LMFDB - Embedded newform 1296.3.q.o.1025.2

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			1025.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1296.1025
Dual form			1296.3.q.o.593.2

$q$-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 $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

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	0.359448	−	0.933165i	$-0.382965\pi$
$5$	−1.34486	−	0.776457i	−0.268973	−	0.155291i	−0.628421	+	0.777874i	$0.716298\pi$
$6$		0			0
$7$	6.19615	+	10.7321i	0.885165	+	1.53315i	0.845525	+	0.533936i	$0.179288\pi$
$7$	6.19615	+	10.7321i	0.885165	+	1.53315i	0.0396398	+	0.999214i	$0.487379\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−12.7279	+	7.34847i	−1.15708	+	0.668043i	−0.950603	−	0.310408i	$-0.899534\pi$
$11$	−12.7279	+	7.34847i	−1.15708	+	0.668043i	−0.206480	+	0.978451i	$0.566201\pi$
$12$		0			0
$13$	5.40192	−	9.35641i	0.415533	−	0.719724i	−0.579952	−	0.814651i	$-0.696929\pi$
$13$	5.40192	−	9.35641i	0.415533	−	0.719724i	0.995484	+	0.0949274i	$0.0302619\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			28.9778i			1.70457i	0.523074	+	0.852287i	$0.324785\pi$
$17$			28.9778i			1.70457i	−0.523074	+	0.852287i	$0.675215\pi$
$18$		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	0.197099	−	0.980384i	$-0.436848\pi$
$23$	−12.7279	−	7.34847i	−0.553388	−	0.319499i	−0.750487	+	0.660885i	$0.770181\pi$
$24$		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	−0.857437	−	0.514588i	$-0.827945\pi$
$29$	−24.3748	+	14.0728i	−0.840510	+	0.485268i	0.0169278	+	0.999857i	$0.494611\pi$
$30$		0			0
$31$	4.00000	−	6.92820i	0.129032	−	0.223490i	−0.794270	−	0.607565i	$-0.792146\pi$
$31$	4.00000	−	6.92820i	0.129032	−	0.223490i	0.923302	+	0.384075i	$0.125480\pi$
$32$		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	0.741529	−	0.670921i	$-0.234101\pi$
$41$	21.7816	+	12.5756i	0.531259	+	0.306722i	−0.210270	+	0.977643i	$0.567434\pi$
$42$		0			0
$43$	−26.5885	−	46.0526i	−0.618336	−	1.07099i	−0.989789	−	0.142538i	$-0.954474\pi$
$43$	−26.5885	−	46.0526i	−0.618336	−	1.07099i	0.371453	−	0.928452i	$-0.378860\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−14.6969	+	8.48528i	−0.312701	+	0.180538i	−0.648134	−	0.761526i	$-0.724451\pi$
$47$	−14.6969	+	8.48528i	−0.312701	+	0.180538i	0.335434	+	0.942064i	$0.391117\pi$
$48$		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	$-0.706090\pi$
$53$		−	84.5482i		−	1.59525i	0.603154	−	0.797625i	$-0.293910\pi$
$54$		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	−0.350367	−	0.936613i	$-0.613943\pi$
$59$	−78.8641	−	45.5322i	−1.33668	−	0.771733i	−0.986314	+	0.164880i	$0.947276\pi$
$60$		0			0
$61$	6.50000	+	11.2583i	0.106557	+	0.184563i	0.914373	−	0.404872i	$-0.132684\pi$
$61$	6.50000	+	11.2583i	0.106557	+	0.184563i	−0.807816	+	0.589435i	$0.799351\pi$
$62$		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	−0.977769	−	0.209687i	$-0.932756\pi$
$67$	−20.5885	+	35.6603i	−0.307290	+	0.532243i	0.670478	+	0.741929i	$0.266089\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			16.3613i			0.230442i	0.993340	+	0.115221i	$0.0367575\pi$
$71$			16.3613i			0.230442i	−0.993340	+	0.115221i	$0.963242\pi$
