LMFDB - Embedded newform 324.3.f.n.55.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(55,324)]

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([3, 4]))

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

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

chi := DirichletCharacter("324.55");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	324.f (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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			55.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	324.55
Dual form			324.3.f.n.271.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.59808 + 7.96410i) q^{5} -8.00000 q^{8} +O(q^{10})$	$q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.59808 + 7.96410i) q^{5} -8.00000 q^{8} +18.3923 q^{10} +(7.89230 + 13.6699i) q^{13} +(-8.00000 + 13.8564i) q^{16} +33.9808 q^{17} +(18.3923 - 31.8564i) q^{20} +(-29.7846 + 51.5885i) q^{25} +31.5692 q^{26} +(8.18653 - 14.1795i) q^{29} +(16.0000 + 27.7128i) q^{32} +(33.9808 - 58.8564i) q^{34} +14.2154 q^{37} +(-36.7846 - 63.7128i) q^{40} +(-40.0000 - 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(59.5692 + 103.177i) q^{50} +(31.5692 - 54.6795i) q^{52} +56.0000 q^{53} +(-16.3731 - 28.3590i) q^{58} +(-57.4615 + 99.5263i) q^{61} +64.0000 q^{64} +(-72.5788 + 125.710i) q^{65} +(-67.9615 - 117.713i) q^{68} -138.138 q^{73} +(14.2154 - 24.6218i) q^{74} -147.138 q^{80} -160.000 q^{82} +(156.246 + 270.626i) q^{85} +12.4500 q^{89} +(65.0000 - 112.583i) q^{97} -98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 8 * q^4 + 8 * q^5 - 32 * q^8	$ 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8} + 32 q^{10} - 10 q^{13} - 32 q^{16} + 32 q^{17} + 32 q^{20} - 36 q^{25} - 40 q^{26} - 40 q^{29} + 64 q^{32} + 32 q^{34} + 140 q^{37} - 64 q^{40} - 160 q^{41} - 98 q^{49} + 72 q^{50} - 40 q^{52} + 224 q^{53} + 80 q^{58} - 22 q^{61} + 256 q^{64} - 176 q^{65} - 64 q^{68} - 220 q^{73} + 140 q^{74} - 256 q^{80} - 640 q^{82} + 334 q^{85} + 320 q^{89} + 260 q^{97} - 392 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 8 * q^4 + 8 * q^5 - 32 * q^8 + 32 * q^10 - 10 * q^13 - 32 * q^16 + 32 * q^17 + 32 * q^20 - 36 * q^25 - 40 * q^26 - 40 * q^29 + 64 * q^32 + 32 * q^34 + 140 * q^37 - 64 * q^40 - 160 * q^41 - 98 * q^49 + 72 * q^50 - 40 * q^52 + 224 * q^53 + 80 * q^58 - 22 * q^61 + 256 * q^64 - 176 * q^65 - 64 * q^68 - 220 * q^73 + 140 * q^74 - 256 * q^80 - 640 * q^82 + 334 * q^85 + 320 * q^89 + 260 * q^97 - 392 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$163$	$245$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	1.00000	−	1.73205i	0.500000	−	0.866025i
$2$	1.00000	−	1.73205i	0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.500000	−	0.866025i
$5$	4.59808	+	7.96410i	0.919615	+	1.59282i	0.800000	+	0.600000i	$0.204833\pi$
$5$	4.59808	+	7.96410i	0.919615	+	1.59282i	0.119615	+	0.992820i	$0.461834\pi$
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−8.00000			−1.00000
$9$		0			0
$10$	18.3923			1.83923
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	7.89230	+	13.6699i	0.607100	+	1.05153i	0.991716	+	0.128452i	$0.0410008\pi$
$13$	7.89230	+	13.6699i	0.607100	+	1.05153i	−0.384615	+	0.923077i	$0.625666\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$17$	33.9808			1.99887			0.999434	−	0.0336351i	$-0.0107084\pi$
$17$	33.9808			1.99887			0.999434	+	0.0336351i	$0.0107084\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	18.3923	−	31.8564i	0.919615	−	1.59282i
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−29.7846	+	51.5885i	−1.19138	+	2.06354i
$26$	31.5692			1.21420
$27$		0			0
$28$		0			0
$29$	8.18653	−	14.1795i	0.282294	−	0.488948i	−0.689655	−	0.724138i	$-0.742238\pi$
$29$	8.18653	−	14.1795i	0.282294	−	0.488948i	0.971949	+	0.235190i	$0.0755712\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	16.0000	+	27.7128i	0.500000	+	0.866025i
$33$		0			0
$34$	33.9808	−	58.8564i	0.999434	−	1.73107i
$35$		0			0
$36$		0			0
$37$	14.2154			0.384200			0.192100	−	0.981375i	$-0.438470\pi$
$37$	14.2154			0.384200			0.192100	+	0.981375i	$0.438470\pi$
$38$		0			0
$39$		0			0
$40$	−36.7846	−	63.7128i	−0.919615	−	1.59282i
$41$	−40.0000	−	69.2820i	−0.975610	−	1.68981i	−0.677908	−	0.735147i	$-0.737113\pi$
