LMFDB - Embedded newform 324.3.f.k.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.k.271.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(0.598076 + 1.03590i) q^{5} +8.00000 q^{8} +O(q^{10})$	$q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(0.598076 + 1.03590i) q^{5} +8.00000 q^{8} -2.39230 q^{10} +(-12.8923 - 22.3301i) q^{13} +(-8.00000 + 13.8564i) q^{16} +17.9808 q^{17} +(2.39230 - 4.14359i) q^{20} +(11.7846 - 20.4115i) q^{25} +51.5692 q^{26} +(28.1865 - 48.8205i) q^{29} +(-16.0000 - 27.7128i) q^{32} +(-17.9808 + 31.1436i) q^{34} +55.7846 q^{37} +(4.78461 + 8.28719i) q^{40} +(40.0000 + 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(23.5692 + 40.8231i) q^{50} +(-51.5692 + 89.3205i) q^{52} -56.0000 q^{53} +(56.3731 + 97.6410i) q^{58} +(46.4615 - 80.4737i) q^{61} +64.0000 q^{64} +(15.4212 - 26.7102i) q^{65} +(-35.9615 - 62.2872i) q^{68} +28.1384 q^{73} +(-55.7846 + 96.6218i) q^{74} -19.1384 q^{80} -160.000 q^{82} +(10.7539 + 18.6262i) q^{85} -147.550 q^{89} +(65.0000 - 112.583i) q^{97} +98.0000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.598076	+	1.03590i	0.119615	+	0.207180i	0.919615	−	0.392820i	$-0.128501\pi$
$5$	0.598076	+	1.03590i	0.119615	+	0.207180i	−0.800000	+	0.600000i	$0.795167\pi$
$6$		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
$10$	−2.39230			−0.239230
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−12.8923	−	22.3301i	−0.991716	−	1.71770i	−0.607100	−	0.794625i	$-0.707667\pi$
$13$	−12.8923	−	22.3301i	−0.991716	−	1.71770i	−0.384615	−	0.923077i	$-0.625666\pi$
$14$		0			0
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$17$	17.9808			1.05769			0.528846	−	0.848718i	$-0.322625\pi$
$17$	17.9808			1.05769			0.528846	+	0.848718i	$0.322625\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	2.39230	−	4.14359i	0.119615	−	0.207180i
$21$		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
$25$	11.7846	−	20.4115i	0.471384	−	0.816462i
$26$	51.5692			1.98343
$27$		0			0
$28$		0			0
$29$	28.1865	−	48.8205i	0.971949	−	1.68347i	0.282294	−	0.959328i	$-0.408905\pi$
$29$	28.1865	−	48.8205i	0.971949	−	1.68347i	0.689655	−	0.724138i	$-0.257762\pi$
$30$		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
$34$	−17.9808	+	31.1436i	−0.528846	+	0.915988i
$35$		0			0
$36$		0			0
$37$	55.7846			1.50769			0.753846	−	0.657051i	$-0.228196\pi$
$37$	55.7846			1.50769			0.753846	+	0.657051i	$0.228196\pi$
$38$		0			0
$39$		0			0
$40$	4.78461	+	8.28719i	0.119615	+	0.207180i
$41$	40.0000	+	69.2820i	0.975610	+	1.68981i	0.677908	+	0.735147i	$0.262887\pi$
$41$	40.0000	+	69.2820i	0.975610	+	1.68981i	0.297702	+	0.954659i	$0.403780\pi$
$42$		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
$50$	23.5692	+	40.8231i	0.471384	+	0.816462i
$51$		0			0
$52$	−51.5692	+	89.3205i	−0.991716	+	1.71770i
$53$	−56.0000			−1.05660			−0.528302	−	0.849057i	$-0.677171\pi$
$53$	−56.0000			−1.05660			−0.528302	+	0.849057i	$0.677171\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	56.3731	+	97.6410i	0.971949	+	1.68347i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	46.4615	−	80.4737i	0.761664	−	1.31924i	−0.180328	−	0.983607i	$-0.557716\pi$
$61$	46.4615	−	80.4737i	0.761664	−	1.31924i	0.941992	−	0.335635i	$-0.108951\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$	15.4212	−	26.7102i	0.237249	−	0.410927i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−35.9615	−	62.2872i	−0.528846	−	0.915988i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	28.1384			0.385458			0.192729	−	0.981252i	$-0.438266\pi$
$73$	28.1384			0.385458			0.192729	+	0.981252i	$0.438266\pi$
