LMFDB - Embedded newform 2151.2.a.k.1.15

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:	$0$
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			1.15
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	$-0.306285\pi$
$2$	1.61701	1.14340	0.571698	+	0.820464i	$0.306285\pi$
$3$	0	0
$3$	0	0
$4$	0.614711	0.307356
$5$	1.46591	0.655573	0.327787	−	0.944752i	$-0.393697\pi$
$5$	1.46591	0.655573	0.327787	+	0.944752i	$0.393697\pi$
$6$	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	$-0.355774\pi$
$11$	2.90374	0.875511	0.437755	+	0.899094i	$0.355774\pi$
$12$	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	$-0.348799\pi$
$17$	3.77139	0.914697	0.457348	+	0.889288i	$0.348799\pi$
$18$	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	$-0.549200\pi$
$23$	−1.47665	−0.307904	−0.153952	+	0.988078i	$0.549200\pi$
$24$	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	$-0.469693\pi$
$29$	1.02391	0.190136	0.0950678	+	0.995471i	$0.469693\pi$
$30$	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	$-0.539382\pi$
$41$	−1.58037	−0.246812	−0.123406	+	0.992356i	$0.539382\pi$
$42$	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	$-0.311337\pi$
$47$	7.65919	1.11721	0.558604	+	0.829434i	$0.311337\pi$
$48$	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	$-0.476484\pi$
$53$	1.07471	0.147623	0.0738117	+	0.997272i	$0.476484\pi$
$54$	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	$-0.601742\pi$
$59$	−4.82708	−0.628432	−0.314216	+	0.949351i	$0.601742\pi$
$60$	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	$-0.275344\pi$
$71$	10.9308	1.29725	0.648626	+	0.761108i	$0.275344\pi$
$72$	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	$-0.755140\pi$
$83$	−13.0905	−1.43686	−0.718432	+	0.695597i	$0.755140\pi$
$84$	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	$-0.604590\pi$
$89$	−6.08868	−0.645398	−0.322699	+	0.946502i	$0.604590\pi$
$90$	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	$-0.286576\pi$
$101$	12.4894	1.24274	0.621371	+	0.783516i	$0.286576\pi$
$102$	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	$-0.200779\pi$
$107$	16.7073	1.61515	0.807577	+	0.589763i	$0.200779\pi$
$108$	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	$-0.346257\pi$
$113$	9.87403	0.928870	0.464435	+	0.885607i	$0.346257\pi$
$114$	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	$-0.641864\pi$
$131$	−9.86768	−0.862143	−0.431072	+	0.902318i	$0.641864\pi$
$132$	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	$-0.455718\pi$
$137$	3.24609	0.277332	0.138666	+	0.990339i	$0.455718\pi$
$138$	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	$-0.448887\pi$
$149$	3.90332	0.319772	0.159886	+	0.987135i	$0.448887\pi$
$150$	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	$-0.304533\pi$
$167$	14.8924	1.15241	0.576205	+	0.817305i	$0.304533\pi$
$168$	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	$-0.646943\pi$
$173$	−11.7169	−0.890823	−0.445411	+	0.895326i	$0.646943\pi$
$174$	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	$-0.540167\pi$
$179$	−3.36761	−0.251707	−0.125854	+	0.992049i	$0.540167\pi$
$180$	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	$-0.693561\pi$
$191$	−15.7911	−1.14260	−0.571300	+	0.820741i	$0.693561\pi$
$192$	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	$-0.553357\pi$
$197$	−4.68346	−0.333683	−0.166842	+	0.985984i	$0.553357\pi$
$198$	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	$-0.470958\pi$
$227$	2.74547	0.182223	0.0911115	+	0.995841i	$0.470958\pi$
$228$	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	$-0.556258\pi$
$233$	−5.36762	−0.351645	−0.175822	+	0.984422i	$0.556258\pi$
$234$	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	$-0.296070\pi$
$251$	18.9396	1.19546	0.597729	+	0.801698i	$0.296070\pi$
$252$	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	$-0.658610\pi$
$257$	−15.3234	−0.955847	−0.477923	+	0.878402i	$0.658610\pi$
$258$	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	$-0.184383\pi$