$41$	−40.0000	−	69.2820i	−0.975610	−	1.68981i	−0.297702	−	0.954659i	$-0.596220\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−24.5000	−	42.4352i	−0.500000	−	0.866025i
$50$	59.5692	+	103.177i	1.19138	+	2.06354i
$51$		0			0
$52$	31.5692	−	54.6795i	0.607100	−	1.05153i
$53$	56.0000			1.05660			0.528302	−	0.849057i	$-0.322829\pi$
$53$	56.0000			1.05660			0.528302	+	0.849057i	$0.322829\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−16.3731	−	28.3590i	−0.282294	−	0.488948i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−57.4615	+	99.5263i	−0.941992	+	1.63158i	−0.180328	+	0.983607i	$0.557716\pi$
$61$	−57.4615	+	99.5263i	−0.941992	+	1.63158i	−0.761664	+	0.647972i	$0.775617\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$	−72.5788	+	125.710i	−1.11660	+	1.93400i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−67.9615	−	117.713i	−0.999434	−	1.73107i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−138.138			−1.89231			−0.946154	−	0.323718i	$-0.895067\pi$
$73$	−138.138			−1.89231			−0.946154	+	0.323718i	$0.895067\pi$
$74$	14.2154	−	24.6218i	0.192100	−	0.332727i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−147.138			−1.83923
$81$		0			0
$82$	−160.000			−1.95122
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	156.246	+	270.626i	1.83819	+	3.18384i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	12.4500			0.139888			0.0699439	−	0.997551i	$-0.477718\pi$
$89$	12.4500			0.139888			0.0699439	+	0.997551i	$0.477718\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	65.0000	−	112.583i	0.670103	−	1.16065i	−0.307771	−	0.951460i	$-0.599583\pi$
$97$	65.0000	−	112.583i	0.670103	−	1.16065i	0.977875	−	0.209192i	$-0.0670835\pi$
$98$	−98.0000			−1.00000
$99$		0			0
$100$	238.277			2.38277
$101$	20.0000	−	34.6410i	0.198020	−	0.342980i	−0.749866	−	0.661589i	$-0.769882\pi$
$101$	20.0000	−	34.6410i	0.198020	−	0.342980i	0.947886	+	0.318609i	$0.103216\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	−63.1384	−	109.359i	−0.607100	−	1.05153i
$105$		0			0
$106$	56.0000	−	96.9948i	0.528302	−	0.915046i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	12.9230			0.118560			0.0592800	−	0.998241i	$-0.481120\pi$
$109$	12.9230			0.118560			0.0592800	+	0.998241i	$0.481120\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	43.0096	+	74.4948i	0.380616	+	0.659246i	0.991150	−	0.132743i	$-0.0423786\pi$
$113$	43.0096	+	74.4948i	0.380616	+	0.659246i	−0.610534	+	0.791990i	$0.709045\pi$
$114$		0			0
$115$		0			0
$116$	−65.4923			−0.564589
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$	114.923	+	199.053i	0.941992	+	1.63158i
$123$		0			0
$124$		0			0
$125$	−317.904			−2.54323
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	64.0000	−	110.851i	0.500000	−	0.866025i
$129$		0			0
$130$	145.158	+	251.420i	1.11660	+	1.93400i
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−271.846			−1.99887
$137$	−46.9327	+	81.2898i	−0.342574	+	0.593356i	−0.984910	−	0.173067i	$-0.944632\pi$
$137$	−46.9327	+	81.2898i	−0.342574	+	0.593356i	0.642336	+	0.766423i	$0.277965\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	150.569			1.03841
$146$	−138.138	+	239.263i	−0.946154	+	1.63879i
$147$		0			0
$148$	−28.4308	−	49.2436i	−0.192100	−	0.332727i
$149$	−114.167	−	197.744i	−0.766223	−	1.32714i	−0.939597	−	0.342282i	$-0.888800\pi$
$149$	−114.167	−	197.744i	−0.766223	−	1.32714i	0.173374	−	0.984856i	$-0.444533\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−71.8154	−	124.388i	−0.457423	−	0.792279i	0.541401	−	0.840764i	$-0.317894\pi$
$157$	−71.8154	−	124.388i	−0.457423	−	0.792279i	−0.998824	+	0.0484851i	$0.984561\pi$
$158$		0			0
$159$		0			0
$160$	−147.138	+	254.851i	−0.919615	+	1.59282i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−160.000	+	277.128i	−0.975610	+	1.68981i
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−40.0770	+	69.4153i	−0.237142	+	0.410742i
$170$	624.985			3.67638
$171$		0			0
$172$		0			0
$173$	168.894	−	292.533i	0.976267	−	1.69094i	0.300578	−	0.953757i	$-0.402820\pi$
$173$	168.894	−	292.533i	0.976267	−	1.69094i	0.675689	−	0.737187i	$-0.263846\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	12.4500	−	21.5641i	0.0699439	−	0.121146i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	38.0000			0.209945			0.104972	−	0.994475i	$-0.466525\pi$
