LMFDB - Embedded newform 2151.2.a.j.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,2,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2151.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:	$17.1758214748$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + \cdots + 1 $ x^20 - 4x^19 - 21x^18 + 96x^17 + 164x^16 - 936x^15 - 540x^14 + 4804x^13 + 229x^12 - 14020x^11 + 3356x^10 + 23404x^9 - 9429x^8 - 21252x^7 + 10479x^6 + 9108x^5 - 4844x^4 - 1184x^3 + 640x^2 - 56*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.61701$ of defining polynomial
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +O(q^{10})$	\(q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +2.37038 q^{10} -2.90374 q^{11} +2.33112 q^{13} -6.20223 q^{14} -4.85155 q^{16} -3.77139 q^{17} -1.21740 q^{19} -0.901109 q^{20} +4.69537 q^{22} +1.47665 q^{23} -2.85112 q^{25} -3.76944 q^{26} +2.35780 q^{28} -1.02391 q^{29} -1.72030 q^{31} +3.36495 q^{32} +6.09837 q^{34} -5.62267 q^{35} -1.08984 q^{37} +1.96855 q^{38} -3.28366 q^{40} +1.58037 q^{41} +0.433251 q^{43} -1.78496 q^{44} -2.38776 q^{46} -7.65919 q^{47} +7.71202 q^{49} +4.61028 q^{50} +1.43297 q^{52} -1.07471 q^{53} +4.25661 q^{55} +8.59188 q^{56} +1.65567 q^{58} +4.82708 q^{59} +10.7610 q^{61} +2.78174 q^{62} +4.26196 q^{64} -3.41720 q^{65} -6.11056 q^{67} -2.31832 q^{68} +9.09189 q^{70} -10.9308 q^{71} -12.1327 q^{73} +1.76228 q^{74} -0.748351 q^{76} -11.1377 q^{77} +14.7398 q^{79} +7.11192 q^{80} -2.55547 q^{82} +13.0905 q^{83} +5.52851 q^{85} -0.700570 q^{86} -6.50444 q^{88} +6.08868 q^{89} +8.94131 q^{91} +0.907716 q^{92} +12.3850 q^{94} +1.78460 q^{95} +7.99478 q^{97} -12.4704 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) $ 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8	$ 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8} + 4 q^{10} - 12 q^{11} - 4 q^{13} - 20 q^{14} + 22 q^{16} - 24 q^{17} - 4 q^{19} - 40 q^{20} - 6 q^{22} - 12 q^{23} + 22 q^{25} - 30 q^{26} - 12 q^{28} - 24 q^{29} - 4 q^{31} - 28 q^{32} + 8 q^{34} - 20 q^{35} - 10 q^{37} - 26 q^{38} + 6 q^{40} - 66 q^{41} + 8 q^{43} - 36 q^{44} - 12 q^{46} - 28 q^{47} + 18 q^{49} - 28 q^{50} - 18 q^{52} - 28 q^{53} - 4 q^{55} - 60 q^{56} - 54 q^{59} - 4 q^{61} - 20 q^{62} + 22 q^{64} - 42 q^{65} + 12 q^{67} - 12 q^{68} + 20 q^{70} - 36 q^{71} + 14 q^{73} - 50 q^{76} - 8 q^{77} - 12 q^{79} - 88 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} - 18 q^{86} - 10 q^{88} - 130 q^{89} - 6 q^{91} + 46 q^{92} - 26 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) $ 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8 + 4 * q^10 - 12 * q^11 - 4 * q^13 - 20 * q^14 + 22 * q^16 - 24 * q^17 - 4 * q^19 - 40 * q^20 - 6 * q^22 - 12 * q^23 + 22 * q^25 - 30 * q^26 - 12 * q^28 - 24 * q^29 - 4 * q^31 - 28 * q^32 + 8 * q^34 - 20 * q^35 - 10 * q^37 - 26 * q^38 + 6 * q^40 - 66 * q^41 + 8 * q^43 - 36 * q^44 - 12 * q^46 - 28 * q^47 + 18 * q^49 - 28 * q^50 - 18 * q^52 - 28 * q^53 - 4 * q^55 - 60 * q^56 - 54 * q^59 - 4 * q^61 - 20 * q^62 + 22 * q^64 - 42 * q^65 + 12 * q^67 - 12 * q^68 + 20 * q^70 - 36 * q^71 + 14 * q^73 - 50 * q^76 - 8 * q^77 - 12 * q^79 - 88 * q^80 - 8 * q^82 - 20 * q^83 + 4 * q^85 - 18 * q^86 - 10 * q^88 - 130 * q^89 - 6 * q^91 + 46 * q^92 - 26 * q^94 - 2 * q^97 - 12 * 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 $

$2$	−1.61701	−1.14340	−0.571698	−	0.820464i	$-0.693715\pi$
$2$	−1.61701	−1.14340	−0.571698	+	0.820464i	$0.693715\pi$
$3$	0	0
$3$	0	0
$4$	0.614711	0.307356
$5$	−1.46591	−0.655573	−0.327787	−	0.944752i	$-0.606303\pi$
$5$	−1.46591	−0.655573	−0.327787	+	0.944752i	$0.606303\pi$
$6$	0	0
$7$	3.83563	1.44973	0.724865	−	0.688891i	$-0.241902\pi$
$7$	3.83563	1.44973	0.724865	+	0.688891i	$0.241902\pi$
$8$	2.24002	0.791967
$9$	0	0
$10$	2.37038	0.749580
$11$	−2.90374	−0.875511	−0.437755	−	0.899094i	$-0.644226\pi$
$11$	−2.90374	−0.875511	−0.437755	+	0.899094i	$0.644226\pi$
$12$	0	0
$13$	2.33112	0.646537	0.323268	−	0.946307i	$-0.395218\pi$
$13$	2.33112	0.646537	0.323268	+	0.946307i	$0.395218\pi$
$14$	−6.20223	−1.65762
$15$	0	0
$16$	−4.85155	−1.21289
$17$	−3.77139	−0.914697	−0.457348	−	0.889288i	$-0.651201\pi$
$17$	−3.77139	−0.914697	−0.457348	+	0.889288i	$0.651201\pi$
$18$	0	0
$19$	−1.21740	−0.279291	−0.139646	−	0.990202i	$-0.544596\pi$
$19$	−1.21740	−0.279291	−0.139646	+	0.990202i	$0.544596\pi$
$20$	−0.901109	−0.201494
$21$	0	0
$22$	4.69537	1.00106
$23$	1.47665	0.307904	0.153952	−	0.988078i	$-0.450800\pi$
$23$	1.47665	0.307904	0.153952	+	0.988078i	$0.450800\pi$
$24$	0	0
$25$	−2.85112	−0.570224
$26$	−3.76944	−0.739248
$27$	0	0
$28$	2.35780	0.445583
$29$	−1.02391	−0.190136	−0.0950678	−	0.995471i	$-0.530307\pi$
$29$	−1.02391	−0.190136	−0.0950678	+	0.995471i	$0.530307\pi$
$30$	0	0
$31$	−1.72030	−0.308975	−0.154487	−	0.987995i	$-0.549373\pi$
$31$	−1.72030	−0.308975	−0.154487	+	0.987995i	$0.549373\pi$
$32$	3.36495	0.594845
$33$	0	0
$34$	6.09837	1.04586
$35$	−5.62267	−0.950404
$36$	0	0
$37$	−1.08984	−0.179169	−0.0895845	−	0.995979i	$-0.528554\pi$
$37$	−1.08984	−0.179169	−0.0895845	+	0.995979i	$0.528554\pi$
$38$	1.96855	0.319341
$39$	0	0
$40$	−3.28366	−0.519192
$41$	1.58037	0.246812	0.123406	−	0.992356i	$-0.460618\pi$
$41$	1.58037	0.246812	0.123406	+	0.992356i	$0.460618\pi$
