LMFDB - Embedded newform 1143.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
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} - 2x^{6} - 8x^{5} + 15x^{4} + 17x^{3} - 28x^{2} - 11x + 15 $ x^7 - 2x^6 - 8x^5 + 15x^4 + 17x^3 - 28x^2 - 11x + 15
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.09124$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09124 q^{2} -0.809198 q^{4} -0.395790 q^{5} +3.35479 q^{7} -3.06551 q^{8} +O(q^{10})$	\(q+1.09124 q^{2} -0.809198 q^{4} -0.395790 q^{5} +3.35479 q^{7} -3.06551 q^{8} -0.431901 q^{10} -5.32343 q^{11} -5.62420 q^{13} +3.66087 q^{14} -1.72680 q^{16} -2.80386 q^{17} +3.32608 q^{19} +0.320272 q^{20} -5.80914 q^{22} +3.48209 q^{23} -4.84335 q^{25} -6.13735 q^{26} -2.71469 q^{28} +1.90498 q^{29} -9.80553 q^{31} +4.24666 q^{32} -3.05968 q^{34} -1.32779 q^{35} +5.47244 q^{37} +3.62954 q^{38} +1.21330 q^{40} -2.35163 q^{41} -10.9476 q^{43} +4.30771 q^{44} +3.79979 q^{46} -11.5300 q^{47} +4.25459 q^{49} -5.28525 q^{50} +4.55109 q^{52} +3.28647 q^{53} +2.10696 q^{55} -10.2841 q^{56} +2.07878 q^{58} +5.59373 q^{59} +0.311791 q^{61} -10.7002 q^{62} +8.08772 q^{64} +2.22600 q^{65} +9.19725 q^{67} +2.26887 q^{68} -1.44894 q^{70} -2.97382 q^{71} +8.53643 q^{73} +5.97174 q^{74} -2.69145 q^{76} -17.8590 q^{77} -9.14495 q^{79} +0.683451 q^{80} -2.56618 q^{82} -11.0246 q^{83} +1.10974 q^{85} -11.9465 q^{86} +16.3190 q^{88} -0.870463 q^{89} -18.8680 q^{91} -2.81770 q^{92} -12.5819 q^{94} -1.31643 q^{95} -8.65673 q^{97} +4.64278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	$ 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * 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 $

$2$	1.09124	0.771622	0.385811	−	0.922578i	$-0.373922\pi$
$2$	1.09124	0.771622	0.385811	+	0.922578i	$0.373922\pi$
$3$	0	0
$3$	0	0
$4$	−0.809198	−0.404599
$5$	−0.395790	−0.177003	−0.0885013	−	0.996076i	$-0.528208\pi$
$5$	−0.395790	−0.177003	−0.0885013	+	0.996076i	$0.528208\pi$
$6$	0	0
$7$	3.35479	1.26799	0.633995	−	0.773337i	$-0.281414\pi$
$7$	3.35479	1.26799	0.633995	+	0.773337i	$0.281414\pi$
$8$	−3.06551	−1.08382
$9$	0	0
$10$	−0.431901	−0.136579
$11$	−5.32343	−1.60508	−0.802538	−	0.596601i	$-0.796517\pi$
$11$	−5.32343	−1.60508	−0.802538	+	0.596601i	$0.796517\pi$
$12$	0	0
$13$	−5.62420	−1.55987	−0.779936	−	0.625859i	$-0.784749\pi$
$13$	−5.62420	−1.55987	−0.779936	+	0.625859i	$0.784749\pi$
$14$	3.66087	0.978409
$15$	0	0
$16$	−1.72680	−0.431701
$17$	−2.80386	−0.680035	−0.340017	−	0.940419i	$-0.610433\pi$
$17$	−2.80386	−0.680035	−0.340017	+	0.940419i	$0.610433\pi$
$18$	0	0
$19$	3.32608	0.763054	0.381527	−	0.924358i	$-0.375398\pi$
$19$	3.32608	0.763054	0.381527	+	0.924358i	$0.375398\pi$
$20$	0.320272	0.0716151
$21$	0	0
$22$	−5.80914	−1.23851
$23$	3.48209	0.726066	0.363033	−	0.931776i	$-0.381741\pi$
$23$	3.48209	0.726066	0.363033	+	0.931776i	$0.381741\pi$
$24$	0	0
$25$	−4.84335	−0.968670
$26$	−6.13735	−1.20363
$27$	0	0
$28$	−2.71469	−0.513028
$29$	1.90498	0.353745	0.176873	−	0.984234i	$-0.443402\pi$
$29$	1.90498	0.353745	0.176873	+	0.984234i	$0.443402\pi$
$30$	0	0
$31$	−9.80553	−1.76112	−0.880562	−	0.473931i	$-0.842835\pi$
$31$	−9.80553	−1.76112	−0.880562	+	0.473931i	$0.842835\pi$
$32$	4.24666	0.750710
$33$	0	0
$34$	−3.05968	−0.524730
$35$	−1.32779	−0.224437
$36$	0	0
$37$	5.47244	0.899664	0.449832	−	0.893113i	$-0.351484\pi$
$37$	5.47244	0.899664	0.449832	+	0.893113i	$0.351484\pi$
$38$	3.62954	0.588790
$39$	0	0
$40$	1.21330	0.191839
$41$	−2.35163	−0.367262	−0.183631	−	0.982995i	$-0.558785\pi$
$41$	−2.35163	−0.367262	−0.183631	+	0.982995i	$0.558785\pi$
$42$	0	0
$43$	−10.9476	−1.66950	−0.834749	−	0.550630i	$-0.814387\pi$
$43$	−10.9476	−1.66950	−0.834749	+	0.550630i	$0.814387\pi$
$44$	4.30771	0.649412
$45$	0	0
$46$	3.79979	0.560249
$47$	−11.5300	−1.68182	−0.840909	−	0.541177i	$-0.817979\pi$
$47$	−11.5300	−1.68182	−0.840909	+	0.541177i	$0.817979\pi$
$48$	0	0
$49$	4.25459	0.607799
$50$	−5.28525	−0.747447
$51$	0	0
$52$	4.55109	0.631123
$53$	3.28647	0.451431	0.225715	−	0.974193i	$-0.427528\pi$
$53$	3.28647	0.451431	0.225715	+	0.974193i	$0.427528\pi$
$54$	0	0
$55$	2.10696	0.284102
$56$	−10.2841	−1.37427
$57$	0	0
$58$	2.07878	0.272958
$59$	5.59373	0.728242	0.364121	−	0.931352i	$-0.381370\pi$
$59$	5.59373	0.728242	0.364121	+	0.931352i	$0.381370\pi$
$60$	0	0
$61$	0.311791	0.0399208	0.0199604	−	0.999801i	$-0.493646\pi$
$61$	0.311791	0.0399208	0.0199604	+	0.999801i	$0.493646\pi$
$62$	−10.7002	−1.35892
$63$	0	0
$64$	8.08772	1.01097
$65$	2.22600	0.276101
$66$	0	0
$67$	9.19725	1.12362	0.561812	−	0.827265i	$-0.310105\pi$
$67$	9.19725	1.12362	0.561812	+	0.827265i	$0.310105\pi$
$68$	2.26887	0.275141
$69$	0	0
$70$	−1.44894	−0.173181
$71$	−2.97382	−0.352927	−0.176464	−	0.984307i	$-0.556466\pi$
$71$	−2.97382	−0.352927	−0.176464	+	0.984307i	$0.556466\pi$
$72$	0	0
$73$	8.53643	0.999114	0.499557	−	0.866281i	$-0.333496\pi$
$73$	8.53643	0.999114	0.499557	+	0.866281i	$0.333496\pi$
$74$	5.97174	0.694201
$75$	0	0
$76$	−2.69145	−0.308731
$77$	−17.8590	−2.03522
$78$	0	0
$79$	−9.14495	−1.02889	−0.514443	−	0.857524i	$-0.672001\pi$
