LMFDB - Embedded newform 6498.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6498, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6498.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.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:	$51.8867912334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 722)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -3.23607 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -3.23607 q^{7} +1.00000 q^{8} +3.61803 q^{10} +3.23607 q^{11} +1.38197 q^{13} -3.23607 q^{14} +1.00000 q^{16} +3.38197 q^{17} +3.61803 q^{20} +3.23607 q^{22} -5.23607 q^{23} +8.09017 q^{25} +1.38197 q^{26} -3.23607 q^{28} +9.09017 q^{29} -1.23607 q^{31} +1.00000 q^{32} +3.38197 q^{34} -11.7082 q^{35} +8.38197 q^{37} +3.61803 q^{40} -0.854102 q^{41} -9.23607 q^{43} +3.23607 q^{44} -5.23607 q^{46} -4.47214 q^{47} +3.47214 q^{49} +8.09017 q^{50} +1.38197 q^{52} +6.09017 q^{53} +11.7082 q^{55} -3.23607 q^{56} +9.09017 q^{58} +0.472136 q^{59} +1.38197 q^{61} -1.23607 q^{62} +1.00000 q^{64} +5.00000 q^{65} +11.7082 q^{67} +3.38197 q^{68} -11.7082 q^{70} -2.94427 q^{71} -5.61803 q^{73} +8.38197 q^{74} -10.4721 q^{77} -2.76393 q^{79} +3.61803 q^{80} -0.854102 q^{82} +0.472136 q^{83} +12.2361 q^{85} -9.23607 q^{86} +3.23607 q^{88} -8.85410 q^{89} -4.47214 q^{91} -5.23607 q^{92} -4.47214 q^{94} +8.61803 q^{97} +3.47214 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8} + 5 q^{10} + 2 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 9 q^{17} + 5 q^{20} + 2 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 9 q^{34} - 10 q^{35} + 19 q^{37} + 5 q^{40} + 5 q^{41} - 14 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{49} + 5 q^{50} + 5 q^{52} + q^{53} + 10 q^{55} - 2 q^{56} + 7 q^{58} - 8 q^{59} + 5 q^{61} + 2 q^{62} + 2 q^{64} + 10 q^{65} + 10 q^{67} + 9 q^{68} - 10 q^{70} + 12 q^{71} - 9 q^{73} + 19 q^{74} - 12 q^{77} - 10 q^{79} + 5 q^{80} + 5 q^{82} - 8 q^{83} + 20 q^{85} - 14 q^{86} + 2 q^{88} - 11 q^{89} - 6 q^{92} + 15 q^{97} - 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8 + 5 * q^10 + 2 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 9 * q^17 + 5 * q^20 + 2 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 9 * q^34 - 10 * q^35 + 19 * q^37 + 5 * q^40 + 5 * q^41 - 14 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^49 + 5 * q^50 + 5 * q^52 + q^53 + 10 * q^55 - 2 * q^56 + 7 * q^58 - 8 * q^59 + 5 * q^61 + 2 * q^62 + 2 * q^64 + 10 * q^65 + 10 * q^67 + 9 * q^68 - 10 * q^70 + 12 * q^71 - 9 * q^73 + 19 * q^74 - 12 * q^77 - 10 * q^79 + 5 * q^80 + 5 * q^82 - 8 * q^83 + 20 * q^85 - 14 * q^86 + 2 * q^88 - 11 * q^89 - 6 * q^92 + 15 * q^97 - 2 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.61803	1.61803	0.809017	−	0.587785i	$-0.200000\pi$
$5$	3.61803	1.61803	0.809017	+	0.587785i	$0.200000\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.61803	1.14412
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	0	0
$13$	1.38197	0.383288	0.191644	−	0.981464i	$-0.438618\pi$
$13$	1.38197	0.383288	0.191644	+	0.981464i	$0.438618\pi$
$14$	−3.23607	−0.864876
$15$	0	0
$16$	1.00000	0.250000
$17$	3.38197	0.820247	0.410124	−	0.912030i	$-0.365486\pi$
$17$	3.38197	0.820247	0.410124	+	0.912030i	$0.365486\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.61803	0.809017
$21$	0	0
$22$	3.23607	0.689932
$23$	−5.23607	−1.09180	−0.545898	−	0.837852i	$-0.683811\pi$
$23$	−5.23607	−1.09180	−0.545898	+	0.837852i	$0.683811\pi$
$24$	0	0
$25$	8.09017	1.61803
$26$	1.38197	0.271026
$27$	0	0
$28$	−3.23607	−0.611559
$29$	9.09017	1.68800	0.844001	−	0.536341i	$-0.180194\pi$
$29$	9.09017	1.68800	0.844001	+	0.536341i	$0.180194\pi$
$30$	0	0
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	3.38197	0.580002
$35$	−11.7082	−1.97905
$36$	0	0
$37$	8.38197	1.37799	0.688993	−	0.724768i	$-0.258053\pi$
$37$	8.38197	1.37799	0.688993	+	0.724768i	$0.258053\pi$
$38$	0	0
$39$	0	0
$40$	3.61803	0.572061
$41$	−0.854102	−0.133388	−0.0666942	−	0.997773i	$-0.521245\pi$
$41$	−0.854102	−0.133388	−0.0666942	+	0.997773i	$0.521245\pi$
$42$	0	0
$43$	−9.23607	−1.40849	−0.704244	−	0.709958i	$-0.748714\pi$
$43$	−9.23607	−1.40849	−0.704244	+	0.709958i	$0.748714\pi$
$44$	3.23607	0.487856
$45$	0	0
$46$	−5.23607	−0.772016
$47$	−4.47214	−0.652328	−0.326164	−	0.945313i	$-0.605756\pi$
$47$	−4.47214	−0.652328	−0.326164	+	0.945313i	$0.605756\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	8.09017	1.14412
$51$	0	0
$52$	1.38197	0.191644
$53$	6.09017	0.836549	0.418275	−	0.908321i	$-0.362635\pi$
$53$	6.09017	0.836549	0.418275	+	0.908321i	$0.362635\pi$
$54$	0	0
$55$	11.7082	1.57873
$56$	−3.23607	−0.432438
$57$	0	0
$58$	9.09017	1.19360
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	1.38197	0.176943	0.0884713	−	0.996079i	$-0.471802\pi$
$61$	1.38197	0.176943	0.0884713	+	0.996079i	$0.471802\pi$
$62$	−1.23607	−0.156981
$63$	0	0
$64$	1.00000	0.125000
$65$	5.00000	0.620174
$66$	0	0
$67$	11.7082	1.43038	0.715192	−	0.698928i	$-0.246339\pi$
$67$	11.7082	1.43038	0.715192	+	0.698928i	$0.246339\pi$
$68$	3.38197	0.410124
$69$	0	0
$70$	−11.7082	−1.39940
$71$	−2.94427	−0.349421	−0.174710	−	0.984620i	$-0.555899\pi$