$42$	0	0
$43$	0.433251	0.0660702	0.0330351	−	0.999454i	$-0.489483\pi$
$43$	0.433251	0.0660702	0.0330351	+	0.999454i	$0.489483\pi$
$44$	−1.78496	−0.269093
$45$	0	0
$46$	−2.38776	−0.352056
$47$	−7.65919	−1.11721	−0.558604	−	0.829434i	$-0.688663\pi$
$47$	−7.65919	−1.11721	−0.558604	+	0.829434i	$0.688663\pi$
$48$	0	0
$49$	7.71202	1.10172
$50$	4.61028	0.651992
$51$	0	0
$52$	1.43297	0.198717
$53$	−1.07471	−0.147623	−0.0738117	−	0.997272i	$-0.523516\pi$
$53$	−1.07471	−0.147623	−0.0738117	+	0.997272i	$0.523516\pi$
$54$	0	0
$55$	4.25661	0.573961
$56$	8.59188	1.14814
$57$	0	0
$58$	1.65567	0.217400
$59$	4.82708	0.628432	0.314216	−	0.949351i	$-0.398258\pi$
$59$	4.82708	0.628432	0.314216	+	0.949351i	$0.398258\pi$
$60$	0	0
$61$	10.7610	1.37780	0.688902	−	0.724854i	$-0.258093\pi$
$61$	10.7610	1.37780	0.688902	+	0.724854i	$0.258093\pi$
$62$	2.78174	0.353281
$63$	0	0
$64$	4.26196	0.532745
$65$	−3.41720	−0.423852
$66$	0	0
$67$	−6.11056	−0.746523	−0.373262	−	0.927726i	$-0.621761\pi$
$67$	−6.11056	−0.746523	−0.373262	+	0.927726i	$0.621761\pi$
$68$	−2.31832	−0.281137
$69$	0	0
$70$	9.09189	1.08669
$71$	−10.9308	−1.29725	−0.648626	−	0.761108i	$-0.724656\pi$
$71$	−10.9308	−1.29725	−0.648626	+	0.761108i	$0.724656\pi$
$72$	0	0
$73$	−12.1327	−1.42003	−0.710015	−	0.704187i	$-0.751312\pi$
$73$	−12.1327	−1.42003	−0.710015	+	0.704187i	$0.751312\pi$
$74$	1.76228	0.204861
$75$	0	0
$76$	−0.748351	−0.0858418
$77$	−11.1377	−1.26925
$78$	0	0
$79$	14.7398	1.65835	0.829177	−	0.558986i	$-0.188810\pi$
$79$	14.7398	1.65835	0.829177	+	0.558986i	$0.188810\pi$
$80$	7.11192	0.795137
$81$	0	0
$82$	−2.55547	−0.282204
$83$	13.0905	1.43686	0.718432	−	0.695597i	$-0.244860\pi$
$83$	13.0905	1.43686	0.718432	+	0.695597i	$0.244860\pi$
$84$	0	0
$85$	5.52851	0.599651
$86$	−0.700570	−0.0755444
$87$	0	0
$88$	−6.50444	−0.693376
$89$	6.08868	0.645398	0.322699	−	0.946502i	$-0.395410\pi$
$89$	6.08868	0.645398	0.322699	+	0.946502i	$0.395410\pi$
$90$	0	0
$91$	8.94131	0.937304
$92$	0.907716	0.0946359
$93$	0	0
$94$	12.3850	1.27741
$95$	1.78460	0.183096
$96$	0	0
$97$	7.99478	0.811747	0.405873	−	0.913929i	$-0.366967\pi$
$97$	7.99478	0.811747	0.405873	+	0.913929i	$0.366967\pi$
$98$	−12.4704	−1.25970
$99$	0	0
$100$	−1.75261	−0.175261
$101$	−12.4894	−1.24274	−0.621371	−	0.783516i	$-0.713424\pi$
$101$	−12.4894	−1.24274	−0.621371	+	0.783516i	$0.713424\pi$
$102$	0	0
$103$	−6.88824	−0.678719	−0.339359	−	0.940657i	$-0.610210\pi$
$103$	−6.88824	−0.678719	−0.339359	+	0.940657i	$0.610210\pi$
$104$	5.22176	0.512036
$105$	0	0
$106$	1.73782	0.168792
$107$	−16.7073	−1.61515	−0.807577	−	0.589763i	$-0.799221\pi$
$107$	−16.7073	−1.61515	−0.807577	+	0.589763i	$0.799221\pi$
$108$	0	0
$109$	−15.9013	−1.52307	−0.761534	−	0.648125i	$-0.775554\pi$
$109$	−15.9013	−1.52307	−0.761534	+	0.648125i	$0.775554\pi$
$110$	−6.88297	−0.656265
$111$	0	0
$112$	−18.6087	−1.75836
$113$	−9.87403	−0.928870	−0.464435	−	0.885607i	$-0.653743\pi$
$113$	−9.87403	−0.928870	−0.464435	+	0.885607i	$0.653743\pi$
$114$	0	0
$115$	−2.16464	−0.201853
$116$	−0.629410	−0.0584393
$117$	0	0
$118$	−7.80542	−0.718548
$119$	−14.4656	−1.32606
$120$	0	0
$121$	−2.56829	−0.233481
$122$	−17.4006	−1.57538
$123$	0	0
$124$	−1.05749	−0.0949652
$125$	11.5090	1.02940
$126$	0	0
$127$	−10.3172	−0.915505	−0.457753	−	0.889080i	$-0.651345\pi$
$127$	−10.3172	−0.915505	−0.457753	+	0.889080i	$0.651345\pi$
$128$	−13.6215	−1.20398
$129$	0	0
$130$	5.52564	0.484631
$131$	9.86768	0.862143	0.431072	−	0.902318i	$-0.358136\pi$
$131$	9.86768	0.862143	0.431072	+	0.902318i	$0.358136\pi$
$132$	0	0
$133$	−4.66950	−0.404897
$134$	9.88081	0.853572
$135$	0	0
$136$	−8.44800	−0.724410
$137$	−3.24609	−0.277332	−0.138666	−	0.990339i	$-0.544282\pi$
$137$	−3.24609	−0.277332	−0.138666	+	0.990339i	$0.544282\pi$
$138$	0	0
$139$	−14.2318	−1.20713	−0.603564	−	0.797315i	$-0.706253\pi$
$139$	−14.2318	−1.20713	−0.603564	+	0.797315i	$0.706253\pi$
$140$	−3.45632	−0.292112
$141$	0	0
$142$	17.6752	1.48327
$143$	−6.76897	−0.566050
$144$	0	0
$145$	1.50096	0.124648
$146$	19.6187	1.62366
$147$	0	0
$148$	−0.669938	−0.0550686
$149$	−3.90332	−0.319772	−0.159886	−	0.987135i	$-0.551113\pi$
$149$	−3.90332	−0.319772	−0.159886	+	0.987135i	$0.551113\pi$
$150$	0	0
$151$	−0.699764	−0.0569460	−0.0284730	−	0.999595i	$-0.509064\pi$
$151$	−0.699764	−0.0569460	−0.0284730	+	0.999595i	$0.509064\pi$
$152$	−2.72701	−0.221190
$153$	0	0
$154$	18.0097	1.45126
$155$	2.52180	0.202556
$156$	0	0
$157$	1.91363	0.152725	0.0763623	−	0.997080i	$-0.475669\pi$
$157$	1.91363	0.152725	0.0763623	+	0.997080i	$0.475669\pi$
$158$	−23.8343	−1.89616
$159$	0	0
$160$	−4.93270	−0.389964
$161$	5.66389	0.446377
$162$	0	0
$163$	−17.1905	−1.34647	−0.673233	−	0.739430i	$-0.735095\pi$
$163$	−17.1905	−1.34647	−0.673233	+	0.739430i	$0.735095\pi$
$164$	0.971471	0.0758592
$165$	0	0
$166$	−21.1673	−1.64290
$167$	−14.8924	−1.15241	−0.576205	−	0.817305i	$-0.695467\pi$
$167$	−14.8924	−1.15241	−0.576205	+	0.817305i	$0.695467\pi$
$168$	0	0
$169$	−7.56588	−0.581990
$170$	−8.93963	−0.685639
$171$	0	0
$172$	0.266324	0.0203070
$173$	11.7169	0.890823	0.445411	−	0.895326i	$-0.353057\pi$
$173$	11.7169	0.890823	0.445411	+	0.895326i	$0.353057\pi$
$174$	0	0
$175$	−10.9358	−0.826671
$176$	14.0876	1.06190
$177$	0	0
