LMFDB - Embedded newform 1728.3.g.k.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, 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("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 864)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.1
Root			$1.20036 - 0.747754i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.k.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.64469 q^{5} -1.12019i q^{7} +O(q^{10})$	$q-9.64469 q^{5} -1.12019i q^{7} -13.8529i q^{11} +8.65671 q^{13} +12.6567 q^{17} -27.6817i q^{19} +27.5298i q^{23} +68.0201 q^{25} -6.15193 q^{29} +13.2481i q^{31} +10.8038i q^{35} +32.9316 q^{37} -60.2500 q^{41} +25.3479i q^{43} -66.7652i q^{47} +47.7452 q^{49} -13.1698 q^{53} +133.607i q^{55} +83.2556i q^{59} -101.413 q^{61} -83.4913 q^{65} -112.006i q^{67} -85.9221i q^{71} +25.0798 q^{73} -15.5178 q^{77} -63.4604i q^{79} -111.443i q^{83} -122.070 q^{85} -22.4278 q^{89} -9.69712i q^{91} +266.981i q^{95} -71.2151 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 16 * q^13 + 48 * q^17 + 48 * q^25 + 32 * q^29 + 96 * q^37 - 128 * q^41 - 168 * q^53 - 32 * q^61 - 112 * q^65 + 24 * q^73 + 440 * q^77 - 144 * q^85 + 624 * q^89 - 136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−9.64469	−1.92894	−0.964469	−	0.264195i	$-0.914894\pi$
$5$	−9.64469	−1.92894	−0.964469	+	0.264195i	$0.914894\pi$
$6$	0	0
$7$	− 1.12019i	− 0.160027i	−0.996794	−	0.0800133i	$-0.974504\pi$
$7$	− 1.12019i	− 0.160027i	0.996794	−	0.0800133i	$-0.0254963\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 13.8529i	− 1.25935i	−0.776858	−	0.629675i	$-0.783188\pi$
$11$	− 13.8529i	− 1.25935i	0.776858	−	0.629675i	$-0.216812\pi$
$12$	0	0
$13$	8.65671	0.665901	0.332950	−	0.942944i	$-0.391956\pi$
$13$	8.65671	0.665901	0.332950	+	0.942944i	$0.391956\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	12.6567	0.744512	0.372256	−	0.928130i	$-0.378584\pi$
$17$	12.6567	0.744512	0.372256	+	0.928130i	$0.378584\pi$
$18$	0	0
$19$	− 27.6817i	− 1.45693i	−0.685082	−	0.728466i	$-0.740234\pi$
$19$	− 27.6817i	− 1.45693i	0.685082	−	0.728466i	$-0.259766\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	27.5298i	1.19695i	0.801143	+	0.598473i	$0.204226\pi$
$23$	27.5298i	1.19695i	−0.801143	+	0.598473i	$0.795774\pi$
$24$	0	0
$25$	68.0201	2.72080
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6.15193	−0.212136	−0.106068	−	0.994359i	$-0.533826\pi$
$29$	−6.15193	−0.212136	−0.106068	+	0.994359i	$0.533826\pi$
$30$	0	0
$31$	13.2481i	0.427358i	0.976904	+	0.213679i	$0.0685446\pi$
$31$	13.2481i	0.427358i	−0.976904	+	0.213679i	$0.931455\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	10.8038i	0.308681i
$36$	0	0
$37$	32.9316	0.890044	0.445022	−	0.895520i	$-0.353196\pi$
$37$	32.9316	0.890044	0.445022	+	0.895520i	$0.353196\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−60.2500	−1.46951	−0.734756	−	0.678332i	$-0.762703\pi$
$41$	−60.2500	−1.46951	−0.734756	+	0.678332i	$0.762703\pi$
$42$	0	0
$43$	25.3479i	0.589487i	0.955576	+	0.294744i	$0.0952342\pi$
$43$	25.3479i	0.589487i	−0.955576	+	0.294744i	$0.904766\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 66.7652i	− 1.42054i	−0.703931	−	0.710268i	$-0.748574\pi$
$47$	− 66.7652i	− 1.42054i	0.703931	−	0.710268i	$-0.251426\pi$
$48$	0	0
$49$	47.7452	0.974392
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.1698	−0.248487	−0.124243	−	0.992252i	$-0.539650\pi$
$53$	−13.1698	−0.248487	−0.124243	+	0.992252i	$0.539650\pi$
$54$	0	0
$55$	133.607i	2.42921i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	83.2556i	1.41111i	0.708655	+	0.705556i	$0.249302\pi$
$59$	83.2556i	1.41111i	−0.708655	+	0.705556i	$0.750698\pi$
$60$	0	0
$61$	−101.413	−1.66251	−0.831257	−	0.555888i	$-0.812378\pi$
$61$	−101.413	−1.66251	−0.831257	+	0.555888i	$0.812378\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−83.4913	−1.28448
$66$	0	0
$67$	− 112.006i	− 1.67173i	−0.548938	−	0.835863i	$-0.684968\pi$
$67$	− 112.006i	− 1.67173i	0.548938	−	0.835863i	$-0.315032\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 85.9221i	− 1.21017i	−0.796161	−	0.605085i	$-0.793139\pi$
$71$	− 85.9221i	− 1.21017i	0.796161	−	0.605085i	$-0.206861\pi$
$72$	0	0
$73$	25.0798	0.343559	0.171780	−	0.985135i	$-0.445048\pi$
$73$	25.0798	0.343559	0.171780	+	0.985135i	$0.445048\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−15.5178	−0.201530
$78$	0	0
$79$	− 63.4604i	− 0.803296i	−0.915794	−	0.401648i	$-0.868437\pi$
$79$	− 63.4604i	− 0.803296i	0.915794	−	0.401648i	$-0.131563\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 111.443i	− 1.34269i	−0.741146	−	0.671344i	$-0.765717\pi$
