LMFDB - Embedded newform 786.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [786,2,Mod(1,786)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(786, 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("786.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 786 = 2 \cdot 3 \cdot 131 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	786.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:	$6.27624159887$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.470683$ of defining polynomial
Character	$\chi$	$=$	786.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24914 q^{5} -1.00000 q^{6} -1.77846 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24914 q^{5} -1.00000 q^{6} -1.77846 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.24914 q^{10} -0.941367 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.77846 q^{14} -2.24914 q^{15} +1.00000 q^{16} +0.941367 q^{17} -1.00000 q^{18} +6.71982 q^{19} -2.24914 q^{20} -1.77846 q^{21} +0.941367 q^{22} +0.941367 q^{23} -1.00000 q^{24} +0.0586332 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.77846 q^{28} +7.33537 q^{29} +2.24914 q^{30} +10.8647 q^{31} -1.00000 q^{32} -0.941367 q^{33} -0.941367 q^{34} +4.00000 q^{35} +1.00000 q^{36} +8.96896 q^{37} -6.71982 q^{38} +2.00000 q^{39} +2.24914 q^{40} -0.221543 q^{41} +1.77846 q^{42} -5.55691 q^{43} -0.941367 q^{44} -2.24914 q^{45} -0.941367 q^{46} -2.11727 q^{47} +1.00000 q^{48} -3.83709 q^{49} -0.0586332 q^{50} +0.941367 q^{51} +2.00000 q^{52} -5.30777 q^{53} -1.00000 q^{54} +2.11727 q^{55} +1.77846 q^{56} +6.71982 q^{57} -7.33537 q^{58} -6.61555 q^{59} -2.24914 q^{60} +5.05863 q^{61} -10.8647 q^{62} -1.77846 q^{63} +1.00000 q^{64} -4.49828 q^{65} +0.941367 q^{66} +8.83709 q^{67} +0.941367 q^{68} +0.941367 q^{69} -4.00000 q^{70} +16.0552 q^{71} -1.00000 q^{72} +2.00000 q^{73} -8.96896 q^{74} +0.0586332 q^{75} +6.71982 q^{76} +1.67418 q^{77} -2.00000 q^{78} -5.68879 q^{79} -2.24914 q^{80} +1.00000 q^{81} +0.221543 q^{82} +0.954357 q^{83} -1.77846 q^{84} -2.11727 q^{85} +5.55691 q^{86} +7.33537 q^{87} +0.941367 q^{88} -10.3940 q^{89} +2.24914 q^{90} -3.55691 q^{91} +0.941367 q^{92} +10.8647 q^{93} +2.11727 q^{94} -15.1138 q^{95} -1.00000 q^{96} -8.17246 q^{97} +3.83709 q^{98} -0.941367 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 2 q^{10} - 2 q^{11} + 3 q^{12} + 6 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} + 2 q^{17} - 3 q^{18} + 11 q^{19} + 2 q^{20} + 3 q^{21} + 2 q^{22} + 2 q^{23} - 3 q^{24} + q^{25} - 6 q^{26} + 3 q^{27} + 3 q^{28} - 3 q^{29} - 2 q^{30} + 8 q^{31} - 3 q^{32} - 2 q^{33} - 2 q^{34} + 12 q^{35} + 3 q^{36} + 9 q^{37} - 11 q^{38} + 6 q^{39} - 2 q^{40} - 9 q^{41} - 3 q^{42} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 8 q^{47} + 3 q^{48} - 4 q^{49} - q^{50} + 2 q^{51} + 6 q^{52} - 8 q^{53} - 3 q^{54} + 8 q^{55} - 3 q^{56} + 11 q^{57} + 3 q^{58} - 4 q^{59} + 2 q^{60} + 16 q^{61} - 8 q^{62} + 3 q^{63} + 3 q^{64} + 4 q^{65} + 2 q^{66} + 19 q^{67} + 2 q^{68} + 2 q^{69} - 12 q^{70} + 14 q^{71} - 3 q^{72} + 6 q^{73} - 9 q^{74} + q^{75} + 11 q^{76} - 10 q^{77} - 6 q^{78} + 10 q^{79} + 2 q^{80} + 3 q^{81} + 9 q^{82} - 3 q^{83} + 3 q^{84} - 8 q^{85} - 3 q^{87} + 2 q^{88} - 7 q^{89} - 2 q^{90} + 6 q^{91} + 2 q^{92} + 8 q^{93} + 8 q^{94} - 12 q^{95} - 3 q^{96} + 8 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 2 * q^10 - 2 * q^11 + 3 * q^12 + 6 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 + 2 * q^17 - 3 * q^18 + 11 * q^19 + 2 * q^20 + 3 * q^21 + 2 * q^22 + 2 * q^23 - 3 * q^24 + q^25 - 6 * q^26 + 3 * q^27 + 3 * q^28 - 3 * q^29 - 2 * q^30 + 8 * q^31 - 3 * q^32 - 2 * q^33 - 2 * q^34 + 12 * q^35 + 3 * q^36 + 9 * q^37 - 11 * q^38 + 6 * q^39 - 2 * q^40 - 9 * q^41 - 3 * q^42 - 2 * q^44 + 2 * q^45 - 2 * q^46 - 8 * q^47 + 3 * q^48 - 4 * q^49 - q^50 + 2 * q^51 + 6 * q^52 - 8 * q^53 - 3 * q^54 + 8 * q^55 - 3 * q^56 + 11 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 + 16 * q^61 - 8 * q^62 + 3 * q^63 + 3 * q^64 + 4 * q^65 + 2 * q^66 + 19 * q^67 + 2 * q^68 + 2 * q^69 - 12 * q^70 + 14 * q^71 - 3 * q^72 + 6 * q^73 - 9 * q^74 + q^75 + 11 * q^76 - 10 * q^77 - 6 * q^78 + 10 * q^79 + 2 * q^80 + 3 * q^81 + 9 * q^82 - 3 * q^83 + 3 * q^84 - 8 * q^85 - 3 * q^87 + 2 * q^88 - 7 * q^89 - 2 * q^90 + 6 * q^91 + 2 * q^92 + 8 * q^93 + 8 * q^94 - 12 * q^95 - 3 * q^96 + 8 * q^97 + 4 * q^98 - 2 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.24914	−1.00585	−0.502923	−	0.864331i	$-0.667742\pi$
$5$	−2.24914	−1.00585	−0.502923	+	0.864331i	$0.667742\pi$
$6$	−1.00000	−0.408248
$7$	−1.77846	−0.672194	−0.336097	−	0.941827i	$-0.609107\pi$
$7$	−1.77846	−0.672194	−0.336097	+	0.941827i	$0.609107\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	2.24914	0.711241
$11$	−0.941367	−0.283833	−0.141916	−	0.989879i	$-0.545326\pi$
$11$	−0.941367	−0.283833	−0.141916	+	0.989879i	$0.545326\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.77846	0.475313
$15$	−2.24914	−0.580726
$16$	1.00000	0.250000
$17$	0.941367	0.228315	0.114157	−	0.993463i	$-0.463583\pi$
$17$	0.941367	0.228315	0.114157	+	0.993463i	$0.463583\pi$
$18$	−1.00000	−0.235702
$19$	6.71982	1.54163	0.770817	−	0.637057i	$-0.219848\pi$
$19$	6.71982	1.54163	0.770817	+	0.637057i	$0.219848\pi$
$20$	−2.24914	−0.502923
$21$	−1.77846	−0.388091
$22$	0.941367	0.200700
$23$	0.941367	0.196289	0.0981443	−	0.995172i	$-0.468709\pi$
$23$	0.941367	0.196289	0.0981443	+	0.995172i	$0.468709\pi$
$24$	−1.00000	−0.204124
$25$	0.0586332	0.0117266
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	−1.77846	−0.336097
$29$	7.33537	1.36214	0.681072	−	0.732216i	$-0.261514\pi$
$29$	7.33537	1.36214	0.681072	+	0.732216i	$0.261514\pi$
$30$	2.24914	0.410635
$31$	10.8647	1.95136	0.975678	−	0.219210i	$-0.0703479\pi$
$31$	10.8647	1.95136	0.975678	+	0.219210i	$0.0703479\pi$
$32$	−1.00000	−0.176777
$33$	−0.941367	−0.163871
$34$	−0.941367	−0.161443
$35$	4.00000	0.676123
$36$	1.00000	0.166667
$37$	8.96896	1.47449	0.737244	−	0.675626i	$-0.236127\pi$
$37$	8.96896	1.47449	0.737244	+	0.675626i	$0.236127\pi$
$38$	−6.71982	−1.09010
$39$	2.00000	0.320256
$40$	2.24914	0.355620
