LMFDB - Embedded newform 1536.2.c.f.1535.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1535,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1536.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.2650217505$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1535.2
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.f.1535.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.70711 + 0.292893i) q^{3} +(2.82843 - 1.00000i) q^{9} +O(q^{10})$	$q+(-1.70711 + 0.292893i) q^{3} +(2.82843 - 1.00000i) q^{9} -2.24264 q^{11} +6.00000i q^{17} -7.41421i q^{19} +5.00000 q^{25} +(-4.53553 + 2.53553i) q^{27} +(3.82843 - 0.656854i) q^{33} -11.3137i q^{41} +13.0711i q^{43} +7.00000 q^{49} +(-1.75736 - 10.2426i) q^{51} +(2.17157 + 12.6569i) q^{57} +5.75736 q^{59} -3.89949i q^{67} +16.9706 q^{73} +(-8.53553 + 1.46447i) q^{75} +(7.00000 - 5.65685i) q^{81} +14.7279 q^{83} +18.0000i q^{89} +16.9706 q^{97} +(-6.34315 + 2.24264i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3}+O(q^{10}) $ 4 * q - 4 * q^3	$ 4 q - 4 q^{3} + 8 q^{11} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 28 q^{49} - 24 q^{51} + 20 q^{57} + 40 q^{59} - 20 q^{75} + 28 q^{81} + 8 q^{83} - 48 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 8 * q^11 + 20 * q^25 - 4 * q^27 + 4 * q^33 + 28 * q^49 - 24 * q^51 + 20 * q^57 + 40 * q^59 - 20 * q^75 + 28 * q^81 + 8 * q^83 - 48 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−1.70711	+	0.292893i	−0.985599	+	0.169102i
$3$	−1.70711	+	0.292893i	−0.985599	+	0.169102i
$4$		0			0
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	2.82843	−	1.00000i	0.942809	−	0.333333i
$10$		0			0
$11$	−2.24264			−0.676182			−0.338091	−	0.941113i	$-0.609781\pi$
$11$	−2.24264			−0.676182			−0.338091	+	0.941113i	$0.609781\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.00000i			1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$			6.00000i			1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$		−	7.41421i		−	1.70094i	−0.526026	−	0.850469i	$-0.676318\pi$
$19$		−	7.41421i		−	1.70094i	0.526026	−	0.850469i	$-0.323682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	−4.53553	+	2.53553i	−0.872864	+	0.487964i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$	3.82843	−	0.656854i	0.666444	−	0.114344i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	11.3137i		−	1.76690i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$		−	11.3137i		−	1.76690i	0.468521	−	0.883452i	$-0.344787\pi$
$42$		0			0
$43$			13.0711i			1.99332i	0.0816682	+	0.996660i	$0.473975\pi$
$43$			13.0711i			1.99332i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−1.75736	−	10.2426i	−0.246080	−	1.43426i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.17157	+	12.6569i	0.287632	+	1.67644i
$58$		0			0
$59$	5.75736			0.749544			0.374772	−	0.927117i	$-0.377721\pi$
$59$	5.75736			0.749544			0.374772	+	0.927117i	$0.377721\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$		−	3.89949i		−	0.476399i	−0.971216	−	0.238200i	$-0.923443\pi$
$67$		−	3.89949i		−	0.476399i	0.971216	−	0.238200i	$-0.0765572\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	16.9706			1.98625			0.993127	−	0.117041i	$-0.0373409\pi$
$73$	16.9706			1.98625			0.993127	+	0.117041i	$0.0373409\pi$
$74$		0			0
$75$	−8.53553	+	1.46447i	−0.985599	+	0.169102i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$		0			0
$83$	14.7279			1.61660			0.808300	−	0.588771i	$-0.200388\pi$
$83$	14.7279			1.61660			0.808300	+	0.588771i	$0.200388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$			18.0000i			1.90800i	0.299813	+	0.953998i	$0.403076\pi$
$89$			18.0000i			1.90800i	−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	16.9706			1.72310			0.861550	−	0.507673i	$-0.169494\pi$
$97$	16.9706			1.72310			0.861550	+	0.507673i	$0.169494\pi$
