LMFDB - Embedded newform 8649.2.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8649,2,Mod(1,8649)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8649, 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("8649.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.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:	$69.0626127082$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.618034 q^{2} -1.61803 q^{4} +2.61803 q^{5} +3.00000 q^{7} -2.23607 q^{8} +1.61803 q^{10} +0.763932 q^{11} +4.85410 q^{13} +1.85410 q^{14} +1.85410 q^{16} -0.236068 q^{17} +5.00000 q^{19} -4.23607 q^{20} +0.472136 q^{22} +5.47214 q^{23} +1.85410 q^{25} +3.00000 q^{26} -4.85410 q^{28} +8.61803 q^{29} +5.61803 q^{32} -0.145898 q^{34} +7.85410 q^{35} +0.236068 q^{37} +3.09017 q^{38} -5.85410 q^{40} -6.47214 q^{41} -4.61803 q^{43} -1.23607 q^{44} +3.38197 q^{46} +3.38197 q^{47} +2.00000 q^{49} +1.14590 q^{50} -7.85410 q^{52} +12.7082 q^{53} +2.00000 q^{55} -6.70820 q^{56} +5.32624 q^{58} -9.47214 q^{59} -6.94427 q^{61} -0.236068 q^{64} +12.7082 q^{65} -4.23607 q^{67} +0.381966 q^{68} +4.85410 q^{70} -0.0901699 q^{71} -8.56231 q^{73} +0.145898 q^{74} -8.09017 q^{76} +2.29180 q^{77} +4.85410 q^{80} -4.00000 q^{82} +4.09017 q^{83} -0.618034 q^{85} -2.85410 q^{86} -1.70820 q^{88} +6.38197 q^{89} +14.5623 q^{91} -8.85410 q^{92} +2.09017 q^{94} +13.0902 q^{95} -5.29180 q^{97} +1.23607 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} + 3 q^{5} + 6 q^{7} + q^{10} + 6 q^{11} + 3 q^{13} - 3 q^{14} - 3 q^{16} + 4 q^{17} + 10 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} - 3 q^{25} + 6 q^{26} - 3 q^{28} + 15 q^{29} + 9 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 + 3 * q^5 + 6 * q^7 + q^10 + 6 * q^11 + 3 * q^13 - 3 * q^14 - 3 * q^16 + 4 * q^17 + 10 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 - 3 * q^25 + 6 * q^26 - 3 * q^28 + 15 * q^29 + 9 * q^32 - 7 * q^34 + 9 * q^35 - 4 * q^37 - 5 * q^38 - 5 * q^40 - 4 * q^41 - 7 * q^43 + 2 * q^44 + 9 * q^46 + 9 * q^47 + 4 * q^49 + 9 * q^50 - 9 * q^52 + 12 * q^53 + 4 * q^55 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 4 * q^64 + 12 * q^65 - 4 * q^67 + 3 * q^68 + 3 * q^70 + 11 * q^71 + 3 * q^73 + 7 * q^74 - 5 * q^76 + 18 * q^77 + 3 * q^80 - 8 * q^82 - 3 * q^83 + q^85 + q^86 + 10 * q^88 + 15 * q^89 + 9 * q^91 - 11 * q^92 - 7 * q^94 + 15 * q^95 - 24 * q^97 - 2 * q^98

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	$-0.429881\pi$
$2$	0.618034	0.437016	0.218508	+	0.975835i	$0.429881\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	2.61803	1.17082	0.585410	−	0.810737i	$-0.300933\pi$
$5$	2.61803	1.17082	0.585410	+	0.810737i	$0.300933\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	1.61803	0.511667
$11$	0.763932	0.230334	0.115167	−	0.993346i	$-0.463260\pi$
$11$	0.763932	0.230334	0.115167	+	0.993346i	$0.463260\pi$
$12$	0	0
$13$	4.85410	1.34629	0.673143	−	0.739512i	$-0.264944\pi$
$13$	4.85410	1.34629	0.673143	+	0.739512i	$0.264944\pi$
$14$	1.85410	0.495530
$15$	0	0
$16$	1.85410	0.463525
$17$	−0.236068	−0.0572549	−0.0286274	−	0.999590i	$-0.509114\pi$
$17$	−0.236068	−0.0572549	−0.0286274	+	0.999590i	$0.509114\pi$
$18$	0	0
$19$	5.00000	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$19$	5.00000	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$20$	−4.23607	−0.947214
$21$	0	0
$22$	0.472136	0.100660
$23$	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
$25$	1.85410	0.370820
$26$	3.00000	0.588348
$27$	0	0
$28$	−4.85410	−0.917339
$29$	8.61803	1.60033	0.800164	−	0.599781i	$-0.204746\pi$
$29$	8.61803	1.60033	0.800164	+	0.599781i	$0.204746\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	5.61803	0.993137
$33$	0	0
$34$	−0.145898	−0.0250213
$35$	7.85410	1.32759
$36$	0	0
$37$	0.236068	0.0388093	0.0194047	−	0.999812i	$-0.493823\pi$
$37$	0.236068	0.0388093	0.0194047	+	0.999812i	$0.493823\pi$
$38$	3.09017	0.501292
$39$	0	0
$40$	−5.85410	−0.925615
$41$	−6.47214	−1.01078	−0.505389	−	0.862892i	$-0.668651\pi$
$41$	−6.47214	−1.01078	−0.505389	+	0.862892i	$0.668651\pi$
$42$	0	0
$43$	−4.61803	−0.704244	−0.352122	−	0.935954i	$-0.614540\pi$
$43$	−4.61803	−0.704244	−0.352122	+	0.935954i	$0.614540\pi$
$44$	−1.23607	−0.186344
$45$	0	0
$46$	3.38197	0.498644
$47$	3.38197	0.493310	0.246655	−	0.969103i	$-0.420668\pi$
$47$	3.38197	0.493310	0.246655	+	0.969103i	$0.420668\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	1.14590	0.162054
$51$	0	0
$52$	−7.85410	−1.08917
$53$	12.7082	1.74561	0.872803	−	0.488073i	$-0.162300\pi$
$53$	12.7082	1.74561	0.872803	+	0.488073i	$0.162300\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	−6.70820	−0.896421
$57$	0	0
$58$	5.32624	0.699369
$59$	−9.47214	−1.23317	−0.616584	−	0.787289i	$-0.711484\pi$
$59$	−9.47214	−1.23317	−0.616584	+	0.787289i	$0.711484\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	0	0
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	12.7082	1.57626
$66$	0	0
$67$	−4.23607	−0.517518	−0.258759	−	0.965942i	$-0.583314\pi$
$67$	−4.23607	−0.517518	−0.258759	+	0.965942i	$0.583314\pi$
$68$	0.381966	0.0463202
$69$	0	0
$70$	4.85410	0.580176
$71$	−0.0901699	−0.0107012	−0.00535060	−	0.999986i	$-0.501703\pi$
$71$	−0.0901699	−0.0107012	−0.00535060	+	0.999986i	$0.501703\pi$
