LMFDB - Embedded newform 1449.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$1$
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			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +3.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +3.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{10} -4.47214 q^{11} +0.236068 q^{13} +0.618034 q^{14} +1.85410 q^{16} -7.23607 q^{19} -5.23607 q^{20} +2.76393 q^{22} +1.00000 q^{23} +5.47214 q^{25} -0.145898 q^{26} +1.61803 q^{28} +1.47214 q^{29} -9.00000 q^{31} -5.61803 q^{32} -3.23607 q^{35} -5.70820 q^{37} +4.47214 q^{38} +7.23607 q^{40} +2.23607 q^{41} +2.47214 q^{43} +7.23607 q^{44} -0.618034 q^{46} +3.47214 q^{47} +1.00000 q^{49} -3.38197 q^{50} -0.381966 q^{52} -11.2361 q^{53} -14.4721 q^{55} -2.23607 q^{56} -0.909830 q^{58} +1.52786 q^{59} +13.4164 q^{61} +5.56231 q^{62} -0.236068 q^{64} +0.763932 q^{65} -12.1803 q^{67} +2.00000 q^{70} +10.2361 q^{71} -6.70820 q^{73} +3.52786 q^{74} +11.7082 q^{76} +4.47214 q^{77} -7.23607 q^{79} +6.00000 q^{80} -1.38197 q^{82} -6.47214 q^{83} -1.52786 q^{86} -10.0000 q^{88} -8.94427 q^{89} -0.236068 q^{91} -1.61803 q^{92} -2.14590 q^{94} -23.4164 q^{95} +3.70820 q^{97} -0.618034 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 2 * q^7	$ 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} - 10 q^{19} - 6 q^{20} + 10 q^{22} + 2 q^{23} + 2 q^{25} - 7 q^{26} + q^{28} - 6 q^{29} - 18 q^{31} - 9 q^{32} - 2 q^{35} + 2 q^{37} + 10 q^{40} - 4 q^{43} + 10 q^{44} + q^{46} - 2 q^{47} + 2 q^{49} - 9 q^{50} - 3 q^{52} - 18 q^{53} - 20 q^{55} - 13 q^{58} + 12 q^{59} - 9 q^{62} + 4 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{70} + 16 q^{71} + 16 q^{74} + 10 q^{76} - 10 q^{79} + 12 q^{80} - 5 q^{82} - 4 q^{83} - 12 q^{86} - 20 q^{88} + 4 q^{91} - q^{92} - 11 q^{94} - 20 q^{95} - 6 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 2 * q^5 - 2 * q^7 - 4 * q^10 - 4 * q^13 - q^14 - 3 * q^16 - 10 * q^19 - 6 * q^20 + 10 * q^22 + 2 * q^23 + 2 * q^25 - 7 * q^26 + q^28 - 6 * q^29 - 18 * q^31 - 9 * q^32 - 2 * q^35 + 2 * q^37 + 10 * q^40 - 4 * q^43 + 10 * q^44 + q^46 - 2 * q^47 + 2 * q^49 - 9 * q^50 - 3 * q^52 - 18 * q^53 - 20 * q^55 - 13 * q^58 + 12 * q^59 - 9 * q^62 + 4 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^70 + 16 * q^71 + 16 * q^74 + 10 * q^76 - 10 * q^79 + 12 * q^80 - 5 * q^82 - 4 * q^83 - 12 * q^86 - 20 * q^88 + 4 * q^91 - q^92 - 11 * q^94 - 20 * q^95 - 6 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	−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	0
$13$	0.236068	0.0654735	0.0327367	−	0.999464i	$-0.489578\pi$
$13$	0.236068	0.0654735	0.0327367	+	0.999464i	$0.489578\pi$
$14$	0.618034	0.165177
$15$	0	0
$16$	1.85410	0.463525
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−7.23607	−1.66007	−0.830034	−	0.557713i	$-0.811679\pi$
$19$	−7.23607	−1.66007	−0.830034	+	0.557713i	$0.811679\pi$
$20$	−5.23607	−1.17082
$21$	0	0
$22$	2.76393	0.589272
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	−0.145898	−0.0286130
$27$	0	0
$28$	1.61803	0.305780
$29$	1.47214	0.273369	0.136684	−	0.990615i	$-0.456355\pi$
$29$	1.47214	0.273369	0.136684	+	0.990615i	$0.456355\pi$
$30$	0	0
$31$	−9.00000	−1.61645	−0.808224	−	0.588875i	$-0.799571\pi$
$31$	−9.00000	−1.61645	−0.808224	+	0.588875i	$0.799571\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−5.70820	−0.938423	−0.469211	−	0.883086i	$-0.655462\pi$
$37$	−5.70820	−0.938423	−0.469211	+	0.883086i	$0.655462\pi$
$38$	4.47214	0.725476
$39$	0	0
$40$	7.23607	1.14412
$41$	2.23607	0.349215	0.174608	−	0.984638i	$-0.444134\pi$
$41$	2.23607	0.349215	0.174608	+	0.984638i	$0.444134\pi$
$42$	0	0
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	7.23607	1.09088
$45$	0	0
$46$	−0.618034	−0.0911241
$47$	3.47214	0.506463	0.253232	−	0.967406i	$-0.418507\pi$
$47$	3.47214	0.506463	0.253232	+	0.967406i	$0.418507\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.38197	−0.478282
$51$	0	0
$52$	−0.381966	−0.0529692
$53$	−11.2361	−1.54339	−0.771696	−	0.635991i	$-0.780591\pi$
$53$	−11.2361	−1.54339	−0.771696	+	0.635991i	$0.780591\pi$
$54$	0	0
$55$	−14.4721	−1.95142
$56$	−2.23607	−0.298807
$57$	0	0
$58$	−0.909830	−0.119467
$59$	1.52786	0.198911	0.0994555	−	0.995042i	$-0.468290\pi$
$59$	1.52786	0.198911	0.0994555	+	0.995042i	$0.468290\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	5.56231	0.706414
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	0.763932	0.0947541
$66$	0	0
$67$	−12.1803	−1.48807	−0.744033	−	0.668143i	$-0.767089\pi$
$67$	−12.1803	−1.48807	−0.744033	+	0.668143i	$0.767089\pi$
$68$	0	0
$69$	0	0
$70$	2.00000	0.239046
$71$	10.2361	1.21480	0.607399	−	0.794397i	$-0.292213\pi$
$71$	10.2361	1.21480	0.607399	+	0.794397i	$0.292213\pi$
$72$	0	0
$73$	−6.70820	−0.785136	−0.392568	−	0.919723i	$-0.628413\pi$
$73$	−6.70820	−0.785136	−0.392568	+	0.919723i	$0.628413\pi$