$83$	− 111.443i	− 1.34269i	0.741146	−	0.671344i	$-0.234283\pi$
$84$	0	0
$85$	−122.070	−1.43612
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−22.4278	−0.251998	−0.125999	−	0.992030i	$-0.540214\pi$
$89$	−22.4278	−0.251998	−0.125999	+	0.992030i	$0.540214\pi$
$90$	0	0
$91$	− 9.69712i	− 0.106562i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	266.981i	2.81033i
$96$	0	0
$97$	−71.2151	−0.734177	−0.367088	−	0.930186i	$-0.619645\pi$
$97$	−71.2151	−0.734177	−0.367088	+	0.930186i	$0.619645\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−138.475	−1.37104	−0.685521	−	0.728052i	$-0.740426\pi$
$101$	−138.475	−1.37104	−0.685521	+	0.728052i	$0.740426\pi$
$102$	0	0
$103$	117.157i	1.13744i	0.822530	+	0.568721i	$0.192562\pi$
$103$	117.157i	1.13744i	−0.822530	+	0.568721i	$0.807438\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	39.2155i	0.366500i	0.983066	+	0.183250i	$0.0586618\pi$
$107$	39.2155i	0.366500i	−0.983066	+	0.183250i	$0.941338\pi$
$108$	0	0
$109$	−153.735	−1.41042	−0.705208	−	0.709000i	$-0.749146\pi$
$109$	−153.735	−1.41042	−0.705208	+	0.709000i	$0.749146\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−113.527	−1.00466	−0.502332	−	0.864675i	$-0.667524\pi$
$113$	−113.527	−1.00466	−0.502332	+	0.864675i	$0.667524\pi$
$114$	0	0
$115$	− 265.516i	− 2.30883i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 14.1779i	− 0.119142i
$120$	0	0
$121$	−70.9017	−0.585965
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−414.915	−3.31932
$126$	0	0
$127$	− 120.302i	− 0.947263i	−0.880723	−	0.473632i	$-0.842943\pi$
$127$	− 120.302i	− 0.947263i	0.880723	−	0.473632i	$-0.157057\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 60.2137i	− 0.459646i	−0.973232	−	0.229823i	$-0.926185\pi$
$131$	− 60.2137i	− 0.459646i	0.973232	−	0.229823i	$-0.0738148\pi$
$132$	0	0
$133$	−31.0086	−0.233148
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.0339	−0.102437	−0.0512186	−	0.998687i	$-0.516311\pi$
$137$	−14.0339	−0.102437	−0.0512186	+	0.998687i	$0.516311\pi$
$138$	0	0
$139$	234.127i	1.68436i	0.539194	+	0.842182i	$0.318729\pi$
$139$	234.127i	1.68436i	−0.539194	+	0.842182i	$0.681271\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 119.920i	− 0.838603i
$144$	0	0
$145$	59.3335	0.409197
$146$	0	0
$147$	0	0
$148$	0	0
$149$	50.9812	0.342155	0.171078	−	0.985258i	$-0.445275\pi$
$149$	50.9812	0.342155	0.171078	+	0.985258i	$0.445275\pi$
$150$	0	0
$151$	158.996i	1.05295i	0.850190	+	0.526475i	$0.176487\pi$
$151$	158.996i	1.05295i	−0.850190	+	0.526475i	$0.823513\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 127.774i	− 0.824347i
$156$	0	0
$157$	2.68658	0.0171120	0.00855600	−	0.999963i	$-0.497277\pi$
$157$	2.68658	0.0171120	0.00855600	+	0.999963i	$0.497277\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	30.8384	0.191543
$162$	0	0
$163$	146.659i	0.899746i	0.893093	+	0.449873i	$0.148531\pi$
$163$	146.659i	0.899746i	−0.893093	+	0.449873i	$0.851469\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 18.7844i	− 0.112481i	−0.998417	−	0.0562407i	$-0.982089\pi$
$167$	− 18.7844i	− 0.112481i	0.998417	−	0.0562407i	$-0.0179114\pi$
$168$	0	0
$169$	−94.0614	−0.556576
$170$	0	0
$171$	0	0
$172$	0	0
$173$	170.434	0.985166	0.492583	−	0.870266i	$-0.336053\pi$
$173$	170.434	0.985166	0.492583	+	0.870266i	$0.336053\pi$
$174$	0	0
$175$	− 76.1951i	− 0.435401i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	186.431i	1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$179$	186.431i	1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$180$	0	0
$181$	−115.886	−0.640255	−0.320127	−	0.947375i	$-0.603726\pi$
$181$	−115.886	−0.640255	−0.320127	+	0.947375i	$0.603726\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−317.615	−1.71684
$186$	0	0
$187$	− 175.332i	− 0.937602i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 46.4872i	− 0.243389i	−0.992568	−	0.121694i	$-0.961167\pi$
$191$	− 46.4872i	− 0.243389i	0.992568	−	0.121694i	$-0.0388328\pi$
$192$	0	0
$193$	62.0795	0.321656	0.160828	−	0.986982i	$-0.448584\pi$
$193$	62.0795	0.321656	0.160828	+	0.986982i	$0.448584\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−98.1691	−0.498320	−0.249160	−	0.968462i	$-0.580155\pi$
$197$	−98.1691	−0.498320	−0.249160	+	0.968462i	$0.580155\pi$
$198$	0	0
$199$	104.609i	0.525671i	0.964841	+	0.262836i	$0.0846577\pi$
$199$	104.609i	0.525671i	−0.964841	+	0.262836i	$0.915342\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.89131i	0.0339473i
$204$	0	0
$205$	581.092	2.83460
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−383.471	−1.83479
$210$	0	0
