LMFDB - Embedded newform 2151.2.a.g.1.3

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:	$8$
Coefficient field:	8.8.2585660609.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 4x^{6} + 15x^{5} + x^{4} - 19x^{3} + 6x^{2} + 3x - 1 $ x^8 - 3x^7 - 4x^6 + 15x^5 + x^4 - 19x^3 + 6x^2 + 3x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.315996$ 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)

$f(q)$	$=$	$q-0.777975 q^{2} -1.39476 q^{4} -0.966236 q^{5} -3.17897 q^{7} +2.64103 q^{8} +O(q^{10})$	\(q-0.777975 q^{2} -1.39476 q^{4} -0.966236 q^{5} -3.17897 q^{7} +2.64103 q^{8} +0.751707 q^{10} +6.34775 q^{11} -5.47993 q^{13} +2.47316 q^{14} +0.734853 q^{16} -3.46788 q^{17} -1.75212 q^{19} +1.34766 q^{20} -4.93839 q^{22} -0.492382 q^{23} -4.06639 q^{25} +4.26325 q^{26} +4.43389 q^{28} -3.20152 q^{29} +9.29370 q^{31} -5.85376 q^{32} +2.69793 q^{34} +3.07164 q^{35} -6.60557 q^{37} +1.36311 q^{38} -2.55186 q^{40} -3.28438 q^{41} +2.44787 q^{43} -8.85356 q^{44} +0.383061 q^{46} +4.94492 q^{47} +3.10586 q^{49} +3.16355 q^{50} +7.64316 q^{52} -6.59818 q^{53} -6.13343 q^{55} -8.39577 q^{56} +2.49071 q^{58} +12.5270 q^{59} -6.41609 q^{61} -7.23026 q^{62} +3.08437 q^{64} +5.29491 q^{65} -10.3793 q^{67} +4.83685 q^{68} -2.38965 q^{70} +8.13711 q^{71} +9.31986 q^{73} +5.13897 q^{74} +2.44378 q^{76} -20.1793 q^{77} +8.53586 q^{79} -0.710041 q^{80} +2.55516 q^{82} +14.6688 q^{83} +3.35079 q^{85} -1.90438 q^{86} +16.7646 q^{88} -6.88742 q^{89} +17.4205 q^{91} +0.686752 q^{92} -3.84702 q^{94} +1.69296 q^{95} +10.7633 q^{97} -2.41628 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	$ 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * 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 $

$2$	−0.777975	−0.550111	−0.275056	−	0.961428i	$-0.588696\pi$
$2$	−0.777975	−0.550111	−0.275056	+	0.961428i	$0.588696\pi$
$3$	0	0
$3$	0	0
$4$	−1.39476	−0.697378
$5$	−0.966236	−0.432114	−0.216057	−	0.976381i	$-0.569320\pi$
$5$	−0.966236	−0.432114	−0.216057	+	0.976381i	$0.569320\pi$
$6$	0	0
$7$	−3.17897	−1.20154	−0.600769	−	0.799423i	$-0.705139\pi$
$7$	−3.17897	−1.20154	−0.600769	+	0.799423i	$0.705139\pi$
$8$	2.64103	0.933746
$9$	0	0
$10$	0.751707	0.237711
$11$	6.34775	1.91392	0.956960	−	0.290220i	$-0.0937284\pi$
$11$	6.34775	1.91392	0.956960	+	0.290220i	$0.0937284\pi$
$12$	0	0
$13$	−5.47993	−1.51986	−0.759930	−	0.650005i	$-0.774767\pi$
$13$	−5.47993	−1.51986	−0.759930	+	0.650005i	$0.774767\pi$
$14$	2.47316	0.660980
$15$	0	0
$16$	0.734853	0.183713
$17$	−3.46788	−0.841085	−0.420543	−	0.907273i	$-0.638160\pi$
$17$	−3.46788	−0.841085	−0.420543	+	0.907273i	$0.638160\pi$
$18$	0	0
$19$	−1.75212	−0.401964	−0.200982	−	0.979595i	$-0.564413\pi$
$19$	−1.75212	−0.401964	−0.200982	+	0.979595i	$0.564413\pi$
$20$	1.34766	0.301346
$21$	0	0
$22$	−4.93839	−1.05287
$23$	−0.492382	−0.102669	−0.0513344	−	0.998682i	$-0.516347\pi$
$23$	−0.492382	−0.102669	−0.0513344	+	0.998682i	$0.516347\pi$
$24$	0	0
$25$	−4.06639	−0.813278
$26$	4.26325	0.836092
$27$	0	0
$28$	4.43389	0.837926
$29$	−3.20152	−0.594508	−0.297254	−	0.954798i	$-0.596071\pi$
$29$	−3.20152	−0.594508	−0.297254	+	0.954798i	$0.596071\pi$
$30$	0	0
$31$	9.29370	1.66920	0.834599	−	0.550859i	$-0.185700\pi$
$31$	9.29370	1.66920	0.834599	+	0.550859i	$0.185700\pi$
$32$	−5.85376	−1.03481
$33$	0	0
$34$	2.69793	0.462691
$35$	3.07164	0.519201
$36$	0	0
$37$	−6.60557	−1.08595	−0.542974	−	0.839749i	$-0.682702\pi$
$37$	−6.60557	−1.08595	−0.542974	+	0.839749i	$0.682702\pi$
$38$	1.36311	0.221125
$39$	0	0
$40$	−2.55186	−0.403485
$41$	−3.28438	−0.512934	−0.256467	−	0.966553i	$-0.582558\pi$
$41$	−3.28438	−0.512934	−0.256467	+	0.966553i	$0.582558\pi$
$42$	0	0
$43$	2.44787	0.373296	0.186648	−	0.982427i	$-0.440238\pi$
$43$	2.44787	0.373296	0.186648	+	0.982427i	$0.440238\pi$
$44$	−8.85356	−1.33473
$45$	0	0
$46$	0.383061	0.0564792
$47$	4.94492	0.721291	0.360645	−	0.932703i	$-0.382556\pi$
$47$	4.94492	0.721291	0.360645	+	0.932703i	$0.382556\pi$
$48$	0	0
$49$	3.10586	0.443694
$50$	3.16355	0.447393
$51$	0	0
$52$	7.64316	1.05992
$53$	−6.59818	−0.906329	−0.453165	−	0.891427i	$-0.649705\pi$
$53$	−6.59818	−0.906329	−0.453165	+	0.891427i	$0.649705\pi$
$54$	0	0
$55$	−6.13343	−0.827031
$56$	−8.39577	−1.12193
$57$	0	0
$58$	2.49071	0.327046
$59$	12.5270	1.63088	0.815440	−	0.578842i	$-0.196495\pi$
$59$	12.5270	1.63088	0.815440	+	0.578842i	$0.196495\pi$
$60$	0	0
$61$	−6.41609	−0.821496	−0.410748	−	0.911749i	$-0.634732\pi$
$61$	−6.41609	−0.821496	−0.410748	+	0.911749i	$0.634732\pi$
$62$	−7.23026	−0.918244
$63$	0	0
$64$	3.08437	0.385547
$65$	5.29491	0.656752
$66$	0	0
$67$	−10.3793	−1.26803	−0.634014	−	0.773321i	$-0.718594\pi$
$67$	−10.3793	−1.26803	−0.634014	+	0.773321i	$0.718594\pi$
$68$	4.83685	0.586554
$69$	0	0
$70$	−2.38965	−0.285618
$71$	8.13711	0.965697	0.482848	−	0.875704i	$-0.339602\pi$
$71$	8.13711	0.965697	0.482848	+	0.875704i	$0.339602\pi$
$72$	0	0
$73$	9.31986	1.09081	0.545404	−	0.838174i	$-0.316376\pi$
$73$	9.31986	1.09081	0.545404	+	0.838174i	$0.316376\pi$
$74$	5.13897	0.597393
$75$	0	0
