LMFDB - Embedded newform 1296.3.q.o.593.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(593,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1296.593");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.q (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:	$35.3134422611$
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^{8} $
Twist minimal:	no (minimal twist has level 162)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1296.593
Dual form			1296.3.q.o.1025.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.34486 + 0.776457i) q^{5} +(6.19615 - 10.7321i) q^{7} +O(q^{10})$	$q+(-1.34486 + 0.776457i) q^{5} +(6.19615 - 10.7321i) q^{7} +(-12.7279 - 7.34847i) q^{11} +(5.40192 + 9.35641i) q^{13} -28.9778i q^{17} -3.60770 q^{19} +(-12.7279 + 7.34847i) q^{23} +(-11.2942 + 19.5622i) q^{25} +(-24.3748 - 14.0728i) q^{29} +(4.00000 + 6.92820i) q^{31} +19.2442i q^{35} +22.5692 q^{37} +(21.7816 - 12.5756i) q^{41} +(-26.5885 + 46.0526i) q^{43} +(-14.6969 - 8.48528i) q^{47} +(-52.2846 - 90.5596i) q^{49} +84.5482i q^{53} +22.8231 q^{55} +(-78.8641 + 45.5322i) q^{59} +(6.50000 - 11.2583i) q^{61} +(-14.5297 - 8.38872i) q^{65} +(-20.5885 - 35.6603i) q^{67} -16.3613i q^{71} +71.5885 q^{73} +(-157.728 + 91.0645i) q^{77} +(-23.3731 + 40.4833i) q^{79} +(13.2555 + 7.65308i) q^{83} +(22.5000 + 38.9711i) q^{85} -78.9756i q^{89} +133.885 q^{91} +(4.85186 - 2.80122i) q^{95} +(-45.5692 + 78.9282i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{7}+O(q^{10}) $ 8 * q + 8 * q^7	$ 8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100}) $ 8 * q + 8 * q^7 + 64 * q^13 - 112 * q^19 - 28 * q^25 + 32 * q^31 - 152 * q^37 - 88 * q^43 - 252 * q^49 + 432 * q^55 + 52 * q^61 - 40 * q^67 + 448 * q^73 + 104 * q^79 + 180 * q^85 - 176 * q^91 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{6}\right)$

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
$5$	−1.34486	+	0.776457i	−0.268973	+	0.155291i	−0.628421	−	0.777874i	$-0.716298\pi$
$5$	−1.34486	+	0.776457i	−0.268973	+	0.155291i	0.359448	+	0.933165i	$0.382965\pi$
$6$		0			0
$7$	6.19615	−	10.7321i	0.885165	−	1.53315i	0.0396398	−	0.999214i	$-0.487379\pi$
$7$	6.19615	−	10.7321i	0.885165	−	1.53315i	0.845525	−	0.533936i	$-0.179288\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−12.7279	−	7.34847i	−1.15708	−	0.668043i	−0.206480	−	0.978451i	$-0.566201\pi$
$11$	−12.7279	−	7.34847i	−1.15708	−	0.668043i	−0.950603	+	0.310408i	$0.899534\pi$
$12$		0			0
$13$	5.40192	+	9.35641i	0.415533	+	0.719724i	0.995484	−	0.0949274i	$-0.0302619\pi$
$13$	5.40192	+	9.35641i	0.415533	+	0.719724i	−0.579952	+	0.814651i	$0.696929\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	28.9778i		−	1.70457i	−0.523074	−	0.852287i	$-0.675215\pi$
$17$		−	28.9778i		−	1.70457i	0.523074	−	0.852287i	$-0.324785\pi$
$18$		0			0
$19$	−3.60770			−0.189879			−0.0949393	−	0.995483i	$-0.530266\pi$
$19$	−3.60770			−0.189879			−0.0949393	+	0.995483i	$0.530266\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−12.7279	+	7.34847i	−0.553388	+	0.319499i	−0.750487	−	0.660885i	$-0.770181\pi$
$23$	−12.7279	+	7.34847i	−0.553388	+	0.319499i	0.197099	+	0.980384i	$0.436848\pi$
$24$		0			0
$25$	−11.2942	+	19.5622i	−0.451769	+	0.782487i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−24.3748	−	14.0728i	−0.840510	−	0.485268i	0.0169278	−	0.999857i	$-0.494611\pi$
$29$	−24.3748	−	14.0728i	−0.840510	−	0.485268i	−0.857437	+	0.514588i	$0.827945\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.129032	+	0.223490i	0.923302	−	0.384075i	$-0.125480\pi$
$31$	4.00000	+	6.92820i	0.129032	+	0.223490i	−0.794270	+	0.607565i	$0.792146\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			19.2442i			0.549834i
$36$		0			0
$37$	22.5692			0.609979			0.304989	−	0.952356i	$-0.401347\pi$
$37$	22.5692			0.609979			0.304989	+	0.952356i	$0.401347\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	21.7816	−	12.5756i	0.531259	−	0.306722i	−0.210270	−	0.977643i	$-0.567434\pi$
$41$	21.7816	−	12.5756i	0.531259	−	0.306722i	0.741529	+	0.670921i	$0.234101\pi$
$42$		0			0
$43$	−26.5885	+	46.0526i	−0.618336	+	1.07099i	0.371453	+	0.928452i	$0.378860\pi$
$43$	−26.5885	+	46.0526i	−0.618336	+	1.07099i	−0.989789	+	0.142538i	$0.954474\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.6969	−	8.48528i	−0.312701	−	0.180538i	0.335434	−	0.942064i	$-0.391117\pi$
$47$	−14.6969	−	8.48528i	−0.312701	−	0.180538i	−0.648134	+	0.761526i	$0.724451\pi$
$48$		0			0
$49$	−52.2846	−	90.5596i	−1.06703	−	1.84816i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			84.5482i			1.59525i	0.603154	+	0.797625i	$0.293910\pi$
$53$			84.5482i			1.59525i	−0.603154	+	0.797625i	$0.706090\pi$
$54$		0			0
$55$	22.8231			0.414965
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−78.8641	+	45.5322i	−1.33668	+	0.771733i	−0.986314	−	0.164880i	$-0.947276\pi$
$59$	−78.8641	+	45.5322i	−1.33668	+	0.771733i	−0.350367	+	0.936613i	$0.613943\pi$
$60$		0			0
$61$	6.50000	−	11.2583i	0.106557	−	0.184563i	−0.807816	−	0.589435i	$-0.799351\pi$
$61$	6.50000	−	11.2583i	0.106557	−	0.184563i	0.914373	+	0.404872i	$0.132684\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−14.5297	−	8.38872i	−0.223534	−	0.129057i
$66$		0			0
$67$	−20.5885	−	35.6603i	−0.307290	−	0.532243i	0.670478	−	0.741929i	$-0.266089\pi$
$67$	−20.5885	−	35.6603i	−0.307290	−	0.532243i	−0.977769	+	0.209687i	$0.932756\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	16.3613i		−	0.230442i	−0.993340	−	0.115221i	$-0.963242\pi$
$71$		−	16.3613i		−	0.230442i	0.993340	−	0.115221i	$-0.0367575\pi$
$72$		0			0
$73$	71.5885			0.980664			0.490332	−	0.871536i	$-0.336876\pi$
$73$	71.5885			0.980664			0.490332	+	0.871536i	$0.336876\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−157.728	+	91.0645i	−2.04842	+	1.18266i
$78$		0			0
