LMFDB - Embedded newform 1127.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1127,2,Mod(1,1127)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1127, 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("1127.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1127 = 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1127.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:	$8.99914030780$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 161)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1127.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} -2.00000 q^{9} +O(q^{10})$	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} -2.00000 q^{9} +2.00000 q^{10} -4.47214 q^{11} +0.618034 q^{12} +4.23607 q^{13} -1.23607 q^{15} -4.85410 q^{16} +3.23607 q^{18} +2.76393 q^{19} -0.763932 q^{20} +7.23607 q^{22} -1.00000 q^{23} +2.23607 q^{24} -3.47214 q^{25} -6.85410 q^{26} -5.00000 q^{27} +7.47214 q^{29} +2.00000 q^{30} +9.00000 q^{31} +3.38197 q^{32} -4.47214 q^{33} -1.23607 q^{36} +7.70820 q^{37} -4.47214 q^{38} +4.23607 q^{39} -2.76393 q^{40} -2.23607 q^{41} -6.47214 q^{43} -2.76393 q^{44} +2.47214 q^{45} +1.61803 q^{46} -5.47214 q^{47} -4.85410 q^{48} +5.61803 q^{50} +2.61803 q^{52} +6.76393 q^{53} +8.09017 q^{54} +5.52786 q^{55} +2.76393 q^{57} -12.0902 q^{58} +10.4721 q^{59} -0.763932 q^{60} +13.4164 q^{61} -14.5623 q^{62} +4.23607 q^{64} -5.23607 q^{65} +7.23607 q^{66} +10.1803 q^{67} -1.00000 q^{69} -5.76393 q^{71} -4.47214 q^{72} -6.70820 q^{73} -12.4721 q^{74} -3.47214 q^{75} +1.70820 q^{76} -6.85410 q^{78} -2.76393 q^{79} +6.00000 q^{80} +1.00000 q^{81} +3.61803 q^{82} +2.47214 q^{83} +10.4721 q^{86} +7.47214 q^{87} -10.0000 q^{88} +8.94427 q^{89} -4.00000 q^{90} -0.618034 q^{92} +9.00000 q^{93} +8.85410 q^{94} -3.41641 q^{95} +3.38197 q^{96} +9.70820 q^{97} +8.94427 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9}+O(q^{10}) $ 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^9	$ 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9} + 4 q^{10} - q^{12} + 4 q^{13} + 2 q^{15} - 3 q^{16} + 2 q^{18} + 10 q^{19} - 6 q^{20} + 10 q^{22} - 2 q^{23} + 2 q^{25} - 7 q^{26} - 10 q^{27} + 6 q^{29} + 4 q^{30} + 18 q^{31} + 9 q^{32} + 2 q^{36} + 2 q^{37} + 4 q^{39} - 10 q^{40} - 4 q^{43} - 10 q^{44} - 4 q^{45} + q^{46} - 2 q^{47} - 3 q^{48} + 9 q^{50} + 3 q^{52} + 18 q^{53} + 5 q^{54} + 20 q^{55} + 10 q^{57} - 13 q^{58} + 12 q^{59} - 6 q^{60} - 9 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{66} - 2 q^{67} - 2 q^{69} - 16 q^{71} - 16 q^{74} + 2 q^{75} - 10 q^{76} - 7 q^{78} - 10 q^{79} + 12 q^{80} + 2 q^{81} + 5 q^{82} - 4 q^{83} + 12 q^{86} + 6 q^{87} - 20 q^{88} - 8 q^{90} + q^{92} + 18 q^{93} + 11 q^{94} + 20 q^{95} + 9 q^{96} + 6 q^{97}+O(q^{100}) $ 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^9 + 4 * q^10 - q^12 + 4 * q^13 + 2 * q^15 - 3 * q^16 + 2 * q^18 + 10 * q^19 - 6 * q^20 + 10 * q^22 - 2 * q^23 + 2 * q^25 - 7 * q^26 - 10 * q^27 + 6 * q^29 + 4 * q^30 + 18 * q^31 + 9 * q^32 + 2 * q^36 + 2 * q^37 + 4 * q^39 - 10 * q^40 - 4 * q^43 - 10 * q^44 - 4 * q^45 + q^46 - 2 * q^47 - 3 * q^48 + 9 * q^50 + 3 * q^52 + 18 * q^53 + 5 * q^54 + 20 * q^55 + 10 * q^57 - 13 * q^58 + 12 * q^59 - 6 * q^60 - 9 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^66 - 2 * q^67 - 2 * q^69 - 16 * q^71 - 16 * q^74 + 2 * q^75 - 10 * q^76 - 7 * q^78 - 10 * q^79 + 12 * q^80 + 2 * q^81 + 5 * q^82 - 4 * q^83 + 12 * q^86 + 6 * q^87 - 20 * q^88 - 8 * q^90 + q^92 + 18 * q^93 + 11 * q^94 + 20 * q^95 + 9 * q^96 + 6 * q^97

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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	1.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	1.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	0.618034	0.309017
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	−1.61803	−0.660560
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	−2.00000	−0.666667
$10$	2.00000	0.632456
$11$	−4.47214	−1.34840	−0.674200	−	0.738549i	$-0.735511\pi$
$11$	−4.47214	−1.34840	−0.674200	+	0.738549i	$0.735511\pi$
$12$	0.618034	0.178411
$13$	4.23607	1.17487	0.587437	−	0.809270i	$-0.300137\pi$
$13$	4.23607	1.17487	0.587437	+	0.809270i	$0.300137\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	−4.85410	−1.21353
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	3.23607	0.762749
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	−0.763932	−0.170820
$21$	0	0
$22$	7.23607	1.54273
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	2.23607	0.456435
$25$	−3.47214	−0.694427
$26$	−6.85410	−1.34420
$27$	−5.00000	−0.962250
$28$	0	0
$29$	7.47214	1.38754	0.693770	−	0.720196i	$-0.255948\pi$
$29$	7.47214	1.38754	0.693770	+	0.720196i	$0.255948\pi$
$30$	2.00000	0.365148
$31$	9.00000	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$31$	9.00000	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$32$	3.38197	0.597853
$33$	−4.47214	−0.778499
$34$	0	0
$35$	0	0
$36$	−1.23607	−0.206011
$37$	7.70820	1.26722	0.633610	−	0.773652i	$-0.281572\pi$
$37$	7.70820	1.26722	0.633610	+	0.773652i	$0.281572\pi$
$38$	−4.47214	−0.725476
$39$	4.23607	0.678314
$40$	−2.76393	−0.437016
$41$	−2.23607	−0.349215	−0.174608	−	0.984638i	$-0.555866\pi$
$41$	−2.23607	−0.349215	−0.174608	+	0.984638i	$0.555866\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	−2.76393	−0.416678
$45$	2.47214	0.368524
$46$	1.61803	0.238566
$47$	−5.47214	−0.798193	−0.399097	−	0.916909i	$-0.630676\pi$