$76$	2.44378	0.280321
$77$	−20.1793	−2.29965
$78$	0	0
$79$	8.53586	0.960359	0.480180	−	0.877170i	$-0.340571\pi$
$79$	8.53586	0.960359	0.480180	+	0.877170i	$0.340571\pi$
$80$	−0.710041	−0.0793850
$81$	0	0
$82$	2.55516	0.282171
$83$	14.6688	1.61011	0.805057	−	0.593197i	$-0.202135\pi$
$83$	14.6688	1.61011	0.805057	+	0.593197i	$0.202135\pi$
$84$	0	0
$85$	3.35079	0.363445
$86$	−1.90438	−0.205355
$87$	0	0
$88$	16.7646	1.78712
$89$	−6.88742	−0.730065	−0.365033	−	0.930995i	$-0.618942\pi$
$89$	−6.88742	−0.730065	−0.365033	+	0.930995i	$0.618942\pi$
$90$	0	0
$91$	17.4205	1.82617
$92$	0.686752	0.0715989
$93$	0	0
$94$	−3.84702	−0.396790
$95$	1.69296	0.173694
$96$	0	0
$97$	10.7633	1.09285	0.546425	−	0.837508i	$-0.315988\pi$
$97$	10.7633	1.09285	0.546425	+	0.837508i	$0.315988\pi$
$98$	−2.41628	−0.244081
$99$	0	0
$100$	5.67162	0.567162
$101$	15.5071	1.54301	0.771506	−	0.636222i	$-0.219504\pi$
$101$	15.5071	1.54301	0.771506	+	0.636222i	$0.219504\pi$
$102$	0	0
$103$	−10.7596	−1.06018	−0.530088	−	0.847942i	$-0.677841\pi$
$103$	−10.7596	−1.06018	−0.530088	+	0.847942i	$0.677841\pi$
$104$	−14.4727	−1.41916
$105$	0	0
$106$	5.13322	0.498582
$107$	0.755696	0.0730559	0.0365279	−	0.999333i	$-0.488370\pi$
$107$	0.755696	0.0730559	0.0365279	+	0.999333i	$0.488370\pi$
$108$	0	0
$109$	4.58706	0.439361	0.219680	−	0.975572i	$-0.429499\pi$
$109$	4.58706	0.439361	0.219680	+	0.975572i	$0.429499\pi$
$110$	4.77165	0.454959
$111$	0	0
$112$	−2.33608	−0.220739
$113$	9.51650	0.895237	0.447618	−	0.894225i	$-0.352272\pi$
$113$	9.51650	0.895237	0.447618	+	0.894225i	$0.352272\pi$
$114$	0	0
$115$	0.475757	0.0443646
$116$	4.46534	0.414597
$117$	0	0
$118$	−9.74571	−0.897165
$119$	11.0243	1.01060
$120$	0	0
$121$	29.2940	2.66309
$122$	4.99155	0.451914
$123$	0	0
$124$	−12.9624	−1.16406
$125$	8.76027	0.783542
$126$	0	0
$127$	11.4064	1.01215	0.506077	−	0.862488i	$-0.331095\pi$
$127$	11.4064	1.01215	0.506077	+	0.862488i	$0.331095\pi$
$128$	9.30796	0.822716
$129$	0	0
$130$	−4.11930	−0.361287
$131$	4.69863	0.410521	0.205261	−	0.978707i	$-0.434196\pi$
$131$	4.69863	0.410521	0.205261	+	0.978707i	$0.434196\pi$
$132$	0	0
$133$	5.56994	0.482975
$134$	8.07480	0.697556
$135$	0	0
$136$	−9.15880	−0.785361
$137$	4.17878	0.357018	0.178509	−	0.983938i	$-0.442873\pi$
$137$	4.17878	0.357018	0.178509	+	0.983938i	$0.442873\pi$
$138$	0	0
$139$	12.5879	1.06769	0.533847	−	0.845581i	$-0.320746\pi$
$139$	12.5879	1.06769	0.533847	+	0.845581i	$0.320746\pi$
$140$	−4.28418	−0.362079
$141$	0	0
$142$	−6.33046	−0.531241
$143$	−34.7853	−2.90889
$144$	0	0
$145$	3.09343	0.256895
$146$	−7.25062	−0.600065
$147$	0	0
$148$	9.21315	0.757316
$149$	−7.62119	−0.624352	−0.312176	−	0.950024i	$-0.601058\pi$
$149$	−7.62119	−0.624352	−0.312176	+	0.950024i	$0.601058\pi$
$150$	0	0
$151$	−5.46506	−0.444740	−0.222370	−	0.974962i	$-0.571379\pi$
$151$	−5.46506	−0.444740	−0.222370	+	0.974962i	$0.571379\pi$
$152$	−4.62741	−0.375333
$153$	0	0
$154$	15.6990	1.26506
$155$	−8.97990	−0.721283
$156$	0	0
$157$	−13.2071	−1.05404	−0.527021	−	0.849852i	$-0.676691\pi$
$157$	−13.2071	−1.05404	−0.527021	+	0.849852i	$0.676691\pi$
$158$	−6.64068	−0.528304
$159$	0	0
$160$	5.65612	0.447155
$161$	1.56527	0.123360
$162$	0	0
$163$	−22.4893	−1.76150	−0.880748	−	0.473585i	$-0.842960\pi$
$163$	−22.4893	−1.76150	−0.880748	+	0.473585i	$0.842960\pi$
$164$	4.58091	0.357709
$165$	0	0
$166$	−11.4120	−0.885742
$167$	−13.0419	−1.00921	−0.504605	−	0.863350i	$-0.668362\pi$
$167$	−13.0419	−1.00921	−0.504605	+	0.863350i	$0.668362\pi$
$168$	0	0
$169$	17.0297	1.30997
$170$	−2.60683	−0.199935
$171$	0	0
$172$	−3.41418	−0.260329
$173$	14.7359	1.12035	0.560177	−	0.828373i	$-0.310733\pi$
$173$	14.7359	1.12035	0.560177	+	0.828373i	$0.310733\pi$
$174$	0	0
$175$	12.9269	0.977184
$176$	4.66467	0.351613
$177$	0	0
$178$	5.35824	0.401617
$179$	17.1724	1.28352	0.641762	−	0.766904i	$-0.278204\pi$
$179$	17.1724	1.28352	0.641762	+	0.766904i	$0.278204\pi$
$180$	0	0
$181$	2.54804	0.189395	0.0946973	−	0.995506i	$-0.469812\pi$
$181$	2.54804	0.189395	0.0946973	+	0.995506i	$0.469812\pi$
$182$	−13.5527	−1.00460
$183$	0	0
$184$	−1.30040	−0.0958666
$185$	6.38254	0.469253
$186$	0	0
$187$	−22.0133	−1.60977
$188$	−6.89695	−0.503012
$189$	0	0
$190$	−1.31708	−0.0955511
$191$	6.22141	0.450165	0.225083	−	0.974340i	$-0.427735\pi$
$191$	6.22141	0.450165	0.225083	+	0.974340i	$0.427735\pi$
$192$	0	0
$193$	3.75474	0.270272	0.135136	−	0.990827i	$-0.456853\pi$
$193$	3.75474	0.270272	0.135136	+	0.990827i	$0.456853\pi$
$194$	−8.37359	−0.601189
$195$	0	0
$196$	−4.33192	−0.309423
$197$	−26.7045	−1.90261	−0.951307	−	0.308244i	$-0.900259\pi$
$197$	−26.7045	−1.90261	−0.951307	+	0.308244i	$0.900259\pi$
$198$	0	0
$199$	17.9670	1.27365	0.636825	−	0.771009i	$-0.280248\pi$
$199$	17.9670	1.27365	0.636825	+	0.771009i	$0.280248\pi$
$200$	−10.7395	−0.759395
$201$	0	0
$202$	−12.0641	−0.848828
$203$	10.1776	0.714324
$204$	0	0
$205$	3.17348	0.221646
$206$	8.37071	0.583215
$207$	0	0
$208$	−4.02695	−0.279218
$209$	−11.1220	−0.769327
$210$	0	0
$211$	10.9090	0.751006	0.375503	−	0.926821i	$-0.377470\pi$
