LMFDB - Embedded newform 240.2.o.a.239.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [240,2,Mod(239,240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(240, 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("240.239");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 240 = 2^{4} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	240.o (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:	$1.91640964851$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 9 $ x^4 + 4*x^2 + 9
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			239.1
Root			$-0.707107 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	240.239
Dual form			240.2.o.a.239.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})$	$q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +(-3.53553 + 1.58114i) q^{15} +(3.00000 + 6.70820i) q^{21} -9.48683i q^{23} -5.00000 q^{25} +(4.94975 + 1.58114i) q^{27} -8.94427i q^{29} +9.48683i q^{35} -4.47214i q^{41} +12.7279 q^{43} +(5.00000 + 4.47214i) q^{45} +9.48683i q^{47} +11.0000 q^{49} -8.00000 q^{61} +(8.48528 - 9.48683i) q^{63} +4.24264 q^{67} +(-15.0000 + 6.70820i) q^{69} +(3.53553 + 7.90569i) q^{75} +(-1.00000 - 8.94427i) q^{81} -9.48683i q^{83} +(-14.1421 + 6.32456i) q^{87} +17.8885i q^{89} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} + 12 q^{21} - 20 q^{25} + 20 q^{45} + 44 q^{49} - 32 q^{61} - 60 q^{69} - 4 q^{81}+O(q^{100}) $ 4 * q - 8 * q^9 + 12 * q^21 - 20 * q^25 + 20 * q^45 + 44 * q^49 - 32 * q^61 - 60 * q^69 - 4 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/240\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$181$
$\chi(n)$	$-1$	$-1$	$-1$	$1$

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
$3$	−0.707107	−	1.58114i	−0.408248	−	0.912871i
$3$	−0.707107	−	1.58114i	−0.408248	−	0.912871i
$4$		0			0
$5$		−	2.23607i		−	1.00000i
$5$		−	2.23607i		−	1.00000i
$6$		0			0
$7$	−4.24264			−1.60357			−0.801784	−	0.597614i	$-0.796115\pi$
$7$	−4.24264			−1.60357			−0.801784	+	0.597614i	$0.796115\pi$
$8$		0			0
$9$	−2.00000	+	2.23607i	−0.666667	+	0.745356i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	−3.53553	+	1.58114i	−0.912871	+	0.408248i
$16$		0			0
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$	3.00000	+	6.70820i	0.654654	+	1.46385i
$22$		0			0
$23$		−	9.48683i		−	1.97814i	−0.147442	−	0.989071i	$-0.547104\pi$
$23$		−	9.48683i		−	1.97814i	0.147442	−	0.989071i	$-0.452896\pi$
$24$		0			0
$25$	−5.00000			−1.00000
$26$		0			0
$27$	4.94975	+	1.58114i	0.952579	+	0.304290i
$28$		0			0
$29$		−	8.94427i		−	1.66091i	−0.557086	−	0.830455i	$-0.688081\pi$
$29$		−	8.94427i		−	1.66091i	0.557086	−	0.830455i	$-0.311919\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			9.48683i			1.60357i
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	4.47214i		−	0.698430i	−0.937043	−	0.349215i	$-0.886448\pi$
$41$		−	4.47214i		−	0.698430i	0.937043	−	0.349215i	$-0.113552\pi$
$42$		0			0
$43$	12.7279			1.94099			0.970495	−	0.241121i	$-0.0775152\pi$
$43$	12.7279			1.94099			0.970495	+	0.241121i	$0.0775152\pi$
$44$		0			0
$45$	5.00000	+	4.47214i	0.745356	+	0.666667i
$46$		0			0
$47$			9.48683i			1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$			9.48683i			1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$		0			0
$49$	11.0000			1.57143
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−8.00000			−1.02430			−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000			−1.02430			−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	8.48528	−	9.48683i	1.06904	−	1.19523i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.24264			0.518321			0.259161	−	0.965834i	$-0.416554\pi$
$67$	4.24264			0.518321			0.259161	+	0.965834i	$0.416554\pi$
$68$		0			0
$69$	−15.0000	+	6.70820i	−1.80579	+	0.807573i
