LMFDB - Embedded newform 588.3.c.j.197.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,3,Mod(197,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("588.197");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	588.c (of order $2$, degree $1$, 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:	$16.0218395444$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.6018425749504.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 18x^{6} + 123x^{4} + 304x^{2} + 441 $ x^8 + 18x^6 + 123x^4 + 304*x^2 + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			197.6
Root			$-0.707107 + 2.83573i$ of defining polynomial
Character	$\chi$	$=$	588.197
Dual form			588.3.c.j.197.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 + 8.63374i) q^{9} +O(q^{10})$	$q+(1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 + 8.63374i) q^{9} +2.45228i q^{11} -6.53953 q^{13} +(2.08276 - 1.55805i) q^{15} +18.7484i q^{17} -13.6106 q^{19} +41.9933i q^{23} +24.2483 q^{25} +(-25.3071 + 9.41009i) q^{27} -22.8358i q^{29} -8.48528 q^{31} +(-5.89094 + 4.40682i) q^{33} -41.7449 q^{37} +(-11.7517 - 15.7094i) q^{39} +71.1998i q^{41} +59.7449 q^{43} +(7.48556 + 2.20341i) q^{45} +69.7916i q^{47} +(-45.0380 + 33.6914i) q^{51} -62.3768i q^{53} +2.12616 q^{55} +(-24.4586 - 32.6957i) q^{57} -57.1123i q^{59} -59.2166 q^{61} +5.66986i q^{65} +66.0000 q^{67} +(-100.877 + 75.4631i) q^{69} -53.7938i q^{71} -95.4546 q^{73} +(43.5748 + 58.2499i) q^{75} -40.9932 q^{79} +(-68.0828 - 43.8832i) q^{81} +23.0835i q^{83} +16.2551 q^{85} +(54.8569 - 41.0367i) q^{87} -5.52795i q^{89} +(-15.2483 - 20.3836i) q^{93} +11.8006i q^{95} +85.5648 q^{97} +(-21.1724 - 6.23219i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 32 q^{15} + 48 q^{25} + 104 q^{37} - 240 q^{39} + 40 q^{43} - 44 q^{51} - 220 q^{57} + 528 q^{67} + 256 q^{79} - 496 q^{81} + 568 q^{85} + 24 q^{93} - 656 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 32 * q^15 + 48 * q^25 + 104 * q^37 - 240 * q^39 + 40 * q^43 - 44 * q^51 - 220 * q^57 + 528 * q^67 + 256 * q^79 - 496 * q^81 + 568 * q^85 + 24 * q^93 - 656 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	1.79703	+	2.40223i	0.599009	+	0.800742i
$3$	1.79703	+	2.40223i	0.599009	+	0.800742i
$4$		0			0
$5$		−	0.867013i		−	0.173403i	−0.996234	−	0.0867013i	$-0.972367\pi$
$5$		−	0.867013i		−	0.173403i	0.996234	−	0.0867013i	$-0.0276326\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−2.54138	+	8.63374i	−0.282376	+	0.959304i
$10$		0			0
$11$			2.45228i			0.222935i	0.993768	+	0.111467i	$0.0355551\pi$
$11$			2.45228i			0.222935i	−0.993768	+	0.111467i	$0.964445\pi$
$12$		0			0
$13$	−6.53953			−0.503041			−0.251520	−	0.967852i	$-0.580931\pi$
$13$	−6.53953			−0.503041			−0.251520	+	0.967852i	$0.580931\pi$
$14$		0			0
$15$	2.08276	−	1.55805i	0.138851	−	0.103870i
$16$		0			0
$17$			18.7484i			1.10285i	0.834225	+	0.551424i	$0.185915\pi$
$17$			18.7484i			1.10285i	−0.834225	+	0.551424i	$0.814085\pi$
$18$		0			0
$19$	−13.6106			−0.716347			−0.358174	−	0.933655i	$-0.616600\pi$
$19$	−13.6106			−0.716347			−0.358174	+	0.933655i	$0.616600\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			41.9933i			1.82579i	0.408189	+	0.912897i	$0.366160\pi$
$23$			41.9933i			1.82579i	−0.408189	+	0.912897i	$0.633840\pi$
$24$		0			0
$25$	24.2483			0.969932
$26$		0			0
$27$	−25.3071	+	9.41009i	−0.937301	+	0.348522i
$28$		0			0
$29$		−	22.8358i		−	0.787443i	−0.919230	−	0.393722i	$-0.871187\pi$
$29$		−	22.8358i		−	0.787443i	0.919230	−	0.393722i	$-0.128813\pi$
$30$		0			0
$31$	−8.48528			−0.273719			−0.136859	−	0.990590i	$-0.543701\pi$
$31$	−8.48528			−0.273719			−0.136859	+	0.990590i	$0.543701\pi$
$32$		0			0
$33$	−5.89094	+	4.40682i	−0.178513	+	0.133540i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−41.7449			−1.12824			−0.564120	−	0.825693i	$-0.690784\pi$
$37$	−41.7449			−1.12824			−0.564120	+	0.825693i	$0.690784\pi$
$38$		0			0
$39$	−11.7517	−	15.7094i	−0.301326	−	0.402806i
$40$		0			0
$41$			71.1998i			1.73658i	0.496057	+	0.868290i	$0.334781\pi$
$41$			71.1998i			1.73658i	−0.496057	+	0.868290i	$0.665219\pi$
$42$		0			0
$43$	59.7449			1.38942			0.694708	−	0.719292i	$-0.255534\pi$
$43$	59.7449			1.38942			0.694708	+	0.719292i	$0.255534\pi$
$44$		0			0
$45$	7.48556	+	2.20341i	0.166346	+	0.0489647i
$46$		0			0
$47$			69.7916i			1.48493i	0.669886	+	0.742464i	$0.266343\pi$
$47$			69.7916i			1.48493i	−0.669886	+	0.742464i	$0.733657\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−45.0380	+	33.6914i	−0.883097	+	0.660617i
$52$		0			0
$53$		−	62.3768i		−	1.17692i	−0.808526	−	0.588461i	$-0.799734\pi$
$53$		−	62.3768i		−	1.17692i	0.808526	−	0.588461i	$-0.200266\pi$
$54$		0			0
$55$	2.12616			0.0386575
$56$		0			0
$57$	−24.4586	−	32.6957i	−0.429099	−	0.573609i
$58$		0			0
$59$		−	57.1123i		−	0.968005i	−0.875067	−	0.484002i	$-0.839183\pi$
$59$		−	57.1123i		−	0.968005i	0.875067	−	0.484002i	$-0.160817\pi$
$60$		0			0
$61$	−59.2166			−0.970763			−0.485382	−	0.874302i	$-0.661319\pi$
$61$	−59.2166			−0.970763			−0.485382	+	0.874302i	$0.661319\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			5.66986i			0.0872286i
$66$		0			0
$67$	66.0000			0.985075			0.492537	−	0.870291i	$-0.336070\pi$
$67$	66.0000			0.985075			0.492537	+	0.870291i	$0.336070\pi$
$68$		0			0
$69$	−100.877	+	75.4631i	−1.46199	+	1.09367i
$70$		0			0
$71$		−	53.7938i		−	0.757660i	−0.925466	−	0.378830i	$-0.876327\pi$
$71$		−	53.7938i		−	0.757660i	0.925466	−	0.378830i	$-0.123673\pi$
$72$		0			0
$73$	−95.4546			−1.30760			−0.653798	−	0.756669i	$-0.726826\pi$