$47$	−5.47214	−0.798193	−0.399097	+	0.916909i	$0.630676\pi$
$48$	−4.85410	−0.700629
$49$	0	0
$50$	5.61803	0.794510
$51$	0	0
$52$	2.61803	0.363056
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	8.09017	1.10093
$55$	5.52786	0.745377
$56$	0	0
$57$	2.76393	0.366092
$58$	−12.0902	−1.58752
$59$	10.4721	1.36336	0.681678	−	0.731652i	$-0.261251\pi$
$59$	10.4721	1.36336	0.681678	+	0.731652i	$0.261251\pi$
$60$	−0.763932	−0.0986232
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	−14.5623	−1.84941
$63$	0	0
$64$	4.23607	0.529508
$65$	−5.23607	−0.649454
$66$	7.23607	0.890698
$67$	10.1803	1.24373	0.621863	−	0.783126i	$-0.286376\pi$
$67$	10.1803	1.24373	0.621863	+	0.783126i	$0.286376\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−5.76393	−0.684053	−0.342026	−	0.939690i	$-0.611113\pi$
$71$	−5.76393	−0.684053	−0.342026	+	0.939690i	$0.611113\pi$
$72$	−4.47214	−0.527046
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	−12.4721	−1.44986
$75$	−3.47214	−0.400928
$76$	1.70820	0.195944
$77$	0	0
$78$	−6.85410	−0.776074
$79$	−2.76393	−0.310967	−0.155483	−	0.987839i	$-0.549693\pi$
$79$	−2.76393	−0.310967	−0.155483	+	0.987839i	$0.549693\pi$
$80$	6.00000	0.670820
$81$	1.00000	0.111111
$82$	3.61803	0.399545
$83$	2.47214	0.271352	0.135676	−	0.990753i	$-0.456679\pi$
$83$	2.47214	0.271352	0.135676	+	0.990753i	$0.456679\pi$
$84$	0	0
$85$	0	0
$86$	10.4721	1.12924
$87$	7.47214	0.801097
$88$	−10.0000	−1.06600
$89$	8.94427	0.948091	0.474045	−	0.880500i	$-0.342793\pi$
$89$	8.94427	0.948091	0.474045	+	0.880500i	$0.342793\pi$
$90$	−4.00000	−0.421637
$91$	0	0
$92$	−0.618034	−0.0644345
$93$	9.00000	0.933257
$94$	8.85410	0.913231
$95$	−3.41641	−0.350516
$96$	3.38197	0.345170
$97$	9.70820	0.985719	0.492859	−	0.870109i	$-0.335952\pi$
$97$	9.70820	0.985719	0.492859	+	0.870109i	$0.335952\pi$
$98$	0	0
$99$	8.94427	0.898933
$100$	−2.14590	−0.214590
$101$	−13.4164	−1.33498	−0.667491	−	0.744618i	$-0.732632\pi$
$101$	−13.4164	−1.33498	−0.667491	+	0.744618i	$0.732632\pi$
$102$	0	0
$103$	2.29180	0.225817	0.112909	−	0.993605i	$-0.463983\pi$
$103$	2.29180	0.225817	0.112909	+	0.993605i	$0.463983\pi$
$104$	9.47214	0.928819
$105$	0	0
$106$	−10.9443	−1.06300
$107$	6.76393	0.653894	0.326947	−	0.945043i	$-0.393980\pi$
$107$	6.76393	0.653894	0.326947	+	0.945043i	$0.393980\pi$
$108$	−3.09017	−0.297352
$109$	11.4164	1.09349	0.546747	−	0.837298i	$-0.315866\pi$
$109$	11.4164	1.09349	0.546747	+	0.837298i	$0.315866\pi$
$110$	−8.94427	−0.852803
$111$	7.70820	0.731630
$112$	0	0
$113$	−6.47214	−0.608847	−0.304424	−	0.952537i	$-0.598464\pi$
$113$	−6.47214	−0.608847	−0.304424	+	0.952537i	$0.598464\pi$
$114$	−4.47214	−0.418854
$115$	1.23607	0.115264
$116$	4.61803	0.428774
$117$	−8.47214	−0.783249
$118$	−16.9443	−1.55985
$119$	0	0
$120$	−2.76393	−0.252311
$121$	9.00000	0.818182
$122$	−21.7082	−1.96537
$123$	−2.23607	−0.201619
$124$	5.56231	0.499510
$125$	10.4721	0.936656
$126$	0	0
$127$	10.7082	0.950199	0.475100	−	0.879932i	$-0.342412\pi$
$127$	10.7082	0.950199	0.475100	+	0.879932i	$0.342412\pi$
$128$	−13.6180	−1.20368
$129$	−6.47214	−0.569840
$130$	8.47214	0.743055
$131$	13.9443	1.21832	0.609158	−	0.793049i	$-0.291507\pi$
$131$	13.9443	1.21832	0.609158	+	0.793049i	$0.291507\pi$
$132$	−2.76393	−0.240569
$133$	0	0
$134$	−16.4721	−1.42298
$135$	6.18034	0.531919
$136$	0	0
$137$	2.29180	0.195801	0.0979007	−	0.995196i	$-0.468787\pi$
$137$	2.29180	0.195801	0.0979007	+	0.995196i	$0.468787\pi$
$138$	1.61803	0.137736
$139$	−11.4721	−0.973054	−0.486527	−	0.873666i	$-0.661736\pi$
$139$	−11.4721	−0.973054	−0.486527	+	0.873666i	$0.661736\pi$
$140$	0	0
$141$	−5.47214	−0.460837
$142$	9.32624	0.782641
$143$	−18.9443	−1.58420
$144$	9.70820	0.809017
$145$	−9.23607	−0.767014
$146$	10.8541	0.898292
$147$	0	0
$148$	4.76393	0.391593
$149$	10.7639	0.881816	0.440908	−	0.897552i	$-0.354657\pi$
$149$	10.7639	0.881816	0.440908	+	0.897552i	$0.354657\pi$
$150$	5.61803	0.458711
$151$	7.18034	0.584328	0.292164	−	0.956368i	$-0.405625\pi$
$151$	7.18034	0.584328	0.292164	+	0.956368i	$0.405625\pi$
$152$	6.18034	0.501292
$153$	0	0
$154$	0	0
$155$	−11.1246	−0.893550
$156$	2.61803	0.209610
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	4.47214	0.355784
$159$	6.76393	0.536415
$160$	−4.18034	−0.330485
$161$	0	0
$162$	−1.61803	−0.127125
$163$	−3.18034	−0.249103	−0.124552	−	0.992213i	$-0.539749\pi$
$163$	−3.18034	−0.249103	−0.124552	+	0.992213i	$0.539749\pi$
$164$	−1.38197	−0.107913
$165$	5.52786	0.430344
$166$	−4.00000	−0.310460
$167$	13.8885	1.07473	0.537364	−	0.843350i	$-0.319420\pi$
$167$	13.8885	1.07473	0.537364	+	0.843350i	$0.319420\pi$
$168$	0	0
$169$	4.94427	0.380329
$170$	0	0
$171$	−5.52786	−0.422726
$172$	−4.00000	−0.304997
$173$	12.4721	0.948239	0.474119	−	0.880461i	$-0.342766\pi$
$173$	12.4721	0.948239	0.474119	+	0.880461i	$0.342766\pi$
$174$	−12.0902	−0.916553
$175$	0	0
$176$	21.7082	1.63632
$177$	10.4721	0.787134
$178$	−14.4721	−1.08473
$179$	−5.29180	−0.395527	−0.197764	−	0.980250i	$-0.563368\pi$
$179$	−5.29180	−0.395527	−0.197764	+	0.980250i	$0.563368\pi$
$180$	1.52786	0.113880
$181$	22.9443	1.70543	0.852717	−	0.522373i	$-0.174953\pi$
$181$	22.9443	1.70543	0.852717	+	0.522373i	$0.174953\pi$
$182$	0	0
$183$	13.4164	0.991769
$184$	−2.23607	−0.164845
$185$	−9.52786	−0.700502
$186$	−14.5623	−1.06776
$187$	0	0
$188$	−3.38197	−0.246655
$189$	0	0
$190$	5.52786	0.401033
$191$	−1.81966	−0.131666	−0.0658330	−	0.997831i	$-0.520970\pi$
$191$	−1.81966	−0.131666	−0.0658330	+	0.997831i	$0.520970\pi$