$41$	−0.221543	−0.0345992	−0.0172996	−	0.999850i	$-0.505507\pi$
$41$	−0.221543	−0.0345992	−0.0172996	+	0.999850i	$0.505507\pi$
$42$	1.77846	0.274422
$43$	−5.55691	−0.847421	−0.423711	−	0.905798i	$-0.639273\pi$
$43$	−5.55691	−0.847421	−0.423711	+	0.905798i	$0.639273\pi$
$44$	−0.941367	−0.141916
$45$	−2.24914	−0.335282
$46$	−0.941367	−0.138797
$47$	−2.11727	−0.308835	−0.154418	−	0.988006i	$-0.549350\pi$
$47$	−2.11727	−0.308835	−0.154418	+	0.988006i	$0.549350\pi$
$48$	1.00000	0.144338
$49$	−3.83709	−0.548156
$50$	−0.0586332	−0.00829198
$51$	0.941367	0.131818
$52$	2.00000	0.277350
$53$	−5.30777	−0.729079	−0.364539	−	0.931188i	$-0.618774\pi$
$53$	−5.30777	−0.729079	−0.364539	+	0.931188i	$0.618774\pi$
$54$	−1.00000	−0.136083
$55$	2.11727	0.285492
$56$	1.77846	0.237656
$57$	6.71982	0.890062
$58$	−7.33537	−0.963181
$59$	−6.61555	−0.861271	−0.430635	−	0.902526i	$-0.641711\pi$
$59$	−6.61555	−0.861271	−0.430635	+	0.902526i	$0.641711\pi$
$60$	−2.24914	−0.290363
$61$	5.05863	0.647692	0.323846	−	0.946110i	$-0.395024\pi$
$61$	5.05863	0.647692	0.323846	+	0.946110i	$0.395024\pi$
$62$	−10.8647	−1.37982
$63$	−1.77846	−0.224065
$64$	1.00000	0.125000
$65$	−4.49828	−0.557943
$66$	0.941367	0.115874
$67$	8.83709	1.07962	0.539811	−	0.841786i	$-0.318496\pi$
$67$	8.83709	1.07962	0.539811	+	0.841786i	$0.318496\pi$
$68$	0.941367	0.114157
$69$	0.941367	0.113327
$70$	−4.00000	−0.478091
$71$	16.0552	1.90540	0.952701	−	0.303911i	$-0.0982924\pi$
$71$	16.0552	1.90540	0.952701	+	0.303911i	$0.0982924\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.96896	−1.04262
$75$	0.0586332	0.00677037
$76$	6.71982	0.770817
$77$	1.67418	0.190791
$78$	−2.00000	−0.226455
$79$	−5.68879	−0.640039	−0.320019	−	0.947411i	$-0.603689\pi$
$79$	−5.68879	−0.640039	−0.320019	+	0.947411i	$0.603689\pi$
$80$	−2.24914	−0.251462
$81$	1.00000	0.111111
$82$	0.221543	0.0244653
$83$	0.954357	0.104754	0.0523771	−	0.998627i	$-0.483320\pi$
$83$	0.954357	0.104754	0.0523771	+	0.998627i	$0.483320\pi$
$84$	−1.77846	−0.194046
$85$	−2.11727	−0.229650
$86$	5.55691	0.599217
$87$	7.33537	0.786434
$88$	0.941367	0.100350
$89$	−10.3940	−1.10176	−0.550881	−	0.834584i	$-0.685708\pi$
$89$	−10.3940	−1.10176	−0.550881	+	0.834584i	$0.685708\pi$
$90$	2.24914	0.237080
$91$	−3.55691	−0.372866
$92$	0.941367	0.0981443
$93$	10.8647	1.12662
$94$	2.11727	0.218379
$95$	−15.1138	−1.55065
$96$	−1.00000	−0.102062
$97$	−8.17246	−0.829788	−0.414894	−	0.909870i	$-0.636181\pi$
$97$	−8.17246	−0.829788	−0.414894	+	0.909870i	$0.636181\pi$
$98$	3.83709	0.387605
$99$	−0.941367	−0.0946109
$100$	0.0586332	0.00586332
$101$	4.36641	0.434474	0.217237	−	0.976119i	$-0.430296\pi$
$101$	4.36641	0.434474	0.217237	+	0.976119i	$0.430296\pi$
$102$	−0.941367	−0.0932092
$103$	7.80605	0.769153	0.384577	−	0.923093i	$-0.374347\pi$
$103$	7.80605	0.769153	0.384577	+	0.923093i	$0.374347\pi$
$104$	−2.00000	−0.196116
$105$	4.00000	0.390360
$106$	5.30777	0.515537
$107$	−8.99656	−0.869730	−0.434865	−	0.900496i	$-0.643204\pi$
$107$	−8.99656	−0.869730	−0.434865	+	0.900496i	$0.643204\pi$
$108$	1.00000	0.0962250
$109$	9.55691	0.915386	0.457693	−	0.889110i	$-0.348676\pi$
$109$	9.55691	0.915386	0.457693	+	0.889110i	$0.348676\pi$
$110$	−2.11727	−0.201873
$111$	8.96896	0.851296
$112$	−1.77846	−0.168048
$113$	13.7164	1.29033	0.645165	−	0.764044i	$-0.276789\pi$
$113$	13.7164	1.29033	0.645165	+	0.764044i	$0.276789\pi$
$114$	−6.71982	−0.629369
$115$	−2.11727	−0.197436
$116$	7.33537	0.681072
$117$	2.00000	0.184900
$118$	6.61555	0.609011
$119$	−1.67418	−0.153472
$120$	2.24914	0.205318
$121$	−10.1138	−0.919439
$122$	−5.05863	−0.457987
$123$	−0.221543	−0.0199758
$124$	10.8647	0.975678
$125$	11.1138	0.994051
$126$	1.77846	0.158438
$127$	5.19051	0.460583	0.230292	−	0.973122i	$-0.426032\pi$
$127$	5.19051	0.460583	0.230292	+	0.973122i	$0.426032\pi$
$128$	−1.00000	−0.0883883
$129$	−5.55691	−0.489259
$130$	4.49828	0.394525
$131$	−1.00000	−0.0873704
$131$	−1.00000	−0.0873704
$132$	−0.941367	−0.0819355
$133$	−11.9509	−1.03628
$134$	−8.83709	−0.763408
$135$	−2.24914	−0.193575
$136$	−0.941367	−0.0807215
$137$	12.6707	1.08253	0.541267	−	0.840851i	$-0.317945\pi$
$137$	12.6707	1.08253	0.541267	+	0.840851i	$0.317945\pi$
$138$	−0.941367	−0.0801345
$139$	0.732814	0.0621564	0.0310782	−	0.999517i	$-0.490106\pi$
$139$	0.732814	0.0621564	0.0310782	+	0.999517i	$0.490106\pi$
$140$	4.00000	0.338062
$141$	−2.11727	−0.178306
$142$	−16.0552	−1.34732
$143$	−1.88273	−0.157442
$144$	1.00000	0.0833333
$145$	−16.4983	−1.37011
$146$	−2.00000	−0.165521
$147$	−3.83709	−0.316478
$148$	8.96896	0.737244
$149$	15.4948	1.26939	0.634694	−	0.772764i	$-0.281126\pi$
$149$	15.4948	1.26939	0.634694	+	0.772764i	$0.281126\pi$
$150$	−0.0586332	−0.00478738
$151$	3.66119	0.297943	0.148972	−	0.988841i	$-0.452404\pi$
$151$	3.66119	0.297943	0.148972	+	0.988841i	$0.452404\pi$
$152$	−6.71982	−0.545050
$153$	0.941367	0.0761050
$154$	−1.67418	−0.134909
$155$	−24.4362	−1.96276
$156$	2.00000	0.160128
$157$	−0.415488	−0.0331596	−0.0165798	−	0.999863i	$-0.505278\pi$
$157$	−0.415488	−0.0331596	−0.0165798	+	0.999863i	$0.505278\pi$
$158$	5.68879	0.452576
$159$	−5.30777	−0.420934
$160$	2.24914	0.177810
$161$	−1.67418	−0.131944
$162$	−1.00000	−0.0785674
$163$	−6.61555	−0.518170	−0.259085	−	0.965855i	$-0.583421\pi$
$163$	−6.61555	−0.518170	−0.259085	+	0.965855i	$0.583421\pi$
$164$	−0.221543	−0.0172996
$165$	2.11727	0.164829
$166$	−0.954357	−0.0740724
$167$	−10.5259	−0.814517	−0.407258	−	0.913313i	$-0.633515\pi$
$167$	−10.5259	−0.814517	−0.407258	+	0.913313i	$0.633515\pi$
$168$	1.77846	0.137211
$169$	−9.00000	−0.692308
$170$	2.11727	0.162387
$171$	6.71982	0.513878
$172$	−5.55691	−0.423711
$173$	−21.2181	−1.61318	−0.806591	−	0.591109i	$-0.798690\pi$
$173$	−21.2181	−1.61318	−0.806591	+	0.591109i	$0.798690\pi$
$174$	−7.33537	−0.556093
$175$	−0.104277	−0.00788257
$176$	−0.941367	−0.0709582
$177$	−6.61555	−0.497255
$178$	10.3940	0.779064
$179$	−8.05520	−0.602074	−0.301037	−	0.953613i	$-0.597333\pi$
$179$	−8.05520	−0.602074	−0.301037	+	0.953613i	$0.597333\pi$
$180$	−2.24914	−0.167641
$181$	−21.5715	−1.60340	−0.801699	−	0.597728i	$-0.796070\pi$