$72$	0	0
$73$	−8.56231	−1.00214	−0.501071	−	0.865406i	$-0.667060\pi$
$73$	−8.56231	−1.00214	−0.501071	+	0.865406i	$0.667060\pi$
$74$	0.145898	0.0169603
$75$	0	0
$76$	−8.09017	−0.928006
$77$	2.29180	0.261174
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	−4.00000	−0.441726
$83$	4.09017	0.448954	0.224477	−	0.974479i	$-0.427933\pi$
$83$	4.09017	0.448954	0.224477	+	0.974479i	$0.427933\pi$
$84$	0	0
$85$	−0.618034	−0.0670352
$86$	−2.85410	−0.307766
$87$	0	0
$88$	−1.70820	−0.182095
$89$	6.38197	0.676487	0.338244	−	0.941059i	$-0.390167\pi$
$89$	6.38197	0.676487	0.338244	+	0.941059i	$0.390167\pi$
$90$	0	0
$91$	14.5623	1.52654
$92$	−8.85410	−0.923104
$93$	0	0
$94$	2.09017	0.215585
$95$	13.0902	1.34302
$96$	0	0
$97$	−5.29180	−0.537300	−0.268650	−	0.963238i	$-0.586578\pi$
$97$	−5.29180	−0.537300	−0.268650	+	0.963238i	$0.586578\pi$
$98$	1.23607	0.124862
$99$	0	0
$100$	−3.00000	−0.300000
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	0	0
$103$	−0.145898	−0.0143758	−0.00718788	−	0.999974i	$-0.502288\pi$
$103$	−0.145898	−0.0143758	−0.00718788	+	0.999974i	$0.502288\pi$
$104$	−10.8541	−1.06433
$105$	0	0
$106$	7.85410	0.762858
$107$	−1.09017	−0.105391	−0.0526954	−	0.998611i	$-0.516781\pi$
$107$	−1.09017	−0.105391	−0.0526954	+	0.998611i	$0.516781\pi$
$108$	0	0
$109$	−8.41641	−0.806146	−0.403073	−	0.915168i	$-0.632058\pi$
$109$	−8.41641	−0.806146	−0.403073	+	0.915168i	$0.632058\pi$
$110$	1.23607	0.117854
$111$	0	0
$112$	5.56231	0.525589
$113$	1.85410	0.174419	0.0872096	−	0.996190i	$-0.472205\pi$
$113$	1.85410	0.174419	0.0872096	+	0.996190i	$0.472205\pi$
$114$	0	0
$115$	14.3262	1.33593
$116$	−13.9443	−1.29469
$117$	0	0
$118$	−5.85410	−0.538914
$119$	−0.708204	−0.0649209
$120$	0	0
$121$	−10.4164	−0.946946
$122$	−4.29180	−0.388561
$123$	0	0
$124$	0	0
$125$	−8.23607	−0.736656
$126$	0	0
$127$	10.2361	0.908304	0.454152	−	0.890924i	$-0.349942\pi$
$127$	10.2361	0.908304	0.454152	+	0.890924i	$0.349942\pi$
$128$	−11.3820	−1.00603
$129$	0	0
$130$	7.85410	0.688850
$131$	−0.0901699	−0.00787818	−0.00393909	−	0.999992i	$-0.501254\pi$
$131$	−0.0901699	−0.00787818	−0.00393909	+	0.999992i	$0.501254\pi$
$132$	0	0
$133$	15.0000	1.30066
$134$	−2.61803	−0.226164
$135$	0	0
$136$	0.527864	0.0452640
$137$	6.47214	0.552952	0.276476	−	0.961021i	$-0.410833\pi$
$137$	6.47214	0.552952	0.276476	+	0.961021i	$0.410833\pi$
$138$	0	0
$139$	5.85410	0.496538	0.248269	−	0.968691i	$-0.420138\pi$
$139$	5.85410	0.496538	0.248269	+	0.968691i	$0.420138\pi$
$140$	−12.7082	−1.07404
$141$	0	0
$142$	−0.0557281	−0.00467660
$143$	3.70820	0.310096
$144$	0	0
$145$	22.5623	1.87370
$146$	−5.29180	−0.437952
$147$	0	0
$148$	−0.381966	−0.0313974
$149$	17.0344	1.39552	0.697758	−	0.716334i	$-0.254181\pi$
$149$	17.0344	1.39552	0.697758	+	0.716334i	$0.254181\pi$
$150$	0	0
$151$	−19.5066	−1.58742	−0.793711	−	0.608295i	$-0.791854\pi$
$151$	−19.5066	−1.58742	−0.793711	+	0.608295i	$0.791854\pi$
$152$	−11.1803	−0.906845
$153$	0	0
$154$	1.41641	0.114137
$155$	0	0
$156$	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	0
$159$	0	0
$160$	14.7082	1.16279
$161$	16.4164	1.29379
$162$	0	0
$163$	−12.7082	−0.995383	−0.497692	−	0.867354i	$-0.665819\pi$
$163$	−12.7082	−0.995383	−0.497692	+	0.867354i	$0.665819\pi$
$164$	10.4721	0.817736
$165$	0	0
$166$	2.52786	0.196200
$167$	9.23607	0.714708	0.357354	−	0.933969i	$-0.383679\pi$
$167$	9.23607	0.714708	0.357354	+	0.933969i	$0.383679\pi$
$168$	0	0
$169$	10.5623	0.812485
$170$	−0.381966	−0.0292955
$171$	0	0
$172$	7.47214	0.569745
$173$	−0.909830	−0.0691731	−0.0345865	−	0.999402i	$-0.511011\pi$
$173$	−0.909830	−0.0691731	−0.0345865	+	0.999402i	$0.511011\pi$
$174$	0	0
$175$	5.56231	0.420471
$176$	1.41641	0.106766
$177$	0	0
$178$	3.94427	0.295636
$179$	−19.7984	−1.47980	−0.739900	−	0.672717i	$-0.765127\pi$
$179$	−19.7984	−1.47980	−0.739900	+	0.672717i	$0.765127\pi$
$180$	0	0
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	9.00000	0.667124
$183$	0	0
$184$	−12.2361	−0.902055
$185$	0.618034	0.0454388
$186$	0	0
$187$	−0.180340	−0.0131878
$188$	−5.47214	−0.399097
$189$	0	0
$190$	8.09017	0.586923
$191$	16.0902	1.16424	0.582122	−	0.813102i	$-0.302223\pi$
$191$	16.0902	1.16424	0.582122	+	0.813102i	$0.302223\pi$
$192$	0	0
$193$	−2.38197	−0.171458	−0.0857288	−	0.996319i	$-0.527322\pi$
$193$	−2.38197	−0.171458	−0.0857288	+	0.996319i	$0.527322\pi$
$194$	−3.27051	−0.234809
$195$	0	0
$196$	−3.23607	−0.231148
$197$	−16.4164	−1.16962	−0.584810	−	0.811170i	$-0.698831\pi$
$197$	−16.4164	−1.16962	−0.584810	+	0.811170i	$0.698831\pi$
$198$	0	0
$199$	−26.7082	−1.89329	−0.946647	−	0.322272i	$-0.895554\pi$
$199$	−26.7082	−1.89329	−0.946647	+	0.322272i	$0.895554\pi$
$200$	−4.14590	−0.293159
$201$	0	0
$202$	−2.94427	−0.207158
$203$	25.8541	1.81460
$204$	0	0
$205$	−16.9443	−1.18344
$206$	−0.0901699	−0.00628244
$207$	0	0
$208$	9.00000	0.624038
$209$	3.81966	0.264211
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	−20.5623	−1.41222
$213$	0	0
$214$	−0.673762	−0.0460574
$215$	−12.0902	−0.824543
$216$	0	0
$217$	0	0
$218$	−5.20163	−0.352299
$219$	0	0
$220$	−3.23607	−0.218176
$221$	−1.14590	−0.0770814
$222$	0	0
$223$	0.708204	0.0474248	0.0237124	−	0.999719i	$-0.492451\pi$
$223$	0.708204	0.0474248	0.0237124	+	0.999719i	$0.492451\pi$
