LMFDB - Embedded newform 4650.2.a.cb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4650,2,Mod(1,4650)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4650, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4650.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4650.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:	$37.1304369399$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	4650.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.60555 q^{11} -1.00000 q^{12} +5.60555 q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -3.00000 q^{21} -1.60555 q^{22} +7.21110 q^{23} +1.00000 q^{24} -5.60555 q^{26} -1.00000 q^{27} +3.00000 q^{28} +7.21110 q^{29} -1.00000 q^{31} -1.00000 q^{32} -1.60555 q^{33} -2.00000 q^{34} +1.00000 q^{36} +9.60555 q^{37} -2.00000 q^{38} -5.60555 q^{39} -3.00000 q^{41} +3.00000 q^{42} +2.39445 q^{43} +1.60555 q^{44} -7.21110 q^{46} +3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} +5.60555 q^{52} -8.81665 q^{53} +1.00000 q^{54} -3.00000 q^{56} -2.00000 q^{57} -7.21110 q^{58} -3.21110 q^{59} -5.60555 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +1.60555 q^{66} +4.00000 q^{67} +2.00000 q^{68} -7.21110 q^{69} -9.00000 q^{71} -1.00000 q^{72} +16.4222 q^{73} -9.60555 q^{74} +2.00000 q^{76} +4.81665 q^{77} +5.60555 q^{78} +13.2111 q^{79} +1.00000 q^{81} +3.00000 q^{82} +6.39445 q^{83} -3.00000 q^{84} -2.39445 q^{86} -7.21110 q^{87} -1.60555 q^{88} -6.00000 q^{89} +16.8167 q^{91} +7.21110 q^{92} +1.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} -12.4222 q^{97} -2.00000 q^{98} +1.60555 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 6 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 4 q^{19} - 6 q^{21} + 4 q^{22} + 2 q^{24} - 4 q^{26} - 2 q^{27} + 6 q^{28} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} + 12 q^{37} - 4 q^{38} - 4 q^{39} - 6 q^{41} + 6 q^{42} + 12 q^{43} - 4 q^{44} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 4 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 6 q^{56} - 4 q^{57} + 8 q^{59} - 4 q^{61} + 2 q^{62} + 6 q^{63} + 2 q^{64} - 4 q^{66} + 8 q^{67} + 4 q^{68} - 18 q^{71} - 2 q^{72} + 4 q^{73} - 12 q^{74} + 4 q^{76} - 12 q^{77} + 4 q^{78} + 12 q^{79} + 2 q^{81} + 6 q^{82} + 20 q^{83} - 6 q^{84} - 12 q^{86} + 4 q^{88} - 12 q^{89} + 12 q^{91} + 2 q^{93} - 6 q^{94} + 2 q^{96} + 4 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 6 * q^14 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 4 * q^19 - 6 * q^21 + 4 * q^22 + 2 * q^24 - 4 * q^26 - 2 * q^27 + 6 * q^28 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^36 + 12 * q^37 - 4 * q^38 - 4 * q^39 - 6 * q^41 + 6 * q^42 + 12 * q^43 - 4 * q^44 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 4 * q^51 + 4 * q^52 + 4 * q^53 + 2 * q^54 - 6 * q^56 - 4 * q^57 + 8 * q^59 - 4 * q^61 + 2 * q^62 + 6 * q^63 + 2 * q^64 - 4 * q^66 + 8 * q^67 + 4 * q^68 - 18 * q^71 - 2 * q^72 + 4 * q^73 - 12 * q^74 + 4 * q^76 - 12 * q^77 + 4 * q^78 + 12 * q^79 + 2 * q^81 + 6 * q^82 + 20 * q^83 - 6 * q^84 - 12 * q^86 + 4 * q^88 - 12 * q^89 + 12 * q^91 + 2 * q^93 - 6 * q^94 + 2 * q^96 + 4 * q^97 - 4 * q^98 - 4 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.60555	0.484092	0.242046	−	0.970265i	$-0.422182\pi$
$11$	1.60555	0.484092	0.242046	+	0.970265i	$0.422182\pi$
$12$	−1.00000	−0.288675
$13$	5.60555	1.55470	0.777350	−	0.629068i	$-0.216563\pi$
$13$	5.60555	1.55470	0.777350	+	0.629068i	$0.216563\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	−3.00000	−0.654654
$22$	−1.60555	−0.342305
$23$	7.21110	1.50362	0.751809	−	0.659380i	$-0.229181\pi$
$23$	7.21110	1.50362	0.751809	+	0.659380i	$0.229181\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−5.60555	−1.09934
$27$	−1.00000	−0.192450
$28$	3.00000	0.566947
$29$	7.21110	1.33907	0.669534	−	0.742781i	$-0.266494\pi$
$29$	7.21110	1.33907	0.669534	+	0.742781i	$0.266494\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−1.00000	−0.176777
$33$	−1.60555	−0.279491
$34$	−2.00000	−0.342997
$35$	0	0
$36$	1.00000	0.166667
$37$	9.60555	1.57914	0.789571	−	0.613659i	$-0.210303\pi$
$37$	9.60555	1.57914	0.789571	+	0.613659i	$0.210303\pi$
$38$	−2.00000	−0.324443
$39$	−5.60555	−0.897607
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	3.00000	0.462910
$43$	2.39445	0.365150	0.182575	−	0.983192i	$-0.441557\pi$
$43$	2.39445	0.365150	0.182575	+	0.983192i	$0.441557\pi$
$44$	1.60555	0.242046
$45$	0	0
$46$	−7.21110	−1.06322
$47$	3.00000	0.437595	0.218797	−	0.975770i	$-0.429787\pi$
$47$	3.00000	0.437595	0.218797	+	0.975770i	$0.429787\pi$
$48$	−1.00000	−0.144338
$49$	2.00000	0.285714
$50$	0	0
$51$	−2.00000	−0.280056
$52$	5.60555	0.777350
$53$	−8.81665	−1.21106	−0.605530	−	0.795822i	$-0.707039\pi$
$53$	−8.81665	−1.21106	−0.605530	+	0.795822i	$0.707039\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	−3.00000	−0.400892
$57$	−2.00000	−0.264906
$58$	−7.21110	−0.946864
$59$	−3.21110	−0.418050	−0.209025	−	0.977910i	$-0.567029\pi$
$59$	−3.21110	−0.418050	−0.209025	+	0.977910i	$0.567029\pi$
$60$	0	0
$61$	−5.60555	−0.717717	−0.358859	−	0.933392i	$-0.616834\pi$
$61$	−5.60555	−0.717717	−0.358859	+	0.933392i	$0.616834\pi$
$62$	1.00000	0.127000
$63$	3.00000	0.377964