$73$	−95.4546			−1.30760			−0.653798	+	0.756669i	$0.726826\pi$
$74$		0			0
$75$	43.5748	+	58.2499i	0.580998	+	0.776665i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−40.9932			−0.518901			−0.259450	−	0.965756i	$-0.583541\pi$
$79$	−40.9932			−0.518901			−0.259450	+	0.965756i	$0.583541\pi$
$80$		0			0
$81$	−68.0828	−	43.8832i	−0.840528	−	0.541768i
$82$		0			0
$83$			23.0835i			0.278114i	0.990284	+	0.139057i	$0.0444072\pi$
$83$			23.0835i			0.278114i	−0.990284	+	0.139057i	$0.955593\pi$
$84$		0			0
$85$	16.2551			0.191237
$86$		0			0
$87$	54.8569	−	41.0367i	0.630539	−	0.471686i
$88$		0			0
$89$		−	5.52795i		−	0.0621118i	−0.999518	−	0.0310559i	$-0.990113\pi$
$89$		−	5.52795i		−	0.0621118i	0.999518	−	0.0310559i	$-0.00988700\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−15.2483	−	20.3836i	−0.163960	−	0.219178i
$94$		0			0
$95$			11.8006i			0.124217i
$96$		0			0
$97$	85.5648			0.882111			0.441055	−	0.897480i	$-0.354604\pi$
$97$	85.5648			0.882111			0.441055	+	0.897480i	$0.354604\pi$
$98$		0			0
$99$	−21.1724	−	6.23219i	−0.213862	−	0.0629514i
$100$		0			0
$101$			89.0812i			0.881992i	0.897509	+	0.440996i	$0.145375\pi$
$101$			89.0812i			0.881992i	−0.897509	+	0.440996i	$0.854625\pi$
$102$		0			0
$103$	−35.0042			−0.339847			−0.169923	−	0.985457i	$-0.554352\pi$
$103$	−35.0042			−0.339847			−0.169923	+	0.985457i	$0.554352\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			88.1258i			0.823606i	0.911273	+	0.411803i	$0.135101\pi$
$107$			88.1258i			0.823606i	−0.911273	+	0.411803i	$0.864899\pi$
$108$		0			0
$109$	133.745			1.22702			0.613509	−	0.789688i	$-0.289758\pi$
$109$	133.745			1.22702			0.613509	+	0.789688i	$0.289758\pi$
$110$		0			0
$111$	−75.0167	−	100.281i	−0.675826	−	0.903429i
$112$		0			0
$113$			70.4990i			0.623885i	0.950101	+	0.311942i	$0.100980\pi$
$113$			70.4990i			0.623885i	−0.950101	+	0.311942i	$0.899020\pi$
$114$		0			0
$115$	36.4087			0.316598
$116$		0			0
$117$	16.6194	−	56.4605i	0.142046	−	0.482569i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	114.986			0.950300
$122$		0			0
$123$	−171.038	+	127.948i	−1.39055	+	1.04023i
$124$		0			0
$125$		−	42.6989i		−	0.341591i
$126$		0			0
$127$	142.993			1.12593			0.562965	−	0.826481i	$-0.309661\pi$
$127$	142.993			1.12593			0.562965	+	0.826481i	$0.309661\pi$
$128$		0			0
$129$	107.363	+	143.521i	0.832273	+	1.11256i
$130$		0			0
$131$		−	116.500i		−	0.889311i	−0.895702	−	0.444656i	$-0.853326\pi$
$131$		−	116.500i		−	0.889311i	0.895702	−	0.444656i	$-0.146674\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	8.15868	+	21.9416i	0.0604346	+	0.162530i
$136$		0			0
$137$			57.4723i			0.419506i	0.977754	+	0.209753i	$0.0672659\pi$
$137$			57.4723i			0.419506i	−0.977754	+	0.209753i	$0.932734\pi$
$138$		0			0
$139$	252.242			1.81469			0.907346	−	0.420385i	$-0.138105\pi$
$139$	252.242			1.81469			0.907346	+	0.420385i	$0.138105\pi$
$140$		0			0
$141$	−167.655	+	125.417i	−1.18904	+	0.889486i
$142$		0			0
$143$		−	16.0368i		−	0.112145i
$144$		0			0
$145$	−19.7990			−0.136545
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	129.197i		−	0.867097i	−0.901130	−	0.433548i	$-0.857261\pi$
$149$		−	129.197i		−	0.867097i	0.901130	−	0.433548i	$-0.142739\pi$
$150$		0			0
$151$	54.5103			0.360995			0.180498	−	0.983575i	$-0.442229\pi$
$151$	54.5103			0.360995			0.180498	+	0.983575i	$0.442229\pi$
$152$		0			0
$153$	−161.869	−	47.6469i	−1.05797	−	0.311418i
$154$		0			0
$155$			7.35685i			0.0474636i
$156$		0			0
$157$	−170.588			−1.08655			−0.543275	−	0.839555i	$-0.682816\pi$
$157$	−170.588			−1.08655			−0.543275	+	0.839555i	$0.682816\pi$
$158$		0			0
$159$	149.843	−	112.093i	0.942411	−	0.704987i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	182.241			1.11805			0.559023	−	0.829152i	$-0.311176\pi$
$163$	182.241			1.11805			0.559023	+	0.829152i	$0.311176\pi$
$164$		0			0
$165$	3.82078	+	5.10753i	0.0231562	+	0.0309547i
$166$		0			0
$167$		−	305.607i		−	1.82998i	−0.403471	−	0.914992i	$-0.632197\pi$
$167$		−	305.607i		−	1.82998i	0.403471	−	0.914992i	$-0.367803\pi$
$168$		0			0
$169$	−126.235			−0.746950
$170$		0			0
$171$	34.5897	−	117.510i	0.202279	−	0.687195i
$172$		0			0
$173$		−	102.302i		−	0.591339i	−0.955290	−	0.295670i	$-0.904457\pi$
$173$		−	102.302i		−	0.591339i	0.955290	−	0.295670i	$-0.0955427\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	137.197	−	102.632i	0.775122	−	0.579844i
$178$		0			0
$179$			190.809i			1.06597i	0.846124	+	0.532986i	$0.178930\pi$
$179$			190.809i			1.06597i	−0.846124	+	0.532986i	$0.821070\pi$
$180$		0			0
$181$	227.859			1.25889			0.629445	−	0.777045i	$-0.283282\pi$
$181$	227.859			1.25889			0.629445	+	0.777045i	$0.283282\pi$
$182$		0			0
$183$	−106.414	−	142.252i	−0.581496	−	0.777331i
$184$		0			0
$185$			36.1934i			0.195640i
$186$		0			0
$187$	−45.9765			−0.245863
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	248.742i		−	1.30231i	−0.758943	−	0.651157i	$-0.774284\pi$
$191$		−	248.742i		−	1.30231i	0.758943	−	0.651157i	$-0.225716\pi$
$192$		0			0
$193$	27.7586			0.143827			0.0719134	−	0.997411i	$-0.477089\pi$
$193$	27.7586			0.143827			0.0719134	+	0.997411i	$0.477089\pi$
$194$		0			0
$195$	−13.6203	+	10.1889i	−0.0698476	+	0.0522507i
$196$		0			0
$197$			351.121i			1.78234i	0.453671	+	0.891169i	$0.350114\pi$
$197$			351.121i			1.78234i	−0.453671	+	0.891169i	$0.649886\pi$
$198$		0			0
$199$	129.386			0.650181			0.325090	−	0.945683i	$-0.394605\pi$
$199$	129.386			0.650181			0.325090	+	0.945683i	$0.394605\pi$
$200$		0			0
