LMFDB - Embedded newform 324.3.g.d.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(53,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("324.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.g (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.1
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	324.269
Dual form			324.3.g.d.53.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.01910 + 2.89778i) q^{5} +(3.09808 - 5.36603i) q^{7} +O(q^{10})$	$q+(-5.01910 + 2.89778i) q^{5} +(3.09808 - 5.36603i) q^{7} +(0.984508 + 0.568406i) q^{11} +(5.69615 + 9.86603i) q^{13} +31.2514i q^{17} -32.9808 q^{19} +(-29.1301 + 16.8183i) q^{23} +(4.29423 - 7.43782i) q^{25} +(-22.1420 - 12.7837i) q^{29} +(11.5885 + 20.0718i) q^{31} +35.9101i q^{35} +8.80385 q^{37} +(-61.2137 + 35.3417i) q^{41} +(29.8827 - 51.7583i) q^{43} +(52.1600 + 30.1146i) q^{47} +(5.30385 + 9.18653i) q^{49} -19.7718i q^{53} -6.58846 q^{55} +(72.2364 - 41.7057i) q^{59} +(-15.3827 + 26.6436i) q^{61} +(-57.1791 - 33.0124i) q^{65} +(5.29423 + 9.16987i) q^{67} +3.63342i q^{71} -17.2346 q^{73} +(6.10016 - 3.52193i) q^{77} +(-56.6865 + 98.1840i) q^{79} +(10.2313 + 5.90705i) q^{83} +(-90.5596 - 156.854i) q^{85} -111.252i q^{89} +70.5885 q^{91} +(165.534 - 95.5709i) q^{95} +(-37.1577 + 64.3590i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7}+O(q^{10}) $ 8 * q + 4 * q^7	$ 8 q + 4 q^{7} + 4 q^{13} - 56 q^{19} - 28 q^{25} - 32 q^{31} + 112 q^{37} + 52 q^{43} + 84 q^{49} + 72 q^{55} + 64 q^{61} - 20 q^{67} - 512 q^{73} - 308 q^{79} - 288 q^{85} + 440 q^{91} + 160 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 + 4 * q^13 - 56 * q^19 - 28 * q^25 - 32 * q^31 + 112 * q^37 + 52 * q^43 + 84 * q^49 + 72 * q^55 + 64 * q^61 - 20 * q^67 - 512 * q^73 - 308 * q^79 - 288 * q^85 + 440 * q^91 + 160 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	−5.01910	+	2.89778i	−1.00382	+	0.579555i	−0.909376	−	0.415975i	$-0.863440\pi$
$5$	−5.01910	+	2.89778i	−1.00382	+	0.579555i	−0.0944434	+	0.995530i	$0.530107\pi$
$6$		0			0
$7$	3.09808	−	5.36603i	0.442582	−	0.766575i	−0.555298	−	0.831651i	$-0.687396\pi$
$7$	3.09808	−	5.36603i	0.442582	−	0.766575i	0.997880	+	0.0650764i	$0.0207291\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.984508	+	0.568406i	0.0895007	+	0.0516733i	0.544082	−	0.839032i	$-0.316878\pi$
$11$	0.984508	+	0.568406i	0.0895007	+	0.0516733i	−0.454582	+	0.890705i	$0.650211\pi$
$12$		0			0
$13$	5.69615	+	9.86603i	0.438166	+	0.758925i	0.997548	−	0.0699846i	$-0.0222950\pi$
$13$	5.69615	+	9.86603i	0.438166	+	0.758925i	−0.559382	+	0.828910i	$0.688962\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			31.2514i			1.83832i	0.393887	+	0.919159i	$0.371130\pi$
$17$			31.2514i			1.83832i	−0.393887	+	0.919159i	$0.628870\pi$
$18$		0			0
$19$	−32.9808			−1.73583			−0.867915	−	0.496713i	$-0.834540\pi$
$19$	−32.9808			−1.73583			−0.867915	+	0.496713i	$0.834540\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−29.1301	+	16.8183i	−1.26653	+	0.731229i	−0.974329	−	0.225130i	$-0.927719\pi$
$23$	−29.1301	+	16.8183i	−1.26653	+	0.731229i	−0.292196	+	0.956358i	$0.594386\pi$
$24$		0			0
$25$	4.29423	−	7.43782i	0.171769	−	0.297513i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−22.1420	−	12.7837i	−0.763516	−	0.440816i	0.0670407	−	0.997750i	$-0.478644\pi$
$29$	−22.1420	−	12.7837i	−0.763516	−	0.440816i	−0.830557	+	0.556934i	$0.811978\pi$
$30$		0			0
$31$	11.5885	+	20.0718i	0.373821	+	0.647477i	0.990150	−	0.140012i	$-0.0447140\pi$
$31$	11.5885	+	20.0718i	0.373821	+	0.647477i	−0.616329	+	0.787489i	$0.711381\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			35.9101i			1.02600i
$36$		0			0
$37$	8.80385			0.237942			0.118971	−	0.992898i	$-0.462040\pi$
$37$	8.80385			0.237942			0.118971	+	0.992898i	$0.462040\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−61.2137	+	35.3417i	−1.49302	+	0.861994i	−0.999968	−	0.00800815i	$-0.997451\pi$
$41$	−61.2137	+	35.3417i	−1.49302	+	0.861994i	−0.493049	+	0.870002i	$0.664118\pi$
$42$		0			0
$43$	29.8827	−	51.7583i	0.694946	−	1.20368i	−0.275253	−	0.961372i	$-0.588762\pi$
$43$	29.8827	−	51.7583i	0.694946	−	1.20368i	0.970199	−	0.242310i	$-0.0779052\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	52.1600	+	30.1146i	1.10979	+	0.640736i	0.938774	−	0.344533i	$-0.111963\pi$
$47$	52.1600	+	30.1146i	1.10979	+	0.640736i	0.171013	+	0.985269i	$0.445296\pi$
$48$		0			0
$49$	5.30385	+	9.18653i	0.108242	+	0.187480i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	19.7718i		−	0.373053i	−0.982450	−	0.186526i	$-0.940277\pi$
$53$		−	19.7718i		−	0.373053i	0.982450	−	0.186526i	$-0.0597229\pi$
$54$		0			0
$55$	−6.58846			−0.119790
$56$		0			0
$57$		0			0
$58$		0			0
$59$	72.2364	−	41.7057i	1.22435	−	0.706876i	0.258504	−	0.966010i	$-0.416770\pi$
$59$	72.2364	−	41.7057i	1.22435	−	0.706876i	0.965841	+	0.259134i	$0.0834371\pi$
$60$		0			0
$61$	−15.3827	+	26.6436i	−0.252175	+	0.436780i	−0.964124	−	0.265451i	$-0.914479\pi$
$61$	−15.3827	+	26.6436i	−0.252175	+	0.436780i	0.711949	+	0.702231i	$0.247813\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−57.1791	−	33.0124i	−0.879678	−	0.507883i
$66$		0			0
$67$	5.29423	+	9.16987i	0.0790183	+	0.136864i	0.902827	−	0.430005i	$-0.141488\pi$
$67$	5.29423	+	9.16987i	0.0790183	+	0.136864i	−0.823808	+	0.566868i	$0.808155\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.63342i			0.0511750i	0.999673	+	0.0255875i	$0.00814564\pi$
$71$			3.63342i			0.0511750i	−0.999673	+	0.0255875i	$0.991854\pi$
$72$		0			0
$73$	−17.2346			−0.236091			−0.118045	−	0.993008i	$-0.537663\pi$
$73$	−17.2346			−0.236091			−0.118045	+	0.993008i	$0.537663\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	6.10016	−	3.52193i	0.0792229	−	0.0457394i
$78$		0			0
$79$	−56.6865	+	98.1840i	−0.717551	+	1.24283i	0.244416	+	0.969670i	$0.421404\pi$
