LMFDB - Embedded newform 1568.3.g.j.687.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(687,1568)]

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, 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("1568.687");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.g (of order $2$, degree $1$, 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:	$42.7249054517$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.15582448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 $ x^6 - 3x^5 + 13x^4 - 21x^3 + 20x^2 - 10*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 56)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			687.5
Root			$0.500000 + 0.148124i$ of defining polynomial
Character	$\chi$	$=$	1568.687
Dual form			1568.3.g.j.687.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.98103 q^{3} -1.88252i q^{5} +6.84860 q^{9} +O(q^{10})$	$q+3.98103 q^{3} -1.88252i q^{5} +6.84860 q^{9} +7.87946 q^{11} -11.4863i q^{13} -7.49437i q^{15} -2.89843 q^{17} +30.0447 q^{19} +38.5483i q^{23} +21.4561 q^{25} -8.56478 q^{27} -27.8701i q^{29} +22.4831i q^{31} +31.3684 q^{33} -45.5552i q^{37} -45.7274i q^{39} -40.6313 q^{41} +47.2806 q^{43} -12.8926i q^{45} -82.5809i q^{47} -11.5387 q^{51} +26.8841i q^{53} -14.8332i q^{55} +119.609 q^{57} +10.4111 q^{59} -22.1624i q^{61} -21.6233 q^{65} +59.3099 q^{67} +153.462i q^{69} -38.2541i q^{71} +13.9778 q^{73} +85.4175 q^{75} -51.1895i q^{79} -95.7341 q^{81} +89.4458 q^{83} +5.45635i q^{85} -110.952i q^{87} +105.258 q^{89} +89.5059i q^{93} -56.5597i q^{95} +55.3301 q^{97} +53.9633 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 40 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 40 * q^9	$ 6 q - 6 q^{3} + 40 q^{9} + 30 q^{11} - 30 q^{17} + 78 q^{19} + 92 q^{25} - 78 q^{27} + 78 q^{33} - 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} - 110 q^{59} + 32 q^{65} + 434 q^{67} - 102 q^{73} - 60 q^{75} + 82 q^{81} + 268 q^{83} - 214 q^{89} - 76 q^{97} - 252 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 40 * q^9 + 30 * q^11 - 30 * q^17 + 78 * q^19 + 92 * q^25 - 78 * q^27 + 78 * q^33 - 116 * q^41 + 100 * q^43 + 10 * q^51 + 166 * q^57 - 110 * q^59 + 32 * q^65 + 434 * q^67 - 102 * q^73 - 60 * q^75 + 82 * q^81 + 268 * q^83 - 214 * q^89 - 76 * q^97 - 252 * q^99

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
$3$	3.98103	1.32701	0.663505	−	0.748172i	$-0.269068\pi$
$3$	3.98103	1.32701	0.663505	+	0.748172i	$0.269068\pi$
$4$	0	0
$5$	− 1.88252i	− 0.376504i	−0.982121	−	0.188252i	$-0.939718\pi$
$5$	− 1.88252i	− 0.376504i	0.982121	−	0.188252i	$-0.0602822\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.84860	0.760956
$10$	0	0
$11$	7.87946	0.716314	0.358157	−	0.933661i	$-0.383405\pi$
$11$	7.87946	0.716314	0.358157	+	0.933661i	$0.383405\pi$
$12$	0	0
$13$	− 11.4863i	− 0.883564i	−0.897122	−	0.441782i	$-0.854346\pi$
$13$	− 11.4863i	− 0.883564i	0.897122	−	0.441782i	$-0.145654\pi$
$14$	0	0
$15$	− 7.49437i	− 0.499625i
$16$	0	0
$17$	−2.89843	−0.170496	−0.0852478	−	0.996360i	$-0.527168\pi$
$17$	−2.89843	−0.170496	−0.0852478	+	0.996360i	$0.527168\pi$
$18$	0	0
$19$	30.0447	1.58130	0.790649	−	0.612270i	$-0.209743\pi$
$19$	30.0447	1.58130	0.790649	+	0.612270i	$0.209743\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	38.5483i	1.67601i	0.545661	+	0.838006i	$0.316279\pi$
$23$	38.5483i	1.67601i	−0.545661	+	0.838006i	$0.683721\pi$
$24$	0	0
$25$	21.4561	0.858245
$26$	0	0
$27$	−8.56478	−0.317214
$28$	0	0
$29$	− 27.8701i	− 0.961039i	−0.876984	−	0.480519i	$-0.840448\pi$
$29$	− 27.8701i	− 0.961039i	0.876984	−	0.480519i	$-0.159552\pi$
$30$	0	0
$31$	22.4831i	0.725262i	0.931933	+	0.362631i	$0.118121\pi$
$31$	22.4831i	0.725262i	−0.931933	+	0.362631i	$0.881879\pi$
$32$	0	0
$33$	31.3684	0.950556
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 45.5552i	− 1.23122i	−0.788050	−	0.615611i	$-0.788909\pi$
$37$	− 45.5552i	− 1.23122i	0.788050	−	0.615611i	$-0.211091\pi$
$38$	0	0
$39$	− 45.7274i	− 1.17250i
$40$	0	0
$41$	−40.6313	−0.991007	−0.495503	−	0.868606i	$-0.665016\pi$
$41$	−40.6313	−0.991007	−0.495503	+	0.868606i	$0.665016\pi$
$42$	0	0
$43$	47.2806	1.09955	0.549774	−	0.835313i	$-0.314714\pi$
$43$	47.2806	1.09955	0.549774	+	0.835313i	$0.314714\pi$
$44$	0	0
$45$	− 12.8926i	− 0.286503i
$46$	0	0
$47$	− 82.5809i	− 1.75704i	−0.477705	−	0.878520i	$-0.658531\pi$
$47$	− 82.5809i	− 1.75704i	0.477705	−	0.878520i	$-0.341469\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−11.5387	−0.226249
$52$	0	0
$53$	26.8841i	0.507247i	0.967303	+	0.253623i	$0.0816224\pi$
$53$	26.8841i	0.507247i	−0.967303	+	0.253623i	$0.918378\pi$
$54$	0	0
$55$	− 14.8332i	− 0.269695i
$56$	0	0
$57$	119.609	2.09840
$58$	0	0
$59$	10.4111	0.176459	0.0882296	−	0.996100i	$-0.471879\pi$
$59$	10.4111	0.176459	0.0882296	+	0.996100i	$0.471879\pi$
$60$	0	0
$61$	− 22.1624i	− 0.363318i	−0.983362	−	0.181659i	$-0.941853\pi$
