LMFDB - Embedded newform 180.2.h.a.179.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [180,2,Mod(179,180)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("180.179"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(180, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 180 = 2^{2} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	180.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$1.43730723638$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			179.4
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	180.179
Dual form			180.2.h.a.179.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +2.82843 q^{8} +(-1.00000 + 3.00000i) q^{10} -6.00000i q^{13} +4.00000 q^{16} -7.07107 q^{17} +(-1.41421 + 4.24264i) q^{20} +(-4.00000 - 3.00000i) q^{25} -8.48528i q^{26} +4.24264i q^{29} +5.65685 q^{32} -10.0000 q^{34} +12.0000i q^{37} +(-2.00000 + 6.00000i) q^{40} -12.7279i q^{41} -7.00000 q^{49} +(-5.65685 - 4.24264i) q^{50} -12.0000i q^{52} +7.07107 q^{53} +6.00000i q^{58} +10.0000 q^{61} +8.00000 q^{64} +(12.7279 + 4.24264i) q^{65} -14.1421 q^{68} +6.00000i q^{73} +16.9706i q^{74} +(-2.82843 + 8.48528i) q^{80} -18.0000i q^{82} +(5.00000 - 15.0000i) q^{85} -4.24264i q^{89} +18.0000i q^{97} -9.89949 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 4 q^{10} + 16 q^{16} - 16 q^{25} - 40 q^{34} - 8 q^{40} - 28 q^{49} + 40 q^{61} + 32 q^{64} + 20 q^{85}+O(q^{100}) $ 4 * q + 8 * q^4 - 4 * q^10 + 16 * q^16 - 16 * q^25 - 40 * q^34 - 8 * q^40 - 28 * q^49 + 40 * q^61 + 32 * q^64 + 20 * q^85

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/180\mathbb{Z}\right)^\times$.

$n$	$37$	$91$	$101$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.41421			1.00000
$2$	1.41421			1.00000
$3$		0			0
$3$		0			0
$4$	2.00000			1.00000
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$5$	−0.707107	+	2.12132i	−0.316228	+	0.948683i
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	2.82843			1.00000
$9$		0			0
$10$	−1.00000	+	3.00000i	−0.316228	+	0.948683i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	6.00000i		−	1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$		−	6.00000i		−	1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$	−7.07107			−1.71499			−0.857493	−	0.514496i	$-0.827979\pi$
$17$	−7.07107			−1.71499			−0.857493	+	0.514496i	$0.827979\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−1.41421	+	4.24264i	−0.316228	+	0.948683i
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−4.00000	−	3.00000i	−0.800000	−	0.600000i
$26$		−	8.48528i		−	1.66410i
$27$		0			0
$28$		0			0
$29$			4.24264i			0.787839i	0.919145	+	0.393919i	$0.128881\pi$
$29$			4.24264i			0.787839i	−0.919145	+	0.393919i	$0.871119\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	5.65685			1.00000
$33$		0			0
$34$	−10.0000			−1.71499
$35$		0			0
$36$		0			0
$37$			12.0000i			1.97279i	0.164399	+	0.986394i	$0.447432\pi$
$37$			12.0000i			1.97279i	−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$	−2.00000	+	6.00000i	−0.316228	+	0.948683i
$41$		−	12.7279i		−	1.98777i	−0.110432	−	0.993884i	$-0.535223\pi$
$41$		−	12.7279i		−	1.98777i	0.110432	−	0.993884i	$-0.464777\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$	−5.65685	−	4.24264i	−0.800000	−	0.600000i
$51$		0			0
$52$		−	12.0000i		−	1.66410i
$53$	7.07107			0.971286			0.485643	−	0.874157i	$-0.338586\pi$
$53$	7.07107			0.971286			0.485643	+	0.874157i	$0.338586\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$			6.00000i			0.787839i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	10.0000			1.28037			0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000			1.28037			0.640184	+	0.768221i	$0.278858\pi$
$62$		0			0
$63$		0			0
