LMFDB - Embedded newform 7098.2.a.bv.1.2

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			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} -5.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} -3.56155 q^{15} +1.00000 q^{16} +6.68466 q^{17} +1.00000 q^{18} -1.56155 q^{19} +3.56155 q^{20} -1.00000 q^{21} -5.56155 q^{22} +6.68466 q^{23} -1.00000 q^{24} +7.68466 q^{25} -1.00000 q^{27} +1.00000 q^{28} +1.56155 q^{29} -3.56155 q^{30} -6.24621 q^{31} +1.00000 q^{32} +5.56155 q^{33} +6.68466 q^{34} +3.56155 q^{35} +1.00000 q^{36} -10.6847 q^{37} -1.56155 q^{38} +3.56155 q^{40} +4.00000 q^{41} -1.00000 q^{42} +6.43845 q^{43} -5.56155 q^{44} +3.56155 q^{45} +6.68466 q^{46} +10.2462 q^{47} -1.00000 q^{48} +1.00000 q^{49} +7.68466 q^{50} -6.68466 q^{51} +4.87689 q^{53} -1.00000 q^{54} -19.8078 q^{55} +1.00000 q^{56} +1.56155 q^{57} +1.56155 q^{58} -4.24621 q^{59} -3.56155 q^{60} -1.56155 q^{61} -6.24621 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.56155 q^{66} -1.12311 q^{67} +6.68466 q^{68} -6.68466 q^{69} +3.56155 q^{70} +9.36932 q^{71} +1.00000 q^{72} -11.5616 q^{73} -10.6847 q^{74} -7.68466 q^{75} -1.56155 q^{76} -5.56155 q^{77} +16.0000 q^{79} +3.56155 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{87} -5.56155 q^{88} +8.00000 q^{89} +3.56155 q^{90} +6.68466 q^{92} +6.24621 q^{93} +10.2462 q^{94} -5.56155 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -5.56155 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	−1.00000	−0.408248
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	3.56155	1.12626
$11$	−5.56155	−1.67687	−0.838436	−	0.545001i	$-0.816529\pi$
$11$	−5.56155	−1.67687	−0.838436	+	0.545001i	$0.816529\pi$
$12$	−1.00000	−0.288675
$13$	0	0
$13$	0	0
$14$	1.00000	0.267261
$15$	−3.56155	−0.919589
$16$	1.00000	0.250000
$17$	6.68466	1.62127	0.810634	−	0.585553i	$-0.199123\pi$
$17$	6.68466	1.62127	0.810634	+	0.585553i	$0.199123\pi$
$18$	1.00000	0.235702
$19$	−1.56155	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$19$	−1.56155	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$20$	3.56155	0.796387
$21$	−1.00000	−0.218218
$22$	−5.56155	−1.18573
$23$	6.68466	1.39385	0.696924	−	0.717145i	$-0.254552\pi$
$23$	6.68466	1.39385	0.696924	+	0.717145i	$0.254552\pi$
$24$	−1.00000	−0.204124
$25$	7.68466	1.53693
$26$	0	0
$27$	−1.00000	−0.192450
$28$	1.00000	0.188982
$29$	1.56155	0.289973	0.144987	−	0.989434i	$-0.453686\pi$
$29$	1.56155	0.289973	0.144987	+	0.989434i	$0.453686\pi$
$30$	−3.56155	−0.650248
$31$	−6.24621	−1.12185	−0.560926	−	0.827866i	$-0.689555\pi$
$31$	−6.24621	−1.12185	−0.560926	+	0.827866i	$0.689555\pi$
$32$	1.00000	0.176777
$33$	5.56155	0.968142
$34$	6.68466	1.14641
$35$	3.56155	0.602012
$36$	1.00000	0.166667
$37$	−10.6847	−1.75655	−0.878274	−	0.478159i	$-0.841304\pi$
$37$	−10.6847	−1.75655	−0.878274	+	0.478159i	$0.841304\pi$
$38$	−1.56155	−0.253317
$39$	0	0
$40$	3.56155	0.563131
$41$	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
$43$	6.43845	0.981854	0.490927	−	0.871201i	$-0.336658\pi$
$43$	6.43845	0.981854	0.490927	+	0.871201i	$0.336658\pi$
$44$	−5.56155	−0.838436
$45$	3.56155	0.530925
$46$	6.68466	0.985599
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	−1.00000	−0.144338
$49$	1.00000	0.142857
$50$	7.68466	1.08677
$51$	−6.68466	−0.936039
$52$	0	0
$53$	4.87689	0.669893	0.334946	−	0.942237i	$-0.391282\pi$
$53$	4.87689	0.669893	0.334946	+	0.942237i	$0.391282\pi$
$54$	−1.00000	−0.136083
$55$	−19.8078	−2.67088
$56$	1.00000	0.133631
$57$	1.56155	0.206833
$58$	1.56155	0.205042
$59$	−4.24621	−0.552810	−0.276405	−	0.961041i	$-0.589143\pi$
$59$	−4.24621	−0.552810	−0.276405	+	0.961041i	$0.589143\pi$
$60$	−3.56155	−0.459794
$61$	−1.56155	−0.199936	−0.0999682	−	0.994991i	$-0.531874\pi$
$61$	−1.56155	−0.199936	−0.0999682	+	0.994991i	$0.531874\pi$
$62$	−6.24621	−0.793270
$63$	1.00000	0.125988
$64$	1.00000	0.125000
$65$	0	0
$66$	5.56155	0.684580
$67$	−1.12311	−0.137209	−0.0686046	−	0.997644i	$-0.521855\pi$
$67$	−1.12311	−0.137209	−0.0686046	+	0.997644i	$0.521855\pi$
$68$	6.68466	0.810634
$69$	−6.68466	−0.804738
$70$	3.56155	0.425687
$71$	9.36932	1.11193	0.555967	−	0.831205i	$-0.312348\pi$
$71$	9.36932	1.11193	0.555967	+	0.831205i	$0.312348\pi$
$72$	1.00000	0.117851
$73$	−11.5616	−1.35318	−0.676589	−	0.736361i	$-0.736542\pi$
$73$	−11.5616	−1.35318	−0.676589	+	0.736361i	$0.736542\pi$
$74$	−10.6847	−1.24207
$75$	−7.68466	−0.887348
$76$	−1.56155	−0.179122
$77$	−5.56155	−0.633798
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	3.56155	0.398194
$81$	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
$85$	23.8078	2.58231
$86$	6.43845	0.694276
$87$	−1.56155	−0.167416
$88$	−5.56155	−0.592864
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	3.56155	0.375421
$91$	0	0
$92$	6.68466	0.696924
$93$	6.24621	0.647702
$94$	10.2462	1.05682
$95$	−5.56155	−0.570603
$96$	−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
$99$	−5.56155	−0.558957
$100$	7.68466	0.768466
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	−6.68466	−0.661880
$103$	−1.80776	−0.178124	−0.0890621	−	0.996026i	$-0.528387\pi$