$79$	−23.3731	+	40.4833i	−0.295862	+	0.512447i	−0.975185	−	0.221392i	$-0.928940\pi$
$79$	−23.3731	+	40.4833i	−0.295862	+	0.512447i	0.679323	+	0.733839i	$0.262273\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.2555	+	7.65308i	0.159705	+	0.0922057i	0.577723	−	0.816233i	$-0.303942\pi$
$83$	13.2555	+	7.65308i	0.159705	+	0.0922057i	−0.418017	+	0.908439i	$0.637275\pi$
$84$		0			0
$85$	22.5000	+	38.9711i	0.264706	+	0.458484i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	78.9756i		−	0.887367i	−0.896184	−	0.443683i	$-0.853671\pi$
$89$		−	78.9756i		−	0.887367i	0.896184	−	0.443683i	$-0.146329\pi$
$90$		0			0
$91$	133.885			1.47126
$92$		0			0
$93$		0			0
$94$		0			0
$95$	4.85186	−	2.80122i	0.0510722	−	0.0294865i
$96$		0			0
$97$	−45.5692	+	78.9282i	−0.469786	+	0.813693i	−0.999403	−	0.0345438i	$-0.989002\pi$
$97$	−45.5692	+	78.9282i	−0.469786	+	0.813693i	0.529617	+	0.848237i	$0.322336\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−76.6313	−	44.2431i	−0.758726	−	0.438051i	0.0701121	−	0.997539i	$-0.477664\pi$
$101$	−76.6313	−	44.2431i	−0.758726	−	0.438051i	−0.828838	+	0.559488i	$0.810998\pi$
$102$		0			0
$103$	−76.3538	−	132.249i	−0.741299	−	1.28397i	−0.951904	−	0.306397i	$-0.900877\pi$
$103$	−76.3538	−	132.249i	−0.741299	−	1.28397i	0.210605	−	0.977571i	$-0.432457\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	60.0062i		−	0.560805i	−0.959882	−	0.280403i	$-0.909532\pi$
$107$		−	60.0062i		−	0.560805i	0.959882	−	0.280403i	$-0.0904680\pi$
$108$		0			0
$109$	−93.9423			−0.861856			−0.430928	−	0.902386i	$-0.641814\pi$
$109$	−93.9423			−0.861856			−0.430928	+	0.902386i	$0.641814\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−76.7279	+	44.2989i	−0.679008	+	0.392025i	−0.799481	−	0.600691i	$-0.794892\pi$
$113$	−76.7279	+	44.2989i	−0.679008	+	0.392025i	0.120473	+	0.992717i	$0.461559\pi$
$114$		0			0
$115$	11.4115	−	19.7654i	0.0992308	−	0.171873i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−310.991	−	179.551i	−2.61337	−	1.50883i
$120$		0			0
$121$	47.5000	+	82.2724i	0.392562	+	0.679937i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	73.9008i		−	0.591206i
$126$		0			0
$127$	−78.8231			−0.620654			−0.310327	−	0.950630i	$-0.600439\pi$
$127$	−78.8231			−0.620654			−0.310327	+	0.950630i	$0.600439\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	71.5157	−	41.2896i	0.545921	−	0.315188i	−0.201554	−	0.979477i	$-0.564599\pi$
$131$	71.5157	−	41.2896i	0.545921	−	0.315188i	0.747475	+	0.664290i	$0.231266\pi$
$132$		0			0
$133$	−22.3538	+	38.7180i	−0.168074	+	0.291113i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−185.970	−	107.370i	−1.35745	−	0.783723i	−0.368169	−	0.929759i	$-0.620015\pi$
$137$	−185.970	−	107.370i	−1.35745	−	0.783723i	−0.989279	+	0.146036i	$0.953349\pi$
$138$		0			0
$139$	−30.3923	−	52.6410i	−0.218650	−	0.378712i	0.735746	−	0.677258i	$-0.236832\pi$
$139$	−30.3923	−	52.6410i	−0.218650	−	0.378712i	−0.954395	+	0.298546i	$0.903499\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	158.783i		−	1.11037i
$144$		0			0
$145$	43.7077			0.301432
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−26.5369	+	15.3211i	−0.178100	+	0.102826i	−0.586400	−	0.810022i	$-0.699455\pi$
$149$	−26.5369	+	15.3211i	−0.178100	+	0.102826i	0.408300	+	0.912848i	$0.366122\pi$
$150$		0			0
$151$	16.0000	−	27.7128i	0.105960	−	0.183529i	−0.808170	−	0.588949i	$-0.799542\pi$
$151$	16.0000	−	27.7128i	0.105960	−	0.183529i	0.914130	+	0.405421i	$0.132875\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.7589	−	6.21166i	−0.0694123	−	0.0400752i
$156$		0			0
$157$	−124.854	−	216.253i	−0.795247	−	1.37741i	−0.922682	−	0.385562i	$-0.874008\pi$
$157$	−124.854	−	216.253i	−0.795247	−	1.37741i	0.127435	−	0.991847i	$-0.459326\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			182.129i			1.13124i
$162$		0			0
$163$	12.7846			0.0784332			0.0392166	−	0.999231i	$-0.487514\pi$
$163$	12.7846			0.0784332			0.0392166	+	0.999231i	$0.487514\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−201.292	+	116.216i	−1.20534	+	0.695902i	−0.961737	−	0.273973i	$-0.911662\pi$
$167$	−201.292	+	116.216i	−1.20534	+	0.695902i	−0.243601	+	0.969876i	$0.578329\pi$
$168$		0			0
$169$	26.1384	−	45.2731i	0.154665	−	0.267888i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−18.6608	−	10.7738i	−0.107866	−	0.0622765i	0.445096	−	0.895483i	$-0.353169\pi$
$173$	−18.6608	−	10.7738i	−0.107866	−	0.0622765i	−0.552963	+	0.833206i	$0.686503\pi$
$174$		0			0
$175$	139.962	+	242.420i	0.799780	+	1.38526i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			277.741i			1.55162i	0.630964	+	0.775812i	$0.282659\pi$
$179$			277.741i			1.55162i	−0.630964	+	0.775812i	$0.717341\pi$
$180$		0			0
$181$	174.277			0.962856			0.481428	−	0.876486i	$-0.340118\pi$
$181$	174.277			0.962856			0.481428	+	0.876486i	$0.340118\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−30.3525	+	17.5240i	−0.164068	+	0.0947245i
$186$		0			0
$187$	−212.942	+	368.827i	−1.13873	+	1.97234i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	131.745	+	76.0629i	0.689764	+	0.398235i	0.803523	−	0.595273i	$-0.202956\pi$
$191$	131.745	+	76.0629i	0.689764	+	0.398235i	−0.113760	+	0.993508i	$0.536289\pi$
$192$		0			0
$193$	27.5000	+	47.6314i	0.142487	+	0.246795i	0.928433	−	0.371501i	$-0.121157\pi$
$193$	27.5000	+	47.6314i	0.142487	+	0.246795i	−0.785946	+	0.618296i	$0.787823\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	133.298i		−	0.676638i	−0.941031	−	0.338319i	$-0.890142\pi$
$197$		−	133.298i		−	0.676638i	0.941031	−	0.338319i	$-0.109858\pi$
$198$		0			0
$199$	208.862			1.04956			0.524778	−	0.851239i	$-0.324148\pi$
$199$	208.862			1.04956			0.524778	+	0.851239i	$0.324148\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−302.060	+	174.394i	−1.48798	+	0.859085i
$204$		0			0