$79$	−9.14495	−1.02889	−0.514443	+	0.857524i	$0.672001\pi$
$80$	0.683451	0.0764121
$81$	0	0
$82$	−2.56618	−0.283388
$83$	−11.0246	−1.21011	−0.605056	−	0.796183i	$-0.706849\pi$
$83$	−11.0246	−1.21011	−0.605056	+	0.796183i	$0.706849\pi$
$84$	0	0
$85$	1.10974	0.120368
$86$	−11.9465	−1.28822
$87$	0	0
$88$	16.3190	1.73961
$89$	−0.870463	−0.0922689	−0.0461344	−	0.998935i	$-0.514690\pi$
$89$	−0.870463	−0.0922689	−0.0461344	+	0.998935i	$0.514690\pi$
$90$	0	0
$91$	−18.8680	−1.97790
$92$	−2.81770	−0.293766
$93$	0	0
$94$	−12.5819	−1.29773
$95$	−1.31643	−0.135063
$96$	0	0
$97$	−8.65673	−0.878958	−0.439479	−	0.898253i	$-0.644837\pi$
$97$	−8.65673	−0.878958	−0.439479	+	0.898253i	$0.644837\pi$
$98$	4.64278	0.468991
$99$	0	0
$100$	3.91923	0.391923
$101$	13.5210	1.34539	0.672696	−	0.739919i	$-0.265136\pi$
$101$	13.5210	1.34539	0.672696	+	0.739919i	$0.265136\pi$
$102$	0	0
$103$	−1.21829	−0.120042	−0.0600210	−	0.998197i	$-0.519117\pi$
$103$	−1.21829	−0.120042	−0.0600210	+	0.998197i	$0.519117\pi$
$104$	17.2410	1.69062
$105$	0	0
$106$	3.58632	0.348334
$107$	6.66014	0.643860	0.321930	−	0.946763i	$-0.395668\pi$
$107$	6.66014	0.643860	0.321930	+	0.946763i	$0.395668\pi$
$108$	0	0
$109$	−6.54109	−0.626522	−0.313261	−	0.949667i	$-0.601422\pi$
$109$	−6.54109	−0.626522	−0.313261	+	0.949667i	$0.601422\pi$
$110$	2.29920	0.219220
$111$	0	0
$112$	−5.79305	−0.547392
$113$	−10.9896	−1.03381	−0.516906	−	0.856042i	$-0.672917\pi$
$113$	−10.9896	−1.03381	−0.516906	+	0.856042i	$0.672917\pi$
$114$	0	0
$115$	−1.37818	−0.128516
$116$	−1.54150	−0.143125
$117$	0	0
$118$	6.10410	0.561928
$119$	−9.40634	−0.862277
$120$	0	0
$121$	17.3389	1.57627
$122$	0.340239	0.0308038
$123$	0	0
$124$	7.93461	0.712549
$125$	3.89590	0.348460
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0.332319	0.0293732
$129$	0	0
$130$	2.42910	0.213046
$131$	4.74610	0.414668	0.207334	−	0.978270i	$-0.433521\pi$
$131$	4.74610	0.414668	0.207334	+	0.978270i	$0.433521\pi$
$132$	0	0
$133$	11.1583	0.967545
$134$	10.0364	0.867013
$135$	0	0
$136$	8.59523	0.737035
$137$	−0.872416	−0.0745356	−0.0372678	−	0.999305i	$-0.511865\pi$
$137$	−0.872416	−0.0745356	−0.0372678	+	0.999305i	$0.511865\pi$
$138$	0	0
$139$	7.26653	0.616339	0.308170	−	0.951331i	$-0.400284\pi$
$139$	7.26653	0.616339	0.308170	+	0.951331i	$0.400284\pi$
$140$	1.07445	0.0908072
$141$	0	0
$142$	−3.24514	−0.272326
$143$	29.9401	2.50371
$144$	0	0
$145$	−0.753970	−0.0626138
$146$	9.31529	0.770939
$147$	0	0
$148$	−4.42829	−0.364003
$149$	−0.974896	−0.0798666	−0.0399333	−	0.999202i	$-0.512715\pi$
$149$	−0.974896	−0.0798666	−0.0399333	+	0.999202i	$0.512715\pi$
$150$	0	0
$151$	−5.51012	−0.448408	−0.224204	−	0.974542i	$-0.571978\pi$
$151$	−5.51012	−0.448408	−0.224204	+	0.974542i	$0.571978\pi$
$152$	−10.1961	−0.827013
$153$	0	0
$154$	−19.4884	−1.57042
$155$	3.88093	0.311723
$156$	0	0
$157$	0.593953	0.0474026	0.0237013	−	0.999719i	$-0.492455\pi$
$157$	0.593953	0.0474026	0.0237013	+	0.999719i	$0.492455\pi$
$158$	−9.97932	−0.793912
$159$	0	0
$160$	−1.68078	−0.132878
$161$	11.6817	0.920645
$162$	0	0
$163$	19.6685	1.54055	0.770277	−	0.637710i	$-0.220118\pi$
$163$	19.6685	1.54055	0.770277	+	0.637710i	$0.220118\pi$
$164$	1.90293	0.148594
$165$	0	0
$166$	−12.0305	−0.933749
$167$	22.5664	1.74624	0.873120	−	0.487505i	$-0.162093\pi$
$167$	22.5664	1.74624	0.873120	+	0.487505i	$0.162093\pi$
$168$	0	0
$169$	18.6316	1.43320
$170$	1.21099	0.0928785
$171$	0	0
$172$	8.85880	0.675477
$173$	−12.4531	−0.946795	−0.473398	−	0.880849i	$-0.656973\pi$
$173$	−12.4531	−0.946795	−0.473398	+	0.880849i	$0.656973\pi$
$174$	0	0
$175$	−16.2484	−1.22826
$176$	9.19252	0.692912
$177$	0	0
$178$	−0.949883	−0.0711967
$179$	21.2425	1.58774	0.793869	−	0.608089i	$-0.208063\pi$
$179$	21.2425	1.58774	0.793869	+	0.608089i	$0.208063\pi$
$180$	0	0
$181$	6.56306	0.487829	0.243914	−	0.969797i	$-0.421568\pi$
$181$	6.56306	0.487829	0.243914	+	0.969797i	$0.421568\pi$
$182$	−20.5895	−1.52619
$183$	0	0
$184$	−10.6744	−0.786925
$185$	−2.16594	−0.159243
$186$	0	0
$187$	14.9261	1.09151
$188$	9.33002	0.680462
$189$	0	0
$190$	−1.43654	−0.104217
$191$	−18.3817	−1.33005	−0.665025	−	0.746821i	$-0.731579\pi$
$191$	−18.3817	−1.33005	−0.665025	+	0.746821i	$0.731579\pi$
$192$	0	0
$193$	11.8417	0.852385	0.426192	−	0.904633i	$-0.359855\pi$
$193$	11.8417	0.852385	0.426192	+	0.904633i	$0.359855\pi$
$194$	−9.44656	−0.678224
$195$	0	0
$196$	−3.44281	−0.245915
$197$	3.45807	0.246377	0.123189	−	0.992383i	$-0.460688\pi$
$197$	3.45807	0.246377	0.123189	+	0.992383i	$0.460688\pi$
$198$	0	0
$199$	11.6119	0.823148	0.411574	−	0.911376i	$-0.364979\pi$
$199$	11.6119	0.823148	0.411574	+	0.911376i	$0.364979\pi$
$200$	14.8473	1.04986
$201$	0	0
$202$	14.7547	1.03813
$203$	6.39079	0.448546
$204$	0	0
$205$	0.930749	0.0650063
$206$	−1.32945	−0.0926271
$207$	0	0
$208$	9.71188	0.673398
$209$	−17.7061	−1.22476
$210$	0	0
$211$	−8.89349	−0.612253	−0.306126	−	0.951991i	$-0.599033\pi$
$211$	−8.89349	−0.612253	−0.306126	+	0.951991i	$0.599033\pi$
$212$	−2.65940	−0.182648
$213$	0	0
$214$	7.26780	0.496817
$215$	4.33296	0.295505
$216$	0	0