$98$		0			0
$99$	−6.34315	+	2.24264i	−0.637510	+	0.225394i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.2426			1.76358			0.881791	−	0.471640i	$-0.156338\pi$
$107$	18.2426			1.76358			0.881791	+	0.471640i	$0.156338\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.97056			−0.542778
$122$		0			0
$123$	3.31371	+	19.3137i	0.298787	+	1.74146i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	−3.82843	−	22.3137i	−0.337074	−	1.96461i
$130$		0			0
$131$	22.7279			1.98575			0.992874	−	0.119170i	$-0.0380233\pi$
$131$	22.7279			1.98575			0.992874	+	0.119170i	$0.0380233\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	22.6274i		−	1.93319i	−0.256307	−	0.966595i	$-0.582506\pi$
$137$		−	22.6274i		−	1.93319i	0.256307	−	0.966595i	$-0.417494\pi$
$138$		0			0
$139$			9.55635i			0.810559i	0.914193	+	0.405279i	$0.132826\pi$
$139$			9.55635i			0.810559i	−0.914193	+	0.405279i	$0.867174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−11.9497	+	2.05025i	−0.985599	+	0.169102i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	6.00000	+	16.9706i	0.485071	+	1.37199i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	16.5858i		−	1.29910i	−0.760319	−	0.649550i	$-0.774958\pi$
$163$		−	16.5858i		−	1.29910i	0.760319	−	0.649550i	$-0.225042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−13.0000			−1.00000
$170$		0			0
$171$	−7.41421	−	20.9706i	−0.566979	−	1.60366i
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−9.82843	+	1.68629i	−0.738750	+	0.126749i
$178$		0			0
$179$	1.27208			0.0950796			0.0475398	−	0.998869i	$-0.484862\pi$
$179$	1.27208			0.0950796			0.0475398	+	0.998869i	$0.484862\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	13.4558i		−	0.983989i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−16.9706			−1.22157			−0.610784	−	0.791797i	$-0.709146\pi$
$193$	−16.9706			−1.22157			−0.610784	+	0.791797i	$0.709146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	1.14214	+	6.65685i	0.0805600	+	0.469538i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			16.6274i			1.15014i
$210$		0			0
$211$			27.8995i			1.92068i	0.278831	+	0.960340i	$0.410053\pi$
$211$			27.8995i			1.92068i	−0.278831	+	0.960340i	$0.589947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−28.9706	+	4.97056i	−1.95765	+	0.335880i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	14.1421	−	5.00000i	0.942809	−	0.333333i
$226$		0			0
$227$	−19.2132			−1.27523			−0.637613	−	0.770357i	$-0.720078\pi$
$227$	−19.2132			−1.27523			−0.637613	+	0.770357i	$0.720078\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	30.0000i		−	1.96537i	−0.185296	−	0.982683i	$-0.559325\pi$
$233$		−	30.0000i		−	1.96537i	0.185296	−	0.982683i	$-0.440675\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	−16.9706			−1.09317			−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706			−1.09317			−0.546585	+	0.837404i	$0.684072\pi$
$242$		0			0
$243$	−10.2929	+	11.7071i	−0.660289	+	0.751011i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−25.1421	+	4.31371i	−1.59332	+	0.273370i
$250$		0			0
$251$	26.2426			1.65642			0.828210	−	0.560417i	$-0.189359\pi$
$251$	26.2426			1.65642			0.828210	+	0.560417i	$0.189359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$			11.3137i			0.705730i	0.935674	+	0.352865i	$0.114792\pi$
$257$			11.3137i			0.705730i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−5.27208	−	30.7279i	−0.322646	−	1.88052i
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−11.2132			−0.676182
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$		−	33.5563i		−	1.99472i	−0.0726300	−	0.997359i	$-0.523139\pi$
$283$		−	33.5563i		−	1.99472i	0.0726300	−	0.997359i	$-0.476861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−19.0000			−1.11765