$64$	8.00000			1.00000
$65$	12.7279	+	4.24264i	1.57870	+	0.526235i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	−14.1421			−1.71499
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$			6.00000i			0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$			6.00000i			0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$			16.9706i			1.97279i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−2.82843	+	8.48528i	−0.316228	+	0.948683i
$81$		0			0
$82$		−	18.0000i		−	1.98777i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$	5.00000	−	15.0000i	0.542326	−	1.62698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.24264i		−	0.449719i	−0.974391	−	0.224860i	$-0.927808\pi$
$89$		−	4.24264i		−	0.449719i	0.974391	−	0.224860i	$-0.0721923\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$			18.0000i			1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$			18.0000i			1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	−9.89949			−1.00000
$99$		0			0
$100$	−8.00000	−	6.00000i	−0.800000	−	0.600000i
$101$			12.7279i			1.26648i	0.773957	+	0.633238i	$0.218274\pi$
$101$			12.7279i			1.26648i	−0.773957	+	0.633238i	$0.781726\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		−	16.9706i		−	1.66410i
$105$		0			0
$106$	10.0000			0.971286
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	20.0000			1.91565			0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000			1.91565			0.957826	+	0.287348i	$0.0927736\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.41421			0.133038			0.0665190	−	0.997785i	$-0.478811\pi$
$113$	1.41421			0.133038			0.0665190	+	0.997785i	$0.478811\pi$
$114$		0			0
$115$		0			0
$116$			8.48528i			0.787839i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−11.0000			−1.00000
$122$	14.1421			1.28037
$123$		0			0
$124$		0			0
$125$	9.19239	−	6.36396i	0.822192	−	0.569210i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	11.3137			1.00000
$129$		0			0
$130$	18.0000	+	6.00000i	1.57870	+	0.526235i
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−20.0000			−1.71499
$137$	−9.89949			−0.845771			−0.422885	−	0.906183i	$-0.638983\pi$
$137$	−9.89949			−0.845771			−0.422885	+	0.906183i	$0.638983\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−9.00000	−	3.00000i	−0.747409	−	0.249136i
$146$			8.48528i			0.702247i
$147$		0			0
$148$			24.0000i			1.97279i
$149$		−	4.24264i		−	0.347571i	−0.984784	−	0.173785i	$-0.944400\pi$
$149$		−	4.24264i		−	0.347571i	0.984784	−	0.173785i	$-0.0555999\pi$
$150$		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
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$		0			0
$159$		0			0
$160$	−4.00000	+	12.0000i	−0.316228	+	0.948683i
$161$		0			0
$162$		0			0
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		−	25.4558i		−	1.98777i
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−23.0000			−1.76923
$170$	7.07107	−	21.2132i	0.542326	−	1.62698i
$171$		0			0
$172$		0			0
$173$	−15.5563			−1.18273			−0.591364	−	0.806405i	$-0.701410\pi$
$173$	−15.5563			−1.18273			−0.591364	+	0.806405i	$0.701410\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		−	6.00000i		−	0.449719i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−20.0000			−1.48659			−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000			−1.48659			−0.743294	+	0.668965i	$0.766738\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−25.4558	−	8.48528i	−1.87155	−	0.623850i
$186$		0			0
$187$		0			0
$188$		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
$193$		−	24.0000i		−	1.72756i	−0.503871	−	0.863779i	$-0.668091\pi$
$193$		−	24.0000i		−	1.72756i	0.503871	−	0.863779i	$-0.331909\pi$
$194$			25.4558i			1.82762i
$195$		0			0
$196$	−14.0000			−1.00000
