LMFDB - Embedded newform 1143.2.a.h.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:	$0$
Dimension:	$5$
Coefficient field:	5.5.81509.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 3x^{2} + 5x - 2 $ x^5 - x^4 - 5x^3 + 3x^2 + 5*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.26835$ 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)

$f(q)$	$=$	$q+2.26835 q^{2} +3.14543 q^{4} +2.63584 q^{5} +1.51754 q^{7} +2.59823 q^{8} +O(q^{10})$	\(q+2.26835 q^{2} +3.14543 q^{4} +2.63584 q^{5} +1.51754 q^{7} +2.59823 q^{8} +5.97902 q^{10} +0.0104875 q^{11} +1.66550 q^{13} +3.44232 q^{14} -0.397144 q^{16} -2.10782 q^{17} -2.77682 q^{19} +8.29085 q^{20} +0.0237894 q^{22} -2.07816 q^{23} +1.94767 q^{25} +3.77794 q^{26} +4.77331 q^{28} +7.41840 q^{29} -4.87819 q^{31} -6.09733 q^{32} -4.78127 q^{34} +4.00000 q^{35} -0.282175 q^{37} -6.29881 q^{38} +6.84854 q^{40} +8.33967 q^{41} -10.8276 q^{43} +0.0329877 q^{44} -4.71401 q^{46} -0.613170 q^{47} -4.69707 q^{49} +4.41801 q^{50} +5.23870 q^{52} +4.98377 q^{53} +0.0276435 q^{55} +3.94292 q^{56} +16.8276 q^{58} -1.48438 q^{59} -3.22612 q^{61} -11.0655 q^{62} -13.0366 q^{64} +4.38999 q^{65} +1.18980 q^{67} -6.62998 q^{68} +9.07341 q^{70} +8.34861 q^{71} +14.7658 q^{73} -0.640073 q^{74} -8.73429 q^{76} +0.0159152 q^{77} +2.10307 q^{79} -1.04681 q^{80} +18.9173 q^{82} +1.86476 q^{83} -5.55587 q^{85} -24.5607 q^{86} +0.0272490 q^{88} -15.7635 q^{89} +2.52746 q^{91} -6.53671 q^{92} -1.39089 q^{94} -7.31927 q^{95} -11.3911 q^{97} -10.6546 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} + q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) $ 5 * q + q^2 + q^4 + 5 * q^5 + 3 * q^8	$ 5 q + q^{2} + q^{4} + 5 q^{5} + 3 q^{8} - 2 q^{10} + 16 q^{11} + 3 q^{13} + 6 q^{14} - 7 q^{16} + 6 q^{17} - 8 q^{19} + 12 q^{20} - 8 q^{22} + 9 q^{23} + 4 q^{25} + 2 q^{26} + 2 q^{28} + 17 q^{29} - 9 q^{31} + 2 q^{32} - q^{34} + 20 q^{35} - q^{37} - q^{38} + 16 q^{40} + 2 q^{41} - 4 q^{43} - 3 q^{44} + 4 q^{46} + 8 q^{47} + 13 q^{49} - 15 q^{50} + 13 q^{52} + 15 q^{53} + 20 q^{55} - 8 q^{56} + 34 q^{58} + 19 q^{59} + q^{61} - 15 q^{62} + q^{64} + 5 q^{65} - 2 q^{67} - 14 q^{68} + 4 q^{70} + 13 q^{73} + 17 q^{74} + 8 q^{76} - 2 q^{77} - 28 q^{79} - 12 q^{80} + 30 q^{82} + q^{83} + 6 q^{85} - 32 q^{86} + 3 q^{88} - q^{89} - 18 q^{91} - 12 q^{92} + 16 q^{94} - 4 q^{95} + 28 q^{97} - 15 q^{98}+O(q^{100}) $ 5 * q + q^2 + q^4 + 5 * q^5 + 3 * q^8 - 2 * q^10 + 16 * q^11 + 3 * q^13 + 6 * q^14 - 7 * q^16 + 6 * q^17 - 8 * q^19 + 12 * q^20 - 8 * q^22 + 9 * q^23 + 4 * q^25 + 2 * q^26 + 2 * q^28 + 17 * q^29 - 9 * q^31 + 2 * q^32 - q^34 + 20 * q^35 - q^37 - q^38 + 16 * q^40 + 2 * q^41 - 4 * q^43 - 3 * q^44 + 4 * q^46 + 8 * q^47 + 13 * q^49 - 15 * q^50 + 13 * q^52 + 15 * q^53 + 20 * q^55 - 8 * q^56 + 34 * q^58 + 19 * q^59 + q^61 - 15 * q^62 + q^64 + 5 * q^65 - 2 * q^67 - 14 * q^68 + 4 * q^70 + 13 * q^73 + 17 * q^74 + 8 * q^76 - 2 * q^77 - 28 * q^79 - 12 * q^80 + 30 * q^82 + q^83 + 6 * q^85 - 32 * q^86 + 3 * q^88 - q^89 - 18 * q^91 - 12 * q^92 + 16 * q^94 - 4 * q^95 + 28 * q^97 - 15 * 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 $

$2$	2.26835	1.60397	0.801984	−	0.597346i	$-0.203778\pi$
$2$	2.26835	1.60397	0.801984	+	0.597346i	$0.203778\pi$
$3$	0	0
$3$	0	0
$4$	3.14543	1.57271
$5$	2.63584	1.17879	0.589393	−	0.807847i	$-0.299367\pi$
$5$	2.63584	1.17879	0.589393	+	0.807847i	$0.299367\pi$
$6$	0	0
$7$	1.51754	0.573576	0.286788	−	0.957994i	$-0.407412\pi$
$7$	1.51754	0.573576	0.286788	+	0.957994i	$0.407412\pi$
$8$	2.59823	0.918614
$9$	0	0
$10$	5.97902	1.89073
$11$	0.0104875	0.00316211	0.00158105	−	0.999999i	$-0.499497\pi$
$11$	0.0104875	0.00316211	0.00158105	+	0.999999i	$0.499497\pi$
$12$	0	0
$13$	1.66550	0.461926	0.230963	−	0.972963i	$-0.425812\pi$
$13$	1.66550	0.461926	0.230963	+	0.972963i	$0.425812\pi$
$14$	3.44232	0.919998
$15$	0	0
$16$	−0.397144	−0.0992860
$17$	−2.10782	−0.511220	−0.255610	−	0.966780i	$-0.582276\pi$
$17$	−2.10782	−0.511220	−0.255610	+	0.966780i	$0.582276\pi$
$18$	0	0
$19$	−2.77682	−0.637046	−0.318523	−	0.947915i	$-0.603187\pi$
$19$	−2.77682	−0.637046	−0.318523	+	0.947915i	$0.603187\pi$
$20$	8.29085	1.85389
$21$	0	0
$22$	0.0237894	0.00507192
$23$	−2.07816	−0.433327	−0.216663	−	0.976246i	$-0.569517\pi$
$23$	−2.07816	−0.433327	−0.216663	+	0.976246i	$0.569517\pi$
$24$	0	0
$25$	1.94767	0.389534
$26$	3.77794	0.740914
$27$	0	0
$28$	4.77331	0.902071
$29$	7.41840	1.37756	0.688781	−	0.724969i	$-0.258146\pi$
$29$	7.41840	1.37756	0.688781	+	0.724969i	$0.258146\pi$
$30$	0	0
$31$	−4.87819	−0.876149	−0.438074	−	0.898939i	$-0.644339\pi$
$31$	−4.87819	−0.876149	−0.438074	+	0.898939i	$0.644339\pi$
$32$	−6.09733	−1.07787
$33$	0	0
$34$	−4.78127	−0.819981
$35$	4.00000	0.676123
$36$	0	0
$37$	−0.282175	−0.0463893	−0.0231946	−	0.999731i	$-0.507384\pi$
$37$	−0.282175	−0.0463893	−0.0231946	+	0.999731i	$0.507384\pi$
$38$	−6.29881	−1.02180
$39$	0	0
$40$	6.84854	1.08285
$41$	8.33967	1.30244	0.651219	−	0.758890i	$-0.274258\pi$
$41$	8.33967	1.30244	0.651219	+	0.758890i	$0.274258\pi$
$42$	0	0
$43$	−10.8276	−1.65119	−0.825594	−	0.564265i	$-0.809160\pi$
$43$	−10.8276	−1.65119	−0.825594	+	0.564265i	$0.809160\pi$
$44$	0.0329877	0.00497309
$45$	0	0
$46$	−4.71401	−0.695042