$201$	118.604	+	158.547i	0.590069	+	0.788791i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	61.7312			0.301128
$206$		0			0
$207$	−362.559	−	106.721i	−1.75149	−	0.515560i
$208$		0			0
$209$		−	33.3771i		−	0.159699i
$210$		0			0
$211$	143.503			0.680111			0.340056	−	0.940405i	$-0.389554\pi$
$211$	143.503			0.680111			0.340056	+	0.940405i	$0.389554\pi$
$212$		0			0
$213$	129.225	−	96.6690i	0.606690	−	0.453845i
$214$		0			0
$215$		−	51.7996i		−	0.240928i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−171.535	−	229.303i	−0.783263	−	1.04705i
$220$		0			0
$221$		−	122.606i		−	0.554777i
$222$		0			0
$223$	303.685			1.36182			0.680909	−	0.732368i	$-0.261585\pi$
$223$	303.685			1.36182			0.680909	+	0.732368i	$0.261585\pi$
$224$		0			0
$225$	−61.6241	+	209.353i	−0.273885	+	0.930459i
$226$		0			0
$227$		−	357.518i		−	1.57497i	−0.616335	−	0.787484i	$-0.711383\pi$
$227$		−	357.518i		−	1.57497i	0.616335	−	0.787484i	$-0.288617\pi$
$228$		0			0
$229$	207.699			0.906984			0.453492	−	0.891260i	$-0.350178\pi$
$229$	207.699			0.906984			0.453492	+	0.891260i	$0.350178\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	346.216i		−	1.48591i	−0.669343	−	0.742953i	$-0.733424\pi$
$233$		−	346.216i		−	1.48591i	0.669343	−	0.742953i	$-0.266576\pi$
$234$		0			0
$235$	60.5103			0.257491
$236$		0			0
$237$	−73.6658	−	98.4748i	−0.310826	−	0.415506i
$238$		0			0
$239$		−	377.331i		−	1.57879i	−0.613886	−	0.789395i	$-0.710395\pi$
$239$		−	377.331i		−	1.57879i	0.613886	−	0.789395i	$-0.289605\pi$
$240$		0			0
$241$	−356.353			−1.47864			−0.739321	−	0.673353i	$-0.764853\pi$
$241$	−356.353			−1.47864			−0.739321	+	0.673353i	$0.764853\pi$
$242$		0			0
$243$	−16.9292	−	242.410i	−0.0696675	−	0.997570i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	89.0068			0.360352
$248$		0			0
$249$	−55.4518	+	41.4817i	−0.222698	+	0.166593i
$250$		0			0
$251$			98.7289i			0.393342i	0.980470	+	0.196671i	$0.0630131\pi$
$251$			98.7289i			0.393342i	−0.980470	+	0.196671i	$0.936987\pi$
$252$		0			0
$253$	−102.979			−0.407033
$254$		0			0
$255$	29.2109	+	39.0485i	0.114553	+	0.153131i
$256$		0			0
$257$			323.274i			1.25787i	0.777456	+	0.628937i	$0.216510\pi$
$257$			323.274i			1.25787i	−0.777456	+	0.628937i	$0.783490\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	197.159	+	58.0346i	0.755397	+	0.222355i
$262$		0			0
$263$		−	71.8815i		−	0.273314i	−0.990618	−	0.136657i	$-0.956364\pi$
$263$		−	71.8815i		−	0.273314i	0.990618	−	0.136657i	$-0.0436358\pi$
$264$		0			0
$265$	−54.0816			−0.204081
$266$		0			0
$267$	13.2794	−	9.93389i	0.0497356	−	0.0372056i
$268$		0			0
$269$			42.4837i			0.157932i	0.996877	+	0.0789659i	$0.0251618\pi$
$269$			42.4837i			0.157932i	−0.996877	+	0.0789659i	$0.974838\pi$
$270$		0			0
$271$	−88.3835			−0.326138			−0.163069	−	0.986615i	$-0.552139\pi$
$271$	−88.3835			−0.326138			−0.163069	+	0.986615i	$0.552139\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			59.4637i			0.216232i
$276$		0			0
$277$	84.9795			0.306785			0.153393	−	0.988165i	$-0.450980\pi$
$277$	84.9795			0.306785			0.153393	+	0.988165i	$0.450980\pi$
$278$		0			0
$279$	21.5643	−	73.2597i	0.0772915	−	0.262579i
$280$		0			0
$281$			237.707i			0.845932i	0.906146	+	0.422966i	$0.139011\pi$
$281$			237.707i			0.845932i	−0.906146	+	0.422966i	$0.860989\pi$
$282$		0			0
$283$	184.721			0.652723			0.326362	−	0.945245i	$-0.394177\pi$
$283$	184.721			0.652723			0.326362	+	0.945245i	$0.394177\pi$
$284$		0			0
$285$	−28.3476	+	21.2060i	−0.0994654	+	0.0744069i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−62.5034			−0.216275
$290$		0			0
$291$	153.762	+	205.546i	0.528393	+	0.706343i
$292$		0			0
$293$			103.384i			0.352846i	0.984314	+	0.176423i	$0.0564527\pi$
$293$			103.384i			0.352846i	−0.984314	+	0.176423i	$0.943547\pi$
$294$		0			0
$295$	−49.5171			−0.167855
$296$		0			0
$297$	−23.0762	−	62.0603i	−0.0776977	−	0.208957i
$298$		0			0
$299$		−	274.616i		−	0.918449i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−213.993	+	160.081i	−0.706248	+	0.528321i
$304$		0			0
$305$			51.3416i			0.168333i
$306$		0			0
$307$	−138.422			−0.450887			−0.225443	−	0.974256i	$-0.572383\pi$
$307$	−138.422			−0.450887			−0.225443	+	0.974256i	$0.572383\pi$
$308$		0			0
$309$	−62.9035	−	84.0880i	−0.203571	−	0.272130i
$310$		0			0
$311$			231.265i			0.743619i	0.928309	+	0.371809i	$0.121263\pi$
$311$			231.265i			0.743619i	−0.928309	+	0.371809i	$0.878737\pi$
$312$		0			0
$313$	−312.522			−0.998472			−0.499236	−	0.866466i	$-0.666386\pi$
$313$	−312.522			−0.998472			−0.499236	+	0.866466i	$0.666386\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			541.164i			1.70714i	0.520976	+	0.853572i	$0.325568\pi$
$317$			541.164i			1.70714i	−0.520976	+	0.853572i	$0.674432\pi$
$318$		0			0
$319$	56.0000			0.175549
$320$		0			0
$321$	−211.698	+	158.365i	−0.659496	+	0.493348i
$322$		0			0
$323$		−	255.177i		−	0.790022i
$324$		0			0
$325$	−158.572			−0.487915
$326$		0			0
$327$	240.343	+	321.285i	0.734995	+	0.982524i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−108.738			−0.328514			−0.164257	−	0.986418i	$-0.552523\pi$
$331$	−108.738			−0.328514			−0.164257	+	0.986418i	$0.552523\pi$
$332$		0			0
$333$	106.090	−	360.414i	0.318587	−	1.08232i
$334$		0			0
$335$		−	57.2229i		−	0.170815i
$336$		0			0
$337$	327.717			0.972455			0.486228	−	0.873832i	$-0.338373\pi$
$337$	327.717			0.972455			0.486228	+	0.873832i	$0.338373\pi$
$338$		0			0
$339$	−169.354	+	126.689i	−0.499571	+	0.373713i
$340$		0			0
$341$		−	20.8083i		−	0.0610215i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	65.4275	+	87.4620i	0.189645	+	0.253513i