$74$	3.52786	0.410106
$75$	0	0
$76$	11.7082	1.34302
$77$	4.47214	0.509647
$78$	0	0
$79$	−7.23607	−0.814121	−0.407061	−	0.913401i	$-0.633446\pi$
$79$	−7.23607	−0.814121	−0.407061	+	0.913401i	$0.633446\pi$
$80$	6.00000	0.670820
$81$	0	0
$82$	−1.38197	−0.152613
$83$	−6.47214	−0.710409	−0.355205	−	0.934789i	$-0.615589\pi$
$83$	−6.47214	−0.710409	−0.355205	+	0.934789i	$0.615589\pi$
$84$	0	0
$85$	0	0
$86$	−1.52786	−0.164754
$87$	0	0
$88$	−10.0000	−1.06600
$89$	−8.94427	−0.948091	−0.474045	−	0.880500i	$-0.657207\pi$
$89$	−8.94427	−0.948091	−0.474045	+	0.880500i	$0.657207\pi$
$90$	0	0
$91$	−0.236068	−0.0247466
$92$	−1.61803	−0.168692
$93$	0	0
$94$	−2.14590	−0.221332
$95$	−23.4164	−2.40247
$96$	0	0
$97$	3.70820	0.376511	0.188256	−	0.982120i	$-0.439717\pi$
$97$	3.70820	0.376511	0.188256	+	0.982120i	$0.439717\pi$
$98$	−0.618034	−0.0624309
$99$	0	0
$100$	−8.85410	−0.885410
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	−15.7082	−1.54778	−0.773888	−	0.633323i	$-0.781691\pi$
$103$	−15.7082	−1.54778	−0.773888	+	0.633323i	$0.781691\pi$
$104$	0.527864	0.0517613
$105$	0	0
$106$	6.94427	0.674487
$107$	−11.2361	−1.08623	−0.543116	−	0.839658i	$-0.682756\pi$
$107$	−11.2361	−1.08623	−0.543116	+	0.839658i	$0.682756\pi$
$108$	0	0
$109$	−15.4164	−1.47662	−0.738312	−	0.674459i	$-0.764377\pi$
$109$	−15.4164	−1.47662	−0.738312	+	0.674459i	$0.764377\pi$
$110$	8.94427	0.852803
$111$	0	0
$112$	−1.85410	−0.175196
$113$	−2.47214	−0.232559	−0.116279	−	0.993217i	$-0.537097\pi$
$113$	−2.47214	−0.232559	−0.116279	+	0.993217i	$0.537097\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	−2.38197	−0.221160
$117$	0	0
$118$	−0.944272	−0.0869273
$119$	0	0
$120$	0	0
$121$	9.00000	0.818182
$122$	−8.29180	−0.750704
$123$	0	0
$124$	14.5623	1.30773
$125$	1.52786	0.136656
$126$	0	0
$127$	−2.70820	−0.240314	−0.120157	−	0.992755i	$-0.538340\pi$
$127$	−2.70820	−0.240314	−0.120157	+	0.992755i	$0.538340\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−0.472136	−0.0414091
$131$	−3.94427	−0.344613	−0.172306	−	0.985043i	$-0.555122\pi$
$131$	−3.94427	−0.344613	−0.172306	+	0.985043i	$0.555122\pi$
$132$	0	0
$133$	7.23607	0.627447
$134$	7.52786	0.650308
$135$	0	0
$136$	0	0
$137$	−15.7082	−1.34204	−0.671021	−	0.741438i	$-0.734144\pi$
$137$	−15.7082	−1.34204	−0.671021	+	0.741438i	$0.734144\pi$
$138$	0	0
$139$	2.52786	0.214411	0.107205	−	0.994237i	$-0.465810\pi$
$139$	2.52786	0.214411	0.107205	+	0.994237i	$0.465810\pi$
$140$	5.23607	0.442529
$141$	0	0
$142$	−6.32624	−0.530886
$143$	−1.05573	−0.0882844
$144$	0	0
$145$	4.76393	0.395623
$146$	4.14590	0.343117
$147$	0	0
$148$	9.23607	0.759200
$149$	−15.2361	−1.24819	−0.624094	−	0.781350i	$-0.714532\pi$
$149$	−15.2361	−1.24819	−0.624094	+	0.781350i	$0.714532\pi$
$150$	0	0
$151$	−15.1803	−1.23536	−0.617679	−	0.786430i	$-0.711927\pi$
$151$	−15.1803	−1.23536	−0.617679	+	0.786430i	$0.711927\pi$
$152$	−16.1803	−1.31240
$153$	0	0
$154$	−2.76393	−0.222724
$155$	−29.1246	−2.33935
$156$	0	0
$157$	15.4164	1.23036	0.615182	−	0.788385i	$-0.289083\pi$
$157$	15.4164	1.23036	0.615182	+	0.788385i	$0.289083\pi$
$158$	4.47214	0.355784
$159$	0	0
$160$	−18.1803	−1.43728
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	19.1803	1.50232	0.751160	−	0.660120i	$-0.229495\pi$
$163$	19.1803	1.50232	0.751160	+	0.660120i	$0.229495\pi$
$164$	−3.61803	−0.282521
$165$	0	0
$166$	4.00000	0.310460
$167$	−21.8885	−1.69379	−0.846893	−	0.531763i	$-0.821530\pi$
$167$	−21.8885	−1.69379	−0.846893	+	0.531763i	$0.821530\pi$
$168$	0	0
$169$	−12.9443	−0.995713
$170$	0	0
$171$	0	0
$172$	−4.00000	−0.304997
$173$	3.52786	0.268219	0.134109	−	0.990967i	$-0.457183\pi$
$173$	3.52786	0.268219	0.134109	+	0.990967i	$0.457183\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	−8.29180	−0.625018
$177$	0	0
$178$	5.52786	0.414331
$179$	18.7082	1.39832	0.699158	−	0.714967i	$-0.253558\pi$
$179$	18.7082	1.39832	0.699158	+	0.714967i	$0.253558\pi$
$180$	0	0
$181$	−5.05573	−0.375789	−0.187895	−	0.982189i	$-0.560166\pi$
$181$	−5.05573	−0.375789	−0.187895	+	0.982189i	$0.560166\pi$
$182$	0.145898	0.0108147
$183$	0	0
$184$	2.23607	0.164845
$185$	−18.4721	−1.35810
$186$	0	0
$187$	0	0
$188$	−5.61803	−0.409737
$189$	0	0
$190$	14.4721	1.04992
$191$	24.1803	1.74963	0.874814	−	0.484459i	$-0.160984\pi$
$191$	24.1803	1.74963	0.874814	+	0.484459i	$0.160984\pi$
$192$	0	0
$193$	4.41641	0.317900	0.158950	−	0.987287i	$-0.449189\pi$
$193$	4.41641	0.317900	0.158950	+	0.987287i	$0.449189\pi$
$194$	−2.29180	−0.164541
$195$	0	0
$196$	−1.61803	−0.115574
$197$	21.4721	1.52983	0.764913	−	0.644133i	$-0.222782\pi$
$197$	21.4721	1.52983	0.764913	+	0.644133i	$0.222782\pi$
$198$	0	0
$199$	15.8885	1.12631	0.563155	−	0.826352i	$-0.309588\pi$
$199$	15.8885	1.12631	0.563155	+	0.826352i	$0.309588\pi$
$200$	12.2361	0.865221
$201$	0	0
$202$	−8.29180	−0.583409
$203$	−1.47214	−0.103324
$204$	0	0
$205$	7.23607	0.505389
$206$	9.70820	0.676403