$224$	16.8541	1.12611
$225$	0	0
$226$	1.14590	0.0762240
$227$	20.7426	1.37674	0.688369	−	0.725361i	$-0.258327\pi$
$227$	20.7426	1.37674	0.688369	+	0.725361i	$0.258327\pi$
$228$	0	0
$229$	−7.23607	−0.478173	−0.239086	−	0.970998i	$-0.576848\pi$
$229$	−7.23607	−0.478173	−0.239086	+	0.970998i	$0.576848\pi$
$230$	8.85410	0.583822
$231$	0	0
$232$	−19.2705	−1.26517
$233$	−18.7984	−1.23152	−0.615761	−	0.787933i	$-0.711151\pi$
$233$	−18.7984	−1.23152	−0.615761	+	0.787933i	$0.711151\pi$
$234$	0	0
$235$	8.85410	0.577578
$236$	15.3262	0.997653
$237$	0	0
$238$	−0.437694	−0.0283715
$239$	13.4164	0.867835	0.433918	−	0.900953i	$-0.357131\pi$
$239$	13.4164	0.867835	0.433918	+	0.900953i	$0.357131\pi$
$240$	0	0
$241$	−8.52786	−0.549328	−0.274664	−	0.961540i	$-0.588567\pi$
$241$	−8.52786	−0.549328	−0.274664	+	0.961540i	$0.588567\pi$
$242$	−6.43769	−0.413831
$243$	0	0
$244$	11.2361	0.719316
$245$	5.23607	0.334520
$246$	0	0
$247$	24.2705	1.54430
$248$	0	0
$249$	0	0
$250$	−5.09017	−0.321931
$251$	−0.944272	−0.0596019	−0.0298010	−	0.999556i	$-0.509487\pi$
$251$	−0.944272	−0.0596019	−0.0298010	+	0.999556i	$0.509487\pi$
$252$	0	0
$253$	4.18034	0.262816
$254$	6.32624	0.396943
$255$	0	0
$256$	−6.56231	−0.410144
$257$	−1.41641	−0.0883531	−0.0441765	−	0.999024i	$-0.514066\pi$
$257$	−1.41641	−0.0883531	−0.0441765	+	0.999024i	$0.514066\pi$
$258$	0	0
$259$	0.708204	0.0440057
$260$	−20.5623	−1.27522
$261$	0	0
$262$	−0.0557281	−0.00344289
$263$	10.7984	0.665856	0.332928	−	0.942952i	$-0.391963\pi$
$263$	10.7984	0.665856	0.332928	+	0.942952i	$0.391963\pi$
$264$	0	0
$265$	33.2705	2.04379
$266$	9.27051	0.568411
$267$	0	0
$268$	6.85410	0.418681
$269$	1.38197	0.0842600	0.0421300	−	0.999112i	$-0.486586\pi$
$269$	1.38197	0.0842600	0.0421300	+	0.999112i	$0.486586\pi$
$270$	0	0
$271$	9.56231	0.580869	0.290434	−	0.956895i	$-0.406200\pi$
$271$	9.56231	0.580869	0.290434	+	0.956895i	$0.406200\pi$
$272$	−0.437694	−0.0265391
$273$	0	0
$274$	4.00000	0.241649
$275$	1.41641	0.0854126
$276$	0	0
$277$	13.3262	0.800696	0.400348	−	0.916363i	$-0.368889\pi$
$277$	13.3262	0.800696	0.400348	+	0.916363i	$0.368889\pi$
$278$	3.61803	0.216995
$279$	0	0
$280$	−17.5623	−1.04955
$281$	−19.0344	−1.13550	−0.567750	−	0.823201i	$-0.692186\pi$
$281$	−19.0344	−1.13550	−0.567750	+	0.823201i	$0.692186\pi$
$282$	0	0
$283$	6.56231	0.390089	0.195044	−	0.980794i	$-0.437515\pi$
$283$	6.56231	0.390089	0.195044	+	0.980794i	$0.437515\pi$
$284$	0.145898	0.00865746
$285$	0	0
$286$	2.29180	0.135517
$287$	−19.4164	−1.14611
$288$	0	0
$289$	−16.9443	−0.996722
$290$	13.9443	0.818836
$291$	0	0
$292$	13.8541	0.810750
$293$	8.23607	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$293$	8.23607	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$294$	0	0
$295$	−24.7984	−1.44382
$296$	−0.527864	−0.0306815
$297$	0	0
$298$	10.5279	0.609863
$299$	26.5623	1.53614
$300$	0	0
$301$	−13.8541	−0.798537
$302$	−12.0557	−0.693729
$303$	0	0
$304$	9.27051	0.531700
$305$	−18.1803	−1.04100
$306$	0	0
$307$	6.09017	0.347584	0.173792	−	0.984782i	$-0.444398\pi$
$307$	6.09017	0.347584	0.173792	+	0.984782i	$0.444398\pi$
$308$	−3.70820	−0.211295
$309$	0	0
$310$	0	0
$311$	−16.4721	−0.934049	−0.467025	−	0.884244i	$-0.654674\pi$
$311$	−16.4721	−0.934049	−0.467025	+	0.884244i	$0.654674\pi$
$312$	0	0
$313$	1.23607	0.0698667	0.0349333	−	0.999390i	$-0.488878\pi$
$313$	1.23607	0.0698667	0.0349333	+	0.999390i	$0.488878\pi$
$314$	6.00000	0.338600
$315$	0	0
$316$	0	0
$317$	−25.8885	−1.45405	−0.727023	−	0.686613i	$-0.759097\pi$
$317$	−25.8885	−1.45405	−0.727023	+	0.686613i	$0.759097\pi$
$318$	0	0
$319$	6.58359	0.368610
$320$	−0.618034	−0.0345492
$321$	0	0
$322$	10.1459	0.565409
$323$	−1.18034	−0.0656759
$324$	0	0
$325$	9.00000	0.499230
$326$	−7.85410	−0.434998
$327$	0	0
$328$	14.4721	0.799090
$329$	10.1459	0.559361
$330$	0	0
$331$	11.2705	0.619483	0.309742	−	0.950821i	$-0.399757\pi$
$331$	11.2705	0.619483	0.309742	+	0.950821i	$0.399757\pi$
$332$	−6.61803	−0.363212
$333$	0	0
$334$	5.70820	0.312339
$335$	−11.0902	−0.605921
$336$	0	0
$337$	18.9787	1.03384	0.516918	−	0.856035i	$-0.327079\pi$
$337$	18.9787	1.03384	0.516918	+	0.856035i	$0.327079\pi$
$338$	6.52786	0.355069
$339$	0	0
$340$	1.00000	0.0542326
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	10.3262	0.556753
$345$	0	0
$346$	−0.562306	−0.0302298
$347$	−8.12461	−0.436152	−0.218076	−	0.975932i	$-0.569978\pi$
$347$	−8.12461	−0.436152	−0.218076	+	0.975932i	$0.569978\pi$
$348$	0	0
$349$	−16.7082	−0.894370	−0.447185	−	0.894442i	$-0.647573\pi$
$349$	−16.7082	−0.894370	−0.447185	+	0.894442i	$0.647573\pi$
$350$	3.43769	0.183752
$351$	0	0
$352$	4.29180	0.228753
$353$	32.3820	1.72352	0.861759	−	0.507318i	$-0.169363\pi$
$353$	32.3820	1.72352	0.861759	+	0.507318i	$0.169363\pi$
$354$	0	0
$355$	−0.236068	−0.0125292
$356$	−10.3262	−0.547290
$357$	0	0
$358$	−12.2361	−0.646696
$359$	25.3262	1.33667	0.668334	−	0.743861i	$-0.267008\pi$
$359$	25.3262	1.33667	0.668334	+	0.743861i	$0.267008\pi$
$360$	0	0
$361$	6.00000	0.315789
$362$	10.5066	0.552213
$363$	0	0
$364$	−23.5623	−1.23500
$365$	−22.4164	−1.17333
$366$	0	0
$367$	−36.2705	−1.89331	−0.946653	−	0.322256i	$-0.895559\pi$
$367$	−36.2705	−1.89331	−0.946653	+	0.322256i	$0.895559\pi$