$346$		0			0
$347$			306.832i			0.884241i	0.896956	+	0.442120i	$0.145774\pi$
$347$			306.832i			0.884241i	−0.896956	+	0.442120i	$0.854226\pi$
$348$		0			0
$349$	−85.7161			−0.245605			−0.122802	−	0.992431i	$-0.539188\pi$
$349$	−85.7161			−0.245605			−0.122802	+	0.992431i	$0.539188\pi$
$350$		0			0
$351$	165.497	−	61.5375i	0.471500	−	0.175321i
$352$		0			0
$353$			450.719i			1.27682i	0.769695	+	0.638412i	$0.220408\pi$
$353$			450.719i			1.27682i	−0.769695	+	0.638412i	$0.779592\pi$
$354$		0			0
$355$	−46.6400			−0.131380
$356$		0			0
$357$		0			0
$358$		0			0
$359$			432.499i			1.20473i	0.798220	+	0.602366i	$0.205775\pi$
$359$			432.499i			1.20473i	−0.798220	+	0.602366i	$0.794225\pi$
$360$		0			0
$361$	−175.752			−0.486847
$362$		0			0
$363$	206.634	+	276.223i	0.569239	+	0.760945i
$364$		0			0
$365$			82.7604i			0.226741i
$366$		0			0
$367$	595.716			1.62320			0.811602	−	0.584211i	$-0.198596\pi$
$367$	595.716			1.62320			0.811602	+	0.584211i	$0.198596\pi$
$368$		0			0
$369$	−614.720	−	180.946i	−1.66591	−	0.490368i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−428.952			−1.15001			−0.575003	−	0.818151i	$-0.694999\pi$
$373$	−428.952			−1.15001			−0.575003	+	0.818151i	$0.694999\pi$
$374$		0			0
$375$	102.572	−	76.7312i	0.273527	−	0.204616i
$376$		0			0
$377$			149.336i			0.396116i
$378$		0			0
$379$	304.228			0.802712			0.401356	−	0.915922i	$-0.368539\pi$
$379$	304.228			0.802712			0.401356	+	0.915922i	$0.368539\pi$
$380$		0			0
$381$	256.963	+	343.502i	0.674443	+	0.901580i
$382$		0			0
$383$			100.783i			0.263141i	0.991307	+	0.131570i	$0.0420019\pi$
$383$			100.783i			0.263141i	−0.991307	+	0.131570i	$0.957998\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−151.834	+	515.821i	−0.392337	+	1.33287i
$388$		0			0
$389$		−	411.811i		−	1.05864i	−0.848423	−	0.529320i	$-0.822447\pi$
$389$		−	411.811i		−	1.05864i	0.848423	−	0.529320i	$-0.177553\pi$
$390$		0			0
$391$	−787.308			−2.01357
$392$		0			0
$393$	279.859	−	209.353i	0.712109	−	0.532706i
$394$		0			0
$395$			35.5416i			0.0899788i
$396$		0			0
$397$	481.696			1.21334			0.606670	−	0.794954i	$-0.292505\pi$
$397$	481.696			1.21334			0.606670	+	0.794954i	$0.292505\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			245.681i			0.612671i	0.951924	+	0.306335i	$0.0991029\pi$
$401$			245.681i			0.612671i	−0.951924	+	0.306335i	$0.900897\pi$
$402$		0			0
$403$	55.4897			0.137692
$404$		0			0
$405$	−38.0473	+	59.0287i	−0.0939441	+	0.145750i
$406$		0			0
$407$		−	102.370i		−	0.251524i
$408$		0			0
$409$	−272.582			−0.666461			−0.333230	−	0.942845i	$-0.608139\pi$
$409$	−272.582			−0.666461			−0.333230	+	0.942845i	$0.608139\pi$
$410$		0			0
$411$	−138.061	+	103.279i	−0.335916	+	0.251288i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	20.0137			0.0482258
$416$		0			0
$417$	453.286	+	605.943i	1.08702	+	1.45310i
$418$		0			0
$419$			477.375i			1.13932i	0.821881	+	0.569660i	$0.192925\pi$
$419$			477.375i			1.13932i	−0.821881	+	0.569660i	$0.807075\pi$
$420$		0			0
$421$	−711.476			−1.68997			−0.844983	−	0.534793i	$-0.820390\pi$
$421$	−711.476			−1.68997			−0.844983	+	0.534793i	$0.820390\pi$
$422$		0			0
$423$	−602.562	−	177.367i	−1.42450	−	0.419308i
$424$		0			0
$425$			454.617i			1.06969i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	38.5240	−	28.8185i	0.0897995	−	0.0671761i
$430$		0			0
$431$		−	581.166i		−	1.34841i	−0.738543	−	0.674207i	$-0.764486\pi$
$431$		−	581.166i		−	1.34841i	0.738543	−	0.674207i	$-0.235514\pi$
$432$		0			0
$433$	−358.137			−0.827107			−0.413554	−	0.910480i	$-0.635713\pi$
$433$	−358.137			−0.827107			−0.413554	+	0.910480i	$0.635713\pi$
$434$		0			0
$435$	−35.5793	−	47.5617i	−0.0817916	−	0.109337i
$436$		0			0
$437$		−	571.553i		−	1.30790i
$438$		0			0
$439$	−620.830			−1.41419			−0.707096	−	0.707118i	$-0.749995\pi$
$439$	−620.830			−1.41419			−0.707096	+	0.707118i	$0.749995\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			162.303i			0.366373i	0.983078	+	0.183186i	$0.0586412\pi$
$443$			162.303i			0.366373i	−0.983078	+	0.183186i	$0.941359\pi$
$444$		0			0
$445$	−4.79281			−0.0107704
$446$		0			0
$447$	310.361	−	232.171i	0.694321	−	0.519399i
$448$		0			0
$449$		−	800.020i		−	1.78178i	−0.454217	−	0.890891i	$-0.650081\pi$
$449$		−	800.020i		−	1.78178i	0.454217	−	0.890891i	$-0.349919\pi$
$450$		0			0
$451$	−174.602			−0.387144
$452$		0			0
$453$	97.9565	+	130.946i	0.216239	+	0.289064i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−682.738			−1.49396			−0.746978	−	0.664849i	$-0.768496\pi$
$457$	−682.738			−1.49396			−0.746978	+	0.664849i	$0.768496\pi$
$458$		0			0
$459$	−176.424	−	474.469i	−0.384367	−	1.03370i
$460$		0			0
$461$			443.241i			0.961478i	0.876864	+	0.480739i	$0.159632\pi$
$461$			443.241i			0.961478i	−0.876864	+	0.480739i	$0.840368\pi$
$462$		0			0
$463$	21.5034			0.0464437			0.0232218	−	0.999730i	$-0.492608\pi$
$463$	21.5034			0.0464437			0.0232218	+	0.999730i	$0.492608\pi$
$464$		0			0
$465$	−17.6728	+	13.2205i	−0.0380061	+	0.0284311i
$466$		0			0
$467$			610.243i			1.30673i	0.757043	+	0.653365i	$0.226643\pi$
$467$			610.243i			1.30673i	−0.757043	+	0.653365i	$0.773357\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−306.552	−	409.792i	−0.650853	−	0.870046i
$472$		0			0
$473$			146.511i			0.309749i
$474$		0			0
$475$	−330.034			−0.694808
$476$		0			0
$477$	538.545	+	158.523i	1.12903	+	0.332334i
$478$		0			0
$479$		−	126.793i		−	0.264704i	−0.991203	−	0.132352i	$-0.957747\pi$
$479$		−	126.793i		−	0.264704i	0.991203	−	0.132352i	$-0.0422530\pi$