$207$	0	0
$208$	0.437694	0.0303486
$209$	32.3607	2.23844
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	18.1803	1.24863
$213$	0	0
$214$	6.94427	0.474701
$215$	8.00000	0.545595
$216$	0	0
$217$	9.00000	0.610960
$218$	9.52786	0.645308
$219$	0	0
$220$	23.4164	1.57873
$221$	0	0
$222$	0	0
$223$	3.41641	0.228780	0.114390	−	0.993436i	$-0.463509\pi$
$223$	3.41641	0.228780	0.114390	+	0.993436i	$0.463509\pi$
$224$	5.61803	0.375371
$225$	0	0
$226$	1.52786	0.101632
$227$	5.88854	0.390836	0.195418	−	0.980720i	$-0.437394\pi$
$227$	5.88854	0.390836	0.195418	+	0.980720i	$0.437394\pi$
$228$	0	0
$229$	14.7639	0.975628	0.487814	−	0.872948i	$-0.337794\pi$
$229$	14.7639	0.975628	0.487814	+	0.872948i	$0.337794\pi$
$230$	−2.00000	−0.131876
$231$	0	0
$232$	3.29180	0.216117
$233$	11.4721	0.751565	0.375782	−	0.926708i	$-0.377374\pi$
$233$	11.4721	0.751565	0.375782	+	0.926708i	$0.377374\pi$
$234$	0	0
$235$	11.2361	0.732960
$236$	−2.47214	−0.160922
$237$	0	0
$238$	0	0
$239$	15.7639	1.01968	0.509842	−	0.860268i	$-0.329704\pi$
$239$	15.7639	1.01968	0.509842	+	0.860268i	$0.329704\pi$
$240$	0	0
$241$	−21.4164	−1.37955	−0.689776	−	0.724023i	$-0.742291\pi$
$241$	−21.4164	−1.37955	−0.689776	+	0.724023i	$0.742291\pi$
$242$	−5.56231	−0.357559
$243$	0	0
$244$	−21.7082	−1.38973
$245$	3.23607	0.206745
$246$	0	0
$247$	−1.70820	−0.108690
$248$	−20.1246	−1.27791
$249$	0	0
$250$	−0.944272	−0.0597210
$251$	−8.29180	−0.523374	−0.261687	−	0.965153i	$-0.584279\pi$
$251$	−8.29180	−0.523374	−0.261687	+	0.965153i	$0.584279\pi$
$252$	0	0
$253$	−4.47214	−0.281161
$254$	1.67376	0.105021
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−4.23607	−0.264239	−0.132119	−	0.991234i	$-0.542178\pi$
$257$	−4.23607	−0.264239	−0.132119	+	0.991234i	$0.542178\pi$
$258$	0	0
$259$	5.70820	0.354691
$260$	−1.23607	−0.0766577
$261$	0	0
$262$	2.43769	0.150601
$263$	26.9443	1.66145	0.830727	−	0.556679i	$-0.187925\pi$
$263$	26.9443	1.66145	0.830727	+	0.556679i	$0.187925\pi$
$264$	0	0
$265$	−36.3607	−2.23362
$266$	−4.47214	−0.274204
$267$	0	0
$268$	19.7082	1.20387
$269$	9.18034	0.559735	0.279868	−	0.960039i	$-0.409709\pi$
$269$	9.18034	0.559735	0.279868	+	0.960039i	$0.409709\pi$
$270$	0	0
$271$	−16.9443	−1.02929	−0.514646	−	0.857403i	$-0.672076\pi$
$271$	−16.9443	−1.02929	−0.514646	+	0.857403i	$0.672076\pi$
$272$	0	0
$273$	0	0
$274$	9.70820	0.586494
$275$	−24.4721	−1.47573
$276$	0	0
$277$	20.4164	1.22670	0.613352	−	0.789810i	$-0.289821\pi$
$277$	20.4164	1.22670	0.613352	+	0.789810i	$0.289821\pi$
$278$	−1.56231	−0.0937009
$279$	0	0
$280$	−7.23607	−0.432438
$281$	−3.70820	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$282$	0	0
$283$	18.9443	1.12612	0.563060	−	0.826416i	$-0.309624\pi$
$283$	18.9443	1.12612	0.563060	+	0.826416i	$0.309624\pi$
$284$	−16.5623	−0.982792
$285$	0	0
$286$	0.652476	0.0385817
$287$	−2.23607	−0.131991
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−2.94427	−0.172894
$291$	0	0
$292$	10.8541	0.635188
$293$	15.7082	0.917683	0.458842	−	0.888518i	$-0.348265\pi$
$293$	15.7082	0.917683	0.458842	+	0.888518i	$0.348265\pi$
$294$	0	0
$295$	4.94427	0.287867
$296$	−12.7639	−0.741888
$297$	0	0
$298$	9.41641	0.545478
$299$	0.236068	0.0136522
$300$	0	0
$301$	−2.47214	−0.142492
$302$	9.38197	0.539871
$303$	0	0
$304$	−13.4164	−0.769484
$305$	43.4164	2.48602
$306$	0	0
$307$	−11.4164	−0.651569	−0.325784	−	0.945444i	$-0.605628\pi$
$307$	−11.4164	−0.651569	−0.325784	+	0.945444i	$0.605628\pi$
$308$	−7.23607	−0.412313
$309$	0	0
$310$	18.0000	1.02233
$311$	2.88854	0.163794	0.0818971	−	0.996641i	$-0.473902\pi$
$311$	2.88854	0.163794	0.0818971	+	0.996641i	$0.473902\pi$
$312$	0	0
$313$	2.76393	0.156227	0.0781133	−	0.996944i	$-0.475110\pi$
$313$	2.76393	0.156227	0.0781133	+	0.996944i	$0.475110\pi$
$314$	−9.52786	−0.537688
$315$	0	0
$316$	11.7082	0.658638
$317$	−31.3050	−1.75826	−0.879131	−	0.476581i	$-0.841876\pi$
$317$	−31.3050	−1.75826	−0.879131	+	0.476581i	$0.841876\pi$
$318$	0	0
$319$	−6.58359	−0.368610
$320$	−0.763932	−0.0427051
$321$	0	0
$322$	0.618034	0.0344417
$323$	0	0
$324$	0	0
$325$	1.29180	0.0716560
$326$	−11.8541	−0.656538
$327$	0	0
$328$	5.00000	0.276079
$329$	−3.47214	−0.191425
$330$	0	0
$331$	0.708204	0.0389264	0.0194632	−	0.999811i	$-0.493804\pi$
$331$	0.708204	0.0389264	0.0194632	+	0.999811i	$0.493804\pi$
$332$	10.4721	0.574733
$333$	0	0
$334$	13.5279	0.740212
$335$	−39.4164	−2.15355
$336$	0	0
$337$	−17.5967	−0.958556	−0.479278	−	0.877663i	$-0.659101\pi$
$337$	−17.5967	−0.958556	−0.479278	+	0.877663i	$0.659101\pi$
$338$	8.00000	0.435143
$339$	0	0
$340$	0	0
$341$	40.2492	2.17962
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.52786	0.298042
$345$	0	0
$346$	−2.18034	−0.117216
$347$	5.88854	0.316114	0.158057	−	0.987430i	$-0.449477\pi$
$347$	5.88854	0.316114	0.158057	+	0.987430i	$0.449477\pi$
$348$	0	0
$349$	1.29180	0.0691483	0.0345741	−	0.999402i	$-0.488993\pi$