$74$	−55.7846	+	96.6218i	−0.753846	+	1.30570i
$75$		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	$0.666667\pi$
$80$	−19.1384			−0.239230
$81$		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
$85$	10.7539	+	18.6262i	0.126516	+	0.219132i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−147.550			−1.65786			−0.828932	−	0.559349i	$-0.811051\pi$
$89$	−147.550			−1.65786			−0.828932	+	0.559349i	$0.811051\pi$
$90$		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
$100$	−94.2769			−0.942769
$101$	−20.0000	+	34.6410i	−0.198020	+	0.342980i	−0.947886	−	0.318609i	$-0.896784\pi$
$101$	−20.0000	+	34.6410i	−0.198020	+	0.342980i	0.749866	+	0.661589i	$0.230118\pi$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	−103.138	−	178.641i	−0.991716	−	1.71770i
$105$		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
$109$	−194.923			−1.78828			−0.894142	−	0.447783i	$-0.852214\pi$
$109$	−194.923			−1.78828			−0.894142	+	0.447783i	$0.852214\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−68.9904	−	119.495i	−0.610534	−	1.05748i	−0.991150	−	0.132743i	$-0.957621\pi$
$113$	−68.9904	−	119.495i	−0.610534	−	1.05748i	0.380616	−	0.924733i	$-0.375712\pi$
$114$		0			0
$115$		0			0
$116$	−225.492			−1.94390
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$	92.9230	+	160.947i	0.761664	+	1.31924i
$123$		0			0
$124$		0			0
$125$	58.0962			0.464770
$126$		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
$130$	30.8423	+	53.4205i	0.237249	+	0.410927i
$131$		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
$136$	143.846			1.05769
$137$	−134.933	+	233.710i	−0.984910	+	1.70591i	−0.342574	+	0.939491i	$0.611299\pi$
$137$	−134.933	+	233.710i	−0.984910	+	1.70591i	−0.642336	+	0.766423i	$0.722035\pi$
$138$		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
$145$	67.4308			0.465040
$146$	−28.1384	+	48.7372i	−0.192729	+	0.333816i
$147$		0			0
$148$	−111.569	−	193.244i	−0.753846	−	1.30570i
$149$	25.8327	+	44.7436i	0.173374	+	0.300292i	0.939597	−	0.342282i	$-0.111200\pi$
$149$	25.8327	+	44.7436i	0.173374	+	0.300292i	−0.766223	+	0.642574i	$0.777866\pi$
$150$		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
$157$	156.815	+	271.612i	0.998824	+	1.73001i	0.541401	+	0.840764i	$0.317894\pi$
$157$	156.815	+	271.612i	0.998824	+	1.73001i	0.457423	+	0.889249i	$0.348773\pi$
$158$		0			0
$159$		0			0
$160$	19.1384	−	33.1487i	0.119615	−	0.207180i
$161$		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
$169$	−247.923	+	429.415i	−1.46700	+	2.54092i
$170$	−43.0155			−0.253032
$171$		0			0
$172$		0			0
$173$	116.894	−	202.467i	0.675689	−	1.17033i	−0.300578	−	0.953757i	$-0.597180\pi$
$173$	116.894	−	202.467i	0.675689	−	1.17033i	0.976267	−	0.216570i	$-0.0694871\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	147.550	−	255.564i	0.828932	−	1.43575i
$179$		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
$185$	33.3634	+	57.7872i	0.180343	+	0.312363i
$186$		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
$193$	97.9923	+	169.728i	0.507732	+	0.879418i	0.999960	+	0.00895123i	$0.00284930\pi$
$193$	97.9923	+	169.728i	0.507732	+	0.879418i	−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
$197$	365.750			1.85660			0.928299	−	0.371834i	$-0.121271\pi$
$197$	365.750			1.85660			0.928299	+	0.371834i	$0.121271\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	94.2769	−	163.292i	0.471384	−	0.816462i
$201$		0			0
$202$	−40.0000	−	69.2820i	−0.198020	−	0.342980i
$203$		0			0
$204$		0			0
$205$	−47.8461	+	82.8719i	−0.233396	+	0.404253i
$206$		0			0