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	3.53553	+	7.90569i	0.408248	+	0.912871i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−1.00000	−	8.94427i	−0.111111	−	0.993808i
$82$		0			0
$83$		−	9.48683i		−	1.04132i	−0.853766	−	0.520658i	$-0.825687\pi$
$83$		−	9.48683i		−	1.04132i	0.853766	−	0.520658i	$-0.174313\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−14.1421	+	6.32456i	−1.51620	+	0.678064i
$88$		0			0
$89$			17.8885i			1.89618i	0.317999	+	0.948091i	$0.396989\pi$
$89$			17.8885i			1.89618i	−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	8.94427i		−	0.889988i	−0.895533	−	0.444994i	$-0.853206\pi$
$101$		−	8.94427i		−	0.889988i	0.895533	−	0.444994i	$-0.146794\pi$
$102$		0			0
$103$	−12.7279			−1.25412			−0.627060	−	0.778971i	$-0.715742\pi$
$103$	−12.7279			−1.25412			−0.627060	+	0.778971i	$0.715742\pi$
$104$		0			0
$105$	15.0000	−	6.70820i	1.46385	−	0.654654i
$106$		0			0
$107$		−	9.48683i		−	0.917127i	−0.888662	−	0.458563i	$-0.848364\pi$
$107$		−	9.48683i		−	0.917127i	0.888662	−	0.458563i	$-0.151636\pi$
$108$		0			0
$109$	16.0000			1.53252			0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000			1.53252			0.766261	+	0.642529i	$0.222115\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−21.2132			−1.97814
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−7.07107	+	3.16228i	−0.637577	+	0.285133i
$124$		0			0
$125$			11.1803i			1.00000i
$126$		0			0
$127$	4.24264			0.376473			0.188237	−	0.982124i	$-0.439723\pi$
$127$	4.24264			0.376473			0.188237	+	0.982124i	$0.439723\pi$
$128$		0			0
$129$	−9.00000	−	20.1246i	−0.792406	−	1.77187i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.53553	−	11.0680i	0.304290	−	0.952579i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	15.0000	−	6.70820i	1.26323	−	0.564933i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−20.0000			−1.66091
$146$		0			0
$147$	−7.77817	−	17.3925i	−0.641533	−	1.43451i
$148$		0			0
$149$		−	4.47214i		−	0.366372i	−0.983078	−	0.183186i	$-0.941359\pi$
$149$		−	4.47214i		−	0.366372i	0.983078	−	0.183186i	$-0.0586410\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			40.2492i			3.17208i
$162$		0			0
$163$	−12.7279			−0.996928			−0.498464	−	0.866910i	$-0.666102\pi$
$163$	−12.7279			−0.996928			−0.498464	+	0.866910i	$0.666102\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.48683i			0.734113i	0.930199	+	0.367057i	$0.119634\pi$
$167$			9.48683i			0.734113i	−0.930199	+	0.367057i	$0.880366\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	21.2132			1.60357
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$	5.65685	+	12.6491i	0.418167	+	0.935049i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−21.0000	−	6.70820i	−1.52753	−	0.487950i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$	−3.00000	−	6.70820i	−0.211604	−	0.473160i
$202$		0			0
$203$			37.9473i			2.66338i
$204$		0			0
$205$	−10.0000			−0.698430
$206$		0			0
$207$	21.2132	+	18.9737i	1.47442	+	1.31876i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	28.4605i		−	1.94099i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	29.6985			1.98876			0.994379	−	0.105881i	$-0.0337662\pi$
$223$	29.6985			1.98876			0.994379	+	0.105881i	$0.0337662\pi$
$224$		0			0
$225$	10.0000	−	11.1803i	0.666667	−	0.745356i
$226$		0			0
$227$		−	28.4605i		−	1.88899i	−0.328526	−	0.944495i	$-0.606552\pi$
$227$		−	28.4605i		−	1.88899i	0.328526	−	0.944495i	$-0.393448\pi$
$228$		0			0
$229$	14.0000			0.925146			0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000			0.925146			0.462573	+	0.886581i	$0.346926\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	21.2132			1.38380
$236$		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
