LMFDB - Embedded newform 4761.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4761,2,Mod(1,4761)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4761, 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("4761.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4761 = 3^{2} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4761.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:	$38.0167764023$
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 23)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4761.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} -3.23607 q^{7} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} -3.23607 q^{7} +2.23607 q^{8} -0.763932 q^{10} -5.23607 q^{11} +3.00000 q^{13} +2.00000 q^{14} +1.85410 q^{16} +0.763932 q^{17} +2.00000 q^{19} -2.00000 q^{20} +3.23607 q^{22} -3.47214 q^{25} -1.85410 q^{26} +5.23607 q^{28} +3.00000 q^{29} +6.70820 q^{31} -5.61803 q^{32} -0.472136 q^{34} -4.00000 q^{35} +1.23607 q^{37} -1.23607 q^{38} +2.76393 q^{40} +3.47214 q^{41} +8.47214 q^{44} +2.23607 q^{47} +3.47214 q^{49} +2.14590 q^{50} -4.85410 q^{52} +0.472136 q^{53} -6.47214 q^{55} -7.23607 q^{56} -1.85410 q^{58} -6.47214 q^{59} +6.94427 q^{61} -4.14590 q^{62} -0.236068 q^{64} +3.70820 q^{65} +2.76393 q^{67} -1.23607 q^{68} +2.47214 q^{70} -12.2361 q^{71} +6.52786 q^{73} -0.763932 q^{74} -3.23607 q^{76} +16.9443 q^{77} +10.9443 q^{79} +2.29180 q^{80} -2.14590 q^{82} -8.76393 q^{83} +0.944272 q^{85} -11.7082 q^{88} -10.4721 q^{89} -9.70820 q^{91} -1.38197 q^{94} +2.47214 q^{95} -17.7082 q^{97} -2.14590 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 4 q^{14} - 3 q^{16} + 6 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 9 q^{32} + 8 q^{34} - 8 q^{35} - 2 q^{37} + 2 q^{38} + 10 q^{40} - 2 q^{41} + 8 q^{44} - 2 q^{49} + 11 q^{50} - 3 q^{52} - 8 q^{53} - 4 q^{55} - 10 q^{56} + 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{67} + 2 q^{68} - 4 q^{70} - 20 q^{71} + 22 q^{73} - 6 q^{74} - 2 q^{76} + 16 q^{77} + 4 q^{79} + 18 q^{80} - 11 q^{82} - 22 q^{83} - 16 q^{85} - 10 q^{88} - 12 q^{89} - 6 q^{91} - 5 q^{94} - 4 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 4 * q^14 - 3 * q^16 + 6 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 9 * q^32 + 8 * q^34 - 8 * q^35 - 2 * q^37 + 2 * q^38 + 10 * q^40 - 2 * q^41 + 8 * q^44 - 2 * q^49 + 11 * q^50 - 3 * q^52 - 8 * q^53 - 4 * q^55 - 10 * q^56 + 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^67 + 2 * q^68 - 4 * q^70 - 20 * q^71 + 22 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 4 * q^79 + 18 * q^80 - 11 * q^82 - 22 * q^83 - 16 * q^85 - 10 * q^88 - 12 * q^89 - 6 * q^91 - 5 * q^94 - 4 * q^95 - 22 * q^97 - 11 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−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
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	2.23607	0.790569
$9$	0	0
$10$	−0.763932	−0.241577
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	1.85410	0.463525
$17$	0.763932	0.185281	0.0926404	−	0.995700i	$-0.470469\pi$
$17$	0.763932	0.185281	0.0926404	+	0.995700i	$0.470469\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	3.23607	0.689932
$23$	0	0
$23$	0	0
$24$	0	0
$25$	−3.47214	−0.694427
$26$	−1.85410	−0.363619
$27$	0	0
$28$	5.23607	0.989524
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	−0.472136	−0.0809706
$35$	−4.00000	−0.676123
$36$	0	0
$37$	1.23607	0.203208	0.101604	−	0.994825i	$-0.467602\pi$
$37$	1.23607	0.203208	0.101604	+	0.994825i	$0.467602\pi$
$38$	−1.23607	−0.200517
$39$	0	0
$40$	2.76393	0.437016
$41$	3.47214	0.542257	0.271128	−	0.962543i	$-0.412603\pi$
$41$	3.47214	0.542257	0.271128	+	0.962543i	$0.412603\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	8.47214	1.27722
$45$	0	0
$46$	0	0
$47$	2.23607	0.326164	0.163082	−	0.986613i	$-0.447856\pi$
$47$	2.23607	0.326164	0.163082	+	0.986613i	$0.447856\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	2.14590	0.303476
$51$	0	0
$52$	−4.85410	−0.673143
$53$	0.472136	0.0648529	0.0324264	−	0.999474i	$-0.489677\pi$
$53$	0.472136	0.0648529	0.0324264	+	0.999474i	$0.489677\pi$
$54$	0	0
$55$	−6.47214	−0.872703
$56$	−7.23607	−0.966960
$57$	0	0
$58$	−1.85410	−0.243456
$59$	−6.47214	−0.842600	−0.421300	−	0.906921i	$-0.638426\pi$
$59$	−6.47214	−0.842600	−0.421300	+	0.906921i	$0.638426\pi$
$60$	0	0
$61$	6.94427	0.889123	0.444561	−	0.895748i	$-0.353360\pi$
$61$	6.94427	0.889123	0.444561	+	0.895748i	$0.353360\pi$
$62$	−4.14590	−0.526530
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	3.70820	0.459946
$66$	0	0
$67$	2.76393	0.337668	0.168834	−	0.985644i	$-0.446000\pi$
$67$	2.76393	0.337668	0.168834	+	0.985644i	$0.446000\pi$
$68$	−1.23607	−0.149895
$69$	0	0
$70$	2.47214	0.295477
$71$	−12.2361	−1.45215	−0.726077	−	0.687613i	$-0.758658\pi$
$71$	−12.2361	−1.45215	−0.726077	+	0.687613i	$0.758658\pi$
$72$	0	0
$73$	6.52786	0.764029	0.382014	−	0.924156i	$-0.375230\pi$
$73$	6.52786	0.764029	0.382014	+	0.924156i	$0.375230\pi$
$74$	−0.763932	−0.0888053
$75$	0	0
$76$	−3.23607	−0.371202
$77$	16.9443	1.93098
$78$	0	0
$79$	10.9443	1.23133	0.615663	−	0.788009i	$-0.288888\pi$
$79$	10.9443	1.23133	0.615663	+	0.788009i	$0.288888\pi$
