LMFDB - Embedded newform 2151.2.a.f.1.5

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 10x^{5} + 8x^{4} + 22x^{3} - 5x^{2} - 7x + 2 $ x^7 - x^6 - 10x^5 + 8x^4 + 22x^3 - 5x^2 - 7*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.727328$ 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)

$f(q)$	$=$	$q+1.51425 q^{2} +0.292958 q^{4} +0.707042 q^{5} +3.59927 q^{7} -2.58489 q^{8} +O(q^{10})$	\(q+1.51425 q^{2} +0.292958 q^{4} +0.707042 q^{5} +3.59927 q^{7} -2.58489 q^{8} +1.07064 q^{10} -6.34294 q^{11} -3.96891 q^{13} +5.45020 q^{14} -4.50009 q^{16} -1.63640 q^{17} -6.14411 q^{19} +0.207133 q^{20} -9.60480 q^{22} +0.675950 q^{23} -4.50009 q^{25} -6.00992 q^{26} +1.05443 q^{28} -7.05440 q^{29} -0.576807 q^{31} -1.64449 q^{32} -2.47792 q^{34} +2.54484 q^{35} +11.1897 q^{37} -9.30372 q^{38} -1.82763 q^{40} -6.00710 q^{41} +10.7021 q^{43} -1.85821 q^{44} +1.02356 q^{46} -4.40157 q^{47} +5.95475 q^{49} -6.81427 q^{50} -1.16272 q^{52} +7.82865 q^{53} -4.48472 q^{55} -9.30372 q^{56} -10.6821 q^{58} +7.08144 q^{59} +9.76093 q^{61} -0.873430 q^{62} +6.51002 q^{64} -2.80619 q^{65} -11.1164 q^{67} -0.479397 q^{68} +3.85352 q^{70} +1.21529 q^{71} -6.32252 q^{73} +16.9440 q^{74} -1.79996 q^{76} -22.8299 q^{77} -3.86062 q^{79} -3.18175 q^{80} -9.09626 q^{82} +1.60480 q^{83} -1.15701 q^{85} +16.2056 q^{86} +16.3958 q^{88} -15.7183 q^{89} -14.2852 q^{91} +0.198025 q^{92} -6.66509 q^{94} -4.34414 q^{95} +1.34319 q^{97} +9.01698 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) $ 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7	$ 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7} - 11 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} - 11 q^{17} - 8 q^{19} - 34 q^{20} - 19 q^{22} - 26 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} - 6 q^{29} - 2 q^{31} + 5 q^{32} - 40 q^{34} + 5 q^{35} + 12 q^{37} + 9 q^{38} - 5 q^{40} - 26 q^{41} + 10 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} + 2 q^{49} + 5 q^{50} - 22 q^{52} - 2 q^{53} - 8 q^{55} + 9 q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} - 18 q^{64} + 17 q^{65} + 24 q^{67} + 9 q^{68} - 3 q^{70} + 25 q^{71} - 16 q^{73} + 9 q^{74} - 32 q^{76} - 24 q^{77} + 19 q^{79} - 18 q^{80} + 39 q^{82} - 37 q^{83} - 20 q^{85} + q^{86} - 21 q^{88} - 29 q^{89} - 19 q^{91} - 52 q^{92} - 22 q^{94} + 24 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) $ 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7 - 11 * q^11 - 5 * q^13 - 6 * q^14 - 4 * q^16 - 11 * q^17 - 8 * q^19 - 34 * q^20 - 19 * q^22 - 26 * q^23 - 4 * q^25 - 6 * q^26 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 5 * q^32 - 40 * q^34 + 5 * q^35 + 12 * q^37 + 9 * q^38 - 5 * q^40 - 26 * q^41 + 10 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 + 2 * q^49 + 5 * q^50 - 22 * q^52 - 2 * q^53 - 8 * q^55 + 9 * q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 - 18 * q^64 + 17 * q^65 + 24 * q^67 + 9 * q^68 - 3 * q^70 + 25 * q^71 - 16 * q^73 + 9 * q^74 - 32 * q^76 - 24 * q^77 + 19 * q^79 - 18 * q^80 + 39 * q^82 - 37 * q^83 - 20 * q^85 + q^86 - 21 * q^88 - 29 * q^89 - 19 * q^91 - 52 * q^92 - 22 * q^94 + 24 * q^95 - 12 * q^97 - 23 * 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 $

$2$	1.51425	1.07074	0.535369	−	0.844618i	$-0.320173\pi$
$2$	1.51425	1.07074	0.535369	+	0.844618i	$0.320173\pi$
$3$	0	0
$3$	0	0
$4$	0.292958	0.146479
$5$	0.707042	0.316199	0.158099	−	0.987423i	$-0.449463\pi$
$5$	0.707042	0.316199	0.158099	+	0.987423i	$0.449463\pi$
$6$	0	0
$7$	3.59927	1.36040	0.680198	−	0.733028i	$-0.261894\pi$
$7$	3.59927	1.36040	0.680198	+	0.733028i	$0.261894\pi$
$8$	−2.58489	−0.913897
$9$	0	0
$10$	1.07064	0.338566
$11$	−6.34294	−1.91247	−0.956234	−	0.292604i	$-0.905478\pi$
$11$	−6.34294	−1.91247	−0.956234	+	0.292604i	$0.905478\pi$
$12$	0	0
$13$	−3.96891	−1.10078	−0.550388	−	0.834909i	$-0.685520\pi$
$13$	−3.96891	−1.10078	−0.550388	+	0.834909i	$0.685520\pi$
$14$	5.45020	1.45663
$15$	0	0
$16$	−4.50009	−1.12502
$17$	−1.63640	−0.396886	−0.198443	−	0.980112i	$-0.563588\pi$
$17$	−1.63640	−0.396886	−0.198443	+	0.980112i	$0.563588\pi$
$18$	0	0
$19$	−6.14411	−1.40955	−0.704777	−	0.709429i	$-0.748953\pi$
$19$	−6.14411	−1.40955	−0.704777	+	0.709429i	$0.748953\pi$
$20$	0.207133	0.0463165
$21$	0	0
$22$	−9.60480	−2.04775
$23$	0.675950	0.140945	0.0704726	−	0.997514i	$-0.477549\pi$
$23$	0.675950	0.140945	0.0704726	+	0.997514i	$0.477549\pi$
$24$	0	0
$25$	−4.50009	−0.900018
$26$	−6.00992	−1.17864
$27$	0	0
$28$	1.05443	0.199269
$29$	−7.05440	−1.30997	−0.654984	−	0.755642i	$-0.727325\pi$
$29$	−7.05440	−1.30997	−0.654984	+	0.755642i	$0.727325\pi$
$30$	0	0
$31$	−0.576807	−0.103598	−0.0517988	−	0.998658i	$-0.516495\pi$
$31$	−0.576807	−0.103598	−0.0517988	+	0.998658i	$0.516495\pi$
$32$	−1.64449	−0.290707
$33$	0	0
$34$	−2.47792	−0.424961
$35$	2.54484	0.430156
$36$	0	0
$37$	11.1897	1.83957	0.919787	−	0.392418i	$-0.128361\pi$
$37$	11.1897	1.83957	0.919787	+	0.392418i	$0.128361\pi$
$38$	−9.30372	−1.50926
$39$	0	0
$40$	−1.82763	−0.288973
$41$	−6.00710	−0.938151	−0.469075	−	0.883158i	$-0.655413\pi$
$41$	−6.00710	−0.938151	−0.469075	+	0.883158i	$0.655413\pi$
$42$	0	0
$43$	10.7021	1.63205	0.816026	−	0.578015i	$-0.196173\pi$
