LMFDB - Embedded newform 6600.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6600,2,Mod(1,6600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6600, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6600.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:	$52.7012653340$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	6600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.438447 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.438447 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.12311 q^{13} -3.12311 q^{17} +3.00000 q^{19} +0.438447 q^{21} +5.68466 q^{23} +1.00000 q^{27} +9.68466 q^{29} +3.00000 q^{31} +1.00000 q^{33} +6.00000 q^{37} +6.12311 q^{39} -6.00000 q^{41} -2.12311 q^{43} -4.00000 q^{47} -6.80776 q^{49} -3.12311 q^{51} -7.12311 q^{53} +3.00000 q^{57} -11.3693 q^{59} -6.68466 q^{61} +0.438447 q^{63} -4.43845 q^{67} +5.68466 q^{69} +1.43845 q^{71} -8.24621 q^{73} +0.438447 q^{77} +14.2462 q^{79} +1.00000 q^{81} -9.68466 q^{83} +9.68466 q^{87} +3.43845 q^{89} +2.68466 q^{91} +3.00000 q^{93} +12.1231 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{21} - q^{23} + 2 q^{27} + 7 q^{29} + 6 q^{31} + 2 q^{33} + 12 q^{37} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 8 q^{47} + 7 q^{49} + 2 q^{51} - 6 q^{53} + 6 q^{57} + 2 q^{59} - q^{61} + 5 q^{63} - 13 q^{67} - q^{69} + 7 q^{71} + 5 q^{77} + 12 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{87} + 11 q^{89} - 7 q^{91} + 6 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^21 - q^23 + 2 * q^27 + 7 * q^29 + 6 * q^31 + 2 * q^33 + 12 * q^37 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^47 + 7 * q^49 + 2 * q^51 - 6 * q^53 + 6 * q^57 + 2 * q^59 - q^61 + 5 * q^63 - 13 * q^67 - q^69 + 7 * q^71 + 5 * q^77 + 12 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^87 + 11 * q^89 - 7 * q^91 + 6 * q^93 + 16 * q^97 + 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.12311	1.69824	0.849122	−	0.528197i	$-0.177132\pi$
$13$	6.12311	1.69824	0.849122	+	0.528197i	$0.177132\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.12311	−0.757464	−0.378732	−	0.925506i	$-0.623640\pi$
$17$	−3.12311	−0.757464	−0.378732	+	0.925506i	$0.623640\pi$
$18$	0	0
$19$	3.00000	0.688247	0.344124	−	0.938924i	$-0.388176\pi$
$19$	3.00000	0.688247	0.344124	+	0.938924i	$0.388176\pi$
$20$	0	0
$21$	0.438447	0.0956770
$22$	0	0
$23$	5.68466	1.18533	0.592667	−	0.805448i	$-0.298075\pi$
$23$	5.68466	1.18533	0.592667	+	0.805448i	$0.298075\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	9.68466	1.79840	0.899198	−	0.437542i	$-0.144151\pi$
$29$	9.68466	1.79840	0.899198	+	0.437542i	$0.144151\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	6.12311	0.980482
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−2.12311	−0.323771	−0.161885	−	0.986810i	$-0.551757\pi$
$43$	−2.12311	−0.323771	−0.161885	+	0.986810i	$0.551757\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	−3.12311	−0.437322
$52$	0	0
$53$	−7.12311	−0.978434	−0.489217	−	0.872162i	$-0.662717\pi$
$53$	−7.12311	−0.978434	−0.489217	+	0.872162i	$0.662717\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.00000	0.397360
$58$	0	0
$59$	−11.3693	−1.48016	−0.740079	−	0.672519i	$-0.765212\pi$
$59$	−11.3693	−1.48016	−0.740079	+	0.672519i	$0.765212\pi$
$60$	0	0
$61$	−6.68466	−0.855883	−0.427941	−	0.903806i	$-0.640761\pi$
$61$	−6.68466	−0.855883	−0.427941	+	0.903806i	$0.640761\pi$
$62$	0	0
$63$	0.438447	0.0552392
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.43845	−0.542243	−0.271121	−	0.962545i	$-0.587394\pi$
$67$	−4.43845	−0.542243	−0.271121	+	0.962545i	$0.587394\pi$
$68$	0	0
$69$	5.68466	0.684352
$70$	0	0
$71$	1.43845	0.170712	0.0853561	−	0.996351i	$-0.472797\pi$
$71$	1.43845	0.170712	0.0853561	+	0.996351i	$0.472797\pi$
$72$	0	0
$73$	−8.24621	−0.965146	−0.482573	−	0.875856i	$-0.660298\pi$
$73$	−8.24621	−0.965146	−0.482573	+	0.875856i	$0.660298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.438447	0.0499657
$78$	0	0
$79$	14.2462	1.60282	0.801412	−	0.598113i	$-0.204082\pi$
$79$	14.2462	1.60282	0.801412	+	0.598113i	$0.204082\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.68466	−1.06303	−0.531515	−	0.847049i	$-0.678377\pi$
$83$	−9.68466	−1.06303	−0.531515	+	0.847049i	$0.678377\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	9.68466	1.03830
$88$	0	0
$89$	3.43845	0.364475	0.182237	−	0.983255i	$-0.441666\pi$