$211$	10.9090	0.751006	0.375503	+	0.926821i	$0.377470\pi$
$212$	9.20284	0.632054
$213$	0	0
$214$	−0.587912	−0.0401889
$215$	−2.36522	−0.161306
$216$	0	0
$217$	−29.5444	−2.00560
$218$	−3.56862	−0.241697
$219$	0	0
$220$	8.55463	0.576753
$221$	19.0038	1.27833
$222$	0	0
$223$	8.51450	0.570173	0.285087	−	0.958502i	$-0.407978\pi$
$223$	8.51450	0.570173	0.285087	+	0.958502i	$0.407978\pi$
$224$	18.6090	1.24336
$225$	0	0
$226$	−7.40359	−0.492480
$227$	14.0333	0.931423	0.465711	−	0.884937i	$-0.345798\pi$
$227$	14.0333	0.931423	0.465711	+	0.884937i	$0.345798\pi$
$228$	0	0
$229$	19.0192	1.25682	0.628411	−	0.777881i	$-0.283706\pi$
$229$	19.0192	1.25682	0.628411	+	0.777881i	$0.283706\pi$
$230$	−0.370127	−0.0244054
$231$	0	0
$232$	−8.45533	−0.555120
$233$	12.2288	0.801133	0.400566	−	0.916268i	$-0.368813\pi$
$233$	12.2288	0.801133	0.400566	+	0.916268i	$0.368813\pi$
$234$	0	0
$235$	−4.77796	−0.311680
$236$	−17.4721	−1.13734
$237$	0	0
$238$	−8.57663	−0.555940
$239$	−1.00000	−0.0646846
$239$	−1.00000	−0.0646846
$240$	0	0
$241$	−26.3453	−1.69705	−0.848524	−	0.529157i	$-0.822508\pi$
$241$	−26.3453	−1.69705	−0.848524	+	0.529157i	$0.822508\pi$
$242$	−22.7900	−1.46500
$243$	0	0
$244$	8.94887	0.572893
$245$	−3.00099	−0.191726
$246$	0	0
$247$	9.60150	0.610929
$248$	24.5450	1.55861
$249$	0	0
$250$	−6.81527	−0.431035
$251$	5.68494	0.358830	0.179415	−	0.983773i	$-0.442579\pi$
$251$	5.68494	0.358830	0.179415	+	0.983773i	$0.442579\pi$
$252$	0	0
$253$	−3.12552	−0.196500
$254$	−8.87389	−0.556798
$255$	0	0
$256$	−13.4101	−0.838132
$257$	4.77257	0.297705	0.148852	−	0.988859i	$-0.452442\pi$
$257$	4.77257	0.297705	0.148852	+	0.988859i	$0.452442\pi$
$258$	0	0
$259$	20.9989	1.30481
$260$	−7.38510	−0.458004
$261$	0	0
$262$	−3.65541	−0.225832
$263$	8.62383	0.531768	0.265884	−	0.964005i	$-0.414336\pi$
$263$	8.62383	0.531768	0.265884	+	0.964005i	$0.414336\pi$
$264$	0	0
$265$	6.37539	0.391637
$266$	−4.33327	−0.265690
$267$	0	0
$268$	14.4765	0.884295
$269$	−17.9665	−1.09544	−0.547718	−	0.836663i	$-0.684503\pi$
$269$	−17.9665	−1.09544	−0.547718	+	0.836663i	$0.684503\pi$
$270$	0	0
$271$	−31.5980	−1.91944	−0.959720	−	0.280957i	$-0.909348\pi$
$271$	−31.5980	−1.91944	−0.959720	+	0.280957i	$0.909348\pi$
$272$	−2.54839	−0.154519
$273$	0	0
$274$	−3.25099	−0.196399
$275$	−25.8124	−1.55655
$276$	0	0
$277$	29.7501	1.78751	0.893756	−	0.448554i	$-0.148061\pi$
$277$	29.7501	1.78751	0.893756	+	0.448554i	$0.148061\pi$
$278$	−9.79309	−0.587350
$279$	0	0
$280$	8.11229	0.484802
$281$	−9.19132	−0.548308	−0.274154	−	0.961686i	$-0.588398\pi$
$281$	−9.19132	−0.548308	−0.274154	+	0.961686i	$0.588398\pi$
$282$	0	0
$283$	−10.9288	−0.649649	−0.324824	−	0.945774i	$-0.605305\pi$
$283$	−10.9288	−0.649649	−0.324824	+	0.945774i	$0.605305\pi$
$284$	−11.3493	−0.673455
$285$	0	0
$286$	27.0621	1.60021
$287$	10.4409	0.616310
$288$	0	0
$289$	−4.97378	−0.292575
$290$	−2.40661	−0.141321
$291$	0	0
$292$	−12.9989	−0.760705
$293$	13.1724	0.769540	0.384770	−	0.923012i	$-0.374281\pi$
$293$	13.1724	0.769540	0.384770	+	0.923012i	$0.374281\pi$
$294$	0	0
$295$	−12.1041	−0.704725
$296$	−17.4455	−1.01400
$297$	0	0
$298$	5.92909	0.343463
$299$	2.69822	0.156042
$300$	0	0
$301$	−7.78170	−0.448530
$302$	4.25168	0.244657
$303$	0	0
$304$	−1.28755	−0.0738462
$305$	6.19945	0.354980
$306$	0	0
$307$	−14.4565	−0.825074	−0.412537	−	0.910941i	$-0.635357\pi$
$307$	−14.4565	−0.825074	−0.412537	+	0.910941i	$0.635357\pi$
$308$	28.1452	1.60372
$309$	0	0
$310$	6.98613	0.396786
$311$	7.43084	0.421365	0.210682	−	0.977555i	$-0.432431\pi$
$311$	7.43084	0.421365	0.210682	+	0.977555i	$0.432431\pi$
$312$	0	0
$313$	19.7060	1.11385	0.556923	−	0.830564i	$-0.311982\pi$
$313$	19.7060	1.11385	0.556923	+	0.830564i	$0.311982\pi$
$314$	10.2748	0.579840
$315$	0	0
$316$	−11.9054	−0.669733
$317$	7.39403	0.415290	0.207645	−	0.978204i	$-0.433420\pi$
$317$	7.39403	0.415290	0.207645	+	0.978204i	$0.433420\pi$
$318$	0	0
$319$	−20.3225	−1.13784
$320$	−2.98023	−0.166600
$321$	0	0
$322$	−1.21774	−0.0678619
$323$	6.07615	0.338086
$324$	0	0
$325$	22.2835	1.23607
$326$	17.4961	0.969019
$327$	0	0
$328$	−8.67416	−0.478950
$329$	−15.7198	−0.866659
$330$	0	0
$331$	23.2832	1.27976	0.639880	−	0.768475i	$-0.278984\pi$
$331$	23.2832	1.27976	0.639880	+	0.768475i	$0.278984\pi$
$332$	−20.4594	−1.12286
$333$	0	0
$334$	10.1463	0.555178
$335$	10.0288	0.547932
$336$	0	0
$337$	31.0930	1.69375	0.846873	−	0.531795i	$-0.178482\pi$
$337$	31.0930	1.69375	0.846873	+	0.531795i	$0.178482\pi$
$338$	−13.2486	−0.720631
$339$	0	0
$340$	−4.67354	−0.253458
$341$	58.9941	3.19471
$342$	0	0
$343$	12.3794	0.668423
$344$	6.46490	0.348564
$345$	0	0
$346$	−11.4642	−0.616319
$347$	−12.0490	−0.646822	−0.323411	−	0.946259i	$-0.604830\pi$
$347$	−12.0490	−0.646822	−0.323411	+	0.946259i	$0.604830\pi$
$348$	0	0
$349$	−14.6081	−0.781952	−0.390976	−	0.920401i	$-0.627862\pi$
$349$	−14.6081	−0.781952	−0.390976	+	0.920401i	$0.627862\pi$
$350$	−10.0568	−0.537560
$351$	0	0
$352$	−37.1583	−1.98054
$353$	12.2315	0.651015	0.325507	−	0.945539i	$-0.394465\pi$
