LMFDB - Embedded newform 1568.3.d.l.1471.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(1471,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1568.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.d (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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.9144576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 12x^{4} + 36x^{2} + 4 $ x^6 + 12x^4 + 36x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.2
Root			$2.60168i$ of defining polynomial
Character	$\chi$	$=$	1568.1471
Dual form			1568.3.d.l.1471.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.76873i q^{3} +8.20336 q^{5} +5.87158 q^{9} +O(q^{10})$	$q-1.76873i q^{3} +8.20336 q^{5} +5.87158 q^{9} +10.5096i q^{11} +5.87158 q^{13} -14.5096i q^{15} -27.0168 q^{17} -12.3085i q^{19} -41.9139i q^{23} +42.2951 q^{25} -26.3039i q^{27} +36.4235 q^{29} +7.64031i q^{31} +18.5886 q^{33} +7.25684 q^{37} -10.3853i q^{39} +48.1667 q^{41} +23.4863i q^{43} +48.1667 q^{45} -59.1173i q^{47} +47.7855i q^{51} -4.06561 q^{53} +86.2137i q^{55} -21.7705 q^{57} +26.8214i q^{59} -0.370024 q^{61} +48.1667 q^{65} -2.08605i q^{67} -74.1346 q^{69} +121.781i q^{71} +77.1452 q^{73} -74.8088i q^{75} -36.8101i q^{79} +6.31965 q^{81} -98.2951i q^{83} -221.628 q^{85} -64.4235i q^{87} +30.3458 q^{89} +13.5137 q^{93} -100.971i q^{95} +5.87158 q^{97} +61.7077i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 18 q^{5}+O(q^{10}) $ 6 * q + 18 * q^5	$ 6 q + 18 q^{5} - 6 q^{17} + 186 q^{33} + 114 q^{37} + 18 q^{53} - 342 q^{57} + 318 q^{61} + 114 q^{69} + 342 q^{73} + 186 q^{81} - 498 q^{85} + 150 q^{89} + 222 q^{93}+O(q^{100}) $ 6 * q + 18 * q^5 - 6 * q^17 + 186 * q^33 + 114 * q^37 + 18 * q^53 - 342 * q^57 + 318 * q^61 + 114 * q^69 + 342 * q^73 + 186 * q^81 - 498 * q^85 + 150 * q^89 + 222 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 1.76873i	− 0.589578i	−0.955562	−	0.294789i	$-0.904751\pi$
$3$	− 1.76873i	− 0.589578i	0.955562	−	0.294789i	$-0.0952494\pi$
$4$	0	0
$5$	8.20336	1.64067	0.820336	−	0.571882i	$-0.193787\pi$
$5$	8.20336	1.64067	0.820336	+	0.571882i	$0.193787\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.87158	0.652398
$10$	0	0
$11$	10.5096i	0.955415i	0.878519	+	0.477707i	$0.158532\pi$
$11$	10.5096i	0.955415i	−0.878519	+	0.477707i	$0.841468\pi$
$12$	0	0
$13$	5.87158	0.451660	0.225830	−	0.974167i	$-0.427491\pi$
$13$	5.87158	0.451660	0.225830	+	0.974167i	$0.427491\pi$
$14$	0	0
$15$	− 14.5096i	− 0.967304i
$16$	0	0
$17$	−27.0168	−1.58922	−0.794612	−	0.607118i	$-0.792325\pi$
$17$	−27.0168	−1.58922	−0.794612	+	0.607118i	$0.792325\pi$
$18$	0	0
$19$	− 12.3085i	− 0.647818i	−0.946088	−	0.323909i	$-0.895003\pi$
$19$	− 12.3085i	− 0.647818i	0.946088	−	0.323909i	$-0.104997\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 41.9139i	− 1.82235i	−0.412025	−	0.911173i	$-0.635178\pi$
$23$	− 41.9139i	− 1.82235i	0.412025	−	0.911173i	$-0.364822\pi$
$24$	0	0
$25$	42.2951	1.69180
$26$	0	0
$27$	− 26.3039i	− 0.974217i
$28$	0	0
$29$	36.4235	1.25598	0.627992	−	0.778220i	$-0.283877\pi$
$29$	36.4235	1.25598	0.627992	+	0.778220i	$0.283877\pi$
$30$	0	0
$31$	7.64031i	0.246462i	0.992378	+	0.123231i	$0.0393256\pi$
$31$	7.64031i	0.246462i	−0.992378	+	0.123231i	$0.960674\pi$
$32$	0	0
$33$	18.5886	0.563292
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.25684	0.196131	0.0980654	−	0.995180i	$-0.468735\pi$
$37$	7.25684	0.196131	0.0980654	+	0.995180i	$0.468735\pi$
$38$	0	0
$39$	− 10.3853i	− 0.266289i
$40$	0	0
$41$	48.1667	1.17480	0.587398	−	0.809298i	$-0.300152\pi$
$41$	48.1667	1.17480	0.587398	+	0.809298i	$0.300152\pi$
$42$	0	0
$43$	23.4863i	0.546193i	0.961987	+	0.273097i	$0.0880479\pi$
$43$	23.4863i	0.546193i	−0.961987	+	0.273097i	$0.911952\pi$
$44$	0	0
$45$	48.1667	1.07037
$46$	0	0
$47$	− 59.1173i	− 1.25781i	−0.777480	−	0.628907i	$-0.783502\pi$
$47$	− 59.1173i	− 1.25781i	0.777480	−	0.628907i	$-0.216498\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	47.7855i	0.936971i
$52$	0	0
$53$	−4.06561	−0.0767097	−0.0383548	−	0.999264i	$-0.512212\pi$
$53$	−4.06561	−0.0767097	−0.0383548	+	0.999264i	$0.512212\pi$
$54$	0	0
$55$	86.2137i	1.56752i
$56$	0	0
$57$	−21.7705	−0.381939
$58$	0	0
$59$	26.8214i	0.454601i	0.973825	+	0.227300i	$0.0729899\pi$
$59$	26.8214i	0.454601i	−0.973825	+	0.227300i	$0.927010\pi$
$60$	0	0
$61$	−0.370024	−0.00606597	−0.00303298	−	0.999995i	$-0.500965\pi$
$61$	−0.370024	−0.00606597	−0.00303298	+	0.999995i	$0.500965\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	48.1667	0.741026
$66$	0	0
$67$	− 2.08605i	− 0.0311351i	−0.999879	−	0.0155676i	$-0.995044\pi$
$67$	− 2.08605i	− 0.0311351i	0.999879	−	0.0155676i	$-0.00495551\pi$
$68$	0	0
$69$	−74.1346	−1.07442
$70$	0	0
$71$	121.781i	1.71523i	0.514291	+	0.857616i	$0.328055\pi$
