LMFDB - Embedded newform 7098.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7098,2,Mod(1,7098)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7098, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7098.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7098.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:	$56.6778153547$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	7098.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -1.43845 q^{11} -1.00000 q^{12} +1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} -5.68466 q^{17} +1.00000 q^{18} +2.56155 q^{19} -0.561553 q^{20} -1.00000 q^{21} -1.43845 q^{22} -5.68466 q^{23} -1.00000 q^{24} -4.68466 q^{25} -1.00000 q^{27} +1.00000 q^{28} -2.56155 q^{29} +0.561553 q^{30} +10.2462 q^{31} +1.00000 q^{32} +1.43845 q^{33} -5.68466 q^{34} -0.561553 q^{35} +1.00000 q^{36} +1.68466 q^{37} +2.56155 q^{38} -0.561553 q^{40} +4.00000 q^{41} -1.00000 q^{42} +10.5616 q^{43} -1.43845 q^{44} -0.561553 q^{45} -5.68466 q^{46} -6.24621 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.68466 q^{50} +5.68466 q^{51} +13.1231 q^{53} -1.00000 q^{54} +0.807764 q^{55} +1.00000 q^{56} -2.56155 q^{57} -2.56155 q^{58} +12.2462 q^{59} +0.561553 q^{60} +2.56155 q^{61} +10.2462 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.43845 q^{66} +7.12311 q^{67} -5.68466 q^{68} +5.68466 q^{69} -0.561553 q^{70} -15.3693 q^{71} +1.00000 q^{72} -7.43845 q^{73} +1.68466 q^{74} +4.68466 q^{75} +2.56155 q^{76} -1.43845 q^{77} +16.0000 q^{79} -0.561553 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{87} -1.43845 q^{88} +8.00000 q^{89} -0.561553 q^{90} -5.68466 q^{92} -10.2462 q^{93} -6.24621 q^{94} -1.43845 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -1.43845 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} - 7 q^{11} - 2 q^{12} + 2 q^{14} - 3 q^{15} + 2 q^{16} + q^{17} + 2 q^{18} + q^{19} + 3 q^{20} - 2 q^{21} - 7 q^{22} + q^{23} - 2 q^{24} + 3 q^{25} - 2 q^{27} + 2 q^{28} - q^{29} - 3 q^{30} + 4 q^{31} + 2 q^{32} + 7 q^{33} + q^{34} + 3 q^{35} + 2 q^{36} - 9 q^{37} + q^{38} + 3 q^{40} + 8 q^{41} - 2 q^{42} + 17 q^{43} - 7 q^{44} + 3 q^{45} + q^{46} + 4 q^{47} - 2 q^{48} + 2 q^{49} + 3 q^{50} - q^{51} + 18 q^{53} - 2 q^{54} - 19 q^{55} + 2 q^{56} - q^{57} - q^{58} + 8 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 7 q^{66} + 6 q^{67} + q^{68} - q^{69} + 3 q^{70} - 6 q^{71} + 2 q^{72} - 19 q^{73} - 9 q^{74} - 3 q^{75} + q^{76} - 7 q^{77} + 32 q^{79} + 3 q^{80} + 2 q^{81} + 8 q^{82} - 4 q^{83} - 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} - 7 q^{88} + 16 q^{89} + 3 q^{90} + q^{92} - 4 q^{93} + 4 q^{94} - 7 q^{95} - 2 q^{96} + 20 q^{97} + 2 q^{98} - 7 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^10 - 7 * q^11 - 2 * q^12 + 2 * q^14 - 3 * q^15 + 2 * q^16 + q^17 + 2 * q^18 + q^19 + 3 * q^20 - 2 * q^21 - 7 * q^22 + q^23 - 2 * q^24 + 3 * q^25 - 2 * q^27 + 2 * q^28 - q^29 - 3 * q^30 + 4 * q^31 + 2 * q^32 + 7 * q^33 + q^34 + 3 * q^35 + 2 * q^36 - 9 * q^37 + q^38 + 3 * q^40 + 8 * q^41 - 2 * q^42 + 17 * q^43 - 7 * q^44 + 3 * q^45 + q^46 + 4 * q^47 - 2 * q^48 + 2 * q^49 + 3 * q^50 - q^51 + 18 * q^53 - 2 * q^54 - 19 * q^55 + 2 * q^56 - q^57 - q^58 + 8 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 + 7 * q^66 + 6 * q^67 + q^68 - q^69 + 3 * q^70 - 6 * q^71 + 2 * q^72 - 19 * q^73 - 9 * q^74 - 3 * q^75 + q^76 - 7 * q^77 + 32 * q^79 + 3 * q^80 + 2 * q^81 + 8 * q^82 - 4 * q^83 - 2 * q^84 + 27 * q^85 + 17 * q^86 + q^87 - 7 * q^88 + 16 * q^89 + 3 * q^90 + q^92 - 4 * q^93 + 4 * q^94 - 7 * q^95 - 2 * q^96 + 20 * q^97 + 2 * q^98 - 7 * 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
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−0.561553	−0.177579
$11$	−1.43845	−0.433708	−0.216854	−	0.976204i	$-0.569580\pi$
$11$	−1.43845	−0.433708	−0.216854	+	0.976204i	$0.569580\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	0.561553	0.144992
$16$	1.00000	0.250000
$17$	−5.68466	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$17$	−5.68466	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$18$	1.00000	0.235702
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	−0.561553	−0.125567
$21$	−1.00000	−0.218218
$22$	−1.43845	−0.306678
$23$	−5.68466	−1.18533	−0.592667	−	0.805448i	$-0.701925\pi$
$23$	−5.68466	−1.18533	−0.592667	+	0.805448i	$0.701925\pi$
$24$	−1.00000	−0.204124
$25$	−4.68466	−0.936932
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	−2.56155	−0.475668	−0.237834	−	0.971306i	$-0.576437\pi$
$29$	−2.56155	−0.475668	−0.237834	+	0.971306i	$0.576437\pi$
$30$	0.561553	0.102525
$31$	10.2462	1.84027	0.920137	−	0.391597i	$-0.128077\pi$
$31$	10.2462	1.84027	0.920137	+	0.391597i	$0.128077\pi$
$32$	1.00000	0.176777
$33$	1.43845	0.250402
$34$	−5.68466	−0.974911
$35$	−0.561553	−0.0949197
$36$	1.00000	0.166667
$37$	1.68466	0.276956	0.138478	−	0.990366i	$-0.455779\pi$
$37$	1.68466	0.276956	0.138478	+	0.990366i	$0.455779\pi$
$38$	2.56155	0.415539
$39$	0	0
$40$	−0.561553	−0.0887893
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	−1.00000	−0.154303
$43$	10.5616	1.61062	0.805311	−	0.592853i	$-0.201998\pi$
$43$	10.5616	1.61062	0.805311	+	0.592853i	$0.201998\pi$
$44$	−1.43845	−0.216854
$45$	−0.561553	−0.0837114
$46$	−5.68466	−0.838157
$47$	−6.24621	−0.911104	−0.455552	−	0.890209i	$-0.650558\pi$
$47$	−6.24621	−0.911104	−0.455552	+	0.890209i	$0.650558\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	−4.68466	−0.662511
$51$	5.68466	0.796011
$52$	0	0
$53$	13.1231	1.80260	0.901299	−	0.433198i	$-0.142615\pi$
$53$	13.1231	1.80260	0.901299	+	0.433198i	$0.142615\pi$
$54$	−1.00000	−0.136083
$55$	0.807764	0.108919