$217$	−32.8954	−2.23309
$218$	−7.13789	−0.483439
$219$	0	0
$220$	−1.70495	−0.114948
$221$	15.7694	1.06077
$222$	0	0
$223$	11.7928	0.789706	0.394853	−	0.918744i	$-0.370796\pi$
$223$	11.7928	0.789706	0.394853	+	0.918744i	$0.370796\pi$
$224$	14.2466	0.951893
$225$	0	0
$226$	−11.9923	−0.797713
$227$	−7.92713	−0.526142	−0.263071	−	0.964776i	$-0.584735\pi$
$227$	−7.92713	−0.526142	−0.263071	+	0.964776i	$0.584735\pi$
$228$	0	0
$229$	−8.58621	−0.567393	−0.283696	−	0.958914i	$-0.591561\pi$
$229$	−8.58621	−0.567393	−0.283696	+	0.958914i	$0.591561\pi$
$230$	−1.50392	−0.0991655
$231$	0	0
$232$	−5.83972	−0.383396
$233$	−26.6758	−1.74759	−0.873795	−	0.486294i	$-0.838348\pi$
$233$	−26.6758	−1.74759	−0.873795	+	0.486294i	$0.838348\pi$
$234$	0	0
$235$	4.56344	0.297686
$236$	−4.52644	−0.294646
$237$	0	0
$238$	−10.2646	−0.665352
$239$	−14.1132	−0.912904	−0.456452	−	0.889748i	$-0.650880\pi$
$239$	−14.1132	−0.912904	−0.456452	+	0.889748i	$0.650880\pi$
$240$	0	0
$241$	9.47718	0.610479	0.305240	−	0.952276i	$-0.401263\pi$
$241$	9.47718	0.610479	0.305240	+	0.952276i	$0.401263\pi$
$242$	18.9209	1.21628
$243$	0	0
$244$	−0.252301	−0.0161519
$245$	−1.68392	−0.107582
$246$	0	0
$247$	−18.7065	−1.19027
$248$	30.0589	1.90874
$249$	0	0
$250$	4.25135	0.268879
$251$	4.67027	0.294785	0.147393	−	0.989078i	$-0.452912\pi$
$251$	4.67027	0.294785	0.147393	+	0.989078i	$0.452912\pi$
$252$	0	0
$253$	−18.5367	−1.16539
$254$	1.09124	0.0684704
$255$	0	0
$256$	−15.8128	−0.988300
$257$	−22.8733	−1.42680	−0.713398	−	0.700759i	$-0.752845\pi$
$257$	−22.8733	−1.42680	−0.713398	+	0.700759i	$0.752845\pi$
$258$	0	0
$259$	18.3589	1.14077
$260$	−1.80128	−0.111710
$261$	0	0
$262$	5.17912	0.319967
$263$	29.0676	1.79239	0.896194	−	0.443663i	$-0.146321\pi$
$263$	29.0676	1.79239	0.896194	+	0.443663i	$0.146321\pi$
$264$	0	0
$265$	−1.30075	−0.0799044
$266$	12.1763	0.746580
$267$	0	0
$268$	−7.44240	−0.454617
$269$	−16.7801	−1.02310	−0.511550	−	0.859254i	$-0.670928\pi$
$269$	−16.7801	−1.02310	−0.511550	+	0.859254i	$0.670928\pi$
$270$	0	0
$271$	−3.78216	−0.229750	−0.114875	−	0.993380i	$-0.536647\pi$
$271$	−3.78216	−0.229750	−0.114875	+	0.993380i	$0.536647\pi$
$272$	4.84170	0.293571
$273$	0	0
$274$	−0.952014	−0.0575133
$275$	25.7833	1.55479
$276$	0	0
$277$	−26.6687	−1.60237	−0.801184	−	0.598419i	$-0.795796\pi$
$277$	−26.6687	−1.60237	−0.801184	+	0.598419i	$0.795796\pi$
$278$	7.92952	0.475581
$279$	0	0
$280$	4.07035	0.243250
$281$	3.12996	0.186717	0.0933587	−	0.995633i	$-0.470240\pi$
$281$	3.12996	0.186717	0.0933587	+	0.995633i	$0.470240\pi$
$282$	0	0
$283$	15.6466	0.930091	0.465046	−	0.885287i	$-0.346038\pi$
$283$	15.6466	0.930091	0.465046	+	0.885287i	$0.346038\pi$
$284$	2.40641	0.142794
$285$	0	0
$286$	32.6717	1.93192
$287$	−7.88920	−0.465685
$288$	0	0
$289$	−9.13840	−0.537553
$290$	−0.822762	−0.0483142
$291$	0	0
$292$	−6.90767	−0.404241
$293$	0.776348	0.0453548	0.0226774	−	0.999743i	$-0.492781\pi$
$293$	0.776348	0.0453548	0.0226774	+	0.999743i	$0.492781\pi$
$294$	0	0
$295$	−2.21394	−0.128901
$296$	−16.7758	−0.975074
$297$	0	0
$298$	−1.06384	−0.0616268
$299$	−19.5840	−1.13257
$300$	0	0
$301$	−36.7270	−2.11691
$302$	−6.01286	−0.346001
$303$	0	0
$304$	−5.74348	−0.329411
$305$	−0.123404	−0.00706608
$306$	0	0
$307$	−3.02568	−0.172685	−0.0863423	−	0.996266i	$-0.527518\pi$
$307$	−3.02568	−0.172685	−0.0863423	+	0.996266i	$0.527518\pi$
$308$	14.4515	0.823448
$309$	0	0
$310$	4.23502	0.240533
$311$	−21.8044	−1.23641	−0.618207	−	0.786016i	$-0.712141\pi$
$311$	−21.8044	−1.23641	−0.618207	+	0.786016i	$0.712141\pi$
$312$	0	0
$313$	26.0058	1.46993	0.734967	−	0.678103i	$-0.237198\pi$
$313$	26.0058	1.46993	0.734967	+	0.678103i	$0.237198\pi$
$314$	0.648145	0.0365769
$315$	0	0
$316$	7.40007	0.416287
$317$	−0.384721	−0.0216081	−0.0108041	−	0.999942i	$-0.503439\pi$
$317$	−0.384721	−0.0216081	−0.0108041	+	0.999942i	$0.503439\pi$
$318$	0	0
$319$	−10.1410	−0.567788
$320$	−3.20104	−0.178943
$321$	0	0
$322$	12.7475	0.710390
$323$	−9.32584	−0.518903
$324$	0	0
$325$	27.2400	1.51100
$326$	21.4630	1.18873
$327$	0	0
$328$	7.20892	0.398046
$329$	−38.6806	−2.13253
$330$	0	0
$331$	20.8671	1.14696	0.573480	−	0.819220i	$-0.305593\pi$
$331$	20.8671	1.14696	0.573480	+	0.819220i	$0.305593\pi$
$332$	8.92112	0.489610
$333$	0	0
$334$	24.6253	1.34744
$335$	−3.64018	−0.198884
$336$	0	0
$337$	16.8976	0.920472	0.460236	−	0.887797i	$-0.347765\pi$
$337$	16.8976	0.920472	0.460236	+	0.887797i	$0.347765\pi$
$338$	20.3316	1.10589
$339$	0	0
$340$	−0.897997	−0.0487007
$341$	52.1991	2.82674
$342$	0	0
$343$	−9.21025	−0.497307
$344$	33.5600	1.80944
$345$	0	0
$346$	−13.5894	−0.730568
$347$	−16.7529	−0.899345	−0.449672	−	0.893194i	$-0.648459\pi$
$347$	−16.7529	−0.899345	−0.449672	+	0.893194i	$0.648459\pi$
$348$	0	0
$349$	−28.3122	−1.51552	−0.757760	−	0.652533i	$-0.773706\pi$
$349$	−28.3122	−1.51552	−0.757760	+	0.652533i	$0.773706\pi$
$350$	−17.7309	−0.947756
$351$	0	0
$352$	−22.6068	−1.20495
$353$	−35.6071	−1.89517	−0.947587	−	0.319498i	$-0.896486\pi$
$353$	−35.6071	−1.89517	−0.947587	+	0.319498i	$0.896486\pi$
$354$	0	0
$355$	1.17701	0.0624690