$71$	−2.94427	−0.349421	−0.174710	+	0.984620i	$0.555899\pi$
$72$	0	0
$73$	−5.61803	−0.657541	−0.328771	−	0.944410i	$-0.606634\pi$
$73$	−5.61803	−0.657541	−0.328771	+	0.944410i	$0.606634\pi$
$74$	8.38197	0.974384
$75$	0	0
$76$	0	0
$77$	−10.4721	−1.19341
$78$	0	0
$79$	−2.76393	−0.310967	−0.155483	−	0.987839i	$-0.549693\pi$
$79$	−2.76393	−0.310967	−0.155483	+	0.987839i	$0.549693\pi$
$80$	3.61803	0.404508
$81$	0	0
$82$	−0.854102	−0.0943198
$83$	0.472136	0.0518237	0.0259118	−	0.999664i	$-0.491751\pi$
$83$	0.472136	0.0518237	0.0259118	+	0.999664i	$0.491751\pi$
$84$	0	0
$85$	12.2361	1.32719
$86$	−9.23607	−0.995951
$87$	0	0
$88$	3.23607	0.344966
$89$	−8.85410	−0.938533	−0.469266	−	0.883057i	$-0.655482\pi$
$89$	−8.85410	−0.938533	−0.469266	+	0.883057i	$0.655482\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	−5.23607	−0.545898
$93$	0	0
$94$	−4.47214	−0.461266
$95$	0	0
$96$	0	0
$97$	8.61803	0.875029	0.437514	−	0.899211i	$-0.355859\pi$
$97$	8.61803	0.875029	0.437514	+	0.899211i	$0.355859\pi$
$98$	3.47214	0.350739
$99$	0	0
$100$	8.09017	0.809017
$101$	15.6180	1.55405	0.777026	−	0.629468i	$-0.216727\pi$
$101$	15.6180	1.55405	0.777026	+	0.629468i	$0.216727\pi$
$102$	0	0
$103$	14.6525	1.44375	0.721876	−	0.692023i	$-0.243280\pi$
$103$	14.6525	1.44375	0.721876	+	0.692023i	$0.243280\pi$
$104$	1.38197	0.135513
$105$	0	0
$106$	6.09017	0.591530
$107$	−9.41641	−0.910319	−0.455159	−	0.890410i	$-0.650418\pi$
$107$	−9.41641	−0.910319	−0.455159	+	0.890410i	$0.650418\pi$
$108$	0	0
$109$	−11.0344	−1.05691	−0.528454	−	0.848962i	$-0.677228\pi$
$109$	−11.0344	−1.05691	−0.528454	+	0.848962i	$0.677228\pi$
$110$	11.7082	1.11633
$111$	0	0
$112$	−3.23607	−0.305780
$113$	9.85410	0.926996	0.463498	−	0.886098i	$-0.346594\pi$
$113$	9.85410	0.926996	0.463498	+	0.886098i	$0.346594\pi$
$114$	0	0
$115$	−18.9443	−1.76656
$116$	9.09017	0.844001
$117$	0	0
$118$	0.472136	0.0434636
$119$	−10.9443	−1.00326
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	1.38197	0.125117
$123$	0	0
$124$	−1.23607	−0.111002
$125$	11.1803	1.00000
$126$	0	0
$127$	−11.4164	−1.01304	−0.506521	−	0.862228i	$-0.669069\pi$
$127$	−11.4164	−1.01304	−0.506521	+	0.862228i	$0.669069\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.00000	0.438529
$131$	−4.47214	−0.390732	−0.195366	−	0.980730i	$-0.562590\pi$
$131$	−4.47214	−0.390732	−0.195366	+	0.980730i	$0.562590\pi$
$132$	0	0
$133$	0	0
$134$	11.7082	1.01143
$135$	0	0
$136$	3.38197	0.290001
$137$	−2.38197	−0.203505	−0.101753	−	0.994810i	$-0.532445\pi$
$137$	−2.38197	−0.203505	−0.101753	+	0.994810i	$0.532445\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−11.7082	−0.989524
$141$	0	0
$142$	−2.94427	−0.247078
$143$	4.47214	0.373979
$144$	0	0
$145$	32.8885	2.73124
$146$	−5.61803	−0.464952
$147$	0	0
$148$	8.38197	0.688993
$149$	13.0902	1.07239	0.536194	−	0.844095i	$-0.319861\pi$
$149$	13.0902	1.07239	0.536194	+	0.844095i	$0.319861\pi$
$150$	0	0
$151$	14.4721	1.17773	0.588863	−	0.808233i	$-0.299576\pi$
$151$	14.4721	1.17773	0.588863	+	0.808233i	$0.299576\pi$
$152$	0	0
$153$	0	0
$154$	−10.4721	−0.843869
$155$	−4.47214	−0.359211
$156$	0	0
$157$	9.38197	0.748762	0.374381	−	0.927275i	$-0.377855\pi$
$157$	9.38197	0.748762	0.374381	+	0.927275i	$0.377855\pi$
$158$	−2.76393	−0.219887
$159$	0	0
$160$	3.61803	0.286031
$161$	16.9443	1.33540
$162$	0	0
$163$	10.6525	0.834366	0.417183	−	0.908822i	$-0.363017\pi$
$163$	10.6525	0.834366	0.417183	+	0.908822i	$0.363017\pi$
$164$	−0.854102	−0.0666942
$165$	0	0
$166$	0.472136	0.0366449
$167$	12.4721	0.965123	0.482561	−	0.875862i	$-0.339707\pi$
$167$	12.4721	0.965123	0.482561	+	0.875862i	$0.339707\pi$
$168$	0	0
$169$	−11.0902	−0.853090
$170$	12.2361	0.938464
$171$	0	0
$172$	−9.23607	−0.704244
$173$	−1.32624	−0.100832	−0.0504160	−	0.998728i	$-0.516055\pi$
$173$	−1.32624	−0.100832	−0.0504160	+	0.998728i	$0.516055\pi$
$174$	0	0
$175$	−26.1803	−1.97905
$176$	3.23607	0.243928
$177$	0	0
$178$	−8.85410	−0.663643
$179$	23.4164	1.75022	0.875112	−	0.483920i	$-0.160787\pi$
$179$	23.4164	1.75022	0.875112	+	0.483920i	$0.160787\pi$
$180$	0	0
$181$	5.41641	0.402598	0.201299	−	0.979530i	$-0.435484\pi$
$181$	5.41641	0.402598	0.201299	+	0.979530i	$0.435484\pi$
$182$	−4.47214	−0.331497
$183$	0	0
$184$	−5.23607	−0.386008
$185$	30.3262	2.22963
$186$	0	0
$187$	10.9443	0.800324
$188$	−4.47214	−0.326164
$189$	0	0
$190$	0	0
$191$	−3.52786	−0.255267	−0.127634	−	0.991821i	$-0.540738\pi$
$191$	−3.52786	−0.255267	−0.127634	+	0.991821i	$0.540738\pi$
$192$	0	0
$193$	−11.5279	−0.829794	−0.414897	−	0.909868i	$-0.636182\pi$
$193$	−11.5279	−0.829794	−0.414897	+	0.909868i	$0.636182\pi$
$194$	8.61803	0.618739
$195$	0	0
$196$	3.47214	0.248010
$197$	4.14590	0.295383	0.147692	−	0.989033i	$-0.452816\pi$
$197$	4.14590	0.295383	0.147692	+	0.989033i	$0.452816\pi$
$198$	0	0
$199$	10.4721	0.742350	0.371175	−	0.928563i	$-0.378955\pi$
$199$	10.4721	0.742350	0.371175	+	0.928563i	$0.378955\pi$
$200$	8.09017	0.572061
$201$	0	0
$202$	15.6180	1.09888
$203$	−29.4164	−2.06463
$204$	0	0
$205$	−3.09017	−0.215827
$206$	14.6525	1.02089
$207$	0	0
$208$	1.38197	0.0958221
$209$	0	0
$210$	0	0