$64$	1.00000	0.125000
$65$	0	0
$66$	1.60555	0.197630
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.00000	0.242536
$69$	−7.21110	−0.868115
$70$	0	0
$71$	−9.00000	−1.06810	−0.534052	−	0.845452i	$-0.679331\pi$
$71$	−9.00000	−1.06810	−0.534052	+	0.845452i	$0.679331\pi$
$72$	−1.00000	−0.117851
$73$	16.4222	1.92207	0.961037	−	0.276420i	$-0.0891482\pi$
$73$	16.4222	1.92207	0.961037	+	0.276420i	$0.0891482\pi$
$74$	−9.60555	−1.11662
$75$	0	0
$76$	2.00000	0.229416
$77$	4.81665	0.548909
$78$	5.60555	0.634704
$79$	13.2111	1.48637	0.743183	−	0.669089i	$-0.233315\pi$
$79$	13.2111	1.48637	0.743183	+	0.669089i	$0.233315\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.00000	0.331295
$83$	6.39445	0.701882	0.350941	−	0.936398i	$-0.385862\pi$
$83$	6.39445	0.701882	0.350941	+	0.936398i	$0.385862\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	−2.39445	−0.258200
$87$	−7.21110	−0.773111
$88$	−1.60555	−0.171152
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	16.8167	1.76286
$92$	7.21110	0.751809
$93$	1.00000	0.103695
$94$	−3.00000	−0.309426
$95$	0	0
$96$	1.00000	0.102062
$97$	−12.4222	−1.26128	−0.630642	−	0.776074i	$-0.717208\pi$
$97$	−12.4222	−1.26128	−0.630642	+	0.776074i	$0.717208\pi$
$98$	−2.00000	−0.202031
$99$	1.60555	0.161364
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	2.00000	0.198030
$103$	11.0000	1.08386	0.541931	−	0.840423i	$-0.317693\pi$
$103$	11.0000	1.08386	0.541931	+	0.840423i	$0.317693\pi$
$104$	−5.60555	−0.549670
$105$	0	0
$106$	8.81665	0.856349
$107$	−18.4222	−1.78094	−0.890471	−	0.455040i	$-0.849625\pi$
$107$	−18.4222	−1.78094	−0.890471	+	0.455040i	$0.849625\pi$
$108$	−1.00000	−0.0962250
$109$	−17.2111	−1.64852	−0.824262	−	0.566208i	$-0.808410\pi$
$109$	−17.2111	−1.64852	−0.824262	+	0.566208i	$0.808410\pi$
$110$	0	0
$111$	−9.60555	−0.911719
$112$	3.00000	0.283473
$113$	−8.42221	−0.792294	−0.396147	−	0.918187i	$-0.629653\pi$
$113$	−8.42221	−0.792294	−0.396147	+	0.918187i	$0.629653\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	7.21110	0.669534
$117$	5.60555	0.518233
$118$	3.21110	0.295606
$119$	6.00000	0.550019
$120$	0	0
$121$	−8.42221	−0.765655
$122$	5.60555	0.507503
$123$	3.00000	0.270501
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−3.00000	−0.267261
$127$	3.21110	0.284939	0.142470	−	0.989799i	$-0.454496\pi$
$127$	3.21110	0.284939	0.142470	+	0.989799i	$0.454496\pi$
$128$	−1.00000	−0.0883883
$129$	−2.39445	−0.210819
$130$	0	0
$131$	−9.21110	−0.804778	−0.402389	−	0.915469i	$-0.631820\pi$
$131$	−9.21110	−0.804778	−0.402389	+	0.915469i	$0.631820\pi$
$132$	−1.60555	−0.139745
$133$	6.00000	0.520266
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−17.2111	−1.47044	−0.735222	−	0.677827i	$-0.762922\pi$
$137$	−17.2111	−1.47044	−0.735222	+	0.677827i	$0.762922\pi$
$138$	7.21110	0.613850
$139$	−8.81665	−0.747819	−0.373909	−	0.927465i	$-0.621983\pi$
$139$	−8.81665	−0.747819	−0.373909	+	0.927465i	$0.621983\pi$
$140$	0	0
$141$	−3.00000	−0.252646
$142$	9.00000	0.755263
$143$	9.00000	0.752618
$144$	1.00000	0.0833333
$145$	0	0
$146$	−16.4222	−1.35911
$147$	−2.00000	−0.164957
$148$	9.60555	0.789571
$149$	15.2111	1.24614	0.623071	−	0.782165i	$-0.285885\pi$
$149$	15.2111	1.24614	0.623071	+	0.782165i	$0.285885\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−2.00000	−0.162221
$153$	2.00000	0.161690
$154$	−4.81665	−0.388137
$155$	0	0
$156$	−5.60555	−0.448803
$157$	−0.788897	−0.0629609	−0.0314804	−	0.999504i	$-0.510022\pi$
$157$	−0.788897	−0.0629609	−0.0314804	+	0.999504i	$0.510022\pi$
$158$	−13.2111	−1.05102
$159$	8.81665	0.699206
$160$	0	0
$161$	21.6333	1.70494
$162$	−1.00000	−0.0785674
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−6.39445	−0.496305
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	3.00000	0.231455
$169$	18.4222	1.41709
$170$	0	0
$171$	2.00000	0.152944
$172$	2.39445	0.182575
$173$	15.2111	1.15648	0.578239	−	0.815867i	$-0.303740\pi$
$173$	15.2111	1.15648	0.578239	+	0.815867i	$0.303740\pi$
$174$	7.21110	0.546672
$175$	0	0
$176$	1.60555	0.121023
$177$	3.21110	0.241361
$178$	6.00000	0.449719
$179$	14.3944	1.07589	0.537946	−	0.842979i	$-0.319200\pi$
$179$	14.3944	1.07589	0.537946	+	0.842979i	$0.319200\pi$
$180$	0	0
$181$	16.8167	1.24997	0.624986	−	0.780636i	$-0.285105\pi$
$181$	16.8167	1.24997	0.624986	+	0.780636i	$0.285105\pi$
$182$	−16.8167	−1.24653
$183$	5.60555	0.414374
$184$	−7.21110	−0.531610
$185$	0	0
$186$	−1.00000	−0.0733236
$187$	3.21110	0.234819
$188$	3.00000	0.218797
$189$	−3.00000	−0.218218
$190$	0	0
$191$	−18.4222	−1.33298	−0.666492	−	0.745512i	$-0.732205\pi$
$191$	−18.4222	−1.33298	−0.666492	+	0.745512i	$0.732205\pi$
$192$	−1.00000	−0.0721688
$193$	−11.4222	−0.822188	−0.411094	−	0.911593i	$-0.634853\pi$
$193$	−11.4222	−0.822188	−0.411094	+	0.911593i	$0.634853\pi$
$194$	12.4222	0.891862
$195$	0	0
$196$	2.00000	0.142857
$197$	−10.3944	−0.740574	−0.370287	−	0.928917i	$-0.620741\pi$
$197$	−10.3944	−0.740574	−0.370287	+	0.928917i	$0.620741\pi$
$198$	−1.60555	−0.114102
$199$	9.21110	0.652958	0.326479	−	0.945204i	$-0.394138\pi$
$199$	9.21110	0.652958	0.326479	+	0.945204i	$0.394138\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	10.0000	0.703598
$203$	21.6333	1.51836
$204$	−2.00000	−0.140028