$181$	38.0000			0.209945			0.104972	+	0.994475i	$0.466525\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	65.3634	+	113.213i	0.353316	+	0.611961i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−192.992	−	334.272i	−0.999960	−	1.73198i	−0.507732	−	0.861515i	$-0.669516\pi$
$193$	−192.992	−	334.272i	−0.999960	−	1.73198i	−0.492228	−	0.870466i	$-0.663817\pi$
$194$	−130.000	−	225.167i	−0.670103	−	1.16065i
$195$		0			0
$196$	−98.0000	+	169.741i	−0.500000	+	0.866025i
$197$	309.750			1.57233			0.786167	−	0.618014i	$-0.212062\pi$
$197$	309.750			1.57233			0.786167	+	0.618014i	$0.212062\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	238.277	−	412.708i	1.19138	−	2.06354i
$201$		0			0
$202$	−40.0000	−	69.2820i	−0.198020	−	0.342980i
$203$		0			0
$204$		0			0
$205$	367.846	−	637.128i	1.79437	−	3.10794i
$206$		0			0
$207$		0			0
$208$	−252.554			−1.21420
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	−112.000	−	193.990i	−0.528302	−	0.915046i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	12.9230	−	22.3834i	0.0592800	−	0.102676i
$219$		0			0
$220$		0			0
$221$	268.187	+	464.513i	1.21351	+	2.10187i
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	172.038			0.761232
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−162.462	−	281.392i	−0.709439	−	1.22878i	−0.965066	−	0.262009i	$-0.915615\pi$
$229$	−162.462	−	281.392i	−0.709439	−	1.22878i	0.255627	−	0.966776i	$-0.417718\pi$
$230$		0			0
$231$		0			0
$232$	−65.4923	+	113.436i	−0.282294	+	0.488948i
$233$	−26.1347			−0.112166			−0.0560830	−	0.998426i	$-0.517861\pi$
$233$	−26.1347			−0.112166			−0.0560830	+	0.998426i	$0.517861\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	−0.576952	+	0.999309i	−0.00239399	+	0.00414651i	−0.867220	−	0.497925i	$-0.834095\pi$
$241$	−0.576952	+	0.999309i	−0.00239399	+	0.00414651i	0.864826	+	0.502072i	$0.167429\pi$
$242$	−242.000			−1.00000
$243$		0			0
$244$	459.692			1.88398
$245$	225.306	−	390.241i	0.919615	−	1.59282i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−317.904	+	550.626i	−1.27162	+	2.20250i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−128.000	−	221.703i	−0.500000	−	0.866025i
$257$	204.836	+	354.787i	0.797029	+	1.38049i	0.921543	+	0.388277i	$0.126930\pi$
$257$	204.836	+	354.787i	0.797029	+	1.38049i	−0.124514	+	0.992218i	$0.539737\pi$
$258$		0			0
$259$		0			0
$260$	580.631			2.23320
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	257.492	+	445.990i	0.971669	+	1.68298i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	379.512			1.41082			0.705412	−	0.708798i	$-0.250762\pi$
$269$	379.512			1.41082			0.705412	+	0.708798i	$0.250762\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−271.846	+	470.851i	−0.999434	+	1.73107i
$273$		0			0
$274$	93.8653	+	162.580i	0.342574	+	0.593356i
$275$		0			0
$276$		0			0
$277$	−115.000	+	199.186i	−0.415162	+	0.719082i	−0.995445	−	0.0953324i	$-0.969609\pi$
$277$	−115.000	+	199.186i	−0.415162	+	0.719082i	0.580283	+	0.814415i	$0.302942\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−120.052	+	207.936i	−0.427231	+	0.739986i	−0.996626	−	0.0820785i	$-0.973844\pi$
$281$	−120.052	+	207.936i	−0.427231	+	0.739986i	0.569395	+	0.822064i	$0.307178\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	865.692			2.99547
$290$	150.569	−	260.794i	0.519204	−	0.899288i
$291$		0			0
$292$	276.277	+	478.526i	0.946154	+	1.63879i
$293$	−280.817	−	486.390i	−0.958421	−	1.66003i	−0.726339	−	0.687337i	$-0.758780\pi$
$293$	−280.817	−	486.390i	−0.958421	−	1.66003i	−0.232082	−	0.972696i	$-0.574554\pi$
$294$		0			0
$295$		0			0
$296$	−113.723			−0.384200
$297$		0			0
$298$	−456.669			−1.53245
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1056.85			−3.46508
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−257.700	+	446.349i	−0.823322	+	1.42604i	0.0798722	+	0.996805i	$0.474549\pi$
$313$	−257.700	+	446.349i	−0.823322	+	1.42604i	−0.903195	+	0.429231i	$0.858785\pi$
$314$	−287.261			−0.914845
$315$		0			0
$316$		0			0
$317$	−89.0481	+	154.236i	−0.280909	+	0.486548i	−0.971609	−	0.236593i	$-0.923969\pi$
$317$	−89.0481	+	154.236i	−0.280909	+	0.486548i	0.690700	+	0.723141i	$0.257303\pi$