$207$		0			0
$208$	412.554			1.98343
$209$		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
$218$	194.923	−	337.617i	0.894142	−	1.54870i
$219$		0			0
$220$		0			0
$221$	−231.813	−	401.513i	−1.04893	−	1.81680i
$222$		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
$226$	275.962			1.22107
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	−58.5385	−	101.392i	−0.255627	−	0.442758i	0.709439	−	0.704767i	$-0.248948\pi$
$229$	−58.5385	−	101.392i	−0.255627	−	0.442758i	−0.965066	+	0.262009i	$0.915615\pi$
$230$		0			0
$231$		0			0
$232$	225.492	−	390.564i	0.971949	−	1.68347i
$233$	389.865			1.67324			0.836621	−	0.547782i	$-0.184528\pi$
$233$	389.865			1.67324			0.836621	+	0.547782i	$0.184528\pi$
$234$		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
$241$	−208.423	+	360.999i	−0.864826	+	1.49792i	0.00239399	+	0.999997i	$0.499238\pi$
$241$	−208.423	+	360.999i	−0.864826	+	1.49792i	−0.867220	+	0.497925i	$0.834095\pi$
$242$	242.000			1.00000
$243$		0			0
$244$	−371.692			−1.52333
$245$	29.3057	−	50.7590i	0.119615	−	0.207180i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−58.0962	+	100.626i	−0.232385	+	0.402502i
$251$		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
$257$	236.836	+	410.213i	0.921543	+	1.59616i	0.797029	+	0.603941i	$0.206404\pi$
$257$	236.836	+	410.213i	0.921543	+	1.59616i	0.124514	+	0.992218i	$0.460263\pi$
$258$		0			0
$259$		0			0
$260$	−123.369			−0.474497
$261$		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
$265$	−33.4923	−	58.0103i	−0.126386	−	0.218907i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−140.488			−0.522262			−0.261131	−	0.965303i	$-0.584095\pi$
$269$	−140.488			−0.522262			−0.261131	+	0.965303i	$0.584095\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−143.846	+	249.149i	−0.528846	+	0.915988i
$273$		0			0
$274$	−269.865	−	467.420i	−0.984910	−	1.70591i
$275$		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
$281$	−280.052	+	485.064i	−0.996626	+	1.72621i	−0.427231	+	0.904143i	$0.640511\pi$
$281$	−280.052	+	485.064i	−0.996626	+	1.72621i	−0.569395	+	0.822064i	$0.692822\pi$
$282$		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
$289$	34.3078			0.118712
$290$	−67.4308	+	116.794i	−0.232520	+	0.402736i
$291$		0			0
$292$	−56.2769	−	97.4744i	−0.192729	−	0.333816i
$293$	−212.817	−	368.610i	−0.726339	−	1.25806i	−0.958421	−	0.285359i	$-0.907887\pi$
$293$	−212.817	−	368.610i	−0.726339	−	1.25806i	0.232082	−	0.972696i	$-0.425446\pi$
$294$		0			0
$295$		0			0
$296$	446.277			1.50769
$297$		0			0
$298$	−103.331			−0.346748
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	111.150			0.364427
$306$		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
$313$	282.700	−	489.651i	0.903195	−	1.56438i	0.0798722	−	0.996805i	$-0.474549\pi$
$313$	282.700	−	489.651i	0.903195	−	1.56438i	0.823322	−	0.567574i	$-0.192118\pi$
$314$	−627.261			−1.99765
$315$		0			0
$316$		0			0
$317$	218.952	−	379.236i	0.690700	−	1.19633i	−0.280909	−	0.959734i	$-0.590636\pi$
$317$	218.952	−	379.236i	0.690700	−	1.19633i	0.971609	−	0.236593i	$-0.0760308\pi$
$318$		0			0
$319$		0			0
$320$	38.2769	+	66.2975i	0.119615	+	0.207180i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−607.723			−1.86992
$326$		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	$-0.340469\pi$
$338$	−495.846	−	858.831i	−1.46700	−	2.54092i
$339$		0			0
$340$	43.0155	−	74.5050i	0.126516	−	0.219132i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	233.788	+	404.933i	0.675689	+	1.17033i
$347$		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	$0.446679\pi$