$211$	44.1261i	0.209128i	0.994518	+	0.104564i	$0.0333447\pi$
$211$	44.1261i	0.209128i	−0.994518	+	0.104564i	$0.966655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 244.473i	− 1.13708i
$216$	0	0
$217$	14.8403	0.0683886
$218$	0	0
$219$	0	0
$220$	0	0
$221$	109.565	0.495771
$222$	0	0
$223$	− 353.445i	− 1.58496i	−0.609901	−	0.792478i	$-0.708791\pi$
$223$	− 353.445i	− 1.58496i	0.609901	−	0.792478i	$-0.291209\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	192.295i	0.847116i	0.905869	+	0.423558i	$0.139219\pi$
$227$	192.295i	0.847116i	−0.905869	+	0.423558i	$0.860781\pi$
$228$	0	0
$229$	356.455	1.55657	0.778286	−	0.627909i	$-0.216089\pi$
$229$	356.455	1.55657	0.778286	+	0.627909i	$0.216089\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−190.754	−0.818687	−0.409344	−	0.912380i	$-0.634242\pi$
$233$	−190.754	−0.818687	−0.409344	+	0.912380i	$0.634242\pi$
$234$	0	0
$235$	643.930i	2.74013i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	230.647i	0.965051i	0.875882	+	0.482526i	$0.160280\pi$
$239$	230.647i	0.965051i	−0.875882	+	0.482526i	$0.839720\pi$
$240$	0	0
$241$	120.413	0.499639	0.249820	−	0.968292i	$-0.419629\pi$
$241$	120.413	0.499639	0.249820	+	0.968292i	$0.419629\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−460.488	−1.87954
$246$	0	0
$247$	− 239.632i	− 0.970171i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	453.709i	1.80761i	0.427948	+	0.903803i	$0.359237\pi$
$251$	453.709i	1.80761i	−0.427948	+	0.903803i	$0.640763\pi$
$252$	0	0
$253$	381.366	1.50737
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−109.467	−0.425940	−0.212970	−	0.977059i	$-0.568314\pi$
$257$	−109.467	−0.425940	−0.212970	+	0.977059i	$0.568314\pi$
$258$	0	0
$259$	− 36.8895i	− 0.142431i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	99.7622i	0.379324i	0.981849	+	0.189662i	$0.0607392\pi$
$263$	99.7622i	0.379324i	−0.981849	+	0.189662i	$0.939261\pi$
$264$	0	0
$265$	127.019	0.479316
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−320.980	−1.19324	−0.596618	−	0.802525i	$-0.703489\pi$
$269$	−320.980	−1.19324	−0.596618	+	0.802525i	$0.703489\pi$
$270$	0	0
$271$	297.292i	1.09702i	0.836145	+	0.548509i	$0.184804\pi$
$271$	297.292i	1.09702i	−0.836145	+	0.548509i	$0.815196\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 942.273i	− 3.42645i
$276$	0	0
$277$	−90.4130	−0.326401	−0.163200	−	0.986593i	$-0.552182\pi$
$277$	−90.4130	−0.326401	−0.163200	+	0.986593i	$0.552182\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	318.187	1.13234	0.566169	−	0.824289i	$-0.308425\pi$
$281$	318.187	1.13234	0.566169	+	0.824289i	$0.308425\pi$
$282$	0	0
$283$	− 365.829i	− 1.29268i	−0.763049	−	0.646340i	$-0.776299\pi$
$283$	− 365.829i	− 1.29268i	0.763049	−	0.646340i	$-0.223701\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	67.4912i	0.235161i
$288$	0	0
$289$	−128.808	−0.445702
$290$	0	0
$291$	0	0
$292$	0	0
$293$	54.6252	0.186434	0.0932171	−	0.995646i	$-0.470285\pi$
$293$	54.6252	0.186434	0.0932171	+	0.995646i	$0.470285\pi$
$294$	0	0
$295$	− 802.974i	− 2.72195i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	238.317i	0.797047i
$300$	0	0
$301$	28.3944	0.0943336
$302$	0	0
$303$	0	0
$304$	0	0
$305$	978.100	3.20689
$306$	0	0
$307$	25.2901i	0.0823781i	0.999151	+	0.0411891i	$0.0131146\pi$
$307$	25.2901i	0.0823781i	−0.999151	+	0.0411891i	$0.986885\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 449.554i	− 1.44551i	−0.691103	−	0.722756i	$-0.742875\pi$
$311$	− 449.554i	− 1.44551i	0.691103	−	0.722756i	$-0.257125\pi$
$312$	0	0
$313$	−278.962	−0.891253	−0.445627	−	0.895219i	$-0.647019\pi$
$313$	−278.962	−0.891253	−0.445627	+	0.895219i	$0.647019\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	363.560	1.14688	0.573439	−	0.819248i	$-0.305609\pi$
$317$	363.560	1.14688	0.573439	+	0.819248i	$0.305609\pi$
$318$	0	0
$319$	85.2219i	0.267153i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 350.359i	− 1.08470i
$324$	0	0
$325$	588.830	1.81178
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−74.7895	−0.227324
$330$	0	0
$331$	209.510i	0.632962i	0.948599	+	0.316481i	$0.102501\pi$
$331$	209.510i	0.632962i	−0.948599	+	0.316481i	$0.897499\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1080.26i	3.22466i
$336$	0	0
$337$	−446.770	−1.32573	−0.662864	−	0.748740i	$-0.730659\pi$
$337$	−446.770	−1.32573	−0.662864	+	0.748740i	$0.730659\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	183.524	0.538193
$342$	0	0
$343$	− 108.373i	− 0.315955i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 109.320i	− 0.315043i	−0.987516	−	0.157522i	$-0.949650\pi$