$211$	−7.23607	−0.498151	−0.249076	−	0.968484i	$-0.580127\pi$
$211$	−7.23607	−0.498151	−0.249076	+	0.968484i	$0.580127\pi$
$212$	6.09017	0.418275
$213$	0	0
$214$	−9.41641	−0.643692
$215$	−33.4164	−2.27898
$216$	0	0
$217$	4.00000	0.271538
$218$	−11.0344	−0.747347
$219$	0	0
$220$	11.7082	0.789367
$221$	4.67376	0.314391
$222$	0	0
$223$	27.7082	1.85548	0.927739	−	0.373229i	$-0.121749\pi$
$223$	27.7082	1.85548	0.927739	+	0.373229i	$0.121749\pi$
$224$	−3.23607	−0.216219
$225$	0	0
$226$	9.85410	0.655485
$227$	8.94427	0.593652	0.296826	−	0.954932i	$-0.404072\pi$
$227$	8.94427	0.593652	0.296826	+	0.954932i	$0.404072\pi$
$228$	0	0
$229$	4.38197	0.289568	0.144784	−	0.989463i	$-0.453751\pi$
$229$	4.38197	0.289568	0.144784	+	0.989463i	$0.453751\pi$
$230$	−18.9443	−1.24915
$231$	0	0
$232$	9.09017	0.596799
$233$	−1.56231	−0.102350	−0.0511750	−	0.998690i	$-0.516297\pi$
$233$	−1.56231	−0.102350	−0.0511750	+	0.998690i	$0.516297\pi$
$234$	0	0
$235$	−16.1803	−1.05549
$236$	0.472136	0.0307334
$237$	0	0
$238$	−10.9443	−0.709412
$239$	−30.1803	−1.95220	−0.976102	−	0.217313i	$-0.930271\pi$
$239$	−30.1803	−1.95220	−0.976102	+	0.217313i	$0.930271\pi$
$240$	0	0
$241$	−25.4164	−1.63721	−0.818607	−	0.574354i	$-0.805253\pi$
$241$	−25.4164	−1.63721	−0.818607	+	0.574354i	$0.805253\pi$
$242$	−0.527864	−0.0339324
$243$	0	0
$244$	1.38197	0.0884713
$245$	12.5623	0.802576
$246$	0	0
$247$	0	0
$248$	−1.23607	−0.0784904
$249$	0	0
$250$	11.1803	0.707107
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	0	0
$253$	−16.9443	−1.06528
$254$	−11.4164	−0.716329
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.3820	−1.14664	−0.573318	−	0.819333i	$-0.694344\pi$
$257$	−18.3820	−1.14664	−0.573318	+	0.819333i	$0.694344\pi$
$258$	0	0
$259$	−27.1246	−1.68544
$260$	5.00000	0.310087
$261$	0	0
$262$	−4.47214	−0.276289
$263$	−2.94427	−0.181552	−0.0907758	−	0.995871i	$-0.528935\pi$
$263$	−2.94427	−0.181552	−0.0907758	+	0.995871i	$0.528935\pi$
$264$	0	0
$265$	22.0344	1.35357
$266$	0	0
$267$	0	0
$268$	11.7082	0.715192
$269$	−14.3262	−0.873486	−0.436743	−	0.899586i	$-0.643868\pi$
$269$	−14.3262	−0.873486	−0.436743	+	0.899586i	$0.643868\pi$
$270$	0	0
$271$	26.0000	1.57939	0.789694	−	0.613501i	$-0.210239\pi$
$271$	26.0000	1.57939	0.789694	+	0.613501i	$0.210239\pi$
$272$	3.38197	0.205062
$273$	0	0
$274$	−2.38197	−0.143900
$275$	26.1803	1.57873
$276$	0	0
$277$	−10.0902	−0.606260	−0.303130	−	0.952949i	$-0.598032\pi$
$277$	−10.0902	−0.606260	−0.303130	+	0.952949i	$0.598032\pi$
$278$	−14.0000	−0.839664
$279$	0	0
$280$	−11.7082	−0.699699
$281$	−20.9787	−1.25149	−0.625743	−	0.780030i	$-0.715204\pi$
$281$	−20.9787	−1.25149	−0.625743	+	0.780030i	$0.715204\pi$
$282$	0	0
$283$	22.4721	1.33583	0.667915	−	0.744238i	$-0.267187\pi$
$283$	22.4721	1.33583	0.667915	+	0.744238i	$0.267187\pi$
$284$	−2.94427	−0.174710
$285$	0	0
$286$	4.47214	0.264443
$287$	2.76393	0.163150
$288$	0	0
$289$	−5.56231	−0.327194
$290$	32.8885	1.93128
$291$	0	0
$292$	−5.61803	−0.328771
$293$	−22.9443	−1.34042	−0.670209	−	0.742172i	$-0.733796\pi$
$293$	−22.9443	−1.34042	−0.670209	+	0.742172i	$0.733796\pi$
$294$	0	0
$295$	1.70820	0.0994555
$296$	8.38197	0.487192
$297$	0	0
$298$	13.0902	0.758293
$299$	−7.23607	−0.418473
$300$	0	0
$301$	29.8885	1.72275
$302$	14.4721	0.832778
$303$	0	0
$304$	0	0
$305$	5.00000	0.286299
$306$	0	0
$307$	11.7082	0.668222	0.334111	−	0.942534i	$-0.391564\pi$
$307$	11.7082	0.668222	0.334111	+	0.942534i	$0.391564\pi$
$308$	−10.4721	−0.596705
$309$	0	0
$310$	−4.47214	−0.254000
$311$	−31.2361	−1.77123	−0.885617	−	0.464415i	$-0.846264\pi$
$311$	−31.2361	−1.77123	−0.885617	+	0.464415i	$0.846264\pi$
$312$	0	0
$313$	−15.5066	−0.876484	−0.438242	−	0.898857i	$-0.644399\pi$
$313$	−15.5066	−0.876484	−0.438242	+	0.898857i	$0.644399\pi$
$314$	9.38197	0.529455
$315$	0	0
$316$	−2.76393	−0.155483
$317$	−1.32624	−0.0744889	−0.0372445	−	0.999306i	$-0.511858\pi$
$317$	−1.32624	−0.0744889	−0.0372445	+	0.999306i	$0.511858\pi$
$318$	0	0
$319$	29.4164	1.64700
$320$	3.61803	0.202254
$321$	0	0
$322$	16.9443	0.944267
$323$	0	0
$324$	0	0
$325$	11.1803	0.620174
$326$	10.6525	0.589986
$327$	0	0
$328$	−0.854102	−0.0471599
$329$	14.4721	0.797875
$330$	0	0
$331$	25.7082	1.41305	0.706525	−	0.707688i	$-0.250262\pi$
$331$	25.7082	1.41305	0.706525	+	0.707688i	$0.250262\pi$
$332$	0.472136	0.0259118
$333$	0	0
$334$	12.4721	0.682445
$335$	42.3607	2.31441
$336$	0	0
$337$	2.38197	0.129754	0.0648770	−	0.997893i	$-0.479335\pi$
$337$	2.38197	0.129754	0.0648770	+	0.997893i	$0.479335\pi$
$338$	−11.0902	−0.603226
$339$	0	0
$340$	12.2361	0.663594
$341$	−4.00000	−0.216612
$342$	0	0
$343$	11.4164	0.616428
$344$	−9.23607	−0.497975
$345$	0	0
$346$	−1.32624	−0.0712990
$347$	2.65248	0.142392	0.0711962	−	0.997462i	$-0.477318\pi$
$347$	2.65248	0.142392	0.0711962	+	0.997462i	$0.477318\pi$
$348$	0	0
$349$	−33.0902	−1.77128	−0.885638	−	0.464376i	$-0.846279\pi$
$349$	−33.0902	−1.77128	−0.885638	+	0.464376i	$0.846279\pi$
$350$	−26.1803	−1.39940
$351$	0	0
$352$	3.23607	0.172483
$353$	23.7426	1.26369	0.631847	−	0.775093i	$-0.282297\pi$
$353$	23.7426	1.26369	0.631847	+	0.775093i	$0.282297\pi$
$354$	0	0
$355$	−10.6525	−0.565375
