LMFDB - Embedded newform 1089.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,2,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, 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("1089.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1089.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:	$8.69570878012$
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 33)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} +0.381966 q^{5} -3.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	$q+1.61803 q^{2} +0.618034 q^{4} +0.381966 q^{5} -3.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -6.23607 q^{13} -4.85410 q^{14} -4.85410 q^{16} -0.618034 q^{17} +0.854102 q^{19} +0.236068 q^{20} +5.47214 q^{23} -4.85410 q^{25} -10.0902 q^{26} -1.85410 q^{28} -4.47214 q^{29} -3.85410 q^{31} -3.38197 q^{32} -1.00000 q^{34} -1.14590 q^{35} -4.23607 q^{37} +1.38197 q^{38} -0.854102 q^{40} +5.94427 q^{41} -1.76393 q^{43} +8.85410 q^{46} +0.618034 q^{47} +2.00000 q^{49} -7.85410 q^{50} -3.85410 q^{52} +7.38197 q^{53} +6.70820 q^{56} -7.23607 q^{58} +5.32624 q^{59} -1.14590 q^{61} -6.23607 q^{62} +4.23607 q^{64} -2.38197 q^{65} +10.5623 q^{67} -0.381966 q^{68} -1.85410 q^{70} -14.5623 q^{71} -1.23607 q^{73} -6.85410 q^{74} +0.527864 q^{76} -0.527864 q^{79} -1.85410 q^{80} +9.61803 q^{82} -12.7082 q^{83} -0.236068 q^{85} -2.85410 q^{86} -9.47214 q^{89} +18.7082 q^{91} +3.38197 q^{92} +1.00000 q^{94} +0.326238 q^{95} +15.0344 q^{97} +3.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7	$ 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7} - q^{10} - 8 q^{13} - 3 q^{14} - 3 q^{16} + q^{17} - 5 q^{19} - 4 q^{20} + 2 q^{23} - 3 q^{25} - 9 q^{26} + 3 q^{28} - q^{31} - 9 q^{32} - 2 q^{34} - 9 q^{35} - 4 q^{37} + 5 q^{38} + 5 q^{40} - 6 q^{41} - 8 q^{43} + 11 q^{46} - q^{47} + 4 q^{49} - 9 q^{50} - q^{52} + 17 q^{53} - 10 q^{58} - 5 q^{59} - 9 q^{61} - 8 q^{62} + 4 q^{64} - 7 q^{65} + q^{67} - 3 q^{68} + 3 q^{70} - 9 q^{71} + 2 q^{73} - 7 q^{74} + 10 q^{76} - 10 q^{79} + 3 q^{80} + 17 q^{82} - 12 q^{83} + 4 q^{85} + q^{86} - 10 q^{89} + 24 q^{91} + 9 q^{92} + 2 q^{94} - 15 q^{95} + q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7 - q^10 - 8 * q^13 - 3 * q^14 - 3 * q^16 + q^17 - 5 * q^19 - 4 * q^20 + 2 * q^23 - 3 * q^25 - 9 * q^26 + 3 * q^28 - q^31 - 9 * q^32 - 2 * q^34 - 9 * q^35 - 4 * q^37 + 5 * q^38 + 5 * q^40 - 6 * q^41 - 8 * q^43 + 11 * q^46 - q^47 + 4 * q^49 - 9 * q^50 - q^52 + 17 * q^53 - 10 * q^58 - 5 * q^59 - 9 * q^61 - 8 * q^62 + 4 * q^64 - 7 * q^65 + q^67 - 3 * q^68 + 3 * q^70 - 9 * q^71 + 2 * q^73 - 7 * q^74 + 10 * q^76 - 10 * q^79 + 3 * q^80 + 17 * q^82 - 12 * q^83 + 4 * q^85 + q^86 - 10 * q^89 + 24 * q^91 + 9 * q^92 + 2 * q^94 - 15 * q^95 + q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	0.381966	0.170820	0.0854102	−	0.996346i	$-0.472780\pi$
$5$	0.381966	0.170820	0.0854102	+	0.996346i	$0.472780\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	0.618034	0.195440
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.23607	−1.72957	−0.864787	−	0.502139i	$-0.832547\pi$
$13$	−6.23607	−1.72957	−0.864787	+	0.502139i	$0.832547\pi$
$14$	−4.85410	−1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	0	0
$19$	0.854102	0.195944	0.0979722	−	0.995189i	$-0.468764\pi$
$19$	0.854102	0.195944	0.0979722	+	0.995189i	$0.468764\pi$
$20$	0.236068	0.0527864
$21$	0	0
$22$	0	0
$23$	5.47214	1.14102	0.570510	−	0.821291i	$-0.306746\pi$
$23$	5.47214	1.14102	0.570510	+	0.821291i	$0.306746\pi$
$24$	0	0
$25$	−4.85410	−0.970820
$26$	−10.0902	−1.97885
$27$	0	0
$28$	−1.85410	−0.350392
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	0	0
$31$	−3.85410	−0.692217	−0.346109	−	0.938194i	$-0.612497\pi$
$31$	−3.85410	−0.692217	−0.346109	+	0.938194i	$0.612497\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	−1.00000	−0.171499
$35$	−1.14590	−0.193692
$36$	0	0
$37$	−4.23607	−0.696405	−0.348203	−	0.937419i	$-0.613208\pi$
$37$	−4.23607	−0.696405	−0.348203	+	0.937419i	$0.613208\pi$
$38$	1.38197	0.224184
$39$	0	0
$40$	−0.854102	−0.135045
$41$	5.94427	0.928339	0.464170	−	0.885746i	$-0.346353\pi$
$41$	5.94427	0.928339	0.464170	+	0.885746i	$0.346353\pi$
$42$	0	0
$43$	−1.76393	−0.268997	−0.134499	−	0.990914i	$-0.542942\pi$
$43$	−1.76393	−0.268997	−0.134499	+	0.990914i	$0.542942\pi$
$44$	0	0
$45$	0	0
$46$	8.85410	1.30547
$47$	0.618034	0.0901495	0.0450748	−	0.998984i	$-0.485647\pi$
$47$	0.618034	0.0901495	0.0450748	+	0.998984i	$0.485647\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−7.85410	−1.11074
$51$	0	0
$52$	−3.85410	−0.534468
$53$	7.38197	1.01399	0.506996	−	0.861949i	$-0.330756\pi$
$53$	7.38197	1.01399	0.506996	+	0.861949i	$0.330756\pi$
$54$	0	0
$55$	0	0
$56$	6.70820	0.896421
$57$	0	0
$58$	−7.23607	−0.950142
$59$	5.32624	0.693417	0.346709	−	0.937973i	$-0.387299\pi$
$59$	5.32624	0.693417	0.346709	+	0.937973i	$0.387299\pi$
$60$	0	0
$61$	−1.14590	−0.146717	−0.0733586	−	0.997306i	$-0.523372\pi$
$61$	−1.14590	−0.146717	−0.0733586	+	0.997306i	$0.523372\pi$
$62$	−6.23607	−0.791981
$63$	0	0
$64$	4.23607	0.529508
$65$	−2.38197	−0.295447
$66$	0	0
$67$	10.5623	1.29039	0.645196	−	0.764017i	$-0.276776\pi$