$241$	28.0000			1.80364			0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000			1.80364			0.901819	+	0.432113i	$0.142232\pi$
$242$		0			0
$243$	−13.4350	+	7.90569i	−0.861858	+	0.507151i
$244$		0			0
$245$		−	24.5967i		−	1.57143i
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−15.0000	+	6.70820i	−0.950586	+	0.425115i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	20.0000	+	17.8885i	1.23797	+	1.10727i
$262$		0			0
$263$			28.4605i			1.75495i	0.479623	+	0.877475i	$0.340774\pi$
$263$			28.4605i			1.75495i	−0.479623	+	0.877475i	$0.659226\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	28.2843	−	12.6491i	1.73097	−	0.774113i
$268$		0			0
$269$			22.3607i			1.36335i	0.731653	+	0.681677i	$0.238749\pi$
$269$			22.3607i			1.36335i	−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	31.3050i		−	1.86750i	−0.357930	−	0.933748i	$-0.616517\pi$
$281$		−	31.3050i		−	1.86750i	0.357930	−	0.933748i	$-0.383483\pi$
$282$		0			0
$283$	−29.6985			−1.76539			−0.882696	−	0.469945i	$-0.844274\pi$
$283$	−29.6985			−1.76539			−0.882696	+	0.469945i	$0.844274\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			18.9737i			1.11998i
$288$		0			0
$289$	−17.0000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−54.0000			−3.11251
$302$		0			0
$303$	−14.1421	+	6.32456i	−0.812444	+	0.363336i
$304$		0			0
$305$			17.8885i			1.02430i
$306$		0			0
$307$	−4.24264			−0.242140			−0.121070	−	0.992644i	$-0.538633\pi$
$307$	−4.24264			−0.242140			−0.121070	+	0.992644i	$0.538633\pi$
$308$		0			0
$309$	9.00000	+	20.1246i	0.511992	+	1.14485i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−21.2132	−	18.9737i	−1.19523	−	1.06904i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−15.0000	+	6.70820i	−0.837218	+	0.374415i
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−11.3137	−	25.2982i	−0.625650	−	1.39899i
$328$		0			0
$329$		−	40.2492i		−	2.21901i
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	9.48683i		−	0.518321i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−16.9706			−0.916324
$344$		0			0
$345$	15.0000	+	33.5410i	0.807573	+	1.80579i
$346$		0			0
$347$		−	28.4605i		−	1.52784i	−0.645311	−	0.763920i	$-0.723272\pi$
$347$		−	28.4605i		−	1.52784i	0.645311	−	0.763920i	$-0.276728\pi$
$348$		0			0
$349$	26.0000			1.39175			0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000			1.39175			0.695874	+	0.718164i	$0.255017\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		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
$361$	19.0000			1.00000
$362$		0			0
$363$	7.77817	+	17.3925i	0.408248	+	0.912871i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−38.1838			−1.99318			−0.996588	−	0.0825348i	$-0.973698\pi$
$367$	−38.1838			−1.99318			−0.996588	+	0.0825348i	$0.973698\pi$
$368$		0			0
$369$	10.0000	+	8.94427i	0.520579	+	0.465620i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	17.6777	−	7.90569i	0.912871	−	0.408248i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−3.00000	−	6.70820i	−0.153695	−	0.343672i
$382$		0			0
$383$		−	28.4605i		−	1.45426i	−0.686498	−	0.727132i	$-0.740853\pi$
$383$		−	28.4605i		−	1.45426i	0.686498	−	0.727132i	$-0.259147\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−25.4558	+	28.4605i	−1.29399	+	1.44673i
$388$		0			0
$389$		−	31.3050i		−	1.58722i	−0.608424	−	0.793612i	$-0.708198\pi$
$389$		−	31.3050i		−	1.58722i	0.608424	−	0.793612i	$-0.291802\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	35.7771i		−	1.78662i	−0.449439	−	0.893311i	$-0.648376\pi$
$401$		−	35.7771i		−	1.78662i	0.449439	−	0.893311i	$-0.351624\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−20.0000	+	2.23607i	−0.993808	+	0.111111i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	4.00000			0.197787			0.0988936	−	0.995098i	$-0.468470\pi$