$197$	−18.3848			−1.30986			−0.654931	−	0.755689i	$-0.727302\pi$
$197$	−18.3848			−1.30986			−0.654931	+	0.755689i	$0.727302\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−11.3137	−	8.48528i	−0.800000	−	0.600000i
$201$		0			0
$202$			18.0000i			1.26648i
$203$		0			0
$204$		0			0
$205$	27.0000	+	9.00000i	1.88576	+	0.628587i
$206$		0			0
$207$		0			0
$208$		−	24.0000i		−	1.66410i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	14.1421			0.971286
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	28.2843			1.91565
$219$		0			0
$220$		0			0
$221$			42.4264i			2.85391i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$	2.00000			0.133038
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	4.00000			0.264327			0.132164	−	0.991228i	$-0.457808\pi$
$229$	4.00000			0.264327			0.132164	+	0.991228i	$0.457808\pi$
$230$		0			0
$231$		0			0
$232$			12.0000i			0.787839i
$233$	7.07107			0.463241			0.231621	−	0.972806i	$-0.425597\pi$
$233$	7.07107			0.463241			0.231621	+	0.972806i	$0.425597\pi$
$234$		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
$241$	8.00000			0.515325			0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$	−15.5563			−1.00000
$243$		0			0
$244$	20.0000			1.28037
$245$	4.94975	−	14.8492i	0.316228	−	0.948683i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	13.0000	−	9.00000i	0.822192	−	0.569210i
$251$		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
$256$	16.0000			1.00000
$257$	24.0416			1.49968			0.749838	−	0.661622i	$-0.230131\pi$
$257$	24.0416			1.49968			0.749838	+	0.661622i	$0.230131\pi$
$258$		0			0
$259$		0			0
$260$	25.4558	+	8.48528i	1.57870	+	0.526235i
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$	−5.00000	+	15.0000i	−0.307148	+	0.921443i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			4.24264i			0.258678i	0.991600	+	0.129339i	$0.0412856\pi$
$269$			4.24264i			0.258678i	−0.991600	+	0.129339i	$0.958714\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−28.2843			−1.71499
$273$		0			0
$274$	−14.0000			−0.845771
$275$		0			0
$276$		0			0
$277$		−	18.0000i		−	1.08152i	−0.841178	−	0.540758i	$-0.818138\pi$
$277$		−	18.0000i		−	1.08152i	0.841178	−	0.540758i	$-0.181862\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	29.6985i		−	1.77166i	−0.464007	−	0.885832i	$-0.653589\pi$
$281$		−	29.6985i		−	1.77166i	0.464007	−	0.885832i	$-0.346411\pi$
$282$		0			0
$283$		0			0			−	1.00000i	$-0.5\pi$
$283$		0			0				1.00000i	$0.5\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	33.0000			1.94118
$290$	−12.7279	−	4.24264i	−0.747409	−	0.249136i
$291$		0			0
$292$			12.0000i			0.702247i
$293$	26.8701			1.56977			0.784883	−	0.619644i	$-0.212723\pi$
$293$	26.8701			1.56977			0.784883	+	0.619644i	$0.212723\pi$
$294$		0			0
$295$		0			0
$296$			33.9411i			1.97279i
$297$		0			0
$298$		−	6.00000i		−	0.347571i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.07107	+	21.2132i	−0.404888	+	1.21466i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		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
$313$			24.0000i			1.35656i	0.734803	+	0.678280i	$0.237274\pi$
$313$			24.0000i			1.35656i	−0.734803	+	0.678280i	$0.762726\pi$
$314$		−	16.9706i		−	0.957704i
$315$		0			0
$316$		0			0
$317$	35.3553			1.98575			0.992877	−	0.119145i	$-0.0380154\pi$
$317$	35.3553			1.98575			0.992877	+	0.119145i	$0.0380154\pi$
$318$		0			0
$319$		0			0
$320$	−5.65685	+	16.9706i	−0.316228	+	0.948683i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−18.0000	+	24.0000i	−0.998460	+	1.33128i
$326$		0			0
$327$		0			0
$328$		−	36.0000i		−	1.98777i