$181$	−21.5715	−1.60340	−0.801699	+	0.597728i	$0.796070\pi$
$182$	3.55691	0.263656
$183$	5.05863	0.373945
$184$	−0.941367	−0.0693985
$185$	−20.1725	−1.48311
$186$	−10.8647	−0.796638
$187$	−0.886172	−0.0648033
$188$	−2.11727	−0.154418
$189$	−1.77846	−0.129364
$190$	15.1138	1.09647
$191$	12.4086	0.897856	0.448928	−	0.893568i	$-0.351806\pi$
$191$	12.4086	0.897856	0.448928	+	0.893568i	$0.351806\pi$
$192$	1.00000	0.0721688
$193$	−4.10428	−0.295432	−0.147716	−	0.989030i	$-0.547192\pi$
$193$	−4.10428	−0.295432	−0.147716	+	0.989030i	$0.547192\pi$
$194$	8.17246	0.586749
$195$	−4.49828	−0.322129
$196$	−3.83709	−0.274078
$197$	−6.60256	−0.470413	−0.235206	−	0.971945i	$-0.575577\pi$
$197$	−6.60256	−0.470413	−0.235206	+	0.971945i	$0.575577\pi$
$198$	0.941367	0.0669000
$199$	18.6854	1.32457	0.662285	−	0.749252i	$-0.269587\pi$
$199$	18.6854	1.32457	0.662285	+	0.749252i	$0.269587\pi$
$200$	−0.0586332	−0.00414599
$201$	8.83709	0.623320
$202$	−4.36641	−0.307219
$203$	−13.0456	−0.915625
$204$	0.941367	0.0659089
$205$	0.498281	0.0348015
$206$	−7.80605	−0.543874
$207$	0.941367	0.0654295
$208$	2.00000	0.138675
$209$	−6.32582	−0.437566
$210$	−4.00000	−0.276026
$211$	−6.99656	−0.481663	−0.240832	−	0.970567i	$-0.577420\pi$
$211$	−6.99656	−0.481663	−0.240832	+	0.970567i	$0.577420\pi$
$212$	−5.30777	−0.364539
$213$	16.0552	1.10008
$214$	8.99656	0.614992
$215$	12.4983	0.852376
$216$	−1.00000	−0.0680414
$217$	−19.3224	−1.31169
$218$	−9.55691	−0.647276
$219$	2.00000	0.135147
$220$	2.11727	0.142746
$221$	1.88273	0.126646
$222$	−8.96896	−0.601957
$223$	−21.9785	−1.47179	−0.735895	−	0.677095i	$-0.763238\pi$
$223$	−21.9785	−1.47179	−0.735895	+	0.677095i	$0.763238\pi$
$224$	1.77846	0.118828
$225$	0.0586332	0.00390888
$226$	−13.7164	−0.912400
$227$	−5.07162	−0.336615	−0.168308	−	0.985735i	$-0.553830\pi$
$227$	−5.07162	−0.336615	−0.168308	+	0.985735i	$0.553830\pi$
$228$	6.71982	0.445031
$229$	−27.6888	−1.82973	−0.914863	−	0.403764i	$-0.867702\pi$
$229$	−27.6888	−1.82973	−0.914863	+	0.403764i	$0.867702\pi$
$230$	2.11727	0.139608
$231$	1.67418	0.110153
$232$	−7.33537	−0.481591
$233$	−9.95092	−0.651906	−0.325953	−	0.945386i	$-0.605685\pi$
$233$	−9.95092	−0.651906	−0.325953	+	0.945386i	$0.605685\pi$
$234$	−2.00000	−0.130744
$235$	4.76203	0.310641
$236$	−6.61555	−0.430635
$237$	−5.68879	−0.369527
$238$	1.67418	0.108521
$239$	18.0828	1.16968	0.584839	−	0.811149i	$-0.301158\pi$
$239$	18.0828	1.16968	0.584839	+	0.811149i	$0.301158\pi$
$240$	−2.24914	−0.145181
$241$	−4.82410	−0.310748	−0.155374	−	0.987856i	$-0.549658\pi$
$241$	−4.82410	−0.310748	−0.155374	+	0.987856i	$0.549658\pi$
$242$	10.1138	0.650142
$243$	1.00000	0.0641500
$244$	5.05863	0.323846
$245$	8.63016	0.551360
$246$	0.221543	0.0141251
$247$	13.4396	0.855144
$248$	−10.8647	−0.689908
$249$	0.954357	0.0604799
$250$	−11.1138	−0.702900
$251$	6.62854	0.418390	0.209195	−	0.977874i	$-0.432916\pi$
$251$	6.62854	0.418390	0.209195	+	0.977874i	$0.432916\pi$
$252$	−1.77846	−0.112032
$253$	−0.886172	−0.0557131
$254$	−5.19051	−0.325681
$255$	−2.11727	−0.132588
$256$	1.00000	0.0625000
$257$	4.28973	0.267586	0.133793	−	0.991009i	$-0.457284\pi$
$257$	4.28973	0.267586	0.133793	+	0.991009i	$0.457284\pi$
$258$	5.55691	0.345958
$259$	−15.9509	−0.991142
$260$	−4.49828	−0.278972
$261$	7.33537	0.454048
$262$	1.00000	0.0617802
$263$	−1.42504	−0.0878717	−0.0439359	−	0.999034i	$-0.513990\pi$
$263$	−1.42504	−0.0878717	−0.0439359	+	0.999034i	$0.513990\pi$
$264$	0.941367	0.0579371
$265$	11.9379	0.733341
$266$	11.9509	0.732758
$267$	−10.3940	−0.636103
$268$	8.83709	0.539811
$269$	−6.80262	−0.414763	−0.207381	−	0.978260i	$-0.566494\pi$
$269$	−6.80262	−0.414763	−0.207381	+	0.978260i	$0.566494\pi$
$270$	2.24914	0.136878
$271$	19.8466	1.20560	0.602799	−	0.797893i	$-0.294052\pi$
$271$	19.8466	1.20560	0.602799	+	0.797893i	$0.294052\pi$
$272$	0.941367	0.0570787
$273$	−3.55691	−0.215274
$274$	−12.6707	−0.765468
$275$	−0.0551953	−0.00332840
$276$	0.941367	0.0566636
$277$	−15.6742	−0.941770	−0.470885	−	0.882195i	$-0.656065\pi$
$277$	−15.6742	−0.941770	−0.470885	+	0.882195i	$0.656065\pi$
$278$	−0.732814	−0.0439512
$279$	10.8647	0.650452
$280$	−4.00000	−0.239046
$281$	16.6448	0.992943	0.496472	−	0.868053i	$-0.334629\pi$
$281$	16.6448	0.992943	0.496472	+	0.868053i	$0.334629\pi$
$282$	2.11727	0.126081
$283$	32.2277	1.91574	0.957868	−	0.287210i	$-0.0927277\pi$
$283$	32.2277	1.91574	0.957868	+	0.287210i	$0.0927277\pi$
$284$	16.0552	0.952701
$285$	−15.1138	−0.895266
$286$	1.88273	0.111328
$287$	0.394005	0.0232574
$288$	−1.00000	−0.0589256
$289$	−16.1138	−0.947872
$290$	16.4983	0.968812
$291$	−8.17246	−0.479078
$292$	2.00000	0.117041
$293$	−7.49484	−0.437853	−0.218927	−	0.975741i	$-0.570256\pi$
$293$	−7.49484	−0.437853	−0.218927	+	0.975741i	$0.570256\pi$
$294$	3.83709	0.223784
$295$	14.8793	0.866306
$296$	−8.96896	−0.521310
$297$	−0.941367	−0.0546236
$298$	−15.4948	−0.897592
$299$	1.88273	0.108881
$300$	0.0586332	0.00338519
$301$	9.88273	0.569631
$302$	−3.66119	−0.210678
$303$	4.36641	0.250844
$304$	6.71982	0.385408
$305$	−11.3776	−0.651478
$306$	−0.941367	−0.0538144
$307$	24.6448	1.40655	0.703275	−	0.710917i	$-0.251720\pi$
$307$	24.6448	1.40655	0.703275	+	0.710917i	$0.251720\pi$
$308$	1.67418	0.0953953
$309$	7.80605	0.444071
$310$	24.4362	1.38788
$311$	22.8095	1.29341	0.646704	−	0.762741i	$-0.276147\pi$
$311$	22.8095	1.29341	0.646704	+	0.762741i	$0.276147\pi$
$312$	−2.00000	−0.113228
$313$	−2.38789	−0.134972	−0.0674858	−	0.997720i	$-0.521498\pi$
$313$	−2.38789	−0.134972	−0.0674858	+	0.997720i	$0.521498\pi$
$314$	0.415488	0.0234474
$315$	4.00000	0.225374
$316$	−5.68879	−0.320019
$317$	−25.5715	−1.43624	−0.718120	−	0.695919i	$-0.754997\pi$
$317$	−25.5715	−1.43624	−0.718120	+	0.695919i	$0.754997\pi$
$318$	5.30777	0.297645
$319$	−6.90528	−0.386621
$320$	−2.24914	−0.125731
$321$	−8.99656	−0.502139
$322$	1.67418	0.0932984
$323$	6.32582	0.351978
$324$	1.00000	0.0555556
$325$	0.117266	0.00650477
$326$	6.61555	0.366401
$327$	9.55691	0.528499
$328$	0.221543	0.0122327
$329$	3.76547	0.207597
$330$	−2.11727	−0.116552
$331$	6.85008	0.376514	0.188257	−	0.982120i	$-0.439716\pi$