$192$	4.23607	0.305712
$193$	−22.4164	−1.61357	−0.806784	−	0.590846i	$-0.798794\pi$
$193$	−22.4164	−1.61357	−0.806784	+	0.590846i	$0.798794\pi$
$194$	−15.7082	−1.12778
$195$	−5.23607	−0.374963
$196$	0	0
$197$	−12.5279	−0.892573	−0.446287	−	0.894890i	$-0.647254\pi$
$197$	−12.5279	−0.892573	−0.446287	+	0.894890i	$0.647254\pi$
$198$	−14.4721	−1.02849
$199$	19.8885	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$199$	19.8885	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$200$	−7.76393	−0.548993
$201$	10.1803	0.718066
$202$	21.7082	1.52738
$203$	0	0
$204$	0	0
$205$	2.76393	0.193041
$206$	−3.70820	−0.258363
$207$	2.00000	0.139010
$208$	−20.5623	−1.42574
$209$	−12.3607	−0.855006
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	4.18034	0.287107
$213$	−5.76393	−0.394938
$214$	−10.9443	−0.748135
$215$	8.00000	0.545595
$216$	−11.1803	−0.760726
$217$	0	0
$218$	−18.4721	−1.25109
$219$	−6.70820	−0.453298
$220$	3.41641	0.230334
$221$	0	0
$222$	−12.4721	−0.837075
$223$	23.4164	1.56808	0.784039	−	0.620711i	$-0.213156\pi$
$223$	23.4164	1.56808	0.784039	+	0.620711i	$0.213156\pi$
$224$	0	0
$225$	6.94427	0.462951
$226$	10.4721	0.696596
$227$	−29.8885	−1.98377	−0.991886	−	0.127129i	$-0.959424\pi$
$227$	−29.8885	−1.98377	−0.991886	+	0.127129i	$0.959424\pi$
$228$	1.70820	0.113129
$229$	−19.2361	−1.27116	−0.635578	−	0.772037i	$-0.719238\pi$
$229$	−19.2361	−1.27116	−0.635578	+	0.772037i	$0.719238\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	16.7082	1.09695
$233$	−2.52786	−0.165606	−0.0828029	−	0.996566i	$-0.526387\pi$
$233$	−2.52786	−0.165606	−0.0828029	+	0.996566i	$0.526387\pi$
$234$	13.7082	0.896133
$235$	6.76393	0.441230
$236$	6.47214	0.421300
$237$	−2.76393	−0.179537
$238$	0	0
$239$	−20.2361	−1.30896	−0.654481	−	0.756078i	$-0.727113\pi$
$239$	−20.2361	−1.30896	−0.654481	+	0.756078i	$0.727113\pi$
$240$	6.00000	0.387298
$241$	−5.41641	−0.348902	−0.174451	−	0.984666i	$-0.555815\pi$
$241$	−5.41641	−0.348902	−0.174451	+	0.984666i	$0.555815\pi$
$242$	−14.5623	−0.936100
$243$	16.0000	1.02640
$244$	8.29180	0.530828
$245$	0	0
$246$	3.61803	0.230677
$247$	11.7082	0.744975
$248$	20.1246	1.27791
$249$	2.47214	0.156665
$250$	−16.9443	−1.07165
$251$	−21.7082	−1.37021	−0.685105	−	0.728444i	$-0.740244\pi$
$251$	−21.7082	−1.37021	−0.685105	+	0.728444i	$0.740244\pi$
$252$	0	0
$253$	4.47214	0.281161
$254$	−17.3262	−1.08714
$255$	0	0
$256$	13.5623	0.847644
$257$	0.236068	0.0147255	0.00736276	−	0.999973i	$-0.497656\pi$
$257$	0.236068	0.0147255	0.00736276	+	0.999973i	$0.497656\pi$
$258$	10.4721	0.651967
$259$	0	0
$260$	−3.23607	−0.200692
$261$	−14.9443	−0.925027
$262$	−22.5623	−1.39390
$263$	−9.05573	−0.558400	−0.279200	−	0.960233i	$-0.590069\pi$
$263$	−9.05573	−0.558400	−0.279200	+	0.960233i	$0.590069\pi$
$264$	−10.0000	−0.615457
$265$	−8.36068	−0.513592
$266$	0	0
$267$	8.94427	0.547381
$268$	6.29180	0.384333
$269$	−13.1803	−0.803620	−0.401810	−	0.915723i	$-0.631619\pi$
$269$	−13.1803	−0.803620	−0.401810	+	0.915723i	$0.631619\pi$
$270$	−10.0000	−0.608581
$271$	−0.944272	−0.0573604	−0.0286802	−	0.999589i	$-0.509130\pi$
$271$	−0.944272	−0.0573604	−0.0286802	+	0.999589i	$0.509130\pi$
$272$	0	0
$273$	0	0
$274$	−3.70820	−0.224021
$275$	15.5279	0.936365
$276$	−0.618034	−0.0372013
$277$	−6.41641	−0.385525	−0.192762	−	0.981245i	$-0.561745\pi$
$277$	−6.41641	−0.385525	−0.192762	+	0.981245i	$0.561745\pi$
$278$	18.5623	1.11329
$279$	−18.0000	−1.07763
$280$	0	0
$281$	−9.70820	−0.579143	−0.289571	−	0.957156i	$-0.593513\pi$
$281$	−9.70820	−0.579143	−0.289571	+	0.957156i	$0.593513\pi$
$282$	8.85410	0.527254
$283$	−1.05573	−0.0627565	−0.0313783	−	0.999508i	$-0.509990\pi$
$283$	−1.05573	−0.0627565	−0.0313783	+	0.999508i	$0.509990\pi$
$284$	−3.56231	−0.211384
$285$	−3.41641	−0.202371
$286$	30.6525	1.81252
$287$	0	0
$288$	−6.76393	−0.398569
$289$	−17.0000	−1.00000
$290$	14.9443	0.877558
$291$	9.70820	0.569105
$292$	−4.14590	−0.242620
$293$	2.29180	0.133888	0.0669441	−	0.997757i	$-0.478675\pi$
$293$	2.29180	0.133888	0.0669441	+	0.997757i	$0.478675\pi$
$294$	0	0
$295$	−12.9443	−0.753645
$296$	17.2361	1.00183
$297$	22.3607	1.29750
$298$	−17.4164	−1.00891
$299$	−4.23607	−0.244978
$300$	−2.14590	−0.123893
$301$	0	0
$302$	−11.6180	−0.668543
$303$	−13.4164	−0.770752
$304$	−13.4164	−0.769484
$305$	−16.5836	−0.949574
$306$	0	0
$307$	−15.4164	−0.879861	−0.439930	−	0.898032i	$-0.644997\pi$
$307$	−15.4164	−0.879861	−0.439930	+	0.898032i	$0.644997\pi$
$308$	0	0
$309$	2.29180	0.130376
$310$	18.0000	1.02233
$311$	−32.8885	−1.86494	−0.932469	−	0.361250i	$-0.882350\pi$
$311$	−32.8885	−1.86494	−0.932469	+	0.361250i	$0.882350\pi$
$312$	9.47214	0.536254
$313$	−7.23607	−0.409007	−0.204503	−	0.978866i	$-0.565558\pi$
$313$	−7.23607	−0.409007	−0.204503	+	0.978866i	$0.565558\pi$
$314$	−18.4721	−1.04244
$315$	0	0
$316$	−1.70820	−0.0960940
$317$	−31.3050	−1.75826	−0.879131	−	0.476581i	$-0.841876\pi$
$317$	−31.3050	−1.75826	−0.879131	+	0.476581i	$0.841876\pi$
$318$	−10.9443	−0.613724
$319$	−33.4164	−1.87096
$320$	−5.23607	−0.292705
$321$	6.76393	0.377526
$322$	0	0
$323$	0	0
$324$	0.618034	0.0343352
$325$	−14.7082	−0.815864
$326$	5.14590	0.285005
$327$	11.4164	0.631329
$328$	−5.00000	−0.276079
$329$	0	0
$330$	−8.94427	−0.492366
$331$	−12.7082	−0.698506	−0.349253	−	0.937028i	$-0.613565\pi$
$331$	−12.7082	−0.698506	−0.349253	+	0.937028i	$0.613565\pi$
$332$	1.52786	0.0838524
$333$	−15.4164	−0.844814
$334$	−22.4721	−1.22962