$43$	10.7021	1.63205	0.816026	+	0.578015i	$0.196173\pi$
$44$	−1.85821	−0.280136
$45$	0	0
$46$	1.02356	0.150915
$47$	−4.40157	−0.642035	−0.321018	−	0.947073i	$-0.604025\pi$
$47$	−4.40157	−0.642035	−0.321018	+	0.947073i	$0.604025\pi$
$48$	0	0
$49$	5.95475	0.850678
$50$	−6.81427	−0.963683
$51$	0	0
$52$	−1.16272	−0.161241
$53$	7.82865	1.07535	0.537674	−	0.843153i	$-0.319303\pi$
$53$	7.82865	1.07535	0.537674	+	0.843153i	$0.319303\pi$
$54$	0	0
$55$	−4.48472	−0.604720
$56$	−9.30372	−1.24326
$57$	0	0
$58$	−10.6821	−1.40263
$59$	7.08144	0.921925	0.460963	−	0.887420i	$-0.347504\pi$
$59$	7.08144	0.921925	0.460963	+	0.887420i	$0.347504\pi$
$60$	0	0
$61$	9.76093	1.24976	0.624880	−	0.780721i	$-0.285148\pi$
$61$	9.76093	1.24976	0.624880	+	0.780721i	$0.285148\pi$
$62$	−0.873430	−0.110926
$63$	0	0
$64$	6.51002	0.813752
$65$	−2.80619	−0.348064
$66$	0	0
$67$	−11.1164	−1.35808	−0.679041	−	0.734100i	$-0.737604\pi$
$67$	−11.1164	−1.35808	−0.679041	+	0.734100i	$0.737604\pi$
$68$	−0.479397	−0.0581354
$69$	0	0
$70$	3.85352	0.460584
$71$	1.21529	0.144228	0.0721142	−	0.997396i	$-0.477025\pi$
$71$	1.21529	0.144228	0.0721142	+	0.997396i	$0.477025\pi$
$72$	0	0
$73$	−6.32252	−0.739995	−0.369997	−	0.929033i	$-0.620641\pi$
$73$	−6.32252	−0.739995	−0.369997	+	0.929033i	$0.620641\pi$
$74$	16.9440	1.96970
$75$	0	0
$76$	−1.79996	−0.206470
$77$	−22.8299	−2.60171
$78$	0	0
$79$	−3.86062	−0.434354	−0.217177	−	0.976132i	$-0.569685\pi$
$79$	−3.86062	−0.434354	−0.217177	+	0.976132i	$0.569685\pi$
$80$	−3.18175	−0.355731
$81$	0	0
$82$	−9.09626	−1.00451
$83$	1.60480	0.176150	0.0880749	−	0.996114i	$-0.471929\pi$
$83$	1.60480	0.176150	0.0880749	+	0.996114i	$0.471929\pi$
$84$	0	0
$85$	−1.15701	−0.125495
$86$	16.2056	1.74750
$87$	0	0
$88$	16.3958	1.74780
$89$	−15.7183	−1.66614	−0.833069	−	0.553168i	$-0.813419\pi$
$89$	−15.7183	−1.66614	−0.833069	+	0.553168i	$0.813419\pi$
$90$	0	0
$91$	−14.2852	−1.49749
$92$	0.198025	0.0206455
$93$	0	0
$94$	−6.66509	−0.687451
$95$	−4.34414	−0.445700
$96$	0	0
$97$	1.34319	0.136380	0.0681901	−	0.997672i	$-0.478278\pi$
$97$	1.34319	0.136380	0.0681901	+	0.997672i	$0.478278\pi$
$98$	9.01698	0.910853
$99$	0	0
$100$	−1.31834	−0.131834
$101$	5.33163	0.530517	0.265258	−	0.964177i	$-0.414543\pi$
$101$	5.33163	0.530517	0.265258	+	0.964177i	$0.414543\pi$
$102$	0	0
$103$	3.09285	0.304747	0.152374	−	0.988323i	$-0.451308\pi$
$103$	3.09285	0.304747	0.152374	+	0.988323i	$0.451308\pi$
$104$	10.2592	1.00600
$105$	0	0
$106$	11.8545	1.15142
$107$	−16.1251	−1.55887	−0.779434	−	0.626484i	$-0.784493\pi$
$107$	−16.1251	−1.55887	−0.779434	+	0.626484i	$0.784493\pi$
$108$	0	0
$109$	−5.29845	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$109$	−5.29845	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$110$	−6.79100	−0.647497
$111$	0	0
$112$	−16.1970	−1.53048
$113$	−2.40870	−0.226592	−0.113296	−	0.993561i	$-0.536141\pi$
$113$	−2.40870	−0.226592	−0.113296	+	0.993561i	$0.536141\pi$
$114$	0	0
$115$	0.477925	0.0445667
$116$	−2.06664	−0.191883
$117$	0	0
$118$	10.7231	0.987140
$119$	−5.88985	−0.539922
$120$	0	0
$121$	29.2328	2.65753
$122$	14.7805	1.33816
$123$	0	0
$124$	−0.168980	−0.0151748
$125$	−6.71697	−0.600784
$126$	0	0
$127$	−17.3266	−1.53749	−0.768745	−	0.639555i	$-0.779119\pi$
$127$	−17.3266	−1.53749	−0.768745	+	0.639555i	$0.779119\pi$
$128$	13.1468	1.16202
$129$	0	0
$130$	−4.24927	−0.372686
$131$	15.8254	1.38267	0.691336	−	0.722534i	$-0.257023\pi$
$131$	15.8254	1.38267	0.691336	+	0.722534i	$0.257023\pi$
$132$	0	0
$133$	−22.1143	−1.91755
$134$	−16.8330	−1.45415
$135$	0	0
$136$	4.22992	0.362713
$137$	5.88685	0.502947	0.251474	−	0.967864i	$-0.419085\pi$
$137$	5.88685	0.502947	0.251474	+	0.967864i	$0.419085\pi$
$138$	0	0
$139$	15.0129	1.27338	0.636690	−	0.771120i	$-0.280303\pi$
$139$	15.0129	1.27338	0.636690	+	0.771120i	$0.280303\pi$
$140$	0.745529	0.0630087
$141$	0	0
$142$	1.84026	0.154431
$143$	25.1745	2.10520
$144$	0	0
$145$	−4.98776	−0.414211
$146$	−9.57388	−0.792340
$147$	0	0
$148$	3.27811	0.269459
$149$	6.07409	0.497609	0.248804	−	0.968554i	$-0.419962\pi$
$149$	6.07409	0.497609	0.248804	+	0.968554i	$0.419962\pi$
$150$	0	0
$151$	−13.7993	−1.12297	−0.561484	−	0.827487i	$-0.689770\pi$
$151$	−13.7993	−1.12297	−0.561484	+	0.827487i	$0.689770\pi$
$152$	15.8818	1.28819
$153$	0	0
$154$	−34.5703	−2.78575
$155$	−0.407827	−0.0327574
$156$	0	0
$157$	−14.7411	−1.17647	−0.588236	−	0.808689i	$-0.700177\pi$
$157$	−14.7411	−1.17647	−0.588236	+	0.808689i	$0.700177\pi$
$158$	−5.84595	−0.465079
$159$	0	0
$160$	−1.16272	−0.0919212
$161$	2.43293	0.191741
$162$	0	0
$163$	11.7451	0.919947	0.459973	−	0.887933i	$-0.347859\pi$
$163$	11.7451	0.919947	0.459973	+	0.887933i	$0.347859\pi$
$164$	−1.75983	−0.137419
$165$	0	0
$166$	2.43007	0.188610
$167$	−12.6938	−0.982273	−0.491137	−	0.871083i	$-0.663418\pi$
$167$	−12.6938	−0.982273	−0.491137	+	0.871083i	$0.663418\pi$
$168$	0	0
$169$	2.75222	0.211710
$170$	−1.75200	−0.134372
$171$	0	0
$172$	3.13526	0.239061
$173$	7.82185	0.594684	0.297342	−	0.954771i	$-0.403900\pi$
$173$	7.82185	0.594684	0.297342	+	0.954771i	$0.403900\pi$
$174$	0	0
$175$	−16.1970	−1.22438
$176$	28.5438	2.15157
$177$	0	0
$178$	−23.8015	−1.78400
$179$	4.64884	0.347471	0.173735	−	0.984792i	$-0.444416\pi$
$179$	4.64884	0.347471	0.173735	+	0.984792i	$0.444416\pi$