$331$	6.85008	0.376514	0.188257	+	0.982120i	$0.439716\pi$
$332$	0.954357	0.0523771
$333$	8.96896	0.491496
$334$	10.5259	0.575950
$335$	−19.8759	−1.08593
$336$	−1.77846	−0.0970228
$337$	−21.4819	−1.17019	−0.585096	−	0.810964i	$-0.698943\pi$
$337$	−21.4819	−1.17019	−0.585096	+	0.810964i	$0.698943\pi$
$338$	9.00000	0.489535
$339$	13.7164	0.744972
$340$	−2.11727	−0.114825
$341$	−10.2277	−0.553859
$342$	−6.71982	−0.363366
$343$	19.2733	1.04066
$344$	5.55691	0.299609
$345$	−2.11727	−0.113990
$346$	21.2181	1.14069
$347$	6.27674	0.336953	0.168476	−	0.985706i	$-0.446115\pi$
$347$	6.27674	0.336953	0.168476	+	0.985706i	$0.446115\pi$
$348$	7.33537	0.393217
$349$	8.58107	0.459334	0.229667	−	0.973269i	$-0.426236\pi$
$349$	8.58107	0.459334	0.229667	+	0.973269i	$0.426236\pi$
$350$	0.104277	0.00557382
$351$	2.00000	0.106752
$352$	0.941367	0.0501750
$353$	27.1560	1.44537	0.722685	−	0.691178i	$-0.242908\pi$
$353$	27.1560	1.44537	0.722685	+	0.691178i	$0.242908\pi$
$354$	6.61555	0.351612
$355$	−36.1104	−1.91654
$356$	−10.3940	−0.550881
$357$	−1.67418	−0.0886070
$358$	8.05520	0.425730
$359$	3.50172	0.184814	0.0924068	−	0.995721i	$-0.470544\pi$
$359$	3.50172	0.184814	0.0924068	+	0.995721i	$0.470544\pi$
$360$	2.24914	0.118540
$361$	26.1560	1.37663
$362$	21.5715	1.13377
$363$	−10.1138	−0.530838
$364$	−3.55691	−0.186433
$365$	−4.49828	−0.235451
$366$	−5.05863	−0.264419
$367$	−23.2733	−1.21486	−0.607428	−	0.794375i	$-0.707799\pi$
$367$	−23.2733	−1.21486	−0.607428	+	0.794375i	$0.707799\pi$
$368$	0.941367	0.0490721
$369$	−0.221543	−0.0115331
$370$	20.1725	1.04872
$371$	9.43965	0.490082
$372$	10.8647	0.563308
$373$	1.46113	0.0756545	0.0378273	−	0.999284i	$-0.487956\pi$
$373$	1.46113	0.0756545	0.0378273	+	0.999284i	$0.487956\pi$
$374$	0.886172	0.0458228
$375$	11.1138	0.573916
$376$	2.11727	0.109190
$377$	14.6707	0.755582
$378$	1.77846	0.0914740
$379$	22.7880	1.17054	0.585271	−	0.810838i	$-0.300988\pi$
$379$	22.7880	1.17054	0.585271	+	0.810838i	$0.300988\pi$
$380$	−15.1138	−0.775323
$381$	5.19051	0.265918
$382$	−12.4086	−0.634880
$383$	24.3043	1.24189	0.620947	−	0.783853i	$-0.286748\pi$
$383$	24.3043	1.24189	0.620947	+	0.783853i	$0.286748\pi$
$384$	−1.00000	−0.0510310
$385$	−3.76547	−0.191906
$386$	4.10428	0.208902
$387$	−5.55691	−0.282474
$388$	−8.17246	−0.414894
$389$	7.07162	0.358546	0.179273	−	0.983799i	$-0.442626\pi$
$389$	7.07162	0.358546	0.179273	+	0.983799i	$0.442626\pi$
$390$	4.49828	0.227779
$391$	0.886172	0.0448156
$392$	3.83709	0.193802
$393$	−1.00000	−0.0504433
$394$	6.60256	0.332632
$395$	12.7949	0.643781
$396$	−0.941367	−0.0473055
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−18.6854	−0.936612
$399$	−11.9509	−0.598294
$400$	0.0586332	0.00293166
$401$	−3.67418	−0.183480	−0.0917399	−	0.995783i	$-0.529243\pi$
$401$	−3.67418	−0.183480	−0.0917399	+	0.995783i	$0.529243\pi$
$402$	−8.83709	−0.440754
$403$	21.7294	1.08242
$404$	4.36641	0.217237
$405$	−2.24914	−0.111761
$406$	13.0456	0.647444
$407$	−8.44309	−0.418508
$408$	−0.941367	−0.0466046
$409$	−5.95092	−0.294254	−0.147127	−	0.989118i	$-0.547003\pi$
$409$	−5.95092	−0.294254	−0.147127	+	0.989118i	$0.547003\pi$
$410$	−0.498281	−0.0246083
$411$	12.6707	0.625002
$412$	7.80605	0.384577
$413$	11.7655	0.578941
$414$	−0.941367	−0.0462657
$415$	−2.14648	−0.105367
$416$	−2.00000	−0.0980581
$417$	0.732814	0.0358860
$418$	6.32582	0.309406
$419$	26.6639	1.30262	0.651308	−	0.758814i	$-0.274221\pi$
$419$	26.6639	1.30262	0.651308	+	0.758814i	$0.274221\pi$
$420$	4.00000	0.195180
$421$	−28.4914	−1.38859	−0.694293	−	0.719692i	$-0.744283\pi$
$421$	−28.4914	−1.38859	−0.694293	+	0.719692i	$0.744283\pi$
$422$	6.99656	0.340587
$423$	−2.11727	−0.102945
$424$	5.30777	0.257768
$425$	0.0551953	0.00267737
$426$	−16.0552	−0.777877
$427$	−8.99656	−0.435374
$428$	−8.99656	−0.434865
$429$	−1.88273	−0.0908992
$430$	−12.4983	−0.602721
$431$	36.8026	1.77272	0.886360	−	0.462997i	$-0.153226\pi$
$431$	36.8026	1.77272	0.886360	+	0.462997i	$0.153226\pi$
$432$	1.00000	0.0481125
$433$	1.26719	0.0608971	0.0304485	−	0.999536i	$-0.490306\pi$
$433$	1.26719	0.0608971	0.0304485	+	0.999536i	$0.490306\pi$
$434$	19.3224	0.927504
$435$	−16.4983	−0.791032
$436$	9.55691	0.457693
$437$	6.32582	0.302605
$438$	−2.00000	−0.0955637
$439$	−17.6543	−0.842594	−0.421297	−	0.906923i	$-0.638425\pi$
$439$	−17.6543	−0.842594	−0.421297	+	0.906923i	$0.638425\pi$
$440$	−2.11727	−0.100937
$441$	−3.83709	−0.182719
$442$	−1.88273	−0.0895525
$443$	−17.3646	−0.825016	−0.412508	−	0.910954i	$-0.635347\pi$
$443$	−17.3646	−0.825016	−0.412508	+	0.910954i	$0.635347\pi$
$444$	8.96896	0.425648
$445$	23.3776	1.10820
$446$	21.9785	1.04071
$447$	15.4948	0.732881
$448$	−1.77846	−0.0840242
$449$	−1.61899	−0.0764046	−0.0382023	−	0.999270i	$-0.512163\pi$
$449$	−1.61899	−0.0764046	−0.0382023	+	0.999270i	$0.512163\pi$
$450$	−0.0586332	−0.00276399
$451$	0.208553	0.00982038
$452$	13.7164	0.645165
$453$	3.66119	0.172018
$454$	5.07162	0.238023
$455$	8.00000	0.375046
$456$	−6.71982	−0.314685
$457$	0.449200	0.0210127	0.0105063	−	0.999945i	$-0.496656\pi$
$457$	0.449200	0.0210127	0.0105063	+	0.999945i	$0.496656\pi$
$458$	27.6888	1.29381
$459$	0.941367	0.0439392
$460$	−2.11727	−0.0987181
$461$	20.4853	0.954095	0.477048	−	0.878877i	$-0.341707\pi$
$461$	20.4853	0.954095	0.477048	+	0.878877i	$0.341707\pi$
$462$	−1.67418	−0.0778899
$463$	−25.3561	−1.17840	−0.589199	−	0.807988i	$-0.700557\pi$
$463$	−25.3561	−1.17840	−0.589199	+	0.807988i	$0.700557\pi$
$464$	7.33537	0.340536
$465$	−24.4362	−1.13320
$466$	9.95092	0.460967
$467$	−5.82754	−0.269666	−0.134833	−	0.990868i	$-0.543050\pi$
$467$	−5.82754	−0.269666	−0.134833	+	0.990868i	$0.543050\pi$
$468$	2.00000	0.0924500
$469$	−15.7164	−0.725715
$470$	−4.76203	−0.219656
$471$	−0.415488	−0.0191447
$472$	6.61555	0.304505
$473$	5.23109	0.240526
$474$	5.68879	0.261295
$475$	0.394005	0.0180782
$476$	−1.67418	−0.0767359
$477$	−5.30777	−0.243026
$478$	−18.0828	−0.827088
$479$	37.4948	1.71318	0.856592	−	0.515995i	$-0.172578\pi$
$479$	37.4948	1.71318	0.856592	+	0.515995i	$0.172578\pi$
$480$	2.24914	0.102659
$481$	17.9379	0.817899
$482$	4.82410	0.219732
