LMFDB - Embedded newform 6498.2.a.bq.1.3

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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +5.06418 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +5.06418 q^{7} +1.00000 q^{8} -2.00000 q^{10} +1.41147 q^{11} +1.30541 q^{13} +5.06418 q^{14} +1.00000 q^{16} -2.38919 q^{17} -2.00000 q^{20} +1.41147 q^{22} +3.06418 q^{23} -1.00000 q^{25} +1.30541 q^{26} +5.06418 q^{28} +8.45336 q^{29} -0.369585 q^{31} +1.00000 q^{32} -2.38919 q^{34} -10.1284 q^{35} -4.82295 q^{37} -2.00000 q^{40} -1.53209 q^{41} -0.758770 q^{43} +1.41147 q^{44} +3.06418 q^{46} +10.2121 q^{47} +18.6459 q^{49} -1.00000 q^{50} +1.30541 q^{52} +1.67499 q^{53} -2.82295 q^{55} +5.06418 q^{56} +8.45336 q^{58} +0.716881 q^{59} +9.75877 q^{61} -0.369585 q^{62} +1.00000 q^{64} -2.61081 q^{65} +1.40373 q^{67} -2.38919 q^{68} -10.1284 q^{70} -6.36959 q^{71} -4.55943 q^{73} -4.82295 q^{74} +7.14796 q^{77} +2.24123 q^{79} -2.00000 q^{80} -1.53209 q^{82} +3.98545 q^{83} +4.77837 q^{85} -0.758770 q^{86} +1.41147 q^{88} -10.6459 q^{89} +6.61081 q^{91} +3.06418 q^{92} +10.2121 q^{94} +1.53209 q^{97} +18.6459 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 6 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} - 6 q^{20} - 6 q^{22} - 3 q^{25} + 6 q^{26} + 6 q^{28} + 12 q^{29} + 6 q^{31} + 3 q^{32} - 3 q^{34} - 12 q^{35} + 6 q^{37} - 6 q^{40} + 9 q^{43} - 6 q^{44} + 6 q^{47} + 15 q^{49} - 3 q^{50} + 6 q^{52} + 12 q^{55} + 6 q^{56} + 12 q^{58} - 6 q^{59} + 18 q^{61} + 6 q^{62} + 3 q^{64} - 12 q^{65} + 18 q^{67} - 3 q^{68} - 12 q^{70} - 12 q^{71} + 12 q^{73} + 6 q^{74} + 6 q^{77} + 18 q^{79} - 6 q^{80} - 6 q^{83} + 6 q^{85} + 9 q^{86} - 6 q^{88} + 9 q^{89} + 24 q^{91} + 6 q^{94} + 15 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 6 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 - 6 * q^20 - 6 * q^22 - 3 * q^25 + 6 * q^26 + 6 * q^28 + 12 * q^29 + 6 * q^31 + 3 * q^32 - 3 * q^34 - 12 * q^35 + 6 * q^37 - 6 * q^40 + 9 * q^43 - 6 * q^44 + 6 * q^47 + 15 * q^49 - 3 * q^50 + 6 * q^52 + 12 * q^55 + 6 * q^56 + 12 * q^58 - 6 * q^59 + 18 * q^61 + 6 * q^62 + 3 * q^64 - 12 * q^65 + 18 * q^67 - 3 * q^68 - 12 * q^70 - 12 * q^71 + 12 * q^73 + 6 * q^74 + 6 * q^77 + 18 * q^79 - 6 * q^80 - 6 * q^83 + 6 * q^85 + 9 * q^86 - 6 * q^88 + 9 * q^89 + 24 * q^91 + 6 * q^94 + 15 * 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
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	5.06418	1.91408	0.957040	−	0.289957i	$-0.0936410\pi$
$7$	5.06418	1.91408	0.957040	+	0.289957i	$0.0936410\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	1.41147	0.425575	0.212788	−	0.977098i	$-0.431746\pi$
$11$	1.41147	0.425575	0.212788	+	0.977098i	$0.431746\pi$
$12$	0	0
$13$	1.30541	0.362055	0.181027	−	0.983478i	$-0.442058\pi$
$13$	1.30541	0.362055	0.181027	+	0.983478i	$0.442058\pi$
$14$	5.06418	1.35346
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.38919	−0.579463	−0.289731	−	0.957108i	$-0.593566\pi$
$17$	−2.38919	−0.579463	−0.289731	+	0.957108i	$0.593566\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−2.00000	−0.447214
$21$	0	0
$22$	1.41147	0.300927
$23$	3.06418	0.638925	0.319463	−	0.947599i	$-0.396498\pi$
$23$	3.06418	0.638925	0.319463	+	0.947599i	$0.396498\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	1.30541	0.256011
$27$	0	0
$28$	5.06418	0.957040
$29$	8.45336	1.56975	0.784875	−	0.619654i	$-0.212727\pi$
$29$	8.45336	1.56975	0.784875	+	0.619654i	$0.212727\pi$
$30$	0	0
$31$	−0.369585	−0.0663794	−0.0331897	−	0.999449i	$-0.510567\pi$
$31$	−0.369585	−0.0663794	−0.0331897	+	0.999449i	$0.510567\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.38919	−0.409742
$35$	−10.1284	−1.71200
$36$	0	0
$37$	−4.82295	−0.792888	−0.396444	−	0.918059i	$-0.629756\pi$
$37$	−4.82295	−0.792888	−0.396444	+	0.918059i	$0.629756\pi$
$38$	0	0
$39$	0	0
$40$	−2.00000	−0.316228
$41$	−1.53209	−0.239272	−0.119636	−	0.992818i	$-0.538173\pi$
$41$	−1.53209	−0.239272	−0.119636	+	0.992818i	$0.538173\pi$
$42$	0	0
$43$	−0.758770	−0.115711	−0.0578557	−	0.998325i	$-0.518426\pi$
$43$	−0.758770	−0.115711	−0.0578557	+	0.998325i	$0.518426\pi$
$44$	1.41147	0.212788
$45$	0	0
$46$	3.06418	0.451788
$47$	10.2121	1.48959	0.744796	−	0.667292i	$-0.232547\pi$
$47$	10.2121	1.48959	0.744796	+	0.667292i	$0.232547\pi$
$48$	0	0
$49$	18.6459	2.66370
$50$	−1.00000	−0.141421
$51$	0	0
$52$	1.30541	0.181027
$53$	1.67499	0.230078	0.115039	−	0.993361i	$-0.463301\pi$
$53$	1.67499	0.230078	0.115039	+	0.993361i	$0.463301\pi$
$54$	0	0
$55$	−2.82295	−0.380646
$56$	5.06418	0.676729
$57$	0	0
$58$	8.45336	1.10998
$59$	0.716881	0.0933300	0.0466650	−	0.998911i	$-0.485141\pi$
$59$	0.716881	0.0933300	0.0466650	+	0.998911i	$0.485141\pi$
$60$	0	0
$61$	9.75877	1.24948	0.624741	−	0.780832i	$-0.285204\pi$
$61$	9.75877	1.24948	0.624741	+	0.780832i	$0.285204\pi$
$62$	−0.369585	−0.0469373
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.61081	−0.323832
$66$	0	0
$67$	1.40373	0.171493	0.0857467	−	0.996317i	$-0.472672\pi$
$67$	1.40373	0.171493	0.0857467	+	0.996317i	$0.472672\pi$
$68$	−2.38919	−0.289731
$69$	0	0
$70$	−10.1284	−1.21057
$71$	−6.36959	−0.755931	−0.377965	−	0.925820i	$-0.623376\pi$
$71$	−6.36959	−0.755931	−0.377965	+	0.925820i	$0.623376\pi$
$72$	0	0