$47$	−0.613170	−0.0894400	−0.0447200	−	0.999000i	$-0.514240\pi$
$47$	−0.613170	−0.0894400	−0.0447200	+	0.999000i	$0.514240\pi$
$48$	0	0
$49$	−4.69707	−0.671010
$50$	4.41801	0.624801
$51$	0	0
$52$	5.23870	0.726477
$53$	4.98377	0.684574	0.342287	−	0.939595i	$-0.388798\pi$
$53$	4.98377	0.684574	0.342287	+	0.939595i	$0.388798\pi$
$54$	0	0
$55$	0.0276435	0.00372745
$56$	3.94292	0.526895
$57$	0	0
$58$	16.8276	2.20957
$59$	−1.48438	−0.193250	−0.0966248	−	0.995321i	$-0.530805\pi$
$59$	−1.48438	−0.193250	−0.0966248	+	0.995321i	$0.530805\pi$
$60$	0	0
$61$	−3.22612	−0.413062	−0.206531	−	0.978440i	$-0.566217\pi$
$61$	−3.22612	−0.413062	−0.206531	+	0.978440i	$0.566217\pi$
$62$	−11.0655	−1.40531
$63$	0	0
$64$	−13.0366	−1.62958
$65$	4.38999	0.544511
$66$	0	0
$67$	1.18980	0.145357	0.0726784	−	0.997355i	$-0.476845\pi$
$67$	1.18980	0.145357	0.0726784	+	0.997355i	$0.476845\pi$
$68$	−6.62998	−0.804003
$69$	0	0
$70$	9.07341	1.08448
$71$	8.34861	0.990798	0.495399	−	0.868666i	$-0.335022\pi$
$71$	8.34861	0.990798	0.495399	+	0.868666i	$0.335022\pi$
$72$	0	0
$73$	14.7658	1.72820	0.864101	−	0.503318i	$-0.167888\pi$
$73$	14.7658	1.72820	0.864101	+	0.503318i	$0.167888\pi$
$74$	−0.640073	−0.0744069
$75$	0	0
$76$	−8.73429	−1.00189
$77$	0.0159152	0.00181371
$78$	0	0
$79$	2.10307	0.236614	0.118307	−	0.992977i	$-0.462253\pi$
$79$	2.10307	0.236614	0.118307	+	0.992977i	$0.462253\pi$
$80$	−1.04681	−0.117037
$81$	0	0
$82$	18.9173	2.08907
$83$	1.86476	0.204684	0.102342	−	0.994749i	$-0.467366\pi$
$83$	1.86476	0.204684	0.102342	+	0.994749i	$0.467366\pi$
$84$	0	0
$85$	−5.55587	−0.602619
$86$	−24.5607	−2.64845
$87$	0	0
$88$	0.0272490	0.00290476
$89$	−15.7635	−1.67093	−0.835464	−	0.549545i	$-0.814801\pi$
$89$	−15.7635	−1.67093	−0.835464	+	0.549545i	$0.814801\pi$
$90$	0	0
$91$	2.52746	0.264950
$92$	−6.53671	−0.681499
$93$	0	0
$94$	−1.39089	−0.143459
$95$	−7.31927	−0.750941
$96$	0	0
$97$	−11.3911	−1.15659	−0.578297	−	0.815827i	$-0.696282\pi$
$97$	−11.3911	−1.15659	−0.578297	+	0.815827i	$0.696282\pi$
$98$	−10.6546	−1.07628
$99$	0	0
$100$	6.12626	0.612626
$101$	−7.16330	−0.712775	−0.356388	−	0.934338i	$-0.615992\pi$
$101$	−7.16330	−0.712775	−0.356388	+	0.934338i	$0.615992\pi$
$102$	0	0
$103$	2.34669	0.231226	0.115613	−	0.993294i	$-0.463117\pi$
$103$	2.34669	0.231226	0.115613	+	0.993294i	$0.463117\pi$
$104$	4.32735	0.424332
$105$	0	0
$106$	11.3050	1.09803
$107$	16.3301	1.57869	0.789346	−	0.613949i	$-0.210420\pi$
$107$	16.3301	1.57869	0.789346	+	0.613949i	$0.210420\pi$
$108$	0	0
$109$	4.23155	0.405309	0.202654	−	0.979250i	$-0.435043\pi$
$109$	4.23155	0.405309	0.202654	+	0.979250i	$0.435043\pi$
$110$	0.0627052	0.00597870
$111$	0	0
$112$	−0.602682	−0.0569481
$113$	0.112564	0.0105892	0.00529459	−	0.999986i	$-0.498315\pi$
$113$	0.112564	0.0105892	0.00529459	+	0.999986i	$0.498315\pi$
$114$	0	0
$115$	−5.47771	−0.510799
$116$	23.3340	2.16651
$117$	0	0
$118$	−3.36710	−0.309966
$119$	−3.19870	−0.293224
$120$	0	0
$121$	−10.9999	−0.999990
$122$	−7.31798	−0.662539
$123$	0	0
$124$	−15.3440	−1.37793
$125$	−8.04546	−0.719608
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−17.3770	−1.53592
$129$	0	0
$130$	9.95805	0.873379
$131$	6.66199	0.582061	0.291030	−	0.956714i	$-0.406002\pi$
$131$	6.66199	0.582061	0.291030	+	0.956714i	$0.406002\pi$
$132$	0	0
$133$	−4.21394	−0.365395
$134$	2.69888	0.233148
$135$	0	0
$136$	−5.47660	−0.469614
$137$	−9.11324	−0.778597	−0.389298	−	0.921112i	$-0.627283\pi$
$137$	−9.11324	−0.778597	−0.389298	+	0.921112i	$0.627283\pi$
$138$	0	0
$139$	−15.9647	−1.35411	−0.677055	−	0.735933i	$-0.736744\pi$
$139$	−15.9647	−1.35411	−0.677055	+	0.735933i	$0.736744\pi$
$140$	12.5817	1.06335
$141$	0	0
$142$	18.9376	1.58921
$143$	0.0174669	0.00146066
$144$	0	0
$145$	19.5538	1.62385
$146$	33.4940	2.77198
$147$	0	0
$148$	−0.887561	−0.0729571
$149$	12.2055	0.999910	0.499955	−	0.866051i	$-0.333350\pi$
$149$	12.2055	0.999910	0.499955	+	0.866051i	$0.333350\pi$
$150$	0	0
$151$	−13.5603	−1.10352	−0.551761	−	0.834002i	$-0.686044\pi$
$151$	−13.5603	−1.10352	−0.551761	+	0.834002i	$0.686044\pi$
$152$	−7.21483	−0.585200
$153$	0	0
$154$	0.0361014	0.00290913
$155$	−12.8581	−1.03279
$156$	0	0
$157$	2.06979	0.165188	0.0825938	−	0.996583i	$-0.473680\pi$
$157$	2.06979	0.165188	0.0825938	+	0.996583i	$0.473680\pi$
$158$	4.77050	0.379520
$159$	0	0
$160$	−16.0716	−1.27057
$161$	−3.15370	−0.248546
$162$	0	0
$163$	−18.2821	−1.43196	−0.715981	−	0.698120i	$-0.754020\pi$
$163$	−18.2821	−1.43196	−0.715981	+	0.698120i	$0.754020\pi$
$164$	26.2318	2.04836
$165$	0	0
$166$	4.22994	0.328307
$167$	4.51573	0.349438	0.174719	−	0.984618i	$-0.444098\pi$
$167$	4.51573	0.349438	0.174719	+	0.984618i	$0.444098\pi$
$168$	0	0
$169$	−10.2261	−0.786625
$170$	−12.6027	−0.966582
$171$	0	0
$172$	−34.0573	−2.59684
$173$	23.3701	1.77680	0.888400	−	0.459071i	$-0.151818\pi$
$173$	23.3701	1.77680	0.888400	+	0.459071i	$0.151818\pi$
$174$	0	0
$175$	2.95567	0.223428
$176$	−0.00416506	−0.000313953 0
$177$	0	0
$178$	−35.7572	−2.68012
$179$	−7.71794	−0.576866	−0.288433	−	0.957500i	$-0.593134\pi$
$179$	−7.71794	−0.576866	−0.288433	+	0.957500i	$0.593134\pi$
$180$	0	0
$181$	17.5777	1.30654	0.653269	−	0.757126i	$-0.273397\pi$
$181$	17.5777	1.30654	0.653269	+	0.757126i	$0.273397\pi$
$182$	5.73317	0.424971
$183$	0	0