$368$	10.1459	0.528891
$369$	0	0
$370$	0.381966	0.0198575
$371$	38.1246	1.97933
$372$	0	0
$373$	−0.347524	−0.0179941	−0.00899706	−	0.999960i	$-0.502864\pi$
$373$	−0.347524	−0.0179941	−0.00899706	+	0.999960i	$0.502864\pi$
$374$	−0.111456	−0.00576326
$375$	0	0
$376$	−7.56231	−0.389996
$377$	41.8328	2.15450
$378$	0	0
$379$	−18.4164	−0.945987	−0.472994	−	0.881066i	$-0.656827\pi$
$379$	−18.4164	−0.945987	−0.472994	+	0.881066i	$0.656827\pi$
$380$	−21.1803	−1.08653
$381$	0	0
$382$	9.94427	0.508793
$383$	16.8541	0.861204	0.430602	−	0.902542i	$-0.358301\pi$
$383$	16.8541	0.861204	0.430602	+	0.902542i	$0.358301\pi$
$384$	0	0
$385$	6.00000	0.305788
$386$	−1.47214	−0.0749297
$387$	0	0
$388$	8.56231	0.434685
$389$	29.0689	1.47385	0.736925	−	0.675974i	$-0.236277\pi$
$389$	29.0689	1.47385	0.736925	+	0.675974i	$0.236277\pi$
$390$	0	0
$391$	−1.29180	−0.0653289
$392$	−4.47214	−0.225877
$393$	0	0
$394$	−10.1459	−0.511143
$395$	0	0
$396$	0	0
$397$	16.2918	0.817662	0.408831	−	0.912610i	$-0.365937\pi$
$397$	16.2918	0.817662	0.408831	+	0.912610i	$0.365937\pi$
$398$	−16.5066	−0.827400
$399$	0	0
$400$	3.43769	0.171885
$401$	29.8328	1.48978	0.744890	−	0.667187i	$-0.232502\pi$
$401$	29.8328	1.48978	0.744890	+	0.667187i	$0.232502\pi$
$402$	0	0
$403$	0	0
$404$	7.70820	0.383497
$405$	0	0
$406$	15.9787	0.793010
$407$	0.180340	0.00893912
$408$	0	0
$409$	−6.18034	−0.305598	−0.152799	−	0.988257i	$-0.548829\pi$
$409$	−6.18034	−0.305598	−0.152799	+	0.988257i	$0.548829\pi$
$410$	−10.4721	−0.517182
$411$	0	0
$412$	0.236068	0.0116302
$413$	−28.4164	−1.39828
$414$	0	0
$415$	10.7082	0.525645
$416$	27.2705	1.33705
$417$	0	0
$418$	2.36068	0.115465
$419$	−4.47214	−0.218478	−0.109239	−	0.994016i	$-0.534841\pi$
$419$	−4.47214	−0.218478	−0.109239	+	0.994016i	$0.534841\pi$
$420$	0	0
$421$	14.7639	0.719550	0.359775	−	0.933039i	$-0.382853\pi$
$421$	14.7639	0.719550	0.359775	+	0.933039i	$0.382853\pi$
$422$	−4.94427	−0.240683
$423$	0	0
$424$	−28.4164	−1.38002
$425$	−0.437694	−0.0212313
$426$	0	0
$427$	−20.8328	−1.00817
$428$	1.76393	0.0852629
$429$	0	0
$430$	−7.47214	−0.360338
$431$	−29.2361	−1.40825	−0.704126	−	0.710075i	$-0.748661\pi$
$431$	−29.2361	−1.40825	−0.704126	+	0.710075i	$0.748661\pi$
$432$	0	0
$433$	0.583592	0.0280456	0.0140228	−	0.999902i	$-0.495536\pi$
$433$	0.583592	0.0280456	0.0140228	+	0.999902i	$0.495536\pi$
$434$	0	0
$435$	0	0
$436$	13.6180	0.652186
$437$	27.3607	1.30884
$438$	0	0
$439$	41.8328	1.99657	0.998286	−	0.0585295i	$-0.0186412\pi$
$439$	41.8328	1.99657	0.998286	+	0.0585295i	$0.0186412\pi$
$440$	−4.47214	−0.213201
$441$	0	0
$442$	−0.708204	−0.0336858
$443$	41.1246	1.95389	0.976945	−	0.213493i	$-0.0684840\pi$
$443$	41.1246	1.95389	0.976945	+	0.213493i	$0.0684840\pi$
$444$	0	0
$445$	16.7082	0.792045
$446$	0.437694	0.0207254
$447$	0	0
$448$	−0.708204	−0.0334595
$449$	−24.0689	−1.13588	−0.567940	−	0.823070i	$-0.692260\pi$
$449$	−24.0689	−1.13588	−0.567940	+	0.823070i	$0.692260\pi$
$450$	0	0
$451$	−4.94427	−0.232817
$452$	−3.00000	−0.141108
$453$	0	0
$454$	12.8197	0.601656
$455$	38.1246	1.78731
$456$	0	0
$457$	−15.7426	−0.736410	−0.368205	−	0.929745i	$-0.620028\pi$
$457$	−15.7426	−0.736410	−0.368205	+	0.929745i	$0.620028\pi$
$458$	−4.47214	−0.208969
$459$	0	0
$460$	−23.1803	−1.08079
$461$	−10.7426	−0.500335	−0.250167	−	0.968203i	$-0.580486\pi$
$461$	−10.7426	−0.500335	−0.250167	+	0.968203i	$0.580486\pi$
$462$	0	0
$463$	−31.1246	−1.44648	−0.723242	−	0.690595i	$-0.757349\pi$
$463$	−31.1246	−1.44648	−0.723242	+	0.690595i	$0.757349\pi$
$464$	15.9787	0.741793
$465$	0	0
$466$	−11.6180	−0.538195
$467$	32.7771	1.51674	0.758371	−	0.651823i	$-0.225995\pi$
$467$	32.7771	1.51674	0.758371	+	0.651823i	$0.225995\pi$
$468$	0	0
$469$	−12.7082	−0.586810
$470$	5.47214	0.252411
$471$	0	0
$472$	21.1803	0.974904
$473$	−3.52786	−0.162211
$474$	0	0
$475$	9.27051	0.425360
$476$	1.14590	0.0525222
$477$	0	0
$478$	8.29180	0.379258
$479$	8.94427	0.408674	0.204337	−	0.978901i	$-0.434496\pi$
$479$	8.94427	0.408674	0.204337	+	0.978901i	$0.434496\pi$
$480$	0	0
$481$	1.14590	0.0522485
$482$	−5.27051	−0.240065
$483$	0	0
$484$	16.8541	0.766096
$485$	−13.8541	−0.629082
$486$	0	0
$487$	22.9230	1.03874	0.519370	−	0.854550i	$-0.326167\pi$
$487$	22.9230	1.03874	0.519370	+	0.854550i	$0.326167\pi$
$488$	15.5279	0.702913
$489$	0	0
$490$	3.23607	0.146191
$491$	27.5967	1.24542	0.622712	−	0.782451i	$-0.286031\pi$
$491$	27.5967	1.24542	0.622712	+	0.782451i	$0.286031\pi$
$492$	0	0
$493$	−2.03444	−0.0916267
$494$	15.0000	0.674882
$495$	0	0
$496$	0	0
$497$	−0.270510	−0.0121340
$498$	0	0
$499$	4.14590	0.185596	0.0927979	−	0.995685i	$-0.470419\pi$
$499$	4.14590	0.185596	0.0927979	+	0.995685i	$0.470419\pi$
$500$	13.3262	0.595967
$501$	0	0
$502$	−0.583592	−0.0260470
$503$	−13.1459	−0.586147	−0.293073	−	0.956090i	$-0.594678\pi$
$503$	−13.1459	−0.586147	−0.293073	+	0.956090i	$0.594678\pi$
$504$	0	0
$505$	−12.4721	−0.555003
$506$	2.58359	0.114855
$507$	0	0
$508$	−16.5623	−0.734833
$509$	1.90983	0.0846517	0.0423259	−	0.999104i	$-0.486523\pi$
$509$	1.90983	0.0846517	0.0423259	+	0.999104i	$0.486523\pi$
$510$	0	0
$511$	−25.6869	−1.13632
$512$	18.7082	0.826794
$513$	0	0
$514$	−0.875388	−0.0386117
$515$	−0.381966	−0.0168314
$516$	0	0
$517$	2.58359	0.113626