$335$	−12.5836	−0.687515
$336$	0	0
$337$	31.5967	1.72118	0.860592	−	0.509295i	$-0.170094\pi$
$337$	31.5967	1.72118	0.860592	+	0.509295i	$0.170094\pi$
$338$	−8.00000	−0.435143
$339$	−6.47214	−0.351518
$340$	0	0
$341$	−40.2492	−2.17962
$342$	8.94427	0.483651
$343$	0	0
$344$	−14.4721	−0.780285
$345$	1.23607	0.0665477
$346$	−20.1803	−1.08490
$347$	29.8885	1.60450	0.802251	−	0.596987i	$-0.203636\pi$
$347$	29.8885	1.60450	0.802251	+	0.596987i	$0.203636\pi$
$348$	4.61803	0.247553
$349$	−14.7082	−0.787312	−0.393656	−	0.919258i	$-0.628790\pi$
$349$	−14.7082	−0.787312	−0.393656	+	0.919258i	$0.628790\pi$
$350$	0	0
$351$	−21.1803	−1.13052
$352$	−15.1246	−0.806145
$353$	6.23607	0.331912	0.165956	−	0.986133i	$-0.446929\pi$
$353$	6.23607	0.331912	0.165956	+	0.986133i	$0.446929\pi$
$354$	−16.9443	−0.900578
$355$	7.12461	0.378135
$356$	5.52786	0.292976
$357$	0	0
$358$	8.56231	0.452532
$359$	9.23607	0.487461	0.243731	−	0.969843i	$-0.421629\pi$
$359$	9.23607	0.487461	0.243731	+	0.969843i	$0.421629\pi$
$360$	5.52786	0.291344
$361$	−11.3607	−0.597931
$362$	−37.1246	−1.95123
$363$	9.00000	0.472377
$364$	0	0
$365$	8.29180	0.434012
$366$	−21.7082	−1.13471
$367$	31.4164	1.63992	0.819962	−	0.572419i	$-0.193995\pi$
$367$	31.4164	1.63992	0.819962	+	0.572419i	$0.193995\pi$
$368$	4.85410	0.253038
$369$	4.47214	0.232810
$370$	15.4164	0.801461
$371$	0	0
$372$	5.56231	0.288392
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	−12.2361	−0.631027
$377$	31.6525	1.63019
$378$	0	0
$379$	21.7082	1.11508	0.557538	−	0.830152i	$-0.311746\pi$
$379$	21.7082	1.11508	0.557538	+	0.830152i	$0.311746\pi$
$380$	−2.11146	−0.108315
$381$	10.7082	0.548598
$382$	2.94427	0.150642
$383$	2.29180	0.117105	0.0585527	−	0.998284i	$-0.481351\pi$
$383$	2.29180	0.117105	0.0585527	+	0.998284i	$0.481351\pi$
$384$	−13.6180	−0.694942
$385$	0	0
$386$	36.2705	1.84612
$387$	12.9443	0.657994
$388$	6.00000	0.304604
$389$	19.4164	0.984451	0.492225	−	0.870468i	$-0.336184\pi$
$389$	19.4164	0.984451	0.492225	+	0.870468i	$0.336184\pi$
$390$	8.47214	0.429003
$391$	0	0
$392$	0	0
$393$	13.9443	0.703395
$394$	20.2705	1.02121
$395$	3.41641	0.171898
$396$	5.52786	0.277786
$397$	−5.65248	−0.283690	−0.141845	−	0.989889i	$-0.545303\pi$
$397$	−5.65248	−0.283690	−0.141845	+	0.989889i	$0.545303\pi$
$398$	−32.1803	−1.61305
$399$	0	0
$400$	16.8541	0.842705
$401$	15.8885	0.793436	0.396718	−	0.917941i	$-0.370149\pi$
$401$	15.8885	0.793436	0.396718	+	0.917941i	$0.370149\pi$
$402$	−16.4721	−0.821555
$403$	38.1246	1.89912
$404$	−8.29180	−0.412532
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	−34.4721	−1.70872
$408$	0	0
$409$	36.1246	1.78625	0.893124	−	0.449811i	$-0.148509\pi$
$409$	36.1246	1.78625	0.893124	+	0.449811i	$0.148509\pi$
$410$	−4.47214	−0.220863
$411$	2.29180	0.113046
$412$	1.41641	0.0697814
$413$	0	0
$414$	−3.23607	−0.159044
$415$	−3.05573	−0.150000
$416$	14.3262	0.702402
$417$	−11.4721	−0.561793
$418$	20.0000	0.978232
$419$	−17.5279	−0.856292	−0.428146	−	0.903710i	$-0.640833\pi$
$419$	−17.5279	−0.856292	−0.428146	+	0.903710i	$0.640833\pi$
$420$	0	0
$421$	−35.4164	−1.72609	−0.863045	−	0.505127i	$-0.831446\pi$
$421$	−35.4164	−1.72609	−0.863045	+	0.505127i	$0.831446\pi$
$422$	19.4164	0.945176
$423$	10.9443	0.532129
$424$	15.1246	0.734516
$425$	0	0
$426$	9.32624	0.451858
$427$	0	0
$428$	4.18034	0.202064
$429$	−18.9443	−0.914638
$430$	−12.9443	−0.624228
$431$	−4.18034	−0.201360	−0.100680	−	0.994919i	$-0.532102\pi$
$431$	−4.18034	−0.201360	−0.100680	+	0.994919i	$0.532102\pi$
$432$	24.2705	1.16772
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	−9.23607	−0.442836
$436$	7.05573	0.337908
$437$	−2.76393	−0.132217
$438$	10.8541	0.518629
$439$	32.3050	1.54183	0.770916	−	0.636937i	$-0.219799\pi$
$439$	32.3050	1.54183	0.770916	+	0.636937i	$0.219799\pi$
$440$	12.3607	0.589272
$441$	0	0
$442$	0	0
$443$	15.1803	0.721240	0.360620	−	0.932713i	$-0.382565\pi$
$443$	15.1803	0.721240	0.360620	+	0.932713i	$0.382565\pi$
$444$	4.76393	0.226086
$445$	−11.0557	−0.524092
$446$	−37.8885	−1.79407
$447$	10.7639	0.509117
$448$	0	0
$449$	−20.8328	−0.983161	−0.491581	−	0.870832i	$-0.663581\pi$
$449$	−20.8328	−0.983161	−0.491581	+	0.870832i	$0.663581\pi$
$450$	−11.2361	−0.529673
$451$	10.0000	0.470882
$452$	−4.00000	−0.188144
$453$	7.18034	0.337362
$454$	48.3607	2.26968
$455$	0	0
$456$	6.18034	0.289421
$457$	5.52786	0.258583	0.129291	−	0.991607i	$-0.458730\pi$
$457$	5.52786	0.258583	0.129291	+	0.991607i	$0.458730\pi$
$458$	31.1246	1.45436
$459$	0	0
$460$	0.763932	0.0356185
$461$	30.2361	1.40823	0.704117	−	0.710084i	$-0.251343\pi$
$461$	30.2361	1.40823	0.704117	+	0.710084i	$0.251343\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	−36.2705	−1.68382
$465$	−11.1246	−0.515892
$466$	4.09017	0.189473
$467$	20.4721	0.947337	0.473669	−	0.880703i	$-0.342929\pi$
$467$	20.4721	0.947337	0.473669	+	0.880703i	$0.342929\pi$
$468$	−5.23607	−0.242037
$469$	0	0
$470$	−10.9443	−0.504822
$471$	11.4164	0.526040
$472$	23.4164	1.07783
$473$	28.9443	1.33086
$474$	4.47214	0.205412
$475$	−9.59675	−0.440329
$476$	0	0
$477$	−13.5279	−0.619398
$478$	32.7426	1.49761
$479$	−17.1246	−0.782443	−0.391222	−	0.920296i	$-0.627947\pi$
$479$	−17.1246	−0.782443	−0.391222	+	0.920296i	$0.627947\pi$
$480$	−4.18034	−0.190806
$481$	32.6525	1.48882
$482$	8.76393	0.399186
$483$	0	0
$484$	5.56231	0.252832
$485$	−12.0000	−0.544892
$486$	−25.8885	−1.17433
$487$	−20.1246	−0.911933	−0.455967	−	0.889997i	$-0.650706\pi$