$79$	−56.6865	+	98.1840i	−0.717551	+	1.24283i	−0.961967	+	0.273164i	$0.911930\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	10.2313	+	5.90705i	0.123269	+	0.0711693i	0.560367	−	0.828245i	$-0.310660\pi$
$83$	10.2313	+	5.90705i	0.123269	+	0.0711693i	−0.437098	+	0.899414i	$0.643994\pi$
$84$		0			0
$85$	−90.5596	−	156.854i	−1.06541	−	1.84534i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	111.252i		−	1.25003i	−0.780614	−	0.625013i	$-0.785094\pi$
$89$		−	111.252i		−	1.25003i	0.780614	−	0.625013i	$-0.214906\pi$
$90$		0			0
$91$	70.5885			0.775697
$92$		0			0
$93$		0			0
$94$		0			0
$95$	165.534	−	95.5709i	1.74246	−	1.00601i
$96$		0			0
$97$	−37.1577	+	64.3590i	−0.383069	+	0.663495i	−0.991499	−	0.130113i	$-0.958466\pi$
$97$	−37.1577	+	64.3590i	−0.383069	+	0.663495i	0.608430	+	0.793607i	$0.291799\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.9607	−	8.63759i	−0.148126	−	0.0855206i	0.424105	−	0.905613i	$-0.360589\pi$
$101$	−14.9607	−	8.63759i	−0.148126	−	0.0855206i	−0.572231	+	0.820092i	$0.693922\pi$
$102$		0			0
$103$	56.0000	+	96.9948i	0.543689	+	0.941698i	0.998688	+	0.0512055i	$0.0163063\pi$
$103$	56.0000	+	96.9948i	0.543689	+	0.941698i	−0.454999	+	0.890492i	$0.650360\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	141.144i		−	1.31910i	−0.751660	−	0.659551i	$-0.770746\pi$
$107$		−	141.144i		−	1.31910i	0.751660	−	0.659551i	$-0.229254\pi$
$108$		0			0
$109$	171.708			1.57530			0.787650	−	0.616123i	$-0.211298\pi$
$109$	171.708			1.57530			0.787650	+	0.616123i	$0.211298\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−4.82598	+	2.78628i	−0.0427078	+	0.0246574i	−0.521202	−	0.853433i	$-0.674516\pi$
$113$	−4.82598	+	2.78628i	−0.0427078	+	0.0246574i	0.478494	+	0.878091i	$0.341183\pi$
$114$		0			0
$115$	97.4711	−	168.825i	0.847575	−	1.46804i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	167.696	+	96.8192i	1.40921	+	0.813607i
$120$		0			0
$121$	−59.8538	−	103.670i	−0.494660	−	0.856776i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	95.1140i		−	0.760912i
$126$		0			0
$127$	−68.2346			−0.537281			−0.268640	−	0.963241i	$-0.586574\pi$
$127$	−68.2346			−0.537281			−0.268640	+	0.963241i	$0.586574\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	133.450	−	77.0474i	1.01870	−	0.588148i	0.104975	−	0.994475i	$-0.466524\pi$
$131$	133.450	−	77.0474i	1.01870	−	0.588148i	0.913728	+	0.406327i	$0.133190\pi$
$132$		0			0
$133$	−102.177	+	176.976i	−0.768247	+	1.33064i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−66.2328	−	38.2395i	−0.483451	−	0.279121i	0.238403	−	0.971166i	$-0.423376\pi$
$137$	−66.2328	−	38.2395i	−0.483451	−	0.279121i	−0.721854	+	0.692046i	$0.756710\pi$
$138$		0			0
$139$	−47.0192	−	81.4397i	−0.338268	−	0.585897i	0.645839	−	0.763474i	$-0.276508\pi$
$139$	−47.0192	−	81.4397i	−0.338268	−	0.585897i	−0.984107	+	0.177576i	$0.943174\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			12.9509i			0.0905658i
$144$		0			0
$145$	148.177			1.02191
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−58.1636	+	33.5808i	−0.390360	+	0.225374i	−0.682316	−	0.731057i	$-0.739027\pi$
$149$	−58.1636	+	33.5808i	−0.390360	+	0.225374i	0.291956	+	0.956432i	$0.405694\pi$
$150$		0			0
$151$	−23.1769	+	40.1436i	−0.153490	+	0.265852i	−0.932508	−	0.361149i	$-0.882384\pi$
$151$	−23.1769	+	40.1436i	−0.153490	+	0.265852i	0.779018	+	0.627001i	$0.215718\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−116.327	−	67.1615i	−0.750498	−	0.433300i
$156$		0			0
$157$	84.7942	+	146.868i	0.540091	+	0.935464i	0.998898	+	0.0469288i	$0.0149434\pi$
$157$	84.7942	+	146.868i	0.540091	+	0.935464i	−0.458808	+	0.888536i	$0.651723\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			208.417i			1.29452i
$162$		0			0
$163$	−243.023			−1.49094			−0.745469	−	0.666540i	$-0.767775\pi$
$163$	−243.023			−1.49094			−0.745469	+	0.666540i	$0.767775\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−97.6215	+	56.3618i	−0.584560	+	0.337496i	−0.762944	−	0.646465i	$-0.776247\pi$
$167$	−97.6215	+	56.3618i	−0.584560	+	0.337496i	0.178383	+	0.983961i	$0.442913\pi$
$168$		0			0
$169$	19.6077	−	33.9615i	0.116022	−	0.200956i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−162.677	−	93.9214i	−0.940328	−	0.542898i	−0.0502647	−	0.998736i	$-0.516007\pi$
$173$	−162.677	−	93.9214i	−0.940328	−	0.542898i	−0.890063	+	0.455837i	$0.849340\pi$
$174$		0			0
$175$	−26.6077	−	46.0859i	−0.152044	−	0.263348i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	16.3613i		−	0.0914042i	−0.998955	−	0.0457021i	$-0.985448\pi$
$179$		−	16.3613i		−	0.0914042i	0.998955	−	0.0457021i	$-0.0145525\pi$
$180$		0			0
$181$	−227.608			−1.25750			−0.628751	−	0.777607i	$-0.716433\pi$
$181$	−227.608			−1.25750			−0.628751	+	0.777607i	$0.716433\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−44.1874	+	25.5116i	−0.238851	+	0.137900i
$186$		0			0
$187$	−17.7635	+	30.7673i	−0.0949919	+	0.164531i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	315.720	+	182.281i	1.65299	+	0.954352i	0.975835	+	0.218508i	$0.0701190\pi$
$191$	315.720	+	182.281i	1.65299	+	0.954352i	0.677151	+	0.735844i	$0.263214\pi$
$192$		0			0
$193$	12.4423	+	21.5507i	0.0644678	+	0.111662i	0.896458	−	0.443129i	$-0.146132\pi$
$193$	12.4423	+	21.5507i	0.0644678	+	0.111662i	−0.831990	+	0.554791i	$0.812798\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			171.259i			0.869333i	0.900592	+	0.434666i	$0.143134\pi$
$197$			171.259i			0.869333i	−0.900592	+	0.434666i	$0.856866\pi$
$198$		0			0
$199$	122.431			0.615230			0.307615	−	0.951511i	$-0.400469\pi$
$199$	122.431			0.615230			0.307615	+	0.951511i	$0.400469\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−137.195	+	79.2096i	−0.675837	+	0.390195i
$204$		0			0
$205$	204.825	−	354.767i	0.999146	−	1.73057i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−32.4698	−	18.7465i	−0.155358	−	0.0896960i