$61$	− 22.1624i	− 0.363318i	0.983362	−	0.181659i	$-0.0581467\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−21.6233	−0.332665
$66$	0	0
$67$	59.3099	0.885222	0.442611	−	0.896714i	$-0.354052\pi$
$67$	59.3099	0.885222	0.442611	+	0.896714i	$0.354052\pi$
$68$	0	0
$69$	153.462i	2.22409i
$70$	0	0
$71$	− 38.2541i	− 0.538791i	−0.963030	−	0.269395i	$-0.913176\pi$
$71$	− 38.2541i	− 0.538791i	0.963030	−	0.269395i	$-0.0868238\pi$
$72$	0	0
$73$	13.9778	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$73$	13.9778	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$74$	0	0
$75$	85.4175	1.13890
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 51.1895i	− 0.647969i	−0.946062	−	0.323984i	$-0.894978\pi$
$79$	− 51.1895i	− 0.647969i	0.946062	−	0.323984i	$-0.105022\pi$
$80$	0	0
$81$	−95.7341	−1.18190
$82$	0	0
$83$	89.4458	1.07766	0.538830	−	0.842414i	$-0.318866\pi$
$83$	89.4458	1.07766	0.538830	+	0.842414i	$0.318866\pi$
$84$	0	0
$85$	5.45635i	0.0641923i
$86$	0	0
$87$	− 110.952i	− 1.27531i
$88$	0	0
$89$	105.258	1.18267	0.591335	−	0.806426i	$-0.298601\pi$
$89$	105.258	1.18267	0.591335	+	0.806426i	$0.298601\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	89.5059i	0.962429i
$94$	0	0
$95$	− 56.5597i	− 0.595365i
$96$	0	0
$97$	55.3301	0.570413	0.285206	−	0.958466i	$-0.407938\pi$
$97$	55.3301	0.570413	0.285206	+	0.958466i	$0.407938\pi$
$98$	0	0
$99$	53.9633	0.545084
$100$	0	0
$101$	− 31.4328i	− 0.311216i	−0.987819	−	0.155608i	$-0.950266\pi$
$101$	− 31.4328i	− 0.311216i	0.987819	−	0.155608i	$-0.0497337\pi$
$102$	0	0
$103$	− 79.9862i	− 0.776565i	−0.921540	−	0.388283i	$-0.873068\pi$
$103$	− 79.9862i	− 0.776565i	0.921540	−	0.388283i	$-0.126932\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−48.7634	−0.455732	−0.227866	−	0.973692i	$-0.573175\pi$
$107$	−48.7634	−0.455732	−0.227866	+	0.973692i	$0.573175\pi$
$108$	0	0
$109$	115.069i	1.05568i	0.849344	+	0.527840i	$0.176998\pi$
$109$	115.069i	1.05568i	−0.849344	+	0.527840i	$0.823002\pi$
$110$	0	0
$111$	− 181.357i	− 1.63384i
$112$	0	0
$113$	−55.7570	−0.493425	−0.246712	−	0.969089i	$-0.579350\pi$
$113$	−55.7570	−0.493425	−0.246712	+	0.969089i	$0.579350\pi$
$114$	0	0
$115$	72.5679	0.631025
$116$	0	0
$117$	− 78.6653i	− 0.672353i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−58.9142	−0.486894
$122$	0	0
$123$	−161.754	−1.31508
$124$	0	0
$125$	− 87.4546i	− 0.699637i
$126$	0	0
$127$	35.6964i	0.281074i	0.990075	+	0.140537i	$0.0448828\pi$
$127$	35.6964i	0.281074i	−0.990075	+	0.140537i	$0.955117\pi$
$128$	0	0
$129$	188.225	1.45911
$130$	0	0
$131$	121.292	0.925896	0.462948	−	0.886385i	$-0.346792\pi$
$131$	121.292	0.925896	0.462948	+	0.886385i	$0.346792\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	16.1234i	0.119432i
$136$	0	0
$137$	−8.49670	−0.0620197	−0.0310099	−	0.999519i	$-0.509872\pi$
$137$	−8.49670	−0.0620197	−0.0310099	+	0.999519i	$0.509872\pi$
$138$	0	0
$139$	3.05942	0.0220102	0.0110051	−	0.999939i	$-0.496497\pi$
$139$	3.05942	0.0220102	0.0110051	+	0.999939i	$0.496497\pi$
$140$	0	0
$141$	− 328.757i	− 2.33161i
$142$	0	0
$143$	− 90.5061i	− 0.632910i
$144$	0	0
$145$	−52.4661	−0.361835
$146$	0	0
$147$	0	0
$148$	0	0
$149$	31.7242i	0.212914i	0.994317	+	0.106457i	$0.0339507\pi$
$149$	31.7242i	0.212914i	−0.994317	+	0.106457i	$0.966049\pi$
$150$	0	0
$151$	253.596i	1.67945i	0.543016	+	0.839723i	$0.317282\pi$
$151$	253.596i	1.67945i	−0.543016	+	0.839723i	$0.682718\pi$
$152$	0	0
$153$	−19.8502	−0.129740
$154$	0	0
$155$	42.3249	0.273064
$156$	0	0
$157$	49.3273i	0.314186i	0.987584	+	0.157093i	$0.0502123\pi$
$157$	49.3273i	0.314186i	−0.987584	+	0.157093i	$0.949788\pi$
$158$	0	0
$159$	107.026i	0.673121i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−115.719	−0.709934	−0.354967	−	0.934879i	$-0.615508\pi$
$163$	−115.719	−0.709934	−0.354967	+	0.934879i	$0.615508\pi$
$164$	0	0
$165$	− 59.0516i	− 0.357888i
$166$	0	0
$167$	32.3859i	0.193928i	0.995288	+	0.0969639i	$0.0309131\pi$
$167$	32.3859i	0.193928i	−0.995288	+	0.0969639i	$0.969087\pi$
$168$	0	0
$169$	37.0641	0.219314
$170$	0	0
$171$	205.764	1.20330
$172$	0	0
$173$	207.245i	1.19795i	0.800769	+	0.598973i	$0.204424\pi$
$173$	207.245i	1.19795i	−0.800769	+	0.598973i	$0.795576\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	41.4469	0.234163
$178$	0	0
$179$	−174.167	−0.973002	−0.486501	−	0.873680i	$-0.661727\pi$
$179$	−174.167	−0.973002	−0.486501	+	0.873680i	$0.661727\pi$
$180$	0	0
$181$	− 204.244i	− 1.12842i	−0.825632	−	0.564209i	$-0.809181\pi$
$181$	− 204.244i	− 1.12842i	0.825632	−	0.564209i	$-0.190819\pi$
$182$	0	0
$183$	− 88.2291i	− 0.482126i
$184$	0	0
$185$	−85.7586	−0.463560
$186$	0	0
$187$	−22.8380	−0.122128
$188$	0	0
$189$	0	0
$190$	0	0
$191$	298.511i	1.56288i	0.623978	+	0.781442i	$0.285515\pi$
$191$	298.511i	1.56288i	−0.623978	+	0.781442i	$0.714485\pi$
$192$	0	0
$193$	−330.762	−1.71379	−0.856896	−	0.515490i	$-0.827610\pi$
$193$	−330.762	−1.71379	−0.856896	+	0.515490i	$0.827610\pi$
$194$	0	0
$195$	−86.0828	−0.441450