$103$	−1.80776	−0.178124	−0.0890621	+	0.996026i	$0.528387\pi$
$104$	0	0
$105$	−3.56155	−0.347572
$106$	4.87689	0.473686
$107$	−4.87689	−0.471467	−0.235734	−	0.971818i	$-0.575749\pi$
$107$	−4.87689	−0.471467	−0.235734	+	0.971818i	$0.575749\pi$
$108$	−1.00000	−0.0962250
$109$	12.9309	1.23855	0.619276	−	0.785173i	$-0.287426\pi$
$109$	12.9309	1.23855	0.619276	+	0.785173i	$0.287426\pi$
$110$	−19.8078	−1.88860
$111$	10.6847	1.01414
$112$	1.00000	0.0944911
$113$	20.2462	1.90460	0.952302	−	0.305158i	$-0.0987093\pi$
$113$	20.2462	1.90460	0.952302	+	0.305158i	$0.0987093\pi$
$114$	1.56155	0.146253
$115$	23.8078	2.22009
$116$	1.56155	0.144987
$117$	0	0
$118$	−4.24621	−0.390895
$119$	6.68466	0.612782
$120$	−3.56155	−0.325124
$121$	19.9309	1.81190
$122$	−1.56155	−0.141376
$123$	−4.00000	−0.360668
$124$	−6.24621	−0.560926
$125$	9.56155	0.855211
$126$	1.00000	0.0890871
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	1.00000	0.0883883
$129$	−6.43845	−0.566874
$130$	0	0
$131$	−1.56155	−0.136434	−0.0682168	−	0.997671i	$-0.521731\pi$
$131$	−1.56155	−0.136434	−0.0682168	+	0.997671i	$0.521731\pi$
$132$	5.56155	0.484071
$133$	−1.56155	−0.135404
$134$	−1.12311	−0.0970215
$135$	−3.56155	−0.306530
$136$	6.68466	0.573205
$137$	−6.68466	−0.571109	−0.285554	−	0.958362i	$-0.592178\pi$
$137$	−6.68466	−0.571109	−0.285554	+	0.958362i	$0.592178\pi$
$138$	−6.68466	−0.569036
$139$	10.2462	0.869072	0.434536	−	0.900654i	$-0.356912\pi$
$139$	10.2462	0.869072	0.434536	+	0.900654i	$0.356912\pi$
$140$	3.56155	0.301006
$141$	−10.2462	−0.862887
$142$	9.36932	0.786256
$143$	0	0
$144$	1.00000	0.0833333
$145$	5.56155	0.461862
$146$	−11.5616	−0.956841
$147$	−1.00000	−0.0824786
$148$	−10.6847	−0.878274
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	−7.68466	−0.627450
$151$	−16.6847	−1.35778	−0.678889	−	0.734241i	$-0.737538\pi$
$151$	−16.6847	−1.35778	−0.678889	+	0.734241i	$0.737538\pi$
$152$	−1.56155	−0.126659
$153$	6.68466	0.540423
$154$	−5.56155	−0.448163
$155$	−22.2462	−1.78686
$156$	0	0
$157$	−11.8078	−0.942362	−0.471181	−	0.882037i	$-0.656172\pi$
$157$	−11.8078	−0.942362	−0.471181	+	0.882037i	$0.656172\pi$
$158$	16.0000	1.27289
$159$	−4.87689	−0.386763
$160$	3.56155	0.281565
$161$	6.68466	0.526825
$162$	1.00000	0.0785674
$163$	−9.12311	−0.714577	−0.357288	−	0.933994i	$-0.616299\pi$
$163$	−9.12311	−0.714577	−0.357288	+	0.933994i	$0.616299\pi$
$164$	4.00000	0.312348
$165$	19.8078	1.54203
$166$	−2.00000	−0.155230
$167$	11.8078	0.913712	0.456856	−	0.889541i	$-0.348975\pi$
$167$	11.8078	0.913712	0.456856	+	0.889541i	$0.348975\pi$
$168$	−1.00000	−0.0771517
$169$	0	0
$170$	23.8078	1.82597
$171$	−1.56155	−0.119415
$172$	6.43845	0.490927
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	−1.56155	−0.118381
$175$	7.68466	0.580906
$176$	−5.56155	−0.419218
$177$	4.24621	0.319165
$178$	8.00000	0.599625
$179$	−11.1231	−0.831380	−0.415690	−	0.909506i	$-0.636460\pi$
$179$	−11.1231	−0.831380	−0.415690	+	0.909506i	$0.636460\pi$
$180$	3.56155	0.265462
$181$	13.3693	0.993733	0.496867	−	0.867827i	$-0.334484\pi$
$181$	13.3693	0.993733	0.496867	+	0.867827i	$0.334484\pi$
$182$	0	0
$183$	1.56155	0.115433
$184$	6.68466	0.492800
$185$	−38.0540	−2.79778
$186$	6.24621	0.457994
$187$	−37.1771	−2.71866
$188$	10.2462	0.747282
$189$	−1.00000	−0.0727393
$190$	−5.56155	−0.403477
$191$	−24.0540	−1.74048	−0.870242	−	0.492624i	$-0.836038\pi$
$191$	−24.0540	−1.74048	−0.870242	+	0.492624i	$0.836038\pi$
$192$	−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	$0.568565\pi$
$198$	−5.56155	−0.395242
$199$	−4.93087	−0.349540	−0.174770	−	0.984609i	$-0.555918\pi$
$199$	−4.93087	−0.349540	−0.174770	+	0.984609i	$0.555918\pi$
$200$	7.68466	0.543387
$201$	1.12311	0.0792178
$202$	−6.00000	−0.422159
$203$	1.56155	0.109600
$204$	−6.68466	−0.468020
$205$	14.2462	0.994999
$206$	−1.80776	−0.125953
$207$	6.68466	0.464616
$208$	0	0
$209$	8.68466	0.600730
$210$	−3.56155	−0.245770
$211$	7.80776	0.537509	0.268754	−	0.963209i	$-0.413388\pi$
$211$	7.80776	0.537509	0.268754	+	0.963209i	$0.413388\pi$
$212$	4.87689	0.334946
$213$	−9.36932	−0.641975
$214$	−4.87689	−0.333378
$215$	22.9309	1.56387
$216$	−1.00000	−0.0680414
$217$	−6.24621	−0.424020
$218$	12.9309	0.875789
$219$	11.5616	0.781257
$220$	−19.8078	−1.33544
$221$	0	0
$222$	10.6847	0.717107
$223$	−1.75379	−0.117442	−0.0587212	−	0.998274i	$-0.518702\pi$
$223$	−1.75379	−0.117442	−0.0587212	+	0.998274i	$0.518702\pi$
$224$	1.00000	0.0668153
$225$	7.68466	0.512311
$226$	20.2462	1.34676
$227$	−17.6155	−1.16918	−0.584592	−	0.811328i	$-0.698745\pi$
$227$	−17.6155	−1.16918	−0.584592	+	0.811328i	$0.698745\pi$
$228$	1.56155	0.103416
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	23.8078	1.56984
$231$	5.56155	0.365923
$232$	1.56155	0.102521
$233$	−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
$235$	36.4924	2.38050
$236$	−4.24621	−0.276405
$237$	−16.0000	−1.03931
$238$	6.68466	0.433302
$239$	14.2462	0.921511	0.460755	−	0.887527i	$-0.347579\pi$
$239$	14.2462	0.921511	0.460755	+	0.887527i	$0.347579\pi$
$240$	−3.56155	−0.229897