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	18.0000i		−	0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$		−	18.0000i		−	0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	−32.5269			−1.76923
$339$		0			0
$340$	10.0000	−	30.0000i	0.542326	−	1.62698i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−22.0000			−1.18273
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−35.3553			−1.88177			−0.940887	−	0.338719i	$-0.890006\pi$
$353$	−35.3553			−1.88177			−0.940887	+	0.338719i	$0.890006\pi$
$354$		0			0
$355$		0			0
$356$		−	8.48528i		−	0.449719i
$357$		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
$362$	−28.2843			−1.48659
$363$		0			0
$364$		0			0
$365$	−12.7279	−	4.24264i	−0.666210	−	0.222070i
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−36.0000	−	12.0000i	−1.87155	−	0.623850i
$371$		0			0
$372$		0			0
$373$			36.0000i			1.86401i	0.362446	+	0.932005i	$0.381942\pi$
$373$			36.0000i			1.86401i	−0.362446	+	0.932005i	$0.618058\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	25.4558			1.31104
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		−	33.9411i		−	1.72756i
$387$		0			0
$388$			36.0000i			1.82762i
$389$		−	38.1838i		−	1.93599i	−0.250962	−	0.967997i	$-0.580747\pi$
$389$		−	38.1838i		−	1.93599i	0.250962	−	0.967997i	$-0.419253\pi$
$390$		0			0
$391$		0			0
$392$	−19.7990			−1.00000
$393$		0			0
$394$	−26.0000			−1.30986
$395$		0			0
$396$		0			0
$397$		−	12.0000i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$397$		−	12.0000i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$398$		0			0
$399$		0			0
$400$	−16.0000	−	12.0000i	−0.800000	−	0.600000i
$401$		−	29.6985i		−	1.48307i	−0.670913	−	0.741536i	$-0.734098\pi$
$401$		−	29.6985i		−	1.48307i	0.670913	−	0.741536i	$-0.265902\pi$
$402$		0			0
$403$		0			0
$404$			25.4558i			1.26648i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−40.0000			−1.97787			−0.988936	−	0.148340i	$-0.952607\pi$
$409$	−40.0000			−1.97787			−0.988936	+	0.148340i	$0.952607\pi$
$410$	38.1838	+	12.7279i	1.88576	+	0.628587i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		−	33.9411i		−	1.66410i
$417$		0			0
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$		0			0
$423$		0			0
$424$	20.0000			0.971286
$425$	28.2843	+	21.2132i	1.37199	+	1.02899i
$426$		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
$433$			24.0000i			1.15337i	0.816968	+	0.576683i	$0.195653\pi$
$433$			24.0000i			1.15337i	−0.816968	+	0.576683i	$0.804347\pi$
$434$		0			0
$435$		0			0
$436$	40.0000			1.91565
$437$		0			0
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$			60.0000i			2.85391i
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	9.00000	+	3.00000i	0.426641	+	0.142214i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			38.1838i			1.80200i	0.433816	+	0.901002i	$0.357167\pi$
$449$			38.1838i			1.80200i	−0.433816	+	0.901002i	$0.642833\pi$
$450$		0			0
$451$		0			0
$452$	2.82843			0.133038
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	42.0000i		−	1.96468i	−0.187112	−	0.982339i	$-0.559913\pi$
$457$		−	42.0000i		−	1.96468i	0.187112	−	0.982339i	$-0.440087\pi$
$458$	5.65685			0.264327
$459$		0			0
$460$		0			0
$461$		−	12.7279i		−	0.592798i	−0.955064	−	0.296399i	$-0.904214\pi$
$461$		−	12.7279i		−	0.592798i	0.955064	−	0.296399i	$-0.0957859\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$			16.9706i			0.787839i
$465$		0			0
$466$	10.0000			0.463241
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		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
$481$	72.0000			3.28292
$482$	11.3137			0.515325
$483$		0			0
$484$	−22.0000			−1.00000
$485$	−38.1838	−	12.7279i	−1.73384	−	0.577945i
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$	28.2843			1.28037
$489$		0			0