$72$		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	0.679323	−	0.733839i	$-0.262273\pi$
$79$	−23.3731	−	40.4833i	−0.295862	−	0.512447i	−0.975185	+	0.221392i	$0.928940\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.2555	−	7.65308i	0.159705	−	0.0922057i	−0.418017	−	0.908439i	$-0.637275\pi$
$83$	13.2555	−	7.65308i	0.159705	−	0.0922057i	0.577723	+	0.816233i	$0.303942\pi$
$84$		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	$0.146329\pi$
$89$			78.9756i			0.887367i	−0.896184	+	0.443683i	$0.853671\pi$
$90$		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	0.529617	−	0.848237i	$-0.322336\pi$
$97$	−45.5692	−	78.9282i	−0.469786	−	0.813693i	−0.999403	+	0.0345438i	$0.989002\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−76.6313	+	44.2431i	−0.758726	+	0.438051i	−0.828838	−	0.559488i	$-0.810998\pi$
$101$	−76.6313	+	44.2431i	−0.758726	+	0.438051i	0.0701121	+	0.997539i	$0.477664\pi$
$102$		0			0
$103$	−76.3538	+	132.249i	−0.741299	+	1.28397i	0.210605	+	0.977571i	$0.432457\pi$
$103$	−76.3538	+	132.249i	−0.741299	+	1.28397i	−0.951904	+	0.306397i	$0.900877\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			60.0062i			0.560805i	0.959882	+	0.280403i	$0.0904680\pi$
$107$			60.0062i			0.560805i	−0.959882	+	0.280403i	$0.909532\pi$
$108$		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	0.120473	−	0.992717i	$-0.461559\pi$
$113$	−76.7279	−	44.2989i	−0.679008	−	0.392025i	−0.799481	+	0.600691i	$0.794892\pi$
$114$		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	0.747475	−	0.664290i	$-0.231266\pi$
$131$	71.5157	+	41.2896i	0.545921	+	0.315188i	−0.201554	+	0.979477i	$0.564599\pi$
$132$		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	−0.989279	−	0.146036i	$-0.953349\pi$
$137$	−185.970	+	107.370i	−1.35745	+	0.783723i	−0.368169	+	0.929759i	$0.620015\pi$
$138$		0			0
$139$	−30.3923	+	52.6410i	−0.218650	+	0.378712i	−0.954395	−	0.298546i	$-0.903499\pi$
$139$	−30.3923	+	52.6410i	−0.218650	+	0.378712i	0.735746	+	0.677258i	$0.236832\pi$
$140$		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	0.408300	−	0.912848i	$-0.366122\pi$
$149$	−26.5369	−	15.3211i	−0.178100	−	0.102826i	−0.586400	+	0.810022i	$0.699455\pi$
$150$		0			0
$151$	16.0000	+	27.7128i	0.105960	+	0.183529i	0.914130	−	0.405421i	$-0.132875\pi$
$151$	16.0000	+	27.7128i	0.105960	+	0.183529i	−0.808170	+	0.588949i	$0.799542\pi$
$152$		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	0.127435	+	0.991847i	$0.459326\pi$
$157$	−124.854	+	216.253i	−0.795247	+	1.37741i	−0.922682	+	0.385562i	$0.874008\pi$
$158$		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	−0.243601	−	0.969876i	$-0.578329\pi$
$167$	−201.292	−	116.216i	−1.20534	−	0.695902i	−0.961737	+	0.273973i	$0.911662\pi$
$168$		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	−0.552963	−	0.833206i	$-0.686503\pi$
$173$	−18.6608	+	10.7738i	−0.107866	+	0.0622765i	0.445096	+	0.895483i	$0.353169\pi$
$174$		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	$-0.717341\pi$
$179$		−	277.741i		−	1.55162i	0.630964	−	0.775812i	$-0.282659\pi$
$180$		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	−0.113760	−	0.993508i	$-0.536289\pi$
$191$	131.745	−	76.0629i	0.689764	−	0.398235i	0.803523	+	0.595273i	$0.202956\pi$