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	19.9309	1.28120
$243$	−1.00000	−0.0641500
$244$	−1.56155	−0.0999682
$245$	3.56155	0.227539
$246$	−4.00000	−0.255031
$247$	0	0
$248$	−6.24621	−0.396635
$249$	2.00000	0.126745
$250$	9.56155	0.604726
$251$	−22.0540	−1.39203	−0.696017	−	0.718025i	$-0.745046\pi$
$251$	−22.0540	−1.39203	−0.696017	+	0.718025i	$0.745046\pi$
$252$	1.00000	0.0629941
$253$	−37.1771	−2.33730
$254$	−10.2462	−0.642904
$255$	−23.8078	−1.49090
$256$	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	$0.198968\pi$
$258$	−6.43845	−0.400840
$259$	−10.6847	−0.663912
$260$	0	0
$261$	1.56155	0.0966577
$262$	−1.56155	−0.0964731
$263$	−23.3693	−1.44101	−0.720507	−	0.693448i	$-0.756091\pi$
$263$	−23.3693	−1.44101	−0.720507	+	0.693448i	$0.756091\pi$
$264$	5.56155	0.342290
$265$	17.3693	1.06699
$266$	−1.56155	−0.0957449
$267$	−8.00000	−0.489592
$268$	−1.12311	−0.0686046
$269$	31.3693	1.91262	0.956311	−	0.292353i	$-0.0944382\pi$
$269$	31.3693	1.91262	0.956311	+	0.292353i	$0.0944382\pi$
$270$	−3.56155	−0.216749
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	6.68466	0.405317
$273$	0	0
$274$	−6.68466	−0.403835
$275$	−42.7386	−2.57724
$276$	−6.68466	−0.402369
$277$	10.8769	0.653529	0.326765	−	0.945106i	$-0.394042\pi$
$277$	10.8769	0.653529	0.326765	+	0.945106i	$0.394042\pi$
$278$	10.2462	0.614527
$279$	−6.24621	−0.373951
$280$	3.56155	0.212843
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−10.2462	−0.610153
$283$	4.87689	0.289901	0.144951	−	0.989439i	$-0.453698\pi$
$283$	4.87689	0.289901	0.144951	+	0.989439i	$0.453698\pi$
$284$	9.36932	0.555967
$285$	5.56155	0.329438
$286$	0	0
$287$	4.00000	0.236113
$288$	1.00000	0.0589256
$289$	27.6847	1.62851
$290$	5.56155	0.326586
$291$	−10.0000	−0.586210
$292$	−11.5616	−0.676589
$293$	7.75379	0.452981	0.226491	−	0.974013i	$-0.427275\pi$
$293$	7.75379	0.452981	0.226491	+	0.974013i	$0.427275\pi$
$294$	−1.00000	−0.0583212
$295$	−15.1231	−0.880501
$296$	−10.6847	−0.621033
$297$	5.56155	0.322714
$298$	10.0000	0.579284
$299$	0	0
$300$	−7.68466	−0.443674
$301$	6.43845	0.371106
$302$	−16.6847	−0.960094
$303$	6.00000	0.344691
$304$	−1.56155	−0.0895612
$305$	−5.56155	−0.318454
$306$	6.68466	0.382136
$307$	−26.2462	−1.49795	−0.748975	−	0.662598i	$-0.769454\pi$
$307$	−26.2462	−1.49795	−0.748975	+	0.662598i	$0.769454\pi$
$308$	−5.56155	−0.316899
$309$	1.80776	0.102840
$310$	−22.2462	−1.26350
$311$	0.492423	0.0279227	0.0139614	−	0.999903i	$-0.495556\pi$
$311$	0.492423	0.0279227	0.0139614	+	0.999903i	$0.495556\pi$
$312$	0	0
$313$	32.2462	1.82266	0.911332	−	0.411673i	$-0.135055\pi$
$313$	32.2462	1.82266	0.911332	+	0.411673i	$0.135055\pi$
$314$	−11.8078	−0.666351
$315$	3.56155	0.200671
$316$	16.0000	0.900070
$317$	−3.36932	−0.189240	−0.0946198	−	0.995513i	$-0.530164\pi$
$317$	−3.36932	−0.189240	−0.0946198	+	0.995513i	$0.530164\pi$
$318$	−4.87689	−0.273483
$319$	−8.68466	−0.486248
$320$	3.56155	0.199097
$321$	4.87689	0.272202
$322$	6.68466	0.372521
$323$	−10.4384	−0.580811
$324$	1.00000	0.0555556
$325$	0	0
$326$	−9.12311	−0.505282
$327$	−12.9309	−0.715079
$328$	4.00000	0.220863
$329$	10.2462	0.564892
$330$	19.8078	1.09038
$331$	−1.12311	−0.0617315	−0.0308657	−	0.999524i	$-0.509826\pi$
$331$	−1.12311	−0.0617315	−0.0308657	+	0.999524i	$0.509826\pi$
$332$	−2.00000	−0.109764
$333$	−10.6847	−0.585516
$334$	11.8078	0.646092
$335$	−4.00000	−0.218543
$336$	−1.00000	−0.0545545
$337$	−20.0540	−1.09241	−0.546205	−	0.837652i	$-0.683928\pi$
$337$	−20.0540	−1.09241	−0.546205	+	0.837652i	$0.683928\pi$
$338$	0	0
$339$	−20.2462	−1.09962
$340$	23.8078	1.29116
$341$	34.7386	1.88120
$342$	−1.56155	−0.0844391
$343$	1.00000	0.0539949
$344$	6.43845	0.347138
$345$	−23.8078	−1.28177
$346$	3.75379	0.201805
$347$	−24.4924	−1.31482	−0.657411	−	0.753532i	$-0.728348\pi$
$347$	−24.4924	−1.31482	−0.657411	+	0.753532i	$0.728348\pi$
$348$	−1.56155	−0.0837080
$349$	−3.36932	−0.180355	−0.0901777	−	0.995926i	$-0.528744\pi$
$349$	−3.36932	−0.180355	−0.0901777	+	0.995926i	$0.528744\pi$
$350$	7.68466	0.410762
$351$	0	0
$352$	−5.56155	−0.296432
$353$	−18.2462	−0.971148	−0.485574	−	0.874196i	$-0.661389\pi$
$353$	−18.2462	−0.971148	−0.485574	+	0.874196i	$0.661389\pi$
$354$	4.24621	0.225684
$355$	33.3693	1.77106
$356$	8.00000	0.423999
$357$	−6.68466	−0.353790
$358$	−11.1231	−0.587874
$359$	−23.6155	−1.24638	−0.623190	−	0.782071i	$-0.714164\pi$
$359$	−23.6155	−1.24638	−0.623190	+	0.782071i	$0.714164\pi$
$360$	3.56155	0.187710
$361$	−16.5616	−0.871661
$362$	13.3693	0.702676
$363$	−19.9309	−1.04610
$364$	0	0
$365$	−41.1771	−2.15531
$366$	1.56155	0.0816237
$367$	0.630683	0.0329214	0.0164607	−	0.999865i	$-0.494760\pi$
$367$	0.630683	0.0329214	0.0164607	+	0.999865i	$0.494760\pi$
$368$	6.68466	0.348462
$369$	4.00000	0.208232
$370$	−38.0540	−1.97833
$371$	4.87689	0.253196
$372$	6.24621	0.323851
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	−37.1771	−1.92238
$375$	−9.56155	−0.493756
$376$	10.2462	0.528408
$377$	0	0
$378$	−1.00000	−0.0514344
$379$	−18.8769	−0.969641	−0.484820	−	0.874614i	$-0.661115\pi$
$379$	−18.8769	−0.969641	−0.484820	+	0.874614i	$0.661115\pi$