$80$	2.29180	0.256231
$81$	0	0
$82$	−2.14590	−0.236975
$83$	−8.76393	−0.961967	−0.480983	−	0.876730i	$-0.659720\pi$
$83$	−8.76393	−0.961967	−0.480983	+	0.876730i	$0.659720\pi$
$84$	0	0
$85$	0.944272	0.102421
$86$	0	0
$87$	0	0
$88$	−11.7082	−1.24810
$89$	−10.4721	−1.11004	−0.555022	−	0.831836i	$-0.687290\pi$
$89$	−10.4721	−1.11004	−0.555022	+	0.831836i	$0.687290\pi$
$90$	0	0
$91$	−9.70820	−1.01770
$92$	0	0
$93$	0	0
$94$	−1.38197	−0.142539
$95$	2.47214	0.253636
$96$	0	0
$97$	−17.7082	−1.79800	−0.898998	−	0.437953i	$-0.855704\pi$
$97$	−17.7082	−1.79800	−0.898998	+	0.437953i	$0.855704\pi$
$98$	−2.14590	−0.216768
$99$	0	0
$100$	5.61803	0.561803
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	4.18034	0.411901	0.205951	−	0.978562i	$-0.433971\pi$
$103$	4.18034	0.411901	0.205951	+	0.978562i	$0.433971\pi$
$104$	6.70820	0.657794
$105$	0	0
$106$	−0.291796	−0.0283417
$107$	13.4164	1.29701	0.648507	−	0.761209i	$-0.275394\pi$
$107$	13.4164	1.29701	0.648507	+	0.761209i	$0.275394\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	4.00000	0.381385
$111$	0	0
$112$	−6.00000	−0.566947
$113$	8.76393	0.824441	0.412221	−	0.911084i	$-0.364753\pi$
$113$	8.76393	0.824441	0.412221	+	0.911084i	$0.364753\pi$
$114$	0	0
$115$	0	0
$116$	−4.85410	−0.450692
$117$	0	0
$118$	4.00000	0.368230
$119$	−2.47214	−0.226620
$120$	0	0
$121$	16.4164	1.49240
$122$	−4.29180	−0.388561
$123$	0	0
$124$	−10.8541	−0.974727
$125$	−10.4721	−0.936656
$126$	0	0
$127$	−7.29180	−0.647042	−0.323521	−	0.946221i	$-0.604867\pi$
$127$	−7.29180	−0.647042	−0.323521	+	0.946221i	$0.604867\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−2.29180	−0.201004
$131$	−18.7082	−1.63454	−0.817272	−	0.576253i	$-0.804514\pi$
$131$	−18.7082	−1.63454	−0.817272	+	0.576253i	$0.804514\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	−1.70820	−0.147566
$135$	0	0
$136$	1.70820	0.146477
$137$	−21.8885	−1.87006	−0.935032	−	0.354563i	$-0.884630\pi$
$137$	−21.8885	−1.87006	−0.935032	+	0.354563i	$0.884630\pi$
$138$	0	0
$139$	−10.7082	−0.908258	−0.454129	−	0.890936i	$-0.650049\pi$
$139$	−10.7082	−0.908258	−0.454129	+	0.890936i	$0.650049\pi$
$140$	6.47214	0.546995
$141$	0	0
$142$	7.56231	0.634615
$143$	−15.7082	−1.31359
$144$	0	0
$145$	3.70820	0.307950
$146$	−4.03444	−0.333893
$147$	0	0
$148$	−2.00000	−0.164399
$149$	23.8885	1.95703	0.978513	−	0.206186i	$-0.0661051\pi$
$149$	23.8885	1.95703	0.978513	+	0.206186i	$0.0661051\pi$
$150$	0	0
$151$	4.23607	0.344726	0.172363	−	0.985033i	$-0.444860\pi$
$151$	4.23607	0.344726	0.172363	+	0.985033i	$0.444860\pi$
$152$	4.47214	0.362738
$153$	0	0
$154$	−10.4721	−0.843869
$155$	8.29180	0.666013
$156$	0	0
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	−6.76393	−0.538110
$159$	0	0
$160$	−6.94427	−0.548993
$161$	0	0
$162$	0	0
$163$	−5.76393	−0.451466	−0.225733	−	0.974189i	$-0.572478\pi$
$163$	−5.76393	−0.451466	−0.225733	+	0.974189i	$0.572478\pi$
$164$	−5.61803	−0.438695
$165$	0	0
$166$	5.41641	0.420395
$167$	−1.52786	−0.118230	−0.0591148	−	0.998251i	$-0.518828\pi$
$167$	−1.52786	−0.118230	−0.0591148	+	0.998251i	$0.518828\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	−0.583592	−0.0447595
$171$	0	0
$172$	0	0
$173$	−22.9443	−1.74442	−0.872210	−	0.489131i	$-0.837314\pi$
$173$	−22.9443	−1.74442	−0.872210	+	0.489131i	$0.837314\pi$
$174$	0	0
$175$	11.2361	0.849367
$176$	−9.70820	−0.731783
$177$	0	0
$178$	6.47214	0.485107
$179$	−0.708204	−0.0529336	−0.0264668	−	0.999650i	$-0.508426\pi$
$179$	−0.708204	−0.0529336	−0.0264668	+	0.999650i	$0.508426\pi$
$180$	0	0
$181$	−16.6525	−1.23777	−0.618884	−	0.785482i	$-0.712415\pi$
$181$	−16.6525	−1.23777	−0.618884	+	0.785482i	$0.712415\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	0	0
$185$	1.52786	0.112331
$186$	0	0
$187$	−4.00000	−0.292509
$188$	−3.61803	−0.263872
$189$	0	0
$190$	−1.52786	−0.110843
$191$	−26.1803	−1.89434	−0.947171	−	0.320728i	$-0.896073\pi$
$191$	−26.1803	−1.89434	−0.947171	+	0.320728i	$0.896073\pi$
$192$	0	0
$193$	9.94427	0.715804	0.357902	−	0.933759i	$-0.383492\pi$
$193$	9.94427	0.715804	0.357902	+	0.933759i	$0.383492\pi$
$194$	10.9443	0.785753
$195$	0	0
$196$	−5.61803	−0.401288
$197$	1.47214	0.104885	0.0524427	−	0.998624i	$-0.483299\pi$
$197$	1.47214	0.104885	0.0524427	+	0.998624i	$0.483299\pi$
$198$	0	0
$199$	12.2918	0.871342	0.435671	−	0.900106i	$-0.356511\pi$
$199$	12.2918	0.871342	0.435671	+	0.900106i	$0.356511\pi$
$200$	−7.76393	−0.548993
$201$	0	0
$202$	2.76393	0.194470
$203$	−9.70820	−0.681382
$204$	0	0
$205$	4.29180	0.299752
$206$	−2.58359	−0.180007
$207$	0	0
$208$	5.56231	0.385677
$209$	−10.4721	−0.724373
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	−0.763932	−0.0524671
$213$	0	0
$214$	−8.29180	−0.566816
$215$	0	0
$216$	0	0
$217$	−21.7082	−1.47365
$218$	0	0
$219$	0	0
$220$	10.4721	0.706031
$221$	2.29180	0.154163
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	18.1803	1.21473
$225$	0	0
$226$	−5.41641	−0.360294
$227$	−12.1803	−0.808438	−0.404219	−	0.914662i	$-0.632457\pi$
$227$	−12.1803	−0.808438	−0.404219	+	0.914662i	$0.632457\pi$
$228$	0	0
$229$	12.0000	0.792982	0.396491	−	0.918039i	$-0.370228\pi$