$178$	−9.84543	−0.737946
$179$	3.36761	0.251707	0.125854	−	0.992049i	$-0.459833\pi$
$179$	3.36761	0.251707	0.125854	+	0.992049i	$0.459833\pi$
$180$	0	0
$181$	−6.66617	−0.495493	−0.247746	−	0.968825i	$-0.579690\pi$
$181$	−6.66617	−0.495493	−0.247746	+	0.968825i	$0.579690\pi$
$182$	−14.4582	−1.07171
$183$	0	0
$184$	3.30774	0.243850
$185$	1.59761	0.117458
$186$	0	0
$187$	10.9511	0.800827
$188$	−4.70819	−0.343380
$189$	0	0
$190$	−2.88571	−0.209351
$191$	15.7911	1.14260	0.571300	−	0.820741i	$-0.306439\pi$
$191$	15.7911	1.14260	0.571300	+	0.820741i	$0.306439\pi$
$192$	0	0
$193$	8.45917	0.608904	0.304452	−	0.952528i	$-0.401527\pi$
$193$	8.45917	0.608904	0.304452	+	0.952528i	$0.401527\pi$
$194$	−12.9276	−0.928149
$195$	0	0
$196$	4.74066	0.338619
$197$	4.68346	0.333683	0.166842	−	0.985984i	$-0.446643\pi$
$197$	4.68346	0.333683	0.166842	+	0.985984i	$0.446643\pi$
$198$	0	0
$199$	−25.8363	−1.83149	−0.915743	−	0.401764i	$-0.868397\pi$
$199$	−25.8363	−1.83149	−0.915743	+	0.401764i	$0.868397\pi$
$200$	−6.38657	−0.451599
$201$	0	0
$202$	20.1955	1.42095
$203$	−3.92734	−0.275645
$204$	0	0
$205$	−2.31667	−0.161804
$206$	11.1383	0.776044
$207$	0	0
$208$	−11.3096	−0.784177
$209$	3.53502	0.244523
$210$	0	0
$211$	4.07485	0.280524	0.140262	−	0.990114i	$-0.455205\pi$
$211$	4.07485	0.280524	0.140262	+	0.990114i	$0.455205\pi$
$212$	−0.660639	−0.0453729
$213$	0	0
$214$	27.0158	1.84676
$215$	−0.635106	−0.0433138
$216$	0	0
$217$	−6.59843	−0.447930
$218$	25.7125	1.74147
$219$	0	0
$220$	2.61659	0.176410
$221$	−8.79157	−0.591385
$222$	0	0
$223$	−2.44657	−0.163835	−0.0819173	−	0.996639i	$-0.526104\pi$
$223$	−2.44657	−0.163835	−0.0819173	+	0.996639i	$0.526104\pi$
$224$	12.9067	0.862364
$225$	0	0
$226$	15.9664	1.06207
$227$	−2.74547	−0.182223	−0.0911115	−	0.995841i	$-0.529042\pi$
$227$	−2.74547	−0.182223	−0.0911115	+	0.995841i	$0.529042\pi$
$228$	0	0
$229$	13.7145	0.906277	0.453139	−	0.891440i	$-0.350304\pi$
$229$	13.7145	0.906277	0.453139	+	0.891440i	$0.350304\pi$
$230$	3.50023	0.230798
$231$	0	0
$232$	−2.29358	−0.150581
$233$	5.36762	0.351645	0.175822	−	0.984422i	$-0.443742\pi$
$233$	5.36762	0.351645	0.175822	+	0.984422i	$0.443742\pi$
$234$	0	0
$235$	11.2277	0.732412
$236$	2.96726	0.193152
$237$	0	0
$238$	23.3910	1.51622
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−12.9180	−0.832120	−0.416060	−	0.909337i	$-0.636589\pi$
$241$	−12.9180	−0.832120	−0.416060	+	0.909337i	$0.636589\pi$
$242$	4.15295	0.266962
$243$	0	0
$244$	6.61491	0.423476
$245$	−11.3051	−0.722256
$246$	0	0
$247$	−2.83791	−0.180572
$248$	−3.85351	−0.244698
$249$	0	0
$250$	−18.6101	−1.17701
$251$	−18.9396	−1.19546	−0.597729	−	0.801698i	$-0.703930\pi$
$251$	−18.9396	−1.19546	−0.597729	+	0.801698i	$0.703930\pi$
$252$	0	0
$253$	−4.28782	−0.269573
$254$	16.6830	1.04679
$255$	0	0
$256$	13.5022	0.843886
$257$	15.3234	0.955847	0.477923	−	0.878402i	$-0.341390\pi$
$257$	15.3234	0.955847	0.477923	+	0.878402i	$0.341390\pi$
$258$	0	0
$259$	−4.18023	−0.259747
$260$	−2.10059	−0.130273
$261$	0	0
$262$	−15.9561	−0.985771
$263$	−27.1435	−1.67374	−0.836870	−	0.547402i	$-0.815617\pi$
$263$	−27.1435	−1.67374	−0.836870	+	0.547402i	$0.815617\pi$
$264$	0	0
$265$	1.57543	0.0967779
$266$	7.55062	0.462958
$267$	0	0
$268$	−3.75623	−0.229448
$269$	−12.7939	−0.780058	−0.390029	−	0.920802i	$-0.627535\pi$
$269$	−12.7939	−0.780058	−0.390029	+	0.920802i	$0.627535\pi$
$270$	0	0
$271$	−28.4970	−1.73107	−0.865533	−	0.500851i	$-0.833020\pi$
$271$	−28.4970	−1.73107	−0.865533	+	0.500851i	$0.833020\pi$
$272$	18.2971	1.10943
$273$	0	0
$274$	5.24895	0.317101
$275$	8.27891	0.499237
$276$	0	0
$277$	−2.41529	−0.145120	−0.0725602	−	0.997364i	$-0.523117\pi$
$277$	−2.41529	−0.145120	−0.0725602	+	0.997364i	$0.523117\pi$
$278$	23.0130	1.38023
$279$	0	0
$280$	−12.5949	−0.752689
$281$	8.27813	0.493832	0.246916	−	0.969037i	$-0.420583\pi$
$281$	8.27813	0.493832	0.246916	+	0.969037i	$0.420583\pi$
$282$	0	0
$283$	−0.142683	−0.00848163	−0.00424082	−	0.999991i	$-0.501350\pi$
$283$	−0.142683	−0.00848163	−0.00424082	+	0.999991i	$0.501350\pi$
$284$	−6.71930	−0.398717
$285$	0	0
$286$	10.9455	0.647219
$287$	6.06171	0.357811
$288$	0	0
$289$	−2.77660	−0.163329
$290$	−2.42706	−0.142522
$291$	0	0
$292$	−7.45813	−0.436454
$293$	−12.9861	−0.758657	−0.379328	−	0.925262i	$-0.623845\pi$
$293$	−12.9861	−0.758657	−0.379328	+	0.925262i	$0.623845\pi$
$294$	0	0
$295$	−7.07605	−0.411983
$296$	−2.44127	−0.141896
$297$	0	0
$298$	6.31169	0.365627
$299$	3.44226	0.199071
$300$	0	0
$301$	1.66179	0.0957839
$302$	1.13152	0.0651119
$303$	0	0
$304$	5.90629	0.338749
$305$	−15.7746	−0.903252
$306$	0	0
$307$	7.69912	0.439412	0.219706	−	0.975566i	$-0.429490\pi$
$307$	7.69912	0.439412	0.219706	+	0.975566i	$0.429490\pi$
$308$	−6.84644	−0.390112
$309$	0	0
$310$	−4.07776	−0.231601
$311$	0.0557085	0.00315894	0.00157947	−	0.999999i	$-0.499497\pi$
$311$	0.0557085	0.00315894	0.00157947	+	0.999999i	$0.499497\pi$
$312$	0	0
$313$	−2.73407	−0.154538	−0.0772692	−	0.997010i	$-0.524620\pi$
$313$	−2.73407	−0.154538	−0.0772692	+	0.997010i	$0.524620\pi$
$314$	−3.09436	−0.174625
$315$	0	0
$316$	9.06070	0.509704
$317$	−5.74313	−0.322566	−0.161283	−	0.986908i	$-0.551563\pi$
$317$	−5.74313	−0.322566	−0.161283	+	0.986908i	$0.551563\pi$
$318$	0	0