$184$	−5.39955	−0.398060
$185$	−0.743769	−0.0546830
$186$	0	0
$187$	−0.0221058	−0.00161653
$188$	−1.92868	−0.140663
$189$	0	0
$190$	−16.6027	−1.20449
$191$	17.6551	1.27748	0.638740	−	0.769423i	$-0.279456\pi$
$191$	17.6551	1.27748	0.638740	+	0.769423i	$0.279456\pi$
$192$	0	0
$193$	17.4512	1.25617	0.628083	−	0.778146i	$-0.283840\pi$
$193$	17.4512	1.25617	0.628083	+	0.778146i	$0.283840\pi$
$194$	−25.8391	−1.85514
$195$	0	0
$196$	−14.7743	−1.05531
$197$	−23.0847	−1.64472	−0.822359	−	0.568969i	$-0.807343\pi$
$197$	−23.0847	−1.64472	−0.822359	+	0.568969i	$0.807343\pi$
$198$	0	0
$199$	10.6745	0.756698	0.378349	−	0.925663i	$-0.376492\pi$
$199$	10.6745	0.756698	0.378349	+	0.925663i	$0.376492\pi$
$200$	5.06051	0.357832
$201$	0	0
$202$	−16.2489	−1.14327
$203$	11.2577	0.790138
$204$	0	0
$205$	21.9821	1.53529
$206$	5.32312	0.370879
$207$	0	0
$208$	−0.661442	−0.0458628
$209$	−0.0291220	−0.00201441
$210$	0	0
$211$	−3.99830	−0.275254	−0.137627	−	0.990484i	$-0.543948\pi$
$211$	−3.99830	−0.275254	−0.137627	+	0.990484i	$0.543948\pi$
$212$	15.6761	1.07664
$213$	0	0
$214$	37.0425	2.53217
$215$	−28.5398	−1.94640
$216$	0	0
$217$	−7.40285	−0.502538
$218$	9.59864	0.650102
$219$	0	0
$220$	0.0869505	0.00586220
$221$	−3.51056	−0.236146
$222$	0	0
$223$	−11.5449	−0.773106	−0.386553	−	0.922267i	$-0.626334\pi$
$223$	−11.5449	−0.773106	−0.386553	+	0.922267i	$0.626334\pi$
$224$	−9.25294	−0.618238
$225$	0	0
$226$	0.255336	0.0169847
$227$	8.43126	0.559603	0.279801	−	0.960058i	$-0.409731\pi$
$227$	8.43126	0.559603	0.279801	+	0.960058i	$0.409731\pi$
$228$	0	0
$229$	19.3760	1.28040	0.640201	−	0.768208i	$-0.278851\pi$
$229$	19.3760	1.28040	0.640201	+	0.768208i	$0.278851\pi$
$230$	−12.4254	−0.819305
$231$	0	0
$232$	19.2747	1.26545
$233$	−14.3938	−0.942971	−0.471486	−	0.881874i	$-0.656282\pi$
$233$	−14.3938	−0.942971	−0.471486	+	0.881874i	$0.656282\pi$
$234$	0	0
$235$	−1.61622	−0.105431
$236$	−4.66901	−0.303926
$237$	0	0
$238$	−7.25577	−0.470322
$239$	−13.1474	−0.850434	−0.425217	−	0.905091i	$-0.639802\pi$
$239$	−13.1474	−0.850434	−0.425217	+	0.905091i	$0.639802\pi$
$240$	0	0
$241$	5.62468	0.362317	0.181159	−	0.983454i	$-0.442015\pi$
$241$	5.62468	0.362317	0.181159	+	0.983454i	$0.442015\pi$
$242$	−24.9516	−1.60395
$243$	0	0
$244$	−10.1475	−0.649628
$245$	−12.3807	−0.790977
$246$	0	0
$247$	−4.62479	−0.294268
$248$	−12.6747	−0.804842
$249$	0	0
$250$	−18.2499	−1.15423
$251$	0.560879	0.0354024	0.0177012	−	0.999843i	$-0.494365\pi$
$251$	0.560879	0.0354024	0.0177012	+	0.999843i	$0.494365\pi$
$252$	0	0
$253$	−0.0217948	−0.00137023
$254$	−2.26835	−0.142329
$255$	0	0
$256$	−13.3439	−0.833994
$257$	−25.2551	−1.57537	−0.787685	−	0.616079i	$-0.788720\pi$
$257$	−25.2551	−1.57537	−0.787685	+	0.616079i	$0.788720\pi$
$258$	0	0
$259$	−0.428212	−0.0266078
$260$	13.8084	0.856360
$261$	0	0
$262$	15.1117	0.933607
$263$	2.36958	0.146115	0.0730574	−	0.997328i	$-0.476724\pi$
$263$	2.36958	0.146115	0.0730574	+	0.997328i	$0.476724\pi$
$264$	0	0
$265$	13.1364	0.806966
$266$	−9.55870	−0.586082
$267$	0	0
$268$	3.74242	0.228605
$269$	−2.34194	−0.142791	−0.0713953	−	0.997448i	$-0.522745\pi$
$269$	−2.34194	−0.142791	−0.0713953	+	0.997448i	$0.522745\pi$
$270$	0	0
$271$	6.17357	0.375018	0.187509	−	0.982263i	$-0.439959\pi$
$271$	6.17357	0.375018	0.187509	+	0.982263i	$0.439959\pi$
$272$	0.837106	0.0507570
$273$	0	0
$274$	−20.6721	−1.24884
$275$	0.0204263	0.00123175
$276$	0	0
$277$	7.16294	0.430379	0.215190	−	0.976572i	$-0.430963\pi$
$277$	7.16294	0.430379	0.215190	+	0.976572i	$0.430963\pi$
$278$	−36.2136	−2.17195
$279$	0	0
$280$	10.3929	0.621096
$281$	23.7997	1.41977	0.709886	−	0.704317i	$-0.248746\pi$
$281$	23.7997	1.41977	0.709886	+	0.704317i	$0.248746\pi$
$282$	0	0
$283$	−29.8299	−1.77321	−0.886603	−	0.462531i	$-0.846941\pi$
$283$	−29.8299	−1.77321	−0.886603	+	0.462531i	$0.846941\pi$
$284$	26.2599	1.55824
$285$	0	0
$286$	0.0396212	0.00234285
$287$	12.6558	0.747048
$288$	0	0
$289$	−12.5571	−0.738654
$290$	44.3548	2.60460
$291$	0	0
$292$	46.4446	2.71797
$293$	14.6787	0.857540	0.428770	−	0.903414i	$-0.358947\pi$
$293$	14.6787	0.857540	0.428770	+	0.903414i	$0.358947\pi$
$294$	0	0
$295$	−3.91259	−0.227800
$296$	−0.733156	−0.0426139
$297$	0	0
$298$	27.6863	1.60382
$299$	−3.46117	−0.200165
$300$	0	0
$301$	−16.4313	−0.947082
$302$	−30.7596	−1.77001
$303$	0	0
$304$	1.10280	0.0632498
$305$	−8.50354	−0.486912
$306$	0	0
$307$	21.9415	1.25227	0.626135	−	0.779715i	$-0.284636\pi$
$307$	21.9415	1.25227	0.626135	+	0.779715i	$0.284636\pi$
$308$	0.0500602	0.00285245
$309$	0	0
$310$	−29.1668	−1.65656
$311$	21.5563	1.22235	0.611174	−	0.791497i	$-0.290698\pi$
$311$	21.5563	1.22235	0.611174	+	0.791497i	$0.290698\pi$
$312$	0	0
$313$	7.85453	0.443964	0.221982	−	0.975051i	$-0.428747\pi$
$313$	7.85453	0.443964	0.221982	+	0.975051i	$0.428747\pi$
$314$	4.69503	0.264956
$315$	0	0
$316$	6.61504	0.372125
$317$	25.5438	1.43468	0.717341	−	0.696722i	$-0.245359\pi$
$317$	25.5438	1.43468	0.717341	+	0.696722i	$0.245359\pi$
$318$	0	0
$319$	0.0778007	0.00435600
$320$	−34.3625	−1.92092
$321$	0	0
$322$	−7.15370	−0.398660
$323$	5.85303	0.325671
$324$	0	0
$325$	3.24384	0.179936
$326$	−41.4702	−2.29682
$327$	0	0
$328$	21.6684	1.19644
$329$	−0.930510	−0.0513007
$330$	0	0