$347$	− 109.320i	− 0.315043i	0.987516	−	0.157522i	$-0.0503504\pi$
$348$	0	0
$349$	−669.523	−1.91840	−0.959201	−	0.282724i	$-0.908762\pi$
$349$	−669.523	−1.91840	−0.959201	+	0.282724i	$0.908762\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−472.139	−1.33751	−0.668753	−	0.743485i	$-0.733171\pi$
$353$	−472.139	−1.33751	−0.668753	+	0.743485i	$0.733171\pi$
$354$	0	0
$355$	828.692i	2.33434i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 148.271i	− 0.413012i	−0.978445	−	0.206506i	$-0.933791\pi$
$359$	− 148.271i	− 0.413012i	0.978445	−	0.206506i	$-0.0662094\pi$
$360$	0	0
$361$	−405.276	−1.12265
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−241.887	−0.662705
$366$	0	0
$367$	− 248.314i	− 0.676604i	−0.941038	−	0.338302i	$-0.890147\pi$
$367$	− 248.314i	− 0.676604i	0.941038	−	0.338302i	$-0.109853\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.7526i	0.0397645i
$372$	0	0
$373$	−168.343	−0.451322	−0.225661	−	0.974206i	$-0.572454\pi$
$373$	−168.343	−0.451322	−0.225661	+	0.974206i	$0.572454\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−53.2555	−0.141261
$378$	0	0
$379$	− 517.108i	− 1.36440i	−0.731165	−	0.682201i	$-0.761023\pi$
$379$	− 517.108i	− 1.36440i	0.731165	−	0.682201i	$-0.238977\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 303.786i	− 0.793175i	−0.917997	−	0.396588i	$-0.870194\pi$
$383$	− 303.786i	− 0.793175i	0.917997	−	0.396588i	$-0.129806\pi$
$384$	0	0
$385$	149.664	0.388738
$386$	0	0
$387$	0	0
$388$	0	0
$389$	114.745	0.294974	0.147487	−	0.989064i	$-0.452881\pi$
$389$	114.745	0.294974	0.147487	+	0.989064i	$0.452881\pi$
$390$	0	0
$391$	348.436i	0.891141i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	612.056i	1.54951i
$396$	0	0
$397$	−602.700	−1.51814	−0.759068	−	0.651011i	$-0.774345\pi$
$397$	−602.700	−1.51814	−0.759068	+	0.651011i	$0.774345\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−336.618	−0.839447	−0.419724	−	0.907652i	$-0.637873\pi$
$401$	−336.618	−0.839447	−0.419724	+	0.907652i	$0.637873\pi$
$402$	0	0
$403$	114.685i	0.284578i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 456.197i	− 1.12088i
$408$	0	0
$409$	−312.211	−0.763353	−0.381676	−	0.924296i	$-0.624653\pi$
$409$	−312.211	−0.763353	−0.381676	+	0.924296i	$0.624653\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	93.2617	0.225815
$414$	0	0
$415$	1074.83i	2.58996i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 40.3367i	− 0.0962690i	−0.998841	−	0.0481345i	$-0.984672\pi$
$419$	− 40.3367i	− 0.0962690i	0.998841	−	0.0481345i	$-0.0153276\pi$
$420$	0	0
$421$	−170.177	−0.404221	−0.202110	−	0.979363i	$-0.564780\pi$
$421$	−170.177	−0.404221	−0.202110	+	0.979363i	$0.564780\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	860.910	2.02567
$426$	0	0
$427$	113.602i	0.266046i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 172.213i	− 0.399565i	−0.979840	−	0.199783i	$-0.935976\pi$
$431$	− 172.213i	− 0.399565i	0.979840	−	0.199783i	$-0.0640236\pi$
$432$	0	0
$433$	−759.997	−1.75519	−0.877595	−	0.479403i	$-0.840853\pi$
$433$	−759.997	−1.75519	−0.877595	+	0.479403i	$0.840853\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	762.070	1.74387
$438$	0	0
$439$	− 558.574i	− 1.27238i	−0.771534	−	0.636189i	$-0.780510\pi$
$439$	− 558.574i	− 1.27238i	0.771534	−	0.636189i	$-0.219490\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	857.477i	1.93562i	0.251691	+	0.967808i	$0.419013\pi$
$443$	857.477i	1.93562i	−0.251691	+	0.967808i	$0.580987\pi$
$444$	0	0
$445$	216.309	0.486089
$446$	0	0
$447$	0	0
$448$	0	0
$449$	64.5889	0.143851	0.0719253	−	0.997410i	$-0.477086\pi$
$449$	64.5889	0.143851	0.0719253	+	0.997410i	$0.477086\pi$
$450$	0	0
$451$	834.634i	1.85063i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	93.5258i	0.205551i
$456$	0	0
$457$	446.676	0.977410	0.488705	−	0.872449i	$-0.337469\pi$
$457$	446.676	0.977410	0.488705	+	0.872449i	$0.337469\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	591.058	1.28212	0.641060	−	0.767491i	$-0.278495\pi$
$461$	591.058	1.28212	0.641060	+	0.767491i	$0.278495\pi$
$462$	0	0
$463$	117.571i	0.253933i	0.991907	+	0.126967i	$0.0405241\pi$
$463$	117.571i	0.253933i	−0.991907	+	0.126967i	$0.959476\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.1484i	0.0559923i	0.999608	+	0.0279962i	$0.00891262\pi$
$467$	26.1484i	0.0559923i	−0.999608	+	0.0279962i	$0.991087\pi$
$468$	0	0
$469$	−125.467	−0.267521
$470$	0	0
$471$	0	0
$472$	0	0
$473$	351.142	0.742371
$474$	0	0
$475$	− 1882.91i	− 3.96402i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 342.053i	− 0.714098i	−0.934086	−	0.357049i	$-0.883783\pi$