$487$	−20.1246	−0.911933	−0.455967	+	0.889997i	$0.650706\pi$
$488$	30.0000	1.35804
$489$	−3.18034	−0.143820
$490$	0	0
$491$	16.2361	0.732723	0.366362	−	0.930472i	$-0.380603\pi$
$491$	16.2361	0.732723	0.366362	+	0.930472i	$0.380603\pi$
$492$	−1.38197	−0.0623038
$493$	0	0
$494$	−18.9443	−0.852343
$495$	−11.0557	−0.496918
$496$	−43.6869	−1.96160
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−23.2918	−1.04268	−0.521342	−	0.853348i	$-0.674568\pi$
$499$	−23.2918	−1.04268	−0.521342	+	0.853348i	$0.674568\pi$
$500$	6.47214	0.289443
$501$	13.8885	0.620494
$502$	35.1246	1.56769
$503$	−33.5967	−1.49800	−0.749002	−	0.662567i	$-0.769467\pi$
$503$	−33.5967	−1.49800	−0.749002	+	0.662567i	$0.769467\pi$
$504$	0	0
$505$	16.5836	0.737960
$506$	−7.23607	−0.321682
$507$	4.94427	0.219583
$508$	6.61803	0.293628
$509$	35.1803	1.55934	0.779671	−	0.626190i	$-0.215387\pi$
$509$	35.1803	1.55934	0.779671	+	0.626190i	$0.215387\pi$
$510$	0	0
$511$	0	0
$512$	5.29180	0.233867
$513$	−13.8197	−0.610153
$514$	−0.381966	−0.0168478
$515$	−2.83282	−0.124829
$516$	−4.00000	−0.176090
$517$	24.4721	1.07628
$518$	0	0
$519$	12.4721	0.547466
$520$	−11.7082	−0.513439
$521$	41.3050	1.80960	0.904801	−	0.425834i	$-0.140019\pi$
$521$	41.3050	1.80960	0.904801	+	0.425834i	$0.140019\pi$
$522$	24.1803	1.05834
$523$	−5.70820	−0.249602	−0.124801	−	0.992182i	$-0.539829\pi$
$523$	−5.70820	−0.249602	−0.124801	+	0.992182i	$0.539829\pi$
$524$	8.61803	0.376481
$525$	0	0
$526$	14.6525	0.638878
$527$	0	0
$528$	21.7082	0.944728
$529$	1.00000	0.0434783
$530$	13.5279	0.587613
$531$	−20.9443	−0.908904
$532$	0	0
$533$	−9.47214	−0.410284
$534$	−14.4721	−0.626271
$535$	−8.36068	−0.361464
$536$	22.7639	0.983252
$537$	−5.29180	−0.228358
$538$	21.3262	0.919439
$539$	0	0
$540$	3.81966	0.164372
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	1.52786	0.0656274
$543$	22.9443	0.984633
$544$	0	0
$545$	−14.1115	−0.604468
$546$	0	0
$547$	20.8197	0.890184	0.445092	−	0.895485i	$-0.353171\pi$
$547$	20.8197	0.890184	0.445092	+	0.895485i	$0.353171\pi$
$548$	1.41641	0.0605059
$549$	−26.8328	−1.14520
$550$	−25.1246	−1.07132
$551$	20.6525	0.879825
$552$	−2.23607	−0.0951734
$553$	0	0
$554$	10.3820	0.441087
$555$	−9.52786	−0.404435
$556$	−7.09017	−0.300690
$557$	−31.2361	−1.32351	−0.661757	−	0.749718i	$-0.730189\pi$
$557$	−31.2361	−1.32351	−0.661757	+	0.749718i	$0.730189\pi$
$558$	29.1246	1.23294
$559$	−27.4164	−1.15959
$560$	0	0
$561$	0	0
$562$	15.7082	0.662611
$563$	−39.5967	−1.66880	−0.834402	−	0.551156i	$-0.814187\pi$
$563$	−39.5967	−1.66880	−0.834402	+	0.551156i	$0.814187\pi$
$564$	−3.38197	−0.142406
$565$	8.00000	0.336563
$566$	1.70820	0.0718012
$567$	0	0
$568$	−12.8885	−0.540791
$569$	14.1803	0.594471	0.297235	−	0.954804i	$-0.403935\pi$
$569$	14.1803	0.594471	0.297235	+	0.954804i	$0.403935\pi$
$570$	5.52786	0.231537
$571$	29.7082	1.24325	0.621625	−	0.783315i	$-0.286473\pi$
$571$	29.7082	1.24325	0.621625	+	0.783315i	$0.286473\pi$
$572$	−11.7082	−0.489545
$573$	−1.81966	−0.0760174
$574$	0	0
$575$	3.47214	0.144798
$576$	−8.47214	−0.353006
$577$	28.7082	1.19514	0.597569	−	0.801817i	$-0.296133\pi$
$577$	28.7082	1.19514	0.597569	+	0.801817i	$0.296133\pi$
$578$	27.5066	1.14412
$579$	−22.4164	−0.931594
$580$	−5.70820	−0.237020
$581$	0	0
$582$	−15.7082	−0.651126
$583$	−30.2492	−1.25279
$584$	−15.0000	−0.620704
$585$	10.4721	0.432970
$586$	−3.70820	−0.153184
$587$	14.8885	0.614516	0.307258	−	0.951626i	$-0.400589\pi$
$587$	14.8885	0.614516	0.307258	+	0.951626i	$0.400589\pi$
$588$	0	0
$589$	24.8754	1.02497
$590$	20.9443	0.862262
$591$	−12.5279	−0.515327
$592$	−37.4164	−1.53780
$593$	10.3607	0.425462	0.212731	−	0.977111i	$-0.431764\pi$
$593$	10.3607	0.425462	0.212731	+	0.977111i	$0.431764\pi$
$594$	−36.1803	−1.48450
$595$	0	0
$596$	6.65248	0.272496
$597$	19.8885	0.813984
$598$	6.85410	0.280285
$599$	6.47214	0.264444	0.132222	−	0.991220i	$-0.457789\pi$
$599$	6.47214	0.264444	0.132222	+	0.991220i	$0.457789\pi$
$600$	−7.76393	−0.316961
$601$	−17.7639	−0.724606	−0.362303	−	0.932060i	$-0.618009\pi$
$601$	−17.7639	−0.724606	−0.362303	+	0.932060i	$0.618009\pi$
$602$	0	0
$603$	−20.3607	−0.829151
$604$	4.43769	0.180567
$605$	−11.1246	−0.452280
$606$	21.7082	0.881836
$607$	−29.3050	−1.18945	−0.594726	−	0.803929i	$-0.702739\pi$
$607$	−29.3050	−1.18945	−0.594726	+	0.803929i	$0.702739\pi$
$608$	9.34752	0.379092
$609$	0	0
$610$	26.8328	1.08643
$611$	−23.1803	−0.937776
$612$	0	0
$613$	0.180340	0.00728386	0.00364193	−	0.999993i	$-0.498841\pi$
$613$	0.180340	0.00728386	0.00364193	+	0.999993i	$0.498841\pi$
$614$	24.9443	1.00667
$615$	2.76393	0.111452
$616$	0	0
$617$	30.7639	1.23851	0.619255	−	0.785190i	$-0.287435\pi$
$617$	30.7639	1.23851	0.619255	+	0.785190i	$0.287435\pi$
$618$	−3.70820	−0.149166
$619$	−37.4853	−1.50666	−0.753331	−	0.657642i	$-0.771554\pi$
$619$	−37.4853	−1.50666	−0.753331	+	0.657642i	$0.771554\pi$
$620$	−6.87539	−0.276122
$621$	5.00000	0.200643
$622$	53.2148	2.13372
$623$	0	0
$624$	−20.5623	−0.823151
$625$	4.41641	0.176656
$626$	11.7082	0.467954
$627$	−12.3607	−0.493638
$628$	7.05573	0.281554
$629$	0	0
$630$	0	0
$631$	−23.4164	−0.932192	−0.466096	−	0.884734i	$-0.654340\pi$
$631$	−23.4164	−0.932192	−0.466096	+	0.884734i	$0.654340\pi$
$632$	−6.18034	−0.245841
$633$	−12.0000	−0.476957
$634$	50.6525	2.01167
$635$	−13.2361	−0.525257
$636$	4.18034	0.165761
$637$	0	0
$638$	54.0689	2.14061
$639$	11.5279	0.456035
$640$	16.8328	0.665375
$641$	24.9443	0.985240	0.492620	−	0.870245i	$-0.336039\pi$