$290$		0			0
$291$	−28.9706	+	4.97056i	−1.69828	+	0.291380i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	10.1716	−	5.68629i	0.590215	−	0.329952i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			30.0416i			1.71457i	0.514845	+	0.857283i	$0.327849\pi$
$307$			30.0416i			1.71457i	−0.514845	+	0.857283i	$0.672151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−31.1421	+	5.34315i	−1.73818	+	0.298225i
$322$		0			0
$323$	44.4853			2.47523
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			0.384776i			0.0211492i	0.999944	+	0.0105746i	$0.00336607\pi$
$331$			0.384776i			0.0211492i	−0.999944	+	0.0105746i	$0.996634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−14.0000			−0.762629			−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000			−0.762629			−0.381314	+	0.924445i	$0.624528\pi$
$338$		0			0
$339$	−3.31371	−	19.3137i	−0.179976	−	1.04898i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.7574			−1.16800			−0.583998	−	0.811755i	$-0.698512\pi$
$347$	−21.7574			−1.16800			−0.583998	+	0.811755i	$0.698512\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			22.6274i			1.20434i	0.798369	+	0.602168i	$0.205696\pi$
$353$			22.6274i			1.20434i	−0.798369	+	0.602168i	$0.794304\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−35.9706			−1.89319
$362$		0			0
$363$	10.1924	−	1.74874i	0.534962	−	0.0917849i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−11.3137	−	32.0000i	−0.588968	−	1.66585i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		−	20.8701i		−	1.07202i	−0.844211	−	0.536011i	$-0.819930\pi$
$379$		−	20.8701i		−	1.07202i	0.844211	−	0.536011i	$-0.180070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	13.0711	+	36.9706i	0.664440	+	1.87932i
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−38.7990	+	6.65685i	−1.95715	+	0.335794i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	6.00000i		−	0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$		−	6.00000i		−	0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	6.62742	+	38.6274i	0.326906	+	1.90535i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.79899	−	16.3137i	−0.137067	−	0.798886i
$418$		0			0
$419$	−38.7279			−1.89198			−0.945991	−	0.324192i	$-0.894908\pi$
$419$	−38.7279			−1.89198			−0.945991	+	0.324192i	$0.894908\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			30.0000i			1.45521i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	16.9706			0.815553			0.407777	−	0.913082i	$-0.366304\pi$
$433$	16.9706			0.815553			0.407777	+	0.913082i	$0.366304\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	19.7990	−	7.00000i	0.942809	−	0.333333i
$442$		0			0
$443$	31.6985			1.50604			0.753020	−	0.657997i	$-0.228596\pi$
$443$	31.6985			1.50604			0.753020	+	0.657997i	$0.228596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	42.0000i		−	1.98210i	−0.133482	−	0.991051i	$-0.542616\pi$
$449$		−	42.0000i		−	1.98210i	0.133482	−	0.991051i	$-0.457384\pi$
$450$		0			0
$451$			25.3726i			1.19475i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$	−15.2132	−	27.2132i	−0.710092	−	1.27020i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	43.2132			1.99967			0.999834	−	0.0182043i	$-0.00579493\pi$
$467$	43.2132			1.99967			0.999834	+	0.0182043i	$0.00579493\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	29.3137i		−	1.34785i
$474$		0			0
$475$		−	37.0711i		−	1.70094i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	4.85786	+	28.3137i	0.219680	+	1.28039i
$490$		0			0
$491$	39.6985			1.79157			0.895784	−	0.444490i	$-0.146615\pi$
$491$	39.6985			1.79157			0.895784	+	0.444490i	$0.146615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	20.1005i		−	0.899822i	−0.893073	−	0.449911i	$-0.851456\pi$