$205$	−19.5289	+	33.8250i	−0.0952627	+	0.165000i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	45.9185	+	26.5110i	0.219706	+	0.126847i
$210$		0			0
$211$	−43.7269	−	75.7372i	−0.207236	−	0.358944i	0.743607	−	0.668617i	$-0.233114\pi$
$211$	−43.7269	−	75.7372i	−0.207236	−	0.358944i	−0.950843	+	0.309673i	$0.899780\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	82.5792i		−	0.384089i
$216$		0			0
$217$	99.1384			0.456859
$218$		0			0
$219$		0			0
$220$		0			0
$221$	271.128	−	156.536i	1.22682	−	0.708306i
$222$		0			0
$223$	111.296	−	192.771i	0.499086	−	0.864442i	−0.500914	−	0.865497i	$-0.667003\pi$
$223$	111.296	−	192.771i	0.499086	−	0.864442i	0.999999	+	0.00105540i	$0.000335944\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	260.079	+	150.157i	1.14572	+	0.661484i	0.947842	−	0.318742i	$-0.103260\pi$
$227$	260.079	+	150.157i	1.14572	+	0.661484i	0.197882	+	0.980226i	$0.436594\pi$
$228$		0			0
$229$	−114.971	−	199.136i	−0.502057	−	0.869589i	−0.999997	−	0.00237731i	$-0.999243\pi$
$229$	−114.971	−	199.136i	−0.502057	−	0.869589i	0.497940	−	0.867212i	$-0.334090\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			116.246i			0.498908i	0.968387	+	0.249454i	$0.0802512\pi$
$233$			116.246i			0.498908i	−0.968387	+	0.249454i	$0.919749\pi$
$234$		0			0
$235$	26.3538			0.112144
$236$		0			0
$237$		0			0
$238$		0			0
$239$	142.362	−	82.1930i	0.595659	−	0.343904i	−0.171673	−	0.985154i	$-0.554917\pi$
$239$	142.362	−	82.1930i	0.595659	−	0.343904i	0.767332	+	0.641250i	$0.221584\pi$
$240$		0			0
$241$	40.6558	−	70.4179i	0.168696	−	0.292190i	−0.769265	−	0.638929i	$-0.779378\pi$
$241$	40.6558	−	70.4179i	0.168696	−	0.292190i	0.937962	+	0.346739i	$0.112711\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	140.631	+	81.1935i	0.574005	+	0.331402i
$246$		0			0
$247$	−19.4885	−	33.7551i	−0.0789008	−	0.136660i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			396.371i			1.57917i	0.613642	+	0.789584i	$0.289704\pi$
$251$			396.371i			1.57917i	−0.613642	+	0.789584i	$0.710296\pi$
$252$		0			0
$253$	216.000			0.853755
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−186.164	+	107.482i	−0.724372	+	0.418216i	−0.816360	−	0.577544i	$-0.804011\pi$
$257$	−186.164	+	107.482i	−0.724372	+	0.418216i	0.0919879	+	0.995760i	$0.470678\pi$
$258$		0			0
$259$	139.842	−	242.214i	0.539932	−	0.935189i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−275.304	−	158.947i	−1.04678	−	0.604360i	−0.125036	−	0.992152i	$-0.539905\pi$
$263$	−275.304	−	158.947i	−1.04678	−	0.604360i	−0.921747	+	0.387792i	$0.873238\pi$
$264$		0			0
$265$	−65.6481	−	113.706i	−0.247729	−	0.429078i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	208.528i		−	0.775199i	−0.921828	−	0.387599i	$-0.873304\pi$
$269$		−	208.528i		−	0.775199i	0.921828	−	0.387599i	$-0.126696\pi$
$270$		0			0
$271$	−409.885			−1.51249			−0.756245	−	0.654289i	$-0.772968\pi$
$271$	−409.885			−1.51249			−0.756245	+	0.654289i	$0.772968\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	287.504	−	165.991i	1.04547	−	0.603602i
$276$		0			0
$277$	−248.708	+	430.774i	−0.897862	+	1.55514i	−0.0676390	+	0.997710i	$0.521547\pi$
$277$	−248.708	+	430.774i	−0.897862	+	1.55514i	−0.830223	+	0.557432i	$0.811787\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	186.427	+	107.634i	0.663442	+	0.383039i	0.793587	−	0.608456i	$-0.208211\pi$
$281$	186.427	+	107.634i	0.663442	+	0.383039i	−0.130145	+	0.991495i	$0.541544\pi$
$282$		0			0
$283$	−148.354	−	256.956i	−0.524218	−	0.907973i	−0.999602	−	0.0281946i	$-0.991024\pi$
$283$	−148.354	−	256.956i	−0.524218	−	0.907973i	0.475384	−	0.879778i	$-0.342309\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	311.682i		−	1.08600i
$288$		0			0
$289$	−550.711			−1.90558
$290$		0			0
$291$		0			0
$292$		0			0
$293$	295.284	−	170.482i	1.00779	−	0.581850i	0.0972490	−	0.995260i	$-0.468996\pi$
$293$	295.284	−	170.482i	1.00779	−	0.581850i	0.910545	+	0.413410i	$0.135662\pi$
$294$		0			0
$295$	70.7077	−	122.469i	0.239687	−	0.415150i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−137.511	−	79.3917i	−0.459901	−	0.265524i
$300$		0			0
$301$	329.492	+	570.697i	1.09466	+	1.89600i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			20.1879i			0.0661898i
$306$		0			0
$307$	114.354			0.372488			0.186244	−	0.982504i	$-0.440368\pi$
$307$	114.354			0.372488			0.186244	+	0.982504i	$0.440368\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	169.929	−	98.1083i	0.546394	−	0.315461i	−0.201272	−	0.979535i	$-0.564508\pi$
$311$	169.929	−	98.1083i	0.546394	−	0.315461i	0.747666	+	0.664074i	$0.231174\pi$
$312$		0			0
$313$	73.3616	−	127.066i	0.234382	−	0.405962i	−0.724711	−	0.689053i	$-0.758027\pi$
$313$	73.3616	−	127.066i	0.234382	−	0.405962i	0.959093	+	0.283092i	$0.0913600\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−41.6908	−	24.0702i	−0.131517	−	0.0759311i	0.432798	−	0.901491i	$-0.357526\pi$
$317$	−41.6908	−	24.0702i	−0.131517	−	0.0759311i	−0.564315	+	0.825560i	$0.690860\pi$
$318$		0			0
$319$	206.827	+	358.235i	0.648360	+	1.12299i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			104.543i			0.323662i
$324$		0			0
$325$	−244.042			−0.750899
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−182.129	+	105.152i	−0.553583	+	0.319612i
$330$		0			0
$331$	−49.7269	+	86.1295i	−0.150232	+	0.260210i	−0.931313	−	0.364220i	$-0.881335\pi$
$331$	−49.7269	+	86.1295i	−0.150232	+	0.260210i	0.781080	+	0.624430i	$0.214669\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	55.3773	+	31.9721i	0.165305	+	0.0954391i
$336$		0			0
$337$	212.631	+	368.287i	0.630952	+	1.09284i	0.987358	+	0.158509i	$0.0506687\pi$
$337$	212.631	+	368.287i	0.630952	+	1.09284i	−0.356406	+	0.934331i	$0.615998\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	117.576i		−	0.344796i
$342$		0			0
$343$	−688.631			−2.00767
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−33.5768	+	19.3856i	−0.0967630	+	0.0558662i	−0.547601	−	0.836740i	$-0.684459\pi$