$263$	27.1435	1.67374	0.836870	+	0.547402i	$0.184383\pi$
$264$	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	$-0.372465\pi$
$269$	12.7939	0.780058	0.390029	+	0.920802i	$0.372465\pi$
$270$	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	$-0.579417\pi$
$281$	−8.27813	−0.493832	−0.246916	+	0.969037i	$0.579417\pi$
$282$	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	$-0.376155\pi$
$293$	12.9861	0.758657	0.379328	+	0.925262i	$0.376155\pi$
$294$	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	$-0.500503\pi$
$311$	−0.0557085	−0.00315894	−0.00157947	+	0.999999i	$0.500503\pi$
$312$	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	$-0.448437\pi$
$317$	5.74313	0.322566	0.161283	+	0.986908i	$0.448437\pi$
$318$	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	$-0.780912\pi$
$347$	−28.7741	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$348$	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	$-0.558673\pi$
$353$	−6.88724	−0.366571	−0.183285	+	0.983060i	$0.558673\pi$
$354$	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	$-0.537558\pi$
$359$	−4.46094	−0.235439	−0.117720	+	0.993047i	$0.537558\pi$
$360$	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	$-0.697340\pi$
$383$	−22.7410	−1.16201	−0.581005	+	0.813900i	$0.697340\pi$
$384$	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	$-0.520481\pi$
$389$	−2.53628	−0.128594	−0.0642972	+	0.997931i	$0.520481\pi$
$390$	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	$-0.506627\pi$
$401$	−0.833758	−0.0416359	−0.0208179	+	0.999783i	$0.506627\pi$
$402$	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	$-0.576614\pi$
$419$	−9.75869	−0.476743	−0.238372	+	0.971174i	$0.576614\pi$
$420$	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	$-0.690236\pi$
$431$	−23.3638	−1.12539	−0.562697	+	0.826663i	$0.690236\pi$
$432$	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	$-0.564241\pi$
$443$	−8.43805	−0.400904	−0.200452	+	0.979704i	$0.564241\pi$
$444$	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	$-0.413630\pi$
$449$	11.3585	0.536043	0.268021	+	0.963413i	$0.413630\pi$
$450$	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	$-0.403484\pi$
$461$	12.8220	0.597179	0.298589	+	0.954382i	$0.403484\pi$
$462$	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	$-0.562849\pi$
$467$	−8.47835	−0.392331	−0.196166	+	0.980571i	$0.562849\pi$
$468$	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	$-0.178228\pi$
$479$	37.0880	1.69459	0.847297	+	0.531119i	$0.178228\pi$
$480$	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	$-0.507757\pi$
$491$	−1.07986	−0.0487335	−0.0243667	+	0.999703i	$0.507757\pi$
$492$	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	$-0.261799\pi$
$503$	30.5203	1.36083	0.680417	+	0.732825i	$0.261799\pi$
$504$	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	$-0.648014\pi$
$509$	−20.2337	−0.896845	−0.448422	+	0.893822i	$0.648014\pi$
$510$	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	$-0.623095\pi$
$521$	−17.2171	−0.754296	−0.377148	+	0.926153i	$0.623095\pi$
$522$	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	$-0.293720\pi$
$557$	28.4924	1.20726	0.603631	+	0.797264i	$0.293720\pi$
$558$	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	$-0.636694\pi$
$563$	−19.7584	−0.832716	−0.416358	+	0.909201i	$0.636694\pi$
$564$	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	$-0.564671\pi$
$569$	−9.62616	−0.403550	−0.201775	+	0.979432i	$0.564671\pi$
$570$	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	$-0.686329\pi$
$587$	−26.7724	−1.10502	−0.552508	+	0.833507i	$0.686329\pi$
$588$	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	$-0.133642\pi$
$593$	44.4733	1.82630	0.913150	+	0.407623i	$0.133642\pi$
$594$	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	$-0.631305\pi$
$599$	−19.6240	−0.801815	−0.400908	+	0.916118i	$0.631305\pi$
$600$	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	$-0.609790\pi$
$617$	−16.7973	−0.676233	−0.338117	+	0.941104i	$0.609790\pi$