$353$	12.2315	0.651015	0.325507	+	0.945539i	$0.394465\pi$
$354$	0	0
$355$	−7.86236	−0.417291
$356$	9.60627	0.509131
$357$	0	0
$358$	−13.3597	−0.706081
$359$	−18.6818	−0.985989	−0.492995	−	0.870032i	$-0.664098\pi$
$359$	−18.6818	−0.985989	−0.492995	+	0.870032i	$0.664098\pi$
$360$	0	0
$361$	−15.9301	−0.838425
$362$	−1.98231	−0.104188
$363$	0	0
$364$	−24.2974	−1.27353
$365$	−9.00518	−0.471353
$366$	0	0
$367$	−4.87248	−0.254341	−0.127171	−	0.991881i	$-0.540590\pi$
$367$	−4.87248	−0.254341	−0.127171	+	0.991881i	$0.540590\pi$
$368$	−0.361828	−0.0188616
$369$	0	0
$370$	−4.96545	−0.258142
$371$	20.9754	1.08899
$372$	0	0
$373$	11.2504	0.582522	0.291261	−	0.956644i	$-0.405925\pi$
$373$	11.2504	0.582522	0.291261	+	0.956644i	$0.405925\pi$
$374$	17.1258	0.885553
$375$	0	0
$376$	13.0597	0.673503
$377$	17.5441	0.903569
$378$	0	0
$379$	−22.5557	−1.15861	−0.579303	−	0.815112i	$-0.696675\pi$
$379$	−22.5557	−1.15861	−0.579303	+	0.815112i	$0.696675\pi$
$380$	−2.36127	−0.121130
$381$	0	0
$382$	−4.84010	−0.247641
$383$	−2.33620	−0.119374	−0.0596871	−	0.998217i	$-0.519010\pi$
$383$	−2.33620	−0.119374	−0.0596871	+	0.998217i	$0.519010\pi$
$384$	0	0
$385$	19.4980	0.993709
$386$	−2.92110	−0.148680
$387$	0	0
$388$	−15.0122	−0.762129
$389$	0.891879	0.0452200	0.0226100	−	0.999744i	$-0.492802\pi$
$389$	0.891879	0.0452200	0.0226100	+	0.999744i	$0.492802\pi$
$390$	0	0
$391$	1.70752	0.0863532
$392$	8.20268	0.414298
$393$	0	0
$394$	20.7754	1.04665
$395$	−8.24765	−0.414984
$396$	0	0
$397$	−9.42748	−0.473151	−0.236576	−	0.971613i	$-0.576025\pi$
$397$	−9.42748	−0.473151	−0.236576	+	0.971613i	$0.576025\pi$
$398$	−13.9779	−0.700649
$399$	0	0
$400$	−2.98820	−0.149410
$401$	28.6734	1.43188	0.715940	−	0.698162i	$-0.245999\pi$
$401$	28.6734	1.43188	0.715940	+	0.698162i	$0.245999\pi$
$402$	0	0
$403$	−50.9288	−2.53695
$404$	−21.6286	−1.07606
$405$	0	0
$406$	−7.91788	−0.392958
$407$	−41.9305	−2.07842
$408$	0	0
$409$	−11.9383	−0.590312	−0.295156	−	0.955449i	$-0.595372\pi$
$409$	−11.9383	−0.590312	−0.295156	+	0.955449i	$0.595372\pi$
$410$	−2.46889	−0.121930
$411$	0	0
$412$	15.0070	0.739344
$413$	−39.8230	−1.95956
$414$	0	0
$415$	−14.1736	−0.695752
$416$	32.0782	1.57276
$417$	0	0
$418$	8.65266	0.423215
$419$	29.7839	1.45504	0.727519	−	0.686087i	$-0.240673\pi$
$419$	29.7839	1.45504	0.727519	+	0.686087i	$0.240673\pi$
$420$	0	0
$421$	−28.4567	−1.38689	−0.693447	−	0.720507i	$-0.743909\pi$
$421$	−28.4567	−1.38689	−0.693447	+	0.720507i	$0.743909\pi$
$422$	−8.48692	−0.413137
$423$	0	0
$424$	−17.4260	−0.846282
$425$	14.1018	0.684036
$426$	0	0
$427$	20.3966	0.987059
$428$	−1.05401	−0.0509475
$429$	0	0
$430$	1.84008	0.0887365
$431$	28.8297	1.38868	0.694340	−	0.719647i	$-0.255696\pi$
$431$	28.8297	1.38868	0.694340	+	0.719647i	$0.255696\pi$
$432$	0	0
$433$	−10.6784	−0.513170	−0.256585	−	0.966522i	$-0.582597\pi$
$433$	−10.6784	−0.513170	−0.256585	+	0.966522i	$0.582597\pi$
$434$	22.9848	1.10331
$435$	0	0
$436$	−6.39783	−0.306400
$437$	0.862713	0.0412691
$438$	0	0
$439$	−3.79946	−0.181338	−0.0906691	−	0.995881i	$-0.528901\pi$
$439$	−3.79946	−0.181338	−0.0906691	+	0.995881i	$0.528901\pi$
$440$	−16.1986	−0.772237
$441$	0	0
$442$	−14.7845	−0.703225
$443$	21.7593	1.03382	0.516908	−	0.856041i	$-0.327083\pi$
$443$	21.7593	1.03382	0.516908	+	0.856041i	$0.327083\pi$
$444$	0	0
$445$	6.65487	0.315471
$446$	−6.62407	−0.313659
$447$	0	0
$448$	−9.80514	−0.463249
$449$	−25.1172	−1.18536	−0.592678	−	0.805439i	$-0.701929\pi$
$449$	−25.1172	−1.18536	−0.592678	+	0.805439i	$0.701929\pi$
$450$	0	0
$451$	−20.8484	−0.981714
$452$	−13.2732	−0.624318
$453$	0	0
$454$	−10.9176	−0.512386
$455$	−16.8324	−0.789113
$456$	0	0
$457$	−30.0100	−1.40381	−0.701904	−	0.712272i	$-0.747666\pi$
$457$	−30.0100	−1.40381	−0.701904	+	0.712272i	$0.747666\pi$
$458$	−14.7964	−0.691392
$459$	0	0
$460$	−0.663564	−0.0309389
$461$	−0.451516	−0.0210292	−0.0105146	−	0.999945i	$-0.503347\pi$
$461$	−0.451516	−0.0210292	−0.0105146	+	0.999945i	$0.503347\pi$
$462$	0	0
$463$	1.62990	0.0757480	0.0378740	−	0.999283i	$-0.487941\pi$
$463$	1.62990	0.0757480	0.0378740	+	0.999283i	$0.487941\pi$
$464$	−2.35265	−0.109219
$465$	0	0
$466$	−9.51367	−0.440712
$467$	12.8200	0.593240	0.296620	−	0.954996i	$-0.404140\pi$
$467$	12.8200	0.593240	0.296620	+	0.954996i	$0.404140\pi$
$468$	0	0
$469$	32.9954	1.52358
$470$	3.71713	0.171458
$471$	0	0
$472$	33.0843	1.52283
$473$	15.5385	0.714459
$474$	0	0
$475$	7.12480	0.326908
$476$	−15.3762	−0.704767
$477$	0	0
$478$	0.777975	0.0355837
$479$	−22.5960	−1.03244	−0.516218	−	0.856457i	$-0.672661\pi$
$479$	−22.5960	−1.03244	−0.516218	+	0.856457i	$0.672661\pi$
$480$	0	0
$481$	36.1981	1.65049
$482$	20.4960	0.933565
$483$	0	0
$484$	−40.8579	−1.85718
$485$	−10.3999	−0.472235
$486$	0	0
$487$	−7.27454	−0.329641	−0.164820	−	0.986324i	$-0.552704\pi$
$487$	−7.27454	−0.329641	−0.164820	+	0.986324i	$0.552704\pi$
$488$	−16.9451	−0.767069
$489$	0	0
$490$	2.33470	0.105471
$491$	−36.0732	−1.62796	−0.813980	−	0.580893i	$-0.802703\pi$
$491$	−36.0732	−1.62796	−0.813980	+	0.580893i	$0.802703\pi$
$492$	0	0