$205$	0	0
$206$	−11.0000	−0.766406
$207$	7.21110	0.501206
$208$	5.60555	0.388675
$209$	3.21110	0.222117
$210$	0	0
$211$	−25.6333	−1.76467	−0.882335	−	0.470622i	$-0.844029\pi$
$211$	−25.6333	−1.76467	−0.882335	+	0.470622i	$0.844029\pi$
$212$	−8.81665	−0.605530
$213$	9.00000	0.616670
$214$	18.4222	1.25932
$215$	0	0
$216$	1.00000	0.0680414
$217$	−3.00000	−0.203653
$218$	17.2111	1.16568
$219$	−16.4222	−1.10971
$220$	0	0
$221$	11.2111	0.754140
$222$	9.60555	0.644682
$223$	−19.2111	−1.28647	−0.643235	−	0.765669i	$-0.722408\pi$
$223$	−19.2111	−1.28647	−0.643235	+	0.765669i	$0.722408\pi$
$224$	−3.00000	−0.200446
$225$	0	0
$226$	8.42221	0.560237
$227$	−10.7889	−0.716084	−0.358042	−	0.933705i	$-0.616556\pi$
$227$	−10.7889	−0.716084	−0.358042	+	0.933705i	$0.616556\pi$
$228$	−2.00000	−0.132453
$229$	−7.21110	−0.476523	−0.238262	−	0.971201i	$-0.576578\pi$
$229$	−7.21110	−0.476523	−0.238262	+	0.971201i	$0.576578\pi$
$230$	0	0
$231$	−4.81665	−0.316913
$232$	−7.21110	−0.473432
$233$	11.4222	0.748294	0.374147	−	0.927370i	$-0.377936\pi$
$233$	11.4222	0.748294	0.374147	+	0.927370i	$0.377936\pi$
$234$	−5.60555	−0.366446
$235$	0	0
$236$	−3.21110	−0.209025
$237$	−13.2111	−0.858153
$238$	−6.00000	−0.388922
$239$	−4.42221	−0.286049	−0.143024	−	0.989719i	$-0.545683\pi$
$239$	−4.42221	−0.286049	−0.143024	+	0.989719i	$0.545683\pi$
$240$	0	0
$241$	10.7889	0.694974	0.347487	−	0.937685i	$-0.387035\pi$
$241$	10.7889	0.694974	0.347487	+	0.937685i	$0.387035\pi$
$242$	8.42221	0.541400
$243$	−1.00000	−0.0641500
$244$	−5.60555	−0.358859
$245$	0	0
$246$	−3.00000	−0.191273
$247$	11.2111	0.713345
$248$	1.00000	0.0635001
$249$	−6.39445	−0.405232
$250$	0	0
$251$	−20.8167	−1.31394	−0.656968	−	0.753919i	$-0.728161\pi$
$251$	−20.8167	−1.31394	−0.656968	+	0.753919i	$0.728161\pi$
$252$	3.00000	0.188982
$253$	11.5778	0.727890
$254$	−3.21110	−0.201482
$255$	0	0
$256$	1.00000	0.0625000
$257$	13.0000	0.810918	0.405459	−	0.914113i	$-0.367112\pi$
$257$	13.0000	0.810918	0.405459	+	0.914113i	$0.367112\pi$
$258$	2.39445	0.149072
$259$	28.8167	1.79058
$260$	0	0
$261$	7.21110	0.446356
$262$	9.21110	0.569064
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	1.60555	0.0988149
$265$	0	0
$266$	−6.00000	−0.367884
$267$	6.00000	0.367194
$268$	4.00000	0.244339
$269$	−21.6333	−1.31901	−0.659503	−	0.751702i	$-0.729233\pi$
$269$	−21.6333	−1.31901	−0.659503	+	0.751702i	$0.729233\pi$
$270$	0	0
$271$	31.6333	1.92159	0.960793	−	0.277266i	$-0.0894282\pi$
$271$	31.6333	1.92159	0.960793	+	0.277266i	$0.0894282\pi$
$272$	2.00000	0.121268
$273$	−16.8167	−1.01779
$274$	17.2111	1.03976
$275$	0	0
$276$	−7.21110	−0.434057
$277$	0.788897	0.0474003	0.0237001	−	0.999719i	$-0.492455\pi$
$277$	0.788897	0.0474003	0.0237001	+	0.999719i	$0.492455\pi$
$278$	8.81665	0.528788
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	11.4222	0.681392	0.340696	−	0.940174i	$-0.389337\pi$
$281$	11.4222	0.681392	0.340696	+	0.940174i	$0.389337\pi$
$282$	3.00000	0.178647
$283$	5.21110	0.309768	0.154884	−	0.987933i	$-0.450500\pi$
$283$	5.21110	0.309768	0.154884	+	0.987933i	$0.450500\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	−9.00000	−0.532181
$287$	−9.00000	−0.531253
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	0	0
$291$	12.4222	0.728203
$292$	16.4222	0.961037
$293$	11.2111	0.654960	0.327480	−	0.944858i	$-0.393801\pi$
$293$	11.2111	0.654960	0.327480	+	0.944858i	$0.393801\pi$
$294$	2.00000	0.116642
$295$	0	0
$296$	−9.60555	−0.558311
$297$	−1.60555	−0.0931635
$298$	−15.2111	−0.881156
$299$	40.4222	2.33768
$300$	0	0
$301$	7.18335	0.414041
$302$	8.00000	0.460348
$303$	10.0000	0.574485
$304$	2.00000	0.114708
$305$	0	0
$306$	−2.00000	−0.114332
$307$	−9.21110	−0.525705	−0.262853	−	0.964836i	$-0.584663\pi$
$307$	−9.21110	−0.525705	−0.262853	+	0.964836i	$0.584663\pi$
$308$	4.81665	0.274454
$309$	−11.0000	−0.625768
$310$	0	0
$311$	11.0000	0.623753	0.311876	−	0.950123i	$-0.399043\pi$
$311$	11.0000	0.623753	0.311876	+	0.950123i	$0.399043\pi$
$312$	5.60555	0.317352
$313$	−28.0000	−1.58265	−0.791327	−	0.611393i	$-0.790609\pi$
$313$	−28.0000	−1.58265	−0.791327	+	0.611393i	$0.790609\pi$
$314$	0.788897	0.0445201
$315$	0	0
$316$	13.2111	0.743183
$317$	8.78890	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$317$	8.78890	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$318$	−8.81665	−0.494413
$319$	11.5778	0.648232
$320$	0	0
$321$	18.4222	1.02823
$322$	−21.6333	−1.20558
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	0	0
$327$	17.2111	0.951776
$328$	3.00000	0.165647
$329$	9.00000	0.496186
$330$	0	0
$331$	6.39445	0.351471	0.175735	−	0.984437i	$-0.443770\pi$
$331$	6.39445	0.351471	0.175735	+	0.984437i	$0.443770\pi$
$332$	6.39445	0.350941
$333$	9.60555	0.526381
$334$	12.0000	0.656611
$335$	0	0
$336$	−3.00000	−0.163663
$337$	−13.6333	−0.742654	−0.371327	−	0.928502i	$-0.621097\pi$
$337$	−13.6333	−0.742654	−0.371327	+	0.928502i	$0.621097\pi$
$338$	−18.4222	−1.00204
$339$	8.42221	0.457431
$340$	0	0
$341$	−1.60555	−0.0869455
$342$	−2.00000	−0.108148
$343$	−15.0000	−0.809924
$344$	−2.39445	−0.129100
$345$	0	0
$346$	−15.2111	−0.817754
$347$	−20.0278	−1.07515	−0.537573	−	0.843217i	$-0.680659\pi$
