LMFDB - Embedded newform 108.6.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(1,108)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.a (trivial)

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:	yes
Analytic conductor:	$17.3214525398$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	108.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+57.6281 q^{5} -188.884 q^{7} +O(q^{10})$	$q+57.6281 q^{5} -188.884 q^{7} -704.025 q^{11} +795.537 q^{13} +180.562 q^{17} -661.769 q^{19} -3632.51 q^{23} +196.000 q^{25} -8021.87 q^{29} -3003.88 q^{31} -10885.0 q^{35} -1520.31 q^{37} -3461.57 q^{41} +11815.8 q^{43} +5979.82 q^{47} +18870.3 q^{49} -13819.2 q^{53} -40571.6 q^{55} -22640.2 q^{59} -37404.2 q^{61} +45845.3 q^{65} +70964.7 q^{67} +67767.0 q^{71} -31562.4 q^{73} +132979. q^{77} +62791.6 q^{79} -93431.5 q^{83} +10405.5 q^{85} +68793.1 q^{89} -150265. q^{91} -38136.5 q^{95} +117774. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 32 q^{7}+O(q^{10}) $ 2 * q - 32 * q^7	$ 2 q - 32 q^{7} - 486 q^{11} + 208 q^{13} - 1944 q^{17} - 632 q^{19} - 6804 q^{23} + 392 q^{25} - 11664 q^{29} + 3328 q^{31} - 19926 q^{35} - 9956 q^{37} - 13608 q^{41} + 4960 q^{43} - 18468 q^{47} + 26676 q^{49} + 11664 q^{53} - 53136 q^{55} - 1944 q^{59} + 8176 q^{61} + 79704 q^{65} + 90064 q^{67} + 44712 q^{71} - 121214 q^{73} + 167184 q^{77} + 28768 q^{79} + 15066 q^{83} + 132840 q^{85} + 178848 q^{89} - 242440 q^{91} - 39852 q^{95} + 88942 q^{97}+O(q^{100}) $ 2 * q - 32 * q^7 - 486 * q^11 + 208 * q^13 - 1944 * q^17 - 632 * q^19 - 6804 * q^23 + 392 * q^25 - 11664 * q^29 + 3328 * q^31 - 19926 * q^35 - 9956 * q^37 - 13608 * q^41 + 4960 * q^43 - 18468 * q^47 + 26676 * q^49 + 11664 * q^53 - 53136 * q^55 - 1944 * q^59 + 8176 * q^61 + 79704 * q^65 + 90064 * q^67 + 44712 * q^71 - 121214 * q^73 + 167184 * q^77 + 28768 * q^79 + 15066 * q^83 + 132840 * q^85 + 178848 * q^89 - 242440 * q^91 - 39852 * q^95 + 88942 * q^97

Display 100 coefficients 10 coefficients

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
$5$	57.6281	1.03088	0.515442	−	0.856925i	$-0.327628\pi$
$5$	57.6281	1.03088	0.515442	+	0.856925i	$0.327628\pi$
$6$	0	0
$7$	−188.884	−1.45697	−0.728485	−	0.685061i	$-0.759775\pi$
$7$	−188.884	−1.45697	−0.728485	+	0.685061i	$0.759775\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−704.025	−1.75431	−0.877155	−	0.480207i	$-0.840561\pi$
$11$	−704.025	−1.75431	−0.877155	+	0.480207i	$0.840561\pi$
$12$	0	0
$13$	795.537	1.30558	0.652788	−	0.757541i	$-0.273599\pi$
$13$	795.537	1.30558	0.652788	+	0.757541i	$0.273599\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	180.562	0.151532	0.0757661	−	0.997126i	$-0.475860\pi$
$17$	180.562	0.151532	0.0757661	+	0.997126i	$0.475860\pi$
$18$	0	0
$19$	−661.769	−0.420554	−0.210277	−	0.977642i	$-0.567437\pi$
$19$	−661.769	−0.420554	−0.210277	+	0.977642i	$0.567437\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3632.51	−1.43182	−0.715909	−	0.698194i	$-0.753987\pi$
$23$	−3632.51	−1.43182	−0.715909	+	0.698194i	$0.753987\pi$
$24$	0	0
$25$	196.000	0.0627200
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8021.87	−1.77125	−0.885626	−	0.464398i	$-0.846271\pi$
$29$	−8021.87	−1.77125	−0.885626	+	0.464398i	$0.846271\pi$
$30$	0	0
$31$	−3003.88	−0.561407	−0.280704	−	0.959795i	$-0.590568\pi$
$31$	−3003.88	−0.561407	−0.280704	+	0.959795i	$0.590568\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10885.0	−1.50197
$36$	0	0
$37$	−1520.31	−0.182570	−0.0912848	−	0.995825i	$-0.529097\pi$
$37$	−1520.31	−0.182570	−0.0912848	+	0.995825i	$0.529097\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3461.57	−0.321598	−0.160799	−	0.986987i	$-0.551407\pi$
$41$	−3461.57	−0.321598	−0.160799	+	0.986987i	$0.551407\pi$
$42$	0	0
$43$	11815.8	0.974519	0.487260	−	0.873257i	$-0.337997\pi$
$43$	11815.8	0.974519	0.487260	+	0.873257i	$0.337997\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5979.82	0.394861	0.197430	−	0.980317i	$-0.436740\pi$
$47$	5979.82	0.394861	0.197430	+	0.980317i	$0.436740\pi$
$48$	0	0
$49$	18870.3	1.12276
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13819.2	−0.675761	−0.337880	−	0.941189i	$-0.609710\pi$
$53$	−13819.2	−0.675761	−0.337880	+	0.941189i	$0.609710\pi$
$54$	0	0
$55$	−40571.6	−1.80849
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−22640.2	−0.846739	−0.423370	−	0.905957i	$-0.639153\pi$
$59$	−22640.2	−0.846739	−0.423370	+	0.905957i	$0.639153\pi$
$60$	0	0
$61$	−37404.2	−1.28705	−0.643526	−	0.765424i	$-0.722529\pi$
$61$	−37404.2	−1.28705	−0.643526	+	0.765424i	$0.722529\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	45845.3	1.34590
$66$	0	0
$67$	70964.7	1.93132	0.965662	−	0.259802i	$-0.0836573\pi$