$73$	−4.55943	−0.533641	−0.266820	−	0.963746i	$-0.585973\pi$
$73$	−4.55943	−0.533641	−0.266820	+	0.963746i	$0.585973\pi$
$74$	−4.82295	−0.560656
$75$	0	0
$76$	0	0
$77$	7.14796	0.814585
$78$	0	0
$79$	2.24123	0.252158	0.126079	−	0.992020i	$-0.459761\pi$
$79$	2.24123	0.252158	0.126079	+	0.992020i	$0.459761\pi$
$80$	−2.00000	−0.223607
$81$	0	0
$82$	−1.53209	−0.169191
$83$	3.98545	0.437460	0.218730	−	0.975785i	$-0.429809\pi$
$83$	3.98545	0.437460	0.218730	+	0.975785i	$0.429809\pi$
$84$	0	0
$85$	4.77837	0.518287
$86$	−0.758770	−0.0818203
$87$	0	0
$88$	1.41147	0.150464
$89$	−10.6459	−1.12846	−0.564231	−	0.825617i	$-0.690827\pi$
$89$	−10.6459	−1.12846	−0.564231	+	0.825617i	$0.690827\pi$
$90$	0	0
$91$	6.61081	0.693002
$92$	3.06418	0.319463
$93$	0	0
$94$	10.2121	1.05330
$95$	0	0
$96$	0	0
$97$	1.53209	0.155560	0.0777800	−	0.996971i	$-0.475217\pi$
$97$	1.53209	0.155560	0.0777800	+	0.996971i	$0.475217\pi$
$98$	18.6459	1.88352
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−8.82295	−0.877916	−0.438958	−	0.898508i	$-0.644652\pi$
$101$	−8.82295	−0.877916	−0.438958	+	0.898508i	$0.644652\pi$
$102$	0	0
$103$	7.14796	0.704309	0.352155	−	0.935942i	$-0.385449\pi$
$103$	7.14796	0.704309	0.352155	+	0.935942i	$0.385449\pi$
$104$	1.30541	0.128006
$105$	0	0
$106$	1.67499	0.162690
$107$	−9.36959	−0.905792	−0.452896	−	0.891563i	$-0.649609\pi$
$107$	−9.36959	−0.905792	−0.452896	+	0.891563i	$0.649609\pi$
$108$	0	0
$109$	−11.0642	−1.05976	−0.529878	−	0.848074i	$-0.677762\pi$
$109$	−11.0642	−1.05976	−0.529878	+	0.848074i	$0.677762\pi$
$110$	−2.82295	−0.269158
$111$	0	0
$112$	5.06418	0.478520
$113$	−13.2986	−1.25103	−0.625514	−	0.780213i	$-0.715110\pi$
$113$	−13.2986	−1.25103	−0.625514	+	0.780213i	$0.715110\pi$
$114$	0	0
$115$	−6.12836	−0.571472
$116$	8.45336	0.784875
$117$	0	0
$118$	0.716881	0.0659943
$119$	−12.0993	−1.10914
$120$	0	0
$121$	−9.00774	−0.818886
$122$	9.75877	0.883518
$123$	0	0
$124$	−0.369585	−0.0331897
$125$	12.0000	1.07331
$126$	0	0
$127$	5.71419	0.507053	0.253526	−	0.967328i	$-0.418410\pi$
$127$	5.71419	0.507053	0.253526	+	0.967328i	$0.418410\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−2.61081	−0.228984
$131$	9.87939	0.863166	0.431583	−	0.902073i	$-0.357955\pi$
$131$	9.87939	0.863166	0.431583	+	0.902073i	$0.357955\pi$
$132$	0	0
$133$	0	0
$134$	1.40373	0.121264
$135$	0	0
$136$	−2.38919	−0.204871
$137$	−5.39693	−0.461091	−0.230545	−	0.973062i	$-0.574051\pi$
$137$	−5.39693	−0.461091	−0.230545	+	0.973062i	$0.574051\pi$
$138$	0	0
$139$	−2.33956	−0.198439	−0.0992193	−	0.995066i	$-0.531635\pi$
$139$	−2.33956	−0.198439	−0.0992193	+	0.995066i	$0.531635\pi$
$140$	−10.1284	−0.856002
$141$	0	0
$142$	−6.36959	−0.534524
$143$	1.84255	0.154082
$144$	0	0
$145$	−16.9067	−1.40403
$146$	−4.55943	−0.377341
$147$	0	0
$148$	−4.82295	−0.396444
$149$	14.3696	1.17720	0.588601	−	0.808424i	$-0.299679\pi$
$149$	14.3696	1.17720	0.588601	+	0.808424i	$0.299679\pi$
$150$	0	0
$151$	20.8384	1.69581	0.847904	−	0.530150i	$-0.177865\pi$
$151$	20.8384	1.69581	0.847904	+	0.530150i	$0.177865\pi$
$152$	0	0
$153$	0	0
$154$	7.14796	0.575999
$155$	0.739170	0.0593716
$156$	0	0
$157$	6.36959	0.508348	0.254174	−	0.967158i	$-0.418196\pi$
$157$	6.36959	0.508348	0.254174	+	0.967158i	$0.418196\pi$
$158$	2.24123	0.178303
$159$	0	0
$160$	−2.00000	−0.158114
$161$	15.5175	1.22295
$162$	0	0
$163$	4.73143	0.370594	0.185297	−	0.982683i	$-0.440675\pi$
$163$	4.73143	0.370594	0.185297	+	0.982683i	$0.440675\pi$
$164$	−1.53209	−0.119636
$165$	0	0
$166$	3.98545	0.309331
$167$	17.2763	1.33688	0.668441	−	0.743766i	$-0.266962\pi$
$167$	17.2763	1.33688	0.668441	+	0.743766i	$0.266962\pi$
$168$	0	0
$169$	−11.2959	−0.868916
$170$	4.77837	0.366484
$171$	0	0
$172$	−0.758770	−0.0578557
$173$	18.8229	1.43108	0.715541	−	0.698571i	$-0.246180\pi$
$173$	18.8229	1.43108	0.715541	+	0.698571i	$0.246180\pi$
$174$	0	0
$175$	−5.06418	−0.382816
$176$	1.41147	0.106394
$177$	0	0
$178$	−10.6459	−0.797944
$179$	−13.8280	−1.03355	−0.516777	−	0.856120i	$-0.672868\pi$
$179$	−13.8280	−1.03355	−0.516777	+	0.856120i	$0.672868\pi$
$180$	0	0
$181$	−12.2567	−0.911034	−0.455517	−	0.890227i	$-0.650546\pi$
$181$	−12.2567	−0.911034	−0.455517	+	0.890227i	$0.650546\pi$
$182$	6.61081	0.490026
$183$	0	0
$184$	3.06418	0.225894
$185$	9.64590	0.709180
$186$	0	0
$187$	−3.37227	−0.246605
$188$	10.2121	0.744796
$189$	0	0
$190$	0	0
$191$	20.0993	1.45433	0.727166	−	0.686462i	$-0.240837\pi$
$191$	20.0993	1.45433	0.727166	+	0.686462i	$0.240837\pi$
$192$	0	0
$193$	−16.6851	−1.20102	−0.600510	−	0.799617i	$-0.705036\pi$
$193$	−16.6851	−1.20102	−0.600510	+	0.799617i	$0.705036\pi$
$194$	1.53209	0.109998
$195$	0	0
$196$	18.6459	1.33185
$197$	−12.9905	−0.925535	−0.462768	−	0.886480i	$-0.653144\pi$
$197$	−12.9905	−0.925535	−0.462768	+	0.886480i	$0.653144\pi$
$198$	0	0
$199$	17.1925	1.21875	0.609373	−	0.792884i	$-0.291421\pi$
$199$	17.1925	1.21875	0.609373	+	0.792884i	$0.291421\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	−8.82295	−0.620780
$203$	42.8093	3.00463
$204$	0	0
$205$	3.06418	0.214011
$206$	7.14796	0.498022
$207$	0	0
$208$	1.30541	0.0905137
$209$	0	0
$210$	0	0
$211$	16.1088	1.10897	0.554486	−	0.832193i	$-0.312915\pi$
$211$	16.1088	1.10897	0.554486	+	0.832193i	$0.312915\pi$
$212$	1.67499	0.115039
$213$	0	0
$214$	−9.36959	−0.640492