$89$	3.43845	0.364475	0.182237	+	0.983255i	$0.441666\pi$
$90$	0	0
$91$	2.68466	0.281429
$92$	0	0
$93$	3.00000	0.311086
$94$	0	0
$95$	0	0
$96$	0	0
$97$	12.1231	1.23091	0.615457	−	0.788170i	$-0.288971\pi$
$97$	12.1231	1.23091	0.615457	+	0.788170i	$0.288971\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	4.80776	0.478390	0.239195	−	0.970972i	$-0.423116\pi$
$101$	4.80776	0.478390	0.239195	+	0.970972i	$0.423116\pi$
$102$	0	0
$103$	19.0540	1.87744	0.938722	−	0.344675i	$-0.112011\pi$
$103$	19.0540	1.87744	0.938722	+	0.344675i	$0.112011\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.68466	0.549557	0.274778	−	0.961508i	$-0.411396\pi$
$107$	5.68466	0.549557	0.274778	+	0.961508i	$0.411396\pi$
$108$	0	0
$109$	−10.1231	−0.969618	−0.484809	−	0.874620i	$-0.661111\pi$
$109$	−10.1231	−0.969618	−0.484809	+	0.874620i	$0.661111\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	16.2462	1.52831	0.764157	−	0.645030i	$-0.223155\pi$
$113$	16.2462	1.52831	0.764157	+	0.645030i	$0.223155\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.12311	0.566081
$118$	0	0
$119$	−1.36932	−0.125525
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.2462	0.909204	0.454602	−	0.890695i	$-0.349781\pi$
$127$	10.2462	0.909204	0.454602	+	0.890695i	$0.349781\pi$
$128$	0	0
$129$	−2.12311	−0.186929
$130$	0	0
$131$	−8.56155	−0.748026	−0.374013	−	0.927423i	$-0.622019\pi$
$131$	−8.56155	−0.748026	−0.374013	+	0.927423i	$0.622019\pi$
$132$	0	0
$133$	1.31534	0.114055
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.43845	0.464638	0.232319	−	0.972640i	$-0.425369\pi$
$137$	5.43845	0.464638	0.232319	+	0.972640i	$0.425369\pi$
$138$	0	0
$139$	20.8078	1.76489	0.882446	−	0.470414i	$-0.155895\pi$
$139$	20.8078	1.76489	0.882446	+	0.470414i	$0.155895\pi$
$140$	0	0
$141$	−4.00000	−0.336861
$142$	0	0
$143$	6.12311	0.512040
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−6.80776	−0.561495
$148$	0	0
$149$	−16.2462	−1.33094	−0.665471	−	0.746424i	$-0.731769\pi$
$149$	−16.2462	−1.33094	−0.665471	+	0.746424i	$0.731769\pi$
$150$	0	0
$151$	−18.4384	−1.50050	−0.750250	−	0.661155i	$-0.770067\pi$
$151$	−18.4384	−1.50050	−0.750250	+	0.661155i	$0.770067\pi$
$152$	0	0
$153$	−3.12311	−0.252488
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−3.80776	−0.303893	−0.151946	−	0.988389i	$-0.548554\pi$
$157$	−3.80776	−0.303893	−0.151946	+	0.988389i	$0.548554\pi$
$158$	0	0
$159$	−7.12311	−0.564899
$160$	0	0
$161$	2.49242	0.196430
$162$	0	0
$163$	7.31534	0.572982	0.286491	−	0.958083i	$-0.407511\pi$
$163$	7.31534	0.572982	0.286491	+	0.958083i	$0.407511\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.12311	−0.0869085	−0.0434543	−	0.999055i	$-0.513836\pi$
$167$	−1.12311	−0.0869085	−0.0434543	+	0.999055i	$0.513836\pi$
$168$	0	0
$169$	24.4924	1.88403
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	−5.93087	−0.450916	−0.225458	−	0.974253i	$-0.572388\pi$
$173$	−5.93087	−0.450916	−0.225458	+	0.974253i	$0.572388\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−11.3693	−0.854570
$178$	0	0
$179$	23.6155	1.76511	0.882554	−	0.470212i	$-0.155822\pi$
$179$	23.6155	1.76511	0.882554	+	0.470212i	$0.155822\pi$
$180$	0	0
$181$	−7.80776	−0.580347	−0.290173	−	0.956974i	$-0.593713\pi$
$181$	−7.80776	−0.580347	−0.290173	+	0.956974i	$0.593713\pi$
$182$	0	0
$183$	−6.68466	−0.494144
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.12311	−0.228384
$188$	0	0
$189$	0.438447	0.0318923
$190$	0	0
$191$	−20.8078	−1.50560	−0.752798	−	0.658251i	$-0.771297\pi$
$191$	−20.8078	−1.50560	−0.752798	+	0.658251i	$0.771297\pi$
$192$	0	0
$193$	6.68466	0.481172	0.240586	−	0.970628i	$-0.422660\pi$
$193$	6.68466	0.481172	0.240586	+	0.970628i	$0.422660\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.93087	0.707545	0.353773	−	0.935331i	$-0.384899\pi$
$197$	9.93087	0.707545	0.353773	+	0.935331i	$0.384899\pi$
$198$	0	0
$199$	19.2462	1.36433	0.682164	−	0.731199i	$-0.261039\pi$
$199$	19.2462	1.36433	0.682164	+	0.731199i	$0.261039\pi$
$200$	0	0
$201$	−4.43845	−0.313064
$202$	0	0
$203$	4.24621	0.298026
$204$	0	0
$205$	0	0
$206$	0	0
$207$	5.68466	0.395111
$208$	0	0
$209$	3.00000	0.207514
$210$	0	0
$211$	3.31534	0.228238	0.114119	−	0.993467i	$-0.463596\pi$
$211$	3.31534	0.228238	0.114119	+	0.993467i	$0.463596\pi$
$212$	0	0
$213$	1.43845	0.0985608
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.31534	0.0892912
$218$	0	0
$219$	−8.24621	−0.557227
$220$	0	0