$356$	−8.85410	−0.469266
$357$	0	0
$358$	23.4164	1.23760
$359$	−11.5279	−0.608417	−0.304209	−	0.952605i	$-0.598392\pi$
$359$	−11.5279	−0.608417	−0.304209	+	0.952605i	$0.598392\pi$
$360$	0	0
$361$	0	0
$362$	5.41641	0.284680
$363$	0	0
$364$	−4.47214	−0.234404
$365$	−20.3262	−1.06392
$366$	0	0
$367$	18.8328	0.983065	0.491532	−	0.870859i	$-0.336437\pi$
$367$	18.8328	0.983065	0.491532	+	0.870859i	$0.336437\pi$
$368$	−5.23607	−0.272949
$369$	0	0
$370$	30.3262	1.57659
$371$	−19.7082	−1.02320
$372$	0	0
$373$	18.3262	0.948897	0.474448	−	0.880283i	$-0.342648\pi$
$373$	18.3262	0.948897	0.474448	+	0.880283i	$0.342648\pi$
$374$	10.9443	0.565915
$375$	0	0
$376$	−4.47214	−0.230633
$377$	12.5623	0.646992
$378$	0	0
$379$	−35.4164	−1.81922	−0.909609	−	0.415465i	$-0.863619\pi$
$379$	−35.4164	−1.81922	−0.909609	+	0.415465i	$0.863619\pi$
$380$	0	0
$381$	0	0
$382$	−3.52786	−0.180501
$383$	−20.7639	−1.06099	−0.530494	−	0.847689i	$-0.677993\pi$
$383$	−20.7639	−1.06099	−0.530494	+	0.847689i	$0.677993\pi$
$384$	0	0
$385$	−37.8885	−1.93098
$386$	−11.5279	−0.586753
$387$	0	0
$388$	8.61803	0.437514
$389$	−12.7984	−0.648903	−0.324452	−	0.945902i	$-0.605180\pi$
$389$	−12.7984	−0.648903	−0.324452	+	0.945902i	$0.605180\pi$
$390$	0	0
$391$	−17.7082	−0.895542
$392$	3.47214	0.175369
$393$	0	0
$394$	4.14590	0.208867
$395$	−10.0000	−0.503155
$396$	0	0
$397$	−22.3607	−1.12225	−0.561125	−	0.827731i	$-0.689631\pi$
$397$	−22.3607	−1.12225	−0.561125	+	0.827731i	$0.689631\pi$
$398$	10.4721	0.524921
$399$	0	0
$400$	8.09017	0.404508
$401$	−18.9443	−0.946032	−0.473016	−	0.881054i	$-0.656835\pi$
$401$	−18.9443	−0.946032	−0.473016	+	0.881054i	$0.656835\pi$
$402$	0	0
$403$	−1.70820	−0.0850917
$404$	15.6180	0.777026
$405$	0	0
$406$	−29.4164	−1.45991
$407$	27.1246	1.34452
$408$	0	0
$409$	16.2705	0.804525	0.402262	−	0.915524i	$-0.368224\pi$
$409$	16.2705	0.804525	0.402262	+	0.915524i	$0.368224\pi$
$410$	−3.09017	−0.152613
$411$	0	0
$412$	14.6525	0.721876
$413$	−1.52786	−0.0751813
$414$	0	0
$415$	1.70820	0.0838524
$416$	1.38197	0.0677565
$417$	0	0
$418$	0	0
$419$	6.47214	0.316185	0.158092	−	0.987424i	$-0.449466\pi$
$419$	6.47214	0.316185	0.158092	+	0.987424i	$0.449466\pi$
$420$	0	0
$421$	13.9098	0.677924	0.338962	−	0.940800i	$-0.389924\pi$
$421$	13.9098	0.677924	0.338962	+	0.940800i	$0.389924\pi$
$422$	−7.23607	−0.352246
$423$	0	0
$424$	6.09017	0.295765
$425$	27.3607	1.32719
$426$	0	0
$427$	−4.47214	−0.216422
$428$	−9.41641	−0.455159
$429$	0	0
$430$	−33.4164	−1.61148
$431$	−6.65248	−0.320438	−0.160219	−	0.987081i	$-0.551220\pi$
$431$	−6.65248	−0.320438	−0.160219	+	0.987081i	$0.551220\pi$
$432$	0	0
$433$	−25.0344	−1.20308	−0.601539	−	0.798843i	$-0.705446\pi$
$433$	−25.0344	−1.20308	−0.601539	+	0.798843i	$0.705446\pi$
$434$	4.00000	0.192006
$435$	0	0
$436$	−11.0344	−0.528454
$437$	0	0
$438$	0	0
$439$	29.2361	1.39536	0.697681	−	0.716409i	$-0.254215\pi$
$439$	29.2361	1.39536	0.697681	+	0.716409i	$0.254215\pi$
$440$	11.7082	0.558167
$441$	0	0
$442$	4.67376	0.222308
$443$	−9.23607	−0.438819	−0.219409	−	0.975633i	$-0.570413\pi$
$443$	−9.23607	−0.438819	−0.219409	+	0.975633i	$0.570413\pi$
$444$	0	0
$445$	−32.0344	−1.51858
$446$	27.7082	1.31202
$447$	0	0
$448$	−3.23607	−0.152890
$449$	22.7984	1.07592	0.537961	−	0.842970i	$-0.319195\pi$
$449$	22.7984	1.07592	0.537961	+	0.842970i	$0.319195\pi$
$450$	0	0
$451$	−2.76393	−0.130148
$452$	9.85410	0.463498
$453$	0	0
$454$	8.94427	0.419775
$455$	−16.1803	−0.758546
$456$	0	0
$457$	25.2148	1.17950	0.589749	−	0.807587i	$-0.299227\pi$
$457$	25.2148	1.17950	0.589749	+	0.807587i	$0.299227\pi$
$458$	4.38197	0.204756
$459$	0	0
$460$	−18.9443	−0.883281
$461$	−21.4164	−0.997462	−0.498731	−	0.866757i	$-0.666200\pi$
$461$	−21.4164	−0.997462	−0.498731	+	0.866757i	$0.666200\pi$
$462$	0	0
$463$	−5.05573	−0.234960	−0.117480	−	0.993075i	$-0.537482\pi$
$463$	−5.05573	−0.234960	−0.117480	+	0.993075i	$0.537482\pi$
$464$	9.09017	0.422001
$465$	0	0
$466$	−1.56231	−0.0723724
$467$	33.3050	1.54117	0.770585	−	0.637338i	$-0.219964\pi$
$467$	33.3050	1.54117	0.770585	+	0.637338i	$0.219964\pi$
$468$	0	0
$469$	−37.8885	−1.74953
$470$	−16.1803	−0.746343
$471$	0	0
$472$	0.472136	0.0217318
$473$	−29.8885	−1.37428
$474$	0	0
$475$	0	0
$476$	−10.9443	−0.501630
$477$	0	0
$478$	−30.1803	−1.38042
$479$	−7.70820	−0.352197	−0.176098	−	0.984373i	$-0.556348\pi$
$479$	−7.70820	−0.352197	−0.176098	+	0.984373i	$0.556348\pi$
$480$	0	0
$481$	11.5836	0.528166
$482$	−25.4164	−1.15769
$483$	0	0
$484$	−0.527864	−0.0239938
$485$	31.1803	1.41583
$486$	0	0
$487$	19.2361	0.871669	0.435835	−	0.900027i	$-0.356453\pi$
$487$	19.2361	0.871669	0.435835	+	0.900027i	$0.356453\pi$
$488$	1.38197	0.0625587
$489$	0	0
$490$	12.5623	0.567507
$491$	1.52786	0.0689515	0.0344758	−	0.999406i	$-0.489024\pi$
$491$	1.52786	0.0689515	0.0344758	+	0.999406i	$0.489024\pi$
$492$	0	0
$493$	30.7426	1.38458
$494$	0	0
$495$	0	0
$496$	−1.23607	−0.0555011
$497$	9.52786	0.427383
$498$	0	0
$499$	−35.3050	−1.58047	−0.790233	−	0.612806i	$-0.790041\pi$
$499$	−35.3050	−1.58047	−0.790233	+	0.612806i	$0.790041\pi$
$500$	11.1803	0.500000
$501$	0	0
$502$	2.29180	0.102288
