LMFDB - Embedded newform 4650.2.a.co.1.1

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
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			1.1
Root			$-0.167449$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.80451 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.80451 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.13941 q^{11} -1.00000 q^{12} -4.13941 q^{13} -3.80451 q^{14} +1.00000 q^{16} +6.80451 q^{17} +1.00000 q^{18} -7.60902 q^{19} +3.80451 q^{21} -6.13941 q^{22} +2.00000 q^{23} -1.00000 q^{24} -4.13941 q^{26} -1.00000 q^{27} -3.80451 q^{28} +6.80451 q^{29} -1.00000 q^{31} +1.00000 q^{32} +6.13941 q^{33} +6.80451 q^{34} +1.00000 q^{36} -8.13941 q^{37} -7.60902 q^{38} +4.13941 q^{39} +2.47431 q^{41} +3.80451 q^{42} +3.33490 q^{43} -6.13941 q^{44} +2.00000 q^{46} +11.2788 q^{47} -1.00000 q^{48} +7.47431 q^{49} -6.80451 q^{51} -4.13941 q^{52} +9.46961 q^{53} -1.00000 q^{54} -3.80451 q^{56} +7.60902 q^{57} +6.80451 q^{58} -8.27882 q^{59} -4.13941 q^{61} -1.00000 q^{62} -3.80451 q^{63} +1.00000 q^{64} +6.13941 q^{66} +10.2788 q^{67} +6.80451 q^{68} -2.00000 q^{69} -4.33020 q^{71} +1.00000 q^{72} +10.0000 q^{73} -8.13941 q^{74} -7.60902 q^{76} +23.3575 q^{77} +4.13941 q^{78} -4.80451 q^{79} +1.00000 q^{81} +2.47431 q^{82} +11.4696 q^{83} +3.80451 q^{84} +3.33490 q^{86} -6.80451 q^{87} -6.13941 q^{88} +18.6924 q^{89} +15.7484 q^{91} +2.00000 q^{92} +1.00000 q^{93} +11.2788 q^{94} -1.00000 q^{96} -13.0833 q^{97} +7.47431 q^{98} -6.13941 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * 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
$7$	−3.80451	−1.43797	−0.718985	−	0.695025i	$-0.755393\pi$
$7$	−3.80451	−1.43797	−0.718985	+	0.695025i	$0.755393\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.13941	−1.85110	−0.925551	−	0.378623i	$-0.876398\pi$
$11$	−6.13941	−1.85110	−0.925551	+	0.378623i	$0.876398\pi$
$12$	−1.00000	−0.288675
$13$	−4.13941	−1.14807	−0.574033	−	0.818832i	$-0.694622\pi$
$13$	−4.13941	−1.14807	−0.574033	+	0.818832i	$0.694622\pi$
$14$	−3.80451	−1.01680
$15$	0	0
$16$	1.00000	0.250000
$17$	6.80451	1.65034	0.825168	−	0.564887i	$-0.191080\pi$
$17$	6.80451	1.65034	0.825168	+	0.564887i	$0.191080\pi$
$18$	1.00000	0.235702
$19$	−7.60902	−1.74563	−0.872815	−	0.488052i	$-0.837708\pi$
$19$	−7.60902	−1.74563	−0.872815	+	0.488052i	$0.837708\pi$
$20$	0	0
$21$	3.80451	0.830212
$22$	−6.13941	−1.30893
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−4.13941	−0.811805
$27$	−1.00000	−0.192450
$28$	−3.80451	−0.718985
$29$	6.80451	1.26357	0.631783	−	0.775145i	$-0.282323\pi$
$29$	6.80451	1.26357	0.631783	+	0.775145i	$0.282323\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	1.00000	0.176777
$33$	6.13941	1.06873
$34$	6.80451	1.16696
$35$	0	0
$36$	1.00000	0.166667
$37$	−8.13941	−1.33811	−0.669055	−	0.743213i	$-0.733301\pi$
$37$	−8.13941	−1.33811	−0.669055	+	0.743213i	$0.733301\pi$
$38$	−7.60902	−1.23435
$39$	4.13941	0.662836
$40$	0	0
$41$	2.47431	0.386422	0.193211	−	0.981157i	$-0.438110\pi$
$41$	2.47431	0.386422	0.193211	+	0.981157i	$0.438110\pi$
$42$	3.80451	0.587049
$43$	3.33490	0.508567	0.254284	−	0.967130i	$-0.418160\pi$
$43$	3.33490	0.508567	0.254284	+	0.967130i	$0.418160\pi$
$44$	−6.13941	−0.925551
$45$	0	0
$46$	2.00000	0.294884
$47$	11.2788	1.64518	0.822592	−	0.568631i	$-0.192527\pi$
$47$	11.2788	1.64518	0.822592	+	0.568631i	$0.192527\pi$
$48$	−1.00000	−0.144338
$49$	7.47431	1.06776
$50$	0	0
$51$	−6.80451	−0.952822
$52$	−4.13941	−0.574033
$53$	9.46961	1.30075	0.650376	−	0.759613i	$-0.274611\pi$
$53$	9.46961	1.30075	0.650376	+	0.759613i	$0.274611\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−3.80451	−0.508399
$57$	7.60902	1.00784
$58$	6.80451	0.893476
$59$	−8.27882	−1.07781	−0.538905	−	0.842366i	$-0.681162\pi$
$59$	−8.27882	−1.07781	−0.538905	+	0.842366i	$0.681162\pi$
$60$	0	0
$61$	−4.13941	−0.529997	−0.264999	−	0.964249i	$-0.585371\pi$
$61$	−4.13941	−0.529997	−0.264999	+	0.964249i	$0.585371\pi$
$62$	−1.00000	−0.127000
$63$	−3.80451	−0.479323
$64$	1.00000	0.125000
$65$	0	0
$66$	6.13941	0.755709
$67$	10.2788	1.25576	0.627879	−	0.778311i	$-0.283923\pi$
$67$	10.2788	1.25576	0.627879	+	0.778311i	$0.283923\pi$
$68$	6.80451	0.825168
$69$	−2.00000	−0.240772
$70$	0	0
$71$	−4.33020	−0.513901	−0.256950	−	0.966425i	$-0.582718\pi$
$71$	−4.33020	−0.513901	−0.256950	+	0.966425i	$0.582718\pi$
$72$	1.00000	0.117851
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	−8.13941	−0.946187
$75$	0	0
$76$	−7.60902	−0.872815
$77$	23.3575	2.66183
$78$	4.13941	0.468696
$79$	−4.80451	−0.540550	−0.270275	−	0.962783i	$-0.587115\pi$
$79$	−4.80451	−0.540550	−0.270275	+	0.962783i	$0.587115\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	2.47431	0.273242
$83$	11.4696	1.25895	0.629477	−	0.777019i	$-0.283269\pi$
$83$	11.4696	1.25895	0.629477	+	0.777019i	$0.283269\pi$
$84$	3.80451	0.415106
$85$	0	0
$86$	3.33490	0.359611
$87$	−6.80451	−0.729520
$88$	−6.13941	−0.654463
$89$	18.6924	1.98139	0.990693	−	0.136117i	$-0.0434622\pi$
$89$	18.6924	1.98139	0.990693	+	0.136117i	$0.0434622\pi$
$90$	0	0
$91$	15.7484	1.65088
$92$	2.00000	0.208514
$93$	1.00000	0.103695
$94$	11.2788	1.16332