$319$	2.97317	0.166466
$320$	−6.24763	−0.349253
$321$	0	0
$322$	−9.15855	−0.510386
$323$	4.59130	0.255467
$324$	0	0
$325$	−6.64630	−0.368671
$326$	27.7972	1.53954
$327$	0	0
$328$	3.54006	0.195467
$329$	−29.3778	−1.61965
$330$	0	0
$331$	−32.3302	−1.77703	−0.888514	−	0.458849i	$-0.848262\pi$
$331$	−32.3302	−1.77703	−0.888514	+	0.458849i	$0.848262\pi$
$332$	8.04685	0.441628
$333$	0	0
$334$	24.0811	1.31766
$335$	8.95750	0.489401
$336$	0	0
$337$	−4.91558	−0.267769	−0.133884	−	0.990997i	$-0.542745\pi$
$337$	−4.91558	−0.267769	−0.133884	+	0.990997i	$0.542745\pi$
$338$	12.2341	0.665446
$339$	0	0
$340$	3.39843	0.184306
$341$	4.99530	0.270511
$342$	0	0
$343$	2.73104	0.147462
$344$	0.970492	0.0523254
$345$	0	0
$346$	−18.9464	−1.01856
$347$	28.7741	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$347$	28.7741	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$348$	0	0
$349$	−12.6879	−0.679168	−0.339584	−	0.940576i	$-0.610286\pi$
$349$	−12.6879	−0.679168	−0.339584	+	0.940576i	$0.610286\pi$
$350$	17.6833	0.945212
$351$	0	0
$352$	−9.77094	−0.520793
$353$	6.88724	0.366571	0.183285	−	0.983060i	$-0.441327\pi$
$353$	6.88724	0.366571	0.183285	+	0.983060i	$0.441327\pi$
$354$	0	0
$355$	16.0236	0.850443
$356$	3.74278	0.198367
$357$	0	0
$358$	−5.44545	−0.287801
$359$	4.46094	0.235439	0.117720	−	0.993047i	$-0.462442\pi$
$359$	4.46094	0.235439	0.117720	+	0.993047i	$0.462442\pi$
$360$	0	0
$361$	−17.5179	−0.921996
$362$	10.7792	0.566545
$363$	0	0
$364$	5.49632	0.288085
$365$	17.7855	0.930934
$366$	0	0
$367$	4.13856	0.216031	0.108016	−	0.994149i	$-0.465550\pi$
$367$	4.13856	0.216031	0.108016	+	0.994149i	$0.465550\pi$
$368$	−7.16406	−0.373453
$369$	0	0
$370$	−2.58334	−0.134302
$371$	−4.12220	−0.214014
$372$	0	0
$373$	−15.9172	−0.824161	−0.412080	−	0.911148i	$-0.635198\pi$
$373$	−15.9172	−0.824161	−0.412080	+	0.911148i	$0.635198\pi$
$374$	−17.7081	−0.915663
$375$	0	0
$376$	−17.1568	−0.884792
$377$	−2.38686	−0.122930
$378$	0	0
$379$	35.7486	1.83628	0.918142	−	0.396252i	$-0.129689\pi$
$379$	35.7486	1.83628	0.918142	+	0.396252i	$0.129689\pi$
$380$	1.09701	0.0562756
$381$	0	0
$382$	−25.5342	−1.30645
$383$	22.7410	1.16201	0.581005	−	0.813900i	$-0.302660\pi$
$383$	22.7410	1.16201	0.581005	+	0.813900i	$0.302660\pi$
$384$	0	0
$385$	16.3268	0.832089
$386$	−13.6785	−0.696219
$387$	0	0
$388$	4.91448	0.249495
$389$	2.53628	0.128594	0.0642972	−	0.997931i	$-0.479519\pi$
$389$	2.53628	0.128594	0.0642972	+	0.997931i	$0.479519\pi$
$390$	0	0
$391$	−5.56904	−0.281639
$392$	17.2751	0.872524
$393$	0	0
$394$	−7.57319	−0.381532
$395$	−21.6071	−1.08717
$396$	0	0
$397$	−13.3410	−0.669564	−0.334782	−	0.942296i	$-0.608663\pi$
$397$	−13.3410	−0.669564	−0.334782	+	0.942296i	$0.608663\pi$
$398$	41.7775	2.09412
$399$	0	0
$400$	13.8324	0.691618
$401$	0.833758	0.0416359	0.0208179	−	0.999783i	$-0.493373\pi$
$401$	0.833758	0.0416359	0.0208179	+	0.999783i	$0.493373\pi$
$402$	0	0
$403$	−4.01023	−0.199764
$404$	−7.67738	−0.381964
$405$	0	0
$406$	6.35054	0.315172
$407$	3.16462	0.156864
$408$	0	0
$409$	12.4602	0.616118	0.308059	−	0.951367i	$-0.400321\pi$
$409$	12.4602	0.616118	0.308059	+	0.951367i	$0.400321\pi$
$410$	3.74608	0.185006
$411$	0	0
$412$	−4.23428	−0.208608
$413$	18.5149	0.911057
$414$	0	0
$415$	−19.1894	−0.941969
$416$	7.84411	0.384589
$417$	0	0
$418$	−5.71615	−0.279586
$419$	9.75869	0.476743	0.238372	−	0.971174i	$-0.423386\pi$
$419$	9.75869	0.476743	0.238372	+	0.971174i	$0.423386\pi$
$420$	0	0
$421$	16.1165	0.785470	0.392735	−	0.919652i	$-0.371529\pi$
$421$	16.1165	0.785470	0.392735	+	0.919652i	$0.371529\pi$
$422$	−6.58906	−0.320750
$423$	0	0
$424$	−2.40738	−0.116913
$425$	10.7527	0.521582
$426$	0	0
$427$	41.2752	1.99744
$428$	−10.2701	−0.496426
$429$	0	0
$430$	1.02697	0.0495249
$431$	23.3638	1.12539	0.562697	−	0.826663i	$-0.309764\pi$
$431$	23.3638	1.12539	0.562697	+	0.826663i	$0.309764\pi$
$432$	0	0
$433$	−16.8084	−0.807758	−0.403879	−	0.914812i	$-0.632338\pi$
$433$	−16.8084	−0.807758	−0.403879	+	0.914812i	$0.632338\pi$
$434$	10.6697	0.512162
$435$	0	0
$436$	−9.77471	−0.468124
$437$	−1.79768	−0.0859948
$438$	0	0
$439$	4.57370	0.218291	0.109145	−	0.994026i	$-0.465189\pi$
$439$	4.57370	0.218291	0.109145	+	0.994026i	$0.465189\pi$
$440$	9.53490	0.454558
$441$	0	0
$442$	14.2160	0.676188
$443$	8.43805	0.400904	0.200452	−	0.979704i	$-0.435759\pi$
$443$	8.43805	0.400904	0.200452	+	0.979704i	$0.435759\pi$
$444$	0	0
$445$	−8.92543	−0.423106
$446$	3.95612	0.187328
$447$	0	0
$448$	16.3473	0.772336
$449$	−11.3585	−0.536043	−0.268021	−	0.963413i	$-0.586370\pi$
$449$	−11.3585	−0.536043	−0.268021	+	0.963413i	$0.586370\pi$
$450$	0	0
$451$	−4.58898	−0.216087
$452$	−6.06967	−0.285493
$453$	0	0
$454$	4.43944	0.208353
$455$	−13.1071	−0.614471
$456$	0	0
$457$	13.8039	0.645720	0.322860	−	0.946447i	$-0.395356\pi$
$457$	13.8039	0.645720	0.322860	+	0.946447i	$0.395356\pi$
$458$	−22.1764	−1.03623
$459$	0	0
$460$	−1.33063	−0.0620408
$461$	−12.8220	−0.597179	−0.298589	−	0.954382i	$-0.596516\pi$
$461$	−12.8220	−0.597179	−0.298589	+	0.954382i	$0.596516\pi$
$462$	0	0
$463$	−5.22053	−0.242618	−0.121309	−	0.992615i	$-0.538709\pi$
$463$	−5.22053	−0.242618	−0.121309	+	0.992615i	$0.538709\pi$
$464$	4.96756	0.230613
$465$	0	0
$466$	−8.67948	−0.402069