$180$	0	0
$181$	−7.07863	−0.526150	−0.263075	−	0.964775i	$-0.584737\pi$
$181$	−7.07863	−0.526150	−0.263075	+	0.964775i	$0.584737\pi$
$182$	−21.6313	−1.60342
$183$	0	0
$184$	−1.74726	−0.128809
$185$	7.91159	0.581671
$186$	0	0
$187$	10.3796	0.759031
$188$	−1.28947	−0.0940446
$189$	0	0
$190$	−6.57813	−0.477227
$191$	3.20600	0.231978	0.115989	−	0.993250i	$-0.462996\pi$
$191$	3.20600	0.231978	0.115989	+	0.993250i	$0.462996\pi$
$192$	0	0
$193$	−5.15343	−0.370952	−0.185476	−	0.982649i	$-0.559383\pi$
$193$	−5.15343	−0.370952	−0.185476	+	0.982649i	$0.559383\pi$
$194$	2.03393	0.146027
$195$	0	0
$196$	1.74449	0.124606
$197$	−24.7178	−1.76107	−0.880536	−	0.473979i	$-0.842817\pi$
$197$	−24.7178	−1.76107	−0.880536	+	0.473979i	$0.842817\pi$
$198$	0	0
$199$	10.4954	0.743999	0.372000	−	0.928233i	$-0.378672\pi$
$199$	10.4954	0.743999	0.372000	+	0.928233i	$0.378672\pi$
$200$	11.6322	0.822524
$201$	0	0
$202$	8.07343	0.568044
$203$	−25.3907	−1.78208
$204$	0	0
$205$	−4.24727	−0.296642
$206$	4.68335	0.326305
$207$	0	0
$208$	17.8604	1.23840
$209$	38.9717	2.69573
$210$	0	0
$211$	20.8921	1.43827	0.719136	−	0.694870i	$-0.244538\pi$
$211$	20.8921	1.43827	0.719136	+	0.694870i	$0.244538\pi$
$212$	2.29346	0.157516
$213$	0	0
$214$	−24.4174	−1.66914
$215$	7.56682	0.516053
$216$	0	0
$217$	−2.07608	−0.140934
$218$	−8.02319	−0.543399
$219$	0	0
$220$	−1.31383	−0.0885787
$221$	6.49473	0.436883
$222$	0	0
$223$	−27.5280	−1.84341	−0.921707	−	0.387887i	$-0.873205\pi$
$223$	−27.5280	−1.84341	−0.921707	+	0.387887i	$0.873205\pi$
$224$	−5.91895	−0.395477
$225$	0	0
$226$	−3.64738	−0.242620
$227$	−13.9417	−0.925341	−0.462670	−	0.886530i	$-0.653109\pi$
$227$	−13.9417	−0.925341	−0.462670	+	0.886530i	$0.653109\pi$
$228$	0	0
$229$	−22.0869	−1.45954	−0.729771	−	0.683692i	$-0.760373\pi$
$229$	−22.0869	−1.45954	−0.729771	+	0.683692i	$0.760373\pi$
$230$	0.723698	0.0477193
$231$	0	0
$232$	18.2349	1.19718
$233$	11.3208	0.741649	0.370824	−	0.928703i	$-0.379075\pi$
$233$	11.3208	0.741649	0.370824	+	0.928703i	$0.379075\pi$
$234$	0	0
$235$	−3.11210	−0.203011
$236$	2.07456	0.135043
$237$	0	0
$238$	−8.91872	−0.578115
$239$	1.00000	0.0646846
$239$	1.00000	0.0646846
$240$	0	0
$241$	−15.7736	−1.01607	−0.508035	−	0.861337i	$-0.669628\pi$
$241$	−15.7736	−1.01607	−0.508035	+	0.861337i	$0.669628\pi$
$242$	44.2659	2.84552
$243$	0	0
$244$	2.85954	0.183063
$245$	4.21026	0.268984
$246$	0	0
$247$	24.3854	1.55161
$248$	1.49098	0.0946775
$249$	0	0
$250$	−10.1712	−0.643282
$251$	9.87166	0.623093	0.311547	−	0.950231i	$-0.399153\pi$
$251$	9.87166	0.623093	0.311547	+	0.950231i	$0.399153\pi$
$252$	0	0
$253$	−4.28750	−0.269553
$254$	−26.2369	−1.64625
$255$	0	0
$256$	6.88749	0.430468
$257$	−14.3276	−0.893729	−0.446865	−	0.894602i	$-0.647459\pi$
$257$	−14.3276	−0.893729	−0.446865	+	0.894602i	$0.647459\pi$
$258$	0	0
$259$	40.2747	2.50255
$260$	−0.822094	−0.0509841
$261$	0	0
$262$	23.9636	1.48048
$263$	3.40077	0.209700	0.104850	−	0.994488i	$-0.466564\pi$
$263$	3.40077	0.209700	0.104850	+	0.994488i	$0.466564\pi$
$264$	0	0
$265$	5.53519	0.340024
$266$	−33.4866	−2.05320
$267$	0	0
$268$	−3.25663	−0.198930
$269$	14.9004	0.908494	0.454247	−	0.890876i	$-0.349908\pi$
$269$	14.9004	0.908494	0.454247	+	0.890876i	$0.349908\pi$
$270$	0	0
$271$	−26.0188	−1.58053	−0.790266	−	0.612764i	$-0.790058\pi$
$271$	−26.0188	−1.58053	−0.790266	+	0.612764i	$0.790058\pi$
$272$	7.36396	0.446506
$273$	0	0
$274$	8.91417	0.538525
$275$	28.5438	1.72126
$276$	0	0
$277$	7.66546	0.460573	0.230286	−	0.973123i	$-0.426034\pi$
$277$	7.66546	0.460573	0.230286	+	0.973123i	$0.426034\pi$
$278$	22.7334	1.36346
$279$	0	0
$280$	−6.57813	−0.393118
$281$	−9.43834	−0.563044	−0.281522	−	0.959555i	$-0.590839\pi$
$281$	−9.43834	−0.563044	−0.281522	+	0.959555i	$0.590839\pi$
$282$	0	0
$283$	21.9058	1.30217	0.651083	−	0.759007i	$-0.274315\pi$
$283$	21.9058	1.30217	0.651083	+	0.759007i	$0.274315\pi$
$284$	0.356029	0.0211264
$285$	0	0
$286$	38.1206	2.25412
$287$	−21.6212	−1.27626
$288$	0	0
$289$	−14.3222	−0.842482
$290$	−7.55272	−0.443511
$291$	0	0
$292$	−1.85223	−0.108394
$293$	26.0757	1.52336	0.761678	−	0.647955i	$-0.224376\pi$
$293$	26.0757	1.52336	0.761678	+	0.647955i	$0.224376\pi$
$294$	0	0
$295$	5.00688	0.291512
$296$	−28.9241	−1.68118
$297$	0	0
$298$	9.19770	0.532808
$299$	−2.68278	−0.155149
$300$	0	0
$301$	38.5197	2.22024
$302$	−20.8956	−1.20240
$303$	0	0
$304$	27.6490	1.58578
$305$	6.90139	0.395173
$306$	0	0
$307$	−3.57846	−0.204234	−0.102117	−	0.994772i	$-0.532562\pi$
$307$	−3.57846	−0.204234	−0.102117	+	0.994772i	$0.532562\pi$
$308$	−6.68821	−0.381096
$309$	0	0
$310$	−0.617552	−0.0350746
$311$	13.2693	0.752430	0.376215	−	0.926532i	$-0.377225\pi$
$311$	13.2693	0.752430	0.376215	+	0.926532i	$0.377225\pi$
$312$	0	0
$313$	5.52409	0.312240	0.156120	−	0.987738i	$-0.450101\pi$
$313$	5.52409	0.312240	0.156120	+	0.987738i	$0.450101\pi$
$314$	−22.3218	−1.25969
$315$	0	0
$316$	−1.13100	−0.0636236
$317$	−13.0599	−0.733519	−0.366760	−	0.930316i	$-0.619533\pi$
$317$	−13.0599	−0.733519	−0.366760	+	0.930316i	$0.619533\pi$
$318$	0	0
$319$	44.7456	2.50527
$320$	4.60286	0.257307
$321$	0	0
$322$	3.68406	0.205305
$323$	10.0542	0.559432
$324$	0	0
$325$	17.8604	0.990719
$326$	17.7850	0.985021