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−13.0833	−1.32841	−0.664206	−	0.747550i	$-0.731230\pi$
$97$	−13.0833	−1.32841	−0.664206	+	0.747550i	$0.731230\pi$
$98$	7.47431	0.755019
$99$	−6.13941	−0.617034
$100$	0	0
$101$	1.33020	0.132360	0.0661801	−	0.997808i	$-0.478919\pi$
$101$	1.33020	0.132360	0.0661801	+	0.997808i	$0.478919\pi$
$102$	−6.80451	−0.673747
$103$	13.8045	1.36020	0.680099	−	0.733120i	$-0.261937\pi$
$103$	13.8045	1.36020	0.680099	+	0.733120i	$0.261937\pi$
$104$	−4.13941	−0.405903
$105$	0	0
$106$	9.46961	0.919770
$107$	−15.0833	−1.45816	−0.729080	−	0.684428i	$-0.760052\pi$
$107$	−15.0833	−1.45816	−0.729080	+	0.684428i	$0.760052\pi$
$108$	−1.00000	−0.0962250
$109$	−7.08333	−0.678460	−0.339230	−	0.940703i	$-0.610167\pi$
$109$	−7.08333	−0.678460	−0.339230	+	0.940703i	$0.610167\pi$
$110$	0	0
$111$	8.13941	0.772559
$112$	−3.80451	−0.359493
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	7.60902	0.712650
$115$	0	0
$116$	6.80451	0.631783
$117$	−4.13941	−0.382689
$118$	−8.27882	−0.762127
$119$	−25.8878	−2.37313
$120$	0	0
$121$	26.6924	2.42658
$122$	−4.13941	−0.374765
$123$	−2.47431	−0.223101
$124$	−1.00000	−0.0898027
$125$	0	0
$126$	−3.80451	−0.338933
$127$	3.47431	0.308295	0.154148	−	0.988048i	$-0.450737\pi$
$127$	3.47431	0.308295	0.154148	+	0.988048i	$0.450737\pi$
$128$	1.00000	0.0883883
$129$	−3.33490	−0.293621
$130$	0	0
$131$	−3.86529	−0.337712	−0.168856	−	0.985641i	$-0.554007\pi$
$131$	−3.86529	−0.337712	−0.168856	+	0.985641i	$0.554007\pi$
$132$	6.13941	0.534367
$133$	28.9486	2.51016
$134$	10.2788	0.887955
$135$	0	0
$136$	6.80451	0.583482
$137$	−20.0226	−1.71064	−0.855321	−	0.518098i	$-0.826640\pi$
$137$	−20.0226	−1.71064	−0.855321	+	0.518098i	$0.826640\pi$
$138$	−2.00000	−0.170251
$139$	−11.6137	−0.985063	−0.492531	−	0.870295i	$-0.663928\pi$
$139$	−11.6137	−0.985063	−0.492531	+	0.870295i	$0.663928\pi$
$140$	0	0
$141$	−11.2788	−0.949848
$142$	−4.33020	−0.363383
$143$	25.4135	2.12519
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	−7.47431	−0.616471
$148$	−8.13941	−0.669055
$149$	8.94862	0.733099	0.366550	−	0.930399i	$-0.380539\pi$
$149$	8.94862	0.733099	0.366550	+	0.930399i	$0.380539\pi$
$150$	0	0
$151$	12.1347	0.987509	0.493755	−	0.869601i	$-0.335624\pi$
$151$	12.1347	0.987509	0.493755	+	0.869601i	$0.335624\pi$
$152$	−7.60902	−0.617173
$153$	6.80451	0.550112
$154$	23.3575	1.88220
$155$	0	0
$156$	4.13941	0.331418
$157$	12.4135	0.990708	0.495354	−	0.868691i	$-0.335038\pi$
$157$	12.4135	0.990708	0.495354	+	0.868691i	$0.335038\pi$
$158$	−4.80451	−0.382226
$159$	−9.46961	−0.750989
$160$	0	0
$161$	−7.60902	−0.599675
$162$	1.00000	0.0785674
$163$	4.66980	0.365767	0.182883	−	0.983135i	$-0.441457\pi$
$163$	4.66980	0.365767	0.182883	+	0.983135i	$0.441457\pi$
$164$	2.47431	0.193211
$165$	0	0
$166$	11.4696	0.890215
$167$	−15.2180	−1.17761	−0.588804	−	0.808276i	$-0.700401\pi$
$167$	−15.2180	−1.17761	−0.588804	+	0.808276i	$0.700401\pi$
$168$	3.80451	0.293524
$169$	4.13471	0.318055
$170$	0	0
$171$	−7.60902	−0.581877
$172$	3.33490	0.254284
$173$	6.66980	0.507095	0.253548	−	0.967323i	$-0.418403\pi$
$173$	6.66980	0.507095	0.253548	+	0.967323i	$0.418403\pi$
$174$	−6.80451	−0.515849
$175$	0	0
$176$	−6.13941	−0.462775
$177$	8.27882	0.622274
$178$	18.6924	1.40105
$179$	−3.20018	−0.239193	−0.119596	−	0.992823i	$-0.538160\pi$
$179$	−3.20018	−0.239193	−0.119596	+	0.992823i	$0.538160\pi$
$180$	0	0
$181$	1.20018	0.0892089	0.0446045	−	0.999005i	$-0.485797\pi$
$181$	1.20018	0.0892089	0.0446045	+	0.999005i	$0.485797\pi$
$182$	15.7484	1.16735
$183$	4.13941	0.305994
$184$	2.00000	0.147442
$185$	0	0
$186$	1.00000	0.0733236
$187$	−41.7757	−3.05494
$188$	11.2788	0.822592
$189$	3.80451	0.276737
$190$	0	0
$191$	7.08333	0.512532	0.256266	−	0.966606i	$-0.417508\pi$
$191$	7.08333	0.512532	0.256266	+	0.966606i	$0.417508\pi$
$192$	−1.00000	−0.0721688
$193$	3.52569	0.253785	0.126892	−	0.991916i	$-0.459500\pi$
$193$	3.52569	0.253785	0.126892	+	0.991916i	$0.459500\pi$
$194$	−13.0833	−0.939328
$195$	0	0
$196$	7.47431	0.533879
$197$	−19.2227	−1.36956	−0.684782	−	0.728748i	$-0.740103\pi$
$197$	−19.2227	−1.36956	−0.684782	+	0.728748i	$0.740103\pi$
$198$	−6.13941	−0.436309
$199$	1.33959	0.0949613	0.0474806	−	0.998872i	$-0.484881\pi$
$199$	1.33959	0.0949613	0.0474806	+	0.998872i	$0.484881\pi$
$200$	0	0
$201$	−10.2788	−0.725012
$202$	1.33020	0.0935928
$203$	−25.8878	−1.81697
$204$	−6.80451	−0.476411
$205$	0	0
$206$	13.8045	0.961806
$207$	2.00000	0.139010
$208$	−4.13941	−0.287016
$209$	46.7149	3.23134
$210$	0	0
$211$	0.390977	0.0269160	0.0134580	−	0.999909i	$-0.495716\pi$
$211$	0.390977	0.0269160	0.0134580	+	0.999909i	$0.495716\pi$
$212$	9.46961	0.650376
$213$	4.33020	0.296701
$214$	−15.0833	−1.03108
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	3.80451	0.258267
$218$	−7.08333	−0.479744
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−28.1667	−1.89469
$222$	8.13941	0.546281
$223$	10.8045	0.723524	0.361762	−	0.932271i	$-0.382175\pi$
$223$	10.8045	0.723524	0.361762	+	0.932271i	$0.382175\pi$
$224$	−3.80451	−0.254200
$225$	0	0
$226$	6.00000	0.399114
$227$	22.4135	1.48764	0.743819	−	0.668381i	$-0.233012\pi$
$227$	22.4135	1.48764	0.743819	+	0.668381i	$0.233012\pi$
