LMFDB - Embedded newform 672.2.j.a.239.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [672,2,Mod(239,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("672.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 672 = 2^{5} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	672.j (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:	$5.36594701583$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	672.239
Dual form			672.2.j.a.239.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.41421i) q^{3} +2.82843 q^{5} +1.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})$	$q+(-1.00000 - 1.41421i) q^{3} +2.82843 q^{5} +1.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +5.65685i q^{11} +4.00000i q^{13} +(-2.82843 - 4.00000i) q^{15} +2.82843i q^{17} +2.00000 q^{19} +(1.41421 - 1.00000i) q^{21} -2.82843 q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} -5.65685 q^{29} +(8.00000 - 5.65685i) q^{33} +2.82843i q^{35} -8.00000i q^{37} +(5.65685 - 4.00000i) q^{39} +2.82843i q^{41} +10.0000 q^{43} +(-2.82843 + 8.00000i) q^{45} -1.00000 q^{49} +(4.00000 - 2.82843i) q^{51} +5.65685 q^{53} +16.0000i q^{55} +(-2.00000 - 2.82843i) q^{57} +8.48528i q^{59} -12.0000i q^{61} +(-2.82843 - 1.00000i) q^{63} +11.3137i q^{65} -6.00000 q^{67} +(2.82843 + 4.00000i) q^{69} +8.48528 q^{71} +10.0000 q^{73} +(-3.00000 - 4.24264i) q^{75} -5.65685 q^{77} -8.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -2.82843i q^{83} +8.00000i q^{85} +(5.65685 + 8.00000i) q^{87} -2.82843i q^{89} -4.00000 q^{91} +5.65685 q^{95} -2.00000 q^{97} +(-16.0000 - 5.65685i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{9} + 8 q^{19} + 12 q^{25} + 20 q^{27} + 32 q^{33} + 40 q^{43} - 4 q^{49} + 16 q^{51} - 8 q^{57} - 24 q^{67} + 40 q^{73} - 12 q^{75} - 28 q^{81} - 16 q^{91} - 8 q^{97} - 64 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^9 + 8 * q^19 + 12 * q^25 + 20 * q^27 + 32 * q^33 + 40 * q^43 - 4 * q^49 + 16 * q^51 - 8 * q^57 - 24 * q^67 + 40 * q^73 - 12 * q^75 - 28 * q^81 - 16 * q^91 - 8 * q^97 - 64 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$.

$n$	$127$	$421$	$449$	$577$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$4$		0			0
$5$	2.82843			1.26491			0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843			1.26491			0.632456	+	0.774597i	$0.282047\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$			5.65685i			1.70561i	0.522233	+	0.852803i	$0.325099\pi$
$11$			5.65685i			1.70561i	−0.522233	+	0.852803i	$0.674901\pi$
$12$		0			0
$13$			4.00000i			1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$			4.00000i			1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$		0			0
$15$	−2.82843	−	4.00000i	−0.730297	−	1.03280i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$	1.41421	−	1.00000i	0.308607	−	0.218218i
$22$		0			0
$23$	−2.82843			−0.589768			−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843			−0.589768			−0.294884	+	0.955533i	$0.595281\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$	−5.65685			−1.05045			−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685			−1.05045			−0.525226	+	0.850963i	$0.676019\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	8.00000	−	5.65685i	1.39262	−	0.984732i
$34$		0			0
$35$			2.82843i			0.478091i
$36$		0			0
$37$		−	8.00000i		−	1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$		−	8.00000i		−	1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$		0			0
$39$	5.65685	−	4.00000i	0.905822	−	0.640513i
$40$		0			0
$41$			2.82843i			0.441726i	0.975305	+	0.220863i	$0.0708874\pi$
$41$			2.82843i			0.441726i	−0.975305	+	0.220863i	$0.929113\pi$
$42$		0			0
$43$	10.0000			1.52499			0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000			1.52499			0.762493	+	0.646997i	$0.223975\pi$
$44$		0			0
$45$	−2.82843	+	8.00000i	−0.421637	+	1.19257i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	4.00000	−	2.82843i	0.560112	−	0.396059i
$52$		0			0
$53$	5.65685			0.777029			0.388514	−	0.921443i	$-0.372988\pi$
$53$	5.65685			0.777029			0.388514	+	0.921443i	$0.372988\pi$
$54$		0			0
$55$			16.0000i			2.15744i
$56$		0			0
$57$	−2.00000	−	2.82843i	−0.264906	−	0.374634i
$58$		0			0
$59$			8.48528i			1.10469i	0.833616	+	0.552345i	$0.186267\pi$
$59$			8.48528i			1.10469i	−0.833616	+	0.552345i	$0.813733\pi$
$60$		0			0
$61$		−	12.0000i		−	1.53644i	−0.640184	−	0.768221i	$-0.721142\pi$
$61$		−	12.0000i		−	1.53644i	0.640184	−	0.768221i	$-0.278858\pi$
$62$		0			0
$63$	−2.82843	−	1.00000i	−0.356348	−	0.125988i
$64$		0			0
$65$			11.3137i			1.40329i
$66$		0			0
$67$	−6.00000			−0.733017			−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000			−0.733017			−0.366508	+	0.930415i	$0.619447\pi$
$68$		0			0
$69$	2.82843	+	4.00000i	0.340503	+	0.481543i
$70$		0			0
$71$	8.48528			1.00702			0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528			1.00702			0.503509	+	0.863990i	$0.332042\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$		0			0
$75$	−3.00000	−	4.24264i	−0.346410	−	0.489898i
$76$		0			0
$77$	−5.65685			−0.644658
$78$		0			0
$79$		−	8.00000i		−	0.900070i	−0.893011	−	0.450035i	$-0.851411\pi$
$79$		−	8.00000i		−	0.900070i	0.893011	−	0.450035i	$-0.148589\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$		−	2.82843i		−	0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$		−	2.82843i		−	0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$		0			0