$67$	70964.7	1.93132	0.965662	+	0.259802i	$0.0836573\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	67767.0	1.59541	0.797705	−	0.603048i	$-0.206047\pi$
$71$	67767.0	1.59541	0.797705	+	0.603048i	$0.206047\pi$
$72$	0	0
$73$	−31562.4	−0.693208	−0.346604	−	0.938012i	$-0.612665\pi$
$73$	−31562.4	−0.693208	−0.346604	+	0.938012i	$0.612665\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	132979.	2.55598
$78$	0	0
$79$	62791.6	1.13197	0.565984	−	0.824416i	$-0.308496\pi$
$79$	62791.6	1.13197	0.565984	+	0.824416i	$0.308496\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−93431.5	−1.48867	−0.744334	−	0.667807i	$-0.767233\pi$
$83$	−93431.5	−1.48867	−0.744334	+	0.667807i	$0.767233\pi$
$84$	0	0
$85$	10405.5	0.156212
$86$	0	0
$87$	0	0
$88$	0	0
$89$	68793.1	0.920598	0.460299	−	0.887764i	$-0.347742\pi$
$89$	68793.1	0.920598	0.460299	+	0.887764i	$0.347742\pi$
$90$	0	0
$91$	−150265.	−1.90219
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−38136.5	−0.433542
$96$	0	0
$97$	117774.	1.27093	0.635463	−	0.772132i	$-0.280809\pi$
$97$	117774.	1.27093	0.635463	+	0.772132i	$0.280809\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	161886.	1.57908	0.789542	−	0.613697i	$-0.210318\pi$
$101$	161886.	1.57908	0.789542	+	0.613697i	$0.210318\pi$
$102$	0	0
$103$	59324.6	0.550987	0.275494	−	0.961303i	$-0.411159\pi$
$103$	59324.6	0.550987	0.275494	+	0.961303i	$0.411159\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−21124.1	−0.178369	−0.0891845	−	0.996015i	$-0.528426\pi$
$107$	−21124.1	−0.178369	−0.0891845	+	0.996015i	$0.528426\pi$
$108$	0	0
$109$	−168898.	−1.36162	−0.680812	−	0.732458i	$-0.738373\pi$
$109$	−168898.	−1.36162	−0.680812	+	0.732458i	$0.738373\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−61274.0	−0.451419	−0.225710	−	0.974195i	$-0.572470\pi$
$113$	−61274.0	−0.451419	−0.225710	+	0.974195i	$0.572470\pi$
$114$	0	0
$115$	−209335.	−1.47604
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−34105.4	−0.220778
$120$	0	0
$121$	334600.	2.07760
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−168793.	−0.966226
$126$	0	0
$127$	23607.7	0.129881	0.0649404	−	0.997889i	$-0.479314\pi$
$127$	23607.7	0.129881	0.0649404	+	0.997889i	$0.479314\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−206849.	−1.05311	−0.526557	−	0.850140i	$-0.676517\pi$
$131$	−206849.	−1.05311	−0.526557	+	0.850140i	$0.676517\pi$
$132$	0	0
$133$	124998.	0.612736
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−191495.	−0.871678	−0.435839	−	0.900025i	$-0.643548\pi$
$137$	−191495.	−0.871678	−0.435839	+	0.900025i	$0.643548\pi$
$138$	0	0
$139$	250418.	1.09933	0.549666	−	0.835385i	$-0.314755\pi$
$139$	250418.	1.09933	0.549666	+	0.835385i	$0.314755\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−560078.	−2.29039
$144$	0	0
$145$	−462285.	−1.82595
$146$	0	0
$147$	0	0
$148$	0	0
$149$	200857.	0.741177	0.370588	−	0.928797i	$-0.379156\pi$
$149$	200857.	0.741177	0.370588	+	0.928797i	$0.379156\pi$
$150$	0	0
$151$	−23263.1	−0.0830283	−0.0415141	−	0.999138i	$-0.513218\pi$
$151$	−23263.1	−0.0830283	−0.0415141	+	0.999138i	$0.513218\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−173108.	−0.578745
$156$	0	0
$157$	−523985.	−1.69656	−0.848281	−	0.529546i	$-0.822362\pi$
$157$	−523985.	−1.69656	−0.848281	+	0.529546i	$0.822362\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	686125.	2.08612
$162$	0	0
$163$	194530.	0.573479	0.286739	−	0.958009i	$-0.407429\pi$
$163$	194530.	0.573479	0.286739	+	0.958009i	$0.407429\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−462757.	−1.28399	−0.641995	−	0.766709i	$-0.721893\pi$
$167$	−462757.	−1.28399	−0.641995	+	0.766709i	$0.721893\pi$
$168$	0	0
$169$	261587.	0.704529
$170$	0	0
$171$	0	0
$172$	0	0
$173$	39891.8	0.101337	0.0506685	−	0.998716i	$-0.483865\pi$
$173$	39891.8	0.101337	0.0506685	+	0.998716i	$0.483865\pi$
$174$	0	0
$175$	−37021.3	−0.0913812
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−221514.	−0.516736	−0.258368	−	0.966047i	$-0.583185\pi$
$179$	−221514.	−0.516736	−0.258368	+	0.966047i	$0.583185\pi$
$180$	0	0
$181$	800588.	1.81640	0.908202	−	0.418532i	$-0.137455\pi$
$181$	800588.	1.81640	0.908202	+	0.418532i	$0.137455\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−87612.8	−0.188208
$186$	0	0
$187$	−127120.	−0.265834
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−70550.6	−0.139932	−0.0699661	−	0.997549i	$-0.522289\pi$
$191$	−70550.6	−0.139932	−0.0699661	+	0.997549i	$0.522289\pi$
$192$	0	0
$193$	−458196.	−0.885437	−0.442719	−	0.896661i	$-0.645986\pi$
$193$	−458196.	−0.885437	−0.442719	+	0.896661i	$0.645986\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	605955.	1.11244	0.556218	−	0.831037i	$-0.312252\pi$
$197$	605955.	1.11244	0.556218	+	0.831037i	$0.312252\pi$
$198$	0	0
$199$	−108129.	−0.193557	−0.0967783	−	0.995306i	$-0.530854\pi$