$480$		0			0
$481$	272.992			0.567550
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	74.1858i		−	0.152960i
$486$		0			0
$487$	−240.497			−0.493833			−0.246916	−	0.969037i	$-0.579417\pi$
$487$	−240.497			−0.493833			−0.246916	+	0.969037i	$0.579417\pi$
$488$		0			0
$489$	327.493	+	437.785i	0.669720	+	0.895266i
$490$		0			0
$491$			602.463i			1.22701i	0.789690	+	0.613506i	$0.210241\pi$
$491$			602.463i			1.22701i	−0.789690	+	0.613506i	$0.789759\pi$
$492$		0			0
$493$	428.136			0.868430
$494$		0			0
$495$	−5.40339	+	18.3567i	−0.0109159	+	0.0370843i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	341.517			0.684403			0.342202	−	0.939627i	$-0.388827\pi$
$499$	341.517			0.684403			0.342202	+	0.939627i	$0.388827\pi$
$500$		0			0
$501$	734.138	−	549.185i	1.46535	−	1.09618i
$502$		0			0
$503$			42.6989i			0.0848885i	0.999099	+	0.0424443i	$0.0135145\pi$
$503$			42.6989i			0.0848885i	−0.999099	+	0.0424443i	$0.986486\pi$
$504$		0			0
$505$	77.2346			0.152940
$506$		0			0
$507$	−226.847	−	303.244i	−0.447430	−	0.598114i
$508$		0			0
$509$		−	777.024i		−	1.52657i	−0.646062	−	0.763285i	$-0.723585\pi$
$509$		−	777.024i		−	1.52657i	0.646062	−	0.763285i	$-0.276415\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	344.445	−	128.077i	0.671433	−	0.249663i
$514$		0			0
$515$			30.3491i			0.0589303i
$516$		0			0
$517$	−171.149			−0.331042
$518$		0			0
$519$	245.752	−	183.839i	0.473510	−	0.354218i
$520$		0			0
$521$			105.659i			0.202801i	0.994846	+	0.101400i	$0.0323323\pi$
$521$			105.659i			0.202801i	−0.994846	+	0.101400i	$0.967668\pi$
$522$		0			0
$523$	415.589			0.794625			0.397312	−	0.917683i	$-0.369943\pi$
$523$	415.589			0.794625			0.397312	+	0.917683i	$0.369943\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	159.086i		−	0.301870i
$528$		0			0
$529$	−1234.43			−2.33353
$530$		0			0
$531$	493.092	+	145.144i	0.928611	+	0.273341i
$532$		0			0
$533$		−	465.613i		−	0.873570i
$534$		0			0
$535$	76.4063			0.142815
$536$		0			0
$537$	−458.366	+	342.889i	−0.853568	+	0.638527i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−283.503			−0.524036			−0.262018	−	0.965063i	$-0.584388\pi$
$541$	−283.503			−0.524036			−0.262018	+	0.965063i	$0.584388\pi$
$542$		0			0
$543$	409.469	+	547.369i	0.754087	+	1.00805i
$544$		0			0
$545$		−	115.959i		−	0.212768i
$546$		0			0
$547$	−398.269			−0.728097			−0.364048	−	0.931380i	$-0.618606\pi$
$547$	−398.269			−0.728097			−0.364048	+	0.931380i	$0.618606\pi$
$548$		0			0
$549$	150.492	−	511.260i	0.274120	−	0.931257i
$550$		0			0
$551$			310.809i			0.564083i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−86.9446	+	65.0405i	−0.156657	+	0.117190i
$556$		0			0
$557$		−	60.8463i		−	0.109239i	−0.998507	−	0.0546196i	$-0.982605\pi$
$557$		−	60.8463i		−	0.109239i	0.998507	−	0.0546196i	$-0.0173946\pi$
$558$		0			0
$559$	−390.703			−0.698932
$560$		0			0
$561$	−82.6210	−	110.446i	−0.147275	−	0.196873i
$562$		0			0
$563$			2.92691i			0.00519878i	0.999997	+	0.00259939i	$0.000827412\pi$
$563$			2.92691i			0.00519878i	−0.999997	+	0.00259939i	$0.999173\pi$
$564$		0			0
$565$	61.1236			0.108183
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	751.131i		−	1.32009i	−0.751226	−	0.660045i	$-0.770537\pi$
$569$		−	751.131i		−	1.32009i	0.751226	−	0.660045i	$-0.229463\pi$
$570$		0			0
$571$	166.724			0.291987			0.145993	−	0.989286i	$-0.453362\pi$
$571$	166.724			0.291987			0.145993	+	0.989286i	$0.453362\pi$
$572$		0			0
$573$	597.535	−	446.996i	1.04282	−	0.780098i
$574$		0			0
$575$			1018.26i			1.77090i
$576$		0			0
$577$	117.682			0.203956			0.101978	−	0.994787i	$-0.467483\pi$
$577$	117.682			0.203956			0.101978	+	0.994787i	$0.467483\pi$
$578$		0			0
$579$	49.8829	+	66.6823i	0.0861536	+	0.115168i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	152.966			0.262377
$584$		0			0
$585$	−48.9521	−	14.4093i	−0.0836787	−	0.0246312i
$586$		0			0
$587$		−	650.556i		−	1.10827i	−0.832426	−	0.554137i	$-0.813049\pi$
$587$		−	650.556i		−	1.10827i	0.832426	−	0.554137i	$-0.186951\pi$
$588$		0			0
$589$	115.490			0.196078
$590$		0			0
$591$	−843.471	+	630.974i	−1.42719	+	1.06764i
$592$		0			0
$593$		−	377.459i		−	0.636524i	−0.948003	−	0.318262i	$-0.896901\pi$
$593$		−	377.459i		−	0.636524i	0.948003	−	0.318262i	$-0.103099\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	232.510	+	310.814i	0.389464	+	0.520627i
$598$		0			0
$599$			76.7861i			0.128190i	0.997944	+	0.0640952i	$0.0204161\pi$
$599$			76.7861i			0.128190i	−0.997944	+	0.0640952i	$0.979584\pi$
$600$		0			0
$601$	436.612			0.726476			0.363238	−	0.931696i	$-0.381671\pi$
$601$	436.612			0.726476			0.363238	+	0.931696i	$0.381671\pi$
$602$		0			0
$603$	−167.731	+	569.827i	−0.278161	+	0.944986i
$604$		0			0
$605$		−	99.6947i		−	0.164785i
$606$		0			0
$607$	751.240			1.23763			0.618814	−	0.785537i	$-0.287613\pi$
$607$	751.240			1.23763			0.618814	+	0.785537i	$0.287613\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	456.404i		−	0.746979i
$612$		0			0
$613$	−188.752			−0.307915			−0.153957	−	0.988077i	$-0.549202\pi$
$613$	−188.752			−0.307915			−0.153957	+	0.988077i	$0.549202\pi$
$614$		0			0
$615$	110.933	+	148.292i	0.180378	+	0.241126i
$616$		0			0
$617$		−	548.513i		−	0.889000i	−0.895779	−	0.444500i	$-0.853381\pi$
$617$		−	548.513i		−	0.889000i	0.895779	−	0.444500i	$-0.146619\pi$
$618$		0			0
$619$	907.716			1.46642			0.733211	−	0.680001i	$-0.238021\pi$
$619$	907.716			1.46642			0.733211	+	0.680001i	$0.238021\pi$
$620$		0			0
$621$	−395.160	−	1062.73i	−0.636329	−	1.71132i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	569.187			0.910699
