LMFDB - Embedded newform 177.2.d.b.176.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,2,Mod(176,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("177.176");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	177.d (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:	$1.41335211578$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.149721291.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{3} + 27 $ x^6 - 7*x^3 + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			176.2
Root			$-1.24353 + 1.20567i$ of defining polynomial
Character	$\chi$	$=$	177.176
Dual form			177.2.d.b.176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.24353 + 1.20567i) q^{3} -2.00000 q^{4} -3.58579i q^{5} +5.15952 q^{7} +(0.0927106 - 2.99857i) q^{9} +O(q^{10})$	$q+(-1.24353 + 1.20567i) q^{3} -2.00000 q^{4} -3.58579i q^{5} +5.15952 q^{7} +(0.0927106 - 2.99857i) q^{9} +(2.48705 - 2.41134i) q^{12} +(4.32329 + 4.45902i) q^{15} +4.00000 q^{16} -7.68115i q^{17} -2.30163 q^{19} +7.17158i q^{20} +(-6.41600 + 6.22069i) q^{21} -7.85789 q^{25} +(3.50000 + 3.84057i) q^{27} -10.3190 q^{28} +3.64824i q^{29} -18.5010i q^{35} +(-0.185421 + 5.99713i) q^{36} -0.0624503i q^{41} +(-10.7522 - 0.332441i) q^{45} +(-4.97410 + 4.82269i) q^{48} +19.6207 q^{49} +(9.26094 + 9.55170i) q^{51} +14.4056i q^{53} +(2.86213 - 2.77501i) q^{57} +7.68115i q^{59} +(-8.64657 - 8.91804i) q^{60} +(0.478343 - 15.4712i) q^{63} -8.00000 q^{64} +15.3623i q^{68} -7.68115i q^{71} +(9.77149 - 9.47404i) q^{75} +4.60326 q^{76} +13.1769 q^{79} -14.3432i q^{80} +(-8.98281 - 0.555998i) q^{81} +(12.8320 - 12.4414i) q^{84} -27.5430 q^{85} +(-4.39858 - 4.53668i) q^{87} +8.25316i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 12 q^{4}+O(q^{10}) $ 6 * q - 12 * q^4	$ 6 q - 12 q^{4} + 3 q^{15} + 24 q^{16} - 15 q^{21} - 30 q^{25} + 21 q^{27} - 33 q^{45} + 42 q^{49} + 39 q^{57} - 6 q^{60} + 12 q^{63} - 48 q^{64} - 24 q^{75} + 30 q^{84} - 51 q^{87}+O(q^{100}) $ 6 * q - 12 * q^4 + 3 * q^15 + 24 * q^16 - 15 * q^21 - 30 * q^25 + 21 * q^27 - 33 * q^45 + 42 * q^49 + 39 * q^57 - 6 * q^60 + 12 * q^63 - 48 * q^64 - 24 * q^75 + 30 * q^84 - 51 * q^87

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	−1.24353	+	1.20567i	−0.717950	+	0.696095i
$3$	−1.24353	+	1.20567i	−0.717950	+	0.696095i
$4$	−2.00000			−1.00000
$5$		−	3.58579i		−	1.60361i	−0.597583	−	0.801807i	$-0.703872\pi$
$5$		−	3.58579i		−	1.60361i	0.597583	−	0.801807i	$-0.296128\pi$
$6$		0			0
$7$	5.15952			1.95012			0.975058	−	0.221950i	$-0.0712420\pi$
$7$	5.15952			1.95012			0.975058	+	0.221950i	$0.0712420\pi$
$8$		0			0
$9$	0.0927106	−	2.99857i	0.0309035	−	0.999522i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	2.48705	−	2.41134i	0.717950	−	0.696095i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$	4.32329	+	4.45902i	1.11627	+	1.15131i
$16$	4.00000			1.00000
$17$		−	7.68115i		−	1.86295i	−0.363803	−	0.931476i	$-0.618522\pi$
$17$		−	7.68115i		−	1.86295i	0.363803	−	0.931476i	$-0.381478\pi$
$18$		0			0
$19$	−2.30163			−0.528030			−0.264015	−	0.964519i	$-0.585047\pi$
$19$	−2.30163			−0.528030			−0.264015	+	0.964519i	$0.585047\pi$
$20$			7.17158i			1.60361i
$21$	−6.41600	+	6.22069i	−1.40009	+	1.35747i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	−7.85789			−1.57158
$26$		0			0
$27$	3.50000	+	3.84057i	0.673575	+	0.739119i
$28$	−10.3190			−1.95012
$29$			3.64824i			0.677461i	0.940883	+	0.338731i	$0.109998\pi$
$29$			3.64824i			0.677461i	−0.940883	+	0.338731i	$0.890002\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	18.5010i		−	3.12723i
$36$	−0.185421	+	5.99713i	−0.0309035	+	0.999522i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	0.0624503i		−	0.00975310i	−0.999988	−	0.00487655i	$-0.998448\pi$
$41$		−	0.0624503i		−	0.00975310i	0.999988	−	0.00487655i	$-0.00155226\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−10.7522	−	0.332441i	−1.60285	−	0.0495574i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−4.97410	+	4.82269i	−0.717950	+	0.696095i