$641$	24.9443	0.985240	0.492620	+	0.870245i	$0.336039\pi$
$642$	−10.9443	−0.431936
$643$	4.29180	0.169252	0.0846260	−	0.996413i	$-0.473030\pi$
$643$	4.29180	0.169252	0.0846260	+	0.996413i	$0.473030\pi$
$644$	0	0
$645$	8.00000	0.315000
$646$	0	0
$647$	27.4721	1.08004	0.540021	−	0.841652i	$-0.318416\pi$
$647$	27.4721	1.08004	0.540021	+	0.841652i	$0.318416\pi$
$648$	2.23607	0.0878410
$649$	−46.8328	−1.83835
$650$	23.7984	0.933449
$651$	0	0
$652$	−1.96556	−0.0769772
$653$	33.4721	1.30987	0.654933	−	0.755687i	$-0.272697\pi$
$653$	33.4721	1.30987	0.654933	+	0.755687i	$0.272697\pi$
$654$	−18.4721	−0.722318
$655$	−17.2361	−0.673469
$656$	10.8541	0.423781
$657$	13.4164	0.523424
$658$	0	0
$659$	11.2361	0.437695	0.218848	−	0.975759i	$-0.429770\pi$
$659$	11.2361	0.437695	0.218848	+	0.975759i	$0.429770\pi$
$660$	3.41641	0.132983
$661$	28.2918	1.10042	0.550212	−	0.835025i	$-0.314547\pi$
$661$	28.2918	1.10042	0.550212	+	0.835025i	$0.314547\pi$
$662$	20.5623	0.799177
$663$	0	0
$664$	5.52786	0.214523
$665$	0	0
$666$	24.9443	0.966571
$667$	−7.47214	−0.289322
$668$	8.58359	0.332109
$669$	23.4164	0.905331
$670$	20.3607	0.786602
$671$	−60.0000	−2.31627
$672$	0	0
$673$	−1.00000	−0.0385472	−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000	−0.0385472	−0.0192736	+	0.999814i	$0.506135\pi$
$674$	−51.1246	−1.96925
$675$	17.3607	0.668213
$676$	3.05573	0.117528
$677$	−18.7639	−0.721156	−0.360578	−	0.932729i	$-0.617421\pi$
$677$	−18.7639	−0.721156	−0.360578	+	0.932729i	$0.617421\pi$
$678$	10.4721	0.402180
$679$	0	0
$680$	0	0
$681$	−29.8885	−1.14533
$682$	65.1246	2.49375
$683$	17.1803	0.657387	0.328694	−	0.944437i	$-0.393392\pi$
$683$	17.1803	0.657387	0.328694	+	0.944437i	$0.393392\pi$
$684$	−3.41641	−0.130630
$685$	−2.83282	−0.108236
$686$	0	0
$687$	−19.2361	−0.733902
$688$	31.4164	1.19774
$689$	28.6525	1.09157
$690$	−2.00000	−0.0761387
$691$	10.8328	0.412100	0.206050	−	0.978541i	$-0.433939\pi$
$691$	10.8328	0.412100	0.206050	+	0.978541i	$0.433939\pi$
$692$	7.70820	0.293022
$693$	0	0
$694$	−48.3607	−1.83575
$695$	14.1803	0.537891
$696$	16.7082	0.633323
$697$	0	0
$698$	23.7984	0.900782
$699$	−2.52786	−0.0956126
$700$	0	0
$701$	−27.2361	−1.02869	−0.514346	−	0.857583i	$-0.671965\pi$
$701$	−27.2361	−1.02869	−0.514346	+	0.857583i	$0.671965\pi$
$702$	34.2705	1.29346
$703$	21.3050	0.803531
$704$	−18.9443	−0.713989
$705$	6.76393	0.254744
$706$	−10.0902	−0.379749
$707$	0	0
$708$	6.47214	0.243238
$709$	−46.2492	−1.73693	−0.868463	−	0.495754i	$-0.834892\pi$
$709$	−46.2492	−1.73693	−0.868463	+	0.495754i	$0.834892\pi$
$710$	−11.5279	−0.432633
$711$	5.52786	0.207311
$712$	20.0000	0.749532
$713$	−9.00000	−0.337053
$714$	0	0
$715$	23.4164	0.875724
$716$	−3.27051	−0.122225
$717$	−20.2361	−0.755730
$718$	−14.9443	−0.557715
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	−12.0000	−0.447214
$721$	0	0
$722$	18.3820	0.684106
$723$	−5.41641	−0.201438
$724$	14.1803	0.527008
$725$	−25.9443	−0.963546
$726$	−14.5623	−0.540458
$727$	1.23607	0.0458432	0.0229216	−	0.999737i	$-0.492703\pi$
$727$	1.23607	0.0458432	0.0229216	+	0.999737i	$0.492703\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	−13.4164	−0.496564
$731$	0	0
$732$	8.29180	0.306474
$733$	4.94427	0.182621	0.0913104	−	0.995822i	$-0.470894\pi$
$733$	4.94427	0.182621	0.0913104	+	0.995822i	$0.470894\pi$
$734$	−50.8328	−1.87627
$735$	0	0
$736$	−3.38197	−0.124661
$737$	−45.5279	−1.67704
$738$	−7.23607	−0.266363
$739$	−4.70820	−0.173194	−0.0865970	−	0.996243i	$-0.527599\pi$
$739$	−4.70820	−0.173194	−0.0865970	+	0.996243i	$0.527599\pi$
$740$	−5.88854	−0.216467
$741$	11.7082	0.430112
$742$	0	0
$743$	−34.4721	−1.26466	−0.632330	−	0.774699i	$-0.717901\pi$
$743$	−34.4721	−1.26466	−0.632330	+	0.774699i	$0.717901\pi$
$744$	20.1246	0.737804
$745$	−13.3050	−0.487456
$746$	−42.0689	−1.54025
$747$	−4.94427	−0.180901
$748$	0	0
$749$	0	0
$750$	−16.9443	−0.618717
$751$	−35.4164	−1.29236	−0.646182	−	0.763184i	$-0.723635\pi$
$751$	−35.4164	−1.29236	−0.646182	+	0.763184i	$0.723635\pi$
$752$	26.5623	0.968628
$753$	−21.7082	−0.791091
$754$	−51.2148	−1.86513
$755$	−8.87539	−0.323008
$756$	0	0
$757$	24.8328	0.902564	0.451282	−	0.892381i	$-0.350967\pi$
$757$	24.8328	0.902564	0.451282	+	0.892381i	$0.350967\pi$
$758$	−35.1246	−1.27578
$759$	4.47214	0.162328
$760$	−7.63932	−0.277107
$761$	0.347524	0.0125977	0.00629887	−	0.999980i	$-0.497995\pi$
$761$	0.347524	0.0125977	0.00629887	+	0.999980i	$0.497995\pi$
$762$	−17.3262	−0.627663
$763$	0	0
$764$	−1.12461	−0.0406870
$765$	0	0
$766$	−3.70820	−0.133983
$767$	44.3607	1.60177
$768$	13.5623	0.489388
$769$	38.7639	1.39786	0.698932	−	0.715189i	$-0.253659\pi$
$769$	38.7639	1.39786	0.698932	+	0.715189i	$0.253659\pi$
$770$	0	0
$771$	0.236068	0.00850178
$772$	−13.8541	−0.498620
$773$	−9.34752	−0.336207	−0.168104	−	0.985769i	$-0.553764\pi$
$773$	−9.34752	−0.336207	−0.168104	+	0.985769i	$0.553764\pi$
$774$	−20.9443	−0.752826
$775$	−31.2492	−1.12251
$776$	21.7082	0.779279
$777$	0	0
$778$	−31.4164	−1.12633
$779$	−6.18034	−0.221434
$780$	−3.23607	−0.115870
$781$	25.7771	0.922377
$782$	0	0
$783$	−37.3607	−1.33516
$784$	0	0
$785$	−14.1115	−0.503659
$786$	−22.5623	−0.804771
$787$	−44.3607	−1.58129	−0.790644	−	0.612276i	$-0.790254\pi$
$787$	−44.3607	−1.58129	−0.790644	+	0.612276i	$0.790254\pi$
$788$	−7.74265	−0.275820
$789$	−9.05573	−0.322392
$790$	−5.52786	−0.196673
$791$	0	0
$792$	20.0000	0.710669
$793$	56.8328	2.01819
$794$	9.14590	0.324576
$795$	−8.36068	−0.296523
$796$	12.2918	0.435671