$490$	7.00000	−	21.0000i	0.316228	−	0.948683i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		0			0
$493$		−	30.0000i		−	1.35113i
$494$		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			$\pi$
$500$	18.3848	−	12.7279i	0.822192	−	0.569210i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−27.0000	−	9.00000i	−1.20148	−	0.400495i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			38.1838i			1.69247i	0.532813	+	0.846233i	$0.321135\pi$
$509$			38.1838i			1.69247i	−0.532813	+	0.846233i	$0.678865\pi$
$510$		0			0
$511$		0			0
$512$	22.6274			1.00000
$513$		0			0
$514$	34.0000			1.49968
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	36.0000	+	12.0000i	1.57870	+	0.526235i
$521$		−	12.7279i		−	0.557620i	−0.960346	−	0.278810i	$-0.910060\pi$
$521$		−	12.7279i		−	0.557620i	0.960346	−	0.278810i	$-0.0899400\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	23.0000			1.00000
$530$	−7.07107	+	21.2132i	−0.307148	+	0.921443i
$531$		0			0
$532$		0			0
$533$	−76.3675			−3.30785
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			6.00000i			0.258678i
$539$		0			0
$540$		0			0
$541$	−20.0000			−0.859867			−0.429934	−	0.902861i	$-0.641463\pi$
$541$	−20.0000			−0.859867			−0.429934	+	0.902861i	$0.641463\pi$
$542$		0			0
$543$		0			0
$544$	−40.0000			−1.71499
$545$	−14.1421	+	42.4264i	−0.605783	+	1.81735i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−19.7990			−0.845771
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	25.4558i		−	1.08152i
$555$		0			0
$556$		0			0
$557$	−7.07107			−0.299611			−0.149805	−	0.988716i	$-0.547865\pi$
$557$	−7.07107			−0.299611			−0.149805	+	0.988716i	$0.547865\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		−	42.0000i		−	1.77166i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	−1.00000	+	3.00000i	−0.0420703	+	0.126211i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			46.6690i			1.95647i	0.207504	+	0.978234i	$0.433466\pi$
$569$			46.6690i			1.95647i	−0.207504	+	0.978234i	$0.566534\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		−	48.0000i		−	1.99827i	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$		−	48.0000i		−	1.99827i	0.0416305	−	0.999133i	$-0.486745\pi$
$578$	46.6690			1.94118
$579$		0			0
$580$	−18.0000	−	6.00000i	−0.747409	−	0.249136i
$581$		0			0
$582$		0			0
$583$		0			0
$584$			16.9706i			0.702247i
$585$		0			0
$586$	38.0000			1.56977
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$			48.0000i			1.97279i
$593$	43.8406			1.80032			0.900159	−	0.435561i	$-0.143450\pi$
$593$	43.8406			1.80032			0.900159	+	0.435561i	$0.143450\pi$
$594$		0			0
$595$		0			0
$596$		−	8.48528i		−	0.347571i
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	10.0000			0.407909			0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000			0.407909			0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	7.77817	−	23.3345i	0.316228	−	0.948683i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$	−10.0000	+	30.0000i	−0.404888	+	1.21466i
$611$		0			0
$612$		0			0
$613$		−	36.0000i		−	1.45403i	−0.686624	−	0.727013i	$-0.740908\pi$
$613$		−	36.0000i		−	1.45403i	0.686624	−	0.727013i	$-0.259092\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−49.4975			−1.99269			−0.996347	−	0.0854011i	$-0.972783\pi$
$617$	−49.4975			−1.99269			−0.996347	+	0.0854011i	$0.972783\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	7.00000	+	24.0000i	0.280000	+	0.960000i
$626$			33.9411i			1.35656i
$627$		0			0
$628$		−	24.0000i		−	0.957704i
$629$		−	84.8528i		−	3.38330i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	50.0000			1.98575
$635$		0			0
$636$		0			0
$637$			42.0000i			1.66410i
$638$		0			0
$639$		0			0
$640$	−8.00000	+	24.0000i	−0.316228	+	0.948683i