$318$		0			0
$319$		0			0
$320$	294.277	+	509.703i	0.919615	+	1.59282i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−940.277			−2.89316
$326$		0			0
$327$		0			0
$328$	320.000	+	554.256i	0.975610	+	1.68981i
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−175.000	−	303.109i	−0.519288	−	0.899433i	−0.999749	−	0.0224168i	$-0.992864\pi$
$337$	−175.000	−	303.109i	−0.519288	−	0.899433i	0.480461	−	0.877016i	$-0.340469\pi$
$338$	80.1539	+	138.831i	0.237142	+	0.410742i
$339$		0			0
$340$	624.985	−	1082.50i	1.83819	−	3.18384i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−337.788	−	585.067i	−0.976267	−	1.69094i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	299.000	−	517.883i	0.856734	−	1.48391i	−0.0182939	−	0.999833i	$-0.505823\pi$
$349$	299.000	−	517.883i	0.856734	−	1.48391i	0.875027	−	0.484073i	$-0.160843\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	272.000	−	471.118i	0.770538	−	1.33461i	−0.166730	−	0.986003i	$-0.553321\pi$
$353$	272.000	−	471.118i	0.770538	−	1.33461i	0.937268	−	0.348609i	$-0.113346\pi$
$354$		0			0
$355$		0			0
$356$	−24.9000	−	43.1281i	−0.0699439	−	0.121146i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	361.000			1.00000
$362$	38.0000	−	65.8179i	0.104972	−	0.181817i
$363$		0			0
$364$		0			0
$365$	−635.171	−	1100.15i	−1.74019	−	3.01411i
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$	261.454			0.706632
$371$		0			0
$372$		0			0
$373$	275.000	+	476.314i	0.737265	+	1.27698i	0.953722	+	0.300689i	$0.0972166\pi$
$373$	275.000	+	476.314i	0.737265	+	1.27698i	−0.216457	+	0.976292i	$0.569450\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	258.442			0.685524
$378$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$	−771.969			−1.99992
$387$		0			0
$388$	−520.000			−1.34021
$389$	−340.000	+	588.897i	−0.874036	+	1.51387i	−0.0162499	+	0.999868i	$0.505173\pi$
$389$	−340.000	+	588.897i	−0.874036	+	1.51387i	−0.857786	+	0.514007i	$0.828161\pi$
$390$		0			0
$391$		0			0
$392$	196.000	+	339.482i	0.500000	+	0.866025i
$393$		0			0
$394$	309.750	−	536.503i	0.786167	−	1.36168i
$395$		0			0
$396$		0			0
$397$	−719.908			−1.81337			−0.906685	−	0.421809i	$-0.861395\pi$
$397$	−719.908			−1.81337			−0.906685	+	0.421809i	$0.861395\pi$
$398$		0			0
$399$		0			0
$400$	−476.554	−	825.415i	−1.19138	−	2.06354i
$401$	365.544	+	633.141i	0.911581	+	1.57891i	0.811831	+	0.583893i	$0.198471\pi$
$401$	365.544	+	633.141i	0.911581	+	1.57891i	0.0997506	+	0.995012i	$0.468195\pi$
$402$		0			0
$403$		0			0
$404$	−160.000			−0.396040
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	299.423	+	518.616i	0.732086	+	1.26801i	0.955990	+	0.293399i	$0.0947863\pi$
$409$	299.423	+	518.616i	0.732086	+	1.26801i	−0.223905	+	0.974611i	$0.571880\pi$
$410$	−735.692	−	1274.26i	−1.79437	−	3.10794i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−252.554	+	437.436i	−0.607100	+	1.05153i
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$	−378.231	+	655.115i	−0.898410	+	1.55609i	−0.0688836	+	0.997625i	$0.521944\pi$
$421$	−378.231	+	655.115i	−0.898410	+	1.55609i	−0.829527	+	0.558467i	$0.811390\pi$
$422$		0			0
$423$		0			0
$424$	−448.000			−1.05660
$425$	−1012.10	+	1753.02i	−2.38142	+	4.12474i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	−851.677			−1.96692			−0.983460	−	0.181123i	$-0.942027\pi$
$433$	−851.677			−1.96692			−0.983460	+	0.181123i	$0.942027\pi$
$434$		0			0
$435$		0			0
$436$	−25.8461	−	44.7668i	−0.0592800	−	0.102676i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$	1072.75			2.42703
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	57.2461	+	99.1532i	0.128643	+	0.222816i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	560.000			1.24722			0.623608	−	0.781737i	$-0.285666\pi$
$449$	560.000			1.24722			0.623608	+	0.781737i	$0.285666\pi$
$450$		0			0
$451$		0			0
$452$	172.038	−	297.979i	0.380616	−	0.659246i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−357.992	+	620.061i	−0.783353	+	1.35681i	0.146625	+	0.989192i	$0.453159\pi$
$457$	−357.992	+	620.061i	−0.783353	+	1.35681i	−0.929978	+	0.367615i	$0.880174\pi$
$458$	−649.846			−1.41888
$459$		0			0
$460$		0			0
$461$	380.000	−	658.179i	0.824295	−	1.42772i	−0.0781619	−	0.996941i	$-0.524905\pi$