$199$	−108129.	−0.193557	−0.0967783	+	0.995306i	$0.530854\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.51521e6	2.58066
$204$	0	0
$205$	−199484.	−0.331530
$206$	0	0
$207$	0	0
$208$	0	0
$209$	465902.	0.737783
$210$	0	0
$211$	274922.	0.425113	0.212556	−	0.977149i	$-0.431821\pi$
$211$	274922.	0.425113	0.212556	+	0.977149i	$0.431821\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	680920.	1.00462
$216$	0	0
$217$	567385.	0.817954
$218$	0	0
$219$	0	0
$220$	0	0
$221$	143644.	0.197837
$222$	0	0
$223$	−503846.	−0.678478	−0.339239	−	0.940700i	$-0.610170\pi$
$223$	−503846.	−0.678478	−0.339239	+	0.940700i	$0.610170\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	389725.	0.501989	0.250994	−	0.967989i	$-0.419242\pi$
$227$	389725.	0.501989	0.250994	+	0.967989i	$0.419242\pi$
$228$	0	0
$229$	−931505.	−1.17381	−0.586903	−	0.809657i	$-0.699653\pi$
$229$	−931505.	−1.17381	−0.586903	+	0.809657i	$0.699653\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−967962.	−1.16807	−0.584034	−	0.811729i	$-0.698527\pi$
$233$	−967962.	−1.16807	−0.584034	+	0.811729i	$0.698527\pi$
$234$	0	0
$235$	344606.	0.407055
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−335896.	−0.380373	−0.190187	−	0.981748i	$-0.560909\pi$
$239$	−335896.	−0.380373	−0.190187	+	0.981748i	$0.560909\pi$
$240$	0	0
$241$	−547142.	−0.606817	−0.303408	−	0.952861i	$-0.598125\pi$
$241$	−547142.	−0.606817	−0.303408	+	0.952861i	$0.598125\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.08746e6	1.15744
$246$	0	0
$247$	−526462.	−0.549066
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−36479.6	−0.0365482	−0.0182741	−	0.999833i	$-0.505817\pi$
$251$	−36479.6	−0.0365482	−0.0182741	+	0.999833i	$0.505817\pi$
$252$	0	0
$253$	2.55738e6	2.51185
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.19968e6	−1.13301	−0.566505	−	0.824058i	$-0.691705\pi$
$257$	−1.19968e6	−1.13301	−0.566505	+	0.824058i	$0.691705\pi$
$258$	0	0
$259$	287163.	0.265999
$260$	0	0
$261$	0	0
$262$	0	0
$263$	878508.	0.783171	0.391585	−	0.920142i	$-0.371927\pi$
$263$	878508.	0.783171	0.391585	+	0.920142i	$0.371927\pi$
$264$	0	0
$265$	−796374.	−0.696630
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−434159.	−0.365820	−0.182910	−	0.983130i	$-0.558552\pi$
$269$	−434159.	−0.365820	−0.182910	+	0.983130i	$0.558552\pi$
$270$	0	0
$271$	799583.	0.661364	0.330682	−	0.943742i	$-0.392721\pi$
$271$	799583.	0.661364	0.330682	+	0.943742i	$0.392721\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−137989.	−0.110030
$276$	0	0
$277$	1.63042e6	1.27674	0.638368	−	0.769731i	$-0.279610\pi$
$277$	1.63042e6	1.27674	0.638368	+	0.769731i	$0.279610\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−2.53698e6	−1.91669	−0.958344	−	0.285617i	$-0.907802\pi$
$281$	−2.53698e6	−1.91669	−0.958344	+	0.285617i	$0.907802\pi$
$282$	0	0
$283$	−2.07324e6	−1.53880	−0.769402	−	0.638765i	$-0.779445\pi$
$283$	−2.07324e6	−1.53880	−0.769402	+	0.638765i	$0.779445\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	653836.	0.468559
$288$	0	0
$289$	−1.38725e6	−0.977038
$290$	0	0
$291$	0	0
$292$	0	0
$293$	666768.	0.453739	0.226869	−	0.973925i	$-0.427151\pi$
$293$	666768.	0.453739	0.226869	+	0.973925i	$0.427151\pi$
$294$	0	0
$295$	−1.30471e6	−0.872889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.88980e6	−1.86935
$300$	0	0
$301$	−2.23181e6	−1.41985
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.15554e6	−1.32680
$306$	0	0
$307$	811069.	0.491148	0.245574	−	0.969378i	$-0.421024\pi$
$307$	811069.	0.491148	0.245574	+	0.969378i	$0.421024\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−985037.	−0.577500	−0.288750	−	0.957405i	$-0.593240\pi$
$311$	−985037.	−0.577500	−0.288750	+	0.957405i	$0.593240\pi$
$312$	0	0
$313$	783184.	0.451859	0.225929	−	0.974144i	$-0.427458\pi$
$313$	783184.	0.451859	0.225929	+	0.974144i	$0.427458\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2.66519e6	−1.48963	−0.744817	−	0.667268i	$-0.767463\pi$
$317$	−2.66519e6	−1.48963	−0.744817	+	0.667268i	$0.767463\pi$
$318$	0	0
$319$	5.64760e6	3.10733
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−119491.	−0.0637275
$324$	0	0
$325$	155925.	0.0818857
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.12950e6	−0.575300
$330$	0	0
$331$	−2.34153e6	−1.17471	−0.587353	−	0.809331i	$-0.699830\pi$
$331$	−2.34153e6	−1.17471	−0.587353	+	0.809331i	$0.699830\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.08956e6	1.99097
$336$	0	0
$337$	−710742.	−0.340908	−0.170454	−	0.985366i	$-0.554523\pi$
$337$	−710742.	−0.340908	−0.170454	+	0.985366i	$0.554523\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.11480e6	0.984882
$342$	0	0
$343$	−389725.	−0.178864
$344$	0	0
$345$	0	0
$346$	0	0