$210$		0			0
$211$	186.256	+	322.604i	0.882729	+	1.52893i	0.848295	+	0.529524i	$0.177629\pi$
$211$	186.256	+	322.604i	0.882729	+	1.52893i	0.0344337	+	0.999407i	$0.489037\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			346.373i			1.61104i
$216$		0			0
$217$	143.608			0.661787
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−308.327	+	178.013i	−1.39515	+	0.805487i
$222$		0			0
$223$	160.060	−	277.231i	0.717756	−	1.24319i	−0.244131	−	0.969742i	$-0.578503\pi$
$223$	160.060	−	277.231i	0.717756	−	1.24319i	0.961887	−	0.273448i	$-0.0881640\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−13.1848	−	7.61227i	−0.0580830	−	0.0335342i	0.470677	−	0.882305i	$-0.344010\pi$
$227$	−13.1848	−	7.61227i	−0.0580830	−	0.0335342i	−0.528760	+	0.848771i	$0.677343\pi$
$228$		0			0
$229$	89.8538	+	155.631i	0.392375	+	0.679613i	0.992762	−	0.120096i	$-0.0383203\pi$
$229$	89.8538	+	155.631i	0.392375	+	0.679613i	−0.600387	+	0.799709i	$0.704987\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	150.796i		−	0.647193i	−0.946195	−	0.323596i	$-0.895108\pi$
$233$		−	150.796i		−	0.647193i	0.946195	−	0.323596i	$-0.104892\pi$
$234$		0			0
$235$	−349.061			−1.48537
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−229.103	+	132.272i	−0.958588	+	0.553441i	−0.895738	−	0.444582i	$-0.853352\pi$
$239$	−229.103	+	132.272i	−0.958588	+	0.553441i	−0.0628500	+	0.998023i	$0.520019\pi$
$240$		0			0
$241$	−114.225	+	197.844i	−0.473963	+	0.820927i	−0.999556	−	0.0298087i	$-0.990510\pi$
$241$	−114.225	+	197.844i	−0.473963	+	0.820927i	0.525593	+	0.850736i	$0.323844\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−53.2411	−	30.7387i	−0.217310	−	0.125464i
$246$		0			0
$247$	−187.863	−	325.389i	−0.760581	−	1.31736i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			47.2783i			0.188360i	0.995555	+	0.0941798i	$0.0300229\pi$
$251$			47.2783i			0.188360i	−0.995555	+	0.0941798i	$0.969977\pi$
$252$		0			0
$253$	−38.2384			−0.151140
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−22.3351	+	12.8952i	−0.0869069	+	0.0501757i	−0.542824	−	0.839847i	$-0.682645\pi$
$257$	−22.3351	+	12.8952i	−0.0869069	+	0.0501757i	0.455917	+	0.890022i	$0.349311\pi$
$258$		0			0
$259$	27.2750	−	47.2417i	0.105309	−	0.182400i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−247.017	−	142.615i	−0.939228	−	0.542263i	−0.0495095	−	0.998774i	$-0.515766\pi$
$263$	−247.017	−	142.615i	−0.939228	−	0.542263i	−0.889718	+	0.456510i	$0.849099\pi$
$264$		0			0
$265$	57.2942	+	99.2365i	0.216205	+	0.374477i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			205.341i			0.763350i	0.924297	+	0.381675i	$0.124653\pi$
$269$			205.341i			0.763350i	−0.924297	+	0.381675i	$0.875347\pi$
$270$		0			0
$271$	282.004			1.04060			0.520302	−	0.853982i	$-0.325819\pi$
$271$	282.004			1.04060			0.520302	+	0.853982i	$0.325819\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	8.45541	−	4.88173i	0.0307469	−	0.0177518i
$276$		0			0
$277$	−24.3538	+	42.1821i	−0.0879200	+	0.152282i	−0.906632	−	0.421923i	$-0.861355\pi$
$277$	−24.3538	+	42.1821i	−0.0879200	+	0.152282i	0.818712	+	0.574205i	$0.194689\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	213.659	+	123.356i	0.760353	+	0.438990i	0.829422	−	0.558622i	$-0.188670\pi$
$281$	213.659	+	123.356i	0.760353	+	0.438990i	−0.0690697	+	0.997612i	$0.522003\pi$
$282$		0			0
$283$	176.473	+	305.660i	0.623580	+	1.08007i	0.988814	+	0.149156i	$0.0476556\pi$
$283$	176.473	+	305.660i	0.623580	+	1.08007i	−0.365234	+	0.930916i	$0.619011\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			437.966i			1.52601i
$288$		0			0
$289$	−687.650			−2.37941
$290$		0			0
$291$		0			0
$292$		0			0
$293$	242.191	−	139.829i	0.826590	−	0.477232i	−0.0260937	−	0.999660i	$-0.508307\pi$
$293$	242.191	−	139.829i	0.826590	−	0.477232i	0.852684	+	0.522428i	$0.174973\pi$
$294$		0			0
$295$	−241.708	+	418.650i	−0.819348	+	1.41915i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−331.859	−	191.599i	−1.10990	−	0.640798i
$300$		0			0
$301$	−185.158	−	320.703i	−0.615142	−	1.06546i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	178.302i		−	0.584598i
$306$		0			0
$307$	234.708			0.764520			0.382260	−	0.924055i	$-0.375146\pi$
$307$	234.708			0.764520			0.382260	+	0.924055i	$0.375146\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	204.123	−	117.850i	0.656343	−	0.378940i	−0.134539	−	0.990908i	$-0.542955\pi$
$311$	204.123	−	117.850i	0.656343	−	0.378940i	0.790882	+	0.611969i	$0.209622\pi$
$312$		0			0
$313$	178.835	−	309.751i	0.571357	−	0.989619i	−0.425070	−	0.905160i	$-0.639751\pi$
$313$	178.835	−	309.751i	0.571357	−	0.989619i	0.996427	−	0.0844583i	$-0.0269160\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	523.067	+	301.993i	1.65005	+	0.952659i	0.977047	+	0.213024i	$0.0683311\pi$
$317$	523.067	+	301.993i	1.65005	+	0.952659i	0.673007	+	0.739636i	$0.265002\pi$
$318$		0			0
$319$	−14.5326	−	25.1713i	−0.0455568	−	0.0789068i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1030.69i		−	3.19101i
$324$		0			0
$325$	97.8423			0.301053
$326$		0			0
$327$		0			0
$328$		0			0
$329$	323.191	−	186.595i	0.982344	−	0.567157i
$330$		0			0
$331$	18.3712	−	31.8198i	0.0555021	−	0.0961324i	−0.836939	−	0.547295i	$-0.815657\pi$
$331$	18.3712	−	31.8198i	0.0555021	−	0.0961324i	0.892442	+	0.451163i	$0.148991\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−53.1445	−	30.6830i	−0.158640	−	0.0915910i
$336$		0			0
$337$	−84.1384	−	145.732i	−0.249669	−	0.432439i	0.713765	−	0.700385i	$-0.246988\pi$
$337$	−84.1384	−	145.732i	−0.249669	−	0.432439i	−0.963434	+	0.267946i	$0.913655\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			26.3478i			0.0772663i
$342$		0			0
$343$	369.338			1.07679
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−152.947	+	88.3041i	−0.440770	+	0.254479i	−0.703924	−	0.710275i	$-0.748570\pi$