$228$	7.60902	0.503920
$229$	−12.2788	−0.811407	−0.405704	−	0.914005i	$-0.632973\pi$
$229$	−12.2788	−0.811407	−0.405704	+	0.914005i	$0.632973\pi$
$230$	0	0
$231$	−23.3575	−1.53681
$232$	6.80451	0.446738
$233$	−0.0833315	−0.00545923	−0.00272961	−	0.999996i	$-0.500869\pi$
$233$	−0.0833315	−0.00545923	−0.00272961	+	0.999996i	$0.500869\pi$
$234$	−4.13941	−0.270602
$235$	0	0
$236$	−8.27882	−0.538905
$237$	4.80451	0.312087
$238$	−25.8878	−1.67806
$239$	−0.660406	−0.0427181	−0.0213591	−	0.999772i	$-0.506799\pi$
$239$	−0.660406	−0.0427181	−0.0213591	+	0.999772i	$0.506799\pi$
$240$	0	0
$241$	−16.5576	−1.06657	−0.533286	−	0.845935i	$-0.679043\pi$
$241$	−16.5576	−1.06657	−0.533286	+	0.845935i	$0.679043\pi$
$242$	26.6924	1.71585
$243$	−1.00000	−0.0641500
$244$	−4.13941	−0.264999
$245$	0	0
$246$	−2.47431	−0.157756
$247$	31.4969	2.00410
$248$	−1.00000	−0.0635001
$249$	−11.4696	−0.726857
$250$	0	0
$251$	13.6363	0.860714	0.430357	−	0.902659i	$-0.358388\pi$
$251$	13.6363	0.860714	0.430357	+	0.902659i	$0.358388\pi$
$252$	−3.80451	−0.239662
$253$	−12.2788	−0.771963
$254$	3.47431	0.217997
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.13471	0.445051	0.222526	−	0.974927i	$-0.428570\pi$
$257$	7.13471	0.445051	0.222526	+	0.974927i	$0.428570\pi$
$258$	−3.33490	−0.207622
$259$	30.9665	1.92416
$260$	0	0
$261$	6.80451	0.421189
$262$	−3.86529	−0.238798
$263$	−7.33020	−0.452000	−0.226000	−	0.974127i	$-0.572565\pi$
$263$	−7.33020	−0.452000	−0.226000	+	0.974127i	$0.572565\pi$
$264$	6.13941	0.377855
$265$	0	0
$266$	28.9486	1.77495
$267$	−18.6924	−1.14395
$268$	10.2788	0.627879
$269$	−2.80451	−0.170994	−0.0854970	−	0.996338i	$-0.527248\pi$
$269$	−2.80451	−0.170994	−0.0854970	+	0.996338i	$0.527248\pi$
$270$	0	0
$271$	−26.9712	−1.63838	−0.819191	−	0.573521i	$-0.805577\pi$
$271$	−26.9712	−1.63838	−0.819191	+	0.573521i	$0.805577\pi$
$272$	6.80451	0.412584
$273$	−15.7484	−0.953139
$274$	−20.0226	−1.20961
$275$	0	0
$276$	−2.00000	−0.120386
$277$	7.49687	0.450443	0.225222	−	0.974308i	$-0.427689\pi$
$277$	7.49687	0.450443	0.225222	+	0.974308i	$0.427689\pi$
$278$	−11.6137	−0.696545
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	4.86529	0.290239	0.145119	−	0.989414i	$-0.453643\pi$
$281$	4.86529	0.290239	0.145119	+	0.989414i	$0.453643\pi$
$282$	−11.2788	−0.671644
$283$	−8.93923	−0.531382	−0.265691	−	0.964058i	$-0.585600\pi$
$283$	−8.93923	−0.531382	−0.265691	+	0.964058i	$0.585600\pi$
$284$	−4.33020	−0.256950
$285$	0	0
$286$	25.4135	1.50273
$287$	−9.41353	−0.555663
$288$	1.00000	0.0589256
$289$	29.3014	1.72361
$290$	0	0
$291$	13.0833	0.766958
$292$	10.0000	0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−7.47431	−0.435911
$295$	0	0
$296$	−8.13941	−0.473094
$297$	6.13941	0.356245
$298$	8.94862	0.518379
$299$	−8.27882	−0.478777
$300$	0	0
$301$	−12.6877	−0.731305
$302$	12.1347	0.698274
$303$	−1.33020	−0.0764182
$304$	−7.60902	−0.436407
$305$	0	0
$306$	6.80451	0.388988
$307$	−22.5482	−1.28690	−0.643448	−	0.765490i	$-0.722497\pi$
$307$	−22.5482	−1.28690	−0.643448	+	0.765490i	$0.722497\pi$
$308$	23.3575	1.33091
$309$	−13.8045	−0.785311
$310$	0	0
$311$	27.1573	1.53995	0.769974	−	0.638075i	$-0.220269\pi$
$311$	27.1573	1.53995	0.769974	+	0.638075i	$0.220269\pi$
$312$	4.13941	0.234348
$313$	34.5482	1.95278	0.976391	−	0.216010i	$-0.0693042\pi$
$313$	34.5482	1.95278	0.976391	+	0.216010i	$0.0693042\pi$
$314$	12.4135	0.700536
$315$	0	0
$316$	−4.80451	−0.270275
$317$	7.33959	0.412233	0.206116	−	0.978527i	$-0.433917\pi$
$317$	7.33959	0.412233	0.206116	+	0.978527i	$0.433917\pi$
$318$	−9.46961	−0.531030
$319$	−41.7757	−2.33899
$320$	0	0
$321$	15.0833	0.841869
$322$	−7.60902	−0.424034
$323$	−51.7757	−2.88088
$324$	1.00000	0.0555556
$325$	0	0
$326$	4.66980	0.258636
$327$	7.08333	0.391709
$328$	2.47431	0.136621
$329$	−42.9104	−2.36573
$330$	0	0
$331$	−3.46961	−0.190707	−0.0953536	−	0.995443i	$-0.530398\pi$
$331$	−3.46961	−0.190707	−0.0953536	+	0.995443i	$0.530398\pi$
$332$	11.4696	0.629477
$333$	−8.13941	−0.446037
$334$	−15.2180	−0.832694
$335$	0	0
$336$	3.80451	0.207553
$337$	17.8878	0.974413	0.487206	−	0.873287i	$-0.338016\pi$
$337$	17.8878	0.974413	0.487206	+	0.873287i	$0.338016\pi$
$338$	4.13471	0.224899
$339$	−6.00000	−0.325875
$340$	0	0
$341$	6.13941	0.332468
$342$	−7.60902	−0.411449
$343$	−1.80451	−0.0974345
$344$	3.33490	0.179806
$345$	0	0
$346$	6.66980	0.358570
$347$	13.0786	0.702098	0.351049	−	0.936357i	$-0.385825\pi$
$347$	13.0786	0.702098	0.351049	+	0.936357i	$0.385825\pi$
$348$	−6.80451	−0.364760
$349$	−14.0226	−0.750610	−0.375305	−	0.926901i	$-0.622462\pi$
$349$	−14.0226	−0.750610	−0.375305	+	0.926901i	$0.622462\pi$
$350$	0	0
$351$	4.13941	0.220945
$352$	−6.13941	−0.327232
$353$	15.2274	0.810475	0.405237	−	0.914212i	$-0.367189\pi$
$353$	15.2274	0.810475	0.405237	+	0.914212i	$0.367189\pi$
$354$	8.27882	0.440014
$355$	0	0
$356$	18.6924	0.990693
$357$	25.8878	1.37013
$358$	−3.20018	−0.169135
$359$	6.66041	0.351523	0.175761	−	0.984433i	$-0.443761\pi$
$359$	6.66041	0.351523	0.175761	+	0.984433i	$0.443761\pi$
$360$	0	0
$361$	38.8972	2.04722
$362$	1.20018	0.0630802
$363$	−26.6924	−1.40099
$364$	15.7484	0.825442
$365$	0	0
$366$	4.13941	0.216370
$367$	21.0833	1.10054	0.550270	−	0.834986i	$-0.314525\pi$