$347$	702187.	0.313061	0.156531	−	0.987673i	$-0.449969\pi$
$347$	702187.	0.313061	0.156531	+	0.987673i	$0.449969\pi$
$348$	0	0
$349$	−138699.	−0.0609553	−0.0304776	−	0.999535i	$-0.509703\pi$
$349$	−138699.	−0.0609553	−0.0304776	+	0.999535i	$0.509703\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	55094.1	0.0235325	0.0117663	−	0.999931i	$-0.496255\pi$
$353$	55094.1	0.0235325	0.0117663	+	0.999931i	$0.496255\pi$
$354$	0	0
$355$	3.90528e6	1.64468
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.94311e6	0.795722	0.397861	−	0.917446i	$-0.369753\pi$
$359$	1.94311e6	0.795722	0.397861	+	0.917446i	$0.369753\pi$
$360$	0	0
$361$	−2.03816e6	−0.823134
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.81888e6	−0.714616
$366$	0	0
$367$	−1.49593e6	−0.579758	−0.289879	−	0.957063i	$-0.593615\pi$
$367$	−1.49593e6	−0.579758	−0.289879	+	0.957063i	$0.593615\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.61023e6	0.984564
$372$	0	0
$373$	−1.71696e6	−0.638982	−0.319491	−	0.947589i	$-0.603512\pi$
$373$	−1.71696e6	−0.638982	−0.319491	+	0.947589i	$0.603512\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.38170e6	−2.31251
$378$	0	0
$379$	−3.54794e6	−1.26876	−0.634378	−	0.773023i	$-0.718744\pi$
$379$	−3.54794e6	−1.26876	−0.634378	+	0.773023i	$0.718744\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.75302e6	0.958987	0.479493	−	0.877545i	$-0.340820\pi$
$383$	2.75302e6	0.958987	0.479493	+	0.877545i	$0.340820\pi$
$384$	0	0
$385$	7.66335e6	2.63492
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.95371e6	0.989679	0.494839	−	0.868984i	$-0.335227\pi$
$389$	2.95371e6	0.989679	0.494839	+	0.868984i	$0.335227\pi$
$390$	0	0
$391$	−655895.	−0.216966
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.61856e6	1.16693
$396$	0	0
$397$	−2.21949e6	−0.706767	−0.353383	−	0.935479i	$-0.614969\pi$
$397$	−2.21949e6	−0.706767	−0.353383	+	0.935479i	$0.614969\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.43822e6	1.37831	0.689157	−	0.724612i	$-0.257981\pi$
$401$	4.43822e6	1.37831	0.689157	+	0.724612i	$0.257981\pi$
$402$	0	0
$403$	−2.38970e6	−0.732960
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.07034e6	0.320284
$408$	0	0
$409$	−331601.	−0.0980184	−0.0490092	−	0.998798i	$-0.515606\pi$
$409$	−331601.	−0.0980184	−0.0490092	+	0.998798i	$0.515606\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.27637e6	1.23367
$414$	0	0
$415$	−5.38428e6	−1.53464
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2.49486e6	0.694243	0.347121	−	0.937820i	$-0.387159\pi$
$419$	2.49486e6	0.694243	0.347121	+	0.937820i	$0.387159\pi$
$420$	0	0
$421$	4.55155e6	1.25157	0.625784	−	0.779997i	$-0.284779\pi$
$421$	4.55155e6	1.25157	0.625784	+	0.779997i	$0.284779\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	35390.2	0.00950410
$426$	0	0
$427$	7.06508e6	1.87520
$428$	0	0
$429$	0	0
$430$	0	0
$431$	614314.	0.159293	0.0796466	−	0.996823i	$-0.474621\pi$
$431$	614314.	0.159293	0.0796466	+	0.996823i	$0.474621\pi$
$432$	0	0
$433$	2.42759e6	0.622236	0.311118	−	0.950371i	$-0.399297\pi$
$433$	2.42759e6	0.622236	0.311118	+	0.950371i	$0.399297\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.40388e6	0.602157
$438$	0	0
$439$	−2.31800e6	−0.574054	−0.287027	−	0.957923i	$-0.592667\pi$
$439$	−2.31800e6	−0.574054	−0.287027	+	0.957923i	$0.592667\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.23648e6	−1.26774	−0.633870	−	0.773439i	$-0.718535\pi$
$443$	−5.23648e6	−1.26774	−0.633870	+	0.773439i	$0.718535\pi$
$444$	0	0
$445$	3.96442e6	0.949029
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−985716.	−0.230747	−0.115373	−	0.993322i	$-0.536806\pi$
$449$	−985716.	−0.230747	−0.115373	+	0.993322i	$0.536806\pi$
$450$	0	0
$451$	2.43703e6	0.564183
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.65946e6	−1.96093
$456$	0	0
$457$	−5.09496e6	−1.14117	−0.570585	−	0.821238i	$-0.693284\pi$
$457$	−5.09496e6	−1.14117	−0.570585	+	0.821238i	$0.693284\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.89512e6	1.94939	0.974697	−	0.223531i	$-0.0717584\pi$
$461$	8.89512e6	1.94939	0.974697	+	0.223531i	$0.0717584\pi$
$462$	0	0
$463$	3.26267e6	0.707327	0.353664	−	0.935373i	$-0.384936\pi$
$463$	3.26267e6	0.707327	0.353664	+	0.935373i	$0.384936\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.63482e6	−0.346879	−0.173439	−	0.984845i	$-0.555488\pi$
$467$	−1.63482e6	−0.346879	−0.173439	+	0.984845i	$0.555488\pi$
$468$	0	0
$469$	−1.34041e7	−2.81388
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8.31859e6	−1.70961
$474$	0	0
$475$	−129707.	−0.0263772
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−8.57113e6	−1.70687	−0.853433	−	0.521203i	$-0.825483\pi$
$479$	−8.57113e6	−1.70687	−0.853433	+	0.521203i	$0.825483\pi$
$480$	0	0
$481$	−1.20947e6	−0.238359
$482$	0	0
$483$	0	0