$347$	−152.947	+	88.3041i	−0.440770	+	0.254479i	0.263154	+	0.964754i	$0.415237\pi$
$348$		0			0
$349$	−65.6077	+	113.636i	−0.187988	+	0.325604i	−0.944579	−	0.328284i	$-0.893530\pi$
$349$	−65.6077	+	113.636i	−0.187988	+	0.325604i	0.756592	+	0.653888i	$0.226863\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	11.7952	+	6.80994i	0.0334140	+	0.0192916i	0.516614	−	0.856218i	$-0.327192\pi$
$353$	11.7952	+	6.80994i	0.0334140	+	0.0192916i	−0.483200	+	0.875510i	$0.660526\pi$
$354$		0			0
$355$	−10.5289	−	18.2365i	−0.0296588	−	0.0513705i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			96.5853i			0.269040i	0.990911	+	0.134520i	$0.0429492\pi$
$359$			96.5853i			0.269040i	−0.990911	+	0.134520i	$0.957051\pi$
$360$		0			0
$361$	726.731			2.01310
$362$		0			0
$363$		0			0
$364$		0			0
$365$	86.5023	−	49.9421i	0.236993	−	0.136828i
$366$		0			0
$367$	−127.473	+	220.790i	−0.347338	+	0.601607i	−0.985776	−	0.168067i	$-0.946248\pi$
$367$	−127.473	+	220.790i	−0.347338	+	0.601607i	0.638438	+	0.769674i	$0.279581\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−106.096	−	61.2545i	−0.285973	−	0.165106i
$372$		0			0
$373$	45.5500	+	78.8949i	0.122118	+	0.211514i	0.920603	−	0.390501i	$-0.127698\pi$
$373$	45.5500	+	78.8949i	0.122118	+	0.211514i	−0.798485	+	0.602015i	$0.794365\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	291.271i		−	0.772602i
$378$		0			0
$379$	193.454			0.510432			0.255216	−	0.966884i	$-0.417853\pi$
$379$	193.454			0.510432			0.255216	+	0.966884i	$0.417853\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	288.347	−	166.477i	0.752865	−	0.434667i	−0.0738632	−	0.997268i	$-0.523533\pi$
$383$	288.347	−	166.477i	0.752865	−	0.434667i	0.826728	+	0.562602i	$0.190199\pi$
$384$		0			0
$385$	−20.4115	+	35.3538i	−0.0530170	+	0.0918281i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	35.6165	+	20.5632i	0.0915590	+	0.0528616i	0.545080	−	0.838384i	$-0.316499\pi$
$389$	35.6165	+	20.5632i	0.0915590	+	0.0528616i	−0.453521	+	0.891245i	$0.649832\pi$
$390$		0			0
$391$	−525.594	−	910.356i	−1.34423	−	2.32828i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	657.060i		−	1.66344i
$396$		0			0
$397$	−176.881			−0.445544			−0.222772	−	0.974871i	$-0.571510\pi$
$397$	−176.881			−0.445544			−0.222772	+	0.974871i	$0.571510\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	378.613	−	218.593i	0.944173	−	0.545119i	0.0529071	−	0.998599i	$-0.483151\pi$
$401$	378.613	−	218.593i	0.944173	−	0.545119i	0.891266	+	0.453481i	$0.149818\pi$
$402$		0			0
$403$	−132.019	+	228.664i	−0.327591	+	0.567405i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.66746	+	5.00416i	0.0212960	+	0.0122952i
$408$		0			0
$409$	72.9519	+	126.356i	0.178367	+	0.308940i	0.941321	−	0.337512i	$-0.109585\pi$
$409$	72.9519	+	126.356i	0.178367	+	0.308940i	−0.762955	+	0.646452i	$0.776252\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	516.830i		−	1.25140i
$414$		0			0
$415$	−68.4693			−0.164986
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−115.555	+	66.7156i	−0.275787	+	0.159226i	−0.631515	−	0.775364i	$-0.717566\pi$
$419$	−115.555	+	66.7156i	−0.275787	+	0.159226i	0.355728	+	0.934590i	$0.384233\pi$
$420$		0			0
$421$	−144.346	+	250.015i	−0.342865	+	0.593859i	−0.984963	−	0.172763i	$-0.944731\pi$
$421$	−144.346	+	250.015i	−0.342865	+	0.593859i	0.642099	+	0.766622i	$0.278064\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	232.442	+	134.201i	0.546923	+	0.315766i
$426$		0			0
$427$	95.3135	+	165.088i	0.223217	+	0.386622i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			669.565i			1.55351i	0.629800	+	0.776757i	$0.283137\pi$
$431$			669.565i			1.55351i	−0.629800	+	0.776757i	$0.716863\pi$
$432$		0			0
$433$	−510.654			−1.17934			−0.589669	−	0.807645i	$-0.700742\pi$
$433$	−510.654			−1.17934			−0.589669	+	0.807645i	$0.700742\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	960.732	−	554.679i	2.19847	−	1.26929i
$438$		0			0
$439$	−38.6499	+	66.9437i	−0.0880409	+	0.152491i	−0.906683	−	0.421813i	$-0.861394\pi$
$439$	−38.6499	+	66.9437i	−0.0880409	+	0.152491i	0.818642	+	0.574304i	$0.194727\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−657.617	−	379.676i	−1.48446	−	0.857055i	−0.484620	−	0.874725i	$-0.661042\pi$
$443$	−657.617	−	379.676i	−1.48446	−	0.857055i	−0.999844	+	0.0176696i	$0.994375\pi$
$444$		0			0
$445$	322.385	+	558.386i	0.724460	+	1.25480i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			541.149i			1.20523i	0.798032	+	0.602616i	$0.205875\pi$
$449$			541.149i			1.20523i	−0.798032	+	0.602616i	$0.794125\pi$
$450$		0			0
$451$	−80.3538			−0.178168
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−354.290	+	204.550i	−0.778660	+	0.449560i
$456$		0			0
$457$	27.7942	−	48.1410i	0.0608189	−	0.105341i	−0.834013	−	0.551745i	$-0.813962\pi$
$457$	27.7942	−	48.1410i	0.0608189	−	0.105341i	0.894832	+	0.446404i	$0.147295\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	217.114	+	125.351i	0.470964	+	0.271911i	0.716643	−	0.697440i	$-0.245678\pi$
$461$	217.114	+	125.351i	0.470964	+	0.271911i	−0.245679	+	0.969351i	$0.579011\pi$
$462$		0			0
$463$	−285.512	−	494.520i	−0.616656	−	1.06808i	−0.990092	−	0.140423i	$-0.955154\pi$
$463$	−285.512	−	494.520i	−0.616656	−	1.06808i	0.373436	−	0.927656i	$-0.378180\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	72.2663i		−	0.154746i	−0.997002	−	0.0773729i	$-0.975347\pi$
$467$		−	72.2663i		−	0.154746i	0.997002	−	0.0773729i	$-0.0246532\pi$
$468$		0			0
$469$	65.6077			0.139888
$470$		0			0
$471$		0			0
$472$		0			0
$473$	58.8395	−	33.9710i	0.124396	−	0.0718203i
$474$		0			0
$475$	−141.627	+	245.305i	−0.298162	+	0.516432i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	612.735	+	353.763i	1.27920	+	0.738544i	0.976701	−	0.214606i	$-0.0688468\pi$
$479$	612.735	+	353.763i	1.27920	+	0.738544i	0.302496	+	0.953151i	$0.402180\pi$
