LMFDB - Embedded newform 2151.2.a.e.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:	$1$
Dimension:	$6$
Coefficient field:	6.6.1767625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 7x^{4} - x^{3} + 11x^{2} + x - 1 $ x^6 - 7x^4 - x^3 + 11x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.94590$ 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)

$f(q)$	$=$	$q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} +O(q^{10})$	\(q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} -0.0449852 q^{10} +1.08824 q^{11} -2.81084 q^{13} -0.288090 q^{14} +3.93506 q^{16} -5.82744 q^{17} -7.25266 q^{19} -0.859312 q^{20} -0.113322 q^{22} +3.48861 q^{23} -4.81338 q^{25} +0.292701 q^{26} -5.50313 q^{28} +5.03919 q^{29} -2.02388 q^{31} -1.24057 q^{32} +0.606828 q^{34} +1.19515 q^{35} +2.08347 q^{37} +0.755240 q^{38} +0.179453 q^{40} +5.40397 q^{41} -5.91283 q^{43} -2.16468 q^{44} -0.363279 q^{46} +0.795216 q^{47} +0.653884 q^{49} +0.501230 q^{50} +5.59121 q^{52} +3.47735 q^{53} +0.470119 q^{55} +1.14924 q^{56} -0.524745 q^{58} +9.03190 q^{59} -10.4206 q^{61} +0.210753 q^{62} -7.74093 q^{64} -1.21428 q^{65} -3.98418 q^{67} +11.5917 q^{68} -0.124454 q^{70} -10.9852 q^{71} +1.10684 q^{73} -0.216957 q^{74} +14.4267 q^{76} +3.01069 q^{77} -0.409081 q^{79} +1.69994 q^{80} -0.562731 q^{82} -15.0992 q^{83} -2.51744 q^{85} +0.615719 q^{86} +0.452058 q^{88} -14.2188 q^{89} -7.77639 q^{91} -6.93940 q^{92} -0.0828081 q^{94} -3.13314 q^{95} +12.5418 q^{97} -0.0680907 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * 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 $

$2$	−0.104133	−0.0736330	−0.0368165	−	0.999322i	$-0.511722\pi$
$2$	−0.104133	−0.0736330	−0.0368165	+	0.999322i	$0.511722\pi$
$3$	0	0
$3$	0	0
$4$	−1.98916	−0.994578
$5$	0.431998	0.193195	0.0965977	−	0.995324i	$-0.469204\pi$
$5$	0.431998	0.193195	0.0965977	+	0.995324i	$0.469204\pi$
$6$	0	0
$7$	2.76657	1.04566	0.522832	−	0.852436i	$-0.324876\pi$
$7$	2.76657	1.04566	0.522832	+	0.852436i	$0.324876\pi$
$8$	0.415402	0.146867
$9$	0	0
$10$	−0.0449852	−0.0142256
$11$	1.08824	0.328117	0.164059	−	0.986451i	$-0.447541\pi$
$11$	1.08824	0.328117	0.164059	+	0.986451i	$0.447541\pi$
$12$	0	0
$13$	−2.81084	−0.779588	−0.389794	−	0.920902i	$-0.627454\pi$
$13$	−2.81084	−0.779588	−0.389794	+	0.920902i	$0.627454\pi$
$14$	−0.288090	−0.0769953
$15$	0	0
$16$	3.93506	0.983764
$17$	−5.82744	−1.41336	−0.706681	−	0.707532i	$-0.749808\pi$
$17$	−5.82744	−1.41336	−0.706681	+	0.707532i	$0.749808\pi$
$18$	0	0
$19$	−7.25266	−1.66388	−0.831938	−	0.554869i	$-0.812768\pi$
$19$	−7.25266	−1.66388	−0.831938	+	0.554869i	$0.812768\pi$
$20$	−0.859312	−0.192148
$21$	0	0
$22$	−0.113322	−0.0241603
$23$	3.48861	0.727426	0.363713	−	0.931511i	$-0.381509\pi$
$23$	3.48861	0.727426	0.363713	+	0.931511i	$0.381509\pi$
$24$	0	0
$25$	−4.81338	−0.962676
$26$	0.292701	0.0574034
$27$	0	0
$28$	−5.50313	−1.03999
$29$	5.03919	0.935754	0.467877	−	0.883794i	$-0.345019\pi$
$29$	5.03919	0.935754	0.467877	+	0.883794i	$0.345019\pi$
$30$	0	0
$31$	−2.02388	−0.363500	−0.181750	−	0.983345i	$-0.558176\pi$
$31$	−2.02388	−0.363500	−0.181750	+	0.983345i	$0.558176\pi$
$32$	−1.24057	−0.219304
$33$	0	0
$34$	0.606828	0.104070
$35$	1.19515	0.202017
$36$	0	0
$37$	2.08347	0.342520	0.171260	−	0.985226i	$-0.445216\pi$
$37$	2.08347	0.342520	0.171260	+	0.985226i	$0.445216\pi$
$38$	0.755240	0.122516
$39$	0	0
$40$	0.179453	0.0283740
$41$	5.40397	0.843959	0.421979	−	0.906605i	$-0.361335\pi$
$41$	5.40397	0.843959	0.421979	+	0.906605i	$0.361335\pi$
$42$	0	0
$43$	−5.91283	−0.901698	−0.450849	−	0.892600i	$-0.648879\pi$
$43$	−5.91283	−0.901698	−0.450849	+	0.892600i	$0.648879\pi$
$44$	−2.16468	−0.326338
$45$	0	0
$46$	−0.363279	−0.0535626
$47$	0.795216	0.115994	0.0579971	−	0.998317i	$-0.481529\pi$
$47$	0.795216	0.115994	0.0579971	+	0.998317i	$0.481529\pi$
$48$	0	0
$49$	0.653884	0.0934120
$50$	0.501230	0.0708847
$51$	0	0
$52$	5.59121	0.775361
$53$	3.47735	0.477650	0.238825	−	0.971063i	$-0.423238\pi$
$53$	3.47735	0.477650	0.238825	+	0.971063i	$0.423238\pi$
$54$	0	0
$55$	0.470119	0.0633908
$56$	1.14924	0.153573
$57$	0	0
$58$	−0.524745	−0.0689024
$59$	9.03190	1.17585	0.587927	−	0.808914i	$-0.299944\pi$
$59$	9.03190	1.17585	0.587927	+	0.808914i	$0.299944\pi$
$60$	0	0
$61$	−10.4206	−1.33422	−0.667109	−	0.744961i	$-0.732468\pi$
$61$	−10.4206	−1.33422	−0.667109	+	0.744961i	$0.732468\pi$
$62$	0.210753	0.0267656
$63$	0	0
$64$	−7.74093	−0.967616
$65$	−1.21428	−0.150613
$66$	0	0
$67$	−3.98418	−0.486745	−0.243373	−	0.969933i	$-0.578254\pi$
$67$	−3.98418	−0.486745	−0.243373	+	0.969933i	$0.578254\pi$
$68$	11.5917	1.40570
$69$	0	0
$70$	−0.124454	−0.0148752
$71$	−10.9852	−1.30370	−0.651851	−	0.758347i	$-0.726007\pi$
$71$	−10.9852	−1.30370	−0.651851	+	0.758347i	$0.726007\pi$
$72$	0	0
$73$	1.10684	0.129546	0.0647729	−	0.997900i	$-0.479368\pi$
$73$	1.10684	0.129546	0.0647729	+	0.997900i	$0.479368\pi$
$74$	−0.216957	−0.0252208
$75$	0	0
$76$	14.4267	1.65485
$77$	3.01069	0.343100
$78$	0	0
$79$	−0.409081	−0.0460252	−0.0230126	−	0.999735i	$-0.507326\pi$