$499$		−	20.1005i		−	0.899822i	0.893073	−	0.449911i	$-0.148544\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	22.1924	−	3.80761i	0.985599	−	0.169102i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	18.7990	+	33.6274i	0.829996	+	1.48469i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			45.2548i			1.98265i	0.131432	+	0.991325i	$0.458042\pi$
$521$			45.2548i			1.98265i	−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$			44.8701i			1.96203i	0.193930	+	0.981015i	$0.437876\pi$
$523$			44.8701i			1.96203i	−0.193930	+	0.981015i	$0.562124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	16.2843	−	5.75736i	0.706677	−	0.249848i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−2.17157	+	0.372583i	−0.0937103	+	0.0160781i
$538$		0			0
$539$	−15.6985			−0.676182
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$			26.5269i			1.13421i	0.823646	+	0.567104i	$0.191936\pi$
$547$			26.5269i			1.13421i	−0.823646	+	0.567104i	$0.808064\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	3.94113	+	22.9706i	0.166394	+	0.969818i
$562$		0			0
$563$	−4.78680			−0.201739			−0.100870	−	0.994900i	$-0.532163\pi$
$563$	−4.78680			−0.201739			−0.100870	+	0.994900i	$0.532163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		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
$571$			14.4437i			0.604448i	0.953237	+	0.302224i	$0.0977291\pi$
$571$			14.4437i			0.604448i	−0.953237	+	0.302224i	$0.902271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$		0			0
$579$	28.9706	−	4.97056i	1.20398	−	0.206570i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−29.7574			−1.22822			−0.614109	−	0.789221i	$-0.710484\pi$
$587$	−29.7574			−1.22822			−0.614109	+	0.789221i	$0.710484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	45.2548i		−	1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$		−	45.2548i		−	1.85839i	0.369586	−	0.929197i	$-0.379500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	16.9706			0.692244			0.346122	−	0.938190i	$-0.387498\pi$
$601$	16.9706			0.692244			0.346122	+	0.938190i	$0.387498\pi$
$602$		0			0
$603$	−3.89949	−	11.0294i	−0.158800	−	0.449153i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			30.0000i			1.20775i	0.797077	+	0.603877i	$0.206378\pi$
$617$			30.0000i			1.20775i	−0.797077	+	0.603877i	$0.793622\pi$
$618$		0			0
$619$			48.3848i			1.94475i	0.233428	+	0.972374i	$0.425006\pi$
$619$			48.3848i			1.94475i	−0.233428	+	0.972374i	$0.574994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	−4.87006	−	28.3848i	−0.194491	−	1.13358i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−8.17157	−	47.6274i	−0.324791	−	1.89302i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			42.0000i			1.65890i	0.558581	+	0.829450i	$0.311346\pi$
$641$			42.0000i			1.65890i	−0.558581	+	0.829450i	$0.688654\pi$
$642$		0			0
$643$		−	41.3553i		−	1.63090i	−0.578831	−	0.815448i	$-0.696491\pi$
$643$		−	41.3553i		−	1.63090i	0.578831	−	0.815448i	$-0.303509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	−12.9117			−0.506828
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	48.0000	−	16.9706i	1.87266	−	0.662085i
$658$		0			0
$659$	−46.7279			−1.82026			−0.910131	−	0.414321i	$-0.864019\pi$
$659$	−46.7279			−1.82026			−0.910131	+	0.414321i	$0.864019\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−50.9117			−1.96250			−0.981251	−	0.192736i	$-0.938264\pi$
$673$	−50.9117			−1.96250			−0.981251	+	0.192736i	$0.938264\pi$
$674$		0			0
$675$	−22.6777	+	12.6777i	−0.872864	+	0.487964i
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	32.7990	−	5.62742i	1.25686	−	0.215643i
$682$		0			0
$683$	−7.69848			−0.294574			−0.147287	−	0.989094i	$-0.547054\pi$
$683$	−7.69848			−0.294574			−0.147287	+	0.989094i	$0.547054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		−	50.5269i		−	1.92213i	−0.276315	−	0.961067i	$-0.589113\pi$