$626$		0			0
$627$	80.1792	−	59.9795i	0.127878	−	0.0956611i
$628$		0			0
$629$		−	782.650i		−	1.24428i
$630$		0			0
$631$	−1032.40			−1.63613			−0.818067	−	0.575123i	$-0.804954\pi$
$631$	−1032.40			−1.63613			−0.818067	+	0.575123i	$0.804954\pi$
$632$		0			0
$633$	257.880	+	344.728i	0.407393	+	0.544593i
$634$		0			0
$635$		−	123.977i		−	0.195239i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	464.442	+	136.711i	0.726826	+	0.213945i
$640$		0			0
$641$			1183.64i			1.84655i	0.384141	+	0.923274i	$0.374498\pi$
$641$			1183.64i			1.84655i	−0.384141	+	0.923274i	$0.625502\pi$
$642$		0			0
$643$	−810.525			−1.26054			−0.630268	−	0.776378i	$-0.717055\pi$
$643$	−810.525			−1.26054			−0.630268	+	0.776378i	$0.717055\pi$
$644$		0			0
$645$	124.434	−	93.0853i	0.192921	−	0.144318i
$646$		0			0
$647$			122.895i			0.189945i	0.995480	+	0.0949727i	$0.0302764\pi$
$647$			122.895i			0.189945i	−0.995480	+	0.0949727i	$0.969724\pi$
$648$		0			0
$649$	140.056			0.215802
$650$		0			0
$651$		0			0
$652$		0			0
$653$			442.160i			0.677121i	0.940945	+	0.338560i	$0.109940\pi$
$653$			442.160i			0.677121i	−0.940945	+	0.338560i	$0.890060\pi$
$654$		0			0
$655$	−101.007			−0.154209
$656$		0			0
$657$	242.586	−	824.129i	0.369234	−	1.25438i
$658$		0			0
$659$			219.158i			0.332562i	0.986078	+	0.166281i	$0.0531758\pi$
$659$			219.158i			0.332562i	−0.986078	+	0.166281i	$0.946824\pi$
$660$		0			0
$661$	−135.984			−0.205724			−0.102862	−	0.994696i	$-0.532800\pi$
$661$	−135.984			−0.205724			−0.102862	+	0.994696i	$0.532800\pi$
$662$		0			0
$663$	294.527	−	220.326i	0.444234	−	0.332317i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	958.952			1.43771
$668$		0			0
$669$	545.731	+	729.521i	0.815742	+	1.09046i
$670$		0			0
$671$		−	145.216i		−	0.216417i
$672$		0			0
$673$	−553.531			−0.822483			−0.411241	−	0.911527i	$-0.634905\pi$
$673$	−553.531			−0.822483			−0.411241	+	0.911527i	$0.634905\pi$
$674$		0			0
$675$	−613.654	+	228.179i	−0.909117	+	0.338042i
$676$		0			0
$677$		−	676.881i		−	0.999824i	−0.866076	−	0.499912i	$-0.833366\pi$
$677$		−	676.881i		−	0.999824i	0.866076	−	0.499912i	$-0.166634\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	858.838	−	642.469i	1.26114	−	0.943420i
$682$		0			0
$683$			257.021i			0.376311i	0.982139	+	0.188156i	$0.0602509\pi$
$683$			257.021i			0.376311i	−0.982139	+	0.188156i	$0.939749\pi$
$684$		0			0
$685$	49.8292			0.0727434
$686$		0			0
$687$	373.241	+	498.941i	0.543292	+	0.726260i
$688$		0			0
$689$			407.915i			0.592039i
$690$		0			0
$691$	−493.039			−0.713515			−0.356757	−	0.934197i	$-0.616118\pi$
$691$	−493.039			−0.713515			−0.356757	+	0.934197i	$0.616118\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	218.697i		−	0.314672i
$696$		0			0
$697$	−1334.88			−1.91518
$698$		0			0
$699$	831.690	−	622.160i	1.18983	−	0.890072i
$700$		0			0
$701$			891.207i			1.27134i	0.771962	+	0.635668i	$0.219275\pi$
$701$			891.207i			1.27134i	−0.771962	+	0.635668i	$0.780725\pi$
$702$		0			0
$703$	568.172			0.808211
$704$		0			0
$705$	108.739	+	145.359i	0.154239	+	0.206183i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−972.241			−1.37129			−0.685643	−	0.727938i	$-0.740479\pi$
$709$	−972.241			−1.37129			−0.685643	+	0.727938i	$0.740479\pi$
$710$		0			0
$711$	104.179	−	353.924i	0.146525	−	0.497783i
$712$		0			0
$713$		−	356.325i		−	0.499754i
$714$		0			0
$715$	−13.9041			−0.0194463
$716$		0			0
$717$	906.433	−	678.074i	1.26420	−	0.945709i
$718$		0			0
$719$		−	708.530i		−	0.985438i	−0.870189	−	0.492719i	$-0.836003\pi$
$719$		−	708.530i		−	0.985438i	0.870189	−	0.492719i	$-0.163997\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−640.376	−	856.040i	−0.885720	−	1.18401i
$724$		0			0
$725$		−	553.730i		−	0.763766i
$726$		0			0
$727$	855.912			1.17732			0.588660	−	0.808381i	$-0.299656\pi$
$727$	855.912			1.17732			0.588660	+	0.808381i	$0.299656\pi$
$728$		0			0
$729$	551.900	−	476.285i	0.757065	−	0.653340i
$730$		0			0
$731$			1120.12i			1.53231i
$732$		0			0
$733$	1272.19			1.73560			0.867799	−	0.496916i	$-0.165534\pi$
$733$	1272.19			1.73560			0.867799	+	0.496916i	$0.165534\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			161.851i			0.219608i
$738$		0			0
$739$	84.2277			0.113975			0.0569876	−	0.998375i	$-0.481850\pi$
$739$	84.2277			0.113975			0.0569876	+	0.998375i	$0.481850\pi$
$740$		0			0
$741$	159.948	+	213.815i	0.215854	+	0.288549i
$742$		0			0
$743$			236.020i			0.317658i	0.987306	+	0.158829i	$0.0507718\pi$
$743$			236.020i			0.317658i	−0.987306	+	0.158829i	$0.949228\pi$
$744$		0			0
$745$	−112.016			−0.150357
$746$		0			0
$747$	−199.297	−	58.6640i	−0.266796	−	0.0785327i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1068.39			1.42262			0.711310	−	0.702879i	$-0.248102\pi$
$751$	1068.39			1.42262			0.711310	+	0.702879i	$0.248102\pi$
$752$		0			0
$753$	−237.169	+	177.419i	−0.314966	+	0.235616i
$754$		0			0
$755$		−	47.2611i		−	0.0625975i
$756$		0			0
$757$	800.187			1.05705			0.528525	−	0.848918i	$-0.322745\pi$
$757$	800.187			1.05705			0.528525	+	0.848918i	$0.322745\pi$
$758$		0			0
$759$	−185.057	−	247.380i	−0.243817	−	0.325929i
$760$		0			0
$761$			801.190i			1.05281i	0.850234	+	0.526406i	$0.176461\pi$
$761$			801.190i			1.05281i	−0.850234	+	0.526406i	$0.823539\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−41.3105	+	140.343i	−0.0540007	+	0.183454i
$766$		0			0
$767$			373.487i			0.486946i
$768$		0			0
$769$	236.096			0.307017			0.153509	−	0.988147i	$-0.450943\pi$