$85$			8.00000i			0.867722i
$86$		0			0
$87$	5.65685	+	8.00000i	0.606478	+	0.857690i
$88$		0			0
$89$		−	2.82843i		−	0.299813i	−0.988700	−	0.149906i	$-0.952103\pi$
$89$		−	2.82843i		−	0.299813i	0.988700	−	0.149906i	$-0.0478972\pi$
$90$		0			0
$91$	−4.00000			−0.419314
$92$		0			0
$93$		0			0
$94$		0			0
$95$	5.65685			0.580381
$96$		0			0
$97$	−2.00000			−0.203069			−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000			−0.203069			−0.101535	+	0.994832i	$0.532375\pi$
$98$		0			0
$99$	−16.0000	−	5.65685i	−1.60806	−	0.568535i
$100$		0			0
$101$	−2.82843			−0.281439			−0.140720	−	0.990050i	$-0.544942\pi$
$101$	−2.82843			−0.281439			−0.140720	+	0.990050i	$0.544942\pi$
$102$		0			0
$103$		−	8.00000i		−	0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$		−	8.00000i		−	0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$		0			0
$105$	4.00000	−	2.82843i	0.390360	−	0.276026i
$106$		0			0
$107$		−	11.3137i		−	1.09374i	−0.837218	−	0.546869i	$-0.815820\pi$
$107$		−	11.3137i		−	1.09374i	0.837218	−	0.546869i	$-0.184180\pi$
$108$		0			0
$109$		−	8.00000i		−	0.766261i	−0.923694	−	0.383131i	$-0.874846\pi$
$109$		−	8.00000i		−	0.766261i	0.923694	−	0.383131i	$-0.125154\pi$
$110$		0			0
$111$	−11.3137	+	8.00000i	−1.07385	+	0.759326i
$112$		0			0
$113$			5.65685i			0.532152i	0.963952	+	0.266076i	$0.0857272\pi$
$113$			5.65685i			0.532152i	−0.963952	+	0.266076i	$0.914273\pi$
$114$		0			0
$115$	−8.00000			−0.746004
$116$		0			0
$117$	−11.3137	−	4.00000i	−1.04595	−	0.369800i
$118$		0			0
$119$	−2.82843			−0.259281
$120$		0			0
$121$	−21.0000			−1.90909
$122$		0			0
$123$	4.00000	−	2.82843i	0.360668	−	0.255031i
$124$		0			0
$125$	−5.65685			−0.505964
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−10.0000	−	14.1421i	−0.880451	−	1.24515i
$130$		0			0
$131$			14.1421i			1.23560i	0.786334	+	0.617802i	$0.211977\pi$
$131$			14.1421i			1.23560i	−0.786334	+	0.617802i	$0.788023\pi$
$132$		0			0
$133$			2.00000i			0.173422i
$134$		0			0
$135$	14.1421	−	4.00000i	1.21716	−	0.344265i
$136$		0			0
$137$			5.65685i			0.483298i	0.970364	+	0.241649i	$0.0776882\pi$
$137$			5.65685i			0.483298i	−0.970364	+	0.241649i	$0.922312\pi$
$138$		0			0
$139$	−14.0000			−1.18746			−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000			−1.18746			−0.593732	+	0.804663i	$0.702346\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−22.6274			−1.89220
$144$		0			0
$145$	−16.0000			−1.32873
$146$		0			0
$147$	1.00000	+	1.41421i	0.0824786	+	0.116642i
$148$		0			0
$149$	5.65685			0.463428			0.231714	−	0.972784i	$-0.425567\pi$
$149$	5.65685			0.463428			0.231714	+	0.972784i	$0.425567\pi$
$150$		0			0
$151$		−	16.0000i		−	1.30206i	−0.759051	−	0.651031i	$-0.774337\pi$
$151$		−	16.0000i		−	1.30206i	0.759051	−	0.651031i	$-0.225663\pi$
$152$		0			0
$153$	−8.00000	−	2.82843i	−0.646762	−	0.228665i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			12.0000i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$			12.0000i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$		0			0
$159$	−5.65685	−	8.00000i	−0.448618	−	0.634441i
$160$		0			0
$161$		−	2.82843i		−	0.222911i
$162$		0			0
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$		0			0
$165$	22.6274	−	16.0000i	1.76154	−	1.24560i
$166$		0			0
$167$	5.65685			0.437741			0.218870	−	0.975754i	$-0.429763\pi$
$167$	5.65685			0.437741			0.218870	+	0.975754i	$0.429763\pi$
$168$		0			0
$169$	−3.00000			−0.230769
$170$		0			0
$171$	−2.00000	+	5.65685i	−0.152944	+	0.432590i
$172$		0			0
$173$	25.4558			1.93537			0.967686	−	0.252158i	$-0.0811404\pi$
$173$	25.4558			1.93537			0.967686	+	0.252158i	$0.0811404\pi$
$174$		0			0
$175$			3.00000i			0.226779i
$176$		0			0
$177$	12.0000	−	8.48528i	0.901975	−	0.637793i
$178$		0			0
$179$		−	11.3137i		−	0.845626i	−0.906217	−	0.422813i	$-0.861043\pi$
$179$		−	11.3137i		−	0.845626i	0.906217	−	0.422813i	$-0.138957\pi$
$180$		0			0
$181$			20.0000i			1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$			20.0000i			1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$		0			0
$183$	−16.9706	+	12.0000i	−1.25450	+	0.887066i
$184$		0			0
$185$		−	22.6274i		−	1.66360i
$186$		0			0
$187$	−16.0000			−1.17004
$188$		0			0
$189$	1.41421	+	5.00000i	0.102869	+	0.363696i
$190$		0			0
$191$	−8.48528			−0.613973			−0.306987	−	0.951714i	$-0.599321\pi$
$191$	−8.48528			−0.613973			−0.306987	+	0.951714i	$0.599321\pi$
$192$		0			0
$193$	18.0000			1.29567			0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000			1.29567			0.647834	+	0.761781i	$0.275675\pi$
$194$		0			0
$195$	16.0000	−	11.3137i	1.14578	−	0.810191i
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$			16.0000i			1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$			16.0000i			1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$		0			0
$201$	6.00000	+	8.48528i	0.423207	+	0.598506i
$202$		0			0
$203$		−	5.65685i		−	0.397033i
$204$		0			0
$205$			8.00000i			0.558744i
$206$		0			0
$207$	2.82843	−	8.00000i	0.196589	−	0.556038i
$208$		0			0
$209$			11.3137i			0.782586i
$210$		0			0
$211$	10.0000			0.688428			0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000			0.688428			0.344214	+	0.938891i	$0.388145\pi$
$212$		0			0
$213$	−8.48528	−	12.0000i	−0.581402	−	0.822226i
$214$		0			0
$215$	28.2843			1.92897
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−10.0000	−	14.1421i	−0.675737	−	0.955637i
