0.806263i	$0.298514\pi$
$824$		0			0
$825$	17.0718	−	38.8401i	0.594364	−	1.35224i
$826$		0			0
$827$			54.0000i			1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$			54.0000i			1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$		0			0
$829$	−46.0000			−1.59765			−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000			−1.59765			−0.798823	+	0.601566i	$0.794544\pi$
$830$		0			0
$831$	14.6969	−	20.7846i	0.509831	−	0.721010i
$832$		0			0
$833$	−24.2487			−0.840168
$834$		0			0
$835$	−8.48528	+	10.3923i	−0.293645	+	0.359641i
$836$		0			0
$837$	−17.3205	−	4.89898i	−0.598684	−	0.169334i
$838$		0			0
$839$	−9.79796			−0.338263			−0.169132	−	0.985593i	$-0.554096\pi$
$839$	−9.79796			−0.338263			−0.169132	+	0.985593i	$0.554096\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$		0			0
$843$	−5.65685	+	8.00000i	−0.194832	+	0.275535i
$844$		0			0
$845$	19.0526	+	15.5563i	0.655428	+	0.535155i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	12.0000	+	8.48528i	0.411839	+	0.291214i
$850$		0			0
$851$	−29.3939			−1.00761
$852$		0			0
$853$		−	44.0908i		−	1.50964i	−0.655932	−	0.754820i	$-0.727724\pi$
$853$		−	44.0908i		−	1.50964i	0.655932	−	0.754820i	$-0.272276\pi$
$854$		0			0
$855$	−21.8695	−	7.85641i	−0.747923	−	0.268683i
$856$		0			0
$857$	−24.2487			−0.828320			−0.414160	−	0.910204i	$-0.635925\pi$
$857$	−24.2487			−0.828320			−0.414160	+	0.910204i	$0.635925\pi$
$858$		0			0
$859$			38.1051i			1.30013i	0.759879	+	0.650065i	$0.225258\pi$
$859$			38.1051i			1.30013i	−0.759879	+	0.650065i	$0.774742\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	6.00000i		−	0.204242i	−0.994772	−	0.102121i	$-0.967437\pi$
$863$		−	6.00000i		−	0.204242i	0.994772	−	0.102121i	$-0.0325630\pi$
$864$		0			0
$865$	6.00000	+	4.89898i	0.204006	+	0.166570i
$866$		0			0
$867$	7.07107	+	5.00000i	0.240146	+	0.169809i
$868$		0			0
$869$			50.9117i			1.72706i
$870$		0			0
$871$			41.5692i			1.40852i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			24.4949i			0.827134i	0.910474	+	0.413567i	$0.135717\pi$
$877$			24.4949i			0.827134i	−0.910474	+	0.413567i	$0.864283\pi$
$878$		0			0
$879$	24.4949	+	17.3205i	0.826192	+	0.584206i
$880$		0			0
$881$		−	22.6274i		−	0.762337i	−0.924506	−	0.381169i	$-0.875522\pi$
$881$		−	22.6274i		−	0.762337i	0.924506	−	0.381169i	$-0.124478\pi$
$882$		0			0
$883$	42.4264			1.42776			0.713881	−	0.700267i	$-0.246936\pi$
$883$	42.4264			1.42776			0.713881	+	0.700267i	$0.246936\pi$
$884$		0			0
$885$	5.07180	+	18.2832i	0.170487	+	0.614584i
$886$		0			0
$887$			42.0000i			1.41022i	0.709097	+	0.705111i	$0.249103\pi$
$887$			42.0000i			1.41022i	−0.709097	+	0.705111i	$0.750897\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−34.2929	+	27.7128i	−1.14885	+	0.928414i
$892$		0			0
$893$	−20.7846			−0.695530
$894$		0			0
$895$	42.4264	+	34.6410i	1.41816	+	1.15792i
$896$		0			0
$897$	−41.5692	−	29.3939i	−1.38796	−	0.981433i
$898$		0			0
$899$	9.79796			0.326780
$900$		0			0
$901$	36.0000			1.19933
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−17.3205	−	14.1421i	−0.575753	−	0.470100i
$906$		0			0
$907$	−8.48528			−0.281749			−0.140875	−	0.990027i	$-0.544991\pi$
$907$	−8.48528			−0.281749			−0.140875	+	0.990027i	$0.544991\pi$
$908$		0			0
$909$	40.0000	−	14.1421i	1.32672	−	0.469065i
$910$		0			0
$911$	19.5959			0.649242			0.324621	−	0.945844i	$-0.394763\pi$