$409$	4.00000			0.197787			0.0988936	+	0.995098i	$0.468470\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−21.2132			−1.04132
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−8.00000			−0.389896			−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000			−0.389896			−0.194948	+	0.980814i	$0.562454\pi$
$422$		0			0
$423$	−21.2132	−	18.9737i	−1.03142	−	0.922531i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	33.9411			1.64253
$428$		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
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	14.1421	+	31.6228i	0.678064	+	1.51620i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−22.0000	+	24.5967i	−1.04762	+	1.17127i
$442$		0			0
$443$			9.48683i			0.450733i	0.974274	+	0.225367i	$0.0723580\pi$
$443$			9.48683i			0.450733i	−0.974274	+	0.225367i	$0.927642\pi$
$444$		0			0
$445$	40.0000			1.89618
$446$		0			0
$447$	−7.07107	+	3.16228i	−0.334450	+	0.149571i
$448$		0			0
$449$			22.3607i			1.05527i	0.849473	+	0.527633i	$0.176920\pi$
$449$			22.3607i			1.05527i	−0.849473	+	0.527633i	$0.823080\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	8.94427i		−	0.416576i	−0.978068	−	0.208288i	$-0.933211\pi$
$461$		−	8.94427i		−	0.416576i	0.978068	−	0.208288i	$-0.0667892\pi$
$462$		0			0
$463$	12.7279			0.591517			0.295758	−	0.955263i	$-0.404428\pi$
$463$	12.7279			0.591517			0.295758	+	0.955263i	$0.404428\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			28.4605i			1.31699i	0.752583	+	0.658497i	$0.228808\pi$
$467$			28.4605i			1.31699i	−0.752583	+	0.658497i	$0.771192\pi$
$468$		0			0
$469$	−18.0000			−0.831163
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		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
$483$	63.6396	−	28.4605i	2.89570	−	1.29500i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	38.1838			1.73027			0.865136	−	0.501538i	$-0.167232\pi$
$487$	38.1838			1.73027			0.865136	+	0.501538i	$0.167232\pi$
$488$		0			0
$489$	9.00000	+	20.1246i	0.406994	+	0.910066i
$490$		0			0
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$		0			0
$501$	15.0000	−	6.70820i	0.670151	−	0.299700i
$502$		0			0
$503$			9.48683i			0.422997i	0.977378	+	0.211498i	$0.0678343\pi$
$503$			9.48683i			0.422997i	−0.977378	+	0.211498i	$0.932166\pi$
$504$		0			0
$505$	−20.0000			−0.889988
$506$		0			0
$507$	−9.19239	−	20.5548i	−0.408248	−	0.912871i
$508$		0			0
$509$			44.7214i			1.98224i	0.132973	+	0.991120i	$0.457548\pi$
$509$			44.7214i			1.98224i	−0.132973	+	0.991120i	$0.542452\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			28.4605i			1.25412i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8885i			0.783711i	0.920027	+	0.391856i	$0.128167\pi$
$521$			17.8885i			0.783711i	−0.920027	+	0.391856i	$0.871833\pi$
$522$		0			0
$523$	29.6985			1.29862			0.649312	−	0.760522i	$-0.275057\pi$
$523$	29.6985			1.29862			0.649312	+	0.760522i	$0.275057\pi$
$524$		0			0
$525$	−15.0000	−	33.5410i	−0.654654	−	1.46385i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−67.0000			−2.91304
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−21.2132			−0.917127
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$		0			0
$543$	−1.41421	−	3.16228i	−0.0606897	−	0.135706i
$544$		0			0
$545$		−	35.7771i		−	1.53252i
$546$		0			0
$547$	46.6690			1.99542			0.997712	−	0.0676046i	$-0.0215356\pi$
$547$	46.6690			1.99542			0.997712	+	0.0676046i	$0.0215356\pi$
$548$		0			0
$549$	16.0000	−	17.8885i	0.682863	−	0.763464i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	47.4342i		−	1.99911i	−0.0298010	−	0.999556i	$-0.509487\pi$
$563$		−	47.4342i		−	1.99911i	0.0298010	−	0.999556i	$-0.490513\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.24264	+	37.9473i	0.178174	+	1.59364i
$568$		0			0
$569$		−	31.3050i		−	1.31237i	−0.754599	−	0.656186i	$-0.772169\pi$