$367$	21.0833	1.10054	0.550270	+	0.834986i	$0.314525\pi$
$368$	2.00000	0.104257
$369$	2.47431	0.128807
$370$	0	0
$371$	−36.0273	−1.87044
$372$	1.00000	0.0518476
$373$	12.1347	0.628312	0.314156	−	0.949371i	$-0.398279\pi$
$373$	12.1347	0.628312	0.314156	+	0.949371i	$0.398279\pi$
$374$	−41.7757	−2.16017
$375$	0	0
$376$	11.2788	0.581661
$377$	−28.1667	−1.45066
$378$	3.80451	0.195683
$379$	19.8878	1.02157	0.510785	−	0.859709i	$-0.329355\pi$
$379$	19.8878	1.02157	0.510785	+	0.859709i	$0.329355\pi$
$380$	0	0
$381$	−3.47431	−0.177994
$382$	7.08333	0.362415
$383$	30.1667	1.54144	0.770722	−	0.637171i	$-0.219896\pi$
$383$	30.1667	1.54144	0.770722	+	0.637171i	$0.219896\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	3.52569	0.179453
$387$	3.33490	0.169522
$388$	−13.0833	−0.664206
$389$	−17.3302	−0.878676	−0.439338	−	0.898322i	$-0.644787\pi$
$389$	−17.3302	−0.878676	−0.439338	+	0.898322i	$0.644787\pi$
$390$	0	0
$391$	13.6090	0.688238
$392$	7.47431	0.377510
$393$	3.86529	0.194978
$394$	−19.2227	−0.968428
$395$	0	0
$396$	−6.13941	−0.308517
$397$	−34.9486	−1.75402	−0.877010	−	0.480471i	$-0.840466\pi$
$397$	−34.9486	−1.75402	−0.877010	+	0.480471i	$0.840466\pi$
$398$	1.33959	0.0671478
$399$	−28.9486	−1.44924
$400$	0	0
$401$	−25.2180	−1.25933	−0.629665	−	0.776867i	$-0.716808\pi$
$401$	−25.2180	−1.25933	−0.629665	+	0.776867i	$0.716808\pi$
$402$	−10.2788	−0.512661
$403$	4.13941	0.206199
$404$	1.33020	0.0661801
$405$	0	0
$406$	−25.8878	−1.28479
$407$	49.9712	2.47698
$408$	−6.80451	−0.336874
$409$	19.3302	0.955817	0.477909	−	0.878410i	$-0.341395\pi$
$409$	19.3302	0.955817	0.477909	+	0.878410i	$0.341395\pi$
$410$	0	0
$411$	20.0226	0.987640
$412$	13.8045	0.680099
$413$	31.4969	1.54986
$414$	2.00000	0.0982946
$415$	0	0
$416$	−4.13941	−0.202951
$417$	11.6137	0.568726
$418$	46.7149	2.28490
$419$	19.7531	0.965003	0.482502	−	0.875895i	$-0.339728\pi$
$419$	19.7531	0.965003	0.482502	+	0.875895i	$0.339728\pi$
$420$	0	0
$421$	17.7437	0.864777	0.432388	−	0.901687i	$-0.357671\pi$
$421$	17.7437	0.864777	0.432388	+	0.901687i	$0.357671\pi$
$422$	0.390977	0.0190325
$423$	11.2788	0.548395
$424$	9.46961	0.459885
$425$	0	0
$426$	4.33020	0.209799
$427$	15.7484	0.762120
$428$	−15.0833	−0.729080
$429$	−25.4135	−1.22698
$430$	0	0
$431$	16.6316	0.801115	0.400558	−	0.916272i	$-0.368816\pi$
$431$	16.6316	0.801115	0.400558	+	0.916272i	$0.368816\pi$
$432$	−1.00000	−0.0481125
$433$	−25.1059	−1.20651	−0.603256	−	0.797548i	$-0.706130\pi$
$433$	−25.1059	−1.20651	−0.603256	+	0.797548i	$0.706130\pi$
$434$	3.80451	0.182622
$435$	0	0
$436$	−7.08333	−0.339230
$437$	−15.2180	−0.727978
$438$	−10.0000	−0.477818
$439$	24.8271	1.18493	0.592466	−	0.805596i	$-0.298155\pi$
$439$	24.8271	1.18493	0.592466	+	0.805596i	$0.298155\pi$
$440$	0	0
$441$	7.47431	0.355919
$442$	−28.1667	−1.33975
$443$	−17.6316	−0.837702	−0.418851	−	0.908055i	$-0.637567\pi$
$443$	−17.6316	−0.837702	−0.418851	+	0.908055i	$0.637567\pi$
$444$	8.13941	0.386279
$445$	0	0
$446$	10.8045	0.511608
$447$	−8.94862	−0.423255
$448$	−3.80451	−0.179746
$449$	11.0833	0.523055	0.261527	−	0.965196i	$-0.415774\pi$
$449$	11.0833	0.523055	0.261527	+	0.965196i	$0.415774\pi$
$450$	0	0
$451$	−15.1908	−0.715307
$452$	6.00000	0.282216
$453$	−12.1347	−0.570139
$454$	22.4135	1.05192
$455$	0	0
$456$	7.60902	0.356325
$457$	−27.4969	−1.28625	−0.643125	−	0.765762i	$-0.722362\pi$
$457$	−27.4969	−1.28625	−0.643125	+	0.765762i	$0.722362\pi$
$458$	−12.2788	−0.573752
$459$	−6.80451	−0.317607
$460$	0	0
$461$	12.1488	0.565826	0.282913	−	0.959146i	$-0.408699\pi$
$461$	12.1488	0.565826	0.282913	+	0.959146i	$0.408699\pi$
$462$	−23.3575	−1.08669
$463$	−29.3622	−1.36457	−0.682287	−	0.731084i	$-0.739015\pi$
$463$	−29.3622	−1.36457	−0.682287	+	0.731084i	$0.739015\pi$
$464$	6.80451	0.315892
$465$	0	0
$466$	−0.0833315	−0.00386026
$467$	36.6924	1.69792	0.848960	−	0.528457i	$-0.177229\pi$
$467$	36.6924	1.69792	0.848960	+	0.528457i	$0.177229\pi$
$468$	−4.13941	−0.191344
$469$	−39.1059	−1.80574
$470$	0	0
$471$	−12.4135	−0.571985
$472$	−8.27882	−0.381064
$473$	−20.4743	−0.941410
$474$	4.80451	0.220679
$475$	0	0
$476$	−25.8878	−1.18657
$477$	9.46961	0.433584
$478$	−0.660406	−0.0302063
$479$	12.4229	0.567618	0.283809	−	0.958881i	$-0.408402\pi$
$479$	12.4229	0.567618	0.283809	+	0.958881i	$0.408402\pi$
$480$	0	0
$481$	33.6924	1.53624
$482$	−16.5576	−0.754180
$483$	7.60902	0.346223
$484$	26.6924	1.21329
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	12.3014	0.557429	0.278714	−	0.960374i	$-0.410092\pi$
$487$	12.3014	0.557429	0.278714	+	0.960374i	$0.410092\pi$
$488$	−4.13941	−0.187382
$489$	−4.66980	−0.211176
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−2.47431	−0.111550
$493$	46.3014	2.08531
$494$	31.4969	1.41711
$495$	0	0
$496$	−1.00000	−0.0449013
$497$	16.4743	0.738974
$498$	−11.4696	−0.513966
$499$	36.9806	1.65548	0.827739	−	0.561114i	$-0.189627\pi$
$499$	36.9806	1.65548	0.827739	+	0.561114i	$0.189627\pi$
$500$	0	0
$501$	15.2180	0.679892
$502$	13.6363	0.608617
$503$	18.4969	0.824735	0.412367	−	0.911018i	$-0.364702\pi$
$503$	18.4969	0.824735	0.412367	+	0.911018i	$0.364702\pi$
$504$	−3.80451	−0.169466
$505$	0	0
$506$	−12.2788	−0.545860