$349$	1.29180	0.0691483	0.0345741	+	0.999402i	$0.488993\pi$
$350$	3.38197	0.180774
$351$	0	0
$352$	25.1246	1.33915
$353$	1.76393	0.0938846	0.0469423	−	0.998898i	$-0.485052\pi$
$353$	1.76393	0.0938846	0.0469423	+	0.998898i	$0.485052\pi$
$354$	0	0
$355$	33.1246	1.75807
$356$	14.4721	0.767022
$357$	0	0
$358$	−11.5623	−0.611087
$359$	−4.76393	−0.251431	−0.125715	−	0.992066i	$-0.540123\pi$
$359$	−4.76393	−0.251431	−0.125715	+	0.992066i	$0.540123\pi$
$360$	0	0
$361$	33.3607	1.75583
$362$	3.12461	0.164226
$363$	0	0
$364$	0.381966	0.0200205
$365$	−21.7082	−1.13626
$366$	0	0
$367$	−4.58359	−0.239262	−0.119631	−	0.992818i	$-0.538171\pi$
$367$	−4.58359	−0.239262	−0.119631	+	0.992818i	$0.538171\pi$
$368$	1.85410	0.0966517
$369$	0	0
$370$	11.4164	0.593511
$371$	11.2361	0.583348
$372$	0	0
$373$	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
$375$	0	0
$376$	7.76393	0.400394
$377$	0.347524	0.0178984
$378$	0	0
$379$	8.29180	0.425921	0.212960	−	0.977061i	$-0.431689\pi$
$379$	8.29180	0.425921	0.212960	+	0.977061i	$0.431689\pi$
$380$	37.8885	1.94364
$381$	0	0
$382$	−14.9443	−0.764615
$383$	15.7082	0.802652	0.401326	−	0.915935i	$-0.368549\pi$
$383$	15.7082	0.802652	0.401326	+	0.915935i	$0.368549\pi$
$384$	0	0
$385$	14.4721	0.737568
$386$	−2.72949	−0.138927
$387$	0	0
$388$	−6.00000	−0.304604
$389$	7.41641	0.376027	0.188013	−	0.982166i	$-0.439795\pi$
$389$	7.41641	0.376027	0.188013	+	0.982166i	$0.439795\pi$
$390$	0	0
$391$	0	0
$392$	2.23607	0.112938
$393$	0	0
$394$	−13.2705	−0.668559
$395$	−23.4164	−1.17821
$396$	0	0
$397$	−25.6525	−1.28746	−0.643730	−	0.765252i	$-0.722614\pi$
$397$	−25.6525	−1.28746	−0.643730	+	0.765252i	$0.722614\pi$
$398$	−9.81966	−0.492215
$399$	0	0
$400$	10.1459	0.507295
$401$	19.8885	0.993186	0.496593	−	0.867983i	$-0.334584\pi$
$401$	19.8885	0.993186	0.496593	+	0.867983i	$0.334584\pi$
$402$	0	0
$403$	−2.12461	−0.105834
$404$	−21.7082	−1.08002
$405$	0	0
$406$	0.909830	0.0451541
$407$	25.5279	1.26537
$408$	0	0
$409$	4.12461	0.203949	0.101974	−	0.994787i	$-0.467484\pi$
$409$	4.12461	0.203949	0.101974	+	0.994787i	$0.467484\pi$
$410$	−4.47214	−0.220863
$411$	0	0
$412$	25.4164	1.25218
$413$	−1.52786	−0.0751813
$414$	0	0
$415$	−20.9443	−1.02811
$416$	−1.32624	−0.0650242
$417$	0	0
$418$	−20.0000	−0.978232
$419$	−26.4721	−1.29325	−0.646624	−	0.762809i	$-0.723820\pi$
$419$	−26.4721	−1.29325	−0.646624	+	0.762809i	$0.723820\pi$
$420$	0	0
$421$	−8.58359	−0.418339	−0.209169	−	0.977879i	$-0.567076\pi$
$421$	−8.58359	−0.418339	−0.209169	+	0.977879i	$0.567076\pi$
$422$	7.41641	0.361025
$423$	0	0
$424$	−25.1246	−1.22016
$425$	0	0
$426$	0	0
$427$	−13.4164	−0.649265
$428$	18.1803	0.878780
$429$	0	0
$430$	−4.94427	−0.238434
$431$	−18.1803	−0.875716	−0.437858	−	0.899044i	$-0.644263\pi$
$431$	−18.1803	−0.875716	−0.437858	+	0.899044i	$0.644263\pi$
$432$	0	0
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	−5.56231	−0.266999
$435$	0	0
$436$	24.9443	1.19461
$437$	−7.23607	−0.346148
$438$	0	0
$439$	30.3050	1.44638	0.723188	−	0.690651i	$-0.242676\pi$
$439$	30.3050	1.44638	0.723188	+	0.690651i	$0.242676\pi$
$440$	−32.3607	−1.54273
$441$	0	0
$442$	0	0
$443$	7.18034	0.341148	0.170574	−	0.985345i	$-0.445438\pi$
$443$	7.18034	0.341148	0.170574	+	0.985345i	$0.445438\pi$
$444$	0	0
$445$	−28.9443	−1.37209
$446$	−2.11146	−0.0999803
$447$	0	0
$448$	0.236068	0.0111532
$449$	−32.8328	−1.54948	−0.774738	−	0.632282i	$-0.782118\pi$
$449$	−32.8328	−1.54948	−0.774738	+	0.632282i	$0.782118\pi$
$450$	0	0
$451$	−10.0000	−0.470882
$452$	4.00000	0.188144
$453$	0	0
$454$	−3.63932	−0.170802
$455$	−0.763932	−0.0358137
$456$	0	0
$457$	14.4721	0.676978	0.338489	−	0.940970i	$-0.390084\pi$
$457$	14.4721	0.676978	0.338489	+	0.940970i	$0.390084\pi$
$458$	−9.12461	−0.426365
$459$	0	0
$460$	−5.23607	−0.244133
$461$	25.7639	1.19995	0.599973	−	0.800020i	$-0.295178\pi$
$461$	25.7639	1.19995	0.599973	+	0.800020i	$0.295178\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	2.72949	0.126713
$465$	0	0
$466$	−7.09017	−0.328446
$467$	11.5279	0.533446	0.266723	−	0.963773i	$-0.414059\pi$
$467$	11.5279	0.533446	0.266723	+	0.963773i	$0.414059\pi$
$468$	0	0
$469$	12.1803	0.562436
$470$	−6.94427	−0.320315
$471$	0	0
$472$	3.41641	0.157253
$473$	−11.0557	−0.508343
$474$	0	0
$475$	−39.5967	−1.81682
$476$	0	0
$477$	0	0
$478$	−9.74265	−0.445618
$479$	23.1246	1.05659	0.528295	−	0.849061i	$-0.322831\pi$
$479$	23.1246	1.05659	0.528295	+	0.849061i	$0.322831\pi$
$480$	0	0
$481$	−1.34752	−0.0614418
$482$	13.2361	0.602886
$483$	0	0
$484$	−14.5623	−0.661923
$485$	12.0000	0.544892
$486$	0	0
$487$	20.1246	0.911933	0.455967	−	0.889997i	$-0.349294\pi$
$487$	20.1246	0.911933	0.455967	+	0.889997i	$0.349294\pi$
$488$	30.0000	1.35804
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−11.7639	−0.530899	−0.265449	−	0.964125i	$-0.585520\pi$
$491$	−11.7639	−0.530899	−0.265449	+	0.964125i	$0.585520\pi$