$347$	−20.0278	−1.07515	−0.537573	+	0.843217i	$0.680659\pi$
$348$	−7.21110	−0.386556
$349$	4.78890	0.256344	0.128172	−	0.991752i	$-0.459089\pi$
$349$	4.78890	0.256344	0.128172	+	0.991752i	$0.459089\pi$
$350$	0	0
$351$	−5.60555	−0.299202
$352$	−1.60555	−0.0855762
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−3.21110	−0.170668
$355$	0	0
$356$	−6.00000	−0.317999
$357$	−6.00000	−0.317554
$358$	−14.3944	−0.760770
$359$	16.0000	0.844448	0.422224	−	0.906492i	$-0.361250\pi$
$359$	16.0000	0.844448	0.422224	+	0.906492i	$0.361250\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−16.8167	−0.883864
$363$	8.42221	0.442051
$364$	16.8167	0.881432
$365$	0	0
$366$	−5.60555	−0.293007
$367$	27.2111	1.42041	0.710204	−	0.703996i	$-0.248603\pi$
$367$	27.2111	1.42041	0.710204	+	0.703996i	$0.248603\pi$
$368$	7.21110	0.375905
$369$	−3.00000	−0.156174
$370$	0	0
$371$	−26.4500	−1.37321
$372$	1.00000	0.0518476
$373$	16.0000	0.828449	0.414224	−	0.910175i	$-0.364053\pi$
$373$	16.0000	0.828449	0.414224	+	0.910175i	$0.364053\pi$
$374$	−3.21110	−0.166042
$375$	0	0
$376$	−3.00000	−0.154713
$377$	40.4222	2.08185
$378$	3.00000	0.154303
$379$	28.0000	1.43826	0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000	1.43826	0.719132	+	0.694874i	$0.244540\pi$
$380$	0	0
$381$	−3.21110	−0.164510
$382$	18.4222	0.942562
$383$	−18.4222	−0.941331	−0.470665	−	0.882312i	$-0.655986\pi$
$383$	−18.4222	−0.941331	−0.470665	+	0.882312i	$0.655986\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	11.4222	0.581375
$387$	2.39445	0.121717
$388$	−12.4222	−0.630642
$389$	11.2111	0.568425	0.284213	−	0.958761i	$-0.408268\pi$
$389$	11.2111	0.568425	0.284213	+	0.958761i	$0.408268\pi$
$390$	0	0
$391$	14.4222	0.729362
$392$	−2.00000	−0.101015
$393$	9.21110	0.464639
$394$	10.3944	0.523665
$395$	0	0
$396$	1.60555	0.0806820
$397$	19.6333	0.985367	0.492684	−	0.870208i	$-0.336016\pi$
$397$	19.6333	0.985367	0.492684	+	0.870208i	$0.336016\pi$
$398$	−9.21110	−0.461711
$399$	−6.00000	−0.300376
$400$	0	0
$401$	−36.0555	−1.80053	−0.900263	−	0.435346i	$-0.856626\pi$
$401$	−36.0555	−1.80053	−0.900263	+	0.435346i	$0.856626\pi$
$402$	4.00000	0.199502
$403$	−5.60555	−0.279232
$404$	−10.0000	−0.497519
$405$	0	0
$406$	−21.6333	−1.07364
$407$	15.4222	0.764450
$408$	2.00000	0.0990148
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	17.2111	0.848961
$412$	11.0000	0.541931
$413$	−9.63331	−0.474024
$414$	−7.21110	−0.354406
$415$	0	0
$416$	−5.60555	−0.274835
$417$	8.81665	0.431753
$418$	−3.21110	−0.157060
$419$	6.78890	0.331659	0.165830	−	0.986154i	$-0.446970\pi$
$419$	6.78890	0.331659	0.165830	+	0.986154i	$0.446970\pi$
$420$	0	0
$421$	13.2111	0.643870	0.321935	−	0.946762i	$-0.395667\pi$
$421$	13.2111	0.643870	0.321935	+	0.946762i	$0.395667\pi$
$422$	25.6333	1.24781
$423$	3.00000	0.145865
$424$	8.81665	0.428175
$425$	0	0
$426$	−9.00000	−0.436051
$427$	−16.8167	−0.813815
$428$	−18.4222	−0.890471
$429$	−9.00000	−0.434524
$430$	0	0
$431$	23.8444	1.14854	0.574272	−	0.818664i	$-0.305285\pi$
$431$	23.8444	1.14854	0.574272	+	0.818664i	$0.305285\pi$
$432$	−1.00000	−0.0481125
$433$	−20.4222	−0.981429	−0.490714	−	0.871321i	$-0.663264\pi$
$433$	−20.4222	−0.981429	−0.490714	+	0.871321i	$0.663264\pi$
$434$	3.00000	0.144005
$435$	0	0
$436$	−17.2111	−0.824262
$437$	14.4222	0.689908
$438$	16.4222	0.784683
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	2.00000	0.0952381
$442$	−11.2111	−0.533258
$443$	9.57779	0.455055	0.227527	−	0.973772i	$-0.426936\pi$
$443$	9.57779	0.455055	0.227527	+	0.973772i	$0.426936\pi$
$444$	−9.60555	−0.455859
$445$	0	0
$446$	19.2111	0.909672
$447$	−15.2111	−0.719460
$448$	3.00000	0.141737
$449$	−29.2111	−1.37856	−0.689279	−	0.724496i	$-0.742072\pi$
$449$	−29.2111	−1.37856	−0.689279	+	0.724496i	$0.742072\pi$
$450$	0	0
$451$	−4.81665	−0.226807
$452$	−8.42221	−0.396147
$453$	8.00000	0.375873
$454$	10.7889	0.506348
$455$	0	0
$456$	2.00000	0.0936586
$457$	−32.4222	−1.51665	−0.758323	−	0.651879i	$-0.773981\pi$
$457$	−32.4222	−1.51665	−0.758323	+	0.651879i	$0.773981\pi$
$458$	7.21110	0.336953
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−15.1833	−0.707159	−0.353579	−	0.935404i	$-0.615036\pi$
$461$	−15.1833	−0.707159	−0.353579	+	0.935404i	$0.615036\pi$
$462$	4.81665	0.224091
$463$	17.2111	0.799868	0.399934	−	0.916544i	$-0.369033\pi$
$463$	17.2111	0.799868	0.399934	+	0.916544i	$0.369033\pi$
$464$	7.21110	0.334767
$465$	0	0
$466$	−11.4222	−0.529123
$467$	2.78890	0.129055	0.0645274	−	0.997916i	$-0.479446\pi$
$467$	2.78890	0.129055	0.0645274	+	0.997916i	$0.479446\pi$
$468$	5.60555	0.259117
$469$	12.0000	0.554109
$470$	0	0
$471$	0.788897	0.0363505
$472$	3.21110	0.147803
$473$	3.84441	0.176766
$474$	13.2111	0.606806
$475$	0	0
$476$	6.00000	0.275010
$477$	−8.81665	−0.403687
$478$	4.42221	0.202267
$479$	12.0000	0.548294	0.274147	−	0.961688i	$-0.411605\pi$
$479$	12.0000	0.548294	0.274147	+	0.961688i	$0.411605\pi$
$480$	0	0
$481$	53.8444	2.45509
$482$	−10.7889	−0.491421
$483$	−21.6333	−0.984350
$484$	−8.42221	−0.382828
$485$	0	0
$486$	1.00000	0.0453609
$487$	7.21110	0.326766	0.163383	−	0.986563i	$-0.447759\pi$
$487$	7.21110	0.326766	0.163383	+	0.986563i	$0.447759\pi$
$488$	5.60555	0.253751
$489$	0	0
$490$	0	0