$507$	−4.13471	−0.183629
$508$	3.47431	0.154148
$509$	−20.9665	−0.929323	−0.464661	−	0.885488i	$-0.653824\pi$
$509$	−20.9665	−0.929323	−0.464661	+	0.885488i	$0.653824\pi$
$510$	0	0
$511$	−38.0451	−1.68302
$512$	1.00000	0.0441942
$513$	7.60902	0.335947
$514$	7.13471	0.314699
$515$	0	0
$516$	−3.33490	−0.146811
$517$	−69.2453	−3.04540
$518$	30.9665	1.36059
$519$	−6.66980	−0.292772
$520$	0	0
$521$	−34.0833	−1.49322	−0.746609	−	0.665263i	$-0.768319\pi$
$521$	−34.0833	−1.49322	−0.746609	+	0.665263i	$0.768319\pi$
$522$	6.80451	0.297825
$523$	21.7437	0.950787	0.475394	−	0.879773i	$-0.342306\pi$
$523$	21.7437	0.950787	0.475394	+	0.879773i	$0.342306\pi$
$524$	−3.86529	−0.168856
$525$	0	0
$526$	−7.33020	−0.319612
$527$	−6.80451	−0.296409
$528$	6.13941	0.267184
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−8.27882	−0.359270
$532$	28.9486	1.25508
$533$	−10.2422	−0.443638
$534$	−18.6924	−0.808897
$535$	0	0
$536$	10.2788	0.443977
$537$	3.20018	0.138098
$538$	−2.80451	−0.120911
$539$	−45.8878	−1.97653
$540$	0	0
$541$	−40.9806	−1.76189	−0.880946	−	0.473217i	$-0.843093\pi$
$541$	−40.9806	−1.76189	−0.880946	+	0.473217i	$0.843093\pi$
$542$	−26.9712	−1.15851
$543$	−1.20018	−0.0515048
$544$	6.80451	0.291741
$545$	0	0
$546$	−15.7484	−0.673971
$547$	−5.33959	−0.228305	−0.114152	−	0.993463i	$-0.536415\pi$
$547$	−5.33959	−0.228305	−0.114152	+	0.993463i	$0.536415\pi$
$548$	−20.0226	−0.855321
$549$	−4.13941	−0.176666
$550$	0	0
$551$	−51.7757	−2.20572
$552$	−2.00000	−0.0851257
$553$	18.2788	0.777294
$554$	7.49687	0.318511
$555$	0	0
$556$	−11.6137	−0.492531
$557$	−17.7531	−0.752224	−0.376112	−	0.926574i	$-0.622739\pi$
$557$	−17.7531	−0.752224	−0.376112	+	0.926574i	$0.622739\pi$
$558$	−1.00000	−0.0423334
$559$	−13.8045	−0.583869
$560$	0	0
$561$	41.7757	1.76377
$562$	4.86529	0.205230
$563$	2.02256	0.0852407	0.0426203	−	0.999091i	$-0.486429\pi$
$563$	2.02256	0.0852407	0.0426203	+	0.999091i	$0.486429\pi$
$564$	−11.2788	−0.474924
$565$	0	0
$566$	−8.93923	−0.375744
$567$	−3.80451	−0.159774
$568$	−4.33020	−0.181691
$569$	−7.73057	−0.324082	−0.162041	−	0.986784i	$-0.551808\pi$
$569$	−7.73057	−0.324082	−0.162041	+	0.986784i	$0.551808\pi$
$570$	0	0
$571$	42.1845	1.76537	0.882684	−	0.469967i	$-0.155734\pi$
$571$	42.1845	1.76537	0.882684	+	0.469967i	$0.155734\pi$
$572$	25.4135	1.06259
$573$	−7.08333	−0.295910
$574$	−9.41353	−0.392913
$575$	0	0
$576$	1.00000	0.0416667
$577$	44.5708	1.85551	0.927753	−	0.373194i	$-0.121737\pi$
$577$	44.5708	1.85551	0.927753	+	0.373194i	$0.121737\pi$
$578$	29.3014	1.21878
$579$	−3.52569	−0.146523
$580$	0	0
$581$	−43.6363	−1.81034
$582$	13.0833	0.542322
$583$	−58.1378	−2.40782
$584$	10.0000	0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	−30.9759	−1.27851	−0.639255	−	0.768994i	$-0.720757\pi$
$587$	−30.9759	−1.27851	−0.639255	+	0.768994i	$0.720757\pi$
$588$	−7.47431	−0.308235
$589$	7.60902	0.313524
$590$	0	0
$591$	19.2227	0.790718
$592$	−8.13941	−0.334528
$593$	−37.0319	−1.52072	−0.760360	−	0.649502i	$-0.774977\pi$
$593$	−37.0319	−1.52072	−0.760360	+	0.649502i	$0.774977\pi$
$594$	6.13941	0.251903
$595$	0	0
$596$	8.94862	0.366550
$597$	−1.33959	−0.0548259
$598$	−8.27882	−0.338546
$599$	0.599633	0.0245003	0.0122502	−	0.999925i	$-0.496101\pi$
$599$	0.599633	0.0245003	0.0122502	+	0.999925i	$0.496101\pi$
$600$	0	0
$601$	−38.8365	−1.58417	−0.792086	−	0.610409i	$-0.791005\pi$
$601$	−38.8365	−1.58417	−0.792086	+	0.610409i	$0.791005\pi$
$602$	−12.6877	−0.517110
$603$	10.2788	0.418586
$604$	12.1347	0.493755
$605$	0	0
$606$	−1.33020	−0.0540358
$607$	−9.41353	−0.382084	−0.191042	−	0.981582i	$-0.561187\pi$
$607$	−9.41353	−0.382084	−0.191042	+	0.981582i	$0.561187\pi$
$608$	−7.60902	−0.308587
$609$	25.8878	1.04903
$610$	0	0
$611$	−46.6877	−1.88878
$612$	6.80451	0.275056
$613$	−1.06077	−0.0428442	−0.0214221	−	0.999771i	$-0.506819\pi$
$613$	−1.06077	−0.0428442	−0.0214221	+	0.999771i	$0.506819\pi$
$614$	−22.5482	−0.909973
$615$	0	0
$616$	23.3575	0.941099
$617$	40.3715	1.62530	0.812649	−	0.582754i	$-0.198025\pi$
$617$	40.3715	1.62530	0.812649	+	0.582754i	$0.198025\pi$
$618$	−13.8045	−0.555299
$619$	20.1620	0.810378	0.405189	−	0.914233i	$-0.367206\pi$
$619$	20.1620	0.810378	0.405189	+	0.914233i	$0.367206\pi$
$620$	0	0
$621$	−2.00000	−0.0802572
$622$	27.1573	1.08891
$623$	−71.1153	−2.84917
$624$	4.13941	0.165709
$625$	0	0
$626$	34.5482	1.38083
$627$	−46.7149	−1.86561
$628$	12.4135	0.495354
$629$	−55.3847	−2.20833
$630$	0	0
$631$	2.79512	0.111272	0.0556360	−	0.998451i	$-0.482281\pi$
$631$	2.79512	0.111272	0.0556360	+	0.998451i	$0.482281\pi$
$632$	−4.80451	−0.191113
$633$	−0.390977	−0.0155399
$634$	7.33959	0.291492
$635$	0	0
$636$	−9.46961	−0.375495
$637$	−30.9392	−1.22586
$638$	−41.7757	−1.65392
$639$	−4.33020	−0.171300
$640$	0	0
$641$	−7.33959	−0.289897	−0.144948	−	0.989439i	$-0.546302\pi$
$641$	−7.33959	−0.289897	−0.144948	+	0.989439i	$0.546302\pi$
$642$	15.0833	0.595292
$643$	25.4922	1.00531	0.502657	−	0.864486i	$-0.332356\pi$
$643$	25.4922	1.00531	0.502657	+	0.864486i	$0.332356\pi$
$644$	−7.60902	−0.299838
$645$	0	0
$646$	−51.7757	−2.03709
$647$	−30.4361	−1.19657	−0.598283	−	0.801285i	$-0.704150\pi$
$647$	−30.4361	−1.19657	−0.598283	+	0.801285i	$0.704150\pi$