$467$	8.47835	0.392331	0.196166	−	0.980571i	$-0.437151\pi$
$467$	8.47835	0.392331	0.196166	+	0.980571i	$0.437151\pi$
$468$	0	0
$469$	−23.4378	−1.08226
$470$	−18.1552	−0.837437
$471$	0	0
$472$	10.8128	0.497698
$473$	−1.25805	−0.0578451
$474$	0	0
$475$	3.47096	0.159259
$476$	−8.89219	−0.407573
$477$	0	0
$478$	1.61701	0.0739602
$479$	−37.0880	−1.69459	−0.847297	−	0.531119i	$-0.821772\pi$
$479$	−37.0880	−1.69459	−0.847297	+	0.531119i	$0.821772\pi$
$480$	0	0
$481$	−2.54056	−0.115839
$482$	20.8885	0.951443
$483$	0	0
$484$	−1.57876	−0.0717618
$485$	−11.7196	−0.532159
$486$	0	0
$487$	34.2956	1.55408	0.777041	−	0.629450i	$-0.216720\pi$
$487$	34.2956	1.55408	0.777041	+	0.629450i	$0.216720\pi$
$488$	24.1049	1.09118
$489$	0	0
$490$	18.2804	0.825825
$491$	1.07986	0.0487335	0.0243667	−	0.999703i	$-0.492243\pi$
$491$	1.07986	0.0487335	0.0243667	+	0.999703i	$0.492243\pi$
$492$	0	0
$493$	3.86157	0.173916
$494$	4.58893	0.206466
$495$	0	0
$496$	8.34613	0.374752
$497$	−41.9266	−1.88066
$498$	0	0
$499$	0.972126	0.0435183	0.0217592	−	0.999763i	$-0.493073\pi$
$499$	0.972126	0.0435183	0.0217592	+	0.999763i	$0.493073\pi$
$500$	7.07471	0.316391
$501$	0	0
$502$	30.6255	1.36688
$503$	−30.5203	−1.36083	−0.680417	−	0.732825i	$-0.738201\pi$
$503$	−30.5203	−1.36083	−0.680417	+	0.732825i	$0.738201\pi$
$504$	0	0
$505$	18.3083	0.814708
$506$	6.93343	0.308229
$507$	0	0
$508$	−6.34211	−0.281386
$509$	20.2337	0.896845	0.448422	−	0.893822i	$-0.351986\pi$
$509$	20.2337	0.896845	0.448422	+	0.893822i	$0.351986\pi$
$510$	0	0
$511$	−46.5367	−2.05866
$512$	5.40993	0.239087
$513$	0	0
$514$	−24.7780	−1.09291
$515$	10.0975	0.444950
$516$	0	0
$517$	22.2403	0.978127
$518$	6.75946	0.296993
$519$	0	0
$520$	−7.65461	−0.335677
$521$	17.2171	0.754296	0.377148	−	0.926153i	$-0.376905\pi$
$521$	17.2171	0.754296	0.377148	+	0.926153i	$0.376905\pi$
$522$	0	0
$523$	−2.23753	−0.0978405	−0.0489202	−	0.998803i	$-0.515578\pi$
$523$	−2.23753	−0.0978405	−0.0489202	+	0.998803i	$0.515578\pi$
$524$	6.06577	0.264985
$525$	0	0
$526$	43.8912	1.91375
$527$	6.48793	0.282618
$528$	0	0
$529$	−20.8195	−0.905195
$530$	−2.54748	−0.110656
$531$	0	0
$532$	−2.87039	−0.124447
$533$	3.68403	0.159573
$534$	0	0
$535$	24.4913	1.05885
$536$	−13.6878	−0.591222
$537$	0	0
$538$	20.6878	0.891916
$539$	−22.3937	−0.964565
$540$	0	0
$541$	−4.02257	−0.172944	−0.0864719	−	0.996254i	$-0.527559\pi$
$541$	−4.02257	−0.172944	−0.0864719	+	0.996254i	$0.527559\pi$
$542$	46.0798	1.97930
$543$	0	0
$544$	−12.6905	−0.544103
$545$	23.3098	0.998483
$546$	0	0
$547$	−16.4728	−0.704327	−0.352163	−	0.935939i	$-0.614554\pi$
$547$	−16.4728	−0.704327	−0.352163	+	0.935939i	$0.614554\pi$
$548$	−1.99541	−0.0852397
$549$	0	0
$550$	−13.3871	−0.570826
$551$	1.24651	0.0531033
$552$	0	0
$553$	56.5362	2.40417
$554$	3.90553	0.165930
$555$	0	0
$556$	−8.74846	−0.371018
$557$	−28.4924	−1.20726	−0.603631	−	0.797264i	$-0.706280\pi$
$557$	−28.4924	−1.20726	−0.603631	+	0.797264i	$0.706280\pi$
$558$	0	0
$559$	1.00996	0.0427168
$560$	27.2787	1.15273
$561$	0	0
$562$	−13.3858	−0.564646
$563$	19.7584	0.832716	0.416358	−	0.909201i	$-0.363306\pi$
$563$	19.7584	0.832716	0.416358	+	0.909201i	$0.363306\pi$
$564$	0	0
$565$	14.4744	0.608942
$566$	0.230720	0.00969787
$567$	0	0
$568$	−24.4853	−1.02738
$569$	9.62616	0.403550	0.201775	−	0.979432i	$-0.435329\pi$
$569$	9.62616	0.403550	0.201775	+	0.979432i	$0.435329\pi$
$570$	0	0
$571$	17.4099	0.728583	0.364292	−	0.931285i	$-0.381311\pi$
$571$	17.4099	0.728583	0.364292	+	0.931285i	$0.381311\pi$
$572$	−4.16096	−0.173979
$573$	0	0
$574$	−9.80182	−0.409120
$575$	−4.21012	−0.175574
$576$	0	0
$577$	39.3271	1.63721	0.818605	−	0.574357i	$-0.194748\pi$
$577$	39.3271	1.63721	0.818605	+	0.574357i	$0.194748\pi$
$578$	4.48978	0.186750
$579$	0	0
$580$	0.922656	0.0383112
$581$	50.2101	2.08306
$582$	0	0
$583$	3.12069	0.129246
$584$	−27.1776	−1.12462
$585$	0	0
$586$	20.9986	0.867445
$587$	26.7724	1.10502	0.552508	−	0.833507i	$-0.313671\pi$
$587$	26.7724	1.10502	0.552508	+	0.833507i	$0.313671\pi$
$588$	0	0
$589$	2.09430	0.0862941
$590$	11.4420	0.471060
$591$	0	0
$592$	5.28743	0.217312
$593$	−44.4733	−1.82630	−0.913150	−	0.407623i	$-0.866358\pi$
$593$	−44.4733	−1.82630	−0.913150	+	0.407623i	$0.866358\pi$
$594$	0	0
$595$	21.2053	0.869332
$596$	−2.39941	−0.0982838
$597$	0	0
$598$	−5.56616	−0.227617
$599$	19.6240	0.801815	0.400908	−	0.916118i	$-0.368695\pi$
$599$	19.6240	0.801815	0.400908	+	0.916118i	$0.368695\pi$
$600$	0	0
$601$	15.4787	0.631389	0.315695	−	0.948861i	$-0.397763\pi$
$601$	15.4787	0.631389	0.315695	+	0.948861i	$0.397763\pi$
$602$	−2.68712	−0.109519
$603$	0	0
$604$	−0.430153	−0.0175027
$605$	3.76488	0.153064
$606$	0	0
$607$	20.5122	0.832566	0.416283	−	0.909235i	$-0.363333\pi$
$607$	20.5122	0.832566	0.416283	+	0.909235i	$0.363333\pi$
$608$	−4.09650	−0.166135
$609$	0	0
$610$	25.5077	1.03277
$611$	−17.8545	−0.722316
$612$	0	0
$613$	32.1077	1.29682	0.648409	−	0.761292i	$-0.275435\pi$
$613$	32.1077	1.29682	0.648409	+	0.761292i	$0.275435\pi$
$614$	−12.4495	−0.502422
$615$	0	0
$616$	−24.9486	−1.00521
$617$	16.7973	0.676233	0.338117	−	0.941104i	$-0.390210\pi$
$617$	16.7973	0.676233	0.338117	+	0.941104i	$0.390210\pi$
$618$	0	0
$619$	21.4017	0.860206	0.430103	−	0.902780i	$-0.358477\pi$