$347$	−33.5768	+	19.3856i	−0.0967630	+	0.0558662i	0.450838	+	0.892606i	$0.351125\pi$
$348$		0			0
$349$	325.985	−	564.622i	0.934053	−	1.61783i	0.157739	−	0.987481i	$-0.449579\pi$
$349$	325.985	−	564.622i	0.934053	−	1.61783i	0.776314	−	0.630347i	$-0.217087\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.03625	−	0.598281i	−0.00293556	−	0.00169485i	0.498532	−	0.866872i	$-0.333873\pi$
$353$	−1.03625	−	0.598281i	−0.00293556	−	0.00169485i	−0.501467	+	0.865177i	$0.667206\pi$
$354$		0			0
$355$	12.7039	+	22.0038i	0.0357856	+	0.0619825i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	534.573i		−	1.48906i	−0.667589	−	0.744530i	$-0.732674\pi$
$359$		−	534.573i		−	1.48906i	0.667589	−	0.744530i	$-0.267326\pi$
$360$		0			0
$361$	−347.985			−0.963946
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−96.2767	+	55.5854i	−0.263772	+	0.152289i
$366$		0			0
$367$	132.354	−	229.244i	0.360637	−	0.624642i	−0.627429	−	0.778674i	$-0.715893\pi$
$367$	132.354	−	229.244i	0.360637	−	0.624642i	0.988066	+	0.154032i	$0.0492259\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	907.376	+	523.874i	2.44576	+	1.41206i
$372$		0			0
$373$	−35.5077	−	61.5012i	−0.0951950	−	0.164883i	0.814495	−	0.580171i	$-0.197014\pi$
$373$	−35.5077	−	61.5012i	−0.0951950	−	0.164883i	−0.909690	+	0.415288i	$0.863681\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	304.080i		−	0.806579i
$378$		0			0
$379$	696.785			1.83848			0.919241	−	0.393696i	$-0.128804\pi$
$379$	696.785			1.83848			0.919241	+	0.393696i	$0.128804\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	376.986	−	217.653i	0.984297	−	0.568284i	0.0807324	−	0.996736i	$-0.474274\pi$
$383$	376.986	−	217.653i	0.984297	−	0.568284i	0.903565	+	0.428452i	$0.140941\pi$
$384$		0			0
$385$	141.415	−	244.939i	0.367313	−	0.636204i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−45.9374	−	26.5220i	−0.118091	−	0.0681799i	0.439791	−	0.898100i	$-0.355053\pi$
$389$	−45.9374	−	26.5220i	−0.118091	−	0.0681799i	−0.557882	+	0.829920i	$0.688386\pi$
$390$		0			0
$391$	212.942	+	368.827i	0.544609	+	0.943291i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	72.5927i		−	0.183779i
$396$		0			0
$397$	63.7077			0.160473			0.0802363	−	0.996776i	$-0.474432\pi$
$397$	63.7077			0.160473			0.0802363	+	0.996776i	$0.474432\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	544.463	−	314.346i	1.35776	−	0.783904i	0.368440	−	0.929651i	$-0.379892\pi$
$401$	544.463	−	314.346i	1.35776	−	0.783904i	0.989322	+	0.145747i	$0.0465586\pi$
$402$		0			0
$403$	−43.2154	+	74.8513i	−0.107234	+	0.185735i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−287.259	−	165.849i	−0.705797	−	0.407492i
$408$		0			0
$409$	−267.640	−	463.567i	−0.654377	−	1.13341i	−0.982050	−	0.188623i	$-0.939597\pi$
$409$	−267.640	−	463.567i	−0.654377	−	1.13341i	0.327672	−	0.944791i	$-0.393736\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1128.50i			2.73244i
$414$		0			0
$415$	−23.7691			−0.0572750
$416$		0			0
$417$		0			0
$418$		0			0
$419$	333.705	−	192.665i	0.796433	−	0.459821i	−0.0457895	−	0.998951i	$-0.514580\pi$
$419$	333.705	−	192.665i	0.796433	−	0.459821i	0.842222	+	0.539130i	$0.181247\pi$
$420$		0			0
$421$	377.660	−	654.126i	0.897054	−	1.55374i	0.0658113	−	0.997832i	$-0.479036\pi$
$421$	377.660	−	654.126i	0.897054	−	1.55374i	0.831242	−	0.555910i	$-0.187630\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	566.868	+	327.282i	1.33381	+	0.770074i
$426$		0			0
$427$	−80.5500	−	139.517i	−0.188642	−	0.326737i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	107.709i		−	0.249904i	−0.992163	−	0.124952i	$-0.960122\pi$
$431$		−	107.709i		−	0.249904i	0.992163	−	0.124952i	$-0.0398777\pi$
$432$		0			0
$433$	655.123			1.51299			0.756493	−	0.654002i	$-0.226911\pi$
$433$	655.123			1.51299			0.756493	+	0.654002i	$0.226911\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	45.9185	−	26.5110i	0.105077	−	0.0606660i
$438$		0			0
$439$	−236.000	+	408.764i	−0.537585	+	0.931125i	0.461448	+	0.887167i	$0.347330\pi$
$439$	−236.000	+	408.764i	−0.537585	+	0.931125i	−0.999033	+	0.0439580i	$0.986003\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	728.516	+	420.609i	1.64451	+	0.949455i	0.979203	+	0.202881i	$0.0650304\pi$
$443$	728.516	+	420.609i	1.64451	+	0.949455i	0.665302	+	0.746575i	$0.268303\pi$
$444$		0			0
$445$	61.3212	+	106.211i	0.137800	+	0.238677i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			382.751i			0.852453i	0.904616	+	0.426227i	$0.140157\pi$
$449$			382.751i			0.852453i	−0.904616	+	0.426227i	$0.859843\pi$
$450$		0			0
$451$	−369.646			−0.819615
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−180.056	+	103.956i	−0.395728	+	0.228474i
$456$		0			0
$457$	−161.852	+	280.336i	−0.354162	+	0.613426i	−0.986974	−	0.160879i	$-0.948567\pi$
$457$	−161.852	+	280.336i	−0.354162	+	0.613426i	0.632812	+	0.774305i	$0.281900\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	555.582	+	320.765i	1.20517	+	0.695803i	0.961700	−	0.274106i	$-0.0883818\pi$
$461$	555.582	+	320.765i	1.20517	+	0.695803i	0.243467	+	0.969909i	$0.421715\pi$
$462$		0			0
$463$	64.7461	+	112.144i	0.139840	+	0.242211i	0.927436	−	0.373982i	$-0.122008\pi$
$463$	64.7461	+	112.144i	0.139840	+	0.242211i	−0.787596	+	0.616192i	$0.788674\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			672.730i			1.44054i	0.693696	+	0.720268i	$0.255981\pi$
$467$			672.730i			1.44054i	−0.693696	+	0.720268i	$0.744019\pi$
$468$		0			0
$469$	−510.277			−1.08801
$470$		0			0
$471$		0			0
$472$		0			0
$473$	676.832	−	390.769i	1.43093	−	0.826150i
$474$		0			0
$475$	40.7461	−	70.5744i	0.0857813	−	0.148578i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−108.786	−	62.8074i	−0.227110	−	0.131122i	0.382128	−	0.924109i	$-0.375191\pi$