$648$	1.00000	0.0392837
$649$	50.8271	1.99514
$650$	0	0
$651$	−3.80451	−0.149111
$652$	4.66980	0.182883
$653$	22.8271	0.893292	0.446646	−	0.894711i	$-0.352618\pi$
$653$	22.8271	0.893292	0.446646	+	0.894711i	$0.352618\pi$
$654$	7.08333	0.276980
$655$	0	0
$656$	2.47431	0.0966055
$657$	10.0000	0.390137
$658$	−42.9104	−1.67282
$659$	−20.8365	−0.811673	−0.405837	−	0.913946i	$-0.633020\pi$
$659$	−20.8365	−0.811673	−0.405837	+	0.913946i	$0.633020\pi$
$660$	0	0
$661$	21.9104	0.852216	0.426108	−	0.904672i	$-0.359884\pi$
$661$	21.9104	0.852216	0.426108	+	0.904672i	$0.359884\pi$
$662$	−3.46961	−0.134850
$663$	28.1667	1.09390
$664$	11.4696	0.445107
$665$	0	0
$666$	−8.13941	−0.315396
$667$	13.6090	0.526943
$668$	−15.2180	−0.588804
$669$	−10.8045	−0.417727
$670$	0	0
$671$	25.4135	0.981079
$672$	3.80451	0.146762
$673$	−31.7663	−1.22450	−0.612250	−	0.790664i	$-0.709736\pi$
$673$	−31.7663	−1.22450	−0.612250	+	0.790664i	$0.709736\pi$
$674$	17.8878	0.689014
$675$	0	0
$676$	4.13471	0.159027
$677$	21.3800	0.821701	0.410850	−	0.911703i	$-0.365232\pi$
$677$	21.3800	0.821701	0.410850	+	0.911703i	$0.365232\pi$
$678$	−6.00000	−0.230429
$679$	49.7757	1.91022
$680$	0	0
$681$	−22.4135	−0.858889
$682$	6.13941	0.235090
$683$	−1.19549	−0.0457441	−0.0228720	−	0.999738i	$-0.507281\pi$
$683$	−1.19549	−0.0457441	−0.0228720	+	0.999738i	$0.507281\pi$
$684$	−7.60902	−0.290938
$685$	0	0
$686$	−1.80451	−0.0688966
$687$	12.2788	0.468466
$688$	3.33490	0.127142
$689$	−39.1986	−1.49335
$690$	0	0
$691$	1.49687	0.0569435	0.0284717	−	0.999595i	$-0.490936\pi$
$691$	1.49687	0.0569435	0.0284717	+	0.999595i	$0.490936\pi$
$692$	6.66980	0.253548
$693$	23.3575	0.887276
$694$	13.0786	0.496458
$695$	0	0
$696$	−6.80451	−0.257924
$697$	16.8365	0.637726
$698$	−14.0226	−0.530762
$699$	0.0833315	0.00315189
$700$	0	0
$701$	45.3847	1.71416	0.857078	−	0.515186i	$-0.172277\pi$
$701$	45.3847	1.71416	0.857078	+	0.515186i	$0.172277\pi$
$702$	4.13941	0.156232
$703$	61.9330	2.33585
$704$	−6.13941	−0.231388
$705$	0	0
$706$	15.2274	0.573092
$707$	−5.06077	−0.190330
$708$	8.27882	0.311137
$709$	33.1059	1.24332	0.621659	−	0.783288i	$-0.286459\pi$
$709$	33.1059	1.24332	0.621659	+	0.783288i	$0.286459\pi$
$710$	0	0
$711$	−4.80451	−0.180183
$712$	18.6924	0.700526
$713$	−2.00000	−0.0749006
$714$	25.8878	0.968828
$715$	0	0
$716$	−3.20018	−0.119596
$717$	0.660406	0.0246633
$718$	6.66041	0.248564
$719$	8.00939	0.298700	0.149350	−	0.988784i	$-0.452282\pi$
$719$	8.00939	0.298700	0.149350	+	0.988784i	$0.452282\pi$
$720$	0	0
$721$	−52.5194	−1.95593
$722$	38.8972	1.44761
$723$	16.5576	0.615785
$724$	1.20018	0.0446045
$725$	0	0
$726$	−26.6924	−0.990646
$727$	1.12532	0.0417360	0.0208680	−	0.999782i	$-0.493357\pi$
$727$	1.12532	0.0417360	0.0208680	+	0.999782i	$0.493357\pi$
$728$	15.7484	0.583676
$729$	1.00000	0.0370370
$730$	0	0
$731$	22.6924	0.839307
$732$	4.13941	0.152997
$733$	18.5351	0.684609	0.342304	−	0.939589i	$-0.388793\pi$
$733$	18.5351	0.684609	0.342304	+	0.939589i	$0.388793\pi$
$734$	21.0833	0.778200
$735$	0	0
$736$	2.00000	0.0737210
$737$	−63.1059	−2.32454
$738$	2.47431	0.0910805
$739$	−24.9618	−0.918234	−0.459117	−	0.888376i	$-0.651834\pi$
$739$	−24.9618	−0.918234	−0.459117	+	0.888376i	$0.651834\pi$
$740$	0	0
$741$	−31.4969	−1.15707
$742$	−36.0273	−1.32260
$743$	4.93923	0.181203	0.0906013	−	0.995887i	$-0.471121\pi$
$743$	4.93923	0.181203	0.0906013	+	0.995887i	$0.471121\pi$
$744$	1.00000	0.0366618
$745$	0	0
$746$	12.1347	0.444283
$747$	11.4696	0.419651
$748$	−41.7757	−1.52747
$749$	57.3847	2.09679
$750$	0	0
$751$	−5.63785	−0.205728	−0.102864	−	0.994695i	$-0.532801\pi$
$751$	−5.63785	−0.205728	−0.102864	+	0.994695i	$0.532801\pi$
$752$	11.2788	0.411296
$753$	−13.6363	−0.496934
$754$	−28.1667	−1.02577
$755$	0	0
$756$	3.80451	0.138369
$757$	−41.2453	−1.49909	−0.749543	−	0.661956i	$-0.769727\pi$
$757$	−41.2453	−1.49909	−0.749543	+	0.661956i	$0.769727\pi$
$758$	19.8878	0.722359
$759$	12.2788	0.445693
$760$	0	0
$761$	31.2500	1.13281	0.566406	−	0.824127i	$-0.308334\pi$
$761$	31.2500	1.13281	0.566406	+	0.824127i	$0.308334\pi$
$762$	−3.47431	−0.125861
$763$	26.9486	0.975605
$764$	7.08333	0.256266
$765$	0	0
$766$	30.1667	1.08997
$767$	34.2694	1.23740
$768$	−1.00000	−0.0360844
$769$	14.7306	0.531198	0.265599	−	0.964084i	$-0.414430\pi$
$769$	14.7306	0.531198	0.265599	+	0.964084i	$0.414430\pi$
$770$	0	0
$771$	−7.13471	−0.256950
$772$	3.52569	0.126892
$773$	30.0226	1.07984	0.539918	−	0.841718i	$-0.318455\pi$
$773$	30.0226	1.07984	0.539918	+	0.841718i	$0.318455\pi$
$774$	3.33490	0.119870
$775$	0	0
$776$	−13.0833	−0.469664
$777$	−30.9665	−1.11092
$778$	−17.3302	−0.621318
$779$	−18.8271	−0.674550
$780$	0	0
$781$	26.5849	0.951282
$782$	13.6090	0.486658
$783$	−6.80451	−0.243173
$784$	7.47431	0.266940
$785$	0	0
$786$	3.86529	0.137870
$787$	−42.6877	−1.52165	−0.760825	−	0.648957i	$-0.775206\pi$
$787$	−42.6877	−1.52165	−0.760825	+	0.648957i	$0.775206\pi$
$788$	−19.2227	−0.684782
$789$	7.33020	0.260962
$790$	0	0
$791$	−22.8271	−0.811637
$792$	−6.13941	−0.218154
$793$	17.1347	0.608472
$794$	−34.9486	−1.24028
$795$	0	0
$796$	1.33959	0.0474806
$797$	42.4455	1.50350	0.751748	−	0.659450i	$-0.229211\pi$