$479$	− 342.053i	− 0.714098i	0.934086	−	0.357049i	$-0.116217\pi$
$480$	0	0
$481$	285.079	0.592680
$482$	0	0
$483$	0	0
$484$	0	0
$485$	686.848	1.41618
$486$	0	0
$487$	58.9556i	0.121059i	0.998166	+	0.0605293i	$0.0192789\pi$
$487$	58.9556i	0.121059i	−0.998166	+	0.0605293i	$0.980721\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 253.074i	− 0.515426i	−0.966222	−	0.257713i	$-0.917031\pi$
$491$	− 253.074i	− 0.515426i	0.966222	−	0.257713i	$-0.0829688\pi$
$492$	0	0
$493$	−77.8632	−0.157938
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−96.2487	−0.193659
$498$	0	0
$499$	− 440.150i	− 0.882063i	−0.897492	−	0.441032i	$-0.854613\pi$
$499$	− 440.150i	− 0.882063i	0.897492	−	0.441032i	$-0.145387\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 299.245i	− 0.594921i	−0.954734	−	0.297461i	$-0.903860\pi$
$503$	− 299.245i	− 0.594921i	0.954734	−	0.297461i	$-0.0961397\pi$
$504$	0	0
$505$	1335.55	2.64466
$506$	0	0
$507$	0	0
$508$	0	0
$509$	380.415	0.747378	0.373689	−	0.927554i	$-0.378093\pi$
$509$	380.415	0.747378	0.373689	+	0.927554i	$0.378093\pi$
$510$	0	0
$511$	− 28.0941i	− 0.0549786i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1129.94i	− 2.19406i
$516$	0	0
$517$	−924.889	−1.78895
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−523.369	−1.00455	−0.502273	−	0.864709i	$-0.667503\pi$
$521$	−523.369	−1.00455	−0.502273	+	0.864709i	$0.667503\pi$
$522$	0	0
$523$	641.527i	1.22663i	0.789839	+	0.613315i	$0.210164\pi$
$523$	641.527i	1.22663i	−0.789839	+	0.613315i	$0.789836\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	167.677i	0.318173i
$528$	0	0
$529$	−228.887	−0.432680
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−521.566	−0.978548
$534$	0	0
$535$	− 378.222i	− 0.706956i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 661.407i	− 1.22710i
$540$	0	0
$541$	−366.964	−0.678307	−0.339154	−	0.940731i	$-0.610141\pi$
$541$	−366.964	−0.678307	−0.339154	+	0.940731i	$0.610141\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1482.73	2.72061
$546$	0	0
$547$	− 550.755i	− 1.00686i	−0.864035	−	0.503432i	$-0.832070\pi$
$547$	− 550.755i	− 1.00686i	0.864035	−	0.503432i	$-0.167930\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	170.296i	0.309067i
$552$	0	0
$553$	−71.0875	−0.128549
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−498.904	−0.895698	−0.447849	−	0.894109i	$-0.647810\pi$
$557$	−498.904	−0.895698	−0.447849	+	0.894109i	$0.647810\pi$
$558$	0	0
$559$	219.430i	0.392540i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	823.441i	1.46260i	0.682058	+	0.731298i	$0.261085\pi$
$563$	823.441i	1.46260i	−0.682058	+	0.731298i	$0.738915\pi$
$564$	0	0
$565$	1094.93	1.93793
$566$	0	0
$567$	0	0
$568$	0	0
$569$	372.094	0.653944	0.326972	−	0.945034i	$-0.393972\pi$
$569$	372.094	0.653944	0.326972	+	0.945034i	$0.393972\pi$
$570$	0	0
$571$	− 800.325i	− 1.40162i	−0.713348	−	0.700810i	$-0.752822\pi$
$571$	− 800.325i	− 1.40162i	0.713348	−	0.700810i	$-0.247178\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1872.58i	3.25665i
$576$	0	0
$577$	404.343	0.700768	0.350384	−	0.936606i	$-0.386051\pi$
$577$	404.343	0.700768	0.350384	+	0.936606i	$0.386051\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−124.837	−0.214866
$582$	0	0
$583$	182.439i	0.312932i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.7837i	0.0405173i	0.999795	+	0.0202586i	$0.00644897\pi$
$587$	23.7837i	0.0405173i	−0.999795	+	0.0202586i	$0.993551\pi$
$588$	0	0
$589$	366.729	0.622631
$590$	0	0
$591$	0	0
$592$	0	0
$593$	568.416	0.958544	0.479272	−	0.877667i	$-0.340901\pi$
$593$	568.416	0.958544	0.479272	+	0.877667i	$0.340901\pi$
$594$	0	0
$595$	136.741i	0.229817i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 244.953i	− 0.408936i	−0.978873	−	0.204468i	$-0.934454\pi$
$599$	− 244.953i	− 0.408936i	0.978873	−	0.204468i	$-0.0655465\pi$
$600$	0	0
$601$	−0.682820	−0.00113614	−0.000568070	−	1.00000i	$-0.500181\pi$
$601$	−0.682820	−0.00113614	−0.000568070		1.00000i	$0.500181\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	683.825	1.13029
$606$	0	0
$607$	1075.29i	1.77149i	0.464173	+	0.885744i	$0.346351\pi$
$607$	1075.29i	1.77149i	−0.464173	+	0.885744i	$0.653649\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 577.967i	− 0.945936i
$612$	0	0
$613$	−284.127	−0.463503	−0.231752	−	0.972775i	$-0.574446\pi$
$613$	−284.127	−0.463503	−0.231752	+	0.972775i	$0.574446\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	740.574	1.20028	0.600141	−	0.799894i	$-0.295111\pi$
$617$	740.574	1.20028	0.600141	+	0.799894i	$0.295111\pi$
$618$	0	0
$619$	− 16.2690i	− 0.0262827i	−0.999914	−	0.0131413i	$-0.995817\pi$