$71$	121.781i	1.71523i	−0.514291	+	0.857616i	$0.671945\pi$
$72$	0	0
$73$	77.1452	1.05678	0.528392	−	0.849001i	$-0.322795\pi$
$73$	77.1452	1.05678	0.528392	+	0.849001i	$0.322795\pi$
$74$	0	0
$75$	− 74.8088i	− 0.997450i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 36.8101i	− 0.465950i	−0.972483	−	0.232975i	$-0.925154\pi$
$79$	− 36.8101i	− 0.465950i	0.972483	−	0.232975i	$-0.0748461\pi$
$80$	0	0
$81$	6.31965	0.0780204
$82$	0	0
$83$	− 98.2951i	− 1.18428i	−0.805836	−	0.592139i	$-0.798284\pi$
$83$	− 98.2951i	− 1.18428i	0.805836	−	0.592139i	$-0.201716\pi$
$84$	0	0
$85$	−221.628	−2.60739
$86$	0	0
$87$	− 64.4235i	− 0.740500i
$88$	0	0
$89$	30.3458	0.340964	0.170482	−	0.985361i	$-0.445468\pi$
$89$	30.3458	0.340964	0.170482	+	0.985361i	$0.445468\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.5137	0.145308
$94$	0	0
$95$	− 100.971i	− 1.06286i
$96$	0	0
$97$	5.87158	0.0605317	0.0302659	−	0.999542i	$-0.490365\pi$
$97$	5.87158	0.0605317	0.0302659	+	0.999542i	$0.490365\pi$
$98$	0	0
$99$	61.7077i	0.623310i
$100$	0	0
$101$	153.317	1.51799	0.758993	−	0.651099i	$-0.225692\pi$
$101$	153.317	1.51799	0.758993	+	0.651099i	$0.225692\pi$
$102$	0	0
$103$	− 77.7269i	− 0.754630i	−0.926085	−	0.377315i	$-0.876847\pi$
$103$	− 77.7269i	− 0.754630i	0.926085	−	0.377315i	$-0.123153\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	164.889i	1.54102i	0.637427	+	0.770511i	$0.279999\pi$
$107$	164.889i	1.54102i	−0.637427	+	0.770511i	$0.720001\pi$
$108$	0	0
$109$	61.2596	0.562015	0.281008	−	0.959706i	$-0.409331\pi$
$109$	61.2596	0.562015	0.281008	+	0.959706i	$0.409331\pi$
$110$	0	0
$111$	− 12.8354i	− 0.115634i
$112$	0	0
$113$	−9.79509	−0.0866822	−0.0433411	−	0.999060i	$-0.513800\pi$
$113$	−9.79509	−0.0866822	−0.0433411	+	0.999060i	$0.513800\pi$
$114$	0	0
$115$	− 343.835i	− 2.98987i
$116$	0	0
$117$	34.4754	0.294662
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.5491	0.0871828
$122$	0	0
$123$	− 85.1940i	− 0.692634i
$124$	0	0
$125$	141.878	1.13502
$126$	0	0
$127$	125.705i	0.989802i	0.868949	+	0.494901i	$0.164796\pi$
$127$	125.705i	0.989802i	−0.868949	+	0.494901i	$0.835204\pi$
$128$	0	0
$129$	41.5411	0.322024
$130$	0	0
$131$	35.6173i	0.271888i	0.990717	+	0.135944i	$0.0434067\pi$
$131$	35.6173i	0.271888i	−0.990717	+	0.135944i	$0.956593\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 215.780i	− 1.59837i
$136$	0	0
$137$	−119.500	−0.872263	−0.436131	−	0.899883i	$-0.643652\pi$
$137$	−119.500	−0.872263	−0.436131	+	0.899883i	$0.643652\pi$
$138$	0	0
$139$	47.3989i	0.341000i	0.985358	+	0.170500i	$0.0545382\pi$
$139$	47.3989i	0.341000i	−0.985358	+	0.170500i	$0.945462\pi$
$140$	0	0
$141$	−104.563	−0.741580
$142$	0	0
$143$	61.7077i	0.431523i
$144$	0	0
$145$	298.795	2.06066
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−189.374	−1.27097	−0.635485	−	0.772114i	$-0.719200\pi$
$149$	−189.374	−1.27097	−0.635485	+	0.772114i	$0.719200\pi$
$150$	0	0
$151$	245.307i	1.62455i	0.583273	+	0.812276i	$0.301772\pi$
$151$	245.307i	1.62455i	−0.583273	+	0.812276i	$0.698228\pi$
$152$	0	0
$153$	−158.631	−1.03681
$154$	0	0
$155$	62.6762i	0.404363i
$156$	0	0
$157$	−220.718	−1.40585	−0.702924	−	0.711264i	$-0.748123\pi$
$157$	−220.718	−1.40585	−0.702924	+	0.711264i	$0.748123\pi$
$158$	0	0
$159$	7.19099i	0.0452263i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 272.034i	− 1.66892i	−0.551067	−	0.834461i	$-0.685779\pi$
$163$	− 272.034i	− 1.66892i	0.551067	−	0.834461i	$-0.314221\pi$
$164$	0	0
$165$	152.489	0.924176
$166$	0	0
$167$	− 88.8674i	− 0.532140i	−0.963954	−	0.266070i	$-0.914275\pi$
$167$	− 88.8674i	− 0.532140i	0.963954	−	0.266070i	$-0.0857252\pi$
$168$	0	0
$169$	−134.525	−0.796003
$170$	0	0
$171$	− 72.2705i	− 0.422635i
$172$	0	0
$173$	319.182	1.84498	0.922491	−	0.386019i	$-0.126150\pi$
$173$	319.182	1.84498	0.922491	+	0.386019i	$0.126150\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	47.4400	0.268023
$178$	0	0
$179$	− 139.436i	− 0.778970i	−0.921033	−	0.389485i	$-0.872653\pi$
$179$	− 139.436i	− 0.778970i	0.921033	−	0.389485i	$-0.127347\pi$
$180$	0	0
$181$	73.1328	0.404048	0.202024	−	0.979381i	$-0.435248\pi$
$181$	73.1328	0.404048	0.202024	+	0.979381i	$0.435248\pi$
$182$	0	0
$183$	0.654474i	0.00357636i
$184$	0	0
$185$	59.5305	0.321786
$186$	0	0
$187$	− 283.935i	− 1.51837i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	128.288i	0.671667i	0.941921	+	0.335833i	$0.109018\pi$
$191$	128.288i	0.671667i	−0.941921	+	0.335833i	$0.890982\pi$
$192$	0	0
$193$	279.538	1.44838	0.724192	−	0.689598i	$-0.242213\pi$
$193$	279.538	1.44838	0.724192	+	0.689598i	$0.242213\pi$
$194$	0	0
$195$	− 85.1940i	− 0.436892i
$196$	0	0
$197$	−99.0628	−0.502857	−0.251428	−	0.967876i	$-0.580900\pi$
$197$	−99.0628	−0.502857	−0.251428	+	0.967876i	$0.580900\pi$