$331$	−13.0625	−0.717979	−0.358989	−	0.933342i	$-0.616879\pi$
$331$	−13.0625	−0.717979	−0.358989	+	0.933342i	$0.616879\pi$
$332$	5.86547	0.321910
$333$	0	0
$334$	10.2433	0.560487
$335$	3.13612	0.171344
$336$	0	0
$337$	24.6134	1.34078	0.670388	−	0.742011i	$-0.266128\pi$
$337$	24.6134	1.34078	0.670388	+	0.742011i	$0.266128\pi$
$338$	−23.1965	−1.26172
$339$	0	0
$340$	−17.4756	−0.947747
$341$	−0.0511601	−0.00277048
$342$	0	0
$343$	−17.7508	−0.958452
$344$	−28.1325	−1.51680
$345$	0	0
$346$	53.0117	2.84993
$347$	−24.7052	−1.32625	−0.663123	−	0.748511i	$-0.730769\pi$
$347$	−24.7052	−1.32625	−0.663123	+	0.748511i	$0.730769\pi$
$348$	0	0
$349$	32.9587	1.76424	0.882120	−	0.471025i	$-0.156116\pi$
$349$	32.9587	1.76424	0.882120	+	0.471025i	$0.156116\pi$
$350$	6.70451	0.358371
$351$	0	0
$352$	−0.0639459	−0.00340833
$353$	7.48664	0.398474	0.199237	−	0.979951i	$-0.436154\pi$
$353$	7.48664	0.398474	0.199237	+	0.979951i	$0.436154\pi$
$354$	0	0
$355$	22.0056	1.16794
$356$	−49.5829	−2.62789
$357$	0	0
$358$	−17.5070	−0.925274
$359$	−29.0163	−1.53142	−0.765712	−	0.643184i	$-0.777613\pi$
$359$	−29.0163	−1.53142	−0.765712	+	0.643184i	$0.777613\pi$
$360$	0	0
$361$	−11.2893	−0.594172
$362$	39.8724	2.09564
$363$	0	0
$364$	7.94994	0.416690
$365$	38.9203	2.03718
$366$	0	0
$367$	29.7552	1.55321	0.776604	−	0.629989i	$-0.216941\pi$
$367$	29.7552	1.55321	0.776604	+	0.629989i	$0.216941\pi$
$368$	0.825330	0.0430233
$369$	0	0
$370$	−1.68713	−0.0877098
$371$	7.56308	0.392656
$372$	0	0
$373$	4.72350	0.244574	0.122287	−	0.992495i	$-0.460977\pi$
$373$	4.72350	0.244574	0.122287	+	0.992495i	$0.460977\pi$
$374$	−0.0501437	−0.00259287
$375$	0	0
$376$	−1.59316	−0.0821608
$377$	12.3553	0.636332
$378$	0	0
$379$	0.240446	0.0123509	0.00617543	−	0.999981i	$-0.498034\pi$
$379$	0.240446	0.0123509	0.00617543	+	0.999981i	$0.498034\pi$
$380$	−23.0222	−1.18101
$381$	0	0
$382$	40.0481	2.04904
$383$	−6.77614	−0.346245	−0.173122	−	0.984900i	$-0.555386\pi$
$383$	−6.77614	−0.346245	−0.173122	+	0.984900i	$0.555386\pi$
$384$	0	0
$385$	0.0419501	0.00213798
$386$	39.5855	2.01485
$387$	0	0
$388$	−35.8299	−1.81899
$389$	−22.6579	−1.14880	−0.574401	−	0.818574i	$-0.694765\pi$
$389$	−22.6579	−1.14880	−0.574401	+	0.818574i	$0.694765\pi$
$390$	0	0
$391$	4.38038	0.221525
$392$	−12.2041	−0.616399
$393$	0	0
$394$	−52.3643	−2.63808
$395$	5.54336	0.278916
$396$	0	0
$397$	26.5016	1.33008	0.665040	−	0.746808i	$-0.268415\pi$
$397$	26.5016	1.33008	0.665040	+	0.746808i	$0.268415\pi$
$398$	24.2136	1.21372
$399$	0	0
$400$	−0.773506	−0.0386753
$401$	24.8855	1.24272	0.621362	−	0.783523i	$-0.286580\pi$
$401$	24.8855	1.24272	0.621362	+	0.783523i	$0.286580\pi$
$402$	0	0
$403$	−8.12461	−0.404716
$404$	−22.5316	−1.12099
$405$	0	0
$406$	25.5365	1.26736
$407$	−0.00295932	−0.000146688 0
$408$	0	0
$409$	37.5198	1.85523	0.927617	−	0.373532i	$-0.121853\pi$
$409$	37.5198	1.85523	0.927617	+	0.373532i	$0.121853\pi$
$410$	49.8631	2.46256
$411$	0	0
$412$	7.38134	0.363652
$413$	−2.25261	−0.110843
$414$	0	0
$415$	4.91522	0.241279
$416$	−10.1551	−0.497894
$417$	0	0
$418$	−0.0660589	−0.00323105
$419$	28.6887	1.40154	0.700768	−	0.713389i	$-0.252841\pi$
$419$	28.6887	1.40154	0.700768	+	0.713389i	$0.252841\pi$
$420$	0	0
$421$	19.1477	0.933200	0.466600	−	0.884468i	$-0.345479\pi$
$421$	19.1477	0.933200	0.466600	+	0.884468i	$0.345479\pi$
$422$	−9.06956	−0.441499
$423$	0	0
$424$	12.9490	0.628859
$425$	−4.10533	−0.199138
$426$	0	0
$427$	−4.89577	−0.236923
$428$	51.3652	2.48283
$429$	0	0
$430$	−64.7383	−3.12196
$431$	−1.55463	−0.0748840	−0.0374420	−	0.999299i	$-0.511921\pi$
$431$	−1.55463	−0.0748840	−0.0374420	+	0.999299i	$0.511921\pi$
$432$	0	0
$433$	−30.9521	−1.48746	−0.743732	−	0.668478i	$-0.766946\pi$
$433$	−30.9521	−1.48746	−0.743732	+	0.668478i	$0.766946\pi$
$434$	−16.7923	−0.806055
$435$	0	0
$436$	13.3100	0.637434
$437$	5.77068	0.276049
$438$	0	0
$439$	9.73798	0.464768	0.232384	−	0.972624i	$-0.425347\pi$
$439$	9.73798	0.464768	0.232384	+	0.972624i	$0.425347\pi$
$440$	0.0718242	0.00342408
$441$	0	0
$442$	−7.96319	−0.378770
$443$	32.5572	1.54684	0.773420	−	0.633894i	$-0.218544\pi$
$443$	32.5572	1.54684	0.773420	+	0.633894i	$0.218544\pi$
$444$	0	0
$445$	−41.5501	−1.96967
$446$	−26.1880	−1.24004
$447$	0	0
$448$	−19.7836	−0.934686
$449$	24.6255	1.16215	0.581074	−	0.813851i	$-0.302633\pi$
$449$	24.6255	1.16215	0.581074	+	0.813851i	$0.302633\pi$
$450$	0	0
$451$	0.0874625	0.00411845
$452$	0.354063	0.0166537
$453$	0	0
$454$	19.1251	0.897585
$455$	6.66199	0.312319
$456$	0	0
$457$	17.8775	0.836275	0.418137	−	0.908384i	$-0.362683\pi$
$457$	17.8775	0.836275	0.418137	+	0.908384i	$0.362683\pi$
$458$	43.9516	2.05372
$459$	0	0
$460$	−17.2297	−0.803341
$461$	19.8380	0.923949	0.461974	−	0.886893i	$-0.347141\pi$
$461$	19.8380	0.923949	0.461974	+	0.886893i	$0.347141\pi$
$462$	0	0
$463$	27.8847	1.29591	0.647955	−	0.761679i	$-0.275624\pi$
$463$	27.8847	1.29591	0.647955	+	0.761679i	$0.275624\pi$
$464$	−2.94617	−0.136773
$465$	0	0
$466$	−32.6503	−1.51250
$467$	−13.8178	−0.639409	−0.319705	−	0.947517i	$-0.603584\pi$
$467$	−13.8178	−0.639409	−0.319705	+	0.947517i	$0.603584\pi$
$468$	0	0
$469$	1.80556	0.0833732
$470$	−3.66616	−0.169107
$471$	0	0
$472$	−3.85676	−0.177522
$473$	−0.113554	−0.00522123
$474$	0	0
$475$	−5.40834	−0.248152