$79$	−0.409081	−0.0460252	−0.0230126	+	0.999735i	$0.507326\pi$
$80$	1.69994	0.190059
$81$	0	0
$82$	−0.562731	−0.0621432
$83$	−15.0992	−1.65736	−0.828679	−	0.559725i	$-0.810907\pi$
$83$	−15.0992	−1.65736	−0.828679	+	0.559725i	$0.810907\pi$
$84$	0	0
$85$	−2.51744	−0.273055
$86$	0.615719	0.0663947
$87$	0	0
$88$	0.452058	0.0481895
$89$	−14.2188	−1.50719	−0.753593	−	0.657342i	$-0.771681\pi$
$89$	−14.2188	−1.50719	−0.753593	+	0.657342i	$0.771681\pi$
$90$	0	0
$91$	−7.77639	−0.815187
$92$	−6.93940	−0.723482
$93$	0	0
$94$	−0.0828081	−0.00854100
$95$	−3.13314	−0.321453
$96$	0	0
$97$	12.5418	1.27342	0.636712	−	0.771101i	$-0.280294\pi$
$97$	12.5418	1.27342	0.636712	+	0.771101i	$0.280294\pi$
$98$	−0.0680907	−0.00687820
$99$	0	0
$100$	9.57456	0.957456
$101$	−4.71052	−0.468714	−0.234357	−	0.972151i	$-0.575298\pi$
$101$	−4.71052	−0.468714	−0.234357	+	0.972151i	$0.575298\pi$
$102$	0	0
$103$	−19.1362	−1.88555	−0.942775	−	0.333431i	$-0.891794\pi$
$103$	−19.1362	−1.88555	−0.942775	+	0.333431i	$0.891794\pi$
$104$	−1.16763	−0.114496
$105$	0	0
$106$	−0.362106	−0.0351708
$107$	−6.40124	−0.618831	−0.309416	−	0.950927i	$-0.600133\pi$
$107$	−6.40124	−0.618831	−0.309416	+	0.950927i	$0.600133\pi$
$108$	0	0
$109$	−2.95365	−0.282908	−0.141454	−	0.989945i	$-0.545178\pi$
$109$	−2.95365	−0.282908	−0.141454	+	0.989945i	$0.545178\pi$
$110$	−0.0489548	−0.00466765
$111$	0	0
$112$	10.8866	1.02869
$113$	−0.888386	−0.0835724	−0.0417862	−	0.999127i	$-0.513305\pi$
$113$	−0.888386	−0.0835724	−0.0417862	+	0.999127i	$0.513305\pi$
$114$	0	0
$115$	1.50707	0.140535
$116$	−10.0237	−0.930680
$117$	0	0
$118$	−0.940517	−0.0865816
$119$	−16.1220	−1.47790
$120$	0	0
$121$	−9.81573	−0.892339
$122$	1.08512	0.0982424
$123$	0	0
$124$	4.02582	0.361530
$125$	−4.23936	−0.379180
$126$	0	0
$127$	−2.46764	−0.218968	−0.109484	−	0.993989i	$-0.534920\pi$
$127$	−2.46764	−0.218968	−0.109484	+	0.993989i	$0.534920\pi$
$128$	3.28723	0.290553
$129$	0	0
$130$	0.126446	0.0110901
$131$	6.69156	0.584644	0.292322	−	0.956320i	$-0.405572\pi$
$131$	6.69156	0.584644	0.292322	+	0.956320i	$0.405572\pi$
$132$	0	0
$133$	−20.0650	−1.73985
$134$	0.414884	0.0358405
$135$	0	0
$136$	−2.42073	−0.207576
$137$	6.91690	0.590951	0.295475	−	0.955350i	$-0.404522\pi$
$137$	6.91690	0.590951	0.295475	+	0.955350i	$0.404522\pi$
$138$	0	0
$139$	−9.48581	−0.804575	−0.402288	−	0.915513i	$-0.631785\pi$
$139$	−9.48581	−0.804575	−0.402288	+	0.915513i	$0.631785\pi$
$140$	−2.37734	−0.200922
$141$	0	0
$142$	1.14392	0.0959955
$143$	−3.05888	−0.255796
$144$	0	0
$145$	2.17692	0.180783
$146$	−0.115258	−0.00953885
$147$	0	0
$148$	−4.14434	−0.340663
$149$	−7.32345	−0.599961	−0.299980	−	0.953945i	$-0.596980\pi$
$149$	−7.32345	−0.599961	−0.299980	+	0.953945i	$0.596980\pi$
$150$	0	0
$151$	−1.83931	−0.149681	−0.0748404	−	0.997196i	$-0.523845\pi$
$151$	−1.83931	−0.149681	−0.0748404	+	0.997196i	$0.523845\pi$
$152$	−3.01277	−0.244368
$153$	0	0
$154$	−0.313512	−0.0252635
$155$	−0.874314	−0.0702266
$156$	0	0
$157$	−1.48477	−0.118498	−0.0592488	−	0.998243i	$-0.518871\pi$
$157$	−1.48477	−0.118498	−0.0592488	+	0.998243i	$0.518871\pi$
$158$	0.0425988	0.00338897
$159$	0	0
$160$	−0.535925	−0.0423686
$161$	9.65148	0.760643
$162$	0	0
$163$	9.95721	0.779909	0.389954	−	0.920834i	$-0.372491\pi$
$163$	9.95721	0.779909	0.389954	+	0.920834i	$0.372491\pi$
$164$	−10.7493	−0.839383
$165$	0	0
$166$	1.57233	0.122036
$167$	1.51314	0.117090	0.0585452	−	0.998285i	$-0.481354\pi$
$167$	1.51314	0.117090	0.0585452	+	0.998285i	$0.481354\pi$
$168$	0	0
$169$	−5.09915	−0.392242
$170$	0.262149	0.0201059
$171$	0	0
$172$	11.7615	0.896809
$173$	−21.1660	−1.60922	−0.804612	−	0.593801i	$-0.797627\pi$
$173$	−21.1660	−1.60922	−0.804612	+	0.593801i	$0.797627\pi$
$174$	0	0
$175$	−13.3165	−1.00663
$176$	4.28229	0.322790
$177$	0	0
$178$	1.48064	0.110979
$179$	23.4094	1.74970	0.874850	−	0.484394i	$-0.160960\pi$
$179$	23.4094	1.74970	0.874850	+	0.484394i	$0.160960\pi$
$180$	0	0
$181$	14.5477	1.08132	0.540660	−	0.841241i	$-0.318175\pi$
$181$	14.5477	1.08132	0.540660	+	0.841241i	$0.318175\pi$
$182$	0.809777	0.0600247
$183$	0	0
$184$	1.44918	0.106835
$185$	0.900055	0.0661733
$186$	0	0
$187$	−6.34167	−0.463749
$188$	−1.58181	−0.115365
$189$	0	0
$190$	0.326262	0.0236696
$191$	−24.9050	−1.80206	−0.901031	−	0.433755i	$-0.857188\pi$
$191$	−24.9050	−1.80206	−0.901031	+	0.433755i	$0.857188\pi$
$192$	0	0
$193$	5.84285	0.420577	0.210289	−	0.977639i	$-0.432560\pi$
$193$	5.84285	0.420577	0.210289	+	0.977639i	$0.432560\pi$
$194$	−1.30601	−0.0937661
$195$	0	0
$196$	−1.30068	−0.0929055
$197$	16.2436	1.15731	0.578655	−	0.815572i	$-0.303578\pi$
$197$	16.2436	1.15731	0.578655	+	0.815572i	$0.303578\pi$
$198$	0	0
$199$	−22.7909	−1.61560	−0.807801	−	0.589455i	$-0.799343\pi$
$199$	−22.7909	−1.61560	−0.807801	+	0.589455i	$0.799343\pi$
$200$	−1.99949	−0.141385
$201$	0	0
$202$	0.490519	0.0345128
$203$	13.9412	0.978484
$204$	0	0
$205$	2.33451	0.163049
$206$	1.99271	0.138839
$207$	0	0
$208$	−11.0608	−0.766931
$209$	−7.89265	−0.545946
$210$	0	0
$211$	18.5913	1.27988	0.639938	−	0.768427i	$-0.278960\pi$