$198$	0	0
$199$	236.318i	1.18753i	0.804640	+	0.593763i	$0.202359\pi$
$199$	236.318i	1.18753i	−0.804640	+	0.593763i	$0.797641\pi$
$200$	0	0
$201$	−3.68967	−0.0183566
$202$	0	0
$203$	0	0
$204$	0	0
$205$	395.128	1.92746
$206$	0	0
$207$	− 246.101i	− 1.18889i
$208$	0	0
$209$	129.357	0.618934
$210$	0	0
$211$	− 389.683i	− 1.84684i	−0.383791	−	0.923420i	$-0.625382\pi$
$211$	− 389.683i	− 1.84684i	0.383791	−	0.923420i	$-0.374618\pi$
$212$	0	0
$213$	215.399	1.01126
$214$	0	0
$215$	192.667i	0.896124i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 136.449i	− 0.623057i
$220$	0	0
$221$	−158.631	−0.717788
$222$	0	0
$223$	102.219i	0.458379i	0.973382	+	0.229190i	$0.0736076\pi$
$223$	102.219i	0.458379i	−0.973382	+	0.229190i	$0.926392\pi$
$224$	0	0
$225$	248.339	1.10373
$226$	0	0
$227$	− 230.982i	− 1.01754i	−0.860901	−	0.508772i	$-0.830100\pi$
$227$	− 230.982i	− 1.01754i	0.860901	−	0.508772i	$-0.169900\pi$
$228$	0	0
$229$	16.8044	0.0733817	0.0366909	−	0.999327i	$-0.488318\pi$
$229$	16.8044	0.0733817	0.0366909	+	0.999327i	$0.488318\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−120.615	−0.517660	−0.258830	−	0.965923i	$-0.583337\pi$
$233$	−120.615	−0.517660	−0.258830	+	0.965923i	$0.583337\pi$
$234$	0	0
$235$	− 484.960i	− 2.06366i
$236$	0	0
$237$	−65.1073	−0.274714
$238$	0	0
$239$	109.863i	0.459679i	0.973229	+	0.229840i	$0.0738202\pi$
$239$	109.863i	0.459679i	−0.973229	+	0.229840i	$0.926180\pi$
$240$	0	0
$241$	−184.899	−0.767214	−0.383607	−	0.923496i	$-0.625318\pi$
$241$	−184.899	−0.767214	−0.383607	+	0.923496i	$0.625318\pi$
$242$	0	0
$243$	− 247.913i	− 1.02022i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 72.2705i	− 0.292593i
$248$	0	0
$249$	−173.858	−0.698224
$250$	0	0
$251$	− 224.827i	− 0.895724i	−0.894103	−	0.447862i	$-0.852186\pi$
$251$	− 224.827i	− 0.895724i	0.894103	−	0.447862i	$-0.147814\pi$
$252$	0	0
$253$	440.497	1.74110
$254$	0	0
$255$	392.002i	1.53726i
$256$	0	0
$257$	−178.414	−0.694219	−0.347109	−	0.937825i	$-0.612837\pi$
$257$	−178.414	−0.694219	−0.347109	+	0.937825i	$0.612837\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	213.864	0.819400
$262$	0	0
$263$	288.613i	1.09739i	0.836023	+	0.548694i	$0.184875\pi$
$263$	288.613i	1.09739i	−0.836023	+	0.548694i	$0.815125\pi$
$264$	0	0
$265$	−33.3517	−0.125855
$266$	0	0
$267$	− 53.6736i	− 0.201025i
$268$	0	0
$269$	−225.297	−0.837536	−0.418768	−	0.908093i	$-0.637538\pi$
$269$	−225.297	−0.837536	−0.418768	+	0.908093i	$0.637538\pi$
$270$	0	0
$271$	− 195.683i	− 0.722079i	−0.932551	−	0.361039i	$-0.882422\pi$
$271$	− 195.683i	− 0.722079i	0.932551	−	0.361039i	$-0.117578\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	444.503i	1.61637i
$276$	0	0
$277$	−135.628	−0.489633	−0.244817	−	0.969569i	$-0.578728\pi$
$277$	−135.628	−0.489633	−0.244817	+	0.969569i	$0.578728\pi$
$278$	0	0
$279$	44.8607i	0.160791i
$280$	0	0
$281$	−372.495	−1.32560	−0.662802	−	0.748795i	$-0.730633\pi$
$281$	−372.495	−1.32560	−0.662802	+	0.748795i	$0.730633\pi$
$282$	0	0
$283$	− 287.409i	− 1.01558i	−0.861481	−	0.507789i	$-0.830463\pi$
$283$	− 287.409i	− 1.01558i	0.861481	−	0.507789i	$-0.169537\pi$
$284$	0	0
$285$	−178.591	−0.626637
$286$	0	0
$287$	0	0
$288$	0	0
$289$	440.907	1.52563
$290$	0	0
$291$	− 10.3853i	− 0.0356882i
$292$	0	0
$293$	−131.947	−0.450332	−0.225166	−	0.974320i	$-0.572292\pi$
$293$	−131.947	−0.450332	−0.225166	+	0.974320i	$0.572292\pi$
$294$	0	0
$295$	220.026i	0.745850i
$296$	0	0
$297$	276.442	0.930782
$298$	0	0
$299$	− 246.101i	− 0.823080i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 271.176i	− 0.894971i
$304$	0	0
$305$	−3.03544	−0.00995226
$306$	0	0
$307$	47.8253i	0.155783i	0.996962	+	0.0778913i	$0.0248187\pi$
$307$	47.8253i	0.155783i	−0.996962	+	0.0778913i	$0.975181\pi$
$308$	0	0
$309$	−137.478	−0.444913
$310$	0	0
$311$	227.499i	0.731508i	0.930712	+	0.365754i	$0.119189\pi$
$311$	227.499i	0.731508i	−0.930712	+	0.365754i	$0.880811\pi$
$312$	0	0
$313$	−123.097	−0.393280	−0.196640	−	0.980476i	$-0.563003\pi$
$313$	−123.097	−0.393280	−0.196640	+	0.980476i	$0.563003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−480.295	−1.51513	−0.757563	−	0.652762i	$-0.773610\pi$
$317$	−480.295	−1.51513	−0.757563	+	0.652762i	$0.773610\pi$
$318$	0	0
$319$	382.795i	1.19998i
$320$	0	0
$321$	291.646	0.908553
$322$	0	0
$323$	332.537i	1.02953i
$324$	0	0
$325$	248.339	0.764120
$326$	0	0
$327$	− 108.352i	− 0.331352i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	474.392i	1.43321i	0.697480	+	0.716604i	$0.254305\pi$
$331$	474.392i	1.43321i	−0.697480	+	0.716604i	$0.745695\pi$
$332$	0	0
$333$	42.6091	0.127955
$334$	0	0
$335$	− 17.1126i	− 0.0510825i
$336$	0	0
$337$	−305.861	−0.907599	−0.453799	−	0.891104i	$-0.649932\pi$
$337$	−305.861	−0.907599	−0.453799	+	0.891104i	$0.649932\pi$
$338$	0	0