$353$	−272.000	+	471.118i	−0.770538	+	1.33461i	−0.937268	+	0.348609i	$0.886654\pi$
$354$		0			0
$355$		0			0
$356$	295.100	+	511.128i	0.828932	+	1.43575i
$357$		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
$365$	16.8289	+	29.1486i	0.0461067	+	0.0798591i
$366$		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
$370$	−133.454			−0.360686
$371$		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
$377$	−1453.56			−3.85559
$378$		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
$386$	−391.969			−1.01546
$387$		0			0
$388$	−520.000			−1.34021
$389$	340.000	−	588.897i	0.874036	−	1.51387i	0.0162499	−	0.999868i	$-0.494827\pi$
$389$	340.000	−	588.897i	0.874036	−	1.51387i	0.857786	−	0.514007i	$-0.171839\pi$
$390$		0			0
$391$		0			0
$392$	−196.000	−	339.482i	−0.500000	−	0.866025i
$393$		0			0
$394$	−365.750	+	633.497i	−0.928299	+	1.60786i
$395$		0			0
$396$		0			0
$397$	69.9076			0.176090			0.0880448	−	0.996117i	$-0.471938\pi$
$397$	69.9076			0.176090			0.0880448	+	0.996117i	$0.471938\pi$
$398$		0			0
$399$		0			0
$400$	188.554	+	326.585i	0.471384	+	0.816462i
$401$	325.544	+	563.859i	0.811831	+	1.40613i	0.911581	+	0.411120i	$0.134862\pi$
$401$	325.544	+	563.859i	0.811831	+	1.40613i	−0.0997506	+	0.995012i	$0.531805\pi$
$402$		0			0
$403$		0			0
$404$	160.000			0.396040
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	91.5770	+	158.616i	0.223905	+	0.387814i	0.955990	−	0.293399i	$-0.0947863\pi$
$409$	91.5770	+	158.616i	0.223905	+	0.387814i	−0.732086	+	0.681213i	$0.761453\pi$
$410$	−95.6922	−	165.744i	−0.233396	−	0.404253i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−412.554	+	714.564i	−0.991716	+	1.71770i
$417$		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
$421$	349.231	−	604.885i	0.829527	−	1.43678i	−0.0688836	−	0.997625i	$-0.521944\pi$
$421$	349.231	−	604.885i	0.829527	−	1.43678i	0.898410	−	0.439157i	$-0.144723\pi$
$422$		0			0
$423$		0			0
$424$	−448.000			−1.05660
$425$	211.896	−	367.015i	0.498579	−	0.863565i
$426$		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
$433$	561.677			1.29717			0.648587	−	0.761140i	$-0.275360\pi$
$433$	561.677			1.29717			0.648587	+	0.761140i	$0.275360\pi$
$434$		0			0
$435$		0			0
$436$	389.846	+	675.233i	0.894142	+	1.54870i
$437$		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
$442$	927.254			2.09786
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	−88.2461	−	152.847i	−0.198306	−	0.343476i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−560.000			−1.24722			−0.623608	−	0.781737i	$-0.714334\pi$
$449$	−560.000			−1.24722			−0.623608	+	0.781737i	$0.714334\pi$
$450$		0			0
$451$		0			0
$452$	−275.962	+	477.979i	−0.610534	+	1.05748i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−67.0077	+	116.061i	−0.146625	+	0.253962i	−0.929978	−	0.367615i	$-0.880174\pi$
$457$	−67.0077	+	116.061i	−0.146625	+	0.253962i	0.783353	+	0.621577i	$0.213508\pi$
$458$	234.154			0.511253
$459$		0			0
$460$		0			0
$461$	−380.000	+	658.179i	−0.824295	+	1.42772i	0.0781619	+	0.996941i	$0.475095\pi$
$461$	−380.000	+	658.179i	−0.824295	+	1.42772i	−0.902457	+	0.430780i	$0.858238\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	450.985	+	781.128i	0.971949	+	1.68347i
$465$		0			0
$466$	−389.865	+	675.267i	−0.836621	+	1.44907i
$467$		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
$481$	−719.192	−	1245.68i	−1.49520	−	2.58977i
$482$	−416.846	−	721.999i	−0.864826	−	1.49792i
$483$		0			0
$484$	−242.000	+	419.156i	−0.500000	+	0.866025i
$485$	155.500			0.320618