$56$	1.00000	0.133631
$57$	−2.56155	−0.339286
$58$	−2.56155	−0.336348
$59$	12.2462	1.59432	0.797160	−	0.603768i	$-0.206334\pi$
$59$	12.2462	1.59432	0.797160	+	0.603768i	$0.206334\pi$
$60$	0.561553	0.0724962
$61$	2.56155	0.327973	0.163987	−	0.986463i	$-0.447565\pi$
$61$	2.56155	0.327973	0.163987	+	0.986463i	$0.447565\pi$
$62$	10.2462	1.30127
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	1.43845	0.177061
$67$	7.12311	0.870226	0.435113	−	0.900376i	$-0.356708\pi$
$67$	7.12311	0.870226	0.435113	+	0.900376i	$0.356708\pi$
$68$	−5.68466	−0.689366
$69$	5.68466	0.684352
$70$	−0.561553	−0.0671184
$71$	−15.3693	−1.82400	−0.912001	−	0.410188i	$-0.865463\pi$
$71$	−15.3693	−1.82400	−0.912001	+	0.410188i	$0.865463\pi$
$72$	1.00000	0.117851
$73$	−7.43845	−0.870604	−0.435302	−	0.900284i	$-0.643358\pi$
$73$	−7.43845	−0.870604	−0.435302	+	0.900284i	$0.643358\pi$
$74$	1.68466	0.195838
$75$	4.68466	0.540938
$76$	2.56155	0.293830
$77$	−1.43845	−0.163926
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	−0.561553	−0.0627835
$81$	1.00000	0.111111
$82$	4.00000	0.441726
$83$	−2.00000	−0.219529	−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000	−0.219529	−0.109764	+	0.993958i	$0.535010\pi$
$84$	−1.00000	−0.109109
$85$	3.19224	0.346247
$86$	10.5616	1.13888
$87$	2.56155	0.274627
$88$	−1.43845	−0.153339
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	−0.561553	−0.0591929
$91$	0	0
$92$	−5.68466	−0.592667
$93$	−10.2462	−1.06248
$94$	−6.24621	−0.644247
$95$	−1.43845	−0.147582
$96$	−1.00000	−0.102062
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	1.00000	0.101015
$99$	−1.43845	−0.144569
$100$	−4.68466	−0.468466
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	5.68466	0.562865
$103$	18.8078	1.85318	0.926592	−	0.376068i	$-0.122724\pi$
$103$	18.8078	1.85318	0.926592	+	0.376068i	$0.122724\pi$
$104$	0	0
$105$	0.561553	0.0548019
$106$	13.1231	1.27463
$107$	−13.1231	−1.26866	−0.634329	−	0.773063i	$-0.718724\pi$
$107$	−13.1231	−1.26866	−0.634329	+	0.773063i	$0.718724\pi$
$108$	−1.00000	−0.0962250
$109$	−15.9309	−1.52590	−0.762950	−	0.646457i	$-0.776250\pi$
$109$	−15.9309	−1.52590	−0.762950	+	0.646457i	$0.776250\pi$
$110$	0.807764	0.0770173
$111$	−1.68466	−0.159901
$112$	1.00000	0.0944911
$113$	3.75379	0.353127	0.176563	−	0.984289i	$-0.443502\pi$
$113$	3.75379	0.353127	0.176563	+	0.984289i	$0.443502\pi$
$114$	−2.56155	−0.239911
$115$	3.19224	0.297678
$116$	−2.56155	−0.237834
$117$	0	0
$118$	12.2462	1.12736
$119$	−5.68466	−0.521112
$120$	0.561553	0.0512625
$121$	−8.93087	−0.811897
$122$	2.56155	0.231912
$123$	−4.00000	−0.360668
$124$	10.2462	0.920137
$125$	5.43845	0.486430
$126$	1.00000	0.0890871
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	1.00000	0.0883883
$129$	−10.5616	−0.929893
$130$	0	0
$131$	2.56155	0.223804	0.111902	−	0.993719i	$-0.464306\pi$
$131$	2.56155	0.223804	0.111902	+	0.993719i	$0.464306\pi$
$132$	1.43845	0.125201
$133$	2.56155	0.222115
$134$	7.12311	0.615343
$135$	0.561553	0.0483308
$136$	−5.68466	−0.487455
$137$	5.68466	0.485673	0.242837	−	0.970067i	$-0.421922\pi$
$137$	5.68466	0.485673	0.242837	+	0.970067i	$0.421922\pi$
$138$	5.68466	0.483910
$139$	−6.24621	−0.529797	−0.264898	−	0.964276i	$-0.585338\pi$
$139$	−6.24621	−0.529797	−0.264898	+	0.964276i	$0.585338\pi$
$140$	−0.561553	−0.0474599
$141$	6.24621	0.526026
$142$	−15.3693	−1.28976
$143$	0	0
$144$	1.00000	0.0833333
$145$	1.43845	0.119457
$146$	−7.43845	−0.615610
$147$	−1.00000	−0.0824786
$148$	1.68466	0.138478
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	4.68466	0.382501
$151$	−4.31534	−0.351178	−0.175589	−	0.984464i	$-0.556183\pi$
$151$	−4.31534	−0.351178	−0.175589	+	0.984464i	$0.556183\pi$
$152$	2.56155	0.207769
$153$	−5.68466	−0.459577
$154$	−1.43845	−0.115913
$155$	−5.75379	−0.462155
$156$	0	0
$157$	8.80776	0.702936	0.351468	−	0.936200i	$-0.385683\pi$
$157$	8.80776	0.702936	0.351468	+	0.936200i	$0.385683\pi$
$158$	16.0000	1.27289
$159$	−13.1231	−1.04073
$160$	−0.561553	−0.0443946
$161$	−5.68466	−0.448014
$162$	1.00000	0.0785674
$163$	−0.876894	−0.0686837	−0.0343418	−	0.999410i	$-0.510933\pi$
$163$	−0.876894	−0.0686837	−0.0343418	+	0.999410i	$0.510933\pi$
$164$	4.00000	0.312348
$165$	−0.807764	−0.0628843
$166$	−2.00000	−0.155230
$167$	−8.80776	−0.681565	−0.340783	−	0.940142i	$-0.610692\pi$
$167$	−8.80776	−0.681565	−0.340783	+	0.940142i	$0.610692\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	3.19224	0.244833
$171$	2.56155	0.195887
$172$	10.5616	0.805311
$173$	20.2462	1.53929	0.769645	−	0.638471i	$-0.220433\pi$
$173$	20.2462	1.53929	0.769645	+	0.638471i	$0.220433\pi$
$174$	2.56155	0.194191
$175$	−4.68466	−0.354127
$176$	−1.43845	−0.108427
$177$	−12.2462	−0.920482
$178$	8.00000	0.599625
$179$	−2.87689	−0.215029	−0.107515	−	0.994204i	$-0.534289\pi$
$179$	−2.87689	−0.215029	−0.107515	+	0.994204i	$0.534289\pi$
$180$	−0.561553	−0.0418557
$181$	−11.3693	−0.845075	−0.422537	−	0.906346i	$-0.638860\pi$
$181$	−11.3693	−0.845075	−0.422537	+	0.906346i	$0.638860\pi$
$182$	0	0
$183$	−2.56155	−0.189355
$184$	−5.68466	−0.419079
$185$	−0.946025	−0.0695531
$186$	−10.2462	−0.751289
$187$	8.17708	0.597967
$188$	−6.24621	−0.455552
$189$	−1.00000	−0.0727393
$190$	−1.43845	−0.104356
$191$	13.0540	0.944553	0.472276	−	0.881451i	$-0.343432\pi$
$191$	13.0540	0.944553	0.472276	+	0.881451i	$0.343432\pi$
$192$	−1.00000	−0.0721688
$193$	12.0000	0.863779	0.431889	−	0.901927i	$-0.357847\pi$
$193$	12.0000	0.863779	0.431889	+	0.901927i	$0.357847\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.43845	−0.102226
$199$	23.9309	1.69641	0.848207	−	0.529665i	$-0.177682\pi$
$199$	23.9309	1.69641	0.848207	+	0.529665i	$0.177682\pi$