$49$	19.6207			2.80295
$50$		0			0
$51$	9.26094	+	9.55170i	1.29679	+	1.33751i
$52$		0			0
$53$			14.4056i			1.97876i	0.145342	+	0.989382i	$0.453572\pi$
$53$			14.4056i			1.97876i	−0.145342	+	0.989382i	$0.546428\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	2.86213	−	2.77501i	0.379099	−	0.367559i
$58$		0			0
$59$			7.68115i			1.00000i
$59$			7.68115i			1.00000i
$60$	−8.64657	−	8.91804i	−1.11627	−	1.15131i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	0.478343	−	15.4712i	0.0602655	−	1.94918i
$64$	−8.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$			15.3623i			1.86295i
$69$		0			0
$70$		0			0
$71$		−	7.68115i		−	0.911584i	−0.890086	−	0.455792i	$-0.849356\pi$
$71$		−	7.68115i		−	0.911584i	0.890086	−	0.455792i	$-0.150644\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	9.77149	−	9.47404i	1.12831	−	1.09397i
$76$	4.60326			0.528030
$77$		0			0
$78$		0			0
$79$	13.1769			1.48252			0.741261	−	0.671217i	$-0.234228\pi$
$79$	13.1769			1.48252			0.741261	+	0.671217i	$0.234228\pi$
$80$		−	14.3432i		−	1.60361i
$81$	−8.98281	−	0.555998i	−0.998090	−	0.0617776i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	12.8320	−	12.4414i	1.40009	−	1.35747i
$85$	−27.5430			−2.98746
$86$		0			0
$87$	−4.39858	−	4.53668i	−0.471577	−	0.486383i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$			8.25316i			0.846756i
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	15.7158			1.57158
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	22.3061	+	23.0064i	2.17685	+	2.24520i
$106$		0			0
$107$			10.8823i			1.05203i	0.850476	+	0.526014i	$0.176314\pi$
$107$			10.8823i			1.05203i	−0.850476	+	0.526014i	$0.823686\pi$
$108$	−7.00000	−	7.68115i	−0.673575	−	0.739119i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	20.6381			1.95012
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	7.29648i		−	0.677461i
$117$		0			0
$118$		0			0
$119$		−	39.6310i		−	3.63297i
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	0.0752946	+	0.0776586i	0.00678908	+	0.00700224i
$124$		0			0
$125$			10.2478i			0.916592i
$126$		0			0
$127$	−18.3365			−1.62710			−0.813549	−	0.581496i	$-0.802467\pi$
$127$	−18.3365			−1.62710			−0.813549	+	0.581496i	$0.802467\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−11.8753			−1.02972
$134$		0			0
$135$	13.7715	−	12.5503i	1.18526	−	1.08015i
$136$		0			0
$137$			10.6949i			0.913729i	0.889536	+	0.456864i	$0.151028\pi$
$137$			10.6949i			0.913729i	−0.889536	+	0.456864i	$0.848972\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$			37.0019i			3.12723i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.370843	−	11.9943i	0.0309035	−	0.999522i
$145$	13.0818			1.08639
$146$		0			0
$147$	−24.3988	+	23.6561i	−2.01238	+	1.95112i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−23.0324	−	0.712124i	−1.86206	−	0.0575718i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$	−17.3684	−	17.9137i	−1.37741	−	1.42065i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	11.0000			0.861586			0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000			0.861586			0.430793	+	0.902451i	$0.358234\pi$
$164$			0.124901i			0.00975310i
$165$		0			0
$166$		0			0
$167$			25.1630i			1.94717i	0.228325	+	0.973585i	$0.426675\pi$
$167$			25.1630i			1.94717i	−0.228325	+	0.973585i	$0.573325\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−0.213386	+	6.90159i	−0.0163180	+	0.527778i
$172$		0			0
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−40.5430			−3.06476
$176$		0			0
$177$	−9.26094	−	9.55170i	−0.696095	−	0.717950i
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	21.5045	+	0.664882i	1.60285	+	0.0495574i
$181$	12.0644			0.896741			0.448370	−	0.893848i	$-0.352005\pi$
$181$	12.0644			0.896741			0.448370	+	0.893848i	$0.352005\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	18.0583	+	19.8155i	1.31355	+	1.44137i
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	9.94820	−	9.64538i	0.717950	−	0.696095i