$67$	10.5623	1.29039	0.645196	+	0.764017i	$0.276776\pi$
$68$	−0.381966	−0.0463202
$69$	0	0
$70$	−1.85410	−0.221608
$71$	−14.5623	−1.72823	−0.864114	−	0.503296i	$-0.832120\pi$
$71$	−14.5623	−1.72823	−0.864114	+	0.503296i	$0.832120\pi$
$72$	0	0
$73$	−1.23607	−0.144671	−0.0723354	−	0.997380i	$-0.523045\pi$
$73$	−1.23607	−0.144671	−0.0723354	+	0.997380i	$0.523045\pi$
$74$	−6.85410	−0.796773
$75$	0	0
$76$	0.527864	0.0605502
$77$	0	0
$78$	0	0
$79$	−0.527864	−0.0593893	−0.0296947	−	0.999559i	$-0.509453\pi$
$79$	−0.527864	−0.0593893	−0.0296947	+	0.999559i	$0.509453\pi$
$80$	−1.85410	−0.207295
$81$	0	0
$82$	9.61803	1.06213
$83$	−12.7082	−1.39491	−0.697453	−	0.716630i	$-0.745684\pi$
$83$	−12.7082	−1.39491	−0.697453	+	0.716630i	$0.745684\pi$
$84$	0	0
$85$	−0.236068	−0.0256052
$86$	−2.85410	−0.307766
$87$	0	0
$88$	0	0
$89$	−9.47214	−1.00404	−0.502022	−	0.864855i	$-0.667410\pi$
$89$	−9.47214	−1.00404	−0.502022	+	0.864855i	$0.667410\pi$
$90$	0	0
$91$	18.7082	1.96115
$92$	3.38197	0.352594
$93$	0	0
$94$	1.00000	0.103142
$95$	0.326238	0.0334713
$96$	0	0
$97$	15.0344	1.52652	0.763258	−	0.646094i	$-0.223598\pi$
$97$	15.0344	1.52652	0.763258	+	0.646094i	$0.223598\pi$
$98$	3.23607	0.326892
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−3.00000	−0.298511	−0.149256	−	0.988799i	$-0.547688\pi$
$101$	−3.00000	−0.298511	−0.149256	+	0.988799i	$0.547688\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	13.9443	1.36735
$105$	0	0
$106$	11.9443	1.16013
$107$	0.236068	0.0228216	0.0114108	−	0.999935i	$-0.496368\pi$
$107$	0.236068	0.0228216	0.0114108	+	0.999935i	$0.496368\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	14.5623	1.37601
$113$	12.7082	1.19549	0.597744	−	0.801687i	$-0.296064\pi$
$113$	12.7082	1.19549	0.597744	+	0.801687i	$0.296064\pi$
$114$	0	0
$115$	2.09017	0.194909
$116$	−2.76393	−0.256625
$117$	0	0
$118$	8.61803	0.793354
$119$	1.85410	0.169965
$120$	0	0
$121$	0	0
$122$	−1.85410	−0.167863
$123$	0	0
$124$	−2.38197	−0.213907
$125$	−3.76393	−0.336656
$126$	0	0
$127$	−9.70820	−0.861464	−0.430732	−	0.902480i	$-0.641745\pi$
$127$	−9.70820	−0.861464	−0.430732	+	0.902480i	$0.641745\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	−3.85410	−0.338027
$131$	−13.8541	−1.21044	−0.605219	−	0.796059i	$-0.706915\pi$
$131$	−13.8541	−1.21044	−0.605219	+	0.796059i	$0.706915\pi$
$132$	0	0
$133$	−2.56231	−0.222180
$134$	17.0902	1.47637
$135$	0	0
$136$	1.38197	0.118503
$137$	1.47214	0.125773	0.0628865	−	0.998021i	$-0.479969\pi$
$137$	1.47214	0.125773	0.0628865	+	0.998021i	$0.479969\pi$
$138$	0	0
$139$	−5.85410	−0.496538	−0.248269	−	0.968691i	$-0.579862\pi$
$139$	−5.85410	−0.496538	−0.248269	+	0.968691i	$0.579862\pi$
$140$	−0.708204	−0.0598542
$141$	0	0
$142$	−23.5623	−1.97730
$143$	0	0
$144$	0	0
$145$	−1.70820	−0.141859
$146$	−2.00000	−0.165521
$147$	0	0
$148$	−2.61803	−0.215201
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	−1.90983	−0.154908
$153$	0	0
$154$	0	0
$155$	−1.47214	−0.118245
$156$	0	0
$157$	9.70820	0.774799	0.387400	−	0.921912i	$-0.373373\pi$
$157$	9.70820	0.774799	0.387400	+	0.921912i	$0.373373\pi$
$158$	−0.854102	−0.0679487
$159$	0	0
$160$	−1.29180	−0.102125
$161$	−16.4164	−1.29379
$162$	0	0
$163$	−15.2705	−1.19608	−0.598039	−	0.801467i	$-0.704053\pi$
$163$	−15.2705	−1.19608	−0.598039	+	0.801467i	$0.704053\pi$
$164$	3.67376	0.286873
$165$	0	0
$166$	−20.5623	−1.59594
$167$	−19.0344	−1.47293	−0.736465	−	0.676476i	$-0.763506\pi$
$167$	−19.0344	−1.47293	−0.736465	+	0.676476i	$0.763506\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	−0.381966	−0.0292955
$171$	0	0
$172$	−1.09017	−0.0831247
$173$	17.6180	1.33947	0.669737	−	0.742598i	$-0.266407\pi$
$173$	17.6180	1.33947	0.669737	+	0.742598i	$0.266407\pi$
$174$	0	0
$175$	14.5623	1.10081
$176$	0	0
$177$	0	0
$178$	−15.3262	−1.14875
$179$	−2.23607	−0.167132	−0.0835658	−	0.996502i	$-0.526631\pi$
$179$	−2.23607	−0.167132	−0.0835658	+	0.996502i	$0.526631\pi$
$180$	0	0
$181$	−8.52786	−0.633871	−0.316936	−	0.948447i	$-0.602654\pi$
$181$	−8.52786	−0.633871	−0.316936	+	0.948447i	$0.602654\pi$
$182$	30.2705	2.24380
$183$	0	0
$184$	−12.2361	−0.902055
$185$	−1.61803	−0.118960
$186$	0	0
$187$	0	0
$188$	0.381966	0.0278577
$189$	0	0
$190$	0.527864	0.0382953
$191$	−1.47214	−0.106520	−0.0532600	−	0.998581i	$-0.516961\pi$
$191$	−1.47214	−0.106520	−0.0532600	+	0.998581i	$0.516961\pi$
$192$	0	0
$193$	−1.56231	−0.112457	−0.0562286	−	0.998418i	$-0.517908\pi$
$193$	−1.56231	−0.112457	−0.0562286	+	0.998418i	$0.517908\pi$
$194$	24.3262	1.74652
$195$	0	0
$196$	1.23607	0.0882906
$197$	26.6180	1.89646	0.948228	−	0.317590i	$-0.102873\pi$
$197$	26.6180	1.89646	0.948228	+	0.317590i	$0.102873\pi$
$198$	0	0
$199$	−3.29180	−0.233349	−0.116675	−	0.993170i	$-0.537223\pi$
$199$	−3.29180	−0.233349	−0.116675	+	0.993170i	$0.537223\pi$
$200$	10.8541	0.767501
$201$	0	0
$202$	−4.85410	−0.341533
$203$	13.4164	0.941647
$204$	0	0
$205$	2.27051	0.158579
$206$	−9.70820	−0.676403
$207$	0	0
$208$	30.2705	2.09888
$209$	0	0
$210$	0	0
$211$	−11.2705	−0.775894	−0.387947	−	0.921682i	$-0.626816\pi$
$211$	−11.2705	−0.775894	−0.387947	+	0.921682i	$0.626816\pi$
$212$	4.56231	0.313340
$213$	0	0
$214$	0.381966	0.0261107