$911$	19.5959			0.649242			0.324621	+	0.945844i	$0.394763\pi$
$912$		0			0
$913$			29.3939i			0.972795i
$914$		0			0
$915$	2.07055	+	7.46410i	0.0684503	+	0.246756i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		−	3.46410i		−	0.114270i	−0.998366	−	0.0571351i	$-0.981803\pi$
$919$		−	3.46410i		−	0.114270i	0.998366	−	0.0571351i	$-0.0181966\pi$
$920$		0			0
$921$	−12.0000	−	8.48528i	−0.395413	−	0.279600i
$922$		0			0
$923$			48.0000i			1.57994i
$924$		0			0
$925$	24.0000	−	4.89898i	0.789115	−	0.161077i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	22.6274i		−	0.742381i	−0.928557	−	0.371191i	$-0.878950\pi$
$929$		−	22.6274i		−	0.742381i	0.928557	−	0.371191i	$-0.121050\pi$
$930$		0			0
$931$			24.2487i			0.794719i
$932$		0			0
$933$	13.8564	+	9.79796i	0.453638	+	0.320771i
$934$		0			0
$935$	−29.3939	−	24.0000i	−0.961283	−	0.784884i
$936$		0			0
$937$			48.9898i			1.60043i	0.599715	+	0.800213i	$0.295280\pi$
$937$			48.9898i			1.60043i	−0.599715	+	0.800213i	$0.704720\pi$
$938$		0			0
$939$	9.79796	−	13.8564i	0.319744	−	0.452187i
$940$		0			0
$941$			2.82843i			0.0922041i	0.998937	+	0.0461020i	$0.0146799\pi$
$941$			2.82843i			0.0922041i	−0.998937	+	0.0461020i	$0.985320\pi$
$942$		0			0
$943$	−33.9411			−1.10528
$944$		0			0
$945$		0			0
$946$		0			0
$947$			54.0000i			1.75476i	0.479792	+	0.877382i	$0.340712\pi$
$947$			54.0000i			1.75476i	−0.479792	+	0.877382i	$0.659288\pi$
$948$		0			0
$949$	48.0000			1.55815
$950$		0			0
$951$	24.4949	+	17.3205i	0.794301	+	0.561656i
$952$		0			0
$953$	58.8897			1.90763			0.953813	−	0.300402i	$-0.0971208\pi$
$953$	58.8897			1.90763			0.953813	+	0.300402i	$0.0971208\pi$
$954$		0			0
$955$	33.9411	+	27.7128i	1.09831	+	0.896766i
$956$		0			0
$957$	13.8564	−	19.5959i	0.447914	−	0.633446i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$	50.9117	−	18.0000i	1.64061	−	0.580042i
$964$		0			0
$965$	27.7128	−	33.9411i	0.892107	−	1.09260i
$966$		0			0
$967$	−33.9411			−1.09147			−0.545737	−	0.837957i	$-0.683750\pi$
$967$	−33.9411			−1.09147			−0.545737	+	0.837957i	$0.683750\pi$
$968$		0			0
$969$	−12.0000	+	16.9706i	−0.385496	+	0.545173i
$970$		0			0
$971$	4.89898			0.157216			0.0786079	−	0.996906i	$-0.474952\pi$
$971$	4.89898			0.157216			0.0786079	+	0.996906i	$0.474952\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	38.8401	+	17.0718i	1.24388	+	0.546735i
$976$		0			0
$977$	−51.9615			−1.66240			−0.831198	−	0.555976i	$-0.812345\pi$
$977$	−51.9615			−1.66240			−0.831198	+	0.555976i	$0.812345\pi$
$978$		0			0
$979$			27.7128i			0.885705i
$980$		0			0
$981$	2.00000	+	5.65685i	0.0638551	+	0.180609i
$982$		0			0
$983$		−	6.00000i		−	0.191370i	−0.995412	−	0.0956851i	$-0.969496\pi$
$983$		−	6.00000i		−	0.191370i	0.995412	−	0.0956851i	$-0.0305042\pi$
$984$		0			0
$985$	−42.0000	−	34.2929i	−1.33823	−	1.09266i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			50.9117i			1.61890i
$990$		0			0
$991$			38.1051i			1.21045i	0.796055	+	0.605224i	$0.206917\pi$
$991$			38.1051i			1.21045i	−0.796055	+	0.605224i	$0.793083\pi$
$992$		0			0
$993$	−17.3205	+	24.4949i	−0.549650	+	0.777322i
$994$		0			0
$995$	14.6969	−	18.0000i	0.465924	−	0.570638i
$996$		0			0
$997$			53.8888i			1.70667i	0.521359	+	0.853337i	$0.325425\pi$
$997$			53.8888i			1.70667i	−0.521359	+	0.853337i	$0.674575\pi$
$998$		0			0
$999$	−24.4949	−	6.92820i	−0.774984	−	0.219199i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	960.2.o.d.959.1		8