$493$	11.1025	0.500032
$494$	−7.46973	−0.336079
$495$	0	0
$496$	6.82950	0.306654
$497$	−25.8676	−1.16032
$498$	0	0
$499$	17.0037	0.761189	0.380594	−	0.924742i	$-0.375719\pi$
$499$	17.0037	0.761189	0.380594	+	0.924742i	$0.375719\pi$
$500$	−12.2184	−0.546425
$501$	0	0
$502$	−4.42274	−0.197397
$503$	−1.77530	−0.0791569	−0.0395785	−	0.999216i	$-0.512602\pi$
$503$	−1.77530	−0.0791569	−0.0395785	+	0.999216i	$0.512602\pi$
$504$	0	0
$505$	−14.9835	−0.666757
$506$	2.43157	0.108097
$507$	0	0
$508$	−15.9091	−0.705854
$509$	−23.5623	−1.04438	−0.522190	−	0.852829i	$-0.674885\pi$
$509$	−23.5623	−1.04438	−0.522190	+	0.852829i	$0.674885\pi$
$510$	0	0
$511$	−29.6276	−1.31065
$512$	−8.18320	−0.361650
$513$	0	0
$514$	−3.71294	−0.163771
$515$	10.3963	0.458117
$516$	0	0
$517$	31.3891	1.38049
$518$	−16.3366	−0.717790
$519$	0	0
$520$	13.9840	0.613240
$521$	39.7505	1.74150	0.870751	−	0.491724i	$-0.163633\pi$
$521$	39.7505	1.74150	0.870751	+	0.491724i	$0.163633\pi$
$522$	0	0
$523$	7.86154	0.343761	0.171880	−	0.985118i	$-0.445016\pi$
$523$	7.86154	0.343761	0.171880	+	0.985118i	$0.445016\pi$
$524$	−6.55344	−0.286288
$525$	0	0
$526$	−6.70912	−0.292532
$527$	−32.2295	−1.40394
$528$	0	0
$529$	−22.7576	−0.989459
$530$	−4.95990	−0.215444
$531$	0	0
$532$	−7.76871	−0.336816
$533$	17.9982	0.779587
$534$	0	0
$535$	−0.730180	−0.0315684
$536$	−27.4120	−1.18402
$537$	0	0
$538$	13.9775	0.602612
$539$	19.7152	0.849195
$540$	0	0
$541$	−8.91742	−0.383390	−0.191695	−	0.981455i	$-0.561398\pi$
$541$	−8.91742	−0.383390	−0.191695	+	0.981455i	$0.561398\pi$
$542$	24.5824	1.05591
$543$	0	0
$544$	20.3002	0.870363
$545$	−4.43218	−0.189854
$546$	0	0
$547$	−2.52775	−0.108079	−0.0540393	−	0.998539i	$-0.517210\pi$
$547$	−2.52775	−0.108079	−0.0540393	+	0.998539i	$0.517210\pi$
$548$	−5.82838	−0.248976
$549$	0	0
$550$	20.0814	0.856275
$551$	5.60946	0.238971
$552$	0	0
$553$	−27.1353	−1.15391
$554$	−23.1448	−0.983330
$555$	0	0
$556$	−17.5571	−0.744586
$557$	6.22030	0.263563	0.131781	−	0.991279i	$-0.457930\pi$
$557$	6.22030	0.263563	0.131781	+	0.991279i	$0.457930\pi$
$558$	0	0
$559$	−13.4142	−0.567358
$560$	2.25720	0.0953842
$561$	0	0
$562$	7.15062	0.301631
$563$	−14.9621	−0.630576	−0.315288	−	0.948996i	$-0.602101\pi$
$563$	−14.9621	−0.630576	−0.315288	+	0.948996i	$0.602101\pi$
$564$	0	0
$565$	−9.19518	−0.386844
$566$	8.50232	0.357379
$567$	0	0
$568$	21.4904	0.901716
$569$	−6.66984	−0.279614	−0.139807	−	0.990179i	$-0.544648\pi$
$569$	−6.66984	−0.279614	−0.139807	+	0.990179i	$0.544648\pi$
$570$	0	0
$571$	41.6646	1.74361	0.871804	−	0.489856i	$-0.162951\pi$
$571$	41.6646	1.74361	0.871804	+	0.489856i	$0.162951\pi$
$572$	48.5169	2.02859
$573$	0	0
$574$	−8.12279	−0.339039
$575$	2.00222	0.0834982
$576$	0	0
$577$	43.0157	1.79077	0.895384	−	0.445295i	$-0.146901\pi$
$577$	43.0157	1.79077	0.895384	+	0.445295i	$0.146901\pi$
$578$	3.86947	0.160949
$579$	0	0
$580$	−4.31457	−0.179153
$581$	−46.6318	−1.93461
$582$	0	0
$583$	−41.8836	−1.73464
$584$	24.6141	1.01854
$585$	0	0
$586$	−10.2478	−0.423333
$587$	10.0818	0.416119	0.208060	−	0.978116i	$-0.433285\pi$
$587$	10.0818	0.416119	0.208060	+	0.978116i	$0.433285\pi$
$588$	0	0
$589$	−16.2837	−0.670957
$590$	9.41665	0.387677
$591$	0	0
$592$	−4.85412	−0.199503
$593$	−23.5623	−0.967587	−0.483794	−	0.875182i	$-0.660741\pi$
$593$	−23.5623	−0.967587	−0.483794	+	0.875182i	$0.660741\pi$
$594$	0	0
$595$	−10.6521	−0.436693
$596$	10.6297	0.435409
$597$	0	0
$598$	−2.09915	−0.0858405
$599$	−29.1880	−1.19259	−0.596295	−	0.802766i	$-0.703361\pi$
$599$	−29.1880	−1.19259	−0.596295	+	0.802766i	$0.703361\pi$
$600$	0	0
$601$	−4.33133	−0.176678	−0.0883392	−	0.996090i	$-0.528156\pi$
$601$	−4.33133	−0.176678	−0.0883392	+	0.996090i	$0.528156\pi$
$602$	6.05397	0.246741
$603$	0	0
$604$	7.62242	0.310152
$605$	−28.3049	−1.15076
$606$	0	0
$607$	30.8460	1.25200	0.626001	−	0.779822i	$-0.284691\pi$
$607$	30.8460	1.25200	0.626001	+	0.779822i	$0.284691\pi$
$608$	10.2565	0.415956
$609$	0	0
$610$	−4.82302	−0.195278
$611$	−27.0978	−1.09626
$612$	0	0
$613$	27.0803	1.09376	0.546882	−	0.837210i	$-0.315815\pi$
$613$	27.0803	1.09376	0.546882	+	0.837210i	$0.315815\pi$
$614$	11.2468	0.453882
$615$	0	0
$616$	−53.2943	−2.14729
$617$	23.1208	0.930807	0.465404	−	0.885099i	$-0.345909\pi$
$617$	23.1208	0.930807	0.465404	+	0.885099i	$0.345909\pi$
$618$	0	0
$619$	14.5490	0.584772	0.292386	−	0.956300i	$-0.405551\pi$
$619$	14.5490	0.584772	0.292386	+	0.956300i	$0.405551\pi$
$620$	12.5248	0.503007
$621$	0	0
$622$	−5.78101	−0.231797
$623$	21.8949	0.877201
$624$	0	0
$625$	11.8675	0.474699
$626$	−15.3307	−0.612740
$627$	0	0
$628$	18.4207	0.735065
$629$	22.9073	0.913376
$630$	0	0
$631$	10.9495	0.435893	0.217947	−	0.975961i	$-0.430064\pi$
$631$	10.9495	0.435893	0.217947	+	0.975961i	$0.430064\pi$
$632$	22.5435	0.896732
$633$	0	0
$634$	−5.75237	−0.228456
$635$	−11.0213	−0.437366
$636$	0	0
$637$	−17.0199	−0.674353
$638$	15.8104	0.625939
$639$	0	0
$640$	−8.99369	−0.355507
$641$	11.2856	0.445755	0.222878	−	0.974846i	$-0.428455\pi$
$641$	11.2856	0.445755	0.222878	+	0.974846i	$0.428455\pi$