$193$	−3.41416			−0.245756			−0.122878	−	0.992422i	$-0.539212\pi$
$193$	−3.41416			−0.245756			−0.122878	+	0.992422i	$0.539212\pi$
$194$		0			0
$195$		0			0
$196$	−39.2413			−2.80295
$197$		−	7.68115i		−	0.547259i	−0.961835	−	0.273629i	$-0.911776\pi$
$197$		−	7.68115i		−	0.547259i	0.961835	−	0.273629i	$-0.0882242\pi$
$198$		0			0
$199$	28.0992			1.99190			0.995951	−	0.0898948i	$-0.0286531\pi$
$199$	28.0992			1.99190			0.995951	+	0.0898948i	$0.0286531\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			18.8232i			1.32113i
$204$	−18.5219	−	19.1034i	−1.29679	−	1.33751i
$205$	−0.223934			−0.0156402
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		−	28.8112i		−	1.97876i
$213$	9.26094	+	9.55170i	0.634549	+	0.654472i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−19.0000			−1.27233			−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000			−1.27233			−0.636167	+	0.771551i	$0.719481\pi$
$224$		0			0
$225$	−0.728510	+	23.5624i	−0.0485674	+	1.57083i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	−5.72427	+	5.55002i	−0.379099	+	0.367559i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		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
$235$		0			0
$236$		−	15.3623i		−	1.00000i
$237$	−16.3859	+	15.8871i	−1.06438	+	1.03198i
$238$		0			0
$239$			18.1163i			1.17185i	0.810367	+	0.585923i	$0.199268\pi$
$239$			18.1163i			1.17185i	−0.810367	+	0.585923i	$0.800732\pi$
$240$	17.2931	+	17.8361i	1.11627	+	1.15131i
$241$	21.1944			1.36525			0.682624	−	0.730770i	$-0.260839\pi$
$241$	21.1944			1.36525			0.682624	+	0.730770i	$0.260839\pi$
$242$		0			0
$243$	11.8407	−	10.1389i	0.759581	−	0.650412i
$244$		0			0
$245$		−	70.3556i		−	4.49486i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	17.8665i		−	1.12772i	−0.825869	−	0.563862i	$-0.809315\pi$
$251$		−	17.8665i		−	1.12772i	0.825869	−	0.563862i	$-0.190685\pi$
$252$	−0.956685	+	30.9424i	−0.0602655	+	1.94918i
$253$		0			0
$254$		0			0
$255$	34.2504	−	33.2078i	2.14484	−	2.07955i
$256$	16.0000			1.00000
$257$		−	25.2879i		−	1.57741i	−0.614769	−	0.788707i	$-0.710751\pi$
$257$		−	25.2879i		−	1.57741i	0.614769	−	0.788707i	$-0.289249\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	10.9395	+	0.338231i	0.677138	+	0.0209360i
$262$		0			0
$263$			32.3970i			1.99769i	0.0480951	+	0.998843i	$0.484685\pi$
$263$			32.3970i			1.99769i	−0.0480951	+	0.998843i	$0.515315\pi$
$264$		0			0
$265$	51.6555			3.17317
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$	13.7332			0.834233			0.417116	−	0.908853i	$-0.363041\pi$
$271$	13.7332			0.834233			0.417116	+	0.908853i	$0.363041\pi$
$272$		−	30.7246i		−	1.86295i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−33.2588			−1.99833			−0.999163	−	0.0409112i	$-0.986974\pi$
$277$	−33.2588			−1.99833			−0.999163	+	0.0409112i	$0.986974\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			28.8737i			1.72246i	0.508216	+	0.861230i	$0.330305\pi$
$281$			28.8737i			1.72246i	−0.508216	+	0.861230i	$0.669695\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			15.3623i			0.911584i
$285$	−9.95060	−	10.2630i	−0.589423	−	0.607929i
$286$		0			0
$287$		−	0.322214i		−	0.0190197i
$288$		0			0
$289$	−42.0000			−2.47059
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	32.3346i		−	1.88901i	−0.328504	−	0.944503i	$-0.606544\pi$
$293$		−	32.3346i		−	1.88901i	0.328504	−	0.944503i	$-0.393456\pi$
$294$		0			0
$295$	27.5430			1.60361
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−19.5430	+	18.9481i	−1.12831	+	1.09397i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−9.20652			−0.528030
$305$		0			0
$306$		0			0
$307$	4.04699			0.230974			0.115487	−	0.993309i	$-0.463157\pi$
$307$	4.04699			0.230974			0.115487	+	0.993309i	$0.463157\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	10.6325i		−	0.602912i	−0.953480	−	0.301456i	$-0.902527\pi$
$311$		−	10.6325i		−	0.602912i	0.953480	−	0.301456i	$-0.0974727\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	−55.4764	−	1.71524i	−3.12574	−	0.0966426i