$339$	17.3249i	0.0511059i
$340$	0	0
$341$	−80.2963	−0.235473
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−608.153	−1.76276
$346$	0	0
$347$	688.668i	1.98463i	0.123722	+	0.992317i	$0.460517\pi$
$347$	688.668i	1.98463i	−0.123722	+	0.992317i	$0.539483\pi$
$348$	0	0
$349$	−128.861	−0.369228	−0.184614	−	0.982811i	$-0.559104\pi$
$349$	−128.861	−0.369228	−0.184614	+	0.982811i	$0.559104\pi$
$350$	0	0
$351$	− 154.445i	− 0.440015i
$352$	0	0
$353$	95.0923	0.269383	0.134692	−	0.990888i	$-0.456996\pi$
$353$	95.0923	0.269383	0.134692	+	0.990888i	$0.456996\pi$
$354$	0	0
$355$	999.016i	2.81413i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 36.2145i	− 0.100876i	−0.998727	−	0.0504380i	$-0.983938\pi$
$359$	− 36.2145i	− 0.100876i	0.998727	−	0.0504380i	$-0.0160617\pi$
$360$	0	0
$361$	209.500	0.580332
$362$	0	0
$363$	− 18.6586i	− 0.0514011i
$364$	0	0
$365$	632.850	1.73384
$366$	0	0
$367$	23.8706i	0.0650424i	0.999471	+	0.0325212i	$0.0103536\pi$
$367$	23.8706i	0.0650424i	−0.999471	+	0.0325212i	$0.989646\pi$
$368$	0	0
$369$	282.814	0.766435
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−171.377	−0.459456	−0.229728	−	0.973255i	$-0.573784\pi$
$373$	−171.377	−0.459456	−0.229728	+	0.973255i	$0.573784\pi$
$374$	0	0
$375$	− 250.944i	− 0.669184i
$376$	0	0
$377$	213.864	0.567277
$378$	0	0
$379$	22.8798i	0.0603689i	0.999544	+	0.0301845i	$0.00960947\pi$
$379$	22.8798i	0.0603689i	−0.999544	+	0.0301845i	$0.990391\pi$
$380$	0	0
$381$	222.339	0.583566
$382$	0	0
$383$	564.774i	1.47460i	0.675563	+	0.737302i	$0.263901\pi$
$383$	564.774i	1.47460i	−0.675563	+	0.737302i	$0.736099\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	137.902i	0.356335i
$388$	0	0
$389$	119.522	0.307254	0.153627	−	0.988129i	$-0.450905\pi$
$389$	119.522	0.307254	0.153627	+	0.988129i	$0.450905\pi$
$390$	0	0
$391$	1132.38i	2.89611i
$392$	0	0
$393$	62.9975	0.160299
$394$	0	0
$395$	− 301.966i	− 0.764472i
$396$	0	0
$397$	−525.469	−1.32360	−0.661799	−	0.749681i	$-0.730207\pi$
$397$	−525.469	−1.32360	−0.661799	+	0.749681i	$0.730207\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−166.506	−0.415226	−0.207613	−	0.978211i	$-0.566569\pi$
$401$	−166.506	−0.415226	−0.207613	+	0.978211i	$0.566569\pi$
$402$	0	0
$403$	44.8607i	0.111317i
$404$	0	0
$405$	51.8424	0.128006
$406$	0	0
$407$	76.2662i	0.187386i
$408$	0	0
$409$	−475.345	−1.16221	−0.581106	−	0.813828i	$-0.697380\pi$
$409$	−475.345	−1.16221	−0.581106	+	0.813828i	$0.697380\pi$
$410$	0	0
$411$	211.364i	0.514267i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 806.350i	− 1.94301i
$416$	0	0
$417$	83.8361	0.201046
$418$	0	0
$419$	− 370.547i	− 0.884360i	−0.896926	−	0.442180i	$-0.854205\pi$
$419$	− 370.547i	− 0.884360i	0.896926	−	0.442180i	$-0.145795\pi$
$420$	0	0
$421$	−248.779	−0.590924	−0.295462	−	0.955355i	$-0.595474\pi$
$421$	−248.779	−0.590924	−0.295462	+	0.955355i	$0.595474\pi$
$422$	0	0
$423$	− 347.112i	− 0.820596i
$424$	0	0
$425$	−1142.68	−2.68865
$426$	0	0
$427$	0	0
$428$	0	0
$429$	109.145	0.254416
$430$	0	0
$431$	66.0180i	0.153174i	0.997063	+	0.0765870i	$0.0244023\pi$
$431$	66.0180i	0.153174i	−0.997063	+	0.0765870i	$0.975598\pi$
$432$	0	0
$433$	−753.932	−1.74118	−0.870591	−	0.492007i	$-0.836263\pi$
$433$	−753.932	−1.74118	−0.870591	+	0.492007i	$0.836263\pi$
$434$	0	0
$435$	− 528.489i	− 1.21492i
$436$	0	0
$437$	−515.899	−1.18055
$438$	0	0
$439$	− 37.2376i	− 0.0848238i	−0.999100	−	0.0424119i	$-0.986496\pi$
$439$	− 37.2376i	− 0.0848238i	0.999100	−	0.0424119i	$-0.0135042\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	562.222i	1.26912i	0.772872	+	0.634562i	$0.218820\pi$
$443$	562.222i	1.26912i	−0.772872	+	0.634562i	$0.781180\pi$
$444$	0	0
$445$	248.937	0.559409
$446$	0	0
$447$	334.953i	0.749335i
$448$	0	0
$449$	265.855	0.592105	0.296052	−	0.955172i	$-0.404330\pi$
$449$	265.855	0.592105	0.296052	+	0.955172i	$0.404330\pi$
$450$	0	0
$451$	506.211i	1.12242i
$452$	0	0
$453$	433.884	0.957801
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−66.0818	−0.144599	−0.0722995	−	0.997383i	$-0.523034\pi$
$457$	−66.0818	−0.144599	−0.0722995	+	0.997383i	$0.523034\pi$
$458$	0	0
$459$	710.646i	1.54825i
$460$	0	0
$461$	−59.0572	−0.128107	−0.0640534	−	0.997946i	$-0.520403\pi$
$461$	−59.0572	−0.128107	−0.0640534	+	0.997946i	$0.520403\pi$
$462$	0	0
$463$	390.109i	0.842569i	0.906929	+	0.421285i	$0.138421\pi$
$463$	390.109i	0.842569i	−0.906929	+	0.421285i	$0.861579\pi$
$464$	0	0
$465$	110.858	0.238403
$466$	0	0
$467$	− 361.203i	− 0.773453i	−0.922194	−	0.386727i	$-0.873606\pi$
$467$	− 361.203i	− 0.773453i	0.922194	−	0.386727i	$-0.126394\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	390.392i	0.828858i
$472$	0	0
$473$	−246.831	−0.521841
$474$	0	0
$475$	− 520.591i	− 1.09598i
$476$	0	0
$477$	−23.8716	−0.0500452