$476$	−10.0613	−0.461157
$477$	0	0
$478$	−29.8229	−1.36407
$479$	−21.1396	−0.965893	−0.482946	−	0.875650i	$-0.660433\pi$
$479$	−21.1396	−0.965893	−0.482946	+	0.875650i	$0.660433\pi$
$480$	0	0
$481$	−0.469962	−0.0214284
$482$	12.7588	0.581145
$483$	0	0
$484$	−34.5993	−1.57270
$485$	−30.0252	−1.36337
$486$	0	0
$487$	−21.7539	−0.985763	−0.492882	−	0.870096i	$-0.664056\pi$
$487$	−21.7539	−0.985763	−0.492882	+	0.870096i	$0.664056\pi$
$488$	−8.38221	−0.379445
$489$	0	0
$490$	−28.0839	−1.26870
$491$	18.3359	0.827486	0.413743	−	0.910394i	$-0.364221\pi$
$491$	18.3359	0.827486	0.413743	+	0.910394i	$0.364221\pi$
$492$	0	0
$493$	−15.6366	−0.704238
$494$	−10.4907	−0.471997
$495$	0	0
$496$	1.93734	0.0869893
$497$	12.6694	0.568298
$498$	0	0
$499$	−6.88268	−0.308111	−0.154056	−	0.988062i	$-0.549233\pi$
$499$	−6.88268	−0.308111	−0.154056	+	0.988062i	$0.549233\pi$
$500$	−25.3064	−1.13174
$501$	0	0
$502$	1.27227	0.0567842
$503$	−33.5050	−1.49392	−0.746958	−	0.664871i	$-0.768486\pi$
$503$	−33.5050	−1.49392	−0.746958	+	0.664871i	$0.768486\pi$
$504$	0	0
$505$	−18.8813	−0.840209
$506$	−0.0494383	−0.00219780
$507$	0	0
$508$	−3.14543	−0.139556
$509$	−0.0324525	−0.00143843	−0.000719215	−	1.00000i	$-0.500229\pi$
$509$	−0.0324525	−0.00143843	−0.000719215		1.00000i	$0.500229\pi$
$510$	0	0
$511$	22.4076	0.991256
$512$	4.48526	0.198222
$513$	0	0
$514$	−57.2875	−2.52684
$515$	6.18551	0.272566
$516$	0	0
$517$	−0.00643063	−0.000282819 0
$518$	−0.971336	−0.0426781
$519$	0	0
$520$	11.4062	0.500196
$521$	11.6901	0.512151	0.256076	−	0.966657i	$-0.417570\pi$
$521$	11.6901	0.512151	0.256076	+	0.966657i	$0.417570\pi$
$522$	0	0
$523$	−23.2907	−1.01843	−0.509217	−	0.860638i	$-0.670065\pi$
$523$	−23.2907	−1.01843	−0.509217	+	0.860638i	$0.670065\pi$
$524$	20.9548	0.915415
$525$	0	0
$526$	5.37505	0.234363
$527$	10.2823	0.447905
$528$	0	0
$529$	−18.6812	−0.812228
$530$	29.7981	1.29435
$531$	0	0
$532$	−13.2546	−0.574661
$533$	13.8897	0.601630
$534$	0	0
$535$	43.0436	1.86094
$536$	3.09137	0.133527
$537$	0	0
$538$	−5.31235	−0.229032
$539$	−0.0492607	−0.00212181
$540$	0	0
$541$	14.7322	0.633385	0.316692	−	0.948528i	$-0.397428\pi$
$541$	14.7322	0.633385	0.316692	+	0.948528i	$0.397428\pi$
$542$	14.0038	0.601516
$543$	0	0
$544$	12.8520	0.551027
$545$	11.1537	0.477772
$546$	0	0
$547$	−11.3142	−0.483761	−0.241880	−	0.970306i	$-0.577764\pi$
$547$	−11.3142	−0.483761	−0.241880	+	0.970306i	$0.577764\pi$
$548$	−28.6650	−1.22451
$549$	0	0
$550$	0.0463340	0.00197569
$551$	−20.5996	−0.877572
$552$	0	0
$553$	3.19149	0.135716
$554$	16.2481	0.690315
$555$	0	0
$556$	−50.2159	−2.12963
$557$	11.1052	0.470544	0.235272	−	0.971930i	$-0.424402\pi$
$557$	11.1052	0.470544	0.235272	+	0.971930i	$0.424402\pi$
$558$	0	0
$559$	−18.0333	−0.762726
$560$	−1.58858	−0.0671296
$561$	0	0
$562$	53.9862	2.27727
$563$	0.683837	0.0288203	0.0144101	−	0.999896i	$-0.495413\pi$
$563$	0.683837	0.0288203	0.0144101	+	0.999896i	$0.495413\pi$
$564$	0	0
$565$	0.296702	0.0124824
$566$	−67.6648	−2.84417
$567$	0	0
$568$	21.6916	0.910160
$569$	−27.5414	−1.15460	−0.577298	−	0.816533i	$-0.695893\pi$
$569$	−27.5414	−1.15460	−0.577298	+	0.816533i	$0.695893\pi$
$570$	0	0
$571$	0.602431	0.0252110	0.0126055	−	0.999921i	$-0.495987\pi$
$571$	0.602431	0.0252110	0.0126055	+	0.999921i	$0.495987\pi$
$572$	0.0549410	0.00229720
$573$	0	0
$574$	28.7078	1.19824
$575$	−4.04758	−0.168796
$576$	0	0
$577$	−8.43262	−0.351055	−0.175527	−	0.984475i	$-0.556163\pi$
$577$	−8.43262	−0.351055	−0.175527	+	0.984475i	$0.556163\pi$
$578$	−28.4840	−1.18478
$579$	0	0
$580$	61.5049	2.55385
$581$	2.82985	0.117402
$582$	0	0
$583$	0.0522675	0.00216470
$584$	38.3649	1.58755
$585$	0	0
$586$	33.2965	1.37547
$587$	−47.4432	−1.95819	−0.979096	−	0.203401i	$-0.934801\pi$
$587$	−47.4432	−1.95819	−0.979096	+	0.203401i	$0.934801\pi$
$588$	0	0
$589$	13.5459	0.558147
$590$	−8.87514	−0.365384
$591$	0	0
$592$	0.112064	0.00460581
$593$	−4.19118	−0.172111	−0.0860555	−	0.996290i	$-0.527426\pi$
$593$	−4.19118	−0.172111	−0.0860555	+	0.996290i	$0.527426\pi$
$594$	0	0
$595$	−8.43126	−0.345648
$596$	38.3914	1.57257
$597$	0	0
$598$	−7.85116	−0.321058
$599$	9.13975	0.373440	0.186720	−	0.982413i	$-0.440214\pi$
$599$	9.13975	0.373440	0.186720	+	0.982413i	$0.440214\pi$
$600$	0	0
$601$	34.8399	1.42115	0.710574	−	0.703623i	$-0.248435\pi$
$601$	34.8399	1.42115	0.710574	+	0.703623i	$0.248435\pi$
$602$	−37.2719	−1.51909
$603$	0	0
$604$	−42.6530	−1.73552
$605$	−28.9940	−1.17877
$606$	0	0
$607$	28.0143	1.13707	0.568533	−	0.822661i	$-0.307511\pi$
$607$	28.0143	1.13707	0.568533	+	0.822661i	$0.307511\pi$
$608$	16.9312	0.686650
$609$	0	0
$610$	−19.2890	−0.780991
$611$	−1.02123	−0.0413146
$612$	0	0
$613$	−24.1731	−0.976342	−0.488171	−	0.872748i	$-0.662336\pi$
$613$	−24.1731	−0.976342	−0.488171	+	0.872748i	$0.662336\pi$
$614$	49.7712	2.00860
$615$	0	0
$616$	0.0413515	0.00166610
$617$	11.8934	0.478810	0.239405	−	0.970920i	$-0.423048\pi$
$617$	11.8934	0.478810	0.239405	+	0.970920i	$0.423048\pi$
$618$	0	0
$619$	−20.6301	−0.829193	−0.414596	−	0.910005i	$-0.636077\pi$
$619$	−20.6301	−0.829193	−0.414596	+	0.910005i	$0.636077\pi$
$620$	−40.4443	−1.62428
$621$	0	0
$622$	48.8974	1.96061
$623$	−23.9218	−0.958405
$624$	0	0
$625$	−30.9449	−1.23780
$626$	17.8169	0.712105
$627$	0	0