$211$	18.5913	1.27988	0.639938	+	0.768427i	$0.278960\pi$
$212$	−6.91698	−0.475060
$213$	0	0
$214$	0.666579	0.0455664
$215$	−2.55433	−0.174204
$216$	0	0
$217$	−5.59921	−0.380099
$218$	0.307572	0.0208314
$219$	0	0
$220$	−0.935140	−0.0630471
$221$	16.3800	1.10184
$222$	0	0
$223$	−12.7025	−0.850622	−0.425311	−	0.905047i	$-0.639835\pi$
$223$	−12.7025	−0.850622	−0.425311	+	0.905047i	$0.639835\pi$
$224$	−3.43212	−0.229318
$225$	0	0
$226$	0.0925102	0.00615369
$227$	11.9490	0.793086	0.396543	−	0.918016i	$-0.370210\pi$
$227$	11.9490	0.793086	0.396543	+	0.918016i	$0.370210\pi$
$228$	0	0
$229$	2.84730	0.188155	0.0940773	−	0.995565i	$-0.470010\pi$
$229$	2.84730	0.188155	0.0940773	+	0.995565i	$0.470010\pi$
$230$	−0.156936	−0.0103480
$231$	0	0
$232$	2.09329	0.137431
$233$	4.49249	0.294313	0.147156	−	0.989113i	$-0.452988\pi$
$233$	4.49249	0.294313	0.147156	+	0.989113i	$0.452988\pi$
$234$	0	0
$235$	0.343532	0.0224096
$236$	−17.9659	−1.16948
$237$	0	0
$238$	1.67883	0.108822
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	13.6633	0.880133	0.440067	−	0.897965i	$-0.354955\pi$
$241$	13.6633	0.880133	0.440067	+	0.897965i	$0.354955\pi$
$242$	1.02214	0.0657056
$243$	0	0
$244$	20.7281	1.32698
$245$	0.282477	0.0180468
$246$	0	0
$247$	20.3861	1.29714
$248$	−0.840726	−0.0533861
$249$	0	0
$250$	0.441457	0.0279202
$251$	18.2147	1.14970	0.574850	−	0.818259i	$-0.305060\pi$
$251$	18.2147	1.14970	0.574850	+	0.818259i	$0.305060\pi$
$252$	0	0
$253$	3.79646	0.238681
$254$	0.256963	0.0161233
$255$	0	0
$256$	15.1395	0.946222
$257$	−4.82747	−0.301129	−0.150565	−	0.988600i	$-0.548109\pi$
$257$	−4.82747	−0.301129	−0.150565	+	0.988600i	$0.548109\pi$
$258$	0	0
$259$	5.76405	0.358161
$260$	2.41539	0.149796
$261$	0	0
$262$	−0.696811	−0.0430491
$263$	0.934556	0.0576272	0.0288136	−	0.999585i	$-0.490827\pi$
$263$	0.934556	0.0576272	0.0288136	+	0.999585i	$0.490827\pi$
$264$	0	0
$265$	1.50221	0.0922799
$266$	2.08942	0.128111
$267$	0	0
$268$	7.92516	0.484106
$269$	−22.8799	−1.39501	−0.697506	−	0.716579i	$-0.745707\pi$
$269$	−22.8799	−1.39501	−0.697506	+	0.716579i	$0.745707\pi$
$270$	0	0
$271$	14.5729	0.885243	0.442621	−	0.896709i	$-0.354049\pi$
$271$	14.5729	0.885243	0.442621	+	0.896709i	$0.354049\pi$
$272$	−22.9313	−1.39041
$273$	0	0
$274$	−0.720276	−0.0435135
$275$	−5.23812	−0.315871
$276$	0	0
$277$	−6.42790	−0.386215	−0.193108	−	0.981178i	$-0.561857\pi$
$277$	−6.42790	−0.386215	−0.193108	+	0.981178i	$0.561857\pi$
$278$	0.987783	0.0592433
$279$	0	0
$280$	0.496468	0.0296697
$281$	11.9766	0.714462	0.357231	−	0.934016i	$-0.383721\pi$
$281$	11.9766	0.714462	0.357231	+	0.934016i	$0.383721\pi$
$282$	0	0
$283$	−18.4031	−1.09395	−0.546975	−	0.837149i	$-0.684221\pi$
$283$	−18.4031	−1.09395	−0.546975	+	0.837149i	$0.684221\pi$
$284$	21.8513	1.29663
$285$	0	0
$286$	0.318530	0.0188351
$287$	14.9504	0.882497
$288$	0	0
$289$	16.9591	0.997592
$290$	−0.226689	−0.0133116
$291$	0	0
$292$	−2.20168	−0.128843
$293$	−24.8968	−1.45449	−0.727244	−	0.686379i	$-0.759199\pi$
$293$	−24.8968	−1.45449	−0.727244	+	0.686379i	$0.759199\pi$
$294$	0	0
$295$	3.90176	0.227170
$296$	0.865477	0.0503048
$297$	0	0
$298$	0.762612	0.0441769
$299$	−9.80595	−0.567093
$300$	0	0
$301$	−16.3582	−0.942872
$302$	0.191532	0.0110214
$303$	0	0
$304$	−28.5396	−1.63686
$305$	−4.50167	−0.257765
$306$	0	0
$307$	−3.67995	−0.210026	−0.105013	−	0.994471i	$-0.533488\pi$
$307$	−3.67995	−0.210026	−0.105013	+	0.994471i	$0.533488\pi$
$308$	−5.98874	−0.341240
$309$	0	0
$310$	0.0910448	0.00517100
$311$	28.6157	1.62265	0.811324	−	0.584596i	$-0.198747\pi$
$311$	28.6157	1.62265	0.811324	+	0.584596i	$0.198747\pi$
$312$	0	0
$313$	14.3414	0.810622	0.405311	−	0.914179i	$-0.367163\pi$
$313$	14.3414	0.810622	0.405311	+	0.914179i	$0.367163\pi$
$314$	0.154613	0.00872534
$315$	0	0
$316$	0.813726	0.0457757
$317$	24.7113	1.38793	0.693964	−	0.720010i	$-0.255863\pi$
$317$	24.7113	1.38793	0.693964	+	0.720010i	$0.255863\pi$
$318$	0	0
$319$	5.48386	0.307037
$320$	−3.34407	−0.186939
$321$	0	0
$322$	−1.00504	−0.0560084
$323$	42.2645	2.35166
$324$	0	0
$325$	13.5297	0.750490
$326$	−1.03687	−0.0574270
$327$	0	0
$328$	2.24482	0.123949
$329$	2.20002	0.121291
$330$	0	0
$331$	−30.5404	−1.67865	−0.839327	−	0.543627i	$-0.817051\pi$
$331$	−30.5404	−1.67865	−0.839327	+	0.543627i	$0.817051\pi$
$332$	30.0348	1.64837
$333$	0	0
$334$	−0.157568	−0.00862171
$335$	−1.72116	−0.0940370
$336$	0	0
$337$	26.8195	1.46095	0.730474	−	0.682940i	$-0.239299\pi$
$337$	26.8195	1.46095	0.730474	+	0.682940i	$0.239299\pi$
$338$	0.530989	0.0288820
$339$	0	0
$340$	5.00759	0.271575
$341$	−2.20248	−0.119271
$342$	0	0
$343$	−17.5569	−0.947986
$344$	−2.45620	−0.132429
$345$	0	0
$346$	2.20408	0.118492
$347$	9.11072	0.489089	0.244544	−	0.969638i	$-0.421362\pi$
$347$	9.11072	0.489089	0.244544	+	0.969638i	$0.421362\pi$
$348$	0	0
$349$	31.8626	1.70557	0.852783	−	0.522265i	$-0.174913\pi$
$349$	31.8626	1.70557	0.852783	+	0.522265i	$0.174913\pi$
$350$	1.38669	0.0741215
$351$	0	0
$352$	−1.35004	−0.0719575
$353$	−13.4752	−0.717214	−0.358607	−	0.933489i	$-0.616748\pi$