$356$	0.704377	0.0373319
$357$	0	0
$358$	23.1806	1.22513
$359$	2.61524	0.138027	0.0690135	−	0.997616i	$-0.478015\pi$
$359$	2.61524	0.138027	0.0690135	+	0.997616i	$0.478015\pi$
$360$	0	0
$361$	−7.93722	−0.417748
$362$	7.16187	0.376419
$363$	0	0
$364$	15.2679	0.800258
$365$	−3.37863	−0.176846
$366$	0	0
$367$	−23.2147	−1.21180	−0.605899	−	0.795541i	$-0.707187\pi$
$367$	−23.2147	−1.21180	−0.605899	+	0.795541i	$0.707187\pi$
$368$	−6.01288	−0.313443
$369$	0	0
$370$	−2.36355	−0.122875
$371$	11.0254	0.572410
$372$	0	0
$373$	8.50541	0.440393	0.220197	−	0.975455i	$-0.429330\pi$
$373$	8.50541	0.440393	0.220197	+	0.975455i	$0.429330\pi$
$374$	16.2880	0.842231
$375$	0	0
$376$	35.3452	1.82279
$377$	−10.7140	−0.551798
$378$	0	0
$379$	−38.3994	−1.97244	−0.986221	−	0.165433i	$-0.947098\pi$
$379$	−38.3994	−1.97244	−0.986221	+	0.165433i	$0.947098\pi$
$380$	1.06525	0.0546462
$381$	0	0
$382$	−20.0588	−1.02630
$383$	−9.86545	−0.504101	−0.252050	−	0.967714i	$-0.581105\pi$
$383$	−9.86545	−0.504101	−0.252050	+	0.967714i	$0.581105\pi$
$384$	0	0
$385$	7.06840	0.360239
$386$	12.9221	0.657719
$387$	0	0
$388$	7.00501	0.355626
$389$	21.8100	1.10581	0.552906	−	0.833244i	$-0.313519\pi$
$389$	21.8100	1.10581	0.552906	+	0.833244i	$0.313519\pi$
$390$	0	0
$391$	−9.76328	−0.493750
$392$	−13.0425	−0.658745
$393$	0	0
$394$	3.77358	0.190110
$395$	3.61948	0.182116
$396$	0	0
$397$	12.7692	0.640867	0.320434	−	0.947271i	$-0.396171\pi$
$397$	12.7692	0.640867	0.320434	+	0.947271i	$0.396171\pi$
$398$	12.6714	0.635160
$399$	0	0
$400$	8.36351	0.418175
$401$	−14.1327	−0.705752	−0.352876	−	0.935670i	$-0.614796\pi$
$401$	−14.1327	−0.705752	−0.352876	+	0.935670i	$0.614796\pi$
$402$	0	0
$403$	55.1482	2.74713
$404$	−10.9412	−0.544344
$405$	0	0
$406$	6.97388	0.346108
$407$	−29.1322	−1.44403
$408$	0	0
$409$	−36.4134	−1.80053	−0.900264	−	0.435345i	$-0.856626\pi$
$409$	−36.4134	−1.80053	−0.900264	+	0.435345i	$0.856626\pi$
$410$	1.01567	0.0501603
$411$	0	0
$412$	0.985841	0.0485689
$413$	18.7658	0.923403
$414$	0	0
$415$	4.36344	0.214193
$416$	−23.8841	−1.17101
$417$	0	0
$418$	−19.3216	−0.945052
$419$	−8.43499	−0.412076	−0.206038	−	0.978544i	$-0.566057\pi$
$419$	−8.43499	−0.412076	−0.206038	+	0.978544i	$0.566057\pi$
$420$	0	0
$421$	−16.8705	−0.822216	−0.411108	−	0.911587i	$-0.634858\pi$
$421$	−16.8705	−0.822216	−0.411108	+	0.911587i	$0.634858\pi$
$422$	−9.70492	−0.472428
$423$	0	0
$424$	−10.0747	−0.489270
$425$	13.5801	0.658729
$426$	0	0
$427$	1.04599	0.0506192
$428$	−5.38937	−0.260505
$429$	0	0
$430$	4.72829	0.228019
$431$	−37.2366	−1.79362	−0.896812	−	0.442413i	$-0.854123\pi$
$431$	−37.2366	−1.79362	−0.896812	+	0.442413i	$0.854123\pi$
$432$	0	0
$433$	−7.27802	−0.349759	−0.174880	−	0.984590i	$-0.555954\pi$
$433$	−7.27802	−0.349759	−0.174880	+	0.984590i	$0.555954\pi$
$434$	−35.8968	−1.72310
$435$	0	0
$436$	5.29303	0.253490
$437$	11.5817	0.554028
$438$	0	0
$439$	−7.85780	−0.375032	−0.187516	−	0.982262i	$-0.560044\pi$
$439$	−7.85780	−0.375032	−0.187516	+	0.982262i	$0.560044\pi$
$440$	−6.45890	−0.307916
$441$	0	0
$442$	17.2082	0.818512
$443$	23.0038	1.09294	0.546471	−	0.837478i	$-0.315971\pi$
$443$	23.0038	1.09294	0.546471	+	0.837478i	$0.315971\pi$
$444$	0	0
$445$	0.344520	0.0163318
$446$	12.8688	0.609355
$447$	0	0
$448$	27.1326	1.28189
$449$	2.71015	0.127900	0.0639498	−	0.997953i	$-0.479630\pi$
$449$	2.71015	0.127900	0.0639498	+	0.997953i	$0.479630\pi$
$450$	0	0
$451$	12.5187	0.589484
$452$	8.89275	0.418280
$453$	0	0
$454$	−8.65039	−0.405983
$455$	7.46776	0.350094
$456$	0	0
$457$	16.0161	0.749202	0.374601	−	0.927186i	$-0.377780\pi$
$457$	16.0161	0.749202	0.374601	+	0.927186i	$0.377780\pi$
$458$	−9.36961	−0.437813
$459$	0	0
$460$	1.11522	0.0519973
$461$	−8.23022	−0.383320	−0.191660	−	0.981461i	$-0.561387\pi$
$461$	−8.23022	−0.383320	−0.191660	+	0.981461i	$0.561387\pi$
$462$	0	0
$463$	−3.03422	−0.141012	−0.0705061	−	0.997511i	$-0.522461\pi$
$463$	−3.03422	−0.141012	−0.0705061	+	0.997511i	$0.522461\pi$
$464$	−3.28952	−0.152712
$465$	0	0
$466$	−29.1097	−1.34848
$467$	18.6263	0.861924	0.430962	−	0.902370i	$-0.358174\pi$
$467$	18.6263	0.861924	0.430962	+	0.902370i	$0.358174\pi$
$468$	0	0
$469$	30.8548	1.42474
$470$	4.97980	0.229701
$471$	0	0
$472$	−17.1476	−0.789283
$473$	58.2790	2.67967
$474$	0	0
$475$	−16.1094	−0.739148
$476$	7.61159	0.348877
$477$	0	0
$478$	−15.4008	−0.704417
$479$	1.84112	0.0841229	0.0420614	−	0.999115i	$-0.486607\pi$
$479$	1.84112	0.0841229	0.0420614	+	0.999115i	$0.486607\pi$
$480$	0	0
$481$	−30.7781	−1.40336
$482$	10.3419	0.471059
$483$	0	0
$484$	−14.0306	−0.637756
$485$	3.42625	0.155578
$486$	0	0
$487$	9.63924	0.436796	0.218398	−	0.975860i	$-0.429917\pi$
$487$	9.63924	0.436796	0.218398	+	0.975860i	$0.429917\pi$
$488$	−0.955798	−0.0432670
$489$	0	0
$490$	−1.83756	−0.0830127
$491$	31.9211	1.44058	0.720291	−	0.693672i	$-0.244008\pi$
$491$	31.9211	1.44058	0.720291	+	0.693672i	$0.244008\pi$
$492$	0	0
$493$	−5.34128	−0.240559
$494$	−20.4133	−0.918437
$495$	0	0
$496$	16.9322	0.760278
$497$	−9.97652	−0.447508
$498$	0	0
$499$	−3.61846	−0.161985	−0.0809923	−	0.996715i	$-0.525809\pi$