$229$	12.0000	0.792982	0.396491	+	0.918039i	$0.370228\pi$
$230$	0	0
$231$	0	0
$232$	6.70820	0.440415
$233$	6.52786	0.427655	0.213827	−	0.976871i	$-0.431407\pi$
$233$	6.52786	0.427655	0.213827	+	0.976871i	$0.431407\pi$
$234$	0	0
$235$	2.76393	0.180299
$236$	10.4721	0.681678
$237$	0	0
$238$	1.52786	0.0990367
$239$	−13.7639	−0.890315	−0.445157	−	0.895452i	$-0.646852\pi$
$239$	−13.7639	−0.890315	−0.445157	+	0.895452i	$0.646852\pi$
$240$	0	0
$241$	23.1246	1.48959	0.744794	−	0.667295i	$-0.232548\pi$
$241$	23.1246	1.48959	0.744794	+	0.667295i	$0.232548\pi$
$242$	−10.1459	−0.652203
$243$	0	0
$244$	−11.2361	−0.719316
$245$	4.29180	0.274193
$246$	0	0
$247$	6.00000	0.381771
$248$	15.0000	0.952501
$249$	0	0
$250$	6.47214	0.409334
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	0	0
$253$	0	0
$254$	4.50658	0.282768
$255$	0	0
$256$	−6.56231	−0.410144
$257$	7.47214	0.466099	0.233050	−	0.972465i	$-0.425130\pi$
$257$	7.47214	0.466099	0.233050	+	0.972465i	$0.425130\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	−6.00000	−0.372104
$261$	0	0
$262$	11.5623	0.714322
$263$	2.94427	0.181552	0.0907758	−	0.995871i	$-0.471065\pi$
$263$	2.94427	0.181552	0.0907758	+	0.995871i	$0.471065\pi$
$264$	0	0
$265$	0.583592	0.0358498
$266$	4.00000	0.245256
$267$	0	0
$268$	−4.47214	−0.273179
$269$	7.94427	0.484371	0.242185	−	0.970230i	$-0.422136\pi$
$269$	7.94427	0.484371	0.242185	+	0.970230i	$0.422136\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.41641	0.0858823
$273$	0	0
$274$	13.5279	0.817248
$275$	18.1803	1.09632
$276$	0	0
$277$	15.4721	0.929631	0.464815	−	0.885408i	$-0.346121\pi$
$277$	15.4721	0.929631	0.464815	+	0.885408i	$0.346121\pi$
$278$	6.61803	0.396923
$279$	0	0
$280$	−8.94427	−0.534522
$281$	−8.76393	−0.522812	−0.261406	−	0.965229i	$-0.584186\pi$
$281$	−8.76393	−0.522812	−0.261406	+	0.965229i	$0.584186\pi$
$282$	0	0
$283$	−27.7082	−1.64708	−0.823541	−	0.567257i	$-0.808005\pi$
$283$	−27.7082	−1.64708	−0.823541	+	0.567257i	$0.808005\pi$
$284$	19.7984	1.17482
$285$	0	0
$286$	9.70820	0.574058
$287$	−11.2361	−0.663244
$288$	0	0
$289$	−16.4164	−0.965671
$290$	−2.29180	−0.134579
$291$	0	0
$292$	−10.5623	−0.618112
$293$	−1.52786	−0.0892588	−0.0446294	−	0.999004i	$-0.514211\pi$
$293$	−1.52786	−0.0892588	−0.0446294	+	0.999004i	$0.514211\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	2.76393	0.160650
$297$	0	0
$298$	−14.7639	−0.855252
$299$	0	0
$300$	0	0
$301$	0	0
$302$	−2.61803	−0.150651
$303$	0	0
$304$	3.70820	0.212680
$305$	8.58359	0.491495
$306$	0	0
$307$	9.52786	0.543784	0.271892	−	0.962328i	$-0.412351\pi$
$307$	9.52786	0.543784	0.271892	+	0.962328i	$0.412351\pi$
$308$	−27.4164	−1.56219
$309$	0	0
$310$	−5.12461	−0.291058
$311$	−13.1803	−0.747389	−0.373694	−	0.927552i	$-0.621909\pi$
$311$	−13.1803	−0.747389	−0.373694	+	0.927552i	$0.621909\pi$
$312$	0	0
$313$	−24.3607	−1.37695	−0.688474	−	0.725261i	$-0.741719\pi$
$313$	−24.3607	−1.37695	−0.688474	+	0.725261i	$0.741719\pi$
$314$	−7.05573	−0.398178
$315$	0	0
$316$	−17.7082	−0.996164
$317$	−25.4164	−1.42753	−0.713764	−	0.700386i	$-0.753011\pi$
$317$	−25.4164	−1.42753	−0.713764	+	0.700386i	$0.753011\pi$
$318$	0	0
$319$	−15.7082	−0.879491
$320$	−0.291796	−0.0163119
$321$	0	0
$322$	0	0
$323$	1.52786	0.0850126
$324$	0	0
$325$	−10.4164	−0.577798
$326$	3.56231	0.197298
$327$	0	0
$328$	7.76393	0.428691
$329$	−7.23607	−0.398937
$330$	0	0
$331$	−19.6525	−1.08020	−0.540099	−	0.841602i	$-0.681613\pi$
$331$	−19.6525	−1.08020	−0.540099	+	0.841602i	$0.681613\pi$
$332$	14.1803	0.778247
$333$	0	0
$334$	0.944272	0.0516683
$335$	3.41641	0.186658
$336$	0	0
$337$	−23.4164	−1.27557	−0.637787	−	0.770213i	$-0.720150\pi$
$337$	−23.4164	−1.27557	−0.637787	+	0.770213i	$0.720150\pi$
$338$	2.47214	0.134466
$339$	0	0
$340$	−1.52786	−0.0828601
$341$	−35.1246	−1.90210
$342$	0	0
$343$	11.4164	0.616428
$344$	0	0
$345$	0	0
$346$	14.1803	0.762340
$347$	9.88854	0.530845	0.265422	−	0.964132i	$-0.414489\pi$
$347$	9.88854	0.530845	0.265422	+	0.964132i	$0.414489\pi$
$348$	0	0
$349$	24.4164	1.30698	0.653490	−	0.756935i	$-0.273304\pi$
$349$	24.4164	1.30698	0.653490	+	0.756935i	$0.273304\pi$
$350$	−6.94427	−0.371187
$351$	0	0
$352$	29.4164	1.56790
$353$	−9.36068	−0.498219	−0.249109	−	0.968475i	$-0.580138\pi$
$353$	−9.36068	−0.498219	−0.249109	+	0.968475i	$0.580138\pi$
$354$	0	0
$355$	−15.1246	−0.802731
$356$	16.9443	0.898045
$357$	0	0
$358$	0.437694	0.0231329
$359$	−19.8885	−1.04968	−0.524839	−	0.851202i	$-0.675874\pi$
$359$	−19.8885	−1.04968	−0.524839	+	0.851202i	$0.675874\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	10.2918	0.540925
$363$	0	0
$364$	15.7082	0.823334
$365$	8.06888	0.422345
$366$	0	0
$367$	4.18034	0.218212	0.109106	−	0.994030i	$-0.465201\pi$
$367$	4.18034	0.218212	0.109106	+	0.994030i	$0.465201\pi$
$368$	0	0
$369$	0	0
$370$	−0.944272	−0.0490904
$371$	−1.52786	−0.0793227
$372$	0	0
$373$	−7.70820	−0.399116	−0.199558	−	0.979886i	$-0.563951\pi$
$373$	−7.70820	−0.399116	−0.199558	+	0.979886i	$0.563951\pi$
$374$	2.47214	0.127831
$375$	0	0
$376$	5.00000	0.257855
$377$	9.00000	0.463524
$378$	0	0
$379$	−24.3607	−1.25132	−0.625662	−	0.780094i	$-0.715171\pi$
$379$	−24.3607	−1.25132	−0.625662	+	0.780094i	$0.715171\pi$