$215$	1.51754	0.103495
$216$	0	0
$217$	−1.87164	−0.127056
$218$	−11.0642	−0.749361
$219$	0	0
$220$	−2.82295	−0.190323
$221$	−3.11886	−0.209797
$222$	0	0
$223$	−4.08378	−0.273470	−0.136735	−	0.990608i	$-0.543661\pi$
$223$	−4.08378	−0.273470	−0.136735	+	0.990608i	$0.543661\pi$
$224$	5.06418	0.338365
$225$	0	0
$226$	−13.2986	−0.884610
$227$	13.6604	0.906676	0.453338	−	0.891339i	$-0.350233\pi$
$227$	13.6604	0.906676	0.453338	+	0.891339i	$0.350233\pi$
$228$	0	0
$229$	−5.22163	−0.345055	−0.172527	−	0.985005i	$-0.555193\pi$
$229$	−5.22163	−0.345055	−0.172527	+	0.985005i	$0.555193\pi$
$230$	−6.12836	−0.404092
$231$	0	0
$232$	8.45336	0.554990
$233$	27.0428	1.77163	0.885817	−	0.464035i	$-0.153599\pi$
$233$	27.0428	1.77163	0.885817	+	0.464035i	$0.153599\pi$
$234$	0	0
$235$	−20.4243	−1.33233
$236$	0.716881	0.0466650
$237$	0	0
$238$	−12.0993	−0.784279
$239$	0.285807	0.0184873	0.00924366	−	0.999957i	$-0.497058\pi$
$239$	0.285807	0.0184873	0.00924366	+	0.999957i	$0.497058\pi$
$240$	0	0
$241$	−3.10101	−0.199754	−0.0998769	−	0.995000i	$-0.531845\pi$
$241$	−3.10101	−0.199754	−0.0998769	+	0.995000i	$0.531845\pi$
$242$	−9.00774	−0.579040
$243$	0	0
$244$	9.75877	0.624741
$245$	−37.2918	−2.38249
$246$	0	0
$247$	0	0
$248$	−0.369585	−0.0234687
$249$	0	0
$250$	12.0000	0.758947
$251$	12.6578	0.798950	0.399475	−	0.916744i	$-0.369192\pi$
$251$	12.6578	0.798950	0.399475	+	0.916744i	$0.369192\pi$
$252$	0	0
$253$	4.32501	0.271911
$254$	5.71419	0.358540
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.3928	−1.89585	−0.947926	−	0.318492i	$-0.896824\pi$
$257$	−30.3928	−1.89585	−0.947926	+	0.318492i	$0.896824\pi$
$258$	0	0
$259$	−24.4243	−1.51765
$260$	−2.61081	−0.161916
$261$	0	0
$262$	9.87939	0.610350
$263$	−21.9026	−1.35057	−0.675286	−	0.737556i	$-0.735980\pi$
$263$	−21.9026	−1.35057	−0.675286	+	0.737556i	$0.735980\pi$
$264$	0	0
$265$	−3.34998	−0.205788
$266$	0	0
$267$	0	0
$268$	1.40373	0.0857467
$269$	−1.43376	−0.0874181	−0.0437090	−	0.999044i	$-0.513917\pi$
$269$	−1.43376	−0.0874181	−0.0437090	+	0.999044i	$0.513917\pi$
$270$	0	0
$271$	16.1729	0.982436	0.491218	−	0.871037i	$-0.336552\pi$
$271$	16.1729	0.982436	0.491218	+	0.871037i	$0.336552\pi$
$272$	−2.38919	−0.144866
$273$	0	0
$274$	−5.39693	−0.326040
$275$	−1.41147	−0.0851151
$276$	0	0
$277$	11.1088	0.667460	0.333730	−	0.942669i	$-0.391693\pi$
$277$	11.1088	0.667460	0.333730	+	0.942669i	$0.391693\pi$
$278$	−2.33956	−0.140317
$279$	0	0
$280$	−10.1284	−0.605285
$281$	4.24897	0.253472	0.126736	−	0.991936i	$-0.459550\pi$
$281$	4.24897	0.253472	0.126736	+	0.991936i	$0.459550\pi$
$282$	0	0
$283$	−5.45605	−0.324329	−0.162164	−	0.986764i	$-0.551847\pi$
$283$	−5.45605	−0.324329	−0.162164	+	0.986764i	$0.551847\pi$
$284$	−6.36959	−0.377965
$285$	0	0
$286$	1.84255	0.108952
$287$	−7.75877	−0.457986
$288$	0	0
$289$	−11.2918	−0.664223
$290$	−16.9067	−0.992797
$291$	0	0
$292$	−4.55943	−0.266820
$293$	17.8135	1.04067	0.520337	−	0.853961i	$-0.325807\pi$
$293$	17.8135	1.04067	0.520337	+	0.853961i	$0.325807\pi$
$294$	0	0
$295$	−1.43376	−0.0834769
$296$	−4.82295	−0.280328
$297$	0	0
$298$	14.3696	0.832408
$299$	4.00000	0.231326
$300$	0	0
$301$	−3.84255	−0.221481
$302$	20.8384	1.19912
$303$	0	0
$304$	0	0
$305$	−19.5175	−1.11757
$306$	0	0
$307$	28.6587	1.63564	0.817819	−	0.575476i	$-0.195183\pi$
$307$	28.6587	1.63564	0.817819	+	0.575476i	$0.195183\pi$
$308$	7.14796	0.407293
$309$	0	0
$310$	0.739170	0.0419820
$311$	−15.8135	−0.896699	−0.448349	−	0.893858i	$-0.647988\pi$
$311$	−15.8135	−0.896699	−0.448349	+	0.893858i	$0.647988\pi$
$312$	0	0
$313$	13.1402	0.742729	0.371364	−	0.928487i	$-0.378890\pi$
$313$	13.1402	0.742729	0.371364	+	0.928487i	$0.378890\pi$
$314$	6.36959	0.359456
$315$	0	0
$316$	2.24123	0.126079
$317$	−21.4047	−1.20221	−0.601103	−	0.799172i	$-0.705272\pi$
$317$	−21.4047	−1.20221	−0.601103	+	0.799172i	$0.705272\pi$
$318$	0	0
$319$	11.9317	0.668047
$320$	−2.00000	−0.111803
$321$	0	0
$322$	15.5175	0.864759
$323$	0	0
$324$	0	0
$325$	−1.30541	−0.0724110
$326$	4.73143	0.262050
$327$	0	0
$328$	−1.53209	−0.0845955
$329$	51.7161	2.85120
$330$	0	0
$331$	25.3979	1.39599	0.697996	−	0.716101i	$-0.254075\pi$
$331$	25.3979	1.39599	0.697996	+	0.716101i	$0.254075\pi$
$332$	3.98545	0.218730
$333$	0	0
$334$	17.2763	0.945318
$335$	−2.80747	−0.153388
$336$	0	0
$337$	−26.3773	−1.43686	−0.718432	−	0.695597i	$-0.755140\pi$
$337$	−26.3773	−1.43686	−0.718432	+	0.695597i	$0.755140\pi$
$338$	−11.2959	−0.614417
$339$	0	0
$340$	4.77837	0.259144
$341$	−0.521660	−0.0282495
$342$	0	0
$343$	58.9769	3.18445
$344$	−0.758770	−0.0409102
$345$	0	0
$346$	18.8229	1.01193
$347$	25.6313	1.37596	0.687981	−	0.725728i	$-0.258497\pi$
$347$	25.6313	1.37596	0.687981	+	0.725728i	$0.258497\pi$
$348$	0	0
$349$	−5.84255	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$349$	−5.84255	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$350$	−5.06418	−0.270692
$351$	0	0
$352$	1.41147	0.0752318
$353$	22.4097	1.19275	0.596375	−	0.802706i	$-0.296607\pi$
$353$	22.4097	1.19275	0.596375	+	0.802706i	$0.296607\pi$
$354$	0	0
$355$	12.7392	0.676125
$356$	−10.6459	−0.564231
$357$	0	0
$358$	−13.8280	−0.730833
$359$	−9.61680	−0.507555	−0.253778	−	0.967263i	$-0.581673\pi$
$359$	−9.61680	−0.507555	−0.253778	+	0.967263i	$0.581673\pi$
$360$	0	0
$361$	0	0