$483$	−1.67418	−0.0761779
$484$	−10.1138	−0.459719
$485$	18.3810	0.834639
$486$	−1.00000	−0.0453609
$487$	−26.4914	−1.20044	−0.600220	−	0.799835i	$-0.704920\pi$
$487$	−26.4914	−1.20044	−0.600220	+	0.799835i	$0.704920\pi$
$488$	−5.05863	−0.228994
$489$	−6.61555	−0.299165
$490$	−8.63016	−0.389871
$491$	−29.1820	−1.31697	−0.658483	−	0.752596i	$-0.728802\pi$
$491$	−29.1820	−1.31697	−0.658483	+	0.752596i	$0.728802\pi$
$492$	−0.221543	−0.00998792
$493$	6.90528	0.310998
$494$	−13.4396	−0.604678
$495$	2.11727	0.0951640
$496$	10.8647	0.487839
$497$	−28.5535	−1.28080
$498$	−0.954357	−0.0427657
$499$	−9.39057	−0.420379	−0.210190	−	0.977661i	$-0.567408\pi$
$499$	−9.39057	−0.420379	−0.210190	+	0.977661i	$0.567408\pi$
$500$	11.1138	0.497026
$501$	−10.5259	−0.470262
$502$	−6.62854	−0.295846
$503$	−15.2242	−0.678814	−0.339407	−	0.940640i	$-0.610226\pi$
$503$	−15.2242	−0.678814	−0.339407	+	0.940640i	$0.610226\pi$
$504$	1.77846	0.0792188
$505$	−9.82066	−0.437014
$506$	0.886172	0.0393951
$507$	−9.00000	−0.399704
$508$	5.19051	0.230292
$509$	−22.9475	−1.01713	−0.508565	−	0.861024i	$-0.669824\pi$
$509$	−22.9475	−1.01713	−0.508565	+	0.861024i	$0.669824\pi$
$510$	2.11727	0.0937541
$511$	−3.55691	−0.157349
$512$	−1.00000	−0.0441942
$513$	6.71982	0.296687
$514$	−4.28973	−0.189212
$515$	−17.5569	−0.773650
$516$	−5.55691	−0.244630
$517$	1.99312	0.0876575
$518$	15.9509	0.700843
$519$	−21.2181	−0.931371
$520$	4.49828	0.197263
$521$	25.3484	1.11053	0.555266	−	0.831673i	$-0.312616\pi$
$521$	25.3484	1.11053	0.555266	+	0.831673i	$0.312616\pi$
$522$	−7.33537	−0.321060
$523$	−18.1173	−0.792213	−0.396106	−	0.918205i	$-0.629639\pi$
$523$	−18.1173	−0.792213	−0.396106	+	0.918205i	$0.629639\pi$
$524$	−1.00000	−0.0436852
$525$	−0.104277	−0.00455100
$526$	1.42504	0.0621347
$527$	10.2277	0.445524
$528$	−0.941367	−0.0409677
$529$	−22.1138	−0.961471
$530$	−11.9379	−0.518550
$531$	−6.61555	−0.287090
$532$	−11.9509	−0.518138
$533$	−0.443086	−0.0191922
$534$	10.3940	0.449793
$535$	20.2345	0.874815
$536$	−8.83709	−0.381704
$537$	−8.05520	−0.347607
$538$	6.80262	0.293282
$539$	3.61211	0.155585
$540$	−2.24914	−0.0967876
$541$	−43.3070	−1.86191	−0.930957	−	0.365129i	$-0.881025\pi$
$541$	−43.3070	−1.86191	−0.930957	+	0.365129i	$0.881025\pi$
$542$	−19.8466	−0.852486
$543$	−21.5715	−0.925723
$544$	−0.941367	−0.0403608
$545$	−21.4948	−0.920738
$546$	3.55691	0.152222
$547$	24.0061	1.02643	0.513214	−	0.858261i	$-0.328455\pi$
$547$	24.0061	1.02643	0.513214	+	0.858261i	$0.328455\pi$
$548$	12.6707	0.541267
$549$	5.05863	0.215897
$550$	0.0551953	0.00235354
$551$	49.2924	2.09993
$552$	−0.941367	−0.0400672
$553$	10.1173	0.430230
$554$	15.6742	0.665932
$555$	−20.1725	−0.856273
$556$	0.732814	0.0310782
$557$	9.80605	0.415496	0.207748	−	0.978182i	$-0.433387\pi$
$557$	9.80605	0.415496	0.207748	+	0.978182i	$0.433387\pi$
$558$	−10.8647	−0.459939
$559$	−11.1138	−0.470065
$560$	4.00000	0.169031
$561$	−0.886172	−0.0374142
$562$	−16.6448	−0.702117
$563$	33.9379	1.43031	0.715157	−	0.698964i	$-0.246355\pi$
$563$	33.9379	1.43031	0.715157	+	0.698964i	$0.246355\pi$
$564$	−2.11727	−0.0891530
$565$	−30.8501	−1.29787
$566$	−32.2277	−1.35463
$567$	−1.77846	−0.0746882
$568$	−16.0552	−0.673661
$569$	−15.3354	−0.642892	−0.321446	−	0.946928i	$-0.604169\pi$
$569$	−15.3354	−0.642892	−0.321446	+	0.946928i	$0.604169\pi$
$570$	15.1138	0.633049
$571$	22.2767	0.932252	0.466126	−	0.884718i	$-0.345649\pi$
$571$	22.2767	0.932252	0.466126	+	0.884718i	$0.345649\pi$
$572$	−1.88273	−0.0787210
$573$	12.4086	0.518377
$574$	−0.394005	−0.0164454
$575$	0.0551953	0.00230180
$576$	1.00000	0.0416667
$577$	−0.338809	−0.0141048	−0.00705241	−	0.999975i	$-0.502245\pi$
$577$	−0.338809	−0.0141048	−0.00705241	+	0.999975i	$0.502245\pi$
$578$	16.1138	0.670247
$579$	−4.10428	−0.170568
$580$	−16.4983	−0.685054
$581$	−1.69728	−0.0704151
$582$	8.17246	0.338759
$583$	4.99656	0.206936
$584$	−2.00000	−0.0827606
$585$	−4.49828	−0.185981
$586$	7.49484	0.309609
$587$	−30.6967	−1.26699	−0.633495	−	0.773747i	$-0.718380\pi$
$587$	−30.6967	−1.26699	−0.633495	+	0.773747i	$0.718380\pi$
$588$	−3.83709	−0.158239
$589$	73.0088	3.00827
$590$	−14.8793	−0.612571
$591$	−6.60256	−0.271593
$592$	8.96896	0.368622
$593$	33.6673	1.38255	0.691275	−	0.722592i	$-0.257049\pi$
$593$	33.6673	1.38255	0.691275	+	0.722592i	$0.257049\pi$
$594$	0.941367	0.0386247
$595$	3.76547	0.154369
$596$	15.4948	0.634694
$597$	18.6854	0.764740
$598$	−1.88273	−0.0769907
$599$	−28.9881	−1.18442	−0.592210	−	0.805784i	$-0.701744\pi$
$599$	−28.9881	−1.18442	−0.592210	+	0.805784i	$0.701744\pi$
$600$	−0.0586332	−0.00239369
$601$	30.3060	1.23621	0.618103	−	0.786097i	$-0.287902\pi$
$601$	30.3060	1.23621	0.618103	+	0.786097i	$0.287902\pi$
$602$	−9.88273	−0.402790
$603$	8.83709	0.359874
$604$	3.66119	0.148972
$605$	22.7474	0.924814
$606$	−4.36641	−0.177373
$607$	29.2749	1.18823	0.594116	−	0.804379i	$-0.297502\pi$
$607$	29.2749	1.18823	0.594116	+	0.804379i	$0.297502\pi$
$608$	−6.71982	−0.272525
$609$	−13.0456	−0.528636
$610$	11.3776	0.460665
$611$	−4.23453	−0.171311
$612$	0.941367	0.0380525
$613$	−10.0812	−0.407175	−0.203587	−	0.979057i	$-0.565260\pi$
$613$	−10.0812	−0.407175	−0.203587	+	0.979057i	$0.565260\pi$
$614$	−24.6448	−0.994582
$615$	0.498281	0.0200926
$616$	−1.67418	−0.0674547
$617$	17.9018	0.720701	0.360350	−	0.932817i	$-0.382657\pi$
$617$	17.9018	0.720701	0.360350	+	0.932817i	$0.382657\pi$
$618$	−7.80605	−0.314006
$619$	−21.3354	−0.857541	−0.428770	−	0.903413i	$-0.641053\pi$
$619$	−21.3354	−0.857541	−0.428770	+	0.903413i	$0.641053\pi$
$620$	−24.4362	−0.981382
$621$	0.941367	0.0377758
$622$	−22.8095	−0.914577
$623$	18.4853	0.740598
$624$	2.00000	0.0800641
$625$	−25.2897	−1.01159
$626$	2.38789	0.0954393
$627$	−6.32582	−0.252629
$628$	−0.415488	−0.0165798
$629$	8.44309	0.336648
$630$	−4.00000	−0.159364
$631$	−7.89572	−0.314324	−0.157162	−	0.987573i	$-0.550234\pi$
$631$	−7.89572	−0.314324	−0.157162	+	0.987573i	$0.550234\pi$
$632$	5.68879	0.226288
$633$	−6.99656	−0.278088
$634$	25.5715	1.01557
$635$	−11.6742	−0.463276
$636$	−5.30777	−0.210467
$637$	−7.67418	−0.304062
$638$	6.90528	0.273382
$639$	16.0552	0.635134