$491$	40.8444	1.84328	0.921641	−	0.388043i	$-0.126849\pi$
$491$	40.8444	1.84328	0.921641	+	0.388043i	$0.126849\pi$
$492$	3.00000	0.135250
$493$	14.4222	0.649543
$494$	−11.2111	−0.504411
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	−27.0000	−1.21112
$498$	6.39445	0.286542
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	20.8167	0.929093
$503$	23.0000	1.02552	0.512760	−	0.858532i	$-0.328623\pi$
$503$	23.0000	1.02552	0.512760	+	0.858532i	$0.328623\pi$
$504$	−3.00000	−0.133631
$505$	0	0
$506$	−11.5778	−0.514696
$507$	−18.4222	−0.818159
$508$	3.21110	0.142470
$509$	−7.18335	−0.318396	−0.159198	−	0.987247i	$-0.550891\pi$
$509$	−7.18335	−0.318396	−0.159198	+	0.987247i	$0.550891\pi$
$510$	0	0
$511$	49.2666	2.17943
$512$	−1.00000	−0.0441942
$513$	−2.00000	−0.0883022
$514$	−13.0000	−0.573405
$515$	0	0
$516$	−2.39445	−0.105410
$517$	4.81665	0.211836
$518$	−28.8167	−1.26613
$519$	−15.2111	−0.667693
$520$	0	0
$521$	−21.4222	−0.938524	−0.469262	−	0.883059i	$-0.655480\pi$
$521$	−21.4222	−0.938524	−0.469262	+	0.883059i	$0.655480\pi$
$522$	−7.21110	−0.315621
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	−9.21110	−0.402389
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−2.00000	−0.0871214
$528$	−1.60555	−0.0698727
$529$	29.0000	1.26087
$530$	0	0
$531$	−3.21110	−0.139350
$532$	6.00000	0.260133
$533$	−16.8167	−0.728410
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−14.3944	−0.621166
$538$	21.6333	0.932678
$539$	3.21110	0.138312
$540$	0	0
$541$	−7.63331	−0.328182	−0.164091	−	0.986445i	$-0.552469\pi$
$541$	−7.63331	−0.328182	−0.164091	+	0.986445i	$0.552469\pi$
$542$	−31.6333	−1.35877
$543$	−16.8167	−0.721672
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	16.8167	0.719686
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	−17.2111	−0.735222
$549$	−5.60555	−0.239239
$550$	0	0
$551$	14.4222	0.614407
$552$	7.21110	0.306925
$553$	39.6333	1.68538
$554$	−0.788897	−0.0335170
$555$	0	0
$556$	−8.81665	−0.373909
$557$	23.2111	0.983486	0.491743	−	0.870740i	$-0.336360\pi$
$557$	23.2111	0.983486	0.491743	+	0.870740i	$0.336360\pi$
$558$	1.00000	0.0423334
$559$	13.4222	0.567699
$560$	0	0
$561$	−3.21110	−0.135573
$562$	−11.4222	−0.481817
$563$	18.0000	0.758610	0.379305	−	0.925272i	$-0.376163\pi$
$563$	18.0000	0.758610	0.379305	+	0.925272i	$0.376163\pi$
$564$	−3.00000	−0.126323
$565$	0	0
$566$	−5.21110	−0.219039
$567$	3.00000	0.125988
$568$	9.00000	0.377632
$569$	10.4222	0.436922	0.218461	−	0.975846i	$-0.429896\pi$
$569$	10.4222	0.436922	0.218461	+	0.975846i	$0.429896\pi$
$570$	0	0
$571$	−0.0277564	−0.00116157	−0.000580784	−	1.00000i	$-0.500185\pi$
$571$	−0.0277564	−0.00116157	−0.000580784		1.00000i	$0.500185\pi$
$572$	9.00000	0.376309
$573$	18.4222	0.769599
$574$	9.00000	0.375653
$575$	0	0
$576$	1.00000	0.0416667
$577$	−14.8444	−0.617981	−0.308990	−	0.951065i	$-0.599991\pi$
$577$	−14.8444	−0.617981	−0.308990	+	0.951065i	$0.599991\pi$
$578$	13.0000	0.540729
$579$	11.4222	0.474691
$580$	0	0
$581$	19.1833	0.795859
$582$	−12.4222	−0.514917
$583$	−14.1556	−0.586265
$584$	−16.4222	−0.679556
$585$	0	0
$586$	−11.2111	−0.463126
$587$	−36.8167	−1.51959	−0.759793	−	0.650165i	$-0.774700\pi$
$587$	−36.8167	−1.51959	−0.759793	+	0.650165i	$0.774700\pi$
$588$	−2.00000	−0.0824786
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	10.3944	0.427570
$592$	9.60555	0.394786
$593$	−33.8444	−1.38982	−0.694912	−	0.719095i	$-0.744557\pi$
$593$	−33.8444	−1.38982	−0.694912	+	0.719095i	$0.744557\pi$
$594$	1.60555	0.0658766
$595$	0	0
$596$	15.2111	0.623071
$597$	−9.21110	−0.376985
$598$	−40.4222	−1.65299
$599$	−43.0000	−1.75693	−0.878466	−	0.477805i	$-0.841433\pi$
$599$	−43.0000	−1.75693	−0.878466	+	0.477805i	$0.841433\pi$
$600$	0	0
$601$	24.8444	1.01342	0.506712	−	0.862115i	$-0.330861\pi$
$601$	24.8444	1.01342	0.506712	+	0.862115i	$0.330861\pi$
$602$	−7.18335	−0.292771
$603$	4.00000	0.162893
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−10.0000	−0.406222
$607$	35.8444	1.45488	0.727440	−	0.686171i	$-0.240710\pi$
$607$	35.8444	1.45488	0.727440	+	0.686171i	$0.240710\pi$
$608$	−2.00000	−0.0811107
$609$	−21.6333	−0.876626
$610$	0	0
$611$	16.8167	0.680329
$612$	2.00000	0.0808452
$613$	36.0555	1.45627	0.728134	−	0.685435i	$-0.240388\pi$
$613$	36.0555	1.45627	0.728134	+	0.685435i	$0.240388\pi$
$614$	9.21110	0.371730
$615$	0	0
$616$	−4.81665	−0.194069
$617$	−39.4222	−1.58708	−0.793539	−	0.608519i	$-0.791764\pi$
$617$	−39.4222	−1.58708	−0.793539	+	0.608519i	$0.791764\pi$
$618$	11.0000	0.442485
$619$	1.60555	0.0645326	0.0322663	−	0.999479i	$-0.489728\pi$
$619$	1.60555	0.0645326	0.0322663	+	0.999479i	$0.489728\pi$
$620$	0	0
$621$	−7.21110	−0.289372
$622$	−11.0000	−0.441060
$623$	−18.0000	−0.721155
$624$	−5.60555	−0.224402
$625$	0	0
$626$	28.0000	1.11911
$627$	−3.21110	−0.128239
$628$	−0.788897	−0.0314804
$629$	19.2111	0.765997
$630$	0	0
$631$	−26.4222	−1.05185	−0.525926	−	0.850531i	$-0.676281\pi$
$631$	−26.4222	−1.05185	−0.525926	+	0.850531i	$0.676281\pi$
$632$	−13.2111	−0.525509
$633$	25.6333	1.01883
$634$	−8.78890	−0.349052
$635$	0	0
$636$	8.81665	0.349603
$637$	11.2111	0.444200
$638$	−11.5778	−0.458369
$639$	−9.00000	−0.356034
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	−18.4222	−0.727066