$461$	380.000	−	658.179i	0.824295	−	1.42772i	0.902457	−	0.430780i	$-0.141762\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	130.985	+	226.872i	0.282294	+	0.488948i
$465$		0			0
$466$	−26.1347	+	45.2666i	−0.0560830	+	0.0971386i
$467$		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		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	112.192	+	194.323i	0.233248	+	0.403997i
$482$	1.15390	+	1.99862i	0.00239399	+	0.00414651i
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$	1195.50			2.46495
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	459.692	−	796.210i	0.941992	−	1.63158i
$489$		0			0
$490$	−450.611	−	780.482i	−0.919615	−	1.59282i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$	278.185	−	481.830i	0.564269	−	0.977343i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	635.808	+	1101.25i	1.27162	+	2.20250i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	367.846			0.728408
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−220.000	−	381.051i	−0.432220	−	0.748627i	0.564844	−	0.825198i	$-0.308936\pi$
$509$	−220.000	−	381.051i	−0.432220	−	0.748627i	−0.997064	+	0.0765706i	$0.975603\pi$
$510$		0			0
$511$		0			0
$512$	−512.000			−1.00000
$513$		0			0
$514$	819.346			1.59406
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	580.631	−	1005.68i	1.11660	−	1.93400i
$521$	−880.000			−1.68906			−0.844530	−	0.535509i	$-0.820120\pi$
$521$	−880.000			−1.68906			−0.844530	+	0.535509i	$0.820120\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$	1029.97			1.94334
$531$		0			0
$532$		0			0
$533$	631.384	−	1093.59i	1.18459	−	2.05176i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	379.512	−	657.333i	0.705412	−	1.22181i
$539$		0			0
$540$		0			0
$541$	1068.46			1.97497			0.987487	−	0.157698i	$-0.0504073\pi$
$541$	1068.46			1.97497			0.987487	+	0.157698i	$0.0504073\pi$
$542$		0			0
$543$		0			0
$544$	543.692	+	941.703i	0.999434	+	1.73107i
$545$	59.4212	+	102.920i	0.109030	+	0.188845i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	375.461			0.685148
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	230.000	+	398.372i	0.415162	+	0.719082i
$555$		0			0
$556$		0			0
$557$	−817.788			−1.46820			−0.734101	−	0.679040i	$-0.762396\pi$
$557$	−817.788			−1.46820			−0.734101	+	0.679040i	$0.762396\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	240.104	+	415.872i	0.427231	+	0.739986i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	−395.523	+	685.066i	−0.700041	+	1.21251i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	59.9481	−	103.833i	0.105357	−	0.182484i	−0.808527	−	0.588459i	$-0.799735\pi$
$569$	59.9481	−	103.833i	0.105357	−	0.182484i	0.913884	+	0.405975i	$0.133068\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	491.862			0.852446			0.426223	−	0.904618i	$-0.359844\pi$
$577$	491.862			0.852446			0.426223	+	0.904618i	$0.359844\pi$
$578$	865.692	−	1499.42i	1.49774	−	2.59416i
$579$		0			0
$580$	−301.138	−	521.587i	−0.519204	−	0.899288i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1105.11			1.89231
$585$		0			0
$586$	−1123.27			−1.91684
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−113.723	+	196.974i	−0.192100	+	0.332727i
$593$	1173.40			1.97876			0.989379	−	0.145358i	$-0.0464334\pi$
$593$	1173.40			1.97876			0.989379	+	0.145358i	$0.0464334\pi$
$594$		0			0
$595$		0			0
$596$	−456.669	+	790.974i	−0.766223	+	1.32714i
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−483.346	+	837.180i	−0.804236	+	1.39298i	0.112569	+	0.993644i	$0.464092\pi$
$601$	−483.346	+	837.180i	−0.804236	+	1.39298i	−0.916805	+	0.399334i	$0.869241\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	556.367	−	963.656i	0.919615	−	1.59282i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$	−1056.85	+	1830.52i	−1.73254	+	3.00085i
$611$		0			0
$612$		0			0
$613$	−70.0000			−0.114192			−0.0570962	−	0.998369i	$-0.518184\pi$
$613$	−70.0000			−0.114192			−0.0570962	+	0.998369i	$0.518184\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−394.933	−	684.043i	−0.640085	−	1.10866i	−0.985413	−	0.170178i	$-0.945566\pi$
$617$	−394.933	−	684.043i	−0.640085	−	1.10866i	0.345328	−	0.938482i	$-0.387768\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−717.131	−	1242.11i	−1.14741	−	1.98737i