$221$	−19.1231	−1.28636
$222$	0	0
$223$	−18.0540	−1.20898	−0.604492	−	0.796611i	$-0.706624\pi$
$223$	−18.0540	−1.20898	−0.604492	+	0.796611i	$0.706624\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.8078	−1.38106	−0.690530	−	0.723304i	$-0.742623\pi$
$227$	−20.8078	−1.38106	−0.690530	+	0.723304i	$0.742623\pi$
$228$	0	0
$229$	−25.5616	−1.68915	−0.844577	−	0.535433i	$-0.820148\pi$
$229$	−25.5616	−1.68915	−0.844577	+	0.535433i	$0.820148\pi$
$230$	0	0
$231$	0.438447	0.0288477
$232$	0	0
$233$	20.4924	1.34250	0.671252	−	0.741230i	$-0.265757\pi$
$233$	20.4924	1.34250	0.671252	+	0.741230i	$0.265757\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.2462	0.925391
$238$	0	0
$239$	19.1231	1.23697	0.618485	−	0.785796i	$-0.287747\pi$
$239$	19.1231	1.23697	0.618485	+	0.785796i	$0.287747\pi$
$240$	0	0
$241$	18.6847	1.20358	0.601792	−	0.798653i	$-0.294453\pi$
$241$	18.6847	1.20358	0.601792	+	0.798653i	$0.294453\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	18.3693	1.16881
$248$	0	0
$249$	−9.68466	−0.613740
$250$	0	0
$251$	5.36932	0.338908	0.169454	−	0.985538i	$-0.445800\pi$
$251$	5.36932	0.338908	0.169454	+	0.985538i	$0.445800\pi$
$252$	0	0
$253$	5.68466	0.357391
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.6847	0.728869	0.364434	−	0.931229i	$-0.381262\pi$
$257$	11.6847	0.728869	0.364434	+	0.931229i	$0.381262\pi$
$258$	0	0
$259$	2.63068	0.163463
$260$	0	0
$261$	9.68466	0.599465
$262$	0	0
$263$	16.8769	1.04067	0.520337	−	0.853961i	$-0.325806\pi$
$263$	16.8769	1.04067	0.520337	+	0.853961i	$0.325806\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.43845	0.210430
$268$	0	0
$269$	0.876894	0.0534652	0.0267326	−	0.999643i	$-0.491490\pi$
$269$	0.876894	0.0534652	0.0267326	+	0.999643i	$0.491490\pi$
$270$	0	0
$271$	−5.75379	−0.349518	−0.174759	−	0.984611i	$-0.555915\pi$
$271$	−5.75379	−0.349518	−0.174759	+	0.984611i	$0.555915\pi$
$272$	0	0
$273$	2.68466	0.162483
$274$	0	0
$275$	0	0
$276$	0	0
$277$	27.5616	1.65601	0.828007	−	0.560718i	$-0.189475\pi$
$277$	27.5616	1.65601	0.828007	+	0.560718i	$0.189475\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−28.0000	−1.67034	−0.835170	−	0.549992i	$-0.814631\pi$
$281$	−28.0000	−1.67034	−0.835170	+	0.549992i	$0.814631\pi$
$282$	0	0
$283$	0.192236	0.0114272	0.00571362	−	0.999984i	$-0.498181\pi$
$283$	0.192236	0.0114272	0.00571362	+	0.999984i	$0.498181\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.63068	−0.155284
$288$	0	0
$289$	−7.24621	−0.426248
$290$	0	0
$291$	12.1231	0.710669
$292$	0	0
$293$	28.7386	1.67893	0.839464	−	0.543415i	$-0.182869\pi$
$293$	28.7386	1.67893	0.839464	+	0.543415i	$0.182869\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	34.8078	2.01298
$300$	0	0
$301$	−0.930870	−0.0536544
$302$	0	0
$303$	4.80776	0.276199
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.192236	0.0109715	0.00548574	−	0.999985i	$-0.498254\pi$
$307$	0.192236	0.0109715	0.00548574	+	0.999985i	$0.498254\pi$
$308$	0	0
$309$	19.0540	1.08394
$310$	0	0
$311$	−27.4384	−1.55589	−0.777946	−	0.628331i	$-0.783738\pi$
$311$	−27.4384	−1.55589	−0.777946	+	0.628331i	$0.783738\pi$
$312$	0	0
$313$	−28.4384	−1.60744	−0.803718	−	0.595010i	$-0.797148\pi$
$313$	−28.4384	−1.60744	−0.803718	+	0.595010i	$0.797148\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.36932	−0.301571	−0.150785	−	0.988567i	$-0.548180\pi$
$317$	−5.36932	−0.301571	−0.150785	+	0.988567i	$0.548180\pi$
$318$	0	0
$319$	9.68466	0.542237
$320$	0	0
$321$	5.68466	0.317287
$322$	0	0
$323$	−9.36932	−0.521323
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.1231	−0.559809
$328$	0	0
$329$	−1.75379	−0.0966895
$330$	0	0
$331$	17.1231	0.941171	0.470586	−	0.882354i	$-0.344043\pi$
$331$	17.1231	0.941171	0.470586	+	0.882354i	$0.344043\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	0	0
$336$	0	0
$337$	25.1771	1.37148	0.685741	−	0.727845i	$-0.259478\pi$
$337$	25.1771	1.37148	0.685741	+	0.727845i	$0.259478\pi$
$338$	0	0
$339$	16.2462	0.882373
$340$	0	0
$341$	3.00000	0.162459
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	−3.43845	−0.184056	−0.0920279	−	0.995756i	$-0.529335\pi$
$349$	−3.43845	−0.184056	−0.0920279	+	0.995756i	$0.529335\pi$
$350$	0	0
$351$	6.12311	0.326827
$352$	0	0
$353$	4.56155	0.242787	0.121393	−	0.992604i	$-0.461264\pi$
$353$	4.56155	0.242787	0.121393	+	0.992604i	$0.461264\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.36932	−0.0724719
$358$	0	0