$691$		−	50.5269i		−	1.92213i	0.276315	−	0.961067i	$-0.410887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	67.8823			2.57122
$698$		0			0
$699$	8.78680	+	51.2132i	0.332347	+	1.93706i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	28.9706	−	4.97056i	1.07743	−	0.184857i
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	14.1421	−	23.0000i	0.523783	−	0.851852i
$730$		0			0
$731$	−78.4264			−2.90071
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			8.74517i			0.322132i
$738$		0			0
$739$		−	54.0416i		−	1.98795i	−0.109591	−	0.993977i	$-0.534954\pi$
$739$		−	54.0416i		−	1.98795i	0.109591	−	0.993977i	$-0.465046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	41.6569	−	14.7279i	1.52414	−	0.538866i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−44.7990	+	7.68629i	−1.63257	+	0.280104i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	11.3137i		−	0.410122i	−0.978749	−	0.205061i	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	50.9117			1.83592			0.917961	−	0.396670i	$-0.129834\pi$
$769$	50.9117			1.83592			0.917961	+	0.396670i	$0.129834\pi$
$770$		0			0
$771$	−3.31371	−	19.3137i	−0.119340	−	0.695566i
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−83.8823			−3.00539
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$			17.3553i			0.618651i	0.950956	+	0.309326i	$0.100103\pi$
$787$			17.3553i			0.618651i	−0.950956	+	0.309326i	$0.899897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	18.0000	+	50.9117i	0.635999	+	1.79888i
$802$		0			0
$803$	−38.0589			−1.34307
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$			56.5685i			1.98884i	0.105474	+	0.994422i	$0.466364\pi$
$809$			56.5685i			1.98884i	−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$		−	3.12994i		−	0.109907i	−0.998489	−	0.0549536i	$-0.982499\pi$
$811$		−	3.12994i		−	0.109907i	0.998489	−	0.0549536i	$-0.0175011\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	96.9117			3.39051
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	19.1421	−	3.28427i	0.666444	−	0.114344i
$826$		0			0
$827$	52.1838			1.81461			0.907304	−	0.420476i	$-0.138137\pi$
$827$	52.1838			1.81461			0.907304	+	0.420476i	$0.138137\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			42.0000i			1.45521i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	29.0000			1.00000
$842$		0			0
$843$	5.27208	+	30.7279i	0.181580	+	1.05833i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	9.82843	+	57.2843i	0.337311	+	1.96599i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	22.6274i		−	0.772938i	−0.922302	−	0.386469i	$-0.873695\pi$
$857$		−	22.6274i		−	0.772938i	0.922302	−	0.386469i	$-0.126305\pi$
$858$		0			0
$859$			47.0122i			1.60404i	0.597300	+	0.802018i	$0.296240\pi$
$859$			47.0122i			1.60404i	−0.597300	+	0.802018i	$0.703760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	32.4350	−	5.56497i	1.10155	−	0.188996i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	48.0000	−	16.9706i	1.62455	−	0.574367i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.5685i		−	1.90584i	−0.303218	−	0.952921i	$-0.598061\pi$
$881$		−	56.5685i		−	1.90584i	0.303218	−	0.952921i	$-0.401939\pi$
$882$		0			0
$883$			40.5858i			1.36582i	0.730502	+	0.682910i	$0.239286\pi$
$883$			40.5858i			1.36582i	−0.730502	+	0.682910i	$0.760714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−15.6985	+	12.6863i	−0.525919	+	0.425007i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		−	34.9289i		−	1.15980i	−0.814689	−	0.579898i	$-0.803092\pi$
$907$		−	34.9289i		−	1.15980i	0.814689	−	0.579898i	$-0.196908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	−33.0294			−1.09311