$626$	515.400	+	892.699i	0.823322	+	1.42604i
$627$		0			0
$628$	−287.261	+	497.551i	−0.457423	+	0.792279i
$629$	483.050			0.767965
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	178.096	+	308.472i	0.280909	+	0.486548i
$635$		0			0
$636$		0			0
$637$	386.723	−	669.824i	0.607100	−	1.05153i
$638$		0			0
$639$		0			0
$640$	1177.11			1.83923
$641$	−627.409	+	1086.71i	−0.978798	+	1.69533i	−0.312012	+	0.950078i	$0.601003\pi$
$641$	−627.409	+	1086.71i	−0.978798	+	1.69533i	−0.666785	+	0.745250i	$0.732330\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		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
$650$	−940.277	+	1628.61i	−1.44658	+	2.50555i
$651$		0			0
$652$		0			0
$653$	572.000	+	990.733i	0.875957	+	1.51720i	0.855740	+	0.517407i	$0.173103\pi$
$653$	572.000	+	990.733i	0.875957	+	1.51720i	0.0202175	+	0.999796i	$0.493564\pi$
$654$		0			0
$655$		0			0
$656$	1280.00			1.95122
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	554.308	+	960.089i	0.838589	+	1.45248i	0.891074	+	0.453858i	$0.149953\pi$
$661$	554.308	+	960.089i	0.838589	+	1.45248i	−0.0524847	+	0.998622i	$0.516714\pi$
$662$		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
$673$	670.546	−	1161.42i	0.996354	−	1.72574i	0.424288	−	0.905527i	$-0.360524\pi$
$673$	670.546	−	1161.42i	0.996354	−	1.72574i	0.572065	−	0.820208i	$-0.306142\pi$
$674$	−700.000			−1.03858
$675$		0			0
$676$	320.616			0.474283
$677$	−52.0000	+	90.0666i	−0.0768095	+	0.133038i	−0.901872	−	0.432004i	$-0.857807\pi$
$677$	−52.0000	+	90.0666i	−0.0768095	+	0.133038i	0.825062	+	0.565042i	$0.191140\pi$
$678$		0			0
$679$		0			0
$680$	−1249.97	−	2165.01i	−1.83819	−	3.18384i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−863.200			−1.26015
$686$		0			0
$687$		0			0
$688$		0			0
$689$	441.969	+	765.513i	0.641465	+	1.11105i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−1351.15			−1.95253
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1359.23	−	2354.26i	−1.95012	−	3.37770i
$698$	−598.000	−	1035.77i	−0.856734	−	1.48391i
$699$		0			0
$700$		0			0
$701$	−867.565			−1.23761			−0.618805	−	0.785544i	$-0.712383\pi$
$701$	−867.565			−1.23761			−0.618805	+	0.785544i	$0.712383\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−544.000	−	942.236i	−0.770538	−	1.33461i
$707$		0			0
$708$		0			0
$709$	701.077	−	1214.30i	0.988825	−	1.71269i	0.365303	−	0.930889i	$-0.380965\pi$
$709$	701.077	−	1214.30i	0.988825	−	1.71269i	0.623522	−	0.781806i	$-0.285701\pi$
$710$		0			0
$711$		0			0
$712$	−99.6001			−0.139888
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	361.000	−	625.270i	0.500000	−	0.866025i
$723$		0			0
$724$	−76.0000	−	131.636i	−0.104972	−	0.181817i
$725$	487.665	+	844.661i	0.672642	+	1.16505i
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$	−2540.68			−3.48039
$731$		0			0
$732$		0			0
$733$	725.000	+	1255.74i	0.989086	+	1.71315i	0.622143	+	0.782904i	$0.286262\pi$
$733$	725.000	+	1255.74i	0.989086	+	1.71315i	0.366943	+	0.930243i	$0.380404\pi$
$734$		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			$\pi$
$740$	261.454	−	452.851i	0.353316	−	0.611961i
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	1049.90	−	1818.48i	1.40926	−	2.44091i
$746$	1100.00			1.47453
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$		0			0
$754$	258.442	−	447.635i	0.342762	−	0.593681i
$755$		0			0
$756$		0			0
$757$	1190.00			1.57199			0.785997	−	0.618230i	$-0.212150\pi$
$757$	1190.00			1.57199			0.785997	+	0.618230i	$0.212150\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	413.775	+	716.679i	0.543725	+	0.941760i	0.998686	+	0.0512484i	$0.0163200\pi$
$761$	413.775	+	716.679i	0.543725	+	0.941760i	−0.454961	+	0.890512i	$0.650347\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−279.115	−	483.442i	−0.362959	−	0.628663i	0.625488	−	0.780234i	$-0.284900\pi$
$769$	−279.115	−	483.442i	−0.362959	−	0.628663i	−0.988446	+	0.151571i	$0.951567\pi$
$770$		0			0
$771$		0			0
$772$	−771.969	+	1337.09i	−0.999960	+	1.73198i
$773$	−410.250			−0.530725			−0.265362	−	0.964149i	$-0.585492\pi$
$773$	−410.250			−0.530725			−0.265362	+	0.964149i	$0.585492\pi$
$774$		0			0
$775$		0			0
$776$	−520.000	+	900.666i	−0.670103	+	1.16065i