$200$	−4.68466	−0.331255
$201$	−7.12311	−0.502425
$202$	−6.00000	−0.422159
$203$	−2.56155	−0.179786
$204$	5.68466	0.398006
$205$	−2.24621	−0.156882
$206$	18.8078	1.31040
$207$	−5.68466	−0.395111
$208$	0	0
$209$	−3.68466	−0.254873
$210$	0.561553	0.0387508
$211$	−12.8078	−0.881723	−0.440861	−	0.897575i	$-0.645327\pi$
$211$	−12.8078	−0.881723	−0.440861	+	0.897575i	$0.645327\pi$
$212$	13.1231	0.901299
$213$	15.3693	1.05309
$214$	−13.1231	−0.897077
$215$	−5.93087	−0.404482
$216$	−1.00000	−0.0680414
$217$	10.2462	0.695558
$218$	−15.9309	−1.07897
$219$	7.43845	0.502644
$220$	0.807764	0.0544594
$221$	0	0
$222$	−1.68466	−0.113067
$223$	−18.2462	−1.22186	−0.610928	−	0.791686i	$-0.709204\pi$
$223$	−18.2462	−1.22186	−0.610928	+	0.791686i	$0.709204\pi$
$224$	1.00000	0.0668153
$225$	−4.68466	−0.312311
$226$	3.75379	0.249698
$227$	23.6155	1.56742	0.783709	−	0.621128i	$-0.213325\pi$
$227$	23.6155	1.56742	0.783709	+	0.621128i	$0.213325\pi$
$228$	−2.56155	−0.169643
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	3.19224	0.210490
$231$	1.43845	0.0946429
$232$	−2.56155	−0.168174
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	3.50758	0.228809
$236$	12.2462	0.797160
$237$	−16.0000	−1.03931
$238$	−5.68466	−0.368482
$239$	−2.24621	−0.145295	−0.0726477	−	0.997358i	$-0.523145\pi$
$239$	−2.24621	−0.145295	−0.0726477	+	0.997358i	$0.523145\pi$
$240$	0.561553	0.0362481
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	−8.93087	−0.574098
$243$	−1.00000	−0.0641500
$244$	2.56155	0.163987
$245$	−0.561553	−0.0358763
$246$	−4.00000	−0.255031
$247$	0	0
$248$	10.2462	0.650635
$249$	2.00000	0.126745
$250$	5.43845	0.343958
$251$	15.0540	0.950198	0.475099	−	0.879932i	$-0.342412\pi$
$251$	15.0540	0.950198	0.475099	+	0.879932i	$0.342412\pi$
$252$	1.00000	0.0629941
$253$	8.17708	0.514089
$254$	6.24621	0.391922
$255$	−3.19224	−0.199906
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	−10.5616	−0.657534
$259$	1.68466	0.104680
$260$	0	0
$261$	−2.56155	−0.158556
$262$	2.56155	0.158253
$263$	1.36932	0.0844357	0.0422178	−	0.999108i	$-0.486558\pi$
$263$	1.36932	0.0844357	0.0422178	+	0.999108i	$0.486558\pi$
$264$	1.43845	0.0885303
$265$	−7.36932	−0.452694
$266$	2.56155	0.157059
$267$	−8.00000	−0.489592
$268$	7.12311	0.435113
$269$	6.63068	0.404280	0.202140	−	0.979357i	$-0.435210\pi$
$269$	6.63068	0.404280	0.202140	+	0.979357i	$0.435210\pi$
$270$	0.561553	0.0341750
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	−5.68466	−0.344683
$273$	0	0
$274$	5.68466	0.343423
$275$	6.73863	0.406355
$276$	5.68466	0.342176
$277$	19.1231	1.14900	0.574498	−	0.818506i	$-0.305197\pi$
$277$	19.1231	1.14900	0.574498	+	0.818506i	$0.305197\pi$
$278$	−6.24621	−0.374623
$279$	10.2462	0.613425
$280$	−0.561553	−0.0335592
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	6.24621	0.371956
$283$	13.1231	0.780088	0.390044	−	0.920796i	$-0.372460\pi$
$283$	13.1231	0.780088	0.390044	+	0.920796i	$0.372460\pi$
$284$	−15.3693	−0.912001
$285$	1.43845	0.0852063
$286$	0	0
$287$	4.00000	0.236113
$288$	1.00000	0.0589256
$289$	15.3153	0.900902
$290$	1.43845	0.0844685
$291$	−10.0000	−0.586210
$292$	−7.43845	−0.435302
$293$	24.2462	1.41648	0.708239	−	0.705972i	$-0.249490\pi$
$293$	24.2462	1.41648	0.708239	+	0.705972i	$0.249490\pi$
$294$	−1.00000	−0.0583212
$295$	−6.87689	−0.400388
$296$	1.68466	0.0979188
$297$	1.43845	0.0834672
$298$	10.0000	0.579284
$299$	0	0
$300$	4.68466	0.270469
$301$	10.5616	0.608758
$302$	−4.31534	−0.248320
$303$	6.00000	0.344691
$304$	2.56155	0.146915
$305$	−1.43845	−0.0823652
$306$	−5.68466	−0.324970
$307$	−9.75379	−0.556678	−0.278339	−	0.960483i	$-0.589784\pi$
$307$	−9.75379	−0.556678	−0.278339	+	0.960483i	$0.589784\pi$
$308$	−1.43845	−0.0819631
$309$	−18.8078	−1.06994
$310$	−5.75379	−0.326793
$311$	−32.4924	−1.84248	−0.921238	−	0.388999i	$-0.872821\pi$
$311$	−32.4924	−1.84248	−0.921238	+	0.388999i	$0.872821\pi$
$312$	0	0
$313$	15.7538	0.890457	0.445228	−	0.895417i	$-0.353122\pi$
$313$	15.7538	0.890457	0.445228	+	0.895417i	$0.353122\pi$
$314$	8.80776	0.497051
$315$	−0.561553	−0.0316399
$316$	16.0000	0.900070
$317$	21.3693	1.20022	0.600110	−	0.799917i	$-0.295123\pi$
$317$	21.3693	1.20022	0.600110	+	0.799917i	$0.295123\pi$
$318$	−13.1231	−0.735907
$319$	3.68466	0.206301
$320$	−0.561553	−0.0313918
$321$	13.1231	0.732460
$322$	−5.68466	−0.316794
$323$	−14.5616	−0.810226
$324$	1.00000	0.0555556
$325$	0	0
$326$	−0.876894	−0.0485667
$327$	15.9309	0.880979
$328$	4.00000	0.220863
$329$	−6.24621	−0.344365
$330$	−0.807764	−0.0444659
$331$	7.12311	0.391521	0.195761	−	0.980652i	$-0.437282\pi$
$331$	7.12311	0.391521	0.195761	+	0.980652i	$0.437282\pi$
$332$	−2.00000	−0.109764
$333$	1.68466	0.0923187
$334$	−8.80776	−0.481939
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	17.0540	0.928989	0.464495	−	0.885576i	$-0.346236\pi$
$337$	17.0540	0.928989	0.464495	+	0.885576i	$0.346236\pi$
$338$	0	0
$339$	−3.75379	−0.203878
$340$	3.19224	0.173123
$341$	−14.7386	−0.798142
$342$	2.56155	0.138513
$343$	1.00000	0.0539949
$344$	10.5616	0.569441
$345$	−3.19224	−0.171864
$346$	20.2462	1.08844
$347$	8.49242	0.455897	0.227949	−	0.973673i	$-0.426798\pi$
$347$	8.49242	0.455897	0.227949	+	0.973673i	$0.426798\pi$
$348$	2.56155	0.137314
$349$	21.3693	1.14387	0.571937	−	0.820298i	$-0.306192\pi$
$349$	21.3693	1.14387	0.571937	+	0.820298i	$0.306192\pi$
$350$	−4.68466	−0.250406
$351$	0	0
$352$	−1.43845	−0.0766695
$353$	−1.75379	−0.0933448	−0.0466724	−	0.998910i	$-0.514862\pi$
$353$	−1.75379	−0.0933448	−0.0466724	+	0.998910i	$0.514862\pi$
$354$	−12.2462	−0.650879
$355$	8.63068	0.458069
$356$	8.00000	0.423999
$357$	5.68466	0.300864
$358$	−2.87689	−0.152049