$569$		−	31.3050i		−	1.31237i	0.754599	−	0.656186i	$-0.227831\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			47.4342i			1.97814i
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			40.2492i			1.66982i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			47.4342i			1.95782i	0.204298	+	0.978909i	$0.434509\pi$
$587$			47.4342i			1.95782i	−0.204298	+	0.978909i	$0.565491\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		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
$601$	28.0000			1.14214			0.571072	−	0.820900i	$-0.306528\pi$
$601$	28.0000			1.14214			0.571072	+	0.820900i	$0.306528\pi$
$602$		0			0
$603$	−8.48528	+	9.48683i	−0.345547	+	0.386334i
$604$		0			0
$605$			24.5967i			1.00000i
$606$		0			0
$607$	−46.6690			−1.89424			−0.947119	−	0.320882i	$-0.896021\pi$
$607$	−46.6690			−1.89424			−0.947119	+	0.320882i	$0.896021\pi$
$608$		0			0
$609$	60.0000	−	26.8328i	2.43132	−	1.08732i
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	7.07107	+	15.8114i	0.285133	+	0.637577i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	15.0000	−	46.9574i	0.601929	−	1.88434i
$622$		0			0
$623$		−	75.8947i		−	3.04066i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	9.48683i		−	0.376473i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			49.1935i			1.94303i	0.236986	+	0.971513i	$0.423841\pi$
$641$			49.1935i			1.94303i	−0.236986	+	0.971513i	$0.576159\pi$
$642$		0			0
$643$	29.6985			1.17119			0.585597	−	0.810602i	$-0.300860\pi$
$643$	29.6985			1.17119			0.585597	+	0.810602i	$0.300860\pi$
$644$		0			0
$645$	−45.0000	+	20.1246i	−1.77187	+	0.792406i
$646$		0			0
$647$		−	47.4342i		−	1.86483i	−0.361390	−	0.932415i	$-0.617698\pi$
$647$		−	47.4342i		−	1.86483i	0.361390	−	0.932415i	$-0.382302\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−32.0000			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$661$	−32.0000			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−84.8528			−3.28551
$668$		0			0
$669$	−21.0000	−	46.9574i	−0.811907	−	1.81548i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−24.7487	−	7.90569i	−0.952579	−	0.304290i
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−45.0000	+	20.1246i	−1.72440	+	0.771177i
$682$		0			0
$683$		−	28.4605i		−	1.08901i	−0.838757	−	0.544505i	$-0.816717\pi$
$683$		−	28.4605i		−	1.08901i	0.838757	−	0.544505i	$-0.183283\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−9.89949	−	22.1359i	−0.377689	−	0.844539i
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$			22.3607i			0.844551i	0.906467	+	0.422276i	$0.138769\pi$
$701$			22.3607i			0.844551i	−0.906467	+	0.422276i	$0.861231\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−15.0000	−	33.5410i	−0.564933	−	1.26323i
$706$		0			0
$707$			37.9473i			1.42716i
$708$		0			0
$709$	−46.0000			−1.72757			−0.863783	−	0.503864i	$-0.831911\pi$
$709$	−46.0000			−1.72757			−0.863783	+	0.503864i	$0.831911\pi$
$710$		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
$721$	54.0000			2.01107
$722$		0			0
$723$	−19.7990	−	44.2719i	−0.736332	−	1.64649i
$724$		0			0
$725$			44.7214i			1.66091i
$726$		0			0
$727$	4.24264			0.157351			0.0786754	−	0.996900i	$-0.474931\pi$
$727$	4.24264			0.157351			0.0786754	+	0.996900i	$0.474931\pi$
$728$		0			0
$729$	22.0000	+	15.6525i	0.814815	+	0.579721i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−38.8909	+	17.3925i	−1.43451	+	0.641533i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			47.4342i			1.74019i	0.492883	+	0.870095i	$0.335943\pi$
$743$			47.4342i			1.74019i	−0.492883	+	0.870095i	$0.664057\pi$
$744$		0			0
$745$	−10.0000			−0.366372
$746$		0			0
$747$	21.2132	+	18.9737i	0.776151	+	0.694210i
$748$		0			0