$380$	−5.56155	−0.285302
$381$	10.2462	0.524929
$382$	−24.0540	−1.23071
$383$	17.5616	0.897353	0.448677	−	0.893694i	$-0.351895\pi$
$383$	17.5616	0.897353	0.448677	+	0.893694i	$0.351895\pi$
$384$	−1.00000	−0.0510310
$385$	−19.8078	−1.00950
$386$	12.0000	0.610784
$387$	6.43845	0.327285
$388$	10.0000	0.507673
$389$	−27.1231	−1.37520	−0.687598	−	0.726092i	$-0.741335\pi$
$389$	−27.1231	−1.37520	−0.687598	+	0.726092i	$0.741335\pi$
$390$	0	0
$391$	44.6847	2.25980
$392$	1.00000	0.0505076
$393$	1.56155	0.0787699
$394$	−6.00000	−0.302276
$395$	56.9848	2.86722
$396$	−5.56155	−0.279479
$397$	−15.3693	−0.771364	−0.385682	−	0.922632i	$-0.626034\pi$
$397$	−15.3693	−0.771364	−0.385682	+	0.922632i	$0.626034\pi$
$398$	−4.93087	−0.247162
$399$	1.56155	0.0781754
$400$	7.68466	0.384233
$401$	30.9848	1.54731	0.773655	−	0.633608i	$-0.218427\pi$
$401$	30.9848	1.54731	0.773655	+	0.633608i	$0.218427\pi$
$402$	1.12311	0.0560154
$403$	0	0
$404$	−6.00000	−0.298511
$405$	3.56155	0.176975
$406$	1.56155	0.0774986
$407$	59.4233	2.94550
$408$	−6.68466	−0.330940
$409$	13.8078	0.682750	0.341375	−	0.939927i	$-0.389107\pi$
$409$	13.8078	0.682750	0.341375	+	0.939927i	$0.389107\pi$
$410$	14.2462	0.703570
$411$	6.68466	0.329730
$412$	−1.80776	−0.0890621
$413$	−4.24621	−0.208942
$414$	6.68466	0.328533
$415$	−7.12311	−0.349660
$416$	0	0
$417$	−10.2462	−0.501759
$418$	8.68466	0.424781
$419$	−20.6847	−1.01051	−0.505256	−	0.862970i	$-0.668602\pi$
$419$	−20.6847	−1.01051	−0.505256	+	0.862970i	$0.668602\pi$
$420$	−3.56155	−0.173786
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	7.80776	0.380076
$423$	10.2462	0.498188
$424$	4.87689	0.236843
$425$	51.3693	2.49178
$426$	−9.36932	−0.453945
$427$	−1.56155	−0.0755688
$428$	−4.87689	−0.235734
$429$	0	0
$430$	22.9309	1.10582
$431$	29.8617	1.43839	0.719195	−	0.694809i	$-0.244511\pi$
$431$	29.8617	1.43839	0.719195	+	0.694809i	$0.244511\pi$
$432$	−1.00000	−0.0481125
$433$	−23.3693	−1.12306	−0.561529	−	0.827457i	$-0.689787\pi$
$433$	−23.3693	−1.12306	−0.561529	+	0.827457i	$0.689787\pi$
$434$	−6.24621	−0.299828
$435$	−5.56155	−0.266656
$436$	12.9309	0.619276
$437$	−10.4384	−0.499339
$438$	11.5616	0.552432
$439$	8.93087	0.426247	0.213124	−	0.977025i	$-0.431636\pi$
$439$	8.93087	0.426247	0.213124	+	0.977025i	$0.431636\pi$
$440$	−19.8078	−0.944298
$441$	1.00000	0.0476190
$442$	0	0
$443$	6.63068	0.315033	0.157517	−	0.987516i	$-0.449651\pi$
$443$	6.63068	0.315033	0.157517	+	0.987516i	$0.449651\pi$
$444$	10.6847	0.507071
$445$	28.4924	1.35067
$446$	−1.75379	−0.0830443
$447$	−10.0000	−0.472984
$448$	1.00000	0.0472456
$449$	9.31534	0.439618	0.219809	−	0.975543i	$-0.429457\pi$
$449$	9.31534	0.439618	0.219809	+	0.975543i	$0.429457\pi$
$450$	7.68466	0.362258
$451$	−22.2462	−1.04753
$452$	20.2462	0.952302
$453$	16.6847	0.783914
$454$	−17.6155	−0.826738
$455$	0	0
$456$	1.56155	0.0731264
$457$	−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
$459$	−6.68466	−0.312013
$460$	23.8078	1.11004
$461$	36.9309	1.72004	0.860021	−	0.510259i	$-0.170450\pi$
$461$	36.9309	1.72004	0.860021	+	0.510259i	$0.170450\pi$
$462$	5.56155	0.258747
$463$	30.5464	1.41961	0.709806	−	0.704397i	$-0.248783\pi$
$463$	30.5464	1.41961	0.709806	+	0.704397i	$0.248783\pi$
$464$	1.56155	0.0724933
$465$	22.2462	1.03164
$466$	−2.00000	−0.0926482
$467$	−1.56155	−0.0722600	−0.0361300	−	0.999347i	$-0.511503\pi$
$467$	−1.56155	−0.0722600	−0.0361300	+	0.999347i	$0.511503\pi$
$468$	0	0
$469$	−1.12311	−0.0518602
$470$	36.4924	1.68327
$471$	11.8078	0.544073
$472$	−4.24621	−0.195448
$473$	−35.8078	−1.64644
$474$	−16.0000	−0.734904
$475$	−12.0000	−0.550598
$476$	6.68466	0.306391
$477$	4.87689	0.223298
$478$	14.2462	0.651607
$479$	−12.3002	−0.562010	−0.281005	−	0.959706i	$-0.590668\pi$
$479$	−12.3002	−0.562010	−0.281005	+	0.959706i	$0.590668\pi$
$480$	−3.56155	−0.162562
$481$	0	0
$482$	−6.00000	−0.273293
$483$	−6.68466	−0.304162
$484$	19.9309	0.905949
$485$	35.6155	1.61722
$486$	−1.00000	−0.0453609
$487$	−13.7538	−0.623244	−0.311622	−	0.950206i	$-0.600872\pi$
$487$	−13.7538	−0.623244	−0.311622	+	0.950206i	$0.600872\pi$
$488$	−1.56155	−0.0706882
$489$	9.12311	0.412561
$490$	3.56155	0.160895
$491$	10.2462	0.462405	0.231203	−	0.972906i	$-0.425734\pi$
$491$	10.2462	0.462405	0.231203	+	0.972906i	$0.425734\pi$
$492$	−4.00000	−0.180334
$493$	10.4384	0.470124
$494$	0	0
$495$	−19.8078	−0.890293
$496$	−6.24621	−0.280463
$497$	9.36932	0.420271
$498$	2.00000	0.0896221
$499$	−1.50758	−0.0674884	−0.0337442	−	0.999431i	$-0.510743\pi$
$499$	−1.50758	−0.0674884	−0.0337442	+	0.999431i	$0.510743\pi$
$500$	9.56155	0.427606
$501$	−11.8078	−0.527532
$502$	−22.0540	−0.984317
$503$	35.6155	1.58802	0.794009	−	0.607906i	$-0.207990\pi$
$503$	35.6155	1.58802	0.794009	+	0.607906i	$0.207990\pi$
$504$	1.00000	0.0445435
$505$	−21.3693	−0.950922
$506$	−37.1771	−1.65272
$507$	0	0
$508$	−10.2462	−0.454602
$509$	2.68466	0.118995	0.0594977	−	0.998228i	$-0.481050\pi$
$509$	2.68466	0.118995	0.0594977	+	0.998228i	$0.481050\pi$
$510$	−23.8078	−1.05423
$511$	−11.5616	−0.511453
$512$	1.00000	0.0441942
$513$	1.56155	0.0689442