$642$	0	0
$643$	−17.5574	−0.692395	−0.346197	−	0.938162i	$-0.612527\pi$
$643$	−17.5574	−0.692395	−0.346197	+	0.938162i	$0.612527\pi$
$644$	−2.18317	−0.0860288
$645$	0	0
$646$	−4.72709	−0.185985
$647$	28.3751	1.11554	0.557769	−	0.829996i	$-0.311657\pi$
$647$	28.3751	1.11554	0.557769	+	0.829996i	$0.311657\pi$
$648$	0	0
$649$	79.5185	3.12137
$650$	−17.3360	−0.679975
$651$	0	0
$652$	31.3670	1.22843
$653$	−47.9825	−1.87770	−0.938850	−	0.344326i	$-0.888108\pi$
$653$	−47.9825	−1.87770	−0.938850	+	0.344326i	$0.888108\pi$
$654$	0	0
$655$	−4.53998	−0.177392
$656$	−2.41354	−0.0942328
$657$	0	0
$658$	12.2296	0.476759
$659$	−20.3814	−0.793948	−0.396974	−	0.917830i	$-0.629940\pi$
$659$	−20.3814	−0.793948	−0.396974	+	0.917830i	$0.629940\pi$
$660$	0	0
$661$	36.2977	1.41182	0.705908	−	0.708304i	$-0.250539\pi$
$661$	36.2977	1.41182	0.705908	+	0.708304i	$0.250539\pi$
$662$	−18.1137	−0.704010
$663$	0	0
$664$	38.7409	1.50344
$665$	−5.38188	−0.208700
$666$	0	0
$667$	1.57637	0.0610374
$668$	18.1902	0.703801
$669$	0	0
$670$	−7.80216	−0.301424
$671$	−40.7277	−1.57228
$672$	0	0
$673$	−32.1749	−1.24025	−0.620125	−	0.784503i	$-0.712918\pi$
$673$	−32.1749	−1.24025	−0.620125	+	0.784503i	$0.712918\pi$
$674$	−24.1896	−0.931749
$675$	0	0
$676$	−23.7522	−0.913546
$677$	−27.1448	−1.04326	−0.521630	−	0.853172i	$-0.674676\pi$
$677$	−27.1448	−1.04326	−0.521630	+	0.853172i	$0.674676\pi$
$678$	0	0
$679$	−34.2163	−1.31310
$680$	8.84956	0.339365
$681$	0	0
$682$	−45.8959	−1.75745
$683$	36.9135	1.41246	0.706228	−	0.707985i	$-0.250395\pi$
$683$	36.9135	1.41246	0.706228	+	0.707985i	$0.250395\pi$
$684$	0	0
$685$	−4.03769	−0.154272
$686$	−9.63083	−0.367707
$687$	0	0
$688$	1.79882	0.0685795
$689$	36.1576	1.37749
$690$	0	0
$691$	6.04891	0.230111	0.115056	−	0.993359i	$-0.463295\pi$
$691$	6.04891	0.230111	0.115056	+	0.993359i	$0.463295\pi$
$692$	−20.5530	−0.781310
$693$	0	0
$694$	9.37379	0.355824
$695$	−12.1629	−0.461365
$696$	0	0
$697$	11.3898	0.431421
$698$	11.3647	0.430160
$699$	0	0
$700$	−18.0299	−0.681467
$701$	−36.9616	−1.39602	−0.698010	−	0.716088i	$-0.745931\pi$
$701$	−36.9616	−1.39602	−0.698010	+	0.716088i	$0.745931\pi$
$702$	0	0
$703$	11.5738	0.436512
$704$	19.5789	0.737906
$705$	0	0
$706$	−9.51577	−0.358131
$707$	−49.2966	−1.85399
$708$	0	0
$709$	23.9030	0.897697	0.448848	−	0.893608i	$-0.351834\pi$
$709$	23.9030	0.897697	0.448848	+	0.893608i	$0.351834\pi$
$710$	6.11672	0.229556
$711$	0	0
$712$	−18.1899	−0.681696
$713$	−4.57605	−0.171374
$714$	0	0
$715$	33.6108	1.25697
$716$	−23.9513	−0.895101
$717$	0	0
$718$	14.5340	0.542404
$719$	9.95495	0.371257	0.185628	−	0.982620i	$-0.440568\pi$
$719$	9.95495	0.371257	0.185628	+	0.982620i	$0.440568\pi$
$720$	0	0
$721$	34.2045	1.27384
$722$	12.3932	0.461227
$723$	0	0
$724$	−3.55390	−0.132080
$725$	13.0186	0.483500
$726$	0	0
$727$	8.99414	0.333574	0.166787	−	0.985993i	$-0.446661\pi$
$727$	8.99414	0.333574	0.166787	+	0.985993i	$0.446661\pi$
$728$	46.0083	1.70518
$729$	0	0
$730$	7.00580	0.259296
$731$	−8.48892	−0.313974
$732$	0	0
$733$	9.95698	0.367769	0.183885	−	0.982948i	$-0.441133\pi$
$733$	9.95698	0.367769	0.183885	+	0.982948i	$0.441133\pi$
$734$	3.79066	0.139916
$735$	0	0
$736$	2.88229	0.106243
$737$	−65.8850	−2.42690
$738$	0	0
$739$	−13.4160	−0.493516	−0.246758	−	0.969077i	$-0.579365\pi$
$739$	−13.4160	−0.493516	−0.246758	+	0.969077i	$0.579365\pi$
$740$	−8.90208	−0.327247
$741$	0	0
$742$	−16.3183	−0.599065
$743$	37.7498	1.38491	0.692453	−	0.721463i	$-0.256530\pi$
$743$	37.7498	1.38491	0.692453	+	0.721463i	$0.256530\pi$
$744$	0	0
$745$	7.36386	0.269791
$746$	−8.75250	−0.320452
$747$	0	0
$748$	30.7031	1.12262
$749$	−2.40234	−0.0877794
$750$	0	0
$751$	−31.0281	−1.13223	−0.566116	−	0.824325i	$-0.691555\pi$
$751$	−31.0281	−1.13223	−0.566116	+	0.824325i	$0.691555\pi$
$752$	3.63379	0.132511
$753$	0	0
$754$	−13.6489	−0.497063
$755$	5.28054	0.192178
$756$	0	0
$757$	−1.55339	−0.0564591	−0.0282296	−	0.999601i	$-0.508987\pi$
$757$	−1.55339	−0.0564591	−0.0282296	+	0.999601i	$0.508987\pi$
$758$	17.5477	0.637363
$759$	0	0
$760$	4.47117	0.162186
$761$	−32.8660	−1.19139	−0.595696	−	0.803210i	$-0.703124\pi$
$761$	−32.8660	−1.19139	−0.595696	+	0.803210i	$0.703124\pi$
$762$	0	0
$763$	−14.5821	−0.527909
$764$	−8.67734	−0.313935
$765$	0	0
$766$	1.81750	0.0656691
$767$	−68.6472	−2.47871
$768$	0	0
$769$	−40.7771	−1.47046	−0.735230	−	0.677818i	$-0.762926\pi$
$769$	−40.7771	−1.47046	−0.735230	+	0.677818i	$0.762926\pi$
$770$	−15.1689	−0.546651
$771$	0	0
$772$	−5.23695	−0.188482
$773$	−0.551240	−0.0198267	−0.00991336	−	0.999951i	$-0.503156\pi$
$773$	−0.551240	−0.0198267	−0.00991336	+	0.999951i	$0.503156\pi$
$774$	0	0
$775$	−37.7918	−1.35752
$776$	28.4263	1.02044
$777$	0	0
$778$	−0.693859	−0.0248760
$779$	5.75463	0.206181
$780$	0	0
$781$	51.6524	1.84827
$782$	−1.32841	−0.0475038
$783$	0	0
$784$	2.28235	0.0815126
$785$	12.7612	0.455466
$786$	0	0
$787$	−1.95513	−0.0696930	−0.0348465	−	0.999393i	$-0.511094\pi$
$787$	−1.95513	−0.0696930	−0.0348465	+	0.999393i	$0.511094\pi$
$788$	37.2462	1.32684
$789$	0	0