$353$	−13.4752	−0.717214	−0.358607	+	0.933489i	$0.616748\pi$
$354$	0	0
$355$	−4.74558	−0.251869
$356$	28.2833	1.49901
$357$	0	0
$358$	−2.43769	−0.128836
$359$	−2.38673	−0.125967	−0.0629835	−	0.998015i	$-0.520062\pi$
$359$	−2.38673	−0.125967	−0.0629835	+	0.998015i	$0.520062\pi$
$360$	0	0
$361$	33.6011	1.76848
$362$	−1.51489	−0.0796208
$363$	0	0
$364$	15.4684	0.810767
$365$	0.478153	0.0250277
$366$	0	0
$367$	1.65472	0.0863755	0.0431877	−	0.999067i	$-0.486249\pi$
$367$	1.65472	0.0863755	0.0431877	+	0.999067i	$0.486249\pi$
$368$	13.7279	0.715616
$369$	0	0
$370$	−0.0937252	−0.00487254
$371$	9.62030	0.499461
$372$	0	0
$373$	−6.13773	−0.317800	−0.158900	−	0.987295i	$-0.550795\pi$
$373$	−6.13773	−0.317800	−0.158900	+	0.987295i	$0.550795\pi$
$374$	0.660376	0.0341472
$375$	0	0
$376$	0.330334	0.0170357
$377$	−14.1644	−0.729503
$378$	0	0
$379$	14.3037	0.734731	0.367365	−	0.930077i	$-0.380260\pi$
$379$	14.3037	0.734731	0.367365	+	0.930077i	$0.380260\pi$
$380$	6.23230	0.319710
$381$	0	0
$382$	2.59343	0.132691
$383$	−18.2784	−0.933981	−0.466990	−	0.884262i	$-0.654662\pi$
$383$	−18.2784	−0.933981	−0.466990	+	0.884262i	$0.654662\pi$
$384$	0	0
$385$	1.30061	0.0662854
$386$	−0.608432	−0.0309684
$387$	0	0
$388$	−24.9476	−1.26652
$389$	−12.9086	−0.654492	−0.327246	−	0.944939i	$-0.606120\pi$
$389$	−12.9086	−0.654492	−0.327246	+	0.944939i	$0.606120\pi$
$390$	0	0
$391$	−20.3297	−1.02812
$392$	0.271625	0.0137191
$393$	0	0
$394$	−1.69149	−0.0852163
$395$	−0.176722	−0.00889186
$396$	0	0
$397$	−17.5435	−0.880484	−0.440242	−	0.897879i	$-0.645107\pi$
$397$	−17.5435	−0.880484	−0.440242	+	0.897879i	$0.645107\pi$
$398$	2.37328	0.118962
$399$	0	0
$400$	−18.9409	−0.947045
$401$	−35.1950	−1.75755	−0.878777	−	0.477233i	$-0.841640\pi$
$401$	−35.1950	−1.75755	−0.878777	+	0.477233i	$0.841640\pi$
$402$	0	0
$403$	5.68883	0.283381
$404$	9.36995	0.466173
$405$	0	0
$406$	−1.45174	−0.0720487
$407$	2.26732	0.112387
$408$	0	0
$409$	6.31266	0.312141	0.156070	−	0.987746i	$-0.450117\pi$
$409$	6.31266	0.312141	0.156070	+	0.987746i	$0.450117\pi$
$410$	−0.243099	−0.0120058
$411$	0	0
$412$	38.0650	1.87533
$413$	24.9873	1.22955
$414$	0	0
$415$	−6.52284	−0.320194
$416$	3.48706	0.170967
$417$	0	0
$418$	0.821884	0.0401997
$419$	−24.6530	−1.20438	−0.602190	−	0.798353i	$-0.705705\pi$
$419$	−24.6530	−1.20438	−0.602190	+	0.798353i	$0.705705\pi$
$420$	0	0
$421$	−25.4233	−1.23906	−0.619529	−	0.784974i	$-0.712676\pi$
$421$	−25.4233	−1.23906	−0.619529	+	0.784974i	$0.712676\pi$
$422$	−1.93596	−0.0942411
$423$	0	0
$424$	1.44450	0.0701510
$425$	28.0497	1.36061
$426$	0	0
$427$	−28.8292	−1.39514
$428$	12.7331	0.615476
$429$	0	0
$430$	0.265990	0.0128272
$431$	35.1475	1.69299	0.846497	−	0.532393i	$-0.178707\pi$
$431$	35.1475	1.69299	0.846497	+	0.532393i	$0.178707\pi$
$432$	0	0
$433$	−28.5125	−1.37022	−0.685111	−	0.728439i	$-0.740246\pi$
$433$	−28.5125	−1.37022	−0.685111	+	0.728439i	$0.740246\pi$
$434$	0.583061	0.0279878
$435$	0	0
$436$	5.87527	0.281374
$437$	−25.3017	−1.21035
$438$	0	0
$439$	19.6250	0.936648	0.468324	−	0.883557i	$-0.344858\pi$
$439$	19.6250	0.936648	0.468324	+	0.883557i	$0.344858\pi$
$440$	0.195288	0.00931000
$441$	0	0
$442$	−1.70570	−0.0811318
$443$	13.5449	0.643539	0.321770	−	0.946818i	$-0.395722\pi$
$443$	13.5449	0.643539	0.321770	+	0.946818i	$0.395722\pi$
$444$	0	0
$445$	−6.14248	−0.291181
$446$	1.32275	0.0626338
$447$	0	0
$448$	−21.4158	−1.01180
$449$	−31.3156	−1.47787	−0.738936	−	0.673775i	$-0.764672\pi$
$449$	−31.3156	−1.47787	−0.738936	+	0.673775i	$0.764672\pi$
$450$	0	0
$451$	5.88083	0.276917
$452$	1.76714	0.0831193
$453$	0	0
$454$	−1.24429	−0.0583973
$455$	−3.35938	−0.157490
$456$	0	0
$457$	−11.7404	−0.549193	−0.274597	−	0.961560i	$-0.588544\pi$
$457$	−11.7404	−0.549193	−0.274597	+	0.961560i	$0.588544\pi$
$458$	−0.296497	−0.0138544
$459$	0	0
$460$	−2.99781	−0.139773
$461$	0.0379172	0.00176598	0.000882990	−	1.00000i	$-0.499719\pi$
$461$	0.0379172	0.00176598	0.000882990		1.00000i	$0.499719\pi$
$462$	0	0
$463$	3.78515	0.175911	0.0879554	−	0.996124i	$-0.471967\pi$
$463$	3.78515	0.175911	0.0879554	+	0.996124i	$0.471967\pi$
$464$	19.8295	0.920561
$465$	0	0
$466$	−0.467816	−0.0216711
$467$	−32.5797	−1.50761	−0.753804	−	0.657099i	$-0.771783\pi$
$467$	−32.5797	−1.50761	−0.753804	+	0.657099i	$0.771783\pi$
$468$	0	0
$469$	−11.0225	−0.508972
$470$	−0.0357729	−0.00165008
$471$	0	0
$472$	3.75187	0.172694
$473$	−6.43459	−0.295863
$474$	0	0
$475$	34.9098	1.60177
$476$	32.0692	1.46989
$477$	0	0
$478$	−0.104133	−0.00476292
$479$	19.3874	0.885833	0.442917	−	0.896563i	$-0.353944\pi$
$479$	19.3874	0.885833	0.442917	+	0.896563i	$0.353944\pi$
$480$	0	0
$481$	−5.85631	−0.267025
$482$	−1.42280	−0.0648069
$483$	0	0
$484$	19.5250	0.887501
$485$	5.41803	0.246020
$486$	0	0
$487$	4.84919	0.219738	0.109869	−	0.993946i	$-0.464957\pi$
$487$	4.84919	0.219738	0.109869	+	0.993946i	$0.464957\pi$
$488$	−4.32873	−0.195952
$489$	0	0
$490$	−0.0294151	−0.00132884
$491$	24.7326	1.11617	0.558084	−	0.829784i	$-0.311537\pi$
$491$	24.7326	1.11617	0.558084	+	0.829784i	$0.311537\pi$
$492$	0	0
$493$	−29.3656	−1.32256