$480$		0			0
$481$	50.1481	+	86.8590i	0.104258	+	0.180580i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	430.699i		−	0.888039i
$486$		0			0
$487$	360.908			0.741083			0.370542	−	0.928816i	$-0.379172\pi$
$487$	360.908			0.741083			0.370542	+	0.928816i	$0.379172\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−588.914	+	340.010i	−1.19942	+	0.692484i	−0.960425	−	0.278538i	$-0.910150\pi$
$491$	−588.914	+	340.010i	−1.19942	+	0.692484i	−0.238992	+	0.971022i	$0.576817\pi$
$492$		0			0
$493$	399.508	−	691.967i	0.810360	−	1.40358i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	19.4971	+	11.2566i	0.0392295	+	0.0226491i
$498$		0			0
$499$	−118.352	−	204.992i	−0.237178	−	0.410805i	0.722725	−	0.691135i	$-0.242889\pi$
$499$	−118.352	−	204.992i	−0.237178	−	0.410805i	−0.959903	+	0.280331i	$0.909556\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	95.6715i		−	0.190202i	−0.995468	−	0.0951009i	$-0.969683\pi$
$503$		−	95.6715i		−	0.190202i	0.995468	−	0.0951009i	$-0.0303174\pi$
$504$		0			0
$505$	100.119			0.198256
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−376.915	+	217.612i	−0.740501	+	0.427529i	−0.822252	−	0.569124i	$-0.807282\pi$
$509$	−376.915	+	217.612i	−0.740501	+	0.427529i	0.0817503	+	0.996653i	$0.473949\pi$
$510$		0			0
$511$	−53.3942	+	92.4815i	−0.104490	+	0.180981i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−562.139	−	324.551i	−1.09153	−	0.630196i
$516$		0			0
$517$	34.2346	+	59.2961i	0.0662178	+	0.114693i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			794.771i			1.52547i	0.646709	+	0.762737i	$0.276145\pi$
$521$			794.771i			1.52547i	−0.646709	+	0.762737i	$0.723855\pi$
$522$		0			0
$523$	−594.481			−1.13667			−0.568337	−	0.822796i	$-0.692413\pi$
$523$	−594.481			−1.13667			−0.568337	+	0.822796i	$0.692413\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−627.272	+	362.156i	−1.19027	+	0.687202i
$528$		0			0
$529$	301.208	−	521.707i	0.569391	−	0.986214i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−697.365	−	402.624i	−1.30838	−	0.755392i
$534$		0			0
$535$	409.004	+	708.415i	0.764493	+	1.32414i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			12.0590i			0.0223728i
$540$		0			0
$541$	219.508			0.405744			0.202872	−	0.979205i	$-0.434972\pi$
$541$	219.508			0.405744			0.202872	+	0.979205i	$0.434972\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−861.818	+	497.571i	−1.58132	+	0.912974i
$546$		0			0
$547$	292.296	−	506.272i	0.534362	−	0.925542i	−0.464832	−	0.885399i	$-0.653885\pi$
$547$	292.296	−	506.272i	0.534362	−	0.925542i	0.999194	−	0.0401434i	$-0.0127815\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	730.259	+	421.615i	1.32533	+	0.765182i
$552$		0			0
$553$	351.238	+	608.363i	0.635151	+	1.10011i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			597.187i			1.07215i	0.844171	+	0.536075i	$0.180093\pi$
$557$			597.187i			1.07215i	−0.844171	+	0.536075i	$0.819907\pi$
$558$		0			0
$559$	680.865			1.21801
$560$		0			0
$561$		0			0
$562$		0			0
$563$	311.957	−	180.108i	0.554097	−	0.319908i	−0.196676	−	0.980469i	$-0.563015\pi$
$563$	311.957	−	180.108i	0.554097	−	0.319908i	0.750773	+	0.660561i	$0.229681\pi$
$564$		0			0
$565$	16.1481	−	27.9693i	0.0285806	−	0.0495031i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	486.466	+	280.861i	0.854949	+	0.493605i	0.862318	−	0.506368i	$-0.169012\pi$
$569$	486.466	+	280.861i	0.854949	+	0.493605i	−0.00736827	+	0.999973i	$0.502345\pi$
$570$		0			0
$571$	19.0385	+	32.9756i	0.0333423	+	0.0577506i	0.882215	−	0.470847i	$-0.156052\pi$
$571$	19.0385	+	32.9756i	0.0333423	+	0.0577506i	−0.848873	+	0.528597i	$0.822718\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			288.886i			0.502410i
$576$		0			0
$577$	−747.008			−1.29464			−0.647320	−	0.762218i	$-0.724110\pi$
$577$	−747.008			−1.29464			−0.647320	+	0.762218i	$0.724110\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	63.3948	−	36.6010i	0.109113	−	0.0629965i
$582$		0			0
$583$	11.2384	−	19.4655i	0.0192768	−	0.0333885i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−519.419	−	299.887i	−0.884870	−	0.510880i	−0.0126092	−	0.999921i	$-0.504014\pi$
$587$	−519.419	−	299.887i	−0.884870	−	0.510880i	−0.872261	+	0.489040i	$0.837347\pi$
$588$		0			0
$589$	−382.196	−	661.983i	−0.648890	−	1.12391i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			212.184i			0.357814i	0.983866	+	0.178907i	$0.0572561\pi$
$593$			212.184i			0.357814i	−0.983866	+	0.178907i	$0.942744\pi$
$594$		0			0
$595$	−1122.24			−1.88612
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−281.649	+	162.610i	−0.470198	+	0.271469i	−0.716323	−	0.697769i	$-0.754176\pi$
$599$	−281.649	+	162.610i	−0.470198	+	0.271469i	0.246124	+	0.969238i	$0.420843\pi$
$600$		0			0
$601$	−373.738	+	647.334i	−0.621861	+	1.07709i	0.367278	+	0.930111i	$0.380290\pi$
$601$	−373.738	+	647.334i	−0.621861	+	1.07709i	−0.989139	+	0.146983i	$0.953044\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	600.824	+	346.886i	0.993098	+	0.573366i
$606$		0			0
$607$	61.8634	+	107.151i	0.101917	+	0.176525i	0.912474	−	0.409134i	$-0.134169\pi$
$607$	61.8634	+	107.151i	0.101917	+	0.176525i	−0.810558	+	0.585659i	$0.800836\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			686.149i			1.12299i
$612$		0			0
$613$	−954.008			−1.55629			−0.778146	−	0.628083i	$-0.783840\pi$
$613$	−954.008			−1.55629			−0.778146	+	0.628083i	$0.783840\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−216.613	+	125.061i	−0.351074	+	0.202693i	−0.665158	−	0.746702i	$-0.731636\pi$
$617$	−216.613	+	125.061i	−0.351074	+	0.202693i	0.314084	+	0.949395i	$0.398303\pi$
$618$		0			0
$619$	508.769	−	881.214i	0.821921	−	1.42361i	−0.0823287	−	0.996605i	$-0.526236\pi$
$619$	508.769	−	881.214i	0.821921	−	1.42361i	0.904250	−	0.427004i	$-0.140431\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−596.983	−	344.668i	−0.958239	−	0.553240i