$192$		0			0
$193$	27.5000	−	47.6314i	0.142487	−	0.246795i	−0.785946	−	0.618296i	$-0.787823\pi$
$193$	27.5000	−	47.6314i	0.142487	−	0.246795i	0.928433	+	0.371501i	$0.121157\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			133.298i			0.676638i	0.941031	+	0.338319i	$0.109858\pi$
$197$			133.298i			0.676638i	−0.941031	+	0.338319i	$0.890142\pi$
$198$		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	−0.950843	−	0.309673i	$-0.899780\pi$
$211$	−43.7269	+	75.7372i	−0.207236	+	0.358944i	0.743607	+	0.668617i	$0.233114\pi$
$212$		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	0.999999	−	0.00105540i	$-0.000335944\pi$
$223$	111.296	+	192.771i	0.499086	+	0.864442i	−0.500914	+	0.865497i	$0.667003\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	260.079	−	150.157i	1.14572	−	0.661484i	0.197882	−	0.980226i	$-0.436594\pi$
$227$	260.079	−	150.157i	1.14572	−	0.661484i	0.947842	+	0.318742i	$0.103260\pi$
$228$		0			0
$229$	−114.971	+	199.136i	−0.502057	+	0.869589i	0.497940	+	0.867212i	$0.334090\pi$
$229$	−114.971	+	199.136i	−0.502057	+	0.869589i	−0.999997	+	0.00237731i	$0.999243\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	116.246i		−	0.498908i	−0.968387	−	0.249454i	$-0.919749\pi$
$233$		−	116.246i		−	0.498908i	0.968387	−	0.249454i	$-0.0802512\pi$
$234$		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	0.767332	−	0.641250i	$-0.221584\pi$
$239$	142.362	+	82.1930i	0.595659	+	0.343904i	−0.171673	+	0.985154i	$0.554917\pi$
$240$		0			0
$241$	40.6558	+	70.4179i	0.168696	+	0.292190i	0.937962	−	0.346739i	$-0.112711\pi$
$241$	40.6558	+	70.4179i	0.168696	+	0.292190i	−0.769265	+	0.638929i	$0.779378\pi$
$242$		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	$-0.710296\pi$
$251$		−	396.371i		−	1.57917i	0.613642	−	0.789584i	$-0.289704\pi$
$252$		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	0.0919879	−	0.995760i	$-0.470678\pi$
$257$	−186.164	−	107.482i	−0.724372	−	0.418216i	−0.816360	+	0.577544i	$0.804011\pi$
$258$		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	−0.921747	−	0.387792i	$-0.873238\pi$
$263$	−275.304	+	158.947i	−1.04678	+	0.604360i	−0.125036	+	0.992152i	$0.539905\pi$
$264$		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	$0.126696\pi$
$269$			208.528i			0.775199i	−0.921828	+	0.387599i	$0.873304\pi$
$270$		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	−0.830223	−	0.557432i	$-0.811787\pi$
$277$	−248.708	−	430.774i	−0.897862	−	1.55514i	−0.0676390	−	0.997710i	$-0.521547\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	186.427	−	107.634i	0.663442	−	0.383039i	−0.130145	−	0.991495i	$-0.541544\pi$
$281$	186.427	−	107.634i	0.663442	−	0.383039i	0.793587	+	0.608456i	$0.208211\pi$
$282$		0			0
$283$	−148.354	+	256.956i	−0.524218	+	0.907973i	0.475384	+	0.879778i	$0.342309\pi$
$283$	−148.354	+	256.956i	−0.524218	+	0.907973i	−0.999602	+	0.0281946i	$0.991024\pi$
$284$		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	0.910545	−	0.413410i	$-0.135662\pi$
$293$	295.284	+	170.482i	1.00779	+	0.581850i	0.0972490	+	0.995260i	$0.468996\pi$
$294$		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	0.747666	−	0.664074i	$-0.231174\pi$
$311$	169.929	+	98.1083i	0.546394	+	0.315461i	−0.201272	+	0.979535i	$0.564508\pi$