$220$		0			0
$221$	−11.3137			−0.761042
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$	−3.00000	+	8.48528i	−0.200000	+	0.565685i
$226$		0			0
$227$		−	2.82843i		−	0.187729i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$		−	2.82843i		−	0.187729i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$		0			0
$229$			20.0000i			1.32164i	0.750546	+	0.660819i	$0.229791\pi$
$229$			20.0000i			1.32164i	−0.750546	+	0.660819i	$0.770209\pi$
$230$		0			0
$231$	5.65685	+	8.00000i	0.372194	+	0.526361i
$232$		0			0
$233$		−	5.65685i		−	0.370593i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$		−	5.65685i		−	0.370593i	0.982683	−	0.185296i	$-0.0593245\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−11.3137	+	8.00000i	−0.734904	+	0.519656i
$238$		0			0
$239$	−19.7990			−1.28069			−0.640345	−	0.768087i	$-0.721209\pi$
$239$	−19.7990			−1.28069			−0.640345	+	0.768087i	$0.721209\pi$
$240$		0			0
$241$	2.00000			0.128831			0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000			0.128831			0.0644157	+	0.997923i	$0.479482\pi$
$242$		0			0
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$		0			0
$245$	−2.82843			−0.180702
$246$		0			0
$247$			8.00000i			0.509028i
$248$		0			0
$249$	−4.00000	+	2.82843i	−0.253490	+	0.179244i
$250$		0			0
$251$			8.48528i			0.535586i	0.963476	+	0.267793i	$0.0862944\pi$
$251$			8.48528i			0.535586i	−0.963476	+	0.267793i	$0.913706\pi$
$252$		0			0
$253$		−	16.0000i		−	1.00591i
$254$		0			0
$255$	11.3137	−	8.00000i	0.708492	−	0.500979i
$256$		0			0
$257$			2.82843i			0.176432i	0.996101	+	0.0882162i	$0.0281166\pi$
$257$			2.82843i			0.176432i	−0.996101	+	0.0882162i	$0.971883\pi$
$258$		0			0
$259$	8.00000			0.497096
$260$		0			0
$261$	5.65685	−	16.0000i	0.350150	−	0.990375i
$262$		0			0
$263$	19.7990			1.22086			0.610429	−	0.792071i	$-0.290997\pi$
$263$	19.7990			1.22086			0.610429	+	0.792071i	$0.290997\pi$
$264$		0			0
$265$	16.0000			0.982872
$266$		0			0
$267$	−4.00000	+	2.82843i	−0.244796	+	0.173097i
$268$		0			0
$269$	2.82843			0.172452			0.0862261	−	0.996276i	$-0.472519\pi$
$269$	2.82843			0.172452			0.0862261	+	0.996276i	$0.472519\pi$
$270$		0			0
$271$		−	8.00000i		−	0.485965i	−0.970031	−	0.242983i	$-0.921874\pi$
$271$		−	8.00000i		−	0.485965i	0.970031	−	0.242983i	$-0.0781258\pi$
$272$		0			0
$273$	4.00000	+	5.65685i	0.242091	+	0.342368i
$274$		0			0
$275$			16.9706i			1.02336i
$276$		0			0
$277$		−	32.0000i		−	1.92269i	−0.275340	−	0.961347i	$-0.588791\pi$
$277$		−	32.0000i		−	1.92269i	0.275340	−	0.961347i	$-0.411209\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5.65685i			0.337460i	0.985662	+	0.168730i	$0.0539665\pi$
$281$			5.65685i			0.337460i	−0.985662	+	0.168730i	$0.946033\pi$
$282$		0			0
$283$	18.0000			1.06999			0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000			1.06999			0.534994	+	0.844856i	$0.320314\pi$
$284$		0			0
$285$	−5.65685	−	8.00000i	−0.335083	−	0.473879i
$286$		0			0
$287$	−2.82843			−0.166957
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	2.00000	+	2.82843i	0.117242	+	0.165805i
$292$		0			0
$293$	−19.7990			−1.15667			−0.578335	−	0.815800i	$-0.696297\pi$
$293$	−19.7990			−1.15667			−0.578335	+	0.815800i	$0.696297\pi$
$294$		0			0
$295$			24.0000i			1.39733i
$296$		0			0
$297$	8.00000	+	28.2843i	0.464207	+	1.64122i
$298$		0			0
$299$		−	11.3137i		−	0.654289i
$300$		0			0
$301$			10.0000i			0.576390i
$302$		0			0
$303$	2.82843	+	4.00000i	0.162489	+	0.229794i
$304$		0			0
$305$		−	33.9411i		−	1.94346i
$306$		0			0
$307$	−18.0000			−1.02731			−0.513657	−	0.857996i	$-0.671710\pi$
$307$	−18.0000			−1.02731			−0.513657	+	0.857996i	$0.671710\pi$
$308$		0			0
$309$	−11.3137	+	8.00000i	−0.643614	+	0.455104i
$310$		0			0
$311$	−11.3137			−0.641542			−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137			−0.641542			−0.320771	+	0.947157i	$0.603942\pi$
$312$		0			0
$313$	6.00000			0.339140			0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000			0.339140			0.169570	+	0.985518i	$0.445762\pi$
$314$		0			0
$315$	−8.00000	−	2.82843i	−0.450749	−	0.159364i
$316$		0			0
$317$	−16.9706			−0.953162			−0.476581	−	0.879131i	$-0.658124\pi$
$317$	−16.9706			−0.953162			−0.476581	+	0.879131i	$0.658124\pi$
$318$		0			0
$319$		−	32.0000i		−	1.79166i
$320$		0			0
$321$	−16.0000	+	11.3137i	−0.893033	+	0.631470i
$322$		0			0
$323$			5.65685i			0.314756i
$324$		0			0
$325$			12.0000i			0.665640i
$326$		0			0
$327$	−11.3137	+	8.00000i	−0.625650	+	0.442401i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	6.00000			0.329790			0.164895	−	0.986311i	$-0.447272\pi$
$331$	6.00000			0.329790			0.164895	+	0.986311i	$0.447272\pi$
$332$		0			0
$333$	22.6274	+	8.00000i	1.23997	+	0.438397i
$334$		0			0
$335$	−16.9706			−0.927201
$336$		0			0
$337$	−2.00000			−0.108947			−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000			−0.108947			−0.0544735	+	0.998515i	$0.517348\pi$
$338$		0			0
$339$	8.00000	−	5.65685i	0.434500	−	0.307238i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	8.00000	+	11.3137i	0.430706	+	0.609110i
$346$		0			0
$347$			16.9706i			0.911028i	0.890229	+	0.455514i	$0.150544\pi$
$347$			16.9706i			0.911028i	−0.890229	+	0.455514i	$0.849456\pi$
$348$		0			0
$349$		−	28.0000i		−	1.49881i	−0.662114	−	0.749403i	$-0.730341\pi$
$349$		−	28.0000i		−	1.49881i	0.662114	−	0.749403i	$-0.269659\pi$
$350$		0			0