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	371.692	−	643.790i	0.761664	−	1.31924i
$489$		0			0
$490$	58.6115	+	101.518i	0.119615	+	0.207180i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$	506.815	−	877.830i	1.02802	−	1.78059i
$494$		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	$0.333333\pi$
$500$	−116.192	−	201.251i	−0.232385	−	0.402502i
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	−47.8461			−0.0947447
$506$		0			0
$507$		0			0
$508$		0			0
$509$	220.000	+	381.051i	0.432220	+	0.748627i	0.997064	−	0.0765706i	$-0.0243970\pi$
$509$	220.000	+	381.051i	0.432220	+	0.748627i	−0.564844	+	0.825198i	$0.691064\pi$
$510$		0			0
$511$		0			0
$512$	512.000			1.00000
$513$		0			0
$514$	−947.346			−1.84309
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	123.369	−	213.682i	0.237249	−	0.410927i
$521$	880.000			1.68906			0.844530	−	0.535509i	$-0.179880\pi$
$521$	880.000			1.68906			0.844530	+	0.535509i	$0.179880\pi$
$522$		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
$530$	133.969			0.252772
$531$		0			0
$532$		0			0
$533$	1031.38	−	1786.41i	1.93506	−	3.35161i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	140.488	−	243.333i	0.261131	−	0.452292i
$539$		0			0
$540$		0			0
$541$	−386.461			−0.714346			−0.357173	−	0.934038i	$-0.616259\pi$
$541$	−386.461			−0.714346			−0.357173	+	0.934038i	$0.616259\pi$
$542$		0			0
$543$		0			0
$544$	−287.692	−	498.297i	−0.528846	−	0.915988i
$545$	−116.579	−	201.920i	−0.213906	−	0.370496i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$	1079.46			1.96982
$549$		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
$557$	246.212			0.442032			0.221016	−	0.975270i	$-0.429063\pi$
$557$	246.212			0.442032			0.221016	+	0.975270i	$0.429063\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−560.104	−	970.128i	−0.996626	−	1.72621i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	82.5230	−	142.934i	0.146058	−	0.252981i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−460.052	+	796.833i	−0.808527	+	1.40041i	0.105357	+	0.994434i	$0.466401\pi$
$569$	−460.052	+	796.833i	−0.808527	+	1.40041i	−0.913884	+	0.405975i	$0.866932\pi$
$570$		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
$577$	658.138			1.14062			0.570311	−	0.821429i	$-0.306823\pi$
$577$	658.138			1.14062			0.570311	+	0.821429i	$0.306823\pi$
$578$	−34.3078	+	59.4229i	−0.0593561	+	0.102808i
$579$		0			0
$580$	−134.862	−	233.587i	−0.232520	−	0.402736i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	225.108			0.385458
$585$		0			0
$586$	851.269			1.45268
$587$		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
$592$	−446.277	+	772.974i	−0.753846	+	1.30570i
$593$	437.404			0.737612			0.368806	−	0.929506i	$-0.379767\pi$
$593$	437.404			0.737612			0.368806	+	0.929506i	$0.379767\pi$
$594$		0			0
$595$		0			0
$596$	103.331	−	178.974i	0.173374	−	0.300292i
$597$		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
$601$	−67.6539	+	117.180i	−0.112569	+	0.194975i	−0.916805	−	0.399334i	$-0.869241\pi$
$601$	−67.6539	+	117.180i	−0.112569	+	0.194975i	0.804236	+	0.594309i	$0.202575\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	72.3672	−	125.344i	0.119615	−	0.207180i
$606$		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
$610$	−111.150	+	192.518i	−0.182213	+	0.315603i
$611$		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
$617$	213.067	+	369.043i	0.345328	+	0.598126i	0.985413	−	0.170178i	$-0.0544344\pi$
$617$	213.067	+	369.043i	0.345328	+	0.598126i	−0.640085	+	0.768304i	$0.721101\pi$
$618$		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