$327$	0	0
$328$	15.5277	0.857373
$329$	−15.8424	−0.873422
$330$	0	0
$331$	−7.63170	−0.419476	−0.209738	−	0.977758i	$-0.567261\pi$
$331$	−7.63170	−0.419476	−0.209738	+	0.977758i	$0.567261\pi$
$332$	0.470139	0.0258022
$333$	0	0
$334$	−19.2216	−1.05176
$335$	−7.85975	−0.429424
$336$	0	0
$337$	−10.6046	−0.577669	−0.288834	−	0.957379i	$-0.593268\pi$
$337$	−10.6046	−0.577669	−0.288834	+	0.957379i	$0.593268\pi$
$338$	4.16756	0.226685
$339$	0	0
$340$	−0.338954	−0.0183823
$341$	3.65865	0.198127
$342$	0	0
$343$	−3.76215	−0.203137
$344$	−27.6637	−1.49153
$345$	0	0
$346$	11.8442	0.636751
$347$	−8.88663	−0.477059	−0.238530	−	0.971135i	$-0.576665\pi$
$347$	−8.88663	−0.477059	−0.238530	+	0.971135i	$0.576665\pi$
$348$	0	0
$349$	−21.3387	−1.14223	−0.571116	−	0.820869i	$-0.693489\pi$
$349$	−21.3387	−1.14223	−0.571116	+	0.820869i	$0.693489\pi$
$350$	−24.5264	−1.31099
$351$	0	0
$352$	10.4309	0.555968
$353$	29.1835	1.55328	0.776641	−	0.629943i	$-0.216922\pi$
$353$	29.1835	1.55328	0.776641	+	0.629943i	$0.216922\pi$
$354$	0	0
$355$	0.859262	0.0456049
$356$	−4.60480	−0.244054
$357$	0	0
$358$	7.03952	0.372050
$359$	31.3770	1.65602	0.828008	−	0.560716i	$-0.189474\pi$
$359$	31.3770	1.65602	0.828008	+	0.560716i	$0.189474\pi$
$360$	0	0
$361$	18.7500	0.986845
$362$	−10.7188	−0.563369
$363$	0	0
$364$	−4.18495	−0.219351
$365$	−4.47029	−0.233986
$366$	0	0
$367$	15.5314	0.810730	0.405365	−	0.914155i	$-0.367144\pi$
$367$	15.5314	0.810730	0.405365	+	0.914155i	$0.367144\pi$
$368$	−3.04183	−0.158567
$369$	0	0
$370$	11.9801	0.622817
$371$	28.1774	1.46290
$372$	0	0
$373$	18.2199	0.943390	0.471695	−	0.881762i	$-0.343642\pi$
$373$	18.2199	0.943390	0.471695	+	0.881762i	$0.343642\pi$
$374$	15.7173	0.812723
$375$	0	0
$376$	11.3776	0.586754
$377$	27.9983	1.44198
$378$	0	0
$379$	−32.7354	−1.68150	−0.840752	−	0.541421i	$-0.817886\pi$
$379$	−32.7354	−1.68150	−0.840752	+	0.541421i	$0.817886\pi$
$380$	−1.27265	−0.0652856
$381$	0	0
$382$	4.85469	0.248388
$383$	11.4585	0.585500	0.292750	−	0.956189i	$-0.405430\pi$
$383$	11.4585	0.585500	0.292750	+	0.956189i	$0.405430\pi$
$384$	0	0
$385$	−16.1417	−0.822659
$386$	−7.80360	−0.397193
$387$	0	0
$388$	0.393498	0.0199768
$389$	−5.82107	−0.295140	−0.147570	−	0.989052i	$-0.547145\pi$
$389$	−5.82107	−0.295140	−0.147570	+	0.989052i	$0.547145\pi$
$390$	0	0
$391$	−1.10613	−0.0559392
$392$	−15.3924	−0.777432
$393$	0	0
$394$	−37.4290	−1.88565
$395$	−2.72962	−0.137342
$396$	0	0
$397$	−0.484118	−0.0242972	−0.0121486	−	0.999926i	$-0.503867\pi$
$397$	−0.484118	−0.0242972	−0.0121486	+	0.999926i	$0.503867\pi$
$398$	15.8927	0.796628
$399$	0	0
$400$	20.2508	1.01254
$401$	−14.5335	−0.725766	−0.362883	−	0.931835i	$-0.618208\pi$
$401$	−14.5335	−0.725766	−0.362883	+	0.931835i	$0.618208\pi$
$402$	0	0
$403$	2.28929	0.114038
$404$	1.56194	0.0777095
$405$	0	0
$406$	−38.4479	−1.90814
$407$	−70.9755	−3.51813
$408$	0	0
$409$	−14.5672	−0.720299	−0.360150	−	0.932895i	$-0.617274\pi$
$409$	−14.5672	−0.720299	−0.360150	+	0.932895i	$0.617274\pi$
$410$	−6.43144	−0.317626
$411$	0	0
$412$	0.906074	0.0446391
$413$	25.4880	1.25418
$414$	0	0
$415$	1.13466	0.0556984
$416$	6.52682	0.320004
$417$	0	0
$418$	59.0129	2.88642
$419$	24.1059	1.17765	0.588826	−	0.808260i	$-0.299590\pi$
$419$	24.1059	1.17765	0.588826	+	0.808260i	$0.299590\pi$
$420$	0	0
$421$	−9.09088	−0.443062	−0.221531	−	0.975153i	$-0.571105\pi$
$421$	−9.09088	−0.443062	−0.221531	+	0.975153i	$0.571105\pi$
$422$	31.6359	1.54001
$423$	0	0
$424$	−20.2362	−0.982757
$425$	7.36396	0.357205
$426$	0	0
$427$	35.1322	1.70017
$428$	−4.72396	−0.228341
$429$	0	0
$430$	11.4581	0.552557
$431$	3.03475	0.146179	0.0730894	−	0.997325i	$-0.476714\pi$
$431$	3.03475	0.146179	0.0730894	+	0.997325i	$0.476714\pi$
$432$	0	0
$433$	−27.2436	−1.30924	−0.654622	−	0.755956i	$-0.727172\pi$
$433$	−27.2436	−1.30924	−0.654622	+	0.755956i	$0.727172\pi$
$434$	−3.14371	−0.150903
$435$	0	0
$436$	−1.55222	−0.0743380
$437$	−4.15311	−0.198670
$438$	0	0
$439$	−19.1125	−0.912188	−0.456094	−	0.889931i	$-0.650752\pi$
$439$	−19.1125	−0.912188	−0.456094	+	0.889931i	$0.650752\pi$
$440$	11.5925	0.552652
$441$	0	0
$442$	9.83465	0.467787
$443$	−39.7932	−1.89063	−0.945316	−	0.326155i	$-0.894247\pi$
$443$	−39.7932	−1.89063	−0.945316	+	0.326155i	$0.894247\pi$
$444$	0	0
$445$	−11.1135	−0.526831
$446$	−41.6844	−1.97381
$447$	0	0
$448$	23.4313	1.10703
$449$	−18.0311	−0.850941	−0.425471	−	0.904972i	$-0.639891\pi$
$449$	−18.0311	−0.850941	−0.425471	+	0.904972i	$0.639891\pi$
$450$	0	0
$451$	38.1026	1.79418
$452$	−0.705648	−0.0331909
$453$	0	0
$454$	−21.1112	−0.990797
$455$	−10.1002	−0.473506
$456$	0	0
$457$	27.1377	1.26945	0.634724	−	0.772739i	$-0.281114\pi$
$457$	27.1377	1.26945	0.634724	+	0.772739i	$0.281114\pi$
$458$	−33.4451	−1.56279
$459$	0	0
$460$	0.140012	0.00652808
$461$	16.3446	0.761246	0.380623	−	0.924730i	$-0.375710\pi$
$461$	16.3446	0.761246	0.380623	+	0.924730i	$0.375710\pi$
$462$	0	0
$463$	4.91423	0.228384	0.114192	−	0.993459i	$-0.463572\pi$
$463$	4.91423	0.228384	0.114192	+	0.993459i	$0.463572\pi$
$464$	31.7454	1.47374
$465$	0	0
$466$	17.1425	0.794111
$467$	4.66234	0.215747	0.107874	−	0.994165i	$-0.465596\pi$
$467$	4.66234	0.215747	0.107874	+	0.994165i	$0.465596\pi$
$468$	0	0
$469$	−40.0109	−1.84753