$514$	26.0000	1.14681
$515$	−6.43845	−0.283712
$516$	−6.43845	−0.283437
$517$	−56.9848	−2.50619
$518$	−10.6847	−0.469457
$519$	−3.75379	−0.164773
$520$	0	0
$521$	8.05398	0.352851	0.176426	−	0.984314i	$-0.443547\pi$
$521$	8.05398	0.352851	0.176426	+	0.984314i	$0.443547\pi$
$522$	1.56155	0.0683473
$523$	−8.49242	−0.371348	−0.185674	−	0.982611i	$-0.559447\pi$
$523$	−8.49242	−0.371348	−0.185674	+	0.982611i	$0.559447\pi$
$524$	−1.56155	−0.0682168
$525$	−7.68466	−0.335386
$526$	−23.3693	−1.01895
$527$	−41.7538	−1.81882
$528$	5.56155	0.242036
$529$	21.6847	0.942811
$530$	17.3693	0.754475
$531$	−4.24621	−0.184270
$532$	−1.56155	−0.0677019
$533$	0	0
$534$	−8.00000	−0.346194
$535$	−17.3693	−0.750941
$536$	−1.12311	−0.0485108
$537$	11.1231	0.479997
$538$	31.3693	1.35243
$539$	−5.56155	−0.239553
$540$	−3.56155	−0.153265
$541$	6.19224	0.266225	0.133113	−	0.991101i	$-0.457503\pi$
$541$	6.19224	0.266225	0.133113	+	0.991101i	$0.457503\pi$
$542$	8.00000	0.343629
$543$	−13.3693	−0.573732
$544$	6.68466	0.286602
$545$	46.0540	1.97274
$546$	0	0
$547$	28.9848	1.23930	0.619651	−	0.784877i	$-0.287274\pi$
$547$	28.9848	1.23930	0.619651	+	0.784877i	$0.287274\pi$
$548$	−6.68466	−0.285554
$549$	−1.56155	−0.0666455
$550$	−42.7386	−1.82238
$551$	−2.43845	−0.103881
$552$	−6.68466	−0.284518
$553$	16.0000	0.680389
$554$	10.8769	0.462115
$555$	38.0540	1.61530
$556$	10.2462	0.434536
$557$	−16.2462	−0.688374	−0.344187	−	0.938901i	$-0.611845\pi$
$557$	−16.2462	−0.688374	−0.344187	+	0.938901i	$0.611845\pi$
$558$	−6.24621	−0.264423
$559$	0	0
$560$	3.56155	0.150503
$561$	37.1771	1.56962
$562$	−6.00000	−0.253095
$563$	−38.0540	−1.60378	−0.801892	−	0.597469i	$-0.796173\pi$
$563$	−38.0540	−1.60378	−0.801892	+	0.597469i	$0.796173\pi$
$564$	−10.2462	−0.431443
$565$	72.1080	3.03360
$566$	4.87689	0.204991
$567$	1.00000	0.0419961
$568$	9.36932	0.393128
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	5.56155	0.232948
$571$	18.2462	0.763580	0.381790	−	0.924249i	$-0.375308\pi$
$571$	18.2462	0.763580	0.381790	+	0.924249i	$0.375308\pi$
$572$	0	0
$573$	24.0540	1.00487
$574$	4.00000	0.166957
$575$	51.3693	2.14225
$576$	1.00000	0.0416667
$577$	7.75379	0.322794	0.161397	−	0.986890i	$-0.448400\pi$
$577$	7.75379	0.322794	0.161397	+	0.986890i	$0.448400\pi$
$578$	27.6847	1.15153
$579$	−12.0000	−0.498703
$580$	5.56155	0.230931
$581$	−2.00000	−0.0829740
$582$	−10.0000	−0.414513
$583$	−27.1231	−1.12332
$584$	−11.5616	−0.478420
$585$	0	0
$586$	7.75379	0.320306
$587$	−37.1231	−1.53223	−0.766117	−	0.642701i	$-0.777814\pi$
$587$	−37.1231	−1.53223	−0.766117	+	0.642701i	$0.777814\pi$
$588$	−1.00000	−0.0412393
$589$	9.75379	0.401898
$590$	−15.1231	−0.622608
$591$	6.00000	0.246807
$592$	−10.6847	−0.439137
$593$	28.0000	1.14982	0.574911	−	0.818216i	$-0.305037\pi$
$593$	28.0000	1.14982	0.574911	+	0.818216i	$0.305037\pi$
$594$	5.56155	0.228193
$595$	23.8078	0.976023
$596$	10.0000	0.409616
$597$	4.93087	0.201807
$598$	0	0
$599$	30.3002	1.23803	0.619016	−	0.785378i	$-0.287532\pi$
$599$	30.3002	1.23803	0.619016	+	0.785378i	$0.287532\pi$
$600$	−7.68466	−0.313725
$601$	−31.8617	−1.29967	−0.649834	−	0.760076i	$-0.725161\pi$
$601$	−31.8617	−1.29967	−0.649834	+	0.760076i	$0.725161\pi$
$602$	6.43845	0.262412
$603$	−1.12311	−0.0457364
$604$	−16.6847	−0.678889
$605$	70.9848	2.88594
$606$	6.00000	0.243733
$607$	−28.5464	−1.15866	−0.579331	−	0.815092i	$-0.696686\pi$
$607$	−28.5464	−1.15866	−0.579331	+	0.815092i	$0.696686\pi$
$608$	−1.56155	−0.0633293
$609$	−1.56155	−0.0632773
$610$	−5.56155	−0.225181
$611$	0	0
$612$	6.68466	0.270211
$613$	−1.80776	−0.0730149	−0.0365075	−	0.999333i	$-0.511623\pi$
$613$	−1.80776	−0.0730149	−0.0365075	+	0.999333i	$0.511623\pi$
$614$	−26.2462	−1.05921
$615$	−14.2462	−0.574463
$616$	−5.56155	−0.224081
$617$	6.19224	0.249290	0.124645	−	0.992201i	$-0.460221\pi$
$617$	6.19224	0.249290	0.124645	+	0.992201i	$0.460221\pi$
$618$	1.80776	0.0727189
$619$	30.9309	1.24322	0.621608	−	0.783328i	$-0.286480\pi$
$619$	30.9309	1.24322	0.621608	+	0.783328i	$0.286480\pi$
$620$	−22.2462	−0.893429
$621$	−6.68466	−0.268246
$622$	0.492423	0.0197443
$623$	8.00000	0.320513
$624$	0	0
$625$	−4.36932	−0.174773
$626$	32.2462	1.28882
$627$	−8.68466	−0.346832
$628$	−11.8078	−0.471181
$629$	−71.4233	−2.84783
$630$	3.56155	0.141896
$631$	−12.6847	−0.504968	−0.252484	−	0.967601i	$-0.581248\pi$
$631$	−12.6847	−0.504968	−0.252484	+	0.967601i	$0.581248\pi$
$632$	16.0000	0.636446
$633$	−7.80776	−0.310331
$634$	−3.36932	−0.133813
$635$	−36.4924	−1.44816
$636$	−4.87689	−0.193381
$637$	0	0
$638$	−8.68466	−0.343829
$639$	9.36932	0.370644
$640$	3.56155	0.140783
$641$	−30.1080	−1.18919	−0.594596	−	0.804024i	$-0.702688\pi$
$641$	−30.1080	−1.18919	−0.594596	+	0.804024i	$0.702688\pi$
$642$	4.87689	0.192476
$643$	−0.192236	−0.00758105	−0.00379052	−	0.999993i	$-0.501207\pi$
$643$	−0.192236	−0.00758105	−0.00379052	+	0.999993i	$0.501207\pi$
$644$	6.68466	0.263412
$645$	−22.9309	−0.902902
$646$	−10.4384	−0.410695
$647$	−10.6307	−0.417935	−0.208968	−	0.977923i	$-0.567010\pi$
$647$	−10.6307	−0.417935	−0.208968	+	0.977923i	$0.567010\pi$