$312$		0			0
$313$	73.3616	+	127.066i	0.234382	+	0.405962i	0.959093	−	0.283092i	$-0.0913600\pi$
$313$	73.3616	+	127.066i	0.234382	+	0.405962i	−0.724711	+	0.689053i	$0.758027\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−41.6908	+	24.0702i	−0.131517	+	0.0759311i	−0.564315	−	0.825560i	$-0.690860\pi$
$317$	−41.6908	+	24.0702i	−0.131517	+	0.0759311i	0.432798	+	0.901491i	$0.357526\pi$
$318$		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	0.781080	−	0.624430i	$-0.214669\pi$
$331$	−49.7269	−	86.1295i	−0.150232	−	0.260210i	−0.931313	+	0.364220i	$0.881335\pi$
$332$		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	−0.356406	−	0.934331i	$-0.615998\pi$
$337$	212.631	−	368.287i	0.630952	−	1.09284i	0.987358	−	0.158509i	$-0.0506687\pi$
$338$		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	0.450838	−	0.892606i	$-0.351125\pi$
$347$	−33.5768	−	19.3856i	−0.0967630	−	0.0558662i	−0.547601	+	0.836740i	$0.684459\pi$
$348$		0			0
$349$	325.985	+	564.622i	0.934053	+	1.61783i	0.776314	+	0.630347i	$0.217087\pi$
$349$	325.985	+	564.622i	0.934053	+	1.61783i	0.157739	+	0.987481i	$0.449579\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.03625	+	0.598281i	−0.00293556	+	0.00169485i	−0.501467	−	0.865177i	$-0.667206\pi$
$353$	−1.03625	+	0.598281i	−0.00293556	+	0.00169485i	0.498532	+	0.866872i	$0.333873\pi$
$354$		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	$0.267326\pi$
$359$			534.573i			1.48906i	−0.667589	+	0.744530i	$0.732674\pi$
$360$		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	0.988066	−	0.154032i	$-0.0492259\pi$
$367$	132.354	+	229.244i	0.360637	+	0.624642i	−0.627429	+	0.778674i	$0.715893\pi$
$368$		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	−0.909690	−	0.415288i	$-0.863681\pi$
$373$	−35.5077	+	61.5012i	−0.0951950	+	0.164883i	0.814495	+	0.580171i	$0.197014\pi$
$374$		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	0.903565	−	0.428452i	$-0.140941\pi$
$383$	376.986	+	217.653i	0.984297	+	0.568284i	0.0807324	+	0.996736i	$0.474274\pi$
$384$		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	−0.557882	−	0.829920i	$-0.688386\pi$
$389$	−45.9374	+	26.5220i	−0.118091	+	0.0681799i	0.439791	+	0.898100i	$0.355053\pi$
$390$		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	0.989322	−	0.145747i	$-0.0465586\pi$
$401$	544.463	+	314.346i	1.35776	+	0.783904i	0.368440	+	0.929651i	$0.379892\pi$
$402$		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	0.327672	+	0.944791i	$0.393736\pi$
$409$	−267.640	+	463.567i	−0.654377	+	1.13341i	−0.982050	+	0.188623i	$0.939597\pi$
$410$		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	0.842222	−	0.539130i	$-0.181247\pi$
$419$	333.705	+	192.665i	0.796433	+	0.459821i	−0.0457895	+	0.998951i	$0.514580\pi$
$420$		0			0
$421$	377.660	+	654.126i	0.897054	+	1.55374i	0.831242	+	0.555910i	$0.187630\pi$
$421$	377.660	+	654.126i	0.897054	+	1.55374i	0.0658113	+	0.997832i	$0.479036\pi$
$422$		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	$0.0398777\pi$
$431$			107.709i			0.249904i	−0.992163	+	0.124952i	$0.960122\pi$
$432$		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	−0.999033	−	0.0439580i	$-0.986003\pi$