$470$	−4.71250	−0.217371
$471$	0	0
$472$	−18.3048	−0.842545
$473$	−67.8826	−3.12125
$474$	0	0
$475$	27.6490	1.26862
$476$	−1.72548	−0.0790872
$477$	0	0
$478$	1.51425	0.0692603
$479$	−8.20170	−0.374745	−0.187373	−	0.982289i	$-0.559997\pi$
$479$	−8.20170	−0.374745	−0.187373	+	0.982289i	$0.559997\pi$
$480$	0	0
$481$	−44.4109	−2.02496
$482$	−23.8852	−1.08794
$483$	0	0
$484$	8.56399	0.389272
$485$	0.949692	0.0431233
$486$	0	0
$487$	−32.5809	−1.47638	−0.738191	−	0.674592i	$-0.764320\pi$
$487$	−32.5809	−1.47638	−0.738191	+	0.674592i	$0.764320\pi$
$488$	−25.2309	−1.14215
$489$	0	0
$490$	6.37539	0.288011
$491$	−6.50037	−0.293358	−0.146679	−	0.989184i	$-0.546858\pi$
$491$	−6.50037	−0.293358	−0.146679	+	0.989184i	$0.546858\pi$
$492$	0	0
$493$	11.5438	0.519908
$494$	36.9256	1.66136
$495$	0	0
$496$	2.59568	0.116550
$497$	4.37416	0.196208
$498$	0	0
$499$	20.9923	0.939744	0.469872	−	0.882735i	$-0.344300\pi$
$499$	20.9923	0.939744	0.469872	+	0.882735i	$0.344300\pi$
$500$	−1.96779	−0.0880021
$501$	0	0
$502$	14.9482	0.667169
$503$	18.5831	0.828578	0.414289	−	0.910145i	$-0.364030\pi$
$503$	18.5831	0.828578	0.414289	+	0.910145i	$0.364030\pi$
$504$	0	0
$505$	3.76969	0.167749
$506$	−6.49236	−0.288621
$507$	0	0
$508$	−5.07597	−0.225210
$509$	17.5745	0.778974	0.389487	−	0.921032i	$-0.372652\pi$
$509$	17.5745	0.778974	0.389487	+	0.921032i	$0.372652\pi$
$510$	0	0
$511$	−22.7565	−1.00669
$512$	−15.8642	−0.701103
$513$	0	0
$514$	−21.6955	−0.956949
$515$	2.18677	0.0963608
$516$	0	0
$517$	27.9189	1.22787
$518$	60.9861	2.67957
$519$	0	0
$520$	7.25368	0.318095
$521$	−15.0306	−0.658501	−0.329251	−	0.944243i	$-0.606796\pi$
$521$	−15.0306	−0.658501	−0.329251	+	0.944243i	$0.606796\pi$
$522$	0	0
$523$	29.9259	1.30857	0.654284	−	0.756249i	$-0.272970\pi$
$523$	29.9259	1.30857	0.654284	+	0.756249i	$0.272970\pi$
$524$	4.63617	0.202532
$525$	0	0
$526$	5.14962	0.224534
$527$	0.943888	0.0411164
$528$	0	0
$529$	−22.5431	−0.980134
$530$	8.38166	0.364076
$531$	0	0
$532$	−6.47856	−0.280881
$533$	23.8416	1.03269
$534$	0	0
$535$	−11.4011	−0.492912
$536$	28.7346	1.24115
$537$	0	0
$538$	22.5630	0.972759
$539$	−37.7706	−1.62689
$540$	0	0
$541$	13.2119	0.568024	0.284012	−	0.958821i	$-0.408334\pi$
$541$	13.2119	0.568024	0.284012	+	0.958821i	$0.408334\pi$
$542$	−39.3991	−1.69233
$543$	0	0
$544$	2.69104	0.115378
$545$	−3.74623	−0.160471
$546$	0	0
$547$	−16.0583	−0.686605	−0.343303	−	0.939225i	$-0.611546\pi$
$547$	−16.0583	−0.686605	−0.343303	+	0.939225i	$0.611546\pi$
$548$	1.72460	0.0736712
$549$	0	0
$550$	43.2225	1.84301
$551$	43.3430	1.84647
$552$	0	0
$553$	−13.8954	−0.590893
$554$	11.6074	0.493153
$555$	0	0
$556$	4.39815	0.186523
$557$	22.7492	0.963916	0.481958	−	0.876194i	$-0.339926\pi$
$557$	22.7492	0.963916	0.481958	+	0.876194i	$0.339926\pi$
$558$	0	0
$559$	−42.4756	−1.79652
$560$	−11.4520	−0.483935
$561$	0	0
$562$	−14.2920	−0.602873
$563$	32.1900	1.35665	0.678324	−	0.734763i	$-0.262707\pi$
$563$	32.1900	1.35665	0.678324	+	0.734763i	$0.262707\pi$
$564$	0	0
$565$	−1.70306	−0.0716481
$566$	33.1709	1.39428
$567$	0	0
$568$	−3.14139	−0.131810
$569$	39.8644	1.67120	0.835601	−	0.549336i	$-0.185119\pi$
$569$	39.8644	1.67120	0.835601	+	0.549336i	$0.185119\pi$
$570$	0	0
$571$	−34.2158	−1.43189	−0.715944	−	0.698158i	$-0.754003\pi$
$571$	−34.2158	−1.43189	−0.715944	+	0.698158i	$0.754003\pi$
$572$	7.37507	0.308367
$573$	0	0
$574$	−32.7399	−1.36654
$575$	−3.04183	−0.126853
$576$	0	0
$577$	−21.9776	−0.914940	−0.457470	−	0.889225i	$-0.651244\pi$
$577$	−21.9776	−0.914940	−0.457470	+	0.889225i	$0.651244\pi$
$578$	−21.6874	−0.902077
$579$	0	0
$580$	−1.46120	−0.0606731
$581$	5.77611	0.239634
$582$	0	0
$583$	−49.6566	−2.05657
$584$	16.3430	0.676279
$585$	0	0
$586$	39.4851	1.63112
$587$	−22.6768	−0.935974	−0.467987	−	0.883735i	$-0.655021\pi$
$587$	−22.6768	−0.935974	−0.467987	+	0.883735i	$0.655021\pi$
$588$	0	0
$589$	3.54396	0.146026
$590$	7.58167	0.312133
$591$	0	0
$592$	−50.3546	−2.06956
$593$	−40.9797	−1.68284	−0.841418	−	0.540384i	$-0.818279\pi$
$593$	−40.9797	−1.68284	−0.841418	+	0.540384i	$0.818279\pi$
$594$	0	0
$595$	−4.16438	−0.170723
$596$	1.77945	0.0728892
$597$	0	0
$598$	−4.06241	−0.166124
$599$	4.90776	0.200526	0.100263	−	0.994961i	$-0.468032\pi$
$599$	4.90776	0.200526	0.100263	+	0.994961i	$0.468032\pi$
$600$	0	0
$601$	−3.78182	−0.154264	−0.0771319	−	0.997021i	$-0.524576\pi$
$601$	−3.78182	−0.154264	−0.0771319	+	0.997021i	$0.524576\pi$
$602$	58.3285	2.37729
$603$	0	0
$604$	−4.04260	−0.164491
$605$	20.6689	0.840309
$606$	0	0
$607$	−21.4560	−0.870873	−0.435437	−	0.900219i	$-0.643406\pi$
$607$	−21.4560	−0.870873	−0.435437	+	0.900219i	$0.643406\pi$
$608$	10.1039	0.409767
$609$	0	0
$610$	10.4504	0.423126
$611$	17.4694	0.706737
$612$	0	0
$613$	2.77004	0.111881	0.0559404	−	0.998434i	$-0.482184\pi$
$613$	2.77004	0.111881	0.0559404	+	0.998434i	$0.482184\pi$
$614$	−5.41869	−0.218681
$615$	0	0
$616$	59.0129	2.37770
$617$	−27.0698	−1.08979	−0.544895	−	0.838504i	$-0.683431\pi$
$617$	−27.0698	−1.08979	−0.544895	+	0.838504i	$0.683431\pi$
$618$	0	0
$619$	39.3426	1.58131	0.790656	−	0.612261i	$-0.209740\pi$
$619$	39.3426	1.58131	0.790656	+	0.612261i	$0.209740\pi$
$620$	−0.119476	−0.00479827
$621$	0	0
$622$	20.0930	0.805655