$640$	2.24914	0.0889051
$641$	−18.2147	−0.719436	−0.359718	−	0.933061i	$-0.617127\pi$
$641$	−18.2147	−0.719436	−0.359718	+	0.933061i	$0.617127\pi$
$642$	8.99656	0.355066
$643$	−8.02922	−0.316641	−0.158321	−	0.987388i	$-0.550608\pi$
$643$	−8.02922	−0.316641	−0.158321	+	0.987388i	$0.550608\pi$
$644$	−1.67418	−0.0659720
$645$	12.4983	0.492119
$646$	−6.32582	−0.248886
$647$	16.6983	0.656479	0.328240	−	0.944594i	$-0.393545\pi$
$647$	16.6983	0.656479	0.328240	+	0.944594i	$0.393545\pi$
$648$	−1.00000	−0.0392837
$649$	6.22766	0.244457
$650$	−0.117266	−0.00459956
$651$	−19.3224	−0.757304
$652$	−6.61555	−0.259085
$653$	−34.1510	−1.33643	−0.668216	−	0.743968i	$-0.732942\pi$
$653$	−34.1510	−1.33643	−0.668216	+	0.743968i	$0.732942\pi$
$654$	−9.55691	−0.373705
$655$	2.24914	0.0878812
$656$	−0.221543	−0.00864980
$657$	2.00000	0.0780274
$658$	−3.76547	−0.146793
$659$	−38.2017	−1.48813	−0.744063	−	0.668109i	$-0.767104\pi$
$659$	−38.2017	−1.48813	−0.744063	+	0.668109i	$0.767104\pi$
$660$	2.11727	0.0824145
$661$	39.3760	1.53155	0.765774	−	0.643110i	$-0.222356\pi$
$661$	39.3760	1.53155	0.765774	+	0.643110i	$0.222356\pi$
$662$	−6.85008	−0.266236
$663$	1.88273	0.0731193
$664$	−0.954357	−0.0370362
$665$	26.8793	1.04233
$666$	−8.96896	−0.347540
$667$	6.90528	0.267373
$668$	−10.5259	−0.407258
$669$	−21.9785	−0.849739
$670$	19.8759	0.767871
$671$	−4.76203	−0.183836
$672$	1.77846	0.0686055
$673$	−18.7068	−0.721095	−0.360548	−	0.932741i	$-0.617410\pi$
$673$	−18.7068	−0.721095	−0.360548	+	0.932741i	$0.617410\pi$
$674$	21.4819	0.827450
$675$	0.0586332	0.00225679
$676$	−9.00000	−0.346154
$677$	−39.3354	−1.51178	−0.755891	−	0.654698i	$-0.772796\pi$
$677$	−39.3354	−1.51178	−0.755891	+	0.654698i	$0.772796\pi$
$678$	−13.7164	−0.526775
$679$	14.5344	0.557778
$680$	2.11727	0.0811935
$681$	−5.07162	−0.194345
$682$	10.2277	0.391637
$683$	−6.67074	−0.255249	−0.127624	−	0.991823i	$-0.540735\pi$
$683$	−6.67074	−0.255249	−0.127624	+	0.991823i	$0.540735\pi$
$684$	6.71982	0.256939
$685$	−28.4983	−1.08886
$686$	−19.2733	−0.735858
$687$	−27.6888	−1.05639
$688$	−5.55691	−0.211855
$689$	−10.6155	−0.404420
$690$	2.11727	0.0806030
$691$	−17.8759	−0.680030	−0.340015	−	0.940420i	$-0.610432\pi$
$691$	−17.8759	−0.680030	−0.340015	+	0.940420i	$0.610432\pi$
$692$	−21.2181	−0.806591
$693$	1.67418	0.0635969
$694$	−6.27674	−0.238262
$695$	−1.64820	−0.0625198
$696$	−7.33537	−0.278047
$697$	−0.208553	−0.00789951
$698$	−8.58107	−0.324798
$699$	−9.95092	−0.376378
$700$	−0.104277	−0.00394128
$701$	13.9594	0.527240	0.263620	−	0.964627i	$-0.415084\pi$
$701$	13.9594	0.527240	0.263620	+	0.964627i	$0.415084\pi$
$702$	−2.00000	−0.0754851
$703$	60.2699	2.27312
$704$	−0.941367	−0.0354791
$705$	4.76203	0.179348
$706$	−27.1560	−1.02203
$707$	−7.76547	−0.292050
$708$	−6.61555	−0.248627
$709$	26.1510	0.982121	0.491060	−	0.871126i	$-0.336609\pi$
$709$	26.1510	0.982121	0.491060	+	0.871126i	$0.336609\pi$
$710$	36.1104	1.35520
$711$	−5.68879	−0.213346
$712$	10.3940	0.389532
$713$	10.2277	0.383029
$714$	1.67418	0.0626546
$715$	4.23453	0.158363
$716$	−8.05520	−0.301037
$717$	18.0828	0.675314
$718$	−3.50172	−0.130683
$719$	14.5811	0.543782	0.271891	−	0.962328i	$-0.412351\pi$
$719$	14.5811	0.543782	0.271891	+	0.962328i	$0.412351\pi$
$720$	−2.24914	−0.0838205
$721$	−13.8827	−0.517020
$722$	−26.1560	−0.973427
$723$	−4.82410	−0.179410
$724$	−21.5715	−0.801699
$725$	0.430096	0.0159734
$726$	10.1138	0.375359
$727$	13.1905	0.489209	0.244604	−	0.969623i	$-0.421342\pi$
$727$	13.1905	0.489209	0.244604	+	0.969623i	$0.421342\pi$
$728$	3.55691	0.131828
$729$	1.00000	0.0370370
$730$	4.49828	0.166489
$731$	−5.23109	−0.193479
$732$	5.05863	0.186972
$733$	1.96553	0.0725984	0.0362992	−	0.999341i	$-0.488443\pi$
$733$	1.96553	0.0725984	0.0362992	+	0.999341i	$0.488443\pi$
$734$	23.2733	0.859033
$735$	8.63016	0.318328
$736$	−0.941367	−0.0346992
$737$	−8.31894	−0.306432
$738$	0.221543	0.00815511
$739$	17.9018	0.658530	0.329265	−	0.944238i	$-0.393199\pi$
$739$	17.9018	0.658530	0.329265	+	0.944238i	$0.393199\pi$
$740$	−20.1725	−0.741554
$741$	13.4396	0.493718
$742$	−9.43965	−0.346540
$743$	−14.4622	−0.530566	−0.265283	−	0.964171i	$-0.585465\pi$
$743$	−14.4622	−0.530566	−0.265283	+	0.964171i	$0.585465\pi$
$744$	−10.8647	−0.398319
$745$	−34.8501	−1.27681
$746$	−1.46113	−0.0534958
$747$	0.954357	0.0349181
$748$	−0.886172	−0.0324016
$749$	16.0000	0.584627
$750$	−11.1138	−0.405820
$751$	−23.1836	−0.845983	−0.422991	−	0.906134i	$-0.639020\pi$
$751$	−23.1836	−0.845983	−0.422991	+	0.906134i	$0.639020\pi$
$752$	−2.11727	−0.0772088
$753$	6.62854	0.241557
$754$	−14.6707	−0.534277
$755$	−8.23453	−0.299685
$756$	−1.77846	−0.0646819
$757$	−39.3155	−1.42895	−0.714473	−	0.699663i	$-0.753334\pi$
$757$	−39.3155	−1.42895	−0.714473	+	0.699663i	$0.753334\pi$
$758$	−22.7880	−0.827698
$759$	−0.886172	−0.0321660
$760$	15.1138	0.548236
$761$	10.2966	0.373252	0.186626	−	0.982431i	$-0.440245\pi$
$761$	10.2966	0.373252	0.186626	+	0.982431i	$0.440245\pi$
$762$	−5.19051	−0.188032
$763$	−16.9966	−0.615317
$764$	12.4086	0.448928
$765$	−2.11727	−0.0765499
$766$	−24.3043	−0.878151
$767$	−13.2311	−0.477747
$768$	1.00000	0.0360844
$769$	11.7233	0.422752	0.211376	−	0.977405i	$-0.432206\pi$
$769$	11.7233	0.422752	0.211376	+	0.977405i	$0.432206\pi$
$770$	3.76547	0.135698
$771$	4.28973	0.154491
$772$	−4.10428	−0.147716
$773$	23.5147	0.845765	0.422883	−	0.906184i	$-0.361018\pi$
$773$	23.5147	0.845765	0.422883	+	0.906184i	$0.361018\pi$
$774$	5.55691	0.199739
$775$	0.637031	0.0228828
$776$	8.17246	0.293374
$777$	−15.9509	−0.572236
$778$	−7.07162	−0.253530
$779$	−1.48873	−0.0533393
$780$	−4.49828	−0.161064
$781$	−15.1138	−0.540815
$782$	−0.886172	−0.0316894
$783$	7.33537	0.262145
$784$	−3.83709	−0.137039
$785$	0.934491	0.0333534
$786$	1.00000	0.0356688
$787$	53.1070	1.89306	0.946529	−	0.322618i	$-0.104563\pi$
$787$	53.1070	1.89306	0.946529	+	0.322618i	$0.104563\pi$
$788$	−6.60256	−0.235206
$789$	−1.42504	−0.0507328
$790$	−12.7949	−0.455222
$791$	−24.3940	−0.867351
$792$	0.941367	0.0334500
$793$	10.1173	0.359275
$794$	22.0000	0.780751
$795$	11.9379	0.423395
$796$	18.6854	0.662285