$484$	0	0
$485$	6.78709e6	1.31018
$486$	0	0
$487$	−5.12742e6	−0.979662	−0.489831	−	0.871817i	$-0.662942\pi$
$487$	−5.12742e6	−0.979662	−0.489831	+	0.871817i	$0.662942\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.33990e6	1.37400	0.687000	−	0.726658i	$-0.258927\pi$
$491$	7.33990e6	1.37400	0.687000	+	0.726658i	$0.258927\pi$
$492$	0	0
$493$	−1.44845e6	−0.268402
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.28001e7	−2.32446
$498$	0	0
$499$	6.08886e6	1.09467	0.547336	−	0.836913i	$-0.315642\pi$
$499$	6.08886e6	1.09467	0.547336	+	0.836913i	$0.315642\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.11508e6	−1.60635	−0.803175	−	0.595743i	$-0.796858\pi$
$503$	−9.11508e6	−1.60635	−0.803175	+	0.595743i	$0.796858\pi$
$504$	0	0
$505$	9.32917e6	1.62785
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.18164e6	0.544323	0.272162	−	0.962252i	$-0.412261\pi$
$509$	3.18164e6	0.544323	0.272162	+	0.962252i	$0.412261\pi$
$510$	0	0
$511$	5.96165e6	1.00998
$512$	0	0
$513$	0	0
$514$	0	0
$515$	3.41876e6	0.568003
$516$	0	0
$517$	−4.20994e6	−0.692708
$518$	0	0
$519$	0	0
$520$	0	0
$521$	27154.5	0.00438276	0.00219138	−	0.999998i	$-0.499302\pi$
$521$	27154.5	0.00438276	0.00219138	+	0.999998i	$0.499302\pi$
$522$	0	0
$523$	−3.93041e6	−0.628324	−0.314162	−	0.949369i	$-0.601724\pi$
$523$	−3.93041e6	−0.628324	−0.314162	+	0.949369i	$0.601724\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−542387.	−0.0850713
$528$	0	0
$529$	6.75880e6	1.05010
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.75381e6	−0.419871
$534$	0	0
$535$	−1.21734e6	−0.183878
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.32852e7	−1.96968
$540$	0	0
$541$	8.38180e6	1.23124	0.615622	−	0.788042i	$-0.288905\pi$
$541$	8.38180e6	1.23124	0.615622	+	0.788042i	$0.288905\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.73325e6	−1.40367
$546$	0	0
$547$	−7.50548e6	−1.07253	−0.536266	−	0.844049i	$-0.680166\pi$
$547$	−7.50548e6	−1.07253	−0.536266	+	0.844049i	$0.680166\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.30862e6	0.744908
$552$	0	0
$553$	−1.18604e7	−1.64924
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.03110e6	−0.140820	−0.0704099	−	0.997518i	$-0.522431\pi$
$557$	−1.03110e6	−0.140820	−0.0704099	+	0.997518i	$0.522431\pi$
$558$	0	0
$559$	9.39988e6	1.27231
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.31962e7	−1.75461	−0.877303	−	0.479937i	$-0.840659\pi$
$563$	−1.31962e7	−1.75461	−0.877303	+	0.479937i	$0.840659\pi$
$564$	0	0
$565$	−3.53110e6	−0.465360
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.49155e6	0.452103	0.226052	−	0.974115i	$-0.427418\pi$
$569$	3.49155e6	0.452103	0.226052	+	0.974115i	$0.427418\pi$
$570$	0	0
$571$	−9.49455e6	−1.21866	−0.609332	−	0.792915i	$-0.708562\pi$
$571$	−9.49455e6	−1.21866	−0.609332	+	0.792915i	$0.708562\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−711972.	−0.0898036
$576$	0	0
$577$	2.60369e6	0.325574	0.162787	−	0.986661i	$-0.447952\pi$
$577$	2.60369e6	0.325574	0.162787	+	0.986661i	$0.447952\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.76477e7	2.16895
$582$	0	0
$583$	9.72905e6	1.18549
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.04099e6	0.603838	0.301919	−	0.953334i	$-0.402373\pi$
$587$	5.04099e6	0.603838	0.301919	+	0.953334i	$0.402373\pi$
$588$	0	0
$589$	1.98787e6	0.236102
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.35192e7	1.57875	0.789374	−	0.613912i	$-0.210405\pi$
$593$	1.35192e7	1.57875	0.789374	+	0.613912i	$0.210405\pi$
$594$	0	0
$595$	−1.96543e6	−0.227596
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−815784.	−0.0928984	−0.0464492	−	0.998921i	$-0.514791\pi$
$599$	−815784.	−0.0928984	−0.0464492	+	0.998921i	$0.514791\pi$
$600$	0	0
$601$	−6.53058e6	−0.737507	−0.368753	−	0.929527i	$-0.620215\pi$
$601$	−6.53058e6	−0.737507	−0.368753	+	0.929527i	$0.620215\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.92824e7	2.14177
$606$	0	0
$607$	−1.10096e6	−0.121284	−0.0606418	−	0.998160i	$-0.519315\pi$
$607$	−1.10096e6	−0.121284	−0.0606418	+	0.998160i	$0.519315\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.75717e6	0.515520
$612$	0	0
$613$	6.09992e6	0.655651	0.327825	−	0.944738i	$-0.393684\pi$
$613$	6.09992e6	0.655651	0.327825	+	0.944738i	$0.393684\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	565825.	0.0598369	0.0299184	−	0.999552i	$-0.490475\pi$
$617$	565825.	0.0598369	0.0299184	+	0.999552i	$0.490475\pi$
$618$	0	0
$619$	−1.37509e7	−1.44246	−0.721232	−	0.692693i	$-0.756424\pi$
$619$	−1.37509e7	−1.44246	−0.721232	+	0.692693i	$0.756424\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.29939e7	−1.34128
$624$	0	0
$625$	−1.03397e7	−1.05879
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−274511.	−0.0276652
$630$	0	0