$359$	17.6155	0.929712	0.464856	−	0.885386i	$-0.346106\pi$
$359$	17.6155	0.929712	0.464856	+	0.885386i	$0.346106\pi$
$360$	−0.561553	−0.0295964
$361$	−12.4384	−0.654655
$362$	−11.3693	−0.597558
$363$	8.93087	0.468749
$364$	0	0
$365$	4.17708	0.218638
$366$	−2.56155	−0.133895
$367$	25.3693	1.32427	0.662134	−	0.749386i	$-0.269651\pi$
$367$	25.3693	1.32427	0.662134	+	0.749386i	$0.269651\pi$
$368$	−5.68466	−0.296333
$369$	4.00000	0.208232
$370$	−0.946025	−0.0491815
$371$	13.1231	0.681318
$372$	−10.2462	−0.531241
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	8.17708	0.422827
$375$	−5.43845	−0.280840
$376$	−6.24621	−0.322124
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	−27.1231	−1.39322	−0.696610	−	0.717450i	$-0.745309\pi$
$379$	−27.1231	−1.39322	−0.696610	+	0.717450i	$0.745309\pi$
$380$	−1.43845	−0.0737908
$381$	−6.24621	−0.320003
$382$	13.0540	0.667899
$383$	13.4384	0.686673	0.343336	−	0.939213i	$-0.388443\pi$
$383$	13.4384	0.686673	0.343336	+	0.939213i	$0.388443\pi$
$384$	−1.00000	−0.0510310
$385$	0.807764	0.0411675
$386$	12.0000	0.610784
$387$	10.5616	0.536874
$388$	10.0000	0.507673
$389$	−18.8769	−0.957097	−0.478548	−	0.878061i	$-0.658837\pi$
$389$	−18.8769	−0.957097	−0.478548	+	0.878061i	$0.658837\pi$
$390$	0	0
$391$	32.3153	1.63426
$392$	1.00000	0.0505076
$393$	−2.56155	−0.129213
$394$	−6.00000	−0.302276
$395$	−8.98485	−0.452077
$396$	−1.43845	−0.0722847
$397$	9.36932	0.470233	0.235116	−	0.971967i	$-0.424453\pi$
$397$	9.36932	0.470233	0.235116	+	0.971967i	$0.424453\pi$
$398$	23.9309	1.19955
$399$	−2.56155	−0.128238
$400$	−4.68466	−0.234233
$401$	−34.9848	−1.74706	−0.873530	−	0.486771i	$-0.838175\pi$
$401$	−34.9848	−1.74706	−0.873530	+	0.486771i	$0.838175\pi$
$402$	−7.12311	−0.355268
$403$	0	0
$404$	−6.00000	−0.298511
$405$	−0.561553	−0.0279038
$406$	−2.56155	−0.127128
$407$	−2.42329	−0.120118
$408$	5.68466	0.281433
$409$	−6.80776	−0.336622	−0.168311	−	0.985734i	$-0.553831\pi$
$409$	−6.80776	−0.336622	−0.168311	+	0.985734i	$0.553831\pi$
$410$	−2.24621	−0.110932
$411$	−5.68466	−0.280404
$412$	18.8078	0.926592
$413$	12.2462	0.602597
$414$	−5.68466	−0.279386
$415$	1.12311	0.0551311
$416$	0	0
$417$	6.24621	0.305878
$418$	−3.68466	−0.180223
$419$	−8.31534	−0.406231	−0.203116	−	0.979155i	$-0.565107\pi$
$419$	−8.31534	−0.406231	−0.203116	+	0.979155i	$0.565107\pi$
$420$	0.561553	0.0274010
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	−12.8078	−0.623472
$423$	−6.24621	−0.303701
$424$	13.1231	0.637314
$425$	26.6307	1.29178
$426$	15.3693	0.744646
$427$	2.56155	0.123962
$428$	−13.1231	−0.634329
$429$	0	0
$430$	−5.93087	−0.286012
$431$	−27.8617	−1.34205	−0.671026	−	0.741433i	$-0.734146\pi$
$431$	−27.8617	−1.34205	−0.671026	+	0.741433i	$0.734146\pi$
$432$	−1.00000	−0.0481125
$433$	1.36932	0.0658052	0.0329026	−	0.999459i	$-0.489525\pi$
$433$	1.36932	0.0658052	0.0329026	+	0.999459i	$0.489525\pi$
$434$	10.2462	0.491834
$435$	−1.43845	−0.0689683
$436$	−15.9309	−0.762950
$437$	−14.5616	−0.696574
$438$	7.43845	0.355423
$439$	−19.9309	−0.951249	−0.475624	−	0.879649i	$-0.657778\pi$
$439$	−19.9309	−0.951249	−0.475624	+	0.879649i	$0.657778\pi$
$440$	0.807764	0.0385086
$441$	1.00000	0.0476190
$442$	0	0
$443$	31.3693	1.49040	0.745201	−	0.666840i	$-0.232354\pi$
$443$	31.3693	1.49040	0.745201	+	0.666840i	$0.232354\pi$
$444$	−1.68466	−0.0799504
$445$	−4.49242	−0.212961
$446$	−18.2462	−0.863983
$447$	−10.0000	−0.472984
$448$	1.00000	0.0472456
$449$	21.6847	1.02336	0.511681	−	0.859175i	$-0.329023\pi$
$449$	21.6847	1.02336	0.511681	+	0.859175i	$0.329023\pi$
$450$	−4.68466	−0.220837
$451$	−5.75379	−0.270935
$452$	3.75379	0.176563
$453$	4.31534	0.202752
$454$	23.6155	1.10833
$455$	0	0
$456$	−2.56155	−0.119956
$457$	−16.0000	−0.748448	−0.374224	−	0.927338i	$-0.622091\pi$
$457$	−16.0000	−0.748448	−0.374224	+	0.927338i	$0.622091\pi$
$458$	−2.00000	−0.0934539
$459$	5.68466	0.265337
$460$	3.19224	0.148839
$461$	8.06913	0.375817	0.187908	−	0.982187i	$-0.439829\pi$
$461$	8.06913	0.375817	0.187908	+	0.982187i	$0.439829\pi$
$462$	1.43845	0.0669226
$463$	−39.5464	−1.83788	−0.918938	−	0.394401i	$-0.870952\pi$
$463$	−39.5464	−1.83788	−0.918938	+	0.394401i	$0.870952\pi$
$464$	−2.56155	−0.118917
$465$	5.75379	0.266826
$466$	−2.00000	−0.0926482
$467$	2.56155	0.118535	0.0592673	−	0.998242i	$-0.481124\pi$
$467$	2.56155	0.118535	0.0592673	+	0.998242i	$0.481124\pi$
$468$	0	0
$469$	7.12311	0.328914
$470$	3.50758	0.161792
$471$	−8.80776	−0.405840
$472$	12.2462	0.563678
$473$	−15.1922	−0.698540
$474$	−16.0000	−0.734904
$475$	−12.0000	−0.550598
$476$	−5.68466	−0.260556
$477$	13.1231	0.600866
$478$	−2.24621	−0.102739
$479$	41.3002	1.88705	0.943527	−	0.331296i	$-0.107486\pi$
$479$	41.3002	1.88705	0.943527	+	0.331296i	$0.107486\pi$
$480$	0.561553	0.0256313
$481$	0	0
$482$	−6.00000	−0.273293
$483$	5.68466	0.258661
$484$	−8.93087	−0.405949
$485$	−5.61553	−0.254988
$486$	−1.00000	−0.0453609
$487$	−30.2462	−1.37059	−0.685293	−	0.728267i	$-0.740326\pi$
$487$	−30.2462	−1.37059	−0.685293	+	0.728267i	$0.740326\pi$
$488$	2.56155	0.115956
$489$	0.876894	0.0396545
$490$	−0.561553	−0.0253684
$491$	−6.24621	−0.281888	−0.140944	−	0.990018i	$-0.545014\pi$
$491$	−6.24621	−0.281888	−0.140944	+	0.990018i	$0.545014\pi$
$492$	−4.00000	−0.180334
$493$	14.5616	0.655819
$494$	0	0
$495$	0.807764	0.0363063
$496$	10.2462	0.460068
$497$	−15.3693	−0.689408
$498$	2.00000	0.0896221
$499$	−34.4924	−1.54409	−0.772046	−	0.635566i	$-0.780767\pi$
$499$	−34.4924	−1.54409	−0.772046	+	0.635566i	$0.780767\pi$
$500$	5.43845	0.243215
$501$	8.80776	0.393502
$502$	15.0540	0.671892
$503$	−5.61553	−0.250384	−0.125192	−	0.992133i	$-0.539955\pi$