$362$	−12.2567	−0.644198
$363$	0	0
$364$	6.61081	0.346501
$365$	9.11886	0.477303
$366$	0	0
$367$	23.3601	1.21939	0.609693	−	0.792637i	$-0.291293\pi$
$367$	23.3601	1.21939	0.609693	+	0.792637i	$0.291293\pi$
$368$	3.06418	0.159731
$369$	0	0
$370$	9.64590	0.501466
$371$	8.48246	0.440387
$372$	0	0
$373$	25.9418	1.34322	0.671608	−	0.740907i	$-0.265604\pi$
$373$	25.9418	1.34322	0.671608	+	0.740907i	$0.265604\pi$
$374$	−3.37227	−0.174376
$375$	0	0
$376$	10.2121	0.526651
$377$	11.0351	0.568336
$378$	0	0
$379$	−9.47834	−0.486870	−0.243435	−	0.969917i	$-0.578274\pi$
$379$	−9.47834	−0.486870	−0.243435	+	0.969917i	$0.578274\pi$
$380$	0	0
$381$	0	0
$382$	20.0993	1.02837
$383$	−13.4884	−0.689227	−0.344614	−	0.938745i	$-0.611990\pi$
$383$	−13.4884	−0.689227	−0.344614	+	0.938745i	$0.611990\pi$
$384$	0	0
$385$	−14.2959	−0.728587
$386$	−16.6851	−0.849249
$387$	0	0
$388$	1.53209	0.0777800
$389$	25.1480	1.27505	0.637526	−	0.770429i	$-0.279958\pi$
$389$	25.1480	1.27505	0.637526	+	0.770429i	$0.279958\pi$
$390$	0	0
$391$	−7.32089	−0.370233
$392$	18.6459	0.941760
$393$	0	0
$394$	−12.9905	−0.654452
$395$	−4.48246	−0.225537
$396$	0	0
$397$	6.56624	0.329550	0.164775	−	0.986331i	$-0.447310\pi$
$397$	6.56624	0.329550	0.164775	+	0.986331i	$0.447310\pi$
$398$	17.1925	0.861784
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−29.5107	−1.47370	−0.736848	−	0.676059i	$-0.763687\pi$
$401$	−29.5107	−1.47370	−0.736848	+	0.676059i	$0.763687\pi$
$402$	0	0
$403$	−0.482459	−0.0240330
$404$	−8.82295	−0.438958
$405$	0	0
$406$	42.8093	2.12459
$407$	−6.80747	−0.337434
$408$	0	0
$409$	−21.1310	−1.04486	−0.522431	−	0.852681i	$-0.674975\pi$
$409$	−21.1310	−1.04486	−0.522431	+	0.852681i	$0.674975\pi$
$410$	3.06418	0.151329
$411$	0	0
$412$	7.14796	0.352155
$413$	3.63041	0.178641
$414$	0	0
$415$	−7.97090	−0.391276
$416$	1.30541	0.0640029
$417$	0	0
$418$	0	0
$419$	27.8830	1.36217	0.681087	−	0.732202i	$-0.261508\pi$
$419$	27.8830	1.36217	0.681087	+	0.732202i	$0.261508\pi$
$420$	0	0
$421$	0.0445774	0.00217257	0.00108629	−	0.999999i	$-0.499654\pi$
$421$	0.0445774	0.00217257	0.00108629	+	0.999999i	$0.499654\pi$
$422$	16.1088	0.784162
$423$	0	0
$424$	1.67499	0.0813448
$425$	2.38919	0.115893
$426$	0	0
$427$	49.4201	2.39161
$428$	−9.36959	−0.452896
$429$	0	0
$430$	1.51754	0.0731823
$431$	−11.7879	−0.567802	−0.283901	−	0.958854i	$-0.591629\pi$
$431$	−11.7879	−0.567802	−0.283901	+	0.958854i	$0.591629\pi$
$432$	0	0
$433$	−27.6459	−1.32858	−0.664289	−	0.747476i	$-0.731265\pi$
$433$	−27.6459	−1.32858	−0.664289	+	0.747476i	$0.731265\pi$
$434$	−1.87164	−0.0898418
$435$	0	0
$436$	−11.0642	−0.529878
$437$	0	0
$438$	0	0
$439$	−37.4492	−1.78735	−0.893677	−	0.448710i	$-0.851884\pi$
$439$	−37.4492	−1.78735	−0.893677	+	0.448710i	$0.851884\pi$
$440$	−2.82295	−0.134579
$441$	0	0
$442$	−3.11886	−0.148349
$443$	−20.9881	−0.997177	−0.498588	−	0.866839i	$-0.666148\pi$
$443$	−20.9881	−0.997177	−0.498588	+	0.866839i	$0.666148\pi$
$444$	0	0
$445$	21.2918	1.00933
$446$	−4.08378	−0.193372
$447$	0	0
$448$	5.06418	0.239260
$449$	−21.8949	−1.03328	−0.516641	−	0.856202i	$-0.672818\pi$
$449$	−21.8949	−1.03328	−0.516641	+	0.856202i	$0.672818\pi$
$450$	0	0
$451$	−2.16250	−0.101828
$452$	−13.2986	−0.625514
$453$	0	0
$454$	13.6604	0.641116
$455$	−13.2216	−0.619840
$456$	0	0
$457$	−13.0496	−0.610436	−0.305218	−	0.952283i	$-0.598729\pi$
$457$	−13.0496	−0.610436	−0.305218	+	0.952283i	$0.598729\pi$
$458$	−5.22163	−0.243991
$459$	0	0
$460$	−6.12836	−0.285736
$461$	−4.40879	−0.205338	−0.102669	−	0.994716i	$-0.532738\pi$
$461$	−4.40879	−0.205338	−0.102669	+	0.994716i	$0.532738\pi$
$462$	0	0
$463$	26.6655	1.23925	0.619625	−	0.784898i	$-0.287285\pi$
$463$	26.6655	1.23925	0.619625	+	0.784898i	$0.287285\pi$
$464$	8.45336	0.392438
$465$	0	0
$466$	27.0428	1.25273
$467$	−30.1138	−1.39350	−0.696750	−	0.717314i	$-0.745371\pi$
$467$	−30.1138	−1.39350	−0.696750	+	0.717314i	$0.745371\pi$
$468$	0	0
$469$	7.10876	0.328252
$470$	−20.4243	−0.942101
$471$	0	0
$472$	0.716881	0.0329971
$473$	−1.07098	−0.0492439
$474$	0	0
$475$	0	0
$476$	−12.0993	−0.554569
$477$	0	0
$478$	0.285807	0.0130725
$479$	−8.16756	−0.373185	−0.186593	−	0.982437i	$-0.559744\pi$
$479$	−8.16756	−0.373185	−0.186593	+	0.982437i	$0.559744\pi$
$480$	0	0
$481$	−6.29591	−0.287069
$482$	−3.10101	−0.141247
$483$	0	0
$484$	−9.00774	−0.409443
$485$	−3.06418	−0.139137
$486$	0	0
$487$	−26.5871	−1.20478	−0.602388	−	0.798203i	$-0.705784\pi$
$487$	−26.5871	−1.20478	−0.602388	+	0.798203i	$0.705784\pi$
$488$	9.75877	0.441759
$489$	0	0
$490$	−37.2918	−1.68467
$491$	−38.8289	−1.75233	−0.876163	−	0.482016i	$-0.839905\pi$
$491$	−38.8289	−1.75233	−0.876163	+	0.482016i	$0.839905\pi$
$492$	0	0
$493$	−20.1967	−0.909611
$494$	0	0
$495$	0	0
$496$	−0.369585	−0.0165949
$497$	−32.2567	−1.44691
$498$	0	0
$499$	20.6587	0.924810	0.462405	−	0.886669i	$-0.346987\pi$
$499$	20.6587	0.924810	0.462405	+	0.886669i	$0.346987\pi$
$500$	12.0000	0.536656
$501$	0	0
$502$	12.6578	0.564943
$503$	−0.0736733	−0.00328493	−0.00164246	−	0.999999i	$-0.500523\pi$
$503$	−0.0736733	−0.00328493	−0.00164246	+	0.999999i	$0.500523\pi$
$504$	0	0
$505$	17.6459	0.785232
$506$	4.32501	0.192270
$507$	0	0
$508$	5.71419	0.253526
$509$	15.0196	0.665732	0.332866	−	0.942974i	$-0.391984\pi$