$518$	0.437694	0.0192312
$519$	0	0
$520$	−28.4164	−1.24614
$521$	−31.0689	−1.36115	−0.680576	−	0.732677i	$-0.738271\pi$
$521$	−31.0689	−1.36115	−0.680576	+	0.732677i	$0.738271\pi$
$522$	0	0
$523$	34.1246	1.49217	0.746083	−	0.665853i	$-0.231932\pi$
$523$	34.1246	1.49217	0.746083	+	0.665853i	$0.231932\pi$
$524$	0.145898	0.00637359
$525$	0	0
$526$	6.67376	0.290990
$527$	0	0
$528$	0	0
$529$	6.94427	0.301925
$530$	20.5623	0.893169
$531$	0	0
$532$	−24.2705	−1.05226
$533$	−31.4164	−1.36080
$534$	0	0
$535$	−2.85410	−0.123394
$536$	9.47214	0.409134
$537$	0	0
$538$	0.854102	0.0368230
$539$	1.52786	0.0658098
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	5.90983	0.253849
$543$	0	0
$544$	−1.32624	−0.0568620
$545$	−22.0344	−0.943852
$546$	0	0
$547$	−23.7082	−1.01369	−0.506845	−	0.862037i	$-0.669188\pi$
$547$	−23.7082	−1.01369	−0.506845	+	0.862037i	$0.669188\pi$
$548$	−10.4721	−0.447347
$549$	0	0
$550$	0.875388	0.0373267
$551$	43.0902	1.83570
$552$	0	0
$553$	0	0
$554$	8.23607	0.349917
$555$	0	0
$556$	−9.47214	−0.401708
$557$	−35.8885	−1.52065	−0.760323	−	0.649545i	$-0.774959\pi$
$557$	−35.8885	−1.52065	−0.760323	+	0.649545i	$0.774959\pi$
$558$	0	0
$559$	−22.4164	−0.948113
$560$	14.5623	0.615370
$561$	0	0
$562$	−11.7639	−0.496232
$563$	8.56231	0.360858	0.180429	−	0.983588i	$-0.442251\pi$
$563$	8.56231	0.360858	0.180429	+	0.983588i	$0.442251\pi$
$564$	0	0
$565$	4.85410	0.204214
$566$	4.05573	0.170475
$567$	0	0
$568$	0.201626	0.00846004
$569$	−15.5279	−0.650962	−0.325481	−	0.945549i	$-0.605526\pi$
$569$	−15.5279	−0.650962	−0.325481	+	0.945549i	$0.605526\pi$
$570$	0	0
$571$	7.00000	0.292941	0.146470	−	0.989215i	$-0.453209\pi$
$571$	7.00000	0.292941	0.146470	+	0.989215i	$0.453209\pi$
$572$	−6.00000	−0.250873
$573$	0	0
$574$	−12.0000	−0.500870
$575$	10.1459	0.423113
$576$	0	0
$577$	38.9787	1.62271	0.811353	−	0.584557i	$-0.198732\pi$
$577$	38.9787	1.62271	0.811353	+	0.584557i	$0.198732\pi$
$578$	−10.4721	−0.435583
$579$	0	0
$580$	−36.5066	−1.51585
$581$	12.2705	0.509067
$582$	0	0
$583$	9.70820	0.402073
$584$	19.1459	0.792263
$585$	0	0
$586$	5.09017	0.210273
$587$	−36.0132	−1.48642	−0.743211	−	0.669057i	$-0.766698\pi$
$587$	−36.0132	−1.48642	−0.743211	+	0.669057i	$0.766698\pi$
$588$	0	0
$589$	0	0
$590$	−15.3262	−0.630971
$591$	0	0
$592$	0.437694	0.0179891
$593$	−6.11146	−0.250967	−0.125484	−	0.992096i	$-0.540048\pi$
$593$	−6.11146	−0.250967	−0.125484	+	0.992096i	$0.540048\pi$
$594$	0	0
$595$	−1.85410	−0.0760108
$596$	−27.5623	−1.12900
$597$	0	0
$598$	16.4164	0.671317
$599$	−29.7984	−1.21753	−0.608764	−	0.793351i	$-0.708334\pi$
$599$	−29.7984	−1.21753	−0.608764	+	0.793351i	$0.708334\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	−8.56231	−0.348974
$603$	0	0
$604$	31.5623	1.28425
$605$	−27.2705	−1.10870
$606$	0	0
$607$	−25.4164	−1.03162	−0.515810	−	0.856703i	$-0.672509\pi$
$607$	−25.4164	−1.03162	−0.515810	+	0.856703i	$0.672509\pi$
$608$	28.0902	1.13921
$609$	0	0
$610$	−11.2361	−0.454935
$611$	16.4164	0.664137
$612$	0	0
$613$	25.0557	1.01199	0.505996	−	0.862536i	$-0.331125\pi$
$613$	25.0557	1.01199	0.505996	+	0.862536i	$0.331125\pi$
$614$	3.76393	0.151900
$615$	0	0
$616$	−5.12461	−0.206476
$617$	14.2361	0.573123	0.286561	−	0.958062i	$-0.407488\pi$
$617$	14.2361	0.573123	0.286561	+	0.958062i	$0.407488\pi$
$618$	0	0
$619$	−40.0000	−1.60774	−0.803868	−	0.594808i	$-0.797228\pi$
$619$	−40.0000	−1.60774	−0.803868	+	0.594808i	$0.797228\pi$
$620$	0	0
$621$	0	0
$622$	−10.1803	−0.408194
$623$	19.1459	0.767064
$624$	0	0
$625$	−30.8328	−1.23331
$626$	0.763932	0.0305329
$627$	0	0
$628$	−15.7082	−0.626826
$629$	−0.0557281	−0.00222202
$630$	0	0
$631$	−8.72949	−0.347516	−0.173758	−	0.984788i	$-0.555591\pi$
$631$	−8.72949	−0.347516	−0.173758	+	0.984788i	$0.555591\pi$
$632$	0	0
$633$	0	0
$634$	−16.0000	−0.635441
$635$	26.7984	1.06346
$636$	0	0
$637$	9.70820	0.384653
$638$	4.06888	0.161089
$639$	0	0
$640$	−29.7984	−1.17788
$641$	41.0902	1.62296	0.811482	−	0.584377i	$-0.198661\pi$
$641$	41.0902	1.62296	0.811482	+	0.584377i	$0.198661\pi$
$642$	0	0
$643$	8.14590	0.321243	0.160621	−	0.987016i	$-0.448650\pi$
$643$	8.14590	0.321243	0.160621	+	0.987016i	$0.448650\pi$
$644$	−26.5623	−1.04670
$645$	0	0
$646$	−0.729490	−0.0287014
$647$	29.8885	1.17504	0.587520	−	0.809210i	$-0.300104\pi$
$647$	29.8885	1.17504	0.587520	+	0.809210i	$0.300104\pi$
$648$	0	0
$649$	−7.23607	−0.284041
$650$	5.56231	0.218172
$651$	0	0
$652$	20.5623	0.805282
$653$	−39.6525	−1.55172	−0.775861	−	0.630904i	$-0.782684\pi$
$653$	−39.6525	−1.55172	−0.775861	+	0.630904i	$0.782684\pi$
$654$	0	0
$655$	−0.236068	−0.00922394
$656$	−12.0000	−0.468521
$657$	0	0
$658$	6.27051	0.244450
$659$	−22.6869	−0.883757	−0.441878	−	0.897075i	$-0.645688\pi$
$659$	−22.6869	−0.883757	−0.441878	+	0.897075i	$0.645688\pi$
$660$	0	0
$661$	−16.6180	−0.646367	−0.323183	−	0.946336i	$-0.604753\pi$
$661$	−16.6180	−0.646367	−0.323183	+	0.946336i	$0.604753\pi$
$662$	6.96556	0.270724
$663$	0	0
$664$	−9.14590	−0.354930
$665$	39.2705	1.52285
$666$	0	0
$667$	47.1591	1.82601
$668$	−14.9443	−0.578211
$669$	0	0
$670$	−6.85410	−0.264797
$671$	−5.30495	−0.204795
$672$	0	0
$673$	−4.41641	−0.170240	−0.0851200	−	0.996371i	$-0.527127\pi$
$673$	−4.41641	−0.170240	−0.0851200	+	0.996371i	$0.527127\pi$