$439$	−236.000	−	408.764i	−0.537585	−	0.931125i	0.461448	−	0.887167i	$-0.347330\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	728.516	−	420.609i	1.64451	−	0.949455i	0.665302	−	0.746575i	$-0.268303\pi$
$443$	728.516	−	420.609i	1.64451	−	0.949455i	0.979203	−	0.202881i	$-0.0650304\pi$
$444$		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	$-0.859843\pi$
$449$		−	382.751i		−	0.852453i	0.904616	−	0.426227i	$-0.140157\pi$
$450$		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	0.632812	−	0.774305i	$-0.281900\pi$
$457$	−161.852	−	280.336i	−0.354162	−	0.613426i	−0.986974	+	0.160879i	$0.948567\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	555.582	−	320.765i	1.20517	−	0.695803i	0.243467	−	0.969909i	$-0.421715\pi$
$461$	555.582	−	320.765i	1.20517	−	0.695803i	0.961700	+	0.274106i	$0.0883818\pi$
$462$		0			0
$463$	64.7461	−	112.144i	0.139840	−	0.242211i	−0.787596	−	0.616192i	$-0.788674\pi$
$463$	64.7461	−	112.144i	0.139840	−	0.242211i	0.927436	+	0.373982i	$0.122008\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	672.730i		−	1.44054i	−0.693696	−	0.720268i	$-0.744019\pi$
$467$		−	672.730i		−	1.44054i	0.693696	−	0.720268i	$-0.255981\pi$
$468$		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	−0.609238	−	0.792987i	$-0.708525\pi$
$479$	−108.786	+	62.8074i	−0.227110	+	0.131122i	0.382128	+	0.924109i	$0.375191\pi$
$480$		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	0.842601	−	0.538538i	$-0.181023\pi$
$491$	391.579	+	226.078i	0.797514	+	0.460445i	−0.0450873	+	0.998983i	$0.514357\pi$
$492$		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	−0.795607	−	0.605813i	$-0.792848\pi$
$499$	63.2961	−	109.632i	0.126846	−	0.219704i	0.922453	+	0.386109i	$0.126181\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			296.822i			0.590103i	0.955481	+	0.295051i	$0.0953367\pi$
$503$			296.822i			0.590103i	−0.955481	+	0.295051i	$0.904663\pi$
$504$		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	−0.174265	−	0.984699i	$-0.555755\pi$
$509$	−567.113	−	327.423i	−1.11417	−	0.643267i	−0.939907	+	0.341432i	$0.889088\pi$
$510$		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	$0.230374\pi$
$521$			690.006i			1.32439i	−0.749333	+	0.662193i	$0.769626\pi$
$522$		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	−0.948143	−	0.317843i	$-0.897041\pi$
$547$	−409.885	−	709.941i	−0.749332	−	1.29788i	0.198811	−	0.980038i	$-0.436292\pi$
$548$		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	$0.0360247\pi$
$557$			125.808i			0.225867i	−0.993603	+	0.112934i	$0.963975\pi$
$558$		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	0.0547261	−	0.998501i	$-0.482571\pi$
$563$	−440.625	−	254.395i	−0.782638	−	0.451856i	−0.837365	+	0.546645i	$0.815905\pi$
$564$		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	−0.424474	−	0.905440i	$-0.639541\pi$
$569$	83.8832	−	48.4300i	0.147422	−	0.0851143i	0.571897	+	0.820326i	$0.306208\pi$
$570$		0			0
$571$	−227.100	+	393.349i	−0.397723	+	0.688877i	−0.993445	−	0.114314i	$-0.963533\pi$
$571$	−227.100	+	393.349i	−0.397723	+	0.688877i	0.595721	+	0.803191i	$0.296866\pi$
$572$		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	−0.776286	−	0.630381i	$-0.782899\pi$