$648$	1.00000	0.0392837
$649$	23.6155	0.926991
$650$	0	0
$651$	6.24621	0.244808
$652$	−9.12311	−0.357288
$653$	−29.5616	−1.15683	−0.578416	−	0.815742i	$-0.696329\pi$
$653$	−29.5616	−1.15683	−0.578416	+	0.815742i	$0.696329\pi$
$654$	−12.9309	−0.505637
$655$	−5.56155	−0.217308
$656$	4.00000	0.156174
$657$	−11.5616	−0.451059
$658$	10.2462	0.399439
$659$	12.8769	0.501613	0.250806	−	0.968037i	$-0.419304\pi$
$659$	12.8769	0.501613	0.250806	+	0.968037i	$0.419304\pi$
$660$	19.8078	0.771016
$661$	44.7386	1.74013	0.870066	−	0.492936i	$-0.164076\pi$
$661$	44.7386	1.74013	0.870066	+	0.492936i	$0.164076\pi$
$662$	−1.12311	−0.0436507
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	−5.56155	−0.215668
$666$	−10.6847	−0.414022
$667$	10.4384	0.404178
$668$	11.8078	0.456856
$669$	1.75379	0.0678054
$670$	−4.00000	−0.154533
$671$	8.68466	0.335268
$672$	−1.00000	−0.0385758
$673$	−37.8078	−1.45738	−0.728691	−	0.684843i	$-0.759871\pi$
$673$	−37.8078	−1.45738	−0.728691	+	0.684843i	$0.759871\pi$
$674$	−20.0540	−0.772450
$675$	−7.68466	−0.295783
$676$	0	0
$677$	31.3693	1.20562	0.602810	−	0.797884i	$-0.294048\pi$
$677$	31.3693	1.20562	0.602810	+	0.797884i	$0.294048\pi$
$678$	−20.2462	−0.777551
$679$	10.0000	0.383765
$680$	23.8078	0.912986
$681$	17.6155	0.675029
$682$	34.7386	1.33021
$683$	−0.684658	−0.0261977	−0.0130989	−	0.999914i	$-0.504170\pi$
$683$	−0.684658	−0.0261977	−0.0130989	+	0.999914i	$0.504170\pi$
$684$	−1.56155	−0.0597075
$685$	−23.8078	−0.909648
$686$	1.00000	0.0381802
$687$	2.00000	0.0763048
$688$	6.43845	0.245463
$689$	0	0
$690$	−23.8078	−0.906346
$691$	36.4924	1.38824	0.694119	−	0.719861i	$-0.255794\pi$
$691$	36.4924	1.38824	0.694119	+	0.719861i	$0.255794\pi$
$692$	3.75379	0.142698
$693$	−5.56155	−0.211266
$694$	−24.4924	−0.929720
$695$	36.4924	1.38424
$696$	−1.56155	−0.0591905
$697$	26.7386	1.01280
$698$	−3.36932	−0.127531
$699$	2.00000	0.0756469
$700$	7.68466	0.290453
$701$	3.61553	0.136557	0.0682783	−	0.997666i	$-0.478249\pi$
$701$	3.61553	0.136557	0.0682783	+	0.997666i	$0.478249\pi$
$702$	0	0
$703$	16.6847	0.629274
$704$	−5.56155	−0.209609
$705$	−36.4924	−1.37438
$706$	−18.2462	−0.686705
$707$	−6.00000	−0.225653
$708$	4.24621	0.159582
$709$	−31.7538	−1.19254	−0.596269	−	0.802784i	$-0.703351\pi$
$709$	−31.7538	−1.19254	−0.596269	+	0.802784i	$0.703351\pi$
$710$	33.3693	1.25233
$711$	16.0000	0.600047
$712$	8.00000	0.299813
$713$	−41.7538	−1.56369
$714$	−6.68466	−0.250167
$715$	0	0
$716$	−11.1231	−0.415690
$717$	−14.2462	−0.532035
$718$	−23.6155	−0.881324
$719$	−13.7538	−0.512930	−0.256465	−	0.966554i	$-0.582558\pi$
$719$	−13.7538	−0.512930	−0.256465	+	0.966554i	$0.582558\pi$
$720$	3.56155	0.132731
$721$	−1.80776	−0.0673247
$722$	−16.5616	−0.616357
$723$	6.00000	0.223142
$724$	13.3693	0.496867
$725$	12.0000	0.445669
$726$	−19.9309	−0.739704
$727$	48.9309	1.81475	0.907373	−	0.420327i	$-0.138085\pi$
$727$	48.9309	1.81475	0.907373	+	0.420327i	$0.138085\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−41.1771	−1.52403
$731$	43.0388	1.59185
$732$	1.56155	0.0577167
$733$	−22.8769	−0.844977	−0.422489	−	0.906368i	$-0.638843\pi$
$733$	−22.8769	−0.844977	−0.422489	+	0.906368i	$0.638843\pi$
$734$	0.630683	0.0232789
$735$	−3.56155	−0.131370
$736$	6.68466	0.246400
$737$	6.24621	0.230082
$738$	4.00000	0.147242
$739$	17.6155	0.647998	0.323999	−	0.946057i	$-0.394973\pi$
$739$	17.6155	0.647998	0.323999	+	0.946057i	$0.394973\pi$
$740$	−38.0540	−1.39889
$741$	0	0
$742$	4.87689	0.179036
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	6.24621	0.228997
$745$	35.6155	1.30485
$746$	−6.00000	−0.219676
$747$	−2.00000	−0.0731762
$748$	−37.1771	−1.35933
$749$	−4.87689	−0.178198
$750$	−9.56155	−0.349139
$751$	−2.24621	−0.0819654	−0.0409827	−	0.999160i	$-0.513049\pi$
$751$	−2.24621	−0.0819654	−0.0409827	+	0.999160i	$0.513049\pi$
$752$	10.2462	0.373641
$753$	22.0540	0.803692
$754$	0	0
$755$	−59.4233	−2.16264
$756$	−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	$0.257361\pi$
$758$	−18.8769	−0.685640
$759$	37.1771	1.34944
$760$	−5.56155	−0.201739
$761$	39.1231	1.41821	0.709106	−	0.705102i	$-0.249099\pi$
$761$	39.1231	1.41821	0.709106	+	0.705102i	$0.249099\pi$
$762$	10.2462	0.371181
$763$	12.9309	0.468129
$764$	−24.0540	−0.870242
$765$	23.8078	0.860772
$766$	17.5616	0.634525
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	−8.43845	−0.304298	−0.152149	−	0.988358i	$-0.548619\pi$
$769$	−8.43845	−0.304298	−0.152149	+	0.988358i	$0.548619\pi$
$770$	−19.8078	−0.713822
$771$	−26.0000	−0.936367
$772$	12.0000	0.431889
$773$	12.9309	0.465091	0.232546	−	0.972586i	$-0.425295\pi$
$773$	12.9309	0.465091	0.232546	+	0.972586i	$0.425295\pi$
$774$	6.43845	0.231425
$775$	−48.0000	−1.72421
$776$	10.0000	0.358979
$777$	10.6847	0.383310
$778$	−27.1231	−0.972410
$779$	−6.24621	−0.223794
$780$	0	0
$781$	−52.1080	−1.86457
$782$	44.6847	1.59792
$783$	−1.56155	−0.0558053
$784$	1.00000	0.0357143
$785$	−42.0540	−1.50097
$786$	1.56155	0.0556987
$787$	−17.0691	−0.608449	−0.304224	−	0.952600i	$-0.598397\pi$
$787$	−17.0691	−0.608449	−0.304224	+	0.952600i	$0.598397\pi$
$788$	−6.00000	−0.213741