$674$	11.7295	0.451803
$675$	0	0
$676$	−17.0902	−0.657314
$677$	−28.6525	−1.10120	−0.550602	−	0.834768i	$-0.685602\pi$
$677$	−28.6525	−1.10120	−0.550602	+	0.834768i	$0.685602\pi$
$678$	0	0
$679$	−15.8754	−0.609241
$680$	1.38197	0.0529960
$681$	0	0
$682$	0	0
$683$	−10.0557	−0.384772	−0.192386	−	0.981319i	$-0.561623\pi$
$683$	−10.0557	−0.384772	−0.192386	+	0.981319i	$0.561623\pi$
$684$	0	0
$685$	16.9443	0.647407
$686$	−9.27051	−0.353950
$687$	0	0
$688$	−8.56231	−0.326435
$689$	61.6869	2.35008
$690$	0	0
$691$	3.83282	0.145807	0.0729036	−	0.997339i	$-0.476773\pi$
$691$	3.83282	0.145807	0.0729036	+	0.997339i	$0.476773\pi$
$692$	1.47214	0.0559622
$693$	0	0
$694$	−5.02129	−0.190605
$695$	15.3262	0.581357
$696$	0	0
$697$	1.52786	0.0578720
$698$	−10.3262	−0.390854
$699$	0	0
$700$	−9.00000	−0.340168
$701$	30.0344	1.13439	0.567193	−	0.823585i	$-0.308030\pi$
$701$	30.0344	1.13439	0.567193	+	0.823585i	$0.308030\pi$
$702$	0	0
$703$	1.18034	0.0445174
$704$	−0.180340	−0.00679682
$705$	0	0
$706$	20.0132	0.753205
$707$	−14.2918	−0.537498
$708$	0	0
$709$	4.14590	0.155702	0.0778512	−	0.996965i	$-0.475194\pi$
$709$	4.14590	0.155702	0.0778512	+	0.996965i	$0.475194\pi$
$710$	−0.145898	−0.00547546
$711$	0	0
$712$	−14.2705	−0.534810
$713$	0	0
$714$	0	0
$715$	9.70820	0.363066
$716$	32.0344	1.19718
$717$	0	0
$718$	15.6525	0.584145
$719$	41.3820	1.54329	0.771643	−	0.636055i	$-0.219435\pi$
$719$	41.3820	1.54329	0.771643	+	0.636055i	$0.219435\pi$
$720$	0	0
$721$	−0.437694	−0.0163006
$722$	3.70820	0.138005
$723$	0	0
$724$	−27.5066	−1.02227
$725$	15.9787	0.593435
$726$	0	0
$727$	−23.8328	−0.883910	−0.441955	−	0.897037i	$-0.645715\pi$
$727$	−23.8328	−0.883910	−0.441955	+	0.897037i	$0.645715\pi$
$728$	−32.5623	−1.20684
$729$	0	0
$730$	−13.8541	−0.512763
$731$	1.09017	0.0403214
$732$	0	0
$733$	27.9443	1.03215	0.516073	−	0.856545i	$-0.327393\pi$
$733$	27.9443	1.03215	0.516073	+	0.856545i	$0.327393\pi$
$734$	−22.4164	−0.827405
$735$	0	0
$736$	30.7426	1.13319
$737$	−3.23607	−0.119202
$738$	0	0
$739$	21.7082	0.798549	0.399275	−	0.916831i	$-0.369262\pi$
$739$	21.7082	0.798549	0.399275	+	0.916831i	$0.369262\pi$
$740$	−1.00000	−0.0367607
$741$	0	0
$742$	23.5623	0.864999
$743$	3.43769	0.126117	0.0630584	−	0.998010i	$-0.479915\pi$
$743$	3.43769	0.126117	0.0630584	+	0.998010i	$0.479915\pi$
$744$	0	0
$745$	44.5967	1.63390
$746$	−0.214782	−0.00786372
$747$	0	0
$748$	0.291796	0.0106691
$749$	−3.27051	−0.119502
$750$	0	0
$751$	−39.8328	−1.45352	−0.726760	−	0.686891i	$-0.758975\pi$
$751$	−39.8328	−1.45352	−0.726760	+	0.686891i	$0.758975\pi$
$752$	6.27051	0.228662
$753$	0	0
$754$	25.8541	0.941551
$755$	−51.0689	−1.85859
$756$	0	0
$757$	43.1246	1.56739	0.783695	−	0.621145i	$-0.213332\pi$
$757$	43.1246	1.56739	0.783695	+	0.621145i	$0.213332\pi$
$758$	−11.3820	−0.413412
$759$	0	0
$760$	−29.2705	−1.06175
$761$	−3.50658	−0.127113	−0.0635567	−	0.997978i	$-0.520244\pi$
$761$	−3.50658	−0.127113	−0.0635567	+	0.997978i	$0.520244\pi$
$762$	0	0
$763$	−25.2492	−0.914083
$764$	−26.0344	−0.941893
$765$	0	0
$766$	10.4164	0.376360
$767$	−45.9787	−1.66020
$768$	0	0
$769$	−53.7426	−1.93801	−0.969005	−	0.247042i	$-0.920541\pi$
$769$	−53.7426	−1.93801	−0.969005	+	0.247042i	$0.920541\pi$
$770$	3.70820	0.133634
$771$	0	0
$772$	3.85410	0.138712
$773$	19.0902	0.686626	0.343313	−	0.939221i	$-0.388451\pi$
$773$	19.0902	0.686626	0.343313	+	0.939221i	$0.388451\pi$
$774$	0	0
$775$	0	0
$776$	11.8328	0.424773
$777$	0	0
$778$	17.9656	0.644096
$779$	−32.3607	−1.15944
$780$	0	0
$781$	−0.0688837	−0.00246485
$782$	−0.798374	−0.0285498
$783$	0	0
$784$	3.70820	0.132436
$785$	25.4164	0.907150
$786$	0	0
$787$	31.2918	1.11543	0.557716	−	0.830032i	$-0.311678\pi$
$787$	31.2918	1.11543	0.557716	+	0.830032i	$0.311678\pi$
$788$	26.5623	0.946243
$789$	0	0
$790$	0	0
$791$	5.56231	0.197773
$792$	0	0
$793$	−33.7082	−1.19701
$794$	10.0689	0.357331
$795$	0	0
$796$	43.2148	1.53171
$797$	−9.05573	−0.320770	−0.160385	−	0.987055i	$-0.551274\pi$
$797$	−9.05573	−0.320770	−0.160385	+	0.987055i	$0.551274\pi$
$798$	0	0
$799$	−0.798374	−0.0282444
$800$	10.4164	0.368276
$801$	0	0
$802$	18.4377	0.651058
$803$	−6.54102	−0.230828
$804$	0	0
$805$	42.9787	1.51480
$806$	0	0
$807$	0	0
$808$	10.6525	0.374753
$809$	−54.7984	−1.92661	−0.963304	−	0.268412i	$-0.913501\pi$
$809$	−54.7984	−1.92661	−0.963304	+	0.268412i	$0.913501\pi$
$810$	0	0
$811$	42.7771	1.50211	0.751053	−	0.660242i	$-0.229546\pi$
$811$	42.7771	1.50211	0.751053	+	0.660242i	$0.229546\pi$
$812$	−41.8328	−1.46804
$813$	0	0
$814$	0.111456	0.00390654
$815$	−33.2705	−1.16542
$816$	0	0
$817$	−23.0902	−0.807823
$818$	−3.81966	−0.133551
$819$	0	0
$820$	27.4164	0.957422
$821$	32.4721	1.13329	0.566643	−	0.823964i	$-0.308242\pi$
$821$	32.4721	1.13329	0.566643	+	0.823964i	$0.308242\pi$
$822$	0	0
$823$	−6.12461	−0.213491	−0.106745	−	0.994286i	$-0.534043\pi$
$823$	−6.12461	−0.213491	−0.106745	+	0.994286i	$0.534043\pi$
$824$	0.326238	0.0113650
$825$	0	0
$826$	−17.5623	−0.611071
$827$	−2.67376	−0.0929758	−0.0464879	−	0.998919i	$-0.514803\pi$
$827$	−2.67376	−0.0929758	−0.0464879	+	0.998919i	$0.514803\pi$
$828$	0	0
$829$	−21.7082	−0.753957	−0.376979	−	0.926222i	$-0.623037\pi$
$829$	−21.7082	−0.753957	−0.376979	+	0.926222i	$0.623037\pi$