$479$	−108.786	−	62.8074i	−0.227110	−	0.131122i	−0.609238	+	0.792987i	$0.708525\pi$
$480$		0			0
$481$	121.917	+	211.167i	0.253466	+	0.439016i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	141.530i		−	0.291815i
$486$		0			0
$487$	448.631			0.921213			0.460606	−	0.887604i	$-0.347632\pi$
$487$	448.631			0.921213			0.460606	+	0.887604i	$0.347632\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	391.579	−	226.078i	0.797514	−	0.460445i	−0.0450873	−	0.998983i	$-0.514357\pi$
$491$	391.579	−	226.078i	0.797514	−	0.460445i	0.842601	+	0.538538i	$0.181023\pi$
$492$		0			0
$493$	−407.798	+	706.327i	−0.827176	+	1.43271i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−175.591	−	101.377i	−0.353301	−	0.203979i
$498$		0			0
$499$	63.2961	+	109.632i	0.126846	+	0.219704i	0.922453	−	0.386109i	$-0.126181\pi$
$499$	63.2961	+	109.632i	0.126846	+	0.219704i	−0.795607	+	0.605813i	$0.792848\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	296.822i		−	0.590103i	−0.955481	−	0.295051i	$-0.904663\pi$
$503$		−	296.822i		−	0.590103i	0.955481	−	0.295051i	$-0.0953367\pi$
$504$		0			0
$505$	137.412			0.272102
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−567.113	+	327.423i	−1.11417	+	0.643267i	−0.939907	−	0.341432i	$-0.889088\pi$
$509$	−567.113	+	327.423i	−1.11417	+	0.643267i	−0.174265	+	0.984699i	$0.555755\pi$
$510$		0			0
$511$	443.573	−	768.291i	0.868049	−	1.50350i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	205.371	+	118.571i	0.398778	+	0.230235i
$516$		0			0
$517$	124.708	+	216.000i	0.241214	+	0.417795i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	690.006i		−	1.32439i	−0.749333	−	0.662193i	$-0.769626\pi$
$521$		−	690.006i		−	1.32439i	0.749333	−	0.662193i	$-0.230374\pi$
$522$		0			0
$523$	−616.238			−1.17828			−0.589138	−	0.808032i	$-0.700533\pi$
$523$	−616.238			−1.17828			−0.589138	+	0.808032i	$0.700533\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	200.764	−	115.911i	0.380956	−	0.219945i
$528$		0			0
$529$	−156.500	+	271.066i	−0.295841	+	0.512412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	235.325	+	135.865i	0.441511	+	0.254906i
$534$		0			0
$535$	46.5922	+	80.7001i	0.0870883	+	0.150841i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1536.85i			2.85129i
$540$		0			0
$541$	−548.734			−1.01430			−0.507148	−	0.861859i	$-0.669300\pi$
$541$	−548.734			−1.01430			−0.507148	+	0.861859i	$0.669300\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	126.340	−	72.9422i	0.231816	−	0.133839i
$546$		0			0
$547$	−409.885	+	709.941i	−0.749332	+	1.29788i	0.198811	+	0.980038i	$0.436292\pi$
$547$	−409.885	+	709.941i	−0.749332	+	1.29788i	−0.948143	+	0.317843i	$0.897041\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	87.9368	+	50.7703i	0.159595	+	0.0921421i
$552$		0			0
$553$	289.646	+	501.682i	0.523772	+	0.907201i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	125.808i		−	0.225867i	−0.993603	−	0.112934i	$-0.963975\pi$
$557$		−	125.808i		−	0.225867i	0.993603	−	0.112934i	$-0.0360247\pi$
$558$		0			0
$559$	−574.515			−1.02776
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−440.625	+	254.395i	−0.782638	+	0.451856i	−0.837365	−	0.546645i	$-0.815905\pi$
$563$	−440.625	+	254.395i	−0.782638	+	0.451856i	0.0547261	+	0.998501i	$0.482571\pi$
$564$		0			0
$565$	68.7923	−	119.152i	0.121756	−	0.210888i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	83.8832	+	48.4300i	0.147422	+	0.0851143i	0.571897	−	0.820326i	$-0.306208\pi$
$569$	83.8832	+	48.4300i	0.147422	+	0.0851143i	−0.424474	+	0.905440i	$0.639541\pi$
$570$		0			0
$571$	−227.100	−	393.349i	−0.397723	−	0.688877i	0.595721	−	0.803191i	$-0.296866\pi$
$571$	−227.100	−	393.349i	−0.397723	−	0.688877i	−0.993445	+	0.114314i	$0.963533\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	331.981i		−	0.577359i
$576$		0			0
$577$	−39.1230			−0.0678041			−0.0339021	−	0.999425i	$-0.510793\pi$
$577$	−39.1230			−0.0678041			−0.0339021	+	0.999425i	$0.510793\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	164.266	−	94.8393i	0.282731	−	0.163235i
$582$		0			0
$583$	621.300	−	1076.12i	1.06569	−	1.84584i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−363.061	−	209.614i	−0.618503	−	0.357093i	0.157783	−	0.987474i	$-0.449565\pi$
$587$	−363.061	−	209.614i	−0.618503	−	0.357093i	−0.776286	+	0.630381i	$0.782899\pi$
$588$		0			0
$589$	−14.4308	−	24.9948i	−0.0245005	−	0.0424361i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	329.210i		−	0.555160i	−0.960703	−	0.277580i	$-0.910468\pi$
$593$		−	329.210i		−	0.555160i	0.960703	−	0.277580i	$-0.0895323\pi$
$594$		0			0
$595$	557.654			0.937233
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−159.556	+	92.1197i	−0.266371	+	0.153789i	−0.627237	−	0.778828i	$-0.715814\pi$
$599$	−159.556	+	92.1197i	−0.266371	+	0.153789i	0.360867	+	0.932617i	$0.382481\pi$
$600$		0			0
$601$	−109.208	+	189.153i	−0.181710	+	0.314731i	−0.942463	−	0.334311i	$-0.891497\pi$
$601$	−109.208	+	189.153i	−0.181710	+	0.314731i	0.760753	+	0.649041i	$0.224830\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−127.762	−	73.7634i	−0.211177	−	0.121923i
$606$		0			0
$607$	5.13467	+	8.89350i	0.00845909	+	0.0146516i	0.870224	−	0.492656i	$-0.163974\pi$
$607$	5.13467	+	8.89350i	0.00845909	+	0.0146516i	−0.861765	+	0.507308i	$0.830641\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	183.347i		−	0.300078i
$612$		0			0
$613$	−180.585			−0.294592			−0.147296	−	0.989092i	$-0.547057\pi$
$613$	−180.585			−0.294592			−0.147296	+	0.989092i	$0.547057\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	784.800	−	453.104i	1.27196	−	0.734367i	0.296604	−	0.955000i	$-0.404146\pi$
$617$	784.800	−	453.104i	1.27196	−	0.734367i	0.975357	+	0.220633i	$0.0708124\pi$
$618$		0			0
$619$	380.823	−	659.605i	0.615223	−	1.06560i	−0.375122	−	0.926975i	$-0.622399\pi$