$196$	0	0
$197$	327.309i	1.66146i	0.556672	+	0.830732i	$0.312078\pi$
$197$	327.309i	1.66146i	−0.556672	+	0.830732i	$0.687922\pi$
$198$	0	0
$199$	12.7358i	0.0639989i	0.999488	+	0.0319994i	$0.0101875\pi$
$199$	12.7358i	0.0639989i	−0.999488	+	0.0319994i	$0.989813\pi$
$200$	0	0
$201$	236.114	1.17470
$202$	0	0
$203$	0	0
$204$	0	0
$205$	76.4892i	0.373118i
$206$	0	0
$207$	264.002i	1.27537i
$208$	0	0
$209$	236.736	1.13271
$210$	0	0
$211$	−120.455	−0.570875	−0.285437	−	0.958397i	$-0.592139\pi$
$211$	−120.455	−0.570875	−0.285437	+	0.958397i	$0.592139\pi$
$212$	0	0
$213$	− 152.291i	− 0.714981i
$214$	0	0
$215$	− 89.0067i	− 0.413984i
$216$	0	0
$217$	0	0
$218$	0	0
$219$	55.6460	0.254092
$220$	0	0
$221$	33.2923i	0.150644i
$222$	0	0
$223$	372.958i	1.67246i	0.548382	+	0.836228i	$0.315244\pi$
$223$	372.958i	1.67246i	−0.548382	+	0.836228i	$0.684756\pi$
$224$	0	0
$225$	146.944	0.653086
$226$	0	0
$227$	−73.4256	−0.323461	−0.161730	−	0.986835i	$-0.551708\pi$
$227$	−73.4256	−0.323461	−0.161730	+	0.986835i	$0.551708\pi$
$228$	0	0
$229$	− 424.453i	− 1.85350i	−0.375673	−	0.926752i	$-0.622588\pi$
$229$	− 424.453i	− 1.85350i	0.375673	−	0.926752i	$-0.377412\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−83.4140	−0.358000	−0.179000	−	0.983849i	$-0.557286\pi$
$233$	−83.4140	−0.358000	−0.179000	+	0.983849i	$0.557286\pi$
$234$	0	0
$235$	−155.460	−0.661533
$236$	0	0
$237$	− 203.787i	− 0.859861i
$238$	0	0
$239$	− 112.561i	− 0.470967i	−0.971878	−	0.235484i	$-0.924333\pi$
$239$	− 112.561i	− 0.470967i	0.971878	−	0.235484i	$-0.0756674\pi$
$240$	0	0
$241$	280.432	1.16362	0.581809	−	0.813325i	$-0.302345\pi$
$241$	280.432	1.16362	0.581809	+	0.813325i	$0.302345\pi$
$242$	0	0
$243$	−304.037	−1.25118
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 345.103i	− 1.39718i
$248$	0	0
$249$	356.086	1.43007
$250$	0	0
$251$	32.9560	0.131299	0.0656493	−	0.997843i	$-0.479088\pi$
$251$	32.9560	0.131299	0.0656493	+	0.997843i	$0.479088\pi$
$252$	0	0
$253$	303.740i	1.20055i
$254$	0	0
$255$	21.7219i	0.0851838i
$256$	0	0
$257$	−224.219	−0.872446	−0.436223	−	0.899839i	$-0.643684\pi$
$257$	−224.219	−0.872446	−0.436223	+	0.899839i	$0.643684\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	− 190.871i	− 0.731308i
$262$	0	0
$263$	− 169.961i	− 0.646239i	−0.946358	−	0.323119i	$-0.895268\pi$
$263$	− 169.961i	− 0.646239i	0.946358	−	0.323119i	$-0.104732\pi$
$264$	0	0
$265$	50.6098	0.190980
$266$	0	0
$267$	419.034	1.56942
$268$	0	0
$269$	− 108.526i	− 0.403442i	−0.979443	−	0.201721i	$-0.935347\pi$
$269$	− 108.526i	− 0.403442i	0.979443	−	0.201721i	$-0.0646535\pi$
$270$	0	0
$271$	− 19.3631i	− 0.0714507i	−0.999362	−	0.0357253i	$-0.988626\pi$
$271$	− 19.3631i	− 0.0714507i	0.999362	−	0.0357253i	$-0.0113741\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	169.063	0.614773
$276$	0	0
$277$	− 129.446i	− 0.467315i	−0.972319	−	0.233658i	$-0.924931\pi$
$277$	− 129.446i	− 0.467315i	0.972319	−	0.233658i	$-0.0750695\pi$
$278$	0	0
$279$	153.978i	0.551892i
$280$	0	0
$281$	83.3608	0.296658	0.148329	−	0.988938i	$-0.452611\pi$
$281$	83.3608	0.296658	0.148329	+	0.988938i	$0.452611\pi$
$282$	0	0
$283$	−28.9293	−0.102224	−0.0511118	−	0.998693i	$-0.516276\pi$
$283$	−28.9293	−0.102224	−0.0511118	+	0.998693i	$0.516276\pi$
$284$	0	0
$285$	− 225.166i	− 0.790055i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−280.599	−0.970931
$290$	0	0
$291$	220.271	0.756944
$292$	0	0
$293$	214.613i	0.732468i	0.930523	+	0.366234i	$0.119353\pi$
$293$	214.613i	0.732468i	−0.930523	+	0.366234i	$0.880647\pi$
$294$	0	0
$295$	− 19.5991i	− 0.0664376i
$296$	0	0
$297$	−67.4858	−0.227225
$298$	0	0
$299$	442.778	1.48086
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 125.135i	− 0.412987i
$304$	0	0
$305$	−41.7211	−0.136791
$306$	0	0
$307$	120.542	0.392644	0.196322	−	0.980539i	$-0.437100\pi$
$307$	120.542	0.392644	0.196322	+	0.980539i	$0.437100\pi$
$308$	0	0
$309$	− 318.427i	− 1.03051i
$310$	0	0
$311$	325.361i	1.04618i	0.852278	+	0.523089i	$0.175220\pi$
$311$	325.361i	1.04618i	−0.852278	+	0.523089i	$0.824780\pi$
$312$	0	0
$313$	456.756	1.45928	0.729642	−	0.683829i	$-0.239687\pi$
$313$	456.756	1.45928	0.729642	+	0.683829i	$0.239687\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	121.220i	0.382399i	0.981551	+	0.191199i	$0.0612377\pi$
$317$	121.220i	0.382399i	−0.981551	+	0.191199i	$0.938762\pi$
$318$	0	0
$319$	− 219.601i	− 0.688406i
$320$	0	0
$321$	−194.128	−0.604761
$322$	0	0
$323$	−87.0822	−0.269604
$324$	0	0
$325$	− 246.452i	− 0.758314i
$326$	0	0
$327$	458.094i	1.40090i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−121.168	−0.366068	−0.183034	−	0.983107i	$-0.558592\pi$
$331$	−121.168	−0.366068	−0.183034	+	0.983107i	$0.558592\pi$
$332$	0	0
$333$	− 311.989i	− 0.936905i
$334$	0	0
$335$	− 111.652i	− 0.333290i
$336$	0	0
$337$	−464.021	−1.37692	−0.688459	−	0.725275i	$-0.741713\pi$
$337$	−464.021	−1.37692	−0.688459	+	0.725275i	$0.741713\pi$
$338$	0	0