$215$	−0.673762	−0.0459502
$216$	0	0
$217$	11.5623	0.784900
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.85410	0.259255
$222$	0	0
$223$	−12.7082	−0.851004	−0.425502	−	0.904957i	$-0.639903\pi$
$223$	−12.7082	−0.851004	−0.425502	+	0.904957i	$0.639903\pi$
$224$	10.1459	0.677901
$225$	0	0
$226$	20.5623	1.36778
$227$	10.8885	0.722698	0.361349	−	0.932431i	$-0.382316\pi$
$227$	10.8885	0.722698	0.361349	+	0.932431i	$0.382316\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	3.38197	0.223000
$231$	0	0
$232$	10.0000	0.656532
$233$	8.67376	0.568237	0.284119	−	0.958789i	$-0.408299\pi$
$233$	8.67376	0.568237	0.284119	+	0.958789i	$0.408299\pi$
$234$	0	0
$235$	0.236068	0.0153994
$236$	3.29180	0.214278
$237$	0	0
$238$	3.00000	0.194461
$239$	−17.5623	−1.13601	−0.568006	−	0.823025i	$-0.692285\pi$
$239$	−17.5623	−1.13601	−0.568006	+	0.823025i	$0.692285\pi$
$240$	0	0
$241$	−17.1246	−1.10309	−0.551547	−	0.834144i	$-0.685962\pi$
$241$	−17.1246	−1.10309	−0.551547	+	0.834144i	$0.685962\pi$
$242$	0	0
$243$	0	0
$244$	−0.708204	−0.0453381
$245$	0.763932	0.0488058
$246$	0	0
$247$	−5.32624	−0.338900
$248$	8.61803	0.547246
$249$	0	0
$250$	−6.09017	−0.385176
$251$	−16.7984	−1.06030	−0.530152	−	0.847903i	$-0.677865\pi$
$251$	−16.7984	−1.06030	−0.530152	+	0.847903i	$0.677865\pi$
$252$	0	0
$253$	0	0
$254$	−15.7082	−0.985620
$255$	0	0
$256$	13.5623	0.847644
$257$	27.3262	1.70456	0.852282	−	0.523083i	$-0.175218\pi$
$257$	27.3262	1.70456	0.852282	+	0.523083i	$0.175218\pi$
$258$	0	0
$259$	12.7082	0.789649
$260$	−1.47214	−0.0912980
$261$	0	0
$262$	−22.4164	−1.38489
$263$	−0.673762	−0.0415459	−0.0207730	−	0.999784i	$-0.506613\pi$
$263$	−0.673762	−0.0415459	−0.0207730	+	0.999784i	$0.506613\pi$
$264$	0	0
$265$	2.81966	0.173210
$266$	−4.14590	−0.254201
$267$	0	0
$268$	6.52786	0.398753
$269$	−24.4721	−1.49209	−0.746046	−	0.665894i	$-0.768050\pi$
$269$	−24.4721	−1.49209	−0.746046	+	0.665894i	$0.768050\pi$
$270$	0	0
$271$	−6.27051	−0.380906	−0.190453	−	0.981696i	$-0.560996\pi$
$271$	−6.27051	−0.380906	−0.190453	+	0.981696i	$0.560996\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	2.38197	0.143900
$275$	0	0
$276$	0	0
$277$	−10.4377	−0.627140	−0.313570	−	0.949565i	$-0.601525\pi$
$277$	−10.4377	−0.627140	−0.313570	+	0.949565i	$0.601525\pi$
$278$	−9.47214	−0.568101
$279$	0	0
$280$	2.56231	0.153127
$281$	−5.23607	−0.312358	−0.156179	−	0.987729i	$-0.549918\pi$
$281$	−5.23607	−0.312358	−0.156179	+	0.987729i	$0.549918\pi$
$282$	0	0
$283$	22.1803	1.31848	0.659242	−	0.751931i	$-0.270877\pi$
$283$	22.1803	1.31848	0.659242	+	0.751931i	$0.270877\pi$
$284$	−9.00000	−0.534052
$285$	0	0
$286$	0	0
$287$	−17.8328	−1.05264
$288$	0	0
$289$	−16.6180	−0.977531
$290$	−2.76393	−0.162304
$291$	0	0
$292$	−0.763932	−0.0447057
$293$	17.9443	1.04832	0.524158	−	0.851621i	$-0.324380\pi$
$293$	17.9443	1.04832	0.524158	+	0.851621i	$0.324380\pi$
$294$	0	0
$295$	2.03444	0.118450
$296$	9.47214	0.550557
$297$	0	0
$298$	24.2705	1.40595
$299$	−34.1246	−1.97348
$300$	0	0
$301$	5.29180	0.305014
$302$	−3.23607	−0.186215
$303$	0	0
$304$	−4.14590	−0.237784
$305$	−0.437694	−0.0250623
$306$	0	0
$307$	19.5623	1.11648	0.558240	−	0.829680i	$-0.311477\pi$
$307$	19.5623	1.11648	0.558240	+	0.829680i	$0.311477\pi$
$308$	0	0
$309$	0	0
$310$	−2.38197	−0.135287
$311$	−11.4721	−0.650525	−0.325263	−	0.945624i	$-0.605453\pi$
$311$	−11.4721	−0.650525	−0.325263	+	0.945624i	$0.605453\pi$
$312$	0	0
$313$	−27.8328	−1.57320	−0.786602	−	0.617461i	$-0.788162\pi$
$313$	−27.8328	−1.57320	−0.786602	+	0.617461i	$0.788162\pi$
$314$	15.7082	0.886465
$315$	0	0
$316$	−0.326238	−0.0183523
$317$	−25.3607	−1.42440	−0.712199	−	0.701978i	$-0.752301\pi$
$317$	−25.3607	−1.42440	−0.712199	+	0.701978i	$0.752301\pi$
$318$	0	0
$319$	0	0
$320$	1.61803	0.0904508
$321$	0	0
$322$	−26.5623	−1.48026
$323$	−0.527864	−0.0293711
$324$	0	0
$325$	30.2705	1.67911
$326$	−24.7082	−1.36846
$327$	0	0
$328$	−13.2918	−0.733917
$329$	−1.85410	−0.102220
$330$	0	0
$331$	−22.5967	−1.24203	−0.621015	−	0.783799i	$-0.713279\pi$
$331$	−22.5967	−1.24203	−0.621015	+	0.783799i	$0.713279\pi$
$332$	−7.85410	−0.431050
$333$	0	0
$334$	−30.7984	−1.68521
$335$	4.03444	0.220425
$336$	0	0
$337$	13.7082	0.746733	0.373367	−	0.927684i	$-0.378203\pi$
$337$	13.7082	0.746733	0.373367	+	0.927684i	$0.378203\pi$
$338$	41.8885	2.27844
$339$	0	0
$340$	−0.145898	−0.00791243
$341$	0	0
$342$	0	0
$343$	15.0000	0.809924
$344$	3.94427	0.212661
$345$	0	0
$346$	28.5066	1.53252
$347$	−3.05573	−0.164040	−0.0820200	−	0.996631i	$-0.526137\pi$
$347$	−3.05573	−0.164040	−0.0820200	+	0.996631i	$0.526137\pi$
$348$	0	0
$349$	−30.1246	−1.61253	−0.806267	−	0.591552i	$-0.798515\pi$
$349$	−30.1246	−1.61253	−0.806267	+	0.591552i	$0.798515\pi$
$350$	23.5623	1.25946
$351$	0	0
$352$	0	0
$353$	1.52786	0.0813200	0.0406600	−	0.999173i	$-0.487054\pi$
$353$	1.52786	0.0813200	0.0406600	+	0.999173i	$0.487054\pi$
$354$	0	0
$355$	−5.56231	−0.295217
$356$	−5.85410	−0.310267
$357$	0	0
$358$	−3.61803	−0.191219
$359$	−17.2361	−0.909685	−0.454842	−	0.890572i	$-0.650304\pi$
$359$	−17.2361	−0.909685	−0.454842	+	0.890572i	$0.650304\pi$
$360$	0	0
$361$	−18.2705	−0.961606
$362$	−13.7984	−0.725226
$363$	0	0