$509$	15.0196	0.665732	0.332866	+	0.942974i	$0.391984\pi$
$510$	0	0
$511$	−23.0898	−1.02143
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.3928	−1.34057
$515$	−14.2959	−0.629953
$516$	0	0
$517$	14.4142	0.633934
$518$	−24.4243	−1.07314
$519$	0	0
$520$	−2.61081	−0.114492
$521$	45.1712	1.97899	0.989493	−	0.144583i	$-0.0461842\pi$
$521$	45.1712	1.97899	0.989493	+	0.144583i	$0.0461842\pi$
$522$	0	0
$523$	−0.167556	−0.00732672	−0.00366336	−	0.999993i	$-0.501166\pi$
$523$	−0.167556	−0.00732672	−0.00366336	+	0.999993i	$0.501166\pi$
$524$	9.87939	0.431583
$525$	0	0
$526$	−21.9026	−0.954999
$527$	0.883007	0.0384644
$528$	0	0
$529$	−13.6108	−0.591775
$530$	−3.34998	−0.145514
$531$	0	0
$532$	0	0
$533$	−2.00000	−0.0866296
$534$	0	0
$535$	18.7392	0.810165
$536$	1.40373	0.0606320
$537$	0	0
$538$	−1.43376	−0.0618139
$539$	26.3182	1.13361
$540$	0	0
$541$	−0.980400	−0.0421507	−0.0210753	−	0.999778i	$-0.506709\pi$
$541$	−0.980400	−0.0421507	−0.0210753	+	0.999778i	$0.506709\pi$
$542$	16.1729	0.694687
$543$	0	0
$544$	−2.38919	−0.102435
$545$	22.1284	0.947875
$546$	0	0
$547$	−28.4047	−1.21450	−0.607248	−	0.794512i	$-0.707727\pi$
$547$	−28.4047	−1.21450	−0.607248	+	0.794512i	$0.707727\pi$
$548$	−5.39693	−0.230545
$549$	0	0
$550$	−1.41147	−0.0601855
$551$	0	0
$552$	0	0
$553$	11.3500	0.482650
$554$	11.1088	0.471966
$555$	0	0
$556$	−2.33956	−0.0992193
$557$	−35.6323	−1.50979	−0.754894	−	0.655847i	$-0.772312\pi$
$557$	−35.6323	−1.50979	−0.754894	+	0.655847i	$0.772312\pi$
$558$	0	0
$559$	−0.990505	−0.0418939
$560$	−10.1284	−0.428001
$561$	0	0
$562$	4.24897	0.179232
$563$	8.62773	0.363615	0.181808	−	0.983334i	$-0.441805\pi$
$563$	8.62773	0.363615	0.181808	+	0.983334i	$0.441805\pi$
$564$	0	0
$565$	26.5972	1.11895
$566$	−5.45605	−0.229335
$567$	0	0
$568$	−6.36959	−0.267262
$569$	22.3310	0.936164	0.468082	−	0.883685i	$-0.344945\pi$
$569$	22.3310	0.936164	0.468082	+	0.883685i	$0.344945\pi$
$570$	0	0
$571$	−9.56448	−0.400261	−0.200131	−	0.979769i	$-0.564137\pi$
$571$	−9.56448	−0.400261	−0.200131	+	0.979769i	$0.564137\pi$
$572$	1.84255	0.0770408
$573$	0	0
$574$	−7.75877	−0.323845
$575$	−3.06418	−0.127785
$576$	0	0
$577$	22.4757	0.935674	0.467837	−	0.883815i	$-0.345033\pi$
$577$	22.4757	0.935674	0.467837	+	0.883815i	$0.345033\pi$
$578$	−11.2918	−0.469677
$579$	0	0
$580$	−16.9067	−0.702014
$581$	20.1830	0.837334
$582$	0	0
$583$	2.36421	0.0979155
$584$	−4.55943	−0.188671
$585$	0	0
$586$	17.8135	0.735867
$587$	−4.14796	−0.171204	−0.0856022	−	0.996329i	$-0.527281\pi$
$587$	−4.14796	−0.171204	−0.0856022	+	0.996329i	$0.527281\pi$
$588$	0	0
$589$	0	0
$590$	−1.43376	−0.0590271
$591$	0	0
$592$	−4.82295	−0.198222
$593$	−10.4175	−0.427794	−0.213897	−	0.976856i	$-0.568616\pi$
$593$	−10.4175	−0.427794	−0.213897	+	0.976856i	$0.568616\pi$
$594$	0	0
$595$	24.1985	0.992043
$596$	14.3696	0.588601
$597$	0	0
$598$	4.00000	0.163572
$599$	−39.6560	−1.62030	−0.810150	−	0.586222i	$-0.800614\pi$
$599$	−39.6560	−1.62030	−0.810150	+	0.586222i	$0.800614\pi$
$600$	0	0
$601$	−23.8648	−0.973467	−0.486734	−	0.873551i	$-0.661812\pi$
$601$	−23.8648	−0.973467	−0.486734	+	0.873551i	$0.661812\pi$
$602$	−3.84255	−0.156611
$603$	0	0
$604$	20.8384	0.847904
$605$	18.0155	0.732433
$606$	0	0
$607$	29.9317	1.21489	0.607445	−	0.794362i	$-0.292194\pi$
$607$	29.9317	1.21489	0.607445	+	0.794362i	$0.292194\pi$
$608$	0	0
$609$	0	0
$610$	−19.5175	−0.790242
$611$	13.3310	0.539314
$612$	0	0
$613$	−35.5776	−1.43697	−0.718483	−	0.695545i	$-0.755163\pi$
$613$	−35.5776	−1.43697	−0.718483	+	0.695545i	$0.755163\pi$
$614$	28.6587	1.15657
$615$	0	0
$616$	7.14796	0.287999
$617$	−12.0324	−0.484406	−0.242203	−	0.970226i	$-0.577870\pi$
$617$	−12.0324	−0.484406	−0.242203	+	0.970226i	$0.577870\pi$
$618$	0	0
$619$	−17.1129	−0.687824	−0.343912	−	0.939002i	$-0.611752\pi$
$619$	−17.1129	−0.687824	−0.343912	+	0.939002i	$0.611752\pi$
$620$	0.739170	0.0296858
$621$	0	0
$622$	−15.8135	−0.634062
$623$	−53.9127	−2.15997
$624$	0	0
$625$	−19.0000	−0.760000
$626$	13.1402	0.525189
$627$	0	0
$628$	6.36959	0.254174
$629$	11.5229	0.459449
$630$	0	0
$631$	18.3405	0.730123	0.365062	−	0.930983i	$-0.381048\pi$
$631$	18.3405	0.730123	0.365062	+	0.930983i	$0.381048\pi$
$632$	2.24123	0.0891513
$633$	0	0
$634$	−21.4047	−0.850088
$635$	−11.4284	−0.453522
$636$	0	0
$637$	24.3405	0.964405
$638$	11.9317	0.472381
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−31.1780	−1.23146	−0.615728	−	0.787959i	$-0.711138\pi$
$641$	−31.1780	−1.23146	−0.615728	+	0.787959i	$0.711138\pi$
$642$	0	0
$643$	−31.9495	−1.25997	−0.629984	−	0.776608i	$-0.716938\pi$
$643$	−31.9495	−1.25997	−0.629984	+	0.776608i	$0.716938\pi$
$644$	15.5175	0.611477
$645$	0	0
$646$	0	0
$647$	−2.99588	−0.117780	−0.0588901	−	0.998264i	$-0.518756\pi$
$647$	−2.99588	−0.117780	−0.0588901	+	0.998264i	$0.518756\pi$
$648$	0	0
$649$	1.01186	0.0397190
$650$	−1.30541	−0.0512023
$651$	0	0
$652$	4.73143	0.185297
$653$	−0.935822	−0.0366216	−0.0183108	−	0.999832i	$-0.505829\pi$
$653$	−0.935822	−0.0366216	−0.0183108	+	0.999832i	$0.505829\pi$
$654$	0	0
$655$	−19.7588	−0.772039
$656$	−1.53209	−0.0598180
$657$	0	0
$658$	51.7161	2.01610
$659$	−12.2371	−0.476690	−0.238345	−	0.971181i	$-0.576605\pi$
$659$	−12.2371	−0.476690	−0.238345	+	0.971181i	$0.576605\pi$
$660$	0	0
$661$	11.9554	0.465012	0.232506	−	0.972595i	$-0.425307\pi$