$618$	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	$-0.255943\pi$
$641$	35.1303	1.38756	0.693782	+	0.720185i	$0.255943\pi$
$642$	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	$-0.426547\pi$
$647$	11.6354	0.457433	0.228716	+	0.973493i	$0.426547\pi$
$648$	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	$-0.518462\pi$
$653$	−2.96261	−0.115936	−0.0579680	+	0.998318i	$0.518462\pi$
$654$	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	$-0.739070\pi$
$659$	−35.0365	−1.36483	−0.682415	+	0.730965i	$0.739070\pi$
$660$	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	$-0.510879\pi$
$677$	−1.77815	−0.0683400	−0.0341700	+	0.999416i	$0.510879\pi$
$678$	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	$-0.728142\pi$
$683$	−34.3363	−1.31384	−0.656922	+	0.753959i	$0.728142\pi$
$684$	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	$-0.488003\pi$
$701$	1.99537	0.0753641	0.0376820	+	0.999290i	$0.488003\pi$
$702$	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	$-0.351187\pi$
$719$	24.1685	0.901331	0.450666	+	0.892693i	$0.351187\pi$
$720$	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	$-0.222214\pi$
$743$	41.7626	1.53212	0.766061	+	0.642768i	$0.222214\pi$
$744$	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	$-0.395958\pi$
$761$	17.7142	0.642139	0.321070	+	0.947056i	$0.395958\pi$
$762$	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	$-0.409926\pi$
$773$	15.5259	0.558428	0.279214	+	0.960229i	$0.409926\pi$
$774$	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	$-0.255528\pi$
$797$	39.2256	1.38944	0.694721	+	0.719279i	$0.255528\pi$
$798$	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	$-0.376198\pi$
$809$	21.5713	0.758405	0.379203	+	0.925314i	$0.376198\pi$
$810$	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	$-0.375416\pi$
$821$	21.8608	0.762949	0.381474	+	0.924379i	$0.375416\pi$
$822$	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	$-0.493612\pi$
$827$	1.15424	0.0401368	0.0200684	+	0.999799i	$0.493612\pi$
$828$	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	$-0.649246\pi$
$839$	−26.1778	−0.903757	−0.451878	+	0.892080i	$0.649246\pi$
$840$	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	$-0.362970\pi$
$857$	24.4336	0.834636	0.417318	+	0.908761i	$0.362970\pi$
$858$	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	$-0.602251\pi$
$863$	−18.5506	−0.631471	−0.315735	+	0.948847i	$0.602251\pi$
$864$	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	$-0.317816\pi$
$881$	32.1516	1.08321	0.541607	+	0.840632i	$0.317816\pi$
$882$	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	$-0.300242\pi$
$887$	34.9748	1.17434	0.587170	+	0.809464i	$0.300242\pi$
$888$	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	$-0.784986\pi$
$911$	−47.1094	−1.56081	−0.780403	+	0.625277i	$0.784986\pi$
$912$	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	$-0.502595\pi$
$929$	−0.496932	−0.0163038	−0.00815190	+	0.999967i	$0.502595\pi$
$930$	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	$-0.195313\pi$
$941$	50.1600	1.63517	0.817584	+	0.575809i	$0.195313\pi$
$942$	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	$-0.439512\pi$
$947$	11.6253	0.377773	0.188886	+	0.981999i	$0.439512\pi$
$948$	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	$-0.352933\pi$
$953$	27.5219	0.891522	0.445761	+	0.895152i	$0.352933\pi$
$954$	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	$-0.755512\pi$
$971$	−44.8246	−1.43849	−0.719244	+	0.694757i	$0.755512\pi$
$972$	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	$-0.699512\pi$
$977$	−36.6672	−1.17309	−0.586544	+	0.809917i	$0.699512\pi$
$978$	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	$-0.743906\pi$
$983$	−43.4827	−1.38688	−0.693441	+	0.720513i	$0.743906\pi$
$984$	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.k.1.15	yes	20
3.2	odd	2		2151.2.a.j.1.6	✓	20

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
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

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