$380$	−4.00000	−0.205196
$381$	0	0
$382$	16.1803	0.827858
$383$	7.05573	0.360531	0.180265	−	0.983618i	$-0.442304\pi$
$383$	7.05573	0.360531	0.180265	+	0.983618i	$0.442304\pi$
$384$	0	0
$385$	20.9443	1.06742
$386$	−6.14590	−0.312818
$387$	0	0
$388$	28.6525	1.45461
$389$	25.5279	1.29431	0.647157	−	0.762357i	$-0.275958\pi$
$389$	25.5279	1.29431	0.647157	+	0.762357i	$0.275958\pi$
$390$	0	0
$391$	0	0
$392$	7.76393	0.392138
$393$	0	0
$394$	−0.909830	−0.0458366
$395$	13.5279	0.680661
$396$	0	0
$397$	−24.4164	−1.22542	−0.612712	−	0.790306i	$-0.709922\pi$
$397$	−24.4164	−1.22542	−0.612712	+	0.790306i	$0.709922\pi$
$398$	−7.59675	−0.380791
$399$	0	0
$400$	−6.43769	−0.321885
$401$	−14.1803	−0.708132	−0.354066	−	0.935220i	$-0.615201\pi$
$401$	−14.1803	−0.708132	−0.354066	+	0.935220i	$0.615201\pi$
$402$	0	0
$403$	20.1246	1.00248
$404$	7.23607	0.360008
$405$	0	0
$406$	6.00000	0.297775
$407$	−6.47214	−0.320812
$408$	0	0
$409$	21.3607	1.05622	0.528109	−	0.849177i	$-0.322901\pi$
$409$	21.3607	1.05622	0.528109	+	0.849177i	$0.322901\pi$
$410$	−2.65248	−0.130996
$411$	0	0
$412$	−6.76393	−0.333235
$413$	20.9443	1.03060
$414$	0	0
$415$	−10.8328	−0.531762
$416$	−16.8541	−0.826340
$417$	0	0
$418$	6.47214	0.316563
$419$	−4.58359	−0.223923	−0.111962	−	0.993713i	$-0.535713\pi$
$419$	−4.58359	−0.223923	−0.111962	+	0.993713i	$0.535713\pi$
$420$	0	0
$421$	10.2918	0.501591	0.250796	−	0.968040i	$-0.419308\pi$
$421$	10.2918	0.501591	0.250796	+	0.968040i	$0.419308\pi$
$422$	14.4721	0.704493
$423$	0	0
$424$	1.05573	0.0512707
$425$	−2.65248	−0.128664
$426$	0	0
$427$	−22.4721	−1.08750
$428$	−21.7082	−1.04931
$429$	0	0
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	0	0
$433$	−17.8197	−0.856358	−0.428179	−	0.903694i	$-0.640845\pi$
$433$	−17.8197	−0.856358	−0.428179	+	0.903694i	$0.640845\pi$
$434$	13.4164	0.644008
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−18.7082	−0.892894	−0.446447	−	0.894810i	$-0.647311\pi$
$439$	−18.7082	−0.892894	−0.446447	+	0.894810i	$0.647311\pi$
$440$	−14.4721	−0.689932
$441$	0	0
$442$	−1.41641	−0.0673717
$443$	−38.1246	−1.81135	−0.905677	−	0.423967i	$-0.860637\pi$
$443$	−38.1246	−1.81135	−0.905677	+	0.423967i	$0.860637\pi$
$444$	0	0
$445$	−12.9443	−0.613617
$446$	−2.47214	−0.117059
$447$	0	0
$448$	0.763932	0.0360924
$449$	14.9443	0.705264	0.352632	−	0.935762i	$-0.385287\pi$
$449$	14.9443	0.705264	0.352632	+	0.935762i	$0.385287\pi$
$450$	0	0
$451$	−18.1803	−0.856079
$452$	−14.1803	−0.666987
$453$	0	0
$454$	7.52786	0.353300
$455$	−12.0000	−0.562569
$456$	0	0
$457$	5.12461	0.239719	0.119860	−	0.992791i	$-0.461756\pi$
$457$	5.12461	0.239719	0.119860	+	0.992791i	$0.461756\pi$
$458$	−7.41641	−0.346546
$459$	0	0
$460$	0	0
$461$	1.47214	0.0685642	0.0342821	−	0.999412i	$-0.489086\pi$
$461$	1.47214	0.0685642	0.0342821	+	0.999412i	$0.489086\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	5.56231	0.258224
$465$	0	0
$466$	−4.03444	−0.186892
$467$	−13.0557	−0.604147	−0.302074	−	0.953285i	$-0.597679\pi$
$467$	−13.0557	−0.604147	−0.302074	+	0.953285i	$0.597679\pi$
$468$	0	0
$469$	−8.94427	−0.413008
$470$	−1.70820	−0.0787936
$471$	0	0
$472$	−14.4721	−0.666134
$473$	0	0
$474$	0	0
$475$	−6.94427	−0.318625
$476$	4.00000	0.183340
$477$	0	0
$478$	8.50658	0.389082
$479$	31.5967	1.44369	0.721846	−	0.692054i	$-0.243294\pi$
$479$	31.5967	1.44369	0.721846	+	0.692054i	$0.243294\pi$
$480$	0	0
$481$	3.70820	0.169080
$482$	−14.2918	−0.650973
$483$	0	0
$484$	−26.5623	−1.20738
$485$	−21.8885	−0.993908
$486$	0	0
$487$	−14.7082	−0.666492	−0.333246	−	0.942840i	$-0.608144\pi$
$487$	−14.7082	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$488$	15.5279	0.702913
$489$	0	0
$490$	−2.65248	−0.119827
$491$	−8.34752	−0.376718	−0.188359	−	0.982100i	$-0.560317\pi$
$491$	−8.34752	−0.376718	−0.188359	+	0.982100i	$0.560317\pi$
$492$	0	0
$493$	2.29180	0.103217
$494$	−3.70820	−0.166840
$495$	0	0
$496$	12.4377	0.558469
$497$	39.5967	1.77616
$498$	0	0
$499$	19.2918	0.863619	0.431810	−	0.901965i	$-0.357875\pi$
$499$	19.2918	0.863619	0.431810	+	0.901965i	$0.357875\pi$
$500$	16.9443	0.757771
$501$	0	0
$502$	−1.41641	−0.0632174
$503$	−26.9443	−1.20139	−0.600693	−	0.799480i	$-0.705109\pi$
$503$	−26.9443	−1.20139	−0.600693	+	0.799480i	$0.705109\pi$
$504$	0	0
$505$	−5.52786	−0.245987
$506$	0	0
$507$	0	0
$508$	11.7984	0.523468
$509$	28.3050	1.25459	0.627297	−	0.778780i	$-0.284161\pi$
$509$	28.3050	1.25459	0.627297	+	0.778780i	$0.284161\pi$
$510$	0	0
$511$	−21.1246	−0.934498
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−4.61803	−0.203693
$515$	5.16718	0.227693
$516$	0	0
$517$	−11.7082	−0.514926
$518$	2.47214	0.108619
$519$	0	0
$520$	8.29180	0.363619
$521$	31.4164	1.37638	0.688189	−	0.725532i	$-0.258406\pi$
$521$	31.4164	1.37638	0.688189	+	0.725532i	$0.258406\pi$
$522$	0	0
$523$	−41.1246	−1.79825	−0.899127	−	0.437688i	$-0.855797\pi$
$523$	−41.1246	−1.79825	−0.899127	+	0.437688i	$0.855797\pi$
$524$	30.2705	1.32237
$525$	0	0
$526$	−1.81966	−0.0793410
$527$	5.12461	0.223232
$528$	0	0
$529$	0	0
$530$	−0.360680	−0.0156669
$531$	0	0
$532$	10.4721	0.454025
$533$	10.4164	0.451185
$534$	0	0
$535$	16.5836	0.716971