$789$	23.3693	0.831970
$790$	56.9848	2.02743
$791$	20.2462	0.719872
$792$	−5.56155	−0.197621
$793$	0	0
$794$	−15.3693	−0.545437
$795$	−17.3693	−0.616026
$796$	−4.93087	−0.174770
$797$	−3.75379	−0.132966	−0.0664830	−	0.997788i	$-0.521178\pi$
$797$	−3.75379	−0.132966	−0.0664830	+	0.997788i	$0.521178\pi$
$798$	1.56155	0.0552784
$799$	68.4924	2.42309
$800$	7.68466	0.271694
$801$	8.00000	0.282666
$802$	30.9848	1.09411
$803$	64.3002	2.26910
$804$	1.12311	0.0396089
$805$	23.8078	0.839113
$806$	0	0
$807$	−31.3693	−1.10425
$808$	−6.00000	−0.211079
$809$	−55.3693	−1.94668	−0.973341	−	0.229364i	$-0.926335\pi$
$809$	−55.3693	−1.94668	−0.973341	+	0.229364i	$0.926335\pi$
$810$	3.56155	0.125140
$811$	−2.05398	−0.0721248	−0.0360624	−	0.999350i	$-0.511482\pi$
$811$	−2.05398	−0.0721248	−0.0360624	+	0.999350i	$0.511482\pi$
$812$	1.56155	0.0547998
$813$	−8.00000	−0.280572
$814$	59.4233	2.08279
$815$	−32.4924	−1.13816
$816$	−6.68466	−0.234010
$817$	−10.0540	−0.351744
$818$	13.8078	0.482777
$819$	0	0
$820$	14.2462	0.497499
$821$	−30.8769	−1.07761	−0.538806	−	0.842430i	$-0.681124\pi$
$821$	−30.8769	−1.07761	−0.538806	+	0.842430i	$0.681124\pi$
$822$	6.68466	0.233154
$823$	30.7386	1.07148	0.535741	−	0.844383i	$-0.320032\pi$
$823$	30.7386	1.07148	0.535741	+	0.844383i	$0.320032\pi$
$824$	−1.80776	−0.0629764
$825$	42.7386	1.48797
$826$	−4.24621	−0.147745
$827$	−22.0540	−0.766892	−0.383446	−	0.923563i	$-0.625263\pi$
$827$	−22.0540	−0.766892	−0.383446	+	0.923563i	$0.625263\pi$
$828$	6.68466	0.232308
$829$	−42.0540	−1.46059	−0.730297	−	0.683129i	$-0.760619\pi$
$829$	−42.0540	−1.46059	−0.730297	+	0.683129i	$0.760619\pi$
$830$	−7.12311	−0.247247
$831$	−10.8769	−0.377315
$832$	0	0
$833$	6.68466	0.231610
$834$	−10.2462	−0.354797
$835$	42.0540	1.45534
$836$	8.68466	0.300365
$837$	6.24621	0.215901
$838$	−20.6847	−0.714540
$839$	−25.7538	−0.889120	−0.444560	−	0.895749i	$-0.646640\pi$
$839$	−25.7538	−0.889120	−0.444560	+	0.895749i	$0.646640\pi$
$840$	−3.56155	−0.122885
$841$	−26.5616	−0.915916
$842$	−10.0000	−0.344623
$843$	6.00000	0.206651
$844$	7.80776	0.268754
$845$	0	0
$846$	10.2462	0.352272
$847$	19.9309	0.684833
$848$	4.87689	0.167473
$849$	−4.87689	−0.167375
$850$	51.3693	1.76195
$851$	−71.4233	−2.44836
$852$	−9.36932	−0.320988
$853$	−10.8769	−0.372418	−0.186209	−	0.982510i	$-0.559620\pi$
$853$	−10.8769	−0.372418	−0.186209	+	0.982510i	$0.559620\pi$
$854$	−1.56155	−0.0534352
$855$	−5.56155	−0.190201
$856$	−4.87689	−0.166689
$857$	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
$859$	0.384472	0.0131180	0.00655901	−	0.999978i	$-0.497912\pi$
$859$	0.384472	0.0131180	0.00655901	+	0.999978i	$0.497912\pi$
$860$	22.9309	0.781936
$861$	−4.00000	−0.136320
$862$	29.8617	1.01709
$863$	−42.7386	−1.45484	−0.727420	−	0.686192i	$-0.759281\pi$
$863$	−42.7386	−1.45484	−0.727420	+	0.686192i	$0.759281\pi$
$864$	−1.00000	−0.0340207
$865$	13.3693	0.454570
$866$	−23.3693	−0.794122
$867$	−27.6847	−0.940220
$868$	−6.24621	−0.212010
$869$	−88.9848	−3.01860
$870$	−5.56155	−0.188554
$871$	0	0
$872$	12.9309	0.437895
$873$	10.0000	0.338449
$874$	−10.4384	−0.353086
$875$	9.56155	0.323239
$876$	11.5616	0.390629
$877$	−22.9848	−0.776143	−0.388072	−	0.921629i	$-0.626859\pi$
$877$	−22.9848	−0.776143	−0.388072	+	0.921629i	$0.626859\pi$
$878$	8.93087	0.301402
$879$	−7.75379	−0.261529
$880$	−19.8078	−0.667720
$881$	−14.3002	−0.481786	−0.240893	−	0.970552i	$-0.577440\pi$
$881$	−14.3002	−0.481786	−0.240893	+	0.970552i	$0.577440\pi$
$882$	1.00000	0.0336718
$883$	14.4384	0.485892	0.242946	−	0.970040i	$-0.421886\pi$
$883$	14.4384	0.485892	0.242946	+	0.970040i	$0.421886\pi$
$884$	0	0
$885$	15.1231	0.508358
$886$	6.63068	0.222762
$887$	−37.8617	−1.27127	−0.635636	−	0.771989i	$-0.719262\pi$
$887$	−37.8617	−1.27127	−0.635636	+	0.771989i	$0.719262\pi$
$888$	10.6847	0.358554
$889$	−10.2462	−0.343647
$890$	28.4924	0.955068
$891$	−5.56155	−0.186319
$892$	−1.75379	−0.0587212
$893$	−16.0000	−0.535420
$894$	−10.0000	−0.334450
$895$	−39.6155	−1.32420
$896$	1.00000	0.0334077
$897$	0	0
$898$	9.31534	0.310857
$899$	−9.75379	−0.325307
$900$	7.68466	0.256155
$901$	32.6004	1.08608
$902$	−22.2462	−0.740718
$903$	−6.43845	−0.214258
$904$	20.2462	0.673379
$905$	47.6155	1.58279
$906$	16.6847	0.554311
$907$	24.4924	0.813258	0.406629	−	0.913593i	$-0.366704\pi$
$907$	24.4924	0.813258	0.406629	+	0.913593i	$0.366704\pi$
$908$	−17.6155	−0.584592
$909$	−6.00000	−0.199007
$910$	0	0
$911$	−4.43845	−0.147052	−0.0735262	−	0.997293i	$-0.523425\pi$
$911$	−4.43845	−0.147052	−0.0735262	+	0.997293i	$0.523425\pi$
$912$	1.56155	0.0517082
$913$	11.1231	0.368121
$914$	−16.0000	−0.529233
$915$	5.56155	0.183859
$916$	−2.00000	−0.0660819
$917$	−1.56155	−0.0515670
$918$	−6.68466	−0.220627
$919$	−17.8617	−0.589204	−0.294602	−	0.955620i	$-0.595187\pi$
$919$	−17.8617	−0.589204	−0.294602	+	0.955620i	$0.595187\pi$
$920$	23.8078	0.784919
$921$	26.2462	0.864842
$922$	36.9309	1.21625
$923$	0	0
$924$	5.56155	0.182962
$925$	−82.1080	−2.69969
$926$	30.5464	1.00382
$927$	−1.80776	−0.0593748
$928$	1.56155	0.0512605
$929$	−40.8769	−1.34113	−0.670564	−	0.741852i	$-0.733948\pi$