$623$	−56.5745	−2.26661
$624$	0	0
$625$	17.7513	0.710051
$626$	8.36487	0.334327
$627$	0	0
$628$	−4.31853	−0.172328
$629$	−18.3108	−0.730101
$630$	0	0
$631$	40.5861	1.61571	0.807854	−	0.589382i	$-0.200629\pi$
$631$	40.5861	1.61571	0.807854	+	0.589382i	$0.200629\pi$
$632$	9.97928	0.396955
$633$	0	0
$634$	−19.7760	−0.785407
$635$	−12.2507	−0.486153
$636$	0	0
$637$	−23.6338	−0.936407
$638$	67.7561	2.68249
$639$	0	0
$640$	9.29533	0.367430
$641$	44.3678	1.75242	0.876212	−	0.481926i	$-0.160063\pi$
$641$	44.3678	1.75242	0.876212	+	0.481926i	$0.160063\pi$
$642$	0	0
$643$	13.2402	0.522144	0.261072	−	0.965319i	$-0.415924\pi$
$643$	13.2402	0.522144	0.261072	+	0.965319i	$0.415924\pi$
$644$	0.712744	0.0280861
$645$	0	0
$646$	15.2246	0.599005
$647$	−40.3033	−1.58449	−0.792243	−	0.610206i	$-0.791087\pi$
$647$	−40.3033	−1.58449	−0.792243	+	0.610206i	$0.791087\pi$
$648$	0	0
$649$	−44.9171	−1.76315
$650$	27.0452	1.06080
$651$	0	0
$652$	3.44082	0.134753
$653$	−28.4567	−1.11360	−0.556799	−	0.830647i	$-0.687971\pi$
$653$	−28.4567	−1.11360	−0.556799	+	0.830647i	$0.687971\pi$
$654$	0	0
$655$	11.1892	0.437199
$656$	27.0325	1.05544
$657$	0	0
$658$	−23.9894	−0.935206
$659$	20.8606	0.812615	0.406308	−	0.913736i	$-0.366816\pi$
$659$	20.8606	0.812615	0.406308	+	0.913736i	$0.366816\pi$
$660$	0	0
$661$	2.83985	0.110457	0.0552286	−	0.998474i	$-0.482411\pi$
$661$	2.83985	0.110457	0.0552286	+	0.998474i	$0.482411\pi$
$662$	−11.5563	−0.449149
$663$	0	0
$664$	−4.14824	−0.160983
$665$	−15.6357	−0.606328
$666$	0	0
$667$	−4.76842	−0.184634
$668$	−3.71874	−0.143882
$669$	0	0
$670$	−11.9016	−0.459801
$671$	−61.9130	−2.39012
$672$	0	0
$673$	20.3077	0.782802	0.391401	−	0.920220i	$-0.371990\pi$
$673$	20.3077	0.782802	0.391401	+	0.920220i	$0.371990\pi$
$674$	−16.0580	−0.618531
$675$	0	0
$676$	0.806285	0.0310110
$677$	−22.4952	−0.864560	−0.432280	−	0.901739i	$-0.642291\pi$
$677$	−22.4952	−0.864560	−0.432280	+	0.901739i	$0.642291\pi$
$678$	0	0
$679$	4.83450	0.185531
$680$	2.99073	0.114689
$681$	0	0
$682$	5.54011	0.212142
$683$	−37.5111	−1.43532	−0.717662	−	0.696391i	$-0.754788\pi$
$683$	−37.5111	−1.43532	−0.717662	+	0.696391i	$0.754788\pi$
$684$	0	0
$685$	4.16225	0.159031
$686$	−5.69684	−0.217506
$687$	0	0
$688$	−48.1603	−1.83610
$689$	−31.0712	−1.18372
$690$	0	0
$691$	−10.7922	−0.410555	−0.205277	−	0.978704i	$-0.565810\pi$
$691$	−10.7922	−0.410555	−0.205277	+	0.978704i	$0.565810\pi$
$692$	2.29147	0.0871087
$693$	0	0
$694$	−13.4566	−0.510805
$695$	10.6148	0.402641
$696$	0	0
$697$	9.83003	0.372339
$698$	−32.3121	−1.22303
$699$	0	0
$700$	−4.74505	−0.179346
$701$	12.9899	0.490622	0.245311	−	0.969444i	$-0.421110\pi$
$701$	12.9899	0.490622	0.245311	+	0.969444i	$0.421110\pi$
$702$	0	0
$703$	−68.7507	−2.59298
$704$	−41.2926	−1.55627
$705$	0	0
$706$	44.1912	1.66316
$707$	19.1900	0.721713
$708$	0	0
$709$	−27.5730	−1.03553	−0.517763	−	0.855524i	$-0.673235\pi$
$709$	−27.5730	−1.03553	−0.517763	+	0.855524i	$0.673235\pi$
$710$	1.30114	0.0488309
$711$	0	0
$712$	40.6302	1.52268
$713$	−0.389892	−0.0146016
$714$	0	0
$715$	17.7995	0.665662
$716$	1.36191	0.0508971
$717$	0	0
$718$	47.5127	1.77316
$719$	−13.5564	−0.505568	−0.252784	−	0.967523i	$-0.581346\pi$
$719$	−13.5564	−0.505568	−0.252784	+	0.967523i	$0.581346\pi$
$720$	0	0
$721$	11.1320	0.414577
$722$	28.3923	1.05665
$723$	0	0
$724$	−2.07374	−0.0770699
$725$	31.7454	1.17900
$726$	0	0
$727$	−12.6431	−0.468907	−0.234453	−	0.972127i	$-0.575330\pi$
$727$	−12.6431	−0.468907	−0.234453	+	0.972127i	$0.575330\pi$
$728$	36.9256	1.36855
$729$	0	0
$730$	−6.76914	−0.250537
$731$	−17.5129	−0.647738
$732$	0	0
$733$	22.9263	0.846801	0.423401	−	0.905943i	$-0.360836\pi$
$733$	22.9263	0.846801	0.423401	+	0.905943i	$0.360836\pi$
$734$	23.5184	0.868080
$735$	0	0
$736$	−1.11159	−0.0409738
$737$	70.5105	2.59729
$738$	0	0
$739$	37.9695	1.39673	0.698366	−	0.715741i	$-0.253911\pi$
$739$	37.9695	1.39673	0.698366	+	0.715741i	$0.253911\pi$
$740$	2.31776	0.0852026
$741$	0	0
$742$	42.6677	1.56638
$743$	−38.6703	−1.41868	−0.709338	−	0.704868i	$-0.751006\pi$
$743$	−38.6703	−1.41868	−0.709338	+	0.704868i	$0.751006\pi$
$744$	0	0
$745$	4.29464	0.157343
$746$	27.5895	1.01012
$747$	0	0
$748$	3.04078	0.111182
$749$	−58.0384	−2.12068
$750$	0	0
$751$	−36.8477	−1.34459	−0.672295	−	0.740283i	$-0.734692\pi$
$751$	−36.8477	−1.34459	−0.672295	+	0.740283i	$0.734692\pi$
$752$	19.8075	0.722304
$753$	0	0
$754$	42.3964	1.54399
$755$	−9.75667	−0.355081
$756$	0	0
$757$	−14.2352	−0.517387	−0.258693	−	0.965959i	$-0.583292\pi$
$757$	−14.2352	−0.517387	−0.258693	+	0.965959i	$0.583292\pi$
$758$	−49.5696	−1.80045
$759$	0	0
$760$	11.2291	0.407324
$761$	7.47087	0.270819	0.135409	−	0.990790i	$-0.456765\pi$
$761$	7.47087	0.270819	0.135409	+	0.990790i	$0.456765\pi$
$762$	0	0
$763$	−19.0706	−0.690401
$764$	0.939223	0.0339799
$765$	0	0
$766$	17.3510	0.626917
$767$	−28.1056	−1.01483
$768$	0	0
$769$	−4.19691	−0.151344	−0.0756721	−	0.997133i	$-0.524110\pi$
$769$	−4.19691	−0.151344	−0.0756721	+	0.997133i	$0.524110\pi$
$770$	−24.4426	−0.880852
$771$	0	0
$772$	−1.50974	−0.0543367
$773$	−41.3393	−1.48687	−0.743436	−	0.668807i	$-0.766805\pi$
$773$	−41.3393	−1.48687	−0.743436	+	0.668807i	$0.766805\pi$
$774$	0	0
$775$	2.59568	0.0932397
$776$	−3.47200	−0.124638