$499$	−3.61846	−0.161985	−0.0809923	+	0.996715i	$0.525809\pi$
$500$	−3.15255	−0.140986
$501$	0	0
$502$	5.09638	0.227463
$503$	−19.2012	−0.856139	−0.428070	−	0.903746i	$-0.640806\pi$
$503$	−19.2012	−0.856139	−0.428070	+	0.903746i	$0.640806\pi$
$504$	0	0
$505$	−5.35148	−0.238138
$506$	−20.2279	−0.899242
$507$	0	0
$508$	−0.809198	−0.0359024
$509$	−25.8768	−1.14697	−0.573485	−	0.819216i	$-0.694409\pi$
$509$	−25.8768	−1.14697	−0.573485	+	0.819216i	$0.694409\pi$
$510$	0	0
$511$	28.6379	1.26687
$512$	−17.9202	−0.791968
$513$	0	0
$514$	−24.9602	−1.10095
$515$	0.482188	0.0212477
$516$	0	0
$517$	61.3790	2.69944
$518$	20.0339	0.880240
$519$	0	0
$520$	−6.82382	−0.299244
$521$	−5.61373	−0.245942	−0.122971	−	0.992410i	$-0.539242\pi$
$521$	−5.61373	−0.245942	−0.122971	+	0.992410i	$0.539242\pi$
$522$	0	0
$523$	−39.7981	−1.74025	−0.870125	−	0.492832i	$-0.835962\pi$
$523$	−39.7981	−1.74025	−0.870125	+	0.492832i	$0.835962\pi$
$524$	−3.84053	−0.167774
$525$	0	0
$526$	31.7197	1.38305
$527$	27.4933	1.19763
$528$	0	0
$529$	−10.8750	−0.472828
$530$	−1.41943	−0.0616560
$531$	0	0
$532$	−9.02926	−0.391468
$533$	13.2260	0.572882
$534$	0	0
$535$	−2.63601	−0.113965
$536$	−28.1942	−1.21781
$537$	0	0
$538$	−18.3111	−0.789446
$539$	−22.6490	−0.975564
$540$	0	0
$541$	−23.8809	−1.02672	−0.513360	−	0.858173i	$-0.671600\pi$
$541$	−23.8809	−1.02672	−0.513360	+	0.858173i	$0.671600\pi$
$542$	−4.12724	−0.177280
$543$	0	0
$544$	−11.9070	−0.510509
$545$	2.58889	0.110896
$546$	0	0
$547$	−27.3795	−1.17066	−0.585331	−	0.810794i	$-0.699036\pi$
$547$	−27.3795	−1.17066	−0.585331	+	0.810794i	$0.699036\pi$
$548$	0.705958	0.0301570
$549$	0	0
$550$	28.1357	1.19971
$551$	6.33610	0.269927
$552$	0	0
$553$	−30.6793	−1.30462
$554$	−29.1019	−1.23642
$555$	0	0
$556$	−5.88006	−0.249370
$557$	21.0351	0.891288	0.445644	−	0.895210i	$-0.352975\pi$
$557$	21.0351	0.891288	0.445644	+	0.895210i	$0.352975\pi$
$558$	0	0
$559$	61.5717	2.60420
$560$	2.29283	0.0968898
$561$	0	0
$562$	3.41553	0.144075
$563$	0.785419	0.0331015	0.0165507	−	0.999863i	$-0.494731\pi$
$563$	0.785419	0.0331015	0.0165507	+	0.999863i	$0.494731\pi$
$564$	0	0
$565$	4.34956	0.182988
$566$	17.0741	0.717679
$567$	0	0
$568$	9.11625	0.382509
$569$	13.6931	0.574046	0.287023	−	0.957924i	$-0.407334\pi$
$569$	13.6931	0.574046	0.287023	+	0.957924i	$0.407334\pi$
$570$	0	0
$571$	−2.85259	−0.119377	−0.0596887	−	0.998217i	$-0.519011\pi$
$571$	−2.85259	−0.119377	−0.0596887	+	0.998217i	$0.519011\pi$
$572$	−24.2274	−1.01300
$573$	0	0
$574$	−8.60900	−0.359333
$575$	−16.8650	−0.703319
$576$	0	0
$577$	5.52540	0.230025	0.115013	−	0.993364i	$-0.463309\pi$
$577$	5.52540	0.230025	0.115013	+	0.993364i	$0.463309\pi$
$578$	−9.97217	−0.414788
$579$	0	0
$580$	0.610111	0.0253335
$581$	−36.9853	−1.53441
$582$	0	0
$583$	−17.4953	−0.724581
$584$	−26.1685	−1.08286
$585$	0	0
$586$	0.847181	0.0349967
$587$	27.4726	1.13392	0.566959	−	0.823746i	$-0.308120\pi$
$587$	27.4726	1.13392	0.566959	+	0.823746i	$0.308120\pi$
$588$	0	0
$589$	−32.6139	−1.34383
$590$	−2.41594	−0.0994626
$591$	0	0
$592$	−9.44983	−0.388385
$593$	−45.2250	−1.85717	−0.928583	−	0.371124i	$-0.878972\pi$
$593$	−45.2250	−1.85717	−0.928583	+	0.371124i	$0.878972\pi$
$594$	0	0
$595$	3.72293	0.152625
$596$	0.788884	0.0323139
$597$	0	0
$598$	−21.3708	−0.873917
$599$	−13.4015	−0.547570	−0.273785	−	0.961791i	$-0.588276\pi$
$599$	−13.4015	−0.547570	−0.273785	+	0.961791i	$0.588276\pi$
$600$	0	0
$601$	−3.17984	−0.129709	−0.0648543	−	0.997895i	$-0.520658\pi$
$601$	−3.17984	−0.129709	−0.0648543	+	0.997895i	$0.520658\pi$
$602$	−40.0779	−1.63345
$603$	0	0
$604$	4.45878	0.181425
$605$	−6.86257	−0.279003
$606$	0	0
$607$	17.4621	0.708764	0.354382	−	0.935101i	$-0.384691\pi$
$607$	17.4621	0.708764	0.354382	+	0.935101i	$0.384691\pi$
$608$	14.1247	0.572833
$609$	0	0
$610$	−0.134663	−0.00545235
$611$	64.8468	2.62342
$612$	0	0
$613$	30.3821	1.22712	0.613561	−	0.789647i	$-0.289736\pi$
$613$	30.3821	1.22712	0.613561	+	0.789647i	$0.289736\pi$
$614$	−3.30174	−0.133247
$615$	0	0
$616$	54.7468	2.20581
$617$	−33.0038	−1.32868	−0.664341	−	0.747429i	$-0.731288\pi$
$617$	−33.0038	−1.32868	−0.664341	+	0.747429i	$0.731288\pi$
$618$	0	0
$619$	28.5040	1.14567	0.572837	−	0.819669i	$-0.305843\pi$
$619$	28.5040	1.14567	0.572837	+	0.819669i	$0.305843\pi$
$620$	−3.14044	−0.126123
$621$	0	0
$622$	−23.7938	−0.954044
$623$	−2.92022	−0.116996
$624$	0	0
$625$	22.6748	0.906992
$626$	28.3785	1.13423
$627$	0	0
$628$	−0.480626	−0.0191791
$629$	−15.3439	−0.611803
$630$	0	0
$631$	−5.85640	−0.233140	−0.116570	−	0.993183i	$-0.537190\pi$
$631$	−5.85640	−0.233140	−0.116570	+	0.993183i	$0.537190\pi$
$632$	28.0339	1.11513
$633$	0	0
$634$	−0.419823	−0.0166733
$635$	−0.395790	−0.0157064
$636$	0	0
$637$	−23.9287	−0.948089
$638$	−11.0663	−0.438118
$639$	0	0
$640$	−0.131529	−0.00519913
$641$	−24.7043	−0.975760	−0.487880	−	0.872911i	$-0.662230\pi$
$641$	−24.7043	−0.975760	−0.487880	+	0.872911i	$0.662230\pi$
$642$	0	0
$643$	−23.7585	−0.936944	−0.468472	−	0.883478i	$-0.655195\pi$
$643$	−23.7585	−0.936944	−0.468472	+	0.883478i	$0.655195\pi$
$644$	−9.45279	−0.372492