$587$	−363.061	+	209.614i	−0.618503	+	0.357093i	0.157783	+	0.987474i	$0.449565\pi$
$588$		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	$0.0895323\pi$
$593$			329.210i			0.555160i	−0.960703	+	0.277580i	$0.910468\pi$
$594$		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	0.360867	−	0.932617i	$-0.382481\pi$
$599$	−159.556	−	92.1197i	−0.266371	−	0.153789i	−0.627237	+	0.778828i	$0.715814\pi$
$600$		0			0
$601$	−109.208	−	189.153i	−0.181710	−	0.314731i	0.760753	−	0.649041i	$-0.224830\pi$
$601$	−109.208	−	189.153i	−0.181710	−	0.314731i	−0.942463	+	0.334311i	$0.891497\pi$
$602$		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	−0.861765	−	0.507308i	$-0.830641\pi$
$607$	5.13467	−	8.89350i	0.00845909	−	0.0146516i	0.870224	+	0.492656i	$0.163974\pi$
$608$		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	0.975357	−	0.220633i	$-0.0708124\pi$
$617$	784.800	+	453.104i	1.27196	+	0.734367i	0.296604	+	0.955000i	$0.404146\pi$
$618$		0			0
$619$	380.823	+	659.605i	0.615223	+	1.06560i	0.990345	+	0.138622i	$0.0442674\pi$
$619$	380.823	+	659.605i	0.615223	+	1.06560i	−0.375122	+	0.926975i	$0.622399\pi$
$620$		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	−0.836491	−	0.547981i	$-0.815396\pi$
$641$	−500.089	+	288.727i	−0.780170	+	0.450431i	0.0563204	+	0.998413i	$0.482063\pi$
$642$		0			0
$643$	65.0615	−	112.690i	0.101184	−	0.175256i	−0.810989	−	0.585062i	$-0.801070\pi$
$643$	65.0615	−	112.690i	0.101184	−	0.175256i	0.912173	+	0.409806i	$0.134403\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			985.467i			1.52313i	0.648087	+	0.761567i	$0.275570\pi$
$647$			985.467i			1.52313i	−0.648087	+	0.761567i	$0.724430\pi$
$648$		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	0.0459644	−	0.998943i	$-0.485364\pi$
$653$	−519.895	−	300.161i	−0.796163	−	0.459665i	−0.842128	+	0.539278i	$0.818697\pi$
$654$		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	−0.510208	−	0.860051i	$-0.670432\pi$
$659$	−13.5004	+	7.79445i	−0.0204862	+	0.0118277i	0.489722	+	0.871879i	$0.337098\pi$
$660$		0			0
$661$	−203.915	+	353.192i	−0.308495	+	0.534329i	−0.978033	−	0.208448i	$-0.933159\pi$
$661$	−203.915	+	353.192i	−0.308495	+	0.534329i	0.669538	+	0.742778i	$0.266492\pi$
$662$		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	0.986700	−	0.162550i	$-0.0519720\pi$
$673$	237.285	+	410.989i	0.352577	+	0.610682i	−0.634123	+	0.773232i	$0.718639\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−244.346	+	141.073i	−0.360925	+	0.208380i	−0.669486	−	0.742824i	$-0.733486\pi$
$677$	−244.346	+	141.073i	−0.360925	+	0.208380i	0.308562	+	0.951204i	$0.400152\pi$
$678$		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	$0.292335\pi$
$683$			1085.46i			1.58926i	−0.607095	+	0.794629i	$0.707665\pi$
$684$		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	−0.957215	−	0.289378i	$-0.906552\pi$
$691$	−503.888	−	872.760i	−0.729216	−	1.26304i	0.227999	−	0.973661i	$-0.426782\pi$
$692$		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	$-0.950595\pi$
$701$		−	216.731i		−	0.309174i	0.987979	−	0.154587i	$-0.0494047\pi$
$702$		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	−0.967997	−	0.250961i	$-0.919253\pi$
$709$	−497.248	−	861.259i	−0.701337	−	1.21475i	0.266660	−	0.963791i	$-0.414080\pi$