$619$	− 16.2690i	− 0.0262827i	0.999914	−	0.0131413i	$-0.00418314\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	25.1233i	0.0403264i
$624$	0	0
$625$	2301.23	3.68197
$626$	0	0
$627$	0	0
$628$	0	0
$629$	416.806	0.662648
$630$	0	0
$631$	600.999i	0.952455i	0.879322	+	0.476227i	$0.157996\pi$
$631$	600.999i	0.952455i	−0.879322	+	0.476227i	$0.842004\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1160.28i	1.82721i
$636$	0	0
$637$	413.316	0.648848
$638$	0	0
$639$	0	0
$640$	0	0
$641$	733.908	1.14494	0.572471	−	0.819925i	$-0.305985\pi$
$641$	733.908	1.14494	0.572471	+	0.819925i	$0.305985\pi$
$642$	0	0
$643$	269.442i	0.419039i	0.977804	+	0.209520i	$0.0671900\pi$
$643$	269.442i	0.419039i	−0.977804	+	0.209520i	$0.932810\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 685.853i	− 1.06005i	−0.847982	−	0.530026i	$-0.822182\pi$
$647$	− 685.853i	− 1.06005i	0.847982	−	0.530026i	$-0.177818\pi$
$648$	0	0
$649$	1153.33	1.77708
$650$	0	0
$651$	0	0
$652$	0	0
$653$	264.611	0.405223	0.202612	−	0.979259i	$-0.435057\pi$
$653$	264.611	0.405223	0.202612	+	0.979259i	$0.435057\pi$
$654$	0	0
$655$	580.742i	0.886629i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 798.019i	− 1.21095i	−0.795863	−	0.605477i	$-0.792982\pi$
$659$	− 798.019i	− 1.21095i	0.795863	−	0.605477i	$-0.207018\pi$
$660$	0	0
$661$	−395.442	−0.598248	−0.299124	−	0.954214i	$-0.596694\pi$
$661$	−395.442	−0.598248	−0.299124	+	0.954214i	$0.596694\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	299.069	0.449727
$666$	0	0
$667$	− 169.361i	− 0.253915i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1404.86i	2.09369i
$672$	0	0
$673$	−36.0196	−0.0535209	−0.0267604	−	0.999642i	$-0.508519\pi$
$673$	−36.0196	−0.0535209	−0.0267604	+	0.999642i	$0.508519\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	107.190	0.158331	0.0791654	−	0.996861i	$-0.474774\pi$
$677$	107.190	0.158331	0.0791654	+	0.996861i	$0.474774\pi$
$678$	0	0
$679$	79.7742i	0.117488i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	529.124i	0.774705i	0.921932	+	0.387353i	$0.126610\pi$
$683$	529.124i	0.774705i	−0.921932	+	0.387353i	$0.873390\pi$
$684$	0	0
$685$	135.353	0.197595
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−114.007	−0.165467
$690$	0	0
$691$	− 1134.76i	− 1.64220i	−0.570788	−	0.821098i	$-0.693362\pi$
$691$	− 1134.76i	− 1.64220i	0.570788	−	0.821098i	$-0.306638\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 2258.08i	− 3.24903i
$696$	0	0
$697$	−762.566	−1.09407
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1092.79	1.55890	0.779451	−	0.626463i	$-0.215498\pi$
$701$	1092.79	1.55890	0.779451	+	0.626463i	$0.215498\pi$
$702$	0	0
$703$	− 911.603i	− 1.29673i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	155.118i	0.219403i
$708$	0	0
$709$	−494.678	−0.697713	−0.348856	−	0.937176i	$-0.613430\pi$
$709$	−494.678	−0.697713	−0.348856	+	0.937176i	$0.613430\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−364.717	−0.511524
$714$	0	0
$715$	1156.59i	1.61761i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 431.893i	− 0.600686i	−0.953831	−	0.300343i	$-0.902899\pi$
$719$	− 431.893i	− 0.600686i	0.953831	−	0.300343i	$-0.0971011\pi$
$720$	0	0
$721$	131.237	0.182021
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−418.455	−0.577179
$726$	0	0
$727$	273.895i	0.376747i	0.982097	+	0.188374i	$0.0603216\pi$
$727$	273.895i	0.376747i	−0.982097	+	0.188374i	$0.939678\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	320.822i	0.438880i
$732$	0	0
$733$	667.160	0.910177	0.455089	−	0.890446i	$-0.349608\pi$
$733$	667.160	0.910177	0.455089	+	0.890446i	$0.349608\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1551.60	−2.10529
$738$	0	0
$739$	202.851i	0.274494i	0.990537	+	0.137247i	$0.0438254\pi$
$739$	202.851i	0.274494i	−0.990537	+	0.137247i	$0.956175\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 23.4215i	− 0.0315229i	−0.999876	−	0.0157614i	$-0.994983\pi$
$743$	− 23.4215i	− 0.0315229i	0.999876	−	0.0157614i	$-0.00501723\pi$
$744$	0	0
$745$	−491.698	−0.659997
$746$	0	0
$747$	0	0
$748$	0	0
$749$	43.9287	0.0586498
$750$	0	0
$751$	− 131.874i	− 0.175598i	−0.996138	−	0.0877988i	$-0.972017\pi$
$751$	− 131.874i	− 0.175598i	0.996138	−	0.0877988i	$-0.0279833\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1533.46i	− 2.03108i
$756$	0	0
$757$	771.905	1.01969	0.509845	−	0.860266i	$-0.329703\pi$
$757$	771.905	1.01969	0.509845	+	0.860266i	$0.329703\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−727.077	−0.955423	−0.477711	−	0.878517i	$-0.658534\pi$
$761$	−727.077	−0.955423	−0.477711	+	0.878517i	$0.658534\pi$