$661$	11.9554	0.465012	0.232506	+	0.972595i	$0.425307\pi$
$662$	25.3979	0.987116
$663$	0	0
$664$	3.98545	0.154666
$665$	0	0
$666$	0	0
$667$	25.9026	1.00295
$668$	17.2763	0.668441
$669$	0	0
$670$	−2.80747	−0.108462
$671$	13.7743	0.531749
$672$	0	0
$673$	−44.8634	−1.72936	−0.864679	−	0.502325i	$-0.832478\pi$
$673$	−44.8634	−1.72936	−0.864679	+	0.502325i	$0.832478\pi$
$674$	−26.3773	−1.01602
$675$	0	0
$676$	−11.2959	−0.434458
$677$	0.945927	0.0363549	0.0181775	−	0.999835i	$-0.494214\pi$
$677$	0.945927	0.0363549	0.0181775	+	0.999835i	$0.494214\pi$
$678$	0	0
$679$	7.75877	0.297754
$680$	4.77837	0.183242
$681$	0	0
$682$	−0.521660	−0.0199754
$683$	−5.92221	−0.226607	−0.113303	−	0.993560i	$-0.536143\pi$
$683$	−5.92221	−0.226607	−0.113303	+	0.993560i	$0.536143\pi$
$684$	0	0
$685$	10.7939	0.412412
$686$	58.9769	2.25175
$687$	0	0
$688$	−0.758770	−0.0289279
$689$	2.18655	0.0833008
$690$	0	0
$691$	0.206148	0.00784222	0.00392111	−	0.999992i	$-0.498752\pi$
$691$	0.206148	0.00784222	0.00392111	+	0.999992i	$0.498752\pi$
$692$	18.8229	0.715541
$693$	0	0
$694$	25.6313	0.972953
$695$	4.67911	0.177489
$696$	0	0
$697$	3.66044	0.138649
$698$	−5.84255	−0.221144
$699$	0	0
$700$	−5.06418	−0.191408
$701$	−4.36959	−0.165037	−0.0825185	−	0.996590i	$-0.526296\pi$
$701$	−4.36959	−0.165037	−0.0825185	+	0.996590i	$0.526296\pi$
$702$	0	0
$703$	0	0
$704$	1.41147	0.0531969
$705$	0	0
$706$	22.4097	0.843401
$707$	−44.6810	−1.68040
$708$	0	0
$709$	8.75465	0.328788	0.164394	−	0.986395i	$-0.447433\pi$
$709$	8.75465	0.328788	0.164394	+	0.986395i	$0.447433\pi$
$710$	12.7392	0.478093
$711$	0	0
$712$	−10.6459	−0.398972
$713$	−1.13247	−0.0424115
$714$	0	0
$715$	−3.68510	−0.137815
$716$	−13.8280	−0.516777
$717$	0	0
$718$	−9.61680	−0.358896
$719$	27.6067	1.02956	0.514778	−	0.857324i	$-0.327874\pi$
$719$	27.6067	1.02956	0.514778	+	0.857324i	$0.327874\pi$
$720$	0	0
$721$	36.1985	1.34810
$722$	0	0
$723$	0	0
$724$	−12.2567	−0.455517
$725$	−8.45336	−0.313950
$726$	0	0
$727$	28.6364	1.06207	0.531033	−	0.847351i	$-0.321804\pi$
$727$	28.6364	1.06207	0.531033	+	0.847351i	$0.321804\pi$
$728$	6.61081	0.245013
$729$	0	0
$730$	9.11886	0.337504
$731$	1.81284	0.0670504
$732$	0	0
$733$	36.9614	1.36520	0.682600	−	0.730792i	$-0.260849\pi$
$733$	36.9614	1.36520	0.682600	+	0.730792i	$0.260849\pi$
$734$	23.3601	0.862237
$735$	0	0
$736$	3.06418	0.112947
$737$	1.98133	0.0729833
$738$	0	0
$739$	46.0265	1.69311	0.846556	−	0.532299i	$-0.178672\pi$
$739$	46.0265	1.69311	0.846556	+	0.532299i	$0.178672\pi$
$740$	9.64590	0.354590
$741$	0	0
$742$	8.48246	0.311401
$743$	26.4107	0.968913	0.484456	−	0.874815i	$-0.339017\pi$
$743$	26.4107	0.968913	0.484456	+	0.874815i	$0.339017\pi$
$744$	0	0
$745$	−28.7392	−1.05292
$746$	25.9418	0.949797
$747$	0	0
$748$	−3.37227	−0.123303
$749$	−47.4492	−1.73376
$750$	0	0
$751$	27.1771	0.991705	0.495852	−	0.868407i	$-0.334856\pi$
$751$	27.1771	0.991705	0.495852	+	0.868407i	$0.334856\pi$
$752$	10.2121	0.372398
$753$	0	0
$754$	11.0351	0.401874
$755$	−41.6769	−1.51678
$756$	0	0
$757$	19.4047	0.705275	0.352637	−	0.935760i	$-0.385285\pi$
$757$	19.4047	0.705275	0.352637	+	0.935760i	$0.385285\pi$
$758$	−9.47834	−0.344269
$759$	0	0
$760$	0	0
$761$	−40.2645	−1.45959	−0.729793	−	0.683669i	$-0.760383\pi$
$761$	−40.2645	−1.45959	−0.729793	+	0.683669i	$0.760383\pi$
$762$	0	0
$763$	−56.0310	−2.02846
$764$	20.0993	0.727166
$765$	0	0
$766$	−13.4884	−0.487357
$767$	0.935822	0.0337906
$768$	0	0
$769$	−38.9418	−1.40428	−0.702139	−	0.712040i	$-0.747771\pi$
$769$	−38.9418	−1.40428	−0.702139	+	0.712040i	$0.747771\pi$
$770$	−14.2959	−0.515189
$771$	0	0
$772$	−16.6851	−0.600510
$773$	−6.76289	−0.243244	−0.121622	−	0.992576i	$-0.538810\pi$
$773$	−6.76289	−0.243244	−0.121622	+	0.992576i	$0.538810\pi$
$774$	0	0
$775$	0.369585	0.0132759
$776$	1.53209	0.0549988
$777$	0	0
$778$	25.1480	0.901598
$779$	0	0
$780$	0	0
$781$	−8.99050	−0.321706
$782$	−7.32089	−0.261794
$783$	0	0
$784$	18.6459	0.665925
$785$	−12.7392	−0.454680
$786$	0	0
$787$	−5.80478	−0.206918	−0.103459	−	0.994634i	$-0.532991\pi$
$787$	−5.80478	−0.206918	−0.103459	+	0.994634i	$0.532991\pi$
$788$	−12.9905	−0.462768
$789$	0	0
$790$	−4.48246	−0.159479
$791$	−67.3465	−2.39456
$792$	0	0
$793$	12.7392	0.452381
$794$	6.56624	0.233027
$795$	0	0
$796$	17.1925	0.609373
$797$	−39.2181	−1.38918	−0.694589	−	0.719407i	$-0.744414\pi$
$797$	−39.2181	−1.38918	−0.694589	+	0.719407i	$0.744414\pi$
$798$	0	0
$799$	−24.3987	−0.863163
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−29.5107	−1.04206
$803$	−6.43552	−0.227104
$804$	0	0
$805$	−31.0351	−1.09384
$806$	−0.482459	−0.0169939
$807$	0	0
$808$	−8.82295	−0.310390
$809$	−7.00269	−0.246201	−0.123101	−	0.992394i	$-0.539284\pi$
$809$	−7.00269	−0.246201	−0.123101	+	0.992394i	$0.539284\pi$
$810$	0	0
$811$	−43.6851	−1.53399	−0.766996	−	0.641652i	$-0.778249\pi$
$811$	−43.6851	−1.53399	−0.766996	+	0.641652i	$0.778249\pi$
$812$	42.8093	1.50231
$813$	0	0
$814$	−6.80747	−0.238602
$815$	−9.46286	−0.331469
$816$	0	0
$817$	0	0
$818$	−21.1310	−0.738830
$819$	0	0
$820$	3.06418	0.107006
$821$	−12.3851	−0.432242	−0.216121	−	0.976367i	$-0.569341\pi$
$821$	−12.3851	−0.432242	−0.216121	+	0.976367i	$0.569341\pi$
$822$	0	0
$823$	−12.8283	−0.447167	−0.223584	−	0.974685i	$-0.571776\pi$