$536$	6.18034	0.266950
$537$	0	0
$538$	−4.90983	−0.211678
$539$	−18.1803	−0.783083
$540$	0	0
$541$	−34.4164	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$541$	−34.4164	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$542$	−4.94427	−0.212375
$543$	0	0
$544$	−4.29180	−0.184009
$545$	0	0
$546$	0	0
$547$	−29.5410	−1.26308	−0.631541	−	0.775342i	$-0.717577\pi$
$547$	−29.5410	−1.26308	−0.631541	+	0.775342i	$0.717577\pi$
$548$	35.4164	1.51291
$549$	0	0
$550$	−11.2361	−0.479108
$551$	6.00000	0.255609
$552$	0	0
$553$	−35.4164	−1.50606
$554$	−9.56231	−0.406263
$555$	0	0
$556$	17.3262	0.734796
$557$	−7.41641	−0.314243	−0.157122	−	0.987579i	$-0.550221\pi$
$557$	−7.41641	−0.314243	−0.157122	+	0.987579i	$0.550221\pi$
$558$	0	0
$559$	0	0
$560$	−7.41641	−0.313400
$561$	0	0
$562$	5.41641	0.228477
$563$	−32.9443	−1.38844	−0.694218	−	0.719765i	$-0.744250\pi$
$563$	−32.9443	−1.38844	−0.694218	+	0.719765i	$0.744250\pi$
$564$	0	0
$565$	10.8328	0.455740
$566$	17.1246	0.719801
$567$	0	0
$568$	−27.3607	−1.14803
$569$	−22.1803	−0.929848	−0.464924	−	0.885351i	$-0.653918\pi$
$569$	−22.1803	−0.929848	−0.464924	+	0.885351i	$0.653918\pi$
$570$	0	0
$571$	14.2918	0.598093	0.299047	−	0.954239i	$-0.403331\pi$
$571$	14.2918	0.598093	0.299047	+	0.954239i	$0.403331\pi$
$572$	25.4164	1.06271
$573$	0	0
$574$	6.94427	0.289848
$575$	0	0
$576$	0	0
$577$	22.8885	0.952863	0.476431	−	0.879212i	$-0.341930\pi$
$577$	22.8885	0.952863	0.476431	+	0.879212i	$0.341930\pi$
$578$	10.1459	0.422014
$579$	0	0
$580$	−6.00000	−0.249136
$581$	28.3607	1.17660
$582$	0	0
$583$	−2.47214	−0.102385
$584$	14.5967	0.604018
$585$	0	0
$586$	0.944272	0.0390075
$587$	24.7082	1.01982	0.509908	−	0.860229i	$-0.329679\pi$
$587$	24.7082	1.01982	0.509908	+	0.860229i	$0.329679\pi$
$588$	0	0
$589$	13.4164	0.552813
$590$	4.94427	0.203552
$591$	0	0
$592$	2.29180	0.0941922
$593$	2.94427	0.120907	0.0604534	−	0.998171i	$-0.480745\pi$
$593$	2.94427	0.120907	0.0604534	+	0.998171i	$0.480745\pi$
$594$	0	0
$595$	−3.05573	−0.125273
$596$	−38.6525	−1.58327
$597$	0	0
$598$	0	0
$599$	−33.8885	−1.38465	−0.692324	−	0.721587i	$-0.743413\pi$
$599$	−33.8885	−1.38465	−0.692324	+	0.721587i	$0.743413\pi$
$600$	0	0
$601$	46.8885	1.91262	0.956312	−	0.292349i	$-0.0944368\pi$
$601$	46.8885	1.91262	0.956312	+	0.292349i	$0.0944368\pi$
$602$	0	0
$603$	0	0
$604$	−6.85410	−0.278889
$605$	20.2918	0.824979
$606$	0	0
$607$	26.4721	1.07447	0.537235	−	0.843432i	$-0.319469\pi$
$607$	26.4721	1.07447	0.537235	+	0.843432i	$0.319469\pi$
$608$	−11.2361	−0.455683
$609$	0	0
$610$	−5.30495	−0.214791
$611$	6.70820	0.271385
$612$	0	0
$613$	−5.70820	−0.230552	−0.115276	−	0.993333i	$-0.536775\pi$
$613$	−5.70820	−0.230552	−0.115276	+	0.993333i	$0.536775\pi$
$614$	−5.88854	−0.237642
$615$	0	0
$616$	37.8885	1.52657
$617$	−7.52786	−0.303060	−0.151530	−	0.988453i	$-0.548420\pi$
$617$	−7.52786	−0.303060	−0.151530	+	0.988453i	$0.548420\pi$
$618$	0	0
$619$	−19.4164	−0.780411	−0.390206	−	0.920728i	$-0.627596\pi$
$619$	−19.4164	−0.780411	−0.390206	+	0.920728i	$0.627596\pi$
$620$	−13.4164	−0.538816
$621$	0	0
$622$	8.14590	0.326621
$623$	33.8885	1.35772
$624$	0	0
$625$	4.41641	0.176656
$626$	15.0557	0.601748
$627$	0	0
$628$	−18.4721	−0.737118
$629$	0.944272	0.0376506
$630$	0	0
$631$	−12.3607	−0.492071	−0.246035	−	0.969261i	$-0.579128\pi$
$631$	−12.3607	−0.492071	−0.246035	+	0.969261i	$0.579128\pi$
$632$	24.4721	0.973449
$633$	0	0
$634$	15.7082	0.623852
$635$	−9.01316	−0.357676
$636$	0	0
$637$	10.4164	0.412713
$638$	9.70820	0.384351
$639$	0	0
$640$	14.0689	0.556121
$641$	−17.3050	−0.683504	−0.341752	−	0.939790i	$-0.611020\pi$
$641$	−17.3050	−0.683504	−0.341752	+	0.939790i	$0.611020\pi$
$642$	0	0
$643$	29.5967	1.16718	0.583591	−	0.812048i	$-0.301647\pi$
$643$	29.5967	1.16718	0.583591	+	0.812048i	$0.301647\pi$
$644$	0	0
$645$	0	0
$646$	−0.944272	−0.0371519
$647$	−6.70820	−0.263727	−0.131863	−	0.991268i	$-0.542096\pi$
$647$	−6.70820	−0.263727	−0.131863	+	0.991268i	$0.542096\pi$
$648$	0	0
$649$	33.8885	1.33024
$650$	6.43769	0.252507
$651$	0	0
$652$	9.32624	0.365244
$653$	38.3050	1.49899	0.749494	−	0.662011i	$-0.230297\pi$
$653$	38.3050	1.49899	0.749494	+	0.662011i	$0.230297\pi$
$654$	0	0
$655$	−23.1246	−0.903553
$656$	6.43769	0.251350
$657$	0	0
$658$	4.47214	0.174342
$659$	−10.6525	−0.414962	−0.207481	−	0.978239i	$-0.566526\pi$
$659$	−10.6525	−0.414962	−0.207481	+	0.978239i	$0.566526\pi$
$660$	0	0
$661$	22.9443	0.892429	0.446214	−	0.894926i	$-0.352772\pi$
$661$	22.9443	0.892429	0.446214	+	0.894926i	$0.352772\pi$
$662$	12.1459	0.472064
$663$	0	0
$664$	−19.5967	−0.760501
$665$	−8.00000	−0.310227
$666$	0	0
$667$	0	0
$668$	2.47214	0.0956498
$669$	0	0
$670$	−2.11146	−0.0815727
$671$	−36.3607	−1.40369
$672$	0	0
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	14.4721	0.557446
$675$	0	0
$676$	6.47214	0.248928
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	57.3050	2.19916
$680$	2.11146	0.0809706
$681$	0	0
$682$	21.7082	0.831250
$683$	−26.5967	−1.01770	−0.508848	−	0.860856i	$-0.669929\pi$
$683$	−26.5967	−1.01770	−0.508848	+	0.860856i	$0.669929\pi$
$684$	0	0
$685$	−27.0557	−1.03375
$686$	−7.05573	−0.269389
$687$	0	0
$688$	0	0