$364$	11.5623	0.606029
$365$	−0.472136	−0.0247127
$366$	0	0
$367$	−14.5623	−0.760146	−0.380073	−	0.924956i	$-0.624101\pi$
$367$	−14.5623	−0.760146	−0.380073	+	0.924956i	$0.624101\pi$
$368$	−26.5623	−1.38466
$369$	0	0
$370$	−2.61803	−0.136105
$371$	−22.1459	−1.14976
$372$	0	0
$373$	−22.4164	−1.16068	−0.580339	−	0.814375i	$-0.697080\pi$
$373$	−22.4164	−1.16068	−0.580339	+	0.814375i	$0.697080\pi$
$374$	0	0
$375$	0	0
$376$	−1.38197	−0.0712695
$377$	27.8885	1.43633
$378$	0	0
$379$	28.4164	1.45965	0.729826	−	0.683633i	$-0.239601\pi$
$379$	28.4164	1.45965	0.729826	+	0.683633i	$0.239601\pi$
$380$	0.201626	0.0103432
$381$	0	0
$382$	−2.38197	−0.121872
$383$	8.88854	0.454183	0.227092	−	0.973873i	$-0.427078\pi$
$383$	8.88854	0.454183	0.227092	+	0.973873i	$0.427078\pi$
$384$	0	0
$385$	0	0
$386$	−2.52786	−0.128665
$387$	0	0
$388$	9.29180	0.471719
$389$	9.27051	0.470034	0.235017	−	0.971991i	$-0.424485\pi$
$389$	9.27051	0.470034	0.235017	+	0.971991i	$0.424485\pi$
$390$	0	0
$391$	−3.38197	−0.171033
$392$	−4.47214	−0.225877
$393$	0	0
$394$	43.0689	2.16978
$395$	−0.201626	−0.0101449
$396$	0	0
$397$	−25.2918	−1.26936	−0.634679	−	0.772776i	$-0.718868\pi$
$397$	−25.2918	−1.26936	−0.634679	+	0.772776i	$0.718868\pi$
$398$	−5.32624	−0.266980
$399$	0	0
$400$	23.5623	1.17812
$401$	14.9098	0.744561	0.372281	−	0.928120i	$-0.378576\pi$
$401$	14.9098	0.744561	0.372281	+	0.928120i	$0.378576\pi$
$402$	0	0
$403$	24.0344	1.19724
$404$	−1.85410	−0.0922450
$405$	0	0
$406$	21.7082	1.07736
$407$	0	0
$408$	0	0
$409$	28.9443	1.43120	0.715601	−	0.698509i	$-0.246153\pi$
$409$	28.9443	1.43120	0.715601	+	0.698509i	$0.246153\pi$
$410$	3.67376	0.181434
$411$	0	0
$412$	−3.70820	−0.182690
$413$	−15.9787	−0.786261
$414$	0	0
$415$	−4.85410	−0.238278
$416$	21.0902	1.03403
$417$	0	0
$418$	0	0
$419$	21.5066	1.05067	0.525333	−	0.850897i	$-0.323941\pi$
$419$	21.5066	1.05067	0.525333	+	0.850897i	$0.323941\pi$
$420$	0	0
$421$	−3.72949	−0.181764	−0.0908821	−	0.995862i	$-0.528969\pi$
$421$	−3.72949	−0.181764	−0.0908821	+	0.995862i	$0.528969\pi$
$422$	−18.2361	−0.887718
$423$	0	0
$424$	−16.5066	−0.801630
$425$	3.00000	0.145521
$426$	0	0
$427$	3.43769	0.166362
$428$	0.145898	0.00705225
$429$	0	0
$430$	−1.09017	−0.0525727
$431$	−1.49342	−0.0719356	−0.0359678	−	0.999353i	$-0.511451\pi$
$431$	−1.49342	−0.0719356	−0.0359678	+	0.999353i	$0.511451\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	18.7082	0.898023
$435$	0	0
$436$	0	0
$437$	4.67376	0.223576
$438$	0	0
$439$	−16.7082	−0.797439	−0.398720	−	0.917073i	$-0.630545\pi$
$439$	−16.7082	−0.797439	−0.398720	+	0.917073i	$0.630545\pi$
$440$	0	0
$441$	0	0
$442$	6.23607	0.296620
$443$	0.875388	0.0415909	0.0207955	−	0.999784i	$-0.493380\pi$
$443$	0.875388	0.0415909	0.0207955	+	0.999784i	$0.493380\pi$
$444$	0	0
$445$	−3.61803	−0.171511
$446$	−20.5623	−0.973653
$447$	0	0
$448$	−12.7082	−0.600406
$449$	15.5279	0.732805	0.366403	−	0.930456i	$-0.380589\pi$
$449$	15.5279	0.732805	0.366403	+	0.930456i	$0.380589\pi$
$450$	0	0
$451$	0	0
$452$	7.85410	0.369426
$453$	0	0
$454$	17.6180	0.826855
$455$	7.14590	0.335005
$456$	0	0
$457$	−32.7984	−1.53424	−0.767122	−	0.641502i	$-0.778312\pi$
$457$	−32.7984	−1.53424	−0.767122	+	0.641502i	$0.778312\pi$
$458$	16.1803	0.756058
$459$	0	0
$460$	1.29180	0.0602303
$461$	−9.90983	−0.461547	−0.230773	−	0.973008i	$-0.574126\pi$
$461$	−9.90983	−0.461547	−0.230773	+	0.973008i	$0.574126\pi$
$462$	0	0
$463$	8.79837	0.408895	0.204448	−	0.978878i	$-0.434460\pi$
$463$	8.79837	0.408895	0.204448	+	0.978878i	$0.434460\pi$
$464$	21.7082	1.00778
$465$	0	0
$466$	14.0344	0.650133
$467$	14.2361	0.658767	0.329383	−	0.944196i	$-0.393159\pi$
$467$	14.2361	0.658767	0.329383	+	0.944196i	$0.393159\pi$
$468$	0	0
$469$	−31.6869	−1.46317
$470$	0.381966	0.0176188
$471$	0	0
$472$	−11.9098	−0.548194
$473$	0	0
$474$	0	0
$475$	−4.14590	−0.190227
$476$	1.14590	0.0525222
$477$	0	0
$478$	−28.4164	−1.29974
$479$	−16.9098	−0.772630	−0.386315	−	0.922367i	$-0.626252\pi$
$479$	−16.9098	−0.772630	−0.386315	+	0.922367i	$0.626252\pi$
$480$	0	0
$481$	26.4164	1.20448
$482$	−27.7082	−1.26207
$483$	0	0
$484$	0	0
$485$	5.74265	0.260760
$486$	0	0
$487$	39.1803	1.77543	0.887715	−	0.460393i	$-0.152291\pi$
$487$	39.1803	1.77543	0.887715	+	0.460393i	$0.152291\pi$
$488$	2.56231	0.115990
$489$	0	0
$490$	1.23607	0.0558399
$491$	−26.2148	−1.18306	−0.591528	−	0.806284i	$-0.701475\pi$
$491$	−26.2148	−1.18306	−0.591528	+	0.806284i	$0.701475\pi$
$492$	0	0
$493$	2.76393	0.124481
$494$	−8.61803	−0.387744
$495$	0	0
$496$	18.7082	0.840023
$497$	43.6869	1.95963
$498$	0	0
$499$	2.56231	0.114705	0.0573523	−	0.998354i	$-0.481734\pi$
$499$	2.56231	0.114705	0.0573523	+	0.998354i	$0.481734\pi$
$500$	−2.32624	−0.104033
$501$	0	0
$502$	−27.1803	−1.21312
$503$	−30.0689	−1.34071	−0.670353	−	0.742043i	$-0.733857\pi$
$503$	−30.0689	−1.34071	−0.670353	+	0.742043i	$0.733857\pi$
$504$	0	0
$505$	−1.14590	−0.0509918
$506$	0	0
$507$	0	0
$508$	−6.00000	−0.266207
$509$	−21.3820	−0.947739	−0.473869	−	0.880595i	$-0.657143\pi$
$509$	−21.3820	−0.947739	−0.473869	+	0.880595i	$0.657143\pi$
$510$	0	0
$511$	3.70820	0.164041
$512$	−5.29180	−0.233867
$513$	0	0
$514$	44.2148	1.95023