$631$	420973.	0.0420902	0.0210451	−	0.999779i	$-0.493301\pi$
$631$	420973.	0.0420902	0.0210451	+	0.999779i	$0.493301\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.36047e6	0.133892
$636$	0	0
$637$	1.50120e7	1.46585
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−314421.	−0.0302251	−0.0151125	−	0.999886i	$-0.504811\pi$
$641$	−314421.	−0.0302251	−0.0151125	+	0.999886i	$0.504811\pi$
$642$	0	0
$643$	1.38039e7	1.31667	0.658333	−	0.752727i	$-0.271262\pi$
$643$	1.38039e7	1.31667	0.658333	+	0.752727i	$0.271262\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.43225e6	0.322343	0.161172	−	0.986926i	$-0.448473\pi$
$647$	3.43225e6	0.322343	0.161172	+	0.986926i	$0.448473\pi$
$648$	0	0
$649$	1.59392e7	1.48544
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.74961e7	1.60568	0.802839	−	0.596196i	$-0.203322\pi$
$653$	1.74961e7	1.60568	0.802839	+	0.596196i	$0.203322\pi$
$654$	0	0
$655$	−1.19203e7	−1.08564
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1.20687e7	−1.08254	−0.541272	−	0.840848i	$-0.682057\pi$
$659$	−1.20687e7	−1.08254	−0.541272	+	0.840848i	$0.682057\pi$
$660$	0	0
$661$	−181872.	−0.0161906	−0.00809529	−	0.999967i	$-0.502577\pi$
$661$	−181872.	−0.0161906	−0.00809529	+	0.999967i	$0.502577\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	7.20339e6	0.631659
$666$	0	0
$667$	2.91395e7	2.53611
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.63335e7	2.25789
$672$	0	0
$673$	−1.06383e7	−0.905388	−0.452694	−	0.891666i	$-0.649537\pi$
$673$	−1.06383e7	−0.905388	−0.452694	+	0.891666i	$0.649537\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8.67911e6	−0.727786	−0.363893	−	0.931441i	$-0.618553\pi$
$677$	−8.67911e6	−0.727786	−0.363893	+	0.931441i	$0.618553\pi$
$678$	0	0
$679$	−2.22457e7	−1.85170
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.67179e7	1.37129	0.685647	−	0.727934i	$-0.259519\pi$
$683$	1.67179e7	1.37129	0.685647	+	0.727934i	$0.259519\pi$
$684$	0	0
$685$	−1.10355e7	−0.898598
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.09937e7	−0.882257
$690$	0	0
$691$	−3.85551e6	−0.307175	−0.153588	−	0.988135i	$-0.549083\pi$
$691$	−3.85551e6	−0.307175	−0.153588	+	0.988135i	$0.549083\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.44311e7	1.13328
$696$	0	0
$697$	−625029.	−0.0487325
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.86020e6	0.142976	0.0714882	−	0.997441i	$-0.477225\pi$
$701$	1.86020e6	0.142976	0.0714882	+	0.997441i	$0.477225\pi$
$702$	0	0
$703$	1.00610e6	0.0767805
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−3.05777e7	−2.30068
$708$	0	0
$709$	1.55608e7	1.16256	0.581281	−	0.813703i	$-0.302552\pi$
$709$	1.55608e7	1.16256	0.581281	+	0.813703i	$0.302552\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.09116e7	0.803832
$714$	0	0
$715$	−3.22763e7	−2.36112
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.35632e6	0.314266	0.157133	−	0.987577i	$-0.449775\pi$
$719$	4.35632e6	0.314266	0.157133	+	0.987577i	$0.449775\pi$
$720$	0	0
$721$	−1.12055e7	−0.802772
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.57229e6	−0.111093
$726$	0	0
$727$	2.96155e6	0.207818	0.103909	−	0.994587i	$-0.466865\pi$
$727$	2.96155e6	0.207818	0.103909	+	0.994587i	$0.466865\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.13348e6	0.147671
$732$	0	0
$733$	−960851.	−0.0660536	−0.0330268	−	0.999454i	$-0.510515\pi$
$733$	−960851.	−0.0660536	−0.0330268	+	0.999454i	$0.510515\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.99609e7	−3.38814
$738$	0	0
$739$	1.36123e7	0.916898	0.458449	−	0.888721i	$-0.348405\pi$
$739$	1.36123e7	0.916898	0.458449	+	0.888721i	$0.348405\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−5.73716e6	−0.381263	−0.190632	−	0.981662i	$-0.561054\pi$
$743$	−5.73716e6	−0.381263	−0.190632	+	0.981662i	$0.561054\pi$
$744$	0	0
$745$	1.15750e7	0.764067
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.99002e6	0.259878
$750$	0	0
$751$	2.44092e7	1.57926	0.789631	−	0.613582i	$-0.210272\pi$
$751$	2.44092e7	1.57926	0.789631	+	0.613582i	$0.210272\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.34061e6	−0.0855924
$756$	0	0
$757$	5.16979e6	0.327894	0.163947	−	0.986469i	$-0.447577\pi$
$757$	5.16979e6	0.327894	0.163947	+	0.986469i	$0.447577\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.08433e7	−0.678732	−0.339366	−	0.940654i	$-0.610212\pi$
$761$	−1.08433e7	−0.678732	−0.339366	+	0.940654i	$0.610212\pi$
$762$	0	0
$763$	3.19021e7	1.98385
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.80111e7	−1.10548
$768$	0	0
$769$	−9.27810e6	−0.565774	−0.282887	−	0.959153i	$-0.591292\pi$
$769$	−9.27810e6	−0.565774	−0.282887	+	0.959153i	$0.591292\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1.33311e7	−0.802450	−0.401225	−	0.915980i	$-0.631415\pi$