$628$	6.51039	0.259793
$629$	0.594773	0.0237152
$630$	0	0
$631$	−4.13649	−0.164671	−0.0823354	−	0.996605i	$-0.526238\pi$
$631$	−4.13649	−0.164671	−0.0823354	+	0.996605i	$0.526238\pi$
$632$	5.46426	0.217356
$633$	0	0
$634$	57.9424	2.30118
$635$	−2.63584	−0.104600
$636$	0	0
$637$	−7.82296	−0.309957
$638$	0.176479	0.00698689
$639$	0	0
$640$	−45.8030	−1.81052
$641$	27.4147	1.08281	0.541407	−	0.840760i	$-0.317892\pi$
$641$	27.4147	1.08281	0.541407	+	0.840760i	$0.317892\pi$
$642$	0	0
$643$	38.0535	1.50068	0.750341	−	0.661051i	$-0.229889\pi$
$643$	38.0535	1.50068	0.750341	+	0.661051i	$0.229889\pi$
$644$	−9.91972	−0.390892
$645$	0	0
$646$	13.2767	0.522366
$647$	46.9425	1.84550	0.922750	−	0.385400i	$-0.125936\pi$
$647$	46.9425	1.84550	0.922750	+	0.385400i	$0.125936\pi$
$648$	0	0
$649$	−0.0155675	−0.000611076 0
$650$	7.35818	0.288612
$651$	0	0
$652$	−57.5049	−2.25207
$653$	32.8959	1.28732	0.643659	−	0.765313i	$-0.277416\pi$
$653$	32.8959	1.28732	0.643659	+	0.765313i	$0.277416\pi$
$654$	0	0
$655$	17.5600	0.686125
$656$	−3.31205	−0.129314
$657$	0	0
$658$	−2.11073	−0.0822846
$659$	−43.1518	−1.68096	−0.840478	−	0.541846i	$-0.817726\pi$
$659$	−43.1518	−1.68096	−0.840478	+	0.541846i	$0.817726\pi$
$660$	0	0
$661$	−8.69294	−0.338116	−0.169058	−	0.985606i	$-0.554073\pi$
$661$	−8.69294	−0.338116	−0.169058	+	0.985606i	$0.554073\pi$
$662$	−29.6303	−1.15161
$663$	0	0
$664$	4.84508	0.188026
$665$	−11.1073	−0.430722
$666$	0	0
$667$	−15.4166	−0.596935
$668$	14.2039	0.549566
$669$	0	0
$670$	7.11382	0.274831
$671$	−0.0338340	−0.00130615
$672$	0	0
$673$	−6.03409	−0.232597	−0.116299	−	0.993214i	$-0.537103\pi$
$673$	−6.03409	−0.232597	−0.116299	+	0.993214i	$0.537103\pi$
$674$	55.8318	2.15056
$675$	0	0
$676$	−32.1655	−1.23713
$677$	3.99616	0.153585	0.0767925	−	0.997047i	$-0.475532\pi$
$677$	3.99616	0.153585	0.0767925	+	0.997047i	$0.475532\pi$
$678$	0	0
$679$	−17.2865	−0.663395
$680$	−14.4354	−0.553574
$681$	0	0
$682$	−0.116049	−0.00444376
$683$	−37.8078	−1.44668	−0.723338	−	0.690494i	$-0.757393\pi$
$683$	−37.8078	−1.44668	−0.723338	+	0.690494i	$0.757393\pi$
$684$	0	0
$685$	−24.0211	−0.917799
$686$	−40.2650	−1.53733
$687$	0	0
$688$	4.30010	0.163940
$689$	8.30046	0.316222
$690$	0	0
$691$	−25.4543	−0.968326	−0.484163	−	0.874978i	$-0.660876\pi$
$691$	−25.4543	−0.968326	−0.484163	+	0.874978i	$0.660876\pi$
$692$	73.5091	2.79440
$693$	0	0
$694$	−56.0402	−2.12726
$695$	−42.0805	−1.59620
$696$	0	0
$697$	−17.5785	−0.665833
$698$	74.7620	2.82978
$699$	0	0
$700$	9.29685	0.351388
$701$	−42.6352	−1.61031	−0.805154	−	0.593065i	$-0.797918\pi$
$701$	−42.6352	−1.61031	−0.805154	+	0.593065i	$0.797918\pi$
$702$	0	0
$703$	0.783550	0.0295521
$704$	−0.136722	−0.00515290
$705$	0	0
$706$	16.9823	0.639139
$707$	−10.8706	−0.408831
$708$	0	0
$709$	−27.6047	−1.03672	−0.518359	−	0.855163i	$-0.673457\pi$
$709$	−27.6047	−1.03672	−0.518359	+	0.855163i	$0.673457\pi$
$710$	49.9165	1.87333
$711$	0	0
$712$	−40.9573	−1.53494
$713$	10.1377	0.379659
$714$	0	0
$715$	0.0460401	0.00172180
$716$	−24.2762	−0.907244
$717$	0	0
$718$	−65.8193	−2.45635
$719$	15.6846	0.584936	0.292468	−	0.956275i	$-0.405523\pi$
$719$	15.6846	0.584936	0.292468	+	0.956275i	$0.405523\pi$
$720$	0	0
$721$	3.56120	0.132626
$722$	−25.6080	−0.953033
$723$	0	0
$724$	55.2893	2.05481
$725$	14.4486	0.536608
$726$	0	0
$727$	47.7864	1.77230	0.886149	−	0.463400i	$-0.153371\pi$
$727$	47.7864	1.77230	0.886149	+	0.463400i	$0.153371\pi$
$728$	6.56693	0.243387
$729$	0	0
$730$	88.2849	3.26757
$731$	22.8225	0.844121
$732$	0	0
$733$	3.47183	0.128235	0.0641175	−	0.997942i	$-0.479577\pi$
$733$	3.47183	0.128235	0.0641175	+	0.997942i	$0.479577\pi$
$734$	67.4953	2.49130
$735$	0	0
$736$	12.6712	0.467068
$737$	0.0124780	0.000459634 0
$738$	0	0
$739$	22.4368	0.825351	0.412676	−	0.910878i	$-0.364594\pi$
$739$	22.4368	0.825351	0.412676	+	0.910878i	$0.364594\pi$
$740$	−2.33947	−0.0860007
$741$	0	0
$742$	17.1557	0.629807
$743$	16.2302	0.595429	0.297714	−	0.954655i	$-0.403776\pi$
$743$	16.2302	0.595429	0.297714	+	0.954655i	$0.403776\pi$
$744$	0	0
$745$	32.1717	1.17868
$746$	10.7146	0.392288
$747$	0	0
$748$	−0.0695321	−0.00254234
$749$	24.7816	0.905500
$750$	0	0
$751$	25.4121	0.927302	0.463651	−	0.886018i	$-0.346539\pi$
$751$	25.4121	0.927302	0.463651	+	0.886018i	$0.346539\pi$
$752$	0.243517	0.00888014
$753$	0	0
$754$	28.0263	1.02066
$755$	−35.7429	−1.30082
$756$	0	0
$757$	38.9444	1.41546	0.707729	−	0.706484i	$-0.249720\pi$
$757$	38.9444	1.41546	0.707729	+	0.706484i	$0.249720\pi$
$758$	0.545415	0.0198104
$759$	0	0
$760$	−19.0172	−0.689825
$761$	−24.0905	−0.873278	−0.436639	−	0.899637i	$-0.643831\pi$
$761$	−24.0905	−0.873278	−0.436639	+	0.899637i	$0.643831\pi$
$762$	0	0
$763$	6.42154	0.232475
$764$	55.5329	2.00911
$765$	0	0
$766$	−15.3707	−0.555365
$767$	−2.47223	−0.0892670
$768$	0	0
$769$	−30.4484	−1.09800	−0.548999	−	0.835823i	$-0.684991\pi$
$769$	−30.4484	−1.09800	−0.548999	+	0.835823i	$0.684991\pi$
$770$	0.0951577	0.00342924
$771$	0	0
$772$	54.8915	1.97559
$773$	3.52407	0.126752	0.0633760	−	0.997990i	$-0.479813\pi$
$773$	3.52407	0.126752	0.0633760	+	0.997990i	$0.479813\pi$
$774$	0	0
$775$	−9.50111	−0.341290
$776$	−29.5968	−1.06246
$777$	0	0
$778$	−51.3962	−1.84264
$779$	−23.1578	−0.829714
$780$	0	0
$781$	0.0875563	0.00313301