$797$	−26.3303	−0.932668	−0.466334	−	0.884609i	$-0.654425\pi$
$797$	−26.3303	−0.932668	−0.466334	+	0.884609i	$0.654425\pi$
$798$	11.9509	0.423058
$799$	−1.99312	−0.0705117
$800$	−0.0586332	−0.00207300
$801$	−10.3940	−0.367254
$802$	3.67418	0.129740
$803$	−1.88273	−0.0664402
$804$	8.83709	0.311660
$805$	3.76547	0.132715
$806$	−21.7294	−0.765385
$807$	−6.80262	−0.239463
$808$	−4.36641	−0.153610
$809$	−26.9085	−0.946053	−0.473026	−	0.881048i	$-0.656838\pi$
$809$	−26.9085	−0.946053	−0.473026	+	0.881048i	$0.656838\pi$
$810$	2.24914	0.0790267
$811$	−12.5795	−0.441724	−0.220862	−	0.975305i	$-0.570887\pi$
$811$	−12.5795	−0.441724	−0.220862	+	0.975305i	$0.570887\pi$
$812$	−13.0456	−0.457812
$813$	19.8466	0.696052
$814$	8.44309	0.295930
$815$	14.8793	0.521199
$816$	0.941367	0.0329544
$817$	−37.3415	−1.30641
$818$	5.95092	0.208069
$819$	−3.55691	−0.124289
$820$	0.498281	0.0174007
$821$	−19.8544	−0.692922	−0.346461	−	0.938064i	$-0.612617\pi$
$821$	−19.8544	−0.692922	−0.346461	+	0.938064i	$0.612617\pi$
$822$	−12.6707	−0.441943
$823$	27.8061	0.969258	0.484629	−	0.874720i	$-0.338955\pi$
$823$	27.8061	0.969258	0.484629	+	0.874720i	$0.338955\pi$
$824$	−7.80605	−0.271937
$825$	−0.0551953	−0.00192165
$826$	−11.7655	−0.409373
$827$	−26.3518	−0.916342	−0.458171	−	0.888864i	$-0.651495\pi$
$827$	−26.3518	−0.916342	−0.458171	+	0.888864i	$0.651495\pi$
$828$	0.941367	0.0327148
$829$	54.8432	1.90478	0.952392	−	0.304877i	$-0.0986155\pi$
$829$	54.8432	1.90478	0.952392	+	0.304877i	$0.0986155\pi$
$830$	2.14648	0.0745055
$831$	−15.6742	−0.543731
$832$	2.00000	0.0693375
$833$	−3.61211	−0.125152
$834$	−0.732814	−0.0253753
$835$	23.6742	0.819279
$836$	−6.32582	−0.218783
$837$	10.8647	0.375539
$838$	−26.6639	−0.921088
$839$	−7.91033	−0.273095	−0.136547	−	0.990634i	$-0.543601\pi$
$839$	−7.91033	−0.273095	−0.136547	+	0.990634i	$0.543601\pi$
$840$	−4.00000	−0.138013
$841$	24.8077	0.855437
$842$	28.4914	0.981879
$843$	16.6448	0.573276
$844$	−6.99656	−0.240832
$845$	20.2423	0.696355
$846$	2.11727	0.0727931
$847$	17.9870	0.618041
$848$	−5.30777	−0.182270
$849$	32.2277	1.10605
$850$	−0.0551953	−0.00189318
$851$	8.44309	0.289425
$852$	16.0552	0.550042
$853$	35.1475	1.20343	0.601714	−	0.798711i	$-0.294485\pi$
$853$	35.1475	1.20343	0.601714	+	0.798711i	$0.294485\pi$
$854$	8.99656	0.307856
$855$	−15.1138	−0.516882
$856$	8.99656	0.307496
$857$	−24.8793	−0.849861	−0.424930	−	0.905226i	$-0.639701\pi$
$857$	−24.8793	−0.849861	−0.424930	+	0.905226i	$0.639701\pi$
$858$	1.88273	0.0642755
$859$	−0.781895	−0.0266779	−0.0133390	−	0.999911i	$-0.504246\pi$
$859$	−0.781895	−0.0266779	−0.0133390	+	0.999911i	$0.504246\pi$
$860$	12.4983	0.426188
$861$	0.394005	0.0134276
$862$	−36.8026	−1.25350
$863$	−2.29135	−0.0779983	−0.0389992	−	0.999239i	$-0.512417\pi$
$863$	−2.29135	−0.0779983	−0.0389992	+	0.999239i	$0.512417\pi$
$864$	−1.00000	−0.0340207
$865$	47.7225	1.62261
$866$	−1.26719	−0.0430607
$867$	−16.1138	−0.547254
$868$	−19.3224	−0.655844
$869$	5.35524	0.181664
$870$	16.4983	0.559344
$871$	17.6742	0.598867
$872$	−9.55691	−0.323638
$873$	−8.17246	−0.276596
$874$	−6.32582	−0.213974
$875$	−19.7655	−0.668195
$876$	2.00000	0.0675737
$877$	35.0518	1.18361	0.591807	−	0.806080i	$-0.298415\pi$
$877$	35.0518	1.18361	0.591807	+	0.806080i	$0.298415\pi$
$878$	17.6543	0.595804
$879$	−7.49484	−0.252795
$880$	2.11727	0.0713730
$881$	43.7225	1.47305	0.736524	−	0.676411i	$-0.236466\pi$
$881$	43.7225	1.47305	0.736524	+	0.676411i	$0.236466\pi$
$882$	3.83709	0.129202
$883$	−20.0613	−0.675117	−0.337558	−	0.941305i	$-0.609601\pi$
$883$	−20.0613	−0.675117	−0.337558	+	0.941305i	$0.609601\pi$
$884$	1.88273	0.0633232
$885$	14.8793	0.500162
$886$	17.3646	0.583375
$887$	−49.7992	−1.67209	−0.836046	−	0.548659i	$-0.815139\pi$
$887$	−49.7992	−1.67209	−0.836046	+	0.548659i	$0.815139\pi$
$888$	−8.96896	−0.300979
$889$	−9.23109	−0.309601
$890$	−23.3776	−0.783618
$891$	−0.941367	−0.0315370
$892$	−21.9785	−0.735895
$893$	−14.2277	−0.476110
$894$	−15.4948	−0.518225
$895$	18.1173	0.605593
$896$	1.77846	0.0594141
$897$	1.88273	0.0628626
$898$	1.61899	0.0540262
$899$	79.6965	2.65803
$900$	0.0586332	0.00195444
$901$	−4.99656	−0.166460
$902$	−0.208553	−0.00694406
$903$	9.88273	0.328877
$904$	−13.7164	−0.456200
$905$	48.5174	1.61277
$906$	−3.66119	−0.121635
$907$	−51.2242	−1.70087	−0.850436	−	0.526078i	$-0.823662\pi$
$907$	−51.2242	−1.70087	−0.850436	+	0.526078i	$0.823662\pi$
$908$	−5.07162	−0.168308
$909$	4.36641	0.144825
$910$	−8.00000	−0.265197
$911$	−27.8551	−0.922882	−0.461441	−	0.887171i	$-0.652667\pi$
$911$	−27.8551	−0.922882	−0.461441	+	0.887171i	$0.652667\pi$
$912$	6.71982	0.222516
$913$	−0.898400	−0.0297327
$914$	−0.449200	−0.0148582
$915$	−11.3776	−0.376131
$916$	−27.6888	−0.914863
$917$	1.77846	0.0587298
$918$	−0.941367	−0.0310697
$919$	26.1319	0.862011	0.431005	−	0.902349i	$-0.358159\pi$
$919$	26.1319	0.862011	0.431005	+	0.902349i	$0.358159\pi$
$920$	2.11727	0.0698042
$921$	24.6448	0.812073
$922$	−20.4853	−0.674647
$923$	32.1104	1.05693
$924$	1.67418	0.0550765
$925$	0.525879	0.0172908
$926$	25.3561	0.833253
$927$	7.80605	0.256384
$928$	−7.33537	−0.240795
$929$	−50.8923	−1.66972	−0.834861	−	0.550461i	$-0.814452\pi$
$929$	−50.8923	−1.66972	−0.834861	+	0.550461i	$0.814452\pi$
$930$	24.4362	0.801295
$931$	−25.7846	−0.845055
$932$	−9.95092	−0.325953
$933$	22.8095	0.746749
$934$	5.82754	0.190683
$935$	1.99312	0.0651821
$936$	−2.00000	−0.0653720
$937$	14.1855	0.463418	0.231709	−	0.972785i	$-0.425568\pi$
$937$	14.1855	0.463418	0.231709	+	0.972785i	$0.425568\pi$
$938$	15.7164	0.513158
$939$	−2.38789	−0.0779259
$940$	4.76203	0.155320
$941$	24.3027	0.792246	0.396123	−	0.918197i	$-0.370355\pi$
$941$	24.3027	0.792246	0.396123	+	0.918197i	$0.370355\pi$
$942$	0.415488	0.0135373
$943$	−0.208553	−0.00679142
$944$	−6.61555	−0.215318
$945$	4.00000	0.130120
$946$	−5.23109	−0.170078
$947$	−26.1595	−0.850069	−0.425034	−	0.905177i	$-0.639738\pi$
$947$	−26.1595	−0.850069	−0.425034	+	0.905177i	$0.639738\pi$
$948$	−5.68879	−0.184763
$949$	4.00000	0.129845
$950$	−0.394005	−0.0127832
$951$	−25.5715	−0.829213
$952$	1.67418	0.0542605