$790$	6.41646	0.228288
$791$	−30.2527	−1.07566
$792$	0	0
$793$	35.1597	1.24856
$794$	7.33434	0.260286
$795$	0	0
$796$	−25.0596	−0.888215
$797$	−0.744348	−0.0263661	−0.0131831	−	0.999913i	$-0.504196\pi$
$797$	−0.744348	−0.0263661	−0.0131831	+	0.999913i	$0.504196\pi$
$798$	0	0
$799$	−17.1484	−0.606667
$800$	23.8037	0.841587
$801$	0	0
$802$	−22.3071	−0.787693
$803$	59.1602	2.08772
$804$	0	0
$805$	−1.51242	−0.0533057
$806$	39.6213	1.39560
$807$	0	0
$808$	40.9547	1.44078
$809$	−56.0726	−1.97141	−0.985705	−	0.168483i	$-0.946113\pi$
$809$	−56.0726	−1.97141	−0.985705	+	0.168483i	$0.946113\pi$
$810$	0	0
$811$	49.4096	1.73501	0.867504	−	0.497431i	$-0.165723\pi$
$811$	49.4096	1.73501	0.867504	+	0.497431i	$0.165723\pi$
$812$	−14.1952	−0.498154
$813$	0	0
$814$	32.6209	1.14336
$815$	21.7299	0.761167
$816$	0	0
$817$	−4.28896	−0.150052
$818$	9.28771	0.324737
$819$	0	0
$820$	−4.42623	−0.154571
$821$	18.3212	0.639415	0.319708	−	0.947516i	$-0.396415\pi$
$821$	18.3212	0.639415	0.319708	+	0.947516i	$0.396415\pi$
$822$	0	0
$823$	−52.3402	−1.82447	−0.912233	−	0.409673i	$-0.865643\pi$
$823$	−52.3402	−1.82447	−0.912233	+	0.409673i	$0.865643\pi$
$824$	−28.4165	−0.989936
$825$	0	0
$826$	30.9813	1.07798
$827$	15.5868	0.542007	0.271004	−	0.962578i	$-0.412644\pi$
$827$	15.5868	0.542007	0.271004	+	0.962578i	$0.412644\pi$
$828$	0	0
$829$	36.2883	1.26035	0.630173	−	0.776455i	$-0.282984\pi$
$829$	36.2883	1.26035	0.630173	+	0.776455i	$0.282984\pi$
$830$	11.0267	0.382741
$831$	0	0
$832$	−16.9022	−0.585977
$833$	−10.7708	−0.373185
$834$	0	0
$835$	12.6015	0.436094
$836$	15.5125	0.536512
$837$	0	0
$838$	−23.1711	−0.800433
$839$	−14.1034	−0.486903	−0.243452	−	0.969913i	$-0.578280\pi$
$839$	−14.1034	−0.486903	−0.243452	+	0.969913i	$0.578280\pi$
$840$	0	0
$841$	−18.7502	−0.646560
$842$	22.1386	0.762946
$843$	0	0
$844$	−15.2154	−0.523735
$845$	−16.4547	−0.566057
$846$	0	0
$847$	−93.1248	−3.19980
$848$	−4.84869	−0.166505
$849$	0	0
$850$	−10.9708	−0.376296
$851$	3.25246	0.111493
$852$	0	0
$853$	−12.6863	−0.434369	−0.217185	−	0.976131i	$-0.569687\pi$
$853$	−12.6863	−0.434369	−0.217185	+	0.976131i	$0.569687\pi$
$854$	−15.8680	−0.542992
$855$	0	0
$856$	1.99582	0.0682157
$857$	34.1805	1.16758	0.583792	−	0.811903i	$-0.301568\pi$
$857$	34.1805	1.16758	0.583792	+	0.811903i	$0.301568\pi$
$858$	0	0
$859$	38.6514	1.31877	0.659384	−	0.751806i	$-0.270817\pi$
$859$	38.6514	1.31877	0.659384	+	0.751806i	$0.270817\pi$
$860$	3.29890	0.112492
$861$	0	0
$862$	−22.4288	−0.763928
$863$	50.3947	1.71546	0.857728	−	0.514104i	$-0.171875\pi$
$863$	50.3947	1.71546	0.857728	+	0.514104i	$0.171875\pi$
$864$	0	0
$865$	−14.2384	−0.484120
$866$	8.30751	0.282301
$867$	0	0
$868$	41.2072	1.39866
$869$	54.1835	1.83805
$870$	0	0
$871$	56.8776	1.92722
$872$	12.1146	0.410251
$873$	0	0
$874$	−0.671169	−0.0227026
$875$	−27.8486	−0.941456
$876$	0	0
$877$	−1.18311	−0.0399507	−0.0199754	−	0.999800i	$-0.506359\pi$
$877$	−1.18311	−0.0399507	−0.0199754	+	0.999800i	$0.506359\pi$
$878$	2.95588	0.0997562
$879$	0	0
$880$	−4.50717	−0.151937
$881$	−41.1512	−1.38642	−0.693210	−	0.720736i	$-0.743804\pi$
$881$	−41.1512	−1.38642	−0.693210	+	0.720736i	$0.743804\pi$
$882$	0	0
$883$	−21.6968	−0.730157	−0.365078	−	0.930977i	$-0.618958\pi$
$883$	−21.6968	−0.730157	−0.365078	+	0.930977i	$0.618958\pi$
$884$	−26.5056	−0.891480
$885$	0	0
$886$	−16.9282	−0.568714
$887$	−53.0183	−1.78018	−0.890091	−	0.455784i	$-0.849359\pi$
$887$	−53.0183	−1.78018	−0.890091	+	0.455784i	$0.849359\pi$
$888$	0	0
$889$	−36.2606	−1.21614
$890$	−5.17732	−0.173544
$891$	0	0
$892$	−11.8756	−0.397626
$893$	−8.66410	−0.289933
$894$	0	0
$895$	−16.5926	−0.554628
$896$	−29.5898	−0.988524
$897$	0	0
$898$	19.5406	0.652078
$899$	−29.7540	−0.992351
$900$	0	0
$901$	22.8817	0.762301
$902$	16.2196	0.540052
$903$	0	0
$904$	25.1334	0.835924
$905$	−2.46201	−0.0818400
$906$	0	0
$907$	−6.97839	−0.231713	−0.115857	−	0.993266i	$-0.536961\pi$
$907$	−6.97839	−0.231713	−0.115857	+	0.993266i	$0.536961\pi$
$908$	−19.5730	−0.649554
$909$	0	0
$910$	13.0951	0.434100
$911$	−15.5047	−0.513693	−0.256846	−	0.966452i	$-0.582683\pi$
$911$	−15.5047	−0.513693	−0.256846	+	0.966452i	$0.582683\pi$
$912$	0	0
$913$	93.1142	3.08163
$914$	23.3470	0.772250
$915$	0	0
$916$	−26.5271	−0.876480
$917$	−14.9368	−0.493257
$918$	0	0
$919$	−34.8493	−1.14957	−0.574786	−	0.818303i	$-0.694915\pi$
$919$	−34.8493	−1.14957	−0.574786	+	0.818303i	$0.694915\pi$
$920$	1.25649	0.0414252
$921$	0	0
$922$	0.351268	0.0115684
$923$	−44.5908	−1.46772
$924$	0	0
$925$	26.8608	0.883178
$926$	−1.26802	−0.0416698
$927$	0	0
$928$	18.7410	0.615203
$929$	49.1760	1.61341	0.806707	−	0.590952i	$-0.201248\pi$
$929$	49.1760	1.61341	0.806707	+	0.590952i	$0.201248\pi$
$930$	0	0
$931$	−5.44184	−0.178349
$932$	−17.0561	−0.558692
$933$	0	0
$934$	−9.97366	−0.326348
$935$	21.2700	0.695604
$936$	0	0
$937$	39.4213	1.28784	0.643918	−	0.765094i	$-0.277308\pi$
$937$	39.4213	1.28784	0.643918	+	0.765094i	$0.277308\pi$
$938$	−25.6696	−0.838141
$939$	0	0
$940$	6.66408	0.217358
$941$	27.9127	0.909928	0.454964	−	0.890510i	$-0.349652\pi$