$619$	21.4017	0.860206	0.430103	+	0.902780i	$0.358477\pi$
$620$	1.55018	0.0622566
$621$	0	0
$622$	−0.0900810	−0.00361192
$623$	23.3539	0.935653
$624$	0	0
$625$	−2.61552	−0.104621
$626$	4.42100	0.176699
$627$	0	0
$628$	1.17633	0.0469407
$629$	4.11022	0.163885
$630$	0	0
$631$	−16.0747	−0.639924	−0.319962	−	0.947430i	$-0.603670\pi$
$631$	−16.0747	−0.639924	−0.319962	+	0.947430i	$0.603670\pi$
$632$	33.0174	1.31336
$633$	0	0
$634$	9.28668	0.368821
$635$	15.1241	0.600181
$636$	0	0
$637$	17.9777	0.712300
$638$	−4.80764	−0.190336
$639$	0	0
$640$	19.9679	0.789299
$641$	−35.1303	−1.38756	−0.693782	−	0.720185i	$-0.744057\pi$
$641$	−35.1303	−1.38756	−0.693782	+	0.720185i	$0.744057\pi$
$642$	0	0
$643$	−1.14336	−0.0450898	−0.0225449	−	0.999746i	$-0.507177\pi$
$643$	−1.14336	−0.0450898	−0.0225449	+	0.999746i	$0.507177\pi$
$644$	3.48166	0.137197
$645$	0	0
$646$	−7.42417	−0.292100
$647$	−11.6354	−0.457433	−0.228716	−	0.973493i	$-0.573453\pi$
$647$	−11.6354	−0.457433	−0.228716	+	0.973493i	$0.573453\pi$
$648$	0	0
$649$	−14.0166	−0.550199
$650$	10.7471	0.421537
$651$	0	0
$652$	−10.5672	−0.413844
$653$	2.96261	0.115936	0.0579680	−	0.998318i	$-0.481538\pi$
$653$	2.96261	0.115936	0.0579680	+	0.998318i	$0.481538\pi$
$654$	0	0
$655$	−14.4651	−0.565198
$656$	−7.66725	−0.299356
$657$	0	0
$658$	47.5041	1.85190
$659$	35.0365	1.36483	0.682415	−	0.730965i	$-0.260930\pi$
$659$	35.0365	1.36483	0.682415	+	0.730965i	$0.260930\pi$
$660$	0	0
$661$	−30.3727	−1.18136	−0.590681	−	0.806905i	$-0.701141\pi$
$661$	−30.3727	−1.18136	−0.590681	+	0.806905i	$0.701141\pi$
$662$	52.2782	2.03185
$663$	0	0
$664$	29.3229	1.13795
$665$	6.84505	0.265440
$666$	0	0
$667$	−1.51196	−0.0585435
$668$	−9.15453	−0.354199
$669$	0	0
$670$	−14.4843	−0.559579
$671$	−31.2471	−1.20628
$672$	0	0
$673$	0.848634	0.0327124	0.0163562	−	0.999866i	$-0.494793\pi$
$673$	0.848634	0.0327124	0.0163562	+	0.999866i	$0.494793\pi$
$674$	7.94852	0.306166
$675$	0	0
$676$	−4.65083	−0.178878
$677$	1.77815	0.0683400	0.0341700	−	0.999416i	$-0.489121\pi$
$677$	1.77815	0.0683400	0.0341700	+	0.999416i	$0.489121\pi$
$678$	0	0
$679$	30.6650	1.17681
$680$	12.3840	0.474904
$681$	0	0
$682$	−8.07744	−0.309301
$683$	34.3363	1.31384	0.656922	−	0.753959i	$-0.271858\pi$
$683$	34.3363	1.31384	0.656922	+	0.753959i	$0.271858\pi$
$684$	0	0
$685$	4.75847	0.181812
$686$	−4.41611	−0.168608
$687$	0	0
$688$	−2.10194	−0.0801357
$689$	−2.50529	−0.0954439
$690$	0	0
$691$	30.6872	1.16740	0.583698	−	0.811971i	$-0.301605\pi$
$691$	30.6872	1.16740	0.583698	+	0.811971i	$0.301605\pi$
$692$	7.20254	0.273799
$693$	0	0
$694$	−46.5279	−1.76618
$695$	20.8625	0.791361
$696$	0	0
$697$	−5.96020	−0.225759
$698$	20.5164	0.776558
$699$	0	0
$700$	−6.72237	−0.254082
$701$	−1.99537	−0.0753641	−0.0376820	−	0.999290i	$-0.511997\pi$
$701$	−1.99537	−0.0753641	−0.0376820	+	0.999290i	$0.511997\pi$
$702$	0	0
$703$	1.32678	0.0500404
$704$	−12.3756	−0.466424
$705$	0	0
$706$	−11.1367	−0.419136
$707$	−47.9047	−1.80164
$708$	0	0
$709$	33.9090	1.27348	0.636739	−	0.771080i	$-0.280283\pi$
$709$	33.9090	1.27348	0.636739	+	0.771080i	$0.280283\pi$
$710$	−25.9102	−0.972394
$711$	0	0
$712$	13.6388	0.511134
$713$	−2.54029	−0.0951345
$714$	0	0
$715$	9.92267	0.371087
$716$	2.07011	0.0773636
$717$	0	0
$718$	−7.21337	−0.269201
$719$	−24.1685	−0.901331	−0.450666	−	0.892693i	$-0.648813\pi$
$719$	−24.1685	−0.901331	−0.450666	+	0.892693i	$0.648813\pi$
$720$	0	0
$721$	−26.4207	−0.983959
$722$	28.3266	1.05421
$723$	0	0
$724$	−4.09777	−0.152292
$725$	2.91929	0.108420
$726$	0	0
$727$	−46.2043	−1.71362	−0.856811	−	0.515631i	$-0.827558\pi$
$727$	−46.2043	−1.71362	−0.856811	+	0.515631i	$0.827558\pi$
$728$	20.0287	0.742314
$729$	0	0
$730$	−28.7592	−1.06443
$731$	−1.63396	−0.0604342
$732$	0	0
$733$	−3.13031	−0.115621	−0.0578104	−	0.998328i	$-0.518412\pi$
$733$	−3.13031	−0.115621	−0.0578104	+	0.998328i	$0.518412\pi$
$734$	−6.69208	−0.247009
$735$	0	0
$736$	4.96887	0.183155
$737$	17.7435	0.653589
$738$	0	0
$739$	−46.7714	−1.72052	−0.860258	−	0.509859i	$-0.829697\pi$
$739$	−46.7714	−1.72052	−0.860258	+	0.509859i	$0.829697\pi$
$740$	0.982067	0.0361015
$741$	0	0
$742$	6.66563	0.244703
$743$	−41.7626	−1.53212	−0.766061	−	0.642768i	$-0.777786\pi$
$743$	−41.7626	−1.53212	−0.766061	+	0.642768i	$0.777786\pi$
$744$	0	0
$745$	5.72190	0.209634
$746$	25.7382	0.942342
$747$	0	0
$748$	6.73179	0.246139
$749$	−64.0828	−2.34154
$750$	0	0
$751$	4.01683	0.146576	0.0732881	−	0.997311i	$-0.476651\pi$
$751$	4.01683	0.146576	0.0732881	+	0.997311i	$0.476651\pi$
$752$	37.1590	1.35505
$753$	0	0
$754$	3.85957	0.140557
$755$	1.02579	0.0373323
$756$	0	0
$757$	34.0626	1.23803	0.619014	−	0.785380i	$-0.287532\pi$
$757$	34.0626	1.23803	0.619014	+	0.785380i	$0.287532\pi$
$758$	−57.8058	−2.09960
$759$	0	0
$760$	3.99754	0.145006
$761$	−17.7142	−0.642139	−0.321070	−	0.947056i	$-0.604042\pi$
$761$	−17.7142	−0.642139	−0.321070	+	0.947056i	$0.604042\pi$
$762$	0	0
$763$	−60.9914	−2.20804
$764$	9.70694	0.351185
$765$	0	0
$766$	−36.7723	−1.32864
$767$	11.2525	0.406305
$768$	0	0
$769$	19.4368	0.700907	0.350454	−	0.936580i	$-0.386027\pi$
$769$	19.4368	0.700907	0.350454	+	0.936580i	$0.386027\pi$
$770$	−26.4005	−0.951407
$771$	0	0
$772$	5.19994	0.187150
$773$	−15.5259	−0.558428	−0.279214	−	0.960229i	$-0.590074\pi$