$351$	5.65685	+	20.0000i	0.301941	+	1.06752i
$352$		0			0
$353$			36.7696i			1.95705i	0.206138	+	0.978523i	$0.433910\pi$
$353$			36.7696i			1.95705i	−0.206138	+	0.978523i	$0.566090\pi$
$354$		0			0
$355$	24.0000			1.27379
$356$		0			0
$357$	2.82843	+	4.00000i	0.149696	+	0.211702i
$358$		0			0
$359$	2.82843			0.149279			0.0746393	−	0.997211i	$-0.476219\pi$
$359$	2.82843			0.149279			0.0746393	+	0.997211i	$0.476219\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	21.0000	+	29.6985i	1.10221	+	1.55877i
$364$		0			0
$365$	28.2843			1.48047
$366$		0			0
$367$			16.0000i			0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$			16.0000i			0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$		0			0
$369$	−8.00000	−	2.82843i	−0.416463	−	0.147242i
$370$		0			0
$371$			5.65685i			0.293689i
$372$		0			0
$373$			16.0000i			0.828449i	0.910175	+	0.414224i	$0.135947\pi$
$373$			16.0000i			0.828449i	−0.910175	+	0.414224i	$0.864053\pi$
$374$		0			0
$375$	5.65685	+	8.00000i	0.292119	+	0.413118i
$376$		0			0
$377$		−	22.6274i		−	1.16537i
$378$		0			0
$379$	−6.00000			−0.308199			−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000			−0.308199			−0.154100	+	0.988055i	$0.549248\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	28.2843			1.44526			0.722629	−	0.691236i	$-0.242933\pi$
$383$	28.2843			1.44526			0.722629	+	0.691236i	$0.242933\pi$
$384$		0			0
$385$	−16.0000			−0.815436
$386$		0			0
$387$	−10.0000	+	28.2843i	−0.508329	+	1.43777i
$388$		0			0
$389$	−5.65685			−0.286814			−0.143407	−	0.989664i	$-0.545806\pi$
$389$	−5.65685			−0.286814			−0.143407	+	0.989664i	$0.545806\pi$
$390$		0			0
$391$		−	8.00000i		−	0.404577i
$392$		0			0
$393$	20.0000	−	14.1421i	1.00887	−	0.713376i
$394$		0			0
$395$		−	22.6274i		−	1.13851i
$396$		0			0
$397$		−	28.0000i		−	1.40528i	−0.711546	−	0.702640i	$-0.752005\pi$
$397$		−	28.0000i		−	1.40528i	0.711546	−	0.702640i	$-0.247995\pi$
$398$		0			0
$399$	2.82843	−	2.00000i	0.141598	−	0.100125i
$400$		0			0
$401$		−	11.3137i		−	0.564980i	−0.959270	−	0.282490i	$-0.908840\pi$
$401$		−	11.3137i		−	0.564980i	0.959270	−	0.282490i	$-0.0911603\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−19.7990	−	16.0000i	−0.983820	−	0.795046i
$406$		0			0
$407$	45.2548			2.24320
$408$		0			0
$409$	−6.00000			−0.296681			−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000			−0.296681			−0.148340	+	0.988936i	$0.547393\pi$
$410$		0			0
$411$	8.00000	−	5.65685i	0.394611	−	0.279032i
$412$		0			0
$413$	−8.48528			−0.417533
$414$		0			0
$415$		−	8.00000i		−	0.392705i
$416$		0			0
$417$	14.0000	+	19.7990i	0.685583	+	0.969561i
$418$		0			0
$419$		−	19.7990i		−	0.967244i	−0.875277	−	0.483622i	$-0.839321\pi$
$419$		−	19.7990i		−	0.967244i	0.875277	−	0.483622i	$-0.160679\pi$
$420$		0			0
$421$			32.0000i			1.55958i	0.626038	+	0.779792i	$0.284675\pi$
$421$			32.0000i			1.55958i	−0.626038	+	0.779792i	$0.715325\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			8.48528i			0.411597i
$426$		0			0
$427$	12.0000			0.580721
$428$		0			0
$429$	22.6274	+	32.0000i	1.09246	+	1.54497i
$430$		0			0
$431$	−31.1127			−1.49865			−0.749323	−	0.662205i	$-0.769621\pi$
$431$	−31.1127			−1.49865			−0.749323	+	0.662205i	$0.769621\pi$
$432$		0			0
$433$	30.0000			1.44171			0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000			1.44171			0.720854	+	0.693087i	$0.243750\pi$
$434$		0			0
$435$	16.0000	+	22.6274i	0.767141	+	1.08490i
$436$		0			0
$437$	−5.65685			−0.270604
$438$		0			0
$439$			8.00000i			0.381819i	0.981608	+	0.190910i	$0.0611437\pi$
$439$			8.00000i			0.381819i	−0.981608	+	0.190910i	$0.938856\pi$
$440$		0			0
$441$	1.00000	−	2.82843i	0.0476190	−	0.134687i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		−	8.00000i		−	0.379236i
$446$		0			0
$447$	−5.65685	−	8.00000i	−0.267560	−	0.378387i
$448$		0			0
$449$		−	33.9411i		−	1.60178i	−0.598811	−	0.800890i	$-0.704360\pi$
$449$		−	33.9411i		−	1.60178i	0.598811	−	0.800890i	$-0.295640\pi$
$450$		0			0
$451$	−16.0000			−0.753411
$452$		0			0
$453$	−22.6274	+	16.0000i	−1.06313	+	0.751746i
$454$		0			0
$455$	−11.3137			−0.530395
$456$		0			0
$457$	10.0000			0.467780			0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000			0.467780			0.233890	+	0.972263i	$0.424854\pi$
$458$		0			0
$459$	4.00000	+	14.1421i	0.186704	+	0.660098i
$460$		0			0
$461$	36.7696			1.71253			0.856264	−	0.516538i	$-0.172779\pi$
$461$	36.7696			1.71253			0.856264	+	0.516538i	$0.172779\pi$
$462$		0			0
$463$		−	8.00000i		−	0.371792i	−0.982569	−	0.185896i	$-0.940481\pi$
$463$		−	8.00000i		−	0.371792i	0.982569	−	0.185896i	$-0.0595187\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	36.7696i		−	1.70149i	−0.525577	−	0.850746i	$-0.676151\pi$
$467$		−	36.7696i		−	1.70149i	0.525577	−	0.850746i	$-0.323849\pi$
$468$		0			0
$469$		−	6.00000i		−	0.277054i
$470$		0			0
$471$	16.9706	−	12.0000i	0.781962	−	0.552931i
$472$		0			0
$473$			56.5685i			2.60102i
$474$		0			0
$475$	6.00000			0.275299
$476$		0			0
$477$	−5.65685	+	16.0000i	−0.259010	+	0.732590i
$478$		0			0
$479$	−11.3137			−0.516937			−0.258468	−	0.966020i	$-0.583218\pi$
$479$	−11.3137			−0.516937			−0.258468	+	0.966020i	$0.583218\pi$
$480$		0			0
$481$	32.0000			1.45907
$482$		0			0
$483$	−4.00000	+	2.82843i	−0.182006	+	0.128698i
$484$		0			0
$485$	−5.65685			−0.256865
$486$		0			0
$487$			16.0000i			0.725029i	0.931978	+	0.362515i	$0.118082\pi$