$823$	−12.8283	−0.447167	−0.223584	+	0.974685i	$0.571776\pi$
$824$	7.14796	0.249011
$825$	0	0
$826$	3.63041	0.126318
$827$	25.0966	0.872693	0.436347	−	0.899779i	$-0.356272\pi$
$827$	25.0966	0.872693	0.436347	+	0.899779i	$0.356272\pi$
$828$	0	0
$829$	−28.3269	−0.983833	−0.491917	−	0.870642i	$-0.663703\pi$
$829$	−28.3269	−0.983833	−0.491917	+	0.870642i	$0.663703\pi$
$830$	−7.97090	−0.276674
$831$	0	0
$832$	1.30541	0.0452569
$833$	−44.5485	−1.54351
$834$	0	0
$835$	−34.5526	−1.19574
$836$	0	0
$837$	0	0
$838$	27.8830	0.963203
$839$	−30.3304	−1.04712	−0.523561	−	0.851988i	$-0.675397\pi$
$839$	−30.3304	−1.04712	−0.523561	+	0.851988i	$0.675397\pi$
$840$	0	0
$841$	42.4593	1.46412
$842$	0.0445774	0.00153624
$843$	0	0
$844$	16.1088	0.554486
$845$	22.5918	0.777182
$846$	0	0
$847$	−45.6168	−1.56741
$848$	1.67499	0.0575195
$849$	0	0
$850$	2.38919	0.0819484
$851$	−14.7784	−0.506596
$852$	0	0
$853$	−43.6323	−1.49394	−0.746970	−	0.664857i	$-0.768492\pi$
$853$	−43.6323	−1.49394	−0.746970	+	0.664857i	$0.768492\pi$
$854$	49.4201	1.69112
$855$	0	0
$856$	−9.36959	−0.320246
$857$	−31.5098	−1.07635	−0.538177	−	0.842832i	$-0.680887\pi$
$857$	−31.5098	−1.07635	−0.538177	+	0.842832i	$0.680887\pi$
$858$	0	0
$859$	−18.6979	−0.637964	−0.318982	−	0.947761i	$-0.603341\pi$
$859$	−18.6979	−0.637964	−0.318982	+	0.947761i	$0.603341\pi$
$860$	1.51754	0.0517477
$861$	0	0
$862$	−11.7879	−0.401496
$863$	10.6263	0.361723	0.180862	−	0.983509i	$-0.442111\pi$
$863$	10.6263	0.361723	0.180862	+	0.983509i	$0.442111\pi$
$864$	0	0
$865$	−37.6459	−1.28000
$866$	−27.6459	−0.939446
$867$	0	0
$868$	−1.87164	−0.0635278
$869$	3.16344	0.107312
$870$	0	0
$871$	1.83244	0.0620900
$872$	−11.0642	−0.374680
$873$	0	0
$874$	0	0
$875$	60.7701	2.05441
$876$	0	0
$877$	−44.8985	−1.51611	−0.758057	−	0.652188i	$-0.773851\pi$
$877$	−44.8985	−1.51611	−0.758057	+	0.652188i	$0.773851\pi$
$878$	−37.4492	−1.26385
$879$	0	0
$880$	−2.82295	−0.0951616
$881$	18.0164	0.606988	0.303494	−	0.952833i	$-0.401847\pi$
$881$	18.0164	0.606988	0.303494	+	0.952833i	$0.401847\pi$
$882$	0	0
$883$	−46.5981	−1.56815	−0.784076	−	0.620665i	$-0.786863\pi$
$883$	−46.5981	−1.56815	−0.784076	+	0.620665i	$0.786863\pi$
$884$	−3.11886	−0.104899
$885$	0	0
$886$	−20.9881	−0.705110
$887$	7.07966	0.237712	0.118856	−	0.992912i	$-0.462077\pi$
$887$	7.07966	0.237712	0.118856	+	0.992912i	$0.462077\pi$
$888$	0	0
$889$	28.9377	0.970539
$890$	21.2918	0.713703
$891$	0	0
$892$	−4.08378	−0.136735
$893$	0	0
$894$	0	0
$895$	27.6560	0.924438
$896$	5.06418	0.169182
$897$	0	0
$898$	−21.8949	−0.730641
$899$	−3.12424	−0.104199
$900$	0	0
$901$	−4.00187	−0.133322
$902$	−2.16250	−0.0720035
$903$	0	0
$904$	−13.2986	−0.442305
$905$	24.5134	0.814854
$906$	0	0
$907$	−15.1156	−0.501904	−0.250952	−	0.968000i	$-0.580744\pi$
$907$	−15.1156	−0.501904	−0.250952	+	0.968000i	$0.580744\pi$
$908$	13.6604	0.453338
$909$	0	0
$910$	−13.2216	−0.438293
$911$	−12.8366	−0.425294	−0.212647	−	0.977129i	$-0.568208\pi$
$911$	−12.8366	−0.425294	−0.212647	+	0.977129i	$0.568208\pi$
$912$	0	0
$913$	5.62536	0.186172
$914$	−13.0496	−0.431643
$915$	0	0
$916$	−5.22163	−0.172527
$917$	50.0310	1.65217
$918$	0	0
$919$	−20.6791	−0.682141	−0.341070	−	0.940038i	$-0.610789\pi$
$919$	−20.6791	−0.682141	−0.341070	+	0.940038i	$0.610789\pi$
$920$	−6.12836	−0.202046
$921$	0	0
$922$	−4.40879	−0.145196
$923$	−8.31490	−0.273688
$924$	0	0
$925$	4.82295	0.158578
$926$	26.6655	0.876283
$927$	0	0
$928$	8.45336	0.277495
$929$	18.2499	0.598760	0.299380	−	0.954134i	$-0.403220\pi$
$929$	18.2499	0.598760	0.299380	+	0.954134i	$0.403220\pi$
$930$	0	0
$931$	0	0
$932$	27.0428	0.885817
$933$	0	0
$934$	−30.1138	−0.985354
$935$	6.74455	0.220570
$936$	0	0
$937$	−4.74691	−0.155075	−0.0775374	−	0.996989i	$-0.524706\pi$
$937$	−4.74691	−0.155075	−0.0775374	+	0.996989i	$0.524706\pi$
$938$	7.10876	0.232109
$939$	0	0
$940$	−20.4243	−0.666166
$941$	47.0215	1.53286	0.766428	−	0.642330i	$-0.222032\pi$
$941$	47.0215	1.53286	0.766428	+	0.642330i	$0.222032\pi$
$942$	0	0
$943$	−4.69459	−0.152877
$944$	0.716881	0.0233325
$945$	0	0
$946$	−1.07098	−0.0348207
$947$	−36.9959	−1.20220	−0.601102	−	0.799172i	$-0.705272\pi$
$947$	−36.9959	−1.20220	−0.601102	+	0.799172i	$0.705272\pi$
$948$	0	0
$949$	−5.95191	−0.193207
$950$	0	0
$951$	0	0
$952$	−12.0993	−0.392139
$953$	−37.7093	−1.22152	−0.610761	−	0.791815i	$-0.709136\pi$
$953$	−37.7093	−1.22152	−0.610761	+	0.791815i	$0.709136\pi$
$954$	0	0
$955$	−40.1985	−1.30079
$956$	0.285807	0.00924366
$957$	0	0
$958$	−8.16756	−0.263882
$959$	−27.3310	−0.882564
$960$	0	0
$961$	−30.8634	−0.995594
$962$	−6.29591	−0.202988
$963$	0	0
$964$	−3.10101	−0.0998769
$965$	33.3702	1.07422
$966$	0	0
$967$	40.5134	1.30282	0.651412	−	0.758724i	$-0.274177\pi$
$967$	40.5134	1.30282	0.651412	+	0.758724i	$0.274177\pi$
$968$	−9.00774	−0.289520
$969$	0	0
$970$	−3.06418	−0.0983848
$971$	−14.0779	−0.451781	−0.225891	−	0.974153i	$-0.572529\pi$
$971$	−14.0779	−0.451781	−0.225891	+	0.974153i	$0.572529\pi$
$972$	0	0
$973$	−11.8479	−0.379827
$974$	−26.5871	−0.851905
$975$	0	0
$976$	9.75877	0.312371
$977$	−35.2057	−1.12633	−0.563164	−	0.826345i	$-0.690416\pi$
$977$	−35.2057	−1.12633	−0.563164	+	0.826345i	$0.690416\pi$
$978$	0	0
$979$	−15.0264	−0.480246
$980$	−37.2918	−1.19124
$981$	0	0
$982$	−38.8289	−1.23908