$777$	0	0
$778$	−8.81457	−0.316018
$779$	36.9082	1.32238
$780$	0	0
$781$	−7.70851	−0.275832
$782$	−1.67495	−0.0598962
$783$	0	0
$784$	−26.7969	−0.957032
$785$	−10.4226	−0.371999
$786$	0	0
$787$	5.75353	0.205091	0.102546	−	0.994728i	$-0.467301\pi$
$787$	5.75353	0.205091	0.102546	+	0.994728i	$0.467301\pi$
$788$	−7.24128	−0.257960
$789$	0	0
$790$	−4.13333	−0.147057
$791$	−8.66958	−0.308255
$792$	0	0
$793$	−38.7402	−1.37571
$794$	−0.733076	−0.0260159
$795$	0	0
$796$	3.07471	0.108980
$797$	−19.4089	−0.687498	−0.343749	−	0.939062i	$-0.611697\pi$
$797$	−19.4089	−0.687498	−0.343749	+	0.939062i	$0.611697\pi$
$798$	0	0
$799$	7.20274	0.254815
$800$	7.40034	0.261642
$801$	0	0
$802$	−22.0073	−0.777105
$803$	40.1033	1.41522
$804$	0	0
$805$	1.72018	0.0606284
$806$	3.46656	0.122104
$807$	0	0
$808$	−13.7817	−0.484838
$809$	50.7810	1.78537	0.892683	−	0.450686i	$-0.148821\pi$
$809$	50.7810	1.78537	0.892683	+	0.450686i	$0.148821\pi$
$810$	0	0
$811$	2.99230	0.105074	0.0525369	−	0.998619i	$-0.483269\pi$
$811$	2.99230	0.105074	0.0525369	+	0.998619i	$0.483269\pi$
$812$	−7.43840	−0.261037
$813$	0	0
$814$	−107.475	−3.76699
$815$	8.30428	0.290886
$816$	0	0
$817$	−65.7547	−2.30047
$818$	−22.0583	−0.771252
$819$	0	0
$820$	−1.24427	−0.0434518
$821$	−25.1621	−0.878165	−0.439083	−	0.898447i	$-0.644696\pi$
$821$	−25.1621	−0.878165	−0.439083	+	0.898447i	$0.644696\pi$
$822$	0	0
$823$	−3.59628	−0.125359	−0.0626793	−	0.998034i	$-0.519965\pi$
$823$	−3.59628	−0.125359	−0.0626793	+	0.998034i	$0.519965\pi$
$824$	−7.99468	−0.278508
$825$	0	0
$826$	38.5953	1.34290
$827$	37.8690	1.31683	0.658417	−	0.752654i	$-0.271226\pi$
$827$	37.8690	1.31683	0.658417	+	0.752654i	$0.271226\pi$
$828$	0	0
$829$	−9.60459	−0.333581	−0.166791	−	0.985992i	$-0.553340\pi$
$829$	−9.60459	−0.333581	−0.166791	+	0.985992i	$0.553340\pi$
$830$	1.71816	0.0596384
$831$	0	0
$832$	−25.8376	−0.895759
$833$	−9.74436	−0.337622
$834$	0	0
$835$	−8.97503	−0.310594
$836$	11.4171	0.394867
$837$	0	0
$838$	36.5024	1.26096
$839$	−39.4044	−1.36039	−0.680196	−	0.733031i	$-0.738105\pi$
$839$	−39.4044	−1.36039	−0.680196	+	0.733031i	$0.738105\pi$
$840$	0	0
$841$	20.7645	0.716018
$842$	−13.7659	−0.474403
$843$	0	0
$844$	6.12050	0.210676
$845$	1.94594	0.0669423
$846$	0	0
$847$	105.217	3.61530
$848$	−35.2296	−1.20979
$849$	0	0
$850$	11.1509	0.382472
$851$	7.56367	0.259279
$852$	0	0
$853$	−21.8759	−0.749015	−0.374508	−	0.927224i	$-0.622188\pi$
$853$	−21.8759	−0.749015	−0.374508	+	0.927224i	$0.622188\pi$
$854$	53.1990	1.82043
$855$	0	0
$856$	41.6815	1.42465
$857$	−8.62119	−0.294494	−0.147247	−	0.989100i	$-0.547041\pi$
$857$	−8.62119	−0.294494	−0.147247	+	0.989100i	$0.547041\pi$
$858$	0	0
$859$	−3.18246	−0.108584	−0.0542921	−	0.998525i	$-0.517290\pi$
$859$	−3.18246	−0.108584	−0.0542921	+	0.998525i	$0.517290\pi$
$860$	2.21676	0.0755909
$861$	0	0
$862$	4.59537	0.156519
$863$	27.6806	0.942259	0.471130	−	0.882064i	$-0.343846\pi$
$863$	27.6806	0.942259	0.471130	+	0.882064i	$0.343846\pi$
$864$	0	0
$865$	5.53038	0.188039
$866$	−41.2537	−1.40186
$867$	0	0
$868$	−0.608204	−0.0206438
$869$	24.4877	0.830687
$870$	0	0
$871$	44.1199	1.49495
$872$	13.6959	0.463803
$873$	0	0
$874$	−6.28885	−0.212723
$875$	−24.1762	−0.817304
$876$	0	0
$877$	1.69952	0.0573889	0.0286944	−	0.999588i	$-0.490865\pi$
$877$	1.69952	0.0573889	0.0286944	+	0.999588i	$0.490865\pi$
$878$	−28.9411	−0.976714
$879$	0	0
$880$	20.1817	0.680324
$881$	−32.5846	−1.09780	−0.548902	−	0.835887i	$-0.684954\pi$
$881$	−32.5846	−1.09780	−0.548902	+	0.835887i	$0.684954\pi$
$882$	0	0
$883$	48.0534	1.61713	0.808563	−	0.588410i	$-0.200246\pi$
$883$	48.0534	1.61713	0.808563	+	0.588410i	$0.200246\pi$
$884$	1.90268	0.0639941
$885$	0	0
$886$	−60.2569	−2.02437
$887$	58.1647	1.95298	0.976490	−	0.215565i	$-0.0691593\pi$
$887$	58.1647	1.95298	0.976490	+	0.215565i	$0.0691593\pi$
$888$	0	0
$889$	−62.3633	−2.09160
$890$	−16.8287	−0.564098
$891$	0	0
$892$	−8.06455	−0.270021
$893$	27.0437	0.904984
$894$	0	0
$895$	3.28693	0.109870
$896$	47.3188	1.58081
$897$	0	0
$898$	−27.3036	−0.911134
$899$	4.06902	0.135710
$900$	0	0
$901$	−12.8108	−0.426790
$902$	57.6970	1.92110
$903$	0	0
$904$	6.22624	0.207082
$905$	−5.00489	−0.166368
$906$	0	0
$907$	56.2731	1.86852	0.934259	−	0.356594i	$-0.116062\pi$
$907$	56.2731	1.86852	0.934259	+	0.356594i	$0.116062\pi$
$908$	−4.08432	−0.135543
$909$	0	0
$910$	−15.2943	−0.507000
$911$	−14.8104	−0.490691	−0.245345	−	0.969436i	$-0.578901\pi$
$911$	−14.8104	−0.490691	−0.245345	+	0.969436i	$0.578901\pi$
$912$	0	0
$913$	−10.1792	−0.336881
$914$	41.0933	1.35925
$915$	0	0
$916$	−6.47052	−0.213792
$917$	56.9599	1.88098
$918$	0	0
$919$	28.9089	0.953617	0.476808	−	0.879007i	$-0.341794\pi$
$919$	28.9089	0.953617	0.476808	+	0.879007i	$0.341794\pi$
$920$	−1.23538	−0.0407294
$921$	0	0
$922$	24.7499	0.815095
$923$	−4.82338	−0.158763
$924$	0	0
$925$	−50.3546	−1.65565
$926$	7.44138	0.244539
$927$	0	0
$928$	11.6009	0.380817
$929$	−11.4430	−0.375431	−0.187716	−	0.982223i	$-0.560108\pi$
$929$	−11.4430	−0.375431	−0.187716	+	0.982223i	$0.560108\pi$
$930$	0	0
$931$	−36.5866	−1.19908
$932$	3.31651	0.108636
$933$	0	0
$934$	7.05995	0.231009
$935$	7.33881	0.240005
$936$	0	0
$937$	−17.1362	−0.559815	−0.279908	−	0.960027i	$-0.590304\pi$