$797$	42.4455	1.50350	0.751748	+	0.659450i	$0.229211\pi$
$798$	−28.9486	−1.02477
$799$	76.7469	2.71511
$800$	0	0
$801$	18.6924	0.660462
$802$	−25.2180	−0.890480
$803$	−61.3941	−2.16655
$804$	−10.2788	−0.362506
$805$	0	0
$806$	4.13941	0.145804
$807$	2.80451	0.0987235
$808$	1.33020	0.0467964
$809$	−16.7017	−0.587202	−0.293601	−	0.955928i	$-0.594854\pi$
$809$	−16.7017	−0.587202	−0.293601	+	0.955928i	$0.594854\pi$
$810$	0	0
$811$	−22.1573	−0.778047	−0.389024	−	0.921228i	$-0.627188\pi$
$811$	−22.1573	−0.778047	−0.389024	+	0.921228i	$0.627188\pi$
$812$	−25.8878	−0.908485
$813$	26.9712	0.945920
$814$	49.9712	1.75149
$815$	0	0
$816$	−6.80451	−0.238206
$817$	−25.3753	−0.887770
$818$	19.3302	0.675865
$819$	15.7484	0.550295
$820$	0	0
$821$	12.4088	0.433071	0.216536	−	0.976275i	$-0.430524\pi$
$821$	12.4088	0.433071	0.216536	+	0.976275i	$0.430524\pi$
$822$	20.0226	0.698367
$823$	−51.2180	−1.78535	−0.892674	−	0.450702i	$-0.851174\pi$
$823$	−51.2180	−1.78535	−0.892674	+	0.450702i	$0.851174\pi$
$824$	13.8045	0.480903
$825$	0	0
$826$	31.4969	1.09592
$827$	−9.83334	−0.341939	−0.170969	−	0.985276i	$-0.554690\pi$
$827$	−9.83334	−0.341939	−0.170969	+	0.985276i	$0.554690\pi$
$828$	2.00000	0.0695048
$829$	−29.2547	−1.01606	−0.508029	−	0.861340i	$-0.669626\pi$
$829$	−29.2547	−1.01606	−0.508029	+	0.861340i	$0.669626\pi$
$830$	0	0
$831$	−7.49687	−0.260063
$832$	−4.13941	−0.143508
$833$	50.8590	1.76216
$834$	11.6137	0.402150
$835$	0	0
$836$	46.7149	1.61567
$837$	1.00000	0.0345651
$838$	19.7531	0.682360
$839$	−11.5031	−0.397132	−0.198566	−	0.980087i	$-0.563628\pi$
$839$	−11.5031	−0.397132	−0.198566	+	0.980087i	$0.563628\pi$
$840$	0	0
$841$	17.3014	0.596599
$842$	17.7437	0.611490
$843$	−4.86529	−0.167569
$844$	0.390977	0.0134580
$845$	0	0
$846$	11.2788	0.387774
$847$	−101.551	−3.48935
$848$	9.46961	0.325188
$849$	8.93923	0.306793
$850$	0	0
$851$	−16.2788	−0.558031
$852$	4.33020	0.148350
$853$	40.1441	1.37451	0.687254	−	0.726417i	$-0.258816\pi$
$853$	40.1441	1.37451	0.687254	+	0.726417i	$0.258816\pi$
$854$	15.7484	0.538900
$855$	0	0
$856$	−15.0833	−0.515538
$857$	−28.7437	−0.981867	−0.490934	−	0.871197i	$-0.663344\pi$
$857$	−28.7437	−0.981867	−0.490934	+	0.871197i	$0.663344\pi$
$858$	−25.4135	−0.867604
$859$	3.04669	0.103952	0.0519758	−	0.998648i	$-0.483448\pi$
$859$	3.04669	0.103952	0.0519758	+	0.998648i	$0.483448\pi$
$860$	0	0
$861$	9.41353	0.320812
$862$	16.6316	0.566474
$863$	−23.8785	−0.812832	−0.406416	−	0.913688i	$-0.633222\pi$
$863$	−23.8785	−0.812832	−0.406416	+	0.913688i	$0.633222\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−25.1059	−0.853133
$867$	−29.3014	−0.995127
$868$	3.80451	0.129134
$869$	29.4969	1.00061
$870$	0	0
$871$	−42.5482	−1.44169
$872$	−7.08333	−0.239872
$873$	−13.0833	−0.442804
$874$	−15.2180	−0.514758
$875$	0	0
$876$	−10.0000	−0.337869
$877$	16.3108	0.550776	0.275388	−	0.961333i	$-0.411194\pi$
$877$	16.3108	0.550776	0.275388	+	0.961333i	$0.411194\pi$
$878$	24.8271	0.837873
$879$	−6.00000	−0.202375
$880$	0	0
$881$	−56.0771	−1.88928	−0.944642	−	0.328102i	$-0.893591\pi$
$881$	−56.0771	−1.88928	−0.944642	+	0.328102i	$0.893591\pi$
$882$	7.47431	0.251673
$883$	17.6457	0.593824	0.296912	−	0.954905i	$-0.404043\pi$
$883$	17.6457	0.593824	0.296912	+	0.954905i	$0.404043\pi$
$884$	−28.1667	−0.947347
$885$	0	0
$886$	−17.6316	−0.592345
$887$	32.8878	1.10427	0.552133	−	0.833756i	$-0.313814\pi$
$887$	32.8878	1.10427	0.552133	+	0.833756i	$0.313814\pi$
$888$	8.13941	0.273141
$889$	−13.2180	−0.443319
$890$	0	0
$891$	−6.13941	−0.205678
$892$	10.8045	0.361762
$893$	−85.8208	−2.87188
$894$	−8.94862	−0.299287
$895$	0	0
$896$	−3.80451	−0.127100
$897$	8.27882	0.276422
$898$	11.0833	0.369856
$899$	−6.80451	−0.226943
$900$	0	0
$901$	64.4361	2.14668
$902$	−15.1908	−0.505798
$903$	12.6877	0.422219
$904$	6.00000	0.199557
$905$	0	0
$906$	−12.1347	−0.403149
$907$	12.5576	0.416970	0.208485	−	0.978026i	$-0.433147\pi$
$907$	12.5576	0.416970	0.208485	+	0.978026i	$0.433147\pi$
$908$	22.4135	0.743819
$909$	1.33020	0.0441201
$910$	0	0
$911$	6.93923	0.229907	0.114953	−	0.993371i	$-0.463328\pi$
$911$	6.93923	0.229907	0.114953	+	0.993371i	$0.463328\pi$
$912$	7.60902	0.251960
$913$	−70.4167	−2.33045
$914$	−27.4969	−0.909515
$915$	0	0
$916$	−12.2788	−0.405704
$917$	14.7055	0.485619
$918$	−6.80451	−0.224582
$919$	−36.5288	−1.20497	−0.602487	−	0.798129i	$-0.705824\pi$
$919$	−36.5288	−1.20497	−0.602487	+	0.798129i	$0.705824\pi$
$920$	0	0
$921$	22.5482	0.742990
$922$	12.1488	0.400099
$923$	17.9245	0.589992
$924$	−23.3575	−0.768404
$925$	0	0
$926$	−29.3622	−0.964900
$927$	13.8045	0.453400
$928$	6.80451	0.223369
$929$	−2.92984	−0.0961248	−0.0480624	−	0.998844i	$-0.515305\pi$
$929$	−2.92984	−0.0961248	−0.0480624	+	0.998844i	$0.515305\pi$
$930$	0	0
$931$	−56.8722	−1.86391
$932$	−0.0833315	−0.00272961
$933$	−27.1573	−0.889089
$934$	36.6924	1.20061
$935$	0	0
$936$	−4.13941	−0.135301
$937$	21.7757	0.711381	0.355690	−	0.934604i	$-0.384246\pi$
$937$	21.7757	0.711381	0.355690	+	0.934604i	$0.384246\pi$
$938$	−39.1059	−1.27685
$939$	−34.5482	−1.12744
$940$	0	0
$941$	17.7531	0.578735	0.289368	−	0.957218i	$-0.406555\pi$
$941$	17.7531	0.578735	0.289368	+	0.957218i	$0.406555\pi$
$942$	−12.4135	−0.404455