$359$	−9.12311	−0.481499	−0.240750	−	0.970587i	$-0.577393\pi$
$359$	−9.12311	−0.481499	−0.240750	+	0.970587i	$0.577393\pi$
$360$	0	0
$361$	−10.0000	−0.526316
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.63068	0.398318	0.199159	−	0.979967i	$-0.436179\pi$
$367$	7.63068	0.398318	0.199159	+	0.979967i	$0.436179\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−3.12311	−0.162144
$372$	0	0
$373$	−7.56155	−0.391522	−0.195761	−	0.980652i	$-0.562718\pi$
$373$	−7.56155	−0.391522	−0.195761	+	0.980652i	$0.562718\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	59.3002	3.05411
$378$	0	0
$379$	−2.93087	−0.150549	−0.0752743	−	0.997163i	$-0.523983\pi$
$379$	−2.93087	−0.150549	−0.0752743	+	0.997163i	$0.523983\pi$
$380$	0	0
$381$	10.2462	0.524929
$382$	0	0
$383$	4.56155	0.233084	0.116542	−	0.993186i	$-0.462819\pi$
$383$	4.56155	0.233084	0.116542	+	0.993186i	$0.462819\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−2.12311	−0.107924
$388$	0	0
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−17.7538	−0.897848
$392$	0	0
$393$	−8.56155	−0.431873
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−22.0540	−1.10686	−0.553429	−	0.832897i	$-0.686681\pi$
$397$	−22.0540	−1.10686	−0.553429	+	0.832897i	$0.686681\pi$
$398$	0	0
$399$	1.31534	0.0658494
$400$	0	0
$401$	−31.3002	−1.56306	−0.781528	−	0.623870i	$-0.785560\pi$
$401$	−31.3002	−1.56306	−0.781528	+	0.623870i	$0.785560\pi$
$402$	0	0
$403$	18.3693	0.915041
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.00000	0.297409
$408$	0	0
$409$	−9.80776	−0.484963	−0.242481	−	0.970156i	$-0.577961\pi$
$409$	−9.80776	−0.484963	−0.242481	+	0.970156i	$0.577961\pi$
$410$	0	0
$411$	5.43845	0.268259
$412$	0	0
$413$	−4.98485	−0.245288
$414$	0	0
$415$	0	0
$416$	0	0
$417$	20.8078	1.01896
$418$	0	0
$419$	−5.12311	−0.250280	−0.125140	−	0.992139i	$-0.539938\pi$
$419$	−5.12311	−0.250280	−0.125140	+	0.992139i	$0.539938\pi$
$420$	0	0
$421$	23.6155	1.15095	0.575475	−	0.817819i	$-0.304817\pi$
$421$	23.6155	1.15095	0.575475	+	0.817819i	$0.304817\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.93087	−0.141835
$428$	0	0
$429$	6.12311	0.295626
$430$	0	0
$431$	−28.7386	−1.38429	−0.692146	−	0.721758i	$-0.743334\pi$
$431$	−28.7386	−1.38429	−0.692146	+	0.721758i	$0.743334\pi$
$432$	0	0
$433$	−35.0000	−1.68199	−0.840996	−	0.541041i	$-0.818030\pi$
$433$	−35.0000	−1.68199	−0.840996	+	0.541041i	$0.818030\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	17.0540	0.815802
$438$	0	0
$439$	−0.930870	−0.0444280	−0.0222140	−	0.999753i	$-0.507072\pi$
$439$	−0.930870	−0.0444280	−0.0222140	+	0.999753i	$0.507072\pi$
$440$	0	0
$441$	−6.80776	−0.324179
$442$	0	0
$443$	29.3693	1.39538	0.697689	−	0.716401i	$-0.254212\pi$
$443$	29.3693	1.39538	0.697689	+	0.716401i	$0.254212\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−16.2462	−0.768419
$448$	0	0
$449$	16.1771	0.763444	0.381722	−	0.924277i	$-0.375331\pi$
$449$	16.1771	0.763444	0.381722	+	0.924277i	$0.375331\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	−18.4384	−0.866314
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.12311	0.146093	0.0730464	−	0.997329i	$-0.476728\pi$
$457$	3.12311	0.146093	0.0730464	+	0.997329i	$0.476728\pi$
$458$	0	0
$459$	−3.12311	−0.145774
$460$	0	0
$461$	−20.2462	−0.942960	−0.471480	−	0.881877i	$-0.656280\pi$
$461$	−20.2462	−0.942960	−0.471480	+	0.881877i	$0.656280\pi$
$462$	0	0
$463$	28.1771	1.30950	0.654750	−	0.755846i	$-0.272774\pi$
$463$	28.1771	1.30950	0.654750	+	0.755846i	$0.272774\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.12311	−0.144520	−0.0722600	−	0.997386i	$-0.523021\pi$
$467$	−3.12311	−0.144520	−0.0722600	+	0.997386i	$0.523021\pi$
$468$	0	0
$469$	−1.94602	−0.0898591
$470$	0	0
$471$	−3.80776	−0.175453
$472$	0	0
$473$	−2.12311	−0.0976205
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.12311	−0.326145
$478$	0	0
$479$	−26.2462	−1.19922	−0.599610	−	0.800292i	$-0.704678\pi$
$479$	−26.2462	−1.19922	−0.599610	+	0.800292i	$0.704678\pi$
$480$	0	0
$481$	36.7386	1.67514
$482$	0	0
$483$	2.49242	0.113409
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.61553	−0.118521	−0.0592604	−	0.998243i	$-0.518874\pi$
$487$	−2.61553	−0.118521	−0.0592604	+	0.998243i	$0.518874\pi$
$488$	0	0
$489$	7.31534	0.330811
$490$	0	0
$491$	28.4233	1.28273	0.641363	−	0.767238i	$-0.278369\pi$
$491$	28.4233	1.28273	0.641363	+	0.767238i	$0.278369\pi$
$492$	0	0
$493$	−30.2462	−1.36222