$494$	−2.12286	−0.0955121
$495$	0	0
$496$	−7.96410	−0.357599
$497$	−30.3912	−1.36323
$498$	0	0
$499$	−13.0377	−0.583647	−0.291823	−	0.956472i	$-0.594262\pi$
$499$	−13.0377	−0.583647	−0.291823	+	0.956472i	$0.594262\pi$
$500$	8.43275	0.377124
$501$	0	0
$502$	−1.89675	−0.0846559
$503$	3.27894	0.146201	0.0731004	−	0.997325i	$-0.476711\pi$
$503$	3.27894	0.146201	0.0731004	+	0.997325i	$0.476711\pi$
$504$	0	0
$505$	−2.03493	−0.0905534
$506$	−0.395336	−0.0175748
$507$	0	0
$508$	4.90853	0.217781
$509$	−19.1552	−0.849040	−0.424520	−	0.905419i	$-0.639557\pi$
$509$	−19.1552	−0.849040	−0.424520	+	0.905419i	$0.639557\pi$
$510$	0	0
$511$	3.06214	0.135461
$512$	−8.15098	−0.360226
$513$	0	0
$514$	0.502698	0.0221731
$515$	−8.26682	−0.364280
$516$	0	0
$517$	0.865388	0.0380597
$518$	−0.600227	−0.0263725
$519$	0	0
$520$	−0.504414	−0.0221200
$521$	9.95719	0.436232	0.218116	−	0.975923i	$-0.430009\pi$
$521$	9.95719	0.436232	0.218116	+	0.975923i	$0.430009\pi$
$522$	0	0
$523$	23.4154	1.02389	0.511943	−	0.859020i	$-0.328926\pi$
$523$	23.4154	1.02389	0.511943	+	0.859020i	$0.328926\pi$
$524$	−13.3106	−0.581474
$525$	0	0
$526$	−0.0973180	−0.00424326
$527$	11.7941	0.513758
$528$	0	0
$529$	−10.8296	−0.470851
$530$	−0.156429	−0.00679484
$531$	0	0
$532$	39.9124	1.73042
$533$	−15.1897	−0.657940
$534$	0	0
$535$	−2.76532	−0.119555
$536$	−1.65504	−0.0714867
$537$	0	0
$538$	2.38255	0.102719
$539$	0.711584	0.0306501
$540$	0	0
$541$	15.9305	0.684904	0.342452	−	0.939535i	$-0.388743\pi$
$541$	15.9305	0.684904	0.342452	+	0.939535i	$0.388743\pi$
$542$	−1.51752	−0.0651831
$543$	0	0
$544$	7.22936	0.309956
$545$	−1.27597	−0.0546566
$546$	0	0
$547$	33.0838	1.41456	0.707280	−	0.706934i	$-0.249922\pi$
$547$	33.0838	1.41456	0.707280	+	0.706934i	$0.249922\pi$
$548$	−13.7588	−0.587746
$549$	0	0
$550$	0.545460	0.0232585
$551$	−36.5475	−1.55698
$552$	0	0
$553$	−1.13175	−0.0481269
$554$	0.669355	0.0284382
$555$	0	0
$556$	18.8687	0.800213
$557$	31.7597	1.34570	0.672851	−	0.739778i	$-0.265069\pi$
$557$	31.7597	1.34570	0.672851	+	0.739778i	$0.265069\pi$
$558$	0	0
$559$	16.6200	0.702953
$560$	4.70299	0.198737
$561$	0	0
$562$	−1.24715	−0.0526080
$563$	33.8109	1.42496	0.712479	−	0.701694i	$-0.247572\pi$
$563$	33.8109	1.42496	0.712479	+	0.701694i	$0.247572\pi$
$564$	0	0
$565$	−0.383781	−0.0161458
$566$	1.91636	0.0805508
$567$	0	0
$568$	−4.56327	−0.191471
$569$	−13.6032	−0.570274	−0.285137	−	0.958487i	$-0.592039\pi$
$569$	−13.6032	−0.570274	−0.285137	+	0.958487i	$0.592039\pi$
$570$	0	0
$571$	19.2244	0.804517	0.402259	−	0.915526i	$-0.368225\pi$
$571$	19.2244	0.804517	0.402259	+	0.915526i	$0.368225\pi$
$572$	6.08459	0.254410
$573$	0	0
$574$	−1.55683	−0.0649809
$575$	−16.7920	−0.700275
$576$	0	0
$577$	−30.1474	−1.25505	−0.627525	−	0.778596i	$-0.715932\pi$
$577$	−30.1474	−1.25505	−0.627525	+	0.778596i	$0.715932\pi$
$578$	−1.76600	−0.0734557
$579$	0	0
$580$	−4.33023	−0.179803
$581$	−41.7730	−1.73304
$582$	0	0
$583$	3.78419	0.156725
$584$	0.459783	0.0190260
$585$	0	0
$586$	2.59258	0.107098
$587$	34.4451	1.42170	0.710850	−	0.703344i	$-0.248311\pi$
$587$	34.4451	1.42170	0.710850	+	0.703344i	$0.248311\pi$
$588$	0	0
$589$	14.6786	0.604819
$590$	−0.406302	−0.0167272
$591$	0	0
$592$	8.19856	0.336959
$593$	18.4600	0.758061	0.379030	−	0.925384i	$-0.376258\pi$
$593$	18.4600	0.758061	0.379030	+	0.925384i	$0.376258\pi$
$594$	0	0
$595$	−6.96467	−0.285524
$596$	14.5675	0.596708
$597$	0	0
$598$	1.02112	0.0417567
$599$	44.8094	1.83086	0.915432	−	0.402473i	$-0.131849\pi$
$599$	44.8094	1.83086	0.915432	+	0.402473i	$0.131849\pi$
$600$	0	0
$601$	−1.25830	−0.0513271	−0.0256636	−	0.999671i	$-0.508170\pi$
$601$	−1.25830	−0.0513271	−0.0256636	+	0.999671i	$0.508170\pi$
$602$	1.70343	0.0694265
$603$	0	0
$604$	3.65867	0.148869
$605$	−4.24038	−0.172396
$606$	0	0
$607$	−24.6395	−1.00009	−0.500044	−	0.866000i	$-0.666683\pi$
$607$	−24.6395	−1.00009	−0.500044	+	0.866000i	$0.666683\pi$
$608$	8.99746	0.364895
$609$	0	0
$610$	0.468771	0.0189800
$611$	−2.23523	−0.0904277
$612$	0	0
$613$	1.92675	0.0778205	0.0389103	−	0.999243i	$-0.487611\pi$
$613$	1.92675	0.0778205	0.0389103	+	0.999243i	$0.487611\pi$
$614$	0.383204	0.0154648
$615$	0	0
$616$	1.25065	0.0503900
$617$	−25.5030	−1.02671	−0.513357	−	0.858175i	$-0.671598\pi$
$617$	−25.5030	−1.02671	−0.513357	+	0.858175i	$0.671598\pi$
$618$	0	0
$619$	−18.7143	−0.752190	−0.376095	−	0.926581i	$-0.622733\pi$
$619$	−18.7143	−0.752190	−0.376095	+	0.926581i	$0.622733\pi$
$620$	1.73915	0.0698459
$621$	0	0
$622$	−2.97984	−0.119481
$623$	−39.3371	−1.57601
$624$	0	0
$625$	22.2355	0.889420
$626$	−1.49341	−0.0596886
$627$	0	0
$628$	2.95344	0.117855
$629$	−12.1413	−0.484105
$630$	0	0
$631$	−2.67027	−0.106302	−0.0531510	−	0.998586i	$-0.516926\pi$
$631$	−2.67027	−0.106302	−0.0531510	+	0.998586i	$0.516926\pi$
$632$	−0.169933	−0.00675957
$633$	0	0
$634$	−2.57326	−0.102197
$635$	−1.06602	−0.0423036
$636$	0	0
$637$	−1.83797	−0.0728229
$638$	−0.571049	−0.0226081
$639$	0	0
$640$	1.42008	0.0561335
$641$	12.0848	0.477320	0.238660	−	0.971103i	$-0.423292\pi$