$487$			16.0000i			0.725029i	−0.931978	+	0.362515i	$0.881918\pi$
$488$		0			0
$489$	−14.0000	−	19.7990i	−0.633102	−	0.895341i
$490$		0			0
$491$		−	22.6274i		−	1.02116i	−0.859830	−	0.510581i	$-0.829431\pi$
$491$		−	22.6274i		−	1.02116i	0.859830	−	0.510581i	$-0.170569\pi$
$492$		0			0
$493$		−	16.0000i		−	0.720604i
$494$		0			0
$495$	−45.2548	−	16.0000i	−2.03405	−	0.719147i
$496$		0			0
$497$			8.48528i			0.380617i
$498$		0			0
$499$	−22.0000			−0.984855			−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000			−0.984855			−0.492428	+	0.870353i	$0.663890\pi$
$500$		0			0
$501$	−5.65685	−	8.00000i	−0.252730	−	0.357414i
$502$		0			0
$503$	−16.9706			−0.756680			−0.378340	−	0.925667i	$-0.623505\pi$
$503$	−16.9706			−0.756680			−0.378340	+	0.925667i	$0.623505\pi$
$504$		0			0
$505$	−8.00000			−0.355995
$506$		0			0
$507$	3.00000	+	4.24264i	0.133235	+	0.188422i
$508$		0			0
$509$	31.1127			1.37905			0.689523	−	0.724264i	$-0.257820\pi$
$509$	31.1127			1.37905			0.689523	+	0.724264i	$0.257820\pi$
$510$		0			0
$511$			10.0000i			0.442374i
$512$		0			0
$513$	10.0000	−	2.82843i	0.441511	−	0.124878i
$514$		0			0
$515$		−	22.6274i		−	0.997083i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−25.4558	−	36.0000i	−1.11739	−	1.58022i
$520$		0			0
$521$		−	14.1421i		−	0.619578i	−0.950805	−	0.309789i	$-0.899742\pi$
$521$		−	14.1421i		−	0.619578i	0.950805	−	0.309789i	$-0.100258\pi$
$522$		0			0
$523$	22.0000			0.961993			0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000			0.961993			0.480996	+	0.876723i	$0.340275\pi$
$524$		0			0
$525$	4.24264	−	3.00000i	0.185164	−	0.130931i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−15.0000			−0.652174
$530$		0			0
$531$	−24.0000	−	8.48528i	−1.04151	−	0.368230i
$532$		0			0
$533$	−11.3137			−0.490051
$534$		0			0
$535$		−	32.0000i		−	1.38348i
$536$		0			0
$537$	−16.0000	+	11.3137i	−0.690451	+	0.488223i
$538$		0			0
$539$		−	5.65685i		−	0.243658i
$540$		0			0
$541$		−	32.0000i		−	1.37579i	−0.725811	−	0.687894i	$-0.758536\pi$
$541$		−	32.0000i		−	1.37579i	0.725811	−	0.687894i	$-0.241464\pi$
$542$		0			0
$543$	28.2843	−	20.0000i	1.21379	−	0.858282i
$544$		0			0
$545$		−	22.6274i		−	0.969252i
$546$		0			0
$547$	−2.00000			−0.0855138			−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000			−0.0855138			−0.0427569	+	0.999086i	$0.513614\pi$
$548$		0			0
$549$	33.9411	+	12.0000i	1.44857	+	0.512148i
$550$		0			0
$551$	−11.3137			−0.481980
$552$		0			0
$553$	8.00000			0.340195
$554$		0			0
$555$	−32.0000	+	22.6274i	−1.35832	+	0.960480i
$556$		0			0
$557$	−5.65685			−0.239689			−0.119844	−	0.992793i	$-0.538240\pi$
$557$	−5.65685			−0.239689			−0.119844	+	0.992793i	$0.538240\pi$
$558$		0			0
$559$			40.0000i			1.69182i
$560$		0			0
$561$	16.0000	+	22.6274i	0.675521	+	0.955330i
$562$		0			0
$563$		−	8.48528i		−	0.357612i	−0.983884	−	0.178806i	$-0.942777\pi$
$563$		−	8.48528i		−	0.357612i	0.983884	−	0.178806i	$-0.0572234\pi$
$564$		0			0
$565$			16.0000i			0.673125i
$566$		0			0
$567$	5.65685	−	7.00000i	0.237566	−	0.293972i
$568$		0			0
$569$		−	22.6274i		−	0.948591i	−0.880366	−	0.474295i	$-0.842703\pi$
$569$		−	22.6274i		−	0.948591i	0.880366	−	0.474295i	$-0.157297\pi$
$570$		0			0
$571$	6.00000			0.251092			0.125546	−	0.992088i	$-0.459932\pi$
$571$	6.00000			0.251092			0.125546	+	0.992088i	$0.459932\pi$
$572$		0			0
$573$	8.48528	+	12.0000i	0.354478	+	0.501307i
$574$		0			0
$575$	−8.48528			−0.353861
$576$		0			0
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		0			0
$579$	−18.0000	−	25.4558i	−0.748054	−	1.05791i
$580$		0			0
$581$	2.82843			0.117343
$582$		0			0
$583$			32.0000i			1.32530i
$584$		0			0
$585$	−32.0000	−	11.3137i	−1.32304	−	0.467764i
$586$		0			0
$587$			36.7696i			1.51764i	0.651299	+	0.758821i	$0.274224\pi$
$587$			36.7696i			1.51764i	−0.651299	+	0.758821i	$0.725776\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	31.1127i		−	1.27765i	−0.769354	−	0.638823i	$-0.779422\pi$
$593$		−	31.1127i		−	1.27765i	0.769354	−	0.638823i	$-0.220578\pi$
$594$		0			0
$595$	−8.00000			−0.327968
$596$		0			0
$597$	22.6274	−	16.0000i	0.926079	−	0.654836i
$598$		0			0
$599$	−25.4558			−1.04010			−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558			−1.04010			−0.520049	+	0.854137i	$0.674086\pi$
$600$		0			0
$601$	26.0000			1.06056			0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000			1.06056			0.530281	+	0.847822i	$0.322086\pi$
$602$		0			0
$603$	6.00000	−	16.9706i	0.244339	−	0.691095i
$604$		0			0
$605$	−59.3970			−2.41483
$606$		0			0
$607$			16.0000i			0.649420i	0.945814	+	0.324710i	$0.105267\pi$
$607$			16.0000i			0.649420i	−0.945814	+	0.324710i	$0.894733\pi$
$608$		0			0
$609$	−8.00000	+	5.65685i	−0.324176	+	0.229227i
$610$		0			0
$611$		0			0
$612$		0			0
$613$			40.0000i			1.61558i	0.589467	+	0.807792i	$0.299338\pi$
$613$			40.0000i			1.61558i	−0.589467	+	0.807792i	$0.700662\pi$
$614$		0			0
$615$	11.3137	−	8.00000i	0.456213	−	0.322591i
$616$		0			0
$617$		−	11.3137i		−	0.455473i	−0.973723	−	0.227736i	$-0.926868\pi$
$617$		−	11.3137i		−	0.455473i	0.973723	−	0.227736i	$-0.0731324\pi$
$618$		0			0
$619$	−22.0000			−0.884255			−0.442127	−	0.896952i	$-0.645776\pi$
$619$	−22.0000			−0.884255			−0.442127	+	0.896952i	$0.645776\pi$
$620$		0			0
$621$	−14.1421	+	4.00000i	−0.567504	+	0.160514i
$622$		0			0
$623$	2.82843			0.113319
$624$		0			0
$625$	−31.0000			−1.24000