$624$		0			0
$625$	382.975	+	663.332i	0.612760	+	1.06133i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			275.133i			0.437413i
$630$		0			0
$631$	−75.6001			−0.119810			−0.0599050	−	0.998204i	$-0.519080\pi$
$631$	−75.6001			−0.119810			−0.0599050	+	0.998204i	$0.519080\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	342.476	−	197.729i	0.539333	−	0.311384i
$636$		0			0
$637$	−60.4230	+	104.656i	−0.0948556	+	0.164295i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−209.895	−	121.183i	−0.327450	−	0.189053i	0.327259	−	0.944935i	$-0.393875\pi$
$641$	−209.895	−	121.183i	−0.327450	−	0.189053i	−0.654708	+	0.755882i	$0.727209\pi$
$642$		0			0
$643$	366.592	+	634.956i	0.570128	+	0.987490i	0.996552	+	0.0829673i	$0.0264397\pi$
$643$	366.592	+	634.956i	0.570128	+	0.987490i	−0.426424	+	0.904523i	$0.640227\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			1177.72i			1.82028i	0.414296	+	0.910142i	$0.364028\pi$
$647$			1177.72i			1.82028i	−0.414296	+	0.910142i	$0.635972\pi$
$648$		0			0
$649$	94.8231			0.146106
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−550.151	+	317.630i	−0.842497	+	0.486416i	−0.858112	−	0.513462i	$-0.828363\pi$
$653$	−550.151	+	317.630i	−0.842497	+	0.486416i	0.0156151	+	0.999878i	$0.495029\pi$
$654$		0			0
$655$	−446.533	+	773.417i	−0.681729	+	1.18079i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−793.616	−	458.194i	−1.20427	−	0.695287i	−0.242770	−	0.970084i	$-0.578056\pi$
$659$	−793.616	−	458.194i	−1.20427	−	0.695287i	−0.961502	+	0.274797i	$0.911389\pi$
$660$		0			0
$661$	−206.679	−	357.978i	−0.312676	−	0.541571i	0.666265	−	0.745715i	$-0.267892\pi$
$661$	−206.679	−	357.978i	−0.312676	−	0.541571i	−0.978941	+	0.204145i	$0.934559\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1184.34i		−	1.78097i
$666$		0			0
$667$	859.996			1.28935
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−30.2888	+	17.4872i	−0.0451397	+	0.0260614i
$672$		0			0
$673$	366.227	−	634.324i	0.544171	−	0.942531i	−0.454488	−	0.890753i	$-0.650178\pi$
$673$	366.227	−	634.324i	0.544171	−	0.942531i	0.998659	−	0.0517784i	$-0.0164890\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	507.219	+	292.843i	0.749215	+	0.432559i	0.825410	−	0.564533i	$-0.190944\pi$
$677$	507.219	+	292.843i	0.749215	+	0.432559i	−0.0761952	+	0.997093i	$0.524277\pi$
$678$		0			0
$679$	230.235	+	398.778i	0.339079	+	0.587302i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1082.97i		−	1.58560i	−0.609480	−	0.792801i	$-0.708622\pi$
$683$		−	1082.97i		−	1.58560i	0.609480	−	0.792801i	$-0.291378\pi$
$684$		0			0
$685$	443.238			0.647063
$686$		0			0
$687$		0			0
$688$		0			0
$689$	195.069	−	112.623i	0.283119	−	0.163459i
$690$		0			0
$691$	89.1788	−	154.462i	0.129058	−	0.223534i	−0.794254	−	0.607586i	$-0.792138\pi$
$691$	89.1788	−	154.462i	0.129058	−	0.223534i	0.923312	+	0.384051i	$0.125471\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	471.988	+	272.503i	0.679120	+	0.392090i
$696$		0			0
$697$	−1104.48	−	1913.01i	−1.58462	−	2.74464i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			638.090i			0.910257i	0.890426	+	0.455129i	$0.150407\pi$
$701$			638.090i			0.910257i	−0.890426	+	0.455129i	$0.849593\pi$
$702$		0			0
$703$	−290.358			−0.413026
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−92.6990	+	53.5198i	−0.131116	+	0.0756999i
$708$		0			0
$709$	−412.423	+	714.338i	−0.581697	+	1.00753i	0.413582	+	0.910467i	$0.364278\pi$
$709$	−412.423	+	714.338i	−0.581697	+	1.00753i	−0.995278	+	0.0970614i	$0.969056\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−675.145	−	389.795i	−0.946908	−	0.546698i
$714$		0			0
$715$	−37.5289	−	65.0019i	−0.0524879	−	0.0909117i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	748.451i		−	1.04096i	−0.853874	−	0.520480i	$-0.825753\pi$
$719$		−	748.451i		−	1.04096i	0.853874	−	0.520480i	$-0.174247\pi$
$720$		0			0
$721$	693.969			0.962509
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−190.165	+	109.792i	−0.262297	+	0.151437i
$726$		0			0
$727$	−13.1788	+	22.8264i	−0.0181276	+	0.0313980i	−0.874947	−	0.484219i	$-0.839104\pi$
$727$	−13.1788	+	22.8264i	−0.0181276	+	0.0313980i	0.856819	+	0.515617i	$0.172437\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1617.52	+	933.876i	2.21275	+	1.27753i
$732$		0			0
$733$	536.473	+	929.199i	0.731887	+	1.26767i	0.956076	+	0.293119i	$0.0946934\pi$
$733$	536.473	+	929.199i	0.731887	+	1.26767i	−0.224189	+	0.974546i	$0.571973\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			12.0371i			0.0163325i
$738$		0			0
$739$	158.831			0.214926			0.107463	−	0.994209i	$-0.465727\pi$
$739$	158.831			0.214926			0.107463	+	0.994209i	$0.465727\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	142.080	−	82.0297i	0.191224	−	0.110403i	−0.401331	−	0.915933i	$-0.631452\pi$
$743$	142.080	−	82.0297i	0.191224	−	0.110403i	0.592556	+	0.805530i	$0.298119\pi$
$744$		0			0
$745$	194.619	−	337.090i	0.261234	−	0.452470i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−757.382	−	437.275i	−1.01119	−	0.583811i
$750$		0			0
$751$	−116.198	−	201.261i	−0.154724	−	0.267991i	0.778234	−	0.627974i	$-0.216116\pi$
$751$	−116.198	−	201.261i	−0.154724	−	0.267991i	−0.932959	+	0.359984i	$0.882782\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	268.646i		−	0.355823i
$756$		0			0
$757$	1012.72			1.33781			0.668905	−	0.743348i	$-0.266763\pi$
$757$	1012.72			1.33781			0.668905	+	0.743348i	$0.266763\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	501.427	−	289.499i	0.658905	−	0.380419i	−0.132954	−	0.991122i	$-0.542446\pi$
$761$	501.427	−	289.499i	0.658905	−	0.380419i	0.791860	+	0.610703i	$0.209113\pi$
$762$		0			0
$763$	531.963	−	921.388i	0.697200	−	1.20759i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	822.939	+	475.124i	1.07293	+	0.619458i
$768$		0			0