$503$	−5.61553	−0.250384	−0.125192	+	0.992133i	$0.539955\pi$
$504$	1.00000	0.0445435
$505$	3.36932	0.149933
$506$	8.17708	0.363516
$507$	0	0
$508$	6.24621	0.277131
$509$	−9.68466	−0.429265	−0.214632	−	0.976695i	$-0.568855\pi$
$509$	−9.68466	−0.429265	−0.214632	+	0.976695i	$0.568855\pi$
$510$	−3.19224	−0.141355
$511$	−7.43845	−0.329058
$512$	1.00000	0.0441942
$513$	−2.56155	−0.113095
$514$	26.0000	1.14681
$515$	−10.5616	−0.465398
$516$	−10.5616	−0.464946
$517$	8.98485	0.395153
$518$	1.68466	0.0740196
$519$	−20.2462	−0.888710
$520$	0	0
$521$	−29.0540	−1.27288	−0.636439	−	0.771327i	$-0.719593\pi$
$521$	−29.0540	−1.27288	−0.636439	+	0.771327i	$0.719593\pi$
$522$	−2.56155	−0.112116
$523$	24.4924	1.07098	0.535489	−	0.844542i	$-0.320127\pi$
$523$	24.4924	1.07098	0.535489	+	0.844542i	$0.320127\pi$
$524$	2.56155	0.111902
$525$	4.68466	0.204455
$526$	1.36932	0.0597051
$527$	−58.2462	−2.53724
$528$	1.43845	0.0626004
$529$	9.31534	0.405015
$530$	−7.36932	−0.320103
$531$	12.2462	0.531440
$532$	2.56155	0.111057
$533$	0	0
$534$	−8.00000	−0.346194
$535$	7.36932	0.318603
$536$	7.12311	0.307671
$537$	2.87689	0.124147
$538$	6.63068	0.285869
$539$	−1.43845	−0.0619583
$540$	0.561553	0.0241654
$541$	26.8078	1.15256	0.576278	−	0.817254i	$-0.304505\pi$
$541$	26.8078	1.15256	0.576278	+	0.817254i	$0.304505\pi$
$542$	8.00000	0.343629
$543$	11.3693	0.487904
$544$	−5.68466	−0.243728
$545$	8.94602	0.383206
$546$	0	0
$547$	−36.9848	−1.58136	−0.790679	−	0.612231i	$-0.790272\pi$
$547$	−36.9848	−1.58136	−0.790679	+	0.612231i	$0.790272\pi$
$548$	5.68466	0.242837
$549$	2.56155	0.109324
$550$	6.73863	0.287336
$551$	−6.56155	−0.279532
$552$	5.68466	0.241955
$553$	16.0000	0.680389
$554$	19.1231	0.812463
$555$	0.946025	0.0401565
$556$	−6.24621	−0.264898
$557$	0.246211	0.0104323	0.00521615	−	0.999986i	$-0.498340\pi$
$557$	0.246211	0.0104323	0.00521615	+	0.999986i	$0.498340\pi$
$558$	10.2462	0.433757
$559$	0	0
$560$	−0.561553	−0.0237299
$561$	−8.17708	−0.345237
$562$	−6.00000	−0.253095
$563$	−0.946025	−0.0398702	−0.0199351	−	0.999801i	$-0.506346\pi$
$563$	−0.946025	−0.0398702	−0.0199351	+	0.999801i	$0.506346\pi$
$564$	6.24621	0.263013
$565$	−2.10795	−0.0886821
$566$	13.1231	0.551605
$567$	1.00000	0.0419961
$568$	−15.3693	−0.644882
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	1.43845	0.0602499
$571$	1.75379	0.0733938	0.0366969	−	0.999326i	$-0.488316\pi$
$571$	1.75379	0.0733938	0.0366969	+	0.999326i	$0.488316\pi$
$572$	0	0
$573$	−13.0540	−0.545338
$574$	4.00000	0.166957
$575$	26.6307	1.11058
$576$	1.00000	0.0416667
$577$	24.2462	1.00938	0.504691	−	0.863300i	$-0.331606\pi$
$577$	24.2462	1.00938	0.504691	+	0.863300i	$0.331606\pi$
$578$	15.3153	0.637034
$579$	−12.0000	−0.498703
$580$	1.43845	0.0597283
$581$	−2.00000	−0.0829740
$582$	−10.0000	−0.414513
$583$	−18.8769	−0.781801
$584$	−7.43845	−0.307805
$585$	0	0
$586$	24.2462	1.00160
$587$	−28.8769	−1.19188	−0.595938	−	0.803030i	$-0.703220\pi$
$587$	−28.8769	−1.19188	−0.595938	+	0.803030i	$0.703220\pi$
$588$	−1.00000	−0.0412393
$589$	26.2462	1.08146
$590$	−6.87689	−0.283117
$591$	6.00000	0.246807
$592$	1.68466	0.0692390
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	1.43845	0.0590202
$595$	3.19224	0.130869
$596$	10.0000	0.409616
$597$	−23.9309	−0.979425
$598$	0	0
$599$	−23.3002	−0.952020	−0.476010	−	0.879440i	$-0.657917\pi$
$599$	−23.3002	−0.952020	−0.476010	+	0.879440i	$0.657917\pi$
$600$	4.68466	0.191250
$601$	25.8617	1.05492	0.527461	−	0.849579i	$-0.323144\pi$
$601$	25.8617	1.05492	0.527461	+	0.849579i	$0.323144\pi$
$602$	10.5616	0.430457
$603$	7.12311	0.290075
$604$	−4.31534	−0.175589
$605$	5.01515	0.203895
$606$	6.00000	0.243733
$607$	41.5464	1.68632	0.843158	−	0.537666i	$-0.180694\pi$
$607$	41.5464	1.68632	0.843158	+	0.537666i	$0.180694\pi$
$608$	2.56155	0.103885
$609$	2.56155	0.103799
$610$	−1.43845	−0.0582410
$611$	0	0
$612$	−5.68466	−0.229789
$613$	18.8078	0.759638	0.379819	−	0.925061i	$-0.375986\pi$
$613$	18.8078	0.759638	0.379819	+	0.925061i	$0.375986\pi$
$614$	−9.75379	−0.393631
$615$	2.24621	0.0905760
$616$	−1.43845	−0.0579567
$617$	26.8078	1.07924	0.539620	−	0.841909i	$-0.318568\pi$
$617$	26.8078	1.07924	0.539620	+	0.841909i	$0.318568\pi$
$618$	−18.8078	−0.756559
$619$	2.06913	0.0831654	0.0415827	−	0.999135i	$-0.486760\pi$
$619$	2.06913	0.0831654	0.0415827	+	0.999135i	$0.486760\pi$
$620$	−5.75379	−0.231078
$621$	5.68466	0.228117
$622$	−32.4924	−1.30283
$623$	8.00000	0.320513
$624$	0	0
$625$	20.3693	0.814773
$626$	15.7538	0.629648
$627$	3.68466	0.147151
$628$	8.80776	0.351468
$629$	−9.57671	−0.381848
$630$	−0.561553	−0.0223728
$631$	−0.315342	−0.0125535	−0.00627677	−	0.999980i	$-0.501998\pi$
$631$	−0.315342	−0.0125535	−0.00627677	+	0.999980i	$0.501998\pi$
$632$	16.0000	0.636446
$633$	12.8078	0.509063
$634$	21.3693	0.848684
$635$	−3.50758	−0.139194
$636$	−13.1231	−0.520365
$637$	0	0
$638$	3.68466	0.145877
$639$	−15.3693	−0.608001
$640$	−0.561553	−0.0221973
$641$	44.1080	1.74216	0.871080	−	0.491142i	$-0.163420\pi$
$641$	44.1080	1.74216	0.871080	+	0.491142i	$0.163420\pi$
$642$	13.1231	0.517928
$643$	−20.8078	−0.820578	−0.410289	−	0.911956i	$-0.634572\pi$
$643$	−20.8078	−0.820578	−0.410289	+	0.911956i	$0.634572\pi$
$644$	−5.68466	−0.224007
$645$	5.93087	0.233528
$646$	−14.5616	−0.572917
$647$	−35.3693	−1.39051	−0.695256	−	0.718763i	$-0.744709\pi$
$647$	−35.3693	−1.39051	−0.695256	+	0.718763i	$0.744709\pi$
$648$	1.00000	0.0392837
$649$	−17.6155	−0.691470
$650$	0	0
$651$	−10.2462	−0.401581
$652$	−0.876894	−0.0343418
$653$	−25.4384	−0.995483	−0.497742	−	0.867325i	$-0.665837\pi$
$653$	−25.4384	−0.995483	−0.497742	+	0.867325i	$0.665837\pi$
$654$	15.9309	0.622946
$655$	−1.43845	−0.0562048