$339$	−221.970	−0.654780
$340$	0	0
$341$	177.155i	0.519515i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	288.895	0.837377
$346$	0	0
$347$	371.186	1.06970	0.534851	−	0.844947i	$-0.320368\pi$
$347$	371.186	1.06970	0.534851	+	0.844947i	$0.320368\pi$
$348$	0	0
$349$	− 207.871i	− 0.595619i	−0.954625	−	0.297809i	$-0.903744\pi$
$349$	− 207.871i	− 0.595619i	0.954625	−	0.297809i	$-0.0962560\pi$
$350$	0	0
$351$	98.3779i	0.280279i
$352$	0	0
$353$	−614.014	−1.73942	−0.869708	−	0.493567i	$-0.835693\pi$
$353$	−614.014	−1.73942	−0.869708	+	0.493567i	$0.835693\pi$
$354$	0	0
$355$	−72.0142	−0.202857
$356$	0	0
$357$	0	0
$358$	0	0
$359$	108.072i	0.301036i	0.988607	+	0.150518i	$0.0480941\pi$
$359$	108.072i	0.301036i	−0.988607	+	0.150518i	$0.951906\pi$
$360$	0	0
$361$	541.682	1.50050
$362$	0	0
$363$	−234.539	−0.646113
$364$	0	0
$365$	− 26.3135i	− 0.0720917i
$366$	0	0
$367$	454.815i	1.23928i	0.784887	+	0.619639i	$0.212721\pi$
$367$	454.815i	1.23928i	−0.784887	+	0.619639i	$0.787279\pi$
$368$	0	0
$369$	−278.268	−0.754113
$370$	0	0
$371$	0	0
$372$	0	0
$373$	271.752i	0.728557i	0.931290	+	0.364279i	$0.118684\pi$
$373$	271.752i	0.728557i	−0.931290	+	0.364279i	$0.881316\pi$
$374$	0	0
$375$	− 348.159i	− 0.928425i
$376$	0	0
$377$	−320.126	−0.849140
$378$	0	0
$379$	−268.351	−0.708051	−0.354026	−	0.935236i	$-0.615187\pi$
$379$	−268.351	−0.708051	−0.354026	+	0.935236i	$0.615187\pi$
$380$	0	0
$381$	142.108i	0.372988i
$382$	0	0
$383$	− 327.610i	− 0.855377i	−0.903926	−	0.427689i	$-0.859328\pi$
$383$	− 327.610i	− 0.855377i	0.903926	−	0.427689i	$-0.140672\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	323.806	0.836708
$388$	0	0
$389$	126.301i	0.324682i	0.986735	+	0.162341i	$0.0519044\pi$
$389$	126.301i	0.324682i	−0.986735	+	0.162341i	$0.948096\pi$
$390$	0	0
$391$	− 111.729i	− 0.285753i
$392$	0	0
$393$	482.869	1.22867
$394$	0	0
$395$	−96.3653	−0.243963
$396$	0	0
$397$	− 127.144i	− 0.320262i	−0.987096	−	0.160131i	$-0.948808\pi$
$397$	− 127.144i	− 0.320262i	0.987096	−	0.160131i	$-0.0511917\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	60.1502	0.150001	0.0750003	−	0.997184i	$-0.476104\pi$
$401$	60.1502	0.150001	0.0750003	+	0.997184i	$0.476104\pi$
$402$	0	0
$403$	258.248	0.640815
$404$	0	0
$405$	180.221i	0.444991i
$406$	0	0
$407$	− 358.950i	− 0.881941i
$408$	0	0
$409$	−69.7747	−0.170598	−0.0852991	−	0.996355i	$-0.527185\pi$
$409$	−69.7747	−0.170598	−0.0852991	+	0.996355i	$0.527185\pi$
$410$	0	0
$411$	−33.8256	−0.0823008
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 168.384i	− 0.405743i
$416$	0	0
$417$	12.1796	0.0292078
$418$	0	0
$419$	−714.794	−1.70595	−0.852976	−	0.521950i	$-0.825205\pi$
$419$	−714.794	−1.70595	−0.852976	+	0.521950i	$0.825205\pi$
$420$	0	0
$421$	− 303.440i	− 0.720759i	−0.932806	−	0.360380i	$-0.882647\pi$
$421$	− 303.440i	− 0.720759i	0.932806	−	0.360380i	$-0.117353\pi$
$422$	0	0
$423$	− 565.564i	− 1.33703i
$424$	0	0
$425$	−62.1890	−0.146327
$426$	0	0
$427$	0	0
$428$	0	0
$429$	− 360.307i	− 0.839877i
$430$	0	0
$431$	431.495i	1.00115i	0.865694	+	0.500574i	$0.166878\pi$
$431$	431.495i	1.00115i	−0.865694	+	0.500574i	$0.833122\pi$
$432$	0	0
$433$	194.875	0.450057	0.225029	−	0.974352i	$-0.427752\pi$
$433$	194.875	0.450057	0.225029	+	0.974352i	$0.427752\pi$
$434$	0	0
$435$	−208.869	−0.480159
$436$	0	0
$437$	1158.17i	2.65028i
$438$	0	0
$439$	305.970i	0.696969i	0.937314	+	0.348485i	$0.113304\pi$
$439$	305.970i	0.696969i	−0.937314	+	0.348485i	$0.886696\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−250.197	−0.564779	−0.282390	−	0.959300i	$-0.591127\pi$
$443$	−250.197	−0.564779	−0.282390	+	0.959300i	$0.591127\pi$
$444$	0	0
$445$	− 198.150i	− 0.445280i
$446$	0	0
$447$	126.295i	0.282539i
$448$	0	0
$449$	−688.681	−1.53381	−0.766905	−	0.641761i	$-0.778204\pi$
$449$	−688.681	−1.53381	−0.766905	+	0.641761i	$0.778204\pi$
$450$	0	0
$451$	−320.152	−0.709872
$452$	0	0
$453$	1009.57i	2.22864i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	804.518	1.76043	0.880217	−	0.474572i	$-0.157397\pi$
$457$	804.518	1.76043	0.880217	+	0.474572i	$0.157397\pi$
$458$	0	0
$459$	24.8244	0.0540836
$460$	0	0
$461$	693.657i	1.50468i	0.658776	+	0.752339i	$0.271074\pi$
$461$	693.657i	1.50468i	−0.658776	+	0.752339i	$0.728926\pi$
$462$	0	0
$463$	321.194i	0.693724i	0.937916	+	0.346862i	$0.112753\pi$
$463$	321.194i	0.693724i	−0.937916	+	0.346862i	$0.887247\pi$
$464$	0	0
$465$	168.497	0.362359
$466$	0	0
$467$	−751.875	−1.61001	−0.805005	−	0.593268i	$-0.797837\pi$
$467$	−751.875	−1.61001	−0.805005	+	0.593268i	$0.797837\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	196.373i	0.416929i
$472$	0	0
$473$	372.545	0.787622
$474$	0	0
$475$	644.642	1.35714
$476$	0	0
$477$	184.118i	0.385992i
$478$	0	0
$479$	− 722.140i	− 1.50760i	−0.657104	−	0.753800i	$-0.728219\pi$
$479$	− 722.140i	− 1.50760i	0.657104	−	0.753800i	$-0.271781\pi$
$480$	0	0
$481$	−523.262	−1.08786