$626$		0			0
$627$	16.0000	−	11.3137i	0.638978	−	0.451826i
$628$		0			0
$629$	22.6274			0.902214
$630$		0			0
$631$		−	24.0000i		−	0.955425i	−0.878516	−	0.477712i	$-0.841466\pi$
$631$		−	24.0000i		−	0.955425i	0.878516	−	0.477712i	$-0.158534\pi$
$632$		0			0
$633$	−10.0000	−	14.1421i	−0.397464	−	0.562099i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		−	4.00000i		−	0.158486i
$638$		0			0
$639$	−8.48528	+	24.0000i	−0.335673	+	0.949425i
$640$		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
$643$	−22.0000			−0.867595			−0.433798	−	0.901010i	$-0.642827\pi$
$643$	−22.0000			−0.867595			−0.433798	+	0.901010i	$0.642827\pi$
$644$		0			0
$645$	−28.2843	−	40.0000i	−1.11369	−	1.57500i
$646$		0			0
$647$	39.5980			1.55676			0.778379	−	0.627795i	$-0.216042\pi$
$647$	39.5980			1.55676			0.778379	+	0.627795i	$0.216042\pi$
$648$		0			0
$649$	−48.0000			−1.88416
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−11.3137			−0.442740			−0.221370	−	0.975190i	$-0.571053\pi$
$653$	−11.3137			−0.442740			−0.221370	+	0.975190i	$0.571053\pi$
$654$		0			0
$655$			40.0000i			1.56293i
$656$		0			0
$657$	−10.0000	+	28.2843i	−0.390137	+	1.10347i
$658$		0			0
$659$			28.2843i			1.10180i	0.834572	+	0.550899i	$0.185715\pi$
$659$			28.2843i			1.10180i	−0.834572	+	0.550899i	$0.814285\pi$
$660$		0			0
$661$		−	12.0000i		−	0.466746i	−0.972387	−	0.233373i	$-0.925024\pi$
$661$		−	12.0000i		−	0.466746i	0.972387	−	0.233373i	$-0.0749763\pi$
$662$		0			0
$663$	11.3137	+	16.0000i	0.439388	+	0.621389i
$664$		0			0
$665$			5.65685i			0.219363i
$666$		0			0
$667$	16.0000			0.619522
$668$		0			0
$669$	−22.6274	+	16.0000i	−0.874826	+	0.618596i
$670$		0			0
$671$	67.8823			2.62057
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$		0			0
$675$	15.0000	−	4.24264i	0.577350	−	0.163299i
$676$		0			0
$677$	25.4558			0.978348			0.489174	−	0.872186i	$-0.337298\pi$
$677$	25.4558			0.978348			0.489174	+	0.872186i	$0.337298\pi$
$678$		0			0
$679$		−	2.00000i		−	0.0767530i
$680$		0			0
$681$	−4.00000	+	2.82843i	−0.153280	+	0.108386i
$682$		0			0
$683$		−	22.6274i		−	0.865814i	−0.901439	−	0.432907i	$-0.857488\pi$
$683$		−	22.6274i		−	0.865814i	0.901439	−	0.432907i	$-0.142512\pi$
$684$		0			0
$685$			16.0000i			0.611329i
$686$		0			0
$687$	28.2843	−	20.0000i	1.07911	−	0.763048i
$688$		0			0
$689$			22.6274i			0.862036i
$690$		0			0
$691$	−6.00000			−0.228251			−0.114125	−	0.993466i	$-0.536407\pi$
$691$	−6.00000			−0.228251			−0.114125	+	0.993466i	$0.536407\pi$
$692$		0			0
$693$	5.65685	−	16.0000i	0.214886	−	0.607790i
$694$		0			0
$695$	−39.5980			−1.50204
$696$		0			0
$697$	−8.00000			−0.303022
$698$		0			0
$699$	−8.00000	+	5.65685i	−0.302588	+	0.213962i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		−	16.0000i		−	0.603451i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	2.82843i		−	0.106374i
$708$		0			0
$709$		−	8.00000i		−	0.300446i	−0.988652	−	0.150223i	$-0.952001\pi$
$709$		−	8.00000i		−	0.300446i	0.988652	−	0.150223i	$-0.0479992\pi$
$710$		0			0
$711$	22.6274	+	8.00000i	0.848594	+	0.300023i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−64.0000			−2.39346
$716$		0			0
$717$	19.7990	+	28.0000i	0.739407	+	1.04568i
$718$		0			0
$719$	39.5980			1.47676			0.738378	−	0.674387i	$-0.235592\pi$
$719$	39.5980			1.47676			0.738378	+	0.674387i	$0.235592\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$		0			0
$723$	−2.00000	−	2.82843i	−0.0743808	−	0.105190i
$724$		0			0
$725$	−16.9706			−0.630271
$726$		0			0
$727$		−	8.00000i		−	0.296704i	−0.988935	−	0.148352i	$-0.952603\pi$
$727$		−	8.00000i		−	0.296704i	0.988935	−	0.148352i	$-0.0473968\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$			28.2843i			1.04613i
$732$		0			0
$733$		−	20.0000i		−	0.738717i	−0.929287	−	0.369358i	$-0.879577\pi$
$733$		−	20.0000i		−	0.738717i	0.929287	−	0.369358i	$-0.120423\pi$
$734$		0			0
$735$	2.82843	+	4.00000i	0.104328	+	0.147542i
$736$		0			0
$737$		−	33.9411i		−	1.25024i
$738$		0			0
$739$	−46.0000			−1.69214			−0.846069	−	0.533074i	$-0.821037\pi$
$739$	−46.0000			−1.69214			−0.846069	+	0.533074i	$0.821037\pi$
$740$		0			0
$741$	11.3137	−	8.00000i	0.415619	−	0.293887i
$742$		0			0
$743$	−14.1421			−0.518825			−0.259412	−	0.965767i	$-0.583529\pi$
$743$	−14.1421			−0.518825			−0.259412	+	0.965767i	$0.583529\pi$
$744$		0			0
$745$	16.0000			0.586195
$746$		0			0
$747$	8.00000	+	2.82843i	0.292705	+	0.103487i
$748$		0			0
$749$	11.3137			0.413394
$750$		0			0
$751$		−	32.0000i		−	1.16770i	−0.811863	−	0.583848i	$-0.801546\pi$
$751$		−	32.0000i		−	1.16770i	0.811863	−	0.583848i	$-0.198454\pi$
$752$		0			0
$753$	12.0000	−	8.48528i	0.437304	−	0.309221i
$754$		0			0
$755$		−	45.2548i		−	1.64699i
$756$		0			0
$757$		−	24.0000i		−	0.872295i	−0.899875	−	0.436147i	$-0.856343\pi$
$757$		−	24.0000i		−	0.872295i	0.899875	−	0.436147i	$-0.143657\pi$
$758$		0			0
$759$	−22.6274	+	16.0000i	−0.821323	+	0.580763i
$760$		0			0
$761$			31.1127i			1.12783i	0.825831	+	0.563917i	$0.190706\pi$
$761$			31.1127i			1.12783i	−0.825831	+	0.563917i	$0.809294\pi$
$762$		0			0
$763$	8.00000			0.289619
$764$		0			0
$765$	−22.6274	−	8.00000i	−0.818096	−	0.289241i
$766$		0			0
$767$	−33.9411			−1.22554
$768$		0			0
$769$	−46.0000			−1.65880			−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000			−1.65880			−0.829401	+	0.558653i	$0.811318\pi$
$770$		0			0