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.630683	0.0282900
$498$	0	0
$499$	−6.05398	−0.271013	−0.135507	−	0.990776i	$-0.543266\pi$
$499$	−6.05398	−0.271013	−0.135507	+	0.990776i	$0.543266\pi$
$500$	0	0
$501$	−1.12311	−0.0501767
$502$	0	0
$503$	29.3693	1.30951	0.654757	−	0.755840i	$-0.272771\pi$
$503$	29.3693	1.30951	0.654757	+	0.755840i	$0.272771\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	24.4924	1.08775
$508$	0	0
$509$	−13.1231	−0.581671	−0.290836	−	0.956773i	$-0.593933\pi$
$509$	−13.1231	−0.581671	−0.290836	+	0.956773i	$0.593933\pi$
$510$	0	0
$511$	−3.61553	−0.159942
$512$	0	0
$513$	3.00000	0.132453
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−5.93087	−0.260336
$520$	0	0
$521$	−23.3002	−1.02080	−0.510400	−	0.859937i	$-0.670503\pi$
$521$	−23.3002	−1.02080	−0.510400	+	0.859937i	$0.670503\pi$
$522$	0	0
$523$	−5.87689	−0.256979	−0.128489	−	0.991711i	$-0.541013\pi$
$523$	−5.87689	−0.256979	−0.128489	+	0.991711i	$0.541013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−9.36932	−0.408134
$528$	0	0
$529$	9.31534	0.405015
$530$	0	0
$531$	−11.3693	−0.493386
$532$	0	0
$533$	−36.7386	−1.59133
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.6155	1.01909
$538$	0	0
$539$	−6.80776	−0.293231
$540$	0	0
$541$	−24.8617	−1.06889	−0.534445	−	0.845203i	$-0.679479\pi$
$541$	−24.8617	−1.06889	−0.534445	+	0.845203i	$0.679479\pi$
$542$	0	0
$543$	−7.80776	−0.335063
$544$	0	0
$545$	0	0
$546$	0	0
$547$	30.4233	1.30081	0.650403	−	0.759589i	$-0.274600\pi$
$547$	30.4233	1.30081	0.650403	+	0.759589i	$0.274600\pi$
$548$	0	0
$549$	−6.68466	−0.285294
$550$	0	0
$551$	29.0540	1.23774
$552$	0	0
$553$	6.24621	0.265616
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−19.0540	−0.807343	−0.403671	−	0.914904i	$-0.632266\pi$
$557$	−19.0540	−0.807343	−0.403671	+	0.914904i	$0.632266\pi$
$558$	0	0
$559$	−13.0000	−0.549841
$560$	0	0
$561$	−3.12311	−0.131858
$562$	0	0
$563$	−5.36932	−0.226290	−0.113145	−	0.993579i	$-0.536092\pi$
$563$	−5.36932	−0.226290	−0.113145	+	0.993579i	$0.536092\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.438447	0.0184131
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	19.0000	0.795125	0.397563	−	0.917575i	$-0.369856\pi$
$571$	19.0000	0.795125	0.397563	+	0.917575i	$0.369856\pi$
$572$	0	0
$573$	−20.8078	−0.869257
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.822919	0.0342586	0.0171293	−	0.999853i	$-0.494547\pi$
$577$	0.822919	0.0342586	0.0171293	+	0.999853i	$0.494547\pi$
$578$	0	0
$579$	6.68466	0.277805
$580$	0	0
$581$	−4.24621	−0.176163
$582$	0	0
$583$	−7.12311	−0.295009
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.63068	0.273678	0.136839	−	0.990593i	$-0.456306\pi$
$587$	6.63068	0.273678	0.136839	+	0.990593i	$0.456306\pi$
$588$	0	0
$589$	9.00000	0.370839
$590$	0	0
$591$	9.93087	0.408501
$592$	0	0
$593$	−19.1231	−0.785292	−0.392646	−	0.919690i	$-0.628440\pi$
$593$	−19.1231	−0.785292	−0.392646	+	0.919690i	$0.628440\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.2462	0.787695
$598$	0	0
$599$	10.7386	0.438769	0.219384	−	0.975639i	$-0.429595\pi$
$599$	10.7386	0.438769	0.219384	+	0.975639i	$0.429595\pi$
$600$	0	0
$601$	−44.7926	−1.82713	−0.913564	−	0.406694i	$-0.866681\pi$
$601$	−44.7926	−1.82713	−0.913564	+	0.406694i	$0.866681\pi$
$602$	0	0
$603$	−4.43845	−0.180748
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.6155	1.68912	0.844561	−	0.535459i	$-0.179861\pi$
$607$	41.6155	1.68912	0.844561	+	0.535459i	$0.179861\pi$
$608$	0	0
$609$	4.24621	0.172065
$610$	0	0
$611$	−24.4924	−0.990857
$612$	0	0
$613$	42.9848	1.73614	0.868071	−	0.496440i	$-0.165360\pi$
$613$	42.9848	1.73614	0.868071	+	0.496440i	$0.165360\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−32.1771	−1.29540	−0.647700	−	0.761895i	$-0.724269\pi$
$617$	−32.1771	−1.29540	−0.647700	+	0.761895i	$0.724269\pi$
$618$	0	0
$619$	13.3153	0.535189	0.267594	−	0.963532i	$-0.413771\pi$
$619$	13.3153	0.535189	0.267594	+	0.963532i	$0.413771\pi$
$620$	0	0
$621$	5.68466	0.228117
$622$	0	0
$623$	1.50758	0.0603998
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.00000	0.119808
$628$	0	0
$629$	−18.7386	−0.747158
$630$	0	0
$631$	22.5464	0.897558	0.448779	−	0.893643i	$-0.351859\pi$
$631$	22.5464	0.897558	0.448779	+	0.893643i	$0.351859\pi$
$632$	0	0
$633$	3.31534	0.131773
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−41.6847	−1.65161
$638$	0	0
$639$	1.43845	0.0569041