$689$	1.41641	0.0539608
$690$	0	0
$691$	7.05573	0.268413	0.134206	−	0.990953i	$-0.457152\pi$
$691$	7.05573	0.268413	0.134206	+	0.990953i	$0.457152\pi$
$692$	37.1246	1.41127
$693$	0	0
$694$	−6.11146	−0.231988
$695$	−13.2361	−0.502073
$696$	0	0
$697$	2.65248	0.100470
$698$	−15.0902	−0.571171
$699$	0	0
$700$	−18.1803	−0.687152
$701$	−3.81966	−0.144267	−0.0721333	−	0.997395i	$-0.522981\pi$
$701$	−3.81966	−0.144267	−0.0721333	+	0.997395i	$0.522981\pi$
$702$	0	0
$703$	2.47214	0.0932384
$704$	1.23607	0.0465861
$705$	0	0
$706$	5.78522	0.217730
$707$	14.4721	0.544281
$708$	0	0
$709$	42.0689	1.57993	0.789965	−	0.613152i	$-0.210099\pi$
$709$	42.0689	1.57993	0.789965	+	0.613152i	$0.210099\pi$
$710$	9.34752	0.350806
$711$	0	0
$712$	−23.4164	−0.877567
$713$	0	0
$714$	0	0
$715$	−19.4164	−0.726132
$716$	1.14590	0.0428242
$717$	0	0
$718$	12.2918	0.458726
$719$	3.05573	0.113959	0.0569797	−	0.998375i	$-0.481853\pi$
$719$	3.05573	0.113959	0.0569797	+	0.998375i	$0.481853\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	9.27051	0.345013
$723$	0	0
$724$	26.9443	1.00138
$725$	−10.4164	−0.386856
$726$	0	0
$727$	27.7082	1.02764	0.513820	−	0.857898i	$-0.328230\pi$
$727$	27.7082	1.02764	0.513820	+	0.857898i	$0.328230\pi$
$728$	−21.7082	−0.804560
$729$	0	0
$730$	−4.98684	−0.184571
$731$	0	0
$732$	0	0
$733$	31.2361	1.15373	0.576865	−	0.816839i	$-0.304276\pi$
$733$	31.2361	1.15373	0.576865	+	0.816839i	$0.304276\pi$
$734$	−2.58359	−0.0953621
$735$	0	0
$736$	0	0
$737$	−14.4721	−0.533088
$738$	0	0
$739$	26.8197	0.986577	0.493289	−	0.869866i	$-0.335795\pi$
$739$	26.8197	0.986577	0.493289	+	0.869866i	$0.335795\pi$
$740$	−2.47214	−0.0908775
$741$	0	0
$742$	0.944272	0.0346653
$743$	41.1246	1.50872	0.754358	−	0.656463i	$-0.227948\pi$
$743$	41.1246	1.50872	0.754358	+	0.656463i	$0.227948\pi$
$744$	0	0
$745$	29.5279	1.08182
$746$	4.76393	0.174420
$747$	0	0
$748$	6.47214	0.236645
$749$	−43.4164	−1.58640
$750$	0	0
$751$	−0.360680	−0.0131614	−0.00658070	−	0.999978i	$-0.502095\pi$
$751$	−0.360680	−0.0131614	−0.00658070	+	0.999978i	$0.502095\pi$
$752$	4.14590	0.151185
$753$	0	0
$754$	−5.56231	−0.202567
$755$	5.23607	0.190560
$756$	0	0
$757$	−1.59675	−0.0580348	−0.0290174	−	0.999579i	$-0.509238\pi$
$757$	−1.59675	−0.0580348	−0.0290174	+	0.999579i	$0.509238\pi$
$758$	15.0557	0.546849
$759$	0	0
$760$	5.52786	0.200517
$761$	−46.3050	−1.67855	−0.839277	−	0.543705i	$-0.817021\pi$
$761$	−46.3050	−1.67855	−0.839277	+	0.543705i	$0.817021\pi$
$762$	0	0
$763$	0	0
$764$	42.3607	1.53256
$765$	0	0
$766$	−4.36068	−0.157558
$767$	−19.4164	−0.701086
$768$	0	0
$769$	23.1246	0.833895	0.416947	−	0.908931i	$-0.363100\pi$
$769$	23.1246	0.833895	0.416947	+	0.908931i	$0.363100\pi$
$770$	−12.9443	−0.466479
$771$	0	0
$772$	−16.0902	−0.579098
$773$	−5.52786	−0.198823	−0.0994117	−	0.995046i	$-0.531696\pi$
$773$	−5.52786	−0.198823	−0.0994117	+	0.995046i	$0.531696\pi$
$774$	0	0
$775$	−23.2918	−0.836666
$776$	−39.5967	−1.42144
$777$	0	0
$778$	−15.7771	−0.565636
$779$	6.94427	0.248804
$780$	0	0
$781$	64.0689	2.29256
$782$	0	0
$783$	0	0
$784$	6.43769	0.229918
$785$	14.1115	0.503659
$786$	0	0
$787$	−24.5836	−0.876310	−0.438155	−	0.898899i	$-0.644368\pi$
$787$	−24.5836	−0.876310	−0.438155	+	0.898899i	$0.644368\pi$
$788$	−2.38197	−0.0848540
$789$	0	0
$790$	−8.36068	−0.297460
$791$	−28.3607	−1.00839
$792$	0	0
$793$	20.8328	0.739795
$794$	15.0902	0.535530
$795$	0	0
$796$	−19.8885	−0.704931
$797$	−34.3607	−1.21712	−0.608559	−	0.793509i	$-0.708252\pi$
$797$	−34.3607	−1.21712	−0.608559	+	0.793509i	$0.708252\pi$
$798$	0	0
$799$	1.70820	0.0604319
$800$	19.5066	0.689662
$801$	0	0
$802$	8.76393	0.309465
$803$	−34.1803	−1.20620
$804$	0	0
$805$	0	0
$806$	−12.4377	−0.438099
$807$	0	0
$808$	−10.0000	−0.351799
$809$	−12.1115	−0.425816	−0.212908	−	0.977072i	$-0.568293\pi$
$809$	−12.1115	−0.425816	−0.212908	+	0.977072i	$0.568293\pi$
$810$	0	0
$811$	−24.3475	−0.854957	−0.427479	−	0.904025i	$-0.640598\pi$
$811$	−24.3475	−0.854957	−0.427479	+	0.904025i	$0.640598\pi$
$812$	15.7082	0.551250
$813$	0	0
$814$	4.00000	0.140200
$815$	−7.12461	−0.249564
$816$	0	0
$817$	0	0
$818$	−13.2016	−0.461584
$819$	0	0
$820$	−6.94427	−0.242504
$821$	38.9443	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$821$	38.9443	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$822$	0	0
$823$	−39.5410	−1.37831	−0.689157	−	0.724612i	$-0.742019\pi$
$823$	−39.5410	−1.37831	−0.689157	+	0.724612i	$0.742019\pi$
$824$	9.34752	0.325636
$825$	0	0
$826$	−12.9443	−0.450389
$827$	1.52786	0.0531290	0.0265645	−	0.999647i	$-0.491543\pi$
$827$	1.52786	0.0531290	0.0265645	+	0.999647i	$0.491543\pi$
$828$	0	0
$829$	40.2492	1.39791	0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492	1.39791	0.698957	+	0.715164i	$0.253648\pi$
$830$	6.69505	0.232389
$831$	0	0
$832$	−0.708204	−0.0245526
$833$	2.65248	0.0919028
$834$	0	0
$835$	−1.88854	−0.0653558
$836$	16.9443	0.586030
$837$	0	0
$838$	2.83282	0.0978580
$839$	−41.1246	−1.41978	−0.709890	−	0.704313i	$-0.751255\pi$
$839$	−41.1246	−1.41978	−0.709890	+	0.704313i	$0.751255\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−6.36068	−0.219204
$843$	0	0
$844$	37.8885	1.30418
$845$	−4.94427	−0.170088