$619$	380.823	−	659.605i	0.615223	−	1.06560i	0.990345	−	0.138622i	$-0.0442674\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−847.571	−	489.345i	−1.36047	−	0.785466i
$624$		0			0
$625$	−224.975	−	389.668i	−0.359960	−	0.623469i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	654.006i		−	1.03975i
$630$		0			0
$631$	−601.108			−0.952627			−0.476313	−	0.879276i	$-0.658027\pi$
$631$	−601.108			−0.952627			−0.476313	+	0.879276i	$0.658027\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	106.006	−	61.2027i	0.166939	−	0.0963823i
$636$		0			0
$637$	564.875	−	978.392i	0.886774	−	1.53594i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−500.089	−	288.727i	−0.780170	−	0.450431i	0.0563204	−	0.998413i	$-0.482063\pi$
$641$	−500.089	−	288.727i	−0.780170	−	0.450431i	−0.836491	+	0.547981i	$0.815396\pi$
$642$		0			0
$643$	65.0615	+	112.690i	0.101184	+	0.175256i	0.912173	−	0.409806i	$-0.134403\pi$
$643$	65.0615	+	112.690i	0.101184	+	0.175256i	−0.810989	+	0.585062i	$0.801070\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	985.467i		−	1.52313i	−0.648087	−	0.761567i	$-0.724430\pi$
$647$		−	985.467i		−	1.52313i	0.648087	−	0.761567i	$-0.275570\pi$
$648$		0			0
$649$	1338.37			2.06220
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−519.895	+	300.161i	−0.796163	+	0.459665i	−0.842128	−	0.539278i	$-0.818697\pi$
$653$	−519.895	+	300.161i	−0.796163	+	0.459665i	0.0459644	+	0.998943i	$0.485364\pi$
$654$		0			0
$655$	−64.1192	+	111.058i	−0.0978919	+	0.169554i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−13.5004	−	7.79445i	−0.0204862	−	0.0118277i	0.489722	−	0.871879i	$-0.337098\pi$
$659$	−13.5004	−	7.79445i	−0.0204862	−	0.0118277i	−0.510208	+	0.860051i	$0.670432\pi$
$660$		0			0
$661$	−203.915	−	353.192i	−0.308495	−	0.534329i	0.669538	−	0.742778i	$-0.266492\pi$
$661$	−203.915	−	353.192i	−0.308495	−	0.534329i	−0.978033	+	0.208448i	$0.933159\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	69.4272i		−	0.104402i
$666$		0			0
$667$	413.654			0.620170
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−165.463	+	95.5301i	−0.246592	+	0.142370i
$672$		0			0
$673$	237.285	−	410.989i	0.352577	−	0.610682i	−0.634123	−	0.773232i	$-0.718639\pi$
$673$	237.285	−	410.989i	0.352577	−	0.610682i	0.986700	+	0.162550i	$0.0519720\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−244.346	−	141.073i	−0.360925	−	0.208380i	0.308562	−	0.951204i	$-0.400152\pi$
$677$	−244.346	−	141.073i	−0.360925	−	0.208380i	−0.669486	+	0.742824i	$0.733486\pi$
$678$		0			0
$679$	564.708	+	978.102i	0.831675	+	1.44050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1085.46i		−	1.58926i	−0.607095	−	0.794629i	$-0.707665\pi$
$683$		−	1085.46i		−	1.58926i	0.607095	−	0.794629i	$-0.292335\pi$
$684$		0			0
$685$	333.473			0.486822
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−791.067	+	456.723i	−1.14814	+	0.662878i
$690$		0			0
$691$	−503.888	+	872.760i	−0.729216	+	1.26304i	0.227999	+	0.973661i	$0.426782\pi$
$691$	−503.888	+	872.760i	−0.729216	+	1.26304i	−0.957215	+	0.289378i	$0.906552\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	81.7470	+	47.1966i	0.117622	+	0.0679088i
$696$		0			0
$697$	−364.413	−	631.183i	−0.522831	−	0.905570i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			216.731i			0.309174i	0.987979	+	0.154587i	$0.0494047\pi$
$701$			216.731i			0.309174i	−0.987979	+	0.154587i	$0.950595\pi$
$702$		0			0
$703$	−81.4229			−0.115822
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−949.639	+	548.274i	−1.34319	+	0.775494i
$708$		0			0
$709$	−497.248	+	861.259i	−0.701337	+	1.21475i	0.266660	+	0.963791i	$0.414080\pi$
$709$	−497.248	+	861.259i	−0.701337	+	1.21475i	−0.967997	+	0.250961i	$0.919253\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−101.823	−	58.7878i	−0.142810	−	0.0824513i
$714$		0			0
$715$	123.289	+	213.542i	0.172432	+	0.298660i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			340.912i			0.474148i	0.971492	+	0.237074i	$0.0761884\pi$
$719$			340.912i			0.474148i	−0.971492	+	0.237074i	$0.923812\pi$
$720$		0			0
$721$	−1892.40			−2.62469
$722$		0			0
$723$		0			0
$724$		0			0
$725$	550.589	−	317.883i	0.759433	−	0.438459i
$726$		0			0
$727$	−485.888	+	841.583i	−0.668347	+	1.15761i	0.310019	+	0.950730i	$0.399665\pi$
$727$	−485.888	+	841.583i	−0.668347	+	1.15761i	−0.978366	+	0.206881i	$0.933669\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1334.50	+	770.474i	1.82558	+	1.05400i
$732$		0			0
$733$	−114.585	−	198.466i	−0.156323	−	0.270759i	0.777217	−	0.629233i	$-0.216631\pi$
$733$	−114.585	−	198.466i	−0.156323	−	0.270759i	−0.933540	+	0.358473i	$0.883297\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			605.175i			0.821132i
$738$		0			0
$739$	629.892			0.852357			0.426179	−	0.904639i	$-0.359859\pi$
$739$	629.892			0.852357			0.426179	+	0.904639i	$0.359859\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	162.825	−	94.0071i	0.219145	−	0.126524i	−0.386409	−	0.922327i	$-0.626285\pi$
$743$	162.825	−	94.0071i	0.219145	−	0.126524i	0.605554	+	0.795804i	$0.292951\pi$
$744$		0			0
$745$	23.7923	−	41.2095i	0.0319360	−	0.0553148i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−643.989	−	371.807i	−0.859799	−	0.496405i
$750$		0			0
$751$	−693.335	−	1200.89i	−0.923215	−	1.59906i	−0.794407	−	0.607386i	$-0.792218\pi$
$751$	−693.335	−	1200.89i	−0.923215	−	1.59906i	−0.128808	−	0.991670i	$-0.541115\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			49.6933i			0.0658189i
$756$		0			0
$757$	1222.12			1.61443			0.807215	−	0.590258i	$-0.200974\pi$
$757$	1222.12			1.61443			0.807215	+	0.590258i	$0.200974\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−489.896	+	282.841i	−0.643752	+	0.371671i	−0.786059	−	0.618152i	$-0.787882\pi$
$761$	−489.896	+	282.841i	−0.643752	+	0.371671i	0.142306	+	0.989823i	$0.454548\pi$
$762$		0			0