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−8.79899	−	51.2843i	−0.289937	−	1.68987i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	54.0000i		−	1.77168i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$		−	54.0000i		−	1.77168i	0.463988	−	0.885841i	$-0.346418\pi$
$930$		0			0
$931$		−	51.8995i		−	1.70094i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	50.9117			1.66321			0.831606	−	0.555366i	$-0.187422\pi$
$937$	50.9117			1.66321			0.831606	+	0.555366i	$0.187422\pi$
$938$		0			0
$939$	17.0711	−	2.92893i	0.557093	−	0.0955821i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−59.2132			−1.92417			−0.962085	−	0.272749i	$-0.912067\pi$
$947$	−59.2132			−1.92417			−0.962085	+	0.272749i	$0.912067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			45.2548i			1.46595i	0.680257	+	0.732974i	$0.261868\pi$
$953$			45.2548i			1.46595i	−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	51.5980	−	18.2426i	1.66272	−	0.587861i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−75.9411	+	13.0294i	−2.43958	+	0.418566i
$970$		0			0
$971$	60.1838			1.93139			0.965694	−	0.259681i	$-0.0836174\pi$
$971$	60.1838			1.93139			0.965694	+	0.259681i	$0.0836174\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			6.00000i			0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$			6.00000i			0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$		0			0
$979$		−	40.3675i		−	1.29015i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−0.112698	−	0.656854i	−0.00357637	−	0.0208446i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.f.1535.2	yes	4
3.2	odd	2		1536.2.c.j.1535.4	yes	4
4.3	odd	2		1536.2.c.j.1535.3	yes	4
8.3	odd	2	CM	1536.2.c.f.1535.2	yes	4
8.5	even	2		1536.2.c.j.1535.3	yes	4
12.11	even	2	inner	1536.2.c.f.1535.1	✓	4
16.3	odd	4		1536.2.f.j.767.2		4
16.5	even	4		1536.2.f.j.767.2		4
16.11	odd	4		1536.2.f.a.767.3		4
16.13	even	4		1536.2.f.a.767.3		4
24.5	odd	2	inner	1536.2.c.f.1535.1	✓	4
24.11	even	2		1536.2.c.j.1535.4	yes	4
48.5	odd	4		1536.2.f.j.767.1		4
48.11	even	4		1536.2.f.a.767.4		4
48.29	odd	4		1536.2.f.a.767.4		4
48.35	even	4		1536.2.f.j.767.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.f.1535.1	✓	4	12.11	even	2	inner
1536.2.c.f.1535.1	✓	4	24.5	odd	2	inner
1536.2.c.f.1535.2	yes	4	1.1	even	1	trivial
1536.2.c.f.1535.2	yes	4	8.3	odd	2	CM
1536.2.c.j.1535.3	yes	4	4.3	odd	2
1536.2.c.j.1535.3	yes	4	8.5	even	2
1536.2.c.j.1535.4	yes	4	3.2	odd	2
1536.2.c.j.1535.4	yes	4	24.11	even	2
1536.2.f.a.767.3		4	16.11	odd	4
1536.2.f.a.767.3		4	16.13	even	4
1536.2.f.a.767.4		4	48.11	even	4
1536.2.f.a.767.4		4	48.29	odd	4
1536.2.f.j.767.1		4	48.5	odd	4
1536.2.f.j.767.1		4	48.35	even	4
1536.2.f.j.767.2		4	16.3	odd	4
1536.2.f.j.767.2		4	16.5	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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.70711 + 0.292893i) q^{3} +(2.82843 - 1.00000i) q^{9} +O(q^{10})\)	\(q+(-1.70711 + 0.292893i) q^{3} +(2.82843 - 1.00000i) q^{9} -2.24264 q^{11} +6.00000i q^{17} -7.41421i q^{19} +5.00000 q^{25} +(-4.53553 + 2.53553i) q^{27} +(3.82843 - 0.656854i) q^{33} -11.3137i q^{41} +13.0711i q^{43} +7.00000 q^{49} +(-1.75736 - 10.2426i) q^{51} +(2.17157 + 12.6569i) q^{57} +5.75736 q^{59} -3.89949i q^{67} +16.9706 q^{73} +(-8.53553 + 1.46447i) q^{75} +(7.00000 - 5.65685i) q^{81} +14.7279 q^{83} +18.0000i q^{89} +16.9706 q^{97} +(-6.34315 + 2.24264i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3}+O(q^{10}) \) 4 * q - 4 * q^3	\( 4 q - 4 q^{3} + 8 q^{11} + 20 q^{25} - 4 q^{27} + 4 q^{33} + 28 q^{49} - 24 q^{51} + 20 q^{57} + 40 q^{59} - 20 q^{75} + 28 q^{81} + 8 q^{83} - 48 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 8 * q^11 + 20 * q^25 - 4 * q^27 + 4 * q^33 + 28 * q^49 - 24 * q^51 + 20 * q^57 + 40 * q^59 - 20 * q^75 + 28 * q^81 + 8 * q^83 - 48 * q^99

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