$937$	−17.1362	−0.559815	−0.279908	+	0.960027i	$0.590304\pi$
$938$	−60.5865	−1.97822
$939$	0	0
$940$	−0.911713	−0.0297368
$941$	35.5269	1.15814	0.579072	−	0.815277i	$-0.303415\pi$
$941$	35.5269	1.15814	0.579072	+	0.815277i	$0.303415\pi$
$942$	0	0
$943$	−4.06049	−0.132228
$944$	−31.8671	−1.03719
$945$	0	0
$946$	−102.791	−3.34203
$947$	−0.584252	−0.0189857	−0.00949283	−	0.999955i	$-0.503022\pi$
$947$	−0.584252	−0.0189857	−0.00949283	+	0.999955i	$0.503022\pi$
$948$	0	0
$949$	25.0935	0.814569
$950$	41.8676	1.35836
$951$	0	0
$952$	15.2246	0.493433
$953$	5.69999	0.184641	0.0923204	−	0.995729i	$-0.470572\pi$
$953$	5.69999	0.184641	0.0923204	+	0.995729i	$0.470572\pi$
$954$	0	0
$955$	2.26678	0.0733512
$956$	0.292958	0.00947493
$957$	0	0
$958$	−12.4194	−0.401254
$959$	21.1884	0.684208
$960$	0	0
$961$	−30.6673	−0.989268
$962$	−67.2492	−2.16820
$963$	0	0
$964$	−4.62101	−0.148833
$965$	−3.64370	−0.117295
$966$	0	0
$967$	41.0538	1.32020	0.660100	−	0.751178i	$-0.270514\pi$
$967$	41.0538	1.32020	0.660100	+	0.751178i	$0.270514\pi$
$968$	−75.5637	−2.42871
$969$	0	0
$970$	1.43807	0.0461737
$971$	33.7420	1.08283	0.541417	−	0.840754i	$-0.317888\pi$
$971$	33.7420	1.08283	0.541417	+	0.840754i	$0.317888\pi$
$972$	0	0
$973$	54.0356	1.73230
$974$	−49.3357	−1.58082
$975$	0	0
$976$	−43.9251	−1.40601
$977$	−3.78798	−0.121188	−0.0605941	−	0.998162i	$-0.519300\pi$
$977$	−3.78798	−0.121188	−0.0605941	+	0.998162i	$0.519300\pi$
$978$	0	0
$979$	99.7003	3.18644
$980$	1.23343	0.0394004
$981$	0	0
$982$	−9.84320	−0.314109
$983$	−46.1483	−1.47190	−0.735951	−	0.677035i	$-0.763265\pi$
$983$	−46.1483	−1.47190	−0.735951	+	0.677035i	$0.763265\pi$
$984$	0	0
$985$	−17.4765	−0.556849
$986$	17.4803	0.556685
$987$	0	0
$988$	7.14389	0.227277
$989$	7.23407	0.230030
$990$	0	0
$991$	5.02951	0.159768	0.0798839	−	0.996804i	$-0.474545\pi$
$991$	5.02951	0.159768	0.0798839	+	0.996804i	$0.474545\pi$
$992$	0.948551	0.0301165
$993$	0	0
$994$	6.62358	0.210087
$995$	7.42069	0.235252
$996$	0	0
$997$	−61.3875	−1.94416	−0.972080	−	0.234649i	$-0.924606\pi$
$997$	−61.3875	−1.94416	−0.972080	+	0.234649i	$0.924606\pi$
$998$	31.7876	1.00622
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.2.a.f.1.5		7
3.2	odd	2		717.2.a.e.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.2.a.e.1.3	✓	7	3.2	odd	2
2151.2.a.f.1.5		7	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)

\(f(q)\)	\(=\)	\(q+1.51425 q^{2} +0.292958 q^{4} +0.707042 q^{5} +3.59927 q^{7} -2.58489 q^{8} +O(q^{10})\)	\(q+1.51425 q^{2} +0.292958 q^{4} +0.707042 q^{5} +3.59927 q^{7} -2.58489 q^{8} +1.07064 q^{10} -6.34294 q^{11} -3.96891 q^{13} +5.45020 q^{14} -4.50009 q^{16} -1.63640 q^{17} -6.14411 q^{19} +0.207133 q^{20} -9.60480 q^{22} +0.675950 q^{23} -4.50009 q^{25} -6.00992 q^{26} +1.05443 q^{28} -7.05440 q^{29} -0.576807 q^{31} -1.64449 q^{32} -2.47792 q^{34} +2.54484 q^{35} +11.1897 q^{37} -9.30372 q^{38} -1.82763 q^{40} -6.00710 q^{41} +10.7021 q^{43} -1.85821 q^{44} +1.02356 q^{46} -4.40157 q^{47} +5.95475 q^{49} -6.81427 q^{50} -1.16272 q^{52} +7.82865 q^{53} -4.48472 q^{55} -9.30372 q^{56} -10.6821 q^{58} +7.08144 q^{59} +9.76093 q^{61} -0.873430 q^{62} +6.51002 q^{64} -2.80619 q^{65} -11.1164 q^{67} -0.479397 q^{68} +3.85352 q^{70} +1.21529 q^{71} -6.32252 q^{73} +16.9440 q^{74} -1.79996 q^{76} -22.8299 q^{77} -3.86062 q^{79} -3.18175 q^{80} -9.09626 q^{82} +1.60480 q^{83} -1.15701 q^{85} +16.2056 q^{86} +16.3958 q^{88} -15.7183 q^{89} -14.2852 q^{91} +0.198025 q^{92} -6.66509 q^{94} -4.34414 q^{95} +1.34319 q^{97} +9.01698 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7}+O(q^{10}) \) 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7	\( 7 q + 10 q^{4} - 3 q^{5} + 3 q^{7} - 11 q^{11} - 5 q^{13} - 6 q^{14} - 4 q^{16} - 11 q^{17} - 8 q^{19} - 34 q^{20} - 19 q^{22} - 26 q^{23} - 4 q^{25} - 6 q^{26} - 2 q^{28} - 6 q^{29} - 2 q^{31} + 5 q^{32} - 40 q^{34} + 5 q^{35} + 12 q^{37} + 9 q^{38} - 5 q^{40} - 26 q^{41} + 10 q^{43} - 3 q^{44} + 6 q^{46} - 7 q^{47} + 2 q^{49} + 5 q^{50} - 22 q^{52} - 2 q^{53} - 8 q^{55} + 9 q^{56} - 6 q^{58} - 6 q^{59} - 8 q^{61} + 2 q^{62} - 18 q^{64} + 17 q^{65} + 24 q^{67} + 9 q^{68} - 3 q^{70} + 25 q^{71} - 16 q^{73} + 9 q^{74} - 32 q^{76} - 24 q^{77} + 19 q^{79} - 18 q^{80} + 39 q^{82} - 37 q^{83} - 20 q^{85} + q^{86} - 21 q^{88} - 29 q^{89} - 19 q^{91} - 52 q^{92} - 22 q^{94} + 24 q^{95} - 12 q^{97} - 23 q^{98}+O(q^{100}) \) 7 * q + 10 * q^4 - 3 * q^5 + 3 * q^7 - 11 * q^11 - 5 * q^13 - 6 * q^14 - 4 * q^16 - 11 * q^17 - 8 * q^19 - 34 * q^20 - 19 * q^22 - 26 * q^23 - 4 * q^25 - 6 * q^26 - 2 * q^28 - 6 * q^29 - 2 * q^31 + 5 * q^32 - 40 * q^34 + 5 * q^35 + 12 * q^37 + 9 * q^38 - 5 * q^40 - 26 * q^41 + 10 * q^43 - 3 * q^44 + 6 * q^46 - 7 * q^47 + 2 * q^49 + 5 * q^50 - 22 * q^52 - 2 * q^53 - 8 * q^55 + 9 * q^56 - 6 * q^58 - 6 * q^59 - 8 * q^61 + 2 * q^62 - 18 * q^64 + 17 * q^65 + 24 * q^67 + 9 * q^68 - 3 * q^70 + 25 * q^71 - 16 * q^73 + 9 * q^74 - 32 * q^76 - 24 * q^77 + 19 * q^79 - 18 * q^80 + 39 * q^82 - 37 * q^83 - 20 * q^85 + q^86 - 21 * q^88 - 29 * q^89 - 19 * q^91 - 52 * q^92 - 22 * q^94 + 24 * q^95 - 12 * q^97 - 23 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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