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 104.160i	− 0.214763i
$486$	0	0
$487$	− 47.8290i	− 0.0982114i	−0.998794	−	0.0491057i	$-0.984363\pi$
$487$	− 47.8290i	− 0.0982114i	0.998794	−	0.0491057i	$-0.0156371\pi$
$488$	0	0
$489$	−460.682	−0.942090
$490$	0	0
$491$	−44.4724	−0.0905752	−0.0452876	−	0.998974i	$-0.514420\pi$
$491$	−44.4724	−0.0905752	−0.0452876	+	0.998974i	$0.514420\pi$
$492$	0	0
$493$	80.7795i	0.163853i
$494$	0	0
$495$	− 101.587i	− 0.205226i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	501.572	1.00515	0.502577	−	0.864532i	$-0.332385\pi$
$499$	501.572	1.00515	0.502577	+	0.864532i	$0.332385\pi$
$500$	0	0
$501$	128.929i	0.257344i
$502$	0	0
$503$	− 462.733i	− 0.919946i	−0.887933	−	0.459973i	$-0.847859\pi$
$503$	− 462.733i	− 0.919946i	0.887933	−	0.459973i	$-0.152141\pi$
$504$	0	0
$505$	−59.1729	−0.117174
$506$	0	0
$507$	147.553	0.291032
$508$	0	0
$509$	471.985i	0.927278i	0.886024	+	0.463639i	$0.153457\pi$
$509$	471.985i	0.927278i	−0.886024	+	0.463639i	$0.846543\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−257.326	−0.501610
$514$	0	0
$515$	−150.576	−0.292380
$516$	0	0
$517$	− 650.693i	− 1.25859i
$518$	0	0
$519$	825.047i	1.58969i
$520$	0	0
$521$	−90.7419	−0.174169	−0.0870843	−	0.996201i	$-0.527755\pi$
$521$	−90.7419	−0.174169	−0.0870843	+	0.996201i	$0.527755\pi$
$522$	0	0
$523$	360.512	0.689316	0.344658	−	0.938728i	$-0.387995\pi$
$523$	360.512	0.689316	0.344658	+	0.938728i	$0.387995\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 65.1656i	− 0.123654i
$528$	0	0
$529$	−956.970	−1.80902
$530$	0	0
$531$	71.3014	0.134278
$532$	0	0
$533$	466.705i	0.875618i
$534$	0	0
$535$	91.7980i	0.171585i
$536$	0	0
$537$	−693.366	−1.29118
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 561.148i	− 1.03724i	−0.855004	−	0.518621i	$-0.826445\pi$
$541$	− 561.148i	− 1.03724i	0.855004	−	0.518621i	$-0.173555\pi$
$542$	0	0
$543$	− 813.101i	− 1.49742i
$544$	0	0
$545$	216.620	0.397468
$546$	0	0
$547$	−1043.62	−1.90790	−0.953952	−	0.299960i	$-0.903027\pi$
$547$	−1043.62	−1.90790	−0.953952	+	0.299960i	$0.903027\pi$
$548$	0	0
$549$	− 151.781i	− 0.276469i
$550$	0	0
$551$	− 837.349i	− 1.51969i
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−341.407	−0.615148
$556$	0	0
$557$	− 48.6444i	− 0.0873328i	−0.999046	−	0.0436664i	$-0.986096\pi$
$557$	− 48.6444i	− 0.0873328i	0.999046	−	0.0436664i	$-0.0139039\pi$
$558$	0	0
$559$	− 543.081i	− 0.971522i
$560$	0	0
$561$	−90.9189	−0.162066
$562$	0	0
$563$	−726.031	−1.28957	−0.644787	−	0.764362i	$-0.723054\pi$
$563$	−726.031	−1.28957	−0.644787	+	0.764362i	$0.723054\pi$
$564$	0	0
$565$	104.964i	0.185776i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−481.392	−0.846032	−0.423016	−	0.906122i	$-0.639029\pi$
$569$	−481.392	−0.846032	−0.423016	+	0.906122i	$0.639029\pi$
$570$	0	0
$571$	−110.127	−0.192866	−0.0964331	−	0.995339i	$-0.530743\pi$
$571$	−110.127	−0.192866	−0.0964331	+	0.995339i	$0.530743\pi$
$572$	0	0
$573$	1188.38i	2.07396i
$574$	0	0
$575$	827.097i	1.43843i
$576$	0	0
$577$	−202.336	−0.350669	−0.175335	−	0.984509i	$-0.556101\pi$
$577$	−202.336	−0.350669	−0.175335	+	0.984509i	$0.556101\pi$
$578$	0	0
$579$	−1316.77	−2.27422
$580$	0	0
$581$	0	0
$582$	0	0
$583$	211.832i	0.363348i
$584$	0	0
$585$	−148.089	−0.253144
$586$	0	0
$587$	568.689	0.968805	0.484403	−	0.874845i	$-0.339037\pi$
$587$	568.689	0.968805	0.484403	+	0.874845i	$0.339037\pi$
$588$	0	0
$589$	675.497i	1.14685i
$590$	0	0
$591$	1303.03i	2.20478i
$592$	0	0
$593$	−685.373	−1.15577	−0.577886	−	0.816117i	$-0.696122\pi$
$593$	−685.373	−1.15577	−0.577886	+	0.816117i	$0.696122\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	50.7015i	0.0849271i
$598$	0	0
$599$	367.283i	0.613161i	0.951845	+	0.306580i	$0.0991848\pi$
$599$	367.283i	0.613161i	−0.951845	+	0.306580i	$0.900815\pi$
$600$	0	0
$601$	412.344	0.686097	0.343049	−	0.939318i	$-0.388540\pi$
$601$	412.344	0.686097	0.343049	+	0.939318i	$0.388540\pi$
$602$	0	0
$603$	406.190	0.673615
$604$	0	0
$605$	110.907i	0.183318i
$606$	0	0
$607$	1166.78i	1.92220i	0.276195	+	0.961102i	$0.410926\pi$
$607$	1166.78i	1.92220i	−0.276195	+	0.961102i	$0.589074\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−948.552	−1.55246
$612$	0	0
$613$	431.743i	0.704311i	0.935942	+	0.352155i	$0.114551\pi$
$613$	431.743i	0.704311i	−0.935942	+	0.352155i	$0.885449\pi$
$614$	0	0
$615$	304.506i	0.495131i
$616$	0	0
$617$	−333.751	−0.540926	−0.270463	−	0.962730i	$-0.587177\pi$
$617$	−333.751	−0.540926	−0.270463	+	0.962730i	$0.587177\pi$
$618$	0	0
$619$	976.316	1.57725	0.788624	−	0.614876i	$-0.210794\pi$
$619$	976.316	1.57725	0.788624	+	0.614876i	$0.210794\pi$
$620$	0	0
$621$	− 330.157i	− 0.531655i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	371.768	0.594829
$626$	0	0
$627$	942.452	1.50311
$628$	0	0
$629$	132.038i	0.209918i
$630$	0	0
$631$	639.885i	1.01408i	0.861922	+	0.507040i	$0.169261\pi$
$631$	639.885i	1.01408i	−0.861922	+	0.507040i	$0.830739\pi$
$632$	0	0