3.2	odd	2	inner	960.2.o.d.959.4		8
4.3	odd	2	inner	960.2.o.d.959.7		8
5.4	even	2	inner	960.2.o.d.959.8		8
8.3	odd	2		240.2.o.b.239.2	yes	8
8.5	even	2		240.2.o.b.239.8	yes	8
12.11	even	2	inner	960.2.o.d.959.6		8
15.14	odd	2	inner	960.2.o.d.959.5		8
20.19	odd	2	inner	960.2.o.d.959.2		8
24.5	odd	2		240.2.o.b.239.5	yes	8
24.11	even	2		240.2.o.b.239.3	yes	8
40.3	even	4		1200.2.h.n.1151.2		4
40.13	odd	4		1200.2.h.j.1151.3		4
40.19	odd	2		240.2.o.b.239.7	yes	8
40.27	even	4		1200.2.h.j.1151.4		4
40.29	even	2		240.2.o.b.239.1	✓	8
40.37	odd	4		1200.2.h.n.1151.1		4
60.59	even	2	inner	960.2.o.d.959.3		8
120.29	odd	2		240.2.o.b.239.4	yes	8
120.53	even	4		1200.2.h.n.1151.4		4
120.59	even	2		240.2.o.b.239.6	yes	8
120.77	even	4		1200.2.h.j.1151.2		4
120.83	odd	4		1200.2.h.j.1151.1		4
120.107	odd	4		1200.2.h.n.1151.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.o.b.239.1	✓	8	40.29	even	2
240.2.o.b.239.2	yes	8	8.3	odd	2
240.2.o.b.239.3	yes	8	24.11	even	2
240.2.o.b.239.4	yes	8	120.29	odd	2
240.2.o.b.239.5	yes	8	24.5	odd	2
240.2.o.b.239.6	yes	8	120.59	even	2
240.2.o.b.239.7	yes	8	40.19	odd	2
240.2.o.b.239.8	yes	8	8.5	even	2
960.2.o.d.959.1		8	1.1	even	1	trivial
960.2.o.d.959.2		8	20.19	odd	2	inner
960.2.o.d.959.3		8	60.59	even	2	inner
960.2.o.d.959.4		8	3.2	odd	2	inner
960.2.o.d.959.5		8	15.14	odd	2	inner
960.2.o.d.959.6		8	12.11	even	2	inner
960.2.o.d.959.7		8	4.3	odd	2	inner
960.2.o.d.959.8		8	5.4	even	2	inner
1200.2.h.j.1151.1		4	120.83	odd	4
1200.2.h.j.1151.2		4	120.77	even	4
1200.2.h.j.1151.3		4	40.13	odd	4
1200.2.h.j.1151.4		4	40.27	even	4
1200.2.h.n.1151.1		4	40.37	odd	4
1200.2.h.n.1151.2		4	40.3	even	4
1200.2.h.n.1151.3		4	120.107	odd	4
1200.2.h.n.1151.4		4	120.53	even	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:	\( N \)	\(=\)	\( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	960.o (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 - 1.00000i) q^{3} +(-1.73205 - 1.41421i) q^{5} +(1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.41421 - 1.00000i) q^{3} +(-1.73205 - 1.41421i) q^{5} +(1.00000 + 2.82843i) q^{9} +4.89898 q^{11} +4.89898i q^{13} +(1.03528 + 3.73205i) q^{15} +3.46410 q^{17} -3.46410i q^{19} -6.00000i q^{23} +(1.00000 + 4.89898i) q^{25} +(1.41421 - 5.00000i) q^{27} +2.82843i q^{29} -3.46410i q^{31} +(-6.92820 - 4.89898i) q^{33} -4.89898i q^{37} +(4.89898 - 6.92820i) q^{39} -5.65685i q^{41} -8.48528 q^{43} +(2.26795 - 6.31319i) q^{45} -6.00000i q^{47} -7.00000 q^{49} +(-4.89898 - 3.46410i) q^{51} +10.3923 q^{53} +(-8.48528 - 6.92820i) q^{55} +(-3.46410 + 4.89898i) q^{57} +4.89898 q^{59} +2.00000 q^{61} +(6.92820 - 8.48528i) q^{65} +8.48528 q^{67} +(-6.00000 + 8.48528i) q^{69} +9.79796 q^{71} -9.79796i q^{73} +(3.48477 - 7.92820i) q^{75} +10.3923i q^{79} +(-7.00000 + 5.65685i) q^{81} +6.00000i q^{83} +(-6.00000 - 4.89898i) q^{85} +(2.82843 - 4.00000i) q^{87} +5.65685i q^{89} +(-3.46410 + 4.89898i) q^{93} +(-4.89898 + 6.00000i) q^{95} +(4.89898 + 13.8564i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 8 q^{25} + 32 q^{45} - 56 q^{49} + 16 q^{61} - 48 q^{69} - 56 q^{81} - 48 q^{85}+O(q^{100}) \) 8 * q + 8 * q^9 + 8 * q^25 + 32 * q^45 - 56 * q^49 + 16 * q^61 - 48 * q^69 - 56 * q^81 - 48 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

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