$777$		0			0
$778$	680.000	+	1177.79i	0.874036	+	1.51387i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	784.000			1.00000
$785$	660.425	−	1143.89i	0.841306	−	1.45718i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−619.500	−	1073.01i	−0.786167	−	1.36168i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1814.02			−2.28754
$794$	−719.908	+	1246.92i	−0.906685	+	1.57042i
$795$		0			0
$796$		0			0
$797$	−766.644	−	1327.87i	−0.961912	−	1.66608i	−0.717691	−	0.696361i	$-0.754801\pi$
$797$	−766.644	−	1327.87i	−0.961912	−	1.66608i	−0.244221	−	0.969720i	$-0.578532\pi$
$798$		0			0
$799$		0			0
$800$	−1906.22			−2.38277
$801$		0			0
$802$	1462.18			1.82316
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−160.000	+	277.128i	−0.198020	+	0.342980i
$809$	−1594.63			−1.97111			−0.985554	−	0.169361i	$-0.945830\pi$
$809$	−1594.63			−1.97111			−0.985554	+	0.169361i	$0.945830\pi$
$810$		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
$818$	1197.69			1.46417
$819$		0			0
$820$	−2942.77			−3.58874
$821$	−21.5249	+	37.2822i	−0.0262179	+	0.0454107i	−0.878837	−	0.477123i	$-0.841680\pi$
$821$	−21.5249	+	37.2822i	−0.0262179	+	0.0454107i	0.852619	+	0.522533i	$0.175013\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−1258.00			−1.51749			−0.758745	−	0.651387i	$-0.774187\pi$
$829$	−1258.00			−1.51749			−0.758745	+	0.651387i	$0.774187\pi$
$830$		0			0
$831$		0			0
$832$	505.108	+	874.872i	0.607100	+	1.05153i
$833$	−832.529	−	1441.98i	−0.999434	−	1.73107i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	286.461	+	496.166i	0.340620	+	0.589971i
$842$	756.461	+	1310.23i	0.898410	+	1.55609i
$843$		0			0
$844$		0			0
$845$	−737.108			−0.872317
$846$		0			0
$847$		0			0
$848$	−448.000	+	775.959i	−0.528302	+	0.915046i
$849$		0			0
$850$	2024.21	+	3506.03i	2.38142	+	4.12474i
$851$		0			0
$852$		0			0
$853$	−205.000	+	355.070i	−0.240328	+	0.416261i	−0.960808	−	0.277215i	$-0.910588\pi$
$853$	−205.000	+	355.070i	−0.240328	+	0.416261i	0.720480	+	0.693476i	$0.243922\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−598.471	+	1036.58i	−0.698333	+	1.20955i	0.270712	+	0.962660i	$0.412741\pi$
$857$	−598.471	+	1036.58i	−0.698333	+	1.20955i	−0.969044	+	0.246887i	$0.920592\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	3106.35			3.59116
$866$	−851.677	+	1475.15i	−0.983460	+	1.70340i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−103.384			−0.118560
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	101.123	+	175.150i	0.115306	+	0.199715i	0.917902	−	0.396807i	$-0.129882\pi$
$877$	101.123	+	175.150i	0.115306	+	0.199715i	−0.802596	+	0.596523i	$0.796549\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1600.00			−1.81612			−0.908059	−	0.418842i	$-0.862436\pi$
$881$	−1600.00			−1.81612			−0.908059	+	0.418842i	$0.862436\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	1072.75	−	1858.05i	1.21351	−	2.10187i
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$		0			0
$890$	228.985			0.257286
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	560.000	−	969.948i	0.623608	−	1.08012i
$899$		0			0
$900$		0			0
$901$	1902.92			2.11201
$902$		0			0
$903$		0			0
$904$	−344.077	−	595.959i	−0.380616	−	0.659246i
$905$	174.727	+	302.636i	0.193068	+	0.334404i
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$	715.985	+	1240.12i	0.783353	+	1.35681i
$915$		0			0
$916$	−649.846	+	1125.57i	−0.709439	+	1.22878i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−760.000	−	1316.36i	−0.824295	−	1.42772i
$923$		0			0
$924$		0			0
$925$	−423.400	+	733.350i	−0.457730	+	0.792811i
$926$		0			0
$927$		0			0
$928$	523.938			0.564589
$929$	−348.283	+	603.243i	−0.374901	+	0.649347i	−0.990312	−	0.138859i	$-0.955657\pi$
$929$	−348.283	+	603.243i	−0.374901	+	0.649347i	0.615411	+	0.788206i	$0.288990\pi$
$930$		0			0
$931$		0			0
$932$	52.2693	+	90.5331i	0.0560830	+	0.0971386i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1364.63			−1.45638			−0.728191	−	0.685374i	$-0.759639\pi$
$937$	−1364.63			−1.45638			−0.728191	+	0.685374i	$0.759639\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	931.725	+	1613.79i	0.990143	+	1.71498i	0.616366	+	0.787460i	$0.288604\pi$