$515$	−2.29180	−0.100989
$516$	0	0
$517$	0	0
$518$	20.5623	0.903456
$519$	0	0
$520$	5.32624	0.233571
$521$	−38.8328	−1.70130	−0.850648	−	0.525735i	$-0.823790\pi$
$521$	−38.8328	−1.70130	−0.850648	+	0.525735i	$0.823790\pi$
$522$	0	0
$523$	−34.9787	−1.52951	−0.764756	−	0.644320i	$-0.777141\pi$
$523$	−34.9787	−1.52951	−0.764756	+	0.644320i	$0.777141\pi$
$524$	−8.56231	−0.374046
$525$	0	0
$526$	−1.09017	−0.0475337
$527$	2.38197	0.103760
$528$	0	0
$529$	6.94427	0.301925
$530$	4.56231	0.198174
$531$	0	0
$532$	−1.58359	−0.0686574
$533$	−37.0689	−1.60563
$534$	0	0
$535$	0.0901699	0.00389839
$536$	−23.6180	−1.02014
$537$	0	0
$538$	−39.5967	−1.70714
$539$	0	0
$540$	0	0
$541$	0.562306	0.0241754	0.0120877	−	0.999927i	$-0.496152\pi$
$541$	0.562306	0.0241754	0.0120877	+	0.999927i	$0.496152\pi$
$542$	−10.1459	−0.435804
$543$	0	0
$544$	2.09017	0.0896153
$545$	0	0
$546$	0	0
$547$	−19.3820	−0.828713	−0.414357	−	0.910115i	$-0.635993\pi$
$547$	−19.3820	−0.828713	−0.414357	+	0.910115i	$0.635993\pi$
$548$	0.909830	0.0388660
$549$	0	0
$550$	0	0
$551$	−3.81966	−0.162723
$552$	0	0
$553$	1.58359	0.0673412
$554$	−16.8885	−0.717525
$555$	0	0
$556$	−3.61803	−0.153439
$557$	26.0902	1.10548	0.552738	−	0.833355i	$-0.313583\pi$
$557$	26.0902	1.10548	0.552738	+	0.833355i	$0.313583\pi$
$558$	0	0
$559$	11.0000	0.465250
$560$	5.56231	0.235050
$561$	0	0
$562$	−8.47214	−0.357375
$563$	26.8885	1.13322	0.566609	−	0.823987i	$-0.308255\pi$
$563$	26.8885	1.13322	0.566609	+	0.823987i	$0.308255\pi$
$564$	0	0
$565$	4.85410	0.204214
$566$	35.8885	1.50851
$567$	0	0
$568$	32.5623	1.36628
$569$	34.0689	1.42824	0.714121	−	0.700022i	$-0.246827\pi$
$569$	34.0689	1.42824	0.714121	+	0.700022i	$0.246827\pi$
$570$	0	0
$571$	25.6869	1.07496	0.537482	−	0.843275i	$-0.319376\pi$
$571$	25.6869	1.07496	0.537482	+	0.843275i	$0.319376\pi$
$572$	0	0
$573$	0	0
$574$	−28.8541	−1.20435
$575$	−26.5623	−1.10772
$576$	0	0
$577$	15.2361	0.634286	0.317143	−	0.948378i	$-0.397277\pi$
$577$	15.2361	0.634286	0.317143	+	0.948378i	$0.397277\pi$
$578$	−26.8885	−1.11842
$579$	0	0
$580$	−1.05573	−0.0438367
$581$	38.1246	1.58168
$582$	0	0
$583$	0	0
$584$	2.76393	0.114372
$585$	0	0
$586$	29.0344	1.19940
$587$	−24.3050	−1.00317	−0.501586	−	0.865108i	$-0.667250\pi$
$587$	−24.3050	−1.00317	−0.501586	+	0.865108i	$0.667250\pi$
$588$	0	0
$589$	−3.29180	−0.135636
$590$	3.29180	0.135521
$591$	0	0
$592$	20.5623	0.845106
$593$	−29.2148	−1.19971	−0.599854	−	0.800110i	$-0.704775\pi$
$593$	−29.2148	−1.19971	−0.599854	+	0.800110i	$0.704775\pi$
$594$	0	0
$595$	0.708204	0.0290335
$596$	9.27051	0.379735
$597$	0	0
$598$	−55.2148	−2.25790
$599$	21.7082	0.886973	0.443487	−	0.896281i	$-0.353741\pi$
$599$	21.7082	0.886973	0.443487	+	0.896281i	$0.353741\pi$
$600$	0	0
$601$	19.8328	0.808997	0.404499	−	0.914539i	$-0.367446\pi$
$601$	19.8328	0.808997	0.404499	+	0.914539i	$0.367446\pi$
$602$	8.56231	0.348974
$603$	0	0
$604$	−1.23607	−0.0502949
$605$	0	0
$606$	0	0
$607$	2.32624	0.0944191	0.0472095	−	0.998885i	$-0.484967\pi$
$607$	2.32624	0.0944191	0.0472095	+	0.998885i	$0.484967\pi$
$608$	−2.88854	−0.117146
$609$	0	0
$610$	−0.708204	−0.0286743
$611$	−3.85410	−0.155920
$612$	0	0
$613$	−3.34752	−0.135205	−0.0676026	−	0.997712i	$-0.521535\pi$
$613$	−3.34752	−0.135205	−0.0676026	+	0.997712i	$0.521535\pi$
$614$	31.6525	1.27739
$615$	0	0
$616$	0	0
$617$	−46.4164	−1.86865	−0.934327	−	0.356417i	$-0.883998\pi$
$617$	−46.4164	−1.86865	−0.934327	+	0.356417i	$0.883998\pi$
$618$	0	0
$619$	−31.1803	−1.25324	−0.626622	−	0.779323i	$-0.715563\pi$
$619$	−31.1803	−1.25324	−0.626622	+	0.779323i	$0.715563\pi$
$620$	−0.909830	−0.0365397
$621$	0	0
$622$	−18.5623	−0.744281
$623$	28.4164	1.13848
$624$	0	0
$625$	22.8328	0.913313
$626$	−45.0344	−1.79994
$627$	0	0
$628$	6.00000	0.239426
$629$	2.61803	0.104388
$630$	0	0
$631$	12.7295	0.506753	0.253377	−	0.967368i	$-0.418459\pi$
$631$	12.7295	0.506753	0.253377	+	0.967368i	$0.418459\pi$
$632$	1.18034	0.0469514
$633$	0	0
$634$	−41.0344	−1.62969
$635$	−3.70820	−0.147156
$636$	0	0
$637$	−12.4721	−0.494164
$638$	0	0
$639$	0	0
$640$	5.20163	0.205612
$641$	6.74265	0.266318	0.133159	−	0.991095i	$-0.457488\pi$
$641$	6.74265	0.266318	0.133159	+	0.991095i	$0.457488\pi$
$642$	0	0
$643$	−18.4377	−0.727112	−0.363556	−	0.931572i	$-0.618437\pi$
$643$	−18.4377	−0.727112	−0.363556	+	0.931572i	$0.618437\pi$
$644$	−10.1459	−0.399804
$645$	0	0
$646$	−0.854102	−0.0336042
$647$	−3.20163	−0.125869	−0.0629345	−	0.998018i	$-0.520046\pi$
$647$	−3.20163	−0.125869	−0.0629345	+	0.998018i	$0.520046\pi$
$648$	0	0
$649$	0	0
$650$	48.9787	1.92110
$651$	0	0
$652$	−9.43769	−0.369609
$653$	−21.9656	−0.859579	−0.429789	−	0.902929i	$-0.641412\pi$
$653$	−21.9656	−0.859579	−0.429789	+	0.902929i	$0.641412\pi$
$654$	0	0
$655$	−5.29180	−0.206768
$656$	−28.8541	−1.12656
$657$	0	0
$658$	−3.00000	−0.116952
$659$	20.6525	0.804506	0.402253	−	0.915528i	$-0.368227\pi$
$659$	20.6525	0.804506	0.402253	+	0.915528i	$0.368227\pi$
$660$	0	0
$661$	−21.0902	−0.820313	−0.410156	−	0.912015i	$-0.634526\pi$
$661$	−21.0902	−0.820313	−0.410156	+	0.912015i	$0.634526\pi$
$662$	−36.5623	−1.42103
$663$	0	0
$664$	28.4164	1.10277
$665$	−0.978714	−0.0379529
$666$	0	0