$771$	4.00000	−	2.82843i	0.144056	−	0.101863i
$772$		0			0
$773$	14.1421			0.508657			0.254329	−	0.967118i	$-0.418146\pi$
$773$	14.1421			0.508657			0.254329	+	0.967118i	$0.418146\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−8.00000	−	11.3137i	−0.286998	−	0.405877i
$778$		0			0
$779$			5.65685i			0.202678i
$780$		0			0
$781$			48.0000i			1.71758i
$782$		0			0
$783$	−28.2843	+	8.00000i	−1.01080	+	0.285897i
$784$		0			0
$785$			33.9411i			1.21141i
$786$		0			0
$787$	42.0000			1.49714			0.748569	−	0.663057i	$-0.230741\pi$
$787$	42.0000			1.49714			0.748569	+	0.663057i	$0.230741\pi$
$788$		0			0
$789$	−19.7990	−	28.0000i	−0.704863	−	0.996826i
$790$		0			0
$791$	−5.65685			−0.201135
$792$		0			0
$793$	48.0000			1.70453
$794$		0			0
$795$	−16.0000	−	22.6274i	−0.567462	−	0.802512i
$796$		0			0
$797$	−36.7696			−1.30244			−0.651222	−	0.758887i	$-0.725743\pi$
$797$	−36.7696			−1.30244			−0.651222	+	0.758887i	$0.725743\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	8.00000	+	2.82843i	0.282666	+	0.0999376i
$802$		0			0
$803$			56.5685i			1.99626i
$804$		0			0
$805$		−	8.00000i		−	0.281963i
$806$		0			0
$807$	−2.82843	−	4.00000i	−0.0995654	−	0.140807i
$808$		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
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$	−11.3137	+	8.00000i	−0.396789	+	0.280572i
$814$		0			0
$815$	39.5980			1.38706
$816$		0			0
$817$	20.0000			0.699711
$818$		0			0
$819$	4.00000	−	11.3137i	0.139771	−	0.395333i
$820$		0			0
$821$	−28.2843			−0.987128			−0.493564	−	0.869710i	$-0.664306\pi$
$821$	−28.2843			−0.987128			−0.493564	+	0.869710i	$0.664306\pi$
$822$		0			0
$823$			24.0000i			0.836587i	0.908312	+	0.418294i	$0.137372\pi$
$823$			24.0000i			0.836587i	−0.908312	+	0.418294i	$0.862628\pi$
$824$		0			0
$825$	24.0000	−	16.9706i	0.835573	−	0.590839i
$826$		0			0
$827$		−	33.9411i		−	1.18025i	−0.807312	−	0.590124i	$-0.799079\pi$
$827$		−	33.9411i		−	1.18025i	0.807312	−	0.590124i	$-0.200921\pi$
$828$		0			0
$829$		−	28.0000i		−	0.972480i	−0.873825	−	0.486240i	$-0.838368\pi$
$829$		−	28.0000i		−	0.972480i	0.873825	−	0.486240i	$-0.161632\pi$
$830$		0			0
$831$	−45.2548	+	32.0000i	−1.56987	+	1.11007i
$832$		0			0
$833$		−	2.82843i		−	0.0979992i
$834$		0			0
$835$	16.0000			0.553703
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−5.65685			−0.195296			−0.0976481	−	0.995221i	$-0.531132\pi$
$839$	−5.65685			−0.195296			−0.0976481	+	0.995221i	$0.531132\pi$
$840$		0			0
$841$	3.00000			0.103448
$842$		0			0
$843$	8.00000	−	5.65685i	0.275535	−	0.194832i
$844$		0			0
$845$	−8.48528			−0.291903
$846$		0			0
$847$		−	21.0000i		−	0.721569i
$848$		0			0
$849$	−18.0000	−	25.4558i	−0.617758	−	0.873642i
$850$		0			0
$851$			22.6274i			0.775658i
$852$		0			0
$853$			4.00000i			0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$			4.00000i			0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$		0			0
$855$	−5.65685	+	16.0000i	−0.193460	+	0.547188i
$856$		0			0
$857$		−	48.0833i		−	1.64249i	−0.570574	−	0.821246i	$-0.693279\pi$
$857$		−	48.0833i		−	1.64249i	0.570574	−	0.821246i	$-0.306721\pi$
$858$		0			0
$859$	−10.0000			−0.341196			−0.170598	−	0.985341i	$-0.554570\pi$
$859$	−10.0000			−0.341196			−0.170598	+	0.985341i	$0.554570\pi$
$860$		0			0
$861$	2.82843	+	4.00000i	0.0963925	+	0.136320i
$862$		0			0
$863$	−2.82843			−0.0962808			−0.0481404	−	0.998841i	$-0.515329\pi$
$863$	−2.82843			−0.0962808			−0.0481404	+	0.998841i	$0.515329\pi$
$864$		0			0
$865$	72.0000			2.44807
$866$		0			0
$867$	−9.00000	−	12.7279i	−0.305656	−	0.432263i
$868$		0			0
$869$	45.2548			1.53517
$870$		0			0
$871$		−	24.0000i		−	0.813209i
$872$		0			0
$873$	2.00000	−	5.65685i	0.0676897	−	0.191456i
$874$		0			0
$875$		−	5.65685i		−	0.191237i
$876$		0			0
$877$			8.00000i			0.270141i	0.990836	+	0.135070i	$0.0431261\pi$
$877$			8.00000i			0.270141i	−0.990836	+	0.135070i	$0.956874\pi$
$878$		0			0
$879$	19.7990	+	28.0000i	0.667803	+	0.944417i
$880$		0			0
$881$			42.4264i			1.42938i	0.699440	+	0.714691i	$0.253433\pi$
$881$			42.4264i			1.42938i	−0.699440	+	0.714691i	$0.746567\pi$
$882$		0			0
$883$	54.0000			1.81724			0.908622	−	0.417619i	$-0.137135\pi$
$883$	54.0000			1.81724			0.908622	+	0.417619i	$0.137135\pi$
$884$		0			0
$885$	33.9411	−	24.0000i	1.14092	−	0.806751i
$886$		0			0
$887$	−28.2843			−0.949693			−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843			−0.949693			−0.474846	+	0.880069i	$0.657496\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	32.0000	−	39.5980i	1.07204	−	1.32658i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	32.0000i		−	1.06964i
$896$		0			0
$897$	−16.0000	+	11.3137i	−0.534224	+	0.377754i
$898$		0			0
$899$		0			0
$900$		0			0
$901$			16.0000i			0.533037i
$902$		0			0
$903$	14.1421	−	10.0000i	0.470621	−	0.332779i
$904$		0			0
$905$			56.5685i			1.88040i
$906$		0			0
$907$	34.0000			1.12895			0.564476	−	0.825450i	$-0.309078\pi$
$907$	34.0000			1.12895			0.564476	+	0.825450i	$0.309078\pi$
$908$		0			0
$909$	2.82843	−	8.00000i	0.0938130	−	0.265343i
$910$		0			0
$911$	31.1127			1.03081			0.515405	−	0.856947i	$-0.327642\pi$
$911$	31.1127			1.03081			0.515405	+	0.856947i	$0.327642\pi$
$912$		0			0
$913$	16.0000			0.529523
$914$		0			0
$915$	−48.0000	+	33.9411i	−1.58683	+	1.12206i
$916$		0			0
$917$	−14.1421			−0.467014
$918$		0			0
$919$		−	40.0000i		−	1.31948i	−0.751495	−	0.659739i	$-0.770667\pi$