$929$	−40.8769	−1.34113	−0.670564	+	0.741852i	$0.733948\pi$
$930$	22.2462	0.729482
$931$	−1.56155	−0.0511778
$932$	−2.00000	−0.0655122
$933$	−0.492423	−0.0161212
$934$	−1.56155	−0.0510956
$935$	−132.408	−4.33021
$936$	0	0
$937$	−19.8617	−0.648855	−0.324427	−	0.945911i	$-0.605172\pi$
$937$	−19.8617	−0.648855	−0.324427	+	0.945911i	$0.605172\pi$
$938$	−1.12311	−0.0366707
$939$	−32.2462	−1.05232
$940$	36.4924	1.19025
$941$	−26.4924	−0.863628	−0.431814	−	0.901963i	$-0.642126\pi$
$941$	−26.4924	−0.863628	−0.431814	+	0.901963i	$0.642126\pi$
$942$	11.8078	0.384718
$943$	26.7386	0.870730
$944$	−4.24621	−0.138202
$945$	−3.56155	−0.115857
$946$	−35.8078	−1.16421
$947$	−27.8078	−0.903631	−0.451815	−	0.892111i	$-0.649223\pi$
$947$	−27.8078	−0.903631	−0.451815	+	0.892111i	$0.649223\pi$
$948$	−16.0000	−0.519656
$949$	0	0
$950$	−12.0000	−0.389331
$951$	3.36932	0.109258
$952$	6.68466	0.216651
$953$	−0.738634	−0.0239267	−0.0119633	−	0.999928i	$-0.503808\pi$
$953$	−0.738634	−0.0239267	−0.0119633	+	0.999928i	$0.503808\pi$
$954$	4.87689	0.157895
$955$	−85.6695	−2.77220
$956$	14.2462	0.460755
$957$	8.68466	0.280735
$958$	−12.3002	−0.397401
$959$	−6.68466	−0.215859
$960$	−3.56155	−0.114949
$961$	8.01515	0.258553
$962$	0	0
$963$	−4.87689	−0.157156
$964$	−6.00000	−0.193247
$965$	42.7386	1.37581
$966$	−6.68466	−0.215075
$967$	2.93087	0.0942504	0.0471252	−	0.998889i	$-0.484994\pi$
$967$	2.93087	0.0942504	0.0471252	+	0.998889i	$0.484994\pi$
$968$	19.9309	0.640602
$969$	10.4384	0.335331
$970$	35.6155	1.14355
$971$	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
$973$	10.2462	0.328478
$974$	−13.7538	−0.440700
$975$	0	0
$976$	−1.56155	−0.0499841
$977$	−46.7926	−1.49703	−0.748514	−	0.663119i	$-0.769232\pi$
$977$	−46.7926	−1.49703	−0.748514	+	0.663119i	$0.769232\pi$
$978$	9.12311	0.291725
$979$	−44.4924	−1.42198
$980$	3.56155	0.113770
$981$	12.9309	0.412851
$982$	10.2462	0.326970
$983$	2.43845	0.0777744	0.0388872	−	0.999244i	$-0.487619\pi$
$983$	2.43845	0.0777744	0.0388872	+	0.999244i	$0.487619\pi$
$984$	−4.00000	−0.127515
$985$	−21.3693	−0.680883
$986$	10.4384	0.332428
$987$	−10.2462	−0.326140
$988$	0	0
$989$	43.0388	1.36855
$990$	−19.8078	−0.629532
$991$	−30.2462	−0.960803	−0.480401	−	0.877049i	$-0.659509\pi$
$991$	−30.2462	−0.960803	−0.480401	+	0.877049i	$0.659509\pi$
$992$	−6.24621	−0.198317
$993$	1.12311	0.0356407
$994$	9.36932	0.297177
$995$	−17.5616	−0.556739
$996$	2.00000	0.0633724
$997$	−34.6307	−1.09676	−0.548382	−	0.836228i	$-0.684756\pi$
$997$	−34.6307	−1.09676	−0.548382	+	0.836228i	$0.684756\pi$
$998$	−1.50758	−0.0477215
$999$	10.6847	0.338048

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7098.2.a.bv.1.2		2
13.5	odd	4		546.2.c.e.337.1	✓	4
13.8	odd	4		546.2.c.e.337.4	yes	4
13.12	even	2		7098.2.a.bg.1.1		2
39.5	even	4		1638.2.c.h.883.4		4
39.8	even	4		1638.2.c.h.883.1		4
52.31	even	4		4368.2.h.n.337.1		4
52.47	even	4		4368.2.h.n.337.4		4
91.34	even	4		3822.2.c.h.883.3		4
91.83	even	4		3822.2.c.h.883.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.2.c.e.337.1	✓	4	13.5	odd	4
546.2.c.e.337.4	yes	4	13.8	odd	4
1638.2.c.h.883.1		4	39.8	even	4
1638.2.c.h.883.4		4	39.5	even	4
3822.2.c.h.883.2		4	91.83	even	4
3822.2.c.h.883.3		4	91.34	even	4
4368.2.h.n.337.1		4	52.31	even	4
4368.2.h.n.337.4		4	52.47	even	4
7098.2.a.bg.1.1		2	13.12	even	2
7098.2.a.bv.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7098.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.56155 q^{5} -1.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.56155 q^{10} -5.56155 q^{11} -1.00000 q^{12} +1.00000 q^{14} -3.56155 q^{15} +1.00000 q^{16} +6.68466 q^{17} +1.00000 q^{18} -1.56155 q^{19} +3.56155 q^{20} -1.00000 q^{21} -5.56155 q^{22} +6.68466 q^{23} -1.00000 q^{24} +7.68466 q^{25} -1.00000 q^{27} +1.00000 q^{28} +1.56155 q^{29} -3.56155 q^{30} -6.24621 q^{31} +1.00000 q^{32} +5.56155 q^{33} +6.68466 q^{34} +3.56155 q^{35} +1.00000 q^{36} -10.6847 q^{37} -1.56155 q^{38} +3.56155 q^{40} +4.00000 q^{41} -1.00000 q^{42} +6.43845 q^{43} -5.56155 q^{44} +3.56155 q^{45} +6.68466 q^{46} +10.2462 q^{47} -1.00000 q^{48} +1.00000 q^{49} +7.68466 q^{50} -6.68466 q^{51} +4.87689 q^{53} -1.00000 q^{54} -19.8078 q^{55} +1.00000 q^{56} +1.56155 q^{57} +1.56155 q^{58} -4.24621 q^{59} -3.56155 q^{60} -1.56155 q^{61} -6.24621 q^{62} +1.00000 q^{63} +1.00000 q^{64} +5.56155 q^{66} -1.12311 q^{67} +6.68466 q^{68} -6.68466 q^{69} +3.56155 q^{70} +9.36932 q^{71} +1.00000 q^{72} -11.5616 q^{73} -10.6847 q^{74} -7.68466 q^{75} -1.56155 q^{76} -5.56155 q^{77} +16.0000 q^{79} +3.56155 q^{80} +1.00000 q^{81} +4.00000 q^{82} -2.00000 q^{83} -1.00000 q^{84} +23.8078 q^{85} +6.43845 q^{86} -1.56155 q^{87} -5.56155 q^{88} +8.00000 q^{89} +3.56155 q^{90} +6.68466 q^{92} +6.24621 q^{93} +10.2462 q^{94} -5.56155 q^{95} -1.00000 q^{96} +10.0000 q^{97} +1.00000 q^{98} -5.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 3 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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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