$710$		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	$-0.923812\pi$
$719$		−	340.912i		−	0.474148i	0.971492	−	0.237074i	$-0.0761884\pi$
$720$		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	−0.978366	−	0.206881i	$-0.933669\pi$
$727$	−485.888	−	841.583i	−0.668347	−	1.15761i	0.310019	−	0.950730i	$-0.399665\pi$
$728$		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	−0.933540	−	0.358473i	$-0.883297\pi$
$733$	−114.585	+	198.466i	−0.156323	+	0.270759i	0.777217	+	0.629233i	$0.216631\pi$
$734$		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	0.605554	−	0.795804i	$-0.292951\pi$
$743$	162.825	+	94.0071i	0.219145	+	0.126524i	−0.386409	+	0.922327i	$0.626285\pi$
$744$		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	−0.128808	+	0.991670i	$0.541115\pi$
$751$	−693.335	+	1200.89i	−0.923215	+	1.59906i	−0.794407	+	0.607386i	$0.792218\pi$
$752$		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	0.142306	−	0.989823i	$-0.454548\pi$
$761$	−489.896	−	282.841i	−0.643752	−	0.371671i	−0.786059	+	0.618152i	$0.787882\pi$
$762$		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	−0.603015	−	0.797730i	$-0.706034\pi$
$769$	299.408	−	518.589i	0.389347	−	0.674368i	0.992362	+	0.123362i	$0.0393675\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	446.970i		−	0.578228i	−0.957295	−	0.289114i	$-0.906639\pi$
$773$		−	446.970i		−	0.578228i	0.957295	−	0.289114i	$-0.0933607\pi$
$774$		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	0.384819	+	0.922992i	$0.374264\pi$
$787$	−477.650	+	827.314i	−0.606925	+	1.05122i	−0.991744	+	0.128233i	$0.959070\pi$
$788$		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	0.615673	−	0.788002i	$-0.288884\pi$
$797$	192.141	+	110.933i	0.241081	+	0.139188i	−0.374593	+	0.927189i	$0.622217\pi$
$798$		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	$-0.739798\pi$
$809$		−	1180.17i		−	1.45881i	0.684084	−	0.729403i	$-0.260202\pi$
$810$		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	−0.691785	−	0.722104i	$-0.743175\pi$
$821$	−338.512	+	195.440i	−0.412317	+	0.238051i	0.279468	+	0.960155i	$0.409842\pi$
$822$		0			0
$823$	163.100	−	282.497i	0.198177	−	0.343253i	−0.749760	−	0.661710i	$-0.769831\pi$
$823$	163.100	−	282.497i	0.198177	−	0.343253i	0.947937	+	0.318456i	$0.103164\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1103.70i		−	1.33458i	−0.744799	−	0.667289i	$-0.767455\pi$
$827$		−	1103.70i		−	1.33458i	0.744799	−	0.667289i	$-0.232545\pi$
$828$		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	−0.216820	−	0.976212i	$-0.569569\pi$
$839$	436.443	−	251.980i	0.520194	−	0.300334i	0.737014	+	0.675878i	$0.236235\pi$
$840$		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	−0.997994	−	0.0633136i	$-0.979833\pi$
$853$	−472.415	−	818.247i	−0.553828	−	0.959258i	0.444166	−	0.895945i	$-0.353500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−642.051	+	370.689i	−0.749185	+	0.432542i	−0.825399	−	0.564549i	$-0.809050\pi$
$857$	−642.051	+	370.689i	−0.749185	+	0.432542i	0.0762145	+	0.997091i	$0.475717\pi$
$858$		0			0
$859$	−680.631	+	1178.89i	−0.792352	+	1.37239i	0.132154	+	0.991229i	$0.457811\pi$