$641$	12.0848	0.477320	0.238660	+	0.971103i	$0.423292\pi$
$642$	0	0
$643$	43.4945	1.71525	0.857627	−	0.514272i	$-0.171938\pi$
$643$	43.4945	1.71525	0.857627	+	0.514272i	$0.171938\pi$
$644$	−19.1983	−0.756519
$645$	0	0
$646$	−4.40112	−0.173160
$647$	−0.330370	−0.0129882	−0.00649409	−	0.999979i	$-0.502067\pi$
$647$	−0.330370	−0.0129882	−0.00649409	+	0.999979i	$0.502067\pi$
$648$	0	0
$649$	9.82889	0.385818
$650$	−1.40888	−0.0552609
$651$	0	0
$652$	−19.8064	−0.775680
$653$	29.4599	1.15285	0.576427	−	0.817148i	$-0.304446\pi$
$653$	29.4599	1.15285	0.576427	+	0.817148i	$0.304446\pi$
$654$	0	0
$655$	2.89074	0.112951
$656$	21.2649	0.830256
$657$	0	0
$658$	−0.229094	−0.00893101
$659$	−42.6534	−1.66154	−0.830770	−	0.556616i	$-0.812100\pi$
$659$	−42.6534	−1.66154	−0.830770	+	0.556616i	$0.812100\pi$
$660$	0	0
$661$	44.7939	1.74228	0.871140	−	0.491034i	$-0.163381\pi$
$661$	44.7939	1.74228	0.871140	+	0.491034i	$0.163381\pi$
$662$	3.18026	0.123604
$663$	0	0
$664$	−6.27226	−0.243411
$665$	−8.66803	−0.336132
$666$	0	0
$667$	17.5798	0.680692
$668$	−3.00987	−0.116455
$669$	0	0
$670$	0.179229	0.00692423
$671$	−11.3401	−0.437780
$672$	0	0
$673$	0.183092	0.00705769	0.00352885	−	0.999994i	$-0.498877\pi$
$673$	0.183092	0.00705769	0.00352885	+	0.999994i	$0.498877\pi$
$674$	−2.79279	−0.107574
$675$	0	0
$676$	10.1430	0.390116
$677$	2.12245	0.0815725	0.0407862	−	0.999168i	$-0.487014\pi$
$677$	2.12245	0.0815725	0.0407862	+	0.999168i	$0.487014\pi$
$678$	0	0
$679$	34.6976	1.33157
$680$	−1.04575	−0.0401027
$681$	0	0
$682$	0.229350	0.00878227
$683$	−6.23555	−0.238597	−0.119298	−	0.992858i	$-0.538064\pi$
$683$	−6.23555	−0.238597	−0.119298	+	0.992858i	$0.538064\pi$
$684$	0	0
$685$	2.98809	0.114169
$686$	1.82825	0.0698031
$687$	0	0
$688$	−23.2673	−0.887057
$689$	−9.77428	−0.372370
$690$	0	0
$691$	29.0740	1.10603	0.553014	−	0.833172i	$-0.313478\pi$
$691$	29.0740	1.10603	0.553014	+	0.833172i	$0.313478\pi$
$692$	42.1025	1.60050
$693$	0	0
$694$	−0.948725	−0.0360131
$695$	−4.09785	−0.155440
$696$	0	0
$697$	−31.4913	−1.19282
$698$	−3.31794	−0.125586
$699$	0	0
$700$	26.4886	1.00118
$701$	−4.01876	−0.151786	−0.0758932	−	0.997116i	$-0.524181\pi$
$701$	−4.01876	−0.151786	−0.0758932	+	0.997116i	$0.524181\pi$
$702$	0	0
$703$	−15.1107	−0.569911
$704$	−8.42400	−0.317492
$705$	0	0
$706$	1.40321	0.0528106
$707$	−13.0320	−0.490117
$708$	0	0
$709$	38.7677	1.45595	0.727975	−	0.685603i	$-0.240462\pi$
$709$	38.7677	1.45595	0.727975	+	0.685603i	$0.240462\pi$
$710$	0.494171	0.0185459
$711$	0	0
$712$	−5.90650	−0.221356
$713$	−7.06055	−0.264420
$714$	0	0
$715$	−1.32143	−0.0494187
$716$	−46.5649	−1.74021
$717$	0	0
$718$	0.248537	0.00927533
$719$	−36.1988	−1.34999	−0.674993	−	0.737824i	$-0.735854\pi$
$719$	−36.1988	−1.34999	−0.674993	+	0.737824i	$0.735854\pi$
$720$	0	0
$721$	−52.9416	−1.97165
$722$	−3.49898	−0.130219
$723$	0	0
$724$	−28.9376	−1.07546
$725$	−24.2555	−0.900827
$726$	0	0
$727$	−9.24566	−0.342902	−0.171451	−	0.985193i	$-0.554846\pi$
$727$	−9.24566	−0.342902	−0.171451	+	0.985193i	$0.554846\pi$
$728$	−3.23033	−0.119724
$729$	0	0
$730$	−0.0497914	−0.00184286
$731$	34.4566	1.27443
$732$	0	0
$733$	8.57554	0.316745	0.158372	−	0.987379i	$-0.449375\pi$
$733$	8.57554	0.316745	0.158372	+	0.987379i	$0.449375\pi$
$734$	−0.172310	−0.00636009
$735$	0	0
$736$	−4.32788	−0.159528
$737$	−4.33576	−0.159710
$738$	0	0
$739$	14.0883	0.518246	0.259123	−	0.965844i	$-0.416566\pi$
$739$	14.0883	0.518246	0.259123	+	0.965844i	$0.416566\pi$
$740$	−1.79035	−0.0658145
$741$	0	0
$742$	−1.00179	−0.0367768
$743$	47.7553	1.75197	0.875986	−	0.482337i	$-0.160212\pi$
$743$	47.7553	1.75197	0.875986	+	0.482337i	$0.160212\pi$
$744$	0	0
$745$	−3.16372	−0.115910
$746$	0.639139	0.0234005
$747$	0	0
$748$	12.6146	0.461234
$749$	−17.7095	−0.647089
$750$	0	0
$751$	24.4902	0.893662	0.446831	−	0.894618i	$-0.352553\pi$
$751$	24.4902	0.893662	0.446831	+	0.894618i	$0.352553\pi$
$752$	3.12922	0.114111
$753$	0	0
$754$	1.47498	0.0537155
$755$	−0.794578	−0.0289176
$756$	0	0
$757$	−18.3094	−0.665465	−0.332733	−	0.943021i	$-0.607971\pi$
$757$	−18.3094	−0.665465	−0.332733	+	0.943021i	$0.607971\pi$
$758$	−1.48948	−0.0541004
$759$	0	0
$760$	−1.30151	−0.0472108
$761$	37.5505	1.36120	0.680602	−	0.732653i	$-0.261718\pi$
$761$	37.5505	1.36120	0.680602	+	0.732653i	$0.261718\pi$
$762$	0	0
$763$	−8.17146	−0.295827
$764$	49.5399	1.79229
$765$	0	0
$766$	1.90338	0.0687718
$767$	−25.3873	−0.916681
$768$	0	0
$769$	−35.3202	−1.27368	−0.636839	−	0.770996i	$-0.719759\pi$
$769$	−35.3202	−1.27368	−0.636839	+	0.770996i	$0.719759\pi$
$770$	−0.135437	−0.00488080
$771$	0	0
$772$	−11.6223	−0.418297
$773$	7.48572	0.269243	0.134621	−	0.990897i	$-0.457018\pi$
$773$	7.48572	0.269243	0.134621	+	0.990897i	$0.457018\pi$
$774$	0	0
$775$	9.74172	0.349933
$776$	5.20988	0.187024
$777$	0	0
$778$	1.34421	0.0481922
$779$	−39.1932	−1.40424
$780$	0	0
$781$	−11.9545	−0.427767
$782$	2.11699	0.0757033
$783$	0	0
$784$	2.57307	0.0918953
$785$	−0.641419	−0.0228932
$786$	0	0
$787$	−4.60088	−0.164004	−0.0820018	−	0.996632i	$-0.526131\pi$
$787$	−4.60088	−0.164004	−0.0820018	+	0.996632i	$0.526131\pi$