$846$	0	0
$847$	−53.1246	−1.82538
$848$	0.875388	0.0300610
$849$	0	0
$850$	1.63932	0.0562282
$851$	0	0
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	13.8885	0.475256
$855$	0	0
$856$	30.0000	1.02538
$857$	−1.47214	−0.0502872	−0.0251436	−	0.999684i	$-0.508004\pi$
$857$	−1.47214	−0.0502872	−0.0251436	+	0.999684i	$0.508004\pi$
$858$	0	0
$859$	−16.7082	−0.570077	−0.285038	−	0.958516i	$-0.592006\pi$
$859$	−16.7082	−0.570077	−0.285038	+	0.958516i	$0.592006\pi$
$860$	0	0
$861$	0	0
$862$	10.8328	0.368967
$863$	21.5410	0.733265	0.366632	−	0.930366i	$-0.380511\pi$
$863$	21.5410	0.733265	0.366632	+	0.930366i	$0.380511\pi$
$864$	0	0
$865$	−28.3607	−0.964292
$866$	11.0132	0.374242
$867$	0	0
$868$	35.1246	1.19221
$869$	−57.3050	−1.94394
$870$	0	0
$871$	8.29180	0.280957
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33.8885	1.14564
$876$	0	0
$877$	−36.4721	−1.23158	−0.615788	−	0.787912i	$-0.711162\pi$
$877$	−36.4721	−1.23158	−0.615788	+	0.787912i	$0.711162\pi$
$878$	11.5623	0.390209
$879$	0	0
$880$	−12.0000	−0.404520
$881$	44.1803	1.48847	0.744237	−	0.667916i	$-0.232813\pi$
$881$	44.1803	1.48847	0.744237	+	0.667916i	$0.232813\pi$
$882$	0	0
$883$	4.00000	0.134611	0.0673054	−	0.997732i	$-0.478560\pi$
$883$	4.00000	0.134611	0.0673054	+	0.997732i	$0.478560\pi$
$884$	−3.70820	−0.124720
$885$	0	0
$886$	23.5623	0.791591
$887$	−23.0689	−0.774577	−0.387289	−	0.921959i	$-0.626588\pi$
$887$	−23.0689	−0.774577	−0.387289	+	0.921959i	$0.626588\pi$
$888$	0	0
$889$	23.5967	0.791410
$890$	8.00000	0.268161
$891$	0	0
$892$	−6.47214	−0.216703
$893$	4.47214	0.149654
$894$	0	0
$895$	−0.875388	−0.0292610
$896$	−36.8328	−1.23050
$897$	0	0
$898$	−9.23607	−0.308212
$899$	20.1246	0.671193
$900$	0	0
$901$	0.360680	0.0120160
$902$	11.2361	0.374120
$903$	0	0
$904$	19.5967	0.651778
$905$	−20.5836	−0.684222
$906$	0	0
$907$	40.2492	1.33645	0.668227	−	0.743958i	$-0.267054\pi$
$907$	40.2492	1.33645	0.668227	+	0.743958i	$0.267054\pi$
$908$	19.7082	0.654040
$909$	0	0
$910$	7.41641	0.245852
$911$	31.3050	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$911$	31.3050	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$912$	0	0
$913$	45.8885	1.51869
$914$	−3.16718	−0.104761
$915$	0	0
$916$	−19.4164	−0.641536
$917$	60.5410	1.99924
$918$	0	0
$919$	−41.1246	−1.35658	−0.678288	−	0.734796i	$-0.737278\pi$
$919$	−41.1246	−1.35658	−0.678288	+	0.734796i	$0.737278\pi$
$920$	0	0
$921$	0	0
$922$	−0.909830	−0.0299637
$923$	−36.7082	−1.20827
$924$	0	0
$925$	−4.29180	−0.141113
$926$	12.3607	0.406197
$927$	0	0
$928$	−16.8541	−0.553263
$929$	24.0557	0.789243	0.394621	−	0.918844i	$-0.370876\pi$
$929$	24.0557	0.789243	0.394621	+	0.918844i	$0.370876\pi$
$930$	0	0
$931$	6.94427	0.227589
$932$	−10.5623	−0.345980
$933$	0	0
$934$	8.06888	0.264022
$935$	−4.94427	−0.161695
$936$	0	0
$937$	−34.1803	−1.11662	−0.558312	−	0.829631i	$-0.688551\pi$
$937$	−34.1803	−1.11662	−0.558312	+	0.829631i	$0.688551\pi$
$938$	5.52786	0.180491
$939$	0	0
$940$	−4.47214	−0.145865
$941$	6.65248	0.216865	0.108432	−	0.994104i	$-0.465417\pi$
$941$	6.65248	0.216865	0.108432	+	0.994104i	$0.465417\pi$
$942$	0	0
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	10.8197	0.351592	0.175796	−	0.984427i	$-0.443750\pi$
$947$	10.8197	0.351592	0.175796	+	0.984427i	$0.443750\pi$
$948$	0	0
$949$	19.5836	0.635710
$950$	4.29180	0.139244
$951$	0	0
$952$	−5.52786	−0.179159
$953$	20.4721	0.663158	0.331579	−	0.943428i	$-0.392419\pi$
$953$	20.4721	0.663158	0.331579	+	0.943428i	$0.392419\pi$
$954$	0	0
$955$	−32.3607	−1.04717
$956$	22.2705	0.720280
$957$	0	0
$958$	−19.5279	−0.630917
$959$	70.8328	2.28731
$960$	0	0
$961$	14.0000	0.451613
$962$	−2.29180	−0.0738905
$963$	0	0
$964$	−37.4164	−1.20510
$965$	12.2918	0.395687
$966$	0	0
$967$	27.5410	0.885659	0.442830	−	0.896606i	$-0.353975\pi$
$967$	27.5410	0.885659	0.442830	+	0.896606i	$0.353975\pi$
$968$	36.7082	1.17985
$969$	0	0
$970$	13.5279	0.434354
$971$	16.4721	0.528616	0.264308	−	0.964438i	$-0.414856\pi$
$971$	16.4721	0.528616	0.264308	+	0.964438i	$0.414856\pi$
$972$	0	0
$973$	34.6525	1.11091
$974$	9.09017	0.291268
$975$	0	0
$976$	12.8754	0.412131
$977$	−23.3475	−0.746953	−0.373477	−	0.927640i	$-0.621834\pi$
$977$	−23.3475	−0.746953	−0.373477	+	0.927640i	$0.621834\pi$
$978$	0	0
$979$	54.8328	1.75246
$980$	−6.94427	−0.221827
$981$	0	0
$982$	5.15905	0.164632
$983$	−40.4721	−1.29086	−0.645430	−	0.763819i	$-0.723322\pi$
$983$	−40.4721	−1.29086	−0.645430	+	0.763819i	$0.723322\pi$
$984$	0	0
$985$	1.81966	0.0579792
$986$	−1.41641	−0.0451076
$987$	0	0
$988$	−9.70820	−0.308859
$989$	0	0
$990$	0	0
$991$	24.0000	0.762385	0.381193	−	0.924496i	$-0.375513\pi$
$991$	24.0000	0.762385	0.381193	+	0.924496i	$0.375513\pi$
$992$	−37.6869	−1.19656
$993$	0	0
$994$	−24.4721	−0.776209
$995$	15.1935	0.481666
$996$	0	0
$997$	16.8328	0.533101	0.266550	−	0.963821i	$-0.414116\pi$
$997$	16.8328	0.533101	0.266550	+	0.963821i	$0.414116\pi$
$998$	−11.9230	−0.377416
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4761.2.a.w.1.1		2
3.2	odd	2		529.2.a.a.1.2		2
12.11	even	2		8464.2.a.bb.1.2		2
23.22	odd	2		207.2.a.d.1.1		2
69.2	odd	22		529.2.c.n.487.2		20
69.5	even	22		529.2.c.o.255.2		20