$797$	41.2361	1.46066	0.730328	−	0.683096i	$-0.239367\pi$
$797$	41.2361	1.46066	0.730328	+	0.683096i	$0.239367\pi$
$798$	0	0
$799$	0	0
$800$	−11.7426	−0.415165
$801$	−17.8885	−0.632061
$802$	−25.7082	−0.907788
$803$	30.0000	1.05868
$804$	6.29180	0.221895
$805$	0	0
$806$	−61.6869	−2.17283
$807$	−13.1803	−0.463970
$808$	−30.0000	−1.05540
$809$	20.8328	0.732443	0.366221	−	0.930528i	$-0.380651\pi$
$809$	20.8328	0.732443	0.366221	+	0.930528i	$0.380651\pi$
$810$	2.00000	0.0702728
$811$	0.639320	0.0224496	0.0112248	−	0.999937i	$-0.496427\pi$
$811$	0.639320	0.0224496	0.0112248	+	0.999937i	$0.496427\pi$
$812$	0	0
$813$	−0.944272	−0.0331171
$814$	55.7771	1.95499
$815$	3.93112	0.137701
$816$	0	0
$817$	−17.8885	−0.625841
$818$	−58.4508	−2.04369
$819$	0	0
$820$	1.70820	0.0596531
$821$	49.4164	1.72464	0.862322	−	0.506360i	$-0.169009\pi$
$821$	49.4164	1.72464	0.862322	+	0.506360i	$0.169009\pi$
$822$	−3.70820	−0.129338
$823$	−41.5410	−1.44803	−0.724014	−	0.689785i	$-0.757705\pi$
$823$	−41.5410	−1.44803	−0.724014	+	0.689785i	$0.757705\pi$
$824$	5.12461	0.178524
$825$	15.5279	0.540611
$826$	0	0
$827$	45.3050	1.57541	0.787704	−	0.616054i	$-0.211270\pi$
$827$	45.3050	1.57541	0.787704	+	0.616054i	$0.211270\pi$
$828$	1.23607	0.0429563
$829$	14.0000	0.486240	0.243120	−	0.969996i	$-0.421829\pi$
$829$	14.0000	0.486240	0.243120	+	0.969996i	$0.421829\pi$
$830$	4.94427	0.171618
$831$	−6.41641	−0.222583
$832$	17.9443	0.622106
$833$	0	0
$834$	18.5623	0.642760
$835$	−17.1672	−0.594095
$836$	−7.63932	−0.264211
$837$	−45.0000	−1.55543
$838$	28.3607	0.979703
$839$	38.0689	1.31428	0.657142	−	0.753767i	$-0.271765\pi$
$839$	38.0689	1.31428	0.657142	+	0.753767i	$0.271765\pi$
$840$	0	0
$841$	26.8328	0.925270
$842$	57.3050	1.97486
$843$	−9.70820	−0.334368
$844$	−7.41641	−0.255283
$845$	−6.11146	−0.210240
$846$	−17.7082	−0.608821
$847$	0	0
$848$	−32.8328	−1.12748
$849$	−1.05573	−0.0362325
$850$	0	0
$851$	−7.70820	−0.264234
$852$	−3.56231	−0.122043
$853$	8.83282	0.302430	0.151215	−	0.988501i	$-0.451681\pi$
$853$	8.83282	0.302430	0.151215	+	0.988501i	$0.451681\pi$
$854$	0	0
$855$	6.83282	0.233677
$856$	15.1246	0.516949
$857$	57.5410	1.96556	0.982782	−	0.184769i	$-0.0591538\pi$
$857$	57.5410	1.96556	0.982782	+	0.184769i	$0.0591538\pi$
$858$	30.6525	1.04646
$859$	−38.4164	−1.31075	−0.655375	−	0.755303i	$-0.727490\pi$
$859$	−38.4164	−1.31075	−0.655375	+	0.755303i	$0.727490\pi$
$860$	4.94427	0.168598
$861$	0	0
$862$	6.76393	0.230380
$863$	−2.23607	−0.0761166	−0.0380583	−	0.999276i	$-0.512117\pi$
$863$	−2.23607	−0.0761166	−0.0380583	+	0.999276i	$0.512117\pi$
$864$	−16.9098	−0.575284
$865$	−15.4164	−0.524174
$866$	22.6525	0.769762
$867$	−17.0000	−0.577350
$868$	0	0
$869$	12.3607	0.419307
$870$	14.9443	0.506658
$871$	43.1246	1.46122
$872$	25.5279	0.864483
$873$	−19.4164	−0.657146
$874$	4.47214	0.151272
$875$	0	0
$876$	−4.14590	−0.140077
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−52.2705	−1.76404
$879$	2.29180	0.0773004
$880$	−26.8328	−0.904534
$881$	23.5967	0.794995	0.397497	−	0.917603i	$-0.369879\pi$
$881$	23.5967	0.794995	0.397497	+	0.917603i	$0.369879\pi$
$882$	0	0
$883$	−25.8885	−0.871219	−0.435609	−	0.900136i	$-0.643467\pi$
$883$	−25.8885	−0.871219	−0.435609	+	0.900136i	$0.643467\pi$
$884$	0	0
$885$	−12.9443	−0.435117
$886$	−24.5623	−0.825187
$887$	18.8885	0.634215	0.317108	−	0.948390i	$-0.397288\pi$
$887$	18.8885	0.634215	0.317108	+	0.948390i	$0.397288\pi$
$888$	17.2361	0.578405
$889$	0	0
$890$	17.8885	0.599625
$891$	−4.47214	−0.149822
$892$	14.4721	0.484563
$893$	−15.1246	−0.506126
$894$	−17.4164	−0.582492
$895$	6.54102	0.218642
$896$	0	0
$897$	−4.23607	−0.141438
$898$	33.7082	1.12486
$899$	67.2492	2.24289
$900$	4.29180	0.143060
$901$	0	0
$902$	−16.1803	−0.538746
$903$	0	0
$904$	−14.4721	−0.481336
$905$	−28.3607	−0.942741
$906$	−11.6180	−0.385983
$907$	−54.5410	−1.81100	−0.905502	−	0.424341i	$-0.860506\pi$
$907$	−54.5410	−1.81100	−0.905502	+	0.424341i	$0.860506\pi$
$908$	−18.4721	−0.613019
$909$	26.8328	0.889988
$910$	0	0
$911$	49.7771	1.64919	0.824594	−	0.565725i	$-0.191403\pi$
$911$	49.7771	1.64919	0.824594	+	0.565725i	$0.191403\pi$
$912$	−13.4164	−0.444262
$913$	−11.0557	−0.365891
$914$	−8.94427	−0.295850
$915$	−16.5836	−0.548237
$916$	−11.8885	−0.392809
$917$	0	0
$918$	0	0
$919$	−9.70820	−0.320244	−0.160122	−	0.987097i	$-0.551189\pi$
$919$	−9.70820	−0.320244	−0.160122	+	0.987097i	$0.551189\pi$
$920$	2.76393	0.0911241
$921$	−15.4164	−0.507988
$922$	−48.9230	−1.61119
$923$	−24.4164	−0.803676
$924$	0	0
$925$	−26.7639	−0.879993
$926$	0	0
$927$	−4.58359	−0.150545
$928$	25.2705	0.829545
$929$	−20.3475	−0.667581	−0.333790	−	0.942647i	$-0.608328\pi$
$929$	−20.3475	−0.667581	−0.333790	+	0.942647i	$0.608328\pi$
$930$	18.0000	0.590243
$931$	0	0
$932$	−1.56231	−0.0511750
$933$	−32.8885	−1.07672
$934$	−33.1246	−1.08387
$935$	0	0
$936$	−18.9443	−0.619213
$937$	16.8754	0.551295	0.275647	−	0.961259i	$-0.411108\pi$
$937$	16.8754	0.551295	0.275647	+	0.961259i	$0.411108\pi$
$938$	0	0
$939$	−7.23607	−0.236140
$940$	4.18034	0.136348
$941$	23.0132	0.750207	0.375104	−	0.926983i	$-0.377607\pi$
$941$	23.0132	0.750207	0.375104	+	0.926983i	$0.377607\pi$
$942$	−18.4721	−0.601855
$943$	2.23607	0.0728164
$944$	−50.8328	−1.65447
$945$	0	0
$946$	−46.8328	−1.52267
$947$	26.5967	0.864278	0.432139	−	0.901807i	$-0.357759\pi$
$947$	26.5967	0.864278	0.432139	+	0.901807i	$0.357759\pi$
$948$	−1.70820	−0.0554799
$949$	−28.4164	−0.922436
$950$	15.5279	0.503790