$643$	−12.8167	−0.505439	−0.252720	−	0.967540i	$-0.581325\pi$
$643$	−12.8167	−0.505439	−0.252720	+	0.967540i	$0.581325\pi$
$644$	21.6333	0.852472
$645$	0	0
$646$	−4.00000	−0.157378
$647$	34.4222	1.35328	0.676638	−	0.736316i	$-0.263436\pi$
$647$	34.4222	1.35328	0.676638	+	0.736316i	$0.263436\pi$
$648$	−1.00000	−0.0392837
$649$	−5.15559	−0.202375
$650$	0	0
$651$	3.00000	0.117579
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−17.2111	−0.673007
$655$	0	0
$656$	−3.00000	−0.117130
$657$	16.4222	0.640691
$658$	−9.00000	−0.350857
$659$	−7.57779	−0.295189	−0.147594	−	0.989048i	$-0.547153\pi$
$659$	−7.57779	−0.295189	−0.147594	+	0.989048i	$0.547153\pi$
$660$	0	0
$661$	−49.6333	−1.93051	−0.965256	−	0.261306i	$-0.915847\pi$
$661$	−49.6333	−1.93051	−0.965256	+	0.261306i	$0.915847\pi$
$662$	−6.39445	−0.248527
$663$	−11.2111	−0.435403
$664$	−6.39445	−0.248153
$665$	0	0
$666$	−9.60555	−0.372208
$667$	52.0000	2.01345
$668$	−12.0000	−0.464294
$669$	19.2111	0.742744
$670$	0	0
$671$	−9.00000	−0.347441
$672$	3.00000	0.115728
$673$	−43.2111	−1.66567	−0.832833	−	0.553525i	$-0.813282\pi$
$673$	−43.2111	−1.66567	−0.832833	+	0.553525i	$0.813282\pi$
$674$	13.6333	0.525135
$675$	0	0
$676$	18.4222	0.708546
$677$	−3.23886	−0.124479	−0.0622397	−	0.998061i	$-0.519824\pi$
$677$	−3.23886	−0.124479	−0.0622397	+	0.998061i	$0.519824\pi$
$678$	−8.42221	−0.323453
$679$	−37.2666	−1.43016
$680$	0	0
$681$	10.7889	0.413431
$682$	1.60555	0.0614797
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	2.00000	0.0764719
$685$	0	0
$686$	15.0000	0.572703
$687$	7.21110	0.275121
$688$	2.39445	0.0912875
$689$	−49.4222	−1.88284
$690$	0	0
$691$	−1.21110	−0.0460725	−0.0230363	−	0.999735i	$-0.507333\pi$
$691$	−1.21110	−0.0460725	−0.0230363	+	0.999735i	$0.507333\pi$
$692$	15.2111	0.578239
$693$	4.81665	0.182970
$694$	20.0278	0.760243
$695$	0	0
$696$	7.21110	0.273336
$697$	−6.00000	−0.227266
$698$	−4.78890	−0.181262
$699$	−11.4222	−0.432027
$700$	0	0
$701$	−6.42221	−0.242563	−0.121282	−	0.992618i	$-0.538700\pi$
$701$	−6.42221	−0.242563	−0.121282	+	0.992618i	$0.538700\pi$
$702$	5.60555	0.211568
$703$	19.2111	0.724560
$704$	1.60555	0.0605115
$705$	0	0
$706$	−18.0000	−0.677439
$707$	−30.0000	−1.12827
$708$	3.21110	0.120681
$709$	−27.2111	−1.02193	−0.510967	−	0.859600i	$-0.670713\pi$
$709$	−27.2111	−1.02193	−0.510967	+	0.859600i	$0.670713\pi$
$710$	0	0
$711$	13.2111	0.495455
$712$	6.00000	0.224860
$713$	−7.21110	−0.270058
$714$	6.00000	0.224544
$715$	0	0
$716$	14.3944	0.537946
$717$	4.42221	0.165150
$718$	−16.0000	−0.597115
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	33.0000	1.22898
$722$	15.0000	0.558242
$723$	−10.7889	−0.401243
$724$	16.8167	0.624986
$725$	0	0
$726$	−8.42221	−0.312577
$727$	37.0000	1.37225	0.686127	−	0.727482i	$-0.259309\pi$
$727$	37.0000	1.37225	0.686127	+	0.727482i	$0.259309\pi$
$728$	−16.8167	−0.623267
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.78890	0.177124
$732$	5.60555	0.207187
$733$	−41.2666	−1.52422	−0.762109	−	0.647449i	$-0.775836\pi$
$733$	−41.2666	−1.52422	−0.762109	+	0.647449i	$0.775836\pi$
$734$	−27.2111	−1.00438
$735$	0	0
$736$	−7.21110	−0.265805
$737$	6.42221	0.236565
$738$	3.00000	0.110432
$739$	32.0000	1.17714	0.588570	−	0.808447i	$-0.299691\pi$
$739$	32.0000	1.17714	0.588570	+	0.808447i	$0.299691\pi$
$740$	0	0
$741$	−11.2111	−0.411850
$742$	26.4500	0.971009
$743$	38.0555	1.39612	0.698061	−	0.716039i	$-0.254047\pi$
$743$	38.0555	1.39612	0.698061	+	0.716039i	$0.254047\pi$
$744$	−1.00000	−0.0366618
$745$	0	0
$746$	−16.0000	−0.585802
$747$	6.39445	0.233961
$748$	3.21110	0.117410
$749$	−55.2666	−2.01940
$750$	0	0
$751$	−3.00000	−0.109472	−0.0547358	−	0.998501i	$-0.517432\pi$
$751$	−3.00000	−0.109472	−0.0547358	+	0.998501i	$0.517432\pi$
$752$	3.00000	0.109399
$753$	20.8167	0.758601
$754$	−40.4222	−1.47209
$755$	0	0
$756$	−3.00000	−0.109109
$757$	−4.02776	−0.146391	−0.0731956	−	0.997318i	$-0.523320\pi$
$757$	−4.02776	−0.146391	−0.0731956	+	0.997318i	$0.523320\pi$
$758$	−28.0000	−1.01701
$759$	−11.5778	−0.420247
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	3.21110	0.116326
$763$	−51.6333	−1.86925
$764$	−18.4222	−0.666492
$765$	0	0
$766$	18.4222	0.665621
$767$	−18.0000	−0.649942
$768$	−1.00000	−0.0360844
$769$	−3.42221	−0.123408	−0.0617039	−	0.998094i	$-0.519653\pi$
$769$	−3.42221	−0.123408	−0.0617039	+	0.998094i	$0.519653\pi$
$770$	0	0
$771$	−13.0000	−0.468184
$772$	−11.4222	−0.411094
$773$	41.6333	1.49745	0.748723	−	0.662883i	$-0.230667\pi$
$773$	41.6333	1.49745	0.748723	+	0.662883i	$0.230667\pi$
$774$	−2.39445	−0.0860667
$775$	0	0
$776$	12.4222	0.445931
$777$	−28.8167	−1.03379
$778$	−11.2111	−0.401937
$779$	−6.00000	−0.214972
$780$	0	0
$781$	−14.4500	−0.517060
$782$	−14.4222	−0.515737
$783$	−7.21110	−0.257704
$784$	2.00000	0.0714286
$785$	0	0
$786$	−9.21110	−0.328549
$787$	8.02776	0.286159	0.143079	−	0.989711i	$-0.454300\pi$
$787$	8.02776	0.286159	0.143079	+	0.989711i	$0.454300\pi$
$788$	−10.3944	−0.370287
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−25.2666	−0.898377
$792$	−1.60555	−0.0570508
$793$	−31.4222	−1.11584
$794$	−19.6333	−0.696760
$795$	0	0
$796$	9.21110	0.326479
$797$	32.0555	1.13546	0.567732	−	0.823213i	$-0.307821\pi$