$640$	0	0
$641$	−35.4384	−1.39973	−0.699867	−	0.714273i	$-0.746758\pi$
$641$	−35.4384	−1.39973	−0.699867	+	0.714273i	$0.746758\pi$
$642$	0	0
$643$	1.61553	0.0637102	0.0318551	−	0.999492i	$-0.489858\pi$
$643$	1.61553	0.0637102	0.0318551	+	0.999492i	$0.489858\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−11.3693	−0.446285
$650$	0	0
$651$	1.31534	0.0515523
$652$	0	0
$653$	11.3693	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$653$	11.3693	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.24621	−0.321715
$658$	0	0
$659$	7.68466	0.299352	0.149676	−	0.988735i	$-0.452177\pi$
$659$	7.68466	0.299352	0.149676	+	0.988735i	$0.452177\pi$
$660$	0	0
$661$	17.3693	0.675588	0.337794	−	0.941220i	$-0.390319\pi$
$661$	17.3693	0.675588	0.337794	+	0.941220i	$0.390319\pi$
$662$	0	0
$663$	−19.1231	−0.742680
$664$	0	0
$665$	0	0
$666$	0	0
$667$	55.0540	2.13170
$668$	0	0
$669$	−18.0540	−0.698007
$670$	0	0
$671$	−6.68466	−0.258058
$672$	0	0
$673$	21.3693	0.823727	0.411863	−	0.911246i	$-0.364878\pi$
$673$	21.3693	0.823727	0.411863	+	0.911246i	$0.364878\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.19224	0.199554	0.0997769	−	0.995010i	$-0.468187\pi$
$677$	5.19224	0.199554	0.0997769	+	0.995010i	$0.468187\pi$
$678$	0	0
$679$	5.31534	0.203984
$680$	0	0
$681$	−20.8078	−0.797355
$682$	0	0
$683$	22.4924	0.860649	0.430324	−	0.902674i	$-0.358399\pi$
$683$	22.4924	0.860649	0.430324	+	0.902674i	$0.358399\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−25.5616	−0.975234
$688$	0	0
$689$	−43.6155	−1.66162
$690$	0	0
$691$	15.5076	0.589936	0.294968	−	0.955507i	$-0.404691\pi$
$691$	15.5076	0.589936	0.294968	+	0.955507i	$0.404691\pi$
$692$	0	0
$693$	0.438447	0.0166552
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.7386	0.709776
$698$	0	0
$699$	20.4924	0.775095
$700$	0	0
$701$	−16.3153	−0.616222	−0.308111	−	0.951350i	$-0.599697\pi$
$701$	−16.3153	−0.616222	−0.308111	+	0.951350i	$0.599697\pi$
$702$	0	0
$703$	18.0000	0.678883
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.10795	0.0792776
$708$	0	0
$709$	36.5464	1.37253	0.686264	−	0.727352i	$-0.259249\pi$
$709$	36.5464	1.37253	0.686264	+	0.727352i	$0.259249\pi$
$710$	0	0
$711$	14.2462	0.534275
$712$	0	0
$713$	17.0540	0.638676
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.1231	0.714165
$718$	0	0
$719$	3.61553	0.134836	0.0674182	−	0.997725i	$-0.478524\pi$
$719$	3.61553	0.134836	0.0674182	+	0.997725i	$0.478524\pi$
$720$	0	0
$721$	8.35416	0.311125
$722$	0	0
$723$	18.6847	0.694890
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33.2462	−1.23303	−0.616517	−	0.787342i	$-0.711457\pi$
$727$	−33.2462	−1.23303	−0.616517	+	0.787342i	$0.711457\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.63068	0.245245
$732$	0	0
$733$	−22.1771	−0.819129	−0.409565	−	0.912281i	$-0.634319\pi$
$733$	−22.1771	−0.819129	−0.409565	+	0.912281i	$0.634319\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.43845	−0.163492
$738$	0	0
$739$	50.2462	1.84834	0.924168	−	0.381985i	$-0.124760\pi$
$739$	50.2462	1.84834	0.924168	+	0.381985i	$0.124760\pi$
$740$	0	0
$741$	18.3693	0.674814
$742$	0	0
$743$	−24.1080	−0.884435	−0.442217	−	0.896908i	$-0.645808\pi$
$743$	−24.1080	−0.884435	−0.442217	+	0.896908i	$0.645808\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.68466	−0.354343
$748$	0	0
$749$	2.49242	0.0910711
$750$	0	0
$751$	16.8078	0.613324	0.306662	−	0.951818i	$-0.400788\pi$
$751$	16.8078	0.613324	0.306662	+	0.951818i	$0.400788\pi$
$752$	0	0
$753$	5.36932	0.195669
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−18.4384	−0.670157	−0.335078	−	0.942190i	$-0.608763\pi$
$757$	−18.4384	−0.670157	−0.335078	+	0.942190i	$0.608763\pi$
$758$	0	0
$759$	5.68466	0.206340
$760$	0	0
$761$	−12.3845	−0.448937	−0.224468	−	0.974481i	$-0.572065\pi$
$761$	−12.3845	−0.448937	−0.224468	+	0.974481i	$0.572065\pi$
$762$	0	0
$763$	−4.43845	−0.160683
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−69.6155	−2.51367
$768$	0	0
$769$	−6.82292	−0.246041	−0.123020	−	0.992404i	$-0.539258\pi$
$769$	−6.82292	−0.246041	−0.123020	+	0.992404i	$0.539258\pi$
$770$	0	0
$771$	11.6847	0.420813
$772$	0	0
$773$	−16.2462	−0.584336	−0.292168	−	0.956367i	$-0.594377\pi$
$773$	−16.2462	−0.584336	−0.292168	+	0.956367i	$0.594377\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.63068	0.0943752
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	1.43845	0.0514717
$782$	0	0