$645$	0	0
$646$	−10.1767	−0.400397
$647$	2.45718	0.0966017	0.0483009	−	0.998833i	$-0.484619\pi$
$647$	2.45718	0.0966017	0.0483009	+	0.998833i	$0.484619\pi$
$648$	0	0
$649$	−29.7779	−1.16888
$650$	29.7253	1.16592
$651$	0	0
$652$	−15.9157	−0.623307
$653$	10.5536	0.412993	0.206496	−	0.978447i	$-0.433794\pi$
$653$	10.5536	0.412993	0.206496	+	0.978447i	$0.433794\pi$
$654$	0	0
$655$	−1.87846	−0.0733973
$656$	4.06079	0.158547
$657$	0	0
$658$	−42.2097	−1.64551
$659$	10.8113	0.421147	0.210573	−	0.977578i	$-0.432467\pi$
$659$	10.8113	0.421147	0.210573	+	0.977578i	$0.432467\pi$
$660$	0	0
$661$	35.1932	1.36886	0.684428	−	0.729081i	$-0.260052\pi$
$661$	35.1932	1.36886	0.684428	+	0.729081i	$0.260052\pi$
$662$	22.7710	0.885019
$663$	0	0
$664$	33.7961	1.31154
$665$	−4.41633	−0.171258
$666$	0	0
$667$	6.63331	0.256843
$668$	−18.2607	−0.706527
$669$	0	0
$670$	−3.97230	−0.153463
$671$	−1.65980	−0.0640759
$672$	0	0
$673$	−21.7639	−0.838936	−0.419468	−	0.907770i	$-0.637783\pi$
$673$	−21.7639	−0.838936	−0.419468	+	0.907770i	$0.637783\pi$
$674$	18.4393	0.710257
$675$	0	0
$676$	−15.0767	−0.579872
$677$	−10.1070	−0.388442	−0.194221	−	0.980958i	$-0.562218\pi$
$677$	−10.1070	−0.388442	−0.194221	+	0.980958i	$0.562218\pi$
$678$	0	0
$679$	−29.0415	−1.11451
$680$	−3.40190	−0.130457
$681$	0	0
$682$	56.9616	2.18117
$683$	5.11288	0.195639	0.0978195	−	0.995204i	$-0.468813\pi$
$683$	5.11288	0.195639	0.0978195	+	0.995204i	$0.468813\pi$
$684$	0	0
$685$	0.345293	0.0131930
$686$	−10.0506	−0.383733
$687$	0	0
$688$	18.9044	0.720723
$689$	−18.4837	−0.704174
$690$	0	0
$691$	−10.5506	−0.401363	−0.200682	−	0.979657i	$-0.564316\pi$
$691$	−10.5506	−0.401363	−0.200682	+	0.979657i	$0.564316\pi$
$692$	10.0771	0.383073
$693$	0	0
$694$	−18.2815	−0.693954
$695$	−2.87602	−0.109094
$696$	0	0
$697$	6.59362	0.249751
$698$	−30.8954	−1.16941
$699$	0	0
$700$	13.1482	0.496955
$701$	50.1760	1.89512	0.947562	−	0.319573i	$-0.103540\pi$
$701$	50.1760	1.89512	0.947562	+	0.319573i	$0.103540\pi$
$702$	0	0
$703$	18.2018	0.686492
$704$	−43.0545	−1.62268
$705$	0	0
$706$	−38.8558	−1.46236
$707$	45.3602	1.70594
$708$	0	0
$709$	40.0936	1.50575	0.752873	−	0.658166i	$-0.228668\pi$
$709$	40.0936	1.50575	0.752873	+	0.658166i	$0.228668\pi$
$710$	1.28439	0.0482025
$711$	0	0
$712$	2.66841	0.100003
$713$	−34.1437	−1.27869
$714$	0	0
$715$	−11.8500	−0.443164
$716$	−17.1894	−0.642397
$717$	0	0
$718$	2.85385	0.106505
$719$	21.9166	0.817352	0.408676	−	0.912680i	$-0.365991\pi$
$719$	21.9166	0.817352	0.408676	+	0.912680i	$0.365991\pi$
$720$	0	0
$721$	−4.08711	−0.152212
$722$	−8.66140	−0.322344
$723$	0	0
$724$	−5.31082	−0.197375
$725$	−9.22647	−0.342663
$726$	0	0
$727$	21.8709	0.811146	0.405573	−	0.914063i	$-0.367072\pi$
$727$	21.8709	0.811146	0.405573	+	0.914063i	$0.367072\pi$
$728$	57.8399	2.14369
$729$	0	0
$730$	−3.68689	−0.136458
$731$	30.6956	1.13532
$732$	0	0
$733$	−16.3017	−0.602117	−0.301058	−	0.953606i	$-0.597340\pi$
$733$	−16.3017	−0.602117	−0.301058	+	0.953606i	$0.597340\pi$
$734$	−25.3328	−0.935051
$735$	0	0
$736$	14.7873	0.545065
$737$	−48.9610	−1.80350
$738$	0	0
$739$	30.1168	1.10786	0.553932	−	0.832562i	$-0.313127\pi$
$739$	30.1168	1.10786	0.553932	+	0.832562i	$0.313127\pi$
$740$	1.75267	0.0644295
$741$	0	0
$742$	12.0313	0.441684
$743$	38.0413	1.39560	0.697799	−	0.716293i	$-0.254163\pi$
$743$	38.0413	1.39560	0.697799	+	0.716293i	$0.254163\pi$
$744$	0	0
$745$	0.385854	0.0141366
$746$	9.28143	0.339817
$747$	0	0
$748$	−12.0782	−0.441623
$749$	22.3434	0.816408
$750$	0	0
$751$	13.6567	0.498340	0.249170	−	0.968460i	$-0.419842\pi$
$751$	13.6567	0.498340	0.249170	+	0.968460i	$0.419842\pi$
$752$	19.9100	0.726042
$753$	0	0
$754$	−11.6915	−0.425779
$755$	2.18085	0.0793693
$756$	0	0
$757$	−36.5562	−1.32866	−0.664328	−	0.747441i	$-0.731282\pi$
$757$	−36.5562	−1.32866	−0.664328	+	0.747441i	$0.731282\pi$
$758$	−41.9029	−1.52198
$759$	0	0
$760$	4.03551	0.146383
$761$	9.14533	0.331518	0.165759	−	0.986166i	$-0.446993\pi$
$761$	9.14533	0.331518	0.165759	+	0.986166i	$0.446993\pi$
$762$	0	0
$763$	−21.9439	−0.794424
$764$	14.8744	0.538137
$765$	0	0
$766$	−10.7656	−0.388975
$767$	−31.4603	−1.13596
$768$	0	0
$769$	13.0316	0.469930	0.234965	−	0.972004i	$-0.424502\pi$
$769$	13.0316	0.469930	0.234965	+	0.972004i	$0.424502\pi$
$770$	7.71331	0.277969
$771$	0	0
$772$	−9.58229	−0.344874
$773$	−24.5496	−0.882989	−0.441495	−	0.897264i	$-0.645552\pi$
$773$	−24.5496	−0.882989	−0.441495	+	0.897264i	$0.645552\pi$
$774$	0	0
$775$	47.4916	1.70595
$776$	26.5373	0.952632
$777$	0	0
$778$	23.7999	0.853269
$779$	−7.82169	−0.280241
$780$	0	0
$781$	15.8309	0.566475
$782$	−10.6541	−0.380989
$783$	0	0
$784$	−7.34684	−0.262387
$785$	−0.235081	−0.00839038
$786$	0	0
$787$	−4.24555	−0.151337	−0.0756687	−	0.997133i	$-0.524109\pi$
$787$	−4.24555	−0.151337	−0.0756687	+	0.997133i	$0.524109\pi$
$788$	−2.79826	−0.0996839
$789$	0	0
$790$	3.94971	0.140524
$791$	−36.8677	−1.31086
$792$	0	0
$793$	−1.75358	−0.0622714
$794$	13.9342	0.494508
$795$	0	0
$796$	−9.39636	−0.333045
$797$	−36.8616	−1.30571	−0.652853	−	0.757484i	$-0.726428\pi$