$633$	−479.533	−0.757556
$634$	0	0
$635$	67.1991	0.105825
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 261.987i	− 0.409996i
$640$	0	0
$641$	20.8591	0.0325415	0.0162707	−	0.999868i	$-0.494821\pi$
$641$	20.8591	0.0325415	0.0162707	+	0.999868i	$0.494821\pi$
$642$	0	0
$643$	69.1348	0.107519	0.0537596	−	0.998554i	$-0.482880\pi$
$643$	69.1348	0.107519	0.0537596	+	0.998554i	$0.482880\pi$
$644$	0	0
$645$	− 354.338i	− 0.549362i
$646$	0	0
$647$	358.959i	0.554806i	0.960754	+	0.277403i	$0.0894737\pi$
$647$	358.959i	0.554806i	−0.960754	+	0.277403i	$0.910526\pi$
$648$	0	0
$649$	82.0338	0.126400
$650$	0	0
$651$	0	0
$652$	0	0
$653$	37.0921i	0.0568026i	0.999597	+	0.0284013i	$0.00904163\pi$
$653$	37.0921i	0.0568026i	−0.999597	+	0.0284013i	$0.990958\pi$
$654$	0	0
$655$	− 228.335i	− 0.348604i
$656$	0	0
$657$	95.7284	0.145705
$658$	0	0
$659$	−197.302	−0.299396	−0.149698	−	0.988732i	$-0.547830\pi$
$659$	−197.302	−0.299396	−0.149698	+	0.988732i	$0.547830\pi$
$660$	0	0
$661$	− 1083.83i	− 1.63969i	−0.572589	−	0.819843i	$-0.694061\pi$
$661$	− 1083.83i	− 1.63969i	0.572589	−	0.819843i	$-0.305939\pi$
$662$	0	0
$663$	132.538i	0.199906i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1074.35	1.61071
$668$	0	0
$669$	1484.76i	2.21937i
$670$	0	0
$671$	− 174.628i	− 0.260250i
$672$	0	0
$673$	−286.066	−0.425061	−0.212530	−	0.977154i	$-0.568170\pi$
$673$	−286.066	−0.425061	−0.212530	+	0.977154i	$0.568170\pi$
$674$	0	0
$675$	−183.767	−0.272247
$676$	0	0
$677$	500.360i	0.739083i	0.929214	+	0.369542i	$0.120485\pi$
$677$	500.360i	0.739083i	−0.929214	+	0.369542i	$0.879515\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−292.310	−0.429236
$682$	0	0
$683$	−946.450	−1.38572	−0.692862	−	0.721070i	$-0.743651\pi$
$683$	−946.450	−1.38572	−0.692862	+	0.721070i	$0.743651\pi$
$684$	0	0
$685$	15.9952i	0.0233507i
$686$	0	0
$687$	− 1689.76i	− 2.45962i
$688$	0	0
$689$	308.799	0.448185
$690$	0	0
$691$	−303.264	−0.438877	−0.219439	−	0.975626i	$-0.570423\pi$
$691$	−303.264	−0.438877	−0.219439	+	0.975626i	$0.570423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 5.75942i	− 0.00828694i
$696$	0	0
$697$	117.767	0.168962
$698$	0	0
$699$	−332.074	−0.475069
$700$	0	0
$701$	390.864i	0.557580i	0.960352	+	0.278790i	$0.0899333\pi$
$701$	390.864i	0.557580i	−0.960352	+	0.278790i	$0.910067\pi$
$702$	0	0
$703$	− 1368.69i	− 1.94693i
$704$	0	0
$705$	−618.892	−0.877861
$706$	0	0
$707$	0	0
$708$	0	0
$709$	961.165i	1.35566i	0.735217	+	0.677831i	$0.237080\pi$
$709$	961.165i	1.35566i	−0.735217	+	0.677831i	$0.762920\pi$
$710$	0	0
$711$	− 350.577i	− 0.493075i
$712$	0	0
$713$	−866.685	−1.21555
$714$	0	0
$715$	−170.379	−0.238293
$716$	0	0
$717$	− 448.109i	− 0.624978i
$718$	0	0
$719$	525.126i	0.730356i	0.930938	+	0.365178i	$0.118992\pi$
$719$	525.126i	0.730356i	−0.930938	+	0.365178i	$0.881008\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1116.41	1.54413
$724$	0	0
$725$	− 597.985i	− 0.824807i
$726$	0	0
$727$	− 108.633i	− 0.149426i	−0.997205	−	0.0747131i	$-0.976196\pi$
$727$	− 108.633i	− 0.149426i	0.997205	−	0.0747131i	$-0.0238041\pi$
$728$	0	0
$729$	−348.775	−0.478429
$730$	0	0
$731$	−137.039	−0.187468
$732$	0	0
$733$	40.2540i	0.0549167i	0.999623	+	0.0274584i	$0.00874137\pi$
$733$	40.2540i	0.0549167i	−0.999623	+	0.0274584i	$0.991259\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	467.330	0.634097
$738$	0	0
$739$	−997.203	−1.34940	−0.674698	−	0.738094i	$-0.735726\pi$
$739$	−997.203	−1.34940	−0.674698	+	0.738094i	$0.735726\pi$
$740$	0	0
$741$	− 1373.87i	− 1.85407i
$742$	0	0
$743$	− 476.575i	− 0.641420i	−0.947177	−	0.320710i	$-0.896078\pi$
$743$	− 476.575i	− 0.641420i	0.947177	−	0.320710i	$-0.103922\pi$
$744$	0	0
$745$	59.7214	0.0801630
$746$	0	0
$747$	612.579	0.820052
$748$	0	0
$749$	0	0
$750$	0	0
$751$	528.496i	0.703723i	0.936052	+	0.351862i	$0.114451\pi$
$751$	528.496i	0.703723i	−0.936052	+	0.351862i	$0.885549\pi$
$752$	0	0
$753$	131.199	0.174235
$754$	0	0
$755$	477.400	0.632318
$756$	0	0
$757$	− 455.964i	− 0.602331i	−0.953572	−	0.301165i	$-0.902624\pi$
$757$	− 455.964i	− 0.602331i	0.953572	−	0.301165i	$-0.0973756\pi$
$758$	0	0
$759$	1209.20i	1.59314i
$760$	0	0
$761$	476.650	0.626347	0.313174	−	0.949696i	$-0.398608\pi$
$761$	476.650	0.626347	0.313174	+	0.949696i	$0.398608\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	37.3683i	0.0488475i
$766$	0	0
$767$	− 119.585i	− 0.155913i
$768$	0	0
$769$	−568.246	−0.738941	−0.369471	−	0.929242i	$-0.620461\pi$
$769$	−568.246	−0.738941	−0.369471	+	0.929242i	$0.620461\pi$
$770$	0	0
$771$	−892.621	−1.15774
$772$	0	0
$773$	1197.03i	1.54855i	0.632849	+	0.774275i	$0.281885\pi$
$773$	1197.03i	1.54855i	−0.632849	+	0.774275i	$0.718115\pi$
$774$	0	0
$775$	482.400i	0.622452i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1220.75	−1.56708
$780$	0	0
$781$	− 301.422i	− 0.385943i
$782$	0	0
$783$	238.701i	0.304855i
$784$	0	0
$785$	92.8596	0.118292
$786$	0	0
$787$	−452.269	−0.574674	−0.287337	−	0.957830i	$-0.592770\pi$