$773$	−1.33311e7	−0.802450	−0.401225	+	0.915980i	$0.631415\pi$
$774$	0	0
$775$	−588760.	−0.0352115
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.29076e6	0.135249
$780$	0	0
$781$	−4.77096e7	−2.79884
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.01963e7	−1.74896
$786$	0	0
$787$	6.35818e6	0.365928	0.182964	−	0.983120i	$-0.441431\pi$
$787$	6.35818e6	0.365928	0.182964	+	0.983120i	$0.441431\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.15737e7	0.657704
$792$	0	0
$793$	−2.97565e7	−1.68035
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−4.59152e6	−0.256042	−0.128021	−	0.991771i	$-0.540862\pi$
$797$	−4.59152e6	−0.256042	−0.128021	+	0.991771i	$0.540862\pi$
$798$	0	0
$799$	1.07973e6	0.0598341
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.22207e7	1.21610
$804$	0	0
$805$	3.95401e7	2.15054
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−8.91492e6	−0.478901	−0.239451	−	0.970909i	$-0.576967\pi$
$809$	−8.91492e6	−0.478901	−0.239451	+	0.970909i	$0.576967\pi$
$810$	0	0
$811$	−3.29527e6	−0.175930	−0.0879648	−	0.996124i	$-0.528036\pi$
$811$	−3.29527e6	−0.175930	−0.0879648	+	0.996124i	$0.528036\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.12104e7	0.591190
$816$	0	0
$817$	−7.81930e6	−0.409838
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.41824e6	0.228766	0.114383	−	0.993437i	$-0.463511\pi$
$821$	4.41824e6	0.228766	0.114383	+	0.993437i	$0.463511\pi$
$822$	0	0
$823$	−5.58259e6	−0.287300	−0.143650	−	0.989629i	$-0.545884\pi$
$823$	−5.58259e6	−0.287300	−0.143650	+	0.989629i	$0.545884\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.18571e7	−1.11130	−0.555648	−	0.831418i	$-0.687530\pi$
$827$	−2.18571e7	−1.11130	−0.555648	+	0.831418i	$0.687530\pi$
$828$	0	0
$829$	1.89089e7	0.955608	0.477804	−	0.878466i	$-0.341433\pi$
$829$	1.89089e7	0.955608	0.477804	+	0.878466i	$0.341433\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.40727e6	0.170135
$834$	0	0
$835$	−2.66678e7	−1.32364
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1.05498e7	−0.517413	−0.258707	−	0.965956i	$-0.583296\pi$
$839$	−1.05498e7	−0.517413	−0.258707	+	0.965956i	$0.583296\pi$
$840$	0	0
$841$	4.38392e7	2.13734
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.50748e7	0.726287
$846$	0	0
$847$	−6.32007e7	−3.02701
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.52256e6	0.261406
$852$	0	0
$853$	−641534.	−0.0301889	−0.0150945	−	0.999886i	$-0.504805\pi$
$853$	−641534.	−0.0301889	−0.0150945	+	0.999886i	$0.504805\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.21711e7	−1.49628	−0.748141	−	0.663540i	$-0.769053\pi$
$857$	−3.21711e7	−1.49628	−0.748141	+	0.663540i	$0.769053\pi$
$858$	0	0
$859$	1.44739e6	0.0669273	0.0334637	−	0.999440i	$-0.489346\pi$
$859$	1.44739e6	0.0669273	0.0334637	+	0.999440i	$0.489346\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9.16338e6	−0.418821	−0.209411	−	0.977828i	$-0.567155\pi$
$863$	−9.16338e6	−0.418821	−0.209411	+	0.977828i	$0.567155\pi$
$864$	0	0
$865$	2.29889e6	0.104467
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−4.42069e7	−1.98582
$870$	0	0
$871$	5.64550e7	2.52149
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.18823e7	1.40776
$876$	0	0
$877$	−2.15236e7	−0.944967	−0.472483	−	0.881340i	$-0.656642\pi$
$877$	−2.15236e7	−0.944967	−0.472483	+	0.881340i	$0.656642\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	4.58447e7	1.98998	0.994991	−	0.0999625i	$-0.0318723\pi$
$881$	4.58447e7	1.98998	0.994991	+	0.0999625i	$0.0318723\pi$
$882$	0	0
$883$	−3.10719e7	−1.34111	−0.670557	−	0.741858i	$-0.733945\pi$
$883$	−3.10719e7	−1.34111	−0.670557	+	0.741858i	$0.733945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.08214e7	−0.461820	−0.230910	−	0.972975i	$-0.574170\pi$
$887$	−1.08214e7	−0.461820	−0.230910	+	0.972975i	$0.574170\pi$
$888$	0	0
$889$	−4.45913e6	−0.189232
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.95726e6	−0.166060
$894$	0	0
$895$	−1.27654e7	−0.532694
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.40967e7	0.994394
$900$	0	0
$901$	−2.49523e6	−0.102399
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.61364e7	1.87250
$906$	0	0
$907$	3.71785e7	1.50063	0.750316	−	0.661080i	$-0.229901\pi$
$907$	3.71785e7	1.50063	0.750316	+	0.661080i	$0.229901\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.76509e7	1.10386	0.551929	−	0.833891i	$-0.313892\pi$
$911$	2.76509e7	1.10386	0.551929	+	0.833891i	$0.313892\pi$
$912$	0	0
$913$	6.57781e7	2.61159
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.90705e7	1.53436
$918$	0	0
$919$	2.24727e6	0.0877743	0.0438872	−	0.999036i	$-0.486026\pi$
$919$	2.24727e6	0.0877743	0.0438872	+	0.999036i	$0.486026\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.39112e7	2.08293
$924$	0	0