$769$	133.635	+	231.462i	0.173777	+	0.300991i	0.939737	−	0.341897i	$-0.111069\pi$
$769$	133.635	+	231.462i	0.173777	+	0.300991i	−0.765960	+	0.642888i	$0.777736\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	1356.98i		−	1.75548i	−0.479140	−	0.877739i	$-0.659051\pi$
$773$		−	1356.98i		−	1.75548i	0.479140	−	0.877739i	$-0.340949\pi$
$774$		0			0
$775$	199.054			0.256844
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2018.87	−	1165.60i	2.59162	−	1.49627i
$780$		0			0
$781$	−2.06526	+	3.57714i	−0.00264438	+	0.00458020i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−851.181	−	491.430i	−1.08431	−	0.626025i
$786$		0			0
$787$	−28.9404	−	50.1262i	−0.0367731	−	0.0636928i	0.847053	−	0.531508i	$-0.178375\pi$
$787$	−28.9404	−	50.1262i	−0.0367731	−	0.0636928i	−0.883826	+	0.467815i	$0.845041\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			34.5285i			0.0436517i
$792$		0			0
$793$	−350.488			−0.441978
$794$		0			0
$795$		0			0
$796$		0			0
$797$	142.793	−	82.4418i	0.179164	−	0.103440i	−0.407736	−	0.913100i	$-0.633682\pi$
$797$	142.793	−	82.4418i	0.179164	−	0.103440i	0.586900	+	0.809660i	$0.300348\pi$
$798$		0			0
$799$	−941.123	+	1630.07i	−1.17788	+	2.04014i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−16.9676	−	9.79627i	−0.0211303	−	0.0121996i
$804$		0			0
$805$	−603.946	−	1046.07i	−0.750244	−	1.29946i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	981.255i		−	1.21292i	−0.795113	−	0.606461i	$-0.792588\pi$
$809$		−	981.255i		−	1.21292i	0.795113	−	0.606461i	$-0.207412\pi$
$810$		0			0
$811$	−289.877			−0.357432			−0.178716	−	0.983901i	$-0.557194\pi$
$811$	−289.877			−0.357432			−0.178716	+	0.983901i	$0.557194\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1219.76	−	704.227i	1.49663	−	0.864082i
$816$		0			0
$817$	−985.554	+	1707.03i	−1.20631	+	2.08939i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	114.436	+	66.0695i	0.139386	+	0.0804745i	0.568071	−	0.822979i	$-0.307690\pi$
$821$	114.436	+	66.0695i	0.139386	+	0.0804745i	−0.428685	+	0.903454i	$0.641023\pi$
$822$		0			0
$823$	−243.096	−	421.055i	−0.295378	−	0.511610i	0.679695	−	0.733495i	$-0.262112\pi$
$823$	−243.096	−	421.055i	−0.295378	−	0.511610i	−0.975073	+	0.221885i	$0.928779\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			258.192i			0.312203i	0.987741	+	0.156101i	$0.0498927\pi$
$827$			258.192i			0.312203i	−0.987741	+	0.156101i	$0.950107\pi$
$828$		0			0
$829$	−376.400			−0.454041			−0.227020	−	0.973890i	$-0.572898\pi$
$829$	−376.400			−0.454041			−0.227020	+	0.973890i	$0.572898\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−287.092	+	165.753i	−0.344648	+	0.198983i
$834$		0			0
$835$	326.648	−	565.771i	0.391195	−	0.677570i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	252.519	+	145.792i	0.300976	+	0.173768i	0.642881	−	0.765966i	$-0.277739\pi$
$839$	252.519	+	145.792i	0.300976	+	0.173768i	−0.341905	+	0.939734i	$0.611072\pi$
$840$		0			0
$841$	−93.6556	−	162.216i	−0.111362	−	0.192885i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			227.275i			0.268964i
$846$		0			0
$847$	−741.727			−0.875711
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−256.457	+	148.065i	−0.301359	+	0.173990i
$852$		0			0
$853$	−96.7691	+	167.609i	−0.113446	+	0.196494i	−0.917157	−	0.398525i	$-0.869522\pi$
$853$	−96.7691	+	167.609i	−0.113446	+	0.196494i	0.803712	+	0.595019i	$0.202855\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	165.998	+	95.8387i	0.193696	+	0.111830i	0.593712	−	0.804678i	$-0.297662\pi$
$857$	165.998	+	95.8387i	0.193696	+	0.111830i	−0.400016	+	0.916508i	$0.630995\pi$
$858$		0			0
$859$	−598.258	−	1036.21i	−0.696458	−	1.20630i	−0.969687	−	0.244352i	$-0.921425\pi$
$859$	−598.258	−	1036.21i	−0.696458	−	1.20630i	0.273228	−	0.961949i	$-0.411909\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	692.323i		−	0.802228i	−0.916028	−	0.401114i	$-0.868623\pi$
$863$		−	692.323i		−	0.802228i	0.916028	−	0.401114i	$-0.131377\pi$
$864$		0			0
$865$	1088.65			1.25856
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−111.617	+	64.4419i	−0.128443	+	0.0741564i
$870$		0			0
$871$	−60.3135	+	104.466i	−0.0692462	+	0.119938i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−510.384	−	294.670i	−0.583296	−	0.336766i
$876$		0			0
$877$	513.306	+	889.072i	0.585297	+	1.01376i	0.994838	+	0.101473i	$0.0323555\pi$
$877$	513.306	+	889.072i	0.585297	+	1.01376i	−0.409541	+	0.912292i	$0.634311\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			984.820i			1.11784i	0.829220	+	0.558922i	$0.188785\pi$
$881$			984.820i			1.11784i	−0.829220	+	0.558922i	$0.811215\pi$
$882$		0			0
$883$	1038.19			1.17576			0.587878	−	0.808950i	$-0.299964\pi$
$883$	1038.19			1.17576			0.587878	+	0.808950i	$0.299964\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	146.847	−	84.7821i	0.165555	−	0.0955830i	−0.414933	−	0.909852i	$-0.636195\pi$
$887$	146.847	−	84.7821i	0.165555	−	0.0955830i	0.580488	+	0.814269i	$0.302862\pi$
$888$		0			0
$889$	−211.396	+	366.149i	−0.237791	+	0.411866i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1720.28	−	993.202i	−1.92640	−	1.11221i
$894$		0			0
$895$	47.4115	+	82.1192i	0.0529738	+	0.0917533i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	592.572i		−	0.659146i
$900$		0			0
$901$	617.896			0.685789
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1142.39	−	659.556i	1.26230	−	0.728792i
$906$		0			0
$907$	−365.981	+	633.897i	−0.403507	+	0.698894i	−0.994146	−	0.108041i	$-0.965542\pi$
$907$	−365.981	+	633.897i	−0.403507	+	0.698894i	0.590640	+	0.806936i	$0.298876\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	853.652	+	492.856i	0.937049	+	0.541006i	0.889034	−	0.457841i	$-0.151377\pi$
$911$	853.652	+	492.856i	0.937049	+	0.541006i	0.0480151	+	0.998847i	$0.484710\pi$
$912$		0			0
$913$	6.71521	+	11.6311i	0.00735510	+	0.0127394i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	954.795i		−	1.04122i