$625$	−259.869	−	450.107i	−0.415791	−	0.720171i
$626$	565.400	+	979.301i	0.903195	+	1.56438i
$627$		0			0
$628$	627.261	−	1086.45i	0.998824	−	1.73001i
$629$	1003.05			1.59467
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	437.904	+	758.472i	0.690700	+	1.19633i
$635$		0			0
$636$		0			0
$637$	−631.723	+	1094.18i	−0.991716	+	1.71770i
$638$		0			0
$639$		0			0
$640$	−153.108			−0.239230
$641$	−427.409	+	740.295i	−0.666785	+	1.15491i	0.312012	+	0.950078i	$0.398997\pi$
$641$	−427.409	+	740.295i	−0.666785	+	1.15491i	−0.978798	+	0.204828i	$0.934336\pi$
$642$		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
$650$	607.723	−	1052.61i	0.934959	−	1.61940i
$651$		0			0
$652$		0			0
$653$	−572.000	−	990.733i	−0.875957	−	1.51720i	−0.855740	−	0.517407i	$-0.826897\pi$
$653$	−572.000	−	990.733i	−0.875957	−	1.51720i	−0.0202175	−	0.999796i	$-0.506436\pi$
$654$		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
$661$	34.6924	+	60.0890i	0.0524847	+	0.0909061i	0.891074	−	0.453858i	$-0.149953\pi$
$661$	34.6924	+	60.0890i	0.0524847	+	0.0909061i	−0.838589	+	0.544764i	$0.816619\pi$
$662$		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
$673$	−285.546	+	494.580i	−0.424288	+	0.734889i	−0.996354	−	0.0853191i	$-0.972809\pi$
$673$	−285.546	+	494.580i	−0.424288	+	0.734889i	0.572065	+	0.820208i	$0.306142\pi$
$674$	700.000			1.03858
$675$		0			0
$676$	1983.38			2.93400
$677$	52.0000	−	90.0666i	0.0768095	−	0.133038i	−0.825062	−	0.565042i	$-0.808860\pi$
$677$	52.0000	−	90.0666i	0.0768095	−	0.133038i	0.901872	+	0.432004i	$0.142193\pi$
$678$		0			0
$679$		0			0
$680$	86.0309	+	149.010i	0.126516	+	0.219132i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−322.800			−0.471241
$686$		0			0
$687$		0			0
$688$		0			0
$689$	721.969	+	1250.49i	1.04785	+	1.81493i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$	−935.154			−1.35138
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	719.230	+	1245.74i	1.03189	+	1.78729i
$698$	598.000	+	1035.77i	0.856734	+	1.48391i
$699$		0			0
$700$		0			0
$701$	−1387.57			−1.97941			−0.989704	−	0.143129i	$-0.954284\pi$
$701$	−1387.57			−1.97941			−0.989704	+	0.143129i	$0.954284\pi$
$702$		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
$709$	−442.077	+	765.699i	−0.623522	+	1.07997i	0.365303	+	0.930889i	$0.380965\pi$
$709$	−442.077	+	765.699i	−0.623522	+	1.07997i	−0.988825	+	0.149082i	$0.952368\pi$
$710$		0			0
$711$		0			0
$712$	−1180.40			−1.65786
$713$		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
$725$	−664.335	−	1150.66i	−0.916324	−	1.58712i
$726$		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
$730$	−67.3157			−0.0922133
$731$		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			$\pi$
$740$	133.454	−	231.149i	0.180343	−	0.312363i
$741$		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
$745$	−30.8999	+	53.5201i	−0.0414763	+	0.0718391i
$746$	−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
$754$	1453.56	−	2517.64i	1.92780	−	3.33904i
$755$		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
$761$	−346.225	−	599.679i	−0.454961	−	0.788015i	0.543725	−	0.839263i	$-0.317013\pi$
$761$	−346.225	−	599.679i	−0.454961	−	0.788015i	−0.998686	+	0.0512484i	$0.983680\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	760.115	+	1316.56i	0.988446	+	1.71204i	0.625488	+	0.780234i	$0.284900\pi$
$769$	760.115	+	1316.56i	0.988446	+	1.71204i	0.362959	+	0.931805i	$0.381767\pi$
$770$		0			0
$771$		0			0
$772$	391.969	−	678.910i	0.507732	−	0.879418i
$773$	1085.75			1.40459			0.702296	−	0.711885i	$-0.252158\pi$
$773$	1085.75			1.40459			0.702296	+	0.711885i	$0.252158\pi$