$478$	0	0
$479$	− 96.4028i	− 0.201258i	−0.994924	−	0.100629i	$-0.967914\pi$
$479$	− 96.4028i	− 0.201258i	0.994924	−	0.100629i	$-0.0320856\pi$
$480$	0	0
$481$	42.6091	0.0885845
$482$	0	0
$483$	0	0
$484$	0	0
$485$	48.1667	0.0993127
$486$	0	0
$487$	− 56.6871i	− 0.116401i	−0.998305	−	0.0582003i	$-0.981464\pi$
$487$	− 56.6871i	− 0.116401i	0.998305	−	0.0582003i	$-0.0185362\pi$
$488$	0	0
$489$	−481.156	−0.983960
$490$	0	0
$491$	− 559.896i	− 1.14032i	−0.821534	−	0.570159i	$-0.806882\pi$
$491$	− 559.896i	− 1.14032i	0.821534	−	0.570159i	$-0.193118\pi$
$492$	0	0
$493$	−984.046	−1.99604
$494$	0	0
$495$	506.211i	1.02265i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	372.698i	0.746890i	0.927652	+	0.373445i	$0.121824\pi$
$499$	372.698i	0.746890i	−0.927652	+	0.373445i	$0.878176\pi$
$500$	0	0
$501$	−157.183	−0.313738
$502$	0	0
$503$	− 648.033i	− 1.28834i	−0.764884	−	0.644168i	$-0.777204\pi$
$503$	− 648.033i	− 1.28834i	0.764884	−	0.644168i	$-0.222796\pi$
$504$	0	0
$505$	1257.71	2.49052
$506$	0	0
$507$	237.938i	0.469306i
$508$	0	0
$509$	−420.562	−0.826251	−0.413125	−	0.910674i	$-0.635563\pi$
$509$	−420.562	−0.826251	−0.413125	+	0.910674i	$0.635563\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−323.762	−0.631115
$514$	0	0
$515$	− 637.622i	− 1.23810i
$516$	0	0
$517$	621.297	1.20173
$518$	0	0
$519$	− 564.548i	− 1.08776i
$520$	0	0
$521$	−277.664	−0.532943	−0.266472	−	0.963843i	$-0.585858\pi$
$521$	−277.664	−0.532943	−0.266472	+	0.963843i	$0.585858\pi$
$522$	0	0
$523$	653.025i	1.24861i	0.781179	+	0.624307i	$0.214618\pi$
$523$	653.025i	1.24861i	−0.781179	+	0.624307i	$0.785382\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 206.417i	− 0.391683i
$528$	0	0
$529$	−1227.78	−2.32094
$530$	0	0
$531$	157.484i	0.296580i
$532$	0	0
$533$	282.814	0.530609
$534$	0	0
$535$	1352.65i	2.52831i
$536$	0	0
$537$	−246.625	−0.459264
$538$	0	0
$539$	0	0
$540$	0	0
$541$	266.287	0.492213	0.246107	−	0.969243i	$-0.420849\pi$
$541$	266.287	0.492213	0.246107	+	0.969243i	$0.420849\pi$
$542$	0	0
$543$	− 129.352i	− 0.238218i
$544$	0	0
$545$	502.535	0.922082
$546$	0	0
$547$	− 291.492i	− 0.532892i	−0.963850	−	0.266446i	$-0.914151\pi$
$547$	− 291.492i	− 0.532892i	0.963850	−	0.266446i	$-0.0858493\pi$
$548$	0	0
$549$	−2.17263	−0.00395742
$550$	0	0
$551$	− 448.320i	− 0.813648i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 105.294i	− 0.189718i
$556$	0	0
$557$	−576.139	−1.03436	−0.517180	−	0.855876i	$-0.673018\pi$
$557$	−576.139	−1.03436	−0.517180	+	0.855876i	$0.673018\pi$
$558$	0	0
$559$	137.902i	0.246694i
$560$	0	0
$561$	−502.205	−0.895196
$562$	0	0
$563$	575.149i	1.02158i	0.859706	+	0.510790i	$0.170647\pi$
$563$	575.149i	1.02158i	−0.859706	+	0.510790i	$0.829353\pi$
$564$	0	0
$565$	−80.3526	−0.142217
$566$	0	0
$567$	0	0
$568$	0	0
$569$	857.219	1.50654	0.753268	−	0.657714i	$-0.228477\pi$
$569$	857.219	1.50654	0.753268	+	0.657714i	$0.228477\pi$
$570$	0	0
$571$	− 295.010i	− 0.516655i	−0.966057	−	0.258327i	$-0.916829\pi$
$571$	− 295.010i	− 0.516655i	0.966057	−	0.258327i	$-0.0831713\pi$
$572$	0	0
$573$	226.908	0.396000
$574$	0	0
$575$	− 1772.75i	− 3.08305i
$576$	0	0
$577$	566.402	0.981633	0.490817	−	0.871263i	$-0.336698\pi$
$577$	566.402	0.981633	0.490817	+	0.871263i	$0.336698\pi$
$578$	0	0
$579$	− 494.429i	− 0.853936i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 42.7278i	− 0.0732896i
$584$	0	0
$585$	282.814	0.483443
$586$	0	0
$587$	882.011i	1.50257i	0.659975	+	0.751287i	$0.270567\pi$
$587$	882.011i	1.50257i	−0.659975	+	0.751287i	$0.729433\pi$
$588$	0	0
$589$	94.0411	0.159662
$590$	0	0
$591$	175.216i	0.296473i
$592$	0	0
$593$	925.083	1.56001	0.780003	−	0.625776i	$-0.215218\pi$
$593$	925.083	1.56001	0.780003	+	0.625776i	$0.215218\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	417.984	0.700140
$598$	0	0
$599$	151.848i	0.253502i	0.991935	+	0.126751i	$0.0404550\pi$
$599$	151.848i	0.253502i	−0.991935	+	0.126751i	$0.959545\pi$
$600$	0	0
$601$	−485.178	−0.807284	−0.403642	−	0.914917i	$-0.632256\pi$
$601$	−485.178	−0.807284	−0.403642	+	0.914917i	$0.632256\pi$
$602$	0	0
$603$	− 12.2484i	− 0.0203125i
$604$	0	0
$605$	86.5382	0.143038
$606$	0	0
$607$	378.194i	0.623054i	0.950237	+	0.311527i	$0.100840\pi$
$607$	378.194i	0.623054i	−0.950237	+	0.311527i	$0.899160\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 347.112i	− 0.568105i
$612$	0	0
$613$	−614.281	−1.00209	−0.501045	−	0.865421i	$-0.667051\pi$
$613$	−614.281	−1.00209	−0.501045	+	0.865421i	$0.667051\pi$
$614$	0	0
$615$	− 698.877i	− 1.13639i
$616$	0	0
$617$	1193.13	1.93377	0.966883	−	0.255220i	$-0.0821479\pi$
$617$	1193.13	1.93377	0.966883	+	0.255220i	$0.0821479\pi$
$618$	0	0
$619$	488.526i	0.789219i	0.918849	+	0.394609i	$0.129120\pi$
$619$	488.526i	0.789219i	−0.918849	+	0.394609i	$0.870880\pi$