$783$	9.68466	0.346101
$784$	0	0
$785$	0	0
$786$	0	0
$787$	7.42329	0.264612	0.132306	−	0.991209i	$-0.457762\pi$
$787$	7.42329	0.264612	0.132306	+	0.991209i	$0.457762\pi$
$788$	0	0
$789$	16.8769	0.600833
$790$	0	0
$791$	7.12311	0.253268
$792$	0	0
$793$	−40.9309	−1.45350
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−29.1231	−1.03159	−0.515797	−	0.856711i	$-0.672504\pi$
$797$	−29.1231	−1.03159	−0.515797	+	0.856711i	$0.672504\pi$
$798$	0	0
$799$	12.4924	0.441950
$800$	0	0
$801$	3.43845	0.121492
$802$	0	0
$803$	−8.24621	−0.291002
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0.876894	0.0308681
$808$	0	0
$809$	31.3693	1.10289	0.551443	−	0.834212i	$-0.314077\pi$
$809$	31.3693	1.10289	0.551443	+	0.834212i	$0.314077\pi$
$810$	0	0
$811$	−18.9309	−0.664753	−0.332376	−	0.943147i	$-0.607850\pi$
$811$	−18.9309	−0.664753	−0.332376	+	0.943147i	$0.607850\pi$
$812$	0	0
$813$	−5.75379	−0.201794
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.36932	−0.222834
$818$	0	0
$819$	2.68466	0.0938096
$820$	0	0
$821$	−25.9309	−0.904994	−0.452497	−	0.891766i	$-0.649467\pi$
$821$	−25.9309	−0.904994	−0.452497	+	0.891766i	$0.649467\pi$
$822$	0	0
$823$	5.17708	0.180462	0.0902308	−	0.995921i	$-0.471240\pi$
$823$	5.17708	0.180462	0.0902308	+	0.995921i	$0.471240\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−16.4233	−0.571094	−0.285547	−	0.958365i	$-0.592175\pi$
$827$	−16.4233	−0.571094	−0.285547	+	0.958365i	$0.592175\pi$
$828$	0	0
$829$	−24.7386	−0.859208	−0.429604	−	0.903017i	$-0.641347\pi$
$829$	−24.7386	−0.859208	−0.429604	+	0.903017i	$0.641347\pi$
$830$	0	0
$831$	27.5616	0.956100
$832$	0	0
$833$	21.2614	0.736663
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.00000	0.103695
$838$	0	0
$839$	−5.36932	−0.185369	−0.0926847	−	0.995696i	$-0.529545\pi$
$839$	−5.36932	−0.185369	−0.0926847	+	0.995696i	$0.529545\pi$
$840$	0	0
$841$	64.7926	2.23423
$842$	0	0
$843$	−28.0000	−0.964371
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.438447	0.0150652
$848$	0	0
$849$	0.192236	0.00659752
$850$	0	0
$851$	34.1080	1.16921
$852$	0	0
$853$	29.4233	1.00743	0.503717	−	0.863869i	$-0.331965\pi$
$853$	29.4233	1.00743	0.503717	+	0.863869i	$0.331965\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20.2462	−0.691597	−0.345799	−	0.938309i	$-0.612392\pi$
$857$	−20.2462	−0.691597	−0.345799	+	0.938309i	$0.612392\pi$
$858$	0	0
$859$	13.6155	0.464556	0.232278	−	0.972649i	$-0.425382\pi$
$859$	13.6155	0.464556	0.232278	+	0.972649i	$0.425382\pi$
$860$	0	0
$861$	−2.63068	−0.0896534
$862$	0	0
$863$	55.6847	1.89553	0.947764	−	0.318973i	$-0.103338\pi$
$863$	55.6847	1.89553	0.947764	+	0.318973i	$0.103338\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.24621	−0.246094
$868$	0	0
$869$	14.2462	0.483270
$870$	0	0
$871$	−27.1771	−0.920860
$872$	0	0
$873$	12.1231	0.410305
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.1080	1.38812	0.694058	−	0.719919i	$-0.255821\pi$
$877$	41.1080	1.38812	0.694058	+	0.719919i	$0.255821\pi$
$878$	0	0
$879$	28.7386	0.969330
$880$	0	0
$881$	35.7926	1.20588	0.602942	−	0.797785i	$-0.293995\pi$
$881$	35.7926	1.20588	0.602942	+	0.797785i	$0.293995\pi$
$882$	0	0
$883$	−34.0540	−1.14601	−0.573004	−	0.819553i	$-0.694222\pi$
$883$	−34.0540	−1.14601	−0.573004	+	0.819553i	$0.694222\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0.738634	0.0248009	0.0124004	−	0.999923i	$-0.496053\pi$
$887$	0.738634	0.0248009	0.0124004	+	0.999923i	$0.496053\pi$
$888$	0	0
$889$	4.49242	0.150671
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−12.0000	−0.401565
$894$	0	0
$895$	0	0
$896$	0	0
$897$	34.8078	1.16220
$898$	0	0
$899$	29.0540	0.969004
$900$	0	0
$901$	22.2462	0.741129
$902$	0	0
$903$	−0.930870	−0.0309774
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−37.6155	−1.24900	−0.624502	−	0.781024i	$-0.714698\pi$
$907$	−37.6155	−1.24900	−0.624502	+	0.781024i	$0.714698\pi$
$908$	0	0
$909$	4.80776	0.159463
$910$	0	0
$911$	−12.8769	−0.426631	−0.213315	−	0.976983i	$-0.568426\pi$
$911$	−12.8769	−0.426631	−0.213315	+	0.976983i	$0.568426\pi$
$912$	0	0
$913$	−9.68466	−0.320515
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.75379	−0.123961
$918$	0	0
$919$	22.0540	0.727494	0.363747	−	0.931498i	$-0.381497\pi$
$919$	22.0540	0.727494	0.363747	+	0.931498i	$0.381497\pi$
$920$	0	0
$921$	0.192236	0.00633439
$922$	0	0
$923$	8.80776	0.289911
$924$	0	0
$925$	0	0
$926$	0	0
$927$	19.0540	0.625815