$774$		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
$785$	−187.575	+	324.890i	−0.238949	+	0.413872i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−731.500	−	1266.99i	−0.928299	−	1.60786i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2395.98			−3.02142
$794$	−69.9076	+	121.083i	−0.0880448	+	0.152498i
$795$		0			0
$796$		0			0
$797$	−194.644	−	337.133i	−0.244221	−	0.423003i	0.717691	−	0.696361i	$-0.245199\pi$
$797$	−194.644	−	337.133i	−0.244221	−	0.423003i	−0.961912	+	0.273358i	$0.911866\pi$
$798$		0			0
$799$		0			0
$800$	−754.215			−0.942769
$801$		0			0
$802$	−1302.18			−1.62366
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−160.000	+	277.128i	−0.198020	+	0.342980i
$809$	−1034.63			−1.27890			−0.639448	−	0.768835i	$-0.720837\pi$
$809$	−1034.63			−1.27890			−0.639448	+	0.768835i	$0.720837\pi$
$810$		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
$818$	−366.308			−0.447809
$819$		0			0
$820$	382.769			0.466791
$821$	−721.525	+	1249.72i	−0.878837	+	1.52219i	−0.0262179	+	0.999656i	$0.508346\pi$
$821$	−721.525	+	1249.72i	−0.878837	+	1.52219i	−0.852619	+	0.522533i	$0.824987\pi$
$822$		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
$832$	−825.108	−	1429.13i	−0.991716	−	1.71770i
$833$	−440.529	−	763.018i	−0.528846	−	0.915988i
$834$		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
$841$	−1168.46	−	2023.83i	−1.38937	−	2.40646i
$842$	698.461	+	1209.77i	0.829527	+	1.43678i
$843$		0			0
$844$		0			0
$845$	−593.108			−0.701902
$846$		0			0
$847$		0			0
$848$	448.000	−	775.959i	0.528302	−	0.915046i
$849$		0			0
$850$	423.793	+	734.030i	0.498579	+	0.863565i
$851$		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
$857$	−830.471	+	1438.42i	−0.969044	+	1.67843i	−0.270712	+	0.962660i	$0.587259\pi$
$857$	−830.471	+	1438.42i	−0.969044	+	1.67843i	−0.698333	+	0.715774i	$0.746074\pi$
$858$		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
$865$	279.647			0.323291
$866$	−561.677	+	972.853i	−0.648587	+	1.12339i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1559.38			−1.78828
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	703.877	+	1219.15i	0.802596	+	1.39014i	0.917902	+	0.396807i	$0.129882\pi$
$877$	703.877	+	1219.15i	0.802596	+	1.39014i	−0.115306	+	0.993330i	$0.536785\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1600.00			1.81612			0.908059	−	0.418842i	$-0.137564\pi$
$881$	1600.00			1.81612			0.908059	+	0.418842i	$0.137564\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−927.254	+	1606.05i	−1.04893	+	1.81680i
$885$		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
$890$	352.985			0.396612
$891$		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
$901$	−1006.92			−1.11756
$902$		0			0
$903$		0			0
$904$	−551.923	−	955.959i	−0.610534	−	1.05748i
$905$	22.7269	+	39.3641i	0.0251126	+	0.0434963i
$906$		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
$914$	−134.015	−	232.122i	−0.146625	−	0.253962i
$915$		0			0
$916$	−234.154	+	405.566i	−0.255627	+	0.442758i
$917$		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
$925$	657.400	−	1138.65i	0.710703	−	1.23097i
$926$		0			0
$927$		0			0
$928$	−1803.94			−1.94390
$929$	571.717	−	990.243i	0.615411	−	1.06592i	−0.374901	−	0.927065i	$-0.622323\pi$
$929$	571.717	−	990.243i	0.615411	−	1.06592i	0.990312	−	0.138859i	$-0.0443435\pi$
$930$		0			0
$931$		0			0
$932$	−779.731	−	1350.53i	−0.836621	−	1.44907i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1794.63			1.91529			0.957647	−	0.287945i	$-0.0929721\pi$