$830$	6.61803	0.229715
$831$	0	0
$832$	−1.14590	−0.0397269
$833$	−0.472136	−0.0163585
$834$	0	0
$835$	24.1803	0.836795
$836$	−6.18034	−0.213752
$837$	0	0
$838$	−2.76393	−0.0954784
$839$	−11.1803	−0.385988	−0.192994	−	0.981200i	$-0.561820\pi$
$839$	−11.1803	−0.385988	−0.192994	+	0.981200i	$0.561820\pi$
$840$	0	0
$841$	45.2705	1.56105
$842$	9.12461	0.314455
$843$	0	0
$844$	12.9443	0.445560
$845$	27.6525	0.951274
$846$	0	0
$847$	−31.2492	−1.07374
$848$	23.5623	0.809133
$849$	0	0
$850$	−0.270510	−0.00927841
$851$	1.29180	0.0442822
$852$	0	0
$853$	4.00000	0.136957	0.0684787	−	0.997653i	$-0.478185\pi$
$853$	4.00000	0.136957	0.0684787	+	0.997653i	$0.478185\pi$
$854$	−12.8754	−0.440587
$855$	0	0
$856$	2.43769	0.0833187
$857$	8.18034	0.279435	0.139718	−	0.990191i	$-0.455381\pi$
$857$	8.18034	0.279435	0.139718	+	0.990191i	$0.455381\pi$
$858$	0	0
$859$	43.2918	1.47710	0.738549	−	0.674200i	$-0.235512\pi$
$859$	43.2918	1.47710	0.738549	+	0.674200i	$0.235512\pi$
$860$	19.5623	0.667069
$861$	0	0
$862$	−18.0689	−0.615429
$863$	−2.49342	−0.0848771	−0.0424385	−	0.999099i	$-0.513513\pi$
$863$	−2.49342	−0.0848771	−0.0424385	+	0.999099i	$0.513513\pi$
$864$	0	0
$865$	−2.38197	−0.0809893
$866$	0.360680	0.0122564
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−20.5623	−0.696727
$872$	18.8197	0.637314
$873$	0	0
$874$	16.9098	0.571984
$875$	−24.7082	−0.835290
$876$	0	0
$877$	16.2918	0.550135	0.275067	−	0.961425i	$-0.411300\pi$
$877$	16.2918	0.550135	0.275067	+	0.961425i	$0.411300\pi$
$878$	25.8541	0.872534
$879$	0	0
$880$	3.70820	0.125004
$881$	15.3607	0.517514	0.258757	−	0.965942i	$-0.416687\pi$
$881$	15.3607	0.517514	0.258757	+	0.965942i	$0.416687\pi$
$882$	0	0
$883$	−1.00000	−0.0336527	−0.0168263	−	0.999858i	$-0.505356\pi$
$883$	−1.00000	−0.0336527	−0.0168263	+	0.999858i	$0.505356\pi$
$884$	1.85410	0.0623602
$885$	0	0
$886$	25.4164	0.853881
$887$	−39.1033	−1.31296	−0.656481	−	0.754343i	$-0.727956\pi$
$887$	−39.1033	−1.31296	−0.656481	+	0.754343i	$0.727956\pi$
$888$	0	0
$889$	30.7082	1.02992
$890$	10.3262	0.346136
$891$	0	0
$892$	−1.14590	−0.0383675
$893$	16.9098	0.565866
$894$	0	0
$895$	−51.8328	−1.73258
$896$	−34.1459	−1.14073
$897$	0	0
$898$	−14.8754	−0.496398
$899$	0	0
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	−3.05573	−0.101745
$903$	0	0
$904$	−4.14590	−0.137891
$905$	44.5066	1.47945
$906$	0	0
$907$	52.9230	1.75728	0.878639	−	0.477486i	$-0.158452\pi$
$907$	52.9230	1.75728	0.878639	+	0.477486i	$0.158452\pi$
$908$	−33.5623	−1.11380
$909$	0	0
$910$	23.5623	0.781083
$911$	−8.18034	−0.271027	−0.135513	−	0.990776i	$-0.543268\pi$
$911$	−8.18034	−0.271027	−0.135513	+	0.990776i	$0.543268\pi$
$912$	0	0
$913$	3.12461	0.103410
$914$	−9.72949	−0.321823
$915$	0	0
$916$	11.7082	0.386850
$917$	−0.270510	−0.00893302
$918$	0	0
$919$	9.87539	0.325759	0.162879	−	0.986646i	$-0.447922\pi$
$919$	9.87539	0.325759	0.162879	+	0.986646i	$0.447922\pi$
$920$	−32.0344	−1.05614
$921$	0	0
$922$	−6.63932	−0.218654
$923$	−0.437694	−0.0144069
$924$	0	0
$925$	0.437694	0.0143913
$926$	−19.2361	−0.632136
$927$	0	0
$928$	48.4164	1.58935
$929$	−33.5410	−1.10045	−0.550223	−	0.835018i	$-0.685457\pi$
$929$	−33.5410	−1.10045	−0.550223	+	0.835018i	$0.685457\pi$
$930$	0	0
$931$	10.0000	0.327737
$932$	30.4164	0.996323
$933$	0	0
$934$	20.2574	0.662841
$935$	−0.472136	−0.0154405
$936$	0	0
$937$	40.4853	1.32260	0.661298	−	0.750123i	$-0.270006\pi$
$937$	40.4853	1.32260	0.661298	+	0.750123i	$0.270006\pi$
$938$	−7.85410	−0.256446
$939$	0	0
$940$	−14.3262	−0.467270
$941$	−28.5836	−0.931798	−0.465899	−	0.884838i	$-0.654269\pi$
$941$	−28.5836	−0.931798	−0.465899	+	0.884838i	$0.654269\pi$
$942$	0	0
$943$	−35.4164	−1.15332
$944$	−17.5623	−0.571604
$945$	0	0
$946$	−2.18034	−0.0708890
$947$	−22.0689	−0.717142	−0.358571	−	0.933502i	$-0.616736\pi$
$947$	−22.0689	−0.717142	−0.358571	+	0.933502i	$0.616736\pi$
$948$	0	0
$949$	−41.5623	−1.34917
$950$	5.72949	0.185889
$951$	0	0
$952$	1.58359	0.0513245
$953$	−42.2148	−1.36747	−0.683735	−	0.729730i	$-0.739646\pi$
$953$	−42.2148	−1.36747	−0.683735	+	0.729730i	$0.739646\pi$
$954$	0	0
$955$	42.1246	1.36312
$956$	−21.7082	−0.702093
$957$	0	0
$958$	5.52786	0.178597
$959$	19.4164	0.626989
$960$	0	0
$961$	0	0
$962$	0.708204	0.0228334
$963$	0	0
$964$	13.7984	0.444416
$965$	−6.23607	−0.200746
$966$	0	0
$967$	43.6525	1.40377	0.701884	−	0.712291i	$-0.252342\pi$
$967$	43.6525	1.40377	0.701884	+	0.712291i	$0.252342\pi$
$968$	23.2918	0.748627
$969$	0	0
$970$	−8.56231	−0.274919
$971$	20.5623	0.659876	0.329938	−	0.944003i	$-0.392972\pi$
$971$	20.5623	0.659876	0.329938	+	0.944003i	$0.392972\pi$
$972$	0	0
$973$	17.5623	0.563022
$974$	14.1672	0.453946
$975$	0	0
$976$	−12.8754	−0.412131
$977$	6.06888	0.194161	0.0970804	−	0.995277i	$-0.469050\pi$
$977$	6.06888	0.194161	0.0970804	+	0.995277i	$0.469050\pi$
$978$	0	0
$979$	4.87539	0.155818
$980$	−8.47214	−0.270632
$981$	0	0
$982$	17.0557	0.544270
$983$	−21.0344	−0.670895	−0.335447	−	0.942059i	$-0.608887\pi$
$983$	−21.0344	−0.670895	−0.335447	+	0.942059i	$0.608887\pi$
$984$	0	0
$985$	−42.9787	−1.36942
$986$	−1.25735	−0.0400423
$987$	0	0
$988$	−39.2705	−1.24936
$989$	−25.2705	−0.803555
$990$	0	0
$991$	−17.2705	−0.548616	−0.274308	−	0.961642i	$-0.588449\pi$