$782$	9.93626	0.355320
$783$	0	0
$784$	1.86541	0.0666219
$785$	5.45566	0.194721
$786$	0	0
$787$	−26.7194	−0.952443	−0.476221	−	0.879325i	$-0.657994\pi$
$787$	−26.7194	−0.952443	−0.476221	+	0.879325i	$0.657994\pi$
$788$	−72.6113	−2.58667
$789$	0	0
$790$	12.5743	0.447373
$791$	0.170821	0.00607370
$792$	0	0
$793$	−5.37309	−0.190804
$794$	60.1151	2.13340
$795$	0	0
$796$	33.5760	1.19007
$797$	29.7776	1.05478	0.527388	−	0.849625i	$-0.323171\pi$
$797$	29.7776	1.05478	0.527388	+	0.849625i	$0.323171\pi$
$798$	0	0
$799$	1.29245	0.0457235
$800$	−11.8756	−0.419866
$801$	0	0
$802$	56.4492	1.99329
$803$	0.154856	0.00546476
$804$	0	0
$805$	−8.31265	−0.292982
$806$	−18.4295	−0.649151
$807$	0	0
$808$	−18.6119	−0.654765
$809$	−23.8830	−0.839680	−0.419840	−	0.907598i	$-0.637914\pi$
$809$	−23.8830	−0.839680	−0.419840	+	0.907598i	$0.637914\pi$
$810$	0	0
$811$	14.5688	0.511578	0.255789	−	0.966733i	$-0.417665\pi$
$811$	14.5688	0.511578	0.255789	+	0.966733i	$0.417665\pi$
$812$	35.4104	1.24266
$813$	0	0
$814$	−0.00671278	−0.000235283 0
$815$	−48.1887	−1.68798
$816$	0	0
$817$	30.0662	1.05188
$818$	85.1082	2.97574
$819$	0	0
$820$	69.1430	2.41458
$821$	−46.7882	−1.63292	−0.816460	−	0.577402i	$-0.804066\pi$
$821$	−46.7882	−1.63292	−0.816460	+	0.577402i	$0.804066\pi$
$822$	0	0
$823$	−25.0104	−0.871809	−0.435904	−	0.899993i	$-0.643571\pi$
$823$	−25.0104	−0.871809	−0.435904	+	0.899993i	$0.643571\pi$
$824$	6.09724	0.212408
$825$	0	0
$826$	−5.10970	−0.177789
$827$	52.7563	1.83452	0.917258	−	0.398292i	$-0.130397\pi$
$827$	52.7563	1.83452	0.917258	+	0.398292i	$0.130397\pi$
$828$	0	0
$829$	−27.0086	−0.938048	−0.469024	−	0.883186i	$-0.655394\pi$
$829$	−27.0086	−0.938048	−0.469024	+	0.883186i	$0.655394\pi$
$830$	11.1495	0.387003
$831$	0	0
$832$	−21.7124	−0.752743
$833$	9.90056	0.343034
$834$	0	0
$835$	11.9028	0.411912
$836$	−0.0916011	−0.00316809
$837$	0	0
$838$	65.0762	2.24802
$839$	10.2793	0.354882	0.177441	−	0.984131i	$-0.443218\pi$
$839$	10.2793	0.354882	0.177441	+	0.984131i	$0.443218\pi$
$840$	0	0
$841$	26.0327	0.897680
$842$	43.4337	1.49682
$843$	0	0
$844$	−12.5764	−0.432896
$845$	−26.9545	−0.927261
$846$	0	0
$847$	−16.6928	−0.573571
$848$	−1.97928	−0.0679686
$849$	0	0
$850$	−9.31235	−0.319411
$851$	0.586406	0.0201017
$852$	0	0
$853$	33.1422	1.13477	0.567383	−	0.823454i	$-0.307956\pi$
$853$	33.1422	1.13477	0.567383	+	0.823454i	$0.307956\pi$
$854$	−11.1053	−0.380016
$855$	0	0
$856$	42.4294	1.45021
$857$	0.539897	0.0184425	0.00922127	−	0.999957i	$-0.497065\pi$
$857$	0.539897	0.0184425	0.00922127	+	0.999957i	$0.497065\pi$
$858$	0	0
$859$	15.7735	0.538184	0.269092	−	0.963114i	$-0.413276\pi$
$859$	15.7735	0.538184	0.269092	+	0.963114i	$0.413276\pi$
$860$	−89.7697	−3.06112
$861$	0	0
$862$	−3.52645	−0.120112
$863$	−11.8283	−0.402640	−0.201320	−	0.979525i	$-0.564523\pi$
$863$	−11.8283	−0.402640	−0.201320	+	0.979525i	$0.564523\pi$
$864$	0	0
$865$	61.6001	2.09446
$866$	−70.2104	−2.38585
$867$	0	0
$868$	−23.2851	−0.790349
$869$	0.0220560	0.000748198 0
$870$	0	0
$871$	1.98160	0.0671441
$872$	10.9945	0.372322
$873$	0	0
$874$	13.0900	0.442774
$875$	−12.2093	−0.412750
$876$	0	0
$877$	−54.3552	−1.83544	−0.917722	−	0.397224i	$-0.869973\pi$
$877$	−54.3552	−1.83544	−0.917722	+	0.397224i	$0.869973\pi$
$878$	22.0892	0.745474
$879$	0	0
$880$	−0.0109784	−0.000370083 0
$881$	43.2780	1.45807	0.729036	−	0.684476i	$-0.239969\pi$
$881$	43.2780	1.45807	0.729036	+	0.684476i	$0.239969\pi$
$882$	0	0
$883$	−29.0340	−0.977072	−0.488536	−	0.872544i	$-0.662469\pi$
$883$	−29.0340	−0.977072	−0.488536	+	0.872544i	$0.662469\pi$
$884$	−11.0422	−0.371390
$885$	0	0
$886$	73.8513	2.48108
$887$	41.5669	1.39568	0.697840	−	0.716253i	$-0.254145\pi$
$887$	41.5669	1.39568	0.697840	+	0.716253i	$0.254145\pi$
$888$	0	0
$889$	−1.51754	−0.0508967
$890$	−94.2504	−3.15928
$891$	0	0
$892$	−36.3137	−1.21587
$893$	1.70266	0.0569774
$894$	0	0
$895$	−20.3433	−0.680001
$896$	−26.3703	−0.880969
$897$	0	0
$898$	55.8593	1.86405
$899$	−36.1884	−1.20695
$900$	0	0
$901$	−10.5049	−0.349968
$902$	0.198396	0.00660586
$903$	0	0
$904$	0.292469	0.00972736
$905$	46.3320	1.54013
$906$	0	0
$907$	−7.49326	−0.248810	−0.124405	−	0.992232i	$-0.539702\pi$
$907$	−7.49326	−0.248810	−0.124405	+	0.992232i	$0.539702\pi$
$908$	26.5199	0.880094
$909$	0	0
$910$	15.1117	0.500949
$911$	8.26482	0.273826	0.136913	−	0.990583i	$-0.456282\pi$
$911$	8.26482	0.273826	0.136913	+	0.990583i	$0.456282\pi$
$912$	0	0
$913$	0.0195567	0.000647233 0
$914$	40.5525	1.34136
$915$	0	0
$916$	60.9458	2.01371
$917$	10.1098	0.333856
$918$	0	0
$919$	−37.8332	−1.24800	−0.624001	−	0.781423i	$-0.714494\pi$
$919$	−37.8332	−1.24800	−0.624001	+	0.781423i	$0.714494\pi$
$920$	−14.2324	−0.469227
$921$	0	0
$922$	44.9996	1.48198
$923$	13.9046	0.457675
$924$	0	0
$925$	−0.549585	−0.0180702
$926$	63.2523	2.07860
$927$	0	0
$928$	−45.2324	−1.48483
$929$	−14.3387	−0.470437	−0.235218	−	0.971943i	$-0.575581\pi$
$929$	−14.3387	−0.470437	−0.235218	+	0.971943i	$0.575581\pi$
$930$	0	0
$931$	13.0429	0.427465
$932$	−45.2747	−1.48302
$933$	0	0
$934$	−31.3435	−1.02559
$935$	−0.0582674	−0.00190555
$936$	0	0
$937$	−31.5539	−1.03082	−0.515410	−	0.856944i	$-0.672360\pi$
$937$	−31.5539	−1.03082	−0.515410	+	0.856944i	$0.672360\pi$
$938$	4.09566	0.133728
$939$	0	0