$641$			29.6985i			1.17302i	0.809942	+	0.586510i	$0.199498\pi$
$641$			29.6985i			1.17302i	−0.809942	+	0.586510i	$0.800502\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$	−25.4558	+	33.9411i	−0.998460	+	1.33128i
$651$		0			0
$652$		0			0
$653$	49.4975			1.93699			0.968493	−	0.249041i	$-0.0801154\pi$
$653$	49.4975			1.93699			0.968493	+	0.249041i	$0.0801154\pi$
$654$		0			0
$655$		0			0
$656$		−	50.9117i		−	1.98777i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−50.0000			−1.94477			−0.972387	−	0.233373i	$-0.925024\pi$
$661$	−50.0000			−1.94477			−0.972387	+	0.233373i	$0.925024\pi$
$662$		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
$673$		−	24.0000i		−	0.925132i	−0.886585	−	0.462566i	$-0.846929\pi$
$673$		−	24.0000i		−	0.925132i	0.886585	−	0.462566i	$-0.153071\pi$
$674$		−	25.4558i		−	0.980522i
$675$		0			0
$676$	−46.0000			−1.76923
$677$	35.3553			1.35882			0.679408	−	0.733761i	$-0.262237\pi$
$677$	35.3553			1.35882			0.679408	+	0.733761i	$0.262237\pi$
$678$		0			0
$679$		0			0
$680$	14.1421	−	42.4264i	0.542326	−	1.62698i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	7.00000	−	21.0000i	0.267456	−	0.802369i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	42.4264i		−	1.61632i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−31.1127			−1.18273
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$			90.0000i			3.40899i
$698$	−14.1421			−0.535288
$699$		0			0
$700$		0			0
$701$			29.6985i			1.12170i	0.827919	+	0.560848i	$0.189525\pi$
$701$			29.6985i			1.12170i	−0.827919	+	0.560848i	$0.810475\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−50.0000			−1.88177
$707$		0			0
$708$		0			0
$709$	−44.0000			−1.65245			−0.826227	−	0.563337i	$-0.809517\pi$
$709$	−44.0000			−1.65245			−0.826227	+	0.563337i	$0.809517\pi$
$710$		0			0
$711$		0			0
$712$		−	12.0000i		−	0.449719i
$713$		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
$722$	26.8701			1.00000
$723$		0			0
$724$	−40.0000			−1.48659
$725$	12.7279	−	16.9706i	0.472703	−	0.630271i
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$	−18.0000	−	6.00000i	−0.666210	−	0.222070i
$731$		0			0
$732$		0			0
$733$			54.0000i			1.99454i	0.0738717	+	0.997268i	$0.476464\pi$
$733$			54.0000i			1.99454i	−0.0738717	+	0.997268i	$0.523536\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−50.9117	−	16.9706i	−1.87155	−	0.623850i
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	9.00000	+	3.00000i	0.329734	+	0.109911i
$746$			50.9117i			1.86401i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$		0			0
$754$	36.0000			1.31104
$755$		0			0
$756$		0			0
$757$			18.0000i			0.654221i	0.944986	+	0.327111i	$0.106075\pi$
$757$			18.0000i			0.654221i	−0.944986	+	0.327111i	$0.893925\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	55.1543i		−	1.99934i	−0.0256326	−	0.999671i	$-0.508160\pi$
$761$		−	55.1543i		−	1.99934i	0.0256326	−	0.999671i	$-0.491840\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	50.0000			1.80305			0.901523	−	0.432731i	$-0.142450\pi$
$769$	50.0000			1.80305			0.901523	+	0.432731i	$0.142450\pi$
$770$		0			0
$771$		0			0
$772$		−	48.0000i		−	1.72756i
$773$	7.07107			0.254329			0.127164	−	0.991882i	$-0.459412\pi$
$773$	7.07107			0.254329			0.127164	+	0.991882i	$0.459412\pi$
$774$		0			0
$775$		0			0
$776$			50.9117i			1.82762i
$777$		0			0
$778$		−	54.0000i		−	1.93599i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$	25.4558	+	8.48528i	0.908558	+	0.302853i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$	−36.7696			−1.30986
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	60.0000i		−	2.13066i
$794$		−	16.9706i		−	0.602263i
$795$		0			0
$796$		0			0