$749$			40.2492i			1.47067i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	35.7771i		−	1.29692i	−0.761249	−	0.648459i	$-0.775414\pi$
$761$		−	35.7771i		−	1.29692i	0.761249	−	0.648459i	$-0.224586\pi$
$762$		0			0
$763$	−67.8823			−2.45750
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	14.1421	−	44.2719i	0.505399	−	1.58215i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−38.1838			−1.36110			−0.680552	−	0.732700i	$-0.738260\pi$
$787$	−38.1838			−1.36110			−0.680552	+	0.732700i	$0.738260\pi$
$788$		0			0
$789$	45.0000	−	20.1246i	1.60204	−	0.716455i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−40.0000	−	35.7771i	−1.41333	−	1.26412i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	90.0000			3.17208
$806$		0			0
$807$	35.3553	−	15.8114i	1.24457	−	0.556587i
$808$		0			0
$809$			17.8885i			0.628928i	0.949269	+	0.314464i	$0.101825\pi$
$809$			17.8885i			0.628928i	−0.949269	+	0.314464i	$0.898175\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			28.4605i			0.996928i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	31.3050i		−	1.09255i	−0.837606	−	0.546275i	$-0.816045\pi$
$821$		−	31.3050i		−	1.09255i	0.837606	−	0.546275i	$-0.183955\pi$
$822$		0			0
$823$	55.1543			1.92256			0.961280	−	0.275575i	$-0.0888683\pi$
$823$	55.1543			1.92256			0.961280	+	0.275575i	$0.0888683\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	47.4342i		−	1.64945i	−0.565536	−	0.824724i	$-0.691331\pi$
$827$		−	47.4342i		−	1.64945i	0.565536	−	0.824724i	$-0.308669\pi$
$828$		0			0
$829$	−56.0000			−1.94496			−0.972480	−	0.232986i	$-0.925151\pi$
$829$	−56.0000			−1.94496			−0.972480	+	0.232986i	$0.925151\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	21.2132			0.734113
$836$		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
$841$	−51.0000			−1.75862
$842$		0			0
$843$	−49.4975	+	22.1359i	−1.70478	+	0.762402i
$844$		0			0
$845$		−	29.0689i		−	1.00000i
$846$		0			0
$847$	46.6690			1.60357
$848$		0			0
$849$	21.0000	+	46.9574i	0.720718	+	1.61157i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$	30.0000	−	13.4164i	1.02240	−	0.457230i
$862$		0			0
$863$			9.48683i			0.322936i	0.986878	+	0.161468i	$0.0516228\pi$
$863$			9.48683i			0.322936i	−0.986878	+	0.161468i	$0.948377\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	12.0208	+	26.8794i	0.408248	+	0.912871i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	47.4342i		−	1.60357i
$876$		0			0
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	58.1378i		−	1.95871i	−0.202145	−	0.979356i	$-0.564791\pi$
$881$		−	58.1378i		−	1.95871i	0.202145	−	0.979356i	$-0.435209\pi$
$882$		0			0
$883$	55.1543			1.85609			0.928045	−	0.372467i	$-0.121488\pi$
$883$	55.1543			1.85609			0.928045	+	0.372467i	$0.121488\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			28.4605i			0.955610i	0.878466	+	0.477805i	$0.158567\pi$
$887$			28.4605i			0.955610i	−0.878466	+	0.477805i	$0.841433\pi$
$888$		0			0
$889$	−18.0000			−0.603701
$890$		0			0
$891$		0			0
$892$		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
$903$	38.1838	+	85.3815i	1.27068	+	2.84132i
$904$		0			0
$905$		−	4.47214i		−	0.148659i
$906$		0			0
$907$	4.24264			0.140875			0.0704373	−	0.997516i	$-0.477561\pi$
$907$	4.24264			0.140875			0.0704373	+	0.997516i	$0.477561\pi$
$908$		0			0
$909$	20.0000	+	17.8885i	0.663358	+	0.593326i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	28.2843	−	12.6491i	0.935049	−	0.418167i
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	3.00000	+	6.70820i	0.0988534	+	0.221043i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	25.4558	−	28.4605i	0.836080	−	0.934765i
$928$		0			0
$929$			49.1935i			1.61399i	0.590561	+	0.806993i	$0.298907\pi$
$929$			49.1935i			1.61399i	−0.590561	+	0.806993i	$0.701093\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			44.7214i			1.45787i	0.684580	+	0.728937i	$0.259985\pi$