$937$	1794.63			1.91529			0.957647	+	0.287945i	$0.0929721\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	351.725	+	609.205i	0.373778	+	0.647402i	0.990143	−	0.140058i	$-0.0447290\pi$
$941$	351.725	+	609.205i	0.373778	+	0.647402i	−0.616366	+	0.787460i	$0.711396\pi$
$942$		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
$949$	−362.769	−	628.335i	−0.382265	−	0.662102i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1793.21			−1.88165			−0.940824	−	0.338895i	$-0.889947\pi$
$953$	−1793.21			−1.88165			−0.940824	+	0.338895i	$0.889947\pi$
$954$		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
$962$	2876.77			2.99040
$963$		0			0
$964$	1667.38			1.72965
$965$	−117.214	+	203.020i	−0.121465	+	0.210383i
$966$		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
$970$	−155.500	+	269.334i	−0.160309	+	0.277663i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	743.384	+	1287.58i	0.761664	+	1.31924i
$977$	−248.000	−	429.549i	−0.253838	−	0.439661i	0.710741	−	0.703454i	$-0.248360\pi$
$977$	−248.000	−	429.549i	−0.253838	−	0.439661i	−0.964579	+	0.263793i	$0.915026\pi$
$978$		0			0
$979$		0			0
$980$	−234.446			−0.239230
$981$		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
$985$	218.746	+	378.880i	0.222077	+	0.384649i
$986$	1013.63	+	1755.66i	1.02802	+	1.78059i
$987$		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
$997$	140.339	−	243.073i	0.140761	−	0.243805i	−0.787023	−	0.616924i	$-0.788378\pi$
$997$	140.339	−	243.073i	0.140761	−	0.243805i	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.k.55.2		4
3.2	odd	2		324.3.f.n.55.1		4
4.3	odd	2	CM	324.3.f.k.55.2		4
9.2	odd	6		324.3.d.a.163.2	✓	2
9.4	even	3	inner	324.3.f.k.271.2		4
9.5	odd	6		324.3.f.n.271.1		4
9.7	even	3		324.3.d.d.163.1	yes	2
12.11	even	2		324.3.f.n.55.1		4
36.7	odd	6		324.3.d.d.163.1	yes	2
36.11	even	6		324.3.d.a.163.2	✓	2
36.23	even	6		324.3.f.n.271.1		4
36.31	odd	6	inner	324.3.f.k.271.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.d.a.163.2	✓	2	9.2	odd	6
324.3.d.a.163.2	✓	2	36.11	even	6
324.3.d.d.163.1	yes	2	9.7	even	3
324.3.d.d.163.1	yes	2	36.7	odd	6
324.3.f.k.55.2		4	1.1	even	1	trivial
324.3.f.k.55.2		4	4.3	odd	2	CM
324.3.f.k.271.2		4	9.4	even	3	inner
324.3.f.k.271.2		4	36.31	odd	6	inner
324.3.f.n.55.1		4	3.2	odd	2
324.3.f.n.55.1		4	12.11	even	2
324.3.f.n.271.1		4	9.5	odd	6
324.3.f.n.271.1		4	36.23	even	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.f (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(0.598076 + 1.03590i) q^{5} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 + 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(0.598076 + 1.03590i) q^{5} +8.00000 q^{8} -2.39230 q^{10} +(-12.8923 - 22.3301i) q^{13} +(-8.00000 + 13.8564i) q^{16} +17.9808 q^{17} +(2.39230 - 4.14359i) q^{20} +(11.7846 - 20.4115i) q^{25} +51.5692 q^{26} +(28.1865 - 48.8205i) q^{29} +(-16.0000 - 27.7128i) q^{32} +(-17.9808 + 31.1436i) q^{34} +55.7846 q^{37} +(4.78461 + 8.28719i) q^{40} +(40.0000 + 69.2820i) q^{41} +(-24.5000 - 42.4352i) q^{49} +(23.5692 + 40.8231i) q^{50} +(-51.5692 + 89.3205i) q^{52} -56.0000 q^{53} +(56.3731 + 97.6410i) q^{58} +(46.4615 - 80.4737i) q^{61} +64.0000 q^{64} +(15.4212 - 26.7102i) q^{65} +(-35.9615 - 62.2872i) q^{68} +28.1384 q^{73} +(-55.7846 + 96.6218i) q^{74} -19.1384 q^{80} -160.000 q^{82} +(10.7539 + 18.6262i) q^{85} -147.550 q^{89} +(65.0000 - 112.583i) q^{97} +98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 8 q^{4} - 8 q^{5} + 32 q^{8}+O(q^{10}) \) 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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