$316$	−26.3539			−1.48252
$317$		−	30.7246i		−	1.72566i	−0.505490	−	0.862832i	$-0.668688\pi$
$317$		−	30.7246i		−	1.72566i	0.505490	−	0.862832i	$-0.331312\pi$
$318$		0			0
$319$		0			0
$320$			28.6863i			1.60361i
$321$	−13.1204	−	13.5324i	−0.732312	−	0.755304i
$322$		0			0
$323$			17.6792i			0.983694i
$324$	17.9656	+	1.11200i	0.998090	+	0.0617776i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−18.8927			−1.03844			−0.519219	−	0.854641i	$-0.673777\pi$
$331$	−18.8927			−1.03844			−0.519219	+	0.854641i	$0.673777\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−25.6640	+	24.8828i	−1.40009	+	1.35747i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	55.0860			2.98746
$341$		0			0
$342$		0			0
$343$	65.1167			3.51597
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$	8.79716	+	9.07336i	0.471577	+	0.486383i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		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
$355$	−27.5430			−1.46183
$356$		0			0
$357$	47.7820	+	49.2822i	2.52889	+	2.60829i
$358$		0			0
$359$		−	36.0453i		−	1.90240i	−0.308581	−	0.951198i	$-0.599854\pi$
$359$		−	36.0453i		−	1.90240i	0.308581	−	0.951198i	$-0.400146\pi$
$360$		0			0
$361$	−13.7025			−0.721184
$362$		0			0
$363$	13.6788	−	13.2624i	0.717950	−	0.696095i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−0.187261	−	0.00578981i	−0.00974844	−	0.000301405i
$370$		0			0
$371$			74.3261i			3.85882i
$372$		0			0
$373$	−31.0000			−1.60512			−0.802560	−	0.596572i	$-0.796529\pi$
$373$	−31.0000			−1.60512			−0.802560	+	0.596572i	$0.796529\pi$
$374$		0			0
$375$	−12.3555	−	12.7434i	−0.638035	−	0.658067i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−24.6851			−1.26799			−0.633994	−	0.773338i	$-0.718586\pi$
$379$	−24.6851			−1.26799			−0.633994	+	0.773338i	$0.718586\pi$
$380$		−	16.5063i		−	0.846756i
$381$	22.8019	−	22.1078i	1.16817	−	1.13261i
$382$		0			0
$383$			15.3623i			0.784976i	0.919757	+	0.392488i	$0.128386\pi$
$383$			15.3623i			0.784976i	−0.919757	+	0.392488i	$0.871614\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$			38.4057i			1.94725i	0.228159	+	0.973624i	$0.426729\pi$
$389$			38.4057i			1.94725i	−0.228159	+	0.973624i	$0.573271\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	47.2497i		−	2.37739i
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$	14.7672	−	14.3177i	0.739287	−	0.716783i
$400$	−31.4316			−1.57158
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.99369	+	32.2105i	−0.0990674	+	1.60055i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	−12.8946	−	13.2994i	−0.636042	−	0.656011i
$412$		0			0
$413$			39.6310i			1.95012i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6.21763	+	6.02836i	−0.304478	+	0.295210i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$	−44.6122	−	46.0128i	−2.17685	−	2.24520i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			60.3576i			2.92777i
$426$		0			0
$427$		0			0
$428$		−	21.7645i		−	1.05203i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	14.0000	+	15.3623i	0.673575	+	0.739119i
$433$	−40.7199			−1.95687			−0.978437	−	0.206545i	$-0.933778\pi$
$433$	−40.7199			−1.95687			−0.978437	+	0.206545i	$0.933778\pi$
$434$		0			0
$435$	−16.2676	+	15.7724i	−0.779971	+	0.756228i
$436$		0			0
$437$		0			0
$438$		0			0
$439$	35.0000			1.67046			0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000			1.67046			0.835229	+	0.549902i	$0.185335\pi$
$440$		0			0
$441$	1.81905	−	58.8339i	0.0866212	−	2.80161i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−41.2762			−1.95012
$449$		−	32.5219i		−	1.53480i	−0.641166	−	0.767402i	$-0.721549\pi$
$449$		−	32.5219i		−	1.53480i	0.641166	−	0.767402i	$-0.278451\pi$
$450$		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
$459$	29.5000	−	26.8840i	1.37694	−	1.25484i
$460$		0			0
$461$		−	30.7246i		−	1.43099i	−0.698620	−	0.715493i	$-0.746202\pi$
$461$		−	30.7246i		−	1.43099i	0.698620	−	0.715493i	$-0.253798\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$			14.5930i			0.677461i