$941$	931.725	+	1613.79i	0.990143	+	1.71498i	0.373778	+	0.927518i	$0.378062\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	−1090.23	−	1888.33i	−1.14882	−	1.98982i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−337.211			−0.353842			−0.176921	−	0.984225i	$-0.556614\pi$
$953$	−337.211			−0.353842			−0.176921	+	0.984225i	$0.556614\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−480.500	+	832.250i	−0.500000	+	0.866025i
$962$	448.769			0.466496
$963$		0			0
$964$	4.61561			0.00478798
$965$	1774.79	−	3074.02i	1.83916	−	3.18551i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	484.000	+	838.313i	0.500000	+	0.866025i
$969$		0			0
$970$	1195.50	−	2070.67i	1.23247	−	2.13471i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−919.384	−	1592.42i	−0.941992	−	1.63158i
$977$	248.000	+	429.549i	0.253838	+	0.439661i	0.964579	−	0.263793i	$-0.0849736\pi$
$977$	248.000	+	429.549i	0.253838	+	0.439661i	−0.710741	+	0.703454i	$0.751640\pi$
$978$		0			0
$979$		0			0
$980$	−1802.45			−1.83923
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	1424.25	+	2466.88i	1.44594	+	2.50445i
$986$	−556.369	−	963.660i	−0.564269	−	0.977343i
$987$		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
$997$	784.661	−	1359.07i	0.787023	−	1.36316i	−0.140761	−	0.990044i	$-0.544955\pi$
$997$	784.661	−	1359.07i	0.787023	−	1.36316i	0.927783	−	0.373119i	$-0.121712\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.f.n.55.2		4
3.2	odd	2		324.3.f.k.55.1		4
4.3	odd	2	CM	324.3.f.n.55.2		4
9.2	odd	6		324.3.d.d.163.2	yes	2
9.4	even	3	inner	324.3.f.n.271.2		4
9.5	odd	6		324.3.f.k.271.1		4
9.7	even	3		324.3.d.a.163.1	✓	2
12.11	even	2		324.3.f.k.55.1		4
36.7	odd	6		324.3.d.a.163.1	✓	2
36.11	even	6		324.3.d.d.163.2	yes	2
36.23	even	6		324.3.f.k.271.1		4
36.31	odd	6	inner	324.3.f.n.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.d.a.163.1	✓	2	9.7	even	3
324.3.d.a.163.1	✓	2	36.7	odd	6
324.3.d.d.163.2	yes	2	9.2	odd	6
324.3.d.d.163.2	yes	2	36.11	even	6
324.3.f.k.55.1		4	3.2	odd	2
324.3.f.k.55.1		4	12.11	even	2
324.3.f.k.271.1		4	9.5	odd	6
324.3.f.k.271.1		4	36.23	even	6
324.3.f.n.55.2		4	1.1	even	1	trivial
324.3.f.n.55.2		4	4.3	odd	2	CM
324.3.f.n.271.2		4	9.4	even	3	inner
324.3.f.n.271.2		4	36.31	odd	6	inner

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.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.59808 + 7.96410i) q^{5} -8.00000 q^{8} +O(q^{10})\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(4.59808 + 7.96410i) q^{5} -8.00000 q^{8} +18.3923 q^{10} +(7.89230 + 13.6699i) q^{13} +(-8.00000 + 13.8564i) q^{16} +33.9808 q^{17} +(18.3923 - 31.8564i) q^{20} +(-29.7846 + 51.5885i) q^{25} +31.5692 q^{26} +(8.18653 - 14.1795i) q^{29} +(16.0000 + 27.7128i) q^{32} +(33.9808 - 58.8564i) q^{34} +14.2154 q^{37} +(-36.7846 - 63.7128i) q^{40} +(-40.0000 - 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(59.5692 + 103.177i) q^{50} +(31.5692 - 54.6795i) q^{52} +56.0000 q^{53} +(-16.3731 - 28.3590i) q^{58} +(-57.4615 + 99.5263i) q^{61} +64.0000 q^{64} +(-72.5788 + 125.710i) q^{65} +(-67.9615 - 117.713i) q^{68} -138.138 q^{73} +(14.2154 - 24.6218i) q^{74} -147.138 q^{80} -160.000 q^{82} +(156.246 + 270.626i) q^{85} +12.4500 q^{89} +(65.0000 - 112.583i) q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 8 * q^4 + 8 * q^5 - 32 * q^8	\( 4 q + 4 q^{2} - 8 q^{4} + 8 q^{5} - 32 q^{8} + 32 q^{10} - 10 q^{13} - 32 q^{16} + 32 q^{17} + 32 q^{20} - 36 q^{25} - 40 q^{26} - 40 q^{29} + 64 q^{32} + 32 q^{34} + 140 q^{37} - 64 q^{40} - 160 q^{41} - 98 q^{49} + 72 q^{50} - 40 q^{52} + 224 q^{53} + 80 q^{58} - 22 q^{61} + 256 q^{64} - 176 q^{65} - 64 q^{68} - 220 q^{73} + 140 q^{74} - 256 q^{80} - 640 q^{82} + 334 q^{85} + 320 q^{89} + 260 q^{97} - 392 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 8 * q^4 + 8 * q^5 - 32 * q^8 + 32 * q^10 - 10 * q^13 - 32 * q^16 + 32 * q^17 + 32 * q^20 - 36 * q^25 - 40 * q^26 - 40 * q^29 + 64 * q^32 + 32 * q^34 + 140 * q^37 - 64 * q^40 - 160 * q^41 - 98 * q^49 + 72 * q^50 - 40 * q^52 + 224 * q^53 + 80 * q^58 - 22 * q^61 + 256 * q^64 - 176 * q^65 - 64 * q^68 - 220 * q^73 + 140 * q^74 - 256 * q^80 - 640 * q^82 + 334 * q^85 + 320 * q^89 + 260 * q^97 - 392 * q^98

Introduction

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