$667$	−24.4721	−0.947565
$668$	−11.7639	−0.455160
$669$	0	0
$670$	6.52786	0.252193
$671$	0	0
$672$	0	0
$673$	14.4164	0.555712	0.277856	−	0.960623i	$-0.410376\pi$
$673$	14.4164	0.555712	0.277856	+	0.960623i	$0.410376\pi$
$674$	22.1803	0.854355
$675$	0	0
$676$	16.0000	0.615385
$677$	22.4721	0.863674	0.431837	−	0.901952i	$-0.357866\pi$
$677$	22.4721	0.863674	0.431837	+	0.901952i	$0.357866\pi$
$678$	0	0
$679$	−45.1033	−1.73091
$680$	0.527864	0.0202427
$681$	0	0
$682$	0	0
$683$	38.8885	1.48803	0.744014	−	0.668164i	$-0.232919\pi$
$683$	38.8885	1.48803	0.744014	+	0.668164i	$0.232919\pi$
$684$	0	0
$685$	0.562306	0.0214846
$686$	24.2705	0.926652
$687$	0	0
$688$	8.56231	0.326435
$689$	−46.0344	−1.75377
$690$	0	0
$691$	−39.7082	−1.51057	−0.755286	−	0.655396i	$-0.772502\pi$
$691$	−39.7082	−1.51057	−0.755286	+	0.655396i	$0.772502\pi$
$692$	10.8885	0.413920
$693$	0	0
$694$	−4.94427	−0.187682
$695$	−2.23607	−0.0848189
$696$	0	0
$697$	−3.67376	−0.139154
$698$	−48.7426	−1.84494
$699$	0	0
$700$	9.00000	0.340168
$701$	49.6869	1.87665	0.938324	−	0.345756i	$-0.112377\pi$
$701$	49.6869	1.87665	0.938324	+	0.345756i	$0.112377\pi$
$702$	0	0
$703$	−3.61803	−0.136457
$704$	0	0
$705$	0	0
$706$	2.47214	0.0930401
$707$	9.00000	0.338480
$708$	0	0
$709$	6.25735	0.235000	0.117500	−	0.993073i	$-0.462512\pi$
$709$	6.25735	0.235000	0.117500	+	0.993073i	$0.462512\pi$
$710$	−9.00000	−0.337764
$711$	0	0
$712$	21.1803	0.793767
$713$	−21.0902	−0.789833
$714$	0	0
$715$	0	0
$716$	−1.38197	−0.0516465
$717$	0	0
$718$	−27.8885	−1.04079
$719$	28.4164	1.05975	0.529877	−	0.848075i	$-0.322238\pi$
$719$	28.4164	1.05975	0.529877	+	0.848075i	$0.322238\pi$
$720$	0	0
$721$	18.0000	0.670355
$722$	−29.5623	−1.10020
$723$	0	0
$724$	−5.27051	−0.195877
$725$	21.7082	0.806222
$726$	0	0
$727$	32.1459	1.19223	0.596113	−	0.802901i	$-0.296711\pi$
$727$	32.1459	1.19223	0.596113	+	0.802901i	$0.296711\pi$
$728$	−41.8328	−1.55043
$729$	0	0
$730$	−0.763932	−0.0282744
$731$	1.09017	0.0403214
$732$	0	0
$733$	24.2918	0.897238	0.448619	−	0.893723i	$-0.351916\pi$
$733$	24.2918	0.897238	0.448619	+	0.893723i	$0.351916\pi$
$734$	−23.5623	−0.869701
$735$	0	0
$736$	−18.5066	−0.682162
$737$	0	0
$738$	0	0
$739$	25.0000	0.919640	0.459820	−	0.888012i	$-0.347914\pi$
$739$	25.0000	0.919640	0.459820	+	0.888012i	$0.347914\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	−35.8328	−1.31546
$743$	12.8197	0.470308	0.235154	−	0.971958i	$-0.424441\pi$
$743$	12.8197	0.470308	0.235154	+	0.971958i	$0.424441\pi$
$744$	0	0
$745$	5.72949	0.209912
$746$	−36.2705	−1.32796
$747$	0	0
$748$	0	0
$749$	−0.708204	−0.0258772
$750$	0	0
$751$	−52.9230	−1.93119	−0.965594	−	0.260056i	$-0.916259\pi$
$751$	−52.9230	−1.93119	−0.965594	+	0.260056i	$0.916259\pi$
$752$	−3.00000	−0.109399
$753$	0	0
$754$	45.1246	1.64334
$755$	−0.763932	−0.0278023
$756$	0	0
$757$	−15.9443	−0.579504	−0.289752	−	0.957102i	$-0.593573\pi$
$757$	−15.9443	−0.579504	−0.289752	+	0.957102i	$0.593573\pi$
$758$	45.9787	1.67002
$759$	0	0
$760$	−0.729490	−0.0264614
$761$	4.88854	0.177210	0.0886048	−	0.996067i	$-0.471759\pi$
$761$	4.88854	0.177210	0.0886048	+	0.996067i	$0.471759\pi$
$762$	0	0
$763$	0	0
$764$	−0.909830	−0.0329165
$765$	0	0
$766$	14.3820	0.519642
$767$	−33.2148	−1.19932
$768$	0	0
$769$	47.6869	1.71963	0.859817	−	0.510602i	$-0.170577\pi$
$769$	47.6869	1.71963	0.859817	+	0.510602i	$0.170577\pi$
$770$	0	0
$771$	0	0
$772$	−0.965558	−0.0347512
$773$	−49.2492	−1.77137	−0.885686	−	0.464285i	$-0.846311\pi$
$773$	−49.2492	−1.77137	−0.885686	+	0.464285i	$0.846311\pi$
$774$	0	0
$775$	18.7082	0.672019
$776$	−33.6180	−1.20682
$777$	0	0
$778$	15.0000	0.537776
$779$	5.07701	0.181903
$780$	0	0
$781$	0	0
$782$	−5.47214	−0.195683
$783$	0	0
$784$	−9.70820	−0.346722
$785$	3.70820	0.132351
$786$	0	0
$787$	23.7082	0.845106	0.422553	−	0.906338i	$-0.361134\pi$
$787$	23.7082	0.845106	0.422553	+	0.906338i	$0.361134\pi$
$788$	16.4508	0.586037
$789$	0	0
$790$	−0.326238	−0.0116070
$791$	−38.1246	−1.35556
$792$	0	0
$793$	7.14590	0.253758
$794$	−40.9230	−1.45230
$795$	0	0
$796$	−2.03444	−0.0721089
$797$	−15.2361	−0.539689	−0.269845	−	0.962904i	$-0.586972\pi$
$797$	−15.2361	−0.539689	−0.269845	+	0.962904i	$0.586972\pi$
$798$	0	0
$799$	−0.381966	−0.0135130
$800$	16.4164	0.580408
$801$	0	0
$802$	24.1246	0.851870
$803$	0	0
$804$	0	0
$805$	−6.27051	−0.221006
$806$	38.8885	1.36979
$807$	0	0
$808$	6.70820	0.235994
$809$	25.3262	0.890423	0.445212	−	0.895425i	$-0.353128\pi$
$809$	25.3262	0.890423	0.445212	+	0.895425i	$0.353128\pi$
$810$	0	0
$811$	−45.7426	−1.60624	−0.803121	−	0.595816i	$-0.796829\pi$
$811$	−45.7426	−1.60624	−0.803121	+	0.595816i	$0.796829\pi$
$812$	8.29180	0.290985
$813$	0	0
$814$	0	0
$815$	−5.83282	−0.204315
$816$	0	0
$817$	−1.50658	−0.0527085
$818$	46.8328	1.63747
$819$	0	0
$820$	1.40325	0.0490037
$821$	13.5836	0.474071	0.237035	−	0.971501i	$-0.423824\pi$
$821$	13.5836	0.474071	0.237035	+	0.971501i	$0.423824\pi$
$822$	0	0
$823$	−26.5279	−0.924703	−0.462352	−	0.886697i	$-0.652994\pi$
$823$	−26.5279	−0.924703	−0.462352	+	0.886697i	$0.652994\pi$
$824$	13.4164	0.467383
$825$	0	0
$826$	−25.8541	−0.899579
$827$	12.8754	0.447721	0.223861	−	0.974621i	$-0.428134\pi$