$797$	32.0555	1.13546	0.567732	+	0.823213i	$0.307821\pi$
$798$	6.00000	0.212398
$799$	6.00000	0.212265
$800$	0	0
$801$	−6.00000	−0.212000
$802$	36.0555	1.27316
$803$	26.3667	0.930460
$804$	−4.00000	−0.141069
$805$	0	0
$806$	5.60555	0.197447
$807$	21.6333	0.761528
$808$	10.0000	0.351799
$809$	−16.7889	−0.590266	−0.295133	−	0.955456i	$-0.595364\pi$
$809$	−16.7889	−0.590266	−0.295133	+	0.955456i	$0.595364\pi$
$810$	0	0
$811$	−23.2111	−0.815052	−0.407526	−	0.913194i	$-0.633608\pi$
$811$	−23.2111	−0.815052	−0.407526	+	0.913194i	$0.633608\pi$
$812$	21.6333	0.759180
$813$	−31.6333	−1.10943
$814$	−15.4222	−0.540548
$815$	0	0
$816$	−2.00000	−0.0700140
$817$	4.78890	0.167542
$818$	−24.0000	−0.839140
$819$	16.8167	0.587621
$820$	0	0
$821$	13.6056	0.474837	0.237419	−	0.971407i	$-0.423699\pi$
$821$	13.6056	0.474837	0.237419	+	0.971407i	$0.423699\pi$
$822$	−17.2111	−0.600306
$823$	−52.8444	−1.84204	−0.921020	−	0.389515i	$-0.872643\pi$
$823$	−52.8444	−1.84204	−0.921020	+	0.389515i	$0.872643\pi$
$824$	−11.0000	−0.383203
$825$	0	0
$826$	9.63331	0.335186
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	7.21110	0.250603
$829$	−32.0278	−1.11237	−0.556185	−	0.831059i	$-0.687735\pi$
$829$	−32.0278	−1.11237	−0.556185	+	0.831059i	$0.687735\pi$
$830$	0	0
$831$	−0.788897	−0.0273665
$832$	5.60555	0.194338
$833$	4.00000	0.138592
$834$	−8.81665	−0.305296
$835$	0	0
$836$	3.21110	0.111058
$837$	1.00000	0.0345651
$838$	−6.78890	−0.234519
$839$	39.8444	1.37558	0.687791	−	0.725909i	$-0.258581\pi$
$839$	39.8444	1.37558	0.687791	+	0.725909i	$0.258581\pi$
$840$	0	0
$841$	23.0000	0.793103
$842$	−13.2111	−0.455285
$843$	−11.4222	−0.393402
$844$	−25.6333	−0.882335
$845$	0	0
$846$	−3.00000	−0.103142
$847$	−25.2666	−0.868171
$848$	−8.81665	−0.302765
$849$	−5.21110	−0.178845
$850$	0	0
$851$	69.2666	2.37443
$852$	9.00000	0.308335
$853$	3.21110	0.109946	0.0549730	−	0.998488i	$-0.482493\pi$
$853$	3.21110	0.109946	0.0549730	+	0.998488i	$0.482493\pi$
$854$	16.8167	0.575454
$855$	0	0
$856$	18.4222	0.629658
$857$	11.8444	0.404597	0.202299	−	0.979324i	$-0.435159\pi$
$857$	11.8444	0.404597	0.202299	+	0.979324i	$0.435159\pi$
$858$	9.00000	0.307255
$859$	38.4500	1.31190	0.655948	−	0.754806i	$-0.272269\pi$
$859$	38.4500	1.31190	0.655948	+	0.754806i	$0.272269\pi$
$860$	0	0
$861$	9.00000	0.306719
$862$	−23.8444	−0.812144
$863$	−56.0555	−1.90815	−0.954076	−	0.299565i	$-0.903158\pi$
$863$	−56.0555	−1.90815	−0.954076	+	0.299565i	$0.903158\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	20.4222	0.693975
$867$	13.0000	0.441503
$868$	−3.00000	−0.101827
$869$	21.2111	0.719537
$870$	0	0
$871$	22.4222	0.759747
$872$	17.2111	0.582841
$873$	−12.4222	−0.420428
$874$	−14.4222	−0.487838
$875$	0	0
$876$	−16.4222	−0.554855
$877$	−1.21110	−0.0408960	−0.0204480	−	0.999791i	$-0.506509\pi$
$877$	−1.21110	−0.0408960	−0.0204480	+	0.999791i	$0.506509\pi$
$878$	−4.00000	−0.134993
$879$	−11.2111	−0.378141
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	−2.00000	−0.0673435
$883$	−26.3944	−0.888244	−0.444122	−	0.895966i	$-0.646484\pi$
$883$	−26.3944	−0.888244	−0.444122	+	0.895966i	$0.646484\pi$
$884$	11.2111	0.377070
$885$	0	0
$886$	−9.57779	−0.321772
$887$	−5.84441	−0.196236	−0.0981180	−	0.995175i	$-0.531282\pi$
$887$	−5.84441	−0.196236	−0.0981180	+	0.995175i	$0.531282\pi$
$888$	9.60555	0.322341
$889$	9.63331	0.323091
$890$	0	0
$891$	1.60555	0.0537880
$892$	−19.2111	−0.643235
$893$	6.00000	0.200782
$894$	15.2111	0.508735
$895$	0	0
$896$	−3.00000	−0.100223
$897$	−40.4222	−1.34966
$898$	29.2111	0.974787
$899$	−7.21110	−0.240504
$900$	0	0
$901$	−17.6333	−0.587451
$902$	4.81665	0.160377
$903$	−7.18335	−0.239047
$904$	8.42221	0.280118
$905$	0	0
$906$	−8.00000	−0.265782
$907$	2.42221	0.0804280	0.0402140	−	0.999191i	$-0.487196\pi$
$907$	2.42221	0.0804280	0.0402140	+	0.999191i	$0.487196\pi$
$908$	−10.7889	−0.358042
$909$	−10.0000	−0.331679
$910$	0	0
$911$	−57.6333	−1.90948	−0.954738	−	0.297447i	$-0.903865\pi$
$911$	−57.6333	−1.90948	−0.954738	+	0.297447i	$0.903865\pi$
$912$	−2.00000	−0.0662266
$913$	10.2666	0.339775
$914$	32.4222	1.07243
$915$	0	0
$916$	−7.21110	−0.238262
$917$	−27.6333	−0.912532
$918$	2.00000	0.0660098
$919$	−37.8444	−1.24837	−0.624186	−	0.781276i	$-0.714569\pi$
$919$	−37.8444	−1.24837	−0.624186	+	0.781276i	$0.714569\pi$
$920$	0	0
$921$	9.21110	0.303516
$922$	15.1833	0.500037
$923$	−50.4500	−1.66058
$924$	−4.81665	−0.158456
$925$	0	0
$926$	−17.2111	−0.565592
$927$	11.0000	0.361287
$928$	−7.21110	−0.236716
$929$	15.6333	0.512912	0.256456	−	0.966556i	$-0.417445\pi$
$929$	15.6333	0.512912	0.256456	+	0.966556i	$0.417445\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	11.4222	0.374147
$933$	−11.0000	−0.360124
$934$	−2.78890	−0.0912555
$935$	0	0
$936$	−5.60555	−0.183223
$937$	−3.57779	−0.116881	−0.0584407	−	0.998291i	$-0.518613\pi$
$937$	−3.57779	−0.116881	−0.0584407	+	0.998291i	$0.518613\pi$
$938$	−12.0000	−0.391814
$939$	28.0000	0.913745
$940$	0	0
$941$	23.2111	0.756660	0.378330	−	0.925671i	$-0.376498\pi$
$941$	23.2111	0.756660	0.378330	+	0.925671i	$0.376498\pi$
$942$	−0.788897	−0.0257037
$943$	−21.6333	−0.704477
$944$	−3.21110	−0.104512
$945$	0	0
$946$	−3.84441	−0.124993