$773$	−15.5259	−0.558428	−0.279214	+	0.960229i	$0.590074\pi$
$774$	0	0
$775$	4.90478	0.176185
$776$	17.9085	0.642877
$777$	0	0
$778$	−4.10118	−0.147034
$779$	−1.92395	−0.0689326
$780$	0	0
$781$	31.7403	1.13576
$782$	9.00518	0.322024
$783$	0	0
$784$	−37.4153	−1.33626
$785$	−2.80521	−0.100122
$786$	0	0
$787$	−25.2433	−0.899828	−0.449914	−	0.893072i	$-0.648545\pi$
$787$	−25.2433	−0.899828	−0.449914	+	0.893072i	$0.648545\pi$
$788$	2.87898	0.102559
$789$	0	0
$790$	34.9389	1.24307
$791$	−37.8731	−1.34661
$792$	0	0
$793$	25.0852	0.890801
$794$	21.5724	0.765577
$795$	0	0
$796$	−15.8819	−0.562918
$797$	−39.2256	−1.38944	−0.694721	−	0.719279i	$-0.744472\pi$
$797$	−39.2256	−1.38944	−0.694721	+	0.719279i	$0.744472\pi$
$798$	0	0
$799$	28.8858	1.02191
$800$	−9.59388	−0.339195
$801$	0	0
$802$	−1.34819	−0.0476063
$803$	35.2303	1.24325
$804$	0	0
$805$	−8.30273	−0.292633
$806$	6.48456	0.228409
$807$	0	0
$808$	−27.9765	−0.984211
$809$	−21.5713	−0.758405	−0.379203	−	0.925314i	$-0.623802\pi$
$809$	−21.5713	−0.758405	−0.379203	+	0.925314i	$0.623802\pi$
$810$	0	0
$811$	2.52560	0.0886858	0.0443429	−	0.999016i	$-0.485881\pi$
$811$	2.52560	0.0886858	0.0443429	+	0.999016i	$0.485881\pi$
$812$	−2.41418	−0.0847211
$813$	0	0
$814$	−5.11721	−0.179358
$815$	25.1997	0.882707
$816$	0	0
$817$	−0.527441	−0.0184528
$818$	−20.1482	−0.704467
$819$	0	0
$820$	−1.42409	−0.0497312
$821$	−21.8608	−0.762949	−0.381474	−	0.924379i	$-0.624584\pi$
$821$	−21.8608	−0.762949	−0.381474	+	0.924379i	$0.624584\pi$
$822$	0	0
$823$	15.6288	0.544785	0.272393	−	0.962186i	$-0.412185\pi$
$823$	15.6288	0.544785	0.272393	+	0.962186i	$0.412185\pi$
$824$	−15.4298	−0.537523
$825$	0	0
$826$	−29.9387	−1.04170
$827$	−1.15424	−0.0401368	−0.0200684	−	0.999799i	$-0.506388\pi$
$827$	−1.15424	−0.0401368	−0.0200684	+	0.999799i	$0.506388\pi$
$828$	0	0
$829$	22.5972	0.784832	0.392416	−	0.919788i	$-0.371639\pi$
$829$	22.5972	0.784832	0.392416	+	0.919788i	$0.371639\pi$
$830$	31.0293	1.07704
$831$	0	0
$832$	9.93514	0.344439
$833$	−29.0851	−1.00774
$834$	0	0
$835$	21.8309	0.755489
$836$	2.17302	0.0751554
$837$	0	0
$838$	−15.7799	−0.545106
$839$	26.1778	0.903757	0.451878	−	0.892080i	$-0.350754\pi$
$839$	26.1778	0.903757	0.451878	+	0.892080i	$0.350754\pi$
$840$	0	0
$841$	−27.9516	−0.963848
$842$	−26.0605	−0.898103
$843$	0	0
$844$	2.50486	0.0862207
$845$	11.0909	0.381537
$846$	0	0
$847$	−9.85102	−0.338485
$848$	5.21403	0.179051
$849$	0	0
$850$	−17.3872	−0.596375
$851$	−1.60932	−0.0551668
$852$	0	0
$853$	20.2252	0.692497	0.346249	−	0.938143i	$-0.387455\pi$
$853$	20.2252	0.692497	0.346249	+	0.938143i	$0.387455\pi$
$854$	−66.7422	−2.28387
$855$	0	0
$856$	−37.4247	−1.27915
$857$	−24.4336	−0.834636	−0.417318	−	0.908761i	$-0.637030\pi$
$857$	−24.4336	−0.834636	−0.417318	+	0.908761i	$0.637030\pi$
$858$	0	0
$859$	36.4278	1.24290	0.621451	−	0.783453i	$-0.286543\pi$
$859$	36.4278	1.24290	0.621451	+	0.783453i	$0.286543\pi$
$860$	−0.390406	−0.0133128
$861$	0	0
$862$	−37.7794	−1.28677
$863$	18.5506	0.631471	0.315735	−	0.948847i	$-0.397749\pi$
$863$	18.5506	0.631471	0.315735	+	0.948847i	$0.397749\pi$
$864$	0	0
$865$	−17.1759	−0.583999
$866$	27.1792	0.923588
$867$	0	0
$868$	−4.05613	−0.137674
$869$	−42.8005	−1.45191
$870$	0	0
$871$	−14.2444	−0.482655
$872$	−35.6193	−1.20622
$873$	0	0
$874$	2.90687	0.0983262
$875$	44.1442	1.49235
$876$	0	0
$877$	39.5945	1.33701	0.668505	−	0.743708i	$-0.266934\pi$
$877$	39.5945	1.33701	0.668505	+	0.743708i	$0.266934\pi$
$878$	−7.39570	−0.249593
$879$	0	0
$880$	−20.6512	−0.696151
$881$	−32.1516	−1.08321	−0.541607	−	0.840632i	$-0.682184\pi$
$881$	−32.1516	−1.08321	−0.541607	+	0.840632i	$0.682184\pi$
$882$	0	0
$883$	20.8275	0.700900	0.350450	−	0.936581i	$-0.386029\pi$
$883$	20.8275	0.700900	0.350450	+	0.936581i	$0.386029\pi$
$884$	−5.40428	−0.181766
$885$	0	0
$886$	−13.6444	−0.458392
$887$	−34.9748	−1.17434	−0.587170	−	0.809464i	$-0.699758\pi$
$887$	−34.9748	−1.17434	−0.587170	+	0.809464i	$0.699758\pi$
$888$	0	0
$889$	−39.5730	−1.32724
$890$	14.4325	0.483778
$891$	0	0
$892$	−1.50394	−0.0503555
$893$	9.32432	0.312027
$894$	0	0
$895$	−4.93660	−0.165012
$896$	−52.2470	−1.74545
$897$	0	0
$898$	18.3668	0.612909
$899$	1.76144	0.0587472
$900$	0	0
$901$	4.05317	0.135031
$902$	7.42042	0.247073
$903$	0	0
$904$	−22.1180	−0.735635
$905$	9.77198	0.324832
$906$	0	0
$907$	−28.3094	−0.939999	−0.470000	−	0.882667i	$-0.655746\pi$
$907$	−28.3094	−0.939999	−0.470000	+	0.882667i	$0.655746\pi$
$908$	−1.68767	−0.0560073
$909$	0	0
$910$	21.1943	0.702584
$911$	47.1094	1.56081	0.780403	−	0.625277i	$-0.215014\pi$
$911$	47.1094	1.56081	0.780403	+	0.625277i	$0.215014\pi$
$912$	0	0
$913$	−38.0113	−1.25799
$914$	−22.3210	−0.738314
$915$	0	0
$916$	8.43044	0.278549
$917$	37.8487	1.24987
$918$	0	0
$919$	−17.0107	−0.561131	−0.280565	−	0.959835i	$-0.590522\pi$
$919$	−17.0107	−0.561131	−0.280565	+	0.959835i	$0.590522\pi$
$920$	−4.84883	−0.159861
$921$	0	0
$922$	20.7332	0.682812
$923$	−25.4811	−0.838720
$924$	0	0
$925$	3.10727	0.102166
$926$	8.44163	0.277409
$927$	0	0
$928$	−3.44541	−0.113101
$929$	0.496932	0.0163038	0.00815190	−	0.999967i	$-0.497405\pi$
$929$	0.496932	0.0163038	0.00815190	+	0.999967i	$0.497405\pi$
$930$	0	0
$931$	−9.38864	−0.307700
$932$	3.29954	0.108080
$933$	0	0