$788$	−32.3111	−1.15104
$789$	0	0
$790$	0.0184026	0.000654735 0
$791$	−2.45778	−0.0873886
$792$	0	0
$793$	29.2906	1.04014
$794$	1.82686	0.0648327
$795$	0	0
$796$	45.3346	1.60684
$797$	4.86780	0.172426	0.0862132	−	0.996277i	$-0.472523\pi$
$797$	4.86780	0.172426	0.0862132	+	0.996277i	$0.472523\pi$
$798$	0	0
$799$	−4.63408	−0.163942
$800$	5.97134	0.211119
$801$	0	0
$802$	3.66495	0.129414
$803$	1.20451	0.0425062
$804$	0	0
$805$	4.16942	0.146953
$806$	−0.592393	−0.0208662
$807$	0	0
$808$	−1.95676	−0.0688385
$809$	29.9137	1.05171	0.525855	−	0.850574i	$-0.323745\pi$
$809$	29.9137	1.05171	0.525855	+	0.850574i	$0.323745\pi$
$810$	0	0
$811$	32.8629	1.15397	0.576986	−	0.816754i	$-0.304229\pi$
$811$	32.8629	1.15397	0.576986	+	0.816754i	$0.304229\pi$
$812$	−27.7313	−0.973178
$813$	0	0
$814$	−0.236102	−0.00827538
$815$	4.30150	0.150675
$816$	0	0
$817$	42.8837	1.50031
$818$	−0.657355	−0.0229839
$819$	0	0
$820$	−4.64370	−0.162165
$821$	3.75765	0.131143	0.0655715	−	0.997848i	$-0.479113\pi$
$821$	3.75765	0.131143	0.0655715	+	0.997848i	$0.479113\pi$
$822$	0	0
$823$	−28.8183	−1.00454	−0.502271	−	0.864710i	$-0.667502\pi$
$823$	−28.8183	−1.00454	−0.502271	+	0.864710i	$0.667502\pi$
$824$	−7.94923	−0.276925
$825$	0	0
$826$	−2.60200	−0.0905352
$827$	55.3088	1.92328	0.961638	−	0.274322i	$-0.0884533\pi$
$827$	55.3088	1.92328	0.961638	+	0.274322i	$0.0884533\pi$
$828$	0	0
$829$	35.2389	1.22390	0.611949	−	0.790897i	$-0.290386\pi$
$829$	35.2389	1.22390	0.611949	+	0.790897i	$0.290386\pi$
$830$	0.679242	0.0235768
$831$	0	0
$832$	21.7585	0.754342
$833$	−3.81047	−0.132025
$834$	0	0
$835$	0.653674	0.0226213
$836$	15.6997	0.542986
$837$	0	0
$838$	2.56719	0.0886821
$839$	−18.9566	−0.654454	−0.327227	−	0.944946i	$-0.606114\pi$
$839$	−18.9566	−0.654454	−0.327227	+	0.944946i	$0.606114\pi$
$840$	0	0
$841$	−3.60658	−0.124365
$842$	2.64740	0.0912355
$843$	0	0
$844$	−36.9809	−1.27294
$845$	−2.20282	−0.0757794
$846$	0	0
$847$	−27.1559	−0.933086
$848$	13.6836	0.469895
$849$	0	0
$850$	−2.92089	−0.100186
$851$	7.26841	0.249158
$852$	0	0
$853$	−5.80817	−0.198868	−0.0994339	−	0.995044i	$-0.531703\pi$
$853$	−5.80817	−0.198868	−0.0994339	+	0.995044i	$0.531703\pi$
$854$	3.00206	0.102728
$855$	0	0
$856$	−2.65909	−0.0908858
$857$	−20.8413	−0.711924	−0.355962	−	0.934500i	$-0.615847\pi$
$857$	−20.8413	−0.711924	−0.355962	+	0.934500i	$0.615847\pi$
$858$	0	0
$859$	20.6840	0.705728	0.352864	−	0.935675i	$-0.385208\pi$
$859$	20.6840	0.705728	0.352864	+	0.935675i	$0.385208\pi$
$860$	5.08096	0.173259
$861$	0	0
$862$	−3.66000	−0.124660
$863$	32.6930	1.11288	0.556441	−	0.830887i	$-0.312166\pi$
$863$	32.6930	1.11288	0.556441	+	0.830887i	$0.312166\pi$
$864$	0	0
$865$	−9.14369	−0.310895
$866$	2.96908	0.100894
$867$	0	0
$868$	11.1377	0.378038
$869$	−0.445179	−0.0151017
$870$	0	0
$871$	11.1989	0.379461
$872$	−1.22695	−0.0415498
$873$	0	0
$874$	2.63474	0.0891214
$875$	−11.7285	−0.396495
$876$	0	0
$877$	31.3932	1.06007	0.530036	−	0.847975i	$-0.322178\pi$
$877$	31.3932	1.06007	0.530036	+	0.847975i	$0.322178\pi$
$878$	−2.04360	−0.0689682
$879$	0	0
$880$	1.84994	0.0623616
$881$	−54.7572	−1.84482	−0.922409	−	0.386214i	$-0.873783\pi$
$881$	−54.7572	−1.84482	−0.922409	+	0.386214i	$0.873783\pi$
$882$	0	0
$883$	−44.9140	−1.51148	−0.755738	−	0.654874i	$-0.772722\pi$
$883$	−44.9140	−1.51148	−0.755738	+	0.654874i	$0.772722\pi$
$884$	−32.5824	−1.09587
$885$	0	0
$886$	−1.41047	−0.0473857
$887$	29.8624	1.00268	0.501340	−	0.865250i	$-0.332841\pi$
$887$	29.8624	1.00268	0.501340	+	0.865250i	$0.332841\pi$
$888$	0	0
$889$	−6.82690	−0.228967
$890$	0.639634	0.0214406
$891$	0	0
$892$	25.2672	0.846010
$893$	−5.76744	−0.193000
$894$	0	0
$895$	10.1128	0.338034
$896$	9.09433	0.303820
$897$	0	0
$898$	3.26098	0.108820
$899$	−10.1987	−0.340147
$900$	0	0
$901$	−20.2640	−0.675093
$902$	−0.612387	−0.0203903
$903$	0	0
$904$	−0.369038	−0.0122740
$905$	6.28456	0.208906
$906$	0	0
$907$	−5.00636	−0.166234	−0.0831168	−	0.996540i	$-0.526487\pi$
$907$	−5.00636	−0.166234	−0.0831168	+	0.996540i	$0.526487\pi$
$908$	−23.7685	−0.788786
$909$	0	0
$910$	0.349822	0.0115965
$911$	−32.0748	−1.06269	−0.531343	−	0.847157i	$-0.678312\pi$
$911$	−32.0748	−1.06269	−0.531343	+	0.847157i	$0.678312\pi$
$912$	0	0
$913$	−16.4316	−0.543808
$914$	1.22256	0.0404388
$915$	0	0
$916$	−5.66371	−0.187134
$917$	18.5126	0.611341
$918$	0	0
$919$	−24.0569	−0.793565	−0.396783	−	0.917913i	$-0.629873\pi$
$919$	−24.0569	−0.793565	−0.396783	+	0.917913i	$0.629873\pi$
$920$	0.626042	0.0206400
$921$	0	0
$922$	−0.00394842	−0.000130034 0
$923$	30.8777	1.01635
$924$	0	0
$925$	−10.0285	−0.329736
$926$	−0.394158	−0.0129528
$927$	0	0
$928$	−6.25148	−0.205215
$929$	39.0700	1.28184	0.640922	−	0.767606i	$-0.278552\pi$
$929$	39.0700	1.28184	0.640922	+	0.767606i	$0.278552\pi$
$930$	0	0
$931$	−4.74240	−0.155426
$932$	−8.93627	−0.292717
$933$	0	0
$934$	3.39262	0.111010
$935$	−2.73959	−0.0895941
$936$	0	0
$937$	−4.37686	−0.142986	−0.0714929	−	0.997441i	$-0.522776\pi$
$937$	−4.37686	−0.142986	−0.0714929	+	0.997441i	$0.522776\pi$
$938$	1.14780	0.0374771