$787$	−452.269	−0.574674	−0.287337	+	0.957830i	$0.592770\pi$
$788$	0	0
$789$	− 676.619i	− 0.857565i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−254.565	−0.321015
$794$	0	0
$795$	201.479	0.253433
$796$	0	0
$797$	− 1047.47i	− 1.31426i	−0.753777	−	0.657130i	$-0.771770\pi$
$797$	− 1047.47i	− 1.31426i	0.753777	−	0.657130i	$-0.228230\pi$
$798$	0	0
$799$	239.355i	0.299568i
$800$	0	0
$801$	720.868	0.899960
$802$	0	0
$803$	110.137	0.137157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 432.045i	− 0.535372i
$808$	0	0
$809$	−298.665	−0.369178	−0.184589	−	0.982816i	$-0.559095\pi$
$809$	−298.665	−0.369178	−0.184589	+	0.982816i	$0.559095\pi$
$810$	0	0
$811$	−633.054	−0.780584	−0.390292	−	0.920691i	$-0.627626\pi$
$811$	−633.054	−0.780584	−0.390292	+	0.920691i	$0.627626\pi$
$812$	0	0
$813$	− 77.0852i	− 0.0948158i
$814$	0	0
$815$	217.844i	0.267293i
$816$	0	0
$817$	1420.53	1.73871
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1251.50i	− 1.52436i	−0.647367	−	0.762178i	$-0.724130\pi$
$821$	− 1251.50i	− 1.52436i	0.647367	−	0.762178i	$-0.275870\pi$
$822$	0	0
$823$	1226.82i	1.49066i	0.666693	+	0.745332i	$0.267709\pi$
$823$	1226.82i	1.49066i	−0.666693	+	0.745332i	$0.732291\pi$
$824$	0	0
$825$	673.043	0.815810
$826$	0	0
$827$	1438.26	1.73913	0.869566	−	0.493816i	$-0.164398\pi$
$827$	1438.26	1.73913	0.869566	+	0.493816i	$0.164398\pi$
$828$	0	0
$829$	− 626.488i	− 0.755715i	−0.925864	−	0.377858i	$-0.876661\pi$
$829$	− 626.488i	− 0.755715i	0.925864	−	0.377858i	$-0.123339\pi$
$830$	0	0
$831$	− 515.330i	− 0.620132i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	60.9672	0.0730146
$836$	0	0
$837$	− 192.563i	− 0.230063i
$838$	0	0
$839$	1062.37i	1.26624i	0.774055	+	0.633118i	$0.218225\pi$
$839$	1062.37i	1.26624i	−0.774055	+	0.633118i	$0.781775\pi$
$840$	0	0
$841$	64.2558	0.0764041
$842$	0	0
$843$	331.862	0.393668
$844$	0	0
$845$	− 69.7739i	− 0.0825727i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−115.168	−0.135652
$850$	0	0
$851$	1756.07	2.06354
$852$	0	0
$853$	698.388i	0.818743i	0.912368	+	0.409372i	$0.134252\pi$
$853$	698.388i	0.818743i	−0.912368	+	0.409372i	$0.865748\pi$
$854$	0	0
$855$	− 387.355i	− 0.453047i
$856$	0	0
$857$	−730.324	−0.852187	−0.426094	−	0.904679i	$-0.640111\pi$
$857$	−730.324	−0.852187	−0.426094	+	0.904679i	$0.640111\pi$
$858$	0	0
$859$	966.259	1.12487	0.562433	−	0.826843i	$-0.309866\pi$
$859$	966.259	1.12487	0.562433	+	0.826843i	$0.309866\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	294.475i	0.341222i	0.985338	+	0.170611i	$0.0545742\pi$
$863$	294.475i	0.341222i	−0.985338	+	0.170611i	$0.945426\pi$
$864$	0	0
$865$	390.142	0.451032
$866$	0	0
$867$	−1117.07	−1.28844
$868$	0	0
$869$	− 403.346i	− 0.464149i
$870$	0	0
$871$	− 681.253i	− 0.782151i
$872$	0	0
$873$	378.934	0.434059
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 944.891i	− 1.07741i	−0.842494	−	0.538706i	$-0.818913\pi$
$877$	− 944.891i	− 1.07741i	0.842494	−	0.538706i	$-0.181087\pi$
$878$	0	0
$879$	854.382i	0.971993i
$880$	0	0
$881$	744.098	0.844606	0.422303	−	0.906455i	$-0.361222\pi$
$881$	744.098	0.844606	0.422303	+	0.906455i	$0.361222\pi$
$882$	0	0
$883$	−23.9032	−0.0270704	−0.0135352	−	0.999908i	$-0.504309\pi$
$883$	−23.9032	−0.0270704	−0.0135352	+	0.999908i	$0.504309\pi$
$884$	0	0
$885$	− 78.0246i	− 0.0881634i
$886$	0	0
$887$	− 947.312i	− 1.06800i	−0.845486	−	0.533998i	$-0.820689\pi$
$887$	− 947.312i	− 1.06800i	0.845486	−	0.533998i	$-0.179311\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−754.332	−0.846613
$892$	0	0
$893$	− 2481.12i	− 2.77840i
$894$	0	0
$895$	327.874i	0.366339i
$896$	0	0
$897$	1762.71	1.96512
$898$	0	0
$899$	626.607	0.697005
$900$	0	0
$901$	− 77.9215i	− 0.0864834i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−384.493	−0.424854
$906$	0	0
$907$	−551.023	−0.607523	−0.303761	−	0.952748i	$-0.598243\pi$
$907$	−551.023	−0.607523	−0.303761	+	0.952748i	$0.598243\pi$
$908$	0	0
$909$	− 215.271i	− 0.236822i
$910$	0	0
$911$	− 827.652i	− 0.908509i	−0.890872	−	0.454254i	$-0.849906\pi$
$911$	− 827.652i	− 0.908509i	0.890872	−	0.454254i	$-0.150094\pi$
$912$	0	0
$913$	704.784	0.771943
$914$	0	0
$915$	−166.093	−0.181523
$916$	0	0
$917$	0	0
$918$	0	0
$919$	356.993i	0.388458i	0.980956	+	0.194229i	$0.0622205\pi$
$919$	356.993i	0.388458i	−0.980956	+	0.194229i	$0.937779\pi$
$920$	0	0
$921$	479.880	0.521043
$922$	0	0
$923$	−439.400	−0.476056
$924$	0	0
$925$	− 977.438i	− 1.05669i
$926$	0	0
$927$	− 547.794i	− 0.590932i
$928$	0	0
$929$	969.745	1.04386	0.521930	−	0.852988i	$-0.325212\pi$
$929$	969.745	1.04386	0.521930	+	0.852988i	$0.325212\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	1295.27i	1.38829i
$934$	0	0
$935$	42.9930i	0.0459819i
$936$	0	0
$937$	1287.55	1.37412	0.687061	−	0.726600i	$-0.258901\pi$
$937$	1287.55	1.37412	0.687061	+	0.726600i	$0.258901\pi$
$938$	0	0
$939$	1818.36	1.93649
$940$	0	0