$656$	4.00000	0.156174
$657$	−7.43845	−0.290201
$658$	−6.24621	−0.243503
$659$	21.1231	0.822839	0.411420	−	0.911446i	$-0.365033\pi$
$659$	21.1231	0.822839	0.411420	+	0.911446i	$0.365033\pi$
$660$	−0.807764	−0.0314422
$661$	−4.73863	−0.184311	−0.0921557	−	0.995745i	$-0.529376\pi$
$661$	−4.73863	−0.184311	−0.0921557	+	0.995745i	$0.529376\pi$
$662$	7.12311	0.276847
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	−1.43845	−0.0557806
$666$	1.68466	0.0652792
$667$	14.5616	0.563826
$668$	−8.80776	−0.340783
$669$	18.2462	0.705439
$670$	−4.00000	−0.154533
$671$	−3.68466	−0.142245
$672$	−1.00000	−0.0385758
$673$	−17.1922	−0.662712	−0.331356	−	0.943506i	$-0.607506\pi$
$673$	−17.1922	−0.662712	−0.331356	+	0.943506i	$0.607506\pi$
$674$	17.0540	0.656895
$675$	4.68466	0.180313
$676$	0	0
$677$	6.63068	0.254838	0.127419	−	0.991849i	$-0.459331\pi$
$677$	6.63068	0.254838	0.127419	+	0.991849i	$0.459331\pi$
$678$	−3.75379	−0.144163
$679$	10.0000	0.383765
$680$	3.19224	0.122417
$681$	−23.6155	−0.904949
$682$	−14.7386	−0.564371
$683$	11.6847	0.447101	0.223551	−	0.974692i	$-0.428235\pi$
$683$	11.6847	0.447101	0.223551	+	0.974692i	$0.428235\pi$
$684$	2.56155	0.0979434
$685$	−3.19224	−0.121969
$686$	1.00000	0.0381802
$687$	2.00000	0.0763048
$688$	10.5616	0.402655
$689$	0	0
$690$	−3.19224	−0.121526
$691$	3.50758	0.133435	0.0667173	−	0.997772i	$-0.478747\pi$
$691$	3.50758	0.133435	0.0667173	+	0.997772i	$0.478747\pi$
$692$	20.2462	0.769645
$693$	−1.43845	−0.0546421
$694$	8.49242	0.322368
$695$	3.50758	0.133050
$696$	2.56155	0.0970954
$697$	−22.7386	−0.861287
$698$	21.3693	0.808841
$699$	2.00000	0.0756469
$700$	−4.68466	−0.177063
$701$	−37.6155	−1.42072	−0.710359	−	0.703839i	$-0.751468\pi$
$701$	−37.6155	−1.42072	−0.710359	+	0.703839i	$0.751468\pi$
$702$	0	0
$703$	4.31534	0.162756
$704$	−1.43845	−0.0542135
$705$	−3.50758	−0.132103
$706$	−1.75379	−0.0660047
$707$	−6.00000	−0.225653
$708$	−12.2462	−0.460241
$709$	−48.2462	−1.81192	−0.905962	−	0.423358i	$-0.860851\pi$
$709$	−48.2462	−1.81192	−0.905962	+	0.423358i	$0.860851\pi$
$710$	8.63068	0.323904
$711$	16.0000	0.600047
$712$	8.00000	0.299813
$713$	−58.2462	−2.18134
$714$	5.68466	0.212743
$715$	0	0
$716$	−2.87689	−0.107515
$717$	2.24621	0.0838863
$718$	17.6155	0.657406
$719$	−30.2462	−1.12799	−0.563997	−	0.825777i	$-0.690737\pi$
$719$	−30.2462	−1.12799	−0.563997	+	0.825777i	$0.690737\pi$
$720$	−0.561553	−0.0209278
$721$	18.8078	0.700438
$722$	−12.4384	−0.462911
$723$	6.00000	0.223142
$724$	−11.3693	−0.422537
$725$	12.0000	0.445669
$726$	8.93087	0.331456
$727$	20.0691	0.744323	0.372161	−	0.928168i	$-0.378617\pi$
$727$	20.0691	0.744323	0.372161	+	0.928168i	$0.378617\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	4.17708	0.154601
$731$	−60.0388	−2.22062
$732$	−2.56155	−0.0946777
$733$	−31.1231	−1.14956	−0.574779	−	0.818309i	$-0.694912\pi$
$733$	−31.1231	−1.14956	−0.574779	+	0.818309i	$0.694912\pi$
$734$	25.3693	0.936399
$735$	0.561553	0.0207132
$736$	−5.68466	−0.209539
$737$	−10.2462	−0.377424
$738$	4.00000	0.147242
$739$	−23.6155	−0.868711	−0.434356	−	0.900741i	$-0.643024\pi$
$739$	−23.6155	−0.868711	−0.434356	+	0.900741i	$0.643024\pi$
$740$	−0.946025	−0.0347766
$741$	0	0
$742$	13.1231	0.481764
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	−10.2462	−0.375644
$745$	−5.61553	−0.205737
$746$	−6.00000	−0.219676
$747$	−2.00000	−0.0731762
$748$	8.17708	0.298984
$749$	−13.1231	−0.479508
$750$	−5.43845	−0.198584
$751$	14.2462	0.519852	0.259926	−	0.965629i	$-0.416302\pi$
$751$	14.2462	0.519852	0.259926	+	0.965629i	$0.416302\pi$
$752$	−6.24621	−0.227776
$753$	−15.0540	−0.548597
$754$	0	0
$755$	2.42329	0.0881926
$756$	−1.00000	−0.0363696
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	−27.1231	−0.985156
$759$	−8.17708	−0.296809
$760$	−1.43845	−0.0521780
$761$	30.8769	1.11929	0.559643	−	0.828734i	$-0.310938\pi$
$761$	30.8769	1.11929	0.559643	+	0.828734i	$0.310938\pi$
$762$	−6.24621	−0.226276
$763$	−15.9309	−0.576736
$764$	13.0540	0.472276
$765$	3.19224	0.115416
$766$	13.4384	0.485551
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−12.5616	−0.452981	−0.226491	−	0.974013i	$-0.572725\pi$
$769$	−12.5616	−0.452981	−0.226491	+	0.974013i	$0.572725\pi$
$770$	0.807764	0.0291098
$771$	−26.0000	−0.936367
$772$	12.0000	0.431889
$773$	−15.9309	−0.572994	−0.286497	−	0.958081i	$-0.592491\pi$
$773$	−15.9309	−0.572994	−0.286497	+	0.958081i	$0.592491\pi$
$774$	10.5616	0.379627
$775$	−48.0000	−1.72421
$776$	10.0000	0.358979
$777$	−1.68466	−0.0604368
$778$	−18.8769	−0.676769
$779$	10.2462	0.367109
$780$	0	0
$781$	22.1080	0.791085
$782$	32.3153	1.15559
$783$	2.56155	0.0915424
$784$	1.00000	0.0357143
$785$	−4.94602	−0.176531
$786$	−2.56155	−0.0913676
$787$	−45.9309	−1.63726	−0.818629	−	0.574322i	$-0.805266\pi$
$787$	−45.9309	−1.63726	−0.818629	+	0.574322i	$0.805266\pi$
$788$	−6.00000	−0.213741
$789$	−1.36932	−0.0487490
$790$	−8.98485	−0.319666
$791$	3.75379	0.133469
$792$	−1.43845	−0.0511130
$793$	0	0
$794$	9.36932	0.332505
$795$	7.36932	0.261363
$796$	23.9309	0.848207
$797$	−20.2462	−0.717158	−0.358579	−	0.933499i	$-0.616739\pi$
$797$	−20.2462	−0.717158	−0.358579	+	0.933499i	$0.616739\pi$
$798$	−2.56155	−0.0906780
$799$	35.5076	1.25617
$800$	−4.68466	−0.165628
$801$	8.00000	0.282666
$802$	−34.9848	−1.23536
$803$	10.6998	0.377588
$804$	−7.12311	−0.251213
$805$	3.19224	0.112512
$806$	0	0
$807$	−6.63068	−0.233411
$808$	−6.00000	−0.211079
$809$	−30.6307	−1.07692	−0.538459	−	0.842652i	$-0.680993\pi$
$809$	−30.6307	−1.07692	−0.538459	+	0.842652i	$0.680993\pi$
$810$	−0.561553	−0.0197310
$811$	35.0540	1.23091	0.615456	−	0.788171i	$-0.288972\pi$
$811$	35.0540	1.23091	0.615456	+	0.788171i	$0.288972\pi$