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	18.0860			0.829841
$476$			79.2621i			3.63297i
$477$	43.1962	+	1.33555i	1.97782	+	0.0611508i
$478$		0			0
$479$			38.4057i			1.75480i	0.479757	+	0.877401i	$0.340725\pi$
$479$			38.4057i			1.75480i	−0.479757	+	0.877401i	$0.659275\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	22.0000			1.00000
$485$		0			0
$486$		0			0
$487$	−16.6677			−0.755284			−0.377642	−	0.925952i	$-0.623265\pi$
$487$	−16.6677			−0.755284			−0.377642	+	0.925952i	$0.623265\pi$
$488$		0			0
$489$	−13.6788	+	13.2624i	−0.618576	+	0.599746i
$490$		0			0
$491$			6.98423i			0.315194i	0.987504	+	0.157597i	$0.0503747\pi$
$491$			6.98423i			0.315194i	−0.987504	+	0.157597i	$0.949625\pi$
$492$	−0.150589	−	0.155317i	−0.00678908	−	0.00700224i
$493$	28.0227			1.26208
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	39.6310i		−	1.77770i
$498$		0			0
$499$	35.0041			1.56700			0.783500	−	0.621392i	$-0.213432\pi$
$499$	35.0041			1.56700			0.783500	+	0.621392i	$0.213432\pi$
$500$		−	20.4956i		−	0.916592i
$501$	−30.3383	−	31.2908i	−1.35542	−	1.39797i
$502$		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
$507$	−16.1658	+	15.6737i	−0.717950	+	0.696095i
$508$	36.6729			1.62710
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−8.05570	−	8.83958i	−0.355668	−	0.390277i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	7.68115i		−	0.336517i	−0.985743	−	0.168259i	$-0.946186\pi$
$521$		−	7.68115i		−	0.336517i	0.985743	−	0.168259i	$-0.0538144\pi$
$522$		0			0
$523$	−3.97042			−0.173614			−0.0868072	−	0.996225i	$-0.527666\pi$
$523$	−3.97042			−0.173614			−0.0868072	+	0.996225i	$0.527666\pi$
$524$		0			0
$525$	50.4162	−	48.8815i	2.20034	−	2.13336i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	23.0324	+	0.712124i	0.999522	+	0.0309035i
$532$	23.7506			1.02972
$533$		0			0
$534$		0			0
$535$	39.0215			1.68705
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−27.5430	+	25.1005i	−1.18526	+	1.08015i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−15.0024	+	14.5457i	−0.643815	+	0.624217i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	8.00000			0.342055			0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000			0.342055			0.171028	+	0.985266i	$0.445291\pi$
$548$		−	21.3898i		−	0.913729i
$549$		0			0
$550$		0			0
$551$		−	8.39690i		−	0.357720i
$552$		0			0
$553$	67.9867			2.89109
$554$		0			0
$555$		0			0
$556$	−10.0000			−0.424094
$557$		−	43.0919i		−	1.82586i	−0.408112	−	0.912932i	$-0.633813\pi$
$557$		−	43.0919i		−	1.82586i	0.408112	−	0.912932i	$-0.366187\pi$
$558$		0			0
$559$		0			0
$560$		−	74.0039i		−	3.12723i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−46.3470	−	2.86869i	−1.94639	−	0.120473i
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		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
$576$	−0.741685	+	23.9885i	−0.0309035	+	0.999522i
$577$	44.1341			1.83733			0.918663	−	0.395043i	$-0.129270\pi$
$577$	44.1341			1.83733			0.918663	+	0.395043i	$0.129270\pi$
$578$		0			0
$579$	4.24559	−	4.11635i	0.176441	−	0.171070i
$580$	−26.1637			−1.08639
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	48.7976	−	47.3122i	2.01238	−	1.95112i
$589$		0			0
$590$		0			0
$591$	9.26094	+	9.55170i	0.380944	+	0.392904i
$592$		0			0
$593$			3.83559i			0.157509i	0.996894	+	0.0787544i	$0.0250943\pi$
$593$			3.83559i			0.157509i	−0.996894	+	0.0787544i	$0.974906\pi$
$594$		0			0
$595$	−142.109			−5.82589
$596$		0			0
$597$	−34.9421	+	33.8785i	−1.43009	+	1.38655i
$598$		0			0
$599$			32.2097i			1.31605i	0.752996	+	0.658026i	$0.228608\pi$
$599$			32.2097i			1.31605i	−0.752996	+	0.658026i	$0.771392\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			39.4437i			1.60361i
$606$		0			0
$607$	−32.1462			−1.30478			−0.652388	−	0.757885i	$-0.726233\pi$
$607$	−32.1462			−1.30478			−0.652388	+	0.757885i	$0.726233\pi$
$608$		0			0
$609$	−22.6946	−	23.4071i	−0.919631	−	0.948504i