$941$	456.796i	0.485436i	0.970097	+	0.242718i	$0.0780390\pi$
$941$	456.796i	0.485436i	−0.970097	+	0.242718i	$0.921961\pi$
$942$	0	0
$943$	− 1566.27i	− 1.66094i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1147.85	1.21209	0.606045	−	0.795430i	$-0.292755\pi$
$947$	1147.85	1.21209	0.606045	+	0.795430i	$0.292755\pi$
$948$	0	0
$949$	− 160.554i	− 0.169182i
$950$	0	0
$951$	482.582i	0.507447i
$952$	0	0
$953$	873.170	0.916232	0.458116	−	0.888892i	$-0.348524\pi$
$953$	873.170	0.916232	0.458116	+	0.888892i	$0.348524\pi$
$954$	0	0
$955$	561.952	0.588432
$956$	0	0
$957$	− 874.240i	− 0.913522i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	455.510	0.473996
$962$	0	0
$963$	−333.961	−0.346792
$964$	0	0
$965$	622.666i	0.645249i
$966$	0	0
$967$	449.047i	0.464372i	0.972671	+	0.232186i	$0.0745878\pi$
$967$	449.047i	0.464372i	−0.972671	+	0.232186i	$0.925412\pi$
$968$	0	0
$969$	−346.677	−0.357768
$970$	0	0
$971$	1237.86	1.27483	0.637414	−	0.770521i	$-0.280004\pi$
$971$	1237.86	1.27483	0.637414	+	0.770521i	$0.280004\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	− 981.134i	− 1.00629i
$976$	0	0
$977$	1706.89	1.74707	0.873536	−	0.486760i	$-0.161821\pi$
$977$	1706.89	1.74707	0.873536	+	0.486760i	$0.161821\pi$
$978$	0	0
$979$	829.373	0.847164
$980$	0	0
$981$	788.063i	0.803326i
$982$	0	0
$983$	− 1152.15i	− 1.17208i	−0.810283	−	0.586039i	$-0.800687\pi$
$983$	− 1152.15i	− 1.17208i	0.810283	−	0.586039i	$-0.199313\pi$
$984$	0	0
$985$	616.165	0.625548
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1822.59i	1.84286i
$990$	0	0
$991$	− 412.818i	− 0.416567i	−0.978068	−	0.208283i	$-0.933212\pi$
$991$	− 412.818i	− 0.416567i	0.978068	−	0.208283i	$-0.0667877\pi$
$992$	0	0
$993$	−482.375	−0.485776
$994$	0	0
$995$	23.9753	0.0240958
$996$	0	0
$997$	− 301.152i	− 0.302058i	−0.988529	−	0.151029i	$-0.951741\pi$
$997$	− 301.152i	− 0.302058i	0.988529	−	0.151029i	$-0.0482587\pi$
$998$	0	0
$999$	390.170i	0.390561i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1568.3.g.j.687.5		6
4.3	odd	2		392.3.g.j.99.6		6
7.2	even	3		224.3.o.d.207.1		12
7.4	even	3		224.3.o.d.79.2		12
7.6	odd	2		1568.3.g.l.687.2		6
8.3	odd	2	inner	1568.3.g.j.687.6		6
8.5	even	2		392.3.g.j.99.5		6
28.3	even	6		392.3.k.l.275.1		12
28.11	odd	6		56.3.k.d.51.1	yes	12
28.19	even	6		392.3.k.l.67.3		12
28.23	odd	6		56.3.k.d.11.3	yes	12
28.27	even	2		392.3.g.i.99.6		6
56.5	odd	6		392.3.k.l.67.1		12
56.11	odd	6		224.3.o.d.79.1		12
56.13	odd	2		392.3.g.i.99.5		6
56.27	even	2		1568.3.g.l.687.1		6
56.37	even	6		56.3.k.d.11.1	✓	12
56.45	odd	6		392.3.k.l.275.3		12
56.51	odd	6		224.3.o.d.207.2		12
56.53	even	6		56.3.k.d.51.3	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.3.k.d.11.1	✓	12	56.37	even	6
56.3.k.d.11.3	yes	12	28.23	odd	6
56.3.k.d.51.1	yes	12	28.11	odd	6
56.3.k.d.51.3	yes	12	56.53	even	6
224.3.o.d.79.1		12	56.11	odd	6
224.3.o.d.79.2		12	7.4	even	3
224.3.o.d.207.1		12	7.2	even	3
224.3.o.d.207.2		12	56.51	odd	6
392.3.g.i.99.5		6	56.13	odd	2
392.3.g.i.99.6		6	28.27	even	2
392.3.g.j.99.5		6	8.5	even	2
392.3.g.j.99.6		6	4.3	odd	2
392.3.k.l.67.1		12	56.5	odd	6
392.3.k.l.67.3		12	28.19	even	6
392.3.k.l.275.1		12	28.3	even	6
392.3.k.l.275.3		12	56.45	odd	6
1568.3.g.j.687.5		6	1.1	even	1	trivial
1568.3.g.j.687.6		6	8.3	odd	2	inner
1568.3.g.l.687.1		6	56.27	even	2
1568.3.g.l.687.2		6	7.6	odd	2

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.98103 q^{3} -1.88252i q^{5} +6.84860 q^{9} +O(q^{10})\)	\(q+3.98103 q^{3} -1.88252i q^{5} +6.84860 q^{9} +7.87946 q^{11} -11.4863i q^{13} -7.49437i q^{15} -2.89843 q^{17} +30.0447 q^{19} +38.5483i q^{23} +21.4561 q^{25} -8.56478 q^{27} -27.8701i q^{29} +22.4831i q^{31} +31.3684 q^{33} -45.5552i q^{37} -45.7274i q^{39} -40.6313 q^{41} +47.2806 q^{43} -12.8926i q^{45} -82.5809i q^{47} -11.5387 q^{51} +26.8841i q^{53} -14.8332i q^{55} +119.609 q^{57} +10.4111 q^{59} -22.1624i q^{61} -21.6233 q^{65} +59.3099 q^{67} +153.462i q^{69} -38.2541i q^{71} +13.9778 q^{73} +85.4175 q^{75} -51.1895i q^{79} -95.7341 q^{81} +89.4458 q^{83} +5.45635i q^{85} -110.952i q^{87} +105.258 q^{89} +89.5059i q^{93} -56.5597i q^{95} +55.3301 q^{97} +53.9633 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 40 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 40 * q^9	\( 6 q - 6 q^{3} + 40 q^{9} + 30 q^{11} - 30 q^{17} + 78 q^{19} + 92 q^{25} - 78 q^{27} + 78 q^{33} - 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} - 110 q^{59} + 32 q^{65} + 434 q^{67} - 102 q^{73} - 60 q^{75} + 82 q^{81} + 268 q^{83} - 214 q^{89} - 76 q^{97} - 252 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 40 * q^9 + 30 * q^11 - 30 * q^17 + 78 * q^19 + 92 * q^25 - 78 * q^27 + 78 * q^33 - 116 * q^41 + 100 * q^43 + 10 * q^51 + 166 * q^57 - 110 * q^59 + 32 * q^65 + 434 * q^67 - 102 * q^73 - 60 * q^75 + 82 * q^81 + 268 * q^83 - 214 * q^89 - 76 * q^97 - 252 * q^99

Introduction

L-functions

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