$859$	−680.631	+	1178.89i	−0.792352	+	1.37239i	−0.924507	+	0.381165i	$0.875523\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			805.003i			0.932796i	0.884575	+	0.466398i	$0.154449\pi$
$863$			805.003i			0.932796i	−0.884575	+	0.466398i	$0.845551\pi$
$864$		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	−0.877939	−	0.478773i	$-0.841082\pi$
$877$	−21.3461	+	36.9725i	−0.0243399	+	0.0421580i	0.853599	+	0.520931i	$0.174415\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	264.567i		−	0.300303i	−0.988663	−	0.150151i	$-0.952024\pi$
$881$		−	264.567i		−	0.300303i	0.988663	−	0.150151i	$-0.0479761\pi$
$882$		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	0.999911	+	0.0133715i	$0.00425642\pi$
$887$	1340.65	+	774.026i	1.51145	+	0.872634i	0.511535	+	0.859262i	$0.329077\pi$
$888$		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	0.943174	+	0.332300i	$0.107825\pi$
$907$	688.746	+	1192.94i	0.759367	+	1.31526i	−0.183806	+	0.982962i	$0.558842\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1217.70	+	703.038i	−1.33666	+	0.771721i	−0.986311	−	0.164898i	$-0.947271\pi$
$911$	−1217.70	+	703.038i	−1.33666	+	0.771721i	−0.350350	+	0.936619i	$0.613937\pi$
$912$		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	−0.149083	−	0.988825i	$-0.547632\pi$
$929$	587.800	−	339.366i	0.632723	−	0.365303i	0.781806	+	0.623522i	$0.214299\pi$
$930$		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	−0.165629	−	0.986188i	$-0.552966\pi$
$941$	−1037.46	−	598.978i	−1.10251	−	0.636533i	−0.936879	+	0.349655i	$0.886299\pi$
$942$		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	−0.693293	−	0.720656i	$-0.743841\pi$
$947$	−393.793	+	227.357i	−0.415832	+	0.240081i	0.277460	+	0.960737i	$0.410507\pi$
$948$		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	$0.187750\pi$
$953$			1060.16i			1.11245i	−0.831033	+	0.556224i	$0.812250\pi$
$954$		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	0.0199032	+	0.999802i	$0.493664\pi$
$967$	−827.657	+	1433.54i	−0.855902	+	1.48247i	−0.875805	+	0.482664i	$0.839669\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			797.780i			0.821606i	0.911724	+	0.410803i	$0.134752\pi$
$971$			797.780i			0.821606i	−0.911724	+	0.410803i	$0.865248\pi$
$972$		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	0.632039	−	0.774936i	$-0.282218\pi$
$977$	270.574	+	156.216i	0.276944	+	0.159894i	−0.355095	+	0.934830i	$0.615552\pi$
$978$		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	−0.654107	−	0.756402i	$-0.726955\pi$
$983$	−320.553	+	185.072i	−0.326097	+	0.188272i	0.328010	+	0.944674i	$0.393622\pi$
$984$		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	0.556814	−	0.830637i	$-0.312024\pi$
$997$	−439.623	−	761.449i	−0.440946	−	0.763741i	−0.997760	+	0.0668966i	$0.978690\pi$
$998$		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.1025.2		8
3.2	odd	2	inner	1296.3.q.o.1025.3		8
4.3	odd	2		162.3.d.c.53.3		8
9.2	odd	6	inner	1296.3.q.o.593.2		8
9.4	even	3		1296.3.e.d.161.3		4
9.5	odd	6		1296.3.e.d.161.2		4
9.7	even	3	inner	1296.3.q.o.593.3		8
12.11	even	2		162.3.d.c.53.2		8
36.7	odd	6		162.3.d.c.107.2		8
36.11	even	6		162.3.d.c.107.3		8
36.23	even	6		162.3.b.b.161.1	✓	4
36.31	odd	6		162.3.b.b.161.4	yes	4

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