$762$	0	0
$763$	172.212i	0.225704i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	720.719i	0.939660i
$768$	0	0
$769$	−609.801	−0.792979	−0.396490	−	0.918039i	$-0.629772\pi$
$769$	−609.801	−0.792979	−0.396490	+	0.918039i	$0.629772\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1108.68	1.43426	0.717130	−	0.696939i	$-0.245455\pi$
$773$	1108.68	1.43426	0.717130	+	0.696939i	$0.245455\pi$
$774$	0	0
$775$	901.136i	1.16276i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1667.82i	2.14098i
$780$	0	0
$781$	−1190.27	−1.52403
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−25.9113	−0.0330080
$786$	0	0
$787$	− 822.206i	− 1.04473i	−0.852721	−	0.522367i	$-0.825049\pi$
$787$	− 822.206i	− 1.04473i	0.852721	−	0.522367i	$-0.174951\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	127.171i	0.160773i
$792$	0	0
$793$	−877.906	−1.10707
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1310.57	−1.64438	−0.822189	−	0.569215i	$-0.807247\pi$
$797$	−1310.57	−1.64438	−0.822189	+	0.569215i	$0.807247\pi$
$798$	0	0
$799$	− 845.028i	− 1.05761i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 347.428i	− 0.432662i
$804$	0	0
$805$	−297.427	−0.369475
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−645.853	−0.798335	−0.399168	−	0.916878i	$-0.630701\pi$
$809$	−645.853	−0.798335	−0.399168	+	0.916878i	$0.630701\pi$
$810$	0	0
$811$	1.06590i	0.00131430i	1.00000		0.000657151i	$0.000209178\pi$
$811$	1.06590i	0.00131430i	−1.00000		0.000657151i	$0.999791\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1414.48i	− 1.73555i
$816$	0	0
$817$	701.674	0.858842
$818$	0	0
$819$	0	0
$820$	0	0
$821$	727.892	0.886592	0.443296	−	0.896375i	$-0.353809\pi$
$821$	727.892	0.886592	0.443296	+	0.896375i	$0.353809\pi$
$822$	0	0
$823$	− 197.736i	− 0.240262i	−0.992758	−	0.120131i	$-0.961669\pi$
$823$	− 197.736i	− 0.240262i	0.992758	−	0.120131i	$-0.0383315\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 287.442i	− 0.347571i	−0.984783	−	0.173786i	$-0.944400\pi$
$827$	− 287.442i	− 0.347571i	0.984783	−	0.173786i	$-0.0556000\pi$
$828$	0	0
$829$	−1080.80	−1.30374	−0.651871	−	0.758330i	$-0.726016\pi$
$829$	−1080.80	−1.30374	−0.651871	+	0.758330i	$0.726016\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	604.297	0.725446
$834$	0	0
$835$	181.170i	0.216970i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 120.216i	− 0.143284i	−0.997430	−	0.0716422i	$-0.977176\pi$
$839$	− 120.216i	− 0.143284i	0.997430	−	0.0716422i	$-0.0228240\pi$
$840$	0	0
$841$	−803.154	−0.954998
$842$	0	0
$843$	0	0
$844$	0	0
$845$	907.193	1.07360
$846$	0	0
$847$	79.4231i	0.0937699i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	906.599i	1.06533i
$852$	0	0
$853$	−1270.01	−1.48888	−0.744438	−	0.667691i	$-0.767283\pi$
$853$	−1270.01	−1.48888	−0.744438	+	0.667691i	$0.767283\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	76.0067	0.0886892	0.0443446	−	0.999016i	$-0.485880\pi$
$857$	76.0067	0.0886892	0.0443446	+	0.999016i	$0.485880\pi$
$858$	0	0
$859$	424.136i	0.493756i	0.969047	+	0.246878i	$0.0794047\pi$
$859$	424.136i	0.493756i	−0.969047	+	0.246878i	$0.920595\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1036.76i	1.20134i	0.799495	+	0.600672i	$0.205100\pi$
$863$	1036.76i	1.20134i	−0.799495	+	0.600672i	$0.794900\pi$
$864$	0	0
$865$	−1643.78	−1.90032
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−879.108	−1.01163
$870$	0	0
$871$	− 969.600i	− 1.11320i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	464.782i	0.531180i
$876$	0	0
$877$	41.6471	0.0474881	0.0237441	−	0.999718i	$-0.492441\pi$
$877$	41.6471	0.0474881	0.0237441	+	0.999718i	$0.492441\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1196.29	1.35788	0.678940	−	0.734194i	$-0.262440\pi$
$881$	1196.29	1.35788	0.678940	+	0.734194i	$0.262440\pi$
$882$	0	0
$883$	− 106.778i	− 0.120926i	−0.998170	−	0.0604629i	$-0.980742\pi$
$883$	− 106.778i	− 0.120926i	0.998170	−	0.0604629i	$-0.0192577\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	216.134i	0.243668i	0.992550	+	0.121834i	$0.0388776\pi$
$887$	216.134i	0.243668i	−0.992550	+	0.121834i	$0.961122\pi$
$888$	0	0
$889$	−134.761	−0.151587
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1848.17	−2.06962
$894$	0	0
$895$	− 1798.07i	− 2.00901i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 81.5014i	− 0.0906578i
$900$	0	0
$901$	−166.686	−0.185001
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1117.69	1.23501
$906$	0	0
$907$	1672.90i	1.84444i	0.386670	+	0.922218i	$0.373625\pi$