$492$	0	0
$493$	0	0
$494$	1.05573	0.0474995
$495$	0	0
$496$	−16.6869	−0.749265
$497$	−10.2361	−0.459150
$498$	0	0
$499$	−36.7082	−1.64328	−0.821642	−	0.570003i	$-0.806942\pi$
$499$	−36.7082	−1.64328	−0.821642	+	0.570003i	$0.806942\pi$
$500$	−2.47214	−0.110557
$501$	0	0
$502$	5.12461	0.228723
$503$	15.5967	0.695425	0.347712	−	0.937601i	$-0.386959\pi$
$503$	15.5967	0.695425	0.347712	+	0.937601i	$0.386959\pi$
$504$	0	0
$505$	43.4164	1.93200
$506$	2.76393	0.122872
$507$	0	0
$508$	4.38197	0.194418
$509$	12.8197	0.568221	0.284111	−	0.958791i	$-0.408302\pi$
$509$	12.8197	0.568221	0.284111	+	0.958791i	$0.408302\pi$
$510$	0	0
$511$	6.70820	0.296753
$512$	−18.7082	−0.826794
$513$	0	0
$514$	2.61803	0.115477
$515$	−50.8328	−2.23996
$516$	0	0
$517$	−15.5279	−0.682915
$518$	−3.52786	−0.155005
$519$	0	0
$520$	1.70820	0.0749097
$521$	−21.3050	−0.933387	−0.466693	−	0.884419i	$-0.654555\pi$
$521$	−21.3050	−0.933387	−0.466693	+	0.884419i	$0.654555\pi$
$522$	0	0
$523$	−7.70820	−0.337056	−0.168528	−	0.985697i	$-0.553901\pi$
$523$	−7.70820	−0.337056	−0.168528	+	0.985697i	$0.553901\pi$
$524$	6.38197	0.278797
$525$	0	0
$526$	−16.6525	−0.726082
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	22.4721	0.976127
$531$	0	0
$532$	−11.7082	−0.507615
$533$	0.527864	0.0228643
$534$	0	0
$535$	−36.3607	−1.57201
$536$	−27.2361	−1.17642
$537$	0	0
$538$	−5.67376	−0.244613
$539$	−4.47214	−0.192629
$540$	0	0
$541$	21.0000	0.902861	0.451430	−	0.892306i	$-0.350914\pi$
$541$	21.0000	0.902861	0.451430	+	0.892306i	$0.350914\pi$
$542$	10.4721	0.449817
$543$	0	0
$544$	0	0
$545$	−49.8885	−2.13699
$546$	0	0
$547$	43.1803	1.84626	0.923129	−	0.384490i	$-0.125623\pi$
$547$	43.1803	1.84626	0.923129	+	0.384490i	$0.125623\pi$
$548$	25.4164	1.08574
$549$	0	0
$550$	15.1246	0.644916
$551$	−10.6525	−0.453811
$552$	0	0
$553$	7.23607	0.307709
$554$	−12.6180	−0.536089
$555$	0	0
$556$	−4.09017	−0.173462
$557$	26.7639	1.13402	0.567012	−	0.823709i	$-0.308099\pi$
$557$	26.7639	1.13402	0.567012	+	0.823709i	$0.308099\pi$
$558$	0	0
$559$	0.583592	0.0246833
$560$	−6.00000	−0.253546
$561$	0	0
$562$	2.29180	0.0966736
$563$	9.59675	0.404455	0.202227	−	0.979339i	$-0.435182\pi$
$563$	9.59675	0.404455	0.202227	+	0.979339i	$0.435182\pi$
$564$	0	0
$565$	−8.00000	−0.336563
$566$	−11.7082	−0.492133
$567$	0	0
$568$	22.8885	0.960382
$569$	8.18034	0.342938	0.171469	−	0.985190i	$-0.445149\pi$
$569$	8.18034	0.342938	0.171469	+	0.985190i	$0.445149\pi$
$570$	0	0
$571$	16.2918	0.681790	0.340895	−	0.940101i	$-0.389270\pi$
$571$	16.2918	0.681790	0.340895	+	0.940101i	$0.389270\pi$
$572$	1.70820	0.0714236
$573$	0	0
$574$	1.38197	0.0576821
$575$	5.47214	0.228204
$576$	0	0
$577$	−15.2918	−0.636606	−0.318303	−	0.947989i	$-0.603113\pi$
$577$	−15.2918	−0.636606	−0.318303	+	0.947989i	$0.603113\pi$
$578$	10.5066	0.437016
$579$	0	0
$580$	−7.70820	−0.320066
$581$	6.47214	0.268509
$582$	0	0
$583$	50.2492	2.08111
$584$	−15.0000	−0.620704
$585$	0	0
$586$	−9.70820	−0.401042
$587$	−20.8885	−0.862162	−0.431081	−	0.902313i	$-0.641868\pi$
$587$	−20.8885	−0.862162	−0.431081	+	0.902313i	$0.641868\pi$
$588$	0	0
$589$	65.1246	2.68341
$590$	−3.05573	−0.125802
$591$	0	0
$592$	−10.5836	−0.434983
$593$	−34.3607	−1.41102	−0.705512	−	0.708698i	$-0.749283\pi$
$593$	−34.3607	−1.41102	−0.705512	+	0.708698i	$0.749283\pi$
$594$	0	0
$595$	0	0
$596$	24.6525	1.00980
$597$	0	0
$598$	−0.145898	−0.00596621
$599$	2.47214	0.101009	0.0505044	−	0.998724i	$-0.483917\pi$
$599$	2.47214	0.101009	0.0505044	+	0.998724i	$0.483917\pi$
$600$	0	0
$601$	22.2361	0.907028	0.453514	−	0.891249i	$-0.350170\pi$
$601$	22.2361	0.907028	0.453514	+	0.891249i	$0.350170\pi$
$602$	1.52786	0.0622711
$603$	0	0
$604$	24.5623	0.999426
$605$	29.1246	1.18408
$606$	0	0
$607$	−33.3050	−1.35181	−0.675903	−	0.736990i	$-0.736246\pi$
$607$	−33.3050	−1.35181	−0.675903	+	0.736990i	$0.736246\pi$
$608$	40.6525	1.64868
$609$	0	0
$610$	−26.8328	−1.08643
$611$	0.819660	0.0331599
$612$	0	0
$613$	−22.1803	−0.895855	−0.447928	−	0.894070i	$-0.647838\pi$
$613$	−22.1803	−0.895855	−0.447928	+	0.894070i	$0.647838\pi$
$614$	7.05573	0.284746
$615$	0	0
$616$	10.0000	0.402911
$617$	−35.2361	−1.41855	−0.709275	−	0.704932i	$-0.750978\pi$
$617$	−35.2361	−1.41855	−0.709275	+	0.704932i	$0.750978\pi$
$618$	0	0
$619$	−47.4853	−1.90860	−0.954298	−	0.298858i	$-0.903394\pi$
$619$	−47.4853	−1.90860	−0.954298	+	0.298858i	$0.903394\pi$
$620$	47.1246	1.89257
$621$	0	0
$622$	−1.78522	−0.0715807
$623$	8.94427	0.358345
$624$	0	0
$625$	−22.4164	−0.896656
$626$	−1.70820	−0.0682736
$627$	0	0
$628$	−24.9443	−0.995385
$629$	0	0
$630$	0	0
$631$	3.41641	0.136005	0.0680025	−	0.997685i	$-0.478337\pi$
$631$	3.41641	0.136005	0.0680025	+	0.997685i	$0.478337\pi$
$632$	−16.1803	−0.643619
$633$	0	0
$634$	19.3475	0.768388
$635$	−8.76393	−0.347786
$636$	0	0
$637$	0.236068	0.00935335
$638$	4.06888	0.161089
$639$	0	0
$640$	36.8328	1.45594