$503$	−6.76393	−0.301589	−0.150794	−	0.988565i	$-0.548183\pi$
$503$	−6.76393	−0.301589	−0.150794	+	0.988565i	$0.548183\pi$
$504$	0	0
$505$	56.5066	2.51451
$506$	−16.9443	−0.753265
$507$	0	0
$508$	−11.4164	−0.506521
$509$	−42.6869	−1.89206	−0.946032	−	0.324073i	$-0.894948\pi$
$509$	−42.6869	−1.89206	−0.946032	+	0.324073i	$0.894948\pi$
$510$	0	0
$511$	18.1803	0.804251
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.3820	−0.810794
$515$	53.0132	2.33604
$516$	0	0
$517$	−14.4721	−0.636484
$518$	−27.1246	−1.19179
$519$	0	0
$520$	5.00000	0.219265
$521$	5.27051	0.230905	0.115453	−	0.993313i	$-0.463168\pi$
$521$	5.27051	0.230905	0.115453	+	0.993313i	$0.463168\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−4.47214	−0.195366
$525$	0	0
$526$	−2.94427	−0.128376
$527$	−4.18034	−0.182098
$528$	0	0
$529$	4.41641	0.192018
$530$	22.0344	0.957115
$531$	0	0
$532$	0	0
$533$	−1.18034	−0.0511262
$534$	0	0
$535$	−34.0689	−1.47293
$536$	11.7082	0.505717
$537$	0	0
$538$	−14.3262	−0.617648
$539$	11.2361	0.483972
$540$	0	0
$541$	−26.6180	−1.14440	−0.572199	−	0.820115i	$-0.693910\pi$
$541$	−26.6180	−1.14440	−0.572199	+	0.820115i	$0.693910\pi$
$542$	26.0000	1.11680
$543$	0	0
$544$	3.38197	0.145001
$545$	−39.9230	−1.71011
$546$	0	0
$547$	5.05573	0.216167	0.108084	−	0.994142i	$-0.465529\pi$
$547$	5.05573	0.216167	0.108084	+	0.994142i	$0.465529\pi$
$548$	−2.38197	−0.101753
$549$	0	0
$550$	26.1803	1.11633
$551$	0	0
$552$	0	0
$553$	8.94427	0.380349
$554$	−10.0902	−0.428690
$555$	0	0
$556$	−14.0000	−0.593732
$557$	−5.05573	−0.214218	−0.107109	−	0.994247i	$-0.534159\pi$
$557$	−5.05573	−0.214218	−0.107109	+	0.994247i	$0.534159\pi$
$558$	0	0
$559$	−12.7639	−0.539857
$560$	−11.7082	−0.494762
$561$	0	0
$562$	−20.9787	−0.884934
$563$	−26.2918	−1.10807	−0.554034	−	0.832494i	$-0.686912\pi$
$563$	−26.2918	−1.10807	−0.554034	+	0.832494i	$0.686912\pi$
$564$	0	0
$565$	35.6525	1.49991
$566$	22.4721	0.944574
$567$	0	0
$568$	−2.94427	−0.123539
$569$	−7.90983	−0.331597	−0.165799	−	0.986160i	$-0.553020\pi$
$569$	−7.90983	−0.331597	−0.165799	+	0.986160i	$0.553020\pi$
$570$	0	0
$571$	−32.5410	−1.36180	−0.680900	−	0.732377i	$-0.738411\pi$
$571$	−32.5410	−1.36180	−0.680900	+	0.732377i	$0.738411\pi$
$572$	4.47214	0.186989
$573$	0	0
$574$	2.76393	0.115364
$575$	−42.3607	−1.76656
$576$	0	0
$577$	45.9230	1.91180	0.955899	−	0.293694i	$-0.0948847\pi$
$577$	45.9230	1.91180	0.955899	+	0.293694i	$0.0948847\pi$
$578$	−5.56231	−0.231361
$579$	0	0
$580$	32.8885	1.36562
$581$	−1.52786	−0.0633865
$582$	0	0
$583$	19.7082	0.816230
$584$	−5.61803	−0.232476
$585$	0	0
$586$	−22.9443	−0.947819
$587$	−2.87539	−0.118680	−0.0593400	−	0.998238i	$-0.518900\pi$
$587$	−2.87539	−0.118680	−0.0593400	+	0.998238i	$0.518900\pi$
$588$	0	0
$589$	0	0
$590$	1.70820	0.0703256
$591$	0	0
$592$	8.38197	0.344497
$593$	34.9230	1.43412	0.717058	−	0.697014i	$-0.245488\pi$
$593$	34.9230	1.43412	0.717058	+	0.697014i	$0.245488\pi$
$594$	0	0
$595$	−39.5967	−1.62331
$596$	13.0902	0.536194
$597$	0	0
$598$	−7.23607	−0.295905
$599$	39.1246	1.59859	0.799294	−	0.600940i	$-0.205207\pi$
$599$	39.1246	1.59859	0.799294	+	0.600940i	$0.205207\pi$
$600$	0	0
$601$	16.4721	0.671912	0.335956	−	0.941878i	$-0.390941\pi$
$601$	16.4721	0.671912	0.335956	+	0.941878i	$0.390941\pi$
$602$	29.8885	1.21817
$603$	0	0
$604$	14.4721	0.588863
$605$	−1.90983	−0.0776456
$606$	0	0
$607$	−47.7771	−1.93921	−0.969606	−	0.244671i	$-0.921320\pi$
$607$	−47.7771	−1.93921	−0.969606	+	0.244671i	$0.921320\pi$
$608$	0	0
$609$	0	0
$610$	5.00000	0.202444
$611$	−6.18034	−0.250030
$612$	0	0
$613$	8.38197	0.338544	0.169272	−	0.985569i	$-0.445858\pi$
$613$	8.38197	0.338544	0.169272	+	0.985569i	$0.445858\pi$
$614$	11.7082	0.472505
$615$	0	0
$616$	−10.4721	−0.421934
$617$	−42.9443	−1.72887	−0.864436	−	0.502743i	$-0.832324\pi$
$617$	−42.9443	−1.72887	−0.864436	+	0.502743i	$0.832324\pi$
$618$	0	0
$619$	22.3607	0.898752	0.449376	−	0.893343i	$-0.351646\pi$
$619$	22.3607	0.898752	0.449376	+	0.893343i	$0.351646\pi$
$620$	−4.47214	−0.179605
$621$	0	0
$622$	−31.2361	−1.25245
$623$	28.6525	1.14794
$624$	0	0
$625$	0	0
$626$	−15.5066	−0.619767
$627$	0	0
$628$	9.38197	0.374381
$629$	28.3475	1.13029
$630$	0	0
$631$	−31.1246	−1.23905	−0.619526	−	0.784976i	$-0.712675\pi$
$631$	−31.1246	−1.23905	−0.619526	+	0.784976i	$0.712675\pi$
$632$	−2.76393	−0.109943
$633$	0	0
$634$	−1.32624	−0.0526716
$635$	−41.3050	−1.63914
$636$	0	0
$637$	4.79837	0.190118
$638$	29.4164	1.16461
$639$	0	0
$640$	3.61803	0.143015
$641$	13.2705	0.524154	0.262077	−	0.965047i	$-0.415593\pi$
$641$	13.2705	0.524154	0.262077	+	0.965047i	$0.415593\pi$
$642$	0	0
$643$	−38.8328	−1.53142	−0.765708	−	0.643188i	$-0.777611\pi$
$643$	−38.8328	−1.53142	−0.765708	+	0.643188i	$0.777611\pi$
$644$	16.9443	0.667698
$645$	0	0
$646$	0	0
$647$	−26.0689	−1.02487	−0.512437	−	0.858725i	$-0.671257\pi$
$647$	−26.0689	−1.02487	−0.512437	+	0.858725i	$0.671257\pi$
$648$	0	0
$649$	1.52786	0.0599739
$650$	11.1803	0.438529
$651$	0	0
$652$	10.6525	0.417183
$653$	35.3951	1.38512	0.692559	−	0.721361i	$-0.256483\pi$
$653$	35.3951	1.38512	0.692559	+	0.721361i	$0.256483\pi$
$654$	0	0
$655$	−16.1803	−0.632218