$827$	12.8754	0.447721	0.223861	+	0.974621i	$0.428134\pi$
$828$	0	0
$829$	−42.6869	−1.48258	−0.741289	−	0.671186i	$-0.765785\pi$
$829$	−42.6869	−1.48258	−0.741289	+	0.671186i	$0.765785\pi$
$830$	−7.85410	−0.272620
$831$	0	0
$832$	−26.4164	−0.915824
$833$	−1.23607	−0.0428272
$834$	0	0
$835$	−7.27051	−0.251606
$836$	0	0
$837$	0	0
$838$	34.7984	1.20209
$839$	23.2918	0.804122	0.402061	−	0.915613i	$-0.368294\pi$
$839$	23.2918	0.804122	0.402061	+	0.915613i	$0.368294\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	−6.03444	−0.207961
$843$	0	0
$844$	−6.96556	−0.239764
$845$	9.88854	0.340176
$846$	0	0
$847$	0	0
$848$	−35.8328	−1.23050
$849$	0	0
$850$	4.85410	0.166494
$851$	−23.1803	−0.794612
$852$	0	0
$853$	−7.94427	−0.272007	−0.136003	−	0.990708i	$-0.543426\pi$
$853$	−7.94427	−0.272007	−0.136003	+	0.990708i	$0.543426\pi$
$854$	5.56231	0.190338
$855$	0	0
$856$	−0.527864	−0.0180420
$857$	−41.7214	−1.42517	−0.712587	−	0.701584i	$-0.752477\pi$
$857$	−41.7214	−1.42517	−0.712587	+	0.701584i	$0.752477\pi$
$858$	0	0
$859$	42.8885	1.46334	0.731669	−	0.681660i	$-0.238742\pi$
$859$	42.8885	1.46334	0.731669	+	0.681660i	$0.238742\pi$
$860$	−0.416408	−0.0141994
$861$	0	0
$862$	−2.41641	−0.0823032
$863$	23.8885	0.813175	0.406588	−	0.913612i	$-0.366719\pi$
$863$	23.8885	0.813175	0.406588	+	0.913612i	$0.366719\pi$
$864$	0	0
$865$	6.72949	0.228810
$866$	−9.70820	−0.329898
$867$	0	0
$868$	7.14590	0.242548
$869$	0	0
$870$	0	0
$871$	−65.8673	−2.23183
$872$	0	0
$873$	0	0
$874$	7.56231	0.255799
$875$	11.2918	0.381732
$876$	0	0
$877$	20.4164	0.689413	0.344707	−	0.938710i	$-0.387978\pi$
$877$	20.4164	0.689413	0.344707	+	0.938710i	$0.387978\pi$
$878$	−27.0344	−0.912368
$879$	0	0
$880$	0	0
$881$	−25.0902	−0.845309	−0.422655	−	0.906291i	$-0.638902\pi$
$881$	−25.0902	−0.845309	−0.422655	+	0.906291i	$0.638902\pi$
$882$	0	0
$883$	37.4164	1.25916	0.629581	−	0.776935i	$-0.283226\pi$
$883$	37.4164	1.25916	0.629581	+	0.776935i	$0.283226\pi$
$884$	2.38197	0.0801142
$885$	0	0
$886$	1.41641	0.0475852
$887$	3.00000	0.100730	0.0503651	−	0.998731i	$-0.483962\pi$
$887$	3.00000	0.100730	0.0503651	+	0.998731i	$0.483962\pi$
$888$	0	0
$889$	29.1246	0.976808
$890$	−5.85410	−0.196230
$891$	0	0
$892$	−7.85410	−0.262975
$893$	0.527864	0.0176643
$894$	0	0
$895$	−0.854102	−0.0285495
$896$	−40.8541	−1.36484
$897$	0	0
$898$	25.1246	0.838419
$899$	17.2361	0.574855
$900$	0	0
$901$	−4.56231	−0.151992
$902$	0	0
$903$	0	0
$904$	−28.4164	−0.945116
$905$	−3.25735	−0.108278
$906$	0	0
$907$	3.97871	0.132111	0.0660555	−	0.997816i	$-0.478959\pi$
$907$	3.97871	0.132111	0.0660555	+	0.997816i	$0.478959\pi$
$908$	6.72949	0.223326
$909$	0	0
$910$	11.5623	0.383287
$911$	−35.9443	−1.19089	−0.595443	−	0.803397i	$-0.703024\pi$
$911$	−35.9443	−1.19089	−0.595443	+	0.803397i	$0.703024\pi$
$912$	0	0
$913$	0	0
$914$	−53.0689	−1.75536
$915$	0	0
$916$	6.18034	0.204204
$917$	41.5623	1.37251
$918$	0	0
$919$	46.9574	1.54898	0.774491	−	0.632585i	$-0.218006\pi$
$919$	46.9574	1.54898	0.774491	+	0.632585i	$0.218006\pi$
$920$	−4.67376	−0.154089
$921$	0	0
$922$	−16.0344	−0.528066
$923$	90.8115	2.98910
$924$	0	0
$925$	20.5623	0.676084
$926$	14.2361	0.467826
$927$	0	0
$928$	15.1246	0.496490
$929$	−2.88854	−0.0947700	−0.0473850	−	0.998877i	$-0.515089\pi$
$929$	−2.88854	−0.0947700	−0.0473850	+	0.998877i	$0.515089\pi$
$930$	0	0
$931$	1.70820	0.0559841
$932$	5.36068	0.175595
$933$	0	0
$934$	23.0344	0.753710
$935$	0	0
$936$	0	0
$937$	−32.5967	−1.06489	−0.532445	−	0.846465i	$-0.678727\pi$
$937$	−32.5967	−1.06489	−0.532445	+	0.846465i	$0.678727\pi$
$938$	−51.2705	−1.67404
$939$	0	0
$940$	0.145898	0.00475867
$941$	33.6312	1.09635	0.548173	−	0.836365i	$-0.315324\pi$
$941$	33.6312	1.09635	0.548173	+	0.836365i	$0.315324\pi$
$942$	0	0
$943$	32.5279	1.05925
$944$	−25.8541	−0.841479
$945$	0	0
$946$	0	0
$947$	−2.67376	−0.0868856	−0.0434428	−	0.999056i	$-0.513833\pi$
$947$	−2.67376	−0.0868856	−0.0434428	+	0.999056i	$0.513833\pi$
$948$	0	0
$949$	7.70820	0.250219
$950$	−6.70820	−0.217643
$951$	0	0
$952$	−4.14590	−0.134369
$953$	60.1803	1.94943	0.974716	−	0.223446i	$-0.0717308\pi$
$953$	60.1803	1.94943	0.974716	+	0.223446i	$0.0717308\pi$
$954$	0	0
$955$	−0.562306	−0.0181958
$956$	−10.8541	−0.351047
$957$	0	0
$958$	−27.3607	−0.883983
$959$	−4.41641	−0.142613
$960$	0	0
$961$	−16.1459	−0.520835
$962$	42.7426	1.37808
$963$	0	0
$964$	−10.5836	−0.340875
$965$	−0.596748	−0.0192100
$966$	0	0
$967$	−25.6869	−0.826036	−0.413018	−	0.910723i	$-0.635525\pi$
$967$	−25.6869	−0.826036	−0.413018	+	0.910723i	$0.635525\pi$
$968$	0	0
$969$	0	0
$970$	9.29180	0.298342
$971$	33.7771	1.08396	0.541979	−	0.840392i	$-0.317675\pi$
$971$	33.7771	1.08396	0.541979	+	0.840392i	$0.317675\pi$
$972$	0	0
$973$	17.5623	0.563022
$974$	63.3951	2.03131
$975$	0	0
$976$	5.56231	0.178045
$977$	49.3607	1.57919	0.789594	−	0.613630i	$-0.210291\pi$
$977$	49.3607	1.57919	0.789594	+	0.613630i	$0.210291\pi$
$978$	0	0
$979$	0	0
$980$	0.472136	0.0150818
$981$	0	0
$982$	−42.4164	−1.35356
$983$	−35.3050	−1.12605	−0.563027	−	0.826439i	$-0.690363\pi$
$983$	−35.3050	−1.12605	−0.563027	+	0.826439i	$0.690363\pi$
$984$	0	0
$985$	10.1672	0.323953
$986$	4.47214	0.142422
$987$	0	0