$763$	−582.081	+	1008.19i	−0.762884	+	1.32135i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−852.036	−	491.923i	−1.11087	−	0.641360i
$768$		0			0
$769$	299.408	+	518.589i	0.389347	+	0.674368i	0.992362	−	0.123362i	$-0.0393675\pi$
$769$	299.408	+	518.589i	0.389347	+	0.674368i	−0.603015	+	0.797730i	$0.706034\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			446.970i			0.578228i	0.957295	+	0.289114i	$0.0933607\pi$
$773$			446.970i			0.578228i	−0.957295	+	0.289114i	$0.906639\pi$
$774$		0			0
$775$	−180.708			−0.233171
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−78.5814	+	45.3690i	−0.100875	+	0.0582400i
$780$		0			0
$781$	−120.231	+	208.246i	−0.153945	+	0.266640i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	335.823	+	193.887i	0.427800	+	0.246990i
$786$		0			0
$787$	−477.650	−	827.314i	−0.606925	−	1.05122i	−0.991744	−	0.128233i	$-0.959070\pi$
$787$	−477.650	−	827.314i	−0.606925	−	1.05122i	0.384819	−	0.922992i	$-0.374264\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			1097.93i			1.38803i
$792$		0			0
$793$	140.450			0.177112
$794$		0			0
$795$		0			0
$796$		0			0
$797$	192.141	−	110.933i	0.241081	−	0.139188i	−0.374593	−	0.927189i	$-0.622217\pi$
$797$	192.141	−	110.933i	0.241081	−	0.139188i	0.615673	+	0.788002i	$0.288884\pi$
$798$		0			0
$799$	−245.885	+	425.885i	−0.307740	+	0.533022i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−911.172	−	526.066i	−1.13471	−	0.655125i
$804$		0			0
$805$	−141.415	−	244.939i	−0.175671	−	0.304271i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1180.17i			1.45881i	0.684084	+	0.729403i	$0.260202\pi$
$809$			1180.17i			1.45881i	−0.684084	+	0.729403i	$0.739798\pi$
$810$		0			0
$811$	627.307			0.773499			0.386749	−	0.922185i	$-0.373598\pi$
$811$	627.307			0.773499			0.386749	+	0.922185i	$0.373598\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−17.1936	+	9.92670i	−0.0210964	+	0.0121800i
$816$		0			0
$817$	95.9230	−	166.144i	0.117409	−	0.203358i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−338.512	−	195.440i	−0.412317	−	0.238051i	0.279468	−	0.960155i	$-0.409842\pi$
$821$	−338.512	−	195.440i	−0.412317	−	0.238051i	−0.691785	+	0.722104i	$0.743175\pi$
$822$		0			0
$823$	163.100	+	282.497i	0.198177	+	0.343253i	0.947937	−	0.318456i	$-0.103164\pi$
$823$	163.100	+	282.497i	0.198177	+	0.343253i	−0.749760	+	0.661710i	$0.769831\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1103.70i			1.33458i	0.744799	+	0.667289i	$0.232545\pi$
$827$			1103.70i			1.33458i	−0.744799	+	0.667289i	$0.767455\pi$
$828$		0			0
$829$	−441.569			−0.532653			−0.266326	−	0.963883i	$-0.585810\pi$
$829$	−441.569			−0.532653			−0.266326	+	0.963883i	$0.585810\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2624.22	+	1515.09i	−3.15032	+	1.81884i
$834$		0			0
$835$	180.473	−	312.588i	0.216135	−	0.374357i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	436.443	+	251.980i	0.520194	+	0.300334i	0.737014	−	0.675878i	$-0.236235\pi$
$839$	436.443	+	251.980i	0.520194	+	0.300334i	−0.216820	+	0.976212i	$0.569569\pi$
$840$		0			0
$841$	−24.4134	−	42.2853i	−0.0290290	−	0.0502798i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			81.1815i			0.0960728i
$846$		0			0
$847$	1177.27			1.38993
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−287.259	+	165.849i	−0.337555	+	0.194887i
$852$		0			0
$853$	−472.415	+	818.247i	−0.553828	+	0.959258i	0.444166	+	0.895945i	$0.353500\pi$
$853$	−472.415	+	818.247i	−0.553828	+	0.959258i	−0.997994	+	0.0633136i	$0.979833\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−642.051	−	370.689i	−0.749185	−	0.432542i	0.0762145	−	0.997091i	$-0.475717\pi$
$857$	−642.051	−	370.689i	−0.749185	−	0.432542i	−0.825399	+	0.564549i	$0.809050\pi$
$858$		0			0
$859$	−680.631	−	1178.89i	−0.792352	−	1.37239i	−0.924507	−	0.381165i	$-0.875523\pi$
$859$	−680.631	−	1178.89i	−0.792352	−	1.37239i	0.132154	−	0.991229i	$-0.457811\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	805.003i		−	0.932796i	−0.884575	−	0.466398i	$-0.845551\pi$
$863$		−	805.003i		−	0.932796i	0.884575	−	0.466398i	$-0.154449\pi$
$864$		0			0
$865$	33.4617			0.0386841
$866$		0			0
$867$		0			0
$868$		0			0
$869$	594.981	−	343.513i	0.684673	−	0.395296i
$870$		0			0
$871$	222.435	−	385.268i	0.255378	−	0.442328i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−793.107	−	457.901i	−0.906408	−	0.523315i
$876$		0			0
$877$	−21.3461	−	36.9725i	−0.0243399	−	0.0421580i	0.853599	−	0.520931i	$-0.174415\pi$
$877$	−21.3461	−	36.9725i	−0.0243399	−	0.0421580i	−0.877939	+	0.478773i	$0.841082\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			264.567i			0.300303i	0.988663	+	0.150151i	$0.0479761\pi$
$881$			264.567i			0.300303i	−0.988663	+	0.150151i	$0.952024\pi$
$882$		0			0
$883$	393.338			0.445457			0.222728	−	0.974881i	$-0.428504\pi$
$883$	393.338			0.445457			0.222728	+	0.974881i	$0.428504\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1340.65	−	774.026i	1.51145	−	0.872634i	0.511535	−	0.859262i	$-0.329077\pi$
$887$	1340.65	−	774.026i	1.51145	−	0.872634i	0.999911	−	0.0133715i	$-0.00425642\pi$
$888$		0			0
$889$	−488.400	+	845.933i	−0.549381	+	0.951556i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	53.0221	+	30.6123i	0.0593752	+	0.0342803i
$894$		0			0
$895$	−215.654	−	373.523i	−0.240954	−	0.417344i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	225.165i		−	0.250461i
$900$		0			0
$901$	2450.02			2.71922
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−234.379	+	135.319i	−0.258982	+	0.149523i
$906$		0			0
$907$	688.746	−	1192.94i	0.759367	−	1.31526i	−0.183806	−	0.982962i	$-0.558842\pi$
$907$	688.746	−	1192.94i	0.759367	−	1.31526i	0.943174	−	0.332300i	$-0.107825\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1217.70	−	703.038i	−1.33666	−	0.771721i	−0.350350	−	0.936619i	$-0.613937\pi$
$911$	−1217.70	−	703.038i	−1.33666	−	0.771721i	−0.986311	+	0.164898i	$0.947271\pi$