$610$		0			0
$611$		0			0
$612$	46.0649	+	1.42425i	1.86206	+	0.0575718i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$	0.278467	−	0.269991i	0.0112289	−	0.0108871i
$616$		0			0
$617$			46.8651i			1.88672i	0.331775	+	0.943358i	$0.392352\pi$
$617$			46.8651i			1.88672i	−0.331775	+	0.943358i	$0.607648\pi$
$618$		0			0
$619$	−26.9101			−1.08161			−0.540805	−	0.841148i	$-0.681880\pi$
$619$	−26.9101			−1.08161			−0.540805	+	0.841148i	$0.681880\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−2.54298			−0.101719
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	20.0000			0.796187			0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000			0.796187			0.398094	+	0.917345i	$0.369672\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			65.7507i			2.60924i
$636$	34.7369	+	35.8275i	1.37741	+	1.42065i
$637$		0			0
$638$		0			0
$639$	−23.0324	−	0.712124i	−0.911149	−	0.0281712i
$640$		0			0
$641$			38.4057i			1.51694i	0.651711	+	0.758468i	$0.274052\pi$
$641$			38.4057i			1.51694i	−0.651711	+	0.758468i	$0.725948\pi$
$642$		0			0
$643$	−48.1811			−1.90008			−0.950038	−	0.312134i	$-0.898956\pi$
$643$	−48.1811			−1.90008			−0.950038	+	0.312134i	$0.898956\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			24.9756i			0.981893i	0.871190	+	0.490947i	$0.163349\pi$
$647$			24.9756i			0.981893i	−0.871190	+	0.490947i	$0.836651\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−22.0000			−0.861586
$653$		−	14.1558i		−	0.553960i	−0.960876	−	0.276980i	$-0.910666\pi$
$653$		−	14.1558i		−	0.553960i	0.960876	−	0.276980i	$-0.0893335\pi$
$654$		0			0
$655$		0			0
$656$		−	0.249801i		−	0.00975310i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	51.0390			1.98519			0.992593	−	0.121489i	$-0.0387669\pi$
$661$	51.0390			1.98519			0.992593	+	0.121489i	$0.0387669\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			42.5824i			1.65127i
$666$		0			0
$667$		0			0
$668$		−	50.3260i		−	1.94717i
$669$	23.6270	−	22.9078i	0.913472	−	0.885665i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−27.5026	−	30.1788i	−1.05858	−	1.16158i
$676$	−26.0000			−1.00000
$677$		−	30.7246i		−	1.18084i	−0.807096	−	0.590421i	$-0.798962\pi$
$677$		−	30.7246i		−	1.18084i	0.807096	−	0.590421i	$-0.201038\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	0.426771	−	13.8032i	0.0163180	−	0.527778i
$685$	38.3497			1.46527
$686$		0			0
$687$		0			0
$688$		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
$695$		−	17.9290i		−	0.680084i
$696$		0			0
$697$	−0.479690			−0.0181696
$698$		0			0
$699$		0			0
$700$	81.0860			3.06476
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	18.5219	+	19.1034i	0.696095	+	0.717950i
$709$	−49.2936			−1.85126			−0.925630	−	0.378430i	$-0.876464\pi$
$709$	−49.2936			−1.85126			−0.925630	+	0.378430i	$0.876464\pi$
$710$		0			0
$711$	1.22164	−	39.5119i	0.0458152	−	1.48181i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−21.8423	−	22.5281i	−0.815716	−	0.841327i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	−43.0089	−	1.32976i	−1.60285	−	0.0495574i
$721$		0			0
$722$		0			0
$723$	−26.3557	+	25.5534i	−0.980180	+	0.950343i
$724$	−24.1288			−0.896741
$725$		−	28.6675i		−	1.06468i
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	−2.50000	+	26.8840i	−0.0925926	+	0.995704i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−49.0000			−1.80986			−0.904928	−	0.425564i	$-0.860076\pi$
$733$	−49.0000			−1.80986			−0.904928	+	0.425564i	$0.860076\pi$
$734$		0			0
$735$	84.8258	+	87.4890i	3.12885	+	3.22708i
$736$		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
$743$		−	53.7680i		−	1.97256i	−0.165089	−	0.986279i	$-0.552791\pi$
$743$		−	53.7680i		−	1.97256i	0.165089	−	0.986279i	$-0.447209\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$			56.1473i			2.05158i
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	21.5411	+	22.2174i	0.785003	+	0.809649i
$754$		0			0
$755$		0			0
$756$	−36.1167	−	39.6310i	−1.31355	−	1.44137i