$797$	−52.3259			−1.85348			−0.926739	−	0.375705i	$-0.877401\pi$
$797$	−52.3259			−1.85348			−0.926739	+	0.375705i	$0.877401\pi$
$798$		0			0
$799$		0			0
$800$	−22.6274	−	16.9706i	−0.800000	−	0.600000i
$801$		0			0
$802$		−	42.0000i		−	1.48307i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$			36.0000i			1.26648i
$809$			46.6690i			1.64080i	0.571793	+	0.820398i	$0.306248\pi$
$809$			46.6690i			1.64080i	−0.571793	+	0.820398i	$0.693752\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−56.5685			−1.97787
$819$		0			0
$820$	54.0000	+	18.0000i	1.88576	+	0.628587i
$821$			55.1543i			1.92490i	0.271460	+	0.962450i	$0.412493\pi$
$821$			55.1543i			1.92490i	−0.271460	+	0.962450i	$0.587507\pi$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$		0			0
$832$		−	48.0000i		−	1.66410i
$833$	49.4975			1.71499
$834$		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
$841$	11.0000			0.379310
$842$	−39.5980			−1.36464
$843$		0			0
$844$		0			0
$845$	16.2635	−	48.7904i	0.559480	−	1.67844i
$846$		0			0
$847$		0			0
$848$	28.2843			0.971286
$849$		0			0
$850$	40.0000	+	30.0000i	1.37199	+	1.02899i
$851$		0			0
$852$		0			0
$853$			36.0000i			1.23262i	0.787505	+	0.616308i	$0.211372\pi$
$853$			36.0000i			1.23262i	−0.787505	+	0.616308i	$0.788628\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	35.3553			1.20772			0.603858	−	0.797092i	$-0.293630\pi$
$857$	35.3553			1.20772			0.603858	+	0.797092i	$0.293630\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	11.0000	−	33.0000i	0.374011	−	1.12203i
$866$			33.9411i			1.15337i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	56.5685			1.91565
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			12.0000i			0.405211i	0.979260	+	0.202606i	$0.0649409\pi$
$877$			12.0000i			0.405211i	−0.979260	+	0.202606i	$0.935059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			12.7279i			0.428815i	0.976744	+	0.214407i	$0.0687820\pi$
$881$			12.7279i			0.428815i	−0.976744	+	0.214407i	$0.931218\pi$
$882$		0			0
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$			84.8528i			2.85391i
$885$		0			0
$886$		0			0
$887$		0			0		1.00000			$0$
$887$		0			0		−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$	12.7279	+	4.24264i	0.426641	+	0.142214i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			54.0000i			1.80200i
$899$		0			0
$900$		0			0
$901$	−50.0000			−1.66574
$902$		0			0
$903$		0			0
$904$	4.00000			0.133038
$905$	14.1421	−	42.4264i	0.470100	−	1.41030i
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		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
$913$		0			0
$914$		−	59.3970i		−	1.96468i
$915$		0			0
$916$	8.00000			0.264327
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		−	18.0000i		−	0.592798i
$923$		0			0
$924$		0			0
$925$	36.0000	−	48.0000i	1.18367	−	1.57823i
$926$		0			0
$927$		0			0
$928$			24.0000i			0.787839i
$929$		−	4.24264i		−	0.139197i	−0.997575	−	0.0695983i	$-0.977828\pi$
$929$		−	4.24264i		−	0.139197i	0.997575	−	0.0695983i	$-0.0221717\pi$
$930$		0			0
$931$		0			0
$932$	14.1421			0.463241
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			48.0000i			1.56809i	0.620703	+	0.784046i	$0.286847\pi$
$937$			48.0000i			1.56809i	−0.620703	+	0.784046i	$0.713153\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			55.1543i			1.79798i	0.437969	+	0.898990i	$0.355698\pi$
$941$			55.1543i			1.79798i	−0.437969	+	0.898990i	$0.644302\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−57.9828			−1.87825			−0.939123	−	0.343582i	$-0.888360\pi$
$953$	−57.9828			−1.87825			−0.939123	+	0.343582i	$0.888360\pi$
$954$		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
$962$	101.823			3.28292
$963$		0			0
$964$	16.0000			0.515325