$988$	−3.29180	−0.104726
$989$	−9.65248	−0.306931
$990$	0	0
$991$	−12.2705	−0.389786	−0.194893	−	0.980825i	$-0.562436\pi$
$991$	−12.2705	−0.389786	−0.194893	+	0.980825i	$0.562436\pi$
$992$	13.0344	0.413844
$993$	0	0
$994$	70.6869	2.24205
$995$	−1.25735	−0.0398608
$996$	0	0
$997$	−33.9787	−1.07612	−0.538058	−	0.842908i	$-0.680842\pi$
$997$	−33.9787	−1.07612	−0.538058	+	0.842908i	$0.680842\pi$
$998$	4.14590	0.131236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.2.a.s.1.2		2
3.2	odd	2		363.2.a.e.1.1		2
11.2	odd	10		99.2.f.b.37.1		4
11.6	odd	10		99.2.f.b.91.1		4
11.10	odd	2		1089.2.a.m.1.1		2
12.11	even	2		5808.2.a.bm.1.2		2
15.14	odd	2		9075.2.a.bv.1.2		2
33.2	even	10		33.2.e.a.4.1	✓	4
33.5	odd	10		363.2.e.j.124.1		4
33.8	even	10		363.2.e.h.130.1		4
33.14	odd	10		363.2.e.c.130.1		4
33.17	even	10		33.2.e.a.25.1	yes	4
33.20	odd	10		363.2.e.j.202.1		4
33.26	odd	10		363.2.e.c.148.1		4
33.29	even	10		363.2.e.h.148.1		4
33.32	even	2		363.2.a.h.1.2		2
99.2	even	30		891.2.n.d.433.1		8
99.13	odd	30		891.2.n.a.136.1		8
99.50	even	30		891.2.n.d.784.1		8
99.61	odd	30		891.2.n.a.190.1		8
99.68	even	30		891.2.n.d.136.1		8
99.79	odd	30		891.2.n.a.433.1		8
99.83	even	30		891.2.n.d.190.1		8
99.94	odd	30		891.2.n.a.784.1		8
132.35	odd	10		528.2.y.f.433.1		4
132.83	odd	10		528.2.y.f.289.1		4
132.131	odd	2		5808.2.a.bl.1.2		2
165.2	odd	20		825.2.bx.b.499.2		8
165.17	odd	20		825.2.bx.b.124.1		8
165.68	odd	20		825.2.bx.b.499.1		8
165.83	odd	20		825.2.bx.b.124.2		8
165.134	even	10		825.2.n.f.301.1		4
165.149	even	10		825.2.n.f.751.1		4
165.164	even	2		9075.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.2.e.a.4.1	✓	4	33.2	even	10
33.2.e.a.25.1	yes	4	33.17	even	10
99.2.f.b.37.1		4	11.2	odd	10
99.2.f.b.91.1		4	11.6	odd	10
363.2.a.e.1.1		2	3.2	odd	2
363.2.a.h.1.2		2	33.32	even	2
363.2.e.c.130.1		4	33.14	odd	10
363.2.e.c.148.1		4	33.26	odd	10
363.2.e.h.130.1		4	33.8	even	10
363.2.e.h.148.1		4	33.29	even	10
363.2.e.j.124.1		4	33.5	odd	10
363.2.e.j.202.1		4	33.20	odd	10
528.2.y.f.289.1		4	132.83	odd	10
528.2.y.f.433.1		4	132.35	odd	10
825.2.n.f.301.1		4	165.134	even	10
825.2.n.f.751.1		4	165.149	even	10
825.2.bx.b.124.1		8	165.17	odd	20
825.2.bx.b.124.2		8	165.83	odd	20
825.2.bx.b.499.1		8	165.68	odd	20
825.2.bx.b.499.2		8	165.2	odd	20
891.2.n.a.136.1		8	99.13	odd	30
891.2.n.a.190.1		8	99.61	odd	30
891.2.n.a.433.1		8	99.79	odd	30
891.2.n.a.784.1		8	99.94	odd	30
891.2.n.d.136.1		8	99.68	even	30
891.2.n.d.190.1		8	99.83	even	30
891.2.n.d.433.1		8	99.2	even	30
891.2.n.d.784.1		8	99.50	even	30
1089.2.a.m.1.1		2	11.10	odd	2
1089.2.a.s.1.2		2	1.1	even	1	trivial
5808.2.a.bl.1.2		2	132.131	odd	2
5808.2.a.bm.1.2		2	12.11	even	2
9075.2.a.x.1.1		2	165.164	even	2
9075.2.a.bv.1.2		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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} +0.381966 q^{5} -3.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} +0.381966 q^{5} -3.00000 q^{7} -2.23607 q^{8} +0.618034 q^{10} -6.23607 q^{13} -4.85410 q^{14} -4.85410 q^{16} -0.618034 q^{17} +0.854102 q^{19} +0.236068 q^{20} +5.47214 q^{23} -4.85410 q^{25} -10.0902 q^{26} -1.85410 q^{28} -4.47214 q^{29} -3.85410 q^{31} -3.38197 q^{32} -1.00000 q^{34} -1.14590 q^{35} -4.23607 q^{37} +1.38197 q^{38} -0.854102 q^{40} +5.94427 q^{41} -1.76393 q^{43} +8.85410 q^{46} +0.618034 q^{47} +2.00000 q^{49} -7.85410 q^{50} -3.85410 q^{52} +7.38197 q^{53} +6.70820 q^{56} -7.23607 q^{58} +5.32624 q^{59} -1.14590 q^{61} -6.23607 q^{62} +4.23607 q^{64} -2.38197 q^{65} +10.5623 q^{67} -0.381966 q^{68} -1.85410 q^{70} -14.5623 q^{71} -1.23607 q^{73} -6.85410 q^{74} +0.527864 q^{76} -0.527864 q^{79} -1.85410 q^{80} +9.61803 q^{82} -12.7082 q^{83} -0.236068 q^{85} -2.85410 q^{86} -9.47214 q^{89} +18.7082 q^{91} +3.38197 q^{92} +1.00000 q^{94} +0.326238 q^{95} +15.0344 q^{97} +3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7	\( 2 q + q^{2} - q^{4} + 3 q^{5} - 6 q^{7} - q^{10} - 8 q^{13} - 3 q^{14} - 3 q^{16} + q^{17} - 5 q^{19} - 4 q^{20} + 2 q^{23} - 3 q^{25} - 9 q^{26} + 3 q^{28} - q^{31} - 9 q^{32} - 2 q^{34} - 9 q^{35} - 4 q^{37} + 5 q^{38} + 5 q^{40} - 6 q^{41} - 8 q^{43} + 11 q^{46} - q^{47} + 4 q^{49} - 9 q^{50} - q^{52} + 17 q^{53} - 10 q^{58} - 5 q^{59} - 9 q^{61} - 8 q^{62} + 4 q^{64} - 7 q^{65} + q^{67} - 3 q^{68} + 3 q^{70} - 9 q^{71} + 2 q^{73} - 7 q^{74} + 10 q^{76} - 10 q^{79} + 3 q^{80} + 17 q^{82} - 12 q^{83} + 4 q^{85} + q^{86} - 10 q^{89} + 24 q^{91} + 9 q^{92} + 2 q^{94} - 15 q^{95} + q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 + 3 * q^5 - 6 * q^7 - q^10 - 8 * q^13 - 3 * q^14 - 3 * q^16 + q^17 - 5 * q^19 - 4 * q^20 + 2 * q^23 - 3 * q^25 - 9 * q^26 + 3 * q^28 - q^31 - 9 * q^32 - 2 * q^34 - 9 * q^35 - 4 * q^37 + 5 * q^38 + 5 * q^40 - 6 * q^41 - 8 * q^43 + 11 * q^46 - q^47 + 4 * q^49 - 9 * q^50 - q^52 + 17 * q^53 - 10 * q^58 - 5 * q^59 - 9 * q^61 - 8 * q^62 + 4 * q^64 - 7 * q^65 + q^67 - 3 * q^68 + 3 * q^70 - 9 * q^71 + 2 * q^73 - 7 * q^74 + 10 * q^76 - 10 * q^79 + 3 * q^80 + 17 * q^82 - 12 * q^83 + 4 * q^85 + q^86 - 10 * q^89 + 24 * q^91 + 9 * q^92 + 2 * q^94 - 15 * q^95 + q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more