$656$	−0.854102	−0.0333471
$657$	0	0
$658$	14.4721	0.564183
$659$	48.5410	1.89089	0.945445	−	0.325782i	$-0.105628\pi$
$659$	48.5410	1.89089	0.945445	+	0.325782i	$0.105628\pi$
$660$	0	0
$661$	−21.4164	−0.833002	−0.416501	−	0.909135i	$-0.636744\pi$
$661$	−21.4164	−0.833002	−0.416501	+	0.909135i	$0.636744\pi$
$662$	25.7082	0.999178
$663$	0	0
$664$	0.472136	0.0183224
$665$	0	0
$666$	0	0
$667$	−47.5967	−1.84295
$668$	12.4721	0.482561
$669$	0	0
$670$	42.3607	1.63654
$671$	4.47214	0.172645
$672$	0	0
$673$	30.3820	1.17114	0.585569	−	0.810622i	$-0.300871\pi$
$673$	30.3820	1.17114	0.585569	+	0.810622i	$0.300871\pi$
$674$	2.38197	0.0917499
$675$	0	0
$676$	−11.0902	−0.426545
$677$	6.61803	0.254352	0.127176	−	0.991880i	$-0.459409\pi$
$677$	6.61803	0.254352	0.127176	+	0.991880i	$0.459409\pi$
$678$	0	0
$679$	−27.8885	−1.07026
$680$	12.2361	0.469232
$681$	0	0
$682$	−4.00000	−0.153168
$683$	20.1803	0.772179	0.386090	−	0.922461i	$-0.373826\pi$
$683$	20.1803	0.772179	0.386090	+	0.922461i	$0.373826\pi$
$684$	0	0
$685$	−8.61803	−0.329278
$686$	11.4164	0.435880
$687$	0	0
$688$	−9.23607	−0.352122
$689$	8.41641	0.320640
$690$	0	0
$691$	−22.3607	−0.850640	−0.425320	−	0.905043i	$-0.639839\pi$
$691$	−22.3607	−0.850640	−0.425320	+	0.905043i	$0.639839\pi$
$692$	−1.32624	−0.0504160
$693$	0	0
$694$	2.65248	0.100687
$695$	−50.6525	−1.92136
$696$	0	0
$697$	−2.88854	−0.109411
$698$	−33.0902	−1.25248
$699$	0	0
$700$	−26.1803	−0.989524
$701$	−38.6312	−1.45908	−0.729540	−	0.683938i	$-0.760266\pi$
$701$	−38.6312	−1.45908	−0.729540	+	0.683938i	$0.760266\pi$
$702$	0	0
$703$	0	0
$704$	3.23607	0.121964
$705$	0	0
$706$	23.7426	0.893566
$707$	−50.5410	−1.90079
$708$	0	0
$709$	6.90983	0.259504	0.129752	−	0.991546i	$-0.458582\pi$
$709$	6.90983	0.259504	0.129752	+	0.991546i	$0.458582\pi$
$710$	−10.6525	−0.399780
$711$	0	0
$712$	−8.85410	−0.331822
$713$	6.47214	0.242383
$714$	0	0
$715$	16.1803	0.605110
$716$	23.4164	0.875112
$717$	0	0
$718$	−11.5279	−0.430216
$719$	−12.2918	−0.458407	−0.229203	−	0.973379i	$-0.573612\pi$
$719$	−12.2918	−0.458407	−0.229203	+	0.973379i	$0.573612\pi$
$720$	0	0
$721$	−47.4164	−1.76588
$722$	0	0
$723$	0	0
$724$	5.41641	0.201299
$725$	73.5410	2.73124
$726$	0	0
$727$	−30.9443	−1.14766	−0.573830	−	0.818975i	$-0.694543\pi$
$727$	−30.9443	−1.14766	−0.573830	+	0.818975i	$0.694543\pi$
$728$	−4.47214	−0.165748
$729$	0	0
$730$	−20.3262	−0.752308
$731$	−31.2361	−1.15531
$732$	0	0
$733$	−1.09017	−0.0402663	−0.0201332	−	0.999797i	$-0.506409\pi$
$733$	−1.09017	−0.0402663	−0.0201332	+	0.999797i	$0.506409\pi$
$734$	18.8328	0.695132
$735$	0	0
$736$	−5.23607	−0.193004
$737$	37.8885	1.39564
$738$	0	0
$739$	9.12461	0.335654	0.167827	−	0.985816i	$-0.446325\pi$
$739$	9.12461	0.335654	0.167827	+	0.985816i	$0.446325\pi$
$740$	30.3262	1.11481
$741$	0	0
$742$	−19.7082	−0.723511
$743$	−6.11146	−0.224208	−0.112104	−	0.993697i	$-0.535759\pi$
$743$	−6.11146	−0.224208	−0.112104	+	0.993697i	$0.535759\pi$
$744$	0	0
$745$	47.3607	1.73516
$746$	18.3262	0.670971
$747$	0	0
$748$	10.9443	0.400162
$749$	30.4721	1.11343
$750$	0	0
$751$	−24.0689	−0.878286	−0.439143	−	0.898417i	$-0.644718\pi$
$751$	−24.0689	−0.878286	−0.439143	+	0.898417i	$0.644718\pi$
$752$	−4.47214	−0.163082
$753$	0	0
$754$	12.5623	0.457492
$755$	52.3607	1.90560
$756$	0	0
$757$	4.43769	0.161291	0.0806454	−	0.996743i	$-0.474302\pi$
$757$	4.43769	0.161291	0.0806454	+	0.996743i	$0.474302\pi$
$758$	−35.4164	−1.28638
$759$	0	0
$760$	0	0
$761$	−26.3607	−0.955574	−0.477787	−	0.878476i	$-0.658561\pi$
$761$	−26.3607	−0.955574	−0.477787	+	0.878476i	$0.658561\pi$
$762$	0	0
$763$	35.7082	1.29272
$764$	−3.52786	−0.127634
$765$	0	0
$766$	−20.7639	−0.750231
$767$	0.652476	0.0235595
$768$	0	0
$769$	38.7426	1.39710	0.698548	−	0.715563i	$-0.253830\pi$
$769$	38.7426	1.39710	0.698548	+	0.715563i	$0.253830\pi$
$770$	−37.8885	−1.36541
$771$	0	0
$772$	−11.5279	−0.414897
$773$	8.14590	0.292988	0.146494	−	0.989212i	$-0.453201\pi$
$773$	8.14590	0.292988	0.146494	+	0.989212i	$0.453201\pi$
$774$	0	0
$775$	−10.0000	−0.359211
$776$	8.61803	0.309369
$777$	0	0
$778$	−12.7984	−0.458844
$779$	0	0
$780$	0	0
$781$	−9.52786	−0.340934
$782$	−17.7082	−0.633244
$783$	0	0
$784$	3.47214	0.124005
$785$	33.9443	1.21152
$786$	0	0
$787$	28.4721	1.01492	0.507461	−	0.861675i	$-0.330584\pi$
$787$	28.4721	1.01492	0.507461	+	0.861675i	$0.330584\pi$
$788$	4.14590	0.147692
$789$	0	0
$790$	−10.0000	−0.355784
$791$	−31.8885	−1.13383
$792$	0	0
$793$	1.90983	0.0678201
$794$	−22.3607	−0.793551
$795$	0	0
$796$	10.4721	0.371175
$797$	−28.4377	−1.00731	−0.503657	−	0.863903i	$-0.668013\pi$
$797$	−28.4377	−1.00731	−0.503657	+	0.863903i	$0.668013\pi$
$798$	0	0
$799$	−15.1246	−0.535070
$800$	8.09017	0.286031
$801$	0	0
$802$	−18.9443	−0.668945
$803$	−18.1803	−0.641570
$804$	0	0
$805$	61.3050	2.16072
$806$	−1.70820	−0.0601689
$807$	0	0
$808$	15.6180	0.549441
$809$	−2.74265	−0.0964263	−0.0482131	−	0.998837i	$-0.515353\pi$
$809$	−2.74265	−0.0964263	−0.0482131	+	0.998837i	$0.515353\pi$
$810$	0	0
$811$	5.59675	0.196528	0.0982642	−	0.995160i	$-0.468671\pi$
$811$	5.59675	0.196528	0.0982642	+	0.995160i	$0.468671\pi$
$812$	−29.4164	−1.03231