$907$	1672.90i	1.84444i	−0.386670	+	0.922218i	$0.626375\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 238.389i	− 0.261678i	−0.991404	−	0.130839i	$-0.958233\pi$
$911$	− 238.389i	− 0.261678i	0.991404	−	0.130839i	$-0.0417672\pi$
$912$	0	0
$913$	−1543.81	−1.69091
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−67.4505	−0.0735556
$918$	0	0
$919$	771.672i	0.839687i	0.907596	+	0.419844i	$0.137915\pi$
$919$	771.672i	0.839687i	−0.907596	+	0.419844i	$0.862085\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 743.802i	− 0.805853i
$924$	0	0
$925$	2240.01	2.42163
$926$	0	0
$927$	0	0
$928$	0	0
$929$	97.4649	0.104914	0.0524569	−	0.998623i	$-0.483295\pi$
$929$	97.4649	0.104914	0.0524569	+	0.998623i	$0.483295\pi$
$930$	0	0
$931$	− 1321.67i	− 1.41962i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1691.02i	1.80858i
$936$	0	0
$937$	−1399.93	−1.49406	−0.747028	−	0.664793i	$-0.768520\pi$
$937$	−1399.93	−1.49406	−0.747028	+	0.664793i	$0.768520\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	253.338	0.269222	0.134611	−	0.990898i	$-0.457021\pi$
$941$	253.338	0.269222	0.134611	+	0.990898i	$0.457021\pi$
$942$	0	0
$943$	− 1658.67i	− 1.75893i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 223.848i	− 0.236376i	−0.992991	−	0.118188i	$-0.962292\pi$
$947$	− 223.848i	− 0.236376i	0.992991	−	0.118188i	$-0.0377085\pi$
$948$	0	0
$949$	217.109	0.228776
$950$	0	0
$951$	0	0
$952$	0	0
$953$	968.543	1.01631	0.508155	−	0.861266i	$-0.330328\pi$
$953$	968.543	1.01631	0.508155	+	0.861266i	$0.330328\pi$
$954$	0	0
$955$	448.355i	0.469482i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.7206i	0.0163927i
$960$	0	0
$961$	785.488	0.817365
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−598.738	−0.620454
$966$	0	0
$967$	1198.55i	1.23945i	0.784818	+	0.619726i	$0.212756\pi$
$967$	1198.55i	1.23945i	−0.784818	+	0.619726i	$0.787244\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1759.74i	− 1.81230i	−0.422961	−	0.906148i	$-0.639009\pi$
$971$	− 1759.74i	− 1.81230i	0.422961	−	0.906148i	$-0.360991\pi$
$972$	0	0
$973$	262.265	0.269543
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1502.13	−1.53749	−0.768746	−	0.639554i	$-0.779119\pi$
$977$	−1502.13	−1.53749	−0.768746	+	0.639554i	$0.779119\pi$
$978$	0	0
$979$	310.689i	0.317354i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	410.455i	0.417554i	0.977963	+	0.208777i	$0.0669483\pi$
$983$	410.455i	0.417554i	−0.977963	+	0.208777i	$0.933052\pi$
$984$	0	0
$985$	946.811	0.961229
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−697.823	−0.705584
$990$	0	0
$991$	210.418i	0.212329i	0.994349	+	0.106164i	$0.0338570\pi$
$991$	210.418i	0.212329i	−0.994349	+	0.106164i	$0.966143\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1008.92i	− 1.01399i
$996$	0	0
$997$	−450.591	−0.451947	−0.225973	−	0.974134i	$-0.572556\pi$
$997$	−450.591	−0.451947	−0.225973	+	0.974134i	$0.572556\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.k.703.1		8
3.2	odd	2		1728.3.g.n.703.7		8
4.3	odd	2	inner	1728.3.g.k.703.2		8
8.3	odd	2		864.3.g.c.703.8	yes	8
8.5	even	2		864.3.g.c.703.7	yes	8
12.11	even	2		1728.3.g.n.703.8		8
24.5	odd	2		864.3.g.a.703.1	✓	8
24.11	even	2		864.3.g.a.703.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.a.703.1	✓	8	24.5	odd	2
864.3.g.a.703.2	yes	8	24.11	even	2
864.3.g.c.703.7	yes	8	8.5	even	2
864.3.g.c.703.8	yes	8	8.3	odd	2
1728.3.g.k.703.1		8	1.1	even	1	trivial
1728.3.g.k.703.2		8	4.3	odd	2	inner
1728.3.g.n.703.7		8	3.2	odd	2
1728.3.g.n.703.8		8	12.11	even	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-9.64469 q^{5} -1.12019i q^{7} +O(q^{10})\)	\(q-9.64469 q^{5} -1.12019i q^{7} -13.8529i q^{11} +8.65671 q^{13} +12.6567 q^{17} -27.6817i q^{19} +27.5298i q^{23} +68.0201 q^{25} -6.15193 q^{29} +13.2481i q^{31} +10.8038i q^{35} +32.9316 q^{37} -60.2500 q^{41} +25.3479i q^{43} -66.7652i q^{47} +47.7452 q^{49} -13.1698 q^{53} +133.607i q^{55} +83.2556i q^{59} -101.413 q^{61} -83.4913 q^{65} -112.006i q^{67} -85.9221i q^{71} +25.0798 q^{73} -15.5178 q^{77} -63.4604i q^{79} -111.443i q^{83} -122.070 q^{85} -22.4278 q^{89} -9.69712i q^{91} +266.981i q^{95} -71.2151 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 16 q^{13} + 48 q^{17} + 48 q^{25} + 32 q^{29} + 96 q^{37} - 128 q^{41} - 168 q^{53} - 32 q^{61} - 112 q^{65} + 24 q^{73} + 440 q^{77} - 144 q^{85} + 624 q^{89} - 136 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 16 * q^13 + 48 * q^17 + 48 * q^25 + 32 * q^29 + 96 * q^37 - 128 * q^41 - 168 * q^53 - 32 * q^61 - 112 * q^65 + 24 * q^73 + 440 * q^77 - 144 * q^85 + 624 * q^89 - 136 * q^97

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