$983$	22.4397	0.715717	0.357858	−	0.933776i	$-0.383507\pi$
$983$	22.4397	0.715717	0.357858	+	0.933776i	$0.383507\pi$
$984$	0	0
$985$	25.9810	0.827824
$986$	−20.1967	−0.643192
$987$	0	0
$988$	0	0
$989$	−2.32501	−0.0739309
$990$	0	0
$991$	−27.1034	−0.860967	−0.430484	−	0.902598i	$-0.641657\pi$
$991$	−27.1034	−0.860967	−0.430484	+	0.902598i	$0.641657\pi$
$992$	−0.369585	−0.0117343
$993$	0	0
$994$	−32.2567	−1.02312
$995$	−34.3851	−1.09008
$996$	0	0
$997$	27.9472	0.885096	0.442548	−	0.896745i	$-0.354075\pi$
$997$	27.9472	0.885096	0.442548	+	0.896745i	$0.354075\pi$
$998$	20.6587	0.653939
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bq.1.3		3
3.2	odd	2		722.2.a.k.1.3		3
12.11	even	2		5776.2.a.bo.1.1		3
19.2	odd	18		342.2.u.c.289.1		6
19.10	odd	18		342.2.u.c.271.1		6
19.18	odd	2		6498.2.a.bl.1.3		3
57.2	even	18		38.2.e.a.23.1	yes	6
57.5	odd	18		722.2.e.a.595.1		6
57.8	even	6		722.2.c.k.653.3		6
57.11	odd	6		722.2.c.l.653.1		6
57.14	even	18		722.2.e.m.595.1		6
57.17	odd	18		722.2.e.k.99.1		6
57.23	odd	18		722.2.e.a.415.1		6
57.26	odd	6		722.2.c.l.429.1		6
57.29	even	18		38.2.e.a.5.1	✓	6
57.32	even	18		722.2.e.b.245.1		6
57.35	odd	18		722.2.e.l.389.1		6
57.41	even	18		722.2.e.b.389.1		6
57.44	odd	18		722.2.e.l.245.1		6
57.47	odd	18		722.2.e.k.423.1		6
57.50	even	6		722.2.c.k.429.3		6
57.53	even	18		722.2.e.m.415.1		6
57.56	even	2		722.2.a.l.1.1		3
228.59	odd	18		304.2.u.c.289.1		6
228.143	odd	18		304.2.u.c.81.1		6
228.227	odd	2		5776.2.a.bn.1.3		3
285.2	odd	36		950.2.u.b.99.2		12
285.29	even	18		950.2.l.d.651.1		6
285.59	even	18		950.2.l.d.251.1		6
285.143	odd	36		950.2.u.b.499.2		12
285.173	odd	36		950.2.u.b.99.1		12
285.257	odd	36		950.2.u.b.499.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.2.e.a.5.1	✓	6	57.29	even	18
38.2.e.a.23.1	yes	6	57.2	even	18
304.2.u.c.81.1		6	228.143	odd	18
304.2.u.c.289.1		6	228.59	odd	18
342.2.u.c.271.1		6	19.10	odd	18
342.2.u.c.289.1		6	19.2	odd	18
722.2.a.k.1.3		3	3.2	odd	2
722.2.a.l.1.1		3	57.56	even	2
722.2.c.k.429.3		6	57.50	even	6
722.2.c.k.653.3		6	57.8	even	6
722.2.c.l.429.1		6	57.26	odd	6
722.2.c.l.653.1		6	57.11	odd	6
722.2.e.a.415.1		6	57.23	odd	18
722.2.e.a.595.1		6	57.5	odd	18
722.2.e.b.245.1		6	57.32	even	18
722.2.e.b.389.1		6	57.41	even	18
722.2.e.k.99.1		6	57.17	odd	18
722.2.e.k.423.1		6	57.47	odd	18
722.2.e.l.245.1		6	57.44	odd	18
722.2.e.l.389.1		6	57.35	odd	18
722.2.e.m.415.1		6	57.53	even	18
722.2.e.m.595.1		6	57.14	even	18
950.2.l.d.251.1		6	285.59	even	18
950.2.l.d.651.1		6	285.29	even	18
950.2.u.b.99.1		12	285.173	odd	36
950.2.u.b.99.2		12	285.2	odd	36
950.2.u.b.499.1		12	285.257	odd	36
950.2.u.b.499.2		12	285.143	odd	36
5776.2.a.bn.1.3		3	228.227	odd	2
5776.2.a.bo.1.1		3	12.11	even	2
6498.2.a.bl.1.3		3	19.18	odd	2
6498.2.a.bq.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +5.06418 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.00000 q^{5} +5.06418 q^{7} +1.00000 q^{8} -2.00000 q^{10} +1.41147 q^{11} +1.30541 q^{13} +5.06418 q^{14} +1.00000 q^{16} -2.38919 q^{17} -2.00000 q^{20} +1.41147 q^{22} +3.06418 q^{23} -1.00000 q^{25} +1.30541 q^{26} +5.06418 q^{28} +8.45336 q^{29} -0.369585 q^{31} +1.00000 q^{32} -2.38919 q^{34} -10.1284 q^{35} -4.82295 q^{37} -2.00000 q^{40} -1.53209 q^{41} -0.758770 q^{43} +1.41147 q^{44} +3.06418 q^{46} +10.2121 q^{47} +18.6459 q^{49} -1.00000 q^{50} +1.30541 q^{52} +1.67499 q^{53} -2.82295 q^{55} +5.06418 q^{56} +8.45336 q^{58} +0.716881 q^{59} +9.75877 q^{61} -0.369585 q^{62} +1.00000 q^{64} -2.61081 q^{65} +1.40373 q^{67} -2.38919 q^{68} -10.1284 q^{70} -6.36959 q^{71} -4.55943 q^{73} -4.82295 q^{74} +7.14796 q^{77} +2.24123 q^{79} -2.00000 q^{80} -1.53209 q^{82} +3.98545 q^{83} +4.77837 q^{85} -0.758770 q^{86} +1.41147 q^{88} -10.6459 q^{89} +6.61081 q^{91} +3.06418 q^{92} +10.2121 q^{94} +1.53209 q^{97} +18.6459 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 6 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 6 q^{5} + 6 q^{7} + 3 q^{8} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 6 q^{14} + 3 q^{16} - 3 q^{17} - 6 q^{20} - 6 q^{22} - 3 q^{25} + 6 q^{26} + 6 q^{28} + 12 q^{29} + 6 q^{31} + 3 q^{32} - 3 q^{34} - 12 q^{35} + 6 q^{37} - 6 q^{40} + 9 q^{43} - 6 q^{44} + 6 q^{47} + 15 q^{49} - 3 q^{50} + 6 q^{52} + 12 q^{55} + 6 q^{56} + 12 q^{58} - 6 q^{59} + 18 q^{61} + 6 q^{62} + 3 q^{64} - 12 q^{65} + 18 q^{67} - 3 q^{68} - 12 q^{70} - 12 q^{71} + 12 q^{73} + 6 q^{74} + 6 q^{77} + 18 q^{79} - 6 q^{80} - 6 q^{83} + 6 q^{85} + 9 q^{86} - 6 q^{88} + 9 q^{89} + 24 q^{91} + 6 q^{94} + 15 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 6 * q^5 + 6 * q^7 + 3 * q^8 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 6 * q^14 + 3 * q^16 - 3 * q^17 - 6 * q^20 - 6 * q^22 - 3 * q^25 + 6 * q^26 + 6 * q^28 + 12 * q^29 + 6 * q^31 + 3 * q^32 - 3 * q^34 - 12 * q^35 + 6 * q^37 - 6 * q^40 + 9 * q^43 - 6 * q^44 + 6 * q^47 + 15 * q^49 - 3 * q^50 + 6 * q^52 + 12 * q^55 + 6 * q^56 + 12 * q^58 - 6 * q^59 + 18 * q^61 + 6 * q^62 + 3 * q^64 - 12 * q^65 + 18 * q^67 - 3 * q^68 - 12 * q^70 - 12 * q^71 + 12 * q^73 + 6 * q^74 + 6 * q^77 + 18 * q^79 - 6 * q^80 - 6 * q^83 + 6 * q^85 + 9 * q^86 - 6 * q^88 + 9 * q^89 + 24 * q^91 + 6 * q^94 + 15 * q^98

Introduction

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