$641$	−7.05573	−0.278685	−0.139342	−	0.990244i	$-0.544499\pi$
$641$	−7.05573	−0.278685	−0.139342	+	0.990244i	$0.544499\pi$
$642$	0	0
$643$	−17.7082	−0.698343	−0.349172	−	0.937059i	$-0.613537\pi$
$643$	−17.7082	−0.698343	−0.349172	+	0.937059i	$0.613537\pi$
$644$	1.61803	0.0637595
$645$	0	0
$646$	0	0
$647$	18.5279	0.728405	0.364203	−	0.931320i	$-0.381342\pi$
$647$	18.5279	0.728405	0.364203	+	0.931320i	$0.381342\pi$
$648$	0	0
$649$	−6.83282	−0.268211
$650$	−0.798374	−0.0313148
$651$	0	0
$652$	−31.0344	−1.21540
$653$	−24.5279	−0.959849	−0.479925	−	0.877310i	$-0.659336\pi$
$653$	−24.5279	−0.959849	−0.479925	+	0.877310i	$0.659336\pi$
$654$	0	0
$655$	−12.7639	−0.498728
$656$	4.14590	0.161870
$657$	0	0
$658$	2.14590	0.0836558
$659$	−6.76393	−0.263485	−0.131743	−	0.991284i	$-0.542057\pi$
$659$	−6.76393	−0.263485	−0.131743	+	0.991284i	$0.542057\pi$
$660$	0	0
$661$	−41.7082	−1.62226	−0.811131	−	0.584865i	$-0.801147\pi$
$661$	−41.7082	−1.62226	−0.811131	+	0.584865i	$0.801147\pi$
$662$	−0.437694	−0.0170115
$663$	0	0
$664$	−14.4721	−0.561628
$665$	23.4164	0.908049
$666$	0	0
$667$	1.47214	0.0570013
$668$	35.4164	1.37030
$669$	0	0
$670$	24.3607	0.941135
$671$	−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	$0.506135\pi$
$674$	10.8754	0.418904
$675$	0	0
$676$	20.9443	0.805549
$677$	−23.2361	−0.893035	−0.446517	−	0.894775i	$-0.647336\pi$
$677$	−23.2361	−0.893035	−0.446517	+	0.894775i	$0.647336\pi$
$678$	0	0
$679$	−3.70820	−0.142308
$680$	0	0
$681$	0	0
$682$	−24.8754	−0.952528
$683$	5.18034	0.198220	0.0991101	−	0.995076i	$-0.468400\pi$
$683$	5.18034	0.198220	0.0991101	+	0.995076i	$0.468400\pi$
$684$	0	0
$685$	−50.8328	−1.94222
$686$	0.618034	0.0235966
$687$	0	0
$688$	4.58359	0.174748
$689$	−2.65248	−0.101051
$690$	0	0
$691$	42.8328	1.62944	0.814719	−	0.579857i	$-0.196891\pi$
$691$	42.8328	1.62944	0.814719	+	0.579857i	$0.196891\pi$
$692$	−5.70820	−0.216993
$693$	0	0
$694$	−3.63932	−0.138147
$695$	8.18034	0.310298
$696$	0	0
$697$	0	0
$698$	−0.798374	−0.0302189
$699$	0	0
$700$	8.85410	0.334654
$701$	22.7639	0.859782	0.429891	−	0.902881i	$-0.358552\pi$
$701$	22.7639	0.859782	0.429891	+	0.902881i	$0.358552\pi$
$702$	0	0
$703$	41.3050	1.55785
$704$	1.05573	0.0397892
$705$	0	0
$706$	−1.09017	−0.0410291
$707$	−13.4164	−0.504576
$708$	0	0
$709$	34.2492	1.28626	0.643128	−	0.765758i	$-0.277636\pi$
$709$	34.2492	1.28626	0.643128	+	0.765758i	$0.277636\pi$
$710$	−20.4721	−0.768306
$711$	0	0
$712$	−20.0000	−0.749532
$713$	−9.00000	−0.337053
$714$	0	0
$715$	−3.41641	−0.127766
$716$	−30.2705	−1.13126
$717$	0	0
$718$	2.94427	0.109879
$719$	−36.0000	−1.34257	−0.671287	−	0.741198i	$-0.734258\pi$
$719$	−36.0000	−1.34257	−0.671287	+	0.741198i	$0.734258\pi$
$720$	0	0
$721$	15.7082	0.585004
$722$	−20.6180	−0.767324
$723$	0	0
$724$	8.18034	0.304020
$725$	8.05573	0.299182
$726$	0	0
$727$	3.23607	0.120019	0.0600096	−	0.998198i	$-0.480887\pi$
$727$	3.23607	0.120019	0.0600096	+	0.998198i	$0.480887\pi$
$728$	−0.527864	−0.0195639
$729$	0	0
$730$	13.4164	0.496564
$731$	0	0
$732$	0	0
$733$	12.9443	0.478108	0.239054	−	0.971006i	$-0.423163\pi$
$733$	12.9443	0.478108	0.239054	+	0.971006i	$0.423163\pi$
$734$	2.83282	0.104561
$735$	0	0
$736$	−5.61803	−0.207083
$737$	54.4721	2.00651
$738$	0	0
$739$	8.70820	0.320336	0.160168	−	0.987090i	$-0.448796\pi$
$739$	8.70820	0.320336	0.160168	+	0.987090i	$0.448796\pi$
$740$	29.8885	1.09872
$741$	0	0
$742$	−6.94427	−0.254932
$743$	25.5279	0.936527	0.468263	−	0.883589i	$-0.344880\pi$
$743$	25.5279	0.936527	0.468263	+	0.883589i	$0.344880\pi$
$744$	0	0
$745$	−49.3050	−1.80639
$746$	−16.0689	−0.588324
$747$	0	0
$748$	0	0
$749$	11.2361	0.410557
$750$	0	0
$751$	−8.58359	−0.313220	−0.156610	−	0.987661i	$-0.550057\pi$
$751$	−8.58359	−0.313220	−0.156610	+	0.987661i	$0.550057\pi$
$752$	6.43769	0.234759
$753$	0	0
$754$	−0.214782	−0.00782189
$755$	−49.1246	−1.78783
$756$	0	0
$757$	−28.8328	−1.04795	−0.523973	−	0.851735i	$-0.675551\pi$
$757$	−28.8328	−1.04795	−0.523973	+	0.851735i	$0.675551\pi$
$758$	−5.12461	−0.186134
$759$	0	0
$760$	−52.3607	−1.89932
$761$	31.6525	1.14740	0.573701	−	0.819065i	$-0.305507\pi$
$761$	31.6525	1.14740	0.573701	+	0.819065i	$0.305507\pi$
$762$	0	0
$763$	15.4164	0.558111
$764$	−39.1246	−1.41548
$765$	0	0
$766$	−9.70820	−0.350772
$767$	0.360680	0.0130234
$768$	0	0
$769$	−43.2361	−1.55913	−0.779566	−	0.626320i	$-0.784560\pi$
$769$	−43.2361	−1.55913	−0.779566	+	0.626320i	$0.784560\pi$
$770$	−8.94427	−0.322329
$771$	0	0
$772$	−7.14590	−0.257186
$773$	−40.6525	−1.46217	−0.731084	−	0.682288i	$-0.760985\pi$
$773$	−40.6525	−1.46217	−0.731084	+	0.682288i	$0.760985\pi$
$774$	0	0
$775$	−49.2492	−1.76908
$776$	8.29180	0.297658
$777$	0	0
$778$	−4.58359	−0.164330
$779$	−16.1803	−0.579721
$780$	0	0
$781$	−45.7771	−1.63803
$782$	0	0
$783$	0	0
$784$	1.85410	0.0662179
$785$	49.8885	1.78060
$786$	0	0
$787$	−0.360680	−0.0128568	−0.00642842	−	0.999979i	$-0.502046\pi$