$813$	0	0
$814$	27.1246	0.950717
$815$	38.5410	1.35003
$816$	0	0
$817$	0	0
$818$	16.2705	0.568885
$819$	0	0
$820$	−3.09017	−0.107913
$821$	−19.2705	−0.672545	−0.336273	−	0.941765i	$-0.609166\pi$
$821$	−19.2705	−0.672545	−0.336273	+	0.941765i	$0.609166\pi$
$822$	0	0
$823$	−20.8328	−0.726186	−0.363093	−	0.931753i	$-0.618279\pi$
$823$	−20.8328	−0.726186	−0.363093	+	0.931753i	$0.618279\pi$
$824$	14.6525	0.510443
$825$	0	0
$826$	−1.52786	−0.0531612
$827$	30.5410	1.06202	0.531008	−	0.847367i	$-0.321814\pi$
$827$	30.5410	1.06202	0.531008	+	0.847367i	$0.321814\pi$
$828$	0	0
$829$	−9.72949	−0.337919	−0.168960	−	0.985623i	$-0.554041\pi$
$829$	−9.72949	−0.337919	−0.168960	+	0.985623i	$0.554041\pi$
$830$	1.70820	0.0592926
$831$	0	0
$832$	1.38197	0.0479111
$833$	11.7426	0.406859
$834$	0	0
$835$	45.1246	1.56160
$836$	0	0
$837$	0	0
$838$	6.47214	0.223576
$839$	−22.0000	−0.759524	−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000	−0.759524	−0.379762	+	0.925084i	$0.623994\pi$
$840$	0	0
$841$	53.6312	1.84935
$842$	13.9098	0.479364
$843$	0	0
$844$	−7.23607	−0.249076
$845$	−40.1246	−1.38033
$846$	0	0
$847$	1.70820	0.0586946
$848$	6.09017	0.209137
$849$	0	0
$850$	27.3607	0.938464
$851$	−43.8885	−1.50448
$852$	0	0
$853$	−29.0902	−0.996028	−0.498014	−	0.867169i	$-0.665937\pi$
$853$	−29.0902	−0.996028	−0.498014	+	0.867169i	$0.665937\pi$
$854$	−4.47214	−0.153033
$855$	0	0
$856$	−9.41641	−0.321846
$857$	−5.56231	−0.190005	−0.0950024	−	0.995477i	$-0.530286\pi$
$857$	−5.56231	−0.190005	−0.0950024	+	0.995477i	$0.530286\pi$
$858$	0	0
$859$	28.0689	0.957698	0.478849	−	0.877897i	$-0.341054\pi$
$859$	28.0689	0.957698	0.478849	+	0.877897i	$0.341054\pi$
$860$	−33.4164	−1.13949
$861$	0	0
$862$	−6.65248	−0.226584
$863$	25.8885	0.881256	0.440628	−	0.897690i	$-0.354756\pi$
$863$	25.8885	0.881256	0.440628	+	0.897690i	$0.354756\pi$
$864$	0	0
$865$	−4.79837	−0.163150
$866$	−25.0344	−0.850705
$867$	0	0
$868$	4.00000	0.135769
$869$	−8.94427	−0.303414
$870$	0	0
$871$	16.1803	0.548250
$872$	−11.0344	−0.373673
$873$	0	0
$874$	0	0
$875$	−36.1803	−1.22312
$876$	0	0
$877$	1.74265	0.0588450	0.0294225	−	0.999567i	$-0.490633\pi$
$877$	1.74265	0.0588450	0.0294225	+	0.999567i	$0.490633\pi$
$878$	29.2361	0.986670
$879$	0	0
$880$	11.7082	0.394683
$881$	−5.21478	−0.175690	−0.0878452	−	0.996134i	$-0.527998\pi$
$881$	−5.21478	−0.175690	−0.0878452	+	0.996134i	$0.527998\pi$
$882$	0	0
$883$	−17.8197	−0.599679	−0.299840	−	0.953990i	$-0.596933\pi$
$883$	−17.8197	−0.599679	−0.299840	+	0.953990i	$0.596933\pi$
$884$	4.67376	0.157196
$885$	0	0
$886$	−9.23607	−0.310292
$887$	−6.94427	−0.233166	−0.116583	−	0.993181i	$-0.537194\pi$
$887$	−6.94427	−0.233166	−0.116583	+	0.993181i	$0.537194\pi$
$888$	0	0
$889$	36.9443	1.23907
$890$	−32.0344	−1.07380
$891$	0	0
$892$	27.7082	0.927739
$893$	0	0
$894$	0	0
$895$	84.7214	2.83192
$896$	−3.23607	−0.108109
$897$	0	0
$898$	22.7984	0.760792
$899$	−11.2361	−0.374744
$900$	0	0
$901$	20.5967	0.686177
$902$	−2.76393	−0.0920289
$903$	0	0
$904$	9.85410	0.327743
$905$	19.5967	0.651418
$906$	0	0
$907$	−0.291796	−0.00968893	−0.00484446	−	0.999988i	$-0.501542\pi$
$907$	−0.291796	−0.00968893	−0.00484446	+	0.999988i	$0.501542\pi$
$908$	8.94427	0.296826
$909$	0	0
$910$	−16.1803	−0.536373
$911$	−12.0689	−0.399860	−0.199930	−	0.979810i	$-0.564071\pi$
$911$	−12.0689	−0.399860	−0.199930	+	0.979810i	$0.564071\pi$
$912$	0	0
$913$	1.52786	0.0505649
$914$	25.2148	0.834031
$915$	0	0
$916$	4.38197	0.144784
$917$	14.4721	0.477912
$918$	0	0
$919$	45.8885	1.51372	0.756862	−	0.653575i	$-0.226732\pi$
$919$	45.8885	1.51372	0.756862	+	0.653575i	$0.226732\pi$
$920$	−18.9443	−0.624574
$921$	0	0
$922$	−21.4164	−0.705312
$923$	−4.06888	−0.133929
$924$	0	0
$925$	67.8115	2.22963
$926$	−5.05573	−0.166142
$927$	0	0
$928$	9.09017	0.298399
$929$	59.1591	1.94095	0.970473	−	0.241211i	$-0.0775445\pi$
$929$	59.1591	1.94095	0.970473	+	0.241211i	$0.0775445\pi$
$930$	0	0
$931$	0	0
$932$	−1.56231	−0.0511750
$933$	0	0
$934$	33.3050	1.08977
$935$	39.5967	1.29495
$936$	0	0
$937$	15.8885	0.519056	0.259528	−	0.965736i	$-0.416433\pi$
$937$	15.8885	0.519056	0.259528	+	0.965736i	$0.416433\pi$
$938$	−37.8885	−1.23710
$939$	0	0
$940$	−16.1803	−0.527744
$941$	−26.0000	−0.847576	−0.423788	−	0.905761i	$-0.639300\pi$
$941$	−26.0000	−0.847576	−0.423788	+	0.905761i	$0.639300\pi$
$942$	0	0
$943$	4.47214	0.145633
$944$	0.472136	0.0153667
$945$	0	0
$946$	−29.8885	−0.971760
$947$	−8.18034	−0.265825	−0.132913	−	0.991128i	$-0.542433\pi$
$947$	−8.18034	−0.265825	−0.132913	+	0.991128i	$0.542433\pi$
$948$	0	0
$949$	−7.76393	−0.252028
$950$	0	0
$951$	0	0
$952$	−10.9443	−0.354706
$953$	−47.8115	−1.54877	−0.774384	−	0.632716i	$-0.781940\pi$
$953$	−47.8115	−1.54877	−0.774384	+	0.632716i	$0.781940\pi$
$954$	0	0
$955$	−12.7639	−0.413031
$956$	−30.1803	−0.976102
$957$	0	0
$958$	−7.70820	−0.249041
$959$	7.70820	0.248911
$960$	0	0
$961$	−29.4721	−0.950714
$962$	11.5836	0.373470
$963$	0	0
$964$	−25.4164	−0.818607
$965$	−41.7082	−1.34263
$966$	0	0
$967$	8.83282	0.284044	0.142022	−	0.989863i	$-0.454640\pi$
$967$	8.83282	0.284044	0.142022	+	0.989863i	$0.454640\pi$
$968$	−0.527864	−0.0169662