$940$	−5.08370	−0.165812
$941$	−21.6114	−0.704510	−0.352255	−	0.935904i	$-0.614585\pi$
$941$	−21.6114	−0.704510	−0.352255	+	0.935904i	$0.614585\pi$
$942$	0	0
$943$	−17.3312	−0.564381
$944$	0.589512	0.0191870
$945$	0	0
$946$	−0.257581	−0.00837469
$947$	−3.91107	−0.127093	−0.0635463	−	0.997979i	$-0.520241\pi$
$947$	−3.91107	−0.127093	−0.0635463	+	0.997979i	$0.520241\pi$
$948$	0	0
$949$	24.5923	0.798301
$950$	−12.2680	−0.398027
$951$	0	0
$952$	−8.31096	−0.269360
$953$	−36.7238	−1.18960	−0.594801	−	0.803873i	$-0.702769\pi$
$953$	−36.7238	−1.18960	−0.594801	+	0.803873i	$0.702769\pi$
$954$	0	0
$955$	46.5361	1.50587
$956$	−41.3541	−1.33749
$957$	0	0
$958$	−47.9521	−1.54926
$959$	−13.8297	−0.446585
$960$	0	0
$961$	−7.20327	−0.232364
$962$	−1.06604	−0.0343705
$963$	0	0
$964$	17.6920	0.569821
$965$	45.9987	1.48075
$966$	0	0
$967$	6.81466	0.219145	0.109572	−	0.993979i	$-0.465052\pi$
$967$	6.81466	0.219145	0.109572	+	0.993979i	$0.465052\pi$
$968$	−28.5803	−0.918605
$969$	0	0
$970$	−68.1078	−2.18681
$971$	56.2769	1.80601	0.903007	−	0.429627i	$-0.141355\pi$
$971$	56.2769	1.80601	0.903007	+	0.429627i	$0.141355\pi$
$972$	0	0
$973$	−24.2271	−0.776685
$974$	−49.3455	−1.58113
$975$	0	0
$976$	1.28123	0.0410113
$977$	−41.4266	−1.32536	−0.662678	−	0.748905i	$-0.730580\pi$
$977$	−41.4266	−1.32536	−0.662678	+	0.748905i	$0.730580\pi$
$978$	0	0
$979$	−0.165320	−0.00528366
$980$	−38.9427	−1.24398
$981$	0	0
$982$	41.5922	1.32726
$983$	−25.1118	−0.800943	−0.400471	−	0.916309i	$-0.631154\pi$
$983$	−25.1118	−0.800943	−0.400471	+	0.916309i	$0.631154\pi$
$984$	0	0
$985$	−60.8477	−1.93877
$986$	−35.4694	−1.12958
$987$	0	0
$988$	−14.5469	−0.462800
$989$	22.5014	0.715504
$990$	0	0
$991$	−60.9601	−1.93646	−0.968230	−	0.250060i	$-0.919550\pi$
$991$	−60.9601	−1.93646	−0.968230	+	0.250060i	$0.919550\pi$
$992$	29.7439	0.944370
$993$	0	0
$994$	28.7386	0.911532
$995$	28.1364	0.891985
$996$	0	0
$997$	51.1625	1.62033	0.810167	−	0.586199i	$-0.199376\pi$
$997$	51.1625	1.62033	0.810167	+	0.586199i	$0.199376\pi$
$998$	−15.6124	−0.494200
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.h.1.5		5
3.2	odd	2		381.2.a.c.1.1	✓	5
12.11	even	2		6096.2.a.be.1.2		5
15.14	odd	2		9525.2.a.k.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.c.1.1	✓	5	3.2	odd	2
1143.2.a.h.1.5		5	1.1	even	1	trivial
6096.2.a.be.1.2		5	12.11	even	2
9525.2.a.k.1.5		5	15.14	odd	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)

\(f(q)\)	\(=\)	\(q+2.26835 q^{2} +3.14543 q^{4} +2.63584 q^{5} +1.51754 q^{7} +2.59823 q^{8} +O(q^{10})\)	\(q+2.26835 q^{2} +3.14543 q^{4} +2.63584 q^{5} +1.51754 q^{7} +2.59823 q^{8} +5.97902 q^{10} +0.0104875 q^{11} +1.66550 q^{13} +3.44232 q^{14} -0.397144 q^{16} -2.10782 q^{17} -2.77682 q^{19} +8.29085 q^{20} +0.0237894 q^{22} -2.07816 q^{23} +1.94767 q^{25} +3.77794 q^{26} +4.77331 q^{28} +7.41840 q^{29} -4.87819 q^{31} -6.09733 q^{32} -4.78127 q^{34} +4.00000 q^{35} -0.282175 q^{37} -6.29881 q^{38} +6.84854 q^{40} +8.33967 q^{41} -10.8276 q^{43} +0.0329877 q^{44} -4.71401 q^{46} -0.613170 q^{47} -4.69707 q^{49} +4.41801 q^{50} +5.23870 q^{52} +4.98377 q^{53} +0.0276435 q^{55} +3.94292 q^{56} +16.8276 q^{58} -1.48438 q^{59} -3.22612 q^{61} -11.0655 q^{62} -13.0366 q^{64} +4.38999 q^{65} +1.18980 q^{67} -6.62998 q^{68} +9.07341 q^{70} +8.34861 q^{71} +14.7658 q^{73} -0.640073 q^{74} -8.73429 q^{76} +0.0159152 q^{77} +2.10307 q^{79} -1.04681 q^{80} +18.9173 q^{82} +1.86476 q^{83} -5.55587 q^{85} -24.5607 q^{86} +0.0272490 q^{88} -15.7635 q^{89} +2.52746 q^{91} -6.53671 q^{92} -1.39089 q^{94} -7.31927 q^{95} -11.3911 q^{97} -10.6546 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} + q^{4} + 5 q^{5} + 3 q^{8}+O(q^{10}) \) 5 * q + q^2 + q^4 + 5 * q^5 + 3 * q^8	\( 5 q + q^{2} + q^{4} + 5 q^{5} + 3 q^{8} - 2 q^{10} + 16 q^{11} + 3 q^{13} + 6 q^{14} - 7 q^{16} + 6 q^{17} - 8 q^{19} + 12 q^{20} - 8 q^{22} + 9 q^{23} + 4 q^{25} + 2 q^{26} + 2 q^{28} + 17 q^{29} - 9 q^{31} + 2 q^{32} - q^{34} + 20 q^{35} - q^{37} - q^{38} + 16 q^{40} + 2 q^{41} - 4 q^{43} - 3 q^{44} + 4 q^{46} + 8 q^{47} + 13 q^{49} - 15 q^{50} + 13 q^{52} + 15 q^{53} + 20 q^{55} - 8 q^{56} + 34 q^{58} + 19 q^{59} + q^{61} - 15 q^{62} + q^{64} + 5 q^{65} - 2 q^{67} - 14 q^{68} + 4 q^{70} + 13 q^{73} + 17 q^{74} + 8 q^{76} - 2 q^{77} - 28 q^{79} - 12 q^{80} + 30 q^{82} + q^{83} + 6 q^{85} - 32 q^{86} + 3 q^{88} - q^{89} - 18 q^{91} - 12 q^{92} + 16 q^{94} - 4 q^{95} + 28 q^{97} - 15 q^{98}+O(q^{100}) \) 5 * q + q^2 + q^4 + 5 * q^5 + 3 * q^8 - 2 * q^10 + 16 * q^11 + 3 * q^13 + 6 * q^14 - 7 * q^16 + 6 * q^17 - 8 * q^19 + 12 * q^20 - 8 * q^22 + 9 * q^23 + 4 * q^25 + 2 * q^26 + 2 * q^28 + 17 * q^29 - 9 * q^31 + 2 * q^32 - q^34 + 20 * q^35 - q^37 - q^38 + 16 * q^40 + 2 * q^41 - 4 * q^43 - 3 * q^44 + 4 * q^46 + 8 * q^47 + 13 * q^49 - 15 * q^50 + 13 * q^52 + 15 * q^53 + 20 * q^55 - 8 * q^56 + 34 * q^58 + 19 * q^59 + q^61 - 15 * q^62 + q^64 + 5 * q^65 - 2 * q^67 - 14 * q^68 + 4 * q^70 + 13 * q^73 + 17 * q^74 + 8 * q^76 - 2 * q^77 - 28 * q^79 - 12 * q^80 + 30 * q^82 + q^83 + 6 * q^85 - 32 * q^86 + 3 * q^88 - q^89 - 18 * q^91 - 12 * q^92 + 16 * q^94 - 4 * q^95 + 28 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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