$919$		−	40.0000i		−	1.31948i	0.751495	−	0.659739i	$-0.229333\pi$
$920$		0			0
$921$	18.0000	+	25.4558i	0.593120	+	0.838799i
$922$		0			0
$923$			33.9411i			1.11719i
$924$		0			0
$925$		−	24.0000i		−	0.789115i
$926$		0			0
$927$	22.6274	+	8.00000i	0.743182	+	0.262754i
$928$		0			0
$929$			25.4558i			0.835179i	0.908636	+	0.417590i	$0.137125\pi$
$929$			25.4558i			0.835179i	−0.908636	+	0.417590i	$0.862875\pi$
$930$		0			0
$931$	−2.00000			−0.0655474
$932$		0			0
$933$	11.3137	+	16.0000i	0.370394	+	0.523816i
$934$		0			0
$935$	−45.2548			−1.47999
$936$		0			0
$937$	6.00000			0.196011			0.0980057	−	0.995186i	$-0.468754\pi$
$937$	6.00000			0.196011			0.0980057	+	0.995186i	$0.468754\pi$
$938$		0			0
$939$	−6.00000	−	8.48528i	−0.195803	−	0.276907i
$940$		0			0
$941$	25.4558			0.829837			0.414918	−	0.909859i	$-0.363810\pi$
$941$	25.4558			0.829837			0.414918	+	0.909859i	$0.363810\pi$
$942$		0			0
$943$		−	8.00000i		−	0.260516i
$944$		0			0
$945$	4.00000	+	14.1421i	0.130120	+	0.460044i
$946$		0			0
$947$			28.2843i			0.919115i	0.888148	+	0.459558i	$0.151992\pi$
$947$			28.2843i			0.919115i	−0.888148	+	0.459558i	$0.848008\pi$
$948$		0			0
$949$			40.0000i			1.29845i
$950$		0			0
$951$	16.9706	+	24.0000i	0.550308	+	0.778253i
$952$		0			0
$953$			33.9411i			1.09946i	0.835342	+	0.549730i	$0.185270\pi$
$953$			33.9411i			1.09946i	−0.835342	+	0.549730i	$0.814730\pi$
$954$		0			0
$955$	−24.0000			−0.776622
$956$		0			0
$957$	−45.2548	+	32.0000i	−1.46288	+	1.03441i
$958$		0			0
$959$	−5.65685			−0.182669
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	32.0000	+	11.3137i	1.03119	+	0.364579i
$964$		0			0
$965$	50.9117			1.63891
$966$		0			0
$967$		−	16.0000i		−	0.514525i	−0.966342	−	0.257263i	$-0.917179\pi$
$967$		−	16.0000i		−	0.514525i	0.966342	−	0.257263i	$-0.0828206\pi$
$968$		0			0
$969$	8.00000	−	5.65685i	0.256997	−	0.181724i
$970$		0			0
$971$			2.82843i			0.0907685i	0.998970	+	0.0453843i	$0.0144512\pi$
$971$			2.82843i			0.0907685i	−0.998970	+	0.0453843i	$0.985549\pi$
$972$		0			0
$973$		−	14.0000i		−	0.448819i
$974$		0			0
$975$	16.9706	−	12.0000i	0.543493	−	0.384308i
$976$		0			0
$977$		−	50.9117i		−	1.62881i	−0.580297	−	0.814405i	$-0.697064\pi$
$977$		−	50.9117i		−	1.62881i	0.580297	−	0.814405i	$-0.302936\pi$
$978$		0			0
$979$	16.0000			0.511362
$980$		0			0
$981$	22.6274	+	8.00000i	0.722438	+	0.255420i
$982$		0			0
$983$	−11.3137			−0.360851			−0.180426	−	0.983589i	$-0.557748\pi$
$983$	−11.3137			−0.360851			−0.180426	+	0.983589i	$0.557748\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−28.2843			−0.899388
$990$		0			0
$991$			40.0000i			1.27064i	0.772248	+	0.635321i	$0.219132\pi$
$991$			40.0000i			1.27064i	−0.772248	+	0.635321i	$0.780868\pi$
$992$		0			0
$993$	−6.00000	−	8.48528i	−0.190404	−	0.269272i
$994$		0			0
$995$			45.2548i			1.43467i
$996$		0			0
$997$			28.0000i			0.886769i	0.896332	+	0.443384i	$0.146222\pi$
$997$			28.0000i			0.886769i	−0.896332	+	0.443384i	$0.853778\pi$
$998$		0			0
$999$	−11.3137	−	40.0000i	−0.357950	−	1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	672.2.j.a.239.2		4
3.2	odd	2	inner	672.2.j.a.239.3		4
4.3	odd	2		168.2.j.b.155.2	yes	4
8.3	odd	2	inner	672.2.j.a.239.1		4
8.5	even	2		168.2.j.b.155.4	yes	4
12.11	even	2		168.2.j.b.155.3	yes	4
24.5	odd	2		168.2.j.b.155.1	✓	4
24.11	even	2	inner	672.2.j.a.239.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.j.b.155.1	✓	4	24.5	odd	2
168.2.j.b.155.2	yes	4	4.3	odd	2
168.2.j.b.155.3	yes	4	12.11	even	2
168.2.j.b.155.4	yes	4	8.5	even	2
672.2.j.a.239.1		4	8.3	odd	2	inner
672.2.j.a.239.2		4	1.1	even	1	trivial
672.2.j.a.239.3		4	3.2	odd	2	inner
672.2.j.a.239.4		4	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	672.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.41421i) q^{3} +2.82843 q^{5} +1.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.41421i) q^{3} +2.82843 q^{5} +1.00000i q^{7} +(-1.00000 + 2.82843i) q^{9} +5.65685i q^{11} +4.00000i q^{13} +(-2.82843 - 4.00000i) q^{15} +2.82843i q^{17} +2.00000 q^{19} +(1.41421 - 1.00000i) q^{21} -2.82843 q^{23} +3.00000 q^{25} +(5.00000 - 1.41421i) q^{27} -5.65685 q^{29} +(8.00000 - 5.65685i) q^{33} +2.82843i q^{35} -8.00000i q^{37} +(5.65685 - 4.00000i) q^{39} +2.82843i q^{41} +10.0000 q^{43} +(-2.82843 + 8.00000i) q^{45} -1.00000 q^{49} +(4.00000 - 2.82843i) q^{51} +5.65685 q^{53} +16.0000i q^{55} +(-2.00000 - 2.82843i) q^{57} +8.48528i q^{59} -12.0000i q^{61} +(-2.82843 - 1.00000i) q^{63} +11.3137i q^{65} -6.00000 q^{67} +(2.82843 + 4.00000i) q^{69} +8.48528 q^{71} +10.0000 q^{73} +(-3.00000 - 4.24264i) q^{75} -5.65685 q^{77} -8.00000i q^{79} +(-7.00000 - 5.65685i) q^{81} -2.82843i q^{83} +8.00000i q^{85} +(5.65685 + 8.00000i) q^{87} -2.82843i q^{89} -4.00000 q^{91} +5.65685 q^{95} -2.00000 q^{97} +(-16.0000 - 5.65685i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{9} + 8 q^{19} + 12 q^{25} + 20 q^{27} + 32 q^{33} + 40 q^{43} - 4 q^{49} + 16 q^{51} - 8 q^{57} - 24 q^{67} + 40 q^{73} - 12 q^{75} - 28 q^{81} - 16 q^{91} - 8 q^{97} - 64 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^9 + 8 * q^19 + 12 * q^25 + 20 * q^27 + 32 * q^33 + 40 * q^43 - 4 * q^49 + 16 * q^51 - 8 * q^57 - 24 * q^67 + 40 * q^73 - 12 * q^75 - 28 * q^81 - 16 * q^91 - 8 * q^97 - 64 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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