$969$	0	0
$970$	31.1803	1.00114
$971$	−60.5410	−1.94285	−0.971427	−	0.237339i	$-0.923725\pi$
$971$	−60.5410	−1.94285	−0.971427	+	0.237339i	$0.923725\pi$
$972$	0	0
$973$	45.3050	1.45241
$974$	19.2361	0.616363
$975$	0	0
$976$	1.38197	0.0442357
$977$	9.03444	0.289037	0.144519	−	0.989502i	$-0.453837\pi$
$977$	9.03444	0.289037	0.144519	+	0.989502i	$0.453837\pi$
$978$	0	0
$979$	−28.6525	−0.915737
$980$	12.5623	0.401288
$981$	0	0
$982$	1.52786	0.0487561
$983$	−19.5967	−0.625039	−0.312520	−	0.949911i	$-0.601173\pi$
$983$	−19.5967	−0.625039	−0.312520	+	0.949911i	$0.601173\pi$
$984$	0	0
$985$	15.0000	0.477940
$986$	30.7426	0.979045
$987$	0	0
$988$	0	0
$989$	48.3607	1.53778
$990$	0	0
$991$	−30.6525	−0.973708	−0.486854	−	0.873483i	$-0.661856\pi$
$991$	−30.6525	−0.973708	−0.486854	+	0.873483i	$0.661856\pi$
$992$	−1.23607	−0.0392452
$993$	0	0
$994$	9.52786	0.302205
$995$	37.8885	1.20115
$996$	0	0
$997$	−13.5066	−0.427758	−0.213879	−	0.976860i	$-0.568610\pi$
$997$	−13.5066	−0.427758	−0.213879	+	0.976860i	$0.568610\pi$
$998$	−35.3050	−1.11756
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bk.1.2		2
3.2	odd	2		722.2.a.h.1.1	✓	2
12.11	even	2		5776.2.a.t.1.2		2
19.18	odd	2		6498.2.a.be.1.2		2
57.2	even	18		722.2.e.p.99.2		12
57.5	odd	18		722.2.e.q.595.2		12
57.8	even	6		722.2.c.h.653.1		4
57.11	odd	6		722.2.c.i.653.2		4
57.14	even	18		722.2.e.p.595.1		12
57.17	odd	18		722.2.e.q.99.1		12
57.23	odd	18		722.2.e.q.415.2		12
57.26	odd	6		722.2.c.i.429.2		4
57.29	even	18		722.2.e.p.423.2		12
57.32	even	18		722.2.e.p.245.2		12
57.35	odd	18		722.2.e.q.389.1		12
57.41	even	18		722.2.e.p.389.2		12
57.44	odd	18		722.2.e.q.245.1		12
57.47	odd	18		722.2.e.q.423.1		12
57.50	even	6		722.2.c.h.429.1		4
57.53	even	18		722.2.e.p.415.1		12
57.56	even	2		722.2.a.i.1.2	yes	2
228.227	odd	2		5776.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
722.2.a.h.1.1	✓	2	3.2	odd	2
722.2.a.i.1.2	yes	2	57.56	even	2
722.2.c.h.429.1		4	57.50	even	6
722.2.c.h.653.1		4	57.8	even	6
722.2.c.i.429.2		4	57.26	odd	6
722.2.c.i.653.2		4	57.11	odd	6
722.2.e.p.99.2		12	57.2	even	18
722.2.e.p.245.2		12	57.32	even	18
722.2.e.p.389.2		12	57.41	even	18
722.2.e.p.415.1		12	57.53	even	18
722.2.e.p.423.2		12	57.29	even	18
722.2.e.p.595.1		12	57.14	even	18
722.2.e.q.99.1		12	57.17	odd	18
722.2.e.q.245.1		12	57.44	odd	18
722.2.e.q.389.1		12	57.35	odd	18
722.2.e.q.415.2		12	57.23	odd	18
722.2.e.q.423.1		12	57.47	odd	18
722.2.e.q.595.2		12	57.5	odd	18
5776.2.a.t.1.2		2	12.11	even	2
5776.2.a.be.1.1		2	228.227	odd	2
6498.2.a.be.1.2		2	19.18	odd	2
6498.2.a.bk.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -3.23607 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.61803 q^{5} -3.23607 q^{7} +1.00000 q^{8} +3.61803 q^{10} +3.23607 q^{11} +1.38197 q^{13} -3.23607 q^{14} +1.00000 q^{16} +3.38197 q^{17} +3.61803 q^{20} +3.23607 q^{22} -5.23607 q^{23} +8.09017 q^{25} +1.38197 q^{26} -3.23607 q^{28} +9.09017 q^{29} -1.23607 q^{31} +1.00000 q^{32} +3.38197 q^{34} -11.7082 q^{35} +8.38197 q^{37} +3.61803 q^{40} -0.854102 q^{41} -9.23607 q^{43} +3.23607 q^{44} -5.23607 q^{46} -4.47214 q^{47} +3.47214 q^{49} +8.09017 q^{50} +1.38197 q^{52} +6.09017 q^{53} +11.7082 q^{55} -3.23607 q^{56} +9.09017 q^{58} +0.472136 q^{59} +1.38197 q^{61} -1.23607 q^{62} +1.00000 q^{64} +5.00000 q^{65} +11.7082 q^{67} +3.38197 q^{68} -11.7082 q^{70} -2.94427 q^{71} -5.61803 q^{73} +8.38197 q^{74} -10.4721 q^{77} -2.76393 q^{79} +3.61803 q^{80} -0.854102 q^{82} +0.472136 q^{83} +12.2361 q^{85} -9.23607 q^{86} +3.23607 q^{88} -8.85410 q^{89} -4.47214 q^{91} -5.23607 q^{92} -4.47214 q^{94} +8.61803 q^{97} +3.47214 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 5 q^{5} - 2 q^{7} + 2 q^{8} + 5 q^{10} + 2 q^{11} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 9 q^{17} + 5 q^{20} + 2 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - 2 q^{28} + 7 q^{29} + 2 q^{31} + 2 q^{32} + 9 q^{34} - 10 q^{35} + 19 q^{37} + 5 q^{40} + 5 q^{41} - 14 q^{43} + 2 q^{44} - 6 q^{46} - 2 q^{49} + 5 q^{50} + 5 q^{52} + q^{53} + 10 q^{55} - 2 q^{56} + 7 q^{58} - 8 q^{59} + 5 q^{61} + 2 q^{62} + 2 q^{64} + 10 q^{65} + 10 q^{67} + 9 q^{68} - 10 q^{70} + 12 q^{71} - 9 q^{73} + 19 q^{74} - 12 q^{77} - 10 q^{79} + 5 q^{80} + 5 q^{82} - 8 q^{83} + 20 q^{85} - 14 q^{86} + 2 q^{88} - 11 q^{89} - 6 q^{92} + 15 q^{97} - 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 5 * q^5 - 2 * q^7 + 2 * q^8 + 5 * q^10 + 2 * q^11 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 9 * q^17 + 5 * q^20 + 2 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - 2 * q^28 + 7 * q^29 + 2 * q^31 + 2 * q^32 + 9 * q^34 - 10 * q^35 + 19 * q^37 + 5 * q^40 + 5 * q^41 - 14 * q^43 + 2 * q^44 - 6 * q^46 - 2 * q^49 + 5 * q^50 + 5 * q^52 + q^53 + 10 * q^55 - 2 * q^56 + 7 * q^58 - 8 * q^59 + 5 * q^61 + 2 * q^62 + 2 * q^64 + 10 * q^65 + 10 * q^67 + 9 * q^68 - 10 * q^70 + 12 * q^71 - 9 * q^73 + 19 * q^74 - 12 * q^77 - 10 * q^79 + 5 * q^80 + 5 * q^82 - 8 * q^83 + 20 * q^85 - 14 * q^86 + 2 * q^88 - 11 * q^89 - 6 * q^92 + 15 * q^97 - 2 * q^98

Introduction

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