$991$	−17.2705	−0.548616	−0.274308	+	0.961642i	$0.588449\pi$
$992$	0	0
$993$	0	0
$994$	−0.167184	−0.00530276
$995$	−69.9230	−2.21671
$996$	0	0
$997$	−27.2492	−0.862992	−0.431496	−	0.902115i	$-0.642014\pi$
$997$	−27.2492	−0.862992	−0.431496	+	0.902115i	$0.642014\pi$
$998$	2.56231	0.0811084
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.g.1.2		2
3.2	odd	2		961.2.a.d.1.1		2
31.2	even	5		279.2.i.a.190.1		4
31.16	even	5		279.2.i.a.163.1		4
31.30	odd	2		8649.2.a.f.1.2		2
93.2	odd	10		31.2.d.a.4.1	✓	4
93.5	odd	6		961.2.c.f.521.1		4
93.8	odd	10		961.2.d.f.374.1		4
93.11	even	30		961.2.g.g.338.1		8
93.14	odd	30		961.2.g.f.816.1		8
93.17	even	30		961.2.g.g.816.1		8
93.20	odd	30		961.2.g.f.338.1		8
93.23	even	10		961.2.d.e.374.1		4
93.26	even	6		961.2.c.d.521.1		4
93.29	even	10		961.2.d.b.531.1		4
93.35	odd	10		961.2.d.f.388.1		4
93.38	odd	30		961.2.g.f.235.1		8
93.41	odd	30		961.2.g.b.844.1		8
93.44	even	30		961.2.g.c.448.1		8
93.47	odd	10		31.2.d.a.8.1	yes	4
93.50	odd	30		961.2.g.b.547.1		8
93.53	even	30		961.2.g.g.732.1		8
93.56	odd	6		961.2.c.f.439.1		4
93.59	odd	30		961.2.g.b.846.1		8
93.65	even	30		961.2.g.c.846.1		8
93.68	even	6		961.2.c.d.439.1		4
93.71	odd	30		961.2.g.f.732.1		8
93.74	even	30		961.2.g.c.547.1		8
93.77	even	10		961.2.d.b.628.1		4
93.80	odd	30		961.2.g.b.448.1		8
93.83	even	30		961.2.g.c.844.1		8
93.86	even	30		961.2.g.g.235.1		8
93.89	even	10		961.2.d.e.388.1		4
93.92	even	2		961.2.a.e.1.1		2
372.47	even	10		496.2.n.b.225.1		4
372.95	even	10		496.2.n.b.97.1		4
465.2	even	20		775.2.bf.a.624.2		8
465.47	even	20		775.2.bf.a.349.1		8
465.188	even	20		775.2.bf.a.624.1		8
465.233	even	20		775.2.bf.a.349.2		8
465.374	odd	10		775.2.k.c.376.1		4
465.419	odd	10		775.2.k.c.101.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.d.a.4.1	✓	4	93.2	odd	10
31.2.d.a.8.1	yes	4	93.47	odd	10
279.2.i.a.163.1		4	31.16	even	5
279.2.i.a.190.1		4	31.2	even	5
496.2.n.b.97.1		4	372.95	even	10
496.2.n.b.225.1		4	372.47	even	10
775.2.k.c.101.1		4	465.419	odd	10
775.2.k.c.376.1		4	465.374	odd	10
775.2.bf.a.349.1		8	465.47	even	20
775.2.bf.a.349.2		8	465.233	even	20
775.2.bf.a.624.1		8	465.188	even	20
775.2.bf.a.624.2		8	465.2	even	20
961.2.a.d.1.1		2	3.2	odd	2
961.2.a.e.1.1		2	93.92	even	2
961.2.c.d.439.1		4	93.68	even	6
961.2.c.d.521.1		4	93.26	even	6
961.2.c.f.439.1		4	93.56	odd	6
961.2.c.f.521.1		4	93.5	odd	6
961.2.d.b.531.1		4	93.29	even	10
961.2.d.b.628.1		4	93.77	even	10
961.2.d.e.374.1		4	93.23	even	10
961.2.d.e.388.1		4	93.89	even	10
961.2.d.f.374.1		4	93.8	odd	10
961.2.d.f.388.1		4	93.35	odd	10
961.2.g.b.448.1		8	93.80	odd	30
961.2.g.b.547.1		8	93.50	odd	30
961.2.g.b.844.1		8	93.41	odd	30
961.2.g.b.846.1		8	93.59	odd	30
961.2.g.c.448.1		8	93.44	even	30
961.2.g.c.547.1		8	93.74	even	30
961.2.g.c.844.1		8	93.83	even	30
961.2.g.c.846.1		8	93.65	even	30
961.2.g.f.235.1		8	93.38	odd	30
961.2.g.f.338.1		8	93.20	odd	30
961.2.g.f.732.1		8	93.71	odd	30
961.2.g.f.816.1		8	93.14	odd	30
961.2.g.g.235.1		8	93.86	even	30
961.2.g.g.338.1		8	93.11	even	30
961.2.g.g.732.1		8	93.53	even	30
961.2.g.g.816.1		8	93.17	even	30
8649.2.a.f.1.2		2	31.30	odd	2
8649.2.a.g.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q+0.618034 q^{2} -1.61803 q^{4} +2.61803 q^{5} +3.00000 q^{7} -2.23607 q^{8} +1.61803 q^{10} +0.763932 q^{11} +4.85410 q^{13} +1.85410 q^{14} +1.85410 q^{16} -0.236068 q^{17} +5.00000 q^{19} -4.23607 q^{20} +0.472136 q^{22} +5.47214 q^{23} +1.85410 q^{25} +3.00000 q^{26} -4.85410 q^{28} +8.61803 q^{29} +5.61803 q^{32} -0.145898 q^{34} +7.85410 q^{35} +0.236068 q^{37} +3.09017 q^{38} -5.85410 q^{40} -6.47214 q^{41} -4.61803 q^{43} -1.23607 q^{44} +3.38197 q^{46} +3.38197 q^{47} +2.00000 q^{49} +1.14590 q^{50} -7.85410 q^{52} +12.7082 q^{53} +2.00000 q^{55} -6.70820 q^{56} +5.32624 q^{58} -9.47214 q^{59} -6.94427 q^{61} -0.236068 q^{64} +12.7082 q^{65} -4.23607 q^{67} +0.381966 q^{68} +4.85410 q^{70} -0.0901699 q^{71} -8.56231 q^{73} +0.145898 q^{74} -8.09017 q^{76} +2.29180 q^{77} +4.85410 q^{80} -4.00000 q^{82} +4.09017 q^{83} -0.618034 q^{85} -2.85410 q^{86} -1.70820 q^{88} +6.38197 q^{89} +14.5623 q^{91} -8.85410 q^{92} +2.09017 q^{94} +13.0902 q^{95} -5.29180 q^{97} +1.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} + 3 q^{5} + 6 q^{7} + q^{10} + 6 q^{11} + 3 q^{13} - 3 q^{14} - 3 q^{16} + 4 q^{17} + 10 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} - 3 q^{25} + 6 q^{26} - 3 q^{28} + 15 q^{29} + 9 q^{32}+ \cdots - 2 q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 + 3 * q^5 + 6 * q^7 + q^10 + 6 * q^11 + 3 * q^13 - 3 * q^14 - 3 * q^16 + 4 * q^17 + 10 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 - 3 * q^25 + 6 * q^26 - 3 * q^28 + 15 * q^29 + 9 * q^32 - 7 * q^34 + 9 * q^35 - 4 * q^37 - 5 * q^38 - 5 * q^40 - 4 * q^41 - 7 * q^43 + 2 * q^44 + 9 * q^46 + 9 * q^47 + 4 * q^49 + 9 * q^50 - 9 * q^52 + 12 * q^53 + 4 * q^55 - 5 * q^58 - 10 * q^59 + 4 * q^61 + 4 * q^64 + 12 * q^65 - 4 * q^67 + 3 * q^68 + 3 * q^70 + 11 * q^71 + 3 * q^73 + 7 * q^74 - 5 * q^76 + 18 * q^77 + 3 * q^80 - 8 * q^82 - 3 * q^83 + q^85 + q^86 + 10 * q^88 + 15 * q^89 + 9 * q^91 - 11 * q^92 - 7 * q^94 + 15 * q^95 - 24 * q^97 - 2 * q^98

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