$934$	−13.7095	−0.448590
$935$	−16.0533	−0.525001
$936$	0	0
$937$	8.22697	0.268763	0.134382	−	0.990930i	$-0.457095\pi$
$937$	8.22697	0.268763	0.134382	+	0.990930i	$0.457095\pi$
$938$	37.8991	1.23745
$939$	0	0
$940$	6.90177	0.225111
$941$	−50.1600	−1.63517	−0.817584	−	0.575809i	$-0.804687\pi$
$941$	−50.1600	−1.63517	−0.817584	+	0.575809i	$0.804687\pi$
$942$	0	0
$943$	2.33366	0.0759944
$944$	−23.4188	−0.762218
$945$	0	0
$946$	2.03427	0.0661399
$947$	−11.6253	−0.377773	−0.188886	−	0.981999i	$-0.560488\pi$
$947$	−11.6253	−0.377773	−0.188886	+	0.981999i	$0.560488\pi$
$948$	0	0
$949$	−28.2829	−0.918101
$950$	−5.61257	−0.182096
$951$	0	0
$952$	−32.4034	−1.05020
$953$	−27.5219	−0.891522	−0.445761	−	0.895152i	$-0.647067\pi$
$953$	−27.5219	−0.891522	−0.445761	+	0.895152i	$0.647067\pi$
$954$	0	0
$955$	−23.1482	−0.749058
$956$	−0.614711	−0.0198812
$957$	0	0
$958$	59.9716	1.93759
$959$	−12.4508	−0.402057
$960$	0	0
$961$	−28.0406	−0.904534
$962$	4.10809	0.132450
$963$	0	0
$964$	−7.94082	−0.255757
$965$	−12.4003	−0.399181
$966$	0	0
$967$	−41.9315	−1.34843	−0.674213	−	0.738537i	$-0.735517\pi$
$967$	−41.9315	−1.34843	−0.674213	+	0.738537i	$0.735517\pi$
$968$	−5.75304	−0.184910
$969$	0	0
$970$	18.9507	0.608469
$971$	44.8246	1.43849	0.719244	−	0.694757i	$-0.244488\pi$
$971$	44.8246	1.43849	0.719244	+	0.694757i	$0.244488\pi$
$972$	0	0
$973$	−54.5880	−1.75001
$974$	−55.4562	−1.77693
$975$	0	0
$976$	−52.2075	−1.67112
$977$	36.6672	1.17309	0.586544	−	0.809917i	$-0.300488\pi$
$977$	36.6672	1.17309	0.586544	+	0.809917i	$0.300488\pi$
$978$	0	0
$979$	−17.6799	−0.565053
$980$	−6.94937	−0.221989
$981$	0	0
$982$	−1.74614	−0.0557217
$983$	43.4827	1.38688	0.693441	−	0.720513i	$-0.256094\pi$
$983$	43.4827	1.38688	0.693441	+	0.720513i	$0.256094\pi$
$984$	0	0
$985$	−6.86552	−0.218754
$986$	−6.24419	−0.198856
$987$	0	0
$988$	−1.74450	−0.0554998
$989$	0.639762	0.0203433
$990$	0	0
$991$	−4.85590	−0.154253	−0.0771263	−	0.997021i	$-0.524574\pi$
$991$	−4.85590	−0.154253	−0.0771263	+	0.997021i	$0.524574\pi$
$992$	−5.78872	−0.183792
$993$	0	0
$994$	67.7956	2.15034
$995$	37.8736	1.20067
$996$	0	0
$997$	−10.2069	−0.323257	−0.161628	−	0.986852i	$-0.551675\pi$
$997$	−10.2069	−0.323257	−0.161628	+	0.986852i	$0.551675\pi$
$998$	−1.57193	−0.0497587
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.j.1.6	✓	20
3.2	odd	2		2151.2.a.k.1.15	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.2.a.j.1.6	✓	20	1.1	even	1	trivial
2151.2.a.k.1.15	yes	20	3.2	odd	2

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +O(q^{10})\)	\(q-1.61701 q^{2} +0.614711 q^{4} -1.46591 q^{5} +3.83563 q^{7} +2.24002 q^{8} +2.37038 q^{10} -2.90374 q^{11} +2.33112 q^{13} -6.20223 q^{14} -4.85155 q^{16} -3.77139 q^{17} -1.21740 q^{19} -0.901109 q^{20} +4.69537 q^{22} +1.47665 q^{23} -2.85112 q^{25} -3.76944 q^{26} +2.35780 q^{28} -1.02391 q^{29} -1.72030 q^{31} +3.36495 q^{32} +6.09837 q^{34} -5.62267 q^{35} -1.08984 q^{37} +1.96855 q^{38} -3.28366 q^{40} +1.58037 q^{41} +0.433251 q^{43} -1.78496 q^{44} -2.38776 q^{46} -7.65919 q^{47} +7.71202 q^{49} +4.61028 q^{50} +1.43297 q^{52} -1.07471 q^{53} +4.25661 q^{55} +8.59188 q^{56} +1.65567 q^{58} +4.82708 q^{59} +10.7610 q^{61} +2.78174 q^{62} +4.26196 q^{64} -3.41720 q^{65} -6.11056 q^{67} -2.31832 q^{68} +9.09189 q^{70} -10.9308 q^{71} -12.1327 q^{73} +1.76228 q^{74} -0.748351 q^{76} -11.1377 q^{77} +14.7398 q^{79} +7.11192 q^{80} -2.55547 q^{82} +13.0905 q^{83} +5.52851 q^{85} -0.700570 q^{86} -6.50444 q^{88} +6.08868 q^{89} +8.94131 q^{91} +0.907716 q^{92} +12.3850 q^{94} +1.78460 q^{95} +7.99478 q^{97} -12.4704 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8	\( 20 q - 4 q^{2} + 18 q^{4} - 16 q^{5} - 4 q^{7} - 12 q^{8} + 4 q^{10} - 12 q^{11} - 4 q^{13} - 20 q^{14} + 22 q^{16} - 24 q^{17} - 4 q^{19} - 40 q^{20} - 6 q^{22} - 12 q^{23} + 22 q^{25} - 30 q^{26} - 12 q^{28} - 24 q^{29} - 4 q^{31} - 28 q^{32} + 8 q^{34} - 20 q^{35} - 10 q^{37} - 26 q^{38} + 6 q^{40} - 66 q^{41} + 8 q^{43} - 36 q^{44} - 12 q^{46} - 28 q^{47} + 18 q^{49} - 28 q^{50} - 18 q^{52} - 28 q^{53} - 4 q^{55} - 60 q^{56} - 54 q^{59} - 4 q^{61} - 20 q^{62} + 22 q^{64} - 42 q^{65} + 12 q^{67} - 12 q^{68} + 20 q^{70} - 36 q^{71} + 14 q^{73} - 50 q^{76} - 8 q^{77} - 12 q^{79} - 88 q^{80} - 8 q^{82} - 20 q^{83} + 4 q^{85} - 18 q^{86} - 10 q^{88} - 130 q^{89} - 6 q^{91} + 46 q^{92} - 26 q^{94} - 2 q^{97} - 12 q^{98}+O(q^{100}) \) 20 * q - 4 * q^2 + 18 * q^4 - 16 * q^5 - 4 * q^7 - 12 * q^8 + 4 * q^10 - 12 * q^11 - 4 * q^13 - 20 * q^14 + 22 * q^16 - 24 * q^17 - 4 * q^19 - 40 * q^20 - 6 * q^22 - 12 * q^23 + 22 * q^25 - 30 * q^26 - 12 * q^28 - 24 * q^29 - 4 * q^31 - 28 * q^32 + 8 * q^34 - 20 * q^35 - 10 * q^37 - 26 * q^38 + 6 * q^40 - 66 * q^41 + 8 * q^43 - 36 * q^44 - 12 * q^46 - 28 * q^47 + 18 * q^49 - 28 * q^50 - 18 * q^52 - 28 * q^53 - 4 * q^55 - 60 * q^56 - 54 * q^59 - 4 * q^61 - 20 * q^62 + 22 * q^64 - 42 * q^65 + 12 * q^67 - 12 * q^68 + 20 * q^70 - 36 * q^71 + 14 * q^73 - 50 * q^76 - 8 * q^77 - 12 * q^79 - 88 * q^80 - 8 * q^82 - 20 * q^83 + 4 * q^85 - 18 * q^86 - 10 * q^88 - 130 * q^89 - 6 * q^91 + 46 * q^92 - 26 * q^94 - 2 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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