69.8	odd	22		529.2.c.n.501.1		20
69.11	even	22		529.2.c.o.466.2		20
69.14	even	22		529.2.c.o.334.2		20
69.17	even	22		529.2.c.o.266.1		20
69.20	even	22		529.2.c.o.170.1		20
69.26	odd	22		529.2.c.n.170.1		20
69.29	odd	22		529.2.c.n.266.1		20
69.32	odd	22		529.2.c.n.334.2		20
69.35	odd	22		529.2.c.n.466.2		20
69.38	even	22		529.2.c.o.501.1		20
69.41	odd	22		529.2.c.n.255.2		20
69.44	even	22		529.2.c.o.487.2		20
69.50	odd	22		529.2.c.n.177.1		20
69.53	even	22		529.2.c.o.118.1		20
69.56	even	22		529.2.c.o.399.1		20
69.59	odd	22		529.2.c.n.399.1		20
69.62	odd	22		529.2.c.n.118.1		20
69.65	even	22		529.2.c.o.177.1		20
69.68	even	2		23.2.a.a.1.2	✓	2
92.91	even	2		3312.2.a.ba.1.1		2
115.114	odd	2		5175.2.a.be.1.2		2
276.275	odd	2		368.2.a.h.1.2		2
345.68	odd	4		575.2.b.d.24.2		4
345.137	odd	4		575.2.b.d.24.3		4
345.344	even	2		575.2.a.f.1.1		2
483.482	odd	2		1127.2.a.c.1.2		2
552.275	odd	2		1472.2.a.s.1.1		2
552.413	even	2		1472.2.a.t.1.2		2
759.758	odd	2		2783.2.a.c.1.1		2
897.896	even	2		3887.2.a.i.1.1		2
1173.1172	even	2		6647.2.a.b.1.2		2
1311.1310	odd	2		8303.2.a.e.1.1		2
1380.1379	odd	2		9200.2.a.bt.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.2	✓	2	69.68	even	2
207.2.a.d.1.1		2	23.22	odd	2
368.2.a.h.1.2		2	276.275	odd	2
529.2.a.a.1.2		2	3.2	odd	2
529.2.c.n.118.1		20	69.62	odd	22
529.2.c.n.170.1		20	69.26	odd	22
529.2.c.n.177.1		20	69.50	odd	22
529.2.c.n.255.2		20	69.41	odd	22
529.2.c.n.266.1		20	69.29	odd	22
529.2.c.n.334.2		20	69.32	odd	22
529.2.c.n.399.1		20	69.59	odd	22
529.2.c.n.466.2		20	69.35	odd	22
529.2.c.n.487.2		20	69.2	odd	22
529.2.c.n.501.1		20	69.8	odd	22
529.2.c.o.118.1		20	69.53	even	22
529.2.c.o.170.1		20	69.20	even	22
529.2.c.o.177.1		20	69.65	even	22
529.2.c.o.255.2		20	69.5	even	22
529.2.c.o.266.1		20	69.17	even	22
529.2.c.o.334.2		20	69.14	even	22
529.2.c.o.399.1		20	69.56	even	22
529.2.c.o.466.2		20	69.11	even	22
529.2.c.o.487.2		20	69.44	even	22
529.2.c.o.501.1		20	69.38	even	22
575.2.a.f.1.1		2	345.344	even	2
575.2.b.d.24.2		4	345.68	odd	4
575.2.b.d.24.3		4	345.137	odd	4
1127.2.a.c.1.2		2	483.482	odd	2
1472.2.a.s.1.1		2	552.275	odd	2
1472.2.a.t.1.2		2	552.413	even	2
2783.2.a.c.1.1		2	759.758	odd	2
3312.2.a.ba.1.1		2	92.91	even	2
3887.2.a.i.1.1		2	897.896	even	2
4761.2.a.w.1.1		2	1.1	even	1	trivial
5175.2.a.be.1.2		2	115.114	odd	2
6647.2.a.b.1.2		2	1173.1172	even	2
8303.2.a.e.1.1		2	1311.1310	odd	2
8464.2.a.bb.1.2		2	12.11	even	2
9200.2.a.bt.1.1		2	1380.1379	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:	\( N \)	\(=\)	\( 4761 = 3^{2} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4761.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} -3.23607 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +1.23607 q^{5} -3.23607 q^{7} +2.23607 q^{8} -0.763932 q^{10} -5.23607 q^{11} +3.00000 q^{13} +2.00000 q^{14} +1.85410 q^{16} +0.763932 q^{17} +2.00000 q^{19} -2.00000 q^{20} +3.23607 q^{22} -3.47214 q^{25} -1.85410 q^{26} +5.23607 q^{28} +3.00000 q^{29} +6.70820 q^{31} -5.61803 q^{32} -0.472136 q^{34} -4.00000 q^{35} +1.23607 q^{37} -1.23607 q^{38} +2.76393 q^{40} +3.47214 q^{41} +8.47214 q^{44} +2.23607 q^{47} +3.47214 q^{49} +2.14590 q^{50} -4.85410 q^{52} +0.472136 q^{53} -6.47214 q^{55} -7.23607 q^{56} -1.85410 q^{58} -6.47214 q^{59} +6.94427 q^{61} -4.14590 q^{62} -0.236068 q^{64} +3.70820 q^{65} +2.76393 q^{67} -1.23607 q^{68} +2.47214 q^{70} -12.2361 q^{71} +6.52786 q^{73} -0.763932 q^{74} -3.23607 q^{76} +16.9443 q^{77} +10.9443 q^{79} +2.29180 q^{80} -2.14590 q^{82} -8.76393 q^{83} +0.944272 q^{85} -11.7082 q^{88} -10.4721 q^{89} -9.70820 q^{91} -1.38197 q^{94} +2.47214 q^{95} -17.7082 q^{97} -2.14590 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7	\( 2 q + q^{2} - q^{4} - 2 q^{5} - 2 q^{7} - 6 q^{10} - 6 q^{11} + 6 q^{13} + 4 q^{14} - 3 q^{16} + 6 q^{17} + 4 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} + 3 q^{26} + 6 q^{28} + 6 q^{29} - 9 q^{32} + 8 q^{34} - 8 q^{35} - 2 q^{37} + 2 q^{38} + 10 q^{40} - 2 q^{41} + 8 q^{44} - 2 q^{49} + 11 q^{50} - 3 q^{52} - 8 q^{53} - 4 q^{55} - 10 q^{56} + 3 q^{58} - 4 q^{59} - 4 q^{61} - 15 q^{62} + 4 q^{64} - 6 q^{65} + 10 q^{67} + 2 q^{68} - 4 q^{70} - 20 q^{71} + 22 q^{73} - 6 q^{74} - 2 q^{76} + 16 q^{77} + 4 q^{79} + 18 q^{80} - 11 q^{82} - 22 q^{83} - 16 q^{85} - 10 q^{88} - 12 q^{89} - 6 q^{91} - 5 q^{94} - 4 q^{95} - 22 q^{97} - 11 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 2 * q^5 - 2 * q^7 - 6 * q^10 - 6 * q^11 + 6 * q^13 + 4 * q^14 - 3 * q^16 + 6 * q^17 + 4 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 + 3 * q^26 + 6 * q^28 + 6 * q^29 - 9 * q^32 + 8 * q^34 - 8 * q^35 - 2 * q^37 + 2 * q^38 + 10 * q^40 - 2 * q^41 + 8 * q^44 - 2 * q^49 + 11 * q^50 - 3 * q^52 - 8 * q^53 - 4 * q^55 - 10 * q^56 + 3 * q^58 - 4 * q^59 - 4 * q^61 - 15 * q^62 + 4 * q^64 - 6 * q^65 + 10 * q^67 + 2 * q^68 - 4 * q^70 - 20 * q^71 + 22 * q^73 - 6 * q^74 - 2 * q^76 + 16 * q^77 + 4 * q^79 + 18 * q^80 - 11 * q^82 - 22 * q^83 - 16 * q^85 - 10 * q^88 - 12 * q^89 - 6 * q^91 - 5 * q^94 - 4 * q^95 - 22 * q^97 - 11 * q^98

Introduction

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