$797$	−36.8616	−1.30571	−0.652853	+	0.757484i	$0.726428\pi$
$798$	0	0
$799$	32.3283	1.14369
$800$	−20.5681	−0.727190
$801$	0	0
$802$	−15.4221	−0.544574
$803$	−45.4431	−1.60365
$804$	0	0
$805$	−4.62349	−0.162957
$806$	60.1799	2.11975
$807$	0	0
$808$	−41.4488	−1.45816
$809$	0.556343	0.0195600	0.00977999	−	0.999952i	$-0.496887\pi$
$809$	0.556343	0.0195600	0.00977999	+	0.999952i	$0.496887\pi$
$810$	0	0
$811$	16.1882	0.568444	0.284222	−	0.958759i	$-0.408265\pi$
$811$	16.1882	0.568444	0.284222	+	0.958759i	$0.408265\pi$
$812$	−5.17142	−0.181481
$813$	0	0
$814$	−31.7902	−1.11424
$815$	−7.78458	−0.272682
$816$	0	0
$817$	−36.4127	−1.27392
$818$	−39.7357	−1.38933
$819$	0	0
$820$	−0.753160	−0.0263015
$821$	40.8717	1.42643	0.713216	−	0.700945i	$-0.247238\pi$
$821$	40.8717	1.42643	0.713216	+	0.700945i	$0.247238\pi$
$822$	0	0
$823$	−18.4396	−0.642763	−0.321381	−	0.946950i	$-0.604147\pi$
$823$	−18.4396	−0.642763	−0.321381	+	0.946950i	$0.604147\pi$
$824$	3.73468	0.130104
$825$	0	0
$826$	20.4779	0.712519
$827$	−2.38526	−0.0829436	−0.0414718	−	0.999140i	$-0.513205\pi$
$827$	−2.38526	−0.0829436	−0.0414718	+	0.999140i	$0.513205\pi$
$828$	0	0
$829$	−28.1260	−0.976856	−0.488428	−	0.872604i	$-0.662430\pi$
$829$	−28.1260	−0.976856	−0.488428	+	0.872604i	$0.662430\pi$
$830$	4.76155	0.165276
$831$	0	0
$832$	−45.4870	−1.57698
$833$	−11.9293	−0.413325
$834$	0	0
$835$	−8.93155	−0.309089
$836$	14.3278	0.495537
$837$	0	0
$838$	−9.20459	−0.317967
$839$	−5.30711	−0.183222	−0.0916109	−	0.995795i	$-0.529202\pi$
$839$	−5.30711	−0.183222	−0.0916109	+	0.995795i	$0.529202\pi$
$840$	0	0
$841$	−25.3711	−0.874864
$842$	−18.4097	−0.634440
$843$	0	0
$844$	7.19659	0.247717
$845$	−7.37421	−0.253680
$846$	0	0
$847$	58.1685	1.99869
$848$	−5.67508	−0.194883
$849$	0	0
$850$	14.8191	0.508290
$851$	19.0555	0.653216
$852$	0	0
$853$	−24.6505	−0.844017	−0.422009	−	0.906592i	$-0.638675\pi$
$853$	−24.6505	−0.844017	−0.422009	+	0.906592i	$0.638675\pi$
$854$	1.14143	0.0390589
$855$	0	0
$856$	−20.4167	−0.697828
$857$	−13.0241	−0.444895	−0.222447	−	0.974945i	$-0.571405\pi$
$857$	−13.0241	−0.444895	−0.222447	+	0.974945i	$0.571405\pi$
$858$	0	0
$859$	−36.3638	−1.24072	−0.620359	−	0.784318i	$-0.713013\pi$
$859$	−36.3638	−1.24072	−0.620359	+	0.784318i	$0.713013\pi$
$860$	−3.50622	−0.119561
$861$	0	0
$862$	−40.6340	−1.38400
$863$	24.2809	0.826530	0.413265	−	0.910611i	$-0.364388\pi$
$863$	24.2809	0.826530	0.413265	+	0.910611i	$0.364388\pi$
$864$	0	0
$865$	4.92883	0.167585
$866$	−7.94206	−0.269882
$867$	0	0
$868$	26.6189	0.903505
$869$	48.6825	1.65144
$870$	0	0
$871$	−51.7272	−1.75271
$872$	20.0517	0.679037
$873$	0	0
$874$	12.6384	0.427500
$875$	13.0699	0.441843
$876$	0	0
$877$	−10.8796	−0.367378	−0.183689	−	0.982984i	$-0.558804\pi$
$877$	−10.8796	−0.367378	−0.183689	+	0.982984i	$0.558804\pi$
$878$	−8.57473	−0.289383
$879$	0	0
$880$	−3.63830	−0.122647
$881$	6.11053	0.205869	0.102935	−	0.994688i	$-0.467177\pi$
$881$	6.11053	0.205869	0.102935	+	0.994688i	$0.467177\pi$
$882$	0	0
$883$	−33.5829	−1.13015	−0.565077	−	0.825038i	$-0.691154\pi$
$883$	−33.5829	−1.13015	−0.565077	+	0.825038i	$0.691154\pi$
$884$	−12.7606	−0.429186
$885$	0	0
$886$	25.1026	0.843339
$887$	−34.1465	−1.14653	−0.573264	−	0.819371i	$-0.694323\pi$
$887$	−34.1465	−1.14653	−0.573264	+	0.819371i	$0.694323\pi$
$888$	0	0
$889$	3.35479	0.112516
$890$	0.375954	0.0126020
$891$	0	0
$892$	−9.54273	−0.319514
$893$	−38.3495	−1.28332
$894$	0	0
$895$	−8.40756	−0.281034
$896$	1.11486	0.0372449
$897$	0	0
$898$	2.95742	0.0986902
$899$	−18.6793	−0.622990
$900$	0	0
$901$	−9.21477	−0.306989
$902$	13.6609	0.454859
$903$	0	0
$904$	33.6886	1.12047
$905$	−2.59759	−0.0863469
$906$	0	0
$907$	4.21238	0.139870	0.0699349	−	0.997552i	$-0.477721\pi$
$907$	4.21238	0.139870	0.0699349	+	0.997552i	$0.477721\pi$
$908$	6.41462	0.212877
$909$	0	0
$910$	8.14910	0.270140
$911$	−22.6700	−0.751091	−0.375546	−	0.926804i	$-0.622545\pi$
$911$	−22.6700	−0.751091	−0.375546	+	0.926804i	$0.622545\pi$
$912$	0	0
$913$	58.6889	1.94232
$914$	17.4774	0.578101
$915$	0	0
$916$	6.94795	0.229567
$917$	15.9221	0.525795
$918$	0	0
$919$	48.2891	1.59291	0.796455	−	0.604698i	$-0.206706\pi$
$919$	48.2891	1.59291	0.796455	+	0.604698i	$0.206706\pi$
$920$	4.22481	0.139288
$921$	0	0
$922$	−8.98114	−0.295778
$923$	16.7253	0.550521
$924$	0	0
$925$	−26.5050	−0.871478
$926$	−3.31106	−0.108808
$927$	0	0
$928$	8.08979	0.265560
$929$	−2.57091	−0.0843488	−0.0421744	−	0.999110i	$-0.513429\pi$
$929$	−2.57091	−0.0843488	−0.0421744	+	0.999110i	$0.513429\pi$
$930$	0	0
$931$	14.1511	0.463784
$932$	21.5860	0.707073
$933$	0	0
$934$	20.3258	0.665080
$935$	−5.90761	−0.193200
$936$	0	0
$937$	−9.98629	−0.326238	−0.163119	−	0.986606i	$-0.552155\pi$
$937$	−9.98629	−0.326238	−0.163119	+	0.986606i	$0.552155\pi$
$938$	33.6700	1.09936
$939$	0	0
$940$	−3.69273	−0.120443
$941$	−37.8410	−1.23358	−0.616791	−	0.787127i	$-0.711567\pi$
$941$	−37.8410	−1.23358	−0.616791	+	0.787127i	$0.711567\pi$
$942$	0	0
$943$	−8.18858	−0.266657
$944$	−9.65927	−0.314382
$945$	0	0
$946$	63.5963	2.06769
$947$	22.2224	0.722130	0.361065	−	0.932541i	$-0.382413\pi$