$953$	13.1921	0.427335	0.213667	−	0.976906i	$-0.431459\pi$
$953$	13.1921	0.427335	0.213667	+	0.976906i	$0.431459\pi$
$954$	5.30777	0.171846
$955$	−27.9087	−0.903105
$956$	18.0828	0.584839
$957$	−6.90528	−0.223216
$958$	−37.4948	−1.21140
$959$	−22.5344	−0.727673
$960$	−2.24914	−0.0725907
$961$	87.0414	2.80779
$962$	−17.9379	−0.578342
$963$	−8.99656	−0.289910
$964$	−4.82410	−0.155374
$965$	9.23109	0.297159
$966$	1.67418	0.0538659
$967$	−22.7113	−0.730347	−0.365174	−	0.930939i	$-0.618990\pi$
$967$	−22.7113	−0.730347	−0.365174	+	0.930939i	$0.618990\pi$
$968$	10.1138	0.325071
$969$	6.32582	0.203215
$970$	−18.3810	−0.590179
$971$	33.2664	1.06757	0.533785	−	0.845620i	$-0.320769\pi$
$971$	33.2664	1.06757	0.533785	+	0.845620i	$0.320769\pi$
$972$	1.00000	0.0320750
$973$	−1.30328	−0.0417812
$974$	26.4914	0.848839
$975$	0.117266	0.00375553
$976$	5.05863	0.161923
$977$	15.2112	0.486650	0.243325	−	0.969945i	$-0.421762\pi$
$977$	15.2112	0.486650	0.243325	+	0.969945i	$0.421762\pi$
$978$	6.61555	0.211542
$979$	9.78457	0.312716
$980$	8.63016	0.275680
$981$	9.55691	0.305129
$982$	29.1820	0.931235
$983$	18.0913	0.577022	0.288511	−	0.957477i	$-0.406840\pi$
$983$	18.0913	0.577022	0.288511	+	0.957477i	$0.406840\pi$
$984$	0.221543	0.00706253
$985$	14.8501	0.473163
$986$	−6.90528	−0.219909
$987$	3.76547	0.119856
$988$	13.4396	0.427572
$989$	−5.23109	−0.166339
$990$	−2.11727	−0.0672911
$991$	−28.1104	−0.892956	−0.446478	−	0.894795i	$-0.647322\pi$
$991$	−28.1104	−0.892956	−0.446478	+	0.894795i	$0.647322\pi$
$992$	−10.8647	−0.344954
$993$	6.85008	0.217381
$994$	28.5535	0.905661
$995$	−42.0260	−1.33231
$996$	0.954357	0.0302399
$997$	−8.43621	−0.267177	−0.133589	−	0.991037i	$-0.542650\pi$
$997$	−8.43621	−0.267177	−0.133589	+	0.991037i	$0.542650\pi$
$998$	9.39057	0.297253
$999$	8.96896	0.283765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	786.2.a.n.1.1	✓	3
3.2	odd	2		2358.2.a.be.1.3		3
4.3	odd	2		6288.2.a.w.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
786.2.a.n.1.1	✓	3	1.1	even	1	trivial
2358.2.a.be.1.3		3	3.2	odd	2
6288.2.a.w.1.1		3	4.3	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:	\( N \)	\(=\)	\( 786 = 2 \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	786.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24914 q^{5} -1.00000 q^{6} -1.77846 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.24914 q^{5} -1.00000 q^{6} -1.77846 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.24914 q^{10} -0.941367 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.77846 q^{14} -2.24914 q^{15} +1.00000 q^{16} +0.941367 q^{17} -1.00000 q^{18} +6.71982 q^{19} -2.24914 q^{20} -1.77846 q^{21} +0.941367 q^{22} +0.941367 q^{23} -1.00000 q^{24} +0.0586332 q^{25} -2.00000 q^{26} +1.00000 q^{27} -1.77846 q^{28} +7.33537 q^{29} +2.24914 q^{30} +10.8647 q^{31} -1.00000 q^{32} -0.941367 q^{33} -0.941367 q^{34} +4.00000 q^{35} +1.00000 q^{36} +8.96896 q^{37} -6.71982 q^{38} +2.00000 q^{39} +2.24914 q^{40} -0.221543 q^{41} +1.77846 q^{42} -5.55691 q^{43} -0.941367 q^{44} -2.24914 q^{45} -0.941367 q^{46} -2.11727 q^{47} +1.00000 q^{48} -3.83709 q^{49} -0.0586332 q^{50} +0.941367 q^{51} +2.00000 q^{52} -5.30777 q^{53} -1.00000 q^{54} +2.11727 q^{55} +1.77846 q^{56} +6.71982 q^{57} -7.33537 q^{58} -6.61555 q^{59} -2.24914 q^{60} +5.05863 q^{61} -10.8647 q^{62} -1.77846 q^{63} +1.00000 q^{64} -4.49828 q^{65} +0.941367 q^{66} +8.83709 q^{67} +0.941367 q^{68} +0.941367 q^{69} -4.00000 q^{70} +16.0552 q^{71} -1.00000 q^{72} +2.00000 q^{73} -8.96896 q^{74} +0.0586332 q^{75} +6.71982 q^{76} +1.67418 q^{77} -2.00000 q^{78} -5.68879 q^{79} -2.24914 q^{80} +1.00000 q^{81} +0.221543 q^{82} +0.954357 q^{83} -1.77846 q^{84} -2.11727 q^{85} +5.55691 q^{86} +7.33537 q^{87} +0.941367 q^{88} -10.3940 q^{89} +2.24914 q^{90} -3.55691 q^{91} +0.941367 q^{92} +10.8647 q^{93} +2.11727 q^{94} -15.1138 q^{95} -1.00000 q^{96} -8.17246 q^{97} +3.83709 q^{98} -0.941367 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 3 q^{8} + 3 q^{9} - 2 q^{10} - 2 q^{11} + 3 q^{12} + 6 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} + 2 q^{17} - 3 q^{18} + 11 q^{19} + 2 q^{20} + 3 q^{21} + 2 q^{22} + 2 q^{23} - 3 q^{24} + q^{25} - 6 q^{26} + 3 q^{27} + 3 q^{28} - 3 q^{29} - 2 q^{30} + 8 q^{31} - 3 q^{32} - 2 q^{33} - 2 q^{34} + 12 q^{35} + 3 q^{36} + 9 q^{37} - 11 q^{38} + 6 q^{39} - 2 q^{40} - 9 q^{41} - 3 q^{42} - 2 q^{44} + 2 q^{45} - 2 q^{46} - 8 q^{47} + 3 q^{48} - 4 q^{49} - q^{50} + 2 q^{51} + 6 q^{52} - 8 q^{53} - 3 q^{54} + 8 q^{55} - 3 q^{56} + 11 q^{57} + 3 q^{58} - 4 q^{59} + 2 q^{60} + 16 q^{61} - 8 q^{62} + 3 q^{63} + 3 q^{64} + 4 q^{65} + 2 q^{66} + 19 q^{67} + 2 q^{68} + 2 q^{69} - 12 q^{70} + 14 q^{71} - 3 q^{72} + 6 q^{73} - 9 q^{74} + q^{75} + 11 q^{76} - 10 q^{77} - 6 q^{78} + 10 q^{79} + 2 q^{80} + 3 q^{81} + 9 q^{82} - 3 q^{83} + 3 q^{84} - 8 q^{85} - 3 q^{87} + 2 q^{88} - 7 q^{89} - 2 q^{90} + 6 q^{91} + 2 q^{92} + 8 q^{93} + 8 q^{94} - 12 q^{95} - 3 q^{96} + 8 q^{97} + 4 q^{98} - 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 3 * q^8 + 3 * q^9 - 2 * q^10 - 2 * q^11 + 3 * q^12 + 6 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 + 2 * q^17 - 3 * q^18 + 11 * q^19 + 2 * q^20 + 3 * q^21 + 2 * q^22 + 2 * q^23 - 3 * q^24 + q^25 - 6 * q^26 + 3 * q^27 + 3 * q^28 - 3 * q^29 - 2 * q^30 + 8 * q^31 - 3 * q^32 - 2 * q^33 - 2 * q^34 + 12 * q^35 + 3 * q^36 + 9 * q^37 - 11 * q^38 + 6 * q^39 - 2 * q^40 - 9 * q^41 - 3 * q^42 - 2 * q^44 + 2 * q^45 - 2 * q^46 - 8 * q^47 + 3 * q^48 - 4 * q^49 - q^50 + 2 * q^51 + 6 * q^52 - 8 * q^53 - 3 * q^54 + 8 * q^55 - 3 * q^56 + 11 * q^57 + 3 * q^58 - 4 * q^59 + 2 * q^60 + 16 * q^61 - 8 * q^62 + 3 * q^63 + 3 * q^64 + 4 * q^65 + 2 * q^66 + 19 * q^67 + 2 * q^68 + 2 * q^69 - 12 * q^70 + 14 * q^71 - 3 * q^72 + 6 * q^73 - 9 * q^74 + q^75 + 11 * q^76 - 10 * q^77 - 6 * q^78 + 10 * q^79 + 2 * q^80 + 3 * q^81 + 9 * q^82 - 3 * q^83 + 3 * q^84 - 8 * q^85 - 3 * q^87 + 2 * q^88 - 7 * q^89 - 2 * q^90 + 6 * q^91 + 2 * q^92 + 8 * q^93 + 8 * q^94 - 12 * q^95 - 3 * q^96 + 8 * q^97 + 4 * q^98 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more