$912$		0			0
$913$	−112.477	−	194.816i	−0.123195	−	0.213380i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	1023.35i		−	1.11597i
$918$		0			0
$919$	−479.992			−0.522299			−0.261149	−	0.965298i	$-0.584101\pi$
$919$	−479.992			−0.522299			−0.261149	+	0.965298i	$0.584101\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	153.083	−	88.3827i	0.165854	−	0.0957560i
$924$		0			0
$925$	−254.902	+	441.503i	−0.275570	+	0.477301i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	587.800	+	339.366i	0.632723	+	0.365303i	0.781806	−	0.623522i	$-0.214299\pi$
$929$	587.800	+	339.366i	0.632723	+	0.365303i	−0.149083	+	0.988825i	$0.547632\pi$
$930$		0			0
$931$	188.627	+	326.711i	0.202607	+	0.350925i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	661.362i		−	0.707339i
$936$		0			0
$937$	−666.600			−0.711420			−0.355710	−	0.934596i	$-0.615761\pi$
$937$	−666.600			−0.711420			−0.355710	+	0.934596i	$0.615761\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1037.46	+	598.978i	−1.10251	+	0.636533i	−0.936879	−	0.349655i	$-0.886299\pi$
$941$	−1037.46	+	598.978i	−1.10251	+	0.636533i	−0.165629	+	0.986188i	$0.552966\pi$
$942$		0			0
$943$	−184.823	+	320.123i	−0.195995	+	0.339473i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−393.793	−	227.357i	−0.415832	−	0.240081i	0.277460	−	0.960737i	$-0.410507\pi$
$947$	−393.793	−	227.357i	−0.415832	−	0.240081i	−0.693293	+	0.720656i	$0.743841\pi$
$948$		0			0
$949$	386.715	+	669.811i	0.407498	+	0.705807i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1060.16i		−	1.11245i	−0.831033	−	0.556224i	$-0.812250\pi$
$953$		−	1060.16i		−	1.11245i	0.831033	−	0.556224i	$-0.187750\pi$
$954$		0			0
$955$	−236.238			−0.247370
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2304.60	+	1330.56i	−2.40313	+	1.38745i
$960$		0			0
$961$	448.500	−	776.825i	0.466701	−	0.808350i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−73.9675	−	42.7051i	−0.0766502	−	0.0442540i
$966$		0			0
$967$	−827.657	−	1433.54i	−0.855902	−	1.48247i	−0.875805	−	0.482664i	$-0.839669\pi$
$967$	−827.657	−	1433.54i	−0.855902	−	1.48247i	0.0199032	−	0.999802i	$-0.493664\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	797.780i		−	0.821606i	−0.911724	−	0.410803i	$-0.865248\pi$
$971$		−	797.780i		−	0.821606i	0.911724	−	0.410803i	$-0.134752\pi$
$972$		0			0
$973$	−753.261			−0.774164
$974$		0			0
$975$		0			0
$976$		0			0
$977$	270.574	−	156.216i	0.276944	−	0.159894i	−0.355095	−	0.934830i	$-0.615552\pi$
$977$	270.574	−	156.216i	0.276944	−	0.159894i	0.632039	+	0.774936i	$0.282218\pi$
$978$		0			0
$979$	−580.350	+	1005.20i	−0.592799	+	1.02676i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−320.553	−	185.072i	−0.326097	−	0.188272i	0.328010	−	0.944674i	$-0.393622\pi$
$983$	−320.553	−	185.072i	−0.326097	−	0.188272i	−0.654107	+	0.756402i	$0.726955\pi$
$984$		0			0
$985$	103.500	+	179.267i	0.105076	+	0.181997i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	781.538i		−	0.790230i
$990$		0			0
$991$	1324.48			1.33651			0.668256	−	0.743931i	$-0.267041\pi$
$991$	1324.48			1.33651			0.668256	+	0.743931i	$0.267041\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−280.890	+	162.172i	−0.282302	+	0.162987i
$996$		0			0
$997$	−439.623	+	761.449i	−0.440946	+	0.763741i	−0.997760	−	0.0668966i	$-0.978690\pi$
$997$	−439.623	+	761.449i	−0.440946	+	0.763741i	0.556814	+	0.830637i	$0.312024\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.q.o.593.2		8
3.2	odd	2	inner	1296.3.q.o.593.3		8
4.3	odd	2		162.3.d.c.107.3		8
9.2	odd	6		1296.3.e.d.161.3		4
9.4	even	3	inner	1296.3.q.o.1025.3		8
9.5	odd	6	inner	1296.3.q.o.1025.2		8
9.7	even	3		1296.3.e.d.161.2		4
12.11	even	2		162.3.d.c.107.2		8
36.7	odd	6		162.3.b.b.161.1	✓	4
36.11	even	6		162.3.b.b.161.4	yes	4
36.23	even	6		162.3.d.c.53.3		8
36.31	odd	6		162.3.d.c.53.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
162.3.b.b.161.1	✓	4	36.7	odd	6
162.3.b.b.161.4	yes	4	36.11	even	6
162.3.d.c.53.2		8	36.31	odd	6
162.3.d.c.53.3		8	36.23	even	6
162.3.d.c.107.2		8	12.11	even	2
162.3.d.c.107.3		8	4.3	odd	2
1296.3.e.d.161.2		4	9.7	even	3
1296.3.e.d.161.3		4	9.2	odd	6
1296.3.q.o.593.2		8	1.1	even	1	trivial
1296.3.q.o.593.3		8	3.2	odd	2	inner
1296.3.q.o.1025.2		8	9.5	odd	6	inner
1296.3.q.o.1025.3		8	9.4	even	3	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.34486 + 0.776457i) q^{5} +(6.19615 - 10.7321i) q^{7} +O(q^{10})\)	\(q+(-1.34486 + 0.776457i) q^{5} +(6.19615 - 10.7321i) q^{7} +(-12.7279 - 7.34847i) q^{11} +(5.40192 + 9.35641i) q^{13} -28.9778i q^{17} -3.60770 q^{19} +(-12.7279 + 7.34847i) q^{23} +(-11.2942 + 19.5622i) q^{25} +(-24.3748 - 14.0728i) q^{29} +(4.00000 + 6.92820i) q^{31} +19.2442i q^{35} +22.5692 q^{37} +(21.7816 - 12.5756i) q^{41} +(-26.5885 + 46.0526i) q^{43} +(-14.6969 - 8.48528i) q^{47} +(-52.2846 - 90.5596i) q^{49} +84.5482i q^{53} +22.8231 q^{55} +(-78.8641 + 45.5322i) q^{59} +(6.50000 - 11.2583i) q^{61} +(-14.5297 - 8.38872i) q^{65} +(-20.5885 - 35.6603i) q^{67} -16.3613i q^{71} +71.5885 q^{73} +(-157.728 + 91.0645i) q^{77} +(-23.3731 + 40.4833i) q^{79} +(13.2555 + 7.65308i) q^{83} +(22.5000 + 38.9711i) q^{85} -78.9756i q^{89} +133.885 q^{91} +(4.85186 - 2.80122i) q^{95} +(-45.5692 + 78.9282i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{7}+O(q^{10}) \) 8 * q + 8 * q^7	\( 8 q + 8 q^{7} + 64 q^{13} - 112 q^{19} - 28 q^{25} + 32 q^{31} - 152 q^{37} - 88 q^{43} - 252 q^{49} + 432 q^{55} + 52 q^{61} - 40 q^{67} + 448 q^{73} + 104 q^{79} + 180 q^{85} - 176 q^{91} - 32 q^{97}+O(q^{100}) \) 8 * q + 8 * q^7 + 64 * q^13 - 112 * q^19 - 28 * q^25 + 32 * q^31 - 152 * q^37 - 88 * q^43 - 252 * q^49 + 432 * q^55 + 52 * q^61 - 40 * q^67 + 448 * q^73 + 104 * q^79 + 180 * q^85 - 176 * q^91 - 32 * q^97

Introduction

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