$757$	42.4653			1.54343			0.771713	−	0.635970i	$-0.219400\pi$
$757$	42.4653			1.54343			0.771713	+	0.635970i	$0.219400\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	54.0367i		−	1.95883i	−0.201863	−	0.979414i	$-0.564700\pi$
$761$		−	54.0367i		−	1.95883i	0.201863	−	0.979414i	$-0.435300\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.55353	+	82.5895i	−0.0923230	+	2.98603i
$766$		0			0
$767$		0			0
$768$	−19.8964	+	19.2908i	−0.717950	+	0.696095i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	30.4889	+	31.4461i	1.09803	+	1.13250i
$772$	6.82831			0.245756
$773$		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
$779$			0.143737i			0.00514993i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−14.0113	+	12.7688i	−0.500724	+	0.456321i
$784$	78.4827			2.80295
$785$		0			0
$786$		0			0
$787$	32.0000			1.14068			0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000			1.14068			0.570338	+	0.821410i	$0.306812\pi$
$788$			15.3623i			0.547259i
$789$	−39.0602	−	40.2865i	−1.39058	−	1.43424i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−64.2349	+	62.2796i	−2.27818	+	2.20883i
$796$	−56.1985			−1.99190
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		−	37.6464i		−	1.32113i
$813$	−17.0776	+	16.5577i	−0.598937	+	0.580705i
$814$		0			0
$815$		−	39.4437i		−	1.38165i
$816$	37.0438	+	38.2068i	1.29679	+	1.33751i
$817$		0			0
$818$		0			0
$819$		0			0
$820$	0.447868			0.0156402
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$		0			0
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	7.68115i		−	0.267100i	−0.991042	−	0.133550i	$-0.957362\pi$
$827$		−	7.68115i		−	0.267100i	0.991042	−	0.133550i	$-0.0426376\pi$
$828$		0			0
$829$	−11.9878			−0.416355			−0.208177	−	0.978091i	$-0.566753\pi$
$829$	−11.9878			−0.416355			−0.208177	+	0.978091i	$0.566753\pi$
$830$		0			0
$831$	41.3581	−	40.0992i	1.43470	−	1.39102i
$832$		0			0
$833$		−	150.709i		−	5.22177i
$834$		0			0
$835$	90.2292			3.12251
$836$		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
$841$	15.6903			0.541046
$842$		0			0
$843$	−34.8122	−	35.9051i	−1.19900	−	1.23664i
$844$		0			0
$845$		−	46.6153i		−	1.60361i
$846$		0			0
$847$	−56.7548			−1.95012
$848$			57.6224i			1.97876i
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−18.5219	−	19.1034i	−0.634549	−	0.654472i
$853$	−47.6248			−1.63064			−0.815321	−	0.579009i	$-0.803440\pi$
$853$	−47.6248			−1.63064			−0.815321	+	0.579009i	$0.803440\pi$
$854$		0			0
$855$	24.7477	+	0.765156i	0.846352	+	0.0261678i
$856$		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
$861$	0.388484	+	0.400681i	0.0132395	+	0.0136552i
$862$		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
$867$	52.2281	−	50.6382i	1.77376	−	1.71976i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$			52.8738i			1.78746i
$876$		0			0
$877$	59.0564			1.99419			0.997096	−	0.0761533i	$-0.0242638\pi$
$877$	59.0564			1.99419			0.997096	+	0.0761533i	$0.0242638\pi$
$878$		0			0
$879$	38.9849	+	40.2089i	1.31493	+	1.35621i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	29.7680			1.00177			0.500887	−	0.865513i	$-0.333007\pi$
$883$	29.7680			1.00177			0.500887	+	0.865513i	$0.333007\pi$
$884$		0			0
$885$	−34.2504	+	33.2078i	−1.15131	+	1.11627i
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−94.6074			−3.17303
$890$		0			0
$891$		0			0
$892$	38.0000			1.27233
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.45702	−	47.1248i	0.0485674	−	1.57083i
$901$	110.652			3.68634
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	43.2605i		−	1.43803i
$906$		0			0
$907$	−55.6422			−1.84757			−0.923785	−	0.382912i	$-0.874921\pi$
$907$	−55.6422			−1.84757			−0.923785	+	0.382912i	$0.874921\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	53.8493i		−	1.78411i	−0.451930	−	0.892054i	$-0.649264\pi$
$911$		−	53.8493i		−	1.78411i	0.451930	−	0.892054i	$-0.350736\pi$
$912$	11.4485	−	11.1000i	0.379099	−	0.367559i
$913$		0			0
$914$		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