$941$			44.7214i			1.45787i	−0.684580	+	0.728937i	$0.740015\pi$
$942$		0			0
$943$	−42.4264			−1.38159
$944$		0			0
$945$	−15.0000	+	46.9574i	−0.487950	+	1.52753i
$946$		0			0
$947$			9.48683i			0.308281i	0.988049	+	0.154140i	$0.0492608\pi$
$947$			9.48683i			0.308281i	−0.988049	+	0.154140i	$0.950739\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		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
$963$	21.2132	+	18.9737i	0.683586	+	0.611418i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	46.6690			1.50078			0.750388	−	0.660998i	$-0.229867\pi$
$967$	46.6690			1.50078			0.750388	+	0.660998i	$0.229867\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−32.0000	+	35.7771i	−1.02168	+	1.14227i
$982$		0			0
$983$			47.4342i			1.51291i	0.654043	+	0.756457i	$0.273072\pi$
$983$			47.4342i			1.51291i	−0.654043	+	0.756457i	$0.726928\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−63.6396	+	28.4605i	−2.02567	+	0.905908i
$988$		0			0
$989$		−	120.748i		−	3.83955i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	240.2.o.a.239.1	✓	4
3.2	odd	2	inner	240.2.o.a.239.2	yes	4
4.3	odd	2	inner	240.2.o.a.239.4	yes	4
5.2	odd	4		1200.2.h.l.1151.2		4
5.3	odd	4		1200.2.h.l.1151.3		4
5.4	even	2	inner	240.2.o.a.239.4	yes	4
8.3	odd	2		960.2.o.b.959.1		4
8.5	even	2		960.2.o.b.959.4		4
12.11	even	2	inner	240.2.o.a.239.3	yes	4
15.2	even	4		1200.2.h.l.1151.4		4
15.8	even	4		1200.2.h.l.1151.1		4
15.14	odd	2	inner	240.2.o.a.239.3	yes	4
20.3	even	4		1200.2.h.l.1151.2		4
20.7	even	4		1200.2.h.l.1151.3		4
20.19	odd	2	CM	240.2.o.a.239.1	✓	4
24.5	odd	2		960.2.o.b.959.3		4
24.11	even	2		960.2.o.b.959.2		4
40.19	odd	2		960.2.o.b.959.4		4
40.29	even	2		960.2.o.b.959.1		4
60.23	odd	4		1200.2.h.l.1151.4		4
60.47	odd	4		1200.2.h.l.1151.1		4
60.59	even	2	inner	240.2.o.a.239.2	yes	4
120.29	odd	2		960.2.o.b.959.2		4
120.59	even	2		960.2.o.b.959.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.o.a.239.1	✓	4	1.1	even	1	trivial
240.2.o.a.239.1	✓	4	20.19	odd	2	CM
240.2.o.a.239.2	yes	4	3.2	odd	2	inner
240.2.o.a.239.2	yes	4	60.59	even	2	inner
240.2.o.a.239.3	yes	4	12.11	even	2	inner
240.2.o.a.239.3	yes	4	15.14	odd	2	inner
240.2.o.a.239.4	yes	4	4.3	odd	2	inner
240.2.o.a.239.4	yes	4	5.4	even	2	inner
960.2.o.b.959.1		4	8.3	odd	2
960.2.o.b.959.1		4	40.29	even	2
960.2.o.b.959.2		4	24.11	even	2
960.2.o.b.959.2		4	120.29	odd	2
960.2.o.b.959.3		4	24.5	odd	2
960.2.o.b.959.3		4	120.59	even	2
960.2.o.b.959.4		4	8.5	even	2
960.2.o.b.959.4		4	40.19	odd	2
1200.2.h.l.1151.1		4	15.8	even	4
1200.2.h.l.1151.1		4	60.47	odd	4
1200.2.h.l.1151.2		4	5.2	odd	4
1200.2.h.l.1151.2		4	20.3	even	4
1200.2.h.l.1151.3		4	5.3	odd	4
1200.2.h.l.1151.3		4	20.7	even	4
1200.2.h.l.1151.4		4	15.2	even	4
1200.2.h.l.1151.4		4	60.23	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +O(q^{10})\)	\(q+(-0.707107 - 1.58114i) q^{3} -2.23607i q^{5} -4.24264 q^{7} +(-2.00000 + 2.23607i) q^{9} +(-3.53553 + 1.58114i) q^{15} +(3.00000 + 6.70820i) q^{21} -9.48683i q^{23} -5.00000 q^{25} +(4.94975 + 1.58114i) q^{27} -8.94427i q^{29} +9.48683i q^{35} -4.47214i q^{41} +12.7279 q^{43} +(5.00000 + 4.47214i) q^{45} +9.48683i q^{47} +11.0000 q^{49} -8.00000 q^{61} +(8.48528 - 9.48683i) q^{63} +4.24264 q^{67} +(-15.0000 + 6.70820i) q^{69} +(3.53553 + 7.90569i) q^{75} +(-1.00000 - 8.94427i) q^{81} -9.48683i q^{83} +(-14.1421 + 6.32456i) q^{87} +17.8885i q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} + 12 q^{21} - 20 q^{25} + 20 q^{45} + 44 q^{49} - 32 q^{61} - 60 q^{69} - 4 q^{81}+O(q^{100}) \) 4 * q - 8 * q^9 + 12 * q^21 - 20 * q^25 + 20 * q^45 + 44 * q^49 - 32 * q^61 - 60 * q^69 - 4 * q^81

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