$620$	0	0
$621$	−1102.50	−1.77536
$622$	0	0
$623$	0	0
$624$	0	0
$625$	106.497	0.170396
$626$	0	0
$627$	− 228.799i	− 0.364910i
$628$	0	0
$629$	−196.057	−0.311696
$630$	0	0
$631$	384.907i	0.609995i	0.952353	+	0.304998i	$0.0986557\pi$
$631$	384.907i	0.609995i	−0.952353	+	0.304998i	$0.901344\pi$
$632$	0	0
$633$	−689.246	−1.08886
$634$	0	0
$635$	1031.20i	1.62394i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	715.049i	1.11901i
$640$	0	0
$641$	1093.96	1.70665	0.853325	−	0.521380i	$-0.174583\pi$
$641$	1093.96	1.70665	0.853325	+	0.521380i	$0.174583\pi$
$642$	0	0
$643$	− 866.317i	− 1.34731i	−0.739048	−	0.673653i	$-0.764724\pi$
$643$	− 866.317i	− 1.34731i	0.739048	−	0.673653i	$-0.235276\pi$
$644$	0	0
$645$	340.776	0.528335
$646$	0	0
$647$	− 37.1894i	− 0.0574798i	−0.999587	−	0.0287399i	$-0.990851\pi$
$647$	− 37.1894i	− 0.0574798i	0.999587	−	0.0287399i	$-0.00914946\pi$
$648$	0	0
$649$	−281.882	−0.434332
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1095.91	−1.67827	−0.839137	−	0.543920i	$-0.816939\pi$
$653$	−1095.91	−1.67827	−0.839137	+	0.543920i	$0.816939\pi$
$654$	0	0
$655$	292.181i	0.446079i
$656$	0	0
$657$	452.964	0.689443
$658$	0	0
$659$	− 13.9021i	− 0.0210957i	−0.999944	−	0.0105479i	$-0.996642\pi$
$659$	− 13.9021i	− 0.0210957i	0.999944	−	0.0105479i	$-0.00335755\pi$
$660$	0	0
$661$	101.621	0.153738	0.0768689	−	0.997041i	$-0.475508\pi$
$661$	101.621	0.153738	0.0768689	+	0.997041i	$0.475508\pi$
$662$	0	0
$663$	280.576i	0.423192i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 1526.65i	− 2.28884i
$668$	0	0
$669$	180.798	0.270250
$670$	0	0
$671$	− 3.88879i	− 0.00579551i
$672$	0	0
$673$	−71.7077	−0.106549	−0.0532747	−	0.998580i	$-0.516966\pi$
$673$	−71.7077	−0.106549	−0.0532747	+	0.998580i	$0.516966\pi$
$674$	0	0
$675$	− 1112.52i	− 1.64818i
$676$	0	0
$677$	1050.00	1.55095	0.775477	−	0.631376i	$-0.217509\pi$
$677$	1050.00	1.55095	0.775477	+	0.631376i	$0.217509\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−408.547	−0.599922
$682$	0	0
$683$	133.996i	0.196188i	0.995177	+	0.0980939i	$0.0312745\pi$
$683$	133.996i	0.196188i	−0.995177	+	0.0980939i	$0.968725\pi$
$684$	0	0
$685$	−980.301	−1.43110
$686$	0	0
$687$	− 29.7225i	− 0.0432643i
$688$	0	0
$689$	−23.8716	−0.0346467
$690$	0	0
$691$	− 450.798i	− 0.652385i	−0.945303	−	0.326193i	$-0.894234\pi$
$691$	− 450.798i	− 0.652385i	0.945303	−	0.326193i	$-0.105766\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	388.831i	0.559468i
$696$	0	0
$697$	−1301.31	−1.86701
$698$	0	0
$699$	213.335i	0.305201i
$700$	0	0
$701$	−524.776	−0.748611	−0.374305	−	0.927306i	$-0.622119\pi$
$701$	−524.776	−0.748611	−0.374305	+	0.927306i	$0.622119\pi$
$702$	0	0
$703$	− 89.3211i	− 0.127057i
$704$	0	0
$705$	−857.766	−1.21669
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−44.4372	−0.0626759	−0.0313379	−	0.999509i	$-0.509977\pi$
$709$	−44.4372	−0.0626759	−0.0313379	+	0.999509i	$0.509977\pi$
$710$	0	0
$711$	− 216.133i	− 0.303985i
$712$	0	0
$713$	320.236	0.449138
$714$	0	0
$715$	506.211i	0.707987i
$716$	0	0
$717$	194.319	0.271017
$718$	0	0
$719$	537.563i	0.747654i	0.927499	+	0.373827i	$0.121955\pi$
$719$	537.563i	0.747654i	−0.927499	+	0.373827i	$0.878045\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	327.037i	0.452333i
$724$	0	0
$725$	1540.54	2.12488
$726$	0	0
$727$	168.754i	0.232124i	0.993242	+	0.116062i	$0.0370271\pi$
$727$	168.754i	0.232124i	−0.993242	+	0.116062i	$0.962973\pi$
$728$	0	0
$729$	−381.615	−0.523477
$730$	0	0
$731$	− 634.525i	− 0.868023i
$732$	0	0
$733$	−201.692	−0.275160	−0.137580	−	0.990491i	$-0.543932\pi$
$733$	−201.692	−0.275160	−0.137580	+	0.990491i	$0.543932\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.9235	0.0297470
$738$	0	0
$739$	221.788i	0.300119i	0.988677	+	0.150059i	$0.0479465\pi$
$739$	221.788i	0.300119i	−0.988677	+	0.150059i	$0.952054\pi$
$740$	0	0
$741$	−127.827	−0.172507
$742$	0	0
$743$	344.416i	0.463547i	0.972770	+	0.231774i	$0.0744529\pi$
$743$	344.416i	0.463547i	−0.972770	+	0.231774i	$0.925547\pi$
$744$	0	0
$745$	−1553.51	−2.08524
$746$	0	0
$747$	− 577.147i	− 0.772620i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1347.61i	1.79442i	0.441603	+	0.897211i	$0.354410\pi$
$751$	1347.61i	1.79442i	−0.441603	+	0.897211i	$0.645590\pi$
$752$	0	0
$753$	−397.659	−0.528099
$754$	0	0
$755$	2012.34i	2.66536i
$756$	0	0
$757$	−796.713	−1.05246	−0.526230	−	0.850342i	$-0.676395\pi$
$757$	−796.713	−1.05246	−0.526230	+	0.850342i	$0.676395\pi$
$758$	0	0
$759$	− 779.122i	− 1.02651i
$760$	0	0
$761$	814.559	1.07038	0.535190	−	0.844732i	$-0.320240\pi$
$761$	814.559	1.07038	0.535190	+	0.844732i	$0.320240\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1301.31	−1.70106
$766$	0	0
$767$	157.484i	0.205325i
$768$	0	0
$769$	−60.5653	−0.0787585	−0.0393792	−	0.999224i	$-0.512538\pi$