$939$	0	0
$940$	−0.683339	−0.0222881
$941$	−42.0947	−1.37225	−0.686123	−	0.727485i	$-0.740689\pi$
$941$	−42.0947	−1.37225	−0.686123	+	0.727485i	$0.740689\pi$
$942$	0	0
$943$	18.8524	0.613918
$944$	35.5410	1.15676
$945$	0	0
$946$	0.670052	0.0217853
$947$	−28.4050	−0.923037	−0.461519	−	0.887131i	$-0.652695\pi$
$947$	−28.4050	−0.923037	−0.461519	+	0.887131i	$0.652695\pi$
$948$	0	0
$949$	−3.11115	−0.100992
$950$	−3.63526	−0.117943
$951$	0	0
$952$	−6.69711	−0.217055
$953$	17.0187	0.551289	0.275645	−	0.961260i	$-0.411109\pi$
$953$	17.0187	0.551289	0.275645	+	0.961260i	$0.411109\pi$
$954$	0	0
$955$	−10.7589	−0.348150
$956$	−1.98916	−0.0643339
$957$	0	0
$958$	−2.01886	−0.0652266
$959$	19.1361	0.617935
$960$	0	0
$961$	−26.9039	−0.867867
$962$	0.609834	0.0196618
$963$	0	0
$964$	−27.1785	−0.875361
$965$	2.52410	0.0812536
$966$	0	0
$967$	9.67491	0.311124	0.155562	−	0.987826i	$-0.450281\pi$
$967$	9.67491	0.311124	0.155562	+	0.987826i	$0.450281\pi$
$968$	−4.07747	−0.131055
$969$	0	0
$970$	−0.564194	−0.0181152
$971$	34.6807	1.11296	0.556478	−	0.830862i	$-0.312152\pi$
$971$	34.6807	1.11296	0.556478	+	0.830862i	$0.312152\pi$
$972$	0	0
$973$	−26.2431	−0.841315
$974$	−0.504960	−0.0161800
$975$	0	0
$976$	−41.0055	−1.31255
$977$	24.2830	0.776882	0.388441	−	0.921474i	$-0.373014\pi$
$977$	24.2830	0.776882	0.388441	+	0.921474i	$0.373014\pi$
$978$	0	0
$979$	−15.4735	−0.494534
$980$	−0.561890	−0.0179489
$981$	0	0
$982$	−2.57548	−0.0821869
$983$	−28.0091	−0.893351	−0.446675	−	0.894696i	$-0.647392\pi$
$983$	−28.0091	−0.893351	−0.446675	+	0.894696i	$0.647392\pi$
$984$	0	0
$985$	7.01722	0.223587
$986$	3.05792	0.0973840
$987$	0	0
$988$	−40.5512	−1.29010
$989$	−20.6276	−0.655918
$990$	0	0
$991$	−0.266294	−0.00845912	−0.00422956	−	0.999991i	$-0.501346\pi$
$991$	−0.266294	−0.00845912	−0.00422956	+	0.999991i	$0.501346\pi$
$992$	2.51078	0.0797172
$993$	0	0
$994$	3.16473	0.100379
$995$	−9.84562	−0.312127
$996$	0	0
$997$	−45.0714	−1.42743	−0.713713	−	0.700439i	$-0.752988\pi$
$997$	−45.0714	−1.42743	−0.713713	+	0.700439i	$0.752988\pi$
$998$	1.35765	0.0429757
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.e.1.3		6
3.2	odd	2		717.2.a.d.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.d.1.4	✓	6	3.2	odd	2
2151.2.a.e.1.3		6	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)

\(f(q)\)	\(=\)	\(q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} +O(q^{10})\)	\(q-0.104133 q^{2} -1.98916 q^{4} +0.431998 q^{5} +2.76657 q^{7} +0.415402 q^{8} -0.0449852 q^{10} +1.08824 q^{11} -2.81084 q^{13} -0.288090 q^{14} +3.93506 q^{16} -5.82744 q^{17} -7.25266 q^{19} -0.859312 q^{20} -0.113322 q^{22} +3.48861 q^{23} -4.81338 q^{25} +0.292701 q^{26} -5.50313 q^{28} +5.03919 q^{29} -2.02388 q^{31} -1.24057 q^{32} +0.606828 q^{34} +1.19515 q^{35} +2.08347 q^{37} +0.755240 q^{38} +0.179453 q^{40} +5.40397 q^{41} -5.91283 q^{43} -2.16468 q^{44} -0.363279 q^{46} +0.795216 q^{47} +0.653884 q^{49} +0.501230 q^{50} +5.59121 q^{52} +3.47735 q^{53} +0.470119 q^{55} +1.14924 q^{56} -0.524745 q^{58} +9.03190 q^{59} -10.4206 q^{61} +0.210753 q^{62} -7.74093 q^{64} -1.21428 q^{65} -3.98418 q^{67} +11.5917 q^{68} -0.124454 q^{70} -10.9852 q^{71} +1.10684 q^{73} -0.216957 q^{74} +14.4267 q^{76} +3.01069 q^{77} -0.409081 q^{79} +1.69994 q^{80} -0.562731 q^{82} -15.0992 q^{83} -2.51744 q^{85} +0.615719 q^{86} +0.452058 q^{88} -14.2188 q^{89} -7.77639 q^{91} -6.93940 q^{92} -0.0828081 q^{94} -3.13314 q^{95} +12.5418 q^{97} -0.0680907 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} - 5 q^{5} - 9 q^{7} + 3 q^{8} - 11 q^{10} + 13 q^{11} - q^{13} - 4 q^{16} - 11 q^{17} - 22 q^{19} + q^{20} - 2 q^{22} + 12 q^{23} - q^{25} - 12 q^{26} - 16 q^{28} - 18 q^{31} - 7 q^{32} - 3 q^{34} + 9 q^{35} - 8 q^{37} + 5 q^{38} - 11 q^{40} - 10 q^{41} - 14 q^{43} + 4 q^{44} - 18 q^{46} + 9 q^{47} + 5 q^{49} - 4 q^{50} - 16 q^{52} + 8 q^{53} - 20 q^{55} - 11 q^{56} - 15 q^{58} + 10 q^{59} - 12 q^{61} + 13 q^{62} - 31 q^{64} + 11 q^{65} - 36 q^{67} - 22 q^{68} + q^{70} + 3 q^{71} - 32 q^{73} - 9 q^{74} - 4 q^{76} - 6 q^{77} - q^{79} + 7 q^{80} + 7 q^{82} + 7 q^{83} - 14 q^{85} - 45 q^{86} - 15 q^{88} - 17 q^{89} - 23 q^{91} + 12 q^{92} + 50 q^{94} - 28 q^{97} - 13 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 - 5 * q^5 - 9 * q^7 + 3 * q^8 - 11 * q^10 + 13 * q^11 - q^13 - 4 * q^16 - 11 * q^17 - 22 * q^19 + q^20 - 2 * q^22 + 12 * q^23 - q^25 - 12 * q^26 - 16 * q^28 - 18 * q^31 - 7 * q^32 - 3 * q^34 + 9 * q^35 - 8 * q^37 + 5 * q^38 - 11 * q^40 - 10 * q^41 - 14 * q^43 + 4 * q^44 - 18 * q^46 + 9 * q^47 + 5 * q^49 - 4 * q^50 - 16 * q^52 + 8 * q^53 - 20 * q^55 - 11 * q^56 - 15 * q^58 + 10 * q^59 - 12 * q^61 + 13 * q^62 - 31 * q^64 + 11 * q^65 - 36 * q^67 - 22 * q^68 + q^70 + 3 * q^71 - 32 * q^73 - 9 * q^74 - 4 * q^76 - 6 * q^77 - q^79 + 7 * q^80 + 7 * q^82 + 7 * q^83 - 14 * q^85 - 45 * q^86 - 15 * q^88 - 17 * q^89 - 23 * q^91 + 12 * q^92 + 50 * q^94 - 28 * q^97 - 13 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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