$947$	22.2224	0.722130	0.361065	+	0.932541i	$0.382413\pi$
$948$	0	0
$949$	−48.0106	−1.55849
$950$	−17.5792	−0.570343
$951$	0	0
$952$	28.8352	0.934553
$953$	59.0351	1.91233	0.956167	−	0.292822i	$-0.0945943\pi$
$953$	59.0351	1.91233	0.956167	+	0.292822i	$0.0945943\pi$
$954$	0	0
$955$	7.27527	0.235422
$956$	11.4203	0.369360
$957$	0	0
$958$	2.00910	0.0649111
$959$	−2.92677	−0.0945104
$960$	0	0
$961$	65.1483	2.10156
$962$	−33.5863	−1.08286
$963$	0	0
$964$	−7.66892	−0.246999
$965$	−4.68683	−0.150874
$966$	0	0
$967$	45.1473	1.45184	0.725920	−	0.687779i	$-0.241414\pi$
$967$	45.1473	1.45184	0.725920	+	0.687779i	$0.241414\pi$
$968$	−53.1526	−1.70839
$969$	0	0
$970$	3.73885	0.120047
$971$	34.6528	1.11206	0.556031	−	0.831161i	$-0.312323\pi$
$971$	34.6528	1.11206	0.556031	+	0.831161i	$0.312323\pi$
$972$	0	0
$973$	24.3777	0.781512
$974$	10.5187	0.337041
$975$	0	0
$976$	−0.538402	−0.0172338
$977$	14.2843	0.456994	0.228497	−	0.973545i	$-0.426619\pi$
$977$	14.2843	0.456994	0.228497	+	0.973545i	$0.426619\pi$
$978$	0	0
$979$	4.63385	0.148098
$980$	1.36263	0.0435276
$981$	0	0
$982$	34.8336	1.11158
$983$	33.6331	1.07273	0.536364	−	0.843987i	$-0.319797\pi$
$983$	33.6331	1.07273	0.536364	+	0.843987i	$0.319797\pi$
$984$	0	0
$985$	−1.36867	−0.0436094
$986$	−5.82861	−0.185621
$987$	0	0
$988$	15.1373	0.481581
$989$	−38.1207	−1.21217
$990$	0	0
$991$	−32.6178	−1.03614	−0.518068	−	0.855339i	$-0.673349\pi$
$991$	−32.6178	−1.03614	−0.518068	+	0.855339i	$0.673349\pi$
$992$	−41.6407	−1.32209
$993$	0	0
$994$	−10.8868	−0.345307
$995$	−4.59588	−0.145699
$996$	0	0
$997$	−2.76427	−0.0875454	−0.0437727	−	0.999042i	$-0.513938\pi$
$997$	−2.76427	−0.0875454	−0.0437727	+	0.999042i	$0.513938\pi$
$998$	−3.94860	−0.124991
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.i.1.5		7
3.2	odd	2		127.2.a.b.1.3	✓	7
12.11	even	2		2032.2.a.p.1.3		7
15.14	odd	2		3175.2.a.j.1.5		7
21.20	even	2		6223.2.a.h.1.3		7
24.5	odd	2		8128.2.a.bi.1.3		7
24.11	even	2		8128.2.a.bj.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.2.a.b.1.3	✓	7	3.2	odd	2
1143.2.a.i.1.5		7	1.1	even	1	trivial
2032.2.a.p.1.3		7	12.11	even	2
3175.2.a.j.1.5		7	15.14	odd	2
6223.2.a.h.1.3		7	21.20	even	2
8128.2.a.bi.1.3		7	24.5	odd	2
8128.2.a.bj.1.5		7	24.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09124 q^{2} -0.809198 q^{4} -0.395790 q^{5} +3.35479 q^{7} -3.06551 q^{8} +O(q^{10})\)	\(q+1.09124 q^{2} -0.809198 q^{4} -0.395790 q^{5} +3.35479 q^{7} -3.06551 q^{8} -0.431901 q^{10} -5.32343 q^{11} -5.62420 q^{13} +3.66087 q^{14} -1.72680 q^{16} -2.80386 q^{17} +3.32608 q^{19} +0.320272 q^{20} -5.80914 q^{22} +3.48209 q^{23} -4.84335 q^{25} -6.13735 q^{26} -2.71469 q^{28} +1.90498 q^{29} -9.80553 q^{31} +4.24666 q^{32} -3.05968 q^{34} -1.32779 q^{35} +5.47244 q^{37} +3.62954 q^{38} +1.21330 q^{40} -2.35163 q^{41} -10.9476 q^{43} +4.30771 q^{44} +3.79979 q^{46} -11.5300 q^{47} +4.25459 q^{49} -5.28525 q^{50} +4.55109 q^{52} +3.28647 q^{53} +2.10696 q^{55} -10.2841 q^{56} +2.07878 q^{58} +5.59373 q^{59} +0.311791 q^{61} -10.7002 q^{62} +8.08772 q^{64} +2.22600 q^{65} +9.19725 q^{67} +2.26887 q^{68} -1.44894 q^{70} -2.97382 q^{71} +8.53643 q^{73} +5.97174 q^{74} -2.69145 q^{76} -17.8590 q^{77} -9.14495 q^{79} +0.683451 q^{80} -2.56618 q^{82} -11.0246 q^{83} +1.10974 q^{85} -11.9465 q^{86} +16.3190 q^{88} -0.870463 q^{89} -18.8680 q^{91} -2.81770 q^{92} -12.5819 q^{94} -1.31643 q^{95} -8.65673 q^{97} +4.64278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8	\( 7 q - 2 q^{2} + 6 q^{4} - 8 q^{5} - 3 q^{7} - 3 q^{8} - 5 q^{10} - q^{13} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 5 q^{19} - 11 q^{20} - 9 q^{22} + q^{23} + 7 q^{25} + 4 q^{26} - 26 q^{28} + 7 q^{29} - 8 q^{31} + 2 q^{32} - q^{34} - 4 q^{35} - 6 q^{37} - 29 q^{38} - 3 q^{40} - 14 q^{41} - q^{43} + 21 q^{44} - 3 q^{46} - 25 q^{47} - 10 q^{50} + 6 q^{52} - 29 q^{53} - 23 q^{55} - 9 q^{56} - 22 q^{58} + 12 q^{59} + 7 q^{61} - 4 q^{62} - 3 q^{64} - 3 q^{65} - 25 q^{67} - 53 q^{68} + 51 q^{70} - 7 q^{71} + 13 q^{73} - 11 q^{74} + 12 q^{76} - 19 q^{77} - 23 q^{79} + 14 q^{80} + 26 q^{82} - 26 q^{83} + 15 q^{85} - 5 q^{86} + 25 q^{88} - 13 q^{89} - 40 q^{91} + 32 q^{92} - 19 q^{94} + 40 q^{95} - 5 q^{97} + 11 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 + 6 * q^4 - 8 * q^5 - 3 * q^7 - 3 * q^8 - 5 * q^10 - q^13 + 4 * q^14 - 8 * q^16 - 24 * q^17 - 5 * q^19 - 11 * q^20 - 9 * q^22 + q^23 + 7 * q^25 + 4 * q^26 - 26 * q^28 + 7 * q^29 - 8 * q^31 + 2 * q^32 - q^34 - 4 * q^35 - 6 * q^37 - 29 * q^38 - 3 * q^40 - 14 * q^41 - q^43 + 21 * q^44 - 3 * q^46 - 25 * q^47 - 10 * q^50 + 6 * q^52 - 29 * q^53 - 23 * q^55 - 9 * q^56 - 22 * q^58 + 12 * q^59 + 7 * q^61 - 4 * q^62 - 3 * q^64 - 3 * q^65 - 25 * q^67 - 53 * q^68 + 51 * q^70 - 7 * q^71 + 13 * q^73 - 11 * q^74 + 12 * q^76 - 19 * q^77 - 23 * q^79 + 14 * q^80 + 26 * q^82 - 26 * q^83 + 15 * q^85 - 5 * q^86 + 25 * q^88 - 13 * q^89 - 40 * q^91 + 32 * q^92 - 19 * q^94 + 40 * q^95 - 5 * q^97 + 11 * q^98

Introduction

L-functions

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