$812$	−2.56155	−0.0898929
$813$	−8.00000	−0.280572
$814$	−2.42329	−0.0849363
$815$	0.492423	0.0172488
$816$	5.68466	0.199003
$817$	27.0540	0.946499
$818$	−6.80776	−0.238028
$819$	0	0
$820$	−2.24621	−0.0784411
$821$	−39.1231	−1.36541	−0.682703	−	0.730696i	$-0.739196\pi$
$821$	−39.1231	−1.36541	−0.682703	+	0.730696i	$0.739196\pi$
$822$	−5.68466	−0.198275
$823$	−18.7386	−0.653188	−0.326594	−	0.945165i	$-0.605901\pi$
$823$	−18.7386	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$824$	18.8078	0.655200
$825$	−6.73863	−0.234609
$826$	12.2462	0.426100
$827$	15.0540	0.523478	0.261739	−	0.965139i	$-0.415704\pi$
$827$	15.0540	0.523478	0.261739	+	0.965139i	$0.415704\pi$
$828$	−5.68466	−0.197556
$829$	−4.94602	−0.171783	−0.0858913	−	0.996305i	$-0.527374\pi$
$829$	−4.94602	−0.171783	−0.0858913	+	0.996305i	$0.527374\pi$
$830$	1.12311	0.0389836
$831$	−19.1231	−0.663373
$832$	0	0
$833$	−5.68466	−0.196962
$834$	6.24621	0.216289
$835$	4.94602	0.171164
$836$	−3.68466	−0.127437
$837$	−10.2462	−0.354161
$838$	−8.31534	−0.287249
$839$	−42.2462	−1.45850	−0.729251	−	0.684247i	$-0.760131\pi$
$839$	−42.2462	−1.45850	−0.729251	+	0.684247i	$0.760131\pi$
$840$	0.561553	0.0193754
$841$	−22.4384	−0.773740
$842$	−10.0000	−0.344623
$843$	6.00000	0.206651
$844$	−12.8078	−0.440861
$845$	0	0
$846$	−6.24621	−0.214749
$847$	−8.93087	−0.306868
$848$	13.1231	0.450649
$849$	−13.1231	−0.450384
$850$	26.6307	0.913425
$851$	−9.57671	−0.328285
$852$	15.3693	0.526544
$853$	−19.1231	−0.654763	−0.327381	−	0.944892i	$-0.606166\pi$
$853$	−19.1231	−0.654763	−0.327381	+	0.944892i	$0.606166\pi$
$854$	2.56155	0.0876545
$855$	−1.43845	−0.0491939
$856$	−13.1231	−0.448539
$857$	38.0000	1.29806	0.649028	−	0.760765i	$-0.275176\pi$
$857$	38.0000	1.29806	0.649028	+	0.760765i	$0.275176\pi$
$858$	0	0
$859$	41.6155	1.41990	0.709952	−	0.704250i	$-0.248717\pi$
$859$	41.6155	1.41990	0.709952	+	0.704250i	$0.248717\pi$
$860$	−5.93087	−0.202241
$861$	−4.00000	−0.136320
$862$	−27.8617	−0.948975
$863$	6.73863	0.229386	0.114693	−	0.993401i	$-0.463412\pi$
$863$	6.73863	0.229386	0.114693	+	0.993401i	$0.463412\pi$
$864$	−1.00000	−0.0340207
$865$	−11.3693	−0.386568
$866$	1.36932	0.0465313
$867$	−15.3153	−0.520136
$868$	10.2462	0.347779
$869$	−23.0152	−0.780736
$870$	−1.43845	−0.0487679
$871$	0	0
$872$	−15.9309	−0.539487
$873$	10.0000	0.338449
$874$	−14.5616	−0.492552
$875$	5.43845	0.183853
$876$	7.43845	0.251322
$877$	42.9848	1.45150	0.725748	−	0.687961i	$-0.241494\pi$
$877$	42.9848	1.45150	0.725748	+	0.687961i	$0.241494\pi$
$878$	−19.9309	−0.672634
$879$	−24.2462	−0.817804
$880$	0.807764	0.0272297
$881$	39.3002	1.32406	0.662028	−	0.749479i	$-0.269696\pi$
$881$	39.3002	1.32406	0.662028	+	0.749479i	$0.269696\pi$
$882$	1.00000	0.0336718
$883$	18.5616	0.624646	0.312323	−	0.949976i	$-0.398893\pi$
$883$	18.5616	0.624646	0.312323	+	0.949976i	$0.398893\pi$
$884$	0	0
$885$	6.87689	0.231164
$886$	31.3693	1.05387
$887$	19.8617	0.666892	0.333446	−	0.942769i	$-0.391789\pi$
$887$	19.8617	0.666892	0.333446	+	0.942769i	$0.391789\pi$
$888$	−1.68466	−0.0565334
$889$	6.24621	0.209491
$890$	−4.49242	−0.150586
$891$	−1.43845	−0.0481898
$892$	−18.2462	−0.610928
$893$	−16.0000	−0.535420
$894$	−10.0000	−0.334450
$895$	1.61553	0.0540011
$896$	1.00000	0.0334077
$897$	0	0
$898$	21.6847	0.723626
$899$	−26.2462	−0.875360
$900$	−4.68466	−0.156155
$901$	−74.6004	−2.48530
$902$	−5.75379	−0.191580
$903$	−10.5616	−0.351466
$904$	3.75379	0.124849
$905$	6.38447	0.212227
$906$	4.31534	0.143368
$907$	−8.49242	−0.281986	−0.140993	−	0.990011i	$-0.545030\pi$
$907$	−8.49242	−0.281986	−0.140993	+	0.990011i	$0.545030\pi$
$908$	23.6155	0.783709
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−8.56155	−0.283657	−0.141828	−	0.989891i	$-0.545298\pi$
$911$	−8.56155	−0.283657	−0.141828	+	0.989891i	$0.545298\pi$
$912$	−2.56155	−0.0848215
$913$	2.87689	0.0952113
$914$	−16.0000	−0.529233
$915$	1.43845	0.0475536
$916$	−2.00000	−0.0660819
$917$	2.56155	0.0845899
$918$	5.68466	0.187622
$919$	39.8617	1.31492	0.657459	−	0.753491i	$-0.271631\pi$
$919$	39.8617	1.31492	0.657459	+	0.753491i	$0.271631\pi$
$920$	3.19224	0.105245
$921$	9.75379	0.321398
$922$	8.06913	0.265743
$923$	0	0
$924$	1.43845	0.0473214
$925$	−7.89205	−0.259489
$926$	−39.5464	−1.29958
$927$	18.8078	0.617728
$928$	−2.56155	−0.0840871
$929$	−49.1231	−1.61168	−0.805838	−	0.592136i	$-0.798285\pi$
$929$	−49.1231	−1.61168	−0.805838	+	0.592136i	$0.798285\pi$
$930$	5.75379	0.188674
$931$	2.56155	0.0839515
$932$	−2.00000	−0.0655122
$933$	32.4924	1.06375
$934$	2.56155	0.0838166
$935$	−4.59186	−0.150170
$936$	0	0
$937$	37.8617	1.23689	0.618445	−	0.785828i	$-0.287763\pi$
$937$	37.8617	1.23689	0.618445	+	0.785828i	$0.287763\pi$
$938$	7.12311	0.232578
$939$	−15.7538	−0.514105
$940$	3.50758	0.114405
$941$	6.49242	0.211647	0.105823	−	0.994385i	$-0.466252\pi$
$941$	6.49242	0.211647	0.105823	+	0.994385i	$0.466252\pi$
$942$	−8.80776	−0.286972
$943$	−22.7386	−0.740472
$944$	12.2462	0.398580
$945$	0.561553	0.0182673
$946$	−15.1922	−0.493942
$947$	−7.19224	−0.233716	−0.116858	−	0.993149i	$-0.537282\pi$
$947$	−7.19224	−0.233716	−0.116858	+	0.993149i	$0.537282\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	−12.0000	−0.389331
$951$	−21.3693	−0.692948
$952$	−5.68466	−0.184241
$953$	48.7386	1.57880	0.789400	−	0.613880i	$-0.210392\pi$
$953$	48.7386	1.57880	0.789400	+	0.613880i	$0.210392\pi$
$954$	13.1231	0.424876
$955$	−7.33050	−0.237209
$956$	−2.24621	−0.0726477
$957$	−3.68466	−0.119108
$958$	41.3002	1.33435
$959$	5.68466	0.183567
$960$	0.561553	0.0181240
$961$	73.9848	2.38661
$962$	0	0
$963$	−13.1231	−0.422886
$964$	−6.00000	−0.193247