$965$	50.9117	+	16.9706i	1.63891	+	0.546302i
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−31.1127			−1.00000
$969$		0			0
$970$	−54.0000	−	18.0000i	−1.73384	−	0.577945i
$971$		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
$976$	40.0000			1.28037
$977$	−49.4975			−1.58356			−0.791782	−	0.610803i	$-0.790847\pi$
$977$	−49.4975			−1.58356			−0.791782	+	0.610803i	$0.790847\pi$
$978$		0			0
$979$		0			0
$980$	9.89949	−	29.6985i	0.316228	−	0.948683i
$981$		0			0
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$	13.0000	−	39.0000i	0.414214	−	1.24264i
$986$		−	42.4264i		−	1.35113i
$987$		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			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$			12.0000i			0.380044i	0.981780	+	0.190022i	$0.0608559\pi$
$997$			12.0000i			0.380044i	−0.981780	+	0.190022i	$0.939144\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	180.2.h.a.179.4	yes	4
3.2	odd	2	inner	180.2.h.a.179.1	✓	4
4.3	odd	2	CM	180.2.h.a.179.4	yes	4
5.2	odd	4		900.2.e.a.251.2		2
5.3	odd	4		900.2.e.c.251.1		2
5.4	even	2	inner	180.2.h.a.179.2	yes	4
8.3	odd	2		2880.2.o.a.2879.3		4
8.5	even	2		2880.2.o.a.2879.3		4
12.11	even	2	inner	180.2.h.a.179.1	✓	4
15.2	even	4		900.2.e.a.251.1		2
15.8	even	4		900.2.e.c.251.2		2
15.14	odd	2	inner	180.2.h.a.179.3	yes	4
20.3	even	4		900.2.e.c.251.1		2
20.7	even	4		900.2.e.a.251.2		2
20.19	odd	2	inner	180.2.h.a.179.2	yes	4
24.5	odd	2		2880.2.o.a.2879.2		4
24.11	even	2		2880.2.o.a.2879.2		4
40.19	odd	2		2880.2.o.a.2879.1		4
40.29	even	2		2880.2.o.a.2879.1		4
60.23	odd	4		900.2.e.c.251.2		2
60.47	odd	4		900.2.e.a.251.1		2
60.59	even	2	inner	180.2.h.a.179.3	yes	4
120.29	odd	2		2880.2.o.a.2879.4		4
120.59	even	2		2880.2.o.a.2879.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
180.2.h.a.179.1	✓	4	3.2	odd	2	inner
180.2.h.a.179.1	✓	4	12.11	even	2	inner
180.2.h.a.179.2	yes	4	5.4	even	2	inner
180.2.h.a.179.2	yes	4	20.19	odd	2	inner
180.2.h.a.179.3	yes	4	15.14	odd	2	inner
180.2.h.a.179.3	yes	4	60.59	even	2	inner
180.2.h.a.179.4	yes	4	1.1	even	1	trivial
180.2.h.a.179.4	yes	4	4.3	odd	2	CM
900.2.e.a.251.1		2	15.2	even	4
900.2.e.a.251.1		2	60.47	odd	4
900.2.e.a.251.2		2	5.2	odd	4
900.2.e.a.251.2		2	20.7	even	4
900.2.e.c.251.1		2	5.3	odd	4
900.2.e.c.251.1		2	20.3	even	4
900.2.e.c.251.2		2	15.8	even	4
900.2.e.c.251.2		2	60.23	odd	4
2880.2.o.a.2879.1		4	40.19	odd	2
2880.2.o.a.2879.1		4	40.29	even	2
2880.2.o.a.2879.2		4	24.5	odd	2
2880.2.o.a.2879.2		4	24.11	even	2
2880.2.o.a.2879.3		4	8.3	odd	2
2880.2.o.a.2879.3		4	8.5	even	2
2880.2.o.a.2879.4		4	120.29	odd	2
2880.2.o.a.2879.4		4	120.59	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	180.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +(-0.707107 + 2.12132i) q^{5} +2.82843 q^{8} +(-1.00000 + 3.00000i) q^{10} -6.00000i q^{13} +4.00000 q^{16} -7.07107 q^{17} +(-1.41421 + 4.24264i) q^{20} +(-4.00000 - 3.00000i) q^{25} -8.48528i q^{26} +4.24264i q^{29} +5.65685 q^{32} -10.0000 q^{34} +12.0000i q^{37} +(-2.00000 + 6.00000i) q^{40} -12.7279i q^{41} -7.00000 q^{49} +(-5.65685 - 4.24264i) q^{50} -12.0000i q^{52} +7.07107 q^{53} +6.00000i q^{58} +10.0000 q^{61} +8.00000 q^{64} +(12.7279 + 4.24264i) q^{65} -14.1421 q^{68} +6.00000i q^{73} +16.9706i q^{74} +(-2.82843 + 8.48528i) q^{80} -18.0000i q^{82} +(5.00000 - 15.0000i) q^{85} -4.24264i q^{89} +18.0000i q^{97} -9.89949 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 4 q^{10} + 16 q^{16} - 16 q^{25} - 40 q^{34} - 8 q^{40} - 28 q^{49} + 40 q^{61} + 32 q^{64} + 20 q^{85}+O(q^{100}) \) 4 * q + 8 * q^4 - 4 * q^10 + 16 * q^16 - 16 * q^25 - 40 * q^34 - 8 * q^40 - 28 * q^49 + 40 * q^61 + 32 * q^64 + 20 * q^85

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