$925$	−297981.	−0.0114508
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−791805.	−0.0301009	−0.0150504	−	0.999887i	$-0.504791\pi$
$929$	−791805.	−0.0301009	−0.0150504	+	0.999887i	$0.504791\pi$
$930$	0	0
$931$	−1.24878e7	−0.472184
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−7.32571e6	−0.274044
$936$	0	0
$937$	−9.61458e6	−0.357751	−0.178876	−	0.983872i	$-0.557246\pi$
$937$	−9.61458e6	−0.357751	−0.178876	+	0.983872i	$0.557246\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.50396e7	−0.553683	−0.276842	−	0.960916i	$-0.589288\pi$
$941$	−1.50396e7	−0.553683	−0.276842	+	0.960916i	$0.589288\pi$
$942$	0	0
$943$	1.25742e7	0.460470
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.55153e7	0.562194	0.281097	−	0.959679i	$-0.409302\pi$
$947$	1.55153e7	0.562194	0.281097	+	0.959679i	$0.409302\pi$
$948$	0	0
$949$	−2.51091e7	−0.905035
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.16599e7	−1.84256	−0.921280	−	0.388899i	$-0.872855\pi$
$953$	−5.16599e7	−1.84256	−0.921280	+	0.388899i	$0.872855\pi$
$954$	0	0
$955$	−4.06570e6	−0.144254
$956$	0	0
$957$	0	0
$958$	0	0
$959$	3.61704e7	1.27001
$960$	0	0
$961$	−1.96059e7	−0.684822
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.64050e7	−0.912782
$966$	0	0
$967$	6.19595e6	0.213079	0.106540	−	0.994308i	$-0.466023\pi$
$967$	6.19595e6	0.213079	0.106540	+	0.994308i	$0.466023\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.67480e7	1.59116	0.795582	−	0.605846i	$-0.207165\pi$
$971$	4.67480e7	1.59116	0.795582	+	0.605846i	$0.207165\pi$
$972$	0	0
$973$	−4.73001e7	−1.60169
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.63907e6	−0.155487	−0.0777435	−	0.996973i	$-0.524772\pi$
$977$	−4.63907e6	−0.155487	−0.0777435	+	0.996973i	$0.524772\pi$
$978$	0	0
$979$	−4.84321e7	−1.61501
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2.75449e7	−0.909197	−0.454598	−	0.890697i	$-0.650217\pi$
$983$	−2.75449e7	−0.909197	−0.454598	+	0.890697i	$0.650217\pi$
$984$	0	0
$985$	3.49200e7	1.14679
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.29209e7	−1.39533
$990$	0	0
$991$	1.83035e7	0.592038	0.296019	−	0.955182i	$-0.404341\pi$
$991$	1.83035e7	0.592038	0.296019	+	0.955182i	$0.404341\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−6.23125e6	−0.199534
$996$	0	0
$997$	−4.44890e7	−1.41747	−0.708736	−	0.705473i	$-0.750734\pi$
$997$	−4.44890e7	−1.41747	−0.708736	+	0.705473i	$0.750734\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.a.b.1.2	✓	2
3.2	odd	2		108.6.a.c.1.1	yes	2
4.3	odd	2		432.6.a.r.1.2		2
9.2	odd	6		324.6.e.f.109.2		4
9.4	even	3		324.6.e.g.217.1		4
9.5	odd	6		324.6.e.f.217.2		4
9.7	even	3		324.6.e.g.109.1		4
12.11	even	2		432.6.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.a.b.1.2	✓	2	1.1	even	1	trivial
108.6.a.c.1.1	yes	2	3.2	odd	2
324.6.e.f.109.2		4	9.2	odd	6
324.6.e.f.217.2		4	9.5	odd	6
324.6.e.g.109.1		4	9.7	even	3
324.6.e.g.217.1		4	9.4	even	3
432.6.a.q.1.1		2	12.11	even	2
432.6.a.r.1.2		2	4.3	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:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.a (trivial)

\(f(q)\)	\(=\)	\(q+57.6281 q^{5} -188.884 q^{7} +O(q^{10})\)	\(q+57.6281 q^{5} -188.884 q^{7} -704.025 q^{11} +795.537 q^{13} +180.562 q^{17} -661.769 q^{19} -3632.51 q^{23} +196.000 q^{25} -8021.87 q^{29} -3003.88 q^{31} -10885.0 q^{35} -1520.31 q^{37} -3461.57 q^{41} +11815.8 q^{43} +5979.82 q^{47} +18870.3 q^{49} -13819.2 q^{53} -40571.6 q^{55} -22640.2 q^{59} -37404.2 q^{61} +45845.3 q^{65} +70964.7 q^{67} +67767.0 q^{71} -31562.4 q^{73} +132979. q^{77} +62791.6 q^{79} -93431.5 q^{83} +10405.5 q^{85} +68793.1 q^{89} -150265. q^{91} -38136.5 q^{95} +117774. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 32 q^{7}+O(q^{10}) \) 2 * q - 32 * q^7	\( 2 q - 32 q^{7} - 486 q^{11} + 208 q^{13} - 1944 q^{17} - 632 q^{19} - 6804 q^{23} + 392 q^{25} - 11664 q^{29} + 3328 q^{31} - 19926 q^{35} - 9956 q^{37} - 13608 q^{41} + 4960 q^{43} - 18468 q^{47} + 26676 q^{49} + 11664 q^{53} - 53136 q^{55} - 1944 q^{59} + 8176 q^{61} + 79704 q^{65} + 90064 q^{67} + 44712 q^{71} - 121214 q^{73} + 167184 q^{77} + 28768 q^{79} + 15066 q^{83} + 132840 q^{85} + 178848 q^{89} - 242440 q^{91} - 39852 q^{95} + 88942 q^{97}+O(q^{100}) \) 2 * q - 32 * q^7 - 486 * q^11 + 208 * q^13 - 1944 * q^17 - 632 * q^19 - 6804 * q^23 + 392 * q^25 - 11664 * q^29 + 3328 * q^31 - 19926 * q^35 - 9956 * q^37 - 13608 * q^41 + 4960 * q^43 - 18468 * q^47 + 26676 * q^49 + 11664 * q^53 - 53136 * q^55 - 1944 * q^59 + 8176 * q^61 + 79704 * q^65 + 90064 * q^67 + 44712 * q^71 - 121214 * q^73 + 167184 * q^77 + 28768 * q^79 + 15066 * q^83 + 132840 * q^85 + 178848 * q^89 - 242440 * q^91 - 39852 * q^95 + 88942 * q^97

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