$769$	−60.5653	−0.0787585	−0.0393792	+	0.999224i	$0.512538\pi$
$770$	0	0
$771$	315.567i	0.409296i
$772$	0	0
$773$	998.420	1.29162	0.645809	−	0.763499i	$-0.276520\pi$
$773$	998.420	1.29162	0.645809	+	0.763499i	$0.276520\pi$
$774$	0	0
$775$	323.148i	0.416965i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 592.861i	− 0.761054i
$780$	0	0
$781$	−1279.87	−1.63876
$782$	0	0
$783$	− 958.079i	− 1.22360i
$784$	0	0
$785$	−1810.63	−2.30654
$786$	0	0
$787$	753.969i	0.958029i	0.877807	+	0.479014i	$0.159006\pi$
$787$	753.969i	0.958029i	−0.877807	+	0.479014i	$0.840994\pi$
$788$	0	0
$789$	510.480	0.646996
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.17263	−0.00273975
$794$	0	0
$795$	58.9903i	0.0742016i
$796$	0	0
$797$	−335.856	−0.421400	−0.210700	−	0.977551i	$-0.567574\pi$
$797$	−335.856	−0.421400	−0.210700	+	0.977551i	$0.567574\pi$
$798$	0	0
$799$	1597.16i	1.99895i
$800$	0	0
$801$	178.178	0.222444
$802$	0	0
$803$	810.762i	1.00967i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	398.491i	0.493793i
$808$	0	0
$809$	749.689	0.926686	0.463343	−	0.886179i	$-0.346650\pi$
$809$	749.689	0.926686	0.463343	+	0.886179i	$0.346650\pi$
$810$	0	0
$811$	790.284i	0.974456i	0.873275	+	0.487228i	$0.161992\pi$
$811$	790.284i	0.974456i	−0.873275	+	0.487228i	$0.838008\pi$
$812$	0	0
$813$	−346.112	−0.425722
$814$	0	0
$815$	− 2231.59i	− 2.73815i
$816$	0	0
$817$	289.082	0.353834
$818$	0	0
$819$	0	0
$820$	0	0
$821$	919.096	1.11948	0.559742	−	0.828667i	$-0.310900\pi$
$821$	919.096	1.11948	0.559742	+	0.828667i	$0.310900\pi$
$822$	0	0
$823$	971.193i	1.18006i	0.807380	+	0.590032i	$0.200885\pi$
$823$	971.193i	1.18006i	−0.807380	+	0.590032i	$0.799115\pi$
$824$	0	0
$825$	786.207	0.952979
$826$	0	0
$827$	110.492i	0.133606i	0.997766	+	0.0668029i	$0.0212799\pi$
$827$	110.492i	0.133606i	−0.997766	+	0.0668029i	$0.978720\pi$
$828$	0	0
$829$	495.972	0.598278	0.299139	−	0.954210i	$-0.403301\pi$
$829$	495.972	0.598278	0.299139	+	0.954210i	$0.403301\pi$
$830$	0	0
$831$	239.891i	0.288677i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 729.011i	− 0.873067i
$836$	0	0
$837$	200.970	0.240107
$838$	0	0
$839$	− 529.377i	− 0.630962i	−0.948932	−	0.315481i	$-0.897834\pi$
$839$	− 529.377i	− 0.630962i	0.948932	−	0.315481i	$-0.102166\pi$
$840$	0	0
$841$	485.672	0.577493
$842$	0	0
$843$	658.844i	0.781547i
$844$	0	0
$845$	−1103.55	−1.30598
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−508.350	−0.598763
$850$	0	0
$851$	− 304.163i	− 0.357418i
$852$	0	0
$853$	−1122.77	−1.31626	−0.658132	−	0.752902i	$-0.728653\pi$
$853$	−1122.77	−1.31626	−0.658132	+	0.752902i	$0.728653\pi$
$854$	0	0
$855$	− 592.861i	− 0.693405i
$856$	0	0
$857$	669.932	0.781717	0.390859	−	0.920451i	$-0.372178\pi$
$857$	669.932	0.781717	0.390859	+	0.920451i	$0.372178\pi$
$858$	0	0
$859$	− 234.046i	− 0.272464i	−0.990677	−	0.136232i	$-0.956501\pi$
$859$	− 234.046i	− 0.272464i	0.990677	−	0.136232i	$-0.0434992\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	51.7036i	0.0599115i	0.999551	+	0.0299557i	$0.00953663\pi$
$863$	51.7036i	0.0599115i	−0.999551	+	0.0299557i	$0.990463\pi$
$864$	0	0
$865$	2618.36	3.02701
$866$	0	0
$867$	− 779.847i	− 0.899478i
$868$	0	0
$869$	386.858	0.445176
$870$	0	0
$871$	− 12.2484i	− 0.0140625i
$872$	0	0
$873$	34.4754	0.0394908
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−108.325	−0.123518	−0.0617590	−	0.998091i	$-0.519671\pi$
$877$	−108.325	−0.123518	−0.0617590	+	0.998091i	$0.519671\pi$
$878$	0	0
$879$	233.379i	0.265506i
$880$	0	0
$881$	−901.643	−1.02343	−0.511716	−	0.859155i	$-0.670990\pi$
$881$	−901.643	−1.02343	−0.511716	+	0.859155i	$0.670990\pi$
$882$	0	0
$883$	391.049i	0.442864i	0.975176	+	0.221432i	$0.0710731\pi$
$883$	391.049i	0.442864i	−0.975176	+	0.221432i	$0.928927\pi$
$884$	0	0
$885$	389.167	0.439737
$886$	0	0
$887$	− 1296.88i	− 1.46209i	−0.682328	−	0.731047i	$-0.739032\pi$
$887$	− 1296.88i	− 1.46209i	0.682328	−	0.731047i	$-0.260968\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	66.4167i	0.0745418i
$892$	0	0
$893$	−727.647	−0.814835
$894$	0	0
$895$	− 1143.84i	− 1.27803i
$896$	0	0
$897$	−435.287	−0.485270
$898$	0	0
$899$	278.287i	0.309552i
$900$	0	0
$901$	109.840	0.121909
$902$	0	0
$903$	0	0
$904$	0	0
$905$	599.934	0.662911
$906$	0	0
$907$	461.509i	0.508831i	0.967095	+	0.254415i	$0.0818830\pi$
$907$	461.509i	0.508831i	−0.967095	+	0.254415i	$0.918117\pi$
$908$	0	0
$909$	900.210	0.990330
$910$	0	0
$911$	119.595i	0.131279i	0.997843	+	0.0656395i	$0.0209087\pi$
$911$	119.595i	0.131279i	−0.997843	+	0.0656395i	$0.979091\pi$
$912$	0	0
$913$	1033.04	1.13148
$914$	0	0
$915$	5.36889i	0.00586764i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	707.698i	0.770074i	0.922901	+	0.385037i	$0.125811\pi$