$947$	−2.39445	−0.0778091	−0.0389046	−	0.999243i	$-0.512387\pi$
$947$	−2.39445	−0.0778091	−0.0389046	+	0.999243i	$0.512387\pi$
$948$	−13.2111	−0.429077
$949$	92.0555	2.98825
$950$	0	0
$951$	−8.78890	−0.285000
$952$	−6.00000	−0.194461
$953$	43.2666	1.40154	0.700772	−	0.713386i	$-0.252839\pi$
$953$	43.2666	1.40154	0.700772	+	0.713386i	$0.252839\pi$
$954$	8.81665	0.285450
$955$	0	0
$956$	−4.42221	−0.143024
$957$	−11.5778	−0.374257
$958$	−12.0000	−0.387702
$959$	−51.6333	−1.66733
$960$	0	0
$961$	1.00000	0.0322581
$962$	−53.8444	−1.73601
$963$	−18.4222	−0.593647
$964$	10.7889	0.347487
$965$	0	0
$966$	21.6333	0.696040
$967$	−33.2666	−1.06978	−0.534891	−	0.844921i	$-0.679647\pi$
$967$	−33.2666	−1.06978	−0.534891	+	0.844921i	$0.679647\pi$
$968$	8.42221	0.270700
$969$	−4.00000	−0.128499
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	−26.4500	−0.847947
$974$	−7.21110	−0.231059
$975$	0	0
$976$	−5.60555	−0.179429
$977$	−23.0000	−0.735835	−0.367918	−	0.929858i	$-0.619929\pi$
$977$	−23.0000	−0.735835	−0.367918	+	0.929858i	$0.619929\pi$
$978$	0	0
$979$	−9.63331	−0.307882
$980$	0	0
$981$	−17.2111	−0.549508
$982$	−40.8444	−1.30340
$983$	10.0555	0.320721	0.160361	−	0.987059i	$-0.448734\pi$
$983$	10.0555	0.320721	0.160361	+	0.987059i	$0.448734\pi$
$984$	−3.00000	−0.0956365
$985$	0	0
$986$	−14.4222	−0.459297
$987$	−9.00000	−0.286473
$988$	11.2111	0.356673
$989$	17.2666	0.549046
$990$	0	0
$991$	17.6333	0.560140	0.280070	−	0.959980i	$-0.409642\pi$
$991$	17.6333	0.560140	0.280070	+	0.959980i	$0.409642\pi$
$992$	1.00000	0.0317500
$993$	−6.39445	−0.202922
$994$	27.0000	0.856388
$995$	0	0
$996$	−6.39445	−0.202616
$997$	12.4222	0.393415	0.196708	−	0.980462i	$-0.436975\pi$
$997$	12.4222	0.393415	0.196708	+	0.980462i	$0.436975\pi$
$998$	−16.0000	−0.506471
$999$	−9.60555	−0.303906

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.cb.1.2	✓	2
5.2	odd	4		4650.2.d.bf.3349.2		4
5.3	odd	4		4650.2.d.bf.3349.4		4
5.4	even	2		4650.2.a.cg.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.cb.1.2	✓	2	1.1	even	1	trivial
4650.2.a.cg.1.2	yes	2	5.4	even	2
4650.2.d.bf.3349.2		4	5.2	odd	4
4650.2.d.bf.3349.4		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:	\( N \)	\(=\)	\( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4650.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.60555 q^{11} -1.00000 q^{12} +5.60555 q^{13} -3.00000 q^{14} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +2.00000 q^{19} -3.00000 q^{21} -1.60555 q^{22} +7.21110 q^{23} +1.00000 q^{24} -5.60555 q^{26} -1.00000 q^{27} +3.00000 q^{28} +7.21110 q^{29} -1.00000 q^{31} -1.00000 q^{32} -1.60555 q^{33} -2.00000 q^{34} +1.00000 q^{36} +9.60555 q^{37} -2.00000 q^{38} -5.60555 q^{39} -3.00000 q^{41} +3.00000 q^{42} +2.39445 q^{43} +1.60555 q^{44} -7.21110 q^{46} +3.00000 q^{47} -1.00000 q^{48} +2.00000 q^{49} -2.00000 q^{51} +5.60555 q^{52} -8.81665 q^{53} +1.00000 q^{54} -3.00000 q^{56} -2.00000 q^{57} -7.21110 q^{58} -3.21110 q^{59} -5.60555 q^{61} +1.00000 q^{62} +3.00000 q^{63} +1.00000 q^{64} +1.60555 q^{66} +4.00000 q^{67} +2.00000 q^{68} -7.21110 q^{69} -9.00000 q^{71} -1.00000 q^{72} +16.4222 q^{73} -9.60555 q^{74} +2.00000 q^{76} +4.81665 q^{77} +5.60555 q^{78} +13.2111 q^{79} +1.00000 q^{81} +3.00000 q^{82} +6.39445 q^{83} -3.00000 q^{84} -2.39445 q^{86} -7.21110 q^{87} -1.60555 q^{88} -6.00000 q^{89} +16.8167 q^{91} +7.21110 q^{92} +1.00000 q^{93} -3.00000 q^{94} +1.00000 q^{96} -12.4222 q^{97} -2.00000 q^{98} +1.60555 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} + 6 q^{7} - 2 q^{8} + 2 q^{9} - 4 q^{11} - 2 q^{12} + 4 q^{13} - 6 q^{14} + 2 q^{16} + 4 q^{17} - 2 q^{18} + 4 q^{19} - 6 q^{21} + 4 q^{22} + 2 q^{24} - 4 q^{26} - 2 q^{27} + 6 q^{28} - 2 q^{31} - 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} + 12 q^{37} - 4 q^{38} - 4 q^{39} - 6 q^{41} + 6 q^{42} + 12 q^{43} - 4 q^{44} + 6 q^{47} - 2 q^{48} + 4 q^{49} - 4 q^{51} + 4 q^{52} + 4 q^{53} + 2 q^{54} - 6 q^{56} - 4 q^{57} + 8 q^{59} - 4 q^{61} + 2 q^{62} + 6 q^{63} + 2 q^{64} - 4 q^{66} + 8 q^{67} + 4 q^{68} - 18 q^{71} - 2 q^{72} + 4 q^{73} - 12 q^{74} + 4 q^{76} - 12 q^{77} + 4 q^{78} + 12 q^{79} + 2 q^{81} + 6 q^{82} + 20 q^{83} - 6 q^{84} - 12 q^{86} + 4 q^{88} - 12 q^{89} + 12 q^{91} + 2 q^{93} - 6 q^{94} + 2 q^{96} + 4 q^{97} - 4 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 + 2 * q^6 + 6 * q^7 - 2 * q^8 + 2 * q^9 - 4 * q^11 - 2 * q^12 + 4 * q^13 - 6 * q^14 + 2 * q^16 + 4 * q^17 - 2 * q^18 + 4 * q^19 - 6 * q^21 + 4 * q^22 + 2 * q^24 - 4 * q^26 - 2 * q^27 + 6 * q^28 - 2 * q^31 - 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^36 + 12 * q^37 - 4 * q^38 - 4 * q^39 - 6 * q^41 + 6 * q^42 + 12 * q^43 - 4 * q^44 + 6 * q^47 - 2 * q^48 + 4 * q^49 - 4 * q^51 + 4 * q^52 + 4 * q^53 + 2 * q^54 - 6 * q^56 - 4 * q^57 + 8 * q^59 - 4 * q^61 + 2 * q^62 + 6 * q^63 + 2 * q^64 - 4 * q^66 + 8 * q^67 + 4 * q^68 - 18 * q^71 - 2 * q^72 + 4 * q^73 - 12 * q^74 + 4 * q^76 - 12 * q^77 + 4 * q^78 + 12 * q^79 + 2 * q^81 + 6 * q^82 + 20 * q^83 - 6 * q^84 - 12 * q^86 + 4 * q^88 - 12 * q^89 + 12 * q^91 + 2 * q^93 - 6 * q^94 + 2 * q^96 + 4 * q^97 - 4 * q^98 - 4 * q^99

Introduction

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