$918$		0			0
$919$	−839.650			−0.913656			−0.456828	−	0.889555i	$-0.651015\pi$
$919$	−839.650			−0.913656			−0.456828	+	0.889555i	$0.651015\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−35.8475	+	20.6965i	−0.0388380	+	0.0224231i
$924$		0			0
$925$	37.8057	−	65.4815i	0.0408711	−	0.0707908i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	519.360	+	299.853i	0.559053	+	0.322769i	0.752765	−	0.658289i	$-0.228720\pi$
$929$	519.360	+	299.853i	0.559053	+	0.322769i	−0.193712	+	0.981058i	$0.562053\pi$
$930$		0			0
$931$	−174.925	−	302.979i	−0.187889	−	0.325434i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	205.899i		−	0.220212i
$936$		0			0
$937$	1610.24			1.71850			0.859252	−	0.511553i	$-0.170930\pi$
$937$	1610.24			1.71850			0.859252	+	0.511553i	$0.170930\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−762.189	+	440.050i	−0.809978	+	0.467641i	−0.846948	−	0.531675i	$-0.821563\pi$
$941$	−762.189	+	440.050i	−0.809978	+	0.467641i	0.0369704	+	0.999316i	$0.488229\pi$
$942$		0			0
$943$	1188.77	−	2059.02i	1.26063	−	2.18347i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	853.072	+	492.522i	0.900816	+	0.520086i	0.877465	−	0.479641i	$-0.159233\pi$
$947$	853.072	+	492.522i	0.900816	+	0.520086i	0.0233509	+	0.999727i	$0.492566\pi$
$948$		0			0
$949$	−98.1711	−	170.037i	−0.103447	−	0.179175i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	845.550i		−	0.887251i	−0.896212	−	0.443625i	$-0.853692\pi$
$953$		−	845.550i		−	0.887251i	0.896212	−	0.443625i	$-0.146308\pi$
$954$		0			0
$955$	−2112.84			−2.21240
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−410.388	+	236.938i	−0.427934	+	0.247068i
$960$		0			0
$961$	211.915	−	367.048i	0.220515	−	0.381944i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−124.898	−	72.1100i	−0.129428	−	0.0747253i
$966$		0			0
$967$	513.471	+	889.358i	0.530994	+	0.919708i	0.999346	+	0.0361664i	$0.0115146\pi$
$967$	513.471	+	889.358i	0.530994	+	0.919708i	−0.468352	+	0.883542i	$0.655152\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			338.171i			0.348271i	0.984722	+	0.174135i	$0.0557130\pi$
$971$			338.171i			0.348271i	−0.984722	+	0.174135i	$0.944287\pi$
$972$		0			0
$973$	−582.677			−0.598846
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−789.485	+	455.809i	−0.808070	+	0.466540i	−0.846285	−	0.532730i	$-0.821166\pi$
$977$	−789.485	+	455.809i	−0.808070	+	0.466540i	0.0382151	+	0.999270i	$0.487833\pi$
$978$		0			0
$979$	63.2365	−	109.529i	0.0645930	−	0.111878i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	829.618	+	478.980i	0.843966	+	0.487264i	0.858610	−	0.512629i	$-0.171328\pi$
$983$	829.618	+	478.980i	0.843966	+	0.487264i	−0.0146445	+	0.999893i	$0.504662\pi$
$984$		0			0
$985$	−496.269	−	859.563i	−0.503827	−	0.872653i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2010.30i			2.03266i
$990$		0			0
$991$	1771.18			1.78727			0.893633	−	0.448798i	$-0.148148\pi$
$991$	1771.18			1.78727			0.893633	+	0.448798i	$0.148148\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−614.492	+	354.777i	−0.617580	+	0.356560i
$996$		0			0
$997$	544.325	−	942.799i	0.545963	−	0.945635i	−0.452583	−	0.891722i	$-0.649497\pi$
$997$	544.325	−	942.799i	0.545963	−	0.945635i	0.998546	−	0.0539130i	$-0.0171694\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.g.d.269.1		8
3.2	odd	2	inner	324.3.g.d.269.4		8
4.3	odd	2		1296.3.q.n.593.1		8
9.2	odd	6		324.3.c.a.161.4	yes	4
9.4	even	3	inner	324.3.g.d.53.4		8
9.5	odd	6	inner	324.3.g.d.53.1		8
9.7	even	3		324.3.c.a.161.1	✓	4
12.11	even	2		1296.3.q.n.593.4		8
36.7	odd	6		1296.3.e.f.161.1		4
36.11	even	6		1296.3.e.f.161.4		4
36.23	even	6		1296.3.q.n.1025.1		8
36.31	odd	6		1296.3.q.n.1025.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.c.a.161.1	✓	4	9.7	even	3
324.3.c.a.161.4	yes	4	9.2	odd	6
324.3.g.d.53.1		8	9.5	odd	6	inner
324.3.g.d.53.4		8	9.4	even	3	inner
324.3.g.d.269.1		8	1.1	even	1	trivial
324.3.g.d.269.4		8	3.2	odd	2	inner
1296.3.e.f.161.1		4	36.7	odd	6
1296.3.e.f.161.4		4	36.11	even	6
1296.3.q.n.593.1		8	4.3	odd	2
1296.3.q.n.593.4		8	12.11	even	2
1296.3.q.n.1025.1		8	36.23	even	6
1296.3.q.n.1025.4		8	36.31	odd	6

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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	324.g (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-5.01910 + 2.89778i) q^{5} +(3.09808 - 5.36603i) q^{7} +O(q^{10})\)	\(q+(-5.01910 + 2.89778i) q^{5} +(3.09808 - 5.36603i) q^{7} +(0.984508 + 0.568406i) q^{11} +(5.69615 + 9.86603i) q^{13} +31.2514i q^{17} -32.9808 q^{19} +(-29.1301 + 16.8183i) q^{23} +(4.29423 - 7.43782i) q^{25} +(-22.1420 - 12.7837i) q^{29} +(11.5885 + 20.0718i) q^{31} +35.9101i q^{35} +8.80385 q^{37} +(-61.2137 + 35.3417i) q^{41} +(29.8827 - 51.7583i) q^{43} +(52.1600 + 30.1146i) q^{47} +(5.30385 + 9.18653i) q^{49} -19.7718i q^{53} -6.58846 q^{55} +(72.2364 - 41.7057i) q^{59} +(-15.3827 + 26.6436i) q^{61} +(-57.1791 - 33.0124i) q^{65} +(5.29423 + 9.16987i) q^{67} +3.63342i q^{71} -17.2346 q^{73} +(6.10016 - 3.52193i) q^{77} +(-56.6865 + 98.1840i) q^{79} +(10.2313 + 5.90705i) q^{83} +(-90.5596 - 156.854i) q^{85} -111.252i q^{89} +70.5885 q^{91} +(165.534 - 95.5709i) q^{95} +(-37.1577 + 64.3590i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7}+O(q^{10}) \) 8 * q + 4 * q^7	\( 8 q + 4 q^{7} + 4 q^{13} - 56 q^{19} - 28 q^{25} - 32 q^{31} + 112 q^{37} + 52 q^{43} + 84 q^{49} + 72 q^{55} + 64 q^{61} - 20 q^{67} - 512 q^{73} - 308 q^{79} - 288 q^{85} + 440 q^{91} + 160 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 + 4 * q^13 - 56 * q^19 - 28 * q^25 - 32 * q^31 + 112 * q^37 + 52 * q^43 + 84 * q^49 + 72 * q^55 + 64 * q^61 - 20 * q^67 - 512 * q^73 - 308 * q^79 - 288 * q^85 + 440 * q^91 + 160 * q^97

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