$787$	−0.360680	−0.0128568	−0.00642842	+	0.999979i	$0.502046\pi$
$788$	−34.7426	−1.23766
$789$	0	0
$790$	14.4721	0.514895
$791$	2.47214	0.0878990
$792$	0	0
$793$	3.16718	0.112470
$794$	15.8541	0.562641
$795$	0	0
$796$	−25.7082	−0.911203
$797$	36.7639	1.30225	0.651123	−	0.758973i	$-0.274298\pi$
$797$	36.7639	1.30225	0.651123	+	0.758973i	$0.274298\pi$
$798$	0	0
$799$	0	0
$800$	−30.7426	−1.08692
$801$	0	0
$802$	−12.2918	−0.434038
$803$	30.0000	1.05868
$804$	0	0
$805$	−3.23607	−0.114056
$806$	1.31308	0.0462514
$807$	0	0
$808$	30.0000	1.05540
$809$	32.8328	1.15434	0.577170	−	0.816624i	$-0.304157\pi$
$809$	32.8328	1.15434	0.577170	+	0.816624i	$0.304157\pi$
$810$	0	0
$811$	−45.3607	−1.59283	−0.796414	−	0.604751i	$-0.793273\pi$
$811$	−45.3607	−1.59283	−0.796414	+	0.604751i	$0.793273\pi$
$812$	2.38197	0.0835906
$813$	0	0
$814$	−15.7771	−0.552987
$815$	62.0689	2.17418
$816$	0	0
$817$	−17.8885	−0.625841
$818$	−2.54915	−0.0891289
$819$	0	0
$820$	−11.7082	−0.408868
$821$	−22.5836	−0.788173	−0.394086	−	0.919073i	$-0.628939\pi$
$821$	−22.5836	−0.788173	−0.394086	+	0.919073i	$0.628939\pi$
$822$	0	0
$823$	25.5410	0.890304	0.445152	−	0.895455i	$-0.353150\pi$
$823$	25.5410	0.890304	0.445152	+	0.895455i	$0.353150\pi$
$824$	−35.1246	−1.22362
$825$	0	0
$826$	0.944272	0.0328554
$827$	17.3050	0.601752	0.300876	−	0.953663i	$-0.402721\pi$
$827$	17.3050	0.601752	0.300876	+	0.953663i	$0.402721\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	12.9443	0.449302
$831$	0	0
$832$	−0.0557281	−0.00193202
$833$	0	0
$834$	0	0
$835$	−70.8328	−2.45127
$836$	−52.3607	−1.81093
$837$	0	0
$838$	16.3607	0.565170
$839$	−20.0689	−0.692855	−0.346427	−	0.938077i	$-0.612605\pi$
$839$	−20.0689	−0.692855	−0.346427	+	0.938077i	$0.612605\pi$
$840$	0	0
$841$	−26.8328	−0.925270
$842$	5.30495	0.182821
$843$	0	0
$844$	19.4164	0.668340
$845$	−41.8885	−1.44101
$846$	0	0
$847$	−9.00000	−0.309244
$848$	−20.8328	−0.715402
$849$	0	0
$850$	0	0
$851$	−5.70820	−0.195675
$852$	0	0
$853$	44.8328	1.53505	0.767523	−	0.641021i	$-0.221489\pi$
$853$	44.8328	1.53505	0.767523	+	0.641021i	$0.221489\pi$
$854$	8.29180	0.283739
$855$	0	0
$856$	−25.1246	−0.858742
$857$	−9.54102	−0.325915	−0.162958	−	0.986633i	$-0.552103\pi$
$857$	−9.54102	−0.325915	−0.162958	+	0.986633i	$0.552103\pi$
$858$	0	0
$859$	11.5836	0.395227	0.197614	−	0.980280i	$-0.436681\pi$
$859$	11.5836	0.395227	0.197614	+	0.980280i	$0.436681\pi$
$860$	−12.9443	−0.441396
$861$	0	0
$862$	11.2361	0.382702
$863$	−2.23607	−0.0761166	−0.0380583	−	0.999276i	$-0.512117\pi$
$863$	−2.23607	−0.0761166	−0.0380583	+	0.999276i	$0.512117\pi$
$864$	0	0
$865$	11.4164	0.388170
$866$	−8.65248	−0.294023
$867$	0	0
$868$	−14.5623	−0.494277
$869$	32.3607	1.09776
$870$	0	0
$871$	−2.87539	−0.0974288
$872$	−34.4721	−1.16737
$873$	0	0
$874$	4.47214	0.151272
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	−42.0000	−1.41824	−0.709120	−	0.705088i	$-0.750907\pi$
$877$	−42.0000	−1.41824	−0.709120	+	0.705088i	$0.750907\pi$
$878$	−18.7295	−0.632090
$879$	0	0
$880$	−26.8328	−0.904534
$881$	−25.5967	−0.862376	−0.431188	−	0.902262i	$-0.641905\pi$
$881$	−25.5967	−0.862376	−0.431188	+	0.902262i	$0.641905\pi$
$882$	0	0
$883$	9.88854	0.332776	0.166388	−	0.986060i	$-0.446790\pi$
$883$	9.88854	0.332776	0.166388	+	0.986060i	$0.446790\pi$
$884$	0	0
$885$	0	0
$886$	−4.43769	−0.149087
$887$	−16.8885	−0.567062	−0.283531	−	0.958963i	$-0.591506\pi$
$887$	−16.8885	−0.567062	−0.283531	+	0.958963i	$0.591506\pi$
$888$	0	0
$889$	2.70820	0.0908302
$890$	17.8885	0.599625
$891$	0	0
$892$	−5.52786	−0.185087
$893$	−25.1246	−0.840763
$894$	0	0
$895$	60.5410	2.02366
$896$	−11.3820	−0.380245
$897$	0	0
$898$	20.2918	0.677146
$899$	−13.2492	−0.441886
$900$	0	0
$901$	0	0
$902$	6.18034	0.205783
$903$	0	0
$904$	−5.52786	−0.183854
$905$	−16.3607	−0.543847
$906$	0	0
$907$	12.5410	0.416418	0.208209	−	0.978084i	$-0.433237\pi$
$907$	12.5410	0.416418	0.208209	+	0.978084i	$0.433237\pi$
$908$	−9.52786	−0.316193
$909$	0	0
$910$	0.472136	0.0156512
$911$	21.7771	0.721507	0.360754	−	0.932661i	$-0.382520\pi$
$911$	21.7771	0.721507	0.360754	+	0.932661i	$0.382520\pi$
$912$	0	0
$913$	28.9443	0.957916
$914$	−8.94427	−0.295850
$915$	0	0
$916$	−23.8885	−0.789300
$917$	3.94427	0.130251
$918$	0	0
$919$	3.70820	0.122322	0.0611612	−	0.998128i	$-0.480520\pi$
$919$	3.70820	0.122322	0.0611612	+	0.998128i	$0.480520\pi$
$920$	7.23607	0.238566
$921$	0	0
$922$	−15.9230	−0.524396
$923$	2.41641	0.0795370
$924$	0	0
$925$	−31.2361	−1.02704
$926$	0	0
$927$	0	0
$928$	−8.27051	−0.271493
$929$	−51.6525	−1.69466	−0.847331	−	0.531065i	$-0.821792\pi$
$929$	−51.6525	−1.69466	−0.847331	+	0.531065i	$0.821792\pi$
$930$	0	0
$931$	−7.23607	−0.237153
$932$	−18.5623	−0.608029
$933$	0	0
$934$	−7.12461	−0.233124
$935$	0	0
$936$	0	0
$937$	−57.1246	−1.86618	−0.933090	−	0.359643i	$-0.882898\pi$
$937$	−57.1246	−1.86618	−0.933090	+	0.359643i	$0.882898\pi$