$769$	236.096			0.307017			0.153509	+	0.988147i	$0.450943\pi$
$770$		0			0
$771$	−776.576	+	580.932i	−1.00723	+	0.753478i
$772$		0			0
$773$		−	1387.16i		−	1.79451i	−0.441513	−	0.897255i	$-0.645558\pi$
$773$		−	1387.16i		−	1.79451i	0.441513	−	0.897255i	$-0.354442\pi$
$774$		0			0
$775$	−205.754			−0.265488
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	969.071i		−	1.24399i
$780$		0			0
$781$	131.918			0.168909
$782$		0			0
$783$	214.887	+	577.910i	0.274441	+	0.738071i
$784$		0			0
$785$			147.902i			0.188411i
$786$		0			0
$787$	−374.225			−0.475509			−0.237754	−	0.971325i	$-0.576411\pi$
$787$	−374.225			−0.475509			−0.237754	+	0.971325i	$0.576411\pi$
$788$		0			0
$789$	172.676	−	129.173i	0.218854	−	0.163718i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	387.248			0.488333
$794$		0			0
$795$	−97.1861	−	129.916i	−0.122247	−	0.163417i
$796$		0			0
$797$		−	1150.47i		−	1.44350i	−0.692155	−	0.721749i	$-0.743339\pi$
$797$		−	1150.47i		−	1.44350i	0.692155	−	0.721749i	$-0.256661\pi$
$798$		0			0
$799$	−1308.48			−1.63765
$800$		0			0
$801$	47.7269	+	14.0486i	0.0595841	+	0.0175389i
$802$		0			0
$803$		−	234.082i		−	0.291509i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−102.055	+	76.3443i	−0.126463	+	0.0946026i
$808$		0			0
$809$			983.620i			1.21585i	0.793996	+	0.607923i	$0.207997\pi$
$809$			983.620i			1.21585i	−0.793996	+	0.607923i	$0.792003\pi$
$810$		0			0
$811$	−1159.05			−1.42916			−0.714578	−	0.699555i	$-0.753381\pi$
$811$	−1159.05			−1.42916			−0.714578	+	0.699555i	$0.753381\pi$
$812$		0			0
$813$	−158.828	−	212.317i	−0.195360	−	0.261153i
$814$		0			0
$815$		−	158.006i		−	0.193872i
$816$		0			0
$817$	−813.163			−0.995304
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	563.548i		−	0.686416i	−0.939259	−	0.343208i	$-0.888486\pi$
$821$		−	563.548i		−	0.686416i	0.939259	−	0.343208i	$-0.111514\pi$
$822$		0			0
$823$	429.503			0.521875			0.260938	−	0.965356i	$-0.415968\pi$
$823$	429.503			0.521875			0.260938	+	0.965356i	$0.415968\pi$
$824$		0			0
$825$	−142.845	+	106.858i	−0.173146	+	0.129525i
$826$		0			0
$827$		−	911.451i		−	1.10212i	−0.834467	−	0.551059i	$-0.814224\pi$
$827$		−	911.451i		−	1.10212i	0.834467	−	0.551059i	$-0.185776\pi$
$828$		0			0
$829$	−59.9769			−0.0723485			−0.0361743	−	0.999345i	$-0.511517\pi$
$829$	−59.9769			−0.0723485			−0.0361743	+	0.999345i	$0.511517\pi$
$830$		0			0
$831$	152.710	+	204.140i	0.183767	+	0.245656i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−264.966			−0.317324
$836$		0			0
$837$	214.738	−	79.8473i	0.256557	−	0.0953970i
$838$		0			0
$839$			1124.25i			1.33999i	0.742364	+	0.669996i	$0.233704\pi$
$839$			1124.25i			1.33999i	−0.742364	+	0.669996i	$0.766296\pi$
$840$		0			0
$841$	319.524			0.379933
$842$		0			0
$843$	−571.025	+	427.166i	−0.677373	+	0.506721i
$844$		0			0
$845$			109.447i			0.129523i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	331.948	+	443.741i	0.390987	+	0.522663i
$850$		0			0
$851$		−	1753.00i		−	2.05993i
$852$		0			0
$853$	776.915			0.910804			0.455402	−	0.890286i	$-0.349496\pi$
$853$	776.915			0.910804			0.455402	+	0.890286i	$0.349496\pi$
$854$		0			0
$855$	−101.883	−	29.9897i	−0.119161	−	0.0350757i
$856$		0			0
$857$		−	1691.79i		−	1.97409i	−0.160455	−	0.987043i	$-0.551296\pi$
$857$		−	1691.79i		−	1.97409i	0.160455	−	0.987043i	$-0.448704\pi$
$858$		0			0
$859$	671.951			0.782248			0.391124	−	0.920338i	$-0.372086\pi$
$859$	671.951			0.782248			0.391124	+	0.920338i	$0.372086\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	674.501i		−	0.781577i	−0.920480	−	0.390789i	$-0.872202\pi$
$863$		−	674.501i		−	0.781577i	0.920480	−	0.390789i	$-0.127798\pi$
$864$		0			0
$865$	−88.6969			−0.102540
$866$		0			0
$867$	−112.320	−	150.147i	−0.129551	−	0.173180i
$868$		0			0
$869$		−	100.527i		−	0.115681i
$870$		0			0
$871$	−431.609			−0.495532
$872$		0			0
$873$	−217.453	+	738.744i	−0.249087	+	0.846212i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−824.159			−0.939748			−0.469874	−	0.882733i	$-0.655701\pi$
$877$	−824.159			−0.939748			−0.469874	+	0.882733i	$0.655701\pi$
$878$		0			0
$879$	−248.352	+	185.784i	−0.282539	+	0.211358i
$880$		0			0
$881$			911.504i			1.03462i	0.855797	+	0.517312i	$0.173067\pi$
$881$			911.504i			1.03462i	−0.855797	+	0.517312i	$0.826933\pi$
$882$		0			0
$883$	917.973			1.03961			0.519803	−	0.854286i	$-0.326005\pi$
$883$	917.973			1.03961			0.519803	+	0.854286i	$0.326005\pi$
$884$		0			0
$885$	−88.9837	−	118.951i	−0.100547	−	0.134408i
$886$		0			0
$887$		−	95.8020i		−	0.108007i	−0.998541	−	0.0540034i	$-0.982802\pi$
$887$		−	95.8020i		−	0.108007i	0.998541	−	0.0540034i	$-0.0171982\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	107.614	−	166.958i	0.120779	−	0.187383i
$892$		0			0
$893$		−	949.905i		−	1.06372i
$894$		0			0
$895$	165.434			0.184842
$896$		0			0
$897$	659.690	−	493.493i	0.735440	−	0.550159i
$898$		0			0
$899$			193.769i			0.215538i
$900$		0			0
$901$	1169.47			1.29797
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	197.557i		−	0.218295i
$906$		0			0
$907$	−26.4555			−0.0291681			−0.0145841	−	0.999894i	$-0.504642\pi$
$907$	−26.4555			−0.0291681			−0.0145841	+	0.999894i	$0.504642\pi$
$908$		0			0
$909$	−769.103	−	226.389i	−0.846098	−	0.249053i
$910$		0			0
$911$			713.573i			0.783285i	0.920117	+	0.391643i	$0.128093\pi$
$911$			713.573i			0.783285i	−0.920117	+	0.391643i	$0.871907\pi$
$912$		0			0
$913$	−56.6073			−0.0620014
$914$		0			0
$915$	−123.334	+	92.2622i	−0.134791	+	0.100833i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−675.408			−0.734937			−0.367469	−	0.930036i	$-0.619776\pi$