$951$	−31.3050	−1.01513
$952$	0	0
$953$	47.8885	1.55126	0.775631	−	0.631187i	$-0.217432\pi$
$953$	47.8885	1.55126	0.775631	+	0.631187i	$0.217432\pi$
$954$	21.8885	0.708668
$955$	2.24922	0.0727832
$956$	−12.5066	−0.404492
$957$	−33.4164	−1.08020
$958$	27.7082	0.895211
$959$	0	0
$960$	−5.23607	−0.168993
$961$	50.0000	1.61290
$962$	−52.8328	−1.70340
$963$	−13.5279	−0.435929
$964$	−3.34752	−0.107816
$965$	27.7082	0.891959
$966$	0	0
$967$	27.0689	0.870477	0.435238	−	0.900315i	$-0.356664\pi$
$967$	27.0689	0.870477	0.435238	+	0.900315i	$0.356664\pi$
$968$	20.1246	0.646830
$969$	0	0
$970$	19.4164	0.623423
$971$	−20.2918	−0.651195	−0.325597	−	0.945509i	$-0.605565\pi$
$971$	−20.2918	−0.651195	−0.325597	+	0.945509i	$0.605565\pi$
$972$	9.88854	0.317175
$973$	0	0
$974$	32.5623	1.04336
$975$	−14.7082	−0.471040
$976$	−65.1246	−2.08459
$977$	12.6525	0.404789	0.202394	−	0.979304i	$-0.435128\pi$
$977$	12.6525	0.404789	0.202394	+	0.979304i	$0.435128\pi$
$978$	5.14590	0.164548
$979$	−40.0000	−1.27841
$980$	0	0
$981$	−22.8328	−0.728996
$982$	−26.2705	−0.838326
$983$	−18.1803	−0.579863	−0.289931	−	0.957047i	$-0.593632\pi$
$983$	−18.1803	−0.579863	−0.289931	+	0.957047i	$0.593632\pi$
$984$	−5.00000	−0.159394
$985$	15.4853	0.493402
$986$	0	0
$987$	0	0
$988$	7.23607	0.230210
$989$	6.47214	0.205802
$990$	17.8885	0.568535
$991$	−31.0557	−0.986518	−0.493259	−	0.869883i	$-0.664194\pi$
$991$	−31.0557	−0.986518	−0.493259	+	0.869883i	$0.664194\pi$
$992$	30.4377	0.966398
$993$	−12.7082	−0.403283
$994$	0	0
$995$	−24.5836	−0.779352
$996$	1.52786	0.0484122
$997$	−35.3050	−1.11812	−0.559060	−	0.829128i	$-0.688838\pi$
$997$	−35.3050	−1.11812	−0.559060	+	0.829128i	$0.688838\pi$
$998$	37.6869	1.19296
$999$	−38.5410	−1.21938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1127.2.a.d.1.1		2
7.6	odd	2		161.2.a.b.1.1	✓	2
21.20	even	2		1449.2.a.i.1.2		2
28.27	even	2		2576.2.a.s.1.2		2
35.34	odd	2		4025.2.a.i.1.2		2
161.160	even	2		3703.2.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.b.1.1	✓	2	7.6	odd	2
1127.2.a.d.1.1		2	1.1	even	1	trivial
1449.2.a.i.1.2		2	21.20	even	2
2576.2.a.s.1.2		2	28.27	even	2
3703.2.a.b.1.1		2	161.160	even	2
4025.2.a.i.1.2		2	35.34	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:	\( N \)	\(=\)	\( 1127 = 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1127.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} -2.00000 q^{9} +O(q^{10})\)	\(q-1.61803 q^{2} +1.00000 q^{3} +0.618034 q^{4} -1.23607 q^{5} -1.61803 q^{6} +2.23607 q^{8} -2.00000 q^{9} +2.00000 q^{10} -4.47214 q^{11} +0.618034 q^{12} +4.23607 q^{13} -1.23607 q^{15} -4.85410 q^{16} +3.23607 q^{18} +2.76393 q^{19} -0.763932 q^{20} +7.23607 q^{22} -1.00000 q^{23} +2.23607 q^{24} -3.47214 q^{25} -6.85410 q^{26} -5.00000 q^{27} +7.47214 q^{29} +2.00000 q^{30} +9.00000 q^{31} +3.38197 q^{32} -4.47214 q^{33} -1.23607 q^{36} +7.70820 q^{37} -4.47214 q^{38} +4.23607 q^{39} -2.76393 q^{40} -2.23607 q^{41} -6.47214 q^{43} -2.76393 q^{44} +2.47214 q^{45} +1.61803 q^{46} -5.47214 q^{47} -4.85410 q^{48} +5.61803 q^{50} +2.61803 q^{52} +6.76393 q^{53} +8.09017 q^{54} +5.52786 q^{55} +2.76393 q^{57} -12.0902 q^{58} +10.4721 q^{59} -0.763932 q^{60} +13.4164 q^{61} -14.5623 q^{62} +4.23607 q^{64} -5.23607 q^{65} +7.23607 q^{66} +10.1803 q^{67} -1.00000 q^{69} -5.76393 q^{71} -4.47214 q^{72} -6.70820 q^{73} -12.4721 q^{74} -3.47214 q^{75} +1.70820 q^{76} -6.85410 q^{78} -2.76393 q^{79} +6.00000 q^{80} +1.00000 q^{81} +3.61803 q^{82} +2.47214 q^{83} +10.4721 q^{86} +7.47214 q^{87} -10.0000 q^{88} +8.94427 q^{89} -4.00000 q^{90} -0.618034 q^{92} +9.00000 q^{93} +8.85410 q^{94} -3.41641 q^{95} +3.38197 q^{96} +9.70820 q^{97} +8.94427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9}+O(q^{10}) \) 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^9	\( 2 q - q^{2} + 2 q^{3} - q^{4} + 2 q^{5} - q^{6} - 4 q^{9} + 4 q^{10} - q^{12} + 4 q^{13} + 2 q^{15} - 3 q^{16} + 2 q^{18} + 10 q^{19} - 6 q^{20} + 10 q^{22} - 2 q^{23} + 2 q^{25} - 7 q^{26} - 10 q^{27} + 6 q^{29} + 4 q^{30} + 18 q^{31} + 9 q^{32} + 2 q^{36} + 2 q^{37} + 4 q^{39} - 10 q^{40} - 4 q^{43} - 10 q^{44} - 4 q^{45} + q^{46} - 2 q^{47} - 3 q^{48} + 9 q^{50} + 3 q^{52} + 18 q^{53} + 5 q^{54} + 20 q^{55} + 10 q^{57} - 13 q^{58} + 12 q^{59} - 6 q^{60} - 9 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{66} - 2 q^{67} - 2 q^{69} - 16 q^{71} - 16 q^{74} + 2 q^{75} - 10 q^{76} - 7 q^{78} - 10 q^{79} + 12 q^{80} + 2 q^{81} + 5 q^{82} - 4 q^{83} + 12 q^{86} + 6 q^{87} - 20 q^{88} - 8 q^{90} + q^{92} + 18 q^{93} + 11 q^{94} + 20 q^{95} + 9 q^{96} + 6 q^{97}+O(q^{100}) \) 2 * q - q^2 + 2 * q^3 - q^4 + 2 * q^5 - q^6 - 4 * q^9 + 4 * q^10 - q^12 + 4 * q^13 + 2 * q^15 - 3 * q^16 + 2 * q^18 + 10 * q^19 - 6 * q^20 + 10 * q^22 - 2 * q^23 + 2 * q^25 - 7 * q^26 - 10 * q^27 + 6 * q^29 + 4 * q^30 + 18 * q^31 + 9 * q^32 + 2 * q^36 + 2 * q^37 + 4 * q^39 - 10 * q^40 - 4 * q^43 - 10 * q^44 - 4 * q^45 + q^46 - 2 * q^47 - 3 * q^48 + 9 * q^50 + 3 * q^52 + 18 * q^53 + 5 * q^54 + 20 * q^55 + 10 * q^57 - 13 * q^58 + 12 * q^59 - 6 * q^60 - 9 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^66 - 2 * q^67 - 2 * q^69 - 16 * q^71 - 16 * q^74 + 2 * q^75 - 10 * q^76 - 7 * q^78 - 10 * q^79 + 12 * q^80 + 2 * q^81 + 5 * q^82 - 4 * q^83 + 12 * q^86 + 6 * q^87 - 20 * q^88 - 8 * q^90 + q^92 + 18 * q^93 + 11 * q^94 + 20 * q^95 + 9 * q^96 + 6 * q^97

Introduction

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