$919$	707.698i	0.770074i	−0.922901	+	0.385037i	$0.874189\pi$
$920$	0	0
$921$	84.5902	0.0918460
$922$	0	0
$923$	715.049i	0.774701i
$924$	0	0
$925$	306.929	0.331815
$926$	0	0
$927$	− 456.380i	− 0.492319i
$928$	0	0
$929$	397.056	0.427401	0.213701	−	0.976899i	$-0.431448\pi$
$929$	397.056	0.427401	0.213701	+	0.976899i	$0.431448\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	402.385	0.431281
$934$	0	0
$935$	− 2329.22i	− 2.49114i
$936$	0	0
$937$	1227.51	1.31004	0.655020	−	0.755612i	$-0.272660\pi$
$937$	1227.51	1.31004	0.655020	+	0.755612i	$0.272660\pi$
$938$	0	0
$939$	217.725i	0.231869i
$940$	0	0
$941$	687.041	0.730118	0.365059	−	0.930984i	$-0.381049\pi$
$941$	687.041	0.730118	0.365059	+	0.930984i	$0.381049\pi$
$942$	0	0
$943$	− 2018.86i	− 2.14089i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 133.564i	− 0.141039i	−0.997510	−	0.0705194i	$-0.977534\pi$
$947$	− 133.564i	− 0.141039i	0.997510	−	0.0705194i	$-0.0224657\pi$
$948$	0	0
$949$	452.964	0.477307
$950$	0	0
$951$	849.514i	0.893285i
$952$	0	0
$953$	1030.34	1.08115	0.540575	−	0.841296i	$-0.318207\pi$
$953$	1030.34	1.08115	0.540575	+	0.841296i	$0.318207\pi$
$954$	0	0
$955$	1052.40i	1.10198i
$956$	0	0
$957$	677.063	0.707485
$958$	0	0
$959$	0	0
$960$	0	0
$961$	902.626	0.939257
$962$	0	0
$963$	968.161i	1.00536i
$964$	0	0
$965$	2293.15	2.37632
$966$	0	0
$967$	− 337.880i	− 0.349410i	−0.984621	−	0.174705i	$-0.944103\pi$
$967$	− 337.880i	− 0.349410i	0.984621	−	0.174705i	$-0.0558972\pi$
$968$	0	0
$969$	588.170	0.606986
$970$	0	0
$971$	− 1679.34i	− 1.72950i	−0.502202	−	0.864750i	$-0.667477\pi$
$971$	− 1679.34i	− 1.72950i	0.502202	−	0.864750i	$-0.332523\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 439.246i	− 0.450508i
$976$	0	0
$977$	−1109.36	−1.13547	−0.567737	−	0.823210i	$-0.692181\pi$
$977$	−1109.36	−1.13547	−0.567737	+	0.823210i	$0.692181\pi$
$978$	0	0
$979$	318.921i	0.325762i
$980$	0	0
$981$	359.691	0.366657
$982$	0	0
$983$	− 416.403i	− 0.423604i	−0.977313	−	0.211802i	$-0.932067\pi$
$983$	− 416.403i	− 0.423604i	0.977313	−	0.211802i	$-0.0679333\pi$
$984$	0	0
$985$	−812.648	−0.825023
$986$	0	0
$987$	0	0
$988$	0	0
$989$	984.404	0.995353
$990$	0	0
$991$	− 1194.84i	− 1.20569i	−0.797858	−	0.602846i	$-0.794033\pi$
$991$	− 1194.84i	− 1.20569i	0.797858	−	0.602846i	$-0.205967\pi$
$992$	0	0
$993$	839.073	0.844988
$994$	0	0
$995$	1938.60i	1.94834i
$996$	0	0
$997$	−1677.13	−1.68218	−0.841090	−	0.540895i	$-0.818086\pi$
$997$	−1677.13	−1.68218	−0.841090	+	0.540895i	$0.818086\pi$
$998$	0	0
$999$	− 190.883i	− 0.191074i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.d.l.1471.2		6
4.3	odd	2	inner	1568.3.d.l.1471.5		6
7.3	odd	6		224.3.r.d.191.5	yes	12
7.5	odd	6		224.3.r.d.95.2	✓	12
7.6	odd	2		1568.3.d.i.1471.5		6
28.3	even	6		224.3.r.d.191.2	yes	12
28.19	even	6		224.3.r.d.95.5	yes	12
28.27	even	2		1568.3.d.i.1471.2		6
56.3	even	6		448.3.r.f.191.5		12
56.5	odd	6		448.3.r.f.319.5		12
56.19	even	6		448.3.r.f.319.2		12
56.45	odd	6		448.3.r.f.191.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.r.d.95.2	✓	12	7.5	odd	6
224.3.r.d.95.5	yes	12	28.19	even	6
224.3.r.d.191.2	yes	12	28.3	even	6
224.3.r.d.191.5	yes	12	7.3	odd	6
448.3.r.f.191.2		12	56.45	odd	6
448.3.r.f.191.5		12	56.3	even	6
448.3.r.f.319.2		12	56.19	even	6
448.3.r.f.319.5		12	56.5	odd	6
1568.3.d.i.1471.2		6	28.27	even	2
1568.3.d.i.1471.5		6	7.6	odd	2
1568.3.d.l.1471.2		6	1.1	even	1	trivial
1568.3.d.l.1471.5		6	4.3	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.76873i q^{3} +8.20336 q^{5} +5.87158 q^{9} +O(q^{10})\)	\(q-1.76873i q^{3} +8.20336 q^{5} +5.87158 q^{9} +10.5096i q^{11} +5.87158 q^{13} -14.5096i q^{15} -27.0168 q^{17} -12.3085i q^{19} -41.9139i q^{23} +42.2951 q^{25} -26.3039i q^{27} +36.4235 q^{29} +7.64031i q^{31} +18.5886 q^{33} +7.25684 q^{37} -10.3853i q^{39} +48.1667 q^{41} +23.4863i q^{43} +48.1667 q^{45} -59.1173i q^{47} +47.7855i q^{51} -4.06561 q^{53} +86.2137i q^{55} -21.7705 q^{57} +26.8214i q^{59} -0.370024 q^{61} +48.1667 q^{65} -2.08605i q^{67} -74.1346 q^{69} +121.781i q^{71} +77.1452 q^{73} -74.8088i q^{75} -36.8101i q^{79} +6.31965 q^{81} -98.2951i q^{83} -221.628 q^{85} -64.4235i q^{87} +30.3458 q^{89} +13.5137 q^{93} -100.971i q^{95} +5.87158 q^{97} +61.7077i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 18 q^{5}+O(q^{10}) \) 6 * q + 18 * q^5	\( 6 q + 18 q^{5} - 6 q^{17} + 186 q^{33} + 114 q^{37} + 18 q^{53} - 342 q^{57} + 318 q^{61} + 114 q^{69} + 342 q^{73} + 186 q^{81} - 498 q^{85} + 150 q^{89} + 222 q^{93}+O(q^{100}) \) 6 * q + 18 * q^5 - 6 * q^17 + 186 * q^33 + 114 * q^37 + 18 * q^53 - 342 * q^57 + 318 * q^61 + 114 * q^69 + 342 * q^73 + 186 * q^81 - 498 * q^85 + 150 * q^89 + 222 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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