$921$	−5.03254	+	4.87935i	−0.165828	+	0.160780i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−45.1595			−1.48004
$932$		0			0
$933$	12.8193	+	13.2217i	0.419684	+	0.432861i
$934$		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
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$			30.7246i			1.00000i
$945$	71.0543	−	64.7534i	2.31140	−	2.10643i
$946$		0			0
$947$			61.1458i			1.98697i	0.113958	+	0.993486i	$0.463647\pi$
$947$			61.1458i			1.98697i	−0.113958	+	0.993486i	$0.536353\pi$
$948$	32.7717	−	31.7741i	1.06438	−	1.03198i
$949$		0			0
$950$		0			0
$951$	37.0438	+	38.2068i	1.20123	+	1.23894i
$952$		0			0
$953$			61.4492i			1.99053i	0.0971795	+	0.995267i	$0.469018\pi$
$953$			61.4492i			1.99053i	−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$		−	36.2326i		−	1.17185i
$957$		0			0
$958$		0			0
$959$			55.1807i			1.78188i
$960$	−34.5863	−	35.6722i	−1.11627	−	1.15131i
$961$	31.0000			1.00000
$962$		0			0
$963$	32.6312	+	1.00890i	1.05153	+	0.0325114i
$964$	−42.3887			−1.36525
$965$			12.2425i			0.394098i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	−21.3153	−	21.9845i	−0.684745	−	0.706243i
$970$		0			0
$971$		−	24.9132i		−	0.799502i	−0.916624	−	0.399751i	$-0.869097\pi$
$971$		−	24.9132i		−	0.799502i	0.916624	−	0.399751i	$-0.130903\pi$
$972$	−23.6814	+	20.2778i	−0.759581	+	0.650412i
$973$	25.7976			0.827034
$974$		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
$980$			140.711i			4.49486i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	−27.5430			−0.877592
$986$		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
$993$	23.4936	−	22.7784i	0.745546	−	0.722852i
$994$		0			0
$995$		−	100.758i		−	3.19424i
$996$		0			0
$997$	−34.9276			−1.10617			−0.553083	−	0.833126i	$-0.686549\pi$
$997$	−34.9276			−1.10617			−0.553083	+	0.833126i	$0.686549\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.2.d.b.176.2	yes	6
3.2	odd	2	inner	177.2.d.b.176.1	✓	6
59.58	odd	2	CM	177.2.d.b.176.2	yes	6
177.176	even	2	inner	177.2.d.b.176.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.d.b.176.1	✓	6	3.2	odd	2	inner
177.2.d.b.176.1	✓	6	177.176	even	2	inner
177.2.d.b.176.2	yes	6	1.1	even	1	trivial
177.2.d.b.176.2	yes	6	59.58	odd	2	CM

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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	177.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.24353 + 1.20567i) q^{3} -2.00000 q^{4} -3.58579i q^{5} +5.15952 q^{7} +(0.0927106 - 2.99857i) q^{9} +O(q^{10})\)	\(q+(-1.24353 + 1.20567i) q^{3} -2.00000 q^{4} -3.58579i q^{5} +5.15952 q^{7} +(0.0927106 - 2.99857i) q^{9} +(2.48705 - 2.41134i) q^{12} +(4.32329 + 4.45902i) q^{15} +4.00000 q^{16} -7.68115i q^{17} -2.30163 q^{19} +7.17158i q^{20} +(-6.41600 + 6.22069i) q^{21} -7.85789 q^{25} +(3.50000 + 3.84057i) q^{27} -10.3190 q^{28} +3.64824i q^{29} -18.5010i q^{35} +(-0.185421 + 5.99713i) q^{36} -0.0624503i q^{41} +(-10.7522 - 0.332441i) q^{45} +(-4.97410 + 4.82269i) q^{48} +19.6207 q^{49} +(9.26094 + 9.55170i) q^{51} +14.4056i q^{53} +(2.86213 - 2.77501i) q^{57} +7.68115i q^{59} +(-8.64657 - 8.91804i) q^{60} +(0.478343 - 15.4712i) q^{63} -8.00000 q^{64} +15.3623i q^{68} -7.68115i q^{71} +(9.77149 - 9.47404i) q^{75} +4.60326 q^{76} +13.1769 q^{79} -14.3432i q^{80} +(-8.98281 - 0.555998i) q^{81} +(12.8320 - 12.4414i) q^{84} -27.5430 q^{85} +(-4.39858 - 4.53668i) q^{87} +8.25316i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 12 q^{4}+O(q^{10}) \) 6 * q - 12 * q^4	\( 6 q - 12 q^{4} + 3 q^{15} + 24 q^{16} - 15 q^{21} - 30 q^{25} + 21 q^{27} - 33 q^{45} + 42 q^{49} + 39 q^{57} - 6 q^{60} + 12 q^{63} - 48 q^{64} - 24 q^{75} + 30 q^{84} - 51 q^{87}+O(q^{100}) \) 6 * q - 12 * q^4 + 3 * q^15 + 24 * q^16 - 15 * q^21 - 30 * q^25 + 21 * q^27 - 33 * q^45 + 42 * q^49 + 39 * q^57 - 6 * q^60 + 12 * q^63 - 48 * q^64 - 24 * q^75 + 30 * q^84 - 51 * q^87

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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