$941$	27.9127	0.909928	0.454964	+	0.890510i	$0.349652\pi$
$942$	0	0
$943$	1.61717	0.0526623
$944$	9.20552	0.299614
$945$	0	0
$946$	−12.0885	−0.393032
$947$	41.8004	1.35833	0.679165	−	0.733986i	$-0.262342\pi$
$947$	41.8004	1.35833	0.679165	+	0.733986i	$0.262342\pi$
$948$	0	0
$949$	−51.0722	−1.65787
$950$	−5.54292	−0.179836
$951$	0	0
$952$	29.1156	0.943641
$953$	2.31226	0.0749014	0.0374507	−	0.999298i	$-0.488076\pi$
$953$	2.31226	0.0749014	0.0374507	+	0.999298i	$0.488076\pi$
$954$	0	0
$955$	−6.01135	−0.194523
$956$	1.39476	0.0451096
$957$	0	0
$958$	17.5791	0.567955
$959$	−13.2842	−0.428970
$960$	0	0
$961$	55.3728	1.78622
$962$	−28.1612	−0.907953
$963$	0	0
$964$	36.7452	1.18348
$965$	−3.62797	−0.116788
$966$	0	0
$967$	−3.06296	−0.0984982	−0.0492491	−	0.998787i	$-0.515683\pi$
$967$	−3.06296	−0.0984982	−0.0492491	+	0.998787i	$0.515683\pi$
$968$	77.3664	2.48665
$969$	0	0
$970$	8.09086	0.259782
$971$	24.5745	0.788635	0.394317	−	0.918974i	$-0.370981\pi$
$971$	24.5745	0.788635	0.394317	+	0.918974i	$0.370981\pi$
$972$	0	0
$973$	−40.0167	−1.28287
$974$	5.65941	0.181339
$975$	0	0
$976$	−4.71488	−0.150920
$977$	51.6619	1.65281	0.826406	−	0.563075i	$-0.190382\pi$
$977$	51.6619	1.65281	0.826406	+	0.563075i	$0.190382\pi$
$978$	0	0
$979$	−43.7197	−1.39729
$980$	4.18565	0.133706
$981$	0	0
$982$	28.0640	0.895559
$983$	47.4875	1.51462	0.757308	−	0.653058i	$-0.226514\pi$
$983$	47.4875	1.51462	0.757308	+	0.653058i	$0.226514\pi$
$984$	0	0
$985$	25.8028	0.822146
$986$	−8.63748	−0.275073
$987$	0	0
$988$	−13.3917	−0.426048
$989$	−1.20529	−0.0383259
$990$	0	0
$991$	12.8892	0.409440	0.204720	−	0.978821i	$-0.434372\pi$
$991$	12.8892	0.409440	0.204720	+	0.978821i	$0.434372\pi$
$992$	−54.4031	−1.72730
$993$	0	0
$994$	20.1244	0.638306
$995$	−17.3604	−0.550361
$996$	0	0
$997$	10.0630	0.318700	0.159350	−	0.987222i	$-0.449060\pi$
$997$	10.0630	0.318700	0.159350	+	0.987222i	$0.449060\pi$
$998$	−13.2284	−0.418738
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.g.1.3		8
3.2	odd	2		717.2.a.f.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.f.1.6	✓	8	3.2	odd	2
2151.2.a.g.1.3		8	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)

\(f(q)\)	\(=\)	\(q-0.777975 q^{2} -1.39476 q^{4} -0.966236 q^{5} -3.17897 q^{7} +2.64103 q^{8} +O(q^{10})\)	\(q-0.777975 q^{2} -1.39476 q^{4} -0.966236 q^{5} -3.17897 q^{7} +2.64103 q^{8} +0.751707 q^{10} +6.34775 q^{11} -5.47993 q^{13} +2.47316 q^{14} +0.734853 q^{16} -3.46788 q^{17} -1.75212 q^{19} +1.34766 q^{20} -4.93839 q^{22} -0.492382 q^{23} -4.06639 q^{25} +4.26325 q^{26} +4.43389 q^{28} -3.20152 q^{29} +9.29370 q^{31} -5.85376 q^{32} +2.69793 q^{34} +3.07164 q^{35} -6.60557 q^{37} +1.36311 q^{38} -2.55186 q^{40} -3.28438 q^{41} +2.44787 q^{43} -8.85356 q^{44} +0.383061 q^{46} +4.94492 q^{47} +3.10586 q^{49} +3.16355 q^{50} +7.64316 q^{52} -6.59818 q^{53} -6.13343 q^{55} -8.39577 q^{56} +2.49071 q^{58} +12.5270 q^{59} -6.41609 q^{61} -7.23026 q^{62} +3.08437 q^{64} +5.29491 q^{65} -10.3793 q^{67} +4.83685 q^{68} -2.38965 q^{70} +8.13711 q^{71} +9.31986 q^{73} +5.13897 q^{74} +2.44378 q^{76} -20.1793 q^{77} +8.53586 q^{79} -0.710041 q^{80} +2.55516 q^{82} +14.6688 q^{83} +3.35079 q^{85} -1.90438 q^{86} +16.7646 q^{88} -6.88742 q^{89} +17.4205 q^{91} +0.686752 q^{92} -3.84702 q^{94} +1.69296 q^{95} +10.7633 q^{97} -2.41628 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8}+O(q^{10}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8	\( 8 q + 5 q^{2} + 7 q^{4} + 13 q^{5} - 7 q^{7} + 15 q^{8} + 4 q^{10} + 19 q^{11} - 3 q^{13} + 8 q^{14} + 9 q^{16} + 13 q^{17} - 6 q^{19} + 18 q^{20} + 3 q^{22} + 18 q^{23} + 7 q^{25} - 2 q^{28} + 10 q^{29} - 2 q^{31} + 20 q^{32} - 4 q^{34} + 7 q^{35} - 8 q^{37} - 9 q^{38} + 29 q^{40} + 22 q^{41} - 24 q^{43} + 7 q^{44} + 30 q^{46} + 17 q^{47} + 15 q^{49} - 24 q^{50} + 22 q^{52} + 32 q^{55} - 19 q^{56} + 18 q^{58} + 24 q^{59} + 10 q^{61} - 30 q^{62} + 33 q^{64} + 17 q^{65} - 48 q^{67} + 21 q^{68} + 31 q^{70} + 17 q^{71} + 2 q^{73} - 9 q^{74} + 10 q^{76} - 10 q^{77} + 17 q^{79} - 8 q^{80} - 17 q^{82} + 37 q^{83} + 28 q^{85} + q^{86} + 15 q^{88} + 41 q^{89} - 39 q^{91} + 38 q^{92} + 2 q^{94} - 16 q^{95} - 20 q^{97} - 46 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 7 * q^4 + 13 * q^5 - 7 * q^7 + 15 * q^8 + 4 * q^10 + 19 * q^11 - 3 * q^13 + 8 * q^14 + 9 * q^16 + 13 * q^17 - 6 * q^19 + 18 * q^20 + 3 * q^22 + 18 * q^23 + 7 * q^25 - 2 * q^28 + 10 * q^29 - 2 * q^31 + 20 * q^32 - 4 * q^34 + 7 * q^35 - 8 * q^37 - 9 * q^38 + 29 * q^40 + 22 * q^41 - 24 * q^43 + 7 * q^44 + 30 * q^46 + 17 * q^47 + 15 * q^49 - 24 * q^50 + 22 * q^52 + 32 * q^55 - 19 * q^56 + 18 * q^58 + 24 * q^59 + 10 * q^61 - 30 * q^62 + 33 * q^64 + 17 * q^65 - 48 * q^67 + 21 * q^68 + 31 * q^70 + 17 * q^71 + 2 * q^73 - 9 * q^74 + 10 * q^76 - 10 * q^77 + 17 * q^79 - 8 * q^80 - 17 * q^82 + 37 * q^83 + 28 * q^85 + q^86 + 15 * q^88 + 41 * q^89 - 39 * q^91 + 38 * q^92 + 2 * q^94 - 16 * q^95 - 20 * q^97 - 46 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more