$943$	4.94862	0.161149
$944$	−8.27882	−0.269453
$945$	0	0
$946$	−20.4743	−0.665677
$947$	26.1845	0.850883	0.425441	−	0.904986i	$-0.360119\pi$
$947$	26.1845	0.850883	0.425441	+	0.904986i	$0.360119\pi$
$948$	4.80451	0.156043
$949$	−41.3941	−1.34371
$950$	0	0
$951$	−7.33959	−0.238003
$952$	−25.8878	−0.839030
$953$	−1.05138	−0.0340576	−0.0170288	−	0.999855i	$-0.505421\pi$
$953$	−1.05138	−0.0340576	−0.0170288	+	0.999855i	$0.505421\pi$
$954$	9.46961	0.306590
$955$	0	0
$956$	−0.660406	−0.0213591
$957$	41.7757	1.35042
$958$	12.4229	0.401366
$959$	76.1761	2.45985
$960$	0	0
$961$	1.00000	0.0322581
$962$	33.6924	1.08629
$963$	−15.0833	−0.486054
$964$	−16.5576	−0.533286
$965$	0	0
$966$	7.60902	0.244816
$967$	11.6090	0.373321	0.186661	−	0.982424i	$-0.440234\pi$
$967$	11.6090	0.373321	0.186661	+	0.982424i	$0.440234\pi$
$968$	26.6924	0.857925
$969$	51.7757	1.66327
$970$	0	0
$971$	−12.9167	−0.414516	−0.207258	−	0.978286i	$-0.566454\pi$
$971$	−12.9167	−0.414516	−0.207258	+	0.978286i	$0.566454\pi$
$972$	−1.00000	−0.0320750
$973$	44.1845	1.41649
$974$	12.3014	0.394162
$975$	0	0
$976$	−4.13941	−0.132499
$977$	−1.64724	−0.0526998	−0.0263499	−	0.999653i	$-0.508388\pi$
$977$	−1.64724	−0.0526998	−0.0263499	+	0.999653i	$0.508388\pi$
$978$	−4.66980	−0.149324
$979$	−114.760	−3.66775
$980$	0	0
$981$	−7.08333	−0.226153
$982$	−20.0000	−0.638226
$983$	−58.4361	−1.86382	−0.931911	−	0.362687i	$-0.881859\pi$
$983$	−58.4361	−1.86382	−0.931911	+	0.362687i	$0.881859\pi$
$984$	−2.47431	−0.0788781
$985$	0	0
$986$	46.3014	1.47454
$987$	42.9104	1.36585
$988$	31.4969	1.00205
$989$	6.66980	0.212087
$990$	0	0
$991$	−24.9712	−0.793236	−0.396618	−	0.917984i	$-0.629816\pi$
$991$	−24.9712	−0.793236	−0.396618	+	0.917984i	$0.629816\pi$
$992$	−1.00000	−0.0317500
$993$	3.46961	0.110105
$994$	16.4743	0.522533
$995$	0	0
$996$	−11.4696	−0.363429
$997$	4.77256	0.151149	0.0755743	−	0.997140i	$-0.475921\pi$
$997$	4.77256	0.151149	0.0755743	+	0.997140i	$0.475921\pi$
$998$	36.9806	1.17060
$999$	8.13941	0.257520

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4650.2.a.co.1.1	yes	3
5.2	odd	4		4650.2.d.bi.3349.4		6
5.3	odd	4		4650.2.d.bi.3349.3		6
5.4	even	2		4650.2.a.cj.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4650.2.a.cj.1.3	✓	3	5.4	even	2
4650.2.a.co.1.1	yes	3	1.1	even	1	trivial
4650.2.d.bi.3349.3		6	5.3	odd	4
4650.2.d.bi.3349.4		6	5.2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.80451 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.80451 q^{7} +1.00000 q^{8} +1.00000 q^{9} -6.13941 q^{11} -1.00000 q^{12} -4.13941 q^{13} -3.80451 q^{14} +1.00000 q^{16} +6.80451 q^{17} +1.00000 q^{18} -7.60902 q^{19} +3.80451 q^{21} -6.13941 q^{22} +2.00000 q^{23} -1.00000 q^{24} -4.13941 q^{26} -1.00000 q^{27} -3.80451 q^{28} +6.80451 q^{29} -1.00000 q^{31} +1.00000 q^{32} +6.13941 q^{33} +6.80451 q^{34} +1.00000 q^{36} -8.13941 q^{37} -7.60902 q^{38} +4.13941 q^{39} +2.47431 q^{41} +3.80451 q^{42} +3.33490 q^{43} -6.13941 q^{44} +2.00000 q^{46} +11.2788 q^{47} -1.00000 q^{48} +7.47431 q^{49} -6.80451 q^{51} -4.13941 q^{52} +9.46961 q^{53} -1.00000 q^{54} -3.80451 q^{56} +7.60902 q^{57} +6.80451 q^{58} -8.27882 q^{59} -4.13941 q^{61} -1.00000 q^{62} -3.80451 q^{63} +1.00000 q^{64} +6.13941 q^{66} +10.2788 q^{67} +6.80451 q^{68} -2.00000 q^{69} -4.33020 q^{71} +1.00000 q^{72} +10.0000 q^{73} -8.13941 q^{74} -7.60902 q^{76} +23.3575 q^{77} +4.13941 q^{78} -4.80451 q^{79} +1.00000 q^{81} +2.47431 q^{82} +11.4696 q^{83} +3.80451 q^{84} +3.33490 q^{86} -6.80451 q^{87} -6.13941 q^{88} +18.6924 q^{89} +15.7484 q^{91} +2.00000 q^{92} +1.00000 q^{93} +11.2788 q^{94} -1.00000 q^{96} -13.0833 q^{97} +7.47431 q^{98} -6.13941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} + 3 q^{8} + 3 q^{9} - 6 q^{11} - 3 q^{12} + 3 q^{16} + 9 q^{17} + 3 q^{18} - 6 q^{22} + 6 q^{23} - 3 q^{24} - 3 q^{27} + 9 q^{29} - 3 q^{31} + 3 q^{32} + 6 q^{33} + 9 q^{34} + 3 q^{36} - 12 q^{37} - 6 q^{41} + 9 q^{43} - 6 q^{44} + 6 q^{46} + 9 q^{47} - 3 q^{48} + 9 q^{49} - 9 q^{51} + 18 q^{53} - 3 q^{54} + 9 q^{58} - 3 q^{62} + 3 q^{64} + 6 q^{66} + 6 q^{67} + 9 q^{68} - 6 q^{69} - 15 q^{71} + 3 q^{72} + 30 q^{73} - 12 q^{74} + 12 q^{77} - 3 q^{79} + 3 q^{81} - 6 q^{82} + 24 q^{83} + 9 q^{86} - 9 q^{87} - 6 q^{88} - 3 q^{89} + 12 q^{91} + 6 q^{92} + 3 q^{93} + 9 q^{94} - 3 q^{96} - 3 q^{97} + 9 q^{98} - 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 + 3 * q^8 + 3 * q^9 - 6 * q^11 - 3 * q^12 + 3 * q^16 + 9 * q^17 + 3 * q^18 - 6 * q^22 + 6 * q^23 - 3 * q^24 - 3 * q^27 + 9 * q^29 - 3 * q^31 + 3 * q^32 + 6 * q^33 + 9 * q^34 + 3 * q^36 - 12 * q^37 - 6 * q^41 + 9 * q^43 - 6 * q^44 + 6 * q^46 + 9 * q^47 - 3 * q^48 + 9 * q^49 - 9 * q^51 + 18 * q^53 - 3 * q^54 + 9 * q^58 - 3 * q^62 + 3 * q^64 + 6 * q^66 + 6 * q^67 + 9 * q^68 - 6 * q^69 - 15 * q^71 + 3 * q^72 + 30 * q^73 - 12 * q^74 + 12 * q^77 - 3 * q^79 + 3 * q^81 - 6 * q^82 + 24 * q^83 + 9 * q^86 - 9 * q^87 - 6 * q^88 - 3 * q^89 + 12 * q^91 + 6 * q^92 + 3 * q^93 + 9 * q^94 - 3 * q^96 - 3 * q^97 + 9 * q^98 - 6 * q^99

Introduction

L-functions

Modular forms

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Database

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