$928$	0	0
$929$	−24.1771	−0.793224	−0.396612	−	0.917986i	$-0.629814\pi$
$929$	−24.1771	−0.793224	−0.396612	+	0.917986i	$0.629814\pi$
$930$	0	0
$931$	−20.4233	−0.669346
$932$	0	0
$933$	−27.4384	−0.898294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.9460	0.520934	0.260467	−	0.965483i	$-0.416123\pi$
$937$	15.9460	0.520934	0.260467	+	0.965483i	$0.416123\pi$
$938$	0	0
$939$	−28.4384	−0.928054
$940$	0	0
$941$	55.8617	1.82104	0.910520	−	0.413464i	$-0.135681\pi$
$941$	55.8617	1.82104	0.910520	+	0.413464i	$0.135681\pi$
$942$	0	0
$943$	−34.1080	−1.11071
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−31.7538	−1.03186	−0.515930	−	0.856631i	$-0.672553\pi$
$947$	−31.7538	−1.03186	−0.515930	+	0.856631i	$0.672553\pi$
$948$	0	0
$949$	−50.4924	−1.63905
$950$	0	0
$951$	−5.36932	−0.174112
$952$	0	0
$953$	−34.4924	−1.11732	−0.558660	−	0.829397i	$-0.688684\pi$
$953$	−34.4924	−1.11732	−0.558660	+	0.829397i	$0.688684\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	9.68466	0.313061
$958$	0	0
$959$	2.38447	0.0769986
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	5.68466	0.183186
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−44.9848	−1.44662	−0.723308	−	0.690526i	$-0.757379\pi$
$967$	−44.9848	−1.44662	−0.723308	+	0.690526i	$0.757379\pi$
$968$	0	0
$969$	−9.36932	−0.300986
$970$	0	0
$971$	−53.8617	−1.72851	−0.864253	−	0.503058i	$-0.832208\pi$
$971$	−53.8617	−1.72851	−0.864253	+	0.503058i	$0.832208\pi$
$972$	0	0
$973$	9.12311	0.292473
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−26.4924	−0.847568	−0.423784	−	0.905763i	$-0.639298\pi$
$977$	−26.4924	−0.847568	−0.423784	+	0.905763i	$0.639298\pi$
$978$	0	0
$979$	3.43845	0.109893
$980$	0	0
$981$	−10.1231	−0.323206
$982$	0	0
$983$	10.0691	0.321155	0.160578	−	0.987023i	$-0.448664\pi$
$983$	10.0691	0.321155	0.160578	+	0.987023i	$0.448664\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.75379	−0.0558237
$988$	0	0
$989$	−12.0691	−0.383776
$990$	0	0
$991$	−43.4233	−1.37939	−0.689693	−	0.724102i	$-0.742255\pi$
$991$	−43.4233	−1.37939	−0.689693	+	0.724102i	$0.742255\pi$
$992$	0	0
$993$	17.1231	0.543385
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−24.2462	−0.767885	−0.383943	−	0.923357i	$-0.625434\pi$
$997$	−24.2462	−0.767885	−0.383943	+	0.923357i	$0.625434\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6600.2.a.bm.1.1	yes	2
5.2	odd	4		6600.2.d.ba.1849.1		4
5.3	odd	4		6600.2.d.ba.1849.4		4
5.4	even	2		6600.2.a.bg.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6600.2.a.bg.1.2	✓	2	5.4	even	2
6600.2.a.bm.1.1	yes	2	1.1	even	1	trivial
6600.2.d.ba.1849.1		4	5.2	odd	4
6600.2.d.ba.1849.4		4	5.3	odd	4

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:	\( N \)	\(=\)	\( 6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.438447 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.438447 q^{7} +1.00000 q^{9} +1.00000 q^{11} +6.12311 q^{13} -3.12311 q^{17} +3.00000 q^{19} +0.438447 q^{21} +5.68466 q^{23} +1.00000 q^{27} +9.68466 q^{29} +3.00000 q^{31} +1.00000 q^{33} +6.00000 q^{37} +6.12311 q^{39} -6.00000 q^{41} -2.12311 q^{43} -4.00000 q^{47} -6.80776 q^{49} -3.12311 q^{51} -7.12311 q^{53} +3.00000 q^{57} -11.3693 q^{59} -6.68466 q^{61} +0.438447 q^{63} -4.43845 q^{67} +5.68466 q^{69} +1.43845 q^{71} -8.24621 q^{73} +0.438447 q^{77} +14.2462 q^{79} +1.00000 q^{81} -9.68466 q^{83} +9.68466 q^{87} +3.43845 q^{89} +2.68466 q^{91} +3.00000 q^{93} +12.1231 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 5 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} + 5 q^{21} - q^{23} + 2 q^{27} + 7 q^{29} + 6 q^{31} + 2 q^{33} + 12 q^{37} + 4 q^{39} - 12 q^{41} + 4 q^{43} - 8 q^{47} + 7 q^{49} + 2 q^{51} - 6 q^{53} + 6 q^{57} + 2 q^{59} - q^{61} + 5 q^{63} - 13 q^{67} - q^{69} + 7 q^{71} + 5 q^{77} + 12 q^{79} + 2 q^{81} - 7 q^{83} + 7 q^{87} + 11 q^{89} - 7 q^{91} + 6 q^{93} + 16 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 5 * q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^17 + 6 * q^19 + 5 * q^21 - q^23 + 2 * q^27 + 7 * q^29 + 6 * q^31 + 2 * q^33 + 12 * q^37 + 4 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^47 + 7 * q^49 + 2 * q^51 - 6 * q^53 + 6 * q^57 + 2 * q^59 - q^61 + 5 * q^63 - 13 * q^67 - q^69 + 7 * q^71 + 5 * q^77 + 12 * q^79 + 2 * q^81 - 7 * q^83 + 7 * q^87 + 11 * q^89 - 7 * q^91 + 6 * q^93 + 16 * q^97 + 2 * q^99

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