$938$	−7.52786	−0.245793
$939$	0	0
$940$	−18.1803	−0.592977
$941$	−53.0132	−1.72818	−0.864090	−	0.503338i	$-0.832105\pi$
$941$	−53.0132	−1.72818	−0.864090	+	0.503338i	$0.832105\pi$
$942$	0	0
$943$	2.23607	0.0728164
$944$	2.83282	0.0922003
$945$	0	0
$946$	6.83282	0.222154
$947$	22.5967	0.734296	0.367148	−	0.930163i	$-0.380334\pi$
$947$	22.5967	0.734296	0.367148	+	0.930163i	$0.380334\pi$
$948$	0	0
$949$	−1.58359	−0.0514056
$950$	24.4721	0.793981
$951$	0	0
$952$	0	0
$953$	−12.1115	−0.392329	−0.196164	−	0.980571i	$-0.562849\pi$
$953$	−12.1115	−0.392329	−0.196164	+	0.980571i	$0.562849\pi$
$954$	0	0
$955$	78.2492	2.53209
$956$	−25.5066	−0.824942
$957$	0	0
$958$	−14.2918	−0.461747
$959$	15.7082	0.507244
$960$	0	0
$961$	50.0000	1.61290
$962$	0.832816	0.0268511
$963$	0	0
$964$	34.6525	1.11608
$965$	14.2918	0.460069
$966$	0	0
$967$	−31.0689	−0.999108	−0.499554	−	0.866283i	$-0.666503\pi$
$967$	−31.0689	−0.999108	−0.499554	+	0.866283i	$0.666503\pi$
$968$	20.1246	0.646830
$969$	0	0
$970$	−7.41641	−0.238127
$971$	−33.7082	−1.08175	−0.540874	−	0.841104i	$-0.681906\pi$
$971$	−33.7082	−1.08175	−0.540874	+	0.841104i	$0.681906\pi$
$972$	0	0
$973$	−2.52786	−0.0810396
$974$	−12.4377	−0.398529
$975$	0	0
$976$	24.8754	0.796242
$977$	18.6525	0.596746	0.298373	−	0.954449i	$-0.403556\pi$
$977$	18.6525	0.596746	0.298373	+	0.954449i	$0.403556\pi$
$978$	0	0
$979$	40.0000	1.27841
$980$	−5.23607	−0.167260
$981$	0	0
$982$	7.27051	0.232011
$983$	4.18034	0.133332	0.0666661	−	0.997775i	$-0.478764\pi$
$983$	4.18034	0.133332	0.0666661	+	0.997775i	$0.478764\pi$
$984$	0	0
$985$	69.4853	2.21399
$986$	0	0
$987$	0	0
$988$	2.76393	0.0879324
$989$	2.47214	0.0786094
$990$	0	0
$991$	−48.9443	−1.55477	−0.777383	−	0.629028i	$-0.783453\pi$
$991$	−48.9443	−1.55477	−0.777383	+	0.629028i	$0.783453\pi$
$992$	50.5623	1.60535
$993$	0	0
$994$	6.32624	0.200656
$995$	51.4164	1.63001
$996$	0	0
$997$	−27.3050	−0.864756	−0.432378	−	0.901692i	$-0.642325\pi$
$997$	−27.3050	−0.864756	−0.432378	+	0.901692i	$0.642325\pi$
$998$	22.6869	0.718142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.i.1.1		2
3.2	odd	2		161.2.a.b.1.2	✓	2
12.11	even	2		2576.2.a.s.1.1		2
15.14	odd	2		4025.2.a.i.1.1		2
21.20	even	2		1127.2.a.d.1.2		2
69.68	even	2		3703.2.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
161.2.a.b.1.2	✓	2	3.2	odd	2
1127.2.a.d.1.2		2	21.20	even	2
1449.2.a.i.1.1		2	1.1	even	1	trivial
2576.2.a.s.1.1		2	12.11	even	2
3703.2.a.b.1.2		2	69.68	even	2
4025.2.a.i.1.1		2	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +3.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +3.23607 q^{5} -1.00000 q^{7} +2.23607 q^{8} -2.00000 q^{10} -4.47214 q^{11} +0.236068 q^{13} +0.618034 q^{14} +1.85410 q^{16} -7.23607 q^{19} -5.23607 q^{20} +2.76393 q^{22} +1.00000 q^{23} +5.47214 q^{25} -0.145898 q^{26} +1.61803 q^{28} +1.47214 q^{29} -9.00000 q^{31} -5.61803 q^{32} -3.23607 q^{35} -5.70820 q^{37} +4.47214 q^{38} +7.23607 q^{40} +2.23607 q^{41} +2.47214 q^{43} +7.23607 q^{44} -0.618034 q^{46} +3.47214 q^{47} +1.00000 q^{49} -3.38197 q^{50} -0.381966 q^{52} -11.2361 q^{53} -14.4721 q^{55} -2.23607 q^{56} -0.909830 q^{58} +1.52786 q^{59} +13.4164 q^{61} +5.56231 q^{62} -0.236068 q^{64} +0.763932 q^{65} -12.1803 q^{67} +2.00000 q^{70} +10.2361 q^{71} -6.70820 q^{73} +3.52786 q^{74} +11.7082 q^{76} +4.47214 q^{77} -7.23607 q^{79} +6.00000 q^{80} -1.38197 q^{82} -6.47214 q^{83} -1.52786 q^{86} -10.0000 q^{88} -8.94427 q^{89} -0.236068 q^{91} -1.61803 q^{92} -2.14590 q^{94} -23.4164 q^{95} +3.70820 q^{97} -0.618034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} + 2 q^{5} - 2 q^{7} - 4 q^{10} - 4 q^{13} - q^{14} - 3 q^{16} - 10 q^{19} - 6 q^{20} + 10 q^{22} + 2 q^{23} + 2 q^{25} - 7 q^{26} + q^{28} - 6 q^{29} - 18 q^{31} - 9 q^{32} - 2 q^{35} + 2 q^{37} + 10 q^{40} - 4 q^{43} + 10 q^{44} + q^{46} - 2 q^{47} + 2 q^{49} - 9 q^{50} - 3 q^{52} - 18 q^{53} - 20 q^{55} - 13 q^{58} + 12 q^{59} - 9 q^{62} + 4 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{70} + 16 q^{71} + 16 q^{74} + 10 q^{76} - 10 q^{79} + 12 q^{80} - 5 q^{82} - 4 q^{83} - 12 q^{86} - 20 q^{88} + 4 q^{91} - q^{92} - 11 q^{94} - 20 q^{95} - 6 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 2 * q^5 - 2 * q^7 - 4 * q^10 - 4 * q^13 - q^14 - 3 * q^16 - 10 * q^19 - 6 * q^20 + 10 * q^22 + 2 * q^23 + 2 * q^25 - 7 * q^26 + q^28 - 6 * q^29 - 18 * q^31 - 9 * q^32 - 2 * q^35 + 2 * q^37 + 10 * q^40 - 4 * q^43 + 10 * q^44 + q^46 - 2 * q^47 + 2 * q^49 - 9 * q^50 - 3 * q^52 - 18 * q^53 - 20 * q^55 - 13 * q^58 + 12 * q^59 - 9 * q^62 + 4 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^70 + 16 * q^71 + 16 * q^74 + 10 * q^76 - 10 * q^79 + 12 * q^80 - 5 * q^82 - 4 * q^83 - 12 * q^86 - 20 * q^88 + 4 * q^91 - q^92 - 11 * q^94 - 20 * q^95 - 6 * q^97 + q^98

Introduction

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