$919$	−675.408			−0.734937			−0.367469	+	0.930036i	$0.619776\pi$
$920$		0			0
$921$	−248.749	−	332.521i	−0.270085	−	0.361044i
$922$		0			0
$923$			351.786i			0.381134i
$924$		0			0
$925$	−1012.24			−1.09432
$926$		0			0
$927$	88.9590	−	302.217i	0.0959644	−	0.326016i
$928$		0			0
$929$			1151.01i			1.23898i	0.785006	+	0.619488i	$0.212660\pi$
$929$			1151.01i			1.23898i	−0.785006	+	0.619488i	$0.787340\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−555.552	+	415.590i	−0.595447	+	0.445435i
$934$		0			0
$935$			39.8622i			0.0426334i
$936$		0			0
$937$	344.395			0.367551			0.183775	−	0.982968i	$-0.441168\pi$
$937$	344.395			0.367551			0.183775	+	0.982968i	$0.441168\pi$
$938$		0			0
$939$	−561.610	−	750.748i	−0.598094	−	0.799519i
$940$		0			0
$941$			667.780i			0.709649i	0.934933	+	0.354825i	$0.115459\pi$
$941$			667.780i			0.709649i	−0.934933	+	0.354825i	$0.884541\pi$
$942$		0			0
$943$	−2989.91			−3.17064
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1436.97i			1.51739i	0.651444	+	0.758697i	$0.274164\pi$
$947$			1436.97i			1.51739i	−0.651444	+	0.758697i	$0.725836\pi$
$948$		0			0
$949$	624.228			0.657774
$950$		0			0
$951$	−1300.00	+	972.488i	−1.36698	+	1.02259i
$952$		0			0
$953$		−	1005.39i		−	1.05497i	−0.849564	−	0.527485i	$-0.823135\pi$
$953$		−	1005.39i		−	1.05497i	0.849564	−	0.527485i	$-0.176865\pi$
$954$		0			0
$955$	−215.663			−0.225825
$956$		0			0
$957$	100.634	+	134.525i	0.105155	+	0.140569i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−889.000			−0.925078
$962$		0			0
$963$	−760.855	−	223.961i	−0.790088	−	0.232566i
$964$		0			0
$965$		−	24.0670i		−	0.0249399i
$966$		0			0
$967$	1695.39			1.75325			0.876626	−	0.481173i	$-0.159789\pi$
$967$	1695.39			1.75325			0.876626	+	0.481173i	$0.159789\pi$
$968$		0			0
$969$	612.993	−	458.561i	0.632604	−	0.473231i
$970$		0			0
$971$			268.430i			0.276447i	0.990401	+	0.138224i	$0.0441393\pi$
$971$			268.430i			0.276447i	−0.990401	+	0.138224i	$0.955861\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−284.959	−	380.927i	−0.292266	−	0.390694i
$976$		0			0
$977$			1321.25i			1.35236i	0.736738	+	0.676179i	$0.236365\pi$
$977$			1321.25i			1.35236i	−0.736738	+	0.676179i	$0.763635\pi$
$978$		0			0
$979$	13.5561			0.0138469
$980$		0			0
$981$	−339.897	+	1154.72i	−0.346480	+	1.17708i
$982$		0			0
$983$			628.543i			0.639413i	0.947517	+	0.319707i	$0.103584\pi$
$983$			628.543i			0.639413i	−0.947517	+	0.319707i	$0.896416\pi$
$984$		0			0
$985$	304.426			0.309062
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2508.88i			2.53679i
$990$		0			0
$991$	908.911			0.917165			0.458583	−	0.888652i	$-0.348357\pi$
$991$	908.911			0.917165			0.458583	+	0.888652i	$0.348357\pi$
$992$		0			0
$993$	−195.405	−	261.213i	−0.196783	−	0.263055i
$994$		0			0
$995$		−	112.179i		−	0.112743i
$996$		0			0
$997$	962.792			0.965690			0.482845	−	0.875706i	$-0.339604\pi$
$997$	962.792			0.965690			0.482845	+	0.875706i	$0.339604\pi$
$998$		0			0
$999$	1056.44	−	392.823i	1.05750	−	0.393216i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.3.c.j.197.6	yes	8
3.2	odd	2	inner	588.3.c.j.197.5	yes	8
7.2	even	3		588.3.p.i.557.1		16
7.3	odd	6		588.3.p.i.569.3		16
7.4	even	3		588.3.p.i.569.6		16
7.5	odd	6		588.3.p.i.557.8		16
7.6	odd	2	inner	588.3.c.j.197.3	✓	8
21.2	odd	6		588.3.p.i.557.6		16
21.5	even	6		588.3.p.i.557.3		16
21.11	odd	6		588.3.p.i.569.1		16
21.17	even	6		588.3.p.i.569.8		16
21.20	even	2	inner	588.3.c.j.197.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
588.3.c.j.197.3	✓	8	7.6	odd	2	inner
588.3.c.j.197.4	yes	8	21.20	even	2	inner
588.3.c.j.197.5	yes	8	3.2	odd	2	inner
588.3.c.j.197.6	yes	8	1.1	even	1	trivial
588.3.p.i.557.1		16	7.2	even	3
588.3.p.i.557.3		16	21.5	even	6
588.3.p.i.557.6		16	21.2	odd	6
588.3.p.i.557.8		16	7.5	odd	6
588.3.p.i.569.1		16	21.11	odd	6
588.3.p.i.569.3		16	7.3	odd	6
588.3.p.i.569.6		16	7.4	even	3
588.3.p.i.569.8		16	21.17	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	588.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 + 8.63374i) q^{9} +O(q^{10})\)	\(q+(1.79703 + 2.40223i) q^{3} -0.867013i q^{5} +(-2.54138 + 8.63374i) q^{9} +2.45228i q^{11} -6.53953 q^{13} +(2.08276 - 1.55805i) q^{15} +18.7484i q^{17} -13.6106 q^{19} +41.9933i q^{23} +24.2483 q^{25} +(-25.3071 + 9.41009i) q^{27} -22.8358i q^{29} -8.48528 q^{31} +(-5.89094 + 4.40682i) q^{33} -41.7449 q^{37} +(-11.7517 - 15.7094i) q^{39} +71.1998i q^{41} +59.7449 q^{43} +(7.48556 + 2.20341i) q^{45} +69.7916i q^{47} +(-45.0380 + 33.6914i) q^{51} -62.3768i q^{53} +2.12616 q^{55} +(-24.4586 - 32.6957i) q^{57} -57.1123i q^{59} -59.2166 q^{61} +5.66986i q^{65} +66.0000 q^{67} +(-100.877 + 75.4631i) q^{69} -53.7938i q^{71} -95.4546 q^{73} +(43.5748 + 58.2499i) q^{75} -40.9932 q^{79} +(-68.0828 - 43.8832i) q^{81} +23.0835i q^{83} +16.2551 q^{85} +(54.8569 - 41.0367i) q^{87} -5.52795i q^{89} +(-15.2483 - 20.3836i) q^{93} +11.8006i q^{95} +85.5648 q^{97} +(-21.1724 - 6.23219i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 32 q^{15} + 48 q^{25} + 104 q^{37} - 240 q^{39} + 40 q^{43} - 44 q^{51} - 220 q^{57} + 528 q^{67} + 256 q^{79} - 496 q^{81} + 568 q^{85} + 24 q^{93} - 656 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 32 * q^15 + 48 * q^25 + 104 * q^37 - 240 * q^39 + 40 * q^43 - 44 * q^51 - 220 * q^57 + 528 * q^67 + 256 * q^79 - 496 * q^81 + 568 * q^85 + 24 * q^93 - 656 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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