$965$	−6.73863	−0.216924
$966$	5.68466	0.182901
$967$	−25.9309	−0.833881	−0.416940	−	0.908934i	$-0.636898\pi$
$967$	−25.9309	−0.833881	−0.416940	+	0.908934i	$0.636898\pi$
$968$	−8.93087	−0.287049
$969$	14.5616	0.467784
$970$	−5.61553	−0.180304
$971$	4.00000	0.128366	0.0641831	−	0.997938i	$-0.479556\pi$
$971$	4.00000	0.128366	0.0641831	+	0.997938i	$0.479556\pi$
$972$	−1.00000	−0.0320750
$973$	−6.24621	−0.200244
$974$	−30.2462	−0.969151
$975$	0	0
$976$	2.56155	0.0819933
$977$	39.7926	1.27308	0.636539	−	0.771244i	$-0.280365\pi$
$977$	39.7926	1.27308	0.636539	+	0.771244i	$0.280365\pi$
$978$	0.876894	0.0280400
$979$	−11.5076	−0.367784
$980$	−0.561553	−0.0179381
$981$	−15.9309	−0.508634
$982$	−6.24621	−0.199325
$983$	6.56155	0.209281	0.104641	−	0.994510i	$-0.466631\pi$
$983$	6.56155	0.209281	0.104641	+	0.994510i	$0.466631\pi$
$984$	−4.00000	−0.127515
$985$	3.36932	0.107355
$986$	14.5616	0.463734
$987$	6.24621	0.198819
$988$	0	0
$989$	−60.0388	−1.90912
$990$	0.807764	0.0256724
$991$	−13.7538	−0.436903	−0.218452	−	0.975848i	$-0.570101\pi$
$991$	−13.7538	−0.436903	−0.218452	+	0.975848i	$0.570101\pi$
$992$	10.2462	0.325318
$993$	−7.12311	−0.226045
$994$	−15.3693	−0.487485
$995$	−13.4384	−0.426027
$996$	2.00000	0.0633724
$997$	−59.3693	−1.88025	−0.940123	−	0.340837i	$-0.889290\pi$
$997$	−59.3693	−1.88025	−0.940123	+	0.340837i	$0.889290\pi$
$998$	−34.4924	−1.09184
$999$	−1.68466	−0.0533002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.bv.1.1		2
13.5	odd	4		546.2.c.e.337.2	✓	4
13.8	odd	4		546.2.c.e.337.3	yes	4
13.12	even	2		7098.2.a.bg.1.2		2
39.5	even	4		1638.2.c.h.883.3		4
39.8	even	4		1638.2.c.h.883.2		4
52.31	even	4		4368.2.h.n.337.3		4
52.47	even	4		4368.2.h.n.337.2		4
91.34	even	4		3822.2.c.h.883.4		4
91.83	even	4		3822.2.c.h.883.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.2	✓	4	13.5	odd	4
546.2.c.e.337.3	yes	4	13.8	odd	4
1638.2.c.h.883.2		4	39.8	even	4
1638.2.c.h.883.3		4	39.5	even	4
3822.2.c.h.883.1		4	91.83	even	4
3822.2.c.h.883.4		4	91.34	even	4
4368.2.h.n.337.2		4	52.47	even	4
4368.2.h.n.337.3		4	52.31	even	4
7098.2.a.bg.1.2		2	13.12	even	2
7098.2.a.bv.1.1		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:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.561553 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.561553 q^{10} -1.43845 q^{11} -1.00000 q^{12} +1.00000 q^{14} +0.561553 q^{15} +1.00000 q^{16} -5.68466 q^{17} +1.00000 q^{18} +2.56155 q^{19} -0.561553 q^{20} -1.00000 q^{21} -1.43845 q^{22} -5.68466 q^{23} -1.00000 q^{24} -4.68466 q^{25} -1.00000 q^{27} +1.00000 q^{28} -2.56155 q^{29} +0.561553 q^{30} +10.2462 q^{31} +1.00000 q^{32} +1.43845 q^{33} -5.68466 q^{34} -0.561553 q^{35} +1.00000 q^{36} +1.68466 q^{37} +2.56155 q^{38} -0.561553 q^{40} +4.00000 q^{41} -1.00000 q^{42} +10.5616 q^{43} -1.43845 q^{44} -0.561553 q^{45} -5.68466 q^{46} -6.24621 q^{47} -1.00000 q^{48} +1.00000 q^{49} -4.68466 q^{50} +5.68466 q^{51} +13.1231 q^{53} -1.00000 q^{54} +0.807764 q^{55} +1.00000 q^{56} -2.56155 q^{57} -2.56155 q^{58} +12.2462 q^{59} +0.561553 q^{60} +2.56155 q^{61} +10.2462 q^{62} +1.00000 q^{63} +1.00000 q^{64} +1.43845 q^{66} +7.12311 q^{67} -5.68466 q^{68} +5.68466 q^{69} -0.561553 q^{70} -15.3693 q^{71} +1.00000 q^{72} -7.43845 q^{73} +1.68466 q^{74} +4.68466 q^{75} +2.56155 q^{76} -1.43845 q^{77} +16.0000 q^{79} -0.561553 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +3.19224 q^{85} +10.5616 q^{86} +2.56155 q^{87} -1.43845 q^{88} +8.00000 q^{89} -0.561553 q^{90} -5.68466 q^{92} -10.2462 q^{93} -6.24621 q^{94} -1.43845 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -1.43845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 3 q^{10} - 7 q^{11} - 2 q^{12} + 2 q^{14} - 3 q^{15} + 2 q^{16} + q^{17} + 2 q^{18} + q^{19} + 3 q^{20} - 2 q^{21} - 7 q^{22} + q^{23} - 2 q^{24} + 3 q^{25} - 2 q^{27} + 2 q^{28} - q^{29} - 3 q^{30} + 4 q^{31} + 2 q^{32} + 7 q^{33} + q^{34} + 3 q^{35} + 2 q^{36} - 9 q^{37} + q^{38} + 3 q^{40} + 8 q^{41} - 2 q^{42} + 17 q^{43} - 7 q^{44} + 3 q^{45} + q^{46} + 4 q^{47} - 2 q^{48} + 2 q^{49} + 3 q^{50} - q^{51} + 18 q^{53} - 2 q^{54} - 19 q^{55} + 2 q^{56} - q^{57} - q^{58} + 8 q^{59} - 3 q^{60} + q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 7 q^{66} + 6 q^{67} + q^{68} - q^{69} + 3 q^{70} - 6 q^{71} + 2 q^{72} - 19 q^{73} - 9 q^{74} - 3 q^{75} + q^{76} - 7 q^{77} + 32 q^{79} + 3 q^{80} + 2 q^{81} + 8 q^{82} - 4 q^{83} - 2 q^{84} + 27 q^{85} + 17 q^{86} + q^{87} - 7 q^{88} + 16 q^{89} + 3 q^{90} + q^{92} - 4 q^{93} + 4 q^{94} - 7 q^{95} - 2 q^{96} + 20 q^{97} + 2 q^{98} - 7 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 + 3 * q^5 - 2 * q^6 + 2 * q^7 + 2 * q^8 + 2 * q^9 + 3 * q^10 - 7 * q^11 - 2 * q^12 + 2 * q^14 - 3 * q^15 + 2 * q^16 + q^17 + 2 * q^18 + q^19 + 3 * q^20 - 2 * q^21 - 7 * q^22 + q^23 - 2 * q^24 + 3 * q^25 - 2 * q^27 + 2 * q^28 - q^29 - 3 * q^30 + 4 * q^31 + 2 * q^32 + 7 * q^33 + q^34 + 3 * q^35 + 2 * q^36 - 9 * q^37 + q^38 + 3 * q^40 + 8 * q^41 - 2 * q^42 + 17 * q^43 - 7 * q^44 + 3 * q^45 + q^46 + 4 * q^47 - 2 * q^48 + 2 * q^49 + 3 * q^50 - q^51 + 18 * q^53 - 2 * q^54 - 19 * q^55 + 2 * q^56 - q^57 - q^58 + 8 * q^59 - 3 * q^60 + q^61 + 4 * q^62 + 2 * q^63 + 2 * q^64 + 7 * q^66 + 6 * q^67 + q^68 - q^69 + 3 * q^70 - 6 * q^71 + 2 * q^72 - 19 * q^73 - 9 * q^74 - 3 * q^75 + q^76 - 7 * q^77 + 32 * q^79 + 3 * q^80 + 2 * q^81 + 8 * q^82 - 4 * q^83 - 2 * q^84 + 27 * q^85 + 17 * q^86 + q^87 - 7 * q^88 + 16 * q^89 + 3 * q^90 + q^92 - 4 * q^93 + 4 * q^94 - 7 * q^95 - 2 * q^96 + 20 * q^97 + 2 * q^98 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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